doc: improve List.toArray doc-string

.github/workflows/awaiting-mathlib.yml

@@ -1,20 +0,0 @@
-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

@@ -137,6 +137,7 @@ jobs:
             let large = ${{ github.repository == 'leanprover/lean4' }};
             let matrix = [
               {
+                // portable release build: use channel with older glibc (2.27)
                 "name": "Linux LLVM",
                 "os": "ubuntu-latest",
                 "release": false,
@@ -151,7 +152,6 @@ jobs:
                 "CMAKE_OPTIONS": "-DLLVM=ON -DLLVM_CONFIG=${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/stackoverflowing tests
-                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest|bv_bitblast_stress|3807'"
+                // exclude seriously slow tests
+                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest|bv_bitblast_stress'"
               },
               // TODO: suddenly started failing in CI
               /*{
@@ -204,8 +204,7 @@ jobs:
                 "os": "macos-14",
                 "CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-darwin_aarch64",
                 "release": true,
-                // special cased below
-                // "check-level": 0,
+                "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*",
@@ -261,21 +260,8 @@ 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${JSON.stringify(matrix, null, 2)}`);
-            const isPr = "${{ 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);
+            console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`)
+            return matrix.filter((job) => level >= job["check-level"])
 
   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@v9 # https://github.com/marketplace/actions/download-workflow-artifact
+        uses: dawidd6/action-download-artifact@v8 # https://github.com/marketplace/actions/download-workflow-artifact
         with:
           run_id: ${{ github.event.workflow_run.id }}
           path: artifacts
@@ -155,20 +155,6 @@ 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"
 
@@ -215,12 +201,7 @@ jobs:
             else
               echo "The message already exists in the comment body."
             fi
-
-            if [[ "$FORCE_CI" == "true" ]]; then
-              echo "mathlib_ready=true" >> "$GITHUB_OUTPUT"
-            else
-              echo "mathlib_ready=false" >> "$GITHUB_OUTPUT"
-            fi
+            echo "mathlib_ready=false" >> "$GITHUB_OUTPUT"
           else
             echo "mathlib_ready=true" >> "$GITHUB_OUTPUT"
           fi
@@ -271,7 +252,7 @@ jobs:
           if git ls-remote --heads --tags --exit-code origin "nightly-testing-${MOST_RECENT_NIGHTLY}" >/dev/null; then
             BASE="nightly-testing-${MOST_RECENT_NIGHTLY}"
           else
-            echo "Couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' tag at Batteries. Falling back to 'nightly-testing'."
+            echo "This shouldn't be possible: couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' tag at Batteries. Falling back to 'nightly-testing'."
             BASE=nightly-testing
           fi
 
@@ -335,7 +316,7 @@ 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 "Couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' branch at Mathlib. Falling back to 'nightly-testing'."
+            echo "This shouldn't be possible: couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' branch at Mathlib. Falling back to 'nightly-testing'."
             BASE=nightly-testing
           fi
 

CMakeLists.txt

@@ -47,11 +47,10 @@ 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-2.1.2
+      GIT_TAG rel-1.9.5
       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

@@ -764,12 +764,11 @@ 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) : Sort u extends Bar, Baz :=
+structure Foo (a : α) extends Bar, Baz : Sort u :=
 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 (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``.
+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``.
 
 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 2.26+
+* x86-64 Linux with glibc 2.27+
 * 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

@@ -1,9 +0,0 @@
-# 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

@@ -1,260 +0,0 @@
-# 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

@@ -1,5 +1,3 @@
-# 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,
@@ -8,515 +6,5 @@ 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.
 
-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
-```
-
+A full style guide and naming convention are currently under construction and
+will be added here soon.

doc/std/vision.md

@@ -13,17 +13,16 @@ 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 (like `Std`). For
-example, the metaprogramming framework is not part of the standard library, but
-basic types like `True` and `Nat` are.
+to a certain directory within the Lean source repository. For example, the
+metaprogramming framework is not part of the standard library.
 
 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.
 
@@ -33,23 +32,23 @@ 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": 1740791350,
-        "narHash": "sha256-igS2Z4tVw5W/x3lCZeeadt0vcU9fxtetZ/RyrqsCRQ0=",
+        "lastModified": 1722221733,
+        "narHash": "sha256-sga9SrrPb+pQJxG1ttJfMPheZvDOxApFfwXCFO0H9xw=",
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "199169a2135e6b864a888e89a2ace345703c025d",
+        "rev": "12bf09802d77264e441f48e25459c10c93eada2e",
         "type": "github"
       },
       "original": {
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "199169a2135e6b864a888e89a2ace345703c025d",
+        "rev": "12bf09802d77264e441f48e25459c10c93eada2e",
         "type": "github"
       }
     },
@@ -67,30 +67,12 @@
         "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-older": "nixpkgs-older"
+        "nixpkgs-old": "nixpkgs-old"
       }
     },
     "systems": {

flake.nix

@@ -5,20 +5,17 @@
   # 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;
-  # 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";
+  # for cadical 1.9.5; sync with CMakeLists.txt
+  inputs.nixpkgs-cadical.url = "github:NixOS/nixpkgs/12bf09802d77264e441f48e25459c10c93eada2e";
   inputs.flake-utils.url = "github:numtide/flake-utils";
 
-  outputs = inputs: inputs.flake-utils.lib.eachDefaultSystem (system:
+  outputs = { self, nixpkgs, nixpkgs-old, flake-utils, ... }@inputs: flake-utils.lib.eachDefaultSystem (system:
     let
-      pkgs = import inputs.nixpkgs { inherit system; };
+      pkgs = import nixpkgs { inherit system; };
       # An old nixpkgs for creating releases with an old glibc
-      pkgsDist-old = import inputs.nixpkgs-older { inherit system; };
+      pkgsDist-old = import nixpkgs-old { inherit system; };
       # An old nixpkgs for creating releases with an old glibc
-      pkgsDist-old-aarch = import inputs.nixpkgs-old { localSystem.config = "aarch64-unknown-linux-gnu"; };
+      pkgsDist-old-aarch = import 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

@@ -25,10 +25,7 @@ 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);
-// 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
+__declspec(dllimport) __stdcall unsigned int SetErrorMode(unsigned int uMode);' > stage1/include/clang/windows.h
 # COFF dependencies
 cp /clang64/lib/{crtbegin,crtend,crt2,dllcrt2}.o stage1/lib/
 # runtime

script/release_notes.py

@@ -65,21 +65,20 @@ 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 commit_types():
-            if first_line.startswith(f'{commit_type}:'):
-                counts[commit_type] += 1
+        for commit_type in ['feat:', 'fix:', 'refactor:', 'doc:', 'chore:']:
+            if first_line.startswith(commit_type):
+                counts[commit_type.rstrip(':')] += 1
                 break
     
     return counts
@@ -159,9 +158,8 @@ 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"{counts['perf']} performance improvements, {counts['test']} improvements to the test suite "
-          f"and {counts['style'] + counts['chore']} other changes.\n")
+          f"there were {counts['refactor']} refactoring changes, {counts['doc']} documentation improvements "
+          f"and {counts['chore']} chores.\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))
@@ -170,12 +168,7 @@ def main():
         section_title = format_section_title(label) if label != "Uncategorised" else "Uncategorised"
         print(f"## {section_title}\n")
         for _, entry in sorted(entries, key=lambda x: x[0]):
-            # Split entry into lines and indent all lines after the first
-            lines = entry.splitlines()
-            print(f"* {lines[0]}")
-            for line in lines[1:]:
-                print(f"  {line}")
-            print()  # Empty line after each entry
+            print(f"* {entry}\n")
 
 if __name__ == "__main__":
     main()

src/CMakeLists.txt

@@ -10,7 +10,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 19)
+set(LEAN_VERSION_MINOR 18)
 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'")
@@ -144,12 +144,11 @@ if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
   # do not import the world from windows.h using appropriately named flag
   string(APPEND LEAN_EXTRA_CXX_FLAGS " -D WIN32_LEAN_AND_MEAN")
   # DLLs must go next to executables on Windows
-  set(CMAKE_RELATIVE_LIBRARY_OUTPUT_DIRECTORY "bin")
+  set(CMAKE_LIBRARY_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/bin")
 else()
-  set(CMAKE_RELATIVE_LIBRARY_OUTPUT_DIRECTORY "lib/lean")
+  set(CMAKE_LIBRARY_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/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.
@@ -455,20 +454,20 @@ if(${CMAKE_SYSTEM_NAME} MATCHES "Linux")
   string(APPEND CMAKE_CXX_FLAGS " -fPIC -ftls-model=initial-exec")
   string(APPEND LEANC_EXTRA_CC_FLAGS " -fPIC")
   string(APPEND TOOLCHAIN_SHARED_LINKER_FLAGS " -Wl,-rpath=\\$$ORIGIN/..:\\$$ORIGIN")
-  string(APPEND LAKESHARED_LINKER_FLAGS " -Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/lean/libLake.a.export -Wl,--no-whole-archive")
-  string(APPEND CMAKE_EXE_LINKER_FLAGS " -Wl,-rpath=$ORIGIN/../lib:$ORIGIN/../lib/lean")
+  string(APPEND LAKESHARED_LINKER_FLAGS " -Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/temp/libLake.a.export -Wl,--no-whole-archive")
+  string(APPEND CMAKE_EXE_LINKER_FLAGS " -Wl,-rpath=\\\$ORIGIN/../lib:\\\$ORIGIN/../lib/lean")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Darwin")
   string(APPEND CMAKE_CXX_FLAGS " -ftls-model=initial-exec")
   string(APPEND INIT_SHARED_LINKER_FLAGS " -install_name @rpath/libInit_shared.dylib")
   string(APPEND LEANSHARED_1_LINKER_FLAGS " -install_name @rpath/libleanshared_1.dylib")
   string(APPEND LEANSHARED_LINKER_FLAGS " -install_name @rpath/libleanshared.dylib")
-  string(APPEND LAKESHARED_LINKER_FLAGS " -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libLake.a.export -install_name @rpath/libLake_shared.dylib")
+  string(APPEND LAKESHARED_LINKER_FLAGS " -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/temp/libLake.a.export -install_name @rpath/libLake_shared.dylib")
   string(APPEND CMAKE_EXE_LINKER_FLAGS " -Wl,-rpath,@executable_path/../lib -Wl,-rpath,@executable_path/../lib/lean")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
   string(APPEND CMAKE_CXX_FLAGS " -fPIC")
   string(APPEND LEANC_EXTRA_CC_FLAGS " -fPIC")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
-  string(APPEND LAKESHARED_LINKER_FLAGS " -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libLake_shared.dll.a -Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/lean/libLake.a.export -Wl,--no-whole-archive")
+  string(APPEND LAKESHARED_LINKER_FLAGS " -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libLake_shared.dll.a -Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/temp/libLake.a.export -Wl,--no-whole-archive")
 endif()
 
 if(${CMAKE_SYSTEM_NAME} MATCHES "Linux")
@@ -542,9 +541,6 @@ include_directories(${LEAN_SOURCE_DIR})
 include_directories(${CMAKE_BINARY_DIR})  # version.h etc., "private" headers
 include_directories(${CMAKE_BINARY_DIR}/include)  # config.h etc., "public" headers
 
-# Lean code only needs this one include
-string(APPEND LEANC_OPTS " -I${CMAKE_BINARY_DIR}/include")
-
 # Use CMake profile C++ flags for building Lean libraries, but do not embed in `leanc`
 string(TOUPPER "${CMAKE_BUILD_TYPE}" uppercase_CMAKE_BUILD_TYPE)
 string(APPEND LEANC_OPTS " ${CMAKE_CXX_FLAGS_${uppercase_CMAKE_BUILD_TYPE}}")
@@ -757,11 +753,7 @@ add_custom_target(clean-olean
   DEPENDS clean-stdlib)
 
 install(DIRECTORY "${CMAKE_BINARY_DIR}/lib/" DESTINATION lib
-  PATTERN temp
-  PATTERN "*.export"
-  PATTERN "*.hash"
-  PATTERN "*.trace"
-  EXCLUDE)
+  PATTERN temp EXCLUDE)
 
 # symlink source into expected installation location for go-to-definition, if file system allows it
 file(MAKE_DIRECTORY ${CMAKE_BINARY_DIR}/src)
@@ -792,28 +784,10 @@ if(LEAN_INSTALL_PREFIX)
 endif()
 
 # Escape for `make`. Yes, twice.
-string(REPLACE "$" "\\\$$" CMAKE_EXE_LINKER_FLAGS_MAKE "${CMAKE_EXE_LINKER_FLAGS}")
+string(REPLACE "$" "$$" CMAKE_EXE_LINKER_FLAGS_MAKE "${CMAKE_EXE_LINKER_FLAGS}")
 string(REPLACE "$" "$$" CMAKE_EXE_LINKER_FLAGS_MAKE_MAKE "${CMAKE_EXE_LINKER_FLAGS_MAKE}")
 configure_file(${LEAN_SOURCE_DIR}/stdlib.make.in ${CMAKE_BINARY_DIR}/stdlib.make)
 
-# hacky
-function(toml_escape IN OUTVAR)
-  if(IN)
-    string(STRIP "${IN}" OUT)
-    string(REPLACE " " "\", \"" OUT "${OUT}")
-    set(${OUTVAR} "\"${OUT}\"" PARENT_SCOPE)
-  endif()
-endfunction()
-string(REPLACE "ROOT" "${CMAKE_BINARY_DIR}" LEANC_CC "${LEANC_CC}")
-string(REPLACE "ROOT" "${CMAKE_BINARY_DIR}" LEANC_INTERNAL_FLAGS "${LEANC_INTERNAL_FLAGS}")
-string(REPLACE "ROOT" "${CMAKE_BINARY_DIR}" LEANC_INTERNAL_LINKER_FLAGS "${LEANC_INTERNAL_LINKER_FLAGS}")
-set(LEANC_OPTS_TOML "${LEANC_OPTS} ${LEANC_EXTRA_CC_FLAGS} ${LEANC_INTERNAL_FLAGS}")
-set(LINK_OPTS_TOML "${LEANC_INTERNAL_LINKER_FLAGS} -L${CMAKE_BINARY_DIR}/lib/lean ${LEAN_EXTRA_LINKER_FLAGS}")
-
-toml_escape("${LEAN_EXTRA_MAKE_OPTS}" LEAN_EXTRA_OPTS_TOML)
-toml_escape("${LEANC_OPTS_TOML}" LEANC_OPTS_TOML)
-toml_escape("${LINK_OPTS_TOML}" LINK_OPTS_TOML)
-
 if(USE_LAKE AND STAGE EQUAL 1)
   configure_file(${LEAN_SOURCE_DIR}/lakefile.toml.in ${LEAN_SOURCE_DIR}/lakefile.toml)
   configure_file(${LEAN_SOURCE_DIR}/lakefile.toml.in ${LEAN_SOURCE_DIR}/../tests/lakefile.toml)

src/Init.lean

@@ -39,5 +39,3 @@ import Init.While
 import Init.Syntax
 import Init.Internal
 import Init.Try
-import Init.BinderNameHint
-import Init.Task

src/Init/BinderNameHint.lean

@@ -1,42 +0,0 @@
-/-
-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

@@ -38,8 +38,7 @@ 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

@@ -175,10 +175,7 @@ theorem or_iff_not_imp_right : a ∨ b ↔ (¬b → a) := Decidable.or_iff_not_i
 
 theorem not_imp_iff_and_not : ¬(a → b) ↔ a ∧ ¬b := Decidable.not_imp_iff_and_not
 
-theorem not_and_iff_not_or_not : ¬(a ∧ b) ↔ ¬a ∨ ¬b := Decidable.not_and_iff_not_or_not
-
-@[deprecated not_and_iff_not_or_not (since := "2025-03-18")]
-abbrev not_and_iff_or_not_not := @not_and_iff_not_or_not
+theorem not_and_iff_or_not_not : ¬(a ∧ b) ↔ ¬a ∨ ¬b := Decidable.not_and_iff_or_not_not
 
 theorem not_iff : ¬(a ↔ b) ↔ (¬a ↔ b) := Decidable.not_iff
 
@@ -198,7 +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 an existential statement, using `Classical.choose`. -/
+/-- Extract an element from a 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

@@ -5,7 +5,6 @@ Author: Leonardo de Moura, Sebastian Ullrich
 -/
 prelude
 import Init.Core
-import Init.BinderNameHint
 
 universe u v w
 
@@ -36,12 +35,6 @@ 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
@@ -51,28 +44,14 @@ def ForInStep.value (x : ForInStep α) : α :=
 @[simp] theorem ForInStep.value_done (b : β) : (ForInStep.done b).value = b := rfl
 @[simp] theorem ForInStep.value_yield (b : β) : (ForInStep.yield b).value = b := rfl
 
-/--
-Maps a function over a functor, with parameters swapped so that the function comes last.
-
-This function is `Functor.map` with the parameters reversed, typically used via the `<&>` operator.
--/
 @[reducible]
 def Functor.mapRev {f : Type u → Type v} [Functor f] {α β : Type u} : f α → (α → β) → f β :=
   fun a f => f <$> a
 
-@[inherit_doc Functor.mapRev]
 infixr:100 " <&> " => Functor.mapRev
 
 recommended_spelling "mapRev" for "<&>" in [Functor.mapRev, «term_<&>_»]
 
-/--
-Discards the value in a functor, retaining the functor's structure.
-
-Discarding values is especially useful when using `Applicative` functors or `Monad`s to implement
-effects, and some operation should be carried out only for its effects. In `do`-notation, statements
-whose values are discarded must return `Unit`, and `discard` can be used to explicitly discard their
-values.
--/
 @[always_inline, inline]
 def Functor.discard {f : Type u → Type v} {α : Type u} [Functor f] (x : f α) : f PUnit :=
   Functor.mapConst PUnit.unit x
@@ -92,7 +71,7 @@ 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) : Type (max (u+1) v) extends Applicative f where
+class Alternative (f : Type u → Type v) extends Applicative f : Type (max (u+1) v) where
   /--
   Produces an empty collection or recoverable failure.  The `<|>` operator collects values or recovers
   from failures. See `Alternative` for more details.

src/Init/Control/Except.lean

@@ -13,20 +13,10 @@ import Init.Coe
 namespace Except
 variable {ε : Type u}
 
-/--
-A successful computation in the `Except ε` monad: `a` is returned, and no exception is thrown.
--/
 @[always_inline, inline]
 protected def pure (a : α) : Except ε α :=
   Except.ok a
 
-/--
-Transforms a successful result with a function, doing nothing when an exception is thrown.
-
-Examples:
- * `(pure 2 : Except String Nat).map toString = pure 2`
- * `(throw "Error" : Except String Nat).map toString = throw "Error"`
--/
 @[always_inline, inline]
 protected def map (f : α → β) : Except ε α → Except ε β
   | Except.error err => Except.error err
@@ -37,78 +27,36 @@ protected def map (f : α → β) : Except ε α → Except ε β
   intro e
   simp [Except.map]; cases e <;> rfl
 
-/--
-Transforms exceptions with a function, doing nothing on successful results.
-
-Examples:
- * `(pure 2 : Except String Nat).mapError (·.length) = pure 2`
- * `(throw "Error" : Except String Nat).mapError (·.length) = throw 5`
--/
 @[always_inline, inline]
 protected def mapError (f : ε → ε') : Except ε α → Except ε' α
   | Except.error err => Except.error <| f err
   | Except.ok v      => Except.ok v
 
-/--
-Sequences two operations that may throw exceptions, allowing the second to depend on the value
-returned by the first.
-
-If the first operation throws an exception, then it is the result of the computation. If the first
-succeeds but the second throws an exception, then that exception is the result. If both succeed,
-then the result is the result of the second computation.
-
-This is the implementation of the `>>=` operator for `Except ε`.
--/
 @[always_inline, inline]
 protected def bind (ma : Except ε α) (f : α → Except ε β) : Except ε β :=
   match ma with
   | Except.error err => Except.error err
   | Except.ok v      => f v
 
-/-- Returns `true` if the value is `Except.ok`, `false` otherwise. -/
+/-- Returns true if the value is `Except.ok`, false otherwise. -/
 @[always_inline, inline]
 protected def toBool : Except ε α → Bool
   | Except.ok _    => true
   | Except.error _ => false
 
-@[inherit_doc Except.toBool]
 abbrev isOk : Except ε α → Bool := Except.toBool
 
-/--
-Returns `none` if an exception was thrown, or `some` around the value on success.
-
-Examples:
- * `(pure 10 : Except String Nat).toOption = some 10`
- * `(throw "Failure" : Except String Nat).toOption = none`
--/
 @[always_inline, inline]
 protected def toOption : Except ε α → Option α
   | Except.ok a    => some a
   | Except.error _ => none
 
-/--
-Handles exceptions thrown in the `Except ε` monad.
-
-If `ma` is successful, its result is returned. If it throws an exception, then `handle` is invoked
-on the exception's value.
-
-Examples:
- * `(pure 2 : Except String Nat).tryCatch (pure ·.length) = pure 2`
- * `(throw "Error" : Except String Nat).tryCatch (pure ·.length) = pure 5`
- * `(throw "Error" : Except String Nat).tryCatch (fun x => throw ("E: " ++ x)) = throw "E: Error"`
--/
 @[always_inline, inline]
 protected def tryCatch (ma : Except ε α) (handle : ε → Except ε α) : Except ε α :=
   match ma with
   | Except.ok a    => Except.ok a
   | Except.error e => handle e
 
-/--
-Recovers from exceptions thrown in the `Except ε` monad. Typically used via the `<|>` operator.
-
-`Except.tryCatch` is a related operator that allows the recovery procedure to depend on _which_
-exception was thrown.
--/
 def orElseLazy (x : Except ε α) (y : Unit → Except ε α) : Except ε α :=
   match x with
   | Except.ok a    => Except.ok a
@@ -122,26 +70,12 @@ instance : Monad (Except ε) where
 
 end Except
 
-/--
-Adds exceptions of type `ε` to a monad `m`.
--/
 def ExceptT (ε : Type u) (m : Type u → Type v) (α : Type u) : Type v :=
   m (Except ε α)
 
-/--
-Use a monadic action that may return an exception's value as an action in the transformed monad that
-may throw the corresponding exception.
-
-This is the inverse of `ExceptT.run`.
--/
 @[always_inline, inline]
 def ExceptT.mk {ε : Type u} {m : Type u → Type v} {α : Type u} (x : m (Except ε α)) : ExceptT ε m α := x
 
-/--
-Use a monadic action that may throw an exception as an action that may return an exception's value.
-
-This is the inverse of `ExceptT.mk`.
--/
 @[always_inline, inline]
 def ExceptT.run {ε : Type u} {m : Type u → Type v} {α : Type u} (x : ExceptT ε m α) : m (Except ε α) := x
 
@@ -149,41 +83,25 @@ namespace ExceptT
 
 variable {ε : Type u} {m : Type u → Type v} [Monad m]
 
-/--
-Returns the value `a` without throwing exceptions or having any other effect.
--/
 @[always_inline, inline]
 protected def pure {α : Type u} (a : α) : ExceptT ε m α :=
   ExceptT.mk <| pure (Except.ok a)
 
-/--
-Handles exceptions thrown by an action that can have no effects _other_ than throwing exceptions.
--/
 @[always_inline, inline]
 protected def bindCont {α β : Type u} (f : α → ExceptT ε m β) : Except ε α → m (Except ε β)
   | Except.ok a    => f a
   | Except.error e => pure (Except.error e)
 
-/--
-Sequences two actions that may throw exceptions. Typically used via `do`-notation or the `>>=`
-operator.
--/
 @[always_inline, inline]
 protected def bind {α β : Type u} (ma : ExceptT ε m α) (f : α → ExceptT ε m β) : ExceptT ε m β :=
   ExceptT.mk <| ma >>= ExceptT.bindCont f
 
-/--
-Transforms a successful computation's value using `f`. Typically used via the `<$>` operator.
--/
 @[always_inline, inline]
 protected def map {α β : Type u} (f : α → β) (x : ExceptT ε m α) : ExceptT ε m β :=
   ExceptT.mk <| x >>= fun a => match a with
     | (Except.ok a)    => pure <| Except.ok (f a)
     | (Except.error e) => pure <| Except.error e
 
-/--
-Runs a computation from an underlying monad in the transformed monad with exceptions.
--/
 @[always_inline, inline]
 protected def lift {α : Type u} (t : m α) : ExceptT ε m α :=
   ExceptT.mk <| Except.ok <$> t
@@ -192,9 +110,6 @@ protected def lift {α : Type u} (t : m α) : ExceptT ε m α :=
 instance : MonadLift (Except ε) (ExceptT ε m) := ⟨fun e => ExceptT.mk <| pure e⟩
 instance : MonadLift m (ExceptT ε m) := ⟨ExceptT.lift⟩
 
-/--
-Handles exceptions produced in the `ExceptT ε` transformer.
--/
 @[always_inline, inline]
 protected def tryCatch {α : Type u} (ma : ExceptT ε m α) (handle : ε → ExceptT ε m α) : ExceptT ε m α :=
   ExceptT.mk <| ma >>= fun res => match res with
@@ -209,11 +124,6 @@ instance : Monad (ExceptT ε m) where
   bind := ExceptT.bind
   map  := ExceptT.map
 
-/--
-Transforms exceptions using the function `f`.
-
-This is the `ExceptT` version of `Except.mapError`.
--/
 @[always_inline, inline]
 protected def adapt {ε' α : Type u} (f : ε → ε') : ExceptT ε m α → ExceptT ε' m α := fun x =>
   ExceptT.mk <| Except.mapError f <$> x
@@ -240,12 +150,8 @@ instance (ε) : MonadExceptOf ε (Except ε) where
 namespace MonadExcept
 variable {ε : Type u} {m : Type v → Type w}
 
-/--
-An alternative unconditional error recovery operator that allows callers to specify which exception
-to throw in cases where both operations throw exceptions.
-
-By default, the first is thrown, because the `<|>` operator throws the second.
--/
+/-- Alternative orelse operator that allows to select which exception should be used.
+    The default is to use the first exception since the standard `orelse` uses the second. -/
 @[always_inline, inline]
 def orelse' [MonadExcept ε m] {α : Type v} (t₁ t₂ : m α) (useFirstEx := true) : m α :=
   tryCatch t₁ fun e₁ => tryCatch t₂ fun e₂ => throw (if useFirstEx then e₁ else e₂)

src/Init/Control/ExceptCps.lean

@@ -10,37 +10,19 @@ import Init.Control.Lawful.Basic
 The Exception monad transformer using CPS style.
 -/
 
-/--
-Adds exceptions of type `ε` to a monad `m`.
-
-Instead of using `Except ε` to model exceptions, this implementation uses continuation passing
-style. This has different performance characteristics from `ExceptT ε`.
--/
 def ExceptCpsT (ε : Type u) (m : Type u → Type v) (α : Type u) := (β : Type u) → (α → m β) → (ε → m β) → m β
 
 namespace ExceptCpsT
 
-/--
-Use a monadic action that may throw an exception as an action that may return an exception's value.
--/
 @[always_inline, inline]
 def run {ε α : Type u} [Monad m] (x : ExceptCpsT ε m α) : m (Except ε α) :=
   x _ (fun a => pure (Except.ok a)) (fun e => pure (Except.error e))
 
 set_option linter.unusedVariables false in  -- `s` unused
-/--
-Use a monadic action that may throw an exception by providing explicit success and failure
-continuations.
--/
 @[always_inline, inline]
 def runK {ε α : Type u} (x : ExceptCpsT ε m α) (s : ε) (ok : α → m β) (error : ε → m β) : m β :=
   x _ ok error
 
-/--
-Returns the value of a computation, forgetting whether it was an exception or a success.
-
-This corresponds to early return.
--/
 @[always_inline, inline]
 def runCatch [Monad m] (x : ExceptCpsT α m α) : m α :=
   x α pure pure
@@ -58,9 +40,6 @@ instance : MonadExceptOf ε (ExceptCpsT ε m) where
   throw e  := fun _ _ k => k e
   tryCatch x handle := fun _ k₁ k₂ => x _ k₁ (fun e => handle e _ k₁ k₂)
 
-/--
-Run an action from the transformed monad in the exception monad.
--/
 @[always_inline, inline]
 def lift [Monad m] (x : m α) : ExceptCpsT ε m α :=
   fun _ k _ => x >>= k

src/Init/Control/Id.lean

@@ -10,28 +10,6 @@ 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
@@ -42,18 +20,9 @@ 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

@@ -13,26 +13,17 @@ open Function
   rfl
 
 /--
-A functor satisfies the functor laws.
-
-The `Functor` class contains the operations of a functor, but does not require that instances
-prove they satisfy the laws of a functor. A `LawfulFunctor` instance includes proofs that the laws
-are satisfied. Because `Functor` instances may provide optimized implementations of `mapConst`,
-`LawfulFunctor` instances must also prove that the optimized implementation is equivalent to the
-standard implementation.
+The `Functor` typeclass only contains the operations of a functor.
+`LawfulFunctor` further asserts that these operations satisfy the laws of a functor,
+including the preservation of the identity and composition laws:
+```
+id <$> x = x
+(h ∘ g) <$> x = h <$> g <$> x
+```
 -/
 class LawfulFunctor (f : Type u → Type v) [Functor f] : Prop where
-  /--
-  The `mapConst` implementation is equivalent to the default implementation.
-  -/
   map_const          : (Functor.mapConst : α → f β → f α) = Functor.map ∘ const β
-  /--
-  The `map` implementation preserves identity.
-  -/
   id_map   (x : f α) : id <$> x = x
-  /--
-  The `map` implementation preserves function composition.
-  -/
   comp_map (g : α → β) (h : β → γ) (x : f α) : (h ∘ g) <$> x = h <$> g <$> x
 
 export LawfulFunctor (map_const id_map comp_map)
@@ -47,48 +38,21 @@ attribute [simp] id_map
   (comp_map _ _ _).symm
 
 /--
-An applicative functor satisfies the laws of an applicative functor.
-
-The `Applicative` class contains the operations of an applicative functor, but does not require that
-instances prove they satisfy the laws of an applicative functor. A `LawfulApplicative` instance
-includes proofs that the laws are satisfied.
-
-Because `Applicative` instances may provide optimized implementations of `seqLeft` and `seqRight`,
-`LawfulApplicative` instances must also prove that the optimized implementation is equivalent to the
-standard implementation.
+The `Applicative` typeclass only contains the operations of an applicative functor.
+`LawfulApplicative` further asserts that these operations satisfy the laws of an applicative functor:
+```
+pure id <*> v = v
+pure (·∘·) <*> u <*> v <*> w = u <*> (v <*> w)
+pure f <*> pure x = pure (f x)
+u <*> pure y = pure (· y) <*> u
+```
 -/
-class LawfulApplicative (f : Type u → Type v) [Applicative f] : Prop extends LawfulFunctor f where
-  /-- `seqLeft` is equivalent to the default implementation. -/
+class LawfulApplicative (f : Type u → Type v) [Applicative f] extends LawfulFunctor f : Prop where
   seqLeft_eq  (x : f α) (y : f β)     : x <* y = const β <$> x <*> y
-  /-- `seqRight` is equivalent to the default implementation. -/
   seqRight_eq (x : f α) (y : f β)     : x *> y = const α id <$> x <*> y
-  /--
-  `pure` before `seq` is equivalent to `Functor.map`.
-
-  This means that `pure` really is pure when occurring immediately prior to `seq`.
-   -/
   pure_seq    (g : α → β) (x : f α)   : pure g <*> x = g <$> x
-
-  /--
-  Mapping a function over the result of `pure` is equivalent to applying the function under `pure`.
-
-  This means that `pure` really is pure with respect to `Functor.map`.
-  -/
   map_pure    (g : α → β) (x : α)     : g <$> (pure x : f α) = pure (g x)
-
-  /--
-  `pure` after `seq` is equivalent to `Functor.map`.
-
-  This means that `pure` really is pure when occurring just after `seq`.
-  -/
   seq_pure    {α β : Type u} (g : f (α → β)) (x : α) : g <*> pure x = (fun h => h x) <$> g
-
-  /--
-  `seq` is associative.
-
-  Changing the nesting of `seq` calls while maintaining the order of computations results in an
-  equivalent computation. This means that `seq` is not doing any more than sequencing.
-  -/
   seq_assoc   {α β γ : Type u} (x : f α) (g : f (α → β)) (h : f (β → γ)) : h <*> (g <*> x) = ((@comp α β γ) <$> h) <*> g <*> x
   comp_map g h x := (by
     repeat rw [← pure_seq]
@@ -102,36 +66,21 @@ attribute [simp] map_pure seq_pure
   simp [pure_seq]
 
 /--
-Lawful monads are those that satisfy a certain behavioral specification. While all instances of
-`Monad` should satisfy these laws, not all implementations are required to prove this.
+The `Monad` typeclass only contains the operations of a monad.
+`LawfulMonad` further asserts that these operations satisfy the laws of a monad,
+including associativity and identity laws for `bind`:
+```
+pure x >>= f = f x
+x >>= pure = x
+x >>= f >>= g = x >>= (fun x => f x >>= g)
+```
 
-`LawfulMonad.mk'` is an alternative constructor that contains useful defaults for many fields.
+`LawfulMonad.mk'` is an alternative constructor containing useful defaults for many fields.
 -/
-class LawfulMonad (m : Type u → Type v) [Monad m] : Prop extends LawfulApplicative m where
-  /--
-  A `bind` followed by `pure` composed with a function is equivalent to a functorial map.
-
-  This means that `pure` really is pure after a `bind` and has no effects.
-  -/
+class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m : Prop where
   bind_pure_comp (f : α → β) (x : m α) : x >>= (fun a => pure (f a)) = f <$> x
-  /--
-  A `bind` followed by a functorial map is equivalent to `Applicative` sequencing.
-
-  This means that the effect sequencing from `Monad` and `Applicative` are the same.
-  -/
   bind_map       {α β : Type u} (f : m (α → β)) (x : m α) : f >>= (. <$> x) = f <*> x
-  /--
-  `pure` followed by `bind` is equivalent to function application.
-
-  This means that `pure` really is pure before a `bind` and has no effects.
-  -/
   pure_bind      (x : α) (f : α → m β) : pure x >>= f = f x
-  /--
-  `bind` is associative.
-
-  Changing the nesting of `bind` calls while maintaining the order of computations results in an
-  equivalent computation. This means that `bind` is not doing more than data-dependent sequencing.
-  -/
   bind_assoc     (x : m α) (f : α → m β) (g : β → m γ) : x >>= f >>= g = x >>= fun x => f x >>= g
   map_pure g x    := (by rw [← bind_pure_comp, pure_bind])
   seq_pure g x    := (by rw [← bind_map]; simp [map_pure, bind_pure_comp])

src/Init/Control/Lawful/Lemmas.lean

@@ -9,49 +9,25 @@ import Init.RCases
 import Init.ByCases
 
 -- Mapping by a function with a left inverse is injective.
-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 <$> ·) h
-    simpa [w] using h
-  · rintro rfl
-    rfl
+theorem map_inj_of_left_inverse [Applicative m] [LawfulApplicative m] {f : α → β}
+    (w : ∃ g : β → α, ∀ x, g (f x) = x) {x y : m α}
+    (h : f <$> x = f <$> y) : x = y := by
+  rcases w with ⟨g, w⟩
+  replace h := congrArg (g <$> ·) h
+  simpa [w] using h
 
 -- Mapping by an injective function is injective, as long as the domain is nonempty.
-@[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']
+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)

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[idx]?.getD ⟨ctx.arbitrary, none⟩
+  ctx.vars.getD idx ⟨ctx.arbitrary, none⟩
 
 instance : ContextInformation (Context α) where
   isNeutral ctx x := ctx.var x |>.neutral.isSome

src/Init/Data/Array.lean

@@ -27,4 +27,3 @@ 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

@@ -9,20 +9,18 @@ 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
 
 /--
-Maps a partially defined function (defined on those terms of `α` that satisfy a predicate `P`) over
-an array `xs : Array α`, given a proof that every element of `xs` in fact satisfies `P`.
+`O(n)`. Partial map. If `f : Π a, P a → β` is a partial function defined on
+`a : α` satisfying `P`, then `pmap f l h` is essentially the same as `map f l`
+but is defined only when all members of `l` satisfy `P`, using the proof
+to apply `f`.
 
-`Array.pmap`, named for “partial map,” is the equivalent of `Array.map` for such partial functions.
+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) (xs : Array α) (H : ∀ a ∈ xs, P a) : Array β :=
-  (xs.toList.pmap f (fun a m => H a (mem_def.mpr m))).toArray
+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
 
 /--
 Unsafe implementation of `attachWith`, taking advantage of the fact that the representation of
@@ -31,27 +29,14 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
 @[inline] private unsafe def attachWithImpl
     (xs : Array α) (P : α → Prop) (_ : ∀ x ∈ xs, P x) : Array {x // P x} := unsafeCast xs
 
-/--
-“Attaches” individual proofs to an array of values that satisfy a predicate `P`, returning an array
-of elements in the corresponding subtype `{ x // P x }`.
-
-`O(1)`.
--/
+/-- `O(1)`. "Attach" a proof `P x` that holds for all the elements of `xs` to produce a new array
+  with the same elements but in the type `{x // P x}`. -/
 @[implemented_by attachWithImpl] def attachWith
     (xs : Array α) (P : α → Prop) (H : ∀ x ∈ xs, P x) : Array {x // P x} :=
   ⟨xs.toList.attachWith P fun x h => H x (Array.Mem.mk h)⟩
 
-/--
-“Attaches” the proof that the elements of `xs` are in fact elements of `xs`, producing a new array with
-the same elements but in the subtype `{ x // x ∈ xs }`.
-
-`O(1)`.
-
-This function is primarily used to allow definitions by [well-founded
-recursion](lean-manual://section/well-founded-recursion) that use higher-order functions (such as
-`Array.map`) to prove that an value taken from a list is smaller than the list. This allows the
-well-founded recursion mechanism to prove that the function terminates.
--/
+/-- `O(1)`. "Attach" the proof that the elements of `xs` are in `xs` to produce a new array
+  with the same elements but in the type `{x // x ∈ xs}`. -/
 @[inline] def attach (xs : Array α) : Array {x // x ∈ xs} := xs.attachWith _ fun _ => id
 
 @[simp] theorem _root_.List.attachWith_toArray {l : List α} {P : α → Prop} {H : ∀ x ∈ l.toArray, P x} :
@@ -66,35 +51,35 @@ well-founded recursion mechanism to prove that the function terminates.
     l.toArray.pmap f H = (l.pmap f (by simpa using H)).toArray := by
   simp [pmap]
 
-@[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] 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 [attachWith]
 
-@[simp] theorem toList_attach {xs : Array α} :
-    xs.attach.toList = xs.toList.attachWith (· ∈ xs) (by simp [mem_toList]) := by
+@[simp] theorem toList_attach {α : Type _} {l : Array α} :
+    l.attach.toList = l.toList.attachWith (· ∈ l) (by simp [mem_toList]) := by
   simp [attach]
 
-@[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] 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 [pmap]
 
 /-- Implementation of `pmap` using the zero-copy version of `attach`. -/
-@[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'
+@[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'
 
 @[csimp] private theorem pmap_eq_pmapImpl : @pmap = @pmapImpl := by
-  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]
+  funext α β p f L h'
+  cases L
+  simp only [pmap, pmapImpl, List.attachWith_toArray, List.map_toArray, mk.injEq, List.map_attachWith]
   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) (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] 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 [pmap]
 
 @[simp] theorem attach_empty : (#[] : Array α).attach = #[] := rfl
@@ -109,158 +94,159 @@ well-founded recursion mechanism to prove that the function terminates.
   simp
 
 @[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_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_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
+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
   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) (xs H) :
-    map g (pmap f xs H) = pmap (fun a h => g (f a h)) xs H := by
-  cases xs
+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
   simp [List.map_pmap]
 
-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
+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
   simp [List.pmap_map]
 
-theorem attach_congr {xs ys : Array α} (h : xs = ys) :
-    xs.attach = ys.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
+theorem attach_congr {l₁ l₂ : Array α} (h : l₁ = l₂) :
+    l₁.attach = l₂.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
   subst h
   simp
 
-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
+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
   subst w
   simp
 
-@[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
+@[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
   rw [attach_congr (List.push_toArray _ _)]
   simp [Function.comp_def]
 
-@[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] 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 [attachWith_congr (List.push_toArray _ _)]
 
-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
+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
   simp [List.pmap_eq_map_attach]
 
 @[simp]
-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
+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
   simp [List.pmap_eq_attachWith]
 
-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
+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
   simp
 
-@[deprecated attach_map_val (since := "2025-02-17")]
-abbrev attach_map_coe := @attach_map_val
+theorem attach_map_val (l : Array α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
+  attach_map_coe _ _
 
-theorem attach_map_subtype_val (xs : Array α) : xs.attach.map Subtype.val = xs := by
-  cases xs; simp
+theorem attach_map_subtype_val (l : Array α) : l.attach.map Subtype.val = l := 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_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
 
-@[deprecated attachWith_map_val (since := "2025-02-17")]
-abbrev attachWith_map_coe := @attachWith_map_val
+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 _ _ _
 
-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
+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
 
 @[simp]
-theorem mem_attach (xs : Array α) : ∀ x, x ∈ xs.attach
+theorem mem_attach (l : Array α) : ∀ x, x ∈ l.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 (xs : Array α) {q : α → Prop} (H) (x : {x // q x}) :
-    x ∈ xs.attachWith q H ↔ x.1 ∈ xs := by
-  cases xs
+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
   simp
 
 @[simp]
-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
+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
   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} {xs H} {a} (h : a ∈ xs) :
-    f a (H a h) ∈ pmap f xs H := by
+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
   rw [mem_pmap]
   exact ⟨a, h, rfl⟩
 
 @[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
+theorem size_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : (pmap f l H).size = l.size := by
+  cases l; simp
 
 @[simp]
-theorem size_attach {xs : Array α} : xs.attach.size = xs.size := by
-  cases xs; simp
+theorem size_attach {L : Array α} : L.attach.size = L.size := by
+  cases L; simp
 
 @[simp]
-theorem size_attachWith {p : α → Prop} {xs : Array α} {H} : (xs.attachWith p H).size = xs.size := by
-  cases xs; simp
+theorem size_attachWith {p : α → Prop} {l : Array α} {H} : (l.attachWith p H).size = l.size := by
+  cases l; simp
 
 @[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_eq_empty_iff {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = #[] ↔ l = #[] := by
+  cases l; 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 {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]
+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]
 
 @[simp]
-theorem attach_eq_empty_iff {xs : Array α} : xs.attach = #[] ↔ xs = #[] := by
-  cases xs; simp
+theorem attach_eq_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
+theorem attach_ne_empty_iff {l : Array α} : l.attach ≠ #[] ↔ l ≠ #[] := by
+  cases l; simp
 
 @[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_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_ne_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
 
 @[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
+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
 
 @[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
+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
 
 @[simp]
 theorem getElem?_attachWith {xs : Array α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
@@ -283,40 +269,40 @@ 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 (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⟩)
+@[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⟩)
         (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 (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⟩)
+@[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⟩)
         (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 (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
+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
   rw [pmap_eq_map_attach, foldl_map]
 
-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
+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
   rw [pmap_eq_map_attach, foldr_map]
 
 @[simp] theorem foldl_attachWith
-    (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
+    (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
   subst w
-  rcases xs with ⟨xs⟩
+  rcases l with ⟨l⟩
   simp [List.foldl_attachWith, List.foldl_map]
 
 @[simp] theorem foldr_attachWith
-    (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
+    (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
   subst w
-  rcases xs with ⟨xs⟩
+  rcases l with ⟨l⟩
   simp [List.foldr_attachWith, List.foldr_map]
 
 /--
@@ -329,10 +315,10 @@ 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 (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,
+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,
     List.length_pmap, List.foldl_toArray', mem_toArray, List.foldl_subtype]
   congr
   ext
@@ -348,101 +334,93 @@ 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 (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,
+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,
     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 {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
+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
   ext <;> simp
 
-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
+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
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
-  cases xs
+  cases l
   simp [List.attachWith_map]
 
-@[simp] theorem map_attachWith {xs : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a}
+theorem map_attachWith {l : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
     (f : { x // P x } → β) :
-    (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
+    (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
   ext <;> simp
 
 /-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
-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
+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
   ext <;> simp
 
-@[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
+theorem attach_filterMap {l : Array 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
-  cases xs
+  cases l
   rw [attach_congr (List.filterMap_toArray f _)]
   simp [List.attach_filterMap, List.map_filterMap, Function.comp_def]
 
-theorem attach_filter {xs : Array α} (p : α → Bool) :
-    (xs.filter p).attach = xs.attach.filterMap
+theorem attach_filter {l : Array α} (p : α → Bool) :
+    (l.filter p).attach = l.attach.filterMap
       fun x => if w : p x.1 then some ⟨x.1, mem_filter.mpr ⟨x.2, w⟩⟩ else none := by
-  cases xs
+  cases l
   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} {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
+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
   subst w
-  cases xs
+  cases l
   simp [Function.comp_def]
 
 @[simp]
-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
+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
   subst w
-  cases xs
+  cases l
   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) (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
+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
         (fun a _ => H₁ a a.2) := by
-  cases xs
+  cases l
   simp [List.pmap_pmap, List.pmap_map]
 
-@[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 : 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 : 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 _
+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 _
 
 @[simp] theorem attach_append (xs ys : Array α) :
     (xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_left ys h⟩) ++
@@ -511,35 +489,35 @@ theorem back?_attach {xs : Array α} :
   simp
 
 @[simp]
-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
+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
   simp [Function.comp_def]
 
 @[simp]
-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
+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
   simp
 
 @[simp]
-theorem count_attach [DecidableEq α] (xs : Array α) (a : {x // x ∈ xs}) :
-    xs.attach.count a = xs.count ↑a := by
-  rcases xs with ⟨xs⟩
+theorem count_attach [DecidableEq α] (l : Array α) (a : {x // x ∈ l}) :
+    l.attach.count a = l.count ↑a := by
+  rcases l with ⟨l⟩
   simp only [List.attach_toArray, List.attachWith_mem_toArray, List.count_toArray]
-  rw [List.map_attach_eq_pmap, List.count_eq_countP]
+  rw [List.map_attach, 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} (xs : Array α) (H : ∀ a ∈ xs, p a) (a : {x // p x}) :
-    (xs.attachWith p H).count a = xs.count ↑a := by
-  cases xs
+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
   simp
 
-@[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] 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 [pmap_eq_map_attach, countP_map, Function.comp_def]
 
 /-! ## unattach
@@ -554,63 +532,49 @@ Further, we provide simp lemmas that push `unattach` inwards.
 -/
 
 /--
-Maps an array of terms in a subtype to the corresponding terms in the type by forgetting that they
-satisfy the predicate.
+A synonym for `l.map (·.val)`. Mostly this should not be needed by users.
+It is introduced as in intermediate step by lemmas such as `map_subtype`,
+and is ideally subsequently simplified away by `unattach_attach`.
 
-This is the inverse of `Array.attachWith` and a synonym for `xs.map (·.val)`.
-
-Mostly this should not be needed by users. It is introduced as an intermediate step by lemmas such
-as `map_subtype`, and is ideally subsequently simplified away by `unattach_attach`.
-
-This function is usually inserted automatically by Lean as an intermediate step while proving
-termination. It is rarely used explicitly in code. It is introduced as an intermediate step during
-the elaboration of definitions by [well-founded
-recursion](lean-manual://section/well-founded-recursion). If this function is encountered in a proof
-state, the right approach is usually the tactic `simp [Array.unattach, -Array.map_subtype]`.
+If not, usually the right approach is `simp [Array.unattach, -Array.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (xs : Array { x // p x }) : Array α := xs.map (·.val)
+def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) : Array α := l.map (·.val)
 
-@[simp] theorem unattach_nil {p : α → Prop} : (#[] : Array { x // p x }).unattach = #[] := by
-  simp [unattach]
-
-@[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] 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 only [unattach, Array.map_push]
 
-@[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
+@[simp] theorem size_unattach {p : α → Prop} {l : Array { x // p x }} :
+    l.unattach.size = l.size := by
   unfold unattach
   simp
 
-@[simp] theorem _root_.List.unattach_toArray {p : α → Prop} {xs : List { x // p x }} :
-    xs.toArray.unattach = xs.unattach.toArray := by
+@[simp] theorem _root_.List.unattach_toArray {p : α → Prop} {l : List { x // p x }} :
+    l.toArray.unattach = l.unattach.toArray := by
   simp only [unattach, List.map_toArray, List.unattach]
 
-@[simp] theorem toList_unattach {p : α → Prop} {xs : Array { x // p x }} :
-    xs.unattach.toList = xs.toList.unattach := by
+@[simp] theorem toList_unattach {p : α → Prop} {l : Array { x // p x }} :
+    l.unattach.toList = l.toList.unattach := by
   simp only [unattach, toList_map, List.unattach]
 
-@[simp] theorem unattach_attach {xs : Array α} : xs.attach.unattach = xs := by
-  cases xs
+@[simp] theorem unattach_attach {l : Array α} : l.attach.unattach = l := by
+  cases l
   simp only [List.attach_toArray, List.unattach_toArray, List.unattach_attachWith]
 
-@[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] theorem unattach_attachWith {p : α → Prop} {l : Array α}
+    {H : ∀ a ∈ l, p a} :
+    (l.attachWith p H).unattach = l := by
+  cases l
   simp
 
-@[simp] theorem getElem?_unattach {p : α → Prop} {xs : Array { x // p x }} (i : Nat) :
-    xs.unattach[i]? = xs[i]?.map Subtype.val := by
+@[simp] theorem getElem?_unattach {p : α → Prop} {l : Array { x // p x }} (i : Nat) :
+    l.unattach[i]? = l[i]?.map Subtype.val := by
   simp [unattach]
 
 @[simp] theorem getElem_unattach
-    {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
+    {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
   simp [unattach]
 
 /-! ### Recognizing higher order functions using a function that only depends on the value. -/
@@ -619,20 +583,20 @@ def unattach {α : Type _} {p : α → Prop} (xs : Array { x // p x }) : Array
 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} {xs : Array { x // p x }}
+theorem foldl_subtype {p : α → Prop} {l : 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) :
-    xs.foldl f x = xs.unattach.foldl g x := by
-  cases xs
+    l.foldl f x = l.unattach.foldl g x := by
+  cases l
   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} {xs : Array { x // p x }}
+@[simp] theorem foldl_subtype' {p : α → Prop} {l : 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 = xs.size) :
-    xs.foldl f x 0 stop = xs.unattach.foldl g x := by
+    (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
   subst h
   rwa [foldl_subtype]
 
@@ -640,20 +604,20 @@ theorem foldl_subtype {p : α → Prop} {xs : 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} {xs : Array { x // p x }}
+theorem foldr_subtype {p : α → Prop} {l : 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) :
-    xs.foldr f x = xs.unattach.foldr g x := by
-  cases xs
+    l.foldr f x = l.unattach.foldr g x := by
+  cases l
   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} {xs : Array { x // p x }}
+@[simp] theorem foldr_subtype' {p : α → Prop} {l : 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 = xs.size) :
-    xs.foldr f x start 0 = xs.unattach.foldr g x := by
+    (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
   subst h
   rwa [foldr_subtype]
 
@@ -661,84 +625,60 @@ theorem foldr_subtype {p : α → Prop} {xs : 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} {xs : Array { x // p x }}
+@[simp] theorem map_subtype {p : α → Prop} {l : Array { x // p x }}
     {f : { x // p x } → β} {g : α → β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    xs.map f = xs.unattach.map g := by
-  cases xs
+    l.map f = l.unattach.map g := by
+  cases l
   simp only [List.map_toArray, List.unattach_toArray]
   rw [List.map_subtype]
   simp [hf]
 
-@[simp] theorem filterMap_subtype {p : α → Prop} {xs : Array { x // p x }}
+@[simp] theorem filterMap_subtype {p : α → Prop} {l : Array { x // p x }}
     {f : { x // p x } → Option β} {g : α → Option β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    xs.filterMap f = xs.unattach.filterMap g := by
-  cases xs
-  simp only [List.size_toArray, List.filterMap_toArray', List.unattach_toArray, List.length_unattach,
+    l.filterMap f = l.unattach.filterMap g := by
+  cases l
+  simp only [size_toArray, List.filterMap_toArray', List.unattach_toArray, List.length_unattach,
     mk.injEq]
   rw [List.filterMap_subtype]
   simp [hf]
 
-
-@[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 }}
+@[simp] theorem findSome?_subtype {p : α → Prop} {l : Array { x // p x }}
     {f : { x // p x } → Option β} {g : α → Option β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    xs.findSome? f = xs.unattach.findSome? g := by
-  cases xs
+    l.findSome? f = l.unattach.findSome? g := by
+  cases l
   simp
   rw [List.findSome?_subtype hf]
 
-@[simp] theorem find?_subtype {p : α → Prop} {xs : Array { x // p x }}
+@[simp] theorem find?_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) :
-    (xs.find? f).map Subtype.val = xs.unattach.find? g := by
-  cases xs
+    (l.find? f).map Subtype.val = l.unattach.find? g := by
+  cases l
   simp
   rw [List.find?_subtype hf]
 
-@[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 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 }}
+@[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) :
-    (xs.filter f).unattach = xs.unattach.filter g := by
-  cases xs
+    (l.filter f).unattach = l.unattach.filter g := by
+  cases l
   simp [hf]
 
-@[simp] theorem unattach_reverse {p : α → Prop} {xs : Array { x // p x }} :
-    xs.reverse.unattach = xs.unattach.reverse := by
-  cases xs
+@[simp] theorem unattach_reverse {p : α → Prop} {l : Array { x // p x }} :
+    l.reverse.unattach = l.unattach.reverse := by
+  cases l
   simp
 
-@[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] 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
 
-@[simp] theorem unattach_flatten {p : α → Prop} {xs : Array (Array { x // p x })} :
-    xs.flatten.unattach = (xs.map unattach).flatten := by
+@[simp] theorem unattach_flatten {p : α → Prop} {l : Array (Array { x // p x })} :
+    l.flatten.unattach = (l.map unattach).flatten := by
   unfold unattach
-  cases xs using array₂_induction
+  cases l 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]
@@ -747,67 +687,4 @@ 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/BasicAux.lean

@@ -8,9 +8,6 @@ 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
@@ -20,13 +17,13 @@ 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 [*]
@@ -38,11 +35,6 @@ private theorem List.of_toArrayAux_eq_toArrayAux {as bs : List α} {cs ds : Arra
 theorem List.toArray_eq_toArray_eq (as bs : List α) : (as.toArray = bs.toArray) = (as = bs) := by
   simp
 
-/--
-Applies the monadic action `f` to every element in the array, left-to-right, and returns the array
-of results. Furthermore, the resulting array's type guarantees that it contains the same number of
-elements as the input array.
--/
 def Array.mapM' [Monad m] (f : α → m β) (as : Array α) : m { bs : Array β // bs.size = as.size } :=
   go 0 ⟨mkEmpty as.size, rfl⟩ (by simp)
 where
@@ -71,19 +63,11 @@ where
       return as
 
 /--
-Applies a monadic function to each element of an array, returning the array of results. The function is
-monomorphic: it is required to return a value of the same type. The internal implementation uses
-pointer equality, and does not allocate a new array if the result of each function call is
-pointer-equal to its argument.
+Monomorphic `Array.mapM`. The internal implementation uses pointer equality, and does not allocate a new array
+if the result of each `f a` is a pointer equal value `a`.
 -/
 @[implemented_by mapMonoMImp] def Array.mapMonoM [Monad m] (as : Array α) (f : α → m α) : m (Array α) :=
   as.mapM f
 
-/--
-Applies a function to each element of an array, returning the array of results. The function is
-monomorphic: it is required to return a value of the same type. The internal implementation uses
-pointer equality, and does not allocate a new array if the result of each function call is
-pointer-equal to its argument.
--/
 @[inline] def Array.mapMono (as : Array α) (f : α → α) : Array α :=
   Id.run <| as.mapMonoM f

src/Init/Data/Array/BinSearch.lean

@@ -5,13 +5,9 @@ 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) :
@@ -29,16 +25,6 @@ namespace Array
     else found (some a)
 termination_by lo hi => hi.1 - lo.1
 
-/--
-Binary search for an element equivalent to `k` in the sorted array `as`. Returns the element from
-the array, if it is found, or `none` otherwise.
-
-The array `as` must be sorted according to the comparison operator `lt`, which should be a total
-order.
-
-The optional parameters `lo` and `hi` determine the region of the array indices to be searched. Both
-are inclusive, and default to searching the entire array.
--/
 @[inline] def binSearch {α : Type} (as : Array α) (k : α) (lt : α → α → Bool) (lo := 0) (hi := as.size - 1) : Option α :=
   if h : lo < as.size then
     let hi := if hi < as.size then hi else as.size - 1
@@ -49,16 +35,6 @@ are inclusive, and default to searching the entire array.
   else
     none
 
-/--
-Binary search for an element equivalent to `k` in the sorted array `as`. Returns `true` if the
-element is found, or `false` otherwise.
-
-The array `as` must be sorted according to the comparison operator `lt`, which should be a total
-order.
-
-The optional parameters `lo` and `hi` determine the region of the array indices to be searched. Both
-are inclusive, and default to searching the entire array.
--/
 @[inline] def binSearchContains {α : Type} (as : Array α) (k : α) (lt : α → α → Bool) (lo := 0) (hi := as.size - 1) : Bool :=
   if h : lo < as.size then
     let hi := if hi < as.size then hi else as.size - 1
@@ -88,16 +64,6 @@ are inclusive, and default to searching the entire array.
       as.modifyM mid <| fun v => merge v
 termination_by lo hi => hi.1 - lo.1
 
-/--
-Inserts an element `k` into a sorted array `as` such that the resulting array is sorted.
-
-The ordering predicate `lt` should be a total order on elements, and the array `as` should be sorted
-with respect to `lt`.
-
-If an element that `lt` equates to `k` is already present in `as`, then `merge` is applied to the
-existing element to determine the value of that position in the resulting array. If no element equal
-to `k` is present, then `add` is used to determine the value to be inserted.
--/
 @[specialize] def binInsertM {α : Type u} {m : Type u → Type v} [Monad m]
     (lt : α → α → Bool)
     (merge : α → m α)
@@ -111,21 +77,6 @@ to `k` is present, then `add` is used to determine the value to be inserted.
   else if !lt k as[as.size - 1] then as.modifyM (as.size - 1) <| merge
   else binInsertAux lt merge add as k ⟨0, by omega⟩ ⟨as.size - 1, by omega⟩ (by simp) (by simpa using h')
 
-/--
-Inserts an element into a sorted array such that the resulting array is sorted. If the element is
-already present in the array, it is not inserted.
-
-The ordering predicate `lt` should be a total order on elements, and the array `as` should be sorted
-with respect to `lt`.
-
-`Array.binInsertM` is a more general operator that provides greater control over the handling of
-duplicate elements in addition to running in a monad.
-
-Examples:
-* `#[0, 1, 3, 5].binInsert (· < ·) 2 = #[0, 1, 2, 3, 5]`
-* `#[0, 1, 3, 5].binInsert (· < ·) 1 = #[0, 1, 3, 5]`
-* `#[].binInsert (· < ·) 1 = #[1]`
--/
 @[inline] def binInsert {α : Type u} (lt : α → α → Bool) (as : Array α) (k : α) : Array α :=
   Id.run <| binInsertM lt (fun _ => k) (fun _ => k) as k
 

src/Init/Data/Array/Bootstrap.lean

@@ -13,151 +13,122 @@ 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) (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
+    (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
   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 xs i (j+1) H]
+    simp [foldlM_toList.aux f arr 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) (xs : Array α) :
-    xs.toList.foldlM f init = xs.foldlM f init := by
+    (f : β → α → m β) (init : β) (arr : Array α) :
+    arr.toList.foldlM f init = arr.foldlM f init := by
   simp [foldlM, foldlM_toList.aux]
 
-@[simp] theorem foldl_toList (f : β → α → β) (init : β) (xs : Array α) :
-    xs.toList.foldl f init = xs.foldl f init :=
+@[simp] theorem foldl_toList (f : β → α → β) (init : β) (arr : Array α) :
+    arr.toList.foldl f init = arr.foldl f init :=
   List.foldl_eq_foldlM .. ▸ foldlM_toList ..
 
 theorem foldrM_eq_reverse_foldlM_toList.aux [Monad m]
-    (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
+    (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
   unfold foldrM.fold
   match i with
   | 0 => simp [List.foldlM, List.take]
-  | i+1 => rw [← List.take_concat_get _ _ h]; simp [← (aux f xs · i)]
+  | i+1 => rw [← List.take_concat_get _ _ h]; simp [← (aux f arr · i)]
 
-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 => ?_
+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 => ?_
   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) (xs : Array α) :
-    xs.toList.foldrM f init = xs.foldrM f init := by
+    (f : α → β → m β) (init : β) (arr : Array α) :
+    arr.toList.foldrM f init = arr.foldrM f init := by
   rw [foldrM_eq_reverse_foldlM_toList, List.foldlM_reverse]
 
-@[simp] theorem foldr_toList (f : α → β → β) (init : β) (xs : Array α) :
-    xs.toList.foldr f init = xs.foldr f init :=
+@[simp] theorem foldr_toList (f : α → β → β) (init : β) (arr : Array α) :
+    arr.toList.foldr f init = arr.foldr f init :=
   List.foldr_eq_foldrM .. ▸ foldrM_toList ..
 
-@[simp] theorem push_toList (xs : Array α) (a : α) : (xs.push a).toList = xs.toList ++ [a] := by
+@[simp] theorem push_toList (arr : Array α) (a : α) : (arr.push a).toList = arr.toList ++ [a] := by
   simp [push, List.concat_eq_append]
 
-@[simp] theorem toListAppend_eq (xs : Array α) (l : List α) : xs.toListAppend l = xs.toList ++ l := by
+@[simp] theorem toListAppend_eq (arr : Array α) (l) : arr.toListAppend l = arr.toList ++ l := by
   simp [toListAppend, ← foldr_toList]
 
-@[simp] theorem toListImpl_eq (xs : Array α) : xs.toListImpl = xs.toList := by
+@[simp] theorem toListImpl_eq (arr : Array α) : arr.toListImpl = arr.toList := by
   simp [toListImpl, ← foldr_toList]
 
-@[simp] theorem toList_pop (xs : Array α) : xs.pop.toList = xs.toList.dropLast := rfl
+@[simp] theorem pop_toList (arr : Array α) : arr.pop.toList = arr.toList.dropLast := rfl
 
-@[deprecated toList_pop (since := "2025-02-17")]
-abbrev pop_toList := @Array.toList_pop
+@[simp] theorem append_eq_append (arr arr' : Array α) : arr.append arr' = arr ++ arr' := rfl
 
-@[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
+@[simp] theorem toList_append (arr arr' : Array α) :
+    (arr ++ arr').toList = arr.toList ++ arr'.toList := by
   rw [← append_eq_append]; unfold Array.append
   rw [← foldl_toList]
-  induction ys.toList generalizing xs <;> simp [*]
+  induction arr'.toList generalizing arr <;> simp [*]
 
 @[simp] theorem toList_empty : (#[] : Array α).toList = [] := rfl
 
-@[simp] theorem append_empty (xs : Array α) : xs ++ #[] = xs := by
+@[simp] theorem append_empty (as : Array α) : as ++ #[] = as := 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 (xs : Array α) : #[] ++ xs = xs := by
+@[simp] theorem empty_append (as : Array α) : #[] ++ as = as := 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 (xs ys zs : Array α) : xs ++ ys ++ zs = xs ++ (ys ++ zs) := by
+@[simp] theorem append_assoc (as bs cs : Array α) : as ++ bs ++ cs = as ++ (bs ++ cs) := by
   apply ext'; simp only [toList_append, List.append_assoc]
 
 @[simp] theorem appendList_eq_append
-    (xs : Array α) (l : List α) : xs.appendList l = xs ++ l := rfl
+    (arr : Array α) (l : List α) : arr.appendList l = arr ++ l := rfl
 
-@[simp] theorem toList_appendList (xs : Array α) (l : List α) :
-    (xs ++ l).toList = xs.toList ++ l := by
+@[simp] theorem toList_appendList (arr : Array α) (l : List α) :
+    (arr ++ l).toList = arr.toList ++ l := by
   rw [← appendList_eq_append]; unfold Array.appendList
-  induction l generalizing xs <;> simp [*]
+  induction l generalizing arr <;> 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) (xs : Array α) :
-    xs.foldrM f init = xs.toList.foldrM f init := by
+    (f : α → β → m β) (init : β) (arr : Array α) :
+    arr.foldrM f init = arr.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) (xs : Array α) :
-    xs.foldlM f init = xs.toList.foldlM f init:= by
+    (f : β → α → m β) (init : β) (arr : Array α) :
+    arr.foldlM f init = arr.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) (xs : Array α) :
-    xs.foldr f init = xs.toList.foldr f init := by
+    (f : α → β → β) (init : β) (arr : Array α) :
+    arr.foldr f init = arr.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) (xs : Array α) :
-    xs.foldl f init = xs.toList.foldl f init:= by
+    (f : β → α → β) (init : β) (arr : Array α) :
+    arr.foldl f init = arr.toList.foldl f init:= by
   simp
 
 @[deprecated foldlM_toList (since := "2024-09-09")]
@@ -182,7 +153,7 @@ abbrev push_data := @push_toList
 abbrev toList_eq := @toListImpl_eq
 
 @[deprecated pop_toList (since := "2024-09-09")]
-abbrev pop_data := @toList_pop
+abbrev pop_data := @pop_toList
 
 @[deprecated toList_append (since := "2024-09-09")]
 abbrev append_data := @toList_append

src/Init/Data/Array/Count.lean

@@ -11,9 +11,6 @@ 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
@@ -23,134 +20,122 @@ 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 (xs) (pa : p a) : countP p (xs.push a) = countP p xs + 1 := by
-  rcases xs with ⟨xs⟩
+@[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_all
 
-@[simp] theorem countP_push_of_neg (xs) (pa : ¬p a) : countP p (xs.push a) = countP p xs := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem countP_push_of_neg (l) (pa : ¬p a) : countP p (l.push a) = countP p l := by
+  rcases l with ⟨l⟩
   simp_all
 
-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⟩
+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⟩
   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 (xs) : xs.size = countP p xs + countP (fun a => ¬p a) xs := by
-  rcases xs with ⟨xs⟩
+theorem size_eq_countP_add_countP (l) : l.size = countP p l + countP (fun a => ¬p a) l := by
+  cases l
   simp [List.length_eq_countP_add_countP (p := p)]
 
-theorem countP_eq_size_filter (xs) : countP p xs = (filter p xs).size := by
-  rcases xs with ⟨xs⟩
+theorem countP_eq_size_filter (l) : countP p l = (filter p l).size := by
+  cases l
   simp [List.countP_eq_length_filter]
 
 theorem countP_eq_size_filter' : countP p = size ∘ filter p := by
-  funext xs
+  funext l
   apply countP_eq_size_filter
 
-theorem countP_le_size : countP p xs ≤ xs.size := by
+theorem countP_le_size : countP p l ≤ l.size := by
   simp only [countP_eq_size_filter]
   apply size_filter_le
 
-@[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] theorem countP_append (l₁ l₂) : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
+  cases l₁
+  cases l₂
   simp
 
-@[simp] theorem countP_pos_iff {p} : 0 < countP p xs ↔ ∃ a ∈ xs, p a := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem countP_pos_iff {p} : 0 < countP p l ↔ ∃ a ∈ l, p a := by
+  cases l
   simp
 
-@[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p xs ↔ ∃ a ∈ xs, p a :=
+@[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p l ↔ ∃ a ∈ l, p a :=
   countP_pos_iff
 
-@[simp] theorem countP_eq_zero {p} : countP p xs = 0 ↔ ∀ a ∈ xs, ¬p a := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem countP_eq_zero {p} : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
+  cases l
   simp
 
-@[simp] theorem countP_eq_size {p} : countP p xs = xs.size ↔ ∀ a ∈ xs, p a := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem countP_eq_size {p} : countP p l = l.size ↔ ∀ a ∈ l, p a := by
+  cases l
   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) (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⟩
+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
   simp [List.boole_getElem_le_countP]
 
-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⟩
+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
   simp [List.countP_set, h]
 
-theorem countP_filter (xs : Array α) :
-    countP p (filter q xs) = countP (fun a => p a && q a) xs := by
-  rcases xs with ⟨xs⟩
+theorem countP_filter (l : Array α) :
+    countP p (filter q l) = countP (fun a => p a && q a) l := by
+  cases l
   simp [List.countP_filter]
 
 @[simp] theorem countP_true : (countP fun (_ : α) => true) = size := by
-  funext xs
+  funext l
   simp
 
 @[simp] theorem countP_false : (countP fun (_ : α) => false) = Function.const _ 0 := by
-  funext xs
+  funext l
   simp
 
-@[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 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 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⟩
+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
   simp [List.length_filterMap_eq_countP]
 
-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⟩
+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
   simp [List.countP_filterMap]
 
-@[simp] theorem countP_flatten (xss : Array (Array α)) :
-    countP p xss.flatten = (xss.map (countP p)).sum := by
-  cases xss using array₂_induction
+@[simp] theorem countP_flatten (l : Array (Array α)) :
+    countP p l.flatten = (l.map (countP p)).sum := by
+  cases l using array₂_induction
   simp [List.countP_flatten, Function.comp_def]
 
-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⟩
+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
   simp [List.countP_flatMap, Function.comp_def]
 
-@[simp] theorem countP_reverse (xs : Array α) : countP p xs.reverse = countP p xs := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem countP_reverse (l : Array α) : countP p l.reverse = countP p l := by
+  cases l
   simp [List.countP_reverse]
 
 variable {p q}
 
-theorem countP_mono_left (h : ∀ x ∈ xs, p x → q x) : countP p xs ≤ countP q xs := by
-  rcases xs with ⟨xs⟩
+theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
+  cases l
   simpa using List.countP_mono_left (by simpa using h)
 
-theorem countP_congr (h : ∀ x ∈ xs, p x ↔ q x) : countP p xs = countP q xs :=
+theorem countP_congr (h : ∀ x ∈ l, p x ↔ q x) : countP p l = countP q l :=
   Nat.le_antisymm
     (countP_mono_left fun x hx => (h x hx).1)
     (countP_mono_left fun x hx => (h x hx).2)
@@ -162,80 +147,73 @@ section count
 
 variable [BEq α]
 
-@[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]
-
-@[simp] theorem count_toList (xs : Array α) (a : α) : xs.toList.count a = xs.count a := by
-  cases xs
-  simp
-
 @[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
+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) (xs : Array α) : count a xs = countP (· == a) xs := rfl
+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 xs
+  funext l
   apply count_eq_countP
 
-theorem count_le_size (a : α) (xs : Array α) : count a xs ≤ xs.size := countP_le_size _
+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) (xs : Array α) : count a xs ≤ count a (xs.push b) := by
+theorem count_le_count_push (a b : α) (l : Array α) : count a l ≤ count a (l.push b) := by
   simp [count_push]
 
-theorem count_singleton (a b : α) : count a #[b] = if b == a then 1 else 0 := by
+@[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) : ∀ xs ys, count a (xs ++ ys) = count a xs + count a ys :=
+@[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) (xss : Array (Array α)) :
-    count a xss.flatten = (xss.map (count a)).sum := by
-  cases xss using array₂_induction
+@[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) (xs : Array α) : count a xs.reverse = count a xs := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem count_reverse (a : α) (l : Array α) : count a l.reverse = count a l := by
+  cases l
   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
+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
   rw [count_eq_countP]
   apply boole_getElem_le_countP (· == a)
 
-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
+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
   simp [count_eq_countP, countP_set, h]
 
 variable [LawfulBEq α]
 
-@[simp] theorem count_push_self (a : α) (xs : Array α) : count a (xs.push a) = count a xs + 1 := by
+@[simp] theorem count_push_self (a : α) (l : Array α) : count a (l.push a) = count a l + 1 := by
   simp [count_push]
 
-@[simp] theorem count_push_of_ne (h : b ≠ a) (xs : Array α) : count a (xs.push b) = count a xs := by
+@[simp] theorem count_push_of_ne (h : b ≠ a) (l : Array α) : count a (l.push b) = count a l := 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} {xs : Array α} : 0 < count a xs ↔ a ∈ xs := by
+theorem count_pos_iff {a : α} {l : Array α} : 0 < count a l ↔ a ∈ l := by
   simp only [count, countP_pos_iff, beq_iff_eq, exists_eq_right]
 
-@[simp] theorem one_le_count_iff {a : α} {xs : Array α} : 1 ≤ count a xs ↔ a ∈ xs :=
+@[simp] theorem one_le_count_iff {a : α} {l : Array α} : 1 ≤ count a l ↔ a ∈ l :=
   count_pos_iff
 
-theorem count_eq_zero_of_not_mem {a : α} {xs : Array α} (h : a ∉ xs) : count a xs = 0 :=
+theorem count_eq_zero_of_not_mem {a : α} {l : Array α} (h : a ∉ l) : count a l = 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} {xs : Array α} (h : count a xs = 0) : a ∉ xs :=
+theorem not_mem_of_count_eq_zero {a : α} {l : Array α} (h : count a l = 0) : a ∉ l :=
   fun h' => Nat.ne_of_lt (count_pos_iff.2 h') h.symm
 
-theorem count_eq_zero {xs : Array α} : count a xs = 0 ↔ a ∉ xs :=
+theorem count_eq_zero {l : Array α} : count a l = 0 ↔ a ∉ l :=
   ⟨not_mem_of_count_eq_zero, count_eq_zero_of_not_mem⟩
 
-theorem count_eq_size {xs : Array α} : count a xs = xs.size ↔ ∀ b ∈ xs, a = b := by
+theorem count_eq_size {l : Array α} : count a l = l.size ↔ ∀ b ∈ l, a = b := by
   rw [count, countP_eq_size]
   refine ⟨fun h b hb => Eq.symm ?_, fun h b hb => ?_⟩
   · simpa using h b hb
@@ -247,37 +225,36 @@ theorem count_eq_size {xs : Array α} : count a xs = xs.size ↔ ∀ b ∈ xs, a
 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 (xs : Array α) (a : α) : xs.filter (· == a) = mkArray (count a xs) a := by
-  rcases xs with ⟨xs⟩
+theorem filter_beq (l : Array α) (a : α) : l.filter (· == a) = mkArray (count a l) a := by
+  cases l
   simp [List.filter_beq]
 
-theorem filter_eq {α} [DecidableEq α] (xs : Array α) (a : α) : xs.filter (· = a) = mkArray (count a xs) a :=
-  filter_beq xs a
+theorem filter_eq {α} [DecidableEq α] (l : Array α) (a : α) : l.filter (· = a) = mkArray (count a l) a :=
+  filter_beq l a
 
-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⟩
+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
   rw [← toList_inj]
   simp [List.replicate_count_eq_of_count_eq_length (by simpa using h)]
 
-@[simp] theorem count_filter {xs : Array α} (h : p a) : count a (filter p xs) = count a xs := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem count_filter {l : Array α} (h : p a) : count a (filter p l) = count a l := by
+  cases l
   simp [List.count_filter, h]
 
-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⟩
+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
   simp [List.count_le_count_map, countP_map]
 
-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⟩
+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
   simp [List.count_filterMap, countP_filterMap]
 
-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]
+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]
 
 -- FIXME these theorems can be restored once `List.erase` and `Array.erase` have been related.
 

src/Init/Data/Array/DecidableEq.lean

@@ -9,15 +9,12 @@ 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} {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
+    {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
   induction i with
   | zero => contradiction
   | succ i ih =>
@@ -30,8 +27,8 @@ private theorem rel_of_isEqvAux
       subst hj'
       exact heqv.left
 
-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
+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
   induction i with
   | zero => simp [Array.isEqvAux]
   | succ i ih =>
@@ -39,23 +36,23 @@ private theorem isEqvAux_of_rel {r : α → α → Bool} {xs ys : 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} {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
+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
   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 {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')) :=
+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')) :=
   ⟨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 (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
+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
   · simp only [h, Bool.true_eq]
     simp only [isEqv_iff_rel] at h
     obtain ⟨h, w⟩ := h
@@ -66,48 +63,48 @@ theorem isEqv_eq_decide (xs ys : Array α) (r) :
       Bool.not_eq_true]
     simpa [isEqv_iff_rel] using h'
 
-@[simp] theorem isEqv_toList [BEq α] (xs ys : Array α) : (xs.toList.isEqv ys.toList r) = (xs.isEqv ys r) := by
+@[simp] theorem isEqv_toList [BEq α] (a b : Array α) : (a.toList.isEqv b.toList r) = (a.isEqv b r) := by
   simp [isEqv_eq_decide, List.isEqv_eq_decide]
 
-theorem eq_of_isEqv [DecidableEq α] (xs ys : Array α) (h : Array.isEqv xs ys (fun x y => x = y)) : xs = ys := by
+theorem eq_of_isEqv [DecidableEq α] (a b : Array α) (h : Array.isEqv a b (fun x y => x = y)) : a = b := 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) (xs : Array α) (i : Nat) (h : i ≤ xs.size) :
-    Array.isEqvAux xs xs rfl r i h = true := by
+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
   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] (xs : Array α) : Array.isEqv xs xs (· == ·) = true := by
+theorem isEqv_self_beq [BEq α] [ReflBEq α] (a : Array α) : Array.isEqv a a (· == ·) = true := by
   simp [isEqv, isEqvAux_self]
 
-theorem isEqv_self [DecidableEq α] (xs : Array α) : Array.isEqv xs xs (· = ·) = true := by
+theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (· = ·) = true := by
   simp [isEqv, isEqvAux_self]
 
 instance [DecidableEq α] : DecidableEq (Array α) :=
-  fun xs ys =>
-    match h:isEqv xs ys (fun a b => a = b) with
-    | true  => isTrue (eq_of_isEqv xs ys h)
+  fun a b =>
+    match h:isEqv a b (fun a b => a = b) with
+    | true  => isTrue (eq_of_isEqv a b h)
     | false => isFalse fun h' => by subst h'; rw [isEqv_self] at h; contradiction
 
-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
+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
   simp [BEq.beq, isEqv_eq_decide]
 
-@[simp] theorem beq_toList [BEq α] (xs ys : Array α) : (xs.toList == ys.toList) = (xs == ys) := by
+@[simp] theorem beq_toList [BEq α] (a b : Array α) : (a.toList == b.toList) = (a == b) := by
   simp [beq_eq_decide, List.beq_eq_decide]
 
 end Array
 
 namespace List
 
-@[simp] theorem isEqv_toArray [BEq α] (as bs : List α) : (as.toArray.isEqv bs.toArray r) = (as.isEqv bs r) := by
+@[simp] theorem isEqv_toArray [BEq α] (a b : List α) : (a.toArray.isEqv b.toArray r) = (a.isEqv b r) := by
   simp [isEqv_eq_decide, Array.isEqv_eq_decide]
 
-@[simp] theorem beq_toArray [BEq α] (as bs : List α) : (as.toArray == bs.toArray) = (as == bs) := by
+@[simp] theorem beq_toArray [BEq α] (a b : List α) : (a.toArray == b.toArray) = (a == b) := by
   simp [beq_eq_decide, Array.beq_eq_decide]
 
 end List
@@ -117,7 +114,7 @@ namespace Array
 instance [BEq α] [LawfulBEq α] : LawfulBEq (Array α) where
   rfl := by simp [BEq.beq, isEqv_self_beq]
   eq_of_beq := by
-    rintro ⟨_⟩ ⟨_⟩ h
+    rintro ⟨a⟩ ⟨b⟩ h
     simpa using h
 
 end Array

src/Init/Data/Array/Erase.lean

@@ -12,22 +12,19 @@ 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
 
 /-! ### eraseP -/
 
-@[simp] theorem eraseP_empty : #[].eraseP p = #[] := by simp
+@[simp] theorem eraseP_empty : #[].eraseP p = #[] := rfl
 
-theorem eraseP_of_forall_mem_not {xs : Array α} (h : ∀ a, a ∈ xs → ¬p a) : xs.eraseP p = xs := by
-  rcases xs with ⟨xs⟩
+theorem eraseP_of_forall_mem_not {l : Array α} (h : ∀ a, a ∈ l → ¬p a) : l.eraseP p = l := by
+  cases l
   simp_all [List.eraseP_of_forall_not]
 
-theorem eraseP_of_forall_getElem_not {xs : Array α} (h : ∀ i, (h : i < xs.size) → ¬p xs[i]) : xs.eraseP p = xs :=
+theorem eraseP_of_forall_getElem_not {l : Array α} (h : ∀ i, (h : i < l.size) → ¬p l[i]) : l.eraseP p = l :=
   eraseP_of_forall_mem_not fun a m => by
     rw [mem_iff_getElem] at m
     obtain ⟨i, w, rfl⟩ := m
@@ -40,86 +37,86 @@ theorem eraseP_of_forall_getElem_not {xs : Array α} (h : ∀ i, (h : i < xs.siz
 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 {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⟩
+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⟩
   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) (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
+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
     let ⟨_, ha, pa⟩ := h
     .inr (exists_of_eraseP ha pa)
   else
     .inl (eraseP_of_forall_mem_not (h ⟨·, ·, ·⟩))
 
-@[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
+@[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
   rw [e₂]; simp [size_append, e₁]; omega
 
-theorem size_eraseP {xs : Array α} : (xs.eraseP p).size = if xs.any p then xs.size - 1 else xs.size := by
+theorem size_eraseP {l : Array α} : (l.eraseP p).size = if l.any p then l.size - 1 else l.size := by
   split <;> rename_i h
   · simp only [any_eq_true] at h
     obtain ⟨i, h, w⟩ := h
-    simp [size_eraseP_of_mem (xs := xs) (by simp) w]
+    simp [size_eraseP_of_mem (l := l) (by simp) w]
   · simp only [any_eq_true] at h
     rw [eraseP_of_forall_getElem_not]
     simp_all
 
-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 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 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 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 mem_of_mem_eraseP {xs : Array α} : a ∈ xs.eraseP p → a ∈ xs := by
-  rcases xs with ⟨xs⟩
+theorem mem_of_mem_eraseP {l : Array α} : a ∈ l.eraseP p → a ∈ l := by
+  rcases l with ⟨l⟩
   simpa using List.mem_of_mem_eraseP
 
-@[simp] theorem mem_eraseP_of_neg {xs : Array α} (pa : ¬p a) : a ∈ xs.eraseP p ↔ a ∈ xs := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem mem_eraseP_of_neg {l : Array α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
+  rcases l with ⟨l⟩
   simpa using List.mem_eraseP_of_neg pa
 
-@[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_eq_self_iff {p} {l : Array α} : l.eraseP p = l ↔ ∀ a ∈ l, ¬ p a := by
+  rcases l with ⟨l⟩
   simp
 
-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_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_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_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_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_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_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_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_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_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 {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]
+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']
   split <;> simp
 
 theorem eraseP_mkArray (n : Nat) (a : α) (p : α → Bool) :
@@ -137,24 +134,24 @@ 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} {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⟩
+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'⟩
   simp [List.eraseP_eq_iff]
   constructor
-  · rintro (h | ⟨a, l₁, h₁, h₂, ⟨l, rfl, rfl⟩⟩)
+  · rintro (h | ⟨a, l₁, h₁, h₂, ⟨x, rfl, rfl⟩⟩)
     · exact Or.inl h
-    · exact Or.inr ⟨a, ⟨l₁⟩, by simpa using h₁, h₂, ⟨⟨l⟩, by simp⟩⟩
-  · rintro (h | ⟨a, ⟨l₁⟩, h₁, h₂, ⟨⟨l⟩, rfl, rfl⟩⟩)
+    · exact Or.inr ⟨a, ⟨l₁⟩, by simpa using h₁, h₂, ⟨⟨x⟩, by simp⟩⟩
+  · rintro (h | ⟨a, ⟨l₁⟩, h₁, h₂, ⟨⟨x⟩, rfl, rfl⟩⟩)
     · exact Or.inl h
-    · exact Or.inr ⟨a, l₁, by simpa using h₁, h₂, ⟨l, by simp⟩⟩
+    · exact Or.inr ⟨a, l₁, by simpa using h₁, h₂, ⟨x, by simp⟩⟩
 
-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⟩
+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⟩
   simpa using List.eraseP_comm (by simpa using h)
 
 /-! ### erase -/
@@ -162,16 +159,16 @@ theorem eraseP_comm {xs : Array α} (h : ∀ a ∈ xs, ¬ p a ∨ ¬ q a) :
 section erase
 variable [BEq α]
 
-theorem erase_of_not_mem [LawfulBEq α] {a : α} {xs : Array α} (h : a ∉ xs) : xs.erase a = xs := by
-  rcases xs with ⟨xs⟩
+theorem erase_of_not_mem [LawfulBEq α] {a : α} {l : Array α} (h : a ∉ l) : l.erase a = l := by
+  rcases l with ⟨l⟩
   simp [List.erase_of_not_mem (by simpa using h)]
 
-theorem erase_eq_eraseP' (a : α) (xs : Array α) : xs.erase a = xs.eraseP (· == a) := by
-  rcases xs with ⟨xs⟩
+theorem erase_eq_eraseP' (a : α) (l : Array α) : l.erase a = l.eraseP (· == a) := by
+  rcases l with ⟨l⟩
   simp [List.erase_eq_eraseP']
 
-theorem erase_eq_eraseP [LawfulBEq α] (a : α) (xs : Array α) : xs.erase a = xs.eraseP (a == ·) := by
-  rcases xs with ⟨xs⟩
+theorem erase_eq_eraseP [LawfulBEq α] (a : α) (l : Array α) : l.erase a = l.eraseP (a == ·) := by
+  rcases l with ⟨l⟩
   simp [List.erase_eq_eraseP]
 
 @[simp] theorem erase_eq_empty_iff [LawfulBEq α] {xs : Array α} {a : α} :
@@ -184,62 +181,62 @@ 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} {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⟩
+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⟩
 
-@[simp] theorem size_erase_of_mem [LawfulBEq α] {a : α} {xs : Array α} (h : a ∈ xs) :
-    (xs.erase a).size = xs.size - 1 := by
+@[simp] theorem size_erase_of_mem [LawfulBEq α] {a : α} {l : Array α} (h : a ∈ l) :
+    (l.erase a).size = l.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) (xs : Array α) :
-    (xs.erase a).size = if a ∈ xs then xs.size - 1 else xs.size := by
+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
   rw [erase_eq_eraseP, size_eraseP]
   congr
   simp [mem_iff_getElem, eq_comm (a := a)]
 
-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 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 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 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 mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a ∈ xs := by
-  rcases xs with ⟨xs⟩
+theorem mem_of_mem_erase {a b : α} {l : Array α} (h : a ∈ l.erase b) : a ∈ l := by
+  rcases l with ⟨l⟩
   simpa using List.mem_of_mem_erase (by simpa using h)
 
-@[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 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 erase_eq_self_iff [LawfulBEq α] {xs : Array α} : xs.erase a = xs ↔ a ∉ xs := by
+@[simp] theorem erase_eq_self_iff [LawfulBEq α] {l : Array α} : l.erase a = l ↔ a ∉ l := by
   rw [erase_eq_eraseP', eraseP_eq_self_iff]
   simp [forall_mem_ne']
 
-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_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_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_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_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_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 [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⟩
+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⟩
   simp only [List.append_toArray, List.erase_toArray, List.erase_append, mem_toArray]
   split <;> simp
 
@@ -249,24 +246,24 @@ 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) (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_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_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
+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
   rw [erase_eq_eraseP', eraseP_eq_iff]
   simp only [beq_iff_eq, forall_mem_ne', exists_and_left]
   constructor
-  · rintro (⟨h, rfl⟩ | ⟨a', as, h, rfl, bs, rfl, rfl⟩)
+  · rintro (⟨h, rfl⟩ | ⟨a', l', h, rfl, x, rfl, rfl⟩)
     · left; simp_all
-    · right; refine ⟨as, h, bs, by simp⟩
-  · rintro (⟨h, rfl⟩ | ⟨as, h, bs, rfl, rfl⟩)
+    · right; refine ⟨l', h, x, by simp⟩
+  · rintro (⟨h, rfl⟩ | ⟨l₁, h, x, rfl, rfl⟩)
     · left; simp_all
-    · right; refine ⟨a, as, h, rfl, bs, by simp⟩
+    · right; refine ⟨a, l₁, h, rfl, x, by simp⟩
 
 @[simp] theorem erase_mkArray_self [LawfulBEq α] {a : α} :
     (mkArray n a).erase a = mkArray (n - 1) a := by
@@ -282,74 +279,70 @@ end erase
 
 /-! ### eraseIdx -/
 
-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
+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
   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,
-    List.size_toArray, List.append_toArray, mk.injEq, List.append_cancel_left_eq]
+    size_toArray, List.append_toArray, mk.injEq, List.append_cancel_left_eq]
   rw [List.take_of_length_le]
   simp
 
-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⟩
+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⟩
   simp [List.getElem?_eraseIdx]
 
-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
+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
   rw [getElem?_eraseIdx]
   simp [h']
 
-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
+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
   rw [getElem?_eraseIdx]
   simp only [dite_eq_ite, ite_eq_right_iff]
   intro h'
   omega
 
-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]
+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]
       else
-        xs[j + 1]'(by rw [size_eraseIdx] at h'; omega) := by
+        l[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 {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,
+@[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,
     or_iff_right_iff_imp]
   rintro rfl
   simp_all
 
-theorem eraseIdx_ne_empty_iff {xs : Array α} {i : Nat} {h} : xs.eraseIdx i ≠ #[] ↔ 2 ≤ xs.size := by
-  rcases xs with ⟨_ | ⟨a, (_ | ⟨b, l⟩)⟩⟩
+theorem eraseIdx_ne_empty_iff {l : Array α} {i : Nat} {h} : eraseIdx l i ≠ #[] ↔ 2 ≤ l.size := by
+  rcases l with ⟨_ | ⟨a, (_ | ⟨b, l⟩)⟩⟩
   · simp
   · simp at h
     simp [h]
   · simp
 
-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⟩
+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⟩
   simpa using List.mem_of_mem_eraseIdx (by simpa using h)
 
-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'⟩
+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'⟩
   simp at hk
   simp [List.eraseIdx_append_of_lt_length, *]
 
-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'⟩
+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'⟩
   simp at hk
   simp [List.eraseIdx_append_of_length_le, *]
 
@@ -359,49 +352,49 @@ 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} {xs : Array α} {k} {h} : x ∈ xs.eraseIdx k h ↔ ∃ i w, i ≠ k ∧ xs[i]'w = x := by
-  rcases xs with ⟨xs⟩
+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⟩
   simp [List.mem_eraseIdx_iff_getElem, *]
 
-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⟩
+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⟩
   simp [List.mem_eraseIdx_iff_getElem?, *]
 
-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⟩
+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⟩
   simp at w
   simp [List.erase_eq_eraseIdx_of_idxOf, *]
 
-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⟩
+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⟩
   simp [List.getElem_eraseIdx_of_lt, *]
 
-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⟩
+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⟩
   simp [List.getElem_eraseIdx_of_ge, *]
 
-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⟩
+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⟩
   simp [List.eraseIdx_set_eq, *]
 
-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⟩
+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⟩
   simp [List.eraseIdx_set_lt, *]
 
-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⟩
+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⟩
   simp [List.eraseIdx_set_gt, *]
 
 @[simp] theorem set_getElem_succ_eraseIdx_succ
-    {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⟩
+    {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⟩
   simp [List.set_getElem_succ_eraseIdx_succ, *]
 
 end Array

src/Init/Data/Array/Extract.lean

@@ -1,430 +0,0 @@
-/-
-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

@@ -5,52 +5,10 @@ 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
 
-/--
-Returns an array of all elements of `Fin n` in order, starting at `0`.
-
-Examples:
- * `Array.finRange 0 = (#[] : Array (Fin 0))`
- * `Array.finRange 2 = (#[0, 1] : Array (Fin 2))`
--/
+/-- `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

@@ -13,92 +13,95 @@ 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 (xs : Array α) (h : (f a).isSome) : (xs.push a).findSomeRev? f = f a := by
-  cases xs; simp_all
+@[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_isNone (xs : Array α) (h : (f a).isNone) : (xs.push a).findSomeRev? f = xs.findSomeRev? f := 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
 
-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]
+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]
 
-@[simp] theorem findSome?_eq_none_iff : findSome? p xs = none ↔ ∀ x ∈ xs, p x = none := by
-  cases xs; simp
+@[simp] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := 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
+@[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
 
-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
+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
   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 ⟨xs, a, ys, h₀, h₁, h₂⟩
-    exact ⟨xs.toList, a, ys.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩
+  · rintro ⟨l₁, a, l₂, h₀, h₁, h₂⟩
+    exact ⟨l₁.toList, a, l₂.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩
 
-@[simp] theorem findSome?_guard (xs : Array α) : findSome? (Option.guard fun x => p x) xs = find? p xs := by
-  cases xs; simp
+@[simp] theorem findSome?_guard (l : Array α) : findSome? (Option.guard fun x => p x) l = find? p l := by
+  cases l; simp
 
-theorem find?_eq_findSome?_guard (xs : Array α) : find? p xs = findSome? (Option.guard fun x => p x) xs :=
-  (findSome?_guard xs).symm
+theorem find?_eq_findSome?_guard (l : Array α) : find? p l = findSome? (Option.guard fun x => p x) l :=
+  (findSome?_guard l).symm
 
-@[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) : (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) (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 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 back?_filterMap (f : α → Option β) (xs : Array α) : (xs.filterMap f).back? = xs.findSomeRev? f := by
-  cases xs; simp
+@[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 [Inhabited β] (f : α → Option β) (xs : Array α) :
-    (xs.filterMap f).back! = (xs.findSomeRev? f).getD default := 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 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
+@[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
 
-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?_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?_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 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 getElem?_zero_flatten (xss : Array (Array α)) :
-    (flatten xss)[0]? = xss.findSome? fun xs => xs[0]? := by
-  cases xss using array₂_induction
+theorem getElem?_zero_flatten (L : Array (Array α)) :
+    (flatten L)[0]? = L.findSome? fun l => l[0]? := by
+  cases L using array₂_induction
   simp [← List.head?_eq_getElem?, List.head?_flatten, List.findSome?_map, Function.comp_def]
 
-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
+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
   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, List.size_toArray, List.length_flatten,
+  simp only [flatten_toArray_map_toArray, 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 {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
+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
   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]
 
@@ -121,16 +124,16 @@ 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 (xs : Array α) (h : p a) :
-    findRev? p (xs.push a) = some a := by
-  cases xs; simp [h]
+@[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?_cons_of_neg (xs : Array α) (h : ¬p a) :
-    findRev? p (xs.push a) = findRev? p xs := 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 find?_eq_none : find? p xs = none ↔ ∀ x ∈ xs, ¬ p x := by
-  cases xs; simp
+@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
+  cases l; 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
@@ -139,10 +142,10 @@ theorem find?_eq_some_iff_append {xs : Array α} :
     Bool.not_true, exists_and_right, and_congr_right_iff]
   intro w
   constructor
-  · rintro ⟨as, ⟨⟨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'⟩,
+  · 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'⟩,
       by simpa using h⟩
 
 @[simp]
@@ -171,22 +174,22 @@ 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) (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) :
+    (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) (h) :
-    (xs.filter p)[0] =
-      (xs.find? p).get (by cases xs; simpa [← List.countP_eq_length_filter] using h) := by
-  cases xs
+@[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 [List.getElem_zero_eq_head]
 
-@[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 (p : α → Bool) (l : Array α) : (l.filter p).back? = l.findRev? p := by
+  cases l; simp
 
-@[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 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 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
@@ -196,19 +199,19 @@ 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 {xs ys : Array α} :
-    (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
-  cases xs
-  cases ys
+@[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
 
-@[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] 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 [List.findSome?_map, Function.comp_def]
 
-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
+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
   simp
 
 @[deprecated find?_flatten_eq_none_iff (since := "2025-02-03")]
@@ -219,12 +222,12 @@ 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 {xss : Array (Array α)} {p : α → Bool} {a : α} :
-    xss.flatten.find? p = some a ↔
+theorem find?_flatten_eq_some_iff {xs : Array (Array α)} {p : α → Bool} {a : α} :
+    xs.flatten.find? p = some a ↔
       p a ∧ ∃ (as : Array (Array α)) (ys zs : Array α) (bs : Array (Array α)),
-        xss = as.push (ys.push a ++ zs) ++ bs ∧
-        (∀ ws ∈ as, ∀ x ∈ ws, !p x) ∧ (∀ x ∈ ys, !p x) := by
-  cases xss using array₂_induction
+        xs = as.push (ys.push a ++ zs) ++ bs ∧
+        (∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
+  cases xs 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
@@ -299,6 +302,24 @@ 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
@@ -374,41 +395,26 @@ 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) (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⟩
+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⟩
   simp [List.findIdx_append]
 
-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⟩
+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⟩
   simp_all [List.findIdx_le_findIdx]
 
-@[simp] theorem findIdx_subtype {p : α → Prop} {xs : Array { x // p x }}
+@[simp] theorem findIdx_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) :
-    xs.findIdx f = xs.unattach.findIdx g := by
-  cases xs
+    l.findIdx f = l.unattach.findIdx g := by
+  cases l
   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 := by simp
+@[simp] theorem findIdx?_empty : (#[] : Array α).findIdx? p = none := rfl
 
 @[simp]
 theorem findIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
@@ -462,8 +468,8 @@ 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) (xs : Array β) : findIdx? p (xs.map f) = xs.findIdx? (p ∘ f) := by
-  rcases xs with ⟨xs⟩
+@[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 [List.findIdx?_map]
 
 @[simp] theorem findIdx?_append :
@@ -473,12 +479,12 @@ 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 {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
+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
   simp [List.findIdx?_flatten, Function.comp_def]
 
 @[simp] theorem findIdx?_mkArray :
@@ -513,66 +519,20 @@ 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} {xs : Array { x // p x }}
+@[simp] theorem findIdx?_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) :
-    xs.findIdx? f = xs.unattach.findIdx? g := by
-  cases xs
+    l.findIdx? f = l.unattach.findIdx? g := by
+  cases l
   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 := by simp
-
--- 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] {xs ys : Array α} {a : α} :
-    (xs ++ ys).idxOf a = if a ∈ xs then xs.idxOf a else ys.idxOf a + xs.size := by
+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
   rw [idxOf, findIdx_append]
   split <;> rename_i h
   · rw [if_pos]
@@ -580,12 +540,12 @@ theorem idxOf_append [BEq α] [LawfulBEq α] {xs ys : Array α} {a : α} :
   · rw [if_neg]
     simpa using h
 
-theorem idxOf_eq_size [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∉ xs) : xs.idxOf a = xs.size := by
-  rcases xs with ⟨xs⟩
+theorem idxOf_eq_size [BEq α] [LawfulBEq α] {l : Array α} (h : a ∉ l) : l.idxOf a = l.size := by
+  rcases l with ⟨l⟩
   simp [List.idxOf_eq_length (by simpa using h)]
 
-theorem idxOf_lt_length [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∈ xs) : xs.idxOf a < xs.size := by
-  rcases xs with ⟨xs⟩
+theorem idxOf_lt_length [BEq α] [LawfulBEq α] {l : Array α} (h : a ∈ l) : l.idxOf a < l.size := by
+  rcases l with ⟨l⟩
   simp [List.idxOf_lt_length (by simpa using h)]
 
 
@@ -595,33 +555,17 @@ The verification API for `idxOf?` is still incomplete.
 The lemmas below should be made consistent with those for `findIdx?` (and proved using them).
 -/
 
-@[simp] theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := by simp
+@[simp] theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := rfl
 
-@[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] 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 [List.idxOf?_eq_none_iff]
 
-/-! ### finIdxOf?
-
-The verification API for `finIdxOf?` is still incomplete.
-The lemmas below should be made consistent with those for `findFinIdx?` (and proved using them).
--/
+/-! ### finIdxOf? -/
 
 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 := by simp
-
-@[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

@@ -7,43 +7,40 @@ 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} (xs : Array α) (i : Nat) (h₁ : xs.size = n) (h₂ : i < n) : α :=
+abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
   have := h₁.symm ▸ h₂
-  xs[i]
+  a[i]
 
 theorem extLit {n : Nat}
-    (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)
+    (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)
 
-def toListLitAux (xs : Array α) (n : Nat) (hsz : xs.size = n) : ∀ (i : Nat), i ≤ xs.size → List α → List α
+def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
   | 0,     _,  acc => 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)
+  | (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)
 
-def toArrayLit (xs : Array α) (n : Nat) (hsz : xs.size = n) : Array α :=
-  List.toArray <| toListLitAux xs n hsz n (hsz ▸ Nat.le_refl _) []
+def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
+  List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.le_refl _) []
 
-theorem toArrayLit_eq (xs : Array α) (n : Nat) (hsz : xs.size = n) : xs = toArrayLit xs n hsz := by
+theorem toArrayLit_eq (as : Array α) (n : Nat) (hsz : as.size = n) : as = toArrayLit as n hsz := by
   apply ext'
-  simp [toArrayLit, List.toList_toArray]
-  have hle : n ≤ xs.size := hsz ▸ Nat.le_refl _
-  have hge : xs.size ≤ n := hsz ▸ Nat.le_refl _
+  simp [toArrayLit, toList_toArray]
+  have hle : n ≤ as.size := hsz ▸ Nat.le_refl _
+  have hge : as.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 (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) :=
+  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) :=
     rfl
-  go (i : Nat) (hi : i ≤ xs.size) : toListLitAux xs n hsz i hi (xs.toList.drop i) = xs.toList := by
+  go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.toList.drop i) = as.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

@@ -13,9 +13,6 @@ 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
@@ -30,23 +27,23 @@ section InsertIdx
 
 variable {a : α}
 
-@[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] 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
 
 @[simp]
-theorem insertIdx_zero (xs : Array α) (x : α) : xs.insertIdx 0 x = #[x] ++ xs := by
-  rcases xs with ⟨xs⟩
+theorem insertIdx_zero (s : Array α) (x : α) : s.insertIdx 0 x = #[x] ++ s := by
+  cases s
   simp
 
-@[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] theorem size_insertIdx {as : Array α} (h : n ≤ as.size) : (as.insertIdx n a).size = as.size + 1 := by
+  cases as
   simp [List.length_insertIdx, h]
 
-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⟩
+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
   simp_all
 
 theorem insertIdx_eraseIdx_of_ge {as : Array α}
@@ -63,68 +60,68 @@ 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) (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⟩
+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
   simpa using List.insertIdx_comm a b i j _ (by simpa) (by simpa)
 
-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⟩
+theorem mem_insertIdx {l : Array α} {h : i ≤ l.size} : a ∈ l.insertIdx i b h ↔ a = b ∨ a ∈ l := by
+  cases l
   simpa using List.mem_insertIdx (by simpa)
 
 @[simp]
-theorem insertIdx_size_self (xs : Array α) (x : α) : xs.insertIdx xs.size x = xs.push x := by
-  rcases xs with ⟨xs⟩
+theorem insertIdx_size_self (l : Array α) (x : α) : l.insertIdx l.size x = l.push x := by
+  cases l
   simp
 
-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] =
+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] =
       if h₁ : k < i then
-        xs[k]'(by simp [size_insertIdx] at h; omega)
+        as[k]'(by simp [size_insertIdx] at h; omega)
       else
         if h₂ : k = i then
           x
         else
-          xs[k-1]'(by simp [size_insertIdx] at h; omega) := by
-  cases xs
+          as[k-1]'(by simp [size_insertIdx] at h; omega) := by
+  cases as
   simp [List.getElem_insertIdx, w]
 
-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
+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
   simp [getElem_insertIdx, w, h]
 
-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
+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
   simp [getElem_insertIdx, w]
 
-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
+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
   simp [getElem_insertIdx, w, h]
   rw [dif_neg (by omega), dif_neg (by omega)]
 
-theorem getElem?_insertIdx {xs : Array α} {x : α} {i k : Nat} (h : i ≤ xs.size) :
-    (xs.insertIdx i x)[k]? =
+theorem getElem?_insertIdx {l : Array α} {x : α} {i k : Nat} (h : i ≤ l.size) :
+    (l.insertIdx i x)[k]? =
       if k < i then
-        xs[k]?
+        l[k]?
       else
         if k = i then
-          if k ≤ xs.size then some x else none
+          if k ≤ l.size then some x else none
         else
-          xs[k-1]? := by
-  cases xs
+          l[k-1]? := by
+  cases l
   simp [List.getElem?_insertIdx, h]
 
-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
+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
   rw [getElem?_insertIdx, if_pos h]
 
-theorem getElem?_insertIdx_self {xs : Array α} {x : α} {i : Nat} (w : i ≤ xs.size) :
-    (xs.insertIdx i x)[i]? = some x := by
+theorem getElem?_insertIdx_self {l : Array α} {x : α} {i : Nat} (w : i ≤ l.size) :
+    (l.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 {xs : Array α} {x : α} {i k : Nat} (w : i < k) (h : k ≤ xs.size) :
-    (xs.insertIdx i x)[k]? = xs[k - 1]? := by
+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
   rw [getElem?_insertIdx, if_neg (by omega), if_neg (by omega)]
 
 end InsertIdx

src/Init/Data/Array/InsertionSort.lean

@@ -6,32 +6,23 @@ 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.
-
-/--
-Sorts an array using insertion sort.
-
-The optional parameter `lt` specifies an ordering predicate. It defaults to `LT.lt`, which must be
-decidable to be used for sorting.
--/
-@[inline] def Array.insertionSort (xs : Array α) (lt : α → α → Bool := by exact (· < ·)) : Array α :=
-  traverse xs 0 xs.size
+@[inline] def Array.insertionSort (a : Array α) (lt : α → α → Bool := by exact (· < ·)) : Array α :=
+  traverse a 0 a.size
 where
-  @[specialize] traverse (xs : Array α) (i : Nat) (fuel : Nat) : Array α :=
+  @[specialize] traverse (a : Array α) (i : Nat) (fuel : Nat) : Array α :=
     match fuel with
-    | 0      => xs
+    | 0      => a
     | fuel+1 =>
-      if h : i < xs.size then
-        traverse (swapLoop xs i h) (i+1) fuel
+      if h : i < a.size then
+        traverse (swapLoop a i h) (i+1) fuel
       else
-        xs
-  @[specialize] swapLoop (xs : Array α) (j : Nat) (h : j < xs.size) : Array α :=
+        a
+  @[specialize] swapLoop (a : Array α) (j : Nat) (h : j < a.size) : Array α :=
     match (generalizing := false) he:j with -- using `generalizing` because we don't want to refine the type of `h`
-    | 0    => xs
+    | 0    => a
     | j'+1 =>
-      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)
+      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)
       else
-        xs
+        a

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

@@ -8,18 +8,14 @@ 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
 
 /--
-Compares arrays lexicographically with respect to a comparison `lt` on their elements.
+Lexicographic comparator for arrays.
 
-Specifically, `Array.lex as bs lt` is true if
-* `bs` is larger than `as` and `as` is pairwise equivalent via `==` to the initial segment of `bs`,
-  or
-* there is an index `i` such that `lt as[i] bs[i]`, and for all `j < i`, `as[j] == bs[j]`.
+`lex as bs lt` is true if
+- `bs` is larger than `as` and `as` is pairwise equivalent via `==` to the initial segment of `bs`, or
+- there is an index `i` such that `lt as[i] bs[i]`, and for all `j < i`, `as[j] == bs[j]`.
 -/
 def lex [BEq α] (as bs : Array α) (lt : α → α → Bool := by exact (· < ·)) : Bool := Id.run do
   for h : i in [0 : min as.size bs.size] do

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

@@ -7,9 +7,6 @@ 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 -/
@@ -17,15 +14,15 @@ 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] (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
+@[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
 
 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} (xs : Array α) : xs.lex #[] lt = false := by
+@[simp] theorem lex_empty [BEq α] {lt : α → α → Bool} (l : Array α) : l.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
@@ -36,7 +33,7 @@ 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, List.size_toArray, List.length_singleton,
+  simp only [Std.Range.forIn'_eq_forIn'_range', size_append, 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]
@@ -55,35 +52,35 @@ 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) (xs ys : Array α) :
-    xs.toList.lex ys.toList lt = xs.lex ys lt := by
-  cases xs <;> cases ys <;> simp
+@[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
 
-protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] (xs : Array α) : ¬ xs < xs :=
-  List.lt_irrefl xs.toList
+protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] (l : Array α) : ¬ l < l :=
+  List.lt_irrefl l.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] (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 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 le_empty [LT α] (xs : Array α) : xs ≤ #[] ↔ xs = #[] := by
-  cases xs
+@[simp] theorem le_empty [LT α] (l : Array α) : l ≤ #[] ↔ l = #[] := by
+  cases l
   simp
 
-@[simp] theorem empty_lt_push [LT α] (xs : Array α) (a : α) : #[] < xs.push a := by
-  rcases xs with (_ | ⟨x, xs⟩) <;> simp
+@[simp] theorem empty_lt_push [LT α] (l : Array α) (a : α) : #[] < l.push a := by
+  rcases l with (_ | ⟨x, l⟩) <;> simp
 
-protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] (xs : Array α) : xs ≤ xs :=
-  List.le_refl xs.toList
+protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] (l : Array α) : l ≤ l :=
+  List.le_refl l.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) (· < ·) (· < ·)]
-    {xs ys zs : Array α} (h₁ : xs < ys) (h₂ : ys < zs) : xs < zs :=
+    {l₁ l₂ l₃ : Array α} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
   List.lt_trans h₁ h₂
 
 instance [LT α] [Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
@@ -95,7 +92,7 @@ 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) (¬ · < ·) (¬ · < ·)]
-    {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys < zs) : xs < zs :=
+    {l₁ l₂ l₃ : Array α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
   List.lt_of_le_of_lt h₁ h₂
 
 protected theorem le_trans [DecidableEq α] [LT α] [DecidableLT α]
@@ -103,7 +100,7 @@ protected theorem le_trans [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
     [Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)]
-    {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys ≤ zs) : xs ≤ zs :=
+    {l₁ l₂ l₃ : Array α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ ≤ l₃) : l₁ ≤ l₃ :=
   fun h₃ => h₁ (Array.lt_of_le_of_lt h₂ h₃)
 
 instance [DecidableEq α] [LT α] [DecidableLT α]
@@ -116,7 +113,7 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
 
 protected theorem lt_asymm [LT α]
     [i : Std.Asymm (· < · : α → α → Prop)]
-    {xs ys : Array α} (h : xs < ys) : ¬ ys < xs := List.lt_asymm h
+    {l₁ l₂ : Array α} (h : l₁ < l₂) : ¬ l₂ < l₁ := List.lt_asymm h
 
 instance [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Asymm (· < · : α → α → Prop)] :
@@ -124,26 +121,26 @@ 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)] (xs ys : Array α) : xs ≤ ys ∨ ys ≤ xs :=
-  List.le_total xs.toList ys.toList
+    [i : Std.Total (¬ · < · : α → α → Prop)] (l₁ l₂ : Array α) : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
+  List.le_total _ _
 
 @[simp] protected theorem not_lt [LT α]
-    {xs ys : Array α} : ¬ xs < ys ↔ ys ≤ xs := Iff.rfl
+    {l₁ l₂ : Array α} : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
 
 @[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
-    {xs ys : Array α} : ¬ ys ≤ xs ↔ xs < ys := Decidable.not_not
+    {l₁ l₂ : Array α} : ¬ l₂ ≤ l₁ ↔ l₁ < l₂ := Decidable.not_not
 
 protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
     [i : Std.Total (¬ · < · : α → α → Prop)]
-    {xs ys : Array α} (h : xs < ys) : xs ≤ ys :=
+    {l₁ l₂ : Array α} (h : l₁ < l₂) : l₁ ≤ l₂ :=
   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)]
-    {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)
+    {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)
 
 instance [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Total (¬ · < · : α → α → Prop)] :
@@ -151,22 +148,22 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
   total := Array.le_total
 
 @[simp] theorem lex_eq_true_iff_lt [DecidableEq α] [LT α] [DecidableLT α]
-    {xs ys : Array α} : lex xs ys = true ↔ xs < ys := by
-  cases xs
-  cases ys
+    {l₁ l₂ : Array α} : lex l₁ l₂ = true ↔ l₁ < l₂ := by
+  cases l₁
+  cases l₂
   simp
 
 @[simp] theorem lex_eq_false_iff_ge [DecidableEq α] [LT α] [DecidableLT α]
-    {xs ys : Array α} : lex xs ys = false ↔ ys ≤ xs := by
-  cases xs
-  cases ys
+    {l₁ l₂ : Array α} : lex l₁ l₂ = false ↔ l₂ ≤ l₁ := by
+  cases l₁
+  cases l₂
   simp [List.not_lt_iff_ge]
 
 instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLT (Array α) :=
-  fun xs ys => decidable_of_iff (lex xs ys = true) lex_eq_true_iff_lt
+  fun l₁ l₂ => decidable_of_iff (lex l₁ l₂ = true) lex_eq_true_iff_lt
 
 instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLE (Array α) :=
-  fun xs ys => decidable_of_iff (lex ys xs = false) lex_eq_false_iff_ge
+  fun l₁ l₂ => decidable_of_iff (lex l₂ l₁ = false) lex_eq_false_iff_ge
 
 /--
 `l₁` is lexicographically less than `l₂` if either
@@ -214,58 +211,58 @@ 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] {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),
+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),
           (∀ j, (hj : j < i) →
-            xs[j]'(Nat.lt_trans hj h₁) = ys[j]'(Nat.lt_trans hj h₂)) ∧ xs[i] < ys[i]) := by
-  cases xs
-  cases ys
+            l₁[j]'(Nat.lt_trans hj h₁) = l₂[j]'(Nat.lt_trans hj h₂)) ∧ l₁[i] < l₂[i]) := by
+  cases l₁
+  cases l₂
   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)] {xs ys : Array α} :
-    xs ≤ ys ↔
-      (xs = ys.take xs.size) ∨
-        (∃ (i : Nat) (h₁ : i < xs.size) (h₂ : i < ys.size),
+    [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),
           (∀ j, (hj : j < i) →
-            xs[j]'(Nat.lt_trans hj h₁) = ys[j]'(Nat.lt_trans hj h₂)) ∧ xs[i] < ys[i]) := by
-  cases xs
-  cases ys
+            l₁[j]'(Nat.lt_trans hj h₁) = l₂[j]'(Nat.lt_trans hj h₂)) ∧ l₁[i] < l₂[i]) := by
+  cases l₁
+  cases l₂
   simp [List.le_iff_exists]
 
-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
+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
   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)]
-    {xs ys zs : Array α} (h : ys ≤ zs) :
-    xs ++ ys ≤ xs ++ zs := by
-  cases xs
-  cases ys
-  cases zs
+    {l₁ l₂ l₃ : Array α} (h : l₂ ≤ l₃) :
+    l₁ ++ l₂ ≤ l₁ ++ l₃ := by
+  cases l₁
+  cases l₂
+  cases l₃
   simpa using List.append_left_le h
 
 theorem le_append_left [LT α] [Std.Irrefl (· < · : α → α → Prop)]
-    {xs ys : Array α} : xs ≤ xs ++ ys := by
-  cases xs
-  cases ys
+    {l₁ l₂ : Array α} : l₁ ≤ l₁ ++ l₂ := by
+  cases l₁
+  cases l₂
   simpa using List.le_append_left
 
 protected theorem map_lt [LT α] [LT β]
-    {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
+    {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
   simpa using List.map_lt w h
 
 protected theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
@@ -275,10 +272,10 @@ 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)]
-    {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
+    {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
   simpa using List.map_le w h
 
 end Array

src/Init/Data/Array/MapIdx.lean

@@ -6,32 +6,28 @@ 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 (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β)
+theorem mapFinIdx_induction (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → β)
     (motive : Nat → Prop) (h0 : motive 0)
-    (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
+    (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
   let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p i bs[i] h) (hm : motive j) :
-    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
+    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
     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 xs[j] (by omega))) (j + 1) (by omega) (by simp; omega)
+      apply @ih (bs.push (f j as[j] (by omega))) (j + 1) (by omega) (by simp; omega)
       · intro i i_lt h'
         rw [getElem_push]
         split
@@ -42,67 +38,67 @@ theorem mapFinIdx_induction (xs : Array α) (f : (i : Nat) → α → (h : i < x
       · exact (hs j (by omega) hm).2
   simp [mapFinIdx, mapFinIdxM]; exact go rfl nofun h0
 
-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 :=
+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 :=
   (mapFinIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
 
-@[simp] theorem size_mapFinIdx (xs : Array α) (f : (i : Nat) → α → (h : i < xs.size) → β) :
-    (xs.mapFinIdx f).size = xs.size :=
+@[simp] theorem size_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) :
+    (a.mapFinIdx f).size = a.size :=
   (mapFinIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
 
-@[simp] theorem size_zipIdx (xs : Array α) (k : Nat) : (xs.zipIdx k).size = xs.size :=
+@[simp] theorem size_zipIdx (as : Array α) (k : Nat) : (as.zipIdx k).size = as.size :=
   Array.size_mapFinIdx _ _
 
 @[deprecated size_zipIdx (since := "2025-01-21")] abbrev size_zipWithIndex := @size_zipIdx
 
-@[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)
+    (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) :
-    (xs.mapFinIdx f)[i]? =
-      xs[i]?.pbind fun b h => f i b (getElem?_eq_some_iff.1 h).1 := by
+@[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 only [getElem?_def, size_mapFinIdx, getElem_mapFinIdx]
   split <;> simp_all
 
-@[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
+@[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
   apply List.ext_getElem <;> simp
 
 /-! ### mapIdx -/
 
-theorem mapIdx_induction (f : Nat → α → β) (xs : Array α)
+theorem mapIdx_induction (f : Nat → α → β) (as : Array α)
     (motive : Nat → Prop) (h0 : motive 0)
-    (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
+    (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
 
-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 :=
+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 :=
   (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
 
-@[simp] theorem size_mapIdx (f : Nat → α → β) (xs : Array α) : (xs.mapIdx f).size = xs.size :=
+@[simp] theorem size_mapIdx (f : Nat → α → β) (as : Array α) : (as.mapIdx f).size = as.size :=
   (mapIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
 
-@[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)
+    (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) :
-    (xs.mapIdx f)[i]? =
-      xs[i]?.map (f i) := by
+@[simp] theorem getElem?_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat) :
+    (as.mapIdx f)[i]? =
+      as[i]?.map (f i) := by
   simp [getElem?_def, size_mapIdx, getElem_mapIdx]
 
-@[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
+@[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
   apply List.ext_getElem <;> simp
 
 end Array
@@ -123,8 +119,8 @@ namespace Array
 
 /-! ### zipIdx -/
 
-@[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] 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 [zipIdx]
 
 @[deprecated getElem_zipIdx (since := "2025-01-21")]
@@ -137,35 +133,35 @@ abbrev getElem_zipWithIndex := @getElem_zipIdx
 @[deprecated zipIdx_toArray (since := "2025-01-21")]
 abbrev zipWithIndex_toArray := @zipIdx_toArray
 
-@[simp] theorem toList_zipIdx (xs : Array α) (k : Nat) :
-    (xs.zipIdx k).toList = xs.toList.zipIdx k := by
-  rcases xs with ⟨xs⟩
+@[simp] theorem toList_zipIdx (a : Array α) (k : Nat) :
+    (a.zipIdx k).toList = a.toList.zipIdx k := by
+  rcases a with ⟨a⟩
   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} {xs : Array α} :
-    (x, i) ∈ xs.zipIdx k ↔ k ≤ i ∧ xs[i - k]? = some x := by
-  rcases xs with ⟨xs⟩
+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⟩
   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} {xs : Array α} :
-    (x, i) ∈ xs.zipIdx ↔ xs[i]? = x := by
+theorem mk_mem_zipIdx_iff_getElem? {x : α} {i : Nat} {l : Array α} :
+    (x, i) ∈ l.zipIdx ↔ l[i]? = x := by
   rw [mk_mem_zipIdx_iff_le_and_getElem?_sub]
   simp
 
-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
+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
   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} {xs : Array α} :
-    x ∈ xs.zipIdx ↔ xs[x.2]? = some x.1 := by
+theorem mem_zipIdx_iff_getElem? {x : α × Nat} {l : Array α} :
+    x ∈ l.zipIdx ↔ l[x.2]? = some x.1 := by
   rw [mk_mem_zipIdx_iff_getElem?]
 
 @[deprecated mk_mem_zipIdx_iff_getElem? (since := "2025-01-21")]
@@ -186,31 +182,31 @@ 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 {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
+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
   simp [List.mapFinIdx_eq_ofFn]
 
-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
+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
   simp [List.mapFinIdx_append]
 
 @[simp]
-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
+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
   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 {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
-    xs.mapFinIdx f = xs.zipIdx.attach.map
+theorem mapFinIdx_eq_zipIdx_map {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = l.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
@@ -219,44 +215,44 @@ theorem mapFinIdx_eq_zipIdx_map {xs : Array α} {f : (i : Nat) → α → (h : i
 abbrev mapFinIdx_eq_zipWithIndex_map := @mapFinIdx_eq_zipIdx_map
 
 @[simp]
-theorem mapFinIdx_eq_empty_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
-    xs.mapFinIdx f = #[] ↔ xs = #[] := by
-  cases xs
+theorem mapFinIdx_eq_empty_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = #[] ↔ l = #[] := by
+  cases l
   simp
 
-theorem mapFinIdx_ne_empty_iff {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
-    xs.mapFinIdx f ≠ #[] ↔ xs ≠ #[] := by
+theorem mapFinIdx_ne_empty_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f ≠ #[] ↔ l ≠ #[] := by
   simp
 
-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⟩
+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⟩
   exact List.exists_of_mem_mapFinIdx (by simpa using h)
 
-@[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 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 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⟩
+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'⟩
   simpa using List.mapFinIdx_eq_iff
 
-@[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_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_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⟩
+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⟩
   simp only [List.mapFinIdx_toArray, List.append_toArray, mk.injEq, List.mapFinIdx_eq_append_iff,
     toArray_eq_append_iff]
   constructor
@@ -268,39 +264,39 @@ theorem mapFinIdx_eq_append_iff {xs : Array α} {f : (i : Nat) → α → (h : i
     obtain rfl := h₂
     refine ⟨l₁, l₂, by simp_all⟩
 
-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
+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
   rw [push_eq_append, mapFinIdx_eq_append_iff]
   constructor
-  · rintro ⟨ys', zs', rfl, rfl, h₂⟩
+  · rintro ⟨l₁, l₂, rfl, rfl, h₂⟩
     simp only [mapFinIdx_eq_singleton_iff, Nat.zero_add] at h₂
     obtain ⟨a, rfl, rfl⟩ := h₂
-    exact ⟨ys', a, by simp⟩
-  · rintro ⟨zs, a, rfl, rfl, rfl⟩
-    exact ⟨zs, #[a], by simp⟩
+    exact ⟨l₁, a, by simp⟩
+  · rintro ⟨l₁, a, rfl, rfl, rfl⟩
+    exact ⟨l₁, #[a], by simp⟩
 
-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
+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
   rw [eq_comm, mapFinIdx_eq_iff]
   simp
 
-@[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] 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 [mapFinIdx_eq_iff]
 
-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⟩
+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⟩
   rw [← toList_inj]
   simp [List.mapFinIdx_eq_replicate_iff]
 
-@[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] 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 [List.mapFinIdx_reverse]
 
 /-! ### mapIdx -/
@@ -309,52 +305,52 @@ theorem mapFinIdx_eq_mkArray_iff {xs : Array α} {f : (i : Nat) → α → (h :
 theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #[] = #[] :=
   rfl
 
-@[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] 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_all [mapFinIdx_eq_iff]
 
-theorem mapIdx_eq_mapFinIdx {xs : Array α} {f : Nat → α → β} :
-    xs.mapIdx f = xs.mapFinIdx (fun i a _ => f i a) := by
+theorem mapIdx_eq_mapFinIdx {l : Array α} {f : Nat → α → β} :
+    l.mapIdx f = l.mapFinIdx (fun i a _ => f i a) := by
   simp [mapFinIdx_eq_mapIdx]
 
-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
+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
   ext <;> simp
 
 @[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-21")]
 abbrev mapIdx_eq_zipWithIndex_map := @mapIdx_eq_zipIdx_map
 
-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⟩
+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⟩
   simp [List.mapIdx_append]
 
 @[simp]
-theorem mapIdx_push {xs : Array α} {a : α} :
-    mapIdx f (xs.push a) = (mapIdx f xs).push (f xs.size a) := by
+theorem mapIdx_push {l : Array α} {a : α} :
+    mapIdx f (l.push a) = (mapIdx f l).push (f l.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 {xs : Array α} : mapIdx f xs = #[] ↔ xs = #[] := by
-  rcases xs with ⟨xs⟩
+theorem mapIdx_eq_empty_iff {l : Array α} : mapIdx f l = #[] ↔ l = #[] := by
+  rcases l with ⟨l⟩
   simp
 
-theorem mapIdx_ne_empty_iff {xs : Array α} :
-    mapIdx f xs ≠ #[] ↔ xs ≠ #[] := by
+theorem mapIdx_ne_empty_iff {l : Array α} :
+    mapIdx f l ≠ #[] ↔ l ≠ #[] := by
   simp
 
-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
+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
   rw [mapIdx_eq_mapFinIdx] at h
   simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
 
-@[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
+@[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
   constructor
   · intro h
     exact exists_of_mem_mapIdx h
@@ -362,138 +358,79 @@ theorem exists_of_mem_mapIdx {b : β} {xs : Array α}
     rw [mem_iff_getElem]
     exact ⟨i, by simpa using h, by simp⟩
 
-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
+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
   rw [mapIdx_eq_mapFinIdx, mapFinIdx_eq_push_iff]
   simp only [mapFinIdx_eq_mapIdx, exists_and_left, exists_prop]
   constructor
-  · rintro ⟨zs, rfl, a, rfl, rfl⟩
-    exact ⟨a, zs, by simp⟩
-  · rintro ⟨a, zs, rfl, rfl, rfl⟩
-    exact ⟨zs, rfl, a, by simp⟩
+  · rintro ⟨l₁, rfl, a, rfl, rfl⟩
+    exact ⟨a, l₁, by simp⟩
+  · rintro ⟨a, l₁, rfl, rfl, rfl⟩
+    exact ⟨l₁, rfl, a, by simp⟩
 
-@[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] 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 [List.mapIdx_eq_singleton_iff]
 
-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⟩
+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⟩
   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, List.size_toArray] at h₁ h₂
+    simp only [List.mapIdx_toArray, mk.injEq, size_toArray] at h₁ h₂
     obtain rfl := h₁
     obtain rfl := h₂
     exact ⟨l₁, l₂, by simp⟩
 
-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⟩
+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'⟩
   simp [List.mapIdx_eq_iff]
 
-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⟩
+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⟩
   simp [List.mapIdx_eq_mapIdx_iff]
 
-@[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] 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 [List.mapIdx_set]
 
-@[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] 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 [List.mapIdx_set]
 
-@[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] 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 [List.getLast?_mapIdx]
 
-@[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] 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 [mapIdx_eq_iff]
 
-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⟩
+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⟩
   rw [← toList_inj]
   simp [List.mapIdx_eq_replicate_iff]
 
-@[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] 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 [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

@@ -8,18 +8,15 @@ 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[i] < sizeOf as := by
+theorem sizeOf_get [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) : sizeOf (as.get i h) < 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
@@ -32,8 +29,8 @@ 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)
@@ -48,7 +45,7 @@ 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

@@ -12,9 +12,6 @@ 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
@@ -23,97 +20,90 @@ open Nat
 
 /-! ### mapM -/
 
-@[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_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_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']
+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']
   rw [List.mapM_eq_reverse_foldlM_cons]
   simp only [bind_pure_comp, Functor.map_map]
-  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 <$> f a) l.reverse.toArray xs by
+  suffices ∀ (k), (fun a => a.reverse.toArray) <$> 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
     exact this []
-  intro l
-  induction xs generalizing l with
+  intro k
+  induction l generalizing k 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) (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
+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
   subst w
-  cases xs
+  cases l
   simp [List.foldlM_map]
 
-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
+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
   subst w
-  cases xs
+  cases l
   simp [List.foldrM_map]
 
 theorem foldlM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : γ → β → m γ)
-    (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
+    (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
   subst w
-  cases xs
+  cases l
   simp [List.foldlM_filterMap]
   rfl
 
 theorem foldrM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : β → γ → m γ)
-    (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
+    (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
   subst w
-  cases xs
+  cases l
   simp [List.foldrM_filterMap]
   rfl
 
 theorem foldlM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : β → α → m β)
-    (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
+    (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
   subst w
-  cases xs
+  cases l
   simp [List.foldlM_filter]
 
 theorem foldrM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : α → β → m β)
-    (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
+    (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
   subst w
-  cases xs
+  cases l
   simp [List.foldrM_filter]
 
 @[simp] theorem foldlM_attachWith [Monad m]
-    (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
+    (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
   subst w
-  rcases xs with ⟨xs⟩
+  rcases l with ⟨l⟩
   simp [List.foldlM_map]
 
 @[simp] theorem foldrM_attachWith [Monad m] [LawfulMonad m]
-    (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
+    (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
   subst w
-  rcases xs with ⟨xs⟩
+  rcases l with ⟨l⟩
   simp [List.foldrM_map]
 
 /-! ### forM -/
@@ -124,15 +114,15 @@ 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] (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] 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
 
-@[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] 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
 
 /-! ### forIn' -/
@@ -152,41 +142,41 @@ 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]
-    (xs : Array α) (f : (a : α) → a ∈ xs → β → m (ForInStep β)) (init : β) :
-    forIn' xs init f = ForInStep.value <$>
-      xs.attach.foldlM (fun b ⟨a, m⟩ => match b with
+    (l : Array α) (f : (a : α) → a ∈ l → β → m (ForInStep β)) (init : β) :
+    forIn' l init f = ForInStep.value <$>
+      l.attach.foldlM (fun b ⟨a, m⟩ => match b with
         | .yield b => f a m b
         | .done b => pure (.done b)) (ForInStep.yield init) := by
-  rcases xs with ⟨xs⟩
+  cases l
   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]
-    (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⟩
+    (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
   simp [List.foldlM_map]
 
-@[simp] theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
-    (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⟩
+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
   simp [List.forIn'_pure_yield_eq_foldl, List.foldl_map]
 
 @[simp] theorem forIn'_yield_eq_foldl
-    (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⟩
+    (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
   simp [List.foldl_map]
 
 @[simp] theorem forIn'_map [Monad m] [LawfulMonad m]
-    (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⟩
+    (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
   simp
 
 /--
@@ -194,290 +184,96 @@ 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) (xs : Array α) :
-    forIn xs init f = ForInStep.value <$>
-      xs.foldlM (fun b a => match b with
+    (f : α → β → m (ForInStep β)) (init : β) (l : Array α) :
+    forIn l init f = ForInStep.value <$>
+      l.foldlM (fun b a => match b with
         | .yield b => f a b
         | .done b => pure (.done b)) (ForInStep.yield init) := by
-  rcases xs with ⟨xs⟩
-  simp only [List.forIn_toArray, List.forIn_eq_foldlM, List.size_toArray, List.foldlM_toArray']
+  cases l
+  simp only [List.forIn_toArray, List.forIn_eq_foldlM, 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]
-    (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⟩
+    (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
   simp [List.foldlM_map]
 
-@[simp] theorem forIn_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
-    (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⟩
+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
   simp [List.forIn_pure_yield_eq_foldl, List.foldl_map]
 
 @[simp] theorem forIn_yield_eq_foldl
-    (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⟩
+    (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
   simp [List.foldl_map]
 
 @[simp] theorem forIn_map [Monad m] [LawfulMonad m]
-    (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⟩
+    (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
   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} {xs : Array { x // p x }}
+@[simp] theorem foldlM_subtype [Monad m] {p : α → Prop} {l : 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 = xs.size) :
-    xs.foldlM f x 0 stop = xs.unattach.foldlM g x 0 stop := by
+    (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
   subst w
-  rcases xs with ⟨l⟩
+  rcases l 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} {xs : Array { x // p x }}
+@[simp] theorem foldrM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {l : 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 = xs.size) :
-    xs.foldrM f x start 0 = xs.unattach.foldrM g x start 0:= by
+    (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
   subst w
-  rcases xs with ⟨xs⟩
+  rcases l with ⟨l⟩
   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} {xs : Array { x // p x }}
+@[simp] theorem mapM_subtype [Monad m] [LawfulMonad m] {p : α → Prop} {l : Array { x // p x }}
     {f : { x // p x } → m β} {g : α → m β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    xs.mapM f = xs.unattach.mapM g := by
-  rcases xs with ⟨xs⟩
+    l.mapM f = l.unattach.mapM g := by
+  rcases l with ⟨l⟩
   simp
   rw [List.mapM_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 `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_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]
+-- 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]
 
 end Array

src/Init/Data/Array/OfFn.lean

@@ -11,31 +11,8 @@ 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 = #[] := by
-  simp [ofFn, ofFn.go]
-
-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

@@ -7,9 +7,6 @@ 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
@@ -30,38 +27,38 @@ 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 (xs : Array α) : xs ~ xs := by
-  cases xs
+@[simp, refl] protected theorem Perm.refl (l : Array α) : l ~ l := by
+  cases l
   simp
 
-protected theorem Perm.rfl {xs : List α} : xs ~ xs := .refl _
+protected theorem Perm.rfl {l : List α} : l ~ l := .refl _
 
-theorem Perm.of_eq {xs ys : Array α} (h : xs = ys) : xs ~ ys := h ▸ .rfl
+theorem Perm.of_eq {l₁ l₂ : Array α} (h : l₁ = l₂) : l₁ ~ l₂ := h ▸ .rfl
 
-protected theorem Perm.symm {xs ys : Array α} (h : xs ~ ys) : ys ~ xs := by
-  cases xs; cases ys
+protected theorem Perm.symm {l₁ l₂ : Array α} (h : l₁ ~ l₂) : l₂ ~ l₁ := by
+  cases l₁; cases l₂
   simp only [perm_toArray] at h
   simpa using h.symm
 
-protected theorem Perm.trans {xs ys zs : Array α} (h₁ : xs ~ ys) (h₂ : ys ~ zs) : xs ~ zs := by
-  cases xs; cases ys; cases zs
+protected theorem Perm.trans {l₁ l₂ l₃ : Array α} (h₁ : l₁ ~ l₂) (h₂ : l₂ ~ l₃) : l₁ ~ l₃ := by
+  cases l₁; cases l₂; cases l₃
   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 {xs ys : Array α} : xs ~ ys ↔ ys ~ xs := ⟨Perm.symm, Perm.symm⟩
+theorem perm_comm {l₁ l₂ : Array α} : l₁ ~ l₂ ↔ l₂ ~ l₁ := ⟨Perm.symm, Perm.symm⟩
 
-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
+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
   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 {xs : Array α} {i j : Nat} (h₁ : i < xs.size) (h₂ : j < xs.size) :
-    xs.swap i j ~ xs := by
+theorem swap_perm {as : Array α} {i j : Nat} (h₁ : i < as.size) (h₂ : j < as.size) :
+    as.swap i j ~ as := by
   simp only [swap, perm_iff_toList_perm, toList_set]
   apply set_set_perm
 

src/Init/Data/Array/QSort.lean

@@ -7,9 +7,6 @@ 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)
@@ -30,16 +27,6 @@ private def qpartition {n} (as : Vector α n) (lt : α → α → Bool) (lo hi :
       (⟨i, ilo⟩, as.swap i hi)
   loop as lo lo
 
-/--
-Sorts an array using the Quicksort algorithm.
-
-The optional parameter `lt` specifies an ordering predicate. It defaults to `LT.lt`, which must be
-decidable to be used for sorting. Use `Array.qsortOrd` to sort the array according to the `Ord α`
-instance.
-
-The optional parameters `low` and `high` delimit the region of the array that is sorted. Both are
-inclusive, and default to sorting the entire array.
--/
 @[inline] def qsort (as : Array α) (lt : α → α → Bool := by exact (· < ·))
     (low := 0) (high := as.size - 1) : Array α :=
   let rec @[specialize] sort {n} (as : Vector α n) (lo hi : Nat)
@@ -60,7 +47,7 @@ inclusive, and default to sorting the entire array.
 
 set_option linter.unusedVariables.funArgs false in
 /--
-Sorts an array using the Quicksort algorithm, using `Ord.compare` to compare elements.
+Sort an array using `compare` to compare elements.
 -/
 def qsortOrd [ord : Ord α] (xs : Array α) : Array α :=
   xs.qsort fun x y => compare x y |>.isLT

src/Init/Data/Array/Range.lean

@@ -15,9 +15,6 @@ 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
@@ -31,7 +28,7 @@ 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_iff, size_range']
+  rw [← size_eq_zero, size_range']
 
 theorem range'_ne_empty_iff (s : Nat) {n step : Nat} : range' s n step ≠ #[] ↔ n ≠ 0 := by
   cases n <;> simp
@@ -39,8 +36,7 @@ theorem range'_ne_empty_iff (s : Nat) {n step : Nat} : range' s n step ≠ #[]
 @[simp] theorem range'_zero : range' s 0 step = #[] := by
   simp
 
-@[simp] theorem range'_one {s step : Nat} : range' s 1 step = #[s] := by
-  simp [range', ofFn, ofFn.go]
+@[simp] theorem range'_one {s step : Nat} : range' s 1 step = #[s] := rfl
 
 @[simp] theorem range'_inj : range' s n = range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s') := by
   rw [← toList_inj]
@@ -78,7 +74,7 @@ theorem range'_append (s m n step : Nat) :
     range' s m ++ range' (s + m) n = range' s (m + n) := by simpa using range'_append s m n 1
 
 theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ #[s + step * n] := by
-  simpa using (range'_append s n 1 step).symm
+  exact (range'_append s n 1 step).symm
 
 theorem range'_1_concat (s n : Nat) : range' s (n + 1) = range' s n ++ #[s + n] := by
   simp [range'_concat]
@@ -128,7 +124,7 @@ 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, List.size_toArray, List.length_cons, List.length_nil,
+  · simp only [getElem_range, getElem_append, 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
@@ -137,7 +133,7 @@ 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_iff, size_range]
+  rw [← size_eq_zero, size_range]
 
 theorem range_ne_empty_iff {n : Nat} : range n ≠ #[] ↔ n ≠ 0 := by
   cases n <;> simp
@@ -150,9 +146,9 @@ theorem range_succ (n : Nat) : range (succ n) = range n ++ #[n] := by
       dite_eq_ite]
     split <;> omega
 
-theorem range_add (n m : Nat) : range (n + m) = range n ++ (range m).map (n + ·) := by
+theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + ·) := by
   rw [← range'_eq_map_range]
-  simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 n m).symm
+  simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm
 
 theorem reverse_range' (s n : Nat) : reverse (range' s n) = map (s + n - 1 - ·) (range n) := by
   simp [← toList_inj, List.reverse_range']
@@ -165,7 +161,7 @@ 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 (i n : Nat) : take (range n) i = range (min i n) := by
+@[simp] theorem take_range (m n : Nat) : take (range n) m = range (min m n) := by
   ext <;> simp
 
 @[simp] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
@@ -183,48 +179,48 @@ theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n -
 /-! ### zipIdx -/
 
 @[simp]
-theorem zipIdx_eq_empty_iff {xs : Array α} {i : Nat} : xs.zipIdx i = #[] ↔ xs = #[] := by
-  cases xs
+theorem zipIdx_eq_empty_iff {l : Array α} {n : Nat} : l.zipIdx n = #[] ↔ l = #[] := by
+  cases l
   simp
 
 @[simp]
-theorem getElem?_zipIdx (xs : Array α) (i j) : (zipIdx xs i)[j]? = xs[j]?.map fun a => (a, i + j) := by
+theorem getElem?_zipIdx (l : Array α) (n m) : (zipIdx l n)[m]? = l[m]?.map fun a => (a, n + m) := by
   simp [getElem?_def]
 
-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) :=
+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) :=
   ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
 
 @[simp]
-theorem zipIdx_map_snd (i) (xs : Array α) : map Prod.snd (zipIdx xs i) = range' i xs.size := by
-  cases xs
+theorem zipIdx_map_snd (n) (l : Array α) : map Prod.snd (zipIdx l n) = range' n l.size := by
+  cases l
   simp
 
 @[simp]
-theorem zipIdx_map_fst (i) (xs : Array α) : map Prod.fst (zipIdx xs i) = xs := by
-  cases xs
+theorem zipIdx_map_fst (n) (l : Array α) : map Prod.fst (zipIdx l n) = l := by
+  cases l
   simp
 
-theorem zipIdx_eq_zip_range' (xs : Array α) {i : Nat} : xs.zipIdx i = xs.zip (range' i xs.size) := by
+theorem zipIdx_eq_zip_range' (l : Array α) {n : Nat} : l.zipIdx n = l.zip (range' n l.size) := by
   simp [zip_of_prod (zipIdx_map_fst _ _) (zipIdx_map_snd _ _)]
 
 @[simp]
-theorem unzip_zipIdx_eq_prod (xs : Array α) {i : Nat} :
-    (xs.zipIdx i).unzip = (xs, range' i xs.size) := by
+theorem unzip_zipIdx_eq_prod (l : Array α) {n : Nat} :
+    (l.zipIdx n).unzip = (l, range' n l.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 (xs : Array α) (i : Nat) :
-    xs.zipIdx (i + 1) = (xs.zipIdx i).map (fun ⟨a, j⟩ => (a, j + 1)) := by
-  cases xs
+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
   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 (xs : Array α) (i : Nat) :
-    xs.zipIdx i = (xs.zipIdx 0).map (fun ⟨a, j⟩ => (a, i + j)) := by
-  cases xs
+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
   simp only [zipIdx_toArray, List.map_toArray, mk.injEq]
   rw [List.zipIdx_eq_map_add]
 
@@ -232,33 +228,33 @@ theorem zipIdx_eq_map_add (xs : Array α) (i : 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} {xs : Array α} :
-    (x, k + i) ∈ zipIdx xs k ↔ xs[i]? = some x := by
+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
   simp [mem_iff_getElem?, and_left_comm]
 
-theorem le_snd_of_mem_zipIdx {x : α × Nat} {k : Nat} {xs : Array α} (h : x ∈ zipIdx xs k) :
+theorem le_snd_of_mem_zipIdx {x : α × Nat} {k : Nat} {l : Array α} (h : x ∈ zipIdx l 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} {k : Nat} {xs : Array α} (h : x ∈ zipIdx xs k) :
-    x.2 < k + xs.size := by
+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
   rcases mem_iff_getElem.1 h with ⟨i, h', rfl⟩
   simpa using h'
 
-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
+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
   simpa [Nat.add_comm] using snd_lt_add_of_mem_zipIdx h
 
-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
+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
   simp [List.map_zipIdx]
 
-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_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_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
+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
   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) :
@@ -271,9 +267,9 @@ 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 (xs : Array α) (k : Nat) (f : α → β) :
-    zipIdx (xs.map f) k = (zipIdx xs k).map (Prod.map f id) := by
-  cases xs
+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
   simp [List.zipIdx_map]
 
 theorem zipIdx_append (xs ys : Array α) (k : Nat) :
@@ -282,19 +278,19 @@ theorem zipIdx_append (xs ys : Array α) (k : Nat) :
   cases ys
   simp [List.zipIdx_append]
 
-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⟩
+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⟩
   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, List.size_toArray] at h
+    simp only [zipIdx_toArray, mk.injEq, size_toArray] at h
     obtain ⟨rfl, rfl⟩ := h
     exact ⟨l₁', l₂', by simp⟩
 

src/Init/Data/Array/Set.lean

@@ -6,40 +6,27 @@ 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.
-
 
 /--
-Replaces the element at a given index in an array.
+Set an element in an array, using a proof that the index is in bounds.
+(This proof can usually be omitted, and will be synthesized automatically.)
 
-No bounds check is performed, but the function requires a proof that the index is in bounds. This
-proof can usually be omitted, and will be synthesized automatically.
-
-The array is modified in-place if there are no other references to it.
-
-Examples:
-* `#[0, 1, 2].set 1 5 = #[0, 5, 2]`
-* `#["orange", "apple"].set 1 "grape" = #["orange", "grape"]`
+This will perform the update destructively provided that `a` has a reference
+count of 1 when called.
 -/
 @[extern "lean_array_fset"]
-def Array.set (xs : Array α) (i : @& Nat) (v : α) (h : i < xs.size := by get_elem_tactic) :
+def Array.set (a : Array α) (i : @& Nat) (v : α) (h : i < a.size := by get_elem_tactic) :
     Array α where
-  toList := xs.toList.set i v
+  toList := a.toList.set i v
 
 /--
-Replaces the element at the provided index in an array. The array is returned unmodified if the
-index is out of bounds.
+Set an element in an array, or do nothing if the index is out of bounds.
 
-The array is modified in-place if there are no other references to it.
-
-Examples:
-* `#[0, 1, 2].setIfInBounds 1 5 = #[0, 5, 2]`
-* `#["orange", "apple"].setIfInBounds 1 "grape" = #["orange", "grape"]`
-* `#["orange", "apple"].setIfInBounds 5 "grape" = #["orange", "apple"]`
+This will perform the update destructively provided that `a` has a reference
+count of 1 when called.
 -/
-@[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)
+@[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)
 
 @[deprecated Array.setIfInBounds (since := "2024-11-24")] abbrev Array.setD := @Array.setIfInBounds
 
@@ -50,5 +37,5 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_set"]
-def Array.set! (xs : Array α) (i : @& Nat) (v : α) : Array α :=
-  Array.setIfInBounds xs i v
+def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
+  Array.setIfInBounds a i v

src/Init/Data/Array/Subarray.lean

@@ -6,33 +6,17 @@ 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
 
-/--
-A region of some underlying array.
-
-A subarray contains an array together with the start and end indices of a region of interest.
-Subarrays can be used to avoid copying or allocating space, while being more convenient than
-tracking the bounds by hand. The region of interest consists of every index that is both greater
-than or equal to `start` and strictly less than `stop`.
--/
-structure Subarray (α : Type u) where
-  /-- The underlying array. -/
+structure Subarray (α : Type u)  where
   array : Array α
-  /-- The starting index of the region of interest (inclusive). -/
   start : Nat
-  /-- The ending index of the region of interest (exclusive). -/
   stop : Nat
   start_le_stop : start ≤ stop
   stop_le_array_size : stop ≤ array.size
 
 namespace Subarray
 
-/--
-Computes the size of the subarray.
--/
 def size (s : Subarray α) : Nat :=
   s.stop - s.start
 
@@ -42,11 +26,6 @@ theorem size_le_array_size {s : Subarray α} : s.size ≤ s.array.size := by
   apply Nat.le_trans (Nat.sub_le stop start)
   assumption
 
-/--
-Extracts an element from the subarray.
-
-The index is relative to the start of the subarray, rather than the underlying array.
--/
 def get (s : Subarray α) (i : Fin s.size) : α :=
   have : s.start + i.val < s.array.size := by
    apply Nat.lt_of_lt_of_le _ s.stop_le_array_size
@@ -59,33 +38,12 @@ def get (s : Subarray α) (i : Fin s.size) : α :=
 instance : GetElem (Subarray α) Nat α fun xs i => i < xs.size where
   getElem xs i h := xs.get ⟨i, h⟩
 
-/--
-Extracts an element from the subarray, or returns a default value `v₀` when the index is out of
-bounds.
-
-The index is relative to the start and end of the subarray, rather than the underlying array.
--/
 @[inline] def getD (s : Subarray α) (i : Nat) (v₀ : α) : α :=
   if h : i < s.size then s[i] else v₀
 
-/--
-Extracts an element from the subarray, or returns a default value when the index is out of bounds.
-
-The index is relative to the start and end of the subarray, rather than the underlying array. The
-default value is that provided by the `Inhabited α` instance.
--/
 abbrev get! [Inhabited α] (s : Subarray α) (i : Nat) : α :=
   getD s i default
 
-/--
-Shrinks the subarray by incrementing its starting index if possible, returning it unchanged if not.
-
-Examples:
- * `#[1,2,3].toSubarray.popFront.toArray = #[2, 3]`
- * `#[1,2,3].toSubarray.popFront.popFront.toArray = #[3]`
- * `#[1,2,3].toSubarray.popFront.popFront.popFront.toArray = #[]`
- * `#[1,2,3].toSubarray.popFront.popFront.popFront.popFront.toArray = #[]`
--/
 def popFront (s : Subarray α) : Subarray α :=
   if h : s.start < s.stop then
     { s with start := s.start + 1, start_le_stop := Nat.le_of_lt_succ (Nat.add_lt_add_right h 1) }
@@ -94,8 +52,6 @@ def popFront (s : Subarray α) : Subarray α :=
 
 /--
 The empty subarray.
-
-This empty subarray is backed by an empty array.
 -/
 protected def empty : Subarray α where
   array := #[]
@@ -130,197 +86,46 @@ protected opaque forIn {α : Type u} {β : Type v} {m : Type v → Type w} [Mona
 instance : ForIn m (Subarray α) α where
   forIn := Subarray.forIn
 
-/--
-Folds a monadic operation from left to right over the elements in a subarray.
-
-An accumulator of type `β` is constructed by starting with `init` and monadically combining each
-element of the subarray with the current accumulator value in turn. The monad in question may permit
-early termination or repetition.
-
-Examples:
-```lean example
-#eval #["red", "green", "blue"].toSubarray.foldlM (init := "") fun acc x => do
-  let l ← Option.guard (· ≠ 0) x.length
-  return s!"{acc}({l}){x} "
-```
-```output
-some "(3)red (5)green (4)blue "
-```
-```lean example
-#eval #["red", "green", "blue"].toSubarray.foldlM (init := 0) fun acc x => do
-  let l ← Option.guard (· ≠ 5) x.length
-  return s!"{acc}({l}){x} "
-```
-```output
-none
-```
--/
 @[inline]
 def foldlM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Subarray α) : m β :=
   as.array.foldlM f (init := init) (start := as.start) (stop := as.stop)
 
-/--
-Folds a monadic operation from right to left over the elements in a subarray.
-
-An accumulator of type `β` is constructed by starting with `init` and monadically combining each
-element of the subarray with the current accumulator value in turn, moving from the end to the
-start. The monad in question may permit early termination or repetition.
-
-Examples:
-```lean example
-#eval #["red", "green", "blue"].toSubarray.foldrM (init := "") fun x acc => do
-  let l ← Option.guard (· ≠ 0) x.length
-  return s!"{acc}({l}){x} "
-```
-```output
-some "(4)blue (5)green (3)red "
-```
-```lean example
-#eval #["red", "green", "blue"].toSubarray.foldrM (init := 0) fun x acc => do
-  let l ← Option.guard (· ≠ 5) x.length
-  return s!"{acc}({l}){x} "
-```
-```output
-none
-```
--/
 @[inline]
 def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Subarray α) : m β :=
   as.array.foldrM f (init := init) (start := as.stop) (stop := as.start)
 
-/--
-Checks whether any of the elements in a subarray satisfy a monadic Boolean predicate.
-
-The elements are tested starting at the lowest index and moving up. The search terminates as soon as
-an element that satisfies the predicate is found.
-
-Example:
-```lean example
-#eval #["red", "green", "blue", "orange"].toSubarray.popFront.anyM fun x => do
-  IO.println x
-  pure (x == "blue")
-```
-```output
-green
-blue
-```
-```output
-true
-```
--/
 @[inline]
 def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Subarray α) : m Bool :=
   as.array.anyM p (start := as.start) (stop := as.stop)
 
-/--
-Checks whether all of the elements in a subarray satisfy a monadic Boolean predicate.
-
-The elements are tested starting at the lowest index and moving up. The search terminates as soon as
-an element that does not satisfy the predicate is found.
-
-Example:
-```lean example
-#eval #["red", "green", "blue", "orange"].toSubarray.popFront.allM fun x => do
-  IO.println x
-  pure (x.length == 5)
-```
-```output
-green
-blue
-```
-```output
-false
-```
--/
 @[inline]
 def allM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Subarray α) : m Bool :=
   as.array.allM p (start := as.start) (stop := as.stop)
 
-/--
-Runs a monadic action on each element of a subarray.
-
-The elements are processed starting at the lowest index and moving up.
--/
 @[inline]
 def forM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Subarray α) : m PUnit :=
   as.array.forM f (start := as.start) (stop := as.stop)
 
-/--
-Runs a monadic action on each element of a subarray, in reverse order.
-
-The elements are processed starting at the highest index and moving down.
--/
 @[inline]
 def forRevM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Subarray α) : m PUnit :=
   as.array.forRevM f (start := as.stop) (stop := as.start)
 
-/--
-Folds an operation from left to right over the elements in a subarray.
-
-An accumulator of type `β` is constructed by starting with `init` and combining each
-element of the subarray with the current accumulator value in turn.
-
-Examples:
- * `#["red", "green", "blue"].toSubarray.foldl (· + ·.length) 0 = 12`
- * `#["red", "green", "blue"].toSubarray.popFront.foldl (· + ·.length) 0 = 9`
--/
 @[inline]
 def foldl {α : Type u} {β : Type v} (f : β → α → β) (init : β) (as : Subarray α) : β :=
   Id.run <| as.foldlM f (init := init)
 
-/--
-Folds an operation from right to left over the elements in a subarray.
-
-An accumulator of type `β` is constructed by starting with `init` and combining each element of the
-subarray with the current accumulator value in turn, moving from the end to the start.
-
-Examples:
- * `#eval #["red", "green", "blue"].toSubarray.foldr (·.length + ·) 0 = 12`
- * `#["red", "green", "blue"].toSubarray.popFront.foldlr (·.length + ·) 0 = 9`
--/
 @[inline]
 def foldr {α : Type u} {β : Type v} (f : α → β → β) (init : β) (as : Subarray α) : β :=
   Id.run <| as.foldrM f (init := init)
 
-/--
-Checks whether any of the elements in a subarray satisfy a Boolean predicate.
-
-The elements are tested starting at the lowest index and moving up. The search terminates as soon as
-an element that satisfies the predicate is found.
--/
 @[inline]
 def any {α : Type u} (p : α → Bool) (as : Subarray α) : Bool :=
   Id.run <| as.anyM p
 
-/--
-Checks whether all of the elements in a subarray satisfy a Boolean predicate.
-
-The elements are tested starting at the lowest index and moving up. The search terminates as soon as
-an element that does not satisfy the predicate is found.
--/
 @[inline]
 def all {α : Type u} (p : α → Bool) (as : Subarray α) : Bool :=
   Id.run <| as.allM p
 
-/--
-Applies a monadic function to each element in a subarray in reverse order, stopping at the first
-element for which the function succeeds by returning a value other than `none`. The succeeding value
-is returned, or `none` if there is no success.
-
-Example:
-```lean example
-#eval #["red", "green", "blue"].toSubarray.findSomeRevM? fun x => do
-  IO.println x
-  return Option.guard (· = 5) x.length
-```
-```output
-blue
-green
-```
-```output
-some 5
-```
--/
 @[inline]
 def findSomeRevM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Subarray α) (f : α → m (Option β)) : m (Option β) :=
   let rec @[specialize] find : (i : Nat) → i ≤ as.size → m (Option β)
@@ -335,39 +140,10 @@ def findSomeRevM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m]
         find i this
   find as.size (Nat.le_refl _)
 
-/--
-Applies a monadic Boolean predicate to each element in a subarray in reverse order, stopping at the
-first element that satisfies the predicate. The element that satisfies the predicate is returned, or
-`none` if no element satisfies it.
-
-Example:
-```lean example
-#eval #["red", "green", "blue"].toSubarray.findRevM? fun x => do
-  IO.println x
-  return (x.length = 5)
-```
-```output
-blue
-green
-```
-```output
-some 5
-```
--/
 @[inline]
 def findRevM? {α : Type} {m : Type → Type w} [Monad m] (as : Subarray α) (p : α → m Bool) : m (Option α) :=
   as.findSomeRevM? fun a => return if (← p a) then some a else none
 
-/--
-Tests each element in a subarray with a Boolean predicate in reverse order, stopping at the first
-element that satisfies the predicate. The element that satisfies the predicate is returned, or
-`none` if no element satisfies the predicate.
-
-Examples:
- * `#["red", "green", "blue"].toSubarray.findRev? (·.length ≠ 4) = some "green"`
- * `#["red", "green", "blue"].toSubarray.findRev? (fun _ => true) = some "blue"`
- * `#["red", "green", "blue"].toSubarray 0 0 |>.findRev? (fun _ => true) = none`
--/
 @[inline]
 def findRev? {α : Type} (as : Subarray α) (p : α → Bool) : Option α :=
   Id.run <| as.findRevM? p
@@ -377,12 +153,6 @@ end Subarray
 namespace Array
 variable {α : Type u}
 
-/--
-Returns a subarray of an array, with the given bounds.
-
-If `start` or `stop` are not valid bounds for a subarray, then they are clamped to array's size.
-Additionally, the starting index is clamped to the ending index.
--/
 def toSubarray (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : Subarray α :=
   if h₂ : stop ≤ as.size then
     if h₁ : start ≤ stop then
@@ -405,9 +175,6 @@ def toSubarray (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : Suba
         start_le_stop := Nat.le_refl _,
         stop_le_array_size := Nat.le_refl _ }
 
-/--
-Allocates a new array that contains the contents of the subarray.
--/
 @[coe]
 def ofSubarray (s : Subarray α) : Array α := Id.run do
   let mut as := mkEmpty (s.stop - s.start)
@@ -417,11 +184,8 @@ def ofSubarray (s : Subarray α) : Array α := Id.run do
 
 instance : Coe (Subarray α) (Array α) := ⟨ofSubarray⟩
 
-/-- A subarray with the provided bounds.-/
 syntax:max term noWs "[" withoutPosition(term ":" term) "]" : term
-/-- A subarray with the provided lower bound that extends to the rest of the array. -/
 syntax:max term noWs "[" withoutPosition(term ":") "]" : term
-/-- A subarray with the provided upper bound, starting at the index 0. -/
 syntax:max term noWs "[" withoutPosition(":" term) "]" : term
 
 macro_rules
@@ -431,7 +195,6 @@ macro_rules
 
 end Array
 
-@[inherit_doc Array.ofSubarray]
 def Subarray.toArray (s : Subarray α) : Array α :=
   Array.ofSubarray s
 

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

@@ -15,13 +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, the first of which contains the first `i` elements and the second
-of which contains the remainder.
+Splits a subarray into two parts.
 -/
 def split (s : Subarray α) (i : Fin s.size.succ) : (Subarray α × Subarray α) :=
   let ⟨i', isLt⟩ := i

src/Init/Data/Array/TakeDrop.lean

@@ -7,28 +7,11 @@ 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 (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
+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
   simpa only [ugetElem_eq_getElem, ← getElem_toList, uset, toList_set] using
     List.exists_of_set _
 

src/Init/Data/Array/Zip.lean

@@ -11,9 +11,6 @@ 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
@@ -22,20 +19,20 @@ open Nat
 
 /-! ### zipWith -/
 
-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
+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
   simpa using List.zipWith_comm _ _ _
 
-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
+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
   rw [zipWith_comm]
   simp only [comm]
 
 @[simp]
-theorem zipWith_self (f : α → α → δ) (xs : Array α) : zipWith f xs xs = xs.map fun a => f a a := by
-  cases xs
+theorem zipWith_self (f : α → α → δ) (l : Array α) : zipWith f l l = l.map fun a => f a a := by
+  cases l
   simp
 
 /--
@@ -57,15 +54,15 @@ theorem getElem?_zipWith' {f : α → β → γ} {i : Nat} :
   cases l₂
   simp [List.getElem?_zipWith']
 
-theorem getElem?_zipWith_eq_some {f : α → β → γ} {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
+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
   simp [List.getElem?_zipWith_eq_some]
 
-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
+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
   cases z
   rw [zip, getElem?_zipWith_eq_some]; constructor
   · rintro ⟨x, y, h₀, h₁, h₂⟩
@@ -74,211 +71,211 @@ theorem getElem?_zip_eq_some {as : Array α} {bs : Array β} {z : α × β} {i :
     exact ⟨_, _, h₀, h₁, rfl⟩
 
 @[simp]
-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
+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
   simp [List.zipWith_map]
 
-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
+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
   simp [List.zipWith_map_left]
 
-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
+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
   simp [List.zipWith_map_right]
 
 theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} (i : δ):
-    (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
+    (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
   simp [List.zipWith_foldr_eq_zip_foldr]
 
 theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} (i : δ):
-    (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
+    (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
   simp [List.zipWith_foldl_eq_zip_foldl]
 
 @[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 zipWith_eq_empty_iff {f : α → β → γ} {l l'} : zipWith f l l' = #[] ↔ l = #[] ∨ l' = #[] := by
+  cases l <;> cases l' <;> simp
 
-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
+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'
   simp [List.map_zipWith]
 
-theorem take_zipWith : (zipWith f as bs).take i = zipWith f (as.take i) (bs.take i) := by
-  cases as
-  cases bs
+theorem take_zipWith : (zipWith f l l').take n = zipWith f (l.take n) (l'.take n) := by
+  cases l
+  cases l'
   simp [List.take_zipWith]
 
-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
+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'
   simp [List.drop_zipWith, List.take_zipWith]
 
-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'
+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
   simp at h
   simp [List.zipWith_append, h]
 
-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
+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'
   simp only [List.zipWith_toArray, List.append_toArray, mk.injEq, List.zipWith_eq_append_iff,
     toArray_eq_append_iff]
   constructor
-  · 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⟩
+  · 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⟩
 
 @[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) (as : Array α) (bs : Array β) :
-    map (Function.uncurry f) (as.zip bs) = zipWith f as bs := by
-  cases as
-  cases bs
+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'
   simp [List.map_uncurry_zip_eq_zipWith]
 
-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
+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'
   simp [List.map_zip_eq_zipWith]
 
-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_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_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 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 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
+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
   simp
   rw [List.zipWith_eq_zipWith_take_min]
 
-theorem reverse_zipWith (h : as.size = bs.size) :
-    (zipWith f as bs).reverse = zipWith f as.reverse bs.reverse := by
-  cases as
-  cases bs
+theorem reverse_zipWith (h : l.size = l'.size) :
+    (zipWith f l l').reverse = zipWith f l.reverse l'.reverse := by
+  cases l
+  cases l'
   simp [List.reverse_zipWith (by simpa using h)]
 
 /-! ### zip -/
 
-theorem lt_size_left_of_zip {i : Nat} {as : Array α} {bs : Array β} (h : i < (zip as bs).size) :
-    i < as.size :=
+theorem lt_size_left_of_zip {i : Nat} {l : Array α} {l' : Array β} (h : i < (zip l l').size) :
+    i < l.size :=
   lt_size_left_of_zipWith h
 
-theorem lt_size_right_of_zip {i : Nat} {as : Array α} {bs : Array β} (h : i < (zip as bs).size) :
-    i < bs.size :=
+theorem lt_size_right_of_zip {i : Nat} {l : Array α} {l' : Array β} (h : i < (zip l l').size) :
+    i < l'.size :=
   lt_size_right_of_zipWith h
 
 @[simp]
-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)) :=
+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)) :=
   getElem_zipWith (hi := by simpa using h)
 
-theorem zip_eq_zipWith (as : Array α) (bs : Array β) : zip as bs = zipWith Prod.mk as bs := by
-  cases as
-  cases bs
+theorem zip_eq_zipWith (l₁ : Array α) (l₂ : Array β) : zip l₁ l₂ = zipWith Prod.mk l₁ l₂ := by
+  cases l₁
+  cases l₂
   simp [List.zip_eq_zipWith]
 
-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
+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
   simp [List.zip_map]
 
-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_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_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_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_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
+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
   simp_all [List.zip_append]
 
-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
+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
   simp [List.zip_map']
 
-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
+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
   simpa using List.of_mem_zip
 
-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
+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
   simp_all [List.map_fst_zip]
 
-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
+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
   simp_all [List.map_snd_zip]
 
-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
+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
   rw [← zip_map']
   congr
   simp
 
-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
+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
   rw [← zip_map']
   congr
   simp
 
-@[simp] theorem zip_eq_empty_iff {as : Array α} {bs : Array β} :
-    zip as bs = #[] ↔ as = #[] ∨ bs = #[] := by
-  cases as
-  cases bs
+@[simp] theorem zip_eq_empty_iff {l₁ : Array α} {l₂ : Array β} :
+    zip l₁ l₂ = #[] ↔ l₁ = #[] ∨ l₂ = #[] := by
+  cases l₁
+  cases l₂
   simp [List.zip_eq_nil_iff]
 
-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
+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
   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 (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]
+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]
   rw [List.zip_eq_zip_take_min]
 
 
@@ -292,30 +289,31 @@ 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) (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
+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
   simp [List.zipWithAll_map]
 
-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 <$> a) b) as bs := by
-  cases as
-  cases bs
+theorem zipWithAll_map_left (l₁ : Array α) (l₂ : Array β) (f : α → α') (g : Option α' → Option β → γ) :
+    zipWithAll g (l₁.map f) l₂ = zipWithAll (fun a b => g (f <$> a) b) l₁ l₂ := by
+  cases l₁
+  cases l₂
   simp [List.zipWithAll_map_left]
 
-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 <$> b)) as bs := by
-  cases as
-  cases bs
+theorem zipWithAll_map_right (l₁ : Array α) (l₂ : Array β) (f : β → β') (g : Option α → Option β' → γ) :
+    zipWithAll g l₁ (l₂.map f) = zipWithAll (fun a b => g a (f <$> b)) l₁ l₂ := by
+  cases l₁
+  cases l₂
   simp [List.zipWithAll_map_right]
 
-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
+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'
   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]
@@ -328,37 +326,37 @@ 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 (xs : Array (α × β)) : unzip xs = (xs.map Prod.fst, xs.map Prod.snd) := by
-  cases xs
+theorem unzip_eq_map (l : Array (α × β)) : unzip l = (l.map Prod.fst, l.map Prod.snd) := by
+  cases l
   simp [List.unzip_eq_map]
 
-theorem zip_unzip (xs : Array (α × β)) : zip (unzip xs).1 (unzip xs).2 = xs := by
-  cases xs
+theorem zip_unzip (l : Array (α × β)) : zip (unzip l).1 (unzip l).2 = l := by
+  cases l
   simp only [List.unzip_toArray, Prod.map_fst, Prod.map_snd, List.zip_toArray, List.zip_unzip]
 
-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,
+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,
     List.unzip_zip_left]
 
-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,
+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,
     List.unzip_zip_right]
 
-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 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 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]
+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]
 
 @[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] : Prop extends PartialEquivBEq α, ReflBEq α
+class EquivBEq (α) [BEq α] extends PartialEquivBEq α, ReflBEq α : Prop
 
 @[simp]
 theorem BEq.refl [BEq α] [ReflBEq α] {a : α} : a == a :=
@@ -49,14 +49,6 @@ 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

@@ -25,17 +25,13 @@ 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, bitvec_to_nat] theorem ofNat_eq_ofNat : @OfNat.ofNat (BitVec n) i _ = .ofNat n i := rfl
+@[simp, bv_toNat] 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
@@ -59,12 +55,12 @@ 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
 
@@ -96,10 +92,16 @@ This will be renamed `getMsb` after the existing deprecated alias is removed.
 @[inline] def getLsbD (x : BitVec w) (i : Nat) : Bool :=
   x.toNat.testBit i
 
+@[deprecated getLsbD (since := "2024-08-29"), inherit_doc getLsbD]
+def getLsb (x : BitVec w) (i : Nat) : Bool := x.getLsbD i
+
 /-- Return the `i`-th most significant bit or `false` if `i ≥ w`. -/
 @[inline] def getMsbD (x : BitVec w) (i : Nat) : Bool :=
   i < w && x.getLsbD (w-1-i)
 
+@[deprecated getMsbD (since := "2024-08-29"), inherit_doc getMsbD]
+def getMsb (x : BitVec w) (i : Nat) : Bool := x.getMsbD i
+
 /-- Return most-significant bit in bitvector. -/
 @[inline] protected def msb (x : BitVec n) : Bool := getMsbD x 0
 
@@ -121,7 +123,6 @@ 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
 
@@ -137,7 +138,7 @@ 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 _)
@@ -166,12 +167,12 @@ 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 ()
 
@@ -355,7 +356,7 @@ 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
@@ -389,7 +390,7 @@ 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_right (by trivial) le)
+    exact Nat.pow_le_pow_of_le_right (by trivial) le)
 
 @[deprecated setWidth' (since := "2024-09-18"), inherit_doc setWidth'] abbrev zeroExtend' := @setWidth'
 

src/Init/Data/BitVec/Bitblast.lean

@@ -109,12 +109,7 @@ open Nat Bool
 
 namespace Bool
 
-/--
-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.
--/
+/-- At least two out of three booleans are true. -/
 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]
@@ -149,7 +144,7 @@ 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_right Nat.zero_lt_two x_lt
+             _ ≤ 2 ^ j := Nat.pow_le_pow_of_le_right Nat.zero_lt_two x_lt
              _ ≤ x := testBit_implies_ge jp
 
 private theorem mod_two_pow_succ (x i : Nat) :
@@ -290,7 +285,7 @@ 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, getElem_add_add_bool]
+    simp [adcb, Prod.mk.injEq, carry_succ, getLsbD_add_add_bool]
 
 theorem add_eq_adc (w : Nat) (x y : BitVec w) : x + y = (adc x y false).snd := by
   simp [adc_spec]
@@ -300,7 +295,7 @@ 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, getElem_add, i_lt, show w - 1 - i < w by omega]
+  simp [getMsbD, getLsbD_add, i_lt, show w - 1 - i < w by omega]
 
 theorem msb_add {w : Nat} {x y: BitVec w} :
     (x + y).msb =
@@ -364,25 +359,24 @@ theorem msb_sub {x y: BitVec w} :
 /-! ### Negation -/
 
 theorem bit_not_testBit (x : BitVec w) (i : Fin w) :
-  (((iunfoldr (fun (i : Fin w) c => (c, !(x[i.val])))) ()).snd)[i.val] = !(getLsbD x i.val) := by
+  getLsbD (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).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[i.val])))) ()).snd + x  = -1 := by
+  ((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd + x  = -1 := by
   simp only [add_eq_adc]
   apply iunfoldr_replace_snd (fun _ => false) (-1) false rfl
-  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]
+  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]
 
 theorem bit_not_eq_not (x : BitVec w) :
-  ((iunfoldr (fun i c => (c, !(x[i])))) ()).snd = ~~~ x := by
+  ((iunfoldr (fun i c => (c, !(x.getLsbD 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[i.val])))) ()).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.getLsbD i)))) ()).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[i.val])))) ()).snd) _ rfl]
+  rw [iunfoldr_replace_snd ((fun _ => ())) (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).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 _]
@@ -483,39 +477,6 @@ 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
-
-@[simp] theorem BitVec.setWidth_neg_of_le {x : BitVec v} (h : w ≤ v) : BitVec.setWidth w (-x) = -BitVec.setWidth w x := by
-  simp [← BitVec.signExtend_eq_setWidth_of_le _ h, BitVec.signExtend_neg_of_le h]
-
 /-! ### abs -/
 
 theorem msb_abs {w : Nat} {x : BitVec w} :
@@ -582,15 +543,6 @@ 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]
@@ -623,18 +575,16 @@ 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 [getElem_setWidth, getLsbD_setWidth, h, getLsbD_eq_getElem, getElem_or, getElem_and,
-      getElem_twoPow]
+    simp only [getLsbD_setWidth, h, decide_true, Bool.true_and, getLsbD_or, getLsbD_and]
     by_cases hik : i = k
     · subst hik
       simp [h]
-    · by_cases hik' : k < (i + 1)
+    · simp only [getLsbD_twoPow, hik, decide_false, Bool.and_false, Bool.or_false]
+      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]
@@ -954,7 +904,7 @@ 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 where
+  extends DivModState.Lawful args qr : Type where
   /-- Only perform a round of shift-subtract if we have dividend bits. -/
   hwn_lt : 0 < qr.wn
 
@@ -1081,10 +1031,11 @@ 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
-  induction hm : qr.wn generalizing qr with
-  | zero =>
+  generalize hm : qr.wn = m
+  induction m generalizing qr
+  case zero =>
     exact h
-  | succ wn' ih =>
+  case succ wn' ih =>
     simp only [divRec_succ]
     apply ih
     · apply lawful_divSubtractShift
@@ -1098,10 +1049,11 @@ 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
-  induction hm : qr.wn generalizing qr with
-  | zero =>
+  generalize hm : qr.wn = m
+  induction m generalizing qr
+  case zero =>
     assumption
-  | succ wn' ih =>
+  case succ wn' ih =>
     apply ih
     simp only [divSubtractShift, hm]
     split <;> rfl
@@ -1291,8 +1243,8 @@ theorem saddOverflow_eq {w : Nat} (x y : BitVec w) :
   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)
+  · have := le_toInt (x := x); have := toInt_lt (x := x)
+    have := le_toInt (x := y); have := 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)]
@@ -1328,17 +1280,4 @@ 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[i.val])) :
+    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value.getLsbD 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[i.val])) :
+    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value.getLsbD i.val)) :
     (iunfoldr f a).snd = value := by
   simp [iunfoldr.eq_test state value a init step]
 

src/Init/Data/Bool.lean

@@ -9,19 +9,7 @@ import Init.NotationExtra
 
 namespace Bool
 
-/--
-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`
--/
+/-- Boolean exclusive or -/
 abbrev xor : Bool → Bool → Bool := bne
 
 @[inherit_doc] infixl:33 " ^^ " => xor
@@ -379,19 +367,17 @@ theorem and_or_inj_left_iff :
 
 /-! ## toNat -/
 
-/--
-Converts `true` to `1` and `false` to `0`.
--/
+/-- convert a `Bool` to a `Nat`, `false -> 0`, `true -> 1` -/
 def toNat (b : Bool) : Nat := cond b 1 0
 
-@[simp, bitvec_to_nat] theorem toNat_false : false.toNat = 0 := rfl
+@[simp, bv_toNat] theorem toNat_false : false.toNat = 0 := rfl
 
-@[simp, bitvec_to_nat] theorem toNat_true : true.toNat = 1 := rfl
+@[simp, bv_toNat] theorem toNat_true : true.toNat = 1 := rfl
 
 theorem toNat_le (c : Bool) : c.toNat ≤ 1 := by
   cases c <;> trivial
 
-@[bitvec_to_nat]
+@[bv_toNat]
 theorem toNat_lt (b : Bool) : b.toNat < 2 :=
   Nat.lt_succ_of_le (toNat_le _)
 
@@ -402,9 +388,7 @@ theorem toNat_lt (b : Bool) : b.toNat < 2 :=
 
 /-! ## toInt -/
 
-/--
-Converts `true` to `1` and `false` to `0`.
--/
+/-- convert a `Bool` to an `Int`, `false -> 0`, `true -> 1` -/
 def toInt (b : Bool) : Int := cond b 1 0
 
 @[simp] theorem toInt_false : false.toInt = 0 := rfl
@@ -555,8 +539,8 @@ 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 : 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
+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
 
 @[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
@@ -596,13 +580,17 @@ 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]
 
-@[bool_to_prop]
-theorem and_eq_decide (p q : Bool) : (p && q) = decide (p ∧ q) := by simp
+@[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 or_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]
+@[boolToPropSimps]
 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

@@ -18,13 +18,10 @@ attribute [extern "lean_byte_array_data"] ByteArray.data
 
 namespace ByteArray
 @[extern "lean_mk_empty_byte_array"]
-def emptyWithCapacity (c : @& Nat) : ByteArray :=
+def mkEmpty (c : @& Nat) : ByteArray :=
   { data := #[] }
 
-@[deprecated emptyWithCapacity (since := "2025-03-12")]
-abbrev mkEmpty := emptyWithCapacity
-
-def empty : ByteArray := emptyWithCapacity 0
+def empty : ByteArray := mkEmpty 0
 
 instance : Inhabited ByteArray where
   default := empty
@@ -50,7 +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[i]!
+  | ⟨bs⟩, i => bs.get! i
 
 @[extern "lean_byte_array_fget"]
 def get : (a : @& ByteArray) → (i : @& Nat) → (h : i < a.size := by get_elem_tactic) → UInt8
@@ -59,7 +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.toFin < xs.size where
+instance : GetElem ByteArray USize UInt8 fun xs i => i.val < xs.size where
   getElem xs i h := xs.uget i h
 
 @[extern "lean_byte_array_set"]
@@ -337,9 +334,6 @@ def prevn : Iterator → Nat → Iterator
 end Iterator
 end ByteArray
 
-/--
-Converts a list of bytes into a `ByteArray`.
--/
 def List.toByteArray (bs : List UInt8) : ByteArray :=
   let rec loop
     | [],    r => r

src/Init/Data/Char/Basic.lean

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

src/Init/Data/Fin/Basic.lean

@@ -14,23 +14,22 @@ instance coeToNat : CoeOut (Fin n) Nat :=
   ⟨fun v => v.val⟩
 
 /--
-The type `Fin 0` is uninhabited, so it can be used to derive any result whatsoever.
-
-This is similar to `Empty.elim`. It can be thought of as a compiler-checked assertion that a code
-path is unreachable, or a logical contradiction from which `False` and thus anything else could be
-derived.
+From the empty type `Fin 0`, any desired result `α` can be derived. This is similar to `Empty.elim`.
 -/
 def elim0.{u} {α : Sort u} : Fin 0 → α
   | ⟨_, h⟩ => absurd h (not_lt_zero _)
 
 /--
-The successor, with an increased bound.
+Returns the successor of the argument.
 
-This differs from adding `1`, which instead wraps around.
-
-Examples:
- * `(2 : Fin 3).succ = (3 : Fin 4)`
- * `(2 : Fin 3) + 1 = (0 : Fin 3)`
+The bound in the result type is increased:
+```
+(2 : Fin 3).succ = (3 : Fin 4)
+```
+This differs from addition, which wraps around:
+```
+(2 : Fin 3) + 1 = (0 : Fin 3)
+```
 -/
 def succ : Fin n → Fin (n + 1)
   | ⟨i, h⟩ => ⟨i+1, Nat.succ_lt_succ h⟩
@@ -52,54 +51,21 @@ 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`.
-/--
-Extracts the underlying `Nat` value.
-
-This function is a synonym for `Fin.val`, which is the simp normal form. `Fin.val` is also a
-coercion, so values of type `Fin n` are automatically converted to `Nat`s as needed.
--/
-@[inline]
-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 =>
     have : n > 0 := Nat.lt_trans (Nat.zero_lt_succ _) h;
     Nat.mod_lt _ this
 
-/--
-Addition modulo `n`, usually invoked via the `+` operator.
-
-Examples:
- * `(2 : Fin 8) + (2 : Fin 8) = (4 : Fin 8)`
- * `(2 : Fin 3) + (2 : Fin 3) = (1 : Fin 3)`
--/
+/-- Addition modulo `n` -/
 protected def add : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a + b) % n, mlt h⟩
 
-/--
-Multiplication modulo `n`, usually invoked via the `*` operator.
-
-Examples:
- * `(2 : Fin 10) * (2 : Fin 10) = (4 : Fin 10)`
- * `(2 : Fin 10) * (7 : Fin 10) = (4 : Fin 10)`
- * `(3 : Fin 10) * (7 : Fin 10) = (1 : Fin 10)`
--/
+/-- Multiplication modulo `n` -/
 protected def mul : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a * b) % n, mlt h⟩
 
-/--
-Subtraction modulo `n`, usually invoked via the `-` operator.
-
-Examples:
- * `(5 : Fin 11) - (3 : Fin 11) = (2 : Fin 11)`
- * `(3 : Fin 11) - (5 : Fin 11) = (9 : Fin 11)`
--/
+/-- Subtraction modulo `n` -/
 protected def sub : Fin n → Fin n → Fin n
   /-
   The definition of `Fin.sub` has been updated to improve performance.
@@ -126,76 +92,27 @@ we are trying to minimize the number of Nat theorems
 needed to bootstrap Lean.
 -/
 
-/--
-Modulus of bounded numbers, usually invoked via the `%` operator.
-
-The resulting value is that computed by the `%` operator on `Nat`.
--/
 protected def mod : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨a % b,  Nat.lt_of_le_of_lt (Nat.mod_le _ _) h⟩
 
-/--
-Division of bounded numbers, usually invoked via the `/` operator.
-
-The resulting value is that computed by the `/` operator on `Nat`. In particular, the result of
-division by `0` is `0`.
-
-Examples:
- * `(5 : Fin 10) / (2 : Fin 10) = (2 : Fin 10)`
- * `(5 : Fin 10) / (0 : Fin 10) = (0 : Fin 10)`
- * `(5 : Fin 10) / (7 : Fin 10) = (0 : Fin 10)`
--/
 protected def div : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨a / b, Nat.lt_of_le_of_lt (Nat.div_le_self _ _) h⟩
 
-/--
-Modulus of bounded numbers with respect to a `Nat`.
-
-The resulting value is that computed by the `%` operator on `Nat`.
--/
 def modn : Fin n → Nat → Fin n
   | ⟨a, h⟩, m => ⟨a % m, Nat.lt_of_le_of_lt (Nat.mod_le _ _) h⟩
 
-/--
-Bitwise and.
--/
 def land : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(Nat.land a b) % n, mlt h⟩
 
-/--
-Bitwise or.
--/
 def lor : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(Nat.lor a b) % n, mlt h⟩
 
-/--
-Bitwise xor (“exclusive or”).
--/
 def xor : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(Nat.xor a b) % n, mlt h⟩
 
-/--
-Bitwise left shift of bounded numbers, with wraparound on overflow.
-
-Examples:
- * `(1 : Fin 10) <<< (1 : Fin 10) = (2 : Fin 10)`
- * `(1 : Fin 10) <<< (3 : Fin 10) = (8 : Fin 10)`
- * `(1 : Fin 10) <<< (4 : Fin 10) = (6 : Fin 10)`
--/
 def shiftLeft : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a <<< b) % n, mlt h⟩
 
-/--
-Bitwise right shift of bounded numbers.
-
-This operator corresponds to logical rather than arithmetic bit shifting. The new bits are always
-`0`.
-
-Examples:
- * `(15 : Fin 16) >>> (1 : Fin 16) = (7 : Fin 16)`
- * `(15 : Fin 16) >>> (2 : Fin 16) = (3 : Fin 16)`
- * `(15 : Fin 17) >>> (2 : Fin 17) = (3 : Fin 17)`
--/
 def shiftRight : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a >>> b) % n, mlt h⟩
 
@@ -249,125 +166,39 @@ theorem val_lt_of_le (i : Fin b) (h : b ≤ n) : i.val < n :=
 protected theorem pos (i : Fin n) : 0 < n :=
   Nat.lt_of_le_of_lt (Nat.zero_le _) i.2
 
-/--
-The greatest value of `Fin (n+1)`, namely `n`.
-
-Examples:
- * `Fin.last 4 = (4 : Fin 5)`
- * `(Fin.last 0).val = (0 : Nat)`
--/
+/-- The greatest value of `Fin (n+1)`. -/
 @[inline] def last (n : Nat) : Fin (n + 1) := ⟨n, n.lt_succ_self⟩
 
-/--
-Replaces the bound with another that is suitable for the value.
-
-The proof embedded in `i` can be used to cast to a larger bound even if the concrete value is not
-known.
-
-Examples:
-```lean example
-example : Fin 12 := (7 : Fin 10).castLT (by decide : 7 < 12)
-```
-```lean example
-example (i : Fin 10) : Fin 12 :=
-  i.castLT <| by
-    cases i; simp; omega
-```
--/
+/-- `castLT i h` embeds `i` into a `Fin` where `h` proves it belongs into.  -/
 @[inline] def castLT (i : Fin m) (h : i.1 < n) : Fin n := ⟨i.1, h⟩
 
-/--
-Coarsens a bound to one at least as large.
-
-See also `Fin.castAdd` for a version that represents the larger bound with addition rather than an
-explicit inequality proof.
--/
+/-- `castLE h i` embeds `i` into a larger `Fin` type.  -/
 @[inline] def castLE (h : n ≤ m) (i : Fin n) : Fin m := ⟨i, Nat.lt_of_lt_of_le i.2 h⟩
 
-/--
-Uses a proof that two bounds are equal to allow a value bounded by one to be used with the other.
-
-In other words, when `eq : n = m`, `Fin.cast eq i` converts `i : Fin n` into a `Fin m`.
--/
+/-- `cast eq i` embeds `i` into an equal `Fin` type. -/
 @[inline] protected def cast (eq : n = m) (i : Fin n) : Fin m := ⟨i, eq ▸ i.2⟩
 
-/--
-Coarsens a bound to one at least as large.
-
-See also `Fin.natAdd` and `Fin.addNat` for addition functions that increase the bound, and
-`Fin.castLE` for a version that uses an explicit inequality proof.
--/
+/-- `castAdd m i` embeds `i : Fin n` in `Fin (n+m)`. See also `Fin.natAdd` and `Fin.addNat`. -/
 @[inline] def castAdd (m) : Fin n → Fin (n + m) :=
   castLE <| Nat.le_add_right n m
 
-/--
-Coarsens a bound by one.
--/
+/-- `castSucc i` embeds `i : Fin n` in `Fin (n+1)`. -/
 @[inline] def castSucc : Fin n → Fin (n + 1) := castAdd 1
 
-/--
-Adds a natural number to a `Fin`, increasing the bound.
-
-This is a generalization of `Fin.succ`.
-
-`Fin.natAdd` is a version of this function that takes its `Nat` parameter first.
-
-Examples:
- * `Fin.addNat (5 : Fin 8) 3 = (8 : Fin 11)`
- * `Fin.addNat (0 : Fin 8) 1 = (1 : Fin 9)`
- * `Fin.addNat (1 : Fin 8) 2 = (3 : Fin 10)`
-
--/
+/-- `addNat m i` adds `m` to `i`, generalizes `Fin.succ`. -/
 def addNat (i : Fin n) (m) : Fin (n + m) := ⟨i + m, Nat.add_lt_add_right i.2 _⟩
 
-/--
-Adds a natural number to a `Fin`, increasing the bound.
-
-This is a generalization of `Fin.succ`.
-
-`Fin.addNat` is a version of this function that takes its `Nat` parameter second.
-
-Examples:
- * `Fin.natAdd 3 (5 : Fin 8) = (8 : Fin 11)`
- * `Fin.natAdd 1 (0 : Fin 8) = (1 : Fin 9)`
- * `Fin.natAdd 1 (2 : Fin 8) = (3 : Fin 9)`
--/
+/-- `natAdd n i` adds `n` to `i` "on the left". -/
 def natAdd (n) (i : Fin m) : Fin (n + m) := ⟨n + i, Nat.add_lt_add_left i.2 _⟩
 
-/--
-Replaces a value with its difference from the largest value in the type.
-
-Considering the values of `Fin n` as a sequence `0`, `1`, …, `n-2`, `n-1`, `Fin.rev` finds the
-corresponding element of the reversed sequence. In other words, it maps `0` to `n-1`, `1` to `n-2`,
-..., and `n-1` to `0`.
-
-Examples:
- * `(5 : Fin 6).rev = (0 : Fin 6)`
- * `(0 : Fin 6).rev = (5 : Fin 6)`
- * `(2 : Fin 5).rev = (2 : Fin 5)`
--/
+/-- Maps `0` to `n-1`, `1` to `n-2`, ..., `n-1` to `0`. -/
 @[inline] def rev (i : Fin n) : Fin n := ⟨n - (i + 1), Nat.sub_lt i.pos (Nat.succ_pos _)⟩
 
-/--
-Subtraction of a natural number from a `Fin`, with the bound narrowed.
-
-This is a generalization of `Fin.pred`. It is guaranteed to not underflow or wrap around.
-
-Examples:
- * `(5 : Fin 9).subNat 2 (by decide) = (3 : Fin 7)`
- * `(5 : Fin 9).subNat 0 (by decide) = (5 : Fin 9)`
- * `(3 : Fin 9).subNat 3 (by decide) = (0 : Fin 6)`
--/
+/-- `subNat i h` subtracts `m` from `i`, generalizes `Fin.pred`. -/
 @[inline] def subNat (m) (i : Fin (n + m)) (h : m ≤ i) : Fin n :=
   ⟨i - m, Nat.sub_lt_right_of_lt_add h i.2⟩
 
-/--
-The predecessor of a non-zero element of `Fin (n+1)`, with the bound decreased.
-
-Examples:
- * `(4 : Fin 8).pred (by decide) = (3 : Fin 7)`
- * `(1 : Fin 2).pred (by decide) = (0 : Fin 1)`
--/
+/-- Predecessor of a nonzero element of `Fin (n+1)`. -/
 @[inline] def pred {n : Nat} (i : Fin (n + 1)) (h : i ≠ 0) : Fin n :=
   subNat 1 i <| Nat.pos_of_ne_zero <| mt (Fin.eq_of_val_eq (j := 0)) h
 

src/Init/Data/Fin/Bitwise.lean

@@ -12,31 +12,4 @@ namespace Fin
 @[simp] theorem and_val (a b : Fin n) : (a &&& b).val = a.val &&& b.val :=
   Nat.mod_eq_of_lt (Nat.lt_of_le_of_lt Nat.and_le_left a.isLt)
 
-@[simp] theorem or_val_of_two_pow {w} (a b : Fin (2 ^ w)) : (a ||| b).val = a.val ||| b.val :=
-  Nat.mod_eq_of_lt (Nat.or_lt_two_pow a.isLt b.isLt)
-
-@[simp] theorem or_val_of_uInt8Size (a b : Fin UInt8.size) : (a ||| b).val = a.val ||| b.val := or_val_of_two_pow (w := 8) a b
-@[simp] theorem or_val_of_uInt16Size (a b : Fin UInt16.size) : (a ||| b).val = a.val ||| b.val := or_val_of_two_pow (w := 16) a b
-@[simp] theorem or_val_of_uInt32Size (a b : Fin UInt32.size) : (a ||| b).val = a.val ||| b.val := or_val_of_two_pow (w := 32) a b
-@[simp] theorem or_val_of_uInt64Size (a b : Fin UInt64.size) : (a ||| b).val = a.val ||| b.val := or_val_of_two_pow (w := 64) a b
-@[simp] theorem or_val_of_uSizeSize (a b : Fin USize.size) : (a ||| b).val = a.val ||| b.val := or_val_of_two_pow a b
-
-theorem or_val (a b : Fin n) : (a ||| b).val = (a.val ||| b.val) % n := rfl
-
-@[simp] theorem xor_val_of_two_pow {w} (a b : Fin (2 ^ w)) : (a ^^^ b).val = a.val ^^^ b.val :=
-  Nat.mod_eq_of_lt (Nat.xor_lt_two_pow a.isLt b.isLt)
-
-@[simp] theorem xor_val_of_uInt8Size (a b : Fin UInt8.size) : (a ^^^ b).val = a.val ^^^ b.val := xor_val_of_two_pow (w := 8) a b
-@[simp] theorem xor_val_of_uInt16Size (a b : Fin UInt16.size) : (a ^^^ b).val = a.val ^^^ b.val := xor_val_of_two_pow (w := 16) a b
-@[simp] theorem xor_val_of_uInt32Size (a b : Fin UInt32.size) : (a ^^^ b).val = a.val ^^^ b.val := xor_val_of_two_pow (w := 32) a b
-@[simp] theorem xor_val_of_uInt64Size (a b : Fin UInt64.size) : (a ^^^ b).val = a.val ^^^ b.val := xor_val_of_two_pow (w := 64) a b
-@[simp] theorem xor_val_of_uSizeSize (a b : Fin USize.size) : (a ^^^ b).val = a.val ^^^ b.val := xor_val_of_two_pow a b
-
-theorem xor_val (a b : Fin n) : (a ^^^ b).val = (a.val ^^^ b.val) % n := rfl
-
-@[simp] theorem shiftLeft_val (a b : Fin n) : (a <<< b).val = (a.val <<< b.val) % n := rfl
-
-@[simp] theorem shiftRight_val (a b : Fin n) : (a >>> b).val = a.val >>> b.val :=
-  Nat.mod_eq_of_lt (Nat.lt_of_le_of_lt (Nat.shiftRight_le _ _) a.isLt)
-
 end Fin

src/Init/Data/Fin/Fold.lean

@@ -10,26 +10,14 @@ import Init.Data.Fin.Lemmas
 
 namespace Fin
 
-/--
-Combine all the values that can be represented by `Fin n` with an initial value, starting at `0` and
-nesting to the left.
-
-Example:
- * `Fin.foldl 3 (· + ·.val) (0 : Nat) = ((0 + (0 : Fin 3).val) + (1 : Fin 3).val) + (2 : Fin 3).val`
--/
+/-- Folds over `Fin n` from the left: `foldl 3 f x = f (f (f x 0) 1) 2`. -/
 @[inline] def foldl (n) (f : α → Fin n → α) (init : α) : α := loop init 0 where
   /-- Inner loop for `Fin.foldl`. `Fin.foldl.loop n f x i = f (f (f x i) ...) (n-1)`  -/
   @[semireducible, specialize] loop (x : α) (i : Nat) : α :=
     if h : i < n then loop (f x ⟨i, h⟩) (i+1) else x
   termination_by n - i
 
-/--
-Combine all the values that can be represented by `Fin n` with an initial value, starting at `n - 1`
-and nesting to the right.
-
-Example:
- * `Fin.foldr 3 (·.val + ·) (0 : Nat) = (0 : Fin 3).val + ((1 : Fin 3).val + ((2 : Fin 3).val + 0))`
--/
+/-- Folds over `Fin n` from the right: `foldr 3 f x = f 0 (f 1 (f 2 x))`. -/
 @[inline] def foldr (n) (f : Fin n → α → α) (init : α) : α := loop n (Nat.le_refl n) init where
   /-- Inner loop for `Fin.foldr`. `Fin.foldr.loop n f i x = f 0 (f ... (f (i-1) x))`  -/
   @[specialize] loop : (i : _) → i ≤ n → α → α
@@ -38,9 +26,7 @@ Example:
   termination_by structural i => i
 
 /--
-Folds a monadic function over all the values in `Fin n` from left to right, starting with `0`.
-
-It is the sequence of steps:
+Folds a monadic function over `Fin n` from left to right:
 ```
 Fin.foldlM n f x₀ = do
   let x₁ ← f x₀ 0
@@ -67,9 +53,7 @@ Fin.foldlM n f x₀ = do
   decreasing_by decreasing_trivial_pre_omega
 
 /--
-Folds a monadic function over `Fin n` from right to left, starting with `n-1`.
-
-It is the sequence of steps:
+Folds a monadic function over `Fin n` from right to left:
 ```
 Fin.foldrM n f xₙ = do
   let xₙ₋₁ ← f (n-1) xₙ
@@ -141,9 +125,7 @@ theorem foldrM_loop [Monad m] [LawfulMonad m] (f : Fin (n+1) → α → m α) (x
   | zero =>
     rw [foldrM_loop_zero, foldrM_loop_succ, pure_bind]
     conv => rhs; rw [←bind_pure (f 0 x)]
-    congr
-    funext
-    try simp only [foldrM.loop] -- the try makes this proof work with and without opaque wf rec
+    congr; funext
   | succ i ih =>
     rw [foldrM_loop_succ, foldrM_loop_succ, bind_assoc]
     congr; funext; exact ih ..

src/Init/Data/Fin/Iterate.lean

@@ -10,18 +10,10 @@ import Init.Data.Fin.Basic
 namespace Fin
 
 /--
-Applies an index-dependent function `f` to all of the values in `[i:n]`, starting at `i` with an
-initial accumulator `a`.
+`hIterateFrom f i bnd a` applies `f` over indices `[i:n]` to compute `P n`
+from `P i`.
 
-Concretely, `Fin.hIterateFrom P f i a` is equal to
-```lean
-  a |> f i |> f (i + 1) |> ... |> f (n - 1)
-```
-
-Theorems about `Fin.hIterateFrom` can be proven using the general theorem `Fin.hIterateFrom_elim` or
-other more specialized theorems.
-
-`Fin.hIterate` is a variant that always starts at `0`.
+See `hIterate` below for more details.
 -/
 def hIterateFrom (P : Nat → Sort _) {n} (f : ∀(i : Fin n), P i.val → P (i.val+1))
       (i : Nat) (ubnd : i ≤ n) (a : P i) : P n :=
@@ -34,18 +26,20 @@ def hIterateFrom (P : Nat → Sort _) {n} (f : ∀(i : Fin n), P i.val → P (i.
   decreasing_by decreasing_trivial_pre_omega
 
 /--
-Applies an index-dependent function to all the values less than the given bound `n`, starting at
-`0` with an accumulator.
+`hIterate` is a heterogeneous iterative operation that applies a
+index-dependent function `f` to a value `init : P start` a total of
+`stop - start` times to produce a value of type `P stop`.
 
-Concretely, `Fin.hIterate P init f` is equal to
+Concretely, `hIterate start stop f init` is equal to
 ```lean
-  init |> f 0 |> f 1 |> ... |> f (n-1)
+  init |> f start _ |> f (start+1) _ ... |> f (end-1) _
 ```
 
-Theorems about `Fin.hIterate` can be proven using the general theorem `Fin.hIterate_elim` or other more
-specialized theorems.
+Because it is heterogeneous and must return a value of type `P stop`,
+`hIterate` requires proof that `start ≤ stop`.
 
-`Fin.hIterateFrom` is a variant that takes a custom starting value instead of `0`.
+One can prove properties of `hIterate` using the general theorem
+`hIterate_elim` or other more specialized theorems.
  -/
 def hIterate (P : Nat → Sort _) {n : Nat} (init : P 0) (f : ∀(i : Fin n), P i.val → P (i.val+1)) :
     P n :=

src/Init/Data/Fin/Lemmas.lean

@@ -45,7 +45,6 @@ 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
 
@@ -56,14 +55,6 @@ 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
 
@@ -670,20 +661,12 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
 @[simp] theorem natAdd_subNat_cast {i : Fin (n + m)} (h : n ≤ i) :
     natAdd n (subNat n (i.cast (Nat.add_comm ..)) h) = i := by simp [← cast_addNat]
 
-/-! ### Recursion and induction principles -/
+/-! ### recursion and induction principles -/
 
-/--
-An induction principle for `Fin` that considers a given `i : Fin n` as given by a sequence of `i`
-applications of `Fin.succ`.
-
-The cases in the induction are:
- * `zero` demonstrates the motive for `(0 : Fin (n + 1))` for all bounds `n`
- * `succ` demonstrates the motive for `Fin.succ` applied to an arbitrary `Fin` for an arbitrary
-   bound `n`
-
-Unlike `Fin.induction`, the motive quantifies over the bound, and the bound varies at each inductive
-step. `Fin.succRecOn` is a version of this induction principle that takes the `Fin` argument first.
--/
+/-- Define `motive n i` by induction on `i : Fin n` interpreted as `(0 : Fin (n - i)).succ.succ…`.
+This function has two arguments: `zero n` defines `0`-th element `motive (n+1) 0` of an
+`(n+1)`-tuple, and `succ n i` defines `(i+1)`-st element of `(n+1)`-tuple based on `n`, `i`, and
+`i`-th element of `n`-tuple. -/
 -- FIXME: Performance review
 @[elab_as_elim] def succRec {motive : ∀ n, Fin n → Sort _}
     (zero : ∀ n, motive n.succ (0 : Fin (n + 1)))
@@ -692,18 +675,13 @@ step. `Fin.succRecOn` is a version of this induction principle that takes the `F
   | Nat.succ n, ⟨0, _⟩ => by rw [mk_zero]; exact zero n
   | Nat.succ _, ⟨Nat.succ i, h⟩ => succ _ _ (succRec zero succ ⟨i, Nat.lt_of_succ_lt_succ h⟩)
 
-/--
-An induction principle for `Fin` that considers a given `i : Fin n` as given by a sequence of `i`
-applications of `Fin.succ`.
+/-- Define `motive n i` by induction on `i : Fin n` interpreted as `(0 : Fin (n - i)).succ.succ…`.
+This function has two arguments:
+`zero n` defines the `0`-th element `motive (n+1) 0` of an `(n+1)`-tuple, and
+`succ n i` defines the `(i+1)`-st element of an `(n+1)`-tuple based on `n`, `i`,
+and the `i`-th element of an `n`-tuple.
 
-The cases in the induction are:
- * `zero` demonstrates the motive for `(0 : Fin (n + 1))` for all bounds `n`
- * `succ` demonstrates the motive for `Fin.succ` applied to an arbitrary `Fin` for an arbitrary
-   bound `n`
-
-Unlike `Fin.induction`, the motive quantifies over the bound, and the bound varies at each inductive
-step. `Fin.succRec` is a version of this induction principle that takes the `Fin` argument last.
--/
+A version of `Fin.succRec` taking `i : Fin n` as the first argument. -/
 -- FIXME: Performance review
 @[elab_as_elim] def succRecOn {n : Nat} (i : Fin n) {motive : ∀ n, Fin n → Sort _}
     (zero : ∀ n, motive (n + 1) 0) (succ : ∀ n i, motive n i → motive (Nat.succ n) i.succ) :
@@ -718,17 +696,9 @@ step. `Fin.succRec` is a version of this induction principle that takes the `Fin
   cases i; rfl
 
 
-/--
-Proves a statement by induction on the underlying `Nat` value in a `Fin (n + 1)`.
-
-For the induction:
- * `zero` is the base case, demonstrating `motive 0`.
- * `succ` is the inductive step, assuming the motive for `i : Fin n` (lifted to `Fin (n + 1)` with
-   `Fin.castSucc`) and demonstrating it for `i.succ`.
-
-`Fin.inductionOn` is a version of this induction principle that takes the `Fin` as its first
-parameter, `Fin.cases` is the corresponding case analysis operator, and `Fin.reverseInduction` is a
-version that starts at the greatest value instead of `0`.
+/-- Define `motive i` by induction on `i : Fin (n + 1)` via induction on the underlying `Nat` value.
+This function has two arguments: `zero` handles the base case on `motive 0`,
+and `succ` defines the inductive step using `motive i.castSucc`.
 -/
 -- FIXME: Performance review
 @[elab_as_elim] def induction {motive : Fin (n + 1) → Sort _} (zero : motive 0)
@@ -749,30 +719,18 @@ where
     (succ : ∀ i : Fin n, motive (castSucc i) → motive i.succ) (i : Fin n) :
     induction (motive := motive) zero succ i.succ = succ i (induction zero succ (castSucc i)) := rfl
 
-/--
-Proves a statement by induction on the underlying `Nat` value in a `Fin (n + 1)`.
+/-- Define `motive i` by induction on `i : Fin (n + 1)` via induction on the underlying `Nat` value.
+This function has two arguments: `zero` handles the base case on `motive 0`,
+and `succ` defines the inductive step using `motive i.castSucc`.
 
-For the induction:
- * `zero` is the base case, demonstrating `motive 0`.
- * `succ` is the inductive step, assuming the motive for `i : Fin n` (lifted to `Fin (n + 1)` with
-   `Fin.castSucc`) and demonstrating it for `i.succ`.
-
-`Fin.induction` is a version of this induction principle that takes the `Fin` as its last
-parameter.
+A version of `Fin.induction` taking `i : Fin (n + 1)` as the first argument.
 -/
 -- FIXME: Performance review
 @[elab_as_elim] def inductionOn (i : Fin (n + 1)) {motive : Fin (n + 1) → Sort _} (zero : motive 0)
     (succ : ∀ i : Fin n, motive (castSucc i) → motive i.succ) : motive i := induction zero succ i
 
-/--
-Proves a statement by cases on the underlying `Nat` value in a `Fin (n + 1)`.
-
-The two cases are:
- * `zero`, used when the value is of the form `(0 : Fin (n + 1))`
- * `succ`, used when the value is of the form `(j : Fin n).succ`
-
-The corresponding induction principle is `Fin.induction`.
--/
+/-- Define `f : Π i : Fin n.succ, motive i` by separately handling the cases `i = 0` and
+`i = j.succ`, `j : Fin n`. -/
 @[elab_as_elim] def cases {motive : Fin (n + 1) → Sort _}
     (zero : motive 0) (succ : ∀ i : Fin n, motive i.succ) :
     ∀ i : Fin (n + 1), motive i := induction zero fun i _ => succ i
@@ -810,14 +768,9 @@ theorem fin_two_eq_of_eq_zero_iff : ∀ {a b : Fin 2}, (a = 0 ↔ b = 0) → a =
   simp only [forall_fin_two]; decide
 
 /--
-Proves a statement by reverse induction on the underlying `Nat` value in a `Fin (n + 1)`.
-
-For the induction:
- * `last` is the base case, demonstrating `motive (Fin.last n)`.
- * `cast` is the inductive step, assuming the motive for `(j : Fin n).succ` and demonstrating it for
-    the predecessor `j.castSucc`.
-
-`Fin.induction` is the non-reverse induction principle.
+Define `motive i` by reverse induction on `i : Fin (n + 1)` via induction on the underlying `Nat`
+value. This function has two arguments: `last` handles the base case on `motive (Fin.last n)`,
+and `cast` defines the inductive step using `motive i.succ`, inducting downwards.
 -/
 @[elab_as_elim] def reverseInduction {motive : Fin (n + 1) → Sort _} (last : motive (Fin.last n))
     (cast : ∀ i : Fin n, motive i.succ → motive (castSucc i)) (i : Fin (n + 1)) : motive i :=
@@ -840,16 +793,8 @@ decreasing_by decreasing_with
       succ i (reverseInduction zero succ i.succ) := by
   rw [reverseInduction, dif_neg (Fin.ne_of_lt (Fin.castSucc_lt_last i))]; rfl
 
-/--
-Proves a statement by cases on the underlying `Nat` value in a `Fin (n + 1)`, checking whether the
-value is the greatest representable or a predecessor of some other.
-
-The two cases are:
- * `last`, used when the value is `Fin.last n`
- * `cast`, used when the value is of the form `(j : Fin n).succ`
-
-The corresponding induction principle is `Fin.reverseInduction`.
--/
+/-- Define `f : Π i : Fin n.succ, motive i` by separately handling the cases `i = Fin.last n` and
+`i = j.castSucc`, `j : Fin n`. -/
 @[elab_as_elim] def lastCases {n : Nat} {motive : Fin (n + 1) → Sort _} (last : motive (Fin.last n))
     (cast : ∀ i : Fin n, motive (castSucc i)) (i : Fin (n + 1)) : motive i :=
   reverseInduction last (fun i _ => cast i) i
@@ -862,16 +807,8 @@ The corresponding induction principle is `Fin.reverseInduction`.
     (i : Fin n) : (Fin.lastCases last cast (Fin.castSucc i) : motive (Fin.castSucc i)) = cast i :=
   reverseInduction_castSucc ..
 
-/--
-A case analysis operator for `i : Fin (m + n)` that separately handles the cases where `i < m` and
-where `m ≤ i < m + n`.
-
-The first case, where `i < m`, is handled by `left`. In this case, `i` can be represented as
-`Fin.castAdd n (j : Fin m)`.
-
-The second case, where `m ≤ i < m + n`, is handled by `right`. In this case, `i` can be represented
-as `Fin.natAdd m (j : Fin n)`.
--/
+/-- Define `f : Π i : Fin (m + n), motive i` by separately handling the cases `i = castAdd n i`,
+`j : Fin m` and `i = natAdd m j`, `j : Fin n`. -/
 @[elab_as_elim] def addCases {m n : Nat} {motive : Fin (m + n) → Sort u}
     (left : ∀ i, motive (castAdd n i)) (right : ∀ i, motive (natAdd m i))
     (i : Fin (m + n)) : motive i :=

src/Init/Data/Fin/Log2.lean

@@ -6,20 +6,4 @@ Authors: Henrik Böving
 prelude
 import Init.Data.Nat.Log2
 
-set_option linter.missingDocs true
-
-/--
-Logarithm base 2 for bounded numbers.
-
-The resulting value is the same as that computed by `Nat.log2`. In particular, the result for `0` is
-`0`.
-
-Examples:
- * `(8 : Fin 10).log2 = (3 : Fin 10)`
- * `(7 : Fin 10).log2 = (2 : Fin 10)`
- * `(4 : Fin 10).log2 = (2 : Fin 10)`
- * `(3 : Fin 10).log2 = (1 : Fin 10)`
- * `(1 : Fin 10).log2 = (0 : Fin 10)`
- * `(0 : Fin 10).log2 = (0 : Fin 10)`
--/
 def Fin.log2 (n : Fin m) : Fin m := ⟨Nat.log2 n.val, Nat.lt_of_le_of_lt (Nat.log2_le_self n.val) n.isLt⟩

src/Init/Data/Float.lean

@@ -134,22 +134,7 @@ 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

@@ -127,25 +127,7 @@ 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

@@ -17,14 +17,11 @@ attribute [extern "lean_float_array_data"] FloatArray.data
 
 namespace FloatArray
 @[extern "lean_mk_empty_float_array"]
-def emptyWithCapacity (c : @& Nat) : FloatArray :=
+def mkEmpty (c : @& Nat) : FloatArray :=
   { data := #[] }
 
-@[deprecated emptyWithCapacity (since := "2025-03-12")]
-abbrev mkEmpty := emptyWithCapacity
-
 def empty : FloatArray :=
-  emptyWithCapacity 0
+  mkEmpty 0
 
 instance : Inhabited FloatArray where
   default := empty
@@ -50,11 +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[i]
+  | ⟨ds⟩, i, h => ds.get i h
 
 @[extern "lean_float_array_get"]
 def get! : (@& FloatArray) → (@& Nat) → Float
-  | ⟨ds⟩, i => ds[i]!
+  | ⟨ds⟩, i => ds.get! i
 
 def get? (ds : FloatArray) (i : Nat) : Option Float :=
   if h : i < ds.size then
@@ -65,7 +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.toNat < xs.size where
+instance : GetElem FloatArray USize Float fun xs i => i.val < xs.size where
   getElem xs i h := xs.uget i h
 
 @[extern "lean_float_array_uset"]
@@ -167,9 +164,6 @@ def foldl {β : Type v} (f : β → Float → β) (init : β) (as : FloatArray)
 
 end FloatArray
 
-/--
-Converts a list of floats into a `FloatArray`.
--/
 def List.toFloatArray (ds : List Float) : FloatArray :=
   let rec loop
     | [],    r => r

src/Init/Data/Format/Instances.lean

@@ -23,12 +23,6 @@ instance [ToFormat α] : ToFormat (List α) where
 instance [ToFormat α] : ToFormat (Array α) where
   format a := "#" ++ format a.toList
 
-/--
-Formats an optional value, with no expectation that the Lean parser should be able to parse the
-result.
-
-This function is usually accessed through the `ToFormat (Option α)` instance.
--/
 def Option.format {α : Type u} [ToFormat α] : Option α → Format
   | none   => "none"
   | some a => "some " ++ Std.format a
@@ -39,10 +33,6 @@ instance {α : Type u} [ToFormat α] : ToFormat (Option α) :=
 instance {α : Type u} {β : Type v} [ToFormat α] [ToFormat β] : ToFormat (Prod α β) where
   format := fun (a, b) => Format.paren <| format a ++ "," ++ Format.line ++ format b
 
-/--
-Converts a string to a pretty-printer document, replacing newlines in the string with
-`Std.Format.line`.
--/
 def String.toFormat (s : String) : Std.Format :=
   Std.Format.joinSep (s.splitOn "\n") Std.Format.line
 

src/Init/Data/Function.lean

@@ -9,23 +9,10 @@ import Init.Core
 
 namespace Function
 
-/--
-Transforms a function from pairs into an equivalent two-parameter function.
-
-Examples:
- * `Function.curry (fun (x, y) => x + y) 3 5 = 8`
- * `Function.curry Prod.swap 3 "five" = ("five", 3)`
--/
 @[inline]
 def curry : (α × β → φ) → α → β → φ := fun f a b => f (a, b)
 
-/--
-Transforms a two-parameter function into an equivalent function from pairs.
-
-Examples:
- * `Function.uncurry List.drop (1, ["a", "b", "c"]) = ["b", "c"]`
- * `[("orange", 2), ("android", 3) ].map (Function.uncurry String.take) = ["or", "and"]`
--/
+/-- Interpret a function with two arguments as a function on `α × β` -/
 @[inline]
 def uncurry : (α → β → φ) → α × β → φ := fun f a => f a.1 a.2
 

src/Init/Data/Hashable.lean

@@ -75,19 +75,13 @@ instance (P : Prop) : Hashable P where
 @[always_inline, inline] def hash64 (u : UInt64) : UInt64 :=
   mixHash u 11
 
-/--
-The `BEq α` and `Hashable α` instances on `α` are compatible. This means that that `a == b` implies
-`hash a = hash b`.
-
-This is automatic if the `BEq` instance is lawful.
+/-- `LawfulHashable α` says that the `BEq α` and `Hashable α` instances on `α` are compatible, i.e.,
+that `a == b` implies `hash a = hash b`. This is automatic if the `BEq` instance is lawful.
 -/
 class LawfulHashable (α : Type u) [BEq α] [Hashable α] where
   /-- If `a == b`, then `hash a = hash b`. -/
   hash_eq (a b : α) : a == b → hash a = hash b
 
-/--
-A lawful hash function respects its Boolean equality test.
--/
 theorem hash_eq [BEq α] [Hashable α] [LawfulHashable α] {a b : α} : a == b → hash a = hash b :=
   LawfulHashable.hash_eq a b
 

src/Init/Data/Int.lean

@@ -7,11 +7,10 @@ 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

@@ -17,12 +17,10 @@ open Nat
 This file defines the `Int` type as well as
 
 * coercions, conversions, and compatibility with numeric literals,
-* basic arithmetic operations add/sub/mul/pow,
+* basic arithmetic operations add/sub/mul/div/mod/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`.
 -/
 
 /--
@@ -293,16 +291,13 @@ def toNat : Int → Nat
   | negSucc _ => 0
 
 /--
-* If `n : Nat`, then `Int.toNat? n = some n`
-* If `n : Int` is negative, then `Int.toNat? n = none`.
+* If `n : Nat`, then `int.toNat' n = some n`
+* If `n : Int` is negative, then `int.toNat' n = none`.
 -/
-def toNat? : Int → Option Nat
+def toNat' : Int → Option Nat
   | (n : Nat) => some n
   | -[_+1] => none
 
-@[deprecated toNat? (since := "2025-03-11"), inherit_doc toNat?]
-abbrev toNat' := toNat?
-
 /-! ## divisibility -/
 
 /--

src/Init/Data/Int/Bitwise.lean

@@ -1,8 +1,50 @@
 /-
-Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Copyright (c) 2022 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Morrison
+Authors: Mario Carneiro
 -/
 prelude
-import Init.Data.Int.Bitwise.Basic
-import Init.Data.Int.Bitwise.Lemmas
+import Init.Data.Int.Basic
+import Init.Data.Nat.Bitwise.Basic
+
+namespace Int
+
+/-! ## bit operations -/
+
+/--
+Bitwise not
+
+Interprets the integer as an infinite sequence of bits in two's complement
+and complements each bit.
+```
+~~~(0:Int) = -1
+~~~(1:Int) = -2
+~~~(-1:Int) = 0
+```
+-/
+protected def not : Int -> Int
+  | Int.ofNat n => Int.negSucc n
+  | Int.negSucc n => Int.ofNat n
+
+instance : Complement Int := ⟨.not⟩
+
+/--
+Bitwise shift right.
+
+Conceptually, this treats the integer as an infinite sequence of bits in two's
+complement and shifts the value to the right.
+
+```lean
+( 0b0111:Int) >>> 1 =  0b0011
+( 0b1000:Int) >>> 1 =  0b0100
+(-0b1000:Int) >>> 1 = -0b0100
+(-0b0111:Int) >>> 1 = -0b0100
+```
+-/
+protected def shiftRight : Int → Nat → Int
+  | Int.ofNat n, s => Int.ofNat (n >>> s)
+  | Int.negSucc n, s => Int.negSucc (n >>> s)
+
+instance : HShiftRight Int Nat Int := ⟨.shiftRight⟩
+
+end Int

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

@@ -1,50 +0,0 @@
-/-
-Copyright (c) 2022 Mario Carneiro. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro
--/
-prelude
-import Init.Data.Int.Basic
-import Init.Data.Nat.Bitwise.Basic
-
-namespace Int
-
-/-! ## bit operations -/
-
-/--
-Bitwise not
-
-Interprets the integer as an infinite sequence of bits in two's complement
-and complements each bit.
-```
-~~~(0:Int) = -1
-~~~(1:Int) = -2
-~~~(-1:Int) = 0
-```
--/
-protected def not : Int → Int
-  | Int.ofNat n => Int.negSucc n
-  | Int.negSucc n => Int.ofNat n
-
-instance : Complement Int := ⟨.not⟩
-
-/--
-Bitwise shift right.
-
-Conceptually, this treats the integer as an infinite sequence of bits in two's
-complement and shifts the value to the right.
-
-```lean
-( 0b0111:Int) >>> 1 =  0b0011
-( 0b1000:Int) >>> 1 =  0b0100
-(-0b1000:Int) >>> 1 = -0b0100
-(-0b0111:Int) >>> 1 = -0b0100
-```
--/
-protected def shiftRight : Int → Nat → Int
-  | Int.ofNat n, s => Int.ofNat (n >>> s)
-  | Int.negSucc n, s => Int.negSucc (n >>> s)
-
-instance : HShiftRight Int Nat Int := ⟨.shiftRight⟩
-
-end Int

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

@@ -5,13 +5,11 @@ Authors: Siddharth Bhat, Jeremy Avigad
 -/
 prelude
 import Init.Data.Nat.Bitwise.Lemmas
-import Init.Data.Int.Bitwise.Basic
-import Init.Data.Int.DivMod.Lemmas
+import Init.Data.Int.Bitwise
 
 namespace Int
 
 theorem shiftRight_eq (n : Int) (s : Nat) : n >>> s = Int.shiftRight n s := rfl
-
 @[simp]
 theorem natCast_shiftRight (n s : Nat) : (n : Int) >>> s = n >>> s := rfl
 
@@ -28,7 +26,7 @@ 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; norm_cast
+  · simp
   · rw [negSucc_ediv _ (by norm_cast; exact Nat.pow_pos (Nat.zero_lt_two))]
     rfl
 
@@ -40,47 +38,4 @@ 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.DivMod.Lemmas
+import Init.Data.Int.DivModLemmas
 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
-  simp [h₁]
+  simpa [h₁] using Int.ofNat_nonneg _
 
 /-- `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) :
@@ -227,4 +227,33 @@ theorem cooper_resolution_dvd_right
     · exact Int.mul_neg _ _ ▸ Int.neg_le_of_neg_le lower
     · exact Int.mul_neg _ _ ▸ Int.neg_mul _ _ ▸ dvd
 
-end Int
+/--
+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

src/Init/Data/Int/DivMod.lean

@@ -1,9 +1,328 @@
 /-
-Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Copyright (c) 2016 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Morrison
+Authors: Jeremy Avigad, Mario Carneiro
 -/
 prelude
-import Init.Data.Int.DivMod.Basic
-import Init.Data.Int.DivMod.Bootstrap
-import Init.Data.Int.DivMod.Lemmas
+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

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

@@ -1,344 +0,0 @@
-/-
-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

@@ -1,333 +0,0 @@
-/-
-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

@@ -7,14 +7,10 @@ prelude
 import Init.Data.Int.Basic
 import Init.Data.Nat.Gcd
 import Init.Data.Nat.Lcm
-import Init.Data.Int.DivMod.Lemmas
+import Init.Data.Int.DivModLemmas
 
 /-!
 Definition and lemmas for gcd and lcm over Int
-
-## Future work
-Most of the material about `Nat.gcd` and `Nat.lcm` from `Init.Data.Nat.Gcd` and `Init.Data.Nat.Lcm`
-has analogues for `Int.gcd` and `Int.lcm` that should be added to this file.
 -/
 namespace Int
 

src/Init/Data/Int/Lemmas.lean

@@ -25,32 +25,31 @@ theorem subNatNat_of_sub_eq_succ {m n k : Nat} (h : n - m = succ k) : subNatNat
 
 @[norm_cast] theorem ofNat_add (n m : Nat) : (↑(n + m) : Int) = n + m := rfl
 @[norm_cast] theorem ofNat_mul (n m : Nat) : (↑(n * m) : Int) = n * m := rfl
-@[norm_cast] theorem ofNat_succ (n : Nat) : (succ n : Int) = n + 1 := rfl
+theorem ofNat_succ (n : Nat) : (succ n : Int) = n + 1 := rfl
 
-theorem neg_ofNat_zero : -((0 : Nat) : Int) = 0 := rfl
-theorem neg_ofNat_succ (n : Nat) : -(succ n : Int) = -[n+1] := rfl
-theorem neg_negSucc (n : Nat) : -(-[n+1]) = succ n := rfl
+@[local simp] theorem neg_ofNat_zero : -((0 : Nat) : Int) = 0 := rfl
+@[local simp] theorem neg_ofNat_succ (n : Nat) : -(succ n : Int) = -[n+1] := rfl
+@[local simp] theorem neg_negSucc (n : Nat) : -(-[n+1]) = succ n := rfl
+
+theorem negSucc_coe (n : Nat) : -[n+1] = -↑(n + 1) := rfl
 
 theorem negOfNat_eq : negOfNat n = -ofNat n := rfl
 
+/-! ## These are only for internal use -/
+
 @[simp] theorem add_def {a b : Int} : Int.add a b = a + b := rfl
-@[simp] theorem mul_def {a b : Int} : Int.mul a b = a * b := rfl
-
-/-!
-## These are only for internal use
-
-Ideally these could all be made private, but they are used in downstream libraries.
--/
 
 @[local simp] theorem ofNat_add_ofNat (m n : Nat) : (↑m + ↑n : Int) = ↑(m + n) := rfl
 @[local simp] theorem ofNat_add_negSucc (m n : Nat) : ↑m + -[n+1] = subNatNat m (succ n) := rfl
 @[local simp] theorem negSucc_add_ofNat (m n : Nat) : -[m+1] + ↑n = subNatNat n (succ m) := rfl
 @[local simp] theorem negSucc_add_negSucc (m n : Nat) : -[m+1] + -[n+1] = -[succ (m + n) +1] := rfl
 
+@[simp] theorem mul_def {a b : Int} : Int.mul a b = a * b := rfl
+
 @[local simp] theorem ofNat_mul_ofNat (m n : Nat) : (↑m * ↑n : Int) = ↑(m * n) := rfl
-@[local simp] private theorem ofNat_mul_negSucc' (m n : Nat) : ↑m * -[n+1] = negOfNat (m * succ n) := rfl
-@[local simp] private theorem negSucc_mul_ofNat' (m n : Nat) : -[m+1] * ↑n = negOfNat (succ m * n) := rfl
-@[local simp] private theorem negSucc_mul_negSucc' (m n : Nat) :
+@[local simp] theorem ofNat_mul_negSucc' (m n : Nat) : ↑m * -[n+1] = negOfNat (m * succ n) := rfl
+@[local simp] theorem negSucc_mul_ofNat' (m n : Nat) : -[m+1] * ↑n = negOfNat (succ m * n) := rfl
+@[local simp] theorem negSucc_mul_negSucc' (m n : Nat) :
     -[m+1] * -[n+1] = ofNat (succ m * succ n) := rfl
 
 /- ## some basic functions and properties -/
@@ -65,14 +64,11 @@ theorem negSucc_inj : negSucc m = negSucc n ↔ m = n := ⟨negSucc.inj, fun H =
 
 theorem negSucc_eq (n : Nat) : -[n+1] = -((n : Int) + 1) := rfl
 
-@[deprecated negSucc_eq (since := "2025-03-11")]
-theorem negSucc_coe (n : Nat) : -[n+1] = -↑(n + 1) := rfl
-
 @[simp] theorem negSucc_ne_zero (n : Nat) : -[n+1] ≠ 0 := nofun
 
 @[simp] theorem zero_ne_negSucc (n : Nat) : 0 ≠ -[n+1] := nofun
 
-@[simp, norm_cast] theorem cast_ofNat_Int :
+@[simp, norm_cast] theorem Nat.cast_ofNat_Int :
   (Nat.cast (no_index (OfNat.ofNat n)) : Int) = OfNat.ofNat n := rfl
 
 /- ## neg -/
@@ -82,7 +78,7 @@ theorem negSucc_coe (n : Nat) : -[n+1] = -↑(n + 1) := rfl
   | succ _ => rfl
   | -[_+1] => rfl
 
-@[simp] protected theorem neg_inj {a b : Int} : -a = -b ↔ a = b :=
+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)
@@ -90,13 +86,12 @@ theorem negSucc_coe (n : Nat) : -[n+1] = -↑(n + 1) := rfl
 protected theorem neg_ne_zero : -a ≠ 0 ↔ a ≠ 0 := not_congr Int.neg_eq_zero
 
 protected theorem sub_eq_add_neg {a b : Int} : a - b = a + -b := rfl
-protected theorem add_neg_eq_sub {a b : Int} : a + -b = a - b := rfl
 
 theorem add_neg_one (i : Int) : i + -1 = i - 1 := rfl
 
 /- ## basic properties of subNatNat -/
 
-@[elab_as_elim]
+-- @[elabAsElim] -- TODO(Mario): unexpected eliminator resulting type
 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]) :
@@ -134,16 +129,28 @@ 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 -/
+
+protected theorem add_comm : ∀ a b : Int, a + b = b + a
+  | ofNat n, ofNat m => by simp [Nat.add_comm]
+  | ofNat _, -[_+1]  => rfl
+  | -[_+1],  ofNat _ => rfl
+  | -[_+1],  -[_+1]  => by simp [Nat.add_comm]
+instance : Std.Commutative (α := Int) (· + ·) := ⟨Int.add_comm⟩
+
+@[simp] protected theorem add_zero : ∀ a : Int, a + 0 = a
+  | ofNat _ => rfl
+  | -[_+1]  => rfl
+
+@[simp] protected theorem zero_add (a : Int) : 0 + a = a := Int.add_comm .. ▸ a.add_zero
+instance : Std.LawfulIdentity (α := Int) (· + ·) 0 where
+  left_id := Int.zero_add
+  right_id := Int.add_zero
+
+theorem ofNat_add_negSucc_of_lt (h : m < n.succ) : ofNat m + -[n+1] = -[n - m+1] :=
+  show subNatNat .. = _ by simp [succ_sub (le_of_lt_succ h), subNatNat]
 
 theorem subNatNat_sub (h : n ≤ m) (k : Nat) : subNatNat (m - n) k = subNatNat m (k + n) := by
   rwa [← subNatNat_add_add _ _ n, Nat.sub_add_cancel]
@@ -173,34 +180,6 @@ theorem subNatNat_add_negSucc (m n k : Nat) :
       ← Nat.add_assoc, succ_sub_succ_eq_sub, Nat.add_comm n,Nat.add_sub_assoc (Nat.le_of_lt h'),
       Nat.add_comm]
 
-theorem subNatNat_self : ∀ n, subNatNat n n = 0
-  | 0      => rfl
-  | succ m => by rw [subNatNat_of_sub_eq_zero (Nat.sub_self ..), Nat.sub_self, ofNat_zero]
-
-/- # Additive group properties -/
-
-/- addition -/
-
-protected theorem add_comm : ∀ a b : Int, a + b = b + a
-  | ofNat n, ofNat m => by simp [Nat.add_comm]
-  | ofNat _, -[_+1]  => rfl
-  | -[_+1],  ofNat _ => rfl
-  | -[_+1],  -[_+1]  => by simp [Nat.add_comm]
-
-instance : Std.Commutative (α := Int) (· + ·) := ⟨Int.add_comm⟩
-
-@[simp] protected theorem add_zero : ∀ a : Int, a + 0 = a
-  | ofNat _ => rfl
-  | -[_+1]  => rfl
-
-@[simp] protected theorem zero_add (a : Int) : 0 + a = a := Int.add_comm .. ▸ a.add_zero
-instance : Std.LawfulIdentity (α := Int) (· + ·) 0 where
-  left_id := Int.zero_add
-  right_id := Int.add_zero
-
-theorem ofNat_add_negSucc_of_lt (h : m < n.succ) : ofNat m + -[n+1] = -[n - m+1] :=
-  show subNatNat .. = _ by simp [succ_sub (le_of_lt_succ h), subNatNat]
-
 protected theorem add_assoc : ∀ a b c : Int, a + b + c = a + (b + c)
   | (m:Nat), (n:Nat), _ => aux1 ..
   | Nat.cast m, b, Nat.cast k => by
@@ -232,15 +211,21 @@ protected theorem add_right_comm (a b c : Int) : a + b + c = a + c + b := by
 
 /- ## negation -/
 
-protected theorem add_left_neg : ∀ a : Int, -a + a = 0
+theorem subNatNat_self : ∀ n, subNatNat n n = 0
   | 0      => rfl
-  | succ m => by simp [neg_ofNat_succ]
-  | -[m+1] => by simp [neg_negSucc]
+  | succ m => by rw [subNatNat_of_sub_eq_zero (Nat.sub_self ..), Nat.sub_self, ofNat_zero]
 
-protected theorem add_right_neg (a : Int) : a + -a = 0 := by
+attribute [local simp] subNatNat_self
+
+@[local simp] protected theorem add_left_neg : ∀ a : Int, -a + a = 0
+  | 0      => rfl
+  | succ m => by simp
+  | -[m+1] => by simp
+
+@[local simp] protected theorem add_right_neg (a : Int) : a + -a = 0 := by
   rw [Int.add_comm, Int.add_left_neg]
 
-protected theorem neg_eq_of_add_eq_zero {a b : Int} (h : a + b = 0) : -a = b := by
+@[simp] 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
@@ -268,22 +253,11 @@ protected theorem add_left_cancel {a b c : Int} (h : a + b = a + c) : b = c := b
   have h₁ : -a + (a + b) = -a + (a + c) := by rw [h]
   simp [← Int.add_assoc, Int.add_left_neg, Int.zero_add] at h₁; exact h₁
 
-protected theorem neg_add {a b : Int} : -(a + b) = -a + -b := by
+@[local simp] protected theorem neg_add {a b : Int} : -(a + b) = -a + -b := by
   apply Int.add_left_cancel (a := a + 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 : ∀ i, P i ↔ P (-i)) (w : ∀ n : Nat, P n) (i : Int) : P i := by
-  cases i 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
@@ -307,13 +281,13 @@ protected theorem sub_eq_zero {a b : Int} : a - b = 0 ↔ a = b :=
   ⟨Int.eq_of_sub_eq_zero, Int.sub_eq_zero_of_eq⟩
 
 protected theorem sub_sub (a b c : Int) : a - b - c = a - (b + c) := by
-  simp [Int.sub_eq_add_neg, Int.add_assoc, Int.neg_add]
+  simp [Int.sub_eq_add_neg, Int.add_assoc]
 
 protected theorem neg_sub (a b : Int) : -(a - b) = b - a := by
-  simp [Int.sub_eq_add_neg, Int.add_comm, Int.neg_add]
+  simp [Int.sub_eq_add_neg, Int.add_comm]
 
 protected theorem sub_sub_self (a b : Int) : a - (a - b) = b := by
-  simp [Int.sub_eq_add_neg, ← Int.add_assoc, Int.neg_add, Int.add_right_neg]
+  simp [Int.sub_eq_add_neg, ← Int.add_assoc]
 
 @[simp] protected theorem sub_neg (a b : Int) : a - -b = a + b := by simp [Int.sub_eq_add_neg]
 
@@ -324,7 +298,7 @@ protected theorem sub_sub_self (a b : Int) : a - (a - b) = b := by
   Int.add_neg_cancel_right a b
 
 protected theorem add_sub_assoc (a b c : Int) : a + b - c = a + (b - c) := by
-  rw [Int.sub_eq_add_neg, Int.add_assoc, Int.add_neg_eq_sub]
+  rw [Int.sub_eq_add_neg, Int.add_assoc, ← Int.sub_eq_add_neg]
 
 @[norm_cast] theorem ofNat_sub (h : m ≤ n) : ((n - m : Nat) : Int) = n - m := by
   match m with
@@ -333,7 +307,6 @@ protected theorem add_sub_assoc (a b c : Int) : a + b - c = a + (b - c) := by
     show ofNat (n - succ m) = subNatNat n (succ m)
     rw [subNatNat, Nat.sub_eq_zero_of_le h]
 
-@[deprecated negSucc_eq (since := "2025-03-11")]
 theorem negSucc_coe' (n : Nat) : -[n+1] = -↑n - 1 := by
   rw [Int.sub_eq_add_neg, ← Int.neg_add]; rfl
 
@@ -343,64 +316,44 @@ protected theorem subNatNat_eq_coe {m n : Nat} : subNatNat m n = ↑m - ↑n :=
     rw [Int.ofNat_add, Int.sub_eq_add_neg, Int.add_assoc, Int.add_left_comm,
       Int.add_right_neg, Int.add_zero]
   · intros i n
-    simp only [negSucc_eq, ofNat_add, ofNat_one, Int.sub_eq_add_neg, Int.neg_add, ← Int.add_assoc]
-    rw [Int.add_neg_eq_sub (a := n), ← ofNat_sub, Nat.sub_self, ofNat_zero, Int.zero_add]
+    simp only [negSucc_coe, ofNat_add, Int.sub_eq_add_neg, Int.neg_add, ← Int.add_assoc]
+    rw [← @Int.sub_eq_add_neg n, ← ofNat_sub, Nat.sub_self, ofNat_zero, Int.zero_add]
     apply Nat.le_refl
 
-@[simp] theorem toNat_sub (m n : Nat) : toNat (m - n) = m - n := by
+theorem toNat_sub (m n : Nat) : toNat (m - n) = m - n := by
   rw [← Int.subNatNat_eq_coe]
   refine subNatNat_elim m n (fun m n i => toNat i = m - n) (fun i n => ?_) (fun i n => ?_)
   · 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 negSucc_eq_neg_ofNat_iff {a b : Nat} : -[a+1] = - (b : Int) ↔ a + 1 = b := by
-  rw [eq_comm, neg_ofNat_eq_negSucc_iff, 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]
-
-@[simp] theorem negSucc_add_one_eq_neg_ofNat_iff {a b : Nat} : -[a+1] + 1 = - (b : Int) ↔ a = b := by
-  rw [eq_comm, neg_ofNat_eq_negSucc_add_one_iff, eq_comm]
-
 /- ## add/sub injectivity -/
 
-@[simp] protected theorem add_left_inj {i j : Int} (k : Int) : (i + k = j + k) ↔ i = j := by
+@[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, Int.add_left_inj]
+@[simp]
+protected theorem add_right_inj {i j : Int} (k : Int) : (k + i = k + j) ↔ i = j := by
+  simp [Int.add_comm k]
 
-@[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, Int.add_right_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] protected theorem sub_left_inj {i j : Int} (k : Int) : (i - k = j - k) ↔ i = j := by
-  simp [Int.sub_eq_add_neg, Int.add_left_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]
 
 /- ## Ring properties -/
 
-theorem ofNat_mul_negSucc (m n : Nat) : (m : Int) * -[n+1] = -↑(m * succ n) := rfl
+@[simp] theorem ofNat_mul_negSucc (m n : Nat) : (m : Int) * -[n+1] = -↑(m * succ n) := rfl
 
-theorem negSucc_mul_ofNat (m n : Nat) : -[m+1] * n = -↑(succ m * n) := rfl
+@[simp] theorem negSucc_mul_ofNat (m n : Nat) : -[m+1] * n = -↑(succ m * n) := rfl
 
-theorem negSucc_mul_negSucc (m n : Nat) : -[m+1] * -[n+1] = succ m * succ n := rfl
+@[simp] theorem negSucc_mul_negSucc (m n : Nat) : -[m+1] * -[n+1] = succ m * succ n := rfl
 
 protected theorem mul_comm (a b : Int) : a * b = b * a := by
   cases a <;> cases b <;> simp [Nat.mul_comm]
@@ -418,10 +371,11 @@ theorem negSucc_mul_negOfNat (m n : Nat) : -[m+1] * negOfNat n = ofNat (succ m *
 theorem negOfNat_mul_negSucc (m n : Nat) : negOfNat n * -[m+1] = ofNat (n * succ m) := by
   rw [Int.mul_comm, negSucc_mul_negOfNat, Nat.mul_comm]
 
-protected theorem mul_assoc (a b c : Int) : a * b * c = a * (b * c) := by
-  cases a <;> cases b <;> cases c <;>
-    simp [Nat.mul_assoc, ofNat_mul_negOfNat, negOfNat_mul_ofNat, negSucc_mul_negOfNat, negOfNat_mul_negSucc]
+attribute [local simp] ofNat_mul_negOfNat negOfNat_mul_ofNat
+  negSucc_mul_negOfNat negOfNat_mul_negSucc
 
+protected theorem mul_assoc (a b c : Int) : a * b * c = a * (b * c) := by
+  cases a <;> cases b <;> cases c <;> simp [Nat.mul_assoc]
 instance : Std.Associative (α := Int) (· * ·) := ⟨Int.mul_assoc⟩
 
 protected theorem mul_left_comm (a b c : Int) : a * (b * c) = b * (a * c) := by
@@ -439,7 +393,7 @@ theorem negOfNat_eq_subNatNat_zero (n) : negOfNat n = subNatNat 0 n := by cases
 theorem ofNat_mul_subNatNat (m n k : Nat) :
     m * subNatNat n k = subNatNat (m * n) (m * k) := by
   cases m with
-  | zero => simp [ofNat_zero, Int.zero_mul, Nat.zero_mul, subNatNat_self]
+  | zero => simp [ofNat_zero, Int.zero_mul, Nat.zero_mul]
   | succ m => cases n.lt_or_ge k with
     | inl h =>
       have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)
@@ -467,7 +421,8 @@ theorem negSucc_mul_subNatNat (m n k : Nat) :
         Nat.mul_sub_left_distrib, ← succ_pred_eq_of_pos (Nat.sub_pos_of_lt h₁)]; rfl
     | inr h' => rw [Nat.le_antisymm h h', subNatNat_self, subNatNat_self, Int.mul_zero]
 
-attribute [local simp] ofNat_mul_subNatNat negOfNat_add negSucc_mul_subNatNat in
+attribute [local simp] ofNat_mul_subNatNat negOfNat_add negSucc_mul_subNatNat
+
 protected theorem mul_add : ∀ a b c : Int, a * (b + c) = a * b + a * c
   | (m:Nat), (n:Nat), (k:Nat) => by simp [Nat.left_distrib]
   | (m:Nat), (n:Nat), -[k+1]  => by
@@ -490,23 +445,21 @@ protected theorem neg_mul_eq_neg_mul (a b : Int) : -(a * b) = -a * b :=
 protected theorem neg_mul_eq_mul_neg (a b : Int) : -(a * b) = a * -b :=
   Int.neg_eq_of_add_eq_zero <| by rw [← Int.mul_add, Int.add_right_neg, Int.mul_zero]
 
--- Note, this is not a `@[simp]` lemma because it interferes with normalization in `simp +arith`.
-protected theorem neg_mul (a b : Int) : -a * b = -(a * b) :=
+@[simp] protected theorem neg_mul (a b : Int) : -a * b = -(a * b) :=
   (Int.neg_mul_eq_neg_mul a b).symm
 
--- Note, this is not a `@[simp]` lemma because it interferes with normalization in `simp +arith`.
-protected theorem mul_neg (a b : Int) : a * -b = -(a * b) :=
+@[simp] protected theorem mul_neg (a b : Int) : a * -b = -(a * b) :=
   (Int.neg_mul_eq_mul_neg a b).symm
 
-protected theorem neg_mul_neg (a b : Int) : -a * -b = a * b := by simp [Int.neg_mul, Int.mul_neg]
+protected theorem neg_mul_neg (a b : Int) : -a * -b = a * b := by simp
 
-protected theorem neg_mul_comm (a b : Int) : -a * b = a * -b := by simp [Int.neg_mul, Int.mul_neg]
+protected theorem neg_mul_comm (a b : Int) : -a * b = a * -b := by simp
 
 protected theorem mul_sub (a b c : Int) : a * (b - c) = a * b - a * c := by
-  simp [Int.sub_eq_add_neg, Int.mul_add, Int.mul_neg]
+  simp [Int.sub_eq_add_neg, Int.mul_add]
 
 protected theorem sub_mul (a b c : Int) : (a - b) * c = a * c - b * c := by
-  simp [Int.sub_eq_add_neg, Int.add_mul, Int.neg_mul]
+  simp [Int.sub_eq_add_neg, Int.add_mul]
 
 @[simp] protected theorem one_mul : ∀ a : Int, 1 * a = a
   | ofNat n => show ofNat (1 * n) = ofNat n by rw [Nat.one_mul]
@@ -517,9 +470,7 @@ instance : Std.LawfulIdentity (α := Int) (· * ·) 1 where
   left_id := Int.one_mul
   right_id := Int.mul_one
 
-@[simp] protected theorem mul_neg_one (a : Int) : a * -1 = -a := by rw [Int.mul_neg, Int.mul_one]
-
-@[simp] protected theorem neg_one_mul (a : Int) : -1 * a = -a := by rw [Int.neg_mul, Int.one_mul]
+protected theorem mul_neg_one (a : Int) : a * -1 = -a := by rw [Int.mul_neg, Int.mul_one]
 
 protected theorem neg_eq_neg_one_mul : ∀ a : Int, -a = -1 * a
   | 0      => rfl
@@ -568,18 +519,16 @@ The following lemmas are later subsumed by e.g. `Nat.cast_add` and `Nat.cast_mul
 but it is convenient to have these earlier, for users who only need `Nat` and `Int`.
 -/
 
-protected theorem natCast_zero : ((0 : Nat) : Int) = (0 : Int) := rfl
+theorem natCast_zero : ((0 : Nat) : Int) = (0 : Int) := rfl
 
-protected theorem natCast_one : ((1 : Nat) : Int) = (1 : Int) := rfl
+theorem natCast_one : ((1 : Nat) : Int) = (1 : Int) := rfl
 
-@[simp] protected theorem natCast_add (a b : Nat) : ((a + b : Nat) : Int) = (a : Int) + (b : Int) := by
+@[simp] theorem natCast_add (a b : Nat) : ((a + b : Nat) : Int) = (a : Int) + (b : Int) := by
   -- Note this only works because of local simp attributes in this file,
   -- so it still makes sense to tag the lemmas with `@[simp]`.
   simp
 
-protected theorem natCast_succ (n : Nat) : ((n + 1 : Nat) : Int) = (n : Int) + 1 := rfl
-
-@[simp] protected theorem natCast_mul (a b : Nat) : ((a * b : Nat) : Int) = (a : Int) * (b : Int) := by
+@[simp] theorem natCast_mul (a b : Nat) : ((a * b : Nat) : Int) = (a : Int) * (b : Int) := by
   simp
 
 end Int

src/Init/Data/Int/LemmasAux.lean

@@ -5,7 +5,6 @@ Authors: Kim Morrison
 -/
 prelude
 import Init.Data.Int.Order
-import Init.Data.Int.DivMod.Lemmas
 import Init.Omega
 
 
@@ -15,32 +14,6 @@ import Init.Omega
 
 namespace Int
 
-/-! ### miscellaneous lemmas -/
-
-@[simp] theorem natCast_le_zero : {n : Nat} → (n : Int) ≤ 0 ↔ n = 0 := by 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
-
-@[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)
-
-/-! ### toNat -/
-
 @[simp] theorem toNat_sub' (a : Int) (b : Nat) : (a - b).toNat = a.toNat - b := by
   symm
   simp only [Int.toNat]
@@ -65,60 +38,6 @@ protected theorem sub_eq_iff_eq_add' {b a c : Int} : a - b = c ↔ a = b + c :=
   simp [toNat]
   split <;> simp_all <;> omega
 
-@[simp] theorem toNat_eq_zero : ∀ {n : Int}, n.toNat = 0 ↔ n ≤ 0 := by omega
-
-@[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
-
-/-! ### natAbs -/
-
-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)
-
-/-! ### min and max -/
-
-@[simp] protected theorem min_assoc : ∀ (a b c : Int), min (min a b) c = min a (min b c) := by omega
-instance : Std.Associative (α := Nat) min := ⟨Nat.min_assoc⟩
-
-@[simp] protected theorem min_self_assoc {m n : Int} : min m (min m n) = min m n := by
-  rw [← Int.min_assoc, Int.min_self]
-
-@[simp] protected theorem min_self_assoc' {m n : Int} : min n (min m n) = min n m := by
-  rw [Int.min_comm m n, ← Int.min_assoc, Int.min_self]
-
-@[simp] protected theorem max_assoc (a b c : Int) : max (max a b) c = max a (max b c) := by omega
-instance : Std.Associative (α := Nat) max := ⟨Nat.max_assoc⟩
-
-@[simp] protected theorem max_self_assoc {m n : Int} : max m (max m n) = max m n := by
-  rw [← Int.max_assoc, Int.max_self]
-
-@[simp] protected theorem max_self_assoc' {m n : Int} : max n (max m n) = max n m := by
-  rw [Int.max_comm m n, ← Int.max_assoc, Int.max_self]
-
-protected theorem max_min_distrib_left (a b c : Int) : max a (min b c) = min (max a b) (max a c) := by omega
-
-protected theorem min_max_distrib_left (a b c : Int) : min a (max b c) = max (min a b) (min a c) := by omega
-
-protected theorem max_min_distrib_right (a b c : Int) :
-    max (min a b) c = min (max a c) (max b c) := by omega
-
-protected theorem min_max_distrib_right (a b c : Int) :
-    min (max a b) c = max (min a c) (min b c) := by omega
-
-protected theorem sub_min_sub_right (a b c : Int) : min (a - c) (b - c) = min a b - c := by omega
-
-protected theorem sub_max_sub_right (a b c : Int) : max (a - c) (b - c) = max a b - c := by omega
-
-protected theorem sub_min_sub_left (a b c : Int) : min (a - b) (a - c) = a - max b c := by omega
-
-protected theorem sub_max_sub_left (a b c : Int) : max (a - b) (a - c) = a - min b c := by omega
-
-/-! ### bmod -/
-
 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]
@@ -126,18 +45,4 @@ theorem bmod_neg_iff {m : Nat} {x : Int} (h2 : -m ≤ x) (h1 : x < m) :
   · 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
 
-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
-
-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]
-
 end Int

src/Init/Data/Int/OfNat.lean

@@ -1,90 +0,0 @@
-/-
-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)
-  deriving BEq
-
-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)
-
-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 of_le (ctx : Context) (lhs rhs : Expr)
-    : lhs.denote ctx ≤ rhs.denote ctx → lhs.denoteAsInt ctx ≤ rhs.denoteAsInt ctx := by
-  rw [Expr.le ctx lhs rhs]; simp
-
-theorem of_not_le (ctx : Context) (lhs rhs : Expr)
-    : ¬ lhs.denote ctx ≤ rhs.denote ctx → ¬ lhs.denoteAsInt ctx ≤ rhs.denoteAsInt ctx := by
-  rw [Expr.le ctx lhs rhs]; simp
-
-theorem of_dvd (ctx : Context) (d : Nat) (e : Expr)
-    : d ∣ e.denote ctx → Int.ofNat d ∣ e.denoteAsInt ctx := by
-  simp [Expr.denoteAsInt_eq, Int.ofNat_dvd]
-
-theorem of_eq (ctx : Context) (lhs rhs : Expr)
-    : lhs.denote ctx = rhs.denote ctx → lhs.denoteAsInt ctx = rhs.denoteAsInt ctx := by
-  rw [Expr.eq ctx lhs rhs]; simp
-
-theorem of_not_eq (ctx : Context) (lhs rhs : Expr)
-    : ¬ lhs.denote ctx = rhs.denote ctx → ¬ lhs.denoteAsInt ctx = rhs.denoteAsInt ctx := by
-  rw [Expr.eq ctx lhs rhs]; simp
-
-theorem ofNat_toNat (a : Int) : (NatCast.natCast a.toNat : Int) = if a ≤ 0 then 0 else a := by
-  split
-  next h =>
-    rw [Int.toNat_of_nonpos h]; rfl
-  next h =>
-    simp at h
-    have := Int.toNat_of_nonneg (Int.le_of_lt h)
-    assumption
-
-theorem Expr.denoteAsInt_nonneg (ctx : Context) (e : Expr) : e.denoteAsInt ctx ≥ 0 := by
-  simp [Expr.denoteAsInt_eq]
-
-end Int.OfNat

src/Init/Data/Int/Order.lean

@@ -33,18 +33,20 @@ theorem lt_iff_add_one_le {a b : Int} : a < b ↔ a + 1 ≤ b := .rfl
 theorem le.intro_sub {a b : Int} (n : Nat) (h : b - a = n) : a ≤ b := by
   simp [le_def, h]; constructor
 
+attribute [local simp] Int.add_left_neg Int.add_right_neg Int.neg_add
+
 theorem le.intro {a b : Int} (n : Nat) (h : a + n = b) : a ≤ b :=
-  le.intro_sub n <| by rw [← h, Int.add_comm]; simp [Int.sub_eq_add_neg, Int.add_assoc, Int.add_right_neg]
+  le.intro_sub n <| by rw [← h, Int.add_comm]; simp [Int.sub_eq_add_neg, Int.add_assoc]
 
 theorem le.dest_sub {a b : Int} (h : a ≤ b) : ∃ n : Nat, b - a = n := nonneg_def.1 h
 
 theorem le.dest {a b : Int} (h : a ≤ b) : ∃ n : Nat, a + n = b :=
   let ⟨n, h₁⟩ := le.dest_sub h
-  ⟨n, by rw [← h₁, Int.add_comm]; simp [Int.sub_eq_add_neg, Int.add_assoc, Int.add_left_neg]⟩
+  ⟨n, by rw [← h₁, Int.add_comm]; simp [Int.sub_eq_add_neg, Int.add_assoc]⟩
 
 protected theorem le_total (a b : Int) : a ≤ b ∨ b ≤ a :=
   (nonneg_or_nonneg_neg (b - a)).imp_right fun H => by
-    rwa [show -(b - a) = a - b by simp [Int.neg_add,Int.add_comm, Int.sub_eq_add_neg]] at H
+    rwa [show -(b - a) = a - b by simp [Int.add_comm, Int.sub_eq_add_neg]] at H
 
 @[simp, norm_cast] theorem ofNat_le {m n : Nat} : (↑m : Int) ≤ ↑n ↔ m ≤ n :=
   ⟨fun h =>
@@ -54,19 +56,19 @@ 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)⟩
 
-@[simp] theorem ofNat_zero_le (n : Nat) : 0 ≤ (↑n : Int) := ofNat_le.2 n.zero_le
+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
 
-theorem eq_succ_of_zero_lt {a : Int} (h : 0 < a) : ∃ n : Nat, a = n + 1 :=
+theorem eq_succ_of_zero_lt {a : Int} (h : 0 < a) : ∃ n : Nat, a = n.succ :=
   let ⟨n, (h : ↑(1 + n) = a)⟩ := le.dest h
   ⟨n, by rw [Nat.add_comm] at h; exact h.symm⟩
 
-theorem lt_add_succ (a : Int) (n : Nat) : a < a + (n + 1) :=
-  le.intro n <| by rw [Int.add_comm, Int.add_left_comm]
+theorem lt_add_succ (a : Int) (n : Nat) : a < a + Nat.succ n :=
+  le.intro n <| by rw [Int.add_comm, Int.add_left_comm]; rfl
 
-theorem lt.intro {a b : Int} {n : Nat} (h : a + (n + 1) = b) : a < b :=
+theorem lt.intro {a b : Int} {n : Nat} (h : a + Nat.succ n = b) : a < b :=
   h ▸ lt_add_succ a n
 
 theorem lt.dest {a b : Int} (h : a < b) : ∃ n : Nat, a + Nat.succ n = b :=
@@ -115,7 +117,7 @@ protected theorem lt_iff_le_and_ne {a b : Int} : a < b ↔ a ≤ b ∧ a ≠ b :
   have : n ≠ 0 := aneb.imp fun eq => by rw [← hn, eq, ofNat_zero, Int.add_zero]
   apply lt.intro; rwa [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero this)] at hn
 
-protected theorem lt_succ (a : Int) : a < a + 1 := Int.le_refl _
+theorem lt_succ (a : Int) : a < a + 1 := Int.le_refl _
 
 protected theorem zero_lt_one : (0 : Int) < 1 := ⟨_⟩
 
@@ -131,15 +133,12 @@ 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
 
-@[simp] protected theorem not_le {a b : Int} : ¬a ≤ b ↔ b < a :=
+protected theorem not_le {a b : Int} : ¬a ≤ b ↔ b < a :=
   Iff.intro Int.lt_of_not_ge Int.not_le_of_gt
 
-@[simp] protected theorem not_lt {a b : Int} : ¬a < b ↔ b ≤ a :=
+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
@@ -184,19 +183,61 @@ instance : Trans (α := Int) (· ≤ ·) (· < ·) (· < ·) := ⟨Int.lt_of_le_
 
 instance : Trans (α := Int) (· < ·) (· < ·) (· < ·) := ⟨Int.lt_trans⟩
 
-theorem eq_natAbs_of_nonneg {a : Int} (h : 0 ≤ a) : a = natAbs a := by
+protected theorem min_def (n m : Int) : min n m = if n ≤ m then n else m := rfl
+
+protected theorem max_def (n m : Int) : max n m = if n ≤ m then m else n := rfl
+
+protected theorem min_comm (a b : Int) : min a b = min b a := by
+  simp [Int.min_def]
+  by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
+  · exact Int.le_antisymm h₁ h₂
+  · cases not_or_intro h₁ h₂ <| Int.le_total ..
+instance : Std.Commutative (α := Int) min := ⟨Int.min_comm⟩
+
+protected theorem min_le_right (a b : Int) : min a b ≤ b := by rw [Int.min_def]; split <;> simp [*]
+
+protected theorem min_le_left (a b : Int) : min a b ≤ a := Int.min_comm .. ▸ Int.min_le_right ..
+
+protected theorem min_eq_left {a b : Int} (h : a ≤ b) : min a b = a := by simp [Int.min_def, h]
+
+protected theorem min_eq_right {a b : Int} (h : b ≤ a) : min a b = b := by
+  rw [Int.min_comm a b]; exact Int.min_eq_left h
+
+protected theorem le_min {a b c : Int} : a ≤ min b c ↔ a ≤ b ∧ a ≤ c :=
+  ⟨fun h => ⟨Int.le_trans h (Int.min_le_left ..), Int.le_trans h (Int.min_le_right ..)⟩,
+   fun ⟨h₁, h₂⟩ => by rw [Int.min_def]; split <;> assumption⟩
+
+protected theorem max_comm (a b : Int) : max a b = max b a := by
+  simp only [Int.max_def]
+  by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
+  · exact Int.le_antisymm h₂ h₁
+  · cases not_or_intro h₁ h₂ <| Int.le_total ..
+instance : Std.Commutative (α := Int) max := ⟨Int.max_comm⟩
+
+protected theorem le_max_left (a b : Int) : a ≤ max a b := by rw [Int.max_def]; split <;> simp [*]
+
+protected theorem le_max_right (a b : Int) : b ≤ max a b := Int.max_comm .. ▸ Int.le_max_left ..
+
+protected theorem max_le {a b c : Int} : max a b ≤ c ↔ a ≤ c ∧ b ≤ c :=
+  ⟨fun h => ⟨Int.le_trans (Int.le_max_left ..) h, Int.le_trans (Int.le_max_right ..) h⟩,
+   fun ⟨h₁, h₂⟩ => by rw [Int.max_def]; split <;> assumption⟩
+
+protected theorem max_eq_right {a b : Int} (h : a ≤ b) : max a b = b := by
+  simp [Int.max_def, h, Int.not_lt.2 h]
+
+protected theorem max_eq_left {a b : Int} (h : b ≤ a) : max a b = a := by
+  rw [← Int.max_comm b a]; exact Int.max_eq_right h
+
+theorem eq_natAbs_of_zero_le {a : Int} (h : 0 ≤ a) : a = natAbs a := by
   let ⟨n, e⟩ := eq_ofNat_of_zero_le h
   rw [e]; rfl
 
-@[deprecated eq_natAbs_of_nonneg (since := "2025-03-11")]
-abbrev eq_natAbs_of_zero_le := @eq_natAbs_of_nonneg
-
 theorem le_natAbs {a : Int} : a ≤ natAbs a :=
   match Int.le_total 0 a with
-  | .inl h => by rw [eq_natAbs_of_nonneg h]; apply Int.le_refl
+  | .inl h => by rw [eq_natAbs_of_zero_le h]; apply Int.le_refl
   | .inr h => Int.le_trans h (ofNat_zero_le _)
 
-@[simp] theorem negSucc_lt_zero (n : Nat) : -[n+1] < 0 :=
+theorem negSucc_lt_zero (n : Nat) : -[n+1] < 0 :=
   Int.not_le.1 fun h => let ⟨_, h⟩ := eq_ofNat_of_zero_le h; nomatch h
 
 theorem negSucc_le_zero (n : Nat) : -[n+1] ≤ 0 :=
@@ -205,6 +246,18 @@ theorem negSucc_le_zero (n : Nat) : -[n+1] ≤ 0 :=
 @[simp] theorem negSucc_not_nonneg (n : Nat) : 0 ≤ -[n+1] ↔ False := by
   simp only [Int.not_le, iff_false]; exact Int.negSucc_lt_zero n
 
+@[simp] theorem ofNat_max_zero (n : Nat) : (max (n : Int) 0) = n := by
+  rw [Int.max_eq_left (ofNat_zero_le n)]
+
+@[simp] theorem zero_max_ofNat (n : Nat) : (max 0 (n : Int)) = n := by
+  rw [Int.max_eq_right (ofNat_zero_le n)]
+
+@[simp] theorem negSucc_max_zero (n : Nat) : (max (Int.negSucc n) 0) = 0 := by
+  rw [Int.max_eq_right (negSucc_le_zero _)]
+
+@[simp] theorem zero_max_negSucc (n : Nat) : (max 0 (Int.negSucc n)) = 0 := by
+  rw [Int.max_eq_left (negSucc_le_zero _)]
+
 protected theorem add_le_add_left {a b : Int} (h : a ≤ b) (c : Int) : c + a ≤ c + b :=
   let ⟨n, hn⟩ := le.dest h; le.intro n <| by rw [Int.add_assoc, hn]
 
@@ -226,10 +279,10 @@ protected theorem le_of_add_le_add_left {a b c : Int} (h : a + b ≤ a + c) : b
 protected theorem le_of_add_le_add_right {a b c : Int} (h : a + b ≤ c + b) : a ≤ c :=
   Int.le_of_add_le_add_left (a := b) <| by rwa [Int.add_comm b a, Int.add_comm b c]
 
-@[simp] protected theorem add_le_add_iff_left (a : Int) : a + b ≤ a + c ↔ b ≤ c :=
+protected theorem add_le_add_iff_left (a : Int) : a + b ≤ a + c ↔ b ≤ c :=
   ⟨Int.le_of_add_le_add_left, (Int.add_le_add_left · _)⟩
 
-@[simp] protected theorem add_le_add_iff_right (c : Int) : a + c ≤ b + c ↔ a ≤ b :=
+protected theorem add_le_add_iff_right (c : Int) : a + c ≤ b + c ↔ a ≤ b :=
   ⟨Int.le_of_add_le_add_right, (Int.add_le_add_right · _)⟩
 
 protected theorem add_le_add {a b c d : Int} (h₁ : a ≤ b) (h₂ : c ≤ d) : a + c ≤ b + d :=
@@ -248,15 +301,6 @@ protected theorem neg_le_neg {a b : Int} (h : a ≤ b) : -b ≤ -a := by
   have : 0 + -b ≤ -a + b + -b := Int.add_le_add_right this (-b)
   rwa [Int.add_neg_cancel_right, Int.zero_add] at this
 
-@[simp] protected theorem neg_le_neg_iff {a b : Int} : -a ≤ -b ↔ b ≤ a :=
-  ⟨fun h => by simpa using Int.neg_le_neg h, Int.neg_le_neg⟩
-
-@[simp] protected theorem neg_le_zero_iff {a : Int} : -a ≤ 0 ↔ 0 ≤ a := by
-  rw [← Int.neg_zero, Int.neg_le_neg_iff, Int.neg_zero]
-
-@[simp] protected theorem zero_le_neg_iff {a : Int} : 0 ≤ -a ↔ a ≤ 0 := by
-  rw [← Int.neg_zero, Int.neg_le_neg_iff, Int.neg_zero]
-
 protected theorem le_of_neg_le_neg {a b : Int} (h : -b ≤ -a) : a ≤ b :=
   suffices - -a ≤ - -b by simp [Int.neg_neg] at this; assumption
   Int.neg_le_neg h
@@ -274,15 +318,6 @@ protected theorem neg_lt_neg {a b : Int} (h : a < b) : -b < -a := by
   have : 0 + -b < -a + b + -b := Int.add_lt_add_right this (-b)
   rwa [Int.add_neg_cancel_right, Int.zero_add] at this
 
-@[simp] protected theorem neg_lt_neg_iff {a b : Int} : -a < -b ↔ b < a :=
-  ⟨fun h => by simpa using Int.neg_lt_neg h, Int.neg_lt_neg⟩
-
-@[simp] protected theorem neg_lt_zero_iff {a : Int} : -a < 0 ↔ 0 < a := by
-  rw [← Int.neg_zero, Int.neg_lt_neg_iff, Int.neg_zero]
-
-@[simp] protected theorem zero_lt_neg_iff {a : Int} : 0 < -a ↔ a < 0 := by
-  rw [← Int.neg_zero, Int.neg_lt_neg_iff, Int.neg_zero]
-
 protected theorem neg_neg_of_pos {a : Int} (h : 0 < a) : -a < 0 := by
   have : -a < -0 := Int.neg_lt_neg h
   rwa [Int.neg_zero] at this
@@ -323,125 +358,9 @@ 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
-
-/- ### min and max -/
-
-protected theorem min_def (n m : Int) : min n m = if n ≤ m then n else m := rfl
-
-protected theorem max_def (n m : Int) : max n m = if n ≤ m then m else n := rfl
-
-@[simp] protected theorem neg_min_neg (a b : Int) : min (-a) (-b) = -max a b := by
-  rw [Int.min_def, Int.max_def]
-  simp
-  split <;> rename_i h₁ <;> split <;> rename_i h₂
-  · simpa using Int.le_antisymm h₂ h₁
-  · simp
-  · simp
-  · simp only [Int.not_le] at h₁ h₂
-    exfalso
-    exact Int.lt_irrefl _ (Int.lt_trans h₁ h₂)
-
-@[simp] protected theorem min_add_right (a b c : Int) : min (a + c) (b + c) = min a b + c := by
-  rw [Int.min_def, Int.min_def]
-  simp only [Int.add_le_add_iff_right]
-  split <;> simp
-
-@[simp] protected theorem min_add_left (a b c : Int) : min (a + b) (a + c) = a + min b c := by
-  rw [Int.min_def, Int.min_def]
-  simp only [Int.add_le_add_iff_left]
-  split <;> simp
-
-protected theorem min_comm (a b : Int) : min a b = min b a := by
-  simp [Int.min_def]
-  by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
-  · exact Int.le_antisymm h₁ h₂
-  · cases not_or_intro h₁ h₂ <| Int.le_total ..
-instance : Std.Commutative (α := Int) min := ⟨Int.min_comm⟩
-
-protected theorem min_le_right (a b : Int) : min a b ≤ b := by rw [Int.min_def]; split <;> simp [*]
-
-protected theorem min_le_left (a b : Int) : min a b ≤ a := Int.min_comm .. ▸ Int.min_le_right ..
-
-protected theorem min_eq_left {a b : Int} (h : a ≤ b) : min a b = a := by simp [Int.min_def, h]
-
-protected theorem min_eq_right {a b : Int} (h : b ≤ a) : min a b = b := by
-  rw [Int.min_comm a b]; exact Int.min_eq_left h
-
-protected theorem le_min {a b c : Int} : a ≤ min b c ↔ a ≤ b ∧ a ≤ c :=
-  ⟨fun h => ⟨Int.le_trans h (Int.min_le_left ..), Int.le_trans h (Int.min_le_right ..)⟩,
-   fun ⟨h₁, h₂⟩ => by rw [Int.min_def]; split <;> assumption⟩
-
-protected theorem lt_min {a b c : Int} : a < min b c ↔ a < b ∧ a < c := Int.le_min
-
-@[simp] protected theorem neg_max_neg (a b : Int) : max (-a) (-b) = -min a b := by
-  rw [Int.min_def, Int.max_def]
-  simp
-  split <;> rename_i h₁ <;> split <;> rename_i h₂
-  · simpa using Int.le_antisymm h₁ h₂
-  · simp
-  · simp
-  · simp only [Int.not_le] at h₁ h₂
-    exfalso
-    exact Int.lt_irrefl _ (Int.lt_trans h₁ h₂)
-
-@[simp] protected theorem max_add_right (a b c : Int) : max (a + c) (b + c) = max a b + c := by
-  rw [Int.max_def, Int.max_def]
-  simp only [Int.add_le_add_iff_right]
-  split <;> simp
-
-@[simp] protected theorem max_add_left (a b c : Int) : max (a + b) (a + c) = a + max b c := by
-  rw [Int.max_def, Int.max_def]
-  simp only [Int.add_le_add_iff_left]
-  split <;> simp
-
-protected theorem max_comm (a b : Int) : max a b = max b a := by
-  simp only [Int.max_def]
-  by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
-  · exact Int.le_antisymm h₂ h₁
-  · cases not_or_intro h₁ h₂ <| Int.le_total ..
-instance : Std.Commutative (α := Int) max := ⟨Int.max_comm⟩
-
-protected theorem le_max_left (a b : Int) : a ≤ max a b := by rw [Int.max_def]; split <;> simp [*]
-
-protected theorem le_max_right (a b : Int) : b ≤ max a b := Int.max_comm .. ▸ Int.le_max_left ..
-
-protected theorem max_eq_right {a b : Int} (h : a ≤ b) : max a b = b := by
-  simp [Int.max_def, h, Int.not_lt.2 h]
-
-protected theorem max_eq_left {a b : Int} (h : b ≤ a) : max a b = a := by
-  rw [← Int.max_comm b a]; exact Int.max_eq_right h
-
-protected theorem max_le {a b c : Int} : max a b ≤ c ↔ a ≤ c ∧ b ≤ c :=
-  ⟨fun h => ⟨Int.le_trans (Int.le_max_left ..) h, Int.le_trans (Int.le_max_right ..) h⟩,
-   fun ⟨h₁, h₂⟩ => by rw [Int.max_def]; split <;> assumption⟩
-
-protected theorem max_lt {a b c : Int} : max a b < c ↔ a < c ∧ b < c := by
-  simp only [Int.lt_iff_add_one_le]
-  simpa using Int.max_le (a := a + 1) (b := b + 1) (c := c)
-
-@[simp] theorem ofNat_max_zero (n : Nat) : (max (n : Int) 0) = n := by
-  rw [Int.max_eq_left (ofNat_zero_le n)]
-
-@[simp] theorem zero_max_ofNat (n : Nat) : (max 0 (n : Int)) = n := by
-  rw [Int.max_eq_right (ofNat_zero_le n)]
-
-@[simp] theorem negSucc_max_zero (n : Nat) : (max (Int.negSucc n) 0) = 0 := by
-  rw [Int.max_eq_right (negSucc_le_zero _)]
-
-@[simp] theorem zero_max_negSucc (n : Nat) : (max 0 (Int.negSucc n)) = 0 := by
-  rw [Int.max_eq_left (negSucc_le_zero _)]
-
-@[simp] protected theorem min_self (a : Int) : min a a = a := Int.min_eq_left (Int.le_refl _)
-instance : Std.IdempotentOp (α := Int) min := ⟨Int.min_self⟩
-
-@[simp] protected theorem max_self (a : Int) : max a a = a := Int.max_eq_right (Int.le_refl _)
-instance : Std.IdempotentOp (α := Int) max := ⟨Int.max_self⟩
-
 /- ### Order properties and multiplication -/
 
+
 protected theorem mul_nonneg {a b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b := by
   let ⟨n, hn⟩ := eq_ofNat_of_zero_le ha
   let ⟨m, hm⟩ := eq_ofNat_of_zero_le hb
@@ -450,8 +369,7 @@ protected theorem mul_nonneg {a b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a
 protected theorem mul_pos {a b : Int} (ha : 0 < a) (hb : 0 < b) : 0 < a * b := by
   let ⟨n, hn⟩ := eq_succ_of_zero_lt ha
   let ⟨m, hm⟩ := eq_succ_of_zero_lt hb
-  rw [hn, hm]
-  apply ofNat_succ_pos
+  rw [hn, hm, ← ofNat_mul]; apply ofNat_succ_pos
 
 protected theorem mul_lt_mul_of_pos_left {a b c : Int}
   (h₁ : a < b) (h₂ : 0 < c) : c * a < c * b := by
@@ -507,7 +425,7 @@ protected theorem mul_le_mul_of_nonpos_left {a b c : Int}
 
 /- ## natAbs -/
 
-@[simp, norm_cast] theorem natAbs_ofNat (n : Nat) : natAbs ↑n = n := rfl
+@[simp] 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
@@ -552,13 +470,6 @@ 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
@@ -581,8 +492,8 @@ theorem toNat_of_nonneg {a : Int} (h : 0 ≤ a) : (toNat a : Int) = a := by
 
 @[simp] theorem ofNat_toNat (a : Int) : (a.toNat : Int) = max a 0 := by
   match a with
-  | (n : Nat) => simp
-  | -(n + 1 : Nat) => norm_cast
+  | Int.ofNat n => simp
+  | Int.negSucc n => simp
 
 theorem self_le_toNat (a : Int) : a ≤ toNat a := by rw [toNat_eq_max]; apply Int.le_max_left
 
@@ -618,15 +529,12 @@ theorem toNat_sub_toNat_neg : ∀ n : Int, ↑n.toNat - ↑(-n).toNat = n
   | 0 => rfl
   | _+1 => rfl
 
-/-! ### toNat? -/
+/-! ### toNat' -/
 
-theorem mem_toNat? : ∀ {a : Int} {n : Nat}, toNat? a = some n ↔ a = n
-  | (m : Nat), n => by simp [toNat?, Int.ofNat_inj]
+theorem mem_toNat' : ∀ {a : Int} {n : Nat}, toNat' a = some n ↔ a = n
+  | (m : Nat), n => by simp [toNat', Int.ofNat_inj]
   | -[m+1], n => by constructor <;> nofun
 
-@[deprecated mem_toNat? (since := "2025-03-11")]
-abbrev mem_toNat' := @mem_toNat?
-
 /-! ## Order properties of the integers -/
 
 protected theorem le_of_not_le {a b : Int} : ¬ a ≤ b → b ≤ a := (Int.le_total a b).resolve_left
@@ -646,10 +554,10 @@ protected theorem lt_of_add_lt_add_left {a b c : Int} (h : a + b < a + c) : b <
 protected theorem lt_of_add_lt_add_right {a b c : Int} (h : a + b < c + b) : a < c :=
   Int.lt_of_add_lt_add_left (a := b) <| by rwa [Int.add_comm b a, Int.add_comm b c]
 
-@[simp] protected theorem add_lt_add_iff_left (a : Int) : a + b < a + c ↔ b < c :=
+protected theorem add_lt_add_iff_left (a : Int) : a + b < a + c ↔ b < c :=
   ⟨Int.lt_of_add_lt_add_left, (Int.add_lt_add_left · _)⟩
 
-@[simp] protected theorem add_lt_add_iff_right (c : Int) : a + c < b + c ↔ a < b :=
+protected theorem add_lt_add_iff_right (c : Int) : a + c < b + c ↔ a < b :=
   ⟨Int.lt_of_add_lt_add_right, (Int.add_lt_add_right · _)⟩
 
 protected theorem add_lt_add {a b c d : Int} (h₁ : a < b) (h₂ : c < d) : a + c < b + d :=
@@ -844,18 +752,6 @@ protected theorem sub_le_sub_right {a b : Int} (h : a ≤ b) (c : Int) : a - c
 protected theorem sub_le_sub {a b c d : Int} (hab : a ≤ b) (hcd : c ≤ d) : a - d ≤ b - c :=
   Int.add_le_add hab (Int.neg_le_neg hcd)
 
-protected theorem le_of_sub_le_sub_left {a b c : Int} (h : c - a ≤ c - b) : b ≤ a :=
-  Int.le_of_neg_le_neg <| Int.le_of_add_le_add_left h
-
-protected theorem le_of_sub_le_sub_right {a b c : Int} (h : a - c ≤ b - c) : a ≤ b :=
-  Int.le_of_add_le_add_right h
-
-@[simp] protected theorem sub_le_sub_left_iff {a b c : Int} : c - a ≤ c - b ↔ b ≤ a :=
-  ⟨Int.le_of_sub_le_sub_left, (Int.sub_le_sub_left · c)⟩
-
-@[simp] protected theorem sub_le_sub_right_iff {a b c : Int} : a - c ≤ b - c ↔ a ≤ b :=
-  ⟨Int.le_of_sub_le_sub_right, (Int.sub_le_sub_right · c)⟩
-
 protected theorem add_lt_of_lt_neg_add {a b c : Int} (h : b < -a + c) : a + b < c := by
   have h := Int.add_lt_add_left h a
   rwa [Int.add_neg_cancel_left] at h
@@ -964,11 +860,11 @@ protected theorem lt_of_sub_lt_sub_right {a b c : Int} (h : a - c < b - c) : a <
   ⟨Int.lt_of_sub_lt_sub_right, (Int.sub_lt_sub_right · c)⟩
 
 protected theorem sub_lt_sub_of_le_of_lt {a b c d : Int}
-    (hab : a ≤ b) (hcd : c < d) : a - d < b - c :=
+  (hab : a ≤ b) (hcd : c < d) : a - d < b - c :=
   Int.add_lt_add_of_le_of_lt hab (Int.neg_lt_neg hcd)
 
 protected theorem sub_lt_sub_of_lt_of_le {a b c d : Int}
-    (hab : a < b) (hcd : c ≤ d) : a - d < b - c :=
+  (hab : a < b) (hcd : c ≤ d) : a - d < b - c :=
   Int.add_lt_add_of_lt_of_le hab (Int.neg_le_neg hcd)
 
 protected theorem add_le_add_three {a b c d e f : Int}
@@ -1042,22 +938,6 @@ 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
@@ -1108,10 +988,10 @@ theorem neg_of_sign_eq_neg_one : ∀ {a : Int}, sign a = -1 → a < 0
   | 0, h => nomatch h
   | -[_+1], _ => negSucc_lt_zero _
 
-@[simp] theorem sign_eq_one_iff_pos {a : Int} : sign a = 1 ↔ 0 < a :=
+theorem sign_eq_one_iff_pos {a : Int} : sign a = 1 ↔ 0 < a :=
   ⟨pos_of_sign_eq_one, sign_eq_one_of_pos⟩
 
-@[simp] theorem sign_eq_neg_one_iff_neg {a : Int} : sign a = -1 ↔ a < 0 :=
+theorem sign_eq_neg_one_iff_neg {a : Int} : sign a = -1 ↔ a < 0 :=
   ⟨neg_of_sign_eq_neg_one, sign_eq_neg_one_of_neg⟩
 
 @[simp] theorem sign_eq_zero_iff_zero {a : Int} : sign a = 0 ↔ a = 0 :=
@@ -1123,7 +1003,7 @@ theorem neg_of_sign_eq_neg_one : ∀ {a : Int}, sign a = -1 → a < 0
   | .ofNat (_ + 1) => rfl
   | .negSucc _ => rfl
 
-@[simp] theorem sign_nonneg_iff : 0 ≤ sign x ↔ 0 ≤ x := by
+@[simp] theorem sign_nonneg : 0 ≤ sign x ↔ 0 ≤ x := by
   match x with
   | 0 => rfl
   | .ofNat (_ + 1) =>
@@ -1131,42 +1011,11 @@ theorem neg_of_sign_eq_neg_one : ∀ {a : Int}, sign a = -1 → a < 0
     exact Int.le_add_one (ofNat_nonneg _)
   | .negSucc _ => simp +decide [sign]
 
-@[deprecated sign_nonneg_iff (since := "2025-03-11")] abbrev sign_nonneg := @sign_nonneg_iff
-
-@[simp] theorem sign_pos_iff : 0 < sign x ↔ 0 < x := by
-  match x with
-  | 0
-  | .ofNat (_ + 1) => simp
-  | .negSucc x => simp
-
-@[simp] theorem sign_nonpos_iff : sign x ≤ 0 ↔ x ≤ 0 := by
-  match x with
-  | 0 => rfl
-  | .ofNat (_ + 1) => simp
-  | .negSucc _ => simpa using negSucc_le_zero _
-
-@[simp] theorem sign_neg_iff : sign x < 0 ↔ x < 0 := by
-  match x with
-  | 0 => simp
-  | .ofNat (_ + 1) => simpa using le.intro_sub _ rfl
-  | .negSucc _ => simp
-
-@[simp] theorem mul_sign_self : ∀ i : Int, i * sign i = natAbs i
+theorem mul_sign : ∀ 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
@@ -1175,7 +1024,7 @@ theorem natAbs_mul_self : ∀ {a : Int}, ↑(natAbs a * natAbs a) = a * a
   | ofNat _ => rfl
   | -[_+1]  => rfl
 
-protected theorem eq_nat_or_neg (a : Int) : ∃ n : Nat, a = n ∨ a = -↑n := ⟨_, natAbs_eq a⟩
+theorem eq_nat_or_neg (a : Int) : ∃ n : Nat, a = n ∨ a = -↑n := ⟨_, natAbs_eq a⟩
 
 theorem natAbs_mul_natAbs_eq {a b : Int} {c : Nat}
     (h : a * b = (c : Int)) : a.natAbs * b.natAbs = c := by rw [← natAbs_mul, h, natAbs.eq_def]
@@ -1189,7 +1038,7 @@ theorem natAbs_eq_iff {a : Int} {n : Nat} : a.natAbs = n ↔ a = n ∨ a = -↑n
 theorem natAbs_add_le (a b : Int) : natAbs (a + b) ≤ natAbs a + natAbs b := by
   suffices ∀ a b : Nat, natAbs (subNatNat a b.succ) ≤ (a + b).succ by
     match a, b with
-    | (a:Nat), (b:Nat) => rw [← ofNat_add, natAbs_ofNat]; apply Nat.le_refl
+    | (a:Nat), (b:Nat) => rw [ofNat_add_ofNat, natAbs_ofNat]; apply Nat.le_refl
     | (a:Nat), -[b+1]  => rw [natAbs_ofNat, natAbs_negSucc]; apply this
     | -[a+1],  (b:Nat) =>
       rw [natAbs_negSucc, natAbs_ofNat, Nat.succ_add, Nat.add_comm a b]; apply this
@@ -1208,7 +1057,6 @@ theorem natAbs_add_le (a b : Int) : natAbs (a + b) ≤ natAbs a + natAbs b := by
 theorem natAbs_sub_le (a b : Int) : natAbs (a - b) ≤ natAbs a + natAbs b := by
   rw [← Int.natAbs_neg b]; apply natAbs_add_le
 
-@[deprecated negSucc_eq (since := "2025-03-11")]
 theorem negSucc_eq' (m : Nat) : -[m+1] = -m - 1 := by simp only [negSucc_eq, Int.neg_add]; rfl
 
 theorem natAbs_lt_natAbs_of_nonneg_of_lt {a b : Int}
@@ -1216,10 +1064,7 @@ theorem natAbs_lt_natAbs_of_nonneg_of_lt {a b : Int}
   match a, b, eq_ofNat_of_zero_le w₁, eq_ofNat_of_zero_le (Int.le_trans w₁ (Int.le_of_lt w₂)) with
   | _, _, ⟨_, rfl⟩, ⟨_, rfl⟩ => ofNat_lt.1 w₂
 
-theorem natAbs_eq_iff_mul_eq_zero : natAbs a = n ↔ (a - n) * (a + n) = 0 := by
+theorem eq_natAbs_iff_mul_eq_zero : natAbs a = n ↔ (a - n) * (a + n) = 0 := by
   rw [natAbs_eq_iff, Int.mul_eq_zero, ← Int.sub_neg, Int.sub_eq_zero, Int.sub_eq_zero]
 
-@[deprecated natAbs_eq_iff_mul_eq_zero (since := "2025-03-11")]
-abbrev eq_natAbs_iff_mul_eq_zero := @natAbs_eq_iff_mul_eq_zero
-
 end Int

src/Init/Data/Int/Pow.lean

@@ -11,27 +11,37 @@ namespace Int
 
 /-! # pow -/
 
-@[simp] protected theorem pow_zero (b : Int) : b^0 = 1 := rfl
+protected theorem pow_zero (b : Int) : b^0 = 1 := rfl
 
 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]
 
-@[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_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_right (since := "2025-02-17")]
-abbrev pow_le_pow_of_le_right := @Nat.pow_le_pow_right
+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_pos (since := "2025-02-17")]
-abbrev pos_pow_of_pos := @Nat.pow_pos
+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 _)
 
 @[norm_cast]
-protected theorem natCast_pow (b n : Nat) : ((b^n : Nat) : Int) = (b : Int) ^ n := by
+theorem natCast_pow (b n : Nat) : ((b^n : Nat) : Int) = (b : Int) ^ n := by
   match n with
   | 0 => rfl
   | n + 1 =>
-    simp only [Nat.pow_succ, Int.pow_succ, Int.natCast_mul, Int.natCast_pow _ n]
+    simp only [Nat.pow_succ, Int.pow_succ, natCast_mul, natCast_pow _ n]
 
 @[simp]
 protected theorem two_pow_pred_sub_two_pow {w : Nat} (h : 0 < w) :

src/Init/Data/List/Attach.lean

@@ -6,51 +6,32 @@ 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
 
-/--
-Maps a partially defined function (defined on those terms of `α` that satisfy a predicate `P`) over
-a list `l : List α`, given a proof that every element of `l` in fact satisfies `P`.
-
-`O(|l|)`. `List.pmap`, named for “partial map,” is the equivalent of `List.map` for such partial
-functions.
--/
+/-- `O(n)`. Partial map. If `f : Π a, P a → β` is a partial function defined on
+  `a : α` satisfying `P`, then `pmap f l h` is essentially the same as `map f l`
+  but is defined only when all members of `l` satisfy `P`, using the proof
+  to apply `f`. -/
 def pmap {P : α → Prop} (f : ∀ a, P a → β) : ∀ l : List α, (H : ∀ a ∈ l, P a) → List β
   | [], _ => []
   | a :: l, H => f a (forall_mem_cons.1 H).1 :: pmap f l (forall_mem_cons.1 H).2
 
 /--
-Unsafe implementation of `attachWith` that takes advantage of the fact that the representation of
+Unsafe implementation of `attachWith`, taking advantage of the fact that the representation of
 `List {x // P x}` is the same as the input `List α`.
+(Someday, the compiler might do this optimization automatically, but until then...)
 -/
 @[inline] private unsafe def attachWithImpl
     (l : List α) (P : α → Prop) (_ : ∀ x ∈ l, P x) : List {x // P x} := unsafeCast l
 
-/--
-“Attaches” individual proofs to a list of values that satisfy a predicate `P`, returning a list of
-elements in the corresponding subtype `{ x // P x }`.
-
-`O(1)`.
--/
+/-- `O(1)`. "Attach" a proof `P x` that holds for all the elements of `l` to produce a new list
+  with the same elements but in the type `{x // P x}`. -/
 @[implemented_by attachWithImpl] def attachWith
     (l : List α) (P : α → Prop) (H : ∀ x ∈ l, P x) : List {x // P x} := pmap Subtype.mk l H
 
-/--
-“Attaches” the proof that the elements of `l` are in fact elements of `l`, producing a new list with
-the same elements but in the subtype `{ x // x ∈ l }`.
-
-`O(1)`.
-
-This function is primarily used to allow definitions by [well-founded
-recursion](lean-manual://section/well-founded-recursion) that use higher-order functions (such as
-`List.map`) to prove that an value taken from a list is smaller than the list. This allows the
-well-founded recursion mechanism to prove that the function terminates.
--/
+/-- `O(1)`. "Attach" the proof that the elements of `l` are in `l` to produce a new list
+  with the same elements but in the type `{x // x ∈ l}`. -/
 @[inline] def attach (l : List α) : List {x // x ∈ l} := attachWith l _ fun _ => id
 
 /-- Implementation of `pmap` using the zero-copy version of `attach`. -/
@@ -58,12 +39,12 @@ well-founded recursion mechanism to prove that the function terminates.
     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
 
@@ -138,26 +119,27 @@ 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_val (l : List α) (f : α → β) :
+theorem attach_map_coe (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 _ _ _ _
 
-@[deprecated attach_map_val (since := "2025-02-17")]
-abbrev attach_map_coe := @attach_map_val
+theorem attach_map_val (l : List α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
+  attach_map_coe _ _
 
 theorem attach_map_subtype_val (l : List α) : l.attach.map Subtype.val = l :=
-  (attach_map_val _ _).trans (List.map_id _)
+  (attach_map_coe _ _).trans (List.map_id _)
 
-theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : List α) (H : ∀ a ∈ l, p a) :
+theorem attachWith_map_coe {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 _ _ _ _
 
-@[deprecated attachWith_map_val (since := "2025-02-17")]
-abbrev attachWith_map_coe := @attachWith_map_val
+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 _ _ _
 
 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_val _ _ _).trans (List.map_id _)
+  (attachWith_map_coe _ _ _).trans (List.map_id _)
 
 @[simp]
 theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
@@ -196,7 +178,7 @@ 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]
@@ -205,7 +187,7 @@ 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_iff, length_pmap, length_eq_zero_iff]
+  rw [← length_eq_zero, length_pmap, length_eq_zero]
 
 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
@@ -240,39 +222,42 @@ 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) (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
+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
   | nil => simp
   | cons hd tl hl =>
-    rcases i with ⟨i⟩
+    rcases n with ⟨n⟩
     · simp only [Option.pmap]
       split <;> simp_all
-    · simp only [pmap, getElem?_cons_succ, hl, Option.pmap]
+    · 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
 
-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) {i : Nat}
-    (hn : i < (pmap f l h).length) :
-    (pmap f l h)[i] =
-      f (l[i]'(@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) {n : Nat}
+    (hn : n < (pmap f l h).length) :
+    (pmap f l h)[n] =
+      f (l[n]'(@length_pmap _ _ p f l h ▸ hn))
         (h _ (getElem_mem (@length_pmap _ _ p f l h ▸ hn))) := by
-  induction l generalizing i with
+  induction l generalizing n with
   | nil =>
     simp only [length, pmap] at hn
-    exact absurd hn (Nat.not_lt_of_le i.zero_le)
+    exact absurd hn (Nat.not_lt_of_le n.zero_le)
   | cons hd tl hl =>
-    cases i
+    cases n
     · 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⟩ =
@@ -431,12 +416,7 @@ theorem attachWith_map {l : List α} (f : α → β) {P : β → Prop} {H : ∀
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
   induction l <;> simp [*]
 
-@[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}
+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.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
@@ -448,7 +428,7 @@ theorem map_attachWith_eq_pmap {l : List α} {P : α → Prop} {H : ∀ (a : α)
     simp
 
 /-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
-theorem map_attach_eq_pmap {l : List α} (f : { x // x ∈ l } → β) :
+theorem map_attach {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
@@ -457,9 +437,6 @@ theorem map_attach_eq_pmap {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
@@ -665,19 +642,11 @@ Further, we provide simp lemmas that push `unattach` inwards.
 -/
 
 /--
-Maps a list of terms in a subtype to the corresponding terms in the type by forgetting that they
-satisfy the predicate.
+A synonym for `l.map (·.val)`. Mostly this should not be needed by users.
+It is introduced as an intermediate step by lemmas such as `map_subtype`,
+and is ideally subsequently simplified away by `unattach_attach`.
 
-This is the inverse of `List.attachWith` and a synonym for `l.map (·.val)`.
-
-Mostly this should not be needed by users. It is introduced as an intermediate step by lemmas such
-as `map_subtype`, and is ideally subsequently simplified away by `unattach_attach`.
-
-This function is usually inserted automatically by Lean as an intermediate step while proving
-termination. It is rarely used explicitly in code. It is introduced as an intermediate step during
-the elaboration of definitions by [well-founded
-recursion](lean-manual://section/well-founded-recursion). If this function is encountered in a proof
-state, the right approach is usually the tactic `simp [List.unattach, -List.map_subtype]`.
+If not, usually the right approach is `simp [List.unattach, -List.map_subtype]` to unfold.
 -/
 def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) : List α := l.map (·.val)
 
@@ -685,10 +654,6 @@ 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
@@ -793,16 +758,6 @@ 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 }}
@@ -833,66 +788,4 @@ 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