.

deprecations
.
2026-03-24 22:04:29 +00:00 · 2025-06-18 12:24:02 +10:00 · 2025-06-18 12:23:59 +10:00 · 2025-06-18 12:23:35 +10:00 · 2025-06-18 12:23:35 +10:00 · 2025-06-18 12:23:32 +10:00
1240 changed files with 19148 additions and 4730 deletions
--- a/.github/workflows/build-template.yml
+++ b/.github/workflows/build-template.yml
@@ -82,7 +82,7 @@ jobs:
      - name: CI Merge Checkout
        run: |
          git fetch --depth=1 origin ${{ github.sha }}
-          git checkout FETCH_HEAD flake.nix flake.lock script/prepare-*
+          git checkout FETCH_HEAD flake.nix flake.lock script/prepare-* tests/lean/run/importStructure.lean
        if: github.event_name == 'pull_request'
      # (needs to be after "Checkout" so files don't get overridden)
      - name: Setup emsdk
@@ -104,7 +104,7 @@ jobs:
          # NOTE: must be in sync with `save` below
          path: |
            .ccache
-            ${{ matrix.name == 'Linux Lake' && 'build/stage1/**/*.trace
+            ${{ matrix.name == 'Linux Lake' && false && 'build/stage1/**/*.trace
            build/stage1/**/*.olean*
            build/stage1/**/*.ilean
            build/stage1/**/*.c
@@ -127,9 +127,12 @@ jobs:
          [ -d build ] || mkdir build
          cd build
          # arguments passed to `cmake`
-          # this also enables githash embedding into stage 1 library
-          OPTIONS=(-DCHECK_OLEAN_VERSION=ON)
-          OPTIONS+=(-DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true)
+          OPTIONS=(-DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true)
+          if [[ -n '${{ matrix.release }}' ]]; then
+            # this also enables githash embedding into stage 1 library, which prohibits reusing
+            # `.olean`s across commits, so we don't do it in the fast non-release CI
+            OPTIONS+=(-DCHECK_OLEAN_VERSION=ON)
+          fi
          if [[ -n '${{ matrix.cross_target }}' ]]; then
            # used by `prepare-llvm`
            export EXTRA_FLAGS=--target=${{ matrix.cross_target }}
@@ -193,7 +196,7 @@ jobs:
        run: |
          ulimit -c unlimited  # coredumps
          time ctest --preset ${{ matrix.CMAKE_PRESET || 'release' }} --test-dir build/stage1 -j$NPROC --output-junit test-results.xml ${{ matrix.CTEST_OPTIONS }}
-        if: (matrix.wasm || !matrix.cross) && (inputs.check-level >= 1 || matrix.name == 'Linux release')
+        if: (matrix.wasm || !matrix.cross) && (inputs.check-level >= 1 || matrix.test)
      - name: Test Summary
        uses: test-summary/action@v2
        with:
@@ -210,7 +213,7 @@ jobs:
      - name: Check Stage 3
        run: |
          make -C build -j$NPROC check-stage3
-        if: matrix.test-speedcenter
+        if: matrix.check-stage3
      - name: Test Speedcenter Benchmarks
        run: |
          # Necessary for some timing metrics but does not work on Namespace runners
@@ -224,7 +227,7 @@ jobs:
        run: |
          # clean rebuild in case of Makefile changes
          make -C build update-stage0 && rm -rf build/stage* && make -C build -j$NPROC
-        if: matrix.name == 'Linux' && inputs.check-level >= 1
+        if: matrix.check-rebootstrap
      - name: CCache stats
        if: always()
        run: ccache -s
@@ -242,7 +245,7 @@ jobs:
          # NOTE: must be in sync with `restore` above
          path: |
            .ccache
-            ${{ matrix.name == 'Linux Lake' && 'build/stage1/**/*.trace
+            ${{ matrix.name == 'Linux Lake' && false && 'build/stage1/**/*.trace
            build/stage1/**/*.olean*
            build/stage1/**/*.ilean
            build/stage1/**/*.c
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -164,9 +164,15 @@ jobs:
              {
                // portable release build: use channel with older glibc (2.26)
                "name": "Linux release",
-                "os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
+                "os": large && level < 2 ? "nscloud-ubuntu-22.04-amd64-4x16" : "ubuntu-latest",
                "release": true,
-                "check-level": 0,
+                // Special handling for release jobs. We want:
+                // 1. To run it in PRs so developrs get PR toolchains (so secondary is sufficient)
+                // 2. To skip it in merge queues as it takes longer than the
+                //    Linux lake build and adds little value in the merge queue
+                // 3. To run it in release (obviously)
+                "check-level": isPr ? 0 : 2,
+                "secondary": isPr,
                "shell": "nix develop .#oldGlibc -c bash -euxo pipefail {0}",
                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/19.1.2/lean-llvm-x86_64-linux-gnu.tar.zst",
                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm*",
@@ -176,21 +182,14 @@ jobs:
              },
              {
                "name": "Linux Lake",
-                "os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
+                "os": large ? "nscloud-ubuntu-22.04-amd64-8x16" : "ubuntu-latest",
                "check-level": 0,
-                // just a secondary build job for now until false positives can be excluded
-                "secondary": true,
-                "CMAKE_OPTIONS": "-DUSE_LAKE=ON",
-                // TODO: importStructure is not compatible with .olean caching
-                // TODO: why does scopedMacros fail?
-                "CTEST_OPTIONS": "-E 'scopedMacros|importStructure'"
-              },
-              {
-                "name": "Linux",
-                "os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
+                "test": true,
+                "check-rebootstrap": level >= 1,
                "check-stage3": level >= 2,
-                "test-speedcenter": level >= 2,
-                "check-level": 1,
+                // NOTE: `test-speedcenter` currently seems to be broken on `ubuntu-latest`
+                "test-speedcenter": large && level >= 2,
+                "CMAKE_OPTIONS": "-DUSE_LAKE=ON",
              },
              {
                "name": "Linux Reldebug",
@@ -223,7 +222,8 @@ jobs:
              },
              {
                "name": "macOS aarch64",
-                "os": "macos-14",
+                // standard GH runner only comes with 7GB so use large runner if possible
+                "os": large ? "nscloud-macos-sonoma-arm64-6x14" : "macos-14",
                "CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-darwin_aarch64",
                "release": true,
                "shell": "bash -euxo pipefail {0}",
@@ -231,11 +231,7 @@ jobs:
                "prepare-llvm": "../script/prepare-llvm-macos.sh lean-llvm*",
                "binary-check": "otool -L",
                "tar": "gtar", // https://github.com/actions/runner-images/issues/2619
-                // Special handling for MacOS aarch64, we want:
-                // 1. To run it in PRs so Mac devs get PR toolchains (so secondary is sufficient)
-                // 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)
+                // See above for release job levels
                "check-level": isPr ? 0 : 2,
                "secondary": isPr,
              },
@@ -254,7 +250,7 @@ jobs:
              },
              {
                "name": "Linux aarch64",
-                "os": "nscloud-ubuntu-22.04-arm64-4x8",
+                "os": "nscloud-ubuntu-22.04-arm64-4x16",
                "CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-linux_aarch64",
                "release": true,
                "check-level": 2,
@@ -364,7 +360,7 @@ jobs:
        with:
          path: artifacts
      - name: Release
-        uses: softprops/action-gh-release@v2
+        uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
        with:
          files: artifacts/*/*
          fail_on_unmatched_files: true
@@ -408,7 +404,7 @@ jobs:
          echo -e "\n*Full commit log*\n" >> diff.md
          git log --oneline "$last_tag"..HEAD | sed 's/^/* /' >> diff.md
      - name: Release Nightly
-        uses: softprops/action-gh-release@v2
+        uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
        with:
          body_path: diff.md
          prerelease: true
--- a/.github/workflows/pr-release.yml
+++ b/.github/workflows/pr-release.yml
@@ -48,19 +48,30 @@ jobs:
          git -C lean4.git remote add origin https://github.com/${{ github.repository_owner }}/lean4.git
          git -C lean4.git fetch -n origin master
          git -C lean4.git fetch -n origin "${{ steps.workflow-info.outputs.sourceHeadSha }}"
+          
+          # Create both the original tag and the SHA-suffixed tag
+          SHORT_SHA="${{ steps.workflow-info.outputs.sourceHeadSha }}"
+          SHORT_SHA="${SHORT_SHA:0:7}"
+          
+          # Export the short SHA for use in subsequent steps
+          echo "SHORT_SHA=${SHORT_SHA}" >> "$GITHUB_ENV"
+
          git -C lean4.git tag -f pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }} "${{ steps.workflow-info.outputs.sourceHeadSha }}"
+          git -C lean4.git tag -f pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-"${SHORT_SHA}" "${{ steps.workflow-info.outputs.sourceHeadSha }}"
+          
          git -C lean4.git remote add pr-releases https://foo:'${{ secrets.PR_RELEASES_TOKEN }}'@github.com/${{ github.repository_owner }}/lean4-pr-releases.git
          git -C lean4.git push -f pr-releases pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}
+          git -C lean4.git push -f pr-releases pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-"${SHORT_SHA}"
      - name: Delete existing release if present
        if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
        run: |
-          # Try to delete any existing release for the current PR.
+          # Try to delete any existing release for the current PR (just the version without the SHA suffix).
          gh release delete --repo ${{ github.repository_owner }}/lean4-pr-releases pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }} -y || true
        env:
          GH_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}
-      - name: Release
+      - name: Release (short format)
        if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
-        uses: softprops/action-gh-release@v2
+        uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
        with:
          name: Release for PR ${{ steps.workflow-info.outputs.pullRequestNumber }}
          # There are coredumps files here as well, but all in deeper subdirectories.
@@ -73,7 +84,22 @@ jobs:
          # The token used here must have `workflow` privileges.
          GITHUB_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}

-      - name: Report release status
+      - name: Release (SHA-suffixed format)
+        if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
+        uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
+        with:
+          name: Release for PR ${{ steps.workflow-info.outputs.pullRequestNumber }} (${{ steps.workflow-info.outputs.sourceHeadSha }})
+          # There are coredumps files here as well, but all in deeper subdirectories.
+          files: artifacts/*/*
+          fail_on_unmatched_files: true
+          draft: false
+          tag_name: pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}
+          repository: ${{ github.repository_owner }}/lean4-pr-releases
+        env:
+          # The token used here must have `workflow` privileges.
+          GITHUB_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}
+
+      - name: Report release status (short format)
        if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
        uses: actions/github-script@v7
        with:
@@ -87,6 +113,20 @@ jobs:
              description: "${{ github.repository_owner }}/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}",
            });

+      - name: Report release status (SHA-suffixed format)
+        if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
+        uses: actions/github-script@v7
+        with:
+          script: |
+            await github.rest.repos.createCommitStatus({
+              owner: context.repo.owner,
+              repo: context.repo.repo,
+              sha: "${{ steps.workflow-info.outputs.sourceHeadSha }}",
+              state: "success",
+              context: "PR toolchain (SHA-suffixed)",
+              description: "${{ github.repository_owner }}/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}",
+            });
+
      - name: Add label
        if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
        uses: actions/github-script@v7
@@ -282,16 +322,18 @@ jobs:
          if [ "$EXISTS" = "0" ]; then
            echo "Branch does not exist, creating it."
            git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
-            echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}" > lean-toolchain
+            echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
            git add lean-toolchain
            git commit -m "Update lean-toolchain for testing https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
          else
-            echo "Branch already exists, pushing an empty commit."
+            echo "Branch already exists, updating lean-toolchain."
            git switch lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}
            # The Batteries `nightly-testing` or `nightly-testing-YYYY-MM-DD` branch may have moved since this branch was created, so merge their changes.
            # (This should no longer be possible once `nightly-testing-YYYY-MM-DD` is a tag, but it is still safe to merge.)
            git merge "$BASE" --strategy-option ours --no-commit --allow-unrelated-histories
-            git commit --allow-empty -m "Trigger CI for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
+            echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
+            git add lean-toolchain
+            git commit -m "Update lean-toolchain for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
          fi

      - name: Push changes
@@ -346,21 +388,23 @@ jobs:
          if [ "$EXISTS" = "0" ]; then
            echo "Branch does not exist, creating it."
            git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
-            echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}" > lean-toolchain
+            echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
            git add lean-toolchain
            sed -i 's,require "leanprover-community" / "batteries" @ git ".\+",require "leanprover-community" / "batteries" @ git "lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}",' lakefile.lean
            lake update batteries
            git add lakefile.lean lake-manifest.json
            git commit -m "Update lean-toolchain for testing https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
          else
-            echo "Branch already exists, merging $BASE and bumping Batteries."
+            echo "Branch already exists, updating lean-toolchain and bumping Batteries."
            git switch lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}
            # The Mathlib `nightly-testing` branch or `nightly-testing-YYYY-MM-DD` tag may have moved since this branch was created, so merge their changes.
            # (This should no longer be possible once `nightly-testing-YYYY-MM-DD` is a tag, but it is still safe to merge.)
            git merge "$BASE" --strategy-option ours --no-commit --allow-unrelated-histories
+            echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
+            git add lean-toolchain
            lake update batteries
            git add lake-manifest.json
-            git commit --allow-empty -m "Trigger CI for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
+            git commit -m "Update lean-toolchain for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
          fi

      - name: Push changes
--- a/doc/dev/release_checklist.md
+++ b/doc/dev/release_checklist.md
@@ -50,7 +50,7 @@ We'll use `v4.6.0` as the intended release version as a running example.
    - Re-running `script/release_checklist.py` will then create the tag `v4.6.0` from `master`/`main` and push it (unless `toolchain-tag: false` in the `release_repos.yml` file)
    - `script/release_checklist.py` will then merge the tag `v4.6.0` into the `stable` branch and push it (unless `stable-branch: false` in the `release_repos.yml` file).
  - Special notes on repositories with exceptional requirements:
-    - `doc-gen4` has addition dependencies which we do not update at each toolchain release, although occasionally these break and need to be updated manually.
+    - `doc-gen4` has additional dependencies which we do not update at each toolchain release, although occasionally these break and need to be updated manually.
    - `verso`:
      - The `subverso` dependency is unusual in that it needs to be compatible with _every_ Lean release simultaneously.
        Usually you don't need to do anything.
@@ -94,6 +94,8 @@ We'll use `v4.6.0` as the intended release version as a running example.

 This checklist walks you through creating the first release candidate for a version of Lean.

+For subsequent release candidates, the process is essentially the same, but we start out with the `releases/v4.7.0` branch already created.
+
 We'll use `v4.7.0-rc1` as the intended release version in this example.

 - Decide which nightly release you want to turn into a release candidate.
@@ -112,7 +114,7 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
    git fetch nightly tag nightly-2024-02-29
    git checkout nightly-2024-02-29
    git checkout -b releases/v4.7.0
-    git push --set-upstream origin releases/v4.18.0
+    git push --set-upstream origin releases/v4.7.0
    ```
 - In `src/CMakeLists.txt`,
  - verify that you see `set(LEAN_VERSION_MINOR 7)` (for whichever `7` is appropriate); this should already have been updated when the development cycle began.
--- a/script/bench.sh
+++ b/script/bench.sh
@@ -0,0 +1,9 @@
+#!/usr/bin/env bash
+set -euo pipefail
+
+# We benchmark against stage 2 to test new optimizations.
+timeout -s KILL 1h time bash -c 'mkdir -p build/release; cd build/release; cmake ../.. && make -j$(nproc) stage2' 1>&2
+export PATH=$PWD/build/release/stage2/bin:$PATH
+cd tests/bench
+timeout -s KILL 1h time temci exec --config speedcenter.yaml --in speedcenter.exec.velcom.yaml 1>&2
+temci report run_output.yaml --reporter codespeed2
--- a/script/release_checklist.py
+++ b/script/release_checklist.py
@@ -53,6 +53,23 @@ def tag_exists(repo_url, tag_name, github_token):
    matching_tags = response.json()
    return any(tag["ref"] == f"refs/tags/{tag_name}" for tag in matching_tags)

+def commit_hash_for_tag(repo_url, tag_name, github_token):
+    # Use /git/matching-refs/tags/ to get all matching tags
+    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/git/matching-refs/tags/{tag_name}"
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+    response = requests.get(api_url, headers=headers)
+
+    if response.status_code != 200:
+        return False
+
+    # Check if any of the returned refs exactly match our tag
+    matching_tags = response.json()
+    matching_commits = [tag["object"]["sha"] for tag in matching_tags if tag["ref"] == f"refs/tags/{tag_name}"]
+    if len(matching_commits) != 1:
+        return None
+    else:
+        return matching_commits[0]
+
 def release_page_exists(repo_url, tag_name, github_token):
    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/releases/tags/{tag_name}"
    headers = {'Authorization': f'token {github_token}'} if github_token else {}
@@ -286,6 +303,14 @@ def main():
        lean4_success = False
    else:
        print(f"  ✅ Tag {toolchain} exists")
+        commit_hash = commit_hash_for_tag(lean_repo_url, toolchain, github_token)
+        SHORT_HASH_LENGTH = 7 # Lake abbreviates the Lean commit to 7 characters.
+        if commit_hash is None:
+            print(f"  ❌ Could not resolve tag {toolchain} to a commit.")
+            lean4_success = False
+        elif commit_hash[0] == '0' and commit_hash[:SHORT_HASH_LENGTH].isnumeric():
+            print(f"  ❌ Short commit hash {commit_hash[:SHORT_HASH_LENGTH]} is numeric and starts with 0, causing issues for version parsing. Try regenerating the last commit to get a new hash.")
+            lean4_success = False

    if not release_page_exists(lean_repo_url, toolchain, github_token):
        print(f"  ❌ Release page for {toolchain} does not exist")
--- a/script/release_steps.py
+++ b/script/release_steps.py
@@ -94,6 +94,7 @@ def generate_script(repo, version, config):
            "echo 'This repo has nightly-testing infrastructure'",
            f"git merge origin/bump/{version.split('-rc')[0]}",
            "echo 'Please resolve any conflicts.'",
+            "grep nightly-testing lakefile.* && echo 'Please ensure the lakefile does not include nightly-testing versions.'",
            ""
        ])
    if re.search(r'rc\d+$', version) and repo_name in ["verso", "reference-manual"]:
--- a/src/CMakeLists.txt
+++ b/src/CMakeLists.txt
@@ -10,7 +10,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 21)
+set(LEAN_VERSION_MINOR 22)
 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'")
--- a/src/Init.lean
+++ b/src/Init.lean
@@ -37,6 +37,7 @@ import Init.Ext
 import Init.Omega
 import Init.MacroTrace
 import Init.Grind
+import Init.GrindInstances
 import Init.While
 import Init.Syntax
 import Init.Internal
--- a/src/Init/Coe.lean
+++ b/src/Init/Coe.lean
@@ -7,6 +7,7 @@ module

 prelude
 import Init.Prelude
+meta import Init.Prelude
 set_option linter.missingDocs true -- keep it documented

 /-!
--- a/src/Init/Control/Lawful/Basic.lean
+++ b/src/Init/Control/Lawful/Basic.lean
@@ -148,7 +148,7 @@ attribute [simp] pure_bind bind_assoc bind_pure_comp
 attribute [grind] pure_bind

@[simp] theorem bind_pure [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x := by
-  show x >>= (fun a => pure (id a)) = x
+  change x >>= (fun a => pure (id a)) = x
  rw [bind_pure_comp, id_map]

 /--
--- a/src/Init/Control/Lawful/Instances.lean
+++ b/src/Init/Control/Lawful/Instances.lean
@@ -58,7 +58,7 @@ protected theorem bind_pure_comp [Monad m] (f : α → β) (x : ExceptT ε m α)
  intros; rfl

 protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x <* y = const β <$> x <*> y := by
-  show (x >>= fun a => y >>= fun _ => pure a) = (const (α := α) β <$> x) >>= fun f => f <$> y
+  change (x >>= fun a => y >>= fun _ => pure a) = (const (α := α) β <$> x) >>= fun f => f <$> y
  rw [← ExceptT.bind_pure_comp]
  apply ext
  simp [run_bind]
@@ -70,7 +70,7 @@ protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad
    cases b <;> simp [comp, Except.map, const]

 protected theorem seqRight_eq [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x *> y = const α id <$> x <*> y := by
-  show (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y
+  change (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y
  rw [← ExceptT.bind_pure_comp]
  apply ext
  simp [run_bind]
@@ -206,15 +206,15 @@ theorem run_bind_lift {α σ : Type u} [Monad m] [LawfulMonad m] (x : m α) (f :
    (monadMap @f x : StateT σ m α).run s = monadMap @f (x.run s) := rfl

@[simp] theorem run_seq {α β σ : Type u} [Monad m] [LawfulMonad m] (f : StateT σ m (α → β)) (x : StateT σ m α) (s : σ) : (f <*> x).run s = (f.run s >>= fun fs => (fun (p : α × σ) => (fs.1 p.1, p.2)) <$> x.run fs.2) := by
-  show (f >>= fun g => g <$> x).run s = _
+  change (f >>= fun g => g <$> x).run s = _
  simp

@[simp] theorem run_seqRight [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x *> y).run s = (x.run s >>= fun p => y.run p.2) := by
-  show (x >>= fun _ => y).run s = _
+  change (x >>= fun _ => y).run s = _
  simp

@[simp] theorem run_seqLeft {α β σ : Type u} [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x <* y).run s = (x.run s >>= fun p => y.run p.2 >>= fun p' => pure (p.1, p'.2)) := by
-  show (x >>= fun a => y >>= fun _ => pure a).run s = _
+  change (x >>= fun a => y >>= fun _ => pure a).run s = _
  simp

 theorem seqRight_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x *> y = const α id <$> x <*> y := by
--- a/src/Init/Conv.lean
+++ b/src/Init/Conv.lean
@@ -9,7 +9,7 @@ module

 prelude
 import Init.Tactics
-import Init.Meta
+meta import Init.Meta

 namespace Lean.Parser.Tactic.Conv

--- a/src/Init/Core.lean
+++ b/src/Init/Core.lean
@@ -8,7 +8,7 @@ notation, basic datatypes and type classes
 module

 prelude
-import Init.Prelude
+meta import Init.Prelude
 import Init.SizeOf
 set_option linter.missingDocs true -- keep it documented

@@ -2252,7 +2252,7 @@ theorem funext {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x}
    Quot.liftOn f
      (fun (f : ∀ (x : α), β x) => f x)
      (fun _ _ h => h x)
-  show extfunApp (Quot.mk eqv f) = extfunApp (Quot.mk eqv g)
+  change extfunApp (Quot.mk eqv f) = extfunApp (Quot.mk eqv g)
  exact congrArg extfunApp (Quot.sound h)

 /--
--- a/src/Init/Data/Array/Basic.lean
+++ b/src/Init/Data/Array/Basic.lean
@@ -246,7 +246,7 @@ def swap (xs : Array α) (i j : @& Nat) (hi : i < xs.size := by get_elem_tactic)
  xs'.set j v₁ (Nat.lt_of_lt_of_eq hj (size_set _).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]
+  change ((xs.set i xs[j]).set j xs[i]
    (Nat.lt_of_lt_of_eq hj (size_set _).symm)).size = xs.size
  rw [size_set, size_set]

--- a/src/Init/Data/Array/Count.lean
+++ b/src/Init/Data/Array/Count.lean
@@ -52,6 +52,7 @@ theorem countP_push {a : α} {xs : Array α} : countP p (xs.push a) = countP p x
  rcases xs with ⟨xs⟩
  simp_all

+@[grind =]
 theorem countP_singleton {a : α} : countP p #[a] = if p a then 1 else 0 := by
  simp

@@ -59,10 +60,12 @@ theorem size_eq_countP_add_countP {xs : Array α} : xs.size = countP p xs + coun
  rcases xs with ⟨xs⟩
  simp [List.length_eq_countP_add_countP (p := p)]

+@[grind _=_]
 theorem countP_eq_size_filter {xs : Array α} : countP p xs = (filter p xs).size := by
  rcases xs with ⟨xs⟩
  simp [List.countP_eq_length_filter]

+@[grind =]
 theorem countP_eq_size_filter' : countP p = size ∘ filter p := by
  funext xs
  apply countP_eq_size_filter
@@ -71,7 +74,7 @@ theorem countP_le_size : countP p xs ≤ xs.size := by
  simp only [countP_eq_size_filter]
  apply size_filter_le

-@[simp] theorem countP_append {xs ys : Array α} : countP p (xs ++ ys) = countP p xs + countP p ys := by
+@[simp, grind =] theorem countP_append {xs ys : Array α} : countP p (xs ++ ys) = countP p xs + countP p ys := by
  rcases xs with ⟨xs⟩
  rcases ys with ⟨ys⟩
  simp
@@ -102,6 +105,7 @@ theorem boole_getElem_le_countP {xs : Array α} {i : Nat} (h : i < xs.size) :
  rcases xs with ⟨xs⟩
  simp [List.boole_getElem_le_countP]

+@[grind =]
 theorem countP_set {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⟩
@@ -146,7 +150,7 @@ theorem countP_flatMap {p : β → Bool} {xs : Array α} {f : α → Array β} :
  rcases xs with ⟨xs⟩
  simp [List.countP_flatMap, Function.comp_def]

-@[simp] theorem countP_reverse {xs : Array α} : countP p xs.reverse = countP p xs := by
+@[simp, grind =] theorem countP_reverse {xs : Array α} : countP p xs.reverse = countP p xs := by
  rcases xs with ⟨xs⟩
  simp [List.countP_reverse]

@@ -173,7 +177,7 @@ variable [BEq α]
  cases xs
  simp

-@[simp] theorem count_empty {a : α} : count a #[] = 0 := rfl
+@[simp, grind =] 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
@@ -186,21 +190,28 @@ theorem count_eq_countP' {a : α} : count a = countP (· == a) := by

 theorem count_le_size {a : α} {xs : Array α} : count a xs ≤ xs.size := countP_le_size

+grind_pattern count_le_size => count a xs
+
+@[grind =]
+theorem count_eq_size_filter {a : α} {xs : Array α} : count a xs = (filter (· == a) xs).size := by
+  simp [count, countP_eq_size_filter]
+
 theorem count_le_count_push {a b : α} {xs : Array α} : count a xs ≤ count a (xs.push b) := by
  simp [count_push]

+@[grind =]
 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 : Array α} : count a (xs ++ ys) = count a xs + count a ys :=
+@[simp, grind =] theorem count_append {a : α} {xs ys : Array α} : count a (xs ++ ys) = count a xs + count a ys :=
  countP_append

-@[simp] theorem count_flatten {a : α} {xss : Array (Array α)} :
+@[simp, grind =] theorem count_flatten {a : α} {xss : Array (Array α)} :
    count a xss.flatten = (xss.map (count a)).sum := by
  cases xss using array₂_induction
  simp [List.count_flatten, Function.comp_def]

-@[simp] theorem count_reverse {a : α} {xs : Array α} : count a xs.reverse = count a xs := by
+@[simp, grind =] theorem count_reverse {a : α} {xs : Array α} : count a xs.reverse = count a xs := by
  rcases xs with ⟨xs⟩
  simp

@@ -209,6 +220,7 @@ theorem boole_getElem_le_count {xs : Array α} {i : Nat} {a : α} (h : i < xs.si
  rw [count_eq_countP]
  apply boole_getElem_le_countP (p := (· == a))

+@[grind =]
 theorem count_set {xs : Array α} {i : Nat} {a b : α} (h : i < xs.size) :
    (xs.set i a).count b = xs.count b - (if xs[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
  simp [count_eq_countP, countP_set, h]
--- a/src/Init/Data/Array/Erase.lean
+++ b/src/Init/Data/Array/Erase.lean
@@ -24,6 +24,7 @@ open Nat

 /-! ### eraseP -/

+@[grind =]
 theorem eraseP_empty : #[].eraseP p = #[] := by simp

 theorem eraseP_of_forall_mem_not {xs : Array α} (h : ∀ a, a ∈ xs → ¬p a) : xs.eraseP p = xs := by
@@ -64,6 +65,7 @@ theorem exists_or_eq_self_of_eraseP (p) (xs : Array α) :
  let ⟨_, ys, zs, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
  rw [e₂]; simp [size_append, e₁]

+@[grind =]
 theorem size_eraseP {xs : Array α} : (xs.eraseP p).size = if xs.any p then xs.size - 1 else xs.size := by
  split <;> rename_i h
  · simp only [any_eq_true] at h
@@ -81,11 +83,12 @@ theorem le_size_eraseP {xs : Array α} : xs.size - 1 ≤ (xs.eraseP p).size := b
  rcases xs with ⟨xs⟩
  simpa using List.le_length_eraseP

+@[grind →]
 theorem mem_of_mem_eraseP {xs : Array α} : a ∈ xs.eraseP p → a ∈ xs := by
  rcases xs with ⟨xs⟩
  simpa using List.mem_of_mem_eraseP

-@[simp] theorem mem_eraseP_of_neg {xs : Array α} (pa : ¬p a) : a ∈ xs.eraseP p ↔ a ∈ xs := by
+@[simp, grind] theorem mem_eraseP_of_neg {xs : Array α} (pa : ¬p a) : a ∈ xs.eraseP p ↔ a ∈ xs := by
  rcases xs with ⟨xs⟩
  simpa using List.mem_eraseP_of_neg pa

@@ -93,15 +96,18 @@ theorem mem_of_mem_eraseP {xs : Array α} : a ∈ xs.eraseP p → a ∈ xs := by
  rcases xs with ⟨xs⟩
  simp

+@[grind _=_]
 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

+@[grind =]
 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

+@[grind =]
 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⟩
@@ -119,6 +125,7 @@ theorem eraseP_append_right {xs : Array α} ys (h : ∀ b ∈ xs, ¬p b) :
  rcases ys with ⟨ys⟩
  simpa using List.eraseP_append_right ys (by simpa using h)

+@[grind =]
 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⟩
@@ -126,6 +133,7 @@ theorem eraseP_append {xs : Array α} {ys : Array α} :
  simp only [List.append_toArray, List.eraseP_toArray, List.eraseP_append, List.any_toArray]
  split <;> simp

+@[grind =]
 theorem eraseP_replicate {n : Nat} {a : α} {p : α → Bool} :
    (replicate n a).eraseP p = if p a then replicate (n - 1) a else replicate n a := by
  simp only [← List.toArray_replicate, List.eraseP_toArray, List.eraseP_replicate]
@@ -165,6 +173,7 @@ theorem eraseP_eq_iff {p} {xs : Array α} :
    · exact Or.inl h
    · exact Or.inr ⟨a, l₁, by simpa using h₁, h₂, ⟨l, by simp⟩⟩

+@[grind =]
 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⟩
@@ -208,6 +217,7 @@ theorem exists_erase_eq [LawfulBEq α] {a : α} {xs : Array α} (h : a ∈ xs) :
    (xs.erase a).size = xs.size - 1 := by
  rw [erase_eq_eraseP]; exact size_eraseP_of_mem h (beq_self_eq_true a)

+@[grind =]
 theorem size_erase [LawfulBEq α] {a : α} {xs : Array α} :
    (xs.erase a).size = if a ∈ xs then xs.size - 1 else xs.size := by
  rw [erase_eq_eraseP, size_eraseP]
@@ -222,11 +232,12 @@ theorem le_size_erase [LawfulBEq α] {a : α} {xs : Array α} : xs.size - 1 ≤
  rcases xs with ⟨xs⟩
  simpa using List.le_length_erase

+@[grind →]
 theorem mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a ∈ xs := by
  rcases xs with ⟨xs⟩
  simpa using List.mem_of_mem_erase (by simpa using h)

-@[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {xs : Array α} (ab : a ≠ b) :
+@[simp, grind] 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)

@@ -234,6 +245,7 @@ theorem mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a
  rw [erase_eq_eraseP', eraseP_eq_self_iff]
  simp [forall_mem_ne']

+@[grind _=_]
 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⟩
@@ -251,6 +263,7 @@ theorem erase_append_right [LawfulBEq α] {a : α} {xs : Array α} (ys : Array
  rcases ys with ⟨ys⟩
  simpa using List.erase_append_right ys (by simpa using h)

+@[grind =]
 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⟩
@@ -258,6 +271,7 @@ theorem erase_append [LawfulBEq α] {a : α} {xs ys : Array α} :
  simp only [List.append_toArray, List.erase_toArray, List.erase_append, mem_toArray]
  split <;> simp

+@[grind =]
 theorem erase_replicate [LawfulBEq α] {n : Nat} {a b : α} :
    (replicate n a).erase b = if b == a then replicate (n - 1) a else replicate n a := by
  simp only [← List.toArray_replicate, List.erase_toArray]
@@ -269,6 +283,7 @@ abbrev erase_mkArray := @erase_replicate

 -- The arguments `a b` are explicit,
 -- so they can be specified to prevent `simp` repeatedly applying the lemma.
+@[grind =]
 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⟩
@@ -312,6 +327,7 @@ theorem eraseIdx_eq_eraseIdxIfInBounds {xs : Array α} {i : Nat} (h : i < xs.siz
    xs.eraseIdx i h = xs.eraseIdxIfInBounds i := by
  simp [eraseIdxIfInBounds, h]

+@[grind =]
 theorem eraseIdx_eq_take_drop_succ {xs : Array α} {i : Nat} (h) :
    xs.eraseIdx i h = xs.take i ++ xs.drop (i + 1) := by
  rcases xs with ⟨xs⟩
@@ -322,6 +338,7 @@ theorem eraseIdx_eq_take_drop_succ {xs : Array α} {i : Nat} (h) :
  rw [List.take_of_length_le]
  simp

+@[grind =]
 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⟩
@@ -339,6 +356,7 @@ theorem getElem?_eraseIdx_of_ge {xs : Array α} {i : Nat} (h : i < xs.size) {j :
  intro h'
  omega

+@[grind =]
 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]
@@ -362,6 +380,7 @@ theorem eraseIdx_ne_empty_iff {xs : Array α} {i : Nat} {h} : xs.eraseIdx i ≠
    simp [h]
  · simp

+@[grind →]
 theorem mem_of_mem_eraseIdx {xs : Array α} {i : Nat} {h} {a : α} (h : a ∈ xs.eraseIdx i) : a ∈ xs := by
  rcases xs with ⟨xs⟩
  simpa using List.mem_of_mem_eraseIdx (by simpa using h)
@@ -373,13 +392,29 @@ theorem eraseIdx_append_of_lt_size {xs : Array α} {k : Nat} (hk : k < xs.size)
  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) :
+theorem eraseIdx_append_of_size_le {xs : Array α} {k : Nat} (hk : xs.size ≤ k) (ys : Array α) (h) :
    eraseIdx (xs ++ ys) k = xs ++ eraseIdx ys (k - xs.size) (by simp at h; omega) := by
  rcases xs with ⟨l⟩
  rcases ys with ⟨l'⟩
  simp at hk
  simp [List.eraseIdx_append_of_length_le, *]

+@[deprecated eraseIdx_append_of_size_le (since := "2025-06-11")]
+abbrev eraseIdx_append_of_length_le := @eraseIdx_append_of_size_le
+
+@[grind =]
+theorem eraseIdx_append {xs ys : Array α} (h : k < (xs ++ ys).size) :
+    eraseIdx (xs ++ ys) k =
+      if h' : k < xs.size then
+        eraseIdx xs k ++ ys
+      else
+        xs ++ eraseIdx ys (k - xs.size) (by simp at h; omega) := by
+  split <;> rename_i h
+  · simp [eraseIdx_append_of_lt_size h]
+  · rw [eraseIdx_append_of_size_le]
+    omega
+
+@[grind =]
 theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} {h} :
    (replicate n a).eraseIdx k = replicate (n - 1) a := by
  simp at h
@@ -428,6 +463,48 @@ theorem eraseIdx_set_gt {xs : Array α} {i : Nat} {j : Nat} {a : α} (h : i < j)
  rcases xs with ⟨xs⟩
  simp [List.eraseIdx_set_gt, *]

+@[grind =]
+theorem eraseIdx_set {xs : Array α} {i : Nat} {a : α} {hi : i < xs.size} {j : Nat} {hj : j < (xs.set i a).size} :
+    (xs.set i a).eraseIdx j =
+      if h' : j < i then
+        (xs.eraseIdx j).set (i - 1) a (by simp; omega)
+      else if h'' : j = i then
+        xs.eraseIdx i
+      else
+        (xs.eraseIdx j (by simp at hj; omega)).set i a (by simp at hj ⊢; omega) := by
+  split <;> rename_i h'
+  · rw [eraseIdx_set_lt]
+    omega
+  · split <;> rename_i h''
+    · subst h''
+      rw [eraseIdx_set_eq]
+    · rw [eraseIdx_set_gt]
+      omega
+
+theorem set_eraseIdx_le {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (h : i ≤ j) (hj : j < (xs.eraseIdx i).size) :
+    (xs.eraseIdx i).set j a = (xs.set (j + 1) a (by simp at hj; omega)).eraseIdx i (by simp at ⊢; omega) := by
+  rw [eraseIdx_set_lt]
+  · simp
+  · omega
+
+theorem set_eraseIdx_gt {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (h : j < i) (hj : j < (xs.eraseIdx i).size) :
+    (xs.eraseIdx i).set j a = (xs.set j a).eraseIdx i (by simp at ⊢; omega) := by
+  rw [eraseIdx_set_gt]
+  omega
+
+@[grind =]
+theorem set_eraseIdx {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (hj : j < (xs.eraseIdx i).size) :
+    (xs.eraseIdx i).set j a =
+      if h' : i ≤ j then
+        (xs.set (j + 1) a (by simp at hj; omega)).eraseIdx i (by simp at ⊢; omega)
+      else
+        (xs.set j a).eraseIdx i (by simp at ⊢; omega) := by
+  split <;> rename_i h'
+  · rw [set_eraseIdx_le]
+    omega
+  · rw [set_eraseIdx_gt]
+    omega
+
@[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
--- a/src/Init/Data/Array/Extract.lean
+++ b/src/Init/Data/Array/Extract.lean
@@ -46,7 +46,7 @@ theorem size_extract_of_le {as : Array α} {i j : Nat} (h : j ≤ as.size) :
  simp
  omega

-@[simp]
+@[simp, grind =]
 theorem extract_push {as : Array α} {b : α} {start stop : Nat} (h : stop ≤ as.size) :
    (as.push b).extract start stop = as.extract start stop := by
  ext i h₁ h₂
@@ -56,7 +56,7 @@ theorem extract_push {as : Array α} {b : α} {start stop : Nat} (h : stop ≤ a
    simp only [getElem_extract, getElem_push]
    rw [dif_pos (by omega)]

-@[simp]
+@[simp, grind =]
 theorem extract_eq_pop {as : Array α} {stop : Nat} (h : stop = as.size - 1) :
    as.extract 0 stop = as.pop := by
  ext i h₁ h₂
@@ -65,7 +65,7 @@ theorem extract_eq_pop {as : Array α} {stop : Nat} (h : stop = as.size - 1) :
  · simp only [size_extract, size_pop] at h₁ h₂
    simp [getElem_extract, getElem_pop]

-@[simp]
+@[simp, grind _=_]
 theorem extract_append_extract {as : Array α} {i j k : Nat} :
    as.extract i j ++ as.extract j k = as.extract (min i j) (max j k) := by
  ext l h₁ h₂
@@ -169,7 +169,7 @@ theorem getElem?_extract_of_succ {as : Array α} {j : Nat} :
  simp [getElem?_extract]
  omega

-@[simp] theorem extract_extract {as : Array α} {i j k l : Nat} :
+@[simp, grind =] theorem extract_extract {as : Array α} {i j k l : Nat} :
    (as.extract i j).extract k l = as.extract (i + k) (min (i + l) j) := by
  ext m h₁ h₂
  · simp
@@ -185,6 +185,7 @@ theorem ne_empty_of_extract_ne_empty {as : Array α} {i j : Nat} (h : as.extract
    as ≠ #[] :=
  mt extract_eq_empty_of_eq_empty h

+@[grind =]
 theorem extract_set {as : Array α} {i j k : Nat} (h : k < as.size) {a : α} :
    (as.set k a).extract i j =
      if _ : k < i then
@@ -211,13 +212,14 @@ theorem extract_set {as : Array α} {i j k : Nat} (h : k < as.size) {a : α} :
        simp [getElem_set]
        omega

+@[grind =]
 theorem set_extract {as : Array α} {i j k : Nat} (h : k < (as.extract i j).size) {a : α} :
    (as.extract i j).set k a = (as.set (i + k) a (by simp at h; omega)).extract i j := by
  ext l h₁ h₂
  · simp
  · simp_all [getElem_set]

-@[simp]
+@[simp, grind =]
 theorem extract_append {as bs : Array α} {i j : Nat} :
    (as ++ bs).extract i j = as.extract i j ++ bs.extract (i - as.size) (j - as.size) := by
  ext l h₁ h₂
@@ -242,14 +244,14 @@ theorem extract_append_right {as bs : Array α} :
    (as ++ bs).extract as.size (as.size + i) = bs.extract 0 i := by
  simp

-@[simp] theorem map_extract {as : Array α} {i j : Nat} :
+@[simp, grind =] theorem map_extract {as : Array α} {i j : Nat} :
    (as.extract i j).map f = (as.map f).extract i j := by
  ext l h₁ h₂
  · simp
  · simp only [size_map, size_extract] at h₁ h₂
    simp only [getElem_map, getElem_extract]

-@[simp] theorem extract_replicate {a : α} {n i j : Nat} :
+@[simp, grind =] theorem extract_replicate {a : α} {n i j : Nat} :
    (replicate n a).extract i j = replicate (min j n - i) a := by
  ext l h₁ h₂
  · simp
@@ -297,6 +299,7 @@ theorem set_eq_push_extract_append_extract {as : Array α} {i : Nat} (h : i < as
  simp at h
  simp [List.set_eq_take_append_cons_drop, h, List.take_of_length_le]

+@[grind =]
 theorem extract_reverse {as : Array α} {i j : Nat} :
    as.reverse.extract i j = (as.extract (as.size - j) (as.size - i)).reverse := by
  ext l h₁ h₂
@@ -307,6 +310,7 @@ theorem extract_reverse {as : Array α} {i j : Nat} :
    congr 1
    omega

+@[grind =]
 theorem reverse_extract {as : Array α} {i j : Nat} :
    (as.extract i j).reverse = as.reverse.extract (as.size - j) (as.size - i) := by
  rw [extract_reverse]
--- a/src/Init/Data/Array/FinRange.lean
+++ b/src/Init/Data/Array/FinRange.lean
@@ -23,10 +23,10 @@ Examples:
 -/
 protected def finRange (n : Nat) : Array (Fin n) := ofFn fun i => i

-@[simp] theorem size_finRange {n} : (Array.finRange n).size = n := by
+@[simp, grind =] theorem size_finRange {n} : (Array.finRange n).size = n := by
  simp [Array.finRange]

-@[simp] theorem getElem_finRange {i : Nat} (h : i < (Array.finRange n).size) :
+@[simp, grind =] theorem getElem_finRange {i : Nat} (h : i < (Array.finRange n).size) :
    (Array.finRange n)[i] = Fin.cast size_finRange ⟨i, h⟩ := by
  simp [Array.finRange]

@@ -49,6 +49,7 @@ theorem finRange_succ_last {n} :
    · simp_all
      omega

+@[grind _=_]
 theorem finRange_reverse {n} : (Array.finRange n).reverse = (Array.finRange n).map Fin.rev := by
  ext i h
  · simp
--- a/src/Init/Data/Array/Find.lean
+++ b/src/Init/Data/Array/Find.lean
@@ -38,11 +38,22 @@ theorem findSome?_singleton {a : α} {f : α → Option β} : #[a].findSome? f =
@[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

+@[grind =]
+theorem findSomeRev?_push {xs : Array α} {a : α} {f : α → Option β} :
+    (xs.push a).findSomeRev? f = (f a).or (xs.findSomeRev? f) := by
+  match h : f a with
+  | some b =>
+    rw [findSomeRev?_push_of_isSome]
+    all_goals simp_all
+  | none =>
+    rw [findSomeRev?_push_of_isNone]
+    all_goals 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 = some b := by
  cases xs; simp_all [List.exists_of_findSome?_eq_some]

-@[simp] theorem findSome?_eq_none_iff : findSome? p xs = none ↔ ∀ x ∈ xs, p x = none := by
+@[simp, grind =] theorem findSome?_eq_none_iff : findSome? p xs = none ↔ ∀ x ∈ xs, p x = none := by
  cases xs; simp

@[simp] theorem findSome?_isSome_iff {f : α → Option β} {xs : Array α} :
@@ -59,36 +70,39 @@ theorem findSome?_eq_some_iff {f : α → Option β} {xs : Array α} {b : β} :
  · rintro ⟨xs, a, ys, h₀, h₁, h₂⟩
    exact ⟨xs.toList, a, ys.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩

-@[simp] theorem findSome?_guard {xs : Array α} : findSome? (Option.guard fun x => p x) xs = find? p xs := by
+@[simp, grind =] theorem findSome?_guard {xs : Array α} : findSome? (Option.guard p) xs = find? p xs := by
  cases xs; simp

-theorem find?_eq_findSome?_guard {xs : Array α} : find? p xs = findSome? (Option.guard fun x => p x) xs :=
+theorem find?_eq_findSome?_guard {xs : Array α} : find? p xs = findSome? (Option.guard p) xs :=
  findSome?_guard.symm

-@[simp] theorem getElem?_zero_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f)[0]? = xs.findSome? f := by
+@[simp, grind =] 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} {xs : Array α} (h) :
+@[simp, grind =] theorem getElem_zero_filterMap {f : α → Option β} {xs : Array α} (h) :
    (xs.filterMap f)[0] = (xs.findSome? f).get (by cases xs; simpa [List.length_filterMap_eq_countP] using h) := by
  cases xs; simp [← List.head_eq_getElem, ← getElem?_zero_filterMap]

-@[simp] theorem back?_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f).back? = xs.findSomeRev? f := by
+@[simp, grind =] theorem back?_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f).back? = xs.findSomeRev? f := by
  cases xs; simp

-@[simp] theorem back!_filterMap [Inhabited β] {f : α → Option β} {xs : Array α} :
+@[simp, grind =] theorem back!_filterMap [Inhabited β] {f : α → Option β} {xs : Array α} :
    (xs.filterMap f).back! = (xs.findSomeRev? f).getD default := by
  cases xs; simp

-@[simp] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Array α} :
+@[simp, grind _=_] 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

+@[grind _=_]
 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]

+@[grind =]
 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]

+@[grind =]
 theorem getElem?_zero_flatten (xss : Array (Array α)) :
    (flatten xss)[0]? = xss.findSome? fun xs => xs[0]? := by
  cases xss using array₂_induction
@@ -104,12 +118,14 @@ 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⟩

+@[grind =]
 theorem getElem_zero_flatten {xss : Array (Array α)} (h) :
    (flatten xss)[0] = (xss.findSome? fun xs => xs[0]?).get (getElem_zero_flatten.proof h) := by
  have t := getElem?_zero_flatten xss
  simp [getElem?_eq_getElem, h] at t
  simp [← t]

+@[grind =]
 theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
  simp [← List.toArray_replicate, List.findSome?_replicate]

@@ -140,8 +156,9 @@ abbrev findSome?_mkArray_of_isNone := @findSome?_replicate_of_isNone

 /-! ### find? -/

-@[simp] theorem find?_empty : find? p #[] = none := rfl
+@[simp, grind =] theorem find?_empty : find? p #[] = none := rfl

+@[grind =]
 theorem find?_singleton {a : α} {p : α → Bool} :
    #[a].find? p = if p a then some a else none := by
  simp
@@ -150,11 +167,26 @@ theorem find?_singleton {a : α} {p : α → Bool} :
    findRev? p (xs.push a) = some a := by
  cases xs; simp [h]

-@[simp] theorem findRev?_cons_of_neg {xs : Array α} (h : ¬p a) :
+@[simp] theorem findRev?_push_of_neg {xs : Array α} (h : ¬p a) :
    findRev? p (xs.push a) = findRev? p xs := by
  cases xs; simp [h]

-@[simp] theorem find?_eq_none : find? p xs = none ↔ ∀ x ∈ xs, ¬ p x := by
+@[deprecated findRev?_push_of_neg (since := "2025-06-12")]
+abbrev findRev?_cons_of_neg := @findRev?_push_of_neg
+
+@[grind =]
+theorem finRev?_push {xs : Array α} :
+    findRev? p (xs.push a) = (Option.guard p a).or (xs.findRev? p) := by
+  cases h : p a
+  · rw [findRev?_push_of_neg, Option.guard_eq_none_iff.mpr h]
+    all_goals simp [h]
+  · rw [findRev?_push_of_pos, Option.guard_eq_some_iff.mpr ⟨rfl, h⟩]
+    all_goals simp [h]
+
+@[deprecated finRev?_push (since := "2025-06-12")]
+abbrev findRev?_cons := @finRev?_push
+
+@[simp, grind =] theorem find?_eq_none : find? p xs = none ↔ ∀ x ∈ xs, ¬ p x := by
  cases xs; simp

 theorem find?_eq_some_iff_append {xs : Array α} :
@@ -178,60 +210,63 @@ theorem find?_push_eq_some {xs : Array α} :
    (xs.push a).find? p = some b ↔ xs.find? p = some b ∨ (xs.find? p = none ∧ (p a ∧ a = b)) := by
  cases xs; simp

-@[simp] theorem find?_isSome {xs : Array α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
+@[simp, grind =] theorem find?_isSome {xs : Array α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
  cases xs; simp

+@[grind →]
 theorem find?_some {xs : Array α} (h : find? p xs = some a) : p a := by
  cases xs
  simp at h
  exact List.find?_some h

+@[grind →]
 theorem mem_of_find?_eq_some {xs : Array α} (h : find? p xs = some a) : a ∈ xs := by
  cases xs
  simp at h
  simpa using List.mem_of_find?_eq_some h

+@[grind]
 theorem get_find?_mem {xs : Array α} (h) : (xs.find? p).get h ∈ xs := by
  cases xs
  simp [List.get_find?_mem]

-@[simp] theorem find?_filter {xs : Array α} (p q : α → Bool) :
+@[simp, grind =] theorem find?_filter {xs : Array α} (p q : α → Bool) :
    (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} :
+@[simp, grind =] 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} {xs : Array α} (h) :
+@[simp, grind =] theorem getElem_zero_filter {p : α → Bool} {xs : Array α} (h) :
    (xs.filter p)[0] =
      (xs.find? p).get (by cases xs; simpa [← List.countP_eq_length_filter] using h) := by
  cases xs
  simp [List.getElem_zero_eq_head]

-@[simp] theorem back?_filter {p : α → Bool} {xs : Array α} : (xs.filter p).back? = xs.findRev? p := by
+@[simp, grind =] theorem back?_filter {p : α → Bool} {xs : Array α} : (xs.filter p).back? = xs.findRev? p := by
  cases xs; simp

-@[simp] theorem back!_filter [Inhabited α] {p : α → Bool} {xs : Array α} :
+@[simp, grind =] theorem back!_filter [Inhabited α] {p : α → Bool} {xs : Array α} :
    (xs.filter p).back! = (xs.findRev? p).get! := by
  cases xs; simp [Option.get!_eq_getD]

-@[simp] theorem find?_filterMap {xs : Array α} {f : α → Option β} {p : β → Bool} :
+@[simp, grind =] 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
  cases xs; simp

-@[simp] theorem find?_map {f : β → α} {xs : Array β} :
+@[simp, grind =] theorem find?_map {f : β → α} {xs : Array β} :
    find? p (xs.map f) = (xs.find? (p ∘ f)).map f := by
  cases xs; simp

-@[simp] theorem find?_append {xs ys : Array α} :
+@[simp, grind =] theorem find?_append {xs ys : Array α} :
    (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
  cases xs
  cases ys
  simp

-@[simp] theorem find?_flatten {xss : Array (Array α)} {p : α → Bool} :
-    xss.flatten.find? p = xss.findSome? (·.find? p) := by
+@[simp, grind _=_] theorem find?_flatten {xss : Array (Array α)} {p : α → Bool} :
+    xss.flatten.find? p = xss.findSome? (find? p) := by
  cases xss using array₂_induction
  simp [List.findSome?_map, Function.comp_def]

@@ -270,7 +305,7 @@ theorem find?_flatten_eq_some_iff {xss : Array (Array α)} {p : α → Bool} {a
@[deprecated find?_flatten_eq_some_iff (since := "2025-02-03")]
 abbrev find?_flatten_eq_some := @find?_flatten_eq_some_iff

-@[simp] theorem find?_flatMap {xs : Array α} {f : α → Array β} {p : β → Bool} :
+@[simp, grind =] theorem find?_flatMap {xs : Array α} {f : α → Array β} {p : β → Bool} :
    (xs.flatMap f).find? p = xs.findSome? (fun x => (f x).find? p) := by
  cases xs
  simp [List.find?_flatMap, Array.flatMap_toArray]
@@ -282,6 +317,7 @@ theorem find?_flatMap_eq_none_iff {xs : Array α} {f : α → Array β} {p : β
@[deprecated find?_flatMap_eq_none_iff (since := "2025-02-03")]
 abbrev find?_flatMap_eq_none := @find?_flatMap_eq_none_iff

+@[grind =]
 theorem find?_replicate :
    find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
  simp [← List.toArray_replicate, List.find?_replicate]
@@ -334,6 +370,7 @@ abbrev find?_mkArray_eq_some := @find?_replicate_eq_some_iff
@[deprecated get_find?_replicate (since := "2025-03-18")]
 abbrev get_find?_mkArray := @get_find?_replicate

+@[grind =]
 theorem find?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
    (H : ∀ (a : α), a ∈ xs → P a) {p : β → Bool} :
    (xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
@@ -347,12 +384,15 @@ theorem find?_eq_some_iff_getElem {xs : Array α} {p : α → Bool} {b : α} :

 /-! ### findIdx -/

+@[grind =]
 theorem findIdx_empty : findIdx p #[] = 0 := rfl

+@[grind =]
 theorem findIdx_singleton {a : α} {p : α → Bool} :
    #[a].findIdx p = if p a then 0 else 1 := by
  simp

+@[grind →]
 theorem findIdx_of_getElem?_eq_some {xs : Array α} (w : xs[xs.findIdx p]? = some y) : p y := by
  rcases xs with ⟨xs⟩
  exact List.findIdx_of_getElem?_eq_some (by simpa using w)
@@ -361,6 +401,8 @@ theorem findIdx_getElem {xs : Array α} {w : xs.findIdx p < xs.size} :
    p xs[xs.findIdx p] :=
  xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)

+grind_pattern findIdx_getElem => xs[xs.findIdx p]
+
 theorem findIdx_lt_size_of_exists {xs : Array α} (h : ∃ x ∈ xs, p x) :
    xs.findIdx p < xs.size := by
  rcases xs with ⟨xs⟩
@@ -387,18 +429,24 @@ theorem findIdx_le_size {p : α → Bool} {xs : Array α} : xs.findIdx p ≤ xs.
  · simp at e
    exact Nat.le_of_eq (findIdx_eq_size.mpr e)

+grind_pattern findIdx_le_size => xs.findIdx p, xs.size
+
@[simp]
 theorem findIdx_lt_size {p : α → Bool} {xs : Array α} :
    xs.findIdx p < xs.size ↔ ∃ x ∈ xs, p x := by
  rcases xs with ⟨xs⟩
  simp

+grind_pattern findIdx_lt_size => xs.findIdx p, xs.size
+
 /-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/
 theorem not_of_lt_findIdx {p : α → Bool} {xs : Array α} {i : Nat} (h : i < xs.findIdx p) :
    p (xs[i]'(Nat.le_trans h findIdx_le_size)) = false := by
  rcases xs with ⟨xs⟩
  simpa using List.not_of_lt_findIdx (by simpa using h)

+grind_pattern not_of_lt_findIdx => xs.findIdx p, xs[i]
+
 /-- If `¬ p xs[j]` for all `j < i`, then `i ≤ xs.findIdx p`. -/
 theorem le_findIdx_of_not {p : α → Bool} {xs : Array α} {i : Nat} (h : i < xs.size)
    (h2 : ∀ j (hji : j < i), p (xs[j]'(Nat.lt_trans hji h)) = false) : i ≤ xs.findIdx p := by
@@ -426,6 +474,7 @@ 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]

+@[grind =]
 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
@@ -433,6 +482,7 @@ theorem findIdx_append {p : α → Bool} {xs ys : Array α} :
  rcases ys with ⟨ys⟩
  simp [List.findIdx_append]

+@[grind =]
 theorem findIdx_push {xs : Array α} {a : α} {p : α → Bool} :
    (xs.push a).findIdx p = if xs.findIdx p < xs.size then xs.findIdx p else xs.size + if p a then 0 else 1 := by
  simp only [push_eq_append, findIdx_append]
@@ -455,7 +505,7 @@ theorem false_of_mem_extract_findIdx {xs : Array α} {p : α → Bool} (h : x
  rcases xs with ⟨xs⟩
  exact List.false_of_mem_take_findIdx (by simpa using h)

-@[simp] theorem findIdx_extract {xs : Array α} {i : Nat} {p : α → Bool} :
+@[simp, grind =] theorem findIdx_extract {xs : Array α} {i : Nat} {p : α → Bool} :
    (xs.extract 0 i).findIdx p = min i (xs.findIdx p) := by
  cases xs
  simp
@@ -467,24 +517,24 @@ theorem false_of_mem_extract_findIdx {xs : Array α} {p : α → Bool} (h : x

 /-! ### findIdx? -/

-@[simp] theorem findIdx?_empty : (#[] : Array α).findIdx? p = none := by simp
-theorem findIdx?_singleton {a : α} {p : α → Bool} :
+@[simp, grind =] theorem findIdx?_empty : (#[] : Array α).findIdx? p = none := by simp
+@[grind =] theorem findIdx?_singleton {a : α} {p : α → Bool} :
    #[a].findIdx? p = if p a then some 0 else none := by
  simp

-@[simp]
+@[simp, grind =]
 theorem findIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
    xs.findIdx? p = none ↔ ∀ x, x ∈ xs → p x = false := by
  rcases xs with ⟨xs⟩
  simp

-@[simp]
+@[simp, grind =]
 theorem findIdx?_isSome {xs : Array α} {p : α → Bool} :
    (xs.findIdx? p).isSome = xs.any p := by
  rcases xs with ⟨xs⟩
  simp [List.findIdx?_isSome]

-@[simp]
+@[simp, grind =]
 theorem findIdx?_isNone {xs : Array α} {p : α → Bool} :
    (xs.findIdx? p).isNone = xs.all (¬p ·) := by
  rcases xs with ⟨xs⟩
@@ -526,18 +576,19 @@ 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} {p : α → Bool} :
+@[simp, grind =] theorem findIdx?_map {f : β → α} {xs : Array β} {p : α → Bool} :
    findIdx? p (xs.map f) = xs.findIdx? (p ∘ f) := by
  rcases xs with ⟨xs⟩
  simp [List.findIdx?_map]

-@[simp] theorem findIdx?_append :
+@[simp, grind =] theorem findIdx?_append :
    (xs ++ ys : Array α).findIdx? p =
      (xs.findIdx? p).or ((ys.findIdx? p).map fun i => i + xs.size) := by
  rcases xs with ⟨xs⟩
  rcases ys with ⟨ys⟩
  simp [List.findIdx?_append]

+@[grind =]
 theorem findIdx?_push {xs : Array α} {a : α} {p : α → Bool} :
    (xs.push a).findIdx? p = (xs.findIdx? p).or (if p a then some xs.size else none) := by
  simp only [push_eq_append, findIdx?_append]
@@ -553,7 +604,7 @@ theorem findIdx?_flatten {xss : Array (Array α)} {p : α → Bool} :
  cases xss using array₂_induction
  simp [List.findIdx?_flatten, Function.comp_def]

-@[simp] theorem findIdx?_replicate :
+@[simp, grind =] theorem findIdx?_replicate :
    (replicate n a).findIdx? p = if 0 < n ∧ p a then some 0 else none := by
  rw [← List.toArray_replicate]
  simp only [List.findIdx?_toArray]
@@ -578,6 +629,7 @@ theorem findIdx?_eq_none_of_findIdx?_eq_none {xs : Array α} {p q : α → Bool}
  rcases xs with ⟨xs⟩
  simpa using List.findIdx?_eq_none_of_findIdx?_eq_none (by simpa using w)

+@[grind =]
 theorem findIdx_eq_getD_findIdx? {xs : Array α} {p : α → Bool} :
    xs.findIdx p = (xs.findIdx? p).getD xs.size := by
  rcases xs with ⟨xs⟩
@@ -594,15 +646,17 @@ theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : Array α} {p q : α → Bo
  cases xs
  simp [hf]

-@[simp] theorem findIdx?_take {xs : Array α} {i : Nat} {p : α → Bool} :
+@[simp, grind =] theorem findIdx?_take {xs : Array α} {i : Nat} {p : α → Bool} :
    (xs.take i).findIdx? p = (xs.findIdx? p).bind (Option.guard (fun j => j < i)) := by
  cases xs
  simp

 /-! ### findFinIdx? -/

+@[grind =]
 theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p #[] = none := by simp

+@[grind =]
 theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
    #[a].findFinIdx? p = if p a then some ⟨0, by simp⟩ else none := by
  simp
@@ -620,7 +674,7 @@ theorem findFinIdx?_eq_pmap_findIdx? {xs : Array α} {p : α → Bool} :
        (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} :
+@[simp, grind =] 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?]

@@ -636,12 +690,14 @@ theorem findFinIdx?_eq_some_iff {xs : Array α} {p : α → Bool} {i : Fin xs.si
  · rintro ⟨h, w⟩
    exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩

+@[grind =]
 theorem findFinIdx?_push {xs : Array α} {a : α} {p : α → Bool} :
    (xs.push a).findFinIdx? p =
      ((xs.findFinIdx? p).map (Fin.castLE (by simp))).or (if p a then some ⟨xs.size, by simp⟩ else none) := by
  simp only [findFinIdx?_eq_pmap_findIdx?, findIdx?_push, Option.pmap_or]
  split <;> rename_i h _ <;> split <;> simp [h]

+@[grind =]
 theorem findFinIdx?_append {xs ys : Array α} {p : α → Bool} :
    (xs ++ ys).findFinIdx? p =
      ((xs.findFinIdx? p).map (Fin.castLE (by simp))).or
@@ -651,13 +707,13 @@ theorem findFinIdx?_append {xs ys : Array α} {p : α → Bool} :
  · simp [h, Option.pmap_map, Option.map_pmap, Nat.add_comm]
  · simp [h]

-@[simp]
+@[simp, grind =]
 theorem isSome_findFinIdx? {xs : Array α} {p : α → Bool} :
    (xs.findFinIdx? p).isSome = xs.any p := by
  rcases xs with ⟨xs⟩
  simp [Array.size]

-@[simp]
+@[simp, grind =]
 theorem isNone_findFinIdx? {xs : Array α} {p : α → Bool} :
    (xs.findFinIdx? p).isNone = xs.all (fun x => ¬ p x) := by
  rcases xs with ⟨xs⟩
@@ -678,6 +734,7 @@ The verification API for `idxOf` is still incomplete.
 The lemmas below should be made consistent with those for `findIdx` (and proved using them).
 -/

+@[grind =]
 theorem idxOf_append [BEq α] [LawfulBEq α] {xs ys : Array α} {a : α} :
    (xs ++ ys).idxOf a = if a ∈ xs then xs.idxOf a else ys.idxOf a + xs.size := by
  rw [idxOf, findIdx_append]
@@ -691,10 +748,23 @@ theorem idxOf_eq_size [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∉ xs) : x
  rcases xs with ⟨xs⟩
  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
+theorem idxOf_lt_length_of_mem [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∈ xs) : xs.idxOf a < xs.size := by
  rcases xs with ⟨xs⟩
-  simp [List.idxOf_lt_length (by simpa using h)]
+  simp [List.idxOf_lt_length_of_mem (by simpa using h)]

+theorem idxOf_le_size [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
+    xs.idxOf a ≤ xs.size := by
+  rcases xs with ⟨xs⟩
+  simp [List.idxOf_le_length]
+
+grind_pattern idxOf_le_size => xs.idxOf a, xs.size
+
+theorem idxOf_lt_size_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
+    xs.idxOf a < xs.size ↔ a ∈ xs := by
+  rcases xs with ⟨xs⟩
+  simp [List.idxOf_lt_length_iff]
+
+grind_pattern idxOf_lt_size_iff => xs.idxOf a, xs.size

 /-! ### idxOf?

@@ -702,19 +772,20 @@ 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?_empty [BEq α] : (#[] : Array α).idxOf? a = none := by simp
+@[grind =] theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := by simp

-@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
+@[simp, grind =] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
    xs.idxOf? a = none ↔ a ∉ xs := by
  rcases xs with ⟨xs⟩
  simp [List.idxOf?_eq_none_iff]

-@[simp]
+@[simp, grind =]
 theorem isSome_idxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
    (xs.idxOf? a).isSome ↔ a ∈ xs := by
  rcases xs with ⟨xs⟩
  simp

+@[grind =]
 theorem isNone_idxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
    (xs.idxOf? a).isNone = ¬ a ∈ xs := by
  simp
@@ -729,9 +800,9 @@ 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]

-theorem finIdxOf?_empty [BEq α] : (#[] : Array α).finIdxOf? a = none := by simp
+@[grind =] theorem finIdxOf?_empty [BEq α] : (#[] : Array α).finIdxOf? a = none := by simp

-@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
+@[simp, grind =] 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, Array.size]
@@ -742,14 +813,16 @@ theorem finIdxOf?_empty [BEq α] : (#[] : Array α).finIdxOf? a = none := by sim
  unfold Array.size at i ⊢
  simp [List.finIdxOf?_eq_some_iff]

-@[simp]
-theorem isSome_finIdxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
-    (xs.finIdxOf? a).isSome ↔ a ∈ xs := by
+@[simp, grind =]
+theorem isSome_finIdxOf? [BEq α] [PartialEquivBEq α] {xs : Array α} {a : α} :
+    (xs.finIdxOf? a).isSome = xs.contains a := by
  rcases xs with ⟨xs⟩
  simp [Array.size]

-theorem isNone_finIdxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
-    (xs.finIdxOf? a).isNone = ¬ a ∈ xs := by
-  simp
+@[simp, grind =]
+theorem isNone_finIdxOf? [BEq α] [PartialEquivBEq α] {xs : Array α} {a : α} :
+    (xs.finIdxOf? a).isNone = !xs.contains a := by
+  rcases xs with ⟨xs⟩
+  simp [Array.size]

 end Array
--- a/src/Init/Data/Array/InsertIdx.lean
+++ b/src/Init/Data/Array/InsertIdx.lean
@@ -47,11 +47,16 @@ theorem insertIdx_zero {xs : Array α} {x : α} : xs.insertIdx 0 x = #[x] ++ xs
  simp at h
  simp [List.length_insertIdx, h]

-theorem eraseIdx_insertIdx {i : Nat} {xs : Array α} (h : i ≤ xs.size) :
+theorem eraseIdx_insertIdx_self {i : Nat} {xs : Array α} (h : i ≤ xs.size) :
    (xs.insertIdx i a).eraseIdx i (by simp; omega) = xs := by
  rcases xs with ⟨xs⟩
  simp_all

+@[deprecated eraseIdx_insertIdx_self (since := "2025-06-15")]
+theorem eraseIdx_insertIdx {i : Nat} {xs : Array α} (h : i ≤ xs.size) :
+    (xs.insertIdx i a).eraseIdx i (by simp; omega) = xs := by
+  simp [eraseIdx_insertIdx_self]
+
 theorem insertIdx_eraseIdx_of_ge {as : Array α}
    (w₁ : i < as.size) (w₂ : j ≤ (as.eraseIdx i).size) (h : i ≤ j) :
    (as.eraseIdx i).insertIdx j a =
@@ -66,6 +71,18 @@ theorem insertIdx_eraseIdx_of_le {as : Array α}
  cases as
  simpa using List.insertIdx_eraseIdx_of_le (by simpa) (by simpa)

+@[grind =]
+theorem insertIdx_eraseIdx {as : Array α} (h₁ : i < as.size) (h₂ : j ≤ (as.eraseIdx i).size) :
+    (as.eraseIdx i).insertIdx j a =
+      if h : i ≤ j then
+        (as.insertIdx (j + 1) a (by simp_all; omega)).eraseIdx i (by simp_all; omega)
+      else
+        (as.insertIdx j a).eraseIdx (i + 1) (by simp_all) := by
+  split <;> rename_i h'
+  · rw [insertIdx_eraseIdx_of_ge] <;> omega
+  · rw [insertIdx_eraseIdx_of_le] <;> omega
+
+@[grind =]
 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
@@ -81,6 +98,7 @@ theorem insertIdx_size_self {xs : Array α} {x : α} : xs.insertIdx xs.size x =
  rcases xs with ⟨xs⟩
  simp

+@[grind =]
 theorem getElem_insertIdx {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.size) (h : k < (xs.insertIdx i x).size) :
    (xs.insertIdx i x)[k] =
      if h₁ : k < i then
@@ -106,6 +124,7 @@ theorem getElem_insertIdx_of_gt {xs : Array α} {x : α} {i k : Nat} (w : k ≤
  simp [getElem_insertIdx, w, h]
  rw [dif_neg (by omega), dif_neg (by omega)]

+@[grind =]
 theorem getElem?_insertIdx {xs : Array α} {x : α} {i k : Nat} (h : i ≤ xs.size) :
    (xs.insertIdx i x)[k]? =
      if k < i then
--- a/src/Init/Data/Array/Lemmas.lean
+++ b/src/Init/Data/Array/Lemmas.lean
@@ -89,6 +89,8 @@ theorem size_pos_of_mem {a : α} {xs : Array α} (h : a ∈ xs) : 0 < xs.size :=
  simp only [mem_toArray] at h
  simpa using List.length_pos_of_mem h

+grind_pattern size_pos_of_mem => a ∈ xs, xs.size
+
 theorem exists_mem_of_size_pos {xs : Array α} (h : 0 < xs.size) : ∃ a, a ∈ xs := by
  cases xs
  simpa using List.exists_mem_of_length_pos h
@@ -1498,6 +1500,19 @@ theorem forall_mem_filter {p : α → Bool} {xs : Array α} {P : α → Prop} :
    (∀ (i) (_ : i ∈ xs.filter p), P i) ↔ ∀ (j) (_ : j ∈ xs), p j → P j := by
  simp

+@[grind] theorem getElem_filter {xs : Array α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).size) :
+    p (xs.filter p)[i] :=
+  (mem_filter.mp (getElem_mem h)).2
+
+theorem getElem?_filter {xs : Array α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).size)
+    (w : (xs.filter p)[i]? = some a) : p a := by
+  rw [getElem?_eq_getElem] at w
+  simp only [Option.some.injEq] at w
+  rw [← w]
+  apply getElem_filter h
+
+grind_pattern getElem?_filter => (xs.filter p)[i]?, some a
+
@[simp] theorem filter_filter {p q : α → Bool} {xs : Array α} :
    filter p (filter q xs) = filter (fun a => p a && q a) xs := by
  apply ext'
@@ -3621,8 +3636,8 @@ We can prove that two folds over the same array are related (by some arbitrary r
 if we know that the initial elements are related and the folding function, for each element of the array,
 preserves the relation.
 -/
-theorem foldl_rel {xs : Array α} {f g : β → α → β} {a b : β} {r : β → β → Prop}
-    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
+theorem foldl_rel {xs : Array α} {f : β → α → β} {g : γ → α → γ} {a : β} {b : γ} {r : β → γ → Prop}
+    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f c a) (g c' a)) :
    r (xs.foldl (fun acc a => f acc a) a) (xs.foldl (fun acc a => g acc a) b) := by
  rcases xs with ⟨xs⟩
  simpa using List.foldl_rel h (by simpa using h')
@@ -3632,8 +3647,8 @@ We can prove that two folds over the same array are related (by some arbitrary r
 if we know that the initial elements are related and the folding function, for each element of the array,
 preserves the relation.
 -/
-theorem foldr_rel {xs : Array α} {f g : α → β → β} {a b : β} {r : β → β → Prop}
-    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
+theorem foldr_rel {xs : Array α} {f : α → β → β} {g : α → γ → γ} {a : β} {b : γ} {r : β → γ → Prop}
+    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f a c) (g a c')) :
    r (xs.foldr (fun a acc => f a acc) a) (xs.foldr (fun a acc => g a acc) b) := by
  rcases xs with ⟨xs⟩
  simpa using List.foldr_rel h (by simpa using h')
@@ -4526,7 +4541,7 @@ abbrev contains_def [DecidableEq α] {a : α} {xs : Array α} : xs.contains a
    (zip xs ys).size = min xs.size ys.size :=
  size_zipWith

-@[simp] theorem getElem_zipWith {xs : Array α} {ys : Array β} {f : α → β → γ} {i : Nat}
+@[simp, grind =] theorem getElem_zipWith {xs : Array α} {ys : Array β} {f : α → β → γ} {i : Nat}
    (hi : i < (zipWith f xs ys).size) :
    (zipWith f xs ys)[i] = f (xs[i]'(by simp at hi; omega)) (ys[i]'(by simp at hi; omega)) := by
  cases xs
--- a/src/Init/Data/Array/MapIdx.lean
+++ b/src/Init/Data/Array/MapIdx.lean
@@ -51,27 +51,27 @@ theorem mapFinIdx_spec {xs : Array α} {f : (i : Nat) → α → (h : i < xs.siz
      ∀ i h, p i ((Array.mapFinIdx xs 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} :
+@[simp, grind =] theorem size_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
    (xs.mapFinIdx f).size = xs.size :=
  (mapFinIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1

-@[simp] theorem size_zipIdx {xs : Array α} {k : Nat} : (xs.zipIdx k).size = xs.size :=
+@[simp, grind =] theorem size_zipIdx {xs : Array α} {k : Nat} : (xs.zipIdx k).size = xs.size :=
  Array.size_mapFinIdx

@[deprecated size_zipIdx (since := "2025-01-21")] abbrev size_zipWithIndex := @size_zipIdx

-@[simp] theorem getElem_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {i : Nat}
+@[simp, grind =] 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 (p := fun i b h => b = f i xs[i] h) fun _ _ => rfl).2 i _

-@[simp] theorem getElem?_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {i : Nat} :
+@[simp, grind =] 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 => some <| 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} :
+@[simp, grind =] theorem toList_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
    (xs.mapFinIdx f).toList = xs.toList.mapFinIdx (fun i a h => f i a (by simpa)) := by
  apply List.ext_getElem <;> simp

@@ -91,20 +91,20 @@ theorem mapIdx_spec {f : Nat → α → β} {xs : Array α}
      ∀ i h, p i ((xs.mapIdx f)[i]) h :=
  (mapIdx_induction (motive := fun _ => True) trivial fun _ _ _ => ⟨hs .., trivial⟩).2

-@[simp] theorem size_mapIdx {f : Nat → α → β} {xs : Array α} : (xs.mapIdx f).size = xs.size :=
+@[simp, grind =] theorem size_mapIdx {f : Nat → α → β} {xs : Array α} : (xs.mapIdx f).size = xs.size :=
  (mapIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1

-@[simp] theorem getElem_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat}
+@[simp, grind =] 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 (p := fun i b h => b = f i xs[i]) fun _ _ => rfl).2 i (by simp_all)

-@[simp] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat} :
+@[simp, grind =] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat} :
    (xs.mapIdx f)[i]? =
      xs[i]?.map (f i) := by
  simp [getElem?_def, size_mapIdx, getElem_mapIdx]

-@[simp] theorem toList_mapIdx {f : Nat → α → β} {xs : Array α} :
+@[simp, grind =] theorem toList_mapIdx {f : Nat → α → β} {xs : Array α} :
    (xs.mapIdx f).toList = xs.toList.mapIdx (fun i a => f i a) := by
  apply List.ext_getElem <;> simp

@@ -126,7 +126,7 @@ namespace Array

 /-! ### zipIdx -/

-@[simp] theorem getElem_zipIdx {xs : Array α} {k : Nat} {i : Nat} (h : i < (xs.zipIdx k).size) :
+@[simp, grind =] theorem getElem_zipIdx {xs : Array α} {k : Nat} {i : Nat} (h : i < (xs.zipIdx k).size) :
    (xs.zipIdx k)[i] = (xs[i]'(by simp_all), k + i) := by
  simp [zipIdx]

@@ -140,7 +140,7 @@ 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} :
+@[simp, grind =] theorem toList_zipIdx {xs : Array α} {k : Nat} :
    (xs.zipIdx k).toList = xs.toList.zipIdx k := by
  rcases xs with ⟨xs⟩
  simp
@@ -185,7 +185,7 @@ abbrev mem_zipWithIndex_iff_getElem? := @mem_zipIdx_iff_getElem?
  subst w
  rfl

-@[simp]
+@[simp, grind =]
 theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #[] f = #[] :=
  rfl

@@ -195,6 +195,7 @@ theorem mapFinIdx_eq_ofFn {xs : Array α} {f : (i : Nat) → α → (h : i < xs.
  simp only [List.mapFinIdx_toArray, List.mapFinIdx_eq_ofFn, Fin.getElem_fin, List.getElem_toArray]
  simp [Array.size]

+@[grind =]
 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)) ++
@@ -203,7 +204,7 @@ theorem mapFinIdx_append {xs ys : Array α} {f : (i : Nat) → α → (h : i < (
  cases ys
  simp [List.mapFinIdx_append, Array.size]

-@[simp]
+@[simp, grind =]
 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
@@ -237,7 +238,7 @@ theorem exists_of_mem_mapFinIdx {b : β} {xs : Array α} {f : (i : Nat) → α
  rcases xs with ⟨xs⟩
  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} :
+@[simp, grind =] 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
@@ -290,7 +291,7 @@ theorem mapFinIdx_eq_mapFinIdx_iff {xs : Array α} {f g : (i : Nat) → α → (
  rw [eq_comm, mapFinIdx_eq_iff]
  simp

-@[simp] theorem mapFinIdx_mapFinIdx {xs : Array α}
+@[simp, grind =] 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
@@ -305,14 +306,14 @@ theorem mapFinIdx_eq_replicate_iff {xs : Array α} {f : (i : Nat) → α → (h
@[deprecated mapFinIdx_eq_replicate_iff (since := "2025-03-18")]
 abbrev mapFinIdx_eq_mkArray_iff := @mapFinIdx_eq_replicate_iff

-@[simp] theorem mapFinIdx_reverse {xs : Array α} {f : (i : Nat) → α → (h : i < xs.reverse.size) → β} :
+@[simp, grind =] theorem mapFinIdx_reverse {xs : Array α} {f : (i : Nat) → α → (h : i < xs.reverse.size) → β} :
    xs.reverse.mapFinIdx f = (xs.mapFinIdx (fun i a h => f (xs.size - 1 - i) a (by simp; omega))).reverse := by
  rcases xs with ⟨l⟩
  simp [List.mapFinIdx_reverse, Array.size]

 /-! ### mapIdx -/

-@[simp]
+@[simp, grind =]
 theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #[] = #[] :=
  rfl

@@ -332,13 +333,14 @@ theorem mapIdx_eq_zipIdx_map {xs : Array α} {f : Nat → α → β} :
@[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-21")]
 abbrev mapIdx_eq_zipWithIndex_map := @mapIdx_eq_zipIdx_map

+@[grind =]
 theorem mapIdx_append {xs ys : Array α} :
    (xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx (fun i => f (i + xs.size)) := by
  rcases xs with ⟨xs⟩
  rcases ys with ⟨ys⟩
  simp [List.mapIdx_append]

-@[simp]
+@[simp, grind =]
 theorem mapIdx_push {xs : Array α} {a : α} :
    mapIdx f (xs.push a) = (mapIdx f xs).push (f xs.size a) := by
  simp [← append_singleton, mapIdx_append]
@@ -360,7 +362,7 @@ theorem exists_of_mem_mapIdx {b : β} {xs : Array α}
  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} :
+@[simp, grind =] theorem mem_mapIdx {b : β} {xs : Array α} :
    b ∈ mapIdx f xs ↔ ∃ (i : Nat) (h : i < xs.size), f i xs[i] = b := by
  constructor
  · intro h
@@ -414,7 +416,7 @@ theorem mapIdx_eq_mapIdx_iff {xs : Array α} :
  rcases xs with ⟨xs⟩
  simp [List.mapIdx_eq_mapIdx_iff]

-@[simp] theorem mapIdx_set {f : Nat → α → β} {xs : Array α} {i : Nat} {h : i < xs.size} {a : α} :
+@[simp, grind =] theorem mapIdx_set {f : Nat → α → β} {xs : Array α} {i : Nat} {h : i < xs.size} {a : α} :
    (xs.set i a).mapIdx f = (xs.mapIdx f).set i (f i a) (by simpa) := by
  rcases xs with ⟨xs⟩
  simp [List.mapIdx_set]
@@ -424,17 +426,17 @@ theorem mapIdx_eq_mapIdx_iff {xs : Array α} :
  rcases xs with ⟨xs⟩
  simp [List.mapIdx_set]

-@[simp] theorem back?_mapIdx {xs : Array α} {f : Nat → α → β} :
+@[simp, grind =] theorem back?_mapIdx {xs : Array α} {f : Nat → α → β} :
    (mapIdx f xs).back? = (xs.back?).map (f (xs.size - 1)) := by
  rcases xs with ⟨xs⟩
  simp [List.getLast?_mapIdx]

-@[simp] theorem back_mapIdx {xs : Array α} {f : Nat → α → β} (h) :
+@[simp, grind =] theorem back_mapIdx {xs : Array α} {f : Nat → α → β} (h) :
    (xs.mapIdx f).back h = f (xs.size - 1) (xs.back (by simpa using h)) := by
  rcases xs with ⟨xs⟩
  simp [List.getLast_mapIdx]

-@[simp] theorem mapIdx_mapIdx {xs : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
+@[simp, grind =] theorem mapIdx_mapIdx {xs : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
    (xs.mapIdx f).mapIdx g = xs.mapIdx (fun i => g i ∘ f i) := by
  simp [mapIdx_eq_iff]

@@ -447,7 +449,7 @@ theorem mapIdx_eq_replicate_iff {xs : Array α} {f : Nat → α → β} {b : β}
@[deprecated mapIdx_eq_replicate_iff (since := "2025-03-18")]
 abbrev mapIdx_eq_mkArray_iff := @mapIdx_eq_replicate_iff

-@[simp] theorem mapIdx_reverse {xs : Array α} {f : Nat → α → β} :
+@[simp, grind =] theorem mapIdx_reverse {xs : Array α} {f : Nat → α → β} :
    xs.reverse.mapIdx f = (mapIdx (fun i => f (xs.size - 1 - i)) xs).reverse := by
  rcases xs with ⟨xs⟩
  simp [List.mapIdx_reverse]
@@ -456,7 +458,7 @@ end Array

 namespace List

-@[grind] theorem mapFinIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
+@[grind =] theorem mapFinIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
    {f : (i : Nat) → α → (h : i < l.length) → m β} :
    l.toArray.mapFinIdxM f = toArray <$> l.mapFinIdxM f := by
  let rec go (i : Nat) (acc : Array β) (inv : i + acc.size = l.length) :
@@ -477,7 +479,7 @@ namespace List
  simp only [Array.mapFinIdxM, mapFinIdxM]
  exact go _ #[] _

-@[grind] theorem mapIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
+@[grind =] theorem mapIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
    {f : Nat → α → m β} :
    l.toArray.mapIdxM f = toArray <$> l.mapIdxM f := by
  let rec go (bs : List α) (acc : Array β) (inv : bs.length + acc.size = l.length) :
--- a/src/Init/Data/Array/OfFn.lean
+++ b/src/Init/Data/Array/OfFn.lean
@@ -23,7 +23,7 @@ namespace Array

 /-! ### ofFn -/

-@[simp] theorem ofFn_zero {f : Fin 0 → α} : ofFn f = #[] := by
+@[simp, grind =] theorem ofFn_zero {f : Fin 0 → α} : ofFn f = #[] := by
  simp [ofFn, ofFn.go]

 theorem ofFn_succ {f : Fin (n+1) → α} :
@@ -42,10 +42,10 @@ theorem ofFn_add {n m} {f : Fin (n + m) → α} :
  | zero => simp
  | succ m ih => simp [ofFn_succ, ih]

-@[simp] theorem _root_.List.toArray_ofFn {f : Fin n → α} : (List.ofFn f).toArray = Array.ofFn f := by
+@[simp, grind =] theorem _root_.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
+@[simp, grind =] theorem toList_ofFn {f : Fin n → α} : (Array.ofFn f).toList = List.ofFn f := by
  apply List.ext_getElem <;> simp

 theorem ofFn_succ' {f : Fin (n+1) → α} :
@@ -58,7 +58,7 @@ theorem ofFn_eq_empty_iff {f : Fin n → α} : ofFn f = #[] ↔ n = 0 := by
  rw [← Array.toList_inj]
  simp

-@[simp 500]
+@[simp 500, grind =]
 theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
  constructor
  · intro w
@@ -73,7 +73,7 @@ theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i =
 def ofFnM {n} [Monad m] (f : Fin n → m α) : m (Array α) :=
  Fin.foldlM n (fun xs i => xs.push <$> f i) (Array.emptyWithCapacity n)

-@[simp]
+@[simp, grind =]
 theorem ofFnM_zero [Monad m] {f : Fin 0 → m α} : ofFnM f = pure #[] := by
  simp [ofFnM]

@@ -109,7 +109,7 @@ theorem ofFnM_add {n m} [Monad m] [LawfulMonad m] {f : Fin (n + k) → m α} :
    funext x
    simp

-@[simp] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
+@[simp, grind =] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
    toList <$> ofFnM f = List.ofFnM f := by
  induction n with
  | zero => simp
--- a/src/Init/Data/Array/Perm.lean
+++ b/src/Init/Data/Array/Perm.lean
@@ -91,17 +91,26 @@ theorem Perm.mem_iff {a : α} {xs ys : Array α} (p : xs ~ ys) : a ∈ xs ↔ a
  simp only [perm_iff_toList_perm] at p
  simpa using p.mem_iff

+grind_pattern Perm.mem_iff => xs ~ ys, a ∈ xs
+grind_pattern Perm.mem_iff => xs ~ ys, a ∈ ys
+
 theorem Perm.append {xs ys as bs : Array α} (p₁ : xs ~ ys) (p₂ : as ~ bs) :
    xs ++ as ~ ys ++ bs := by
  cases xs; cases ys; cases as; cases bs
  simp only [append_toArray, perm_iff_toList_perm] at p₁ p₂ ⊢
  exact p₁.append p₂

+grind_pattern Perm.append => xs ~ ys, as ~ bs, xs ++ as
+grind_pattern Perm.append => xs ~ ys, as ~ bs, ys ++ bs
+
 theorem Perm.push (x : α) {xs ys : Array α} (p : xs ~ ys) :
    xs.push x ~ ys.push x := by
  rw [push_eq_append_singleton]
  exact p.append .rfl

+grind_pattern Perm.push => xs ~ ys, xs.push x
+grind_pattern Perm.push => xs ~ ys, ys.push x
+
 theorem Perm.push_comm (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
--- a/src/Init/Data/Array/Range.lean
+++ b/src/Init/Data/Array/Range.lean
@@ -128,6 +128,16 @@ theorem erase_range' :
  simp only [← List.toArray_range', List.erase_toArray]
  simp [List.erase_range']

+@[simp, grind =]
+theorem count_range' {a s n step} (h : 0 < step := by simp) :
+    count a (range' s n step) = if ∃ i, i < n ∧ a = s + step * i then 1 else 0 := by
+  rw [← List.toArray_range', List.count_toArray, ← List.count_range' h]
+
+@[simp, grind =]
+theorem count_range_1' {a s n} :
+    count a (range' s n) = if s ≤ a ∧ a < s + n then 1 else 0 := by
+  rw [← List.toArray_range', List.count_toArray, ← List.count_range_1']
+
 /-! ### range -/

@[grind _=_]
@@ -179,11 +189,11 @@ theorem self_mem_range_succ {n : Nat} : n ∈ range (n + 1) := by simp
@[simp, grind =] theorem take_range {i n : Nat} : take (range n) i = range (min i n) := by
  ext <;> simp

-@[simp] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
+@[simp, grind =] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
    (range n).find? p = some i ↔ p i ∧ i ∈ range n ∧ ∀ j, j < i → !p j := by
  simp [range_eq_range']

-@[simp] theorem find?_range_eq_none {n : Nat} {p : Nat → Bool} :
+@[simp, grind =] theorem find?_range_eq_none {n : Nat} {p : Nat → Bool} :
    (range n).find? p = none ↔ ∀ i, i < n → !p i := by
  simp only [← List.toArray_range, List.find?_toArray, List.find?_range_eq_none]

@@ -191,6 +201,10 @@ theorem self_mem_range_succ {n : Nat} : n ∈ range (n + 1) := by simp
 theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n - (i + 1)) := by
  simp [range_eq_range', erase_range']

+@[simp, grind =]
+theorem count_range {a n} :
+    count a (range n) = if a < n then 1 else 0 := by
+  rw [← List.toArray_range, List.count_toArray, ← List.count_range]

 /-! ### zipIdx -/

@@ -199,7 +213,7 @@ theorem zipIdx_eq_empty_iff {xs : Array α} {i : Nat} : xs.zipIdx i = #[] ↔ xs
  cases xs
  simp

-@[simp]
+@[simp, grind =]
 theorem getElem?_zipIdx {xs : Array α} {i j} : (zipIdx xs i)[j]? = xs[j]?.map fun a => (a, i + j) := by
  simp [getElem?_def]

@@ -242,7 +256,7 @@ theorem zipIdx_eq_map_add {xs : Array α} {i : Nat} :
  simp only [zipIdx_toArray, List.map_toArray, mk.injEq]
  rw [List.zipIdx_eq_map_add]

-@[simp]
+@[simp, grind =]
 theorem zipIdx_singleton {x : α} {k : Nat} : zipIdx #[x] k = #[(x, k)] :=
  rfl

@@ -290,6 +304,7 @@ theorem zipIdx_map {xs : Array α} {k : Nat} {f : α → β} :
  cases xs
  simp [List.zipIdx_map]

+@[grind =]
 theorem zipIdx_append {xs ys : Array α} {k : Nat} :
    zipIdx (xs ++ ys) k = zipIdx xs k ++ zipIdx ys (k + xs.size) := by
  cases xs
--- a/src/Init/Data/Array/Zip.lean
+++ b/src/Init/Data/Array/Zip.lean
@@ -45,6 +45,7 @@ theorem zipWith_self {f : α → α → δ} {xs : Array α} : zipWith f xs xs =
 See also `getElem?_zipWith'` for a variant
 using `Option.map` and `Option.bind` rather than a `match`.
 -/
+@[grind =]
 theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
    (zipWith f as bs)[i]? = match as[i]?, bs[i]? with
      | some a, some b => some (f a b) | _, _ => none := by
@@ -76,31 +77,35 @@ theorem getElem?_zip_eq_some {as : Array α} {bs : Array β} {z : α × β} {i :
  · rintro ⟨h₀, h₁⟩
    exact ⟨_, _, h₀, h₁, rfl⟩

-@[simp]
+@[simp, grind =]
 theorem zipWith_map {μ} {f : γ → δ → μ} {g : α → γ} {h : β → δ} {as : Array α} {bs : Array β} :
    zipWith f (as.map g) (bs.map h) = zipWith (fun a b => f (g a) (h b)) as bs := by
  cases as
  cases bs
  simp [List.zipWith_map]

+@[grind =]
 theorem zipWith_map_left {as : Array α} {bs : Array β} {f : α → α'} {g : α' → β → γ} :
    zipWith g (as.map f) bs = zipWith (fun a b => g (f a) b) as bs := by
  cases as
  cases bs
  simp [List.zipWith_map_left]

+@[grind =]
 theorem zipWith_map_right {as : Array α} {bs : Array β} {f : β → β'} {g : α → β' → γ} :
    zipWith g as (bs.map f) = zipWith (fun a b => g a (f b)) as bs := by
  cases as
  cases bs
  simp [List.zipWith_map_right]

+@[grind =]
 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
  simp [List.zipWith_foldr_eq_zip_foldr]

+@[grind =]
 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
@@ -111,22 +116,26 @@ theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} :
 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

+@[grind =]
 theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {cs : Array γ} {ds : Array δ} :
    map f (zipWith g cs ds) = zipWith (fun x y => f (g x y)) cs ds := by
  cases cs
  cases ds
  simp [List.map_zipWith]

+@[grind =]
 theorem take_zipWith : (zipWith f as bs).take i = zipWith f (as.take i) (bs.take i) := by
  cases as
  cases bs
  simp [List.take_zipWith]

+@[grind =]
 theorem extract_zipWith : (zipWith f as bs).extract i j = zipWith f (as.extract i j) (bs.extract i j) := by
  cases as
  cases bs
  simp [List.drop_zipWith, List.take_zipWith]

+@[grind =]
 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
@@ -152,7 +161,7 @@ theorem zipWith_eq_append_iff {f : α → β → γ} {as : Array α} {bs : Array
  · rintro ⟨⟨ws⟩, ⟨xs⟩, ⟨ys⟩, ⟨zs⟩, h, rfl, rfl, h₁, h₂⟩
    exact ⟨ws, xs, ys, zs, by simp_all⟩

-@[simp] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
+@[simp, grind =] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
    zipWith f (replicate m a) (replicate n b) = replicate (min m n) (f a b) := by
  simp [← List.toArray_replicate]

@@ -184,6 +193,7 @@ theorem zipWith_eq_zipWith_take_min (as : Array α) (bs : Array β) :
  simp
  rw [List.zipWith_eq_zipWith_take_min]

+@[grind =]
 theorem reverse_zipWith (h : as.size = bs.size) :
    (zipWith f as bs).reverse = zipWith f as.reverse bs.reverse := by
  cases as
@@ -200,7 +210,7 @@ theorem lt_size_right_of_zip {i : Nat} {as : Array α} {bs : Array β} (h : i <
    i < bs.size :=
  lt_size_right_of_zipWith h

-@[simp]
+@[simp, grind =]
 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)) :=
@@ -211,18 +221,22 @@ theorem zip_eq_zipWith {as : Array α} {bs : Array β} : zip as bs = zipWith Pro
  cases bs
  simp [List.zip_eq_zipWith]

+@[grind _=_]
 theorem zip_map {f : α → γ} {g : β → δ} {as : Array α} {bs : Array β} :
    zip (as.map f) (bs.map g) = (zip as bs).map (Prod.map f g) := by
  cases as
  cases bs
  simp [List.zip_map]

+@[grind _=_]
 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]

+@[grind _=_]
 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]

+@[grind =]
 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
@@ -231,6 +245,7 @@ theorem zip_append {as bs : Array α} {cs ds : Array β} (_h : as.size = cs.size
  cases ds
  simp_all [List.zip_append]

+@[grind =]
 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
@@ -276,7 +291,7 @@ theorem zip_eq_append_iff {as : Array α} {bs : Array β} :
      ∃ as₁ as₂ bs₁ bs₂, as₁.size = bs₁.size ∧ as = as₁ ++ as₂ ∧ bs = bs₁ ++ bs₂ ∧ xs = zip as₁ bs₁ ∧ ys = zip as₂ bs₂ := by
  simp [zip_eq_zipWith, zipWith_eq_append_iff]

-@[simp] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
+@[simp, grind =] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
    zip (replicate m a) (replicate n b) = replicate (min m n) (a, b) := by
  simp [← List.toArray_replicate]

@@ -293,6 +308,7 @@ theorem zip_eq_zip_take_min {as : Array α} {bs : Array β} :

 /-! ### zipWithAll -/

+@[grind =]
 theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
    (zipWithAll f as bs)[i]? = match as[i]?, bs[i]? with
      | none, none => .none | a?, b? => some (f a? b?) := by
@@ -301,31 +317,35 @@ theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
  simp [List.getElem?_zipWithAll]
  rfl

+@[grind =]
 theorem zipWithAll_map {μ} {f : Option γ → Option δ → μ} {g : α → γ} {h : β → δ} {as : Array α} {bs : Array β} :
    zipWithAll f (as.map g) (bs.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) as bs := by
  cases as
  cases bs
  simp [List.zipWithAll_map]

+@[grind =]
 theorem zipWithAll_map_left {as : Array α} {bs : Array β} {f : α → α'} {g : Option α' → Option β → γ} :
    zipWithAll g (as.map f) bs = zipWithAll (fun a b => g (f <$> a) b) as bs := by
  cases as
  cases bs
  simp [List.zipWithAll_map_left]

+@[grind =]
 theorem zipWithAll_map_right {as : Array α} {bs : Array β} {f : β → β'} {g : Option α → Option β' → γ} :
    zipWithAll g as (bs.map f) = zipWithAll (fun a b => g a (f <$> b)) as bs := by
  cases as
  cases bs
  simp [List.zipWithAll_map_right]

+@[grind =]
 theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option δ → α} {cs : Array γ} {ds : Array δ} :
    map f (zipWithAll g cs ds) = zipWithAll (fun x y => f (g x y)) cs ds := by
  cases cs
  cases ds
  simp [List.map_zipWithAll]

-@[simp] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
+@[simp, grind =] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
    zipWithAll f (replicate n a) (replicate n b) = replicate n (f (some a) (some b)) := by
  simp [← List.toArray_replicate]

@@ -342,6 +362,7 @@ theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
 theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
  simp

+@[grind =]
 theorem unzip_eq_map {xs : Array (α × β)} : unzip xs = (xs.map Prod.fst, xs.map Prod.snd) := by
  cases xs
  simp [List.unzip_eq_map]
@@ -375,9 +396,11 @@ theorem zip_of_prod {as : Array α} {bs : Array β} {xs : Array (α × β)} (hl
    (hr : xs.map Prod.snd = bs) : xs = as.zip bs := by
  rw [← hl, ← hr, ← zip_unzip xs, ← fst_unzip, ← snd_unzip, zip_unzip, zip_unzip]

-@[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
+@[simp, grind =] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
    unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
  ext1 <;> simp

@[deprecated unzip_replicate (since := "2025-03-18")]
 abbrev unzip_mkArray := @unzip_replicate
+
+end Array
--- a/src/Init/Data/BitVec.lean
+++ b/src/Init/Data/BitVec.lean
@@ -6,7 +6,10 @@ Authors: Kim Morrison
 module

 prelude
+import Init.Data.BitVec.BasicAux
 import Init.Data.BitVec.Basic
+import Init.Data.BitVec.Bootstrap
 import Init.Data.BitVec.Bitblast
-import Init.Data.BitVec.Folds
+import Init.Data.BitVec.Decidable
 import Init.Data.BitVec.Lemmas
+import Init.Data.BitVec.Folds
--- a/src/Init/Data/BitVec/Basic.lean
+++ b/src/Init/Data/BitVec/Basic.lean
@@ -74,25 +74,27 @@ section getXsb

 /--
 Returns the `i`th least significant bit.
-
-This will be renamed `getLsb` after the existing deprecated alias is removed.
 -/
-@[inline, expose] def getLsb' (x : BitVec w) (i : Fin w) : Bool := x.toNat.testBit i
+@[inline, expose] def getLsb (x : BitVec w) (i : Fin w) : Bool := x.toNat.testBit i
+
+@[deprecated getLsb (since := "2025-06-17"), inherit_doc getLsb]
+abbrev getLsb' := @getLsb

 /-- Returns the `i`th least significant bit, or `none` if `i ≥ w`. -/
@[inline, expose] def getLsb? (x : BitVec w) (i : Nat) : Option Bool :=
-  if h : i < w then some (getLsb' x ⟨i, h⟩) else none
+  if h : i < w then some (getLsb x ⟨i, h⟩) else none

 /--
 Returns the `i`th most significant bit.
-
-This will be renamed `BitVec.getMsb` after the existing deprecated alias is removed.
 -/
-@[inline] def getMsb' (x : BitVec w) (i : Fin w) : Bool := x.getLsb' ⟨w-1-i, by omega⟩
+@[inline] def getMsb (x : BitVec w) (i : Fin w) : Bool := x.getLsb ⟨w-1-i, by omega⟩
+
+@[deprecated getMsb (since := "2025-06-17"), inherit_doc getMsb]
+abbrev getMsb' := @getMsb

 /-- Returns the `i`th most significant bit or `none` if `i ≥ w`. -/
@[inline] def getMsb? (x : BitVec w) (i : Nat) : Option Bool :=
-  if h : i < w then some (getMsb' x ⟨i, h⟩) else none
+  if h : i < w then some (getMsb x ⟨i, h⟩) else none

 /-- Returns the `i`th least significant bit or `false` if `i ≥ w`. -/
@[inline, expose] def getLsbD (x : BitVec w) (i : Nat) : Bool :=
@@ -110,11 +112,11 @@ end getXsb
 section getElem

 instance : GetElem (BitVec w) Nat Bool fun _ i => i < w where
-  getElem xs i h := xs.getLsb' ⟨i, h⟩
+  getElem xs i h := xs.getLsb ⟨i, h⟩

 /-- We prefer `x[i]` as the simp normal form for `getLsb'` -/
-@[simp] theorem getLsb'_eq_getElem (x : BitVec w) (i : Fin w) :
-    x.getLsb' i = x[i] := rfl
+@[simp] theorem getLsb_eq_getElem (x : BitVec w) (i : Fin w) :
+    x.getLsb i = x[i] := rfl

 /-- We prefer `x[i]?` as the simp normal form for `getLsb?` -/
@[simp] theorem getLsb?_eq_getElem? (x : BitVec w) (i : Nat) :
@@ -174,7 +176,7 @@ recommended_spelling "zero" for "0#n" in [BitVec.ofNat, «term__#__»]
 recommended_spelling "one" for "1#n" in [BitVec.ofNat, «term__#__»]

 /-- Unexpander for bitvector literals. -/
-@[app_unexpander BitVec.ofNat] def unexpandBitVecOfNat : Lean.PrettyPrinter.Unexpander
+@[app_unexpander BitVec.ofNat] meta def unexpandBitVecOfNat : Lean.PrettyPrinter.Unexpander
  | `($(_) $n $i:num) => `($i:num#$n)
  | _ => throw ()

@@ -183,7 +185,7 @@ scoped syntax:max term:max noWs "#'" noWs term:max : term
 macro_rules | `($i#'$p) => `(BitVec.ofNatLT $i $p)

 /-- Unexpander for bitvector literals without truncation. -/
-@[app_unexpander BitVec.ofNatLT] def unexpandBitVecOfNatLt : Lean.PrettyPrinter.Unexpander
+@[app_unexpander BitVec.ofNatLT] meta def unexpandBitVecOfNatLt : Lean.PrettyPrinter.Unexpander
  | `($(_) $i $p) => `($i#'$p)
  | _ => throw ()

@@ -723,6 +725,12 @@ def twoPow (w : Nat) (i : Nat) : BitVec w := 1#w <<< i

 end bitwise

+/-- The bitvector of width `w` that has the smallest value when interpreted as an integer. -/
+def intMin (w : Nat) := twoPow w (w - 1)
+
+/-- The bitvector of width `w` that has the largest value when interpreted as an integer. -/
+def intMax (w : Nat) := (twoPow w (w - 1)) - 1
+
 /--
 Computes a hash of a bitvector, combining 64-bit words using `mixHash`.
 -/
--- a/src/Init/Data/BitVec/Bitblast.lean
+++ b/src/Init/Data/BitVec/Bitblast.lean
@@ -6,12 +6,14 @@ Authors: Harun Khan, Abdalrhman M Mohamed, Joe Hendrix, Siddharth Bhat
 module

 prelude
-import Init.Data.BitVec.Folds
 import all Init.Data.Nat.Bitwise.Basic
 import Init.Data.Nat.Mod
 import all Init.Data.Int.DivMod
 import Init.Data.Int.LemmasAux
-import all Init.Data.BitVec.Lemmas
+import all Init.Data.BitVec.Basic
+import Init.Data.BitVec.Decidable
+import Init.Data.BitVec.Lemmas
+import Init.Data.BitVec.Folds

 /-!
 # Bit blasting of bitvectors
@@ -518,9 +520,6 @@ theorem msb_neg {w : Nat} {x : BitVec w} :
  rw [(show w = w - 1 + 1 by omega), Int.pow_succ] at this
  omega

-@[simp] theorem setWidth_neg_of_le {x : BitVec v} (h : w ≤ v) : BitVec.setWidth w (-x) = -BitVec.setWidth w x := by
-  simp [← BitVec.signExtend_eq_setWidth_of_le _ h, BitVec.signExtend_neg_of_le h]
-
 /-! ### abs -/

 theorem msb_abs {w : Nat} {x : BitVec w} :
@@ -548,54 +547,14 @@ theorem ult_eq_not_carry (x y : BitVec w) : x.ult y = !carry w x (~~~y) true :=
  rw [Nat.mod_eq_of_lt (by omega)]
  omega

-theorem ule_eq_not_ult (x y : BitVec w) : x.ule y = !y.ult x := by
-  simp [BitVec.ule, BitVec.ult, ← decide_not]
-
 theorem ule_eq_carry (x y : BitVec w) : x.ule y = carry w y (~~~x) true := by
  simp [ule_eq_not_ult, ult_eq_not_carry]

-/-- If two bitvectors have the same `msb`, then signed and unsigned comparisons coincide -/
-theorem slt_eq_ult_of_msb_eq {x y : BitVec w} (h : x.msb = y.msb) :
-    x.slt y = x.ult y := by
-  simp only [BitVec.slt, toInt_eq_msb_cond, BitVec.ult, decide_eq_decide, h]
-  cases y.msb <;> simp
-
-/-- If two bitvectors have different `msb`s, then unsigned comparison is determined by this bit -/
-theorem ult_eq_msb_of_msb_neq {x y : BitVec w} (h : x.msb ≠ y.msb) :
-    x.ult y = y.msb := by
-  simp only [BitVec.ult, msb_eq_decide, ne_eq, decide_eq_decide] at *
-  omega
-
-/-- If two bitvectors have different `msb`s, then signed and unsigned comparisons are opposites -/
-theorem slt_eq_not_ult_of_msb_neq {x y : BitVec w} (h : x.msb ≠ y.msb) :
-    x.slt y = !x.ult y := by
-  simp only [BitVec.slt, toInt_eq_msb_cond, Bool.eq_not_of_ne h, ult_eq_msb_of_msb_neq h]
-  cases y.msb <;> (simp [-Int.natCast_pow]; omega)
-
-theorem slt_eq_ult {x y : BitVec w} :
-    x.slt y = (x.msb != y.msb).xor (x.ult y) := by
-  by_cases h : x.msb = y.msb
-  · simp [h, slt_eq_ult_of_msb_eq]
-  · have h' : x.msb != y.msb := by simp_all
-    simp [slt_eq_not_ult_of_msb_neq h, h']
-
 theorem slt_eq_not_carry {x y : BitVec w} :
    x.slt y = (x.msb == y.msb).xor (carry w x (~~~y) true) := by
  simp only [slt_eq_ult, bne, ult_eq_not_carry]
  cases x.msb == y.msb <;> simp

-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} {x : BitVec w} (hx : BitVec.sle 0#w x) : x.toInt.toNat = x.toNat :=
-  toNat_toInt_of_msb x (zero_sle_iff_msb_eq_false.1 hx)
-
 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]
@@ -618,12 +577,6 @@ theorem neg_sle_zero (h : 0 < w) {x : BitVec w} :
  rw [sle_eq_slt_or_eq, neg_slt_zero h, sle_eq_slt_or_eq]
  simp [Bool.beq_eq_decide_eq (-x), Bool.beq_eq_decide_eq _ x, Eq.comm (a := x), Bool.or_assoc]

-theorem sle_eq_ule {x y : BitVec w} : x.sle y = (x.msb != y.msb ^^ x.ule y) := by
-  rw [sle_eq_not_slt, slt_eq_ult, ← Bool.xor_not, ← ule_eq_not_ult, bne_comm]
-
-theorem sle_eq_ule_of_msb_eq {x y : BitVec w} (h : x.msb = y.msb) : x.sle y = x.ule y := by
-  simp [BitVec.sle_eq_ule, h]
-
 /-! ### mul recurrence for bit blasting -/

 /--
@@ -1023,7 +976,7 @@ theorem DivModState.toNat_shiftRight_sub_one_eq
    {args : DivModArgs w} {qr : DivModState w} (h : qr.Poised args) :
    args.n.toNat >>> (qr.wn - 1)
    = (args.n.toNat >>> qr.wn) * 2 + (args.n.getLsbD (qr.wn - 1)).toNat := by
-  show BitVec.toNat (args.n >>> (qr.wn - 1)) = _
+  change BitVec.toNat (args.n >>> (qr.wn - 1)) = _
  have {..} := h -- break the structure down for `omega`
  rw [shiftRight_sub_one_eq_shiftConcat args.n h.hwn_lt]
  rw [toNat_shiftConcat_eq_of_lt (k := w - qr.wn)]
--- a/src/Init/Data/BitVec/Bootstrap.lean
+++ b/src/Init/Data/BitVec/Bootstrap.lean
@@ -0,0 +1,146 @@
+/-
+Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joe Hendrix, Harun Khan, Alex Keizer, Abdalrhman M Mohamed, Siddharth Bhat
+-/
+module
+
+prelude
+import all Init.Data.BitVec.Basic
+
+namespace BitVec
+
+theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsbD i := rfl
+
+@[simp] theorem getLsbD_ofFin (x : Fin (2^n)) (i : Nat) :
+    getLsbD (BitVec.ofFin x) i = x.val.testBit i := rfl
+
+@[simp] theorem getLsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false := by
+  let ⟨x, x_lt⟩ := x
+  simp only [getLsbD_ofFin]
+  apply Nat.testBit_lt_two_pow
+  have p : 2^w ≤ 2^i := Nat.pow_le_pow_right (by omega) ge
+  omega
+
+/-- Prove equality of bitvectors in terms of nat operations. -/
+theorem eq_of_toNat_eq {n} : ∀ {x y : BitVec n}, x.toNat = y.toNat → x = y
+  | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
+
+theorem eq_of_getLsbD_eq {x y : BitVec w}
+    (pred : ∀ i, i < w → x.getLsbD i = y.getLsbD i) : x = y := by
+  apply eq_of_toNat_eq
+  apply Nat.eq_of_testBit_eq
+  intro i
+  if i_lt : i < w then
+    exact pred i i_lt
+  else
+    have p : i ≥ w := Nat.le_of_not_gt i_lt
+    simp [testBit_toNat, getLsbD_of_ge _ _ p]
+
+@[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]
+
+@[ext] theorem eq_of_getElem_eq {x y : BitVec n} :
+        (∀ i (hi : i < n), x[i] = y[i]) → x = y :=
+  fun h => BitVec.eq_of_getLsbD_eq (h ↑·)
+
+@[simp] theorem toNat_append (x : BitVec m) (y : BitVec n) :
+    (x ++ y).toNat = x.toNat <<< n ||| y.toNat :=
+  rfl
+
+@[simp] theorem toNat_ofBool (b : Bool) : (ofBool b).toNat = b.toNat := by
+  cases b <;> rfl
+
+@[simp, bitvec_to_nat] theorem toNat_cast (h : w = v) (x : BitVec w) : (x.cast h).toNat = x.toNat := rfl
+
+@[simp, bitvec_to_nat] theorem toNat_ofFin (x : Fin (2^n)) : (BitVec.ofFin x).toNat = x.val := rfl
+
+@[simp] theorem toNat_ofNatLT (x : Nat) (p : x < 2^w) : (x#'p).toNat = x := rfl
+
+@[simp] theorem toNat_cons (b : Bool) (x : BitVec w) :
+    (cons b x).toNat = (b.toNat <<< w) ||| x.toNat := by
+  let ⟨x, _⟩ := x
+  simp only [cons, toNat_cast, toNat_append, toNat_ofBool, toNat_ofFin]
+
+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
+  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
+  · have p1 : ¬(n ≤ i) := by omega
+    have p2 : i ≠ n := by omega
+    simp [p1, p2]
+  · simp only [i_eq_n, ge_iff_le, Nat.le_refl, decide_true, Nat.sub_self, Nat.testBit_zero,
+    Bool.true_and, testBit_toNat, getLsbD_of_ge, Bool.or_false, ↓reduceIte]
+    cases b <;> trivial
+  · have p1 : i ≠ n := by omega
+    have p2 : i - n ≠ 0 := by omega
+    simp [p1, p2, Nat.testBit_bool_to_nat]
+
+private theorem lt_two_pow_of_le {x m n : Nat} (lt : x < 2 ^ m) (le : m ≤ n) : x < 2 ^ n :=
+  Nat.lt_of_lt_of_le lt (Nat.pow_le_pow_right (by trivial : 0 < 2) le)
+
+@[simp, bitvec_to_nat] theorem toNat_setWidth' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
+    (setWidth' p x).toNat = x.toNat := by
+  simp only [setWidth', toNat_ofNatLT]
+
+@[simp, bitvec_to_nat] theorem toNat_setWidth (i : Nat) (x : BitVec n) :
+    BitVec.toNat (setWidth i x) = x.toNat % 2^i := by
+  let ⟨x, lt_n⟩ := x
+  simp only [setWidth]
+  if n_le_i : n ≤ i then
+    have x_lt_two_i : x < 2 ^ i := lt_two_pow_of_le lt_n n_le_i
+    simp [n_le_i, Nat.mod_eq_of_lt, x_lt_two_i]
+  else
+    simp [n_le_i, toNat_ofNat]
+
+@[simp] theorem ofNat_toNat (m : Nat) (x : BitVec n) : BitVec.ofNat m x.toNat = setWidth m x := by
+  apply eq_of_toNat_eq
+  simp only [toNat_ofNat, toNat_setWidth]
+
+theorem getElem_setWidth' (x : BitVec w) (i : Nat) (h : w ≤ v) (hi : i < v) :
+    (setWidth' h x)[i] = x.getLsbD i := by
+  rw [getElem_eq_testBit_toNat, toNat_setWidth', getLsbD]
+
+@[simp]
+theorem getElem_setWidth (m : Nat) (x : BitVec n) (i : Nat) (h : i < m) :
+    (setWidth m x)[i] = x.getLsbD i := by
+  rw [setWidth]
+  split
+  · rw [getElem_setWidth']
+  · simp only [ofNat_toNat, getElem_eq_testBit_toNat, toNat_setWidth, Nat.testBit_mod_two_pow,
+      getLsbD, Bool.and_eq_right_iff_imp, decide_eq_true_eq]
+    omega
+
+@[simp] theorem cons_msb_setWidth (x : BitVec (w+1)) : (cons x.msb (x.setWidth w)) = x := by
+  ext i
+  simp only [getElem_cons]
+  split <;> rename_i h
+  · simp [BitVec.msb, getMsbD, h]
+  · by_cases h' : i < w
+    · simp_all only [getElem_setWidth, getLsbD_eq_getElem]
+    · omega
+
+@[simp, bitvec_to_nat] theorem toNat_neg (x : BitVec n) : (- x).toNat = (2^n - x.toNat) % 2^n := by
+  simp [Neg.neg, BitVec.neg]
+
+@[simp] theorem setWidth_neg_of_le {x : BitVec v} (h : w ≤ v) : BitVec.setWidth w (-x) = -BitVec.setWidth w x := by
+  apply BitVec.eq_of_toNat_eq
+  simp only [toNat_setWidth, toNat_neg]
+  rw [Nat.mod_mod_of_dvd _ (Nat.pow_dvd_pow 2 h)]
+  rw [Nat.mod_eq_mod_iff]
+  rw [Nat.mod_def]
+  refine ⟨1 + x.toNat / 2^w, 2^(v-w), ?_⟩
+  rw [← Nat.pow_add]
+  have : v - w + w = v := by omega
+  rw [this]
+  rw [Nat.add_mul, Nat.one_mul, Nat.mul_comm (2^w)]
+  have sub_sub : ∀ (a : Nat) {b c : Nat} (h : c ≤ b), a - (b - c) = a + c - b := by omega
+  rw [sub_sub _ (Nat.div_mul_le_self x.toNat (2 ^ w))]
+  have : x.toNat / 2 ^ w * 2 ^ w ≤ x.toNat := Nat.div_mul_le_self x.toNat (2 ^ w)
+  have : x.toNat < 2 ^w ∨ x.toNat - 2 ^ w < x.toNat / 2 ^ w * 2 ^ w := by
+    have := Nat.lt_div_mul_add (a := x.toNat) (b := 2 ^ w) (Nat.two_pow_pos w)
+    omega
+  omega
+
+end BitVec
--- a/src/Init/Data/BitVec/Decidable.lean
+++ b/src/Init/Data/BitVec/Decidable.lean
@@ -0,0 +1,79 @@
+/-
+Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joe Hendrix, Harun Khan, Alex Keizer, Abdalrhman M Mohamed, Siddharth Bhat
+
+-/
+module
+
+prelude
+import Init.Data.BitVec.Bootstrap
+
+set_option linter.missingDocs true
+
+namespace BitVec
+
+/-! ### Decidable quantifiers -/
+
+theorem forall_zero_iff {P : BitVec 0 → Prop} :
+    (∀ v, P v) ↔ P 0#0 := by
+  constructor
+  · intro h
+    apply h
+  · intro h v
+    obtain (rfl : v = 0#0) := (by ext i ⟨⟩)
+    apply h
+
+theorem forall_cons_iff {P : BitVec (n + 1) → Prop} :
+    (∀ v : BitVec (n + 1), P v) ↔ (∀ (x : Bool) (v : BitVec n), P (v.cons x)) := by
+  constructor
+  · intro h _ _
+    apply h
+  · intro h v
+    have w : v = (v.setWidth n).cons v.msb := by simp only [cons_msb_setWidth]
+    rw [w]
+    apply h
+
+instance instDecidableForallBitVecZero (P : BitVec 0 → Prop) :
+    ∀ [Decidable (P 0#0)], Decidable (∀ v, P v)
+  | .isTrue h => .isTrue fun v => by
+    obtain (rfl : v = 0#0) := (by ext i ⟨⟩)
+    exact h
+  | .isFalse h => .isFalse (fun w => h (w _))
+
+instance instDecidableForallBitVecSucc (P : BitVec (n+1) → Prop) [DecidablePred P]
+    [Decidable (∀ (x : Bool) (v : BitVec n), P (v.cons x))] : Decidable (∀ v, P v) :=
+  decidable_of_iff' (∀ x (v : BitVec n), P (v.cons x)) forall_cons_iff
+
+instance instDecidableExistsBitVecZero (P : BitVec 0 → Prop) [Decidable (P 0#0)] :
+    Decidable (∃ v, P v) :=
+  decidable_of_iff (¬ ∀ v, ¬ P v) Classical.not_forall_not
+
+instance instDecidableExistsBitVecSucc (P : BitVec (n+1) → Prop) [DecidablePred P]
+    [Decidable (∀ (x : Bool) (v : BitVec n), ¬ P (v.cons x))] : Decidable (∃ v, P v) :=
+  decidable_of_iff (¬ ∀ v, ¬ P v) Classical.not_forall_not
+
+/--
+For small numerals this isn't necessary (as typeclass search can use the above two instances),
+but for large numerals this provides a shortcut.
+Note, however, that for large numerals the decision procedure may be very slow,
+and you should use `bv_decide` if possible.
+-/
+instance instDecidableForallBitVec :
+    ∀ (n : Nat) (P : BitVec n → Prop) [DecidablePred P], Decidable (∀ v, P v)
+  | 0, _, _ => inferInstance
+  | n + 1, _, _ =>
+    have := instDecidableForallBitVec n
+    inferInstance
+
+/--
+For small numerals this isn't necessary (as typeclass search can use the above two instances),
+but for large numerals this provides a shortcut.
+Note, however, that for large numerals the decision procedure may be very slow.
+-/
+instance instDecidableExistsBitVec :
+    ∀ (n : Nat) (P : BitVec n → Prop) [DecidablePred P], Decidable (∃ v, P v)
+  | 0, _, _ => inferInstance
+  | _ + 1, _, _ => inferInstance
+
+end BitVec
--- a/src/Init/Data/BitVec/Lemmas.lean
+++ b/src/Init/Data/BitVec/Lemmas.lean
@@ -2,7 +2,6 @@
 Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joe Hendrix, Harun Khan, Alex Keizer, Abdalrhman M Mohamed, Siddharth Bhat
-
 -/
 module

@@ -19,6 +18,7 @@ import Init.Data.Int.Bitwise.Lemmas
 import Init.Data.Int.LemmasAux
 import Init.Data.Int.Pow
 import Init.Data.Int.LemmasAux
+import Init.Data.BitVec.Bootstrap

 set_option linter.missingDocs true

@@ -27,19 +27,9 @@ namespace BitVec
@[simp] theorem mk_zero : BitVec.ofFin (w := w) ⟨0, h⟩ = 0#w := rfl
@[simp] theorem ofNatLT_zero : BitVec.ofNatLT (w := w) 0 h = 0#w := rfl

-@[simp] theorem getLsbD_ofFin (x : Fin (2^n)) (i : Nat) :
-    getLsbD (BitVec.ofFin x) i = x.val.testBit i := rfl
-
@[simp] theorem getElem_ofFin (x : Fin (2^n)) (i : Nat) (h : i < n) :
    (BitVec.ofFin x)[i] = x.val.testBit i := rfl

-@[simp] theorem getLsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false := by
-  let ⟨x, x_lt⟩ := x
-  simp only [getLsbD_ofFin]
-  apply Nat.testBit_lt_two_pow
-  have p : 2^w ≤ 2^i := Nat.pow_le_pow_right (by omega) ge
-  omega
-
@[simp] theorem getMsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getMsbD x i = false := by
  rw [getMsbD]
  simp only [Bool.and_eq_false_imp, decide_eq_true_eq]
@@ -127,10 +117,6 @@ This normalized a bitvec using `ofFin` to `ofNat`.
 theorem ofFin_eq_ofNat : @BitVec.ofFin w (Fin.mk x lt) = BitVec.ofNat w x := by
  simp only [BitVec.ofNat, Fin.ofNat, lt, Nat.mod_eq_of_lt]

-/-- Prove equality of bitvectors in terms of nat operations. -/
-theorem eq_of_toNat_eq {n} : ∀ {x y : BitVec n}, x.toNat = y.toNat → x = y
-  | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
-
 /-- Prove nonequality of bitvectors in terms of nat operations. -/
 theorem toNat_ne_iff_ne {n} {x y : BitVec n} : x.toNat ≠ y.toNat ↔ x ≠ y := by
  constructor
@@ -153,26 +139,28 @@ protected theorem toNat_lt_twoPow_of_le (h : m ≤ n) {x : BitVec m} :
  apply Nat.pow_le_pow_of_le
  <;> omega

-theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsbD i := rfl
-
 theorem two_pow_le_toNat_of_getElem_eq_true {i : Nat} {x : BitVec w}
    (hi : i < w) (hx : x[i] = true) : 2^i ≤ x.toNat := by
  apply Nat.ge_two_pow_of_testBit
  rw [← getElem_eq_testBit_toNat x i hi]
  exact hx

-theorem getMsb'_eq_getLsb' (x : BitVec w) (i : Fin w) :
-    x.getMsb' i = x.getLsb' ⟨w - 1 - i, by omega⟩ := by
-  simp only [getMsb', getLsb']
+theorem getMsb_eq_getLsb (x : BitVec w) (i : Fin w) :
+    x.getMsb i = x.getLsb ⟨w - 1 - i, by omega⟩ := by
+  simp only [getMsb, getLsb]

 theorem getMsb?_eq_getLsb? (x : BitVec w) (i : Nat) :
    x.getMsb? i = if i < w then x.getLsb? (w - 1 - i) else none := by
  simp only [getMsb?, getLsb?_eq_getElem?]
-  split <;> simp [getMsb'_eq_getLsb']
+  split <;> simp [getMsb_eq_getLsb]

 theorem getMsbD_eq_getLsbD (x : BitVec w) (i : Nat) : x.getMsbD i = (decide (i < w) && x.getLsbD (w - 1 - i)) := by
  rw [getMsbD, getLsbD]

+@[deprecated getMsb_eq_getLsb (since := "2025-06-17")]
+theorem getMsb'_eq_getLsb' (x : BitVec w) (i : Nat) : x.getMsbD i = (decide (i < w) && x.getLsbD (w - 1 - i)) := by
+  rw [getMsbD, getLsbD]
+
 theorem getLsbD_eq_getMsbD (x : BitVec w) (i : Nat) : x.getLsbD i = (decide (i < w) && x.getMsbD (w - 1 - i)) := by
  rw [getMsbD]
  by_cases h₁ : i < w <;> by_cases h₂ : w - 1 - i < w <;>
@@ -241,21 +229,6 @@ theorem getMsbD_eq_getMsb?_getD (x : BitVec w) (i : Nat) :
      intros
      omega

-theorem eq_of_getLsbD_eq {x y : BitVec w}
-    (pred : ∀ i, i < w → x.getLsbD i = y.getLsbD i) : x = y := by
-  apply eq_of_toNat_eq
-  apply Nat.eq_of_testBit_eq
-  intro i
-  if i_lt : i < w then
-    exact pred i i_lt
-  else
-    have p : i ≥ w := Nat.le_of_not_gt i_lt
-    simp [testBit_toNat, getLsbD_of_ge _ _ p]
-
-@[ext] theorem eq_of_getElem_eq {x y : BitVec n} :
-        (∀ i (hi : i < n), x[i] = y[i]) → x = y :=
-  fun h => BitVec.eq_of_getLsbD_eq (h ↑·)
-
 theorem eq_of_getLsbD_eq_iff {w : Nat} {x y : BitVec w} :
    x = y ↔ ∀ (i : Nat), i < w → x.getLsbD i = y.getLsbD i := by
  have iff := @BitVec.eq_of_getElem_eq_iff w x y
@@ -342,9 +315,6 @@ open Fin.NatCast in
@[simp, norm_cast] theorem toFin_natCast (n : Nat) : toFin (n : BitVec w) = (n : Fin (2^w)) := by
  rfl

-@[simp] theorem toNat_ofBool (b : Bool) : (ofBool b).toNat = b.toNat := by
-  cases b <;> rfl
-
@[simp] theorem toInt_ofBool (b : Bool) : (ofBool b).toInt = -b.toInt := by
  cases b <;> simp

@@ -368,10 +338,6 @@ theorem ofBool_eq_iff_eq : ∀ {b b' : Bool}, BitVec.ofBool b = BitVec.ofBool b'
@[simp] theorem ofBool_xor_ofBool : ofBool b ^^^ ofBool b' = ofBool (b ^^ b') := by
  cases b <;> cases b' <;> rfl

-@[simp, bitvec_to_nat] theorem toNat_ofFin (x : Fin (2^n)) : (BitVec.ofFin x).toNat = x.val := rfl
-
-@[simp] theorem toNat_ofNatLT (x : Nat) (p : x < 2^w) : (x#'p).toNat = x := rfl
-
@[deprecated toNat_ofNatLT (since := "2025-02-13")]
 theorem toNat_ofNatLt (x : Nat) (p : x < 2^w) : (x#'p).toNat = x := rfl

@@ -391,9 +357,6 @@ theorem getLsbD_ofNatLt {n : Nat} (x : Nat) (lt : x < 2^n) (i : Nat) :
 theorem getMsbD_ofNatLt {n x i : Nat} (h : x < 2^n) :
    getMsbD (x#'h) i = (decide (i < n) && x.testBit (n - 1 - i)) := getMsbD_ofNatLT h

-@[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])

@@ -581,7 +544,6 @@ theorem msb_eq_getMsbD_zero (x : BitVec w) : x.msb = x.getMsbD 0 := by

 /-! ### cast -/

-@[simp, bitvec_to_nat] theorem toNat_cast (h : w = v) (x : BitVec w) : (x.cast h).toNat = x.toNat := rfl
@[simp] theorem toFin_cast (h : w = v) (x : BitVec w) :
    (x.cast h).toFin = x.toFin.cast (by rw [h]) :=
  rfl
@@ -882,6 +844,19 @@ theorem slt_eq_sle_and_ne {x y : BitVec w} : x.slt y = (x.sle y && x != y) := by
  apply Bool.eq_iff_iff.2
  simp [BitVec.slt, BitVec.sle, Int.lt_iff_le_and_ne, BitVec.toInt_inj]

+/-- For all bitvectors `x, y`, either `x` is signed less than `y`,
+or is equal to `y`, or is signed greater than `y`. -/
+theorem slt_trichotomy (x y : BitVec w) : x.slt y ∨ x = y ∨ y.slt x := by
+  simpa [slt_iff_toInt_lt, ← toInt_inj]
+    using Int.lt_trichotomy x.toInt y.toInt
+
+/-- For all bitvectors `x, y`, either `x` is unsigned less than `y`,
+or is equal to `y`, or is unsigned greater than `y`. -/
+theorem lt_trichotomy (x y : BitVec w) :
+    x < y ∨ x = y ∨ y < x := by
+  simpa [← ult_iff_lt, ult_eq_decide, decide_eq_true_eq, ← toNat_inj]
+    using Nat.lt_trichotomy x.toNat y.toNat
+
 /-! ### setWidth, zeroExtend and truncate -/

@[simp]
@@ -892,20 +867,6 @@ theorem truncate_eq_setWidth {v : Nat} {x : BitVec w} :
 theorem zeroExtend_eq_setWidth {v : Nat} {x : BitVec w} :
  zeroExtend v x = setWidth v x := rfl

-@[simp, bitvec_to_nat] theorem toNat_setWidth' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
-    (setWidth' p x).toNat = x.toNat := by
-  simp [setWidth']
-
-@[simp, bitvec_to_nat] theorem toNat_setWidth (i : Nat) (x : BitVec n) :
-    BitVec.toNat (setWidth i x) = x.toNat % 2^i := by
-  let ⟨x, lt_n⟩ := x
-  simp only [setWidth]
-  if n_le_i : n ≤ i then
-    have x_lt_two_i : x < 2 ^ i := lt_two_pow_of_le lt_n n_le_i
-    simp [n_le_i, Nat.mod_eq_of_lt, x_lt_two_i]
-  else
-    simp [n_le_i, toNat_ofNat]
-
@[simp] theorem toInt_setWidth (x : BitVec w) :
    (x.setWidth v).toInt = Int.bmod x.toNat (2^v) := by
  simp [toInt_eq_toNat_bmod, toNat_setWidth, Int.emod_bmod, -Int.natCast_pow]
@@ -923,10 +884,6 @@ theorem zeroExtend_eq_setWidth {v : Nat} {x : BitVec w} :
  apply eq_of_toNat_eq
  simp [toNat_setWidth]

-@[simp] theorem ofNat_toNat (m : Nat) (x : BitVec n) : BitVec.ofNat m x.toNat = setWidth m x := by
-  apply eq_of_toNat_eq
-  simp
-
 /-- Moves one-sided left toNat equality to BitVec equality. -/
 theorem toNat_eq_nat {x : BitVec w} {y : Nat}
  : (x.toNat = y) ↔ (y < 2^w ∧ (x = BitVec.ofNat w y)) := by
@@ -942,19 +899,6 @@ theorem nat_eq_toNat {x : BitVec w} {y : Nat}
  rw [@eq_comm _ _ x.toNat]
  apply toNat_eq_nat

-theorem getElem_setWidth' (x : BitVec w) (i : Nat) (h : w ≤ v) (hi : i < v) :
-    (setWidth' h x)[i] = x.getLsbD i := by
-  rw [getElem_eq_testBit_toNat, toNat_setWidth', getLsbD]
-
-@[simp]
-theorem getElem_setWidth (m : Nat) (x : BitVec n) (i : Nat) (h : i < m) :
-    (setWidth m x)[i] = x.getLsbD i := by
-  rw [setWidth]
-  split
-  · rw [getElem_setWidth']
-  · simp [getElem_eq_testBit_toNat, getLsbD]
-    omega
-
 theorem getElem?_setWidth' (x : BitVec w) (i : Nat) (h : w ≤ v) :
    (setWidth' h x)[i]? = if i < v then some (x.getLsbD i) else none := by
  simp [getElem?_eq, getElem_setWidth']
@@ -1919,6 +1863,63 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
    exact Nat.mul_lt_mul_of_pos_right x.isLt (Nat.two_pow_pos _)
  · omega

+@[simp]
+theorem toNat_shiftLeftZeroExtend {x : BitVec w} :
+    (shiftLeftZeroExtend x n).toNat = x.toNat <<< n := by
+  rcases n with _|n
+  · simp [shiftLeftZeroExtend]
+  · simp only [shiftLeftZeroExtend_eq, toNat_shiftLeft, toNat_setWidth]
+    have :=  Nat.pow_lt_pow_of_lt (a := 2) (n := w) (m := w + (n + 1)) (by omega) (by omega)
+    have : x.toNat <<< (n + 1) < 2 ^ (w + (n + 1)) := by
+      rw [Nat.shiftLeft_eq, Nat.pow_add (m := w) (n := n + 1), Nat.mul_lt_mul_right (by apply Nat.two_pow_pos (w := n + 1))]
+      omega
+    rw [Nat.mod_eq_of_lt (by rw [Nat.mod_eq_of_lt (by omega)]; omega), Nat.mod_eq_of_lt (by omega)]
+
+@[simp]
+theorem toInt_shiftLeftZeroExtend {x : BitVec w} :
+    (shiftLeftZeroExtend x n).toInt = x.toInt * 2 ^ n := by
+  rw [shiftLeftZeroExtend_eq]
+  rcases w with _|w
+  · simp [of_length_zero, shiftLeftZeroExtend_eq]
+  · rcases n with _|n
+    · simp [shiftLeftZeroExtend_eq]
+    · have := Nat.pow_pos (a := 2) (n := n + 1) (by omega)
+      have : x.toNat <<< (n + 1) < 2 ^ (w + 1 + (n + 1)) := by
+        rw [Nat.shiftLeft_eq, Nat.pow_add (a := 2) (m := w + 1) (n := n + 1), Nat.mul_lt_mul_right (by omega)]
+        omega
+      simp only [shiftLeftZeroExtend_eq, toInt_shiftLeft, toNat_setWidth, Nat.lt_add_right_iff_pos,
+        Nat.zero_lt_succ, toNat_mod_cancel_of_lt, Int.bmod_def]
+      by_cases hmsb : x.msb
+      · have hge := toNat_ge_of_msb_true hmsb
+        simp only [Nat.add_one_sub_one, ge_iff_le] at hge
+        rw [Int.emod_eq_of_lt (by norm_cast; rw [Nat.shiftLeft_eq]; omega) (by omega)]
+        rw_mod_cast [← Nat.add_assoc]
+        rw [show (2 ^ (w + 1 + n + 1) + 1) / 2 = 2 ^ (w + 1 + n) by omega, Int.natCast_pow,
+          Int.cast_ofNat_Int, Nat.shiftLeft_eq, Nat.add_assoc, Nat.pow_add (a := 2) (m := w) (n := 1 + n),
+          Nat.add_comm 1 n]
+        simp only [Nat.mul_lt_mul_right (by omega), show ¬x.toNat < 2 ^ w by omega, reduceIte,
+          Int.natCast_mul, Int.natCast_pow, Int.cast_ofNat_Int, toInt_eq_toNat_cond,
+          show ¬2 * x.toNat < 2 ^ (w + 1) by simp [Nat.pow_add, Nat.mul_comm (2 ^ w) 2, hge]]
+        norm_cast
+        simp [Int.natCast_mul, Int.natCast_pow, Int.cast_ofNat_Int, Int.sub_mul,
+          Int.sub_right_inj, show w + (n + 1) + 1 = (w + 1) + (n + 1) by omega, Nat.pow_add]
+      · simp only [Bool.not_eq_true] at hmsb
+        have hle := toNat_lt_of_msb_false (x := x) hmsb
+        simp only [Nat.add_one_sub_one] at hle
+        rw [Int.emod_eq_of_lt (by norm_cast; rw [Nat.shiftLeft_eq]; omega) (by omega)]
+        rw_mod_cast [← Nat.add_assoc]
+        rw [show (2 ^ (w + 1 + n + 1) + 1) / 2 = 2 ^ (w + 1 + n) by omega, Int.natCast_pow,
+          Int.cast_ofNat_Int, Nat.shiftLeft_eq, Nat.add_assoc,  Nat.pow_add (a := 2) (m := w) (n := 1 + n), Nat.add_comm 1 n]
+        simp [Nat.mul_lt_mul_right (b := x.toNat) (c := 2 ^ w) (a := 2 ^ (n + 1)) (by omega), hle,
+          reduceIte, Int.natCast_mul, Int.natCast_pow, Int.cast_ofNat_Int, toInt_eq_toNat_of_msb hmsb]
+
+theorem toFin_shiftLeftZeroExtend {x : BitVec w} :
+    (shiftLeftZeroExtend x n).toFin = Fin.ofNat (2 ^ (w + n)) (x.toNat * 2 ^ n) := by
+  rcases w with _|w
+  · simp [of_length_zero, shiftLeftZeroExtend_eq]
+  · have := Nat.pow_le_pow_of_le (a := 2) (n := w + 1) (m := w + 1 + n) (by omega) (by omega)
+    rw [shiftLeftZeroExtend_eq, toFin_shiftLeft, toNat_setWidth, Nat.mod_eq_of_lt (by omega), Nat.shiftLeft_eq]
+
@[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]
@@ -2660,10 +2661,6 @@ theorem toFin_signExtend (x : BitVec w) :
 theorem append_def (x : BitVec v) (y : BitVec w) :
    x ++ y = (shiftLeftZeroExtend x w ||| setWidth' (Nat.le_add_left w v) y) := rfl

-@[simp] theorem toNat_append (x : BitVec m) (y : BitVec n) :
-    (x ++ y).toNat = x.toNat <<< n ||| y.toNat :=
-  rfl
-
 theorem getLsbD_append {x : BitVec n} {y : BitVec m} :
    getLsbD (x ++ y) i = if i < m then getLsbD y i else getLsbD x (i - m) := by
  simp only [append_def, getLsbD_or, getLsbD_shiftLeftZeroExtend, getLsbD_setWidth']
@@ -3048,11 +3045,6 @@ theorem getMsbD_rev (x : BitVec w) (i : Fin w) :

 /-! ### cons -/

-@[simp] theorem toNat_cons (b : Bool) (x : BitVec w) :
-    (cons b x).toNat = (b.toNat <<< w) ||| x.toNat := by
-  let ⟨x, _⟩ := x
-  simp [cons, toNat_append, toNat_ofBool]
-
 /-- Variant of `toNat_cons` using `+` instead of `|||`. -/
 theorem toNat_cons' {x : BitVec w} :
    (cons a x).toNat = (a.toNat <<< w) + x.toNat := by
@@ -3072,21 +3064,6 @@ theorem getLsbD_cons (b : Bool) {n} (x : BitVec n) (i : Nat) :
    have p2 : i - n ≠ 0 := by omega
    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
-  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
-  · have p1 : ¬(n ≤ i) := by omega
-    have p2 : i ≠ n := by omega
-    simp [p1, p2]
-  · simp only [i_eq_n, ge_iff_le, Nat.le_refl, decide_true, Nat.sub_self, Nat.testBit_zero,
-    Bool.true_and, testBit_toNat, getLsbD_of_ge, Bool.or_false, ↓reduceIte]
-    cases b <;> trivial
-  · have p1 : i ≠ n := by omega
-    have p2 : i - n ≠ 0 := by omega
-    simp [p1, p2, Nat.testBit_bool_to_nat]
-
@[simp] theorem msb_cons : (cons a x).msb = a := by
  simp [cons, msb_cast, msb_append]

@@ -3106,15 +3083,6 @@ theorem setWidth_succ (x : BitVec w) :
    have j_lt : j < i := Nat.lt_of_le_of_ne (Nat.le_of_succ_le_succ h) j_eq
    simp [j_eq, j_lt]

-@[simp] theorem cons_msb_setWidth (x : BitVec (w+1)) : (cons x.msb (x.setWidth w)) = x := by
-  ext i
-  simp only [getElem_cons]
-  split <;> rename_i h
-  · simp [BitVec.msb, getMsbD, h]
-  · by_cases h' : i < w
-    · simp_all
-    · omega
-
@[simp] theorem not_cons (x : BitVec w) (b : Bool) : ~~~(cons b x) = cons (!b) (~~~x) := by
  simp [cons]

@@ -3341,6 +3309,17 @@ theorem toNat_add_of_not_uaddOverflow {x y : BitVec w} (h : ¬ uaddOverflow x y)
  · simp only [uaddOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h
    rw [toNat_add, Nat.mod_eq_of_lt h]

+/--
+Unsigned addition overflow reassociation.
+If `(x + y)` and `(y + z)` do not overflow, then `(x + y) + z` overflows iff `x + (y + z)` overflows.
+-/
+theorem uaddOverflow_assoc {x y z : BitVec w} (h : ¬ x.uaddOverflow y) (h' : ¬ y.uaddOverflow z) :
+    (x + y).uaddOverflow z = x.uaddOverflow (y + z) := by
+  simp only [uaddOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h h'
+  simp only [uaddOverflow, toNat_add, ge_iff_le, decide_eq_decide]
+  repeat rw [Nat.mod_eq_of_lt (by omega)]
+  omega
+
 protected theorem add_assoc (x y z : BitVec n) : x + y + z = x + (y + z) := by
  apply eq_of_toNat_eq ; simp [Nat.add_assoc]
 instance : Std.Associative (α := BitVec n) (· + ·) := ⟨BitVec.add_assoc⟩
@@ -3379,6 +3358,20 @@ theorem toInt_add_of_not_saddOverflow {x y : BitVec w} (h : ¬ saddOverflow x y)
      _root_.not_or, Int.not_le, Int.not_lt] at h
    rw [toInt_add, Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]

+/--
+Signed addition overflow reassociation.
+If `(x + y)` and `(y + z)` do not overflow, then `(x + y) + z` overflows iff `x + (y + z)` overflows.
+-/
+theorem saddOverflow_assoc {x y z : BitVec w} (h : ¬ x.saddOverflow y) (h' : ¬ y.saddOverflow z) :
+    (x + y).saddOverflow z = x.saddOverflow (y + z) := by
+  rcases w with _|w
+  · simp [of_length_zero]
+  · simp only [saddOverflow, Nat.add_one_sub_one, ge_iff_le, Bool.or_eq_true, decide_eq_true_eq,
+      _root_.not_or, Int.not_le, Int.not_lt] at h h'
+    simp only [bool_to_prop, saddOverflow, toInt_add, ge_iff_le, Nat.add_one_sub_one]
+    repeat rw [Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]
+    omega
+
@[simp]
 theorem shiftLeft_add_distrib {x y : BitVec w} {n : Nat} :
    (x + y) <<< n = x <<< n + y <<< n := by
@@ -3486,9 +3479,6 @@ theorem ofNat_sub_ofNat {n} (x y : Nat) : BitVec.ofNat n x - BitVec.ofNat n y =
  · simp
  · exact Nat.le_of_lt x.isLt

-@[simp, bitvec_to_nat] theorem toNat_neg (x : BitVec n) : (- x).toNat = (2^n - x.toNat) % 2^n := by
-  simp [Neg.neg, BitVec.neg]
-
 theorem toNat_neg_of_pos {x : BitVec n} (h : 0#n < x) :
    (- x).toNat = 2^n - x.toNat := by
  change 0 < x.toNat at h
@@ -3807,6 +3797,18 @@ theorem toNat_mul_of_not_umulOverflow {x y : BitVec w} (h : ¬ umulOverflow x y)
  · simp only [umulOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h
    rw [toNat_mul, Nat.mod_eq_of_lt h]

+/--
+Unsigned multiplication overflow reassociation.
+If `(x * y)` and `(y * z)` do not overflow, then `(x * y) * z` overflows iff `x * (y * z)` overflows.
+-/
+theorem umulOverflow_assoc {x y z : BitVec w} (h : ¬ x.umulOverflow y) (h' : ¬ y.umulOverflow z) :
+
+    (x * y).umulOverflow z = x.umulOverflow (y * z) := by
+  simp only [umulOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h h'
+  simp only [umulOverflow, toNat_mul, ge_iff_le, decide_eq_decide]
+  repeat rw [Nat.mod_eq_of_lt (by omega)]
+  rw [Nat.mul_assoc]
+
@[simp]
 theorem toInt_mul_of_not_smulOverflow {x y : BitVec w} (h : ¬ smulOverflow x y) :
    (x * y).toInt = x.toInt * y.toInt := by
@@ -3816,6 +3818,20 @@ theorem toInt_mul_of_not_smulOverflow {x y : BitVec w} (h : ¬ smulOverflow x y)
      _root_.not_or, Int.not_le, Int.not_lt] at h
    rw [toInt_mul, Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]

+/--
+Signed multiplication overflow reassociation.
+If `(x * y)` and `(y * z)` do not overflow, then `(x * y) * z` overflows iff `x * (y * z)` overflows.
+-/
+theorem smulOverflow_assoc {x y z : BitVec w} (h : ¬ x.smulOverflow y) (h' : ¬ y.smulOverflow z) :
+    (x * y).smulOverflow z = x.smulOverflow (y * z) := by
+  rcases w with _|w
+  · simp [of_length_zero]
+  · simp only [smulOverflow, Nat.add_one_sub_one, ge_iff_le, Bool.or_eq_true, decide_eq_true_eq,
+      _root_.not_or, Int.not_le, Int.not_lt] at h h'
+    simp only [smulOverflow, toInt_mul, Nat.add_one_sub_one, ge_iff_le, bool_to_prop]
+    repeat rw [Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]
+    rw [Int.mul_assoc]
+
 theorem ofInt_mul {n} (x y : Int) : BitVec.ofInt n (x * y) =
    BitVec.ofInt n x * BitVec.ofInt n y := by
  apply eq_of_toInt_eq
@@ -5108,9 +5124,6 @@ theorem BitVec.setWidth_add_eq_mod {x y : BitVec w} : BitVec.setWidth i (x + y)

 /-! ### intMin -/

-/-- The bitvector of width `w` that has the smallest value when interpreted as an integer. -/
-def intMin (w : Nat) := twoPow w (w - 1)
-
 theorem getLsbD_intMin (w : Nat) : (intMin w).getLsbD i = decide (i + 1 = w) := by
  simp only [intMin, getLsbD_twoPow, bool_to_prop]
  omega
@@ -5261,9 +5274,6 @@ theorem neg_le_intMin_of_msb_eq_true {x : BitVec w} (hx : x.msb = true) : -x ≤

 /-! ### intMax -/

-/-- The bitvector of width `w` that has the largest value when interpreted as an integer. -/
-def intMax (w : Nat) := (twoPow w (w - 1)) - 1
-
@[simp, bitvec_to_nat]
 theorem toNat_intMax : (intMax w).toNat = 2 ^ (w - 1) - 1 := by
  simp only [intMax]
@@ -5615,68 +5625,54 @@ theorem msb_replicate {n w : Nat} {x : BitVec w} :
  simp only [BitVec.msb, getMsbD_replicate, Nat.zero_mod]
  cases n <;> cases w <;> simp

-/-! ### Decidable quantifiers -/

-theorem forall_zero_iff {P : BitVec 0 → Prop} :
-    (∀ v, P v) ↔ P 0#0 := by
-  constructor
-  · intro h
-    apply h
-  · intro h v
-    obtain (rfl : v = 0#0) := (by ext i ⟨⟩)
-    apply h
+/-! ### Inequalities (le / lt) -/

-theorem forall_cons_iff {P : BitVec (n + 1) → Prop} :
-    (∀ v : BitVec (n + 1), P v) ↔ (∀ (x : Bool) (v : BitVec n), P (v.cons x)) := by
-  constructor
-  · intro h _ _
-    apply h
-  · intro h v
-    have w : v = (v.setWidth n).cons v.msb := by simp
-    rw [w]
-    apply h
+theorem ule_eq_not_ult (x y : BitVec w) : x.ule y = !y.ult x := by
+  simp [BitVec.ule, BitVec.ult, ← decide_not]

-instance instDecidableForallBitVecZero (P : BitVec 0 → Prop) :
-    ∀ [Decidable (P 0#0)], Decidable (∀ v, P v)
-  | .isTrue h => .isTrue fun v => by
-    obtain (rfl : v = 0#0) := (by ext i ⟨⟩)
-    exact h
-  | .isFalse h => .isFalse (fun w => h (w _))
+/-- If two bitvectors have the same `msb`, then signed and unsigned comparisons coincide -/
+theorem slt_eq_ult_of_msb_eq {x y : BitVec w} (h : x.msb = y.msb) :
+    x.slt y = x.ult y := by
+  simp only [BitVec.slt, toInt_eq_msb_cond, BitVec.ult, decide_eq_decide, h]
+  cases y.msb <;> simp

-instance instDecidableForallBitVecSucc (P : BitVec (n+1) → Prop) [DecidablePred P]
-    [Decidable (∀ (x : Bool) (v : BitVec n), P (v.cons x))] : Decidable (∀ v, P v) :=
-  decidable_of_iff' (∀ x (v : BitVec n), P (v.cons x)) forall_cons_iff
+/-- If two bitvectors have different `msb`s, then unsigned comparison is determined by this bit -/
+theorem ult_eq_msb_of_msb_neq {x y : BitVec w} (h : x.msb ≠ y.msb) :
+    x.ult y = y.msb := by
+  simp only [BitVec.ult, msb_eq_decide, ne_eq, decide_eq_decide] at *
+  omega

-instance instDecidableExistsBitVecZero (P : BitVec 0 → Prop) [Decidable (P 0#0)] :
-    Decidable (∃ v, P v) :=
-  decidable_of_iff (¬ ∀ v, ¬ P v) Classical.not_forall_not
+/-- If two bitvectors have different `msb`s, then signed and unsigned comparisons are opposites -/
+theorem slt_eq_not_ult_of_msb_neq {x y : BitVec w} (h : x.msb ≠ y.msb) :
+    x.slt y = !x.ult y := by
+  simp only [BitVec.slt, toInt_eq_msb_cond, Bool.eq_not_of_ne h, ult_eq_msb_of_msb_neq h]
+  cases y.msb <;> (simp [-Int.natCast_pow]; omega)

-instance instDecidableExistsBitVecSucc (P : BitVec (n+1) → Prop) [DecidablePred P]
-    [Decidable (∀ (x : Bool) (v : BitVec n), ¬ P (v.cons x))] : Decidable (∃ v, P v) :=
-  decidable_of_iff (¬ ∀ v, ¬ P v) Classical.not_forall_not
+theorem slt_eq_ult {x y : BitVec w} :
+    x.slt y = (x.msb != y.msb).xor (x.ult y) := by
+  by_cases h : x.msb = y.msb
+  · simp [h, slt_eq_ult_of_msb_eq]
+  · have h' : x.msb != y.msb := by simp_all
+    simp [slt_eq_not_ult_of_msb_neq h, h']

-/--
-For small numerals this isn't necessary (as typeclass search can use the above two instances),
-but for large numerals this provides a shortcut.
-Note, however, that for large numerals the decision procedure may be very slow,
-and you should use `bv_decide` if possible.
-/
-instance instDecidableForallBitVec :
-    ∀ (n : Nat) (P : BitVec n → Prop) [DecidablePred P], Decidable (∀ v, P v)
-  | 0, _, _ => inferInstance
-  | n + 1, _, _ =>
-    have := instDecidableForallBitVec n
-    inferInstance
+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

-/--
-For small numerals this isn't necessary (as typeclass search can use the above two instances),
-but for large numerals this provides a shortcut.
-Note, however, that for large numerals the decision procedure may be very slow.
-/
-instance instDecidableExistsBitVec :
-    ∀ (n : Nat) (P : BitVec n → Prop) [DecidablePred P], Decidable (∃ v, P v)
-  | 0, _, _ => inferInstance
-  | _ + 1, _, _ => inferInstance
+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} {x : BitVec w} (hx : BitVec.sle 0#w x) : x.toInt.toNat = x.toNat :=
+  toNat_toInt_of_msb x (zero_sle_iff_msb_eq_false.1 hx)
+
+theorem sle_eq_ule {x y : BitVec w} : x.sle y = (x.msb != y.msb ^^ x.ule y) := by
+  rw [sle_eq_not_slt, slt_eq_ult, ← Bool.xor_not, ← ule_eq_not_ult, bne_comm]
+
+theorem sle_eq_ule_of_msb_eq {x y : BitVec w} (h : x.msb = y.msb) : x.sle y = x.ule y := by
+  simp [BitVec.sle_eq_ule, h]

 /-! ### Deprecations -/

--- a/src/Init/Data/Fin/Fold.lean
+++ b/src/Init/Data/Fin/Fold.lean
@@ -183,10 +183,7 @@ theorem foldrM_loop [Monad m] [LawfulMonad m] (f : Fin (n+1) → α → m α) (x
  | zero =>
    rw [foldrM_loop_zero, foldrM_loop_succ, pure_bind]
    conv => rhs; rw [←bind_pure (f 0 x)]
-    congr
-    try  -- TODO: block can be deleted after bootstrapping
-      funext
-      simp [foldrM_loop_zero]
+    rfl
  | succ i ih =>
    rw [foldrM_loop_succ, foldrM_loop_succ, bind_assoc]
    congr; funext; exact ih ..
--- a/src/Init/Data/Fin/Lemmas.lean
+++ b/src/Init/Data/Fin/Lemmas.lean
@@ -1079,6 +1079,17 @@ theorem val_neg {n : Nat} [NeZero n] (x : Fin n) :
    have := Fin.val_ne_zero_iff.mpr h
    omega

+protected theorem sub_eq_add_neg {n : Nat} (x y : Fin n) : x - y = x + -y := by
+  by_cases h : n = 0
+  · subst h
+    apply elim0 x
+  · replace h : NeZero n := ⟨h⟩
+    ext
+    rw [Fin.coe_sub, Fin.val_add, val_neg]
+    split
+    · simp_all
+    · simp [Nat.add_comm]
+
 /-! ### mul -/

 theorem ofNat_mul [NeZero n] (x : Nat) (y : Fin n) :
--- a/src/Init/Data/Int/DivMod/Bootstrap.lean
+++ b/src/Init/Data/Int/DivMod/Bootstrap.lean
@@ -3,7 +3,6 @@ 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
 -/
-
 module

 prelude
@@ -99,7 +98,7 @@ theorem ofNat_emod (m n : Nat) : (↑(m % n) : Int) = m % n := natCast_emod m n
 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
+    change (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
@@ -149,7 +148,7 @@ theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c
  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)
+    change ((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
@@ -158,7 +157,7 @@ theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c
      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)
+      change 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_of_le]; rwa [Nat.mul_comm]
--- a/src/Init/Data/Int/DivMod/Lemmas.lean
+++ b/src/Init/Data/Int/DivMod/Lemmas.lean
@@ -3,7 +3,6 @@ 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, Kim Morrison, Markus Himmel
 -/
-
 module

 prelude
@@ -203,6 +202,9 @@ theorem tdiv_eq_ediv_of_nonneg : ∀ {a b : Int}, 0 ≤ a → a.tdiv b = a / b
  | succ _, succ _, _ => rfl
  | succ _, -[_+1], _ => rfl

+@[simp] theorem natCast_tdiv_eq_ediv {a : Nat} {b : Int} : (a : Int).tdiv b = a / b :=
+    tdiv_eq_ediv_of_nonneg (by simp)
+
 theorem tdiv_eq_ediv {a b : Int} :
    a.tdiv b = a / b + if 0 ≤ a ∨ b ∣ a then 0 else sign b := by
  simp only [dvd_iff_emod_eq_zero]
@@ -329,17 +331,17 @@ theorem fdiv_eq_ediv_of_dvd {a b : Int} (h : b ∣ a) : a.fdiv b = a / b := by
 theorem tmod_add_tdiv : ∀ a b : Int, tmod a b + b * (a.tdiv 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
+    change (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
    rw [Int.neg_mul_neg]; exact congrArg ofNat (Nat.mod_add_div ..)
  | -[m+1], 0 => by
-    show -(↑((succ m) % 0) : Int) + 0 * -↑(succ m / 0) = -↑(succ m)
+    change -(↑((succ m) % 0) : Int) + 0 * -↑(succ m / 0) = -↑(succ m)
    rw [Nat.mod_zero, Int.zero_mul, Int.add_zero]
  | -[m+1], ofNat n => by
-    show -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m)
+    change -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m)
    rw [Int.mul_neg, ← Int.neg_add]
    exact congrArg (-ofNat ·) (Nat.mod_add_div ..)
  | -[m+1], -[n+1] => by
-    show -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)
+    change -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)
    rw [Int.neg_mul, ← Int.neg_add]
    exact congrArg (-ofNat ·) (Nat.mod_add_div ..)

@@ -361,17 +363,17 @@ theorem fmod_add_fdiv : ∀ a b : Int, a.fmod b + b * a.fdiv b = a
  | 0, ofNat _ | 0, -[_+1] => congrArg ofNat <| by simp
  | succ _, ofNat _ => congrArg ofNat <| Nat.mod_add_div ..
  | succ m, -[n+1] => by
-    show subNatNat (m % succ n) n + (↑(succ n * (m / succ n)) + n + 1) = (m + 1)
+    change subNatNat (m % succ n) n + (↑(succ n * (m / succ n)) + n + 1) = (m + 1)
    rw [Int.add_comm _ n, ← Int.add_assoc, ← Int.add_assoc,
      Int.subNatNat_eq_coe, Int.sub_add_cancel]
    exact congrArg (ofNat · + 1) <| Nat.mod_add_div ..
  | -[_+1], 0 => by rw [fmod_zero]; rfl
  | -[m+1], succ n => by
-    show subNatNat .. - (↑(succ n * (m / succ n)) + ↑(succ n)) = -↑(succ m)
+    change subNatNat .. - (↑(succ n * (m / succ n)) + ↑(succ n)) = -↑(succ m)
    rw [Int.subNatNat_eq_coe, ← Int.sub_sub, ← Int.neg_sub, Int.sub_sub, Int.sub_sub_self]
    exact congrArg (-ofNat ·) <| Nat.succ_add .. ▸ Nat.mod_add_div .. ▸ rfl
  | -[m+1], -[n+1] => by
-    show -(↑(succ m % succ n) : Int) + -↑(succ n * (succ m / succ n)) = -↑(succ m)
+    change -(↑(succ m % succ n) : Int) + -↑(succ n * (succ m / succ n)) = -↑(succ m)
    rw [← Int.neg_add]; exact congrArg (-ofNat ·) <| Nat.mod_add_div ..

 /-- Variant of `fmod_add_fdiv` with the multiplication written the other way around. -/
@@ -572,7 +574,7 @@ theorem neg_one_ediv (b : Int) : -1 / b = -b.sign :=
      · refine Nat.le_trans ?_ (Nat.le_add_right _ _)
        rw [← Nat.mul_div_mul_left _ _ m.succ_pos]
        apply Nat.div_mul_le_self
-      · show m.succ * n.succ ≤ _
+      · change m.succ * n.succ ≤ _
        rw [Nat.mul_left_comm]
        apply Nat.mul_le_mul_left
        apply (Nat.div_lt_iff_lt_mul k.succ_pos).1
@@ -2745,7 +2747,7 @@ theorem bmod_lt {x : Int} {m : Nat} (h : 0 < m) : bmod x m < (m + 1) / 2 := by
  split
  · assumption
  · apply Int.lt_of_lt_of_le
-    · show _ < 0
+    · change _ < 0
      have : x % m < m := emod_lt_of_pos x (natCast_pos.mpr h)
      exact Int.sub_neg_of_lt this
    · exact Int.le.intro_sub _ rfl
--- a/src/Init/Data/Int/Lemmas.lean
+++ b/src/Init/Data/Int/Lemmas.lean
@@ -339,7 +339,7 @@ protected theorem add_sub_assoc (a b c : Int) : a + b - c = a + (b - c) := by
  match m with
  | 0 => rfl
  | succ m =>
-    show ofNat (n - succ m) = subNatNat n (succ m)
+    change ofNat (n - succ m) = subNatNat n (succ m)
    rw [subNatNat, Nat.sub_eq_zero_of_le h]

@[deprecated negSucc_eq (since := "2025-03-11")]
--- a/src/Init/Data/Int/Linear.lean
+++ b/src/Init/Data/Int/Linear.lean
@@ -1665,7 +1665,7 @@ theorem natCast_sub (x y : Nat)
        (NatCast.natCast x : Int) + -1*NatCast.natCast y
      else
        (0 : Int) := by
-  show (↑(x - y) : Int) = if (↑y : Int) + (-1)*↑x ≤ 0 then ↑x + (-1)*↑y else 0
+  change (↑(x - y) : Int) = if (↑y : Int) + (-1)*↑x ≤ 0 then (↑x : Int) + (-1)*↑y else 0
  rw [Int.neg_mul, ← Int.sub_eq_add_neg, Int.one_mul]
  rw [Int.neg_mul, ← Int.sub_eq_add_neg, Int.one_mul]
  split
--- a/src/Init/Data/Int/Pow.lean
+++ b/src/Init/Data/Int/Pow.lean
@@ -19,6 +19,13 @@ protected theorem pow_succ (b : Int) (e : Nat) : b ^ (e+1) = (b ^ e) * b := rfl
 protected theorem pow_succ' (b : Int) (e : Nat) : b ^ (e+1) = b * (b ^ e) := by
  rw [Int.mul_comm, Int.pow_succ]

+protected theorem zero_pow {n : Nat} (h : n ≠ 0) : (0 : Int) ^ n = 0 := by
+  match n, h with
+  | n + 1, _ => simp [Int.pow_succ]
+
+protected theorem one_pow {n : Nat} : (1 : Int) ^ n = 1 := by
+  induction n with simp_all [Int.pow_succ]
+
 protected theorem pow_pos {n : Int} {m : Nat} : 0 < n → 0 < n ^ m := by
  induction m with
  | zero => simp
--- a/src/Init/Data/List/Basic.lean
+++ b/src/Init/Data/List/Basic.lean
@@ -9,6 +9,7 @@ prelude
 import Init.SimpLemmas
 import Init.Data.Nat.Basic
 import Init.Data.List.Notation
+import Init.Data.Nat.Div.Basic

@[expose] section

@@ -672,7 +673,7 @@ instance : Std.Associative (α := List α) (· ++ ·) := ⟨append_assoc⟩
 theorem append_cons (as : List α) (b : α) (bs : List α) : as ++ b :: bs = as ++ [b] ++ bs := by
  simp

-@[simp] theorem concat_eq_append {as : List α} {a : α} : as.concat a = as ++ [a] := by
+@[simp, grind =] theorem concat_eq_append {as : List α} {a : α} : as.concat a = as ++ [a] := by
  induction as <;> simp [concat, *]

 theorem reverseAux_eq_append {as bs : List α} : reverseAux as bs = reverseAux as [] ++ bs := by
@@ -1624,8 +1625,8 @@ def find? (p : α → Bool) : List α → Option α
    | true  => some a
    | false => find? p as

-@[simp] theorem find?_nil : ([] : List α).find? p = none := rfl
-theorem find?_cons : (a::as).find? p = match p a with | true => some a | false => as.find? p :=
+@[simp, grind =] theorem find?_nil : ([] : List α).find? p = none := rfl
+@[grind =]theorem find?_cons : (a::as).find? p = match p a with | true => some a | false => as.find? p :=
  rfl

 /-! ### findSome? -/
@@ -1845,8 +1846,8 @@ def lookup [BEq α] : α → List (α × β) → Option β
    | true  => some b
    | false => lookup a as

-@[simp] theorem lookup_nil [BEq α] : ([] : List (α × β)).lookup a = none := rfl
-theorem lookup_cons [BEq α] {k : α} :
+@[simp, grind =] theorem lookup_nil [BEq α] : ([] : List (α × β)).lookup a = none := rfl
+@[grind =] theorem lookup_cons [BEq α] {k : α} :
    ((k, b)::as).lookup a = match a == k with | true => some b | false => as.lookup a :=
  rfl

--- a/src/Init/Data/List/Count.lean
+++ b/src/Init/Data/List/Count.lean
@@ -64,8 +64,8 @@ theorem length_eq_countP_add_countP (p : α → Bool) {l : List α} : length l =
      · rfl
      · simp [h]

-@[grind =]
-theorem countP_eq_length_filter {l : List α} : countP p l = length (filter p l) := by
+@[grind _=_]  -- This to quite aggressive, as it introduces `filter` based reasoning whenever we see `countP`.
+theorem countP_eq_length_filter {l : List α} : countP p l = (filter p l).length := by
  induction l with
  | nil => rfl
  | cons x l ih =>
@@ -82,7 +82,7 @@ theorem countP_le_length : countP p l ≤ l.length := by
  simp only [countP_eq_length_filter]
  apply length_filter_le

-@[simp] theorem countP_append {l₁ l₂ : List α} : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
+@[simp, grind =] theorem countP_append {l₁ l₂ : List α} : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
  simp only [countP_eq_length_filter, filter_append, length_append]

@[simp] theorem countP_pos_iff {p} : 0 < countP p l ↔ ∃ a ∈ l, p a := by
@@ -120,10 +120,24 @@ theorem Sublist.countP_le (s : l₁ <+ l₂) : countP p l₁ ≤ countP p l₂ :
  simp only [countP_eq_length_filter]
  apply s.filter _ |>.length_le

+grind_pattern Sublist.countP_le => l₁ <+ l₂, countP p l₁
+grind_pattern Sublist.countP_le => l₁ <+ l₂, countP p l₂
+
 theorem IsPrefix.countP_le (s : l₁ <+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le
+
+grind_pattern IsPrefix.countP_le => l₁ <+: l₂, countP p l₁
+grind_pattern IsPrefix.countP_le => l₁ <+: l₂, countP p l₂
+
 theorem IsSuffix.countP_le (s : l₁ <:+ l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le
+
+grind_pattern IsSuffix.countP_le => l₁ <:+ l₂, countP p l₁
+grind_pattern IsSuffix.countP_le => l₁ <:+ l₂, countP p l₂
+
 theorem IsInfix.countP_le (s : l₁ <:+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le

+grind_pattern IsInfix.countP_le => l₁ <:+: l₂, countP p l₁
+grind_pattern IsInfix.countP_le => l₁ <:+: l₂, countP p l₂
+
 -- See `Init.Data.List.Nat.Count` for `Sublist.le_countP : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁`.

@[grind]
@@ -174,7 +188,7 @@ theorem countP_flatMap {p : β → Bool} {l : List α} {f : α → List β} :
    countP p (l.flatMap f) = sum (map (countP p ∘ f) l) := by
  rw [List.flatMap, countP_flatten, map_map]

-@[simp] theorem countP_reverse {l : List α} : countP p l.reverse = countP p l := by
+@[simp, grind =] theorem countP_reverse {l : List α} : countP p l.reverse = countP p l := by
  simp [countP_eq_length_filter, filter_reverse]

 theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
@@ -203,18 +217,22 @@ section count

 variable [BEq α]

-@[simp] theorem count_nil {a : α} : count a [] = 0 := rfl
+@[simp, grind =] theorem count_nil {a : α} : count a [] = 0 := rfl

@[grind]
 theorem count_cons {a b : α} {l : List α} :
    count a (b :: l) = count a l + if b == a then 1 else 0 := by
  simp [count, countP_cons]

-@[grind =] theorem count_eq_countP {a : α} {l : List α} : count a l = countP (· == a) l := rfl
+theorem count_eq_countP {a : α} {l : List α} : count a l = countP (· == a) l := rfl
 theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
  funext l
  apply count_eq_countP

+@[grind =]
+theorem count_eq_length_filter {a : α} {l : List α} : count a l = (filter (· == a) l).length := by
+  simp [count, countP_eq_length_filter]
+
@[grind]
 theorem count_tail : ∀ {l : List α} {a : α},
      l.tail.count a = l.count a - if l.head? == some a then 1 else 0
@@ -223,12 +241,28 @@ theorem count_tail : ∀ {l : List α} {a : α},

 theorem count_le_length {a : α} {l : List α} : count a l ≤ l.length := countP_le_length

+grind_pattern count_le_length => count a l
+
 theorem Sublist.count_le (a : α) (h : l₁ <+ l₂) : count a l₁ ≤ count a l₂ := h.countP_le

+grind_pattern Sublist.count_le => l₁ <+ l₂, count a l₁
+grind_pattern Sublist.count_le => l₁ <+ l₂, count a l₂
+
 theorem IsPrefix.count_le (a : α) (h : l₁ <+: l₂) : count a l₁ ≤ count a l₂ := h.sublist.count_le a
+
+grind_pattern IsPrefix.count_le => l₁ <+: l₂, count a l₁
+grind_pattern IsPrefix.count_le => l₁ <+: l₂, count a l₂
+
 theorem IsSuffix.count_le (a : α) (h : l₁ <:+ l₂) : count a l₁ ≤ count a l₂ := h.sublist.count_le a
+
+grind_pattern IsSuffix.count_le => l₁ <:+ l₂, count a l₁
+grind_pattern IsSuffix.count_le => l₁ <:+ l₂, count a l₂
+
 theorem IsInfix.count_le (a : α) (h : l₁ <:+: l₂) : count a l₁ ≤ count a l₂ := h.sublist.count_le a

+grind_pattern IsInfix.count_le => l₁ <:+: l₂, count a l₁
+grind_pattern IsInfix.count_le => l₁ <:+: l₂, count a l₂
+
 -- See `Init.Data.List.Nat.Count` for `Sublist.le_count : count a l₂ - (l₂.length - l₁.length) ≤ countP a l₁`.

 theorem count_tail_le {a : α} {l : List α} : count a l.tail ≤ count a l :=
@@ -245,10 +279,11 @@ theorem count_singleton {a b : α} : count a [b] = if b == a then 1 else 0 := by
@[simp, grind =] theorem count_append {a : α} {l₁ l₂ : List α} : count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
  countP_append

+@[grind =]
 theorem count_flatten {a : α} {l : List (List α)} : count a l.flatten = (l.map (count a)).sum := by
  simp only [count_eq_countP, countP_flatten, count_eq_countP']

-@[simp] theorem count_reverse {a : α} {l : List α} : count a l.reverse = count a l := by
+@[simp, grind =] theorem count_reverse {a : α} {l : List α} : count a l.reverse = count a l := by
  simp only [count_eq_countP, countP_eq_length_filter, filter_reverse, length_reverse]

@[grind]
--- a/src/Init/Data/List/Erase.lean
+++ b/src/Init/Data/List/Erase.lean
@@ -23,9 +23,9 @@ open Nat

 /-! ### eraseP -/

-@[simp] theorem eraseP_nil : [].eraseP p = [] := rfl
+@[simp, grind =] theorem eraseP_nil : [].eraseP p = [] := rfl

-theorem eraseP_cons {a : α} {l : List α} :
+@[grind =] theorem eraseP_cons {a : α} {l : List α} :
    (a :: l).eraseP p = bif p a then l else a :: l.eraseP p := rfl

@[simp] theorem eraseP_cons_of_pos {l : List α} {p} (h : p a) : (a :: l).eraseP p = l := by
@@ -92,7 +92,7 @@ theorem exists_or_eq_self_of_eraseP (p) (l : List α) :
  let ⟨_, l₁, l₂, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
  rw [e₂]; simp [length_append, e₁]

-theorem length_eraseP {l : List α} : (l.eraseP p).length = if l.any p then l.length - 1 else l.length := by
+@[grind =] theorem length_eraseP {l : List α} : (l.eraseP p).length = if l.any p then l.length - 1 else l.length := by
  split <;> rename_i h
  · simp only [any_eq_true] at h
    obtain ⟨x, m, h⟩ := h
@@ -106,8 +106,13 @@ theorem eraseP_sublist {l : List α} : l.eraseP p <+ l := by
  | .inl h => rw [h]; apply Sublist.refl
  | .inr ⟨c, l₁, l₂, _, _, h₃, h₄⟩ => rw [h₄, h₃]; simp

+grind_pattern eraseP_sublist => l.eraseP p, List.Sublist
+
 theorem eraseP_subset {l : List α} : l.eraseP p ⊆ l := eraseP_sublist.subset

+grind_pattern eraseP_subset => l.eraseP p, List.Subset
+
+@[grind ←]
 protected theorem Sublist.eraseP : l₁ <+ l₂ → l₁.eraseP p <+ l₂.eraseP p
  | .slnil => Sublist.refl _
  | .cons a s => by
@@ -147,10 +152,12 @@ theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (erase
    · intro; obtain ⟨x, m, h⟩ := h; simp_all
  · simp_all

+@[grind _=_]
 theorem eraseP_map {f : β → α} : ∀ {l : List β}, (map f l).eraseP p = map f (l.eraseP (p ∘ f))
  | [] => rfl
  | b::l => by by_cases h : p (f b) <;> simp [h, eraseP_map, eraseP_cons_of_pos]

+@[grind =]
 theorem eraseP_filterMap {f : α → Option β} : ∀ {l : List α},
    (filterMap f l).eraseP p = filterMap f (l.eraseP (fun x => match f x with | some y => p y | none => false))
  | [] => rfl
@@ -165,6 +172,7 @@ theorem eraseP_filterMap {f : α → Option β} : ∀ {l : List α},
      · simp only [w, cond_false]
        rw [filterMap_cons_some h, eraseP_filterMap]

+@[grind =]
 theorem eraseP_filter {f : α → Bool} {l : List α} :
    (filter f l).eraseP p = filter f (l.eraseP (fun x => p x && f x)) := by
  rw [← filterMap_eq_filter, eraseP_filterMap]
@@ -174,18 +182,19 @@ theorem eraseP_filter {f : α → Bool} {l : List α} :
  split <;> split at * <;> simp_all

 theorem eraseP_append_left {a : α} (pa : p a) :
-    ∀ {l₁ : List α} l₂, a ∈ l₁ → (l₁++l₂).eraseP p = l₁.eraseP p ++ l₂
+    ∀ {l₁ : List α} l₂, a ∈ l₁ → (l₁ ++ l₂).eraseP p = l₁.eraseP p ++ l₂
  | x :: xs, l₂, h => by
    by_cases h' : p x <;> simp [h']
    rw [eraseP_append_left pa l₂ ((mem_cons.1 h).resolve_left (mt _ h'))]
    intro | rfl => exact pa

 theorem eraseP_append_right :
-    ∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁++l₂) = l₁ ++ l₂.eraseP p
+    ∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁ ++ l₂) = l₁ ++ l₂.eraseP p
  | [],     _, _ => rfl
  | _ :: _, _, h => by
    simp [(forall_mem_cons.1 h).1, eraseP_append_right _ (forall_mem_cons.1 h).2]

+@[grind =]
 theorem eraseP_append {l₁ l₂ : List α} :
    (l₁ ++ l₂).eraseP p = if l₁.any p then l₁.eraseP p ++ l₂ else l₁ ++ l₂.eraseP p := by
  split <;> rename_i h
@@ -196,6 +205,7 @@ theorem eraseP_append {l₁ l₂ : List α} :
    rw [eraseP_append_right _]
    simp_all

+@[grind =]
 theorem eraseP_replicate {n : Nat} {a : α} {p : α → Bool} :
    (replicate n a).eraseP p = if p a then replicate (n - 1) a else replicate n a := by
  induction n with
@@ -212,6 +222,7 @@ theorem eraseP_replicate {n : Nat} {a : α} {p : α → Bool} :
    (replicate n a).eraseP p = replicate n a := by
  rw [eraseP_of_forall_not (by simp_all)]

+@[grind ←]
 protected theorem IsPrefix.eraseP (h : l₁ <+: l₂) : l₁.eraseP p <+: l₂.eraseP p := by
  rw [IsPrefix] at h
  obtain ⟨t, rfl⟩ := h
@@ -258,13 +269,15 @@ theorem eraseP_eq_iff {p} {l : List α} :
        subst p
        simp_all

+@[grind ←]
 theorem Pairwise.eraseP (q) : Pairwise p l → Pairwise p (l.eraseP q) :=
  Pairwise.sublist <| eraseP_sublist

-@[grind]
+@[grind ←]
 theorem Nodup.eraseP (p) : Nodup l → Nodup (l.eraseP p) :=
  Pairwise.eraseP p

+@[grind =]
 theorem eraseP_comm {l : List α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
    (l.eraseP p).eraseP q = (l.eraseP q).eraseP p := by
  induction l with
@@ -357,6 +370,7 @@ theorem exists_erase_eq [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
    length (l.erase a) = length l - 1 := by
  rw [erase_eq_eraseP]; exact length_eraseP_of_mem h (beq_self_eq_true a)

+@[grind =]
 theorem length_erase [LawfulBEq α] {a : α} {l : List α} :
    length (l.erase a) = if a ∈ l then length l - 1 else length l := by
  rw [erase_eq_eraseP, length_eraseP]
@@ -365,11 +379,17 @@ theorem length_erase [LawfulBEq α] {a : α} {l : List α} :
 theorem erase_sublist {a : α} {l : List α} : l.erase a <+ l :=
  erase_eq_eraseP' a l ▸ eraseP_sublist ..

+grind_pattern length_erase => l.erase a, List.Sublist
+
 theorem erase_subset {a : α} {l : List α} : l.erase a ⊆ l := erase_sublist.subset

+grind_pattern erase_subset => l.erase a, List.Subset
+
+@[grind ←]
 theorem Sublist.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₁.erase a <+ l₂.erase a := by
  simp only [erase_eq_eraseP']; exact h.eraseP

+@[grind ←]
 theorem IsPrefix.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+: l₂) : l₁.erase a <+: l₂.erase a := by
  simp only [erase_eq_eraseP']; exact h.eraseP

@@ -391,6 +411,7 @@ theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈
  rw [erase_eq_eraseP', eraseP_eq_self_iff]
  simp [forall_mem_ne']

+@[grind _=_]
 theorem erase_filter [LawfulBEq α] {f : α → Bool} {l : List α} :
    (filter f l).erase a = filter f (l.erase a) := by
  induction l with
@@ -418,10 +439,12 @@ theorem erase_append_right [LawfulBEq α] {a : α} {l₁ : List α} (l₂ : List
  rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right]
  intros b h' h''; rw [eq_of_beq h''] at h; exact h h'

+@[grind =]
 theorem erase_append [LawfulBEq α] {a : α} {l₁ l₂ : List α} :
    (l₁ ++ l₂).erase a = if a ∈ l₁ then l₁.erase a ++ l₂ else l₁ ++ l₂.erase a := by
  simp [erase_eq_eraseP, eraseP_append]

+@[grind =]
 theorem erase_replicate [LawfulBEq α] {n : Nat} {a b : α} :
    (replicate n a).erase b = if b == a then replicate (n - 1) a else replicate n a := by
  rw [erase_eq_eraseP]
@@ -429,6 +452,7 @@ theorem erase_replicate [LawfulBEq α] {n : Nat} {a b : α} :

 -- The arguments `a b` are explicit,
 -- so they can be specified to prevent `simp` repeatedly applying the lemma.
+@[grind =]
 theorem erase_comm [LawfulBEq α] (a b : α) {l : List α} :
    (l.erase a).erase b = (l.erase b).erase a := by
  if ab : a == b then rw [eq_of_beq ab] else ?_
@@ -468,6 +492,7 @@ theorem erase_eq_iff [LawfulBEq α] {a : α} {l : List α} :
  rw [erase_of_not_mem]
  simp_all

+@[grind ←]
 theorem Pairwise.erase [LawfulBEq α] {l : List α} (a) : Pairwise p l → Pairwise p (l.erase a) :=
  Pairwise.sublist <| erase_sublist

@@ -520,6 +545,7 @@ end erase

 /-! ### eraseIdx -/

+@[grind =]
 theorem length_eraseIdx {l : List α} {i : Nat} :
    (l.eraseIdx i).length = if i < l.length then l.length - 1 else l.length := by
  induction l generalizing i with
@@ -537,8 +563,9 @@ theorem length_eraseIdx_of_lt {l : List α} {i} (h : i < length l) :
    (l.eraseIdx i).length = length l - 1 := by
  simp [length_eraseIdx, h]

-@[simp] theorem eraseIdx_zero {l : List α} : eraseIdx l 0 = l.tail := by cases l <;> rfl
+@[simp, grind =] theorem eraseIdx_zero {l : List α} : eraseIdx l 0 = l.tail := by cases l <;> rfl

+@[grind =]
 theorem eraseIdx_eq_take_drop_succ :
    ∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
  | nil, _ => by simp
@@ -565,6 +592,7 @@ theorem eraseIdx_ne_nil_iff {l : List α} {i : Nat} : eraseIdx l i ≠ [] ↔ 2
@[deprecated eraseIdx_ne_nil_iff (since := "2025-01-30")]
 abbrev eraseIdx_ne_nil := @eraseIdx_ne_nil_iff

+@[grind]
 theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
  | [], _ => by simp
  | a::l, 0 => by simp
@@ -573,6 +601,7 @@ theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
 theorem mem_of_mem_eraseIdx {l : List α} {i : Nat} {a : α} (h : a ∈ l.eraseIdx i) : a ∈ l :=
  (eraseIdx_sublist _ _).mem h

+@[grind]
 theorem eraseIdx_subset {l : List α} {k : Nat} : eraseIdx l k ⊆ l :=
  (eraseIdx_sublist _ _).subset

@@ -612,6 +641,15 @@ theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤
    | zero => simp_all
    | succ k => simp_all [eraseIdx_cons_succ, Nat.succ_sub_succ]

+@[grind =]
+theorem eraseIdx_append :
+    eraseIdx (l ++ l') k = if k < length l then eraseIdx l k ++ l' else l ++ eraseIdx l' (k - length l) := by
+  split <;> rename_i h
+  · simp [eraseIdx_append_of_lt_length h]
+  · rw [eraseIdx_append_of_length_le]
+    omega
+
+@[grind =]
 theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
    (replicate n a).eraseIdx k = if k < n then replicate (n - 1) a else replicate n a := by
  split <;> rename_i h
@@ -623,12 +661,15 @@ theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
    exact m.2
  · rw [eraseIdx_of_length_le (by simpa using h)]

+@[grind ←]
 theorem Pairwise.eraseIdx {l : List α} (k) : Pairwise p l → Pairwise p (l.eraseIdx k) :=
  Pairwise.sublist <| eraseIdx_sublist _ _

+@[grind ←]
 theorem Nodup.eraseIdx {l : List α} (k) : Nodup l → Nodup (l.eraseIdx k) :=
  Pairwise.eraseIdx k

+@[grind ←]
 protected theorem IsPrefix.eraseIdx {l l' : List α} (h : l <+: l') (k : Nat) :
    eraseIdx l k <+: eraseIdx l' k := by
  rcases h with ⟨t, rfl⟩
--- a/src/Init/Data/List/FinRange.lean
+++ b/src/Init/Data/List/FinRange.lean
@@ -23,14 +23,14 @@ Examples:
 -/
 def finRange (n : Nat) : List (Fin n) := ofFn fun i => i

-@[simp] theorem length_finRange {n : Nat} : (List.finRange n).length = n := by
+@[simp, grind =] theorem length_finRange {n : Nat} : (List.finRange n).length = n := by
  simp [List.finRange]

-@[simp] theorem getElem_finRange {i : Nat} (h : i < (List.finRange n).length) :
+@[simp, grind =] theorem getElem_finRange {i : Nat} (h : i < (List.finRange n).length) :
    (finRange n)[i] = Fin.cast length_finRange ⟨i, h⟩ := by
  simp [List.finRange]

-@[simp] theorem finRange_zero : finRange 0 = [] := by simp [finRange]
+@[simp, grind =] theorem finRange_zero : finRange 0 = [] := by simp [finRange]

 theorem finRange_succ {n} : finRange (n+1) = 0 :: (finRange n).map Fin.succ := by
  apply List.ext_getElem; simp; intro i; cases i <;> simp
@@ -46,6 +46,7 @@ theorem finRange_succ_last {n} :
    · rfl
    · next h => exact Fin.eq_last_of_not_lt h

+@[grind _=_]
 theorem finRange_reverse {n} : (finRange n).reverse = (finRange n).map Fin.rev := by
  induction n with
  | zero => simp
--- a/src/Init/Data/List/Find.lean
+++ b/src/Init/Data/List/Find.lean
@@ -45,7 +45,7 @@ theorem exists_of_findSome?_eq_some {l : List α} {f : α → Option β} (w : l.
    simp_all only [findSome?_cons, mem_cons, exists_eq_or_imp]
    split at w <;> simp_all

-@[simp] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
+@[simp, grind =] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
  induction l <;> simp [findSome?_cons]; split <;> simp [*]

@[simp] theorem findSome?_isSome_iff {f : α → Option β} {l : List α} :
@@ -91,7 +91,7 @@ theorem findSome?_eq_some_iff {f : α → Option β} {l : List α} {b : β} :
          obtain ⟨⟨rfl, rfl⟩, rfl⟩ := h₁
          exact ⟨l₁, a, l₂, rfl, h₂, fun a' w => h₃ a' (mem_cons_of_mem p w)⟩

-@[simp] theorem findSome?_guard {l : List α} : findSome? (Option.guard fun x => p x) l = find? p l := by
+@[simp, grind =] theorem findSome?_guard {l : List α} : findSome? (Option.guard p) l = find? p l := by
  induction l with
  | nil => simp
  | cons x xs ih =>
@@ -103,32 +103,33 @@ theorem findSome?_eq_some_iff {f : α → Option β} {l : List α} {b : β} :
    · simp only [Option.guard_eq_none_iff] at h
      simp [ih, h]

-theorem find?_eq_findSome?_guard {l : List α} : find? p l = findSome? (Option.guard fun x => p x) l :=
+theorem find?_eq_findSome?_guard {l : List α} : find? p l = findSome? (Option.guard p) l :=
  findSome?_guard.symm

-@[simp] theorem head?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).head? = l.findSome? f := by
+@[simp, grind =] theorem head?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).head? = l.findSome? f := by
  induction l with
  | nil => simp
  | cons x xs ih =>
    simp only [filterMap_cons, findSome?_cons]
    split <;> simp [*]

-@[simp] theorem head_filterMap {f : α → Option β} {l : List α} (h) :
+@[simp, grind =] theorem head_filterMap {f : α → Option β} {l : List α} (h) :
    (l.filterMap f).head h = (l.findSome? f).get (by simp_all [Option.isSome_iff_ne_none]) := by
  simp [head_eq_iff_head?_eq_some]

-@[simp] theorem getLast?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).getLast? = l.reverse.findSome? f := by
+@[simp, grind =] theorem getLast?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).getLast? = l.reverse.findSome? f := by
  rw [getLast?_eq_head?_reverse]
  simp [← filterMap_reverse]

-@[simp] theorem getLast_filterMap {f : α → Option β} {l : List α} (h) :
+@[simp, grind =] theorem getLast_filterMap {f : α → Option β} {l : List α} (h) :
    (l.filterMap f).getLast h = (l.reverse.findSome? f).get (by simp_all [Option.isSome_iff_ne_none]) := by
  simp [getLast_eq_iff_getLast?_eq_some]

-@[simp] theorem map_findSome? {f : α → Option β} {g : β → γ} {l : List α} :
+@[simp, grind _=_] theorem map_findSome? {f : α → Option β} {g : β → γ} {l : List α} :
    (l.findSome? f).map g = l.findSome? (Option.map g ∘ f) := by
  induction l <;> simp [findSome?_cons]; split <;> simp [*]

+@[grind _=_]
 theorem findSome?_map {f : β → γ} {l : List β} : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
  induction l with
  | nil => simp
@@ -136,15 +137,18 @@ theorem findSome?_map {f : β → γ} {l : List β} : findSome? p (l.map f) = l.
    simp only [map_cons, findSome?]
    split <;> simp_all

+@[grind =]
 theorem head_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
-    (flatten L).head (by simpa using h) = (L.findSome? fun l => l.head?).get (by simpa using h) := by
+    (flatten L).head (by simpa using h) = (L.findSome? head?).get (by simpa using h) := by
  simp [head_eq_iff_head?_eq_some, head?_flatten]

+@[grind =]
 theorem getLast_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
    (flatten L).getLast (by simpa using h) =
-      (L.reverse.findSome? fun l => l.getLast?).get (by simpa using h) := by
+      (L.reverse.findSome? getLast?).get (by simpa using h) := by
  simp [getLast_eq_iff_getLast?_eq_some, getLast?_flatten]

+@[grind =]
 theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
  cases n with
  | zero => simp
@@ -174,6 +178,9 @@ theorem Sublist.findSome?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
    · simp_all
    · exact ih

+grind_pattern Sublist.findSome?_isSome => l₁ <+ l₂, l₁.findSome? f
+grind_pattern Sublist.findSome?_isSome => l₁ <+ l₂, l₂.findSome? f
+
 theorem Sublist.findSome?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) :
    l₂.findSome? f = none → l₁.findSome? f = none := by
  simp only [List.findSome?_eq_none_iff, Bool.not_eq_true]
@@ -185,16 +192,30 @@ theorem IsPrefix.findSome?_eq_some {l₁ l₂ : List α} {f : α → Option β}
  obtain ⟨t, rfl⟩ := h
  simp +contextual [findSome?_append]

+grind_pattern IsPrefix.findSome?_eq_some => l₁ <+: l₂, l₁.findSome? f, some b
+grind_pattern IsPrefix.findSome?_eq_some => l₁ <+: l₂, l₂.findSome? f, some b
+
 theorem IsPrefix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <+: l₂) :
    List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
  h.sublist.findSome?_eq_none
+
+grind_pattern IsPrefix.findSome?_eq_none => l₁ <+: l₂, l₂.findSome? f
+grind_pattern IsPrefix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
+
 theorem IsSuffix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <:+ l₂) :
    List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
  h.sublist.findSome?_eq_none
+
+grind_pattern IsSuffix.findSome?_eq_none => l₁ <+: l₂, l₂.findSome? f
+grind_pattern IsSuffix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
+
 theorem IsInfix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <:+: l₂) :
    List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
  h.sublist.findSome?_eq_none

+grind_pattern IsInfix.findSome?_eq_none => l₁ <+: l₂, l₂.findSome? f
+grind_pattern IsInfix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
+
 /-! ### find? -/

@[simp] theorem find?_cons_of_pos {l} (h : p a) : find? p (a :: l) = some a := by
@@ -203,7 +224,7 @@ theorem IsInfix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (
@[simp] theorem find?_cons_of_neg {l} (h : ¬p a) : find? p (a :: l) = find? p l := by
  simp [find?, h]

-@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
+@[simp, grind =] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
  induction l <;> simp [find?_cons]; split <;> simp [*]

 theorem find?_eq_some_iff_append :
@@ -248,25 +269,28 @@ theorem find?_cons_eq_some : (a :: xs).find? p = some b ↔ (p a ∧ a = b) ∨
  rw [find?_cons]
  split <;> simp_all

-@[simp] theorem find?_isSome {xs : List α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
+@[simp, grind =] theorem find?_isSome {xs : List α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
  induction xs with
  | nil => simp
  | cons x xs ih =>
    simp only [find?_cons, mem_cons, exists_eq_or_imp]
    split <;> simp_all

+@[grind →]
 theorem find?_some : ∀ {l}, find? p l = some a → p a
  | b :: l, H => by
    by_cases h : p b <;> simp [find?, h] at H
    · exact H ▸ h
    · exact find?_some H

+@[grind →]
 theorem mem_of_find?_eq_some : ∀ {l}, find? p l = some a → a ∈ l
  | b :: l, H => by
    by_cases h : p b <;> simp [find?, h] at H
    · exact H ▸ .head _
    · exact .tail _ (mem_of_find?_eq_some H)

+@[grind]
 theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h ∈ xs := by
  induction xs with
  | nil => simp at h
@@ -278,7 +302,7 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
      right
      apply ih

-@[simp] theorem find?_filter {xs : List α} {p : α → Bool} {q : α → Bool} :
+@[simp, grind =] theorem find?_filter {xs : List α} {p : α → Bool} {q : α → Bool} :
    (xs.filter p).find? q = xs.find? (fun a => p a ∧ q a) := by
  induction xs with
  | nil => simp
@@ -288,22 +312,22 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
    · simp only [find?_cons]
      split <;> simp_all

-@[simp] theorem head?_filter {p : α → Bool} {l : List α} : (l.filter p).head? = l.find? p := by
+@[simp, grind =] theorem head?_filter {p : α → Bool} {l : List α} : (l.filter p).head? = l.find? p := by
  rw [← filterMap_eq_filter, head?_filterMap, findSome?_guard]

-@[simp] theorem head_filter {p : α → Bool} {l : List α} (h) :
+@[simp, grind =] theorem head_filter {p : α → Bool} {l : List α} (h) :
    (l.filter p).head h = (l.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
  simp [head_eq_iff_head?_eq_some]

-@[simp] theorem getLast?_filter {p : α → Bool} {l : List α} : (l.filter p).getLast? = l.reverse.find? p := by
+@[simp, grind =] theorem getLast?_filter {p : α → Bool} {l : List α} : (l.filter p).getLast? = l.reverse.find? p := by
  rw [getLast?_eq_head?_reverse]
  simp [← filter_reverse]

-@[simp] theorem getLast_filter {p : α → Bool} {l : List α} (h) :
+@[simp, grind =] theorem getLast_filter {p : α → Bool} {l : List α} (h) :
    (l.filter p).getLast h = (l.reverse.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
  simp [getLast_eq_iff_getLast?_eq_some]

-@[simp] theorem find?_filterMap {xs : List α} {f : α → Option β} {p : β → Bool} :
+@[simp, grind =] theorem find?_filterMap {xs : List α} {f : α → Option β} {p : β → Bool} :
    (xs.filterMap f).find? p = (xs.find? (fun a => (f a).any p)).bind f := by
  induction xs with
  | nil => simp
@@ -313,15 +337,15 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
    · simp only [find?_cons]
      split <;> simp_all

-@[simp] theorem find?_map {f : β → α} {l : List β} : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
+@[simp, grind =] theorem find?_map {f : β → α} {l : List β} : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
  induction l with
  | nil => simp
  | cons x xs ih =>
    simp only [map_cons, find?]
    by_cases h : p (f x) <;> simp [h, ih]

-@[simp] theorem find?_flatten {xss : List (List α)} {p : α → Bool} :
-    xss.flatten.find? p = xss.findSome? (·.find? p) := by
+@[simp, grind _=_] theorem find?_flatten {xss : List (List α)} {p : α → Bool} :
+    xss.flatten.find? p = xss.findSome? (find? p) := by
  induction xss with
  | nil => simp
  | cons _ _ ih =>
@@ -378,7 +402,7 @@ theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a :
@[deprecated find?_flatten_eq_some_iff (since := "2025-02-03")]
 abbrev find?_flatten_eq_some := @find?_flatten_eq_some_iff

-@[simp] theorem find?_flatMap {xs : List α} {f : α → List β} {p : β → Bool} :
+@[simp, grind =] theorem find?_flatMap {xs : List α} {f : α → List β} {p : β → Bool} :
    (xs.flatMap f).find? p = xs.findSome? (fun x => (f x).find? p) := by
  simp [flatMap_def, findSome?_map]; rfl

@@ -386,6 +410,7 @@ theorem find?_flatMap_eq_none_iff {xs : List α} {f : α → List β} {p : β
    (xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
  simp

+@[grind =]
 theorem find?_replicate : find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
  cases n
  · simp
@@ -430,6 +455,9 @@ theorem Sublist.find?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) : (l₁.fi
    · simp
    · simpa using ih

+grind_pattern Sublist.find?_isSome => l₁ <+ l₂, l₁.find? p
+grind_pattern Sublist.find?_isSome => l₁ <+ l₂, l₂.find? p
+
 theorem Sublist.find?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₂.find? p = none → l₁.find? p = none := by
  simp only [List.find?_eq_none, Bool.not_eq_true]
  exact fun w x m => w x (Sublist.mem m h)
@@ -440,16 +468,31 @@ theorem IsPrefix.find?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁
  obtain ⟨t, rfl⟩ := h
  simp +contextual [find?_append]

+grind_pattern IsPrefix.find?_eq_some => l₁ <+: l₂, l₁.find? p, some b
+grind_pattern IsPrefix.find?_eq_some => l₁ <+: l₂, l₂.find? p, some b
+
 theorem IsPrefix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
    List.find? p l₂ = none → List.find? p l₁ = none :=
  h.sublist.find?_eq_none
+
+grind_pattern Sublist.find?_eq_none => l₁ <+ l₂, l₂.find? p
+grind_pattern Sublist.find?_eq_none => l₁ <+ l₂, l₁.find? p
+
 theorem IsSuffix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+ l₂) :
    List.find? p l₂ = none → List.find? p l₁ = none :=
  h.sublist.find?_eq_none
+
+grind_pattern IsPrefix.find?_eq_none => l₁ <+: l₂, l₂.find? p
+grind_pattern IsPrefix.find?_eq_none => l₁ <+: l₂, l₁.find? p
+
 theorem IsInfix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+: l₂) :
    List.find? p l₂ = none → List.find? p l₁ = none :=
  h.sublist.find?_eq_none

+grind_pattern IsSuffix.find?_eq_none => l₁ <:+ l₂, l₂.find? p
+grind_pattern IsSuffix.find?_eq_none => l₁ <:+ l₂, l₁.find? p
+
+@[grind =]
 theorem find?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
    (H : ∀ (a : α), a ∈ xs → P a) {p : β → Bool} :
    (xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
@@ -482,9 +525,9 @@ private theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
      ext
      simp only [Nat.add_comm i, Function.comp_apply, Nat.add_assoc]

-@[simp] theorem findIdx?_nil : ([] : List α).findIdx? p = none := rfl
+@[simp, grind =] theorem findIdx?_nil : ([] : List α).findIdx? p = none := rfl

-theorem findIdx?_cons :
+@[grind =] theorem findIdx?_cons :
    (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]

@@ -493,6 +536,7 @@ theorem findIdx?_cons :

 /-! ### findIdx -/

+@[grind =]
 theorem findIdx_cons {p : α → Bool} {b : α} {l : List α} :
    (b :: l).findIdx p = bif p b then 0 else (l.findIdx p) + 1 := by
  cases H : p b with
@@ -511,6 +555,7 @@ where
@[simp] theorem findIdx_singleton {a : α} {p : α → Bool} : [a].findIdx p = if p a then 0 else 1 := by
  simp [findIdx_cons, findIdx_nil]

+@[grind →]
 theorem findIdx_of_getElem?_eq_some {xs : List α} (w : xs[xs.findIdx p]? = some y) : p y := by
  induction xs with
  | nil => simp_all
@@ -520,6 +565,8 @@ theorem findIdx_getElem {xs : List α} {w : xs.findIdx p < xs.length} :
    p xs[xs.findIdx p] :=
  xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)

+grind_pattern findIdx_getElem => xs[xs.findIdx p]
+
 theorem findIdx_lt_length_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
    xs.findIdx p < xs.length := by
  induction xs with
@@ -558,6 +605,8 @@ theorem findIdx_le_length {p : α → Bool} {xs : List α} : xs.findIdx p ≤ xs
  · simp at e
    exact Nat.le_of_eq (findIdx_eq_length.mpr e)

+grind_pattern findIdx_le_length => xs.findIdx p, xs.length
+
@[simp]
 theorem findIdx_lt_length {p : α → Bool} {xs : List α} :
    xs.findIdx p < xs.length ↔ ∃ x ∈ xs, p x := by
@@ -567,6 +616,8 @@ theorem findIdx_lt_length {p : α → Bool} {xs : List α} :
  rw [← this, findIdx_eq_length, not_exists]
  simp only [Bool.not_eq_true, not_and]

+grind_pattern findIdx_lt_length => xs.findIdx p, xs.length
+
 /-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/
 theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.findIdx p) :
    p (xs[i]'(Nat.le_trans h findIdx_le_length)) = false := by
@@ -591,6 +642,8 @@ theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs
      rw [← ipm, Nat.succ_lt_succ_iff] at h
      simpa using ih h

+grind_pattern not_of_lt_findIdx => xs.findIdx p, xs[i]
+
 /-- If `¬ p xs[j]` for all `j < i`, then `i ≤ xs.findIdx p`. -/
 theorem le_findIdx_of_not {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length)
    (h2 : ∀ j (hji : j < i), p (xs[j]'(Nat.lt_trans hji h)) = false) : i ≤ xs.findIdx p := by
@@ -618,6 +671,7 @@ theorem findIdx_eq {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length
  simp at h3
  simp_all [not_of_lt_findIdx h3]

+@[grind =]
 theorem findIdx_append {p : α → Bool} {l₁ l₂ : List α} :
    (l₁ ++ l₂).findIdx p =
      if l₁.findIdx p < l₁.length then l₁.findIdx p else l₂.findIdx p + l₁.length := by
@@ -639,6 +693,9 @@ theorem IsPrefix.findIdx_le {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+
  · exact Nat.le_refl ..
  · simp_all [findIdx_eq_length_of_false]

+grind_pattern IsPrefix.findIdx_le => l₁ <:+ l₂, l₁.findIdx p
+grind_pattern IsPrefix.findIdx_le => l₁ <:+ l₂, l₂.findIdx p
+
 theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂)
    (lt : l₁.findIdx p < l₁.length) : l₂.findIdx p = l₁.findIdx p := by
  rw [IsPrefix] at h
@@ -648,6 +705,8 @@ theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α
  · rfl
  · simp_all

+grind_pattern IsPrefix.findIdx_eq_of_findIdx_lt_length => l₁ <:+ l₂, l₁.findIdx p, l₂.findIdx p
+
 theorem findIdx_le_findIdx {l : List α} {p q : α → Bool} (h : ∀ x ∈ l, p x → q x) : l.findIdx q ≤ l.findIdx p := by
  induction l with
  | nil => simp
@@ -671,7 +730,7 @@ theorem findIdx_le_findIdx {l : List α} {p q : α → Bool} (h : ∀ x ∈ l, p

 /-! ### findIdx? -/

-@[simp]
+@[simp, grind =]
 theorem findIdx?_eq_none_iff {xs : List α} {p : α → Bool} :
    xs.findIdx? p = none ↔ ∀ x, x ∈ xs → p x = false := by
  induction xs with
@@ -680,7 +739,7 @@ theorem findIdx?_eq_none_iff {xs : List α} {p : α → Bool} :
    simp only [findIdx?_cons]
    split <;> simp_all [cond_eq_if]

-@[simp]
+@[simp, grind =]
 theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
    (xs.findIdx? p).isSome = xs.any p := by
  induction xs with
@@ -689,7 +748,7 @@ theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
    simp only [findIdx?_cons]
    split <;> simp_all

-@[simp]
+@[simp, grind =]
 theorem findIdx?_isNone {xs : List α} {p : α → Bool} :
    (xs.findIdx? p).isNone = xs.all (¬p ·) := by
  induction xs with
@@ -795,14 +854,14 @@ theorem of_findIdx?_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p
@[deprecated of_findIdx?_eq_none (since := "2025-02-02")]
 abbrev findIdx?_of_eq_none := @of_findIdx?_eq_none

-@[simp] theorem findIdx?_map {f : β → α} {l : List β} : findIdx? p (l.map f) = l.findIdx? (p ∘ f) := by
+@[simp, grind _=_] theorem findIdx?_map {f : β → α} {l : List β} : findIdx? p (l.map f) = l.findIdx? (p ∘ f) := by
  induction l with
  | nil => simp
  | cons x xs ih =>
    simp only [map_cons, findIdx?_cons]
    split <;> simp_all

-@[simp] theorem findIdx?_append :
+@[simp, grind =] theorem findIdx?_append :
    (xs ++ ys : List α).findIdx? p =
      (xs.findIdx? p).or ((ys.findIdx? p).map fun i => i + xs.length) := by
  induction xs with simp [findIdx?_cons]
@@ -824,7 +883,7 @@ theorem findIdx?_flatten {l : List (List α)} {p : α → Bool} :
    · rw [Option.or_of_isNone (by simp_all [findIdx?_isNone])]
      simp [Function.comp_def, Nat.add_comm, Nat.add_assoc]

-@[simp] theorem findIdx?_replicate :
+@[simp, grind =] theorem findIdx?_replicate :
    (replicate n a).findIdx? p = if 0 < n ∧ p a then some 0 else none := by
  cases n with
  | zero => simp
@@ -878,22 +937,38 @@ theorem Sublist.findIdx?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) :
  simp only [findIdx?_eq_none_iff]
  exact fun w x m => w x (h.mem m)

+grind_pattern Sublist.findIdx?_eq_none => l₁ <+ l₂, l₁.findIdx? p
+grind_pattern Sublist.findIdx?_eq_none => l₁ <+ l₂, l₂.findIdx? p
+
 theorem IsPrefix.findIdx?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
    List.findIdx? p l₁ = some i → List.findIdx? p l₂ = some i := by
  rw [IsPrefix] at h
  obtain ⟨t, rfl⟩ := h
  intro h
  simp [findIdx?_append, h]
+
 theorem IsPrefix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
    List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
  h.sublist.findIdx?_eq_none
+
+grind_pattern IsPrefix.findIdx?_eq_none => l₁ <+: l₂, l₁.findIdx? p
+grind_pattern IsPrefix.findIdx?_eq_none => l₁ <+: l₂, l₂.findIdx? p
+
 theorem IsSuffix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+ l₂) :
    List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
  h.sublist.findIdx?_eq_none
+
+grind_pattern IsSuffix.findIdx?_eq_none => l₁ <:+ l₂, l₁.findIdx? p
+grind_pattern IsSuffix.findIdx?_eq_none => l₁ <:+ l₂, l₂.findIdx? p
+
 theorem IsInfix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+: l₂) :
    List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
  h.sublist.findIdx?_eq_none

+grind_pattern IsInfix.findIdx?_eq_none => l₁ <:+: l₂, l₁.findIdx? p
+grind_pattern IsInfix.findIdx?_eq_none => l₁ <:+: l₂, l₂.findIdx? p
+
+@[grind =]
 theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :
    xs.findIdx p = (xs.findIdx? p).getD xs.length := by
  induction xs with
@@ -914,7 +989,7 @@ theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :

 /-! ### findFinIdx? -/

-@[simp] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
+@[simp, grind =] 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 =
@@ -940,6 +1015,7 @@ theorem findFinIdx?_eq_pmap_findIdx? {xs : List α} {p : α → Bool} :
        (fun i h => h) := by
  simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]

+@[grind =]
 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)]
@@ -950,6 +1026,7 @@ theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
  · rw [findIdx?_eq_map_findFinIdx?_val]
    simp [Function.comp_def]

+@[grind =]
 theorem findFinIdx?_append {xs ys : List α} {p : α → Bool} :
    (xs ++ ys).findFinIdx? p =
      ((xs.findFinIdx? p).map (Fin.castLE (by simp))).or
@@ -959,11 +1036,11 @@ theorem findFinIdx?_append {xs ys : List α} {p : α → Bool} :
  · simp [h, Option.pmap_map, Option.map_pmap, Nat.add_comm]
  · simp [h]

-@[simp] theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
+@[simp, grind =] theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
    [a].findFinIdx? p = if p a then some ⟨0, by simp⟩ else none := by
  simp [findFinIdx?_cons, findFinIdx?_nil]

-@[simp] theorem findFinIdx?_eq_none_iff {l : List α} {p : α → Bool} :
+@[simp, grind =] 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?]

@@ -979,7 +1056,7 @@ theorem findFinIdx?_eq_some_iff {xs : List α} {p : α → Bool} {i : Fin xs.len
  · rintro ⟨h, w⟩
    exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩

-@[simp]
+@[simp, grind =]
 theorem isSome_findFinIdx? {l : List α} {p : α → Bool} :
    (l.findFinIdx? p).isSome = l.any p := by
  induction l with
@@ -988,7 +1065,7 @@ theorem isSome_findFinIdx? {l : List α} {p : α → Bool} :
    simp only [findFinIdx?_cons]
    split <;> simp_all

-@[simp]
+@[simp, grind =]
 theorem isNone_findFinIdx? {l : List α} {p : α → Bool} :
    (l.findFinIdx? p).isNone = l.all (fun x => ¬ p x) := by
  induction l with
@@ -1013,6 +1090,7 @@ The verification API for `idxOf` is still incomplete.
 The lemmas below should be made consistent with those for `findIdx` (and proved using them).
 -/

+@[grind =]
 theorem idxOf_cons [BEq α] :
    (x :: xs : List α).idxOf y = bif x == y then 0 else xs.idxOf y + 1 := by
  dsimp [idxOf]
@@ -1027,6 +1105,7 @@ abbrev indexOf_cons := @idxOf_cons
@[deprecated idxOf_cons_self (since := "2025-01-29")]
 abbrev indexOf_cons_self := @idxOf_cons_self

+@[grind =]
 theorem idxOf_append [BEq α] [LawfulBEq α] {l₁ l₂ : List α} {a : α} :
    (l₁ ++ l₂).idxOf a = if a ∈ l₁ then l₁.idxOf a else l₂.idxOf a + l₁.length := by
  rw [idxOf, findIdx_append]
@@ -1050,7 +1129,7 @@ theorem idxOf_eq_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∉ l) : l.
@[deprecated idxOf_eq_length (since := "2025-01-29")]
 abbrev indexOf_eq_length := @idxOf_eq_length

-theorem idxOf_lt_length [BEq α] [EquivBEq α] {l : List α} (h : a ∈ l) : l.idxOf a < l.length := by
+theorem idxOf_lt_length_of_mem [BEq α] [EquivBEq α] {l : List α} (h : a ∈ l) : l.idxOf a < l.length := by
  induction l with
  | nil => simp at h
  | cons x xs ih =>
@@ -1063,8 +1142,23 @@ theorem idxOf_lt_length [BEq α] [EquivBEq α] {l : List α} (h : a ∈ l) : l.i
      · exact zero_lt_succ xs.length
      · exact Nat.add_lt_add_right ih 1

-@[deprecated idxOf_lt_length (since := "2025-01-29")]
-abbrev indexOf_lt_length := @idxOf_lt_length
+theorem idxOf_le_length [BEq α] [LawfulBEq α] {l : List α} {a : α} :
+    l.idxOf a ≤ l.length := by
+  simpa [idxOf] using findIdx_le_length
+
+grind_pattern idxOf_le_length => l.idxOf a, l.length
+
+theorem idxOf_lt_length_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
+    l.idxOf a < l.length ↔ a ∈ l := by
+  constructor
+  · intro h
+    simpa [idxOf] using h
+  · exact idxOf_lt_length_of_mem
+
+grind_pattern idxOf_lt_length_iff => l.idxOf a, l.length
+
+@[deprecated idxOf_lt_length_of_mem (since := "2025-01-29")]
+abbrev indexOf_lt_length := @idxOf_lt_length_of_mem

 /-! ### finIdxOf?

@@ -1076,14 +1170,14 @@ 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, grind =] theorem finIdxOf?_nil [BEq α] : ([] : List α).finIdxOf? a = none := rfl

-theorem finIdxOf?_cons [BEq α] {a : α} {xs : List α} :
+@[grind =] 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?, findFinIdx?_cons]

-@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
+@[simp, grind =] 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
@@ -1096,18 +1190,19 @@ theorem finIdxOf?_cons [BEq α] {a : α} {xs : List α} :
    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]

-@[simp]
-theorem isSome_finIdxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
-    (l.finIdxOf? a).isSome ↔ a ∈ l := by
+@[simp, grind =]
+theorem isSome_finIdxOf? [BEq α] [PartialEquivBEq α] {l : List α} {a : α} :
+    (l.finIdxOf? a).isSome = l.contains a := by
  induction l with
  | nil => simp
  | cons x xs ih =>
    simp only [finIdxOf?_cons]
-    split <;> simp_all [@eq_comm _ x a]
+    split <;> simp_all [BEq.comm]

-theorem isNone_finIdxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
-    (l.finIdxOf? a).isNone = ¬ a ∈ l := by
-  simp
+@[simp]
+theorem isNone_finIdxOf? [BEq α] [PartialEquivBEq α] {l : List α} {a : α} :
+    (l.finIdxOf? a).isNone = !l.contains a := by
+  rw [← isSome_finIdxOf?, Option.not_isSome]

 /-! ### idxOf?

@@ -1115,16 +1210,16 @@ The verification API for `idxOf?` is still incomplete.
 The lemmas below should be made consistent with those for `findIdx?` (and proved using them).
 -/

-@[simp] theorem idxOf?_nil [BEq α] : ([] : List α).idxOf? a = none := rfl
+@[simp, grind =] theorem idxOf?_nil [BEq α] : ([] : List α).idxOf? a = none := rfl

-theorem idxOf?_cons [BEq α] {a : α} {xs : List α} {b : α} :
+@[grind =] theorem idxOf?_cons [BEq α] {a : α} {xs : List α} {b : α} :
    (a :: xs).idxOf? b = if a == b then some 0 else (xs.idxOf? b).map (· + 1) := by
  simp [idxOf?, findIdx?_cons]

@[simp] theorem idxOf?_singleton [BEq α] {a b : α} : [a].idxOf? b = if a == b then some 0 else none := by
  simp [idxOf?_cons, idxOf?_nil]

-@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
+@[simp, grind =] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
    l.idxOf? a = none ↔ a ∉ l := by
  simp only [idxOf?, findIdx?_eq_none_iff, beq_eq_false_iff_ne, ne_eq]
  constructor
@@ -1137,7 +1232,7 @@ 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

-@[simp]
+@[simp, grind =]
 theorem isSome_idxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
    (l.idxOf? a).isSome ↔ a ∈ l := by
  induction l with
@@ -1146,6 +1241,7 @@ theorem isSome_idxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
    simp only [idxOf?_cons]
    split <;> simp_all [@eq_comm _ x a]

+@[grind =]
 theorem isNone_idxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
    (l.idxOf? a).isNone = ¬ a ∈ l := by
  simp
@@ -1172,7 +1268,7 @@ theorem lookup_eq_findSome? {l : List (α × β)} {k : α} :
      simp only [lookup_cons, findSome?_cons]
      split <;> simp_all

-@[simp] theorem lookup_eq_none_iff {l : List (α × β)} {k : α} :
+@[simp, grind =] theorem lookup_eq_none_iff {l : List (α × β)} {k : α} :
    l.lookup k = none ↔ ∀ p ∈ l, k != p.1 := by
  simp [lookup_eq_findSome?]

@@ -1192,10 +1288,12 @@ theorem lookup_eq_some_iff {l : List (α × β)} {k : α} {b : β} :
  · rintro ⟨l₁, l₂, rfl, h⟩
    exact ⟨l₁, (k, b), l₂, rfl, by simp, by simpa using h⟩

+@[grind =]
 theorem lookup_append {l₁ l₂ : List (α × β)} {k : α} :
    (l₁ ++ l₂).lookup k = (l₁.lookup k).or (l₂.lookup k) := by
  simp [lookup_eq_findSome?, findSome?_append]

+@[grind =]
 theorem lookup_replicate {k : α} :
    (replicate n (a,b)).lookup k = if n = 0 then none else if k == a then some b else none := by
  induction n with
@@ -1230,6 +1328,9 @@ theorem Sublist.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <+ l₂)
  simp only [lookup_eq_findSome?]
  exact h.findSome?_eq_none

+grind_pattern Sublist.lookup_isSome => l₁ <+ l₂, l₁.lookup k
+grind_pattern Sublist.lookup_isSome => l₁ <+ l₂, l₂.lookup k
+
 theorem IsPrefix.lookup_eq_some {l₁ l₂ : List (α × β)} (h : l₁ <+: l₂) :
    List.lookup k l₁ = some b → List.lookup k l₂ = some b := by
  simp only [lookup_eq_findSome?]
@@ -1238,13 +1339,24 @@ theorem IsPrefix.lookup_eq_some {l₁ l₂ : List (α × β)} (h : l₁ <+: l₂
 theorem IsPrefix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <+: l₂) :
    List.lookup k l₂ = none → List.lookup k l₁ = none :=
  h.sublist.lookup_eq_none
+
+grind_pattern IsPrefix.lookup_eq_none => l₁ <+: l₂, l₁.lookup k
+grind_pattern IsPrefix.lookup_eq_none => l₁ <+: l₂, l₂.lookup k
+
 theorem IsSuffix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <:+ l₂) :
    List.lookup k l₂ = none → List.lookup k l₁ = none :=
  h.sublist.lookup_eq_none
+
+grind_pattern IsSuffix.lookup_eq_none => l₁ <:+ l₂, l₁.lookup k
+grind_pattern IsSuffix.lookup_eq_none => l₁ <:+ l₂, l₂.lookup k
+
 theorem IsInfix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <:+: l₂) :
    List.lookup k l₂ = none → List.lookup k l₁ = none :=
  h.sublist.lookup_eq_none

+grind_pattern IsInfix.lookup_eq_none => l₁ <:+: l₂, l₁.lookup k
+grind_pattern IsInfix.lookup_eq_none => l₁ <:+: l₂, l₂.lookup k
+
 end lookup

 end List
--- a/src/Init/Data/List/Impl.lean
+++ b/src/Init/Data/List/Impl.lean
@@ -261,11 +261,11 @@ Examples:
 /-- Tail recursive implementation of `findRev?`. This is only used at runtime. -/
 def findRev?TR (p : α → Bool) (l : List α) : Option α := l.reverse.find? p

-@[simp] theorem find?_singleton {a : α} : [a].find? p = if p a then some a else none := by
+@[simp, grind =] theorem find?_singleton {a : α} : [a].find? p = if p a then some a else none := by
  simp only [find?]
  split <;> simp_all

-@[simp] theorem find?_append {xs ys : List α} : (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
+@[simp, grind =] theorem find?_append {xs ys : List α} : (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
  induction xs with
  | nil => simp [find?]
  | cons x xs ih =>
@@ -287,12 +287,12 @@ def findRev?TR (p : α → Bool) (l : List α) : Option α := l.reverse.find? p
 /-- Tail recursive implementation of `finSomedRev?`. This is only used at runtime. -/
 def findSomeRev?TR (f : α → Option β) (l : List α) : Option β := l.reverse.findSome? f

-@[simp] theorem findSome?_singleton {a : α} :
+@[simp, grind =] theorem findSome?_singleton {a : α} :
    [a].findSome? f = f a := by
  simp only [findSome?_cons, findSome?_nil]
  split <;> simp_all

-@[simp] theorem findSome?_append {xs ys : List α} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
+@[simp, grind =] theorem findSome?_append {xs ys : List α} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
  induction xs with
  | nil => simp [findSome?]
  | cons x xs ih =>
--- a/src/Init/Data/List/Lemmas.lean
+++ b/src/Init/Data/List/Lemmas.lean
@@ -109,9 +109,11 @@ abbrev length_eq_zero := @length_eq_zero_iff
 theorem eq_nil_iff_length_eq_zero : l = [] ↔ length l = 0 :=
  length_eq_zero_iff.symm

-@[grind →] theorem length_pos_of_mem {a : α} : ∀ {l : List α}, a ∈ l → 0 < length l
+theorem length_pos_of_mem {a : α} : ∀ {l : List α}, a ∈ l → 0 < length l
  | _::_, _ => Nat.zero_lt_succ _

+grind_pattern length_pos_of_mem => a ∈ l, length l
+
 theorem exists_mem_of_length_pos : ∀ {l : List α}, 0 < length l → ∃ a, a ∈ l
  | _::_, _ => ⟨_, .head ..⟩

@@ -1318,6 +1320,19 @@ theorem forall_mem_filter {l : List α} {p : α → Bool} {P : α → Prop} :
    (∀ (i) (_ : i ∈ l.filter p), P i) ↔ ∀ (j) (_ : j ∈ l), p j → P j := by
  simp

+@[grind] theorem getElem_filter {xs : List α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).length) :
+    p (xs.filter p)[i] :=
+  (mem_filter.mp (getElem_mem h)).2
+
+theorem getElem?_filter {xs : List α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).length)
+    (w : (xs.filter p)[i]? = some a) : p a := by
+  rw [getElem?_eq_getElem] at w
+  simp only [Option.some.injEq] at w
+  rw [← w]
+  apply getElem_filter h
+
+grind_pattern getElem?_filter => (xs.filter p)[i]?, some a
+
@[simp] theorem filter_filter : ∀ {l}, filter p (filter q l) = filter (fun a => p a && q a) l
  | [] => rfl
  | a :: l => by by_cases hp : p a <;> by_cases hq : q a <;> simp [hp, hq, filter_filter]
@@ -2727,11 +2742,11 @@ def foldlRecOn {motive : β → Sort _} : ∀ (l : List α) (op : β → α →
    foldlRecOn tl op (hl b hb hd mem_cons_self)
      fun y hy x hx => hl y hy x (mem_cons_of_mem hd hx)

-@[simp] theorem foldlRecOn_nil {motive : β → Sort _} {op : β → α → β} (hb : motive b)
+@[simp, grind =] theorem foldlRecOn_nil {motive : β → Sort _} {op : β → α → β} (hb : motive b)
    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ []), motive (op b a)) :
    foldlRecOn [] op hb hl = hb := rfl

-@[simp] theorem foldlRecOn_cons {motive : β → Sort _} {op : β → α → β} (hb : motive b)
+@[simp, grind =] theorem foldlRecOn_cons {motive : β → Sort _} {op : β → α → β} (hb : motive b)
    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ x :: l), motive (op b a)) :
    foldlRecOn (x :: l) op hb hl =
      foldlRecOn l op (hl b hb x mem_cons_self)
@@ -2762,11 +2777,11 @@ def foldrRecOn {motive : β → Sort _} : ∀ (l : List α) (op : α → β →
    hl (foldr op b l)
      (foldrRecOn l op hb fun b c a m => hl b c a (mem_cons_of_mem x m)) x mem_cons_self

-@[simp] theorem foldrRecOn_nil {motive : β → Sort _} {op : α → β → β} (hb : motive b)
+@[simp, grind =] theorem foldrRecOn_nil {motive : β → Sort _} {op : α → β → β} (hb : motive b)
    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ []), motive (op a b)) :
    foldrRecOn [] op hb hl = hb := rfl

-@[simp] theorem foldrRecOn_cons {motive : β → Sort _} {op : α → β → β} (hb : motive b)
+@[simp, grind =] theorem foldrRecOn_cons {motive : β → Sort _} {op : α → β → β} (hb : motive b)
    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ x :: l), motive (op a b)) :
    foldrRecOn (x :: l) op hb hl =
      hl _ (foldrRecOn l op hb fun b c a m => hl b c a (mem_cons_of_mem x m))
@@ -2778,8 +2793,8 @@ We can prove that two folds over the same list are related (by some arbitrary re
 if we know that the initial elements are related and the folding function, for each element of the list,
 preserves the relation.
 -/
-theorem foldl_rel {l : List α} {f g : β → α → β} {a b : β} {r : β → β → Prop}
-    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
+theorem foldl_rel {l : List α} {f : β → α → β} {g : γ → α → γ} {a : β} {b : γ} {r : β → γ → Prop}
+    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c : β) (c' : γ), r c c' → r (f c a) (g c' a)) :
    r (l.foldl (fun acc a => f acc a) a) (l.foldl (fun acc a => g acc a) b) := by
  induction l generalizing a b with
  | nil => simp_all
@@ -2794,8 +2809,8 @@ We can prove that two folds over the same list are related (by some arbitrary re
 if we know that the initial elements are related and the folding function, for each element of the list,
 preserves the relation.
 -/
-theorem foldr_rel {l : List α} {f g : α → β → β} {a b : β} {r : β → β → Prop}
-    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
+theorem foldr_rel {l : List α} {f : α → β → β} {g : α → γ → γ} {a : β} {b : γ} {r : β → γ → Prop}
+    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c : β) (c' : γ), r c c' → r (f a c) (g a c')) :
    r (l.foldr (fun a acc => f a acc) a) (l.foldr (fun a acc => g a acc) b) := by
  induction l generalizing a b with
  | nil => simp_all
@@ -2902,13 +2917,13 @@ theorem getLast_filterMap_of_eq_some {f : α → Option β} {l : List α} (w : l
  rw [head_filterMap_of_eq_some (by simp_all)]
  simp_all

-theorem getLast?_flatMap {l : List α} {f : α → List β} :
+@[grind =] theorem getLast?_flatMap {l : List α} {f : α → List β} :
    (l.flatMap f).getLast? = l.reverse.findSome? fun a => (f a).getLast? := by
  simp only [← head?_reverse, reverse_flatMap]
  rw [head?_flatMap]
  rfl

-theorem getLast?_flatten {L : List (List α)} :
+@[grind =] theorem getLast?_flatten {L : List (List α)} :
    (flatten L).getLast? = L.reverse.findSome? fun l => l.getLast? := by
  simp [← flatMap_id, getLast?_flatMap]

@@ -2923,7 +2938,7 @@ theorem getLast?_replicate {a : α} {n : Nat} : (replicate n a).getLast? = if n
 /-! ### leftpad -/

 -- We unfold `leftpad` and `rightpad` for verification purposes.
-attribute [simp] leftpad rightpad
+attribute [simp, grind] leftpad rightpad

 -- `length_leftpad` and `length_rightpad` are in `Init.Data.List.Nat.Basic`.

@@ -3027,6 +3042,9 @@ we do not separately develop much theory about it.
 theorem mem_partition : a ∈ l ↔ a ∈ (partition p l).1 ∨ a ∈ (partition p l).2 := by
  by_cases p a <;> simp_all

+grind_pattern mem_partition => a ∈ (partition p l).1
+grind_pattern mem_partition => a ∈ (partition p l).2
+
 /-! ### dropLast

 `dropLast` is the specification for `Array.pop`, so theorems about `List.dropLast`
@@ -3098,7 +3116,7 @@ theorem dropLast_concat_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l +
    congr
    exact dropLast_concat_getLast (cons_ne_nil b l)

-@[simp] theorem map_dropLast {f : α → β} {l : List α} : l.dropLast.map f = (l.map f).dropLast := by
+@[simp, grind _=_] theorem map_dropLast {f : α → β} {l : List α} : l.dropLast.map f = (l.map f).dropLast := by
  induction l with
  | nil => rfl
  | cons x xs ih => cases xs <;> simp [ih]
@@ -3110,6 +3128,7 @@ theorem dropLast_concat_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l +
    rw [cons_append, dropLast, dropLast_append_of_ne_nil h, cons_append]
    simp [h]

+@[grind =]
 theorem dropLast_append {l₁ l₂ : List α} :
    (l₁ ++ l₂).dropLast = if l₂.isEmpty then l₁.dropLast else l₁ ++ l₂.dropLast := by
  split <;> simp_all
@@ -3117,9 +3136,9 @@ theorem dropLast_append {l₁ l₂ : List α} :
 theorem dropLast_append_cons : dropLast (l₁ ++ b :: l₂) = l₁ ++ dropLast (b :: l₂) := by
  simp

-@[simp] theorem dropLast_concat : dropLast (l₁ ++ [b]) = l₁ := by simp
+@[simp, grind =] theorem dropLast_concat : dropLast (l₁ ++ [b]) = l₁ := by simp

-@[simp] theorem dropLast_replicate {n : Nat} {a : α} : dropLast (replicate n a) = replicate (n - 1) a := by
+@[simp, grind =] theorem dropLast_replicate {n : Nat} {a : α} : dropLast (replicate n a) = replicate (n - 1) a := by
  match n with
  | 0 => simp
  | 1 => simp [replicate_succ]
@@ -3132,7 +3151,7 @@ theorem dropLast_append_cons : dropLast (l₁ ++ b :: l₂) = l₁ ++ dropLast (
    dropLast (a :: replicate n a) = replicate n a := by
  rw [← replicate_succ, dropLast_replicate, Nat.add_sub_cancel]

-@[simp] theorem tail_reverse {l : List α} : l.reverse.tail = l.dropLast.reverse := by
+@[simp, grind _=_] theorem tail_reverse {l : List α} : l.reverse.tail = l.dropLast.reverse := by
  apply ext_getElem
  · simp
  · intro i h₁ h₂
@@ -3372,6 +3391,7 @@ theorem replace_append_right [LawfulBEq α] {l₁ l₂ : List α} (h : ¬ a ∈
    (l₁ ++ l₂).replace a b = l₁ ++ l₂.replace a b := by
  simp [replace_append, h]

+@[grind _=_]
 theorem replace_take {l : List α} {i : Nat} :
    (l.take i).replace a b = (l.replace a b).take i := by
  induction l generalizing i with
@@ -3538,10 +3558,10 @@ end insert

 /-! ### `removeAll` -/

-@[simp] theorem removeAll_nil [BEq α] {xs : List α} : xs.removeAll [] = xs := by
+@[simp, grind =] theorem removeAll_nil [BEq α] {xs : List α} : xs.removeAll [] = xs := by
  simp [removeAll]

-theorem cons_removeAll [BEq α] {x : α} {xs ys : List α} :
+@[grind =] theorem cons_removeAll [BEq α] {x : α} {xs ys : List α} :
    (x :: xs).removeAll ys =
      if ys.contains x = false then
        x :: xs.removeAll ys
@@ -3549,6 +3569,7 @@ theorem cons_removeAll [BEq α] {x : α} {xs ys : List α} :
        xs.removeAll ys := by
  simp [removeAll, filter_cons]

+@[grind =]
 theorem removeAll_cons [BEq α] {xs : List α} {y : α} {ys : List α} :
    xs.removeAll (y :: ys) = (xs.filter fun x => !x == y).removeAll ys := by
  simp [removeAll, Bool.and_comm]
@@ -3568,7 +3589,7 @@ theorem removeAll_cons [BEq α] {xs : List α} {y : α} {ys : List α} :

 /-! ### `eraseDupsBy` and `eraseDups` -/

-@[simp] theorem eraseDupsBy_nil : ([] : List α).eraseDupsBy r = [] := rfl
+@[simp, grind =] theorem eraseDupsBy_nil : ([] : List α).eraseDupsBy r = [] := rfl

 private theorem eraseDupsBy_loop_cons {as bs : List α} {r : α → α → Bool} :
    eraseDupsBy.loop r as bs = bs.reverse ++ eraseDupsBy.loop r (as.filter fun a => !bs.any (r a)) [] := by
@@ -3588,17 +3609,19 @@ private theorem eraseDupsBy_loop_cons {as bs : List α} {r : α → α → Bool}
      simp
 termination_by as.length

+@[grind =]
 theorem eraseDupsBy_cons :
    (a :: as).eraseDupsBy r = a :: (as.filter fun b => r b a = false).eraseDupsBy r := by
  simp only [eraseDupsBy, eraseDupsBy.loop, any_nil]
  rw [eraseDupsBy_loop_cons]
  simp

-@[simp] theorem eraseDups_nil [BEq α] : ([] : List α).eraseDups = [] := rfl
-theorem eraseDups_cons [BEq α] {a : α} {as : List α} :
+@[simp, grind =] theorem eraseDups_nil [BEq α] : ([] : List α).eraseDups = [] := rfl
+@[grind =] theorem eraseDups_cons [BEq α] {a : α} {as : List α} :
    (a :: as).eraseDups = a :: (as.filter fun b => !b == a).eraseDups := by
  simp [eraseDups, eraseDupsBy_cons]

+@[grind =]
 theorem eraseDups_append [BEq α] [LawfulBEq α] {as bs : List α} :
    (as ++ bs).eraseDups = as.eraseDups ++ (bs.removeAll as).eraseDups := by
  match as with
--- a/src/Init/Data/List/MapIdx.lean
+++ b/src/Init/Data/List/MapIdx.lean
@@ -91,7 +91,7 @@ is valid.
  subst w
  rfl

-@[simp]
+@[simp, grind =]
 theorem mapFinIdx_nil {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx [] f = [] :=
  rfl

@@ -101,7 +101,7 @@ theorem mapFinIdx_nil {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx
  | nil => simpa using h
  | cons _ _ ih => simp [mapFinIdx.go, ih]

-@[simp] theorem length_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} :
+@[simp, grind =] theorem length_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} :
    (as.mapFinIdx f).length = as.length := by
  simp [mapFinIdx, length_mapFinIdx_go]

@@ -129,7 +129,7 @@ theorem getElem_mapFinIdx_go {as : List α} {f : (i : Nat) → α → (h : i < a
    · have h₃ : i - acc.size = (i - (acc.size + 1)) + 1 := by omega
      simp [h₃]

-@[simp] theorem getElem_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} {i : Nat} {h} :
+@[simp, grind =] theorem getElem_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} {i : Nat} {h} :
    (as.mapFinIdx f)[i] = f i (as[i]'(by simp at h; omega)) (by simp at h; omega) := by
  simp [mapFinIdx, getElem_mapFinIdx_go]

@@ -137,18 +137,19 @@ theorem mapFinIdx_eq_ofFn {as : List α} {f : (i : Nat) → α → (h : i < as.l
    as.mapFinIdx f = List.ofFn fun i : Fin as.length => f i as[i] i.2 := by
  apply ext_getElem <;> simp

-@[simp] theorem getElem?_mapFinIdx {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {i : Nat} :
+@[simp, grind =] theorem getElem?_mapFinIdx {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {i : Nat} :
    (l.mapFinIdx f)[i]? = l[i]?.pbind fun x m => some <| f i x (by simp [getElem?_eq_some_iff] at m; exact m.1) := by
  simp only [getElem?_def, length_mapFinIdx, getElem_mapFinIdx]
  split <;> simp

-@[simp]
+@[simp, grind =]
 theorem mapFinIdx_cons {l : List α} {a : α} {f : (i : Nat) → α → (h : i < l.length + 1) → β} :
    mapFinIdx (a :: l) f = f 0 a (by omega) :: mapFinIdx l (fun i a h => f (i + 1) a (by omega)) := by
  apply ext_getElem
  · simp
  · rintro (_|i) h₁ h₂ <;> simp

+@[grind =]
 theorem mapFinIdx_append {xs ys : List α} {f : (i : Nat) → α → (h : i < (xs ++ ys).length) → β} :
    (xs ++ ys).mapFinIdx f =
      xs.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
@@ -165,7 +166,7 @@ theorem mapFinIdx_append {xs ys : List α} {f : (i : Nat) → α → (h : i < (x
      congr
      omega

-@[simp] theorem mapFinIdx_concat {l : List α} {e : α} {f : (i : Nat) → α → (h : i < (l ++ [e]).length) → β}:
+@[simp, grind =] theorem mapFinIdx_concat {l : List α} {e : α} {f : (i : Nat) → α → (h : i < (l ++ [e]).length) → β}:
    (l ++ [e]).mapFinIdx f = l.mapFinIdx (fun i a h => f i a (by simp; omega)) ++ [f l.length e (by simp)] := by
  simp [mapFinIdx_append]

@@ -201,7 +202,7 @@ theorem exists_of_mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α →
  obtain ⟨h', rfl⟩ := h
  exact ⟨i, h', rfl⟩

-@[simp] theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
+@[simp, grind =] theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
    b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] h = b := by
  constructor
  · intro h
@@ -287,7 +288,7 @@ theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : (i : Nat) → α → (h
  rw [eq_comm, mapFinIdx_eq_iff]
  simp [Fin.forall_iff]

-@[simp] theorem mapFinIdx_mapFinIdx {l : List α}
+@[simp, grind =] theorem mapFinIdx_mapFinIdx {l : List α}
    {f : (i : Nat) → α → (h : i < l.length) → β}
    {g : (i : Nat) → β → (h : i < (l.mapFinIdx f).length) → γ} :
    (l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i a h => g i (f i a h) (by simpa)) := by
@@ -303,7 +304,7 @@ theorem mapFinIdx_eq_replicate_iff {l : List α} {f : (i : Nat) → α → (h :
  · rintro w b i h rfl
    exact w i h

-@[simp] theorem mapFinIdx_reverse {l : List α} {f : (i : Nat) → α → (h : i < l.reverse.length) → β} :
+@[simp, grind =] theorem mapFinIdx_reverse {l : List α} {f : (i : Nat) → α → (h : i < l.reverse.length) → β} :
    l.reverse.mapFinIdx f =
      (l.mapFinIdx (fun i a h => f (l.length - 1 - i) a (by simp; omega))).reverse := by
  simp [mapFinIdx_eq_iff]
@@ -313,7 +314,7 @@ theorem mapFinIdx_eq_replicate_iff {l : List α} {f : (i : Nat) → α → (h :

 /-! ### mapIdx -/

-@[simp]
+@[simp, grind =]
 theorem mapIdx_nil {f : Nat → α → β} : mapIdx f [] = [] :=
  rfl

@@ -333,7 +334,7 @@ theorem length_mapIdx_go : ∀ {l : List α} {acc : Array β},
    simp
    omega

-@[simp] theorem length_mapIdx {l : List α} : (l.mapIdx f).length = l.length := by
+@[simp, grind =] theorem length_mapIdx {l : List α} : (l.mapIdx f).length = l.length := by
  simp [mapIdx, length_mapIdx_go]

 theorem getElem?_mapIdx_go : ∀ {l : List α} {acc : Array β} {i : Nat},
@@ -356,11 +357,11 @@ theorem getElem?_mapIdx_go : ∀ {l : List α} {acc : Array β} {i : Nat},
    · have : i - acc.size = i - (acc.size + 1) + 1 := by omega
      simp_all

-@[simp] theorem getElem?_mapIdx {l : List α} {i : Nat} :
+@[simp, grind =] theorem getElem?_mapIdx {l : List α} {i : Nat} :
    (l.mapIdx f)[i]? = Option.map (f i) l[i]? := by
  simp [mapIdx, getElem?_mapIdx_go]

-@[simp] theorem getElem_mapIdx {l : List α} {f : Nat → α → β} {i : Nat} {h : i < (l.mapIdx f).length} :
+@[simp, grind =] theorem getElem_mapIdx {l : List α} {f : Nat → α → β} {i : Nat} {h : i < (l.mapIdx f).length} :
    (l.mapIdx f)[i] = f i (l[i]'(by simpa using h)) := by
  apply Option.some_inj.mp
  rw [← getElem?_eq_getElem, getElem?_mapIdx, getElem?_eq_getElem (by simpa using h)]
@@ -384,18 +385,19 @@ theorem mapIdx_eq_zipIdx_map {l : List α} {f : Nat → α → β} :
@[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-21")]
 abbrev mapIdx_eq_enum_map := @mapIdx_eq_zipIdx_map

-@[simp]
+@[simp, grind =]
 theorem mapIdx_cons {l : List α} {a : α} :
    mapIdx f (a :: l) = f 0 a :: mapIdx (fun i => f (i + 1)) l := by
  simp [mapIdx_eq_zipIdx_map, List.zipIdx_succ]

+@[grind =]
 theorem mapIdx_append {xs ys : List α} :
    (xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx fun i => f (i + xs.length) := by
  induction xs generalizing f with
  | nil => rfl
  | cons _ _ ih => simp [ih (f := fun i => f (i + 1)), Nat.add_assoc]

-@[simp] theorem mapIdx_concat {l : List α} {e : α} :
+@[simp, grind =] theorem mapIdx_concat {l : List α} {e : α} :
    mapIdx f (l ++ [e]) = mapIdx f l ++ [f l.length e] := by
  simp [mapIdx_append]

@@ -415,7 +417,7 @@ theorem exists_of_mem_mapIdx {b : β} {l : List α}
  rw [mapIdx_eq_mapFinIdx] at h
  simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h

-@[simp] theorem mem_mapIdx {b : β} {l : List α} :
+@[simp, grind =] theorem mem_mapIdx {b : β} {l : List α} :
    b ∈ mapIdx f l ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] = b := by
  constructor
  · intro h
@@ -470,7 +472,7 @@ theorem mapIdx_eq_mapIdx_iff {l : List α} :
    · intro i h₁ h₂
      simp [w]

-@[simp] theorem mapIdx_set {l : List α} {i : Nat} {a : α} :
+@[simp, grind =] theorem mapIdx_set {l : List α} {i : Nat} {a : α} :
    (l.set i a).mapIdx f = (l.mapIdx f).set i (f i a) := by
  simp only [mapIdx_eq_iff, getElem?_set, length_mapIdx, getElem?_mapIdx]
  intro i
@@ -478,16 +480,16 @@ theorem mapIdx_eq_mapIdx_iff {l : List α} :
  · split <;> simp_all
  · rfl

-@[simp] theorem head_mapIdx {l : List α} {f : Nat → α → β} {w : mapIdx f l ≠ []} :
+@[simp, grind =] theorem head_mapIdx {l : List α} {f : Nat → α → β} {w : mapIdx f l ≠ []} :
    (mapIdx f l).head w = f 0 (l.head (by simpa using w)) := by
  cases l with
  | nil => simp at w
  | cons _ _ => simp

-@[simp] theorem head?_mapIdx {l : List α} {f : Nat → α → β} : (mapIdx f l).head? = l.head?.map (f 0) := by
+@[simp, grind =] theorem head?_mapIdx {l : List α} {f : Nat → α → β} : (mapIdx f l).head? = l.head?.map (f 0) := by
  cases l <;> simp

-@[simp] theorem getLast_mapIdx {l : List α} {f : Nat → α → β} {h} :
+@[simp, grind =] theorem getLast_mapIdx {l : List α} {f : Nat → α → β} {h} :
    (mapIdx f l).getLast h = f (l.length - 1) (l.getLast (by simpa using h)) := by
  cases l with
  | nil => simp at h
@@ -498,13 +500,13 @@ theorem mapIdx_eq_mapIdx_iff {l : List α} :
    simp only [← mapIdx_cons, getElem_mapIdx]
    simp

-@[simp] theorem getLast?_mapIdx {l : List α} {f : Nat → α → β} :
+@[simp, grind =] theorem getLast?_mapIdx {l : List α} {f : Nat → α → β} :
    (mapIdx f l).getLast? = (getLast? l).map (f (l.length - 1)) := by
  cases l
  · simp
  · rw [getLast?_eq_getLast, getLast?_eq_getLast, getLast_mapIdx] <;> simp

-@[simp] theorem mapIdx_mapIdx {l : List α} {f : Nat → α → β} {g : Nat → β → γ} :
+@[simp, grind =] theorem mapIdx_mapIdx {l : List 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]

@@ -517,7 +519,7 @@ theorem mapIdx_eq_replicate_iff {l : List α} {f : Nat → α → β} {b : β} :
  · rintro w _ i h rfl
    exact w i h

-@[simp] theorem mapIdx_reverse {l : List α} {f : Nat → α → β} :
+@[simp, grind =] theorem mapIdx_reverse {l : List α} {f : Nat → α → β} :
    l.reverse.mapIdx f = (mapIdx (fun i => f (l.length - 1 - i)) l).reverse := by
  simp [mapIdx_eq_iff]
  intro i
--- a/src/Init/Data/List/Nat/Basic.lean
+++ b/src/Init/Data/List/Nat/Basic.lean
@@ -26,6 +26,7 @@ namespace List

 /-! ### dropLast -/

+@[grind _=_]
 theorem tail_dropLast {l : List α} : tail (dropLast l) = dropLast (tail l) := by
  ext1
  simp only [getElem?_tail, getElem?_dropLast, length_tail]
@@ -35,7 +36,7 @@ theorem tail_dropLast {l : List α} : tail (dropLast l) = dropLast (tail l) := b
  · omega
  · rfl

-@[simp] theorem dropLast_reverse {l : List α} : l.reverse.dropLast = l.tail.reverse := by
+@[simp, grind _=_] theorem dropLast_reverse {l : List α} : l.reverse.dropLast = l.tail.reverse := by
  apply ext_getElem
  · simp
  · intro i h₁ h₂
@@ -114,7 +115,7 @@ section intersperse

 variable {l : List α} {sep : α} {i : Nat}

-@[simp] theorem length_intersperse : (l.intersperse sep).length = 2 * l.length - 1 := by
+@[simp, grind =] theorem length_intersperse : (l.intersperse sep).length = 2 * l.length - 1 := by
  fun_induction intersperse <;> simp only [intersperse, length_cons, length_nil] at *
  rename_i h _
  have := length_pos_iff.mpr h
--- a/src/Init/Data/List/Nat/Count.lean
+++ b/src/Init/Data/List/Nat/Count.lean
@@ -16,6 +16,7 @@ namespace List

 open Nat

+@[grind =]
 theorem countP_set {p : α → Bool} {l : List α} {i : Nat} {a : α} (h : i < l.length) :
    (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
  induction l generalizing i with
@@ -29,10 +30,12 @@ theorem countP_set {p : α → Bool} {l : List α} {i : Nat} {a : α} (h : i < l
      have : (if p l[i] = true then 1 else 0) ≤ l.countP p := boole_getElem_le_countP (p := p) h
      omega

+@[grind =]
 theorem count_set [BEq α] {a b : α} {l : List α} {i : Nat} (h : i < l.length) :
    (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]

+@[grind =]
 theorem countP_replace [BEq α] [LawfulBEq α] {a b : α} {l : List α} {p : α → Bool} :
    (l.replace a b).countP p =
      if l.contains a then l.countP p + (if p b then 1 else 0) - (if p a then 1 else 0) else l.countP p := by
@@ -55,11 +58,31 @@ theorem countP_replace [BEq α] [LawfulBEq α] {a b : α} {l : List α} {p : α
          omega
      · omega

+@[grind =]
 theorem count_replace [BEq α] [LawfulBEq α] {a b c : α} {l : List α} :
    (l.replace a b).count c =
      if l.contains a then l.count c + (if b == c then 1 else 0) - (if a == c then 1 else 0) else l.count c := by
  simp [count_eq_countP, countP_replace]

+@[grind =] theorem count_insert [BEq α] [LawfulBEq α] {a b : α} {l : List α} :
+    count a (List.insert b l) = max (count a l) (if b == a then 1 else 0) := by
+  simp only [List.insert, contains_eq_mem, decide_eq_true_eq, beq_iff_eq]
+  split <;> rename_i h
+  · split <;> rename_i h'
+    · rw [Nat.max_def]
+      simp only [beq_iff_eq] at h'
+      split
+      · have := List.count_pos_iff.mpr (h' ▸ h)
+        omega
+      · rfl
+    · simp [h']
+  · rw [count_cons]
+    split <;> rename_i h'
+    · simp only [beq_iff_eq] at h'
+      rw [count_eq_zero.mpr (h' ▸ h)]
+      simp [h']
+    · simp
+
 /--
 The number of elements satisfying a predicate in a sublist is at least the number of elements satisfying the predicate in the list,
 minus the difference in the lengths.
@@ -98,6 +121,8 @@ theorem le_countP_tail {l} : countP p l - 1 ≤ countP p l.tail := by
  simp only [length_tail] at this
  omega

+grind_pattern le_countP_tail => countP p l.tail
+
 variable [BEq α]

 theorem Sublist.le_count (s : l₁ <+ l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
@@ -115,4 +140,6 @@ theorem IsInfix.le_count (s : l₁ <:+: l₂) (a : α) : count a l₂ - (l₂.le
 theorem le_count_tail {a : α} {l : List α} : count a l - 1 ≤ count a l.tail :=
  le_countP_tail

+grind_pattern le_count_tail => count a l.tail
+
 end List
--- a/src/Init/Data/List/Nat/Erase.lean
+++ b/src/Init/Data/List/Nat/Erase.lean
@@ -14,6 +14,7 @@ set_option linter.indexVariables true -- Enforce naming conventions for index va

 namespace List

+@[grind =]
 theorem getElem?_eraseIdx {l : List α} {i : Nat} {j : Nat} :
    (l.eraseIdx i)[j]? = if j < i then l[j]? else l[j + 1]? := by
  rw [eraseIdx_eq_take_drop_succ, getElem?_append]
@@ -49,6 +50,7 @@ theorem getElem?_eraseIdx_of_ge {l : List α} {i : Nat} {j : Nat} (h : i ≤ j)
  intro h'
  omega

+@[grind =]
 theorem getElem_eraseIdx {l : List α} {i : Nat} {j : Nat} (h : j < (l.eraseIdx i).length) :
    (l.eraseIdx i)[j] = if h' : j < i then
        l[j]'(by have := length_eraseIdx_le l i; omega)
@@ -123,6 +125,48 @@ theorem eraseIdx_set_gt {l : List α} {i : Nat} {j : Nat} {a : α} (h : i < j) :
      · have t : i ≠ n := by omega
        simp [t]

+@[grind =]
+theorem eraseIdx_set {xs : List α} {i : Nat} {a : α} {j : Nat} :
+    (xs.set i a).eraseIdx j =
+      if j < i then
+        (xs.eraseIdx j).set (i - 1) a
+      else if j = i then
+        xs.eraseIdx i
+      else
+        (xs.eraseIdx j).set i a := by
+  split <;> rename_i h'
+  · rw [eraseIdx_set_lt]
+    omega
+  · split <;> rename_i h''
+    · subst h''
+      rw [eraseIdx_set_eq]
+    · rw [eraseIdx_set_gt]
+      omega
+
+theorem set_eraseIdx_le {xs : List α} {i : Nat} {j : Nat} {a : α} (h : i ≤ j) :
+    (xs.eraseIdx i).set j a = (xs.set (j + 1) a).eraseIdx i := by
+  rw [eraseIdx_set_lt]
+  · simp
+  · omega
+
+theorem set_eraseIdx_gt {xs : List α} {i : Nat} {j : Nat} {a : α} (h : j < i) :
+    (xs.eraseIdx i).set j a = (xs.set j a).eraseIdx i := by
+  rw [eraseIdx_set_gt]
+  omega
+
+@[grind =]
+theorem set_eraseIdx {xs : List α} {i : Nat} {j : Nat} {a : α} :
+    (xs.eraseIdx i).set j a =
+      if i ≤ j then
+        (xs.set (j + 1) a).eraseIdx i
+      else
+        (xs.set j a).eraseIdx i := by
+  split <;> rename_i h'
+  · rw [set_eraseIdx_le]
+    omega
+  · rw [set_eraseIdx_gt]
+    omega
+
@[simp] theorem set_getElem_succ_eraseIdx_succ
    {l : List α} {i : Nat} (h : i + 1 < l.length) :
    (l.eraseIdx (i + 1)).set i l[i + 1] = l.eraseIdx i := by
@@ -143,7 +187,7 @@ theorem eraseIdx_set_gt {l : List α} {i : Nat} {j : Nat} {a : α} (h : i < j) :
      · have t : ¬ n < i := by omega
        simp [t]

-@[simp] theorem eraseIdx_length_sub_one {l : List α} :
+@[simp, grind =] theorem eraseIdx_length_sub_one {l : List α} :
    (l.eraseIdx (l.length - 1)) = l.dropLast := by
  apply ext_getElem
  · simp [length_eraseIdx]
--- a/src/Init/Data/List/Nat/InsertIdx.lean
+++ b/src/Init/Data/List/Nat/InsertIdx.lean
@@ -30,19 +30,20 @@ section InsertIdx

 variable {a : α}

-@[simp]
+@[simp, grind =]
 theorem insertIdx_zero {xs : List α} {x : α} : xs.insertIdx 0 x = x :: xs :=
  rfl

-@[simp]
+@[simp, grind =]
 theorem insertIdx_succ_nil {n : Nat} {a : α} : ([] : List α).insertIdx (n + 1) a = [] :=
  rfl

-@[simp]
+@[simp, grind =]
 theorem insertIdx_succ_cons {xs : List α} {hd x : α} {i : Nat} :
    (hd :: xs).insertIdx (i + 1) x = hd :: xs.insertIdx i x :=
  rfl

+@[grind =]
 theorem length_insertIdx : ∀ {i} {as : List α}, (as.insertIdx i a).length = if i ≤ as.length then as.length + 1 else as.length
  | 0, _ => by simp
  | n + 1, [] => by simp
@@ -56,14 +57,9 @@ theorem length_insertIdx_of_le_length (h : i ≤ length as) (a : α) : (as.inser
 theorem length_insertIdx_of_length_lt (h : length as < i) (a : α) : (as.insertIdx i a).length = as.length := by
  simp [length_insertIdx, h]

-@[simp]
-theorem eraseIdx_insertIdx {i : Nat} {l : List α} (a : α) : (l.insertIdx i a).eraseIdx i = l := by
-  rw [eraseIdx_eq_modifyTailIdx, insertIdx, modifyTailIdx_modifyTailIdx_self]
-  exact modifyTailIdx_id _ _
-
 theorem insertIdx_eraseIdx_of_ge :
-    ∀ {i m as},
-      i < length as → i ≤ m → (as.eraseIdx i).insertIdx m a = (as.insertIdx (m + 1) a).eraseIdx i
+    ∀ {i j as},
+      i < length as → i ≤ j → (as.eraseIdx i).insertIdx j a = (as.insertIdx (j + 1) a).eraseIdx i
  | 0, 0, [], has, _ => (Nat.lt_irrefl _ has).elim
  | 0, 0, _ :: as, _, _ => by simp [eraseIdx, insertIdx]
  | 0, _ + 1, _ :: _, _, _ => rfl
@@ -79,6 +75,15 @@ theorem insertIdx_eraseIdx_of_le :
    congrArg (cons a) <|
      insertIdx_eraseIdx_of_le (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)

+@[grind =]
+theorem insertIdx_eraseIdx (h : i < length as) :
+    (as.eraseIdx i).insertIdx j a =
+      if i ≤ j then (as.insertIdx (j + 1) a).eraseIdx i else (as.insertIdx j a).eraseIdx (i + 1) := by
+  split <;> rename_i h'
+  · rw [insertIdx_eraseIdx_of_ge h h']
+  · rw [insertIdx_eraseIdx_of_le h (by omega)]
+
+@[grind =]
 theorem insertIdx_comm (a b : α) :
    ∀ {i j : Nat} {l : List α} (_ : i ≤ j) (_ : j ≤ length l),
      (l.insertIdx i a).insertIdx (j + 1) b = (l.insertIdx j b).insertIdx i a
@@ -110,6 +115,11 @@ theorem insertIdx_of_length_lt {l : List α} {x : α} {i : Nat} (h : l.length <
    · simp only [Nat.succ_lt_succ_iff, length] at h
      simpa using ih h

+@[simp, grind =]
+theorem eraseIdx_insertIdx_self {i : Nat} {l : List α} (a : α) : (l.insertIdx i a).eraseIdx i = l := by
+  rw [eraseIdx_eq_modifyTailIdx, insertIdx, modifyTailIdx_modifyTailIdx_self]
+  exact modifyTailIdx_id _ _
+
@[simp]
 theorem insertIdx_length_self {l : List α} {x : α} : l.insertIdx l.length x = l ++ [x] := by
  induction l with
@@ -185,6 +195,7 @@ theorem getElem_insertIdx_of_gt {l : List α} {x : α} {i j : Nat} (hn : i < j)
@[deprecated getElem_insertIdx_of_gt (since := "2025-02-04")]
 abbrev getElem_insertIdx_of_ge := @getElem_insertIdx_of_gt

+@[grind =]
 theorem getElem_insertIdx {l : List α} {x : α} {i j : Nat} (h : j < (l.insertIdx i x).length) :
    (l.insertIdx i x)[j] =
      if h₁ : j < i then
@@ -201,6 +212,7 @@ theorem getElem_insertIdx {l : List α} {x : α} {i j : Nat} (h : j < (l.insertI
      rw [getElem_insertIdx_self h]
    · rw [getElem_insertIdx_of_gt (by omega)]

+@[grind =]
 theorem getElem?_insertIdx {l : List α} {x : α} {i j : Nat} :
    (l.insertIdx i x)[j]? =
      if j < i then
--- a/src/Init/Data/List/Nat/Modify.lean
+++ b/src/Init/Data/List/Nat/Modify.lean
@@ -17,7 +17,7 @@ namespace List

 /-! ### modifyHead -/

-@[simp] theorem length_modifyHead {f : α → α} {l : List α} : (l.modifyHead f).length = l.length := by
+@[simp, grind =] theorem length_modifyHead {f : α → α} {l : List α} : (l.modifyHead f).length = l.length := by
  cases l <;> simp [modifyHead]

 theorem modifyHead_eq_set [Inhabited α] (f : α → α) (l : List α) :
@@ -26,9 +26,10 @@ theorem modifyHead_eq_set [Inhabited α] (f : α → α) (l : List α) :
@[simp] theorem modifyHead_eq_nil_iff {f : α → α} {l : List α} :
    l.modifyHead f = [] ↔ l = [] := by cases l <;> simp [modifyHead]

-@[simp] theorem modifyHead_modifyHead {l : List α} {f g : α → α} :
+@[simp, grind =] theorem modifyHead_modifyHead {l : List α} {f g : α → α} :
    (l.modifyHead f).modifyHead g = l.modifyHead (g ∘ f) := by cases l <;> simp [modifyHead]

+@[grind =]
 theorem getElem_modifyHead {l : List α} {f : α → α} {i} (h : i < (l.modifyHead f).length) :
    (l.modifyHead f)[i] = if h' : i = 0 then f (l[0]'(by simp at h; omega)) else l[i]'(by simpa using h) := by
  cases l with
@@ -41,6 +42,7 @@ theorem getElem_modifyHead {l : List α} {f : α → α} {i} (h : i < (l.modifyH
@[simp] theorem getElem_modifyHead_succ {l : List α} {f : α → α} {n} (h : n + 1 < (l.modifyHead f).length) :
    (l.modifyHead f)[n + 1] = l[n + 1]'(by simpa using h) := by simp [getElem_modifyHead]

+@[grind =]
 theorem getElem?_modifyHead {l : List α} {f : α → α} {i} :
    (l.modifyHead f)[i]? = if i = 0 then l[i]?.map f else l[i]? := by
  cases l with
@@ -53,19 +55,19 @@ theorem getElem?_modifyHead {l : List α} {f : α → α} {i} :
@[simp] theorem getElem?_modifyHead_succ {l : List α} {f : α → α} {n} :
    (l.modifyHead f)[n + 1]? = l[n + 1]? := by simp [getElem?_modifyHead]

-@[simp] theorem head_modifyHead (f : α → α) (l : List α) (h) :
+@[simp, grind =] theorem head_modifyHead (f : α → α) (l : List α) (h) :
    (l.modifyHead f).head h = f (l.head (by simpa using h)) := by
  cases l with
  | nil => simp at h
  | cons hd tl => simp

-@[simp] theorem head?_modifyHead {l : List α} {f : α → α} :
+@[simp, grind =] theorem head?_modifyHead {l : List α} {f : α → α} :
    (l.modifyHead f).head? = l.head?.map f := by cases l <;> simp

-@[simp] theorem tail_modifyHead {f : α → α} {l : List α} :
+@[simp, grind =] theorem tail_modifyHead {f : α → α} {l : List α} :
    (l.modifyHead f).tail = l.tail := by cases l <;> simp

-@[simp] theorem take_modifyHead {f : α → α} {l : List α} {i} :
+@[simp, grind =] theorem take_modifyHead {f : α → α} {l : List α} {i} :
    (l.modifyHead f).take i = (l.take i).modifyHead f := by
  cases l <;> cases i <;> simp

@@ -73,6 +75,7 @@ theorem getElem?_modifyHead {l : List α} {f : α → α} {i} :
    (l.modifyHead f).drop i = l.drop i := by
  cases l <;> cases i <;> simp_all

+@[grind =]
 theorem eraseIdx_modifyHead_zero {f : α → α} {l : List α} :
    (l.modifyHead f).eraseIdx 0 = l.eraseIdx 0 := by simp

@@ -81,7 +84,7 @@ theorem eraseIdx_modifyHead_zero {f : α → α} {l : List α} :

@[simp] theorem modifyHead_id : modifyHead (id : α → α) = id := by funext l; cases l <;> simp

-@[simp] theorem modifyHead_dropLast {l : List α} {f : α → α} :
+@[simp, grind _=_] theorem modifyHead_dropLast {l : List α} {f : α → α} :
    l.dropLast.modifyHead f = (l.modifyHead f).dropLast := by
  rcases l with _|⟨a, l⟩
  · simp
@@ -99,7 +102,7 @@ theorem eraseIdx_eq_modifyTailIdx : ∀ i (l : List α), eraseIdx l i = l.modify
  | _+1, [] => rfl
  | _+1, _ :: _ => congrArg (cons _) (eraseIdx_eq_modifyTailIdx _ _)

-@[simp] theorem length_modifyTailIdx (f : List α → List α) (H : ∀ l, (f l).length = l.length) :
+@[simp, grind =] theorem length_modifyTailIdx (f : List α → List α) (H : ∀ l, (f l).length = l.length) :
    ∀ (l : List α) i, (l.modifyTailIdx i f).length = l.length
  | _, 0 => H _
  | [], _+1 => rfl
@@ -142,7 +145,7 @@ theorem modifyTailIdx_modifyTailIdx_self {f g : List α → List α} (i : Nat) (

 /-! ### modify -/

-@[simp] theorem modify_nil (f : α → α) (i) : [].modify i f = [] := by cases i <;> rfl
+@[simp, grind =] theorem modify_nil (f : α → α) (i) : [].modify i f = [] := by cases i <;> rfl

@[simp] theorem modify_zero_cons (f : α → α) (a : α) (l : List α) :
    (a :: l).modify 0 f = f a :: l := rfl
@@ -150,6 +153,15 @@ theorem modifyTailIdx_modifyTailIdx_self {f g : List α → List α} (i : Nat) (
@[simp] theorem modify_succ_cons (f : α → α) (a : α) (l : List α) (i) :
    (a :: l).modify (i + 1) f = a :: l.modify i f := rfl

+@[grind =]
+theorem modify_cons {f : α → α} {a : α} {l : List α} {i : Nat} :
+    (a :: l).modify i f =
+      if i = 0 then f a :: l else a :: l.modify (i - 1) f := by
+  split <;> rename_i h
+  · subst h
+    simp
+  · match i, h with | i + 1, _ => simp
+
 theorem modifyHead_eq_modify_zero (f : α → α) (l : List α) :
    l.modifyHead f = l.modify 0 f := by cases l <;> simp

@@ -200,6 +212,7 @@ theorem modify_eq_self {f : α → α} {i} {l : List α} (h : l.length ≤ i) :
    intro h
    omega

+@[grind =]
 theorem modify_modify_eq (f g : α → α) (i) (l : List α) :
    (l.modify i f).modify i g = l.modify i (g ∘ f) := by
  apply ext_getElem
@@ -245,7 +258,7 @@ theorem exists_of_modify (f : α → α) {i} {l : List α} (h : i < l.length) :
@[simp] theorem modify_id (i) (l : List α) : l.modify i id = l := by
  simp [modify]

-@[grind =]
+@[grind _=_]
 theorem take_modify (f : α → α) (i j) (l : List α) :
    (l.modify i f).take j = (l.take j).modify i f := by
  induction j generalizing l i with
--- a/src/Init/Data/List/Nat/Range.lean
+++ b/src/Init/Data/List/Nat/Range.lean
@@ -27,11 +27,12 @@ open Nat

 /-! ### range' -/

-@[simp] theorem mem_range'_1 : m ∈ range' s n ↔ s ≤ m ∧ m < s + n := by
+@[simp, grind =] theorem mem_range'_1 : m ∈ range' s n ↔ s ≤ m ∧ m < s + n := by
  simp [mem_range']; exact ⟨
    fun ⟨i, h, e⟩ => e ▸ ⟨Nat.le_add_right .., Nat.add_lt_add_left h _⟩,
    fun ⟨h₁, h₂⟩ => ⟨m - s, Nat.sub_lt_left_of_lt_add h₁ h₂, (Nat.add_sub_cancel' h₁).symm⟩⟩

+@[grind =]
 theorem getLast?_range' {n : Nat} : (range' s n).getLast? = if n = 0 then none else some (s + n - 1) := by
  induction n generalizing s with
  | zero => simp
@@ -43,7 +44,7 @@ theorem getLast?_range' {n : Nat} : (range' s n).getLast? = if n = 0 then none e
    · rw [if_neg h]
      simp

-@[simp] theorem getLast_range' {n : Nat} (h) : (range' s n).getLast h = s + n - 1 := by
+@[simp, grind =] theorem getLast_range' {n : Nat} (h) : (range' s n).getLast h = s + n - 1 := by
  cases n with
  | zero => simp at h
  | succ n => simp [getLast?_range', getLast_eq_iff_getLast?_eq_some]
@@ -158,6 +159,26 @@ theorem erase_range' :
      simp [p]
      omega

+@[simp, grind =]
+theorem count_range' {a s n step} (h : 0 < step := by simp) :
+    count a (range' s n step) = if ∃ i, i < n ∧ a = s + step * i then 1 else 0 := by
+  rw [(nodup_range' step h).count]
+  simp only [mem_range']
+
+@[simp, grind =]
+theorem count_range_1' {a s n} :
+    count a (range' s n) = if s ≤ a ∧ a < s + n then 1 else 0 := by
+  rw [count_range' (by simp)]
+  split <;> rename_i h
+  · obtain ⟨i, h, rfl⟩ := h
+    simp [h]
+  · simp at h
+    rw [if_neg]
+    simp only [not_and, Nat.not_lt]
+    intro w
+    specialize h (a - s)
+    omega
+
 /-! ### range -/

 theorem reverse_range' : ∀ {s n : Nat}, reverse (range' s n) = map (s + n - 1 - ·) (range n)
@@ -167,7 +188,7 @@ theorem reverse_range' : ∀ {s n : Nat}, reverse (range' s n) = map (s + n - 1
      show s + (n + 1) - 1 = s + n from rfl, map, map_map]
    simp [reverse_range', Nat.sub_right_comm, Nat.sub_sub]

-@[simp]
+@[simp, grind =]
 theorem mem_range {m n : Nat} : m ∈ range n ↔ m < n := by
  simp only [range_eq_range', mem_range'_1, Nat.zero_le, true_and, Nat.zero_add]

@@ -181,7 +202,7 @@ theorem pairwise_lt_range {n : Nat} : Pairwise (· < ·) (range n) := by
 theorem pairwise_le_range {n : Nat} : Pairwise (· ≤ ·) (range n) :=
  Pairwise.imp Nat.le_of_lt pairwise_lt_range

-@[simp] theorem take_range {i n : Nat} : take i (range n) = range (min i n) := by
+@[simp, grind =] theorem take_range {i n : Nat} : take i (range n) = range (min i n) := by
  apply List.ext_getElem
  · simp
  · simp +contextual [getElem_take, Nat.lt_min]
@@ -189,10 +210,11 @@ theorem pairwise_le_range {n : Nat} : Pairwise (· ≤ ·) (range n) :=
 theorem nodup_range {n : Nat} : Nodup (range n) := by
  simp +decide only [range_eq_range', nodup_range']

-@[simp] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
+@[simp, grind] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
    (range n).find? p = some i ↔ p i ∧ i ∈ range n ∧ ∀ j, j < i → !p j := by
  simp [range_eq_range']

+@[grind]
 theorem find?_range_eq_none {n : Nat} {p : Nat → Bool} :
    (range n).find? p = none ↔ ∀ i, i < n → !p i := by
  simp
@@ -200,6 +222,12 @@ theorem find?_range_eq_none {n : Nat} {p : Nat → Bool} :
 theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n - (i + 1)) := by
  simp [range_eq_range', erase_range']

+@[simp, grind =]
+theorem count_range {a n} :
+    count a (range n) = if a < n then 1 else 0 := by
+  rw [range_eq_range', count_range_1']
+  simp
+
 /-! ### iota -/

 section
@@ -348,15 +376,15 @@ end

 /-! ### zipIdx -/

-@[simp]
+@[simp, grind =]
 theorem zipIdx_singleton {x : α} {k : Nat} : zipIdx [x] k = [(x, k)] :=
  rfl

-@[simp] theorem head?_zipIdx {l : List α} {k : Nat} :
+@[simp, grind =] theorem head?_zipIdx {l : List α} {k : Nat} :
    (zipIdx l k).head? = l.head?.map fun a => (a, k) := by
  simp [head?_eq_getElem?]

-@[simp] theorem getLast?_zipIdx {l : List α} {k : Nat} :
+@[simp, grind =] theorem getLast?_zipIdx {l : List α} {k : Nat} :
    (zipIdx l k).getLast? = l.getLast?.map fun a => (a, k + l.length - 1) := by
  simp [getLast?_eq_getElem?]
  cases l <;> simp
@@ -379,6 +407,7 @@ to avoid the inequality and the subtraction. -/
 theorem mk_mem_zipIdx_iff_getElem? {i : Nat} {x : α} {l : List α} : (x, i) ∈ zipIdx l ↔ l[i]? = some x := by
  simp [mk_mem_zipIdx_iff_le_and_getElem?_sub]

+@[grind =]
 theorem mem_zipIdx_iff_le_and_getElem?_sub {x : α × Nat} {l : List α} {k : Nat} :
    x ∈ zipIdx l k ↔ k ≤ x.2 ∧ l[x.2 - k]? = some x.1 := by
  cases x
@@ -441,6 +470,7 @@ theorem zipIdx_map {l : List α} {k : Nat} {f : α → β} :
    rw [map_cons, zipIdx_cons', zipIdx_cons', map_cons, map_map, IH, map_map]
    rfl

+@[grind =]
 theorem zipIdx_append {xs ys : List α} {k : Nat} :
    zipIdx (xs ++ ys) k = zipIdx xs k ++ zipIdx ys (k + xs.length) := by
  induction xs generalizing ys k with
--- a/src/Init/Data/List/Nat/Sublist.lean
+++ b/src/Init/Data/List/Nat/Sublist.lean
@@ -118,6 +118,7 @@ theorem suffix_iff_eq_append : l₁ <:+ l₂ ↔ take (length l₂ - length l₁
  ⟨by rintro ⟨r, rfl⟩; simp only [length_append, Nat.add_sub_cancel_right, take_left], fun e =>
    ⟨_, e⟩⟩

+@[grind =]
 theorem prefix_take_iff {xs ys : List α} {i : Nat} : xs <+: ys.take i ↔ xs <+: ys ∧ xs.length ≤ i := by
  constructor
  · intro h
--- a/src/Init/Data/List/Nat/TakeDrop.lean
+++ b/src/Init/Data/List/Nat/TakeDrop.lean
@@ -99,6 +99,7 @@ theorem getLast_take {l : List α} (h : l.take i ≠ []) :
  · rw [getElem?_eq_none (by omega), getLast_eq_getElem]
    simp

+@[grind =]
 theorem take_take : ∀ {i j} {l : List α}, take i (take j l) = take (min i j) l
  | n, 0, l => by rw [Nat.min_zero, take_zero, take_nil]
  | 0, m, l => by rw [Nat.zero_min, take_zero, take_zero]
@@ -117,19 +118,19 @@ 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, grind =] theorem take_replicate {a : α} : ∀ {i n : Nat}, take i (replicate n a) = replicate (min i n) 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, grind =] theorem drop_replicate {a : α} : ∀ {i n : Nat}, drop i (replicate n a) = replicate (n - i) a
  | n, 0 => by simp
  | 0, m => by simp
  | succ n, succ m => by simp [replicate_succ, succ_sub_succ, drop_replicate]

 /-- Taking the first `i` elements in `l₁ ++ l₂` is the same as appending the first `i` elements
 of `l₁` to the first `n - l₁.length` elements of `l₂`. -/
-theorem take_append_eq_append_take {l₁ l₂ : List α} {i : Nat} :
+theorem take_append {l₁ l₂ : List α} {i : Nat} :
    take i (l₁ ++ l₂) = take i l₁ ++ take (i - l₁.length) l₂ := by
  induction l₁ generalizing i
  · simp
@@ -140,15 +141,18 @@ theorem take_append_eq_append_take {l₁ l₂ : List α} {i : Nat} :
      congr 1
      omega

+@[deprecated take_append (since := "2025-06-16")]
+abbrev take_append_eq_append_take := @take_append
+
 theorem take_append_of_le_length {l₁ l₂ : List α} {i : Nat} (h : i ≤ l₁.length) :
    (l₁ ++ l₂).take i = l₁.take i := by
-  simp [take_append_eq_append_take, Nat.sub_eq_zero_of_le h]
+  simp [take_append, Nat.sub_eq_zero_of_le h]

 /-- Taking the first `l₁.length + i` elements in `l₁ ++ l₂` is the same as appending the first
 `i` elements of `l₂` to `l₁`. -/
-theorem take_append {l₁ l₂ : List α} (i : Nat) :
+theorem take_length_add_append {l₁ l₂ : List α} (i : Nat) :
    take (l₁.length + i) (l₁ ++ l₂) = l₁ ++ take i l₂ := by
-  rw [take_append_eq_append_take, take_of_length_le (Nat.le_add_right _ _), Nat.add_sub_cancel_left]
+  rw [take_append, take_of_length_le (Nat.le_add_right _ _), Nat.add_sub_cancel_left]

@[simp]
 theorem take_eq_take_iff :
@@ -162,11 +166,12 @@ theorem take_eq_take_iff :
@[deprecated take_eq_take_iff (since := "2025-02-16")]
 abbrev take_eq_take := @take_eq_take_iff

+@[grind =]
 theorem take_add {l : List α} {i j : Nat} : l.take (i + j) = l.take i ++ (l.drop i).take j := by
  suffices take (i + j) (take i l ++ drop i l) = take i l ++ take j (drop i l) by
    rw [take_append_drop] at this
    assumption
-  rw [take_append_eq_append_take, take_of_length_le, append_right_inj]
+  rw [take_append, take_of_length_le, append_right_inj]
  · simp only [take_eq_take_iff, length_take, length_drop]
    omega
  apply Nat.le_trans (m := i)
@@ -236,7 +241,7 @@ dropping the first `i` elements. Version designed to rewrite from the small list
      exact Nat.add_lt_of_lt_sub (length_drop ▸ h)) := by
  rw [getElem_drop']

-@[simp]
+@[simp, grind =]
 theorem getElem?_drop {xs : List α} {i j : Nat} : (xs.drop i)[j]? = xs[i + j]? := by
  ext
  simp only [getElem?_eq_some_iff, getElem_drop]
@@ -285,7 +290,7 @@ theorem getLast?_drop {l : List α} : (l.drop i).getLast? = if l.length ≤ i th
    congr
    omega

-@[simp] theorem getLast_drop {l : List α} (h : l.drop i ≠ []) :
+@[simp, grind =] theorem getLast_drop {l : List α} (h : l.drop i ≠ []) :
    (l.drop i).getLast h = l.getLast (ne_nil_of_length_pos (by simp at h; omega)) := by
  simp only [ne_eq, drop_eq_nil_iff] at h
  apply Option.some_inj.1
@@ -306,7 +311,8 @@ theorem drop_length_cons {l : List α} (h : l ≠ []) (a : α) :

 /-- Dropping the elements up to `i` in `l₁ ++ l₂` is the same as dropping the elements up to `i`
 in `l₁`, dropping the elements up to `i - l₁.length` in `l₂`, and appending them. -/
-theorem drop_append_eq_append_drop {l₁ l₂ : List α} {i : Nat} :
+@[grind =]
+theorem drop_append {l₁ l₂ : List α} {i : Nat} :
    drop i (l₁ ++ l₂) = drop i l₁ ++ drop (i - l₁.length) l₂ := by
  induction l₁ generalizing i
  · simp
@@ -316,15 +322,18 @@ theorem drop_append_eq_append_drop {l₁ l₂ : List α} {i : Nat} :
      congr 1
      omega

+@[deprecated drop_append (since := "2025-06-16")]
+abbrev drop_append_eq_append_drop := @drop_append
+
 theorem drop_append_of_le_length {l₁ l₂ : List α} {i : Nat} (h : i ≤ l₁.length) :
    (l₁ ++ l₂).drop i = l₁.drop i ++ l₂ := by
-  simp [drop_append_eq_append_drop, Nat.sub_eq_zero_of_le h]
+  simp [drop_append, Nat.sub_eq_zero_of_le h]

 /-- Dropping the elements up to `l₁.length + i` in `l₁ + l₂` is the same as dropping the elements
 up to `i` in `l₂`. -/
@[simp]
-theorem drop_append {l₁ l₂ : List α} (i : Nat) : drop (l₁.length + i) (l₁ ++ l₂) = drop i l₂ := by
-  rw [drop_append_eq_append_drop, drop_eq_nil_of_le] <;>
+theorem drop_length_add_append {l₁ l₂ : List α} (i : Nat) : drop (l₁.length + i) (l₁ ++ l₂) = drop i l₂ := by
+  rw [drop_append, drop_eq_nil_of_le] <;>
    simp [Nat.add_sub_cancel_left, Nat.le_add_right]

 theorem set_eq_take_append_cons_drop {l : List α} {i : Nat} {a : α} :
@@ -458,7 +467,7 @@ theorem false_of_mem_take_findIdx {xs : List α} {p : α → Bool} (h : x ∈ xs
  obtain ⟨i, h, rfl⟩ := h
  exact not_of_lt_findIdx (by omega)

-@[simp] theorem findIdx_take {xs : List α} {i : Nat} {p : α → Bool} :
+@[simp, grind =] theorem findIdx_take {xs : List α} {i : Nat} {p : α → Bool} :
    (xs.take i).findIdx p = min i (xs.findIdx p) := by
  induction xs generalizing i with
  | nil => simp
@@ -470,7 +479,7 @@ theorem false_of_mem_take_findIdx {xs : List α} {p : α → Bool} (h : x ∈ xs
      · simp
      · rw [Nat.add_min_add_right]

-@[simp] theorem min_findIdx_findIdx {xs : List α} {p q : α → Bool} :
+@[simp, grind =] theorem min_findIdx_findIdx {xs : List α} {p q : α → Bool} :
    min (xs.findIdx p) (xs.findIdx q) = xs.findIdx (fun a => p a || q a) := by
  induction xs with
  | nil => simp
@@ -512,7 +521,7 @@ theorem dropWhile_eq_drop_findIdx_not {xs : List α} {p : α → Bool} :

 /-! ### rotateLeft -/

-@[simp] theorem rotateLeft_replicate {n} {a : α} : rotateLeft (replicate m a) n = replicate m a := by
+@[simp, grind =] theorem rotateLeft_replicate {n} {a : α} : rotateLeft (replicate m a) n = replicate m a := by
  cases n with
  | zero => simp
  | succ n =>
@@ -525,7 +534,7 @@ theorem dropWhile_eq_drop_findIdx_not {xs : List α} {p : α → Bool} :

 /-! ### rotateRight -/

-@[simp] theorem rotateRight_replicate {n} {a : α} : rotateRight (replicate m a) n = replicate m a := by
+@[simp, grind =] theorem rotateRight_replicate {n} {a : α} : rotateRight (replicate m a) n = replicate m a := by
  cases n with
  | zero => simp
  | succ n =>
@@ -538,7 +547,7 @@ theorem dropWhile_eq_drop_findIdx_not {xs : List α} {p : α → Bool} :

 /-! ### zipWith -/

-@[simp] theorem length_zipWith {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
+@[simp, grind =] theorem length_zipWith {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
    length (zipWith f l₁ l₂) = min (length l₁) (length l₂) := by
  induction l₁ generalizing l₂ <;> cases l₂ <;>
    simp_all [succ_min_succ, Nat.zero_min, Nat.min_zero]
@@ -549,7 +558,7 @@ theorem lt_length_left_of_zipWith {f : α → β → γ} {i : Nat} {l : List α}
 theorem lt_length_right_of_zipWith {f : α → β → γ} {i : Nat} {l : List α} {l' : List β}
    (h : i < (zipWith f l l').length) : i < l'.length := by rw [length_zipWith] at h; omega

-@[simp]
+@[simp, grind =]
 theorem getElem_zipWith {f : α → β → γ} {l : List α} {l' : List β}
    {i : Nat} {h : i < (zipWith f l l').length} :
    (zipWith f l l')[i] =
@@ -566,6 +575,7 @@ theorem zipWith_eq_zipWith_take_min : ∀ {l₁ : List α} {l₂ : List β},
  | _, [] => by simp
  | a :: l₁, b :: l₂ => by simp [succ_min_succ, zipWith_eq_zipWith_take_min (l₁ := l₁) (l₂ := l₂)]

+@[grind =]
 theorem reverse_zipWith (h : l.length = l'.length) :
    (zipWith f l l').reverse = zipWith f l.reverse l'.reverse := by
  induction l generalizing l' with
@@ -578,14 +588,14 @@ theorem reverse_zipWith (h : l.length = l'.length) :
      have : tl.reverse.length = tl'.reverse.length := by simp [h]
      simp [hl h, zipWith_append this]

-@[simp] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
+@[simp, grind =] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
    zipWith f (replicate m a) (replicate n b) = replicate (min m n) (f a b) := by
  rw [zipWith_eq_zipWith_take_min]
  simp

 /-! ### zip -/

-@[simp] theorem length_zip {l₁ : List α} {l₂ : List β} :
+@[simp, grind =] theorem length_zip {l₁ : List α} {l₂ : List β} :
    length (zip l₁ l₂) = min (length l₁) (length l₂) := by
  simp [zip]

@@ -597,7 +607,7 @@ theorem lt_length_right_of_zip {i : Nat} {l : List α} {l' : List β} (h : i < (
    i < l'.length :=
  lt_length_right_of_zipWith h

-@[simp]
+@[simp, grind =]
 theorem getElem_zip {l : List α} {l' : List β} {i : Nat} {h : i < (zip l l').length} :
    (zip l l')[i] =
      (l[i]'(lt_length_left_of_zip h), l'[i]'(lt_length_right_of_zip h)) :=
@@ -609,7 +619,7 @@ theorem zip_eq_zip_take_min : ∀ {l₁ : List α} {l₂ : List β},
  | _, [] => by simp
  | a :: l₁, b :: l₂ => by simp [succ_min_succ, zip_eq_zip_take_min (l₁ := l₁) (l₂ := l₂)]

-@[simp] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
+@[simp, grind =] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
    zip (replicate m a) (replicate n b) = replicate (min m n) (a, b) := by
  rw [zip_eq_zip_take_min]
  simp
--- a/src/Init/Data/List/Notation.lean
+++ b/src/Init/Data/List/Notation.lean
@@ -6,7 +6,7 @@ Author: Leonardo de Moura
 module

 prelude
-import Init.Data.Nat.Div.Basic
+meta import Init.Data.Nat.Div.Basic

 /-!
 # Notation for `List` literals.
--- a/src/Init/Data/List/OfFn.lean
+++ b/src/Init/Data/List/OfFn.lean
@@ -34,14 +34,14 @@ to each potential index in order, starting at `0`.
 def ofFnM {n} [Monad m] (f : Fin n → m α) : m (List α) :=
  List.reverse <$> Fin.foldlM n (fun xs i => (· :: xs) <$> f i) []

-@[simp]
+@[simp, grind =]
 theorem length_ofFn {f : Fin n → α} : (ofFn f).length = n := by
  simp only [ofFn]
  induction n with
  | zero => simp
  | succ n ih => simp [Fin.foldr_succ, ih]

-@[simp]
+@[simp, grind =]
 protected theorem getElem_ofFn {f : Fin n → α} (h : i < (ofFn f).length) :
    (ofFn f)[i] = f ⟨i, by simp_all⟩ := by
  simp only [ofFn]
@@ -55,7 +55,7 @@ protected theorem getElem_ofFn {f : Fin n → α} (h : i < (ofFn f).length) :
      apply ih
      simp_all

-@[simp]
+@[simp, grind =]
 protected theorem getElem?_ofFn {f : Fin n → α} :
    (ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none :=
  if h : i < (ofFn f).length
@@ -67,7 +67,7 @@ protected theorem getElem?_ofFn {f : Fin n → α} :
    simpa using h

 /-- `ofFn` on an empty domain is the empty list. -/
-@[simp]
+@[simp, grind =]
 theorem ofFn_zero {f : Fin 0 → α} : ofFn f = [] := by
  rw [ofFn, Fin.foldr_zero]

@@ -98,7 +98,7 @@ theorem ofFn_add {n m} {f : Fin (n + m) → α} :
 theorem ofFn_eq_nil_iff {f : Fin n → α} : ofFn f = [] ↔ n = 0 := by
  cases n <;> simp only [ofFn_zero, ofFn_succ, eq_self_iff_true, Nat.succ_ne_zero, reduceCtorEq]

-@[simp 500]
+@[simp 500, grind =]
 theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
  constructor
  · intro w
@@ -107,17 +107,17 @@ theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i =
  · rintro ⟨i, rfl⟩
    apply mem_of_getElem (i := i) <;> simp

-theorem head_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
+@[grind =] theorem head_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
    (ofFn f).head h = f ⟨0, Nat.pos_of_ne_zero (mt ofFn_eq_nil_iff.2 h)⟩ := by
  rw [← getElem_zero (length_ofFn ▸ Nat.pos_of_ne_zero (mt ofFn_eq_nil_iff.2 h)),
    List.getElem_ofFn]

-theorem getLast_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
+@[grind =]theorem getLast_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
    (ofFn f).getLast h = f ⟨n - 1, Nat.sub_one_lt (mt ofFn_eq_nil_iff.2 h)⟩ := by
  simp [getLast_eq_getElem, length_ofFn, List.getElem_ofFn]

 /-- `ofFnM` on an empty domain is the empty list. -/
-@[simp]
+@[simp, grind =]
 theorem ofFnM_zero [Monad m] [LawfulMonad m] {f : Fin 0 → m α} : ofFnM f = pure [] := by
  simp [ofFnM]

--- a/src/Init/Data/List/Pairwise.lean
+++ b/src/Init/Data/List/Pairwise.lean
@@ -159,7 +159,7 @@ theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y →

@[grind =] theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {a : α} {l₁ l₂ : List α} :
    Pairwise R (l₁ ++ a :: l₂) ↔ Pairwise R (a :: (l₁ ++ l₂)) := by
-  show Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
+  change Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
  rw [← append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l₁), pairwise_append_comm s]
  simp only [mem_append, or_comm]

@@ -279,7 +279,11 @@ theorem nodup_nil : @Nodup α [] :=
 theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by
  simp only [Nodup, pairwise_cons, forall_mem_ne]

-@[grind →] theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
+@[grind =] theorem nodup_append {l₁ l₂ : List α} :
+    (l₁ ++ l₂).Nodup ↔ l₁.Nodup ∧ l₂.Nodup ∧ ∀ a ∈ l₁, ∀ b ∈ l₂, a ≠ b :=
+  pairwise_append
+
+theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
  Pairwise.sublist

 grind_pattern Nodup.sublist => l₁ <+ l₂, Nodup l₁
@@ -312,4 +316,48 @@ theorem getElem?_inj {xs : List α}
@[simp, grind =] theorem nodup_replicate {n : Nat} {a : α} :
    (replicate n a).Nodup ↔ n ≤ 1 := by simp [Nodup]

+theorem Nodup.count [BEq α] [LawfulBEq α] {a : α} {l : List α} (h : Nodup l) : count a l = if a ∈ l then 1 else 0 := by
+  split <;> rename_i h'
+  · obtain ⟨s, t, rfl⟩ := List.append_of_mem h'
+    rw [nodup_append] at h
+    simp_all
+    rw [count_eq_zero.mpr ?_, count_eq_zero.mpr ?_]
+    · exact h.2.1.1
+    · intro w
+      simpa using h.2.2 _ w
+  · rw [count_eq_zero_of_not_mem h']
+
+grind_pattern Nodup.count => count a l, Nodup l
+
+@[grind =]
+theorem nodup_iff_count [BEq α] [LawfulBEq α] {l : List α} : l.Nodup ↔ ∀ a, count a l ≤ 1 := by
+  induction l with
+  | nil => simp
+  | cons x l ih =>
+    constructor
+    · intro h a
+      simp at h
+      rw [count_cons]
+      split <;> rename_i h'
+      · simp at h'
+        rw [count_eq_zero.mpr ?_]
+        · exact Nat.le_refl _
+        · exact h' ▸ h.1
+      · simp at h'
+        refine ih.mp h.2 a
+    · intro h
+      simp only [count_cons] at h
+      simp only [nodup_cons]
+      constructor
+      · intro w
+        specialize h x
+        simp at h
+        have := count_pos_iff.mpr w
+        replace h := le_of_lt_succ h
+        apply Nat.lt_irrefl _ (Nat.lt_of_lt_of_le this h)
+      · rw [ih]
+        intro a
+        specialize h a
+        exact le_of_add_right_le h
+
 end List
--- a/src/Init/Data/List/Perm.lean
+++ b/src/Init/Data/List/Perm.lean
@@ -23,8 +23,7 @@ The notation `~` is used for permutation equivalence.
 -/

 set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
-- TODO: restore after an update-stage0
-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.

 open Nat

@@ -90,6 +89,9 @@ theorem Perm.mem_iff {a : α} {l₁ l₂ : List α} (p : l₁ ~ l₂) : a ∈ l
  | swap => simp only [mem_cons, or_left_comm]
  | trans _ _ ih₁ ih₂ => simp only [ih₁, ih₂]

+grind_pattern Perm.mem_iff => l₁ ~ l₂, a ∈ l₁
+grind_pattern Perm.mem_iff => l₁ ~ l₂, a ∈ l₂
+
 theorem Perm.subset {l₁ l₂ : List α} (p : l₁ ~ l₂) : l₁ ⊆ l₂ := fun _ => p.mem_iff.mp

 theorem Perm.append_right {l₁ l₂ : List α} (t₁ : List α) (p : l₁ ~ l₂) : l₁ ++ t₁ ~ l₂ ++ t₁ := by
@@ -106,9 +108,15 @@ theorem Perm.append_left {t₁ t₂ : List α} : ∀ l : List α, t₁ ~ t₂
 theorem Perm.append {l₁ l₂ t₁ t₂ : List α} (p₁ : l₁ ~ l₂) (p₂ : t₁ ~ t₂) : l₁ ++ t₁ ~ l₂ ++ t₂ :=
  (p₁.append_right t₁).trans (p₂.append_left l₂)

+grind_pattern Perm.append => l₁ ~ l₂, t₁ ~ t₂, l₁ ++ t₁
+grind_pattern Perm.append => l₁ ~ l₂, t₁ ~ t₂, l₂ ++ t₂
+
 theorem Perm.append_cons (a : α) {l₁ l₂ r₁ r₂ : List α} (p₁ : l₁ ~ l₂) (p₂ : r₁ ~ r₂) :
    l₁ ++ a :: r₁ ~ l₂ ++ a :: r₂ := p₁.append (p₂.cons a)

+grind_pattern Perm.append_cons => l₁ ~ l₂, r₁ ~ r₂, l₁ ++ a :: r₁
+grind_pattern Perm.append_cons => l₁ ~ l₂, r₁ ~ r₂, l₂ ++ a :: r₂
+
@[simp] theorem perm_middle {a : α} : ∀ {l₁ l₂ : List α}, l₁ ++ a :: l₂ ~ a :: (l₁ ++ l₂)
  | [], _ => .refl _
  | b :: _, _ => (Perm.cons _ perm_middle).trans (swap a b _)
@@ -194,9 +202,15 @@ theorem Perm.filterMap (f : α → Option β) {l₁ l₂ : List α} (p : l₁ ~
  | swap x y l₂ => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap_cons, swap]
  | trans _p₁ _p₂ IH₁ IH₂ => exact IH₁.trans IH₂

+grind_pattern Perm.filterMap => l₁ ~ l₂, filterMap f l₁
+grind_pattern Perm.filterMap => l₁ ~ l₂, filterMap f l₂
+
 theorem Perm.map (f : α → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) : map f l₁ ~ map f l₂ :=
  filterMap_eq_map ▸ p.filterMap _

+grind_pattern Perm.map => l₁ ~ l₂, map f l₁
+grind_pattern Perm.map => l₁ ~ l₂, map f l₂
+
 theorem Perm.pmap {p : α → Prop} (f : ∀ a, p a → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) {H₁ H₂} :
    pmap f l₁ H₁ ~ pmap f l₂ H₂ := by
  induction p with
@@ -205,12 +219,18 @@ theorem Perm.pmap {p : α → Prop} (f : ∀ a, p a → β) {l₁ l₂ : List α
  | swap x y => simp [swap]
  | trans _p₁ p₂ IH₁ IH₂ => exact IH₁.trans (IH₂ (H₁ := fun a m => H₂ a (p₂.subset m)))

+grind_pattern Perm.pmap => l₁ ~ l₂, pmap f l₁ H₁
+grind_pattern Perm.pmap => l₁ ~ l₂, pmap f l₂ H₂
+
 theorem Perm.unattach {α : Type u} {p : α → Prop} {l₁ l₂ : List { x // p x }} (h : l₁ ~ l₂) :
    l₁.unattach.Perm l₂.unattach := h.map _

 theorem Perm.filter (p : α → Bool) {l₁ l₂ : List α} (s : l₁ ~ l₂) :
    filter p l₁ ~ filter p l₂ := by rw [← filterMap_eq_filter]; apply s.filterMap

+grind_pattern Perm.filter => l₁ ~ l₂, filter p l₁
+grind_pattern Perm.filter => l₁ ~ l₂, filter p l₂
+
 theorem filter_append_perm (p : α → Bool) (l : List α) :
    filter p l ++ filter (fun x => !p x) l ~ l := by
  induction l with
@@ -388,12 +408,16 @@ theorem Perm.erase (a : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) : l₁.erase
    have h₂ : a ∉ l₂ := mt p.mem_iff.2 h₁
    rw [erase_of_not_mem h₁, erase_of_not_mem h₂]; exact p

+grind_pattern Perm.erase => l₁ ~ l₂, l₁.erase a
+grind_pattern Perm.erase => l₁ ~ l₂, l₂.erase a
+
 theorem cons_perm_iff_perm_erase {a : α} {l₁ l₂ : List α} :
    a :: l₁ ~ l₂ ↔ a ∈ l₂ ∧ l₁ ~ l₂.erase a := by
  refine ⟨fun h => ?_, fun ⟨m, h⟩ => (h.cons a).trans (perm_cons_erase m).symm⟩
  have : a ∈ l₂ := h.subset mem_cons_self
  exact ⟨this, (h.trans <| perm_cons_erase this).cons_inv⟩

+@[grind =]
 theorem perm_iff_count {l₁ l₂ : List α} : l₁ ~ l₂ ↔ ∀ a, count a l₁ = count a l₂ := by
  refine ⟨Perm.count_eq, fun H => ?_⟩
  induction l₁ generalizing l₂ with
@@ -410,6 +434,12 @@ theorem perm_iff_count {l₁ l₂ : List α} : l₁ ~ l₂ ↔ ∀ a, count a l
    rw [(perm_cons_erase this).count_eq] at H
    by_cases h : b = a <;> simpa [h, count_cons, Nat.succ_inj] using H

+theorem Perm.count (h : l₁ ~ l₂) (a : α) : count a l₁ = count a l₂ := by
+  rw [perm_iff_count.mp h]
+
+grind_pattern Perm.count => l₁ ~ l₂, count a l₁
+grind_pattern Perm.count => l₁ ~ l₂, count a l₂
+
 theorem isPerm_iff : ∀ {l₁ l₂ : List α}, l₁.isPerm l₂ ↔ l₁ ~ l₂
  | [], [] => by simp [isPerm, isEmpty]
  | [], _ :: _ => by simp [isPerm, isEmpty, Perm.nil_eq]
@@ -425,6 +455,9 @@ protected theorem Perm.insert (a : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) :
    have := p.cons a
    simpa [h, mt p.mem_iff.2 h] using this

+grind_pattern Perm.insert => l₁ ~ l₂, l₁.insert a
+grind_pattern Perm.insert => l₁ ~ l₂, l₂.insert a
+
 theorem perm_insert_swap (x y : α) (l : List α) :
    List.insert x (List.insert y l) ~ List.insert y (List.insert x l) := by
  by_cases xl : x ∈ l <;> by_cases yl : y ∈ l <;> simp [xl, yl]
@@ -491,6 +524,9 @@ theorem Perm.nodup {l l' : List α} (hl : l ~ l') (hR : l.Nodup) : l'.Nodup := h
 theorem Perm.nodup_iff {l₁ l₂ : List α} : l₁ ~ l₂ → (Nodup l₁ ↔ Nodup l₂) :=
  Perm.pairwise_iff <| @Ne.symm α

+grind_pattern Perm.nodup_iff => l₁ ~ l₂, Nodup l₁
+grind_pattern Perm.nodup_iff => l₁ ~ l₂, Nodup l₂
+
 theorem Perm.flatten {l₁ l₂ : List (List α)} (h : l₁ ~ l₂) : l₁.flatten ~ l₂.flatten := by
  induction h with
  | nil => rfl
@@ -541,20 +577,30 @@ theorem perm_insertIdx {α} (x : α) (l : List α) {i} (h : i ≤ l.length) :

 namespace Perm

-theorem take {l₁ l₂ : List α} (h : l₁ ~ l₂) {n : Nat} (w : l₁.drop n ~ l₂.drop n) :
-    l₁.take n ~ l₂.take n := by
+theorem take {l₁ l₂ : List α} (h : l₁ ~ l₂) {i : Nat} (w : l₁.drop i ~ l₂.drop i) :
+    l₁.take i ~ l₂.take i := by
  classical
  rw [perm_iff_count] at h w ⊢
-  rw [← take_append_drop n l₁, ← take_append_drop n l₂] at h
+  rw [← take_append_drop i l₁, ← take_append_drop i l₂] at h
  simpa only [count_append, w, Nat.add_right_cancel_iff] using h

-theorem drop {l₁ l₂ : List α} (h : l₁ ~ l₂) {n : Nat} (w : l₁.take n ~ l₂.take n) :
-    l₁.drop n ~ l₂.drop n := by
+theorem drop {l₁ l₂ : List α} (h : l₁ ~ l₂) {i : Nat} (w : l₁.take i ~ l₂.take i) :
+    l₁.drop i ~ l₂.drop i := by
  classical
  rw [perm_iff_count] at h w ⊢
-  rw [← take_append_drop n l₁, ← take_append_drop n l₂] at h
+  rw [← take_append_drop i l₁, ← take_append_drop i l₂] at h
  simpa only [count_append, w, Nat.add_left_cancel_iff] using h

+theorem sum_nat {l₁ l₂ : List Nat} (h : l₁ ~ l₂) : l₁.sum = l₂.sum := by
+  induction h with
+  | nil => simp
+  | cons _ _ ih => simp [ih]
+  | swap => simpa [List.sum_cons] using Nat.add_left_comm ..
+  | trans _ _ ih₁ ih₂ => simp [ih₁, ih₂]
+
+grind_pattern Perm.sum_nat => l₁ ~ l₂, l₁.sum
+grind_pattern Perm.sum_nat => l₁ ~ l₂, l₂.sum
+
 end Perm

 end List
--- a/src/Init/Data/List/Range.lean
+++ b/src/Init/Data/List/Range.lean
@@ -225,7 +225,7 @@ theorem zipIdx_eq_nil_iff {l : List α} {i : Nat} : List.zipIdx l i = [] ↔ l =
  | [], _ => rfl
  | _ :: _, _ => congrArg Nat.succ length_zipIdx

-@[simp]
+@[simp, grind =]
 theorem getElem?_zipIdx :
    ∀ {l : List α} {i j}, (zipIdx l i)[j]? = l[j]?.map fun a => (a, i + j)
  | [], _, _ => rfl
@@ -234,7 +234,7 @@ theorem getElem?_zipIdx :
    simp only [zipIdx_cons, getElem?_cons_succ]
    exact getElem?_zipIdx.trans <| by rw [Nat.add_right_comm]; rfl

-@[simp]
+@[simp, grind =]
 theorem getElem_zipIdx {l : List α} (h : i < (l.zipIdx j).length) :
    (l.zipIdx j)[i] = (l[i]'(by simpa [length_zipIdx] using h), j + i) := by
  simp only [length_zipIdx] at h
@@ -242,7 +242,7 @@ theorem getElem_zipIdx {l : List α} (h : i < (l.zipIdx j).length) :
  simp only [getElem?_zipIdx, getElem?_eq_getElem h]
  simp

-@[simp]
+@[simp, grind =]
 theorem tail_zipIdx {l : List α} {i : Nat} : (zipIdx l i).tail = zipIdx l.tail (i + 1) := by
  induction l generalizing i with
  | nil => simp
--- a/src/Init/Data/List/TakeDrop.lean
+++ b/src/Init/Data/List/TakeDrop.lean
@@ -467,7 +467,7 @@ theorem replace_takeWhile [BEq α] [LawfulBEq α] {l : List α} {p : α → Bool

 /-! ### splitAt -/

-@[simp] theorem splitAt_eq {i : Nat} {l : List α} : splitAt i l = (l.take i, l.drop i) := by
+@[simp, grind =] theorem splitAt_eq {i : Nat} {l : List α} : splitAt i l = (l.take i, l.drop i) := by
  rw [splitAt, splitAt_go, reverse_nil, nil_append]
  split <;> simp_all [take_of_length_le, drop_of_length_le]

--- a/src/Init/Data/List/ToArray.lean
+++ b/src/Init/Data/List/ToArray.lean
@@ -685,7 +685,7 @@ theorem replace_toArray [BEq α] [LawfulBEq α] (l : List α) (a b : α) :
        · rw [if_pos (by omega), if_pos, if_neg]
          · simp only [mem_take_iff_getElem, not_exists]
            intro k hk
-            simpa using h.2 ⟨k, by omega⟩ (by show k < i.1; omega)
+            simpa using h.2 ⟨k, by omega⟩ (by change k < i.1; omega)
          · subst h₃
            simpa using h.1
        · rw [if_neg (by omega)]
--- a/src/Init/Data/List/Zip.lean
+++ b/src/Init/Data/List/Zip.lean
@@ -46,6 +46,7 @@ theorem zipWith_self {f : α → α → δ} : ∀ {l : List α}, zipWith f l l =
 See also `getElem?_zipWith'` for a variant
 using `Option.map` and `Option.bind` rather than a `match`.
 -/
+@[grind =]
 theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
    (zipWith f as bs)[i]? = match as[i]?, bs[i]? with
      | some a, some b => some (f a b) | _, _ => none := by
@@ -83,33 +84,39 @@ theorem getElem?_zip_eq_some {l₁ : List α} {l₂ : List β} {z : α × β} {i
  · rintro ⟨h₀, h₁⟩
    exact ⟨_, _, h₀, h₁, rfl⟩

+@[grind =]
 theorem head?_zipWith {f : α → β → γ} :
    (List.zipWith f as bs).head? = match as.head?, bs.head? with
      | some a, some b => some (f a b) | _, _ => none := by
  simp [head?_eq_getElem?, getElem?_zipWith]

+@[grind =]
 theorem head_zipWith {f : α → β → γ} (h):
    (List.zipWith f as bs).head h = f (as.head (by rintro rfl; simp_all)) (bs.head (by rintro rfl; simp_all)) := by
  apply Option.some.inj
  rw [← head?_eq_head, head?_zipWith, head?_eq_head, head?_eq_head]

-@[simp]
+@[simp, grind =]
 theorem zipWith_map {μ} {f : γ → δ → μ} {g : α → γ} {h : β → δ} {l₁ : List α} {l₂ : List β} :
    zipWith f (l₁.map g) (l₂.map h) = zipWith (fun a b => f (g a) (h b)) l₁ l₂ := by
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem zipWith_map_left {l₁ : List α} {l₂ : List 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
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem zipWith_map_right {l₁ : List α} {l₂ : List 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
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} {i : δ} {g : γ → δ → δ} :
    (zipWith f l₁ l₂).foldr g i = (zip l₁ l₂).foldr (fun p r => g (f p.1 p.2) r) i := by
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} {g : δ → γ → δ} :
    (zipWith f l₁ l₂).foldl g i = (zip l₁ l₂).foldl (fun r p => g r (f p.1 p.2)) i := by
  induction l₁ generalizing i l₂ <;> cases l₂ <;> simp_all
@@ -118,6 +125,7 @@ theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} {g : δ →
 theorem zipWith_eq_nil_iff {f : α → β → γ} {l l'} : zipWith f l l' = [] ↔ l = [] ∨ l' = [] := by
  cases l <;> cases l' <;> simp

+@[grind =]
 theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {l : List γ} {l' : List δ} :
    map f (zipWith g l l') = zipWith (fun x y => f (g x y)) l l' := by
  induction l generalizing l' with
@@ -127,6 +135,7 @@ theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {l : Li
      · simp
      · simp [hl]

+@[grind =]
 theorem take_zipWith : (zipWith f l l').take i = zipWith f (l.take i) (l'.take i) := by
  induction l generalizing l' i with
  | nil => simp
@@ -137,6 +146,7 @@ theorem take_zipWith : (zipWith f l l').take i = zipWith f (l.take i) (l'.take i
      · simp
      · simp [hl]

+@[grind =]
 theorem drop_zipWith : (zipWith f l l').drop i = zipWith f (l.drop i) (l'.drop i) := by
  induction l generalizing l' i with
  | nil => simp
@@ -147,10 +157,11 @@ theorem drop_zipWith : (zipWith f l l').drop i = zipWith f (l.drop i) (l'.drop i
        · simp
        · simp [hl]

-@[simp]
+@[simp, grind =]
 theorem tail_zipWith : (zipWith f l l').tail = zipWith f l.tail l'.tail := by
  rw [← drop_one]; simp [drop_zipWith]

+@[grind =]
 theorem zipWith_append {f : α → β → γ} {l₁ l₁' : List α} {l₂ l₂' : List β}
    (h : l₁.length = l₂.length) :
    zipWith f (l₁ ++ l₁') (l₂ ++ l₂') = zipWith f l₁ l₂ ++ zipWith f l₁' l₂' := by
@@ -254,22 +265,26 @@ theorem zip_eq_zipWith : ∀ {l₁ : List α} {l₂ : List β}, zip l₁ l₂ =
  | _, [] => rfl
  | a :: l₁, b :: l₂ => by simp [zip_cons_cons, zip_eq_zipWith (l₁ := l₁)]

+@[grind _=_]
 theorem zip_map {f : α → γ} {g : β → δ} :
    ∀ {l₁ : List α} {l₂ : List β}, zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
  | [], _ => rfl
  | _, [] => by simp only [map, zip_nil_right]
  | _ :: _, _ :: _ => by simp only [map, zip_cons_cons, zip_map, Prod.map]

+@[grind _=_]
 theorem zip_map_left {f : α → γ} {l₁ : List α} {l₂ : List β} :
    zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]

+@[grind _=_]
 theorem zip_map_right {f : β → γ} {l₁ : List α} {l₂ : List β} :
    zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]

-@[simp] theorem tail_zip {l₁ : List α} {l₂ : List β} :
+@[simp, grind =] theorem tail_zip {l₁ : List α} {l₂ : List β} :
    (zip l₁ l₂).tail = zip l₁.tail l₂.tail := by
  cases l₁ <;> cases l₂ <;> simp

+@[grind =]
 theorem zip_append :
    ∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
      zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
@@ -278,6 +293,7 @@ theorem zip_append :
  | _ :: _, _, _ :: _, _, h => by
    simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]

+@[grind =]
 theorem zip_map' {f : α → β} {g : α → γ} :
    ∀ {l : List α}, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
  | [] => rfl
@@ -296,7 +312,7 @@ theorem map_fst_zip :
  | [], _, _ => rfl
  | _ :: as, _ :: bs, h => by
    simp [Nat.succ_le_succ_iff] at h
-    show _ :: map Prod.fst (zip as bs) = _ :: as
+    change _ :: map Prod.fst (zip as bs) = _ :: as
    rw [map_fst_zip (l₁ := as) h]
  | _ :: _, [], h => by simp at h

@@ -308,7 +324,7 @@ theorem map_snd_zip :
  | [], b :: bs, h => by simp at h
  | a :: as, b :: bs, h => by
    simp [Nat.succ_le_succ_iff] at h
-    show _ :: map Prod.snd (zip as bs) = _ :: bs
+    change _ :: map Prod.snd (zip as bs) = _ :: bs
    rw [map_snd_zip (l₂ := bs) h]

 theorem map_prod_left_eq_zip {l : List α} {f : α → β} :
@@ -353,6 +369,7 @@ theorem zip_eq_append_iff {l₁ : List α} {l₂ : List β} :

 /-! ### zipWithAll -/

+@[grind =]
 theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
    (zipWithAll f as bs)[i]? = match as[i]?, bs[i]? with
      | none, none => .none | a?, b? => some (f a? b?) := by
@@ -366,33 +383,38 @@ theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
      cases i <;> simp_all
    | cons b bs => cases i <;> simp_all

+@[grind =]
 theorem head?_zipWithAll {f : Option α → Option β → γ} :
    (zipWithAll f as bs).head? = match as.head?, bs.head? with
      | none, none => .none | a?, b? => some (f a? b?) := by
  simp [head?_eq_getElem?, getElem?_zipWithAll]

-@[simp] theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
+@[simp, grind =] theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
    (zipWithAll f as bs).head h = f as.head? bs.head? := by
  apply Option.some.inj
  rw [← head?_eq_head, head?_zipWithAll]
  split <;> simp_all

-@[simp] theorem tail_zipWithAll {f : Option α → Option β → γ} :
+@[simp, grind =] theorem tail_zipWithAll {f : Option α → Option β → γ} :
    (zipWithAll f as bs).tail = zipWithAll f as.tail bs.tail := by
  cases as <;> cases bs <;> simp

+@[grind =]
 theorem zipWithAll_map {μ} {f : Option γ → Option δ → μ} {g : α → γ} {h : β → δ} {l₁ : List α} {l₂ : List β} :
    zipWithAll f (l₁.map g) (l₂.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) l₁ l₂ := by
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem zipWithAll_map_left {l₁ : List α} {l₂ : List 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
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem zipWithAll_map_right {l₁ : List α} {l₂ : List 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
  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all

+@[grind =]
 theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option δ → α} {l : List γ} {l' : List δ} :
    map f (zipWithAll g l l') = zipWithAll (fun x y => f (g x y)) l l' := by
  induction l generalizing l' with
@@ -400,7 +422,7 @@ theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option
  | cons hd tl hl =>
    cases l' <;> simp_all

-@[simp] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
+@[simp, grind =] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
    zipWithAll f (replicate n a) (replicate n b) = replicate n (f (some a) (some b)) := by
  induction n with
  | zero => rfl
@@ -408,12 +430,13 @@ theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option

 /-! ### unzip -/

-@[simp] theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
+@[simp, grind =] theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
  induction l <;> simp_all

-@[simp] theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
+@[simp, grind =] theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
  induction l <;> simp_all

+@[grind =]
 theorem unzip_eq_map : ∀ {l : List (α × β)}, unzip l = (l.map Prod.fst, l.map Prod.snd)
  | [] => rfl
  | (a, b) :: l => by simp only [unzip_cons, map_cons, unzip_eq_map (l := l)]
@@ -453,6 +476,6 @@ theorem tail_zip_fst {l : List (α × β)} : l.unzip.1.tail = l.tail.unzip.1 :=
 theorem tail_zip_snd {l : List (α × β)} : l.unzip.2.tail = l.tail.unzip.2 := by
  simp

-@[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
+@[simp, grind =] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
    unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
  ext1 <;> simp
--- a/src/Init/Data/Nat/Basic.lean
+++ b/src/Init/Data/Nat/Basic.lean
@@ -406,6 +406,12 @@ theorem le_of_add_right_le {n m k : Nat} (h : n + k ≤ m) : n ≤ m :=
 theorem le_add_right_of_le {n m k : Nat} (h : n ≤ m) : n ≤ m + k :=
  Nat.le_trans h (le_add_right m k)

+theorem le_of_add_left_le {n m k : Nat} (h : k + n ≤ m) : n ≤ m :=
+  Nat.le_trans (le_add_left n k) h
+
+theorem le_add_left_of_le {n m k : Nat} (h : n ≤ m) : n ≤ k + m :=
+  Nat.le_trans h (le_add_left m k)
+
 theorem lt_of_add_one_le {n m : Nat} (h : n + 1 ≤ m) : n < m := h

 theorem add_one_le_of_lt {n m : Nat} (h : n < m) : n + 1 ≤ m := h
@@ -1069,7 +1075,7 @@ protected theorem sub_lt_sub_right : ∀ {a b c : Nat}, c ≤ a → a < b → a
    exact Nat.sub_lt_sub_right (le_of_succ_le_succ hle) (lt_of_succ_lt_succ h)

 protected theorem sub_self_add (n m : Nat) : n - (n + m) = 0 := by
-  show (n + 0) - (n + m) = 0
+  change (n + 0) - (n + m) = 0
  rw [Nat.add_sub_add_left, Nat.zero_sub]

@[simp] protected theorem sub_eq_zero_of_le {n m : Nat} (h : n ≤ m) : n - m = 0 := by
--- a/src/Init/Data/Nat/Bitwise/Lemmas.lean
+++ b/src/Init/Data/Nat/Bitwise/Lemmas.lean
@@ -51,24 +51,24 @@ noncomputable def div2Induction {motive : Nat → Sort u}
      apply hyp
      exact Nat.div_lt_self n_pos (Nat.le_refl _)

-@[simp] theorem zero_and (x : Nat) : 0 &&& x = 0 := by
+@[simp, grind =] theorem zero_and (x : Nat) : 0 &&& x = 0 := by
  simp only [HAnd.hAnd, AndOp.and, land]
  unfold bitwise
  simp

-@[simp] theorem and_zero (x : Nat) : x &&& 0 = 0 := by
+@[simp, grind =] theorem and_zero (x : Nat) : x &&& 0 = 0 := by
  simp only [HAnd.hAnd, AndOp.and, land]
  unfold bitwise
  simp

-@[simp] theorem one_and_eq_mod_two (n : Nat) : 1 &&& n = n % 2 := by
+@[simp, grind =] theorem one_and_eq_mod_two (n : Nat) : 1 &&& n = n % 2 := by
  if n0 : n = 0 then
    subst n0; decide
  else
    simp only [HAnd.hAnd, AndOp.and, land]
    cases mod_two_eq_zero_or_one n with | _ h => simp [bitwise, n0, h]

-@[simp] theorem and_one_is_mod (x : Nat) : x &&& 1 = x % 2 := by
+@[simp, grind =] theorem and_one_is_mod (x : Nat) : x &&& 1 = x % 2 := by
  if xz : x = 0 then
    simp [xz, zero_and]
  else
@@ -102,11 +102,12 @@ Depending on use cases either `testBit_add_one` or `testBit_div_two`
 may be more useful as a `simp` lemma, so neither is a global `simp` lemma.
 -/
 -- We turn `testBit_add_one` on as a `local simp` for this file.
-@[local simp]
+@[local simp, grind _=_]
 theorem testBit_add_one (x i : Nat) : testBit x (i + 1) = testBit (x/2) i := by
  unfold testBit
  simp [shiftRight_succ_inside]

+@[grind _=_]
 theorem testBit_add (x i n : Nat) : testBit x (i + n) = testBit (x / 2 ^ n) i := by
  revert x
  induction n with
@@ -122,6 +123,7 @@ theorem testBit_div_two (x i : Nat) : testBit (x / 2) i = testBit x (i + 1) := b
 theorem testBit_div_two_pow (x i : Nat) : testBit (x / 2 ^ n) i = testBit x (i + n) :=
  testBit_add .. |>.symm

+@[grind =]
 theorem testBit_eq_decide_div_mod_eq {x : Nat} : testBit x i = decide (x / 2^i % 2 = 1) := by
  induction i generalizing x with
  | zero =>
@@ -290,7 +292,7 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
  | d+1 =>
    simp [Nat.pow_succ, Nat.mul_comm _ 2, Nat.mul_add_mod]

-@[simp] theorem testBit_mod_two_pow (x j i : Nat) :
+@[simp, grind =] theorem testBit_mod_two_pow (x j i : Nat) :
    testBit (x % 2^j) i = (decide (i < j) && testBit x i) := by
  induction x using Nat.strongRecOn generalizing j i with
  | ind x hyp =>
@@ -322,6 +324,7 @@ theorem not_decide_mod_two_eq_one (x : Nat)
    : (!decide (x % 2 = 1)) = decide (x % 2 = 0) := by
  cases Nat.mod_two_eq_zero_or_one x <;> (rename_i p; simp [p])

+@[grind =]
 theorem testBit_two_pow_sub_succ (h₂ : x < 2 ^ n) (i : Nat) :
    testBit (2^n - (x + 1)) i = (decide (i < n) && ! testBit x i) := by
  induction i generalizing n x with
@@ -357,6 +360,7 @@ theorem testBit_one_eq_true_iff_self_eq_zero {i : Nat} :
    Nat.testBit 1 i = true ↔ i = 0 := by
  cases i <;> simp

+@[grind =]
 theorem testBit_two_pow {n m : Nat} : testBit (2 ^ n) m = decide (n = m) := by
  rw [testBit, shiftRight_eq_div_pow]
  by_cases h : n = m
@@ -482,18 +486,20 @@ theorem bitwise_mod_two_pow (of_false_false : f false false = false := by rfl) :

 /-! ### and -/

-@[simp] theorem testBit_and (x y i : Nat) : (x &&& y).testBit i = (x.testBit i && y.testBit i) := by
+@[simp, grind =] theorem testBit_and (x y i : Nat) : (x &&& y).testBit i = (x.testBit i && y.testBit i) := by
  simp [HAnd.hAnd, AndOp.and, land, testBit_bitwise ]


-@[simp] protected theorem and_self (x : Nat) : x &&& x = x := by
+@[simp, grind =] protected theorem and_self (x : Nat) : x &&& x = x := by
   apply Nat.eq_of_testBit_eq
   simp

+@[grind =]
 protected theorem and_comm (x y : Nat) : x &&& y = y &&& x := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.and_comm]

+@[grind _=_]
 protected theorem and_assoc (x y z : Nat) : (x &&& y) &&& z = x &&& (y &&& z) := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.and_assoc]
@@ -537,54 +543,63 @@ abbrev and_pow_two_sub_one_of_lt_two_pow := @and_two_pow_sub_one_of_lt_two_pow
  rw [testBit_and]
  simp

+@[grind _=_]
 theorem and_div_two_pow : (a &&& b) / 2 ^ n = a / 2 ^ n &&& b / 2 ^ n :=
  bitwise_div_two_pow

+@[grind _=_]
 theorem and_div_two : (a &&& b) / 2 = a / 2 &&& b / 2 :=
  and_div_two_pow (n := 1)

+@[grind _=_]
 theorem and_mod_two_pow : (a &&& b) % 2 ^ n = (a % 2 ^ n) &&& (b % 2 ^ n) :=
  bitwise_mod_two_pow

 /-! ### lor -/

-@[simp] theorem zero_or (x : Nat) : 0 ||| x = x := by
+@[simp, grind =] theorem zero_or (x : Nat) : 0 ||| x = x := by
  simp only [HOr.hOr, OrOp.or, lor]
  unfold bitwise
  simp [@eq_comm _ 0]

-@[simp] theorem or_zero (x : Nat) : x ||| 0 = x := by
+@[simp, grind =] theorem or_zero (x : Nat) : x ||| 0 = x := by
  simp only [HOr.hOr, OrOp.or, lor]
  unfold bitwise
  simp [@eq_comm _ 0]

-@[simp] theorem testBit_or (x y i : Nat) : (x ||| y).testBit i = (x.testBit i || y.testBit i) := by
+@[simp, grind =] theorem testBit_or (x y i : Nat) : (x ||| y).testBit i = (x.testBit i || y.testBit i) := by
  simp [HOr.hOr, OrOp.or, lor, testBit_bitwise ]

-@[simp] protected theorem or_self (x : Nat) : x ||| x = x := by
+@[simp, grind =] protected theorem or_self (x : Nat) : x ||| x = x := by
   apply Nat.eq_of_testBit_eq
   simp

+@[grind =]
 protected theorem or_comm (x y : Nat) : x ||| y = y ||| x := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.or_comm]

+@[grind _=_]
 protected theorem or_assoc (x y z : Nat) : (x ||| y) ||| z = x ||| (y ||| z) := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.or_assoc]

+@[grind _=_]
 theorem and_or_distrib_left (x y z : Nat) : x &&& (y ||| z) = (x &&& y) ||| (x &&& z) := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.and_or_distrib_left]

+@[grind _=_]
 theorem and_distrib_right (x y z : Nat) : (x ||| y) &&& z = (x &&& z) ||| (y &&& z) := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.and_or_distrib_right]

+@[grind _=_]
 theorem or_and_distrib_left (x y z : Nat) : x ||| (y &&& z) = (x ||| y) &&& (x ||| z) := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.or_and_distrib_left]

+@[grind _=_]
 theorem or_and_distrib_right (x y z : Nat) : (x &&& y) ||| z = (x ||| z) &&& (y ||| z) := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.or_and_distrib_right]
@@ -610,37 +625,42 @@ theorem or_lt_two_pow {x y n : Nat} (left : x < 2^n) (right : y < 2^n) : x ||| y
  rw [testBit_or]
  simp

+@[grind _=_]
 theorem or_div_two_pow : (a ||| b) / 2 ^ n = a / 2 ^ n ||| b / 2 ^ n :=
  bitwise_div_two_pow

+@[grind _=_]
 theorem or_div_two : (a ||| b) / 2 = a / 2 ||| b / 2 :=
  or_div_two_pow (n := 1)

+@[grind _=_]
 theorem or_mod_two_pow : (a ||| b) % 2 ^ n = a % 2 ^ n ||| b % 2 ^ n :=
  bitwise_mod_two_pow

 /-! ### xor -/

-@[simp] theorem testBit_xor (x y i : Nat) :
+@[simp, grind =] theorem testBit_xor (x y i : Nat) :
    (x ^^^ y).testBit i = ((x.testBit i) ^^ (y.testBit i)) := by
  simp [HXor.hXor, Xor.xor, xor, testBit_bitwise ]

-@[simp] theorem zero_xor (x : Nat) : 0 ^^^ x = x := by
+@[simp, grind =] theorem zero_xor (x : Nat) : 0 ^^^ x = x := by
   apply Nat.eq_of_testBit_eq
   simp

-@[simp] theorem xor_zero (x : Nat) : x ^^^ 0 = x := by
+@[simp, grind =] theorem xor_zero (x : Nat) : x ^^^ 0 = x := by
   apply Nat.eq_of_testBit_eq
   simp

-@[simp] protected theorem xor_self (x : Nat) : x ^^^ x = 0 := by
+@[simp, grind =] protected theorem xor_self (x : Nat) : x ^^^ x = 0 := by
   apply Nat.eq_of_testBit_eq
   simp

+@[grind =]
 protected theorem xor_comm (x y : Nat) : x ^^^ y = y ^^^ x := by
   apply Nat.eq_of_testBit_eq
   simp [Bool.xor_comm]

+@[grind _=_]
 protected theorem xor_assoc (x y z : Nat) : (x ^^^ y) ^^^ z = x ^^^ (y ^^^ z) := by
   apply Nat.eq_of_testBit_eq
   simp
@@ -658,10 +678,12 @@ instance : Std.LawfulCommIdentity (α := Nat) (· ^^^ ·) 0 where
 theorem xor_lt_two_pow {x y n : Nat} (left : x < 2^n) (right : y < 2^n) : x ^^^ y < 2^n :=
  bitwise_lt_two_pow left right

+@[grind _=_]
 theorem and_xor_distrib_right {a b c : Nat} : (a ^^^ b) &&& c = (a &&& c) ^^^ (b &&& c) := by
  apply Nat.eq_of_testBit_eq
  simp [Bool.and_xor_distrib_right]

+@[grind _=_]
 theorem and_xor_distrib_left {a b c : Nat} : a &&& (b ^^^ c) = (a &&& b) ^^^ (a &&& c) := by
  apply Nat.eq_of_testBit_eq
  simp [Bool.and_xor_distrib_left]
@@ -671,12 +693,15 @@ theorem and_xor_distrib_left {a b c : Nat} : a &&& (b ^^^ c) = (a &&& b) ^^^ (a
  rw [testBit_xor]
  simp

+@[grind _=_]
 theorem xor_div_two_pow : (a ^^^ b) / 2 ^ n = a / 2 ^ n ^^^ b / 2 ^ n :=
  bitwise_div_two_pow

+@[grind _=_]
 theorem xor_div_two : (a ^^^ b) / 2 = a / 2 ^^^ b / 2 :=
  xor_div_two_pow (n := 1)

+@[grind _=_]
 theorem xor_mod_two_pow : (a ^^^ b) % 2 ^ n = a % 2 ^ n ^^^ b % 2 ^ n :=
  bitwise_mod_two_pow

@@ -713,6 +738,7 @@ theorem testBit_two_pow_mul_add (a : Nat) {b i : Nat} (b_lt : b < 2^i) (j : Nat)
@[deprecated testBit_two_pow_mul_add (since := "2025-03-18")]
 abbrev testBit_mul_pow_two_add := @testBit_two_pow_mul_add

+@[grind =]
 theorem testBit_two_pow_mul :
    testBit (2 ^ i * a) j = (decide (j ≥ i) && testBit a (j-i)) := by
  have gen := testBit_two_pow_mul_add a (Nat.two_pow_pos i) j
@@ -721,6 +747,11 @@ theorem testBit_two_pow_mul :
  cases Nat.lt_or_ge j i with
  | _ p => simp [p, Nat.not_le_of_lt, Nat.not_lt_of_le]

+@[grind =] -- Ideally `grind` could do this just with `testBit_two_pow_mul`.
+theorem testBit_mul_two_pow (x j i : Nat) :
+    (x * 2 ^ i).testBit j = (decide (i ≤ j) && x.testBit (j - i)) := by
+  rw [Nat.mul_comm, testBit_two_pow_mul]
+
@[deprecated testBit_two_pow_mul (since := "2025-03-18")]
 abbrev testBit_mul_pow_two := @testBit_two_pow_mul

@@ -744,21 +775,17 @@ abbrev mul_add_lt_is_or := @two_pow_add_eq_or_of_lt

 /-! ### shiftLeft and shiftRight -/

-@[simp] theorem testBit_shiftLeft (x : Nat) : testBit (x <<< i) j =
+@[simp, grind =] theorem testBit_shiftLeft (x : Nat) : testBit (x <<< i) j =
    (decide (j ≥ i) && testBit x (j-i)) := by
  simp [shiftLeft_eq, Nat.mul_comm _ (2^_), testBit_two_pow_mul]

-@[simp] theorem testBit_shiftRight (x : Nat) : testBit (x >>> i) j = testBit x (i+j) := by
+@[simp, grind =] theorem testBit_shiftRight (x : Nat) : testBit (x >>> i) j = testBit x (i+j) := by
  simp [testBit, ←shiftRight_add]

@[simp] theorem shiftLeft_mod_two_eq_one : x <<< i % 2 = 1 ↔ i = 0 ∧ x % 2 = 1 := by
  rw [mod_two_eq_one_iff_testBit_zero, testBit_shiftLeft]
  simp

-theorem testBit_mul_two_pow (x i n : Nat) :
-    (x * 2 ^ n).testBit i = (decide (n ≤ i) && x.testBit (i - n)) := by
-  rw [← testBit_shiftLeft, shiftLeft_eq]
-
 theorem bitwise_mul_two_pow (of_false_false : f false false = false := by rfl) :
    (bitwise f x y) * 2 ^ n = bitwise f (x * 2 ^ n) (y * 2 ^ n) := by
  apply Nat.eq_of_testBit_eq
@@ -768,16 +795,20 @@ theorem bitwise_mul_two_pow (of_false_false : f false false = false := by rfl) :
  · simp [hn]
  · simp [hn, of_false_false]

+@[grind _=_]
 theorem shiftLeft_bitwise_distrib {a b : Nat} (of_false_false : f false false = false := by rfl) :
    (bitwise f a b) <<< i = bitwise f (a <<< i) (b <<< i) := by
  simp [shiftLeft_eq, bitwise_mul_two_pow of_false_false]

+@[grind _=_]
 theorem shiftLeft_and_distrib {a b : Nat} : (a &&& b) <<< i = a <<< i &&& b <<< i :=
  shiftLeft_bitwise_distrib

+@[grind _=_]
 theorem shiftLeft_or_distrib {a b : Nat} : (a ||| b) <<< i = a <<< i ||| b <<< i :=
  shiftLeft_bitwise_distrib

+@[grind _=_]
 theorem shiftLeft_xor_distrib {a b : Nat} : (a ^^^ b) <<< i = a <<< i ^^^ b <<< i :=
  shiftLeft_bitwise_distrib

@@ -786,16 +817,20 @@ theorem shiftLeft_xor_distrib {a b : Nat} : (a ^^^ b) <<< i = a <<< i ^^^ b <<<
  simp only [testBit, one_and_eq_mod_two, mod_two_bne_zero]
  exact (Bool.beq_eq_decide_eq _ _).symm

+@[grind _=_]
 theorem shiftRight_bitwise_distrib {a b : Nat} (of_false_false : f false false = false := by rfl) :
    (bitwise f a b) >>> i = bitwise f (a >>> i) (b >>> i) := by
  simp [shiftRight_eq_div_pow, bitwise_div_two_pow of_false_false]

+@[grind _=_]
 theorem shiftRight_and_distrib {a b : Nat} : (a &&& b) >>> i = a >>> i &&& b >>> i :=
  shiftRight_bitwise_distrib

+@[grind _=_]
 theorem shiftRight_or_distrib {a b : Nat} : (a ||| b) >>> i = a >>> i ||| b >>> i :=
  shiftRight_bitwise_distrib

+@[grind _=_]
 theorem shiftRight_xor_distrib {a b : Nat} : (a ^^^ b) >>> i = a >>> i ^^^ b >>> i :=
  shiftRight_bitwise_distrib

--- a/src/Init/Data/Nat/Div/Basic.lean
+++ b/src/Init/Data/Nat/Div/Basic.lean
@@ -9,6 +9,7 @@ prelude
 import Init.WF
 import Init.WFTactics
 import Init.Data.Nat.Basic
+meta import Init.MetaTypes

@[expose] section

@@ -75,7 +76,7 @@ private theorem div.go.fuel_congr (x y fuel1 fuel2 : Nat) (hy : 0 < y) (h1 : x <
 termination_by structural fuel1

 theorem div_eq (x y : Nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 else 0 := by
-  show Nat.div _ _ = ite _ (Nat.div _ _ + 1) _
+  change Nat.div _ _ = ite _ (Nat.div _ _ + 1) _
  unfold Nat.div
  split
  next =>
@@ -257,7 +258,7 @@ protected def mod : @& Nat → @& Nat → Nat
 instance instMod : Mod Nat := ⟨Nat.mod⟩

 protected theorem modCore_eq_mod (n m : Nat) : Nat.modCore n m = n % m := by
-  show Nat.modCore n m = Nat.mod n m
+  change Nat.modCore n m = Nat.mod n m
  match n, m with
  | 0, _ =>
    rw [Nat.modCore_eq]
@@ -521,7 +522,7 @@ theorem mul_sub_div (x n p : Nat) (h₁ : x < n*p) : (n * p - (x + 1)) / n = p -
    rw [Nat.mul_sub_right_distrib, Nat.mul_comm]
    exact Nat.sub_le_sub_left ((div_lt_iff_lt_mul npos).1 (lt_succ_self _)) _
  focus
-    show succ (pred (n * p - x)) ≤ (succ (pred (p - x / n))) * n
+    change succ (pred (n * p - x)) ≤ (succ (pred (p - x / n))) * n
    rw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h₁),
      fun h => succ_pred_eq_of_pos (Nat.sub_pos_of_lt h)] -- TODO: why is the function needed?
    focus
--- a/src/Init/Data/Nat/Div/Lemmas.lean
+++ b/src/Init/Data/Nat/Div/Lemmas.lean
@@ -210,4 +210,19 @@ theorem mod_mod_eq_mod_mod_mod_of_dvd {a b c : Nat} (hb : b ∣ c) :
  have : b < c := Nat.lt_of_le_of_ne (Nat.le_of_dvd hc hb) hb'
  rw [Nat.mod_mod_of_dvd' hb, Nat.mod_eq_of_lt this, Nat.mod_mod_of_dvd _ hb]

+theorem mod_eq_mod_iff {x y z : Nat} :
+    x % z = y % z ↔ ∃ k₁ k₂, x + k₁ * z = y + k₂ * z := by
+  constructor
+  · rw [Nat.mod_def, Nat.mod_def]
+    rw [Nat.sub_eq_iff_eq_add, Nat.add_comm, ← Nat.add_sub_assoc, eq_comm, Nat.sub_eq_iff_eq_add, eq_comm]
+    · intro h
+      refine ⟨(y / z), (x / z), ?_⟩
+      rwa [Nat.mul_comm z, Nat.add_comm _ y, Nat.mul_comm z] at h
+    · exact le_add_left_of_le (mul_div_le y z)
+    · exact mul_div_le y z
+    · exact mul_div_le x z
+  · rintro ⟨k₁, k₂, h⟩
+    replace h := congrArg (· % z) h
+    simpa using h
+
 end Nat
--- a/src/Init/Data/Nat/Lemmas.lean
+++ b/src/Init/Data/Nat/Lemmas.lean
@@ -1772,6 +1772,12 @@ instance decidableExistsLE' {p : (m : Nat) → m ≤ k → Prop} [I : ∀ m h, D
    intro
    exact ⟨fun ⟨h, w⟩ => ⟨le_of_lt_succ h, w⟩, fun ⟨h, w⟩ => ⟨lt_add_one_of_le h, w⟩⟩

+instance decidableExistsFin (P : Fin n → Prop) [DecidablePred P] : Decidable (∃ i, P i) :=
+  decidable_of_iff (∃ k, k < n ∧ ((h: k < n) → P ⟨k, h⟩))
+    ⟨fun ⟨k, a⟩ => Exists.intro ⟨k, a.left⟩ (a.right a.left),
+    fun ⟨i, e⟩ => Exists.intro i.val ⟨i.isLt, fun _ => e⟩⟩
+
+
 /-! ### Results about `List.sum` specialized to `Nat` -/

 protected theorem sum_pos_iff_exists_pos {l : List Nat} : 0 < l.sum ↔ ∃ x ∈ l, 0 < x := by
--- a/src/Init/Data/Nat/Linear.lean
+++ b/src/Init/Data/Nat/Linear.lean
@@ -149,7 +149,7 @@ instance : LawfulBEq PolyCnstr where
    rw [h₁, h₂, h₃]
  rfl {a} := by
    cases a; rename_i eq lhs rhs
-    show (eq == eq && (lhs == lhs && rhs == rhs)) = true
+    change (eq == eq && (lhs == lhs && rhs == rhs)) = true
    simp

 structure ExprCnstr where
--- a/src/Init/Data/Range/Basic.lean
+++ b/src/Init/Data/Range/Basic.lean
@@ -85,4 +85,4 @@ theorem Membership.get_elem_helper {i n : Nat} {r : Std.Range} (h₁ : i ∈ r)
    i < n := h₂ ▸ h₁.2.1

 macro_rules
-  | `(tactic| get_elem_tactic_trivial) => `(tactic| apply Membership.get_elem_helper; assumption; rfl)
+  | `(tactic| get_elem_tactic_extensible) => `(tactic| apply Membership.get_elem_helper; assumption; rfl)
--- a/src/Init/Data/Repr.lean
+++ b/src/Init/Data/Repr.lean
@@ -342,11 +342,12 @@ instance : Repr Int where
 def hexDigitRepr (n : Nat) : String :=
  String.singleton <| Nat.digitChar n

-def Char.quoteCore (c : Char) : String :=
+def Char.quoteCore (c : Char) (inString : Bool := false) : String :=
  if       c = '\n' then "\\n"
  else if  c = '\t' then "\\t"
  else if  c = '\\' then "\\\\"
  else if  c = '\"' then "\\\""
+  else if !inString && c = '\'' then "\\\'"
  else if  c.toNat <= 31 ∨ c = '\x7f' then "\\x" ++ smallCharToHex c
  else String.singleton c
 where
@@ -383,7 +384,7 @@ Examples:
 -/
 def String.quote (s : String) : String :=
  if s.isEmpty then "\"\""
-  else s.foldl (fun s c => s ++ c.quoteCore) "\"" ++ "\""
+  else s.foldl (fun s c => s ++ c.quoteCore (inString := true)) "\"" ++ "\""

 instance : Repr String where
  reprPrec s _ := s.quote
--- a/src/Init/Data/SInt/Lemmas.lean
+++ b/src/Init/Data/SInt/Lemmas.lean
@@ -9,8 +9,8 @@ prelude
 import all Init.Data.Nat.Bitwise.Basic
 import all Init.Data.SInt.Basic
 import all Init.Data.BitVec.Basic
+import Init.Data.BitVec.Lemmas
 import Init.Data.BitVec.Bitblast
-import all Init.Data.BitVec.Lemmas
 import Init.Data.Int.LemmasAux
 import all Init.Data.UInt.Basic
 import Init.Data.UInt.Lemmas
--- a/src/Init/Data/String/Basic.lean
+++ b/src/Init/Data/String/Basic.lean
@@ -586,7 +586,7 @@ decreasing_by
  focus
    rename_i i₀ j₀ _ eq h'
    rw [show (s.next i₀ - sep.next j₀).1 = (i₀ - j₀).1 by
-      show (_ + Char.utf8Size _) - (_ + Char.utf8Size _) = _
+      change (_ + Char.utf8Size _) - (_ + Char.utf8Size _) = _
      rw [(beq_iff_eq ..).1 eq, Nat.add_sub_add_right]; rfl]
    right; exact Nat.sub_lt_sub_left
      (Nat.lt_of_le_of_lt (Nat.le_add_right ..) (Nat.gt_of_not_le (mt decide_eq_true h')))
--- a/src/Init/Data/String/Extra.lean
+++ b/src/Init/Data/String/Extra.lean
@@ -295,11 +295,11 @@ where
  termination_by text.utf8ByteSize - pos.byteIdx
  decreasing_by
    decreasing_with
-      show text.utf8ByteSize - (text.next (text.next pos)).byteIdx < text.utf8ByteSize - pos.byteIdx
+      change text.utf8ByteSize - (text.next (text.next pos)).byteIdx < text.utf8ByteSize - pos.byteIdx
      have k := Nat.gt_of_not_le <| mt decide_eq_true h
      exact Nat.sub_lt_sub_left k (Nat.lt_trans (String.lt_next text pos) (String.lt_next _ _))
    decreasing_with
-      show text.utf8ByteSize - (text.next pos).byteIdx < text.utf8ByteSize - pos.byteIdx
+      change text.utf8ByteSize - (text.next pos).byteIdx < text.utf8ByteSize - pos.byteIdx
      have k := Nat.gt_of_not_le <| mt decide_eq_true h
      exact Nat.sub_lt_sub_left k (String.lt_next _ _)

--- a/src/Init/Data/ToString/Macro.lean
+++ b/src/Init/Data/ToString/Macro.lean
@@ -6,8 +6,7 @@ Author: Leonardo de Moura
 module

 prelude
-import Init.Meta
-import Init.Data.ToString.Basic
+meta import Init.Meta

 syntax:max "s!" interpolatedStr(term) : term

--- a/src/Init/Data/UInt/Lemmas.lean
+++ b/src/Init/Data/UInt/Lemmas.lean
@@ -10,10 +10,9 @@ import all Init.Data.UInt.Basic
 import all Init.Data.UInt.BasicAux
 import Init.Data.Fin.Lemmas
 import all Init.Data.Fin.Bitwise
-import all Init.Data.BitVec.Basic
 import all Init.Data.BitVec.BasicAux
+import all Init.Data.BitVec.Basic
 import Init.Data.BitVec.Lemmas
-import Init.Data.BitVec.Bitblast
 import Init.Data.Nat.Div.Lemmas
 import Init.System.Platform

--- a/src/Init/Data/Vector/Basic.lean
+++ b/src/Init/Data/Vector/Basic.lean
@@ -7,6 +7,7 @@ Authors: Shreyas Srinivas, François G. Dorais, Kim Morrison
 module

 prelude
+meta import Init.Coe
 import Init.Data.Array.Lemmas
 import Init.Data.Array.MapIdx
 import Init.Data.Array.InsertIdx
--- a/src/Init/Data/Vector/Count.lean
+++ b/src/Init/Data/Vector/Count.lean
@@ -40,6 +40,7 @@ theorem countP_push {a : α} {xs : Vector α n} : countP p (xs.push a) = countP
  rcases xs with ⟨xs, rfl⟩
  simp [Array.countP_push]

+@[grind =]
 theorem countP_singleton {a : α} : countP p #v[a] = if p a then 1 else 0 := by
  simp

@@ -51,7 +52,7 @@ theorem countP_le_size {xs : Vector α n} : countP p xs ≤ n := by
  rcases xs with ⟨xs, rfl⟩
  simp [Array.countP_le_size (p := p)]

-@[simp] theorem countP_append {xs : Vector α n} {ys : Vector α m} : countP p (xs ++ ys) = countP p xs + countP p ys := by
+@[simp, grind =] theorem countP_append {xs : Vector α n} {ys : Vector α m} : countP p (xs ++ ys) = countP p xs + countP p ys := by
  cases xs
  cases ys
  simp
@@ -116,7 +117,7 @@ theorem countP_flatMap {p : β → Bool} {xs : Vector α n} {f : α → Vector
  rcases xs with ⟨xs, rfl⟩
  simp [Array.countP_flatMap, Function.comp_def]

-@[simp] theorem countP_reverse {xs : Vector α n} : countP p xs.reverse = countP p xs := by
+@[simp, grind =] theorem countP_reverse {xs : Vector α n} : countP p xs.reverse = countP p xs := by
  rcases xs with ⟨xs, rfl⟩
  simp

@@ -136,7 +137,7 @@ section count

 variable [BEq α]

-@[simp] theorem count_empty {a : α} : count a #v[] = 0 := rfl
+@[simp, grind =] theorem count_empty {a : α} : count a #v[] = 0 := rfl

 theorem count_push {a b : α} {xs : Vector α n} :
    count a (xs.push b) = count a xs + if b == a then 1 else 0 := by
@@ -151,23 +152,25 @@ theorem count_eq_countP' {a : α} : count (n := n) a = countP (· == a) := by

 theorem count_le_size {a : α} {xs : Vector α n} : count a xs ≤ n := countP_le_size

+grind_pattern count_le_size => count a xs
+
 theorem count_le_count_push {a b : α} {xs : Vector α n} : count a xs ≤ count a (xs.push b) := by
  rcases xs with ⟨xs, rfl⟩
  simp [Array.count_push]

-@[simp] theorem count_singleton {a b : α} : count a #v[b] = if b == a then 1 else 0 := by
+@[simp, grind =] theorem count_singleton {a b : α} : count a #v[b] = if b == a then 1 else 0 := by
  simp [count_eq_countP]

-@[simp] theorem count_append {a : α} {xs : Vector α n} {ys : Vector α m} :
+@[simp, grind =] theorem count_append {a : α} {xs : Vector α n} {ys : Vector α m} :
    count a (xs ++ ys) = count a xs + count a ys :=
  countP_append ..

-@[simp] theorem count_flatten {a : α} {xss : Vector (Vector α m) n} :
+@[simp, grind =] theorem count_flatten {a : α} {xss : Vector (Vector α m) n} :
    count a xss.flatten = (xss.map (count a)).sum := by
  rcases xss with ⟨xss, rfl⟩
  simp [Array.count_flatten, Function.comp_def]

-@[simp] theorem count_reverse {a : α} {xs : Vector α n} : count a xs.reverse = count a xs := by
+@[simp, grind =] theorem count_reverse {a : α} {xs : Vector α n} : count a xs.reverse = count a xs := by
  rcases xs with ⟨xs, rfl⟩
  simp

--- a/src/Init/Data/Vector/Erase.lean
+++ b/src/Init/Data/Vector/Erase.lean
@@ -22,11 +22,13 @@ open Nat

 /-! ### eraseIdx -/

+@[grind =]
 theorem eraseIdx_eq_take_drop_succ {xs : Vector α n} {i : Nat} (h) :
    xs.eraseIdx i = (xs.take i ++ xs.drop (i + 1)).cast (by omega) := by
  rcases xs with ⟨xs, rfl⟩
  simp [Array.eraseIdx_eq_take_drop_succ, *]

+@[grind =]
 theorem getElem?_eraseIdx {xs : Vector α n} {i : Nat} (h : i < n) {j : Nat} :
    (xs.eraseIdx i)[j]? = if j < i then xs[j]? else xs[j + 1]? := by
  rcases xs with ⟨xs, rfl⟩
@@ -44,12 +46,14 @@ theorem getElem?_eraseIdx_of_ge {xs : Vector α n} {i : Nat} (h : i < n) {j : Na
  intro h'
  omega

+@[grind =]
 theorem getElem_eraseIdx {xs : Vector α n} {i : Nat} (h : i < n) {j : Nat} (h' : j < n - 1) :
    (xs.eraseIdx i)[j] = if h'' : j < i then xs[j] else xs[j + 1] := by
  apply Option.some.inj
  rw [← getElem?_eq_getElem, getElem?_eraseIdx]
  split <;> simp

+@[grind →]
 theorem mem_of_mem_eraseIdx {xs : Vector α n} {i : Nat} {h} {a : α} (h : a ∈ xs.eraseIdx i) : a ∈ xs := by
  rcases xs with ⟨xs, rfl⟩
  simpa using Array.mem_of_mem_eraseIdx (by simpa using h)
@@ -64,13 +68,23 @@ theorem eraseIdx_append_of_length_le {xs : Vector α n} {k : Nat} (hk : n ≤ k)
    eraseIdx (xs ++ xs') k = (xs ++ eraseIdx xs' (k - n)).cast (by omega) := by
  rcases xs with ⟨xs⟩
  rcases xs' with ⟨xs'⟩
-  simp [Array.eraseIdx_append_of_length_le, *]
+  simp [Array.eraseIdx_append_of_size_le, *]

+@[grind =]
+theorem eraseIdx_append {xs : Vector α n} {ys : Vector α m} {k : Nat} (h : k < n + m) :
+    eraseIdx (xs ++ ys) k = if h' : k < n then (eraseIdx xs k ++ ys).cast (by omega) else (xs ++ eraseIdx ys (k - n) (by omega)).cast (by omega) := by
+  rcases xs with ⟨xs, rfl⟩
+  rcases ys with ⟨ys, rfl⟩
+  simp only [mk_append_mk, eraseIdx_mk, Array.eraseIdx_append]
+  split <;> simp
+
+@[grind = ]
 theorem eraseIdx_cast {xs : Vector α n} {k : Nat} (h : k < m) :
    eraseIdx (xs.cast w) k h = (eraseIdx xs k).cast (by omega) := by
  rcases xs with ⟨xs⟩
  simp

+@[grind =]
 theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} {h} :
    (replicate n a).eraseIdx k = replicate (n - 1) a := by
  rw [replicate_eq_mk_replicate, eraseIdx_mk]
@@ -112,6 +126,45 @@ theorem eraseIdx_set_gt {xs : Vector α n} {i : Nat} {j : Nat} {a : α} (h : i <
  rcases xs with ⟨xs⟩
  simp [Array.eraseIdx_set_gt, *]

+@[grind =]
+theorem eraseIdx_set {xs : Vector α n} {i : Nat} {a : α} {hi : i < n} {j : Nat} {hj : j < n} :
+    (xs.set i a).eraseIdx j =
+      if h' : j < i then
+        (xs.eraseIdx j).set (i - 1) a
+      else if h'' : j = i then
+        xs.eraseIdx i
+      else
+        (xs.eraseIdx j).set i a := by
+  rcases xs with ⟨xs⟩
+  simp only [set_mk, eraseIdx_mk, Array.eraseIdx_set]
+  split
+  · simp
+  · split <;> simp
+
+theorem set_eraseIdx_le {xs : Vector α n} {i : Nat} {w : i < n} {j : Nat} {a : α} (h : i ≤ j) (hj : j < n - 1) :
+    (xs.eraseIdx i).set j a = (xs.set (j + 1) a).eraseIdx i := by
+  rw [eraseIdx_set_lt]
+  · simp
+  · omega
+
+theorem set_eraseIdx_gt {xs : Vector α n} {i : Nat} {w : i < n} {j : Nat} {a : α} (h : j < i) (hj : j < n - 1) :
+    (xs.eraseIdx i).set j a = (xs.set j a).eraseIdx i := by
+  rw [eraseIdx_set_gt]
+  omega
+
+@[grind =]
+theorem set_eraseIdx {xs : Vector α n} {i : Nat} {w : i < n} {j : Nat} {a : α} (hj : j < n - 1) :
+    (xs.eraseIdx i).set j a =
+      if h' : i ≤ j then
+        (xs.set (j + 1) a).eraseIdx i
+      else
+        (xs.set j a).eraseIdx i := by
+  split <;> rename_i h'
+  · rw [set_eraseIdx_le]
+    omega
+  · rw [set_eraseIdx_gt]
+    omega
+
@[simp] theorem set_getElem_succ_eraseIdx_succ
    {xs : Vector α n} {i : Nat} (h : i + 1 < n) :
    (xs.eraseIdx (i + 1)).set i xs[i + 1] = xs.eraseIdx i := by
--- a/src/Init/Data/Vector/Extract.lean
+++ b/src/Init/Data/Vector/Extract.lean
@@ -28,19 +28,19 @@ set_option linter.indexVariables false
  rcases xs with ⟨as, rfl⟩
  simp [h]

-@[simp]
+@[simp, grind =]
 theorem extract_push {xs : Vector α n} {b : α} {start stop : Nat} (h : stop ≤ n) :
    (xs.push b).extract start stop = (xs.extract start stop).cast (by omega) := by
  rcases xs with ⟨xs, rfl⟩
  simp [h]

-@[simp]
+@[simp, grind =]
 theorem extract_eq_pop {xs : Vector α n} {stop : Nat} (h : stop = n - 1) :
    xs.extract 0 stop = xs.pop.cast (by omega) := by
  rcases xs with ⟨xs, rfl⟩
  simp [h]

-@[simp]
+@[simp, grind _=_]
 theorem extract_append_extract {xs : Vector α n} {i j k : Nat} :
    xs.extract i j ++ xs.extract j k =
      (xs.extract (min i j) (max j k)).cast (by omega) := by
@@ -79,11 +79,12 @@ theorem getElem?_extract_of_succ {xs : Vector α n} {j : Nat} :
  · rw [if_neg (by omega)]
    simp_all

-@[simp] theorem extract_extract {xs : Vector α n} {i j k l : Nat} :
+@[simp, grind =] theorem extract_extract {xs : Vector α n} {i j k l : Nat} :
    (xs.extract i j).extract k l = (xs.extract (i + k) (min (i + l) j)).cast (by omega) := by
  rcases xs with ⟨xs, rfl⟩
  simp

+@[grind =]
 theorem extract_set {xs : Vector α n} {i j k : Nat} (h : k < n) {a : α} :
    (xs.set k a).extract i j =
      if _ : k < i then
@@ -97,12 +98,13 @@ theorem extract_set {xs : Vector α n} {i j k : Nat} (h : k < n) {a : α} :
  · simp
  · split <;> simp

+@[grind =]
 theorem set_extract {xs : Vector α n} {i j k : Nat} (h : k < min j n - i) {a : α} :
    (xs.extract i j).set k a = (xs.set (i + k) a).extract i j := by
  rcases xs with ⟨xs, rfl⟩
  simp [Array.set_extract]

-@[simp]
+@[simp, grind =]
 theorem extract_append {xs : Vector α n} {ys : Vector α m} {i j : Nat} :
    (xs ++ ys).extract i j =
      (xs.extract i j ++ ys.extract (i - n) (j - n)).cast (by omega) := by
@@ -128,12 +130,12 @@ theorem extract_append_left {xs : Vector α n} {ys : Vector α m} :
  congr 1
  omega

-@[simp] theorem map_extract {xs : Vector α n} {i j : Nat} :
+@[simp, grind =] theorem map_extract {xs : Vector α n} {i j : Nat} :
    (xs.extract i j).map f = (xs.map f).extract i j := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem extract_replicate {a : α} {n i j : Nat} :
+@[simp, grind =] theorem extract_replicate {a : α} {n i j : Nat} :
    (replicate n a).extract i j = replicate (min j n - i) a := by
  ext i h
  simp
@@ -161,6 +163,7 @@ theorem set_eq_push_extract_append_extract {xs : Vector α n} {i : Nat} (h : i <
  rcases xs with ⟨as, rfl⟩
  simp [Array.set_eq_push_extract_append_extract, h]

+@[grind =]
 theorem extract_reverse {xs : Vector α n} {i j : Nat} :
    xs.reverse.extract i j = (xs.extract (n - j) (n - i)).reverse.cast (by omega) := by
  ext i h
@@ -168,6 +171,7 @@ theorem extract_reverse {xs : Vector α n} {i j : Nat} :
  congr 1
  omega

+@[grind =]
 theorem reverse_extract {xs : Vector α n} {i j : Nat} :
    (xs.extract i j).reverse = (xs.reverse.extract (n - j) (n - i)).cast (by omega) := by
  rcases xs with ⟨xs, rfl⟩
--- a/src/Init/Data/Vector/FinRange.lean
+++ b/src/Init/Data/Vector/FinRange.lean
@@ -17,7 +17,7 @@ namespace Vector
 /-- `finRange n` is the vector of all elements of `Fin n` in order. -/
 protected def finRange (n : Nat) : Vector (Fin n) n := ofFn fun i => i

-@[simp] theorem getElem_finRange {i : Nat} (h : i < n) :
+@[simp, grind =] theorem getElem_finRange {i : Nat} (h : i < n) :
    (Vector.finRange n)[i] = ⟨i, h⟩ := by
  simp [Vector.finRange]

@@ -39,6 +39,7 @@ theorem finRange_succ_last {n} :
    · simp_all
      omega

+@[grind _=_]
 theorem finRange_reverse {n} : (Vector.finRange n).reverse = (Vector.finRange n).map Fin.rev := by
  ext i h
  simp
--- a/src/Init/Data/Vector/Find.lean
+++ b/src/Init/Data/Vector/Find.lean
@@ -44,12 +44,23 @@ theorem findSome?_singleton {a : α} {f : α → Option β} : #v[a].findSome? f
  rcases xs with ⟨xs, rfl⟩
  simp only [push_mk, findSomeRev?_mk, Array.findSomeRev?_push_of_isNone, h]

+@[grind =]
+theorem findSomeRev?_push {xs : Vector α n} {a : α} {f : α → Option β} :
+    (xs.push a).findSomeRev? f = (f a).or (xs.findSomeRev? f) := by
+  match h : f a with
+  | some b =>
+    rw [findSomeRev?_push_of_isSome]
+    all_goals simp_all
+  | none =>
+    rw [findSomeRev?_push_of_isNone]
+    all_goals simp_all
+
 theorem exists_of_findSome?_eq_some {f : α → Option β} {xs : Vector α n} (w : xs.findSome? f = some b) :
    ∃ a, a ∈ xs ∧ f a = some b := by
  rcases xs with ⟨xs, rfl⟩
  simpa using Array.exists_of_findSome?_eq_some (by simpa using w)

-@[simp] theorem findSome?_eq_none_iff {f : α → Option β} {xs : Vector α n} :
+@[simp, grind =] theorem findSome?_eq_none_iff {f : α → Option β} {xs : Vector α n} :
    xs.findSome? f = none ↔ ∀ x ∈ xs, f x = none := by
  rcases xs with ⟨xs, rfl⟩
  simp
@@ -72,24 +83,27 @@ theorem findSome?_eq_some_iff {f : α → Option β} {xs : Vector α n} {b : β}
  · rintro ⟨k₁, k₂, h, ys, a, zs, w, h₁, h₂⟩
    exact ⟨ys.toArray, a, zs.toArray, by simp [w], h₁, by simpa using h₂⟩

-@[simp] theorem findSome?_guard {xs : Vector α n} : findSome? (Option.guard fun x => p x) xs = find? p xs := by
+@[simp, grind =] theorem findSome?_guard {xs : Vector α n} : findSome? (Option.guard p) xs = find? p xs := by
  rcases xs with ⟨xs, rfl⟩
  simp

-theorem find?_eq_findSome?_guard {xs : Vector α n} : find? p xs = findSome? (Option.guard fun x => p x) xs :=
+theorem find?_eq_findSome?_guard {xs : Vector α n} : find? p xs = findSome? (Option.guard p) xs :=
  findSome?_guard.symm

-@[simp] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Vector α n} :
+@[simp, grind =] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Vector α n} :
    (xs.findSome? f).map g = xs.findSome? (Option.map g ∘ f) := by
  cases xs; simp

+@[grind _=_]
 theorem findSome?_map {f : β → γ} {xs : Vector β n} : findSome? p (xs.map f) = xs.findSome? (p ∘ f) := by
  rcases xs with ⟨xs, rfl⟩
  simp [Array.findSome?_map]

+@[grind =]
 theorem findSome?_append {xs : Vector α n₁} {ys : Vector α n₂} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
  cases xs; cases ys; simp [Array.findSome?_append]

+@[grind =]
 theorem getElem?_zero_flatten {xss : Vector (Vector α m) n} :
    (flatten xss)[0]? = xss.findSome? fun xs => xs[0]? := by
  cases xss using vector₂_induction
@@ -106,12 +120,14 @@ theorem getElem_zero_flatten.proof {xss : Vector (Vector α m) n} (h : 0 < n * m
      Option.isSome_some, and_true]
    exact ⟨⟨xss[0], h₂ _ (by simp)⟩, by simp⟩

+@[grind =]
 theorem getElem_zero_flatten {xss : Vector (Vector α m) n} (h : 0 < n * m) :
    (flatten xss)[0] = (xss.findSome? fun xs => xs[0]?).get (getElem_zero_flatten.proof h) := by
  have t := getElem?_zero_flatten (xss := xss)
  simp [getElem?_eq_getElem, h] at t
  simp [← t]

+@[grind =]
 theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
  rw [replicate_eq_mk_replicate, findSome?_mk, Array.findSome?_replicate]

@@ -142,9 +158,9 @@ abbrev findSome?_mkVector_of_isNone := @findSome?_replicate_of_isNone

 /-! ### find? -/

-@[simp] theorem find?_empty : find? p #v[] = none := rfl
+@[simp, grind =] theorem find?_empty : find? p #v[] = none := rfl

-theorem find?_singleton {a : α} {p : α → Bool} :
+@[grind =]theorem find?_singleton {a : α} {p : α → Bool} :
    #v[a].find? p = if p a then some a else none := by
  simp

@@ -152,11 +168,23 @@ theorem find?_singleton {a : α} {p : α → Bool} :
    findRev? p (xs.push a) = some a := by
  cases xs; simp [h]

-@[simp] theorem findRev?_cons_of_neg {xs : Vector α n} (h : ¬p a) :
+@[simp] theorem findRev?_push_of_neg {xs : Vector α n} (h : ¬p a) :
    findRev? p (xs.push a) = findRev? p xs := by
  cases xs; simp [h]

-@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
+@[deprecated findRev?_push_of_neg (since := "2025-06-12")]
+abbrev findRev?_cons_of_neg := @findRev?_push_of_neg
+
+@[grind =]
+theorem finRev?_push {xs : Vector α n} :
+    findRev? p (xs.push a) = (Option.guard p a).or (xs.findRev? p) := by
+  cases h : p a
+  · rw [findRev?_push_of_neg, Option.guard_eq_none_iff.mpr h]
+    all_goals simp [h]
+  · rw [findRev?_push_of_pos, Option.guard_eq_some_iff.mpr ⟨rfl, h⟩]
+    all_goals simp [h]
+
+@[simp, grind =] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
  cases l; simp

 theorem find?_eq_some_iff_append {xs : Vector α n} :
@@ -181,35 +209,38 @@ theorem find?_push_eq_some {xs : Vector α n} :
    (xs.push a).find? p = some b ↔ xs.find? p = some b ∨ (xs.find? p = none ∧ (p a ∧ a = b)) := by
  cases xs; simp

-@[simp] theorem find?_isSome {xs : Vector α n} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
+@[simp, grind =] theorem find?_isSome {xs : Vector α n} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
  cases xs; simp

+@[grind →]
 theorem find?_some {xs : Vector α n} (h : find? p xs = some a) : p a := by
  rcases xs with ⟨xs, rfl⟩
  simp at h
  exact Array.find?_some h

+@[grind →]
 theorem mem_of_find?_eq_some {xs : Vector α n} (h : find? p xs = some a) : a ∈ xs := by
  cases xs
  simp at h
  simpa using Array.mem_of_find?_eq_some h

+@[grind]
 theorem get_find?_mem {xs : Vector α n} (h) : (xs.find? p).get h ∈ xs := by
  cases xs
  simp [Array.get_find?_mem]

-@[simp] theorem find?_map {f : β → α} {xs : Vector β n} :
+@[simp, grind =] theorem find?_map {f : β → α} {xs : Vector β n} :
    find? p (xs.map f) = (xs.find? (p ∘ f)).map f := by
  cases xs; simp

-@[simp] theorem find?_append {xs : Vector α n₁} {ys : Vector α n₂} :
+@[simp, grind =] theorem find?_append {xs : Vector α n₁} {ys : Vector α n₂} :
    (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
  cases xs
  cases ys
  simp

-@[simp] theorem find?_flatten {xs : Vector (Vector α m) n} {p : α → Bool} :
-    xs.flatten.find? p = xs.findSome? (·.find? p) := by
+@[simp, grind =] theorem find?_flatten {xs : Vector (Vector α m) n} {p : α → Bool} :
+    xs.flatten.find? p = xs.findSome? (find? p) := by
  cases xs using vector₂_induction
  simp [Array.findSome?_map, Function.comp_def]

@@ -217,7 +248,7 @@ theorem find?_flatten_eq_none_iff {xs : Vector (Vector α m) n} {p : α → Bool
    xs.flatten.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
  simp

-@[simp] theorem find?_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
+@[simp, grind =] theorem find?_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
    (xs.flatMap f).find? p = xs.findSome? (fun x => (f x).find? p) := by
  cases xs
  simp [Array.find?_flatMap, Array.flatMap_toArray]
@@ -227,6 +258,7 @@ theorem find?_flatMap_eq_none_iff {xs : Vector α n} {f : α → Vector β m} {p
    (xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
  simp

+@[grind =]
 theorem find?_replicate :
    find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
  rw [replicate_eq_mk_replicate, find?_mk, Array.find?_replicate]
@@ -278,6 +310,7 @@ abbrev find?_mkVector_eq_some_iff := @find?_replicate_eq_some_iff
@[deprecated get_find?_replicate (since := "2025-03-18")]
 abbrev get_find?_mkVector := @get_find?_replicate

+@[grind =]
 theorem find?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Vector α n}
    (H : ∀ (a : α), a ∈ xs → P a) {p : β → Bool} :
    (xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
@@ -291,8 +324,10 @@ theorem find?_eq_some_iff_getElem {xs : Vector α n} {p : α → Bool} {b : α}

 /-! ### findFinIdx? -/

+@[grind =]
 theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p (#v[] : Vector α 0) = none := by simp

+@[grind =]
 theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
    #[a].findFinIdx? p = if p a then some ⟨0, by simp⟩ else none := by
  simp
@@ -302,6 +337,12 @@ theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
  subst w
  simp

+@[simp, grind =] theorem findFinIdx?_eq_none_iff {xs : Vector α n} {p : α → Bool} :
+    xs.findFinIdx? p = none ↔ ∀ x, x ∈ xs → ¬ p x := by
+  rcases xs with ⟨xs, rfl⟩
+  simp [Array.findFinIdx?_eq_none_iff]
+
+@[grind =]
 theorem findFinIdx?_push {xs : Vector α n} {a : α} {p : α → Bool} :
    (xs.push a).findFinIdx? p =
      ((xs.findFinIdx? p).map Fin.castSucc).or (if p a then some ⟨n, by simp⟩ else none) := by
@@ -309,6 +350,7 @@ theorem findFinIdx?_push {xs : Vector α n} {a : α} {p : α → Bool} :
  simp [Array.findFinIdx?_push, Option.map_or, Function.comp_def]
  congr

+@[grind =]
 theorem findFinIdx?_append {xs : Vector α n₁} {ys : Vector α n₂} {p : α → Bool} :
    (xs ++ ys).findFinIdx? p =
      ((xs.findFinIdx? p).map (Fin.castLE (by simp))).or
@@ -317,13 +359,13 @@ theorem findFinIdx?_append {xs : Vector α n₁} {ys : Vector α n₂} {p : α
  rcases ys with ⟨ys, rfl⟩
  simp [Array.findFinIdx?_append, Option.map_or, Function.comp_def]

-@[simp]
+@[simp, grind =]
 theorem isSome_findFinIdx? {xs : Vector α n} {p : α → Bool} :
    (xs.findFinIdx? p).isSome = xs.any p := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp]
+@[simp, grind =]
 theorem isNone_findFinIdx? {xs : Vector α n} {p : α → Bool} :
    (xs.findFinIdx? p).isNone = xs.all (fun x => ¬ p x) := by
  rcases xs with ⟨xs, rfl⟩
--- a/src/Init/Data/Vector/InsertIdx.lean
+++ b/src/Init/Data/Vector/InsertIdx.lean
@@ -30,15 +30,20 @@ section InsertIdx

 variable {a : α}

-@[simp]
+@[simp, grind =]
 theorem insertIdx_zero {xs : Vector α n} {x : α} : xs.insertIdx 0 x = (#v[x] ++ xs).cast (by omega) := by
  cases xs
  simp

-theorem eraseIdx_insertIdx {i : Nat} {xs : Vector α n} {h : i ≤ n} :
+theorem eraseIdx_insertIdx_self {i : Nat} {xs : Vector α n} {h : i ≤ n} :
    (xs.insertIdx i a).eraseIdx i = xs := by
  rcases xs with ⟨xs, rfl⟩
-  simp_all [Array.eraseIdx_insertIdx]
+  simp_all [Array.eraseIdx_insertIdx_self]
+
+@[deprecated eraseIdx_insertIdx_self (since := "2025-06-15")]
+theorem eraseIdx_insertIdx {i : Nat} {xs : Vector α n} {h : i ≤ n} :
+    (xs.insertIdx i a).eraseIdx i = xs := by
+  simp [eraseIdx_insertIdx_self]

 theorem insertIdx_eraseIdx_of_ge {xs : Vector α n}
    (w₁ : i < n) (w₂ : j ≤ n - 1) (h : i ≤ j) :
@@ -54,6 +59,18 @@ theorem insertIdx_eraseIdx_of_le {xs : Vector α n}
  rcases xs with ⟨as, rfl⟩
  simpa using Array.insertIdx_eraseIdx_of_le (by simpa) (by simpa) (by simpa)

+@[grind =]
+theorem insertIdx_eraseIdx {as : Vector α n} (h₁ : i < n) (h₂ : j ≤ n - 1) :
+    (as.eraseIdx i).insertIdx j a =
+      if h : i ≤ j then
+        ((as.insertIdx (j + 1) a).eraseIdx i).cast (by omega)
+      else
+        ((as.insertIdx j a).eraseIdx (i + 1) (by simp_all)).cast (by omega) := by
+  split <;> rename_i h'
+  · rw [insertIdx_eraseIdx_of_ge] <;> omega
+  · rw [insertIdx_eraseIdx_of_le] <;> omega
+
+@[grind =]
 theorem insertIdx_comm (a b : α) {i j : Nat} {xs : Vector α n} (_ : i ≤ j) (_ : j ≤ n) :
    (xs.insertIdx i a).insertIdx (j + 1) b =
      (xs.insertIdx j b).insertIdx i a := by
@@ -70,6 +87,7 @@ theorem insertIdx_size_self {xs : Vector α n} {x : α} : xs.insertIdx n x = xs.
  rcases xs with ⟨as, rfl⟩
  simp

+@[grind =]
 theorem getElem_insertIdx {xs : Vector α n} {x : α} {i k : Nat} (w : i ≤ n) (h : k < n + 1) :
    (xs.insertIdx i x)[k] =
      if h₁ : k < i then
@@ -98,6 +116,7 @@ theorem getElem_insertIdx_of_gt {xs : Vector α n} {x : α} {i k : Nat} (w : k
  simp [Array.getElem_insertIdx, w, h]
  rw [dif_neg (by omega), dif_neg (by omega)]

+@[grind =]
 theorem getElem?_insertIdx {xs : Vector α n} {x : α} {i k : Nat} (h : i ≤ n) :
    (xs.insertIdx i x)[k]? =
      if k < i then
--- a/src/Init/Data/Vector/Lemmas.lean
+++ b/src/Init/Data/Vector/Lemmas.lean
@@ -2538,23 +2538,23 @@ theorem foldr_hom (f : β₁ → β₂) {g₁ : α → β₁ → β₁} {g₂ :
  rw [Array.foldr_hom _ H]

 /--
-We can prove that two folds over the same array are related (by some arbitrary relation)
-if we know that the initial elements are related and the folding function, for each element of the array,
-preserves the relation.
+We can prove that two folds over the same vector are related (by some arbitrary relation)
+if we know that the initial elements are related and the folding function, for each element of the
+vector, preserves the relation.
 -/
-theorem foldl_rel {xs : Vector α n} {f g : β → α → β} {a b : β} {r : β → β → Prop}
-    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
+theorem foldl_rel {xs : Vector α n} {f : β → α → β} {g : γ → α → γ} {a : β} {b : γ} {r : β → γ → Prop}
+    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f c a) (g c' a)) :
    r (xs.foldl (fun acc a => f acc a) a) (xs.foldl (fun acc a => g acc a) b) := by
  rcases xs with ⟨xs, rfl⟩
  simpa using Array.foldl_rel h (by simpa using h')

 /--
-We can prove that two folds over the same array are related (by some arbitrary relation)
-if we know that the initial elements are related and the folding function, for each element of the array,
-preserves the relation.
+We can prove that two folds over the same vector are related (by some arbitrary relation)
+if we know that the initial elements are related and the folding function, for each element of the
+vector, preserves the relation.
 -/
-theorem foldr_rel {xs : Vector α n} {f g : α → β → β} {a b : β} {r : β → β → Prop}
-    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
+theorem foldr_rel {xs : Vector α n} {f : α → β → β} {g : α → γ → γ} {a : β} {b : γ} {r : β → γ → Prop}
+    (h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f a c) (g a c')) :
    r (xs.foldr (fun a acc => f a acc) a) (xs.foldr (fun a acc => g a acc) b) := by
  rcases xs with ⟨xs, rfl⟩
  simpa using Array.foldr_rel h (by simpa using h')
@@ -3076,7 +3076,7 @@ theorem getElem_push_last {xs : Vector α n} {x : α} : (xs.push x)[n] = x := by

 /-! ### zipWith -/

-@[simp] theorem getElem_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} {i : Nat}
+@[simp, grind =] theorem getElem_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} {i : Nat}
    (hi : i < n) : (zipWith f as bs)[i] = f as[i] bs[i] := by
  cases as
  cases bs
--- a/src/Init/Data/Vector/MapIdx.lean
+++ b/src/Init/Data/Vector/MapIdx.lean
@@ -19,13 +19,13 @@ namespace Vector

 /-! ### mapFinIdx -/

-@[simp] theorem getElem_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat}
+@[simp, grind =] theorem getElem_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat}
    (h : i < n) :
    (xs.mapFinIdx f)[i] = f i xs[i] h := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem getElem?_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat} :
+@[simp, grind =] theorem getElem?_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat} :
    (xs.mapFinIdx f)[i]? =
      xs[i]?.pbind fun b h => some <| f i b (getElem?_eq_some_iff.1 h).1 := by
  simp only [getElem?_def, getElem_mapFinIdx]
@@ -33,12 +33,12 @@ namespace Vector

 /-! ### mapIdx -/

-@[simp] theorem getElem_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} (h : i < n) :
+@[simp, grind =] theorem getElem_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} (h : i < n) :
    (xs.mapIdx f)[i] = f i (xs[i]'(by simp_all)) := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} :
+@[simp, grind =] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} :
    (xs.mapIdx f)[i]? = xs[i]?.map (f i) := by
  rcases xs with ⟨xs, rfl⟩
  simp
@@ -47,11 +47,11 @@ end Vector

 namespace Array

-@[simp] theorem mapFinIdx_toVector {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
+@[simp, grind =] theorem mapFinIdx_toVector {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
    xs.toVector.mapFinIdx f = (xs.mapFinIdx f).toVector.cast (by simp) := by
  ext <;> simp

-@[simp] theorem mapIdx_toVector {f : Nat → α → β} {xs : Array α} :
+@[simp, grind =] theorem mapIdx_toVector {f : Nat → α → β} {xs : Array α} :
    xs.toVector.mapIdx f = (xs.mapIdx f).toVector.cast (by simp) := by
  ext <;> simp

@@ -61,12 +61,12 @@ namespace Vector

 /-! ### zipIdx -/

-@[simp] theorem toList_zipIdx {xs : Vector α n} (k : Nat := 0) :
+@[simp, grind =] theorem toList_zipIdx {xs : Vector α n} (k : Nat := 0) :
    (xs.zipIdx k).toList = xs.toList.zipIdx k := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem getElem_zipIdx {xs : Vector α n} {i : Nat} {h : i < n} :
+@[simp, grind =] theorem getElem_zipIdx {xs : Vector α n} {i : Nat} {h : i < n} :
    (xs.zipIdx k)[i] = (xs[i]'(by simp_all), k + i) := by
  rcases xs with ⟨xs, rfl⟩
  simp
@@ -116,7 +116,7 @@ abbrev mem_zipWithIndex_iff_getElem? := @mem_zipIdx_iff_getElem?
  subst w
  rfl

-@[simp]
+@[simp, grind =]
 theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #v[] f = #v[] :=
  rfl

@@ -125,6 +125,7 @@ theorem mapFinIdx_eq_ofFn {as : Vector α n} {f : (i : Nat) → α → (h : i <
  rcases as with ⟨as, rfl⟩
  simp [Array.mapFinIdx_eq_ofFn]

+@[grind =]
 theorem mapFinIdx_append {xs : Vector α n} {ys : Vector α m} {f : (i : Nat) → α → (h : i < n + m) → β} :
    (xs ++ ys).mapFinIdx f =
      xs.mapFinIdx (fun i a h => f i a (by omega)) ++
@@ -133,7 +134,7 @@ theorem mapFinIdx_append {xs : Vector α n} {ys : Vector α m} {f : (i : Nat)
  rcases ys with ⟨ys, rfl⟩
  simp [Array.mapFinIdx_append]

-@[simp]
+@[simp, grind =]
 theorem mapFinIdx_push {xs : Vector α n} {a : α} {f : (i : Nat) → α → (h : i < n + 1) → β} :
    mapFinIdx (xs.push a) f =
      (mapFinIdx xs (fun i a h => f i a (by omega))).push (f n a (by simp)) := by
@@ -154,7 +155,7 @@ theorem exists_of_mem_mapFinIdx {b : β} {xs : Vector α n} {f : (i : Nat) →
  rcases xs with ⟨xs, rfl⟩
  exact List.exists_of_mem_mapFinIdx (by simpa using h)

-@[simp] theorem mem_mapFinIdx {b : β} {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+@[simp, grind =] theorem mem_mapFinIdx {b : β} {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
    b ∈ xs.mapFinIdx f ↔ ∃ (i : Nat) (h : i < n), f i xs[i] h = b := by
  rcases xs with ⟨xs, rfl⟩
  simp
@@ -215,7 +216,7 @@ theorem mapFinIdx_eq_mapFinIdx_iff {xs : Vector α n} {f g : (i : Nat) → α
  rw [eq_comm, mapFinIdx_eq_iff]
  simp

-@[simp] theorem mapFinIdx_mapFinIdx {xs : Vector α n}
+@[simp, grind =] theorem mapFinIdx_mapFinIdx {xs : Vector α n}
    {f : (i : Nat) → α → (h : i < n) → β}
    {g : (i : Nat) → β → (h : i < n) → γ} :
    (xs.mapFinIdx f).mapFinIdx g = xs.mapFinIdx (fun i a h => g i (f i a h) h) := by
@@ -229,14 +230,14 @@ theorem mapFinIdx_eq_replicate_iff {xs : Vector α n} {f : (i : Nat) → α →
@[deprecated mapFinIdx_eq_replicate_iff (since := "2025-03-18")]
 abbrev mapFinIdx_eq_mkVector_iff := @mapFinIdx_eq_replicate_iff

-@[simp] theorem mapFinIdx_reverse {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+@[simp, grind =] theorem mapFinIdx_reverse {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
    xs.reverse.mapFinIdx f = (xs.mapFinIdx (fun i a h => f (n - 1 - i) a (by omega))).reverse := by
  rcases xs with ⟨xs, rfl⟩
  simp

 /-! ### mapIdx -/

-@[simp]
+@[simp, grind =]
 theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #v[] = #v[] :=
  rfl

@@ -256,13 +257,14 @@ theorem mapIdx_eq_zipIdx_map {xs : Vector α n} {f : Nat → α → β} :
@[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-27")]
 abbrev mapIdx_eq_zipWithIndex_map := @mapIdx_eq_zipIdx_map

+@[grind =]
 theorem mapIdx_append {xs : Vector α n} {ys : Vector α m} :
    (xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx fun i => f (i + n) := by
  rcases xs with ⟨xs, rfl⟩
  rcases ys with ⟨ys, rfl⟩
  simp [Array.mapIdx_append]

-@[simp]
+@[simp, grind =]
 theorem mapIdx_push {xs : Vector α n} {a : α} :
    mapIdx f (xs.push a) = (mapIdx f xs).push (f n a) := by
  simp [← append_singleton, mapIdx_append]
@@ -275,7 +277,7 @@ theorem exists_of_mem_mapIdx {b : β} {xs : Vector α n}
  rw [mapIdx_eq_mapFinIdx] at h
  simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h

-@[simp] theorem mem_mapIdx {b : β} {xs : Vector α n} :
+@[simp, grind =] theorem mem_mapIdx {b : β} {xs : Vector α n} :
    b ∈ xs.mapIdx f ↔ ∃ (i : Nat) (h : i < n), f i xs[i] = b := by
  constructor
  · intro h
@@ -332,7 +334,7 @@ theorem mapIdx_eq_mapIdx_iff {xs : Vector α n} :
  rcases xs with ⟨xs, rfl⟩
  simp [Array.mapIdx_eq_mapIdx_iff]

-@[simp] theorem mapIdx_set {xs : Vector α n} {i : Nat} {h : i < n} {a : α} :
+@[simp, grind =] theorem mapIdx_set {xs : Vector α n} {i : Nat} {h : i < n} {a : α} :
    (xs.set i a).mapIdx f = (xs.mapIdx f).set i (f i a) (by simpa) := by
  rcases xs with ⟨xs, rfl⟩
  simp
@@ -342,17 +344,17 @@ theorem mapIdx_eq_mapIdx_iff {xs : Vector α n} :
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem back?_mapIdx {xs : Vector α n} {f : Nat → α → β} :
+@[simp, grind =] theorem back?_mapIdx {xs : Vector α n} {f : Nat → α → β} :
    (mapIdx f xs).back? = (xs.back?).map (f (n - 1)) := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem back_mapIdx [NeZero n] {xs : Vector α n} {f : Nat → α → β} :
+@[simp, grind =] theorem back_mapIdx [NeZero n] {xs : Vector α n} {f : Nat → α → β} :
    (mapIdx f xs).back = f (n - 1) (xs.back) := by
  rcases xs with ⟨xs, rfl⟩
  simp

-@[simp] theorem mapIdx_mapIdx {xs : Vector α n} {f : Nat → α → β} {g : Nat → β → γ} :
+@[simp, grind =] theorem mapIdx_mapIdx {xs : Vector α n} {f : Nat → α → β} {g : Nat → β → γ} :
    (xs.mapIdx f).mapIdx g = xs.mapIdx (fun i => g i ∘ f i) := by
  simp [mapIdx_eq_iff]

@@ -364,7 +366,7 @@ theorem mapIdx_eq_replicate_iff {xs : Vector α n} {f : Nat → α → β} {b :
@[deprecated mapIdx_eq_replicate_iff (since := "2025-03-18")]
 abbrev mapIdx_eq_mkVector_iff := @mapIdx_eq_replicate_iff

-@[simp] theorem mapIdx_reverse {xs : Vector α n} {f : Nat → α → β} :
+@[simp, grind =] theorem mapIdx_reverse {xs : Vector α n} {f : Nat → α → β} :
    xs.reverse.mapIdx f = (mapIdx (fun i => f (n - 1 - i)) xs).reverse := by
  rcases xs with ⟨xs, rfl⟩
  simp [Array.mapIdx_reverse]
--- a/src/Init/Data/Vector/OfFn.lean
+++ b/src/Init/Data/Vector/OfFn.lean
@@ -20,15 +20,15 @@ set_option linter.indexVariables true -- Enforce naming conventions for index va

 namespace Vector

-@[simp] theorem getElem_ofFn {α n} {f : Fin n → α} (h : i < n) :
+@[simp, grind =] theorem getElem_ofFn {α n} {f : Fin n → α} (h : i < n) :
    (Vector.ofFn f)[i] = f ⟨i, by simpa using h⟩ := by
  simp [ofFn]

-theorem getElem?_ofFn {α n} {f : Fin n → α} :
+@[simp, grind =] theorem getElem?_ofFn {α n} {f : Fin n → α} :
    (ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none := by
  simp [getElem?_def]

-@[simp 500]
+@[simp 500, grind =]
 theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
  constructor
  · intro w
@@ -37,7 +37,7 @@ theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i =
  · rintro ⟨i, rfl⟩
    apply mem_of_getElem (i := i) <;> simp

-theorem back_ofFn {n} [NeZero n] {f : Fin n → α} :
+@[grind =] theorem back_ofFn {n} [NeZero n] {f : Fin n → α} :
    (ofFn f).back = f ⟨n - 1, by have := NeZero.ne n; omega⟩ := by
  simp [back]

@@ -71,7 +71,7 @@ def ofFnM {n} [Monad m] (f : Fin n → m α) : m (Vector α n) :=
    else
      pure ⟨acc, by omega⟩

-@[simp]
+@[simp, grind =]
 theorem ofFnM_zero [Monad m] {f : Fin 0 → m α} : Vector.ofFnM f = pure #v[] := by
  simp [ofFnM, ofFnM.go]

@@ -114,13 +114,13 @@ theorem ofFnM_add {n m} [Monad m] [LawfulMonad m] {f : Fin (n + k) → m α} :
  | zero => simp
  | succ k ih => simp [ofFnM_succ, ih, ← push_append]

-@[simp, grind] theorem toArray_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
+@[simp, grind =] theorem toArray_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
    toArray <$> ofFnM f = Array.ofFnM f := by
  induction n with
  | zero => simp
  | succ n ih => simp [ofFnM_succ, Array.ofFnM_succ, ← ih]

-@[simp, grind] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
+@[simp, grind =] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
    toList <$> Vector.ofFnM f = List.ofFnM f := by
  unfold toList
  suffices Array.toList <$> (toArray <$> ofFnM f) = List.ofFnM f by
--- a/src/Init/Data/Vector/Range.lean
+++ b/src/Init/Data/Vector/Range.lean
@@ -112,14 +112,25 @@ theorem range'_eq_append_iff : range' s (n + m) = xs ++ ys ↔ xs = range' s n
  · rintro ⟨h₁, h₂⟩
    exact ⟨n, by omega, by simp_all⟩

-@[simp] theorem find?_range'_eq_some {s n : Nat} {i : Nat} {p : Nat → Bool} :
+@[simp, grind =] theorem find?_range'_eq_some {s n : Nat} {i : Nat} {p : Nat → Bool} :
    (range' s n).find? p = some i ↔ p i ∧ i ∈ range' s n ∧ ∀ j, s ≤ j → j < i → !p j := by
  simp [range'_eq_mk_range']

-@[simp] theorem find?_range'_eq_none {s n : Nat} {p : Nat → Bool} :
+@[simp, grind =] theorem find?_range'_eq_none {s n : Nat} {p : Nat → Bool} :
    (range' s n).find? p = none ↔ ∀ i, s ≤ i → i < s + n → !p i := by
  simp [range'_eq_mk_range']

+@[simp, grind =]
+theorem count_range' {a s n step} (h : 0 < step := by simp) :
+    count a (range' s n step) = if ∃ i, i < n ∧ a = s + step * i then 1 else 0 := by
+  rw [range'_eq_mk_range', count_mk, ← Array.count_range' h]
+
+@[simp, grind =]
+theorem count_range_1' {a s n} :
+    count a (range' s n) = if s ≤ a ∧ a < s + n then 1 else 0 := by
+  rw [range'_eq_mk_range', count_mk, ← Array.count_range_1']
+
+
 /-! ### range -/

@[simp, grind =] theorem getElem_range {i : Nat} (hi : i < n) : (Vector.range n)[i] = i := by
@@ -171,9 +182,15 @@ theorem self_mem_range_succ {n : Nat} : n ∈ range (n + 1) := by simp
    (range n).find? p = none ↔ ∀ i, i < n → !p i := by
  simp [range_eq_range']

+@[simp, grind =]
+theorem count_range {a n} :
+    count a (range n) = if a < n then 1 else 0 := by
+  rw [range_eq_range', count_range_1']
+  simp
+
 /-! ### zipIdx -/

-@[simp]
+@[simp, grind =]
 theorem getElem?_zipIdx {xs : Vector α n} {i j} : (zipIdx xs i)[j]? = xs[j]?.map fun a => (a, i + j) := by
  simp [getElem?_def]

@@ -216,7 +233,7 @@ theorem zipIdx_eq_map_add {xs : Vector α n} {i : Nat} :
  simp only [zipIdx_mk, map_mk, eq_mk]
  rw [Array.zipIdx_eq_map_add]

-@[simp]
+@[simp, grind =]
 theorem zipIdx_singleton {x : α} {k : Nat} : zipIdx (#v[x]) k = #v[(x, k)] :=
  rfl

@@ -265,6 +282,7 @@ theorem zipIdx_map {f : α → β} {xs : Vector α n} {k : Nat} :
  rcases xs with ⟨xs, rfl⟩
  simp [Array.zipIdx_map]

+@[grind =]
 theorem zipIdx_append {xs : Vector α n} {ys : Vector α m} {k : Nat} :
    zipIdx (xs ++ ys) k = zipIdx xs k ++ zipIdx ys (k + n) := by
  rcases xs with ⟨xs, rfl⟩
--- a/src/Init/Data/Vector/Zip.lean
+++ b/src/Init/Data/Vector/Zip.lean
@@ -46,6 +46,7 @@ theorem zipWith_self {f : α → α → δ} {xs : Vector α n} : zipWith f xs xs
 See also `getElem?_zipWith'` for a variant
 using `Option.map` and `Option.bind` rather than a `match`.
 -/
+@[grind =]
 theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
    (zipWith f as bs)[i]? = match as[i]?, bs[i]? with
      | some a, some b => some (f a b) | _, _ => none := by
@@ -74,53 +75,61 @@ theorem getElem?_zip_eq_some {as : Vector α n} {bs : Vector β n} {z : α × β
  rcases bs with ⟨bs, h⟩
  simp [Array.getElem?_zip_eq_some]

-@[simp]
+@[simp, grind =]
 theorem zipWith_map {μ} {f : γ → δ → μ} {g : α → γ} {h : β → δ} {as : Vector α n} {bs : Vector β n} :
    zipWith f (as.map g) (bs.map h) = zipWith (fun a b => f (g a) (h b)) as bs := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.zipWith_map]

+@[grind =]
 theorem zipWith_map_left {as : Vector α n} {bs : Vector β n} {f : α → α'} {g : α' → β → γ} :
    zipWith g (as.map f) bs = zipWith (fun a b => g (f a) b) as bs := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.zipWith_map_left]

+@[grind =]
 theorem zipWith_map_right {as : Vector α n} {bs : Vector β n} {f : β → β'} {g : α → β' → γ} :
    zipWith g as (bs.map f) = zipWith (fun a b => g a (f b)) as bs := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.zipWith_map_right]

+@[grind =]
 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
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simpa using Array.zipWith_foldr_eq_zip_foldr

+@[grind =]
 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
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simpa using Array.zipWith_foldl_eq_zip_foldl

+@[grind =]
 theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {as : Vector γ n} {bs : Vector δ n} :
    map f (zipWith g as bs) = zipWith (fun x y => f (g x y)) as bs := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.map_zipWith]

+@[grind =]
 theorem take_zipWith : (zipWith f as bs).take i = zipWith f (as.take i) (bs.take i) := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.take_zipWith]

+@[grind =]
 theorem extract_zipWith : (zipWith f as bs).extract i j = zipWith f (as.extract i j) (bs.extract i j) := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.extract_zipWith]

+@[grind =]
 theorem zipWith_append {f : α → β → γ}
    {as : Vector α n} {as' : Vector α m} {bs : Vector β n} {bs' : Vector β m} :
    zipWith f (as ++ as') (bs ++ bs') = zipWith f as bs ++ zipWith f as' bs' := by
@@ -147,7 +156,8 @@ theorem zipWith_eq_append_iff {f : α → β → γ} {as : Vector α (n + m)} {b
    simp only at w₁ w₂
    exact ⟨as₁, as₂, bs₁, bs₂, by simpa [hw, hy] using ⟨w₁, w₂⟩⟩

-@[simp] theorem zipWith_replicate {a : α} {b : β} {n : Nat} :
+@[simp, grind =]
+theorem zipWith_replicate {a : α} {b : β} {n : Nat} :
    zipWith f (replicate n a) (replicate n b) = replicate n (f a b) := by
  ext
  simp
@@ -167,6 +177,7 @@ theorem map_zip_eq_zipWith {f : α × β → γ} {as : Vector α n} {bs : Vector
  rcases bs with ⟨bs, h⟩
  simp [Array.map_zip_eq_zipWith]

+@[grind =]
 theorem reverse_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} :
    (zipWith f as bs).reverse = zipWith f as.reverse bs.reverse := by
  rcases as with ⟨as, rfl⟩
@@ -175,7 +186,7 @@ theorem reverse_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector

 /-! ### zip -/

-@[simp]
+@[simp, grind =]
 theorem getElem_zip {as : Vector α n} {bs : Vector β n} {i : Nat} {h : i < n} :
    (zip as bs)[i] = (as[i], bs[i]) :=
  getElem_zipWith ..
@@ -185,18 +196,22 @@ theorem zip_eq_zipWith {as : Vector α n} {bs : Vector β n} : zip as bs = zipWi
  rcases bs with ⟨bs, h⟩
  simp [Array.zip_eq_zipWith, h]

+@[grind _=_]
 theorem zip_map {f : α → γ} {g : β → δ} {as : Vector α n} {bs : Vector β n} :
    zip (as.map f) (bs.map g) = (zip as bs).map (Prod.map f g) := by
  rcases as with ⟨as, rfl⟩
  rcases bs with ⟨bs, h⟩
  simp [Array.zip_map, h]

+@[grind _=_]
 theorem zip_map_left {f : α → γ} {as : Vector α n} {bs : Vector β n} :
    zip (as.map f) bs = (zip as bs).map (Prod.map f id) := by rw [← zip_map, map_id]

+@[grind _=_]
 theorem zip_map_right {f : β → γ} {as : Vector α n} {bs : Vector β n} :
    zip as (bs.map f) = (zip as bs).map (Prod.map id f) := by rw [← zip_map, map_id]

+@[grind =]
 theorem zip_append {as : Vector α n} {bs : Vector β n} {as' : Vector α m} {bs' : Vector β m} :
    zip (as ++ as') (bs ++ bs') = zip as bs ++ zip as' bs' := by
  rcases as with ⟨as, rfl⟩
@@ -205,6 +220,7 @@ theorem zip_append {as : Vector α n} {bs : Vector β n} {as' : Vector α m} {bs
  rcases bs' with ⟨bs', h'⟩
  simp [Array.zip_append, h, h']

+@[grind =]
 theorem zip_map' {f : α → β} {g : α → γ} {xs : Vector α n} :
    zip (xs.map f) (xs.map g) = xs.map fun a => (f a, g a) := by
  rcases xs with ⟨xs, rfl⟩
@@ -248,7 +264,8 @@ theorem zip_eq_append_iff {as : Vector α (n + m)} {bs : Vector β (n + m)} {xs
      ∃ as₁ as₂ bs₁ bs₂, as = as₁ ++ as₂ ∧ bs = bs₁ ++ bs₂ ∧ xs = zip as₁ bs₁ ∧ ys = zip as₂ bs₂ := by
  simp [zip_eq_zipWith, zipWith_eq_append_iff]

-@[simp] theorem zip_replicate {a : α} {b : β} {n : Nat} :
+@[simp, grind =]
+theorem zip_replicate {a : α} {b : β} {n : Nat} :
    zip (replicate n a) (replicate n b) = replicate n (a, b) := by
  ext <;> simp

@@ -257,14 +274,17 @@ abbrev zip_mkVector := @zip_replicate

 /-! ### unzip -/

-@[simp] theorem unzip_fst : (unzip xs).fst = xs.map Prod.fst := by
+@[simp, grind =]
+theorem unzip_fst : (unzip xs).fst = xs.map Prod.fst := by
  cases xs
  simp_all

-@[simp] theorem unzip_snd : (unzip xs).snd = xs.map Prod.snd := by
+@[simp, grind =]
+theorem unzip_snd : (unzip xs).snd = xs.map Prod.snd := by
  cases xs
  simp_all

+@[grind =]
 theorem unzip_eq_map {xs : Vector (α × β) n} : unzip xs = (xs.map Prod.fst, xs.map Prod.snd) := by
  cases xs
  simp [List.unzip_eq_map]
@@ -296,7 +316,8 @@ theorem zip_of_prod {as : Vector α n} {bs : Vector β n} {xs : Vector (α × β
    (hr : xs.map Prod.snd = bs) : xs = as.zip bs := by
  rw [← hl, ← hr, ← zip_unzip xs, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]

-@[simp] theorem unzip_replicate {a : α} {b : β} {n : Nat} :
+@[simp, grind =]
+theorem unzip_replicate {a : α} {b : β} {n : Nat} :
    unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
  ext1 <;> simp

--- a/src/Init/GetElem.lean
+++ b/src/Init/GetElem.lean
@@ -47,7 +47,7 @@ proof in the context using `have`, because `get_elem_tactic` tries

 The proof side-condition `valid xs i` is automatically dispatched by the
 `get_elem_tactic` tactic; this tactic can be extended by adding more clauses to
-`get_elem_tactic_trivial` using `macro_rules`.
+`get_elem_tactic_extensible` using `macro_rules`.

 `xs[i]?` and `xs[i]!` do not impose a proof obligation; the former returns
 an `Option elem`, with `none` signalling that the value isn't present, and
@@ -281,7 +281,7 @@ instance [GetElem? cont Nat elem dom] [h : LawfulGetElem cont Nat elem dom] :
@[simp, grind =] theorem getElem!_fin [GetElem? Cont Nat Elem Dom] (a : Cont) (i : Fin n) [Inhabited Elem] : a[i]! = a[i.1]! := rfl

 macro_rules
-  | `(tactic| get_elem_tactic_trivial) => `(tactic| (with_reducible apply Fin.val_lt_of_le); get_elem_tactic_trivial; done)
+  | `(tactic| get_elem_tactic_extensible) => `(tactic| (with_reducible apply Fin.val_lt_of_le); get_elem_tactic_extensible; done)

 end Fin

--- a/src/Init/Grind.lean
+++ b/src/Init/Grind.lean
@@ -14,7 +14,7 @@ import Init.Grind.Propagator
 import Init.Grind.Util
 import Init.Grind.Offset
 import Init.Grind.PP
-import Init.Grind.CommRing
+import Init.Grind.Ring
 import Init.Grind.Module
 import Init.Grind.Ordered
 import Init.Grind.Ext
--- a/src/Init/Grind/CommRing.lean
+++ b/src/Init/Grind/CommRing.lean
@@ -1,16 +0,0 @@
-/-
-Copyright (c) 2025 Lean FRO, LLC. or its affiliates. All Rights Reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Morrison
-/
-module
-
-prelude
-import Init.Grind.CommRing.Basic
-import Init.Grind.CommRing.Int
-import Init.Grind.CommRing.UInt
-import Init.Grind.CommRing.SInt
-import Init.Grind.CommRing.Fin
-import Init.Grind.CommRing.BitVec
-import Init.Grind.CommRing.Poly
-import Init.Grind.CommRing.Field
--- a/src/Init/Grind/Module/Basic.lean
+++ b/src/Init/Grind/Module/Basic.lean
@@ -7,12 +7,15 @@ module

 prelude
 import Init.Data.Int.Order
+import Init.Grind.ToInt

 namespace Lean.Grind

+class AddRightCancel (M : Type u) [Add M] where
+  add_right_cancel : ∀ a b c : M, a + c = b + c → a = b
+
 class NatModule (M : Type u) extends Zero M, Add M, HMul Nat M M where
  add_zero : ∀ a : M, a + 0 = a
-  zero_add : ∀ a : M, 0 + a = a
  add_comm : ∀ a b : M, a + b = b + a
  add_assoc : ∀ a b c : M, a + b + c = a + (b + c)
  zero_hmul : ∀ a : M, 0 * a = 0
@@ -26,7 +29,6 @@ attribute [instance 100] NatModule.toZero NatModule.toAdd NatModule.toHMul

 class IntModule (M : Type u) extends Zero M, Add M, Neg M, Sub M, HMul Int M M where
  add_zero : ∀ a : M, a + 0 = a
-  zero_add : ∀ a : M, 0 + a = a
  add_comm : ∀ a b : M, a + b = b + a
  add_assoc : ∀ a b c : M, a + b + c = a + (b + c)
  zero_hmul : ∀ a : M, (0 : Int) * a = 0
@@ -38,6 +40,15 @@ class IntModule (M : Type u) extends Zero M, Add M, Neg M, Sub M, HMul Int M M w
  neg_add_cancel : ∀ a : M, -a + a = 0
  sub_eq_add_neg : ∀ a b : M, a - b = a + -b

+namespace NatModule
+
+variable {M : Type u} [NatModule M]
+
+theorem zero_add (a : M) : 0 + a = a := by
+  rw [add_comm, add_zero]
+
+end NatModule
+
 namespace IntModule

 attribute [instance 100] IntModule.toZero IntModule.toAdd IntModule.toNeg IntModule.toSub IntModule.toHMul
@@ -52,9 +63,21 @@ instance toNatModule (M : Type u) [i : IntModule M] : NatModule M :=

 variable {M : Type u} [IntModule M]

+instance (priority := 100) (M : Type u) [IntModule M] : SMul Nat M where
+  smul a x := (a : Int) * x
+
+instance (priority := 100) (M : Type u) [IntModule M] : SMul Int M where
+  smul a x := a * x
+
+theorem zero_add (a : M) : 0 + a = a := by
+  rw [add_comm, add_zero]
+
 theorem add_neg_cancel (a : M) : a + -a = 0 := by
  rw [add_comm, neg_add_cancel]

+theorem add_left_comm (a b c : M) : a + (b + c) = b + (a + c) := by
+  rw [← add_assoc, ← add_assoc, add_comm a]
+
 theorem add_left_inj {a b : M} (c : M) : a + c = b + c ↔ a = b :=
  ⟨fun h => by simpa [add_assoc, add_neg_cancel, add_zero] using (congrArg (· + -c) h),
   fun g => congrArg (· + c) g⟩
@@ -93,6 +116,12 @@ theorem sub_eq_iff {a b c : M} : a - b = c ↔ a = c + b := by
 theorem sub_eq_zero_iff {a b : M} : a - b = 0 ↔ a = b := by
  simp [sub_eq_iff, zero_add]

+theorem add_sub_cancel {a b : M} : a + b - b = a := by
+  rw [sub_eq_add_neg, add_assoc, add_neg_cancel, add_zero]
+
+theorem sub_add_cancel {a b : M} : a - b + b = a := by
+  rw [sub_eq_add_neg, add_assoc, neg_add_cancel, add_zero]
+
 theorem neg_hmul (n : Int) (a : M) : (-n) * a = - (n * a) := by
  apply (add_left_inj (n * a)).mp
  rw [← add_hmul, Int.add_left_neg, zero_hmul, neg_add_cancel]
@@ -101,6 +130,12 @@ theorem hmul_neg (n : Int) (a : M) : n * (-a) = - (n * a) := by
  apply (add_left_inj (n * a)).mp
  rw [← hmul_add, neg_add_cancel, neg_add_cancel, hmul_zero]

+theorem hmul_sub (k : Int) (a b : M) : k * (a - b) = k * a - k * b := by
+  rw [sub_eq_add_neg, hmul_add, hmul_neg, ← sub_eq_add_neg]
+
+theorem sub_hmul (k₁ k₂ : Int) (a : M) : (k₁ - k₂) * a = k₁ * a - k₂ * a := by
+  rw [Int.sub_eq_add_neg, add_hmul, neg_hmul, ← sub_eq_add_neg]
+
 end IntModule

 /--
@@ -111,4 +146,17 @@ class NoNatZeroDivisors (α : Type u) [Zero α] [HMul Nat α α] where

 export NoNatZeroDivisors (no_nat_zero_divisors)

+instance [ToInt α (some lo) (some hi)] [IntModule α] [ToInt.Zero α (some lo) (some hi)] [ToInt.Add α (some lo) (some hi)] : ToInt.Neg α (some lo) (some hi) where
+  toInt_neg x := by
+    have := (ToInt.Add.toInt_add (-x) x).symm
+    rw [IntModule.neg_add_cancel, ToInt.Zero.toInt_zero] at this
+    rw [ToInt.wrap_eq_wrap_iff] at this
+    simp at this
+    rw [← ToInt.wrap_toInt]
+    rw [ToInt.wrap_eq_wrap_iff]
+    simpa
+
+instance [ToInt α (some lo) (some hi)] [IntModule α] [ToInt.Add α (some lo) (some hi)] [ToInt.Neg α (some lo) (some hi)] : ToInt.Sub α (some lo) (some hi) :=
+  ToInt.Sub.of_sub_eq_add_neg IntModule.sub_eq_add_neg
+
 end Lean.Grind
--- a/src/Init/Grind/Norm.lean
+++ b/src/Init/Grind/Norm.lean
@@ -12,6 +12,7 @@ import Init.Classical
 import Init.ByCases
 import Init.Data.Int.Linear
 import Init.Data.Int.Pow
+import Init.Grind.Ring.Field

 namespace Lean.Grind
 /-!
@@ -177,15 +178,16 @@ init_grind_norm
  Nat.add_eq Nat.sub_eq Nat.mul_eq Nat.zero_eq Nat.le_eq
  Nat.div_zero Nat.mod_zero Nat.div_one Nat.mod_one
  Nat.sub_sub Nat.pow_zero Nat.pow_one Nat.sub_self
+  Nat.one_pow
  -- Int
  Int.lt_eq
  Int.emod_neg Int.ediv_neg
  Int.ediv_zero Int.emod_zero
  Int.ediv_one Int.emod_one
-
+  Int.negSucc_eq
  natCast_eq natCast_div natCast_mod
  natCast_add natCast_mul
-
+  Int.one_pow
  Int.pow_zero Int.pow_one
  -- GT GE
  ge_eq gt_eq
@@ -197,5 +199,7 @@ init_grind_norm
  -- Function composition
  Function.const_apply Function.comp_apply Function.const_comp
  Function.comp_const Function.true_comp Function.false_comp
+  -- Field
+  Field.div_eq_mul_inv Field.inv_zero Field.inv_inv Field.inv_one Field.inv_neg

 end Lean.Grind
--- a/src/Init/Grind/Ordered.lean
+++ b/src/Init/Grind/Ordered.lean
@@ -11,3 +11,4 @@ import Init.Grind.Ordered.Module
 import Init.Grind.Ordered.Ring
 import Init.Grind.Ordered.Field
 import Init.Grind.Ordered.Int
+import Init.Grind.Ordered.Linarith
--- a/src/Init/Grind/Ordered/Field.lean
+++ b/src/Init/Grind/Ordered/Field.lean
@@ -6,7 +6,7 @@ Authors: Kim Morrison
 module

 prelude
-import Init.Grind.CommRing.Field
+import Init.Grind.Ring.Field
 import Init.Grind.Ordered.Ring

 namespace Lean.Grind
@@ -185,4 +185,54 @@ theorem mul_le_mul_iff_of_neg_left {a b c : R} (h : c < 0) : c * a ≤ c * b ↔

 end Field.IsOrdered

+theorem Field.zero_lt_one [Field α] [LinearOrder α] [Ring.IsOrdered α] : (0 : α) < 1 := by
+  cases LinearOrder.trichotomy (0:α) 1
+  next => assumption
+  next h =>
+    cases h
+    next => have := Field.zero_ne_one (α := α); contradiction
+    next h =>
+      have := Ring.IsOrdered.mul_pos_of_neg_of_neg h h
+      simp [Semiring.one_mul] at this
+      assumption
+
+theorem Field.minus_one_lt_zero [Field α] [LinearOrder α] [Ring.IsOrdered α] : (-1 : α) < 0 := by
+  have h := Field.zero_lt_one (α := α)
+  have := IntModule.IsOrdered.add_lt_left h (-1)
+  rw [Semiring.zero_add, Ring.add_neg_cancel] at this
+  assumption
+
+theorem ofNat_nonneg [Field α] [LinearOrder α] [Ring.IsOrdered α] (x : Nat) : (OfNat.ofNat x : α) ≥ 0 := by
+  induction x
+  next => simp [OfNat.ofNat, Zero.zero]; apply Preorder.le_refl
+  next n ih =>
+    have := Field.zero_lt_one (α := α)
+    rw [Semiring.ofNat_succ]
+    replace ih := IntModule.IsOrdered.add_le_left ih 1
+    rw [Semiring.zero_add] at ih
+    have := Preorder.lt_of_lt_of_le this ih
+    exact Preorder.le_of_lt this
+
+instance [Field α] [LinearOrder α] [Ring.IsOrdered α] : IsCharP α 0 where
+  ofNat_eq_zero_iff := by
+    intro x
+    simp only [Nat.mod_zero]; constructor
+    next =>
+      intro h
+      cases x
+      next => rfl
+      next x =>
+        rw [Semiring.ofNat_succ] at h
+        replace h := congrArg (· - 1) h; simp at h
+        rw [Ring.sub_eq_add_neg, Semiring.add_assoc, Ring.add_neg_cancel,
+            Ring.sub_eq_add_neg, Semiring.zero_add, Semiring.add_zero] at h
+        have h₁ : (OfNat.ofNat x : α) < 0 := by
+          have := Field.minus_one_lt_zero (α := α)
+          rw [h]; assumption
+        have h₂ := ofNat_nonneg (α := α) x
+        have : (0 : α) < 0 := Preorder.lt_of_le_of_lt h₂ h₁
+        simp
+        exact (Preorder.lt_irrefl 0) this
+    next => intro h; rw [OfNat.ofNat, h]; rfl
+
 end Lean.Grind
--- a/src/Init/Grind/Ordered/Int.lean
+++ b/src/Init/Grind/Ordered/Int.lean
@@ -7,7 +7,7 @@ module

 prelude
 import Init.Grind.Ordered.Ring
-import Init.Grind.CommRing.Int
+import Init.GrindInstances.Ring.Int
 import Init.Omega

 /-!
@@ -24,7 +24,7 @@ instance : Preorder Int where
 instance : IntModule.IsOrdered Int where
  neg_le_iff := by omega
  add_le_left := by omega
-  hmul_pos k a ha := ⟨fun hk => Int.mul_pos hk ha, fun h => Int.pos_of_mul_pos_left h ha⟩
+  hmul_pos_iff k a ha := ⟨fun h => Int.pos_of_mul_pos_left h ha, fun hk => Int.mul_pos hk ha⟩
  hmul_nonneg hk ha := Int.mul_nonneg hk ha

 instance : Ring.IsOrdered Int where
--- a/src/Init/Grind/Ordered/Linarith.lean
+++ b/src/Init/Grind/Ordered/Linarith.lean
@@ -0,0 +1,564 @@
+/-
+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
+-/
+module
+prelude
+import Init.Grind.Ordered.Module
+import Init.Grind.Ordered.Ring
+import Init.Grind.Ring.Field
+import all Init.Data.Ord
+import all Init.Data.AC
+import Init.Data.RArray
+
+/-!
+Support for the linear arithmetic module for `IntModule` in `grind`
+-/
+
+namespace Lean.Grind.Linarith
+abbrev Var := Nat
+open IntModule
+
+attribute [local simp] add_zero zero_add zero_hmul hmul_zero one_hmul
+
+inductive Expr where
+  | zero
+  | var  (i : Var)
+  | add  (a b : Expr)
+  | sub  (a b : Expr)
+  | neg  (a : Expr)
+  | mul  (k : Int) (a : Expr)
+  deriving Inhabited, BEq
+
+abbrev Context (α : Type u) := RArray α
+
+def Var.denote {α} (ctx : Context α) (v : Var) : α :=
+  ctx.get v
+
+def Expr.denote {α} [IntModule α] (ctx : Context α) : Expr → α
+  | zero      => 0
+  | .var v    => v.denote ctx
+  | .add a b  => denote ctx a + denote ctx b
+  | .sub a b  => denote ctx a - denote ctx b
+  | .mul k a  => k * denote ctx a
+  | .neg a    => -denote ctx a
+
+inductive Poly where
+  | nil
+  | add (k : Int) (v : Var) (p : Poly)
+  deriving BEq
+
+def Poly.denote {α} [IntModule α] (ctx : Context α) (p : Poly) : α :=
+  match p with
+  | .nil => 0
+  | .add k v p => k * v.denote ctx + denote ctx p
+
+/--
+Similar to `Poly.denote`, but produces a denotation better for normalization.
+-/
+def Poly.denote' {α} [IntModule α] (ctx : Context α) (p : Poly) : α :=
+  match p with
+  | .nil => 0
+  | .add 1 v p => go (v.denote ctx) p
+  | .add k v p => go (k * v.denote ctx) p
+where
+  go (r : α)  (p : Poly) : α :=
+    match p with
+    | .nil => r
+    | .add 1 v p => go (r + v.denote ctx) p
+    | .add k v p => go (r + k * v.denote ctx) p
+
+-- Helper instance for `ac_rfl`
+local instance {α} [IntModule α] : Std.Associative (· + · : α → α → α) where
+  assoc := IntModule.add_assoc
+-- Helper instance for `ac_rfl`
+local instance {α} [IntModule α] : Std.Commutative (· + · : α → α → α) where
+  comm := IntModule.add_comm
+
+theorem Poly.denote'_go_eq_denote {α} [IntModule α] (ctx : Context α) (p : Poly) (r : α) : denote'.go ctx r p = p.denote ctx + r := by
+  induction r, p using denote'.go.induct ctx <;> simp [denote'.go, denote]
+  next ih => rw [ih]; ac_rfl
+  next ih => rw [ih]; ac_rfl
+
+theorem Poly.denote'_eq_denote {α} [IntModule α] (ctx : Context α) (p : Poly) : p.denote' ctx = p.denote ctx := by
+  unfold denote' <;> split <;> simp [denote, denote'_go_eq_denote] <;> ac_rfl
+
+def Poly.coeff (p : Poly) (x : Var) : Int :=
+  match p with
+  | .add a y p => bif x == y then a else coeff p x
+  | .nil => 0
+
+def Poly.insert (k : Int) (v : Var) (p : Poly) : Poly :=
+  match p with
+  | .nil => .add k v .nil
+  | .add k' v' p =>
+    bif Nat.blt v' v then
+      .add k v <| .add k' v' p
+    else bif Nat.beq v v' then
+      if Int.add k k' == 0 then
+        p
+      else
+        .add (Int.add k k') v' p
+    else
+      .add k' v' (insert k v p)
+
+/-- Normalizes the given polynomial by fusing monomial and constants. -/
+def Poly.norm (p : Poly) : Poly :=
+  match p with
+  | .nil => .nil
+  | .add k v p => (norm p).insert k v
+
+def Poly.append (p₁ p₂ : Poly) : Poly :=
+  match p₁ with
+  | .nil => p₂
+  | .add k x p₁ => .add k x (append p₁ p₂)
+
+def Poly.combine' (fuel : Nat) (p₁ p₂ : Poly) : Poly :=
+  match fuel with
+  | 0 => p₁.append p₂
+  | fuel + 1 => match p₁, p₂ with
+    | .nil, p₂ => p₂
+    | p₁, .nil => p₁
+    | .add a₁ x₁ p₁, .add a₂ x₂ p₂ =>
+      bif Nat.beq x₁ x₂ then
+        let a := a₁ + a₂
+        bif a == 0 then
+          combine' fuel p₁ p₂
+        else
+          .add a x₁ (combine' fuel p₁ p₂)
+      else bif Nat.blt x₂ x₁ then
+        .add a₁ x₁ (combine' fuel p₁ (.add a₂ x₂ p₂))
+      else
+        .add a₂ x₂ (combine' fuel (.add a₁ x₁ p₁) p₂)
+
+def Poly.combine (p₁ p₂ : Poly) : Poly :=
+  combine' 100000000 p₁ p₂
+
+/-- Converts the given expression into a polynomial. -/
+def Expr.toPoly' (e : Expr) : Poly :=
+  go 1 e .nil
+where
+  go (coeff : Int) : Expr → (Poly → Poly)
+    | .zero     => id
+    | .var v    => (.add coeff v ·)
+    | .add a b  => go coeff a ∘ go coeff b
+    | .sub a b  => go coeff a ∘ go (-coeff) b
+    | .mul k a  => bif k == 0 then id else go (Int.mul coeff k) a
+    | .neg a    => go (-coeff) a
+
+/-- Converts the given expression into a polynomial, and then normalizes it. -/
+def Expr.norm (e : Expr) : Poly :=
+  e.toPoly'.norm
+
+/--
+`p.mul k` multiplies all coefficients and constant of the polynomial `p` by `k`.
+-/
+def Poly.mul' (p : Poly) (k : Int) : Poly :=
+  match p with
+  | .nil => .nil
+  | .add k' v p => .add (k*k') v (mul' p k)
+
+def Poly.mul (p : Poly) (k : Int) : Poly :=
+  if k == 0 then
+    .nil
+  else
+    p.mul' k
+
+@[simp] theorem Poly.denote_mul {α} [IntModule α] (ctx : Context α) (p : Poly) (k : Int) : (p.mul k).denote ctx = k * p.denote ctx := by
+  simp [mul]
+  split
+  next => simp [*, denote]
+  next =>
+    induction p <;> simp [mul', denote, *]
+    rw [mul_hmul, hmul_add]
+
+theorem Poly.denote_insert {α} [IntModule α] (ctx : Context α) (k : Int) (v : Var) (p : Poly) :
+    (p.insert k v).denote ctx = p.denote ctx + k * v.denote ctx := by
+  fun_induction p.insert k v <;> simp [denote]
+  next => ac_rfl
+  next h₁ h₂ h₃ =>
+    simp at h₃; simp at h₂; subst h₂
+    rw [add_comm, ← add_assoc, ← add_hmul, h₃, zero_hmul, zero_add]
+  next h _ => simp at h; subst h; rw [add_hmul]; ac_rfl
+  next ih => rw [ih]; ac_rfl
+
+attribute [local simp] Poly.denote_insert
+
+theorem Poly.denote_norm {α} [IntModule α] (ctx : Context α) (p : Poly) : p.norm.denote ctx = p.denote ctx := by
+  induction p <;> simp [denote, norm, add_comm, *]
+
+attribute [local simp] Poly.denote_norm
+
+theorem Poly.denote_append {α} [IntModule α] (ctx : Context α) (p₁ p₂ : Poly) : (p₁.append p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
+  induction p₁ <;> simp [append, denote, *]; ac_rfl
+
+attribute [local simp] Poly.denote_append
+
+theorem Poly.denote_combine' {α} [IntModule α] (ctx : Context α) (fuel : Nat) (p₁ p₂ : Poly) : (p₁.combine' fuel p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
+  fun_induction p₁.combine' fuel p₂ <;>
+    simp_all +zetaDelta [denote, ← Int.add_mul]
+  next h _ =>
+    rw [Int.add_comm] at h
+    rw [add_left_comm, add_assoc, ← add_assoc, ← add_hmul, h, zero_hmul, zero_add]
+  next => rw [add_hmul]; ac_rfl
+  all_goals ac_rfl
+
+theorem Poly.denote_combine {α} [IntModule α] (ctx : Context α) (p₁ p₂ : Poly) : (p₁.combine p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
+  simp [combine, denote_combine']
+
+attribute [local simp] Poly.denote_combine
+
+theorem Expr.denote_toPoly'_go {α} [IntModule α] {k p} (ctx : Context α) (e : Expr)
+    : (toPoly'.go k e p).denote ctx = k * e.denote ctx + p.denote ctx := by
+  induction k, e using Expr.toPoly'.go.induct generalizing p <;> simp [toPoly'.go, denote, Poly.denote, *, hmul_add]
+  next => ac_rfl
+  next => rw [sub_eq_add_neg, neg_hmul, hmul_add, hmul_neg]; ac_rfl
+  next h => simp at h; subst h; simp
+  next ih => simp at ih; rw [ih, mul_hmul]
+  next => rw [hmul_neg, neg_hmul]
+
+theorem Expr.denote_norm {α} [IntModule α] (ctx : Context α) (e : Expr) : e.norm.denote ctx = e.denote ctx := by
+  simp [norm, toPoly', Expr.denote_toPoly'_go, Poly.denote]
+
+attribute [local simp] Expr.denote_norm
+
+instance : LawfulBEq Poly where
+  eq_of_beq {a} := by
+    induction a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
+    next ih =>
+      intro _ _ h
+      exact ih h
+  rfl := by
+    intro a
+    induction a <;> simp! [BEq.beq]
+    assumption
+
+attribute [local simp] Poly.denote'_eq_denote
+
+def Poly.leadCoeff (p : Poly) : Int :=
+  match p with
+  | .add a _ _ => a
+  | _ => 1
+
+open IntModule.IsOrdered
+
+/-!
+Helper theorems for conflict resolution during model construction.
+-/
+
+private theorem le_add_le {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] {a b : α}
+    (h₁ : a ≤ 0) (h₂ : b ≤ 0) : a + b ≤ 0 := by
+  replace h₁ := add_le_left h₁ b; simp at h₁
+  exact Preorder.le_trans h₁ h₂
+
+private theorem le_add_lt {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] {a b : α}
+    (h₁ : a ≤ 0) (h₂ : b < 0) : a + b < 0 := by
+  replace h₁ := add_le_left h₁ b; simp at h₁
+  exact Preorder.lt_of_le_of_lt h₁ h₂
+
+private theorem lt_add_lt {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] {a b : α}
+    (h₁ : a < 0) (h₂ : b < 0) : a + b < 0 := by
+  replace h₁ := add_lt_left h₁ b; simp at h₁
+  exact Preorder.lt_trans h₁ h₂
+
+private theorem coe_natAbs_nonneg (a : Int) : (a.natAbs : Int) ≥ 0 := by
+  exact Int.natCast_nonneg a.natAbs
+
+def le_le_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
+  let a₁ := p₁.leadCoeff.natAbs
+  let a₂ := p₂.leadCoeff.natAbs
+  p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
+
+theorem le_le_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
+    : le_le_combine_cert p₁ p₂ p₃ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 → p₃.denote' ctx ≤ 0 := by
+  simp [le_le_combine_cert]; intro _ h₁ h₂; subst p₃; simp
+  replace h₁ := hmul_nonpos (coe_natAbs_nonneg p₂.leadCoeff) h₁
+  replace h₂ := hmul_nonpos (coe_natAbs_nonneg p₁.leadCoeff) h₂
+  exact le_add_le h₁ h₂
+
+def le_lt_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
+  let a₁ := p₁.leadCoeff.natAbs
+  let a₂ := p₂.leadCoeff.natAbs
+  a₁ > (0 : Int) && p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
+
+theorem le_lt_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
+    : le_lt_combine_cert p₁ p₂ p₃ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx < 0 → p₃.denote' ctx < 0 := by
+  simp [-Int.natAbs_pos, -Int.ofNat_pos, le_lt_combine_cert]; intro hp _ h₁ h₂; subst p₃; simp
+  replace h₁ := hmul_nonpos (coe_natAbs_nonneg p₂.leadCoeff) h₁
+  replace h₂ := hmul_neg_iff (↑p₁.leadCoeff.natAbs) h₂ |>.mpr hp
+  exact le_add_lt h₁ h₂
+
+def lt_lt_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
+  let a₁ := p₁.leadCoeff.natAbs
+  let a₂ := p₂.leadCoeff.natAbs
+  a₂ > (0 : Int) && a₁ > (0 : Int) && p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
+
+theorem lt_lt_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
+    : lt_lt_combine_cert p₁ p₂ p₃ → p₁.denote' ctx < 0 → p₂.denote' ctx < 0 → p₃.denote' ctx < 0 := by
+  simp [-Int.natAbs_pos, -Int.ofNat_pos, lt_lt_combine_cert]; intro hp₁ hp₂ _ h₁ h₂; subst p₃; simp
+  replace h₁ := hmul_neg_iff (↑p₂.leadCoeff.natAbs) h₁ |>.mpr hp₁
+  replace h₂ := hmul_neg_iff (↑p₁.leadCoeff.natAbs) h₂ |>.mpr hp₂
+  exact lt_add_lt h₁ h₂
+
+def diseq_split_cert (p₁ p₂ : Poly) : Bool :=
+  p₂ == p₁.mul (-1)
+
+-- We need `LinearOrder` to use `trichotomy`
+theorem diseq_split {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly)
+    : diseq_split_cert p₁ p₂ → p₁.denote' ctx ≠ 0 → p₁.denote' ctx < 0 ∨ p₂.denote' ctx < 0 := by
+  simp [diseq_split_cert]; intro _ h₁; subst p₂; simp
+  cases LinearOrder.trichotomy (p₁.denote ctx) 0
+  next h => exact Or.inl h
+  next h =>
+    apply Or.inr
+    simp [h₁] at h
+    rw [← neg_pos_iff, neg_hmul, neg_neg, one_hmul]; assumption
+
+theorem diseq_split_resolve {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly)
+    : diseq_split_cert p₁ p₂ → p₁.denote' ctx ≠ 0 → ¬p₁.denote' ctx < 0 → p₂.denote' ctx < 0 := by
+  intro h₁ h₂ h₃
+  exact (diseq_split ctx p₁ p₂ h₁ h₂).resolve_left h₃
+
+/-!
+Helper theorems for internalizing facts into the linear arithmetic procedure
+-/
+
+def norm_cert (lhs rhs : Expr) (p : Poly) :=
+  p == (lhs.sub rhs).norm
+
+theorem eq_norm {α} [IntModule α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → lhs.denote ctx = rhs.denote ctx → p.denote' ctx = 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+
+theorem le_of_eq {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → lhs.denote ctx = rhs.denote ctx → p.denote' ctx ≤ 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+  apply Preorder.le_refl
+
+theorem diseq_norm {α} [IntModule α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → lhs.denote ctx ≠ rhs.denote ctx → p.denote' ctx ≠ 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+  intro h
+  replace h := congrArg (rhs.denote ctx + ·) h; simp [sub_eq_add_neg] at h
+  rw [add_left_comm, ← sub_eq_add_neg, sub_self, add_zero] at h
+  contradiction
+
+theorem le_norm {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → lhs.denote ctx ≤ rhs.denote ctx → p.denote' ctx ≤ 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+  replace h₁ := add_le_left h₁ (-rhs.denote ctx)
+  simp [← sub_eq_add_neg, sub_self] at h₁
+  assumption
+
+theorem lt_norm {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → lhs.denote ctx < rhs.denote ctx → p.denote' ctx < 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+  replace h₁ := add_lt_left h₁ (-rhs.denote ctx)
+  simp [← sub_eq_add_neg, sub_self] at h₁
+  assumption
+
+theorem not_le_norm {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert rhs lhs p → ¬ lhs.denote ctx ≤ rhs.denote ctx → p.denote' ctx < 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+  replace h₁ := LinearOrder.lt_of_not_le h₁
+  replace h₁ := add_lt_left h₁ (-lhs.denote ctx)
+  simp [← sub_eq_add_neg, sub_self] at h₁
+  assumption
+
+theorem not_lt_norm {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert rhs lhs p → ¬ lhs.denote ctx < rhs.denote ctx → p.denote' ctx ≤ 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
+  replace h₁ := LinearOrder.le_of_not_lt h₁
+  replace h₁ := add_le_left h₁ (-lhs.denote ctx)
+  simp [← sub_eq_add_neg, sub_self] at h₁
+  assumption
+
+-- If the module does not have a linear order, we can still put the expressions in polynomial forms
+
+theorem not_le_norm' {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → ¬ lhs.denote ctx ≤ rhs.denote ctx → ¬ p.denote' ctx ≤ 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]; intro h
+  replace h := add_le_right (rhs.denote ctx) h
+  rw [sub_eq_add_neg, add_left_comm, ← sub_eq_add_neg, sub_self] at h; simp at h
+  contradiction
+
+theorem not_lt_norm' {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
+    : norm_cert lhs rhs p → ¬ lhs.denote ctx < rhs.denote ctx → ¬ p.denote' ctx < 0 := by
+  simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]; intro h
+  replace h := add_lt_right (rhs.denote ctx) h
+  rw [sub_eq_add_neg, add_left_comm, ← sub_eq_add_neg, sub_self] at h; simp at h
+  contradiction
+
+/-!
+Equality detection
+-/
+def eq_of_le_ge_cert (p₁ p₂ : Poly) : Bool :=
+  p₂ == p₁.mul (-1)
+
+theorem eq_of_le_ge {α} [IntModule α] [PartialOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ : Poly) (p₂ : Poly)
+    : eq_of_le_ge_cert p₁ p₂ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 → p₁.denote' ctx = 0 := by
+  simp [eq_of_le_ge_cert]
+  intro; subst p₂; simp
+  intro h₁ h₂
+  replace h₂ := add_le_left h₂ (p₁.denote ctx)
+  rw [add_comm, neg_hmul, one_hmul, ← sub_eq_add_neg, sub_self, zero_add] at h₂
+  exact PartialOrder.le_antisymm h₁ h₂
+
+/-!
+Helper theorems for closing the goal
+-/
+
+theorem diseq_unsat {α} [IntModule α] (ctx : Context α) : (Poly.nil).denote ctx ≠ 0 → False := by
+  simp [Poly.denote]
+
+theorem lt_unsat {α} [IntModule α] [Preorder α] (ctx : Context α) : (Poly.nil).denote ctx < 0 → False := by
+  simp [Poly.denote]; intro h
+  have := Preorder.lt_iff_le_not_le.mp h
+  simp at this
+
+def zero_lt_one_cert (p : Poly) : Bool :=
+  p == .add (-1) 0 .nil
+
+theorem zero_lt_one {α} [Ring α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (p : Poly)
+    : zero_lt_one_cert p → (0 : Var).denote ctx = One.one → p.denote' ctx < 0 := by
+  simp [zero_lt_one_cert]; intro _ h; subst p; simp [Poly.denote, h, One.one, neg_hmul]
+  rw [neg_lt_iff, neg_zero]; apply Ring.IsOrdered.zero_lt_one
+
+def zero_ne_one_cert (p : Poly) : Bool :=
+  p == .add 1 0 .nil
+
+theorem zero_ne_one_of_ord_ring {α} [Ring α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (p : Poly)
+    : zero_ne_one_cert p → (0 : Var).denote ctx = One.one → p.denote' ctx ≠ 0 := by
+  simp [zero_ne_one_cert]; intro _ h; subst p; simp [Poly.denote, h, One.one]
+  intro h; have := Ring.IsOrdered.zero_lt_one (R := α); simp [h, Preorder.lt_irrefl] at this
+
+theorem zero_ne_one_of_field {α} [Field α] (ctx : Context α) (p : Poly)
+    : zero_ne_one_cert p → (0 : Var).denote ctx = One.one → p.denote' ctx ≠ 0 := by
+  simp [zero_ne_one_cert]; intro _ h; subst p; simp [Poly.denote, h, One.one]
+  intro h; have := Field.zero_ne_one (α := α); simp [h] at this
+
+theorem zero_ne_one_of_char0 {α} [Ring α] [IsCharP α 0] (ctx : Context α) (p : Poly)
+    : zero_ne_one_cert p → (0 : Var).denote ctx = One.one → p.denote' ctx ≠ 0 := by
+  simp [zero_ne_one_cert]; intro _ h; subst p; simp [Poly.denote, h, One.one]
+  intro h; have := IsCharP.intCast_eq_zero_iff (α := α) 0 1; simp [Ring.intCast_one] at this
+  contradiction
+
+def zero_ne_one_of_charC_cert (c : Nat) (p : Poly) : Bool :=
+  (c:Int) > 1 && p == .add 1 0 .nil
+
+theorem zero_ne_one_of_charC {α c} [Ring α] [IsCharP α c] (ctx : Context α) (p : Poly)
+    : zero_ne_one_of_charC_cert c p → (0 : Var).denote ctx = One.one → p.denote' ctx ≠ 0 := by
+  simp [zero_ne_one_of_charC_cert]; intro hc _ h; subst p; simp [Poly.denote, h, One.one]
+  intro h; have h' := IsCharP.intCast_eq_zero_iff (α := α) c 1; simp [Ring.intCast_one] at h'
+  replace h' := h'.mp h
+  have := Int.emod_eq_of_lt (by decide) hc
+  simp [this] at h'
+
+/-!
+Coefficient normalization
+-/
+
+def eq_neg_cert (p₁ p₂ : Poly) :=
+  p₂ == p₁.mul (-1)
+
+theorem eq_neg {α} [IntModule α] (ctx : Context α) (p₁ p₂ : Poly)
+    : eq_neg_cert p₁ p₂ → p₁.denote' ctx = 0 → p₂.denote' ctx = 0 := by
+  simp [eq_neg_cert]; intros; simp [*]
+
+def eq_coeff_cert (p₁ p₂ : Poly) (k : Nat) :=
+  k != 0 && p₁ == p₂.mul k
+
+theorem eq_coeff {α} [IntModule α] [NoNatZeroDivisors α] (ctx : Context α) (p₁ p₂ : Poly) (k : Nat)
+    : eq_coeff_cert p₁ p₂ k → p₁.denote' ctx = 0 → p₂.denote' ctx = 0 := by
+  simp [eq_coeff_cert]; intro h _; subst p₁; simp [*]
+  exact no_nat_zero_divisors k (p₂.denote ctx) h
+
+def coeff_cert (p₁ p₂ : Poly) (k : Nat) :=
+  k > 0 && p₁ == p₂.mul k
+
+theorem le_coeff {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly) (k : Nat)
+    : coeff_cert p₁ p₂ k → p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 := by
+  simp [coeff_cert]; intro h _; subst p₁; simp
+  have : ↑k > (0 : Int) := Int.natCast_pos.mpr h
+  intro h₁; apply Classical.byContradiction
+  intro h₂; replace h₂ := LinearOrder.lt_of_not_le h₂
+  replace h₂ := IsOrdered.hmul_pos_iff (↑k) h₂ |>.mpr this
+  exact Preorder.lt_irrefl 0 (Preorder.lt_of_lt_of_le h₂ h₁)
+
+theorem lt_coeff {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly) (k : Nat)
+    : coeff_cert p₁ p₂ k → p₁.denote' ctx < 0 → p₂.denote' ctx < 0 := by
+  simp [coeff_cert]; intro h _; subst p₁; simp
+  have : ↑k > (0 : Int) := Int.natCast_pos.mpr h
+  intro h₁; apply Classical.byContradiction
+  intro h₂; replace h₂ := LinearOrder.le_of_not_lt h₂
+  replace h₂ := IsOrdered.hmul_nonneg (Int.le_of_lt this) h₂
+  exact Preorder.lt_irrefl 0 (Preorder.lt_of_le_of_lt h₂ h₁)
+
+theorem diseq_neg {α} [IntModule α] (ctx : Context α) (p p' : Poly) : p' == p.mul (-1) → p.denote' ctx ≠ 0 → p'.denote' ctx ≠ 0 := by
+  simp; intro _ _; subst p'; simp [neg_hmul]
+  intro h; replace h := congrArg (- ·) h; simp [neg_neg, neg_zero] at h
+  contradiction
+
+/-!
+Substitution
+-/
+
+def eq_diseq_subst_cert (k₁ k₂ : Int) (p₁ p₂ p₃ : Poly) : Bool :=
+  k₁.natAbs ≠ 0 && p₃ == (p₁.mul k₂ |>.combine (p₂.mul k₁))
+
+theorem eq_diseq_subst {α} [IntModule α] [NoNatZeroDivisors α] (ctx : Context α) (k₁ k₂ : Int) (p₁ p₂ p₃ : Poly)
+    : eq_diseq_subst_cert k₁ k₂ p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx ≠ 0 → p₃.denote' ctx ≠ 0 := by
+  simp [eq_diseq_subst_cert, - Int.natAbs_eq_zero, -Int.natCast_eq_zero]; intro hne _ h₁ h₂; subst p₃
+  simp [h₁, h₂]; intro h₃
+  have :  k₁.natAbs * Poly.denote ctx p₂ = 0 := by
+    have : (k₁.natAbs : Int) * Poly.denote ctx p₂ = 0 := by
+      cases Int.natAbs_eq_iff.mp (Eq.refl k₁.natAbs)
+      next h => rw [← h]; assumption
+      next h => replace h := congrArg (- ·) h; simp at h; rw [← h, IntModule.neg_hmul, h₃, IntModule.neg_zero]
+    exact this
+  have := no_nat_zero_divisors (k₁.natAbs) (p₂.denote ctx) hne this
+  contradiction
+
+def eq_diseq_subst1_cert (k : Int) (p₁ p₂ p₃ : Poly) : Bool :=
+  p₃ == (p₁.mul k |>.combine p₂)
+
+/-
+Special case of `diseq_eq_subst` where leading coefficient `c₁` of `p₁` is `-k*c₂`, where
+`c₂` is the leading coefficient of `p₂`.
+-/
+theorem eq_diseq_subst1 {α} [IntModule α] (ctx : Context α) (k : Int) (p₁ p₂ p₃ : Poly)
+    : eq_diseq_subst1_cert k p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx ≠ 0 → p₃.denote' ctx ≠ 0 := by
+  simp [eq_diseq_subst1_cert]; intro _ h₁ h₂; subst p₃
+  simp [h₁, h₂]
+
+def eq_le_subst_cert (x : Var) (p₁ p₂ p₃ : Poly) :=
+  let a := p₁.coeff x
+  let b := p₂.coeff x
+  a ≥ 0 && p₃ == (p₂.mul a |>.combine (p₁.mul (-b)))
+
+theorem eq_le_subst {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (x : Var) (p₁ p₂ p₃ : Poly)
+    : eq_le_subst_cert x p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx ≤ 0 → p₃.denote' ctx ≤ 0 := by
+  simp [eq_le_subst_cert]; intro h _ h₁ h₂; subst p₃; simp [h₁]
+  exact hmul_nonpos h h₂
+
+def eq_lt_subst_cert (x : Var) (p₁ p₂ p₃ : Poly) :=
+  let a := p₁.coeff x
+  let b := p₂.coeff x
+  a > 0 && p₃ == (p₂.mul a |>.combine (p₁.mul (-b)))
+
+theorem eq_lt_subst {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (x : Var) (p₁ p₂ p₃ : Poly)
+    : eq_lt_subst_cert x p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx < 0 → p₃.denote' ctx < 0 := by
+  simp [eq_lt_subst_cert]; intro h _ h₁ h₂; subst p₃; simp [h₁]
+  exact IsOrdered.hmul_neg_iff (p₁.coeff x) h₂ |>.mpr h
+
+def eq_eq_subst_cert (x : Var) (p₁ p₂ p₃ : Poly) :=
+  let a := p₁.coeff x
+  let b := p₂.coeff x
+  p₃ == (p₂.mul a |>.combine (p₁.mul (-b)))
+
+theorem eq_eq_subst {α} [IntModule α] (ctx : Context α) (x : Var) (p₁ p₂ p₃ : Poly)
+    : eq_eq_subst_cert x p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx = 0 → p₃.denote' ctx = 0 := by
+  simp [eq_eq_subst_cert]; intro _ h₁ h₂; subst p₃; simp [h₁, h₂]
+
+end Lean.Grind.Linarith
--- a/src/Init/Grind/Ordered/Module.lean
+++ b/src/Init/Grind/Ordered/Module.lean
@@ -12,12 +12,79 @@ import Init.Grind.Ordered.Order

 namespace Lean.Grind

+class NatModule.IsOrdered (M : Type u) [Preorder M] [NatModule M] where
+  add_le_left_iff : ∀ {a b : M} (c : M), a ≤ b ↔ a + c ≤ b + c
+  hmul_lt_hmul_iff : ∀ (k : Nat) {a b : M}, a < b → (k * a < k * b ↔ 0 < k)
+  hmul_le_hmul : ∀ {k : Nat} {a b : M}, a ≤ b → k * a ≤ k * b
+
 class IntModule.IsOrdered (M : Type u) [Preorder M] [IntModule M] where
  neg_le_iff : ∀ a b : M, -a ≤ b ↔ -b ≤ a
  add_le_left : ∀ {a b : M}, a ≤ b → (c : M) → a + c ≤ b + c
-  hmul_pos : ∀ (k : Int) {a : M}, 0 < a → (0 < k ↔ 0 < k * a)
+  hmul_pos_iff : ∀ (k : Int) {a : M}, 0 < a → (0 < k * a ↔ 0 < k)
  hmul_nonneg : ∀ {k : Int} {a : M}, 0 ≤ k → 0 ≤ a → 0 ≤ k * a

+namespace NatModule.IsOrdered
+
+variable {M : Type u} [Preorder M] [NatModule M] [NatModule.IsOrdered M]
+
+theorem add_le_right_iff {a b : M} (c : M) : a ≤ b ↔ c + a ≤ c + b := by
+  rw [add_comm c a, add_comm c b, add_le_left_iff]
+
+theorem add_le_left {a b : M} (h : a ≤ b) (c : M) : a + c ≤ b + c :=
+  (add_le_left_iff c).mp h
+
+theorem add_le_right {a b : M} (c : M) (h : a ≤ b) : c + a ≤ c + b :=
+  (add_le_right_iff c).mp h
+
+theorem add_lt_left {a b : M} (h : a < b) (c : M) : a + c < b + c := by
+  simp only [Preorder.lt_iff_le_not_le] at h ⊢
+  constructor
+  · exact add_le_left h.1 _
+  · intro w
+    apply h.2
+    exact (add_le_left_iff c).mpr w
+
+theorem add_lt_right {a b : M} (c : M) (h : a < b) : c + a < c + b := by
+  rw [add_comm c a, add_comm c b]
+  exact add_lt_left h c
+
+theorem add_lt_left_iff {a b : M} (c : M) : a < b ↔ a + c < b + c := by
+  constructor
+  · exact fun h => add_lt_left h c
+  · intro w
+    simp only [Preorder.lt_iff_le_not_le] at w ⊢
+    constructor
+    · exact (add_le_left_iff c).mpr w.1
+    · intro h
+      exact w.2 ((add_le_left_iff c).mp h)
+
+theorem add_lt_right_iff {a b : M} (c : M) : a < b ↔ c + a < c + b := by
+  rw [add_comm c a, add_comm c b, add_lt_left_iff]
+
+theorem hmul_pos_iff {k : Nat} {a : M} (h : 0 < a) : 0 < k * a ↔ 0 < k:= by
+  rw [← hmul_lt_hmul_iff k h, hmul_zero]
+
+theorem hmul_nonneg {k : Nat} {a : M} (h : 0 ≤ a) : 0 ≤ k * a := by
+  have := hmul_le_hmul (k := k) h
+  rwa [hmul_zero] at this
+
+theorem hmul_le_hmul_of_le_of_le_of_nonneg
+    {k₁ k₂ : Nat} {x y : M} (hk : k₁ ≤ k₂) (h : x ≤ y) (w : 0 ≤ x) :
+    k₁ * x ≤ k₂ * y := by
+  apply Preorder.le_trans
+  · change k₁ * x ≤ k₂ * x
+    obtain ⟨k', rfl⟩ := Nat.exists_eq_add_of_le hk
+    rw [add_hmul]
+    conv => lhs; rw [← add_zero (k₁ * x)]
+    rw [← add_le_right_iff]
+    exact hmul_nonneg w
+  · exact hmul_le_hmul h
+
+theorem add_le_add {a b c d : M} (hab : a ≤ b) (hcd : c ≤ d) : a + c ≤ b + d :=
+  Preorder.le_trans (add_le_right a hcd) (add_le_left hab d)
+
+end NatModule.IsOrdered
+
 namespace IntModule.IsOrdered

 variable {M : Type u} [Preorder M] [IntModule M] [IntModule.IsOrdered M]
@@ -41,7 +108,7 @@ theorem neg_pos_iff {a : M} : 0 < -a ↔ a < 0 := by
  rw [lt_neg_iff, neg_zero]

 theorem add_lt_left {a b : M} (h : a < b) (c : M) : a + c < b + c := by
-  simp [Preorder.lt_iff_le_not_le] at h ⊢
+  simp only [Preorder.lt_iff_le_not_le] at h ⊢
  constructor
  · exact add_le_left h.1 _
  · intro w
@@ -58,12 +125,59 @@ theorem add_lt_right (a : M) {b c : M} (h : b < c) : a + b < a + c := by
  rw [add_comm a b, add_comm a c]
  exact add_lt_left h a

-theorem hmul_neg (k : Int) {a : M} (h : a < 0) : 0 < k ↔ k * a < 0 := by
-  simpa [IntModule.hmul_neg, neg_pos_iff] using hmul_pos k (neg_pos_iff.mpr h)
+theorem add_le_left_iff {a b : M} (c : M) : a ≤ b ↔ a + c ≤ b + c := by
+  constructor
+  · intro w
+    exact add_le_left w c
+  · intro w
+    have := add_le_left w (-c)
+    rwa [add_assoc, add_neg_cancel, add_zero, add_assoc, add_neg_cancel, add_zero] at this
+
+theorem add_le_right_iff {a b : M} (c : M) : a ≤ b ↔ c + a ≤ c + b := by
+  constructor
+  · intro w
+    exact add_le_right c w
+  · intro w
+    have := add_le_right (-c) w
+    rwa [← add_assoc, neg_add_cancel, zero_add, ← add_assoc, neg_add_cancel, zero_add] at this
+
+theorem add_lt_left_iff {a b : M} (c : M) : a < b ↔ a + c < b + c := by
+  constructor
+  · intro w
+    exact add_lt_left w c
+  · intro w
+    have := add_lt_left w (-c)
+    rwa [add_assoc, add_neg_cancel, add_zero, add_assoc, add_neg_cancel, add_zero] at this
+
+theorem add_lt_right_iff {a b : M} (c : M) : a < b ↔ c + a < c + b := by
+  constructor
+  · intro w
+    exact add_lt_right c w
+  · intro w
+    have := add_lt_right (-c) w
+    rwa [← add_assoc, neg_add_cancel, zero_add, ← add_assoc, neg_add_cancel, zero_add] at this
+
+theorem sub_nonneg_iff {a b : M} : 0 ≤ a - b ↔ b ≤ a := by
+  rw [add_le_left_iff b, zero_add, sub_add_cancel]
+
+theorem hmul_neg_iff (k : Int) {a : M} (h : a < 0) : k * a < 0 ↔ 0 < k := by
+  simpa [IntModule.hmul_neg, neg_pos_iff] using hmul_pos_iff k (neg_pos_iff.mpr h)

 theorem hmul_nonpos {k : Int} {a : M} (hk : 0 ≤ k) (ha : a ≤ 0) : k * a ≤ 0 := by
  simpa [IntModule.hmul_neg, neg_nonneg_iff] using hmul_nonneg hk (neg_nonneg_iff.mpr ha)

+theorem hmul_le_hmul_of_le_of_le_of_nonneg_of_nonneg
+    {k₁ k₂ : Int} {x y : M} (hk : k₁ ≤ k₂) (h : x ≤ y) (w : 0 ≤ k₁) (w' : 0 ≤ x) :
+    k₁ * x ≤ k₂ * y := by
+  apply Preorder.le_trans
+  · have : 0 ≤ k₁ * (y - x) := hmul_nonneg w (sub_nonneg_iff.mpr h)
+    rwa [IntModule.hmul_sub, sub_nonneg_iff] at this
+  · have : 0 ≤ (k₂ - k₁) * y := hmul_nonneg (Int.sub_nonneg.mpr hk) (Preorder.le_trans w' h)
+    rwa [IntModule.sub_hmul, sub_nonneg_iff] at this
+
+theorem add_le_add {a b c d : M} (hab : a ≤ b) (hcd : c ≤ d) : a + c ≤ b + d :=
+  Preorder.le_trans (add_le_right a hcd) (add_le_left hab d)
+
 end IntModule.IsOrdered

 end Lean.Grind
--- a/src/Init/Grind/Ordered/Order.lean
+++ b/src/Init/Grind/Ordered/Order.lean
@@ -91,6 +91,19 @@ theorem trichotomy (a b : α) : a < b ∨ a = b ∨ b < a := by
    | inl h => right; right; exact h
    | inr h => right; left; exact h.symm

+theorem le_of_not_lt {α} [LinearOrder α] {a b : α} (h : ¬ a < b) : b ≤ a := by
+  cases LinearOrder.trichotomy a b
+  next => contradiction
+  next h => apply PartialOrder.le_iff_lt_or_eq.mpr; cases h <;> simp [*]
+
+theorem lt_of_not_le {α} [LinearOrder α] {a b : α} (h : ¬ a ≤ b) : b < a := by
+  cases LinearOrder.trichotomy a b
+  next h₁ h₂ => have := Preorder.lt_iff_le_not_le.mp h₂; simp [h] at this
+  next h =>
+    cases h
+    next h => subst a; exact False.elim <| h (Preorder.le_refl b)
+    next => assumption
+
 end LinearOrder

 end Lean.Grind
--- a/Show More
+++ b/Show More