fix test

This PR adds the BV_CONCAT_CONST, BV_CONCAT_EXTRACT and ELIM_ZERO_EXTEND
rule from Bitwuzla to bv_decide.

.github/workflows/awaiting-mathlib.yml

@@ -0,0 +1,20 @@
+name: Check awaiting-mathlib label
+
+on:
+  merge_group:
+  pull_request:
+    types: [opened, labeled]
+
+jobs:
+  check-awaiting-mathlib:
+    runs-on: ubuntu-latest
+    steps:
+      - name: Check awaiting-mathlib label
+        if: github.event_name == 'pull_request'
+        uses: actions/github-script@v7
+        with:
+          script: |
+            const { labels } = context.payload.pull_request;
+            if (labels.some(label => label.name == "awaiting-mathlib") && !labels.some(label => label.name == "builds-mathlib")) {
+              core.setFailed('PR is marked "awaiting-mathlib" but "builds-mathlib" label has not been applied yet by the bot');
+            }

.github/workflows/ci.yml

@@ -137,7 +137,6 @@ jobs:
             let large = ${{ github.repository == 'leanprover/lean4' }};
             let matrix = [
               {
-                // portable release build: use channel with older glibc (2.27)
                 "name": "Linux LLVM",
                 "os": "ubuntu-latest",
                 "release": false,
@@ -152,6 +151,7 @@ jobs:
                 "CMAKE_OPTIONS": "-DLLVM=ON -DLLVM_CONFIG=${GITHUB_WORKSPACE}/build/llvm-host/bin/llvm-config"
               },
               {
+                // portable release build: use channel with older glibc (2.26)
                 "name": "Linux release",
                 "os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
                 "release": true,
@@ -175,8 +175,8 @@ jobs:
                 "os": "ubuntu-latest",
                 "check-level": 2,
                 "CMAKE_PRESET": "debug",
-                // exclude seriously slow tests
-                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest|bv_bitblast_stress'"
+                // exclude seriously slow/stackoverflowing tests
+                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest|bv_bitblast_stress|3807'"
               },
               // TODO: suddenly started failing in CI
               /*{
@@ -204,7 +204,8 @@ jobs:
                 "os": "macos-14",
                 "CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-darwin_aarch64",
                 "release": true,
-                "check-level": 0,
+                // special cased below
+                // "check-level": 0,
                 "shell": "bash -euxo pipefail {0}",
                 "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-aarch64-apple-darwin.tar.zst",
                 "prepare-llvm": "../script/prepare-llvm-macos.sh lean-llvm*",
@@ -260,8 +261,21 @@ jobs:
               //   "CTEST_OPTIONS": "-R \"leantest_1007\\.lean|leantest_Format\\.lean|leanruntest\\_1037.lean|leanruntest_ac_rfl\\.lean|leanruntest_tempfile.lean\\.|leanruntest_libuv\\.lean\""
               // }
             ];
-            console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`)
-            return matrix.filter((job) => level >= job["check-level"])
+            console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`);
+            const isPr = "${{ github.event_name }}" == "pull_request";
+            const filter = (job) => {
+              if (job["name"] === "macOS aarch64") {
+                // Special handling for MacOS aarch64, we want:
+                // 1. To run it in PRs so Mac devs get PR toolchains
+                // 2. To skip it in merge queues as it takes longer than the Linux build and adds
+                //    little value in the merge queue
+                // 3. To run it in release (obviously)
+                return isPr || level >= 2;
+              } else {
+                return level >= job["check-level"];
+              }
+            };
+            return matrix.filter(filter);
 
   build:
     needs: [configure]

.github/workflows/pr-release.yml

@@ -34,7 +34,7 @@ jobs:
       - name: Download artifact from the previous workflow.
         if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
         id: download-artifact
-        uses: dawidd6/action-download-artifact@v8 # https://github.com/marketplace/actions/download-workflow-artifact
+        uses: dawidd6/action-download-artifact@v9 # https://github.com/marketplace/actions/download-workflow-artifact
         with:
           run_id: ${{ github.event.workflow_run.id }}
           path: artifacts
@@ -155,6 +155,20 @@ jobs:
           fi
 
           if [[ -n "$MESSAGE" ]]; then
+            # Check if force-mathlib-ci label is present
+            LABELS="$(curl --retry 3 --location --silent \
+                          -H "Authorization: token ${{ secrets.MATHLIB4_COMMENT_BOT }}" \
+                          -H "Accept: application/vnd.github.v3+json" \
+                          "https://api.github.com/repos/leanprover/lean4/issues/${{ steps.workflow-info.outputs.pullRequestNumber }}/labels" \
+                          | jq -r '.[].name')"
+            
+            if echo "$LABELS" | grep -q "^force-mathlib-ci$"; then
+              echo "force-mathlib-ci label detected, forcing CI despite issues"
+              MESSAGE="Forcing Mathlib CI because the \`force-mathlib-ci\` label is present, despite problem: $MESSAGE"
+              FORCE_CI=true
+            else
+              MESSAGE="$MESSAGE You can force Mathlib CI using the \`force-mathlib-ci\` label."
+            fi
 
             echo "Checking existing messages"
 
@@ -201,7 +215,12 @@ jobs:
             else
               echo "The message already exists in the comment body."
             fi
-            echo "mathlib_ready=false" >> "$GITHUB_OUTPUT"
+
+            if [[ "$FORCE_CI" == "true" ]]; then
+              echo "mathlib_ready=true" >> "$GITHUB_OUTPUT"
+            else
+              echo "mathlib_ready=false" >> "$GITHUB_OUTPUT"
+            fi
           else
             echo "mathlib_ready=true" >> "$GITHUB_OUTPUT"
           fi
@@ -252,7 +271,7 @@ jobs:
           if git ls-remote --heads --tags --exit-code origin "nightly-testing-${MOST_RECENT_NIGHTLY}" >/dev/null; then
             BASE="nightly-testing-${MOST_RECENT_NIGHTLY}"
           else
-            echo "This shouldn't be possible: couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' tag at Batteries. Falling back to 'nightly-testing'."
+            echo "Couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' tag at Batteries. Falling back to 'nightly-testing'."
             BASE=nightly-testing
           fi
 
@@ -316,7 +335,7 @@ jobs:
           if git ls-remote --heads --tags --exit-code origin "nightly-testing-${MOST_RECENT_NIGHTLY}" >/dev/null; then
             BASE="nightly-testing-${MOST_RECENT_NIGHTLY}"
           else
-            echo "This shouldn't be possible: couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' branch at Mathlib. Falling back to 'nightly-testing'."
+            echo "Couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' branch at Mathlib. Falling back to 'nightly-testing'."
             BASE=nightly-testing
           fi
 

CMakeLists.txt

@@ -47,10 +47,11 @@ if (NOT ${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
     if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
       string(APPEND CADICAL_CXXFLAGS " -DNUNLOCKED")
     endif()
+    string(APPEND CADICAL_CXXFLAGS " -DNCLOSEFROM")
     ExternalProject_add(cadical
       PREFIX cadical
       GIT_REPOSITORY https://github.com/arminbiere/cadical
-      GIT_TAG rel-1.9.5
+      GIT_TAG rel-2.1.2
       CONFIGURE_COMMAND ""
       # https://github.com/arminbiere/cadical/blob/master/BUILD.md#manual-build
       BUILD_COMMAND $(MAKE) -f ${CMAKE_SOURCE_DIR}/src/cadical.mk CMAKE_EXECUTABLE_SUFFIX=${CMAKE_EXECUTABLE_SUFFIX} CXX=${CADICAL_CXX} CXXFLAGS=${CADICAL_CXXFLAGS}

doc/declarations.md

@@ -764,11 +764,12 @@ Structures and Records
 The ``structure`` command in Lean is used to define an inductive data type with a single constructor and to define its projections at the same time. The syntax is as follows:
 
 ```
-structure Foo (a : α) extends Bar, Baz : Sort u :=
+structure Foo (a : α) : Sort u extends Bar, Baz :=
 constructor :: (field₁ : β₁) ... (fieldₙ : βₙ)
 ```
 
-Here ``(a : α)`` is a telescope, that is, the parameters to the inductive definition. The name ``constructor`` followed by the double colon is optional; if it is not present, the name ``mk`` is used by default. The keyword ``extends`` followed by a list of previously defined structures is also optional; if it is present, an instance of each of these structures is included among the fields to ``Foo``, and the types ``βᵢ`` can refer to their fields as well. The output type, ``Sort u``, can be omitted, in which case Lean infers to smallest non-``Prop`` sort possible. Finally, ``(field₁ : β₁) ... (fieldₙ : βₙ)`` is a telescope relative to ``(a : α)`` and the fields in ``bar`` and ``baz``.
+Here ``(a : α)`` is a telescope, that is, the parameters to the inductive definition. The name ``constructor`` followed by the double colon is optional; if it is not present, the name ``mk`` is used by default. The keyword ``extends`` followed by a list of previously defined structures is also optional; if it is present, an instance of each of these structures is included among the fields to ``Foo``, and the types ``βᵢ`` can refer to their fields as well. The output type, ``Sort u``, can be omitted, in which case Lean infers to smallest non-``Prop`` sort possible (unless all the fields are ``Prop``, in which case it infers ``Prop``).
+Finally, ``(field₁ : β₁) ... (fieldₙ : βₙ)`` is a telescope relative to ``(a : α)`` and the fields in ``bar`` and ``baz``.
 
 The declaration above is syntactic sugar for an inductive type declaration, and so results in the addition of the following constants to the environment:
 

doc/examples/bintree.lean

@@ -179,7 +179,7 @@ local macro "have_eq " lhs:term:max rhs:term:max : tactic =>
   `(tactic|
     (have h : $lhs = $rhs :=
        -- TODO: replace with linarith
-       by simp_arith at *; apply Nat.le_antisymm <;> assumption
+       by simp +arith at *; apply Nat.le_antisymm <;> assumption
      try subst $lhs))
 
 /-!

doc/setup.md

@@ -4,7 +4,7 @@
 
 Platforms built & tested by our CI, available as binary releases via elan (see below)
 
-* x86-64 Linux with glibc 2.27+
+* x86-64 Linux with glibc 2.26+
 * x86-64 macOS 10.15+
 * aarch64 (Apple Silicon) macOS 10.15+
 * x86-64 Windows 11 (any version), Windows 10 (version 1903 or higher), Windows Server 2022

doc/std/README.md

@@ -0,0 +1,9 @@
+# The Lean standard library
+
+This directory contains development information about the Lean standard library. The user-facing documentation of the standard library
+is part of the [Lean Language Reference](https://lean-lang.org/doc/reference/latest/).
+
+Here you will find
+* the [standard library vision document](./vision.md), including the call for contributions,
+* the [standard library style guide](./style.md), and
+* the [standard library naming conventions](./naming.md).

doc/std/naming.md

@@ -0,0 +1,260 @@
+# Standard library naming conventions
+
+The easiest way to access a result in the standard library is to correctly guess the name of the declaration (possibly with the help of identifier autocompletion). This is faster and has lower friction than more sophisticated search tools, so easily guessable names (which are still reasonably short) make Lean users more productive.
+
+The guide that follows contains very few hard rules, many heuristics and a selection of examples. It cannot and does not present a deterministic algorithm for choosing good names in all situations. It is intended as a living document that gets clarified and expanded as situations arise during code reviews for the standard library. If applying one of the suggestions in this guide leads to nonsensical results in a certain situation, it is
+probably safe to ignore the suggestion (or even better, suggest a way to improve the suggestion).
+
+## Prelude
+
+Identifiers use a mix of `UpperCamelCase`, `lowerCamelCase` and `snake_case`, used for types, data, and theorems, respectively.
+
+Structure fields should be named such that the projections have the correct names.
+
+## Naming convention for types
+
+When defining a type, i.e., a (possibly 0-ary) function whose codomain is Sort u for some u, it should be named in UpperCamelCase. Examples include `List`, and `List.IsPrefix`.
+
+When defining a predicate, prefix the name by `Is`, like in `List.IsPrefix`. The `Is` prefix may be omitted if
+* the resulting name would be ungrammatical, or
+* the predicate depends on additional data in a way where the `Is` prefix would be confusing (like `List.Pairwise`), or
+* the name is an adjective (like `Std.Time.Month.Ordinal.Valid`)
+
+## Namespaces and generalized projection notation
+
+Almost always, definitions and theorems relating to a type should be placed in a namespace with the same name as the type. For example, operations and theorems about lists should be placed in the `List` namespace, and operations and theorems about `Std.Time.PlainDate` should be placed in the `Std.Time.PlainDate` namespace.
+
+Declarations in the root namespace will be relatively rare. The most common type of declaration in the root namespace are declarations about data and properties exported by notation type classes, as long as they are not about a specific type implementing that type class. For example, we have
+
+```lean
+theorem beq_iff_eq [BEq α] [LawfulBEq α] {a b : α} : a == b ↔ a = b := sorry
+```
+
+in the root namespace, but
+
+```lean
+theorem List.cons_beq_cons [BEq α] {a b : α} {l₁ l₂ : List α} :
+    (a :: l₁ == b :: l₂) = (a == b && l₁ == l₂) := rfl
+```
+
+belongs in the `List` namespace.
+
+Subtleties arise when multiple namespaces are in play. Generally, place your theorem in the most specific namespace that appears in one of the hypotheses of the theorem. The following names are both correct according to this convention:
+
+```lean
+theorem List.Sublist.reverse : l₁ <+ l₂ → l₁.reverse <+ l₂.reverse := sorry
+theorem List.reverse_sublist : l₁.reverse <+ l₂.reverse ↔ l₁ <+ l₂ := sorry
+```
+
+Notice that the second theorem does not have a hypothesis of type `List.Sublist l` for some `l`, so the name `List.Sublist.reverse_iff` would be incorrect.
+
+The advantage of placing results in a namespace like `List.Sublist` is that it enables generalized projection notation, i.e., given `h : l₁ <+ l₂`,
+one can write `h.reverse` to obtain a proof of `l₁.reverse <+ l₂.reverse`. Thinking about which dot notations are convenient can act as a guideline
+for deciding where to place a theorem, and is, on occasion, a good reason to duplicate a theorem into multiple namespaces.
+
+### The `Std` namespace
+
+New types that are added will usually be placed in the `Std` namespace and in the `Std/` source directory, unless there are good reasons to place
+them elsewhere.
+
+Inside the `Std` namespace, all internal declarations should be `private` or else have a name component that clearly marks them as internal, preferably
+`Internal`.
+
+
+## Naming convention for data
+
+When defining data, i.e., a (possibly 0-ary) function whose codomain is not Sort u, but has type Type u for some u, it should be named in lowerCamelCase. Examples include `List.append` and `List.isPrefixOf`.
+If your data is morally fully specified by its type, then use the naming procedure for theorems described below and convert the result to lower camel case.
+
+If your function returns an `Option`, consider adding `?` as a suffix. If your function may panic, consider adding `!` as a suffix. In many cases, there will be multiple variants of a function; one returning an option, one that may panic and possibly one that takes a proof argument.
+
+## Naming algorithm for theorems and some definitions
+
+There is, in principle, a general algorithm for naming a theorem. The problem with this algorithm is that it produces very long and unwieldy names which need to be shortened. So choosing a name for a declaration can be thought of as consisting of a mechanical part and a creative part.
+
+Usually the first part is to decide which namespace the result should live in, according to the guidelines described above.
+
+Next, consider the type of your declaration as a tree. Inner nodes of this tree are function types or function applications. Leaves of the tree are 0-ary functions or bound variables.
+
+As an example, consider the following result from the standard library:
+
+```lean
+example {α : Type u} {β : Type v} [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α]
+    [Inhabited β] {m : Std.HashMap α β} {a : α} {h' : a ∈ m} : m[a]? = some (m[a]'h') :=
+  sorry
+```
+
+The correct namespace is clearly `Std.HashMap`. The corresponding tree looks like this:
+
+![](naming-tree.svg)
+
+The preferred spelling of a notation can be looked up by hovering over the notation.
+
+Now traverse the tree and build a name according to the following rules:
+
+* When encountering a function type, first turn the result type into a name, then all of the argument types from left to right, and join the names using `_of_`.
+* When encountering a function that is neither an infix notation nor a structure projection, first put the function name and then the arguments, joined by an underscore.
+* When encountering an infix notation, join the arguments using the name of the notation, separated by underscores.
+* When encountering a structure projection, proceed as for normal functions, but put the name of the projection last.
+* When encountering a name, put it in lower camel case.
+* Skip bound variables and proofs.
+* Type class arguments are also generally skipped.
+
+When encountering namespaces names, concatenate them in lower camel case.
+
+Applying this algorithm to our example yields the name `Std.HashMap.getElem?_eq_optionSome_getElem_of_mem`.
+
+From there, the name should be shortened, using the following heuristics:
+
+* The namespace of functions can be omitted if it is clear from context or if the namespace is the current one. This is almost always the case.
+* For infix operators, it is possible to leave out the RHS or the name of the notation and the RHS if they are clear from context.
+* Hypotheses can be left out if it is clear that they are required or if they appear in the conclusion.
+
+Based on this, here are some possible names for our example:
+
+1. `Std.HashMap.getElem?_eq`
+2. `Std.HashMap.getElem?_eq_of_mem`
+3. `Std.HashMap.getElem?_eq_some`
+4. `Std.HashMap.getElem?_eq_some_of_mem`
+5. `Std.HashMap.getElem?_eq_some_getElem`
+6. `Std.Hashmap.getElem?_eq_some_getElem_of_mem`
+
+Choosing a good name among these then requires considering the context of the lemma. In this case it turns out that the first four options are underspecified as there is also a lemma relating `m[a]?` and `m[a]!` which could have the same name. This leaves the last two options, the first of which is shorter, and this is how the lemma is called in the Lean standard library.
+
+Here are some additional examples:
+
+```lean
+example {x y : List α} (h : x <+: y) (hx : x ≠ []) :
+  x.head hx = y.head (h.ne_nil hx) := sorry
+```
+
+Since we have an `IsPrefix` parameter, this should live in the `List.IsPrefix` namespace, and the algorithm suggests `List.IsPrefix.head_eq_head_of_ne_nil`, which is shortened to `List.IsPrefix.head`. Note here the difference between the namespace name (`IsPrefix`) and the recommended spelling of the corresponding notation (`prefix`).
+
+```lean
+example : l₁ <+: l₂ → reverse l₁ <:+ reverse l₂ := sorry
+```
+
+Again, this result should be in the `List.IsPrefix` namespace; the algorithm suggests `List.IsPrefix.reverse_prefix_reverse`, which becomes `List.IsPrefix.reverse`.
+
+The following examples show how the traversal order often matters.
+
+```lean
+theorem Nat.mul_zero (n : Nat) : n * 0 = 0 := sorry
+theorem Nat.zero_mul (n : Nat) : 0 * n = 0 := sorry
+```
+
+Here we see that one name may be a prefix of another name:
+
+```lean
+theorem Int.mul_ne_zero {a b : Int} (a0 : a ≠ 0) (b0 : b ≠ 0) : a * b ≠ 0 := sorry
+theorem Int.mul_ne_zero_iff {a b : Int} : a * b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 := sorry
+```
+
+It is usually a good idea to include the `iff` in a theorem name even if the name would still be unique without the name. For example,
+
+```lean
+theorem List.head?_eq_none_iff : l.head? = none ↔ l = [] := sorry
+```
+
+is a good name: if the lemma was simply called `List.head?_eq_none`, users might try to `apply` it when the goal is `l.head? = none`, leading
+to confusion.
+
+The more common you expect (or want) a theorem to be, the shorter you should try to make the name. For example, we have both
+
+```lean
+theorem Std.HashMap.getElem?_eq_none_of_contains_eq_false {a : α} : m.contains a = false → m[a]? = none := sorry
+theorem Std.HashMap.getElem?_eq_none {a : α} : ¬a ∈ m → m[a]? = none := sorry
+```
+
+As users of the hash map are encouraged to use ∈ rather than contains, the second lemma gets the shorter name.
+
+## Special cases
+
+There are certain special “keywords” that may appear in identifiers.
+
+| Keyword | Meaning | Example |
+| :---- | :---- | :---- |
+| `def` | Unfold a definition. Avoid this for public APIs. | `Nat.max_def` |
+| `refl` | Theorems of the form `a R a`, where R is a reflexive relation and `a` is an explicit parameter | `Nat.le_refl` |
+| `rfl` | Like `refl`, but with `a` implicit | `Nat.le_rfl` |
+| `irrefl` | Theorems of the form `¬a R a`, where R is an irreflexive relation | `Nat.lt_irrefl` |
+| `symm` | Theorems of the form `a R b → b R a`, where R is a symmetric relation (compare `comm` below) | `Eq.symm` |
+| `trans` | Theorems of the form `a R b → b R c → a R c`, where R is a transitive relation (R may carry data) | `Eq.trans` |
+| `antisymmm` | Theorems of the form `a R b → b R a → a = b`, where R is an antisymmetric relation | `Nat.le_antisymm` |
+| `congr` | Theorems of the form `a R b → f a S f b`, where R and S are usually equivalence relations | `Std.HashMap.mem_congr` |
+| `comm` | Theorems of the form `f a b = f b a` (compare `symm` above) | `Eq.comm`, `Nat.add_comm` |
+| `assoc` | Theorems of the form `g (f a b) c = f a (g b c)` (note the order! In most cases, we have f = g) | `Nat.add_sub_assoc` |
+| `distrib` | Theorems of the form `f (g a b) = g (f a) (f b)` | `Nat.add_left_distrib` |
+| `self` | May be used if a variable appears multiple times in the conclusion | `List.mem_cons_self` |
+| `inj` | Theorems of the form `f a = f b ↔ a = b`. | `Int.neg_inj`, `Nat.add_left_inj` |
+| `cancel` | Theorems which have one of the forms `f a = f b → a = b` or `g (f a) = a`, where `f` and `g` usually involve a binary operator | `Nat.add_sub_cancel` |
+| `cancel_iff` | Same as `inj`, but with different conventions for left and right (see below) | `Nat.add_right_cancel_iff` |
+| `ext` | Theorems of the form `f a = f b → a = b`, where `f` usually involves some kind of projection | `List.ext_getElem`
+| `mono` | Theorems of the form `a R b → f a R f b`, where `R` is a transitive relation | `List.countP_mono_left`
+
+### Left and right
+
+The keywords left and right are useful to disambiguate symmetric variants of theorems.
+
+```lean
+theorem imp_congr_left (h : a ↔ b) : (a → c) ↔ (b → c) := sorry
+theorem imp_congr_right (h : a → (b ↔ c)) : (a → b) ↔ (a → c) := sorry
+```
+
+It is not always obvious which version of a theorem should be “left” and which should be “right”.
+Heuristically, the theorem should name the side which is “more variable”, but there are exceptions. For some of the special keywords discussed in this section, there are conventions which should be followed, as laid out in the following examples:
+
+```lean
+theorem Nat.left_distrib (n m k : Nat) : n * (m + k) = n * m + n * k := sorry
+theorem Nat.right_distrib (n m k : Nat) : (n + m) * k = n * k + m * k := sorry
+theorem Nat.add_left_cancel {n m k : Nat} : n + m = n + k → m = k := sorry
+theorem Nat.add_right_cancel {n m k : Nat} : n + m = k + m → n = k := sorry
+theorem Nat.add_left_cancel_iff {m k n : Nat} : n + m = n + k ↔ m = k := sorry
+theorem Nat.add_right_cancel_iff {m k n : Nat} : m + n = k + n ↔ m = k := sorry
+theorem Nat.add_left_inj {m k n : Nat} : m + n = k + n ↔ m = k := sorry
+theorem Nat.add_right_inj {m k n : Nat} : n + m = n + k ↔ m = k := sorry
+```
+
+Note in particular that the convention is opposite for `cancel_iff` and `inj`.
+
+```lean
+theorem Nat.add_sub_self_left (a b : Nat) : (a + b) - a = b := sorry
+theorem Nat.add_sub_self_right (a b : Nat) : (a + b) - b = a := sorry
+theorem Nat.add_sub_cancel (n m : Nat) : (n + m) - m = n := sorry
+```
+
+## Primed names
+
+Avoid disambiguating variants of a concept by appending the `'` character (e.g., introducing both `BitVec.sshiftRight` and `BitVec.sshiftRight'`), as it is impossible to tell the difference without looking at the type signature, the documentation or even the code, and even if you know what the two variants are there is no way to tell which is which. Prefer descriptive pairs `BitVec.sshiftRightNat`/`BitVec.sshiftRight`.
+
+## Acronyms
+
+For acronyms which are three letters or shorter, all letters should use the same case as dictated by the convention. For example, `IO` is a correct name for a type and the name `IO.Ref` may become `IORef` when used as part of a definition name and `ioRef` when used as part of a theorem name.
+
+For acronyms which are at least four letters long, switch to lower case starting from the second letter. For example, `Json` is a correct name for a type, as is `JsonRPC`.
+
+If an acronym is typically spelled using mixed case, this mixed spelling may be used in identifiers (for example `Std.Net.IPv4Addr`).
+
+## Simp sets
+
+Simp sets centered around a conversion function should be called `source_to_target`. For example, a simp set for the `BitVec.toNat` function, which goes from `BitVec` to
+`Nat`, should be called `bitvec_to_nat`.
+
+## Variable names
+
+We make the following recommendations for variable names, but without insisting on them:
+* Simple hypotheses should be named `h`, `h'`, or using a numerical sequence `h₁`, `h₂`, etc.
+* Another common name for a simple hypothesis is `w` (for "witness").
+* `List`s should be named `l`, `l'`, `l₁`, etc, or `as`, `bs`, etc.
+  (Use of `as`, `bs` is encouraged when the lists are of different types, e.g. `as : List α` and `bs : List β`.)
+  `xs`, `ys`, `zs` are allowed, but it is better if these are reserved for `Array` and `Vector`.
+  A list of lists may be named `L`.
+* `Array`s should be named `xs`, `ys`, `zs`, although `as`, `bs` are encouraged when the arrays are of different types, e.g. `as : Array α` and `bs : Array β`.
+  An array of arrays may be named `xss`.
+* `Vector`s should be named `xs`, `ys`, `zs`, although `as`, `bs` are encouraged when the vectors are of different types, e.g. `as : Vector α n` and `bs : Vector β n`.
+  A vector of vectors may be named `xss`.
+* A common exception for `List` / `Array` / `Vector` is to use `acc` for an accumulator in a recursive function.
+* `i`, `j`, `k` are preferred for numerical indices.
+  Descriptive names such as `start`, `stop`, `lo`, and `hi` are encouraged when they increase readability.
+* `n`, `m` are preferred for sizes, e.g. in `Vector α n` or `xs.size = n`.
+* `w` is preferred for the width of a `BitVec`.

doc/std/style.md

@@ -1,3 +1,5 @@
+# Standard library style
+
 Please take some time to familiarize yourself with the stylistic conventions of
 the project and the specific part of the library you are planning to contribute
 to. While the Lean compiler may not enforce strict formatting rules,
@@ -6,5 +8,515 @@ Attention to formatting is more than a cosmetic concern—it reflects the same
 level of precision and care required to meet the deeper standards of the Lean 4
 standard library.
 
-A full style guide and naming convention are currently under construction and
-will be added here soon.
+Below we will give specific formatting prescriptions for various language constructs. Note that this style guide only applies to the Lean standard library, even though some examples in the guide are taken from other parts of the Lean code base.
+
+## Basic whitespace rules
+
+Syntactic elements (like `:`, `:=`, `|`, `::`) are surrounded by single spaces, with the exception of `,` and `;`, which are followed by a space but not preceded by one. Delimiters (like `()`, `{}`) do not have spaces on the inside, with the exceptions of subtype notation and structure instance notation.
+
+Examples of correctly formatted function parameters:
+
+* `{α : Type u}`
+* `[BEq α]`
+* `(cmp : α → α → Ordering)`
+* `(hab : a = b)`
+* `{d : { l : List ((n : Nat) × Vector Nat n) // l.length % 2 = 0 }}`
+
+Examples of correctly formatted terms:
+
+* `1 :: [2, 3]`
+* `letI : Ord α := ⟨cmp⟩; True`
+* `(⟨2, 3⟩ : Nat × Nat)`
+* `((2, 3) : Nat × Nat)`
+* `{ x with fst := f (4 + f 0), snd := 4, .. }`
+* `match 1 with | 0 => 0 | _ => 0`
+* `fun ⟨a, b⟩ _ _ => by cases hab <;> apply id; rw [hbc]`
+
+Configure your editor to remove trailing whitespace. If you have set up Visual Studio Code for Lean development in the recommended way then the correct setting is applied automatically.
+
+## Splitting terms across multiple lines
+
+When splitting a term across multiple lines, increase indentation by two spaces starting from the second line. When splitting a function application, try to split at argument boundaries. If an argument itself needs to be split, increase indentation further as appropriate.
+
+When splitting at an infix operator, the operator goes at the end of the first line, not at the beginning of the second line. When splitting at an infix operator, you may or may not increase indentation depth, depending on what is more readable.
+
+When splitting an `if`-`then`-`else` expression, the `then` keyword wants to stay with the condition and the `else` keyword wants to stay with the alternative term. Otherwise, indent as if the `if` and `else` keywords were arguments to the same function.
+
+When splitting a comma-separated bracketed sequence (i.e., anonymous constructor application, list/array/vector literal, tuple) it is allowed to indent subsequent lines for alignment, but indenting by two spaces is also allowed.
+
+Do not orphan parentheses.
+
+Correct:
+```lean
+def MacroScopesView.isPrefixOf (v₁ v₂ : MacroScopesView) : Bool :=
+  v₁.name.isPrefixOf v₂.name &&
+  v₁.scopes == v₂.scopes &&
+  v₁.mainModule == v₂.mainModule &&
+  v₁.imported == v₂.imported
+```
+
+Correct:
+```lean
+theorem eraseP_eq_iff {p} {l : List α} :
+    l.eraseP p = l' ↔
+      ((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
+        ∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧
+          l = l₁ ++ a :: l₂ ∧ l' = l₁ ++ l₂ :=
+  sorry
+```
+
+Correct:
+```lean
+example : Nat :=
+  functionWithAVeryLongNameSoThatSomeArgumentsWillNotFit firstArgument secondArgument
+    (firstArgumentWithAnEquallyLongNameAndThatFunctionDoesHaveMoreArguments firstArgument
+      secondArgument)
+    secondArgument
+```
+
+Correct:
+```lean
+theorem size_alter [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} (h : m.WF) :
+    (m.alter k f).size =
+      if m.contains k && (f (m.get? k)).isNone then
+        m.size - 1
+      else if !m.contains k && (f (m.get? k)).isSome then
+        m.size + 1
+      else
+        m.size := by
+ simp_to_raw using Raw₀.size_alter
+```
+
+Correct:
+```lean
+theorem get?_alter [LawfulBEq α] {k k' : α} {f : Option (β k) → Option (β k)} (h : m.WF) :
+    (m.alter k f).get? k' =
+      if h : k == k' then
+        cast (congrArg (Option ∘ β) (eq_of_beq h)) (f (m.get? k))
+      else m.get? k' := by
+ simp_to_raw using Raw₀.get?_alter
+```
+
+Correct:
+```lean
+example : Nat × Nat :=
+  ⟨imagineThisWasALongTerm,
+   imagineThisWasAnotherLongTerm⟩
+```
+
+Correct:
+```lean
+example : Nat × Nat :=
+  ⟨imagineThisWasALongTerm,
+    imagineThisWasAnotherLongTerm⟩
+```
+
+Correct:
+```lean
+example : Vector Nat :=
+  #v[imagineThisWasALongTerm,
+     imagineThisWasAnotherLongTerm]
+```
+
+## Basic file structure
+
+Every file should start with a copyright header, imports (in the standard library, this always includes a `prelude` declaration) and a module documentation string. There should not be a blank line between the copyright header and the imports. There should be a blank line between the imports and the module documentation string.
+
+If you explicitly declare universe variables, do so at the top of the file, after the module documentation.
+
+Correct:
+```lean
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
+  Yury Kudryashov
+-/
+prelude
+import Init.Data.List.Pairwise
+import Init.Data.List.Find
+
+/-!
+**# Lemmas about `List.eraseP` and `List.erase`.**
+-/
+
+universe u u'
+```
+
+Syntax that is not supposed to be user-facing must be scoped. New public syntax must always be discussed explicitly in an RFC.
+
+## Top-level commands and declarations
+
+All top-level commands are unindented. Sectioning commands like `section` and `namespace` do not increase the indentation level.
+
+Attributes may be placed on the same line as the rest of the command or on a separate line.
+
+Multi-line declaration headers are indented by four spaces starting from the second line. The colon that indicates the type of a declaration may not be placed at the start of a line or on its own line.
+
+Declaration bodies are indented by two spaces. Short declaration bodies may be placed on the same line as the declaration type.
+
+Correct:
+```lean
+theorem eraseP_eq_iff {p} {l : List α} :
+    l.eraseP p = l' ↔
+      ((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
+        ∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧
+          l = l₁ ++ a :: l₂ ∧ l' = l₁ ++ l₂ :=
+  sorry
+```
+
+Correct:
+```lean
+@[simp] theorem eraseP_nil : [].eraseP p = [] := rfl
+```
+
+Correct:
+```lean
+@[simp]
+theorem eraseP_nil : [].eraseP p = [] := rfl
+```
+
+### Documentation comments
+
+Note to external contributors: this is a section where the Lean style and the mathlib style are different.
+
+Declarations should be documented as required by the `docBlame` linter, which may be activated in a file using
+`set_option linter.missingDocs true` (we allow these to stay in the file).
+
+Single-line documentation comments should go on the same line as `/--`/`-/`, while multi-line documentation strings
+should have these delimiters on their own line, with the documentation comment itself unindented.
+
+Documentation comments must be written in the indicative mood. Use American orthography.
+
+Correct:
+```lean
+/-- Carries out a monadic action on each mapping in the hash map in some order. -/
+@[inline] def forM (f : (a : α) → β a → m PUnit) (b : Raw α β) : m PUnit :=
+  b.buckets.forM (AssocList.forM f)
+```
+
+Correct:
+```lean
+/--
+Monadically computes a value by folding the given function over the mappings in the hash
+map in some order.
+-/
+@[inline] def foldM (f : δ → (a : α) → β a → m δ) (init : δ) (b : Raw α β) : m δ :=
+  b.buckets.foldlM (fun acc l => l.foldlM f acc) init
+```
+
+### Where clauses
+
+The `where` keyword should be unindented, and all declarations bound by it should be indented with two spaces.
+
+Blank lines before and after `where` and between declarations bound by `where` are optional and should be chosen
+to maximize readability.
+
+Correct:
+```lean
+@[simp] theorem partition_eq_filter_filter (p : α → Bool) (l : List α) :
+    partition p l = (filter p l, filter (not ∘ p) l) := by
+  simp [partition, aux]
+where
+  aux (l) {as bs} : partition.loop p l (as, bs) =
+      (as.reverse ++ filter p l, bs.reverse ++ filter (not ∘ p) l) :=
+    match l with
+    | [] => by simp [partition.loop, filter]
+    | a :: l => by cases pa : p a <;> simp [partition.loop, pa, aux, filter, append_assoc]
+```
+
+### Termination arguments
+
+The `termination_by`, `decreasing_by`, `partial_fixpoint` keywords should be unindented. The associated terms should be indented like declaration bodies.
+
+Correct:
+```lean
+@[inline] def multiShortOption (handle : Char → m PUnit) (opt : String) : m PUnit := do
+  let rec loop (p : String.Pos) := do
+    if h : opt.atEnd p then
+      return
+    else
+      handle (opt.get' p h)
+      loop (opt.next' p h)
+  termination_by opt.utf8ByteSize - p.byteIdx
+  decreasing_by
+    simp [String.atEnd] at h
+    apply Nat.sub_lt_sub_left h
+    simp [String.lt_next opt p]
+  loop ⟨1⟩
+```
+
+Correct:
+```lean
+def substrEq (s1 : String) (off1 : String.Pos) (s2 : String) (off2 : String.Pos) (sz : Nat) : Bool :=
+  off1.byteIdx + sz ≤ s1.endPos.byteIdx && off2.byteIdx + sz ≤ s2.endPos.byteIdx && loop off1 off2 { byteIdx := off1.byteIdx + sz }
+where
+  loop (off1 off2 stop1 : Pos) :=
+    if _h : off1.byteIdx < stop1.byteIdx then
+      let c₁ := s1.get off1
+      let c₂ := s2.get off2
+      c₁ == c₂ && loop (off1 + c₁) (off2 + c₂) stop1
+    else true
+  termination_by stop1.1 - off1.1
+  decreasing_by
+    have := Nat.sub_lt_sub_left _h (Nat.add_lt_add_left c₁.utf8Size_pos off1.1)
+    decreasing_tactic
+```
+
+Correct:
+```lean
+theorem div_add_mod (m n : Nat) : n * (m / n) + m % n = m := by
+  rw [div_eq, mod_eq]
+  have h : Decidable (0 < n ∧ n ≤ m) := inferInstance
+  cases h with
+  | isFalse h => simp [h]
+  | isTrue h =>
+    simp [h]
+    have ih := div_add_mod (m - n) n
+    rw [Nat.left_distrib, Nat.mul_one, Nat.add_assoc, Nat.add_left_comm, ih, Nat.add_comm, Nat.sub_add_cancel h.2]
+decreasing_by apply div_rec_lemma; assumption
+```
+
+### Deriving
+
+The `deriving` clause should be unindented.
+
+Correct:
+```lean
+structure Iterator where
+  array : ByteArray
+  idx : Nat
+deriving Inhabited
+```
+
+## Notation and Unicode
+
+We generally prefer to use notation as available. We usually prefer the Unicode versions of notations over non-Unicode alternatives.
+
+There are some rules and exceptions regarding specific notations which are listed below:
+
+* Sigma types: use `(a : α) × β a` instead of `Σ a, β a` or `Sigma β`.
+* Function arrows: use `fun a => f x` instead of `fun x ↦ f x` or `λ x => f x` or any other variant.
+
+## Language constructs
+
+### Pattern matching, induction etc.
+
+Match arms are indented at the indentation level that the match statement would have if it was on its own line. If the match is implicit, then the arms should be indented as if the match was explicitly given. The content of match arms is indented two spaces, so that it appears on the same level as the match pattern.
+
+Correct:
+```lean
+def alter [BEq α] {β : Type v} (a : α) (f : Option β → Option β) :
+    AssocList α (fun _ => β) → AssocList α (fun _ => β)
+  | nil => match f none with
+    | none => nil
+    | some b => AssocList.cons a b nil
+  | cons k v l =>
+    if k == a then
+      match f v with
+      | none => l
+      | some b => cons a b l
+    else
+      cons k v (alter a f l)
+```
+
+Correct:
+```lean
+theorem eq_append_cons_of_mem {a : α} {xs : List α} (h : a ∈ xs) :
+    ∃ as bs, xs = as ++ a :: bs ∧ a ∉ as := by
+  induction xs with
+  | nil => cases h
+  | cons x xs ih =>
+    simp at h
+    cases h with
+    | inl h => exact ⟨[], xs, by simp_all⟩
+    | inr h =>
+      by_cases h' : a = x
+      · subst h'
+        exact ⟨[], xs, by simp⟩
+      · obtain ⟨as, bs, rfl, h⟩ := ih h
+        exact ⟨x :: as, bs, rfl, by simp_all⟩
+```
+
+Aligning match arms is allowed, but not required.
+
+Correct:
+```lean
+def mkEqTrans? (h₁? h₂? : Option Expr) : MetaM (Option Expr) :=
+  match h₁?, h₂? with
+  | none, none       => return none
+  | none, some h     => return h
+  | some h, none     => return h
+  | some h₁, some h₂ => mkEqTrans h₁ h₂
+```
+
+Correct:
+```lean
+def mkEqTrans? (h₁? h₂? : Option Expr) : MetaM (Option Expr) :=
+  match h₁?, h₂? with
+  | none, none => return none
+  | none, some h => return h
+  | some h, none => return h
+  | some h₁, some h₂ => mkEqTrans h₁ h₂
+```
+
+Correct:
+```lean
+def mkEqTrans? (h₁? h₂? : Option Expr) : MetaM (Option Expr) :=
+  match h₁?, h₂? with
+  | none,    none    => return none
+  | none,    some h  => return h
+  | some h,  none    => return h
+  | some h₁, some h₂ => mkEqTrans h₁ h₂
+```
+
+### Structures
+
+Note to external contributors: this is a section where the Lean style and the mathlib style are different.
+
+When using structure instance syntax over multiple lines, the opening brace should go on the preceding line, while the closing brace should go on its own line. The rest of the syntax should be indented by one level. During structure updates, the `with` clause goes on the same line as the opening brace. Aligning at the assignment symbol is allowed but not required.
+
+Correct:
+```lean
+def addConstAsync (env : Environment) (constName : Name) (kind : ConstantKind) (reportExts := true) :
+    IO AddConstAsyncResult := do
+  let sigPromise ← IO.Promise.new
+  let infoPromise ← IO.Promise.new
+  let extensionsPromise ← IO.Promise.new
+  let checkedEnvPromise ← IO.Promise.new
+  let asyncConst := {
+    constInfo := {
+      name := constName
+      kind
+      sig := sigPromise.result
+      constInfo := infoPromise.result
+    }
+    exts? := guard reportExts *> some extensionsPromise.result
+  }
+  return {
+    constName, kind
+    mainEnv := { env with
+      asyncConsts := env.asyncConsts.add asyncConst
+      checked := checkedEnvPromise.result }
+    asyncEnv := { env with
+      asyncCtx? := some { declPrefix := privateToUserName constName.eraseMacroScopes }
+    }
+    sigPromise, infoPromise, extensionsPromise, checkedEnvPromise
+  }
+```
+
+Correct:
+```lean
+instance [Inhabited α] : Inhabited (Descr α β σ) where
+  default := {
+    name         := default
+    mkInitial    := default
+    ofOLeanEntry := default
+    toOLeanEntry := default
+    addEntry     := fun s _ => s
+  }
+```
+
+### Declaring structures
+
+When defining structure types, do not parenthesize structure fields.
+
+When declaring a structure type with a custom constructor name, put the custom name on its own line, indented like the
+structure fields, and add a documentation comment.
+
+Correct:
+
+```lean
+/--
+A bitvector of the specified width.
+
+This is represented as the underlying `Nat` number in both the runtime
+and the kernel, inheriting all the special support for `Nat`.
+-/
+structure BitVec (w : Nat) where
+  /--
+  Constructs a `BitVec w` from a number less than `2^w`.
+  O(1), because we use `Fin` as the internal representation of a bitvector.
+  -/
+  ofFin ::
+  /--
+  Interprets a bitvector as a number less than `2^w`.
+  O(1), because we use `Fin` as the internal representation of a bitvector.
+  -/
+  toFin : Fin (2 ^ w)
+```
+
+## Tactic proofs
+
+Tactic proofs are the most common thing to break during any kind of upgrade, so it is important to write them in a way that minimizes the likelihood of proofs breaking and that makes it easy to debug breakages if they do occur.
+
+If there are multiple goals, either use a tactic combinator (like `all_goals`) to operate on all of them or a clearly specified subset, or use focus dots to work on goals one at a time. Using structured proofs (e.g., `induction … with`) is encouraged but not mandatory.
+
+Squeeze non-terminal `simp`s (i.e., calls to `simp` which do not close the goal). Squeezing terminal `simp`s is generally discouraged, although there are exceptions (for example if squeezing yields a noticeable performance improvement).
+
+Do not over-golf proofs in ways that are likely to lead to hard-to-debug breakage. Examples of things to avoid include complex multi-goal manipulation using lots of tactic combinators, complex uses of the substitution operator (`▸`) and clever point-free expressions (possibly involving anonymous function notation for multiple arguments).
+
+Do not under-golf proofs: for routine tasks, use the most powerful tactics available.
+
+Do not use `erw`. Avoid using `rfl` after `simp` or `rw`, as this usually indicates a missing lemma that should be used instead of `rfl`.
+
+Use `(d)simp` or `rw` instead of `delta` or `unfold`. Use `refine` instead of `refine’`. Use `haveI` and `letI` only if they are actually required.
+
+Prefer highly automated tactics (like `grind` and `omega`) over low-level proofs, unless the automated tactic requires unacceptable additional imports or has bad performance. If you decide against using a highly automated tactic, leave a comment explaining the decision.
+
+## `do` notation
+
+The `do` keyword goes on the same line as the corresponding `:=` (or `=>`, or similar). `Id.run do` should be treated as if it was a bare `do`.
+
+Use early `return` statements to reduce nesting depth and make the non-exceptional control flow of a function easier to see.
+
+Alternatives for `let` matches may be placed in the same line or in the next line, indented by two spaces. If the term that is
+being matched on is itself more than one line and there is an alternative present, consider breaking immediately after `←` and indent
+as far as necessary to ensure readability.
+
+Correct:
+```lean
+def getFunDecl (fvarId : FVarId) : CompilerM FunDecl := do
+  let some decl ← findFunDecl? fvarId | throwError "unknown local function {fvarId.name}"
+  return decl
+```
+
+Correct:
+```lean
+def getFunDecl (fvarId : FVarId) : CompilerM FunDecl := do
+  let some decl ←
+        findFunDecl? fvarId
+    | throwError "unknown local function {fvarId.name}"
+  return decl
+```
+
+Correct:
+```lean
+def getFunDecl (fvarId : FVarId) : CompilerM FunDecl := do
+  let some decl ← findFunDecl?
+      fvarId
+    | throwError "unknown local function {fvarId.name}"
+  return decl
+```
+
+Correct:
+```lean
+def tagUntaggedGoals (parentTag : Name) (newSuffix : Name) (newGoals : List MVarId) : TacticM Unit := do
+  let mctx ← getMCtx
+  let mut numAnonymous := 0
+  for g in newGoals do
+    if mctx.isAnonymousMVar g then
+      numAnonymous := numAnonymous + 1
+  modifyMCtx fun mctx => Id.run do
+    let mut mctx := mctx
+    let mut idx  := 1
+    for g in newGoals do
+      if mctx.isAnonymousMVar g then
+        if numAnonymous == 1 then
+          mctx := mctx.setMVarUserName g parentTag
+        else
+          mctx := mctx.setMVarUserName g (parentTag ++ newSuffix.appendIndexAfter idx)
+        idx := idx + 1
+    pure mctx
+```
+

doc/std/vision.md

@@ -13,16 +13,17 @@ as part of verified applications.
 The standard library is a public API that contains the components listed in the
 standard library outline below. Not all public APIs in the Lean distribution
 are part of the standard library, and the standard library does not correspond
-to a certain directory within the Lean source repository. For example, the
-metaprogramming framework is not part of the standard library.
+to a certain directory within the Lean source repository (like `Std`). For
+example, the metaprogramming framework is not part of the standard library, but
+basic types like `True` and `Nat` are.
 
 The standard library is under active development. Our guiding principles are:
 
-* Provide comprehensive, verified building blocks for real-world software.  
-* Build a public API of the highest quality with excellent internal consistency.  
-* Carefully optimize components that may be used in performance-critical software.  
-* Ensure smooth adoption and maintenance for users.  
-* Offer excellent documentation, example projects, and guides.  
+* Provide comprehensive, verified building blocks for real-world software.
+* Build a public API of the highest quality with excellent internal consistency.
+* Carefully optimize components that may be used in performance-critical software.
+* Ensure smooth adoption and maintenance for users.
+* Offer excellent documentation, example projects, and guides.
 * Provide a reliable and extensible basis that libraries for software
   development, software verification and mathematics can build on.
 
@@ -32,23 +33,23 @@ call for contributions below.
 
 ### Standard library outline
 
-1. Core types and operations  
-   1. Basic types  
-   2. Numeric types, including floating point numbers  
-   3. Containers  
-   4. Strings and formatting  
-2. Language constructs  
-   1. Ranges and iterators  
-   2. Comparison, ordering, hashing and related type classes  
-   3. Basic monad infrastructure  
-3. Libraries  
-   1. Random numbers  
-   2. Dates and times  
-4. Operating system abstractions  
-   1. Concurrency and parallelism primitives  
-   2. Asynchronous I/O  
-   3. FFI helpers  
-   4. Environment, file system, processes  
+1. Core types and operations
+   1. Basic types
+   2. Numeric types, including floating point numbers
+   3. Containers
+   4. Strings and formatting
+2. Language constructs
+   1. Ranges and iterators
+   2. Comparison, ordering, hashing and related type classes
+   3. Basic monad infrastructure
+3. Libraries
+   1. Random numbers
+   2. Dates and times
+4. Operating system abstractions
+   1. Concurrency and parallelism primitives
+   2. Asynchronous I/O
+   3. FFI helpers
+   4. Environment, file system, processes
    5. Locales
 
 The material covered in the first three sections (core types and operations,

flake.lock

@@ -36,17 +36,17 @@
     },
     "nixpkgs-cadical": {
       "locked": {
-        "lastModified": 1722221733,
-        "narHash": "sha256-sga9SrrPb+pQJxG1ttJfMPheZvDOxApFfwXCFO0H9xw=",
+        "lastModified": 1740791350,
+        "narHash": "sha256-igS2Z4tVw5W/x3lCZeeadt0vcU9fxtetZ/RyrqsCRQ0=",
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "12bf09802d77264e441f48e25459c10c93eada2e",
+        "rev": "199169a2135e6b864a888e89a2ace345703c025d",
         "type": "github"
       },
       "original": {
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "12bf09802d77264e441f48e25459c10c93eada2e",
+        "rev": "199169a2135e6b864a888e89a2ace345703c025d",
         "type": "github"
       }
     },
@@ -67,12 +67,30 @@
         "type": "github"
       }
     },
+    "nixpkgs-older": {
+      "flake": false,
+      "locked": {
+        "lastModified": 1523316493,
+        "narHash": "sha256-5qJS+i5ECICPAKA6FhPLIWkhPKDnOZsZbh2PHYF1Kbs=",
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "rev": "0b307aa73804bbd7a7172899e59ae0b8c347a62d",
+        "type": "github"
+      },
+      "original": {
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "rev": "0b307aa73804bbd7a7172899e59ae0b8c347a62d",
+        "type": "github"
+      }
+    },
     "root": {
       "inputs": {
         "flake-utils": "flake-utils",
         "nixpkgs": "nixpkgs",
         "nixpkgs-cadical": "nixpkgs-cadical",
-        "nixpkgs-old": "nixpkgs-old"
+        "nixpkgs-old": "nixpkgs-old",
+        "nixpkgs-older": "nixpkgs-older"
       }
     },
     "systems": {

flake.nix

@@ -5,17 +5,20 @@
   # old nixpkgs used for portable release with older glibc (2.27)
   inputs.nixpkgs-old.url = "github:NixOS/nixpkgs/nixos-19.03";
   inputs.nixpkgs-old.flake = false;
-  # for cadical 1.9.5; sync with CMakeLists.txt
-  inputs.nixpkgs-cadical.url = "github:NixOS/nixpkgs/12bf09802d77264e441f48e25459c10c93eada2e";
+  # old nixpkgs used for portable release with older glibc (2.26)
+  inputs.nixpkgs-older.url = "github:NixOS/nixpkgs/0b307aa73804bbd7a7172899e59ae0b8c347a62d";
+  inputs.nixpkgs-older.flake = false;
+  # for cadical 2.1.2; sync with CMakeLists.txt by taking commit from https://www.nixhub.io/packages/cadical
+  inputs.nixpkgs-cadical.url = "github:NixOS/nixpkgs/199169a2135e6b864a888e89a2ace345703c025d";
   inputs.flake-utils.url = "github:numtide/flake-utils";
 
-  outputs = { self, nixpkgs, nixpkgs-old, flake-utils, ... }@inputs: flake-utils.lib.eachDefaultSystem (system:
+  outputs = inputs: inputs.flake-utils.lib.eachDefaultSystem (system:
     let
-      pkgs = import nixpkgs { inherit system; };
+      pkgs = import inputs.nixpkgs { inherit system; };
       # An old nixpkgs for creating releases with an old glibc
-      pkgsDist-old = import nixpkgs-old { inherit system; };
+      pkgsDist-old = import inputs.nixpkgs-older { inherit system; };
       # An old nixpkgs for creating releases with an old glibc
-      pkgsDist-old-aarch = import nixpkgs-old { localSystem.config = "aarch64-unknown-linux-gnu"; };
+      pkgsDist-old-aarch = import inputs.nixpkgs-old { localSystem.config = "aarch64-unknown-linux-gnu"; };
       pkgsCadical = import inputs.nixpkgs-cadical { inherit system; };
       cadical = if pkgs.stdenv.isLinux then
         # use statically-linked cadical on Linux to avoid glibc versioning troubles

script/prepare-llvm-mingw.sh

@@ -25,7 +25,10 @@ cp llvm/lib/clang/*/include/{std*,__std*,limits}.h stage1/include/clang
 echo '
 // https://docs.microsoft.com/en-us/windows/win32/api/errhandlingapi/nf-errhandlingapi-seterrormode
 #define SEM_FAILCRITICALERRORS 0x0001
-__declspec(dllimport) __stdcall unsigned int SetErrorMode(unsigned int uMode);' > stage1/include/clang/windows.h
+__declspec(dllimport) __stdcall unsigned int SetErrorMode(unsigned int uMode);
+// https://docs.microsoft.com/en-us/windows/console/setconsoleoutputcp
+#define CP_UTF8 65001
+__declspec(dllimport) __stdcall int SetConsoleOutputCP(unsigned int wCodePageID);' > stage1/include/clang/windows.h
 # COFF dependencies
 cp /clang64/lib/{crtbegin,crtend,crt2,dllcrt2}.o stage1/lib/
 # runtime

script/release_notes.py

@@ -65,20 +65,21 @@ def format_markdown_description(pr_number, description):
     link = f"[#{pr_number}](https://github.com/leanprover/lean4/pull/{pr_number})"
     return f"{link} {description}"
 
+def commit_types():
+    # see doc/dev/commit_convention.md
+    return ['feat', 'fix', 'doc', 'style', 'refactor', 'test', 'chore', 'perf']
+
 def count_commit_types(commits):
     counts = {
         'total': len(commits),
-        'feat': 0,
-        'fix': 0,
-        'refactor': 0,
-        'doc': 0,
-        'chore': 0
     }
+    for commit_type in commit_types():
+        counts[commit_type] = 0
     
     for _, first_line, _ in commits:
-        for commit_type in ['feat:', 'fix:', 'refactor:', 'doc:', 'chore:']:
-            if first_line.startswith(commit_type):
-                counts[commit_type.rstrip(':')] += 1
+        for commit_type in commit_types():
+            if first_line.startswith(f'{commit_type}:'):
+                counts[commit_type] += 1
                 break
     
     return counts
@@ -158,8 +159,9 @@ def main():
     counts = count_commit_types(commits)
     print(f"For this release, {counts['total']} changes landed. "
           f"In addition to the {counts['feat']} feature additions and {counts['fix']} fixes listed below "
-          f"there were {counts['refactor']} refactoring changes, {counts['doc']} documentation improvements "
-          f"and {counts['chore']} chores.\n")
+          f"there were {counts['refactor']} refactoring changes, {counts['doc']} documentation improvements, "
+          f"{counts['perf']} performance improvements, {counts['test']} improvements to the test suite "
+          f"and {counts['style'] + counts['chore']} other changes.\n")
 
     section_order = sort_sections_order()
     sorted_changelog = sorted(changelog.items(), key=lambda item: section_order.index(format_section_title(item[0])) if format_section_title(item[0]) in section_order else len(section_order))

src/CMakeLists.txt

@@ -10,7 +10,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 18)
+set(LEAN_VERSION_MINOR 19)
 set(LEAN_VERSION_PATCH 0)
 set(LEAN_VERSION_IS_RELEASE 0)  # This number is 1 in the release revision, and 0 otherwise.
 set(LEAN_SPECIAL_VERSION_DESC "" CACHE STRING "Additional version description like 'nightly-2018-03-11'")
@@ -144,11 +144,12 @@ if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
   # do not import the world from windows.h using appropriately named flag
   string(APPEND LEAN_EXTRA_CXX_FLAGS " -D WIN32_LEAN_AND_MEAN")
   # DLLs must go next to executables on Windows
-  set(CMAKE_LIBRARY_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/bin")
+  set(CMAKE_RELATIVE_LIBRARY_OUTPUT_DIRECTORY "bin")
 else()
-  set(CMAKE_LIBRARY_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/lib/lean")
+  set(CMAKE_RELATIVE_LIBRARY_OUTPUT_DIRECTORY "lib/lean")
 endif()
 
+set(CMAKE_LIBRARY_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/${CMAKE_RELATIVE_LIBRARY_OUTPUT_DIRECTORY}")
 set(CMAKE_ARCHIVE_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/lib/lean")
 
 # OSX default thread stack size is very small. Moreover, in Debug mode, each new stack frame consumes a lot of extra memory.

src/Init.lean

@@ -39,3 +39,4 @@ import Init.While
 import Init.Syntax
 import Init.Internal
 import Init.Try
+import Init.BinderNameHint

src/Init/BinderNameHint.lean

@@ -0,0 +1,42 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+
+prelude
+import Init.Prelude
+import Init.Tactics
+
+set_option linter.unusedVariables false in
+/--
+The expression `binderNameHint v binder e` defined to be `e`.
+
+If it is used on the right-hand side of an equation that is used for rewriting by `rw` or `simp`,
+and `v` is a local variable, and `binder` is an expression that (after beta-reduction) is a binder
+(`fun w => …` or `∀ w, …`), then it will rename `v` to the name used in that binder, and remove
+the `binderNameHint`.
+
+A typical use of this gadget would be as follows; the gadget ensures that after rewriting, the local
+variable is still `name`, and not `x`:
+```
+theorem all_eq_not_any_not (l : List α) (p : α → Bool) :
+    l.all p = !l.any fun x => binderNameHint x p (!p x) := sorry
+
+example (names : List String) : names.all (fun name => "Waldo".isPrefixOf name) = true := by
+  rw [all_eq_not_any_not]
+  -- ⊢ (!names.any fun name => !"Waldo".isPrefixOf name) = true
+```
+
+If `binder` is not a binder, then the name of `v` attains a macro scope. This only matters when the
+resulting term is used in a non-hygienic way, e.g. in termination proofs for well-founded recursion.
+
+This gadget is supported by
+* `simp`, `dsimp` and `rw` in the right-hand-side of an equation
+* `simp` in the assumptions of congruence rules
+
+It is ineffective in other positions (hyptheses of rewrite rules) or when used by other tactics
+(e.g. `apply`).
+-/
+@[simp ↓]
+def binderNameHint {α : Sort u} {β : Sort v} {γ : Sort w} (v : α) (binder : β) (e : γ) : γ := e

src/Init/ByCases.lean

@@ -38,7 +38,8 @@ theorem apply_ite (f : α → β) (P : Prop) [Decidable P] (x y : α) :
   apply_dite f P (fun _ => x) (fun _ => y)
 
 /-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/
-@[simp] theorem dite_eq_ite [Decidable P] : (dite P (fun _ => a) fun _ => b) = ite P a b := rfl
+@[simp] theorem dite_eq_ite [Decidable P] :
+  (dite P (fun _ => a) (fun _ => b)) = ite P a b := rfl
 
 @[deprecated "Use `ite_eq_right_iff`" (since := "2024-09-18")]
 theorem ite_some_none_eq_none [Decidable P] :

src/Init/Classical.lean

@@ -195,7 +195,7 @@ end Classical
 /- Export for Mathlib compat. -/
 export Classical (imp_iff_right_iff imp_and_neg_imp_iff and_or_imp not_imp)
 
-/-- Extract an element from a existential statement, using `Classical.choose`. -/
+/-- Extract an element from an existential statement, using `Classical.choose`. -/
 -- This enables projection notation.
 @[reducible] noncomputable def Exists.choose {p : α → Prop} (P : ∃ a, p a) : α := Classical.choose P
 

src/Init/Control/Basic.lean

@@ -5,6 +5,7 @@ Author: Leonardo de Moura, Sebastian Ullrich
 -/
 prelude
 import Init.Core
+import Init.BinderNameHint
 
 universe u v w
 
@@ -35,6 +36,12 @@ instance (priority := 500) instForInOfForIn' [ForIn' m ρ α d] : ForIn m ρ α
   simp [h]
   rfl
 
+@[wf_preprocess] theorem forIn_eq_forin' [d : Membership α ρ] [ForIn' m ρ α d] {β} [Monad m]
+    (x : ρ) (b : β) (f : (a : α) → β → m (ForInStep β)) :
+    forIn x b f = forIn' x b (fun x h => binderNameHint x f <| binderNameHint h () <| f x) := by
+  simp [binderNameHint]
+  rfl -- very strange why `simp` did not close it
+
 /-- Extract the value from a `ForInStep`, ignoring whether it is `done` or `yield`. -/
 def ForInStep.value (x : ForInStep α) : α :=
   match x with
@@ -71,7 +78,7 @@ Error recovery and state can interact subtly. For example, the implementation of
 -/
 -- NB: List instance is in mathlib. Once upstreamed, add
 -- * `List`, where `failure` is the empty list and `<|>` concatenates.
-class Alternative (f : Type u → Type v) extends Applicative f : Type (max (u+1) v) where
+class Alternative (f : Type u → Type v) : Type (max (u+1) v) extends Applicative f where
   /--
   Produces an empty collection or recoverable failure.  The `<|>` operator collects values or recovers
   from failures. See `Alternative` for more details.

src/Init/Control/Id.lean

@@ -10,6 +10,28 @@ import Init.Core
 
 universe u
 
+/--
+The identity function on types, used primarily for its `Monad` instance.
+
+The identity monad is useful together with monad transformers to construct monads for particular
+purposes. Additionally, it can be used with `do`-notation in order to use control structures such as
+local mutability, `for`-loops, and early returns in code that does not otherwise use monads.
+
+Examples:
+```lean example
+def containsFive (xs : List Nat) : Bool := Id.run do
+  for x in xs do
+    if x == 5 then return true
+  return false
+```
+
+```lean example
+#eval containsFive [1, 3, 5, 7]
+```
+```output
+true
+```
+-/
 def Id (type : Type u) : Type u := type
 
 namespace Id
@@ -20,9 +42,18 @@ instance : Monad Id where
   bind x f := f x
   map f x := f x
 
+/--
+The identity monad has a `bind` operator.
+-/
 def hasBind : Bind Id :=
   inferInstance
 
+/--
+Runs a computation in the identity monad.
+
+This function is the identity function. Because its parameter has type `Id α`, it causes
+`do`-notation in its arguments to use the `Monad Id` instance.
+-/
 @[always_inline, inline]
 protected def run (x : Id α) : α := x
 

src/Init/Control/Lawful/Basic.lean

@@ -47,7 +47,7 @@ pure f <*> pure x = pure (f x)
 u <*> pure y = pure (· y) <*> u
 ```
 -/
-class LawfulApplicative (f : Type u → Type v) [Applicative f] extends LawfulFunctor f : Prop where
+class LawfulApplicative (f : Type u → Type v) [Applicative f] : Prop extends LawfulFunctor f where
   seqLeft_eq  (x : f α) (y : f β)     : x <* y = const β <$> x <*> y
   seqRight_eq (x : f α) (y : f β)     : x *> y = const α id <$> x <*> y
   pure_seq    (g : α → β) (x : f α)   : pure g <*> x = g <$> x
@@ -77,7 +77,7 @@ x >>= f >>= g = x >>= (fun x => f x >>= g)
 
 `LawfulMonad.mk'` is an alternative constructor containing useful defaults for many fields.
 -/
-class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m : Prop where
+class LawfulMonad (m : Type u → Type v) [Monad m] : Prop extends LawfulApplicative m where
   bind_pure_comp (f : α → β) (x : m α) : x >>= (fun a => pure (f a)) = f <$> x
   bind_map       {α β : Type u} (f : m (α → β)) (x : m α) : f >>= (. <$> x) = f <*> x
   pure_bind      (x : α) (f : α → m β) : pure x >>= f = f x

src/Init/Control/Lawful/Lemmas.lean

@@ -9,25 +9,49 @@ import Init.RCases
 import Init.ByCases
 
 -- Mapping by a function with a left inverse is injective.
-theorem map_inj_of_left_inverse [Applicative m] [LawfulApplicative m] {f : α → β}
-    (w : ∃ g : β → α, ∀ x, g (f x) = x) {x y : m α}
-    (h : f <$> x = f <$> y) : x = y := by
-  rcases w with ⟨g, w⟩
-  replace h := congrArg (g <$> ·) h
-  simpa [w] using h
+theorem map_inj_of_left_inverse [Functor m] [LawfulFunctor m] {f : α → β}
+    (w : ∃ g : β → α, ∀ x, g (f x) = x) {x y : m α} :
+    f <$> x = f <$> y ↔ x = y := by
+  constructor
+  · intro h
+    rcases w with ⟨g, w⟩
+    replace h := congrArg (g <$> ·) h
+    simpa [w] using h
+  · rintro rfl
+    rfl
 
 -- Mapping by an injective function is injective, as long as the domain is nonempty.
-theorem map_inj_of_inj [Applicative m] [LawfulApplicative m] [Nonempty α] {f : α → β}
-    (w : ∀ x y, f x = f y → x = y) {x y : m α}
-    (h : f <$> x = f <$> y) : x = y := by
-  apply map_inj_of_left_inverse ?_ h
-  let ⟨a⟩ := ‹Nonempty α›
-  refine ⟨?_, ?_⟩
-  · intro b
-    by_cases p : ∃ a, f a = b
-    · exact Exists.choose p
-    · exact a
-  · intro b
-    simp only [exists_apply_eq_apply, ↓reduceDIte]
-    apply w
-    apply Exists.choose_spec (p := fun a => f a = f b)
+@[simp] theorem map_inj_right_of_nonempty [Functor m] [LawfulFunctor m] [Nonempty α] {f : α → β}
+    (w : ∀ {x y}, f x = f y → x = y) {x y : m α} :
+    f <$> x = f <$> y ↔ x = y := by
+  constructor
+  · intro h
+    apply (map_inj_of_left_inverse ?_).mp h
+    let ⟨a⟩ := ‹Nonempty α›
+    refine ⟨?_, ?_⟩
+    · intro b
+      by_cases p : ∃ a, f a = b
+      · exact Exists.choose p
+      · exact a
+    · intro b
+      simp only [exists_apply_eq_apply, ↓reduceDIte]
+      apply w
+      apply Exists.choose_spec (p := fun a => f a = f b)
+  · rintro rfl
+    rfl
+
+@[simp] theorem map_inj_right [Monad m] [LawfulMonad m]
+    {f : α → β} (h : ∀ {x y : α}, f x = f y → x = y) {x y : m α} :
+    f <$> x = f <$> y ↔ x = y := by
+  by_cases hempty : Nonempty α
+  · exact map_inj_right_of_nonempty h
+  · constructor
+    · intro h'
+      have (z : m α) : z = (do let a ← z; let b ← pure (f a); x) := by
+        conv => lhs; rw [← bind_pure z]
+        congr; funext a
+        exact (hempty ⟨a⟩).elim
+      rw [this x, this y]
+      rw [← bind_assoc, ← map_eq_pure_bind, h', map_eq_pure_bind, bind_assoc]
+    · intro h'
+      rw [h']

src/Init/Core.lean

@@ -226,9 +226,9 @@ structure PSigma {α : Sort u} (β : α → Sort v) where
   (This will usually require a type ascription to determine `β`
   since it is not determined from `a` and `b` alone.) -/
   mk ::
-  /-- The first component of a dependent pair. If `p : @Sigma α β` then `p.1 : α`. -/
+  /-- The first component of a dependent pair. If `p : @PSigma α β` then `p.1 : α`. -/
   fst : α
-  /-- The second component of a dependent pair. If `p : Sigma β` then `p.2 : β p.1`. -/
+  /-- The second component of a dependent pair. If `p : PSigma β` then `p.2 : β p.1`. -/
   snd : β fst
 
 /--
@@ -593,7 +593,9 @@ set_option linter.unusedVariables.funArgs false in
 be available and then calls `f` on the result.
 
 `prio`, if provided, is the priority of the task.
-If `sync` is set to true, `f` is executed on the current thread if `x` has already finished.
+If `sync` is set to true, `f` is executed on the current thread if `x` has already finished and
+otherwise on the thread that `x` finished on. `prio` is ignored in this case. This should only be
+done when executing `f` is cheap and non-blocking.
 -/
 @[noinline, extern "lean_task_map"]
 protected def map (f : α → β) (x : Task α) (prio := Priority.default) (sync := false) : Task β :=
@@ -607,7 +609,9 @@ for the value of `x` to be available and then calls `f` on the result,
 resulting in a new task which is then run for a result.
 
 `prio`, if provided, is the priority of the task.
-If `sync` is set to true, `f` is executed on the current thread if `x` has already finished.
+If `sync` is set to true, `f` is executed on the current thread if `x` has already finished and
+otherwise on the thread that `x` finished on. `prio` is ignored in this case. This should only be
+done when executing `f` is cheap and non-blocking.
 -/
 @[noinline, extern "lean_task_bind"]
 protected def bind (x : Task α) (f : α → Task β) (prio := Priority.default) (sync := false) :
@@ -1921,10 +1925,6 @@ protected abbrev recOnSubsingleton₂
 end
 end Quotient
 
-section
-variable {α : Type u}
-variable (r : α → α → Prop)
-
 instance Quotient.decidableEq {α : Sort u} {s : Setoid α} [d : ∀ (a b : α), Decidable (a ≈ b)]
     : DecidableEq (Quotient s) :=
   fun (q₁ q₂ : Quotient s) =>
@@ -2016,7 +2016,7 @@ free variables. The frontend automatically declares a fresh auxiliary constant `
 
 Warning: by using this feature, the Lean compiler and interpreter become part of your trusted code base.
 This is extra 30k lines of code. More importantly, you will probably not be able to check your development using
-external type checkers (e.g., Trepplein) that do not implement this feature.
+external type checkers that do not implement this feature.
 Keep in mind that if you are using Lean as programming language, you are already trusting the Lean compiler and interpreter.
 So, you are mainly losing the capability of type checking your development using external checkers.
 
@@ -2051,7 +2051,7 @@ decidability instance can be evaluated to `true` using the lean compiler / inter
 
 Warning: by using this feature, the Lean compiler and interpreter become part of your trusted code base.
 This is extra 30k lines of code. More importantly, you will probably not be able to check your development using
-external type checkers (e.g., Trepplein) that do not implement this feature.
+external type checkers that do not implement this feature.
 Keep in mind that if you are using Lean as programming language, you are already trusting the Lean compiler and interpreter.
 So, you are mainly losing the capability of type checking your development using external checkers.
 -/
@@ -2062,7 +2062,7 @@ The axiom `ofReduceNat` is used to perform proofs by reflection. See `reduceBool
 
 Warning: by using this feature, the Lean compiler and interpreter become part of your trusted code base.
 This is extra 30k lines of code. More importantly, you will probably not be able to check your development using
-external type checkers (e.g., Trepplein) that do not implement this feature.
+external type checkers that do not implement this feature.
 Keep in mind that if you are using Lean as programming language, you are already trusting the Lean compiler and interpreter.
 So, you are mainly losing the capability of type checking your development using external checkers.
 -/
@@ -2121,7 +2121,7 @@ class LeftIdentity (op : α → β → β) (o : outParam α) : Prop
 `LawfulLeftIdentify op o` indicates `o` is a verified left identity of
 `op`.
 -/
-class LawfulLeftIdentity (op : α → β → β) (o : outParam α) extends LeftIdentity op o : Prop where
+class LawfulLeftIdentity (op : α → β → β) (o : outParam α) : Prop extends LeftIdentity op o where
   /-- Left identity `o` is an identity. -/
   left_id : ∀ a, op o a = a
 
@@ -2137,7 +2137,7 @@ class RightIdentity (op : α → β → α) (o : outParam β) : Prop
 `LawfulRightIdentify op o` indicates `o` is a verified right identity of
 `op`.
 -/
-class LawfulRightIdentity (op : α → β → α) (o : outParam β) extends RightIdentity op o : Prop where
+class LawfulRightIdentity (op : α → β → α) (o : outParam β) : Prop extends RightIdentity op o where
   /-- Right identity `o` is an identity. -/
   right_id : ∀ a, op a o = a
 
@@ -2147,13 +2147,13 @@ class LawfulRightIdentity (op : α → β → α) (o : outParam β) extends Righ
 This class does not require a proof that `o` is an identity, and is used
 primarily for inferring the identity using class resolution.
 -/
-class Identity (op : α → α → α) (o : outParam α) extends LeftIdentity op o, RightIdentity op o : Prop
+class Identity (op : α → α → α) (o : outParam α) : Prop extends LeftIdentity op o, RightIdentity op o
 
 /--
 `LawfulIdentity op o` indicates `o` is a verified left and right
 identity of `op`.
 -/
-class LawfulIdentity (op : α → α → α) (o : outParam α) extends Identity op o, LawfulLeftIdentity op o, LawfulRightIdentity op o : Prop
+class LawfulIdentity (op : α → α → α) (o : outParam α) : Prop extends Identity op o, LawfulLeftIdentity op o, LawfulRightIdentity op o
 
 /--
 `LawfulCommIdentity` can simplify defining instances of `LawfulIdentity`
@@ -2164,7 +2164,7 @@ This class is intended for simplifying defining instances of
 `LawfulIdentity` and functions needed commutative operations with
 identity should just add a `LawfulIdentity` constraint.
 -/
-class LawfulCommIdentity (op : α → α → α) (o : outParam α) [hc : Commutative op] extends LawfulIdentity op o : Prop where
+class LawfulCommIdentity (op : α → α → α) (o : outParam α) [hc : Commutative op] : Prop extends LawfulIdentity op o where
   left_id a := Eq.trans (hc.comm o a) (right_id a)
   right_id a := Eq.trans (hc.comm a o) (left_id a)
 

src/Init/Data/AC.lean

@@ -39,7 +39,7 @@ class EvalInformation (α : Sort u) (β : Sort v) where
   evalVar : α → Nat → β
 
 def Context.var (ctx : Context α) (idx : Nat) : Variable ctx.op :=
-  ctx.vars.getD idx ⟨ctx.arbitrary, none⟩
+  ctx.vars[idx]?.getD ⟨ctx.arbitrary, none⟩
 
 instance : ContextInformation (Context α) where
   isNeutral ctx x := ctx.var x |>.neutral.isSome

src/Init/Data/Array.lean

@@ -27,3 +27,4 @@ import Init.Data.Array.Range
 import Init.Data.Array.Erase
 import Init.Data.Array.Zip
 import Init.Data.Array.InsertIdx
+import Init.Data.Array.Extract

src/Init/Data/Array/Attach.lean

@@ -9,6 +9,9 @@ import Init.Data.Array.Lemmas
 import Init.Data.Array.Count
 import Init.Data.List.Attach
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 /--
@@ -19,8 +22,8 @@ to apply `f`.
 
 We replace this at runtime with a more efficient version via the `csimp` lemma `pmap_eq_pmapImpl`.
 -/
-def pmap {P : α → Prop} (f : ∀ a, P a → β) (l : Array α) (H : ∀ a ∈ l, P a) : Array β :=
-  (l.toList.pmap f (fun a m => H a (mem_def.mpr m))).toArray
+def pmap {P : α → Prop} (f : ∀ a, P a → β) (xs : Array α) (H : ∀ a ∈ xs, P a) : Array β :=
+  (xs.toList.pmap f (fun a m => H a (mem_def.mpr m))).toArray
 
 /--
 Unsafe implementation of `attachWith`, taking advantage of the fact that the representation of
@@ -51,35 +54,35 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
     l.toArray.pmap f H = (l.pmap f (by simpa using H)).toArray := by
   simp [pmap]
 
-@[simp] theorem toList_attachWith {l : Array α} {P : α → Prop} {H : ∀ x ∈ l, P x} :
-   (l.attachWith P H).toList = l.toList.attachWith P (by simpa [mem_toList] using H) := by
+@[simp] theorem toList_attachWith {xs : Array α} {P : α → Prop} {H : ∀ x ∈ xs, P x} :
+   (xs.attachWith P H).toList = xs.toList.attachWith P (by simpa [mem_toList] using H) := by
   simp [attachWith]
 
-@[simp] theorem toList_attach {α : Type _} {l : Array α} :
-    l.attach.toList = l.toList.attachWith (· ∈ l) (by simp [mem_toList]) := by
+@[simp] theorem toList_attach {xs : Array α} :
+    xs.attach.toList = xs.toList.attachWith (· ∈ xs) (by simp [mem_toList]) := by
   simp [attach]
 
-@[simp] theorem toList_pmap {l : Array α} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ l, P a} :
-    (l.pmap f H).toList = l.toList.pmap f (fun a m => H a (mem_def.mpr m)) := by
+@[simp] theorem toList_pmap {xs : Array α} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ xs, P a} :
+    (xs.pmap f H).toList = xs.toList.pmap f (fun a m => H a (mem_def.mpr m)) := by
   simp [pmap]
 
 /-- Implementation of `pmap` using the zero-copy version of `attach`. -/
-@[inline] private def pmapImpl {P : α → Prop} (f : ∀ a, P a → β) (l : Array α) (H : ∀ a ∈ l, P a) :
-    Array β := (l.attachWith _ H).map fun ⟨x, h'⟩ => f x h'
+@[inline] private def pmapImpl {P : α → Prop} (f : ∀ a, P a → β) (xs : Array α) (H : ∀ a ∈ xs, P a) :
+    Array β := (xs.attachWith _ H).map fun ⟨x, h'⟩ => f x h'
 
 @[csimp] private theorem pmap_eq_pmapImpl : @pmap = @pmapImpl := by
-  funext α β p f L h'
-  cases L
-  simp only [pmap, pmapImpl, List.attachWith_toArray, List.map_toArray, mk.injEq, List.map_attachWith]
+  funext α β p f xs H
+  cases xs
+  simp only [pmap, pmapImpl, List.attachWith_toArray, List.map_toArray, mk.injEq, List.map_attachWith_eq_pmap]
   apply List.pmap_congr_left
   intro a m h₁ h₂
   congr
 
 @[simp] theorem pmap_empty {P : α → Prop} (f : ∀ a, P a → β) : pmap f #[] (by simp) = #[] := rfl
 
-@[simp] theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (l : Array α) (h : ∀ b ∈ l.push a, P b) :
-    pmap f (l.push a) h =
-      (pmap f l (fun a m => by simp at h; exact h a (.inl m))).push (f a (h a (by simp))) := by
+@[simp] theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (xs : Array α) (h : ∀ b ∈ xs.push a, P b) :
+    pmap f (xs.push a) h =
+      (pmap f xs (fun a m => by simp at h; exact h a (.inl m))).push (f a (h a (by simp))) := by
   simp [pmap]
 
 @[simp] theorem attach_empty : (#[] : Array α).attach = #[] := rfl
@@ -94,159 +97,158 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
   simp
 
 @[simp]
-theorem pmap_eq_map (p : α → Prop) (f : α → β) (l : Array α) (H) :
-    @pmap _ _ p (fun a _ => f a) l H = map f l := by
-  cases l; simp
+theorem pmap_eq_map (p : α → Prop) (f : α → β) (xs : Array α) (H) :
+    @pmap _ _ p (fun a _ => f a) xs H = map f xs := by
+  cases xs; simp
 
-theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (l : Array α) {H₁ H₂}
-    (h : ∀ a ∈ l, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f l H₁ = pmap g l H₂ := by
-  cases l
+theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (xs : Array α) {H₁ H₂}
+    (h : ∀ a ∈ xs, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f xs H₁ = pmap g xs H₂ := by
+  cases xs
   simp only [mem_toArray] at h
   simp only [List.pmap_toArray, mk.injEq]
   rw [List.pmap_congr_left _ h]
 
-theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l H) :
-    map g (pmap f l H) = pmap (fun a h => g (f a h)) l H := by
-  cases l
+theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (xs H) :
+    map g (pmap f xs H) = pmap (fun a h => g (f a h)) xs H := by
+  cases xs
   simp [List.map_pmap]
 
-theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (l H) :
-    pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun _ h => H _ (mem_map_of_mem _ h) := by
-  cases l
+theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (xs H) :
+    pmap g (map f xs) H = pmap (fun a h => g (f a) h) xs fun _ h => H _ (mem_map_of_mem _ h) := by
+  cases xs
   simp [List.pmap_map]
 
-theorem attach_congr {l₁ l₂ : Array α} (h : l₁ = l₂) :
-    l₁.attach = l₂.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
+theorem attach_congr {xs ys : Array α} (h : xs = ys) :
+    xs.attach = ys.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
   subst h
   simp
 
-theorem attachWith_congr {l₁ l₂ : Array α} (w : l₁ = l₂) {P : α → Prop} {H : ∀ x ∈ l₁, P x} :
-    l₁.attachWith P H = l₂.attachWith P fun _ h => H _ (w ▸ h) := by
+theorem attachWith_congr {xs ys : Array α} (w : xs = ys) {P : α → Prop} {H : ∀ x ∈ xs, P x} :
+    xs.attachWith P H = ys.attachWith P fun _ h => H _ (w ▸ h) := by
   subst w
   simp
 
-@[simp] theorem attach_push {a : α} {l : Array α} :
-    (l.push a).attach =
-      (l.attach.map (fun ⟨x, h⟩ => ⟨x, mem_push_of_mem a h⟩)).push ⟨a, by simp⟩ := by
-  cases l
+@[simp] theorem attach_push {a : α} {xs : Array α} :
+    (xs.push a).attach =
+      (xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_push_of_mem a h⟩)).push ⟨a, by simp⟩ := by
+  cases xs
   rw [attach_congr (List.push_toArray _ _)]
   simp [Function.comp_def]
 
-@[simp] theorem attachWith_push {a : α} {l : Array α} {P : α → Prop} {H : ∀ x ∈ l.push a, P x} :
-    (l.push a).attachWith P H =
-      (l.attachWith P (fun x h => by simp at H; exact H x (.inl h))).push ⟨a, H a (by simp)⟩ := by
-  cases l
+@[simp] theorem attachWith_push {a : α} {xs : Array α} {P : α → Prop} {H : ∀ x ∈ xs.push a, P x} :
+    (xs.push a).attachWith P H =
+      (xs.attachWith P (fun x h => by simp at H; exact H x (.inl h))).push ⟨a, H a (by simp)⟩ := by
+  cases xs
   simp [attachWith_congr (List.push_toArray _ _)]
 
-theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
-    pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
-  cases l
+theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (xs H) :
+    pmap f xs H = xs.attach.map fun x => f x.1 (H _ x.2) := by
+  cases xs
   simp [List.pmap_eq_map_attach]
 
 @[simp]
-theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l H) :
-    pmap (fun a h => ⟨a, f a h⟩) l H = l.attachWith q (fun x h => f x (H x h)) := by
-  cases l
+theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (xs H) :
+    pmap (fun a h => ⟨a, f a h⟩) xs H = xs.attachWith q (fun x h => f x (H x h)) := by
+  cases xs
   simp [List.pmap_eq_attachWith]
 
-theorem attach_map_coe (l : Array α) (f : α → β) :
-    (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
-  cases l
+theorem attach_map_val (xs : Array α) (f : α → β) :
+    (xs.attach.map fun (i : {i // i ∈ xs}) => f i) = xs.map f := by
+  cases xs
   simp
 
-theorem attach_map_val (l : Array α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
-  attach_map_coe _ _
+@[deprecated attach_map_val (since := "2025-02-17")]
+abbrev attach_map_coe := @attach_map_val
 
-theorem attach_map_subtype_val (l : Array α) : l.attach.map Subtype.val = l := by
-  cases l; simp
+theorem attach_map_subtype_val (xs : Array α) : xs.attach.map Subtype.val = xs := by
+  cases xs; simp
 
-theorem attachWith_map_coe {p : α → Prop} (f : α → β) (l : Array α) (H : ∀ a ∈ l, p a) :
-    ((l.attachWith p H).map fun (i : { i // p i}) => f i) = l.map f := by
-  cases l; simp
+theorem attachWith_map_val {p : α → Prop} (f : α → β) (xs : Array α) (H : ∀ a ∈ xs, p a) :
+    ((xs.attachWith p H).map fun (i : { i // p i}) => f i) = xs.map f := by
+  cases xs; simp
 
-theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : Array α) (H : ∀ a ∈ l, p a) :
-    ((l.attachWith p H).map fun i => f i.val) = l.map f :=
-  attachWith_map_coe _ _ _
+@[deprecated attachWith_map_val (since := "2025-02-17")]
+abbrev attachWith_map_coe := @attachWith_map_val
 
-theorem attachWith_map_subtype_val {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) :
-    (l.attachWith p H).map Subtype.val = l := by
-  cases l; simp
+theorem attachWith_map_subtype_val {p : α → Prop} (xs : Array α) (H : ∀ a ∈ xs, p a) :
+    (xs.attachWith p H).map Subtype.val = xs := by
+  cases xs; simp
 
 @[simp]
-theorem mem_attach (l : Array α) : ∀ x, x ∈ l.attach
+theorem mem_attach (xs : Array α) : ∀ x, x ∈ xs.attach
   | ⟨a, h⟩ => by
     have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
 @[simp]
-theorem mem_attachWith (l : Array α) {q : α → Prop} (H) (x : {x // q x}) :
-    x ∈ l.attachWith q H ↔ x.1 ∈ l := by
-  cases l
+theorem mem_attachWith (xs : Array α) {q : α → Prop} (H) (x : {x // q x}) :
+    x ∈ xs.attachWith q H ↔ x.1 ∈ xs := by
+  cases xs
   simp
 
 @[simp]
-theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
-    b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
+theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {xs H b} :
+    b ∈ pmap f xs H ↔ ∃ (a : _) (h : a ∈ xs), f a (H a h) = b := by
   simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, Subtype.exists, eq_comm]
 
-theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {l H} {a} (h : a ∈ l) :
-    f a (H a h) ∈ pmap f l H := by
+theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {xs H} {a} (h : a ∈ xs) :
+    f a (H a h) ∈ pmap f xs H := by
   rw [mem_pmap]
   exact ⟨a, h, rfl⟩
 
 @[simp]
-theorem size_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : (pmap f l H).size = l.size := by
-  cases l; simp
+theorem size_pmap {p : α → Prop} {f : ∀ a, p a → β} {xs H} : (pmap f xs H).size = xs.size := by
+  cases xs; simp
 
 @[simp]
-theorem size_attach {L : Array α} : L.attach.size = L.size := by
-  cases L; simp
+theorem size_attach {xs : Array α} : xs.attach.size = xs.size := by
+  cases xs; simp
 
 @[simp]
-theorem size_attachWith {p : α → Prop} {l : Array α} {H} : (l.attachWith p H).size = l.size := by
-  cases l; simp
+theorem size_attachWith {p : α → Prop} {xs : Array α} {H} : (xs.attachWith p H).size = xs.size := by
+  cases xs; simp
 
 @[simp]
-theorem pmap_eq_empty_iff {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = #[] ↔ l = #[] := by
-  cases l; simp
+theorem pmap_eq_empty_iff {p : α → Prop} {f : ∀ a, p a → β} {xs H} : pmap f xs H = #[] ↔ xs = #[] := by
+  cases xs; simp
 
 theorem pmap_ne_empty_iff {P : α → Prop} (f : (a : α) → P a → β) {xs : Array α}
     (H : ∀ (a : α), a ∈ xs → P a) : xs.pmap f H ≠ #[] ↔ xs ≠ #[] := by
   cases xs; simp
 
-theorem pmap_eq_self {l : Array α} {p : α → Prop} {hp : ∀ (a : α), a ∈ l → p a}
-    {f : (a : α) → p a → α} : l.pmap f hp = l ↔ ∀ a (h : a ∈ l), f a (hp a h) = a := by
-  cases l; simp [List.pmap_eq_self]
+theorem pmap_eq_self {xs : Array α} {p : α → Prop} {hp : ∀ (a : α), a ∈ xs → p a}
+    {f : (a : α) → p a → α} : xs.pmap f hp = xs ↔ ∀ a (h : a ∈ xs), f a (hp a h) = a := by
+  cases xs; simp [List.pmap_eq_self]
 
 @[simp]
-theorem attach_eq_empty_iff {l : Array α} : l.attach = #[] ↔ l = #[] := by
-  cases l; simp
+theorem attach_eq_empty_iff {xs : Array α} : xs.attach = #[] ↔ xs = #[] := by
+  cases xs; simp
 
-theorem attach_ne_empty_iff {l : Array α} : l.attach ≠ #[] ↔ l ≠ #[] := by
-  cases l; simp
+theorem attach_ne_empty_iff {xs : Array α} : xs.attach ≠ #[] ↔ xs ≠ #[] := by
+  cases xs; simp
 
 @[simp]
-theorem attachWith_eq_empty_iff {l : Array α} {P : α → Prop} {H : ∀ a ∈ l, P a} :
-    l.attachWith P H = #[] ↔ l = #[] := by
-  cases l; simp
+theorem attachWith_eq_empty_iff {xs : Array α} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
+    xs.attachWith P H = #[] ↔ xs = #[] := by
+  cases xs; simp
 
-theorem attachWith_ne_empty_iff {l : Array α} {P : α → Prop} {H : ∀ a ∈ l, P a} :
-    l.attachWith P H ≠ #[] ↔ l ≠ #[] := by
-  cases l; simp
+theorem attachWith_ne_empty_iff {xs : Array α} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
+    xs.attachWith P H ≠ #[] ↔ xs ≠ #[] := by
+  cases xs; simp
 
 @[simp]
-theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) (i : Nat) :
-    (pmap f l h)[i]? = Option.pmap f l[i]? fun x H => h x (mem_of_getElem? H) := by
-  cases l; simp
+theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {xs : Array α} (h : ∀ a ∈ xs, p a) (i : Nat) :
+    (pmap f xs h)[i]? = Option.pmap f xs[i]? fun x H => h x (mem_of_getElem? H) := by
+  cases xs; simp
 
 @[simp]
-theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) {i : Nat}
-    (hi : i < (pmap f l h).size) :
-    (pmap f l h)[i] =
-      f (l[i]'(@size_pmap _ _ p f l h ▸ hi))
-        (h _ (getElem_mem (@size_pmap _ _ p f l h ▸ hi))) := by
-  cases l; simp
+theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {xs : Array α} (h : ∀ a ∈ xs, p a) {i : Nat}
+    (hi : i < (pmap f xs h).size) :
+    (pmap f xs h)[i] =
+      f (xs[i]'(@size_pmap _ _ p f xs h ▸ hi))
+        (h _ (getElem_mem (@size_pmap _ _ p f xs h ▸ hi))) := by
+  cases xs; simp
 
 @[simp]
 theorem getElem?_attachWith {xs : Array α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
@@ -269,40 +271,40 @@ theorem getElem_attach {xs : Array α} {i : Nat} (h : i < xs.attach.size) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
-@[simp] theorem pmap_attach (l : Array α) {p : {x // x ∈ l} → Prop} (f : ∀ a, p a → β) (H) :
-    pmap f l.attach H =
-      l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
+@[simp] theorem pmap_attach (xs : Array α) {p : {x // x ∈ xs} → Prop} (f : ∀ a, p a → β) (H) :
+    pmap f xs.attach H =
+      xs.pmap (P := fun a => ∃ h : a ∈ xs, p ⟨a, h⟩)
         (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
   ext <;> simp
 
-@[simp] theorem pmap_attachWith (l : Array α) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
-    pmap f (l.attachWith q H₁) H₂ =
-      l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
+@[simp] theorem pmap_attachWith (xs : Array α) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
+    pmap f (xs.attachWith q H₁) H₂ =
+      xs.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
         (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
   ext <;> simp
 
-theorem foldl_pmap (l : Array α) {P : α → Prop} (f : (a : α) → P a → β)
-  (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
-    (l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
+theorem foldl_pmap (xs : Array α) {P : α → Prop} (f : (a : α) → P a → β)
+  (H : ∀ (a : α), a ∈ xs → P a) (g : γ → β → γ) (x : γ) :
+    (xs.pmap f H).foldl g x = xs.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
   rw [pmap_eq_map_attach, foldl_map]
 
-theorem foldr_pmap (l : Array α) {P : α → Prop} (f : (a : α) → P a → β)
-  (H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) :
-    (l.pmap f H).foldr g x = l.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
+theorem foldr_pmap (xs : Array α) {P : α → Prop} (f : (a : α) → P a → β)
+  (H : ∀ (a : α), a ∈ xs → P a) (g : β → γ → γ) (x : γ) :
+    (xs.pmap f H).foldr g x = xs.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
   rw [pmap_eq_map_attach, foldr_map]
 
 @[simp] theorem foldl_attachWith
-    (l : Array α) {q : α → Prop} (H : ∀ a, a ∈ l → q a) {f : β → { x // q x} → β} {b} (w : stop = l.size) :
-    (l.attachWith q H).foldl f b 0 stop = l.attach.foldl (fun b ⟨a, h⟩ => f b ⟨a, H _ h⟩) b := by
+    (xs : Array α) {q : α → Prop} (H : ∀ a, a ∈ xs → q a) {f : β → { x // q x} → β} {b} (w : stop = xs.size) :
+    (xs.attachWith q H).foldl f b 0 stop = xs.attach.foldl (fun b ⟨a, h⟩ => f b ⟨a, H _ h⟩) b := by
   subst w
-  rcases l with ⟨l⟩
+  rcases xs with ⟨xs⟩
   simp [List.foldl_attachWith, List.foldl_map]
 
 @[simp] theorem foldr_attachWith
-    (l : Array α) {q : α → Prop} (H : ∀ a, a ∈ l → q a) {f : { x // q x} → β → β} {b} (w : start = l.size) :
-    (l.attachWith q H).foldr f b start 0 = l.attach.foldr (fun a acc => f ⟨a.1, H _ a.2⟩ acc) b := by
+    (xs : Array α) {q : α → Prop} (H : ∀ a, a ∈ xs → q a) {f : { x // q x} → β → β} {b} (w : start = xs.size) :
+    (xs.attachWith q H).foldr f b start 0 = xs.attach.foldr (fun a acc => f ⟨a.1, H _ a.2⟩ acc) b := by
   subst w
-  rcases l with ⟨l⟩
+  rcases xs with ⟨xs⟩
   simp [List.foldr_attachWith, List.foldr_map]
 
 /--
@@ -315,10 +317,10 @@ Unfortunately this can't be applied by `simp` because of the higher order unific
 and even when rewriting we need to specify the function explicitly.
 See however `foldl_subtype` below.
 -/
-theorem foldl_attach (l : Array α) (f : β → α → β) (b : β) :
-    l.attach.foldl (fun acc t => f acc t.1) b = l.foldl f b := by
-  rcases l with ⟨l⟩
-  simp only [List.attach_toArray, List.attachWith_mem_toArray, List.map_attach, size_toArray,
+theorem foldl_attach (xs : Array α) (f : β → α → β) (b : β) :
+    xs.attach.foldl (fun acc t => f acc t.1) b = xs.foldl f b := by
+  rcases xs with ⟨xs⟩
+  simp only [List.attach_toArray, List.attachWith_mem_toArray, List.size_toArray,
     List.length_pmap, List.foldl_toArray', mem_toArray, List.foldl_subtype]
   congr
   ext
@@ -334,93 +336,101 @@ Unfortunately this can't be applied by `simp` because of the higher order unific
 and even when rewriting we need to specify the function explicitly.
 See however `foldr_subtype` below.
 -/
-theorem foldr_attach (l : Array α) (f : α → β → β) (b : β) :
-    l.attach.foldr (fun t acc => f t.1 acc) b = l.foldr f b := by
-  rcases l with ⟨l⟩
-  simp only [List.attach_toArray, List.attachWith_mem_toArray, List.map_attach, size_toArray,
+theorem foldr_attach (xs : Array α) (f : α → β → β) (b : β) :
+    xs.attach.foldr (fun t acc => f t.1 acc) b = xs.foldr f b := by
+  rcases xs with ⟨xs⟩
+  simp only [List.attach_toArray, List.attachWith_mem_toArray, List.size_toArray,
     List.length_pmap, List.foldr_toArray', mem_toArray, List.foldr_subtype]
   congr
   ext
   simpa using fun a => List.mem_of_getElem? a
 
-theorem attach_map {l : Array α} (f : α → β) :
-    (l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
-  cases l
+theorem attach_map {xs : Array α} (f : α → β) :
+    (xs.map f).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
+  cases xs
   ext <;> simp
 
-theorem attachWith_map {l : Array α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ l.map f → P b} :
-    (l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
+theorem attachWith_map {xs : Array α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ xs.map f → P b} :
+    (xs.map f).attachWith P H = (xs.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
-  cases l
+  cases xs
   simp [List.attachWith_map]
 
-theorem map_attachWith {l : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
+@[simp] theorem map_attachWith {xs : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a}
     (f : { x // P x } → β) :
-    (l.attachWith P H).map f =
-      l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
-  cases l
+    (xs.attachWith P H).map f = xs.attach.map fun ⟨x, h⟩ => f ⟨x, H _ h⟩ := by
+  cases xs <;> simp_all
+
+theorem map_attachWith_eq_pmap {xs : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a}
+    (f : { x // P x } → β) :
+    (xs.attachWith P H).map f =
+      xs.pmap (fun a (h : a ∈ xs ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
+  cases xs
   ext <;> simp
 
 /-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
-theorem map_attach {l : Array α} (f : { x // x ∈ l } → β) :
-    l.attach.map f = l.pmap (fun a h => f ⟨a, h⟩) (fun _ => id) := by
-  cases l
+theorem map_attach_eq_pmap {xs : Array α} (f : { x // x ∈ xs } → β) :
+    xs.attach.map f = xs.pmap (fun a h => f ⟨a, h⟩) (fun _ => id) := by
+  cases xs
   ext <;> simp
 
-theorem attach_filterMap {l : Array α} {f : α → Option β} :
-    (l.filterMap f).attach = l.attach.filterMap
+@[deprecated map_attach_eq_pmap (since := "2025-02-09")]
+abbrev map_attach := @map_attach_eq_pmap
+
+theorem attach_filterMap {xs : Array α} {f : α → Option β} :
+    (xs.filterMap f).attach = xs.attach.filterMap
       fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩) := by
-  cases l
+  cases xs
   rw [attach_congr (List.filterMap_toArray f _)]
   simp [List.attach_filterMap, List.map_filterMap, Function.comp_def]
 
-theorem attach_filter {l : Array α} (p : α → Bool) :
-    (l.filter p).attach = l.attach.filterMap
+theorem attach_filter {xs : Array α} (p : α → Bool) :
+    (xs.filter p).attach = xs.attach.filterMap
       fun x => if w : p x.1 then some ⟨x.1, mem_filter.mpr ⟨x.2, w⟩⟩ else none := by
-  cases l
+  cases xs
   rw [attach_congr (List.filter_toArray p _)]
   simp [List.attach_filter, List.map_filterMap, Function.comp_def]
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
 
 @[simp]
-theorem filterMap_attachWith {q : α → Prop} {l : Array α} {f : {x // q x} → Option β} (H)
-    (w : stop = (l.attachWith q H).size) :
-    (l.attachWith q H).filterMap f 0 stop = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
+theorem filterMap_attachWith {q : α → Prop} {xs : Array α} {f : {x // q x} → Option β} (H)
+    (w : stop = (xs.attachWith q H).size) :
+    (xs.attachWith q H).filterMap f 0 stop = xs.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
   subst w
-  cases l
+  cases xs
   simp [Function.comp_def]
 
 @[simp]
-theorem filter_attachWith {q : α → Prop} {l : Array α} {p : {x // q x} → Bool} (H)
-    (w : stop = (l.attachWith q H).size) :
-    (l.attachWith q H).filter p 0 stop =
-      (l.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
+theorem filter_attachWith {q : α → Prop} {xs : Array α} {p : {x // q x} → Bool} (H)
+    (w : stop = (xs.attachWith q H).size) :
+    (xs.attachWith q H).filter p 0 stop =
+      (xs.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
   subst w
-  cases l
+  cases xs
   simp [Function.comp_def, List.filter_map]
 
-theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
-    pmap f (pmap g l H₁) H₂ =
-      pmap (α := { x // x ∈ l }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) l.attach
+theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (xs H₁ H₂) :
+    pmap f (pmap g xs H₁) H₂ =
+      pmap (α := { x // x ∈ xs }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) xs.attach
         (fun a _ => H₁ a a.2) := by
-  cases l
+  cases xs
   simp [List.pmap_pmap, List.pmap_map]
 
-@[simp] theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (l₁ l₂ : Array ι)
-    (h : ∀ a ∈ l₁ ++ l₂, p a) :
-    (l₁ ++ l₂).pmap f h =
-      (l₁.pmap f fun a ha => h a (mem_append_left l₂ ha)) ++
-        l₂.pmap f fun a ha => h a (mem_append_right l₁ ha) := by
-  cases l₁
-  cases l₂
+@[simp] theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (xs ys : Array ι)
+    (h : ∀ a ∈ xs ++ ys, p a) :
+    (xs ++ ys).pmap f h =
+      (xs.pmap f fun a ha => h a (mem_append_left ys ha)) ++
+        ys.pmap f fun a ha => h a (mem_append_right xs ha) := by
+  cases xs
+  cases ys
   simp
 
-theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ l₂ : Array α)
-    (h₁ : ∀ a ∈ l₁, p a) (h₂ : ∀ a ∈ l₂, p a) :
-    ((l₁ ++ l₂).pmap f fun a ha => (mem_append.1 ha).elim (h₁ a) (h₂ a)) =
-      l₁.pmap f h₁ ++ l₂.pmap f h₂ :=
-  pmap_append f l₁ l₂ _
+theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (xs ys : Array α)
+    (h₁ : ∀ a ∈ xs, p a) (h₂ : ∀ a ∈ ys, p a) :
+    ((xs ++ ys).pmap f fun a ha => (mem_append.1 ha).elim (h₁ a) (h₂ a)) =
+      xs.pmap f h₁ ++ ys.pmap f h₂ :=
+  pmap_append f xs ys _
 
 @[simp] theorem attach_append (xs ys : Array α) :
     (xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_left ys h⟩) ++
@@ -489,35 +499,35 @@ theorem back?_attach {xs : Array α} :
   simp
 
 @[simp]
-theorem countP_attach (l : Array α) (p : α → Bool) :
-    l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
-  cases l
+theorem countP_attach (xs : Array α) (p : α → Bool) :
+    xs.attach.countP (fun a : {x // x ∈ xs} => p a) = xs.countP p := by
+  cases xs
   simp [Function.comp_def]
 
 @[simp]
-theorem countP_attachWith {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) (q : α → Bool) :
-    (l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
-  cases l
+theorem countP_attachWith {p : α → Prop} (xs : Array α) (H : ∀ a ∈ xs, p a) (q : α → Bool) :
+    (xs.attachWith p H).countP (fun a : {x // p x} => q a) = xs.countP q := by
+  cases xs
   simp
 
 @[simp]
-theorem count_attach [DecidableEq α] (l : Array α) (a : {x // x ∈ l}) :
-    l.attach.count a = l.count ↑a := by
-  rcases l with ⟨l⟩
+theorem count_attach [DecidableEq α] (xs : Array α) (a : {x // x ∈ xs}) :
+    xs.attach.count a = xs.count ↑a := by
+  rcases xs with ⟨xs⟩
   simp only [List.attach_toArray, List.attachWith_mem_toArray, List.count_toArray]
-  rw [List.map_attach, List.count_eq_countP]
+  rw [List.map_attach_eq_pmap, List.count_eq_countP]
   simp only [Subtype.beq_iff]
   rw [List.countP_pmap, List.countP_attach (p := (fun x => x == a.1)), List.count]
 
 @[simp]
-theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
-    (l.attachWith p H).count a = l.count ↑a := by
-  cases l
+theorem count_attachWith [DecidableEq α] {p : α → Prop} (xs : Array α) (H : ∀ a ∈ xs, p a) (a : {x // p x}) :
+    (xs.attachWith p H).count a = xs.count ↑a := by
+  cases xs
   simp
 
-@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (l : Array α) (H₁) :
-    (l.pmap g H₁).countP f =
-      l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
+@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (xs : Array α) (H₁) :
+    (xs.pmap g H₁).countP f =
+      xs.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
   simp [pmap_eq_map_attach, countP_map, Function.comp_def]
 
 /-! ## unattach
@@ -538,43 +548,47 @@ and is ideally subsequently simplified away by `unattach_attach`.
 
 If not, usually the right approach is `simp [Array.unattach, -Array.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) : Array α := l.map (·.val)
+def unattach {α : Type _} {p : α → Prop} (xs : Array { x // p x }) : Array α := xs.map (·.val)
 
 @[simp] theorem unattach_nil {p : α → Prop} : (#[] : Array { x // p x }).unattach = #[] := rfl
-@[simp] theorem unattach_push {p : α → Prop} {a : { x // p x }} {l : Array { x // p x }} :
-    (l.push a).unattach = l.unattach.push a.1 := by
+@[simp] theorem unattach_push {p : α → Prop} {a : { x // p x }} {xs : Array { x // p x }} :
+    (xs.push a).unattach = xs.unattach.push a.1 := by
   simp only [unattach, Array.map_push]
 
-@[simp] theorem size_unattach {p : α → Prop} {l : Array { x // p x }} :
-    l.unattach.size = l.size := by
+@[simp] theorem mem_unattach {p : α → Prop} {xs : Array { x // p x }} {a} :
+    a ∈ xs.unattach ↔ ∃ h : p a, ⟨a, h⟩ ∈ xs := by
+  simp only [unattach, mem_map, Subtype.exists, exists_and_right, exists_eq_right]
+
+@[simp] theorem size_unattach {p : α → Prop} {xs : Array { x // p x }} :
+    xs.unattach.size = xs.size := by
   unfold unattach
   simp
 
-@[simp] theorem _root_.List.unattach_toArray {p : α → Prop} {l : List { x // p x }} :
-    l.toArray.unattach = l.unattach.toArray := by
+@[simp] theorem _root_.List.unattach_toArray {p : α → Prop} {xs : List { x // p x }} :
+    xs.toArray.unattach = xs.unattach.toArray := by
   simp only [unattach, List.map_toArray, List.unattach]
 
-@[simp] theorem toList_unattach {p : α → Prop} {l : Array { x // p x }} :
-    l.unattach.toList = l.toList.unattach := by
+@[simp] theorem toList_unattach {p : α → Prop} {xs : Array { x // p x }} :
+    xs.unattach.toList = xs.toList.unattach := by
   simp only [unattach, toList_map, List.unattach]
 
-@[simp] theorem unattach_attach {l : Array α} : l.attach.unattach = l := by
-  cases l
+@[simp] theorem unattach_attach {xs : Array α} : xs.attach.unattach = xs := by
+  cases xs
   simp only [List.attach_toArray, List.unattach_toArray, List.unattach_attachWith]
 
-@[simp] theorem unattach_attachWith {p : α → Prop} {l : Array α}
-    {H : ∀ a ∈ l, p a} :
-    (l.attachWith p H).unattach = l := by
-  cases l
+@[simp] theorem unattach_attachWith {p : α → Prop} {xs : Array α}
+    {H : ∀ a ∈ xs, p a} :
+    (xs.attachWith p H).unattach = xs := by
+  cases xs
   simp
 
-@[simp] theorem getElem?_unattach {p : α → Prop} {l : Array { x // p x }} (i : Nat) :
-    l.unattach[i]? = l[i]?.map Subtype.val := by
+@[simp] theorem getElem?_unattach {p : α → Prop} {xs : Array { x // p x }} (i : Nat) :
+    xs.unattach[i]? = xs[i]?.map Subtype.val := by
   simp [unattach]
 
 @[simp] theorem getElem_unattach
-    {p : α → Prop} {l : Array { x // p x }} (i : Nat) (h : i < l.unattach.size) :
-    l.unattach[i] = (l[i]'(by simpa using h)).1 := by
+    {p : α → Prop} {xs : Array { x // p x }} (i : Nat) (h : i < xs.unattach.size) :
+    xs.unattach[i] = (xs[i]'(by simpa using h)).1 := by
   simp [unattach]
 
 /-! ### Recognizing higher order functions using a function that only depends on the value. -/
@@ -583,20 +597,20 @@ def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) : Array α
 This lemma identifies folds over arrays of subtypes, where the function only depends on the value, not the proposition,
 and simplifies these to the function directly taking the value.
 -/
-theorem foldl_subtype {p : α → Prop} {l : Array { x // p x }}
+theorem foldl_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : β → { x // p x } → β} {g : β → α → β} {x : β}
     (hf : ∀ b x h, f b ⟨x, h⟩ = g b x) :
-    l.foldl f x = l.unattach.foldl g x := by
-  cases l
+    xs.foldl f x = xs.unattach.foldl g x := by
+  cases xs
   simp only [List.foldl_toArray', List.unattach_toArray]
   rw [List.foldl_subtype] -- Why can't simp do this?
   simp [hf]
 
 /-- Variant of `foldl_subtype` with side condition to check `stop = l.size`. -/
-@[simp] theorem foldl_subtype' {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem foldl_subtype' {p : α → Prop} {xs : Array { x // p x }}
     {f : β → { x // p x } → β} {g : β → α → β} {x : β}
-    (hf : ∀ b x h, f b ⟨x, h⟩ = g b x) (h : stop = l.size) :
-    l.foldl f x 0 stop = l.unattach.foldl g x := by
+    (hf : ∀ b x h, f b ⟨x, h⟩ = g b x) (h : stop = xs.size) :
+    xs.foldl f x 0 stop = xs.unattach.foldl g x := by
   subst h
   rwa [foldl_subtype]
 
@@ -604,20 +618,20 @@ theorem foldl_subtype {p : α → Prop} {l : Array { x // p x }}
 This lemma identifies folds over arrays of subtypes, where the function only depends on the value, not the proposition,
 and simplifies these to the function directly taking the value.
 -/
-theorem foldr_subtype {p : α → Prop} {l : Array { x // p x }}
+theorem foldr_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → β → β} {g : α → β → β} {x : β}
     (hf : ∀ x h b, f ⟨x, h⟩ b = g x b) :
-    l.foldr f x = l.unattach.foldr g x := by
-  cases l
+    xs.foldr f x = xs.unattach.foldr g x := by
+  cases xs
   simp only [List.foldr_toArray', List.unattach_toArray]
   rw [List.foldr_subtype]
   simp [hf]
 
 /-- Variant of `foldr_subtype` with side condition to check `stop = l.size`. -/
-@[simp] theorem foldr_subtype' {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem foldr_subtype' {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → β → β} {g : α → β → β} {x : β}
-    (hf : ∀ x h b, f ⟨x, h⟩ b = g x b) (h : start = l.size) :
-    l.foldr f x start 0 = l.unattach.foldr g x := by
+    (hf : ∀ x h b, f ⟨x, h⟩ b = g x b) (h : start = xs.size) :
+    xs.foldr f x start 0 = xs.unattach.foldr g x := by
   subst h
   rwa [foldr_subtype]
 
@@ -625,60 +639,84 @@ theorem foldr_subtype {p : α → Prop} {l : Array { x // p x }}
 This lemma identifies maps over arrays of subtypes, where the function only depends on the value, not the proposition,
 and simplifies these to the function directly taking the value.
 -/
-@[simp] theorem map_subtype {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem map_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → β} {g : α → β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.map f = l.unattach.map g := by
-  cases l
+    xs.map f = xs.unattach.map g := by
+  cases xs
   simp only [List.map_toArray, List.unattach_toArray]
   rw [List.map_subtype]
   simp [hf]
 
-@[simp] theorem filterMap_subtype {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem filterMap_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → Option β} {g : α → Option β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.filterMap f = l.unattach.filterMap g := by
-  cases l
-  simp only [size_toArray, List.filterMap_toArray', List.unattach_toArray, List.length_unattach,
+    xs.filterMap f = xs.unattach.filterMap g := by
+  cases xs
+  simp only [List.size_toArray, List.filterMap_toArray', List.unattach_toArray, List.length_unattach,
     mk.injEq]
   rw [List.filterMap_subtype]
   simp [hf]
 
-@[simp] theorem findSome?_subtype {p : α → Prop} {l : Array { x // p x }}
+
+@[simp] theorem flatMap_subtype {p : α → Prop} {xs : Array { x // p x }}
+    {f : { x // p x } → Array β} {g : α → Array β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    (xs.flatMap f) = xs.unattach.flatMap g := by
+  cases xs
+  simp only [List.size_toArray, List.flatMap_toArray, List.unattach_toArray, List.length_unattach,
+    mk.injEq]
+  rw [List.flatMap_subtype]
+  simp [hf]
+
+@[simp] theorem findSome?_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → Option β} {g : α → Option β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.findSome? f = l.unattach.findSome? g := by
-  cases l
+    xs.findSome? f = xs.unattach.findSome? g := by
+  cases xs
   simp
   rw [List.findSome?_subtype hf]
 
-@[simp] theorem find?_subtype {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem find?_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    (l.find? f).map Subtype.val = l.unattach.find? g := by
-  cases l
+    (xs.find? f).map Subtype.val = xs.unattach.find? g := by
+  cases xs
   simp
   rw [List.find?_subtype hf]
 
-/-! ### Simp lemmas pushing `unattach` inwards. -/
-
-@[simp] theorem unattach_filter {p : α → Prop} {l : Array { x // p x }}
-    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    (l.filter f).unattach = l.unattach.filter g := by
-  cases l
+@[simp] theorem all_subtype {p : α → Prop} {xs : Array { x // p x }} {f : { x // p x } → Bool} {g : α → Bool}
+    (hf : ∀ x h, f ⟨x, h⟩ = g x) (w : stop = xs.size) :
+    xs.all f 0 stop = xs.unattach.all g := by
+  subst w
+  rcases xs with ⟨xs⟩
   simp [hf]
 
-@[simp] theorem unattach_reverse {p : α → Prop} {l : Array { x // p x }} :
-    l.reverse.unattach = l.unattach.reverse := by
-  cases l
+@[simp] theorem any_subtype {p : α → Prop} {xs : Array { x // p x }} {f : { x // p x } → Bool} {g : α → Bool}
+    (hf : ∀ x h, f ⟨x, h⟩ = g x) (w : stop = xs.size) :
+    xs.any f 0 stop = xs.unattach.any g := by
+  subst w
+  rcases xs with ⟨xs⟩
+  simp [hf]
+
+/-! ### Simp lemmas pushing `unattach` inwards. -/
+
+@[simp] theorem unattach_filter {p : α → Prop} {xs : Array { x // p x }}
+    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    (xs.filter f).unattach = xs.unattach.filter g := by
+  cases xs
+  simp [hf]
+
+@[simp] theorem unattach_reverse {p : α → Prop} {xs : Array { x // p x }} :
+    xs.reverse.unattach = xs.unattach.reverse := by
+  cases xs
   simp
 
-@[simp] theorem unattach_append {p : α → Prop} {l₁ l₂ : Array { x // p x }} :
-    (l₁ ++ l₂).unattach = l₁.unattach ++ l₂.unattach := by
-  cases l₁
-  cases l₂
+@[simp] theorem unattach_append {p : α → Prop} {xs₁ xs₂ : Array { x // p x }} :
+    (xs₁ ++ xs₂).unattach = xs₁.unattach ++ xs₂.unattach := by
+  cases xs₁
+  cases xs₂
   simp
 
-@[simp] theorem unattach_flatten {p : α → Prop} {l : Array (Array { x // p x })} :
-    l.flatten.unattach = (l.map unattach).flatten := by
+@[simp] theorem unattach_flatten {p : α → Prop} {xs : Array (Array { x // p x })} :
+    xs.flatten.unattach = (xs.map unattach).flatten := by
   unfold unattach
-  cases l using array₂_induction
+  cases xs using array₂_induction
   simp only [flatten_toArray, List.map_map, Function.comp_def, List.map_id_fun', id_eq,
     List.map_toArray, List.map_flatten, map_subtype, map_id_fun', List.unattach_toArray, mk.injEq]
   simp only [List.unattach]
@@ -687,4 +725,67 @@ and simplifies these to the function directly taking the value.
     (Array.mkArray n x).unattach = Array.mkArray n x.1 := by
   simp [unattach]
 
+/-! ### Well-founded recursion preprocessing setup -/
+
+@[wf_preprocess] theorem Array.map_wfParam (xs : Array α) (f : α → β) :
+    (wfParam xs).map f = xs.attach.unattach.map f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem Array.map_unattach (P : α → Prop) (xs : Array (Subtype P)) (f : α → β) :
+    xs.unattach.map f = xs.map fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldl_wfParam (xs : Array α) (f : β → α → β) (x : β) :
+    (wfParam xs).foldl f x = xs.attach.unattach.foldl f x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldl_unattach (P : α → Prop) (xs : Array (Subtype P)) (f : β → α → β) (x : β):
+    xs.unattach.foldl f x = xs.foldl (fun s ⟨x, h⟩ =>
+      binderNameHint s f <| binderNameHint x (f s) <| binderNameHint h () <| f s (wfParam x)) x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldr_wfParam (xs : Array α) (f : α → β → β) (x : β) :
+    (wfParam xs).foldr f x = xs.attach.unattach.foldr f x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldr_unattach (P : α → Prop) (xs : Array (Subtype P)) (f : α → β → β) (x : β):
+    xs.unattach.foldr f x = xs.foldr (fun ⟨x, h⟩ s =>
+      binderNameHint x f <| binderNameHint s (f x) <| binderNameHint h () <| f (wfParam x) s) x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filter_wfParam (xs : Array α) (f : α → Bool) :
+    (wfParam xs).filter f = xs.attach.unattach.filter f:= by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filter_unattach (P : α → Prop) (xs : Array (Subtype P)) (f : α → Bool) :
+    xs.unattach.filter f = (xs.filter (fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x))).unattach := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem reverse_wfParam (xs : Array α) :
+    (wfParam xs).reverse = xs.attach.unattach.reverse := by simp [wfParam]
+
+@[wf_preprocess] theorem reverse_unattach (P : α → Prop) (xs : Array (Subtype P)) :
+    xs.unattach.reverse = xs.reverse.unattach := by simp
+
+@[wf_preprocess] theorem filterMap_wfParam (xs : Array α) (f : α → Option β) :
+    (wfParam xs).filterMap f = xs.attach.unattach.filterMap f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filterMap_unattach (P : α → Prop) (xs : Array (Subtype P)) (f : α → Option β) :
+    xs.unattach.filterMap f = xs.filterMap fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem flatMap_wfParam (xs : Array α) (f : α → Array β) :
+    (wfParam xs).flatMap f = xs.attach.unattach.flatMap f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem flatMap_unattach (P : α → Prop) (xs : Array (Subtype P)) (f : α → Array β) :
+    xs.unattach.flatMap f = xs.flatMap fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+
 end Array

src/Init/Data/Array/Basic.lean

@@ -14,12 +14,15 @@ import Init.GetElem
 import Init.Data.List.ToArrayImpl
 import Init.Data.Array.Set
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 universe u v w
 
 /-! ### Array literal syntax -/
 
 /-- Syntax for `Array α`. -/
-syntax "#[" withoutPosition(sepBy(term, ", ")) "]" : term
+syntax (name := «term#[_,]») "#[" withoutPosition(term,*,?) "]" : term
 
 macro_rules
   | `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
@@ -35,62 +38,60 @@ namespace Array
 
 /-! ### Preliminary theorems -/
 
-@[simp] theorem size_set (a : Array α) (i : Nat) (v : α) (h : i < a.size) :
-    (set a i v h).size = a.size :=
+@[simp] theorem size_set (xs : Array α) (i : Nat) (v : α) (h : i < xs.size) :
+    (set xs i v h).size = xs.size :=
   List.length_set ..
 
-@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
+@[simp] theorem size_push (xs : Array α) (v : α) : (push xs v).size = xs.size + 1 :=
   List.length_concat ..
 
-theorem ext (a b : Array α)
-    (h₁ : a.size = b.size)
-    (h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
-    : a = b := by
-  let rec extAux (a b : List α)
-      (h₁ : a.length = b.length)
-      (h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
-      : a = b := by
-    induction a generalizing b with
+theorem ext (xs ys : Array α)
+    (h₁ : xs.size = ys.size)
+    (h₂ : (i : Nat) → (hi₁ : i < xs.size) → (hi₂ : i < ys.size) → xs[i] = ys[i])
+    : xs = ys := by
+  let rec extAux (as bs : List α)
+      (h₁ : as.length = bs.length)
+      (h₂ : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as[i] = bs[i])
+      : as = bs := by
+    induction as generalizing bs with
     | nil =>
-      cases b with
+      cases bs with
       | nil       => rfl
       | cons b bs => rw [List.length_cons] at h₁; injection h₁
     | cons a as ih =>
-      cases b with
+      cases bs with
       | nil => rw [List.length_cons] at h₁; injection h₁
       | cons b bs =>
         have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
         have hz₂ : 0 < (b::bs).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
         have headEq : a = b := h₂ 0 hz₁ hz₂
         have h₁' : as.length = bs.length := by rw [List.length_cons, List.length_cons] at h₁; injection h₁
-        have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get ⟨i, hi₁⟩ = bs.get ⟨i, hi₂⟩ := by
+        have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as[i] = bs[i] := by
           intro i hi₁ hi₂
           have hi₁' : i+1 < (a::as).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
           have hi₂' : i+1 < (b::bs).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
-          have : (a::as).get ⟨i+1, hi₁'⟩ = (b::bs).get ⟨i+1, hi₂'⟩ := h₂ (i+1) hi₁' hi₂'
+          have : (a::as)[i+1] = (b::bs)[i+1] := h₂ (i+1) hi₁' hi₂'
           apply this
         have tailEq : as = bs := ih bs h₁' h₂'
         rw [headEq, tailEq]
-  cases a; cases b
+  cases xs; cases ys
   apply congrArg
   apply extAux
   assumption
   assumption
 
-theorem ext' {as bs : Array α} (h : as.toList = bs.toList) : as = bs := by
-  cases as; cases bs; simp at h; rw [h]
+theorem ext' {xs ys : Array α} (h : xs.toList = ys.toList) : xs = ys := by
+  cases xs; cases ys; simp at h; rw [h]
 
 @[simp] theorem toArrayAux_eq (as : List α) (acc : Array α) : (as.toArrayAux acc).toList = acc.toList ++ as := by
   induction as generalizing acc <;> simp [*, List.toArrayAux, Array.push, List.append_assoc, List.concat_eq_append]
 
--- This does not need to be a simp lemma, as already after the `whnfR` the right hand side is `as`.
-theorem toList_toArray (as : List α) : as.toArray.toList = as := rfl
+@[simp] theorem toArray_toList (xs : Array α) : xs.toList.toArray = xs := rfl
 
-@[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [size]
+@[simp] theorem getElem_toList {xs : Array α} {i : Nat} (h : i < xs.size) : xs.toList[i] = xs[i] := rfl
 
-@[simp] theorem getElem_toList {a : Array α} {i : Nat} (h : i < a.size) : a.toList[i] = a[i] := rfl
-
-@[simp] theorem getElem?_toList {a : Array α} {i : Nat} : a.toList[i]? = a[i]? := rfl
+@[simp] theorem getElem?_toList {xs : Array α} {i : Nat} : xs.toList[i]? = xs[i]? := by
+  simp [getElem?_def]
 
 /-- `a ∈ as` is a predicate which asserts that `a` is in the array `as`. -/
 -- NB: This is defined as a structure rather than a plain def so that a lemma
@@ -107,7 +108,7 @@ theorem mem_def {a : α} {as : Array α} : a ∈ as ↔ a ∈ as.toList :=
 @[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
   simp [mem_def]
 
-@[simp] theorem getElem_mem {l : Array α} {i : Nat} (h : i < l.size) : l[i] ∈ l := by
+@[simp] theorem getElem_mem {xs : Array α} {i : Nat} (h : i < xs.size) : xs[i] ∈ xs := by
   rw [Array.mem_def, ← getElem_toList]
   apply List.getElem_mem
 
@@ -115,21 +116,37 @@ end Array
 
 namespace List
 
-@[simp] theorem toArray_toList (a : Array α) : a.toList.toArray = a := rfl
+@[deprecated Array.toArray_toList (since := "2025-02-17")]
+abbrev toArray_toList := @Array.toArray_toList
 
-@[simp] theorem getElem_toArray {a : List α} {i : Nat} (h : i < a.toArray.size) :
-    a.toArray[i] = a[i]'(by simpa using h) := rfl
+-- This does not need to be a simp lemma, as already after the `whnfR` the right hand side is `as`.
+theorem toList_toArray (as : List α) : as.toArray.toList = as := rfl
 
-@[simp] theorem getElem?_toArray {a : List α} {i : Nat} : a.toArray[i]? = a[i]? := rfl
+@[deprecated toList_toArray (since := "2025-02-17")]
+abbrev _root_.Array.toList_toArray := @List.toList_toArray
 
-@[simp] theorem getElem!_toArray [Inhabited α] {a : List α} {i : Nat} :
-    a.toArray[i]! = a[i]! := rfl
+@[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [Array.size]
+
+@[deprecated size_toArray (since := "2025-02-17")]
+abbrev _root_.Array.size_toArray := @List.size_toArray
+
+@[simp] theorem getElem_toArray {xs : List α} {i : Nat} (h : i < xs.toArray.size) :
+    xs.toArray[i] = xs[i]'(by simpa using h) := rfl
+
+@[simp] theorem getElem?_toArray {xs : List α} {i : Nat} : xs.toArray[i]? = xs[i]? := by
+  simp [getElem?_def]
+
+@[simp] theorem getElem!_toArray [Inhabited α] {xs : List α} {i : Nat} :
+    xs.toArray[i]! = xs[i]! := by
+  simp [getElem!_def]
 
 end List
 
 namespace Array
 
-@[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @toList_toArray
+theorem size_eq_length_toList (xs : Array α) : xs.size = xs.toList.length := rfl
+
+@[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @List.toList_toArray
 
 @[deprecated Array.toList (since := "2024-09-10")] abbrev Array.data := @Array.toList
 
@@ -153,15 +170,15 @@ def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
    `Fin` values are represented as tag pointers in the Lean runtime. Thus,
    `fset` may be slightly slower than `uset`. -/
 @[extern "lean_array_uset"]
-def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
-  a.set i.toNat v h
+def uset (xs : Array α) (i : USize) (v : α) (h : i.toNat < xs.size) : Array α :=
+  xs.set i.toNat v h
 
 @[extern "lean_array_pop"]
-def pop (a : Array α) : Array α where
-  toList := a.toList.dropLast
+def pop (xs : Array α) : Array α where
+  toList := xs.toList.dropLast
 
-@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 := by
-  match a with
+@[simp] theorem size_pop (xs : Array α) : xs.pop.size = xs.size - 1 := by
+  match xs with
   | ⟨[]⟩ => rfl
   | ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
 
@@ -176,15 +193,15 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_fswap"]
-def swap (a : Array α) (i j : @& Nat) (hi : i < a.size := by get_elem_tactic) (hj : j < a.size := by get_elem_tactic) : Array α :=
-  let v₁ := a[i]
-  let v₂ := a[j]
-  let a'  := a.set i v₂
-  a'.set j v₁ (Nat.lt_of_lt_of_eq hj (size_set a i v₂ _).symm)
+def swap (xs : Array α) (i j : @& Nat) (hi : i < xs.size := by get_elem_tactic) (hj : j < xs.size := by get_elem_tactic) : Array α :=
+  let v₁ := xs[i]
+  let v₂ := xs[j]
+  let xs'  := xs.set i v₂
+  xs'.set j v₁ (Nat.lt_of_lt_of_eq hj (size_set xs i v₂ _).symm)
 
-@[simp] theorem size_swap (a : Array α) (i j : Nat) {hi hj} : (a.swap i j hi hj).size = a.size := by
-  show ((a.set i a[j]).set j a[i]
-    (Nat.lt_of_lt_of_eq hj (size_set a i a[j] _).symm)).size = a.size
+@[simp] theorem size_swap (xs : Array α) (i j : Nat) {hi hj} : (xs.swap i j hi hj).size = xs.size := by
+  show ((xs.set i xs[j]).set j xs[i]
+    (Nat.lt_of_lt_of_eq hj (size_set xs i xs[j] _).symm)).size = xs.size
   rw [size_set, size_set]
 
 /--
@@ -194,11 +211,11 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_swap"]
-def swapIfInBounds (a : Array α) (i j : @& Nat) : Array α :=
-  if h₁ : i < a.size then
-  if h₂ : j < a.size then swap a i j
-  else a
-  else a
+def swapIfInBounds (xs : Array α) (i j : @& Nat) : Array α :=
+  if h₁ : i < xs.size then
+  if h₂ : j < xs.size then swap xs i j
+  else xs
+  else xs
 
 @[deprecated swapIfInBounds (since := "2024-11-24")] abbrev swap! := @swapIfInBounds
 
@@ -213,24 +230,24 @@ instance : EmptyCollection (Array α) := ⟨Array.empty⟩
 instance : Inhabited (Array α) where
   default := Array.empty
 
-def isEmpty (a : Array α) : Bool :=
-  a.size = 0
+def isEmpty (xs : Array α) : Bool :=
+  xs.size = 0
 
 @[specialize]
-def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) :
-    ∀ (i : Nat) (_ : i ≤ a.size), Bool
+def isEqvAux (xs ys : Array α) (hsz : xs.size = ys.size) (p : α → α → Bool) :
+    ∀ (i : Nat) (_ : i ≤ xs.size), Bool
   | 0, _ => true
   | i+1, h =>
-    p a[i] (b[i]'(hsz ▸ h)) && isEqvAux a b hsz p i (Nat.le_trans (Nat.le_add_right i 1) h)
+    p xs[i] (ys[i]'(hsz ▸ h)) && isEqvAux xs ys hsz p i (Nat.le_trans (Nat.le_add_right i 1) h)
 
-@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
-  if h : a.size = b.size then
-    isEqvAux a b h p a.size (Nat.le_refl a.size)
+@[inline] def isEqv (xs ys : Array α) (p : α → α → Bool) : Bool :=
+  if h : xs.size = ys.size then
+    isEqvAux xs ys h p xs.size (Nat.le_refl xs.size)
   else
     false
 
 instance [BEq α] : BEq (Array α) :=
-  ⟨fun a b => isEqv a b BEq.beq⟩
+  ⟨fun xs ys => isEqv xs ys BEq.beq⟩
 
 /--
 `ofFn f` with `f : Fin n → α` returns the list whose ith element is `f i`.
@@ -254,76 +271,97 @@ def range' (start size : Nat) (step : Nat := 1) : Array Nat :=
 
 @[inline] protected def singleton (v : α) : Array α := #[v]
 
-def back! [Inhabited α] (a : Array α) : α :=
-  a[a.size - 1]!
+/--
+Return the last element of an array, or panic if the array is empty.
 
-@[deprecated back! (since := "2024-10-31")] abbrev back := @back!
+See `back` for the version that requires a proof the array is non-empty,
+or `back?` for the version that returns an option.
+-/
+def back! [Inhabited α] (xs : Array α) : α :=
+  xs[xs.size - 1]!
 
-def get? (a : Array α) (i : Nat) : Option α :=
-  if h : i < a.size then some a[i] else none
+/--
+Return the last element of an array, given a proof that the array is not empty.
 
-def back? (a : Array α) : Option α :=
-  a[a.size - 1]?
+See `back!` for the version that panics if the array is empty,
+or `back?` for the version that returns an option.
+-/
+def back (xs : Array α) (h : 0 < xs.size := by get_elem_tactic) : α :=
+  xs[xs.size - 1]'(Nat.sub_one_lt_of_lt h)
 
-@[inline] def swapAt (a : Array α) (i : Nat) (v : α) (hi : i < a.size := by get_elem_tactic) : α × Array α :=
-  let e := a[i]
-  let a := a.set i v
-  (e, a)
+/--
+Return the last element of an array, or `none` if the array is empty.
+
+See `back!` for the version that panics if the array is empty,
+or `back` for the version that requires a proof the array is non-empty.
+-/
+def back? (xs : Array α) : Option α :=
+  xs[xs.size - 1]?
+
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
+def get? (xs : Array α) (i : Nat) : Option α :=
+  if h : i < xs.size then some xs[i] else none
+
+@[inline] def swapAt (xs : Array α) (i : Nat) (v : α) (hi : i < xs.size := by get_elem_tactic) : α × Array α :=
+  let e := xs[i]
+  let xs' := xs.set i v
+  (e, xs')
 
 @[inline]
-def swapAt! (a : Array α) (i : Nat) (v : α) : α × Array α :=
-  if h : i < a.size then
-    swapAt a i v
+def swapAt! (xs : Array α) (i : Nat) (v : α) : α × Array α :=
+  if h : i < xs.size then
+    swapAt xs i v
   else
-    have : Inhabited (α × Array α) := ⟨(v, a)⟩
+    have : Inhabited (α × Array α) := ⟨(v, xs)⟩
     panic! ("index " ++ toString i ++ " out of bounds")
 
 /-- `shrink a n` returns the first `n` elements of `a`, implemented by repeatedly popping the last element. -/
-def shrink (a : Array α) (n : Nat) : Array α :=
+def shrink (xs : Array α) (n : Nat) : Array α :=
   let rec loop
-    | 0,   a => a
-    | n+1, a => loop n a.pop
-  loop (a.size - n) a
+    | 0,   xs => xs
+    | n+1, xs => loop n xs.pop
+  loop (xs.size - n) xs
 
 /-- `take a n` returns the first `n` elements of `a`, implemented by copying the first `n` elements. -/
-abbrev take (a : Array α) (n : Nat) : Array α := extract a 0 n
+abbrev take (xs : Array α) (i : Nat) : Array α := extract xs 0 i
 
-@[simp] theorem take_eq_extract (a : Array α) (n : Nat) : a.take n = a.extract 0 n := rfl
+@[simp] theorem take_eq_extract (xs : Array α) (i : Nat) : xs.take i = xs.extract 0 i := rfl
 
 /-- `drop a n` removes the first `n` elements of `a`, implemented by copying the remaining elements. -/
-abbrev drop (a : Array α) (n : Nat) : Array α := extract a n a.size
+abbrev drop (xs : Array α) (i : Nat) : Array α := extract xs i xs.size
 
-@[simp] theorem drop_eq_extract (a : Array α) (n : Nat) : a.drop n = a.extract n a.size := rfl
+@[simp] theorem drop_eq_extract (xs : Array α) (i : Nat) : xs.drop i = xs.extract i xs.size := rfl
 
 @[inline]
-unsafe def modifyMUnsafe [Monad m] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
-  if h : i < a.size then
-    let v                := a[i]
+unsafe def modifyMUnsafe [Monad m] (xs : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
+  if h : i < xs.size then
+    let v                := xs[i]
     -- Replace a[i] by `box(0)`.  This ensures that `v` remains unshared if possible.
     -- Note: we assume that arrays have a uniform representation irrespective
     -- of the element type, and that it is valid to store `box(0)` in any array.
-    let a'               := a.set i (unsafeCast ())
+    let xs'               := xs.set i (unsafeCast ())
     let v ← f v
-    pure <| a'.set i v (Nat.lt_of_lt_of_eq h (size_set a ..).symm)
+    pure <| xs'.set i v (Nat.lt_of_lt_of_eq h (size_set xs ..).symm)
   else
-    pure a
+    pure xs
 
 @[implemented_by modifyMUnsafe]
-def modifyM [Monad m] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
-  if h : i < a.size then
-    let v   := a[i]
+def modifyM [Monad m] (xs : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
+  if h : i < xs.size then
+    let v   := xs[i]
     let v ← f v
-    pure <| a.set i v
+    pure <| xs.set i v
   else
-    pure a
+    pure xs
 
 @[inline]
-def modify (a : Array α) (i : Nat) (f : α → α) : Array α :=
-  Id.run <| modifyM a i f
+def modify (xs : Array α) (i : Nat) (f : α → α) : Array α :=
+  Id.run <| modifyM xs i f
 
+set_option linter.indexVariables false in -- Changing `idx` causes bootstrapping issues, haven't investigated.
 @[inline]
-def modifyOp (self : Array α) (idx : Nat) (f : α → α) : Array α :=
-  self.modify idx f
+def modifyOp (xs : Array α) (idx : Nat) (f : α → α) : Array α :=
+  xs.modify idx f
 
 /--
   We claim this unsafe implementation is correct because an array cannot have more than `usizeSz` elements in our runtime.
@@ -442,17 +480,17 @@ def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
 @[inline]
 unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
   let sz := as.usize
-  let rec @[specialize] map (i : USize) (r : Array NonScalar) : m (Array PNonScalar.{v}) := do
+  let rec @[specialize] map (i : USize) (bs : Array NonScalar) : m (Array PNonScalar.{v}) := do
     if i < sz then
-     let v    := r.uget i lcProof
-     -- Replace r[i] by `box(0)`.  This ensures that `v` remains unshared if possible.
+     let v    := bs.uget i lcProof
+     -- Replace bs[i] by `box(0)`.  This ensures that `v` remains unshared if possible.
      -- Note: we assume that arrays have a uniform representation irrespective
      -- of the element type, and that it is valid to store `box(0)` in any array.
-     let r    := r.uset i default lcProof
+     let bs'    := bs.uset i default lcProof
      let vNew ← f (unsafeCast v)
-     map (i+1) (r.uset i (unsafeCast vNew) lcProof)
+     map (i+1) (bs'.uset i (unsafeCast vNew) lcProof)
     else
-     pure (unsafeCast r)
+     pure (unsafeCast bs)
   unsafeCast <| map 0 (unsafeCast as)
 
 /-- Reference implementation for `mapM` -/
@@ -461,11 +499,11 @@ def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
   -- Note: we cannot use `foldlM` here for the reference implementation because this calls
   -- `bind` and `pure` too many times. (We are not assuming `m` is a `LawfulMonad`)
   let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-    map (i : Nat) (r : Array β) : m (Array β) := do
+    map (i : Nat) (bs : Array β) : m (Array β) := do
       if hlt : i < as.size then
-        map (i+1) (r.push (← f as[i]))
+        map (i+1) (bs.push (← f as[i]))
       else
-        pure r
+        pure bs
   decreasing_by simp_wf; decreasing_trivial_pre_omega
   map 0 (mkEmpty as.size)
 
@@ -646,8 +684,8 @@ def mapIdx {α : Type u} {β : Type v} (f : Nat → α → β) (as : Array α) :
   Id.run <| as.mapIdxM f
 
 /-- Turns `#[a, b]` into `#[(a, 0), (b, 1)]`. -/
-def zipIdx (arr : Array α) (start := 0) : Array (α × Nat) :=
-  arr.mapIdx fun i a => (a, start + i)
+def zipIdx (xs : Array α) (start := 0) : Array (α × Nat) :=
+  xs.mapIdx fun i a => (a, start + i)
 
 @[deprecated zipIdx (since := "2025-01-21")] abbrev zipWithIndex := @zipIdx
 
@@ -664,8 +702,8 @@ def findSome? {α : Type u} {β : Type v} (f : α → Option β) (as : Array α)
   Id.run <| as.findSomeM? f
 
 @[inline]
-def findSome! {α : Type u} {β : Type v} [Inhabited β] (f : α → Option β) (a : Array α) : β :=
-  match a.findSome? f with
+def findSome! {α : Type u} {β : Type v} [Inhabited β] (f : α → Option β) (xs : Array α) : β :=
+  match xs.findSome? f with
   | some b => b
   | none   => panic! "failed to find element"
 
@@ -719,18 +757,18 @@ theorem findIdx?_eq_map_findFinIdx?_val {xs : Array α} {p : α → Bool} :
 def findIdx (p : α → Bool) (as : Array α) : Nat := (as.findIdx? p).getD as.size
 
 @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-def idxOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
-  if h : i < a.size then
-    if a[i] == v then some ⟨i, h⟩
-    else idxOfAux a v (i+1)
+def idxOfAux [BEq α] (xs : Array α) (v : α) (i : Nat) : Option (Fin xs.size) :=
+  if h : i < xs.size then
+    if xs[i] == v then some ⟨i, h⟩
+    else idxOfAux xs v (i+1)
   else none
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 @[deprecated idxOfAux (since := "2025-01-29")]
 abbrev indexOfAux := @idxOfAux
 
-def finIdxOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
-  idxOfAux a v 0
+def finIdxOf? [BEq α] (xs : Array α) (v : α) : Option (Fin xs.size) :=
+  idxOfAux xs v 0
 
 @[deprecated "`Array.indexOf?` has been deprecated, use `idxOf?` or `finIdxOf?` instead." (since := "2025-01-29")]
 abbrev indexOf? := @finIdxOf?
@@ -738,12 +776,12 @@ abbrev indexOf? := @finIdxOf?
 /-- Returns the index of the first element equal to `a`, or the length of the array otherwise. -/
 def idxOf [BEq α] (a : α) : Array α → Nat := findIdx (· == a)
 
-def idxOf? [BEq α] (a : Array α) (v : α) : Option Nat :=
-  (a.finIdxOf? v).map (·.val)
+def idxOf? [BEq α] (xs : Array α) (v : α) : Option Nat :=
+  (xs.finIdxOf? v).map (·.val)
 
 @[deprecated idxOf? (since := "2024-11-20")]
-def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
-  a.findIdx? fun a => a == v
+def getIdx? [BEq α] (xs : Array α) (v : α) : Option Nat :=
+  xs.findIdx? fun a => a == v
 
 @[inline]
 def any (as : Array α) (p : α → Bool) (start := 0) (stop := as.size) : Bool :=
@@ -778,12 +816,12 @@ def toListAppend (as : Array α) (l : List α) : List α :=
   as.foldr List.cons l
 
 protected def append (as : Array α) (bs : Array α) : Array α :=
-  bs.foldl (init := as) fun r v => r.push v
+  bs.foldl (init := as) fun xs v => xs.push v
 
 instance : Append (Array α) := ⟨Array.append⟩
 
 protected def appendList (as : Array α) (bs : List α) : Array α :=
-  bs.foldl (init := as) fun r v => r.push v
+  bs.foldl (init := as) fun xs v => xs.push v
 
 instance : HAppend (Array α) (List α) (Array α) := ⟨Array.appendList⟩
 
@@ -803,8 +841,8 @@ def flatMap (f : α → Array β) (as : Array α) : Array β :=
 
 `flatten #[#[a₁, a₂, ⋯], #[b₁, b₂, ⋯], ⋯]` = `#[a₁, a₂, ⋯, b₁, b₂, ⋯]`
 -/
-@[inline] def flatten (as : Array (Array α)) : Array α :=
-  as.foldl (init := empty) fun r a => r ++ a
+@[inline] def flatten (xss : Array (Array α)) : Array α :=
+  xss.foldl (init := empty) fun acc xs => acc ++ xs
 
 def reverse (as : Array α) : Array α :=
   if h : as.size ≤ 1 then
@@ -826,18 +864,18 @@ where
 
 @[inline]
 def filter (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Array α :=
-  as.foldl (init := #[]) (start := start) (stop := stop) fun r a =>
-    if p a then r.push a else r
+  as.foldl (init := #[]) (start := start) (stop := stop) fun acc a =>
+    if p a then acc.push a else acc
 
 @[inline]
 def filterM {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
-  as.foldlM (init := #[]) (start := start) (stop := stop) fun r a => do
-    if (← p a) then return r.push a else return r
+  as.foldlM (init := #[]) (start := start) (stop := stop) fun acc a => do
+    if (← p a) then return acc.push a else return acc
 
 @[inline]
 def filterRevM {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start := as.size) (stop := 0) : m (Array α) :=
-  reverse <$> as.foldrM (init := #[]) (start := start) (stop := stop) fun a r => do
-    if (← p a) then return r.push a else return r
+  reverse <$> as.foldrM (init := #[]) (start := start) (stop := stop) fun a acc => do
+    if (← p a) then return acc.push a else return acc
 
 @[specialize]
 def filterMapM [Monad m] (f : α → m (Option β)) (as : Array α) (start := 0) (stop := as.size) : m (Array β) :=
@@ -881,17 +919,21 @@ def popWhile (p : α → Bool) (as : Array α) : Array α :=
     as
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
+@[simp] theorem popWhile_empty (p : α → Bool) :
+    popWhile p #[] = #[] := by
+  simp [popWhile]
+
 def takeWhile (p : α → Bool) (as : Array α) : Array α :=
   let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-  go (i : Nat) (r : Array α) : Array α :=
+  go (i : Nat) (acc : Array α) : Array α :=
     if h : i < as.size then
       let a := as[i]
       if p a then
-        go (i+1) (r.push a)
+        go (i+1) (acc.push a)
       else
-        r
+        acc
     else
-      r
+      acc
     decreasing_by simp_wf; decreasing_trivial_pre_omega
   go 0 #[]
 
@@ -902,22 +944,22 @@ using a `Nat` index and a tactic-provided bound.
 This function takes worst case O(n) time because
 it has to backshift all elements at positions greater than `i`.-/
 @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-def eraseIdx (a : Array α) (i : Nat) (h : i < a.size := by get_elem_tactic) : Array α :=
-  if h' : i + 1 < a.size then
-    let a' := a.swap (i + 1) i
-    a'.eraseIdx (i + 1) (by simp [a', h'])
+def eraseIdx (xs : Array α) (i : Nat) (h : i < xs.size := by get_elem_tactic) : Array α :=
+  if h' : i + 1 < xs.size then
+    let xs' := xs.swap (i + 1) i
+    xs'.eraseIdx (i + 1) (by simp [xs', h'])
   else
-    a.pop
-termination_by a.size - i
+    xs.pop
+termination_by xs.size - i
 decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ h
 
 -- This is required in `Lean.Data.PersistentHashMap`.
-@[simp] theorem size_eraseIdx (a : Array α) (i : Nat) (h) : (a.eraseIdx i h).size = a.size - 1 := by
-  induction a, i, h using Array.eraseIdx.induct with
-  | @case1 a i h h' a' ih =>
+@[simp] theorem size_eraseIdx (xs : Array α) (i : Nat) (h) : (xs.eraseIdx i h).size = xs.size - 1 := by
+  induction xs, i, h using Array.eraseIdx.induct with
+  | @case1 xs i h h' xs' ih =>
     unfold eraseIdx
-    simp +zetaDelta [h', a', ih]
-  | case2 a i h h' =>
+    simp +zetaDelta [h', xs', ih]
+  | case2 xs i h h' =>
     unfold eraseIdx
     simp [h']
 
@@ -925,15 +967,15 @@ decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ h
 
   This function takes worst case O(n) time because
   it has to backshift all elements at positions greater than `i`.-/
-def eraseIdxIfInBounds (a : Array α) (i : Nat) : Array α :=
-  if h : i < a.size then a.eraseIdx i h else a
+def eraseIdxIfInBounds (xs : Array α) (i : Nat) : Array α :=
+  if h : i < xs.size then xs.eraseIdx i h else xs
 
 /-- Remove the element at a given index from an array, or panic if the index is out of bounds.
 
 This function takes worst case O(n) time because
 it has to backshift all elements at positions greater than `i`. -/
-def eraseIdx! (a : Array α) (i : Nat) : Array α :=
-  if h : i < a.size then a.eraseIdx i h else panic! "invalid index"
+def eraseIdx! (xs : Array α) (i : Nat) : Array α :=
+  if h : i < xs.size then xs.eraseIdx i h else panic! "invalid index"
 
 /-- Remove a specified element from an array, or do nothing if it is not present.
 
@@ -1050,6 +1092,11 @@ def split (as : Array α) (p : α → Bool) : Array α × Array α :=
   as.foldl (init := (#[], #[])) fun (as, bs) a =>
     if p a then (as.push a, bs) else (as, bs.push a)
 
+def replace [BEq α] (xs : Array α) (a b : α) : Array α :=
+  match xs.finIdxOf? a with
+  | none => xs
+  | some i => xs.set i b
+
 /-! ### Lexicographic ordering -/
 
 instance instLT [LT α] : LT (Array α) := ⟨fun as bs => as.toList < bs.toList⟩
@@ -1062,6 +1109,20 @@ instance instLE [LT α] : LE (Array α) := ⟨fun as bs => as.toList ≤ bs.toLi
 We do not currently intend to provide verification theorems for these functions.
 -/
 
+/-! ### leftpad and rightpad -/
+
+/--
+Pads `l : Array α` on the left with repeated occurrences of `a : α` until it is of size `n`.
+If `l` is initially larger than `n`, just return `l`.
+-/
+def leftpad (n : Nat) (a : α) (xs : Array α) : Array α := mkArray (n - xs.size) a ++ xs
+
+/--
+Pads `l : Array α` on the right with repeated occurrences of `a : α` until it is of size `n`.
+If `l` is initially larger than `n`, just return `l`.
+-/
+def rightpad (n : Nat) (a : α) (xs : Array α) : Array α := xs ++ mkArray (n - xs.size) a
+
 /- ### reduceOption -/
 
 /-- Drop `none`s from a Array, and replace each remaining `some a` with `a`. -/
@@ -1076,9 +1137,9 @@ We do not currently intend to provide verification theorems for these functions.
 -/
 def eraseReps {α} [BEq α] (as : Array α) : Array α :=
   if h : 0 < as.size then
-    let ⟨last, r⟩ := as.foldl (init := (as[0], #[])) fun ⟨last, r⟩ a =>
-      if a == last then ⟨last, r⟩ else ⟨a, r.push last⟩
-    r.push last
+    let ⟨last, acc⟩ := as.foldl (init := (as[0], #[])) fun ⟨last, acc⟩ a =>
+      if a == last then ⟨last, acc⟩ else ⟨a, acc.push last⟩
+    acc.push last
   else
     #[]
 
@@ -1104,24 +1165,24 @@ def allDiff [BEq α] (as : Array α) : Bool :=
 /-! ### getEvenElems -/
 
 @[inline] def getEvenElems (as : Array α) : Array α :=
-  (·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
+  (·.2) <| as.foldl (init := (true, Array.empty)) fun (even, acc) a =>
     if even then
-      (false, r.push a)
+      (false, acc.push a)
     else
-      (true, r)
+      (true, acc)
 
 /-! ### Repr and ToString -/
 
 instance {α : Type u} [Repr α] : Repr (Array α) where
-  reprPrec a _ :=
+  reprPrec xs _ :=
     let _ : Std.ToFormat α := ⟨repr⟩
-    if a.size == 0 then
+    if xs.size == 0 then
       "#[]"
     else
-      Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
+      Std.Format.bracketFill "#[" (Std.Format.joinSep (toList xs) ("," ++ Std.Format.line)) "]"
 
 instance [ToString α] : ToString (Array α) where
-  toString a := "#" ++ toString a.toList
+  toString xs := "#" ++ toString xs.toList
 
 end Array
 

src/Init/Data/Array/BasicAux.lean

@@ -8,6 +8,9 @@ import Init.Data.Array.Basic
 import Init.Data.Nat.Linear
 import Init.NotationExtra
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 theorem Array.of_push_eq_push {as bs : Array α} (h : as.push a = bs.push b) : as = bs ∧ a = b := by
   simp only [push, mk.injEq] at h
   have ⟨h₁, h₂⟩ := List.of_concat_eq_concat h
@@ -17,13 +20,13 @@ theorem Array.of_push_eq_push {as bs : Array α} (h : as.push a = bs.push b) : a
 private theorem List.size_toArrayAux (as : List α) (bs : Array α) : (as.toArrayAux bs).size = as.length + bs.size := by
   induction as generalizing bs with
   | nil => simp [toArrayAux]
-  | cons a as ih => simp_arith [toArrayAux, *]
+  | cons a as ih => simp +arith [toArrayAux, *]
 
 private theorem List.of_toArrayAux_eq_toArrayAux {as bs : List α} {cs ds : Array α} (h : as.toArrayAux cs = bs.toArrayAux ds) (hlen : cs.size = ds.size) : as = bs ∧ cs = ds := by
   match as, bs with
   | [], []    => simp [toArrayAux] at h; simp [h]
-  | a::as, [] => simp [toArrayAux] at h; rw [← h] at hlen; simp_arith [size_toArrayAux] at hlen
-  | [], b::bs => simp [toArrayAux] at h; rw [h] at hlen; simp_arith [size_toArrayAux] at hlen
+  | a::as, [] => simp [toArrayAux] at h; rw [← h] at hlen; simp +arith [size_toArrayAux] at hlen
+  | [], b::bs => simp [toArrayAux] at h; rw [h] at hlen; simp +arith [size_toArrayAux] at hlen
   | a::as, b::bs =>
     simp [toArrayAux] at h
     have : (cs.push a).size = (ds.push b).size := by simp [*]

src/Init/Data/Array/BinSearch.lean

@@ -5,9 +5,13 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.Data.Array.Basic
+import Init.Data.Int.DivMod.Lemmas
 import Init.Omega
 universe u v
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- We do not use `linter.indexVariables` here as it is helpful to name the index variables as `lo`, `mid`, and `hi`.
+
 namespace Array
 
 @[specialize] def binSearchAux {α : Type u} {β : Type v} (lt : α → α → Bool) (found : Option α → β) (as : Array α) (k : α) :

src/Init/Data/Array/Bootstrap.lean

@@ -13,122 +13,151 @@ import Init.Data.List.TakeDrop
 This file contains some theorems about `Array` and `List` needed for `Init.Data.List.Impl`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
+/--
+Use the indexing notation `a[i]` instead.
+
+Access an element from an array without needing a runtime bounds checks,
+using a `Nat` index and a proof that it is in bounds.
+
+This function does not use `get_elem_tactic` to automatically find the proof that
+the index is in bounds. This is because the tactic itself needs to look up values in
+arrays.
+-/
+@[deprecated "Use indexing notation `as[i]` instead" (since := "2025-02-17")]
+def get {α : Type u} (a : @& Array α) (i : @& Nat) (h : LT.lt i a.size) : α :=
+  a.toList.get ⟨i, h⟩
+
+/--
+Use the indexing notation `a[i]!` instead.
+
+Access an element from an array, or panic if the index is out of bounds.
+-/
+@[deprecated "Use indexing notation `as[i]!` instead" (since := "2025-02-17")]
+def get! {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
+  Array.getD a i default
+
 theorem foldlM_toList.aux [Monad m]
-    (f : β → α → m β) (arr : Array α) (i j) (H : arr.size ≤ i + j) (b) :
-    foldlM.loop f arr arr.size (Nat.le_refl _) i j b = (arr.toList.drop j).foldlM f b := by
+    (f : β → α → m β) (xs : Array α) (i j) (H : xs.size ≤ i + j) (b) :
+    foldlM.loop f xs xs.size (Nat.le_refl _) i j b = (xs.toList.drop j).foldlM f b := by
   unfold foldlM.loop
   split; split
   · cases Nat.not_le_of_gt ‹_› (Nat.zero_add _ ▸ H)
   · rename_i i; rw [Nat.succ_add] at H
-    simp [foldlM_toList.aux f arr i (j+1) H]
+    simp [foldlM_toList.aux f xs i (j+1) H]
     rw (occs := [2]) [← List.getElem_cons_drop_succ_eq_drop ‹_›]
     rfl
   · rw [List.drop_of_length_le (Nat.ge_of_not_lt ‹_›)]; rfl
 
 @[simp] theorem foldlM_toList [Monad m]
-    (f : β → α → m β) (init : β) (arr : Array α) :
-    arr.toList.foldlM f init = arr.foldlM f init := by
+    (f : β → α → m β) (init : β) (xs : Array α) :
+    xs.toList.foldlM f init = xs.foldlM f init := by
   simp [foldlM, foldlM_toList.aux]
 
-@[simp] theorem foldl_toList (f : β → α → β) (init : β) (arr : Array α) :
-    arr.toList.foldl f init = arr.foldl f init :=
+@[simp] theorem foldl_toList (f : β → α → β) (init : β) (xs : Array α) :
+    xs.toList.foldl f init = xs.foldl f init :=
   List.foldl_eq_foldlM .. ▸ foldlM_toList ..
 
 theorem foldrM_eq_reverse_foldlM_toList.aux [Monad m]
-    (f : α → β → m β) (arr : Array α) (init : β) (i h) :
-    (arr.toList.take i).reverse.foldlM (fun x y => f y x) init = foldrM.fold f arr 0 i h init := by
+    (f : α → β → m β) (xs : Array α) (init : β) (i h) :
+    (xs.toList.take i).reverse.foldlM (fun x y => f y x) init = foldrM.fold f xs 0 i h init := by
   unfold foldrM.fold
   match i with
   | 0 => simp [List.foldlM, List.take]
-  | i+1 => rw [← List.take_concat_get _ _ h]; simp [← (aux f arr · i)]
+  | i+1 => rw [← List.take_concat_get _ _ h]; simp [← (aux f xs · i)]
 
-theorem foldrM_eq_reverse_foldlM_toList [Monad m] (f : α → β → m β) (init : β) (arr : Array α) :
-    arr.foldrM f init = arr.toList.reverse.foldlM (fun x y => f y x) init := by
-  have : arr = #[] ∨ 0 < arr.size :=
-    match arr with | ⟨[]⟩ => .inl rfl | ⟨a::l⟩ => .inr (Nat.zero_lt_succ _)
-  match arr, this with | _, .inl rfl => rfl | arr, .inr h => ?_
+theorem foldrM_eq_reverse_foldlM_toList [Monad m] (f : α → β → m β) (init : β) (xs : Array α) :
+    xs.foldrM f init = xs.toList.reverse.foldlM (fun x y => f y x) init := by
+  have : xs = #[] ∨ 0 < xs.size :=
+    match xs with | ⟨[]⟩ => .inl rfl | ⟨a::l⟩ => .inr (Nat.zero_lt_succ _)
+  match xs, this with | _, .inl rfl => rfl | xs, .inr h => ?_
   simp [foldrM, h, ← foldrM_eq_reverse_foldlM_toList.aux, List.take_length]
 
 @[simp] theorem foldrM_toList [Monad m]
-    (f : α → β → m β) (init : β) (arr : Array α) :
-    arr.toList.foldrM f init = arr.foldrM f init := by
+    (f : α → β → m β) (init : β) (xs : Array α) :
+    xs.toList.foldrM f init = xs.foldrM f init := by
   rw [foldrM_eq_reverse_foldlM_toList, List.foldlM_reverse]
 
-@[simp] theorem foldr_toList (f : α → β → β) (init : β) (arr : Array α) :
-    arr.toList.foldr f init = arr.foldr f init :=
+@[simp] theorem foldr_toList (f : α → β → β) (init : β) (xs : Array α) :
+    xs.toList.foldr f init = xs.foldr f init :=
   List.foldr_eq_foldrM .. ▸ foldrM_toList ..
 
-@[simp] theorem push_toList (arr : Array α) (a : α) : (arr.push a).toList = arr.toList ++ [a] := by
+@[simp] theorem push_toList (xs : Array α) (a : α) : (xs.push a).toList = xs.toList ++ [a] := by
   simp [push, List.concat_eq_append]
 
-@[simp] theorem toListAppend_eq (arr : Array α) (l) : arr.toListAppend l = arr.toList ++ l := by
+@[simp] theorem toListAppend_eq (xs : Array α) (l : List α) : xs.toListAppend l = xs.toList ++ l := by
   simp [toListAppend, ← foldr_toList]
 
-@[simp] theorem toListImpl_eq (arr : Array α) : arr.toListImpl = arr.toList := by
+@[simp] theorem toListImpl_eq (xs : Array α) : xs.toListImpl = xs.toList := by
   simp [toListImpl, ← foldr_toList]
 
-@[simp] theorem pop_toList (arr : Array α) : arr.pop.toList = arr.toList.dropLast := rfl
+@[simp] theorem toList_pop (xs : Array α) : xs.pop.toList = xs.toList.dropLast := rfl
 
-@[simp] theorem append_eq_append (arr arr' : Array α) : arr.append arr' = arr ++ arr' := rfl
+@[deprecated toList_pop (since := "2025-02-17")]
+abbrev pop_toList := @Array.toList_pop
 
-@[simp] theorem toList_append (arr arr' : Array α) :
-    (arr ++ arr').toList = arr.toList ++ arr'.toList := by
+@[simp] theorem append_eq_append (xs ys : Array α) : xs.append ys = xs ++ ys := rfl
+
+@[simp] theorem toList_append (xs ys : Array α) :
+    (xs ++ ys).toList = xs.toList ++ ys.toList := by
   rw [← append_eq_append]; unfold Array.append
   rw [← foldl_toList]
-  induction arr'.toList generalizing arr <;> simp [*]
+  induction ys.toList generalizing xs <;> simp [*]
 
 @[simp] theorem toList_empty : (#[] : Array α).toList = [] := rfl
 
-@[simp] theorem append_empty (as : Array α) : as ++ #[] = as := by
+@[simp] theorem append_empty (xs : Array α) : xs ++ #[] = xs := by
   apply ext'; simp only [toList_append, toList_empty, List.append_nil]
 
 @[deprecated append_empty (since := "2025-01-13")]
 abbrev append_nil := @append_empty
 
-@[simp] theorem empty_append (as : Array α) : #[] ++ as = as := by
+@[simp] theorem empty_append (xs : Array α) : #[] ++ xs = xs := by
   apply ext'; simp only [toList_append, toList_empty, List.nil_append]
 
 @[deprecated empty_append (since := "2025-01-13")]
 abbrev nil_append := @empty_append
 
-@[simp] theorem append_assoc (as bs cs : Array α) : as ++ bs ++ cs = as ++ (bs ++ cs) := by
+@[simp] theorem append_assoc (xs ys zs : Array α) : xs ++ ys ++ zs = xs ++ (ys ++ zs) := by
   apply ext'; simp only [toList_append, List.append_assoc]
 
 @[simp] theorem appendList_eq_append
-    (arr : Array α) (l : List α) : arr.appendList l = arr ++ l := rfl
+    (xs : Array α) (l : List α) : xs.appendList l = xs ++ l := rfl
 
-@[simp] theorem toList_appendList (arr : Array α) (l : List α) :
-    (arr ++ l).toList = arr.toList ++ l := by
+@[simp] theorem toList_appendList (xs : Array α) (l : List α) :
+    (xs ++ l).toList = xs.toList ++ l := by
   rw [← appendList_eq_append]; unfold Array.appendList
-  induction l generalizing arr <;> simp [*]
+  induction l generalizing xs <;> simp [*]
 
 @[deprecated toList_appendList (since := "2024-12-11")]
 abbrev appendList_toList := @toList_appendList
 
 @[deprecated "Use the reverse direction of `foldrM_toList`." (since := "2024-11-13")]
 theorem foldrM_eq_foldrM_toList [Monad m]
-    (f : α → β → m β) (init : β) (arr : Array α) :
-    arr.foldrM f init = arr.toList.foldrM f init := by
+    (f : α → β → m β) (init : β) (xs : Array α) :
+    xs.foldrM f init = xs.toList.foldrM f init := by
   simp
 
 @[deprecated "Use the reverse direction of `foldlM_toList`." (since := "2024-11-13")]
 theorem foldlM_eq_foldlM_toList [Monad m]
-    (f : β → α → m β) (init : β) (arr : Array α) :
-    arr.foldlM f init = arr.toList.foldlM f init:= by
+    (f : β → α → m β) (init : β) (xs : Array α) :
+    xs.foldlM f init = xs.toList.foldlM f init:= by
   simp
 
 @[deprecated "Use the reverse direction of `foldr_toList`." (since := "2024-11-13")]
 theorem foldr_eq_foldr_toList
-    (f : α → β → β) (init : β) (arr : Array α) :
-    arr.foldr f init = arr.toList.foldr f init := by
+    (f : α → β → β) (init : β) (xs : Array α) :
+    xs.foldr f init = xs.toList.foldr f init := by
   simp
 
 @[deprecated "Use the reverse direction of `foldl_toList`." (since := "2024-11-13")]
 theorem foldl_eq_foldl_toList
-    (f : β → α → β) (init : β) (arr : Array α) :
-    arr.foldl f init = arr.toList.foldl f init:= by
+    (f : β → α → β) (init : β) (xs : Array α) :
+    xs.foldl f init = xs.toList.foldl f init:= by
   simp
 
 @[deprecated foldlM_toList (since := "2024-09-09")]
@@ -153,7 +182,7 @@ abbrev push_data := @push_toList
 abbrev toList_eq := @toListImpl_eq
 
 @[deprecated pop_toList (since := "2024-09-09")]
-abbrev pop_data := @pop_toList
+abbrev pop_data := @toList_pop
 
 @[deprecated toList_append (since := "2024-09-09")]
 abbrev append_data := @toList_append

src/Init/Data/Array/Count.lean

@@ -11,6 +11,9 @@ import Init.Data.List.Nat.Count
 # Lemmas about `Array.countP` and `Array.count`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 open Nat
@@ -20,122 +23,134 @@ section countP
 
 variable (p q : α → Bool)
 
+@[simp] theorem _root_.List.countP_toArray (l : List α) : countP p l.toArray = l.countP p := by
+  simp [countP]
+  induction l with
+  | nil => rfl
+  | cons hd tl ih =>
+    simp only [List.foldr_cons, ih, List.countP_cons]
+    split <;> simp_all
+
+@[simp] theorem countP_toList (xs : Array α) : xs.toList.countP p = countP p xs := by
+  cases xs
+  simp
+
 @[simp] theorem countP_empty : countP p #[] = 0 := rfl
 
-@[simp] theorem countP_push_of_pos (l) (pa : p a) : countP p (l.push a) = countP p l + 1 := by
-  rcases l with ⟨l⟩
+@[simp] theorem countP_push_of_pos (xs) (pa : p a) : countP p (xs.push a) = countP p xs + 1 := by
+  rcases xs with ⟨xs⟩
   simp_all
 
-@[simp] theorem countP_push_of_neg (l) (pa : ¬p a) : countP p (l.push a) = countP p l := by
-  rcases l with ⟨l⟩
+@[simp] theorem countP_push_of_neg (xs) (pa : ¬p a) : countP p (xs.push a) = countP p xs := by
+  rcases xs with ⟨xs⟩
   simp_all
 
-theorem countP_push (a : α) (l) : countP p (l.push a) = countP p l + if p a then 1 else 0 := by
-  rcases l with ⟨l⟩
+theorem countP_push (a : α) (xs) : countP p (xs.push a) = countP p xs + if p a then 1 else 0 := by
+  rcases xs with ⟨xs⟩
   simp_all
 
 @[simp] theorem countP_singleton (a : α) : countP p #[a] = if p a then 1 else 0 := by
   simp [countP_push]
 
-theorem size_eq_countP_add_countP (l) : l.size = countP p l + countP (fun a => ¬p a) l := by
-  cases l
+theorem size_eq_countP_add_countP (xs) : xs.size = countP p xs + countP (fun a => ¬p a) xs := by
+  rcases xs with ⟨xs⟩
   simp [List.length_eq_countP_add_countP (p := p)]
 
-theorem countP_eq_size_filter (l) : countP p l = (filter p l).size := by
-  cases l
+theorem countP_eq_size_filter (xs) : countP p xs = (filter p xs).size := by
+  rcases xs with ⟨xs⟩
   simp [List.countP_eq_length_filter]
 
 theorem countP_eq_size_filter' : countP p = size ∘ filter p := by
-  funext l
+  funext xs
   apply countP_eq_size_filter
 
-theorem countP_le_size : countP p l ≤ l.size := by
+theorem countP_le_size : countP p xs ≤ xs.size := by
   simp only [countP_eq_size_filter]
   apply size_filter_le
 
-@[simp] theorem countP_append (l₁ l₂) : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
-  cases l₁
-  cases l₂
+@[simp] theorem countP_append (xs ys) : countP p (xs ++ ys) = countP p xs + countP p ys := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp
 
-@[simp] theorem countP_pos_iff {p} : 0 < countP p l ↔ ∃ a ∈ l, p a := by
-  cases l
+@[simp] theorem countP_pos_iff {p} : 0 < countP p xs ↔ ∃ a ∈ xs, p a := by
+  rcases xs with ⟨xs⟩
   simp
 
-@[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p l ↔ ∃ a ∈ l, p a :=
+@[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p xs ↔ ∃ a ∈ xs, p a :=
   countP_pos_iff
 
-@[simp] theorem countP_eq_zero {p} : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
-  cases l
+@[simp] theorem countP_eq_zero {p} : countP p xs = 0 ↔ ∀ a ∈ xs, ¬p a := by
+  rcases xs with ⟨xs⟩
   simp
 
-@[simp] theorem countP_eq_size {p} : countP p l = l.size ↔ ∀ a ∈ l, p a := by
-  cases l
+@[simp] theorem countP_eq_size {p} : countP p xs = xs.size ↔ ∀ a ∈ xs, p a := by
+  rcases xs with ⟨xs⟩
   simp
 
 theorem countP_mkArray (p : α → Bool) (a : α) (n : Nat) :
     countP p (mkArray n a) = if p a then n else 0 := by
   simp [← List.toArray_replicate, List.countP_replicate]
 
-theorem boole_getElem_le_countP (p : α → Bool) (l : Array α) (i : Nat) (h : i < l.size) :
-    (if p l[i] then 1 else 0) ≤ l.countP p := by
-  cases l
+theorem boole_getElem_le_countP (p : α → Bool) (xs : Array α) (i : Nat) (h : i < xs.size) :
+    (if p xs[i] then 1 else 0) ≤ xs.countP p := by
+  rcases xs with ⟨xs⟩
   simp [List.boole_getElem_le_countP]
 
-theorem countP_set (p : α → Bool) (l : Array α) (i : Nat) (a : α) (h : i < l.size) :
-    (l.set i a).countP p = l.countP p - (if p l[i] then 1 else 0) + (if p a then 1 else 0) := by
-  cases l
+theorem countP_set (p : α → Bool) (xs : Array α) (i : Nat) (a : α) (h : i < xs.size) :
+    (xs.set i a).countP p = xs.countP p - (if p xs[i] then 1 else 0) + (if p a then 1 else 0) := by
+  rcases xs with ⟨xs⟩
   simp [List.countP_set, h]
 
-theorem countP_filter (l : Array α) :
-    countP p (filter q l) = countP (fun a => p a && q a) l := by
-  cases l
+theorem countP_filter (xs : Array α) :
+    countP p (filter q xs) = countP (fun a => p a && q a) xs := by
+  rcases xs with ⟨xs⟩
   simp [List.countP_filter]
 
 @[simp] theorem countP_true : (countP fun (_ : α) => true) = size := by
-  funext l
+  funext xs
   simp
 
 @[simp] theorem countP_false : (countP fun (_ : α) => false) = Function.const _ 0 := by
-  funext l
+  funext xs
   simp
 
-@[simp] theorem countP_map (p : β → Bool) (f : α → β) (l : Array α) :
-    countP p (map f l) = countP (p ∘ f) l := by
-  cases l
+@[simp] theorem countP_map (p : β → Bool) (f : α → β) (xs : Array α) :
+    countP p (map f xs) = countP (p ∘ f) xs := by
+  rcases xs with ⟨xs⟩
   simp
 
-theorem size_filterMap_eq_countP (f : α → Option β) (l : Array α) :
-    (filterMap f l).size = countP (fun a => (f a).isSome) l := by
-  cases l
+theorem size_filterMap_eq_countP (f : α → Option β) (xs : Array α) :
+    (filterMap f xs).size = countP (fun a => (f a).isSome) xs := by
+  rcases xs with ⟨xs⟩
   simp [List.length_filterMap_eq_countP]
 
-theorem countP_filterMap (p : β → Bool) (f : α → Option β) (l : Array α) :
-    countP p (filterMap f l) = countP (fun a => ((f a).map p).getD false) l := by
-  cases l
+theorem countP_filterMap (p : β → Bool) (f : α → Option β) (xs : Array α) :
+    countP p (filterMap f xs) = countP (fun a => ((f a).map p).getD false) xs := by
+  rcases xs with ⟨xs⟩
   simp [List.countP_filterMap]
 
-@[simp] theorem countP_flatten (l : Array (Array α)) :
-    countP p l.flatten = (l.map (countP p)).sum := by
-  cases l using array₂_induction
+@[simp] theorem countP_flatten (xss : Array (Array α)) :
+    countP p xss.flatten = (xss.map (countP p)).sum := by
+  cases xss using array₂_induction
   simp [List.countP_flatten, Function.comp_def]
 
-theorem countP_flatMap (p : β → Bool) (l : Array α) (f : α → Array β) :
-    countP p (l.flatMap f) = sum (map (countP p ∘ f) l) := by
-  cases l
+theorem countP_flatMap (p : β → Bool) (xs : Array α) (f : α → Array β) :
+    countP p (xs.flatMap f) = sum (map (countP p ∘ f) xs) := by
+  rcases xs with ⟨xs⟩
   simp [List.countP_flatMap, Function.comp_def]
 
-@[simp] theorem countP_reverse (l : Array α) : countP p l.reverse = countP p l := by
-  cases l
+@[simp] theorem countP_reverse (xs : Array α) : countP p xs.reverse = countP p xs := by
+  rcases xs with ⟨xs⟩
   simp [List.countP_reverse]
 
 variable {p q}
 
-theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
-  cases l
+theorem countP_mono_left (h : ∀ x ∈ xs, p x → q x) : countP p xs ≤ countP q xs := by
+  rcases xs with ⟨xs⟩
   simpa using List.countP_mono_left (by simpa using h)
 
-theorem countP_congr (h : ∀ x ∈ l, p x ↔ q x) : countP p l = countP q l :=
+theorem countP_congr (h : ∀ x ∈ xs, p x ↔ q x) : countP p xs = countP q xs :=
   Nat.le_antisymm
     (countP_mono_left fun x hx => (h x hx).1)
     (countP_mono_left fun x hx => (h x hx).2)
@@ -147,73 +162,80 @@ section count
 
 variable [BEq α]
 
-@[simp] theorem count_empty (a : α) : count a #[] = 0 := rfl
+@[simp] theorem _root_.List.count_toArray (l : List α) (a : α) : count a l.toArray = l.count a := by
+  simp [count, List.count_eq_countP]
 
-theorem count_push (a b : α) (l : Array α) :
-    count a (l.push b) = count a l + if b == a then 1 else 0 := by
-  simp [count, countP_push]
-
-theorem count_eq_countP (a : α) (l : Array α) : count a l = countP (· == a) l := rfl
-theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
-  funext l
-  apply count_eq_countP
-
-theorem count_le_size (a : α) (l : Array α) : count a l ≤ l.size := countP_le_size _
-
-theorem count_le_count_push (a b : α) (l : Array α) : count a l ≤ count a (l.push b) := by
-  simp [count_push]
-
-@[simp] theorem count_singleton (a b : α) : count a #[b] = if b == a then 1 else 0 := by
-  simp [count_eq_countP]
-
-@[simp] theorem count_append (a : α) : ∀ l₁ l₂, count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
-  countP_append _
-
-@[simp] theorem count_flatten (a : α) (l : Array (Array α)) :
-    count a l.flatten = (l.map (count a)).sum := by
-  cases l using array₂_induction
-  simp [List.count_flatten, Function.comp_def]
-
-@[simp] theorem count_reverse (a : α) (l : Array α) : count a l.reverse = count a l := by
-  cases l
+@[simp] theorem count_toList (xs : Array α) (a : α) : xs.toList.count a = xs.count a := by
+  cases xs
   simp
 
-theorem boole_getElem_le_count (a : α) (l : Array α) (i : Nat) (h : i < l.size) :
-    (if l[i] == a then 1 else 0) ≤ l.count a := by
+@[simp] theorem count_empty (a : α) : count a #[] = 0 := rfl
+
+theorem count_push (a b : α) (xs : Array α) :
+    count a (xs.push b) = count a xs + if b == a then 1 else 0 := by
+  simp [count, countP_push]
+
+theorem count_eq_countP (a : α) (xs : Array α) : count a xs = countP (· == a) xs := rfl
+theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
+  funext xs
+  apply count_eq_countP
+
+theorem count_le_size (a : α) (xs : Array α) : count a xs ≤ xs.size := countP_le_size _
+
+theorem count_le_count_push (a b : α) (xs : Array α) : count a xs ≤ count a (xs.push b) := by
+  simp [count_push]
+
+theorem count_singleton (a b : α) : count a #[b] = if b == a then 1 else 0 := by
+  simp [count_eq_countP]
+
+@[simp] theorem count_append (a : α) : ∀ xs ys, count a (xs ++ ys) = count a xs + count a ys :=
+  countP_append _
+
+@[simp] theorem count_flatten (a : α) (xss : Array (Array α)) :
+    count a xss.flatten = (xss.map (count a)).sum := by
+  cases xss using array₂_induction
+  simp [List.count_flatten, Function.comp_def]
+
+@[simp] theorem count_reverse (a : α) (xs : Array α) : count a xs.reverse = count a xs := by
+  rcases xs with ⟨xs⟩
+  simp
+
+theorem boole_getElem_le_count (a : α) (xs : Array α) (i : Nat) (h : i < xs.size) :
+    (if xs[i] == a then 1 else 0) ≤ xs.count a := by
   rw [count_eq_countP]
   apply boole_getElem_le_countP (· == a)
 
-theorem count_set (a b : α) (l : Array α) (i : Nat) (h : i < l.size) :
-    (l.set i a).count b = l.count b - (if l[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
+theorem count_set (a b : α) (xs : Array α) (i : Nat) (h : i < xs.size) :
+    (xs.set i a).count b = xs.count b - (if xs[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
   simp [count_eq_countP, countP_set, h]
 
 variable [LawfulBEq α]
 
-@[simp] theorem count_push_self (a : α) (l : Array α) : count a (l.push a) = count a l + 1 := by
+@[simp] theorem count_push_self (a : α) (xs : Array α) : count a (xs.push a) = count a xs + 1 := by
   simp [count_push]
 
-@[simp] theorem count_push_of_ne (h : b ≠ a) (l : Array α) : count a (l.push b) = count a l := by
+@[simp] theorem count_push_of_ne (h : b ≠ a) (xs : Array α) : count a (xs.push b) = count a xs := by
   simp_all [count_push, h]
 
 theorem count_singleton_self (a : α) : count a #[a] = 1 := by simp
 
 @[simp]
-theorem count_pos_iff {a : α} {l : Array α} : 0 < count a l ↔ a ∈ l := by
+theorem count_pos_iff {a : α} {xs : Array α} : 0 < count a xs ↔ a ∈ xs := by
   simp only [count, countP_pos_iff, beq_iff_eq, exists_eq_right]
 
-@[simp] theorem one_le_count_iff {a : α} {l : Array α} : 1 ≤ count a l ↔ a ∈ l :=
+@[simp] theorem one_le_count_iff {a : α} {xs : Array α} : 1 ≤ count a xs ↔ a ∈ xs :=
   count_pos_iff
 
-theorem count_eq_zero_of_not_mem {a : α} {l : Array α} (h : a ∉ l) : count a l = 0 :=
+theorem count_eq_zero_of_not_mem {a : α} {xs : Array α} (h : a ∉ xs) : count a xs = 0 :=
   Decidable.byContradiction fun h' => h <| count_pos_iff.1 (Nat.pos_of_ne_zero h')
 
-theorem not_mem_of_count_eq_zero {a : α} {l : Array α} (h : count a l = 0) : a ∉ l :=
+theorem not_mem_of_count_eq_zero {a : α} {xs : Array α} (h : count a xs = 0) : a ∉ xs :=
   fun h' => Nat.ne_of_lt (count_pos_iff.2 h') h.symm
 
-theorem count_eq_zero {l : Array α} : count a l = 0 ↔ a ∉ l :=
+theorem count_eq_zero {xs : Array α} : count a xs = 0 ↔ a ∉ xs :=
   ⟨not_mem_of_count_eq_zero, count_eq_zero_of_not_mem⟩
 
-theorem count_eq_size {l : Array α} : count a l = l.size ↔ ∀ b ∈ l, a = b := by
+theorem count_eq_size {xs : Array α} : count a xs = xs.size ↔ ∀ b ∈ xs, a = b := by
   rw [count, countP_eq_size]
   refine ⟨fun h b hb => Eq.symm ?_, fun h b hb => ?_⟩
   · simpa using h b hb
@@ -225,36 +247,37 @@ theorem count_eq_size {l : Array α} : count a l = l.size ↔ ∀ b ∈ l, a = b
 theorem count_mkArray (a b : α) (n : Nat) : count a (mkArray n b) = if b == a then n else 0 := by
   simp [← List.toArray_replicate, List.count_replicate]
 
-theorem filter_beq (l : Array α) (a : α) : l.filter (· == a) = mkArray (count a l) a := by
-  cases l
+theorem filter_beq (xs : Array α) (a : α) : xs.filter (· == a) = mkArray (count a xs) a := by
+  rcases xs with ⟨xs⟩
   simp [List.filter_beq]
 
-theorem filter_eq {α} [DecidableEq α] (l : Array α) (a : α) : l.filter (· = a) = mkArray (count a l) a :=
-  filter_beq l a
+theorem filter_eq {α} [DecidableEq α] (xs : Array α) (a : α) : xs.filter (· = a) = mkArray (count a xs) a :=
+  filter_beq xs a
 
-theorem mkArray_count_eq_of_count_eq_size {l : Array α} (h : count a l = l.size) :
-    mkArray (count a l) a = l := by
-  cases l
+theorem mkArray_count_eq_of_count_eq_size {xs : Array α} (h : count a xs = xs.size) :
+    mkArray (count a xs) a = xs := by
+  rcases xs with ⟨xs⟩
   rw [← toList_inj]
   simp [List.replicate_count_eq_of_count_eq_length (by simpa using h)]
 
-@[simp] theorem count_filter {l : Array α} (h : p a) : count a (filter p l) = count a l := by
-  cases l
+@[simp] theorem count_filter {xs : Array α} (h : p a) : count a (filter p xs) = count a xs := by
+  rcases xs with ⟨xs⟩
   simp [List.count_filter, h]
 
-theorem count_le_count_map [DecidableEq β] (l : Array α) (f : α → β) (x : α) :
-    count x l ≤ count (f x) (map f l) := by
-  cases l
+theorem count_le_count_map [DecidableEq β] (xs : Array α) (f : α → β) (x : α) :
+    count x xs ≤ count (f x) (map f xs) := by
+  rcases xs with ⟨xs⟩
   simp [List.count_le_count_map, countP_map]
 
-theorem count_filterMap {α} [BEq β] (b : β) (f : α → Option β) (l : Array α) :
-    count b (filterMap f l) = countP (fun a => f a == some b) l := by
-  cases l
+theorem count_filterMap {α} [BEq β] (b : β) (f : α → Option β) (xs : Array α) :
+    count b (filterMap f xs) = countP (fun a => f a == some b) xs := by
+  rcases xs with ⟨xs⟩
   simp [List.count_filterMap, countP_filterMap]
 
-theorem count_flatMap {α} [BEq β] (l : Array α) (f : α → Array β) (x : β) :
-    count x (l.flatMap f) = sum (map (count x ∘ f) l) := by
-  simp [count_eq_countP, countP_flatMap, Function.comp_def]
+theorem count_flatMap {α} [BEq β] (xs : Array α) (f : α → Array β) (x : β) :
+    count x (xs.flatMap f) = sum (map (count x ∘ f) xs) := by
+  rcases xs with ⟨xs⟩
+  simp [List.count_flatMap, countP_flatMap, Function.comp_def]
 
 -- FIXME these theorems can be restored once `List.erase` and `Array.erase` have been related.
 

src/Init/Data/Array/DecidableEq.lean

@@ -9,12 +9,15 @@ import Init.Data.BEq
 import Init.Data.List.Nat.BEq
 import Init.ByCases
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 private theorem rel_of_isEqvAux
-    {r : α → α → Bool} {a b : Array α} (hsz : a.size = b.size) {i : Nat} (hi : i ≤ a.size)
-    (heqv : Array.isEqvAux a b hsz r i hi)
-    {j : Nat} (hj : j < i) : r (a[j]'(Nat.lt_of_lt_of_le hj hi)) (b[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi))) := by
+    {r : α → α → Bool} {xs ys : Array α} (hsz : xs.size = ys.size) {i : Nat} (hi : i ≤ xs.size)
+    (heqv : Array.isEqvAux xs ys hsz r i hi)
+    {j : Nat} (hj : j < i) : r (xs[j]'(Nat.lt_of_lt_of_le hj hi)) (ys[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi))) := by
   induction i with
   | zero => contradiction
   | succ i ih =>
@@ -27,8 +30,8 @@ private theorem rel_of_isEqvAux
       subst hj'
       exact heqv.left
 
-private theorem isEqvAux_of_rel {r : α → α → Bool} {a b : Array α} (hsz : a.size = b.size) {i : Nat} (hi : i ≤ a.size)
-    (w : ∀ j, (hj : j < i) → r (a[j]'(Nat.lt_of_lt_of_le hj hi)) (b[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi)))) : Array.isEqvAux a b hsz r i hi := by
+private theorem isEqvAux_of_rel {r : α → α → Bool} {xs ys : Array α} (hsz : xs.size = ys.size) {i : Nat} (hi : i ≤ xs.size)
+    (w : ∀ j, (hj : j < i) → r (xs[j]'(Nat.lt_of_lt_of_le hj hi)) (ys[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi)))) : Array.isEqvAux xs ys hsz r i hi := by
   induction i with
   | zero => simp [Array.isEqvAux]
   | succ i ih =>
@@ -36,23 +39,23 @@ private theorem isEqvAux_of_rel {r : α → α → Bool} {a b : Array α} (hsz :
     exact ⟨w i (Nat.lt_add_one i), ih _ fun j hj => w j (Nat.lt_add_right 1 hj)⟩
 
 -- This is private as the forward direction of `isEqv_iff_rel` may be used.
-private theorem rel_of_isEqv {r : α → α → Bool} {a b : Array α} :
-    Array.isEqv a b r → ∃ h : a.size = b.size, ∀ (i : Nat) (h' : i < a.size), r (a[i]) (b[i]'(h ▸ h')) := by
+private theorem rel_of_isEqv {r : α → α → Bool} {xs ys : Array α} :
+    Array.isEqv xs ys r → ∃ h : xs.size = ys.size, ∀ (i : Nat) (h' : i < xs.size), r (xs[i]) (ys[i]'(h ▸ h')) := by
   simp only [isEqv]
   split <;> rename_i h
   · exact fun h' => ⟨h, fun i => rel_of_isEqvAux h (Nat.le_refl ..) h'⟩
   · intro; contradiction
 
-theorem isEqv_iff_rel {a b : Array α} {r} :
-    Array.isEqv a b r ↔ ∃ h : a.size = b.size, ∀ (i : Nat) (h' : i < a.size), r (a[i]) (b[i]'(h ▸ h')) :=
+theorem isEqv_iff_rel {xs ys : Array α} {r} :
+    Array.isEqv xs ys r ↔ ∃ h : xs.size = ys.size, ∀ (i : Nat) (h' : i < xs.size), r (xs[i]) (ys[i]'(h ▸ h')) :=
   ⟨rel_of_isEqv, fun ⟨h, w⟩ => by
     simp only [isEqv, ← h, ↓reduceDIte]
     exact isEqvAux_of_rel h (by simp [h]) w⟩
 
-theorem isEqv_eq_decide (a b : Array α) (r) :
-    Array.isEqv a b r =
-      if h : a.size = b.size then decide (∀ (i : Nat) (h' : i < a.size), r (a[i]) (b[i]'(h ▸ h'))) else false := by
-  by_cases h : Array.isEqv a b r
+theorem isEqv_eq_decide (xs ys : Array α) (r) :
+    Array.isEqv xs ys r =
+      if h : xs.size = ys.size then decide (∀ (i : Nat) (h' : i < xs.size), r (xs[i]) (ys[i]'(h ▸ h'))) else false := by
+  by_cases h : Array.isEqv xs ys r
   · simp only [h, Bool.true_eq]
     simp only [isEqv_iff_rel] at h
     obtain ⟨h, w⟩ := h
@@ -63,48 +66,48 @@ theorem isEqv_eq_decide (a b : Array α) (r) :
       Bool.not_eq_true]
     simpa [isEqv_iff_rel] using h'
 
-@[simp] theorem isEqv_toList [BEq α] (a b : Array α) : (a.toList.isEqv b.toList r) = (a.isEqv b r) := by
+@[simp] theorem isEqv_toList [BEq α] (xs ys : Array α) : (xs.toList.isEqv ys.toList r) = (xs.isEqv ys r) := by
   simp [isEqv_eq_decide, List.isEqv_eq_decide]
 
-theorem eq_of_isEqv [DecidableEq α] (a b : Array α) (h : Array.isEqv a b (fun x y => x = y)) : a = b := by
+theorem eq_of_isEqv [DecidableEq α] (xs ys : Array α) (h : Array.isEqv xs ys (fun x y => x = y)) : xs = ys := by
   have ⟨h, h'⟩ := rel_of_isEqv h
   exact ext _ _ h (fun i lt _ => by simpa using h' i lt)
 
-private theorem isEqvAux_self (r : α → α → Bool) (hr : ∀ a, r a a) (a : Array α) (i : Nat) (h : i ≤ a.size) :
-    Array.isEqvAux a a rfl r i h = true := by
+private theorem isEqvAux_self (r : α → α → Bool) (hr : ∀ a, r a a) (xs : Array α) (i : Nat) (h : i ≤ xs.size) :
+    Array.isEqvAux xs xs rfl r i h = true := by
   induction i with
   | zero => simp [Array.isEqvAux]
   | succ i ih =>
     simp_all only [isEqvAux, Bool.and_self]
 
-theorem isEqv_self_beq [BEq α] [ReflBEq α] (a : Array α) : Array.isEqv a a (· == ·) = true := by
+theorem isEqv_self_beq [BEq α] [ReflBEq α] (xs : Array α) : Array.isEqv xs xs (· == ·) = true := by
   simp [isEqv, isEqvAux_self]
 
-theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (· = ·) = true := by
+theorem isEqv_self [DecidableEq α] (xs : Array α) : Array.isEqv xs xs (· = ·) = true := by
   simp [isEqv, isEqvAux_self]
 
 instance [DecidableEq α] : DecidableEq (Array α) :=
-  fun a b =>
-    match h:isEqv a b (fun a b => a = b) with
-    | true  => isTrue (eq_of_isEqv a b h)
+  fun xs ys =>
+    match h:isEqv xs ys (fun a b => a = b) with
+    | true  => isTrue (eq_of_isEqv xs ys h)
     | false => isFalse fun h' => by subst h'; rw [isEqv_self] at h; contradiction
 
-theorem beq_eq_decide [BEq α] (a b : Array α) :
-    (a == b) = if h : a.size = b.size then
-      decide (∀ (i : Nat) (h' : i < a.size), a[i] == b[i]'(h ▸ h')) else false := by
+theorem beq_eq_decide [BEq α] (xs ys : Array α) :
+    (xs == ys) = if h : xs.size = ys.size then
+      decide (∀ (i : Nat) (h' : i < xs.size), xs[i] == ys[i]'(h ▸ h')) else false := by
   simp [BEq.beq, isEqv_eq_decide]
 
-@[simp] theorem beq_toList [BEq α] (a b : Array α) : (a.toList == b.toList) = (a == b) := by
+@[simp] theorem beq_toList [BEq α] (xs ys : Array α) : (xs.toList == ys.toList) = (xs == ys) := by
   simp [beq_eq_decide, List.beq_eq_decide]
 
 end Array
 
 namespace List
 
-@[simp] theorem isEqv_toArray [BEq α] (a b : List α) : (a.toArray.isEqv b.toArray r) = (a.isEqv b r) := by
+@[simp] theorem isEqv_toArray [BEq α] (as bs : List α) : (as.toArray.isEqv bs.toArray r) = (as.isEqv bs r) := by
   simp [isEqv_eq_decide, Array.isEqv_eq_decide]
 
-@[simp] theorem beq_toArray [BEq α] (a b : List α) : (a.toArray == b.toArray) = (a == b) := by
+@[simp] theorem beq_toArray [BEq α] (as bs : List α) : (as.toArray == bs.toArray) = (as == bs) := by
   simp [beq_eq_decide, Array.beq_eq_decide]
 
 end List
@@ -114,7 +117,7 @@ namespace Array
 instance [BEq α] [LawfulBEq α] : LawfulBEq (Array α) where
   rfl := by simp [BEq.beq, isEqv_self_beq]
   eq_of_beq := by
-    rintro ⟨a⟩ ⟨b⟩ h
+    rintro ⟨_⟩ ⟨_⟩ h
     simpa using h
 
 end Array

src/Init/Data/Array/Erase.lean

@@ -12,6 +12,9 @@ import Init.Data.List.Nat.Basic
 # Lemmas about `Array.eraseP`, `Array.erase`, and `Array.eraseIdx`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 open Nat
@@ -20,11 +23,11 @@ open Nat
 
 @[simp] theorem eraseP_empty : #[].eraseP p = #[] := rfl
 
-theorem eraseP_of_forall_mem_not {l : Array α} (h : ∀ a, a ∈ l → ¬p a) : l.eraseP p = l := by
-  cases l
+theorem eraseP_of_forall_mem_not {xs : Array α} (h : ∀ a, a ∈ xs → ¬p a) : xs.eraseP p = xs := by
+  rcases xs with ⟨xs⟩
   simp_all [List.eraseP_of_forall_not]
 
-theorem eraseP_of_forall_getElem_not {l : Array α} (h : ∀ i, (h : i < l.size) → ¬p l[i]) : l.eraseP p = l :=
+theorem eraseP_of_forall_getElem_not {xs : Array α} (h : ∀ i, (h : i < xs.size) → ¬p xs[i]) : xs.eraseP p = xs :=
   eraseP_of_forall_mem_not fun a m => by
     rw [mem_iff_getElem] at m
     obtain ⟨i, w, rfl⟩ := m
@@ -37,86 +40,86 @@ theorem eraseP_of_forall_getElem_not {l : Array α} (h : ∀ i, (h : i < l.size)
 theorem eraseP_ne_empty_iff {xs : Array α} {p : α → Bool} : xs.eraseP p ≠ #[] ↔ xs ≠ #[] ∧ ∀ x, p x → xs ≠ #[x] := by
   simp
 
-theorem exists_of_eraseP {l : Array α} {a} (hm : a ∈ l) (hp : p a) :
-    ∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁.push a ++ l₂ ∧ l.eraseP p = l₁ ++ l₂ := by
-  rcases l with ⟨l⟩
+theorem exists_of_eraseP {xs : Array α} {a} (hm : a ∈ xs) (hp : p a) :
+    ∃ a ys zs, (∀ b ∈ ys, ¬p b) ∧ p a ∧ xs = ys.push a ++ zs ∧ xs.eraseP p = ys ++ zs := by
+  rcases xs with ⟨xs⟩
   obtain ⟨a, l₁, l₂, h₁, h₂, rfl, h₃⟩ := List.exists_of_eraseP (by simpa using hm) (hp)
   refine ⟨a, ⟨l₁⟩, ⟨l₂⟩, by simpa using h₁, h₂, by simp, by simpa using h₃⟩
 
-theorem exists_or_eq_self_of_eraseP (p) (l : Array α) :
-    l.eraseP p = l ∨
-    ∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁.push a ++ l₂ ∧ l.eraseP p = l₁ ++ l₂ :=
-  if h : ∃ a ∈ l, p a then
+theorem exists_or_eq_self_of_eraseP (p) (xs : Array α) :
+    xs.eraseP p = xs ∨
+    ∃ a ys zs, (∀ b ∈ ys, ¬p b) ∧ p a ∧ xs = ys.push a ++ zs ∧ xs.eraseP p = ys ++ zs :=
+  if h : ∃ a ∈ xs, p a then
     let ⟨_, ha, pa⟩ := h
     .inr (exists_of_eraseP ha pa)
   else
     .inl (eraseP_of_forall_mem_not (h ⟨·, ·, ·⟩))
 
-@[simp] theorem size_eraseP_of_mem {l : Array α} (al : a ∈ l) (pa : p a) :
-    (l.eraseP p).size = l.size - 1 := by
-  let ⟨_, l₁, l₂, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
+@[simp] theorem size_eraseP_of_mem {xs : Array α} (al : a ∈ xs) (pa : p a) :
+    (xs.eraseP p).size = xs.size - 1 := by
+  let ⟨_, ys, zs, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
   rw [e₂]; simp [size_append, e₁]; omega
 
-theorem size_eraseP {l : Array α} : (l.eraseP p).size = if l.any p then l.size - 1 else l.size := by
+theorem size_eraseP {xs : Array α} : (xs.eraseP p).size = if xs.any p then xs.size - 1 else xs.size := by
   split <;> rename_i h
   · simp only [any_eq_true] at h
     obtain ⟨i, h, w⟩ := h
-    simp [size_eraseP_of_mem (l := l) (by simp) w]
+    simp [size_eraseP_of_mem (xs := xs) (by simp) w]
   · simp only [any_eq_true] at h
     rw [eraseP_of_forall_getElem_not]
     simp_all
 
-theorem size_eraseP_le (l : Array α) : (l.eraseP p).size ≤ l.size := by
-  rcases l with ⟨l⟩
-  simpa using List.length_eraseP_le l
+theorem size_eraseP_le (xs : Array α) : (xs.eraseP p).size ≤ xs.size := by
+  rcases xs with ⟨xs⟩
+  simpa using List.length_eraseP_le xs
 
-theorem le_size_eraseP (l : Array α) : l.size - 1 ≤ (l.eraseP p).size := by
-  rcases l with ⟨l⟩
-  simpa using List.le_length_eraseP l
+theorem le_size_eraseP (xs : Array α) : xs.size - 1 ≤ (xs.eraseP p).size := by
+  rcases xs with ⟨xs⟩
+  simpa using List.le_length_eraseP xs
 
-theorem mem_of_mem_eraseP {l : Array α} : a ∈ l.eraseP p → a ∈ l := by
-  rcases l with ⟨l⟩
+theorem mem_of_mem_eraseP {xs : Array α} : a ∈ xs.eraseP p → a ∈ xs := by
+  rcases xs with ⟨xs⟩
   simpa using List.mem_of_mem_eraseP
 
-@[simp] theorem mem_eraseP_of_neg {l : Array α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
-  rcases l with ⟨l⟩
+@[simp] theorem mem_eraseP_of_neg {xs : Array α} (pa : ¬p a) : a ∈ xs.eraseP p ↔ a ∈ xs := by
+  rcases xs with ⟨xs⟩
   simpa using List.mem_eraseP_of_neg pa
 
-@[simp] theorem eraseP_eq_self_iff {p} {l : Array α} : l.eraseP p = l ↔ ∀ a ∈ l, ¬ p a := by
-  rcases l with ⟨l⟩
+@[simp] theorem eraseP_eq_self_iff {xs : Array α} : xs.eraseP p = xs ↔ ∀ a ∈ xs, ¬ p a := by
+  rcases xs with ⟨xs⟩
   simp
 
-theorem eraseP_map (f : β → α) (l : Array β) : (map f l).eraseP p = map f (l.eraseP (p ∘ f)) := by
-  rcases l with ⟨l⟩
-  simpa using List.eraseP_map f l
+theorem eraseP_map (f : β → α) (xs : Array β) : (xs.map f).eraseP p = (xs.eraseP (p ∘ f)).map f := by
+  rcases xs with ⟨xs⟩
+  simpa using List.eraseP_map f xs
 
-theorem eraseP_filterMap (f : α → Option β) (l : Array α) :
-    (filterMap f l).eraseP p = filterMap f (l.eraseP (fun x => match f x with | some y => p y | none => false)) := by
-  rcases l with ⟨l⟩
-  simpa using List.eraseP_filterMap f l
+theorem eraseP_filterMap (f : α → Option β) (xs : Array α) :
+    (filterMap f xs).eraseP p = filterMap f (xs.eraseP (fun x => match f x with | some y => p y | none => false)) := by
+  rcases xs with ⟨xs⟩
+  simpa using List.eraseP_filterMap f xs
 
-theorem eraseP_filter (f : α → Bool) (l : Array α) :
-    (filter f l).eraseP p = filter f (l.eraseP (fun x => p x && f x)) := by
-  rcases l with ⟨l⟩
-  simpa using List.eraseP_filter f l
+theorem eraseP_filter (f : α → Bool) (xs : Array α) :
+    (filter f xs).eraseP p = filter f (xs.eraseP (fun x => p x && f x)) := by
+  rcases xs with ⟨xs⟩
+  simpa using List.eraseP_filter f xs
 
-theorem eraseP_append_left {a : α} (pa : p a) {l₁ : Array α} l₂ (h : a ∈ l₁) :
-    (l₁ ++ l₂).eraseP p = l₁.eraseP p ++ l₂ := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
-  simpa using List.eraseP_append_left pa l₂ (by simpa using h)
+theorem eraseP_append_left {a : α} (pa : p a) {xs : Array α} {ys : Array α} (h : a ∈ xs) :
+    (xs ++ ys).eraseP p = xs.eraseP p ++ ys := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simpa using List.eraseP_append_left pa ys (by simpa using h)
 
-theorem eraseP_append_right {l₁ : Array α} l₂ (h : ∀ b ∈ l₁, ¬p b) :
-    (l₁ ++ l₂).eraseP p = l₁ ++ l₂.eraseP p := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
-  simpa using List.eraseP_append_right l₂ (by simpa using h)
+theorem eraseP_append_right {xs : Array α} ys (h : ∀ b ∈ xs, ¬p b) :
+    (xs ++ ys).eraseP p = xs ++ ys.eraseP p := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simpa using List.eraseP_append_right ys (by simpa using h)
 
-theorem eraseP_append (l₁ l₂ : Array α) :
-    (l₁ ++ l₂).eraseP p = if l₁.any p then l₁.eraseP p ++ l₂ else l₁ ++ l₂.eraseP p := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
-  simp only [List.append_toArray, List.eraseP_toArray, List.eraseP_append l₁ l₂, List.any_toArray']
+theorem eraseP_append {xs : Array α} {ys : Array α} :
+    (xs ++ ys).eraseP p = if xs.any p then xs.eraseP p ++ ys else xs ++ ys.eraseP p := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simp only [List.append_toArray, List.eraseP_toArray, List.eraseP_append, List.any_toArray]
   split <;> simp
 
 theorem eraseP_mkArray (n : Nat) (a : α) (p : α → Bool) :
@@ -134,24 +137,24 @@ theorem eraseP_mkArray (n : Nat) (a : α) (p : α → Bool) :
   simp only [← List.toArray_replicate, List.eraseP_toArray]
   simp [h]
 
-theorem eraseP_eq_iff {p} {l : Array α} :
-    l.eraseP p = l' ↔
-      ((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
-        ∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧ l = l₁.push a ++ l₂ ∧ l' = l₁ ++ l₂ := by
-  rcases l with ⟨l⟩
-  rcases l' with ⟨l'⟩
+theorem eraseP_eq_iff {p} {xs : Array α} :
+    xs.eraseP p = ys ↔
+      ((∀ a ∈ xs, ¬ p a) ∧ xs = ys) ∨
+        ∃ a as bs, (∀ b ∈ as, ¬ p b) ∧ p a ∧ xs = as.push a ++ bs ∧ ys = as ++ bs := by
+  rcases xs with ⟨l⟩
+  rcases ys with ⟨ys⟩
   simp [List.eraseP_eq_iff]
   constructor
-  · rintro (h | ⟨a, l₁, h₁, h₂, ⟨x, rfl, rfl⟩⟩)
+  · rintro (h | ⟨a, l₁, h₁, h₂, ⟨l, rfl, rfl⟩⟩)
     · exact Or.inl h
-    · exact Or.inr ⟨a, ⟨l₁⟩, by simpa using h₁, h₂, ⟨⟨x⟩, by simp⟩⟩
-  · rintro (h | ⟨a, ⟨l₁⟩, h₁, h₂, ⟨⟨x⟩, rfl, rfl⟩⟩)
+    · exact Or.inr ⟨a, ⟨l₁⟩, by simpa using h₁, h₂, ⟨⟨l⟩, by simp⟩⟩
+  · rintro (h | ⟨a, ⟨l₁⟩, h₁, h₂, ⟨⟨l⟩, rfl, rfl⟩⟩)
     · exact Or.inl h
-    · exact Or.inr ⟨a, l₁, by simpa using h₁, h₂, ⟨x, by simp⟩⟩
+    · exact Or.inr ⟨a, l₁, by simpa using h₁, h₂, ⟨l, by simp⟩⟩
 
-theorem eraseP_comm {l : Array α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
-    (l.eraseP p).eraseP q = (l.eraseP q).eraseP p := by
-  rcases l with ⟨l⟩
+theorem eraseP_comm {xs : Array α} (h : ∀ a ∈ xs, ¬ p a ∨ ¬ q a) :
+    (xs.eraseP p).eraseP q = (xs.eraseP q).eraseP p := by
+  rcases xs with ⟨xs⟩
   simpa using List.eraseP_comm (by simpa using h)
 
 /-! ### erase -/
@@ -159,16 +162,16 @@ theorem eraseP_comm {l : Array α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
 section erase
 variable [BEq α]
 
-theorem erase_of_not_mem [LawfulBEq α] {a : α} {l : Array α} (h : a ∉ l) : l.erase a = l := by
-  rcases l with ⟨l⟩
+theorem erase_of_not_mem [LawfulBEq α] {a : α} {xs : Array α} (h : a ∉ xs) : xs.erase a = xs := by
+  rcases xs with ⟨xs⟩
   simp [List.erase_of_not_mem (by simpa using h)]
 
-theorem erase_eq_eraseP' (a : α) (l : Array α) : l.erase a = l.eraseP (· == a) := by
-  rcases l with ⟨l⟩
+theorem erase_eq_eraseP' (a : α) (xs : Array α) : xs.erase a = xs.eraseP (· == a) := by
+  rcases xs with ⟨xs⟩
   simp [List.erase_eq_eraseP']
 
-theorem erase_eq_eraseP [LawfulBEq α] (a : α) (l : Array α) : l.erase a = l.eraseP (a == ·) := by
-  rcases l with ⟨l⟩
+theorem erase_eq_eraseP [LawfulBEq α] (a : α) (xs : Array α) : xs.erase a = xs.eraseP (a == ·) := by
+  rcases xs with ⟨xs⟩
   simp [List.erase_eq_eraseP]
 
 @[simp] theorem erase_eq_empty_iff [LawfulBEq α] {xs : Array α} {a : α} :
@@ -181,62 +184,62 @@ theorem erase_ne_empty_iff [LawfulBEq α] {xs : Array α} {a : α} :
   rcases xs with ⟨xs⟩
   simp [List.erase_ne_nil_iff]
 
-theorem exists_erase_eq [LawfulBEq α] {a : α} {l : Array α} (h : a ∈ l) :
-    ∃ l₁ l₂, a ∉ l₁ ∧ l = l₁.push a ++ l₂ ∧ l.erase a = l₁ ++ l₂ := by
-  let ⟨_, l₁, l₂, h₁, e, h₂, h₃⟩ := exists_of_eraseP h (beq_self_eq_true _)
-  rw [erase_eq_eraseP]; exact ⟨l₁, l₂, fun h => h₁ _ h (beq_self_eq_true _), eq_of_beq e ▸ h₂, h₃⟩
+theorem exists_erase_eq [LawfulBEq α] {a : α} {xs : Array α} (h : a ∈ xs) :
+    ∃ ys zs, a ∉ ys ∧ xs = ys.push a ++ zs ∧ xs.erase a = ys ++ zs := by
+  let ⟨_, ys, zs, h₁, e, h₂, h₃⟩ := exists_of_eraseP h (beq_self_eq_true _)
+  rw [erase_eq_eraseP]; exact ⟨ys, zs, fun h => h₁ _ h (beq_self_eq_true _), eq_of_beq e ▸ h₂, h₃⟩
 
-@[simp] theorem size_erase_of_mem [LawfulBEq α] {a : α} {l : Array α} (h : a ∈ l) :
-    (l.erase a).size = l.size - 1 := by
+@[simp] theorem size_erase_of_mem [LawfulBEq α] {a : α} {xs : Array α} (h : a ∈ xs) :
+    (xs.erase a).size = xs.size - 1 := by
   rw [erase_eq_eraseP]; exact size_eraseP_of_mem h (beq_self_eq_true a)
 
-theorem size_erase [LawfulBEq α] (a : α) (l : Array α) :
-    (l.erase a).size = if a ∈ l then l.size - 1 else l.size := by
+theorem size_erase [LawfulBEq α] (a : α) (xs : Array α) :
+    (xs.erase a).size = if a ∈ xs then xs.size - 1 else xs.size := by
   rw [erase_eq_eraseP, size_eraseP]
   congr
   simp [mem_iff_getElem, eq_comm (a := a)]
 
-theorem size_erase_le (a : α) (l : Array α) : (l.erase a).size ≤ l.size := by
-  rcases l with ⟨l⟩
-  simpa using List.length_erase_le a l
+theorem size_erase_le (a : α) (xs : Array α) : (xs.erase a).size ≤ xs.size := by
+  rcases xs with ⟨xs⟩
+  simpa using List.length_erase_le a xs
 
-theorem le_size_erase [LawfulBEq α] (a : α) (l : Array α) : l.size - 1 ≤ (l.erase a).size := by
-  rcases l with ⟨l⟩
-  simpa using List.le_length_erase a l
+theorem le_size_erase [LawfulBEq α] (a : α) (xs : Array α) : xs.size - 1 ≤ (xs.erase a).size := by
+  rcases xs with ⟨xs⟩
+  simpa using List.le_length_erase a xs
 
-theorem mem_of_mem_erase {a b : α} {l : Array α} (h : a ∈ l.erase b) : a ∈ l := by
-  rcases l with ⟨l⟩
+theorem mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a ∈ xs := by
+  rcases xs with ⟨xs⟩
   simpa using List.mem_of_mem_erase (by simpa using h)
 
-@[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {l : Array α} (ab : a ≠ b) :
-    a ∈ l.erase b ↔ a ∈ l :=
-  erase_eq_eraseP b l ▸ mem_eraseP_of_neg (mt eq_of_beq ab.symm)
+@[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {xs : Array α} (ab : a ≠ b) :
+    a ∈ xs.erase b ↔ a ∈ xs :=
+  erase_eq_eraseP b xs ▸ mem_eraseP_of_neg (mt eq_of_beq ab.symm)
 
-@[simp] theorem erase_eq_self_iff [LawfulBEq α] {l : Array α} : l.erase a = l ↔ a ∉ l := by
+@[simp] theorem erase_eq_self_iff [LawfulBEq α] {xs : Array α} : xs.erase a = xs ↔ a ∉ xs := by
   rw [erase_eq_eraseP', eraseP_eq_self_iff]
   simp [forall_mem_ne']
 
-theorem erase_filter [LawfulBEq α] (f : α → Bool) (l : Array α) :
-    (filter f l).erase a = filter f (l.erase a) := by
-  rcases l with ⟨l⟩
-  simpa using List.erase_filter f l
+theorem erase_filter [LawfulBEq α] (f : α → Bool) (xs : Array α) :
+    (filter f xs).erase a = filter f (xs.erase a) := by
+  rcases xs with ⟨xs⟩
+  simpa using List.erase_filter f xs
 
-theorem erase_append_left [LawfulBEq α] {l₁ : Array α} (l₂) (h : a ∈ l₁) :
-    (l₁ ++ l₂).erase a = l₁.erase a ++ l₂ := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
-  simpa using List.erase_append_left l₂ (by simpa using h)
+theorem erase_append_left [LawfulBEq α] {xs : Array α} (ys) (h : a ∈ xs) :
+    (xs ++ ys).erase a = xs.erase a ++ ys := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simpa using List.erase_append_left ys (by simpa using h)
 
-theorem erase_append_right [LawfulBEq α] {a : α} {l₁ : Array α} (l₂ : Array α) (h : a ∉ l₁) :
-    (l₁ ++ l₂).erase a = (l₁ ++ l₂.erase a) := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
-  simpa using List.erase_append_right l₂ (by simpa using h)
+theorem erase_append_right [LawfulBEq α] {a : α} {xs : Array α} (ys : Array α) (h : a ∉ xs) :
+    (xs ++ ys).erase a = (xs ++ ys.erase a) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simpa using List.erase_append_right ys (by simpa using h)
 
-theorem erase_append [LawfulBEq α] {a : α} {l₁ l₂ : Array α} :
-    (l₁ ++ l₂).erase a = if a ∈ l₁ then l₁.erase a ++ l₂ else l₁ ++ l₂.erase a := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+theorem erase_append [LawfulBEq α] {a : α} {xs ys : Array α} :
+    (xs ++ ys).erase a = if a ∈ xs then xs.erase a ++ ys else xs ++ ys.erase a := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp only [List.append_toArray, List.erase_toArray, List.erase_append, mem_toArray]
   split <;> simp
 
@@ -246,24 +249,24 @@ theorem erase_mkArray [LawfulBEq α] (n : Nat) (a b : α) :
   simp only [List.erase_replicate, beq_iff_eq, List.toArray_replicate]
   split <;> simp
 
-theorem erase_comm [LawfulBEq α] (a b : α) (l : Array α) :
-    (l.erase a).erase b = (l.erase b).erase a := by
-  rcases l with ⟨l⟩
-  simpa using List.erase_comm a b l
+theorem erase_comm [LawfulBEq α] (a b : α) (xs : Array α) :
+    (xs.erase a).erase b = (xs.erase b).erase a := by
+  rcases xs with ⟨xs⟩
+  simpa using List.erase_comm a b xs
 
-theorem erase_eq_iff [LawfulBEq α] {a : α} {l : Array α} :
-    l.erase a = l' ↔
-      (a ∉ l ∧ l = l') ∨
-        ∃ l₁ l₂, a ∉ l₁ ∧ l = l₁.push a ++ l₂ ∧ l' = l₁ ++ l₂ := by
+theorem erase_eq_iff [LawfulBEq α] {a : α} {xs : Array α} :
+    xs.erase a = ys ↔
+      (a ∉ xs ∧ xs = ys) ∨
+        ∃ as bs, a ∉ as ∧ xs = as.push a ++ bs ∧ ys = as ++ bs := by
   rw [erase_eq_eraseP', eraseP_eq_iff]
   simp only [beq_iff_eq, forall_mem_ne', exists_and_left]
   constructor
-  · rintro (⟨h, rfl⟩ | ⟨a', l', h, rfl, x, rfl, rfl⟩)
+  · rintro (⟨h, rfl⟩ | ⟨a', as, h, rfl, bs, rfl, rfl⟩)
     · left; simp_all
-    · right; refine ⟨l', h, x, by simp⟩
-  · rintro (⟨h, rfl⟩ | ⟨l₁, h, x, rfl, rfl⟩)
+    · right; refine ⟨as, h, bs, by simp⟩
+  · rintro (⟨h, rfl⟩ | ⟨as, h, bs, rfl, rfl⟩)
     · left; simp_all
-    · right; refine ⟨a, l₁, h, rfl, x, by simp⟩
+    · right; refine ⟨a, as, h, rfl, bs, by simp⟩
 
 @[simp] theorem erase_mkArray_self [LawfulBEq α] {a : α} :
     (mkArray n a).erase a = mkArray (n - 1) a := by
@@ -279,70 +282,74 @@ end erase
 
 /-! ### eraseIdx -/
 
-theorem eraseIdx_eq_take_drop_succ (l : Array α) (i : Nat) (h) : l.eraseIdx i = l.take i ++ l.drop (i + 1) := by
-  rcases l with ⟨l⟩
-  simp only [size_toArray] at h
+theorem eraseIdx_eq_eraseIdxIfInBounds {xs : Array α} {i : Nat} (h : i < xs.size) :
+    xs.eraseIdx i h = xs.eraseIdxIfInBounds i := by
+  simp [eraseIdxIfInBounds, h]
+
+theorem eraseIdx_eq_take_drop_succ (xs : Array α) (i : Nat) (h) : xs.eraseIdx i = xs.take i ++ xs.drop (i + 1) := by
+  rcases xs with ⟨xs⟩
+  simp only [List.size_toArray] at h
   simp only [List.eraseIdx_toArray, List.eraseIdx_eq_take_drop_succ, take_eq_extract,
     List.extract_toArray, List.extract_eq_drop_take, Nat.sub_zero, List.drop_zero, drop_eq_extract,
-    size_toArray, List.append_toArray, mk.injEq, List.append_cancel_left_eq]
+    List.size_toArray, List.append_toArray, mk.injEq, List.append_cancel_left_eq]
   rw [List.take_of_length_le]
   simp
 
-theorem getElem?_eraseIdx (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) :
-    (l.eraseIdx i)[j]? = if j < i then l[j]? else l[j + 1]? := by
-  rcases l with ⟨l⟩
+theorem getElem?_eraseIdx (xs : Array α) (i : Nat) (h : i < xs.size) (j : Nat) :
+    (xs.eraseIdx i)[j]? = if j < i then xs[j]? else xs[j + 1]? := by
+  rcases xs with ⟨xs⟩
   simp [List.getElem?_eraseIdx]
 
-theorem getElem?_eraseIdx_of_lt (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) (h' : j < i) :
-    (l.eraseIdx i)[j]? = l[j]? := by
+theorem getElem?_eraseIdx_of_lt (xs : Array α) (i : Nat) (h : i < xs.size) (j : Nat) (h' : j < i) :
+    (xs.eraseIdx i)[j]? = xs[j]? := by
   rw [getElem?_eraseIdx]
   simp [h']
 
-theorem getElem?_eraseIdx_of_ge (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) (h' : i ≤ j) :
-    (l.eraseIdx i)[j]? = l[j + 1]? := by
+theorem getElem?_eraseIdx_of_ge (xs : Array α) (i : Nat) (h : i < xs.size) (j : Nat) (h' : i ≤ j) :
+    (xs.eraseIdx i)[j]? = xs[j + 1]? := by
   rw [getElem?_eraseIdx]
   simp only [dite_eq_ite, ite_eq_right_iff]
   intro h'
   omega
 
-theorem getElem_eraseIdx (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) (h' : j < (l.eraseIdx i).size) :
-    (l.eraseIdx i)[j] = if h'' : j < i then
-        l[j]
+theorem getElem_eraseIdx (xs : Array α) (i : Nat) (h : i < xs.size) (j : Nat) (h' : j < (xs.eraseIdx i).size) :
+    (xs.eraseIdx i)[j] = if h'' : j < i then
+        xs[j]
       else
-        l[j + 1]'(by rw [size_eraseIdx] at h'; omega) := by
+        xs[j + 1]'(by rw [size_eraseIdx] at h'; omega) := by
   apply Option.some.inj
   rw [← getElem?_eq_getElem, getElem?_eraseIdx]
   split <;> simp
 
-@[simp] theorem eraseIdx_eq_empty_iff {l : Array α} {i : Nat} {h} : eraseIdx l i = #[] ↔ l.size = 1 ∧ i = 0 := by
-  rcases l with ⟨l⟩
-  simp only [List.eraseIdx_toArray, mk.injEq, List.eraseIdx_eq_nil_iff, size_toArray,
+@[simp] theorem eraseIdx_eq_empty_iff {xs : Array α} {i : Nat} {h} : xs.eraseIdx i = #[] ↔ xs.size = 1 ∧ i = 0 := by
+  rcases xs with ⟨xs⟩
+  simp only [List.eraseIdx_toArray, mk.injEq, List.eraseIdx_eq_nil_iff, List.size_toArray,
     or_iff_right_iff_imp]
   rintro rfl
   simp_all
 
-theorem eraseIdx_ne_empty_iff {l : Array α} {i : Nat} {h} : eraseIdx l i ≠ #[] ↔ 2 ≤ l.size := by
-  rcases l with ⟨_ | ⟨a, (_ | ⟨b, l⟩)⟩⟩
+theorem eraseIdx_ne_empty_iff {xs : Array α} {i : Nat} {h} : xs.eraseIdx i ≠ #[] ↔ 2 ≤ xs.size := by
+  rcases xs with ⟨_ | ⟨a, (_ | ⟨b, l⟩)⟩⟩
   · simp
   · simp at h
     simp [h]
   · simp
 
-theorem mem_of_mem_eraseIdx {l : Array α} {i : Nat} {h} {a : α} (h : a ∈ l.eraseIdx i) : a ∈ l := by
-  rcases l with ⟨l⟩
+theorem mem_of_mem_eraseIdx {xs : Array α} {i : Nat} {h} {a : α} (h : a ∈ xs.eraseIdx i) : a ∈ xs := by
+  rcases xs with ⟨xs⟩
   simpa using List.mem_of_mem_eraseIdx (by simpa using h)
 
-theorem eraseIdx_append_of_lt_size {l : Array α} {k : Nat} (hk : k < l.size) (l' : Array α) (h) :
-    eraseIdx (l ++ l') k = eraseIdx l k ++ l' := by
-  rcases l with ⟨l⟩
-  rcases l' with ⟨l'⟩
+theorem eraseIdx_append_of_lt_size {xs : Array α} {k : Nat} (hk : k < xs.size) (ys : Array α) (h) :
+    eraseIdx (xs ++ ys) k = eraseIdx xs k ++ ys := by
+  rcases xs with ⟨l⟩
+  rcases ys with ⟨l'⟩
   simp at hk
   simp [List.eraseIdx_append_of_lt_length, *]
 
-theorem eraseIdx_append_of_length_le {l : Array α} {k : Nat} (hk : l.size ≤ k) (l' : Array α) (h) :
-    eraseIdx (l ++ l') k = l ++ eraseIdx l' (k - l.size) (by simp at h; omega) := by
-  rcases l with ⟨l⟩
-  rcases l' with ⟨l'⟩
+theorem eraseIdx_append_of_length_le {xs : Array α} {k : Nat} (hk : xs.size ≤ k) (ys : Array α) (h) :
+    eraseIdx (xs ++ ys) k = xs ++ eraseIdx ys (k - xs.size) (by simp at h; omega) := by
+  rcases xs with ⟨l⟩
+  rcases ys with ⟨l'⟩
   simp at hk
   simp [List.eraseIdx_append_of_length_le, *]
 
@@ -352,49 +359,49 @@ theorem eraseIdx_mkArray {n : Nat} {a : α} {k : Nat} {h} :
   simp only [← List.toArray_replicate, List.eraseIdx_toArray]
   simp [List.eraseIdx_replicate, h]
 
-theorem mem_eraseIdx_iff_getElem {x : α} {l} {k} {h} : x ∈ eraseIdx l k h ↔ ∃ i w, i ≠ k ∧ l[i]'w = x := by
-  rcases l with ⟨l⟩
+theorem mem_eraseIdx_iff_getElem {x : α} {xs : Array α} {k} {h} : x ∈ xs.eraseIdx k h ↔ ∃ i w, i ≠ k ∧ xs[i]'w = x := by
+  rcases xs with ⟨xs⟩
   simp [List.mem_eraseIdx_iff_getElem, *]
 
-theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} {h} : x ∈ eraseIdx l k h ↔ ∃ i ≠ k, l[i]? = some x := by
-  rcases l with ⟨l⟩
+theorem mem_eraseIdx_iff_getElem? {x : α} {xs : Array α} {k} {h} : x ∈ xs.eraseIdx k h ↔ ∃ i ≠ k, xs[i]? = some x := by
+  rcases xs with ⟨xs⟩
   simp [List.mem_eraseIdx_iff_getElem?, *]
 
-theorem erase_eq_eraseIdx_of_idxOf [BEq α] [LawfulBEq α] (l : Array α) (a : α) (i : Nat) (w : l.idxOf a = i) (h : i < l.size) :
-    l.erase a = l.eraseIdx i := by
-  rcases l with ⟨l⟩
+theorem erase_eq_eraseIdx_of_idxOf [BEq α] [LawfulBEq α] (xs : Array α) (a : α) (i : Nat) (w : xs.idxOf a = i) (h : i < xs.size) :
+    xs.erase a = xs.eraseIdx i := by
+  rcases xs with ⟨xs⟩
   simp at w
   simp [List.erase_eq_eraseIdx_of_idxOf, *]
 
-theorem getElem_eraseIdx_of_lt (l : Array α) (i : Nat) (w : i < l.size) (j : Nat) (h : j < (l.eraseIdx i).size) (h' : j < i) :
-    (l.eraseIdx i)[j] = l[j] := by
-  rcases l with ⟨l⟩
+theorem getElem_eraseIdx_of_lt (xs : Array α) (i : Nat) (w : i < xs.size) (j : Nat) (h : j < (xs.eraseIdx i).size) (h' : j < i) :
+    (xs.eraseIdx i)[j] = xs[j] := by
+  rcases xs with ⟨xs⟩
   simp [List.getElem_eraseIdx_of_lt, *]
 
-theorem getElem_eraseIdx_of_ge (l : Array α) (i : Nat) (w : i < l.size) (j : Nat) (h : j < (l.eraseIdx i).size) (h' : i ≤ j) :
-    (l.eraseIdx i)[j] = l[j + 1]'(by simp at h; omega) := by
-  rcases l with ⟨l⟩
+theorem getElem_eraseIdx_of_ge (xs : Array α) (i : Nat) (w : i < xs.size) (j : Nat) (h : j < (xs.eraseIdx i).size) (h' : i ≤ j) :
+    (xs.eraseIdx i)[j] = xs[j + 1]'(by simp at h; omega) := by
+  rcases xs with ⟨xs⟩
   simp [List.getElem_eraseIdx_of_ge, *]
 
-theorem eraseIdx_set_eq {l : Array α} {i : Nat} {a : α} {h : i < l.size} :
-    (l.set i a).eraseIdx i (by simp; omega) = l.eraseIdx i := by
-  rcases l with ⟨l⟩
+theorem eraseIdx_set_eq {xs : Array α} {i : Nat} {a : α} {h : i < xs.size} :
+    (xs.set i a).eraseIdx i (by simp; omega) = xs.eraseIdx i := by
+  rcases xs with ⟨xs⟩
   simp [List.eraseIdx_set_eq, *]
 
-theorem eraseIdx_set_lt {l : Array α} {i : Nat} {w : i < l.size} {j : Nat} {a : α} (h : j < i) :
-    (l.set i a).eraseIdx j (by simp; omega) = (l.eraseIdx j).set (i - 1) a (by simp; omega) := by
-  rcases l with ⟨l⟩
+theorem eraseIdx_set_lt {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (h : j < i) :
+    (xs.set i a).eraseIdx j (by simp; omega) = (xs.eraseIdx j).set (i - 1) a (by simp; omega) := by
+  rcases xs with ⟨xs⟩
   simp [List.eraseIdx_set_lt, *]
 
-theorem eraseIdx_set_gt {l : Array α} {i : Nat} {j : Nat} {a : α} (h : i < j) {w : j < l.size} :
-    (l.set i a).eraseIdx j (by simp; omega) = (l.eraseIdx j).set i a (by simp; omega) := by
-  rcases l with ⟨l⟩
+theorem eraseIdx_set_gt {xs : Array α} {i : Nat} {j : Nat} {a : α} (h : i < j) {w : j < xs.size} :
+    (xs.set i a).eraseIdx j (by simp; omega) = (xs.eraseIdx j).set i a (by simp; omega) := by
+  rcases xs with ⟨xs⟩
   simp [List.eraseIdx_set_gt, *]
 
 @[simp] theorem set_getElem_succ_eraseIdx_succ
-    {l : Array α} {i : Nat} (h : i + 1 < l.size) :
-    (l.eraseIdx (i + 1)).set i l[i + 1] (by simp; omega) = l.eraseIdx i := by
-  rcases l with ⟨l⟩
+    {xs : Array α} {i : Nat} (h : i + 1 < xs.size) :
+    (xs.eraseIdx (i + 1)).set i xs[i + 1] (by simp; omega) = xs.eraseIdx i := by
+  rcases xs with ⟨xs⟩
   simp [List.set_getElem_succ_eraseIdx_succ, *]
 
 end Array

src/Init/Data/Array/Extract.lean

@@ -0,0 +1,430 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.Array.Lemmas
+import Init.Data.List.Nat.TakeDrop
+
+/-!
+# Lemmas about `Array.extract`
+
+This file follows the contents of `Init.Data.List.TakeDrop` and `Init.Data.List.Nat.TakeDrop`.
+-/
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
+open Nat
+namespace Array
+
+/-! ### extract -/
+
+@[simp] theorem extract_of_size_lt {as : Array α} {i j : Nat} (h : as.size < j) :
+    as.extract i j = as.extract i as.size := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract] at h₁ h₂
+    simp [h]
+
+theorem size_extract_le {as : Array α} {i j : Nat} :
+    (as.extract i j).size ≤ j - i := by
+  simp
+  omega
+
+theorem size_extract_le' {as : Array α} {i j : Nat} :
+    (as.extract i j).size ≤ as.size - i := by
+  simp
+  omega
+
+theorem size_extract_of_le {as : Array α} {i j : Nat} (h : j ≤ as.size) :
+    (as.extract i j).size = j - i := by
+  simp
+  omega
+
+@[simp]
+theorem extract_push {as : Array α} {b : α} {start stop : Nat} (h : stop ≤ as.size) :
+    (as.push b).extract start stop = as.extract start stop := by
+  ext i h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract, size_push] at h₁ h₂
+    simp only [getElem_extract, getElem_push]
+    rw [dif_pos (by omega)]
+
+@[simp]
+theorem extract_eq_pop {as : Array α} {stop : Nat} (h : stop = as.size - 1) :
+    as.extract 0 stop = as.pop := by
+  ext i h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract, size_pop] at h₁ h₂
+    simp [getElem_extract, getElem_pop]
+
+@[simp]
+theorem extract_append_extract {as : Array α} {i j k : Nat} :
+    as.extract i j ++ as.extract j k = as.extract (min i j) (max j k) := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_append, size_extract] at h₁ h₂
+    simp only [getElem_append, size_extract, getElem_extract]
+    split <;>
+    · congr 1
+      omega
+
+@[simp]
+theorem extract_eq_empty_iff {as : Array α} :
+    as.extract i j = #[] ↔ min j as.size ≤ i := by
+  constructor
+  · intro h
+    replace h := congrArg Array.size h
+    simp at h
+    omega
+  · intro h
+    exact eq_empty_of_size_eq_zero (by simp; omega)
+
+theorem extract_eq_empty_of_le {as : Array α} (h : min j as.size ≤ i) :
+    as.extract i j = #[] :=
+  extract_eq_empty_iff.2 h
+
+theorem lt_of_extract_ne_empty {as : Array α} (h : as.extract i j ≠ #[]) :
+    i < min j as.size :=
+  gt_of_not_le (mt extract_eq_empty_of_le h)
+
+@[simp]
+theorem extract_eq_self_iff {as : Array α} :
+    as.extract i j = as ↔ as.size = 0 ∨ i = 0 ∧ as.size ≤ j := by
+  constructor
+  · intro h
+    replace h := congrArg Array.size h
+    simp at h
+    omega
+  · intro h
+    ext l h₁ h₂
+    · simp
+      omega
+    · simp only [size_extract] at h₁
+      simp only [getElem_extract]
+      congr 1
+      omega
+
+theorem extract_eq_self_of_le {as : Array α} (h : as.size ≤ j) :
+    as.extract 0 j = as :=
+  extract_eq_self_iff.2 (.inr ⟨rfl, h⟩)
+
+theorem le_of_extract_eq_self {as : Array α} (h : as.extract i j = as) :
+    as.size ≤ j := by
+  replace h := congrArg Array.size h
+  simp at h
+  omega
+
+@[simp]
+theorem extract_size_left {as : Array α} :
+    as.extract as.size j = #[] := by
+  simp
+  omega
+
+@[simp]
+theorem push_extract_getElem {as : Array α} {i j : Nat} (h : j < as.size) :
+    (as.extract i j).push as[j] = as.extract (min i j) (j + 1) := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_push, size_extract] at h₁ h₂
+    simp only [getElem_push, size_extract, getElem_extract]
+    split <;>
+    · congr
+      omega
+
+theorem extract_succ_right {as : Array α} {i j : Nat} (w : i < j + 1) (h : j < as.size) :
+    as.extract i (j + 1) = (as.extract i j).push as[j] := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract, push_extract_getElem] at h₁ h₂
+    simp only [getElem_extract, push_extract_getElem]
+    congr
+    omega
+
+theorem extract_sub_one {as : Array α} {i j : Nat} (h : j < as.size) :
+    as.extract i (j - 1) = (as.extract i j).pop := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract, size_pop] at h₁ h₂
+    simp only [getElem_extract, getElem_pop]
+
+@[simp]
+theorem getElem?_extract_of_lt {as : Array α} {i j k : Nat} (h : k < min j as.size - i) :
+    (as.extract i j)[k]? = some (as[i + k]'(by omega)) := by
+  simp [getElem?_extract, h]
+
+theorem getElem?_extract_of_succ {as : Array α} {j : Nat} :
+    (as.extract 0 (j + 1))[j]? = as[j]? := by
+  simp [getElem?_extract]
+  omega
+
+@[simp] theorem extract_extract {as : Array α} {i j k l : Nat} :
+    (as.extract i j).extract k l = as.extract (i + k) (min (i + l) j) := by
+  ext m h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract] at h₁ h₂
+    simp [Nat.add_assoc]
+
+theorem extract_eq_empty_of_eq_empty {as : Array α} {i j : Nat} (h : as = #[]) :
+    as.extract i j = #[] := by
+  simp [h]
+
+theorem ne_empty_of_extract_ne_empty {as : Array α} {i j : Nat} (h : as.extract i j ≠ #[]) :
+    as ≠ #[] :=
+  mt extract_eq_empty_of_eq_empty h
+
+theorem extract_set {as : Array α} {i j k : Nat} (h : k < as.size) {a : α} :
+    (as.set k a).extract i j =
+      if _ : k < i then
+        as.extract i j
+      else if _ : k < min j as.size then
+        (as.extract i j).set (k - i) a (by simp; omega)
+      else as.extract i j := by
+  split
+  · ext l h₁ h₂
+    · simp
+    · simp at h₁ h₂
+      simp [getElem_set]
+      omega
+  · split
+    · ext l h₁ h₂
+      · simp
+      · simp only [getElem_extract, getElem_set]
+        split
+        · rw [if_pos]; omega
+        · rw [if_neg]; omega
+    · ext l h₁ h₂
+      · simp
+      · simp at h₁ h₂
+        simp [getElem_set]
+        omega
+
+theorem set_extract {as : Array α} {i j k : Nat} (h : k < (as.extract i j).size) {a : α} :
+    (as.extract i j).set k a = (as.set (i + k) a (by simp at h; omega)).extract i j := by
+  ext l h₁ h₂
+  · simp
+  · simp_all [getElem_set]
+
+@[simp]
+theorem extract_append {as bs : Array α} {i j : Nat} :
+    (as ++ bs).extract i j = as.extract i j ++ bs.extract (i - as.size) (j - as.size) := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract, size_append] at h₁ h₂
+    simp only [getElem_extract, getElem_append, size_extract]
+    split
+    · split
+      · rfl
+      · omega
+    · split
+      · omega
+      · congr 1
+        omega
+
+theorem extract_append_left {as bs : Array α} :
+    (as ++ bs).extract 0 as.size = as.extract 0 as.size := by
+  simp
+
+@[simp] theorem extract_append_right {as bs : Array α} :
+    (as ++ bs).extract as.size (as.size + i) = bs.extract 0 i := by
+  simp only [extract_append, extract_size_left, Nat.sub_self, empty_append]
+  congr 1
+  omega
+
+@[simp] theorem map_extract {as : Array α} {i j : Nat} :
+    (as.extract i j).map f = (as.map f).extract i j := by
+  ext l h₁ h₂
+  · simp
+  · simp only [size_map, size_extract] at h₁ h₂
+    simp only [getElem_map, getElem_extract]
+
+@[simp] theorem extract_mkArray {a : α} {n i j : Nat} :
+    (mkArray n a).extract i j = mkArray (min j n - i) a := by
+  ext l h₁ h₂
+  · simp
+  · simp only [size_extract, size_mkArray] at h₁ h₂
+    simp only [getElem_extract, getElem_mkArray]
+
+theorem extract_eq_extract_right {as : Array α} {i j j' : Nat} :
+    as.extract i j = as.extract i j' ↔ min (j - i) (as.size - i) = min (j' - i) (as.size - i) := by
+  rcases as with ⟨as⟩
+  simp
+
+theorem extract_eq_extract_left {as : Array α} {i i' j : Nat} :
+    as.extract i j = as.extract i' j ↔ min j as.size - i = min j as.size - i' := by
+  constructor
+  · intro h
+    replace h := congrArg Array.size h
+    simpa using h
+  · intro h
+    ext l h₁ h₂
+    · simpa
+    · simp only [size_extract] at h₁ h₂
+      simp only [getElem_extract]
+      congr 1
+      omega
+
+theorem extract_add_left {as : Array α} {i j k : Nat} :
+    as.extract (i + j) k = (as.extract i k).extract j (k - i) := by
+  simp [extract_eq_extract_right]
+  omega
+
+theorem mem_extract_iff_getElem {as : Array α} {a : α} {i j : Nat} :
+    a ∈ as.extract i j ↔ ∃ (k : Nat) (hm : k < min j as.size - i), as[i + k] = a := by
+  rcases as with ⟨as⟩
+  simp [List.mem_take_iff_getElem]
+  constructor <;>
+  · rintro ⟨k, h, rfl⟩
+    exact ⟨k, by omega, rfl⟩
+
+theorem set_eq_push_extract_append_extract {as : Array α} {i : Nat} (h : i < as.size) {a : α} :
+    as.set i a = (as.extract 0 i).push a ++ (as.extract (i + 1) as.size) := by
+  rcases as with ⟨as⟩
+  simp at h
+  simp [List.set_eq_take_append_cons_drop, h, List.take_of_length_le]
+
+theorem extract_reverse {as : Array α} {i j : Nat} :
+    as.reverse.extract i j = (as.extract (as.size - j) (as.size - i)).reverse := by
+  ext l h₁ h₂
+  · simp
+    omega
+  · simp only [size_extract, size_reverse] at h₁ h₂
+    simp only [getElem_extract, getElem_reverse, size_extract]
+    congr 1
+    omega
+
+theorem reverse_extract {as : Array α} {i j : Nat} :
+    (as.extract i j).reverse = as.reverse.extract (as.size - j) (as.size - i) := by
+  rw [extract_reverse]
+  simp
+  by_cases h : j ≤ as.size
+  · have : as.size - (as.size - j) = j := by omega
+    simp [this, extract_eq_extract_left]
+    omega
+  · have : as.size - (as.size - j) = as.size := by omega
+    simp only [Nat.not_le] at h
+    simp [h, this, extract_eq_extract_left]
+    omega
+
+/-! ### takeWhile -/
+
+theorem takeWhile_map (f : α → β) (p : β → Bool) (as : Array α) :
+    (as.map f).takeWhile p = (as.takeWhile (p ∘ f)).map f := by
+  rcases as with ⟨as⟩
+  simp [List.takeWhile_map]
+
+theorem popWhile_map (f : α → β) (p : β → Bool) (as : Array α) :
+    (as.map f).popWhile p = (as.popWhile (p ∘ f)).map f := by
+  rcases as with ⟨as⟩
+  simp [List.dropWhile_map, ← List.map_reverse]
+
+theorem takeWhile_filterMap (f : α → Option β) (p : β → Bool) (as : Array α) :
+    (as.filterMap f).takeWhile p = (as.takeWhile fun a => (f a).all p).filterMap f := by
+  rcases as with ⟨as⟩
+  simp [List.takeWhile_filterMap]
+
+theorem popWhile_filterMap (f : α → Option β) (p : β → Bool) (as : Array α) :
+    (as.filterMap f).popWhile p = (as.popWhile fun a => (f a).all p).filterMap f := by
+  rcases as with ⟨as⟩
+  simp [List.dropWhile_filterMap, ← List.filterMap_reverse]
+
+theorem takeWhile_filter (p q : α → Bool) (as : Array α) :
+    (as.filter p).takeWhile q = (as.takeWhile fun a => !p a || q a).filter p := by
+  rcases as with ⟨as⟩
+  simp [List.takeWhile_filter]
+
+theorem popWhile_filter (p q : α → Bool) (as : Array α) :
+    (as.filter p).popWhile q = (as.popWhile fun a => !p a || q a).filter p := by
+  rcases as with ⟨as⟩
+  simp [List.dropWhile_filter, ← List.filter_reverse]
+
+theorem takeWhile_append {xs ys : Array α} :
+    (xs ++ ys).takeWhile p =
+      if (xs.takeWhile p).size = xs.size then xs ++ ys.takeWhile p else xs.takeWhile p := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simp only [List.append_toArray, List.takeWhile_toArray, List.takeWhile_append, List.size_toArray]
+  split <;> rfl
+
+@[simp] theorem takeWhile_append_of_pos {p : α → Bool} {xs ys : Array α} (h : ∀ a ∈ xs, p a) :
+    (xs ++ ys).takeWhile p = xs ++ ys.takeWhile p := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simp at h
+  simp [List.takeWhile_append_of_pos h]
+
+theorem popWhile_append {xs ys : Array α} :
+    (xs ++ ys).popWhile p =
+      if (ys.popWhile p).isEmpty then xs.popWhile p else xs ++ ys.popWhile p := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simp only [List.append_toArray, List.popWhile_toArray, List.reverse_append, List.dropWhile_append,
+    List.isEmpty_iff, List.isEmpty_toArray, List.isEmpty_reverse]
+  -- Why do these not fire with `simp`?
+  rw [List.popWhile_toArray, List.isEmpty_toArray, List.isEmpty_reverse]
+  split
+  · rfl
+  · simp
+
+@[simp] theorem popWhile_append_of_pos {p : α → Bool} {xs ys : Array α} (h : ∀ a ∈ ys, p a) :
+    (xs ++ ys).popWhile p = xs.popWhile p := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simp at h
+  simp only [List.append_toArray, List.popWhile_toArray, List.reverse_append, mk.injEq,
+    List.reverse_inj]
+  rw [List.dropWhile_append_of_pos]
+  simpa
+
+@[simp] theorem takeWhile_mkArray_eq_filter (p : α → Bool) :
+    (mkArray n a).takeWhile p = (mkArray n a).filter p := by
+  simp [← List.toArray_replicate]
+
+theorem takeWhile_mkArray (p : α → Bool) :
+    (mkArray n a).takeWhile p = if p a then mkArray n a else #[] := by
+  simp [takeWhile_mkArray_eq_filter, filter_mkArray]
+
+@[simp] theorem popWhile_mkArray_eq_filter_not (p : α → Bool) :
+    (mkArray n a).popWhile p = (mkArray n a).filter (fun a => !p a) := by
+  simp [← List.toArray_replicate, ← List.filter_reverse]
+
+theorem popWhile_mkArray (p : α → Bool) :
+    (mkArray n a).popWhile p = if p a then #[] else mkArray n a := by
+  simp only [popWhile_mkArray_eq_filter_not, size_mkArray, filter_mkArray, Bool.not_eq_eq_eq_not,
+    Bool.not_true]
+  split <;> simp_all
+
+theorem extract_takeWhile {as : Array α} {i : Nat} :
+    (as.takeWhile p).extract 0 i = (as.extract 0 i).takeWhile p := by
+  rcases as with ⟨as⟩
+  simp [List.take_takeWhile]
+
+@[simp] theorem all_takeWhile {as : Array α} :
+    (as.takeWhile p).all p = true := by
+  rcases as with ⟨as⟩
+  rw [List.takeWhile_toArray] -- Not sure why this doesn't fire with `simp`.
+  simp
+
+@[simp] theorem any_popWhile {as : Array α} :
+    (as.popWhile p).any (fun a => !p a) = !as.all p := by
+  rcases as with ⟨as⟩
+  rw [List.popWhile_toArray] -- Not sure why this doesn't fire with `simp`.
+  simp
+
+theorem takeWhile_eq_extract_findIdx_not {xs : Array α} {p : α → Bool} :
+    takeWhile p xs = xs.extract 0 (xs.findIdx (fun a => !p a)) := by
+  rcases xs with ⟨xs⟩
+  simp [List.takeWhile_eq_take_findIdx_not]
+
+end Array

src/Init/Data/Array/FinRange.lean

@@ -5,10 +5,46 @@ Authors: François G. Dorais
 -/
 prelude
 import Init.Data.List.FinRange
+import Init.Data.Array.OfFn
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 namespace Array
 
 /-- `finRange n` is the array of all elements of `Fin n` in order. -/
 protected def finRange (n : Nat) : Array (Fin n) := ofFn fun i => i
 
+@[simp] theorem size_finRange (n) : (Array.finRange n).size = n := by
+  simp [Array.finRange]
+
+@[simp] theorem getElem_finRange (i : Nat) (h : i < (Array.finRange n).size) :
+    (Array.finRange n)[i] = Fin.cast (size_finRange n) ⟨i, h⟩ := by
+  simp [Array.finRange]
+
+@[simp] theorem finRange_zero : Array.finRange 0 = #[] := by simp [Array.finRange]
+
+theorem finRange_succ (n) : Array.finRange (n+1) = #[0] ++ (Array.finRange n).map Fin.succ := by
+  ext
+  · simp [Nat.add_comm]
+  · simp [getElem_append]
+    split <;>
+    · simp; omega
+
+theorem finRange_succ_last (n) :
+    Array.finRange (n+1) = (Array.finRange n).map Fin.castSucc ++ #[Fin.last n] := by
+  ext
+  · simp
+  · simp [getElem_push]
+    split
+    · simp
+    · simp_all
+      omega
+
+theorem finRange_reverse (n) : (Array.finRange n).reverse = (Array.finRange n).map Fin.rev := by
+  ext i h
+  · simp
+  · simp
+    omega
+
 end Array

src/Init/Data/Array/Find.lean

@@ -13,95 +13,92 @@ import Init.Data.Array.Range
 # Lemmas about `Array.findSome?`, `Array.find?, `Array.findIdx`, `Array.findIdx?`, `Array.idxOf`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 namespace Array
 
 open Nat
 
 /-! ### findSome? -/
 
-@[simp] theorem findSomeRev?_push_of_isSome (l : Array α) (h : (f a).isSome) : (l.push a).findSomeRev? f = f a := by
-  cases l; simp_all
+@[simp] theorem findSomeRev?_push_of_isSome (xs : Array α) (h : (f a).isSome) : (xs.push a).findSomeRev? f = f a := by
+  cases xs; simp_all
 
-@[simp] theorem findSomeRev?_push_of_isNone (l : Array α) (h : (f a).isNone) : (l.push a).findSomeRev? f = l.findSomeRev? f := by
-  cases l; simp_all
+@[simp] theorem findSomeRev?_push_of_isNone (xs : Array α) (h : (f a).isNone) : (xs.push a).findSomeRev? f = xs.findSomeRev? f := by
+  cases xs; simp_all
 
-theorem exists_of_findSome?_eq_some {f : α → Option β} {l : Array α} (w : l.findSome? f = some b) :
-    ∃ a, a ∈ l ∧ f a = b := by
-  cases l; simp_all [List.exists_of_findSome?_eq_some]
+theorem exists_of_findSome?_eq_some {f : α → Option β} {xs : Array α} (w : xs.findSome? f = some b) :
+    ∃ a, a ∈ xs ∧ f a = b := by
+  cases xs; simp_all [List.exists_of_findSome?_eq_some]
 
-@[simp] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
-  cases l; simp
+@[simp] theorem findSome?_eq_none_iff : findSome? p xs = none ↔ ∀ x ∈ xs, p x = none := by
+  cases xs; simp
 
-@[simp] theorem findSome?_isSome_iff {f : α → Option β} {l : Array α} :
-    (l.findSome? f).isSome ↔ ∃ x, x ∈ l ∧ (f x).isSome := by
-  cases l; simp
+@[simp] theorem findSome?_isSome_iff {f : α → Option β} {xs : Array α} :
+    (xs.findSome? f).isSome ↔ ∃ x, x ∈ xs ∧ (f x).isSome := by
+  cases xs; simp
 
-theorem findSome?_eq_some_iff {f : α → Option β} {l : Array α} {b : β} :
-    l.findSome? f = some b ↔ ∃ (l₁ : Array α) (a : α) (l₂ : Array α), l = l₁.push a ++ l₂ ∧ f a = some b ∧ ∀ x ∈ l₁, f x = none := by
-  cases l
+theorem findSome?_eq_some_iff {f : α → Option β} {xs : Array α} {b : β} :
+    xs.findSome? f = some b ↔ ∃ (ys : Array α) (a : α) (zs : Array α), xs = ys.push a ++ zs ∧ f a = some b ∧ ∀ x ∈ ys, f x = none := by
+  cases xs
   simp only [List.findSome?_toArray, List.findSome?_eq_some_iff]
   constructor
   · rintro ⟨l₁, a, l₂, rfl, h₁, h₂⟩
     exact ⟨l₁.toArray, a, l₂.toArray, by simp_all⟩
-  · rintro ⟨l₁, a, l₂, h₀, h₁, h₂⟩
-    exact ⟨l₁.toList, a, l₂.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩
+  · rintro ⟨xs, a, ys, h₀, h₁, h₂⟩
+    exact ⟨xs.toList, a, ys.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩
 
-@[simp] theorem findSome?_guard (l : Array α) : findSome? (Option.guard fun x => p x) l = find? p l := by
-  cases l; simp
+@[simp] theorem findSome?_guard (xs : Array α) : findSome? (Option.guard fun x => p x) xs = find? p xs := by
+  cases xs; simp
 
-theorem find?_eq_findSome?_guard (l : Array α) : find? p l = findSome? (Option.guard fun x => p x) l :=
-  (findSome?_guard l).symm
+theorem find?_eq_findSome?_guard (xs : Array α) : find? p xs = findSome? (Option.guard fun x => p x) xs :=
+  (findSome?_guard xs).symm
 
-@[simp] theorem getElem?_zero_filterMap (f : α → Option β) (l : Array α) : (l.filterMap f)[0]? = l.findSome? f := by
-  cases l; simp [← List.head?_eq_getElem?]
+@[simp] theorem getElem?_zero_filterMap (f : α → Option β) (xs : Array α) : (xs.filterMap f)[0]? = xs.findSome? f := by
+  cases xs; simp [← List.head?_eq_getElem?]
 
-@[simp] theorem getElem_zero_filterMap (f : α → Option β) (l : Array α) (h) :
-    (l.filterMap f)[0] = (l.findSome? f).get (by cases l; simpa [List.length_filterMap_eq_countP] using h) := by
-  cases l; simp [← List.head_eq_getElem, ← getElem?_zero_filterMap]
+@[simp] theorem getElem_zero_filterMap (f : α → Option β) (xs : Array α) (h) :
+    (xs.filterMap f)[0] = (xs.findSome? f).get (by cases xs; simpa [List.length_filterMap_eq_countP] using h) := by
+  cases xs; simp [← List.head_eq_getElem, ← getElem?_zero_filterMap]
 
-@[simp] theorem back?_filterMap (f : α → Option β) (l : Array α) : (l.filterMap f).back? = l.findSomeRev? f := by
-  cases l; simp
+@[simp] theorem back?_filterMap (f : α → Option β) (xs : Array α) : (xs.filterMap f).back? = xs.findSomeRev? f := by
+  cases xs; simp
 
-@[simp] theorem back!_filterMap [Inhabited β] (f : α → Option β) (l : Array α) :
-    (l.filterMap f).back! = (l.findSomeRev? f).getD default := by
-  cases l; simp
+@[simp] theorem back!_filterMap [Inhabited β] (f : α → Option β) (xs : Array α) :
+    (xs.filterMap f).back! = (xs.findSomeRev? f).getD default := by
+  cases xs; simp
 
-@[simp] theorem map_findSome? (f : α → Option β) (g : β → γ) (l : Array α) :
-    (l.findSome? f).map g = l.findSome? (Option.map g ∘ f) := by
-  cases l; simp
+@[simp] theorem map_findSome? (f : α → Option β) (g : β → γ) (xs : Array α) :
+    (xs.findSome? f).map g = xs.findSome? (Option.map g ∘ f) := by
+  cases xs; simp
 
-theorem findSome?_map (f : β → γ) (l : Array β) : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
-  cases l; simp [List.findSome?_map]
+theorem findSome?_map (f : β → γ) (xs : Array β) : findSome? p (xs.map f) = xs.findSome? (p ∘ f) := by
+  cases xs; simp [List.findSome?_map]
 
-theorem findSome?_append {l₁ l₂ : Array α} : (l₁ ++ l₂).findSome? f = (l₁.findSome? f).or (l₂.findSome? f) := by
-  cases l₁; cases l₂; simp [List.findSome?_append]
+theorem findSome?_append {xs ys : Array α} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
+  cases xs; cases ys; simp [List.findSome?_append]
 
-theorem getElem?_zero_flatten (L : Array (Array α)) :
-    (flatten L)[0]? = L.findSome? fun l => l[0]? := by
-  cases L using array₂_induction
+theorem getElem?_zero_flatten (xss : Array (Array α)) :
+    (flatten xss)[0]? = xss.findSome? fun xs => xs[0]? := by
+  cases xss using array₂_induction
   simp [← List.head?_eq_getElem?, List.head?_flatten, List.findSome?_map, Function.comp_def]
 
-theorem getElem_zero_flatten.proof {L : Array (Array α)} (h : 0 < L.flatten.size) :
-    (L.findSome? fun l => l[0]?).isSome := by
-  cases L using array₂_induction
+theorem getElem_zero_flatten.proof {xss : Array (Array α)} (h : 0 < xss.flatten.size) :
+    (xss.findSome? fun xs => xs[0]?).isSome := by
+  cases xss using array₂_induction
   simp only [List.findSome?_toArray, List.findSome?_map, Function.comp_def, List.getElem?_toArray,
     List.findSome?_isSome_iff, isSome_getElem?]
-  simp only [flatten_toArray_map_toArray, size_toArray, List.length_flatten,
+  simp only [flatten_toArray_map_toArray, List.size_toArray, List.length_flatten,
     Nat.sum_pos_iff_exists_pos, List.mem_map] at h
   obtain ⟨_, ⟨xs, m, rfl⟩, h⟩ := h
   exact ⟨xs, m, by simpa using h⟩
 
-theorem getElem_zero_flatten {L : Array (Array α)} (h) :
-    (flatten L)[0] = (L.findSome? fun l => l[0]?).get (getElem_zero_flatten.proof h) := by
-  have t := getElem?_zero_flatten L
+theorem getElem_zero_flatten {xss : Array (Array α)} (h) :
+    (flatten xss)[0] = (xss.findSome? fun xs => xs[0]?).get (getElem_zero_flatten.proof h) := by
+  have t := getElem?_zero_flatten xss
   simp [getElem?_eq_getElem, h] at t
   simp [← t]
 
-theorem back?_flatten {L : Array (Array α)} :
-    (flatten L).back? = (L.findSomeRev? fun l => l.back?) := by
-  cases L using array₂_induction
-  simp [List.getLast?_flatten, ← List.map_reverse, List.findSome?_map, Function.comp_def]
-
 theorem findSome?_mkArray : findSome? f (mkArray n a) = if n = 0 then none else f a := by
   simp [← List.toArray_replicate, List.findSome?_replicate]
 
@@ -124,16 +121,16 @@ theorem findSome?_mkArray : findSome? f (mkArray n a) = if n = 0 then none else
     #[a].find? p = if p a then some a else none := by
   simp [singleton_eq_toArray_singleton]
 
-@[simp] theorem findRev?_push_of_pos (l : Array α) (h : p a) :
-    findRev? p (l.push a) = some a := by
-  cases l; simp [h]
+@[simp] theorem findRev?_push_of_pos (xs : Array α) (h : p a) :
+    findRev? p (xs.push a) = some a := by
+  cases xs; simp [h]
 
-@[simp] theorem findRev?_cons_of_neg (l : Array α) (h : ¬p a) :
-    findRev? p (l.push a) = findRev? p l := by
-  cases l; simp [h]
+@[simp] theorem findRev?_cons_of_neg (xs : Array α) (h : ¬p a) :
+    findRev? p (xs.push a) = findRev? p xs := by
+  cases xs; simp [h]
 
-@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
-  cases l; simp
+@[simp] theorem find?_eq_none : find? p xs = none ↔ ∀ x ∈ xs, ¬ p x := by
+  cases xs; simp
 
 theorem find?_eq_some_iff_append {xs : Array α} :
     xs.find? p = some b ↔ p b ∧ ∃ (as bs : Array α), xs = as.push b ++ bs ∧ ∀ a ∈ as, !p a := by
@@ -142,10 +139,10 @@ theorem find?_eq_some_iff_append {xs : Array α} :
     Bool.not_true, exists_and_right, and_congr_right_iff]
   intro w
   constructor
-  · rintro ⟨as, ⟨⟨x, rfl⟩, h⟩⟩
-    exact ⟨as.toArray, ⟨x.toArray, by simp⟩ , by simpa using h⟩
-  · rintro ⟨as, ⟨⟨x, h'⟩, h⟩⟩
-    exact ⟨as.toList, ⟨x.toList, by simpa using congrArg Array.toList h'⟩,
+  · rintro ⟨as, ⟨⟨xs, rfl⟩, h⟩⟩
+    exact ⟨as.toArray, ⟨xs.toArray, by simp⟩ , by simpa using h⟩
+  · rintro ⟨as, ⟨⟨⟨l⟩, h'⟩, h⟩⟩
+    exact ⟨as.toList, ⟨l, by simpa using congrArg Array.toList h'⟩,
       by simpa using h⟩
 
 @[simp]
@@ -174,22 +171,22 @@ theorem get_find?_mem {xs : Array α} (h) : (xs.find? p).get h ∈ xs := by
     (xs.filter p).find? q = xs.find? (fun a => p a ∧ q a) := by
   cases xs; simp
 
-@[simp] theorem getElem?_zero_filter (p : α → Bool) (l : Array α) :
-    (l.filter p)[0]? = l.find? p := by
-  cases l; simp [← List.head?_eq_getElem?]
+@[simp] theorem getElem?_zero_filter (p : α → Bool) (xs : Array α) :
+    (xs.filter p)[0]? = xs.find? p := by
+  cases xs; simp [← List.head?_eq_getElem?]
 
-@[simp] theorem getElem_zero_filter (p : α → Bool) (l : Array α) (h) :
-    (l.filter p)[0] =
-      (l.find? p).get (by cases l; simpa [← List.countP_eq_length_filter] using h) := by
-  cases l
+@[simp] theorem getElem_zero_filter (p : α → Bool) (xs : Array α) (h) :
+    (xs.filter p)[0] =
+      (xs.find? p).get (by cases xs; simpa [← List.countP_eq_length_filter] using h) := by
+  cases xs
   simp [List.getElem_zero_eq_head]
 
-@[simp] theorem back?_filter (p : α → Bool) (l : Array α) : (l.filter p).back? = l.findRev? p := by
-  cases l; simp
+@[simp] theorem back?_filter (p : α → Bool) (xs : Array α) : (xs.filter p).back? = xs.findRev? p := by
+  cases xs; simp
 
-@[simp] theorem back!_filter [Inhabited α] (p : α → Bool) (l : Array α) :
-    (l.filter p).back! = (l.findRev? p).get! := by
-  cases l; simp [Option.get!_eq_getD]
+@[simp] theorem back!_filter [Inhabited α] (p : α → Bool) (xs : Array α) :
+    (xs.filter p).back! = (xs.findRev? p).get! := by
+  cases xs; simp [Option.get!_eq_getD]
 
 @[simp] theorem find?_filterMap (xs : Array α) (f : α → Option β) (p : β → Bool) :
     (xs.filterMap f).find? p = (xs.find? (fun a => (f a).any p)).bind f := by
@@ -199,19 +196,19 @@ theorem get_find?_mem {xs : Array α} (h) : (xs.find? p).get h ∈ xs := by
     find? p (xs.map f) = (xs.find? (p ∘ f)).map f := by
   cases xs; simp
 
-@[simp] theorem find?_append {l₁ l₂ : Array α} :
-    (l₁ ++ l₂).find? p = (l₁.find? p).or (l₂.find? p) := by
-  cases l₁
-  cases l₂
+@[simp] theorem find?_append {xs ys : Array α} :
+    (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
+  cases xs
+  cases ys
   simp
 
-@[simp] theorem find?_flatten (xs : Array (Array α)) (p : α → Bool) :
-    xs.flatten.find? p = xs.findSome? (·.find? p) := by
-  cases xs using array₂_induction
+@[simp] theorem find?_flatten (xss : Array (Array α)) (p : α → Bool) :
+    xss.flatten.find? p = xss.findSome? (·.find? p) := by
+  cases xss using array₂_induction
   simp [List.findSome?_map, Function.comp_def]
 
-theorem find?_flatten_eq_none_iff {xs : Array (Array α)} {p : α → Bool} :
-    xs.flatten.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
+theorem find?_flatten_eq_none_iff {xss : Array (Array α)} {p : α → Bool} :
+    xss.flatten.find? p = none ↔ ∀ ys ∈ xss, ∀ x ∈ ys, !p x := by
   simp
 
 @[deprecated find?_flatten_eq_none_iff (since := "2025-02-03")]
@@ -222,12 +219,12 @@ If `find? p` returns `some a` from `xs.flatten`, then `p a` holds, and
 some array in `xs` contains `a`, and no earlier element of that array satisfies `p`.
 Moreover, no earlier array in `xs` has an element satisfying `p`.
 -/
-theorem find?_flatten_eq_some_iff {xs : Array (Array α)} {p : α → Bool} {a : α} :
-    xs.flatten.find? p = some a ↔
+theorem find?_flatten_eq_some_iff {xss : Array (Array α)} {p : α → Bool} {a : α} :
+    xss.flatten.find? p = some a ↔
       p a ∧ ∃ (as : Array (Array α)) (ys zs : Array α) (bs : Array (Array α)),
-        xs = as.push (ys.push a ++ zs) ++ bs ∧
-        (∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
-  cases xs using array₂_induction
+        xss = as.push (ys.push a ++ zs) ++ bs ∧
+        (∀ ws ∈ as, ∀ x ∈ ws, !p x) ∧ (∀ x ∈ ys, !p x) := by
+  cases xss using array₂_induction
   simp only [flatten_toArray_map_toArray, List.find?_toArray, List.find?_flatten_eq_some_iff]
   simp only [Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, and_congr_right_iff]
   intro w
@@ -302,24 +299,6 @@ theorem find?_eq_some_iff_getElem {xs : Array α} {p : α → Bool} {b : α} :
   rcases xs with ⟨xs⟩
   simp [List.find?_eq_some_iff_getElem]
 
-/-! ### findFinIdx? -/
-
-@[simp] theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p #[] = none := rfl
-
--- We can't mark this as a `@[congr]` lemma since the head of the RHS is not `findFinIdx?`.
-theorem findFinIdx?_congr {p : α → Bool} {l₁ : Array α} {l₂ : Array α} (w : l₁ = l₂) :
-    findFinIdx? p l₁ = (findFinIdx? p l₂).map (fun i => i.cast (by simp [w])) := by
-  subst w
-  simp
-
-@[simp] theorem findFinIdx?_subtype {p : α → Prop} {l : Array { x // p x }}
-    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.findFinIdx? f = (l.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
-  cases l
-  simp only [List.findFinIdx?_toArray, hf, List.findFinIdx?_subtype]
-  rw [findFinIdx?_congr List.unattach_toArray]
-  simp [Function.comp_def]
-
 /-! ### findIdx -/
 
 theorem findIdx_of_getElem?_eq_some {xs : Array α} (w : xs[xs.findIdx p]? = some y) : p y := by
@@ -395,23 +374,38 @@ theorem findIdx_eq {p : α → Bool} {xs : Array α} {i : Nat} (h : i < xs.size)
   simp at h3
   simp_all [not_of_lt_findIdx h3]
 
-theorem findIdx_append (p : α → Bool) (l₁ l₂ : Array α) :
-    (l₁ ++ l₂).findIdx p =
-      if l₁.findIdx p < l₁.size then l₁.findIdx p else l₂.findIdx p + l₁.size := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+theorem findIdx_append (p : α → Bool) (xs ys : Array α) :
+    (xs ++ ys).findIdx p =
+      if xs.findIdx p < xs.size then xs.findIdx p else ys.findIdx p + xs.size := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp [List.findIdx_append]
 
-theorem findIdx_le_findIdx {l : Array α} {p q : α → Bool} (h : ∀ x ∈ l, p x → q x) : l.findIdx q ≤ l.findIdx p := by
-  rcases l with ⟨l⟩
+theorem findIdx_le_findIdx {xs : Array α} {p q : α → Bool} (h : ∀ x ∈ xs, p x → q x) : xs.findIdx q ≤ xs.findIdx p := by
+  rcases xs with ⟨xs⟩
   simp_all [List.findIdx_le_findIdx]
 
-@[simp] theorem findIdx_subtype {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem findIdx_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.findIdx f = l.unattach.findIdx g := by
-  cases l
+    xs.findIdx f = xs.unattach.findIdx g := by
+  cases xs
   simp [hf]
 
+theorem false_of_mem_extract_findIdx {xs : Array α} {p : α → Bool} (h : x ∈ xs.extract 0 (xs.findIdx p)) :
+    p x = false := by
+  rcases xs with ⟨xs⟩
+  exact List.false_of_mem_take_findIdx (by simpa using h)
+
+@[simp] theorem findIdx_extract {xs : Array α} {i : Nat} {p : α → Bool} :
+    (xs.extract 0 i).findIdx p = min i (xs.findIdx p) := by
+  cases xs
+  simp
+
+@[simp] theorem min_findIdx_findIdx {xs : Array α} {p q : α → Bool} :
+    min (xs.findIdx p) (xs.findIdx q) = xs.findIdx (fun a => p a || q a) := by
+  cases xs
+  simp
+
 /-! ### findIdx? -/
 
 @[simp] theorem findIdx?_empty : (#[] : Array α).findIdx? p = none := rfl
@@ -468,8 +462,8 @@ theorem of_findIdx?_eq_none {xs : Array α} {p : α → Bool} (w : xs.findIdx? p
   rcases xs with ⟨xs⟩
   simpa using List.of_findIdx?_eq_none (by simpa using w)
 
-@[simp] theorem findIdx?_map (f : β → α) (l : Array β) : findIdx? p (l.map f) = l.findIdx? (p ∘ f) := by
-  rcases l with ⟨l⟩
+@[simp] theorem findIdx?_map (f : β → α) (xs : Array β) : findIdx? p (xs.map f) = xs.findIdx? (p ∘ f) := by
+  rcases xs with ⟨xs⟩
   simp [List.findIdx?_map]
 
 @[simp] theorem findIdx?_append :
@@ -479,12 +473,12 @@ theorem of_findIdx?_eq_none {xs : Array α} {p : α → Bool} (w : xs.findIdx? p
   rcases ys with ⟨ys⟩
   simp [List.findIdx?_append]
 
-theorem findIdx?_flatten {l : Array (Array α)} {p : α → Bool} :
-    l.flatten.findIdx? p =
-      (l.findIdx? (·.any p)).map
-        fun i => ((l.take i).map Array.size).sum +
-          (l[i]?.map fun xs => xs.findIdx p).getD 0 := by
-  cases l using array₂_induction
+theorem findIdx?_flatten {xss : Array (Array α)} {p : α → Bool} :
+    xss.flatten.findIdx? p =
+      (xss.findIdx? (·.any p)).map
+        fun i => ((xss.take i).map Array.size).sum +
+          (xss[i]?.map fun xs => xs.findIdx p).getD 0 := by
+  cases xss using array₂_induction
   simp [List.findIdx?_flatten, Function.comp_def]
 
 @[simp] theorem findIdx?_mkArray :
@@ -519,20 +513,66 @@ theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : Array α} {p q : α → Bo
   rcases xs with ⟨xs⟩
   simp [List.findIdx?_eq_some_le_of_findIdx?_eq_some (by simpa using w) (by simpa using h)]
 
-@[simp] theorem findIdx?_subtype {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem findIdx?_subtype {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.findIdx? f = l.unattach.findIdx? g := by
-  cases l
+    xs.findIdx? f = xs.unattach.findIdx? g := by
+  cases xs
   simp [hf]
 
+@[simp] theorem findIdx?_take {xs : Array α} {i : Nat} {p : α → Bool} :
+    (xs.take i).findIdx? p = (xs.findIdx? p).bind (Option.guard (fun j => j < i)) := by
+  cases xs
+  simp
+
+/-! ### findFinIdx? -/
+
+@[simp] theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p #[] = none := rfl
+
+-- We can't mark this as a `@[congr]` lemma since the head of the RHS is not `findFinIdx?`.
+theorem findFinIdx?_congr {p : α → Bool} {xs ys : Array α} (w : xs = ys) :
+    findFinIdx? p xs = (findFinIdx? p ys).map (fun i => i.cast (by simp [w])) := by
+  subst w
+  simp
+
+theorem findFinIdx?_eq_pmap_findIdx? {xs : Array α} {p : α → Bool} :
+    xs.findFinIdx? p =
+      (xs.findIdx? p).pmap
+        (fun i m => by simp [findIdx?_eq_some_iff_getElem] at m; exact ⟨i, m.choose⟩)
+        (fun i h => h) := by
+  simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]
+
+@[simp] theorem findFinIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
+    xs.findFinIdx? p = none ↔ ∀ x, x ∈ xs → ¬ p x := by
+  simp [findFinIdx?_eq_pmap_findIdx?]
+
+@[simp]
+theorem findFinIdx?_eq_some_iff {xs : Array α} {p : α → Bool} {i : Fin xs.size} :
+    xs.findFinIdx? p = some i ↔
+      p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji i.2)) := by
+  simp only [findFinIdx?_eq_pmap_findIdx?, Option.pmap_eq_some_iff, findIdx?_eq_some_iff_getElem,
+    Bool.not_eq_true, Option.mem_def, exists_and_left, and_exists_self, Fin.getElem_fin]
+  constructor
+  · rintro ⟨a, ⟨h, w₁, w₂⟩, rfl⟩
+    exact ⟨w₁, fun j hji => by simpa using w₂ j hji⟩
+  · rintro ⟨h, w⟩
+    exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩
+
+@[simp] theorem findFinIdx?_subtype {p : α → Prop} {xs : Array { x // p x }}
+    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    xs.findFinIdx? f = (xs.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
+  cases xs
+  simp only [List.findFinIdx?_toArray, hf, List.findFinIdx?_subtype]
+  rw [findFinIdx?_congr List.unattach_toArray]
+  simp [Function.comp_def]
+
 /-! ### idxOf
 
 The verification API for `idxOf` is still incomplete.
 The lemmas below should be made consistent with those for `findIdx` (and proved using them).
 -/
 
-theorem idxOf_append [BEq α] [LawfulBEq α] {l₁ l₂ : Array α} {a : α} :
-    (l₁ ++ l₂).idxOf a = if a ∈ l₁ then l₁.idxOf a else l₂.idxOf a + l₁.size := by
+theorem idxOf_append [BEq α] [LawfulBEq α] {xs ys : Array α} {a : α} :
+    (xs ++ ys).idxOf a = if a ∈ xs then xs.idxOf a else ys.idxOf a + xs.size := by
   rw [idxOf, findIdx_append]
   split <;> rename_i h
   · rw [if_pos]
@@ -540,12 +580,12 @@ theorem idxOf_append [BEq α] [LawfulBEq α] {l₁ l₂ : Array α} {a : α} :
   · rw [if_neg]
     simpa using h
 
-theorem idxOf_eq_size [BEq α] [LawfulBEq α] {l : Array α} (h : a ∉ l) : l.idxOf a = l.size := by
-  rcases l with ⟨l⟩
+theorem idxOf_eq_size [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∉ xs) : xs.idxOf a = xs.size := by
+  rcases xs with ⟨xs⟩
   simp [List.idxOf_eq_length (by simpa using h)]
 
-theorem idxOf_lt_length [BEq α] [LawfulBEq α] {l : Array α} (h : a ∈ l) : l.idxOf a < l.size := by
-  rcases l with ⟨l⟩
+theorem idxOf_lt_length [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∈ xs) : xs.idxOf a < xs.size := by
+  rcases xs with ⟨xs⟩
   simp [List.idxOf_lt_length (by simpa using h)]
 
 
@@ -557,15 +597,31 @@ The lemmas below should be made consistent with those for `findIdx?` (and proved
 
 @[simp] theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := rfl
 
-@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : Array α} {a : α} :
-    l.idxOf? a = none ↔ a ∉ l := by
-  rcases l with ⟨l⟩
+@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
+    xs.idxOf? a = none ↔ a ∉ xs := by
+  rcases xs with ⟨xs⟩
   simp [List.idxOf?_eq_none_iff]
 
-/-! ### finIdxOf? -/
+/-! ### finIdxOf?
+
+The verification API for `finIdxOf?` is still incomplete.
+The lemmas below should be made consistent with those for `findFinIdx?` (and proved using them).
+-/
 
 theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : Array α} {a : α} :
     xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
   simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
 
+@[simp] theorem finIdxOf?_empty [BEq α] : (#[] : Array α).finIdxOf? a = none := rfl
+
+@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
+    xs.finIdxOf? a = none ↔ a ∉ xs := by
+  rcases xs with ⟨xs⟩
+  simp [List.finIdxOf?_eq_none_iff]
+
+@[simp] theorem finIdxOf?_eq_some_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} {i : Fin xs.size} :
+    xs.finIdxOf? a = some i ↔ xs[i] = a ∧ ∀ j (_ : j < i), ¬xs[j] = a := by
+  rcases xs with ⟨xs⟩
+  simp [List.finIdxOf?_eq_some_iff]
+
 end Array

src/Init/Data/Array/GetLit.lean

@@ -7,40 +7,43 @@ Authors: Leonardo de Moura
 prelude
 import Init.Data.Array.Basic
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 /-! ### getLit -/
 
 -- auxiliary declaration used in the equation compiler when pattern matching array literals.
-abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
+abbrev getLit {α : Type u} {n : Nat} (xs : Array α) (i : Nat) (h₁ : xs.size = n) (h₂ : i < n) : α :=
   have := h₁.symm ▸ h₂
-  a[i]
+  xs[i]
 
 theorem extLit {n : Nat}
-    (a b : Array α)
-    (hsz₁ : a.size = n) (hsz₂ : b.size = n)
-    (h : (i : Nat) → (hi : i < n) → a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
-  Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ _ => h i (hsz₁ ▸ hi₁)
+    (xs ys : Array α)
+    (hsz₁ : xs.size = n) (hsz₂ : ys.size = n)
+    (h : (i : Nat) → (hi : i < n) → xs.getLit i hsz₁ hi = ys.getLit i hsz₂ hi) : xs = ys :=
+  Array.ext xs ys (hsz₁.trans hsz₂.symm) fun i hi₁ _ => h i (hsz₁ ▸ hi₁)
 
-def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
+def toListLitAux (xs : Array α) (n : Nat) (hsz : xs.size = n) : ∀ (i : Nat), i ≤ xs.size → List α → List α
   | 0,     _,  acc => acc
-  | (i+1), hi, acc => toListLitAux a n hsz i (Nat.le_of_succ_le hi) (a.getLit i hsz (Nat.lt_of_lt_of_eq (Nat.lt_of_lt_of_le (Nat.lt_succ_self i) hi) hsz) :: acc)
+  | (i+1), hi, acc => toListLitAux xs n hsz i (Nat.le_of_succ_le hi) (xs.getLit i hsz (Nat.lt_of_lt_of_eq (Nat.lt_of_lt_of_le (Nat.lt_succ_self i) hi) hsz) :: acc)
 
-def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
-  List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.le_refl _) []
+def toArrayLit (xs : Array α) (n : Nat) (hsz : xs.size = n) : Array α :=
+  List.toArray <| toListLitAux xs n hsz n (hsz ▸ Nat.le_refl _) []
 
-theorem toArrayLit_eq (as : Array α) (n : Nat) (hsz : as.size = n) : as = toArrayLit as n hsz := by
+theorem toArrayLit_eq (xs : Array α) (n : Nat) (hsz : xs.size = n) : xs = toArrayLit xs n hsz := by
   apply ext'
-  simp [toArrayLit, toList_toArray]
-  have hle : n ≤ as.size := hsz ▸ Nat.le_refl _
-  have hge : as.size ≤ n := hsz ▸ Nat.le_refl _
+  simp [toArrayLit, List.toList_toArray]
+  have hle : n ≤ xs.size := hsz ▸ Nat.le_refl _
+  have hge : xs.size ≤ n := hsz ▸ Nat.le_refl _
   have := go n hle
   rw [List.drop_eq_nil_of_le hge] at this
   rw [this]
 where
-  getLit_eq (as : Array α) (i : Nat) (h₁ : as.size = n) (h₂ : i < n) : as.getLit i h₁ h₂ = getElem as.toList i ((id (α := as.toList.length = n) h₁) ▸ h₂) :=
+  getLit_eq (xs : Array α) (i : Nat) (h₁ : xs.size = n) (h₂ : i < n) : xs.getLit i h₁ h₂ = getElem xs.toList i ((id (α := xs.toList.length = n) h₁) ▸ h₂) :=
     rfl
-  go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.toList.drop i) = as.toList := by
+  go (i : Nat) (hi : i ≤ xs.size) : toListLitAux xs n hsz i hi (xs.toList.drop i) = xs.toList := by
     induction i <;> simp only [List.drop, toListLitAux, getLit_eq, List.getElem_cons_drop_succ_eq_drop, *]
 
 end Array

src/Init/Data/Array/InsertIdx.lean

@@ -13,6 +13,9 @@ import Init.Data.List.Nat.InsertIdx
 Proves various lemmas about `Array.insertIdx`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 open Function
 
 open Nat
@@ -27,23 +30,23 @@ section InsertIdx
 
 variable {a : α}
 
-@[simp] theorem toList_insertIdx (a : Array α) (i x) (h) :
-    (a.insertIdx i x h).toList = a.toList.insertIdx i x := by
-  rcases a with ⟨a⟩
+@[simp] theorem toList_insertIdx (xs : Array α) (i x) (h) :
+    (xs.insertIdx i x h).toList = xs.toList.insertIdx i x := by
+  rcases xs with ⟨xs⟩
   simp
 
 @[simp]
-theorem insertIdx_zero (s : Array α) (x : α) : s.insertIdx 0 x = #[x] ++ s := by
-  cases s
+theorem insertIdx_zero (xs : Array α) (x : α) : xs.insertIdx 0 x = #[x] ++ xs := by
+  rcases xs with ⟨xs⟩
   simp
 
-@[simp] theorem size_insertIdx {as : Array α} (h : n ≤ as.size) : (as.insertIdx n a).size = as.size + 1 := by
-  cases as
+@[simp] theorem size_insertIdx {xs : Array α} (h : i ≤ xs.size) : (xs.insertIdx i a).size = xs.size + 1 := by
+  rcases xs with ⟨xs⟩
   simp [List.length_insertIdx, h]
 
-theorem eraseIdx_insertIdx (i : Nat) (l : Array α) (h : i ≤ l.size) :
-    (l.insertIdx i a).eraseIdx i (by simp; omega) = l := by
-  cases l
+theorem eraseIdx_insertIdx (i : Nat) (xs : Array α) (h : i ≤ xs.size) :
+    (xs.insertIdx i a).eraseIdx i (by simp; omega) = xs := by
+  rcases xs with ⟨xs⟩
   simp_all
 
 theorem insertIdx_eraseIdx_of_ge {as : Array α}
@@ -60,68 +63,68 @@ theorem insertIdx_eraseIdx_of_le {as : Array α}
   cases as
   simpa using List.insertIdx_eraseIdx_of_le _ _ _ (by simpa) (by simpa)
 
-theorem insertIdx_comm (a b : α) (i j : Nat) (l : Array α) (_ : i ≤ j) (_ : j ≤ l.size) :
-    (l.insertIdx i a).insertIdx (j + 1) b (by simpa) =
-      (l.insertIdx j b).insertIdx i a (by simp; omega) := by
-  cases l
+theorem insertIdx_comm (a b : α) (i j : Nat) (xs : Array α) (_ : i ≤ j) (_ : j ≤ xs.size) :
+    (xs.insertIdx i a).insertIdx (j + 1) b (by simpa) =
+      (xs.insertIdx j b).insertIdx i a (by simp; omega) := by
+  rcases xs with ⟨xs⟩
   simpa using List.insertIdx_comm a b i j _ (by simpa) (by simpa)
 
-theorem mem_insertIdx {l : Array α} {h : i ≤ l.size} : a ∈ l.insertIdx i b h ↔ a = b ∨ a ∈ l := by
-  cases l
+theorem mem_insertIdx {xs : Array α} {h : i ≤ xs.size} : a ∈ xs.insertIdx i b h ↔ a = b ∨ a ∈ xs := by
+  rcases xs with ⟨xs⟩
   simpa using List.mem_insertIdx (by simpa)
 
 @[simp]
-theorem insertIdx_size_self (l : Array α) (x : α) : l.insertIdx l.size x = l.push x := by
-  cases l
+theorem insertIdx_size_self (xs : Array α) (x : α) : xs.insertIdx xs.size x = xs.push x := by
+  rcases xs with ⟨xs⟩
   simp
 
-theorem getElem_insertIdx {as : Array α} {x : α} {i k : Nat} (w : i ≤ as.size) (h : k < (as.insertIdx i x).size) :
-    (as.insertIdx i x)[k] =
+theorem getElem_insertIdx {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.size) (h : k < (xs.insertIdx i x).size) :
+    (xs.insertIdx i x)[k] =
       if h₁ : k < i then
-        as[k]'(by simp [size_insertIdx] at h; omega)
+        xs[k]'(by simp [size_insertIdx] at h; omega)
       else
         if h₂ : k = i then
           x
         else
-          as[k-1]'(by simp [size_insertIdx] at h; omega) := by
-  cases as
+          xs[k-1]'(by simp [size_insertIdx] at h; omega) := by
+  cases xs
   simp [List.getElem_insertIdx, w]
 
-theorem getElem_insertIdx_of_lt {as : Array α} {x : α} {i k : Nat} (w : i ≤ as.size) (h : k < i) :
-    (as.insertIdx i x)[k]'(by simp; omega) = as[k] := by
+theorem getElem_insertIdx_of_lt {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.size) (h : k < i) :
+    (xs.insertIdx i x)[k]'(by simp; omega) = xs[k] := by
   simp [getElem_insertIdx, w, h]
 
-theorem getElem_insertIdx_self {as : Array α} {x : α} {i : Nat} (w : i ≤ as.size) :
-    (as.insertIdx i x)[i]'(by simp; omega) = x := by
+theorem getElem_insertIdx_self {xs : Array α} {x : α} {i : Nat} (w : i ≤ xs.size) :
+    (xs.insertIdx i x)[i]'(by simp; omega) = x := by
   simp [getElem_insertIdx, w]
 
-theorem getElem_insertIdx_of_gt {as : Array α} {x : α} {i k : Nat} (w : k ≤ as.size) (h : k > i) :
-    (as.insertIdx i x)[k]'(by simp; omega) = as[k - 1]'(by omega) := by
+theorem getElem_insertIdx_of_gt {xs : Array α} {x : α} {i k : Nat} (w : k ≤ xs.size) (h : k > i) :
+    (xs.insertIdx i x)[k]'(by simp; omega) = xs[k - 1]'(by omega) := by
   simp [getElem_insertIdx, w, h]
   rw [dif_neg (by omega), dif_neg (by omega)]
 
-theorem getElem?_insertIdx {l : Array α} {x : α} {i k : Nat} (h : i ≤ l.size) :
-    (l.insertIdx i x)[k]? =
+theorem getElem?_insertIdx {xs : Array α} {x : α} {i k : Nat} (h : i ≤ xs.size) :
+    (xs.insertIdx i x)[k]? =
       if k < i then
-        l[k]?
+        xs[k]?
       else
         if k = i then
-          if k ≤ l.size then some x else none
+          if k ≤ xs.size then some x else none
         else
-          l[k-1]? := by
-  cases l
+          xs[k-1]? := by
+  cases xs
   simp [List.getElem?_insertIdx, h]
 
-theorem getElem?_insertIdx_of_lt {l : Array α} {x : α} {i k : Nat} (w : i ≤ l.size) (h : k < i) :
-    (l.insertIdx i x)[k]? = l[k]? := by
+theorem getElem?_insertIdx_of_lt {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.size) (h : k < i) :
+    (xs.insertIdx i x)[k]? = xs[k]? := by
   rw [getElem?_insertIdx, if_pos h]
 
-theorem getElem?_insertIdx_self {l : Array α} {x : α} {i : Nat} (w : i ≤ l.size) :
-    (l.insertIdx i x)[i]? = some x := by
+theorem getElem?_insertIdx_self {xs : Array α} {x : α} {i : Nat} (w : i ≤ xs.size) :
+    (xs.insertIdx i x)[i]? = some x := by
   rw [getElem?_insertIdx, if_neg (by omega), if_pos rfl, if_pos w]
 
-theorem getElem?_insertIdx_of_ge {l : Array α} {x : α} {i k : Nat} (w : i < k) (h : k ≤ l.size) :
-    (l.insertIdx i x)[k]? = l[k - 1]? := by
+theorem getElem?_insertIdx_of_ge {xs : Array α} {x : α} {i k : Nat} (w : i < k) (h : k ≤ xs.size) :
+    (xs.insertIdx i x)[k]? = xs[k - 1]? := by
   rw [getElem?_insertIdx, if_neg (by omega), if_neg (by omega)]
 
 end InsertIdx

src/Init/Data/Array/InsertionSort.lean

@@ -6,23 +6,26 @@ Authors: Leonardo de Moura
 prelude
 import Init.Data.Array.Basic
 
-@[inline] def Array.insertionSort (a : Array α) (lt : α → α → Bool := by exact (· < ·)) : Array α :=
-  traverse a 0 a.size
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
+@[inline] def Array.insertionSort (xs : Array α) (lt : α → α → Bool := by exact (· < ·)) : Array α :=
+  traverse xs 0 xs.size
 where
-  @[specialize] traverse (a : Array α) (i : Nat) (fuel : Nat) : Array α :=
+  @[specialize] traverse (xs : Array α) (i : Nat) (fuel : Nat) : Array α :=
     match fuel with
-    | 0      => a
+    | 0      => xs
     | fuel+1 =>
-      if h : i < a.size then
-        traverse (swapLoop a i h) (i+1) fuel
+      if h : i < xs.size then
+        traverse (swapLoop xs i h) (i+1) fuel
       else
-        a
-  @[specialize] swapLoop (a : Array α) (j : Nat) (h : j < a.size) : Array α :=
+        xs
+  @[specialize] swapLoop (xs : Array α) (j : Nat) (h : j < xs.size) : Array α :=
     match (generalizing := false) he:j with -- using `generalizing` because we don't want to refine the type of `h`
-    | 0    => a
+    | 0    => xs
     | j'+1 =>
-      have h' : j' < a.size := by subst j; exact Nat.lt_trans (Nat.lt_succ_self _) h
-      if lt a[j] a[j'] then
-        swapLoop (a.swap j j') j' (by rw [size_swap]; assumption; done)
+      have h' : j' < xs.size := by subst j; exact Nat.lt_trans (Nat.lt_succ_self _) h
+      if lt xs[j] xs[j'] then
+        swapLoop (xs.swap j j') j' (by rw [size_swap]; assumption; done)
       else
-        a
+        xs

src/Init/Data/Array/Lex/Basic.lean

@@ -8,6 +8,9 @@ import Init.Data.Array.Basic
 import Init.Data.Nat.Lemmas
 import Init.Data.Range
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 /--

src/Init/Data/Array/Lex/Lemmas.lean

@@ -7,6 +7,9 @@ prelude
 import Init.Data.Array.Lemmas
 import Init.Data.List.Lex
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 /-! ### Lexicographic ordering -/
@@ -14,15 +17,15 @@ namespace Array
 @[simp] theorem _root_.List.lt_toArray [LT α] (l₁ l₂ : List α) : l₁.toArray < l₂.toArray ↔ l₁ < l₂ := Iff.rfl
 @[simp] theorem _root_.List.le_toArray [LT α] (l₁ l₂ : List α) : l₁.toArray ≤ l₂.toArray ↔ l₁ ≤ l₂ := Iff.rfl
 
-@[simp] theorem lt_toList [LT α] (l₁ l₂ : Array α) : l₁.toList < l₂.toList ↔ l₁ < l₂ := Iff.rfl
-@[simp] theorem le_toList [LT α] (l₁ l₂ : Array α) : l₁.toList ≤ l₂.toList ↔ l₁ ≤ l₂ := Iff.rfl
+@[simp] theorem lt_toList [LT α] (xs ys : Array α) : xs.toList < ys.toList ↔ xs < ys := Iff.rfl
+@[simp] theorem le_toList [LT α] (xs ys : Array α) : xs.toList ≤ ys.toList ↔ xs ≤ ys := Iff.rfl
 
 protected theorem not_lt_iff_ge [LT α] (l₁ l₂ : List α) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
 protected theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) :
     ¬ l₁ ≤ l₂ ↔ l₂ < l₁ :=
   Decidable.not_not
 
-@[simp] theorem lex_empty [BEq α] {lt : α → α → Bool} (l : Array α) : l.lex #[] lt = false := by
+@[simp] theorem lex_empty [BEq α] {lt : α → α → Bool} (xs : Array α) : xs.lex #[] lt = false := by
   simp [lex, Id.run]
 
 @[simp] theorem singleton_lex_singleton [BEq α] {lt : α → α → Bool} : #[a].lex #[b] lt = lt a b := by
@@ -33,7 +36,7 @@ private theorem cons_lex_cons [BEq α] {lt : α → α → Bool} {a b : α} {xs
      (#[a] ++ xs).lex (#[b] ++ ys) lt =
        (lt a b || a == b && xs.lex ys lt) := by
   simp only [lex, Id.run]
-  simp only [Std.Range.forIn'_eq_forIn'_range', size_append, size_toArray, List.length_singleton,
+  simp only [Std.Range.forIn'_eq_forIn'_range', size_append, List.size_toArray, List.length_singleton,
     Nat.add_comm 1]
   simp [Nat.add_min_add_right, List.range'_succ, getElem_append_left, List.range'_succ_left,
     getElem_append_right]
@@ -52,35 +55,35 @@ private theorem cons_lex_cons [BEq α] {lt : α → α → Bool} {a b : α} {xs
     | cons y l₂ =>
       rw [List.toArray_cons, List.toArray_cons y, cons_lex_cons, List.lex, ih]
 
-@[simp] theorem lex_toList [BEq α] (lt : α → α → Bool) (l₁ l₂ : Array α) :
-    l₁.toList.lex l₂.toList lt = l₁.lex l₂ lt := by
-  cases l₁ <;> cases l₂ <;> simp
+@[simp] theorem lex_toList [BEq α] (lt : α → α → Bool) (xs ys : Array α) :
+    xs.toList.lex ys.toList lt = xs.lex ys lt := by
+  cases xs <;> cases ys <;> simp
 
-protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] (l : Array α) : ¬ l < l :=
-  List.lt_irrefl l.toList
+protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] (xs : Array α) : ¬ xs < xs :=
+  List.lt_irrefl xs.toList
 
 instance ltIrrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Irrefl (α := Array α) (· < ·) where
   irrefl := Array.lt_irrefl
 
-@[simp] theorem not_lt_empty [LT α] (l : Array α) : ¬ l < #[] := List.not_lt_nil l.toList
-@[simp] theorem empty_le [LT α] (l : Array α) : #[] ≤ l := List.nil_le l.toList
+@[simp] theorem not_lt_empty [LT α] (xs : Array α) : ¬ xs < #[] := List.not_lt_nil xs.toList
+@[simp] theorem empty_le [LT α] (xs : Array α) : #[] ≤ xs := List.nil_le xs.toList
 
-@[simp] theorem le_empty [LT α] (l : Array α) : l ≤ #[] ↔ l = #[] := by
-  cases l
+@[simp] theorem le_empty [LT α] (xs : Array α) : xs ≤ #[] ↔ xs = #[] := by
+  cases xs
   simp
 
-@[simp] theorem empty_lt_push [LT α] (l : Array α) (a : α) : #[] < l.push a := by
-  rcases l with (_ | ⟨x, l⟩) <;> simp
+@[simp] theorem empty_lt_push [LT α] (xs : Array α) (a : α) : #[] < xs.push a := by
+  rcases xs with (_ | ⟨x, xs⟩) <;> simp
 
-protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] (l : Array α) : l ≤ l :=
-  List.le_refl l.toList
+protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] (xs : Array α) : xs ≤ xs :=
+  List.le_refl xs.toList
 
 instance [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Refl (· ≤ · : Array α → Array α → Prop) where
   refl := Array.le_refl
 
 protected theorem lt_trans [LT α]
     [i₁ : Trans (· < · : α → α → Prop) (· < ·) (· < ·)]
-    {l₁ l₂ l₃ : Array α} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
+    {xs ys zs : Array α} (h₁ : xs < ys) (h₂ : ys < zs) : xs < zs :=
   List.lt_trans h₁ h₂
 
 instance [LT α] [Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
@@ -92,7 +95,7 @@ protected theorem lt_of_le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
     [i₁ : Std.Asymm (· < · : α → α → Prop)]
     [i₂ : Std.Antisymm (¬ · < · : α → α → Prop)]
     [i₃ : Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)]
-    {l₁ l₂ l₃ : Array α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
+    {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys < zs) : xs < zs :=
   List.lt_of_le_of_lt h₁ h₂
 
 protected theorem le_trans [DecidableEq α] [LT α] [DecidableLT α]
@@ -100,7 +103,7 @@ protected theorem le_trans [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
     [Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)]
-    {l₁ l₂ l₃ : Array α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ ≤ l₃) : l₁ ≤ l₃ :=
+    {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys ≤ zs) : xs ≤ zs :=
   fun h₃ => h₁ (Array.lt_of_le_of_lt h₂ h₃)
 
 instance [DecidableEq α] [LT α] [DecidableLT α]
@@ -113,7 +116,7 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
 
 protected theorem lt_asymm [LT α]
     [i : Std.Asymm (· < · : α → α → Prop)]
-    {l₁ l₂ : Array α} (h : l₁ < l₂) : ¬ l₂ < l₁ := List.lt_asymm h
+    {xs ys : Array α} (h : xs < ys) : ¬ ys < xs := List.lt_asymm h
 
 instance [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Asymm (· < · : α → α → Prop)] :
@@ -121,26 +124,26 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
   asymm _ _ := Array.lt_asymm
 
 protected theorem le_total [DecidableEq α] [LT α] [DecidableLT α]
-    [i : Std.Total (¬ · < · : α → α → Prop)] (l₁ l₂ : Array α) : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
-  List.le_total _ _
+    [i : Std.Total (¬ · < · : α → α → Prop)] (xs ys : Array α) : xs ≤ ys ∨ ys ≤ xs :=
+  List.le_total xs.toList ys.toList
 
 @[simp] protected theorem not_lt [LT α]
-    {l₁ l₂ : Array α} : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
+    {xs ys : Array α} : ¬ xs < ys ↔ ys ≤ xs := Iff.rfl
 
 @[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
-    {l₁ l₂ : Array α} : ¬ l₂ ≤ l₁ ↔ l₁ < l₂ := Decidable.not_not
+    {xs ys : Array α} : ¬ ys ≤ xs ↔ xs < ys := Decidable.not_not
 
 protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
     [i : Std.Total (¬ · < · : α → α → Prop)]
-    {l₁ l₂ : Array α} (h : l₁ < l₂) : l₁ ≤ l₂ :=
+    {xs ys : Array α} (h : xs < ys) : xs ≤ ys :=
   List.le_of_lt h
 
 protected theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
     [Std.Total (¬ · < · : α → α → Prop)]
-    {l₁ l₂ : Array α} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂ := by
-  simpa using List.le_iff_lt_or_eq (l₁ := l₁.toList) (l₂ := l₂.toList)
+    {xs ys : Array α} : xs ≤ ys ↔ xs < ys ∨ xs = ys := by
+  simpa using List.le_iff_lt_or_eq (l₁ := xs.toList) (l₂ := ys.toList)
 
 instance [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Total (¬ · < · : α → α → Prop)] :
@@ -148,22 +151,22 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
   total := Array.le_total
 
 @[simp] theorem lex_eq_true_iff_lt [DecidableEq α] [LT α] [DecidableLT α]
-    {l₁ l₂ : Array α} : lex l₁ l₂ = true ↔ l₁ < l₂ := by
-  cases l₁
-  cases l₂
+    {xs ys : Array α} : lex xs ys = true ↔ xs < ys := by
+  cases xs
+  cases ys
   simp
 
 @[simp] theorem lex_eq_false_iff_ge [DecidableEq α] [LT α] [DecidableLT α]
-    {l₁ l₂ : Array α} : lex l₁ l₂ = false ↔ l₂ ≤ l₁ := by
-  cases l₁
-  cases l₂
+    {xs ys : Array α} : lex xs ys = false ↔ ys ≤ xs := by
+  cases xs
+  cases ys
   simp [List.not_lt_iff_ge]
 
 instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLT (Array α) :=
-  fun l₁ l₂ => decidable_of_iff (lex l₁ l₂ = true) lex_eq_true_iff_lt
+  fun xs ys => decidable_of_iff (lex xs ys = true) lex_eq_true_iff_lt
 
 instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLE (Array α) :=
-  fun l₁ l₂ => decidable_of_iff (lex l₂ l₁ = false) lex_eq_false_iff_ge
+  fun xs ys => decidable_of_iff (lex ys xs = false) lex_eq_false_iff_ge
 
 /--
 `l₁` is lexicographically less than `l₂` if either
@@ -211,58 +214,58 @@ theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α
   cases l₂
   simp_all [List.lex_eq_false_iff_exists]
 
-protected theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} :
-    l₁ < l₂ ↔
-      (l₁ = l₂.take l₁.size ∧ l₁.size < l₂.size) ∨
-        (∃ (i : Nat) (h₁ : i < l₁.size) (h₂ : i < l₂.size),
+protected theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {xs ys : Array α} :
+    xs < ys ↔
+      (xs = ys.take xs.size ∧ xs.size < ys.size) ∨
+        (∃ (i : Nat) (h₁ : i < xs.size) (h₂ : i < ys.size),
           (∀ j, (hj : j < i) →
-            l₁[j]'(Nat.lt_trans hj h₁) = l₂[j]'(Nat.lt_trans hj h₂)) ∧ l₁[i] < l₂[i]) := by
-  cases l₁
-  cases l₂
+            xs[j]'(Nat.lt_trans hj h₁) = ys[j]'(Nat.lt_trans hj h₂)) ∧ xs[i] < ys[i]) := by
+  cases xs
+  cases ys
   simp [List.lt_iff_exists]
 
 protected theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
-    [Std.Antisymm (¬ · < · : α → α → Prop)] {l₁ l₂ : Array α} :
-    l₁ ≤ l₂ ↔
-      (l₁ = l₂.take l₁.size) ∨
-        (∃ (i : Nat) (h₁ : i < l₁.size) (h₂ : i < l₂.size),
+    [Std.Antisymm (¬ · < · : α → α → Prop)] {xs ys : Array α} :
+    xs ≤ ys ↔
+      (xs = ys.take xs.size) ∨
+        (∃ (i : Nat) (h₁ : i < xs.size) (h₂ : i < ys.size),
           (∀ j, (hj : j < i) →
-            l₁[j]'(Nat.lt_trans hj h₁) = l₂[j]'(Nat.lt_trans hj h₂)) ∧ l₁[i] < l₂[i]) := by
-  cases l₁
-  cases l₂
+            xs[j]'(Nat.lt_trans hj h₁) = ys[j]'(Nat.lt_trans hj h₂)) ∧ xs[i] < ys[i]) := by
+  cases xs
+  cases ys
   simp [List.le_iff_exists]
 
-theorem append_left_lt [LT α] {l₁ l₂ l₃ : Array α} (h : l₂ < l₃) :
-    l₁ ++ l₂ < l₁ ++ l₃ := by
-  cases l₁
-  cases l₂
-  cases l₃
+theorem append_left_lt [LT α] {xs ys zs : Array α} (h : ys < zs) :
+    xs ++ ys < xs ++ zs := by
+  cases xs
+  cases ys
+  cases zs
   simpa using List.append_left_lt h
 
 theorem append_left_le [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
-    {l₁ l₂ l₃ : Array α} (h : l₂ ≤ l₃) :
-    l₁ ++ l₂ ≤ l₁ ++ l₃ := by
-  cases l₁
-  cases l₂
-  cases l₃
+    {xs ys zs : Array α} (h : ys ≤ zs) :
+    xs ++ ys ≤ xs ++ zs := by
+  cases xs
+  cases ys
+  cases zs
   simpa using List.append_left_le h
 
 theorem le_append_left [LT α] [Std.Irrefl (· < · : α → α → Prop)]
-    {l₁ l₂ : Array α} : l₁ ≤ l₁ ++ l₂ := by
-  cases l₁
-  cases l₂
+    {xs ys : Array α} : xs ≤ xs ++ ys := by
+  cases xs
+  cases ys
   simpa using List.le_append_left
 
 protected theorem map_lt [LT α] [LT β]
-    {l₁ l₂ : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
-    map f l₁ < map f l₂ := by
-  cases l₁
-  cases l₂
+    {xs ys : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : xs < ys) :
+    map f xs < map f ys := by
+  cases xs
+  cases ys
   simpa using List.map_lt w h
 
 protected theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
@@ -272,10 +275,10 @@ protected theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq
     [Std.Irrefl (· < · : β → β → Prop)]
     [Std.Asymm (· < · : β → β → Prop)]
     [Std.Antisymm (¬ · < · : β → β → Prop)]
-    {l₁ l₂ : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ ≤ l₂) :
-    map f l₁ ≤ map f l₂ := by
-  cases l₁
-  cases l₂
+    {xs ys : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : xs ≤ ys) :
+    map f xs ≤ map f ys := by
+  cases xs
+  cases ys
   simpa using List.map_le w h
 
 end Array

src/Init/Data/Array/MapIdx.lean

@@ -6,28 +6,32 @@ Authors: Mario Carneiro, Kim Morrison
 prelude
 import Init.Data.Array.Lemmas
 import Init.Data.Array.Attach
+import Init.Data.Array.OfFn
 import Init.Data.List.MapIdx
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 /-! ### mapFinIdx -/
 
 -- This could also be proved from `SatisfiesM_mapIdxM` in Batteries.
-theorem mapFinIdx_induction (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → β)
+theorem mapFinIdx_induction (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β)
     (motive : Nat → Prop) (h0 : motive 0)
-    (p : (i : Nat) → β → (h : i < as.size) → Prop)
-    (hs : ∀ i h, motive i → p i (f i as[i] h) h ∧ motive (i + 1)) :
-    motive as.size ∧ ∃ eq : (Array.mapFinIdx as f).size = as.size,
-      ∀ i h, p i ((Array.mapFinIdx as f)[i]) h := by
+    (p : (i : Nat) → β → (h : i < xs.size) → Prop)
+    (hs : ∀ i h, motive i → p i (f i xs[i] h) h ∧ motive (i + 1)) :
+    motive xs.size ∧ ∃ eq : (Array.mapFinIdx xs f).size = xs.size,
+      ∀ i h, p i ((Array.mapFinIdx xs f)[i]) h := by
   let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p i bs[i] h) (hm : motive j) :
-    let arr : Array β := Array.mapFinIdxM.map (m := Id) as f i j h bs
-    motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p i arr[i] h := by
+    let as : Array β := Array.mapFinIdxM.map (m := Id) xs f i j h bs
+    motive xs.size ∧ ∃ eq : as.size = xs.size, ∀ i h, p i as[i] h := by
     induction i generalizing j bs with simp [mapFinIdxM.map]
     | zero =>
       have := (Nat.zero_add _).symm.trans h
       exact ⟨this ▸ hm, h₁ ▸ this, fun _ _ => h₂ ..⟩
     | succ i ih =>
-      apply @ih (bs.push (f j as[j] (by omega))) (j + 1) (by omega) (by simp; omega)
+      apply @ih (bs.push (f j xs[j] (by omega))) (j + 1) (by omega) (by simp; omega)
       · intro i i_lt h'
         rw [getElem_push]
         split
@@ -38,67 +42,67 @@ theorem mapFinIdx_induction (as : Array α) (f : (i : Nat) → α → (h : i < a
       · exact (hs j (by omega) hm).2
   simp [mapFinIdx, mapFinIdxM]; exact go rfl nofun h0
 
-theorem mapFinIdx_spec (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → β)
-    (p : (i : Nat) → β → (h : i < as.size) → Prop) (hs : ∀ i h, p i (f i as[i] h) h) :
-    ∃ eq : (Array.mapFinIdx as f).size = as.size,
-      ∀ i h, p i ((Array.mapFinIdx as f)[i]) h :=
+theorem mapFinIdx_spec (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β)
+    (p : (i : Nat) → β → (h : i < xs.size) → Prop) (hs : ∀ i h, p i (f i xs[i] h) h) :
+    ∃ eq : (Array.mapFinIdx xs f).size = xs.size,
+      ∀ i h, p i ((Array.mapFinIdx xs f)[i]) h :=
   (mapFinIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
 
-@[simp] theorem size_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) :
-    (a.mapFinIdx f).size = a.size :=
+@[simp] theorem size_mapFinIdx (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β) :
+    (xs.mapFinIdx f).size = xs.size :=
   (mapFinIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
 
-@[simp] theorem size_zipIdx (as : Array α) (k : Nat) : (as.zipIdx k).size = as.size :=
+@[simp] theorem size_zipIdx (xs : Array α) (k : Nat) : (xs.zipIdx k).size = xs.size :=
   Array.size_mapFinIdx _ _
 
 @[deprecated size_zipIdx (since := "2025-01-21")] abbrev size_zipWithIndex := @size_zipIdx
 
-@[simp] theorem getElem_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) (i : Nat)
-    (h : i < (mapFinIdx a f).size) :
-    (a.mapFinIdx f)[i] = f i (a[i]'(by simp_all))  (by simp_all) :=
-  (mapFinIdx_spec _ _ (fun i b h => b = f i a[i] h) fun _ _ => rfl).2 i _
+@[simp] theorem getElem_mapFinIdx (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β) (i : Nat)
+    (h : i < (xs.mapFinIdx f).size) :
+    (xs.mapFinIdx f)[i] = f i (xs[i]'(by simp_all)) (by simp_all) :=
+  (mapFinIdx_spec _ _ (fun i b h => b = f i xs[i] h) fun _ _ => rfl).2 i _
 
-@[simp] theorem getElem?_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) (i : Nat) :
-    (a.mapFinIdx f)[i]? =
-      a[i]?.pbind fun b h => f i b (getElem?_eq_some_iff.1 h).1 := by
+@[simp] theorem getElem?_mapFinIdx (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β) (i : Nat) :
+    (xs.mapFinIdx f)[i]? =
+      xs[i]?.pbind fun b h => f i b (getElem?_eq_some_iff.1 h).1 := by
   simp only [getElem?_def, size_mapFinIdx, getElem_mapFinIdx]
   split <;> simp_all
 
-@[simp] theorem toList_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) :
-    (a.mapFinIdx f).toList = a.toList.mapFinIdx (fun i a h => f i a (by simpa)) := by
+@[simp] theorem toList_mapFinIdx (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β) :
+    (xs.mapFinIdx f).toList = xs.toList.mapFinIdx (fun i a h => f i a (by simpa)) := by
   apply List.ext_getElem <;> simp
 
 /-! ### mapIdx -/
 
-theorem mapIdx_induction (f : Nat → α → β) (as : Array α)
+theorem mapIdx_induction (f : Nat → α → β) (xs : Array α)
     (motive : Nat → Prop) (h0 : motive 0)
-    (p : (i : Nat) → β → (h : i < as.size) → Prop)
-    (hs : ∀ i h, motive i → p i (f i as[i]) h ∧ motive (i + 1)) :
-    motive as.size ∧ ∃ eq : (as.mapIdx f).size = as.size,
-      ∀ i h, p i ((as.mapIdx f)[i]) h :=
-  mapFinIdx_induction as (fun i a _ => f i a) motive h0 p hs
+    (p : (i : Nat) → β → (h : i < xs.size) → Prop)
+    (hs : ∀ i h, motive i → p i (f i xs[i]) h ∧ motive (i + 1)) :
+    motive xs.size ∧ ∃ eq : (xs.mapIdx f).size = xs.size,
+      ∀ i h, p i ((xs.mapIdx f)[i]) h :=
+  mapFinIdx_induction xs (fun i a _ => f i a) motive h0 p hs
 
-theorem mapIdx_spec (f : Nat → α → β) (as : Array α)
-    (p : (i : Nat) → β → (h : i < as.size) → Prop) (hs : ∀ i h, p i (f i as[i]) h) :
-    ∃ eq : (as.mapIdx f).size = as.size,
-      ∀ i h, p i ((as.mapIdx f)[i]) h :=
+theorem mapIdx_spec (f : Nat → α → β) (xs : Array α)
+    (p : (i : Nat) → β → (h : i < xs.size) → Prop) (hs : ∀ i h, p i (f i xs[i]) h) :
+    ∃ eq : (xs.mapIdx f).size = xs.size,
+      ∀ i h, p i ((xs.mapIdx f)[i]) h :=
   (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
 
-@[simp] theorem size_mapIdx (f : Nat → α → β) (as : Array α) : (as.mapIdx f).size = as.size :=
+@[simp] theorem size_mapIdx (f : Nat → α → β) (xs : Array α) : (xs.mapIdx f).size = xs.size :=
   (mapIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
 
-@[simp] theorem getElem_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat)
-    (h : i < (as.mapIdx f).size) :
-    (as.mapIdx f)[i] = f i (as[i]'(by simp_all)) :=
-  (mapIdx_spec _ _ (fun i b h => b = f i as[i]) fun _ _ => rfl).2 i (by simp_all)
+@[simp] theorem getElem_mapIdx (f : Nat → α → β) (xs : Array α) (i : Nat)
+    (h : i < (xs.mapIdx f).size) :
+    (xs.mapIdx f)[i] = f i (xs[i]'(by simp_all)) :=
+  (mapIdx_spec _ _ (fun i b h => b = f i xs[i]) fun _ _ => rfl).2 i (by simp_all)
 
-@[simp] theorem getElem?_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat) :
-    (as.mapIdx f)[i]? =
-      as[i]?.map (f i) := by
+@[simp] theorem getElem?_mapIdx (f : Nat → α → β) (xs : Array α) (i : Nat) :
+    (xs.mapIdx f)[i]? =
+      xs[i]?.map (f i) := by
   simp [getElem?_def, size_mapIdx, getElem_mapIdx]
 
-@[simp] theorem toList_mapIdx (f : Nat → α → β) (as : Array α) :
-    (as.mapIdx f).toList = as.toList.mapIdx (fun i a => f i a) := by
+@[simp] theorem toList_mapIdx (f : Nat → α → β) (xs : Array α) :
+    (xs.mapIdx f).toList = xs.toList.mapIdx (fun i a => f i a) := by
   apply List.ext_getElem <;> simp
 
 end Array
@@ -119,8 +123,8 @@ namespace Array
 
 /-! ### zipIdx -/
 
-@[simp] theorem getElem_zipIdx (a : Array α) (k : Nat) (i : Nat) (h : i < (a.zipIdx k).size) :
-    (a.zipIdx k)[i] = (a[i]'(by simp_all), k + i) := by
+@[simp] theorem getElem_zipIdx (xs : Array α) (k : Nat) (i : Nat) (h : i < (xs.zipIdx k).size) :
+    (xs.zipIdx k)[i] = (xs[i]'(by simp_all), k + i) := by
   simp [zipIdx]
 
 @[deprecated getElem_zipIdx (since := "2025-01-21")]
@@ -133,35 +137,35 @@ abbrev getElem_zipWithIndex := @getElem_zipIdx
 @[deprecated zipIdx_toArray (since := "2025-01-21")]
 abbrev zipWithIndex_toArray := @zipIdx_toArray
 
-@[simp] theorem toList_zipIdx (a : Array α) (k : Nat) :
-    (a.zipIdx k).toList = a.toList.zipIdx k := by
-  rcases a with ⟨a⟩
+@[simp] theorem toList_zipIdx (xs : Array α) (k : Nat) :
+    (xs.zipIdx k).toList = xs.toList.zipIdx k := by
+  rcases xs with ⟨xs⟩
   simp
 
 @[deprecated toList_zipIdx (since := "2025-01-21")]
 abbrev toList_zipWithIndex := @toList_zipIdx
 
-theorem mk_mem_zipIdx_iff_le_and_getElem?_sub {k i : Nat} {x : α} {l : Array α} :
-    (x, i) ∈ zipIdx l k ↔ k ≤ i ∧ l[i - k]? = some x := by
-  rcases l with ⟨l⟩
+theorem mk_mem_zipIdx_iff_le_and_getElem?_sub {k i : Nat} {x : α} {xs : Array α} :
+    (x, i) ∈ xs.zipIdx k ↔ k ≤ i ∧ xs[i - k]? = some x := by
+  rcases xs with ⟨xs⟩
   simp [List.mk_mem_zipIdx_iff_le_and_getElem?_sub]
 
 /-- Variant of `mk_mem_zipIdx_iff_le_and_getElem?_sub` specialized at `k = 0`,
 to avoid the inequality and the subtraction. -/
-theorem mk_mem_zipIdx_iff_getElem? {x : α} {i : Nat} {l : Array α} :
-    (x, i) ∈ l.zipIdx ↔ l[i]? = x := by
+theorem mk_mem_zipIdx_iff_getElem? {x : α} {i : Nat} {xs : Array α} :
+    (x, i) ∈ xs.zipIdx ↔ xs[i]? = x := by
   rw [mk_mem_zipIdx_iff_le_and_getElem?_sub]
   simp
 
-theorem mem_zipIdx_iff_le_and_getElem?_sub {x : α × Nat} {l : Array α} {k : Nat} :
-    x ∈ zipIdx l k ↔ k ≤ x.2 ∧ l[x.2 - k]? = some x.1 := by
+theorem mem_zipIdx_iff_le_and_getElem?_sub {x : α × Nat} {xs : Array α} {k : Nat} :
+    x ∈ xs.zipIdx k ↔ k ≤ x.2 ∧ xs[x.2 - k]? = some x.1 := by
   cases x
   simp [mk_mem_zipIdx_iff_le_and_getElem?_sub]
 
 /-- Variant of `mem_zipIdx_iff_le_and_getElem?_sub` specialized at `k = 0`,
 to avoid the inequality and the subtraction. -/
-theorem mem_zipIdx_iff_getElem? {x : α × Nat} {l : Array α} :
-    x ∈ l.zipIdx ↔ l[x.2]? = some x.1 := by
+theorem mem_zipIdx_iff_getElem? {x : α × Nat} {xs : Array α} :
+    x ∈ xs.zipIdx ↔ xs[x.2]? = some x.1 := by
   rw [mk_mem_zipIdx_iff_getElem?]
 
 @[deprecated mk_mem_zipIdx_iff_getElem? (since := "2025-01-21")]
@@ -182,31 +186,31 @@ abbrev mem_zipWithIndex_iff_getElem? := @mem_zipIdx_iff_getElem?
 theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #[] f = #[] :=
   rfl
 
-theorem mapFinIdx_eq_ofFn {as : Array α} {f : (i : Nat) → α → (h : i < as.size) → β} :
-    as.mapFinIdx f = Array.ofFn fun i : Fin as.size => f i as[i] i.2 := by
-  cases as
+theorem mapFinIdx_eq_ofFn {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+    xs.mapFinIdx f = Array.ofFn fun i : Fin xs.size => f i xs[i] i.2 := by
+  cases xs
   simp [List.mapFinIdx_eq_ofFn]
 
-theorem mapFinIdx_append {K L : Array α} {f : (i : Nat) → α → (h : i < (K ++ L).size) → β} :
-    (K ++ L).mapFinIdx f =
-      K.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
-        L.mapFinIdx (fun i a h => f (i + K.size) a (by simp; omega)) := by
-  cases K
-  cases L
+theorem mapFinIdx_append {xs ys : Array α} {f : (i : Nat) → α → (h : i < (xs ++ ys).size) → β} :
+    (xs ++ ys).mapFinIdx f =
+      xs.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
+        ys.mapFinIdx (fun i a h => f (i + xs.size) a (by simp; omega)) := by
+  cases xs
+  cases ys
   simp [List.mapFinIdx_append]
 
 @[simp]
-theorem mapFinIdx_push {l : Array α} {a : α} {f : (i : Nat) → α → (h : i < (l.push a).size) → β} :
-    mapFinIdx (l.push a) f =
-      (mapFinIdx l (fun i a h => f i a (by simp; omega))).push (f l.size a (by simp)) := by
+theorem mapFinIdx_push {xs : Array α} {a : α} {f : (i : Nat) → α → (h : i < (xs.push a).size) → β} :
+    mapFinIdx (xs.push a) f =
+      (mapFinIdx xs (fun i a h => f i a (by simp; omega))).push (f xs.size a (by simp)) := by
   simp [← append_singleton, mapFinIdx_append]
 
 theorem mapFinIdx_singleton {a : α} {f : (i : Nat) → α → (h : i < 1) → β} :
     #[a].mapFinIdx f = #[f 0 a (by simp)] := by
   simp
 
-theorem mapFinIdx_eq_zipIdx_map {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
-    l.mapFinIdx f = l.zipIdx.attach.map
+theorem mapFinIdx_eq_zipIdx_map {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+    xs.mapFinIdx f = xs.zipIdx.attach.map
       fun ⟨⟨x, i⟩, m⟩ =>
         f i x (by simp [mk_mem_zipIdx_iff_getElem?, getElem?_eq_some_iff] at m; exact m.1) := by
   ext <;> simp
@@ -215,44 +219,44 @@ theorem mapFinIdx_eq_zipIdx_map {l : Array α} {f : (i : Nat) → α → (h : i
 abbrev mapFinIdx_eq_zipWithIndex_map := @mapFinIdx_eq_zipIdx_map
 
 @[simp]
-theorem mapFinIdx_eq_empty_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
-    l.mapFinIdx f = #[] ↔ l = #[] := by
-  cases l
+theorem mapFinIdx_eq_empty_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+    xs.mapFinIdx f = #[] ↔ xs = #[] := by
+  cases xs
   simp
 
-theorem mapFinIdx_ne_empty_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
-    l.mapFinIdx f ≠ #[] ↔ l ≠ #[] := by
+theorem mapFinIdx_ne_empty_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+    xs.mapFinIdx f ≠ #[] ↔ xs ≠ #[] := by
   simp
 
-theorem exists_of_mem_mapFinIdx {b : β} {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β}
-    (h : b ∈ l.mapFinIdx f) : ∃ (i : Nat) (h : i < l.size), f i l[i] h = b := by
-  rcases l with ⟨l⟩
+theorem exists_of_mem_mapFinIdx {b : β} {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β}
+    (h : b ∈ xs.mapFinIdx f) : ∃ (i : Nat) (h : i < xs.size), f i xs[i] h = b := by
+  rcases xs with ⟨xs⟩
   exact List.exists_of_mem_mapFinIdx (by simpa using h)
 
-@[simp] theorem mem_mapFinIdx {b : β} {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
-    b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.size), f i l[i] h = b := by
-  rcases l with ⟨l⟩
+@[simp] theorem mem_mapFinIdx {b : β} {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+    b ∈ xs.mapFinIdx f ↔ ∃ (i : Nat) (h : i < xs.size), f i xs[i] h = b := by
+  rcases xs with ⟨xs⟩
   simp
 
-theorem mapFinIdx_eq_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
-    l.mapFinIdx f = l' ↔ ∃ h : l'.size = l.size, ∀ (i : Nat) (h : i < l.size), l'[i] = f i l[i] h := by
-  rcases l with ⟨l⟩
-  rcases l' with ⟨l'⟩
+theorem mapFinIdx_eq_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {ys : Array β} :
+    xs.mapFinIdx f = ys ↔ ∃ h : ys.size = xs.size, ∀ (i : Nat) (h : i < xs.size), ys[i] = f i xs[i] h := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simpa using List.mapFinIdx_eq_iff
 
-@[simp] theorem mapFinIdx_eq_singleton_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {b : β} :
-    l.mapFinIdx f = #[b] ↔ ∃ (a : α) (w : l = #[a]), f 0 a (by simp [w]) = b := by
-  rcases l with ⟨l⟩
+@[simp] theorem mapFinIdx_eq_singleton_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {b : β} :
+    xs.mapFinIdx f = #[b] ↔ ∃ (a : α) (w : xs = #[a]), f 0 a (by simp [w]) = b := by
+  rcases xs with ⟨xs⟩
   simp
 
-theorem mapFinIdx_eq_append_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {l₁ l₂ : Array β} :
-    l.mapFinIdx f = l₁ ++ l₂ ↔
-      ∃ (l₁' : Array α) (l₂' : Array α) (w : l = l₁' ++ l₂'),
-        l₁'.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₁ ∧
-        l₂'.mapFinIdx (fun i a h => f (i + l₁'.size) a (by simp [w]; omega)) = l₂ := by
-  rcases l with ⟨l⟩
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+theorem mapFinIdx_eq_append_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {ys zs : Array β} :
+    xs.mapFinIdx f = ys ++ zs ↔
+      ∃ (ys' : Array α) (zs' : Array α) (w : xs = ys' ++ zs'),
+        ys'.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = ys ∧
+        zs'.mapFinIdx (fun i a h => f (i + ys'.size) a (by simp [w]; omega)) = zs := by
+  rcases xs with ⟨l⟩
+  rcases ys with ⟨l₁⟩
+  rcases zs with ⟨l₂⟩
   simp only [List.mapFinIdx_toArray, List.append_toArray, mk.injEq, List.mapFinIdx_eq_append_iff,
     toArray_eq_append_iff]
   constructor
@@ -264,39 +268,39 @@ theorem mapFinIdx_eq_append_iff {l : Array α} {f : (i : Nat) → α → (h : i
     obtain rfl := h₂
     refine ⟨l₁, l₂, by simp_all⟩
 
-theorem mapFinIdx_eq_push_iff {l : Array α} {b : β} {f : (i : Nat) → α → (h : i < l.size) → β} :
-    l.mapFinIdx f = l₂.push b ↔
-      ∃ (l₁ : Array α) (a : α) (w : l = l₁.push a),
-        l₁.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₂ ∧ b = f (l.size - 1) a (by simp [w]) := by
+theorem mapFinIdx_eq_push_iff {xs : Array α} {b : β} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+    xs.mapFinIdx f = ys.push b ↔
+      ∃ (zs : Array α) (a : α) (w : xs = zs.push a),
+        zs.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = ys ∧ b = f (xs.size - 1) a (by simp [w]) := by
   rw [push_eq_append, mapFinIdx_eq_append_iff]
   constructor
-  · rintro ⟨l₁, l₂, rfl, rfl, h₂⟩
+  · rintro ⟨ys', zs', rfl, rfl, h₂⟩
     simp only [mapFinIdx_eq_singleton_iff, Nat.zero_add] at h₂
     obtain ⟨a, rfl, rfl⟩ := h₂
-    exact ⟨l₁, a, by simp⟩
-  · rintro ⟨l₁, a, rfl, rfl, rfl⟩
-    exact ⟨l₁, #[a], by simp⟩
+    exact ⟨ys', a, by simp⟩
+  · rintro ⟨zs, a, rfl, rfl, rfl⟩
+    exact ⟨zs, #[a], by simp⟩
 
-theorem mapFinIdx_eq_mapFinIdx_iff {l : Array α} {f g : (i : Nat) → α → (h : i < l.size) → β} :
-    l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Nat) (h : i < l.size), f i l[i] h = g i l[i] h := by
+theorem mapFinIdx_eq_mapFinIdx_iff {xs : Array α} {f g : (i : Nat) → α → (h : i < xs.size) → β} :
+    xs.mapFinIdx f = xs.mapFinIdx g ↔ ∀ (i : Nat) (h : i < xs.size), f i xs[i] h = g i xs[i] h := by
   rw [eq_comm, mapFinIdx_eq_iff]
   simp
 
-@[simp] theorem mapFinIdx_mapFinIdx {l : Array α}
-    {f : (i : Nat) → α → (h : i < l.size) → β}
-    {g : (i : Nat) → β → (h : i < (l.mapFinIdx f).size) → γ} :
-    (l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i a h => g i (f i a h) (by simpa using h)) := by
+@[simp] theorem mapFinIdx_mapFinIdx {xs : Array α}
+    {f : (i : Nat) → α → (h : i < xs.size) → β}
+    {g : (i : Nat) → β → (h : i < (xs.mapFinIdx f).size) → γ} :
+    (xs.mapFinIdx f).mapFinIdx g = xs.mapFinIdx (fun i a h => g i (f i a h) (by simpa using h)) := by
   simp [mapFinIdx_eq_iff]
 
-theorem mapFinIdx_eq_mkArray_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {b : β} :
-    l.mapFinIdx f = mkArray l.size b ↔ ∀ (i : Nat) (h : i < l.size), f i l[i] h = b := by
-  rcases l with ⟨l⟩
+theorem mapFinIdx_eq_mkArray_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {b : β} :
+    xs.mapFinIdx f = mkArray xs.size b ↔ ∀ (i : Nat) (h : i < xs.size), f i xs[i] h = b := by
+  rcases xs with ⟨l⟩
   rw [← toList_inj]
   simp [List.mapFinIdx_eq_replicate_iff]
 
-@[simp] theorem mapFinIdx_reverse {l : Array α} {f : (i : Nat) → α → (h : i < l.reverse.size) → β} :
-    l.reverse.mapFinIdx f = (l.mapFinIdx (fun i a h => f (l.size - 1 - i) a (by simp; omega))).reverse := by
-  rcases l with ⟨l⟩
+@[simp] theorem mapFinIdx_reverse {xs : Array α} {f : (i : Nat) → α → (h : i < xs.reverse.size) → β} :
+    xs.reverse.mapFinIdx f = (xs.mapFinIdx (fun i a h => f (xs.size - 1 - i) a (by simp; omega))).reverse := by
+  rcases xs with ⟨l⟩
   simp [List.mapFinIdx_reverse]
 
 /-! ### mapIdx -/
@@ -305,52 +309,52 @@ theorem mapFinIdx_eq_mkArray_iff {l : Array α} {f : (i : Nat) → α → (h : i
 theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #[] = #[] :=
   rfl
 
-@[simp] theorem mapFinIdx_eq_mapIdx {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {g : Nat → α → β}
-    (h : ∀ (i : Nat) (h : i < l.size), f i l[i] h = g i l[i]) :
-    l.mapFinIdx f = l.mapIdx g := by
+@[simp] theorem mapFinIdx_eq_mapIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {g : Nat → α → β}
+    (h : ∀ (i : Nat) (h : i < xs.size), f i xs[i] h = g i xs[i]) :
+    xs.mapFinIdx f = xs.mapIdx g := by
   simp_all [mapFinIdx_eq_iff]
 
-theorem mapIdx_eq_mapFinIdx {l : Array α} {f : Nat → α → β} :
-    l.mapIdx f = l.mapFinIdx (fun i a _ => f i a) := by
+theorem mapIdx_eq_mapFinIdx {xs : Array α} {f : Nat → α → β} :
+    xs.mapIdx f = xs.mapFinIdx (fun i a _ => f i a) := by
   simp [mapFinIdx_eq_mapIdx]
 
-theorem mapIdx_eq_zipIdx_map {l : Array α} {f : Nat → α → β} :
-    l.mapIdx f = l.zipIdx.map fun ⟨a, i⟩ => f i a := by
+theorem mapIdx_eq_zipIdx_map {xs : Array α} {f : Nat → α → β} :
+    xs.mapIdx f = xs.zipIdx.map fun ⟨a, i⟩ => f i a := by
   ext <;> simp
 
 @[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-21")]
 abbrev mapIdx_eq_zipWithIndex_map := @mapIdx_eq_zipIdx_map
 
-theorem mapIdx_append {K L : Array α} :
-    (K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i => f (i + K.size) := by
-  rcases K with ⟨K⟩
-  rcases L with ⟨L⟩
+theorem mapIdx_append {xs ys : Array α} :
+    (xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx (fun i => f (i + xs.size)) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp [List.mapIdx_append]
 
 @[simp]
-theorem mapIdx_push {l : Array α} {a : α} :
-    mapIdx f (l.push a) = (mapIdx f l).push (f l.size a) := by
+theorem mapIdx_push {xs : Array α} {a : α} :
+    mapIdx f (xs.push a) = (mapIdx f xs).push (f xs.size a) := by
   simp [← append_singleton, mapIdx_append]
 
 theorem mapIdx_singleton {a : α} : mapIdx f #[a] = #[f 0 a] := by
   simp
 
 @[simp]
-theorem mapIdx_eq_empty_iff {l : Array α} : mapIdx f l = #[] ↔ l = #[] := by
-  rcases l with ⟨l⟩
+theorem mapIdx_eq_empty_iff {xs : Array α} : mapIdx f xs = #[] ↔ xs = #[] := by
+  rcases xs with ⟨xs⟩
   simp
 
-theorem mapIdx_ne_empty_iff {l : Array α} :
-    mapIdx f l ≠ #[] ↔ l ≠ #[] := by
+theorem mapIdx_ne_empty_iff {xs : Array α} :
+    mapIdx f xs ≠ #[] ↔ xs ≠ #[] := by
   simp
 
-theorem exists_of_mem_mapIdx {b : β} {l : Array α}
-    (h : b ∈ mapIdx f l) : ∃ (i : Nat) (h : i < l.size), f i l[i] = b := by
+theorem exists_of_mem_mapIdx {b : β} {xs : Array α}
+    (h : b ∈ mapIdx f xs) : ∃ (i : Nat) (h : i < xs.size), f i xs[i] = b := by
   rw [mapIdx_eq_mapFinIdx] at h
   simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
 
-@[simp] theorem mem_mapIdx {b : β} {l : Array α} :
-    b ∈ mapIdx f l ↔ ∃ (i : Nat) (h : i < l.size), f i l[i] = b := by
+@[simp] theorem mem_mapIdx {b : β} {xs : Array α} :
+    b ∈ mapIdx f xs ↔ ∃ (i : Nat) (h : i < xs.size), f i xs[i] = b := by
   constructor
   · intro h
     exact exists_of_mem_mapIdx h
@@ -358,79 +362,138 @@ theorem exists_of_mem_mapIdx {b : β} {l : Array α}
     rw [mem_iff_getElem]
     exact ⟨i, by simpa using h, by simp⟩
 
-theorem mapIdx_eq_push_iff {l : Array α} {b : β} :
-    mapIdx f l = l₂.push b ↔
-      ∃ (a : α) (l₁ : Array α), l = l₁.push a ∧ mapIdx f l₁ = l₂ ∧ f l₁.size a = b := by
+theorem mapIdx_eq_push_iff {xs : Array α} {b : β} :
+    mapIdx f xs = ys.push b ↔
+      ∃ (a : α) (zs : Array α), xs = zs.push a ∧ mapIdx f zs = ys ∧ f zs.size a = b := by
   rw [mapIdx_eq_mapFinIdx, mapFinIdx_eq_push_iff]
   simp only [mapFinIdx_eq_mapIdx, exists_and_left, exists_prop]
   constructor
-  · rintro ⟨l₁, rfl, a, rfl, rfl⟩
-    exact ⟨a, l₁, by simp⟩
-  · rintro ⟨a, l₁, rfl, rfl, rfl⟩
-    exact ⟨l₁, rfl, a, by simp⟩
+  · rintro ⟨zs, rfl, a, rfl, rfl⟩
+    exact ⟨a, zs, by simp⟩
+  · rintro ⟨a, zs, rfl, rfl, rfl⟩
+    exact ⟨zs, rfl, a, by simp⟩
 
-@[simp] theorem mapIdx_eq_singleton_iff {l : Array α} {f : Nat → α → β} {b : β} :
-    mapIdx f l = #[b] ↔ ∃ (a : α), l = #[a] ∧ f 0 a = b := by
-  rcases l with ⟨l⟩
+@[simp] theorem mapIdx_eq_singleton_iff {xs : Array α} {f : Nat → α → β} {b : β} :
+    mapIdx f xs = #[b] ↔ ∃ (a : α), xs = #[a] ∧ f 0 a = b := by
+  rcases xs with ⟨xs⟩
   simp [List.mapIdx_eq_singleton_iff]
 
-theorem mapIdx_eq_append_iff {l : Array α} {f : Nat → α → β} {l₁ l₂ : Array β} :
-    mapIdx f l = l₁ ++ l₂ ↔
-      ∃ (l₁' : Array α) (l₂' : Array α), l = l₁' ++ l₂' ∧
-        l₁'.mapIdx f = l₁ ∧
-        l₂'.mapIdx (fun i => f (i + l₁'.size)) = l₂ := by
-  rcases l with ⟨l⟩
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+theorem mapIdx_eq_append_iff {xs : Array α} {f : Nat → α → β} {ys zs : Array β} :
+    mapIdx f xs = ys ++ zs ↔
+      ∃ (xs' : Array α) (zs' : Array α), xs = xs' ++ zs' ∧
+        xs'.mapIdx f = ys ∧
+        zs'.mapIdx (fun i => f (i + xs'.size)) = zs := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  rcases zs with ⟨zs⟩
   simp only [List.mapIdx_toArray, List.append_toArray, mk.injEq, List.mapIdx_eq_append_iff,
     toArray_eq_append_iff]
   constructor
   · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
     exact ⟨l₁.toArray, l₂.toArray, by simp⟩
   · rintro ⟨⟨l₁⟩, ⟨l₂⟩, rfl, h₁, h₂⟩
-    simp only [List.mapIdx_toArray, mk.injEq, size_toArray] at h₁ h₂
+    simp only [List.mapIdx_toArray, mk.injEq, List.size_toArray] at h₁ h₂
     obtain rfl := h₁
     obtain rfl := h₂
     exact ⟨l₁, l₂, by simp⟩
 
-theorem mapIdx_eq_iff {l : Array α} : mapIdx f l = l' ↔ ∀ i : Nat, l'[i]? = l[i]?.map (f i) := by
-  rcases l with ⟨l⟩
-  rcases l' with ⟨l'⟩
+theorem mapIdx_eq_iff {xs : Array α} : mapIdx f xs = ys ↔ ∀ i : Nat, ys[i]? = xs[i]?.map (f i) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp [List.mapIdx_eq_iff]
 
-theorem mapIdx_eq_mapIdx_iff {l : Array α} :
-    mapIdx f l = mapIdx g l ↔ ∀ i : Nat, (h : i < l.size) → f i l[i] = g i l[i] := by
-  rcases l with ⟨l⟩
+theorem mapIdx_eq_mapIdx_iff {xs : Array α} :
+    mapIdx f xs = mapIdx g xs ↔ ∀ i : Nat, (h : i < xs.size) → f i xs[i] = g i xs[i] := by
+  rcases xs with ⟨xs⟩
   simp [List.mapIdx_eq_mapIdx_iff]
 
-@[simp] theorem mapIdx_set {l : Array α} {i : Nat} {h : i < l.size} {a : α} :
-    (l.set i a).mapIdx f = (l.mapIdx f).set i (f i a) (by simpa) := by
-  rcases l with ⟨l⟩
+@[simp] theorem mapIdx_set {xs : Array α} {i : Nat} {h : i < xs.size} {a : α} :
+    (xs.set i a).mapIdx f = (xs.mapIdx f).set i (f i a) (by simpa) := by
+  rcases xs with ⟨xs⟩
   simp [List.mapIdx_set]
 
-@[simp] theorem mapIdx_setIfInBounds {l : Array α} {i : Nat} {a : α} :
-    (l.setIfInBounds i a).mapIdx f = (l.mapIdx f).setIfInBounds i (f i a) := by
-  rcases l with ⟨l⟩
+@[simp] theorem mapIdx_setIfInBounds {xs : Array α} {i : Nat} {a : α} :
+    (xs.setIfInBounds i a).mapIdx f = (xs.mapIdx f).setIfInBounds i (f i a) := by
+  rcases xs with ⟨xs⟩
   simp [List.mapIdx_set]
 
-@[simp] theorem back?_mapIdx {l : Array α} {f : Nat → α → β} :
-    (mapIdx f l).back? = (l.back?).map (f (l.size - 1)) := by
-  rcases l with ⟨l⟩
+@[simp] theorem back?_mapIdx {xs : Array α} {f : Nat → α → β} :
+    (mapIdx f xs).back? = (xs.back?).map (f (xs.size - 1)) := by
+  rcases xs with ⟨xs⟩
   simp [List.getLast?_mapIdx]
 
-@[simp] theorem mapIdx_mapIdx {l : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
-    (l.mapIdx f).mapIdx g = l.mapIdx (fun i => g i ∘ f i) := by
+@[simp] theorem back_mapIdx {xs : Array α} {f : Nat → α → β} (h) :
+    (xs.mapIdx f).back h = f (xs.size - 1) (xs.back (by simpa using h)) := by
+  rcases xs with ⟨xs⟩
+  simp [List.getLast_mapIdx]
+
+@[simp] theorem mapIdx_mapIdx {xs : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
+    (xs.mapIdx f).mapIdx g = xs.mapIdx (fun i => g i ∘ f i) := by
   simp [mapIdx_eq_iff]
 
-theorem mapIdx_eq_mkArray_iff {l : Array α} {f : Nat → α → β} {b : β} :
-    mapIdx f l = mkArray l.size b ↔ ∀ (i : Nat) (h : i < l.size), f i l[i] = b := by
-  rcases l with ⟨l⟩
+theorem mapIdx_eq_mkArray_iff {xs : Array α} {f : Nat → α → β} {b : β} :
+    mapIdx f xs = mkArray xs.size b ↔ ∀ (i : Nat) (h : i < xs.size), f i xs[i] = b := by
+  rcases xs with ⟨xs⟩
   rw [← toList_inj]
   simp [List.mapIdx_eq_replicate_iff]
 
-@[simp] theorem mapIdx_reverse {l : Array α} {f : Nat → α → β} :
-    l.reverse.mapIdx f = (mapIdx (fun i => f (l.size - 1 - i)) l).reverse := by
-  rcases l with ⟨l⟩
+@[simp] theorem mapIdx_reverse {xs : Array α} {f : Nat → α → β} :
+    xs.reverse.mapIdx f = (mapIdx (fun i => f (xs.size - 1 - i)) xs).reverse := by
+  rcases xs with ⟨xs⟩
   simp [List.mapIdx_reverse]
 
 end Array
+
+namespace List
+
+theorem mapFinIdxM_toArray [Monad m] [LawfulMonad m] (l : List α)
+    (f : (i : Nat) → α → (h : i < l.length) → m β) :
+    l.toArray.mapFinIdxM f = toArray <$> l.mapFinIdxM f := by
+  let rec go (i : Nat) (acc : Array β) (inv : i + acc.size = l.length) :
+      Array.mapFinIdxM.map l.toArray f i acc.size inv acc
+      = toArray <$> mapFinIdxM.go l f (l.drop acc.size) acc
+        (by simp [Nat.sub_add_cancel (Nat.le.intro (Nat.add_comm _ _ ▸ inv))]) := by
+    match i with
+    | 0 =>
+      rw [Nat.zero_add] at inv
+      simp only [Array.mapFinIdxM.map, inv, drop_length, mapFinIdxM.go, map_pure]
+    | k + 1 =>
+      conv => enter [2, 2, 3]; rw [← getElem_cons_drop l acc.size (by omega)]
+      simp only [Array.mapFinIdxM.map, mapFinIdxM.go, _root_.map_bind]
+      congr; funext x
+      conv => enter [1, 4]; rw [← Array.size_push _ x]
+      conv => enter [2, 2, 3]; rw [← Array.size_push _ x]
+      refine go k (acc.push x) _
+  simp only [Array.mapFinIdxM, mapFinIdxM]
+  exact go _ #[] _
+
+theorem mapIdxM_toArray [Monad m] [LawfulMonad m] (l : List α)
+    (f : Nat → α → m β) :
+    l.toArray.mapIdxM f = toArray <$> l.mapIdxM f := by
+  let rec go (bs : List α) (acc : Array β) (inv : bs.length + acc.size = l.length) :
+      mapFinIdxM.go l (fun i a h => f i a) bs acc inv = mapIdxM.go f bs acc := by
+    match bs with
+    | [] => simp only [mapFinIdxM.go, mapIdxM.go]
+    | x :: xs => simp only [mapFinIdxM.go, mapIdxM.go, go]
+  unfold Array.mapIdxM
+  rw [mapFinIdxM_toArray]
+  simp only [mapFinIdxM, mapIdxM]
+  rw [go]
+
+end List
+
+namespace Array
+
+theorem toList_mapFinIdxM [Monad m] [LawfulMonad m] (xs : Array α)
+    (f : (i : Nat) → α → (h : i < xs.size) → m β) :
+    toList <$> xs.mapFinIdxM f = xs.toList.mapFinIdxM f := by
+  rw [List.mapFinIdxM_toArray]
+  simp only [Functor.map_map, id_map']
+
+theorem toList_mapIdxM [Monad m] [LawfulMonad m] (xs : Array α)
+    (f : Nat → α → m β) :
+    toList <$> xs.mapIdxM f = xs.toList.mapIdxM f := by
+  rw [List.mapIdxM_toArray]
+  simp only [Functor.map_map, id_map']
+
+end Array

src/Init/Data/Array/Mem.lean

@@ -8,15 +8,18 @@ import Init.Data.Array.Basic
 import Init.Data.Nat.Linear
 import Init.Data.List.BasicAux
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 theorem sizeOf_lt_of_mem [SizeOf α] {as : Array α} (h : a ∈ as) : sizeOf a < sizeOf as := by
   cases as with | _ as =>
-  exact Nat.lt_trans (List.sizeOf_lt_of_mem h.val) (by simp_arith)
+  exact Nat.lt_trans (List.sizeOf_lt_of_mem h.val) (by simp +arith)
 
-theorem sizeOf_get [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) : sizeOf (as.get i h) < sizeOf as := by
+theorem sizeOf_get [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) : sizeOf as[i] < sizeOf as := by
   cases as with | _ as =>
-  simpa using Nat.lt_trans (List.sizeOf_get _ ⟨i, h⟩) (by simp_arith)
+  simpa using Nat.lt_trans (List.sizeOf_get _ ⟨i, h⟩) (by simp +arith)
 
 @[simp] theorem sizeOf_getElem [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) :
   sizeOf (as[i]'h) < sizeOf as := sizeOf_get _ _ h
@@ -29,8 +32,8 @@ macro "array_get_dec" : tactic =>
     -- subsumed by simp
     -- | with_reducible apply sizeOf_get
     -- | with_reducible apply sizeOf_getElem
-    | (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_get ..)); simp_arith
-    | (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_getElem ..)); simp_arith
+    | (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_get ..)); simp +arith
+    | (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_getElem ..)); simp +arith
     )
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_get_dec)
@@ -45,7 +48,7 @@ macro "array_mem_dec" : tactic =>
     | with_reducible
         apply Nat.lt_of_lt_of_le (Array.sizeOf_lt_of_mem ?h)
         case' h => assumption
-      simp_arith)
+      simp +arith)
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_mem_dec)
 

src/Init/Data/Array/Monadic.lean

@@ -12,6 +12,9 @@ import Init.Data.List.Monadic
 # Lemmas about `Array.forIn'` and `Array.forIn`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 open Nat
@@ -20,90 +23,97 @@ open Nat
 
 /-! ### mapM -/
 
-@[simp] theorem mapM_append [Monad m] [LawfulMonad m] (f : α → m β) {l₁ l₂ : Array α} :
-    (l₁ ++ l₂).mapM f = (return (← l₁.mapM f) ++ (← l₂.mapM f)) := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+@[simp] theorem mapM_pure [Monad m] [LawfulMonad m] (xs : Array α) (f : α → β) :
+    xs.mapM (m := m) (pure <| f ·) = pure (xs.map f) := by
+  induction xs; simp_all
+
+@[simp] theorem mapM_id {xs : Array α} {f : α → Id β} : xs.mapM f = xs.map f :=
+  mapM_pure _ _
+
+@[simp] theorem mapM_append [Monad m] [LawfulMonad m] (f : α → m β) {xs ys : Array α} :
+    (xs ++ ys).mapM f = (return (← xs.mapM f) ++ (← ys.mapM f)) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp
 
-theorem mapM_eq_foldlM_push [Monad m] [LawfulMonad m] (f : α → m β) (l : Array α) :
-    mapM f l = l.foldlM (fun acc a => return (acc.push (← f a))) #[] := by
-  rcases l with ⟨l⟩
-  simp only [List.mapM_toArray, bind_pure_comp, size_toArray, List.foldlM_toArray']
+theorem mapM_eq_foldlM_push [Monad m] [LawfulMonad m] (f : α → m β) (xs : Array α) :
+    mapM f xs = xs.foldlM (fun acc a => return (acc.push (← f a))) #[] := by
+  rcases xs with ⟨xs⟩
+  simp only [List.mapM_toArray, bind_pure_comp, List.size_toArray, List.foldlM_toArray']
   rw [List.mapM_eq_reverse_foldlM_cons]
   simp only [bind_pure_comp, Functor.map_map]
-  suffices ∀ (k), (fun a => a.reverse.toArray) <$> List.foldlM (fun acc a => (fun a => a :: acc) <$> f a) k l =
-      List.foldlM (fun acc a => acc.push <$> f a) k.reverse.toArray l by
+  suffices ∀ (l), (fun l' => l'.reverse.toArray) <$> List.foldlM (fun acc a => (fun a => a :: acc) <$> f a) l xs =
+      List.foldlM (fun acc a => acc.push <$> f a) l.reverse.toArray xs by
     exact this []
-  intro k
-  induction l generalizing k with
+  intro l
+  induction xs generalizing l with
   | nil => simp
   | cons a as ih =>
     simp [ih, List.foldlM_cons]
 
 /-! ### foldlM and foldrM -/
 
-theorem foldlM_map [Monad m] (f : β₁ → β₂) (g : α → β₂ → m α) (l : Array β₁) (init : α) (w : stop = l.size) :
-    (l.map f).foldlM g init 0 stop = l.foldlM (fun x y => g x (f y)) init 0 stop := by
+theorem foldlM_map [Monad m] (f : β₁ → β₂) (g : α → β₂ → m α) (xs : Array β₁) (init : α) (w : stop = xs.size) :
+    (xs.map f).foldlM g init 0 stop = xs.foldlM (fun x y => g x (f y)) init 0 stop := by
   subst w
-  cases l
+  cases xs
   simp [List.foldlM_map]
 
-theorem foldrM_map [Monad m] [LawfulMonad m] (f : β₁ → β₂) (g : β₂ → α → m α) (l : Array β₁)
-    (init : α) (w : start = l.size) :
-    (l.map f).foldrM g init start 0 = l.foldrM (fun x y => g (f x) y) init start 0 := by
+theorem foldrM_map [Monad m] [LawfulMonad m] (f : β₁ → β₂) (g : β₂ → α → m α) (xs : Array β₁)
+    (init : α) (w : start = xs.size) :
+    (xs.map f).foldrM g init start 0 = xs.foldrM (fun x y => g (f x) y) init start 0 := by
   subst w
-  cases l
+  cases xs
   simp [List.foldrM_map]
 
 theorem foldlM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : γ → β → m γ)
-    (l : Array α) (init : γ) (w : stop = (l.filterMap f).size) :
-    (l.filterMap f).foldlM g init 0 stop =
-      l.foldlM (fun x y => match f y with | some b => g x b | none => pure x) init := by
+    (xs : Array α) (init : γ) (w : stop = (xs.filterMap f).size) :
+    (xs.filterMap f).foldlM g init 0 stop =
+      xs.foldlM (fun x y => match f y with | some b => g x b | none => pure x) init := by
   subst w
-  cases l
+  cases xs
   simp [List.foldlM_filterMap]
   rfl
 
 theorem foldrM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : β → γ → m γ)
-    (l : Array α) (init : γ) (w : start = (l.filterMap f).size) :
-    (l.filterMap f).foldrM g init start 0 =
-      l.foldrM (fun x y => match f x with | some b => g b y | none => pure y) init := by
+    (xs : Array α) (init : γ) (w : start = (xs.filterMap f).size) :
+    (xs.filterMap f).foldrM g init start 0 =
+      xs.foldrM (fun x y => match f x with | some b => g b y | none => pure y) init := by
   subst w
-  cases l
+  cases xs
   simp [List.foldrM_filterMap]
   rfl
 
 theorem foldlM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : β → α → m β)
-    (l : Array α) (init : β) (w : stop = (l.filter p).size) :
-    (l.filter p).foldlM g init 0 stop =
-      l.foldlM (fun x y => if p y then g x y else pure x) init := by
+    (xs : Array α) (init : β) (w : stop = (xs.filter p).size) :
+    (xs.filter p).foldlM g init 0 stop =
+      xs.foldlM (fun x y => if p y then g x y else pure x) init := by
   subst w
-  cases l
+  cases xs
   simp [List.foldlM_filter]
 
 theorem foldrM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : α → β → m β)
-    (l : Array α) (init : β) (w : start = (l.filter p).size) :
-    (l.filter p).foldrM g init start 0 =
-      l.foldrM (fun x y => if p x then g x y else pure y) init := by
+    (xs : Array α) (init : β) (w : start = (xs.filter p).size) :
+    (xs.filter p).foldrM g init start 0 =
+      xs.foldrM (fun x y => if p x then g x y else pure y) init := by
   subst w
-  cases l
+  cases xs
   simp [List.foldrM_filter]
 
 @[simp] theorem foldlM_attachWith [Monad m]
-    (l : Array α) {q : α → Prop} (H : ∀ a, a ∈ l → q a) {f : β → { x // q x} → m β} {b} (w : stop = l.size):
-    (l.attachWith q H).foldlM f b 0 stop =
-      l.attach.foldlM (fun b ⟨a, h⟩ => f b ⟨a, H _ h⟩) b := by
+    (xs : Array α) {q : α → Prop} (H : ∀ a, a ∈ xs → q a) {f : β → { x // q x} → m β} {b} (w : stop = xs.size):
+    (xs.attachWith q H).foldlM f b 0 stop =
+      xs.attach.foldlM (fun b ⟨a, h⟩ => f b ⟨a, H _ h⟩) b := by
   subst w
-  rcases l with ⟨l⟩
+  rcases xs with ⟨xs⟩
   simp [List.foldlM_map]
 
 @[simp] theorem foldrM_attachWith [Monad m] [LawfulMonad m]
-    (l : Array α) {q : α → Prop} (H : ∀ a, a ∈ l → q a) {f : { x // q x} → β → m β} {b} (w : start = l.size):
-    (l.attachWith q H).foldrM f b start 0 =
-      l.attach.foldrM (fun a acc => f ⟨a.1, H _ a.2⟩ acc) b := by
+    (xs : Array α) {q : α → Prop} (H : ∀ a, a ∈ xs → q a) {f : { x // q x} → β → m β} {b} (w : start = xs.size):
+    (xs.attachWith q H).foldrM f b start 0 =
+      xs.attach.foldrM (fun a acc => f ⟨a.1, H _ a.2⟩ acc) b := by
   subst w
-  rcases l with ⟨l⟩
+  rcases xs with ⟨xs⟩
   simp [List.foldrM_map]
 
 /-! ### forM -/
@@ -114,15 +124,15 @@ theorem foldrM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : α → β
   cases as <;> cases bs
   simp_all
 
-@[simp] theorem forM_append [Monad m] [LawfulMonad m] (l₁ l₂ : Array α) (f : α → m PUnit) :
-    forM (l₁ ++ l₂) f = (do forM l₁ f; forM l₂ f) := by
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+@[simp] theorem forM_append [Monad m] [LawfulMonad m] (xs ys : Array α) (f : α → m PUnit) :
+    forM (xs ++ ys) f = (do forM xs f; forM ys f) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
   simp
 
-@[simp] theorem forM_map [Monad m] [LawfulMonad m] (l : Array α) (g : α → β) (f : β → m PUnit) :
-    forM (l.map g) f = forM l (fun a => f (g a)) := by
-  cases l
+@[simp] theorem forM_map [Monad m] [LawfulMonad m] (xs : Array α) (g : α → β) (f : β → m PUnit) :
+    forM (xs.map g) f = forM xs (fun a => f (g a)) := by
+  rcases xs with ⟨xs⟩
   simp
 
 /-! ### forIn' -/
@@ -142,41 +152,41 @@ We can express a for loop over an array as a fold,
 in which whenever we reach `.done b` we keep that value through the rest of the fold.
 -/
 theorem forIn'_eq_foldlM [Monad m] [LawfulMonad m]
-    (l : Array α) (f : (a : α) → a ∈ l → β → m (ForInStep β)) (init : β) :
-    forIn' l init f = ForInStep.value <$>
-      l.attach.foldlM (fun b ⟨a, m⟩ => match b with
+    (xs : Array α) (f : (a : α) → a ∈ xs → β → m (ForInStep β)) (init : β) :
+    forIn' xs init f = ForInStep.value <$>
+      xs.attach.foldlM (fun b ⟨a, m⟩ => match b with
         | .yield b => f a m b
         | .done b => pure (.done b)) (ForInStep.yield init) := by
-  cases l
+  rcases xs with ⟨xs⟩
   simp [List.forIn'_eq_foldlM, List.foldlM_map]
   congr
 
 /-- We can express a for loop over an array which always yields as a fold. -/
 @[simp] theorem forIn'_yield_eq_foldlM [Monad m] [LawfulMonad m]
-    (l : Array α) (f : (a : α) → a ∈ l → β → m γ) (g : (a : α) → a ∈ l → β → γ → β) (init : β) :
-    forIn' l init (fun a m b => (fun c => .yield (g a m b c)) <$> f a m b) =
-      l.attach.foldlM (fun b ⟨a, m⟩ => g a m b <$> f a m b) init := by
-  cases l
+    (xs : Array α) (f : (a : α) → a ∈ xs → β → m γ) (g : (a : α) → a ∈ xs → β → γ → β) (init : β) :
+    forIn' xs init (fun a m b => (fun c => .yield (g a m b c)) <$> f a m b) =
+      xs.attach.foldlM (fun b ⟨a, m⟩ => g a m b <$> f a m b) init := by
+  rcases xs with ⟨xs⟩
   simp [List.foldlM_map]
 
 theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
-    (l : Array α) (f : (a : α) → a ∈ l → β → β) (init : β) :
-    forIn' l init (fun a m b => pure (.yield (f a m b))) =
-      pure (f := m) (l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init) := by
-  cases l
+    (xs : Array α) (f : (a : α) → a ∈ xs → β → β) (init : β) :
+    forIn' xs init (fun a m b => pure (.yield (f a m b))) =
+      pure (f := m) (xs.attach.foldl (fun b ⟨a, h⟩ => f a h b) init) := by
+  rcases xs with ⟨xs⟩
   simp [List.forIn'_pure_yield_eq_foldl, List.foldl_map]
 
 @[simp] theorem forIn'_yield_eq_foldl
-    (l : Array α) (f : (a : α) → a ∈ l → β → β) (init : β) :
-    forIn' (m := Id) l init (fun a m b => .yield (f a m b)) =
-      l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init := by
-  cases l
+    (xs : Array α) (f : (a : α) → a ∈ xs → β → β) (init : β) :
+    forIn' (m := Id) xs init (fun a m b => .yield (f a m b)) =
+      xs.attach.foldl (fun b ⟨a, h⟩ => f a h b) init := by
+  rcases xs with ⟨xs⟩
   simp [List.foldl_map]
 
 @[simp] theorem forIn'_map [Monad m] [LawfulMonad m]
-    (l : Array α) (g : α → β) (f : (b : β) → b ∈ l.map g → γ → m (ForInStep γ)) :
-    forIn' (l.map g) init f = forIn' l init fun a h y => f (g a) (mem_map_of_mem g h) y := by
-  cases l
+    (xs : Array α) (g : α → β) (f : (b : β) → b ∈ xs.map g → γ → m (ForInStep γ)) :
+    forIn' (xs.map g) init f = forIn' xs init fun a h y => f (g a) (mem_map_of_mem g h) y := by
+  rcases xs with ⟨xs⟩
   simp
 
 /--
@@ -184,96 +194,290 @@ We can express a for loop over an array as a fold,
 in which whenever we reach `.done b` we keep that value through the rest of the fold.
 -/
 theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
-    (f : α → β → m (ForInStep β)) (init : β) (l : Array α) :
-    forIn l init f = ForInStep.value <$>
-      l.foldlM (fun b a => match b with
+    (f : α → β → m (ForInStep β)) (init : β) (xs : Array α) :
+    forIn xs init f = ForInStep.value <$>
+      xs.foldlM (fun b a => match b with
         | .yield b => f a b
         | .done b => pure (.done b)) (ForInStep.yield init) := by
-  cases l
-  simp only [List.forIn_toArray, List.forIn_eq_foldlM, size_toArray, List.foldlM_toArray']
+  rcases xs with ⟨xs⟩
+  simp only [List.forIn_toArray, List.forIn_eq_foldlM, List.size_toArray, List.foldlM_toArray']
   congr
 
 /-- We can express a for loop over an array which always yields as a fold. -/
 @[simp] theorem forIn_yield_eq_foldlM [Monad m] [LawfulMonad m]
-    (l : Array α) (f : α → β → m γ) (g : α → β → γ → β) (init : β) :
-    forIn l init (fun a b => (fun c => .yield (g a b c)) <$> f a b) =
-      l.foldlM (fun b a => g a b <$> f a b) init := by
-  cases l
+    (xs : Array α) (f : α → β → m γ) (g : α → β → γ → β) (init : β) :
+    forIn xs init (fun a b => (fun c => .yield (g a b c)) <$> f a b) =
+      xs.foldlM (fun b a => g a b <$> f a b) init := by
+  rcases xs with ⟨xs⟩
   simp [List.foldlM_map]
 
 theorem forIn_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
-    (l : Array α) (f : α → β → β) (init : β) :
-    forIn l init (fun a b => pure (.yield (f a b))) =
-      pure (f := m) (l.foldl (fun b a => f a b) init) := by
-  cases l
+    (xs : Array α) (f : α → β → β) (init : β) :
+    forIn xs init (fun a b => pure (.yield (f a b))) =
+      pure (f := m) (xs.foldl (fun b a => f a b) init) := by
+  rcases xs with ⟨xs⟩
   simp [List.forIn_pure_yield_eq_foldl, List.foldl_map]
 
 @[simp] theorem forIn_yield_eq_foldl
-    (l : Array α) (f : α → β → β) (init : β) :
-    forIn (m := Id) l init (fun a b => .yield (f a b)) =
-      l.foldl (fun b a => f a b) init := by
-  cases l
+    (xs : Array α) (f : α → β → β) (init : β) :
+    forIn (m := Id) xs init (fun a b => .yield (f a b)) =
+      xs.foldl (fun b a => f a b) init := by
+  rcases xs with ⟨xs⟩
   simp [List.foldl_map]
 
 @[simp] theorem forIn_map [Monad m] [LawfulMonad m]
-    (l : Array α) (g : α → β) (f : β → γ → m (ForInStep γ)) :
-    forIn (l.map g) init f = forIn l init fun a y => f (g a) y := by
-  cases l
+    (xs : Array α) (g : α → β) (f : β → γ → m (ForInStep γ)) :
+    forIn (xs.map g) init f = forIn xs init fun a y => f (g a) y := by
+  rcases xs with ⟨xs⟩
   simp
 
+/-! ### allM and anyM -/
+
+@[simp] theorem anyM_pure [Monad m] [LawfulMonad m] (p : α → Bool) (xs : Array α) :
+    xs.anyM (m := m) (pure <| p ·) = pure (xs.any p) := by
+  cases xs
+  simp
+
+@[simp] theorem allM_pure [Monad m] [LawfulMonad m] (p : α → Bool) (xs : Array α) :
+    xs.allM (m := m) (pure <| p ·) = pure (xs.all p) := by
+  cases xs
+  simp
+
+/-! ### findM? and findSomeM? -/
+
+@[simp]
+theorem findM?_pure {m} [Monad m] [LawfulMonad m] (p : α → Bool) (xs : Array α) :
+    findM? (m := m) (pure <| p ·) xs = pure (xs.find? p) := by
+  cases xs
+  simp
+
+@[simp]
+theorem findSomeM?_pure [Monad m] [LawfulMonad m] (f : α → Option β) (xs : Array α) :
+    findSomeM? (m := m) (pure <| f ·) xs = pure (xs.findSome? f) := by
+  cases xs
+  simp
+
+end Array
+
+namespace List
+
+theorem filterM_toArray [Monad m] [LawfulMonad m] (l : List α) (p : α → m Bool) :
+    l.toArray.filterM p = toArray <$> l.filterM p := by
+  simp only [Array.filterM, filterM, foldlM_toArray, bind_pure_comp, Functor.map_map]
+  conv => lhs; rw [← reverse_nil]
+  generalize [] = acc
+  induction l generalizing acc with simp
+  | cons x xs ih =>
+    congr; funext b
+    cases b
+    · simp only [Bool.false_eq_true, ↓reduceIte, pure_bind, cond_false]
+      exact ih acc
+    · simp only [↓reduceIte, ← reverse_cons, pure_bind, cond_true]
+      exact ih (x :: acc)
+
+/-- Variant of `filterM_toArray` with a side condition for the stop position. -/
+@[simp] theorem filterM_toArray' [Monad m] [LawfulMonad m] (l : List α) (p : α → m Bool) (w : stop = l.length) :
+    l.toArray.filterM p 0 stop = toArray <$> l.filterM p := by
+  subst w
+  rw [filterM_toArray]
+
+theorem filterRevM_toArray [Monad m] [LawfulMonad m] (l : List α) (p : α → m Bool) :
+    l.toArray.filterRevM p = toArray <$> l.filterRevM p := by
+  simp [Array.filterRevM, filterRevM]
+  rw [← foldlM_reverse, ← foldlM_toArray, ← Array.filterM, filterM_toArray]
+  simp only [filterM, bind_pure_comp, Functor.map_map, reverse_toArray, reverse_reverse]
+
+/-- Variant of `filterRevM_toArray` with a side condition for the start position. -/
+@[simp] theorem filterRevM_toArray' [Monad m] [LawfulMonad m] (l : List α) (p : α → m Bool) (w : start = l.length) :
+    l.toArray.filterRevM p start 0 = toArray <$> l.filterRevM p := by
+  subst w
+  rw [filterRevM_toArray]
+
+theorem filterMapM_toArray [Monad m] [LawfulMonad m] (l : List α) (f : α → m (Option β)) :
+    l.toArray.filterMapM f = toArray <$> l.filterMapM f := by
+  simp [Array.filterMapM, filterMapM]
+  conv => lhs; rw [← reverse_nil]
+  generalize [] = acc
+  induction l generalizing acc with simp [filterMapM.loop]
+  | cons x xs ih =>
+    congr; funext o
+    cases o
+    · simp only [pure_bind]; exact ih acc
+    · simp only [pure_bind]; rw [← List.reverse_cons]; exact ih _
+
+/-- Variant of `filterMapM_toArray` with a side condition for the stop position. -/
+@[simp] theorem filterMapM_toArray' [Monad m] [LawfulMonad m] (l : List α) (f : α → m (Option β)) (w : stop = l.length) :
+    l.toArray.filterMapM f 0 stop = toArray <$> l.filterMapM f := by
+  subst w
+  rw [filterMapM_toArray]
+
+@[simp] theorem flatMapM_toArray [Monad m] [LawfulMonad m] (l : List α) (f : α → m (Array β)) :
+    l.toArray.flatMapM f = toArray <$> l.flatMapM (fun a => Array.toList <$> f a) := by
+  simp only [Array.flatMapM, bind_pure_comp, foldlM_toArray, flatMapM]
+  conv => lhs; arg 2; change [].reverse.flatten.toArray
+  generalize [] = acc
+  induction l generalizing acc with
+  | nil => simp only [foldlM_nil, flatMapM.loop, map_pure]
+  | cons x xs ih =>
+    simp only [foldlM_cons, bind_map_left, flatMapM.loop, _root_.map_bind]
+    congr; funext xs
+    conv => lhs; rw [Array.toArray_append, ← flatten_concat, ← reverse_cons]
+    exact ih _
+
+end List
+
+namespace Array
+
+@[congr] theorem filterM_congr [Monad m] {as bs : Array α} (w : as = bs)
+    {p : α → m Bool} {q : α → m Bool} (h : ∀ a, p a = q a) :
+    as.filterM p = bs.filterM q := by
+  subst w
+  simp [filterM, h]
+
+@[congr] theorem filterRevM_congr [Monad m] {as bs : Array α} (w : as = bs)
+    {p : α → m Bool} {q : α → m Bool} (h : ∀ a, p a = q a) :
+    as.filterRevM p = bs.filterRevM q := by
+  subst w
+  simp [filterRevM, h]
+
+@[congr] theorem filterMapM_congr [Monad m] {as bs : Array α} (w : as = bs)
+    {f : α → m (Option β)} {g : α → m (Option β)} (h : ∀ a, f a = g a) :
+    as.filterMapM f = bs.filterMapM g := by
+  subst w
+  simp [filterMapM, h]
+
+@[congr] theorem flatMapM_congr [Monad m] {as bs : Array α} (w : as = bs)
+    {f : α → m (Array β)} {g : α → m (Array β)} (h : ∀ a, f a = g a) :
+    as.flatMapM f = bs.flatMapM g := by
+  subst w
+  simp [flatMapM, h]
+
+theorem toList_filterM [Monad m] [LawfulMonad m] (xs : Array α) (p : α → m Bool) :
+    toList <$> xs.filterM p = xs.toList.filterM p := by
+  rw [List.filterM_toArray]
+  simp only [Functor.map_map, id_map']
+
+theorem toList_filterRevM [Monad m] [LawfulMonad m] (xs : Array α) (p : α → m Bool) :
+    toList <$> xs.filterRevM p = xs.toList.filterRevM p := by
+  rw [List.filterRevM_toArray]
+  simp only [Functor.map_map, id_map']
+
+theorem toList_filterMapM [Monad m] [LawfulMonad m] (xs : Array α) (f : α → m (Option β)) :
+    toList <$> xs.filterMapM f = xs.toList.filterMapM f := by
+  rw [List.filterMapM_toArray]
+  simp only [Functor.map_map, id_map']
+
+theorem toList_flatMapM [Monad m] [LawfulMonad m] (xs : Array α) (f : α → m (Array β)) :
+    toList <$> xs.flatMapM f = xs.toList.flatMapM (fun a => toList <$> f a) := by
+  rw [List.flatMapM_toArray]
+  simp only [Functor.map_map, id_map']
+
 /-! ### Recognizing higher order functions using a function that only depends on the value. -/
 
 /--
 This lemma identifies monadic folds over lists of subtypes, where the function only depends on the value, not the proposition,
 and simplifies these to the function directly taking the value.
 -/
-@[simp] theorem foldlM_subtype [Monad m] {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem foldlM_subtype [Monad m] {p : α → Prop} {xs : Array { x // p x }}
     {f : β → { x // p x } → m β} {g : β → α → m β} {x : β}
-    (hf : ∀ b x h, f b ⟨x, h⟩ = g b x) (w : stop = l.size) :
-    l.foldlM f x 0 stop = l.unattach.foldlM g x 0 stop := by
+    (hf : ∀ b x h, f b ⟨x, h⟩ = g b x) (w : stop = xs.size) :
+    xs.foldlM f x 0 stop = xs.unattach.foldlM g x 0 stop := by
   subst w
-  rcases l with ⟨l⟩
+  rcases xs with ⟨l⟩
   simp
   rw [List.foldlM_subtype hf]
 
+@[wf_preprocess] theorem foldlM_wfParam [Monad m] (xs : Array α) (f : β → α → m β) (init : β) :
+    (wfParam xs).foldlM f init = xs.attach.unattach.foldlM f init := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldlM_unattach [Monad m] (P : α → Prop) (xs : Array (Subtype P)) (f : β → α → m β) (init : β) :
+    xs.unattach.foldlM f init = xs.foldlM (init := init) fun b ⟨x, h⟩ =>
+      binderNameHint b f <| binderNameHint x (f b) <| binderNameHint h () <|
+      f b (wfParam x) := by
+  simp [wfParam]
+
 /--
 This lemma identifies monadic folds over lists of subtypes, where the function only depends on the value, not the proposition,
 and simplifies these to the function directly taking the value.
 -/
-@[simp] theorem foldrM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem foldrM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → β → m β} {g : α → β → m β} {x : β}
-    (hf : ∀ x h b, f ⟨x, h⟩ b = g x b) (w : start = l.size) :
-    l.foldrM f x start 0 = l.unattach.foldrM g x start 0:= by
+    (hf : ∀ x h b, f ⟨x, h⟩ b = g x b) (w : start = xs.size) :
+    xs.foldrM f x start 0 = xs.unattach.foldrM g x start 0:= by
   subst w
-  rcases l with ⟨l⟩
+  rcases xs with ⟨xs⟩
   simp
   rw [List.foldrM_subtype hf]
 
+
+@[wf_preprocess] theorem foldrM_wfParam [Monad m] [LawfulMonad m] (xs : Array α) (f : α → β → m β) (init : β) :
+    (wfParam xs).foldrM f init = xs.attach.unattach.foldrM f init := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldrM_unattach [Monad m] [LawfulMonad m] (P : α → Prop) (xs : Array (Subtype P)) (f : α → β → m β) (init : β):
+    xs.unattach.foldrM f init = xs.foldrM (init := init) fun ⟨x, h⟩ b =>
+      binderNameHint x f <| binderNameHint h () <| binderNameHint b (f x) <|
+      f (wfParam x) b := by
+  simp [wfParam]
+
 /--
 This lemma identifies monadic maps over lists of subtypes, where the function only depends on the value, not the proposition,
 and simplifies these to the function directly taking the value.
 -/
-@[simp] theorem mapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {l : Array { x // p x }}
+@[simp] theorem mapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {xs : Array { x // p x }}
     {f : { x // p x } → m β} {g : α → m β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.mapM f = l.unattach.mapM g := by
-  rcases l with ⟨l⟩
+    xs.mapM f = xs.unattach.mapM g := by
+  rcases xs with ⟨xs⟩
   simp
   rw [List.mapM_subtype hf]
 
--- Without `filterMapM_toArray` relating `filterMapM` on `List` and `Array` we can't prove this yet:
--- @[simp] theorem filterMapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {l : Array { x // p x }}
---     {f : { x // p x } → m (Option β)} {g : α → m (Option β)} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
---     l.filterMapM f = l.unattach.filterMapM g := by
---   rcases l with ⟨l⟩
---   simp
---   rw [List.filterMapM_subtype hf]
+@[wf_preprocess] theorem mapM_wfParam [Monad m] [LawfulMonad m] (xs : Array α) (f : α → m β) :
+    (wfParam xs).mapM f = xs.attach.unattach.mapM f := by
+  simp [wfParam]
 
--- Without `flatMapM_toArray` relating `flatMapM` on `List` and `Array` we can't prove this yet:
--- @[simp] theorem flatMapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {l : Array { x // p x }}
---     {f : { x // p x } → m (Array β)} {g : α → m (Array β)} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
---     (l.flatMapM f) = l.unattach.flatMapM g := by
---   rcases l with ⟨l⟩
---   simp
---   rw [List.flatMapM_subtype hf]
+@[wf_preprocess] theorem mapM_unattach [Monad m] [LawfulMonad m] (P : α → Prop) (xs : Array (Subtype P)) (f : α → m β) :
+    xs.unattach.mapM f = xs.mapM fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+@[simp] theorem filterMapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {xs : Array { x // p x }}
+    {f : { x // p x } → m (Option β)} {g : α → m (Option β)} (hf : ∀ x h, f ⟨x, h⟩ = g x) (w : stop = xs.size) :
+    xs.filterMapM f 0 stop = xs.unattach.filterMapM g := by
+  subst w
+  rcases xs with ⟨xs⟩
+  simp
+  rw [List.filterMapM_subtype hf]
+
+
+@[wf_preprocess] theorem filterMapM_wfParam [Monad m] [LawfulMonad m]
+    (xs : Array α) (f : α → m (Option β)) :
+    (wfParam xs).filterMapM f = xs.attach.unattach.filterMapM f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filterMapM_unattach [Monad m] [LawfulMonad m]
+    (P : α → Prop) (xs : Array (Subtype P)) (f : α → m (Option β)) :
+    xs.unattach.filterMapM f = xs.filterMapM fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+@[simp] theorem flatMapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {xs : Array { x // p x }}
+    {f : { x // p x } → m (Array β)} {g : α → m (Array β)} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    (xs.flatMapM f) = xs.unattach.flatMapM g := by
+  rcases xs with ⟨xs⟩
+  simp
+  rw [List.flatMapM_subtype]
+  simp [hf]
+
+@[wf_preprocess] theorem flatMapM_wfParam [Monad m] [LawfulMonad m]
+    (xs : Array α) (f : α → m (Array β)) :
+    (wfParam xs).flatMapM f = xs.attach.unattach.flatMapM f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem flatMapM_unattach [Monad m] [LawfulMonad m]
+    (P : α → Prop) (xs : Array (Subtype P)) (f : α → m (Array β)) :
+    xs.unattach.flatMapM f = xs.flatMapM fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
 
 end Array

src/Init/Data/Array/OfFn.lean

@@ -11,8 +11,30 @@ import Init.Data.List.OfFn
 # Theorems about `Array.ofFn`
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
+@[simp] theorem ofFn_zero (f : Fin 0 → α) : ofFn f = #[] := rfl
+
+theorem ofFn_succ (f : Fin (n+1) → α) :
+    ofFn f = (ofFn (fun (i : Fin n) => f i.castSucc)).push (f ⟨n, by omega⟩) := by
+  ext i h₁ h₂
+  · simp
+  · simp [getElem_push]
+    split <;> rename_i h₃
+    · rfl
+    · congr
+      simp at h₁ h₂
+      omega
+
+@[simp] theorem _rooy_.List.toArray_ofFn (f : Fin n → α) : (List.ofFn f).toArray = Array.ofFn f := by
+  ext <;> simp
+
+@[simp] theorem toList_ofFn (f : Fin n → α) : (Array.ofFn f).toList = List.ofFn f := by
+  apply List.ext_getElem <;> simp
+
 @[simp]
 theorem ofFn_eq_empty_iff {f : Fin n → α} : ofFn f = #[] ↔ n = 0 := by
   rw [← Array.toList_inj]

src/Init/Data/Array/Perm.lean

@@ -7,6 +7,9 @@ prelude
 import Init.Data.List.Nat.Perm
 import Init.Data.Array.Lemmas
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 open List
@@ -27,38 +30,38 @@ theorem perm_iff_toList_perm {as bs : Array α} : as ~ bs ↔ as.toList ~ bs.toL
 @[simp] theorem perm_toArray (as bs : List α) : as.toArray ~ bs.toArray ↔ as ~ bs := by
   simp [perm_iff_toList_perm]
 
-@[simp, refl] protected theorem Perm.refl (l : Array α) : l ~ l := by
-  cases l
+@[simp, refl] protected theorem Perm.refl (xs : Array α) : xs ~ xs := by
+  cases xs
   simp
 
-protected theorem Perm.rfl {l : List α} : l ~ l := .refl _
+protected theorem Perm.rfl {xs : List α} : xs ~ xs := .refl _
 
-theorem Perm.of_eq {l₁ l₂ : Array α} (h : l₁ = l₂) : l₁ ~ l₂ := h ▸ .rfl
+theorem Perm.of_eq {xs ys : Array α} (h : xs = ys) : xs ~ ys := h ▸ .rfl
 
-protected theorem Perm.symm {l₁ l₂ : Array α} (h : l₁ ~ l₂) : l₂ ~ l₁ := by
-  cases l₁; cases l₂
+protected theorem Perm.symm {xs ys : Array α} (h : xs ~ ys) : ys ~ xs := by
+  cases xs; cases ys
   simp only [perm_toArray] at h
   simpa using h.symm
 
-protected theorem Perm.trans {l₁ l₂ l₃ : Array α} (h₁ : l₁ ~ l₂) (h₂ : l₂ ~ l₃) : l₁ ~ l₃ := by
-  cases l₁; cases l₂; cases l₃
+protected theorem Perm.trans {xs ys zs : Array α} (h₁ : xs ~ ys) (h₂ : ys ~ zs) : xs ~ zs := by
+  cases xs; cases ys; cases zs
   simp only [perm_toArray] at h₁ h₂
   simpa using h₁.trans h₂
 
 instance : Trans (Perm (α := α)) (Perm (α := α)) (Perm (α := α)) where
   trans h₁ h₂ := Perm.trans h₁ h₂
 
-theorem perm_comm {l₁ l₂ : Array α} : l₁ ~ l₂ ↔ l₂ ~ l₁ := ⟨Perm.symm, Perm.symm⟩
+theorem perm_comm {xs ys : Array α} : xs ~ ys ↔ ys ~ xs := ⟨Perm.symm, Perm.symm⟩
 
-theorem Perm.push (x y : α) {l₁ l₂ : Array α} (p : l₁ ~ l₂) :
-    (l₁.push x).push y ~ (l₂.push y).push x := by
-  cases l₁; cases l₂
+theorem Perm.push (x y : α) {xs ys : Array α} (p : xs ~ ys) :
+    (xs.push x).push y ~ (ys.push y).push x := by
+  cases xs; cases ys
   simp only [perm_toArray] at p
   simp only [push_toArray, List.append_assoc, singleton_append, perm_toArray]
   exact p.append (Perm.swap' _ _ Perm.nil)
 
-theorem swap_perm {as : Array α} {i j : Nat} (h₁ : i < as.size) (h₂ : j < as.size) :
-    as.swap i j ~ as := by
+theorem swap_perm {xs : Array α} {i j : Nat} (h₁ : i < xs.size) (h₂ : j < xs.size) :
+    xs.swap i j ~ xs := by
   simp only [swap, perm_iff_toList_perm, toList_set]
   apply set_set_perm
 

src/Init/Data/Array/QSort.lean

@@ -7,6 +7,9 @@ prelude
 import Init.Data.Vector.Basic
 import Init.Data.Ord
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- We do not enable `linter.indexVariables` because it is helpful to name index variables `lo`, `mid`, `hi`, etc.
+
 namespace Array
 
 private def qpartition {n} (as : Vector α n) (lt : α → α → Bool) (lo hi : Nat)

src/Init/Data/Array/Range.lean

@@ -15,6 +15,9 @@ import Init.Data.List.Nat.Range
 
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 open Nat
@@ -28,7 +31,7 @@ theorem range'_succ (s n step) : range' s (n + 1) step = #[s] ++ range' (s + ste
   simp [List.range'_succ]
 
 @[simp] theorem range'_eq_empty_iff : range' s n step = #[] ↔ n = 0 := by
-  rw [← size_eq_zero, size_range']
+  rw [← size_eq_zero_iff, size_range']
 
 theorem range'_ne_empty_iff (s : Nat) {n step : Nat} : range' s n step ≠ #[] ↔ n ≠ 0 := by
   cases n <;> simp
@@ -124,7 +127,7 @@ theorem range_succ_eq_map (n : Nat) : range (n + 1) = #[0] ++ map succ (range n)
   ext i h₁ h₂
   · simp
     omega
-  · simp only [getElem_range, getElem_append, size_toArray, List.length_cons, List.length_nil,
+  · simp only [getElem_range, getElem_append, List.size_toArray, List.length_cons, List.length_nil,
       Nat.zero_add, lt_one_iff, List.getElem_toArray, List.getElem_singleton, getElem_map,
       succ_eq_add_one, dite_eq_ite]
     split <;> omega
@@ -133,7 +136,7 @@ theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) :=
   rw [range_eq_range', map_add_range']; rfl
 
 @[simp] theorem range_eq_empty_iff {n : Nat} : range n = #[] ↔ n = 0 := by
-  rw [← size_eq_zero, size_range]
+  rw [← size_eq_zero_iff, size_range]
 
 theorem range_ne_empty_iff {n : Nat} : range n ≠ #[] ↔ n ≠ 0 := by
   cases n <;> simp
@@ -146,9 +149,9 @@ theorem range_succ (n : Nat) : range (succ n) = range n ++ #[n] := by
       dite_eq_ite]
     split <;> omega
 
-theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + ·) := by
+theorem range_add (n m : Nat) : range (n + m) = range n ++ (range m).map (n + ·) := by
   rw [← range'_eq_map_range]
-  simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm
+  simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 n m).symm
 
 theorem reverse_range' (s n : Nat) : reverse (range' s n) = map (s + n - 1 - ·) (range n) := by
   simp [← toList_inj, List.reverse_range']
@@ -161,7 +164,7 @@ theorem not_mem_range_self {n : Nat} : n ∉ range n := by simp
 
 theorem self_mem_range_succ (n : Nat) : n ∈ range (n + 1) := by simp
 
-@[simp] theorem take_range (m n : Nat) : take (range n) m = range (min m n) := by
+@[simp] theorem take_range (i n : Nat) : take (range n) i = range (min i n) := by
   ext <;> simp
 
 @[simp] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
@@ -179,48 +182,48 @@ theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n -
 /-! ### zipIdx -/
 
 @[simp]
-theorem zipIdx_eq_empty_iff {l : Array α} {n : Nat} : l.zipIdx n = #[] ↔ l = #[] := by
-  cases l
+theorem zipIdx_eq_empty_iff {xs : Array α} {i : Nat} : xs.zipIdx i = #[] ↔ xs = #[] := by
+  cases xs
   simp
 
 @[simp]
-theorem getElem?_zipIdx (l : Array α) (n m) : (zipIdx l n)[m]? = l[m]?.map fun a => (a, n + m) := by
+theorem getElem?_zipIdx (xs : Array α) (i j) : (zipIdx xs i)[j]? = xs[j]?.map fun a => (a, i + j) := by
   simp [getElem?_def]
 
-theorem map_snd_add_zipIdx_eq_zipIdx (l : Array α) (n k : Nat) :
-    map (Prod.map id (· + n)) (zipIdx l k) = zipIdx l (n + k) :=
+theorem map_snd_add_zipIdx_eq_zipIdx (xs : Array α) (n k : Nat) :
+    map (Prod.map id (· + n)) (zipIdx xs k) = zipIdx xs (n + k) :=
   ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
 
 @[simp]
-theorem zipIdx_map_snd (n) (l : Array α) : map Prod.snd (zipIdx l n) = range' n l.size := by
-  cases l
+theorem zipIdx_map_snd (i) (xs : Array α) : map Prod.snd (zipIdx xs i) = range' i xs.size := by
+  cases xs
   simp
 
 @[simp]
-theorem zipIdx_map_fst (n) (l : Array α) : map Prod.fst (zipIdx l n) = l := by
-  cases l
+theorem zipIdx_map_fst (i) (xs : Array α) : map Prod.fst (zipIdx xs i) = xs := by
+  cases xs
   simp
 
-theorem zipIdx_eq_zip_range' (l : Array α) {n : Nat} : l.zipIdx n = l.zip (range' n l.size) := by
+theorem zipIdx_eq_zip_range' (xs : Array α) {i : Nat} : xs.zipIdx i = xs.zip (range' i xs.size) := by
   simp [zip_of_prod (zipIdx_map_fst _ _) (zipIdx_map_snd _ _)]
 
 @[simp]
-theorem unzip_zipIdx_eq_prod (l : Array α) {n : Nat} :
-    (l.zipIdx n).unzip = (l, range' n l.size) := by
+theorem unzip_zipIdx_eq_prod (xs : Array α) {i : Nat} :
+    (xs.zipIdx i).unzip = (xs, range' i xs.size) := by
   simp only [zipIdx_eq_zip_range', unzip_zip, size_range']
 
 /-- Replace `zipIdx` with a starting index `n+1` with `zipIdx` starting from `n`,
 followed by a `map` increasing the indices by one. -/
-theorem zipIdx_succ (l : Array α) (n : Nat) :
-    l.zipIdx (n + 1) = (l.zipIdx n).map (fun ⟨a, i⟩ => (a, i + 1)) := by
-  cases l
+theorem zipIdx_succ (xs : Array α) (i : Nat) :
+    xs.zipIdx (i + 1) = (xs.zipIdx i).map (fun ⟨a, j⟩ => (a, j + 1)) := by
+  cases xs
   simp [List.zipIdx_succ]
 
 /-- Replace `zipIdx` with a starting index with `zipIdx` starting from 0,
 followed by a `map` increasing the indices. -/
-theorem zipIdx_eq_map_add (l : Array α) (n : Nat) :
-    l.zipIdx n = l.zipIdx.map (fun ⟨a, i⟩ => (a, n + i)) := by
-  cases l
+theorem zipIdx_eq_map_add (xs : Array α) (i : Nat) :
+    xs.zipIdx i = (xs.zipIdx 0).map (fun ⟨a, j⟩ => (a, i + j)) := by
+  cases xs
   simp only [zipIdx_toArray, List.map_toArray, mk.injEq]
   rw [List.zipIdx_eq_map_add]
 
@@ -228,33 +231,33 @@ theorem zipIdx_eq_map_add (l : Array α) (n : Nat) :
 theorem zipIdx_singleton (x : α) (k : Nat) : zipIdx #[x] k = #[(x, k)] :=
   rfl
 
-theorem mk_add_mem_zipIdx_iff_getElem? {k i : Nat} {x : α} {l : Array α} :
-    (x, k + i) ∈ zipIdx l k ↔ l[i]? = some x := by
+theorem mk_add_mem_zipIdx_iff_getElem? {k i : Nat} {x : α} {xs : Array α} :
+    (x, k + i) ∈ zipIdx xs k ↔ xs[i]? = some x := by
   simp [mem_iff_getElem?, and_left_comm]
 
-theorem le_snd_of_mem_zipIdx {x : α × Nat} {k : Nat} {l : Array α} (h : x ∈ zipIdx l k) :
+theorem le_snd_of_mem_zipIdx {x : α × Nat} {k : Nat} {xs : Array α} (h : x ∈ zipIdx xs k) :
     k ≤ x.2 :=
   (mk_mem_zipIdx_iff_le_and_getElem?_sub.1 h).1
 
-theorem snd_lt_add_of_mem_zipIdx {x : α × Nat} {l : Array α} {k : Nat} (h : x ∈ zipIdx l k) :
-    x.2 < k + l.size := by
+theorem snd_lt_add_of_mem_zipIdx {x : α × Nat} {k : Nat} {xs : Array α} (h : x ∈ zipIdx xs k) :
+    x.2 < k + xs.size := by
   rcases mem_iff_getElem.1 h with ⟨i, h', rfl⟩
   simpa using h'
 
-theorem snd_lt_of_mem_zipIdx {x : α × Nat} {l : Array α} {k : Nat} (h : x ∈ l.zipIdx k) : x.2 < l.size + k := by
+theorem snd_lt_of_mem_zipIdx {x : α × Nat} {k : Nat} {xs : Array α} (h : x ∈ zipIdx xs k) : x.2 < xs.size + k := by
   simpa [Nat.add_comm] using snd_lt_add_of_mem_zipIdx h
 
-theorem map_zipIdx (f : α → β) (l : Array α) (k : Nat) :
-    map (Prod.map f id) (zipIdx l k) = zipIdx (l.map f) k := by
-  cases l
+theorem map_zipIdx (f : α → β) (xs : Array α) (k : Nat) :
+    map (Prod.map f id) (zipIdx xs k) = zipIdx (xs.map f) k := by
+  cases xs
   simp [List.map_zipIdx]
 
-theorem fst_mem_of_mem_zipIdx {x : α × Nat} {l : Array α} {k : Nat} (h : x ∈ zipIdx l k) : x.1 ∈ l :=
-  zipIdx_map_fst k l ▸ mem_map_of_mem _ h
+theorem fst_mem_of_mem_zipIdx {x : α × Nat} {xs : Array α} {k : Nat} (h : x ∈ zipIdx xs k) : x.1 ∈ xs :=
+  zipIdx_map_fst k xs ▸ mem_map_of_mem _ h
 
-theorem fst_eq_of_mem_zipIdx {x : α × Nat} {l : Array α} {k : Nat} (h : x ∈ zipIdx l k) :
-    x.1 = l[x.2 - k]'(by have := le_snd_of_mem_zipIdx h; have := snd_lt_add_of_mem_zipIdx h; omega) := by
-  cases l
+theorem fst_eq_of_mem_zipIdx {x : α × Nat} {xs : Array α} {k : Nat} (h : x ∈ zipIdx xs k) :
+    x.1 = xs[x.2 - k]'(by have := le_snd_of_mem_zipIdx h; have := snd_lt_add_of_mem_zipIdx h; omega) := by
+  cases xs
   exact List.fst_eq_of_mem_zipIdx (by simpa using h)
 
 theorem mem_zipIdx {x : α} {i : Nat} {xs : Array α} {k : Nat} (h : (x, i) ∈ xs.zipIdx k) :
@@ -267,9 +270,9 @@ theorem mem_zipIdx' {x : α} {i : Nat} {xs : Array α} (h : (x, i) ∈ xs.zipIdx
     i < xs.size ∧ x = xs[i]'(by have := le_snd_of_mem_zipIdx h; have := snd_lt_add_of_mem_zipIdx h; omega) :=
   ⟨by simpa using snd_lt_add_of_mem_zipIdx h, fst_eq_of_mem_zipIdx h⟩
 
-theorem zipIdx_map (l : Array α) (k : Nat) (f : α → β) :
-    zipIdx (l.map f) k = (zipIdx l k).map (Prod.map f id) := by
-  cases l
+theorem zipIdx_map (xs : Array α) (k : Nat) (f : α → β) :
+    zipIdx (xs.map f) k = (zipIdx xs k).map (Prod.map f id) := by
+  cases xs
   simp [List.zipIdx_map]
 
 theorem zipIdx_append (xs ys : Array α) (k : Nat) :
@@ -278,19 +281,19 @@ theorem zipIdx_append (xs ys : Array α) (k : Nat) :
   cases ys
   simp [List.zipIdx_append]
 
-theorem zipIdx_eq_append_iff {l : Array α} {k : Nat} :
-    zipIdx l k = l₁ ++ l₂ ↔
-      ∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = zipIdx l₁' k ∧ l₂ = zipIdx l₂' (k + l₁'.size) := by
-  rcases l with ⟨l⟩
-  rcases l₁ with ⟨l₁⟩
-  rcases l₂ with ⟨l₂⟩
+theorem zipIdx_eq_append_iff {xs : Array α} {k : Nat} :
+    zipIdx xs k = ys ++ zs ↔
+      ∃ ys' zs', xs = ys' ++ zs' ∧ ys = zipIdx ys' k ∧ zs = zipIdx zs' (k + ys'.size) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  rcases zs with ⟨zs⟩
   simp only [zipIdx_toArray, List.append_toArray, mk.injEq, List.zipIdx_eq_append_iff,
     toArray_eq_append_iff]
   constructor
   · rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
     exact ⟨⟨l₁'⟩, ⟨l₂'⟩, by simp⟩
   · rintro ⟨⟨l₁'⟩, ⟨l₂'⟩, rfl, h⟩
-    simp only [zipIdx_toArray, mk.injEq, size_toArray] at h
+    simp only [zipIdx_toArray, mk.injEq, List.size_toArray] at h
     obtain ⟨rfl, rfl⟩ := h
     exact ⟨l₁', l₂', by simp⟩
 

src/Init/Data/Array/Set.lean

@@ -6,6 +6,9 @@ Authors: Leonardo de Moura, Mario Carneiro
 prelude
 import Init.Tactics
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 
 /--
 Set an element in an array, using a proof that the index is in bounds.
@@ -15,9 +18,9 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_fset"]
-def Array.set (a : Array α) (i : @& Nat) (v : α) (h : i < a.size := by get_elem_tactic) :
+def Array.set (xs : Array α) (i : @& Nat) (v : α) (h : i < xs.size := by get_elem_tactic) :
     Array α where
-  toList := a.toList.set i v
+  toList := xs.toList.set i v
 
 /--
 Set an element in an array, or do nothing if the index is out of bounds.
@@ -25,8 +28,8 @@ Set an element in an array, or do nothing if the index is out of bounds.
 This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
-@[inline] def Array.setIfInBounds (a : Array α) (i : Nat) (v : α) : Array α :=
-  dite (LT.lt i a.size) (fun h => a.set i v h) (fun _ => a)
+@[inline] def Array.setIfInBounds (xs : Array α) (i : Nat) (v : α) : Array α :=
+  dite (LT.lt i xs.size) (fun h => xs.set i v h) (fun _ => xs)
 
 @[deprecated Array.setIfInBounds (since := "2024-11-24")] abbrev Array.setD := @Array.setIfInBounds
 
@@ -37,5 +40,5 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_set"]
-def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
-  Array.setIfInBounds a i v
+def Array.set! (xs : Array α) (i : @& Nat) (v : α) : Array α :=
+  Array.setIfInBounds xs i v

src/Init/Data/Array/Subarray.lean

@@ -6,6 +6,8 @@ Authors: Leonardo de Moura
 prelude
 import Init.Data.Array.Basic
 
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 universe u v w
 
 structure Subarray (α : Type u)  where

src/Init/Data/Array/Subarray/Split.lean

@@ -15,6 +15,9 @@ automation. Placing them in another module breaks an import cycle, because `omeg
 array library.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Subarray
 /--
 Splits a subarray into two parts.

src/Init/Data/Array/TakeDrop.lean

@@ -7,11 +7,28 @@ prelude
 import Init.Data.Array.Lemmas
 import Init.Data.List.Nat.TakeDrop
 
+/-!
+These lemmas are used in the internals of HashMap.
+They should find a new home and/or be reformulated.
+-/
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
+namespace List
+
+theorem exists_of_set {i : Nat} {a' : α} {l : List α} (h : i < l.length) :
+    ∃ l₁ l₂, l = l₁ ++ l[i] :: l₂ ∧ l₁.length = i ∧ l.set i a' = l₁ ++ a' :: l₂ := by
+  refine ⟨l.take i, l.drop (i + 1), ⟨by simp, ⟨length_take_of_le (Nat.le_of_lt h), ?_⟩⟩⟩
+  simp [set_eq_take_append_cons_drop, h]
+
+end List
+
 namespace Array
 
-theorem exists_of_uset (self : Array α) (i d h) :
-    ∃ l₁ l₂, self.toList = l₁ ++ self[i] :: l₂ ∧ List.length l₁ = i.toNat ∧
-      (self.uset i d h).toList = l₁ ++ d :: l₂ := by
+theorem exists_of_uset (xs : Array α) (i d h) :
+    ∃ l₁ l₂, xs.toList = l₁ ++ xs[i] :: l₂ ∧ List.length l₁ = i.toNat ∧
+      (xs.uset i d h).toList = l₁ ++ d :: l₂ := by
   simpa only [ugetElem_eq_getElem, ← getElem_toList, uset, toList_set] using
     List.exists_of_set _
 

src/Init/Data/Array/Zip.lean

@@ -11,6 +11,9 @@ import Init.Data.List.Zip
 # Lemmas about `Array.zip`, `Array.zipWith`, `Array.zipWithAll`, and `Array.unzip`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace Array
 
 open Nat
@@ -19,20 +22,20 @@ open Nat
 
 /-! ### zipWith -/
 
-theorem zipWith_comm (f : α → β → γ) (la : Array α) (lb : Array β) :
-    zipWith f la lb = zipWith (fun b a => f a b) lb la := by
-  cases la
-  cases lb
+theorem zipWith_comm (f : α → β → γ) (as : Array α) (bs : Array β) :
+    zipWith f as bs = zipWith (fun b a => f a b) bs as := by
+  cases as
+  cases bs
   simpa using List.zipWith_comm _ _ _
 
-theorem zipWith_comm_of_comm (f : α → α → β) (comm : ∀ x y : α, f x y = f y x) (l l' : Array α) :
-    zipWith f l l' = zipWith f l' l := by
+theorem zipWith_comm_of_comm (f : α → α → β) (comm : ∀ x y : α, f x y = f y x) (xs ys : Array α) :
+    zipWith f xs ys = zipWith f ys xs := by
   rw [zipWith_comm]
   simp only [comm]
 
 @[simp]
-theorem zipWith_self (f : α → α → δ) (l : Array α) : zipWith f l l = l.map fun a => f a a := by
-  cases l
+theorem zipWith_self (f : α → α → δ) (xs : Array α) : zipWith f xs xs = xs.map fun a => f a a := by
+  cases xs
   simp
 
 /--
@@ -54,15 +57,15 @@ theorem getElem?_zipWith' {f : α → β → γ} {i : Nat} :
   cases l₂
   simp [List.getElem?_zipWith']
 
-theorem getElem?_zipWith_eq_some {f : α → β → γ} {l₁ : Array α} {l₂ : Array β} {z : γ} {i : Nat} :
-    (zipWith f l₁ l₂)[i]? = some z ↔
-      ∃ x y, l₁[i]? = some x ∧ l₂[i]? = some y ∧ f x y = z := by
-  cases l₁
-  cases l₂
+theorem getElem?_zipWith_eq_some {f : α → β → γ} {as : Array α} {bs : Array β} {z : γ} {i : Nat} :
+    (zipWith f as bs)[i]? = some z ↔
+      ∃ x y, as[i]? = some x ∧ bs[i]? = some y ∧ f x y = z := by
+  cases as
+  cases bs
   simp [List.getElem?_zipWith_eq_some]
 
-theorem getElem?_zip_eq_some {l₁ : Array α} {l₂ : Array β} {z : α × β} {i : Nat} :
-    (zip l₁ l₂)[i]? = some z ↔ l₁[i]? = some z.1 ∧ l₂[i]? = some z.2 := by
+theorem getElem?_zip_eq_some {as : Array α} {bs : Array β} {z : α × β} {i : Nat} :
+    (zip as bs)[i]? = some z ↔ as[i]? = some z.1 ∧ bs[i]? = some z.2 := by
   cases z
   rw [zip, getElem?_zipWith_eq_some]; constructor
   · rintro ⟨x, y, h₀, h₁, h₂⟩
@@ -71,211 +74,211 @@ theorem getElem?_zip_eq_some {l₁ : Array α} {l₂ : Array β} {z : α × β}
     exact ⟨_, _, h₀, h₁, rfl⟩
 
 @[simp]
-theorem zipWith_map {μ} (f : γ → δ → μ) (g : α → γ) (h : β → δ) (l₁ : Array α) (l₂ : Array β) :
-    zipWith f (l₁.map g) (l₂.map h) = zipWith (fun a b => f (g a) (h b)) l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zipWith_map {μ} (f : γ → δ → μ) (g : α → γ) (h : β → δ) (as : Array α) (bs : Array β) :
+    zipWith f (as.map g) (bs.map h) = zipWith (fun a b => f (g a) (h b)) as bs := by
+  cases as
+  cases bs
   simp [List.zipWith_map]
 
-theorem zipWith_map_left (l₁ : Array α) (l₂ : Array β) (f : α → α') (g : α' → β → γ) :
-    zipWith g (l₁.map f) l₂ = zipWith (fun a b => g (f a) b) l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zipWith_map_left (as : Array α) (bs : Array β) (f : α → α') (g : α' → β → γ) :
+    zipWith g (as.map f) bs = zipWith (fun a b => g (f a) b) as bs := by
+  cases as
+  cases bs
   simp [List.zipWith_map_left]
 
-theorem zipWith_map_right (l₁ : Array α) (l₂ : Array β) (f : β → β') (g : α → β' → γ) :
-    zipWith g l₁ (l₂.map f) = zipWith (fun a b => g a (f b)) l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zipWith_map_right (as : Array α) (bs : Array β) (f : β → β') (g : α → β' → γ) :
+    zipWith g as (bs.map f) = zipWith (fun a b => g a (f b)) as bs := by
+  cases as
+  cases bs
   simp [List.zipWith_map_right]
 
 theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} (i : δ):
-    (zipWith f l₁ l₂).foldr g i = (zip l₁ l₂).foldr (fun p r => g (f p.1 p.2) r) i := by
-  cases l₁
-  cases l₂
+    (zipWith f as bs).foldr g i = (zip as bs).foldr (fun p r => g (f p.1 p.2) r) i := by
+  cases as
+  cases bs
   simp [List.zipWith_foldr_eq_zip_foldr]
 
 theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} (i : δ):
-    (zipWith f l₁ l₂).foldl g i = (zip l₁ l₂).foldl (fun r p => g r (f p.1 p.2)) i := by
-  cases l₁
-  cases l₂
+    (zipWith f as bs).foldl g i = (zip as bs).foldl (fun r p => g r (f p.1 p.2)) i := by
+  cases as
+  cases bs
   simp [List.zipWith_foldl_eq_zip_foldl]
 
 @[simp]
-theorem zipWith_eq_empty_iff {f : α → β → γ} {l l'} : zipWith f l l' = #[] ↔ l = #[] ∨ l' = #[] := by
-  cases l <;> cases l' <;> simp
+theorem zipWith_eq_empty_iff {f : α → β → γ} {as : Array α} {bs : Array β} : zipWith f as bs = #[] ↔ as = #[] ∨ bs = #[] := by
+  cases as <;> cases bs <;> simp
 
-theorem map_zipWith {δ : Type _} (f : α → β) (g : γ → δ → α) (l : Array γ) (l' : Array δ) :
-    map f (zipWith g l l') = zipWith (fun x y => f (g x y)) l l' := by
-  cases l
-  cases l'
+theorem map_zipWith {δ : Type _} (f : α → β) (g : γ → δ → α) (cs : Array γ) (ds : Array δ) :
+    map f (zipWith g cs ds) = zipWith (fun x y => f (g x y)) cs ds := by
+  cases cs
+  cases ds
   simp [List.map_zipWith]
 
-theorem take_zipWith : (zipWith f l l').take n = zipWith f (l.take n) (l'.take n) := by
-  cases l
-  cases l'
+theorem take_zipWith : (zipWith f as bs).take i = zipWith f (as.take i) (bs.take i) := by
+  cases as
+  cases bs
   simp [List.take_zipWith]
 
-theorem extract_zipWith : (zipWith f l l').extract m n = zipWith f (l.extract m n) (l'.extract m n) := by
-  cases l
-  cases l'
+theorem extract_zipWith : (zipWith f as bs).extract i j = zipWith f (as.extract i j) (bs.extract i j) := by
+  cases as
+  cases bs
   simp [List.drop_zipWith, List.take_zipWith]
 
-theorem zipWith_append (f : α → β → γ) (l la : Array α) (l' lb : Array β)
-    (h : l.size = l'.size) :
-    zipWith f (l ++ la) (l' ++ lb) = zipWith f l l' ++ zipWith f la lb := by
-  cases l
-  cases l'
-  cases la
-  cases lb
+theorem zipWith_append (f : α → β → γ) (as as' : Array α) (bs bs' : Array β)
+    (h : as.size = bs.size) :
+    zipWith f (as ++ as') (bs ++ bs') = zipWith f as bs ++ zipWith f as' bs' := by
+  cases as
+  cases bs
+  cases as'
+  cases bs'
   simp at h
   simp [List.zipWith_append, h]
 
-theorem zipWith_eq_append_iff {f : α → β → γ} {l₁ : Array α} {l₂ : Array β} :
-    zipWith f l₁ l₂ = l₁' ++ l₂' ↔
-      ∃ w x y z, w.size = y.size ∧ l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zipWith f w y ∧ l₂' = zipWith f x z := by
-  cases l₁
-  cases l₂
-  cases l₁'
-  cases l₂'
+theorem zipWith_eq_append_iff {f : α → β → γ} {as : Array α} {bs : Array β} :
+    zipWith f as bs = xs ++ ys ↔
+      ∃ as₁ as₂ bs₁ bs₂, as₁.size = bs₁.size ∧ as = as₁ ++ as₂ ∧ bs = bs₁ ++ bs₂ ∧ xs = zipWith f as₁ bs₁ ∧ ys = zipWith f as₂ bs₂ := by
+  cases as
+  cases bs
+  cases xs
+  cases ys
   simp only [List.zipWith_toArray, List.append_toArray, mk.injEq, List.zipWith_eq_append_iff,
     toArray_eq_append_iff]
   constructor
-  · rintro ⟨w, x, y, z, h, rfl, rfl, rfl, rfl⟩
-    exact ⟨w.toArray, x.toArray, y.toArray, z.toArray, by simp [h]⟩
-  · rintro ⟨⟨w⟩, ⟨x⟩, ⟨y⟩, ⟨z⟩, h, rfl, rfl, h₁, h₂⟩
-    exact ⟨w, x, y, z, by simp_all⟩
+  · rintro ⟨ws, xs, ys, zs, h, rfl, rfl, rfl, rfl⟩
+    exact ⟨ws.toArray, xs.toArray, ys.toArray, zs.toArray, by simp [h]⟩
+  · rintro ⟨⟨ws⟩, ⟨xs⟩, ⟨ys⟩, ⟨zs⟩, h, rfl, rfl, h₁, h₂⟩
+    exact ⟨ws, xs, ys, zs, by simp_all⟩
 
 @[simp] theorem zipWith_mkArray {a : α} {b : β} {m n : Nat} :
     zipWith f (mkArray m a) (mkArray n b) = mkArray (min m n) (f a b) := by
   simp [← List.toArray_replicate]
 
-theorem map_uncurry_zip_eq_zipWith (f : α → β → γ) (l : Array α) (l' : Array β) :
-    map (Function.uncurry f) (l.zip l') = zipWith f l l' := by
-  cases l
-  cases l'
+theorem map_uncurry_zip_eq_zipWith (f : α → β → γ) (as : Array α) (bs : Array β) :
+    map (Function.uncurry f) (as.zip bs) = zipWith f as bs := by
+  cases as
+  cases bs
   simp [List.map_uncurry_zip_eq_zipWith]
 
-theorem map_zip_eq_zipWith (f : α × β → γ) (l : Array α) (l' : Array β) :
-    map f (l.zip l') = zipWith (Function.curry f) l l' := by
-  cases l
-  cases l'
+theorem map_zip_eq_zipWith (f : α × β → γ) (as : Array α) (bs : Array β) :
+    map f (as.zip bs) = zipWith (Function.curry f) as bs := by
+  cases as
+  cases bs
   simp [List.map_zip_eq_zipWith]
 
-theorem lt_size_left_of_zipWith {f : α → β → γ} {i : Nat} {l : Array α} {l' : Array β}
-    (h : i < (zipWith f l l').size) : i < l.size := by rw [size_zipWith] at h; omega
+theorem lt_size_left_of_zipWith {f : α → β → γ} {i : Nat} {as : Array α} {bs : Array β}
+    (h : i < (zipWith f as bs).size) : i < as.size := by rw [size_zipWith] at h; omega
 
-theorem lt_size_right_of_zipWith {f : α → β → γ} {i : Nat} {l : Array α} {l' : Array β}
-    (h : i < (zipWith f l l').size) : i < l'.size := by rw [size_zipWith] at h; omega
+theorem lt_size_right_of_zipWith {f : α → β → γ} {i : Nat} {as : Array α} {bs : Array β}
+    (h : i < (zipWith f as bs).size) : i < bs.size := by rw [size_zipWith] at h; omega
 
-theorem zipWith_eq_zipWith_take_min (l₁ : Array α) (l₂ : Array β) :
-    zipWith f l₁ l₂ = zipWith f (l₁.take (min l₁.size l₂.size)) (l₂.take (min l₁.size l₂.size)) := by
-  cases l₁
-  cases l₂
+theorem zipWith_eq_zipWith_take_min (as : Array α) (bs : Array β) :
+    zipWith f as bs = zipWith f (as.take (min as.size bs.size)) (bs.take (min as.size bs.size)) := by
+  cases as
+  cases bs
   simp
   rw [List.zipWith_eq_zipWith_take_min]
 
-theorem reverse_zipWith (h : l.size = l'.size) :
-    (zipWith f l l').reverse = zipWith f l.reverse l'.reverse := by
-  cases l
-  cases l'
+theorem reverse_zipWith (h : as.size = bs.size) :
+    (zipWith f as bs).reverse = zipWith f as.reverse bs.reverse := by
+  cases as
+  cases bs
   simp [List.reverse_zipWith (by simpa using h)]
 
 /-! ### zip -/
 
-theorem lt_size_left_of_zip {i : Nat} {l : Array α} {l' : Array β} (h : i < (zip l l').size) :
-    i < l.size :=
+theorem lt_size_left_of_zip {i : Nat} {as : Array α} {bs : Array β} (h : i < (zip as bs).size) :
+    i < as.size :=
   lt_size_left_of_zipWith h
 
-theorem lt_size_right_of_zip {i : Nat} {l : Array α} {l' : Array β} (h : i < (zip l l').size) :
-    i < l'.size :=
+theorem lt_size_right_of_zip {i : Nat} {as : Array α} {bs : Array β} (h : i < (zip as bs).size) :
+    i < bs.size :=
   lt_size_right_of_zipWith h
 
 @[simp]
-theorem getElem_zip {l : Array α} {l' : Array β} {i : Nat} {h : i < (zip l l').size} :
-    (zip l l')[i] =
-      (l[i]'(lt_size_left_of_zip h), l'[i]'(lt_size_right_of_zip h)) :=
+theorem getElem_zip {as : Array α} {bs : Array β} {i : Nat} {h : i < (zip as bs).size} :
+    (zip as bs)[i] =
+      (as[i]'(lt_size_left_of_zip h), bs[i]'(lt_size_right_of_zip h)) :=
   getElem_zipWith (hi := by simpa using h)
 
-theorem zip_eq_zipWith (l₁ : Array α) (l₂ : Array β) : zip l₁ l₂ = zipWith Prod.mk l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zip_eq_zipWith (as : Array α) (bs : Array β) : zip as bs = zipWith Prod.mk as bs := by
+  cases as
+  cases bs
   simp [List.zip_eq_zipWith]
 
-theorem zip_map (f : α → γ) (g : β → δ) (l₁ : Array α) (l₂ : Array β) :
-    zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g) := by
-  cases l₁
-  cases l₂
+theorem zip_map (f : α → γ) (g : β → δ) (as : Array α) (bs : Array β) :
+    zip (as.map f) (bs.map g) = (zip as bs).map (Prod.map f g) := by
+  cases as
+  cases bs
   simp [List.zip_map]
 
-theorem zip_map_left (f : α → γ) (l₁ : Array α) (l₂ : Array β) :
-    zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]
+theorem zip_map_left (f : α → γ) (as : Array α) (bs : Array β) :
+    zip (as.map f) bs = (zip as bs).map (Prod.map f id) := by rw [← zip_map, map_id]
 
-theorem zip_map_right (f : β → γ) (l₁ : Array α) (l₂ : Array β) :
-    zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]
+theorem zip_map_right (f : β → γ) (as : Array α) (bs : Array β) :
+    zip as (bs.map f) = (zip as bs).map (Prod.map id f) := by rw [← zip_map, map_id]
 
-theorem zip_append {l₁ r₁ : Array α} {l₂ r₂ : Array β} (_h : l₁.size = l₂.size) :
-    zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂ := by
-  cases l₁
-  cases l₂
-  cases r₁
-  cases r₂
+theorem zip_append {as bs : Array α} {cs ds : Array β} (_h : as.size = cs.size) :
+    zip (as ++ bs) (cs ++ ds) = zip as cs ++ zip bs ds := by
+  cases as
+  cases cs
+  cases bs
+  cases ds
   simp_all [List.zip_append]
 
-theorem zip_map' (f : α → β) (g : α → γ) (l : Array α) :
-    zip (l.map f) (l.map g) = l.map fun a => (f a, g a) := by
-  cases l
+theorem zip_map' (f : α → β) (g : α → γ) (xs : Array α) :
+    zip (xs.map f) (xs.map g) = xs.map fun a => (f a, g a) := by
+  cases xs
   simp [List.zip_map']
 
-theorem of_mem_zip {a b} {l₁ : Array α} {l₂ : Array β} : (a, b) ∈ zip l₁ l₂ → a ∈ l₁ ∧ b ∈ l₂ := by
-  cases l₁
-  cases l₂
+theorem of_mem_zip {a b} {as : Array α} {bs : Array β} : (a, b) ∈ zip as bs → a ∈ as ∧ b ∈ bs := by
+  cases as
+  cases bs
   simpa using List.of_mem_zip
 
-theorem map_fst_zip (l₁ : Array α) (l₂ : Array β) (h : l₁.size ≤ l₂.size) :
-    map Prod.fst (zip l₁ l₂) = l₁ := by
-  cases l₁
-  cases l₂
+theorem map_fst_zip (as : Array α) (bs : Array β) (h : as.size ≤ bs.size) :
+    map Prod.fst (zip as bs) = as := by
+  cases as
+  cases bs
   simp_all [List.map_fst_zip]
 
-theorem map_snd_zip (l₁ : Array α) (l₂ : Array β) (h : l₂.size ≤ l₁.size) :
-    map Prod.snd (zip l₁ l₂) = l₂ := by
-  cases l₁
-  cases l₂
+theorem map_snd_zip (as : Array α) (bs : Array β) (h : bs.size ≤ as.size) :
+    map Prod.snd (zip as bs) = bs := by
+  cases as
+  cases bs
   simp_all [List.map_snd_zip]
 
-theorem map_prod_left_eq_zip {l : Array α} (f : α → β) :
-    (l.map fun x => (x, f x)) = l.zip (l.map f) := by
+theorem map_prod_left_eq_zip {xs : Array α} (f : α → β) :
+    (xs.map fun x => (x, f x)) = xs.zip (xs.map f) := by
   rw [← zip_map']
   congr
   simp
 
-theorem map_prod_right_eq_zip {l : Array α} (f : α → β) :
-    (l.map fun x => (f x, x)) = (l.map f).zip l := by
+theorem map_prod_right_eq_zip {xs : Array α} (f : α → β) :
+    (xs.map fun x => (f x, x)) = (xs.map f).zip xs := by
   rw [← zip_map']
   congr
   simp
 
-@[simp] theorem zip_eq_empty_iff {l₁ : Array α} {l₂ : Array β} :
-    zip l₁ l₂ = #[] ↔ l₁ = #[] ∨ l₂ = #[] := by
-  cases l₁
-  cases l₂
+@[simp] theorem zip_eq_empty_iff {as : Array α} {bs : Array β} :
+    zip as bs = #[] ↔ as = #[] ∨ bs = #[] := by
+  cases as
+  cases bs
   simp [List.zip_eq_nil_iff]
 
-theorem zip_eq_append_iff {l₁ : Array α} {l₂ : Array β} :
-    zip l₁ l₂ = l₁' ++ l₂' ↔
-      ∃ w x y z, w.size = y.size ∧ l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zip w y ∧ l₂' = zip x z := by
+theorem zip_eq_append_iff {as : Array α} {bs : Array β} :
+    zip as bs = xs ++ ys ↔
+      ∃ as₁ as₂ bs₁ bs₂, as₁.size = bs₁.size ∧ as = as₁ ++ as₂ ∧ bs = bs₁ ++ bs₂ ∧ xs = zip as₁ bs₁ ∧ ys = zip as₂ bs₂ := by
   simp [zip_eq_zipWith, zipWith_eq_append_iff]
 
 @[simp] theorem zip_mkArray {a : α} {b : β} {m n : Nat} :
     zip (mkArray m a) (mkArray n b) = mkArray (min m n) (a, b) := by
   simp [← List.toArray_replicate]
 
-theorem zip_eq_zip_take_min (l₁ : Array α) (l₂ : Array β) :
-    zip l₁ l₂ = zip (l₁.take (min l₁.size l₂.size)) (l₂.take (min l₁.size l₂.size)) := by
-  cases l₁
-  cases l₂
-  simp only [List.zip_toArray, size_toArray, List.take_toArray, mk.injEq]
+theorem zip_eq_zip_take_min (as : Array α) (bs : Array β) :
+    zip as bs = zip (as.take (min as.size bs.size)) (bs.take (min as.size bs.size)) := by
+  cases as
+  cases bs
+  simp only [List.zip_toArray, List.size_toArray, List.take_toArray, mk.injEq]
   rw [List.zip_eq_zip_take_min]
 
 
@@ -289,31 +292,30 @@ theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
   simp [List.getElem?_zipWithAll]
   rfl
 
-theorem zipWithAll_map {μ} (f : Option γ → Option δ → μ) (g : α → γ) (h : β → δ) (l₁ : Array α) (l₂ : Array β) :
-    zipWithAll f (l₁.map g) (l₂.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zipWithAll_map {μ} (f : Option γ → Option δ → μ) (g : α → γ) (h : β → δ) (as : Array α) (bs : Array β) :
+    zipWithAll f (as.map g) (bs.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) as bs := by
+  cases as
+  cases bs
   simp [List.zipWithAll_map]
 
-theorem zipWithAll_map_left (l₁ : Array α) (l₂ : Array β) (f : α → α') (g : Option α' → Option β → γ) :
-    zipWithAll g (l₁.map f) l₂ = zipWithAll (fun a b => g (f <$> a) b) l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zipWithAll_map_left (as : Array α) (bs : Array β) (f : α → α') (g : Option α' → Option β → γ) :
+    zipWithAll g (as.map f) bs = zipWithAll (fun a b => g (f <$> a) b) as bs := by
+  cases as
+  cases bs
   simp [List.zipWithAll_map_left]
 
-theorem zipWithAll_map_right (l₁ : Array α) (l₂ : Array β) (f : β → β') (g : Option α → Option β' → γ) :
-    zipWithAll g l₁ (l₂.map f) = zipWithAll (fun a b => g a (f <$> b)) l₁ l₂ := by
-  cases l₁
-  cases l₂
+theorem zipWithAll_map_right (as : Array α) (bs : Array β) (f : β → β') (g : Option α → Option β' → γ) :
+    zipWithAll g as (bs.map f) = zipWithAll (fun a b => g a (f <$> b)) as bs := by
+  cases as
+  cases bs
   simp [List.zipWithAll_map_right]
 
-theorem map_zipWithAll {δ : Type _} (f : α → β) (g : Option γ → Option δ → α) (l : Array γ) (l' : Array δ) :
-    map f (zipWithAll g l l') = zipWithAll (fun x y => f (g x y)) l l' := by
-  cases l
-  cases l'
+theorem map_zipWithAll {δ : Type _} (f : α → β) (g : Option γ → Option δ → α) (cs : Array γ) (ds : Array δ) :
+    map f (zipWithAll g cs ds) = zipWithAll (fun x y => f (g x y)) cs ds := by
+  cases cs
+  cases ds
   simp [List.map_zipWithAll]
 
-
 @[simp] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
     zipWithAll f (mkArray n a) (mkArray n b) = mkArray n (f a b) := by
   simp [← List.toArray_replicate]
@@ -326,37 +328,37 @@ theorem map_zipWithAll {δ : Type _} (f : α → β) (g : Option γ → Option
 @[simp] theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
   induction l <;> simp_all
 
-theorem unzip_eq_map (l : Array (α × β)) : unzip l = (l.map Prod.fst, l.map Prod.snd) := by
-  cases l
+theorem unzip_eq_map (xs : Array (α × β)) : unzip xs = (xs.map Prod.fst, xs.map Prod.snd) := by
+  cases xs
   simp [List.unzip_eq_map]
 
-theorem zip_unzip (l : Array (α × β)) : zip (unzip l).1 (unzip l).2 = l := by
-  cases l
+theorem zip_unzip (xs : Array (α × β)) : zip (unzip xs).1 (unzip xs).2 = xs := by
+  cases xs
   simp only [List.unzip_toArray, Prod.map_fst, Prod.map_snd, List.zip_toArray, List.zip_unzip]
 
-theorem unzip_zip_left {l₁ : Array α} {l₂ : Array β} (h : l₁.size ≤ l₂.size) :
-    (unzip (zip l₁ l₂)).1 = l₁ := by
-  cases l₁
-  cases l₂
-  simp_all only [size_toArray, List.zip_toArray, List.unzip_toArray, Prod.map_fst,
+theorem unzip_zip_left {as : Array α} {bs : Array β} (h : as.size ≤ bs.size) :
+    (unzip (zip as bs)).1 = as := by
+  cases as
+  cases bs
+  simp_all only [List.size_toArray, List.zip_toArray, List.unzip_toArray, Prod.map_fst,
     List.unzip_zip_left]
 
-theorem unzip_zip_right {l₁ : Array α} {l₂ : Array β} (h : l₂.size ≤ l₁.size) :
-    (unzip (zip l₁ l₂)).2 = l₂ := by
-  cases l₁
-  cases l₂
-  simp_all only [size_toArray, List.zip_toArray, List.unzip_toArray, Prod.map_snd,
+theorem unzip_zip_right {as : Array α} {bs : Array β} (h : bs.size ≤ as.size) :
+    (unzip (zip as bs)).2 = bs := by
+  cases as
+  cases bs
+  simp_all only [List.size_toArray, List.zip_toArray, List.unzip_toArray, Prod.map_snd,
     List.unzip_zip_right]
 
-theorem unzip_zip {l₁ : Array α} {l₂ : Array β} (h : l₁.size = l₂.size) :
-    unzip (zip l₁ l₂) = (l₁, l₂) := by
-  cases l₁
-  cases l₂
-  simp_all only [size_toArray, List.zip_toArray, List.unzip_toArray, List.unzip_zip, Prod.map_apply]
+theorem unzip_zip {as : Array α} {bs : Array β} (h : as.size = bs.size) :
+    unzip (zip as bs) = (as, bs) := by
+  cases as
+  cases bs
+  simp_all only [List.size_toArray, List.zip_toArray, List.unzip_toArray, List.unzip_zip, Prod.map_apply]
 
-theorem zip_of_prod {l : Array α} {l' : Array β} {lp : Array (α × β)} (hl : lp.map Prod.fst = l)
-    (hr : lp.map Prod.snd = l') : lp = l.zip l' := by
-  rw [← hl, ← hr, ← zip_unzip lp, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]
+theorem zip_of_prod {as : Array α} {bs : Array β} {xs : Array (α × β)} (hl : xs.map Prod.fst = as)
+    (hr : xs.map Prod.snd = bs) : xs = as.zip bs := by
+  rw [← hl, ← hr, ← zip_unzip xs, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]
 
 @[simp] theorem unzip_mkArray {n : Nat} {a : α} {b : β} :
     unzip (mkArray n (a, b)) = (mkArray n a, mkArray n b) := by

src/Init/Data/BEq.lean

@@ -25,7 +25,7 @@ class ReflBEq (α) [BEq α] : Prop where
   refl : (a : α) == a
 
 /-- `EquivBEq` says that the `BEq` implementation is an equivalence relation. -/
-class EquivBEq (α) [BEq α] extends PartialEquivBEq α, ReflBEq α : Prop
+class EquivBEq (α) [BEq α] : Prop extends PartialEquivBEq α, ReflBEq α
 
 @[simp]
 theorem BEq.refl [BEq α] [ReflBEq α] {a : α} : a == a :=
@@ -49,6 +49,14 @@ theorem BEq.symm_false [BEq α] [PartialEquivBEq α] {a b : α} : (a == b) = fal
 theorem BEq.trans [BEq α] [PartialEquivBEq α] {a b c : α} : a == b → b == c → a == c :=
   PartialEquivBEq.trans
 
+theorem BEq.congr_left [BEq α] [PartialEquivBEq α] {a b c : α} (h : a == b) :
+    (a == c) = (b == c) :=
+  Bool.eq_iff_iff.mpr ⟨BEq.trans (BEq.symm h), BEq.trans h⟩
+
+theorem BEq.congr_right [BEq α] [PartialEquivBEq α] {a b c : α} (h : b == c) :
+    (a == b) = (a == c) :=
+  Bool.eq_iff_iff.mpr ⟨fun h' => BEq.trans h' h, fun h' => BEq.trans h' (BEq.symm h)⟩
+
 theorem BEq.neq_of_neq_of_beq [BEq α] [PartialEquivBEq α] {a b c : α} :
     (a == b) = false → b == c → (a == c) = false :=
   fun h₁ h₂ => Bool.eq_false_iff.2 fun h₃ => Bool.eq_false_iff.1 h₁ (BEq.trans h₃ (BEq.symm h₂))

src/Init/Data/BitVec/Basic.lean

@@ -25,13 +25,17 @@ set_option linter.missingDocs true
 
 namespace BitVec
 
+@[inline, deprecated BitVec.ofNatLT (since := "2025-02-13"), inherit_doc BitVec.ofNatLT]
+protected def ofNatLt {n : Nat} (i : Nat) (p : i < 2 ^ n) : BitVec n :=
+  BitVec.ofNatLT i p
+
 section Nat
 
 instance natCastInst : NatCast (BitVec w) := ⟨BitVec.ofNat w⟩
 
 /-- Theorem for normalizing the bit vector literal representation. -/
 -- TODO: This needs more usage data to assess which direction the simp should go.
-@[simp, bv_toNat] theorem ofNat_eq_ofNat : @OfNat.ofNat (BitVec n) i _ = .ofNat n i := rfl
+@[simp, bitvec_to_nat] theorem ofNat_eq_ofNat : @OfNat.ofNat (BitVec n) i _ = .ofNat n i := rfl
 
 -- Note. Mathlib would like this to go the other direction.
 @[simp] theorem natCast_eq_ofNat (w x : Nat) : @Nat.cast (BitVec w) _ x = .ofNat w x := rfl
@@ -55,12 +59,12 @@ end subsingleton
 section zero_allOnes
 
 /-- Return a bitvector `0` of size `n`. This is the bitvector with all zero bits. -/
-protected def zero (n : Nat) : BitVec n := .ofNatLt 0 (Nat.two_pow_pos n)
+protected def zero (n : Nat) : BitVec n := .ofNatLT 0 (Nat.two_pow_pos n)
 instance : Inhabited (BitVec n) where default := .zero n
 
 /-- Bit vector of size `n` where all bits are `1`s -/
 def allOnes (n : Nat) : BitVec n :=
-  .ofNatLt (2^n - 1) (Nat.le_of_eq (Nat.sub_add_cancel (Nat.two_pow_pos n)))
+  .ofNatLT (2^n - 1) (Nat.le_of_eq (Nat.sub_add_cancel (Nat.two_pow_pos n)))
 
 end zero_allOnes
 
@@ -123,6 +127,7 @@ instance : GetElem (BitVec w) Nat Bool fun _ i => i < w where
 theorem getElem_eq_testBit_toNat (x : BitVec w) (i : Nat) (h : i < w) :
   x[i] = x.toNat.testBit i := rfl
 
+@[simp]
 theorem getLsbD_eq_getElem {x : BitVec w} {i : Nat} (h : i < w) :
     x.getLsbD i = x[i] := rfl
 
@@ -138,7 +143,7 @@ protected def toInt (x : BitVec n) : Int :=
     (x.toNat : Int) - (2^n : Nat)
 
 /-- The `BitVec` with value `(2^n + (i mod 2^n)) mod 2^n`.  -/
-protected def ofInt (n : Nat) (i : Int) : BitVec n := .ofNatLt (i % (Int.ofNat (2^n))).toNat (by
+protected def ofInt (n : Nat) (i : Int) : BitVec n := .ofNatLT (i % (Int.ofNat (2^n))).toNat (by
   apply (Int.toNat_lt _).mpr
   · apply Int.emod_lt_of_pos
     exact Int.ofNat_pos.mpr (Nat.two_pow_pos _)
@@ -167,12 +172,12 @@ recommended_spelling "one" for "1#n" in [BitVec.ofNat, «term__#__»]
   | `($(_) $n $i:num) => `($i:num#$n)
   | _ => throw ()
 
-/-- Notation for bit vector literals without truncation. `i#'lt` is a shorthand for `BitVec.ofNatLt i lt`. -/
+/-- Notation for bit vector literals without truncation. `i#'lt` is a shorthand for `BitVec.ofNatLT i lt`. -/
 scoped syntax:max term:max noWs "#'" noWs term:max : term
-macro_rules | `($i#'$p) => `(BitVec.ofNatLt $i $p)
+macro_rules | `($i#'$p) => `(BitVec.ofNatLT $i $p)
 
 /-- Unexpander for bit vector literals without truncation. -/
-@[app_unexpander BitVec.ofNatLt] def unexpandBitVecOfNatLt : Lean.PrettyPrinter.Unexpander
+@[app_unexpander BitVec.ofNatLT] def unexpandBitVecOfNatLt : Lean.PrettyPrinter.Unexpander
   | `($(_) $i $p) => `($i#'$p)
   | _ => throw ()
 
@@ -356,7 +361,7 @@ end relations
 section cast
 
 /-- `cast eq x` embeds `x` into an equal `BitVec` type. -/
-@[inline] protected def cast (eq : n = m) (x : BitVec n) : BitVec m := .ofNatLt x.toNat (eq ▸ x.isLt)
+@[inline] protected def cast (eq : n = m) (x : BitVec n) : BitVec m := .ofNatLT x.toNat (eq ▸ x.isLt)
 
 @[simp] theorem cast_ofNat {n m : Nat} (h : n = m) (x : Nat) :
     (BitVec.ofNat n x).cast h = BitVec.ofNat m x := by
@@ -390,7 +395,7 @@ and is a computational noop.
 def setWidth' {n w : Nat} (le : n ≤ w) (x : BitVec n) : BitVec w :=
   x.toNat#'(by
     apply Nat.lt_of_lt_of_le x.isLt
-    exact Nat.pow_le_pow_of_le_right (by trivial) le)
+    exact Nat.pow_le_pow_right (by trivial) le)
 
 @[deprecated setWidth' (since := "2024-09-18"), inherit_doc setWidth'] abbrev zeroExtend' := @setWidth'
 
@@ -675,6 +680,22 @@ def ofBoolListLE : (bs : List Bool) → BitVec bs.length
 | [] => 0#0
 | b :: bs => concat (ofBoolListLE bs) b
 
+/-! ## Overflow -/
+
+/-- `uaddOverflow x y` returns `true` if addition of `x` and `y` results in *unsigned* overflow.
+
+  SMT-Lib name: `bvuaddo`.
+-/
+def uaddOverflow {w : Nat} (x y : BitVec w) : Bool := x.toNat + y.toNat ≥ 2 ^ w
+
+/-- `saddOverflow x y` returns `true` if addition of `x` and `y` results in *signed* overflow,
+treating `x` and `y` as 2's complement signed bitvectors.
+
+  SMT-Lib name: `bvsaddo`.
+-/
+def saddOverflow {w : Nat} (x y : BitVec w) : Bool :=
+  (x.toInt + y.toInt ≥ 2 ^ (w - 1)) || (x.toInt + y.toInt < - 2 ^ (w - 1))
+
 /- ### reverse -/
 
 /-- Reverse the bits in a bitvector. -/

src/Init/Data/BitVec/Bitblast.lean

@@ -6,6 +6,7 @@ Authors: Harun Khan, Abdalrhman M Mohamed, Joe Hendrix, Siddharth Bhat
 prelude
 import Init.Data.BitVec.Folds
 import Init.Data.Nat.Mod
+import Init.Data.Int.LemmasAux
 
 /-!
 # Bitblasting of bitvectors
@@ -108,7 +109,12 @@ open Nat Bool
 
 namespace Bool
 
-/-- At least two out of three booleans are true. -/
+/--
+At least two out of three Booleans are true.
+
+This function is typically used to model addition of binary numbers, to combine a carry bit with two
+addend bits.
+-/
 abbrev atLeastTwo (a b c : Bool) : Bool := a && b || a && c || b && c
 
 @[simp] theorem atLeastTwo_false_left  : atLeastTwo false b c = (b && c) := by simp [atLeastTwo]
@@ -143,7 +149,7 @@ private theorem testBit_limit {x i : Nat} (x_lt_succ : x < 2^(i+1)) :
         exfalso
         apply Nat.lt_irrefl
         calc x < 2^(i+1) := x_lt_succ
-             _ ≤ 2 ^ j := Nat.pow_le_pow_of_le_right Nat.zero_lt_two x_lt
+             _ ≤ 2 ^ j := Nat.pow_le_pow_right Nat.zero_lt_two x_lt
              _ ≤ x := testBit_implies_ge jp
 
 private theorem mod_two_pow_succ (x i : Nat) :
@@ -284,7 +290,7 @@ theorem adc_spec (x y : BitVec w) (c : Bool) :
     simp [carry, Nat.mod_one]
     cases c <;> rfl
   case step =>
-    simp [adcb, Prod.mk.injEq, carry_succ, getLsbD_add_add_bool]
+    simp [adcb, Prod.mk.injEq, carry_succ, getElem_add_add_bool]
 
 theorem add_eq_adc (w : Nat) (x y : BitVec w) : x + y = (adc x y false).snd := by
   simp [adc_spec]
@@ -294,7 +300,7 @@ theorem add_eq_adc (w : Nat) (x y : BitVec w) : x + y = (adc x y false).snd := b
 theorem getMsbD_add {i : Nat} {i_lt : i < w} {x y : BitVec w} :
     getMsbD (x + y) i =
       Bool.xor (getMsbD x i) (Bool.xor (getMsbD y i) (carry (w - 1 - i) x y false)) := by
-  simp [getMsbD, getLsbD_add, i_lt, show w - 1 - i < w by omega]
+  simp [getMsbD, getElem_add, i_lt, show w - 1 - i < w by omega]
 
 theorem msb_add {w : Nat} {x y: BitVec w} :
     (x + y).msb =
@@ -358,24 +364,25 @@ theorem msb_sub {x y: BitVec w} :
 /-! ### Negation -/
 
 theorem bit_not_testBit (x : BitVec w) (i : Fin w) :
-  getLsbD (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd) i.val = !(getLsbD x i.val) := by
+  (((iunfoldr (fun (i : Fin w) c => (c, !(x[i.val])))) ()).snd)[i.val] = !(getLsbD x i.val) := by
   apply iunfoldr_getLsbD (fun _ => ()) i (by simp)
 
 theorem bit_not_add_self (x : BitVec w) :
-  ((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd + x  = -1 := by
+  ((iunfoldr (fun (i : Fin w) c => (c, !(x[i.val])))) ()).snd + x  = -1 := by
   simp only [add_eq_adc]
   apply iunfoldr_replace_snd (fun _ => false) (-1) false rfl
-  intro i; simp only [ BitVec.not, adcb, testBit_toNat]
-  rw [iunfoldr_replace_snd (fun _ => ()) (((iunfoldr (fun i c => (c, !(x.getLsbD i)))) ()).snd)]
-  <;> simp [bit_not_testBit, negOne_eq_allOnes, getLsbD_allOnes]
+  intro i; simp only [adcb, Fin.is_lt, getLsbD_eq_getElem, atLeastTwo_false_right, bne_false,
+    ofNat_eq_ofNat, Fin.getElem_fin, Prod.mk.injEq, and_eq_false_imp]
+  rw [iunfoldr_replace_snd (fun _ => ()) (((iunfoldr (fun i c => (c, !(x[i.val])))) ()).snd)]
+  <;> simp [bit_not_testBit, negOne_eq_allOnes, getElem_allOnes]
 
 theorem bit_not_eq_not (x : BitVec w) :
-  ((iunfoldr (fun i c => (c, !(x.getLsbD i)))) ()).snd = ~~~ x := by
+  ((iunfoldr (fun i c => (c, !(x[i])))) ()).snd = ~~~ x := by
   simp [←allOnes_sub_eq_not, BitVec.eq_sub_iff_add_eq.mpr (bit_not_add_self x), ←negOne_eq_allOnes]
 
-theorem bit_neg_eq_neg (x : BitVec w) : -x = (adc (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd) (BitVec.ofNat w 1) false).snd:= by
+theorem bit_neg_eq_neg (x : BitVec w) : -x = (adc (((iunfoldr (fun (i : Fin w) c => (c, !(x[i.val])))) ()).snd) (BitVec.ofNat w 1) false).snd:= by
   simp only [← add_eq_adc]
-  rw [iunfoldr_replace_snd ((fun _ => ())) (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd) _ rfl]
+  rw [iunfoldr_replace_snd ((fun _ => ())) (((iunfoldr (fun (i : Fin w) c => (c, !(x[i.val])))) ()).snd) _ rfl]
   · rw [BitVec.eq_sub_iff_add_eq.mpr (bit_not_add_self x), sub_toAdd, BitVec.add_comm _ (-x)]
     simp [← sub_toAdd, BitVec.sub_add_cancel]
   · simp [bit_not_testBit x _]
@@ -476,6 +483,36 @@ theorem msb_neg {w : Nat} {x : BitVec w} :
         case zero => exact hmsb
         case succ => exact getMsbD_x _ hi (by omega)
 
+/-- This is false if `v < w` and `b = intMin`. See also `signExtend_neg_of_ne_intMin`. -/
+@[simp] theorem signExtend_neg_of_le {v w : Nat} (h : w ≤ v) (b : BitVec v) :
+    (-b).signExtend w = -b.signExtend w := by
+  apply BitVec.eq_of_getElem_eq
+  intro i hi
+  simp only [getElem_signExtend, getElem_neg]
+  rw [dif_pos (by omega), dif_pos (by omega)]
+  simp only [getLsbD_signExtend, Bool.and_eq_true, decide_eq_true_eq, Bool.ite_eq_true_distrib,
+    Bool.bne_right_inj, decide_eq_decide]
+  exact ⟨fun ⟨j, hj₁, hj₂⟩ => ⟨j, ⟨hj₁, ⟨by omega, by rwa [if_pos (by omega)]⟩⟩⟩,
+    fun ⟨j, hj₁, hj₂, hj₃⟩ => ⟨j, hj₁, by rwa [if_pos (by omega)] at hj₃⟩⟩
+
+/-- This is false if `v < w` and `b = intMin`. See also `signExtend_neg_of_le`. -/
+@[simp] theorem signExtend_neg_of_ne_intMin {v w : Nat} (b : BitVec v) (hb : b ≠ intMin v) :
+    (-b).signExtend w = -b.signExtend w := by
+  refine (by omega : w ≤ v ∨ v < w).elim (fun h => signExtend_neg_of_le h b) (fun h => ?_)
+  apply BitVec.eq_of_toInt_eq
+  rw [toInt_signExtend_of_le (by omega), toInt_neg_of_ne_intMin hb, toInt_neg_of_ne_intMin,
+    toInt_signExtend_of_le (by omega)]
+  apply ne_of_apply_ne BitVec.toInt
+  rw [toInt_signExtend_of_le (by omega), toInt_intMin_of_pos (by omega)]
+  have := b.le_two_mul_toInt
+  have : -2 ^ w < -2 ^ v := by
+    apply Int.neg_lt_neg
+    norm_cast
+    rwa [Nat.pow_lt_pow_iff_right (by omega)]
+  have : 2 * b.toInt ≠ -2 ^ w := by omega
+  rw [(show w = w - 1 + 1 by omega), Int.pow_succ] at this
+  omega
+
 /-! ### abs -/
 
 theorem msb_abs {w : Nat} {x : BitVec w} :
@@ -542,6 +579,15 @@ theorem slt_eq_not_carry (x y : BitVec w) :
 theorem sle_eq_not_slt (x y : BitVec w) : x.sle y = !y.slt x := by
   simp only [BitVec.sle, BitVec.slt, ← decide_not, decide_eq_decide]; omega
 
+theorem zero_sle_eq_not_msb {w : Nat} {x : BitVec w} : BitVec.sle 0#w x = !x.msb := by
+  rw [sle_eq_not_slt, BitVec.slt_zero_eq_msb]
+
+theorem zero_sle_iff_msb_eq_false {w : Nat} {x : BitVec w} : BitVec.sle 0#w x ↔ x.msb = false := by
+  simp [zero_sle_eq_not_msb]
+
+theorem toNat_toInt_of_sle {w : Nat} (b : BitVec w) (hb : BitVec.sle 0#w b) : b.toInt.toNat = b.toNat :=
+  toNat_toInt_of_msb b (zero_sle_iff_msb_eq_false.1 hb)
+
 theorem sle_eq_carry (x y : BitVec w) :
     x.sle y = !((x.msb == y.msb).xor (carry w y (~~~x) true)) := by
   rw [sle_eq_not_slt, slt_eq_not_carry, beq_comm]
@@ -574,16 +620,18 @@ theorem setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (x : BitVec w) (i
       setWidth w (x.setWidth i) + (x &&& twoPow w i) := by
   rw [add_eq_or_of_and_eq_zero]
   · ext k h
-    simp only [getLsbD_setWidth, h, decide_true, Bool.true_and, getLsbD_or, getLsbD_and]
+    simp only [getElem_setWidth, getLsbD_setWidth, h, getLsbD_eq_getElem, getElem_or, getElem_and,
+      getElem_twoPow]
     by_cases hik : i = k
     · subst hik
       simp [h]
-    · simp only [getLsbD_twoPow, hik, decide_false, Bool.and_false, Bool.or_false]
-      by_cases hik' : k < (i + 1)
+    · by_cases hik' : k < (i + 1)
       · have hik'' : k < i := by omega
         simp [hik', hik'']
+        omega
       · have hik'' : ¬ (k < i) := by omega
         simp [hik', hik'']
+        omega
   · ext k
     simp only [and_twoPow, getLsbD_and, getLsbD_setWidth, Fin.is_lt, decide_true, Bool.true_and,
       getLsbD_zero, and_eq_false_imp, and_eq_true, decide_eq_true_eq, and_imp]
@@ -903,7 +951,7 @@ The input to the shift subtractor is a legal input to `divrem`, and we also need
 input bit to perform shift subtraction on, and thus we need `0 < wn`.
 -/
 structure DivModState.Poised {w : Nat} (args : DivModArgs w) (qr : DivModState w)
-  extends DivModState.Lawful args qr : Type where
+  extends DivModState.Lawful args qr where
   /-- Only perform a round of shift-subtract if we have dividend bits. -/
   hwn_lt : 0 < qr.wn
 
@@ -1030,11 +1078,10 @@ theorem divRec_succ (m : Nat) (args : DivModArgs w) (qr : DivModState w) :
 theorem lawful_divRec {args : DivModArgs w} {qr : DivModState w}
     (h : DivModState.Lawful args qr) :
     DivModState.Lawful args (divRec qr.wn args qr) := by
-  generalize hm : qr.wn = m
-  induction m generalizing qr
-  case zero =>
+  induction hm : qr.wn generalizing qr with
+  | zero =>
     exact h
-  case succ wn' ih =>
+  | succ wn' ih =>
     simp only [divRec_succ]
     apply ih
     · apply lawful_divSubtractShift
@@ -1048,11 +1095,10 @@ theorem lawful_divRec {args : DivModArgs w} {qr : DivModState w}
 @[simp]
 theorem wn_divRec (args : DivModArgs w) (qr : DivModState w) :
     (divRec qr.wn args qr).wn = 0 := by
-  generalize hm : qr.wn = m
-  induction m generalizing qr
-  case zero =>
+  induction hm : qr.wn generalizing qr with
+  | zero =>
     assumption
-  case succ wn' ih =>
+  | succ wn' ih =>
     apply ih
     simp only [divSubtractShift, hm]
     split <;> rfl
@@ -1230,6 +1276,26 @@ theorem shiftRight_eq_ushiftRightRec (x : BitVec w₁) (y : BitVec w₂) :
   · simp [of_length_zero]
   · simp [ushiftRightRec_eq]
 
+/-! ### Overflow definitions -/
+
+/-- Unsigned addition overflows iff the final carry bit of the addition circuit is `true`. -/
+theorem uaddOverflow_eq {w : Nat} (x y : BitVec w) :
+    uaddOverflow x y = (x.setWidth (w + 1) + y.setWidth (w + 1)).msb := by
+  simp [uaddOverflow, msb_add, msb_setWidth, carry]
+
+theorem saddOverflow_eq {w : Nat} (x y : BitVec w) :
+    saddOverflow x y = (x.msb == y.msb && !((x + y).msb == x.msb)) := by
+  simp only [saddOverflow]
+  rcases w with _|w
+  · revert x y; decide
+  · have := le_two_mul_toInt (x := x); have := two_mul_toInt_lt (x := x)
+    have := le_two_mul_toInt (x := y); have := two_mul_toInt_lt (x := y)
+    simp only [← decide_or, msb_eq_toInt, decide_beq_decide, toInt_add, ← decide_not, ← decide_and,
+      decide_eq_decide]
+    rw_mod_cast [Int.bmod_neg_iff (by omega) (by omega)]
+    simp
+    omega
+
 /- ### umod -/
 
 theorem getElem_umod {n d : BitVec w} (hi : i < w) :
@@ -1259,4 +1325,17 @@ theorem getMsbD_umod {n d : BitVec w}:
     simp [BitVec.getMsbD_eq_getLsbD, hi]
   · simp [show w ≤ i by omega]
 
+
+/-! ### Mappings to and from BitVec -/
+
+theorem eq_iff_eq_of_inv (f : α → BitVec w) (g : BitVec w → α) (h : ∀ x, g (f x) = x) :
+    ∀ x y, x = y ↔ f x = f y := by
+  intro x y
+  constructor
+  · intro h'
+    rw [h']
+  · intro h'
+    have := congrArg g h'
+    simpa [h] using this
+
 end BitVec

src/Init/Data/BitVec/Folds.lean

@@ -101,14 +101,14 @@ Correctness theorem for `iunfoldr`.
 theorem iunfoldr_replace
     {f : Fin w → α → α × Bool} (state : Nat → α) (value : BitVec w) (a : α)
     (init : state 0 = a)
-    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value.getLsbD i.val)) :
+    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value[i.val])) :
     iunfoldr f a = (state w, value) := by
   simp [iunfoldr.eq_test state value a init step]
 
 theorem iunfoldr_replace_snd
   {f : Fin w → α → α × Bool} (state : Nat → α) (value : BitVec w) (a : α)
     (init : state 0 = a)
-    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value.getLsbD i.val)) :
+    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value[i.val])) :
     (iunfoldr f a).snd = value := by
   simp [iunfoldr.eq_test state value a init step]
 

src/Init/Data/Bool.lean

@@ -9,7 +9,19 @@ import Init.NotationExtra
 
 namespace Bool
 
-/-- Boolean exclusive or -/
+/--
+Boolean “exclusive or”. `xor x y` can be written `x ^^ y`.
+
+`x ^^ y` is `true` when precisely one of `x` or `y` is `true`. Unlike `and` and `or`, it does not
+have short-circuiting behavior, because one argument's value never determines the final value. Also
+unlike `and` and `or`, there is no commonly-used corresponding propositional connective.
+
+Examples:
+ * `false ^^ false = false`
+ * `true ^^ false = true`
+ * `false ^^ true = true`
+ * `true ^^ true = false`
+-/
 abbrev xor : Bool → Bool → Bool := bne
 
 @[inherit_doc] infixl:33 " ^^ " => xor
@@ -367,17 +379,19 @@ theorem and_or_inj_left_iff :
 
 /-! ## toNat -/
 
-/-- convert a `Bool` to a `Nat`, `false -> 0`, `true -> 1` -/
+/--
+Converts `true` to `1` and `false` to `0`.
+-/
 def toNat (b : Bool) : Nat := cond b 1 0
 
-@[simp, bv_toNat] theorem toNat_false : false.toNat = 0 := rfl
+@[simp, bitvec_to_nat] theorem toNat_false : false.toNat = 0 := rfl
 
-@[simp, bv_toNat] theorem toNat_true : true.toNat = 1 := rfl
+@[simp, bitvec_to_nat] theorem toNat_true : true.toNat = 1 := rfl
 
 theorem toNat_le (c : Bool) : c.toNat ≤ 1 := by
   cases c <;> trivial
 
-@[bv_toNat]
+@[bitvec_to_nat]
 theorem toNat_lt (b : Bool) : b.toNat < 2 :=
   Nat.lt_succ_of_le (toNat_le _)
 
@@ -388,7 +402,9 @@ theorem toNat_lt (b : Bool) : b.toNat < 2 :=
 
 /-! ## toInt -/
 
-/-- convert a `Bool` to an `Int`, `false -> 0`, `true -> 1` -/
+/--
+Converts `true` to `1` and `false` to `0`.
+-/
 def toInt (b : Bool) : Int := cond b 1 0
 
 @[simp] theorem toInt_false : false.toInt = 0 := rfl
@@ -539,8 +555,8 @@ theorem cond_decide {α} (p : Prop) [Decidable p] (t e : α) :
 @[simp] theorem cond_eq_false_distrib : ∀(c t f : Bool),
     (cond c t f = false) = ite (c = true) (t = false) (f = false) := by decide
 
-protected theorem cond_true  {α : Type u} {a b : α} : cond true  a b = a := cond_true  a b
-protected theorem cond_false {α : Type u} {a b : α} : cond false a b = b := cond_false a b
+protected theorem cond_true  {α : Sort u} {a b : α} : cond true  a b = a := cond_true  a b
+protected theorem cond_false {α : Sort u} {a b : α} : cond false a b = b := cond_false a b
 
 @[simp] theorem cond_true_left   : ∀(c f : Bool), cond c true f  = ( c || f) := by decide
 @[simp] theorem cond_false_left  : ∀(c f : Bool), cond c false f = (!c && f) := by decide
@@ -580,17 +596,13 @@ protected theorem decide_coe (b : Bool) [Decidable (b = true)] : decide (b = tru
     decide (p ↔ q) = (decide p == decide q) := by
   cases dp with | _ p => simp [p]
 
-@[boolToPropSimps]
-theorem and_eq_decide (p q : Prop) [dpq : Decidable (p ∧ q)] [dp : Decidable p] [dq : Decidable q] :
-    (p && q) = decide (p ∧ q) := by
-  cases dp with | _ p => simp [p]
+@[bool_to_prop]
+theorem and_eq_decide (p q : Bool) : (p && q) = decide (p ∧ q) := by simp
 
-@[boolToPropSimps]
-theorem or_eq_decide (p q : Prop) [dpq : Decidable (p ∨ q)] [dp : Decidable p] [dq : Decidable q] :
-    (p || q) = decide (p ∨ q) := by
-  cases dp with | _ p => simp [p]
+@[bool_to_prop]
+theorem or_eq_decide (p q : Bool) : (p || q) = decide (p ∨ q) := by simp
 
-@[boolToPropSimps]
+@[bool_to_prop]
 theorem decide_beq_decide (p q : Prop) [dpq : Decidable (p ↔ q)] [dp : Decidable p] [dq : Decidable q] :
     (decide p == decide q) = decide (p ↔ q) := by
   cases dp with | _ p => simp [p]

src/Init/Data/ByteArray/Basic.lean

@@ -47,7 +47,7 @@ def uget : (a : @& ByteArray) → (i : USize) → (h : i.toNat < a.size := by ge
 
 @[extern "lean_byte_array_get"]
 def get! : (@& ByteArray) → (@& Nat) → UInt8
-  | ⟨bs⟩, i => bs.get! i
+  | ⟨bs⟩, i => bs[i]!
 
 @[extern "lean_byte_array_fget"]
 def get : (a : @& ByteArray) → (i : @& Nat) → (h : i < a.size := by get_elem_tactic) → UInt8
@@ -56,7 +56,7 @@ def get : (a : @& ByteArray) → (i : @& Nat) → (h : i < a.size := by get_elem
 instance : GetElem ByteArray Nat UInt8 fun xs i => i < xs.size where
   getElem xs i h := xs.get i
 
-instance : GetElem ByteArray USize UInt8 fun xs i => i.val < xs.size where
+instance : GetElem ByteArray USize UInt8 fun xs i => i.toFin < xs.size where
   getElem xs i h := xs.uget i h
 
 @[extern "lean_byte_array_set"]

src/Init/Data/Char/Basic.lean

@@ -15,7 +15,15 @@ Note that values in `[0xd800, 0xdfff]` are reserved for [UTF-16 surrogate pairs]
 
 namespace Char
 
+/--
+One character is less than another if its code point is strictly less than the other's.
+-/
 protected def lt (a b : Char) : Prop := a.val < b.val
+
+/--
+One character is less than or equal to another if its code point is less than or equal to the
+other's.
+-/
 protected def le (a b : Char) : Prop := a.val ≤ b.val
 
 instance : LT Char := ⟨Char.lt⟩
@@ -27,7 +35,10 @@ instance (a b : Char) :  Decidable (a < b) :=
 instance (a b : Char) : Decidable (a ≤ b) :=
   UInt32.decLe _ _
 
-/-- Determines if the given nat is a valid [Unicode scalar value](https://www.unicode.org/glossary/#unicode_scalar_value).-/
+/--
+True for natural numbers that are valid [Unicode scalar
+values](https://www.unicode.org/glossary/#unicode_scalar_value).
+-/
 abbrev isValidCharNat (n : Nat) : Prop :=
   n < 0xd800 ∨ (0xdfff < n ∧ n < 0x110000)
 
@@ -40,65 +51,103 @@ theorem isValidUInt32 (n : Nat) (h : isValidCharNat n) : n < UInt32.size := by
     apply Nat.lt_trans h₂
     decide
 
-theorem isValidChar_of_isValidCharNat (n : Nat) (h : isValidCharNat n) : isValidChar (UInt32.ofNat' n (isValidUInt32 n h)) :=
+theorem isValidChar_of_isValidCharNat (n : Nat) (h : isValidCharNat n) : isValidChar (UInt32.ofNatLT n (isValidUInt32 n h)) :=
   match h with
   | Or.inl h =>
-    Or.inl (UInt32.ofNat'_lt_of_lt _ (by decide) h)
+    Or.inl (UInt32.ofNatLT_lt_of_lt _ (by decide) h)
   | Or.inr ⟨h₁, h₂⟩ =>
-    Or.inr ⟨UInt32.lt_ofNat'_of_lt _ (by decide) h₁, UInt32.ofNat'_lt_of_lt _ (by decide) h₂⟩
+    Or.inr ⟨UInt32.lt_ofNatLT_of_lt _ (by decide) h₁, UInt32.ofNatLT_lt_of_lt _ (by decide) h₂⟩
 
 theorem isValidChar_zero : isValidChar 0 :=
   Or.inl (by decide)
 
-/-- Underlying unicode code point as a `Nat`. -/
+/--
+The character's Unicode code point as a `Nat`.
+-/
 @[inline] def toNat (c : Char) : Nat :=
   c.val.toNat
 
-/-- Convert a character into a `UInt8`, by truncating (reducing modulo 256) if necessary. -/
+/--
+Converts a character into a `UInt8` that contains its code point.
+
+If the code point is larger than 255, it is truncated (reduced modulo 256).
+-/
 @[inline] def toUInt8 (c : Char) : UInt8 :=
   c.val.toUInt8
 
-/-- The numbers from 0 to 256 are all valid UTF-8 characters, so we can embed one in the other. -/
+/--
+Converts an 8-bit unsigned integer into a character.
+
+The integer's value is interpreted as a Unicode code point.
+-/
 def ofUInt8 (n : UInt8) : Char := ⟨n.toUInt32, .inl (Nat.lt_trans n.toBitVec.isLt (by decide))⟩
 
 instance : Inhabited Char where
   default := 'A'
 
-/-- Is the character a space (U+0020) a tab (U+0009), a carriage return (U+000D) or a newline (U+000A)? -/
+/--
+Returns `true` if the character is a space `(' ', U+0020)`, a tab `('\t', U+0009)`, a carriage
+return `('\r', U+000D)`, or a newline `('\n', U+000A)`.
+-/
 @[inline] def isWhitespace (c : Char) : Bool :=
   c = ' ' || c = '\t' || c = '\r' || c = '\n'
 
-/-- Is the character in `ABCDEFGHIJKLMNOPQRSTUVWXYZ`? -/
+/--
+Returns `true` if the character is a uppercase ASCII letter.
+
+The uppercase ASCII letters are the following: `ABCDEFGHIJKLMNOPQRSTUVWXYZ`.
+-/
 @[inline] def isUpper (c : Char) : Bool :=
   c.val ≥ 65 && c.val ≤ 90
 
-/-- Is the character in `abcdefghijklmnopqrstuvwxyz`? -/
+/--
+Returns `true` if the character is a lowercase ASCII letter.
+
+The lowercase ASCII letters are the following: `abcdefghijklmnopqrstuvwxyz`.
+-/
 @[inline] def isLower (c : Char) : Bool :=
   c.val ≥ 97 && c.val ≤ 122
 
-/-- Is the character in `ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz`? -/
+/--
+Returns `true` if the character is an ASCII letter.
+
+The ASCII letters are the following: `ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz`.
+-/
 @[inline] def isAlpha (c : Char) : Bool :=
   c.isUpper || c.isLower
 
-/-- Is the character in `0123456789`? -/
+/--
+Returns `true` if the character is an ASCII digit.
+
+The ASCII digits are the following: `0123456789`.
+-/
 @[inline] def isDigit (c : Char) : Bool :=
   c.val ≥ 48 && c.val ≤ 57
 
-/-- Is the character in `ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789`? -/
+/--
+Returns `true` if the character is an ASCII letter or digit.
+
+The ASCII letters are the following: `ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz`.
+The ASCII digits are the following: `0123456789`.
+-/
 @[inline] def isAlphanum (c : Char) : Bool :=
   c.isAlpha || c.isDigit
 
-/-- Convert an upper case character to its lower case character.
+/--
+Converts an uppercase ASCII letter to the corresponding lowercase letter. Letters outside the ASCII
+alphabet are returned unchanged.
 
-Only works on basic latin letters.
+The uppercase ASCII letters are the following: `ABCDEFGHIJKLMNOPQRSTUVWXYZ`.
 -/
 def toLower (c : Char) : Char :=
   let n := toNat c;
   if n >= 65 ∧ n <= 90 then ofNat (n + 32) else c
 
-/-- Convert a lower case character to its upper case character.
+/--
+Converts a lowercase ASCII letter to the corresponding uppercase letter. Letters outside the ASCII
+alphabet are returned unchanged.
 
-Only works on basic latin letters.
+The lowercase ASCII letters are the following: `abcdefghijklmnopqrstuvwxyz`.
 -/
 def toUpper (c : Char) : Char :=
   let n := toNat c;

src/Init/Data/Fin/Basic.lean

@@ -51,6 +51,14 @@ Returns `a` modulo `n + 1` as a `Fin n.succ`.
 protected def ofNat {n : Nat} (a : Nat) : Fin (n + 1) :=
   ⟨a % (n+1), Nat.mod_lt _ (Nat.zero_lt_succ _)⟩
 
+-- We provide this because other similar types have a `toNat` function, but `simp` rewrites
+-- `i.toNat` to `i.val`.
+@[inline, inherit_doc val]
+protected def toNat (i : Fin n) : Nat :=
+  i.val
+
+@[simp] theorem toNat_eq_val {i : Fin n} : i.toNat = i.val := rfl
+
 private theorem mlt {b : Nat} : {a : Nat} → a < n → b % n < n
   | 0,   h => Nat.mod_lt _ h
   | _+1, h =>

src/Init/Data/Fin/Lemmas.lean

@@ -45,6 +45,7 @@ theorem val_ne_iff {a b : Fin n} : a.1 ≠ b.1 ↔ a ≠ b := not_congr val_inj
 theorem forall_iff {p : Fin n → Prop} : (∀ i, p i) ↔ ∀ i h, p ⟨i, h⟩ :=
   ⟨fun h i hi => h ⟨i, hi⟩, fun h ⟨i, hi⟩ => h i hi⟩
 
+/-- Restatement of `Fin.mk.injEq` as an `iff`. -/
 protected theorem mk.inj_iff {n a b : Nat} {ha : a < n} {hb : b < n} :
     (⟨a, ha⟩ : Fin n) = ⟨b, hb⟩ ↔ a = b := Fin.ext_iff
 
@@ -55,6 +56,14 @@ theorem eq_mk_iff_val_eq {a : Fin n} {k : Nat} {hk : k < n} :
 
 theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
 
+@[simp] theorem mk_eq_zero {n a : Nat} {ha : a < n} [NeZero n] :
+    (⟨a, ha⟩ : Fin n) = 0 ↔ a = 0 :=
+  mk.inj_iff
+
+@[simp] theorem zero_eq_mk {n a : Nat} {ha : a < n} [NeZero n] :
+    0 = (⟨a, ha⟩ : Fin n) ↔ a = 0 := by
+  simp [eq_comm]
+
 @[simp] theorem val_ofNat' (n : Nat) [NeZero n] (a : Nat) :
   (Fin.ofNat' n a).val = a % n := rfl
 

src/Init/Data/Float.lean

@@ -134,7 +134,22 @@ Returns an undefined value if `x` is not finite.
 instance : ToString Float where
   toString := Float.toString
 
+/-- Obtains the `Float` whose value is the same as the given `UInt8`. -/
+@[extern "lean_uint8_to_float"] opaque UInt8.toFloat (n : UInt8) : Float
+/-- Obtains the `Float` whose value is the same as the given `UInt16`. -/
+@[extern "lean_uint16_to_float"] opaque UInt16.toFloat (n : UInt16) : Float
+/-- Obtains the `Float` whose value is the same as the given `UInt32`. -/
+@[extern "lean_uint32_to_float"] opaque UInt32.toFloat (n : UInt32) : Float
+/-- Obtains a `Float` whose value is near the given `UInt64`. It will be exactly the value of the
+given `UInt64` if such a `Float` exists. If no such `Float` exists, the returned value will either
+be the smallest `Float` this is larger than the given value, or the largest `Float` this is smaller
+than the given value. -/
 @[extern "lean_uint64_to_float"] opaque UInt64.toFloat (n : UInt64) : Float
+/-- Obtains a `Float` whose value is near the given `USize`. It will be exactly the value of the
+given `USize` if such a `Float` exists. If no such `Float` exists, the returned value will either
+be the smallest `Float` this is larger than the given value, or the largest `Float` this is smaller
+than the given value. -/
+@[extern "lean_usize_to_float"] opaque USize.toFloat (n : USize) : Float
 
 instance : Inhabited Float where
   default := UInt64.toFloat 0

src/Init/Data/Float32.lean

@@ -127,7 +127,25 @@ Returns an undefined value if `x` is not finite.
 instance : ToString Float32 where
   toString := Float32.toString
 
+/-- Obtains the `Float32` whose value is the same as the given `UInt8`. -/
+@[extern "lean_uint8_to_float32"] opaque UInt8.toFloat32 (n : UInt8) : Float32
+/-- Obtains the `Float32` whose value is the same as the given `UInt16`. -/
+@[extern "lean_uint16_to_float32"] opaque UInt16.toFloat32 (n : UInt16) : Float32
+/-- Obtains a `Float32` whose value is near the given `UInt32`. It will be exactly the value of the
+given `UInt32` if such a `Float32` exists. If no such `Float32` exists, the returned value will either
+be the smallest `Float32` this is larger than the given value, or the largest `Float32` this is smaller
+than the given value. -/
+@[extern "lean_uint32_to_float32"] opaque UInt32.toFloat32 (n : UInt32) : Float32
+/-- Obtains a `Float32` whose value is near the given `UInt64`. It will be exactly the value of the
+given `UInt64` if such a `Float32` exists. If no such `Float32` exists, the returned value will either
+be the smallest `Float32` this is larger than the given value, or the largest `Float32` this is smaller
+than the given value. -/
 @[extern "lean_uint64_to_float32"] opaque UInt64.toFloat32 (n : UInt64) : Float32
+/-- Obtains a `Float32` whose value is near the given `USize`. It will be exactly the value of the
+given `USize` if such a `Float32` exists. If no such `Float32` exists, the returned value will either
+be the smallest `Float32` this is larger than the given value, or the largest `Float32` this is smaller
+than the given value. -/
+@[extern "lean_usize_to_float32"] opaque USize.toFloat32 (n : USize) : Float32
 
 instance : Inhabited Float32 where
   default := UInt64.toFloat32 0

src/Init/Data/FloatArray/Basic.lean

@@ -47,11 +47,11 @@ def uget : (a : @& FloatArray) → (i : USize) → i.toNat < a.size → Float
 
 @[extern "lean_float_array_fget"]
 def get : (ds : @& FloatArray) → (i : @& Nat) → (h : i < ds.size := by get_elem_tactic) → Float
-  | ⟨ds⟩, i, h => ds.get i h
+  | ⟨ds⟩, i, h => ds[i]
 
 @[extern "lean_float_array_get"]
 def get! : (@& FloatArray) → (@& Nat) → Float
-  | ⟨ds⟩, i => ds.get! i
+  | ⟨ds⟩, i => ds[i]!
 
 def get? (ds : FloatArray) (i : Nat) : Option Float :=
   if h : i < ds.size then
@@ -62,7 +62,7 @@ def get? (ds : FloatArray) (i : Nat) : Option Float :=
 instance : GetElem FloatArray Nat Float fun xs i => i < xs.size where
   getElem xs i h := xs.get i h
 
-instance : GetElem FloatArray USize Float fun xs i => i.val < xs.size where
+instance : GetElem FloatArray USize Float fun xs i => i.toNat < xs.size where
   getElem xs i h := xs.uget i h
 
 @[extern "lean_float_array_uset"]

src/Init/Data/Int.lean

@@ -7,10 +7,11 @@ prelude
 import Init.Data.Int.Basic
 import Init.Data.Int.Bitwise
 import Init.Data.Int.DivMod
-import Init.Data.Int.DivModLemmas
 import Init.Data.Int.Gcd
 import Init.Data.Int.Lemmas
 import Init.Data.Int.LemmasAux
 import Init.Data.Int.Order
 import Init.Data.Int.Pow
 import Init.Data.Int.Cooper
+import Init.Data.Int.Linear
+import Init.Data.Int.OfNat

src/Init/Data/Int/Basic.lean

@@ -17,10 +17,12 @@ open Nat
 This file defines the `Int` type as well as
 
 * coercions, conversions, and compatibility with numeric literals,
-* basic arithmetic operations add/sub/mul/div/mod/pow,
+* basic arithmetic operations add/sub/mul/pow,
 * a few `Nat`-related operations such as `negOfNat` and `subNatNat`,
 * relations `<`/`≤`/`≥`/`>`, the `NonNeg` property and `min`/`max`,
 * decidability of equality, relations and `NonNeg`.
+
+Division and modulus operations are defined in `Init.Data.Int.DivMod.Basic`.
 -/
 
 /--

src/Init/Data/Int/Bitwise/Lemmas.lean

@@ -6,6 +6,7 @@ Authors: Siddharth Bhat, Jeremy Avigad
 prelude
 import Init.Data.Nat.Bitwise.Lemmas
 import Init.Data.Int.Bitwise
+import Init.Data.Int.DivMod.Lemmas
 
 namespace Int
 
@@ -26,7 +27,7 @@ theorem shiftRight_eq_div_pow (m : Int) (n : Nat) :
     m >>> n = m / ((2 ^ n) : Nat) := by
   simp only [shiftRight_eq, Int.shiftRight, Nat.shiftRight_eq_div_pow]
   split
-  · simp
+  · simp; norm_cast
   · rw [negSucc_ediv _ (by norm_cast; exact Nat.pow_pos (Nat.zero_lt_two))]
     rfl
 
@@ -38,4 +39,47 @@ theorem zero_shiftRight (n : Nat) : (0 : Int) >>> n = 0 := by
 theorem shiftRight_zero (n : Int) : n >>> 0 = n := by
   simp [Int.shiftRight_eq_div_pow]
 
+theorem le_shiftRight_of_nonpos {n : Int} {s : Nat} (h : n ≤ 0) : n ≤ n >>> s := by
+  simp only [Int.shiftRight_eq, Int.shiftRight, Int.ofNat_eq_coe]
+  split
+  case _ _ _ m =>
+    simp only [ofNat_eq_coe] at h
+    by_cases hm : m = 0
+    · simp [hm]
+    · omega
+  case _ _ _ m =>
+    by_cases hm : m = 0
+    · simp [hm]
+    · have := Nat.shiftRight_le m s
+      omega
+
+theorem shiftRight_le_of_nonneg {n : Int} {s : Nat} (h : 0 ≤ n) : n >>> s ≤ n := by
+  simp only [Int.shiftRight_eq, Int.shiftRight, Int.ofNat_eq_coe]
+  split
+  case _ _ _ m =>
+    simp only [Int.ofNat_eq_coe] at h
+    by_cases hm : m = 0
+    · simp [hm]
+    · have := Nat.shiftRight_le m s
+      simp
+      omega
+  case _ _ _ m =>
+    omega
+
+theorem le_shiftRight_of_nonneg {n : Int} {s : Nat} (h : 0 ≤ n) : 0 ≤ (n >>> s) := by
+  rw [Int.shiftRight_eq_div_pow]
+  by_cases h' : s = 0
+  · simp [h', h]
+  · have := @Nat.pow_pos 2 s (by omega)
+    have := @Int.ediv_nonneg n (2^s) h (by norm_cast at *; omega)
+    norm_cast at *
+
+theorem shiftRight_le_of_nonpos {n : Int} {s : Nat} (h : n ≤ 0) : (n >>> s) ≤ 0 := by
+  rw [Int.shiftRight_eq_div_pow]
+  by_cases h' : s = 0
+  · simp [h', h]
+  · have : 1 < 2 ^ s := Nat.one_lt_two_pow (by omega)
+    have rl : n / 2 ^ s ≤ 0 := Int.ediv_nonpos_of_nonpos_of_neg (by omega) (by norm_cast at *; omega)
+    norm_cast at *
+
 end Int

src/Init/Data/Int/Cooper.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kim Morrison
 -/
 prelude
-import Init.Data.Int.DivModLemmas
+import Init.Data.Int.DivMod.Lemmas
 import Init.Data.Int.Gcd
 
 /-!
@@ -99,7 +99,7 @@ def resolve_left' (a c d p x : Int) (h₁ : p ≤ a * x) : Nat := (add_of_le h
 /-- `resolve_left` is nonnegative when `p ≤ a * x`. -/
 theorem le_zero_resolve_left (a c d p x : Int) (h₁ : p ≤ a * x) :
     0 ≤ resolve_left a c d p x := by
-  simpa [h₁] using Int.ofNat_nonneg _
+  simp [h₁]
 
 /-- `resolve_left` is bounded above by `lcm a (a * d / gcd (a * d) c)`. -/
 theorem resolve_left_lt_lcm (a c d p x : Int) (a_pos : 0 < a) (d_pos : 0 < d) (h₁ : p ≤ a * x) :
@@ -227,33 +227,4 @@ theorem cooper_resolution_dvd_right
     · exact Int.mul_neg _ _ ▸ Int.neg_le_of_neg_le lower
     · exact Int.mul_neg _ _ ▸ Int.neg_mul _ _ ▸ dvd
 
-/--
-Left Cooper resolution of an upper and lower bound.
--/
-theorem cooper_resolution_left
-    {a b p q : Int} (a_pos : 0 < a) (b_pos : 0 < b) :
-    (∃ x, p ≤ a * x ∧ b * x ≤ q) ↔
-    (∃ k : Int, 0 ≤ k ∧ k < a ∧ b * k + b * p ≤ a * q ∧ a ∣ k + p) := by
-  have h := cooper_resolution_dvd_left
-    a_pos b_pos Int.zero_lt_one (c := 1) (s := 0) (p := p) (q := q)
-  simp only [Int.mul_one, Int.one_mul, Int.mul_zero, Int.add_zero, gcd_one, Int.ofNat_one,
-    Int.ediv_one, lcm_self, Int.natAbs_of_nonneg (Int.le_of_lt a_pos), Int.one_dvd, and_true,
-    and_self] at h
-  exact h
-
-/--
-Right Cooper resolution of an upper and lower bound.
--/
-theorem cooper_resolution_right
-    {a b p q : Int} (a_pos : 0 < a) (b_pos : 0 < b) :
-    (∃ x, p ≤ a * x ∧ b * x ≤ q) ↔
-    (∃ k : Int, 0 ≤ k ∧ k < b ∧ a * k + b * p ≤ a * q ∧ b ∣ k - q) := by
-  have h := cooper_resolution_dvd_right
-    a_pos b_pos Int.zero_lt_one (c := 1) (s := 0) (p := p) (q := q)
-  have : ∀ k : Int, (b ∣ -k + q) ↔ (b ∣ k - q) := by
-    intro k
-    rw [← Int.dvd_neg, Int.neg_add, Int.neg_neg, Int.sub_eq_add_neg]
-  simp only [Int.mul_one, Int.one_mul, Int.mul_zero, Int.add_zero, gcd_one, Int.ofNat_one,
-    Int.ediv_one, lcm_self, Int.natAbs_of_nonneg (Int.le_of_lt b_pos), Int.one_dvd, and_true,
-    and_self, ← Int.neg_eq_neg_one_mul, this] at h
-  exact h
+end Int

src/Init/Data/Int/DivMod.lean

@@ -1,328 +1,9 @@
 /-
-Copyright (c) 2016 Jeremy Avigad. All rights reserved.
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jeremy Avigad, Mario Carneiro
+Authors: Kim Morrison
 -/
 prelude
-import Init.Data.Int.Basic
-
-open Nat
-
-namespace Int
-
-/-! ## Quotient and remainder
-
-There are three main conventions for integer division,
-referred here as the E, F, T rounding conventions.
-All three pairs satisfy the identity `x % y + (x / y) * y = x` unconditionally,
-and satisfy `x / 0 = 0` and `x % 0 = x`.
-
-### Historical notes
-In early versions of Lean, the typeclasses provided by `/` and `%`
-were defined in terms of `tdiv` and `tmod`, and these were named simply as `div` and `mod`.
-
-However we decided it was better to use `ediv` and `emod`,
-as they are consistent with the conventions used in SMTLib, and Mathlib,
-and often mathematical reasoning is easier with these conventions.
-
-At that time, we did not rename `div` and `mod` to `tdiv` and `tmod` (along with all their lemma).
-In September 2024, we decided to do this rename (with deprecations in place),
-and later we intend to rename `ediv` and `emod` to `div` and `mod`, as nearly all users will only
-ever need to use these functions and their associated lemmas.
-
-In December 2024, we removed `tdiv` and `tmod`, but have not yet renamed `ediv` and `emod`.
--/
-
-/-! ### T-rounding division -/
-
-/--
-`tdiv` uses the [*"T-rounding"*][t-rounding]
-(**T**runcation-rounding) convention, meaning that it rounds toward
-zero. Also note that division by zero is defined to equal zero.
-
-  The relation between integer division and modulo is found in
-  `Int.tmod_add_tdiv` which states that
-  `tmod a b + b * (tdiv a b) = a`, unconditionally.
-
-  [t-rounding]: https://dl.acm.org/doi/pdf/10.1145/128861.128862
-  [theo tmod_add_tdiv]: https://leanprover-community.github.io/mathlib4_docs/find/?pattern=Int.tmod_add_tdiv#doc
-
-  Examples:
-
-  ```
-  #eval (7 : Int).tdiv (0 : Int) -- 0
-  #eval (0 : Int).tdiv (7 : Int) -- 0
-
-  #eval (12 : Int).tdiv (6 : Int) -- 2
-  #eval (12 : Int).tdiv (-6 : Int) -- -2
-  #eval (-12 : Int).tdiv (6 : Int) -- -2
-  #eval (-12 : Int).tdiv (-6 : Int) -- 2
-
-  #eval (12 : Int).tdiv (7 : Int) -- 1
-  #eval (12 : Int).tdiv (-7 : Int) -- -1
-  #eval (-12 : Int).tdiv (7 : Int) -- -1
-  #eval (-12 : Int).tdiv (-7 : Int) -- 1
-  ```
-
-  Implemented by efficient native code.
--/
-@[extern "lean_int_div"]
-def tdiv : (@& Int) → (@& Int) → Int
-  | ofNat m, ofNat n =>  ofNat (m / n)
-  | ofNat m, -[n +1] => -ofNat (m / succ n)
-  | -[m +1], ofNat n => -ofNat (succ m / n)
-  | -[m +1], -[n +1] =>  ofNat (succ m / succ n)
-
-/-- Integer modulo. This function uses the
-  [*"T-rounding"*][t-rounding] (**T**runcation-rounding) convention
-  to pair with `Int.tdiv`, meaning that `tmod a b + b * (tdiv a b) = a`
-  unconditionally (see [`Int.tmod_add_tdiv`][theo tmod_add_tdiv]). In
-  particular, `a % 0 = a`.
-
-  [t-rounding]: https://dl.acm.org/doi/pdf/10.1145/128861.128862
-  [theo tmod_add_tdiv]: https://leanprover-community.github.io/mathlib4_docs/find/?pattern=Int.tmod_add_tdiv#doc
-
-  Examples:
-
-  ```
-  #eval (7 : Int).tmod (0 : Int) -- 7
-  #eval (0 : Int).tmod (7 : Int) -- 0
-
-  #eval (12 : Int).tmod (6 : Int) -- 0
-  #eval (12 : Int).tmod (-6 : Int) -- 0
-  #eval (-12 : Int).tmod (6 : Int) -- 0
-  #eval (-12 : Int).tmod (-6 : Int) -- 0
-
-  #eval (12 : Int).tmod (7 : Int) -- 5
-  #eval (12 : Int).tmod (-7 : Int) -- 5
-  #eval (-12 : Int).tmod (7 : Int) -- -5
-  #eval (-12 : Int).tmod (-7 : Int) -- -5
-  ```
-
-  Implemented by efficient native code. -/
-@[extern "lean_int_mod"]
-def tmod : (@& Int) → (@& Int) → Int
-  | ofNat m, ofNat n =>  ofNat (m % n)
-  | ofNat m, -[n +1] =>  ofNat (m % succ n)
-  | -[m +1], ofNat n => -ofNat (succ m % n)
-  | -[m +1], -[n +1] => -ofNat (succ m % succ n)
-
-/-! ### F-rounding division
-This pair satisfies `fdiv x y = floor (x / y)`.
--/
-
-/--
-Integer division. This version of division uses the F-rounding convention
-(flooring division), in which `Int.fdiv x y` satisfies `fdiv x y = floor (x / y)`
-and `Int.fmod` is the unique function satisfying `fmod x y + (fdiv x y) * y = x`.
-
-Examples:
-```
-#eval (7 : Int).fdiv (0 : Int) -- 0
-#eval (0 : Int).fdiv (7 : Int) -- 0
-
-#eval (12 : Int).fdiv (6 : Int) -- 2
-#eval (12 : Int).fdiv (-6 : Int) -- -2
-#eval (-12 : Int).fdiv (6 : Int) -- -2
-#eval (-12 : Int).fdiv (-6 : Int) -- 2
-
-#eval (12 : Int).fdiv (7 : Int) -- 1
-#eval (12 : Int).fdiv (-7 : Int) -- -2
-#eval (-12 : Int).fdiv (7 : Int) -- -2
-#eval (-12 : Int).fdiv (-7 : Int) -- 1
-```
--/
-def fdiv : Int → Int → Int
-  | 0,       _       => 0
-  | ofNat m, ofNat n => ofNat (m / n)
-  | ofNat (succ m), -[n+1] => -[m / succ n +1]
-  | -[_+1],  0       => 0
-  | -[m+1],  ofNat (succ n) => -[m / succ n +1]
-  | -[m+1],  -[n+1]  => ofNat (succ m / succ n)
-
-/--
-Integer modulus. This version of `Int.mod` uses the F-rounding convention
-(flooring division), in which `Int.fdiv x y` satisfies `fdiv x y = floor (x / y)`
-and `Int.fmod` is the unique function satisfying `fmod x y + (fdiv x y) * y = x`.
-
-Examples:
-
-```
-#eval (7 : Int).fmod (0 : Int) -- 7
-#eval (0 : Int).fmod (7 : Int) -- 0
-
-#eval (12 : Int).fmod (6 : Int) -- 0
-#eval (12 : Int).fmod (-6 : Int) -- 0
-#eval (-12 : Int).fmod (6 : Int) -- 0
-#eval (-12 : Int).fmod (-6 : Int) -- 0
-
-#eval (12 : Int).fmod (7 : Int) -- 5
-#eval (12 : Int).fmod (-7 : Int) -- -2
-#eval (-12 : Int).fmod (7 : Int) -- 2
-#eval (-12 : Int).fmod (-7 : Int) -- -5
-```
--/
-def fmod : Int → Int → Int
-  | 0,       _       => 0
-  | ofNat m, ofNat n => ofNat (m % n)
-  | ofNat (succ m),  -[n+1]  => subNatNat (m % succ n) n
-  | -[m+1],  ofNat n => subNatNat n (succ (m % n))
-  | -[m+1],  -[n+1]  => -ofNat (succ m % succ n)
-
-/-! ### E-rounding division
-This pair satisfies `0 ≤ mod x y < natAbs y` for `y ≠ 0`.
--/
-
-/--
-Integer division. This version of `Int.div` uses the E-rounding convention
-(euclidean division), in which `Int.emod x y` satisfies `0 ≤ mod x y < natAbs y` for `y ≠ 0`
-and `Int.ediv` is the unique function satisfying `emod x y + (ediv x y) * y = x`.
-
-This is the function powering the `/` notation on integers.
-
-Examples:
-```
-#eval (7 : Int) / (0 : Int) -- 0
-#eval (0 : Int) / (7 : Int) -- 0
-
-#eval (12 : Int) / (6 : Int) -- 2
-#eval (12 : Int) / (-6 : Int) -- -2
-#eval (-12 : Int) / (6 : Int) -- -2
-#eval (-12 : Int) / (-6 : Int) -- 2
-
-#eval (12 : Int) / (7 : Int) -- 1
-#eval (12 : Int) / (-7 : Int) -- -1
-#eval (-12 : Int) / (7 : Int) -- -2
-#eval (-12 : Int) / (-7 : Int) -- 2
-```
-
-Implemented by efficient native code.
--/
-@[extern "lean_int_ediv"]
-def ediv : (@& Int) → (@& Int) → Int
-  | ofNat m, ofNat n => ofNat (m / n)
-  | ofNat m, -[n+1]  => -ofNat (m / succ n)
-  | -[_+1],  0       => 0
-  | -[m+1],  ofNat (succ n) => -[m / succ n +1]
-  | -[m+1],  -[n+1]  => ofNat (succ (m / succ n))
-
-/--
-Integer modulus. This version of `Int.mod` uses the E-rounding convention
-(euclidean division), in which `Int.emod x y` satisfies `0 ≤ emod x y < natAbs y` for `y ≠ 0`
-and `Int.ediv` is the unique function satisfying `emod x y + (ediv x y) * y = x`.
-
-This is the function powering the `%` notation on integers.
-
-Examples:
-```
-#eval (7 : Int) % (0 : Int) -- 7
-#eval (0 : Int) % (7 : Int) -- 0
-
-#eval (12 : Int) % (6 : Int) -- 0
-#eval (12 : Int) % (-6 : Int) -- 0
-#eval (-12 : Int) % (6 : Int) -- 0
-#eval (-12 : Int) % (-6 : Int) -- 0
-
-#eval (12 : Int) % (7 : Int) -- 5
-#eval (12 : Int) % (-7 : Int) -- 5
-#eval (-12 : Int) % (7 : Int) -- 2
-#eval (-12 : Int) % (-7 : Int) -- 2
-```
-
-Implemented by efficient native code.
--/
-@[extern "lean_int_emod"]
-def emod : (@& Int) → (@& Int) → Int
-  | ofNat m, n => ofNat (m % natAbs n)
-  | -[m+1],  n => subNatNat (natAbs n) (succ (m % natAbs n))
-
-/--
-The Div and Mod syntax uses ediv and emod for compatibility with SMTLIb and mathematical
-reasoning tends to be easier.
--/
-instance : Div Int where
-  div := Int.ediv
-instance : Mod Int where
-  mod := Int.emod
-
-@[simp, norm_cast] theorem ofNat_ediv (m n : Nat) : (↑(m / n) : Int) = ↑m / ↑n := rfl
-
-theorem ofNat_tdiv (m n : Nat) : ↑(m / n) = tdiv ↑m ↑n := rfl
-
-theorem ofNat_fdiv : ∀ m n : Nat, ↑(m / n) = fdiv ↑m ↑n
-  | 0, _ => by simp [fdiv]
-  | succ _, _ => rfl
-
-/-!
-# `bmod` ("balanced" mod)
-
-Balanced mod (and balanced div) are a division and modulus pair such
-that `b * (Int.bdiv a b) + Int.bmod a b = a` and `-b/2 ≤ Int.bmod a b <
-b/2` for all `a : Int` and `b > 0`.
-
-This is used in Omega as well as signed bitvectors.
--/
-
-/--
-Balanced modulus.  This version of Integer modulus uses the
-balanced rounding convention, which guarantees that
-`-m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
-to `x` modulo `m`.
-
-If `m = 0`, then `bmod x m = x`.
-
-Examples:
-```
-#eval (7 : Int).bdiv 0 -- 0
-#eval (0 : Int).bdiv 7 -- 0
-
-#eval (12 : Int).bdiv 6 -- 2
-#eval (12 : Int).bdiv 7 -- 2
-#eval (12 : Int).bdiv 8 -- 2
-#eval (12 : Int).bdiv 9 -- 1
-
-#eval (-12 : Int).bdiv 6 -- -2
-#eval (-12 : Int).bdiv 7 -- -2
-#eval (-12 : Int).bdiv 8 -- -1
-#eval (-12 : Int).bdiv 9 -- -1
-```
--/
-def bmod (x : Int) (m : Nat) : Int :=
-  let r := x % m
-  if r < (m + 1) / 2 then
-    r
-  else
-    r - m
-
-/--
-Balanced division.  This returns the unique integer so that
-`b * (Int.bdiv a b) + Int.bmod a b = a`.
-
-Examples:
-```
-#eval (7 : Int).bmod 0 -- 7
-#eval (0 : Int).bmod 7 -- 0
-
-#eval (12 : Int).bmod 6 -- 0
-#eval (12 : Int).bmod 7 -- -2
-#eval (12 : Int).bmod 8 -- -4
-#eval (12 : Int).bmod 9 -- 3
-
-#eval (-12 : Int).bmod 6 -- 0
-#eval (-12 : Int).bmod 7 -- 2
-#eval (-12 : Int).bmod 8 -- -4
-#eval (-12 : Int).bmod 9 -- -3
-```
--/
-def bdiv (x : Int) (m : Nat) : Int :=
-  if m = 0 then
-    0
-  else
-    let q := x / m
-    let r := x % m
-    if r < (m + 1) / 2 then
-      q
-    else
-      q + 1
-
-end Int
+import Init.Data.Int.DivMod.Basic
+import Init.Data.Int.DivMod.Bootstrap
+import Init.Data.Int.DivMod.Lemmas

src/Init/Data/Int/DivMod/Basic.lean

@@ -0,0 +1,344 @@
+/-
+Copyright (c) 2016 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Mario Carneiro
+-/
+prelude
+import Init.Data.Int.Basic
+
+open Nat
+
+namespace Int
+
+/-! ## Quotient and remainder
+
+There are three main conventions for integer division,
+referred here as the E, F, T rounding conventions.
+All three pairs satisfy the identity `x % y + (x / y) * y = x` unconditionally,
+and satisfy `x / 0 = 0` and `x % 0 = x`.
+
+### Historical notes
+In early versions of Lean, the typeclasses provided by `/` and `%`
+were defined in terms of `tdiv` and `tmod`, and these were named simply as `div` and `mod`.
+
+However we decided it was better to use `ediv` and `emod` for the default typeclass instances,
+as they are consistent with the conventions used in SMTLib, and Mathlib,
+and often mathematical reasoning is easier with these conventions.
+At that time, we did not rename `div` and `mod` to `tdiv` and `tmod` (along with all their lemma).
+
+In September 2024, we decided to do this rename (with deprecations in place),
+and later we intend to rename `ediv` and `emod` to `div` and `mod`, as nearly all users will only
+ever need to use these functions and their associated lemmas.
+
+In December 2024, we removed `div` and `mod`, but have not yet renamed `ediv` and `emod`.
+-/
+
+/-! ### E-rounding division
+This pair satisfies `0 ≤ emod x y < natAbs y` for `y ≠ 0`.
+-/
+
+/--
+Integer division. This version of integer division uses the E-rounding convention
+(euclidean division), in which `Int.emod x y` satisfies `0 ≤ emod x y < natAbs y` for `y ≠ 0`
+and `Int.ediv` is the unique function satisfying `emod x y + (ediv x y) * y = x` for `y ≠ 0`.
+
+This means that `Int.ediv x y = floor (x / y)` when `y > 0` and `Int.ediv x y = ceil (x / y)` when `y < 0`.
+
+This is the function powering the `/` notation on integers.
+
+Examples:
+```
+#eval (7 : Int) / (0 : Int) -- 0
+#eval (0 : Int) / (7 : Int) -- 0
+
+#eval (12 : Int) / (6 : Int) -- 2
+#eval (12 : Int) / (-6 : Int) -- -2
+#eval (-12 : Int) / (6 : Int) -- -2
+#eval (-12 : Int) / (-6 : Int) -- 2
+
+#eval (12 : Int) / (7 : Int) -- 1
+#eval (12 : Int) / (-7 : Int) -- -1
+#eval (-12 : Int) / (7 : Int) -- -2
+#eval (-12 : Int) / (-7 : Int) -- 2
+```
+
+Implemented by efficient native code.
+-/
+@[extern "lean_int_ediv"]
+def ediv : (@& Int) → (@& Int) → Int
+  | ofNat m, ofNat n => ofNat (m / n)
+  | ofNat m, -[n+1]  => -ofNat (m / succ n)
+  | -[_+1],  0       => 0
+  | -[m+1],  ofNat (succ n) => -[m / succ n +1]
+  | -[m+1],  -[n+1]  => ofNat (succ (m / succ n))
+
+/--
+Integer modulus. This version of integer modulus uses the E-rounding convention
+(euclidean division), in which `Int.emod x y` satisfies `0 ≤ emod x y < natAbs y` for `y ≠ 0`
+and `Int.ediv` is the unique function satisfying `emod x y + (ediv x y) * y = x`.
+
+This is the function powering the `%` notation on integers.
+
+Examples:
+```
+#eval (7 : Int) % (0 : Int) -- 7
+#eval (0 : Int) % (7 : Int) -- 0
+
+#eval (12 : Int) % (6 : Int) -- 0
+#eval (12 : Int) % (-6 : Int) -- 0
+#eval (-12 : Int) % (6 : Int) -- 0
+#eval (-12 : Int) % (-6 : Int) -- 0
+
+#eval (12 : Int) % (7 : Int) -- 5
+#eval (12 : Int) % (-7 : Int) -- 5
+#eval (-12 : Int) % (7 : Int) -- 2
+#eval (-12 : Int) % (-7 : Int) -- 2
+```
+
+Implemented by efficient native code.
+-/
+@[extern "lean_int_emod"]
+def emod : (@& Int) → (@& Int) → Int
+  | ofNat m, n => ofNat (m % natAbs n)
+  | -[m+1],  n => subNatNat (natAbs n) (succ (m % natAbs n))
+
+/--
+The Div and Mod syntax uses ediv and emod for compatibility with SMTLIb and mathematical
+reasoning tends to be easier.
+-/
+instance : Div Int where
+  div := Int.ediv
+instance : Mod Int where
+  mod := Int.emod
+
+@[norm_cast] theorem ofNat_ediv (m n : Nat) : (↑(m / n) : Int) = ↑m / ↑n := rfl
+
+theorem ofNat_ediv_ofNat {a b : Nat} : (↑a / ↑b : Int) = (a / b : Nat) := rfl
+@[norm_cast]
+theorem negSucc_ediv_ofNat_succ {a b : Nat} : ((-[a+1]) / ↑(b+1) : Int) = -[a / succ b +1] := rfl
+theorem negSucc_ediv_negSucc {a b : Nat} : ((-[a+1]) / (-[b+1]) : Int) = ((a / (b + 1)) + 1 : Nat) := rfl
+theorem ofNat_ediv_negSucc {a b : Nat} : (ofNat a / (-[b+1])) = -(a / (b + 1) : Nat) := rfl
+theorem negSucc_emod_ofNat {a b : Nat} : -[a+1] % (b : Int) = subNatNat b (succ (a % b)) := rfl
+theorem negSucc_emod_negSucc {a b : Nat} : -[a+1] % -[b+1] = subNatNat (b + 1) (succ (a % (b + 1))) := rfl
+
+/-! ### T-rounding division -/
+
+/--
+`tdiv` uses the [*"T-rounding"*][t-rounding]
+(**T**runcation-rounding) convention, meaning that it rounds toward
+zero. Also note that division by zero is defined to equal zero.
+
+  The relation between integer division and modulo is found in
+  `Int.tmod_add_tdiv` which states that
+  `tmod a b + b * (tdiv a b) = a`, unconditionally.
+
+  [t-rounding]: https://dl.acm.org/doi/pdf/10.1145/128861.128862
+  [theo tmod_add_tdiv]: https://leanprover-community.github.io/mathlib4_docs/find/?pattern=Int.tmod_add_tdiv#doc
+
+  Examples:
+
+  ```
+  #eval (7 : Int).tdiv (0 : Int) -- 0
+  #eval (0 : Int).tdiv (7 : Int) -- 0
+
+  #eval (12 : Int).tdiv (6 : Int) -- 2
+  #eval (12 : Int).tdiv (-6 : Int) -- -2
+  #eval (-12 : Int).tdiv (6 : Int) -- -2
+  #eval (-12 : Int).tdiv (-6 : Int) -- 2
+
+  #eval (12 : Int).tdiv (7 : Int) -- 1
+  #eval (12 : Int).tdiv (-7 : Int) -- -1
+  #eval (-12 : Int).tdiv (7 : Int) -- -1
+  #eval (-12 : Int).tdiv (-7 : Int) -- 1
+  ```
+
+  Implemented by efficient native code.
+-/
+@[extern "lean_int_div"]
+def tdiv : (@& Int) → (@& Int) → Int
+  | ofNat m, ofNat n =>  ofNat (m / n)
+  | ofNat m, -[n +1] => -ofNat (m / succ n)
+  | -[m +1], ofNat n => -ofNat (succ m / n)
+  | -[m +1], -[n +1] =>  ofNat (succ m / succ n)
+
+/-- Integer modulo. This function uses the
+  [*"T-rounding"*][t-rounding] (**T**runcation-rounding) convention
+  to pair with `Int.tdiv`, meaning that `tmod a b + b * (tdiv a b) = a`
+  unconditionally (see [`Int.tmod_add_tdiv`][theo tmod_add_tdiv]). In
+  particular, `a % 0 = a`.
+
+  `tmod` satisfies `natAbs (tmod a b) = natAbs a % natAbs b`,
+  and when `b` does not divide `a`, `tmod a b` has the same sign as `a`.
+
+  [t-rounding]: https://dl.acm.org/doi/pdf/10.1145/128861.128862
+  [theo tmod_add_tdiv]: https://leanprover-community.github.io/mathlib4_docs/find/?pattern=Int.tmod_add_tdiv#doc
+
+  Examples:
+
+  ```
+  #eval (7 : Int).tmod (0 : Int) -- 7
+  #eval (0 : Int).tmod (7 : Int) -- 0
+
+  #eval (12 : Int).tmod (6 : Int) -- 0
+  #eval (12 : Int).tmod (-6 : Int) -- 0
+  #eval (-12 : Int).tmod (6 : Int) -- 0
+  #eval (-12 : Int).tmod (-6 : Int) -- 0
+
+  #eval (12 : Int).tmod (7 : Int) -- 5
+  #eval (12 : Int).tmod (-7 : Int) -- 5
+  #eval (-12 : Int).tmod (7 : Int) -- -5
+  #eval (-12 : Int).tmod (-7 : Int) -- -5
+  ```
+
+  Implemented by efficient native code. -/
+@[extern "lean_int_mod"]
+def tmod : (@& Int) → (@& Int) → Int
+  | ofNat m, ofNat n =>  ofNat (m % n)
+  | ofNat m, -[n +1] =>  ofNat (m % succ n)
+  | -[m +1], ofNat n => -ofNat (succ m % n)
+  | -[m +1], -[n +1] => -ofNat (succ m % succ n)
+
+theorem ofNat_tdiv (m n : Nat) : ↑(m / n) = tdiv ↑m ↑n := rfl
+
+/-! ### F-rounding division
+This pair satisfies `fdiv x y = floor (x / y)`.
+-/
+
+/--
+Integer division. This version of division uses the F-rounding convention
+(flooring division), in which `Int.fdiv x y` satisfies `fdiv x y = floor (x / y)`
+and `Int.fmod` is the unique function satisfying `fmod x y + (fdiv x y) * y = x`.
+
+Examples:
+```
+#eval (7 : Int).fdiv (0 : Int) -- 0
+#eval (0 : Int).fdiv (7 : Int) -- 0
+
+#eval (12 : Int).fdiv (6 : Int) -- 2
+#eval (12 : Int).fdiv (-6 : Int) -- -2
+#eval (-12 : Int).fdiv (6 : Int) -- -2
+#eval (-12 : Int).fdiv (-6 : Int) -- 2
+
+#eval (12 : Int).fdiv (7 : Int) -- 1
+#eval (12 : Int).fdiv (-7 : Int) -- -2
+#eval (-12 : Int).fdiv (7 : Int) -- -2
+#eval (-12 : Int).fdiv (-7 : Int) -- 1
+```
+-/
+def fdiv : Int → Int → Int
+  | 0,       _       => 0
+  | ofNat m, ofNat n => ofNat (m / n)
+  | ofNat (succ m), -[n+1] => -[m / succ n +1]
+  | -[_+1],  0       => 0
+  | -[m+1],  ofNat (succ n) => -[m / succ n +1]
+  | -[m+1],  -[n+1]  => ofNat (succ m / succ n)
+
+/--
+Integer modulus. This version of integer modulus uses the F-rounding convention
+(flooring division), in which `Int.fdiv x y` satisfies `fdiv x y = floor (x / y)`
+and `Int.fmod` is the unique function satisfying `fmod x y + (fdiv x y) * y = x`.
+
+Examples:
+
+```
+#eval (7 : Int).fmod (0 : Int) -- 7
+#eval (0 : Int).fmod (7 : Int) -- 0
+
+#eval (12 : Int).fmod (6 : Int) -- 0
+#eval (12 : Int).fmod (-6 : Int) -- 0
+#eval (-12 : Int).fmod (6 : Int) -- 0
+#eval (-12 : Int).fmod (-6 : Int) -- 0
+
+#eval (12 : Int).fmod (7 : Int) -- 5
+#eval (12 : Int).fmod (-7 : Int) -- -2
+#eval (-12 : Int).fmod (7 : Int) -- 2
+#eval (-12 : Int).fmod (-7 : Int) -- -5
+```
+-/
+def fmod : Int → Int → Int
+  | 0,       _       => 0
+  | ofNat m, ofNat n => ofNat (m % n)
+  | ofNat (succ m),  -[n+1]  => subNatNat (m % succ n) n
+  | -[m+1],  ofNat n => subNatNat n (succ (m % n))
+  | -[m+1],  -[n+1]  => -ofNat (succ m % succ n)
+
+theorem ofNat_fdiv : ∀ m n : Nat, ↑(m / n) = fdiv ↑m ↑n
+  | 0, _ => by simp [fdiv]
+  | succ _, _ => rfl
+
+/-!
+# `bmod` ("balanced" mod)
+
+Balanced mod (and balanced div) are a division and modulus pair such
+that `b * (Int.bdiv a b) + Int.bmod a b = a` and
+`-b/2 ≤ Int.bmod a b < b/2` for all `a : Int` and `b > 0`.
+
+Note that unlike `emod`, `fmod`, and `tmod`,
+`bmod` takes a natural number as the second argument, rather than an integer.
+
+This function is used in `omega` as well as signed bitvectors.
+-/
+
+/--
+Balanced modulus.  This version of integer modulus uses the
+balanced rounding convention, which guarantees that
+`-m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
+to `x` modulo `m`.
+
+If `m = 0`, then `bmod x m = x`.
+
+Examples:
+```
+#eval (7 : Int).bdiv 0 -- 0
+#eval (0 : Int).bdiv 7 -- 0
+
+#eval (12 : Int).bdiv 6 -- 2
+#eval (12 : Int).bdiv 7 -- 2
+#eval (12 : Int).bdiv 8 -- 2
+#eval (12 : Int).bdiv 9 -- 1
+
+#eval (-12 : Int).bdiv 6 -- -2
+#eval (-12 : Int).bdiv 7 -- -2
+#eval (-12 : Int).bdiv 8 -- -1
+#eval (-12 : Int).bdiv 9 -- -1
+```
+-/
+def bmod (x : Int) (m : Nat) : Int :=
+  let r := x % m
+  if r < (m + 1) / 2 then
+    r
+  else
+    r - m
+
+/--
+Balanced division.  This returns the unique integer so that
+`b * (Int.bdiv a b) + Int.bmod a b = a`.
+
+Examples:
+```
+#eval (7 : Int).bmod 0 -- 7
+#eval (0 : Int).bmod 7 -- 0
+
+#eval (12 : Int).bmod 6 -- 0
+#eval (12 : Int).bmod 7 -- -2
+#eval (12 : Int).bmod 8 -- -4
+#eval (12 : Int).bmod 9 -- 3
+
+#eval (-12 : Int).bmod 6 -- 0
+#eval (-12 : Int).bmod 7 -- 2
+#eval (-12 : Int).bmod 8 -- -4
+#eval (-12 : Int).bmod 9 -- -3
+```
+-/
+def bdiv (x : Int) (m : Nat) : Int :=
+  if m = 0 then
+    0
+  else
+    let q := x / m
+    let r := x % m
+    if r < (m + 1) / 2 then
+      q
+    else
+      q + 1
+
+end Int

src/Init/Data/Int/DivMod/Bootstrap.lean

@@ -0,0 +1,333 @@
+/-
+Copyright (c) 2016 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Mario Carneiro
+-/
+
+prelude
+import Init.Data.Int.DivMod.Basic
+import Init.Data.Int.Order
+import Init.Data.Nat.Dvd
+import Init.RCases
+
+/-!
+# Lemmas about integer division needed to bootstrap `omega`.
+-/
+
+open Nat (succ)
+
+namespace Int
+
+/-! ### dvd  -/
+
+protected theorem dvd_def (a b : Int) : (a ∣ b) = Exists (fun c => b = a * c) := rfl
+
+@[simp] protected theorem dvd_zero (n : Int) : n ∣ 0 := ⟨0, (Int.mul_zero _).symm⟩
+
+@[simp] protected theorem dvd_refl (n : Int) : n ∣ n := ⟨1, (Int.mul_one _).symm⟩
+
+@[simp] protected theorem one_dvd (n : Int) : 1 ∣ n := ⟨n, (Int.one_mul n).symm⟩
+
+protected theorem dvd_trans : ∀ {a b c : Int}, a ∣ b → b ∣ c → a ∣ c
+  | _, _, _, ⟨d, rfl⟩, ⟨e, rfl⟩ => Exists.intro (d * e) (by rw [Int.mul_assoc])
+
+@[norm_cast] theorem ofNat_dvd {m n : Nat} : (↑m : Int) ∣ ↑n ↔ m ∣ n := by
+  refine ⟨fun ⟨a, ae⟩ => ?_, fun ⟨k, e⟩ => ⟨k, by rw [e, Int.ofNat_mul]⟩⟩
+  match Int.le_total a 0 with
+  | .inl h =>
+    have := ae.symm ▸ Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h
+    rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)]
+    apply Nat.dvd_zero
+  | .inr h => match a, eq_ofNat_of_zero_le h with
+    | _, ⟨k, rfl⟩ => exact ⟨k, Int.ofNat.inj ae⟩
+
+@[simp] protected theorem zero_dvd {n : Int} : 0 ∣ n ↔ n = 0 :=
+  Iff.intro (fun ⟨k, e⟩ => by rw [e, Int.zero_mul])
+            (fun h => h.symm ▸ Int.dvd_refl _)
+
+protected theorem dvd_mul_right (a b : Int) : a ∣ a * b := ⟨_, rfl⟩
+
+protected theorem dvd_mul_left (a b : Int) : b ∣ a * b := ⟨_, Int.mul_comm ..⟩
+
+@[simp] protected theorem neg_dvd {a b : Int} : -a ∣ b ↔ a ∣ b := by
+  constructor <;> exact fun ⟨k, e⟩ =>
+    ⟨-k, by simp [e, Int.neg_mul, Int.mul_neg, Int.neg_neg]⟩
+
+@[simp] protected theorem dvd_neg {a b : Int} : a ∣ -b ↔ a ∣ b := by
+  constructor <;> exact fun ⟨k, e⟩ =>
+    ⟨-k, by simp [← e, Int.neg_mul, Int.mul_neg, Int.neg_neg]⟩
+
+@[simp] theorem natAbs_dvd_natAbs {a b : Int} : natAbs a ∣ natAbs b ↔ a ∣ b := by
+  refine ⟨fun ⟨k, hk⟩ => ?_, fun ⟨k, hk⟩ => ⟨natAbs k, hk.symm ▸ natAbs_mul a k⟩⟩
+  rw [← natAbs_ofNat k, ← natAbs_mul, natAbs_eq_natAbs_iff] at hk
+  cases hk <;> subst b
+  · apply Int.dvd_mul_right
+  · rw [← Int.mul_neg]; apply Int.dvd_mul_right
+
+theorem ofNat_dvd_left {n : Nat} {z : Int} : (↑n : Int) ∣ z ↔ n ∣ z.natAbs := by
+  rw [← natAbs_dvd_natAbs, natAbs_ofNat]
+
+/-! ### ediv zero  -/
+
+@[simp] theorem zero_ediv : ∀ b : Int, 0 / b = 0
+  | ofNat _ => show ofNat _ = _ by simp
+  | -[_+1] => show -ofNat _ = _ by simp
+
+@[simp] protected theorem ediv_zero : ∀ a : Int, a / 0 = 0
+  | ofNat _ => show ofNat _ = _ by simp
+  | -[_+1] => rfl
+
+/-! ### emod zero -/
+
+@[simp] theorem zero_emod (b : Int) : 0 % b = 0 := rfl
+
+@[simp] theorem emod_zero : ∀ a : Int, a % 0 = a
+  | ofNat _ => congrArg ofNat <| Nat.mod_zero _
+  | -[_+1]  => congrArg negSucc <| Nat.mod_zero _
+
+/-! ### ofNat mod -/
+
+@[simp, norm_cast] theorem ofNat_emod (m n : Nat) : (↑(m % n) : Int) = m % n := rfl
+
+/-! ### mod definitions -/
+
+theorem emod_add_ediv : ∀ a b : Int, a % b + b * (a / b) = a
+  | ofNat _, ofNat _ => congrArg ofNat <| Nat.mod_add_div ..
+  | ofNat m, -[n+1] => by
+    show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
+    rw [Int.neg_mul_neg]; exact congrArg ofNat <| Nat.mod_add_div ..
+  | -[_+1], 0 => by rw [emod_zero]; rfl
+  | -[m+1], succ n => aux m n.succ
+  | -[m+1], -[n+1] => aux m n.succ
+where
+  aux (m n : Nat) : n - (m % n + 1) - (n * (m / n) + n) = -[m+1] := by
+    rw [← ofNat_emod, ← ofNat_ediv, ← Int.sub_sub, negSucc_eq, Int.sub_sub n,
+      ← Int.neg_neg (_-_), Int.neg_sub, Int.sub_sub_self, Int.add_right_comm]
+    exact congrArg (fun x => -(ofNat x + 1)) (Nat.mod_add_div ..)
+
+/-- Variant of `emod_add_ediv` with the multiplication written the other way around. -/
+theorem emod_add_ediv' (a b : Int) : a % b + a / b * b = a := by
+  rw [Int.mul_comm]; exact emod_add_ediv ..
+
+theorem ediv_add_emod (a b : Int) : b * (a / b) + a % b = a := by
+  rw [Int.add_comm]; exact emod_add_ediv ..
+
+/-- Variant of `ediv_add_emod` with the multiplication written the other way around. -/
+theorem ediv_add_emod' (a b : Int) : a / b * b + a % b = a := by
+  rw [Int.mul_comm]; exact ediv_add_emod ..
+
+theorem emod_def (a b : Int) : a % b = a - b * (a / b) := by
+  rw [← Int.add_sub_cancel (a % b), emod_add_ediv]
+
+/-! ### `/` ediv -/
+
+@[simp] theorem ediv_neg : ∀ a b : Int, a / (-b) = -(a / b)
+  | ofNat m, 0 => show ofNat (m / 0) = -↑(m / 0) by rw [Nat.div_zero]; rfl
+  | ofNat _, -[_+1] => (Int.neg_neg _).symm
+  | ofNat _, succ _ | -[_+1], 0 | -[_+1], succ _ | -[_+1], -[_+1] => rfl
+
+protected theorem div_def (a b : Int) : a / b = Int.ediv a b := rfl
+
+theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c = a / c + b :=
+  suffices ∀ {{a b c : Int}}, 0 < c → (a + b * c).ediv c = a.ediv c + b from
+    match Int.lt_trichotomy c 0 with
+    | Or.inl hlt => by
+      rw [← Int.neg_inj, ← Int.ediv_neg, Int.neg_add, ← Int.ediv_neg, ← Int.neg_mul_neg]
+      exact this (Int.neg_pos_of_neg hlt)
+    | Or.inr (Or.inl HEq) => absurd HEq H
+    | Or.inr (Or.inr hgt) => this hgt
+  suffices ∀ {k n : Nat} {a : Int}, (a + n * k.succ).ediv k.succ = a.ediv k.succ + n from
+    fun a b c H => match c, eq_succ_of_zero_lt H, b with
+      | _, ⟨_, rfl⟩, ofNat _ => this
+      | _, ⟨k, rfl⟩, -[n+1] => show (a - n.succ * k.succ).ediv k.succ = a.ediv k.succ - n.succ by
+        rw [← Int.add_sub_cancel (ediv ..), ← this, Int.sub_add_cancel]
+  fun {k n} => @fun
+  | ofNat _ => congrArg ofNat <| Nat.add_mul_div_right _ _ k.succ_pos
+  | -[m+1] => by
+    show ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat)
+    by_cases h : m < n * k.succ
+    · rw [← Int.ofNat_sub h, ← Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)]
+      apply congrArg ofNat
+      rw [Nat.mul_comm, Nat.mul_sub_div]; rwa [Nat.mul_comm]
+    · have h := Nat.not_lt.1 h
+      have H {a b : Nat} (h : a ≤ b) : (a : Int) + -((b : Int) + 1) = -[b - a +1] := by
+        rw [negSucc_eq, Int.ofNat_sub h]
+        simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc]
+      show ediv (↑(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(↑(m / succ k) + 1 : Int)
+      rw [H h, H ((Nat.le_div_iff_mul_le k.succ_pos).2 h)]
+      apply congrArg negSucc
+      rw [Nat.mul_comm, Nat.sub_mul_div]; rwa [Nat.mul_comm]
+
+theorem add_mul_ediv_left (a : Int) {b : Int}
+    (c : Int) (H : b ≠ 0) : (a + b * c) / b = a / b + c :=
+  Int.mul_comm .. ▸ Int.add_mul_ediv_right _ _ H
+
+theorem add_ediv_of_dvd_right {a b c : Int} (H : c ∣ b) : (a + b) / c = a / c + b / c :=
+  if h : c = 0 then by simp [h] else by
+    let ⟨k, hk⟩ := H
+    rw [hk, Int.mul_comm c k, Int.add_mul_ediv_right _ _ h,
+      ← Int.zero_add (k * c), Int.add_mul_ediv_right _ _ h, Int.zero_ediv, Int.zero_add]
+
+theorem add_ediv_of_dvd_left {a b c : Int} (H : c ∣ a) : (a + b) / c = a / c + b / c := by
+  rw [Int.add_comm, Int.add_ediv_of_dvd_right H, Int.add_comm]
+
+@[simp] theorem mul_ediv_cancel (a : Int) {b : Int} (H : b ≠ 0) : (a * b) / b = a := by
+  have := Int.add_mul_ediv_right 0 a H
+  rwa [Int.zero_add, Int.zero_ediv, Int.zero_add] at this
+
+@[simp] theorem mul_ediv_cancel_left (b : Int) (H : a ≠ 0) : (a * b) / a = b :=
+  Int.mul_comm .. ▸ Int.mul_ediv_cancel _ H
+
+theorem ediv_nonneg_iff_of_pos {a b : Int} (h : 0 < b) : 0 ≤ a / b ↔ 0 ≤ a := by
+  rw [Int.div_def]
+  match b, h with
+  | Int.ofNat (b+1), _ =>
+    rcases a with ⟨a⟩ <;> simp [Int.ediv]
+
+@[deprecated ediv_nonneg_iff_of_pos (since := "2025-02-28")]
+abbrev div_nonneg_iff_of_pos := @ediv_nonneg_iff_of_pos
+
+/-! ### emod -/
+
+theorem emod_nonneg : ∀ (a : Int) {b : Int}, b ≠ 0 → 0 ≤ a % b
+  | ofNat _, _, _ => ofNat_zero_le _
+  | -[_+1], _, H => Int.sub_nonneg_of_le <| ofNat_le.2 <| Nat.mod_lt _ (natAbs_pos.2 H)
+
+theorem emod_lt_of_pos (a : Int) {b : Int} (H : 0 < b) : a % b < b :=
+  match a, b, eq_succ_of_zero_lt H with
+  | ofNat _, _, ⟨_, rfl⟩ => ofNat_lt.2 (Nat.mod_lt _ (Nat.succ_pos _))
+  | -[_+1], _, ⟨_, rfl⟩ => Int.sub_lt_self _ (ofNat_lt.2 <| Nat.succ_pos _)
+
+@[simp] theorem add_mul_emod_self {a b c : Int} : (a + b * c) % c = a % c :=
+  if cz : c = 0 then by
+    rw [cz, Int.mul_zero, Int.add_zero]
+  else by
+    rw [Int.emod_def, Int.emod_def, Int.add_mul_ediv_right _ _ cz, Int.add_comm _ b,
+      Int.mul_add, Int.mul_comm, ← Int.sub_sub, Int.add_sub_cancel]
+
+@[simp] theorem add_mul_emod_self_left (a b c : Int) : (a + b * c) % b = a % b := by
+  rw [Int.mul_comm, Int.add_mul_emod_self]
+
+@[simp] theorem emod_add_emod (m n k : Int) : (m % n + k) % n = (m + k) % n := by
+  have := (add_mul_emod_self_left (m % n + k) n (m / n)).symm
+  rwa [Int.add_right_comm, emod_add_ediv] at this
+
+@[simp] theorem add_emod_emod (m n k : Int) : (m + n % k) % k = (m + n) % k := by
+  rw [Int.add_comm, emod_add_emod, Int.add_comm]
+
+theorem add_emod (a b n : Int) : (a + b) % n = (a % n + b % n) % n := by
+  rw [add_emod_emod, emod_add_emod]
+
+theorem add_emod_eq_add_emod_right {m n k : Int} (i : Int)
+    (H : m % n = k % n) : (m + i) % n = (k + i) % n := by
+  rw [← emod_add_emod, ← emod_add_emod k, H]
+
+theorem emod_add_cancel_right {m n k : Int} (i) : (m + i) % n = (k + i) % n ↔ m % n = k % n :=
+  ⟨fun H => by
+    have := add_emod_eq_add_emod_right (-i) H
+    rwa [Int.add_neg_cancel_right, Int.add_neg_cancel_right] at this,
+  add_emod_eq_add_emod_right _⟩
+
+@[simp] theorem mul_emod_left (a b : Int) : (a * b) % b = 0 := by
+  rw [← Int.zero_add (a * b), Int.add_mul_emod_self, Int.zero_emod]
+
+@[simp] theorem mul_emod_right (a b : Int) : (a * b) % a = 0 := by
+  rw [Int.mul_comm, mul_emod_left]
+
+theorem mul_emod (a b n : Int) : (a * b) % n = (a % n) * (b % n) % n := by
+  conv => lhs; rw [
+    ← emod_add_ediv a n, ← emod_add_ediv' b n, Int.add_mul, Int.mul_add, Int.mul_add,
+    Int.mul_assoc, Int.mul_assoc, ← Int.mul_add n _ _, add_mul_emod_self_left,
+    ← Int.mul_assoc, add_mul_emod_self]
+
+@[simp] theorem emod_self {a : Int} : a % a = 0 := by
+  have := mul_emod_left 1 a; rwa [Int.one_mul] at this
+
+@[simp] theorem emod_emod_of_dvd (n : Int) {m k : Int}
+    (h : m ∣ k) : (n % k) % m = n % m := by
+  conv => rhs; rw [← emod_add_ediv n k]
+  match k, h with
+  | _, ⟨t, rfl⟩ => rw [Int.mul_assoc, add_mul_emod_self_left]
+
+@[simp] theorem emod_emod (a b : Int) : (a % b) % b = a % b := by
+  conv => rhs; rw [← emod_add_ediv a b, add_mul_emod_self_left]
+
+theorem sub_emod (a b n : Int) : (a - b) % n = (a % n - b % n) % n := by
+  apply (emod_add_cancel_right b).mp
+  rw [Int.sub_add_cancel, ← Int.add_emod_emod, Int.sub_add_cancel, emod_emod]
+
+/-! ### properties of `/` and `%` -/
+
+theorem mul_ediv_cancel_of_emod_eq_zero {a b : Int} (H : a % b = 0) : b * (a / b) = a := by
+  have := emod_add_ediv a b; rwa [H, Int.zero_add] at this
+
+theorem ediv_mul_cancel_of_emod_eq_zero {a b : Int} (H : a % b = 0) : a / b * b = a := by
+  rw [Int.mul_comm, mul_ediv_cancel_of_emod_eq_zero H]
+
+theorem dvd_of_emod_eq_zero {a b : Int} (H : b % a = 0) : a ∣ b :=
+  ⟨b / a, (mul_ediv_cancel_of_emod_eq_zero H).symm⟩
+
+theorem emod_eq_zero_of_dvd : ∀ {a b : Int}, a ∣ b → b % a = 0
+  | _, _, ⟨_, rfl⟩ => mul_emod_right ..
+
+theorem dvd_iff_emod_eq_zero {a b : Int} : a ∣ b ↔ b % a = 0 :=
+  ⟨emod_eq_zero_of_dvd, dvd_of_emod_eq_zero⟩
+
+protected theorem mul_ediv_assoc (a : Int) : ∀ {b c : Int}, c ∣ b → (a * b) / c = a * (b / c)
+  | _, c, ⟨d, rfl⟩ =>
+    if cz : c = 0 then by simp [cz, Int.mul_zero] else by
+      rw [Int.mul_left_comm, Int.mul_ediv_cancel_left _ cz, Int.mul_ediv_cancel_left _ cz]
+
+protected theorem mul_ediv_assoc' (b : Int) {a c : Int}
+    (h : c ∣ a) : (a * b) / c = a / c * b := by
+  rw [Int.mul_comm, Int.mul_ediv_assoc _ h, Int.mul_comm]
+
+theorem neg_ediv_of_dvd : ∀ {a b : Int}, b ∣ a → (-a) / b = -(a / b)
+  | _, b, ⟨c, rfl⟩ => by
+    by_cases bz : b = 0
+    · simp [bz]
+    · rw [Int.neg_mul_eq_mul_neg, Int.mul_ediv_cancel_left _ bz, Int.mul_ediv_cancel_left _ bz]
+
+theorem sub_ediv_of_dvd (a : Int) {b c : Int}
+    (hcb : c ∣ b) : (a - b) / c = a / c - b / c := by
+  rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.add_ediv_of_dvd_right (Int.dvd_neg.2 hcb)]
+  congr; exact Int.neg_ediv_of_dvd hcb
+
+protected theorem ediv_mul_cancel {a b : Int} (H : b ∣ a) : a / b * b = a :=
+  ediv_mul_cancel_of_emod_eq_zero (emod_eq_zero_of_dvd H)
+
+protected theorem mul_ediv_cancel' {a b : Int} (H : a ∣ b) : a * (b / a) = b := by
+  rw [Int.mul_comm, Int.ediv_mul_cancel H]
+
+theorem emod_pos_of_not_dvd {a b : Int} (h : ¬ a ∣ b) : a = 0 ∨ 0 < b % a := by
+  rw [dvd_iff_emod_eq_zero] at h
+  by_cases w : a = 0
+  · simp_all
+  · exact Or.inr (Int.lt_iff_le_and_ne.mpr ⟨emod_nonneg b w, Ne.symm h⟩)
+
+/-! ### `/` and ordering -/
+
+theorem mul_ediv_self_le {x k : Int} (h : k ≠ 0) : k * (x / k) ≤ x :=
+  calc k * (x / k)
+    _ ≤ k * (x / k) + x % k := Int.le_add_of_nonneg_right (emod_nonneg x h)
+    _ = x                   := ediv_add_emod _ _
+
+theorem lt_mul_ediv_self_add {x k : Int} (h : 0 < k) : x < k * (x / k) + k :=
+  calc x
+    _ = k * (x / k) + x % k := (ediv_add_emod _ _).symm
+    _ < k * (x / k) + k     := Int.add_lt_add_left (emod_lt_of_pos x h) _
+
+/-! ### bmod -/
+
+@[simp] theorem bmod_emod : bmod x m % m = x % m := by
+  dsimp [bmod]
+  split <;> simp [Int.sub_emod]
+
+theorem bmod_def (x : Int) (m : Nat) : bmod x m =
+  if (x % m) < (m + 1) / 2 then
+    x % m
+  else
+    (x % m) - m :=
+  rfl
+
+end Int

src/Init/Data/Int/Gcd.lean

@@ -7,7 +7,7 @@ prelude
 import Init.Data.Int.Basic
 import Init.Data.Nat.Gcd
 import Init.Data.Nat.Lcm
-import Init.Data.Int.DivModLemmas
+import Init.Data.Int.DivMod.Lemmas
 
 /-!
 Definition and lemmas for gcd and lcm over Int

src/Init/Data/Int/Lemmas.lean

@@ -78,7 +78,7 @@ theorem negSucc_eq (n : Nat) : -[n+1] = -((n : Int) + 1) := rfl
   | succ _ => rfl
   | -[_+1] => rfl
 
-protected theorem neg_inj {a b : Int} : -a = -b ↔ a = b :=
+@[simp] protected theorem neg_inj {a b : Int} : -a = -b ↔ a = b :=
   ⟨fun h => by rw [← Int.neg_neg a, ← Int.neg_neg b, h], congrArg _⟩
 
 @[simp] protected theorem neg_eq_zero : -a = 0 ↔ a = 0 := Int.neg_inj (b := 0)
@@ -91,7 +91,7 @@ theorem add_neg_one (i : Int) : i + -1 = i - 1 := rfl
 
 /- ## basic properties of subNatNat -/
 
--- @[elabAsElim] -- TODO(Mario): unexpected eliminator resulting type
+@[elab_as_elim]
 theorem subNatNat_elim (m n : Nat) (motive : Nat → Nat → Int → Prop)
     (hp : ∀ i n, motive (n + i) n i)
     (hn : ∀ i m, motive m (m + i + 1) -[i+1]) :
@@ -129,6 +129,17 @@ theorem subNatNat_of_le {m n : Nat} (h : n ≤ m) : subNatNat m n = ↑(m - n) :
 theorem subNatNat_of_lt {m n : Nat} (h : m < n) : subNatNat m n = -[pred (n - m) +1] :=
   subNatNat_of_sub_eq_succ <| (Nat.succ_pred_eq_of_pos (Nat.sub_pos_of_lt h)).symm
 
+@[simp] theorem subNat_eq_zero_iff {a b : Nat} : subNatNat a b = 0 ↔ a = b := by
+  cases Nat.lt_or_ge a b with
+  | inl h =>
+    rw [subNatNat_of_lt h]
+    simpa using ne_of_lt h
+  | inr h =>
+    rw [subNatNat_of_le h]
+    norm_cast
+    rw [Nat.sub_eq_iff_eq_add' h]
+    simp
+
 /- # Additive group properties -/
 
 /- addition -/
@@ -225,7 +236,7 @@ attribute [local simp] subNatNat_self
 @[local simp] protected theorem add_right_neg (a : Int) : a + -a = 0 := by
   rw [Int.add_comm, Int.add_left_neg]
 
-@[simp] protected theorem neg_eq_of_add_eq_zero {a b : Int} (h : a + b = 0) : -a = b := by
+protected theorem neg_eq_of_add_eq_zero {a b : Int} (h : a + b = 0) : -a = b := by
   rw [← Int.add_zero (-a), ← h, ← Int.add_assoc, Int.add_left_neg, Int.zero_add]
 
 protected theorem eq_neg_of_eq_neg {a b : Int} (h : a = -b) : b = -a := by
@@ -258,6 +269,17 @@ protected theorem add_left_cancel {a b c : Int} (h : a + b = a + c) : b = c := b
   rw [Int.add_right_neg, Int.add_comm a, ← Int.add_assoc, Int.add_assoc b,
     Int.add_right_neg, Int.add_zero, Int.add_right_neg]
 
+/--
+If a predicate on the integers is invariant under negation,
+then it is sufficient to prove it for the nonnegative integers.
+-/
+theorem wlog_sign {P : Int → Prop} (inv : ∀ a, P a ↔ P (-a)) (w : ∀ n : Nat, P n) (a : Int) : P a := by
+  cases a with
+  | ofNat n => exact w n
+  | negSucc n =>
+    rw [negSucc_eq, ← inv, ← ofNat_succ]
+    apply w
+
 /- ## subtraction -/
 
 @[simp] theorem negSucc_sub_one (n : Nat) : -[n+1] - 1 = -[n + 1 +1] := rfl
@@ -326,26 +348,40 @@ theorem toNat_sub (m n : Nat) : toNat (m - n) = m - n := by
   · exact (Nat.add_sub_cancel_left ..).symm
   · dsimp; rw [Nat.add_assoc, Nat.sub_eq_zero_of_le (Nat.le_add_right ..)]; rfl
 
+theorem toNat_of_nonpos : ∀ {z : Int}, z ≤ 0 → z.toNat = 0
+  | 0, _ => rfl
+  | -[_+1], _ => rfl
+
+@[simp] theorem neg_ofNat_eq_negSucc_iff {a b : Nat} : - (a : Int) = -[b+1] ↔ a = b + 1 := by
+  rw [Int.neg_eq_comm]
+  rw [Int.neg_negSucc]
+  norm_cast
+  simp [eq_comm]
+
+@[simp] theorem neg_ofNat_eq_negSucc_add_one_iff {a b : Nat} : - (a : Int) = -[b+1] + 1 ↔ a = b := by
+  cases b with
+  | zero => simp; norm_cast
+  | succ b =>
+    rw [Int.neg_eq_comm, ← Int.negSucc_sub_one, Int.sub_add_cancel, Int.neg_negSucc]
+    norm_cast
+    simp [eq_comm]
+
 /- ## add/sub injectivity -/
 
-@[simp]
 protected theorem add_left_inj {i j : Int} (k : Int) : (i + k = j + k) ↔ i = j := by
   apply Iff.intro
   · intro p
     rw [←Int.add_sub_cancel i k, ←Int.add_sub_cancel j k, p]
   · exact congrArg (· + k)
 
-@[simp]
 protected theorem add_right_inj {i j : Int} (k : Int) : (k + i = k + j) ↔ i = j := by
-  simp [Int.add_comm k]
+  simp [Int.add_comm k, Int.add_left_inj]
 
-@[simp]
 protected theorem sub_right_inj {i j : Int} (k : Int) : (k - i = k - j) ↔ i = j := by
-  simp [Int.sub_eq_add_neg, Int.neg_inj]
+  simp [Int.sub_eq_add_neg, Int.neg_inj, Int.add_right_inj]
 
-@[simp]
 protected theorem sub_left_inj {i j : Int} (k : Int) : (i - k = j - k) ↔ i = j := by
-  simp [Int.sub_eq_add_neg]
+  simp [Int.sub_eq_add_neg, Int.add_left_inj]
 
 /- ## Ring properties -/
 

src/Init/Data/Int/LemmasAux.lean

@@ -5,6 +5,7 @@ Authors: Kim Morrison
 -/
 prelude
 import Init.Data.Int.Order
+import Init.Data.Int.DivMod.Lemmas
 import Init.Omega
 
 
@@ -38,4 +39,59 @@ namespace Int
   simp [toNat]
   split <;> simp_all <;> omega
 
+theorem bmod_neg_iff {m : Nat} {x : Int} (h2 : -m ≤ x) (h1 : x < m) :
+    (x.bmod m) < 0 ↔ (-(m / 2) ≤ x ∧ x < 0) ∨ ((m + 1) / 2 ≤ x) := by
+  simp only [Int.bmod_def]
+  by_cases xpos : 0 ≤ x
+  · rw [Int.emod_eq_of_lt xpos (by omega)]; omega
+  · rw [Int.add_emod_self.symm, Int.emod_eq_of_lt (by omega) (by omega)]; omega
+
+@[simp] theorem natCast_le_zero : {n : Nat} → (n : Int) ≤ 0 ↔ n = 0 := by omega
+
+@[simp] theorem toNat_eq_zero : ∀ {n : Int}, n.toNat = 0 ↔ n ≤ 0 := by omega
+
+theorem eq_zero_of_dvd_of_natAbs_lt_natAbs {d n : Int} (h : d ∣ n) (h₁ : n.natAbs < d.natAbs) :
+    n = 0 := by
+  obtain ⟨a, rfl⟩ := h
+  rw [natAbs_mul] at h₁
+  suffices ¬ 0 < a.natAbs by simp [Int.natAbs_eq_zero.1 (Nat.eq_zero_of_not_pos this)]
+  exact fun h => Nat.lt_irrefl _ (Nat.lt_of_le_of_lt (Nat.le_mul_of_pos_right d.natAbs h) h₁)
+
+theorem bmod_eq_self_of_le {n : Int} {m : Nat} (hn' : -(m / 2) ≤ n) (hn : n < (m + 1) / 2) :
+    n.bmod m = n := by
+  rw [← Int.sub_eq_zero]
+  have := le_bmod (x := n) (m := m) (by omega)
+  have := bmod_lt (x := n) (m := m) (by omega)
+  apply eq_zero_of_dvd_of_natAbs_lt_natAbs Int.dvd_bmod_sub_self
+  omega
+
+protected theorem sub_eq_iff_eq_add {b a c : Int} : a - b = c ↔ a = c + b := by omega
+protected theorem sub_eq_iff_eq_add' {b a c : Int} : a - b = c ↔ a = b + c := by omega
+
+theorem bmod_bmod_of_dvd {a : Int} {n m : Nat} (hnm : n ∣ m) :
+    (a.bmod m).bmod n = a.bmod n := by
+  rw [← Int.sub_eq_iff_eq_add.2 (bmod_add_bdiv a m).symm]
+  obtain ⟨k, rfl⟩ := hnm
+  simp [Int.mul_assoc]
+
+@[simp] theorem toNat_le {m : Int} {n : Nat} : m.toNat ≤ n ↔ m ≤ n := by omega
+@[simp] theorem toNat_lt' {m : Int} {n : Nat} (hn : 0 < n) : m.toNat < n ↔ m < n := by omega
+
+@[simp] protected theorem neg_nonpos_iff (i : Int) : -i ≤ 0 ↔ 0 ≤ i := by omega
+
+@[simp] theorem zero_le_ofNat (n : Nat) : 0 ≤ ((no_index (OfNat.ofNat n)) : Int) :=
+  ofNat_nonneg _
+
+@[simp] theorem neg_natCast_le_natCast (n m : Nat) : -(n : Int) ≤ (m : Int) :=
+  Int.le_trans (by simp) (ofNat_zero_le m)
+
+@[simp] theorem neg_natCast_le_ofNat (n m : Nat) : -(n : Int) ≤ (no_index (OfNat.ofNat m)) :=
+  Int.le_trans (by simp) (ofNat_zero_le m)
+
+@[simp] theorem neg_ofNat_le_ofNat (n m : Nat) : -(no_index (OfNat.ofNat n)) ≤ (no_index (OfNat.ofNat m)) :=
+  Int.le_trans (by simp) (ofNat_zero_le m)
+
+@[simp] theorem neg_ofNat_le_natCast (n m : Nat) : -(no_index (OfNat.ofNat n)) ≤ (m : Int) :=
+  Int.le_trans (by simp) (ofNat_zero_le m)
+
 end Int

src/Init/Data/Int/OfNat.lean

@@ -0,0 +1,64 @@
+/-
+Copyright (c) 2025 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Data.Int.Lemmas
+import Init.Data.Int.DivMod
+import Init.Data.RArray
+
+namespace Int.OfNat
+/-!
+Helper definitions and theorems for converting `Nat` expressions into `Int` one.
+We use them to implement the arithmetic theories in `grind`
+-/
+
+abbrev Var := Nat
+abbrev Context := Lean.RArray Nat
+def Var.denote (ctx : Context) (v : Var) : Nat :=
+  ctx.get v
+
+inductive Expr where
+  | num  (v : Nat)
+  | var  (i : Var)
+  | add  (a b : Expr)
+  | mul  (a b : Expr)
+  | div  (a b : Expr)
+  | mod  (a b : Expr)
+
+def Expr.denote (ctx : Context) : Expr → Nat
+  | .num k    => k
+  | .var v    => v.denote ctx
+  | .add a b  => Nat.add (denote ctx a) (denote ctx b)
+  | .mul a b  => Nat.mul (denote ctx a) (denote ctx b)
+  | .div a b  => Nat.div (denote ctx a) (denote ctx b)
+  | .mod a b  => Nat.mod (denote ctx a) (denote ctx b)
+
+def Expr.denoteAsInt (ctx : Context) : Expr → Int
+  | .num k    => Int.ofNat k
+  | .var v    => Int.ofNat (v.denote ctx)
+  | .add a b  => Int.add (denoteAsInt ctx a) (denoteAsInt ctx b)
+  | .mul a b  => Int.mul (denoteAsInt ctx a) (denoteAsInt ctx b)
+  | .div a b  => Int.ediv (denoteAsInt ctx a) (denoteAsInt ctx b)
+  | .mod a b  => Int.emod (denoteAsInt ctx a) (denoteAsInt ctx b)
+
+@[local simp] private theorem fold_div (a b : Nat) : a.div b = a / b := rfl
+@[local simp] private theorem fold_mod (a b : Nat) : a.mod b = a % b := rfl
+
+theorem Expr.denoteAsInt_eq (ctx : Context) (e : Expr) : e.denoteAsInt ctx = e.denote ctx := by
+  induction e <;> simp [denote, denoteAsInt, Int.ofNat_ediv, *] <;> rfl
+
+theorem Expr.eq (ctx : Context) (lhs rhs : Expr)
+    : (lhs.denote ctx = rhs.denote ctx) = (lhs.denoteAsInt ctx = rhs.denoteAsInt ctx) := by
+  simp [denoteAsInt_eq, Int.ofNat_inj]
+
+theorem Expr.le (ctx : Context) (lhs rhs : Expr)
+    : (lhs.denote ctx ≤ rhs.denote ctx) = (lhs.denoteAsInt ctx ≤ rhs.denoteAsInt ctx) := by
+  simp [denoteAsInt_eq, Int.ofNat_le]
+
+theorem Expr.dvd (ctx : Context) (lhs rhs : Expr)
+    : (lhs.denote ctx ∣ rhs.denote ctx) = (lhs.denoteAsInt ctx ∣ rhs.denoteAsInt ctx) := by
+  simp [denoteAsInt_eq, Int.ofNat_dvd]
+
+end Int.OfNat

src/Init/Data/Int/Order.lean

@@ -56,7 +56,7 @@ protected theorem le_total (a b : Int) : a ≤ b ∨ b ≤ a :=
     let ⟨k, (hk : m + k = n)⟩ := Nat.le.dest h
     le.intro k (by rw [← hk]; rfl)⟩
 
-theorem ofNat_zero_le (n : Nat) : 0 ≤ (↑n : Int) := ofNat_le.2 n.zero_le
+@[simp] theorem ofNat_zero_le (n : Nat) : 0 ≤ (↑n : Int) := ofNat_le.2 n.zero_le
 
 theorem eq_ofNat_of_zero_le {a : Int} (h : 0 ≤ a) : ∃ n : Nat, a = n := by
   have t := le.dest_sub h; rwa [Int.sub_zero] at t
@@ -133,12 +133,15 @@ protected theorem lt_of_not_ge {a b : Int} (h : ¬a ≤ b) : b < a :=
 protected theorem not_le_of_gt {a b : Int} (h : b < a) : ¬a ≤ b :=
   (Int.lt_iff_le_not_le.mp h).right
 
-protected theorem not_le {a b : Int} : ¬a ≤ b ↔ b < a :=
+@[simp] protected theorem not_le {a b : Int} : ¬a ≤ b ↔ b < a :=
   Iff.intro Int.lt_of_not_ge Int.not_le_of_gt
 
-protected theorem not_lt {a b : Int} : ¬a < b ↔ b ≤ a :=
+@[simp] protected theorem not_lt {a b : Int} : ¬a < b ↔ b ≤ a :=
   by rw [← Int.not_le, Decidable.not_not]
 
+protected theorem le_of_not_gt {a b : Int} (h : ¬ a > b) : a ≤ b :=
+  Int.not_lt.mp h
+
 protected theorem lt_trichotomy (a b : Int) : a < b ∨ a = b ∨ b < a :=
   if eq : a = b then .inr <| .inl eq else
   if le : a ≤ b then .inl <| Int.lt_iff_le_and_ne.2 ⟨le, eq⟩ else
@@ -358,6 +361,10 @@ protected theorem sub_lt_self (a : Int) {b : Int} (h : 0 < b) : a - b < a :=
 
 theorem add_one_le_of_lt {a b : Int} (H : a < b) : a + 1 ≤ b := H
 
+protected theorem le_iff_lt_add_one {a b : Int} : a ≤ b ↔ a < b + 1 := by
+  rw [Int.lt_iff_add_one_le]
+  exact (Int.add_le_add_iff_right 1).symm
+
 /- ### Order properties and multiplication -/
 
 
@@ -425,7 +432,7 @@ protected theorem mul_le_mul_of_nonpos_left {a b c : Int}
 
 /- ## natAbs -/
 
-@[simp] theorem natAbs_ofNat (n : Nat) : natAbs ↑n = n := rfl
+@[simp, norm_cast] theorem natAbs_ofNat (n : Nat) : natAbs ↑n = n := rfl
 @[simp] theorem natAbs_negSucc (n : Nat) : natAbs -[n+1] = n.succ := rfl
 @[simp] theorem natAbs_zero : natAbs (0 : Int) = (0 : Nat) := rfl
 @[simp] theorem natAbs_one : natAbs (1 : Int) = (1 : Nat) := rfl
@@ -470,6 +477,13 @@ theorem natAbs_of_nonneg {a : Int} (H : 0 ≤ a) : (natAbs a : Int) = a :=
 theorem ofNat_natAbs_of_nonpos {a : Int} (H : a ≤ 0) : (natAbs a : Int) = -a := by
   rw [← natAbs_neg, natAbs_of_nonneg (Int.neg_nonneg_of_nonpos H)]
 
+theorem natAbs_sub_of_nonneg_of_le {a b : Int} (h₁ : 0 ≤ b) (h₂ : b ≤ a) :
+    (a - b).natAbs = a.natAbs - b.natAbs := by
+  rw [← Int.ofNat_inj]
+  rw [natAbs_of_nonneg, ofNat_sub, natAbs_of_nonneg (Int.le_trans h₁ h₂), natAbs_of_nonneg h₁]
+  · rwa [← Int.ofNat_le, natAbs_of_nonneg h₁, natAbs_of_nonneg (Int.le_trans h₁ h₂)]
+  · exact Int.sub_nonneg_of_le h₂
+
 /-! ### toNat -/
 
 theorem toNat_eq_max : ∀ a : Int, (toNat a : Int) = max a 0
@@ -938,6 +952,22 @@ protected theorem mul_self_le_mul_self {a b : Int} (h1 : 0 ≤ a) (h2 : a ≤ b)
 protected theorem mul_self_lt_mul_self {a b : Int} (h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b :=
   Int.mul_lt_mul' (Int.le_of_lt h2) h2 h1 (Int.lt_of_le_of_lt h1 h2)
 
+protected theorem nonneg_of_mul_nonneg_left {a b : Int}
+    (h : 0 ≤ a * b) (hb : 0 < b) : 0 ≤ a :=
+  Int.le_of_not_gt fun ha => Int.not_le_of_gt (Int.mul_neg_of_neg_of_pos ha hb) h
+
+protected theorem nonneg_of_mul_nonneg_right {a b : Int}
+    (h : 0 ≤ a * b) (ha : 0 < a) : 0 ≤ b :=
+  Int.le_of_not_gt fun hb => Int.not_le_of_gt (Int.mul_neg_of_pos_of_neg ha hb) h
+
+protected theorem nonpos_of_mul_nonpos_left {a b : Int}
+    (h : a * b ≤ 0) (hb : 0 < b) : a ≤ 0 :=
+  Int.le_of_not_gt fun ha : a > 0 => Int.not_le_of_gt (Int.mul_pos ha hb) h
+
+protected theorem nonpos_of_mul_nonpos_right {a b : Int}
+    (h : a * b ≤ 0) (ha : 0 < a) : b ≤ 0 :=
+  Int.le_of_not_gt fun hb : b > 0 => Int.not_le_of_gt (Int.mul_pos ha hb) h
+
 /- ## sign -/
 
 @[simp] theorem sign_zero : sign 0 = 0 := rfl
@@ -1011,11 +1041,22 @@ theorem sign_eq_neg_one_iff_neg {a : Int} : sign a = -1 ↔ a < 0 :=
     exact Int.le_add_one (ofNat_nonneg _)
   | .negSucc _ => simp +decide [sign]
 
-theorem mul_sign : ∀ i : Int, i * sign i = natAbs i
+@[simp] theorem mul_sign_self : ∀ i : Int, i * sign i = natAbs i
   | succ _ => Int.mul_one _
   | 0 => Int.mul_zero _
   | -[_+1] => Int.mul_neg_one _
 
+@[deprecated mul_sign_self (since := "2025-02-24")] abbrev mul_sign := @mul_sign_self
+
+@[simp] theorem sign_mul_self : sign i * i = natAbs i := by
+  rw [Int.mul_comm, mul_sign_self]
+
+theorem sign_trichotomy (a : Int) : sign a = 1 ∨ sign a = 0 ∨ sign a = -1 := by
+  match a with
+  | 0 => simp
+  | .ofNat (_ + 1) => simp
+  | .negSucc _ => simp
+
 /- ## natAbs -/
 
 theorem natAbs_ne_zero {a : Int} : a.natAbs ≠ 0 ↔ a ≠ 0 := not_congr Int.natAbs_eq_zero

src/Init/Data/Int/Pow.lean

@@ -17,24 +17,14 @@ protected theorem pow_succ (b : Int) (e : Nat) : b ^ (e+1) = (b ^ e) * b := rfl
 protected theorem pow_succ' (b : Int) (e : Nat) : b ^ (e+1) = b * (b ^ e) := by
   rw [Int.mul_comm, Int.pow_succ]
 
-theorem pow_le_pow_of_le_left {n m : Nat} (h : n ≤ m) : ∀ (i : Nat), n^i ≤ m^i
-  | 0      => Nat.le_refl _
-  | i + 1 => Nat.mul_le_mul (pow_le_pow_of_le_left h i) h
+@[deprecated Nat.pow_le_pow_left (since := "2025-02-17")]
+abbrev pow_le_pow_of_le_left := @Nat.pow_le_pow_left
 
-theorem pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} : ∀ {j}, i ≤ j → n^i ≤ n^j
-  | 0,      h =>
-    have : i = 0 := Nat.eq_zero_of_le_zero h
-    this.symm ▸ Nat.le_refl _
-  | j + 1, h =>
-    match Nat.le_or_eq_of_le_succ h with
-    | Or.inl h => show n^i ≤ n^j * n from
-      have : n^i * 1 ≤ n^j * n := Nat.mul_le_mul (pow_le_pow_of_le_right hx h) hx
-      Nat.mul_one (n^i) ▸ this
-    | Or.inr h =>
-      h.symm ▸ Nat.le_refl _
+@[deprecated Nat.pow_le_pow_right (since := "2025-02-17")]
+abbrev pow_le_pow_of_le_right := @Nat.pow_le_pow_right
 
-theorem pos_pow_of_pos {n : Nat} (m : Nat) (h : 0 < n) : 0 < n^m :=
-  pow_le_pow_of_le_right h (Nat.zero_le _)
+@[deprecated Nat.pow_pos (since := "2025-02-17")]
+abbrev pos_pow_of_pos := @Nat.pow_pos
 
 @[norm_cast]
 theorem natCast_pow (b n : Nat) : ((b^n : Nat) : Int) = (b : Int) ^ n := by

src/Init/Data/List/Attach.lean

@@ -6,6 +6,10 @@ Authors: Mario Carneiro
 prelude
 import Init.Data.List.Count
 import Init.Data.Subtype
+import Init.BinderNameHint
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 namespace List
 
@@ -39,12 +43,12 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
     List β := (l.attachWith _ H).map fun ⟨x, h'⟩ => f x h'
 
 @[csimp] private theorem pmap_eq_pmapImpl : @pmap = @pmapImpl := by
-  funext α β p f L h'
-  let rec go : ∀ L' (hL' : ∀ ⦃x⦄, x ∈ L' → p x),
-      pmap f L' hL' = map (fun ⟨x, hx⟩ => f x hx) (pmap Subtype.mk L' hL')
+  funext α β p f l h'
+  let rec go : ∀ l' (hL' : ∀ ⦃x⦄, x ∈ l' → p x),
+      pmap f l' hL' = map (fun ⟨x, hx⟩ => f x hx) (pmap Subtype.mk l' hL')
   | nil, hL' => rfl
-  | cons _ L', hL' => congrArg _ <| go L' fun _ hx => hL' (.tail _ hx)
-  exact go L h'
+  | cons _ l', hL' => congrArg _ <| go l' fun _ hx => hL' (.tail _ hx)
+  exact go l h'
 
 @[simp] theorem pmap_nil {P : α → Prop} (f : ∀ a, P a → β) : pmap f [] (by simp) = [] := rfl
 
@@ -119,27 +123,26 @@ theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l H) :
   | cons a l ih =>
     simp [pmap, attachWith, ih]
 
-theorem attach_map_coe (l : List α) (f : α → β) :
+theorem attach_map_val (l : List α) (f : α → β) :
     (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
   rw [attach, attachWith, map_pmap]; exact pmap_eq_map _ _ _ _
 
-theorem attach_map_val (l : List α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
-  attach_map_coe _ _
+@[deprecated attach_map_val (since := "2025-02-17")]
+abbrev attach_map_coe := @attach_map_val
 
 theorem attach_map_subtype_val (l : List α) : l.attach.map Subtype.val = l :=
-  (attach_map_coe _ _).trans (List.map_id _)
+  (attach_map_val _ _).trans (List.map_id _)
 
-theorem attachWith_map_coe {p : α → Prop} (f : α → β) (l : List α) (H : ∀ a ∈ l, p a) :
+theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : List α) (H : ∀ a ∈ l, p a) :
     ((l.attachWith p H).map fun (i : { i // p i}) => f i) = l.map f := by
   rw [attachWith, map_pmap]; exact pmap_eq_map _ _ _ _
 
-theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : List α) (H : ∀ a ∈ l, p a) :
-    ((l.attachWith p H).map fun i => f i.val) = l.map f :=
-  attachWith_map_coe _ _ _
+@[deprecated attachWith_map_val (since := "2025-02-17")]
+abbrev attachWith_map_coe := @attachWith_map_val
 
 theorem attachWith_map_subtype_val {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) :
     (l.attachWith p H).map Subtype.val = l :=
-  (attachWith_map_coe _ _ _).trans (List.map_id _)
+  (attachWith_map_val _ _ _).trans (List.map_id _)
 
 @[simp]
 theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
@@ -178,7 +181,7 @@ theorem length_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : (pmap f l
   · simp only [*, pmap, length]
 
 @[simp]
-theorem length_attach {L : List α} : L.attach.length = L.length :=
+theorem length_attach {l : List α} : l.attach.length = l.length :=
   length_pmap
 
 @[simp]
@@ -187,7 +190,7 @@ theorem length_attachWith {p : α → Prop} {l H} : length (l.attachWith p H) =
 
 @[simp]
 theorem pmap_eq_nil_iff {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = [] ↔ l = [] := by
-  rw [← length_eq_zero, length_pmap, length_eq_zero]
+  rw [← length_eq_zero_iff, length_pmap, length_eq_zero_iff]
 
 theorem pmap_ne_nil_iff {P : α → Prop} (f : (a : α) → P a → β) {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) : xs.pmap f H ≠ [] ↔ xs ≠ [] := by
@@ -222,42 +225,39 @@ theorem attachWith_ne_nil_iff {l : List α} {P : α → Prop} {H : ∀ a ∈ l,
 @[deprecated attach_ne_nil_iff (since := "2024-09-06")] abbrev attach_ne_nil := @attach_ne_nil_iff
 
 @[simp]
-theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
-    (pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (mem_of_getElem? H) := by
-  induction l generalizing n with
+theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (i : Nat) :
+    (pmap f l h)[i]? = Option.pmap f l[i]? fun x H => h x (mem_of_getElem? H) := by
+  induction l generalizing i with
   | nil => simp
   | cons hd tl hl =>
-    rcases n with ⟨n⟩
+    rcases i with ⟨i⟩
     · simp only [Option.pmap]
       split <;> simp_all
-    · simp only [hl, pmap, Option.pmap, getElem?_cons_succ]
-      split <;> rename_i h₁ _ <;> split <;> rename_i h₂ _
-      · simp_all
-      · simp at h₂
-        simp_all
-      · simp_all
-      · simp_all
+    · simp only [pmap, getElem?_cons_succ, hl, Option.pmap]
 
+set_option linter.deprecated false in
+@[deprecated List.getElem?_pmap (since := "2025-02-12")]
 theorem get?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
     get? (pmap f l h) n = Option.pmap f (get? l n) fun x H => h x (mem_of_get? H) := by
   simp only [get?_eq_getElem?]
   simp [getElem?_pmap, h]
 
 @[simp]
-theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) {n : Nat}
-    (hn : n < (pmap f l h).length) :
-    (pmap f l h)[n] =
-      f (l[n]'(@length_pmap _ _ p f l h ▸ hn))
+theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) {i : Nat}
+    (hn : i < (pmap f l h).length) :
+    (pmap f l h)[i] =
+      f (l[i]'(@length_pmap _ _ p f l h ▸ hn))
         (h _ (getElem_mem (@length_pmap _ _ p f l h ▸ hn))) := by
-  induction l generalizing n with
+  induction l generalizing i with
   | nil =>
     simp only [length, pmap] at hn
-    exact absurd hn (Nat.not_lt_of_le n.zero_le)
+    exact absurd hn (Nat.not_lt_of_le i.zero_le)
   | cons hd tl hl =>
-    cases n
+    cases i
     · simp
     · simp [hl]
 
+@[deprecated getElem_pmap (since := "2025-02-13")]
 theorem get_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) {n : Nat}
     (hn : n < (pmap f l h).length) :
     get (pmap f l h) ⟨n, hn⟩ =
@@ -416,7 +416,12 @@ theorem attachWith_map {l : List α} (f : α → β) {P : β → Prop} {H : ∀
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
   induction l <;> simp [*]
 
-theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
+@[simp] theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
+    (f : { x // P x } → β) :
+    (l.attachWith P H).map f = l.attach.map fun ⟨x, h⟩ => f ⟨x, H _ h⟩ := by
+  induction l <;> simp_all
+
+theorem map_attachWith_eq_pmap {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
     (f : { x // P x } → β) :
     (l.attachWith P H).map f =
       l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
@@ -428,7 +433,7 @@ theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈
     simp
 
 /-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
-theorem map_attach {l : List α} (f : { x // x ∈ l } → β) :
+theorem map_attach_eq_pmap {l : List α} (f : { x // x ∈ l } → β) :
     l.attach.map f = l.pmap (fun a h => f ⟨a, h⟩) (fun _ => id) := by
   induction l with
   | nil => rfl
@@ -437,6 +442,9 @@ theorem map_attach {l : List α} (f : { x // x ∈ l } → β) :
     apply pmap_congr_left
     simp
 
+@[deprecated map_attach_eq_pmap (since := "2025-02-09")]
+abbrev map_attach := @map_attach_eq_pmap
+
 theorem attach_filterMap {l : List α} {f : α → Option β} :
     (l.filterMap f).attach = l.attach.filterMap
       fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩) := by
@@ -654,6 +662,10 @@ def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) : List α :
 @[simp] theorem unattach_cons {p : α → Prop} {a : { x // p x }} {l : List { x // p x }} :
   (a :: l).unattach = a.val :: l.unattach := rfl
 
+@[simp] theorem mem_unattach {p : α → Prop} {l : List { x // p x }} {a} :
+    a ∈ l.unattach ↔ ∃ h : p a, ⟨a, h⟩ ∈ l := by
+  simp only [unattach, mem_map, Subtype.exists, exists_and_right, exists_eq_right]
+
 @[simp] theorem length_unattach {p : α → Prop} {l : List { x // p x }} :
     l.unattach.length = l.length := by
   unfold unattach
@@ -758,6 +770,16 @@ and simplifies these to the function directly taking the value.
     simp [hf, find?_cons]
     split <;> simp [ih]
 
+@[simp] theorem all_subtype {p : α → Prop} {l : List { x // p x }} {f : { x // p x } → Bool} {g : α → Bool}
+    (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    l.all f = l.unattach.all g := by
+  simp [all_eq, hf]
+
+@[simp] theorem any_subtype {p : α → Prop} {l : List { x // p x }} {f : { x // p x } → Bool} {g : α → Bool}
+    (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    l.any f = l.unattach.any g := by
+  simp [any_eq, hf]
+
 /-! ### Simp lemmas pushing `unattach` inwards. -/
 
 @[simp] theorem unattach_filter {p : α → Prop} {l : List { x // p x }}
@@ -788,4 +810,66 @@ and simplifies these to the function directly taking the value.
     (List.replicate n x).unattach = List.replicate n x.1 := by
   simp [unattach, -map_subtype]
 
+/-! ### Well-founded recursion preprocessing setup -/
+
+@[wf_preprocess] theorem map_wfParam (xs : List α) (f : α → β) :
+    (wfParam xs).map f = xs.attach.unattach.map f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem map_unattach (P : α → Prop) (xs : List (Subtype P)) (f : α → β) :
+    xs.unattach.map f = xs.map fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldl_wfParam (xs : List α) (f : β → α → β) (x : β) :
+    (wfParam xs).foldl f x = xs.attach.unattach.foldl f x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldl_unattach (P : α → Prop) (xs : List (Subtype P)) (f : β → α → β) (x : β):
+    xs.unattach.foldl f x = xs.foldl (fun s ⟨x, h⟩ =>
+      binderNameHint s f <| binderNameHint x (f s) <| binderNameHint h () <| f s (wfParam x)) x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldr_wfParam (xs : List α) (f : α → β → β) (x : β) :
+    (wfParam xs).foldr f x = xs.attach.unattach.foldr f x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem foldr_unattach (P : α → Prop) (xs : List (Subtype P)) (f : α → β → β) (x : β):
+    xs.unattach.foldr f x = xs.foldr (fun ⟨x, h⟩ s =>
+      binderNameHint x f <| binderNameHint s (f x) <| binderNameHint h () <| f (wfParam x) s) x := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filter_wfParam (xs : List α) (f : α → Bool) :
+    (wfParam xs).filter f = xs.attach.unattach.filter f:= by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filter_unattach (P : α → Prop) (xs : List (Subtype P)) (f : α → Bool) :
+    xs.unattach.filter f = (xs.filter (fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x))).unattach := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem reverse_wfParam (xs : List α) :
+    (wfParam xs).reverse = xs.attach.unattach.reverse := by simp [wfParam]
+
+@[wf_preprocess] theorem reverse_unattach (P : α → Prop) (xs : List (Subtype P)) :
+    xs.unattach.reverse = xs.reverse.unattach := by simp
+
+@[wf_preprocess] theorem filterMap_wfParam (xs : List α) (f : α → Option β) :
+    (wfParam xs).filterMap f = xs.attach.unattach.filterMap f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem filterMap_unattach (P : α → Prop) (xs : List (Subtype P)) (f : α → Option β) :
+    xs.unattach.filterMap f = xs.filterMap fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem flatMap_wfParam (xs : List α) (f : α → List β) :
+    (wfParam xs).flatMap f = xs.attach.unattach.flatMap f := by
+  simp [wfParam]
+
+@[wf_preprocess] theorem flatMap_unattach (P : α → Prop) (xs : List (Subtype P)) (f : α → List β) :
+    xs.unattach.flatMap f = xs.flatMap fun ⟨x, h⟩ =>
+      binderNameHint x f <| binderNameHint h () <| f (wfParam x) := by
+  simp [wfParam]
+
 end List

src/Init/Data/List/Basic.lean

@@ -58,6 +58,8 @@ Further operations are defined in `Init.Data.List.BasicAux`
 -/
 
 set_option linter.missingDocs true -- keep it documented
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 open Decidable List
 
@@ -204,7 +206,7 @@ instance decidableLT [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List
 abbrev hasDecidableLt := @decidableLT
 
 /-- The lexicographic order on lists. -/
-@[reducible] protected def le [LT α] (a b : List α) : Prop := ¬ b < a
+@[reducible] protected def le [LT α] (as bs : List α) : Prop := ¬ bs < as
 
 instance instLE [LT α] : LE (List α) := ⟨List.le⟩
 
@@ -225,54 +227,27 @@ def lex [BEq α] (l₁ l₂ : List α) (lt : α → α → Bool := by exact (·
   | _,      []       => false
   | a :: as, b :: bs => lt a b || (a == b && lex as bs lt)
 
-@[simp] theorem lex_nil_nil [BEq α] : lex ([] : List α) [] lt = false := rfl
-@[simp] theorem lex_nil_cons [BEq α] {b} {bs : List α} : lex [] (b :: bs) lt = true := rfl
-@[simp] theorem lex_cons_nil [BEq α] {a} {as : List α} : lex (a :: as) [] lt = false := rfl
-@[simp] theorem lex_cons_cons [BEq α] {a b} {as bs : List α} :
+theorem nil_lex_nil [BEq α] : lex ([] : List α) [] lt = false := rfl
+@[simp] theorem nil_lex_cons [BEq α] {b} {bs : List α} : lex [] (b :: bs) lt = true := rfl
+theorem cons_lex_nil [BEq α] {a} {as : List α} : lex (a :: as) [] lt = false := rfl
+@[simp] theorem cons_lex_cons [BEq α] {a b} {as bs : List α} :
+    lex (a :: as) (b :: bs) lt = (lt a b || (a == b && lex as bs lt)) := rfl
+
+@[simp] theorem lex_nil [BEq α] {as : List α} : lex as [] lt = false := by
+  cases as <;> simp [nil_lex_nil, cons_lex_nil]
+
+@[deprecated nil_lex_nil (since := "2025-02-10")]
+theorem lex_nil_nil [BEq α] : lex ([] : List α) [] lt = false := rfl
+@[deprecated nil_lex_cons (since := "2025-02-10")]
+theorem lex_nil_cons [BEq α] {b} {bs : List α} : lex [] (b :: bs) lt = true := rfl
+@[deprecated cons_lex_nil (since := "2025-02-10")]
+theorem lex_cons_nil [BEq α] {a} {as : List α} : lex (a :: as) [] lt = false := rfl
+@[deprecated cons_lex_cons (since := "2025-02-10")]
+theorem lex_cons_cons [BEq α] {a b} {as bs : List α} :
     lex (a :: as) (b :: bs) lt = (lt a b || (a == b && lex as bs lt)) := rfl
 
 /-! ## Alternative getters -/
 
-/-! ### get? -/
-
-/--
-Returns the `i`-th element in the list (zero-based).
-
-If the index is out of bounds (`i ≥ as.length`), this function returns `none`.
-Also see `get`, `getD` and `get!`.
--/
-def get? : (as : List α) → (i : Nat) → Option α
-  | a::_,  0   => some a
-  | _::as, n+1 => get? as n
-  | _,     _   => none
-
-@[simp] theorem get?_nil : @get? α [] n = none := rfl
-@[simp] theorem get?_cons_zero : @get? α (a::l) 0 = some a := rfl
-@[simp] theorem get?_cons_succ : @get? α (a::l) (n+1) = get? l n := rfl
-
-theorem ext_get? : ∀ {l₁ l₂ : List α}, (∀ n, l₁.get? n = l₂.get? n) → l₁ = l₂
-  | [], [], _ => rfl
-  | _ :: _, [], h => nomatch h 0
-  | [], _ :: _, h => nomatch h 0
-  | a :: l₁, a' :: l₂, h => by
-    have h0 : some a = some a' := h 0
-    injection h0 with aa; simp only [aa, ext_get? fun n => h (n+1)]
-
-/-! ### getD -/
-
-/--
-Returns the `i`-th element in the list (zero-based).
-
-If the index is out of bounds (`i ≥ as.length`), this function returns `fallback`.
-See also `get?` and `get!`.
--/
-def getD (as : List α) (i : Nat) (fallback : α) : α :=
-  (as.get? i).getD fallback
-
-@[simp] theorem getD_nil : getD [] n d = d := rfl
-@[simp] theorem getD_cons_zero : getD (x :: xs) 0 d = x := rfl
-@[simp] theorem getD_cons_succ : getD (x :: xs) (n + 1) d = getD xs n d := rfl
-
 /-! ### getLast -/
 
 /--
@@ -382,14 +357,15 @@ def tail? : List α → Option (List α)
 
 /-! ### tailD -/
 
+set_option linter.listVariables false in
 /--
 Drops the first element of the list.
 
 If the list is empty, this function returns `fallback`.
 Also see `head?` and `head!`.
 -/
-def tailD (list fallback : List α) : List α :=
-  match list with
+def tailD (l fallback : List α) : List α :=
+  match l with
   | [] => fallback
   | _ :: tl => tl
 
@@ -581,10 +557,10 @@ theorem reverseAux_eq_append (as bs : List α) : reverseAux as bs = reverseAux a
 -/
 def flatten : List (List α) → List α
   | []      => []
-  | a :: as => a ++ flatten as
+  | l :: L => l ++ flatten L
 
 @[simp] theorem flatten_nil : List.flatten ([] : List (List α)) = [] := rfl
-@[simp] theorem flatten_cons : (l :: ls).flatten = l ++ ls.flatten := rfl
+@[simp] theorem flatten_cons : (l :: L).flatten = l ++ L.flatten := rfl
 
 @[deprecated flatten (since := "2024-10-14"), inherit_doc flatten] abbrev join := @flatten
 
@@ -603,7 +579,7 @@ set_option linter.missingDocs false in
 to get a list of lists, and then concatenates them all together.
 * `[2, 3, 2].bind range = [0, 1, 0, 1, 2, 0, 1]`
 -/
-@[inline] def flatMap {α : Type u} {β : Type v} (b : α → List β) (a : List α) : List β := flatten (map b a)
+@[inline] def flatMap {α : Type u} {β : Type v} (b : α → List β) (as : List α) : List β := flatten (map b as)
 
 @[simp] theorem flatMap_nil (f : α → List β) : List.flatMap f [] = [] := by simp [flatten, List.flatMap]
 @[simp] theorem flatMap_cons x xs (f : α → List β) :
@@ -808,14 +784,14 @@ def take : Nat → List α → List α
 * `drop 6 [a, b, c, d, e] = []`
 -/
 def drop : Nat → List α → List α
-  | 0,   a     => a
+  | 0,   as     => as
   | _+1, []    => []
   | n+1, _::as => drop n as
 
 @[simp] theorem drop_nil : ([] : List α).drop i = [] := by
   cases i <;> rfl
 @[simp] theorem drop_zero (l : List α) : l.drop 0 = l := rfl
-@[simp] theorem drop_succ_cons : (a :: l).drop (n + 1) = l.drop n := rfl
+@[simp] theorem drop_succ_cons : (a :: l).drop (i + 1) = l.drop i := rfl
 
 theorem drop_eq_nil_of_le {as : List α} {i : Nat} (h : as.length ≤ i) : as.drop i = [] := by
   match as, i with
@@ -1049,15 +1025,15 @@ def splitAt (n : Nat) (l : List α) : List α × List α := go l n [] where
 * `rotateLeft [1, 2, 3, 4, 5] 5 = [1, 2, 3, 4, 5]`
 * `rotateLeft [1, 2, 3, 4, 5] = [2, 3, 4, 5, 1]`
 -/
-def rotateLeft (xs : List α) (n : Nat := 1) : List α :=
+def rotateLeft (xs : List α) (i : Nat := 1) : List α :=
   let len := xs.length
   if len ≤ 1 then
     xs
   else
-    let n := n % len
-    let b := xs.take n
-    let e := xs.drop n
-    e ++ b
+    let i := i % len
+    let ys := xs.take i
+    let zs := xs.drop i
+    zs ++ ys
 
 @[simp] theorem rotateLeft_nil : ([] : List α).rotateLeft n = [] := rfl
 
@@ -1070,15 +1046,15 @@ def rotateLeft (xs : List α) (n : Nat := 1) : List α :=
 * `rotateRight [1, 2, 3, 4, 5] 5 = [1, 2, 3, 4, 5]`
 * `rotateRight [1, 2, 3, 4, 5] = [5, 1, 2, 3, 4]`
 -/
-def rotateRight (xs : List α) (n : Nat := 1) : List α :=
+def rotateRight (xs : List α) (i : Nat := 1) : List α :=
   let len := xs.length
   if len ≤ 1 then
     xs
   else
-    let n := len - n % len
-    let b := xs.take n
-    let e := xs.drop n
-    e ++ b
+    let i := len - i % len
+    let ys := xs.take i
+    let zs := xs.drop i
+    zs ++ ys
 
 @[simp] theorem rotateRight_nil : ([] : List α).rotateRight n = [] := rfl
 
@@ -1193,8 +1169,8 @@ def modify (f : α → α) : Nat → List α → List α :=
 insertIdx 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]
 ```
 -/
-def insertIdx (n : Nat) (a : α) : List α → List α :=
-  modifyTailIdx (cons a) n
+def insertIdx (i : Nat) (a : α) : List α → List α :=
+  modifyTailIdx (cons a) i
 
 /-! ### erase -/
 
@@ -1367,13 +1343,13 @@ and returns the first `β` value corresponding to an `α` value in the list equa
 -/
 def lookup [BEq α] : α → List (α × β) → Option β
   | _, []        => none
-  | a, (k,b)::es => match a == k with
+  | a, (k,b)::as => match a == k with
     | true  => some b
-    | false => lookup a es
+    | false => lookup a as
 
 @[simp] theorem lookup_nil [BEq α] : ([] : List (α × β)).lookup a = none := rfl
 theorem lookup_cons [BEq α] {k : α} :
-    ((k,b)::es).lookup a = match a == k with | true => some b | false => es.lookup a :=
+    ((k,b)::as).lookup a = match a == k with | true => some b | false => as.lookup a :=
   rfl
 
 /-! ## Permutations -/
@@ -1519,11 +1495,11 @@ def zipWithAll (f : Option α → Option β → γ) : List α → List β → Li
 -/
 def unzip : List (α × β) → List α × List β
   | []          => ([], [])
-  | (a, b) :: t => match unzip t with | (al, bl) => (a::al, b::bl)
+  | (a, b) :: t => match unzip t with | (as, bs) => (a::as, b::bs)
 
 @[simp] theorem unzip_nil : ([] : List (α × β)).unzip = ([], []) := rfl
 @[simp] theorem unzip_cons {h : α × β} :
-    (h :: t).unzip = match unzip t with | (al, bl) => (h.1::al, h.2::bl) := rfl
+    (h :: t).unzip = match unzip t with | (as, bs) => (h.1::as, h.2::bs) := rfl
 
 /-! ## Ranges and enumeration -/
 
@@ -1558,8 +1534,8 @@ def range (n : Nat) : List Nat :=
   loop n []
 where
   loop : Nat → List Nat → List Nat
-  | 0,   ns => ns
-  | n+1, ns => loop n (n::ns)
+  | 0,   acc => acc
+  | n+1, acc => loop n (n::acc)
 
 @[simp] theorem range_zero : range 0 = [] := rfl
 
@@ -1690,6 +1666,7 @@ def intersperse (sep : α) : List α → List α
 
 /-! ### intercalate -/
 
+set_option linter.listVariables false in
 /--
 `O(|xs|)`. `intercalate sep xs` alternates `sep` and the elements of `xs`:
 * `intercalate sep [] = []`
@@ -1726,10 +1703,10 @@ def eraseReps {α} [BEq α] : List α → List α
   | a::as => loop a as []
 where
   loop {α} [BEq α] : α → List α → List α → List α
-  | a, [], rs => (a::rs).reverse
-  | a, a'::as, rs => match a == a' with
-    | true  => loop a as rs
-    | false => loop a' as (a::rs)
+  | a, [], acc => (a::acc).reverse
+  | a, a'::as, acc => match a == a' with
+    | true  => loop a as acc
+    | false => loop a' as (a::acc)
 
 /-! ### span -/
 
@@ -1745,10 +1722,10 @@ and the second part is everything else.
   loop as []
 where
   @[specialize] loop : List α → List α → List α × List α
-  | [],    rs => (rs.reverse, [])
-  | a::as, rs => match p a with
-    | true  => loop as (a::rs)
-    | false => (rs.reverse, a::as)
+  | [],    acc => (acc.reverse, [])
+  | a::as, acc => match p a with
+    | true  => loop as (a::acc)
+    | false => (acc.reverse, a::as)
 
 /-! ### splitBy -/
 
@@ -1764,27 +1741,27 @@ such that adjacent elements are related by `R`.
   | a::as => loop as a [] []
 where
   /--
-  The arguments of `splitBy.loop l ag g gs` represent the following:
+  The arguments of `splitBy.loop l b g gs` represent the following:
 
   - `l : List α` are the elements which we still need to split.
-  - `ag : α` is the previous element for which a comparison was performed.
-  - `g : List α` is the group currently being assembled, in **reverse order**.
-  - `gs : List (List α)` is all of the groups that have been completed, in **reverse order**.
+  - `b : α` is the previous element for which a comparison was performed.
+  - `r : List α` is the group currently being assembled, in **reverse order**.
+  - `acc : List (List α)` is all of the groups that have been completed, in **reverse order**.
   -/
   @[specialize] loop : List α → α → List α → List (List α) → List (List α)
-  | a::as, ag, g, gs => match R ag a with
-    | true  => loop as a (ag::g) gs
-    | false => loop as a [] ((ag::g).reverse::gs)
-  | [], ag, g, gs => ((ag::g).reverse::gs).reverse
+  | a::as, b, r, acc => match R b a with
+    | true  => loop as a (b::r) acc
+    | false => loop as a [] ((b::r).reverse::acc)
+  | [], ag, r, acc => ((ag::r).reverse::acc).reverse
 
 @[deprecated splitBy (since := "2024-10-30"), inherit_doc splitBy] abbrev groupBy := @splitBy
 
 /-! ### removeAll -/
 
-/-- `O(|xs|)`. Computes the "set difference" of lists,
+/-- `O(|xs| * |ys|)`. Computes the "set difference" of lists,
 by filtering out all elements of `xs` which are also in `ys`.
 * `removeAll [1, 1, 5, 1, 2, 4, 5] [1, 2, 2] = [5, 4, 5]`
- -/
+-/
 def removeAll [BEq α] (xs ys : List α) : List α :=
   xs.filter (fun x => !ys.elem x)
 
@@ -1841,10 +1818,10 @@ theorem mapTR_loop_eq (f : α → β) (as : List α) (bs : List β) :
   loop as []
 where
   @[specialize] loop : List α → List α → List α
-  | [],    rs => rs.reverse
-  | a::as, rs => match p a with
-     | true  => loop as (a::rs)
-     | false => loop as rs
+  | [],    acc => acc.reverse
+  | a::as, acc => match p a with
+     | true  => loop as (a::acc)
+     | false => loop as acc
 
 theorem filterTR_loop_eq (p : α → Bool) (as bs : List α) :
     filterTR.loop p as bs = bs.reverse ++ filter p as := by
@@ -1900,7 +1877,7 @@ theorem replicateTR_loop_eq : ∀ n, replicateTR.loop a n acc = replicate n a ++
 
 /-- Tail recursive version of `List.unzip`. -/
 def unzipTR (l : List (α × β)) : List α × List β :=
-  l.foldr (fun (a, b) (al, bl) => (a::al, b::bl)) ([], [])
+  l.foldr (fun (a, b) (as, bs) => (a::as, b::bs)) ([], [])
 
 @[csimp] theorem unzip_eq_unzipTR : @unzip = @unzipTR := by
   apply funext; intro α; apply funext; intro β; apply funext; intro l

src/Init/Data/List/BasicAux.lean

@@ -6,6 +6,9 @@ Author: Leonardo de Moura
 prelude
 import Init.Data.Nat.Linear
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 universe u
 
 namespace List
@@ -14,6 +17,40 @@ namespace List
 
 /-! ## Alternative getters -/
 
+/-! ### get? -/
+
+/--
+Returns the `i`-th element in the list (zero-based).
+
+If the index is out of bounds (`i ≥ as.length`), this function returns `none`.
+Also see `get`, `getD` and `get!`.
+-/
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
+def get? : (as : List α) → (i : Nat) → Option α
+  | a::_,  0   => some a
+  | _::as, n+1 => get? as n
+  | _,     _   => none
+
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12"), simp]
+theorem get?_nil : @get? α [] n = none := rfl
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12"), simp]
+theorem get?_cons_zero : @get? α (a::l) 0 = some a := rfl
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12"), simp]
+theorem get?_cons_succ : @get? α (a::l) (n+1) = get? l n := rfl
+
+set_option linter.deprecated false in
+@[deprecated "Use `List.ext_getElem?`." (since := "2025-02-12")]
+theorem ext_get? : ∀ {l₁ l₂ : List α}, (∀ n, l₁.get? n = l₂.get? n) → l₁ = l₂
+  | [], [], _ => rfl
+  | _ :: _, [], h => nomatch h 0
+  | [], _ :: _, h => nomatch h 0
+  | a :: l₁, a' :: l₂, h => by
+    have h0 : some a = some a' := h 0
+    injection h0 with aa; simp only [aa, ext_get? fun n => h (n+1)]
+
 /-! ### get! -/
 
 /--
@@ -22,16 +59,36 @@ Returns the `i`-th element in the list (zero-based).
 If the index is out of bounds (`i ≥ as.length`), this function panics when executed, and returns
 `default`. See `get?` and `getD` for safer alternatives.
 -/
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12")]
 def get! [Inhabited α] : (as : List α) → (i : Nat) → α
   | a::_,  0   => a
   | _::as, n+1 => get! as n
   | _,     _   => panic! "invalid index"
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12")]
 theorem get!_nil [Inhabited α] (n : Nat) : [].get! n = (default : α) := rfl
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12")]
 theorem get!_cons_succ [Inhabited α] (l : List α) (a : α) (n : Nat) :
     (a::l).get! (n+1) = get! l n := rfl
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12")]
 theorem get!_cons_zero [Inhabited α] (l : List α) (a : α) : (a::l).get! 0 = a := rfl
 
+/-! ### getD -/
+
+/--
+Returns the `i`-th element in the list (zero-based).
+
+If the index is out of bounds (`i ≥ as.length`), this function returns `fallback`.
+See also `get?` and `get!`.
+-/
+def getD (as : List α) (i : Nat) (fallback : α) : α :=
+  as[i]?.getD fallback
+
+@[simp] theorem getD_nil : getD [] n d = d := rfl
+
 /-! ### getLast! -/
 
 /--
@@ -155,6 +212,7 @@ def mapMono (as : List α) (f : α → α) : List α :=
 
 /-! ## Additional lemmas required for bootstrapping `Array`. -/
 
+@[simp]
 theorem getElem_append_left {as bs : List α} (h : i < as.length) {h' : i < (as ++ bs).length} :
     (as ++ bs)[i] = as[i] := by
   induction as generalizing i with
@@ -164,29 +222,31 @@ theorem getElem_append_left {as bs : List α} (h : i < as.length) {h' : i < (as
     | zero => rfl
     | succ i => apply ih
 
+@[simp]
 theorem getElem_append_right {as bs : List α} {i : Nat} (h₁ : as.length ≤ i) {h₂} :
     (as ++ bs)[i]'h₂ =
       bs[i - as.length]'(by rw [length_append] at h₂; exact Nat.sub_lt_left_of_lt_add h₁ h₂) := by
   induction as generalizing i with
   | nil => trivial
   | cons a as ih =>
-    cases i with simp [get, Nat.succ_sub_succ] <;> simp [Nat.succ_sub_succ] at h₁
+    cases i with simp [Nat.succ_sub_succ] <;> simp [Nat.succ_sub_succ] at h₁
     | succ i => apply ih; simp [h₁]
 
+@[deprecated "Deprecated without replacement." (since := "2025-02-13")]
 theorem get_last {as : List α} {i : Fin (length (as ++ [a]))} (h : ¬ i.1 < as.length) : (as ++ [a] : List _).get i = a := by
   cases i; rename_i i h'
   induction as generalizing i with
   | nil => cases i with
     | zero => simp [List.get]
-    | succ => simp_arith at h'
+    | succ => simp +arith at h'
   | cons a as ih =>
     cases i with simp at h
     | succ i => apply ih; simp [h]
 
 theorem sizeOf_lt_of_mem [SizeOf α] {as : List α} (h : a ∈ as) : sizeOf a < sizeOf as := by
   induction h with
-  | head => simp_arith
-  | tail _ _ ih => exact Nat.lt_trans ih (by simp_arith)
+  | head => simp +arith
+  | tail _ _ ih => exact Nat.lt_trans ih (by simp +arith)
 
 /-- This tactic, added to the `decreasing_trivial` toolbox, proves that
 `sizeOf a < sizeOf as` when `a ∈ as`, which is useful for well founded recursions
@@ -197,7 +257,7 @@ macro "sizeOf_list_dec" : tactic =>
     | with_reducible
         apply Nat.lt_of_lt_of_le (sizeOf_lt_of_mem ?h)
         case' h => assumption
-      simp_arith)
+      simp +arith)
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| sizeOf_list_dec)
 
@@ -211,8 +271,8 @@ theorem append_cancel_left {as bs cs : List α} (h : as ++ bs = as ++ cs) : bs =
 theorem append_cancel_right {as bs cs : List α} (h : as ++ bs = cs ++ bs) : as = cs := by
   match as, cs with
   | [], []       => rfl
-  | [], c::cs    => have aux := congrArg length h; simp_arith at aux
-  | a::as, []    => have aux := congrArg length h; simp_arith at aux
+  | [], c::cs    => have aux := congrArg length h; simp +arith at aux
+  | a::as, []    => have aux := congrArg length h; simp +arith at aux
   | a::as, c::cs => injection h with h₁ h₂; subst h₁; rw [append_cancel_right h₂]
 
 @[simp] theorem append_cancel_left_eq (as bs cs : List α) : (as ++ bs = as ++ cs) = (bs = cs) := by
@@ -227,11 +287,11 @@ theorem append_cancel_right {as bs cs : List α} (h : as ++ bs = cs ++ bs) : as
 
 theorem sizeOf_get [SizeOf α] (as : List α) (i : Fin as.length) : sizeOf (as.get i) < sizeOf as := by
   match as, i with
-  | a::as, ⟨0, _⟩  => simp_arith [get]
+  | a::as, ⟨0, _⟩  => simp +arith [get]
   | a::as, ⟨i+1, h⟩ =>
     have ih := sizeOf_get as ⟨i, Nat.le_of_succ_le_succ h⟩
     apply Nat.lt_trans ih
-    simp_arith
+    simp +arith
 
 theorem not_lex_antisymm [DecidableEq α] {r : α → α → Prop} [DecidableRel r]
     (antisymm : ∀ x y : α, ¬ r x y → ¬ r y x → x = y)

src/Init/Data/List/Control.lean

@@ -9,6 +9,9 @@ import Init.Control.Id
 import Init.Control.Lawful
 import Init.Data.List.Basic
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 universe u v w u₁ u₂
 
@@ -128,7 +131,7 @@ Applies the monadic function `f` on every element `x` in the list, left-to-right
 results `y` for which `f x` returns `some y`.
 -/
 @[inline]
-def filterMapM {m : Type u → Type v} [Monad m] {α β : Type u} (f : α → m (Option β)) (as : List α) : m (List β) :=
+def filterMapM {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f : α → m (Option β)) (as : List α) : m (List β) :=
   let rec @[specialize] loop
     | [],     bs => pure bs.reverse
     | a :: as, bs => do
@@ -161,7 +164,7 @@ foldlM f x₀ [a, b, c] = do
 ```
 -/
 @[specialize]
-protected def foldlM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w} : (f : s → α → m s) → (init : s) → List α → m s
+def foldlM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w} : (f : s → α → m s) → (init : s) → List α → m s
   | _, s, []      => pure s
   | f, s, a :: as => do
     let s' ← f s a
@@ -224,14 +227,19 @@ def findM? {m : Type → Type u} [Monad m] {α : Type} (p : α → m Bool) : Lis
     | false => findM? p as
 
 @[simp]
-theorem findM?_id (p : α → Bool) (as : List α) : findM? (m := Id) p as = as.find? p := by
+theorem findM?_pure {m} [Monad m] [LawfulMonad m] (p : α → Bool) (as : List α) :
+    findM? (m := m) (pure <| p ·) as = pure (as.find? p) := by
   induction as with
   | nil => rfl
   | cons a as ih =>
     simp only [findM?, find?]
     cases p a with
-    | true  => rfl
-    | false => rw [ih]; rfl
+    | true  => simp
+    | false => simp [ih]
+
+@[simp]
+theorem findM?_id (p : α → Bool) (as : List α) : findM? (m := Id) p as = as.find? p :=
+  findM?_pure _ _
 
 @[specialize]
 def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f : α → m (Option β)) : List α → m (Option β)
@@ -242,14 +250,19 @@ def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f
     | none   => findSomeM? f as
 
 @[simp]
-theorem findSomeM?_id (f : α → Option β) (as : List α) : findSomeM? (m := Id) f as = as.findSome? f := by
+theorem findSomeM?_pure [Monad m] [LawfulMonad m] (f : α → Option β) (as : List α) :
+    findSomeM? (m := m) (pure <| f ·) as = pure (as.findSome? f) := by
   induction as with
   | nil => rfl
   | cons a as ih =>
     simp only [findSomeM?, findSome?]
     cases f a with
-    | some b => rfl
-    | none   => rw [ih]; rfl
+    | some b => simp
+    | none   => simp [ih]
+
+@[simp]
+theorem findSomeM?_id (f : α → Option β) (as : List α) : findSomeM? (m := Id) f as = as.findSome? f :=
+  findSomeM?_pure _ _
 
 theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
     as.findM? p = as.findSomeM? fun a => return if (← p a) then some a else none := by

src/Init/Data/List/Count.lean

@@ -10,6 +10,9 @@ import Init.Data.List.Sublist
 # Lemmas about `List.countP` and `List.count`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 open Nat
@@ -24,10 +27,10 @@ variable (p q : α → Bool)
 protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
   induction l generalizing n with
   | nil => rfl
-  | cons head tail ih =>
+  | cons hd _ ih =>
     unfold countP.go
     rw [ih (n := n + 1), ih (n := n), ih (n := 1)]
-    if h : p head then simp [h, Nat.add_assoc] else simp [h]
+    if h : p hd then simp [h, Nat.add_assoc] else simp [h]
 
 @[simp] theorem countP_cons_of_pos (l) (pa : p a) : countP p (a :: l) = countP p l + 1 := by
   have : countP.go p (a :: l) 0 = countP.go p l 1 := show cond .. = _ by rw [pa]; rfl
@@ -46,8 +49,8 @@ theorem countP_cons (a : α) (l) : countP p (a :: l) = countP p l + if p a then
 theorem length_eq_countP_add_countP (l) : length l = countP p l + countP (fun a => ¬p a) l := by
   induction l with
   | nil => rfl
-  | cons x h ih =>
-    if h : p x then
+  | cons hd _ ih =>
+    if h : p hd then
       rw [countP_cons_of_pos _ _ h, countP_cons_of_neg _ _ _, length, ih]
       · rw [Nat.add_assoc, Nat.add_comm _ 1, Nat.add_assoc]
       · simp [h]
@@ -84,7 +87,7 @@ theorem countP_le_length : countP p l ≤ l.length := by
   countP_pos_iff
 
 @[simp] theorem countP_eq_zero {p} : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
-  simp only [countP_eq_length_filter, length_eq_zero, filter_eq_nil_iff]
+  simp only [countP_eq_length_filter, length_eq_zero_iff, filter_eq_nil_iff]
 
 @[simp] theorem countP_eq_length {p} : countP p l = l.length ↔ ∀ a ∈ l, p a := by
   rw [countP_eq_length_filter, filter_length_eq_length]
@@ -210,7 +213,7 @@ theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
 
 theorem count_tail : ∀ (l : List α) (a : α) (h : l ≠ []),
       l.tail.count a = l.count a - if l.head h == a then 1 else 0
-  | head :: tail, a, _ => by simp [count_cons]
+  | _ :: _, a, _ => by simp [count_cons]
 
 theorem count_le_length (a : α) (l : List α) : count a l ≤ l.length := countP_le_length _
 

src/Init/Data/List/Erase.lean

@@ -12,6 +12,9 @@ import Init.Data.List.Find
 # Lemmas about `List.eraseP`, `List.erase`, and `List.eraseIdx`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 open Nat
@@ -134,7 +137,7 @@ theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (erase
 @[simp] theorem eraseP_eq_self_iff {p} {l : List α} : l.eraseP p = l ↔ ∀ a ∈ l, ¬ p a := by
   rw [← Sublist.length_eq (eraseP_sublist l), length_eraseP]
   split <;> rename_i h
-  · simp only [any_eq_true, length_eq_zero] at h
+  · simp only [any_eq_true, length_eq_zero_iff] at h
     constructor
     · intro; simp_all [Nat.sub_one_eq_self]
     · intro; obtain ⟨x, m, h⟩ := h; simp_all
@@ -437,10 +440,10 @@ theorem erase_eq_iff [LawfulBEq α] {a : α} {l : List α} :
   rw [erase_eq_eraseP', eraseP_eq_iff]
   simp only [beq_iff_eq, forall_mem_ne', exists_and_left]
   constructor
-  · rintro (⟨h, rfl⟩ | ⟨a', l', h, rfl, x, rfl, rfl⟩)
+  · rintro (⟨h, rfl⟩ | ⟨a', l', h, rfl, xs, rfl, rfl⟩)
     · left; simp_all
-    · right; refine ⟨l', h, x, by simp⟩
-  · rintro (⟨h, rfl⟩ | ⟨l₁, h, x, rfl, rfl⟩)
+    · right; refine ⟨l', h, xs, by simp⟩
+  · rintro (⟨h, rfl⟩ | ⟨l₁, h, xs, rfl, rfl⟩)
     · left; simp_all
     · right; refine ⟨a, l₁, h, by simp⟩
 

src/Init/Data/List/FinRange.lean

@@ -6,6 +6,9 @@ Authors: François G. Dorais
 prelude
 import Init.Data.List.OfFn
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 /-- `finRange n` lists all elements of `Fin n` in order -/

src/Init/Data/List/Find.lean

@@ -15,6 +15,10 @@ Lemmas about `List.findSome?`, `List.find?`, `List.findIdx`, `List.findIdx?`, `L
 and `List.lookup`.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
+
 namespace List
 
 open Nat
@@ -117,7 +121,7 @@ theorem find?_eq_findSome?_guard (l : List α) : find? p l = findSome? (Option.g
 
 @[simp] theorem getLast_filterMap (f : α → Option β) (l : List α) (h) :
     (l.filterMap f).getLast h = (l.reverse.findSome? f).get (by simp_all [Option.isSome_iff_ne_none]) := by
-  simp [getLast_eq_iff_getLast_eq_some]
+  simp [getLast_eq_iff_getLast?_eq_some]
 
 @[simp] theorem map_findSome? (f : α → Option β) (g : β → γ) (l : List α) :
     (l.findSome? f).map g = l.findSome? (Option.map g ∘ f) := by
@@ -144,7 +148,7 @@ theorem head_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
 theorem getLast_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
     (flatten L).getLast (by simpa using h) =
       (L.reverse.findSome? fun l => l.getLast?).get (by simpa using h) := by
-  simp [getLast_eq_iff_getLast_eq_some, getLast?_flatten]
+  simp [getLast_eq_iff_getLast?_eq_some, getLast?_flatten]
 
 theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
   cases n with
@@ -309,7 +313,7 @@ theorem get_find?_mem (xs : List α) (p : α → Bool) (h) : (xs.find? p).get h
 
 @[simp] theorem getLast_filter (p : α → Bool) (l : List α) (h) :
     (l.filter p).getLast h = (l.reverse.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
-  simp [getLast_eq_iff_getLast_eq_some]
+  simp [getLast_eq_iff_getLast?_eq_some]
 
 @[simp] theorem find?_filterMap (xs : List α) (f : α → Option β) (p : β → Bool) :
     (xs.filterMap f).find? p = (xs.find? (fun a => (f a).any p)).bind f := by
@@ -335,11 +339,11 @@ theorem get_find?_mem (xs : List α) (p : α → Bool) (h) : (xs.find? p).get h
     simp only [cons_append, find?]
     by_cases h : p x <;> simp [h, ih]
 
-@[simp] theorem find?_flatten (xs : List (List α)) (p : α → Bool) :
-    xs.flatten.find? p = xs.findSome? (·.find? p) := by
-  induction xs with
+@[simp] theorem find?_flatten (xss : List (List α)) (p : α → Bool) :
+    xss.flatten.find? p = xss.findSome? (·.find? p) := by
+  induction xss with
   | nil => simp
-  | cons x xs ih =>
+  | cons _ _ ih =>
     simp only [flatten_cons, find?_append, findSome?_cons, ih]
     split <;> simp [*]
 
@@ -358,7 +362,7 @@ Moreover, no earlier list in `xs` has an element satisfying `p`.
 theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a : α} :
     xs.flatten.find? p = some a ↔
       p a ∧ ∃ as ys zs bs, xs = as ++ (ys ++ a :: zs) :: bs ∧
-        (∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
+        (∀ l ∈ as, ∀ x ∈ l, !p x) ∧ (∀ x ∈ ys, !p x) := by
   rw [find?_eq_some_iff_append]
   constructor
   · rintro ⟨h, ⟨ys, zs, h₁, h₂⟩⟩
@@ -370,8 +374,8 @@ theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a :
       obtain ⟨bs, cs, ds, rfl, h₁, rfl⟩ := h₁
       refine ⟨as ++ bs, [], cs, ds, by simp, ?_⟩
       simp
-      rintro a (ma | mb) x m
-      · simpa using h₂ x (by simpa using ⟨a, ma, m⟩)
+      rintro l (ma | mb) x m
+      · simpa using h₂ x (by simpa using ⟨l, ma, m⟩)
       · specialize h₁ _ mb
         simp_all
     · simp [h₁]
@@ -510,47 +514,6 @@ private theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
     (x :: xs).findIdx? p = if p x then some 0 else (xs.findIdx? p).map fun i => i + 1 := by
   simp [findIdx?, findIdx?_go_eq]
 
-/-! ### findFinIdx? -/
-
-@[simp] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
-
-theorem findIdx?_go_eq_map_findFinIdx?_go_val {xs : List α} {p : α → Bool} {i : Nat} {h} :
-    List.findIdx?.go p xs i =
-      (List.findFinIdx?.go p l xs i h).map (·.val) := by
-  unfold findIdx?.go
-  unfold findFinIdx?.go
-  split <;> rename_i a xs
-  · simp_all
-  · simp only
-    split
-    · simp
-    · rw [findIdx?_go_eq_map_findFinIdx?_go_val]
-
-theorem findIdx?_eq_map_findFinIdx?_val {xs : List α} {p : α → Bool} :
-    xs.findIdx? p = (xs.findFinIdx? p).map (·.val) := by
-  simp [findIdx?, findFinIdx?]
-  rw [findIdx?_go_eq_map_findFinIdx?_go_val]
-
-@[simp] theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
-    findFinIdx? p (x :: xs) = if p x then some 0 else (findFinIdx? p xs).map Fin.succ := by
-  rw [← Option.map_inj_right (f := Fin.val) (fun a b => Fin.eq_of_val_eq)]
-  rw [← findIdx?_eq_map_findFinIdx?_val]
-  rw [findIdx?_cons]
-  split
-  · simp
-  · rw [findIdx?_eq_map_findFinIdx?_val]
-    simp [Function.comp_def]
-
-@[simp] theorem findFinIdx?_subtype {p : α → Prop} {l : List { x // p x }}
-    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.findFinIdx? f = (l.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
-  unfold unattach
-  induction l with
-  | nil => simp
-  | cons a l ih =>
-    simp [hf, findFinIdx?_cons]
-    split <;> simp [ih, Function.comp_def]
-
 /-! ### findIdx -/
 
 theorem findIdx_cons (p : α → Bool) (b : α) (l : List α) :
@@ -563,10 +526,10 @@ where
       List.findIdx.go p l (n + 1) = (findIdx.go p l n) + 1 := by
     cases l with
     | nil => unfold findIdx.go; exact Nat.succ_eq_add_one n
-    | cons head tail =>
+    | cons hd tl =>
       unfold findIdx.go
-      cases p head <;> simp only [cond_false, cond_true]
-      exact findIdx_go_succ p tail (n + 1)
+      cases p hd <;> simp only [cond_false, cond_true]
+      exact findIdx_go_succ p tl (n + 1)
 
 theorem findIdx_of_getElem?_eq_some {xs : List α} (w : xs[xs.findIdx p]? = some y) : p y := by
   induction xs with
@@ -577,10 +540,6 @@ theorem findIdx_getElem {xs : List α} {w : xs.findIdx p < xs.length} :
     p xs[xs.findIdx p] :=
   xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)
 
-@[deprecated findIdx_of_getElem?_eq_some (since := "2024-08-12")]
-theorem findIdx_of_get?_eq_some {xs : List α} (w : xs.get? (xs.findIdx p) = some y) : p y :=
-  findIdx_of_getElem?_eq_some (by simpa using w)
-
 @[deprecated findIdx_getElem (since := "2024-08-12")]
 theorem findIdx_get {xs : List α} {w : xs.findIdx p < xs.length} :
     p (xs.get ⟨xs.findIdx p, w⟩) :=
@@ -603,11 +562,6 @@ theorem findIdx_getElem?_eq_getElem_of_exists {xs : List α} (h : ∃ x ∈ xs,
     xs[xs.findIdx p]? = some (xs[xs.findIdx p]'(xs.findIdx_lt_length_of_exists h)) :=
   getElem?_eq_getElem (findIdx_lt_length_of_exists h)
 
-@[deprecated findIdx_getElem?_eq_getElem_of_exists (since := "2024-08-12")]
-theorem findIdx_get?_eq_get_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
-    xs.get? (xs.findIdx p) = some (xs.get ⟨xs.findIdx p, xs.findIdx_lt_length_of_exists h⟩) :=
-  get?_eq_get (findIdx_lt_length_of_exists h)
-
 @[simp]
 theorem findIdx_eq_length {p : α → Bool} {xs : List α} :
     xs.findIdx p = xs.length ↔ ∀ x ∈ xs, p x = false := by
@@ -981,6 +935,71 @@ theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :
     simp [hf, findIdx?_cons]
     split <;> simp [ih, Function.comp_def]
 
+/-! ### findFinIdx? -/
+
+@[simp] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
+
+theorem findIdx?_go_eq_map_findFinIdx?_go_val {xs : List α} {p : α → Bool} {i : Nat} {h} :
+    List.findIdx?.go p xs i =
+      (List.findFinIdx?.go p l xs i h).map (·.val) := by
+  unfold findIdx?.go
+  unfold findFinIdx?.go
+  split
+  · simp_all
+  · simp only
+    split
+    · simp
+    · rw [findIdx?_go_eq_map_findFinIdx?_go_val]
+
+theorem findIdx?_eq_map_findFinIdx?_val {xs : List α} {p : α → Bool} :
+    xs.findIdx? p = (xs.findFinIdx? p).map (·.val) := by
+  simp [findIdx?, findFinIdx?]
+  rw [findIdx?_go_eq_map_findFinIdx?_go_val]
+
+theorem findFinIdx?_eq_pmap_findIdx? {xs : List α} {p : α → Bool} :
+    xs.findFinIdx? p =
+      (xs.findIdx? p).pmap
+        (fun i m => by simp [findIdx?_eq_some_iff_getElem] at m; exact ⟨i, m.choose⟩)
+        (fun i h => h) := by
+  simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]
+
+@[simp] theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
+    findFinIdx? p (x :: xs) = if p x then some 0 else (findFinIdx? p xs).map Fin.succ := by
+  rw [← Option.map_inj_right (f := Fin.val) (fun a b => Fin.eq_of_val_eq)]
+  rw [← findIdx?_eq_map_findFinIdx?_val]
+  rw [findIdx?_cons]
+  split
+  · simp
+  · rw [findIdx?_eq_map_findFinIdx?_val]
+    simp [Function.comp_def]
+
+@[simp] theorem findFinIdx?_eq_none_iff {l : List α} {p : α → Bool} :
+    l.findFinIdx? p = none ↔ ∀ x ∈ l, ¬ p x := by
+  simp [findFinIdx?_eq_pmap_findIdx?]
+
+@[simp]
+theorem findFinIdx?_eq_some_iff {xs : List α} {p : α → Bool} {i : Fin xs.length} :
+    xs.findFinIdx? p = some i ↔
+      p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji i.2)) := by
+  simp only [findFinIdx?_eq_pmap_findIdx?, Option.pmap_eq_some_iff, findIdx?_eq_some_iff_getElem,
+    Bool.not_eq_true, Option.mem_def, exists_and_left, and_exists_self, Fin.getElem_fin]
+  constructor
+  · rintro ⟨a, ⟨h, w₁, w₂⟩, rfl⟩
+    exact ⟨w₁, fun j hji => by simpa using w₂ j hji⟩
+  · rintro ⟨h, w⟩
+    exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩
+
+@[simp] theorem findFinIdx?_subtype {p : α → Prop} {l : List { x // p x }}
+    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    l.findFinIdx? f = (l.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
+  unfold unattach
+  induction l with
+  | nil => simp
+  | cons a l ih =>
+    simp [hf, findFinIdx?_cons]
+    split <;> simp [ih, Function.comp_def]
+
+
 /-! ### idxOf
 
 The verification API for `idxOf` is still incomplete.
@@ -1040,6 +1059,36 @@ theorem idxOf_lt_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∈ l) : l.
 @[deprecated idxOf_lt_length (since := "2025-01-29")]
 abbrev indexOf_lt_length := @idxOf_lt_length
 
+/-! ### finIdxOf?
+
+The verification API for `finIdxOf?` is still incomplete.
+The lemmas below should be made consistent with those for `findFinIdx?` (and proved using them).
+-/
+
+theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : List α} {a : α} :
+    xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
+  simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
+
+@[simp] theorem finIdxOf?_nil [BEq α] : ([] : List α).finIdxOf? a = none := rfl
+
+@[simp] theorem finIdxOf?_cons [BEq α] (a : α) (xs : List α) :
+    (a :: xs).finIdxOf? b =
+      if a == b then some ⟨0, by simp⟩ else (xs.finIdxOf? b).map (·.succ) := by
+  simp [finIdxOf?]
+
+@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
+    l.finIdxOf? a = none ↔ a ∉ l := by
+  simp only [finIdxOf?, findFinIdx?_eq_none_iff, beq_iff_eq]
+  constructor
+  · intro w m
+    exact w a m rfl
+  · rintro h a m rfl
+    exact h m
+
+@[simp] theorem finIdxOf?_eq_some_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} {i : Fin l.length} :
+    l.finIdxOf? a = some i ↔ l[i] = a ∧ ∀ j (_ : j < i), ¬l[j] = a := by
+  simp only [finIdxOf?, findFinIdx?_eq_some_iff, beq_iff_eq]
+
 /-! ### idxOf?
 
 The verification API for `idxOf?` is still incomplete.
@@ -1065,12 +1114,6 @@ theorem idxOf?_cons [BEq α] (a : α) (xs : List α) (b : α) :
 @[deprecated idxOf?_eq_none_iff (since := "2025-01-29")]
 abbrev indexOf?_eq_none_iff := @idxOf?_eq_none_iff
 
-/-! ### finIdxOf? -/
-
-theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : List α} {a : α} :
-    xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
-  simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
-
 /-! ### lookup -/
 
 section lookup

src/Init/Data/List/Impl.lean

@@ -16,6 +16,9 @@ If you import `Init.Data.List.Basic` but do not import this file,
 then at runtime you will get non-tail recursive versions of the following definitions.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 /-! ## Basic `List` operations.
@@ -57,8 +60,8 @@ The following operations are given `@[csimp]` replacements below:
 
 @[csimp] theorem set_eq_setTR : @set = @setTR := by
   funext α l n a; simp [setTR]
-  let rec go (acc) : ∀ xs n, l = acc.toList ++ xs →
-    setTR.go l a xs n acc = acc.toList ++ xs.set n a
+  let rec go (acc) : ∀ xs i, l = acc.toList ++ xs →
+    setTR.go l a xs i acc = acc.toList ++ xs.set i a
   | [], _ => fun h => by simp [setTR.go, set, h]
   | x::xs, 0 => by simp [setTR.go, set]
   | x::xs, n+1 => fun h => by simp only [setTR.go, set]; rw [go _ xs] <;> simp [h]
@@ -91,7 +94,7 @@ The following operations are given `@[csimp]` replacements below:
 @[specialize] def foldrTR (f : α → β → β) (init : β) (l : List α) : β := l.toArray.foldr f init
 
 @[csimp] theorem foldr_eq_foldrTR : @foldr = @foldrTR := by
-  funext α β f init l; simp [foldrTR, ← Array.foldr_toList, -Array.size_toArray]
+  funext α β f init l; simp only [foldrTR, ← Array.foldr_toList]
 
 /-! ### flatMap  -/
 
@@ -131,13 +134,13 @@ The following operations are given `@[csimp]` replacements below:
   | a::as, n+1, acc => go as n (acc.push a)
 
 @[csimp] theorem take_eq_takeTR : @take = @takeTR := by
-  funext α n l; simp [takeTR]
-  suffices ∀ xs acc, l = acc.toList ++ xs → takeTR.go l xs n acc = acc.toList ++ xs.take n from
+  funext α i l; simp [takeTR]
+  suffices ∀ xs acc, l = acc.toList ++ xs → takeTR.go l xs i acc = acc.toList ++ xs.take i from
     (this l #[] (by simp)).symm
-  intro xs; induction xs generalizing n with intro acc
-  | nil => cases n <;> simp [take, takeTR.go]
+  intro xs; induction xs generalizing i with intro acc
+  | nil => cases i <;> simp [take, takeTR.go]
   | cons x xs IH =>
-    cases n with simp only [take, takeTR.go]
+    cases i with simp only [take, takeTR.go]
     | zero => simp
     | succ n => intro h; rw [IH] <;> simp_all
 
@@ -207,7 +210,7 @@ def modifyTR (f : α → α) (n : Nat) (l : List α) : List α := go l n #[] whe
   | a :: l, 0, acc => acc.toListAppend (f a :: l)
   | a :: l, n+1, acc => go l n (acc.push a)
 
-theorem modifyTR_go_eq : ∀ l n, modifyTR.go f l n acc = acc.toList ++ modify f n l
+theorem modifyTR_go_eq : ∀ l i, modifyTR.go f l i acc = acc.toList ++ modify f i l
   | [], n => by cases n <;> simp [modifyTR.go, modify]
   | a :: l, 0 => by simp [modifyTR.go, modify]
   | a :: l, n+1 => by simp [modifyTR.go, modify, modifyTR_go_eq l]
@@ -225,7 +228,7 @@ theorem modifyTR_go_eq : ∀ l n, modifyTR.go f l n acc = acc.toList ++ modify f
   | _, [], acc => acc.toList
   | n+1, a :: l, acc => go n l (acc.push a)
 
-theorem insertIdxTR_go_eq : ∀ n l, insertIdxTR.go a n l acc = acc.toList ++ insertIdx n a l
+theorem insertIdxTR_go_eq : ∀ i l, insertIdxTR.go a i l acc = acc.toList ++ insertIdx i a l
   | 0, l | _+1, [] => by simp [insertIdxTR.go, insertIdx]
   | n+1, a :: l => by simp [insertIdxTR.go, insertIdx, insertIdxTR_go_eq n l]
 
@@ -284,15 +287,15 @@ theorem insertIdxTR_go_eq : ∀ n l, insertIdxTR.go a n l acc = acc.toList ++ in
   | a::as, n+1, acc => go as n (acc.push a)
 
 @[csimp] theorem eraseIdx_eq_eraseIdxTR : @eraseIdx = @eraseIdxTR := by
-  funext α l n; simp [eraseIdxTR]
-  suffices ∀ xs acc, l = acc.toList ++ xs → eraseIdxTR.go l xs n acc = acc.toList ++ xs.eraseIdx n from
+  funext α l i; simp [eraseIdxTR]
+  suffices ∀ xs acc, l = acc.toList ++ xs → eraseIdxTR.go l xs i acc = acc.toList ++ xs.eraseIdx i from
     (this l #[] (by simp)).symm
-  intro xs; induction xs generalizing n with intro acc h
+  intro xs; induction xs generalizing i with intro acc h
   | nil => simp [eraseIdx, eraseIdxTR.go, h]
   | cons x xs IH =>
-    match n with
+    match i with
     | 0 => simp [eraseIdx, eraseIdxTR.go]
-    | n+1 =>
+    | i+1 =>
       simp only [eraseIdxTR.go, eraseIdx]
       rw [IH]; simp; simp; exact h
 
@@ -320,13 +323,13 @@ theorem insertIdxTR_go_eq : ∀ n l, insertIdxTR.go a n l acc = acc.toList ++ in
 
 /-- Tail recursive version of `List.zipIdx`. -/
 def zipIdxTR (l : List α) (n : Nat := 0) : List (α × Nat) :=
-  let arr := l.toArray
-  (arr.foldr (fun a (n, acc) => (n-1, (a, n-1) :: acc)) (n + arr.size, [])).2
+  let as := l.toArray
+  (as.foldr (fun a (n, acc) => (n-1, (a, n-1) :: acc)) (n + as.size, [])).2
 
 @[csimp] theorem zipIdx_eq_zipIdxTR : @zipIdx = @zipIdxTR := by
-  funext α l n; simp [zipIdxTR, -Array.size_toArray]
+  funext α l n; simp only [zipIdxTR, size_toArray]
   let f := fun (a : α) (n, acc) => (n-1, (a, n-1) :: acc)
-  let rec go : ∀ l n, l.foldr f (n + l.length, []) = (n, zipIdx l n)
+  let rec go : ∀ l i, l.foldr f (i + l.length, []) = (i, zipIdx l i)
     | [], n => rfl
     | a::as, n => by
       rw [← show _ + as.length = n + (a::as).length from Nat.succ_add .., foldr, go as]
@@ -339,13 +342,13 @@ def zipIdxTR (l : List α) (n : Nat := 0) : List (α × Nat) :=
 /-- Tail recursive version of `List.enumFrom`. -/
 @[deprecated zipIdxTR (since := "2025-01-21")]
 def enumFromTR (n : Nat) (l : List α) : List (Nat × α) :=
-  let arr := l.toArray
-  (arr.foldr (fun a (n, acc) => (n-1, (n-1, a) :: acc)) (n + arr.size, [])).2
+  let as := l.toArray
+  (as.foldr (fun a (n, acc) => (n-1, (n-1, a) :: acc)) (n + as.size, [])).2
 
 set_option linter.deprecated false in
 @[deprecated zipIdx_eq_zipIdxTR (since := "2025-01-21"), csimp]
 theorem enumFrom_eq_enumFromTR : @enumFrom = @enumFromTR := by
-  funext α n l; simp [enumFromTR, -Array.size_toArray]
+  funext α n l; simp only [enumFromTR, size_toArray]
   let f := fun (a : α) (n, acc) => (n-1, (n-1, a) :: acc)
   let rec go : ∀ l n, l.foldr f (n + l.length, []) = (n, enumFrom n l)
     | [], n => rfl
@@ -359,6 +362,7 @@ theorem enumFrom_eq_enumFromTR : @enumFrom = @enumFromTR := by
 
 /-! ### intercalate -/
 
+set_option linter.listVariables false in
 /-- Tail recursive version of `List.intercalate`. -/
 def intercalateTR (sep : List α) : List (List α) → List α
   | [] => []
@@ -371,6 +375,7 @@ where
   | x, [], acc => acc.toListAppend x
   | x, y::xs, acc => go sep y xs (acc ++ x ++ sep)
 
+set_option linter.listVariables false in
 @[csimp] theorem intercalate_eq_intercalateTR : @intercalate = @intercalateTR := by
   funext α sep l; simp [intercalate, intercalateTR]
   match l with

src/Init/Data/List/Lemmas.lean

@@ -73,6 +73,10 @@ Also
 * `Init.Data.List.Monadic` for addiation lemmas about `List.mapM` and `List.forM`.
 
 -/
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 open Nat
@@ -92,9 +96,15 @@ theorem ne_nil_of_length_eq_add_one (_ : length l = n + 1) : l ≠ [] := fun _ =
 
 theorem ne_nil_of_length_pos (_ : 0 < length l) : l ≠ [] := fun _ => nomatch l
 
-@[simp] theorem length_eq_zero : length l = 0 ↔ l = [] :=
+@[simp] theorem length_eq_zero_iff : length l = 0 ↔ l = [] :=
   ⟨eq_nil_of_length_eq_zero, fun h => h ▸ rfl⟩
 
+@[deprecated length_eq_zero_iff (since := "2025-02-24")]
+abbrev length_eq_zero := @length_eq_zero_iff
+
+theorem eq_nil_iff_length_eq_zero : l = [] ↔ length l = 0 :=
+  length_eq_zero_iff.symm
+
 theorem length_pos_of_mem {a : α} : ∀ {l : List α}, a ∈ l → 0 < length l
   | _::_, _ => Nat.zero_lt_succ _
 
@@ -119,12 +129,21 @@ theorem exists_cons_of_length_eq_add_one :
     ∀ {l : List α}, l.length = n + 1 → ∃ h t, l = h :: t
   | _::_, _ => ⟨_, _, rfl⟩
 
-theorem length_pos {l : List α} : 0 < length l ↔ l ≠ [] :=
-  Nat.pos_iff_ne_zero.trans (not_congr length_eq_zero)
+theorem length_pos_iff {l : List α} : 0 < length l ↔ l ≠ [] :=
+  Nat.pos_iff_ne_zero.trans (not_congr length_eq_zero_iff)
 
-theorem length_eq_one {l : List α} : length l = 1 ↔ ∃ a, l = [a] :=
+@[deprecated length_pos_iff (since := "2025-02-24")]
+abbrev length_pos := @length_pos_iff
+
+theorem ne_nil_iff_length_pos {l : List α} : l ≠ [] ↔ 0 < length l :=
+  length_pos_iff.symm
+
+theorem length_eq_one_iff {l : List α} : length l = 1 ↔ ∃ a, l = [a] :=
   ⟨fun h => match l, h with | [_], _ => ⟨_, rfl⟩, fun ⟨_, h⟩ => by simp [h]⟩
 
+@[deprecated length_eq_one_iff (since := "2025-02-24")]
+abbrev length_eq_one := @length_eq_one_iff
+
 /-! ### cons -/
 
 theorem cons_ne_nil (a : α) (l : List α) : a :: l ≠ [] := nofun
@@ -146,10 +165,10 @@ theorem cons_inj_right (a : α) {l l' : List α} : a :: l = a :: l' ↔ l = l' :
 theorem cons_eq_cons {a b : α} {l l' : List α} : a :: l = b :: l' ↔ a = b ∧ l = l' :=
   List.cons.injEq .. ▸ .rfl
 
-theorem exists_cons_of_ne_nil : ∀ {l : List α}, l ≠ [] → ∃ b L, l = b :: L
+theorem exists_cons_of_ne_nil : ∀ {l : List α}, l ≠ [] → ∃ b l', l = b :: l'
   | c :: l', _ => ⟨c, l', rfl⟩
 
-theorem ne_nil_iff_exists_cons {l : List α} : l ≠ [] ↔ ∃ b L, l = b :: L :=
+theorem ne_nil_iff_exists_cons {l : List α} : l ≠ [] ↔ ∃ b l', l = b :: l' :=
   ⟨exists_cons_of_ne_nil, fun ⟨_, _, eq⟩ => eq.symm ▸ cons_ne_nil _ _⟩
 
 theorem singleton_inj {α : Type _} {a b : α} : [a] = [b] ↔ a = b := by
@@ -167,51 +186,38 @@ We simplify `l.get i` to `l[i.1]'i.2` and `l.get? i` to `l[i]?`.
 
 @[simp] theorem get_eq_getElem (l : List α) (i : Fin l.length) : l.get i = l[i.1]'i.2 := rfl
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
 theorem get?_eq_none : ∀ {l : List α} {n}, length l ≤ n → l.get? n = none
   | [], _, _ => rfl
   | _ :: l, _+1, h => get?_eq_none (l := l) <| Nat.le_of_succ_le_succ h
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
 theorem get?_eq_get : ∀ {l : List α} {n} (h : n < l.length), l.get? n = some (get l ⟨n, h⟩)
   | _ :: _, 0, _ => rfl
   | _ :: l, _+1, _ => get?_eq_get (l := l) _
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
 theorem get?_eq_some_iff : l.get? n = some a ↔ ∃ h, get l ⟨n, h⟩ = a :=
   ⟨fun e =>
     have : n < length l := Nat.gt_of_not_le fun hn => by cases get?_eq_none hn ▸ e
     ⟨this, by rwa [get?_eq_get this, Option.some.injEq] at e⟩,
   fun ⟨_, e⟩ => e ▸ get?_eq_get _⟩
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
 theorem get?_eq_none_iff : l.get? n = none ↔ length l ≤ n :=
   ⟨fun e => Nat.ge_of_not_lt (fun h' => by cases e ▸ get?_eq_some_iff.2 ⟨h', rfl⟩), get?_eq_none⟩
 
-@[simp] theorem get?_eq_getElem? (l : List α) (i : Nat) : l.get? i = l[i]? := by
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12"), simp]
+theorem get?_eq_getElem? (l : List α) (i : Nat) : l.get? i = l[i]? := by
   simp only [getElem?_def]; split
   · exact (get?_eq_get ‹_›)
   · exact (get?_eq_none_iff.2 <| Nat.not_lt.1 ‹_›)
 
-/-! ### getD
-
-We simplify away `getD`, replacing `getD l n a` with `(l[n]?).getD a`.
-Because of this, there is only minimal API for `getD`.
--/
-
-@[simp] theorem getD_eq_getElem?_getD (l) (i) (a : α) : getD l i a = (l[i]?).getD a := by
-  simp [getD]
-
-/-! ### get!
-
-We simplify `l.get! i` to `l[i]!`.
--/
-
-theorem get!_eq_getD [Inhabited α] : ∀ (l : List α) i, l.get! i = l.getD i default
-  | [], _      => rfl
-  | _a::_, 0   => rfl
-  | _a::l, n+1 => get!_eq_getD l n
-
-@[simp] theorem get!_eq_getElem! [Inhabited α] (l : List α) (i) : l.get! i = l[i]! := by
-  simp [get!_eq_getD]
-  rfl
-
 /-! ### getElem!
 
 We simplify `l[i]!` to `(l[i]?).getD default`.
@@ -226,19 +232,29 @@ We simplify `l[i]!` to `(l[i]?).getD default`.
 
 /-! ### getElem? and getElem -/
 
-@[simp] theorem getElem?_eq_none_iff : l[i]? = none ↔ length l ≤ i := by
-  simp only [← get?_eq_getElem?, get?_eq_none_iff]
+@[simp] theorem getElem?_nil {i : Nat} : ([] : List α)[i]? = none := rfl
 
-@[simp] theorem none_eq_getElem?_iff {l : List α} {i : Nat} : none = l[i]? ↔ length l ≤ i := by
-  simp [eq_comm (a := none)]
+theorem getElem_cons {l : List α} (w : i < (a :: l).length) :
+    (a :: l)[i] =
+      if h : i = 0 then a else l[i-1]'(match i, h with | i+1, _ => succ_lt_succ_iff.mp w) := by
+  cases i <;> simp
 
-theorem getElem?_eq_none (h : length l ≤ i) : l[i]? = none := getElem?_eq_none_iff.mpr h
+theorem getElem?_cons_zero {l : List α} : (a::l)[0]? = some a := rfl
 
-@[simp] theorem getElem?_eq_getElem {l : List α} {i} (h : i < l.length) : l[i]? = some l[i] :=
-  getElem?_pos ..
+@[simp] theorem getElem?_cons_succ {l : List α} : (a::l)[i+1]? = l[i]? := rfl
 
-theorem getElem?_eq_some_iff {l : List α} : l[i]? = some a ↔ ∃ h : i < l.length, l[i] = a := by
-  simp only [← get?_eq_getElem?, get?_eq_some_iff, get_eq_getElem]
+theorem getElem?_cons : (a :: l)[i]? = if i = 0 then some a else l[i-1]? := by
+  cases i <;> simp [getElem?_cons_zero]
+
+theorem getElem?_eq_some_iff {l : List α} : l[i]? = some a ↔ ∃ h : i < l.length, l[i] = a :=
+  match l with
+  | [] => by simp
+  | _ :: l => by
+    simp only [getElem?_cons, length_cons]
+    split <;> rename_i h
+    · simp_all
+    · match i, h with
+      | i + 1, h => simp [getElem?_eq_some_iff, Nat.succ_lt_succ_iff]
 
 theorem some_eq_getElem?_iff {l : List α} : some a = l[i]? ↔ ∃ h : i < l.length, l[i] = a := by
   rw [eq_comm, getElem?_eq_some_iff]
@@ -267,22 +283,6 @@ theorem getD_getElem? (l : List α) (i : Nat) (d : α) :
     have p : i ≥ l.length := Nat.le_of_not_gt h
     simp [getElem?_eq_none p, h]
 
-@[simp] theorem getElem?_nil {i : Nat} : ([] : List α)[i]? = none := rfl
-
-theorem getElem_cons {l : List α} (w : i < (a :: l).length) :
-    (a :: l)[i] =
-      if h : i = 0 then a else l[i-1]'(match i, h with | i+1, _ => succ_lt_succ_iff.mp w) := by
-  cases i <;> simp
-
-theorem getElem?_cons_zero {l : List α} : (a::l)[0]? = some a := by simp
-
-@[simp] theorem getElem?_cons_succ {l : List α} : (a::l)[i+1]? = l[i]? := by
-  simp only [← get?_eq_getElem?]
-  rfl
-
-theorem getElem?_cons : (a :: l)[i]? = if i = 0 then some a else l[i-1]? := by
-  cases i <;> simp
-
 @[simp] theorem getElem_singleton (a : α) (h : i < 1) : [a][i] = a :=
   match i, h with
   | 0, _ => rfl
@@ -299,12 +299,18 @@ such a rewrite, with `rw [getElem_of_eq h]`.
 theorem getElem_of_eq {l l' : List α} (h : l = l') {i : Nat} (w : i < l.length) :
     l[i] = l'[i]'(h ▸ w) := by cases h; rfl
 
-theorem getElem_zero {l : List α} (h : 0 < l.length) : l[0] = l.head (length_pos.mp h) :=
+theorem getElem_zero {l : List α} (h : 0 < l.length) : l[0] = l.head (length_pos_iff.mp h) :=
   match l, h with
   | _ :: _, _ => rfl
 
 @[ext] theorem ext_getElem? {l₁ l₂ : List α} (h : ∀ i : Nat, l₁[i]? = l₂[i]?) : l₁ = l₂ :=
-  ext_get? fun n => by simp_all
+  match l₁, l₂, h with
+  | [], [], _ => rfl
+  | _ :: _, [], h => by simpa using h 0
+  | [], _ :: _, h => by simpa using h 0
+  | a :: l₁, a' :: l₂, h => by
+    have h0 : some a = some a' := by simpa using h 0
+    injection h0 with aa; simp only [aa, ext_getElem? fun n => by simpa using h (n+1)]
 
 theorem ext_getElem {l₁ l₂ : List α} (hl : length l₁ = length l₂)
     (h : ∀ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length), l₁[i]'h₁ = l₂[i]'h₂) : l₁ = l₂ :=
@@ -322,6 +328,35 @@ theorem ext_getElem {l₁ l₂ : List α} (hl : length l₁ = length l₂)
 theorem getElem?_concat_length (l : List α) (a : α) : (l ++ [a])[l.length]? = some a := by
   simp
 
+/-! ### getD
+
+We simplify away `getD`, replacing `getD l n a` with `(l[n]?).getD a`.
+Because of this, there is only minimal API for `getD`.
+-/
+
+@[simp] theorem getD_eq_getElem?_getD (l) (i) (a : α) : getD l i a = (l[i]?).getD a := by
+  simp [getD]
+
+theorem getD_cons_zero : getD (x :: xs) 0 d = x := by simp
+theorem getD_cons_succ : getD (x :: xs) (n + 1) d = getD xs n d := by simp
+
+/-! ### get!
+
+We simplify `l.get! i` to `l[i]!`.
+-/
+
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12")]
+theorem get!_eq_getD [Inhabited α] : ∀ (l : List α) i, l.get! i = l.getD i default
+  | [], _      => rfl
+  | _a::_, 0   => by simp [get!]
+  | _a::l, n+1 => by simpa using get!_eq_getD l n
+
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12"), simp]
+theorem get!_eq_getElem! [Inhabited α] (l : List α) (i) : l.get! i = l[i]! := by
+  simp [get!_eq_getD]
+
 /-! ### mem -/
 
 @[simp] theorem not_mem_nil (a : α) : ¬ a ∈ [] := nofun
@@ -355,7 +390,7 @@ theorem eq_append_cons_of_mem {a : α} {xs : List α} (h : a ∈ xs) :
 theorem mem_cons_of_mem (y : α) {a : α} {l : List α} : a ∈ l → a ∈ y :: l := .tail _
 
 theorem exists_mem_of_ne_nil (l : List α) (h : l ≠ []) : ∃ x, x ∈ l :=
-  exists_mem_of_length_pos (length_pos.2 h)
+  exists_mem_of_length_pos (length_pos_iff.2 h)
 
 theorem eq_nil_iff_forall_not_mem {l : List α} : l = [] ↔ ∀ a, a ∉ l := by
   cases l <;> simp [-not_or]
@@ -496,21 +531,24 @@ theorem elem_eq_mem [BEq α] [LawfulBEq α] (a : α) (as : List α) :
 
 /-! ### `isEmpty` -/
 
-theorem isEmpty_iff {l : List α} : l.isEmpty ↔ l = [] := by
+@[simp] theorem isEmpty_iff {l : List α} : l.isEmpty ↔ l = [] := by
   cases l <;> simp
 
+@[deprecated isEmpty_iff (since := "2025-02-17")]
+abbrev isEmpty_eq_true := @isEmpty_iff
+
+@[simp] theorem isEmpty_eq_false_iff {l : List α} : l.isEmpty = false ↔ l ≠ [] := by
+  cases l <;> simp
+
+@[deprecated isEmpty_eq_false_iff (since := "2025-02-17")]
+abbrev isEmpty_eq_false := @isEmpty_eq_false_iff
+
 theorem isEmpty_eq_false_iff_exists_mem {xs : List α} :
     xs.isEmpty = false ↔ ∃ x, x ∈ xs := by
   cases xs <;> simp
 
 theorem isEmpty_iff_length_eq_zero {l : List α} : l.isEmpty ↔ l.length = 0 := by
-  rw [isEmpty_iff, length_eq_zero]
-
-@[simp] theorem isEmpty_eq_true {l : List α} : l.isEmpty ↔ l = [] := by
-  cases l <;> simp
-
-@[simp] theorem isEmpty_eq_false {l : List α} : l.isEmpty = false ↔ l ≠ [] := by
-  cases l <;> simp
+  rw [isEmpty_iff, length_eq_zero_iff]
 
 /-! ### any / all -/
 
@@ -557,11 +595,11 @@ theorem all_bne' [BEq α] [PartialEquivBEq α] {l : List α} :
 /-! ### set -/
 
 -- As `List.set` is defined in `Init.Prelude`, we write the basic simplification lemmas here.
-@[simp] theorem set_nil (n : Nat) (a : α) : [].set n a = [] := rfl
+@[simp] theorem set_nil (i : Nat) (a : α) : [].set i a = [] := rfl
 @[simp] theorem set_cons_zero (x : α) (xs : List α) (a : α) :
   (x :: xs).set 0 a = a :: xs := rfl
-@[simp] theorem set_cons_succ (x : α) (xs : List α) (n : Nat) (a : α) :
-  (x :: xs).set (n + 1) a = x :: xs.set n a := rfl
+@[simp] theorem set_cons_succ (x : α) (xs : List α) (i : Nat) (a : α) :
+  (x :: xs).set (i + 1) a = x :: xs.set i a := rfl
 
 @[simp] theorem getElem_set_self {l : List α} {i : Nat} {a : α} (h : i < (l.set i a).length) :
     (l.set i a)[i] = a :=
@@ -637,22 +675,22 @@ theorem getElem?_set' {l : List α} {i j : Nat} {a : α} :
     rw [getElem_set]
     split <;> simp_all
 
-theorem set_eq_of_length_le {l : List α} {n : Nat} (h : l.length ≤ n) {a : α} :
-    l.set n a = l := by
-  induction l generalizing n with
+theorem set_eq_of_length_le {l : List α} {i : Nat} (h : l.length ≤ i) {a : α} :
+    l.set i a = l := by
+  induction l generalizing i with
   | nil => simp_all
   | cons a l ih =>
-    induction n
+    induction i
     · simp_all
     · simp only [set_cons_succ, cons.injEq, true_and]
       rw [ih]
       exact Nat.succ_le_succ_iff.mp h
 
-@[simp] theorem set_eq_nil_iff {l : List α} (n : Nat) (a : α) : l.set n a = [] ↔ l = [] := by
-  cases l <;> cases n <;> simp [set]
+@[simp] theorem set_eq_nil_iff {l : List α} (i : Nat) (a : α) : l.set i a = [] ↔ l = [] := by
+  cases l <;> cases i <;> simp [set]
 
-theorem set_comm (a b : α) : ∀ {n m : Nat} (l : List α), n ≠ m →
-    (l.set n a).set m b = (l.set m b).set n a
+theorem set_comm (a b : α) : ∀ {i j : Nat} (l : List α), i ≠ j →
+    (l.set i a).set j b = (l.set j b).set i a
   | _, _, [], _ => by simp
   | _+1, 0, _ :: _, _ => by simp [set]
   | 0, _+1, _ :: _, _ => by simp [set]
@@ -660,17 +698,17 @@ theorem set_comm (a b : α) : ∀ {n m : Nat} (l : List α), n ≠ m →
     congrArg _ <| set_comm a b t fun h' => h <| Nat.succ_inj'.mpr h'
 
 @[simp]
-theorem set_set (a b : α) : ∀ (l : List α) (n : Nat), (l.set n a).set n b = l.set n b
+theorem set_set (a b : α) : ∀ (l : List α) (i : Nat), (l.set i a).set i b = l.set i b
   | [], _ => by simp
   | _ :: _, 0 => by simp [set]
   | _ :: _, _+1 => by simp [set, set_set]
 
-theorem mem_set (l : List α) (n : Nat) (h : n < l.length) (a : α) :
-    a ∈ l.set n a := by
+theorem mem_set (l : List α) (i : Nat) (h : i < l.length) (a : α) :
+    a ∈ l.set i a := by
   simp [mem_iff_getElem]
-  exact ⟨n, (by simpa using h), by simp⟩
+  exact ⟨i, (by simpa using h), by simp⟩
 
-theorem mem_or_eq_of_mem_set : ∀ {l : List α} {n : Nat} {a b : α}, a ∈ l.set n b → a ∈ l ∨ a = b
+theorem mem_or_eq_of_mem_set : ∀ {l : List α} {i : Nat} {a b : α}, a ∈ l.set i b → a ∈ l ∨ a = b
   | _ :: _, 0, _, _, h => ((mem_cons ..).1 h).symm.imp_left (.tail _)
   | _ :: _, _+1, _, _, .head .. => .inl (.head ..)
   | _ :: _, _+1, _, _, .tail _ h => (mem_or_eq_of_mem_set h).imp_left (.tail _)
@@ -725,10 +763,10 @@ theorem length_eq_of_beq [BEq α] {l₁ l₂ : List α} (h : l₁ == l₂) : l
     simp
   · intro h
     constructor
-    intro a
-    induction a with
+    intro l
+    induction l with
     | nil => simp only [List.instBEq, List.beq]
-    | cons a as ih =>
+    | cons _ _ ih =>
       simp [List.instBEq, List.beq]
       exact ih
 
@@ -747,9 +785,9 @@ theorem length_eq_of_beq [BEq α] {l₁ l₂ : List α} (h : l₁ == l₂) : l
       simp
   · intro h
     constructor
-    · intro a b h
+    · intro _ _ h
       simpa using h
-    · intro a
+    · intro _
       simp
 
 /-! ### isEqv -/
@@ -771,7 +809,7 @@ theorem getLast_eq_getElem : ∀ (l : List α) (h : l ≠ []),
       | a :: l => exact Nat.le_refl _)
   | [_], _ => rfl
   | _ :: _ :: _, _ => by
-    simp [getLast, get, Nat.succ_sub_succ, getLast_eq_getElem]
+    simp [getLast, Nat.succ_sub_succ, getLast_eq_getElem]
 
 theorem getElem_length_sub_one_eq_getLast (l : List α) (h : l.length - 1 < l.length) :
     l[l.length - 1] = getLast l (by cases l; simp at h; simp) := by
@@ -844,10 +882,6 @@ theorem getLast?_cons {a : α} : (a::l).getLast? = l.getLast?.getD a := by
 @[simp] theorem getLast?_cons_cons : (a :: b :: l).getLast? = (b :: l).getLast? := by
   simp [getLast?_cons]
 
-@[deprecated getLast?_eq_getElem? (since := "2024-07-07")]
-theorem getLast?_eq_get? (l : List α) : getLast? l = l.get? (l.length - 1) := by
-  simp [getLast?_eq_getElem?]
-
 theorem getLast?_concat (l : List α) : getLast? (l ++ [a]) = some a := by
   simp [getLast?_eq_getElem?, Nat.succ_sub_succ]
 
@@ -891,13 +925,13 @@ theorem head?_eq_getElem? : ∀ l : List α, head? l = l[0]?
   | [] => rfl
   | a :: l => by simp
 
-theorem head_eq_getElem (l : List α) (h : l ≠ []) : head l h = l[0]'(length_pos.mpr h) := by
+theorem head_eq_getElem (l : List α) (h : l ≠ []) : head l h = l[0]'(length_pos_iff.mpr h) := by
   cases l with
   | nil => simp at h
   | cons _ _ => simp
 
 theorem getElem_zero_eq_head (l : List α) (h : 0 < l.length) :
-    l[0] = head l (by simpa [length_pos] using h) := by
+    l[0] = head l (by simpa [length_pos_iff] using h) := by
   cases l with
   | nil => simp at h
   | cons _ _ => simp
@@ -984,7 +1018,7 @@ theorem one_lt_length_of_tail_ne_nil {l : List α} (h : l.tail ≠ []) : 1 < l.l
   | nil => simp at h
   | cons _ l =>
     simp only [tail_cons, ne_eq] at h
-    exact Nat.lt_add_of_pos_left (length_pos.mpr h)
+    exact Nat.lt_add_of_pos_left (length_pos_iff.mpr h)
 
 @[simp] theorem head_tail (l : List α) (h : l.tail ≠ []) :
     (tail l).head h = l[1]'(one_lt_length_of_tail_ne_nil h) := by
@@ -1144,8 +1178,8 @@ theorem map_eq_foldr (f : α → β) (l : List α) : map f l = foldr (fun a bs =
   | cons b l ih => cases i <;> simp_all
 
 @[deprecated "Use the reverse direction of `map_set`." (since := "2024-09-20")]
-theorem set_map {f : α → β} {l : List α} {n : Nat} {a : α} :
-    (map f l).set n (f a) = map f (l.set n a) := by
+theorem set_map {f : α → β} {l : List α} {i : Nat} {a : α} :
+    (map f l).set i (f a) = map f (l.set i a) := by
   simp
 
 @[simp] theorem head_map (f : α → β) (l : List α) (w) :
@@ -1344,6 +1378,11 @@ theorem head_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p
 theorem filterMap_eq_map (f : α → β) : filterMap (some ∘ f) = map f := by
   funext l; induction l <;> simp [*, filterMap_cons]
 
+/-- Variant of `filterMap_eq_map` with `some ∘ f` expanded out to a lambda. -/
+@[simp]
+theorem filterMap_eq_map' (f : α → β) : filterMap (fun x => some (f x)) = map f :=
+  filterMap_eq_map f
+
 @[simp] theorem filterMap_some_fun : filterMap (some : α → Option α) = id := by
   funext l
   erw [filterMap_eq_map]
@@ -1558,7 +1597,7 @@ theorem getElem_append_right' (l₁ : List α) {l₂ : List α} {i : Nat} (hi :
   rw [getElem_append_right] <;> simp [*, le_add_left]
 
 theorem getElem_of_append {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i) :
-    l[i]'(eq ▸ h ▸ by simp_arith) = a := Option.some.inj <| by
+    l[i]'(eq ▸ h ▸ by simp +arith) = a := Option.some.inj <| by
   rw [← getElem?_eq_getElem, eq, getElem?_append_right (h ▸ Nat.le_refl _), h]
   simp
 
@@ -1595,16 +1634,16 @@ theorem append_right_inj {t₁ t₂ : List α} (s) : s ++ t₁ = s ++ t₂ ↔ t
 theorem append_left_inj {s₁ s₂ : List α} (t) : s₁ ++ t = s₂ ++ t ↔ s₁ = s₂ :=
   ⟨fun h => append_inj_left' h rfl, congrArg (· ++ _)⟩
 
-@[simp] theorem append_left_eq_self {x y : List α} : x ++ y = y ↔ x = [] := by
-  rw [← append_left_inj (s₁ := x), nil_append]
+@[simp] theorem append_left_eq_self {xs ys : List α} : xs ++ ys = ys ↔ xs = [] := by
+  rw [← append_left_inj (s₁ := xs), nil_append]
 
-@[simp] theorem self_eq_append_left {x y : List α} : y = x ++ y ↔ x = [] := by
+@[simp] theorem self_eq_append_left {xs ys : List α} : ys = xs ++ ys ↔ xs = [] := by
   rw [eq_comm, append_left_eq_self]
 
-@[simp] theorem append_right_eq_self {x y : List α} : x ++ y = x ↔ y = [] := by
-  rw [← append_right_inj (t₁ := y), append_nil]
+@[simp] theorem append_right_eq_self {xs ys : List α} : xs ++ ys = xs ↔ ys = [] := by
+  rw [← append_right_inj (t₁ := ys), append_nil]
 
-@[simp] theorem self_eq_append_right {x y : List α} : x = x ++ y ↔ y = [] := by
+@[simp] theorem self_eq_append_right {xs ys : List α} : xs = xs ++ ys ↔ ys = [] := by
   rw [eq_comm, append_right_eq_self]
 
 theorem getLast_concat {a : α} : ∀ (l : List α), getLast (l ++ [a]) (by simp) = a
@@ -1631,14 +1670,14 @@ theorem append_ne_nil_of_ne_nil_left {s : List α} (h : s ≠ []) (t : List α)
 theorem append_ne_nil_of_ne_nil_right (s : List α) : t ≠ [] → s ++ t ≠ [] := by simp_all
 
 theorem append_eq_cons_iff :
-    a ++ b = x :: c ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
-  cases a with simp | cons a as => ?_
-  exact ⟨fun h => ⟨as, by simp [h]⟩, fun ⟨a', ⟨aeq, aseq⟩, h⟩ => ⟨aeq, by rw [aseq, h]⟩⟩
+    as ++ bs = x :: c ↔ (as = [] ∧ bs = x :: c) ∨ (∃ as', as = x :: as' ∧ c = as' ++ bs) := by
+  cases as with simp | cons a as => ?_
+  exact ⟨fun h => ⟨as, by simp [h]⟩, fun ⟨as', ⟨aeq, aseq⟩, h⟩ => ⟨aeq, by rw [aseq, h]⟩⟩
 
 @[deprecated append_eq_cons_iff (since := "2024-07-24")] abbrev append_eq_cons := @append_eq_cons_iff
 
 theorem cons_eq_append_iff :
-    x :: c = a ++ b ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
+    x :: cs = as ++ bs ↔ (as = [] ∧ bs = x :: cs) ∨ (∃ as', as = x :: as' ∧ cs = as' ++ bs) := by
   rw [eq_comm, append_eq_cons_iff]
 
 @[deprecated cons_eq_append_iff (since := "2024-07-24")] abbrev cons_eq_append := @cons_eq_append_iff
@@ -1651,11 +1690,11 @@ theorem singleton_eq_append_iff :
     [x] = a ++ b ↔ (a = [] ∧ b = [x]) ∨ (a = [x] ∧ b = []) := by
   cases a <;> cases b <;> simp [eq_comm]
 
-theorem append_eq_append_iff {a b c d : List α} :
-    a ++ b = c ++ d ↔ (∃ a', c = a ++ a' ∧ b = a' ++ d) ∨ ∃ c', a = c ++ c' ∧ d = c' ++ b := by
-  induction a generalizing c with
+theorem append_eq_append_iff {ws xs ys zs : List α} :
+    ws ++ xs = ys ++ zs ↔ (∃ as, ys = ws ++ as ∧ xs = as ++ zs) ∨ ∃ bs, ws = ys ++ bs ∧ zs = bs ++ xs := by
+  induction ws generalizing ys with
   | nil => simp_all
-  | cons a as ih => cases c <;> simp [eq_comm, and_assoc, ih, and_or_left]
+  | cons a as ih => cases ys <;> simp [eq_comm, and_assoc, ih, and_or_left]
 
 @[deprecated append_inj (since := "2024-07-24")] abbrev append_inj_of_length_left := @append_inj
 @[deprecated append_inj' (since := "2024-07-24")] abbrev append_inj_of_length_right := @append_inj'
@@ -1746,7 +1785,7 @@ theorem filterMap_eq_append_iff {f : α → Option β} :
         simp_all
       · rename_i b w
         intro h
-        rcases cons_eq_append_iff.mp h with (⟨rfl, rfl⟩ | ⟨L₁, ⟨rfl, h⟩⟩)
+        rcases cons_eq_append_iff.mp h with (⟨rfl, rfl⟩ | ⟨_, ⟨rfl, h⟩⟩)
         · refine ⟨[], x :: l, ?_⟩
           simp [filterMap_cons, w]
         · obtain ⟨l₁, l₂, rfl, rfl, rfl⟩ := ih ‹_›
@@ -1829,11 +1868,11 @@ theorem map_concat (f : α → β) (a : α) (l : List α) : map f (concat l a) =
   | nil => rfl
   | cons x xs ih => simp [ih]
 
-theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ L b, l = concat L b
+theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ l' b, l = concat l' b
   | [] => .inl rfl
   | a::l => match l, eq_nil_or_concat l with
     | _, .inl rfl => .inr ⟨[], a, rfl⟩
-    | _, .inr ⟨L, b, rfl⟩ => .inr ⟨a::L, b, rfl⟩
+    | _, .inr ⟨l', b, rfl⟩ => .inr ⟨a::l', b, rfl⟩
 
 /-! ### flatten -/
 
@@ -1847,7 +1886,7 @@ theorem flatten_singleton (l : List α) : [l].flatten = l := by simp
 
 @[simp] theorem mem_flatten : ∀ {L : List (List α)}, a ∈ L.flatten ↔ ∃ l, l ∈ L ∧ a ∈ l
   | [] => by simp
-  | b :: l => by simp [mem_flatten, or_and_right, exists_or]
+  | _ :: _ => by simp [mem_flatten, or_and_right, exists_or]
 
 @[simp] theorem flatten_eq_nil_iff {L : List (List α)} : L.flatten = [] ↔ ∀ l ∈ L, l = [] := by
   induction L <;> simp_all
@@ -1855,7 +1894,7 @@ theorem flatten_singleton (l : List α) : [l].flatten = l := by simp
 @[simp] theorem nil_eq_flatten_iff {L : List (List α)} : [] = L.flatten ↔ ∀ l ∈ L, l = [] := by
   rw [eq_comm, flatten_eq_nil_iff]
 
-theorem flatten_ne_nil_iff {xs : List (List α)} : xs.flatten ≠ [] ↔ ∃ x, x ∈ xs ∧ x ≠ [] := by
+theorem flatten_ne_nil_iff {xss : List (List α)} : xss.flatten ≠ [] ↔ ∃ xs, xs ∈ xss ∧ xs ≠ [] := by
   simp
 
 theorem exists_of_mem_flatten : a ∈ flatten L → ∃ l, l ∈ L ∧ a ∈ l := mem_flatten.1
@@ -1913,13 +1952,13 @@ theorem flatten_concat (L : List (List α)) (l : List α) : flatten (L ++ [l]) =
 theorem flatten_flatten {L : List (List (List α))} : flatten (flatten L) = flatten (map flatten L) := by
   induction L <;> simp_all
 
-theorem flatten_eq_cons_iff {xs : List (List α)} {y : α} {ys : List α} :
-    xs.flatten = y :: ys ↔
-      ∃ as bs cs, xs = as ++ (y :: bs) :: cs ∧ (∀ l, l ∈ as → l = []) ∧ ys = bs ++ cs.flatten := by
+theorem flatten_eq_cons_iff {xss : List (List α)} {y : α} {ys : List α} :
+    xss.flatten = y :: ys ↔
+      ∃ as bs cs, xss = as ++ (y :: bs) :: cs ∧ (∀ l, l ∈ as → l = []) ∧ ys = bs ++ cs.flatten := by
   constructor
-  · induction xs with
+  · induction xss with
     | nil => simp
-    | cons x xs ih =>
+    | cons xs xss ih =>
       intro h
       simp only [flatten_cons] at h
       replace h := h.symm
@@ -1928,8 +1967,8 @@ theorem flatten_eq_cons_iff {xs : List (List α)} {y : α} {ys : List α} :
       · obtain ⟨as, bs, cs, rfl, _, rfl⟩ := ih h
         refine ⟨[] :: as, bs, cs, ?_⟩
         simpa
-      · obtain ⟨a', rfl, rfl⟩ := z
-        refine ⟨[], a', xs, ?_⟩
+      · obtain ⟨as', rfl, rfl⟩ := z
+        refine ⟨[], as', xss, ?_⟩
         simp
   · rintro ⟨as, bs, cs, rfl, h₁, rfl⟩
     simp [flatten_eq_nil_iff.mpr h₁]
@@ -1954,30 +1993,30 @@ theorem singleton_eq_flatten_iff {xs : List (List α)} {y : α} :
     [y] = xs.flatten ↔ ∃ as bs, xs = as ++ [y] :: bs ∧ (∀ l, l ∈ as → l = []) ∧ (∀ l, l ∈ bs → l = []) := by
   rw [eq_comm, flatten_eq_singleton_iff]
 
-theorem flatten_eq_append_iff {xs : List (List α)} {ys zs : List α} :
-    xs.flatten = ys ++ zs ↔
-      (∃ as bs, xs = as ++ bs ∧ ys = as.flatten ∧ zs = bs.flatten) ∨
-        ∃ as bs c cs ds, xs = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.flatten ++ bs ∧
+theorem flatten_eq_append_iff {xss : List (List α)} {ys zs : List α} :
+    xss.flatten = ys ++ zs ↔
+      (∃ as bs, xss = as ++ bs ∧ ys = as.flatten ∧ zs = bs.flatten) ∨
+        ∃ as bs c cs ds, xss = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.flatten ++ bs ∧
           zs = c :: cs ++ ds.flatten := by
   constructor
-  · induction xs generalizing ys with
+  · induction xss generalizing ys with
     | nil =>
       simp only [flatten_nil, nil_eq, append_eq_nil_iff, and_false, cons_append, false_and,
         exists_const, exists_false, or_false, and_imp, List.cons_ne_nil]
       rintro rfl rfl
       exact ⟨[], [], by simp⟩
-    | cons x xs ih =>
+    | cons xs xss ih =>
       intro h
       simp only [flatten_cons] at h
       rw [append_eq_append_iff] at h
-      obtain (⟨ys, rfl, h⟩ | ⟨c', rfl, h⟩) := h
+      obtain (⟨ys, rfl, h⟩ | ⟨bs, rfl, h⟩) := h
       · obtain (⟨as, bs, rfl, rfl, rfl⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, rfl⟩) := ih h
-        · exact .inl ⟨x :: as, bs, by simp⟩
-        · exact .inr ⟨x :: as, bs, c, cs, ds, by simp⟩
+        · exact .inl ⟨xs :: as, bs, by simp⟩
+        · exact .inr ⟨xs :: as, bs, c, cs, ds, by simp⟩
       · simp only [h]
-        cases c' with
-        | nil => exact .inl ⟨[ys], xs, by simp⟩
-        | cons x c' => exact .inr ⟨[], ys, x, c', xs, by simp⟩
+        cases bs with
+        | nil => exact .inl ⟨[ys], xss, by simp⟩
+        | cons b bs => exact .inr ⟨[], ys, b, bs, xss, by simp⟩
   · rintro (⟨as, bs, rfl, rfl, rfl⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, rfl⟩)
     · simp
     · simp
@@ -1994,8 +2033,8 @@ sublists. -/
 theorem eq_iff_flatten_eq : ∀ {L L' : List (List α)},
     L = L' ↔ L.flatten = L'.flatten ∧ map length L = map length L'
   | _, [] => by simp_all
-  | [], x' :: L' => by simp_all
-  | x :: L, x' :: L' => by
+  | [], _ :: _ => by simp_all
+  | _ :: _, _ :: _ => by
     simp
     rw [eq_iff_flatten_eq]
     constructor
@@ -2009,9 +2048,9 @@ theorem eq_iff_flatten_eq : ∀ {L L' : List (List α)},
 
 theorem flatMap_def (l : List α) (f : α → List β) : l.flatMap f = flatten (map f l) := by rfl
 
-@[simp] theorem flatMap_id (l : List (List α)) : l.flatMap id = l.flatten := by simp [flatMap_def]
+@[simp] theorem flatMap_id (L : List (List α)) : L.flatMap id = L.flatten := by simp [flatMap_def]
 
-@[simp] theorem flatMap_id' (l : List (List α)) : l.flatMap (fun a => a) = l.flatten := by simp [flatMap_def]
+@[simp] theorem flatMap_id' (L : List (List α)) : L.flatMap (fun as => as) = L.flatten := by simp [flatMap_def]
 
 @[simp]
 theorem length_flatMap (l : List α) (f : α → List β) :
@@ -2134,16 +2173,16 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
 
 @[deprecated replicate_eq_nil_iff (since := "2024-09-05")] abbrev replicate_eq_nil := @replicate_eq_nil_iff
 
-@[simp] theorem getElem_replicate (a : α) {n : Nat} {m} (h : m < (replicate n a).length) :
-    (replicate n a)[m] = a :=
+@[simp] theorem getElem_replicate (a : α) {n : Nat} {i} (h : i < (replicate n a).length) :
+    (replicate n a)[i] = a :=
   eq_of_mem_replicate (getElem_mem _)
 
-theorem getElem?_replicate : (replicate n a)[m]? = if m < n then some a else none := by
-  by_cases h : m < n
+theorem getElem?_replicate : (replicate n a)[i]? = if i < n then some a else none := by
+  by_cases h : i < n
   · rw [getElem?_eq_getElem (by simpa), getElem_replicate, if_pos h]
   · rw [getElem?_eq_none (by simpa using h), if_neg h]
 
-@[simp] theorem getElem?_replicate_of_lt {n : Nat} {m : Nat} (h : m < n) : (replicate n a)[m]? = some a := by
+@[simp] theorem getElem?_replicate_of_lt {n : Nat} {i : Nat} (h : i < n) : (replicate n a)[i]? = some a := by
   simp [getElem?_replicate, h]
 
 theorem head?_replicate (a : α) (n : Nat) : (replicate n a).head? = if n = 0 then none else some a := by
@@ -2325,18 +2364,18 @@ theorem eq_replicate_or_eq_replicate_append_cons {α : Type _} (l : List α) :
 
 /-- An induction principle for lists based on contiguous runs of identical elements. -/
 -- A `Sort _` valued version would require a different design. (And associated `@[simp]` lemmas.)
-theorem replicateRecOn {α : Type _} {p : List α → Prop} (m : List α)
+theorem replicateRecOn {α : Type _} {p : List α → Prop} (l : List α)
     (h0 : p []) (hr : ∀ a n, 0 < n → p (replicate n a))
-    (hi : ∀ a b n l, a ≠ b → 0 < n → p (b :: l) → p (replicate n a ++ b :: l)) : p m := by
-  rcases eq_replicate_or_eq_replicate_append_cons m with
+    (hi : ∀ a b n l, a ≠ b → 0 < n → p (b :: l) → p (replicate n a ++ b :: l)) : p l := by
+  rcases eq_replicate_or_eq_replicate_append_cons l with
     rfl | ⟨n, a, rfl, hn⟩ | ⟨n, a, b, l', w, hn, h⟩
   · exact h0
   · exact hr _ _ hn
-  · have : (b :: l').length < m.length := by
+  · have : (b :: l').length < l.length := by
       simpa [w] using Nat.lt_add_of_pos_left hn
     subst w
     exact hi _ _ _ _ h hn (replicateRecOn (b :: l') h0 hr hi)
-termination_by m.length
+termination_by l.length
 
 @[simp] theorem sum_replicate_nat (n : Nat) (a : Nat) : (replicate n a).sum = n * a := by
   induction n <;> simp_all [replicate_succ, Nat.add_mul, Nat.add_comm]
@@ -2363,6 +2402,9 @@ theorem mem_reverseAux {x : α} : ∀ {as bs}, x ∈ reverseAux as bs ↔ x ∈
 theorem reverse_ne_nil_iff {xs : List α} : xs.reverse ≠ [] ↔ xs ≠ [] :=
   not_congr reverse_eq_nil_iff
 
+@[simp] theorem isEmpty_reverse {xs : List α} : xs.reverse.isEmpty = xs.isEmpty := by
+  cases xs <;> simp
+
 /-- Variant of `getElem?_reverse` with a hypothesis giving the linear relation between the indices. -/
 theorem getElem?_reverse' : ∀ {l : List α} (i j), i + j + 1 = length l →
     l.reverse[i]? = l[j]?
@@ -2493,6 +2535,14 @@ theorem flatMap_reverse {β} (l : List α) (f : α → List β) : (l.reverse.fla
   simp only [foldrM]
   induction l <;> simp_all
 
+@[simp] theorem foldlM_pure [Monad m] [LawfulMonad m] (f : β → α → β) (b) (l : List α) :
+    l.foldlM (m := m) (pure <| f · ·) b = pure (l.foldl f b) := by
+  induction l generalizing b <;> simp [*]
+
+@[simp] theorem foldrM_pure [Monad m] [LawfulMonad m] (f : α → β → β) (b) (l : List α) :
+    l.foldrM (m := m) (pure <| f · ·) b = pure (l.foldr f b) := by
+  induction l generalizing b <;> simp [*]
+
 theorem foldl_eq_foldlM (f : β → α → β) (b) (l : List α) :
     l.foldl f b = l.foldlM (m := Id) f b := by
   induction l generalizing b <;> simp [*, foldl]
@@ -2516,12 +2566,35 @@ theorem foldr_eq_foldrM (f : α → β → β) (b) (l : List α) :
 
 /-! ### foldl and foldr -/
 
-@[simp] theorem foldr_cons_eq_append (l : List α) : l.foldr cons l' = l ++ l' := by
+@[simp] theorem foldr_cons_eq_append (l : List α) (f : α → β) (l' : List β) :
+    l.foldr (fun x ys => f x :: ys) l' = l.map f ++ l' := by
+  induction l <;> simp [*]
+
+/-- Variant of `foldr_cons_eq_append` specalized to `f = id`. -/
+@[simp] theorem foldr_cons_eq_append' (l l' : List β) :
+    l.foldr cons l' = l ++ l' := by
   induction l <;> simp [*]
 
 @[deprecated foldr_cons_eq_append (since := "2024-08-22")] abbrev foldr_self_append := @foldr_cons_eq_append
 
-@[simp] theorem foldl_flip_cons_eq_append (l : List α) : l.foldl (fun x y => y :: x) l' = l.reverse ++ l' := by
+@[simp] theorem foldl_flip_cons_eq_append (l : List α) (f : α → β) (l' : List β) :
+    l.foldl (fun xs y => f y :: xs) l' = (l.map f).reverse ++ l' := by
+  induction l generalizing l' <;> simp [*]
+
+@[simp] theorem foldr_append_eq_append (l : List α) (f : α → List β) (l' : List β) :
+    l.foldr (f · ++ ·) l' = (l.map f).flatten ++ l' := by
+  induction l <;> simp [*]
+
+@[simp] theorem foldl_append_eq_append (l : List α) (f : α → List β) (l' : List β) :
+    l.foldl (· ++ f ·) l' = l' ++ (l.map f).flatten := by
+  induction l generalizing l'<;> simp [*]
+
+@[simp] theorem foldr_flip_append_eq_append (l : List α) (f : α → List β) (l' : List β) :
+    l.foldr (fun x ys => ys ++ f x) l' = l' ++ (l.map f).reverse.flatten := by
+  induction l generalizing l' <;> simp [*]
+
+@[simp] theorem foldl_flip_append_eq_append (l : List α) (f : α → List β) (l' : List β) :
+    l.foldl (fun xs y => f y ++ xs) l' = (l.map f).reverse.flatten ++ l' := by
   induction l generalizing l' <;> simp [*]
 
 theorem foldr_cons_nil (l : List α) : l.foldr cons [] = l := by simp
@@ -2740,9 +2813,8 @@ theorem getLast_eq_head_reverse {l : List α} (h : l ≠ []) :
     l.getLast h = l.reverse.head (by simp_all) := by
   rw [← head_reverse]
 
-theorem getLast_eq_iff_getLast_eq_some {xs : List α} (h) : xs.getLast h = a ↔ xs.getLast? = some a := by
-  rw [getLast_eq_head_reverse, head_eq_iff_head?_eq_some]
-  simp
+@[deprecated getLast_eq_iff_getLast?_eq_some (since := "2025-02-17")]
+abbrev getLast_eq_iff_getLast_eq_some := @getLast_eq_iff_getLast?_eq_some
 
 @[simp] theorem getLast?_eq_none_iff {xs : List α} : xs.getLast? = none ↔ xs = [] := by
   rw [getLast?_eq_head?_reverse, head?_eq_none_iff, reverse_eq_nil_iff]
@@ -2811,8 +2883,8 @@ theorem getLast_filterMap_of_eq_some {f : α → Option β} {l : List α} {w : l
   rw [head_filterMap_of_eq_some (by simp_all)]
   simp_all
 
-theorem getLast?_flatMap {L : List α} {f : α → List β} :
-    (L.flatMap f).getLast? = L.reverse.findSome? fun a => (f a).getLast? := by
+theorem getLast?_flatMap {l : List α} {f : α → List β} :
+    (l.flatMap f).getLast? = l.reverse.findSome? fun a => (f a).getLast? := by
   simp only [← head?_reverse, reverse_flatMap]
   rw [head?_flatMap]
   rfl
@@ -2834,7 +2906,7 @@ theorem getLast?_replicate (a : α) (n : Nat) : (replicate n a).getLast? = if n
 -- We unfold `leftpad` and `rightpad` for verification purposes.
 attribute [simp] leftpad rightpad
 
--- `length_leftpad` is in `Init.Data.List.Nat.Basic`.
+-- `length_leftpad` and `length_rightpad` are in `Init.Data.List.Nat.Basic`.
 
 theorem leftpad_prefix (n : Nat) (a : α) (l : List α) :
     replicate (n - length l) a <+: leftpad n a l := by
@@ -3000,16 +3072,16 @@ We don't provide any API for `splitAt`, beyond the `@[simp]` lemma
 which is proved in `Init.Data.List.TakeDrop`.
 -/
 
-theorem splitAt_go (n : Nat) (l acc : List α) :
-    splitAt.go l xs n acc =
-      if n < xs.length then (acc.reverse ++ xs.take n, xs.drop n) else (l, []) := by
-  induction xs generalizing n acc with
+theorem splitAt_go (i : Nat) (l acc : List α) :
+    splitAt.go l xs i acc =
+      if i < xs.length then (acc.reverse ++ xs.take i, xs.drop i) else (l, []) := by
+  induction xs generalizing i acc with
   | nil => simp [splitAt.go]
   | cons x xs ih =>
-    cases n with
+    cases i with
     | zero => simp [splitAt.go]
-    | succ n =>
-      rw [splitAt.go, take_succ_cons, drop_succ_cons, ih n (x :: acc),
+    | succ i =>
+      rw [splitAt.go, take_succ_cons, drop_succ_cons, ih i (x :: acc),
         reverse_cons, append_assoc, singleton_append, length_cons]
       simp only [Nat.succ_lt_succ_iff]
 
@@ -3022,8 +3094,12 @@ variable [BEq α]
 @[simp] theorem replace_cons_self [LawfulBEq α] {a : α} : (a::as).replace a b = b::as := by
   simp [replace_cons]
 
-@[simp] theorem replace_of_not_mem {l : List α} (h : !l.elem a) : l.replace a b = l := by
-  induction l <;> simp_all [replace_cons]
+@[simp] theorem replace_of_not_mem [LawfulBEq α] {l : List α} (h : a ∉ l) : l.replace a b = l := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [replace_cons]
+    split <;> simp_all
 
 @[simp] theorem length_replace {l : List α} : (l.replace a b).length = l.length := by
   induction l with
@@ -3091,14 +3167,14 @@ theorem replace_append_right [LawfulBEq α] {l₁ l₂ : List α} (h : ¬ a ∈
     (l₁ ++ l₂).replace a b = l₁ ++ l₂.replace a b := by
   simp [replace_append, h]
 
-theorem replace_take {l : List α} {n : Nat} :
-    (l.take n).replace a b = (l.replace a b).take n := by
-  induction l generalizing n with
+theorem replace_take {l : List α} {i : Nat} :
+    (l.take i).replace a b = (l.replace a b).take i := by
+  induction l generalizing i with
   | nil => simp
   | cons x xs ih =>
-    cases n with
+    cases i with
     | zero => simp [ih]
-    | succ n =>
+    | succ i =>
       simp only [replace_cons, take_succ_cons]
       split <;> simp_all
 
@@ -3106,7 +3182,7 @@ theorem replace_take {l : List α} {n : Nat} :
     (replicate n a).replace a b = b :: replicate (n - 1) a := by
   cases n <;> simp_all [replicate_succ, replace_cons]
 
-@[simp] theorem replace_replicate_ne {a b c : α} (h : !b == a) :
+@[simp] theorem replace_replicate_ne [LawfulBEq α] {a b c : α} (h : !b == a) :
     (replicate n a).replace b c = replicate n a := by
   rw [replace_of_not_mem]
   simp_all
@@ -3302,13 +3378,13 @@ theorem all_eq_not_any_not (l : List α) (p : α → Bool) : l.all p = !l.any (!
     simp only [filterMap_cons]
     split <;> simp_all
 
-@[simp] theorem any_append {x y : List α} : (x ++ y).any f = (x.any f || y.any f) := by
-  induction x with
+@[simp] theorem any_append {xs ys : List α} : (xs ++ ys).any f = (xs.any f || ys.any f) := by
+  induction xs with
   | nil => rfl
   | cons h t ih => simp_all [Bool.or_assoc]
 
-@[simp] theorem all_append {x y : List α} : (x ++ y).all f = (x.all f && y.all f) := by
-  induction x with
+@[simp] theorem all_append {xs ys : List α} : (xs ++ ys).all f = (xs.all f && ys.all f) := by
+  induction xs with
   | nil => rfl
   | cons h t ih => simp_all [Bool.and_assoc]
 
@@ -3368,9 +3444,11 @@ theorem get_cons_succ {as : List α} {h : i + 1 < (a :: as).length} :
 theorem get_cons_succ' {as : List α} {i : Fin as.length} :
   (a :: as).get i.succ = as.get i := rfl
 
-theorem get_mk_zero : ∀ {l : List α} (h : 0 < l.length), l.get ⟨0, h⟩ = l.head (length_pos.mp h)
+theorem get_mk_zero : ∀ {l : List α} (h : 0 < l.length), l.get ⟨0, h⟩ = l.head (length_pos_iff.mp h)
   | _::_, _ => rfl
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[0]?` instead." (since := "2025-02-12")]
 theorem get?_zero (l : List α) : l.get? 0 = l.head? := by cases l <;> rfl
 
 /--
@@ -3382,10 +3460,14 @@ such a rewrite, with `rw [get_of_eq h]`.
 theorem get_of_eq {l l' : List α} (h : l = l') (i : Fin l.length) :
     get l i = get l' ⟨i, h ▸ i.2⟩ := by cases h; rfl
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
 theorem get!_of_get? [Inhabited α] : ∀ {l : List α} {n}, get? l n = some a → get! l n = a
   | _a::_, 0, rfl => rfl
   | _::l, _+1, e => get!_of_get? (l := l) e
 
+set_option linter.deprecated false in
+@[deprecated "Use `a[i]!` instead." (since := "2025-02-12")]
 theorem get!_len_le [Inhabited α] : ∀ {l : List α} {n}, length l ≤ n → l.get! n = (default : α)
   | [], _, _ => rfl
   | _ :: l, _+1, h => get!_len_le (l := l) <| Nat.le_of_succ_le_succ h
@@ -3415,6 +3497,8 @@ theorem get_of_mem {a} {l : List α} (h : a ∈ l) : ∃ n, get l n = a := by
   obtain ⟨n, h, e⟩ := getElem_of_mem h
   exact ⟨⟨n, h⟩, e⟩
 
+set_option linter.deprecated false in
+@[deprecated getElem?_of_mem (since := "2025-02-12")]
 theorem get?_of_mem {a} {l : List α} (h : a ∈ l) : ∃ n, l.get? n = some a :=
   let ⟨⟨n, _⟩, e⟩ := get_of_mem h; ⟨n, e ▸ get?_eq_get _⟩
 
@@ -3422,12 +3506,16 @@ theorem get_mem : ∀ (l : List α) n, get l n ∈ l
   | _ :: _, ⟨0, _⟩ => .head ..
   | _ :: l, ⟨_+1, _⟩ => .tail _ (get_mem l ..)
 
+set_option linter.deprecated false in
+@[deprecated mem_of_getElem? (since := "2025-02-12")]
 theorem mem_of_get? {l : List α} {n a} (e : l.get? n = some a) : a ∈ l :=
   let ⟨_, e⟩ := get?_eq_some_iff.1 e; e ▸ get_mem ..
 
 theorem mem_iff_get {a} {l : List α} : a ∈ l ↔ ∃ n, get l n = a :=
   ⟨get_of_mem, fun ⟨_, e⟩ => e ▸ get_mem ..⟩
 
+set_option linter.deprecated false in
+@[deprecated mem_iff_getElem? (since := "2025-02-12")]
 theorem mem_iff_get? {a} {l : List α} : a ∈ l ↔ ∃ n, l.get? n = some a := by
   simp [getElem?_eq_some_iff, Fin.exists_iff, mem_iff_get]
 
@@ -3449,7 +3537,6 @@ theorem join_map_filter (p : α → Bool) (l : List (List α)) :
 @[deprecated flatten_eq_cons_iff (since := "2024-09-05")] abbrev join_eq_cons := @flatten_eq_cons_iff
 @[deprecated flatten_eq_append_iff (since := "2024-09-05")] abbrev join_eq_append := @flatten_eq_append_iff
 @[deprecated mem_of_getElem? (since := "2024-09-06")] abbrev getElem?_mem := @mem_of_getElem?
-@[deprecated mem_of_get? (since := "2024-09-06")] abbrev get?_mem := @mem_of_get?
 @[deprecated getElem_set_self (since := "2024-09-04")] abbrev getElem_set_eq := @getElem_set_self
 @[deprecated getElem?_set_self (since := "2024-09-04")] abbrev getElem?_set_eq := @getElem?_set_self
 @[deprecated set_eq_nil_iff (since := "2024-09-05")] abbrev set_eq_nil := @set_eq_nil_iff
@@ -3510,11 +3597,11 @@ theorem join_map_filter (p : α → Bool) (l : List (List α)) :
 @[deprecated any_flatMap (since := "2024-10-16")] abbrev any_bind := @any_flatMap
 @[deprecated all_flatMap (since := "2024-10-16")] abbrev all_bind := @all_flatMap
 
-@[deprecated get?_eq_none (since := "2024-11-29")] abbrev get?_len_le := @get?_eq_none
+@[deprecated get?_eq_none (since := "2024-11-29")] abbrev get?_len_le := @getElem?_eq_none
 @[deprecated getElem?_eq_some_iff (since := "2024-11-29")]
 abbrev getElem?_eq_some := @getElem?_eq_some_iff
 @[deprecated get?_eq_some_iff (since := "2024-11-29")]
-abbrev get?_eq_some := @get?_eq_some_iff
+abbrev get?_eq_some := @getElem?_eq_some_iff
 @[deprecated LawfulGetElem.getElem?_def (since := "2024-11-29")]
 theorem getElem?_eq (l : List α) (i : Nat) :
     l[i]? = if h : i < l.length then some l[i] else none :=
@@ -3522,7 +3609,7 @@ theorem getElem?_eq (l : List α) (i : Nat) :
 @[deprecated getElem?_eq_none (since := "2024-11-29")] abbrev getElem?_len_le := @getElem?_eq_none
 
 @[deprecated _root_.isSome_getElem? (since := "2024-12-09")]
-theorem isSome_getElem? {l : List α} {n : Nat} : l[n]?.isSome ↔ n < l.length := by
+theorem isSome_getElem? {l : List α} {i : Nat} : l[i]?.isSome ↔ i < l.length := by
   simp
 
 @[deprecated _root_.isNone_getElem? (since := "2024-12-09")]

src/Init/Data/List/Lex.lean

@@ -7,6 +7,9 @@ prelude
 import Init.Data.List.Lemmas
 import Init.Data.List.Nat.TakeDrop
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 /-! ### Lexicographic ordering -/
@@ -48,7 +51,9 @@ instance ltIrrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Irre
 
 @[simp] theorem le_nil [LT α] (l : List α) : l ≤ [] ↔ l = [] := not_nil_lex_iff
 
-@[simp] theorem nil_lex_cons : Lex r [] (a :: l) := Lex.nil
+-- This is named with a prime to avoid conflict with `lex [] (b :: bs) lt = true`.
+-- Better naming for the `Lex` vs `lex` distinction would be welcome.
+@[simp] theorem nil_lex_cons' : Lex r [] (a :: l) := Lex.nil
 
 @[simp] theorem nil_lt_cons [LT α] (a : α) (l : List α) : [] < a :: l := Lex.nil
 
@@ -165,7 +170,7 @@ protected theorem lt_of_le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
   induction h₂ generalizing l₁ with
   | nil => simp_all
   | rel hab =>
-    rename_i a b
+    rename_i a xs
     cases l₁ with
     | nil => simp_all
     | cons c l₁ =>
@@ -333,7 +338,7 @@ theorem lex_eq_true_iff_exists [BEq α] (lt : α → α → Bool) :
     cases l₂ with
     | nil => simp [lex]
     | cons b l₂ =>
-      simp [lex_cons_cons, Bool.or_eq_true, Bool.and_eq_true, ih, isEqv, length_cons]
+      simp [cons_lex_cons, Bool.or_eq_true, Bool.and_eq_true, ih, isEqv, length_cons]
       constructor
       · rintro (hab | ⟨hab, ⟨h₁, h₂⟩ | ⟨i, h₁, h₂, w₁, w₂⟩⟩)
         · exact .inr ⟨0, by simp [hab]⟩
@@ -397,7 +402,7 @@ theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α
     cases l₂ with
     | nil => simp [lex]
     | cons b l₂ =>
-      simp [lex_cons_cons, Bool.or_eq_false_iff, Bool.and_eq_false_imp, ih, isEqv,
+      simp [cons_lex_cons, Bool.or_eq_false_iff, Bool.and_eq_false_imp, ih, isEqv,
         Bool.and_eq_true, length_cons]
       constructor
       · rintro ⟨hab, h⟩

src/Init/Data/List/MapIdx.lean

@@ -11,6 +11,9 @@ import Init.Data.List.OfFn
 import Init.Data.Fin.Lemmas
 import Init.Data.Option.Attach
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 /-! ## Operations using indexes -/
@@ -131,10 +134,10 @@ theorem mapFinIdx_cons {l : List α} {a : α} {f : (i : Nat) → α → (h : i <
   · simp
   · rintro (_|i) h₁ h₂ <;> simp
 
-theorem mapFinIdx_append {K L : List α} {f : (i : Nat) → α → (h : i < (K ++ L).length) → β} :
-    (K ++ L).mapFinIdx f =
-      K.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
-        L.mapFinIdx (fun i a h => f (i + K.length) a (by simp; omega)) := by
+theorem mapFinIdx_append {xs ys : List α} {f : (i : Nat) → α → (h : i < (xs ++ ys).length) → β} :
+    (xs ++ ys).mapFinIdx f =
+      xs.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
+        ys.mapFinIdx (fun i a h => f (i + xs.length) a (by simp; omega)) := by
   apply ext_getElem
   · simp
   · intro i h₁ h₂
@@ -299,15 +302,15 @@ theorem mapFinIdx_eq_replicate_iff {l : List α} {f : (i : Nat) → α → (h :
 theorem mapIdx_nil {f : Nat → α → β} : mapIdx f [] = [] :=
   rfl
 
-theorem mapIdx_go_length {arr : Array β} :
-    length (mapIdx.go f l arr) = length l + arr.size := by
-  induction l generalizing arr with
+theorem mapIdx_go_length {acc : Array β} :
+    length (mapIdx.go f l acc) = length l + acc.size := by
+  induction l generalizing acc with
   | nil => simp only [mapIdx.go, length_nil, Nat.zero_add]
   | cons _ _ ih =>
     simp only [mapIdx.go, ih, Array.size_push, Nat.add_succ, length_cons, Nat.add_comm]
 
-theorem length_mapIdx_go : ∀ {l : List α} {arr : Array β},
-    (mapIdx.go f l arr).length = l.length + arr.size
+theorem length_mapIdx_go : ∀ {l : List α} {acc : Array β},
+    (mapIdx.go f l acc).length = l.length + acc.size
   | [], _ => by simp [mapIdx.go]
   | a :: l, _ => by
     simp only [mapIdx.go, length_cons]
@@ -318,13 +321,13 @@ theorem length_mapIdx_go : ∀ {l : List α} {arr : Array β},
 @[simp] theorem length_mapIdx {l : List α} : (l.mapIdx f).length = l.length := by
   simp [mapIdx, length_mapIdx_go]
 
-theorem getElem?_mapIdx_go : ∀ {l : List α} {arr : Array β} {i : Nat},
-    (mapIdx.go f l arr)[i]? =
-      if h : i < arr.size then some arr[i] else Option.map (f i) l[i - arr.size]?
-  | [], arr, i => by
+theorem getElem?_mapIdx_go : ∀ {l : List α} {acc : Array β} {i : Nat},
+    (mapIdx.go f l acc)[i]? =
+      if h : i < acc.size then some acc[i] else Option.map (f i) l[i - acc.size]?
+  | [], acc, i => by
     simp only [mapIdx.go, Array.toListImpl_eq, getElem?_def, Array.length_toList,
       ← Array.getElem_toList, length_nil, Nat.not_lt_zero, ↓reduceDIte, Option.map_none']
-  | a :: l, arr, i => by
+  | a :: l, acc, i => by
     rw [mapIdx.go, getElem?_mapIdx_go]
     simp only [Array.size_push]
     split <;> split
@@ -332,10 +335,10 @@ theorem getElem?_mapIdx_go : ∀ {l : List α} {arr : Array β} {i : Nat},
       rw [← Array.getElem_toList]
       simp only [Array.push_toList]
       rw [getElem_append_left, ← Array.getElem_toList]
-    · have : i = arr.size := by omega
+    · have : i = acc.size := by omega
       simp_all
     · omega
-    · have : i - arr.size = i - (arr.size + 1) + 1 := by omega
+    · have : i - acc.size = i - (acc.size + 1) + 1 := by omega
       simp_all
 
 @[simp] theorem getElem?_mapIdx {l : List α} {i : Nat} :
@@ -371,9 +374,9 @@ theorem mapIdx_cons {l : List α} {a : α} :
     mapIdx f (a :: l) = f 0 a :: mapIdx (fun i => f (i + 1)) l := by
   simp [mapIdx_eq_zipIdx_map, List.zipIdx_succ]
 
-theorem mapIdx_append {K L : List α} :
-    (K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i => f (i + K.length) := by
-  induction K generalizing f with
+theorem mapIdx_append {xs ys : List α} :
+    (xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx fun i => f (i + xs.length) := by
+  induction xs generalizing f with
   | nil => rfl
   | cons _ _ ih => simp [ih (f := fun i => f (i + 1)), Nat.add_assoc]
 

src/Init/Data/List/MinMax.lean

@@ -10,6 +10,9 @@ import Init.Data.List.Lemmas
 # Lemmas about `List.min?` and `List.max?.
 -/
 
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
 namespace List
 
 open Nat