chore: deprecate Array.get

fix test

.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

@@ -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/std/naming.md

@@ -239,22 +239,3 @@ If an acronym is typically spelled using mixed case, this mixed spelling may be
 
 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`.

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"
       }
     },

flake.nix

@@ -8,8 +8,8 @@
   # 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:

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))

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.

src/Init/Control/Basic.lean

@@ -78,7 +78,7 @@ Error recovery and state can interact subtly. For example, the implementation of
 -/
 -- NB: List instance is in mathlib. Once upstreamed, add
 -- * `List`, where `failure` is the empty list and `<|>` concatenates.
-class Alternative (f : Type u → Type v) : 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/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

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

src/Init/Core.lean

@@ -226,9 +226,9 @@ structure PSigma {α : Sort u} (β : α → Sort v) where
   (This will usually require a type ascription to determine `β`
   since it is not determined from `a` and `b` alone.) -/
   mk ::
-  /-- The first component of a dependent pair. If `p : @PSigma α β` then `p.1 : α`. -/
+  /-- The first component of a dependent pair. If `p : @Sigma α β` then `p.1 : α`. -/
   fst : α
-  /-- The second component of a dependent pair. If `p : PSigma β` then `p.2 : β p.1`. -/
+  /-- The second component of a dependent pair. If `p : Sigma β` then `p.2 : β p.1`. -/
   snd : β fst
 
 /--
@@ -1925,6 +1925,10 @@ protected abbrev recOnSubsingleton₂
 end
 end Quotient
 
+section
+variable {α : Type u}
+variable (r : α → α → Prop)
+
 instance Quotient.decidableEq {α : Sort u} {s : Setoid α} [d : ∀ (a b : α), Decidable (a ≈ b)]
     : DecidableEq (Quotient s) :=
   fun (q₁ q₂ : Quotient s) =>
@@ -2016,7 +2020,7 @@ free variables. The frontend automatically declares a fresh auxiliary constant `
 
 Warning: by using this feature, the Lean compiler and interpreter become part of your trusted code base.
 This is extra 30k lines of code. More importantly, you will probably not be able to check your development using
-external type checkers that do not implement this feature.
+external type checkers (e.g., Trepplein) that do not implement this feature.
 Keep in mind that if you are using Lean as programming language, you are already trusting the Lean compiler and interpreter.
 So, you are mainly losing the capability of type checking your development using external checkers.
 
@@ -2051,7 +2055,7 @@ decidability instance can be evaluated to `true` using the lean compiler / inter
 
 Warning: by using this feature, the Lean compiler and interpreter become part of your trusted code base.
 This is extra 30k lines of code. More importantly, you will probably not be able to check your development using
-external type checkers that do not implement this feature.
+external type checkers (e.g., Trepplein) that do not implement this feature.
 Keep in mind that if you are using Lean as programming language, you are already trusting the Lean compiler and interpreter.
 So, you are mainly losing the capability of type checking your development using external checkers.
 -/
@@ -2062,7 +2066,7 @@ The axiom `ofReduceNat` is used to perform proofs by reflection. See `reduceBool
 
 Warning: by using this feature, the Lean compiler and interpreter become part of your trusted code base.
 This is extra 30k lines of code. More importantly, you will probably not be able to check your development using
-external type checkers that do not implement this feature.
+external type checkers (e.g., Trepplein) that do not implement this feature.
 Keep in mind that if you are using Lean as programming language, you are already trusting the Lean compiler and interpreter.
 So, you are mainly losing the capability of type checking your development using external checkers.
 -/
@@ -2121,7 +2125,7 @@ class LeftIdentity (op : α → β → β) (o : outParam α) : Prop
 `LawfulLeftIdentify op o` indicates `o` is a verified left identity of
 `op`.
 -/
-class LawfulLeftIdentity (op : α → β → β) (o : outParam α) : Prop extends LeftIdentity op o where
+class LawfulLeftIdentity (op : α → β → β) (o : outParam α) extends LeftIdentity op o : Prop where
   /-- Left identity `o` is an identity. -/
   left_id : ∀ a, op o a = a
 
@@ -2137,7 +2141,7 @@ class RightIdentity (op : α → β → α) (o : outParam β) : Prop
 `LawfulRightIdentify op o` indicates `o` is a verified right identity of
 `op`.
 -/
-class LawfulRightIdentity (op : α → β → α) (o : outParam β) : Prop extends RightIdentity op o where
+class LawfulRightIdentity (op : α → β → α) (o : outParam β) extends RightIdentity op o : Prop where
   /-- Right identity `o` is an identity. -/
   right_id : ∀ a, op a o = a
 
@@ -2147,13 +2151,13 @@ class LawfulRightIdentity (op : α → β → α) (o : outParam β) : Prop exten
 This class does not require a proof that `o` is an identity, and is used
 primarily for inferring the identity using class resolution.
 -/
-class Identity (op : α → α → α) (o : outParam α) : Prop extends LeftIdentity op o, RightIdentity op o
+class Identity (op : α → α → α) (o : outParam α) extends LeftIdentity op o, RightIdentity op o : Prop
 
 /--
 `LawfulIdentity op o` indicates `o` is a verified left and right
 identity of `op`.
 -/
-class LawfulIdentity (op : α → α → α) (o : outParam α) : Prop extends Identity op o, LawfulLeftIdentity op o, LawfulRightIdentity op o
+class LawfulIdentity (op : α → α → α) (o : outParam α) extends Identity op o, LawfulLeftIdentity op o, LawfulRightIdentity op o : Prop
 
 /--
 `LawfulCommIdentity` can simplify defining instances of `LawfulIdentity`
@@ -2164,7 +2168,7 @@ This class is intended for simplifying defining instances of
 `LawfulIdentity` and functions needed commutative operations with
 identity should just add a `LawfulIdentity` constraint.
 -/
-class LawfulCommIdentity (op : α → α → α) (o : outParam α) [hc : Commutative op] : Prop extends LawfulIdentity op o where
+class LawfulCommIdentity (op : α → α → α) (o : outParam α) [hc : Commutative op] extends LawfulIdentity op o : Prop where
   left_id a := Eq.trans (hc.comm o a) (right_id a)
   right_id a := Eq.trans (hc.comm a o) (left_id a)
 

src/Init/Data/Array/Attach.lean

@@ -9,9 +9,6 @@ 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
 
 /--
@@ -22,8 +19,8 @@ to apply `f`.
 
 We replace this at runtime with a more efficient version via the `csimp` lemma `pmap_eq_pmapImpl`.
 -/
-def pmap {P : α → Prop} (f : ∀ a, P a → β) (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
@@ -54,25 +51,25 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
     l.toArray.pmap f H = (l.pmap f (by simpa using H)).toArray := by
   simp [pmap]
 
-@[simp] theorem toList_attachWith {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
+  funext α β p f L h'
+  cases L
   simp only [pmap, pmapImpl, List.attachWith_toArray, List.map_toArray, mk.injEq, List.map_attachWith_eq_pmap]
   apply List.pmap_congr_left
   intro a m h₁ h₂
@@ -80,9 +77,9 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
 
 @[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
@@ -97,158 +94,158 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
   simp
 
 @[simp]
-theorem pmap_eq_map (p : α → Prop) (f : α → β) (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_val (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_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_val {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_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} :
@@ -271,40 +268,40 @@ 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]
 
 /--
@@ -317,9 +314,9 @@ 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⟩
+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.size_toArray,
     List.length_pmap, List.foldl_toArray', mem_toArray, List.foldl_subtype]
   congr
@@ -336,101 +333,101 @@ Unfortunately this can't be applied by `simp` because of the higher order unific
 and even when rewriting we need to specify the function explicitly.
 See however `foldr_subtype` below.
 -/
-theorem foldr_attach (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⟩
+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.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}
+@[simp] 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
+    (l.attachWith P H).map f = l.attach.map fun ⟨x, h⟩ => f ⟨x, H _ h⟩ := by
+  cases l <;> simp_all
 
-theorem map_attachWith_eq_pmap {xs : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a}
+theorem map_attachWith_eq_pmap {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.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_eq_pmap {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⟩) ++
@@ -499,35 +496,35 @@ 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]
   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
@@ -548,47 +545,43 @@ and is ideally subsequently simplified away by `unattach_attach`.
 
 If not, usually the right approach is `simp [Array.unattach, -Array.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (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 = #[] := rfl
-@[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_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. -/
@@ -597,20 +590,20 @@ 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]
 
@@ -618,20 +611,20 @@ 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]
 
@@ -639,84 +632,70 @@ 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
+    l.filterMap f = l.unattach.filterMap g := by
+  cases l
   simp only [List.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 }}
+@[simp] theorem flatMap_subtype {p : α → Prop} {l : 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
+    (l.flatMap f) = l.unattach.flatMap g := by
+  cases l
   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]

src/Init/Data/Array/Basic.lean

@@ -14,9 +14,6 @@ import Init.GetElem
 import Init.Data.List.ToArrayImpl
 import Init.Data.Array.Set
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
-
 universe u v w
 
 /-! ### Array literal syntax -/
@@ -38,28 +35,28 @@ namespace Array
 
 /-! ### Preliminary theorems -/
 
-@[simp] theorem size_set (xs : Array α) (i : Nat) (v : α) (h : i < xs.size) :
-    (set xs i v h).size = xs.size :=
+@[simp] theorem size_set (a : Array α) (i : Nat) (v : α) (h : i < a.size) :
+    (set a i v h).size = a.size :=
   List.length_set ..
 
-@[simp] theorem size_push (xs : Array α) (v : α) : (push xs v).size = xs.size + 1 :=
+@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
   List.length_concat ..
 
-theorem ext (xs ys : Array α)
-    (h₁ : xs.size = ys.size)
-    (h₂ : (i : Nat) → (hi₁ : i < xs.size) → (hi₂ : i < ys.size) → xs[i] = ys[i])
-    : xs = ys := by
-  let rec extAux (as bs : List α)
-      (h₁ : as.length = bs.length)
-      (h₂ : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as[i] = bs[i])
-      : as = bs := by
-    induction as generalizing bs with
+theorem ext (a b : Array α)
+    (h₁ : a.size = b.size)
+    (h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
+    : a = b := by
+  let rec extAux (a b : List α)
+      (h₁ : a.length = b.length)
+      (h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a[i] = b[i])
+      : a = b := by
+    induction a generalizing b with
     | nil =>
-      cases bs with
+      cases b with
       | nil       => rfl
       | cons b bs => rw [List.length_cons] at h₁; injection h₁
     | cons a as ih =>
-      cases bs with
+      cases b with
       | nil => rw [List.length_cons] at h₁; injection h₁
       | cons b bs =>
         have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
@@ -74,23 +71,23 @@ theorem ext (xs ys : Array α)
           apply this
         have tailEq : as = bs := ih bs h₁' h₂'
         rw [headEq, tailEq]
-  cases xs; cases ys
+  cases a; cases b
   apply congrArg
   apply extAux
   assumption
   assumption
 
-theorem ext' {xs ys : Array α} (h : xs.toList = ys.toList) : xs = ys := by
-  cases xs; cases ys; simp at h; rw [h]
+theorem ext' {as bs : Array α} (h : as.toList = bs.toList) : as = bs := by
+  cases as; cases bs; simp at h; rw [h]
 
 @[simp] theorem toArrayAux_eq (as : List α) (acc : Array α) : (as.toArrayAux acc).toList = acc.toList ++ as := by
   induction as generalizing acc <;> simp [*, List.toArrayAux, Array.push, List.append_assoc, List.concat_eq_append]
 
-@[simp] theorem toArray_toList (xs : Array α) : xs.toList.toArray = xs := rfl
+@[simp] theorem toArray_toList (a : Array α) : a.toList.toArray = a := rfl
 
-@[simp] theorem getElem_toList {xs : Array α} {i : Nat} (h : i < xs.size) : xs.toList[i] = xs[i] := rfl
+@[simp] theorem getElem_toList {a : Array α} {i : Nat} (h : i < a.size) : a.toList[i] = a[i] := rfl
 
-@[simp] theorem getElem?_toList {xs : Array α} {i : Nat} : xs.toList[i]? = xs[i]? := by
+@[simp] theorem getElem?_toList {a : Array α} {i : Nat} : a.toList[i]? = a[i]? := by
   simp [getElem?_def]
 
 /-- `a ∈ as` is a predicate which asserts that `a` is in the array `as`. -/
@@ -108,7 +105,7 @@ theorem mem_def {a : α} {as : Array α} : a ∈ as ↔ a ∈ as.toList :=
 @[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
   simp [mem_def]
 
-@[simp] theorem getElem_mem {xs : Array α} {i : Nat} (h : i < xs.size) : xs[i] ∈ xs := by
+@[simp] theorem getElem_mem {l : Array α} {i : Nat} (h : i < l.size) : l[i] ∈ l := by
   rw [Array.mem_def, ← getElem_toList]
   apply List.getElem_mem
 
@@ -130,22 +127,20 @@ abbrev _root_.Array.toList_toArray := @List.toList_toArray
 @[deprecated size_toArray (since := "2025-02-17")]
 abbrev _root_.Array.size_toArray := @List.size_toArray
 
-@[simp] theorem getElem_toArray {xs : List α} {i : Nat} (h : i < xs.toArray.size) :
-    xs.toArray[i] = xs[i]'(by simpa using h) := rfl
+@[simp] theorem getElem_toArray {a : List α} {i : Nat} (h : i < a.toArray.size) :
+    a.toArray[i] = a[i]'(by simpa using h) := rfl
 
-@[simp] theorem getElem?_toArray {xs : List α} {i : Nat} : xs.toArray[i]? = xs[i]? := by
+@[simp] theorem getElem?_toArray {a : List α} {i : Nat} : a.toArray[i]? = a[i]? := by
   simp [getElem?_def]
 
-@[simp] theorem getElem!_toArray [Inhabited α] {xs : List α} {i : Nat} :
-    xs.toArray[i]! = xs[i]! := by
+@[simp] theorem getElem!_toArray [Inhabited α] {a : List α} {i : Nat} :
+    a.toArray[i]! = a[i]! := by
   simp [getElem!_def]
 
 end List
 
 namespace Array
 
-theorem size_eq_length_toList (xs : Array α) : xs.size = xs.toList.length := rfl
-
 @[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @List.toList_toArray
 
 @[deprecated Array.toList (since := "2024-09-10")] abbrev Array.data := @Array.toList
@@ -170,15 +165,15 @@ def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
    `Fin` values are represented as tag pointers in the Lean runtime. Thus,
    `fset` may be slightly slower than `uset`. -/
 @[extern "lean_array_uset"]
-def uset (xs : Array α) (i : USize) (v : α) (h : i.toNat < xs.size) : Array α :=
-  xs.set i.toNat v h
+def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
+  a.set i.toNat v h
 
 @[extern "lean_array_pop"]
-def pop (xs : Array α) : Array α where
-  toList := xs.toList.dropLast
+def pop (a : Array α) : Array α where
+  toList := a.toList.dropLast
 
-@[simp] theorem size_pop (xs : Array α) : xs.pop.size = xs.size - 1 := by
-  match xs with
+@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 := by
+  match a with
   | ⟨[]⟩ => rfl
   | ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
 
@@ -193,15 +188,15 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_fswap"]
-def swap (xs : Array α) (i j : @& Nat) (hi : i < xs.size := by get_elem_tactic) (hj : j < xs.size := by get_elem_tactic) : Array α :=
-  let v₁ := xs[i]
-  let v₂ := xs[j]
-  let xs'  := xs.set i v₂
-  xs'.set j v₁ (Nat.lt_of_lt_of_eq hj (size_set xs i v₂ _).symm)
+def swap (a : Array α) (i j : @& Nat) (hi : i < a.size := by get_elem_tactic) (hj : j < a.size := by get_elem_tactic) : Array α :=
+  let v₁ := a[i]
+  let v₂ := a[j]
+  let a'  := a.set i v₂
+  a'.set j v₁ (Nat.lt_of_lt_of_eq hj (size_set a i v₂ _).symm)
 
-@[simp] theorem size_swap (xs : Array α) (i j : Nat) {hi hj} : (xs.swap i j hi hj).size = xs.size := by
-  show ((xs.set i xs[j]).set j xs[i]
-    (Nat.lt_of_lt_of_eq hj (size_set xs i xs[j] _).symm)).size = xs.size
+@[simp] theorem size_swap (a : Array α) (i j : Nat) {hi hj} : (a.swap i j hi hj).size = a.size := by
+  show ((a.set i a[j]).set j a[i]
+    (Nat.lt_of_lt_of_eq hj (size_set a i a[j] _).symm)).size = a.size
   rw [size_set, size_set]
 
 /--
@@ -211,11 +206,11 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_swap"]
-def swapIfInBounds (xs : Array α) (i j : @& Nat) : Array α :=
-  if h₁ : i < xs.size then
-  if h₂ : j < xs.size then swap xs i j
-  else xs
-  else xs
+def swapIfInBounds (a : Array α) (i j : @& Nat) : Array α :=
+  if h₁ : i < a.size then
+  if h₂ : j < a.size then swap a i j
+  else a
+  else a
 
 @[deprecated swapIfInBounds (since := "2024-11-24")] abbrev swap! := @swapIfInBounds
 
@@ -230,31 +225,31 @@ instance : EmptyCollection (Array α) := ⟨Array.empty⟩
 instance : Inhabited (Array α) where
   default := Array.empty
 
-def isEmpty (xs : Array α) : Bool :=
-  xs.size = 0
+def isEmpty (a : Array α) : Bool :=
+  a.size = 0
 
 @[specialize]
-def isEqvAux (xs ys : Array α) (hsz : xs.size = ys.size) (p : α → α → Bool) :
-    ∀ (i : Nat) (_ : i ≤ xs.size), Bool
+def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) :
+    ∀ (i : Nat) (_ : i ≤ a.size), Bool
   | 0, _ => true
   | i+1, h =>
-    p xs[i] (ys[i]'(hsz ▸ h)) && isEqvAux xs ys hsz p i (Nat.le_trans (Nat.le_add_right i 1) h)
+    p a[i] (b[i]'(hsz ▸ h)) && isEqvAux a b hsz p i (Nat.le_trans (Nat.le_add_right i 1) h)
 
-@[inline] def isEqv (xs ys : Array α) (p : α → α → Bool) : Bool :=
-  if h : xs.size = ys.size then
-    isEqvAux xs ys h p xs.size (Nat.le_refl xs.size)
+@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
+  if h : a.size = b.size then
+    isEqvAux a b h p a.size (Nat.le_refl a.size)
   else
     false
 
 instance [BEq α] : BEq (Array α) :=
-  ⟨fun xs ys => isEqv xs ys BEq.beq⟩
+  ⟨fun a b => isEqv a b BEq.beq⟩
 
 /--
 `ofFn f` with `f : Fin n → α` returns the list whose ith element is `f i`.
 ```
 ofFn f = #[f 0, f 1, ... , f(n - 1)]
 ``` -/
-def ofFn {n} (f : Fin n → α) : Array α := go 0 (emptyWithCapacity n) where
+def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
   /-- Auxiliary for `ofFn`. `ofFn.go f i acc = acc ++ #[f i, ..., f(n - 1)]` -/
   @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
   go (i : Nat) (acc : Array α) : Array α :=
@@ -277,8 +272,8 @@ Return the last element of an array, or panic if the array is empty.
 See `back` for the version that requires a proof the array is non-empty,
 or `back?` for the version that returns an option.
 -/
-def back! [Inhabited α] (xs : Array α) : α :=
-  xs[xs.size - 1]!
+def back! [Inhabited α] (a : Array α) : α :=
+  a[a.size - 1]!
 
 /--
 Return the last element of an array, given a proof that the array is not empty.
@@ -286,8 +281,8 @@ Return the last element of an array, given a proof that the array is not empty.
 See `back!` for the version that panics if the array is empty,
 or `back?` for the version that returns an option.
 -/
-def back (xs : Array α) (h : 0 < xs.size := by get_elem_tactic) : α :=
-  xs[xs.size - 1]'(Nat.sub_one_lt_of_lt h)
+def back (a : Array α) (h : 0 < a.size := by get_elem_tactic) : α :=
+  a[a.size - 1]'(Nat.sub_one_lt_of_lt h)
 
 /--
 Return the last element of an array, or `none` if the array is empty.
@@ -295,73 +290,72 @@ Return the last element of an array, or `none` if the array is empty.
 See `back!` for the version that panics if the array is empty,
 or `back` for the version that requires a proof the array is non-empty.
 -/
-def back? (xs : Array α) : Option α :=
-  xs[xs.size - 1]?
+def back? (a : Array α) : Option α :=
+  a[a.size - 1]?
 
 @[deprecated "Use `a[i]?` instead." (since := "2025-02-12")]
-def get? (xs : Array α) (i : Nat) : Option α :=
-  if h : i < xs.size then some xs[i] else none
+def get? (a : Array α) (i : Nat) : Option α :=
+  if h : i < a.size then some a[i] else none
 
-@[inline] def swapAt (xs : Array α) (i : Nat) (v : α) (hi : i < xs.size := by get_elem_tactic) : α × Array α :=
-  let e := xs[i]
-  let xs' := xs.set i v
-  (e, xs')
+@[inline] def swapAt (a : Array α) (i : Nat) (v : α) (hi : i < a.size := by get_elem_tactic) : α × Array α :=
+  let e := a[i]
+  let a := a.set i v
+  (e, a)
 
 @[inline]
-def swapAt! (xs : Array α) (i : Nat) (v : α) : α × Array α :=
-  if h : i < xs.size then
-    swapAt xs i v
+def swapAt! (a : Array α) (i : Nat) (v : α) : α × Array α :=
+  if h : i < a.size then
+    swapAt a i v
   else
-    have : Inhabited (α × Array α) := ⟨(v, xs)⟩
+    have : Inhabited (α × Array α) := ⟨(v, a)⟩
     panic! ("index " ++ toString i ++ " out of bounds")
 
 /-- `shrink a n` returns the first `n` elements of `a`, implemented by repeatedly popping the last element. -/
-def shrink (xs : Array α) (n : Nat) : Array α :=
+def shrink (a : Array α) (n : Nat) : Array α :=
   let rec loop
-    | 0,   xs => xs
-    | n+1, xs => loop n xs.pop
-  loop (xs.size - n) xs
+    | 0,   a => a
+    | n+1, a => loop n a.pop
+  loop (a.size - n) a
 
 /-- `take a n` returns the first `n` elements of `a`, implemented by copying the first `n` elements. -/
-abbrev take (xs : Array α) (i : Nat) : Array α := extract xs 0 i
+abbrev take (a : Array α) (n : Nat) : Array α := extract a 0 n
 
-@[simp] theorem take_eq_extract (xs : Array α) (i : Nat) : xs.take i = xs.extract 0 i := rfl
+@[simp] theorem take_eq_extract (a : Array α) (n : Nat) : a.take n = a.extract 0 n := rfl
 
 /-- `drop a n` removes the first `n` elements of `a`, implemented by copying the remaining elements. -/
-abbrev drop (xs : Array α) (i : Nat) : Array α := extract xs i xs.size
+abbrev drop (a : Array α) (n : Nat) : Array α := extract a n a.size
 
-@[simp] theorem drop_eq_extract (xs : Array α) (i : Nat) : xs.drop i = xs.extract i xs.size := rfl
+@[simp] theorem drop_eq_extract (a : Array α) (n : Nat) : a.drop n = a.extract n a.size := rfl
 
 @[inline]
-unsafe def modifyMUnsafe [Monad m] (xs : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
-  if h : i < xs.size then
-    let v                := xs[i]
+unsafe def modifyMUnsafe [Monad m] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
+  if h : i < a.size then
+    let v                := a[i]
     -- Replace a[i] by `box(0)`.  This ensures that `v` remains unshared if possible.
     -- Note: we assume that arrays have a uniform representation irrespective
     -- of the element type, and that it is valid to store `box(0)` in any array.
-    let xs'               := xs.set i (unsafeCast ())
+    let a'               := a.set i (unsafeCast ())
     let v ← f v
-    pure <| xs'.set i v (Nat.lt_of_lt_of_eq h (size_set xs ..).symm)
+    pure <| a'.set i v (Nat.lt_of_lt_of_eq h (size_set a ..).symm)
   else
-    pure xs
+    pure a
 
 @[implemented_by modifyMUnsafe]
-def modifyM [Monad m] (xs : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
-  if h : i < xs.size then
-    let v   := xs[i]
+def modifyM [Monad m] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
+  if h : i < a.size then
+    let v   := a[i]
     let v ← f v
-    pure <| xs.set i v
+    pure <| a.set i v
   else
-    pure xs
+    pure a
 
 @[inline]
-def modify (xs : Array α) (i : Nat) (f : α → α) : Array α :=
-  Id.run <| modifyM xs i f
+def modify (a : Array α) (i : Nat) (f : α → α) : Array α :=
+  Id.run <| modifyM a i f
 
-set_option linter.indexVariables false in -- Changing `idx` causes bootstrapping issues, haven't investigated.
 @[inline]
-def modifyOp (xs : Array α) (idx : Nat) (f : α → α) : Array α :=
-  xs.modify idx f
+def modifyOp (self : Array α) (idx : Nat) (f : α → α) : Array α :=
+  self.modify idx f
 
 /--
   We claim this unsafe implementation is correct because an array cannot have more than `usizeSz` elements in our runtime.
@@ -480,17 +474,17 @@ def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
 @[inline]
 unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
   let sz := as.usize
-  let rec @[specialize] map (i : USize) (bs : Array NonScalar) : m (Array PNonScalar.{v}) := do
+  let rec @[specialize] map (i : USize) (r : Array NonScalar) : m (Array PNonScalar.{v}) := do
     if i < sz then
-     let v    := bs.uget i lcProof
-     -- Replace bs[i] by `box(0)`.  This ensures that `v` remains unshared if possible.
+     let v    := r.uget i lcProof
+     -- Replace r[i] by `box(0)`.  This ensures that `v` remains unshared if possible.
      -- Note: we assume that arrays have a uniform representation irrespective
      -- of the element type, and that it is valid to store `box(0)` in any array.
-     let bs'    := bs.uset i default lcProof
+     let r    := r.uset i default lcProof
      let vNew ← f (unsafeCast v)
-     map (i+1) (bs'.uset i (unsafeCast vNew) lcProof)
+     map (i+1) (r.uset i (unsafeCast vNew) lcProof)
     else
-     pure (unsafeCast bs)
+     pure (unsafeCast r)
   unsafeCast <| map 0 (unsafeCast as)
 
 /-- Reference implementation for `mapM` -/
@@ -499,13 +493,13 @@ def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
   -- Note: we cannot use `foldlM` here for the reference implementation because this calls
   -- `bind` and `pure` too many times. (We are not assuming `m` is a `LawfulMonad`)
   let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-    map (i : Nat) (bs : Array β) : m (Array β) := do
+    map (i : Nat) (r : Array β) : m (Array β) := do
       if hlt : i < as.size then
-        map (i+1) (bs.push (← f as[i]))
+        map (i+1) (r.push (← f as[i]))
       else
-        pure bs
+        pure r
   decreasing_by simp_wf; decreasing_trivial_pre_omega
-  map 0 (emptyWithCapacity as.size)
+  map 0 (mkEmpty as.size)
 
 @[deprecated mapM (since := "2024-11-11")] abbrev sequenceMap := @mapM
 
@@ -522,7 +516,7 @@ def mapFinIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m]
         apply Nat.le_add_right
       have : i + (j + 1) = as.size := by rw [← inv, Nat.add_comm j 1, Nat.add_assoc]
       map i (j+1) this (bs.push (← f j as[j] j_lt))
-  map as.size 0 rfl (emptyWithCapacity as.size)
+  map as.size 0 rfl (mkEmpty as.size)
 
 @[inline]
 def mapIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : Nat → α → m β) (as : Array α) : m (Array β) :=
@@ -684,8 +678,8 @@ def mapIdx {α : Type u} {β : Type v} (f : Nat → α → β) (as : Array α) :
   Id.run <| as.mapIdxM f
 
 /-- Turns `#[a, b]` into `#[(a, 0), (b, 1)]`. -/
-def zipIdx (xs : Array α) (start := 0) : Array (α × Nat) :=
-  xs.mapIdx fun i a => (a, start + i)
+def zipIdx (arr : Array α) (start := 0) : Array (α × Nat) :=
+  arr.mapIdx fun i a => (a, start + i)
 
 @[deprecated zipIdx (since := "2025-01-21")] abbrev zipWithIndex := @zipIdx
 
@@ -702,8 +696,8 @@ def findSome? {α : Type u} {β : Type v} (f : α → Option β) (as : Array α)
   Id.run <| as.findSomeM? f
 
 @[inline]
-def findSome! {α : Type u} {β : Type v} [Inhabited β] (f : α → Option β) (xs : Array α) : β :=
-  match xs.findSome? f with
+def findSome! {α : Type u} {β : Type v} [Inhabited β] (f : α → Option β) (a : Array α) : β :=
+  match a.findSome? f with
   | some b => b
   | none   => panic! "failed to find element"
 
@@ -757,18 +751,18 @@ theorem findIdx?_eq_map_findFinIdx?_val {xs : Array α} {p : α → Bool} :
 def findIdx (p : α → Bool) (as : Array α) : Nat := (as.findIdx? p).getD as.size
 
 @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-def idxOfAux [BEq α] (xs : Array α) (v : α) (i : Nat) : Option (Fin xs.size) :=
-  if h : i < xs.size then
-    if xs[i] == v then some ⟨i, h⟩
-    else idxOfAux xs v (i+1)
+def idxOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
+  if h : i < a.size then
+    if a[i] == v then some ⟨i, h⟩
+    else idxOfAux a v (i+1)
   else none
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 @[deprecated idxOfAux (since := "2025-01-29")]
 abbrev indexOfAux := @idxOfAux
 
-def finIdxOf? [BEq α] (xs : Array α) (v : α) : Option (Fin xs.size) :=
-  idxOfAux xs v 0
+def finIdxOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
+  idxOfAux a v 0
 
 @[deprecated "`Array.indexOf?` has been deprecated, use `idxOf?` or `finIdxOf?` instead." (since := "2025-01-29")]
 abbrev indexOf? := @finIdxOf?
@@ -776,12 +770,12 @@ abbrev indexOf? := @finIdxOf?
 /-- Returns the index of the first element equal to `a`, or the length of the array otherwise. -/
 def idxOf [BEq α] (a : α) : Array α → Nat := findIdx (· == a)
 
-def idxOf? [BEq α] (xs : Array α) (v : α) : Option Nat :=
-  (xs.finIdxOf? v).map (·.val)
+def idxOf? [BEq α] (a : Array α) (v : α) : Option Nat :=
+  (a.finIdxOf? v).map (·.val)
 
 @[deprecated idxOf? (since := "2024-11-20")]
-def getIdx? [BEq α] (xs : Array α) (v : α) : Option Nat :=
-  xs.findIdx? fun a => a == v
+def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
+  a.findIdx? fun a => a == v
 
 @[inline]
 def any (as : Array α) (p : α → Bool) (start := 0) (stop := as.size) : Bool :=
@@ -816,12 +810,12 @@ def toListAppend (as : Array α) (l : List α) : List α :=
   as.foldr List.cons l
 
 protected def append (as : Array α) (bs : Array α) : Array α :=
-  bs.foldl (init := as) fun xs v => xs.push v
+  bs.foldl (init := as) fun r v => r.push v
 
 instance : Append (Array α) := ⟨Array.append⟩
 
 protected def appendList (as : Array α) (bs : List α) : Array α :=
-  bs.foldl (init := as) fun xs v => xs.push v
+  bs.foldl (init := as) fun r v => r.push v
 
 instance : HAppend (Array α) (List α) (Array α) := ⟨Array.appendList⟩
 
@@ -841,8 +835,8 @@ def flatMap (f : α → Array β) (as : Array α) : Array β :=
 
 `flatten #[#[a₁, a₂, ⋯], #[b₁, b₂, ⋯], ⋯]` = `#[a₁, a₂, ⋯, b₁, b₂, ⋯]`
 -/
-@[inline] def flatten (xss : Array (Array α)) : Array α :=
-  xss.foldl (init := empty) fun acc xs => acc ++ xs
+@[inline] def flatten (as : Array (Array α)) : Array α :=
+  as.foldl (init := empty) fun r a => r ++ a
 
 def reverse (as : Array α) : Array α :=
   if h : as.size ≤ 1 then
@@ -864,18 +858,18 @@ where
 
 @[inline]
 def filter (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Array α :=
-  as.foldl (init := #[]) (start := start) (stop := stop) fun acc a =>
-    if p a then acc.push a else acc
+  as.foldl (init := #[]) (start := start) (stop := stop) fun r a =>
+    if p a then r.push a else r
 
 @[inline]
 def filterM {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
-  as.foldlM (init := #[]) (start := start) (stop := stop) fun acc a => do
-    if (← p a) then return acc.push a else return acc
+  as.foldlM (init := #[]) (start := start) (stop := stop) fun r a => do
+    if (← p a) then return r.push a else return r
 
 @[inline]
 def filterRevM {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start := as.size) (stop := 0) : m (Array α) :=
-  reverse <$> as.foldrM (init := #[]) (start := start) (stop := stop) fun a acc => do
-    if (← p a) then return acc.push a else return acc
+  reverse <$> as.foldrM (init := #[]) (start := start) (stop := stop) fun a r => do
+    if (← p a) then return r.push a else return r
 
 @[specialize]
 def filterMapM [Monad m] (f : α → m (Option β)) (as : Array α) (start := 0) (stop := as.size) : m (Array β) :=
@@ -925,15 +919,15 @@ decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def takeWhile (p : α → Bool) (as : Array α) : Array α :=
   let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-  go (i : Nat) (acc : Array α) : Array α :=
+  go (i : Nat) (r : Array α) : Array α :=
     if h : i < as.size then
       let a := as[i]
       if p a then
-        go (i+1) (acc.push a)
+        go (i+1) (r.push a)
       else
-        acc
+        r
     else
-      acc
+      r
     decreasing_by simp_wf; decreasing_trivial_pre_omega
   go 0 #[]
 
@@ -944,22 +938,22 @@ using a `Nat` index and a tactic-provided bound.
 This function takes worst case O(n) time because
 it has to backshift all elements at positions greater than `i`.-/
 @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
-def eraseIdx (xs : Array α) (i : Nat) (h : i < xs.size := by get_elem_tactic) : Array α :=
-  if h' : i + 1 < xs.size then
-    let xs' := xs.swap (i + 1) i
-    xs'.eraseIdx (i + 1) (by simp [xs', h'])
+def eraseIdx (a : Array α) (i : Nat) (h : i < a.size := by get_elem_tactic) : Array α :=
+  if h' : i + 1 < a.size then
+    let a' := a.swap (i + 1) i
+    a'.eraseIdx (i + 1) (by simp [a', h'])
   else
-    xs.pop
-termination_by xs.size - i
+    a.pop
+termination_by a.size - i
 decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ h
 
 -- This is required in `Lean.Data.PersistentHashMap`.
-@[simp] theorem size_eraseIdx (xs : Array α) (i : Nat) (h) : (xs.eraseIdx i h).size = xs.size - 1 := by
-  induction xs, i, h using Array.eraseIdx.induct with
-  | @case1 xs i h h' xs' ih =>
+@[simp] theorem size_eraseIdx (a : Array α) (i : Nat) (h) : (a.eraseIdx i h).size = a.size - 1 := by
+  induction a, i, h using Array.eraseIdx.induct with
+  | @case1 a i h h' a' ih =>
     unfold eraseIdx
-    simp +zetaDelta [h', xs', ih]
-  | case2 xs i h h' =>
+    simp +zetaDelta [h', a', ih]
+  | case2 a i h h' =>
     unfold eraseIdx
     simp [h']
 
@@ -967,15 +961,15 @@ decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ h
 
   This function takes worst case O(n) time because
   it has to backshift all elements at positions greater than `i`.-/
-def eraseIdxIfInBounds (xs : Array α) (i : Nat) : Array α :=
-  if h : i < xs.size then xs.eraseIdx i h else xs
+def eraseIdxIfInBounds (a : Array α) (i : Nat) : Array α :=
+  if h : i < a.size then a.eraseIdx i h else a
 
 /-- Remove the element at a given index from an array, or panic if the index is out of bounds.
 
 This function takes worst case O(n) time because
 it has to backshift all elements at positions greater than `i`. -/
-def eraseIdx! (xs : Array α) (i : Nat) : Array α :=
-  if h : i < xs.size then xs.eraseIdx i h else panic! "invalid index"
+def eraseIdx! (a : Array α) (i : Nat) : Array α :=
+  if h : i < a.size then a.eraseIdx i h else panic! "invalid index"
 
 /-- Remove a specified element from an array, or do nothing if it is not present.
 
@@ -1092,11 +1086,6 @@ def split (as : Array α) (p : α → Bool) : Array α × Array α :=
   as.foldl (init := (#[], #[])) fun (as, bs) a =>
     if p a then (as.push a, bs) else (as, bs.push a)
 
-def replace [BEq α] (xs : Array α) (a b : α) : Array α :=
-  match xs.finIdxOf? a with
-  | none => xs
-  | some i => xs.set i b
-
 /-! ### Lexicographic ordering -/
 
 instance instLT [LT α] : LT (Array α) := ⟨fun as bs => as.toList < bs.toList⟩
@@ -1109,20 +1098,6 @@ instance instLE [LT α] : LE (Array α) := ⟨fun as bs => as.toList ≤ bs.toLi
 We do not currently intend to provide verification theorems for these functions.
 -/
 
-/-! ### leftpad and rightpad -/
-
-/--
-Pads `l : Array α` on the left with repeated occurrences of `a : α` until it is of size `n`.
-If `l` is initially larger than `n`, just return `l`.
--/
-def leftpad (n : Nat) (a : α) (xs : Array α) : Array α := mkArray (n - xs.size) a ++ xs
-
-/--
-Pads `l : Array α` on the right with repeated occurrences of `a : α` until it is of size `n`.
-If `l` is initially larger than `n`, just return `l`.
--/
-def rightpad (n : Nat) (a : α) (xs : Array α) : Array α := xs ++ mkArray (n - xs.size) a
-
 /- ### reduceOption -/
 
 /-- Drop `none`s from a Array, and replace each remaining `some a` with `a`. -/
@@ -1137,9 +1112,9 @@ def rightpad (n : Nat) (a : α) (xs : Array α) : Array α := xs ++ mkArray (n -
 -/
 def eraseReps {α} [BEq α] (as : Array α) : Array α :=
   if h : 0 < as.size then
-    let ⟨last, acc⟩ := as.foldl (init := (as[0], #[])) fun ⟨last, acc⟩ a =>
-      if a == last then ⟨last, acc⟩ else ⟨a, acc.push last⟩
-    acc.push last
+    let ⟨last, r⟩ := as.foldl (init := (as[0], #[])) fun ⟨last, r⟩ a =>
+      if a == last then ⟨last, r⟩ else ⟨a, r.push last⟩
+    r.push last
   else
     #[]
 
@@ -1165,24 +1140,24 @@ def allDiff [BEq α] (as : Array α) : Bool :=
 /-! ### getEvenElems -/
 
 @[inline] def getEvenElems (as : Array α) : Array α :=
-  (·.2) <| as.foldl (init := (true, Array.empty)) fun (even, acc) a =>
+  (·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
     if even then
-      (false, acc.push a)
+      (false, r.push a)
     else
-      (true, acc)
+      (true, r)
 
 /-! ### Repr and ToString -/
 
 instance {α : Type u} [Repr α] : Repr (Array α) where
-  reprPrec xs _ :=
+  reprPrec a _ :=
     let _ : Std.ToFormat α := ⟨repr⟩
-    if xs.size == 0 then
+    if a.size == 0 then
       "#[]"
     else
-      Std.Format.bracketFill "#[" (Std.Format.joinSep (toList xs) ("," ++ Std.Format.line)) "]"
+      Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
 
 instance [ToString α] : ToString (Array α) where
-  toString xs := "#" ++ toString xs.toList
+  toString a := "#" ++ toString a.toList
 
 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

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) :

src/Init/Data/Array/Bootstrap.lean

@@ -13,9 +13,6 @@ 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
 
 /--
@@ -42,122 +39,122 @@ 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 toList_pop (a : Array α) : a.pop.toList = a.toList.dropLast := rfl
 
 @[deprecated toList_pop (since := "2025-02-17")]
 abbrev pop_toList := @Array.toList_pop
 
-@[simp] theorem append_eq_append (xs ys : Array α) : xs.append ys = xs ++ ys := rfl
+@[simp] theorem append_eq_append (arr arr' : Array α) : arr.append arr' = arr ++ arr' := 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")]

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 [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

@@ -9,9 +9,6 @@ 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
@@ -87,9 +84,9 @@ theorem isEqv_self [DecidableEq α] (xs : Array α) : Array.isEqv xs xs (· = ·
   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) :

src/Init/Data/Array/Erase.lean

@@ -12,9 +12,6 @@ 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
@@ -23,11 +20,11 @@ open Nat
 
 @[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
@@ -282,12 +279,8 @@ 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⟩
+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 [List.size_toArray] at h
   simp only [List.eraseIdx_toArray, List.eraseIdx_eq_take_drop_succ, take_eq_extract,
     List.extract_toArray, List.extract_eq_drop_take, Nat.sub_zero, List.drop_zero, drop_eq_extract,
@@ -295,61 +288,61 @@ theorem eraseIdx_eq_take_drop_succ (xs : Array α) (i : Nat) (h) : xs.eraseIdx i
   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] 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, List.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

@@ -13,9 +13,6 @@ import Init.Data.List.Nat.TakeDrop
 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
 
@@ -357,10 +354,10 @@ theorem takeWhile_append {xs ys : Array α} :
   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] theorem takeWhile_append_of_pos {p : α → Bool} {l₁ l₂ : Array α} (h : ∀ a ∈ l₁, p a) :
+    (l₁ ++ l₂).takeWhile p = l₁ ++ l₂.takeWhile p := by
+  rcases l₁ with ⟨l₁⟩
+  rcases l₂ with ⟨l₂⟩
   simp at h
   simp [List.takeWhile_append_of_pos h]
 
@@ -377,10 +374,10 @@ theorem popWhile_append {xs ys : Array α} :
   · 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] theorem popWhile_append_of_pos {p : α → Bool} {l₁ l₂ : Array α} (h : ∀ a ∈ l₂, p a) :
+    (l₁ ++ l₂).popWhile p = l₁.popWhile p := by
+  rcases l₁ with ⟨l₁⟩
+  rcases l₂ with ⟨l₂⟩
   simp at h
   simp only [List.append_toArray, List.popWhile_toArray, List.reverse_append, mk.injEq,
     List.reverse_inj]

src/Init/Data/Array/FinRange.lean

@@ -7,9 +7,6 @@ prelude
 import Init.Data.List.FinRange
 import Init.Data.Array.OfFn
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
-
 namespace Array
 
 /-- `finRange n` is the array of all elements of `Fin n` in order. -/

src/Init/Data/Array/Find.lean

@@ -13,79 +13,77 @@ 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,
@@ -93,12 +91,17 @@ theorem getElem_zero_flatten.proof {xss : Array (Array α)} (h : 0 < xss.flatten
   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
@@ -374,21 +395,21 @@ 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)) :
@@ -462,8 +483,8 @@ 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 +494,12 @@ 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 :
@@ -513,10 +534,10 @@ 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} :
@@ -524,55 +545,14 @@ theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : Array α} {p q : α → Bo
   cases xs
   simp
 
-/-! ### findFinIdx? -/
-
-@[simp] theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p #[] = none := rfl
-
--- We can't mark this as a `@[congr]` lemma since the head of the RHS is not `findFinIdx?`.
-theorem findFinIdx?_congr {p : α → Bool} {xs ys : Array α} (w : xs = ys) :
-    findFinIdx? p xs = (findFinIdx? p ys).map (fun i => i.cast (by simp [w])) := by
-  subst w
-  simp
-
-theorem findFinIdx?_eq_pmap_findIdx? {xs : Array α} {p : α → Bool} :
-    xs.findFinIdx? p =
-      (xs.findIdx? p).pmap
-        (fun i m => by simp [findIdx?_eq_some_iff_getElem] at m; exact ⟨i, m.choose⟩)
-        (fun i h => h) := by
-  simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]
-
-@[simp] theorem findFinIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
-    xs.findFinIdx? p = none ↔ ∀ x, x ∈ xs → ¬ p x := by
-  simp [findFinIdx?_eq_pmap_findIdx?]
-
-@[simp]
-theorem findFinIdx?_eq_some_iff {xs : Array α} {p : α → Bool} {i : Fin xs.size} :
-    xs.findFinIdx? p = some i ↔
-      p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji i.2)) := by
-  simp only [findFinIdx?_eq_pmap_findIdx?, Option.pmap_eq_some_iff, findIdx?_eq_some_iff_getElem,
-    Bool.not_eq_true, Option.mem_def, exists_and_left, and_exists_self, Fin.getElem_fin]
-  constructor
-  · rintro ⟨a, ⟨h, w₁, w₂⟩, rfl⟩
-    exact ⟨w₁, fun j hji => by simpa using w₂ j hji⟩
-  · rintro ⟨h, w⟩
-    exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩
-
-@[simp] theorem findFinIdx?_subtype {p : α → Prop} {xs : Array { x // p x }}
-    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    xs.findFinIdx? f = (xs.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
-  cases xs
-  simp only [List.findFinIdx?_toArray, hf, List.findFinIdx?_subtype]
-  rw [findFinIdx?_congr List.unattach_toArray]
-  simp [Function.comp_def]
-
 /-! ### idxOf
 
 The verification API for `idxOf` is still incomplete.
 The lemmas below should be made consistent with those for `findIdx` (and proved using them).
 -/
 
-theorem idxOf_append [BEq α] [LawfulBEq α] {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 +560,12 @@ 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)]
 
 
@@ -597,31 +577,15 @@ The lemmas below should be made consistent with those for `findIdx?` (and proved
 
 @[simp] theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := rfl
 
-@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {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 := rfl
-
-@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
-    xs.finIdxOf? a = none ↔ a ∉ xs := by
-  rcases xs with ⟨xs⟩
-  simp [List.finIdxOf?_eq_none_iff]
-
-@[simp] theorem finIdxOf?_eq_some_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} {i : Fin xs.size} :
-    xs.finIdxOf? a = some i ↔ xs[i] = a ∧ ∀ j (_ : j < i), ¬xs[j] = a := by
-  rcases xs with ⟨xs⟩
-  simp [List.finIdxOf?_eq_some_iff]
-
 end Array

src/Init/Data/Array/GetLit.lean

@@ -7,43 +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 _
+  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

@@ -6,26 +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.
-
-@[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

@@ -8,8 +8,8 @@ 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.
+-- set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 namespace Array
 

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

@@ -7,9 +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
@@ -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
@@ -362,30 +358,30 @@ 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
@@ -397,49 +393,49 @@ theorem mapIdx_eq_append_iff {xs : Array α} {f : Nat → α → β} {ys zs : Ar
     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] theorem back_mapIdx {l : Array α} {f : Nat → α → β} (h) :
+    (l.mapIdx f).back h = f (l.size - 1) (l.back (by simpa using h)) := by
+  rcases l with ⟨l⟩
   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
@@ -484,15 +480,15 @@ 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 <$> xs.mapFinIdxM f = xs.toList.mapFinIdxM f := by
+theorem toList_mapFinIdxM [Monad m] [LawfulMonad m] (l : Array α)
+    (f : (i : Nat) → α → (h : i < l.size) → m β) :
+    toList <$> l.mapFinIdxM f = l.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)
+theorem toList_mapIdxM [Monad m] [LawfulMonad m] (l : Array α)
     (f : Nat → α → m β) :
-    toList <$> xs.mapIdxM f = xs.toList.mapIdxM f := by
+    toList <$> l.mapIdxM f = l.toList.mapIdxM f := by
   rw [List.mapIdxM_toArray]
   simp only [Functor.map_map, id_map']
 

src/Init/Data/Array/Mem.lean

@@ -8,9 +8,6 @@ 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

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⟩
+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, List.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]
 
 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⟩
+    (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
 
 /--
@@ -194,67 +184,41 @@ 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]
-    (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⟩
+  cases l
   simp only [List.forIn_toArray, List.forIn_eq_foldlM, List.size_toArray, List.foldlM_toArray']
   congr
 
 /-- We can express a for loop over an array which always yields as a fold. -/
 @[simp] theorem forIn_yield_eq_foldlM [Monad m] [LawfulMonad m]
-    (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]
 
 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⟩
+    (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⟩
-  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
+    (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
 
 end Array
@@ -320,7 +284,7 @@ theorem filterMapM_toArray [Monad m] [LawfulMonad m] (l : List α) (f : α → m
   | 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
+    congr; funext a
     conv => lhs; rw [Array.toArray_append, ← flatten_concat, ← reverse_cons]
     exact ih _
 
@@ -352,23 +316,23 @@ namespace Array
   subst w
   simp [flatMapM, h]
 
-theorem toList_filterM [Monad m] [LawfulMonad m] (xs : Array α) (p : α → m Bool) :
-    toList <$> xs.filterM p = xs.toList.filterM p := by
+theorem toList_filterM [Monad m] [LawfulMonad m] (a : Array α) (p : α → m Bool) :
+    toList <$> a.filterM p = a.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 <$> xs.filterRevM p = xs.toList.filterRevM p := by
+theorem toList_filterRevM [Monad m] [LawfulMonad m] (a : Array α) (p : α → m Bool) :
+    toList <$> a.filterRevM p = a.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 <$> xs.filterMapM f = xs.toList.filterMapM f := by
+theorem toList_filterMapM [Monad m] [LawfulMonad m] (a : Array α) (f : α → m (Option β)) :
+    toList <$> a.filterMapM f = a.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
+theorem toList_flatMapM [Monad m] [LawfulMonad m] (a : Array α) (f : α → m (Array β)) :
+    toList <$> a.flatMapM f = a.toList.flatMapM (fun a => toList <$> f a) := by
   rw [List.flatMapM_toArray]
   simp only [Functor.map_map, id_map']
 
@@ -378,21 +342,21 @@ theorem toList_flatMapM [Monad m] [LawfulMonad m] (xs : Array α) (f : α → m
 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
+@[wf_preprocess] theorem foldlM_wfParam [Monad m] (xs : Array α) (f : β → α → m β) :
+    (wfParam xs).foldlM f = xs.attach.unattach.foldlM f := 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⟩ =>
+@[wf_preprocess] theorem foldlM_unattach [Monad m] (P : α → Prop) (xs : Array (Subtype P)) (f : β → α → m β) :
+    xs.unattach.foldlM f = xs.foldlM fun b ⟨x, h⟩ =>
       binderNameHint b f <| binderNameHint x (f b) <| binderNameHint h () <|
       f b (wfParam x) := by
   simp [wfParam]
@@ -401,22 +365,22 @@ and simplifies these to the function directly taking 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 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
+@[wf_preprocess] theorem foldrM_wfParam [Monad m] [LawfulMonad m] (xs : Array α) (f : α → β → m β) :
+    (wfParam xs).foldrM f = xs.attach.unattach.foldrM f := 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 =>
+@[wf_preprocess] theorem foldrM_unattach [Monad m] [LawfulMonad m] (P : α → Prop) (xs : Array (Subtype P)) (f : α → β → m β) :
+    xs.unattach.foldrM f = xs.foldrM fun ⟨x, h⟩ b =>
       binderNameHint x f <| binderNameHint h () <| binderNameHint b (f x) <|
       f (wfParam x) b := by
   simp [wfParam]
@@ -425,10 +389,10 @@ and simplifies these to the function directly taking the value.
 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]
 
@@ -441,11 +405,11 @@ and simplifies these to the function directly taking the value.
       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
+@[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) (w : stop = l.size) :
+    l.filterMapM f 0 stop = l.unattach.filterMapM g := by
   subst w
-  rcases xs with ⟨xs⟩
+  rcases l with ⟨l⟩
   simp
   rw [List.filterMapM_subtype hf]
 
@@ -461,14 +425,15 @@ and simplifies these to the function directly taking the value.
       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 }}
+@[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) :
-    (xs.flatMapM f) = xs.unattach.flatMapM g := by
-  rcases xs with ⟨xs⟩
+    (l.flatMapM f) = l.unattach.flatMapM g := by
+  rcases l with ⟨l⟩
   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

src/Init/Data/Array/OfFn.lean

@@ -11,30 +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 = #[] := rfl
-
-theorem ofFn_succ (f : Fin (n+1) → α) :
-    ofFn f = (ofFn (fun (i : Fin n) => f i.castSucc)).push (f ⟨n, by omega⟩) := by
-  ext i h₁ h₂
-  · simp
-  · simp [getElem_push]
-    split <;> rename_i h₃
-    · rfl
-    · congr
-      simp at h₁ h₂
-      omega
-
-@[simp] theorem _rooy_.List.toArray_ofFn (f : Fin n → α) : (List.ofFn f).toArray = Array.ofFn f := by
-  ext <;> simp
-
-@[simp] theorem toList_ofFn (f : Fin n → α) : (Array.ofFn f).toList = List.ofFn f := by
-  apply List.ext_getElem <;> simp
-
 @[simp]
 theorem ofFn_eq_empty_iff {f : Fin n → α} : ofFn f = #[] ↔ n = 0 := by
   rw [← Array.toList_inj]

src/Init/Data/Array/Perm.lean

@@ -7,9 +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)

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
@@ -136,7 +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
@@ -149,9 +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']
@@ -164,7 +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} :
@@ -182,48 +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]
 
@@ -231,33 +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) :
@@ -270,9 +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) :
@@ -281,12 +278,12 @@ 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

src/Init/Data/Array/Set.lean

@@ -6,9 +6,6 @@ Authors: Leonardo de Moura, Mario Carneiro
 prelude
 import Init.Tactics
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
-
 
 /--
 Set an element in an array, using a proof that the index is in bounds.
@@ -18,9 +15,9 @@ This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
 @[extern "lean_array_fset"]
-def Array.set (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
 
 /--
 Set an element in an array, or do nothing if the index is out of bounds.
@@ -28,8 +25,8 @@ Set an element in an array, or do nothing if the index is out of bounds.
 This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
 -/
-@[inline] def Array.setIfInBounds (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
 
@@ -40,5 +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

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

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

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

src/Init/Data/Array/TakeDrop.lean

@@ -12,9 +12,6 @@ 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) :
@@ -26,9 +23,9 @@ 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⟩
@@ -74,210 +71,210 @@ 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
+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, List.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
+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 [List.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
+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 [List.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
+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 [List.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/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]
@@ -483,36 +478,6 @@ 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
-
 /-! ### abs -/
 
 theorem msb_abs {w : Nat} {x : BitVec w} :
@@ -579,15 +544,6 @@ 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]
@@ -951,7 +907,7 @@ 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
 
@@ -1078,10 +1034,11 @@ 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
@@ -1095,10 +1052,11 @@ 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
@@ -1288,8 +1246,8 @@ 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)]

src/Init/Data/BitVec/Lemmas.lean

@@ -13,9 +13,7 @@ import Init.Data.Nat.Div.Lemmas
 import Init.Data.Nat.Mod
 import Init.Data.Nat.Div.Lemmas
 import Init.Data.Int.Bitwise.Lemmas
-import Init.Data.Int.LemmasAux
 import Init.Data.Int.Pow
-import Init.Data.Int.LemmasAux
 
 set_option linter.missingDocs true
 
@@ -241,16 +239,12 @@ theorem eq_of_getMsbD_eq {x y : BitVec w}
 theorem of_length_zero {x : BitVec 0} : x = 0#0 := by ext; simp [← getLsbD_eq_getElem]
 
 theorem toNat_zero_length (x : BitVec 0) : x.toNat = 0 := by simp [of_length_zero]
-theorem toInt_zero_length (x : BitVec 0) : x.toInt = 0 := by simp [of_length_zero]
-
 theorem getLsbD_zero_length (x : BitVec 0) : x.getLsbD i = false := by simp
 theorem getMsbD_zero_length (x : BitVec 0) : x.getMsbD i = false := by simp
 theorem msb_zero_length (x : BitVec 0) : x.msb = false := by simp [BitVec.msb, of_length_zero]
 
 theorem toNat_of_zero_length (h : w = 0) (x : BitVec w) : x.toNat = 0 := by
   subst h; simp [toNat_zero_length]
-theorem toInt_of_zero_length (h : w = 0) (x : BitVec w) : x.toInt = 0 := by
-  subst h; simp [toInt_zero_length]
 theorem getLsbD_of_zero_length (h : w = 0) (x : BitVec w) : x.getLsbD i = false := by
   subst h; simp [getLsbD_zero_length]
 theorem getMsbD_of_zero_length (h : w = 0) (x : BitVec w) : x.getMsbD i = false := by
@@ -329,25 +323,8 @@ theorem getMsbD_ofNatLt {n x i : Nat} (h : x < 2^n) :
 @[simp, bitvec_to_nat] theorem toNat_ofNat (x w : Nat) : (BitVec.ofNat w x).toNat = x % 2^w := by
   simp [BitVec.toNat, BitVec.ofNat, Fin.ofNat']
 
-theorem ofNatLT_eq_ofNat {w : Nat} {n : Nat} (hn) : BitVec.ofNatLT n hn = BitVec.ofNat w n :=
-  eq_of_toNat_eq (by simp [Nat.mod_eq_of_lt hn])
-
 @[simp] theorem toFin_ofNat (x : Nat) : toFin (BitVec.ofNat w x) = Fin.ofNat' (2^w) x := rfl
 
-@[simp] theorem finMk_toNat (x : BitVec w) : Fin.mk x.toNat x.isLt = x.toFin := rfl
-
-@[simp] theorem toFin_ofNatLT {n : Nat} (h : n < 2 ^ w) : (BitVec.ofNatLT n h).toFin = Fin.mk n h := rfl
-
-@[simp] theorem toFin_ofFin (n : Fin (2 ^ w)) : (BitVec.ofFin n).toFin = n := rfl
-@[simp] theorem ofFin_toFin (x : BitVec w) : BitVec.ofFin x.toFin = x := rfl
-
-@[simp] theorem ofNatLT_finVal (n : Fin (2 ^ w)) : BitVec.ofNatLT n.val n.isLt = BitVec.ofFin n := rfl
-
-@[simp] theorem ofNatLT_toNat (x : BitVec w) : BitVec.ofNatLT x.toNat x.isLt = x := rfl
-
-@[simp] theorem ofNat_finVal (n : Fin (2 ^ w)) : BitVec.ofNat w n.val = BitVec.ofFin n := by
-  rw [← BitVec.ofNatLT_eq_ofNat n.isLt, ofNatLT_finVal]
-
 -- Remark: we don't use `[simp]` here because simproc` subsumes it for literals.
 -- If `x` and `n` are not literals, applying this theorem eagerly may not be a good idea.
 theorem getLsbD_ofNat (n : Nat) (x : Nat) (i : Nat) :
@@ -551,9 +528,6 @@ theorem toInt_eq_toNat_of_msb {x : BitVec w} (h : x.msb = false) :
     x.toInt = x.toNat := by
   simp [toInt_eq_msb_cond, h]
 
-theorem toNat_toInt_of_msb {w : Nat} (b : BitVec w) (hb : b.msb = false) : b.toInt.toNat = b.toNat := by
-  simp [b.toInt_eq_toNat_of_msb hb]
-
 theorem toInt_eq_toNat_bmod (x : BitVec n) : x.toInt = Int.bmod x.toNat (2^n) := by
   simp only [toInt_eq_toNat_cond]
   split
@@ -564,16 +538,6 @@ theorem toInt_eq_toNat_bmod (x : BitVec n) : x.toInt = Int.bmod x.toNat (2^n) :=
     rw [Int.bmod_neg] <;> simp only [←Int.ofNat_emod, toNat_mod_cancel]
     omega
 
-theorem toInt_neg_of_msb_true {x : BitVec w} (h : x.msb = true) : x.toInt < 0 := by
-  simp only [BitVec.toInt]
-  have : 2 * x.toNat ≥ 2 ^ w := msb_eq_true_iff_two_mul_ge.mp h
-  omega
-
-theorem toInt_nonneg_of_msb_false {x : BitVec w} (h : x.msb = false) : 0 ≤ x.toInt := by
-  simp only [BitVec.toInt]
-  have : 2 * x.toNat < 2 ^ w := msb_eq_false_iff_two_mul_lt.mp h
-  omega
-
 /-- Prove equality of bitvectors in terms of nat operations. -/
 theorem eq_of_toInt_eq {x y : BitVec n} : x.toInt = y.toInt → x = y := by
   intro eq
@@ -605,11 +569,6 @@ theorem toInt_ofNat {n : Nat} (x : Nat) :
   have p : 0 ≤ i % (2^n : Nat) := by omega
   simp [toInt_eq_toNat_bmod, Int.toNat_of_nonneg p]
 
-theorem toInt_ofInt_eq_self {w : Nat} (hw : 0 < w) {n : Int}
-    (h : -2 ^ (w - 1) ≤ n) (h' : n < 2 ^ (w - 1)) : (BitVec.ofInt w n).toInt = n := by
-  have hw : w = (w - 1) + 1 := by omega
-  rw [toInt_ofInt, Int.bmod_eq_self_of_le] <;> (rw [hw]; simp [Int.natCast_pow]; omega)
-
 @[simp] theorem ofInt_natCast (w n : Nat) :
   BitVec.ofInt w (n : Int) = BitVec.ofNat w n := rfl
 
@@ -645,7 +604,7 @@ theorem toInt_zero {w : Nat} : (0#w).toInt = 0 := by
 `x.toInt` is less than `2^(w-1)`.
 We phrase the fact in terms of `2^w` to prevent a case split on `w=0` when the lemma is used.
 -/
-theorem two_mul_toInt_lt {w : Nat} {x : BitVec w} : 2 * x.toInt < 2 ^ w := by
+theorem toInt_lt {w : Nat} {x : BitVec w} : 2 * x.toInt < 2 ^ w := by
   simp only [BitVec.toInt]
   rcases w with _|w'
   · omega
@@ -653,25 +612,11 @@ theorem two_mul_toInt_lt {w : Nat} {x : BitVec w} : 2 * x.toInt < 2 ^ w := by
     simp only [Nat.zero_lt_succ, Nat.mul_lt_mul_left, Int.natCast_mul, Int.Nat.cast_ofNat_Int]
     norm_cast; omega
 
-theorem two_mul_toInt_le {w : Nat} {x : BitVec w} : 2 * x.toInt ≤ 2 ^ w - 1 :=
-  Int.le_sub_one_of_lt two_mul_toInt_lt
-
-theorem toInt_lt {w : Nat} {x : BitVec w} : x.toInt < 2 ^ (w - 1) := by
-  by_cases h : w = 0
-  · subst h
-    simp [eq_nil x]
-  · have := @two_mul_toInt_lt w x
-    rw_mod_cast [← Nat.two_pow_pred_add_two_pow_pred (by omega), Int.mul_comm, Int.natCast_add] at this
-    omega
-
-theorem toInt_le {w : Nat} {x : BitVec w} : x.toInt ≤ 2 ^ (w - 1) - 1 :=
-  Int.le_sub_one_of_lt toInt_lt
-
 /--
 `x.toInt` is greater than or equal to `-2^(w-1)`.
 We phrase the fact in terms of `2^w` to prevent a case split on `w=0` when the lemma is used.
 -/
-theorem le_two_mul_toInt {w : Nat} {x : BitVec w} : -2 ^ w ≤ 2 * x.toInt := by
+theorem le_toInt {w : Nat} {x : BitVec w} : -2 ^ w ≤ 2 * x.toInt := by
   simp only [BitVec.toInt]
   rcases w with _|w'
   · omega
@@ -679,16 +624,6 @@ theorem le_two_mul_toInt {w : Nat} {x : BitVec w} : -2 ^ w ≤ 2 * x.toInt := by
     simp only [Nat.zero_lt_succ, Nat.mul_lt_mul_left, Int.natCast_mul, Int.Nat.cast_ofNat_Int]
     norm_cast; omega
 
-
-theorem le_toInt {w : Nat} (x : BitVec w) : -2 ^ (w - 1) ≤ x.toInt := by
-  by_cases h : w = 0
-  · subst h
-    simp [BitVec.eq_nil x]
-  · have := le_two_mul_toInt (w := w) (x := x)
-    generalize x.toInt = x at *
-    rw [(show w = w - 1 + 1 by omega), Int.pow_succ] at this
-    omega
-
 /-! ### slt -/
 
 /--
@@ -710,12 +645,6 @@ theorem slt_zero_iff_msb_cond {x : BitVec w} : x.slt 0#w ↔ x.msb = true := by
     simp [BitVec.slt, this]
     omega
 
-theorem slt_zero_eq_msb {w : Nat} {x : BitVec  w} : x.slt 0#w = x.msb := by
-  rw [Bool.eq_iff_iff, BitVec.slt_zero_iff_msb_cond]
-
-theorem sle_iff_toInt_le {w : Nat} {b b' : BitVec w} : b.sle b' ↔ b.toInt ≤ b'.toInt :=
-  decide_eq_true_iff
-
 /-! ### setWidth, zeroExtend and truncate -/
 
 @[simp]
@@ -1067,11 +996,6 @@ theorem extractLsb'_eq_extractLsb {w : Nat} (x : BitVec w) (start len : Nat) (h
   apply eq_of_toNat_eq
   simp [extractLsb, show len - 1 + 1 = len by omega]
 
-/-- Extracting all the bits of a bitvector is an identity operation. -/
-@[simp] theorem extractLsb'_eq_self {x : BitVec w} : x.extractLsb' 0 w = x := by
-  apply eq_of_toNat_eq
-  simp [extractLsb']
-
 /-! ### allOnes -/
 
 @[simp] theorem toNat_allOnes : (allOnes v).toNat = 2^v - 1 := by
@@ -1543,16 +1467,6 @@ theorem extractLsb_not_of_lt {x : BitVec w} {hi lo : Nat} (hlo : lo ≤ hi) (hhi
   simp [hk, show k ≤ hi - lo by omega]
   omega
 
-@[simp]
-theorem ne_not_self {a : BitVec w} (h : 0 < w) : a ≠ ~~~a := by
-  have : ∃ x, x < w := ⟨w - 1, by omega⟩
-  simp [BitVec.eq_of_getElem_eq_iff, this]
-
-@[simp]
-theorem not_self_ne {a : BitVec w} (h : 0 < w) : ~~~a ≠ a := by
-  rw [ne_comm]
-  simp [h]
-
 /-! ### cast -/
 
 @[simp] theorem not_cast {x : BitVec w} (h : w = w') : ~~~(x.cast h) = (~~~x).cast h := by
@@ -1603,8 +1517,8 @@ theorem zero_shiftLeft (n : Nat) : 0#w <<< n = 0#w := by
   all_goals { simp_all <;> omega }
 
 @[simp] theorem getElem_shiftLeft {x : BitVec m} {n : Nat} (h : i < m) :
-    (x <<< n)[i] = (!decide (i < n) && x[i - n]) := by
-  rw [getElem_eq_testBit_toNat, getElem_eq_testBit_toNat]
+    (x <<< n)[i] = (!decide (i < n) && getLsbD x (i - n)) := by
+  rw [← testBit_toNat, getElem_eq_testBit_toNat]
   simp only [toNat_shiftLeft, Nat.testBit_mod_two_pow, Nat.testBit_shiftLeft, ge_iff_le]
   -- This step could be a case bashing tactic.
   cases h₁ : decide (i < m) <;> cases h₂ : decide (n ≤ i) <;> cases h₃ : decide (i < n)
@@ -1654,8 +1568,8 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
   · omega
 
 @[simp] theorem getElem_shiftLeftZeroExtend {x : BitVec m} {n : Nat} (h : i < m + n) :
-    (shiftLeftZeroExtend x n)[i] = if h' : i < n then false else x[i - n] := by
-  rw [shiftLeftZeroExtend_eq]
+    (shiftLeftZeroExtend x n)[i] = ((! decide (i < n)) && getLsbD x (i - n)) := by
+  rw [shiftLeftZeroExtend_eq, getLsbD]
   simp only [getElem_eq_testBit_toNat, getLsbD_shiftLeft, getLsbD_setWidth]
   cases h₁ : decide (i < n) <;> cases h₂ : decide (i - n < m + n)
     <;> simp_all [h]
@@ -1684,8 +1598,8 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
 theorem shiftLeft_add {w : Nat} (x : BitVec w) (n m : Nat) :
     x <<< (n + m) = (x <<< n) <<< m := by
   ext i
-  simp only [getElem_shiftLeft]
-  rw [show x[i - (n + m)] = x[i - m - n] by congr 1; omega]
+  simp only [getElem_shiftLeft, Fin.is_lt, decide_true, Bool.true_and]
+  rw [show i - (n + m) = (i - m - n) by omega]
   cases h₂ : decide (i < m) <;>
   cases h₃ : decide (i - m < w) <;>
   cases h₄ : decide (i - m < n) <;>
@@ -1718,7 +1632,7 @@ theorem getLsbD_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {i : Nat} :
   simp [shiftLeft_eq', getLsbD_shiftLeft]
 
 theorem getElem_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {i : Nat} (h : i < w₁) :
-    (x <<< y)[i] = (!decide (i < y.toNat) && x[i - y.toNat]) := by
+    (x <<< y)[i] = (!decide (i < y.toNat) && x.getLsbD (i - y.toNat)) := by
   simp
 
 @[simp] theorem shiftLeft_eq_zero {x : BitVec w} {n : Nat} (hn : w ≤ n) : x <<< n = 0#w := by
@@ -1821,7 +1735,7 @@ theorem toInt_ushiftRight {x : BitVec w} {n : Nat} :
     simp [hn]
 
 @[simp]
-theorem toFin_ushiftRight {x : BitVec w} {n : Nat} :
+theorem toFin_uShiftRight {x : BitVec w} {n : Nat} :
     (x >>> n).toFin = x.toFin / (Fin.ofNat' (2^w) (2^n)) := by
   apply Fin.eq_of_val_eq
   by_cases hn : n < w
@@ -1930,10 +1844,13 @@ theorem getLsbD_sshiftRight (x : BitVec w) (s i : Nat) :
       omega
 
 theorem getElem_sshiftRight {x : BitVec w} {s i : Nat} (h : i < w) :
-    (x.sshiftRight s)[i] = (if h : s + i < w then x[s + i] else x.msb) := by
-  rw [← getLsbD_eq_getElem, getLsbD_sshiftRight]
-  simp only [show ¬(w ≤ i) by omega, decide_false, Bool.not_false, Bool.true_and]
-  by_cases h' : s + i < w <;> simp [h']
+    (x.sshiftRight s)[i] = (if s + i < w then x.getLsbD (s + i) else x.msb) := by
+  rcases hmsb : x.msb with rfl | rfl
+  · simp only [sshiftRight_eq_of_msb_false hmsb, getElem_ushiftRight, Bool.if_false_right,
+    Bool.iff_and_self, decide_eq_true_eq]
+    intros hlsb
+    apply BitVec.lt_of_getLsbD hlsb
+  · simp [sshiftRight_eq_of_msb_true hmsb]
 
 theorem sshiftRight_xor_distrib (x y : BitVec w) (n : Nat) :
     (x ^^^ y).sshiftRight n = (x.sshiftRight n) ^^^ (y.sshiftRight n) := by
@@ -2027,118 +1944,11 @@ theorem getMsbD_sshiftRight {x : BitVec w} {i n : Nat} :
         by_cases h₄ : n + (w - 1 - i) < w <;> (simp only [h₄, ↓reduceIte]; congr; omega)
   · simp [h]
 
-theorem toInt_shiftRight_lt {x : BitVec w} {n : Nat} :
-    x.toInt >>> n < 2 ^ (w - 1) := by
-  have := @Int.shiftRight_le_of_nonneg x.toInt n
-  have := @Int.shiftRight_le_of_nonpos x.toInt n
-  have := @BitVec.toInt_lt w x
-  have := @Nat.one_le_two_pow (w-1)
-  norm_cast at *
-  omega
-
-theorem le_toInt_shiftRight {x : BitVec w} {n : Nat} :
-    -(2 ^ (w - 1)) ≤ x.toInt >>> n := by
-  have := @Int.le_shiftRight_of_nonpos x.toInt n
-  have := @Int.le_shiftRight_of_nonneg x.toInt n
-  have := @BitVec.le_toInt w x
-  have := @Nat.one_le_two_pow (w-1)
-  norm_cast at *
-  omega
-
-theorem toNat_sshiftRight_of_msb_true {x : BitVec w} {n : Nat} (h : x.msb = true) :
-    (x.sshiftRight n).toNat = 2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> n := by
-  simp [sshiftRight_eq_of_msb_true, h]
-
-theorem toNat_sshiftRight_of_msb_false {x : BitVec w} {n : Nat} (h : x.msb = false) :
-    (x.sshiftRight n).toNat = x.toNat >>> n := by
-  simp [sshiftRight_eq_of_msb_false, h]
-
-theorem toNat_sshiftRight {x : BitVec w} {n : Nat} :
-    (x.sshiftRight n).toNat =
-      if x.msb
-      then 2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> n
-      else x.toNat >>> n := by
-  by_cases h : x.msb
-  · simp [toNat_sshiftRight_of_msb_true, h]
-  · rw [Bool.not_eq_true] at h
-    simp [toNat_sshiftRight_of_msb_false, h]
-
-theorem toFin_sshiftRight_of_msb_true {x : BitVec w} {n : Nat} (h : x.msb = true) :
-    (x.sshiftRight n).toFin = Fin.ofNat' (2^w) (2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> n) := by
-  apply Fin.eq_of_val_eq
-  simp only [val_toFin, toNat_sshiftRight, h, ↓reduceIte, Fin.val_ofNat']
-  rw [Nat.mod_eq_of_lt]
-  have := x.isLt
-  have ineq : ∀ y, 2 ^ w - 1 - y < 2 ^ w := by omega
-  exact ineq ((2 ^ w - 1 - x.toNat) >>> n)
-
-theorem toFin_sshiftRight_of_msb_false {x : BitVec w} {n : Nat} (h : x.msb = false) :
-    (x.sshiftRight n).toFin = Fin.ofNat' (2^w) (x.toNat >>> n) := by
-  apply Fin.eq_of_val_eq
-  simp only [val_toFin, toNat_sshiftRight, h, Bool.false_eq_true, ↓reduceIte, Fin.val_ofNat']
-  have := Nat.shiftRight_le x.toNat n
-  rw [Nat.mod_eq_of_lt (by omega)]
-
-theorem toFin_sshiftRight {x : BitVec w} {n : Nat} :
-    (x.sshiftRight n).toFin =
-      if x.msb
-      then Fin.ofNat' (2^w) (2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> n)
-      else Fin.ofNat' (2^w) (x.toNat >>> n) := by
-  by_cases h : x.msb
-  · simp [toFin_sshiftRight_of_msb_true, h]
-  · simp [toFin_sshiftRight_of_msb_false, h]
-
-@[simp]
-theorem toInt_sshiftRight {x : BitVec w} {n : Nat} :
-    (x.sshiftRight n).toInt = x.toInt >>> n := by
-  by_cases h : w = 0
-  · subst h
-    simp [BitVec.eq_nil x]
-  · rw [sshiftRight, toInt_ofInt, ←Nat.two_pow_pred_add_two_pow_pred (by omega)]
-    have := @toInt_shiftRight_lt w x n
-    have := @le_toInt_shiftRight w x n
-    norm_cast at *
-    exact Int.bmod_eq_self_of_le (by omega) (by omega)
-
 /-! ### sshiftRight reductions from BitVec to Nat -/
 
 @[simp]
 theorem sshiftRight_eq' (x : BitVec w) : x.sshiftRight' y = x.sshiftRight y.toNat := rfl
 
-theorem toNat_sshiftRight'_of_msb_true {x y : BitVec w} (h : x.msb = true) :
-    (x.sshiftRight' y).toNat = 2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> y.toNat := by
-  rw [sshiftRight_eq', toNat_sshiftRight_of_msb_true h]
-
-theorem toNat_sshiftRight'_of_msb_false {x y : BitVec w} (h : x.msb = false) :
-    (x.sshiftRight' y).toNat = x.toNat >>> y.toNat := by
-  rw [sshiftRight_eq', toNat_sshiftRight_of_msb_false h]
-
-theorem toNat_sshiftRight' {x y : BitVec w} :
-    (x.sshiftRight' y).toNat =
-      if x.msb
-      then 2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> y.toNat
-      else x.toNat >>> y.toNat := by
-  rw [sshiftRight_eq', toNat_sshiftRight]
-
-theorem toFin_sshiftRight'_of_msb_true {x y : BitVec w} (h : x.msb = true) :
-    (x.sshiftRight' y).toFin = Fin.ofNat' (2^w) (2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> y.toNat) := by
-  rw [sshiftRight_eq', toFin_sshiftRight_of_msb_true h]
-
-theorem toFin_sshiftRight'_of_msb_false {x y : BitVec w} (h : x.msb = false) :
-    (x.sshiftRight' y).toFin = Fin.ofNat' (2^w) (x.toNat >>> y.toNat) := by
-  rw [sshiftRight_eq', toFin_sshiftRight_of_msb_false h]
-
-theorem toFin_sshiftRight' {x y : BitVec w} :
-    (x.sshiftRight' y).toFin =
-      if x.msb
-      then Fin.ofNat' (2^w) (2 ^ w - 1 - (2 ^ w - 1 - x.toNat) >>> y.toNat)
-      else Fin.ofNat' (2^w) (x.toNat >>> y.toNat) := by
-  rw [sshiftRight_eq', toFin_sshiftRight]
-
-theorem toInt_sshiftRight' {x y : BitVec w} :
-    (x.sshiftRight' y).toInt = x.toInt >>> y.toNat := by
-  rw [sshiftRight_eq', toInt_sshiftRight]
-
 -- This should not be a `@[simp]` lemma as the left hand side is not in simp normal form.
 theorem getLsbD_sshiftRight' {x y : BitVec w} {i : Nat} :
     getLsbD (x.sshiftRight' y) i =
@@ -2147,8 +1957,9 @@ theorem getLsbD_sshiftRight' {x y : BitVec w} {i : Nat} :
 
 -- This should not be a `@[simp]` lemma as the left hand side is not in simp normal form.
 theorem getElem_sshiftRight' {x y : BitVec w} {i : Nat} (h : i < w) :
-    (x.sshiftRight' y)[i] = (if h : y.toNat + i < w then x[y.toNat + i] else x.msb) := by
-  simp [show ¬ w ≤ i by omega, getElem_sshiftRight]
+    (x.sshiftRight' y)[i] =
+      (!decide (w ≤ i) && if y.toNat + i < w then x.getLsbD (y.toNat + i) else x.msb) := by
+  simp only [← getLsbD_eq_getElem, BitVec.sshiftRight', BitVec.getLsbD_sshiftRight]
 
 theorem getMsbD_sshiftRight' {x y: BitVec w} {i : Nat} :
     (x.sshiftRight y.toNat).getMsbD i =
@@ -2219,8 +2030,9 @@ theorem getMsbD_signExtend {x : BitVec w} {v i : Nat} :
     by_cases h : i < v <;> by_cases h' : v - w ≤ i <;> simp [h, h'] <;> omega
 
 theorem getElem_signExtend {x  : BitVec w} {v i : Nat} (h : i < v) :
-    (x.signExtend v)[i] = if h : i < w then x[i] else x.msb := by
-  simp [←getLsbD_eq_getElem, getLsbD_signExtend, h]
+    (x.signExtend v)[i] = if i < w then x.getLsbD i else x.msb := by
+  rw [←getLsbD_eq_getElem, getLsbD_signExtend]
+  simp [h]
 
 theorem msb_signExtend {x : BitVec w} :
     (x.signExtend v).msb = (decide (0 < v) && if w ≥ v then x.getMsbD (w - v) else x.msb) := by
@@ -2229,18 +2041,16 @@ theorem msb_signExtend {x : BitVec w} :
   · simp [h, BitVec.msb, getMsbD_signExtend, show ¬ (v - w = 0) by omega]
 
 /-- Sign extending to a width smaller than the starting width is a truncation. -/
-theorem signExtend_eq_setWidth_of_le (x : BitVec w) {v : Nat} (hv : v ≤ w) :
+theorem signExtend_eq_setWidth_of_lt (x : BitVec w) {v : Nat} (hv : v ≤ w):
   x.signExtend v = x.setWidth v := by
   ext i h
-  simp [getElem_signExtend, show i < w by omega]
-
-@[deprecated signExtend_eq_setWidth_of_le (since := "2025-03-07")]
-theorem signExtend_eq_setWidth_of_lt (x : BitVec w) {v : Nat} (hv : v ≤ w) :
-  x.signExtend v = x.setWidth v := signExtend_eq_setWidth_of_le x hv
+  simp only [getElem_signExtend, h, decide_true, Bool.true_and, getElem_setWidth,
+    ite_eq_left_iff, Nat.not_lt]
+  omega
 
 /-- Sign extending to the same bitwidth is a no op. -/
-@[simp] theorem signExtend_eq (x : BitVec w) : x.signExtend w = x := by
-  rw [signExtend_eq_setWidth_of_le _ (Nat.le_refl _), setWidth_eq]
+theorem signExtend_eq (x : BitVec w) : x.signExtend w = x := by
+  rw [signExtend_eq_setWidth_of_lt _ (Nat.le_refl _), setWidth_eq]
 
 /-- Sign extending to a larger bitwidth depends on the msb.
 If the msb is false, then the result equals the original value.
@@ -2277,65 +2087,46 @@ theorem toNat_signExtend (x : BitVec w) {v : Nat} :
     (x.signExtend v).toNat = (x.setWidth v).toNat + if x.msb then 2^v - 2^w else 0 := by
   by_cases h : v ≤ w
   · have : 2^v ≤ 2^w := Nat.pow_le_pow_right Nat.two_pos h
-    simp [signExtend_eq_setWidth_of_le x h, toNat_setWidth, Nat.sub_eq_zero_of_le this]
+    simp [signExtend_eq_setWidth_of_lt x h, toNat_setWidth, Nat.sub_eq_zero_of_le this]
   · have : 2^w ≤ 2^v := Nat.pow_le_pow_right Nat.two_pos (by omega)
     rw [toNat_signExtend_of_le x (by omega), toNat_setWidth, Nat.mod_eq_of_lt (by omega)]
 
-/--
+/-
 If the current width `w` is smaller than the extended width `v`,
 then the value when interpreted as an integer does not change.
 -/
-theorem toInt_signExtend_of_le {x : BitVec w} (h : w ≤ v) :
+theorem toInt_signExtend_of_lt {x : BitVec w} (hv : w < v):
     (x.signExtend v).toInt = x.toInt := by
-  by_cases hlt : w < v
-  · rw [toInt_signExtend_of_lt hlt]
-  · obtain rfl : w = v := by omega
-    simp
-where
-  toInt_signExtend_of_lt {x : BitVec w} (hv : w < v):
-      (x.signExtend v).toInt = x.toInt := by
-    simp only [toInt_eq_msb_cond, toNat_signExtend]
-    have : (x.signExtend v).msb = x.msb := by
-      rw [msb_eq_getLsbD_last, getLsbD_eq_getElem (Nat.sub_one_lt_of_lt hv)]
-      simp [getElem_signExtend, Nat.le_sub_one_of_lt hv]
-      omega
-    have H : 2^w ≤ 2^v := Nat.pow_le_pow_right (by omega) (by omega)
-    simp only [this, toNat_setWidth, Int.natCast_add, Int.ofNat_emod, Int.natCast_mul]
-    by_cases h : x.msb
-    <;> norm_cast
-    <;> simp [h, Nat.mod_eq_of_lt (Nat.lt_of_lt_of_le x.isLt H)]
-    omega
+  simp only [toInt_eq_msb_cond, toNat_signExtend]
+  have : (x.signExtend v).msb = x.msb := by
+    rw [msb_eq_getLsbD_last, getLsbD_eq_getElem (Nat.sub_one_lt_of_lt hv)]
+    simp [getElem_signExtend, Nat.le_sub_one_of_lt hv]
+  have H : 2^w ≤ 2^v := Nat.pow_le_pow_right (by omega) (by omega)
+  simp only [this, toNat_setWidth, Int.natCast_add, Int.ofNat_emod, Int.natCast_mul]
+  by_cases h : x.msb
+  <;> norm_cast
+  <;> simp [h, Nat.mod_eq_of_lt (Nat.lt_of_lt_of_le x.isLt H)]
+  omega
 
-/--
+/-
 If the current width `w` is larger than the extended width `v`,
 then the value when interpreted as an integer is truncated,
 and we compute a modulo by `2^v`.
 -/
-theorem toInt_signExtend_eq_toNat_bmod_of_le {x : BitVec w} (hv : v ≤ w) :
+theorem toInt_signExtend_of_le {x : BitVec w} (hv : v ≤ w) :
     (x.signExtend v).toInt = Int.bmod x.toNat (2^v) := by
-  simp [signExtend_eq_setWidth_of_le _ hv]
+  simp [signExtend_eq_setWidth_of_lt _ hv]
 
-/--
+/-
 Interpreting the sign extension of `(x : BitVec w)` to width `v`
-computes `x % 2^v` (where `%` is the balanced mod). See `toInt_signExtend` for a version stated
-in terms of `toInt` instead of `toNat`.
+computes `x % 2^v` (where `%` is the balanced mod).
 -/
-theorem toInt_signExtend_eq_toNat_bmod (x : BitVec w) :
-    (x.signExtend v).toInt = Int.bmod x.toNat (2 ^ min v w) := by
-  by_cases hv : v ≤ w
-  · simp [toInt_signExtend_eq_toNat_bmod_of_le hv, Nat.min_eq_left hv]
-  · simp only [Nat.not_le] at hv
-    rw [toInt_signExtend_of_le (Nat.le_of_lt hv),
-      Nat.min_eq_right (by omega), toInt_eq_toNat_bmod]
-
 theorem toInt_signExtend (x : BitVec w) :
-    (x.signExtend v).toInt = x.toInt.bmod (2 ^ min v w) := by
-  rw [toInt_signExtend_eq_toNat_bmod, BitVec.toInt_eq_toNat_bmod, Int.bmod_bmod_of_dvd]
-  exact Nat.pow_dvd_pow _ (Nat.min_le_right v w)
-
-theorem toInt_signExtend_eq_toInt_bmod_of_le (x : BitVec w) (h : v ≤ w) :
-    (x.signExtend v).toInt = x.toInt.bmod (2 ^ v) := by
-  rw [BitVec.toInt_signExtend, Nat.min_eq_left h]
+    (x.signExtend v).toInt = Int.bmod x.toNat (2^(min v w)) := by
+  by_cases hv : v ≤ w
+  · simp [toInt_signExtend_of_le hv, Nat.min_eq_left hv]
+  · simp only [Nat.not_le] at hv
+    rw [toInt_signExtend_of_lt hv, Nat.min_eq_right (by omega), toInt_eq_toNat_bmod]
 
 /-! ### append -/
 
@@ -2480,42 +2271,6 @@ theorem msb_shiftLeft {x : BitVec w} {n : Nat} :
     (x <<< n).msb = x.getMsbD n := by
   simp [BitVec.msb]
 
-/--
-A `(x : BitVec v)` set to width `w` equals `(v - w)` zeros,
-followed by the low `(min v w) bits of `x`
--/
-theorem setWidth_eq_append_extractLsb' {v : Nat} {x : BitVec v} {w : Nat} :
-    x.setWidth w = ((0#(w - v)) ++ x.extractLsb' 0 (min v w)).cast (by omega) := by
-  ext i hi
-  simp only [getElem_cast, getElem_append]
-  by_cases hiv : i < v
-  · simp [hi]
-    omega
-  · simp [getLsbD_ge x i (by omega)]
-
-/--
-A `(x : BitVec v)` set to a width `w ≥ v` equals `(w - v)` zeros, followed by `x`.
--/
-theorem setWidth_eq_append {v : Nat} {x : BitVec v} {w : Nat} (h : v ≤ w) :
-    x.setWidth w = ((0#(w - v)) ++ x).cast (by omega) := by
-  rw [setWidth_eq_append_extractLsb']
-  ext i hi
-  simp only [getElem_cast, getElem_append]
-  by_cases hiv : i < v
-  · simp [hiv]
-    omega
-  · simp [hiv, getLsbD_ge x i (by omega)]
-
-theorem setWidth_eq_extractLsb' {v : Nat} {x : BitVec v} {w : Nat} (h : w ≤ v) :
-    x.setWidth w = (x.extractLsb' 0 w).cast (by omega) := by
-  rw [setWidth_eq_append_extractLsb']
-  ext i hi
-  simp only [getElem_cast, getElem_append]
-  by_cases hiv : i < v
-  · simp [hi]
-    omega
-  · simp [getLsbD_ge x i (by omega)]
-
 theorem ushiftRight_eq_extractLsb'_of_lt {x : BitVec w} {n : Nat} (hn : n < w) :
     x >>> n = ((0#n) ++ (x.extractLsb' n (w - n))).cast (by omega) := by
   ext i hi
@@ -2527,34 +2282,11 @@ theorem ushiftRight_eq_extractLsb'_of_lt {x : BitVec w} {n : Nat} (hn : n < w) :
 theorem shiftLeft_eq_concat_of_lt {x : BitVec w} {n : Nat} (hn : n < w) :
     x <<< n = (x.extractLsb' 0 (w - n) ++ 0#n).cast (by omega) := by
   ext i hi
-  simp only [getElem_shiftLeft, getElem_cast, getElem_append, getElem_zero, getElem_extractLsb',
+  simp only [getElem_shiftLeft, getElem_cast, getElem_append, getLsbD_zero, getLsbD_extractLsb',
     Nat.zero_add, Bool.if_false_left]
   by_cases hi' : i < n
   · simp [hi']
-  · simp [hi', show i - n < w by omega]
-
-/-- Combine adjacent `extractLsb'` operations into a single `extractLsb'`. -/
-theorem extractLsb'_append_extractLsb'_eq_extractLsb' {x : BitVec w} (h : start₂ = start₁ + len₁) :
-    ((x.extractLsb' start₂ len₂) ++ (x.extractLsb' start₁ len₁)) =
-    (x.extractLsb' start₁ (len₁ + len₂)).cast (by omega) := by
-  ext i h
-  simp only [getElem_append, getElem_extractLsb', dite_eq_ite, getElem_cast, ite_eq_left_iff,
-    Nat.not_lt]
-  intros hi
-  congr 1
-  omega
-
-/-- Combine adjacent `~~~ (extractLsb _)'` operations into a single `~~~ (extractLsb _)'`. -/
-theorem not_extractLsb'_append_not_extractLsb'_eq_not_extractLsb' {x : BitVec w} (h : start₂ = start₁ + len₁) :
-    (~~~ (x.extractLsb' start₂ len₂) ++ ~~~ (x.extractLsb' start₁ len₁)) =
-    (~~~ x.extractLsb' start₁ (len₁ + len₂)).cast (by omega) := by
-  ext i h
-  simp only [getElem_cast, getElem_not, getElem_extractLsb', getElem_append]
-  by_cases hi : i < len₁
-  · simp [hi]
-  · simp only [hi, ↓reduceDIte, Bool.not_eq_eq_eq_not, Bool.not_not]
-    congr 1
-    omega
+  · simp [hi']
 
 /-! ### rev -/
 
@@ -2604,7 +2336,7 @@ theorem getLsbD_cons (b : Bool) {n} (x : BitVec n) (i : Nat) :
     simp [p1, p2, Nat.testBit_bool_to_nat]
 
 theorem getElem_cons {b : Bool} {n} {x : BitVec n} {i : Nat} (h : i < n + 1) :
-    (cons b x)[i] = if h : i = n then b else x[i] := by
+    (cons b x)[i] = if i = n then b else getLsbD x i := by
   simp only [getElem_eq_testBit_toNat, toNat_cons, Nat.testBit_or, getLsbD]
   rw [Nat.testBit_shiftLeft]
   rcases Nat.lt_trichotomy i n with i_lt_n | i_eq_n | n_lt_i
@@ -2712,7 +2444,7 @@ theorem getLsbD_concat (x : BitVec w) (b : Bool) (i : Nat) :
   · simp [Nat.div_eq_of_lt b.toNat_lt, Nat.testBit_add_one]
 
 theorem getElem_concat (x : BitVec w) (b : Bool) (i : Nat) (h : i < w + 1) :
-    (concat x b)[i] = if h : i = 0 then b else x[i - 1] := by
+    (concat x b)[i] = if i = 0 then b else x.getLsbD (i - 1) := by
   simp only [concat, getElem_eq_testBit_toNat, getLsbD, toNat_append,
     toNat_ofBool, Nat.testBit_or, Nat.shiftLeft_eq]
   cases i
@@ -2752,7 +2484,10 @@ theorem msb_concat {w : Nat} {b : Bool} {x : BitVec w} :
   simp only [BitVec.msb, getMsbD_eq_getLsbD, Nat.zero_lt_succ, decide_true, Nat.add_one_sub_one,
     Nat.sub_zero, Bool.true_and]
   by_cases h₀ : 0 < w
-  · simp [getElem_concat, h₀, show ¬ w = 0 by omega, show w - 1 < w by omega]
+  · simp only [Nat.lt_add_one, getLsbD_eq_getElem, getElem_concat, h₀, ↓reduceIte, decide_true,
+      Bool.true_and, ite_eq_right_iff]
+    intro
+    omega
   · simp [h₀, show w = 0 by omega]
 
 @[simp] theorem toInt_concat (x : BitVec w) (b : Bool) :
@@ -2967,9 +2702,6 @@ theorem toInt_neg {x : BitVec w} :
   rw [← BitVec.zero_sub, toInt_sub]
   simp [BitVec.toInt_ofNat]
 
-theorem ofInt_neg {w : Nat} {n : Int} : BitVec.ofInt w (-n) = -BitVec.ofInt w n :=
-  eq_of_toInt_eq (by simp [toInt_neg])
-
 @[simp] theorem toFin_neg (x : BitVec n) :
     (-x).toFin = Fin.ofNat' (2^n) (2^n - x.toNat) :=
   rfl
@@ -3428,7 +3160,6 @@ then `x / y` is nonnegative, thus `toInt` and `toNat` coincide.
 theorem toInt_udiv_of_msb {x : BitVec w} (h : x.msb = false) (y : BitVec w) :
     (x / y).toInt = x.toNat / y.toNat := by
   simp [toInt_eq_msb_cond, msb_udiv_eq_false_of h]
-  norm_cast
 
 /-! ### umod -/
 
@@ -4228,6 +3959,7 @@ theorem toNat_intMin : (intMin w).toNat = 2 ^ (w - 1) % 2 ^ w := by
 /--
 The RHS is zero in case `w = 0` which is modeled by wrapping the expression in `... % 2 ^ w`.
 -/
+@[simp]
 theorem toInt_intMin {w : Nat} :
     (intMin w).toInt = -((2 ^ (w - 1) % 2 ^ w) : Nat) := by
   by_cases h : w = 0
@@ -4239,16 +3971,10 @@ theorem toInt_intMin {w : Nat} :
     rw [Nat.mul_comm]
     simp [w_pos]
 
-theorem toInt_intMin_of_pos {v : Nat} (hv : 0 < v) : (intMin v).toInt = -2 ^ (v - 1) := by
-  rw [toInt_intMin, Nat.mod_eq_of_lt]
-  · simp [Int.natCast_pow]
-  · rw [Nat.pow_lt_pow_iff_right (by omega)]
-    omega
-
 theorem toInt_intMin_le (x : BitVec w) :
     (intMin w).toInt ≤ x.toInt := by
   cases w
-  case zero => simp [toInt_intMin, @of_length_zero x]
+  case zero => simp [@of_length_zero x]
   case succ w =>
     simp only [toInt_intMin, Nat.add_one_sub_one, Int.ofNat_emod]
     have : 0 < 2 ^ w := Nat.two_pow_pos w
@@ -4392,7 +4118,9 @@ theorem sub_le_sub_iff_le {x y z : BitVec w} (hxz : z ≤ x) (hyz : z ≤ y) :
 
 theorem msb_eq_toInt {x : BitVec w}:
     x.msb = decide (x.toInt < 0) := by
-  by_cases h : x.msb <;> simp [h, toInt_eq_msb_cond] <;> omega
+  by_cases h : x.msb <;>
+  · simp [h, toInt_eq_msb_cond]
+    omega
 
 theorem msb_eq_toNat {x : BitVec w}:
     x.msb = decide (x.toNat ≥ 2 ^ (w - 1)) := by
@@ -4489,10 +4217,6 @@ theorem toInt_abs_eq_natAbs_of_ne_intMin {x : BitVec w} (hx : x ≠ intMin w) :
     x.abs.toInt = x.toInt.natAbs := by
   simp [toInt_abs_eq_natAbs, hx]
 
-theorem toFin_abs {x : BitVec w} :
-    x.abs.toFin = if x.msb then Fin.ofNat' (2 ^ w) (2 ^ w - x.toNat) else x.toFin := by
-  by_cases h : x.msb <;> simp [BitVec.abs, h]
-
 /-! ### Reverse -/
 
 theorem getLsbD_reverse {i : Nat} {x : BitVec w} :
@@ -4627,9 +4351,6 @@ instance instDecidableExistsBitVec :
 
 set_option linter.missingDocs false
 
-@[deprecated toFin_uShiftRight (since := "2025-02-18")]
-abbrev toFin_uShiftRight := @toFin_ushiftRight
-
 @[deprecated signExtend_eq_setWidth_of_msb_false (since := "2024-12-08")]
 abbrev signExtend_eq_not_setWidth_not_of_msb_false := @signExtend_eq_setWidth_of_msb_false
 
@@ -4742,7 +4463,7 @@ abbrev signExtend_eq_not_zeroExtend_not_of_msb_false  := @signExtend_eq_setWidth
 abbrev signExtend_eq_not_zeroExtend_not_of_msb_true := @signExtend_eq_not_setWidth_not_of_msb_true
 
 @[deprecated signExtend_eq_setWidth_of_lt (since := "2024-09-18")]
-abbrev signExtend_eq_truncate_of_lt := @signExtend_eq_setWidth_of_le
+abbrev signExtend_eq_truncate_of_lt := @signExtend_eq_setWidth_of_lt
 
 @[deprecated truncate_append (since := "2024-09-18")]
 abbrev truncate_append := @setWidth_append

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
@@ -379,9 +367,7 @@ 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
@@ -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

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

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)
 
@@ -61,93 +50,55 @@ theorem isValidChar_of_isValidCharNat (n : Nat) (h : isValidCharNat n) : isValid
 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/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
 

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

src/Init/Data/Int.lean

@@ -7,6 +7,7 @@ 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
@@ -14,4 +15,3 @@ 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`.
 -/
 
 /--

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

@@ -6,7 +6,6 @@ Authors: Siddharth Bhat, Jeremy Avigad
 prelude
 import Init.Data.Nat.Bitwise.Lemmas
 import Init.Data.Int.Bitwise
-import Init.Data.Int.DivMod.Lemmas
 
 namespace Int
 
@@ -27,7 +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
 
@@ -39,47 +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

@@ -7,7 +7,7 @@ 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

src/Init/Data/Int/Lemmas.lean

@@ -78,7 +78,7 @@ theorem negSucc_eq (n : Nat) : -[n+1] = -((n : Int) + 1) := rfl
   | succ _ => rfl
   | -[_+1] => rfl
 
-@[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)
@@ -91,7 +91,7 @@ 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]) :
@@ -129,17 +129,6 @@ 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 -/
@@ -269,17 +258,6 @@ protected theorem add_left_cancel {a b c : Int} (h : a + b = a + c) : b = c := b
   rw [Int.add_right_neg, Int.add_comm a, ← Int.add_assoc, Int.add_assoc b,
     Int.add_right_neg, Int.add_zero, Int.add_right_neg]
 
-/--
-If a predicate on the integers is invariant under negation,
-then it is sufficient to prove it for the nonnegative integers.
--/
-theorem wlog_sign {P : Int → Prop} (inv : ∀ a, P a ↔ P (-a)) (w : ∀ n : Nat, P n) (a : Int) : P a := by
-  cases a with
-  | ofNat n => exact w n
-  | negSucc n =>
-    rw [negSucc_eq, ← inv, ← ofNat_succ]
-    apply w
-
 /- ## subtraction -/
 
 @[simp] theorem negSucc_sub_one (n : Nat) : -[n+1] - 1 = -[n + 1 +1] := rfl
@@ -352,20 +330,6 @@ theorem toNat_of_nonpos : ∀ {z : Int}, z ≤ 0 → z.toNat = 0
   | 0, _ => rfl
   | -[_+1], _ => rfl
 
-@[simp] theorem neg_ofNat_eq_negSucc_iff {a b : Nat} : - (a : Int) = -[b+1] ↔ a = b + 1 := by
-  rw [Int.neg_eq_comm]
-  rw [Int.neg_negSucc]
-  norm_cast
-  simp [eq_comm]
-
-@[simp] theorem neg_ofNat_eq_negSucc_add_one_iff {a b : Nat} : - (a : Int) = -[b+1] + 1 ↔ a = b := by
-  cases b with
-  | zero => simp; norm_cast
-  | succ b =>
-    rw [Int.neg_eq_comm, ← Int.negSucc_sub_one, Int.sub_add_cancel, Int.neg_negSucc]
-    norm_cast
-    simp [eq_comm]
-
 /- ## add/sub injectivity -/
 
 protected theorem add_left_inj {i j : Int} (k : Int) : (i + k = j + k) ↔ i = j := by

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
 
 
@@ -46,52 +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
 
-@[simp] theorem natCast_le_zero : {n : Nat} → (n : Int) ≤ 0 ↔ n = 0 := by omega
-
-@[simp] theorem toNat_eq_zero : ∀ {n : Int}, n.toNat = 0 ↔ n ≤ 0 := by omega
-
-theorem eq_zero_of_dvd_of_natAbs_lt_natAbs {d n : Int} (h : d ∣ n) (h₁ : n.natAbs < d.natAbs) :
-    n = 0 := by
-  obtain ⟨a, rfl⟩ := h
-  rw [natAbs_mul] at h₁
-  suffices ¬ 0 < a.natAbs by simp [Int.natAbs_eq_zero.1 (Nat.eq_zero_of_not_pos this)]
-  exact fun h => Nat.lt_irrefl _ (Nat.lt_of_le_of_lt (Nat.le_mul_of_pos_right d.natAbs h) h₁)
-
-theorem bmod_eq_self_of_le {n : Int} {m : Nat} (hn' : -(m / 2) ≤ n) (hn : n < (m + 1) / 2) :
-    n.bmod m = n := by
-  rw [← Int.sub_eq_zero]
-  have := le_bmod (x := n) (m := m) (by omega)
-  have := bmod_lt (x := n) (m := m) (by omega)
-  apply eq_zero_of_dvd_of_natAbs_lt_natAbs Int.dvd_bmod_sub_self
-  omega
-
-protected theorem sub_eq_iff_eq_add {b a c : Int} : a - b = c ↔ a = c + b := by omega
-protected theorem sub_eq_iff_eq_add' {b a c : Int} : a - b = c ↔ a = b + c := by omega
-
-theorem bmod_bmod_of_dvd {a : Int} {n m : Nat} (hnm : n ∣ m) :
-    (a.bmod m).bmod n = a.bmod n := by
-  rw [← Int.sub_eq_iff_eq_add.2 (bmod_add_bdiv a m).symm]
-  obtain ⟨k, rfl⟩ := hnm
-  simp [Int.mul_assoc]
-
-@[simp] theorem toNat_le {m : Int} {n : Nat} : m.toNat ≤ n ↔ m ≤ n := by omega
-@[simp] theorem toNat_lt' {m : Int} {n : Nat} (hn : 0 < n) : m.toNat < n ↔ m < n := by omega
-
-@[simp] protected theorem neg_nonpos_iff (i : Int) : -i ≤ 0 ↔ 0 ≤ i := by omega
-
-@[simp] theorem zero_le_ofNat (n : Nat) : 0 ≤ ((no_index (OfNat.ofNat n)) : Int) :=
-  ofNat_nonneg _
-
-@[simp] theorem neg_natCast_le_natCast (n m : Nat) : -(n : Int) ≤ (m : Int) :=
-  Int.le_trans (by simp) (ofNat_zero_le m)
-
-@[simp] theorem neg_natCast_le_ofNat (n m : Nat) : -(n : Int) ≤ (no_index (OfNat.ofNat m)) :=
-  Int.le_trans (by simp) (ofNat_zero_le m)
-
-@[simp] theorem neg_ofNat_le_ofNat (n m : Nat) : -(no_index (OfNat.ofNat n)) ≤ (no_index (OfNat.ofNat m)) :=
-  Int.le_trans (by simp) (ofNat_zero_le m)
-
-@[simp] theorem neg_ofNat_le_natCast (n m : Nat) : -(no_index (OfNat.ofNat n)) ≤ (m : Int) :=
-  Int.le_trans (by simp) (ofNat_zero_le m)
-
 end Int

src/Init/Data/Int/OfNat.lean

@@ -1,64 +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)
-
-def Expr.denote (ctx : Context) : Expr → Nat
-  | .num k    => k
-  | .var v    => v.denote ctx
-  | .add a b  => Nat.add (denote ctx a) (denote ctx b)
-  | .mul a b  => Nat.mul (denote ctx a) (denote ctx b)
-  | .div a b  => Nat.div (denote ctx a) (denote ctx b)
-  | .mod a b  => Nat.mod (denote ctx a) (denote ctx b)
-
-def Expr.denoteAsInt (ctx : Context) : Expr → Int
-  | .num k    => Int.ofNat k
-  | .var v    => Int.ofNat (v.denote ctx)
-  | .add a b  => Int.add (denoteAsInt ctx a) (denoteAsInt ctx b)
-  | .mul a b  => Int.mul (denoteAsInt ctx a) (denoteAsInt ctx b)
-  | .div a b  => Int.ediv (denoteAsInt ctx a) (denoteAsInt ctx b)
-  | .mod a b  => Int.emod (denoteAsInt ctx a) (denoteAsInt ctx b)
-
-@[local simp] private theorem fold_div (a b : Nat) : a.div b = a / b := rfl
-@[local simp] private theorem fold_mod (a b : Nat) : a.mod b = a % b := rfl
-
-theorem Expr.denoteAsInt_eq (ctx : Context) (e : Expr) : e.denoteAsInt ctx = e.denote ctx := by
-  induction e <;> simp [denote, denoteAsInt, Int.ofNat_ediv, *] <;> rfl
-
-theorem Expr.eq (ctx : Context) (lhs rhs : Expr)
-    : (lhs.denote ctx = rhs.denote ctx) = (lhs.denoteAsInt ctx = rhs.denoteAsInt ctx) := by
-  simp [denoteAsInt_eq, Int.ofNat_inj]
-
-theorem Expr.le (ctx : Context) (lhs rhs : Expr)
-    : (lhs.denote ctx ≤ rhs.denote ctx) = (lhs.denoteAsInt ctx ≤ rhs.denoteAsInt ctx) := by
-  simp [denoteAsInt_eq, Int.ofNat_le]
-
-theorem Expr.dvd (ctx : Context) (lhs rhs : Expr)
-    : (lhs.denote ctx ∣ rhs.denote ctx) = (lhs.denoteAsInt ctx ∣ rhs.denoteAsInt ctx) := by
-  simp [denoteAsInt_eq, Int.ofNat_dvd]
-
-end Int.OfNat

src/Init/Data/Int/Order.lean

@@ -56,7 +56,7 @@ protected theorem le_total (a b : Int) : a ≤ b ∨ b ≤ a :=
     let ⟨k, (hk : m + k = n)⟩ := Nat.le.dest h
     le.intro k (by rw [← hk]; rfl)⟩
 
-@[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
@@ -133,15 +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
@@ -361,10 +358,6 @@ protected theorem sub_lt_self (a : Int) {b : Int} (h : 0 < b) : a - b < a :=
 
 theorem add_one_le_of_lt {a b : Int} (H : a < b) : a + 1 ≤ b := H
 
-protected theorem le_iff_lt_add_one {a b : Int} : a ≤ b ↔ a < b + 1 := by
-  rw [Int.lt_iff_add_one_le]
-  exact (Int.add_le_add_iff_right 1).symm
-
 /- ### Order properties and multiplication -/
 
 
@@ -432,7 +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
@@ -477,13 +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
@@ -952,22 +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
@@ -1041,22 +1011,11 @@ theorem sign_eq_neg_one_iff_neg {a : Int} : sign a = -1 ↔ a < 0 :=
     exact Int.le_add_one (ofNat_nonneg _)
   | .negSucc _ => simp +decide [sign]
 
-@[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

src/Init/Data/List/Attach.lean

@@ -8,8 +8,8 @@ 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.
+-- 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
 
@@ -190,7 +190,7 @@ theorem length_attachWith {p : α → Prop} {l H} : length (l.attachWith p H) =
 
 @[simp]
 theorem pmap_eq_nil_iff {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = [] ↔ l = [] := by
-  rw [← length_eq_zero_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
@@ -662,10 +662,6 @@ 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
@@ -770,16 +766,6 @@ 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 }}

src/Init/Data/List/Basic.lean

@@ -58,8 +58,8 @@ Further operations are defined in `Init.Data.List.BasicAux`
 -/
 
 set_option linter.missingDocs true -- keep it documented
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 open Decidable List
 
@@ -1758,10 +1758,10 @@ where
 
 /-! ### removeAll -/
 
-/-- `O(|xs| * |ys|)`. Computes the "set difference" of lists,
+/-- `O(|xs|)`. Computes the "set difference" of lists,
 by filtering out all elements of `xs` which are also in `ys`.
 * `removeAll [1, 1, 5, 1, 2, 4, 5] [1, 2, 2] = [5, 4, 5]`
--/
+ -/
 def removeAll [BEq α] (xs ys : List α) : List α :=
   xs.filter (fun x => !ys.elem x)
 

src/Init/Data/List/BasicAux.lean

@@ -6,8 +6,8 @@ Author: Leonardo de Moura
 prelude
 import Init.Data.Nat.Linear
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 universe u
 
@@ -212,7 +212,6 @@ def mapMono (as : List α) (f : α → α) : List α :=
 
 /-! ## Additional lemmas required for bootstrapping `Array`. -/
 
-@[simp]
 theorem getElem_append_left {as bs : List α} (h : i < as.length) {h' : i < (as ++ bs).length} :
     (as ++ bs)[i] = as[i] := by
   induction as generalizing i with
@@ -222,7 +221,6 @@ theorem getElem_append_left {as bs : List α} (h : i < as.length) {h' : i < (as
     | zero => rfl
     | succ i => apply ih
 
-@[simp]
 theorem getElem_append_right {as bs : List α} {i : Nat} (h₁ : as.length ≤ i) {h₂} :
     (as ++ bs)[i]'h₂ =
       bs[i - as.length]'(by rw [length_append] at h₂; exact Nat.sub_lt_left_of_lt_add h₁ h₂) := by

src/Init/Data/List/Control.lean

@@ -9,8 +9,8 @@ import Init.Control.Id
 import Init.Control.Lawful
 import Init.Data.List.Basic
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 namespace List
 universe u v w u₁ u₂
@@ -227,19 +227,14 @@ def findM? {m : Type → Type u} [Monad m] {α : Type} (p : α → m Bool) : Lis
     | false => findM? p as
 
 @[simp]
-theorem findM?_pure {m} [Monad m] [LawfulMonad m] (p : α → Bool) (as : List α) :
-    findM? (m := m) (pure <| p ·) as = pure (as.find? p) := by
+theorem findM?_id (p : α → Bool) (as : List α) : findM? (m := Id) p as = as.find? p := by
   induction as with
   | nil => rfl
   | cons a as ih =>
     simp only [findM?, find?]
     cases p a with
-    | true  => simp
-    | false => simp [ih]
-
-@[simp]
-theorem findM?_id (p : α → Bool) (as : List α) : findM? (m := Id) p as = as.find? p :=
-  findM?_pure _ _
+    | true  => rfl
+    | false => rw [ih]; rfl
 
 @[specialize]
 def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f : α → m (Option β)) : List α → m (Option β)
@@ -250,19 +245,14 @@ def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f
     | none   => findSomeM? f as
 
 @[simp]
-theorem findSomeM?_pure [Monad m] [LawfulMonad m] (f : α → Option β) (as : List α) :
-    findSomeM? (m := m) (pure <| f ·) as = pure (as.findSome? f) := by
+theorem findSomeM?_id (f : α → Option β) (as : List α) : findSomeM? (m := Id) f as = as.findSome? f := by
   induction as with
   | nil => rfl
   | cons a as ih =>
     simp only [findSomeM?, findSome?]
     cases f a with
-    | some b => simp
-    | none   => simp [ih]
-
-@[simp]
-theorem findSomeM?_id (f : α → Option β) (as : List α) : findSomeM? (m := Id) f as = as.findSome? f :=
-  findSomeM?_pure _ _
+    | some b => rfl
+    | none   => rw [ih]; rfl
 
 theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
     as.findM? p = as.findSomeM? fun a => return if (← p a) then some a else none := by

src/Init/Data/List/Count.lean

@@ -10,8 +10,8 @@ import Init.Data.List.Sublist
 # Lemmas about `List.countP` and `List.count`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 
@@ -87,7 +87,7 @@ theorem countP_le_length : countP p l ≤ l.length := by
   countP_pos_iff
 
 @[simp] theorem countP_eq_zero {p} : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
-  simp only [countP_eq_length_filter, length_eq_zero_iff, filter_eq_nil_iff]
+  simp only [countP_eq_length_filter, length_eq_zero, filter_eq_nil_iff]
 
 @[simp] theorem countP_eq_length {p} : countP p l = l.length ↔ ∀ a ∈ l, p a := by
   rw [countP_eq_length_filter, filter_length_eq_length]

src/Init/Data/List/Erase.lean

@@ -12,8 +12,8 @@ import Init.Data.List.Find
 # Lemmas about `List.eraseP`, `List.erase`, and `List.eraseIdx`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 
@@ -137,7 +137,7 @@ theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (erase
 @[simp] theorem eraseP_eq_self_iff {p} {l : List α} : l.eraseP p = l ↔ ∀ a ∈ l, ¬ p a := by
   rw [← Sublist.length_eq (eraseP_sublist l), length_eraseP]
   split <;> rename_i h
-  · simp only [any_eq_true, length_eq_zero_iff] at h
+  · simp only [any_eq_true, length_eq_zero] at h
     constructor
     · intro; simp_all [Nat.sub_one_eq_self]
     · intro; obtain ⟨x, m, h⟩ := h; simp_all

src/Init/Data/List/FinRange.lean

@@ -6,8 +6,8 @@ Authors: François G. Dorais
 prelude
 import Init.Data.List.OfFn
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 

src/Init/Data/List/Find.lean

@@ -15,8 +15,8 @@ Lemmas about `List.findSome?`, `List.find?`, `List.findIdx`, `List.findIdx?`, `L
 and `List.lookup`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
@@ -514,6 +514,47 @@ private theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
     (x :: xs).findIdx? p = if p x then some 0 else (xs.findIdx? p).map fun i => i + 1 := by
   simp [findIdx?, findIdx?_go_eq]
 
+/-! ### findFinIdx? -/
+
+@[simp] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
+
+theorem findIdx?_go_eq_map_findFinIdx?_go_val {xs : List α} {p : α → Bool} {i : Nat} {h} :
+    List.findIdx?.go p xs i =
+      (List.findFinIdx?.go p l xs i h).map (·.val) := by
+  unfold findIdx?.go
+  unfold findFinIdx?.go
+  split
+  · simp_all
+  · simp only
+    split
+    · simp
+    · rw [findIdx?_go_eq_map_findFinIdx?_go_val]
+
+theorem findIdx?_eq_map_findFinIdx?_val {xs : List α} {p : α → Bool} :
+    xs.findIdx? p = (xs.findFinIdx? p).map (·.val) := by
+  simp [findIdx?, findFinIdx?]
+  rw [findIdx?_go_eq_map_findFinIdx?_go_val]
+
+@[simp] theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
+    findFinIdx? p (x :: xs) = if p x then some 0 else (findFinIdx? p xs).map Fin.succ := by
+  rw [← Option.map_inj_right (f := Fin.val) (fun a b => Fin.eq_of_val_eq)]
+  rw [← findIdx?_eq_map_findFinIdx?_val]
+  rw [findIdx?_cons]
+  split
+  · simp
+  · rw [findIdx?_eq_map_findFinIdx?_val]
+    simp [Function.comp_def]
+
+@[simp] theorem findFinIdx?_subtype {p : α → Prop} {l : List { x // p x }}
+    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
+    l.findFinIdx? f = (l.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
+  unfold unattach
+  induction l with
+  | nil => simp
+  | cons a l ih =>
+    simp [hf, findFinIdx?_cons]
+    split <;> simp [ih, Function.comp_def]
+
 /-! ### findIdx -/
 
 theorem findIdx_cons (p : α → Bool) (b : α) (l : List α) :
@@ -935,71 +976,6 @@ theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :
     simp [hf, findIdx?_cons]
     split <;> simp [ih, Function.comp_def]
 
-/-! ### findFinIdx? -/
-
-@[simp] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
-
-theorem findIdx?_go_eq_map_findFinIdx?_go_val {xs : List α} {p : α → Bool} {i : Nat} {h} :
-    List.findIdx?.go p xs i =
-      (List.findFinIdx?.go p l xs i h).map (·.val) := by
-  unfold findIdx?.go
-  unfold findFinIdx?.go
-  split
-  · simp_all
-  · simp only
-    split
-    · simp
-    · rw [findIdx?_go_eq_map_findFinIdx?_go_val]
-
-theorem findIdx?_eq_map_findFinIdx?_val {xs : List α} {p : α → Bool} :
-    xs.findIdx? p = (xs.findFinIdx? p).map (·.val) := by
-  simp [findIdx?, findFinIdx?]
-  rw [findIdx?_go_eq_map_findFinIdx?_go_val]
-
-theorem findFinIdx?_eq_pmap_findIdx? {xs : List α} {p : α → Bool} :
-    xs.findFinIdx? p =
-      (xs.findIdx? p).pmap
-        (fun i m => by simp [findIdx?_eq_some_iff_getElem] at m; exact ⟨i, m.choose⟩)
-        (fun i h => h) := by
-  simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]
-
-@[simp] theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
-    findFinIdx? p (x :: xs) = if p x then some 0 else (findFinIdx? p xs).map Fin.succ := by
-  rw [← Option.map_inj_right (f := Fin.val) (fun a b => Fin.eq_of_val_eq)]
-  rw [← findIdx?_eq_map_findFinIdx?_val]
-  rw [findIdx?_cons]
-  split
-  · simp
-  · rw [findIdx?_eq_map_findFinIdx?_val]
-    simp [Function.comp_def]
-
-@[simp] theorem findFinIdx?_eq_none_iff {l : List α} {p : α → Bool} :
-    l.findFinIdx? p = none ↔ ∀ x ∈ l, ¬ p x := by
-  simp [findFinIdx?_eq_pmap_findIdx?]
-
-@[simp]
-theorem findFinIdx?_eq_some_iff {xs : List α} {p : α → Bool} {i : Fin xs.length} :
-    xs.findFinIdx? p = some i ↔
-      p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji i.2)) := by
-  simp only [findFinIdx?_eq_pmap_findIdx?, Option.pmap_eq_some_iff, findIdx?_eq_some_iff_getElem,
-    Bool.not_eq_true, Option.mem_def, exists_and_left, and_exists_self, Fin.getElem_fin]
-  constructor
-  · rintro ⟨a, ⟨h, w₁, w₂⟩, rfl⟩
-    exact ⟨w₁, fun j hji => by simpa using w₂ j hji⟩
-  · rintro ⟨h, w⟩
-    exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩
-
-@[simp] theorem findFinIdx?_subtype {p : α → Prop} {l : List { x // p x }}
-    {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
-    l.findFinIdx? f = (l.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
-  unfold unattach
-  induction l with
-  | nil => simp
-  | cons a l ih =>
-    simp [hf, findFinIdx?_cons]
-    split <;> simp [ih, Function.comp_def]
-
-
 /-! ### idxOf
 
 The verification API for `idxOf` is still incomplete.
@@ -1059,36 +1035,6 @@ theorem idxOf_lt_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∈ l) : l.
 @[deprecated idxOf_lt_length (since := "2025-01-29")]
 abbrev indexOf_lt_length := @idxOf_lt_length
 
-/-! ### finIdxOf?
-
-The verification API for `finIdxOf?` is still incomplete.
-The lemmas below should be made consistent with those for `findFinIdx?` (and proved using them).
--/
-
-theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : List α} {a : α} :
-    xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
-  simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
-
-@[simp] theorem finIdxOf?_nil [BEq α] : ([] : List α).finIdxOf? a = none := rfl
-
-@[simp] theorem finIdxOf?_cons [BEq α] (a : α) (xs : List α) :
-    (a :: xs).finIdxOf? b =
-      if a == b then some ⟨0, by simp⟩ else (xs.finIdxOf? b).map (·.succ) := by
-  simp [finIdxOf?]
-
-@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
-    l.finIdxOf? a = none ↔ a ∉ l := by
-  simp only [finIdxOf?, findFinIdx?_eq_none_iff, beq_iff_eq]
-  constructor
-  · intro w m
-    exact w a m rfl
-  · rintro h a m rfl
-    exact h m
-
-@[simp] theorem finIdxOf?_eq_some_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} {i : Fin l.length} :
-    l.finIdxOf? a = some i ↔ l[i] = a ∧ ∀ j (_ : j < i), ¬l[j] = a := by
-  simp only [finIdxOf?, findFinIdx?_eq_some_iff, beq_iff_eq]
-
 /-! ### idxOf?
 
 The verification API for `idxOf?` is still incomplete.
@@ -1114,6 +1060,12 @@ theorem idxOf?_cons [BEq α] (a : α) (xs : List α) (b : α) :
 @[deprecated idxOf?_eq_none_iff (since := "2025-01-29")]
 abbrev indexOf?_eq_none_iff := @idxOf?_eq_none_iff
 
+/-! ### finIdxOf? -/
+
+theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : List α} {a : α} :
+    xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
+  simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
+
 /-! ### lookup -/
 
 section lookup

src/Init/Data/List/Impl.lean

@@ -16,8 +16,8 @@ If you import `Init.Data.List.Basic` but do not import this file,
 then at runtime you will get non-tail recursive versions of the following definitions.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 
@@ -210,7 +210,7 @@ def modifyTR (f : α → α) (n : Nat) (l : List α) : List α := go l n #[] whe
   | a :: l, 0, acc => acc.toListAppend (f a :: l)
   | a :: l, n+1, acc => go l n (acc.push a)
 
-theorem modifyTR_go_eq : ∀ l i, modifyTR.go f l i acc = acc.toList ++ modify f i l
+theorem modifyTR_go_eq : ∀ l n, modifyTR.go f l n acc = acc.toList ++ modify f n l
   | [], n => by cases n <;> simp [modifyTR.go, modify]
   | a :: l, 0 => by simp [modifyTR.go, modify]
   | a :: l, n+1 => by simp [modifyTR.go, modify, modifyTR_go_eq l]

src/Init/Data/List/Lemmas.lean

@@ -74,8 +74,8 @@ Also
 
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -96,15 +96,9 @@ theorem ne_nil_of_length_eq_add_one (_ : length l = n + 1) : l ≠ [] := fun _ =
 
 theorem ne_nil_of_length_pos (_ : 0 < length l) : l ≠ [] := fun _ => nomatch l
 
-@[simp] theorem length_eq_zero_iff : length l = 0 ↔ l = [] :=
+@[simp] theorem length_eq_zero : length l = 0 ↔ l = [] :=
   ⟨eq_nil_of_length_eq_zero, fun h => h ▸ rfl⟩
 
-@[deprecated length_eq_zero_iff (since := "2025-02-24")]
-abbrev length_eq_zero := @length_eq_zero_iff
-
-theorem eq_nil_iff_length_eq_zero : l = [] ↔ length l = 0 :=
-  length_eq_zero_iff.symm
-
 theorem length_pos_of_mem {a : α} : ∀ {l : List α}, a ∈ l → 0 < length l
   | _::_, _ => Nat.zero_lt_succ _
 
@@ -129,21 +123,12 @@ theorem exists_cons_of_length_eq_add_one :
     ∀ {l : List α}, l.length = n + 1 → ∃ h t, l = h :: t
   | _::_, _ => ⟨_, _, rfl⟩
 
-theorem length_pos_iff {l : List α} : 0 < length l ↔ l ≠ [] :=
-  Nat.pos_iff_ne_zero.trans (not_congr length_eq_zero_iff)
+theorem length_pos {l : List α} : 0 < length l ↔ l ≠ [] :=
+  Nat.pos_iff_ne_zero.trans (not_congr length_eq_zero)
 
-@[deprecated length_pos_iff (since := "2025-02-24")]
-abbrev length_pos := @length_pos_iff
-
-theorem ne_nil_iff_length_pos {l : List α} : l ≠ [] ↔ 0 < length l :=
-  length_pos_iff.symm
-
-theorem length_eq_one_iff {l : List α} : length l = 1 ↔ ∃ a, l = [a] :=
+theorem length_eq_one {l : List α} : length l = 1 ↔ ∃ a, l = [a] :=
   ⟨fun h => match l, h with | [_], _ => ⟨_, rfl⟩, fun ⟨_, h⟩ => by simp [h]⟩
 
-@[deprecated length_eq_one_iff (since := "2025-02-24")]
-abbrev length_eq_one := @length_eq_one_iff
-
 /-! ### cons -/
 
 theorem cons_ne_nil (a : α) (l : List α) : a :: l ≠ [] := nofun
@@ -299,7 +284,7 @@ such a rewrite, with `rw [getElem_of_eq h]`.
 theorem getElem_of_eq {l l' : List α} (h : l = l') {i : Nat} (w : i < l.length) :
     l[i] = l'[i]'(h ▸ w) := by cases h; rfl
 
-theorem getElem_zero {l : List α} (h : 0 < l.length) : l[0] = l.head (length_pos_iff.mp h) :=
+theorem getElem_zero {l : List α} (h : 0 < l.length) : l[0] = l.head (length_pos.mp h) :=
   match l, h with
   | _ :: _, _ => rfl
 
@@ -390,7 +375,7 @@ theorem eq_append_cons_of_mem {a : α} {xs : List α} (h : a ∈ xs) :
 theorem mem_cons_of_mem (y : α) {a : α} {l : List α} : a ∈ l → a ∈ y :: l := .tail _
 
 theorem exists_mem_of_ne_nil (l : List α) (h : l ≠ []) : ∃ x, x ∈ l :=
-  exists_mem_of_length_pos (length_pos_iff.2 h)
+  exists_mem_of_length_pos (length_pos.2 h)
 
 theorem eq_nil_iff_forall_not_mem {l : List α} : l = [] ↔ ∀ a, a ∉ l := by
   cases l <;> simp [-not_or]
@@ -548,7 +533,7 @@ theorem isEmpty_eq_false_iff_exists_mem {xs : List α} :
   cases xs <;> simp
 
 theorem isEmpty_iff_length_eq_zero {l : List α} : l.isEmpty ↔ l.length = 0 := by
-  rw [isEmpty_iff, length_eq_zero_iff]
+  rw [isEmpty_iff, length_eq_zero]
 
 /-! ### any / all -/
 
@@ -925,13 +910,13 @@ theorem head?_eq_getElem? : ∀ l : List α, head? l = l[0]?
   | [] => rfl
   | a :: l => by simp
 
-theorem head_eq_getElem (l : List α) (h : l ≠ []) : head l h = l[0]'(length_pos_iff.mpr h) := by
+theorem head_eq_getElem (l : List α) (h : l ≠ []) : head l h = l[0]'(length_pos.mpr h) := by
   cases l with
   | nil => simp at h
   | cons _ _ => simp
 
 theorem getElem_zero_eq_head (l : List α) (h : 0 < l.length) :
-    l[0] = head l (by simpa [length_pos_iff] using h) := by
+    l[0] = head l (by simpa [length_pos] using h) := by
   cases l with
   | nil => simp at h
   | cons _ _ => simp
@@ -1018,7 +1003,7 @@ theorem one_lt_length_of_tail_ne_nil {l : List α} (h : l.tail ≠ []) : 1 < l.l
   | nil => simp at h
   | cons _ l =>
     simp only [tail_cons, ne_eq] at h
-    exact Nat.lt_add_of_pos_left (length_pos_iff.mpr h)
+    exact Nat.lt_add_of_pos_left (length_pos.mpr h)
 
 @[simp] theorem head_tail (l : List α) (h : l.tail ≠ []) :
     (tail l).head h = l[1]'(one_lt_length_of_tail_ne_nil h) := by
@@ -2535,14 +2520,6 @@ theorem flatMap_reverse {β} (l : List α) (f : α → List β) : (l.reverse.fla
   simp only [foldrM]
   induction l <;> simp_all
 
-@[simp] theorem foldlM_pure [Monad m] [LawfulMonad m] (f : β → α → β) (b) (l : List α) :
-    l.foldlM (m := m) (pure <| f · ·) b = pure (l.foldl f b) := by
-  induction l generalizing b <;> simp [*]
-
-@[simp] theorem foldrM_pure [Monad m] [LawfulMonad m] (f : α → β → β) (b) (l : List α) :
-    l.foldrM (m := m) (pure <| f · ·) b = pure (l.foldr f b) := by
-  induction l generalizing b <;> simp [*]
-
 theorem foldl_eq_foldlM (f : β → α → β) (b) (l : List α) :
     l.foldl f b = l.foldlM (m := Id) f b := by
   induction l generalizing b <;> simp [*, foldl]
@@ -2906,7 +2883,7 @@ theorem getLast?_replicate (a : α) (n : Nat) : (replicate n a).getLast? = if n
 -- We unfold `leftpad` and `rightpad` for verification purposes.
 attribute [simp] leftpad rightpad
 
--- `length_leftpad` and `length_rightpad` are in `Init.Data.List.Nat.Basic`.
+-- `length_leftpad` is in `Init.Data.List.Nat.Basic`.
 
 theorem leftpad_prefix (n : Nat) (a : α) (l : List α) :
     replicate (n - length l) a <+: leftpad n a l := by
@@ -3094,12 +3071,8 @@ variable [BEq α]
 @[simp] theorem replace_cons_self [LawfulBEq α] {a : α} : (a::as).replace a b = b::as := by
   simp [replace_cons]
 
-@[simp] theorem replace_of_not_mem [LawfulBEq α] {l : List α} (h : a ∉ l) : l.replace a b = l := by
-  induction l with
-  | nil => rfl
-  | cons x xs ih =>
-    simp only [replace_cons]
-    split <;> simp_all
+@[simp] theorem replace_of_not_mem {l : List α} (h : !l.elem a) : l.replace a b = l := by
+  induction l <;> simp_all [replace_cons]
 
 @[simp] theorem length_replace {l : List α} : (l.replace a b).length = l.length := by
   induction l with
@@ -3182,7 +3155,7 @@ theorem replace_take {l : List α} {i : Nat} :
     (replicate n a).replace a b = b :: replicate (n - 1) a := by
   cases n <;> simp_all [replicate_succ, replace_cons]
 
-@[simp] theorem replace_replicate_ne [LawfulBEq α] {a b c : α} (h : !b == a) :
+@[simp] theorem replace_replicate_ne {a b c : α} (h : !b == a) :
     (replicate n a).replace b c = replicate n a := by
   rw [replace_of_not_mem]
   simp_all
@@ -3444,7 +3417,7 @@ theorem get_cons_succ {as : List α} {h : i + 1 < (a :: as).length} :
 theorem get_cons_succ' {as : List α} {i : Fin as.length} :
   (a :: as).get i.succ = as.get i := rfl
 
-theorem get_mk_zero : ∀ {l : List α} (h : 0 < l.length), l.get ⟨0, h⟩ = l.head (length_pos_iff.mp h)
+theorem get_mk_zero : ∀ {l : List α} (h : 0 < l.length), l.get ⟨0, h⟩ = l.head (length_pos.mp h)
   | _::_, _ => rfl
 
 set_option linter.deprecated false in

src/Init/Data/List/Lex.lean

@@ -7,8 +7,8 @@ prelude
 import Init.Data.List.Lemmas
 import Init.Data.List.Nat.TakeDrop
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 

src/Init/Data/List/MapIdx.lean

@@ -11,8 +11,8 @@ import Init.Data.List.OfFn
 import Init.Data.Fin.Lemmas
 import Init.Data.Option.Attach
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 

src/Init/Data/List/MinMax.lean

@@ -10,8 +10,8 @@ import Init.Data.List.Lemmas
 # Lemmas about `List.min?` and `List.max?.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 

src/Init/Data/List/Monadic.lean

@@ -11,8 +11,8 @@ import Init.Data.List.Attach
 # Lemmas about `List.mapM` and `List.forM`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 
@@ -56,13 +56,9 @@ theorem mapM'_eq_mapM [Monad m] [LawfulMonad m] (f : α → m β) (l : List α)
 @[simp] theorem mapM_cons [Monad m] [LawfulMonad m] (f : α → m β) :
     (a :: l).mapM f = (return (← f a) :: (← l.mapM f)) := by simp [← mapM'_eq_mapM, mapM']
 
-@[simp] theorem mapM_pure [Monad m] [LawfulMonad m] (l : List α) (f : α → β) :
-    l.mapM (m := m) (pure <| f ·) = pure (l.map f) := by
+@[simp] theorem mapM_id {l : List α} {f : α → Id β} : l.mapM f = l.map f := by
   induction l <;> simp_all
 
-@[simp] theorem mapM_id {l : List α} {f : α → Id β} : l.mapM f = l.map f :=
-  mapM_pure _ _
-
 @[simp] theorem mapM_append [Monad m] [LawfulMonad m] (f : α → m β) {l₁ l₂ : List α} :
     (l₁ ++ l₂).mapM f = (return (← l₁.mapM f) ++ (← l₂.mapM f)) := by induction l₁ <;> simp [*]
 
@@ -325,21 +321,24 @@ theorem forIn'_eq_foldlM [Monad m] [LawfulMonad m]
     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
   simp only [forIn'_eq_foldlM]
-  induction l.attach generalizing init <;> simp_all
+  generalize l.attach = l'
+  induction l' generalizing init <;> simp_all
 
 theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
     (l : List 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
   simp only [forIn'_eq_foldlM]
-  induction l.attach generalizing init <;> simp_all
+  generalize l.attach = l'
+  induction l' generalizing init <;> simp_all
 
 @[simp] theorem forIn'_yield_eq_foldl
     (l : List 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
   simp only [forIn'_eq_foldlM]
-  induction l.attach generalizing init <;> simp_all
+  generalize l.attach = l'
+  induction l' generalizing init <;> simp_all
 
 @[simp] theorem forIn'_map [Monad m] [LawfulMonad m]
     (l : List α) (g : α → β) (f : (b : β) → b ∈ l.map g → γ → m (ForInStep γ)) :
@@ -399,7 +398,7 @@ theorem forIn_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
     forIn (l.map g) init f = forIn l init fun a y => f (g a) y := by
   induction l generalizing init <;> simp_all
 
-/-! ### allM and anyM -/
+/-! ### allM -/
 
 theorem allM_eq_not_anyM_not [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
     allM p as = (! ·) <$> anyM ((! ·) <$> p ·) as := by
@@ -411,18 +410,6 @@ theorem allM_eq_not_anyM_not [Monad m] [LawfulMonad m] (p : α → m Bool) (as :
     funext b
     split <;> simp_all
 
-@[simp] theorem anyM_pure [Monad m] [LawfulMonad m] (p : α → Bool) (as : List α) :
-    as.anyM (m := m) (pure <| p ·) = pure (as.any p) := by
-  induction as with
-  | nil => simp
-  | cons a as ih =>
-    simp only [anyM, ih, pure_bind, all_cons]
-    split <;> simp_all
-
-@[simp] theorem allM_pure [Monad m] [LawfulMonad m] (p : α → Bool) (as : List α) :
-    as.allM (m := m) (pure <| p ·) = pure (as.all p) := by
-  simp [allM_eq_not_anyM_not, all_eq_not_any_not]
-
 /-! ### Recognizing higher order functions using a function that only depends on the value. -/
 
 /--
@@ -438,12 +425,12 @@ and simplifies these to the function directly taking the value.
   | nil => simp
   | cons a l ih => simp [ih, hf]
 
-@[wf_preprocess] theorem foldlM_wfParam [Monad m] (xs : List α) (f : β → α → m β) (init : β) :
-    (wfParam xs).foldlM f init = xs.attach.unattach.foldlM f init := by
+@[wf_preprocess] theorem foldlM_wfParam [Monad m] (xs : List α) (f : β → α → m β) :
+    (wfParam xs).foldlM f = xs.attach.unattach.foldlM f := by
   simp [wfParam]
 
-@[wf_preprocess] theorem foldlM_unattach [Monad m] (P : α → Prop) (xs : List (Subtype P)) (f : β → α → m β) (init : β):
-    xs.unattach.foldlM f init = xs.foldlM (init := init) fun b ⟨x, h⟩ =>
+@[wf_preprocess] theorem foldlM_unattach [Monad m] (P : α → Prop) (xs : List (Subtype P)) (f : β → α → m β) :
+    xs.unattach.foldlM f = xs.foldlM fun b ⟨x, h⟩ =>
       binderNameHint b f <| binderNameHint x (f b) <| binderNameHint h () <|
       f b (wfParam x) := by
   simp [wfParam]
@@ -465,12 +452,12 @@ and simplifies these to the function directly taking the value.
     funext b
     simp [hf]
 
-@[wf_preprocess] theorem foldrM_wfParam [Monad m] [LawfulMonad m] (xs : List α) (f : α → β → m β) (init : β) :
-    (wfParam xs).foldrM f init = xs.attach.unattach.foldrM f init := by
+@[wf_preprocess] theorem foldrM_wfParam [Monad m] [LawfulMonad m] (xs : List α) (f : α → β → m β) :
+    (wfParam xs).foldrM f = xs.attach.unattach.foldrM f := by
   simp [wfParam]
 
-@[wf_preprocess] theorem foldrM_unattach [Monad m] [LawfulMonad m] (P : α → Prop) (xs : List (Subtype P)) (f : α → β → m β) (init : β) :
-    xs.unattach.foldrM f init = xs.foldrM (init := init) fun ⟨x, h⟩ b =>
+@[wf_preprocess] theorem foldrM_unattach [Monad m] [LawfulMonad m] (P : α → Prop) (xs : List (Subtype P)) (f : α → β → m β) :
+    xs.unattach.foldrM f = xs.foldrM fun ⟨x, h⟩ b =>
       binderNameHint x f <| binderNameHint h () <| binderNameHint b (f x) <|
       f (wfParam x) b := by
   simp [wfParam]

src/Init/Data/List/Nat/BEq.lean

@@ -7,8 +7,8 @@ prelude
 import Init.Data.Nat.Lemmas
 import Init.Data.List.Basic
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 

src/Init/Data/List/Nat/Basic.lean

@@ -15,8 +15,8 @@ import Init.Data.Nat.Lemmas
 In particular, `omega` is available here.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -44,42 +44,10 @@ theorem tail_dropLast (l : List α) : tail (dropLast l) = dropLast (tail l) := b
 
 /-! ### filter -/
 
-@[simp]
-theorem length_filter_pos_iff {l : List α} {p : α → Bool} :
-    0 < (filter p l).length ↔ ∃ x ∈ l, p x := by
-  simpa [length_eq_countP_add_countP p l, countP_eq_length_filter] using
-    countP_pos_iff (p := p)
-
-@[simp]
 theorem length_filter_lt_length_iff_exists {l} :
-    (filter p l).length < l.length ↔ ∃ x ∈ l, ¬p x := by
-  simp [length_eq_countP_add_countP p l, countP_eq_length_filter]
-
-/-! ### filterMap -/
-
-@[simp]
-theorem length_filterMap_pos_iff {xs : List α} {f : α → Option β} :
-    0 < (filterMap f xs).length ↔ ∃ (x : α) (_ : x ∈ xs) (b : β), f x = some b := by
-  induction xs with
-  | nil => simp
-  | cons x xs ih =>
-    simp only [filterMap, mem_cons, exists_prop, exists_eq_or_imp]
-    split
-    · simp_all [ih]
-    · simp_all
-
-@[simp]
-theorem length_filterMap_lt_length_iff_exists {xs : List α} {f : α → Option β} :
-    (filterMap f xs).length < xs.length ↔ ∃ (x : α) (_ : x ∈ xs), f x = none := by
-  induction xs with
-  | nil => simp
-  | cons x xs ih =>
-    simp only [filterMap, mem_cons, exists_prop, exists_eq_or_imp]
-    split
-    · simp_all only [exists_prop, length_cons, true_or, iff_true]
-      have := length_filterMap_le f xs
-      omega
-    · simp_all
+    length (filter p l) < length l ↔ ∃ x ∈ l, ¬p x := by
+  simpa [length_eq_countP_add_countP p l, countP_eq_length_filter] using
+    countP_pos_iff (p := fun x => ¬p x)
 
 /-! ### reverse -/
 
@@ -95,18 +63,10 @@ theorem getElem_eq_getElem_reverse {l : List α} {i} (h : i < l.length) :
   to the larger of `n` and `l.length` -/
 -- We don't mark this as a `@[simp]` lemma since we allow `simp` to unfold `leftpad`,
 -- so the left hand side simplifies directly to `n - l.length + l.length`.
-theorem length_leftpad (n : Nat) (a : α) (l : List α) :
+theorem leftpad_length (n : Nat) (a : α) (l : List α) :
     (leftpad n a l).length = max n l.length := by
   simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max]
 
-@[deprecated length_leftpad (since := "2025-02-24")]
-abbrev leftpad_length := @length_leftpad
-
-theorem length_rightpad (n : Nat) (a : α) (l : List α) :
-    (rightpad n a l).length = max n l.length := by
-  simp [rightpad]
-  omega
-
 /-! ### eraseIdx -/
 
 theorem mem_eraseIdx_iff_getElem {x : α} :

src/Init/Data/List/Nat/Count.lean

@@ -7,8 +7,8 @@ prelude
 import Init.Data.List.Count
 import Init.Data.Nat.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.
+-- 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
 

src/Init/Data/List/Nat/Erase.lean

@@ -7,8 +7,8 @@ prelude
 import Init.Data.List.Nat.TakeDrop
 import Init.Data.List.Erase
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 

src/Init/Data/List/Nat/Find.lean

@@ -7,8 +7,8 @@ prelude
 import Init.Data.List.Nat.Range
 import Init.Data.List.Find
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 

src/Init/Data/List/Nat/InsertIdx.lean

@@ -12,8 +12,8 @@ import Init.Data.List.Nat.Modify
 Proves various lemmas about `List.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.
+-- 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 Nat
 

src/Init/Data/List/Nat/Modify.lean

@@ -8,8 +8,8 @@ prelude
 import Init.Data.List.Nat.TakeDrop
 import Init.Data.List.Nat.Erase
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -140,19 +140,19 @@ theorem modifyTailIdx_modifyTailIdx_self {f g : List α → List α} (n : Nat) (
 
 /-! ### modify -/
 
-@[simp] theorem modify_nil (f : α → α) (i) : [].modify f i = [] := by cases i <;> rfl
+@[simp] theorem modify_nil (f : α → α) (n) : [].modify f n = [] := by cases n <;> rfl
 
 @[simp] theorem modify_zero_cons (f : α → α) (a : α) (l : List α) :
     (a :: l).modify f 0 = f a :: l := rfl
 
-@[simp] theorem modify_succ_cons (f : α → α) (a : α) (l : List α) (i) :
-    (a :: l).modify f (i + 1) = a :: l.modify f i := by rfl
+@[simp] theorem modify_succ_cons (f : α → α) (a : α) (l : List α) (n) :
+    (a :: l).modify f (n + 1) = a :: l.modify f n := by rfl
 
 theorem modifyHead_eq_modify_zero (f : α → α) (l : List α) :
     l.modifyHead f = l.modify f 0 := by cases l <;> simp
 
-@[simp] theorem modify_eq_nil_iff {f : α → α} {i} {l : List α} :
-    l.modify f i = [] ↔ l = [] := by cases l <;> cases i <;> simp
+@[simp] theorem modify_eq_nil_iff {f : α → α} {n} {l : List α} :
+    l.modify f n = [] ↔ l = [] := by cases l <;> cases n <;> simp
 
 theorem getElem?_modify (f : α → α) :
     ∀ i (l : List α) j, (modify f i l)[j]? = (fun a => if i = j then f a else a) <$> l[j]?
@@ -189,8 +189,8 @@ theorem getElem_modify (f : α → α) (i) (l : List α) (j) (h : j < (modify f
 @[simp] theorem getElem_modify_ne (f : α → α) {i j} (l : List α) (h : i ≠ j) (h') :
     (modify f i l)[j] = l[j]'(by simpa using h') := by simp [getElem_modify, h]
 
-theorem modify_eq_self {f : α → α} {i} {l : List α} (h : l.length ≤ i) :
-    l.modify f i = l := by
+theorem modify_eq_self {f : α → α} {n} {l : List α} (h : l.length ≤ n) :
+    l.modify f n = l := by
   apply ext_getElem
   · simp
   · intro m h₁ h₂
@@ -198,16 +198,16 @@ theorem modify_eq_self {f : α → α} {i} {l : List α} (h : l.length ≤ i) :
     intro h
     omega
 
-theorem modify_modify_eq (f g : α → α) (i) (l : List α) :
-    (modify f i l).modify g i = modify (g ∘ f) i l := by
+theorem modify_modify_eq (f g : α → α) (n) (l : List α) :
+    (modify f n l).modify g n = modify (g ∘ f) n l := by
   apply ext_getElem
   · simp
   · intro m h₁ h₂
     simp only [getElem_modify, Function.comp_apply]
     split <;> simp
 
-theorem modify_modify_ne (f g : α → α) {i j} (l : List α) (h : i ≠ j) :
-    (modify f i l).modify g j = (l.modify g j).modify f i := by
+theorem modify_modify_ne (f g : α → α) {m n} (l : List α) (h : m ≠ n) :
+    (modify f m l).modify g n = (l.modify g n).modify f m := by
   apply ext_getElem
   · simp
   · intro m' h₁ h₂
@@ -234,13 +234,13 @@ theorem modify_eq_take_cons_drop {f : α → α} {i} {l : List α} (h : i < l.le
     modify f i l = take i l ++ f l[i] :: drop (i + 1) l := by
   rw [modify_eq_take_drop, drop_eq_getElem_cons h]; rfl
 
-theorem exists_of_modify (f : α → α) {i} {l : List α} (h : i < l.length) :
-    ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = i ∧ modify f i l = l₁ ++ f a :: l₂ :=
+theorem exists_of_modify (f : α → α) {n} {l : List α} (h : n < l.length) :
+    ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ modify f n l = l₁ ++ f a :: l₂ :=
   match exists_of_modifyTailIdx _ (Nat.le_of_lt h) with
   | ⟨_, _::_, eq, hl, H⟩ => ⟨_, _, _, eq, hl, H⟩
   | ⟨_, [], eq, hl, _⟩ => nomatch Nat.ne_of_gt h (eq ▸ append_nil _ ▸ hl)
 
-@[simp] theorem modify_id (i) (l : List α) : l.modify id i = l := by
+@[simp] theorem modify_id (n) (l : List α) : l.modify id n = l := by
   simp [modify]
 
 theorem take_modify (f : α → α) (i j) (l : List α) :

src/Init/Data/List/Nat/Pairwise.lean

@@ -12,8 +12,8 @@ import Init.Data.List.Pairwise
 # Lemmas about `List.Pairwise`
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 

src/Init/Data/List/Nat/Perm.lean

@@ -7,8 +7,8 @@ prelude
 import Init.Data.List.Nat.TakeDrop
 import Init.Data.List.Perm
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 

src/Init/Data/List/Nat/Range.lean

@@ -14,8 +14,8 @@ import Init.Data.List.Erase
 # Lemmas about `List.range` and `List.enum`
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 

src/Init/Data/List/Nat/Sublist.lean

@@ -16,8 +16,8 @@ These are in a separate file from most of the lemmas about `List.IsSuffix`
 as they required importing more lemmas about natural numbers, and use `omega`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -156,7 +156,7 @@ theorem append_sublist_of_sublist_left {xs ys zs : List α} (h : zs <+ xs) :
     have hl' := h'.length_le
     simp only [length_append] at hl'
     have : ys.length = 0 := by omega
-    simp_all only [Nat.add_zero, length_eq_zero_iff, true_and, append_nil]
+    simp_all only [Nat.add_zero, length_eq_zero, true_and, append_nil]
     exact Sublist.eq_of_length_le h' hl
   · rintro ⟨rfl, rfl⟩
     simp
@@ -169,7 +169,7 @@ theorem append_sublist_of_sublist_right {xs ys zs : List α} (h : zs <+ ys) :
     have hl' := h'.length_le
     simp only [length_append] at hl'
     have : xs.length = 0 := by omega
-    simp_all only [Nat.zero_add, length_eq_zero_iff, true_and, append_nil]
+    simp_all only [Nat.zero_add, length_eq_zero, true_and, append_nil]
     exact Sublist.eq_of_length_le h' hl
   · rintro ⟨rfl, rfl⟩
     simp

src/Init/Data/List/Nat/TakeDrop.lean

@@ -16,8 +16,8 @@ These are in a separate file from most of the list lemmas
 as they required importing more lemmas about natural numbers, and use `omega`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -115,12 +115,12 @@ theorem take_set_of_le (a : α) {i j : Nat} (l : List α) (h : j ≤ i) :
 @[deprecated take_set_of_le (since := "2025-02-04")]
 abbrev take_set_of_lt := @take_set_of_le
 
-@[simp] theorem take_replicate (a : α) : ∀ i n : Nat, take i (replicate n a) = replicate (min i n) a
+@[simp] theorem take_replicate (a : α) : ∀ i j : Nat, take i (replicate j a) = replicate (min i j) a
   | n, 0 => by simp [Nat.min_zero]
   | 0, m => by simp [Nat.zero_min]
   | succ n, succ m => by simp [replicate_succ, succ_min_succ, take_replicate]
 
-@[simp] theorem drop_replicate (a : α) : ∀ i n : Nat, drop i (replicate n a) = replicate (n - i) a
+@[simp] theorem drop_replicate (a : α) : ∀ i j : Nat, drop i (replicate j a) = replicate (j - i) a
   | n, 0 => by simp
   | 0, m => by simp
   | succ n, succ m => by simp [replicate_succ, succ_sub_succ, drop_replicate]

src/Init/Data/List/Notation.lean

@@ -11,8 +11,8 @@ import Init.Data.Nat.Div.Basic
 -/
 
 set_option linter.missingDocs true -- keep it documented
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
 
 open Decidable List
 

src/Init/Data/List/OfFn.lean

@@ -11,8 +11,8 @@ import Init.Data.Fin.Fold
 # Theorems about `List.ofFn`
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- 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
 

src/Init/Data/List/Pairwise.lean

@@ -11,8 +11,8 @@ import Init.Data.List.Attach
 # Lemmas about `List.Pairwise` and `List.Nodup`.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 

src/Init/Data/List/Perm.lean

@@ -18,8 +18,8 @@ another.
 The notation `~` is used for permutation equivalence.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -47,14 +47,6 @@ instance : Trans (Perm (α := α)) (Perm (α := α)) (Perm (α := α)) where
 
 theorem perm_comm {l₁ l₂ : List α} : l₁ ~ l₂ ↔ l₂ ~ l₁ := ⟨Perm.symm, Perm.symm⟩
 
-protected theorem Perm.congr_left {l₁ l₂ : List α} (h : l₁ ~ l₂) (l₃ : List α) :
-    l₁ ~ l₃ ↔ l₂ ~ l₃ :=
-  ⟨h.symm.trans, h.trans⟩
-
-protected theorem Perm.congr_right {l₁ l₂ : List α} (h : l₁ ~ l₂) (l₃ : List α) :
-    l₃ ~ l₁ ↔ l₃ ~ l₂ :=
-  ⟨fun h' => h'.trans h, fun h' => h'.trans h.symm⟩
-
 theorem Perm.swap' (x y : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) : y :: x :: l₁ ~ x :: y :: l₂ :=
   (swap ..).trans <| p.cons _ |>.cons _
 

src/Init/Data/List/Range.lean

@@ -14,8 +14,8 @@ Most of the results are deferred to `Data.Init.List.Nat.Range`, where more resul
 natural arithmetic are available.
 -/
 
-set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+-- set_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
 
@@ -33,7 +33,7 @@ theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step)
   | _ + 1 => congrArg succ (length_range' _ _ _)
 
 @[simp] theorem range'_eq_nil_iff : range' s n step = [] ↔ n = 0 := by
-  rw [← length_eq_zero_iff, length_range']
+  rw [← length_eq_zero, length_range']
 
 @[deprecated range'_eq_nil_iff (since := "2025-01-29")] abbrev range'_eq_nil := @range'_eq_nil_iff
 
@@ -74,7 +74,7 @@ theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step *
     rw [exists_comm]; simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
 
 theorem getElem?_range' (s step) :
-    ∀ {i n : Nat}, i < n → (range' s n step)[i]? = some (s + step * i)
+    ∀ {i j : Nat}, i < j → (range' s j step)[i]? = some (s + step * i)
   | 0, n + 1, _ => by simp [range'_succ]
   | m + 1, n + 1, h => by
     simp only [range'_succ, getElem?_cons_succ]
@@ -147,10 +147,10 @@ theorem range_loop_range' : ∀ s n : Nat, range.loop s (range' s n) = range' 0
 theorem range_eq_range' (n : Nat) : range n = range' 0 n :=
   (range_loop_range' n 0).trans <| by rw [Nat.zero_add]
 
-theorem getElem?_range {i n : Nat} (h : i < n) : (range n)[i]? = some i := by
+theorem getElem?_range {i j : Nat} (h : i < j) : (range j)[i]? = some i := by
   simp [range_eq_range', getElem?_range' _ _ h]
 
-@[simp] theorem getElem_range {n : Nat} (j) (h : j < (range n).length) : (range n)[j] = j := by
+@[simp] theorem getElem_range {i : Nat} (j) (h : j < (range i).length) : (range i)[j] = j := by
   simp [range_eq_range']
 
 theorem range_succ_eq_map (n : Nat) : range (n + 1) = 0 :: map succ (range n) := by
@@ -164,7 +164,7 @@ theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) :=
   simp only [range_eq_range', length_range']
 
 @[simp] theorem range_eq_nil {n : Nat} : range n = [] ↔ n = 0 := by
-  rw [← length_eq_zero_iff, length_range]
+  rw [← length_eq_zero, length_range]
 
 theorem range_ne_nil {n : Nat} : range n ≠ [] ↔ n ≠ 0 := by
   cases n <;> simp
@@ -183,9 +183,9 @@ theorem range_subset {m n : Nat} : range m ⊆ range n ↔ m ≤ n := by
 theorem range_succ (n : Nat) : range (succ n) = range n ++ [n] := by
   simp only [range_eq_range', range'_1_concat, Nat.zero_add]
 
-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 head?_range (n : Nat) : (range n).head? = if n = 0 then none else some 0 := by
   induction n with

src/Init/Data/List/Sort/Basic.lean

@@ -14,8 +14,8 @@ These definitions are intended for verification purposes,
 and are replaced at runtime by efficient versions in `Init.Data.List.Sort.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.
+-- 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