doc-string

This PR refactors the `NoNatZeroDivisors` to make sure it will work with
the new `Semiring` support.

.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

.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

.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

.gitignore

@@ -6,7 +6,6 @@
 lake-manifest.json
 /build
 /src/lakefile.toml
-/tests/lakefile.toml
 /lakefile.toml
 GPATH
 GRTAGS

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.

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

script/prepare-llvm-mingw.sh

@@ -50,5 +50,4 @@ echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='--sysroot ROOT -L ROOT/lib -Wl,-Bstatic
 # when not using the above flags, link GMP dynamically/as usual. Always link ICU dynamically.
 echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp $(pkg-config --libs libuv) -lucrtbase'"
 # do not set `LEAN_CC` for tests
-echo -n " -DAUTO_THREAD_FINALIZATION=OFF -DSTAGE0_AUTO_THREAD_FINALIZATION=OFF"
 echo -n " -DLEAN_TEST_VARS=''"

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")

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"]:

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'")
@@ -58,9 +58,6 @@ option(USE_GITHASH        "GIT_HASH"           ON)
 option(INSTALL_LICENSE "INSTALL_LICENSE" ON)
 # When ON we install a copy of cadical
 option(INSTALL_CADICAL "Install a copy of cadical" ON)
-# When ON thread storage is automatically finalized, it assumes platform support pthreads.
-# This option is important when using Lean as library that is invoked from a different programming language (e.g., Haskell).
-option(AUTO_THREAD_FINALIZATION "AUTO_THREAD_FINALIZATION" ON)
 
 # FLAGS for disabling optimizations and debugging
 option(FREE_VAR_RANGE_OPT  "FREE_VAR_RANGE_OPT"   ON)
@@ -182,10 +179,6 @@ else()
   string(APPEND LEAN_EXTRA_CXX_FLAGS " -D LEAN_MULTI_THREAD")
 endif()
 
-if(AUTO_THREAD_FINALIZATION AND NOT MSVC)
-  string(APPEND LEAN_EXTRA_CXX_FLAGS " -D LEAN_AUTO_THREAD_FINALIZATION")
-endif()
-
 # Set Module Path
 set(CMAKE_MODULE_PATH ${CMAKE_MODULE_PATH} "${CMAKE_SOURCE_DIR}/cmake/Modules")
 

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

src/Init/Classical.lean

@@ -45,7 +45,7 @@ theorem em (p : Prop) : p ∨ ¬p :=
     | Or.inr h, _ => Or.inr h
     | _, Or.inr h => Or.inr h
     | Or.inl hut, Or.inl hvf =>
-      have hne : u ≠ v := by simp [hvf, hut, true_ne_false]
+      have hne : u ≠ v := by simp [hvf, hut]
       Or.inl hne
   have p_implies_uv : p → u = v :=
     fun hp =>

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
 
 /-!

src/Init/Control/Lawful/Basic.lean

@@ -50,7 +50,7 @@ attribute [simp] id_map
   (comp_map _ _ _).symm
 
 theorem Functor.map_unit [Functor f] [LawfulFunctor f] {a : f PUnit} : (fun _ => PUnit.unit) <$> a = a := by
-  simp [map]
+  simp
 
 /--
 An applicative functor satisfies the laws of an applicative functor.
@@ -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]
 
 /--

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]
@@ -67,10 +67,10 @@ protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad
   | Except.error _ => simp
   | Except.ok _ =>
     simp [←bind_pure_comp]; apply bind_congr; intro b;
-    cases b <;> simp [comp, Except.map, const]
+    cases b <;> simp [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

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
 

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)
 
 /--

src/Init/Data.lean

@@ -46,3 +46,4 @@ import Init.Data.NeZero
 import Init.Data.Function
 import Init.Data.RArray
 import Init.Data.Vector
+import Init.Data.Iterators

src/Init/Data/AC.lean

@@ -209,7 +209,7 @@ theorem Context.evalList_sort_congr
   induction c generalizing a b with
   | nil => simp [sort.loop, h₂]
   | cons c _  ih =>
-    simp [sort.loop]; apply ih; simp [evalList_insert ctx h, evalList]
+    simp [sort.loop]; apply ih; simp [evalList_insert ctx h]
     cases a with
     | nil => apply absurd h₃; simp
     | cons a as =>
@@ -282,7 +282,7 @@ theorem Context.toList_nonEmpty (e : Expr) : e.toList ≠ [] := by
     simp [Expr.toList]
     cases h : l.toList with
     | nil => contradiction
-    | cons => simp [List.append]
+    | cons => simp
 
 theorem Context.unwrap_isNeutral
   {ctx : Context α}
@@ -328,13 +328,13 @@ theorem Context.eval_toList (ctx : Context α) (e : Expr) : evalList α ctx e.to
   induction e with
   | var x => rfl
   | op l r ih₁ ih₂ =>
-    simp [evalList, Expr.toList, eval, ←ih₁, ←ih₂]
+    simp [Expr.toList, eval, ←ih₁, ←ih₂]
     apply evalList_append <;> apply toList_nonEmpty
 
 theorem Context.eval_norm (ctx : Context α) (e : Expr) : evalList α ctx (norm ctx e) = eval α ctx e := by
   simp [norm]
   cases h₁ : ContextInformation.isIdem ctx <;> cases h₂ : ContextInformation.isComm ctx <;>
-  simp_all [evalList_removeNeutrals, eval_toList, toList_nonEmpty, evalList_mergeIdem, evalList_sort]
+  simp_all [evalList_removeNeutrals, eval_toList, evalList_mergeIdem, evalList_sort]
 
 theorem Context.eq_of_norm (ctx : Context α) (a b : Expr) (h : norm ctx a == norm ctx b) : eval α ctx a = eval α ctx b := by
   have h := congrArg (evalList α ctx) (eq_of_beq h)

src/Init/Data/Array/Attach.lean

@@ -68,15 +68,15 @@ well-founded recursion mechanism to prove that the function terminates.
     l.toArray.pmap f H = (l.pmap f (by simpa using H)).toArray := by
   simp [pmap]
 
-@[simp] theorem toList_attachWith {xs : Array α} {P : α → Prop} {H : ∀ x ∈ xs, P x} :
+@[simp, grind =] theorem toList_attachWith {xs : Array α} {P : α → Prop} {H : ∀ x ∈ xs, P x} :
    (xs.attachWith P H).toList = xs.toList.attachWith P (by simpa [mem_toList_iff] using H) := by
   simp [attachWith]
 
-@[simp] theorem toList_attach {xs : Array α} :
+@[simp, grind =] theorem toList_attach {xs : Array α} :
     xs.attach.toList = xs.toList.attachWith (· ∈ xs) (by simp [mem_toList_iff]) := by
   simp [attach]
 
-@[simp] theorem toList_pmap {xs : Array α} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ xs, P a} :
+@[simp, grind =] theorem toList_pmap {xs : Array α} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ xs, P a} :
     (xs.pmap f H).toList = xs.toList.pmap f (fun a m => H a (mem_def.mpr m)) := by
   simp [pmap]
 
@@ -92,16 +92,16 @@ well-founded recursion mechanism to prove that the function terminates.
   intro a m h₁ h₂
   congr
 
-@[simp] theorem pmap_empty {P : α → Prop} (f : ∀ a, P a → β) : pmap f #[] (by simp) = #[] := rfl
+@[simp, grind =] theorem pmap_empty {P : α → Prop} (f : ∀ a, P a → β) : pmap f #[] (by simp) = #[] := rfl
 
-@[simp] theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (xs : Array α) (h : ∀ b ∈ xs.push a, P b) :
+@[simp, grind =] theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (xs : Array α) (h : ∀ b ∈ xs.push a, P b) :
     pmap f (xs.push a) h =
       (pmap f xs (fun a m => by simp at h; exact h a (.inl m))).push (f a (h a (by simp))) := by
   simp [pmap]
 
-@[simp] theorem attach_empty : (#[] : Array α).attach = #[] := rfl
+@[simp, grind =] theorem attach_empty : (#[] : Array α).attach = #[] := rfl
 
-@[simp] theorem attachWith_empty {P : α → Prop} (H : ∀ x ∈ #[], P x) : (#[] : Array α).attachWith P H = #[] := rfl
+@[simp, grind =] theorem attachWith_empty {P : α → Prop} (H : ∀ x ∈ #[], P x) : (#[] : Array α).attachWith P H = #[] := rfl
 
 @[simp] theorem _root_.List.attachWith_mem_toArray {l : List α} :
     l.attachWith (fun x => x ∈ l.toArray) (fun x h => by simpa using h) =
@@ -122,11 +122,13 @@ theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a,
   simp only [List.pmap_toArray, mk.injEq]
   rw [List.pmap_congr_left _ h]
 
+@[grind =]
 theorem map_pmap {p : α → Prop} {g : β → γ} {f : ∀ a, p a → β} {xs : Array α} (H) :
     map g (pmap f xs H) = pmap (fun a h => g (f a h)) xs H := by
   cases xs
   simp [List.map_pmap]
 
+@[grind =]
 theorem pmap_map {p : β → Prop} {g : ∀ b, p b → γ} {f : α → β} {xs : Array α} (H) :
     pmap g (map f xs) H = pmap (fun a h => g (f a) h) xs fun _ h => H _ (mem_map_of_mem h) := by
   cases xs
@@ -142,18 +144,18 @@ theorem attachWith_congr {xs ys : Array α} (w : xs = ys) {P : α → Prop} {H :
   subst w
   simp
 
-@[simp] theorem attach_push {a : α} {xs : Array α} :
+@[simp, grind =] theorem attach_push {a : α} {xs : Array α} :
     (xs.push a).attach =
       (xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_push_of_mem a h⟩)).push ⟨a, by simp⟩ := by
   cases xs
   rw [attach_congr (List.push_toArray _ _)]
   simp [Function.comp_def]
 
-@[simp] theorem attachWith_push {a : α} {xs : Array α} {P : α → Prop} {H : ∀ x ∈ xs.push a, P x} :
+@[simp, grind =] theorem attachWith_push {a : α} {xs : Array α} {P : α → Prop} {H : ∀ x ∈ xs.push a, P x} :
     (xs.push a).attachWith P H =
       (xs.attachWith P (fun x h => by simp at H; exact H x (.inl h))).push ⟨a, H a (by simp)⟩ := by
   cases xs
-  simp [attachWith_congr (List.push_toArray _ _)]
+  simp
 
 theorem pmap_eq_map_attach {p : α → Prop} {f : ∀ a, p a → β} {xs : Array α} (H) :
     pmap f xs H = xs.attach.map fun x => f x.1 (H _ x.2) := by
@@ -189,38 +191,39 @@ theorem attachWith_map_subtype_val {p : α → Prop} {xs : Array α} (H : ∀ a
     (xs.attachWith p H).map Subtype.val = xs := by
   cases xs; simp
 
-@[simp]
+@[simp, grind]
 theorem mem_attach (xs : Array α) : ∀ x, x ∈ xs.attach
   | ⟨a, h⟩ => by
     have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
-@[simp]
+@[simp, grind]
 theorem mem_attachWith {xs : Array α} {q : α → Prop} (H) (x : {x // q x}) :
     x ∈ xs.attachWith q H ↔ x.1 ∈ xs := by
   cases xs
   simp
 
-@[simp]
+@[simp, grind =]
 theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {xs H b} :
     b ∈ pmap f xs H ↔ ∃ (a : _) (h : a ∈ xs), f a (H a h) = b := by
   simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, Subtype.exists, eq_comm]
 
+@[grind]
 theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {xs H} {a} (h : a ∈ xs) :
     f a (H a h) ∈ pmap f xs H := by
   rw [mem_pmap]
   exact ⟨a, h, rfl⟩
 
-@[simp]
+@[simp, grind =]
 theorem size_pmap {p : α → Prop} {f : ∀ a, p a → β} {xs H} : (pmap f xs H).size = xs.size := by
   cases xs; simp
 
-@[simp]
+@[simp, grind =]
 theorem size_attach {xs : Array α} : xs.attach.size = xs.size := by
   cases xs; simp
 
-@[simp]
+@[simp, grind =]
 theorem size_attachWith {p : α → Prop} {xs : Array α} {H} : (xs.attachWith p H).size = xs.size := by
   cases xs; simp
 
@@ -252,13 +255,13 @@ theorem attachWith_ne_empty_iff {xs : Array α} {P : α → Prop} {H : ∀ a ∈
     xs.attachWith P H ≠ #[] ↔ xs ≠ #[] := by
   cases xs; simp
 
-@[simp]
+@[simp, grind =]
 theorem getElem?_pmap {p : α → Prop} {f : ∀ a, p a → β} {xs : Array α} (h : ∀ a ∈ xs, p a) (i : Nat) :
     (pmap f xs h)[i]? = Option.pmap f xs[i]? fun x H => h x (mem_of_getElem? H) := by
   cases xs; simp
 
 -- The argument `f` is explicit to allow rewriting from right to left.
-@[simp]
+@[simp, grind =]
 theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {xs : Array α} (h : ∀ a ∈ xs, p a) {i : Nat}
     (hi : i < (pmap f xs h).size) :
     (pmap f xs h)[i] =
@@ -266,57 +269,59 @@ theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {xs : Array α} (
         (h _ (getElem_mem (@size_pmap _ _ p f xs h ▸ hi))) := by
   cases xs; simp
 
-@[simp]
+@[simp, grind =]
 theorem getElem?_attachWith {xs : Array α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
     (xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (mem_of_getElem? a)) :=
   getElem?_pmap ..
 
-@[simp]
+@[simp, grind =]
 theorem getElem?_attach {xs : Array α} {i : Nat} :
     xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => mem_of_getElem? a) :=
   getElem?_attachWith
 
-@[simp]
+@[simp, grind =]
 theorem getElem_attachWith {xs : Array α} {P : α → Prop} {H : ∀ a ∈ xs, P a}
     {i : Nat} (h : i < (xs.attachWith P H).size) :
     (xs.attachWith P H)[i] = ⟨xs[i]'(by simpa using h), H _ (getElem_mem (by simpa using h))⟩ :=
   getElem_pmap _ _ h
 
-@[simp]
+@[simp, grind =]
 theorem getElem_attach {xs : Array α} {i : Nat} (h : i < xs.attach.size) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
-@[simp] theorem pmap_attach {xs : Array α} {p : {x // x ∈ xs} → Prop} {f : ∀ a, p a → β} (H) :
+@[simp, grind =] theorem pmap_attach {xs : Array α} {p : {x // x ∈ xs} → Prop} {f : ∀ a, p a → β} (H) :
     pmap f xs.attach H =
       xs.pmap (P := fun a => ∃ h : a ∈ xs, p ⟨a, h⟩)
         (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
   ext <;> simp
 
-@[simp] theorem pmap_attachWith {xs : Array α} {p : {x // q x} → Prop} {f : ∀ a, p a → β} (H₁ H₂) :
+@[simp, grind =] theorem pmap_attachWith {xs : Array α} {p : {x // q x} → Prop} {f : ∀ a, p a → β} (H₁ H₂) :
     pmap f (xs.attachWith q H₁) H₂ =
       xs.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
         (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
   ext <;> simp
 
+@[grind =]
 theorem foldl_pmap {xs : Array α} {P : α → Prop} {f : (a : α) → P a → β}
     (H : ∀ (a : α), a ∈ xs → P a) (g : γ → β → γ) (x : γ) :
     (xs.pmap f H).foldl g x = xs.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
   rw [pmap_eq_map_attach, foldl_map]
 
+@[grind =]
 theorem foldr_pmap {xs : Array α} {P : α → Prop} {f : (a : α) → P a → β}
     (H : ∀ (a : α), a ∈ xs → P a) (g : β → γ → γ) (x : γ) :
     (xs.pmap f H).foldr g x = xs.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
   rw [pmap_eq_map_attach, foldr_map]
 
-@[simp] theorem foldl_attachWith
+@[simp, grind =] theorem foldl_attachWith
     {xs : Array α} {q : α → Prop} (H : ∀ a, a ∈ xs → q a) {f : β → { x // q x} → β} {b} (w : stop = xs.size) :
     (xs.attachWith q H).foldl f b 0 stop = xs.attach.foldl (fun b ⟨a, h⟩ => f b ⟨a, H _ h⟩) b := by
   subst w
   rcases xs with ⟨xs⟩
   simp [List.foldl_attachWith, List.foldl_map]
 
-@[simp] theorem foldr_attachWith
+@[simp, grind =] theorem foldr_attachWith
     {xs : Array α} {q : α → Prop} (H : ∀ a, a ∈ xs → q a) {f : { x // q x} → β → β} {b} (w : start = xs.size) :
     (xs.attachWith q H).foldr f b start 0 = xs.attach.foldr (fun a acc => f ⟨a.1, H _ a.2⟩ acc) b := by
   subst w
@@ -337,7 +342,7 @@ theorem foldl_attach {xs : Array α} {f : β → α → β} {b : β} :
     xs.attach.foldl (fun acc t => f acc t.1) b = xs.foldl f b := by
   rcases xs with ⟨xs⟩
   simp only [List.attach_toArray, List.attachWith_mem_toArray, List.size_toArray,
-    List.length_pmap, List.foldl_toArray', mem_toArray, List.foldl_subtype]
+    List.foldl_toArray', mem_toArray, List.foldl_subtype]
   congr
   ext
   simpa using fun a => List.mem_of_getElem? a
@@ -356,23 +361,25 @@ theorem foldr_attach {xs : Array α} {f : α → β → β} {b : β} :
     xs.attach.foldr (fun t acc => f t.1 acc) b = xs.foldr f b := by
   rcases xs with ⟨xs⟩
   simp only [List.attach_toArray, List.attachWith_mem_toArray, List.size_toArray,
-    List.length_pmap, List.foldr_toArray', mem_toArray, List.foldr_subtype]
+    List.foldr_toArray', mem_toArray, List.foldr_subtype]
   congr
   ext
   simpa using fun a => List.mem_of_getElem? a
 
+@[grind =]
 theorem attach_map {xs : Array α} {f : α → β} :
     (xs.map f).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem h⟩) := by
   cases xs
   ext <;> simp
 
+@[grind =]
 theorem attachWith_map {xs : Array α} {f : α → β} {P : β → Prop} (H : ∀ (b : β), b ∈ xs.map f → P b) :
     (xs.map f).attachWith P H = (xs.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem h))).map
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
   cases xs
   simp [List.attachWith_map]
 
-@[simp] theorem map_attachWith {xs : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a}
+@[simp, grind =] theorem map_attachWith {xs : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a}
     {f : { x // P x } → β} :
     (xs.attachWith P H).map f = xs.attach.map fun ⟨x, h⟩ => f ⟨x, H _ h⟩ := by
   cases xs <;> simp_all
@@ -393,6 +400,7 @@ theorem map_attach_eq_pmap {xs : Array α} {f : { x // x ∈ xs } → β} :
 @[deprecated map_attach_eq_pmap (since := "2025-02-09")]
 abbrev map_attach := @map_attach_eq_pmap
 
+@[grind =]
 theorem attach_filterMap {xs : Array α} {f : α → Option β} :
     (xs.filterMap f).attach = xs.attach.filterMap
       fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩) := by
@@ -400,6 +408,7 @@ theorem attach_filterMap {xs : Array α} {f : α → Option β} :
   rw [attach_congr List.filterMap_toArray]
   simp [List.attach_filterMap, List.map_filterMap, Function.comp_def]
 
+@[grind =]
 theorem attach_filter {xs : Array α} (p : α → Bool) :
     (xs.filter p).attach = xs.attach.filterMap
       fun x => if w : p x.1 then some ⟨x.1, mem_filter.mpr ⟨x.2, w⟩⟩ else none := by
@@ -409,7 +418,7 @@ theorem attach_filter {xs : Array α} (p : α → Bool) :
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
 
-@[simp]
+@[simp, grind =]
 theorem filterMap_attachWith {q : α → Prop} {xs : Array α} {f : {x // q x} → Option β} (H)
     (w : stop = (xs.attachWith q H).size) :
     (xs.attachWith q H).filterMap f 0 stop = xs.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
@@ -417,7 +426,7 @@ theorem filterMap_attachWith {q : α → Prop} {xs : Array α} {f : {x // q x}
   cases xs
   simp [Function.comp_def]
 
-@[simp]
+@[simp, grind =]
 theorem filter_attachWith {q : α → Prop} {xs : Array α} {p : {x // q x} → Bool} (H)
     (w : stop = (xs.attachWith q H).size) :
     (xs.attachWith q H).filter p 0 stop =
@@ -426,6 +435,7 @@ theorem filter_attachWith {q : α → Prop} {xs : Array α} {p : {x // q x} →
   cases xs
   simp [Function.comp_def, List.filter_map]
 
+@[grind =]
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} {g : ∀ a, p a → β} {f : ∀ b, q b → γ} {xs} (H₁ H₂) :
     pmap f (pmap g xs H₁) H₂ =
       pmap (α := { x // x ∈ xs }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) xs.attach
@@ -433,7 +443,7 @@ theorem pmap_pmap {p : α → Prop} {q : β → Prop} {g : ∀ a, p a → β} {f
   cases xs
   simp [List.pmap_pmap, List.pmap_map]
 
-@[simp] theorem pmap_append {p : ι → Prop} {f : ∀ a : ι, p a → α} {xs ys : Array ι}
+@[simp, grind =] theorem pmap_append {p : ι → Prop} {f : ∀ a : ι, p a → α} {xs ys : Array ι}
     (h : ∀ a ∈ xs ++ ys, p a) :
     (xs ++ ys).pmap f h =
       (xs.pmap f fun a ha => h a (mem_append_left ys ha)) ++
@@ -448,7 +458,7 @@ theorem pmap_append' {p : α → Prop} {f : ∀ a : α, p a → β} {xs ys : Arr
       xs.pmap f h₁ ++ ys.pmap f h₂ :=
   pmap_append _
 
-@[simp] theorem attach_append {xs ys : Array α} :
+@[simp, grind =] theorem attach_append {xs ys : Array α} :
     (xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_left ys h⟩) ++
       ys.attach.map fun ⟨x, h⟩ => ⟨x, mem_append_right xs h⟩ := by
   cases xs
@@ -456,59 +466,62 @@ theorem pmap_append' {p : α → Prop} {f : ∀ a : α, p a → β} {xs ys : Arr
   rw [attach_congr (List.append_toArray _ _)]
   simp [List.attach_append, Function.comp_def]
 
-@[simp] theorem attachWith_append {P : α → Prop} {xs ys : Array α}
+@[simp, grind =] theorem attachWith_append {P : α → Prop} {xs ys : Array α}
     {H : ∀ (a : α), a ∈ xs ++ ys → P a} :
     (xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_left ys h)) ++
       ys.attachWith P (fun a h => H a (mem_append_right xs h)) := by
-  simp [attachWith, attach_append, map_pmap, pmap_append]
+  simp [attachWith]
 
-@[simp] theorem pmap_reverse {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
+@[simp, grind =] theorem pmap_reverse {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
     (H : ∀ (a : α), a ∈ xs.reverse → P a) :
     xs.reverse.pmap f H = (xs.pmap f (fun a h => H a (by simpa using h))).reverse := by
   induction xs <;> simp_all
 
+@[grind =]
 theorem reverse_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h)) := by
   rw [pmap_reverse]
 
-@[simp] theorem attachWith_reverse {P : α → Prop} {xs : Array α}
+@[simp, grind =] theorem attachWith_reverse {P : α → Prop} {xs : Array α}
     {H : ∀ (a : α), a ∈ xs.reverse → P a} :
     xs.reverse.attachWith P H =
       (xs.attachWith P (fun a h => H a (by simpa using h))).reverse := by
   cases xs
   simp
 
+@[grind =]
 theorem reverse_attachWith {P : α → Prop} {xs : Array α}
     {H : ∀ (a : α), a ∈ xs → P a} :
     (xs.attachWith P H).reverse = (xs.reverse.attachWith P (fun a h => H a (by simpa using h))) := by
   cases xs
   simp
 
-@[simp] theorem attach_reverse {xs : Array α} :
+@[simp, grind =] theorem attach_reverse {xs : Array α} :
     xs.reverse.attach = xs.attach.reverse.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
   cases xs
   rw [attach_congr List.reverse_toArray]
   simp
 
+@[grind =]
 theorem reverse_attach {xs : Array α} :
     xs.attach.reverse = xs.reverse.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
   cases xs
   simp
 
-@[simp] theorem back?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
+@[simp, grind =] theorem back?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).back? = xs.attach.back?.map fun ⟨a, m⟩ => f a (H a m) := by
   cases xs
   simp
 
-@[simp] theorem back?_attachWith {P : α → Prop} {xs : Array α}
+@[simp, grind =] theorem back?_attachWith {P : α → Prop} {xs : Array α}
     {H : ∀ (a : α), a ∈ xs → P a} :
     (xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back? h)⟩) := by
   cases xs
   simp
 
-@[simp]
+@[simp, grind =]
 theorem back?_attach {xs : Array α} :
     xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back? h⟩ := by
   cases xs
@@ -526,7 +539,7 @@ theorem countP_attachWith {p : α → Prop} {q : α → Bool} {xs : Array α} {H
   cases xs
   simp
 
-@[simp]
+@[simp, grind =]
 theorem count_attach [BEq α] {xs : Array α} {a : {x // x ∈ xs}} :
     xs.attach.count a = xs.count ↑a := by
   rcases xs with ⟨xs⟩
@@ -535,13 +548,13 @@ theorem count_attach [BEq α] {xs : Array α} {a : {x // x ∈ xs}} :
   simp only [Subtype.beq_iff]
   rw [List.countP_pmap, List.countP_attach (p := (fun x => x == a.1)), List.count]
 
-@[simp]
+@[simp, grind =]
 theorem count_attachWith [BEq α] {p : α → Prop} {xs : Array α} (H : ∀ a ∈ xs, p a) {a : {x // p x}} :
     (xs.attachWith p H).count a = xs.count ↑a := by
   cases xs
   simp
 
-@[simp] theorem countP_pmap {p : α → Prop} {g : ∀ a, p a → β} {f : β → Bool} {xs : Array α} (H₁) :
+@[simp, grind =] theorem countP_pmap {p : α → Prop} {g : ∀ a, p a → β} {f : β → Bool} {xs : Array α} (H₁) :
     (xs.pmap g H₁).countP f =
       xs.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
   simp [pmap_eq_map_attach, countP_map, Function.comp_def]
@@ -690,7 +703,7 @@ and simplifies these to the function directly taking the value.
     {f : { x // p x } → Array β} {g : α → Array β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
     (xs.flatMap f) = xs.unattach.flatMap g := by
   cases xs
-  simp only [List.size_toArray, List.flatMap_toArray, List.unattach_toArray, List.length_unattach,
+  simp only [List.flatMap_toArray, List.unattach_toArray, 
     mk.injEq]
   rw [List.flatMap_subtype]
   simp [hf]

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]
 
@@ -1788,7 +1788,7 @@ decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ h
   induction xs, i, h using Array.eraseIdx.induct with
   | @case1 xs i h h' xs' ih =>
     unfold eraseIdx
-    simp +zetaDelta [h', xs', ih]
+    simp +zetaDelta [h', ih]
   | case2 xs i h h' =>
     unfold eraseIdx
     simp [h']

src/Init/Data/Array/Bootstrap.lean

@@ -119,13 +119,13 @@ abbrev pop_toList := @Array.toList_pop
 @[simp] theorem toList_empty : (#[] : Array α).toList = [] := rfl
 
 @[simp, grind =] theorem append_empty {xs : Array α} : xs ++ #[] = xs := by
-  apply ext'; simp only [toList_append, toList_empty, List.append_nil]
+  apply ext'; simp only [toList_append, List.append_nil]
 
 @[deprecated append_empty (since := "2025-01-13")]
 abbrev append_nil := @append_empty
 
 @[simp, grind =] theorem empty_append {xs : Array α} : #[] ++ xs = xs := by
-  apply ext'; simp only [toList_append, toList_empty, List.nil_append]
+  apply ext'; simp only [toList_append, List.nil_append]
 
 @[deprecated empty_append (since := "2025-01-13")]
 abbrev nil_append := @empty_append

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
 
@@ -209,9 +220,10 @@ 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]
+  simp [count_eq_countP, countP_set]
 
 variable [LawfulBEq α]
 
@@ -219,7 +231,7 @@ variable [LawfulBEq α]
   simp [count_push]
 
 @[simp] theorem count_push_of_ne {xs : Array α} (h : b ≠ a) : count a (xs.push b) = count a xs := by
-  simp_all [count_push, h]
+  simp_all [count_push]
 
 theorem count_singleton_self {a : α} : count a #[a] = 1 := by simp
 
@@ -280,17 +292,17 @@ abbrev mkArray_count_eq_of_count_eq_size := @replicate_count_eq_of_count_eq_size
 theorem count_le_count_map [BEq β] [LawfulBEq β] {xs : Array α} {f : α → β} {x : α} :
     count x xs ≤ count (f x) (map f xs) := by
   rcases xs with ⟨xs⟩
-  simp [List.count_le_count_map, countP_map]
+  simp [List.count_le_count_map]
 
 theorem count_filterMap {α} [BEq β] {b : β} {f : α → Option β} {xs : Array α} :
     count b (filterMap f xs) = countP (fun a => f a == some b) xs := by
   rcases xs with ⟨xs⟩
-  simp [List.count_filterMap, countP_filterMap]
+  simp [List.count_filterMap]
 
 theorem count_flatMap {α} [BEq β] {xs : Array α} {f : α → Array β} {x : β} :
     count x (xs.flatMap f) = sum (map (count x ∘ f) xs) := by
   rcases xs with ⟨xs⟩
-  simp [List.count_flatMap, countP_flatMap, Function.comp_def]
+  simp [List.count_flatMap, Function.comp_def]
 
 theorem countP_replace {a b : α} {xs : Array α} {p : α → Bool} :
     (xs.replace a b).countP p =

src/Init/Data/Array/DecidableEq.lean

@@ -23,7 +23,7 @@ private theorem rel_of_isEqvAux
   induction i with
   | zero => contradiction
   | succ i ih =>
-    simp only [Array.isEqvAux, Bool.and_eq_true, decide_eq_true_eq] at heqv
+    simp only [Array.isEqvAux, Bool.and_eq_true] at heqv
     by_cases hj' : j < i
     next =>
       exact ih _ heqv.right hj'

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⟩
@@ -197,7 +206,7 @@ theorem erase_eq_eraseP [LawfulBEq α] (a : α) (xs : Array α) : xs.erase a = x
 theorem erase_ne_empty_iff [LawfulBEq α] {xs : Array α} {a : α} :
     xs.erase a ≠ #[] ↔ xs ≠ #[] ∧ xs ≠ #[a] := by
   rcases xs with ⟨xs⟩
-  simp [List.erase_ne_nil_iff]
+  simp
 
 theorem exists_erase_eq [LawfulBEq α] {a : α} {xs : Array α} (h : a ∈ xs) :
     ∃ ys zs, a ∉ ys ∧ xs = ys.push a ++ zs ∧ xs.erase a = ys ++ zs := by
@@ -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⟩
@@ -291,7 +306,7 @@ theorem erase_eq_iff [LawfulBEq α] {a : α} {xs : Array α} :
 @[simp] theorem erase_replicate_self [LawfulBEq α] {a : α} :
     (replicate n a).erase a = replicate (n - 1) a := by
   simp only [← List.toArray_replicate, List.erase_toArray]
-  simp [List.erase_replicate]
+  simp
 
 @[deprecated erase_replicate_self (since := "2025-03-18")]
 abbrev erase_mkArray_self := @erase_replicate_self
@@ -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⟩
@@ -335,10 +352,11 @@ theorem getElem?_eraseIdx_of_lt {xs : Array α} {i : Nat} (h : i < xs.size) {j :
 theorem getElem?_eraseIdx_of_ge {xs : Array α} {i : Nat} (h : i < xs.size) {j : Nat} (h' : i ≤ j) :
     (xs.eraseIdx i)[j]? = xs[j + 1]? := by
   rw [getElem?_eraseIdx]
-  simp only [dite_eq_ite, ite_eq_right_iff]
+  simp only [ite_eq_right_iff]
   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

src/Init/Data/Array/Extract.lean

@@ -29,7 +29,7 @@ namespace Array
   · simp
     omega
   · simp only [size_extract] at h₁ h₂
-    simp [h]
+    simp
 
 theorem size_extract_le {as : Array α} {i j : Nat} :
     (as.extract i j).size ≤ j - i := by
@@ -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
@@ -162,14 +162,14 @@ theorem extract_sub_one {as : Array α} {i j : Nat} (h : j < as.size) :
 @[simp]
 theorem getElem?_extract_of_lt {as : Array α} {i j k : Nat} (h : k < min j as.size - i) :
     (as.extract i j)[k]? = some (as[i + k]'(by omega)) := by
-  simp [getElem?_extract, h]
+  simp [h]
 
 theorem getElem?_extract_of_succ {as : Array α} {j : Nat} :
     (as.extract 0 (j + 1))[j]? = as[j]? := by
   simp [getElem?_extract]
   omega
 
-@[simp] theorem extract_extract {as : Array α} {i j k l : Nat} :
+@[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]

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

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]
+  cases xs; simp [← 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 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]
 
@@ -270,10 +305,10 @@ 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]
+  simp [List.find?_flatMap]
 
 theorem find?_flatMap_eq_none_iff {xs : Array α} {f : α → Array β} {p : β → Bool} :
     (xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
@@ -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]
@@ -312,7 +348,7 @@ abbrev find?_mkArray_of_neg := @find?_replicate_of_neg
 -- This isn't a `@[simp]` lemma since there is already a lemma for `l.find? p = none` for any `l`.
 theorem find?_replicate_eq_none_iff {n : Nat} {a : α} {p : α → Bool} :
     (replicate n a).find? p = none ↔ n = 0 ∨ !p a := by
-  simp [← List.toArray_replicate, List.find?_replicate_eq_none_iff, Classical.or_iff_not_imp_left]
+  simp [← List.toArray_replicate, Classical.or_iff_not_imp_left]
 
 @[deprecated find?_replicate_eq_none_iff (since := "2025-03-18")]
 abbrev find?_mkArray_eq_none_iff := @find?_replicate_eq_none_iff
@@ -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
@@ -433,12 +482,13 @@ 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]
   split <;> rename_i h
   · rfl
-  · simp [findIdx_singleton, Nat.add_comm]
+  · simp [Nat.add_comm]
 
 theorem findIdx_le_findIdx {xs : Array α} {p q : α → Bool} (h : ∀ x ∈ xs, p x → q x) : xs.findIdx q ≤ xs.findIdx p := by
   rcases xs with ⟨xs⟩
@@ -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⟩
@@ -503,7 +553,7 @@ theorem findIdx?_eq_some_of_exists {xs : Array α} {p : α → Bool} (h : ∃ x,
 theorem findIdx?_eq_none_iff_findIdx_eq {xs : Array α} {p : α → Bool} :
     xs.findIdx? p = none ↔ xs.findIdx p = xs.size := by
   rcases xs with ⟨xs⟩
-  simp [List.findIdx?_eq_none_iff_findIdx_eq]
+  simp
 
 theorem findIdx?_eq_guard_findIdx_lt {xs : Array α} {p : α → Bool} :
     xs.findIdx? p = Option.guard (fun i => i < xs.size) (xs.findIdx p) := by
@@ -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
@@ -727,11 +798,11 @@ The lemmas below should be made consistent with those for `findFinIdx?` (and pro
 
 theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : Array α} {a : α} :
     xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
-  simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
+  simp [idxOf?, finIdxOf?]
 
-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

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
@@ -91,21 +109,22 @@ theorem getElem_insertIdx {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.siz
         else
           xs[k-1]'(by simp [size_insertIdx] at h; omega) := by
   cases xs
-  simp [List.getElem_insertIdx, w]
+  simp [List.getElem_insertIdx]
 
 theorem getElem_insertIdx_of_lt {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.size) (h : k < i) :
     (xs.insertIdx i x)[k]'(by simp; omega) = xs[k] := by
-  simp [getElem_insertIdx, w, h]
+  simp [getElem_insertIdx, h]
 
 theorem getElem_insertIdx_self {xs : Array α} {x : α} {i : Nat} (w : i ≤ xs.size) :
     (xs.insertIdx i x)[i]'(by simp; omega) = x := by
-  simp [getElem_insertIdx, w]
+  simp [getElem_insertIdx]
 
 theorem getElem_insertIdx_of_gt {xs : Array α} {x : α} {i k : Nat} (w : k ≤ xs.size) (h : k > i) :
     (xs.insertIdx i x)[k]'(by simp; omega) = xs[k - 1]'(by omega) := by
-  simp [getElem_insertIdx, w, h]
+  simp [getElem_insertIdx]
   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
@@ -116,7 +135,7 @@ theorem getElem?_insertIdx {xs : Array α} {x : α} {i k : Nat} (h : i ≤ xs.si
         else
           xs[k-1]? := by
   cases xs
-  simp [List.getElem?_insertIdx, h]
+  simp [List.getElem?_insertIdx]
 
 theorem getElem?_insertIdx_of_lt {xs : Array α} {x : α} {i k : Nat} (w : i ≤ xs.size) (h : k < i) :
     (xs.insertIdx i x)[k]? = xs[k]? := by

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
@@ -123,7 +125,7 @@ theorem none_eq_getElem?_iff {xs : Array α} {i : Nat} : none = xs[i]? ↔ xs.si
   simp
 
 theorem getElem?_eq_none {xs : Array α} (h : xs.size ≤ i) : xs[i]? = none := by
-  simp [getElem?_eq_none_iff, h]
+  simp [h]
 
 grind_pattern Array.getElem?_eq_none => xs.size ≤ i, xs[i]?
 
@@ -133,7 +135,6 @@ grind_pattern Array.getElem?_eq_none => xs.size ≤ i, xs[i]?
 theorem getElem?_eq_some_iff {xs : Array α} : xs[i]? = some b ↔ ∃ h : i < xs.size, xs[i] = b :=
   _root_.getElem?_eq_some_iff
 
-@[grind →]
 theorem getElem_of_getElem? {xs : Array α} : xs[i]? = some a → ∃ h : i < xs.size, xs[i] = a :=
   getElem?_eq_some_iff.mp
 
@@ -153,16 +154,16 @@ theorem getElem_eq_iff {xs : Array α} {i : Nat} {h : i < xs.size} : xs[i] = x
   exact ⟨fun w => ⟨h, w⟩, fun h => h.2⟩
 
 theorem getElem_eq_getElem?_get {xs : Array α} {i : Nat} (h : i < xs.size) :
-    xs[i] = xs[i]?.get (by simp [getElem?_eq_getElem, h]) := by
-  simp [getElem_eq_iff]
+    xs[i] = xs[i]?.get (by simp [h]) := by
+  simp
 
 theorem getD_getElem? {xs : Array α} {i : Nat} {d : α} :
     xs[i]?.getD d = if p : i < xs.size then xs[i]'p else d := by
   if h : i < xs.size then
-    simp [h, getElem?_def]
+    simp [h]
   else
     have p : i ≥ xs.size := Nat.le_of_not_gt h
-    simp [getElem?_eq_none p, h]
+    simp [h]
 
 @[simp] theorem getElem?_empty {i : Nat} : (#[] : Array α)[i]? = none := rfl
 
@@ -174,14 +175,14 @@ theorem getElem_push_lt {xs : Array α} {x : α} {i : Nat} (h : i < xs.size) :
 
 @[simp] theorem getElem_push_eq {xs : Array α} {x : α} : (xs.push x)[xs.size] = x := by
   simp only [push, ← getElem_toList, List.concat_eq_append]
-  rw [List.getElem_append_right] <;> simp [← getElem_toList, Nat.zero_lt_one]
+  rw [List.getElem_append_right] <;> simp
 
-theorem getElem_push {xs : Array α} {x : α} {i : Nat} (h : i < (xs.push x).size) :
+@[grind =] theorem getElem_push {xs : Array α} {x : α} {i : Nat} (h : i < (xs.push x).size) :
     (xs.push x)[i] = if h : i < xs.size then xs[i] else x := by
   by_cases h' : i < xs.size
   · simp [getElem_push_lt, h']
   · simp at h
-    simp [getElem_push_lt, Nat.le_antisymm (Nat.le_of_lt_succ h) (Nat.ge_of_not_lt h')]
+    simp [Nat.le_antisymm (Nat.le_of_lt_succ h) (Nat.ge_of_not_lt h')]
 
 @[grind =] theorem getElem?_push {xs : Array α} {x} : (xs.push x)[i]? = if i = xs.size then some x else xs[i]? := by
   simp [getElem?_def, getElem_push]
@@ -905,7 +906,7 @@ theorem all_push [BEq α] {xs : Array α} {a : α} {p : α → Bool} :
 abbrev getElem_set_eq := @getElem_set_self
 
 @[simp] theorem getElem?_set_self {xs : Array α} {i : Nat} (h : i < xs.size) {v : α} :
-    (xs.set i v)[i]? = some v := by simp [getElem?_eq_getElem, h]
+    (xs.set i v)[i]? = some v := by simp [h]
 
 @[deprecated getElem?_set_self (since := "2024-12-11")]
 abbrev getElem?_set_eq := @getElem?_set_self
@@ -917,7 +918,7 @@ abbrev getElem?_set_eq := @getElem?_set_self
 
 @[simp] theorem getElem?_set_ne {xs : Array α} {i : Nat} (h : i < xs.size) {v : α} {j : Nat}
     (ne : i ≠ j) : (xs.set i v)[j]? = xs[j]? := by
-  by_cases h : j < xs.size <;> simp [getElem?_eq_getElem, getElem?_eq_none, Nat.ge_of_not_lt, ne, h]
+  by_cases h : j < xs.size <;> simp [ne, h]
 
 @[grind] theorem getElem_set {xs : Array α} {i : Nat} (h' : i < xs.size) {v : α} {j : Nat}
     (h : j < (xs.set i v).size) :
@@ -954,6 +955,13 @@ theorem set_push {xs : Array α} {x y : α} {h} :
         · simp at h
           omega
 
+@[grind _=_]
+theorem set_pop {xs : Array α} {x : α} {i : Nat} (h : i < xs.pop.size) :
+    xs.pop.set i x h = (xs.set i x (by simp at h; omega)).pop := by
+  ext i h₁ h₂
+  · simp
+  · simp [getElem_set]
+
 @[simp] theorem set_eq_empty_iff {xs : Array α} {i : Nat} {a : α} {h : i < xs.size} :
     xs.set i a = #[] ↔ xs = #[] := by
   cases xs <;> cases i <;> simp [set]
@@ -986,7 +994,11 @@ theorem mem_or_eq_of_mem_set
 @[simp, grind] theorem setIfInBounds_empty {i : Nat} {a : α} :
     #[].setIfInBounds i a = #[] := rfl
 
-@[simp] theorem set!_eq_setIfInBounds : @set! = @setIfInBounds := rfl
+@[simp, grind =] theorem set!_eq_setIfInBounds : set! xs i v = setIfInBounds xs i v := rfl
+
+@[grind]
+theorem setIfInBounds_def (xs : Array α) (i : Nat) (a : α) :
+    xs.setIfInBounds i a = if h : i < xs.size then xs.set i a else xs := rfl
 
 @[deprecated set!_eq_setIfInBounds (since := "2024-12-12")]
 abbrev set!_is_setIfInBounds := @set!_eq_setIfInBounds
@@ -1032,7 +1044,7 @@ theorem getElem?_setIfInBounds_self {xs : Array α} {i : Nat} {a : α} :
 @[simp]
 theorem getElem?_setIfInBounds_self_of_lt {xs : Array α} {i : Nat} {a : α} (h : i < xs.size) :
     (xs.setIfInBounds i a)[i]? = some a := by
-  simp [getElem?_setIfInBounds, h]
+  simp [h]
 
 @[deprecated getElem?_setIfInBounds_self (since := "2024-12-11")]
 abbrev getElem?_setIfInBounds_eq := @getElem?_setIfInBounds_self
@@ -1076,9 +1088,9 @@ theorem mem_or_eq_of_mem_setIfInBounds
 @[simp] theorem getD_getElem?_setIfInBounds {xs : Array α} {i : Nat} {v d : α} :
     (xs.setIfInBounds i v)[i]?.getD d = if i < xs.size then v else d := by
   by_cases h : i < xs.size <;>
-    simp [setIfInBounds, Nat.not_lt_of_le, h,  getD_getElem?]
+    simp [setIfInBounds, h,  ]
 
-@[simp] theorem toList_setIfInBounds {xs : Array α} {i : Nat} {x : α} :
+@[simp, grind =] theorem toList_setIfInBounds {xs : Array α} {i : Nat} {x : α} :
     (xs.setIfInBounds i x).toList = xs.toList.set i x := by
   simp only [setIfInBounds]
   split <;> rename_i h
@@ -1188,7 +1200,7 @@ where
       mapM.map f xs i bs = (xs.toList.drop i).foldlM (fun bs a => bs.push <$> f a) bs := by
     unfold mapM.map; split
     · rw [← List.getElem_cons_drop_succ_eq_drop ‹_›]
-      simp only [aux (i + 1), map_eq_pure_bind, length_toList, List.foldlM_cons, bind_assoc,
+      simp only [aux (i + 1), map_eq_pure_bind, List.foldlM_cons, bind_assoc,
         pure_bind]
       rfl
     · rw [List.drop_of_length_le (Nat.ge_of_not_lt ‹_›)]; rfl
@@ -1488,6 +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'
@@ -1730,7 +1755,7 @@ theorem forall_mem_filterMap {f : α → Option β} {xs : Array α} {P : β →
 
 theorem map_filterMap_of_inv {f : α → Option β} {g : β → α} (H : ∀ x : α, (f x).map g = some x) {xs : Array α} :
     map g (filterMap f xs) = xs := by
-  simp only [map_filterMap, H, filterMap_some, id]
+  simp only [map_filterMap, H, filterMap_some]
 
 @[grind →]
 theorem forall_none_of_filterMap_eq_empty (h : filterMap f xs = #[]) : ∀ x ∈ xs, f x = none := by
@@ -1869,7 +1894,7 @@ theorem getElem?_append_left {xs ys : Array α} {i : Nat} (hn : i < xs.size) :
     (xs ++ ys)[i]? = xs[i]? := by
   have hn' : i < (xs ++ ys).size := Nat.lt_of_lt_of_le hn <|
     size_append .. ▸ Nat.le_add_right ..
-  simp_all [getElem?_eq_getElem, getElem_append]
+  simp_all
 
 theorem getElem?_append_right {xs ys : Array α} {i : Nat} (h : xs.size ≤ i) :
     (xs ++ ys)[i]? = ys[i - xs.size]? := by
@@ -2000,7 +2025,7 @@ theorem append_eq_singleton_iff {xs ys : Array α} {x : α} :
     xs ++ ys = #[x] ↔ (xs = #[] ∧ ys = #[x]) ∨ (xs = #[x] ∧ ys = #[]) := by
   rcases xs with ⟨xs⟩
   rcases ys with ⟨ys⟩
-  simp only [List.append_toArray, mk.injEq, List.append_eq_singleton_iff, toArray_eq_append_iff]
+  simp only [List.append_toArray, mk.injEq, List.append_eq_singleton_iff]
 
 theorem singleton_eq_append_iff {xs ys : Array α} {x : α} :
     #[x] = xs ++ ys ↔ (xs = #[] ∧ ys = #[x]) ∨ (xs = #[x] ∧ ys = #[]) := by
@@ -2326,7 +2351,7 @@ theorem flatMap_toArray {β} {f : α → Array β} {as : List α} :
 theorem size_flatMap {xs : Array α} {f : α → Array β} :
     (xs.flatMap f).size = sum (map (fun a => (f a).size) xs) := by
   rcases xs with ⟨l⟩
-  simp [Function.comp_def]
+  simp
 
 @[simp, grind] theorem mem_flatMap {f : α → Array β} {b} {xs : Array α} : b ∈ xs.flatMap f ↔ ∃ a, a ∈ xs ∧ b ∈ f a := by
   simp [flatMap_def, mem_flatten]
@@ -2544,7 +2569,7 @@ abbrev map_mkArray := @map_replicate
 @[grind] theorem filter_replicate (w : stop = n) :
     (replicate n a).filter p 0 stop = if p a then replicate n a else #[] := by
   apply Array.ext'
-  simp only [w, toList_filter', toList_replicate, List.filter_replicate]
+  simp only [w]
   split <;> simp_all
 
 @[deprecated filter_replicate (since := "2025-03-18")]
@@ -2584,7 +2609,7 @@ abbrev filterMap_mkArray_of_some := @filterMap_replicate_of_some
 @[simp] theorem filterMap_replicate_of_isSome {f : α → Option β} (h : (f a).isSome) :
     (replicate n a).filterMap f = replicate n (Option.get _ h) := by
   match w : f a, h with
-  | some b, _ => simp [filterMap_replicate, h, w]
+  | some b, _ => simp [filterMap_replicate, w]
 
 @[deprecated filterMap_replicate_of_isSome (since := "2025-03-18")]
 abbrev filterMap_mkArray_of_isSome := @filterMap_replicate_of_isSome
@@ -2619,7 +2644,7 @@ abbrev flatten_mkArray_replicate := @flatten_replicate_replicate
 
 theorem flatMap_replicate {f : α → Array β} : (replicate n a).flatMap f = (replicate n (f a)).flatten := by
   rw [← toList_inj]
-  simp [flatMap_toList, List.flatMap_replicate]
+  simp [List.flatMap_replicate]
 
 @[deprecated flatMap_replicate (since := "2025-03-18")]
 abbrev flatMap_mkArray := @flatMap_replicate
@@ -2997,6 +3022,10 @@ theorem extract_empty_of_size_le_start {xs : Array α} {start stop : Nat} (h : x
   apply ext'
   simp
 
+theorem _root_.List.toArray_drop {l : List α} {k : Nat} :
+    (l.drop k).toArray = l.toArray.extract k := by
+  rw [List.drop_eq_extract, List.extract_toArray, List.size_toArray]
+
 @[deprecated extract_size (since := "2025-02-27")]
 theorem take_size {xs : Array α} : xs.take xs.size = xs := by
   cases xs
@@ -3607,8 +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')
@@ -3618,8 +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')
@@ -4113,7 +4142,7 @@ theorem getElem_modify_of_ne {xs : Array α} {i : Nat} (h : i ≠ j)
 
 @[simp] theorem getElem_swap_right {xs : Array α} {i j : Nat} {hi hj} :
     (xs.swap i j hi hj)[j]'(by simpa using hj) = xs[i] := by
-  simp [swap_def, getElem_set]
+  simp [swap_def]
 
 @[simp] theorem getElem_swap_left {xs : Array α} {i j : Nat} {hi hj} :
     (xs.swap i j hi hj)[i]'(by simpa using hi) = xs[j] := by
@@ -4410,7 +4439,7 @@ theorem size_uset {xs : Array α} {v : α} {i : USize} (h : i.toNat < xs.size) :
 
 @[simp] theorem getD_eq_getD_getElem? {xs : Array α} {i : Nat} {d : α} :
     xs.getD i d = xs[i]?.getD d := by
-  simp only [getD]; split <;> simp [getD_getElem?, *]
+  simp only [getD]; split <;> simp [*]
 
 theorem getElem!_eq_getD [Inhabited α] {xs : Array α} {i} : xs[i]! = xs.getD i default := by
   rfl
@@ -4438,13 +4467,13 @@ theorem getElem?_size_le {xs : Array α} {i : Nat} (h : xs.size ≤ i) : xs[i]?
   simp [getElem?_neg, h]
 
 theorem getElem_mem_toList {xs : Array α} {i : Nat} (h : i < xs.size) : xs[i] ∈ xs.toList := by
-  simp only [← getElem_toList, List.getElem_mem, ugetElem_eq_getElem]
+  simp only [← getElem_toList, List.getElem_mem]
 
 theorem back!_eq_back? [Inhabited α] {xs : Array α} : xs.back! = xs.back?.getD default := by
   simp [back!, back?, getElem!_def, Option.getD]; rfl
 
 @[simp, grind] theorem back?_push {xs : Array α} {x : α} : (xs.push x).back? = some x := by
-  simp [back?, ← getElem?_toList]
+  simp [back?]
 
 @[simp] theorem back!_push [Inhabited α] {xs : Array α} {x : α} : (xs.push x).back! = x := by
   simp [back!_eq_back?]
@@ -4512,7 +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
@@ -4593,7 +4622,7 @@ theorem uset_toArray {l : List α} {i : USize} {a : α} {h : i.toNat < l.toArray
 @[simp, grind =] theorem flatten_toArray {L : List (List α)} :
     (L.toArray.map List.toArray).flatten = L.flatten.toArray := by
   apply ext'
-  simp [Function.comp_def]
+  simp
 
 end List
 
@@ -4685,7 +4714,7 @@ namespace List
       intro h'
       specialize ih (by omega)
       have : as.length - (i + 1) + 1 = as.length - i := by omega
-      simp_all [ih]
+      simp_all
     · simp only [size_toArray, Nat.not_lt] at h
       have : as.length = 0 := by omega
       simp_all
@@ -4696,7 +4725,7 @@ end List
 namespace Array
 
 @[deprecated size_toArray (since := "2024-12-11")]
-theorem size_mk (as : List α) : (Array.mk as).size = as.length := by simp [size]
+theorem size_mk (as : List α) : (Array.mk as).size = as.length := by simp
 
 @[deprecated getElem?_eq_getElem (since := "2024-12-11")]
 theorem getElem?_lt
@@ -4712,7 +4741,7 @@ theorem get?_eq_getElem? (xs : Array α) (i : Nat) : xs.get? i = xs[i]? := rfl
 
 @[deprecated getElem?_eq_none (since := "2024-12-11")]
 theorem getElem?_len_le (xs : Array α) {i : Nat} (h : xs.size ≤ i) : xs[i]? = none := by
-  simp [getElem?_eq_none, h]
+  simp [h]
 
 @[deprecated getD_getElem? (since := "2024-12-11")] abbrev getD_get? := @getD_getElem?
 
@@ -4727,7 +4756,7 @@ set_option linter.deprecated false in
 theorem get!_eq_getD_getElem? [Inhabited α] (xs : Array α) (i : Nat) :
     xs.get! i = xs[i]?.getD default := by
   by_cases p : i < xs.size <;>
-  simp [get!, getElem!_eq_getD, getD_eq_getD_getElem?, getD_getElem?, p]
+  simp [get!, getD_eq_getD_getElem?, p]
 
 set_option linter.deprecated false in
 @[deprecated get!_eq_getD_getElem? (since := "2025-02-12")] abbrev get!_eq_getElem? := @get!_eq_getD_getElem?

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

@@ -162,7 +162,7 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
     {xs ys : Array α} : lex xs ys = false ↔ ys ≤ xs := by
   cases xs
   cases ys
-  simp [List.not_lt_iff_ge]
+  simp
 
 instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLT (Array α) :=
   fun xs ys => decidable_of_iff (lex xs ys = true) lex_eq_true_iff_lt

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]
 
@@ -135,12 +135,12 @@ abbrev getElem_zipWithIndex := @getElem_zipIdx
 
 @[simp, grind =] theorem zipIdx_toArray {l : List α} {k : Nat} :
     l.toArray.zipIdx k = (l.zipIdx k).toArray := by
-  ext i hi₁ hi₂ <;> simp [Nat.add_comm]
+  ext i hi₁ hi₂ <;> simp
 
 @[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) :

src/Init/Data/Array/Monadic.lean

@@ -36,19 +36,19 @@ theorem map_toList_inj [Monad m] [LawfulMonad m]
     xs.mapM (m := m) (pure <| f ·) = pure (xs.map f) := by
   induction xs; simp_all
 
-@[simp] theorem idRun_mapM {xs : Array α} {f : α → Id β} : (xs.mapM f).run = xs.map (f · |>.run) :=
+@[simp, grind =] theorem idRun_mapM {xs : Array α} {f : α → Id β} : (xs.mapM f).run = xs.map (f · |>.run) :=
   mapM_pure
 
 @[deprecated idRun_mapM (since := "2025-05-21")]
 theorem mapM_id {xs : Array α} {f : α → Id β} : xs.mapM f = xs.map f :=
   mapM_pure
 
-@[simp] theorem mapM_map [Monad m] [LawfulMonad m] {f : α → β} {g : β → m γ} {xs : Array α} :
+@[simp, grind =] theorem mapM_map [Monad m] [LawfulMonad m] {f : α → β} {g : β → m γ} {xs : Array α} :
     (xs.map f).mapM g = xs.mapM (g ∘ f) := by
   rcases xs with ⟨xs⟩
   simp
 
-@[simp] theorem mapM_append [Monad m] [LawfulMonad m] {f : α → m β} {xs ys : Array α} :
+@[simp, grind =] theorem mapM_append [Monad m] [LawfulMonad m] {f : α → m β} {xs ys : Array α} :
     (xs ++ ys).mapM f = (return (← xs.mapM f) ++ (← ys.mapM f)) := by
   rcases xs with ⟨xs⟩
   rcases ys with ⟨ys⟩
@@ -59,7 +59,7 @@ theorem mapM_eq_foldlM_push [Monad m] [LawfulMonad m] {f : α → m β} {xs : Ar
   rcases xs with ⟨xs⟩
   simp only [List.mapM_toArray, bind_pure_comp, List.size_toArray, List.foldlM_toArray']
   rw [List.mapM_eq_reverse_foldlM_cons]
-  simp only [bind_pure_comp, Functor.map_map]
+  simp only [Functor.map_map]
   suffices ∀ (l), (fun l' => l'.reverse.toArray) <$> List.foldlM (fun acc a => (fun a => a :: acc) <$> f a) l xs =
       List.foldlM (fun acc a => acc.push <$> f a) l.reverse.toArray xs by
     exact this []
@@ -143,13 +143,13 @@ theorem foldrM_filter [Monad m] [LawfulMonad m] {p : α → Bool} {g : α → β
   cases as <;> cases bs
   simp_all
 
-@[simp] theorem forM_append [Monad m] [LawfulMonad m] {xs ys : Array α} {f : α → m PUnit} :
+@[simp, grind =] theorem forM_append [Monad m] [LawfulMonad m] {xs ys : Array α} {f : α → m PUnit} :
     forM (xs ++ ys) f = (do forM xs f; forM ys f) := by
   rcases xs with ⟨xs⟩
   rcases ys with ⟨ys⟩
   simp
 
-@[simp] theorem forM_map [Monad m] [LawfulMonad m] {xs : Array α} {g : α → β} {f : β → m PUnit} :
+@[simp, grind =] theorem forM_map [Monad m] [LawfulMonad m] {xs : Array α} {g : α → β} {f : β → m PUnit} :
     forM (xs.map g) f = forM xs (fun a => f (g a)) := by
   rcases xs with ⟨xs⟩
   simp
@@ -208,7 +208,7 @@ theorem forIn'_yield_eq_foldl
       xs.attach.foldl (fun b ⟨a, h⟩ => f a h b) init :=
   forIn'_pure_yield_eq_foldl _ _
 
-@[simp] theorem forIn'_map [Monad m] [LawfulMonad m]
+@[simp, grind =] theorem forIn'_map [Monad m] [LawfulMonad m]
     {xs : Array α} (g : α → β) (f : (b : β) → b ∈ xs.map g → γ → m (ForInStep γ)) :
     forIn' (xs.map g) init f = forIn' xs init fun a h y => f (g a) (mem_map_of_mem h) y := by
   rcases xs with ⟨xs⟩
@@ -234,14 +234,14 @@ theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
     forIn xs init (fun a b => (fun c => .yield (g a b c)) <$> f a b) =
       xs.foldlM (fun b a => g a b <$> f a b) init := by
   rcases xs with ⟨xs⟩
-  simp [List.foldlM_map]
+  simp
 
 @[simp] theorem forIn_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
     {xs : Array α} (f : α → β → β) (init : β) :
     forIn xs init (fun a b => pure (.yield (f a b))) =
       pure (f := m) (xs.foldl (fun b a => f a b) init) := by
   rcases xs with ⟨xs⟩
-  simp [List.forIn_pure_yield_eq_foldl, List.foldl_map]
+  simp [List.forIn_pure_yield_eq_foldl]
 
 theorem idRun_forIn_yield_eq_foldl
     {xs : Array α} (f : α → β → Id β) (init : β) :
@@ -256,7 +256,7 @@ theorem forIn_yield_eq_foldl
       xs.foldl (fun b a => f a b) init :=
   forIn_pure_yield_eq_foldl _ _
 
-@[simp] theorem forIn_map [Monad m] [LawfulMonad m]
+@[simp, grind =] theorem forIn_map [Monad m] [LawfulMonad m]
     {xs : Array α} {g : α → β} {f : β → γ → m (ForInStep γ)} :
     forIn (xs.map g) init f = forIn xs init fun a y => f (g a) y := by
   rcases xs with ⟨xs⟩

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

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

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 -/
 
@@ -199,13 +213,13 @@ 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]
 
 theorem map_snd_add_zipIdx_eq_zipIdx {xs : Array α} {n k : Nat} :
     map (Prod.map id (· + n)) (zipIdx xs k) = zipIdx xs (n + k) :=
-  ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
+  ext_getElem? fun i ↦ by simp [Nat.add_comm, Nat.add_left_comm]; rfl
 
 -- Arguments are explicit for parity with `zipIdx_map_fst`.
 @[simp]
@@ -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

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

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

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`.
 -/
@@ -842,4 +850,15 @@ treating `x` and `y` as 2's complement signed bitvectors.
 def smulOverflow {w : Nat} (x y : BitVec w) : Bool :=
   (x.toInt * y.toInt ≥ 2 ^ (w - 1)) || (x.toInt * y.toInt < - 2 ^ (w - 1))
 
+/-- Count the number of leading zeros downward from the `n`-th bit to the `0`-th bit for the bitblaster.
+  This builds a tree of `if-then-else` lookups whose length is linear in the bitwidth,
+  and an efficient circuit for bitblasting `clz`. -/
+def clzAuxRec {w : Nat} (x : BitVec w) (n : Nat) : BitVec w :=
+  match n with
+  | 0 => if x.getLsbD 0 then BitVec.ofNat w (w - 1) else BitVec.ofNat w w
+  | n' + 1 => if x.getLsbD n then BitVec.ofNat w (w - 1 - n) else clzAuxRec x n'
+
+/-- Count the number of leading zeros. -/
+def clz (x : BitVec w) : BitVec w := clzAuxRec x (w - 1)
+
 end BitVec

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
@@ -238,7 +240,7 @@ theorem toNat_add_of_and_eq_zero {x y : BitVec w} (h : x &&& y = 0#w) :
     simp only [decide_eq_true_eq] at this
     omega
   rw [← carry_width]
-  simp [not_eq_true, carry_of_and_eq_zero h]
+  simp [carry_of_and_eq_zero h]
 
 /-- Carry function for bitwise addition. -/
 def adcb (x y c : Bool) : Bool × Bool := (atLeastTwo x y c, x ^^ (y ^^ c))
@@ -252,7 +254,7 @@ theorem getLsbD_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool
       (getLsbD x i ^^ (getLsbD y i ^^ carry i x y c)) := by
   let ⟨x, x_lt⟩ := x
   let ⟨y, y_lt⟩ := y
-  simp only [getLsbD, toNat_add, toNat_setWidth, i_lt, toNat_ofFin, toNat_ofBool,
+  simp only [getLsbD, toNat_add, toNat_setWidth, toNat_ofFin, toNat_ofBool,
     Nat.mod_add_mod, Nat.add_mod_mod]
   apply Eq.trans
   rw [← Nat.div_add_mod x (2^i), ← Nat.div_add_mod y (2^i)]
@@ -295,7 +297,7 @@ theorem adc_spec (x y : BitVec w) (c : Bool) :
     simp [carry, Nat.mod_one]
     cases c <;> rfl
   case step =>
-    simp [adcb, Prod.mk.injEq, carry_succ, getElem_add_add_bool]
+    simp [adcb, carry_succ, getElem_add_add_bool]
 
 theorem add_eq_adc (w : Nat) (x y : BitVec w) : x + y = (adc x y false).snd := by
   simp [adc_spec]
@@ -312,7 +314,7 @@ theorem msb_add {w : Nat} {x y: BitVec w} :
       Bool.xor x.msb (Bool.xor y.msb (carry (w - 1) x y false)) := by
   simp only [BitVec.msb, BitVec.getMsbD]
   by_cases h : w ≤ 0
-  · simp [h, show w = 0 by omega]
+  · simp [show w = 0 by omega]
   · rw [getLsbD_add (x := x)]
     simp [show w > 0 by omega]
     omega
@@ -332,15 +334,15 @@ theorem add_eq_or_of_and_eq_zero {w : Nat} (x y : BitVec w)
     (h : x &&& y = 0#w) : x + y = x ||| y := by
   rw [add_eq_adc, adc, iunfoldr_replace (fun _ => false) (x ||| y)]
   · rfl
-  · simp only [adcb, atLeastTwo, Bool.and_false, Bool.or_false, bne_false, getLsbD_or,
+  · simp only [adcb, atLeastTwo, Bool.and_false, Bool.or_false, bne_false, 
     Prod.mk.injEq, and_eq_false_imp]
     intros i
     replace h : (x &&& y).getLsbD i = (0#w).getLsbD i := by rw [h]
     simp only [getLsbD_and, getLsbD_zero, and_eq_false_imp] at h
     constructor
     · intros hx
-      simp_all [hx]
-    · by_cases hx : x.getLsbD i <;> simp_all [hx]
+      simp_all
+    · by_cases hx : x.getLsbD i <;> simp_all
 
 /-! ### Sub-/
 
@@ -377,7 +379,7 @@ theorem bit_not_add_self (x : BitVec w) :
   simp only [add_eq_adc]
   apply iunfoldr_replace_snd (fun _ => false) (-1) false rfl
   intro i; simp only [adcb, Fin.is_lt, getLsbD_eq_getElem, atLeastTwo_false_right, bne_false,
-    ofNat_eq_ofNat, Fin.getElem_fin, Prod.mk.injEq, and_eq_false_imp]
+    ofNat_eq_ofNat, Prod.mk.injEq, and_eq_false_imp]
   rw [iunfoldr_replace_snd (fun _ => ()) (((iunfoldr (fun i c => (c, !(x[i.val])))) ()).snd)]
   <;> simp [bit_not_testBit, neg_one_eq_allOnes, getElem_allOnes]
 
@@ -409,7 +411,7 @@ theorem getLsbD_neg {i : Nat} {x : BitVec w} :
   · rw [getLsbD_add hi]
     have : 0 < w := by omega
     simp only [getLsbD_not, hi, decide_true, Bool.true_and, getLsbD_one, this, not_bne,
-      _root_.true_and, not_eq_eq_eq_not]
+      not_eq_eq_eq_not]
     cases i with
     | zero =>
       have carry_zero : carry 0 ?x ?y false = false := by
@@ -424,7 +426,7 @@ theorem getLsbD_neg {i : Nat} {x : BitVec w} :
       · rintro h j hj; exact And.right <| h j (by omega)
       · rintro h j hj; exact ⟨by omega, h j (by omega)⟩
   · have h_ge : w ≤ i := by omega
-    simp [getLsbD_of_ge _ _ h_ge, h_ge, hi]
+    simp [h_ge, hi]
 
 theorem getElem_neg {i : Nat} {x : BitVec w} (h : i < w) :
     (-x)[i] = (x[i] ^^ decide (∃ j < i, x.getLsbD j = true)) := by
@@ -433,7 +435,7 @@ theorem getElem_neg {i : Nat} {x : BitVec w} (h : i < w) :
 theorem getMsbD_neg {i : Nat} {x : BitVec w} :
     getMsbD (-x) i =
       (getMsbD x i ^^ decide (∃ j < w, i < j ∧ getMsbD x j = true)) := by
-  simp only [getMsbD, getLsbD_neg, Bool.decide_and, Bool.and_eq_true, decide_eq_true_eq]
+  simp only [getMsbD, getLsbD_neg, Bool.and_eq_true, decide_eq_true_eq]
   by_cases hi : i < w
   case neg =>
     simp [hi]; omega
@@ -518,14 +520,11 @@ 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} :
     x.abs.msb = (decide (x = intMin w) && decide (0 < w)) := by
-  simp only [BitVec.abs, getMsbD_neg, ne_eq, decide_not, Bool.not_bne]
+  simp only [BitVec.abs]
   by_cases h₀ : 0 < w
   · by_cases h₁ : x = intMin w
     · simp [h₁, msb_intMin]
@@ -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 -/
 
 /--
@@ -658,7 +611,7 @@ theorem setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (x : BitVec w) (i
       getElem_twoPow]
     by_cases hik : i = k
     · subst hik
-      simp [h]
+      simp
     · by_cases hik' : k < (i + 1)
       · have hik'' : k < i := by omega
         simp [hik', hik'']
@@ -667,8 +620,8 @@ theorem setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (x : BitVec w) (i
         simp [hik', hik'']
         omega
   · ext k
-    simp only [and_twoPow, getLsbD_and, getLsbD_setWidth, Fin.is_lt, decide_true, Bool.true_and,
-      getLsbD_zero, and_eq_false_imp, and_eq_true, decide_eq_true_eq, and_imp]
+    simp only [and_twoPow, 
+      ]
     by_cases hi : x.getLsbD i <;> simp [hi] <;> omega
 
 /--
@@ -825,7 +778,7 @@ private theorem Nat.div_add_eq_left_of_lt {x y z : Nat} (hx : z ∣ x) (hy : y <
   · apply Nat.le_trans
     · exact div_mul_le_self x z
     · omega
-  · simp only [succ_eq_add_one, Nat.add_mul, Nat.one_mul]
+  · simp only [Nat.add_mul, Nat.one_mul]
     apply Nat.add_lt_add_of_le_of_lt
     · apply Nat.le_of_eq
       exact (Nat.div_eq_iff_eq_mul_left hz hx).mp rfl
@@ -938,10 +891,10 @@ def DivModState.lawful_init {w : Nat} (args : DivModArgs w) (hd : 0#w < args.d)
     hwrn := by simp only; omega,
     hdPos := by assumption
     hrLtDivisor := by simp [BitVec.lt_def] at hd ⊢; assumption
-    hrWidth := by simp [DivModState.init],
-    hqWidth := by simp [DivModState.init],
+    hrWidth := by simp,
+    hqWidth := by simp,
     hdiv := by
-      simp only [DivModState.init, toNat_ofNat, zero_mod, Nat.mul_zero, Nat.add_zero];
+      simp only [toNat_ofNat, zero_mod, Nat.mul_zero, Nat.add_zero];
       rw [Nat.shiftRight_eq_div_pow]
       apply Nat.div_eq_of_lt args.n.isLt
   }
@@ -969,7 +922,7 @@ theorem DivModState.umod_eq_of_lawful {qr : DivModState w}
     n % d = qr.r := by
   apply umod_eq_of_mul_add_toNat h.hrLtDivisor
   have hdiv := h.hdiv
-  simp only [shiftRight_zero] at hdiv
+  simp only at hdiv
   simp only [h_final] at *
   exact hdiv.symm
 
@@ -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)]
@@ -1047,7 +1000,7 @@ obeys the division equation. -/
 theorem lawful_divSubtractShift (qr : DivModState w) (h : qr.Poised args) :
     DivModState.Lawful args (divSubtractShift args qr) := by
   rcases args with ⟨n, d⟩
-  simp only [divSubtractShift, decide_eq_true_eq]
+  simp only [divSubtractShift]
   -- We add these hypotheses for `omega` to find them later.
   have ⟨⟨hrwn, hd, hrd, hr, hn, hrnd⟩, hwn_lt⟩ := h
   have : d.toNat * (qr.q.toNat * 2) = d.toNat * qr.q.toNat * 2 := by rw [Nat.mul_assoc]
@@ -1184,7 +1137,7 @@ theorem getLsbD_udiv (n d : BitVec w) (hy : 0#w < d)  (i : Nat) :
 
 theorem getMsbD_udiv (n d : BitVec w) (hd : 0#w < d)  (i : Nat) :
     (n / d).getMsbD i = (decide (i < w) && (divRec w {n, d} (DivModState.init w)).q.getMsbD i) := by
-  simp [getMsbD_eq_getLsbD, getLsbD_udiv, udiv_eq_divRec (by assumption)]
+  simp [getMsbD_eq_getLsbD, udiv_eq_divRec (by assumption)]
 
 /- ### Arithmetic shift right (sshiftRight) recurrence -/
 
@@ -1351,7 +1304,7 @@ theorem negOverflow_eq {w : Nat} (x : BitVec w) :
     (negOverflow x) = (decide (0 < w) && (x == intMin w)) := by
   simp only [negOverflow]
   rcases w with _|w
-  · simp [toInt_of_zero_length, Int.min_eq_right]
+  · simp [toInt_of_zero_length]
   · suffices - 2 ^ w = (intMin (w + 1)).toInt by simp [beq_eq_decide_eq, ← toInt_inj, this]
     simp only [toInt_intMin, Nat.add_one_sub_one, Int.natCast_emod, Int.neg_inj]
     rw_mod_cast [Nat.mod_eq_of_lt (by simp [Nat.pow_lt_pow_succ])]
@@ -1393,7 +1346,7 @@ theorem umulOverflow_eq {w : Nat} (x y : BitVec w) :
       (0 < w && BitVec.twoPow (w * 2) w ≤ x.zeroExtend (w * 2) * y.zeroExtend (w * 2)) := by
   simp only [umulOverflow, toNat_twoPow, le_def, toNat_mul, toNat_setWidth, mod_mul_mod]
   rcases w with _|w
-  · simp [of_length_zero, toInt_zero, mul_mod_mod]
+  · simp [of_length_zero]
   · simp only [ge_iff_le, show 0 < w + 1 by omega, decide_true, mul_mod_mod, Bool.true_and,
       decide_eq_decide]
     rw [Nat.mod_eq_of_lt BitVec.toNat_mul_toNat_lt, Nat.mod_eq_of_lt]
@@ -1629,11 +1582,11 @@ theorem toInt_sdiv_of_ne_or_ne (a b : BitVec w) (h : a ≠ intMin w ∨ b ≠ -1
   have := Nat.two_pow_pos (w - 1)
 
   by_cases hbintMin : b = intMin w
-  · simp only [ne_eq, Decidable.not_not] at hbintMin
+  · simp only at hbintMin
     subst hbintMin
     have toIntA_lt := @BitVec.toInt_lt w a; norm_cast at toIntA_lt
     have le_toIntA := @BitVec.le_toInt w a; norm_cast at le_toIntA
-    simp only [sdiv_intMin, h, ↓reduceIte, toInt_zero, toInt_intMin, wpos,
+    simp only [sdiv_intMin, toInt_intMin, wpos,
       Nat.two_pow_pred_mod_two_pow, Int.tdiv_neg]
     · by_cases ha_intMin : a = intMin w
       · simp only [ha_intMin, ↓reduceIte, show 1 < w by omega, toInt_one, toInt_intMin, wpos,
@@ -1709,6 +1662,88 @@ theorem toInt_sdiv (a b : BitVec w) : (a.sdiv b).toInt = (a.toInt.tdiv b.toInt).
   · rw [← toInt_bmod_cancel]
     rw [BitVec.toInt_sdiv_of_ne_or_ne _ _ (by simpa only [Decidable.not_and_iff_not_or_not] using h)]
 
+private theorem neg_udiv_eq_intMin_iff_eq_intMin_eq_one_of_msb_eq_true
+    {x y : BitVec w} (hx : x.msb = true) (hy : y.msb = false) :
+    -x / y = intMin w ↔ (x = intMin w ∧ y = 1#w) := by
+  constructor
+  · intros h
+    rcases w with _ | w; decide +revert
+    have : (-x / y).msb = true := by simp [h, msb_intMin]
+    rw [msb_udiv] at this
+    simp only [bool_to_prop] at this
+    obtain ⟨hx, hy⟩ := this
+    simp only [beq_iff_eq] at hy
+    subst hy
+    simp only [udiv_one, zero_lt_succ, neg_eq_intMin] at h
+    simp [h]
+  · rintro ⟨hx, hy⟩
+    subst hx hy
+    simp
+
+/--
+the most significant bit of the signed division `x.sdiv y` can be computed
+by the following cases:
+(1) x nonneg, y nonneg: never neg.
+(2) x nonneg, y neg: neg when result nonzero.
+   We know that y is nonzero since it is negative, so we only check `|x| ≥ |y|`.
+(3) x neg, y nonneg: neg when result nonzero.
+  We check that `y ≠ 0` and `|x| ≥ |y|`.
+(4) x neg, y neg: neg when `x = intMin, `y = -1`, since `intMin / -1 = intMin`.
+
+The proof strategy is to perform a case analysis on the sign of `x` and `y`,
+followed by unfolding the `sdiv` into `udiv`.
+-/
+theorem msb_sdiv_eq_decide {x y : BitVec w} :
+    (x.sdiv y).msb = (decide (0 < w) &&
+      (!x.msb && y.msb && decide (-y ≤ x)) ||
+      (x.msb && !y.msb && decide (y ≤ -x) && !decide (y = 0#w)) ||
+      (x.msb && y.msb && decide (x = intMin w) && decide (y = -1#w)))
+     := by
+  rcases w; decide +revert
+  case succ w =>
+    simp only [decide_true, ne_eq, decide_and, decide_not, Bool.true_and,
+      sdiv_eq, udiv_eq]
+    rcases hxmsb : x.msb <;> rcases hymsb : y.msb
+    · simp [hxmsb, hymsb, msb_udiv_eq_false_of, Bool.not_false, Bool.and_false, Bool.false_and,
+        Bool.and_true, Bool.or_self, Bool.and_self]
+    · simp only [hxmsb, hymsb, msb_neg, msb_udiv_eq_false_of, bne_false, Bool.not_false,
+        Bool.and_self, ne_zero_of_msb_true, decide_false, Bool.and_true, Bool.true_and, Bool.not_true,
+        Bool.false_and, Bool.or_false, bool_to_prop]
+      have : x / -y ≠ intMin (w + 1) := by
+        intros h
+        have : (x / -y).msb = (intMin (w + 1)).msb := by simp only [h]
+        simp only [msb_udiv, msb_intMin, show 0 < w + 1 by omega, decide_true, and_eq_true, beq_iff_eq] at this
+        obtain ⟨hcontra, _⟩ := this
+        simp only [hcontra, true_eq_false] at hxmsb
+      simp [this, hymsb, udiv_ne_zero_iff_ne_zero_and_le]
+    · simp only [hxmsb, hymsb, Bool.not_true, Bool.and_self, Bool.false_and, Bool.not_false,
+        Bool.true_and, Bool.false_or, Bool.and_false, Bool.or_false]
+      by_cases hx₁ : x = 0#(w + 1)
+      · simp [hx₁, neg_zero, zero_udiv, msb_zero, le_zero_iff, Bool.and_not_self]
+      · by_cases hy₁ : y = 0#(w + 1)
+        · simp [hy₁, udiv_zero, neg_zero, msb_zero, decide_true, Bool.not_true, Bool.and_false]
+        · simp only [hy₁, decide_false, Bool.not_false, Bool.and_true]
+          by_cases hxy₁ : (- x / y) = 0#(w + 1)
+          · simp only [hxy₁, neg_zero, msb_zero, false_eq_decide_iff, BitVec.not_le,
+              decide_eq_true_eq, BitVec.not_le]
+            simp only [udiv_eq_zero_iff_eq_zero_or_lt, hy₁, _root_.false_or] at hxy₁
+            bv_omega
+          · simp only [udiv_eq_zero_iff_eq_zero_or_lt, _root_.not_or, BitVec.not_lt,
+              hy₁, not_false_eq_true, _root_.true_and] at hxy₁
+            simp only [hxy₁, decide_true, msb_neg, bne_iff_ne, ne_eq,
+              bool_to_prop,
+              bne_iff_ne, ne_eq, udiv_eq_zero_iff_eq_zero_or_lt, hy₁, _root_.false_or,
+              BitVec.not_lt, hxy₁, _root_.true_and, decide_not, not_eq_eq_eq_not, not_eq_not,
+              msb_udiv, msb_neg]
+            simp only [hx₁, not_false_eq_true, _root_.true_and, decide_not, hxmsb, not_eq_eq_eq_not,
+              Bool.not_true, decide_eq_false_iff_not, Decidable.not_not, beq_iff_eq]
+            rw [neg_udiv_eq_intMin_iff_eq_intMin_eq_one_of_msb_eq_true hxmsb hymsb]
+    · simp only [msb_udiv, msb_neg, hxmsb, bne_true, Bool.not_and, Bool.not_true, Bool.and_true,
+        Bool.false_and, Bool.and_false, hymsb, ne_zero_of_msb_true, decide_false, Bool.not_false,
+        Bool.or_self, Bool.and_self, Bool.true_and, Bool.false_or]
+      simp only [bool_to_prop]
+      simp [BitVec.ne_zero_of_msb_true (x := x) hxmsb, neg_eq_iff_eq_neg]
+
 theorem msb_umod_eq_false_of_left {x : BitVec w} (hx : x.msb = false) (y : BitVec w) : (x % y).msb = false := by
   rw [msb_eq_false_iff_two_mul_lt] at hx ⊢
   rw [toNat_umod]
@@ -1728,7 +1763,7 @@ theorem msb_umod_of_le_of_ne_zero_of_le {x y : BitVec w}
 theorem toInt_srem (x y : BitVec w) : (x.srem y).toInt = x.toInt.tmod y.toInt := by
   rw [srem_eq]
   by_cases hyz : y = 0#w
-  · simp only [hyz, ofNat_eq_ofNat, msb_zero, umod_zero, neg_zero, neg_neg, toInt_zero, Int.tmod_zero]
+  · simp only [hyz, msb_zero, umod_zero, neg_zero, neg_neg, toInt_zero, Int.tmod_zero]
     cases x.msb <;> rfl
   cases h : x.msb
   · cases h' : y.msb
@@ -1807,7 +1842,7 @@ theorem toInt_umod_neg_add {x y : BitVec w} (hymsb : y.msb = true) (hxmsb : x.ms
   have hylt : (-y).toNat ≤ 2 ^ (w) := toNat_neg_lt_of_msb y hymsb
   have hmodlt := Nat.mod_lt x.toNat (y := (-y).toNat)
       (by rw [toNat_neg, Nat.mod_eq_of_lt (by omega)]; omega)
-  simp only [hdvd, reduceIte, toInt_add, hxnonneg, show ¬0 ≤ y.toInt by omega]
+  simp only [toInt_add]
   rw [toInt_umod, toInt_eq_neg_toNat_neg_of_msb_true hymsb, Int.bmod_add_bmod,
     Int.bmod_eq_of_le (by omega) (by omega),
     toInt_eq_toNat_of_msb hxmsb, Int.emod_neg]
@@ -1822,7 +1857,7 @@ theorem toInt_sub_neg_umod {x y : BitVec w} (hxmsb : x.msb = true) (hymsb : y.ms
     · subst hyzero; simp
     · simp only [toNat_eq, toNat_ofNat, zero_mod] at hyzero
       have hypos : 0 < y.toNat := by omega
-      simp only [reduceIte, toInt_sub, toInt_eq_toNat_of_msb hymsb, toInt_umod,
+      simp only [toInt_sub, toInt_eq_toNat_of_msb hymsb, toInt_umod,
         Int.sub_bmod_bmod, toInt_eq_neg_toNat_neg_of_msb_true hxmsb, Int.neg_emod]
       have hmodlt := Nat.mod_lt (x := (-x).toNat) (y := y.toNat) hypos
       rw [Int.bmod_eq_of_le (by omega) (by omega)]
@@ -1860,7 +1895,7 @@ theorem toInt_smod {x y : BitVec w} :
         · simp [show ¬-x % y = 0#(w + 1) by simp_all, toInt_sub_neg_umod hxmsb hymsb hx_dvd_y]
       · rw [←Int.neg_inj, neg_toInt_neg_umod_eq_of_msb_true_msb_true hxmsb hymsb]
         simp [BitVec.toInt_eq_neg_toNat_neg_of_msb_true, hxmsb, hymsb,
-          Int.fmod_eq_emod_of_nonneg _, show 0 ≤ (-y).toNat by omega]
+          Int.fmod_eq_emod_of_nonneg _]
 
 /-! ### Lemmas that use bit blasting circuits -/
 
@@ -1894,7 +1929,7 @@ theorem carry_extractLsb'_eq_carry {w i len : Nat} (hi : i < len)
     {x y : BitVec w} {b : Bool}:
     (carry i (extractLsb' 0 len x) (extractLsb' 0 len y) b)
     = (carry i x y b) := by
-  simp only [carry, extractLsb'_toNat, shiftRight_zero, toNat_false, Nat.add_zero, ge_iff_le,
+  simp only [carry, extractLsb'_toNat, shiftRight_zero, ge_iff_le,
     decide_eq_decide]
   have : 2 ^ i ∣ 2^len := by
     apply Nat.pow_dvd_pow

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]
+  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]
+    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

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

src/Init/Data/BitVec/Folds.lean

@@ -82,9 +82,9 @@ theorem iunfoldr_getLsbD' {f : Fin w → α → α × Bool} (state : Nat → α)
       simp only [getLsbD_cons]
       have hj2 : j.val ≤ w := by simp
       cases (Nat.lt_or_eq_of_le (Nat.lt_succ.mp i.isLt)) with
-      | inl h3 => simp [if_neg, (Nat.ne_of_lt h3)]
+      | inl h3 => simp [(Nat.ne_of_lt h3)]
                   exact (ih hj2).1 ⟨i.val, h3⟩
-      | inr h3 => simp [h3, if_pos]
+      | inr h3 => simp [h3]
                   cases (Nat.eq_zero_or_pos j.val) with
                   | inl hj3 => congr
                                rw [← (ih hj2).2]

src/Init/Data/Bool.lean

@@ -488,7 +488,7 @@ Converts `true` to `1` and `false` to `0`.
 
 @[simp] theorem ite_eq_true_else_eq_false {q : Prop} :
     (if b = true then q else b = false) ↔ (b = true → q) := by
-  cases b <;> simp [not_eq_self]
+  cases b <;> simp
 
 /-
 `not_ite_eq_true_eq_true` and related theorems below are added for

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 ..
@@ -255,7 +252,7 @@ theorem foldl_succ_last (f : α → Fin (n+1) → α) (x) :
     foldl (n+1) f x = f (foldl n (f · ·.castSucc) x) (last n) := by
   rw [foldl_succ]
   induction n generalizing x with
-  | zero => simp [foldl_succ, Fin.last]
+  | zero => simp [Fin.last]
   | succ n ih => rw [foldl_succ, ih (f · ·.succ), foldl_succ]; simp
 
 theorem foldl_add (f : α → Fin (n + m) → α) (x) :

src/Init/Data/Fin/Lemmas.lean

@@ -102,9 +102,30 @@ theorem dite_val {n : Nat} {c : Prop} [Decidable c] {x y : Fin n} :
     (if c then x else y).val = if c then x.val else y.val := by
   by_cases c <;> simp [*]
 
-instance (n : Nat) [NeZero n] : NatCast (Fin n) where
+namespace NatCast
+
+/--
+This is not a global instance, but may be activated locally via `open Fin.NatCast in ...`.
+
+This is not an instance because the `binop%` elaborator assumes that
+there are no non-trivial coercion loops,
+but this introduces a coercion from `Nat` to `Fin n` and back.
+
+Non-trivial loops lead to undesirable and counterintuitive elaboration behavior.
+For example, for `x : Fin k` and `n : Nat`,
+it causes `x < n` to be elaborated as `x < ↑n` rather than `↑x < n`,
+silently introducing wraparound arithmetic.
+
+Note: as of 2025-06-03, Mathlib has such a coercion for `Fin n` anyway!
+-/
+@[expose]
+def instNatCast (n : Nat) [NeZero n] : NatCast (Fin n) where
   natCast a := Fin.ofNat n a
 
+attribute [scoped instance] instNatCast
+
+end NatCast
+
 @[expose]
 def intCast [NeZero n] (a : Int) : Fin n :=
   if 0 ≤ a then
@@ -112,9 +133,22 @@ def intCast [NeZero n] (a : Int) : Fin n :=
   else
     - Fin.ofNat n a.natAbs
 
-instance (n : Nat) [NeZero n] : IntCast (Fin n) where
+namespace IntCast
+
+/--
+This is not a global instance, but may be activated locally via `open Fin.IntCast in ...`.
+
+See the doc-string for `Fin.NatCast.instNatCast` for more details.
+-/
+@[expose]
+def instIntCast (n : Nat) [NeZero n] : IntCast (Fin n) where
   intCast := Fin.intCast
 
+attribute [scoped instance] instIntCast
+
+end IntCast
+
+open IntCast in
 theorem intCast_def {n : Nat} [NeZero n] (x : Int) :
     (x : Fin n) = if 0 ≤ x then Fin.ofNat n x.natAbs else -Fin.ofNat n x.natAbs := rfl
 
@@ -347,7 +381,7 @@ theorem zero_ne_one : (0 : Fin (n + 2)) ≠ 1 := Fin.ne_of_lt one_pos
 @[simp] theorem val_succ (j : Fin n) : (j.succ : Nat) = j + 1 := rfl
 
 @[simp] theorem succ_pos (a : Fin n) : (0 : Fin (n + 1)) < a.succ := by
-  simp [Fin.lt_def, Nat.succ_pos]
+  simp [Fin.lt_def]
 
 @[simp] theorem succ_le_succ_iff {a b : Fin n} : a.succ ≤ b.succ ↔ a ≤ b := Nat.succ_le_succ_iff
 
@@ -380,7 +414,7 @@ theorem one_lt_succ_succ (a : Fin n) : (1 : Fin (n + 2)) < a.succ.succ := by
   simp only [lt_def, val_add, val_last, Fin.ext_iff]
   let ⟨k, hk⟩ := k
   match Nat.eq_or_lt_of_le (Nat.le_of_lt_succ hk) with
-  | .inl h => cases h; simp [Nat.succ_pos]
+  | .inl h => cases h; simp
   | .inr hk' => simp [Nat.ne_of_lt hk', Nat.mod_eq_of_lt (Nat.succ_lt_succ hk'), Nat.le_succ]
 
 @[simp] theorem add_one_le_iff {n : Nat} : ∀ {k : Fin (n + 1)}, k + 1 ≤ k ↔ k = last _ := by
@@ -392,7 +426,7 @@ theorem one_lt_succ_succ (a : Fin n) : (1 : Fin (n + 2)) < a.succ.succ := by
     intro (k : Fin (n+2))
     rw [← add_one_lt_iff, lt_def, le_def, Nat.lt_iff_le_and_ne, and_iff_left]
     rw [val_add_one]
-    split <;> simp [*, (Nat.succ_ne_zero _).symm, Nat.ne_of_gt (Nat.lt_succ_self _)]
+    split <;> simp [*, Nat.ne_of_gt (Nat.lt_succ_self _)]
 
 @[simp] theorem last_le_iff {n : Nat} {k : Fin (n + 1)} : last n ≤ k ↔ k = last n := by
   rw [Fin.ext_iff, Nat.le_antisymm_iff, le_def, and_iff_right (by apply le_last)]
@@ -704,7 +738,7 @@ theorem pred_mk {n : Nat} (i : Nat) (h : i < n + 1) (w) : Fin.pred ⟨i, h⟩ w
     ∀ {a b : Fin (n + 1)} {ha : a ≠ 0} {hb : b ≠ 0}, a.pred ha = b.pred hb ↔ a = b
   | ⟨0, _⟩, _, ha, _ => by simp only [mk_zero, ne_eq, not_true] at ha
   | ⟨i + 1, _⟩, ⟨0, _⟩, _, hb => by simp only [mk_zero, ne_eq, not_true] at hb
-  | ⟨i + 1, hi⟩, ⟨j + 1, hj⟩, ha, hb => by simp [Fin.ext_iff, Nat.succ.injEq]
+  | ⟨i + 1, hi⟩, ⟨j + 1, hj⟩, ha, hb => by simp [Fin.ext_iff]
 
 @[simp] theorem pred_one {n : Nat} :
     Fin.pred (1 : Fin (n + 2)) (Ne.symm (Fin.ne_of_lt one_pos)) = 0 := rfl
@@ -1045,6 +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) :
@@ -1072,7 +1117,7 @@ protected theorem mul_one [i : NeZero n] (k : Fin n) : k * 1 = k := by
   | n + 1, _ =>
     match n with
     | 0 => exact Subsingleton.elim (α := Fin 1) ..
-    | n+1 => simp [Fin.ext_iff, mul_def, Nat.mod_eq_of_lt (is_lt k)]
+    | n+1 => simp [mul_def, Nat.mod_eq_of_lt (is_lt k)]
 
 protected theorem mul_comm (a b : Fin n) : a * b = b * a :=
   Fin.ext <| by rw [mul_def, mul_def, Nat.mul_comm]

src/Init/Data/Function.lean

@@ -31,19 +31,19 @@ Examples:
 @[inline, expose]
 def uncurry : (α → β → φ) → α × β → φ := fun f a => f a.1 a.2
 
-@[simp]
+@[simp, grind]
 theorem curry_uncurry (f : α → β → φ) : curry (uncurry f) = f :=
   rfl
 
-@[simp]
+@[simp, grind]
 theorem uncurry_curry (f : α × β → φ) : uncurry (curry f) = f :=
   funext fun ⟨_a, _b⟩ => rfl
 
-@[simp]
+@[simp, grind]
 theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
   rfl
 
-@[simp]
+@[simp, grind]
 theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
   rfl
 

src/Init/Data/Int/Compare.lean

@@ -37,7 +37,7 @@ theorem compare_eq_ite_le (a b : Int) :
   · next hlt => simp [Int.le_of_lt hlt, Int.not_le.2 hlt]
   · next hge =>
     split
-    · next hgt => simp [Int.le_of_lt hgt, Int.not_le.2 hgt]
+    · next hgt => simp [Int.not_le.2 hgt]
     · next hle => simp [Int.not_lt.1 hge, Int.not_lt.1 hle]
 
 protected theorem compare_swap (a b : Int) : (compare a b).swap = compare b a := by

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
@@ -57,7 +56,7 @@ protected theorem dvd_trans : ∀ {a b c : Int}, a ∣ b → b ∣ c → a ∣ c
 
 @[simp] protected theorem dvd_neg {a b : Int} : a ∣ -b ↔ a ∣ b := by
   constructor <;> exact fun ⟨k, e⟩ =>
-    ⟨-k, by simp [← e, Int.neg_mul, Int.mul_neg, Int.neg_neg]⟩
+    ⟨-k, by simp [← e, Int.mul_neg, Int.neg_neg]⟩
 
 @[simp] theorem natAbs_dvd_natAbs {a b : Int} : natAbs a ∣ natAbs b ↔ a ∣ b := by
   refine ⟨fun ⟨k, hk⟩ => ?_, fun ⟨k, hk⟩ => ⟨natAbs k, hk.symm ▸ natAbs_mul a k⟩⟩
@@ -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]

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]
@@ -215,7 +217,7 @@ theorem tdiv_eq_ediv {a b : Int} :
       negSucc_not_nonneg, sign_of_add_one]
     simp only [negSucc_emod_ofNat_succ_eq_zero_iff]
     norm_cast
-    simp only [subNat_eq_zero_iff, Nat.succ_eq_add_one, sign_negSucc, Int.sub_neg, false_or]
+    simp only [Nat.succ_eq_add_one, false_or]
     split <;> rename_i h
     · rw [Int.add_zero, neg_ofNat_eq_negSucc_iff]
       exact Nat.succ_div_of_mod_eq_zero h
@@ -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
@@ -1315,7 +1317,7 @@ protected theorem eq_tdiv_of_mul_eq_left {a b c : Int}
   | 0, n => by simp [Int.neg_zero]
   | succ _, (n:Nat) => by simp [tdiv, ← Int.negSucc_eq]
   | -[_+1], 0 | -[_+1], -[_+1] => by
-    simp only [tdiv, neg_negSucc, ← Int.natCast_succ, Int.neg_neg]
+    simp only [tdiv, neg_negSucc, Int.neg_neg]
   | succ _, -[_+1] | -[_+1], succ _ => (Int.neg_neg _).symm
 
 protected theorem neg_tdiv_neg (a b : Int) : (-a).tdiv (-b) = a.tdiv b := by
@@ -1406,7 +1408,7 @@ theorem mul_tmod (a b n : Int) : (a * b).tmod n = (a.tmod n * b.tmod n).tmod n :
   case inv => simp [Int.dvd_neg]
   induction m using wlog_sign
   case inv => simp
-  simp only [← Int.natCast_mul, ← ofNat_tmod]
+  simp only [← ofNat_tmod]
   norm_cast at h
   rw [Nat.mod_mod_of_dvd _ h]
 
@@ -1574,7 +1576,7 @@ theorem neg_mul_tmod_left (a b : Int) : (-(a * b)).tmod b = 0 := by
 
 @[simp] protected theorem tdiv_one : ∀ a : Int, a.tdiv 1 = a
   | (n:Nat) => congrArg ofNat (Nat.div_one _)
-  | -[n+1] => by simp [Int.tdiv, neg_ofNat_succ]; rfl
+  | -[n+1] => by simp [Int.tdiv]; rfl
 
 @[simp] theorem tmod_one (a : Int) : tmod a 1 = 0 := by
   simp [tmod_def, Int.tdiv_one, Int.one_mul, Int.sub_self]
@@ -1696,7 +1698,7 @@ theorem lt_ediv_iff_of_dvd_of_neg {a b c : Int} (hc : c < 0) (hcb : c ∣ b) :
 theorem ediv_le_ediv_iff_of_dvd_of_pos_of_pos {a b c d : Int} (hb : 0 < b) (hd : 0 < d)
     (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ d * a ≤ c * b := by
   obtain ⟨⟨x, rfl⟩, y, rfl⟩ := hba, hdc
-  simp [*, Int.ne_of_lt, Int.ne_of_gt, d.mul_assoc, b.mul_comm]
+  simp [*, Int.ne_of_gt, d.mul_assoc, b.mul_comm]
 
 theorem ediv_le_ediv_iff_of_dvd_of_pos_of_neg {a b c d : Int} (hb : 0 < b) (hd : d < 0)
     (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ c * b ≤ d * a := by
@@ -1711,12 +1713,12 @@ theorem ediv_le_ediv_iff_of_dvd_of_neg_of_pos {a b c d : Int} (hb : b < 0) (hd :
 theorem ediv_le_ediv_iff_of_dvd_of_neg_of_neg {a b c d : Int} (hb : b < 0) (hd : d < 0)
     (hba : b ∣ a) (hdc : d ∣ c) : a / b ≤ c / d ↔ d * a ≤ c * b := by
   obtain ⟨⟨x, rfl⟩, y, rfl⟩ := hba, hdc
-  simp [*, Int.ne_of_lt, Int.ne_of_gt, d.mul_assoc, b.mul_comm]
+  simp [*, Int.ne_of_lt, d.mul_assoc, b.mul_comm]
 
 theorem ediv_lt_ediv_iff_of_dvd_of_pos {a b c d : Int} (hb : 0 < b) (hd : 0 < d) (hba : b ∣ a)
     (hdc : d ∣ c) : a / b < c / d ↔ d * a < c * b := by
   obtain ⟨⟨x, rfl⟩, y, rfl⟩ := hba, hdc
-  simp [*, Int.ne_of_lt, Int.ne_of_gt, d.mul_assoc, b.mul_comm]
+  simp [*, Int.ne_of_gt, d.mul_assoc, b.mul_comm]
 
 theorem ediv_lt_ediv_iff_of_dvd_of_pos_of_neg {a b c d : Int} (hb : 0 < b) (hd : d < 0)
     (hba : b ∣ a) (hdc : d ∣ c) : a / b < c / d ↔ c * b < d * a := by
@@ -1731,7 +1733,7 @@ theorem ediv_lt_ediv_iff_of_dvd_of_neg_of_pos {a b c d : Int} (hb : b < 0) (hd :
 theorem ediv_lt_ediv_iff_of_dvd_of_neg_of_neg {a b c d : Int} (hb : b < 0) (hd : d < 0)
     (hba : b ∣ a) (hdc : d ∣ c) : a / b < c / d ↔ d * a < c * b := by
   obtain ⟨⟨x, rfl⟩, y, rfl⟩ := hba, hdc
-  simp [*, Int.ne_of_lt, Int.ne_of_gt, d.mul_assoc, b.mul_comm]
+  simp [*, Int.ne_of_lt, d.mul_assoc, b.mul_comm]
 
 /-! ### `tdiv` and ordering -/
 
@@ -2444,7 +2446,7 @@ theorem lt_mul_fdiv_self_add {x k : Int} (h : 0 < k) : x < k * (x.fdiv k) + k :=
 
 @[simp]
 theorem emod_bmod (x : Int) (n : Nat) : Int.bmod (x%n) n = Int.bmod x n := by
-  simp [bmod, Int.emod_emod]
+  simp [bmod]
 
 @[deprecated emod_bmod (since := "2025-04-11")]
 theorem emod_bmod_congr (x : Int) (n : Nat) : Int.bmod (x%n) n = Int.bmod x n :=
@@ -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
@@ -2985,7 +2987,7 @@ theorem self_le_ediv_of_nonpos_of_nonneg {x y : Int} (hx : x ≤ 0) (hy : 0 ≤
   · simp [hx', zero_ediv]
   · by_cases hy : y = 0
     · simp [hy]; omega
-    · simp only [ge_iff_le, Int.le_ediv_iff_mul_le (c := y) (a := x) (b := x) (by omega),
+    · simp only [Int.le_ediv_iff_mul_le (c := y) (a := x) (b := x) (by omega),
         show (x * y ≤ x) = (x * y ≤ x * 1) by rw [Int.mul_one], Int.mul_one]
       apply Int.mul_le_mul_of_nonpos_left (a := x) (b := y) (c  := (1 : Int)) (by omega) (by omega)
 

src/Init/Data/Int/Gcd.lean

@@ -631,7 +631,7 @@ theorem lcm_mul_left_dvd_mul_lcm (k m n : Nat) : lcm (m * n) k ∣ lcm m k * lcm
   simpa [lcm_comm, Nat.mul_comm] using lcm_mul_right_dvd_mul_lcm _ _ _
 
 theorem lcm_dvd_mul_self_left_iff_dvd_mul {k n m : Nat} : lcm k n ∣ k * m ↔ n ∣ k * m := by
-  simp [← natAbs_dvd_natAbs, natAbs_mul, Nat.lcm_dvd_mul_self_left_iff_dvd_mul,
+  simp [Nat.lcm_dvd_mul_self_left_iff_dvd_mul,
     lcm_eq_natAbs_lcm_natAbs]
 
 theorem lcm_dvd_mul_self_right_iff_dvd_mul {k m n : Nat} : lcm n k ∣ m * k ↔ n ∣ m * k := by

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")]
@@ -454,7 +454,7 @@ theorem negOfNat_eq_subNatNat_zero (n) : negOfNat n = subNatNat 0 n := by cases
 theorem ofNat_mul_subNatNat (m n k : Nat) :
     m * subNatNat n k = subNatNat (m * n) (m * k) := by
   cases m with
-  | zero => simp [ofNat_zero, Int.zero_mul, Nat.zero_mul, subNatNat_self]
+  | zero => simp [Int.zero_mul, Nat.zero_mul, subNatNat_self]
   | succ m => cases n.lt_or_ge k with
     | inl h =>
       have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)

src/Init/Data/Int/Linear.lean

@@ -23,6 +23,7 @@ namespace Int.Linear
 abbrev Var := Nat
 abbrev Context := Lean.RArray Int
 
+@[expose]
 def Var.denote (ctx : Context) (v : Var) : Int :=
   ctx.get v
 
@@ -36,6 +37,7 @@ inductive Expr where
   | mulR (a : Expr) (k : Int)
   deriving Inhabited, BEq
 
+@[expose]
 def Expr.denote (ctx : Context) : Expr → Int
   | .add a b  => Int.add (denote ctx a) (denote ctx b)
   | .sub a b  => Int.sub (denote ctx a) (denote ctx b)
@@ -50,6 +52,7 @@ inductive Poly where
   | add (k : Int) (v : Var) (p : Poly)
   deriving BEq
 
+@[expose]
 def Poly.denote (ctx : Context) (p : Poly) : Int :=
   match p with
   | .num k => k
@@ -59,6 +62,7 @@ def Poly.denote (ctx : Context) (p : Poly) : Int :=
 Similar to `Poly.denote`, but produces a denotation better for `simp +arith`.
 Remark: we used to convert `Poly` back into `Expr` to achieve that.
 -/
+@[expose]
 def Poly.denote' (ctx : Context) (p : Poly) : Int :=
   match p with
   | .num k => k
@@ -75,8 +79,8 @@ where
 theorem Poly.denote'_go_eq_denote (ctx : Context) (p : Poly) (r : Int) : denote'.go ctx r p = p.denote ctx + r := by
   induction r, p using denote'.go.induct ctx <;> simp [denote'.go, denote]
   next => rw [Int.add_comm]
-  next ih => simp [denote'.go] at ih; rw [ih]; ac_rfl
-  next ih => simp [denote'.go] at ih; rw [ih]; ac_rfl
+  next ih => simp at ih; rw [ih]; ac_rfl
+  next ih => simp at ih; rw [ih]; ac_rfl
 
 theorem Poly.denote'_eq_denote (ctx : Context) (p : Poly) : p.denote' ctx = p.denote ctx := by
   unfold denote' <;> split <;> simp [denote, denote'_go_eq_denote] <;> ac_rfl
@@ -84,11 +88,13 @@ theorem Poly.denote'_eq_denote (ctx : Context) (p : Poly) : p.denote' ctx = p.de
 theorem Poly.denote'_add (ctx : Context) (a : Int) (x : Var) (p : Poly) : (Poly.add a x p).denote' ctx = a * x.denote ctx + p.denote ctx := by
   simp [Poly.denote'_eq_denote, denote]
 
+@[expose]
 def Poly.addConst (p : Poly) (k : Int) : Poly :=
   match p with
   | .num k' => .num (k+k')
   | .add k' v' p => .add k' v' (addConst p k)
 
+@[expose]
 def Poly.insert (k : Int) (v : Var) (p : Poly) : Poly :=
   match p with
   | .num k' => .add k v (.num k')
@@ -104,16 +110,19 @@ def Poly.insert (k : Int) (v : Var) (p : Poly) : Poly :=
       .add k' v' (insert k v p)
 
 /-- Normalizes the given polynomial by fusing monomial and constants. -/
+@[expose]
 def Poly.norm (p : Poly) : Poly :=
   match p with
   | .num k => .num k
   | .add k v p => (norm p).insert k v
 
+@[expose]
 def Poly.append (p₁ p₂ : Poly) : Poly :=
   match p₁ with
   | .num k₁ => p₂.addConst k₁
   | .add k x p₁ => .add k x (append p₁ p₂)
 
+@[expose]
 def Poly.combine' (fuel : Nat) (p₁ p₂ : Poly) : Poly :=
   match fuel with
   | 0 => p₁.append p₂
@@ -133,10 +142,12 @@ def Poly.combine' (fuel : Nat) (p₁ p₂ : Poly) : Poly :=
     else
       .add a₂ x₂ (combine' fuel (.add a₁ x₁ p₁) p₂)
 
+@[expose]
 def Poly.combine (p₁ p₂ : Poly) : Poly :=
   combine' 100000000 p₁ p₂
 
 /-- Converts the given expression into a polynomial. -/
+@[expose]
 def Expr.toPoly' (e : Expr) : Poly :=
   go 1 e (.num 0)
 where
@@ -150,6 +161,7 @@ where
     | .neg a    => go (-coeff) a
 
 /-- Converts the given expression into a polynomial, and then normalizes it. -/
+@[expose]
 def Expr.norm (e : Expr) : Poly :=
   e.toPoly'.norm
 
@@ -159,6 +171,7 @@ Examples:
 - `cdiv 7 3` returns `3`
 - `cdiv (-7) 3` returns `-2`.
 -/
+@[expose]
 def cdiv (a b : Int) : Int :=
   -((-a)/b)
 
@@ -173,6 +186,7 @@ See theorem `cdiv_add_cmod`. We also have
 -b < cmod a b ≤ 0
 ```
 -/
+@[expose]
 def cmod (a b : Int) : Int :=
   -((-a)%b)
 
@@ -219,6 +233,7 @@ theorem cdiv_eq_div_of_divides {a b : Int} (h : a % b = 0) : a/b = cdiv a b := b
   next => rw [Int.mul_eq_mul_right_iff h] at this; assumption
 
 /-- Returns the constant of the given linear polynomial. -/
+@[expose]
 def Poly.getConst : Poly → Int
   | .num k => k
   | .add _ _ p => getConst p
@@ -230,6 +245,7 @@ Notes:
 - We only use this function with `k`s that divides all coefficients.
 - We use `cdiv` for the constant to implement the inequality tightening rule.
 -/
+@[expose]
 def Poly.div (k : Int) : Poly → Poly
   | .num k' => .num (cdiv k' k)
   | .add k' x p => .add (k'/k) x (div k p)
@@ -238,6 +254,7 @@ def Poly.div (k : Int) : Poly → Poly
 Returns `true` if `k` divides all coefficients and the constant of the given
 linear polynomial.
 -/
+@[expose]
 def Poly.divAll (k : Int) : Poly → Bool
   | .num k' => k' % k == 0
   | .add k' _ p => k' % k == 0 && divAll k p
@@ -245,6 +262,7 @@ def Poly.divAll (k : Int) : Poly → Bool
 /--
 Returns `true` if `k` divides all coefficients of the given linear polynomial.
 -/
+@[expose]
 def Poly.divCoeffs (k : Int) : Poly → Bool
   | .num _ => true
   | .add k' _ p => k' % k == 0 && divCoeffs k p
@@ -252,11 +270,13 @@ def Poly.divCoeffs (k : Int) : Poly → Bool
 /--
 `p.mul k` multiplies all coefficients and constant of the polynomial `p` by `k`.
 -/
+@[expose]
 def Poly.mul' (p : Poly) (k : Int) : Poly :=
   match p with
   | .num k' => .num (k*k')
   | .add k' v p => .add (k*k') v (mul' p k)
 
+@[expose]
 def Poly.mul (p : Poly) (k : Int) : Poly :=
   if k == 0 then
     .num 0
@@ -343,7 +363,7 @@ theorem Expr.denote_toPoly'_go (ctx : Context) (e : Expr) :
     simp [eq_of_beq h]
   | case2 k k' h =>
     simp only [toPoly'.go, h, cond_false]
-    simp [Var.denote]
+    simp
   | case3 k i => simp [toPoly'.go]
   | case4 k a b iha ihb => simp [toPoly'.go, iha, ihb]
   | case5 k a b iha ihb =>
@@ -351,7 +371,7 @@ theorem Expr.denote_toPoly'_go (ctx : Context) (e : Expr) :
     rw [Int.sub_eq_add_neg, ←Int.neg_mul, Int.add_assoc]
   | case6 k k' a h
   | case8 k a k' h =>
-    simp only [toPoly'.go, h, cond_false]
+    simp only [toPoly'.go, h]
     simp [eq_of_beq h]
   | case7 k a k' h ih =>
     simp only [toPoly'.go, h, cond_false]
@@ -383,9 +403,10 @@ attribute [local simp] Poly.denote'_eq_denote
 
 theorem Expr.eq_of_norm_eq (ctx : Context) (e : Expr) (p : Poly) (h : e.norm == p) : e.denote ctx = p.denote' ctx := by
   have h := congrArg (Poly.denote ctx) (eq_of_beq h)
-  simp [Poly.norm] at h
+  simp at h
   simp [*]
 
+@[expose]
 def norm_eq_cert (lhs rhs : Expr) (p : Poly) : Bool :=
   p == (lhs.sub rhs).norm
 
@@ -401,6 +422,7 @@ theorem norm_le (ctx : Context) (lhs rhs : Expr) (p : Poly) (h : norm_eq_cert lh
   · exact Int.sub_nonpos_of_le
   · exact Int.le_of_sub_nonpos
 
+@[expose]
 def norm_eq_var_cert (lhs rhs : Expr) (x y : Var) : Bool :=
   (lhs.sub rhs).norm == .add 1 x (.add (-1) y (.num 0))
 
@@ -411,6 +433,7 @@ theorem norm_eq_var (ctx : Context) (lhs rhs : Expr) (x y : Var) (h : norm_eq_va
   simp at h
   rw [←Int.sub_eq_zero, h, ← @Int.sub_eq_zero (Var.denote ctx x), Int.sub_eq_add_neg]
 
+@[expose]
 def norm_eq_var_const_cert (lhs rhs : Expr) (x : Var) (k : Int) : Bool :=
   (lhs.sub rhs).norm == .add 1 x (.num (-k))
 
@@ -429,6 +452,7 @@ private theorem mul_eq_zero_iff (a k : Int) (h₁ : k > 0) : k * a = 0 ↔ a = 0
 theorem norm_eq_coeff' (ctx : Context) (p p' : Poly) (k : Int) : p = p'.mul k → k > 0 → (p.denote ctx = 0 ↔ p'.denote ctx = 0) := by
   intro; subst p; intro h; simp [mul_eq_zero_iff, *]
 
+@[expose]
 def norm_eq_coeff_cert (lhs rhs : Expr) (p : Poly) (k : Int) : Bool :=
   (lhs.sub rhs).norm == p.mul k && k > 0
 
@@ -448,7 +472,7 @@ private theorem mul_le_zero_iff (a k : Int) (h₁ : k > 0) : k * a ≤ 0 ↔ a
     simp at h; assumption
 
 private theorem norm_le_coeff' (ctx : Context) (p p' : Poly) (k : Int) : p = p'.mul k → k > 0 → (p.denote ctx ≤ 0 ↔ p'.denote ctx ≤ 0) := by
-  simp [norm_eq_coeff_cert]
+  simp
   intro; subst p; intro h; simp [mul_le_zero_iff, *]
 
 theorem norm_le_coeff (ctx : Context) (lhs rhs : Expr) (p : Poly) (k : Int)
@@ -492,6 +516,7 @@ private theorem eq_of_norm_eq_of_divCoeffs {ctx : Context} {p₁ p₂ : Poly} {k
   apply mul_add_cmod_le_iff
   assumption
 
+@[expose]
 def norm_le_coeff_tight_cert (lhs rhs : Expr) (p : Poly) (k : Int) : Bool :=
   let p' := lhs.sub rhs |>.norm
   k > 0 && (p'.divCoeffs k && p == p'.div k)
@@ -502,11 +527,13 @@ theorem norm_le_coeff_tight (ctx : Context) (lhs rhs : Expr) (p : Poly) (k : Int
   rw [norm_le ctx lhs rhs (lhs.sub rhs).norm BEq.rfl, Poly.denote'_eq_denote]
   apply eq_of_norm_eq_of_divCoeffs
 
+@[expose]
 def Poly.isUnsatEq (p : Poly) : Bool :=
   match p with
   | .num k => k != 0
   | _ => false
 
+@[expose]
 def Poly.isValidEq (p : Poly) : Bool :=
   match p with
   | .num k => k == 0
@@ -530,11 +557,13 @@ theorem eq_eq_true (ctx : Context) (lhs rhs : Expr) : (lhs.sub rhs).norm.isValid
     rw [← Int.sub_eq_zero, h]
     assumption
 
+@[expose]
 def Poly.isUnsatLe (p : Poly) : Bool :=
   match p with
   | .num k => k > 0
   | _ => false
 
+@[expose]
 def Poly.isValidLe (p : Poly) : Bool :=
   match p with
   | .num k => k ≤ 0
@@ -595,6 +624,7 @@ private theorem poly_eq_zero_eq_false (ctx : Context) {p : Poly} {k : Int} : p.d
   have high := h₃
   exact contra h₂ low high this
 
+@[expose]
 def unsatEqDivCoeffCert (lhs rhs : Expr) (k : Int) : Bool :=
   let p := (lhs.sub rhs).norm
   p.divCoeffs k && k > 0 && cmod p.getConst k < 0
@@ -621,6 +651,7 @@ private theorem gcd_dvd_step {k a b x : Int} (h : k ∣ a*x + b) : gcd a k ∣ b
   have h₂ : gcd a k ∣ a*x := Int.dvd_trans (gcd_dvd_left a k) (Int.dvd_mul_right a x)
   exact Int.dvd_iff_dvd_of_dvd_add h₁ |>.mp h₂
 
+@[expose]
 def Poly.gcdCoeffs : Poly → Int → Int
   | .num _, k => k
   | .add k' _ p, k => gcdCoeffs p (gcd k' k)
@@ -631,6 +662,7 @@ theorem Poly.gcd_dvd_const {ctx : Context} {p : Poly} {k : Int} (h : k ∣ p.den
     rw [Int.add_comm] at h
     exact ih (gcd_dvd_step h)
 
+@[expose]
 def Poly.isUnsatDvd (k : Int) (p : Poly) : Bool :=
   p.getConst % p.gcdCoeffs k != 0
 
@@ -668,9 +700,11 @@ theorem dvd_eq_false (ctx : Context) (k : Int) (e : Expr) (h : e.norm.isUnsatDvd
   rw [norm_dvd ctx k e e.norm BEq.rfl]
   apply dvd_eq_false' ctx k e.norm h
 
+@[expose]
 def dvd_coeff_cert (k₁ : Int) (p₁ : Poly) (k₂ : Int) (p₂ : Poly) (k : Int) : Bool :=
   k != 0 && (k₁ == k*k₂ && p₁ == p₂.mul k)
 
+@[expose]
 def norm_dvd_gcd_cert (k₁ : Int) (e₁ : Expr) (k₂ : Int) (p₂ : Poly) (k : Int) : Bool :=
   dvd_coeff_cert k₁ e₁.norm k₂ p₂ k
 
@@ -702,6 +736,7 @@ private theorem dvd_gcd_of_dvd (d a x p : Int) (h : d ∣ a * x + p) : gcd d a
   rw [Int.mul_assoc, Int.mul_assoc, ← Int.mul_sub] at h
   exists k₁ * k - k₂ * x
 
+@[expose]
 def dvd_elim_cert (k₁ : Int) (p₁ : Poly) (k₂ : Int) (p₂ : Poly) : Bool :=
   match p₁ with
   | .add a _ p => k₂ == gcd k₁ a && p₂ == p
@@ -764,6 +799,7 @@ private theorem dvd_solve_elim' {x : Int} {d₁ a₁ p₁ : Int} {d₂ a₂ p₂
  rw [h₃, h₄, Int.mul_assoc, Int.mul_assoc, ←Int.mul_sub] at this
  exact ⟨k₄ * k₁ - k₃ * k₂, this⟩
 
+@[expose]
 def dvd_solve_combine_cert (d₁ : Int) (p₁ : Poly) (d₂ : Int) (p₂ : Poly) (d : Int) (p : Poly) (g α β : Int) : Bool :=
   match p₁, p₂ with
   | .add a₁ x₁ p₁, .add a₂ x₂ p₂ =>
@@ -779,12 +815,13 @@ theorem dvd_solve_combine (ctx : Context) (d₁ : Int) (p₁ : Poly) (d₂ : Int
   split <;> simp
   next a₁ x₁ p₁ a₂ x₂ p₂ =>
   intro _ hg hd hp; subst x₁ p
-  simp [Poly.denote'_add]
+  simp
   intro h₁ h₂
   rw [Int.add_comm] at h₁ h₂
   rw [Int.add_comm _ (g * x₂.denote ctx), Int.add_left_comm, ← Int.add_assoc, hd]
   exact dvd_solve_combine' hg.symm h₁ h₂
 
+@[expose]
 def dvd_solve_elim_cert (d₁ : Int) (p₁ : Poly) (d₂ : Int) (p₂ : Poly) (d : Int) (p : Poly) : Bool :=
   match p₁, p₂ with
   | .add a₁ x₁ p₁, .add a₂ x₂ p₂ =>
@@ -816,6 +853,7 @@ theorem le_norm (ctx : Context) (p₁ p₂ : Poly) (h : p₁.norm == p₂) : p
   simp at h
   simp [*]
 
+@[expose]
 def le_coeff_cert (p₁ p₂ : Poly) (k : Int) : Bool :=
   k > 0 && (p₁.divCoeffs k && p₂ == p₁.div k)
 
@@ -824,6 +862,7 @@ theorem le_coeff (ctx : Context) (p₁ p₂ : Poly) (k : Int) : le_coeff_cert p
   intro h₁ h₂ h₃
   exact eq_of_norm_eq_of_divCoeffs h₁ h₂ h₃ |>.mp
 
+@[expose]
 def le_neg_cert (p₁ p₂ : Poly) : Bool :=
   p₂ == (p₁.mul (-1) |>.addConst 1)
 
@@ -834,11 +873,13 @@ theorem le_neg (ctx : Context) (p₁ p₂ : Poly) : le_neg_cert p₁ p₂ → ¬
   simp at h
   exact h
 
+@[expose]
 def Poly.leadCoeff (p : Poly) : Int :=
   match p with
   | .add a _ _ => a
   | _ => 1
 
+@[expose]
 def le_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
   let a₁ := p₁.leadCoeff.natAbs
   let a₂ := p₂.leadCoeff.natAbs
@@ -854,6 +895,7 @@ theorem le_combine (ctx : Context) (p₁ p₂ p₃ : Poly)
   · rw [← Int.zero_mul (Poly.denote ctx p₂)]; apply Int.mul_le_mul_of_nonpos_right <;> simp [*]
   · rw [← Int.zero_mul (Poly.denote ctx p₁)]; apply Int.mul_le_mul_of_nonpos_right <;> simp [*]
 
+@[expose]
 def le_combine_coeff_cert (p₁ p₂ p₃ : Poly) (k : Int) : Bool :=
   let a₁ := p₁.leadCoeff.natAbs
   let a₂ := p₂.leadCoeff.natAbs
@@ -883,6 +925,7 @@ theorem eq_norm (ctx : Context) (p₁ p₂ : Poly) (h : p₁.norm == p₂) : p
   simp at h
   simp [*]
 
+@[expose]
 def eq_coeff_cert (p p' : Poly) (k : Int) : Bool :=
   p == p'.mul k && k > 0
 
@@ -893,6 +936,7 @@ theorem eq_coeff (ctx : Context) (p p' : Poly) (k : Int) : eq_coeff_cert p p' k
 theorem eq_unsat (ctx : Context) (p : Poly) : p.isUnsatEq → p.denote' ctx = 0 → False := by
   simp [Poly.isUnsatEq] <;> split <;> simp
 
+@[expose]
 def eq_unsat_coeff_cert (p : Poly) (k : Int) : Bool :=
   p.divCoeffs k && k > 0 && cmod p.getConst k < 0
 
@@ -902,6 +946,7 @@ theorem eq_unsat_coeff (ctx : Context) (p : Poly) (k : Int) : eq_unsat_coeff_cer
   have h := poly_eq_zero_eq_false ctx h₁ h₂ h₃; clear h₁ h₂ h₃
   simp [h]
 
+@[expose]
 def Poly.coeff (p : Poly) (x : Var) : Int :=
   match p with
   | .add a y p => bif x == y then a else coeff p x
@@ -916,7 +961,8 @@ private theorem dvd_of_eq' {a x p : Int} : a*x + p = 0 → a ∣ p := by
   rw [Int.mul_comm, ← Int.neg_mul, Eq.comm, Int.mul_comm] at h
   exact ⟨-x, h⟩
 
-private def abs (x : Int) : Int :=
+@[expose]
+def abs (x : Int) : Int :=
   Int.ofNat x.natAbs
 
 private theorem abs_dvd {a p : Int} (h : a ∣ p) : abs a ∣ p := by
@@ -924,6 +970,7 @@ private theorem abs_dvd {a p : Int} (h : a ∣ p) : abs a ∣ p := by
   · simp at h; assumption
   · simp [Int.negSucc_eq] at h; assumption
 
+@[expose]
 def dvd_of_eq_cert (x : Var) (p₁ : Poly) (d₂ : Int) (p₂ : Poly) : Bool :=
   let a := p₁.coeff x
   d₂ == abs a && p₂ == p₁.insert (-a) x
@@ -950,6 +997,7 @@ private theorem eq_dvd_subst' {a x p d b q : Int} : a*x + p = 0 → d ∣ b*x +
   rw [← Int.mul_assoc] at h
   exact ⟨z, h⟩
 
+@[expose]
 def eq_dvd_subst_cert (x : Var) (p₁ : Poly) (d₂ : Int) (p₂ : Poly) (d₃ : Int) (p₃ : Poly) : Bool :=
   let a := p₁.coeff x
   let b := p₂.coeff x
@@ -979,6 +1027,7 @@ theorem eq_dvd_subst (ctx : Context) (x : Var) (p₁ : Poly) (d₂ : Int) (p₂
   apply abs_dvd
   simp [this, Int.neg_mul]
 
+@[expose]
 def eq_eq_subst_cert (x : Var) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly) : Bool :=
   let a := p₁.coeff x
   let b := p₂.coeff x
@@ -991,6 +1040,7 @@ theorem eq_eq_subst (ctx : Context) (x : Var) (p₁ : Poly) (p₂ : Poly) (p₃
   intro h₁ h₂
   simp [*]
 
+@[expose]
 def eq_le_subst_nonneg_cert (x : Var) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly) : Bool :=
   let a := p₁.coeff x
   let b := p₂.coeff x
@@ -1006,6 +1056,7 @@ theorem eq_le_subst_nonneg (ctx : Context) (x : Var) (p₁ : Poly) (p₂ : Poly)
   simp at h₂
   simp [*]
 
+@[expose]
 def eq_le_subst_nonpos_cert (x : Var) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly) : Bool :=
   let a := p₁.coeff x
   let b := p₂.coeff x
@@ -1022,6 +1073,7 @@ theorem eq_le_subst_nonpos (ctx : Context) (x : Var) (p₁ : Poly) (p₂ : Poly)
   rw [Int.mul_comm]
   assumption
 
+@[expose]
 def eq_of_core_cert (p₁ : Poly) (p₂ : Poly) (p₃ : Poly) : Bool :=
   p₃ == p₁.combine (p₂.mul (-1))
 
@@ -1031,6 +1083,7 @@ theorem eq_of_core (ctx : Context) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly)
   intro; subst p₃; simp
   intro h; rw [h, Int.add_neg_eq_sub, Int.sub_self]
 
+@[expose]
 def Poly.isUnsatDiseq (p : Poly) : Bool :=
   match p with
   | .num 0 => true
@@ -1047,11 +1100,12 @@ theorem diseq_coeff (ctx : Context) (p p' : Poly) (k : Int) : eq_coeff_cert p p'
   intro _ _; simp [mul_eq_zero_iff, *]
 
 theorem diseq_neg (ctx : Context) (p p' : Poly) : p' == p.mul (-1) → p.denote' ctx ≠ 0 → p'.denote' ctx ≠ 0 := by
-  simp; intro _ _; simp [mul_eq_zero_iff, *]
+  simp; intro _ _; simp [*]
 
 theorem diseq_unsat (ctx : Context) (p : Poly) : p.isUnsatDiseq → p.denote' ctx ≠ 0 → False := by
   simp [Poly.isUnsatDiseq] <;> split <;> simp
 
+@[expose]
 def diseq_eq_subst_cert (x : Var) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly) : Bool :=
   let a := p₁.coeff x
   let b := p₂.coeff x
@@ -1071,6 +1125,7 @@ theorem diseq_of_core (ctx : Context) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly)
   intro h; rw [← Int.sub_eq_zero] at h
   rw [Int.add_neg_eq_sub]; assumption
 
+@[expose]
 def eq_of_le_ge_cert (p₁ p₂ : Poly) : Bool :=
   p₂ == p₁.mul (-1)
 
@@ -1081,6 +1136,7 @@ theorem eq_of_le_ge (ctx : Context) (p₁ : Poly) (p₂ : Poly)
   intro h₁ h₂
   simp [Int.eq_iff_le_and_ge, *]
 
+@[expose]
 def le_of_le_diseq_cert (p₁ : Poly) (p₂ : Poly) (p₃ : Poly) : Bool :=
   -- Remark: we can generate two different certificates in the future, and avoid the `||` in the certificate.
   (p₂ == p₁ || p₂ == p₁.mul (-1)) &&
@@ -1095,6 +1151,7 @@ theorem le_of_le_diseq (ctx : Context) (p₁ : Poly) (p₂ : Poly) (p₃ : Poly)
     next h => have := Int.lt_of_le_of_lt h₁ h; simp at this
   intro h; cases h <;> intro <;> subst p₂ p₃ <;> simp <;> apply this
 
+@[expose]
 def diseq_split_cert (p₁ p₂ p₃ : Poly) : Bool :=
   p₂ == p₁.addConst 1 &&
   p₃ == (p₁.mul (-1)).addConst 1
@@ -1113,6 +1170,7 @@ theorem diseq_split_resolve (ctx : Context) (p₁ p₂ p₃ : Poly)
   intro h₁ h₂ h₃
   exact (diseq_split ctx p₁ p₂ p₃ h₁ h₂).resolve_left h₃
 
+@[expose]
 def OrOver (n : Nat) (p : Nat → Prop) : Prop :=
   match n with
   | 0 => False
@@ -1127,6 +1185,7 @@ theorem orOver_resolve {n p} : OrOver (n+1) p → ¬ p n → OrOver n p := by
   · contradiction
   · assumption
 
+@[expose]
 def OrOver_cases_type (n : Nat) (p : Nat → Prop) : Prop :=
   match n with
   | 0 => p 0
@@ -1186,6 +1245,7 @@ private theorem cooper_dvd_left_core
   rw [this] at h₃
   exists k.toNat
 
+@[expose]
 def cooper_dvd_left_cert (p₁ p₂ p₃ : Poly) (d : Int) (n : Nat) : Bool :=
   p₁.casesOn (fun _ => false) fun a x _ =>
   p₂.casesOn (fun _ => false) fun b y _ =>
@@ -1194,11 +1254,13 @@ def cooper_dvd_left_cert (p₁ p₂ p₃ : Poly) (d : Int) (n : Nat) : Bool :=
    .and (a < 0)  <| .and (b > 0)  <|
    .and (d > 0)  <| n == Int.lcm a (a * d / Int.gcd (a * d) c)
 
+@[expose]
 def Poly.tail (p : Poly) : Poly :=
   match p with
   | .add _ _ p => p
   | _ => p
 
+@[expose]
 def cooper_dvd_left_split (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (k : Nat) : Prop :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1238,6 +1300,7 @@ theorem cooper_dvd_left (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (n : N
  simp only [denote'_addConst_eq]
  exact cooper_dvd_left_core ha hb hd h₁ h₂ h₃
 
+@[expose]
 def cooper_dvd_left_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (b : Int) (p' : Poly) : Bool :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1248,8 +1311,9 @@ def cooper_dvd_left_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (b : Int) (p' :
 theorem cooper_dvd_left_split_ineq (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (k : Nat) (b : Int) (p' : Poly)
     : cooper_dvd_left_split ctx p₁ p₂ p₃ d k → cooper_dvd_left_split_ineq_cert p₁ p₂ k b p' → p'.denote' ctx ≤ 0 := by
   simp [cooper_dvd_left_split_ineq_cert, cooper_dvd_left_split]
-  intros; subst p' b; simp [denote'_mul_combine_mul_addConst_eq]; assumption
+  intros; subst p' b; simp; assumption
 
+@[expose]
 def cooper_dvd_left_split_dvd1_cert (p₁ p' : Poly) (a : Int) (k : Int) : Bool :=
   a == p₁.leadCoeff && p' == p₁.tail.addConst k
 
@@ -1258,6 +1322,7 @@ theorem cooper_dvd_left_split_dvd1 (ctx : Context) (p₁ p₂ p₃ : Poly) (d :
   simp [cooper_dvd_left_split_dvd1_cert, cooper_dvd_left_split]
   intros; subst a p'; simp; assumption
 
+@[expose]
 def cooper_dvd_left_split_dvd2_cert (p₁ p₃ : Poly) (d : Int) (k : Nat) (d' : Int) (p' : Poly): Bool :=
   let p  := p₁.tail
   let s  := p₃.tail
@@ -1283,16 +1348,18 @@ private theorem cooper_left_core
   have h := cooper_dvd_left_core a_neg b_pos d_pos h₁ h₂ h₃
   simp only [Int.mul_one, gcd_zero, ofNat_natAbs_of_nonpos (Int.le_of_lt a_neg), Int.ediv_neg,
     Int.ediv_self (Int.ne_of_lt a_neg), Int.reduceNeg, lcm_neg_right, lcm_one,
-    Int.add_left_comm, Int.zero_mul, Int.mul_zero, Int.add_zero, Int.dvd_zero,
+    Int.zero_mul, Int.mul_zero, Int.add_zero, Int.dvd_zero,
     and_true] at h
   assumption
 
+@[expose]
 def cooper_left_cert (p₁ p₂ : Poly) (n : Nat) : Bool :=
   p₁.casesOn (fun _ => false) fun a x _ =>
   p₂.casesOn (fun _ => false) fun b y _ =>
    .and (x == y) <| .and (a < 0)  <| .and (b > 0)  <|
    n == a.natAbs
 
+@[expose]
 def cooper_left_split (ctx : Context) (p₁ p₂ : Poly) (k : Nat) : Prop :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1320,6 +1387,7 @@ theorem cooper_left (ctx : Context) (p₁ p₂ : Poly) (n : Nat)
  simp only [denote'_addConst_eq]
  assumption
 
+@[expose]
 def cooper_left_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (b : Int) (p' : Poly) : Bool :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1330,8 +1398,9 @@ def cooper_left_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (b : Int) (p' : Pol
 theorem cooper_left_split_ineq (ctx : Context) (p₁ p₂ : Poly) (k : Nat) (b : Int) (p' : Poly)
     : cooper_left_split ctx p₁ p₂ k → cooper_left_split_ineq_cert p₁ p₂ k b p' → p'.denote' ctx ≤ 0 := by
   simp [cooper_left_split_ineq_cert, cooper_left_split]
-  intros; subst p' b; simp [denote'_mul_combine_mul_addConst_eq]; assumption
+  intros; subst p' b; simp; assumption
 
+@[expose]
 def cooper_left_split_dvd_cert (p₁ p' : Poly) (a : Int) (k : Int) : Bool :=
   a == p₁.leadCoeff && p' == p₁.tail.addConst k
 
@@ -1353,7 +1422,7 @@ private theorem cooper_dvd_right_core
   have h₁'    : p ≤ (-a)*x := by rw [Int.neg_mul, ← Lean.Omega.Int.add_le_zero_iff_le_neg']; assumption
   have h₂'    : b * x ≤ -q := by rw [← Lean.Omega.Int.add_le_zero_iff_le_neg', Int.add_comm]; assumption
   have ⟨k, h₁, h₂, h₃, h₄, h₅⟩ := Int.cooper_resolution_dvd_right a_pos' b_pos d_pos |>.mp ⟨x, h₁', h₂', h₃⟩
-  simp only [Int.neg_mul, neg_gcd, lcm_neg_left, Int.mul_neg, Int.neg_neg, Int.neg_dvd] at *
+  simp only [Int.neg_mul, Int.mul_neg, Int.neg_neg] at *
   apply orOver_of_exists
   have hlt := ofNat_lt h₁ h₂
   replace h₃ := Int.add_le_add_right h₃ (-(a*q)); rw [Int.add_right_neg] at h₃
@@ -1363,8 +1432,9 @@ private theorem cooper_dvd_right_core
   have : -(c * k) + -(c * q) + b * s = -(c * q) + b * s + -(c * k) := by ac_rfl
   rw [this] at h₅; clear this
   exists k.toNat
-  simp only [hlt, true_and, and_true, cast_toNat h₁, h₃, h₄, h₅]
+  simp only [hlt, and_true, cast_toNat h₁, h₃, h₄, h₅]
 
+@[expose]
 def cooper_dvd_right_cert (p₁ p₂ p₃ : Poly) (d : Int) (n : Nat) : Bool :=
   p₁.casesOn (fun _ => false) fun a x _ =>
   p₂.casesOn (fun _ => false) fun b y _ =>
@@ -1373,6 +1443,7 @@ def cooper_dvd_right_cert (p₁ p₂ p₃ : Poly) (d : Int) (n : Nat) : Bool :=
    .and (a < 0)  <| .and (b > 0)  <|
    .and (d > 0)  <| n == Int.lcm b (b * d / Int.gcd (b * d) c)
 
+@[expose]
 def cooper_dvd_right_split (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (k : Nat) : Prop :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1402,9 +1473,10 @@ theorem cooper_dvd_right (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (n :
  intro h₁ h₂ h₃
  have := cooper_dvd_right_core ha hb hd h₁ h₂ h₃
  simp only [denote'_mul_combine_mul_addConst_eq]
- simp only [denote'_addConst_eq, ←Int.neg_mul]
+ simp only [denote'_addConst_eq]
  exact cooper_dvd_right_core ha hb hd h₁ h₂ h₃
 
+@[expose]
 def cooper_dvd_right_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (a : Int) (p' : Poly) : Bool :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1415,8 +1487,9 @@ def cooper_dvd_right_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (a : Int) (p'
 theorem cooper_dvd_right_split_ineq (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (k : Nat) (a : Int) (p' : Poly)
     : cooper_dvd_right_split ctx p₁ p₂ p₃ d k → cooper_dvd_right_split_ineq_cert p₁ p₂ k a p' → p'.denote' ctx ≤ 0 := by
   simp [cooper_dvd_right_split_ineq_cert, cooper_dvd_right_split]
-  intros; subst a p'; simp [denote'_mul_combine_mul_addConst_eq]; assumption
+  intros; subst a p'; simp; assumption
 
+@[expose]
 def cooper_dvd_right_split_dvd1_cert (p₂ p' : Poly) (b : Int) (k : Int) : Bool :=
   b == p₂.leadCoeff && p' == p₂.tail.addConst k
 
@@ -1425,6 +1498,7 @@ theorem cooper_dvd_right_split_dvd1 (ctx : Context) (p₁ p₂ p₃ : Poly) (d :
   simp [cooper_dvd_right_split_dvd1_cert, cooper_dvd_right_split]
   intros; subst b p'; simp; assumption
 
+@[expose]
 def cooper_dvd_right_split_dvd2_cert (p₂ p₃ : Poly) (d : Int) (k : Nat) (d' : Int) (p' : Poly): Bool :=
   let q  := p₂.tail
   let s  := p₃.tail
@@ -1448,17 +1522,19 @@ private theorem cooper_right_core
   have d_pos : (0 : Int) < 1 := by decide
   have h₃ : 1 ∣ 0*x + 0 := Int.one_dvd _
   have h := cooper_dvd_right_core a_neg b_pos d_pos h₁ h₂ h₃
-  simp only [Int.mul_one, gcd_zero, Int.natAbs_of_nonneg (Int.le_of_lt b_pos), Int.ediv_neg,
-    Int.ediv_self (Int.ne_of_gt b_pos), Int.reduceNeg, lcm_neg_right, lcm_one,
-    Int.add_left_comm, Int.zero_mul, Int.mul_zero, Int.add_zero, Int.dvd_zero,
+  simp only [Int.mul_one, gcd_zero, Int.natAbs_of_nonneg (Int.le_of_lt b_pos), 
+    Int.ediv_self (Int.ne_of_gt b_pos), lcm_one,
+    Int.zero_mul, Int.mul_zero, Int.add_zero, Int.dvd_zero,
     and_true, Int.neg_zero] at h
   assumption
 
+@[expose]
 def cooper_right_cert (p₁ p₂ : Poly) (n : Nat) : Bool :=
   p₁.casesOn (fun _ => false) fun a x _ =>
   p₂.casesOn (fun _ => false) fun b y _ =>
   .and (x == y) <| .and (a < 0)  <| .and (b > 0) <| n == b.natAbs
 
+@[expose]
 def cooper_right_split (ctx : Context) (p₁ p₂ : Poly) (k : Nat) : Prop :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1483,9 +1559,10 @@ theorem cooper_right (ctx : Context) (p₁ p₂ : Poly) (n : Nat)
  intro h₁ h₂
  have := cooper_right_core ha hb h₁ h₂
  simp only [denote'_mul_combine_mul_addConst_eq]
- simp only [denote'_addConst_eq, ←Int.neg_mul]
+ simp only [denote'_addConst_eq]
  assumption
 
+@[expose]
 def cooper_right_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (a : Int) (p' : Poly) : Bool :=
   let p  := p₁.tail
   let q  := p₂.tail
@@ -1496,8 +1573,9 @@ def cooper_right_split_ineq_cert (p₁ p₂ : Poly) (k : Int) (a : Int) (p' : Po
 theorem cooper_right_split_ineq (ctx : Context) (p₁ p₂ : Poly) (k : Nat) (a : Int) (p' : Poly)
     : cooper_right_split ctx p₁ p₂ k → cooper_right_split_ineq_cert p₁ p₂ k a p' → p'.denote' ctx ≤ 0 := by
   simp [cooper_right_split_ineq_cert, cooper_right_split]
-  intros; subst a p'; simp [denote'_mul_combine_mul_addConst_eq]; assumption
+  intros; subst a p'; simp; assumption
 
+@[expose]
 def cooper_right_split_dvd_cert (p₂ p' : Poly) (b : Int) (k : Int) : Bool :=
   b == p₂.leadCoeff && p' == p₂.tail.addConst k
 
@@ -1587,6 +1665,7 @@ abbrev Poly.casesOnAdd (p : Poly) (k : Int → Var → Poly → Bool) : Bool :=
 abbrev Poly.casesOnNum (p : Poly) (k : Int → Bool) : Bool :=
   p.casesOn k (fun _ _ _ => false)
 
+@[expose]
 def cooper_unsat_cert (p₁ p₂ p₃ : Poly) (d : Int) (α β : Int) : Bool :=
   p₁.casesOnAdd fun k₁ x p₁ =>
   p₂.casesOnAdd fun k₂ y p₂ =>
@@ -1603,7 +1682,7 @@ theorem cooper_unsat (ctx : Context) (p₁ p₂ p₃ : Poly) (d : Int) (α β :
    : cooper_unsat_cert p₁ p₂ p₃ d α β →
      p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 → d ∣ p₃.denote' ctx → False := by
   unfold cooper_unsat_cert <;> cases p₁ <;> cases p₂ <;> cases p₃ <;> simp only [Poly.casesOnAdd,
-    Bool.false_eq_true, Poly.denote'_add, mul_def, add_def, false_implies]
+    Bool.false_eq_true, Poly.denote'_add, false_implies]
   next k₁ x p₁ k₂ y p₂ c z p₃ =>
   cases p₁ <;> cases p₂ <;> cases p₃ <;> simp only [Poly.casesOnNum, Int.reduceNeg,
     Bool.and_eq_true, beq_iff_eq, decide_eq_true_eq, and_imp, Bool.false_eq_true,
@@ -1626,6 +1705,7 @@ theorem emod_nonneg (x y : Int) : y != 0 → -1 * (x % y) ≤ 0 := by
   simp at this
   assumption
 
+@[expose]
 def emod_le_cert (y n : Int) : Bool :=
   y != 0 && n == 1 - y.natAbs
 
@@ -1665,7 +1745,7 @@ 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
@@ -1708,6 +1788,7 @@ private theorem eq_neg_addConst_add (ctx : Context) (p : Poly)
   rw [Int.add_right_neg]
   simp
 
+@[expose]
 def dvd_le_tight_cert (d : Int) (p₁ p₂ p₃ : Poly) : Bool :=
   let b₁ := p₁.getConst
   let b₂ := p₂.getConst
@@ -1728,6 +1809,7 @@ theorem dvd_le_tight (ctx : Context) (d : Int) (p₁ p₂ p₃ : Poly)
   simp only [Poly.denote'_eq_denote]
   exact dvd_le_tight' hd
 
+@[expose]
 def dvd_neg_le_tight_cert (d : Int) (p₁ p₂ p₃ : Poly) : Bool :=
   let b₁ := p₁.getConst
   let b₂ := p₂.getConst
@@ -1737,7 +1819,7 @@ def dvd_neg_le_tight_cert (d : Int) (p₁ p₂ p₃ : Poly) : Bool :=
   d > 0 && (p₂ == p.addConst b₂ && p₃ == p.addConst (b₁ - d*((b₁ - b₂)/d)))
 
 theorem Poly.mul_minus_one_getConst_eq (p : Poly) : (p.mul (-1)).getConst = -p.getConst := by
-  simp [Poly.mul, Poly.getConst]
+  simp [Poly.mul]
   induction p <;> simp [Poly.mul', Poly.getConst, *]
 
 theorem dvd_neg_le_tight (ctx : Context) (d : Int) (p₁ p₂ p₃ : Poly)
@@ -1764,6 +1846,7 @@ theorem le_norm_expr (ctx : Context) (lhs rhs : Expr) (p : Poly)
     : norm_eq_cert lhs rhs p → lhs.denote ctx ≤ rhs.denote ctx → p.denote' ctx ≤ 0 := by
   intro h₁ h₂; rwa [norm_le ctx lhs rhs p h₁] at h₂
 
+@[expose]
 def not_le_norm_expr_cert (lhs rhs : Expr) (p : Poly) : Bool :=
   p == (((lhs.sub rhs).norm).mul (-1)).addConst 1
 
@@ -1796,6 +1879,7 @@ theorem of_not_dvd (a b : Int) : a != 0 → ¬ (a ∣ b) → b % a > 0 := by
   simp [h₁] at h₂
   assumption
 
+@[expose]
 def le_of_le_cert (p q : Poly) (k : Nat) : Bool :=
   q == p.addConst (- k)
 
@@ -1806,6 +1890,7 @@ theorem le_of_le (ctx : Context) (p q : Poly) (k : Nat)
   simp [Lean.Omega.Int.add_le_zero_iff_le_neg']
   exact Int.le_trans h (Int.ofNat_zero_le _)
 
+@[expose]
 def not_le_of_le_cert (p q : Poly) (k : Nat) : Bool :=
   q == (p.mul (-1)).addConst (1 + k)
 
@@ -1815,10 +1900,11 @@ theorem not_le_of_le (ctx : Context) (p q : Poly) (k : Nat)
   intro h
   apply Int.pos_of_neg_neg
   apply Int.lt_of_add_one_le
-  simp [Int.neg_add, Int.neg_sub]
+  simp [Int.neg_add]
   rw [← Int.add_assoc, ← Int.add_assoc, Int.add_neg_cancel_right, Lean.Omega.Int.add_le_zero_iff_le_neg']
   simp; exact Int.le_trans h (Int.ofNat_zero_le _)
 
+@[expose]
 def eq_def_cert (x : Var) (xPoly : Poly) (p : Poly) : Bool :=
   p == .add (-1) x xPoly
 
@@ -1827,6 +1913,7 @@ theorem eq_def (ctx : Context) (x : Var) (xPoly : Poly) (p : Poly)
   simp [eq_def_cert]; intro _ h; subst p; simp [h]
   rw [← Int.sub_eq_add_neg, Int.sub_self]
 
+@[expose]
 def eq_def'_cert (x : Var) (e : Expr) (p : Poly) : Bool :=
   p == .add (-1) x e.norm
 

src/Init/Data/Int/OfNat.lean

@@ -19,6 +19,7 @@ We use them to implement the arithmetic theories in `grind`
 
 abbrev Var := Nat
 abbrev Context := Lean.RArray Nat
+@[expose]
 def Var.denote (ctx : Context) (v : Var) : Nat :=
   ctx.get v
 
@@ -31,6 +32,7 @@ inductive Expr where
   | mod  (a b : Expr)
   deriving BEq
 
+@[expose]
 def Expr.denote (ctx : Context) : Expr → Nat
   | .num k    => k
   | .var v    => v.denote ctx
@@ -39,6 +41,7 @@ def Expr.denote (ctx : Context) : Expr → Nat
   | .div a b  => Nat.div (denote ctx a) (denote ctx b)
   | .mod a b  => Nat.mod (denote ctx a) (denote ctx b)
 
+@[expose]
 def Expr.denoteAsInt (ctx : Context) : Expr → Int
   | .num k    => Int.ofNat k
   | .var v    => Int.ofNat (v.denote ctx)
@@ -48,7 +51,7 @@ def Expr.denoteAsInt (ctx : Context) : Expr → Int
   | .mod a b  => Int.emod (denoteAsInt ctx a) (denoteAsInt ctx b)
 
 theorem Expr.denoteAsInt_eq (ctx : Context) (e : Expr) : e.denoteAsInt ctx = e.denote ctx := by
-  induction e <;> simp [denote, denoteAsInt, Int.natCast_ediv, *] <;> rfl
+  induction e <;> simp [denote, denoteAsInt, *] <;> rfl
 
 theorem Expr.eq_denoteAsInt (ctx : Context) (e : Expr) : e.denote ctx = e.denoteAsInt ctx := by
   apply Eq.symm; apply denoteAsInt_eq

src/Init/Data/Int/Order.lean

@@ -448,7 +448,7 @@ protected theorem le_max_left (a b : Int) : a ≤ max a b := by rw [Int.max_def]
 protected theorem le_max_right (a b : Int) : b ≤ max a b := Int.max_comm .. ▸ Int.le_max_left ..
 
 protected theorem max_eq_right {a b : Int} (h : a ≤ b) : max a b = b := by
-  simp [Int.max_def, h, Int.not_lt.2 h]
+  simp [Int.max_def, h]
 
 protected theorem max_eq_left {a b : Int} (h : b ≤ a) : max a b = a := by
   rw [← Int.max_comm b a]; exact Int.max_eq_right h

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

src/Init/Data/Iterators.lean

@@ -0,0 +1,19 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.PostconditionMonad
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.Lemmas
+import Init.Data.Iterators.Internal
+
+/-!
+# Iterators
+
+See `Std.Data.Iterators` for an overview over the iterator API.
+-/

src/Init/Data/Iterators/Basic.lean

@@ -3,9 +3,12 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
 import Init.Core
 import Init.Classical
+import Init.Ext
 import Init.NotationExtra
 import Init.TacticsExtra
 
@@ -30,7 +33,7 @@ See `Std.Data.Iterators.Producers` for ways to iterate over common data structur
 By convention, the monadic iterator associated with an object can be obtained via dot notation.
 For example, `List.iterM IO` creates an iterator over a list in the monad `IO`.
 
-See `Std.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
+See `Init.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
 convert a provably finite iterator `it` into a list and `it.allowNontermination.toList` will
 do so even if finiteness cannot be proved. It is also always possible to manually iterate using
 `it.step`, relying on the termination measures `it.finitelyManySteps` and `it.finitelyManySkips`.
@@ -58,6 +61,7 @@ def x := [1, 2, 3].iterM IO
 def x := ([1, 2, 3].iterM IO : IterM IO Nat)
 ```
 -/
+@[ext]
 structure IterM {α : Type w} (m : Type w → Type w') (β : Type w) where
   /-- Internal implementation detail of the iterator. -/
   internalState : α
@@ -73,7 +77,7 @@ See `Std.Data.Iterators.Producers` for ways to iterate over common data structur
 By convention, the monadic iterator associated with an object can be obtained via dot notation.
 For example, `List.iterM IO` creates an iterator over a list in the monad `IO`.
 
-See `Std.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
+See `Init.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
 convert a provably finite iterator `it` into a list and `it.allowNontermination.toList` will
 do so even if finiteness cannot be proved. It is also always possible to manually iterate using
 `it.step`, relying on the termination measures `it.finitelyManySteps` and `it.finitelyManySkips`.
@@ -109,12 +113,14 @@ structure Iter {α : Type w} (β : Type w) where
 Converts a pure iterator (`Iter β`) into a monadic iterator (`IterM Id β`) in the
 identity monad `Id`.
 -/
+@[expose]
 def Iter.toIterM {α : Type w} {β : Type w} (it : Iter (α := α) β) : IterM (α := α) Id β :=
   ⟨it.internalState⟩
 
 /--
 Converts a monadic iterator (`IterM Id β`) over `Id` into a pure iterator (`Iter β`).
 -/
+@[expose]
 def IterM.toIter {α : Type w} {β : Type w} (it : IterM (α := α) Id β) : Iter (α := α) β :=
   ⟨it.internalState⟩
 
@@ -168,6 +174,7 @@ inductive IterStep (α β) where
 Returns the succeeding iterator stored in an iterator step or `none` if the step is `.done`
 and the iterator has finished.
 -/
+@[expose]
 def IterStep.successor : IterStep α β → Option α
   | .yield it _ => some it
   | .skip it => some it
@@ -177,7 +184,7 @@ def IterStep.successor : IterStep α β → Option α
 If present, applies `f` to the iterator of an `IterStep` and replaces the iterator
 with the result of the application of `f`.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def IterStep.mapIterator {α' : Type u'} (f : α → α') : IterStep α β → IterStep α' β
   | .yield it out => .yield (f it) out
   | .skip it => .skip (f it)
@@ -204,6 +211,12 @@ theorem IterStep.mapIterator_mapIterator {α' : Type u'} {α'' : Type u''}
     (step.mapIterator f).mapIterator g = step.mapIterator (g ∘ f) := by
   cases step <;> rfl
 
+theorem IterStep.mapIterator_comp {α' : Type u'} {α'' : Type u''}
+    {f : α → α'} {g : α' → α''} :
+    IterStep.mapIterator (β := β) (g ∘ f) = mapIterator g ∘ mapIterator f := by
+  apply funext
+  exact fun _ => mapIterator_mapIterator.symm
+
 @[simp]
 theorem IterStep.mapIterator_id {step : IterStep α β} :
     step.mapIterator id = step := by
@@ -216,12 +229,13 @@ of another state. Having this proof bundled up with the step is important for te
 
 See `IterM.Step` and `Iter.Step` for the concrete choice of the plausibility predicate.
 -/
+@[expose]
 def PlausibleIterStep (IsPlausibleStep : IterStep α β → Prop) := Subtype IsPlausibleStep
 
 /--
 Match pattern for the `yield` case. See also `IterStep.yield`.
 -/
-@[match_pattern, simp]
+@[match_pattern, simp, expose]
 def PlausibleIterStep.yield {IsPlausibleStep : IterStep α β → Prop}
     (it' : α) (out : β) (h : IsPlausibleStep (.yield it' out)) :
     PlausibleIterStep IsPlausibleStep :=
@@ -230,7 +244,7 @@ def PlausibleIterStep.yield {IsPlausibleStep : IterStep α β → Prop}
 /--
 Match pattern for the `skip` case. See also `IterStep.skip`.
 -/
-@[match_pattern, simp]
+@[match_pattern, simp, expose]
 def PlausibleIterStep.skip {IsPlausibleStep : IterStep α β → Prop}
     (it' : α) (h : IsPlausibleStep (.skip it')) : PlausibleIterStep IsPlausibleStep :=
   ⟨.skip it', h⟩
@@ -238,7 +252,7 @@ def PlausibleIterStep.skip {IsPlausibleStep : IterStep α β → Prop}
 /--
 Match pattern for the `done` case. See also `IterStep.done`.
 -/
-@[match_pattern, simp]
+@[match_pattern, simp, expose]
 def PlausibleIterStep.done {IsPlausibleStep : IterStep α β → Prop}
     (h : IsPlausibleStep .done) : PlausibleIterStep IsPlausibleStep :=
   ⟨.done, h⟩
@@ -275,7 +289,7 @@ section Monadic
 /--
 Converts wraps the state of an iterator into an `IterM` object.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def toIterM {α : Type w} (it : α) (m : Type w → Type w') (β : Type w) :
     IterM (α := α) m β :=
   ⟨it⟩
@@ -294,6 +308,7 @@ theorem internalState_toIterM {α m β} (it : α) :
 Asserts that certain step is plausibly the successor of a given iterator. What "plausible" means
 is up to the `Iterator` instance but it should be strong enough to allow termination proofs.
 -/
+@[expose]
 abbrev IterM.IsPlausibleStep {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] :
     IterM (α := α) m β → IterStep (IterM (α := α) m β) β → Prop :=
   Iterator.IsPlausibleStep (α := α) (m := m)
@@ -302,6 +317,7 @@ abbrev IterM.IsPlausibleStep {α : Type w} {m : Type w → Type w'} {β : Type w
 The type of the step object returned by `IterM.step`, containing an `IterStep`
 and a proof that this is a plausible step for the given iterator.
 -/
+@[expose]
 abbrev IterM.Step {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     (it : IterM (α := α) m β) :=
   PlausibleIterStep it.IsPlausibleStep
@@ -310,6 +326,7 @@ abbrev IterM.Step {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator
 Asserts that a certain output value could plausibly be emitted by the given iterator in its next
 step.
 -/
+@[expose]
 def IterM.IsPlausibleOutput {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     (it : IterM (α := α) m β) (out : β) : Prop :=
   ∃ it', it.IsPlausibleStep (.yield it' out)
@@ -318,6 +335,7 @@ def IterM.IsPlausibleOutput {α : Type w} {m : Type w → Type w'} {β : Type w}
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it`.
 -/
+@[expose]
 def IterM.IsPlausibleSuccessorOf {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     (it' it : IterM (α := α) m β) : Prop :=
   ∃ step, step.successor = some it' ∧ it.IsPlausibleStep step
@@ -326,6 +344,7 @@ def IterM.IsPlausibleSuccessorOf {α : Type w} {m : Type w → Type w'} {β : Ty
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it` while no value is emitted (see `IterStep.skip`).
 -/
+@[expose]
 def IterM.IsPlausibleSkipSuccessorOf {α : Type w} {m : Type w → Type w'} {β : Type w}
     [Iterator α m β] (it' it : IterM (α := α) m β) : Prop :=
   it.IsPlausibleStep (.skip it')
@@ -348,14 +367,27 @@ section Pure
 Asserts that certain step is plausibly the successor of a given iterator. What "plausible" means
 is up to the `Iterator` instance but it should be strong enough to allow termination proofs.
 -/
+@[expose]
 def Iter.IsPlausibleStep {α : Type w} {β : Type w} [Iterator α Id β]
     (it : Iter (α := α) β) (step : IterStep (Iter (α := α) β) β) : Prop :=
   it.toIterM.IsPlausibleStep (step.mapIterator Iter.toIterM)
 
+/--
+Asserts that a certain iterator `it` could plausibly yield the value `out` after an arbitrary
+number of steps.
+-/
+inductive IterM.IsPlausibleIndirectOutput {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    : IterM (α := α) m β → β → Prop where
+  | direct {it : IterM (α := α) m β} {out : β} : it.IsPlausibleOutput out →
+      it.IsPlausibleIndirectOutput out
+  | indirect {it it' : IterM (α := α) m β} {out : β} : it'.IsPlausibleSuccessorOf it →
+      it'.IsPlausibleIndirectOutput out → it.IsPlausibleIndirectOutput out
+
 /--
 The type of the step object returned by `Iter.step`, containing an `IterStep`
 and a proof that this is a plausible step for the given iterator.
 -/
+@[expose]
 def Iter.Step {α : Type w} {β : Type w} [Iterator α Id β] (it : Iter (α := α) β) :=
   PlausibleIterStep (Iter.IsPlausibleStep it)
 
@@ -370,7 +402,7 @@ def Iter.Step.toMonadic {α : Type w} {β : Type w} [Iterator α Id β] {it : It
 /--
 Converts an `IterM.Step` into an `Iter.Step`.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def IterM.Step.toPure {α : Type w} {β : Type w} [Iterator α Id β] {it : IterM (α := α) Id β}
     (step : it.Step) : it.toIter.Step :=
   ⟨step.val.mapIterator IterM.toIter, (by simp [Iter.IsPlausibleStep, step.property])⟩
@@ -394,6 +426,7 @@ theorem IterM.Step.toPure_done {α β : Type w} [Iterator α Id β] {it : IterM
 Asserts that a certain output value could plausibly be emitted by the given iterator in its next
 step.
 -/
+@[expose]
 def Iter.IsPlausibleOutput {α : Type w} {β : Type w} [Iterator α Id β]
     (it : Iter (α := α) β) (out : β) : Prop :=
   it.toIterM.IsPlausibleOutput out
@@ -402,10 +435,43 @@ def Iter.IsPlausibleOutput {α : Type w} {β : Type w} [Iterator α Id β]
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it`.
 -/
+@[expose]
 def Iter.IsPlausibleSuccessorOf {α : Type w} {β : Type w} [Iterator α Id β]
     (it' it : Iter (α := α) β) : Prop :=
   it'.toIterM.IsPlausibleSuccessorOf it.toIterM
 
+/--
+Asserts that a certain iterator `it` could plausibly yield the value `out` after an arbitrary
+number of steps.
+-/
+inductive Iter.IsPlausibleIndirectOutput {α β : Type w} [Iterator α Id β] :
+    Iter (α := α) β → β → Prop where
+  | direct {it : Iter (α := α) β} {out : β} : it.IsPlausibleOutput out →
+      it.IsPlausibleIndirectOutput out
+  | indirect {it it' : Iter (α := α) β} {out : β} : it'.IsPlausibleSuccessorOf it →
+      it'.IsPlausibleIndirectOutput out → it.IsPlausibleIndirectOutput out
+
+theorem Iter.isPlausibleIndirectOutput_iff_isPlausibleIndirectOutput_toIterM {α β : Type w}
+    [Iterator α Id β] {it : Iter (α := α) β} {out : β} :
+    it.IsPlausibleIndirectOutput out ↔ it.toIterM.IsPlausibleIndirectOutput out := by
+  constructor
+  · intro h
+    induction h with
+    | direct h =>
+      exact .direct h
+    | indirect h _ ih =>
+      exact .indirect h ih
+  · intro h
+    rw [← Iter.toIter_toIterM (it := it)]
+    generalize it.toIterM = it at ⊢ h
+    induction h with
+    | direct h =>
+      exact .direct h
+    | indirect h h' ih =>
+      rename_i it it' out
+      replace h : it'.toIter.IsPlausibleSuccessorOf it.toIter := h
+      exact .indirect (α := α) h ih
+
 /--
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it` while no value is emitted (see `IterStep.skip`).
@@ -419,7 +485,7 @@ Makes a single step with the given iterator `it`, potentially emitting a value a
 succeeding iterator. If this function is used recursively, termination can sometimes be proved with
 the termination measures `it.finitelyManySteps` and `it.finitelyManySkips`.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def Iter.step {α β : Type w} [Iterator α Id β] (it : Iter (α := α) β) : it.Step :=
   it.toIterM.step.run.toPure
 
@@ -448,6 +514,7 @@ structure IterM.TerminationMeasures.Finite
 The relation of plausible successors on `IterM.TerminationMeasures.Finite`. It is well-founded
 if there is a `Finite` instance.
 -/
+@[expose]
 def IterM.TerminationMeasures.Finite.Rel
     {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] :
     TerminationMeasures.Finite α m → TerminationMeasures.Finite α m → Prop :=
@@ -456,12 +523,13 @@ def IterM.TerminationMeasures.Finite.Rel
 instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Finite α m] : WellFoundedRelation (IterM.TerminationMeasures.Finite α m) where
   rel := IterM.TerminationMeasures.Finite.Rel
-  wf := (InvImage.wf _ Finite.wf).transGen
+  wf := by exact (InvImage.wf _ Finite.wf).transGen
 
 /--
 Termination measure to be used in well-founded recursive functions recursing over a finite iterator
 (see also `Finite`).
 -/
+@[expose]
 def IterM.finitelyManySteps {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Finite α m] (it : IterM (α := α) m β) : IterM.TerminationMeasures.Finite α m :=
   ⟨it⟩
@@ -486,9 +554,10 @@ theorem IterM.TerminationMeasures.Finite.rel_of_skip
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
   first
   | exact IterM.TerminationMeasures.Finite.rel_of_yield ‹_›
-  | exact IterM.TerminationMeasures.Finite.rel_of_skip ‹_›)
+  | exact IterM.TerminationMeasures.Finite.rel_of_skip ‹_›
+  | fail)
 
-@[inherit_doc IterM.finitelyManySteps]
+@[inherit_doc IterM.finitelyManySteps, expose]
 def Iter.finitelyManySteps {α : Type w} {β : Type w} [Iterator α Id β] [Finite α Id]
     (it : Iter (α := α) β) : IterM.TerminationMeasures.Finite α Id :=
   it.toIterM.finitelyManySteps
@@ -513,7 +582,8 @@ theorem Iter.TerminationMeasures.Finite.rel_of_skip
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
   first
   | exact Iter.TerminationMeasures.Finite.rel_of_yield ‹_›
-  | exact Iter.TerminationMeasures.Finite.rel_of_skip ‹_›)
+  | exact Iter.TerminationMeasures.Finite.rel_of_skip ‹_›
+  | fail)
 
 theorem IterM.isPlausibleSuccessorOf_of_yield
     {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
@@ -553,6 +623,7 @@ structure IterM.TerminationMeasures.Productive
 The relation of plausible successors while skipping on `IterM.TerminationMeasures.Productive`.
 It is well-founded if there is a `Productive` instance.
 -/
+@[expose]
 def IterM.TerminationMeasures.Productive.Rel
     {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] :
     TerminationMeasures.Productive α m → TerminationMeasures.Productive α m → Prop :=
@@ -561,12 +632,13 @@ def IterM.TerminationMeasures.Productive.Rel
 instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Productive α m] : WellFoundedRelation (IterM.TerminationMeasures.Productive α m) where
   rel := IterM.TerminationMeasures.Productive.Rel
-  wf := (InvImage.wf _ Productive.wf).transGen
+  wf := by exact (InvImage.wf _ Productive.wf).transGen
 
 /--
 Termination measure to be used in well-founded recursive functions recursing over a productive
 iterator (see also `Productive`).
 -/
+@[expose]
 def IterM.finitelyManySkips {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Productive α m] (it : IterM (α := α) m β) : IterM.TerminationMeasures.Productive α m :=
   ⟨it⟩
@@ -582,9 +654,11 @@ theorem IterM.TerminationMeasures.Productive.rel_of_skip
   .single h
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
-  exact IterM.TerminationMeasures.Productive.rel_of_skip ‹_›)
+  first
+    | exact IterM.TerminationMeasures.Productive.rel_of_skip ‹_›
+    | fail)
 
-@[inherit_doc IterM.finitelyManySkips]
+@[inherit_doc IterM.finitelyManySkips, expose]
 def Iter.finitelyManySkips {α : Type w} {β : Type w} [Iterator α Id β] [Productive α Id]
     (it : Iter (α := α) β) : IterM.TerminationMeasures.Productive α Id :=
   it.toIterM.finitelyManySkips
@@ -600,7 +674,9 @@ theorem Iter.TerminationMeasures.Productive.rel_of_skip
   IterM.TerminationMeasures.Productive.rel_of_skip h
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
-  exact Iter.TerminationMeasures.Productive.rel_of_skip ‹_›)
+  first
+    | exact Iter.TerminationMeasures.Productive.rel_of_skip ‹_›
+    | fail)
 
 instance [Iterator α m β] [Finite α m] : Productive α m where
   wf := by

src/Init/Data/Iterators/Consumers.lean

@@ -0,0 +1,13 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Consumers.Monadic
+import Init.Data.Iterators.Consumers.Access
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.Consumers.Partial

src/Init/Data/Iterators/Consumers/Access.lean

@@ -3,8 +3,10 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Consumers.Partial
+import Init.Data.Iterators.Consumers.Partial
 
 namespace Std.Iterators
 

src/Init/Data/Iterators/Consumers/Collect.lean

@@ -3,10 +3,12 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Partial
-import Std.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Partial
+import Init.Data.Iterators.Consumers.Monadic.Collect
 
 /-!
 # Collectors
@@ -54,4 +56,18 @@ def Iter.Partial.toList {α : Type w} {β : Type w}
     [Iterator α Id β] [IteratorCollectPartial α Id Id] (it : Iter.Partial (α := α) β) : List β :=
   it.it.toIterM.allowNontermination.toList.run
 
+/--
+This class charaterizes how the plausibility behavior (`IsPlausibleStep`) and the actual iteration
+behavior (`it.step`) should relate to each other for pure iterators. Intuitively, a step should
+only be plausible if it is possible. For simplicity's sake, the actual definition is weaker but
+presupposes that the iterator is finite.
+
+This is an experimental instance and it should not be explicitly used downstream of the standard
+library.
+-/
+class LawfulPureIterator (α : Type w) [Iterator α Id β]
+    [Finite α Id] [IteratorCollect α Id Id] where
+  mem_toList_iff_isPlausibleIndirectOutput {it : Iter (α := α) β} {out : β} :
+    out ∈ it.toList ↔ it.IsPlausibleIndirectOutput out
+
 end Std.Iterators

src/Init/Data/Iterators/Consumers/Loop.lean

@@ -3,9 +3,11 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Consumers.Partial
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Consumers.Partial
 
 /-!
 # Loop consumers
@@ -22,11 +24,24 @@ These operations are implemented using the `IteratorLoop` and `IteratorLoopParti
 
 namespace Std.Iterators
 
+/--
+A `ForIn'` instance for iterators. Its generic membership relation is not easy to use,
+so this is not marked as `instance`. This way, more convenient instances can be built on top of it
+or future library improvements will make it more comfortable.
+-/
+def Iter.instForIn' {α : Type w} {β : Type w} {n : Type w → Type w'} [Monad n]
+    [Iterator α Id β] [Finite α Id] [IteratorLoop α Id n] :
+    ForIn' n (Iter (α := α) β) β ⟨fun it out => it.IsPlausibleIndirectOutput out⟩ where
+  forIn' it init f :=
+    IteratorLoop.finiteForIn' (fun δ (c : Id δ) => pure c.run) |>.forIn' it.toIterM init
+        fun out h acc =>
+          f out (Iter.isPlausibleIndirectOutput_iff_isPlausibleIndirectOutput_toIterM.mpr h) acc
+
 instance (α : Type w) (β : Type w) (n : Type w → Type w') [Monad n]
     [Iterator α Id β] [Finite α Id] [IteratorLoop α Id n] :
-    ForIn n (Iter (α := α) β) β where
-  forIn it init f :=
-    IteratorLoop.finiteForIn (fun δ (c : Id δ) => pure c.run) |>.forIn it.toIterM init f
+    ForIn n (Iter (α := α) β) β :=
+  haveI : ForIn' n (Iter (α := α) β) β _ := Iter.instForIn'
+  instForInOfForIn'
 
 instance (α : Type w) (β : Type w) (n : Type w → Type w') [Monad n]
     [Iterator α Id β] [IteratorLoopPartial α Id n] :
@@ -111,4 +126,14 @@ def Iter.Partial.fold {α : Type w} {β : Type w} {γ : Type w} [Iterator α Id
     (init : γ) (it : Iter.Partial (α := α) β) : γ :=
   ForIn.forIn (m := Id) it init (fun x acc => ForInStep.yield (f acc x))
 
+@[always_inline, inline, inherit_doc IterM.size]
+def Iter.size {α : Type w} {β : Type w} [Iterator α Id β] [IteratorSize α Id]
+    (it : Iter (α := α) β) : Nat :=
+  (IteratorSize.size it.toIterM).run.down
+
+@[always_inline, inline, inherit_doc IterM.Partial.size]
+def Iter.Partial.size {α : Type w} {β : Type w} [Iterator α Id β] [IteratorSizePartial α Id]
+    (it : Iter (α := α) β) : Nat :=
+  (IteratorSizePartial.size it.toIterM).run.down
+
 end Std.Iterators

src/Init/Data/Iterators/Consumers/Monadic.lean

@@ -0,0 +1,11 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Consumers.Monadic.Partial

src/Init/Data/Iterators/Consumers/Monadic/Collect.lean

@@ -3,9 +3,11 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Consumers.Monadic.Partial
-import Std.Data.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Consumers.Monadic.Partial
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
 
 /-!
 # Collectors

src/Init/Data/Iterators/Consumers/Monadic/Loop.lean

@@ -3,10 +3,12 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
 import Init.RCases
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Monadic.Partial
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Monadic.Partial
 
 /-!
 # Loop-based consumers
@@ -62,8 +64,9 @@ class IteratorLoop (α : Type w) (m : Type w → Type w') {β : Type w} [Iterato
   forIn : ∀ (_lift : (γ : Type w) → m γ → n γ) (γ : Type w),
       (plausible_forInStep : β → γ → ForInStep γ → Prop) →
       IteratorLoop.WellFounded α m plausible_forInStep →
-      IterM (α := α) m β → γ →
-      ((b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))) → n γ
+      (it : IterM (α := α) m β) → γ →
+      ((b : β) → it.IsPlausibleIndirectOutput b → (c : γ) → n (Subtype (plausible_forInStep b c))) →
+      n γ
 
 /--
 `IteratorLoopPartial α m` provides efficient implementations of loop-based consumers for `α`-based
@@ -76,7 +79,29 @@ provided by the standard library.
 class IteratorLoopPartial (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β]
     (n : Type w → Type w'') where
   forInPartial : ∀ (_lift : (γ : Type w) → m γ → n γ) {γ : Type w},
-      IterM (α := α) m β → γ → ((b : β) → (c : γ) → n (ForInStep γ)) → n γ
+      (it : IterM (α := α) m β) → γ →
+      ((b : β) → it.IsPlausibleIndirectOutput b → (c : γ) → n (ForInStep γ)) → n γ
+
+/--
+`IteratorSize α m` provides an implementation of the `IterM.size` function.
+
+This class is experimental and users of the iterator API should not explicitly depend on it.
+They can, however, assume that consumers that require an instance will work for all iterators
+provided by the standard library.
+-/
+class IteratorSize (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β] where
+  size : IterM (α := α) m β → m (ULift Nat)
+
+/--
+`IteratorSizePartial α m` provides an implementation of the `IterM.Partial.size` function that
+can be used as `it.allowTermination.size`.
+
+This class is experimental and users of the iterator API should not explicitly depend on it.
+They can, however, assume that consumers that require an instance will work for all iterators
+provided by the standard library.
+-/
+class IteratorSizePartial (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β] where
+  size : IterM (α := α) m β → m (ULift Nat)
 
 end Typeclasses
 
@@ -91,7 +116,7 @@ private def IteratorLoop.WFRel.mk {α : Type w} {m : Type w → Type w'} {β : T
     IteratorLoop.WFRel wf :=
   (it, c)
 
-instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
+private instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     {γ : Type x} {plausible_forInStep : β → γ → ForInStep γ → Prop}
     (wf : IteratorLoop.WellFounded α m plausible_forInStep) :
     WellFoundedRelation (IteratorLoop.WFRel wf) where
@@ -102,23 +127,27 @@ instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
 This is the loop implementation of the default instance `IteratorLoop.defaultImplementation`.
 -/
 @[specialize]
-def IterM.DefaultConsumers.forIn {m : Type w → Type w'} {α : Type w} {β : Type w}
+def IterM.DefaultConsumers.forIn' {m : Type w → Type w'} {α : Type w} {β : Type w}
     [Iterator α m β]
     {n : Type w → Type w''} [Monad n]
     (lift : ∀ γ, m γ → n γ) (γ : Type w)
     (plausible_forInStep : β → γ → ForInStep γ → Prop)
     (wf : IteratorLoop.WellFounded α m plausible_forInStep)
     (it : IterM (α := α) m β) (init : γ)
-    (f : (b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))) : n γ :=
+    (f : (b : β) → it.IsPlausibleIndirectOutput b → (c : γ) → n (Subtype (plausible_forInStep b c))) : n γ :=
   haveI : WellFounded _ := wf
   letI : MonadLift m n := ⟨fun {γ} => lift γ⟩
   do
     match ← it.step with
-    | .yield it' out _ =>
-      match ← f out init with
-      | ⟨.yield c, _⟩ => IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' c f
+    | .yield it' out h =>
+      match ← f out (.direct ⟨_, h⟩) init with
+      | ⟨.yield c, _⟩ =>
+        IterM.DefaultConsumers.forIn' lift _ plausible_forInStep wf it' c
+          (fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc)
       | ⟨.done c, _⟩ => return c
-    | .skip it' _ => IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' init f
+    | .skip it' h =>
+      IterM.DefaultConsumers.forIn' lift _ plausible_forInStep wf it' init
+        (fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc)
     | .done _ => return init
 termination_by IteratorLoop.WFRel.mk wf it init
 decreasing_by
@@ -134,7 +163,7 @@ implementations are possible and should be used instead.
 def IteratorLoop.defaultImplementation {α : Type w} {m : Type w → Type w'} {n : Type w → Type w''}
     [Monad n] [Iterator α m β] :
     IteratorLoop α m n where
-  forIn lift := IterM.DefaultConsumers.forIn lift
+  forIn lift := IterM.DefaultConsumers.forIn' lift
 
 /--
 Asserts that a given `IteratorLoop` instance is equal to `IteratorLoop.defaultImplementation`.
@@ -153,15 +182,19 @@ partial def IterM.DefaultConsumers.forInPartial {m : Type w → Type w'} {α : T
     {n : Type w → Type w''} [Monad n]
     (lift : ∀ γ, m γ → n γ) (γ : Type w)
     (it : IterM (α := α) m β) (init : γ)
-    (f : (b : β) → (c : γ) → n (ForInStep γ)) : n γ :=
+    (f : (b : β) → it.IsPlausibleIndirectOutput b → (c : γ) → n (ForInStep γ)) : n γ :=
   letI : MonadLift m n := ⟨fun {γ} => lift γ⟩
   do
     match ← it.step with
-    | .yield it' out _ =>
-      match ← f out init with
-      | .yield c => IterM.DefaultConsumers.forInPartial lift _ it' c f
+    | .yield it' out h =>
+      match ← f out (.direct ⟨_, h⟩) init with
+      | .yield c =>
+        IterM.DefaultConsumers.forInPartial lift _ it' c
+          fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc
       | .done c => return c
-    | .skip it' _ => IterM.DefaultConsumers.forInPartial lift _ it' init f
+    | .skip it' h =>
+      IterM.DefaultConsumers.forInPartial lift _ it' init
+        fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc
     | .done _ => return init
 
 /--
@@ -175,7 +208,7 @@ def IteratorLoopPartial.defaultImplementation {α : Type w} {m : Type w → Type
     IteratorLoopPartial α m n where
   forInPartial lift := IterM.DefaultConsumers.forInPartial lift _
 
-instance (α : Type w) (m : Type w → Type w') (n : Type w → Type w')
+instance (α : Type w) (m : Type w → Type w') (n : Type w → Type w'')
     [Monad m] [Monad n] [Iterator α m β] [Finite α m] :
     letI : IteratorLoop α m n := .defaultImplementation
     LawfulIteratorLoop α m n :=
@@ -196,27 +229,40 @@ theorem IteratorLoop.wellFounded_of_finite {m : Type w → Type w'}
     exact WellFoundedRelation.wf
 
 /--
-This `ForIn`-style loop construct traverses a finite iterator using an `IteratorLoop` instance.
+This `ForIn'`-style loop construct traverses a finite iterator using an `IteratorLoop` instance.
 -/
 @[always_inline, inline]
-def IteratorLoop.finiteForIn {m : Type w → Type w'} {n : Type w → Type w''}
+def IteratorLoop.finiteForIn' {m : Type w → Type w'} {n : Type w → Type w''}
     {α : Type w} {β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n]
     (lift : ∀ γ, m γ → n γ) :
-    ForIn n (IterM (α := α) m β) β where
-  forIn {γ} [Monad n] it init f :=
+    ForIn' n (IterM (α := α) m β) β ⟨fun it out => it.IsPlausibleIndirectOutput out⟩ where
+  forIn' {γ} [Monad n] it init f :=
     IteratorLoop.forIn (α := α) (m := m) lift γ (fun _ _ _ => True)
       wellFounded_of_finite
-      it init ((⟨·, .intro⟩) <$> f · ·)
+      it init (fun out h acc => (⟨·, .intro⟩) <$> f out h acc)
+
+/--
+A `ForIn'` instance for iterators. Its generic membership relation is not easy to use,
+so this is not marked as `instance`. This way, more convenient instances can be built on top of it
+or future library improvements will make it more comfortable.
+-/
+def IterM.instForIn' {m : Type w → Type w'} {n : Type w → Type w''}
+    {α : Type w} {β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n]
+    [MonadLiftT m n] :
+    ForIn' n (IterM (α := α) m β) β ⟨fun it out => it.IsPlausibleIndirectOutput out⟩ :=
+  IteratorLoop.finiteForIn' (fun _ => monadLift)
 
 instance {m : Type w → Type w'} {n : Type w → Type w''}
     {α : Type w} {β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n]
     [MonadLiftT m n] :
-    ForIn n (IterM (α := α) m β) β := IteratorLoop.finiteForIn (fun _ => monadLift)
+    ForIn n (IterM (α := α) m β) β :=
+  haveI : ForIn' n (IterM (α := α) m β) β _ := IterM.instForIn'
+  instForInOfForIn'
 
 instance {m : Type w → Type w'} {n : Type w → Type w''}
     {α : Type w} {β : Type w} [Iterator α m β] [IteratorLoopPartial α m n] [MonadLiftT m n] :
-    ForIn n (IterM.Partial (α := α) m β) β where
-  forIn it init f :=
+    ForIn' n (IterM.Partial (α := α) m β) β ⟨fun it out => it.it.IsPlausibleIndirectOutput out⟩ where
+  forIn' it init f :=
     IteratorLoopPartial.forInPartial (α := α) (m := m) (fun _ => monadLift) it.it init f
 
 instance {m : Type w → Type w'} {n : Type w → Type w''}
@@ -327,4 +373,86 @@ def IterM.Partial.drain {α : Type w} {m : Type w → Type w'} [Monad m] {β : T
     m PUnit :=
   it.fold (γ := PUnit) (fun _ _ => .unit) .unit
 
+section Size
+
+/--
+This is the implementation of the default instance `IteratorSize.defaultImplementation`.
+-/
+@[always_inline, inline]
+def IterM.DefaultConsumers.size {α : Type w} {m : Type w → Type w'} [Monad m] {β : Type w}
+    [Iterator α m β] [IteratorLoop α m m] [Finite α m] (it : IterM (α := α) m β) :
+    m (ULift Nat) :=
+  it.fold (init := .up 0) fun acc _ => .up (acc.down + 1)
+
+/--
+This is the implementation of the default instance `IteratorSizePartial.defaultImplementation`.
+-/
+@[always_inline, inline]
+def IterM.DefaultConsumers.sizePartial {α : Type w} {m : Type w → Type w'} [Monad m] {β : Type w}
+    [Iterator α m β] [IteratorLoopPartial α m m] (it : IterM (α := α) m β) :
+    m (ULift Nat) :=
+  it.allowNontermination.fold (init := .up 0) fun acc _ => .up (acc.down + 1)
+
+/--
+This is the default implementation of the `IteratorSize` class.
+It simply iterates using `IteratorLoop` and counts the elements.
+For certain iterators, more efficient implementations are possible and should be used instead.
+-/
+@[always_inline, inline]
+def IteratorSize.defaultImplementation {α β : Type w} {m : Type w → Type w'} [Monad m]
+    [Iterator α m β] [Finite α m] [IteratorLoop α m m] :
+    IteratorSize α m where
+  size := IterM.DefaultConsumers.size
+
+
+/--
+This is the default implementation of the `IteratorSizePartial` class.
+It simply iterates using `IteratorLoopPartial` and counts the elements.
+For certain iterators, more efficient implementations are possible and should be used instead.
+-/
+@[always_inline, inline]
+instance IteratorSizePartial.defaultImplementation {α β : Type w} {m : Type w → Type w'} [Monad m]
+    [Iterator α m β] [IteratorLoopPartial α m m] :
+    IteratorSizePartial α m where
+  size := IterM.DefaultConsumers.sizePartial
+
+/--
+Computes how many elements the iterator returns. In monadic situations, it is unclear which effects
+are caused by calling `size`, and if the monad is nondeterministic, it is also unclear what the
+returned value should be. The reference implementation, `IteratorSize.defaultImplementation`,
+simply iterates over the whole iterator monadically, counting the number of emitted values.
+An `IteratorSize` instance is considered lawful if it is equal to the reference implementation.
+
+**Performance**:
+
+Default performance is linear in the number of steps taken by the iterator.
+-/
+@[always_inline, inline]
+def IterM.size {α : Type} {m : Type → Type w'} {β : Type} [Iterator α m β] [Monad m]
+    (it : IterM (α := α) m β) [IteratorSize α m] : m Nat :=
+  ULift.down <$> IteratorSize.size it
+
+/--
+Computes how many elements the iterator emits.
+
+With monadic iterators (`IterM`), it is unclear which effects
+are caused by calling `size`, and if the monad is nondeterministic, it is also unclear what the
+returned value should be. The reference implementation, `IteratorSize.defaultImplementation`,
+simply iterates over the whole iterator monadically, counting the number of emitted values.
+An `IteratorSize` instance is considered lawful if it is equal to the reference implementation.
+
+This is the partial version of `size`. It does not require a proof of finiteness and might loop
+forever. It is not possible to verify the behavior in Lean because it uses `partial`.
+
+**Performance**:
+
+Default performance is linear in the number of steps taken by the iterator.
+-/
+@[always_inline, inline]
+def IterM.Partial.size {α : Type} {m : Type → Type w'} {β : Type} [Iterator α m β] [Monad m]
+    (it : IterM.Partial (α := α) m β) [IteratorSizePartial α m] : m Nat :=
+  ULift.down <$> IteratorSizePartial.size it.it
+
+end Size
+
 end Std.Iterators

src/Init/Data/Iterators/Consumers/Monadic/Partial.lean

@@ -3,8 +3,10 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 

src/Init/Data/Iterators/Consumers/Partial.lean

@@ -3,8 +3,10 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 

src/Init/Data/Iterators/Internal.lean

@@ -0,0 +1,10 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Internal.Termination

src/Init/Data/Iterators/Internal/LawfulMonadLiftFunction.lean

@@ -3,6 +3,8 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
 import Init.Control.Basic
 import Init.Control.Lawful.Basic
@@ -10,7 +12,7 @@ import Init.NotationExtra
 import Init.Control.Lawful.MonadLift
 
 /-!
-# Typeclass for lawfule monad lifting functions
+# Typeclass for lawful monad lifting functions
 
 This module provides a typeclass `LawfulMonadLiftFunction f` that asserts that a function `f`
 mapping values from one monad to another monad commutes with `pure` and `bind`. This equivalent to
@@ -60,9 +62,13 @@ theorem LawfulMonadLiftFunction.lift_seqRight [LawfulMonad m] [LawfulMonad n]
     lift (x *> y) = (lift x : n α) *> (lift y : n β) := by
   simp only [seqRight_eq, lift_map, lift_seq]
 
-def instMonadLiftOfFunction {lift : ⦃α : Type u⦄ -> m α → n α} :
-    MonadLift m n where
-  monadLift := lift (α := _)
+abbrev idToMonad [Monad m] ⦃α : Type u⦄ (x : Id α) : m α :=
+    pure x.run
+
+def LawfulMonadLiftFunction.idToMonad [Monad m] [LawfulMonad m] :
+    LawfulMonadLiftFunction (m := Id) (n := m) idToMonad where
+  lift_pure := by simp [Internal.idToMonad]
+  lift_bind := by simp [Internal.idToMonad]
 
 instance [LawfulMonadLiftFunction lift] :
     letI : MonadLift m n := ⟨lift (α := _)⟩

src/Init/Data/Iterators/Internal/Termination.lean

@@ -3,8 +3,10 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 /-!
 This is an internal module used by iterator implementations.

src/Init/Data/Iterators/Lemmas.lean

@@ -3,5 +3,7 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Lemmas.Consumers

src/Init/Data/Iterators/Lemmas/Basic.lean

@@ -3,8 +3,10 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 

src/Init/Data/Iterators/Lemmas/Consumers.lean

@@ -0,0 +1,11 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Collect
+import Init.Data.Iterators.Lemmas.Consumers.Loop

src/Init/Data/Iterators/Lemmas/Consumers/Collect.lean

@@ -0,0 +1,114 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Lemmas.Basic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import all Init.Data.Iterators.Consumers.Access
+import all Init.Data.Iterators.Consumers.Collect
+
+namespace Std.Iterators
+
+theorem Iter.toArray_eq_toArray_toIterM {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toArray = it.toIterM.toArray.run :=
+  (rfl)
+
+theorem Iter.toList_eq_toList_toIterM {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toList = it.toIterM.toList.run :=
+  (rfl)
+
+theorem Iter.toListRev_eq_toListRev_toIterM {α β} [Iterator α Id β] [Finite α Id]
+    {it : Iter (α := α) β} :
+    it.toListRev = it.toIterM.toListRev.run :=
+  (rfl)
+
+@[simp]
+theorem IterM.toList_toIter {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    {it : IterM (α := α) Id β} :
+    it.toIter.toList = it.toList.run :=
+  (rfl)
+
+@[simp]
+theorem IterM.toListRev_toIter {α β} [Iterator α Id β] [Finite α Id]
+    {it : IterM (α := α) Id β} :
+    it.toIter.toListRev = it.toListRev.run :=
+  (rfl)
+
+theorem Iter.toList_toArray {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toArray.toList = it.toList := by
+  simp [toArray_eq_toArray_toIterM, toList_eq_toList_toIterM, ← IterM.toList_toArray]
+
+theorem Iter.toArray_toList {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toList.toArray = it.toArray := by
+  simp [toArray_eq_toArray_toIterM, toList_eq_toList_toIterM, ← IterM.toArray_toList]
+
+@[simp]
+theorem Iter.reverse_toListRev [Iterator α Id β] [Finite α Id]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {it : Iter (α := α) β} :
+    it.toListRev.reverse = it.toList := by
+  simp [toListRev_eq_toListRev_toIterM, toList_eq_toList_toIterM, ← IterM.reverse_toListRev]
+
+theorem Iter.toListRev_eq {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toListRev = it.toList.reverse := by
+  simp [Iter.toListRev_eq_toListRev_toIterM, Iter.toList_eq_toList_toIterM, IterM.toListRev_eq]
+
+theorem Iter.toArray_eq_match_step {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toArray = match it.step with
+      | .yield it' out _ => #[out] ++ it'.toArray
+      | .skip it' _ => it'.toArray
+      | .done _ => #[] := by
+  simp only [Iter.toArray_eq_toArray_toIterM, Iter.step]
+  rw [IterM.toArray_eq_match_step, Id.run_bind]
+  generalize it.toIterM.step.run = step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem Iter.toList_eq_match_step {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toList = match it.step with
+      | .yield it' out _ => out :: it'.toList
+      | .skip it' _ => it'.toList
+      | .done _ => [] := by
+  rw [← Iter.toList_toArray, Iter.toArray_eq_match_step]
+  split <;> simp [Iter.toList_toArray]
+
+theorem Iter.toListRev_eq_match_step {α β} [Iterator α Id β] [Finite α Id] {it : Iter (α := α) β} :
+    it.toListRev = match it.step with
+      | .yield it' out _ => it'.toListRev ++ [out]
+      | .skip it' _ => it'.toListRev
+      | .done _ => [] := by
+  rw [Iter.toListRev_eq_toListRev_toIterM, IterM.toListRev_eq_match_step, Iter.step, Id.run_bind]
+  generalize it.toIterM.step.run = step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem Iter.getElem?_toList_eq_atIdxSlow? {α β}
+    [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {it : Iter (α := α) β} {k : Nat} :
+    it.toList[k]? = it.atIdxSlow? k := by
+  induction it using Iter.inductSteps generalizing k with | step it ihy ihs =>
+  rw [toList_eq_match_step, atIdxSlow?]
+  obtain ⟨step, h⟩ := it.step
+  cases step
+  · cases k <;> simp [ihy h]
+  · simp [ihs h]
+  · simp
+
+theorem Iter.toList_eq_of_atIdxSlow?_eq {α₁ α₂ β}
+    [Iterator α₁ Id β] [Finite α₁ Id] [IteratorCollect α₁ Id Id] [LawfulIteratorCollect α₁ Id Id]
+    [Iterator α₂ Id β] [Finite α₂ Id] [IteratorCollect α₂ Id Id] [LawfulIteratorCollect α₂ Id Id]
+    {it₁ : Iter (α := α₁) β} {it₂ : Iter (α := α₂) β}
+    (h : ∀ k, it₁.atIdxSlow? k = it₂.atIdxSlow? k) :
+    it₁.toList = it₂.toList := by
+  ext; simp [getElem?_toList_eq_atIdxSlow?, h]
+
+end Std.Iterators

src/Init/Data/Iterators/Lemmas/Consumers/Loop.lean

@@ -0,0 +1,285 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Lemmas.Consumers.Collect
+import all Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop
+import all Init.Data.Iterators.Consumers.Loop
+
+namespace Std.Iterators
+
+theorem Iter.forIn'_eq {α β : Type w} [Iterator α Id β] [Finite α Id]
+    {m : Type w → Type w''} [Monad m] [IteratorLoop α Id m] [hl : LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (b : β) → it.IsPlausibleIndirectOutput b → γ → m (ForInStep γ)} :
+    letI : ForIn' m (Iter (α := α) β) β _ := Iter.instForIn'
+    ForIn'.forIn' it init f =
+      IterM.DefaultConsumers.forIn' (fun _ c => pure c.run) γ (fun _ _ _ => True)
+        IteratorLoop.wellFounded_of_finite it.toIterM init
+          (fun out h acc => (⟨·, .intro⟩) <$>
+            f out (Iter.isPlausibleIndirectOutput_iff_isPlausibleIndirectOutput_toIterM.mpr h) acc) := by
+  cases hl.lawful; rfl
+
+theorem Iter.forIn_eq {α β : Type w} [Iterator α Id β] [Finite α Id]
+    {m : Type w → Type w''} [Monad m] [IteratorLoop α Id m] [hl : LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (b : β) → γ → m (ForInStep γ)} :
+    ForIn.forIn it init f =
+      IterM.DefaultConsumers.forIn' (fun _ c => pure c.run) γ (fun _ _ _ => True)
+        IteratorLoop.wellFounded_of_finite it.toIterM init
+          (fun out _ acc => (⟨·, .intro⟩) <$>
+            f out acc) := by
+  cases hl.lawful; rfl
+
+theorem Iter.forIn'_eq_forIn'_toIterM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (out : β) → _ → γ → m (ForInStep γ)} :
+    letI : ForIn' m (Iter (α := α) β) β _ := Iter.instForIn'
+    ForIn'.forIn' it init f =
+      letI : MonadLift Id m := ⟨Std.Internal.idToMonad (α := _)⟩
+      letI : ForIn' m (IterM (α := α) Id β) β _ := IterM.instForIn'
+      ForIn'.forIn' it.toIterM init
+        (fun out h acc => f out (isPlausibleIndirectOutput_iff_isPlausibleIndirectOutput_toIterM.mpr h) acc) := by
+  rfl
+
+theorem Iter.forIn_eq_forIn_toIterM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it init f =
+      letI : MonadLift Id m := ⟨Std.Internal.idToMonad (α := _)⟩
+      ForIn.forIn it.toIterM init f := by
+  rfl
+
+theorem Iter.forIn'_eq_match_step {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (out : β) → _ → γ → m (ForInStep γ)} :
+    letI : ForIn' m (Iter (α := α) β) β _ := Iter.instForIn'
+    ForIn'.forIn' it init f = (do
+      match it.step with
+      | .yield it' out h =>
+        match ← f out (.direct ⟨_, h⟩) init with
+        | .yield c =>
+          ForIn'.forIn' it' c
+            fun out h'' acc => f out (.indirect ⟨_, rfl, h⟩ h'') acc
+        | .done c => return c
+      | .skip it' h =>
+          ForIn'.forIn' it' init
+            fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc
+      | .done _ => return init) := by
+  rw [Iter.forIn'_eq_forIn'_toIterM, @IterM.forIn'_eq_match_step, Iter.step]
+  simp only [liftM, monadLift, pure_bind]
+  generalize it.toIterM.step = step
+  cases step using PlausibleIterStep.casesOn
+  · apply bind_congr
+    intro forInStep
+    rfl
+  · rfl
+  · rfl
+
+theorem Iter.forIn_eq_match_step {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it init f = (do
+      match it.step with
+      | .yield it' out _ =>
+        match ← f out init with
+        | .yield c => ForIn.forIn it' c f
+        | .done c => return c
+      | .skip it' _ => ForIn.forIn it' init f
+      | .done _ => return init) := by
+  rw [Iter.forIn_eq_forIn_toIterM, @IterM.forIn_eq_match_step, Iter.step]
+  simp only [liftM, monadLift, pure_bind]
+  generalize it.toIterM.step = step
+  cases step using PlausibleIterStep.casesOn
+  · apply bind_congr
+    intro forInStep
+    rfl
+  · rfl
+  · rfl
+
+private theorem Iter.forIn'_toList.aux {ρ : Type u} {α : Type v} {γ : Type w} {m : Type w → Type w'}
+    [Monad m] {_ : Membership α ρ} [ForIn' m ρ α inferInstance]
+    {r s : ρ} {init : γ} {f : (a : α) → _ → γ → m (ForInStep γ)} (h : r = s) :
+    forIn' r init f = forIn' s init (fun a h' acc => f a (h ▸ h') acc) := by
+  cases h; rfl
+
+theorem Iter.forIn'_toList {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    [LawfulPureIterator α]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (out : β) → _ → γ → m (ForInStep γ)} :
+    letI : ForIn' m (Iter (α := α) β) β _ := Iter.instForIn'
+    ForIn'.forIn' it.toList init f = ForIn'.forIn' it init (fun out h acc => f out (LawfulPureIterator.mem_toList_iff_isPlausibleIndirectOutput.mpr h) acc) := by
+  induction it using Iter.inductSteps generalizing init with case step it ihy ihs =>
+  have := it.toList_eq_match_step
+  generalize hs : it.step = step at this
+  rw [forIn'_toList.aux this]
+  rw [forIn'_eq_match_step]
+  rw [List.forIn'_eq_foldlM] at *
+  simp only [map_eq_pure_bind, List.foldlM_map, hs]
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    simp only [List.attach_cons, List.foldlM_cons, bind_pure_comp, map_bind]
+    apply bind_congr
+    intro forInStep
+    cases forInStep
+    · induction it'.toList.attach <;> simp [*]
+    · simp only [List.foldlM_map]
+      simp only [List.forIn'_eq_foldlM] at ihy
+      simp only at this
+      simp only [ihy h (f := fun out h acc => f out (by rw [this]; exact List.mem_cons_of_mem _ h) acc)]
+  · rename_i it' h
+    simp only [bind_pure_comp]
+    simp only [List.forIn'_eq_foldlM] at ihs
+    simp only at this
+    simp only [ihs h (f := fun out h acc => f out (this ▸ h) acc)]
+  · simp
+
+theorem Iter.forIn'_eq_forIn'_toList {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    [LawfulPureIterator α]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (out : β) → _ → γ → m (ForInStep γ)} :
+    letI : ForIn' m (Iter (α := α) β) β _ := Iter.instForIn'
+    ForIn'.forIn' it init f = ForIn'.forIn' it.toList init (fun out h acc => f out (LawfulPureIterator.mem_toList_iff_isPlausibleIndirectOutput.mp h) acc) := by
+  simp only [forIn'_toList]
+  congr
+
+theorem Iter.forIn_toList {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it.toList init f = ForIn.forIn it init f := by
+  rw [List.forIn_eq_foldlM]
+  induction it using Iter.inductSteps generalizing init with case step it ihy ihs =>
+  rw [forIn_eq_match_step, Iter.toList_eq_match_step]
+  simp only [map_eq_pure_bind]
+  generalize it.step = step
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    simp only [List.foldlM_cons, bind_pure_comp, map_bind]
+    apply bind_congr
+    intro forInStep
+    cases forInStep
+    · induction it'.toList <;> simp [*]
+    · simp only [ForIn.forIn, forIn', List.forIn'] at ihy
+      simp [ihy h, forIn_eq_forIn_toIterM]
+  · rename_i it' h
+    simp only [bind_pure_comp]
+    rw [ihs h]
+  · simp
+
+theorem Iter.foldM_eq_forIn {α β γ : Type w} [Iterator α Id β] [Finite α Id] {m : Type w → Type w'}
+    [Monad m] [IteratorLoop α Id m] {f : γ → β → m γ}
+    {init : γ} {it : Iter (α := α) β} :
+    it.foldM (init := init) f = ForIn.forIn it init (fun x acc => ForInStep.yield <$> f acc x) :=
+  (rfl)
+
+theorem Iter.foldM_eq_foldM_toIterM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ} {f : γ → β → m γ} :
+    it.foldM (init := init) f = letI : MonadLift Id m := ⟨pure⟩; it.toIterM.foldM (init := init) f :=
+  (rfl)
+
+theorem Iter.forIn_yield_eq_foldM {α β γ δ : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m] [IteratorLoop α Id m]
+    [LawfulIteratorLoop α Id m] {f : β → γ → m δ} {g : β → γ → δ → γ} {init : γ}
+    {it : Iter (α := α) β} :
+    ForIn.forIn it init (fun c b => (fun d => .yield (g c b d)) <$> f c b) =
+      it.foldM (fun b c => g c b <$> f c b) init := by
+  simp [Iter.foldM_eq_forIn]
+
+theorem Iter.foldM_eq_match_step {α β γ : Type w} [Iterator α Id β] [Finite α Id]
+    {m : Type w → Type w'} [Monad m] [LawfulMonad m] [IteratorLoop α Id m]
+    [LawfulIteratorLoop α Id m] {f : γ → β → m γ} {init : γ} {it : Iter (α := α) β} :
+    it.foldM (init := init) f = (do
+      match it.step with
+      | .yield it' out _ => it'.foldM (init := ← f init out) f
+      | .skip it' _ => it'.foldM (init := init) f
+      | .done _ => return init) := by
+  rw [Iter.foldM_eq_forIn, Iter.forIn_eq_match_step]
+  generalize it.step = step
+  cases step using PlausibleIterStep.casesOn <;> simp [foldM_eq_forIn]
+
+theorem Iter.foldlM_toList {α β γ : Type w} [Iterator α Id β] [Finite α Id] {m : Type w → Type w'}
+    [Monad m] [LawfulMonad m] [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {f : γ → β → m γ}
+    {init : γ} {it : Iter (α := α) β} :
+    it.toList.foldlM (init := init) f = it.foldM (init := init) f := by
+  rw [Iter.foldM_eq_forIn, ← Iter.forIn_toList]
+  simp only [List.forIn_yield_eq_foldlM, id_map']
+
+theorem IterM.forIn_eq_foldM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    forIn it init f = ForInStep.value <$>
+      it.foldM (fun c b => match c with
+        | .yield c => f b c
+        | .done c => pure (.done c)) (ForInStep.yield init) := by
+  simp only [← Iter.forIn_toList, List.forIn_eq_foldlM, ← Iter.foldlM_toList]; rfl
+
+theorem Iter.fold_eq_forIn {α β γ : Type w} [Iterator α Id β]
+    [Finite α Id] [IteratorLoop α Id Id] {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
+    it.fold (init := init) f =
+      (ForIn.forIn (m := Id) it init (fun x acc => pure (ForInStep.yield (f acc x)))).run := by
+  rfl
+
+theorem Iter.fold_eq_foldM {α β γ : Type w} [Iterator α Id β]
+    [Finite α Id] [IteratorLoop α Id Id] {f : γ → β → γ} {init : γ}
+    {it : Iter (α := α) β} :
+    it.fold (init := init) f = (it.foldM (m := Id) (init := init) (pure <| f · ·)).run := by
+  simp [foldM_eq_forIn, fold_eq_forIn]
+
+@[simp]
+theorem Iter.forIn_pure_yield_eq_fold {α β γ : Type w} [Iterator α Id β]
+    [Finite α Id] [IteratorLoop α Id Id]
+    [LawfulIteratorLoop α Id Id] {f : β → γ → γ} {init : γ}
+    {it : Iter (α := α) β} :
+    ForIn.forIn (m := Id) it init (fun c b => pure (.yield (f c b))) =
+      pure (it.fold (fun b c => f c b) init) := by
+  simp only [fold_eq_forIn]
+  rfl
+
+theorem Iter.fold_eq_match_step {α β γ : Type w} [Iterator α Id β] [Finite α Id]
+    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
+    {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
+    it.fold (init := init) f = (match it.step with
+      | .yield it' out _ => it'.fold (init := f init out) f
+      | .skip it' _ => it'.fold (init := init) f
+      | .done _ => init) := by
+  rw [fold_eq_foldM, foldM_eq_match_step]
+  simp only [fold_eq_foldM]
+  generalize it.step = step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem Iter.foldl_toList {α β γ : Type w} [Iterator α Id β] [Finite α Id]
+    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
+    it.toList.foldl (init := init) f = it.fold (init := init) f := by
+  rw [fold_eq_foldM, List.foldl_eq_foldlM, ← Iter.foldlM_toList]
+
+end Std.Iterators

src/Init/Data/Iterators/Lemmas/Consumers/Monadic.lean

@@ -3,7 +3,8 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Consumers.Monadic.Partial
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop

src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean

@@ -0,0 +1,157 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Array.Lemmas
+import Init.Data.Iterators.Lemmas.Monadic.Basic
+import all Init.Data.Iterators.Consumers.Monadic.Collect
+
+namespace Std.Iterators
+
+variable {α β γ : Type w} {m : Type w → Type w'} {n : Type w → Type w''}
+  {lift : ⦃δ : Type w⦄ → m δ → n δ} {f : β → n γ} {it : IterM (α := α) m β}
+
+theorem IterM.DefaultConsumers.toArrayMapped.go.aux₁ [Monad n] [LawfulMonad n] [Iterator α m β]
+    [Finite α m] {b : γ} {bs : Array γ} :
+    IterM.DefaultConsumers.toArrayMapped.go lift f it (#[b] ++ bs) (m := m) =
+      (#[b] ++ ·) <$> IterM.DefaultConsumers.toArrayMapped.go lift f it bs (m := m) := by
+  induction it, bs using IterM.DefaultConsumers.toArrayMapped.go.induct
+  next it bs ih₁ ih₂ =>
+  rw [go, map_eq_pure_bind, go, bind_assoc]
+  apply bind_congr
+  intro step
+  split
+  · simp [ih₁ _ _ ‹_›]
+  · simp [ih₂ _ ‹_›]
+  · simp
+
+theorem IterM.DefaultConsumers.toArrayMapped.go.aux₂ [Monad n] [LawfulMonad n] [Iterator α m β]
+    [Finite α m] {acc : Array γ} :
+    IterM.DefaultConsumers.toArrayMapped.go lift f it acc (m := m) =
+      (acc ++ ·) <$> IterM.DefaultConsumers.toArrayMapped lift f it (m := m) := by
+  rw [← Array.toArray_toList (xs := acc)]
+  generalize acc.toList = acc
+  induction acc with
+  | nil => simp [toArrayMapped]
+  | cons x xs ih =>
+    rw [List.toArray_cons, IterM.DefaultConsumers.toArrayMapped.go.aux₁, ih]
+    simp only [Functor.map_map, Array.append_assoc]
+
+theorem IterM.DefaultConsumers.toArrayMapped_eq_match_step [Monad n] [LawfulMonad n]
+    [Iterator α m β] [Finite α m] :
+    IterM.DefaultConsumers.toArrayMapped lift f it (m := m) = letI : MonadLift m n := ⟨lift (δ := _)⟩; (do
+      match ← it.step with
+      | .yield it' out _ =>
+        return #[← f out] ++ (← IterM.DefaultConsumers.toArrayMapped lift f it' (m := m))
+      | .skip it' _ => IterM.DefaultConsumers.toArrayMapped lift f it' (m := m)
+      | .done _ => return #[]) := by
+  rw [IterM.DefaultConsumers.toArrayMapped, IterM.DefaultConsumers.toArrayMapped.go]
+  apply bind_congr
+  intro step
+  split <;> simp [IterM.DefaultConsumers.toArrayMapped.go.aux₂]
+
+theorem IterM.toArray_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m] :
+    it.toArray = (do
+      match ← it.step with
+      | .yield it' out _ => return #[out] ++ (← it'.toArray)
+      | .skip it' _ => it'.toArray
+      | .done _ => return #[]) := by
+  simp only [IterM.toArray, LawfulIteratorCollect.toArrayMapped_eq]
+  rw [IterM.DefaultConsumers.toArrayMapped_eq_match_step]
+  simp [bind_pure_comp, pure_bind, toArray]
+
+theorem IterM.toList_toArray [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    Array.toList <$> it.toArray = it.toList := by
+  simp [IterM.toList]
+
+theorem IterM.toArray_toList [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] {it : IterM (α := α) m β} :
+    List.toArray <$> it.toList = it.toArray := by
+  simp [IterM.toList]
+
+theorem IterM.toList_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m] {it : IterM (α := α) m β} :
+    it.toList = (do
+      match ← it.step with
+      | .yield it' out _ => return out :: (← it'.toList)
+      | .skip it' _ => it'.toList
+      | .done _ => return []) := by
+  simp [← IterM.toList_toArray]
+  rw [IterM.toArray_eq_match_step, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  split <;> simp
+
+theorem IterM.toListRev.go.aux₁ [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    {it : IterM (α := α) m β} {b : β} {bs : List β} :
+    IterM.toListRev.go it (bs ++ [b]) = (· ++ [b]) <$> IterM.toListRev.go it bs:= by
+  induction it, bs using IterM.toListRev.go.induct
+  next it bs ih₁ ih₂ =>
+  rw [go, go, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  simp only [List.cons_append] at ih₁
+  split <;> simp [*]
+
+theorem IterM.toListRev.go.aux₂ [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    {it : IterM (α := α) m β} {acc : List β} :
+    IterM.toListRev.go it acc = (· ++ acc) <$> it.toListRev := by
+  rw [← List.reverse_reverse (as := acc)]
+  generalize acc.reverse = acc
+  induction acc with
+  | nil => simp [toListRev]
+  | cons x xs ih => simp [IterM.toListRev.go.aux₁, ih]
+
+theorem IterM.toListRev_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    {it : IterM (α := α) m β} :
+    it.toListRev = (do
+      match ← it.step with
+      | .yield it' out _ => return (← it'.toListRev) ++ [out]
+      | .skip it' _ => it'.toListRev
+      | .done _ => return []) := by
+  simp [IterM.toListRev]
+  rw [toListRev.go]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp [IterM.toListRev.go.aux₂]
+
+theorem IterM.reverse_toListRev [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    List.reverse <$> it.toListRev = it.toList := by
+  apply Eq.symm
+  induction it using IterM.inductSteps
+  rename_i it ihy ihs
+  rw [toListRev_eq_match_step, toList_eq_match_step, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  split <;> simp (discharger := assumption) [ihy, ihs]
+
+theorem IterM.toListRev_eq [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toListRev = List.reverse <$> it.toList := by
+  rw [← IterM.reverse_toListRev]
+  simp
+
+theorem LawfulIteratorCollect.toArray_eq {α β : Type w} {m : Type w → Type w'}
+    [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
+    [hl : LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toArray = (letI : IteratorCollect α m m := .defaultImplementation; it.toArray) := by
+  simp only [IterM.toArray, toArrayMapped_eq]
+
+theorem LawfulIteratorCollect.toList_eq {α β : Type w} {m : Type w → Type w'}
+    [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
+    [hl : LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toList = (letI : IteratorCollect α m m := .defaultImplementation; it.toList) := by
+  simp [IterM.toList, toArray_eq]
+
+end Std.Iterators

src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean

@@ -0,0 +1,277 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import all Init.Data.Iterators.Consumers.Monadic.Loop
+
+namespace Std.Iterators
+
+theorem IterM.DefaultConsumers.forIn'_eq_match_step {α β : Type w} {m : Type w → Type w'}
+    [Iterator α m β]
+    {n : Type w → Type w''} [Monad n]
+    {lift : ∀ γ, m γ → n γ} {γ : Type w}
+    {plausible_forInStep : β → γ → ForInStep γ → Prop}
+    {wf : IteratorLoop.WellFounded α m plausible_forInStep}
+    {it : IterM (α := α) m β} {init : γ}
+    {f : (b : β) → it.IsPlausibleIndirectOutput b → (c : γ) → n (Subtype (plausible_forInStep b c))} :
+    IterM.DefaultConsumers.forIn' lift γ plausible_forInStep wf it init f = (do
+      match ← lift _ it.step with
+      | .yield it' out h =>
+        match ← f out (.direct ⟨_, h⟩) init with
+        | ⟨.yield c, _⟩ =>
+          IterM.DefaultConsumers.forIn' lift _ plausible_forInStep wf it' c
+            fun out h'' acc => f out (.indirect ⟨_, rfl, h⟩ h'') acc
+        | ⟨.done c, _⟩ => return c
+      | .skip it' h =>
+        IterM.DefaultConsumers.forIn' lift _ plausible_forInStep wf it' init
+          fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc
+      | .done _ => return init) := by
+  rw [forIn']
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> rfl
+
+theorem IterM.forIn'_eq {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [hl : LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
+    {f : (b : β) → it.IsPlausibleIndirectOutput b → γ → n (ForInStep γ)} :
+    letI : ForIn' n (IterM (α := α) m β) β _ := IterM.instForIn'
+    ForIn'.forIn' it init f = IterM.DefaultConsumers.forIn' (fun _ => monadLift) γ (fun _ _ _ => True)
+        IteratorLoop.wellFounded_of_finite it init ((⟨·, .intro⟩) <$> f · · ·) := by
+  cases hl.lawful; rfl
+
+theorem IterM.forIn_eq {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [hl : LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
+    {f : β → γ → n (ForInStep γ)} :
+    ForIn.forIn it init f = IterM.DefaultConsumers.forIn' (fun _ => monadLift) γ (fun _ _ _ => True)
+        IteratorLoop.wellFounded_of_finite it init (fun out _ acc => (⟨·, .intro⟩) <$> f out acc) := by
+  cases hl.lawful; rfl
+
+theorem IterM.forIn'_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
+    {f : (out : β) → _ → γ → n (ForInStep γ)} :
+    letI : ForIn' n (IterM (α := α) m β) β _ := IterM.instForIn'
+    ForIn'.forIn' it init f = (do
+      match ← it.step with
+      | .yield it' out h =>
+        match ← f out (.direct ⟨_, h⟩) init with
+        | .yield c =>
+          ForIn'.forIn' it' c
+            fun out h'' acc => f out (.indirect ⟨_, rfl, h⟩ h'') acc
+        | .done c => return c
+      | .skip it' h =>
+        ForIn'.forIn' it' init
+          fun out h' acc => f out (.indirect ⟨_, rfl, h⟩ h') acc
+      | .done _ => return init) := by
+  rw [IterM.forIn'_eq, DefaultConsumers.forIn'_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · simp only [map_eq_pure_bind, bind_assoc]
+    apply bind_congr
+    intro forInStep
+    cases forInStep <;> simp [IterM.forIn'_eq]
+  · simp [IterM.forIn'_eq]
+  · simp
+
+theorem IterM.forIn_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
+    {f : β → γ → n (ForInStep γ)} :
+    ForIn.forIn it init f = (do
+      match ← it.step with
+      | .yield it' out _ =>
+        match ← f out init with
+        | .yield c => ForIn.forIn it' c f
+        | .done c => return c
+      | .skip it' _ => ForIn.forIn it' init f
+      | .done _ => return init) := by
+  rw [IterM.forIn_eq, DefaultConsumers.forIn'_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · simp only [map_eq_pure_bind, bind_assoc]
+    apply bind_congr
+    intro forInStep
+    cases forInStep <;> simp [IterM.forIn_eq]
+  · simp [IterM.forIn_eq]
+  · simp
+
+theorem IterM.forM_eq_forIn {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {it : IterM (α := α) m β}
+    {f : β → n PUnit} :
+    ForM.forM it f = ForIn.forIn it PUnit.unit (fun out _ => do f out; return .yield .unit) :=
+  rfl
+
+theorem IterM.forM_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {it : IterM (α := α) m β}
+    {f : β → n PUnit} :
+    ForM.forM it f = (do
+      match ← it.step with
+      | .yield it' out _ =>
+        f out
+        ForM.forM it' f
+      | .skip it' _ => ForM.forM it' f
+      | .done _ => return) := by
+  rw [forM_eq_forIn, forIn_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp [forM_eq_forIn]
+
+theorem IterM.foldM_eq_forIn {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [MonadLiftT m n] {f : γ → β → n γ}
+    {init : γ} {it : IterM (α := α) m β} :
+    it.foldM (init := init) f = ForIn.forIn it init (fun x acc => ForInStep.yield <$> f acc x) :=
+  (rfl)
+
+theorem IterM.forIn_yield_eq_foldM {α β γ δ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n] [IteratorLoop α m n]
+    [LawfulIteratorLoop α m n] [MonadLiftT m n] {f : β → γ → n δ} {g : β → γ → δ → γ} {init : γ}
+    {it : IterM (α := α) m β} :
+    ForIn.forIn it init (fun c b => (fun d => .yield (g c b d)) <$> f c b) =
+      it.foldM (fun b c => g c b <$> f c b) init := by
+  simp [IterM.foldM_eq_forIn]
+
+theorem IterM.foldM_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [LawfulMonad n] [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {f : γ → β → n γ} {init : γ} {it : IterM (α := α) m β} :
+    it.foldM (init := init) f = (do
+      match ← it.step with
+      | .yield it' out _ => it'.foldM (init := ← f init out) f
+      | .skip it' _ => it'.foldM (init := init) f
+      | .done _ => return init) := by
+  rw [IterM.foldM_eq_forIn, IterM.forIn_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp [foldM_eq_forIn]
+
+theorem IterM.fold_eq_forIn {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m]
+    [IteratorLoop α m m] {f : γ → β → γ} {init : γ} {it : IterM (α := α) m β} :
+    it.fold (init := init) f =
+      ForIn.forIn (m := m) it init (fun x acc => pure (ForInStep.yield (f acc x))) := by
+  rfl
+
+theorem IterM.fold_eq_foldM {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] {f : γ → β → γ} {init : γ}
+    {it : IterM (α := α) m β} :
+    it.fold (init := init) f = it.foldM (init := init) (pure <| f · ·) := by
+  simp [foldM_eq_forIn, fold_eq_forIn]
+
+@[simp]
+theorem IterM.forIn_pure_yield_eq_fold {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m]
+    [LawfulIteratorLoop α m m] {f : β → γ → γ} {init : γ}
+    {it : IterM (α := α) m β} :
+    ForIn.forIn it init (fun c b => pure (.yield (f c b))) =
+      it.fold (fun b c => f c b) init := by
+  simp [IterM.fold_eq_forIn]
+
+theorem IterM.fold_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    {f : γ → β → γ} {init : γ} {it : IterM (α := α) m β} :
+    it.fold (init := init) f = (do
+      match ← it.step with
+      | .yield it' out _ => it'.fold (init := f init out) f
+      | .skip it' _ => it'.fold (init := init) f
+      | .done _ => return init) := by
+  rw [fold_eq_foldM, foldM_eq_match_step]
+  simp only [fold_eq_foldM]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem IterM.toList_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toList = it.fold (init := []) (fun l out => l ++ [out]) := by
+  suffices h : ∀ l' : List β, (l' ++ ·) <$> it.toList =
+      it.fold (init := l') (fun l out => l ++ [out]) by
+    specialize h []
+    simpa using h
+  induction it using IterM.inductSteps with | step it ihy ihs =>
+  intro l'
+  rw [IterM.toList_eq_match_step, IterM.fold_eq_match_step]
+  simp only [map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    specialize ihy h (l' ++ [out])
+    simpa using ihy
+  · rename_i it' h
+    simp [ihs h]
+  · simp
+
+theorem IterM.drain_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
+    it.drain = it.fold (init := PUnit.unit) (fun _ _ => .unit) :=
+  (rfl)
+
+theorem IterM.drain_eq_foldM {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [LawfulMonad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
+    it.drain = it.foldM (init := PUnit.unit) (fun _ _ => pure .unit) := by
+  simp [IterM.drain_eq_fold, IterM.fold_eq_foldM]
+
+theorem IterM.drain_eq_forIn {α β : Type w} {m : Type w → Type w'}  [Iterator α m β] [Finite α m]
+    [Monad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
+    it.drain = ForIn.forIn (m := m) it PUnit.unit (fun _ _ => pure (ForInStep.yield .unit)) := by
+  simp [IterM.drain_eq_fold, IterM.fold_eq_forIn]
+
+theorem IterM.drain_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (do
+      match ← it.step with
+      | .yield it' _ _ => it'.drain
+      | .skip it' _ => it'.drain
+      | .done _ => return .unit) := by
+  rw [IterM.drain_eq_fold, IterM.fold_eq_match_step]
+  simp [IterM.drain_eq_fold]
+
+theorem IterM.drain_eq_map_toList {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (fun _ => .unit) <$> it.toList := by
+  induction it using IterM.inductSteps with | step it ihy ihs =>
+  rw [IterM.drain_eq_match_step, IterM.toList_eq_match_step]
+  simp only [map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    simp [ihy h]
+  · rename_i it' h
+    simp [ihs h]
+  · simp
+
+theorem IterM.drain_eq_map_toListRev {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (fun _ => .unit) <$> it.toListRev := by
+  simp [IterM.drain_eq_map_toList, IterM.toListRev_eq]
+
+theorem IterM.drain_eq_map_toArray {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (fun _ => .unit) <$> it.toList := by
+  simp [IterM.drain_eq_map_toList]
+
+end Std.Iterators

src/Init/Data/Iterators/Lemmas/Monadic/Basic.lean

@@ -3,8 +3,10 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 

src/Init/Data/Iterators/PostconditionMonad.lean

@@ -3,6 +3,8 @@ Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
+module
+
 prelude
 import Init.Control.Lawful.Basic
 import Init.Data.Subtype
@@ -48,6 +50,11 @@ def PostconditionT.lift {α : Type w} {m : Type w → Type w'} [Functor m] (x :
     PostconditionT m α :=
   ⟨fun _ => True, (⟨·, .intro⟩) <$> x⟩
 
+@[always_inline, inline]
+protected def PostconditionT.pure {m : Type w → Type w'} [Pure m] {α : Type w}
+    (a : α) : PostconditionT m α :=
+  ⟨fun y => a = y, pure <| ⟨a, rfl⟩⟩
+
 /--
 Lifts a monadic value from `m { a : α // P a }` to a value `PostconditionT m α`.
 -/
@@ -107,52 +114,93 @@ def PostconditionT.run {m : Type w → Type w'} [Monad m] {α : Type w} (x : Pos
   (fun a => a.val) <$> x.operation
 
 instance {m : Type w → Type w'} [Functor m] : Functor (PostconditionT m) where
-  map f x := ⟨fun b => ∃ a, f a = b, (fun b => ⟨f b.1, b.1, rfl⟩) <$> x.operation ⟩
+  map := PostconditionT.map
 
 instance {m : Type w → Type w'} [Monad m] : Monad (PostconditionT m) where
-  pure x := ⟨fun y => x = y, pure <| ⟨x, rfl⟩⟩
-  bind x f := ⟨fun b => ∃ a, (f a).Property b,
-      x.operation >>= fun a => (fun b => ⟨b.val, a, b.property⟩) <$> (f a).operation⟩
+  pure := PostconditionT.pure
+  bind := PostconditionT.bind
 
-@[simp]
-theorem PostconditionT.computation_pure {m : Type w → Type w'} [Monad m] {α : Type w}
-    {x : α} :
-    (pure x : PostconditionT m α).operation = pure ⟨x, rfl⟩ :=
+theorem PostconditionT.pure_eq_pure {m : Type w → Type w'} [Monad m] {α} {a : α} :
+    pure a = PostconditionT.pure (m := m) a :=
   rfl
 
 @[simp]
 theorem PostconditionT.property_pure {m : Type w → Type w'} [Monad m] {α : Type w}
     {x : α} :
-    (pure x : PostconditionT m α).Property = (x = ·) :=
+    (pure x : PostconditionT m α).Property = (x = ·) := by
   rfl
 
-private theorem congrArg₂ {α : Sort u} {β : α → Type v} {γ : (a : α) → (b : β a) → Sort w}
-    (f : (a : α) → (b : β a) → γ a b)
-    {α α' a a'} (h : α = α') (h' : HEq a a') : HEq (f α a) (f α' a') := by
-  cases h; cases h'; rfl
+@[simp]
+theorem PostconditionT.operation_pure {m : Type w → Type w'} [Monad m] {α : Type w}
+    {x : α} :
+    (pure x : PostconditionT m α).operation = pure ⟨x, property_pure (m := m) ▸ rfl⟩ := by
+  rfl
 
-private theorem congrArg₄ {α : Sort u} {β : (a : α) → Sort v} {γ : (a : α) → (b : β a) → Sort w}
-    {δ : (a : α) → (b : β a) → (c : γ a b) → Sort x} {ε : (a : α) → (b : β a) → (c : γ a b) →
-      (d : δ a b c) → Sort y}
-    (f : (a : α) → (b : β a) → (c : γ a b) → (d : δ a b c) → ε a b c d)
-    {a a' b b' c c' d d'} (h₁ : a = a') (h₂ : HEq b b') (h₃ : HEq c c') (h₄ : HEq d d') :
-    HEq (f a b c d) (f a' b' c' d') := by
-  cases h₁; cases h₂; cases h₃; cases h₄; rfl
+theorem PostconditionT.ext {m : Type w → Type w'} [Monad m] [LawfulMonad m]
+    {α : Type w} {x y : PostconditionT m α}
+    (h : x.Property = y.Property) (h' : (fun p => ⟨p.1, h ▸ p.2⟩) <$> x.operation = y.operation) :
+    x = y := by
+  cases x; cases y; cases h
+  simpa using h'
+
+theorem PostconditionT.ext_iff {m : Type w → Type w'} [Monad m] [LawfulMonad m]
+    {α : Type w} {x y : PostconditionT m α} :
+    x = y ↔ ∃ h : x.Property = y.Property, (fun p => ⟨p.1, h ▸ p.2⟩) <$> x.operation = y.operation := by
+  constructor
+  · rintro rfl
+    exact ⟨rfl, by simp⟩
+  · rintro ⟨h, h'⟩
+    exact PostconditionT.ext h h'
+
+@[simp]
+protected theorem PostconditionT.map_eq_pure_bind {m : Type w → Type w'} [Monad m] [LawfulMonad m]
+    {α : Type w} {β : Type w} {f : α → β} {x : PostconditionT m α} :
+    x.map f = x.bind (pure ∘ f) := by
+  apply PostconditionT.ext <;> simp [PostconditionT.map, PostconditionT.bind]
+
+@[simp]
+protected theorem PostconditionT.pure_bind {m : Type w → Type w'} [Monad m] [LawfulMonad m]
+    {α : Type w} {β : Type w} {f : α → PostconditionT m β} {a : α} :
+    (pure a : PostconditionT m α).bind f = f a := by
+  apply PostconditionT.ext <;> simp [pure, PostconditionT.pure, PostconditionT.bind]
+
+@[simp]
+protected theorem PostconditionT.bind_pure {m : Type w → Type w'} [Monad m] [LawfulMonad m]
+    {α : Type w} {x : PostconditionT m α} :
+    x.bind pure = x := by
+  apply PostconditionT.ext <;> simp [pure, PostconditionT.pure, PostconditionT.bind]
+
+@[simp]
+protected theorem PostconditionT.bind_assoc {m : Type w → Type w'} [Monad m] [LawfulMonad m]
+    {α β γ: Type w} {x : PostconditionT m α} {f : α → PostconditionT m β} {g : β → PostconditionT m γ} :
+    (x.bind f).bind g = x.bind (fun a => (f a).bind g) := by
+  apply PostconditionT.ext
+  · simp [PostconditionT.bind]
+  · simp only [PostconditionT.bind, bind_assoc, bind_map_left, map_bind, Functor.map_map]
+    ext c
+    constructor
+    · rintro ⟨b, ⟨a, ha, hb⟩, h⟩
+      exact ⟨a, ha, b, hb, h⟩
+    · rintro ⟨a, ha, b, hb, h⟩
+      exact ⟨b, ⟨a, ha, hb⟩, h⟩
 
 @[simp]
 protected theorem PostconditionT.map_pure {m : Type w → Type w'} [Monad m] [LawfulMonad m]
     {α : Type w} {β : Type w} {f : α → β} {a : α} :
     (pure a : PostconditionT m α).map f = pure (f a) := by
-  simp only [PostconditionT.map, pure, map_pure, mk.injEq, Subtype.exists, exists_prop,
-    exists_eq_left', true_and]
-  apply congrArg₂ (f := fun α (a : α) => (pure a : m _)) (by simp)
-  apply congrArg₂ (f := fun α (a : α) => a)
-  · simp
-  · apply congrArg₄ fun β (p : β → Prop) (x : β) (h : p x) => Subtype.mk x h
-    · rfl
-    · simp
-    · rfl
-    · simp
+  apply PostconditionT.ext <;> simp [pure, PostconditionT.map, PostconditionT.pure]
+
+instance [Monad m] [LawfulMonad m] : LawfulMonad (PostconditionT m) where
+  map_const {α β} := by ext a x; simp [Functor.mapConst, Functor.map]
+  id_map {α} x := by simp [Functor.map]
+  comp_map {α β γ} g h := by intro x; simp [Functor.map]; rfl
+  seqLeft_eq {α β} x y := by simp [SeqLeft.seqLeft, Functor.map, Seq.seq]; rfl
+  seqRight_eq {α β} x y := by simp [Seq.seq, SeqRight.seqRight, Functor.map]
+  pure_seq g x := by simp [Seq.seq, Functor.map]
+  bind_pure_comp f x := by simp [Functor.map, Bind.bind]; rfl
+  bind_map f x := by simp [Seq.seq, Functor.map]; rfl
+  pure_bind x f := PostconditionT.pure_bind
+  bind_assoc x f g := PostconditionT.bind_assoc
 
 @[simp]
 theorem PostconditionT.property_map {m : Type w → Type w'} [Functor m] {α : Type w} {β : Type w}
@@ -168,12 +216,19 @@ theorem PostconditionT.property_map {m : Type w → Type w'} [Functor m] {α : T
 @[simp]
 theorem PostconditionT.operation_map {m : Type w → Type w'} [Functor m] {α : Type w} {β : Type w}
     {x : PostconditionT m α} {f : α → β} :
-    (x.map f).operation = (fun a => ⟨_, a, rfl⟩) <$> x.operation :=
+    (x.map f).operation =
+      (fun a => ⟨_, (property_map (m := m)).mpr ⟨a.1, rfl, a.2⟩⟩) <$> x.operation := by
   rfl
 
 @[simp]
-theorem PostconditionT.operation_lift {m : Type w →Type w'} [Functor m] {α : Type w}
-    {x : m α} : (lift x : PostconditionT m α).operation = (⟨·, True.intro⟩) <$> x :=
+theorem PostconditionT.property_lift {m : Type w → Type w'} [Functor m] {α : Type w}
+    {x : m α} : (lift x : PostconditionT m α).Property = (fun _ => True) := by
+  rfl
+
+@[simp]
+theorem PostconditionT.operation_lift {m : Type w → Type w'} [Functor m] {α : Type w}
+    {x : m α} : (lift x : PostconditionT m α).operation =
+      (⟨·, property_lift (m := m) ▸ True.intro⟩) <$> x := by
   rfl
 
 end Std.Iterators

src/Init/Data/List/Attach.lean

@@ -69,14 +69,14 @@ well-founded recursion mechanism to prove that the function terminates.
   | cons _ l', hL' => congrArg _ <| go l' fun _ hx => hL' (.tail _ hx)
   exact go l h'
 
-@[simp] theorem pmap_nil {P : α → Prop} {f : ∀ a, P a → β} : pmap f [] (by simp) = [] := rfl
+@[simp, grind =] theorem pmap_nil {P : α → Prop} {f : ∀ a, P a → β} : pmap f [] (by simp) = [] := rfl
 
-@[simp] theorem pmap_cons {P : α → Prop} {f : ∀ a, P a → β} {a : α} {l : List α} (h : ∀ b ∈ a :: l, P b) :
+@[simp, grind =] theorem pmap_cons {P : α → Prop} {f : ∀ a, P a → β} {a : α} {l : List α} (h : ∀ b ∈ a :: l, P b) :
     pmap f (a :: l) h = f a (forall_mem_cons.1 h).1 :: pmap f l (forall_mem_cons.1 h).2 := rfl
 
-@[simp] theorem attach_nil : ([] : List α).attach = [] := rfl
+@[simp, grind =] theorem attach_nil : ([] : List α).attach = [] := rfl
 
-@[simp] theorem attachWith_nil : ([] : List α).attachWith P H = [] := rfl
+@[simp, grind =] theorem attachWith_nil : ([] : List α).attachWith P H = [] := rfl
 
 @[simp]
 theorem pmap_eq_map {p : α → Prop} {f : α → β} {l : List α} (H) :
@@ -92,12 +92,14 @@ theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a,
   | cons x l ih =>
     rw [pmap, pmap, h _ mem_cons_self, ih fun a ha => h a (mem_cons_of_mem _ ha)]
 
+@[grind =]
 theorem map_pmap {p : α → Prop} {g : β → γ} {f : ∀ a, p a → β} {l : List α} (H) :
     map g (pmap f l H) = pmap (fun a h => g (f a h)) l H := by
   induction l
   · rfl
   · simp only [*, pmap, map]
 
+@[grind =]
 theorem pmap_map {p : β → Prop} {g : ∀ b, p b → γ} {f : α → β} {l : List α} (H) :
     pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun _ h => H _ (mem_map_of_mem h) := by
   induction l
@@ -114,7 +116,7 @@ theorem attachWith_congr {l₁ l₂ : List α} (w : l₁ = l₂) {P : α → Pro
   subst w
   simp
 
-@[simp] theorem attach_cons {x : α} {xs : List α} :
+@[simp, grind =] theorem attach_cons {x : α} {xs : List α} :
     (x :: xs).attach =
       ⟨x, mem_cons_self⟩ :: xs.attach.map fun ⟨y, h⟩ => ⟨y, mem_cons_of_mem x h⟩ := by
   simp only [attach, attachWith, pmap, map_pmap, cons.injEq, true_and]
@@ -122,7 +124,7 @@ theorem attachWith_congr {l₁ l₂ : List α} (w : l₁ = l₂) {P : α → Pro
   intros a _ m' _
   rfl
 
-@[simp]
+@[simp, grind =]
 theorem attachWith_cons {x : α} {xs : List α} {p : α → Prop} (h : ∀ a ∈ x :: xs, p a) :
     (x :: xs).attachWith p h = ⟨x, h x (mem_cons_self)⟩ ::
       xs.attachWith p (fun a ha ↦ h a (mem_cons_of_mem x ha)) :=
@@ -162,14 +164,14 @@ theorem attachWith_map_subtype_val {p : α → Prop} {l : List α} (H : ∀ a
     (l.attachWith p H).map Subtype.val = l :=
   (attachWith_map_val _).trans (List.map_id _)
 
-@[simp]
+@[simp, grind]
 theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
   | ⟨a, h⟩ => by
     have := mem_map.1 (by rw [attach_map_subtype_val]; exact h)
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
-@[simp]
+@[simp, grind]
 theorem mem_attachWith {l : List α} {q : α → Prop} (H) (x : {x // q x}) :
     x ∈ l.attachWith q H ↔ x.1 ∈ l := by
   induction l with
@@ -182,27 +184,28 @@ theorem mem_attachWith {l : List α} {q : α → Prop} (H) (x : {x // q x}) :
       · simp [← h]
       · simp_all
 
-@[simp]
+@[simp, grind =]
 theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
     b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
   simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, Subtype.exists, eq_comm]
 
+@[grind]
 theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {l H} {a} (h : a ∈ l) :
     f a (H a h) ∈ pmap f l H := by
   rw [mem_pmap]
   exact ⟨a, h, rfl⟩
 
-@[simp]
+@[simp, grind =]
 theorem length_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : (pmap f l H).length = l.length := by
   induction l
   · rfl
   · simp only [*, pmap, length]
 
-@[simp]
+@[simp, grind =]
 theorem length_attach {l : List α} : l.attach.length = l.length :=
   length_pmap
 
-@[simp]
+@[simp, grind =]
 theorem length_attachWith {p : α → Prop} {l H} : length (l.attachWith p H) = length l :=
   length_pmap
 
@@ -237,7 +240,7 @@ theorem attachWith_ne_nil_iff {l : List α} {P : α → Prop} {H : ∀ a ∈ l,
     l.attachWith P H ≠ [] ↔ l ≠ [] :=
   pmap_ne_nil_iff _ _
 
-@[simp]
+@[simp, grind =]
 theorem getElem?_pmap {p : α → Prop} {f : ∀ a, p a → β} {l : List α} (h : ∀ a ∈ l, p a) (i : Nat) :
     (pmap f l h)[i]? = Option.pmap f l[i]? fun x H => h x (mem_of_getElem? H) := by
   induction l generalizing i with
@@ -252,10 +255,10 @@ set_option linter.deprecated false in
 theorem get?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
     get? (pmap f l h) n = Option.pmap f (get? l n) fun x H => h x (mem_of_get? H) := by
   simp only [get?_eq_getElem?]
-  simp [getElem?_pmap, h]
+  simp [getElem?_pmap]
 
 -- The argument `f` is explicit to allow rewriting from right to left.
-@[simp]
+@[simp, grind =]
 theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) {i : Nat}
     (hn : i < (pmap f l h).length) :
     (pmap f l h)[i] =
@@ -279,109 +282,111 @@ theorem get_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h :
   simp only [get_eq_getElem]
   simp [getElem_pmap]
 
-@[simp]
+@[simp, grind =]
 theorem getElem?_attachWith {xs : List α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
     (xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (mem_of_getElem? a)) :=
   getElem?_pmap ..
 
-@[simp]
+@[simp, grind =]
 theorem getElem?_attach {xs : List α} {i : Nat} :
     xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => mem_of_getElem? a) :=
   getElem?_attachWith
 
-@[simp]
+@[simp, grind =]
 theorem getElem_attachWith {xs : List α} {P : α → Prop} {H : ∀ a ∈ xs, P a}
     {i : Nat} (h : i < (xs.attachWith P H).length) :
     (xs.attachWith P H)[i] = ⟨xs[i]'(by simpa using h), H _ (getElem_mem (by simpa using h))⟩ :=
   getElem_pmap ..
 
-@[simp]
+@[simp, grind =]
 theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
-@[simp] theorem pmap_attach {l : List α} {p : {x // x ∈ l} → Prop} {f : ∀ a, p a → β} (H) :
+@[simp, grind =] theorem pmap_attach {l : List α} {p : {x // x ∈ l} → Prop} {f : ∀ a, p a → β} (H) :
     pmap f l.attach H =
       l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
         (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
   apply ext_getElem <;> simp
 
-@[simp] theorem pmap_attachWith {l : List α} {p : {x // q x} → Prop} {f : ∀ a, p a → β} (H₁ H₂) :
+@[simp, grind =] theorem pmap_attachWith {l : List α} {p : {x // q x} → Prop} {f : ∀ a, p a → β} (H₁ H₂) :
     pmap f (l.attachWith q H₁) H₂ =
       l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
         (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
   apply ext_getElem <;> simp
 
-@[simp] theorem head?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
+@[simp, grind =] theorem head?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).head? = xs.attach.head?.map fun ⟨a, m⟩ => f a (H a m) := by
   induction xs with
   | nil => simp
   | cons x xs ih =>
     simp at ih
-    simp [head?_pmap, ih]
+    simp
 
-@[simp] theorem head_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
+@[simp, grind =] theorem head_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) (h : xs.pmap f H ≠ []) :
     (xs.pmap f H).head h = f (xs.head (by simpa using h)) (H _ (head_mem _)) := by
   induction xs with
   | nil => simp at h
-  | cons x xs ih => simp [head_pmap, ih]
+  | cons x xs ih => simp
 
-@[simp] theorem head?_attachWith {P : α → Prop} {xs : List α}
+@[simp, grind =] theorem head?_attachWith {P : α → Prop} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.attachWith P H).head? = xs.head?.pbind (fun a h => some ⟨a, H _ (mem_of_head? h)⟩) := by
   cases xs <;> simp_all
 
-@[simp] theorem head_attachWith {P : α → Prop} {xs : List α}
+@[simp, grind =] theorem head_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} (h : xs.attachWith P H ≠ []) :
     (xs.attachWith P H).head h = ⟨xs.head (by simpa using h), H _ (head_mem _)⟩ := by
   cases xs with
   | nil => simp at h
-  | cons x xs => simp [head_attachWith, h]
+  | cons x xs => simp
 
-@[simp] theorem head?_attach {xs : List α} :
+@[simp, grind =] theorem head?_attach {xs : List α} :
     xs.attach.head? = xs.head?.pbind (fun a h => some ⟨a, mem_of_head? h⟩) := by
   cases xs <;> simp_all
 
-@[simp] theorem head_attach {xs : List α} (h) :
+@[simp, grind =] theorem head_attach {xs : List α} (h) :
     xs.attach.head h = ⟨xs.head (by simpa using h), head_mem (by simpa using h)⟩ := by
   cases xs with
   | nil => simp at h
-  | cons x xs => simp [head_attach, h]
+  | cons x xs => simp
 
-@[simp] theorem tail_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
+@[simp, grind =] theorem tail_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).tail = xs.tail.pmap f (fun a h => H a (mem_of_mem_tail h)) := by
   cases xs <;> simp
 
-@[simp] theorem tail_attachWith {P : α → Prop} {xs : List α}
+@[simp, grind =] theorem tail_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} :
     (xs.attachWith P H).tail = xs.tail.attachWith P (fun a h => H a (mem_of_mem_tail h)) := by
   cases xs <;> simp
 
-@[simp] theorem tail_attach {xs : List α} :
+@[simp, grind =] theorem tail_attach {xs : List α} :
     xs.attach.tail = xs.tail.attach.map (fun ⟨x, h⟩ => ⟨x, mem_of_mem_tail h⟩) := by
   cases xs <;> simp
 
+@[grind]
 theorem foldl_pmap {l : List α} {P : α → Prop} {f : (a : α) → P a → β}
     (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
     (l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
   rw [pmap_eq_map_attach, foldl_map]
 
+@[grind]
 theorem foldr_pmap {l : List α} {P : α → Prop} {f : (a : α) → P a → β}
     (H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) :
     (l.pmap f H).foldr g x = l.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
   rw [pmap_eq_map_attach, foldr_map]
 
-@[simp] theorem foldl_attachWith
+@[simp, grind =] theorem foldl_attachWith
     {l : List α} {q : α → Prop} (H : ∀ a, a ∈ l → q a) {f : β → { x // q x } → β} {b} :
     (l.attachWith q H).foldl f b = l.attach.foldl (fun b ⟨a, h⟩ => f b ⟨a, H _ h⟩) b := by
   induction l generalizing b with
   | nil => simp
   | cons a l ih => simp [ih, foldl_map]
 
-@[simp] theorem foldr_attachWith
+@[simp, grind =] theorem foldr_attachWith
     {l : List α} {q : α → Prop} (H : ∀ a, a ∈ l → q a) {f : { x // q x } → β → β} {b} :
     (l.attachWith q H).foldr f b = l.attach.foldr (fun a acc => f ⟨a.1, H _ a.2⟩ acc) b := by
   induction l generalizing b with
@@ -420,16 +425,18 @@ theorem foldr_attach {l : List α} {f : α → β → β} {b : β} :
   | nil => simp
   | cons a l ih => rw [foldr_cons, attach_cons, foldr_cons, foldr_map, ih]
 
+@[grind =]
 theorem attach_map {l : List α} {f : α → β} :
     (l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem h⟩) := by
   induction l <;> simp [*]
 
+@[grind =]
 theorem attachWith_map {l : List α} {f : α → β} {P : β → Prop} (H : ∀ (b : β), b ∈ l.map f → P b) :
     (l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem h))).map
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
   induction l <;> simp [*]
 
-@[simp] theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
+@[simp, grind =] theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
     {f : { x // P x } → β} :
     (l.attachWith P H).map f = l.attach.map fun ⟨x, h⟩ => f ⟨x, H _ h⟩ := by
   induction l <;> simp_all
@@ -458,13 +465,14 @@ theorem map_attach_eq_pmap {l : List α} {f : { x // x ∈ l } → β} :
 @[deprecated map_attach_eq_pmap (since := "2025-02-09")]
 abbrev map_attach := @map_attach_eq_pmap
 
+@[grind =]
 theorem attach_filterMap {l : List α} {f : α → Option β} :
     (l.filterMap f).attach = l.attach.filterMap
       fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩) := by
   induction l with
   | nil => rfl
   | cons x xs ih =>
-    simp only [filterMap_cons, attach_cons, ih, filterMap_map]
+    simp only [filterMap_cons, attach_cons, filterMap_map]
     split <;> rename_i h
     · simp only [Option.pbind_eq_none_iff, reduceCtorEq, exists_false,
         or_false] at h
@@ -488,6 +496,7 @@ theorem attach_filterMap {l : List α} {f : α → Option β} :
       ext
       simp
 
+@[grind =]
 theorem attach_filter {l : List α} (p : α → Bool) :
     (l.filter p).attach = l.attach.filterMap
       fun x => if w : p x.1 then some ⟨x.1, mem_filter.mpr ⟨x.2, w⟩⟩ else none := by
@@ -499,7 +508,7 @@ theorem attach_filter {l : List α} (p : α → Bool) :
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
 
-@[simp]
+@[simp, grind =]
 theorem filterMap_attachWith {q : α → Prop} {l : List α} {f : {x // q x} → Option β} (H) :
     (l.attachWith q H).filterMap f = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
   induction l with
@@ -508,7 +517,7 @@ theorem filterMap_attachWith {q : α → Prop} {l : List α} {f : {x // q x} →
     simp only [attachWith_cons, filterMap_cons]
     split <;> simp_all [Function.comp_def]
 
-@[simp]
+@[simp, grind =]
 theorem filter_attachWith {q : α → Prop} {l : List α} {p : {x // q x} → Bool} (H) :
     (l.attachWith q H).filter p =
       (l.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
@@ -518,13 +527,14 @@ theorem filter_attachWith {q : α → Prop} {l : List α} {p : {x // q x} → Bo
     simp only [attachWith_cons, filter_cons]
     split <;> simp_all [Function.comp_def, filter_map]
 
+@[grind =]
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} {g : ∀ a, p a → β} {f : ∀ b, q b → γ} {l} (H₁ H₂) :
     pmap f (pmap g l H₁) H₂ =
       pmap (α := { x // x ∈ l }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) l.attach
         (fun a _ => H₁ a a.2) := by
   simp [pmap_eq_map_attach, attach_map]
 
-@[simp] theorem pmap_append {p : ι → Prop} {f : ∀ a : ι, p a → α} {l₁ l₂ : List ι}
+@[simp, grind =] theorem pmap_append {p : ι → Prop} {f : ∀ a : ι, p a → α} {l₁ l₂ : List ι}
     (h : ∀ a ∈ l₁ ++ l₂, p a) :
     (l₁ ++ l₂).pmap f h =
       (l₁.pmap f fun a ha => h a (mem_append_left l₂ ha)) ++
@@ -541,47 +551,50 @@ theorem pmap_append' {p : α → Prop} {f : ∀ a : α, p a → β} {l₁ l₂ :
       l₁.pmap f h₁ ++ l₂.pmap f h₂ :=
   pmap_append _
 
-@[simp] theorem attach_append {xs ys : List α} :
+@[simp, grind =] theorem attach_append {xs ys : List α} :
     (xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_left ys h⟩) ++
       ys.attach.map fun ⟨x, h⟩ => ⟨x, mem_append_right xs h⟩ := by
-  simp only [attach, attachWith, pmap, map_pmap, pmap_append]
+  simp only [attach, attachWith, map_pmap, pmap_append]
   congr 1 <;>
   exact pmap_congr_left _ fun _ _ _ _ => rfl
 
-@[simp] theorem attachWith_append {P : α → Prop} {xs ys : List α}
+@[simp, grind =] theorem attachWith_append {P : α → Prop} {xs ys : List α}
     {H : ∀ (a : α), a ∈ xs ++ ys → P a} :
     (xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_left ys h)) ++
       ys.attachWith P (fun a h => H a (mem_append_right xs h)) := by
-  simp only [attachWith, attach_append, map_pmap, pmap_append]
+  simp only [attachWith, pmap_append]
 
-@[simp] theorem pmap_reverse {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
+@[simp, grind =] theorem pmap_reverse {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs.reverse → P a) :
     xs.reverse.pmap f H = (xs.pmap f (fun a h => H a (by simpa using h))).reverse := by
   induction xs <;> simp_all
 
+@[grind =]
 theorem reverse_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h)) := by
   rw [pmap_reverse]
 
-@[simp] theorem attachWith_reverse {P : α → Prop} {xs : List α}
+@[simp, grind =] theorem attachWith_reverse {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs.reverse → P a} :
     xs.reverse.attachWith P H =
       (xs.attachWith P (fun a h => H a (by simpa using h))).reverse :=
   pmap_reverse ..
 
+@[grind =]
 theorem reverse_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} :
     (xs.attachWith P H).reverse = (xs.reverse.attachWith P (fun a h => H a (by simpa using h))) :=
   reverse_pmap ..
 
-@[simp] theorem attach_reverse {xs : List α} :
+@[simp, grind =] theorem attach_reverse {xs : List α} :
     xs.reverse.attach = xs.attach.reverse.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
   simp only [attach, attachWith, reverse_pmap, map_pmap]
   apply pmap_congr_left
   intros
   rfl
 
+@[grind =]
 theorem reverse_attach {xs : List α} :
     xs.attach.reverse = xs.reverse.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
   simp only [attach, attachWith, reverse_pmap, map_pmap]
@@ -589,7 +602,7 @@ theorem reverse_attach {xs : List α} :
   intros
   rfl
 
-@[simp] theorem getLast?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
+@[simp, grind =] theorem getLast?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).getLast? = xs.attach.getLast?.map fun ⟨a, m⟩ => f a (H a m) := by
   simp only [getLast?_eq_head?_reverse]
@@ -597,30 +610,30 @@ theorem reverse_attach {xs : List α} :
   simp only [Option.map_map]
   congr
 
-@[simp] theorem getLast_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
+@[simp, grind =] theorem getLast_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
     (H : ∀ (a : α), a ∈ xs → P a) (h : xs.pmap f H ≠ []) :
     (xs.pmap f H).getLast h = f (xs.getLast (by simpa using h)) (H _ (getLast_mem _)) := by
   simp only [getLast_eq_head_reverse]
   simp only [reverse_pmap, head_pmap, head_reverse]
 
-@[simp] theorem getLast?_attachWith {P : α → Prop} {xs : List α}
+@[simp, grind =] theorem getLast?_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} :
     (xs.attachWith P H).getLast? = xs.getLast?.pbind (fun a h => some ⟨a, H _ (mem_of_getLast? h)⟩) := by
   rw [getLast?_eq_head?_reverse, reverse_attachWith, head?_attachWith]
   simp
 
-@[simp] theorem getLast_attachWith {P : α → Prop} {xs : List α}
+@[simp, grind =] theorem getLast_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} (h : xs.attachWith P H ≠ []) :
     (xs.attachWith P H).getLast h = ⟨xs.getLast (by simpa using h), H _ (getLast_mem _)⟩ := by
-  simp only [getLast_eq_head_reverse, reverse_attachWith, head_attachWith, head_map]
+  simp only [getLast_eq_head_reverse, reverse_attachWith, head_attachWith]
 
-@[simp]
+@[simp, grind =]
 theorem getLast?_attach {xs : List α} :
     xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast? h⟩ := by
   rw [getLast?_eq_head?_reverse, reverse_attach, head?_map, head?_attach]
   simp
 
-@[simp]
+@[simp, grind =]
 theorem getLast_attach {xs : List α} (h : xs.attach ≠ []) :
     xs.attach.getLast h = ⟨xs.getLast (by simpa using h), getLast_mem (by simpa using h)⟩ := by
   simp only [getLast_eq_head_reverse, reverse_attach, head_map, head_attach]
@@ -638,14 +651,14 @@ theorem countP_attachWith {p : α → Prop} {q : α → Bool} {l : List α} (H :
 @[simp]
 theorem count_attach [BEq α] {l : List α} {a : {x // x ∈ l}} :
     l.attach.count a = l.count ↑a :=
-  Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attach
+  Eq.trans (countP_congr fun _ _ => by simp) <| countP_attach
 
-@[simp]
+@[simp, grind =]
 theorem count_attachWith [BEq α] {p : α → Prop} {l : List α} (H : ∀ a ∈ l, p a) {a : {x // p x}} :
     (l.attachWith p H).count a = l.count ↑a :=
-  Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attachWith _
+  Eq.trans (countP_congr fun _ _ => by simp) <| countP_attachWith _
 
-@[simp] theorem countP_pmap {p : α → Prop} {g : ∀ a, p a → β} {f : β → Bool} {l : List α} (H₁) :
+@[simp, grind =] theorem countP_pmap {p : α → Prop} {g : ∀ a, p a → β} {f : β → Bool} {l : List α} (H₁) :
     (l.pmap g H₁).countP f =
       l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
   simp [pmap_eq_map_attach, countP_map, Function.comp_def]
@@ -704,7 +717,7 @@ def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) : List α :
   unfold unattach
   induction l with
   | nil => simp
-  | cons a l ih => simp [ih, Function.comp_def]
+  | cons a l ih => simp [ih]
 
 @[simp] theorem getElem?_unattach {p : α → Prop} {l : List { x // p x }} (i : Nat) :
     l.unattach[i]? = l[i]?.map Subtype.val := by

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
@@ -729,9 +730,9 @@ Examples:
 -/
 @[inline] def flatMap {α : Type u} {β : Type v} (b : α → List β) (as : List α) : List β := flatten (map b as)
 
-@[simp, grind] theorem flatMap_nil {f : α → List β} : List.flatMap f [] = [] := by simp [flatten, List.flatMap]
+@[simp, grind] theorem flatMap_nil {f : α → List β} : List.flatMap f [] = [] := by simp [List.flatMap]
 @[simp, grind] theorem flatMap_cons {x : α} {xs : List α} {f : α → List β} :
-  List.flatMap f (x :: xs) = f x ++ List.flatMap f xs := by simp [flatten, List.flatMap]
+  List.flatMap f (x :: xs) = f x ++ List.flatMap f xs := by simp [List.flatMap]
 
 /-! ### replicate -/
 
@@ -752,7 +753,7 @@ def replicate : (n : Nat) → (a : α) → List α
 @[simp, grind] theorem length_replicate {n : Nat} {a : α} : (replicate n a).length = n := by
   induction n with
   | zero => simp
-  | succ n ih => simp only [ih, replicate_succ, length_cons, Nat.succ_eq_add_one]
+  | succ n ih => simp only [ih, replicate_succ, length_cons]
 
 /-! ## Additional functions -/
 
@@ -891,7 +892,7 @@ theorem mem_of_elem_eq_true [BEq α] [LawfulBEq α] {a : α} {as : List α} : el
   | a'::as =>
     simp [elem]
     split
-    next h => intros; simp [BEq.beq] at h; subst h; apply Mem.head
+    next h => intros; simp at h; subst h; apply Mem.head
     next _ => intro h; exact Mem.tail _ (mem_of_elem_eq_true h)
 
 theorem elem_eq_true_of_mem [BEq α] [ReflBEq α] {a : α} {as : List α} (h : a ∈ as) : elem a as = true := 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? -/
@@ -1779,7 +1780,7 @@ where
   | a :: l, i, h =>
     if p a then
       some ⟨i, by
-        simp only [Nat.add_comm _ i, ← Nat.add_assoc] at h
+        simp only [Nat.add_comm _ i] at h
         exact Nat.lt_of_add_right_lt (Nat.lt_of_succ_le (Nat.le_of_eq h))⟩
     else
       go l (i + 1) (by simp at h; simpa [← Nat.add_assoc, Nat.add_right_comm] using h)
@@ -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
 
@@ -2014,7 +2015,7 @@ def zip : List α → List β → List (Prod α β) :=
   zipWith Prod.mk
 
 @[simp] theorem zip_nil_left : zip ([] : List α) (l : List β)  = [] := rfl
-@[simp] theorem zip_nil_right : zip (l : List α) ([] : List β)  = [] := by simp [zip, zipWith]
+@[simp] theorem zip_nil_right : zip (l : List α) ([] : List β)  = [] := by simp [zip]
 @[simp] theorem zip_cons_cons : zip (a :: as) (b :: bs) = (a, b) :: zip as bs := rfl
 
 /-! ### zipWithAll -/

src/Init/Data/List/BasicAux.lean

@@ -276,7 +276,7 @@ theorem getElem_append_right {as bs : List α} {i : Nat} (h₁ : as.length ≤ i
   induction as generalizing i with
   | nil => trivial
   | cons a as ih =>
-    cases i with simp [Nat.succ_sub_succ] <;> simp [Nat.succ_sub_succ] at h₁
+    cases i with simp [Nat.succ_sub_succ] <;> simp at h₁
     | succ i => apply ih; simp [h₁]
 
 @[deprecated "Deprecated without replacement." (since := "2025-02-13")]

src/Init/Data/List/Control.lean

@@ -237,8 +237,8 @@ def foldlM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w} : (f : s
     let s' ← f s a
     List.foldlM f s' as
 
-@[simp, grind] theorem foldlM_nil [Monad m] {f : β → α → m β} {b : β} : [].foldlM f b = pure b := rfl
-@[simp, grind] theorem foldlM_cons [Monad m] {f : β → α → m β} {b : β} {a : α} {l : List α} :
+@[simp, grind =] theorem foldlM_nil [Monad m] {f : β → α → m β} {b : β} : [].foldlM f b = pure b := rfl
+@[simp, grind =] theorem foldlM_cons [Monad m] {f : β → α → m β} {b : β} {a : α} {l : List α} :
     (a :: l).foldlM f b = f b a >>= l.foldlM f := by
   simp [List.foldlM]
 
@@ -261,7 +261,7 @@ example [Monad m] (f : α → β → m β) :
 def foldrM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w} (f : α → s → m s) (init : s) (l : List α) : m s :=
   l.reverse.foldlM (fun s a => f a s) init
 
-@[simp, grind] theorem foldrM_nil [Monad m] {f : α → β → m β} {b : β} : [].foldrM f b = pure b := rfl
+@[simp, grind =] theorem foldrM_nil [Monad m] {f : α → β → m β} {b : β} : [].foldrM f b = pure b := rfl
 
 /--
 Maps `f` over the list and collects the results with `<|>`. The result for the end of the list is
@@ -347,7 +347,7 @@ theorem findM?_pure {m} [Monad m] [LawfulMonad m] (p : α → Bool) (as : List
     | true  => simp
     | false => simp [ih]
 
-@[simp]
+@[simp, grind =]
 theorem idRun_findM? (p : α → Id Bool) (as : List α) :
     (findM? p as).run = as.find? (p · |>.run) :=
   findM?_pure _ _
@@ -400,7 +400,7 @@ theorem findSomeM?_pure [Monad m] [LawfulMonad m] {f : α → Option β} {as : L
     | some b => simp
     | none   => simp [ih]
 
-@[simp]
+@[simp, grind =]
 theorem idRun_findSomeM? (f : α → Id (Option β)) (as : List α) :
     (findSomeM? f as).run = as.findSome? (f · |>.run) :=
   findSomeM?_pure
@@ -444,23 +444,23 @@ instance : ForIn' m (List α) α inferInstance where
 -- No separate `ForIn` instance is required because it can be derived from `ForIn'`.
 
 -- We simplify `List.forIn'` to `forIn'`.
-@[simp] theorem forIn'_eq_forIn' [Monad m] : @List.forIn' α β m _ = forIn' := rfl
+@[simp, grind =] theorem forIn'_eq_forIn' [Monad m] : @List.forIn' α β m _ = forIn' := rfl
 
-@[simp] theorem forIn'_nil [Monad m] {f : (a : α) → a ∈ [] → β → m (ForInStep β)} {b : β} : forIn' [] b f = pure b :=
+@[simp, grind =] theorem forIn'_nil [Monad m] {f : (a : α) → a ∈ [] → β → m (ForInStep β)} {b : β} : forIn' [] b f = pure b :=
   rfl
 
-@[simp] theorem forIn_nil [Monad m] {f : α → β → m (ForInStep β)} {b : β} : forIn [] b f = pure b :=
+@[simp, grind =] theorem forIn_nil [Monad m] {f : α → β → m (ForInStep β)} {b : β} : forIn [] b f = pure b :=
   rfl
 
 instance : ForM m (List α) α where
   forM := List.forM
 
 -- We simplify `List.forM` to `forM`.
-@[simp] theorem forM_eq_forM [Monad m] : @List.forM m _ α = forM := rfl
+@[simp, grind =] theorem forM_eq_forM [Monad m] : @List.forM m _ α = forM := rfl
 
-@[simp] theorem forM_nil [Monad m] {f : α → m PUnit} : forM [] f = pure ⟨⟩ :=
+@[simp, grind =] theorem forM_nil [Monad m] {f : α → m PUnit} : forM [] f = pure ⟨⟩ :=
   rfl
-@[simp] theorem forM_cons [Monad m] {f : α → m PUnit} {a : α} {as : List α} : forM (a::as) f = f a >>= fun _ => forM as f :=
+@[simp, grind =] theorem forM_cons [Monad m] {f : α → m PUnit} {a : α} {as : List α} : forM (a::as) f = f a >>= fun _ => forM as f :=
   rfl
 
 instance : Functor List where

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 =>
@@ -82,11 +82,11 @@ 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
-  simp only [countP_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]
+  simp only [countP_eq_length_filter, length_pos_iff_exists_mem, mem_filter]
 
 @[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p l ↔ ∃ a ∈ l, p a :=
   countP_pos_iff
@@ -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]
@@ -155,7 +169,7 @@ theorem length_filterMap_eq_countP {f : α → Option β} {l : List α} :
   | nil => rfl
   | cons x l ih =>
     simp only [filterMap_cons, countP_cons]
-    split <;> simp [ih, *]
+    split <;> simp [*]
 
 theorem countP_filterMap {p : β → Bool} {f : α → Option β} {l : List α} :
     countP p (filterMap f l) = countP (fun a => ((f a).map p).getD false) l := by
@@ -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]

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
@@ -141,16 +146,18 @@ theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (erase
 @[simp] theorem eraseP_eq_self_iff {p} {l : List α} : l.eraseP p = l ↔ ∀ a ∈ l, ¬ p a := by
   rw [← Sublist.length_eq eraseP_sublist, length_eraseP]
   split <;> rename_i h
-  · simp only [any_eq_true, length_eq_zero_iff] at h
+  · simp only [any_eq_true] at h
     constructor
     · intro; simp_all [Nat.sub_one_eq_self]
     · 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
@@ -274,9 +287,9 @@ theorem eraseP_comm {l : List α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
     by_cases h₁ : p a
     · by_cases h₂ : q a
       · simp_all
-      · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
+      · simp [h₁, h₂]
     · by_cases h₂ : q a
-      · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
+      · simp [h₁, h₂]
       · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
 
 theorem head_eraseP_mem {xs : List α} {p : α → Bool} (h) : (xs.eraseP p).head h ∈ xs :=
@@ -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
@@ -551,7 +578,7 @@ theorem eraseIdx_eq_take_drop_succ :
   match l, i with
   | [], _
   | a::l, 0
-  | a::l, i + 1 => simp [Nat.succ_inj]
+  | a::l, i + 1 => simp
 
 @[deprecated eraseIdx_eq_nil_iff (since := "2025-01-30")]
 abbrev eraseIdx_eq_nil := @eraseIdx_eq_nil_iff
@@ -560,11 +587,12 @@ theorem eraseIdx_ne_nil_iff {l : List α} {i : Nat} : eraseIdx l i ≠ [] ↔ 2
   match l with
   | []
   | [a]
-  | a::b::l => simp [Nat.succ_inj]
+  | a::b::l => simp
 
 @[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⟩

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
@@ -61,7 +62,7 @@ end List
 
 namespace Fin
 
-theorem foldlM_eq_foldlM_finRange [Monad m] (f : α → Fin n → m α) (x : α) :
+@[grind =] theorem foldlM_eq_foldlM_finRange [Monad m] (f : α → Fin n → m α) (x : α) :
     foldlM n f x = (List.finRange n).foldlM f x := by
   induction n generalizing x with
   | zero => simp
@@ -71,21 +72,21 @@ theorem foldlM_eq_foldlM_finRange [Monad m] (f : α → Fin n → m α) (x : α)
     funext y
     simp [ih, List.foldlM_map]
 
-theorem foldrM_eq_foldrM_finRange [Monad m] [LawfulMonad m] (f : Fin n → α → m α) (x : α) :
+@[grind =] theorem foldrM_eq_foldrM_finRange [Monad m] [LawfulMonad m] (f : Fin n → α → m α) (x : α) :
     foldrM n f x = (List.finRange n).foldrM f x := by
   induction n generalizing x with
   | zero => simp
   | succ n ih =>
     simp [foldrM_succ, List.finRange_succ, ih, List.foldrM_map]
 
-theorem foldl_eq_finRange_foldl (f : α → Fin n → α) (x : α) :
+@[grind =] theorem foldl_eq_finRange_foldl (f : α → Fin n → α) (x : α) :
     foldl n f x = (List.finRange n).foldl f x := by
   induction n generalizing x with
   | zero => simp
   | succ n ih =>
     simp [foldl_succ, List.finRange_succ, ih, List.foldl_map]
 
-theorem foldr_eq_finRange_foldr (f : Fin n → α → α) (x : α) :
+@[grind =] theorem foldr_eq_finRange_foldr (f : Fin n → α → α) (x : α) :
     foldr n f x = (List.finRange n).foldr f x := by
   induction n generalizing x with
   | zero => simp

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
@@ -174,9 +178,12 @@ 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]
+  simp only [List.findSome?_eq_none_iff]
   exact fun w x m => w x (Sublist.mem m h)
 
 theorem IsPrefix.findSome?_eq_some {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <+: l₂) :
@@ -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 =>
@@ -359,7 +383,7 @@ theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a :
       · simpa using h₂ x (by simpa using ⟨l, ma, m⟩)
       · specialize h₁ _ mb
         simp_all
-    · simp [h₁]
+    · simp
       refine ⟨as, bs, ?_⟩
       refine ⟨?_, ?_, ?_⟩
       · simp_all
@@ -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
@@ -545,7 +592,7 @@ theorem findIdx_eq_length {p : α → Bool} {xs : List α} :
   | cons x xs ih =>
     rw [findIdx_cons, length_cons]
     simp only [cond_eq_if]
-    split <;> simp_all [Nat.succ.injEq]
+    split <;> simp_all
 
 theorem findIdx_eq_length_of_false {p : α → Bool} {xs : List α} (h : ∀ x ∈ xs, p x = false) :
     xs.findIdx p = xs.length := by
@@ -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
@@ -671,16 +730,16 @@ 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
   | nil => simp_all
   | cons x xs ih =>
     simp only [findIdx?_cons]
-    split <;> simp_all [cond_eq_if]
+    split <;> simp_all
 
-@[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
@@ -740,13 +799,13 @@ theorem findIdx?_eq_some_iff_getElem {xs : List α} {p : α → Bool} {i : Nat}
   induction xs generalizing i with
   | nil => simp
   | cons x xs ih =>
-    simp only [findIdx?_cons, Nat.zero_add]
+    simp only [findIdx?_cons]
     split
     · simp only [Option.some.injEq, Bool.not_eq_true, length_cons]
       cases i with
       | zero => simp_all
       | succ i =>
-        simp only [Bool.not_eq_true, zero_ne_add_one, getElem_cons_succ, false_iff, not_exists,
+        simp only [zero_ne_add_one, getElem_cons_succ, false_iff, not_exists,
           not_and, Classical.not_forall, Bool.not_eq_false]
         intros
         refine ⟨0, zero_lt_succ i, ‹_›⟩
@@ -771,8 +830,8 @@ theorem of_findIdx?_eq_some {xs : List α} {p : α → Bool} (w : xs.findIdx? p
   induction xs generalizing i with
   | nil => simp_all
   | cons x xs ih =>
-    simp_all only [findIdx?_cons, Nat.zero_add]
-    split at w <;> cases i <;> simp_all [succ_inj]
+    simp_all only [findIdx?_cons]
+    split at w <;> cases i <;> simp_all
 
 @[deprecated of_findIdx?_eq_some (since := "2025-02-02")]
 abbrev findIdx?_of_eq_some := @of_findIdx?_eq_some
@@ -783,7 +842,7 @@ theorem of_findIdx?_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p
   induction xs generalizing i with
   | nil => simp_all
   | cons x xs ih =>
-    simp_all only [Bool.not_eq_true, findIdx?_cons, Nat.zero_add]
+    simp_all only [Bool.not_eq_true, findIdx?_cons]
     cases i with
     | zero =>
       split at w <;> simp_all
@@ -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]
@@ -824,12 +883,12 @@ 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
   | succ n =>
-    simp only [replicate, findIdx?_cons, Nat.zero_add, zero_lt_succ, true_and]
+    simp only [replicate, findIdx?_cons, zero_lt_succ, true_and]
     split <;> simp_all
 
 theorem findIdx?_eq_findSome?_zipIdx {xs : List α} {p : α → Bool} :
@@ -840,7 +899,7 @@ theorem findIdx?_eq_findSome?_zipIdx {xs : List α} {p : α → Bool} :
     simp only [findIdx?_cons, Nat.zero_add, zipIdx]
     split
     · simp_all
-    · simp_all only [zipIdx_cons, ite_false, Option.isNone_none, findSome?_cons_of_isNone, reduceCtorEq]
+    · simp_all only [ite_false, Option.isNone_none, findSome?_cons_of_isNone, reduceCtorEq]
       rw [← map_snd_add_zipIdx_eq_zipIdx (n := 1) (k := 0)]
       simp [Function.comp_def, findSome?_map]
 
@@ -878,29 +937,45 @@ 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
   | nil => simp
   | cons x xs ih =>
     simp only [findIdx_cons, findIdx?_cons]
-    split <;> simp_all [ih]
+    split <;> simp_all
 
 @[simp] theorem findIdx?_subtype {p : α → Prop} {l : List { x // p x }}
     {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
@@ -910,11 +985,11 @@ theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :
   | nil => simp
   | cons a l ih =>
     simp [hf, findIdx?_cons]
-    split <;> simp [ih, Function.comp_def]
+    split <;> simp [ih]
 
 /-! ### 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
@@ -1001,7 +1078,7 @@ theorem isNone_findFinIdx? {l : List α} {p : α → Bool} :
     {f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
     l.findFinIdx? f = (l.unattach.findFinIdx? g).map (fun i => i.cast (by simp)) := by
   induction l with
-  | nil => simp [unattach]
+  | nil => simp
   | cons a l ih =>
     simp [hf, findFinIdx?_cons]
     split <;> simp [ih, Function.comp_def]
@@ -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]
@@ -1050,21 +1129,36 @@ 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 =>
     simp only [mem_cons] at h
     obtain rfl | h := h
     · simp
-    · simp only [idxOf_cons, cond_eq_if, beq_iff_eq, length_cons]
+    · simp only [idxOf_cons, cond_eq_if, length_cons]
       specialize ih h
       split
       · 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

src/Init/Data/List/Impl.lean

@@ -234,7 +234,7 @@ Examples:
   intro xs; induction xs with intro acc
   | nil => simp [takeWhile, takeWhileTR.go]
   | cons x xs IH =>
-    simp only [takeWhileTR.go, Array.toListImpl_eq, takeWhile]
+    simp only [takeWhileTR.go, takeWhile]
     split
     · intro h; rw [IH] <;> simp_all
     · simp [*]
@@ -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 =>

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 ..⟩
 
@@ -253,7 +255,7 @@ theorem getElem?_cons_zero {l : List α} : (a::l)[0]? = some a := rfl
 
 @[grind =]
 theorem getElem?_cons : (a :: l)[i]? = if i = 0 then some a else l[i-1]? := by
-  cases i <;> simp [getElem?_cons_zero]
+  cases i <;> simp
 
 theorem getElem?_eq_some_iff {l : List α} : l[i]? = some a ↔ ∃ h : i < l.length, l[i] = a :=
   match l with
@@ -285,16 +287,16 @@ theorem getElem_eq_iff {l : List α} {i : Nat} (h : i < l.length) : l[i] = x ↔
   exact ⟨fun w => ⟨h, w⟩, fun h => h.2⟩
 
 theorem getElem_eq_getElem?_get {l : List α} {i : Nat} (h : i < l.length) :
-    l[i] = l[i]?.get (by simp [getElem?_eq_getElem, h]) := by
-  simp [getElem_eq_iff]
+    l[i] = l[i]?.get (by simp [h]) := by
+  simp
 
 theorem getD_getElem? {l : List α} {i : Nat} {d : α} :
     l[i]?.getD d = if p : i < l.length then l[i]'p else d := by
   if h : i < l.length then
-    simp [h, getElem?_def]
+    simp [h]
   else
     have p : i ≥ l.length := Nat.le_of_not_gt h
-    simp [getElem?_eq_none p, h]
+    simp [h]
 
 @[simp] theorem getElem_singleton {a : α} {i : Nat} (h : i < 1) : [a][i] = a := by
   match i, h with
@@ -330,7 +332,7 @@ theorem ext_getElem {l₁ l₂ : List α} (hl : length l₁ = length l₂)
     (h : ∀ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length), l₁[i]'h₁ = l₂[i]'h₂) : l₁ = l₂ :=
   ext_getElem? fun n =>
     if h₁ : n < length l₁ then by
-      simp_all [getElem?_eq_getElem]
+      simp_all
     else by
       have h₁ := Nat.le_of_not_lt h₁
       rw [getElem?_eq_none h₁, getElem?_eq_none]; rwa [← hl]
@@ -634,7 +636,7 @@ theorem all_bne' [BEq α] [PartialEquivBEq α] {l : List α} :
 
 @[simp] theorem getElem?_set_self {l : List α} {i : Nat} {a : α} (h : i < l.length) :
     (l.set i a)[i]? = some a := by
-  simp_all [getElem?_eq_some_iff]
+  simp_all
 
 /-- This differs from `getElem?_set_self` by monadically mapping `Function.const _ a` over the `Option`
 returned by `l[i]?`. -/
@@ -677,7 +679,7 @@ theorem getElem?_set_self' {l : List α} {i : Nat} {a : α} :
     subst h
     rw [if_pos rfl]
     split <;> rename_i h
-    · simp only [getElem?_set_self (by simpa), h]
+    · simp only [getElem?_set_self (by simpa)]
     · simp_all
   else
     simp [h]
@@ -1274,9 +1276,9 @@ theorem length_filter_le (p : α → Bool) (l : List α) :
   induction l with
   | nil => simp
   | cons a l ih =>
-    simp only [filter_cons, length_cons, succ_eq_add_one]
+    simp only [filter_cons, length_cons]
     split
-    · simp only [length_cons, succ_eq_add_one]
+    · simp only [length_cons]
       exact Nat.succ_le_succ ih
     · exact Nat.le_trans ih (Nat.le_add_right _ _)
 
@@ -1294,7 +1296,7 @@ theorem length_filter_eq_length_iff {l} : (filter p l).length = l.length ↔ ∀
   induction l with
   | nil => simp
   | cons a l ih =>
-    simp only [filter_cons, length_cons, succ_eq_add_one, mem_cons, forall_eq_or_imp]
+    simp only [filter_cons, length_cons, mem_cons, forall_eq_or_imp]
     split <;> rename_i h
     · simp_all [Nat.add_one_inj] -- Why does the simproc not fire here?
     · have := Nat.ne_of_lt (Nat.lt_succ.mpr (length_filter_le p l))
@@ -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]
@@ -1373,7 +1388,7 @@ theorem filter_eq_cons_iff {l} {a} {as} :
       · obtain ⟨l₁, l₂, rfl, w₁, w₂, w₃⟩ := ih h
         exact ⟨x :: l₁, l₂, by simp_all⟩
   · rintro ⟨l₁, l₂, rfl, h₁, h, h₂⟩
-    simp [h₂, filter_cons, filter_eq_nil_iff.mpr h₁, h]
+    simp [h₂, filter_eq_nil_iff.mpr h₁, h]
 
 theorem filter_congr {p q : α → Bool} :
     ∀ {l : List α}, (∀ x ∈ l, p x = q x) → filter p l = filter q l
@@ -1389,7 +1404,7 @@ theorem head_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p
   | nil => simp
   | cons =>
     simp only [head_cons] at h
-    simp [filter_cons, h]
+    simp [h]
 
 @[simp] theorem filter_sublist {p : α → Bool} : ∀ {l : List α}, filter p l <+ l
   | [] => .slnil
@@ -1405,7 +1420,7 @@ theorem head_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p
 
 @[simp]
 theorem filterMap_eq_map {f : α → β} : filterMap (some ∘ f) = map f := by
-  funext l; induction l <;> simp [*, filterMap_cons]
+  funext l; induction l <;> simp [*]
 
 /-- Variant of `filterMap_eq_map` with `some ∘ f` expanded out to a lambda. -/
 @[simp]
@@ -1438,7 +1453,7 @@ theorem filterMap_length_eq_length {l} :
   induction l with
   | nil => simp
   | cons a l ih =>
-    simp only [filterMap_cons, length_cons, succ_eq_add_one, mem_cons, forall_eq_or_imp]
+    simp only [filterMap_cons, length_cons, mem_cons, forall_eq_or_imp]
     split <;> rename_i h
     · have := Nat.ne_of_lt (Nat.lt_succ.mpr (length_filterMap_le f l))
       simp_all
@@ -1450,7 +1465,7 @@ theorem filterMap_eq_filter {p : α → Bool} :
   funext l
   induction l with
   | nil => rfl
-  | cons a l IH => by_cases pa : p a <;> simp [filterMap_cons, Option.guard, pa, ← IH]
+  | cons a l IH => by_cases pa : p a <;> simp [Option.guard, pa, ← IH]
 
 @[grind]
 theorem filterMap_filterMap {f : α → Option β} {g : β → Option γ} {l : List α} :
@@ -1497,7 +1512,7 @@ theorem forall_mem_filterMap {f : α → Option β} {l : List α} {P : β → Pr
 
 theorem map_filterMap_of_inv
     {f : α → Option β} {g : β → α} (H : ∀ x : α, (f x).map g = some x) {l : List α} :
-    map g (filterMap f l) = l := by simp only [map_filterMap, H, filterMap_some, id]
+    map g (filterMap f l) = l := by simp only [map_filterMap, H, filterMap_some]
 
 theorem head_filterMap_of_eq_some {f : α → Option β} {l : List α} (w : l ≠ []) {b : β} (h : f (l.head w) = some b) :
     (filterMap f l).head ((ne_nil_of_mem (mem_filterMap.2 ⟨_, head_mem w, h⟩))) =
@@ -1506,7 +1521,7 @@ theorem head_filterMap_of_eq_some {f : α → Option β} {l : List α} (w : l
   | nil => simp at w
   | cons a l =>
     simp only [head_cons] at h
-    simp [filterMap_cons, h]
+    simp [h]
 
 @[grind →]
 theorem forall_none_of_filterMap_eq_nil (h : filterMap f xs = []) : ∀ x ∈ xs, f x = none := by
@@ -1598,7 +1613,7 @@ theorem getElem?_append_left {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.leng
     (l₁ ++ l₂)[i]? = l₁[i]? := by
   have hn' : i < (l₁ ++ l₂).length := Nat.lt_of_lt_of_le hn <|
     length_append .. ▸ Nat.le_add_right ..
-  simp_all [getElem?_eq_getElem, getElem_append]
+  simp_all
 
 theorem getElem?_append_right : ∀ {l₁ l₂ : List α} {i : Nat}, l₁.length ≤ i →
   (l₁ ++ l₂)[i]? = l₂[i - l₁.length]?
@@ -1802,10 +1817,10 @@ theorem filterMap_eq_append_iff {f : α → Option β} :
         intro h
         rcases cons_eq_append_iff.mp h with (⟨rfl, rfl⟩ | ⟨_, ⟨rfl, h⟩⟩)
         · refine ⟨[], x :: l, ?_⟩
-          simp [filterMap_cons, w]
+          simp [w]
         · obtain ⟨l₁, l₂, rfl, rfl, rfl⟩ := ih ‹_›
           refine ⟨x :: l₁, l₂, ?_⟩
-          simp [filterMap_cons, w]
+          simp [w]
   · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
     simp
 
@@ -1870,7 +1885,7 @@ theorem append_concat {a : α} {l₁ l₂ : List α} : l₁ ++ concat l₂ a = c
 theorem map_concat {f : α → β} {a : α} {l : List α} : map f (concat l a) = concat (map f l) (f a) := by
   induction l with
   | nil => rfl
-  | cons x xs ih => simp [ih]
+  | cons x xs ih => simp
 
 theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ l' b, l = concat l' b
   | [] => .inl rfl
@@ -2176,7 +2191,7 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
   · rw [getElem?_eq_none (by simpa using h), if_neg h]
 
 @[simp] theorem getElem?_replicate_of_lt {n : Nat} {i : Nat} (h : i < n) : (replicate n a)[i]? = some a := by
-  simp [getElem?_replicate, h]
+  simp [h]
 
 @[grind] theorem head?_replicate {a : α} {n : Nat} : (replicate n a).head? = if n = 0 then none else some a := by
   cases n <;> simp [replicate_succ]
@@ -2314,8 +2329,8 @@ theorem filterMap_replicate_of_some {f : α → Option β} (h : f a = some b) :
   induction n with
   | zero => simp
   | succ n ih =>
-    simp only [replicate_succ, flatten_cons, ih, replicate_append_replicate, replicate_inj, or_true,
-      and_true, add_one_mul, Nat.add_comm]
+    simp only [replicate_succ, flatten_cons, ih, replicate_append_replicate, 
+      add_one_mul, Nat.add_comm]
 
 theorem flatMap_replicate {β} {f : α → List β} : (replicate n a).flatMap f = (replicate n (f a)).flatten := by
   induction n with
@@ -2398,7 +2413,7 @@ theorem reverse_ne_nil_iff {xs : List α} : xs.reverse ≠ [] ↔ xs ≠ [] :=
 theorem getElem?_reverse' : ∀ {l : List α} {i j}, i + j + 1 = length l →
     l.reverse[i]? = l[j]?
   | [], _, _, _ => rfl
-  | a::l, i, 0, h => by simp [Nat.succ.injEq] at h; simp [h, getElem?_append_right, Nat.succ.injEq]
+  | a::l, i, 0, h => by simp [Nat.succ.injEq] at h; simp [h]
   | a::l, i, j+1, h => by
     have := Nat.succ.inj h; simp at this ⊢
     rw [getElem?_append_left, getElem?_reverse' this]
@@ -2634,7 +2649,7 @@ theorem foldl_map_hom {g : α → β} {f : α → α → α} {f' : β → β →
     (l.map g).foldl f' (g a) = g (l.foldl f a) := by
   induction l generalizing a
   · simp
-  · simp [*, h]
+  · simp [*]
 
 @[deprecated foldl_map_hom (since := "2025-01-20")] abbrev foldl_map' := @foldl_map_hom
 
@@ -2643,7 +2658,7 @@ theorem foldr_map_hom {g : α → β} {f : α → α → α} {f' : β → β →
     (l.map g).foldr f' (g a) = g (l.foldr f a) := by
   induction l generalizing a
   · simp
-  · simp [*, h]
+  · simp [*]
 
 @[deprecated foldr_map_hom (since := "2025-01-20")] abbrev foldr_map' := @foldr_map_hom
 
@@ -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
@@ -2832,7 +2847,7 @@ theorem foldr_rel {l : List α} {f g : α → β → β} {a b : β} {r : β →
     by_cases h' : l = []
     · simp_all
     · simp only [head_eq_iff_head?_eq_some, head?_reverse] at ih
-      simp [ih, h, h', getLast_cons, head_eq_iff_head?_eq_some]
+      simp [ih, h', getLast_cons, head_eq_iff_head?_eq_some]
 
 theorem getLast_eq_head_reverse {l : List α} (h : l ≠ []) :
     l.getLast h = l.reverse.head (by simp_all) := by
@@ -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
@@ -3372,20 +3391,21 @@ 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
   | nil => simp
   | cons x xs ih =>
     cases i with
-    | zero => simp [ih]
+    | zero => simp
     | succ i =>
       simp only [replace_cons, take_succ_cons]
       split <;> simp_all
 
 @[simp] theorem replace_replicate_self [LawfulBEq α] {a : α} (h : 0 < n) :
     (replicate n a).replace a b = b :: replicate (n - 1) a := by
-  cases n <;> simp_all [replicate_succ, replace_cons]
+  cases n <;> simp_all [replicate_succ]
 
 @[simp] theorem replace_replicate_ne [LawfulBEq α] {a b c : α} (h : !b == a) :
     (replicate n a).replace b c = replicate n a := by
@@ -3487,11 +3507,11 @@ theorem getElem?_insert_succ {l : List α} {a : α} {i : Nat} :
   apply Option.some.inj
   rw [← getElem?_eq_getElem, getElem?_insert]
   split
-  · simp [getElem?_eq_getElem, h]
+  · simp [h]
   · split
     · rfl
     · have h' : i - 1 < l.length := Nat.lt_of_le_of_lt (Nat.pred_le _) h
-      simp [getElem?_eq_getElem, h']
+      simp [h']
 
 theorem head?_insert {l : List α} {a : α} :
     (l.insert a).head? = some (if h : a ∈ l then l.head (ne_nil_of_mem h) else a) := by
@@ -3512,7 +3532,7 @@ theorem head_insert {l : List α} {a : α} (w) :
 
 theorem insert_append_of_mem_left {l₁ l₂ : List α} (h : a ∈ l₂) :
     (l₁ ++ l₂).insert a = l₁ ++ l₂ := by
-  simp [insert_append, h]
+  simp [h]
 
 theorem insert_append_of_not_mem_left {l₁ l₂ : List α} (h : ¬ a ∈ l₂) :
     (l₁ ++ l₂).insert a = l₁.insert a ++ l₂ := by
@@ -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
@@ -3670,7 +3693,7 @@ theorem getElem!_of_getElem? [Inhabited α] : ∀ {l : List α} {i : Nat}, l[i]?
     rw [getElem!_pos] <;> simp_all
   | _::l, _+1, e => by
     simp at e
-    simp_all [getElem!_of_getElem? (l := l) e]
+    simp_all
 
 theorem ext_get {l₁ l₂ : List α} (hl : length l₁ = length l₂)
     (h : ∀ n h₁ h₂, get l₁ ⟨n, h₁⟩ = get l₂ ⟨n, h₂⟩) : l₁ = l₂ :=

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
@@ -217,8 +218,8 @@ theorem mapFinIdx_eq_cons_iff {l : List α} {b : β} {f : (i : Nat) → α → (
   cases l with
   | nil => simp
   | cons x l' =>
-    simp only [mapFinIdx_cons, cons.injEq, length_cons, Fin.zero_eta, Fin.cast_succ_eq,
-      exists_and_left]
+    simp only [mapFinIdx_cons, cons.injEq, 
+      ]
     constructor
     · rintro ⟨rfl, rfl⟩
       refine ⟨x, l', ⟨rfl, rfl⟩, by simp⟩
@@ -266,7 +267,7 @@ theorem mapFinIdx_eq_append_iff {l : List α} {f : (i : Nat) → α → (h : i <
       · simp
         omega
       · intro i hi₁ hi₂
-        simp only [getElem_mapFinIdx, getElem_take]
+        simp only [getElem_mapFinIdx]
         simp only [length_take, getElem_drop]
         have : l₁.length ≤ l.length := by omega
         simp only [Nat.min_eq_left this, Nat.add_comm]
@@ -285,9 +286,9 @@ theorem mapFinIdx_eq_append_iff {l : List α} {f : (i : Nat) → α → (h : i <
 theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : (i : Nat) → α → (h : i < l.length) → β} :
     l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Nat) (h : i < l.length), f i l[i] h = g i l[i] h := by
   rw [eq_comm, mapFinIdx_eq_iff]
-  simp [Fin.forall_iff]
+  simp
 
-@[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
 
@@ -333,14 +334,14 @@ 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},
     (mapIdx.go f l acc)[i]? =
       if h : i < acc.size then some acc[i] else Option.map (f i) l[i - acc.size]?
   | [], acc, i => by
-    simp only [mapIdx.go, Array.toListImpl_eq, getElem?_def, Array.length_toList,
+    simp only [mapIdx.go, getElem?_def, Array.length_toList,
       ← Array.getElem_toList, length_nil, Nat.not_lt_zero, ↓reduceDIte, Option.map_none]
   | a :: l, acc, i => by
     rw [mapIdx.go, getElem?_mapIdx_go]
@@ -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]
 
@@ -517,12 +519,12 @@ 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
   by_cases h : i < l.length
-  · simp [getElem?_reverse, h]
+  · simp [h]
     congr
     omega
   · simp at h

src/Init/Data/List/Monadic.lean

@@ -60,27 +60,27 @@ theorem mapM'_eq_mapM [Monad m] [LawfulMonad m] {f : α → m β} {l : List α}
     | [], acc => by simp [mapM.loop, mapM']
     | a::l, acc => by simp [go l, mapM.loop, mapM']
 
-@[simp] theorem mapM_nil [Monad m] {f : α → m β} : [].mapM f = pure [] := rfl
+@[simp, grind =] theorem mapM_nil [Monad m] {f : α → m β} : [].mapM f = pure [] := rfl
 
-@[simp] theorem mapM_cons [Monad m] [LawfulMonad m] {f : α → m β} :
+@[simp, grind =] theorem mapM_cons [Monad m] [LawfulMonad m] {f : α → m β} :
     (a :: l).mapM f = (return (← f a) :: (← l.mapM f)) := by simp [← mapM'_eq_mapM, mapM']
 
 @[simp] theorem mapM_pure [Monad m] [LawfulMonad m] {l : List α} {f : α → β} :
     l.mapM (m := m) (pure <| f ·) = pure (l.map f) := by
   induction l <;> simp_all
 
-@[simp] theorem idRun_mapM {l : List α} {f : α → Id β} : (l.mapM f).run = l.map (f · |>.run) :=
+@[simp, grind =] theorem idRun_mapM {l : List α} {f : α → Id β} : (l.mapM f).run = l.map (f · |>.run) :=
   mapM_pure
 
 @[deprecated idRun_mapM (since := "2025-05-21")]
 theorem mapM_id {l : List α} {f : α → Id β} : (l.mapM f).run = l.map (f · |>.run) :=
   mapM_pure
 
-@[simp] theorem mapM_map [Monad m] [LawfulMonad m] {f : α → β} {g : β → m γ} {l : List α} :
+@[simp, grind =] theorem mapM_map [Monad m] [LawfulMonad m] {f : α → β} {g : β → m γ} {l : List α} :
     (l.map f).mapM g = l.mapM (g ∘ f) := by
   induction l <;> simp_all
 
-@[simp] theorem mapM_append [Monad m] [LawfulMonad m] {f : α → m β} {l₁ l₂ : List α} :
+@[simp, grind =] theorem mapM_append [Monad m] [LawfulMonad m] {f : α → m β} {l₁ l₂ : List α} :
     (l₁ ++ l₂).mapM f = (return (← l₁.mapM f) ++ (← l₂.mapM f)) := by induction l₁ <;> simp [*]
 
 /-- Auxiliary lemma for `mapM_eq_reverse_foldlM_cons`. -/
@@ -90,8 +90,8 @@ theorem foldlM_cons_eq_append [Monad m] [LawfulMonad m] {f : α → m β} {as :
   induction as generalizing b bs with
   | nil => simp
   | cons a as ih =>
-    simp only [bind_pure_comp] at ih
-    simp [ih, _root_.map_bind, Functor.map_map, Function.comp_def]
+    simp only at ih
+    simp [ih, _root_.map_bind, Functor.map_map]
 
 theorem mapM_eq_reverse_foldlM_cons [Monad m] [LawfulMonad m] {f : α → m β} {l : List α} :
     mapM f l = reverse <$> (l.foldlM (fun acc a => (· :: acc) <$> f a) []) := by
@@ -99,14 +99,14 @@ theorem mapM_eq_reverse_foldlM_cons [Monad m] [LawfulMonad m] {f : α → m β}
   induction l with
   | nil => simp
   | cons a as ih =>
-    simp only [mapM'_cons, ih, bind_map_left, foldlM_cons, LawfulMonad.bind_assoc, pure_bind,
-      foldlM_cons_eq_append, _root_.map_bind, Functor.map_map, Function.comp_def, reverse_append,
+    simp only [mapM'_cons, ih, bind_map_left, foldlM_cons, 
+      foldlM_cons_eq_append, _root_.map_bind, Functor.map_map, reverse_append,
       reverse_cons, reverse_nil, nil_append, singleton_append]
     simp [bind_pure_comp]
 
 /-! ### filterMapM -/
 
-@[simp] theorem filterMapM_nil [Monad m] {f : α → m (Option β)} : [].filterMapM f = pure [] := rfl
+@[simp, grind =] theorem filterMapM_nil [Monad m] {f : α → m (Option β)} : [].filterMapM f = pure [] := rfl
 
 theorem filterMapM_loop_eq [Monad m] [LawfulMonad m] {f : α → m (Option β)} {l : List α} {acc : List β} :
     filterMapM.loop f l acc = (acc.reverse ++ ·) <$> filterMapM.loop f l [] := by
@@ -121,7 +121,7 @@ theorem filterMapM_loop_eq [Monad m] [LawfulMonad m] {f : α → m (Option β)}
     · rw [ih, ih (acc := [b])]
       simp
 
-@[simp] theorem filterMapM_cons [Monad m] [LawfulMonad m] {f : α → m (Option β)} :
+@[simp, grind =] theorem filterMapM_cons [Monad m] [LawfulMonad m] {f : α → m (Option β)} :
     (a :: l).filterMapM f = do
       match (← f a) with
       | none => filterMapM f l
@@ -137,20 +137,20 @@ theorem filterMapM_loop_eq [Monad m] [LawfulMonad m] {f : α → m (Option β)}
 
 /-! ### flatMapM -/
 
-@[simp] theorem flatMapM_nil [Monad m] {f : α → m (List β)} : [].flatMapM f = pure [] := rfl
+@[simp, grind =] theorem flatMapM_nil [Monad m] {f : α → m (List β)} : [].flatMapM f = pure [] := rfl
 
 theorem flatMapM_loop_eq [Monad m] [LawfulMonad m] {f : α → m (List β)} {l : List α} {acc : List (List β)} :
     flatMapM.loop f l acc = (acc.reverse.flatten ++ ·) <$> flatMapM.loop f l [] := by
   induction l generalizing acc with
   | nil => simp [flatMapM.loop]
   | cons a l ih =>
-    simp only [flatMapM.loop, append_nil, _root_.map_bind]
+    simp only [flatMapM.loop, _root_.map_bind]
     congr
     funext bs
     rw [ih, ih (acc := [bs])]
     simp
 
-@[simp] theorem flatMapM_cons [Monad m] [LawfulMonad m] {f : α → m (List β)} :
+@[simp, grind =] theorem flatMapM_cons [Monad m] [LawfulMonad m] {f : α → m (List β)} :
     (a :: l).flatMapM f = do
       let bs ← f a
       return (bs ++ (← l.flatMapM f)) := by
@@ -230,11 +230,11 @@ theorem forM_cons' [Monad m] :
     (a::as).forM f = (f a >>= fun _ => as.forM f : m PUnit) :=
   List.forM_cons
 
-@[simp] theorem forM_append [Monad m] [LawfulMonad m] {l₁ l₂ : List α} {f : α → m PUnit} :
+@[simp, grind =] theorem forM_append [Monad m] [LawfulMonad m] {l₁ l₂ : List α} {f : α → m PUnit} :
     forM (l₁ ++ l₂) f = (do forM l₁ f; forM l₂ f) := by
   induction l₁ <;> simp [*]
 
-@[simp] theorem forM_map [Monad m] [LawfulMonad m] {l : List α} {g : α → β} {f : β → m PUnit} :
+@[simp, grind =] theorem forM_map [Monad m] [LawfulMonad m] {l : List α} {g : α → β} {f : β → m PUnit} :
     forM (l.map g) f = forM l (fun a => f (g a)) := by
   induction l <;> simp [*]
 
@@ -257,7 +257,7 @@ theorem forIn'_loop_congr [Monad m] {as bs : List α}
       · simp
         rw [ih]
 
-@[simp] theorem forIn'_cons [Monad m] {a : α} {as : List α}
+@[simp, grind =] theorem forIn'_cons [Monad m] {a : α} {as : List α}
     (f : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)) (b : β) :
     forIn' (a::as) b f = f a mem_cons_self b >>=
       fun | ForInStep.done b => pure b | ForInStep.yield b => forIn' as b fun a' m b => f a' (mem_cons_of_mem a m) b := by
@@ -270,7 +270,7 @@ theorem forIn'_loop_congr [Monad m] {as bs : List α}
     intros
     rfl
 
-@[simp] theorem forIn_cons [Monad m] (f : α → β → m (ForInStep β)) (a : α) (as : List α) (b : β) :
+@[simp, grind =] theorem forIn_cons [Monad m] (f : α → β → m (ForInStep β)) (a : α) (as : List α) (b : β) :
     forIn (a::as) b f = f a b >>= fun | ForInStep.done b => pure b | ForInStep.yield b => forIn as b f := by
   have := forIn'_cons (a := a) (as := as) (fun a' _ b => f a' b) b
   simpa only [forIn'_eq_forIn]
@@ -363,7 +363,7 @@ theorem forIn'_yield_eq_foldl
       l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init :=
   forIn'_pure_yield_eq_foldl _ _
 
-@[simp] theorem forIn'_map [Monad m] [LawfulMonad m]
+@[simp, grind =] theorem forIn'_map [Monad m] [LawfulMonad m]
     {l : List α} (g : α → β) (f : (b : β) → b ∈ l.map g → γ → m (ForInStep γ)) :
     forIn' (l.map g) init f = forIn' l init fun a h y => f (g a) (mem_map_of_mem h) y := by
   induction l generalizing init <;> simp_all
@@ -381,7 +381,7 @@ theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
   induction l generalizing init with
   | nil => simp
   | cons a as ih =>
-    simp only [foldlM_cons, bind_pure_comp, forIn_cons, _root_.map_bind]
+    simp only [foldlM_cons, forIn_cons, _root_.map_bind]
     congr 1
     funext x
     match x with
@@ -422,7 +422,7 @@ theorem forIn_yield_eq_foldl
       l.foldl (fun b a => f a b) init :=
   forIn_pure_yield_eq_foldl _ _
 
-@[simp] theorem forIn_map [Monad m] [LawfulMonad m]
+@[simp, grind =] theorem forIn_map [Monad m] [LawfulMonad m]
     {l : List α} {g : α → β} {f : β → γ → m (ForInStep γ)} :
     forIn (l.map g) init f = forIn l init fun a y => f (g a) y := by
   induction l generalizing init <;> simp_all
@@ -444,7 +444,7 @@ theorem allM_eq_not_anyM_not [Monad m] [LawfulMonad m] {p : α → m Bool} {as :
   induction as with
   | nil => simp
   | cons a as ih =>
-    simp only [anyM, ih, pure_bind, all_cons]
+    simp only [anyM, ih, pure_bind]
     split <;> simp_all
 
 @[simp] theorem allM_pure [Monad m] [LawfulMonad m] {p : α → Bool} {as : List α} :
@@ -486,7 +486,7 @@ and simplifies these to the function directly taking the value.
   induction l generalizing x with
   | nil => simp
   | cons a l ih =>
-    simp [ih, hf, foldrM_cons]
+    simp [ih, foldrM_cons]
     congr
     funext b
     simp [hf]

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
@@ -67,7 +68,7 @@ theorem length_filterMap_pos_iff {xs : List α} {f : α → Option β} :
   | cons x xs ih =>
     simp only [filterMap, mem_cons, exists_prop, exists_eq_or_imp]
     split
-    · simp_all [ih]
+    · simp_all
     · simp_all
 
 @[simp]
@@ -114,8 +115,8 @@ section intersperse
 
 variable {l : List α} {sep : α} {i : Nat}
 
-@[simp] theorem length_intersperse : (l.intersperse sep).length = 2 * l.length - 1 := by
-  fun_induction intersperse <;> simp only [intersperse, length_cons, length_nil] at *
+@[simp, grind =] theorem length_intersperse : (l.intersperse sep).length = 2 * l.length - 1 := by
+  fun_induction intersperse <;> simp only [length_cons, length_nil] at *
   rename_i h _
   have := length_pos_iff.mpr h
   omega
@@ -193,7 +194,7 @@ theorem mem_eraseIdx_iff_getElem {x : α} :
   | a::l, 0 => by simp [mem_iff_getElem, Nat.succ_lt_succ_iff]
   | a::l, k+1 => by
     rw [← Nat.or_exists_add_one]
-    simp [mem_eraseIdx_iff_getElem, @eq_comm _ a, succ_inj, Nat.succ_lt_succ_iff]
+    simp [mem_eraseIdx_iff_getElem, @eq_comm _ a, Nat.succ_lt_succ_iff]
 
 theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l[i]? = some x := by
   simp only [mem_eraseIdx_iff_getElem, getElem_eq_iff, exists_and_left]

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
@@ -42,7 +45,7 @@ theorem countP_replace [BEq α] [LawfulBEq α] {a b : α} {l : List α} {p : α
     simp [replace_cons]
     split <;> rename_i h
     · simp at h
-      simp [h, ih, countP_cons]
+      simp [h, countP_cons]
       omega
     · simp only [beq_eq_false_iff_ne, ne_eq] at h
       simp only [countP_cons, ih, contains_eq_mem, decide_eq_true_eq, mem_cons, h, false_or]
@@ -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
+  · rw [count_cons]
+    split <;> rename_i h'
+    · simp only [beq_iff_eq] at h'
+      rw [count_eq_zero.mpr (h' ▸ h)]
+      simp
+    · 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

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]
@@ -28,7 +29,7 @@ theorem getElem?_eraseIdx {l : List α} {i : Nat} {j : Nat} :
     · simp only [length_take, Nat.min_def, Nat.not_lt] at h
       split at h
       · omega
-      · simp_all [getElem?_eq_none]
+      · simp_all
         omega
     · simp only [length_take]
       simp only [length_take, Nat.min_def, Nat.not_lt] at h
@@ -45,10 +46,11 @@ theorem getElem?_eraseIdx_of_lt {l : List α} {i : Nat} {j : Nat} (h : j < i) :
 theorem getElem?_eraseIdx_of_ge {l : List α} {i : Nat} {j : Nat} (h : i ≤ j) :
     (l.eraseIdx i)[j]? = l[j + 1]? := by
   rw [getElem?_eraseIdx]
-  simp only [dite_eq_ite, ite_eq_right_iff]
+  simp only [ite_eq_right_iff]
   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]

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

@@ -15,8 +15,7 @@ Proves various lemmas about `List.insertIdx`.
 -/
 
 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 Function Nat
 
@@ -30,19 +29,20 @@ 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 +56,9 @@ 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 +74,15 @@ 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 +114,14 @@ 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 _ _
+
+@[deprecated eraseIdx_insertIdx_self (since := "2025-06-18")]
+abbrev eraseIdx_insertIdx := @eraseIdx_insertIdx_self
+
 @[simp]
 theorem insertIdx_length_self {l : List α} {x : α} : l.insertIdx l.length x = l ++ [x] := by
   induction l with
@@ -185,6 +197,7 @@ 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 +214,7 @@ 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

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
 
@@ -160,12 +172,12 @@ theorem modifyHead_eq_modify_zero (f : α → α) (l : List α) :
     ∀ i (l : List α) j, (l.modify i f)[j]? = (fun a => if i = j then f a else a) <$> l[j]?
   | n, l, 0 => by cases l <;> cases n <;> simp
   | n, [], _+1 => by cases n <;> rfl
-  | 0, _ :: l, j+1 => by cases h : l[j]? <;> simp [h, modify, j.succ_ne_zero.symm]
+  | 0, _ :: l, j+1 => by cases h : l[j]? <;> simp [h, modify]
   | i+1, a :: l, j+1 => by
     simp only [modify_succ_cons, getElem?_cons_succ, Nat.reduceEqDiff, Option.map_eq_map]
     refine (getElem?_modify f i l j).trans ?_
     cases h' : l[j]? <;> by_cases h : i = j <;>
-      simp [h, if_pos, if_neg, Option.map, mt Nat.succ.inj, not_false_iff, h']
+      simp [h, Option.map]
 
 @[simp, grind =] theorem length_modify (f : α → α) : ∀ (l : List α) i, (l.modify i f).length = l.length :=
   length_modifyTailIdx _ fun l => by cases l <;> rfl
@@ -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
@@ -213,7 +226,7 @@ theorem modify_modify_ne (f g : α → α) {i j} (l : List α) (h : i ≠ j) :
   apply ext_getElem
   · simp
   · intro m' h₁ h₂
-    simp only [getElem_modify, getElem_modify_ne, h₂]
+    simp only [getElem_modify]
     split <;> split <;> first | rfl | omega
 
 theorem modify_eq_set [Inhabited α] (f : α → α) (i) (l : List α) :
@@ -221,7 +234,7 @@ theorem modify_eq_set [Inhabited α] (f : α → α) (i) (l : List α) :
   apply ext_getElem
   · simp
   · intro m h₁ h₂
-    simp [getElem_modify, getElem_set, h₂]
+    simp [getElem_modify, getElem_set]
     split <;> rename_i h
     · subst h
       simp only [length_modify] at h₁
@@ -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
@@ -274,7 +287,7 @@ theorem drop_modify_of_ge (f : α → α) (i j) (l : List α) (h : i ≥ j) :
   apply ext_getElem
   · simp
   · intro m' h₁ h₂
-    simp [getElem_drop, getElem_modify, ite_eq_right_iff]
+    simp [getElem_drop, getElem_modify]
     split <;> split <;> first | rfl | omega
 
 theorem eraseIdx_modify_of_eq (f : α → α) (i) (l : List α) :

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
@@ -478,7 +508,7 @@ theorem zipIdx_eq_append_iff {l : List α} {k : Nat} :
   · rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
     simp only [zipIdx_eq_zip_range']
     refine ⟨l₁', l₂', range' k l₁'.length, range' (k + l₁'.length) l₂'.length, ?_⟩
-    simp [Nat.add_comm]
+    simp
 
 /-! ### enumFrom -/
 
@@ -605,7 +635,7 @@ theorem enumFrom_eq_append_iff {l : List α} {n : Nat} :
   · rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
     simp only [enumFrom_eq_zip_range']
     refine ⟨range' n l₁'.length, range' (n + l₁'.length) l₂'.length, l₁', l₂', ?_⟩
-    simp [Nat.add_comm]
+    simp
 
 end
 

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
@@ -140,7 +141,7 @@ theorem suffix_iff_eq_drop : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length
     xs.take i <+: xs.take j ↔ i ≤ j := by
   simp only [prefix_iff_eq_take, length_take]
   induction i generalizing xs j with
-  | zero => simp [Nat.min_eq_left, eq_self_iff_true, Nat.zero_le, take]
+  | zero => simp [Nat.min_eq_left, Nat.zero_le, take]
   | succ i IH =>
     cases xs with
     | nil => simp_all
@@ -149,7 +150,7 @@ theorem suffix_iff_eq_drop : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length
       | zero =>
         simp
       | succ j =>
-        simp only [length_cons, Nat.succ_eq_add_one, Nat.add_lt_add_iff_right] at hm
+        simp only [length_cons, Nat.add_lt_add_iff_right] at hm
         simp [← @IH j xs hm, Nat.min_eq_left, Nat.le_of_lt hm]
 
 @[simp] theorem append_left_sublist_self {xs : List α} (ys : List α) : xs ++ ys <+ ys ↔ xs = [] := by
@@ -192,7 +193,7 @@ theorem append_sublist_of_sublist_right {xs ys zs : List α} (h : zs <+ ys) :
     have hl' := h'.length_le
     simp only [length_append] at hl'
     have : xs.length = 0 := by omega
-    simp_all only [Nat.zero_add, length_eq_zero_iff, true_and, append_nil]
+    simp_all only [Nat.zero_add, length_eq_zero_iff, true_and]
     exact Sublist.eq_of_length_le h' hl
   · rintro ⟨rfl, rfl⟩
     simp