chore: missing modifier

This PR adds the `reduceCtorIdx` simproc which recognizes and reduces
`ctorIdx` applications. This is not on by default yet because it does
not use the discrimination tree (yet).

.gitattributes

@@ -4,3 +4,9 @@ RELEASES.md merge=union
 stage0/** binary linguist-generated
 # The following file is often manually edited, so do show it in diffs
 stage0/src/stdlib_flags.h -binary -linguist-generated
+# These files should not have line endings translated on Windows, because
+# it throws off parser tests. Later lines override earlier ones, so the
+# runner code is still treated as ordinary text.
+tests/lean/docparse/* eol=lf
+tests/lean/docparse/*.lean eol=auto
+tests/lean/docparse/*.sh eol=auto

.github/workflows/actionlint.yml

@@ -15,7 +15,7 @@ jobs:
     runs-on: ubuntu-latest
     steps:
     - name: Checkout
-      uses: actions/checkout@v4
+      uses: actions/checkout@v5
     - name: actionlint
       uses: raven-actions/actionlint@v2
       with:

.github/workflows/build-template.yml

@@ -70,7 +70,7 @@ jobs:
         if: runner.os == 'macOS'
       - name: Checkout
         if: (!endsWith(matrix.os, '-with-cache'))
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         with:
           # the default is to use a virtual merge commit between the PR and master: just use the PR
           ref: ${{ github.event.pull_request.head.sha }}

.github/workflows/check-prelude.yml

@@ -7,7 +7,7 @@ jobs:
     runs-on: ubuntu-latest
     steps:
       - name: Checkout
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         with:
           # the default is to use a virtual merge commit between the PR and master: just use the PR
           ref: ${{ github.event.pull_request.head.sha }}

.github/workflows/check-stage0.yml

@@ -8,7 +8,7 @@ jobs:
   check-stage0-on-queue:
     runs-on: ubuntu-latest
     steps:
-    - uses: actions/checkout@v4
+    - uses: actions/checkout@v5
       with:
         ref: ${{ github.event.pull_request.head.sha }}
         filter: blob:none

.github/workflows/ci.yml

@@ -54,7 +54,7 @@ jobs:
 
     steps:
       - name: Checkout
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         # don't schedule nightlies on forks
         if: github.event_name == 'schedule' && github.repository == 'leanprover/lean4' || inputs.action == 'release nightly'
       - name: Set Nightly
@@ -363,7 +363,7 @@ jobs:
     runs-on: ubuntu-latest
     needs: build
     steps:
-      - uses: actions/download-artifact@v4
+      - uses: actions/download-artifact@v5
         with:
           path: artifacts
       - name: Release
@@ -388,12 +388,12 @@ jobs:
     runs-on: ubuntu-latest
     steps:
       - name: Checkout
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         with:
           # needed for tagging
           fetch-depth: 0
           token: ${{ secrets.PUSH_NIGHTLY_TOKEN }}
-      - uses: actions/download-artifact@v4
+      - uses: actions/download-artifact@v5
         with:
           path: artifacts
       - name: Prepare Nightly Release

.github/workflows/copyright-header.yml

@@ -6,7 +6,7 @@ jobs:
   check-lean-files:
     runs-on: ubuntu-latest
     steps:
-    - uses: actions/checkout@v4
+    - uses: actions/checkout@v5
 
     - name: Verify .lean files start with a copyright header.
       run: |

.github/workflows/pr-release.yml

@@ -395,7 +395,7 @@ jobs:
       # Checkout the Batteries repository with all branches
       - name: Checkout Batteries repository
         if: steps.workflow-info.outputs.pullRequestNumber != '' && steps.ready.outputs.mathlib_ready == 'true'
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         with:
           repository: leanprover-community/batteries
           token: ${{ secrets.MATHLIB4_BOT }}
@@ -454,7 +454,7 @@ jobs:
       # Checkout the mathlib4 repository with all branches
       - name: Checkout mathlib4 repository
         if: steps.workflow-info.outputs.pullRequestNumber != '' && steps.ready.outputs.mathlib_ready == 'true'
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         with:
           repository: leanprover-community/mathlib4-nightly-testing
           token: ${{ secrets.MATHLIB4_BOT }}
@@ -524,7 +524,7 @@ jobs:
       # Checkout the reference manual repository with all branches
       - name: Checkout mathlib4 repository
         if: steps.workflow-info.outputs.pullRequestNumber != '' && steps.reference-manual-ready.outputs.manual_ready == 'true'
-        uses: actions/checkout@v4
+        uses: actions/checkout@v5
         with:
           repository: leanprover/reference-manual
           token: ${{ secrets.MANUAL_PR_BOT }}

.github/workflows/update-stage0.yml

@@ -21,11 +21,13 @@ jobs:
     runs-on: nscloud-ubuntu-22.04-amd64-8x16
     env:
       CCACHE_DIR: ${{ github.workspace }}/.ccache
+      CCACHE_COMPRESS: true
+      CCACHE_MAXSIZE: 400M
     steps:
     # This action should push to an otherwise protected branch, so it
     # uses a deploy key with write permissions, as suggested at
     # https://stackoverflow.com/a/76135647/946226
-    - uses: actions/checkout@v4
+    - uses: actions/checkout@v5
       with:
         ssh-key: ${{secrets.STAGE0_SSH_KEY}}
     - run: echo "should_update_stage0=yes" >> "$GITHUB_ENV"
@@ -72,10 +74,14 @@ jobs:
         restore-keys: |
           Linux Lake-build-v3
     - if: env.should_update_stage0 == 'yes'
-      run: cmake --preset release
+      # sync options with `Linux Lake` to ensure cache reuse
+      run: |
+        mkdir -p build
+        cmake --preset release -B build -DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true
       shell: 'nix develop -c bash -euxo pipefail {0}'
     - if: env.should_update_stage0 == 'yes'
-      run: make -j$NPROC -C build/release  update-stage0-commit
+      run: |
+        make -j$NPROC -C build update-stage0-commit
       shell: 'nix develop -c bash -euxo pipefail {0}'
     - if: env.should_update_stage0 == 'yes'
       run: git show --stat

README.md

@@ -2,19 +2,19 @@ This is the repository for **Lean 4**.
 
 # About
 
-- [Quickstart](https://lean-lang.org/documentation/setup/)
+- [Quickstart](https://lean-lang.org/install/)
 - [Homepage](https://lean-lang.org)
 - [Theorem Proving Tutorial](https://lean-lang.org/theorem_proving_in_lean4/)
 - [Functional Programming in Lean](https://lean-lang.org/functional_programming_in_lean/)
-- [Documentation Overview](https://lean-lang.org/documentation/)
+- [Documentation Overview](https://lean-lang.org/learn/)
 - [Language Reference](https://lean-lang.org/doc/reference/latest/)
 - [Release notes](RELEASES.md) starting at v4.0.0-m3
-- [Examples](https://lean-lang.org/lean4/doc/examples.html)
+- [Examples](https://lean-lang.org/examples/)
 - [External Contribution Guidelines](CONTRIBUTING.md)
 
 # Installation
 
-See [Setting Up Lean](https://lean-lang.org/documentation/setup/).
+See [Install Lean](https://lean-lang.org/install/).
 
 # Contributing
 

doc/dev/index.md

@@ -99,3 +99,19 @@ on to `nightly-with-manual` branch. (It is fine to force push after rebasing.)
 CI will generate a branch of the reference manual called `lean-pr-testing-NNNN`
 in `leanprover/reference-manual`. This branch uses the toolchain for your PR,
 and will report back to the Lean PR with results from Mathlib CI.
+
+### Avoiding rebuilds for downstream projects
+
+If you want to test changes to Lean on downstream projects and would like to avoid rebuilding modules you have already built/fetched using the project's configured Lean toolchain, you can often do so as long as your build of Lean is close enough to that Lean toolchain (compatible .olean format including structure of all relevant environment extensions).
+
+To override the toolchain without rebuilding for a single command, for example `lake build` or `lake lean`, you can use the prefix
+```
+LEAN_GITHASH=$(lean --githash) lake +lean4 ...
+```
+Alternatively, use
+```
+export LEAN_GITHASH=$(lean --githash)
+export ELAN_TOOLCHAIN=lean4
+```
+to persist these changes for the lifetime of the current shell, which will affect any processes spawned from it such as VS Code started via `code .`.
+If you use a setup where you cannot directly start your editor from the command line, such as VS Code Remote, you might want to consider using [direnv](https://direnv.net/) together with an editor extension for it instead so that you can put the lines above into `.envrc`.

lean.code-workspace

@@ -37,6 +37,15 @@
 					"isDefault": true
 				}
 			},
+			{
+				"label": "build-old",
+				"type": "shell",
+				"command": "make -C build/release -j$(nproc 2>/dev/null || sysctl -n hw.logicalcpu 2>/dev/null || echo 4) LAKE_EXTRA_ARGS=--old",
+				"problemMatcher": [],
+				"group": {
+					"kind": "build"
+				}
+			},
 			{
 				"label": "test",
 				"type": "shell",

script/merge_remote.py

@@ -5,6 +5,7 @@ Merge a tag into a branch on a GitHub repository.
 
 This script checks if a specified tag can be merged cleanly into a branch and performs
 the merge if possible. If the merge cannot be done cleanly, it prints a helpful message.
+Merge conflicts in the lean-toolchain file are automatically resolved by accepting the incoming changes.
 
 Usage:
     python3 merge_remote.py <org/repo> <branch> <tag>
@@ -58,6 +59,32 @@ def clone_repo(repo, temp_dir):
     return True
 
 
+def get_conflicted_files():
+    """Get list of files with merge conflicts."""
+    result = run_command("git diff --name-only --diff-filter=U", check=False)
+    if result.returncode == 0:
+        return result.stdout.strip().split('\n') if result.stdout.strip() else []
+    return []
+
+
+def resolve_lean_toolchain_conflict(tag):
+    """Resolve lean-toolchain conflict by accepting incoming (tag) changes."""
+    print("Resolving lean-toolchain conflict by accepting incoming changes...")
+    # Accept theirs (incoming) version for lean-toolchain
+    result = run_command(f"git checkout --theirs lean-toolchain", check=False)
+    if result.returncode != 0:
+        print("Failed to resolve lean-toolchain conflict")
+        return False
+
+    # Add the resolved file
+    add_result = run_command("git add lean-toolchain", check=False)
+    if add_result.returncode != 0:
+        print("Failed to stage resolved lean-toolchain")
+        return False
+
+    return True
+
+
 def check_and_merge(repo, branch, tag, temp_dir):
     """Check if tag can be merged into branch and perform the merge if possible."""
     # Change to the temporary directory
@@ -98,12 +125,37 @@ def check_and_merge(repo, branch, tag, temp_dir):
     # Try merging the tag directly
     print(f"Merging {tag} into {branch}...")
     merge_result = run_command(f"git merge {tag} --no-edit", check=False)
-    
+
     if merge_result.returncode != 0:
-        print(f"Cannot merge {tag} cleanly into {branch}.")
-        print("Merge conflicts would occur. Aborting merge.")
-        run_command("git merge --abort")
-        return False
+        # Check which files have conflicts
+        conflicted_files = get_conflicted_files()
+
+        if conflicted_files == ['lean-toolchain']:
+            # Only lean-toolchain has conflicts, resolve it
+            print("Merge conflict detected only in lean-toolchain.")
+            if resolve_lean_toolchain_conflict(tag):
+                # Continue the merge with the resolved conflict
+                print("Continuing merge with resolved lean-toolchain...")
+                continue_result = run_command(f"git commit --no-edit", check=False)
+                if continue_result.returncode != 0:
+                    print("Failed to complete merge after resolving lean-toolchain")
+                    run_command("git merge --abort")
+                    return False
+            else:
+                print("Failed to resolve lean-toolchain conflict")
+                run_command("git merge --abort")
+                return False
+        else:
+            # Other files have conflicts, or unable to determine
+            if conflicted_files:
+                print(f"Cannot merge {tag} cleanly into {branch}.")
+                print(f"Merge conflicts in: {', '.join(conflicted_files)}")
+            else:
+                print(f"Cannot merge {tag} cleanly into {branch}.")
+                print("Merge conflicts would occur.")
+            print("Aborting merge.")
+            run_command("git merge --abort")
+            return False
     
     print(f"Pushing changes to remote...")
     push_result = run_command(f"git push origin {branch}")

script/release_notes.py

@@ -52,6 +52,7 @@ def sort_sections_order():
     return [
         "Language",
         "Library",
+        "Tactics",
         "Compiler",
         "Pretty Printing",
         "Documentation",

script/release_repos.yml

@@ -1,4 +1,11 @@
 repositories:
+  - name: lean4-cli
+    url: https://github.com/leanprover/lean4-cli
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
   - name: batteries
     url: https://github.com/leanprover-community/batteries
     toolchain-tag: true
@@ -7,6 +14,13 @@ repositories:
     bump-branch: true
     dependencies: []
 
+  - name: verso
+    url: https://github.com/leanprover/verso
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
   - name: lean4checker
     url: https://github.com/leanprover/lean4checker
     toolchain-tag: true
@@ -21,20 +35,6 @@ repositories:
     branch: master
     dependencies: []
 
-  - name: lean4-cli
-    url: https://github.com/leanprover/lean4-cli
-    toolchain-tag: true
-    stable-branch: false
-    branch: main
-    dependencies: []
-
-  - name: verso
-    url: https://github.com/leanprover/verso
-    toolchain-tag: true
-    stable-branch: false
-    branch: main
-    dependencies: []
-
   - name: plausible
     url: https://github.com/leanprover-community/plausible
     toolchain-tag: true
@@ -96,6 +96,15 @@ repositories:
       - import-graph
       - plausible
 
+  - name: cslib
+    url: https://github.com/leanprover/cslib
+    toolchain-tag: true
+    stable-branch: true
+    branch: main
+    bump-branch: true
+    dependencies:
+      - mathlib4
+
   - name: repl
     url: https://github.com/leanprover-community/repl
     toolchain-tag: true

script/release_steps.py

@@ -377,6 +377,21 @@ def execute_release_steps(repo, version, config):
         except subprocess.CalledProcessError as e:
             print(red("Tests failed, but continuing with PR creation..."))
             print(red(f"Test error: {e}"))
+    elif repo_name == "cslib":
+        print(blue("Updating lakefile.toml..."))
+        run_command(f'perl -pi -e \'s/"v4\\.[0-9]+(\\.[0-9]+)?(-rc[0-9]+)?"/"' + version + '"/g\' lakefile.*', cwd=repo_path)
+        
+        print(blue("Updating docs/lakefile.toml..."))
+        run_command(f'perl -pi -e \'s/"v4\\.[0-9]+(\\.[0-9]+)?(-rc[0-9]+)?"/"' + version + '"/g\' lakefile.*', cwd=repo_path / "docs")
+
+        # Update lean-toolchain in docs
+        print(blue("Updating docs/lean-toolchain..."))
+        docs_toolchain = repo_path / "docs" / "lean-toolchain"
+        with open(docs_toolchain, "w") as f:
+            f.write(f"leanprover/lean4:{version}\n")
+        print(green(f"Updated docs/lean-toolchain to leanprover/lean4:{version}"))
+
+        run_command("lake update", cwd=repo_path, stream_output=True)
     elif dependencies:
         run_command(f'perl -pi -e \'s/"v4\\.[0-9]+(\\.[0-9]+)?(-rc[0-9]+)?"/"' + version + '"/g\' lakefile.*', cwd=repo_path)
         run_command("lake update", cwd=repo_path, stream_output=True)

src/CMakeLists.txt

@@ -10,7 +10,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 24)
+set(LEAN_VERSION_MINOR 25)
 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'")

src/Init.lean

@@ -42,5 +42,8 @@ public import Init.While
 public import Init.Syntax
 public import Init.Internal
 public import Init.Try
+public meta import Init.Try  -- make sure `Try.Config` can be evaluated anywhere
 public import Init.BinderNameHint
 public import Init.Task
+public import Init.MethodSpecsSimp
+public import Init.LawfulBEqTactics

src/Init/Control/Lawful/Basic.lean

@@ -147,7 +147,7 @@ class LawfulMonad (m : Type u → Type v) [Monad m] : Prop extends LawfulApplica
 
 export LawfulMonad (bind_pure_comp bind_map pure_bind bind_assoc)
 attribute [simp] pure_bind bind_assoc bind_pure_comp
-attribute [grind] pure_bind
+attribute [grind <=] pure_bind
 
 @[simp] theorem bind_pure [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x := by
   change x >>= (fun a => pure (id a)) = x

src/Init/Core.lean

@@ -1580,6 +1580,7 @@ instance {p q : Prop} [d : Decidable (p ↔ q)] : Decidable (p = q) :=
 
 gen_injective_theorems% Array
 gen_injective_theorems% BitVec
+gen_injective_theorems% ByteArray
 gen_injective_theorems% Char
 gen_injective_theorems% DoResultBC
 gen_injective_theorems% DoResultPR
@@ -2546,7 +2547,3 @@ class Irrefl (r : α → α → Prop) : Prop where
   irrefl : ∀ a, ¬r a a
 
 end Std
-
-/-- Deprecated alias for `XorOp`. -/
-@[deprecated XorOp (since := "2025-07-30")]
-abbrev Xor := XorOp

src/Init/Data/Array/Attach.lean

@@ -121,7 +121,7 @@ theorem pmap_eq_map {p : α → Prop} {f : α → β} {xs : Array α} (H) :
 theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (xs : Array α) {H₁ H₂}
     (h : ∀ a ∈ xs, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f xs H₁ = pmap g xs H₂ := by
   cases xs
-  simp only [mem_toArray] at h
+  simp only [List.mem_toArray] at h
   simp only [List.pmap_toArray, mk.injEq]
   rw [List.pmap_congr_left _ h]
 
@@ -194,14 +194,14 @@ 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, grind]
+@[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, grind]
+@[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
@@ -212,12 +212,13 @@ 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⟩
 
+grind_pattern mem_pmap_of_mem => _ ∈ pmap f xs H, a ∈ xs
+
 @[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
@@ -345,7 +346,7 @@ 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.foldl_toArray', mem_toArray, List.foldl_subtype]
+    List.foldl_toArray', List.mem_toArray, List.foldl_subtype]
   congr
   ext
   simpa using fun a => List.mem_of_getElem? a
@@ -364,7 +365,7 @@ 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.foldr_toArray', mem_toArray, List.foldr_subtype]
+    List.foldr_toArray', List.mem_toArray, List.foldr_subtype]
   congr
   ext
   simpa using fun a => List.mem_of_getElem? a
@@ -706,7 +707,7 @@ 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.flatMap_toArray, List.unattach_toArray, 
+  simp only [List.flatMap_toArray, List.unattach_toArray,
     mk.injEq]
   rw [List.flatMap_subtype]
   simp [hf]

src/Init/Data/Array/Basic.lean

@@ -40,11 +40,11 @@ namespace Array
 
 /-! ### Preliminary theorems -/
 
-@[simp, grind] theorem size_set {xs : Array α} {i : Nat} {v : α} (h : i < xs.size) :
+@[simp, grind =] theorem size_set {xs : Array α} {i : Nat} {v : α} (h : i < xs.size) :
     (set xs i v h).size = xs.size :=
   List.length_set ..
 
-@[simp, grind] theorem size_push {xs : Array α} (v : α) : (push xs v).size = xs.size + 1 :=
+@[simp, grind =] theorem size_push {xs : Array α} (v : α) : (push xs v).size = xs.size + 1 :=
   List.length_concat ..
 
 theorem ext {xs ys : Array α}
@@ -108,13 +108,19 @@ instance : Membership α (Array α) where
 theorem mem_def {a : α} {as : Array α} : a ∈ as ↔ a ∈ as.toList :=
   ⟨fun | .mk h => h, Array.Mem.mk⟩
 
-@[simp, grind =] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
+@[simp, grind =] theorem _root_.List.mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
   simp [mem_def]
 
-@[simp, grind] theorem getElem_mem {xs : Array α} {i : Nat} (h : i < xs.size) : xs[i] ∈ xs := by
+@[deprecated List.mem_toArray (since := "2025-09-04")]
+theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l :=
+  List.mem_toArray
+
+@[simp] theorem getElem_mem {xs : Array α} {i : Nat} (h : i < xs.size) : xs[i] ∈ xs := by
   rw [Array.mem_def, ← getElem_toList]
   apply List.getElem_mem
 
+grind_pattern getElem_mem => xs[i] ∈ xs
+
 @[simp, grind =] theorem emptyWithCapacity_eq {α n} : @emptyWithCapacity α n = #[] := rfl
 
 @[simp] theorem mkEmpty_eq {α n} : @mkEmpty α n = #[] := rfl
@@ -132,7 +138,7 @@ theorem toList_toArray {as : List α} : as.toArray.toList = as := rfl
 @[deprecated toList_toArray (since := "2025-02-17")]
 abbrev _root_.Array.toList_toArray := @List.toList_toArray
 
-@[simp, grind] theorem size_toArray {as : List α} : as.toArray.size = as.length := by simp [Array.size]
+@[simp, grind =] theorem size_toArray {as : List α} : as.toArray.size = as.length := by simp [Array.size]
 
 @[deprecated size_toArray (since := "2025-02-17")]
 abbrev _root_.Array.size_toArray := @List.size_toArray
@@ -197,7 +203,7 @@ Examples:
 def pop (xs : Array α) : Array α where
   toList := xs.toList.dropLast
 
-@[simp, grind] theorem size_pop {xs : Array α} : xs.pop.size = xs.size - 1 := by
+@[simp, grind =] theorem size_pop {xs : Array α} : xs.pop.size = xs.size - 1 := by
   match xs with
   | ⟨[]⟩ => rfl
   | ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]

src/Init/Data/Array/Erase.lean

@@ -91,7 +91,7 @@ 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, grind] 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
 
@@ -240,7 +240,7 @@ theorem mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a
   rcases xs with ⟨xs⟩
   simpa using List.mem_of_mem_erase (by simpa using h)
 
-@[simp, grind] 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)
 
@@ -271,7 +271,7 @@ theorem erase_append [LawfulBEq α] {a : α} {xs ys : Array α} :
     (xs ++ ys).erase a = if a ∈ xs then xs.erase a ++ ys else xs ++ ys.erase a := by
   rcases xs with ⟨xs⟩
   rcases ys with ⟨ys⟩
-  simp only [List.append_toArray, List.erase_toArray, List.erase_append, mem_toArray]
+  simp only [List.append_toArray, List.erase_toArray, List.erase_append, List.mem_toArray]
   split <;> simp
 
 @[grind =]

src/Init/Data/Array/Find.lean

@@ -27,11 +27,11 @@ open Nat
 
 /-! ### findSome? -/
 
-@[simp, grind] theorem findSome?_empty : (#[] : Array α).findSome? f = none := rfl
-@[simp, grind] theorem findSome?_push {xs : Array α} : (xs.push a).findSome? f = (xs.findSome? f).or (f a) := by
+@[simp, grind =] theorem findSome?_empty : (#[] : Array α).findSome? f = none := rfl
+@[simp, grind =] theorem findSome?_push {xs : Array α} : (xs.push a).findSome? f = (xs.findSome? f).or (f a) := by
   cases xs; simp [List.findSome?_append]
 
-@[grind]
+@[grind =]
 theorem findSome?_singleton {a : α} {f : α → Option β} : #[a].findSome? f = f a := by
   simp
 
@@ -228,11 +228,12 @@ theorem mem_of_find?_eq_some {xs : Array α} (h : find? p xs = some a) : a ∈ x
   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]
 
+grind_pattern get_find?_mem => (xs.find? p).get h
+
 @[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
@@ -395,7 +396,6 @@ 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)

src/Init/Data/Array/Monadic.lean

@@ -167,7 +167,7 @@ theorem foldrM_filter [Monad m] [LawfulMonad m] {p : α → Bool} {g : α → β
     (h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
     forIn' as b f = forIn' bs b' g := by
   cases as <;> cases bs
-  simp only [mk.injEq, mem_toArray, List.forIn'_toArray] at w h ⊢
+  simp only [mk.injEq, List.mem_toArray, List.forIn'_toArray] at w h ⊢
   exact List.forIn'_congr w hb h
 
 /--

src/Init/Data/Array/Range.lean

@@ -116,7 +116,7 @@ theorem range'_eq_append_iff : range' s n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs =
 @[simp] theorem find?_range'_eq_some {s n : Nat} {i : Nat} {p : Nat → Bool} :
     (range' s n).find? p = some i ↔ p i ∧ i ∈ range' s n ∧ ∀ j, s ≤ j → j < i → !p j := by
   rw [← List.toArray_range']
-  simp only [List.find?_toArray, mem_toArray]
+  simp only [List.find?_toArray, List.mem_toArray]
   simp [List.find?_range'_eq_some]
 
 @[simp] theorem find?_range'_eq_none {s n : Nat} {p : Nat → Bool} :

src/Init/Data/BitVec/Basic.lean

@@ -874,4 +874,7 @@ def clzAuxRec {w : Nat} (x : BitVec w) (n : Nat) : BitVec w :=
 /-- Count the number of leading zeros. -/
 def clz (x : BitVec w) : BitVec w := clzAuxRec x (w - 1)
 
+/-- Count the number of trailing zeros. -/
+def ctz (x : BitVec w) : BitVec w := (x.reverse).clz
+
 end BitVec

src/Init/Data/BitVec/Bootstrap.lean

@@ -19,7 +19,7 @@ theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsbD i := rfl
 @[simp, grind =] theorem getLsbD_ofFin (x : Fin (2^n)) (i : Nat) :
     getLsbD (BitVec.ofFin x) i = x.val.testBit i := rfl
 
-@[simp, grind] theorem getLsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false := by
+@[simp, grind =] 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

src/Init/Data/BitVec/Lemmas.lean

@@ -37,7 +37,7 @@ namespace BitVec
 @[simp] theorem getElem_ofFin (x : Fin (2^n)) (i : Nat) (h : i < n) :
     (BitVec.ofFin x)[i] = x.val.testBit i := rfl
 
-@[simp, grind] theorem getMsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getMsbD x i = false := by
+@[simp, grind =] theorem getMsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getMsbD x i = false := by
   rw [getMsbD]
   simp only [Bool.and_eq_false_imp, decide_eq_true_eq]
   omega
@@ -1100,6 +1100,10 @@ theorem toInt_setWidth' {m n : Nat} (p : m ≤ n) {x : BitVec m} :
   rw [setWidth'_eq, toFin_setWidth, Fin.val_ofNat, Fin.coe_castLE, val_toFin,
     Nat.mod_eq_of_lt (by apply BitVec.toNat_lt_twoPow_of_le p)]
 
+theorem toNat_setWidth_of_le {w w' : Nat} {b : BitVec w} (h : w ≤ w') : (b.setWidth w').toNat = b.toNat := by
+  rw [BitVec.toNat_setWidth, Nat.mod_eq_of_lt]
+  exact BitVec.toNat_lt_twoPow_of_le h
+
 /-! ## extractLsb -/
 
 @[simp, grind =]
@@ -1287,6 +1291,17 @@ theorem extractLsb'_eq_zero {x : BitVec w} {start : Nat} :
   ext i hi
   omega
 
+theorem extractLsb'_setWidth_of_le {b : BitVec w} {start len w' : Nat} (h : start + len ≤ w') :
+    (b.setWidth w').extractLsb' start len = b.extractLsb' start len := by
+  ext i h_i
+  simp
+  omega
+
+theorem setWidth_extractLsb'_of_le {c : BitVec w} (h : len₁ ≤ len₂) :
+    (c.extractLsb' start len₂).setWidth len₁ = c.extractLsb' start len₁ := by
+  ext i hi
+  simp [show i < len₂ by omega]
+
 /-! ### allOnes -/
 
 @[simp, grind =] theorem toNat_allOnes : (allOnes v).toNat = 2^v - 1 := by
@@ -1530,6 +1545,12 @@ theorem extractLsb_and {x : BitVec w} {hi lo : Nat} :
 @[simp, grind =] theorem ofNat_and {x y : Nat} : BitVec.ofNat w (x &&& y) = BitVec.ofNat w x &&& BitVec.ofNat w y :=
   eq_of_toNat_eq (by simp [Nat.and_mod_two_pow])
 
+theorem and_or_distrib_left {x y z : BitVec w} : x &&& (y ||| z) = (x &&& y) ||| (x &&& z) :=
+  BitVec.eq_of_getElem_eq (by simp [Bool.and_or_distrib_left])
+
+theorem and_or_distrib_right {x y z : BitVec w} : (x ||| y) &&& z = (x &&& z) ||| (y &&& z) :=
+  BitVec.eq_of_getElem_eq (by simp [Bool.and_or_distrib_right])
+
 /-! ### xor -/
 
 @[simp, grind =] theorem toNat_xor (x y : BitVec v) :
@@ -2180,6 +2201,10 @@ theorem msb_ushiftRight {x : BitVec w} {n : Nat} :
   have := lt_of_getLsbD ha
   omega
 
+theorem setWidth_ushiftRight_eq_extractLsb {b : BitVec w} : (b >>> w').setWidth w'' = b.extractLsb' w' w'' := by
+  ext i hi
+  simp
+
 /-! ### ushiftRight reductions from BitVec to Nat -/
 
 @[simp, grind =]
@@ -2970,10 +2995,9 @@ theorem shiftLeft_eq_concat_of_lt {x : BitVec w} {n : Nat} (hn : n < w) :
 /-- Combine adjacent `extractLsb'` operations into a single `extractLsb'`. -/
 theorem extractLsb'_append_extractLsb'_eq_extractLsb' {x : BitVec w} (h : start₂ = start₁ + len₁) :
     ((x.extractLsb' start₂ len₂) ++ (x.extractLsb' start₁ len₁)) =
-    (x.extractLsb' start₁ (len₁ + len₂)).cast (by omega) := by
+      x.extractLsb' start₁ (len₂ + len₁) := by
   ext i h
-  simp only [getElem_append, getElem_extractLsb', dite_eq_ite, getElem_cast, ite_eq_left_iff,
-    Nat.not_lt]
+  simp only [getElem_append, getElem_extractLsb', dite_eq_ite, ite_eq_left_iff, Nat.not_lt]
   intro hi
   congr 1
   omega
@@ -3085,6 +3109,51 @@ theorem extractLsb'_append_eq_of_le {v w} {xhi : BitVec v} {xlo : BitVec w}
     extractLsb' start len (xhi ++ xlo) = extractLsb' (start - w) len xhi := by
   simp [extractLsb'_append_eq_ite, show ¬ start < w by omega]
 
+theorem extractLsb'_append_eq_left {a : BitVec w} {b : BitVec w'} : (a ++ b).extractLsb' w' w = a := by
+  simp [BitVec.extractLsb'_append_eq_of_le]
+
+theorem extractLsb'_append_eq_right {a : BitVec w} {b : BitVec w'} : (a ++ b).extractLsb' 0 w' = b := by
+  simp [BitVec.extractLsb'_append_eq_of_add_le]
+
+theorem setWidth_append_eq_right {a : BitVec w} {b : BitVec w'} : (a ++ b).setWidth w' = b := by
+  ext i hi
+  simp [getLsbD_append, hi]
+
+theorem append_left_inj {s₁ s₂ : BitVec w} (t : BitVec w') : s₁ ++ t = s₂ ++ t ↔ s₁ = s₂ := by
+  refine ⟨fun h => ?_, fun h => h ▸ rfl⟩
+  ext i hi
+  simpa [getElem_append, dif_neg] using congrArg (·[i + w']'(by omega)) h
+
+theorem append_right_inj (s : BitVec w) {t₁ t₂ : BitVec w'} : s ++ t₁ = s ++ t₂ ↔ t₁ = t₂ := by
+  refine ⟨fun h => ?_, fun h => h ▸ rfl⟩
+  ext i hi
+  simpa [getElem_append, hi] using congrArg (·[i]) h
+
+theorem setWidth_append_eq_shiftLeft_setWidth_or {b : BitVec w} {b' : BitVec w'} :
+    (b ++ b').setWidth w'' = (b.setWidth w'' <<< w') ||| b'.setWidth w'' := by
+  ext i hi
+  simp only [getElem_setWidth, getElem_or, getElem_shiftLeft]
+  rw [getLsbD_append]
+  split <;> simp_all
+
+theorem setWidth_append_append_eq_shiftLeft_setWidth_or {b : BitVec w} {b' : BitVec w'} {b'' : BitVec w''} :
+    (b ++ b' ++ b'').setWidth w''' = (b.setWidth w''' <<< (w' + w'')) ||| (b'.setWidth w''' <<< w'') ||| b''.setWidth w''' := by
+  rw [BitVec.setWidth_append_eq_shiftLeft_setWidth_or,
+    BitVec.setWidth_append_eq_shiftLeft_setWidth_or,
+    BitVec.shiftLeft_or_distrib, BitVec.shiftLeft_add]
+
+theorem setWidth_append_append_append_eq_shiftLeft_setWidth_or {b : BitVec w} {b' : BitVec w'} {b'' : BitVec w''} {b''' : BitVec w'''} :
+    (b ++ b' ++ b'' ++ b''').setWidth w'''' = (b.setWidth w'''' <<< (w' + w'' + w''')) ||| (b'.setWidth w'''' <<< (w'' + w''')) |||
+      (b''.setWidth w'''' <<< w''') ||| b'''.setWidth w'''' := by
+  simp only [BitVec.setWidth_append_eq_shiftLeft_setWidth_or, BitVec.shiftLeft_or_distrib, BitVec.shiftLeft_add]
+
+theorem and_setWidth_allOnes (w' w : Nat) (b : BitVec (w' + w)) :
+    b &&& (BitVec.allOnes w).setWidth (w' + w) = 0#w' ++ b.setWidth w := by
+  ext i hi
+  simp only [getElem_and, getElem_setWidth, getLsbD_allOnes]
+  rw [BitVec.getElem_append]
+  split <;> simp_all
+
 /-! ### rev -/
 
 @[grind =]
@@ -4041,6 +4110,9 @@ instance instLawfulOrderLT : LawfulOrderLT (BitVec n) := by
   apply LawfulOrderLT.of_le
   simpa using fun _ _ => BitVec.lt_asymm
 
+theorem length_pos_of_lt {b b' : BitVec w} (h : b < b') : 0 < w :=
+  length_pos_of_ne (BitVec.ne_of_lt h)
+
 protected theorem umod_lt (x : BitVec n) {y : BitVec n} : 0 < y → x % y < y := by
   simp only [ofNat_eq_ofNat, lt_def, toNat_ofNat, Nat.zero_mod]
   apply Nat.mod_lt
@@ -4112,6 +4184,14 @@ theorem lt_of_msb_false_of_msb_true {x y : BitVec w} (hx : x.msb = false) (hy :
   simp
   omega
 
+theorem lt_add_one {b : BitVec w} (h : b ≠ allOnes w) : b < b + 1 := by
+  simp only [ne_eq, ← toNat_inj, toNat_allOnes] at h
+  simp only [BitVec.lt_def, ofNat_eq_ofNat, toNat_add, toNat_ofNat, Nat.add_mod_mod]
+  rw [Nat.mod_eq_of_lt]
+  · exact Nat.lt_add_one _
+  · have := b.toNat_lt_twoPow_of_le (Nat.le_refl _)
+    omega
+
 /-! ### udiv -/
 
 theorem udiv_def {x y : BitVec n} : x / y = BitVec.ofNat n (x.toNat / y.toNat) := by
@@ -5257,7 +5337,7 @@ theorem replicate_succ' {x : BitVec w} :
     (replicate n x ++ x).cast (by rw [Nat.mul_succ]) := by
   simp [replicate_append_self]
 
-theorem BitVec.setWidth_add_eq_mod {x y : BitVec w} : BitVec.setWidth i (x + y) = (BitVec.setWidth i x + BitVec.setWidth i y) % (BitVec.twoPow i w) := by
+theorem setWidth_add_eq_mod {x y : BitVec w} : BitVec.setWidth i (x + y) = (BitVec.setWidth i x + BitVec.setWidth i y) % (BitVec.twoPow i w) := by
   apply BitVec.eq_of_toNat_eq
   rw [toNat_setWidth]
   simp only [toNat_setWidth, toNat_add, toNat_umod, Nat.add_mod_mod, Nat.mod_add_mod, toNat_twoPow]
@@ -5266,6 +5346,14 @@ theorem BitVec.setWidth_add_eq_mod {x y : BitVec w} : BitVec.setWidth i (x + y)
   · have hk : 2 ^ w < 2 ^ i := Nat.pow_lt_pow_of_lt (by decide) (Nat.lt_of_not_le h)
     rw [Nat.mod_eq_of_lt hk, Nat.mod_mod_eq_mod_mod_of_dvd (Nat.pow_dvd_pow _ (Nat.le_of_not_le h))]
 
+theorem setWidth_setWidth_eq_self {a : BitVec w} {w' : Nat} (h : a < BitVec.twoPow w w') : (a.setWidth w').setWidth w = a := by
+  by_cases hw : w' < w
+  · simp only [toNat_eq, toNat_setWidth]
+    rw [Nat.mod_mod_of_dvd' (Nat.pow_dvd_pow _ (Nat.le_of_lt hw)), Nat.mod_eq_of_lt]
+    rwa [BitVec.lt_def, BitVec.toNat_twoPow_of_lt hw] at h
+  · rw [BitVec.lt_def, BitVec.toNat_twoPow_of_le (by omega)] at h
+    simp at h
+
 /-! ### intMin -/
 
 @[grind =]
@@ -5779,6 +5867,25 @@ theorem msb_replicate {n w : Nat} {x : BitVec w} :
   simp only [BitVec.msb, getMsbD_replicate, Nat.zero_mod]
   cases n <;> cases w <;> simp
 
+@[simp]
+theorem reverse_eq_zero_iff {x : BitVec w} :
+    x.reverse = 0#w ↔ x = 0#w := by
+  constructor
+  · intro hrev
+    ext i hi
+    rw [← getLsbD_eq_getElem, getLsbD_eq_getMsbD, ← getLsbD_reverse]
+    simp [hrev]
+  · intro hzero
+    ext i hi
+    rw [← getLsbD_eq_getElem, getLsbD_eq_getMsbD, getMsbD_reverse]
+    simp [hi, hzero]
+
+@[simp]
+theorem reverse_reverse_eq {x : BitVec w} :
+    x.reverse.reverse = x := by
+  ext k hk
+  rw [getElem_reverse, getMsbD_reverse, getLsbD_eq_getElem]
+
 /-! ### Inequalities (le / lt) -/
 
 theorem ule_eq_not_ult (x y : BitVec w) : x.ule y = !y.ult x := by
@@ -5909,6 +6016,12 @@ theorem getElem_eq_true_of_lt_of_le {x : BitVec w} (hk' : k < w) (hlt: x.toNat <
       omega
   · simp [show w ≤ k + k' by omega] at hk'
 
+theorem not_lt_iff {b : BitVec w} : ~~~b < b ↔ 0 < w ∧ b.msb = true := by
+  refine ⟨fun h => ?_, fun ⟨hw, hb⟩ => ?_⟩
+  · have := length_pos_of_lt h
+    exact ⟨this, by rwa [← ult_iff_lt, ult_eq_msb_of_msb_neq (by simp_all)] at h⟩
+  · rwa [← ult_iff_lt, ult_eq_msb_of_msb_neq (by simp_all)]
+
 /-! ### Count leading zeros -/
 
 theorem clzAuxRec_zero (x : BitVec w) :
@@ -6182,6 +6295,71 @@ theorem toNat_lt_two_pow_sub_clz {x : BitVec w} :
         · simp [show w + 1 ≤ i by omega]
       · simp; omega
 
+theorem clz_eq_reverse_ctz {x : BitVec w} :
+    x.clz = (x.reverse).ctz := by
+  simp [ctz]
+
+/-! ### Count trailing zeros -/
+
+theorem ctz_eq_reverse_clz {x : BitVec w} :
+    x.ctz = (x.reverse).clz := by
+  simp [ctz]
+
+/-- The number of trailing zeroes is strictly less than the bitwidth iff the bitvector is nonzero. -/
+@[simp]
+theorem ctz_lt_iff_ne_zero {x : BitVec w} :
+    ctz x < w ↔ x ≠ 0#w := by
+  simp only [ctz_eq_reverse_clz, natCast_eq_ofNat, ne_eq]
+  rw [show BitVec.ofNat w w = w by simp, ← reverse_eq_zero_iff (x := x)]
+  apply clz_lt_iff_ne_zero (x := x.reverse)
+
+/-- If a bitvec is different than zero the bits at indexes lower than `ctz x` are false. -/
+theorem getLsbD_false_of_lt_ctz {x : BitVec w} (hi : i < x.ctz.toNat) :
+    x.getLsbD i = false := by
+  rw [getLsbD_eq_getMsbD, ← getLsbD_reverse]
+  have hiff := ctz_lt_iff_ne_zero (x := x)
+  by_cases hzero : x = 0#w
+  · simp [hzero, getLsbD_reverse]
+  · simp only [ctz_eq_reverse_clz, natCast_eq_ofNat, ne_eq, hzero, not_false_eq_true,
+      iff_true] at hiff
+    simp only [ctz] at hi
+    have hi' : i < w := by simp [BitVec.lt_def] at hiff; omega
+    simp only [hi', decide_true, Bool.true_and]
+    have : (x.reverse.clzAuxRec (w - 1)).toNat ≤ w := by
+      rw [show ((x.reverse.clzAuxRec (w - 1)).toNat ≤ w) =
+            ((x.reverse.clzAuxRec (w - 1)).toNat ≤ (BitVec.ofNat w w).toNat) by simp, ← le_def]
+      apply clzAuxRec_le (x := x.reverse) (n := w - 1)
+    let j := (x.reverse.clzAuxRec (w - 1)).toNat - 1 - i
+    rw [show w - 1 - i = w - (x.reverse.clzAuxRec (w - 1)).toNat + j by
+      subst j
+      rw [Nat.sub_sub (n := (x.reverse.clzAuxRec (w - 1)).toNat),
+        ← Nat.add_sub_assoc (by exact Nat.one_add_le_iff.mpr hi)]
+      omega]
+    have hfalse : ∀ (i : Nat), w - 1 < i → x.reverse.getLsbD i = false := by
+      intros i hj
+      simp [show w ≤ i by omega]
+    exact getLsbD_false_of_clzAuxRec (x := x.reverse) (n := w - 1) hfalse (j := j)
+
+/-- If a bitvec is different than zero, the bit at index `ctz x`, i.e., the first bit after the
+  trailing zeros, is true. -/
+theorem getLsbD_true_ctz_of_ne_zero {x : BitVec w} (hx : x ≠ 0#w) :
+    x.getLsbD (ctz x).toNat = true := by
+  simp only [ctz_eq_reverse_clz, clz]
+  rw [getLsbD_eq_getMsbD, ← getLsbD_reverse]
+  have := ctz_lt_iff_ne_zero (x := x)
+  simp only [ctz_eq_reverse_clz, clz, natCast_eq_ofNat, lt_def, toNat_ofNat, Nat.mod_two_pow_self,
+    ne_eq] at this
+  simp only [this, hx, not_false_eq_true, decide_true, Bool.true_and]
+  have hnotrev : ¬x.reverse = 0#w := by simp [reverse_eq_zero_iff, hx]
+  apply getLsbD_true_of_eq_clzAuxRec_of_ne_zero (x := x.reverse) (n := w - 1) hnotrev
+  intro i hi
+  simp [show w ≤ i by omega]
+
+/-- A nonzero bitvector is lower-bounded by its trailing zeroes. -/
+theorem two_pow_ctz_le_toNat_of_ne_zero {x : BitVec w} (hx : x ≠ 0#w) :
+    2 ^ (ctz x).toNat ≤ x.toNat := by
+  have hclz := getLsbD_true_ctz_of_ne_zero (x := x) hx
+  exact Nat.ge_two_pow_of_testBit hclz
 
 /-! ### Deprecations -/
 

src/Init/Data/ByteArray.lean

@@ -7,6 +7,8 @@ module
 
 prelude
 public import Init.Data.ByteArray.Basic
+public import Init.Data.ByteArray.Bootstrap
 public import Init.Data.ByteArray.Extra
+public import Init.Data.ByteArray.Lemmas
 
 public section

src/Init/Data/ByteArray/Basic.lean

@@ -11,16 +11,11 @@ public import Init.Data.UInt.Basic
 public import Init.Data.UInt.BasicAux
 import all Init.Data.UInt.BasicAux
 public import Init.Data.Option.Basic
+public import Init.Data.Array.Extract
 
 @[expose] public section
 universe u
 
-structure ByteArray where
-  data : Array UInt8
-
-attribute [extern "lean_byte_array_mk"] ByteArray.mk
-attribute [extern "lean_byte_array_data"] ByteArray.data
-
 namespace ByteArray
 
 deriving instance BEq for ByteArray
@@ -30,29 +25,15 @@ attribute [ext] ByteArray
 instance : DecidableEq ByteArray :=
   fun _ _ => decidable_of_decidable_of_iff ByteArray.ext_iff.symm
 
-@[extern "lean_mk_empty_byte_array"]
-def emptyWithCapacity (c : @& Nat) : ByteArray :=
-  { data := #[] }
-
 @[deprecated emptyWithCapacity (since := "2025-03-12")]
 abbrev mkEmpty := emptyWithCapacity
 
-def empty : ByteArray := emptyWithCapacity 0
-
 instance : Inhabited ByteArray where
   default := empty
 
 instance : EmptyCollection ByteArray where
   emptyCollection := ByteArray.empty
 
-@[extern "lean_byte_array_push"]
-def push : ByteArray → UInt8 → ByteArray
-  | ⟨bs⟩, b => ⟨bs.push b⟩
-
-@[extern "lean_byte_array_size"]
-def size : (@& ByteArray) → Nat
-  | ⟨bs⟩ => bs.size
-
 @[extern "lean_sarray_size", simp]
 def usize (a : @& ByteArray) : USize :=
   a.size.toUSize
@@ -106,11 +87,31 @@ def copySlice (src : @& ByteArray) (srcOff : Nat) (dest : ByteArray) (destOff le
 def extract (a : ByteArray) (b e : Nat) : ByteArray :=
   a.copySlice b empty 0 (e - b)
 
-protected def append (a : ByteArray) (b : ByteArray) : ByteArray :=
+protected def fastAppend (a : ByteArray) (b : ByteArray) : ByteArray :=
   -- we assume that `append`s may be repeated, so use asymptotic growing; use `copySlice` directly to customize
   b.copySlice 0 a a.size b.size false
 
-instance : Append ByteArray := ⟨ByteArray.append⟩
+@[simp]
+theorem size_data {a : ByteArray} :
+  a.data.size = a.size := rfl
+
+@[csimp]
+theorem append_eq_fastAppend : @ByteArray.append = @ByteArray.fastAppend := by
+  funext a b
+  ext1
+  apply Array.ext'
+  simp [ByteArray.fastAppend, copySlice, ← size_data, - Array.append_assoc]
+
+-- Needs to come after the `csimp` lemma
+instance : Append ByteArray where
+  append := ByteArray.append
+
+@[simp]
+theorem append_eq {a b : ByteArray} : a.append b = a ++ b := rfl
+
+@[simp]
+theorem fastAppend_eq {a b : ByteArray} : a.fastAppend b = a ++ b := by
+  simp [← append_eq_fastAppend]
 
 def toList (bs : ByteArray) : List UInt8 :=
   let rec loop (i : Nat) (r : List UInt8) :=
@@ -350,13 +351,4 @@ def prevn : Iterator → Nat → Iterator
 end Iterator
 end ByteArray
 
-/--
-Converts a list of bytes into a `ByteArray`.
--/
-def List.toByteArray (bs : List UInt8) : ByteArray :=
-  let rec loop
-    | [],    r => r
-    | b::bs, r => loop bs (r.push b)
-  loop bs ByteArray.empty
-
 instance : ToString ByteArray := ⟨fun bs => bs.toList.toString⟩

src/Init/Data/ByteArray/Bootstrap.lean

@@ -0,0 +1,53 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Markus Himmel
+-/
+module
+
+prelude
+public import Init.Prelude
+public import Init.Data.List.Basic
+
+public section
+
+namespace ByteArray
+
+@[simp]
+theorem data_push {a : ByteArray} {b : UInt8} : (a.push b).data = a.data.push b := rfl
+
+@[expose]
+protected def append (a b : ByteArray) : ByteArray :=
+  ⟨⟨a.data.toList ++ b.data.toList⟩⟩
+
+@[simp]
+theorem toList_data_append' {a b : ByteArray} :
+    (a.append b).data.toList = a.data.toList ++ b.data.toList := by
+  have ⟨⟨a⟩⟩ := a
+  have ⟨⟨b⟩⟩ := b
+  rfl
+
+theorem ext : {x y : ByteArray} → x.data = y.data → x = y
+  | ⟨_⟩, ⟨_⟩, rfl => rfl
+
+end ByteArray
+
+@[simp]
+theorem List.toList_data_toByteArray {l : List UInt8} :
+    l.toByteArray.data.toList = l := by
+  rw [List.toByteArray]
+  suffices ∀ a b, (List.toByteArray.loop a b).data.toList = b.data.toList ++ a by
+    simpa using this l ByteArray.empty
+  intro a b
+  fun_induction List.toByteArray.loop a b with simp_all [toList_push]
+where
+  toList_push {xs : Array UInt8} {x : UInt8} : (xs.push x).toList = xs.toList ++ [x] := by
+    have ⟨xs⟩ := xs
+    simp [Array.push, List.concat_eq_append]
+
+theorem List.toByteArray_append' {l l' : List UInt8} :
+    (l ++ l').toByteArray = l.toByteArray.append l'.toByteArray :=
+  ByteArray.ext (ext (by simp))
+where
+  ext : {x y : Array UInt8} → x.toList = y.toList → x = y
+  | ⟨_⟩, ⟨_⟩, rfl => rfl

src/Init/Data/ByteArray/Lemmas.lean

@@ -0,0 +1,252 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Markus Himmel
+-/
+module
+
+prelude
+public import Init.Data.ByteArray.Basic
+public import Init.Data.Array.Extract
+
+public section
+
+@[simp]
+theorem ByteArray.data_empty : ByteArray.empty.data = #[] := rfl
+
+@[simp]
+theorem ByteArray.data_extract {a : ByteArray} {b e : Nat} :
+    (a.extract b e).data = a.data.extract b e := by
+  simp [extract, copySlice]
+  by_cases b ≤ e
+  · rw [(by omega : b + (e - b) = e)]
+  · rw [Array.extract_eq_empty_of_le (by omega), Array.extract_eq_empty_of_le (by omega)]
+
+@[simp]
+theorem ByteArray.extract_zero_size {b : ByteArray} : b.extract 0 b.size = b := by
+  ext1
+  simp
+
+@[simp]
+theorem ByteArray.extract_same {b : ByteArray} {i : Nat} : b.extract i i = ByteArray.empty := by
+  ext1
+  simp [Nat.min_le_left]
+
+theorem ByteArray.fastAppend_eq_copySlice {a b : ByteArray} :
+  a.fastAppend b = b.copySlice 0 a a.size b.size false := rfl
+
+@[simp]
+theorem List.toByteArray_append {l l' : List UInt8} :
+    (l ++ l').toByteArray = l.toByteArray ++ l'.toByteArray := by
+  simp [List.toByteArray_append']
+
+@[simp]
+theorem ByteArray.toList_data_append {l l' : ByteArray} :
+    (l ++ l').data.toList = l.data.toList ++ l'.data.toList := by
+  simp [← append_eq]
+
+@[simp]
+theorem ByteArray.data_append {l l' : ByteArray} :
+    (l ++ l').data = l.data ++ l'.data := by
+  simp [← Array.toList_inj]
+
+@[simp]
+theorem ByteArray.size_empty : ByteArray.empty.size = 0 := by
+  simp [← ByteArray.size_data]
+
+@[simp]
+theorem List.data_toByteArray {l : List UInt8} :
+    l.toByteArray.data = l.toArray := by
+  rw [List.toByteArray]
+  suffices ∀ a b, (List.toByteArray.loop a b).data = b.data ++ a.toArray by
+    simpa using this l ByteArray.empty
+  intro a b
+  fun_induction List.toByteArray.loop a b with simp_all
+
+@[simp]
+theorem List.size_toByteArray {l : List UInt8} :
+    l.toByteArray.size = l.length := by
+  simp [← ByteArray.size_data]
+
+@[simp]
+theorem List.toByteArray_nil : List.toByteArray [] = ByteArray.empty := rfl
+
+@[simp]
+theorem ByteArray.empty_append {b : ByteArray} : ByteArray.empty ++ b = b := by
+  ext1
+  simp
+
+@[simp]
+theorem ByteArray.append_empty {b : ByteArray} : b ++ ByteArray.empty = b := by
+  ext1
+  simp
+
+@[simp, grind =]
+theorem ByteArray.size_append {a b : ByteArray} : (a ++ b).size = a.size + b.size := by
+  simp [← size_data]
+
+@[simp]
+theorem ByteArray.size_eq_zero_iff {a : ByteArray} : a.size = 0 ↔ a = ByteArray.empty := by
+  refine ⟨fun h => ?_, fun h => h ▸ ByteArray.size_empty⟩
+  ext1
+  simp [← Array.size_eq_zero_iff, h]
+
+theorem ByteArray.getElem_eq_getElem_data {a : ByteArray} {i : Nat} {h : i < a.size} :
+    a[i] = a.data[i]'(by simpa [← size_data]) := rfl
+
+@[simp]
+theorem ByteArray.getElem_append_left {i : Nat} {a b : ByteArray} {h : i < (a ++ b).size}
+    (hlt : i < a.size) : (a ++ b)[i] = a[i] := by
+  simp only [getElem_eq_getElem_data, data_append]
+  rw [Array.getElem_append_left (by simpa)]
+
+theorem ByteArray.getElem_append_right {i : Nat} {a b : ByteArray} {h : i < (a ++ b).size}
+    (hle : a.size ≤ i) : (a ++ b)[i] = b[i - a.size]'(by simp_all; omega) := by
+  simp only [getElem_eq_getElem_data, data_append]
+  rw [Array.getElem_append_right (by simpa)]
+  simp
+
+@[simp]
+theorem List.getElem_toByteArray {l : List UInt8} {i : Nat} {h : i < l.toByteArray.size} :
+    l.toByteArray[i]'h = l[i]'(by simp_all) := by
+  simp [ByteArray.getElem_eq_getElem_data]
+
+theorem List.getElem_eq_getElem_toByteArray {l : List UInt8} {i : Nat} {h : i < l.length} :
+    l[i]'h = l.toByteArray[i]'(by simp_all) := by
+  simp
+
+@[simp]
+theorem ByteArray.size_extract {a : ByteArray} {b e : Nat} :
+    (a.extract b e).size = min e a.size - b := by
+  simp [← size_data]
+
+@[simp]
+theorem ByteArray.extract_eq_empty_iff {b : ByteArray} {i j : Nat} : b.extract i j = ByteArray.empty ↔ min j b.size ≤ i := by
+  rw [← size_eq_zero_iff, size_extract]
+  omega
+
+@[simp]
+theorem ByteArray.extract_add_left {b : ByteArray} {i j : Nat} : b.extract (i + j) i = ByteArray.empty := by
+  simp only [extract_eq_empty_iff]
+  exact Nat.le_trans (Nat.min_le_left _ _) (by simp)
+
+@[simp]
+theorem ByteArray.append_eq_empty_iff {a b : ByteArray} :
+    a ++ b = ByteArray.empty ↔ a = ByteArray.empty ∧ b = ByteArray.empty := by
+  simp [← size_eq_zero_iff, size_append]
+
+@[simp]
+theorem List.toByteArray_eq_empty {l : List UInt8} :
+    l.toByteArray = ByteArray.empty ↔ l = [] := by
+  simp [← ByteArray.size_eq_zero_iff]
+
+theorem ByteArray.append_right_inj {ys₁ ys₂ : ByteArray} (xs : ByteArray) :
+    xs ++ ys₁ = xs ++ ys₂ ↔ ys₁ = ys₂ := by
+  simp [ByteArray.ext_iff, Array.append_right_inj]
+
+@[simp]
+theorem ByteArray.extract_append_extract {a : ByteArray} {i j k : Nat} :
+    a.extract i j ++ a.extract j k = a.extract (min i j) (max j k) := by
+  ext1
+  simp
+
+theorem ByteArray.extract_eq_extract_append_extract {a : ByteArray} {i k : Nat} (j : Nat)
+    (hi : i ≤ j) (hk : j ≤ k) :
+    a.extract i k = a.extract i j ++ a.extract j k := by
+  simp
+  rw [Nat.min_eq_left hi, Nat.max_eq_right hk]
+
+theorem ByteArray.append_inj_left {xs₁ xs₂ ys₁ ys₂ : ByteArray} (h : xs₁ ++ ys₁ = xs₂ ++ ys₂) (hl : xs₁.size = xs₂.size) : xs₁ = xs₂ := by
+  simp only [ByteArray.ext_iff, ← ByteArray.size_data, ByteArray.data_append] at *
+  exact Array.append_inj_left h hl
+
+theorem ByteArray.extract_append_eq_right {a b : ByteArray} {i : Nat} (hi : i = a.size) :
+    (a ++ b).extract i (a ++ b).size = b := by
+  subst hi
+  ext1
+  simp [← size_data]
+
+theorem ByteArray.extract_append_eq_left {a b : ByteArray} {i : Nat} (hi : i = a.size) :
+    (a ++ b).extract 0 i = a := by
+  subst hi
+  ext1
+  simp
+
+theorem ByteArray.extract_append_size_left {a b : ByteArray} {i : Nat} :
+    (a ++ b).extract i a.size = a.extract i a.size := by
+  ext1
+  simp
+
+theorem ByteArray.extract_append_size_add {a b : ByteArray} {i j : Nat} :
+    (a ++ b).extract (a.size + i) (a.size + j) = b.extract i j := by
+  ext1
+  simp
+
+theorem ByteArray.extract_append  {as bs : ByteArray} {i j : Nat} :
+    (as ++ bs).extract i j = as.extract i j ++ bs.extract (i - as.size) (j - as.size) := by
+  ext1
+  simp
+
+theorem ByteArray.extract_append_size_add' {a b : ByteArray} {i j k : Nat} (h : k = a.size) :
+    (a ++ b).extract (k + i) (k + j) = b.extract i j := by
+  cases h
+  rw [extract_append_size_add]
+
+theorem ByteArray.extract_extract {a : ByteArray} {i j k l : Nat} :
+    (a.extract i j).extract k l = a.extract (i + k) (min (i + l) j) := by
+  ext1
+  simp
+
+theorem ByteArray.getElem_extract_aux {xs : ByteArray} {start stop : Nat} (h : i < (xs.extract start stop).size) :
+    start + i < xs.size := by
+  rw [size_extract] at h; apply Nat.add_lt_of_lt_sub'; apply Nat.lt_of_lt_of_le h
+  apply Nat.sub_le_sub_right; apply Nat.min_le_right
+
+theorem ByteArray.getElem_extract {i : Nat} {b : ByteArray} {start stop : Nat}
+    (h) : (b.extract start stop)[i]'h = b[start + i]'(getElem_extract_aux h) := by
+  simp [getElem_eq_getElem_data]
+
+theorem ByteArray.extract_eq_extract_left {a : ByteArray} {i i' j : Nat} :
+    a.extract i j = a.extract i' j ↔ min j a.size - i = min j a.size - i' := by
+  simp [ByteArray.ext_iff, Array.extract_eq_extract_left]
+
+theorem ByteArray.extract_add_one {a : ByteArray} {i : Nat} (ha : i + 1 ≤ a.size) :
+    a.extract i (i + 1) = [a[i]].toByteArray := by
+  ext
+  · simp
+    omega
+  · rename_i j hj hj'
+    obtain rfl : j = 0 := by simpa using hj'
+    simp [ByteArray.getElem_eq_getElem_data]
+
+theorem ByteArray.extract_add_two {a : ByteArray} {i : Nat} (ha : i + 2 ≤ a.size) :
+    a.extract i (i + 2) = [a[i], a[i + 1]].toByteArray := by
+  rw [extract_eq_extract_append_extract (i + 1) (by simp) (by omega),
+    extract_add_one (by omega), extract_add_one (by omega)]
+  simp [← List.toByteArray_append]
+
+theorem ByteArray.extract_add_three {a : ByteArray} {i : Nat} (ha : i + 3 ≤ a.size) :
+    a.extract i (i + 3) = [a[i], a[i + 1], a[i + 2]].toByteArray := by
+  rw [extract_eq_extract_append_extract (i + 1) (by simp) (by omega),
+    extract_add_one (by omega), extract_add_two (by omega)]
+  simp [← List.toByteArray_append]
+
+theorem ByteArray.extract_add_four {a : ByteArray} {i : Nat} (ha : i + 4 ≤ a.size) :
+    a.extract i (i + 4) = [a[i], a[i + 1], a[i + 2], a[i + 3]].toByteArray := by
+  rw [extract_eq_extract_append_extract (i + 1) (by simp) (by omega),
+    extract_add_one (by omega), extract_add_three (by omega)]
+  simp [← List.toByteArray_append]
+
+theorem ByteArray.append_assoc {a b c : ByteArray} : a ++ b ++ c = a ++ (b ++ c) := by
+  ext1
+  simp
+
+@[simp]
+theorem ByteArray.toList_empty : ByteArray.empty.toList = [] := by
+  simp [ByteArray.toList, ByteArray.toList.loop]
+
+theorem ByteArray.copySlice_eq_append {src : ByteArray} {srcOff : Nat} {dest : ByteArray} {destOff len : Nat} {exact : Bool} :
+    ByteArray.copySlice src srcOff dest destOff len exact =
+      dest.extract 0 destOff ++ src.extract srcOff (srcOff +len) ++ dest.extract (destOff + min len (src.data.size - srcOff)) dest.data.size := by
+  ext1
+  simp [copySlice]

src/Init/Data/Fin/Fold.lean

@@ -122,7 +122,7 @@ private theorem foldlM_loop [Monad m] (f : α → Fin (n+1) → m α) (x) (h : i
     rw [foldlM_loop_lt _ _ h', foldlM_loop]; rfl
   else
     cases Nat.le_antisymm (Nat.le_of_lt_succ h) (Nat.not_lt.1 h')
-    rw [foldlM_loop_lt]
+    rw [foldlM_loop_lt _ _ h]
     congr; funext
     rw [foldlM_loop_eq, foldlM_loop_eq]
 termination_by n - i

src/Init/Data/Function.lean

@@ -34,20 +34,104 @@ Examples:
 @[inline, expose]
 def uncurry : (α → β → φ) → α × β → φ := fun f a => f a.1 a.2
 
-@[simp, grind]
+@[simp, grind =]
 theorem curry_uncurry (f : α → β → φ) : curry (uncurry f) = f :=
   rfl
 
-@[simp, grind]
+@[simp, grind =]
 theorem uncurry_curry (f : α × β → φ) : uncurry (curry f) = f :=
   funext fun ⟨_a, _b⟩ => rfl
 
-@[simp, grind]
+@[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, grind]
+@[simp, grind =]
 theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
   rfl
 
+/-- A function `f : α → β` is called injective if `f x = f y` implies `x = y`. -/
+def Injective (f : α → β) : Prop :=
+  ∀ ⦃a₁ a₂⦄, f a₁ = f a₂ → a₁ = a₂
+
+theorem Injective.comp {α β γ} {g : β → γ} {f : α → β} (hg : Injective g) (hf : Injective f) :
+    Injective (g ∘ f) := fun _a₁ _a₂ => fun h => hf (hg h)
+
+/-- A function `f : α → β` is called surjective if every `b : β` is equal to `f a`
+for some `a : α`. -/
+def Surjective (f : α → β) : Prop :=
+  ∀ b, Exists fun a => f a = b
+
+theorem Surjective.comp {α β γ} {g : β → γ} {f : α → β} (hg : Surjective g) (hf : Surjective f) :
+    Surjective (g ∘ f) := fun c : γ =>
+  Exists.elim (hg c) fun b hb =>
+    Exists.elim (hf b) fun a ha =>
+      Exists.intro a (show g (f a) = c from Eq.trans (congrArg g ha) hb)
+
+/-- `LeftInverse g f` means that `g` is a left inverse to `f`. That is, `g ∘ f = id`. -/
+@[grind]
+def LeftInverse {α β} (g : β → α) (f : α → β) : Prop :=
+  ∀ x, g (f x) = x
+
+/-- `HasLeftInverse f` means that `f` has an unspecified left inverse. -/
+def HasLeftInverse {α β} (f : α → β) : Prop :=
+  Exists fun finv : β → α => LeftInverse finv f
+
+/-- `RightInverse g f` means that `g` is a right inverse to `f`. That is, `f ∘ g = id`. -/
+@[grind]
+def RightInverse {α β} (g : β → α) (f : α → β) : Prop :=
+  LeftInverse f g
+
+/-- `HasRightInverse f` means that `f` has an unspecified right inverse. -/
+def HasRightInverse {α β} (f : α → β) : Prop :=
+  Exists fun finv : β → α => RightInverse finv f
+
+theorem LeftInverse.injective {α β} {g : β → α} {f : α → β} : LeftInverse g f → Injective f :=
+  fun h a b faeqfb => ((h a).symm.trans (congrArg g faeqfb)).trans (h b)
+
+theorem HasLeftInverse.injective {α β} {f : α → β} : HasLeftInverse f → Injective f := fun h =>
+  Exists.elim h fun _finv inv => inv.injective
+
+theorem rightInverse_of_injective_of_leftInverse {α β} {f : α → β} {g : β → α} (injf : Injective f)
+    (lfg : LeftInverse f g) : RightInverse f g := fun x =>
+  have h : f (g (f x)) = f x := lfg (f x)
+  injf h
+
+theorem RightInverse.surjective {α β} {f : α → β} {g : β → α} (h : RightInverse g f) : Surjective f :=
+  fun y => ⟨g y, h y⟩
+
+theorem HasRightInverse.surjective {α β} {f : α → β} : HasRightInverse f → Surjective f
+  | ⟨_finv, inv⟩ => inv.surjective
+
+theorem leftInverse_of_surjective_of_rightInverse {α β} {f : α → β} {g : β → α} (surjf : Surjective f)
+    (rfg : RightInverse f g) : LeftInverse f g := fun y =>
+  Exists.elim (surjf y) fun x hx => ((hx ▸ rfl : f (g y) = f (g (f x))).trans (Eq.symm (rfg x) ▸ rfl)).trans hx
+
+theorem injective_id : Injective (@id α) := fun _a₁ _a₂ h => h
+
+theorem surjective_id : Surjective (@id α) := fun a => ⟨a, rfl⟩
+
+variable {f : α → β}
+
+theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
+  ⟨@I _ _, congrArg f⟩
+
+theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
+  h ▸ I.eq_iff
+
+theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
+  mt fun h ↦ hf h
+
+theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
+  ⟨mt <| congrArg f, hf.ne⟩
+
+theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
+  h ▸ hf.ne_iff
+
+protected theorem LeftInverse.id {α β} {g : β → α} {f : α → β} (h : LeftInverse g f) : g ∘ f = id :=
+  funext h
+
+protected theorem RightInverse.id {α β} {g : β → α} {f : α → β} (h : RightInverse g f) : f ∘ g = id :=
+  funext h
+
 end Function

src/Init/Data/Int/Linear.lean

@@ -17,6 +17,7 @@ import all Init.Data.Int.Gcd
 public import Init.Data.RArray
 public import Init.Data.AC
 import all Init.Data.AC
+import Init.LawfulBEqTactics
 
 public section
 
@@ -54,7 +55,7 @@ def Expr.denote (ctx : Context) : Expr → Int
 inductive Poly where
   | num (k : Int)
   | add (k : Int) (v : Var) (p : Poly)
-  deriving @[expose] BEq
+  deriving @[expose] BEq, ReflBEq, LawfulBEq
 
 @[expose]
 protected noncomputable def Poly.beq' (p₁ : Poly) : Poly → Bool :=
@@ -525,18 +526,6 @@ theorem Expr.denote_norm (ctx : Context) (e : Expr) : e.norm.denote ctx = e.deno
   simp [norm, toPoly', Expr.denote_toPoly'_go]
 
 attribute [local simp] Expr.denote_norm
-
-instance : LawfulBEq Poly where
-  eq_of_beq {a} := by
-    induction a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
-    next ih =>
-      intro _ _ h
-      exact ih h
-  rfl := by
-    intro a
-    induction a <;> simp! [BEq.beq]
-    assumption
-
 attribute [local simp] Poly.denote'_eq_denote
 
 theorem Expr.eq_of_norm_eq (ctx : Context) (e : Expr) (p : Poly) (h : e.norm.beq' p) : e.denote ctx = p.denote' ctx := by

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

@@ -139,7 +139,7 @@ def Iter.Partial.fold {α : Type w} {β : Type w} {γ : Type x} [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]
+@[always_inline, inline, expose, 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

src/Init/Data/Iterators/Lemmas/Combinators/Monadic/Attach.lean

@@ -57,6 +57,6 @@ theorem IterM.map_unattach_toArray_attachWith [Iterator α m β] [Monad m] [Mona
     [LawfulMonad m] [LawfulIteratorCollect α m m] :
     (·.map Subtype.val) <$> (it.attachWith P hP).toArray = it.toArray := by
   rw [← toArray_toList, ← toArray_toList, ← map_unattach_toList_attachWith (it := it) (hP := hP)]
-  simp [-map_unattach_toList_attachWith]
+  simp [-map_unattach_toList_attachWith, -IterM.toArray_toList]
 
 end Std.Iterators

src/Init/Data/Iterators/Lemmas/Combinators/ULift.lean

@@ -53,6 +53,6 @@ theorem Iter.toArray_uLift [Iterator α Id β] {it : Iter (α := α) β}
     [LawfulIteratorCollect α Id Id] :
     it.uLift.toArray = it.toArray.map ULift.up := by
   rw [← toArray_toList, ← toArray_toList, toList_uLift]
-  simp
+  simp [-toArray_toList]
 
 end Std.Iterators

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

@@ -44,11 +44,13 @@ theorem IterM.toListRev_toIter {α β} [Iterator α Id β] [Finite α Id]
     it.toIter.toListRev = it.toListRev.run :=
   (rfl)
 
+@[simp]
 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]
 
+@[simp]
 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

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

@@ -14,6 +14,7 @@ public import Init.Data.Iterators.Consumers.Loop
 import all Init.Data.Iterators.Consumers.Loop
 public import Init.Data.Iterators.Consumers.Monadic.Collect
 import all Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Array.Monadic
 
 public section
 
@@ -43,6 +44,20 @@ theorem Iter.forIn_eq {α β : Type w} [Iterator α Id β] [Finite α Id]
             f out acc) := by
   simp [ForIn.forIn, forIn'_eq, -forIn'_eq_forIn]
 
+@[congr] theorem Iter.forIn'_congr {α β : Type w}
+    [Iterator α Id β] [Finite α Id] [IteratorLoop α Id Id]
+    {ita itb : Iter (α := α) β} (w : ita = itb)
+    {b b' : γ} (hb : b = b')
+    {f : (a' : β) → _ → γ → Id (ForInStep γ)}
+    {g : (a' : β) → _ → γ → Id (ForInStep γ)}
+    (h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
+    letI : ForIn' Id (Iter (α := α) β) β _ := Iter.instForIn'
+    forIn' ita b f = forIn' itb b' g := by
+  subst_eqs
+  simp only [← funext_iff] at h
+  rw [← h]
+  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]
@@ -188,6 +203,13 @@ theorem Iter.mem_toList_iff_isPlausibleIndirectOutput {α β} [Iterator α Id β
         obtain ⟨step, h₁, rfl⟩ := h₁
         simp [heq, IterStep.successor] at h₁
 
+theorem Iter.mem_toArray_iff_isPlausibleIndirectOutput {α β} [Iterator α Id β]
+    [IteratorCollect α Id Id] [Finite α Id]
+    [LawfulIteratorCollect α Id Id] [LawfulDeterministicIterator α Id]
+    {it : Iter (α := α) β} {out : β} :
+    out ∈ it.toArray ↔ it.IsPlausibleIndirectOutput out := by
+  rw [← Iter.toArray_toList, List.mem_toArray, mem_toList_iff_isPlausibleIndirectOutput]
+
 theorem Iter.forIn'_toList {α β : Type w} [Iterator α Id β]
     [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
     [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
@@ -222,6 +244,17 @@ theorem Iter.forIn'_toList {α β : Type w} [Iterator α Id β]
     simp only [ihs h (f := fun out h acc => f out (this ▸ h) acc)]
   · simp
 
+theorem Iter.forIn'_toArray {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    [LawfulDeterministicIterator α Id]
+    {γ : Type x} {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.toArray init f = ForIn'.forIn' it init (fun out h acc => f out (Iter.mem_toArray_iff_isPlausibleIndirectOutput.mpr h) acc) := by
+  simp only [← Iter.toArray_toList (it := it), List.forIn'_toArray, Iter.forIn'_toList]
+
 theorem Iter.forIn'_eq_forIn'_toList {α β : Type w} [Iterator α Id β]
     [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
     [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
@@ -234,6 +267,18 @@ theorem Iter.forIn'_eq_forIn'_toList {α β : Type w} [Iterator α Id β]
   simp only [forIn'_toList]
   congr
 
+theorem Iter.forIn'_eq_forIn'_toArray {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    [LawfulDeterministicIterator α Id]
+    {γ : Type x} {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.toArray init (fun out h acc => f out (Iter.mem_toArray_iff_isPlausibleIndirectOutput.mp h) acc) := by
+  simp only [forIn'_toArray]
+  congr
+
 theorem Iter.forIn_toList {α β : Type w} [Iterator α Id β]
     [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
     [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
@@ -260,6 +305,15 @@ theorem Iter.forIn_toList {α β : Type w} [Iterator α Id β]
     rw [ihs h]
   · simp
 
+theorem Iter.forIn_toArray {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {γ : Type x} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it.toArray init f = ForIn.forIn it init f := by
+  simp only [← Iter.toArray_toList, List.forIn_toArray, forIn_toList]
+
 theorem Iter.foldM_eq_forIn {α β : Type w} {γ : Type x} [Iterator α Id β] [Finite α Id]
     {m : Type x → Type x'} [Monad m] [IteratorLoop α Id m] {f : γ → β → m γ}
     {init : γ} {it : Iter (α := α) β} :
@@ -301,6 +355,14 @@ theorem Iter.foldlM_toList {α β : Type w} {γ : Type x} [Iterator α Id β] [F
   rw [Iter.foldM_eq_forIn, ← Iter.forIn_toList]
   simp only [List.forIn_yield_eq_foldlM, id_map']
 
+theorem Iter.foldlM_toArray {α β : Type w} {γ : Type x} [Iterator α Id β] [Finite α Id]
+    {m : Type x → Type x'} [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.toArray.foldlM (init := init) f = it.foldM (init := init) f := by
+  rw [Iter.foldM_eq_forIn, ← Iter.forIn_toArray]
+  simp only [Array.forIn_yield_eq_foldlM, id_map']
+
 theorem IterM.forIn_eq_foldM {α β : Type w} [Iterator α Id β]
     [Finite α Id] {m : Type x → Type x'} [Monad m] [LawfulMonad m]
     [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
@@ -324,6 +386,12 @@ theorem Iter.fold_eq_foldM {α β : Type w} {γ : Type x} [Iterator α Id β]
     it.fold (init := init) f = (it.foldM (m := Id) (init := init) (pure <| f · ·)).run := by
   simp [foldM_eq_forIn, fold_eq_forIn]
 
+theorem Iter.fold_eq_fold_toIterM {α β : Type w} {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 = (it.toIterM.fold (init := init) f).run := by
+  rw [fold_eq_foldM, foldM_eq_foldM_toIterM, IterM.fold_eq_foldM]
+
 @[simp]
 theorem Iter.forIn_pure_yield_eq_fold {α β : Type w} {γ : Type x} [Iterator α Id β]
     [Finite α Id] [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id] {f : β → γ → γ} {init : γ}
@@ -344,6 +412,38 @@ theorem Iter.fold_eq_match_step {α β : Type w} {γ : Type x} [Iterator α Id
   generalize it.step = step
   cases step using PlausibleIterStep.casesOn <;> simp
 
+-- The argument `f : γ₁ → γ₂` is intentionally explicit, as it is sometimes not found by unification.
+theorem Iter.fold_hom [Iterator α Id β] [Finite α Id]
+    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
+    {it : Iter (α := α) β}
+    (f : γ₁ → γ₂) {g₁ : γ₁ → β → γ₁} {g₂ : γ₂ → β → γ₂} {init : γ₁}
+    (H : ∀ x y, g₂ (f x) y = f (g₁ x y)) :
+    it.fold g₂ (f init) = f (it.fold g₁ init) := by
+  -- We cannot reduce to `IterM.fold_hom` because `IterM.fold` is necessarily more restrictive
+  -- w.r.t. the universe of the output.
+  induction it using Iter.inductSteps generalizing init with | step it ihy ihs =>
+  rw [fold_eq_match_step, fold_eq_match_step]
+  split
+  · rw [H, ihy ‹_›]
+  · rw [ihs ‹_›]
+  · simp
+
+theorem Iter.toList_eq_fold {α β : 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]
+    {it : Iter (α := α) β} :
+    it.toList = it.fold (init := []) (fun l out => l ++ [out]) := by
+  rw [Iter.toList_eq_toList_toIterM, IterM.toList_eq_fold, Iter.fold_eq_fold_toIterM]
+
+theorem Iter.toArray_eq_fold {α β : 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]
+    {it : Iter (α := α) β} :
+    it.toArray = it.fold (init := #[]) (fun xs out => xs.push out) := by
+  simp only [← toArray_toList, toList_eq_fold]
+  rw [← fold_hom (List.toArray)]
+  simp
+
 @[simp]
 theorem Iter.foldl_toList {α β : Type w} {γ : Type x} [Iterator α Id β] [Finite α Id]
     [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
@@ -352,6 +452,14 @@ theorem Iter.foldl_toList {α β : Type w} {γ : Type x} [Iterator α Id β] [Fi
     it.toList.foldl (init := init) f = it.fold (init := init) f := by
   rw [fold_eq_foldM, List.foldl_eq_foldlM, ← Iter.foldlM_toList]
 
+@[simp]
+theorem Iter.foldl_toArray {α β : Type w} {γ : Type x} [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.toArray.foldl (init := init) f = it.fold (init := init) f := by
+  rw [fold_eq_foldM, Array.foldl_eq_foldlM, ← Iter.foldlM_toArray]
+
 @[simp]
 theorem Iter.size_toArray_eq_size {α β : Type w} [Iterator α Id β] [Finite α Id]
     [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]

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

@@ -67,15 +67,17 @@ theorem IterM.toArray_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β]
   rw [IterM.DefaultConsumers.toArrayMapped_eq_match_step]
   simp [bind_pure_comp, pure_bind]
 
+@[simp]
 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]
 
+@[simp]
 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]
+  simp [IterM.toList, -toList_toArray]
 
 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} :
@@ -153,6 +155,6 @@ theorem LawfulIteratorCollect.toList_eq {α β : Type w} {m : Type w → Type w'
     [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]
+  simp [IterM.toList, toArray_eq, -IterM.toList_toArray]
 
 end Std.Iterators

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

@@ -60,6 +60,20 @@ theorem IterM.forIn_eq {α β : Type w} {m : Type w → Type w'} [Iterator α m
         IteratorLoop.wellFounded_of_finite it init _ (fun _ => id) (fun out _ acc => (⟨·, .intro⟩) <$> f out acc) := by
   simp only [ForIn.forIn, forIn'_eq]
 
+@[congr] theorem IterM.forIn'_congr {α β : Type w} {m : Type w → Type w'} [Monad m]
+    [Iterator α m β] [Finite α m] [IteratorLoop α m m]
+    {ita itb : IterM (α := α) m β} (w : ita = itb)
+    {b b' : γ} (hb : b = b')
+    {f : (a' : β) → _ → γ → m (ForInStep γ)}
+    {g : (a' : β) → _ → γ → m (ForInStep γ)}
+    (h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
+    letI : ForIn' m (IterM (α := α) m β) β _ := IterM.instForIn'
+    forIn' ita b f = forIn' itb b' g := by
+  subst_eqs
+  simp only [← funext_iff] at h
+  rw [← h]
+  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 m] [Monad n] [LawfulMonad n]
     [IteratorLoop α m n] [LawfulIteratorLoop α m n]
@@ -200,6 +214,23 @@ theorem IterM.fold_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [I
   intro step
   cases step using PlausibleIterStep.casesOn <;> simp
 
+-- The argument `f : γ₁ → γ₂` is intentionally explicit, as it is sometimes not found by unification.
+theorem IterM.fold_hom {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}
+    (f : γ₁ → γ₂) {g₁ : γ₁ → β → γ₁} {g₂ : γ₂ → β → γ₂} {init : γ₁}
+    (H : ∀ x y, g₂ (f x) y = f (g₁ x y)) :
+    it.fold g₂ (f init) = f <$> (it.fold g₁ init) := by
+  induction it using IterM.inductSteps generalizing init with | step it ihy ihs =>
+  rw [fold_eq_match_step, fold_eq_match_step, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  rw [bind_pure_comp]
+  split
+  · rw [H, ihy ‹_›]
+  · rw [ihs ‹_›]
+  · 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]
@@ -223,6 +254,15 @@ theorem IterM.toList_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator
     simp [ihs h]
   · simp
 
+theorem IterM.toArray_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.toArray = it.fold (init := #[]) (fun xs out => xs.push out) := by
+  simp only [← toArray_toList, toList_eq_fold]
+  rw [← fold_hom]
+  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) :=

src/Init/Data/List/Attach.lean

@@ -167,14 +167,14 @@ 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, grind]
+@[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, grind]
+@[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
@@ -192,12 +192,13 @@ 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⟩
 
+grind_pattern mem_pmap_of_mem => _ ∈ pmap f l H, a ∈ l
+
 @[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
@@ -370,13 +371,13 @@ theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
     xs.attach.tail = xs.tail.attach.map (fun ⟨x, h⟩ => ⟨x, mem_of_mem_tail h⟩) := by
   cases xs <;> simp
 
-@[grind]
+@[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]
+@[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

src/Init/Data/List/Basic.lean

@@ -80,17 +80,17 @@ namespace List
 
 /-! ### length -/
 
-@[simp, grind] theorem length_nil : length ([] : List α) = 0 :=
+@[simp, grind =] theorem length_nil : length ([] : List α) = 0 :=
   rfl
 
 @[simp] theorem length_singleton {a : α} : length [a] = 1 := rfl
 
-@[simp, grind] theorem length_cons {a : α} {as : List α} : (cons a as).length = as.length + 1 :=
+@[simp, grind =] theorem length_cons {a : α} {as : List α} : (cons a as).length = as.length + 1 :=
   rfl
 
 /-! ### set -/
 
-@[simp, grind] theorem length_set {as : List α} {i : Nat} {a : α} : (as.set i a).length = as.length := by
+@[simp, grind =] theorem length_set {as : List α} {i : Nat} {a : α} : (as.set i a).length = as.length := by
   induction as generalizing i with
   | nil => rfl
   | cons x xs ih =>
@@ -101,8 +101,8 @@ namespace List
 /-! ### foldl -/
 
 -- As `List.foldl` is defined in `Init.Prelude`, we write the basic simplification lemmas here.
-@[simp, grind] theorem foldl_nil : [].foldl f b = b := rfl
-@[simp, grind] theorem foldl_cons {l : List α} {f : β → α → β} {b : β} : (a :: l).foldl f b = l.foldl f (f b a) := rfl
+@[simp, grind =] theorem foldl_nil : [].foldl f b = b := rfl
+@[simp, grind =] theorem foldl_cons {l : List α} {f : β → α → β} {b : β} : (a :: l).foldl f b = l.foldl f (f b a) := rfl
 
 /-! ### concat -/
 
@@ -332,7 +332,7 @@ def getLast? : List α → Option α
   | []    => none
   | a::as => some (getLast (a::as) (fun h => List.noConfusion h))
 
-@[simp, grind] theorem getLast?_nil : @getLast? α [] = none := rfl
+@[simp, grind =] theorem getLast?_nil : @getLast? α [] = none := rfl
 
 /-! ### getLastD -/
 
@@ -365,7 +365,7 @@ Returns the first element of a non-empty list.
 def head : (as : List α) → as ≠ [] → α
   | a::_, _ => a
 
-@[simp, grind] theorem head_cons {a : α} {l : List α} {h} : head (a::l) h = a := rfl
+@[simp, grind =] theorem head_cons {a : α} {l : List α} {h} : head (a::l) h = a := rfl
 
 /-! ### head? -/
 
@@ -383,8 +383,8 @@ def head? : List α → Option α
   | []   => none
   | a::_ => some a
 
-@[simp, grind] theorem head?_nil : head? ([] : List α) = none := rfl
-@[simp, grind] theorem head?_cons {a : α} {l : List α} : head? (a::l) = some a := rfl
+@[simp, grind =] theorem head?_nil : head? ([] : List α) = none := rfl
+@[simp, grind =] theorem head?_cons {a : α} {l : List α} : head? (a::l) = some a := rfl
 
 /-! ### headD -/
 
@@ -420,8 +420,8 @@ def tail : List α → List α
   | []    => []
   | _::as => as
 
-@[simp, grind] theorem tail_nil : tail ([] : List α) = [] := rfl
-@[simp, grind] theorem tail_cons {a : α} {as : List α} : tail (a::as) = as := rfl
+@[simp, grind =] theorem tail_nil : tail ([] : List α) = [] := rfl
+@[simp, grind =] theorem tail_cons {a : α} {as : List α} : tail (a::as) = as := rfl
 
 /-! ### tail? -/
 
@@ -441,8 +441,8 @@ def tail? : List α → Option (List α)
   | []    => none
   | _::as => some as
 
-@[simp, grind] theorem tail?_nil : tail? ([] : List α) = none := rfl
-@[simp, grind] theorem tail?_cons {a : α} {l : List α} : tail? (a::l) = some l := rfl
+@[simp, grind =] theorem tail?_nil : tail? ([] : List α) = none := rfl
+@[simp, grind =] theorem tail?_cons {a : α} {l : List α} : tail? (a::l) = some l := rfl
 
 /-! ### tailD -/
 
@@ -475,23 +475,8 @@ We define the basic functional programming operations on `List`:
 
 /-! ### map -/
 
-/--
-Applies a function to each element of the list, returning the resulting list of values.
-
-`O(|l|)`.
-
-Examples:
-* `[a, b, c].map f = [f a, f b, f c]`
-* `[].map Nat.succ = []`
-* `["one", "two", "three"].map (·.length) = [3, 3, 5]`
-* `["one", "two", "three"].map (·.reverse) = ["eno", "owt", "eerht"]`
--/
-@[specialize] def map (f : α → β) : (l : List α) → List β
-  | []    => []
-  | a::as => f a :: map f as
-
-@[simp, grind] theorem map_nil {f : α → β} : map f [] = [] := rfl
-@[simp, grind] theorem map_cons {f : α → β} {a : α} {l : List α} : map f (a :: l) = f a :: map f l := rfl
+@[simp, grind =] theorem map_nil {f : α → β} : map f [] = [] := rfl
+@[simp, grind =] theorem map_cons {f : α → β} {a : α} {l : List α} : map f (a :: l) = f a :: map f l := rfl
 
 /-! ### filter -/
 
@@ -511,7 +496,7 @@ def filter (p : α → Bool) : (l : List α) → List α
     | true => a :: filter p as
     | false => filter p as
 
-@[simp, grind] theorem filter_nil {p : α → Bool} : filter p [] = [] := rfl
+@[simp, grind =] theorem filter_nil {p : α → Bool} : filter p [] = [] := rfl
 
 /-! ### filterMap -/
 
@@ -537,8 +522,8 @@ Example:
     | none   => filterMap f as
     | some b => b :: filterMap f as
 
-@[simp, grind] theorem filterMap_nil {f : α → Option β} : filterMap f [] = [] := rfl
-@[grind] theorem filterMap_cons {f : α → Option β} {a : α} {l : List α} :
+@[simp, grind =] theorem filterMap_nil {f : α → Option β} : filterMap f [] = [] := rfl
+@[grind =] theorem filterMap_cons {f : α → Option β} {a : α} {l : List α} :
     filterMap f (a :: l) =
       match f a with
       | none => filterMap f l
@@ -561,8 +546,8 @@ Examples:
   | []     => init
   | a :: l => f a (foldr f init l)
 
-@[simp, grind] theorem foldr_nil : [].foldr f b = b := rfl
-@[simp, grind] theorem foldr_cons {a} {l : List α} {f : α → β → β} {b} :
+@[simp, grind =] theorem foldr_nil : [].foldr f b = b := rfl
+@[simp, grind =] theorem foldr_cons {a} {l : List α} {f : α → β → β} {b} :
     (a :: l).foldr f b = f a (l.foldr f b) := rfl
 
 /-! ### reverse -/
@@ -591,7 +576,7 @@ Examples:
 @[expose] def reverse (as : List α) : List α :=
   reverseAux as []
 
-@[simp, grind] theorem reverse_nil : reverse ([] : List α) = [] := rfl
+@[simp, grind =] theorem reverse_nil : reverse ([] : List α) = [] := rfl
 
 theorem reverseAux_reverseAux {as bs cs : List α} :
     reverseAux (reverseAux as bs) cs = reverseAux bs (reverseAux (reverseAux as []) cs) := by
@@ -606,20 +591,6 @@ Appends two lists. Normally used via the `++` operator.
 
 Appending lists takes time proportional to the length of the first list: `O(|xs|)`.
 
-Examples:
-  * `[1, 2, 3] ++ [4, 5] = [1, 2, 3, 4, 5]`.
-  * `[] ++ [4, 5] = [4, 5]`.
-  * `[1, 2, 3] ++ [] = [1, 2, 3]`.
--/
-protected def append : (xs ys : List α) → List α
-  | [],    bs => bs
-  | a::as, bs => a :: List.append as bs
-
-/--
-Appends two lists. Normally used via the `++` operator.
-
-Appending lists takes time proportional to the length of the first list: `O(|xs|)`.
-
 This is a tail-recursive version of `List.append`.
 
 Examples:
@@ -645,10 +616,10 @@ instance : Append (List α) := ⟨List.append⟩
 
 @[simp] theorem append_eq {as bs : List α} : List.append as bs = as ++ bs := rfl
 
-@[simp, grind] theorem nil_append (as : List α) : [] ++ as = as := rfl
+@[simp, grind =] theorem nil_append (as : List α) : [] ++ as = as := rfl
 @[simp, grind _=_] theorem cons_append {a : α} {as bs : List α} : (a::as) ++ bs = a::(as ++ bs) := rfl
 
-@[simp, grind] theorem append_nil (as : List α) : as ++ [] = as := by
+@[simp, grind =] theorem append_nil (as : List α) : as ++ [] = as := by
   induction as with
   | nil => rfl
   | cons a as ih =>
@@ -658,7 +629,7 @@ instance : Std.LawfulIdentity (α := List α) (· ++ ·) [] where
   left_id := nil_append
   right_id := append_nil
 
-@[simp, grind] theorem length_append {as bs : List α} : (as ++ bs).length = as.length + bs.length := by
+@[simp, grind =] theorem length_append {as bs : List α} : (as ++ bs).length = as.length + bs.length := by
   induction as with
   | nil => simp
   | cons _ as ih => simp [ih, Nat.succ_add]
@@ -685,27 +656,15 @@ theorem reverseAux_eq_append {as bs : List α} : reverseAux as bs = reverseAux a
     rw [ih (bs := a :: bs), ih (bs := [a]), append_assoc]
     rfl
 
-@[simp, grind] theorem reverse_cons {a : α} {as : List α} : reverse (a :: as) = reverse as ++ [a] := by
+@[simp, grind =] theorem reverse_cons {a : α} {as : List α} : reverse (a :: as) = reverse as ++ [a] := by
   simp [reverse, reverseAux]
   rw [← reverseAux_eq_append]
 
 /-! ### flatten -/
 
-/--
-Concatenates a list of lists into a single list, preserving the order of the elements.
 
-`O(|flatten L|)`.
-
-Examples:
-* `[["a"], ["b", "c"]].flatten = ["a", "b", "c"]`
-* `[["a"], [], ["b", "c"], ["d", "e", "f"]].flatten = ["a", "b", "c", "d", "e", "f"]`
--/
-def flatten : List (List α) → List α
-  | []      => []
-  | l :: L => l ++ flatten L
-
-@[simp, grind] theorem flatten_nil : List.flatten ([] : List (List α)) = [] := rfl
-@[simp, grind] theorem flatten_cons : (l :: L).flatten = l ++ L.flatten := rfl
+@[simp, grind =] theorem flatten_nil : List.flatten ([] : List (List α)) = [] := rfl
+@[simp, grind =] theorem flatten_cons : (l :: L).flatten = l ++ L.flatten := rfl
 
 /-! ### singleton -/
 
@@ -721,20 +680,14 @@ Examples:
 
 /-! ### flatMap -/
 
-/--
-Applies a function that returns a list to each element of a list, and concatenates the resulting
-lists.
-
-Examples:
-* `[2, 3, 2].flatMap List.range = [0, 1, 0, 1, 2, 0, 1]`
-* `["red", "blue"].flatMap String.toList = ['r', 'e', 'd', 'b', 'l', 'u', 'e']`
--/
-@[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 [List.flatMap]
-@[simp, grind] theorem flatMap_cons {x : α} {xs : List α} {f : α → List β} :
+@[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 [List.flatMap]
 
+@[simp, grind _=_] theorem flatMap_append {xs ys : List α} {f : α → List β} :
+    (xs ++ ys).flatMap f = xs.flatMap f ++ ys.flatMap f := by
+  induction xs; {rfl}; simp_all [flatMap_cons, append_assoc]
+
 /-! ### replicate -/
 
 /--
@@ -748,10 +701,10 @@ def replicate : (n : Nat) → (a : α) → List α
   | 0,   _ => []
   | n+1, a => a :: replicate n a
 
-@[simp, grind] theorem replicate_zero {a : α} : replicate 0 a = [] := rfl
-@[grind] theorem replicate_succ {a : α} {n : Nat} : replicate (n+1) a = a :: replicate n a := rfl
+@[simp, grind =] theorem replicate_zero {a : α} : replicate 0 a = [] := rfl
+@[grind =] theorem replicate_succ {a : α} {n : Nat} : replicate (n+1) a = a :: replicate n a := rfl
 
-@[simp, grind] theorem length_replicate {n : Nat} {a : α} : (replicate n a).length = n := by
+@[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]
@@ -819,8 +772,8 @@ def isEmpty : List α → Bool
   | []     => true
   | _ :: _ => false
 
-@[simp, grind] theorem isEmpty_nil : ([] : List α).isEmpty = true := rfl
-@[simp, grind] theorem isEmpty_cons : (x :: xs : List α).isEmpty = false := rfl
+@[simp, grind =] theorem isEmpty_nil : ([] : List α).isEmpty = true := rfl
+@[simp, grind =] theorem isEmpty_cons : (x :: xs : List α).isEmpty = false := rfl
 
 /-! ### elem -/
 
@@ -842,7 +795,7 @@ def elem [BEq α] (a : α) : (l : List α) → Bool
     | true  => true
     | false => elem a bs
 
-@[simp, grind] theorem elem_nil [BEq α] : ([] : List α).elem a = false := rfl
+@[simp, grind =] theorem elem_nil [BEq α] : ([] : List α).elem a = false := rfl
 theorem elem_cons [BEq α] {a : α} :
     (b::bs).elem a = match a == b with | true => true | false => bs.elem a := rfl
 
@@ -958,9 +911,9 @@ def take : (n : Nat) → (xs : List α) → List α
   | _+1, []    => []
   | n+1, a::as => a :: take n as
 
-@[simp, grind] theorem take_nil {i : Nat} : ([] : List α).take i = [] := by cases i <;> rfl
-@[simp, grind] theorem take_zero {l : List α} : l.take 0 = [] := rfl
-@[simp, grind] theorem take_succ_cons {a : α} {as : List α} {i : Nat} : (a::as).take (i+1) = a :: as.take i := rfl
+@[simp, grind =] theorem take_nil {i : Nat} : ([] : List α).take i = [] := by cases i <;> rfl
+@[simp, grind =] theorem take_zero {l : List α} : l.take 0 = [] := rfl
+@[simp, grind =] theorem take_succ_cons {a : α} {as : List α} {i : Nat} : (a::as).take (i+1) = a :: as.take i := rfl
 
 /-! ### drop -/
 
@@ -980,10 +933,10 @@ def drop : (n : Nat) → (xs : List α) → List α
   | _+1, []    => []
   | n+1, _::as => drop n as
 
-@[simp, grind] theorem drop_nil : ([] : List α).drop i = [] := by
+@[simp, grind =] theorem drop_nil : ([] : List α).drop i = [] := by
   cases i <;> rfl
-@[simp, grind] theorem drop_zero {l : List α} : l.drop 0 = l := rfl
-@[simp, grind] theorem drop_succ_cons {a : α} {l : List α} {i : Nat} : (a :: l).drop (i + 1) = l.drop i := rfl
+@[simp, grind =] theorem drop_zero {l : List α} : l.drop 0 = l := rfl
+@[simp, grind =] theorem drop_succ_cons {a : α} {l : List α} {i : Nat} : (a :: l).drop (i + 1) = l.drop i := rfl
 
 theorem drop_eq_nil_of_le {as : List α} {i : Nat} (h : as.length ≤ i) : as.drop i = [] := by
   match as, i with
@@ -1094,13 +1047,13 @@ def dropLast {α} : List α → List α
   | [_]   => []
   | a::as => a :: dropLast as
 
-@[simp, grind] theorem dropLast_nil : ([] : List α).dropLast = [] := rfl
-@[simp, grind] theorem dropLast_singleton : [x].dropLast = [] := rfl
+@[simp, grind =] theorem dropLast_nil : ([] : List α).dropLast = [] := rfl
+@[simp, grind =] theorem dropLast_singleton : [x].dropLast = [] := rfl
 
 @[deprecated dropLast_singleton (since := "2025-04-16")]
 theorem dropLast_single : [x].dropLast = [] := dropLast_singleton
 
-@[simp, grind] theorem dropLast_cons₂ :
+@[simp, grind =] theorem dropLast_cons₂ :
     (x::y::zs).dropLast = x :: (y::zs).dropLast := rfl
 
 -- Later this can be proved by `simp` via `[List.length_dropLast, List.length_cons, Nat.add_sub_cancel]`,
@@ -1439,8 +1392,8 @@ def replace [BEq α] : (l : List α) → (a : α) → (b : α) → List α
     | true  => c::as
     | false => a :: replace as b c
 
-@[simp, grind] theorem replace_nil [BEq α] : ([] : List α).replace a b = [] := rfl
-@[grind] theorem replace_cons [BEq α] {a : α} :
+@[simp, grind =] theorem replace_nil [BEq α] : ([] : List α).replace a b = [] := rfl
+@[grind =] theorem replace_cons [BEq α] {a : α} :
     (a::as).replace b c = match b == a with | true => c::as | false => a :: replace as b c :=
   rfl
 
@@ -1648,8 +1601,8 @@ def findSome? (f : α → Option β) : List α → Option β
     | some b => some b
     | none   => findSome? f as
 
-@[simp, grind] theorem findSome?_nil : ([] : List α).findSome? f = none := rfl
-@[grind] theorem findSome?_cons {f : α → Option β} :
+@[simp, grind =] theorem findSome?_nil : ([] : List α).findSome? f = none := rfl
+@[grind =] theorem findSome?_cons {f : α → Option β} :
     (a::as).findSome? f = match f a with | some b => some b | none => as.findSome? f :=
   rfl
 
@@ -1906,8 +1859,8 @@ def any : (l : List α) → (p : α → Bool) → Bool
   | [], _ => false
   | h :: t, p => p h || any t p
 
-@[simp, grind] theorem any_nil : [].any f = false := rfl
-@[simp, grind] theorem any_cons : (a::l).any f = (f a || l.any f) := rfl
+@[simp, grind =] theorem any_nil : [].any f = false := rfl
+@[simp, grind =] theorem any_cons : (a::l).any f = (f a || l.any f) := rfl
 
 /-! ### all -/
 
@@ -1925,8 +1878,8 @@ def all : List α → (α → Bool) → Bool
   | [], _ => true
   | h :: t, p => p h && all t p
 
-@[simp, grind] theorem all_nil : [].all f = true := rfl
-@[simp, grind] theorem all_cons : (a::l).all f = (f a && l.all f) := rfl
+@[simp, grind =] theorem all_nil : [].all f = true := rfl
+@[simp, grind =] theorem all_cons : (a::l).all f = (f a && l.all f) := rfl
 
 /-! ### or -/
 
@@ -2066,8 +2019,8 @@ Examples:
 def sum {α} [Add α] [Zero α] : List α → α :=
   foldr (· + ·) 0
 
-@[simp, grind] theorem sum_nil [Add α] [Zero α] : ([] : List α).sum = 0 := rfl
-@[simp, grind] theorem sum_cons [Add α] [Zero α] {a : α} {l : List α} : (a::l).sum = a + l.sum := rfl
+@[simp, grind =] theorem sum_nil [Add α] [Zero α] : ([] : List α).sum = 0 := rfl
+@[simp, grind =] theorem sum_cons [Add α] [Zero α] {a : α} {l : List α} : (a::l).sum = a + l.sum := rfl
 
 /-! ### range -/
 

src/Init/Data/List/Count.lean

@@ -223,7 +223,7 @@ variable [BEq α]
 
 @[simp, grind =] theorem count_nil {a : α} : count a [] = 0 := rfl
 
-@[grind]
+@[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]
@@ -237,7 +237,7 @@ theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
 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]
+@[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
   | [], a => by simp
@@ -380,7 +380,7 @@ theorem count_filterMap {α} [BEq β] {b : β} {f : α → Option β} {l : List
 theorem count_flatMap {α} [BEq β] {l : List α} {f : α → List β} {x : β} :
     count x (l.flatMap f) = sum (map (count x ∘ f) l) := countP_flatMap
 
-@[grind]
+@[grind =]
 theorem count_erase {a b : α} :
     ∀ {l : List α}, count a (l.erase b) = count a l - if b == a then 1 else 0
   | [] => by simp

src/Init/Data/List/Erase.lean

@@ -130,7 +130,7 @@ theorem le_length_eraseP {l : List α} : l.length - 1 ≤ (l.eraseP p).length :=
 @[grind →]
 theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (eraseP_subset ·)
 
-@[simp, grind] theorem mem_eraseP_of_neg {l : List α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
+@[simp, grind =] theorem mem_eraseP_of_neg {l : List α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
   refine ⟨mem_of_mem_eraseP, fun al => ?_⟩
   match exists_or_eq_self_of_eraseP p l with
   | .inl h => rw [h]; assumption
@@ -265,14 +265,18 @@ 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_pattern Pairwise.eraseP => Pairwise p (l.eraseP q)
+grind_pattern Pairwise.eraseP => Pairwise p l, l.eraseP q
+
 theorem Nodup.eraseP (p) : Nodup l → Nodup (l.eraseP p) :=
   Pairwise.eraseP p
 
+grind_pattern Nodup.eraseP => Nodup (l.eraseP p)
+grind_pattern Nodup.eraseP => Nodup l, l.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
@@ -393,7 +397,7 @@ theorem le_length_erase [LawfulBEq α] {a : α} {l : List α} : l.length - 1 ≤
 @[grind →]
 theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈ l := erase_subset h
 
-@[simp, grind] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {l : List α} (ab : a ≠ b) :
+@[simp, grind =] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {l : List α} (ab : a ≠ b) :
     a ∈ l.erase b ↔ a ∈ l :=
   erase_eq_eraseP b l ▸ mem_eraseP_of_neg (mt eq_of_beq ab.symm)
 
@@ -508,10 +512,12 @@ theorem Nodup.not_mem_erase [LawfulBEq α] {a : α} (h : Nodup l) : a ∉ l.eras
 -- Only activate `not_mem_erase` when `l.Nodup` is already available.
 grind_pattern List.Nodup.not_mem_erase => a ∈ l.erase a, l.Nodup
 
-@[grind]
 theorem Nodup.erase [LawfulBEq α] (a : α) : Nodup l → Nodup (l.erase a) :=
   Pairwise.erase a
 
+grind_pattern Nodup.erase => Nodup (l.erase a)
+grind_pattern Nodup.erase => Nodup l, l.erase a
+
 theorem head_erase_mem (xs : List α) (a : α) (h) : (xs.erase a).head h ∈ xs :=
   erase_sublist.head_mem h
 
@@ -578,21 +584,21 @@ theorem eraseIdx_ne_nil_iff {l : List α} {i : Nat} : eraseIdx l i ≠ [] ↔ 2
   | [a]
   | a::b::l => simp
 
-
-
-@[grind]
 theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
   | [], _ => by simp
   | a::l, 0 => by simp
   | a::l, k + 1 => by simp [eraseIdx_sublist]
 
+grind_pattern eraseIdx_sublist => l.eraseIdx 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
 
+grind_pattern eraseIdx_sublist => l.eraseIdx k, _ ⊆ l
+
 @[simp]
 theorem eraseIdx_eq_self : ∀ {l : List α} {k : Nat}, eraseIdx l k = l ↔ length l ≤ k
   | [], _ => by simp
@@ -649,15 +655,18 @@ 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 ←]
+grind_pattern Pairwise.eraseIdx => Pairwise p (l.eraseIdx k)
+grind_pattern Pairwise.eraseIdx => Pairwise p l, l.eraseIdx k
+
 theorem Nodup.eraseIdx {l : List α} (k) : Nodup l → Nodup (l.eraseIdx k) :=
   Pairwise.eraseIdx k
 
-@[grind ←]
+grind_pattern Nodup.eraseIdx => Nodup (l.eraseIdx k)
+grind_pattern Nodup.eraseIdx => Nodup l, l.eraseIdx k
+
 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⟩
@@ -667,6 +676,10 @@ protected theorem IsPrefix.eraseIdx {l l' : List α} (h : l <+: l') (k : Nat) :
     rw [Nat.not_lt] at hkl
     simp [eraseIdx_append_of_length_le hkl, eraseIdx_of_length_le hkl]
 
+grind_pattern IsPrefix.eraseIdx => eraseIdx l k <+: eraseIdx l' k
+grind_pattern IsPrefix.eraseIdx => eraseIdx l k, l <+: l'
+grind_pattern IsPrefix.eraseIdx => eraseIdx l' k, l <+: l'
+
 -- See also `mem_eraseIdx_iff_getElem` and `mem_eraseIdx_iff_getElem?` in
 -- `Init/Data/List/Nat/Basic.lean`.
 
@@ -686,6 +699,4 @@ theorem erase_eq_eraseIdx_of_idxOf [BEq α] [LawfulBEq α]
     rw [eq_comm, eraseIdx_eq_self]
     exact Nat.le_of_eq (idxOf_eq_length h).symm
 
-
-
 end List

src/Init/Data/List/Find.lean

@@ -293,7 +293,6 @@ theorem mem_of_find?_eq_some : ∀ {l}, find? p l = some a → a ∈ l
     · 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
@@ -305,6 +304,8 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
       right
       apply ih
 
+grind_pattern get_find?_mem => (xs.find? p).get h
+
 @[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
@@ -558,7 +559,6 @@ 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

src/Init/Data/List/Lemmas.lean

@@ -28,14 +28,14 @@ For each `List` operation, we would like theorems describing the following, when
 * the length of the result `(f L).length`
 * the `i`-th element, described via `(f L)[i]` and/or `(f L)[i]?` (these should typically be `@[simp]`)
 * consequences for `f L` of the fact `x ∈ L` or `x ∉ L`
-* conditions characterising `x ∈ f L` (often but not always `@[simp]`)
+* conditions characterizing `x ∈ f L` (often but not always `@[simp]`)
 * injectivity statements, or congruence statements of the form `p L M → f L = f M`.
-* conditions characterising the result, i.e. of the form `f L = M ↔ p M` for some predicate `p`,
+* conditions characterizing the result, i.e. of the form `f L = M ↔ p M` for some predicate `p`,
   along with special cases of `M` (e.g. `List.append_eq_nil : L ++ M = [] ↔ L = [] ∧ M = []`)
-* negative characterisations are also useful, e.g. `List.cons_ne_nil`
+* negative characterizations are also useful, e.g. `List.cons_ne_nil`
 * interactions with all previously described `List` operations where possible
   (some of these should be `@[simp]`, particularly if the result can be described by a single operation)
-* characterising `(∀ (i) (_ : i ∈ f L), P i)`, for some predicate `P`
+* characterizing `(∀ (i) (_ : i ∈ f L), P i)`, for some predicate `P`
 
 Of course for any individual operation, not all of these will be relevant or helpful, so some judgement is required.
 
@@ -306,7 +306,7 @@ theorem getD_getElem? {l : List α} {i : Nat} {d : α} :
   match i, h with
   | 0, _ => rfl
 
-@[grind]
+@[grind =]
 theorem getElem?_singleton {a : α} {i : Nat} : [a][i]? = if i = 0 then some a else none := by
   simp [getElem?_cons]
 
@@ -348,6 +348,18 @@ theorem ext_getElem {l₁ l₂ : List α} (hl : length l₁ = length l₂)
 theorem getElem?_concat_length {l : List α} {a : α} : (l ++ [a])[l.length]? = some a := by
   simp
 
+theorem eq_getElem_of_length_eq_one : (l : List α) → (hl : l.length = 1) → l = [l[0]'(hl ▸ by decide)]
+  | [_], _ => rfl
+
+theorem eq_getElem_of_length_eq_two : (l : List α) → (hl : l.length = 2) → l = [l[0]'(hl ▸ by decide), l[1]'(hl ▸ by decide)]
+  | [_, _], _ => rfl
+
+theorem eq_getElem_of_length_eq_three : (l : List α) → (hl : l.length = 3) → l = [l[0]'(hl ▸ by decide), l[1]'(hl ▸ by decide), l[2]'(hl ▸ by decide)]
+  | [_, _, _], _ => rfl
+
+theorem eq_getElem_of_length_eq_four : (l : List α) → (hl : l.length = 4) → l = [l[0]'(hl ▸ by decide), l[1]'(hl ▸ by decide), l[2]'(hl ▸ by decide), l[3]'(hl ▸ by decide)]
+  | [_, _, _, _], _ => rfl
+
 /-! ### getD
 
 We simplify away `getD`, replacing `getD l n a` with `(l[n]?).getD a`.
@@ -382,14 +394,20 @@ theorem get!_eq_getElem! [Inhabited α] (l : List α) (i) : l.get! i = l[i]! :=
 
 @[simp] theorem not_mem_nil {a : α} : ¬ a ∈ [] := nofun
 
-@[simp] theorem mem_cons : a ∈ b :: l ↔ a = b ∨ a ∈ l :=
+@[simp, grind =] theorem mem_cons : a ∈ b :: l ↔ a = b ∨ a ∈ l :=
   ⟨fun h => by cases h <;> simp [Membership.mem, *],
    fun | Or.inl rfl => by constructor | Or.inr h => by constructor; assumption⟩
 
-@[grind] theorem eq_or_mem_of_mem_cons {a b : α} {l : List α} :
+theorem eq_or_mem_of_mem_cons {a b : α} {l : List α} :
     a ∈ b :: l → a = b ∨ a ∈ l := List.mem_cons.mp
 
-@[grind] theorem mem_cons_self {a : α} {l : List α} : a ∈ a :: l := .head ..
+-- This pattern may be excessively general:
+-- it fires anytime we ae thinking about membership of lists,
+-- and constructing a list via `cons`, even if the elements are unrelated.
+-- Nevertheless in practice it is quite helpful!
+grind_pattern eq_or_mem_of_mem_cons => b :: l, a ∈ l
+
+theorem mem_cons_self {a : α} {l : List α} : a ∈ a :: l := .head ..
 
 theorem mem_concat_self {xs : List α} {a : α} : a ∈ xs ++ [a] :=
   mem_append_right xs mem_cons_self
@@ -411,7 +429,7 @@ theorem eq_append_cons_of_mem {a : α} {xs : List α} (h : a ∈ xs) :
       · obtain ⟨as, bs, rfl, h⟩ := ih h
         exact ⟨x :: as, bs, rfl, by simp_all⟩
 
-@[grind] theorem mem_cons_of_mem (y : α) {a : α} {l : List α} : a ∈ l → a ∈ y :: l := .tail _
+theorem mem_cons_of_mem (y : α) {a : α} {l : List α} : a ∈ l → a ∈ y :: l := .tail _
 
 -- The argument `l : List α` is intentionally explicit,
 -- as a tactic may generate `h` without determining `l`.
@@ -547,10 +565,10 @@ theorem contains_iff [BEq α] [LawfulBEq α] {a : α} {as : List α} :
 theorem elem_eq_mem [BEq α] [LawfulBEq α] (a : α) (as : List α) :
     elem a as = decide (a ∈ as) := by rw [Bool.eq_iff_iff, elem_iff, decide_eq_true_iff]
 
-@[simp, grind] theorem contains_eq_mem [BEq α] [LawfulBEq α] (a : α) (as : List α) :
+@[simp, grind =] theorem contains_eq_mem [BEq α] [LawfulBEq α] (a : α) (as : List α) :
     as.contains a = decide (a ∈ as) := by rw [Bool.eq_iff_iff, elem_iff, decide_eq_true_iff]
 
-@[simp, grind] theorem contains_cons [BEq α] {a : α} {b : α} {l : List α} :
+@[simp, grind =] theorem contains_cons [BEq α] {a : α} {b : α} {l : List α} :
     (a :: l).contains b = (b == a || l.contains b) := by
   simp only [contains, elem_cons]
   split <;> simp_all
@@ -605,7 +623,7 @@ theorem decide_forall_mem {l : List α} {p : α → Prop} [DecidablePred p] :
 @[simp] theorem all_eq_false {l : List α} : l.all p = false ↔ ∃ x, x ∈ l ∧ ¬p x := by
   simp [all_eq]
 
-@[grind] theorem any_beq [BEq α] {l : List α} {a : α} : (l.any fun x => a == x) = l.contains a := by
+theorem any_beq [BEq α] {l : List α} {a : α} : (l.any fun x => a == x) = l.contains a := by
   induction l <;> simp_all [contains_cons]
 
 /-- Variant of `any_beq` with `==` reversed. -/
@@ -613,7 +631,7 @@ theorem any_beq' [BEq α] [PartialEquivBEq α] {l : List α} :
     (l.any fun x => x == a) = l.contains a := by
   simp only [BEq.comm, any_beq]
 
-@[grind] theorem all_bne [BEq α] {l : List α} : (l.all fun x => a != x) = !l.contains a := by
+theorem all_bne [BEq α] {l : List α} : (l.all fun x => a != x) = !l.contains a := by
   induction l <;> simp_all [bne]
 
 /-- Variant of `all_bne` with `!=` reversed. -/
@@ -624,10 +642,10 @@ theorem all_bne' [BEq α] [PartialEquivBEq α] {l : List α} :
 /-! ### set -/
 
 -- As `List.set` is defined in `Init.Prelude`, we write the basic simplification lemmas here.
-@[simp, grind] theorem set_nil {i : Nat} {a : α} : [].set i a = [] := rfl
-@[simp, grind] theorem set_cons_zero {x : α} {xs : List α} {a : α} :
+@[simp, grind =] theorem set_nil {i : Nat} {a : α} : [].set i a = [] := rfl
+@[simp, grind =] theorem set_cons_zero {x : α} {xs : List α} {a : α} :
   (x :: xs).set 0 a = a :: xs := rfl
-@[simp, grind] theorem set_cons_succ {x : α} {xs : List α} {i : Nat} {a : α} :
+@[simp, grind =] theorem set_cons_succ {x : α} {xs : List α} {i : Nat} {a : α} :
   (x :: xs).set (i + 1) a = x :: xs.set i a := rfl
 
 @[simp] theorem getElem_set_self {l : List α} {i : Nat} {a : α} (h : i < (l.set i a).length) :
@@ -670,14 +688,14 @@ theorem getElem?_set_self' {l : List α} {i : Nat} {a : α} :
     simp_all
   · rw [getElem?_eq_none (by simp_all), getElem?_eq_none (by simp_all)]
 
-@[grind] theorem getElem_set {l : List α} {i j} {a} (h) :
+@[grind =] theorem getElem_set {l : List α} {i j} {a} (h) :
     (set l i a)[j]'h = if i = j then a else l[j]'(length_set .. ▸ h) := by
   if h : i = j then
     subst h; simp only [getElem_set_self, ↓reduceIte]
   else
     simp [h]
 
-@[grind] theorem getElem?_set {l : List α} {i j : Nat} {a : α} :
+@[grind =] theorem getElem?_set {l : List α} {i j : Nat} {a : α} :
     (l.set i a)[j]? = if i = j then if i < l.length then some a else none else l[j]? := by
   if h : i = j then
     subst h
@@ -747,10 +765,10 @@ theorem mem_or_eq_of_mem_set : ∀ {l : List α} {i : Nat} {a b : α}, a ∈ l.s
 
 /-! ### BEq -/
 
-@[simp, grind] theorem beq_nil_eq [BEq α] {l : List α} : (l == []) = l.isEmpty := by
+@[simp, grind =] theorem beq_nil_eq [BEq α] {l : List α} : (l == []) = l.isEmpty := by
   cases l <;> rfl
 
-@[simp, grind] theorem nil_beq_eq [BEq α] {l : List α} : ([] == l) = l.isEmpty := by
+@[simp, grind =] theorem nil_beq_eq [BEq α] {l : List α} : ([] == l) = l.isEmpty := by
   cases l <;> rfl
 
 @[deprecated beq_nil_eq (since := "2025-04-04")]
@@ -759,7 +777,7 @@ abbrev beq_nil_iff := @beq_nil_eq
 @[deprecated nil_beq_eq (since := "2025-04-04")]
 abbrev nil_beq_iff := @nil_beq_eq
 
-@[simp, grind] theorem cons_beq_cons [BEq α] {a b : α} {l₁ l₂ : List α} :
+@[simp, grind =] theorem cons_beq_cons [BEq α] {a b : α} {l₁ l₂ : List α} :
     (a :: l₁ == b :: l₂) = (a == b && l₁ == l₂) := rfl
 
 @[simp] theorem concat_beq_concat [BEq α] {a b : α} {l₁ l₂ : List α} :
@@ -825,7 +843,7 @@ theorem length_eq_of_beq [BEq α] {l₁ l₂ : List α} (h : l₁ == l₂) : l
 
 /-! ### getLast -/
 
-@[grind]
+@[grind =]
 theorem getLast_eq_getElem : ∀ {l : List α} (h : l ≠ []),
     getLast l h = l[l.length - 1]'(by
       match l with
@@ -839,7 +857,7 @@ theorem getElem_length_sub_one_eq_getLast {l : List α} (h : l.length - 1 < l.le
     l[l.length - 1] = getLast l (by cases l; simp at h; simp) := by
   rw [← getLast_eq_getElem]
 
-@[simp, grind] theorem getLast_cons_cons {a : α} {l : List α} :
+@[simp, grind =] theorem getLast_cons_cons {a : α} {l : List α} :
     getLast (a :: b :: l) (by simp) = getLast (b :: l) (by simp) :=
   rfl
 
@@ -852,10 +870,10 @@ theorem getLast_cons {a : α} {l : List α} : ∀ (h : l ≠ nil),
 theorem getLast_eq_getLastD {a l} (h) : @getLast α (a::l) h = getLastD l a := by
   cases l <;> rfl
 
-@[simp, grind] theorem getLastD_eq_getLast? {a l} : @getLastD α l a = (getLast? l).getD a := by
+@[simp, grind =] theorem getLastD_eq_getLast? {a l} : @getLastD α l a = (getLast? l).getD a := by
   cases l <;> rfl
 
-@[simp, grind] theorem getLast_singleton {a} (h) : @getLast α [a] h = a := rfl
+@[simp, grind =] theorem getLast_singleton {a} (h) : @getLast α [a] h = a := rfl
 
 theorem getLast!_cons_eq_getLastD [Inhabited α] : @getLast! α _ (a::l) = getLastD l a := by
   simp [getLast!, getLast_eq_getLastD]
@@ -888,7 +906,7 @@ theorem getLast?_eq_getLast : ∀ {l : List α} h, l.getLast? = some (l.getLast
   | [], h => nomatch h rfl
   | _ :: _, _ => rfl
 
-@[grind] theorem getLast?_eq_getElem? : ∀ {l : List α}, l.getLast? = l[l.length - 1]?
+@[grind =] theorem getLast?_eq_getElem? : ∀ {l : List α}, l.getLast? = l[l.length - 1]?
   | [] => rfl
   | a::l => by
     rw [getLast?_eq_getLast (l := a :: l) nofun, getLast_eq_getElem, getElem?_eq_getElem]
@@ -901,14 +919,14 @@ theorem getLast_eq_iff_getLast?_eq_some {xs : List α} (h) :
 -- `getLast?_eq_none_iff`, `getLast?_eq_some_iff`, `getLast?_isSome`, and `getLast_mem`
 -- are proved later once more `reverse` theorems are available.
 
-@[grind]
+@[grind =]
 theorem getLast?_cons {a : α} : (a::l).getLast? = some (l.getLast?.getD a) := by
   cases l <;> simp [getLast?, getLast]
 
 @[simp] theorem getLast?_cons_cons : (a :: b :: l).getLast? = (b :: l).getLast? := by
   simp [getLast?_cons]
 
-@[grind]
+@[grind =]
 theorem getLast?_concat {l : List α} {a : α} : (l ++ [a]).getLast? = some a := by
   simp [getLast?_eq_getElem?, Nat.succ_sub_succ]
 
@@ -927,14 +945,14 @@ theorem getLast!_nil [Inhabited α] : ([] : List α).getLast! = default := rfl
 theorem getLast!_of_getLast? [Inhabited α] : ∀ {l : List α}, getLast? l = some a → getLast! l = a
   | _ :: _, rfl => rfl
 
-@[grind]
+@[grind =]
 theorem getLast!_eq_getElem! [Inhabited α] {l : List α} : l.getLast! = l[l.length - 1]! := by
   cases l with
   | nil => simp
   | cons _ _ =>
     apply getLast!_of_getLast?
-    rw [getElem!_pos, getElem_cons_length (h := by simp)]
-    rfl
+    rw [getLast?_eq_getElem?]
+    simp
 
 /-! ## Head and tail -/
 
@@ -955,7 +973,7 @@ theorem head?_eq_getElem? : ∀ {l : List α}, l.head? = l[0]?
 
 theorem head_singleton {a : α} : head [a] (by simp) = a := by simp
 
-@[grind]
+@[grind =]
 theorem head_eq_getElem {l : List α} (h : l ≠ []) : head l h = l[0]'(length_pos_iff.mpr h) := by
   cases l with
   | nil => simp at h
@@ -1017,18 +1035,18 @@ theorem head_of_mem_head? {l : List α} {x} (hx : x ∈ l.head?) :
 /-! ### headD -/
 
 /-- `simp` unfolds `headD` in terms of `head?` and `Option.getD`. -/
-@[simp, grind] theorem headD_eq_head?_getD {l : List α} : headD l a = (head? l).getD a := by
+@[simp, grind =] theorem headD_eq_head?_getD {l : List α} : headD l a = (head? l).getD a := by
   cases l <;> simp [headD]
 
 /-! ### tailD -/
 
 /-- `simp` unfolds `tailD` in terms of `tail?` and `Option.getD`. -/
-@[simp, grind] theorem tailD_eq_tail? {l l' : List α} : tailD l l' = (tail? l).getD l' := by
+@[simp, grind =] theorem tailD_eq_tail? {l l' : List α} : tailD l l' = (tail? l).getD l' := by
   cases l <;> rfl
 
 /-! ### tail -/
 
-@[simp, grind] theorem length_tail {l : List α} : l.tail.length = l.length - 1 := by cases l <;> rfl
+@[simp, grind =] theorem length_tail {l : List α} : l.tail.length = l.length - 1 := by cases l <;> rfl
 
 theorem tail_eq_tailD {l : List α} : l.tail = tailD l [] := by cases l <;> rfl
 
@@ -1040,13 +1058,13 @@ theorem mem_of_mem_tail {a : α} {l : List α} (h : a ∈ tail l) : a ∈ l := b
 theorem ne_nil_of_tail_ne_nil {l : List α} : l.tail ≠ [] → l ≠ [] := by
   cases l <;> simp
 
-@[simp, grind] theorem getElem_tail {l : List α} {i : Nat} (h : i < l.tail.length) :
+@[simp, grind =] theorem getElem_tail {l : List α} {i : Nat} (h : i < l.tail.length) :
     (tail l)[i] = l[i + 1]'(add_lt_of_lt_sub (by simpa using h)) := by
   cases l with
   | nil => simp at h
   | cons _ l => simp
 
-@[simp, grind] theorem getElem?_tail {l : List α} {i : Nat} :
+@[simp, grind =] theorem getElem?_tail {l : List α} {i : Nat} :
     (tail l)[i]? = l[i + 1]? := by
   cases l <;> simp
 
@@ -1070,7 +1088,7 @@ theorem one_lt_length_of_tail_ne_nil {l : List α} (h : l.tail ≠ []) : 1 < l.l
 @[simp] theorem head?_tail {l : List α} : (tail l).head? = l[1]? := by
   simp [head?_eq_getElem?]
 
-@[simp, grind] theorem getLast_tail {l : List α} (h : l.tail ≠ []) :
+@[simp, grind =] theorem getLast_tail {l : List α} (h : l.tail ≠ []) :
     (tail l).getLast h = l.getLast (ne_nil_of_tail_ne_nil h) := by
   simp only [getLast_eq_getElem, length_tail, getElem_tail]
   congr
@@ -1096,7 +1114,7 @@ theorem cons_head_tail (h : l ≠ []) : l.head h :: l.tail = l := by
 
 /-! ### map -/
 
-@[simp, grind] theorem length_map {as : List α} (f : α → β) : (as.map f).length = as.length := by
+@[simp, grind =] theorem length_map {as : List α} (f : α → β) : (as.map f).length = as.length := by
   induction as with
   | nil => simp [List.map]
   | cons _ as ih => simp [List.map, ih]
@@ -1104,13 +1122,13 @@ theorem cons_head_tail (h : l ≠ []) : l.head h :: l.tail = l := by
 @[simp] theorem isEmpty_map {l : List α} {f : α → β} : (l.map f).isEmpty = l.isEmpty := by
   cases l <;> simp
 
-@[simp, grind] theorem getElem?_map {f : α → β} : ∀ {l : List α} {i : Nat}, (map f l)[i]? = Option.map f l[i]?
+@[simp, grind =] theorem getElem?_map {f : α → β} : ∀ {l : List α} {i : Nat}, (map f l)[i]? = Option.map f l[i]?
   | [], _ => rfl
   | _ :: _, 0 => by simp
   | _ :: l, i+1 => by simp [getElem?_map]
 
 -- The argument `f : α → β` is explicit, to facilitate rewriting from right to left.
-@[simp, grind] theorem getElem_map (f : α → β) {l} {i : Nat} {h : i < (map f l).length} :
+@[simp, grind =] theorem getElem_map (f : α → β) {l} {i : Nat} {h : i < (map f l).length} :
     (map f l)[i] = f (l[i]'(length_map f ▸ h)) :=
   Option.some.inj <| by rw [← getElem?_eq_getElem, getElem?_map, getElem?_eq_getElem]; rfl
 
@@ -1156,7 +1174,9 @@ theorem forall_mem_map {f : α → β} {l : List α} {P : β → Prop} :
 @[simp] theorem map_eq_nil_iff {f : α → β} {l : List α} : map f l = [] ↔ l = [] := by
   constructor <;> exact fun _ => match l with | [] => rfl
 
-@[grind →]
+-- This would be helpful as a `grind` lemma if
+-- we could have it fire only once `map f l` and `[]` are the same equivalence class.
+-- Otherwise it is too aggressive.
 theorem eq_nil_of_map_eq_nil {f : α → β} {l : List α} (h : map f l = []) : l = [] :=
   map_eq_nil_iff.mp h
 
@@ -1276,7 +1296,7 @@ theorem getLastD_map {f : α → β} {l : List α} {a : α} : (map f l).getLastD
 @[simp] theorem filter_cons_of_neg {p : α → Bool} {a : α} {l} (pa : ¬ p a) :
     filter p (a :: l) = filter p l := by rw [filter, eq_false_of_ne_true pa]
 
-@[grind] theorem filter_cons :
+@[grind =] theorem filter_cons :
     (x :: xs : List α).filter p = if p x then x :: (xs.filter p) else xs.filter p := by
   split <;> simp [*]
 
@@ -1315,7 +1335,7 @@ theorem length_filter_eq_length_iff {l} : (filter p l).length = l.length ↔ ∀
 @[deprecated length_filter_eq_length_iff (since := "2025-04-04")]
 abbrev filter_length_eq_length := @length_filter_eq_length_iff
 
-@[simp, grind] theorem mem_filter : x ∈ filter p as ↔ x ∈ as ∧ p x := by
+@[simp, grind =] theorem mem_filter : x ∈ filter p as ↔ x ∈ as ∧ p x := by
   induction as with
   | nil => simp
   | cons a as ih =>
@@ -1330,13 +1350,15 @@ 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) :
+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
 
+grind_pattern getElem_filter => (xs.filter p)[i]
+
 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
+  rw [getElem?_eq_getElem h] at w
   simp only [Option.some.injEq] at w
   rw [← w]
   apply getElem_filter h
@@ -1377,7 +1399,7 @@ theorem map_filter_eq_foldr {f : α → β} {p : α → Bool} {as : List α} :
     simp only [foldr]
     cases hp : p head <;> simp [filter, *]
 
-@[simp, grind] theorem filter_append {p : α → Bool} :
+@[simp, grind =] theorem filter_append {p : α → Bool} :
     ∀ (l₁ l₂ : List α), filter p (l₁ ++ l₂) = filter p l₁ ++ filter p l₂
   | [], _ => rfl
   | a :: l₁, l₂ => by simp only [cons_append, filter]; split <;> simp [filter_append l₁]
@@ -1442,7 +1464,7 @@ theorem filterMap_some_fun : filterMap (some : α → Option α) = id := by
   erw [filterMap_eq_map]
   simp
 
-@[simp, grind] theorem filterMap_some {l : List α} : filterMap some l = l := by
+@[simp, grind =] theorem filterMap_some {l : List α} : filterMap some l = l := by
   rw [filterMap_some_fun, id]
 
 theorem map_filterMap_some_eq_filter_map_isSome {f : α → Option β} {l : List α} :
@@ -1477,19 +1499,19 @@ theorem filterMap_eq_filter {p : α → Bool} :
   | nil => rfl
   | cons a l IH => by_cases pa : p a <;> simp [Option.guard, pa, ← IH]
 
-@[grind]
+@[grind =]
 theorem filterMap_filterMap {f : α → Option β} {g : β → Option γ} {l : List α} :
     filterMap g (filterMap f l) = filterMap (fun x => (f x).bind g) l := by
   induction l with
   | nil => rfl
   | cons a l IH => cases h : f a <;> simp [filterMap_cons, *]
 
-@[grind]
+@[grind =]
 theorem map_filterMap {f : α → Option β} {g : β → γ} {l : List α} :
     map g (filterMap f l) = filterMap (fun x => (f x).map g) l := by
   simp only [← filterMap_eq_map, filterMap_filterMap, Option.map_eq_bind]
 
-@[simp, grind]
+@[simp, grind =]
 theorem filterMap_map {f : α → β} {g : β → Option γ} {l : List α} :
     filterMap g (map f l) = filterMap (g ∘ f) l := by
   rw [← filterMap_eq_map, filterMap_filterMap]; rfl
@@ -1504,7 +1526,7 @@ theorem filterMap_filter {p : α → Bool} {f : α → Option β} {l : List α}
   rw [← filterMap_eq_filter, filterMap_filterMap]
   congr; funext x; by_cases h : p x <;> simp [Option.guard, h]
 
-@[simp, grind] theorem mem_filterMap {f : α → Option β} {l : List α} {b : β} :
+@[simp, grind =] theorem mem_filterMap {f : α → Option β} {l : List α} {b : β} :
     b ∈ filterMap f l ↔ ∃ a, a ∈ l ∧ f a = some b := by
   induction l <;> simp [filterMap_cons]; split <;> simp [*, eq_comm]
 
@@ -1516,7 +1538,7 @@ theorem forall_mem_filterMap {f : α → Option β} {l : List α} {P : β → Pr
   intro a
   rw [forall_comm]
 
-@[simp, grind] theorem filterMap_append {l l' : List α} {f : α → Option β} :
+@[simp, grind =] theorem filterMap_append {l l' : List α} {f : α → Option β} :
     filterMap f (l ++ l') = filterMap f l ++ filterMap f l' := by
   induction l <;> simp [filterMap_cons]; split <;> simp [*]
 
@@ -1588,7 +1610,7 @@ theorem filterMap_eq_cons_iff {l} {b} {bs} :
 @[simp] theorem cons_append_fun {a : α} {as : List α} :
     (fun bs => ((a :: as) ++ bs)) = fun bs => a :: (as ++ bs) := rfl
 
-@[simp, grind] theorem mem_append {a : α} {s t : List α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
+@[simp, grind =] theorem mem_append {a : α} {s t : List α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
   induction s <;> simp_all [or_assoc]
 
 theorem not_mem_append {a : α} {s t : List α} (h₁ : a ∉ s) (h₂ : a ∉ t) : a ∉ s ++ t :=
@@ -1611,7 +1633,7 @@ theorem forall_mem_append {p : α → Prop} {l₁ l₂ : List α} :
     (∀ (x) (_ : x ∈ l₁ ++ l₂), p x) ↔ (∀ (x) (_ : x ∈ l₁), p x) ∧ (∀ (x) (_ : x ∈ l₂), p x) := by
   simp only [mem_append, or_imp, forall_and]
 
-@[grind] theorem getElem_append {l₁ l₂ : List α} {i : Nat} (h : i < (l₁ ++ l₂).length) :
+@[grind =] theorem getElem_append {l₁ l₂ : List α} {i : Nat} (h : i < (l₁ ++ l₂).length) :
     (l₁ ++ l₂)[i] = if h' : i < l₁.length then l₁[i] else l₂[i - l₁.length]'(by simp at h h'; exact Nat.sub_lt_left_of_lt_add h' h) := by
   split <;> rename_i h'
   · rw [getElem_append_left h']
@@ -1630,7 +1652,7 @@ theorem getElem?_append_right : ∀ {l₁ l₂ : List α} {i : Nat}, l₁.length
   rw [cons_append]
   simp [Nat.succ_sub_succ_eq_sub, getElem?_append_right (Nat.lt_succ.1 h₁)]
 
-@[grind] theorem getElem?_append {l₁ l₂ : List α} {i : Nat} :
+@[grind =] theorem getElem?_append {l₁ l₂ : List α} {i : Nat} :
     (l₁ ++ l₂)[i]? = if i < l₁.length then l₁[i]? else l₂[i - l₁.length]? := by
   split <;> rename_i h
   · exact getElem?_append_left h
@@ -1709,7 +1731,6 @@ theorem getLast_concat {a : α} : ∀ {l : List α}, getLast (l ++ [a]) (by simp
 theorem nil_eq_append_iff : [] = a ++ b ↔ a = [] ∧ b = [] := by
   simp
 
-@[grind →]
 theorem eq_nil_of_append_eq_nil {l₁ l₂ : List α} (h : l₁ ++ l₂ = []) : l₁ = [] ∧ l₂ = [] :=
   append_eq_nil_iff.mp h
 
@@ -1739,12 +1760,12 @@ theorem append_eq_append_iff {ws xs ys zs : List α} :
   | nil => simp_all
   | cons a as ih => cases ys <;> simp [eq_comm, and_assoc, ih, and_or_left]
 
-@[simp, grind] theorem head_append_of_ne_nil {l : List α} {w₁} (w₂) :
+@[simp, grind =] theorem head_append_of_ne_nil {l : List α} {w₁} (w₂) :
     head (l ++ l') w₁ = head l w₂ := by
   match l, w₂ with
   | a :: l, _ => rfl
 
-@[grind] theorem head_append {l₁ l₂ : List α} (w : l₁ ++ l₂ ≠ []) :
+@[grind =] theorem head_append {l₁ l₂ : List α} (w : l₁ ++ l₂ ≠ []) :
     head (l₁ ++ l₂) w =
       if h : l₁.isEmpty then
         head l₂ (by simp_all [isEmpty_iff])
@@ -1765,28 +1786,28 @@ theorem head_append_right {l₁ l₂ : List α} (w : l₁ ++ l₂ ≠ []) (h : l
     head (l₁ ++ l₂) w = head l₂ (by simp_all) := by
   rw [head_append, dif_pos (by simp_all)]
 
-@[simp, grind] theorem head?_append {l : List α} : (l ++ l').head? = l.head?.or l'.head? := by
+@[simp, grind =] theorem head?_append {l : List α} : (l ++ l').head? = l.head?.or l'.head? := by
   cases l <;> simp
 
 -- Note:
 -- `getLast_append_of_ne_nil`, `getLast_append` and `getLast?_append`
 -- are stated and proved later in the `reverse` section.
 
-@[grind] theorem tail?_append {l l' : List α} : (l ++ l').tail? = (l.tail?.map (· ++ l')).or l'.tail? := by
+@[grind =] theorem tail?_append {l l' : List α} : (l ++ l').tail? = (l.tail?.map (· ++ l')).or l'.tail? := by
   cases l <;> simp
 
 theorem tail?_append_of_ne_nil {l l' : List α} (_ : l ≠ []) : (l ++ l').tail? = some (l.tail ++ l') :=
   match l with
   | _ :: _ => by simp
 
-@[grind] theorem tail_append {l l' : List α} : (l ++ l').tail = if l.isEmpty then l'.tail else l.tail ++ l' := by
+@[grind =] theorem tail_append {l l' : List α} : (l ++ l').tail = if l.isEmpty then l'.tail else l.tail ++ l' := by
   cases l <;> simp
 
 @[simp] theorem tail_append_of_ne_nil {xs ys : List α} (h : xs ≠ []) :
     (xs ++ ys).tail = xs.tail ++ ys := by
   simp_all [tail_append]
 
-@[grind] theorem set_append {s t : List α} :
+@[grind =] theorem set_append {s t : List α} :
     (s ++ t).set i x = if i < s.length then s.set i x ++ t else s ++ t.set (i - s.length) x := by
   induction s generalizing i with
   | nil => simp
@@ -1844,7 +1865,7 @@ theorem append_eq_filter_iff {p : α → Bool} :
     L₁ ++ L₂ = filter p l ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filter p l₁ = L₁ ∧ filter p l₂ = L₂ := by
   rw [eq_comm, filter_eq_append_iff]
 
-@[simp, grind] theorem map_append {f : α → β} : ∀ {l₁ l₂}, map f (l₁ ++ l₂) = map f l₁ ++ map f l₂ := by
+@[simp, grind =] theorem map_append {f : α → β} : ∀ {l₁ l₂}, map f (l₁ ++ l₂) = map f l₁ ++ map f l₂ := by
   intro l₁; induction l₁ <;> intros <;> simp_all
 
 theorem map_eq_append_iff {f : α → β} :
@@ -1917,7 +1938,7 @@ theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ l' b, l = concat l' b
   | cons =>
     simp [flatten, length_append, *]
 
-@[grind] theorem flatten_singleton {l : List α} : [l].flatten = l := by simp
+@[grind =] theorem flatten_singleton {l : List α} : [l].flatten = l := by simp
 
 @[simp] theorem mem_flatten : ∀ {L : List (List α)}, a ∈ L.flatten ↔ ∃ l, l ∈ L ∧ a ∈ l
   | [] => by simp
@@ -2092,7 +2113,7 @@ theorem length_flatMap {l : List α} {f : α → List β} :
     length (l.flatMap f) = sum (map (fun a => (f a).length) l) := by
   rw [List.flatMap, length_flatten, map_map, Function.comp_def]
 
-@[simp, grind] theorem mem_flatMap {f : α → List β} {b} {l : List α} : b ∈ l.flatMap f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by
+@[simp, grind =] theorem mem_flatMap {f : α → List β} {b} {l : List α} : b ∈ l.flatMap f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by
   simp [flatMap_def, mem_flatten]
   exact ⟨fun ⟨_, ⟨a, h₁, rfl⟩, h₂⟩ => ⟨a, h₁, h₂⟩, fun ⟨a, h₁, h₂⟩ => ⟨_, ⟨a, h₁, rfl⟩, h₂⟩⟩
 
@@ -2119,7 +2140,7 @@ theorem flatMap_singleton (f : α → List β) (x : α) : [x].flatMap f = f x :=
 @[simp] theorem flatMap_singleton' (l : List α) : (l.flatMap fun x => [x]) = l := by
   induction l <;> simp [*]
 
-@[grind] theorem head?_flatMap {l : List α} {f : α → List β} :
+@[grind =] theorem head?_flatMap {l : List α} {f : α → List β} :
     (l.flatMap f).head? = l.findSome? fun a => (f a).head? := by
   induction l with
   | nil => rfl
@@ -2127,10 +2148,6 @@ theorem flatMap_singleton (f : α → List β) (x : α) : [x].flatMap f = f x :=
     simp only [findSome?_cons]
     split <;> simp_all
 
-@[simp, grind _=_] theorem flatMap_append {xs ys : List α} {f : α → List β} :
-    (xs ++ ys).flatMap f = xs.flatMap f ++ ys.flatMap f := by
-  induction xs; {rfl}; simp_all [flatMap_cons, append_assoc]
-
 theorem flatMap_assoc {l : List α} {f : α → List β} {g : β → List γ} :
     (l.flatMap f).flatMap g = l.flatMap fun x => (f x).flatMap g := by
   induction l <;> simp [*]
@@ -2172,7 +2189,7 @@ theorem flatMap_eq_foldl {f : α → List β} {l : List α} :
 theorem replicate_succ' : replicate (n + 1) a = replicate n a ++ [a] := by
   induction n <;> simp_all [replicate_succ, ← cons_append]
 
-@[simp, grind] theorem mem_replicate {a b : α} : ∀ {n}, b ∈ replicate n a ↔ n ≠ 0 ∧ b = a
+@[simp, grind =] theorem mem_replicate {a b : α} : ∀ {n}, b ∈ replicate n a ↔ n ≠ 0 ∧ b = a
   | 0 => by simp
   | n+1 => by simp [replicate_succ, mem_replicate, Nat.succ_ne_zero]
 
@@ -2197,11 +2214,11 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
 @[simp] theorem replicate_eq_nil_iff {n : Nat} (a : α) : replicate n a = [] ↔ n = 0 := by
   cases n <;> simp
 
-@[simp, grind] theorem getElem_replicate {a : α} {n : Nat} {i : Nat} (h : i < (replicate n a).length) :
+@[simp, grind =] theorem getElem_replicate {a : α} {n : Nat} {i : Nat} (h : i < (replicate n a).length) :
     (replicate n a)[i] = a :=
   eq_of_mem_replicate (getElem_mem _)
 
-@[grind] theorem getElem?_replicate : (replicate n a)[i]? = if i < n then some a else none := by
+@[grind =] theorem getElem?_replicate : (replicate n a)[i]? = if i < n then some a else none := by
   by_cases h : i < n
   · rw [getElem?_eq_getElem (by simpa), getElem_replicate, if_pos h]
   · rw [getElem?_eq_none (by simpa using h), if_neg h]
@@ -2209,7 +2226,7 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
 @[simp] theorem getElem?_replicate_of_lt {n : Nat} {i : Nat} (h : i < n) : (replicate n a)[i]? = some a := by
   simp [h]
 
-@[grind] theorem head?_replicate {a : α} {n : Nat} : (replicate n a).head? = if n = 0 then none else some a := by
+@[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]
 
 @[simp] theorem head_replicate (w : replicate n a ≠ []) : (replicate n a).head w = a := by
@@ -2298,7 +2315,7 @@ theorem replicate_eq_append_iff {l₁ l₂ : List α} {a : α} :
   simp only [getElem?_map, getElem?_replicate]
   split <;> simp
 
-@[grind] theorem filter_replicate : (replicate n a).filter p = if p a then replicate n a else [] := by
+ @[grind =] theorem filter_replicate : (replicate n a).filter p = if p a then replicate n a else [] := by
   cases n with
   | zero => simp
   | succ n =>
@@ -2401,7 +2418,7 @@ termination_by l.length
 
 /-! ### reverse -/
 
-@[simp, grind] theorem length_reverse {as : List α} : (as.reverse).length = as.length := by
+@[simp, grind =] theorem length_reverse {as : List α} : (as.reverse).length = as.length := by
   induction as with
   | nil => rfl
   | cons a as ih => simp [ih]
@@ -2410,7 +2427,7 @@ theorem mem_reverseAux {x : α} : ∀ {as bs}, x ∈ reverseAux as bs ↔ x ∈
   | [], _ => ⟨.inr, fun | .inr h => h⟩
   | a :: _, _ => by rw [reverseAux, mem_cons, or_assoc, or_left_comm, mem_reverseAux, mem_cons]
 
-@[simp, grind] theorem mem_reverse {x : α} {as : List α} : x ∈ reverse as ↔ x ∈ as := by
+@[simp, grind =] theorem mem_reverse {x : α} {as : List α} : x ∈ reverse as ↔ x ∈ as := by
   simp [reverse, mem_reverseAux]
 
 @[simp] theorem reverse_eq_nil_iff {xs : List α} : xs.reverse = [] ↔ xs = [] := by
@@ -2434,14 +2451,14 @@ theorem getElem?_reverse' : ∀ {l : List α} {i j}, i + j + 1 = length l →
     rw [getElem?_append_left, getElem?_reverse' this]
     rw [length_reverse, ← this]; apply Nat.lt_add_of_pos_right (Nat.succ_pos _)
 
-@[simp, grind]
+@[simp, grind =]
 theorem getElem?_reverse {l : List α} {i} (h : i < length l) :
     l.reverse[i]? = l[l.length - 1 - i]? :=
   getElem?_reverse' <| by
     rw [Nat.add_sub_of_le (Nat.le_sub_one_of_lt h),
       Nat.sub_add_cancel (Nat.lt_of_le_of_lt (Nat.zero_le _) h)]
 
-@[simp, grind]
+@[simp, grind =]
 theorem getElem_reverse {l : List α} {i} (h : i < l.reverse.length) :
     l.reverse[i] = l[l.length - 1 - i]'(Nat.sub_one_sub_lt_of_lt (by simpa using h)) := by
   apply Option.some.inj
@@ -2454,7 +2471,7 @@ theorem reverseAux_reverseAux_nil {as bs : List α} : reverseAux (reverseAux as
   | cons a as ih => simp [reverseAux, ih]
 
 -- The argument `as : List α` is explicit to allow rewriting from right to left.
-@[simp, grind] theorem reverse_reverse (as : List α) : as.reverse.reverse = as := by
+@[simp, grind =] theorem reverse_reverse (as : List α) : as.reverse.reverse = as := by
   simp only [reverse]; rw [reverseAux_reverseAux_nil]; rfl
 
 theorem reverse_eq_iff {as bs : List α} : as.reverse = bs ↔ as = bs.reverse := by
@@ -2467,10 +2484,10 @@ theorem reverse_eq_iff {as bs : List α} : as.reverse = bs ↔ as = bs.reverse :
     xs.reverse = a :: ys ↔ xs = ys.reverse ++ [a] := by
   rw [reverse_eq_iff, reverse_cons]
 
-@[simp, grind] theorem getLast?_reverse {l : List α} : l.reverse.getLast? = l.head? := by
+@[simp, grind =] theorem getLast?_reverse {l : List α} : l.reverse.getLast? = l.head? := by
   cases l <;> simp [getLast?_concat]
 
-@[simp, grind] theorem head?_reverse {l : List α} : l.reverse.head? = l.getLast? := by
+@[simp, grind =] theorem head?_reverse {l : List α} : l.reverse.head? = l.getLast? := by
   rw [← getLast?_reverse, reverse_reverse]
 
 theorem getLast?_eq_head?_reverse {xs : List α} : xs.getLast? = xs.reverse.head? := by
@@ -2534,16 +2551,16 @@ theorem flatten_reverse {L : List (List α)} :
     L.reverse.flatten = (L.map reverse).flatten.reverse := by
   induction L <;> simp_all
 
-@[grind] theorem reverse_flatMap {β} {l : List α} {f : α → List β} : (l.flatMap f).reverse = l.reverse.flatMap (reverse ∘ f) := by
+@[grind =] theorem reverse_flatMap {β} {l : List α} {f : α → List β} : (l.flatMap f).reverse = l.reverse.flatMap (reverse ∘ f) := by
   induction l <;> simp_all
 
-@[grind] theorem flatMap_reverse {β} {l : List α} {f : α → List β} : (l.reverse.flatMap f) = (l.flatMap (reverse ∘ f)).reverse := by
+@[grind =] theorem flatMap_reverse {β} {l : List α} {f : α → List β} : (l.reverse.flatMap f) = (l.flatMap (reverse ∘ f)).reverse := by
   induction l <;> simp_all
 
 @[simp] theorem reverseAux_eq {as bs : List α} : reverseAux as bs = reverse as ++ bs :=
   reverseAux_eq_append ..
 
-@[simp, grind] theorem reverse_replicate {n : Nat} {a : α} : (replicate n a).reverse = replicate n a :=
+@[simp, grind =] theorem reverse_replicate {n : Nat} {a : α} : (replicate n a).reverse = replicate n a :=
   eq_replicate_iff.2
     ⟨by rw [length_reverse, length_replicate],
      fun _ h => eq_of_mem_replicate (mem_reverse.1 h)⟩
@@ -2555,7 +2572,7 @@ theorem flatten_reverse {L : List (List α)} :
     (l ++ l').foldlM f b = l.foldlM f b >>= l'.foldlM f := by
   induction l generalizing b <;> simp [*]
 
-@[simp, grind] theorem foldrM_cons [Monad m] [LawfulMonad m] {a : α} {l : List α} {f : α → β → m β} {b : β} :
+@[simp, grind =] theorem foldrM_cons [Monad m] [LawfulMonad m] {a : α} {l : List α} {f : α → β → m β} {b : β} :
     (a :: l).foldrM f b = l.foldrM f b >>= f a := by
   simp only [foldrM]
   induction l <;> simp_all
@@ -2599,37 +2616,37 @@ theorem id_run_foldrM {f : α → β → Id β} {b : β} {l : List α} :
 
 /-! ### foldl and foldr -/
 
-@[simp, grind] theorem foldr_cons_eq_append {l : List α} {f : α → β} {l' : List β} :
+@[simp] theorem foldr_cons_eq_append {l : List α} {f : α → β} {l' : List β} :
     l.foldr (fun x ys => f x :: ys) l' = l.map f ++ l' := by
   induction l <;> simp [*]
 
 /-- Variant of `foldr_cons_eq_append` specalized to `f = id`. -/
-@[simp, grind] theorem foldr_cons_eq_append' {l l' : List β} :
+@[simp, grind =] theorem foldr_cons_eq_append' {l l' : List β} :
     l.foldr cons l' = l ++ l' := by
   induction l <;> simp [*]
 
-@[simp, grind] theorem foldl_flip_cons_eq_append {l : List α} {f : α → β} {l' : List β} :
+@[simp] theorem foldl_flip_cons_eq_append {l : List α} {f : α → β} {l' : List β} :
     l.foldl (fun xs y => f y :: xs) l' = (l.map f).reverse ++ l' := by
   induction l generalizing l' <;> simp [*]
 
 /-- Variant of `foldl_flip_cons_eq_append` specalized to `f = id`. -/
-@[grind] theorem foldl_flip_cons_eq_append' {l l' : List α} :
+theorem foldl_flip_cons_eq_append' {l l' : List α} :
     l.foldl (fun xs y => y :: xs) l' = l.reverse ++ l' := by
   simp
 
-@[simp, grind] theorem foldr_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
+@[simp] theorem foldr_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
     l.foldr (f · ++ ·) l' = (l.map f).flatten ++ l' := by
   induction l <;> simp [*]
 
-@[simp, grind] theorem foldl_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
+@[simp] theorem foldl_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
     l.foldl (· ++ f ·) l' = l' ++ (l.map f).flatten := by
   induction l generalizing l'<;> simp [*]
 
-@[simp, grind] theorem foldr_flip_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
+@[simp] theorem foldr_flip_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
     l.foldr (fun x ys => ys ++ f x) l' = l' ++ (l.map f).reverse.flatten := by
   induction l generalizing l' <;> simp [*]
 
-@[simp, grind] theorem foldl_flip_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
+@[simp] theorem foldl_flip_append_eq_append {l : List α} {f : α → List β} {l' : List β} :
     l.foldl (fun xs y => f y ++ xs) l' = (l.map f).reverse.flatten ++ l' := by
   induction l generalizing l' <;> simp [*]
 
@@ -2683,19 +2700,19 @@ theorem foldr_map_hom {g : α → β} {f : α → α → α} {f' : β → β →
 @[simp, grind _=_] theorem foldr_append {f : α → β → β} {b : β} {l l' : List α} :
     (l ++ l').foldr f b = l.foldr f (l'.foldr f b) := by simp [foldr_eq_foldrM, -foldrM_pure]
 
-@[grind] theorem foldl_flatten {f : β → α → β} {b : β} {L : List (List α)} :
+@[grind =] theorem foldl_flatten {f : β → α → β} {b : β} {L : List (List α)} :
     (flatten L).foldl f b = L.foldl (fun b l => l.foldl f b) b := by
   induction L generalizing b <;> simp_all
 
-@[grind] theorem foldr_flatten {f : α → β → β} {b : β} {L : List (List α)} :
+@[grind =] theorem foldr_flatten {f : α → β → β} {b : β} {L : List (List α)} :
     (flatten L).foldr f b = L.foldr (fun l b => l.foldr f b) b := by
   induction L <;> simp_all
 
-@[simp, grind] theorem foldl_reverse {l : List α} {f : β → α → β} {b : β} :
+@[simp, grind =] theorem foldl_reverse {l : List α} {f : β → α → β} {b : β} :
     l.reverse.foldl f b = l.foldr (fun x y => f y x) b := by
   simp [foldl_eq_foldlM, foldr_eq_foldrM, -foldrM_pure]
 
-@[simp, grind] theorem foldr_reverse {l : List α} {f : α → β → β} {b : β} :
+@[simp, grind =] theorem foldr_reverse {l : List α} {f : α → β → β} {b : β} :
     l.reverse.foldr f b = l.foldl (fun x y => f y x) b :=
   (foldl_reverse ..).symm.trans <| by simp
 
@@ -2849,7 +2866,7 @@ theorem foldr_rel {l : List α} {f : α → β → β} {g : α → γ → γ} {a
 
 /-! #### Further results about `getLast` and `getLast?` -/
 
-@[simp, grind] theorem head_reverse {l : List α} (h : l.reverse ≠ []) :
+@[simp, grind =] theorem head_reverse {l : List α} (h : l.reverse ≠ []) :
     l.reverse.head h = getLast l (by simp_all) := by
   induction l with
   | nil => contradiction
@@ -2879,7 +2896,7 @@ theorem getLast?_eq_some_iff {xs : List α} {a : α} : xs.getLast? = some a ↔
   rw [getLast?_eq_head?_reverse, isSome_head?]
   simp
 
-@[simp, grind] theorem getLast_reverse {l : List α} (h : l.reverse ≠ []) :
+@[simp, grind =] theorem getLast_reverse {l : List α} (h : l.reverse ≠ []) :
     l.reverse.getLast h = l.head (by simp_all) := by
   simp [getLast_eq_head_reverse]
 
@@ -2892,7 +2909,7 @@ theorem head_eq_getLast_reverse {l : List α} (h : l ≠ []) :
   simp only [getLast_eq_head_reverse, reverse_append]
   rw [head_append_of_ne_nil]
 
-@[grind] theorem getLast_append {l : List α} (h : l ++ l' ≠ []) :
+@[grind =] theorem getLast_append {l : List α} (h : l ++ l' ≠ []) :
     (l ++ l').getLast h =
       if h' : l'.isEmpty then
         l.getLast (by simp_all [isEmpty_iff])
@@ -2913,7 +2930,7 @@ theorem getLast_append_left {l : List α} (w : l ++ l' ≠ []) (h : l' = []) :
     (l ++ l').getLast w = l.getLast (by simp_all) := by
   rw [getLast_append, dif_pos (by simp_all)]
 
-@[simp, grind] theorem getLast?_append {l l' : List α} : (l ++ l').getLast? = l'.getLast?.or l.getLast? := by
+@[simp, grind =] theorem getLast?_append {l l' : List α} : (l ++ l').getLast? = l'.getLast?.or l.getLast? := by
   simp [← head?_reverse]
 
 theorem getLast_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p (getLast l w) = true) :
@@ -2949,7 +2966,7 @@ theorem getLast?_replicate {a : α} {n : Nat} : (replicate n a).getLast? = if n
 /-! ### leftpad -/
 
 -- We unfold `leftpad` and `rightpad` for verification purposes.
-attribute [simp, grind] leftpad rightpad
+attribute [simp, grind =] leftpad rightpad
 
 -- `length_leftpad` and `length_rightpad` are in `Init.Data.List.Nat.Basic`.
 
@@ -2978,17 +2995,21 @@ theorem contains_iff_exists_mem_beq [BEq α] {l : List α} {a : α} :
     l.contains a ↔ ∃ a' ∈ l, a == a' := by
   induction l <;> simp_all
 
+-- We add this as a `grind` lemma because it is useful without `LawfulBEq α`.
+-- With `LawfulBEq α`, it would be better to use `contains_iff_mem` directly.
+grind_pattern contains_iff_exists_mem_beq => l.contains a
+
 @[grind _=_]
 theorem contains_iff_mem [BEq α] [LawfulBEq α] {l : List α} {a : α} :
     l.contains a ↔ a ∈ l := by
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_map [BEq β] {l : List α} {x : β} {f : α → β} :
     (l.map f).contains x = l.any (fun a => x == f a) := by
   induction l with simp_all
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_filter [BEq α] {l : List α} {x : α} {p : α → Bool} :
     (l.filter p).contains x = l.any (fun a => x == a && p a) := by
   induction l with
@@ -2997,7 +3018,7 @@ theorem contains_filter [BEq α] {l : List α} {x : α} {p : α → Bool} :
     simp only [filter_cons, any_cons]
     split <;> simp_all
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_filterMap [BEq β] {l : List α} {x : β} {f : α → Option β} :
     (l.filterMap f).contains x = l.any (fun a => (f a).any fun b => x == b) := by
   induction l with
@@ -3013,21 +3034,21 @@ theorem contains_append [BEq α] {l₁ l₂ : List α} {x : α} :
   | nil => simp
   | cons a l ih => simp [ih, Bool.or_assoc]
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_flatten [BEq α] {l : List (List α)} {x : α} :
     l.flatten.contains x = l.any fun l => l.contains x := by
   induction l with
   | nil => simp
   | cons _ l ih => simp [ih]
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_reverse [BEq α] {l : List α} {x : α} :
     (l.reverse).contains x = l.contains x := by
   induction l with
   | nil => simp
   | cons a l ih => simp [ih, Bool.or_comm]
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_flatMap [BEq β] {l : List α} {f : α → List β} {x : β} :
     (l.flatMap f).contains x = l.any fun a => (f a).contains x := by
   induction l with
@@ -3042,7 +3063,7 @@ Because we immediately simplify `partition` into two `filter`s for verification
 we do not separately develop much theory about it.
 -/
 
-@[simp, grind] theorem partition_eq_filter_filter {p : α → Bool} {l : List α} :
+@[simp, grind =] theorem partition_eq_filter_filter {p : α → Bool} {l : List α} :
     partition p l = (filter p l, filter (not ∘ p) l) := by simp [partition, aux]
   where
     aux : ∀ l {as bs}, partition.loop p l (as, bs) =
@@ -3062,16 +3083,16 @@ grind_pattern mem_partition => a ∈ (partition p l).2
 are often used for theorems about `Array.pop`.
 -/
 
-@[simp, grind] theorem length_dropLast : ∀ {xs : List α}, xs.dropLast.length = xs.length - 1
+@[simp, grind =] theorem length_dropLast : ∀ {xs : List α}, xs.dropLast.length = xs.length - 1
   | [] => rfl
   | x::xs => by simp
 
-@[simp, grind] theorem getElem_dropLast : ∀ {xs : List α} {i : Nat} (h : i < xs.dropLast.length),
+@[simp, grind =] theorem getElem_dropLast : ∀ {xs : List α} {i : Nat} (h : i < xs.dropLast.length),
     xs.dropLast[i] = xs[i]'(Nat.lt_of_lt_of_le h (length_dropLast .. ▸ Nat.pred_le _))
   | _ :: _ :: _, 0, _ => rfl
   | _ :: _ :: _, _ + 1, h => getElem_dropLast (Nat.add_one_lt_add_one_iff.mp h)
 
-@[grind] theorem getElem?_dropLast {xs : List α} {i : Nat} :
+@[grind =] theorem getElem?_dropLast {xs : List α} {i : Nat} :
     xs.dropLast[i]? = if i < xs.length - 1 then xs[i]? else none := by
   split
   · rw [getElem?_eq_getElem, getElem?_eq_getElem, getElem_dropLast]
@@ -3269,24 +3290,24 @@ theorem all_eq_not_any_not {l : List α} {p : α → Bool} : l.all p = !l.any (!
   | nil => rfl
   | cons h t ih => simp_all [Bool.and_assoc]
 
-@[simp, grind] theorem any_flatten {l : List (List α)} : l.flatten.any f = l.any (any · f) := by
+@[simp, grind =] theorem any_flatten {l : List (List α)} : l.flatten.any f = l.any (any · f) := by
   induction l <;> simp_all
 
-@[simp, grind] theorem all_flatten {l : List (List α)} : l.flatten.all f = l.all (all · f) := by
+@[simp, grind =] theorem all_flatten {l : List (List α)} : l.flatten.all f = l.all (all · f) := by
   induction l <;> simp_all
 
-@[simp, grind] theorem any_flatMap {l : List α} {f : α → List β} :
+@[simp, grind =] theorem any_flatMap {l : List α} {f : α → List β} :
     (l.flatMap f).any p = l.any fun a => (f a).any p := by
   induction l <;> simp_all
 
-@[simp, grind] theorem all_flatMap {l : List α} {f : α → List β} :
+@[simp, grind =] theorem all_flatMap {l : List α} {f : α → List β} :
     (l.flatMap f).all p = l.all fun a => (f a).all p := by
   induction l <;> simp_all
 
-@[simp, grind] theorem any_reverse {l : List α} : l.reverse.any f = l.any f := by
+@[simp, grind =] theorem any_reverse {l : List α} : l.reverse.any f = l.any f := by
   induction l <;> simp_all [Bool.or_comm]
 
-@[simp, grind] theorem all_reverse {l : List α} : l.reverse.all f = l.all f := by
+@[simp, grind =] theorem all_reverse {l : List α} : l.reverse.all f = l.all f := by
   induction l <;> simp_all [Bool.and_comm]
 
 @[simp] theorem any_replicate {n : Nat} {a : α} :
@@ -3336,14 +3357,14 @@ variable [BEq α]
     simp only [replace_cons]
     split <;> simp_all
 
-@[simp, grind] theorem length_replace {l : List α} : (l.replace a b).length = l.length := by
+@[simp, grind =] theorem length_replace {l : List α} : (l.replace a b).length = l.length := by
   induction l with
   | nil => simp
   | cons x l ih =>
     simp only [replace_cons]
     split <;> simp_all
 
-@[grind] theorem getElem?_replace [LawfulBEq α] {l : List α} {i : Nat} :
+@[grind =] theorem getElem?_replace [LawfulBEq α] {l : List α} {i : Nat} :
     (l.replace a b)[i]? = if l[i]? == some a then if a ∈ l.take i then some a else some b else l[i]? := by
   induction l generalizing i with
   | nil => cases i <;> simp
@@ -3356,7 +3377,7 @@ theorem getElem?_replace_of_ne [LawfulBEq α] {l : List α} {i : Nat} (h : l[i]?
     (l.replace a b)[i]? = l[i]? := by
   simp_all [getElem?_replace]
 
-@[grind] theorem getElem_replace [LawfulBEq α] {l : List α} {i : Nat} (h : i < l.length) :
+@[grind =] theorem getElem_replace [LawfulBEq α] {l : List α} {i : Nat} (h : i < l.length) :
     (l.replace a b)[i]'(by simpa) = if l[i] == a then if a ∈ l.take i then a else b else l[i] := by
   apply Option.some.inj
   rw [← getElem?_eq_getElem, getElem?_replace]
@@ -3386,7 +3407,7 @@ theorem head_replace {l : List α} {a b : α} (w) :
   apply Option.some.inj
   rw [← head?_eq_head, head?_replace, head?_eq_head]
 
-@[grind] theorem replace_append [LawfulBEq α] {l₁ l₂ : List α} :
+@[grind =] theorem replace_append [LawfulBEq α] {l₁ l₂ : List α} :
     (l₁ ++ l₂).replace a b = if a ∈ l₁ then l₁.replace a b ++ l₂ else l₁ ++ l₂.replace a b := by
   induction l₁ with
   | nil => simp
@@ -3430,9 +3451,9 @@ end replace
 section insert
 variable [BEq α]
 
-@[simp, grind] theorem insert_nil (a : α) : [].insert a = [a] := rfl
+@[simp, grind =] theorem insert_nil (a : α) : [].insert a = [a] := rfl
 
-@[simp, grind] theorem contains_insert [PartialEquivBEq α] {l : List α} {a : α} {x : α} :
+@[simp, grind =] theorem contains_insert [PartialEquivBEq α] {l : List α} {a : α} {x : α} :
     (l.insert a).contains x = (x == a || l.contains x) := by
   simp only [List.insert]
   split <;> rename_i h
@@ -3449,7 +3470,7 @@ variable [LawfulBEq α]
 @[simp] theorem insert_of_not_mem {l : List α} (h : a ∉ l) : l.insert a = a :: l := by
   simp [List.insert, h]
 
-@[simp, grind] theorem mem_insert_iff {l : List α} : a ∈ l.insert b ↔ a = b ∨ a ∈ l := by
+@[simp, grind =] theorem mem_insert_iff {l : List α} : a ∈ l.insert b ↔ a = b ∨ a ∈ l := by
   if h : b ∈ l then
     rw [insert_of_mem h]
     constructor; {apply Or.inr}
@@ -3473,7 +3494,7 @@ theorem eq_or_mem_of_mem_insert {l : List α} (h : a ∈ l.insert b) : a = b ∨
 @[simp] theorem length_insert_of_not_mem {l : List α} (h : a ∉ l) :
     length (l.insert a) = length l + 1 := by rw [insert_of_not_mem h]; rfl
 
-@[grind] theorem length_insert {l : List α} :
+@[grind =] theorem length_insert {l : List α} :
     (l.insert a).length = l.length + if a ∈ l then 0 else 1 := by
   split <;> simp_all
 
@@ -3508,13 +3529,13 @@ theorem getElem?_insert_succ {l : List α} {a : α} {i : Nat} :
   simp only [insert_eq]
   split <;> simp
 
-@[grind] theorem getElem?_insert {l : List α} {a : α} {i : Nat} :
+@[grind =] theorem getElem?_insert {l : List α} {a : α} {i : Nat} :
     (l.insert a)[i]? = if a ∈ l then l[i]? else if i = 0 then some a else l[i-1]? := by
   cases i
   · simp [getElem?_insert_zero]
   · simp [getElem?_insert_succ]
 
-@[grind] theorem getElem_insert {l : List α} {a : α} {i : Nat} (h : i < l.length) :
+@[grind =] theorem getElem_insert {l : List α} {a : α} {i : Nat} (h : i < l.length) :
     (l.insert a)[i]'(Nat.lt_of_lt_of_le h length_le_length_insert) =
       if a ∈ l then l[i] else if i = 0 then a else l[i-1]'(Nat.lt_of_le_of_lt (Nat.pred_le _) h) := by
   apply Option.some.inj
@@ -3538,7 +3559,7 @@ theorem head_insert {l : List α} {a : α} (w) :
   apply Option.some.inj
   rw [← head?_eq_head, head?_insert]
 
-@[grind] theorem insert_append {l₁ l₂ : List α} {a : α} :
+@[grind =] theorem insert_append {l₁ l₂ : List α} {a : α} :
     (l₁ ++ l₂).insert a = if a ∈ l₂ then l₁ ++ l₂ else l₁.insert a ++ l₂ := by
   simp only [insert_eq, mem_append]
   (repeat split) <;> simp_all
@@ -3551,7 +3572,7 @@ theorem insert_append_of_not_mem_left {l₁ l₂ : List α} (h : ¬ a ∈ l₂)
     (l₁ ++ l₂).insert a = l₁.insert a ++ l₂ := by
   simp [insert_append, h]
 
-@[simp, grind] theorem insert_replicate_self {a : α} (h : 0 < n) : (replicate n a).insert a = replicate n a := by
+@[simp, grind =] theorem insert_replicate_self {a : α} (h : 0 < n) : (replicate n a).insert a = replicate n a := by
   cases n <;> simp_all
 
 @[simp] theorem insert_replicate_ne {a b : α} (h : !b == a) :

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

@@ -248,11 +248,10 @@ 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, grind] theorem find?_range_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
+@[simp] 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

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

@@ -567,9 +567,10 @@ theorem getElem_zipWith {f : α → β → γ} {l : List α} {l' : List β}
       f (l[i]'(lt_length_left_of_zipWith h))
         (l'[i]'(lt_length_right_of_zipWith h)) := by
   rw [← Option.some_inj, ← getElem?_eq_getElem, getElem?_zipWith_eq_some]
+  have := lt_length_right_of_zipWith h
   exact
-    ⟨l[i]'(lt_length_left_of_zipWith h), l'[i]'(lt_length_right_of_zipWith h),
-      by rw [getElem?_eq_getElem], by rw [getElem?_eq_getElem]; exact ⟨rfl, rfl⟩⟩
+    ⟨l[i]'(lt_length_left_of_zipWith h), l'[i],
+      by rw [getElem?_eq_getElem], by rw [getElem?_eq_getElem this]; exact ⟨rfl, rfl⟩⟩
 
 theorem zipWith_eq_zipWith_take_min : ∀ {l₁ : List α} {l₂ : List β},
     zipWith f l₁ l₂ = zipWith f (l₁.take (min l₁.length l₂.length)) (l₂.take (min l₁.length l₂.length))

src/Init/Data/List/Pairwise.lean

@@ -43,7 +43,7 @@ theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l →
   (pairwise_cons.1 p).2
 
 set_option linter.unusedVariables false in
-@[grind] theorem Pairwise.tail : ∀ {l : List α} (h : Pairwise R l), Pairwise R l.tail
+@[grind ←] theorem Pairwise.tail : ∀ {l : List α} (h : Pairwise R l), Pairwise R l.tail
   | [], h => h
   | _ :: _, h => h.of_cons
 
@@ -103,7 +103,7 @@ theorem Pairwise.forall_of_forall_of_flip (h₁ : ∀ x ∈ l, R x x) (h₂ : Pa
     · exact h₃.1 _ hx
     · exact ih (fun x hx => h₁ _ <| mem_cons_of_mem _ hx) h₂.2 h₃.2 hx hy
 
-@[grind] theorem pairwise_singleton (R) (a : α) : Pairwise R [a] := by simp
+@[grind ←] theorem pairwise_singleton (R) (a : α) : Pairwise R [a] := by simp
 
 @[grind =] theorem pairwise_pair {a b : α} : Pairwise R [a, b] ↔ R a b := by simp
 
@@ -117,7 +117,7 @@ theorem Pairwise.of_map {S : β → β → Prop} (f : α → β) (H : ∀ a b :
     (p : Pairwise S (map f l)) : Pairwise R l :=
   (pairwise_map.1 p).imp (H _ _)
 
-@[grind] theorem Pairwise.map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
+@[grind <=] theorem Pairwise.map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
     (p : Pairwise R l) : Pairwise S (map f l) :=
   pairwise_map.2 <| p.imp (H _ _)
 
@@ -136,7 +136,7 @@ theorem Pairwise.of_map {S : β → β → Prop} (f : α → β) (H : ∀ a b :
     simpa [IH, e] using fun _ =>
       ⟨fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab⟩
 
-@[grind] theorem Pairwise.filterMap {S : β → β → Prop} (f : α → Option β)
+@[grind <=] theorem Pairwise.filterMap {S : β → β → Prop} (f : α → Option β)
     (H : ∀ a a' : α, R a a' → ∀ b, f a = some b → ∀ b', f a' = some b' → S b b') {l : List α} (p : Pairwise R l) :
     Pairwise S (filterMap f l) :=
   pairwise_filterMap.2 <| p.imp (H _ _)
@@ -146,7 +146,7 @@ theorem Pairwise.of_map {S : β → β → Prop} (f : α → β) (H : ∀ a b :
   rw [← filterMap_eq_filter, pairwise_filterMap]
   simp
 
-@[grind] theorem Pairwise.filter (p : α → Bool) : Pairwise R l → Pairwise R (filter p l) :=
+@[grind ←] theorem Pairwise.filter (p : α → Bool) : Pairwise R l → Pairwise R (filter p l) :=
   Pairwise.sublist filter_sublist
 
 @[grind =] theorem pairwise_append {l₁ l₂ : List α} :
@@ -171,7 +171,7 @@ theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y →
   induction L with
   | nil => simp
   | cons l L IH =>
-    simp only [flatten, pairwise_append, IH, mem_flatten, exists_imp, and_imp, forall_mem_cons,
+    simp only [flatten_cons, pairwise_append, IH, mem_flatten, exists_imp, and_imp, forall_mem_cons,
       pairwise_cons, and_assoc, and_congr_right_iff]
     rw [and_comm, and_congr_left_iff]
     intros; exact ⟨fun h l' b c d e => h c d e l' b, fun h c d e l' b => h l' b c d e⟩
@@ -207,10 +207,10 @@ theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y →
         simp
       · exact ⟨fun _ => h, Or.inr h⟩
 
-@[grind] theorem Pairwise.drop {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.drop i) :=
+@[grind ←] theorem Pairwise.drop {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.drop i) :=
   h.sublist (drop_sublist _ _)
 
-@[grind] theorem Pairwise.take {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.take i) :=
+@[grind ←] theorem Pairwise.take {l : List α} {i : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.take i) :=
   h.sublist (take_sublist _ _)
 
 -- This theorem is not annotated with `grind` because it leads to a loop of instantiations with `Pairwise.sublist`.
@@ -266,7 +266,7 @@ theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h :
     rintro H _ b hb rfl
     exact H b hb _ _
 
-@[grind] theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
+@[grind <=] theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
     (h : ∀ x ∈ l, p x) {S : β → β → Prop}
     (hS : ∀ ⦃x⦄ (hx : p x) ⦃y⦄ (hy : p y), R x y → S (f x hx) (f y hy)) :
     Pairwise S (l.pmap f h) := by
@@ -277,10 +277,12 @@ theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h :
 
 @[grind =] theorem nodup_iff_pairwise_ne : List.Nodup l ↔ List.Pairwise (· ≠ ·) l := Iff.rfl
 
-@[simp, grind]
+@[simp]
 theorem nodup_nil : @Nodup α [] :=
   Pairwise.nil
 
+grind_pattern nodup_nil => @Nodup α []
+
 @[simp, grind =]
 theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by
   simp only [Nodup, pairwise_cons, forall_mem_ne]

src/Init/Data/List/Sublist.lean

@@ -151,11 +151,11 @@ theorem subset_replicate {n : Nat} {a : α} {l : List α} (h : n ≠ 0) : l ⊆
 
 /-! ### Sublist and isSublist -/
 
-@[simp, grind] theorem nil_sublist : ∀ l : List α, [] <+ l
+@[simp, grind ←] theorem nil_sublist : ∀ l : List α, [] <+ l
   | [] => .slnil
   | a :: l => (nil_sublist l).cons a
 
-@[simp, grind] theorem Sublist.refl : ∀ l : List α, l <+ l
+@[simp, grind ←] theorem Sublist.refl : ∀ l : List α, l <+ l
   | [] => .slnil
   | a :: l => (Sublist.refl l).cons₂ a
 
@@ -172,7 +172,7 @@ theorem Sublist.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ <+ l₂) (h₂ : l
 
 instance : Trans (@Sublist α) Sublist Sublist := ⟨Sublist.trans⟩
 
-attribute [simp, grind] Sublist.cons
+attribute [simp, grind ←] Sublist.cons
 
 theorem sublist_cons_self (a : α) (l : List α) : l <+ a :: l := (Sublist.refl l).cons _
 
@@ -202,12 +202,18 @@ theorem sublist_or_mem_of_sublist (h : l <+ l₁ ++ a :: l₂) : l <+ l₁ ++ l
 protected theorem Sublist.mem (hx : a ∈ l₁) (hl : l₁ <+ l₂) : a ∈ l₂ :=
   hl.subset hx
 
-@[grind] theorem Sublist.head_mem (s : ys <+ xs) (h) : ys.head h ∈ xs :=
+theorem Sublist.head_mem (s : ys <+ xs) (h) : ys.head h ∈ xs :=
   s.mem (List.head_mem h)
 
-@[grind] theorem Sublist.getLast_mem (s : ys <+ xs) (h) : ys.getLast h ∈ xs :=
+grind_pattern Sublist.head_mem => ys <+ xs, ys.head h
+grind_pattern Sublist.head_mem => ys.head h ∈ xs -- This is somewhat aggressive, as it initiates sublist based reasoning.
+
+theorem Sublist.getLast_mem (s : ys <+ xs) (h) : ys.getLast h ∈ xs :=
   s.mem (List.getLast_mem h)
 
+grind_pattern Sublist.getLast_mem => ys <+ xs, ys.getLast h
+grind_pattern Sublist.getLast_mem => ys.getLast h ∈ xs -- This is somewhat aggressive, as it initiates sublist based reasoning.
+
 instance : Trans (@Sublist α) Subset Subset :=
   ⟨fun h₁ h₂ => trans h₁.subset h₂⟩
 
@@ -248,12 +254,13 @@ theorem Sublist.eq_of_length_le (s : l₁ <+ l₂) (h : length l₂ ≤ length l
 theorem Sublist.length_eq (s : l₁ <+ l₂) : length l₁ = length l₂ ↔ l₁ = l₂ :=
   ⟨s.eq_of_length, congrArg _⟩
 
-@[grind]
 theorem tail_sublist : ∀ l : List α, tail l <+ l
   | [] => .slnil
   | a::l => sublist_cons_self a l
 
-@[grind]
+grind_pattern tail_sublist => tail l <+ _
+
+@[grind ←]
 protected theorem Sublist.tail : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → tail l₁ <+ tail l₂
   | _, _, slnil => .slnil
   | _, _, Sublist.cons _ h => (tail_sublist _).trans h
@@ -263,7 +270,7 @@ protected theorem Sublist.tail : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → tai
 theorem Sublist.of_cons_cons {l₁ l₂ : List α} {a b : α} (h : a :: l₁ <+ b :: l₂) : l₁ <+ l₂ :=
   h.tail
 
-@[grind]
+@[grind ←]
 protected theorem Sublist.map (f : α → β) {l₁ l₂} (s : l₁ <+ l₂) : map f l₁ <+ map f l₂ := by
   induction s with
   | slnil => simp
@@ -275,7 +282,7 @@ protected theorem Sublist.map (f : α → β) {l₁ l₂} (s : l₁ <+ l₂) : m
 grind_pattern Sublist.map => l₁ <+ l₂, map f l₁
 grind_pattern Sublist.map => l₁ <+ l₂, map f l₂
 
-@[grind]
+@[grind ←]
 protected theorem Sublist.filterMap (f : α → Option β) (s : l₁ <+ l₂) :
     filterMap f l₁ <+ filterMap f l₂ := by
   induction s <;> simp [filterMap_cons] <;> split <;> simp [*, cons]
@@ -283,7 +290,7 @@ protected theorem Sublist.filterMap (f : α → Option β) (s : l₁ <+ l₂) :
 grind_pattern Sublist.filterMap => l₁ <+ l₂, filterMap f l₁
 grind_pattern Sublist.filterMap => l₁ <+ l₂, filterMap f l₂
 
-@[grind]
+@[grind ←]
 protected theorem Sublist.filter (p : α → Bool) {l₁ l₂} (s : l₁ <+ l₂) : filter p l₁ <+ filter p l₂ := by
   rw [← filterMap_eq_filter]; apply s.filterMap
 
@@ -481,7 +488,7 @@ theorem Sublist.of_sublist_append_right (w : ∀ a, a ∈ l → a ∉ l₁) (h :
     exact fun x m => w x (mem_append_left l₂' m) (h₁.mem m)
   simp_all
 
-@[grind]
+@[grind ←]
 theorem Sublist.middle {l : List α} (h : l <+ l₁ ++ l₂) (a : α) : l <+ l₁ ++ a :: l₂ := by
   rw [sublist_append_iff] at h
   obtain ⟨l₁', l₂', rfl, h₁, h₂⟩ := h
@@ -624,22 +631,28 @@ theorem flatten_sublist_iff {L : List (List α)} {l} :
 instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <+ l₂) :=
   decidable_of_iff (l₁.isSublist l₂) isSublist_iff_sublist
 
-@[grind]
+@[grind ←]
 protected theorem Sublist.drop : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → ∀ i, l₁.drop i <+ l₂.drop i
   | _, _, h, 0 => h
   | _, _, h, i + 1 => by rw [← drop_tail, ← drop_tail]; exact h.tail.drop i
 
 /-! ### IsPrefix / IsSuffix / IsInfix -/
 
-@[simp, grind] theorem prefix_append (l₁ l₂ : List α) : l₁ <+: l₁ ++ l₂ := ⟨l₂, rfl⟩
+@[simp] theorem prefix_append (l₁ l₂ : List α) : l₁ <+: l₁ ++ l₂ := ⟨l₂, rfl⟩
 
-@[simp, grind] theorem suffix_append (l₁ l₂ : List α) : l₂ <:+ l₁ ++ l₂ := ⟨l₁, rfl⟩
+grind_pattern prefix_append => l₁ <+: l₁ ++ l₂
+
+@[simp] theorem suffix_append (l₁ l₂ : List α) : l₂ <:+ l₁ ++ l₂ := ⟨l₁, rfl⟩
+
+grind_pattern suffix_append => l₂ <:+ l₁ ++ l₂
 
 theorem infix_append (l₁ l₂ l₃ : List α) : l₂ <:+: l₁ ++ l₂ ++ l₃ := ⟨l₁, l₃, rfl⟩
 
-@[simp, grind] theorem infix_append' (l₁ l₂ l₃ : List α) : l₂ <:+: l₁ ++ (l₂ ++ l₃) := by
+@[simp] theorem infix_append' (l₁ l₂ l₃ : List α) : l₂ <:+: l₁ ++ (l₂ ++ l₃) := by
   rw [← List.append_assoc]; apply infix_append
 
+grind_pattern infix_append' => l₂ <:+: l₁ ++ (l₂ ++ l₃)
+
 theorem infix_append_left : l₁ <:+: l₁ ++ l₂ := ⟨[], l₂, rfl⟩
 theorem infix_append_right : l₂ <:+: l₁ ++ l₂ := ⟨l₁, [], by simp⟩
 
@@ -651,22 +664,24 @@ theorem IsSuffix.isInfix : l₁ <:+ l₂ → l₁ <:+: l₂ := fun ⟨t, h⟩ =>
 
 grind_pattern IsSuffix.isInfix => l₁ <:+ l₂, IsInfix
 
-@[simp, grind] theorem nil_prefix {l : List α} : [] <+: l := ⟨l, rfl⟩
+@[simp, grind ←] theorem nil_prefix {l : List α} : [] <+: l := ⟨l, rfl⟩
 
-@[simp, grind] theorem nil_suffix {l : List α} : [] <:+ l := ⟨l, append_nil _⟩
+@[simp, grind ←] theorem nil_suffix {l : List α} : [] <:+ l := ⟨l, append_nil _⟩
 
-@[simp, grind] theorem nil_infix {l : List α} : [] <:+: l := nil_prefix.isInfix
+@[simp, grind ←] theorem nil_infix {l : List α} : [] <:+: l := nil_prefix.isInfix
 
 theorem prefix_refl (l : List α) : l <+: l := ⟨[], append_nil _⟩
-@[simp, grind] theorem prefix_rfl {l : List α} : l <+: l := prefix_refl l
+@[simp, grind ←] theorem prefix_rfl {l : List α} : l <+: l := prefix_refl l
 
 theorem suffix_refl (l : List α) : l <:+ l := ⟨[], rfl⟩
-@[simp, grind] theorem suffix_rfl {l : List α} : l <:+ l := suffix_refl l
+@[simp, grind ←] theorem suffix_rfl {l : List α} : l <:+ l := suffix_refl l
 
 theorem infix_refl (l : List α) : l <:+: l := prefix_rfl.isInfix
-@[simp, grind] theorem infix_rfl {l : List α} : l <:+: l := infix_refl l
+@[simp, grind ←] theorem infix_rfl {l : List α} : l <:+: l := infix_refl l
 
-@[simp, grind] theorem suffix_cons (a : α) : ∀ l, l <:+ a :: l := suffix_append [a]
+@[simp] theorem suffix_cons (a : α) : ∀ l, l <:+ a :: l := suffix_append [a]
+
+grind_pattern suffix_cons => _ <:+ a :: l
 
 theorem infix_cons : l₁ <:+: l₂ → l₁ <:+: a :: l₂ := fun ⟨l₁', l₂', h⟩ => ⟨a :: l₁', l₂', h ▸ rfl⟩
 
@@ -1108,24 +1123,36 @@ theorem infix_of_mem_flatten : ∀ {L : List (List α)}, l ∈ L → l <:+: flat
 theorem prefix_cons_inj (a) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ :=
   prefix_append_right_inj [a]
 
-@[grind] theorem take_prefix (i) (l : List α) : take i l <+: l :=
+theorem take_prefix (i) (l : List α) : take i l <+: l :=
   ⟨_, take_append_drop _ _⟩
 
-@[grind] theorem drop_suffix (i) (l : List α) : drop i l <:+ l :=
+grind_pattern take_prefix => take i l <+: _
+
+theorem drop_suffix (i) (l : List α) : drop i l <:+ l :=
   ⟨_, take_append_drop _ _⟩
 
-@[grind] theorem take_sublist (i) (l : List α) : take i l <+ l :=
+grind_pattern drop_suffix => drop i l <+: _
+
+theorem take_sublist (i) (l : List α) : take i l <+ l :=
   (take_prefix i l).sublist
 
-@[grind] theorem drop_sublist (i) (l : List α) : drop i l <+ l :=
+grind_pattern take_sublist => take i l <+ l
+
+theorem drop_sublist (i) (l : List α) : drop i l <+ l :=
   (drop_suffix i l).sublist
 
+grind_pattern drop_sublist => drop i l <+ l
+
 theorem take_subset (i) (l : List α) : take i l ⊆ l :=
   (take_sublist i l).subset
 
+grind_pattern take_subset => take i l ⊆ l
+
 theorem drop_subset (i) (l : List α) : drop i l ⊆ l :=
   (drop_sublist i l).subset
 
+grind_pattern drop_subset => drop i l ⊆ l
+
 theorem mem_of_mem_take {l : List α} (h : a ∈ l.take i) : a ∈ l :=
   take_subset _ _ h
 
@@ -1138,64 +1165,84 @@ theorem drop_suffix_drop_left (l : List α) {i j : Nat} (h : i ≤ j) : drop j l
 
 -- See `Init.Data.List.Nat.TakeDrop` for `take_prefix_take_left`.
 
-@[grind] theorem drop_sublist_drop_left (l : List α) {i j : Nat} (h : i ≤ j) : drop j l <+ drop i l :=
+@[grind ←] theorem drop_sublist_drop_left (l : List α) {i j : Nat} (h : i ≤ j) : drop j l <+ drop i l :=
   (drop_suffix_drop_left l h).sublist
 
-@[grind] theorem drop_subset_drop_left (l : List α) {i j : Nat} (h : i ≤ j) : drop j l ⊆ drop i l :=
+@[grind ←] theorem drop_subset_drop_left (l : List α) {i j : Nat} (h : i ≤ j) : drop j l ⊆ drop i l :=
   (drop_sublist_drop_left l h).subset
 
-@[grind] theorem takeWhile_prefix (p : α → Bool) : l.takeWhile p <+: l :=
+theorem takeWhile_prefix (p : α → Bool) : l.takeWhile p <+: l :=
   ⟨l.dropWhile p, takeWhile_append_dropWhile⟩
 
-@[grind] theorem dropWhile_suffix (p : α → Bool) : l.dropWhile p <:+ l :=
+grind_pattern takeWhile_prefix => l.takeWhile p <+: _
+
+theorem dropWhile_suffix (p : α → Bool) : l.dropWhile p <:+ l :=
   ⟨l.takeWhile p, takeWhile_append_dropWhile⟩
 
-@[grind] theorem takeWhile_sublist (p : α → Bool) : l.takeWhile p <+ l :=
+grind_pattern dropWhile_suffix => l.dropWhile p <+: _
+
+theorem takeWhile_sublist (p : α → Bool) : l.takeWhile p <+ l :=
   (takeWhile_prefix p).sublist
 
-@[grind] theorem dropWhile_sublist (p : α → Bool) : l.dropWhile p <+ l :=
+grind_pattern takeWhile_sublist => l.takeWhile p <+ _
+
+theorem dropWhile_sublist (p : α → Bool) : l.dropWhile p <+ l :=
   (dropWhile_suffix p).sublist
 
+grind_pattern dropWhile_sublist => l.dropWhile p <+ _
+
 theorem takeWhile_subset {l : List α} (p : α → Bool) : l.takeWhile p ⊆ l :=
   (takeWhile_sublist p).subset
 
+grind_pattern takeWhile_subset => l.takeWhile p ⊆ _
+
 theorem dropWhile_subset {l : List α} (p : α → Bool) : l.dropWhile p ⊆ l :=
   (dropWhile_sublist p).subset
 
-@[grind] theorem dropLast_prefix : ∀ l : List α, l.dropLast <+: l
+grind_pattern dropWhile_subset => l.dropWhile p ⊆ _
+
+theorem dropLast_prefix : ∀ l : List α, l.dropLast <+: l
   | [] => ⟨nil, by rw [dropLast, List.append_nil]⟩
   | a :: l => ⟨_, dropLast_concat_getLast (cons_ne_nil a l)⟩
 
-@[grind] theorem dropLast_sublist (l : List α) : l.dropLast <+ l :=
+grind_pattern dropLast_prefix => l.dropLast <+: _
+
+theorem dropLast_sublist (l : List α) : l.dropLast <+ l :=
   (dropLast_prefix l).sublist
 
+grind_pattern dropLast_sublist => l.dropLast <+ _
+
 theorem dropLast_subset (l : List α) : l.dropLast ⊆ l :=
   (dropLast_sublist l).subset
 
-@[grind] theorem tail_suffix (l : List α) : tail l <:+ l := by rw [← drop_one]; apply drop_suffix
+grind_pattern dropLast_subset => l.dropLast ⊆ _
 
-@[grind] theorem IsPrefix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) : l₁.map f <+: l₂.map f := by
+theorem tail_suffix (l : List α) : tail l <:+ l := by rw [← drop_one]; apply drop_suffix
+
+grind_pattern tail_suffix => tail l <+: _
+
+@[grind ←] theorem IsPrefix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) : l₁.map f <+: l₂.map f := by
   obtain ⟨r, rfl⟩ := h
   rw [map_append]; apply prefix_append
 
 grind_pattern IsPrefix.map => l₁ <+: l₂, l₁.map f
 grind_pattern IsPrefix.map => l₁ <+: l₂, l₂.map f
 
-@[grind] theorem IsSuffix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) : l₁.map f <:+ l₂.map f := by
+@[grind ←] theorem IsSuffix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) : l₁.map f <:+ l₂.map f := by
   obtain ⟨r, rfl⟩ := h
   rw [map_append]; apply suffix_append
 
 grind_pattern IsSuffix.map => l₁ <:+ l₂, l₁.map f
 grind_pattern IsSuffix.map => l₁ <:+ l₂, l₂.map f
 
-@[grind] theorem IsInfix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) : l₁.map f <:+: l₂.map f := by
+@[grind ←] theorem IsInfix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) : l₁.map f <:+: l₂.map f := by
   obtain ⟨r₁, r₂, rfl⟩ := h
   rw [map_append, map_append]; apply infix_append
 
 grind_pattern IsInfix.map => l₁ <:+: l₂, l₁.map f
 grind_pattern IsInfix.map => l₁ <:+: l₂, l₂.map f
 
-@[grind] theorem IsPrefix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
+@[grind ←] theorem IsPrefix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
     l₁.filter p <+: l₂.filter p := by
   obtain ⟨xs, rfl⟩ := h
   rw [filter_append]; apply prefix_append
@@ -1203,7 +1250,7 @@ grind_pattern IsInfix.map => l₁ <:+: l₂, l₂.map f
 grind_pattern IsPrefix.filter => l₁ <+: l₂, l₁.filter p
 grind_pattern IsPrefix.filter => l₁ <+: l₂, l₂.filter p
 
-@[grind] theorem IsSuffix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) :
+@[grind ←] theorem IsSuffix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) :
     l₁.filter p <:+ l₂.filter p := by
   obtain ⟨xs, rfl⟩ := h
   rw [filter_append]; apply suffix_append
@@ -1211,7 +1258,7 @@ grind_pattern IsPrefix.filter => l₁ <+: l₂, l₂.filter p
 grind_pattern IsSuffix.filter => l₁ <:+ l₂, l₁.filter p
 grind_pattern IsSuffix.filter => l₁ <:+ l₂, l₂.filter p
 
-@[grind] theorem IsInfix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) :
+@[grind ←] theorem IsInfix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) :
     l₁.filter p <:+: l₂.filter p := by
   obtain ⟨xs, ys, rfl⟩ := h
   rw [filter_append, filter_append]; apply infix_append _
@@ -1219,7 +1266,7 @@ grind_pattern IsSuffix.filter => l₁ <:+ l₂, l₂.filter p
 grind_pattern IsInfix.filter => l₁ <:+: l₂, l₁.filter p
 grind_pattern IsInfix.filter => l₁ <:+: l₂, l₂.filter p
 
-@[grind] theorem IsPrefix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
+@[grind ←] theorem IsPrefix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
     filterMap f l₁ <+: filterMap f l₂ := by
   obtain ⟨xs, rfl⟩ := h
   rw [filterMap_append]; apply prefix_append
@@ -1227,7 +1274,7 @@ grind_pattern IsInfix.filter => l₁ <:+: l₂, l₂.filter p
 grind_pattern IsPrefix.filterMap => l₁ <+: l₂, filterMap f l₁
 grind_pattern IsPrefix.filterMap => l₁ <+: l₂, filterMap f l₂
 
-@[grind] theorem IsSuffix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) :
+@[grind ←] theorem IsSuffix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) :
     filterMap f l₁ <:+ filterMap f l₂ := by
   obtain ⟨xs, rfl⟩ := h
   rw [filterMap_append]; apply suffix_append
@@ -1235,7 +1282,7 @@ grind_pattern IsPrefix.filterMap => l₁ <+: l₂, filterMap f l₂
 grind_pattern IsSuffix.filterMap => l₁ <:+ l₂, filterMap f l₁
 grind_pattern IsSuffix.filterMap => l₁ <:+ l₂, filterMap f l₂
 
-@[grind] theorem IsInfix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) :
+@[grind ←] theorem IsInfix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) :
     filterMap f l₁ <:+: filterMap f l₂ := by
   obtain ⟨xs, ys, rfl⟩ := h
   rw [filterMap_append, filterMap_append]; apply infix_append

src/Init/Data/Nat/Linear.lean

@@ -9,6 +9,7 @@ prelude
 public import Init.ByCases
 public import Init.Data.Prod
 public import Init.Data.RArray
+import Init.LawfulBEqTactics
 
 public section
 
@@ -138,21 +139,7 @@ structure PolyCnstr  where
   eq  : Bool
   lhs : Poly
   rhs : Poly
-  deriving BEq
-
--- TODO: implement LawfulBEq generator companion for BEq
-instance : LawfulBEq PolyCnstr where
-  eq_of_beq {a b} h := by
-    cases a; rename_i eq₁ lhs₁ rhs₁
-    cases b; rename_i eq₂ lhs₂ rhs₂
-    have h : eq₁ == eq₂ && (lhs₁ == lhs₂ && rhs₁ == rhs₂) := h
-    simp at h
-    have ⟨h₁, h₂, h₃⟩ := h
-    rw [h₁, h₂, h₃]
-  rfl {a} := by
-    cases a; rename_i eq lhs rhs
-    change (eq == eq && (lhs == lhs && rhs == rhs)) = true
-    simp
+  deriving BEq, ReflBEq, LawfulBEq
 
 structure ExprCnstr where
   eq  : Bool

src/Init/Data/Option/Array.lean

@@ -15,30 +15,30 @@ public section
 
 namespace Option
 
-@[simp, grind] theorem mem_toArray {a : α} {o : Option α} : a ∈ o.toArray ↔ o = some a := by
+@[simp, grind =] theorem mem_toArray {a : α} {o : Option α} : a ∈ o.toArray ↔ o = some a := by
   cases o <;> simp [eq_comm]
 
-@[simp, grind] theorem forIn'_toArray [Monad m] (o : Option α) (b : β) (f : (a : α) → a ∈ o.toArray → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn'_toArray [Monad m] (o : Option α) (b : β) (f : (a : α) → a ∈ o.toArray → β → m (ForInStep β)) :
     forIn' o.toArray b f = forIn' o b fun a m b => f a (by simpa using m) b := by
   cases o <;> simp <;> rfl
 
-@[simp, grind] theorem forIn_toArray [Monad m] (o : Option α) (b : β) (f : α → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn_toArray [Monad m] (o : Option α) (b : β) (f : α → β → m (ForInStep β)) :
     forIn o.toArray b f = forIn o b f := by
   cases o <;> simp <;> rfl
 
-@[simp, grind] theorem foldlM_toArray [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : α → β → m α) :
+@[simp, grind =] theorem foldlM_toArray [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : α → β → m α) :
     o.toArray.foldlM f a = o.elim (pure a) (fun b => f a b) := by
   cases o <;> simp
 
-@[simp, grind] theorem foldrM_toArray [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : β → α → m α) :
+@[simp, grind =] theorem foldrM_toArray [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : β → α → m α) :
     o.toArray.foldrM f a = o.elim (pure a) (fun b => f b a) := by
   cases o <;> simp
 
-@[simp, grind] theorem foldl_toArray (o : Option β) (a : α) (f : α → β → α) :
+@[simp, grind =] theorem foldl_toArray (o : Option β) (a : α) (f : α → β → α) :
     o.toArray.foldl f a = o.elim a (fun b => f a b) := by
   cases o <;> simp
 
-@[simp, grind] theorem foldr_toArray (o : Option β) (a : α) (f : β → α → α) :
+@[simp, grind =] theorem foldr_toArray (o : Option β) (a : α) (f : β → α → α) :
     o.toArray.foldr f a = o.elim a (fun b => f b a) := by
   cases o <;> simp
 

src/Init/Data/Option/Attach.lean

@@ -97,7 +97,7 @@ theorem attach_eq_some : ∀ (o : Option α) (x : {x // o = some x}), o.attach =
   | none, ⟨x, h⟩ => by simp at h
   | some a, ⟨x, h⟩ => by simpa using h
 
-@[grind]
+@[grind ←]
 theorem mem_attach : ∀ (o : Option α) (x : {x // o = some x}), x ∈ o.attach :=
   attach_eq_some
 

src/Init/Data/Option/Basic.lean

@@ -18,27 +18,27 @@ namespace Option
 deriving instance DecidableEq for Option
 deriving instance BEq for Option
 
-@[simp, grind] theorem getD_none : getD none a = a := rfl
-@[simp, grind] theorem getD_some : getD (some a) b = a := rfl
+@[simp, grind =] theorem getD_none : getD none a = a := rfl
+@[simp, grind =] theorem getD_some : getD (some a) b = a := rfl
 
-@[simp, grind] theorem map_none (f : α → β) : none.map f = none := rfl
-@[simp, grind] theorem map_some (a) (f : α → β) : (some a).map f = some (f a) := rfl
+@[simp, grind =] theorem map_none (f : α → β) : none.map f = none := rfl
+@[simp, grind =] theorem map_some (a) (f : α → β) : (some a).map f = some (f a) := rfl
 
 /-- Lifts an optional value to any `Alternative`, sending `none` to `failure`. -/
 def getM [Alternative m] : Option α → m α
   | none     => failure
   | some a   => pure a
 
-@[simp, grind] theorem getM_none [Alternative m] : getM none = (failure : m α) := rfl
-@[simp, grind] theorem getM_some [Alternative m] {a : α} : getM (some a) = (pure a : m α) := rfl
+@[simp, grind =] theorem getM_none [Alternative m] : getM none = (failure : m α) := rfl
+@[simp, grind =] theorem getM_some [Alternative m] {a : α} : getM (some a) = (pure a : m α) := rfl
 
 /-- Returns `true` on `some x` and `false` on `none`. -/
 @[inline] def isSome : Option α → Bool
   | some _ => true
   | none   => false
 
-@[simp, grind] theorem isSome_none : @isSome α none = false := rfl
-@[simp, grind] theorem isSome_some : isSome (some a) = true := rfl
+@[simp, grind =] theorem isSome_none : @isSome α none = false := rfl
+@[simp, grind =] theorem isSome_some : isSome (some a) = true := rfl
 
 /--
 Returns `true` on `none` and `false` on `some x`.
@@ -53,8 +53,8 @@ Examples:
   | some _ => false
   | none   => true
 
-@[simp, grind] theorem isNone_none : @isNone α none = true := rfl
-@[simp, grind] theorem isNone_some : isNone (some a) = false := rfl
+@[simp, grind =] theorem isNone_none : @isNone α none = true := rfl
+@[simp, grind =] theorem isNone_some : isNone (some a) = false := rfl
 
 /--
 Checks whether an optional value is both present and equal to some other value.
@@ -89,8 +89,8 @@ Examples:
   | none,   _ => none
   | some a, f => f a
 
-@[simp, grind] theorem bind_none (f : α → Option β) : none.bind f = none := rfl
-@[simp, grind] theorem bind_some (a) (f : α → Option β) : (some a).bind f = f a := rfl
+@[simp, grind =] theorem bind_none (f : α → Option β) : none.bind f = none := rfl
+@[simp, grind =] theorem bind_some (a) (f : α → Option β) : (some a).bind f = f a := rfl
 
 @[deprecated bind_none (since := "2025-05-03")]
 abbrev none_bind := @bind_none
@@ -125,8 +125,8 @@ This function only requires `m` to be an applicative functor. An alias `Option.m
   | none => pure none
   | some x => some <$> f x
 
-@[simp, grind] theorem mapM_none [Applicative m] (f : α → m β) : none.mapM f = pure none := rfl
-@[simp, grind] theorem mapM_some [Applicative m] (x) (f : α → m β) : (some x).mapM f = some <$> f x := rfl
+@[simp, grind =] theorem mapM_none [Applicative m] (f : α → m β) : none.mapM f = pure none := rfl
+@[simp, grind =] theorem mapM_some [Applicative m] (x) (f : α → m β) : (some x).mapM f = some <$> f x := rfl
 
 /--
 Applies a function in some applicative functor to an optional value, returning `none` with no
@@ -138,9 +138,9 @@ This is an alias for `Option.mapM`, which already works for applicative functors
   Option.mapM f
 
 /-- For verification purposes, we replace `mapA` with `mapM`. -/
-@[simp, grind] theorem mapA_eq_mapM [Applicative m] {f : α → m β} : Option.mapA f o = Option.mapM f o := rfl
+@[simp, grind =] theorem mapA_eq_mapM [Applicative m] {f : α → m β} : Option.mapA f o = Option.mapM f o := rfl
 
-@[simp, grind]
+@[simp, grind =]
 theorem map_id : (Option.map id : Option α → Option α) = id :=
   funext (fun o => match o with | none => rfl | some _ => rfl)
 
@@ -182,8 +182,8 @@ Examples:
   | some a => p a
   | none   => true
 
-@[simp, grind] theorem all_none : Option.all p none = true := rfl
-@[simp, grind] theorem all_some : Option.all p (some x) = p x := rfl
+@[simp, grind =] theorem all_none : Option.all p none = true := rfl
+@[simp, grind =] theorem all_some : Option.all p (some x) = p x := rfl
 
 /--
 Checks whether an optional value is not `none` and satisfies a Boolean predicate.
@@ -197,8 +197,8 @@ Examples:
   | some a => p a
   | none   => false
 
-@[simp, grind] theorem any_none : Option.any p none = false := rfl
-@[simp, grind] theorem any_some : Option.any p (some x) = p x := rfl
+@[simp, grind =] theorem any_none : Option.any p none = false := rfl
+@[simp, grind =] theorem any_some : Option.any p (some x) = p x := rfl
 
 /--
 Implementation of `OrElse`'s `<|>` syntax for `Option`. If the first argument is `some a`, returns
@@ -210,8 +210,8 @@ See also `or` for a version that is strict in the second argument.
   | some a, _ => some a
   | none,   b => b ()
 
-@[simp, grind] theorem orElse_some : (some a).orElse b = some a := rfl
-@[simp, grind] theorem orElse_none : none.orElse b = b () := rfl
+@[simp, grind =] theorem orElse_some : (some a).orElse b = some a := rfl
+@[simp, grind =] theorem orElse_none : none.orElse b = b () := rfl
 
 instance : OrElse (Option α) where
   orElse := Option.orElse
@@ -351,9 +351,9 @@ Extracts the value from an option that can be proven to be `some`.
 @[inline] def get {α : Type u} : (o : Option α) → isSome o → α
   | some x, _ => x
 
-@[simp, grind] theorem some_get : ∀ {x : Option α} (h : isSome x), some (x.get h) = x
+@[simp, grind =] theorem some_get : ∀ {x : Option α} (h : isSome x), some (x.get h) = x
 | some _, _ => rfl
-@[simp, grind] theorem get_some (x : α) (h : isSome (some x)) : (some x).get h = x := rfl
+@[simp, grind =] theorem get_some (x : α) (h : isSome (some x)) : (some x).get h = x := rfl
 
 /--
 Returns `none` if a value doesn't satisfy a Boolean predicate, or the value itself otherwise.
@@ -431,8 +431,8 @@ Examples:
 -/
 @[inline] def join (x : Option (Option α)) : Option α := x.bind id
 
-@[simp, grind] theorem join_none : (none : Option (Option α)).join = none := rfl
-@[simp, grind] theorem join_some : (some o).join = o := rfl
+@[simp, grind =] theorem join_none : (none : Option (Option α)).join = none := rfl
+@[simp, grind =] theorem join_some : (some o).join = o := rfl
 
 /--
 Converts an optional monadic computation into a monadic computation of an optional value.
@@ -457,8 +457,8 @@ some "world"
   | none => pure none
   | some f => some <$> f
 
-@[simp, grind] theorem sequence_none [Applicative m] : (none : Option (m α)).sequence = pure none := rfl
-@[simp, grind] theorem sequence_some [Applicative m] (f : m α) : (some f).sequence = some <$> f := rfl
+@[simp, grind =] theorem sequence_none [Applicative m] : (none : Option (m α)).sequence = pure none := rfl
+@[simp, grind =] theorem sequence_some [Applicative m] (f : m α) : (some f).sequence = some <$> f := rfl
 
 /--
 A monadic case analysis function for `Option`.
@@ -483,8 +483,8 @@ This is the monadic analogue of `Option.getD`.
   | some a => pure a
   | none => y
 
-@[simp, grind] theorem getDM_none [Pure m] (y : m α) : (none : Option α).getDM y = y := rfl
-@[simp, grind] theorem getDM_some [Pure m] (a : α) (y : m α) : (some a).getDM y = pure a := rfl
+@[simp, grind =] theorem getDM_none [Pure m] (y : m α) : (none : Option α).getDM y = y := rfl
+@[simp, grind =] theorem getDM_some [Pure m] (a : α) (y : m α) : (some a).getDM y = pure a := rfl
 
 instance (α) [BEq α] [ReflBEq α] : ReflBEq (Option α) where
   rfl {x} := private
@@ -520,10 +520,10 @@ protected def min [Min α] : Option α → Option α → Option α
 
 instance [Min α] : Min (Option α) where min := Option.min
 
-@[simp, grind] theorem min_some_some [Min α] {a b : α} : min (some a) (some b) = some (min a b) := rfl
-@[simp, grind] theorem min_none_left [Min α] {o : Option α} : min none o = none := by
+@[simp, grind =] theorem min_some_some [Min α] {a b : α} : min (some a) (some b) = some (min a b) := rfl
+@[simp, grind =] theorem min_none_left [Min α] {o : Option α} : min none o = none := by
   cases o <;> rfl
-@[simp, grind] theorem min_none_right [Min α] {o : Option α} : min o none = none := by
+@[simp, grind =] theorem min_none_right [Min α] {o : Option α} : min o none = none := by
   cases o <;> rfl
 
 @[deprecated min_none_right (since := "2025-05-12")]
@@ -553,10 +553,10 @@ protected def max [Max α] : Option α → Option α → Option α
 
 instance [Max α] : Max (Option α) where max := Option.max
 
-@[simp, grind] theorem max_some_some [Max α] {a b : α} : max (some a) (some b) = some (max a b) := rfl
-@[simp, grind] theorem max_none_left [Max α] {o : Option α} : max none o = o := by
+@[simp, grind =] theorem max_some_some [Max α] {a b : α} : max (some a) (some b) = some (max a b) := rfl
+@[simp, grind =] theorem max_none_left [Max α] {o : Option α} : max none o = o := by
   cases o <;> rfl
-@[simp, grind] theorem max_none_right [Max α] {o : Option α} : max o none = o := by
+@[simp, grind =] theorem max_none_right [Max α] {o : Option α} : max o none = o := by
   cases o <;> rfl
 
 @[deprecated max_none_right (since := "2025-05-12")]

src/Init/Data/Option/Lemmas.lean

@@ -24,7 +24,7 @@ namespace Option
 @[deprecated mem_def (since := "2025-04-07")]
 theorem mem_iff {a : α} {b : Option α} : a ∈ b ↔ b = some a := .rfl
 
-@[grind] theorem mem_some {a b : α} : a ∈ some b ↔ b = a := by simp
+@[grind =] theorem mem_some {a b : α} : a ∈ some b ↔ b = a := by simp
 
 theorem mem_some_iff {a b : α} : a ∈ some b ↔ b = a := mem_some
 
@@ -52,7 +52,7 @@ theorem get_of_mem : ∀ {o : Option α} (h : isSome o), a ∈ o → o.get h = a
 theorem get_of_eq_some : ∀ {o : Option α} (h : isSome o), o = some a → o.get h = a
   | _, _, rfl => rfl
 
-@[simp, grind] theorem not_mem_none (a : α) : a ∉ (none : Option α) := nofun
+@[simp, grind ←] theorem not_mem_none (a : α) : a ∉ (none : Option α) := nofun
 
 theorem getD_of_ne_none {x : Option α} (hx : x ≠ none) (y : α) : some (x.getD y) = x := by
   cases x; {contradiction}; rw [getD_some]

src/Init/Data/Option/List.lean

@@ -16,38 +16,38 @@ public section
 
 namespace Option
 
-@[simp, grind] theorem mem_toList {a : α} {o : Option α} : a ∈ o.toList ↔ o = some a := by
+@[simp, grind =] theorem mem_toList {a : α} {o : Option α} : a ∈ o.toList ↔ o = some a := by
   cases o <;> simp [eq_comm]
 
-@[simp, grind] theorem forIn'_toList [Monad m] (o : Option α) (b : β) (f : (a : α) → a ∈ o.toList → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn'_toList [Monad m] (o : Option α) (b : β) (f : (a : α) → a ∈ o.toList → β → m (ForInStep β)) :
     forIn' o.toList b f = forIn' o b fun a m b => f a (by simpa using m) b := by
   cases o <;> rfl
 
-@[simp, grind] theorem forIn_toList [Monad m] (o : Option α) (b : β) (f : α → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn_toList [Monad m] (o : Option α) (b : β) (f : α → β → m (ForInStep β)) :
     forIn o.toList b f = forIn o b f := by
   cases o <;> rfl
 
-@[simp, grind] theorem foldlM_toList [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : α → β → m α) :
+@[simp, grind =] theorem foldlM_toList [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : α → β → m α) :
     o.toList.foldlM f a = o.elim (pure a) (fun b => f a b) := by
   cases o <;> simp
 
-@[simp, grind] theorem foldrM_toList [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : β → α → m α) :
+@[simp, grind =] theorem foldrM_toList [Monad m] [LawfulMonad m] (o : Option β) (a : α) (f : β → α → m α) :
     o.toList.foldrM f a = o.elim (pure a) (fun b => f b a) := by
   cases o <;> simp
 
-@[simp, grind] theorem foldl_toList (o : Option β) (a : α) (f : α → β → α) :
+@[simp, grind =] theorem foldl_toList (o : Option β) (a : α) (f : α → β → α) :
     o.toList.foldl f a = o.elim a (fun b => f a b) := by
   cases o <;> simp
 
-@[simp, grind] theorem foldr_toList (o : Option β) (a : α) (f : β → α → α) :
+@[simp, grind =] theorem foldr_toList (o : Option β) (a : α) (f : β → α → α) :
     o.toList.foldr f a = o.elim a (fun b => f b a) := by
   cases o <;> simp
 
-@[simp, grind]
+@[simp, grind ←]
 theorem pairwise_toList {P : α → α → Prop} {o : Option α} : o.toList.Pairwise P := by
   cases o <;> simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem head?_toList {o : Option α} : o.toList.head? = o := by
   cases o <;> simp
 

src/Init/Data/Option/Monadic.lean

@@ -16,20 +16,20 @@ public section
 
 namespace Option
 
-@[simp, grind] theorem bindM_none [Pure m] (f : α → m (Option β)) : none.bindM f = pure none := rfl
-@[simp, grind] theorem bindM_some [Pure m] (a) (f : α → m (Option β)) : (some a).bindM f = f a := by
+@[simp, grind =] theorem bindM_none [Pure m] (f : α → m (Option β)) : none.bindM f = pure none := rfl
+@[simp, grind =] theorem bindM_some [Pure m] (a) (f : α → m (Option β)) : (some a).bindM f = f a := by
   simp [Option.bindM]
 
 -- We simplify `Option.forM` to `forM`.
 @[simp] theorem forM_eq_forM [Monad m] : @Option.forM m α _ = forM := rfl
 
-@[simp, grind] theorem forM_none [Monad m] (f : α → m PUnit) :
+@[simp, grind =] theorem forM_none [Monad m] (f : α → m PUnit) :
     forM none f = pure .unit := rfl
 
-@[simp, grind] theorem forM_some [Monad m] (f : α → m PUnit) (a : α) :
+@[simp, grind =] theorem forM_some [Monad m] (f : α → m PUnit) (a : α) :
     forM (some a) f = f a := rfl
 
-@[simp, grind] theorem forM_map [Monad m] [LawfulMonad m] (o : Option α) (g : α → β) (f : β → m PUnit) :
+@[simp, grind =] theorem forM_map [Monad m] [LawfulMonad m] (o : Option α) (g : α → β) (f : β → m PUnit) :
     forM (o.map g) f = forM o (fun a => f (g a)) := by
   cases o <;> simp
 
@@ -37,11 +37,11 @@ theorem forM_join [Monad m] [LawfulMonad m] (o : Option (Option α)) (f : α →
     forM o.join f = forM o (forM · f) := by
   cases o <;> simp
 
-@[simp, grind] theorem forIn'_none [Monad m] (b : β) (f : (a : α) → a ∈ none → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn'_none [Monad m] (b : β) (f : (a : α) → a ∈ none → β → m (ForInStep β)) :
     forIn' none b f = pure b := by
   rfl
 
-@[simp, grind] theorem forIn'_some [Monad m] [LawfulMonad m] (a : α) (b : β) (f : (a' : α) → a' ∈ some a → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn'_some [Monad m] [LawfulMonad m] (a : α) (b : β) (f : (a' : α) → a' ∈ some a → β → m (ForInStep β)) :
     forIn' (some a) b f = bind (f a rfl b) (fun r => pure (ForInStep.value r)) := by
   simp only [forIn', bind_pure_comp]
   rw [map_eq_pure_bind]
@@ -49,11 +49,11 @@ theorem forM_join [Monad m] [LawfulMonad m] (o : Option (Option α)) (f : α →
   funext x
   split <;> simp
 
-@[simp, grind] theorem forIn_none [Monad m] (b : β) (f : α → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn_none [Monad m] (b : β) (f : α → β → m (ForInStep β)) :
     forIn none b f = pure b := by
   rfl
 
-@[simp, grind] theorem forIn_some [Monad m] [LawfulMonad m] (a : α) (b : β) (f : α → β → m (ForInStep β)) :
+@[simp, grind =] theorem forIn_some [Monad m] [LawfulMonad m] (a : α) (b : β) (f : α → β → m (ForInStep β)) :
     forIn (some a) b f = bind (f a b) (fun r => pure (ForInStep.value r)) := by
   simp only [forIn, forIn', bind_pure_comp]
   rw [map_eq_pure_bind]
@@ -106,7 +106,7 @@ theorem forIn'_id_yield_eq_pelim
       o.pelim b (fun a h => f a h b) :=
   forIn'_pure_yield_eq_pelim _ _ _
 
-@[simp, grind] theorem forIn'_map [Monad m] [LawfulMonad m]
+@[simp, grind =] theorem forIn'_map [Monad m] [LawfulMonad m]
     (o : Option α) (g : α → β) (f : (b : β) → b ∈ o.map g → γ → m (ForInStep γ)) :
     forIn' (o.map g) init f = forIn' o init fun a h y => f (g a) (mem_map_of_mem g h) y := by
   cases o <;> simp
@@ -149,7 +149,7 @@ theorem forIn_id_yield_eq_elim
       o.elim b (fun a => f a b) :=
   forIn_pure_yield_eq_elim _ _ _
 
-@[simp, grind] theorem forIn_map [Monad m] [LawfulMonad m]
+@[simp, grind =] theorem forIn_map [Monad m] [LawfulMonad m]
     (o : Option α) (g : α → β) (f : β → γ → m (ForInStep γ)) :
     forIn (o.map g) init f = forIn o init fun a y => f (g a) y := by
   cases o <;> simp

src/Init/Data/Order/Ord.lean

@@ -349,13 +349,13 @@ theorem LawfulEqCmp.compare_beq_iff_eq {a b : α} : cmp a b == .eq ↔ a = b :=
   beq_iff_eq.trans compare_eq_iff_eq
 
 /-- The corresponding lemma for `LawfulEqCmp` is `LawfulEqCmp.compare_eq_iff_eq` -/
-@[simp, grind]
+@[simp, grind =]
 theorem LawfulEqOrd.compare_eq_iff_eq [Ord α] [LawfulEqOrd α] {a b : α} :
     compare a b = .eq ↔ a = b :=
   LawfulEqCmp.compare_eq_iff_eq
 
 /-- The corresponding lemma for `LawfulEqCmp` is `LawfulEqCmp.compare_beq_iff_eq` -/
-@[grind]
+@[grind =]
 theorem LawfulEqOrd.compare_beq_iff_eq [Ord α] [LawfulEqOrd α] {a b : α} :
     compare a b == .eq ↔ a = b :=
   LawfulEqCmp.compare_beq_iff_eq

src/Init/Data/Range/Polymorphic.lean

@@ -12,6 +12,8 @@ public import Init.Data.Range.Polymorphic.Stream
 public import Init.Data.Range.Polymorphic.Lemmas
 public import Init.Data.Range.Polymorphic.Nat
 public import Init.Data.Range.Polymorphic.Int
+public import Init.Data.Range.Polymorphic.BitVec
+public import Init.Data.Range.Polymorphic.UInt
 public import Init.Data.Range.Polymorphic.NatLemmas
 public import Init.Data.Range.Polymorphic.GetElemTactic
 

src/Init/Data/Range/Polymorphic/BitVec.lean

@@ -0,0 +1,88 @@
+/-
+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
+public import Init.Data.Range.Polymorphic.Instances
+public import Init.Data.Order.Lemmas
+public import Init.Data.UInt
+import Init.Omega
+
+public section
+
+open Std Std.PRange
+
+namespace BitVec
+
+variable {n : Nat}
+
+instance : UpwardEnumerable (BitVec n) where
+  succ? i := if i + 1 = 0 then none else some (i + 1)
+  succMany? m i := if h : i.toNat + m < 2 ^ n then some (.ofNatLT _ h) else none
+
+instance : LawfulUpwardEnumerable (BitVec n) where
+  ne_of_lt := by
+    simp +contextual [UpwardEnumerable.LT, ← BitVec.toNat_inj, succMany?] at ⊢
+    omega
+  succMany?_zero := by simp [UpwardEnumerable.succMany?, BitVec.toNat_lt_twoPow_of_le]
+  succMany?_succ? a b := by
+    simp +contextual [← BitVec.toNat_inj, succMany?, succ?]
+    split <;> split
+    · rename_i h
+      simp [← BitVec.toNat_inj, Nat.mod_eq_of_lt (a := b.toNat + a + 1) ‹_›]
+      all_goals omega
+    · omega
+    · have : b.toNat + a + 1 = 2 ^ n := by omega
+      simp [this]
+    · simp
+
+instance : LawfulUpwardEnumerableLE (BitVec n) where
+  le_iff x y := by
+    simp [UpwardEnumerable.LE, UpwardEnumerable.succMany?, BitVec.le_def]
+    apply Iff.intro
+    · intro hle
+      refine ⟨y.toNat - x.toNat, ?_⟩
+      apply Exists.intro <;> simp [Nat.add_sub_cancel' hle, BitVec.toNat_lt_twoPow_of_le]
+    · rintro ⟨n, hn, rfl⟩
+      simp [BitVec.ofNatLT]
+
+instance : LawfulOrderLT (BitVec n) := inferInstance
+instance : LawfulUpwardEnumerableLT (BitVec n) := inferInstance
+instance : LawfulUpwardEnumerableLT (BitVec n) := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed (BitVec n) := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed (BitVec n) := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open (BitVec n) := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open (BitVec n) := inferInstance
+
+instance : RangeSize .closed (BitVec n) where
+  size bound a := bound.toNat + 1 - a.toNat
+
+instance : RangeSize .open (BitVec n) := RangeSize.openOfClosed
+
+instance : LawfulRangeSize .closed (BitVec n) where
+  size_eq_zero_of_not_isSatisfied bound x := by
+    simp [SupportsUpperBound.IsSatisfied, BitVec.not_le, RangeSize.size, BitVec.lt_def]
+    omega
+  size_eq_one_of_succ?_eq_none bound x := by
+    have := BitVec.toNat_lt_twoPow_of_le (Nat.le_refl _) (x := bound)
+    have (h : (x.toNat + 1) % 2 ^ n = 0) : x.toNat = 2 ^ n - 1 := by
+      apply Classical.not_not.mp
+      intro _
+      simp [Nat.mod_eq_of_lt (a := x.toNat + 1) (b := 2 ^ n) (by omega)] at h
+    simp [RangeSize.size, BitVec.le_def, ← BitVec.toNat_inj, succ?]
+    omega
+  size_eq_succ_of_succ?_eq_some bound init x := by
+    have (h : ¬ (init.toNat + 1) % 2 ^ n = 0) : ¬ (init.toNat + 1 ≥ 2 ^ n) := by
+      intro _
+      have : init.toNat + 1 = 2 ^ n := by omega
+      simp_all
+    simp_all +contextual [RangeSize.size, BitVec.le_def, ← BitVec.toNat_inj,
+      Nat.mod_eq_of_lt (a := init.toNat + 1) (b := 2 ^ n), succ?]
+    omega
+
+instance : LawfulRangeSize .open (BitVec n) := inferInstance
+
+end BitVec

src/Init/Data/Range/Polymorphic/Int.lean

@@ -8,6 +8,7 @@ module
 prelude
 public import Init.Data.Range.Polymorphic.Instances
 public import Init.Data.Order.Classes
+public import Init.Data.Int.Order
 import Init.Omega
 
 public section
@@ -23,7 +24,7 @@ instance : LawfulUpwardEnumerable Int where
     simp only [UpwardEnumerable.LT, UpwardEnumerable.succMany?, Option.some.injEq]
     omega
   succMany?_zero := by simp [UpwardEnumerable.succMany?]
-  succMany?_succ := by
+  succMany?_succ? := by
     simp only [UpwardEnumerable.succMany?, UpwardEnumerable.succ?,
       Option.bind_some, Option.some.injEq]
     omega
@@ -36,6 +37,14 @@ instance : LawfulUpwardEnumerableLE Int where
     simp [UpwardEnumerable.LE, UpwardEnumerable.succMany?, Int.le_def, Int.nonneg_def,
       Int.sub_eq_iff_eq_add', eq_comm (a := y)]
 
+instance : LawfulOrderLT Int := inferInstance
+instance : LawfulUpwardEnumerableLT Int := inferInstance
+instance : LawfulUpwardEnumerableLT Int := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed Int := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed Int := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open Int := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open Int := inferInstance
+
 instance : RangeSize .closed Int where
   size bound a := (bound + 1 - a).toNat
 

src/Init/Data/Range/Polymorphic/Iterators.lean

@@ -27,7 +27,7 @@ def Internal.iter {sl su α} [UpwardEnumerable α] [BoundedUpwardEnumerable sl
 
 /--
 Returns the elements of the given range as a list in ascending order, given that ranges of the given
-type and shape support this function and the range is finite.
+type and shape are finite and support this function.
 -/
 @[always_inline, inline, expose]
 def toList {sl su α} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
@@ -37,6 +37,18 @@ def toList {sl su α} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
     [IteratorCollect (RangeIterator su α) Id Id] : List α :=
   PRange.Internal.iter r |>.toList
 
+/--
+Returns the elements of the given range as an array in ascending order, given that ranges of the
+given type and shape are finite and support this function.
+-/
+@[always_inline, inline, expose]
+def toArray {sl su α} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
+    [SupportsUpperBound su α]
+    (r : PRange ⟨sl, su⟩ α)
+    [Iterator (RangeIterator su α) Id α] [Finite (RangeIterator su α) Id]
+    [IteratorCollect (RangeIterator su α) Id Id] : Array α :=
+  PRange.Internal.iter r |>.toArray
+
 /--
 Iterators for ranges implementing `RangeSize` support the `size` function.
 -/

src/Init/Data/Range/Polymorphic/Lemmas.lean

@@ -16,6 +16,7 @@ public import Init.Data.Range.Polymorphic.Iterators
 import all Init.Data.Range.Polymorphic.Iterators
 public import Init.Data.Iterators.Consumers.Loop
 import all Init.Data.Iterators.Consumers.Loop
+import Init.Data.Array.Monadic
 
 public section
 
@@ -44,6 +45,12 @@ private theorem Internal.toList_eq_toList_iter {sl su} [UpwardEnumerable α]
     r.toList = (Internal.iter r).toList := by
   rfl
 
+private theorem Internal.toArray_eq_toArray_iter {sl su} [UpwardEnumerable α]
+    [BoundedUpwardEnumerable sl α] [SupportsUpperBound su α] [HasFiniteRanges su α]
+    [LawfulUpwardEnumerable α] {r : PRange ⟨sl, su⟩ α} :
+    r.toArray = (Internal.iter r).toArray := by
+  rfl
+
 public theorem RangeIterator.toList_eq_match {su} [UpwardEnumerable α]
     [SupportsUpperBound su α] [HasFiniteRanges su α]
     [LawfulUpwardEnumerable α]
@@ -61,6 +68,35 @@ public theorem RangeIterator.toList_eq_match {su} [UpwardEnumerable α]
   · simp [*]
   · split <;> rename_i heq' <;> simp [*]
 
+public theorem RangeIterator.toArray_eq_match {su} [UpwardEnumerable α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α]
+    [LawfulUpwardEnumerable α]
+    {it : Iter (α := RangeIterator su α) α} :
+    it.toArray =  match it.internalState.next with
+      | none => #[]
+      | some a => if SupportsUpperBound.IsSatisfied it.internalState.upperBound a then
+          #[a] ++ (⟨⟨UpwardEnumerable.succ? a, it.internalState.upperBound⟩⟩ : Iter (α := RangeIterator su α) α).toArray
+        else
+          #[] := by
+  rw [← Iter.toArray_toList, toList_eq_match]
+  split
+  · rfl
+  · split <;> simp
+
+@[simp]
+public theorem toList_toArray {sl su} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α] [LawfulUpwardEnumerable α]
+    {r : PRange ⟨sl, su⟩ α} :
+    r.toArray.toList = r.toList := by
+  simp [Internal.toArray_eq_toArray_iter, Internal.toList_eq_toList_iter]
+
+@[simp]
+public theorem toArray_toList {sl su} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α] [LawfulUpwardEnumerable α]
+    {r : PRange ⟨sl, su⟩ α} :
+    r.toList.toArray = r.toArray := by
+  simp [Internal.toArray_eq_toArray_iter, Internal.toList_eq_toList_iter]
+
 public theorem toList_eq_match {sl su} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
     [SupportsUpperBound su α] [HasFiniteRanges su α]
     [LawfulUpwardEnumerable α]
@@ -73,6 +109,18 @@ public theorem toList_eq_match {sl su} [UpwardEnumerable α] [BoundedUpwardEnume
         [] := by
   rw [Internal.toList_eq_toList_iter, RangeIterator.toList_eq_match]; rfl
 
+public theorem toArray_eq_match {sl su} [UpwardEnumerable α] [BoundedUpwardEnumerable sl α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α]
+    [LawfulUpwardEnumerable α]
+    {r : PRange ⟨sl, su⟩ α} :
+    r.toArray = match init? r.lower with
+      | none => #[]
+      | some a => if SupportsUpperBound.IsSatisfied r.upper a then
+        #[a] ++ (PRange.mk (shape := ⟨.open, su⟩) a r.upper).toArray
+      else
+        #[] := by
+  rw [Internal.toArray_eq_toArray_iter, RangeIterator.toArray_eq_match]; rfl
+
 public theorem toList_Rox_eq_toList_Rcx_of_isSome_succ? {su} [UpwardEnumerable α]
     [SupportsUpperBound su α] [HasFiniteRanges su α]
     [LawfulUpwardEnumerable α]
@@ -90,6 +138,14 @@ public theorem toList_open_eq_toList_closed_of_isSome_succ? {su} [UpwardEnumerab
       (PRange.mk (shape := ⟨.closed, su⟩) (UpwardEnumerable.succ? lo |>.get h) hi).toList :=
   toList_Rox_eq_toList_Rcx_of_isSome_succ? h
 
+public theorem toArray_Rox_eq_toList_Rcx_of_isSome_succ? {su} [UpwardEnumerable α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α]
+    [LawfulUpwardEnumerable α]
+    {lo : Bound .open α} {hi} (h : (UpwardEnumerable.succ? lo).isSome) :
+    (PRange.mk (shape := ⟨.open, su⟩) lo hi).toArray =
+      (PRange.mk (shape := ⟨.closed, su⟩) (UpwardEnumerable.succ? lo |>.get h) hi).toArray := by
+  simp [Internal.toArray_eq_toArray_iter, Internal.iter_Rox_eq_iter_Rcx_of_isSome_succ?, h]
+
 public theorem toList_eq_nil_iff {sl su} [UpwardEnumerable α]
     [SupportsUpperBound su α] [HasFiniteRanges su α] [BoundedUpwardEnumerable sl α]
     [LawfulUpwardEnumerable α]
@@ -101,6 +157,14 @@ public theorem toList_eq_nil_iff {sl su} [UpwardEnumerable α]
   simp only
   split <;> rename_i heq <;> simp [heq]
 
+public theorem toArray_eq_empty_iff {sl su} [UpwardEnumerable α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α] [BoundedUpwardEnumerable sl α]
+    [LawfulUpwardEnumerable α]
+    {r : PRange ⟨sl, su⟩ α} :
+    r.toArray = #[] ↔
+      ¬ (∃ a, init? r.lower = some a ∧ SupportsUpperBound.IsSatisfied r.upper a) := by
+  rw [← toArray_toList, List.toArray_eq_iff, Array.toList_empty, toList_eq_nil_iff]
+
 public theorem mem_toList_iff_mem {sl su} [UpwardEnumerable α]
     [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
     [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
@@ -110,6 +174,15 @@ public theorem mem_toList_iff_mem {sl su} [UpwardEnumerable α]
   rw [Internal.toList_eq_toList_iter, Iter.mem_toList_iff_isPlausibleIndirectOutput,
     Internal.isPlausibleIndirectOutput_iter_iff]
 
+public theorem mem_toArray_iff_mem {sl su} [UpwardEnumerable α]
+    [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
+    [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
+    [LawfulUpwardEnumerableLowerBound sl α] [LawfulUpwardEnumerableUpperBound su α]
+    {r : PRange ⟨sl, su⟩ α}
+    {a : α} : a ∈ r.toArray ↔ a ∈ r := by
+  rw [Internal.toArray_eq_toArray_iter, Iter.mem_toArray_iff_isPlausibleIndirectOutput,
+    Internal.isPlausibleIndirectOutput_iter_iff]
+
 public theorem BoundedUpwardEnumerable.init?_succ?_closed [UpwardEnumerable α]
     [LawfulUpwardEnumerable α] {lower lower' : Bound .closed α}
     (h : UpwardEnumerable.succ? lower = some lower') :
@@ -301,6 +374,17 @@ public theorem ClosedOpen.toList_succ_succ_eq_map [UpwardEnumerable α] [Support
       (lower...upper).toList.map succ :=
   toList_Rco_succ_succ_eq_map
 
+public theorem toArray_Rco_succ_succ_eq_map [UpwardEnumerable α] [SupportsLowerBound .closed α]
+    [LinearlyUpwardEnumerable α] [InfinitelyUpwardEnumerable α] [SupportsUpperBound .open α]
+    [HasFiniteRanges .open α] [LawfulUpwardEnumerable α] [LawfulOpenUpperBound α]
+    [LawfulUpwardEnumerableLowerBound .closed α] [LawfulUpwardEnumerableUpperBound .open α]
+    {lower : Bound .closed α} {upper : Bound .open α} :
+    ((succ lower)...(succ upper)).toArray =
+      (lower...upper).toArray.map succ := by
+  simp only [← toArray_toList]
+  rw [toList_Rco_succ_succ_eq_map]
+  simp only [List.map_toArray]
+
 private theorem Internal.forIn'_eq_forIn'_iter [UpwardEnumerable α]
     [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
     [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
@@ -324,6 +408,18 @@ public theorem forIn'_eq_forIn'_toList [UpwardEnumerable α]
   simp [Internal.forIn'_eq_forIn'_iter, Internal.toList_eq_toList_iter,
     Iter.forIn'_eq_forIn'_toList]
 
+public theorem forIn'_eq_forIn'_toArray [UpwardEnumerable α]
+    [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
+    [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
+    [LawfulUpwardEnumerableLowerBound sl α] [LawfulUpwardEnumerableUpperBound su α]
+    {r : PRange ⟨sl, su⟩ α}
+    {γ : Type u} {init : γ} {m : Type u → Type w} [Monad m] [LawfulMonad m]
+    {f : (a : α) → a ∈ r → γ → m (ForInStep γ)} :
+    ForIn'.forIn' r init f =
+      ForIn'.forIn' r.toArray init (fun a ha acc => f a (mem_toArray_iff_mem.mp ha) acc) := by
+  simp [Internal.forIn'_eq_forIn'_iter, Internal.toArray_eq_toArray_iter,
+    Iter.forIn'_eq_forIn'_toArray]
+
 public theorem forIn'_toList_eq_forIn' [UpwardEnumerable α]
     [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
     [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
@@ -335,6 +431,17 @@ public theorem forIn'_toList_eq_forIn' [UpwardEnumerable α]
       ForIn'.forIn' r init (fun a ha acc => f a (mem_toList_iff_mem.mpr ha) acc) := by
   simp [forIn'_eq_forIn'_toList]
 
+public theorem forIn'_toArray_eq_forIn' [UpwardEnumerable α]
+    [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
+    [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
+    [LawfulUpwardEnumerableLowerBound sl α] [LawfulUpwardEnumerableUpperBound su α]
+    {r : PRange ⟨sl, su⟩ α}
+    {γ : Type u} {init : γ} {m : Type u → Type w} [Monad m] [LawfulMonad m]
+    {f : (a : α) → _ → γ → m (ForInStep γ)} :
+    ForIn'.forIn' r.toArray init f =
+      ForIn'.forIn' r init (fun a ha acc => f a (mem_toArray_iff_mem.mpr ha) acc) := by
+  simp [forIn'_eq_forIn'_toArray]
+
 public theorem mem_of_mem_open [UpwardEnumerable α]
     [SupportsUpperBound su α] [SupportsLowerBound sl α] [HasFiniteRanges su α]
     [BoundedUpwardEnumerable sl α] [LawfulUpwardEnumerable α]
@@ -431,6 +538,20 @@ public instance {su} [UpwardEnumerable α] [SupportsUpperBound su α] [RangeSize
         · have := LawfulRangeSize.size_eq_zero_of_not_isSatisfied _ _ h'
           simp [*] at this
 
+public theorem length_toList {sl su} [UpwardEnumerable α] [SupportsUpperBound su α]
+    [RangeSize su α] [LawfulUpwardEnumerable α] [BoundedUpwardEnumerable sl α]
+    [HasFiniteRanges su α] [LawfulRangeSize su α]
+    {r : PRange ⟨sl, su⟩ α} :
+    r.toList.length = r.size := by
+  simp [PRange.toList, PRange.size]
+
+public theorem size_toArray {sl su} [UpwardEnumerable α] [SupportsUpperBound su α]
+    [RangeSize su α] [LawfulUpwardEnumerable α] [BoundedUpwardEnumerable sl α]
+    [HasFiniteRanges su α] [LawfulRangeSize su α]
+    {r : PRange ⟨sl, su⟩ α} :
+    r.toArray.size = r.size := by
+  simp [PRange.toArray, PRange.size]
+
 public theorem isEmpty_iff_forall_not_mem {sl su} [UpwardEnumerable α] [LawfulUpwardEnumerable α]
     [BoundedUpwardEnumerable sl α] [SupportsLowerBound sl α] [SupportsUpperBound su α]
     [LawfulUpwardEnumerableLowerBound sl α] [LawfulUpwardEnumerableUpperBound su α]
@@ -455,4 +576,94 @@ public theorem isEmpty_iff_forall_not_mem {sl su} [UpwardEnumerable α] [LawfulU
       (Option.some_get hi).symm
     exact h ((init? r.lower).get hi) ⟨hl, hu⟩
 
+theorem Std.PRange.getElem?_toList_Rcx_eq [LE α] [UpwardEnumerable α] [LawfulUpwardEnumerable α]
+    [SupportsUpperBound su α] [LawfulUpwardEnumerableUpperBound su α]
+    [LawfulUpwardEnumerableLE α] [HasFiniteRanges su α]
+    {r : PRange ⟨.closed, su⟩ α} {i} :
+    r.toList[i]? = (UpwardEnumerable.succMany? i r.lower).filter (SupportsUpperBound.IsSatisfied r.upper) := by
+  induction i generalizing r
+  · rw [PRange.toList_eq_match, UpwardEnumerable.succMany?_zero]
+    simp only [Option.filter_some, decide_eq_true_eq]
+    split <;> simp
+  · rename_i n ih
+    rw [PRange.toList_eq_match]
+    simp only
+    split
+    · simp [UpwardEnumerable.succMany?_succ?_eq_succ?_bind_succMany?]
+      cases hs : UpwardEnumerable.succ? r.lower
+      · rw [PRange.toList_eq_match]
+        simp [BoundedUpwardEnumerable.init?, hs]
+      · rw [toList_Rox_eq_toList_Rcx_of_isSome_succ? (by simp [hs])]
+        rw [ih]
+        simp [hs]
+    · simp only [List.length_nil, Nat.not_lt_zero, not_false_eq_true, getElem?_neg]
+      cases hs : UpwardEnumerable.succMany? (n + 1) r.lower
+      · simp
+      · rename_i hl a
+        simp only [Option.filter_some, decide_eq_true_eq, right_eq_ite_iff]
+        have : UpwardEnumerable.LE r.lower a := ⟨n + 1, hs⟩
+        intro ha
+        exact hl.elim <| LawfulUpwardEnumerableUpperBound.isSatisfied_of_le r.upper _ _ ha this (α := α)
+
+theorem Std.PRange.getElem?_toArray_Rcx_eq [LE α] [UpwardEnumerable α] [LawfulUpwardEnumerable α]
+    [SupportsUpperBound su α] [LawfulUpwardEnumerableUpperBound su α]
+    [LawfulUpwardEnumerableLE α] [HasFiniteRanges su α]
+    {r : PRange ⟨.closed, su⟩ α} {i} :
+    r.toArray[i]? = (UpwardEnumerable.succMany? i r.lower).filter (SupportsUpperBound.IsSatisfied r.upper) := by
+  rw [← toArray_toList, List.getElem?_toArray, getElem?_toList_Rcx_eq]
+
+theorem Std.PRange.isSome_succMany?_of_lt_length_toList_Rcx [LE α] [UpwardEnumerable α]
+    [LawfulUpwardEnumerable α] [SupportsUpperBound su α] [LawfulUpwardEnumerableUpperBound su α]
+    [LawfulUpwardEnumerableLE α] [HasFiniteRanges su α]
+    {r : PRange ⟨.closed, su⟩ α} {i} (h : i < r.toList.length) :
+    (UpwardEnumerable.succMany? i r.lower).isSome := by
+  have : r.toList[i]?.isSome := by simp [h]
+  simp only [getElem?_toList_Rcx_eq, Option.isSome_filter] at this
+  exact Option.isSome_of_any this
+
+theorem Std.PRange.isSome_succMany?_of_lt_size_toArray_Rcx [LE α] [UpwardEnumerable α]
+    [LawfulUpwardEnumerable α] [SupportsUpperBound su α] [LawfulUpwardEnumerableUpperBound su α]
+    [LawfulUpwardEnumerableLE α] [HasFiniteRanges su α]
+    {r : PRange ⟨.closed, su⟩ α} {i} (h : i < r.toArray.size) :
+    (UpwardEnumerable.succMany? i r.lower).isSome := by
+  have : r.toArray[i]?.isSome := by simp [h]
+  simp only [getElem?_toArray_Rcx_eq, Option.isSome_filter] at this
+  exact Option.isSome_of_any this
+
+theorem Std.PRange.getElem_toList_Rcx_eq [LE α] [UpwardEnumerable α] [LawfulUpwardEnumerable α]
+    [SupportsUpperBound su α] [LawfulUpwardEnumerableUpperBound su α]
+    [LawfulUpwardEnumerableLE α] [HasFiniteRanges su α]
+    {r : PRange ⟨.closed, su⟩ α} {i h} :
+    r.toList[i]'h = (UpwardEnumerable.succMany? i r.lower).get
+        (isSome_succMany?_of_lt_length_toList_Rcx h) := by
+  simp [List.getElem_eq_getElem?_get, getElem?_toList_Rcx_eq]
+
+theorem Std.PRange.getElem_toArray_Rcx_eq [LE α] [UpwardEnumerable α] [LawfulUpwardEnumerable α]
+    [SupportsUpperBound su α] [LawfulUpwardEnumerableUpperBound su α]
+    [LawfulUpwardEnumerableLE α] [HasFiniteRanges su α]
+    {r : PRange ⟨.closed, su⟩ α} {i h} :
+    r.toArray[i]'h = (UpwardEnumerable.succMany? i r.lower).get
+        (isSome_succMany?_of_lt_size_toArray_Rcx h) := by
+  simp [Array.getElem_eq_getElem?_get, getElem?_toArray_Rcx_eq]
+
+theorem Std.PRange.eq_succMany?_of_toList_Rcx_eq_append_cons [LE α]
+    [UpwardEnumerable α] [LawfulUpwardEnumerable α] [LawfulUpwardEnumerableLE α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α] [LawfulUpwardEnumerableUpperBound su α]
+    {r : PRange ⟨.closed, su⟩ α} {pref suff : List α} {cur : α} (h : r.toList = pref ++ cur :: suff) :
+    cur = (UpwardEnumerable.succMany? pref.length r.lower).get
+        (isSome_succMany?_of_lt_length_toList_Rcx (by simp [h])) := by
+  have : cur = (pref ++ cur :: suff)[pref.length] := by simp
+  simp only [← h] at this
+  simp [this, getElem_toList_Rcx_eq]
+
+theorem Std.PRange.eq_succMany?_of_toArray_Rcx_eq_append_append [LE α]
+    [UpwardEnumerable α] [LawfulUpwardEnumerable α] [LawfulUpwardEnumerableLE α]
+    [SupportsUpperBound su α] [HasFiniteRanges su α] [LawfulUpwardEnumerableUpperBound su α]
+    {r : PRange ⟨.closed, su⟩ α} {pref suff : Array α} {cur : α} (h : r.toArray = pref ++ #[cur] ++ suff) :
+    cur = (UpwardEnumerable.succMany? pref.size r.lower).get
+        (isSome_succMany?_of_lt_size_toArray_Rcx (by simp [h, Nat.add_assoc, Nat.add_comm 1])) := by
+  have : cur = (pref ++ #[cur] ++ suff)[pref.size] := by simp
+  simp only [← h] at this
+  simp [this, getElem_toArray_Rcx_eq]
+
 end Std.PRange

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

@@ -10,10 +10,12 @@ import Init.Data.Nat.Lemmas
 public import Init.Data.Nat.Order
 public import Init.Data.Range.Polymorphic.Instances
 public import Init.Data.Order.Classes
-import Init.Data.Order.Lemmas
+public import Init.Data.Order.Lemmas
 
 public section
 
+open Std PRange
+
 namespace Std.PRange
 
 instance : UpwardEnumerable Nat where
@@ -39,7 +41,7 @@ instance : LawfulUpwardEnumerableLE Nat where
 
 instance : LawfulUpwardEnumerable Nat where
   succMany?_zero := by simp [UpwardEnumerable.succMany?]
-  succMany?_succ := by simp [UpwardEnumerable.succMany?, UpwardEnumerable.succ?, Nat.add_assoc]
+  succMany?_succ? := by simp [UpwardEnumerable.succMany?, UpwardEnumerable.succ?, Nat.add_assoc]
   ne_of_lt a b hlt := by
     have hn := hlt.choose_spec
     simp only [UpwardEnumerable.succMany?, Option.some.injEq] at hn
@@ -76,8 +78,7 @@ instance : LawfulRangeSize .closed Nat where
 instance : LawfulRangeSize .open Nat := inferInstance
 instance : HasFiniteRanges .closed Nat := inferInstance
 instance : HasFiniteRanges .open Nat := inferInstance
-instance : LinearlyUpwardEnumerable Nat := by
-  exact instLinearlyUpwardEnumerableOfTotalLeOfLawfulUpwardEnumerableOfLawfulUpwardEnumerableLE
+instance : LinearlyUpwardEnumerable Nat := inferInstance
 
 /-!
 The following instances are used for the implementation of array slices a.k.a. `Subarray`.

src/Init/Data/Range/Polymorphic/NatLemmas.lean

@@ -25,4 +25,17 @@ theorem toList_Rco_succ_succ {m n : Nat} :
 theorem ClosedOpen.toList_succ_succ {m n : Nat} :
     ((m+1)...(n+1)).toList = (m...n).toList.map (· + 1) := toList_Rco_succ_succ
 
+@[simp]
+theorem Nat.size_Rco {a b : Nat} :
+    (a...b).size = b - a := by
+  simp only [size, Iterators.Iter.size, Iterators.IteratorSize.size, Iterators.Iter.toIterM,
+    Internal.iter, init?, RangeSize.size, Id.run_pure]
+  omega
+
+@[simp]
+theorem Nat.size_Rcc {a b : Nat} :
+    (a...=b).size = b + 1- a := by
+  simp [Std.PRange.size, Std.Iterators.Iter.size, Std.Iterators.IteratorSize.size,
+    Std.PRange.Internal.iter, Std.Iterators.Iter.toIterM, Std.PRange.RangeSize.size]
+
 end Std.PRange.Nat

src/Init/Data/Range/Polymorphic/UInt.lean

@@ -0,0 +1,382 @@
+/-
+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
+public import Init.Data.Range.Polymorphic.Instances
+public import Init.Data.Order.Lemmas
+public import Init.Data.UInt
+import Init.Omega
+public import Init.Data.Range.Polymorphic.BitVec
+
+public section
+
+open Std Std.PRange
+
+namespace UInt8
+
+instance : UpwardEnumerable UInt8 where
+  succ? i := if i + 1 = 0 then none else some (i + 1)
+  succMany? n i := if h : i.toNat + n < UInt8.size then some (.ofNatLT _ h) else none
+
+theorem succ?_ofBitVec {x : BitVec 8} :
+    UpwardEnumerable.succ? (UInt8.ofBitVec x) = UInt8.ofBitVec <$> UpwardEnumerable.succ? x := by
+  simp only [succ?, BitVec.ofNat_eq_ofNat, Option.map_eq_map, ← UInt8.toBitVec_inj]
+  split <;> simp_all
+
+theorem succMany?_ofBitVec {k : Nat} {x : BitVec 8} :
+    UpwardEnumerable.succMany? k (UInt8.ofBitVec x) = UInt8.ofBitVec <$> UpwardEnumerable.succMany? k x := by
+  simp [succMany?]
+
+theorem upwardEnumerableLE_ofBitVec {x y : BitVec 8} :
+    UpwardEnumerable.LE (UInt8.ofBitVec x) (UInt8.ofBitVec y) ↔ UpwardEnumerable.LE x y := by
+  simp [UpwardEnumerable.LE, succMany?_ofBitVec]
+
+theorem upwardEnumerableLT_ofBitVec {x y : BitVec 8} :
+    UpwardEnumerable.LT (UInt8.ofBitVec x) (UInt8.ofBitVec y) ↔ UpwardEnumerable.LT x y := by
+  simp [UpwardEnumerable.LT, succMany?_ofBitVec]
+
+instance : LawfulUpwardEnumerable UInt8 where
+  ne_of_lt x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLT_ofBitVec] using LawfulUpwardEnumerable.ne_of_lt _ _
+  succMany?_zero x := by
+    cases x
+    simpa [succMany?_ofBitVec] using succMany?_zero
+  succMany?_succ? n x := by
+    cases x
+    simp [succMany?_ofBitVec, succMany?_succ?, Option.bind_map, Function.comp_def,
+      succ?_ofBitVec]
+
+instance : LawfulUpwardEnumerableLE UInt8 where
+  le_iff x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLE_ofBitVec, UInt8.le_iff_toBitVec_le] using
+      LawfulUpwardEnumerableLE.le_iff _ _
+
+instance : LawfulOrderLT UInt8 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt8 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt8 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed UInt8 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed UInt8 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open UInt8 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open UInt8 := inferInstance
+
+instance : RangeSize .closed UInt8 where
+  size bound a := bound.toNat + 1 - a.toNat
+
+theorem rangeSizeSize_eq_toBitVec {bound : Bound .closed UInt8} {x : BitVec 8} :
+    RangeSize.size bound (UInt8.ofBitVec x) = RangeSize.size (shape := .closed) bound.toBitVec x := by
+  simp [RangeSize.size]
+
+instance : LawfulRangeSize .closed UInt8 where
+  size_eq_zero_of_not_isSatisfied bound x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt8.lt_iff_toBitVec_lt] using
+      LawfulRangeSize.size_eq_zero_of_not_isSatisfied (su := .closed) (α := BitVec 8) _ _
+  size_eq_one_of_succ?_eq_none bound x := by
+    cases x
+    simpa [rangeSizeSize_eq_toBitVec, UInt8.le_iff_toBitVec_le, succ?_ofBitVec] using
+      LawfulRangeSize.size_eq_one_of_succ?_eq_none (su := .closed) (α := BitVec 8) _ _
+  size_eq_succ_of_succ?_eq_some bound init x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt8.le_iff_toBitVec_le, ← UInt8.toBitVec_inj, succ?] using
+      LawfulRangeSize.size_eq_succ_of_succ?_eq_some (su := .closed) (α := BitVec 8) _ _ _
+
+instance : RangeSize .open UInt8 := RangeSize.openOfClosed
+instance : LawfulRangeSize .open UInt8 := inferInstance
+
+end UInt8
+
+namespace UInt16
+
+instance : UpwardEnumerable UInt16 where
+  succ? i := if i + 1 = 0 then none else some (i + 1)
+  succMany? n i := if h : i.toNat + n < UInt16.size then some (.ofNatLT _ h) else none
+
+theorem succ?_ofBitVec {x : BitVec 16} :
+    UpwardEnumerable.succ? (UInt16.ofBitVec x) = UInt16.ofBitVec <$> UpwardEnumerable.succ? x := by
+  simp only [succ?, BitVec.ofNat_eq_ofNat, Option.map_eq_map, ← UInt16.toBitVec_inj]
+  split <;> simp_all
+
+theorem succMany?_ofBitVec {k : Nat} {x : BitVec 16} :
+    UpwardEnumerable.succMany? k (UInt16.ofBitVec x) = UInt16.ofBitVec <$> UpwardEnumerable.succMany? k x := by
+  simp [succMany?]
+
+theorem upwardEnumerableLE_ofBitVec {x y : BitVec 16} :
+    UpwardEnumerable.LE (UInt16.ofBitVec x) (UInt16.ofBitVec y) ↔ UpwardEnumerable.LE x y := by
+  simp [UpwardEnumerable.LE, succMany?_ofBitVec]
+
+theorem upwardEnumerableLT_ofBitVec {x y : BitVec 16} :
+    UpwardEnumerable.LT (UInt16.ofBitVec x) (UInt16.ofBitVec y) ↔ UpwardEnumerable.LT x y := by
+  simp [UpwardEnumerable.LT, succMany?_ofBitVec]
+
+instance : LawfulUpwardEnumerable UInt16 where
+  ne_of_lt x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLT_ofBitVec] using LawfulUpwardEnumerable.ne_of_lt _ _
+  succMany?_zero x := by
+    cases x
+    simpa [succMany?_ofBitVec] using succMany?_zero
+  succMany?_succ? n x := by
+    cases x
+    simp [succMany?_ofBitVec, succMany?_succ?, Option.bind_map, Function.comp_def,
+      succ?_ofBitVec]
+
+instance : LawfulUpwardEnumerableLE UInt16 where
+  le_iff x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLE_ofBitVec, UInt16.le_iff_toBitVec_le] using
+      LawfulUpwardEnumerableLE.le_iff _ _
+
+instance : LawfulOrderLT UInt16 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt16 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt16 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed UInt16 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed UInt16 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open UInt16 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open UInt16 := inferInstance
+
+instance : RangeSize .closed UInt16 where
+  size bound a := bound.toNat + 1 - a.toNat
+
+theorem rangeSizeSize_eq_toBitVec {bound : Bound .closed UInt16} {x : BitVec 16} :
+    RangeSize.size bound (UInt16.ofBitVec x) = RangeSize.size (shape := .closed) bound.toBitVec x := by
+  simp [RangeSize.size]
+
+instance : LawfulRangeSize .closed UInt16 where
+  size_eq_zero_of_not_isSatisfied bound x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt16.lt_iff_toBitVec_lt] using
+      LawfulRangeSize.size_eq_zero_of_not_isSatisfied (su := .closed) (α := BitVec 16) _ _
+  size_eq_one_of_succ?_eq_none bound x := by
+    cases x
+    simpa [rangeSizeSize_eq_toBitVec, UInt16.le_iff_toBitVec_le, succ?_ofBitVec] using
+      LawfulRangeSize.size_eq_one_of_succ?_eq_none (su := .closed) (α := BitVec 16) _ _
+  size_eq_succ_of_succ?_eq_some bound init x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt16.le_iff_toBitVec_le, ← UInt16.toBitVec_inj, succ?] using
+      LawfulRangeSize.size_eq_succ_of_succ?_eq_some (su := .closed) (α := BitVec 16) _ _ _
+
+instance : RangeSize .open UInt16 := RangeSize.openOfClosed
+instance : LawfulRangeSize .open UInt16 := inferInstance
+
+end UInt16
+
+namespace UInt32
+
+instance : UpwardEnumerable UInt32 where
+  succ? i := if i + 1 = 0 then none else some (i + 1)
+  succMany? n i := if h : i.toNat + n < UInt32.size then some (.ofNatLT _ h) else none
+
+theorem succ?_ofBitVec {x : BitVec 32} :
+    UpwardEnumerable.succ? (UInt32.ofBitVec x) = UInt32.ofBitVec <$> UpwardEnumerable.succ? x := by
+  simp only [succ?, BitVec.ofNat_eq_ofNat, Option.map_eq_map, ← UInt32.toBitVec_inj]
+  split <;> simp_all
+
+theorem succMany?_ofBitVec {k : Nat} {x : BitVec 32} :
+    UpwardEnumerable.succMany? k (UInt32.ofBitVec x) = UInt32.ofBitVec <$> UpwardEnumerable.succMany? k x := by
+  simp [succMany?]
+
+theorem upwardEnumerableLE_ofBitVec {x y : BitVec 32} :
+    UpwardEnumerable.LE (UInt32.ofBitVec x) (UInt32.ofBitVec y) ↔ UpwardEnumerable.LE x y := by
+  simp [UpwardEnumerable.LE, succMany?_ofBitVec]
+
+theorem upwardEnumerableLT_ofBitVec {x y : BitVec 32} :
+    UpwardEnumerable.LT (UInt32.ofBitVec x) (UInt32.ofBitVec y) ↔ UpwardEnumerable.LT x y := by
+  simp [UpwardEnumerable.LT, succMany?_ofBitVec]
+
+instance : LawfulUpwardEnumerable UInt32 where
+  ne_of_lt x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLT_ofBitVec] using LawfulUpwardEnumerable.ne_of_lt _ _
+  succMany?_zero x := by
+    cases x
+    simpa [succMany?_ofBitVec] using succMany?_zero
+  succMany?_succ? n x := by
+    cases x
+    simp [succMany?_ofBitVec, succMany?_succ?, Option.bind_map, Function.comp_def,
+      succ?_ofBitVec]
+
+instance : LawfulUpwardEnumerableLE UInt32 where
+  le_iff x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLE_ofBitVec, UInt32.le_iff_toBitVec_le] using
+      LawfulUpwardEnumerableLE.le_iff _ _
+
+instance : LawfulOrderLT UInt32 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt32 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt32 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed UInt32 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed UInt32 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open UInt32 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open UInt32 := inferInstance
+
+instance : RangeSize .closed UInt32 where
+  size bound a := bound.toNat + 1 - a.toNat
+
+theorem rangeSizeSize_eq_toBitVec {bound : Bound .closed UInt32} {x : BitVec 32} :
+    RangeSize.size bound (UInt32.ofBitVec x) = RangeSize.size (shape := .closed) bound.toBitVec x := by
+  simp [RangeSize.size]
+
+instance : LawfulRangeSize .closed UInt32 where
+  size_eq_zero_of_not_isSatisfied bound x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt32.lt_iff_toBitVec_lt] using
+      LawfulRangeSize.size_eq_zero_of_not_isSatisfied (su := .closed) (α := BitVec 32) _ _
+  size_eq_one_of_succ?_eq_none bound x := by
+    cases x
+    simpa [rangeSizeSize_eq_toBitVec, UInt32.le_iff_toBitVec_le, succ?_ofBitVec] using
+      LawfulRangeSize.size_eq_one_of_succ?_eq_none (su := .closed) (α := BitVec 32) _ _
+  size_eq_succ_of_succ?_eq_some bound init x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt32.le_iff_toBitVec_le, ← UInt32.toBitVec_inj, succ?] using
+      LawfulRangeSize.size_eq_succ_of_succ?_eq_some (su := .closed) (α := BitVec 32) _ _ _
+
+instance : RangeSize .open UInt32 := RangeSize.openOfClosed
+instance : LawfulRangeSize .open UInt32 := inferInstance
+
+end UInt32
+
+namespace UInt64
+
+instance : UpwardEnumerable UInt64 where
+  succ? i := if i + 1 = 0 then none else some (i + 1)
+  succMany? n i := if h : i.toNat + n < UInt64.size then some (.ofNatLT _ h) else none
+
+theorem succ?_ofBitVec {x : BitVec 64} :
+    UpwardEnumerable.succ? (UInt64.ofBitVec x) = UInt64.ofBitVec <$> UpwardEnumerable.succ? x := by
+  simp only [succ?, BitVec.ofNat_eq_ofNat, Option.map_eq_map, ← UInt64.toBitVec_inj]
+  split <;> simp_all
+
+theorem succMany?_ofBitVec {k : Nat} {x : BitVec 64} :
+    UpwardEnumerable.succMany? k (UInt64.ofBitVec x) = UInt64.ofBitVec <$> UpwardEnumerable.succMany? k x := by
+  simp [succMany?]
+
+theorem upwardEnumerableLE_ofBitVec {x y : BitVec 64} :
+    UpwardEnumerable.LE (UInt64.ofBitVec x) (UInt64.ofBitVec y) ↔ UpwardEnumerable.LE x y := by
+  simp [UpwardEnumerable.LE, succMany?_ofBitVec]
+
+theorem upwardEnumerableLT_ofBitVec {x y : BitVec 64} :
+    UpwardEnumerable.LT (UInt64.ofBitVec x) (UInt64.ofBitVec y) ↔ UpwardEnumerable.LT x y := by
+  simp [UpwardEnumerable.LT, succMany?_ofBitVec]
+
+instance : LawfulUpwardEnumerable UInt64 where
+  ne_of_lt x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLT_ofBitVec] using LawfulUpwardEnumerable.ne_of_lt _ _
+  succMany?_zero x := by
+    cases x
+    simpa [succMany?_ofBitVec] using succMany?_zero
+  succMany?_succ? n x := by
+    cases x
+    simp [succMany?_ofBitVec, succMany?_succ?, Option.bind_map, Function.comp_def,
+      succ?_ofBitVec]
+
+instance : LawfulUpwardEnumerableLE UInt64 where
+  le_iff x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLE_ofBitVec, UInt64.le_iff_toBitVec_le] using
+      LawfulUpwardEnumerableLE.le_iff _ _
+
+instance : LawfulOrderLT UInt64 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt64 := inferInstance
+instance : LawfulUpwardEnumerableLT UInt64 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed UInt64 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed UInt64 := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open UInt64 := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open UInt64 := inferInstance
+
+instance : RangeSize .closed UInt64 where
+  size bound a := bound.toNat + 1 - a.toNat
+
+theorem rangeSizeSize_eq_toBitVec {bound : Bound .closed UInt64} {x : BitVec 64} :
+    RangeSize.size bound (UInt64.ofBitVec x) = RangeSize.size (shape := .closed) bound.toBitVec x := by
+  simp [RangeSize.size]
+
+instance : LawfulRangeSize .closed UInt64 where
+  size_eq_zero_of_not_isSatisfied bound x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt64.lt_iff_toBitVec_lt] using
+      LawfulRangeSize.size_eq_zero_of_not_isSatisfied (su := .closed) (α := BitVec 64) _ _
+  size_eq_one_of_succ?_eq_none bound x := by
+    cases x
+    simpa [rangeSizeSize_eq_toBitVec, UInt64.le_iff_toBitVec_le, succ?_ofBitVec] using
+      LawfulRangeSize.size_eq_one_of_succ?_eq_none (su := .closed) (α := BitVec 64) _ _
+  size_eq_succ_of_succ?_eq_some bound init x := by
+    simpa [rangeSizeSize_eq_toBitVec, UInt64.le_iff_toBitVec_le, ← UInt64.toBitVec_inj, succ?] using
+      LawfulRangeSize.size_eq_succ_of_succ?_eq_some (su := .closed) (α := BitVec 64) _ _ _
+
+instance : RangeSize .open UInt64 := RangeSize.openOfClosed
+instance : LawfulRangeSize .open UInt64 := inferInstance
+
+end UInt64
+
+namespace USize
+
+instance : UpwardEnumerable USize where
+  succ? i := if i + 1 = 0 then none else some (i + 1)
+  succMany? n i := if h : i.toNat + n < USize.size then some (.ofNatLT _ h) else none
+
+theorem succ?_ofBitVec {x : BitVec System.Platform.numBits} :
+    UpwardEnumerable.succ? (USize.ofBitVec x) = USize.ofBitVec <$> UpwardEnumerable.succ? x := by
+  simp only [succ?, BitVec.ofNat_eq_ofNat, Option.map_eq_map, ← USize.toBitVec_inj]
+  split <;> simp_all
+
+theorem succMany?_ofBitVec {k : Nat} {x : BitVec System.Platform.numBits} :
+    UpwardEnumerable.succMany? k (USize.ofBitVec x) = USize.ofBitVec <$> UpwardEnumerable.succMany? k x := by
+  simp [succMany?]
+
+theorem upwardEnumerableLE_ofBitVec {x y : BitVec System.Platform.numBits} :
+    UpwardEnumerable.LE (USize.ofBitVec x) (USize.ofBitVec y) ↔ UpwardEnumerable.LE x y := by
+  simp [UpwardEnumerable.LE, succMany?_ofBitVec]
+
+theorem upwardEnumerableLT_ofBitVec {x y : BitVec System.Platform.numBits} :
+    UpwardEnumerable.LT (USize.ofBitVec x) (USize.ofBitVec y) ↔ UpwardEnumerable.LT x y := by
+  simp [UpwardEnumerable.LT, succMany?_ofBitVec]
+
+instance : LawfulUpwardEnumerable USize where
+  ne_of_lt x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLT_ofBitVec] using LawfulUpwardEnumerable.ne_of_lt _ _
+  succMany?_zero x := by
+    cases x
+    simpa [succMany?_ofBitVec] using succMany?_zero
+  succMany?_succ? n x := by
+    cases x
+    simp [succMany?_ofBitVec, succMany?_succ?, Option.bind_map, Function.comp_def,
+      succ?_ofBitVec]
+
+instance : LawfulUpwardEnumerableLE USize where
+  le_iff x y := by
+    cases x; cases y
+    simpa [upwardEnumerableLE_ofBitVec, USize.le_iff_toBitVec_le] using
+      LawfulUpwardEnumerableLE.le_iff _ _
+
+instance : LawfulOrderLT USize := inferInstance
+instance : LawfulUpwardEnumerableLT USize := inferInstance
+instance : LawfulUpwardEnumerableLT USize := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .closed USize := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .closed USize := inferInstance
+instance : LawfulUpwardEnumerableLowerBound .open USize := inferInstance
+instance : LawfulUpwardEnumerableUpperBound .open USize := inferInstance
+
+instance : RangeSize .closed USize where
+  size bound a := bound.toNat + 1 - a.toNat
+
+theorem rangeSizeSize_eq_toBitVec {bound : Bound .closed USize} {x : BitVec System.Platform.numBits} :
+    RangeSize.size bound (USize.ofBitVec x) = RangeSize.size (shape := .closed) bound.toBitVec x := by
+  simp [RangeSize.size]
+
+instance : LawfulRangeSize .closed USize where
+  size_eq_zero_of_not_isSatisfied bound x := by
+    simpa [rangeSizeSize_eq_toBitVec, USize.lt_iff_toBitVec_lt] using
+      LawfulRangeSize.size_eq_zero_of_not_isSatisfied (su := .closed) (α := BitVec System.Platform.numBits) _ _
+  size_eq_one_of_succ?_eq_none bound x := by
+    cases x
+    simpa [rangeSizeSize_eq_toBitVec, USize.le_iff_toBitVec_le, succ?_ofBitVec] using
+      LawfulRangeSize.size_eq_one_of_succ?_eq_none (su := .closed) (α := BitVec System.Platform.numBits) _ _
+  size_eq_succ_of_succ?_eq_some bound init x := by
+    simpa [rangeSizeSize_eq_toBitVec, USize.le_iff_toBitVec_le, ← USize.toBitVec_inj, succ?] using
+      LawfulRangeSize.size_eq_succ_of_succ?_eq_some (su := .closed) (α := BitVec System.Platform.numBits) _ _ _
+
+instance : RangeSize .open USize := RangeSize.openOfClosed
+instance : LawfulRangeSize .open USize := inferInstance
+
+end USize

src/Init/Data/Range/Polymorphic/UpwardEnumerable.lean

@@ -40,7 +40,6 @@ class UpwardEnumerable (α : Type u) where
   -/
   succMany? (n : Nat) (a : α) : Option α := Nat.repeat (· >>= succ?) n (some a)
 
-attribute [simp] UpwardEnumerable.succ? UpwardEnumerable.succMany?
 export UpwardEnumerable (succ? succMany?)
 
 /--
@@ -80,7 +79,6 @@ class Least? (α : Type u) where
   -/
   least? : Option α
 
-attribute [simp] Least?.least?
 export Least? (least?)
 
 /--
@@ -95,7 +93,7 @@ class LawfulUpwardEnumerable (α : Type u) [UpwardEnumerable α] where
   The `n + 1`-th successor of `a` is the successor of the `n`-th successor, given that said
   successors actually exist.
   -/
-  succMany?_succ (n : Nat) (a : α) :
+  succMany?_succ? (n : Nat) (a : α) :
     succMany? (n + 1) a = (succMany? n a).bind succ?
 
 theorem UpwardEnumerable.succMany?_zero [UpwardEnumerable α] [LawfulUpwardEnumerable α] {a : α} :
@@ -105,7 +103,7 @@ theorem UpwardEnumerable.succMany?_zero [UpwardEnumerable α] [LawfulUpwardEnume
 theorem UpwardEnumerable.succMany?_succ? [UpwardEnumerable α] [LawfulUpwardEnumerable α]
     {n : Nat} {a : α} :
     succMany? (n + 1) a = (succMany? n a).bind succ? :=
-  LawfulUpwardEnumerable.succMany?_succ n a
+  LawfulUpwardEnumerable.succMany?_succ? n a
 
 @[deprecated succMany?_succ? (since := "2025-09-03")]
 theorem UpwardEnumerable.succMany?_succ [UpwardEnumerable α] [LawfulUpwardEnumerable α]

src/Init/Data/String.lean

@@ -8,6 +8,7 @@ module
 prelude
 public import Init.Data.String.Basic
 public import Init.Data.String.Bootstrap
+public import Init.Data.String.Decode
 public import Init.Data.String.Extra
 public import Init.Data.String.Lemmas
 public import Init.Data.String.Repr

src/Init/Data/String/Basic.lean

@@ -9,11 +9,370 @@ prelude
 public import Init.Data.List.Basic
 public import Init.Data.Char.Basic
 public import Init.Data.String.Bootstrap
+public import Init.Data.ByteArray.Basic
+public import Init.Data.String.Decode
+import Init.Data.ByteArray.Lemmas
 
 public section
 
 universe u
 
+section
+
+@[simp]
+theorem List.utf8Encode_nil : List.utf8Encode [] = ByteArray.empty := by simp [utf8Encode]
+
+theorem List.utf8Encode_singleton {c : Char} : [c].utf8Encode = (String.utf8EncodeChar c).toByteArray := by
+  simp [utf8Encode]
+
+@[simp]
+theorem List.utf8Encode_append {l l' : List Char} :
+    (l ++ l').utf8Encode = l.utf8Encode ++ l'.utf8Encode := by
+  simp [utf8Encode]
+
+theorem List.utf8Encode_cons {c : Char} {l : List Char} : (c :: l).utf8Encode = [c].utf8Encode ++ l.utf8Encode := by
+  rw [← singleton_append, List.utf8Encode_append]
+
+@[simp]
+theorem String.utf8EncodeChar_ne_nil {c : Char} : String.utf8EncodeChar c ≠ [] := by
+  fun_cases String.utf8EncodeChar with simp
+
+@[simp]
+theorem List.utf8Encode_eq_empty {l : List Char} : l.utf8Encode = ByteArray.empty ↔ l = [] := by
+  simp [utf8Encode, ← List.eq_nil_iff_forall_not_mem]
+
+theorem ByteArray.isValidUtf8_utf8Encode {l : List Char} : IsValidUtf8 l.utf8Encode :=
+  .intro l rfl
+
+@[simp]
+theorem ByteArray.isValidUtf8_empty : IsValidUtf8 ByteArray.empty :=
+  .intro [] (by simp)
+
+theorem Char.isValidUtf8_toByteArray_utf8EncodeChar {c : Char} :
+    ByteArray.IsValidUtf8 (String.utf8EncodeChar c).toByteArray :=
+  .intro [c] (by simp [List.utf8Encode_singleton])
+
+theorem ByteArray.IsValidUtf8.append {b b' : ByteArray} (h : IsValidUtf8 b) (h' : IsValidUtf8 b') :
+    IsValidUtf8 (b ++ b') := by
+  rcases h with ⟨m, rfl⟩
+  rcases h' with ⟨m', rfl⟩
+  exact .intro (m ++ m') (by simp)
+
+theorem ByteArray.isValidUtf8_utf8Encode_singleton_append_iff {b : ByteArray} {c : Char} :
+    IsValidUtf8 ([c].utf8Encode ++ b) ↔ IsValidUtf8 b := by
+  refine ⟨?_, fun h => IsValidUtf8.append isValidUtf8_utf8Encode h⟩
+  rintro ⟨l, hl⟩
+  match l with
+  | [] => simp at hl
+  | d::l =>
+    obtain rfl : c = d := by
+      replace hl := congrArg (fun l => utf8DecodeChar? l 0) hl
+      simpa [List.utf8DecodeChar?_utf8Encode_singleton_append,
+        List.utf8DecodeChar?_utf8Encode_cons] using hl
+    rw [← List.singleton_append (l := l), List.utf8Encode_append,
+      ByteArray.append_right_inj] at hl
+    exact hl ▸ isValidUtf8_utf8Encode
+
+@[expose]
+def ByteArray.utf8Decode? (b : ByteArray) : Option (Array Char) :=
+  go (b.size + 1) 0 #[] (by simp) (by simp)
+where
+  go (fuel : Nat) (i : Nat) (acc : Array Char) (hi : i ≤ b.size) (hf : b.size - i < fuel) : Option (Array Char) :=
+    match fuel, hf with
+    | fuel + 1, _ =>
+      if i = b.size then
+        some acc
+      else
+        match h : utf8DecodeChar? b i with
+        | none => none
+        | some c => go fuel (i + c.utf8Size) (acc.push c)
+            (le_size_of_utf8DecodeChar?_eq_some h)
+            (have := c.utf8Size_pos; have := le_size_of_utf8DecodeChar?_eq_some h; by omega)
+  termination_by structural fuel
+
+theorem ByteArray.utf8Decode?.go.congr {b b' : ByteArray} {fuel fuel' i i' : Nat} {acc acc' : Array Char} {hi hi' hf hf'}
+    (hbb' : b = b') (hii' : i = i') (hacc : acc = acc') :
+    ByteArray.utf8Decode?.go b fuel i acc hi hf = ByteArray.utf8Decode?.go b' fuel' i' acc' hi' hf' := by
+  subst hbb' hii' hacc
+  fun_induction ByteArray.utf8Decode?.go b fuel i acc hi hf generalizing fuel' with
+  | case1 =>
+    rw [go.eq_def]
+    split
+    simp
+  | case2 =>
+    rw [go.eq_def]
+    split <;> split
+    · simp_all
+    · split <;> simp_all
+  | case3 =>
+    conv => rhs; rw [go.eq_def]
+    split <;> split
+    · simp_all
+    · split
+      · simp_all
+      · rename_i c₁ hc₁ ih _ _ _ _ _ c₂ hc₂
+        obtain rfl : c₁ = c₂ := by rw [← Option.some_inj, ← hc₁, ← hc₂]
+        apply ih
+
+@[simp]
+theorem ByteArray.utf8Decode?_empty : ByteArray.empty.utf8Decode? = some #[] := by
+  simp [utf8Decode?, utf8Decode?.go]
+
+private theorem ByteArray.isSome_utf8Decode?go_iff {b : ByteArray} {fuel i : Nat} {hi : i ≤ b.size} {hf} {acc : Array Char} :
+    (ByteArray.utf8Decode?.go b fuel i acc hi hf).isSome ↔ IsValidUtf8 (b.extract i b.size) := by
+  fun_induction ByteArray.utf8Decode?.go with
+  | case1 => simp
+  | case2 fuel i hi hf acc h₁ h₂ =>
+    simp only [Option.isSome_none, Bool.false_eq_true, false_iff]
+    rintro ⟨l, hl⟩
+    have : l ≠ [] := by
+      rintro rfl
+      simp at hl
+      omega
+    rw [← l.cons_head_tail this] at hl
+    rw [utf8DecodeChar?_eq_utf8DecodeChar?_extract, hl, List.utf8DecodeChar?_utf8Encode_cons] at h₂
+    simp at h₂
+  | case3 i acc hi fuel hf h₁ c h₂ ih =>
+    rw [ih]
+    have h₂' := h₂
+    rw [utf8DecodeChar?_eq_utf8DecodeChar?_extract] at h₂'
+    obtain ⟨l, hl⟩ := exists_of_utf8DecodeChar?_eq_some h₂'
+    rw [ByteArray.extract_eq_extract_append_extract (i := i) (i + c.utf8Size) (by omega)
+      (le_size_of_utf8DecodeChar?_eq_some h₂)] at hl ⊢
+    rw [ByteArray.append_inj_left hl (by have := le_size_of_utf8DecodeChar?_eq_some h₂; simp; omega),
+      ← List.utf8Encode_singleton, isValidUtf8_utf8Encode_singleton_append_iff]
+
+theorem ByteArray.isSome_utf8Decode?_iff {b : ByteArray} :
+    b.utf8Decode?.isSome ↔ IsValidUtf8 b := by
+  rw [utf8Decode?, isSome_utf8Decode?go_iff, extract_zero_size]
+
+@[simp]
+theorem String.bytes_empty : "".bytes = ByteArray.empty := (rfl)
+
+/--
+Appends two strings. Usually accessed via the `++` operator.
+
+The internal implementation will perform destructive updates if the string is not shared.
+
+Examples:
+ * `"abc".append "def" = "abcdef"`
+ * `"abc" ++ "def" = "abcdef"`
+ * `"" ++ "" = ""`
+-/
+@[extern "lean_string_append", expose]
+def String.append (s t : String) : String where
+  bytes := s.bytes ++ t.bytes
+  isValidUtf8 := s.isValidUtf8.append t.isValidUtf8
+
+instance : Append String where
+  append s t := s.append t
+
+@[simp]
+theorem String.bytes_append {s t : String} : (s ++ t).bytes = s.bytes ++ t.bytes := (rfl)
+
+theorem String.bytes_inj {s t : String} : s.bytes = t.bytes ↔ s = t := by
+  refine ⟨fun h => ?_, (· ▸ rfl)⟩
+  rcases s with ⟨s⟩
+  rcases t with ⟨t⟩
+  subst h
+  rfl
+
+@[simp]
+theorem String.empty_append {s : String} : "" ++ s = s := by
+  simp [← String.bytes_inj]
+
+@[simp]
+theorem String.append_empty {s : String} : s ++ "" = s := by
+  simp [← String.bytes_inj]
+
+@[simp] theorem List.bytes_asString {l : List Char} : l.asString.bytes = l.utf8Encode := by
+  simp [List.asString, String.mk]
+
+@[simp]
+theorem List.asString_nil : List.asString [] = "" := by
+  simp [← String.bytes_inj]
+
+@[simp]
+theorem List.asString_append {l₁ l₂ : List Char} : (l₁ ++ l₂).asString = l₁.asString ++ l₂.asString := by
+  simp [← String.bytes_inj]
+
+@[expose]
+def String.Internal.toArray (b : String) : Array Char :=
+  b.bytes.utf8Decode?.get (b.bytes.isSome_utf8Decode?_iff.2 b.isValidUtf8)
+
+@[simp]
+theorem String.Internal.toArray_empty : String.Internal.toArray "" = #[] := by
+  simp [toArray]
+
+@[extern "lean_string_data", expose]
+def String.data (b : String) : List Char :=
+  (String.Internal.toArray b).toList
+
+@[simp]
+theorem String.data_empty : "".data = [] := by
+  simp [data]
+
+/--
+Returns the length of a string in Unicode code points.
+
+Examples:
+* `"".length = 0`
+* `"abc".length = 3`
+* `"L∃∀N".length = 4`
+-/
+@[extern "lean_string_length"]
+def String.length (b : String) : Nat :=
+  b.data.length
+
+@[simp]
+theorem String.Internal.size_toArray {b : String} : (String.Internal.toArray b).size = b.length :=
+  (rfl)
+
+@[simp]
+theorem String.length_data {b : String} : b.data.length = b.length := (rfl)
+
+theorem String.exists_eq_asString (s : String) :
+    ∃ l : List Char, s = l.asString := by
+  rcases s with ⟨_, ⟨l, rfl⟩⟩
+  refine ⟨l, by simp [← String.bytes_inj]⟩
+
+private theorem ByteArray.utf8Decode?go_eq_utf8Decode?go_extract {b : ByteArray} {fuel i : Nat} {hi : i ≤ b.size} {hf} {acc : Array Char} :
+    utf8Decode?.go b fuel i acc hi hf = (utf8Decode?.go (b.extract i b.size) fuel 0 #[] (by simp) (by simp [hf])).map (acc ++ ·) := by
+  fun_cases utf8Decode?.go b fuel i acc hi hf with
+  | case1 =>
+      rw [utf8Decode?.go]
+      simp only [size_extract, Nat.le_refl, Nat.min_eq_left, Nat.zero_add, List.push_toArray,
+        List.nil_append]
+      rw [if_pos (by omega)]
+      simp
+  | case2 fuel hf₁ h₁ h₂ hf₂ =>
+    rw [utf8Decode?.go]
+    simp only [size_extract, Nat.le_refl, Nat.min_eq_left, Nat.zero_add, List.push_toArray,
+      List.nil_append]
+    rw [if_neg (by omega)]
+    rw [utf8DecodeChar?_eq_utf8DecodeChar?_extract] at h₂
+    split <;> simp_all
+  | case3 fuel hf₁ h₁ c h₂ hf₂ =>
+    conv => rhs; rw [utf8Decode?.go]
+    simp only [size_extract, Nat.le_refl, Nat.min_eq_left, Nat.zero_add, List.push_toArray,
+      List.nil_append]
+    rw [if_neg (by omega)]
+    rw [utf8DecodeChar?_eq_utf8DecodeChar?_extract] at h₂
+    split
+    · simp_all
+    · rename_i c' hc'
+      obtain rfl : c = c' := by
+        rw [← Option.some_inj, ← h₂, hc']
+      have := c.utf8Size_pos
+      conv => lhs; rw [ByteArray.utf8Decode?go_eq_utf8Decode?go_extract]
+      conv => rhs; rw [ByteArray.utf8Decode?go_eq_utf8Decode?go_extract]
+      simp only [size_extract, Nat.le_refl, Nat.min_eq_left, Option.map_map, ByteArray.extract_extract]
+      have : (fun x => acc ++ x) ∘ (fun x => #[c] ++ x) = fun x => acc.push c ++ x := by funext; simp
+      simp [(by omega : i + (b.size - i) = b.size), this]
+
+theorem ByteArray.utf8Decode?_utf8Encode_singleton_append {l : ByteArray} {c : Char} :
+    ([c].utf8Encode ++ l).utf8Decode? = l.utf8Decode?.map (#[c] ++ ·) := by
+  rw [utf8Decode?, utf8Decode?.go,
+    if_neg (by simp [List.utf8Encode_singleton]; have := c.utf8Size_pos; omega)]
+  split
+  · simp_all [List.utf8DecodeChar?_utf8Encode_singleton_append]
+  · rename_i d h
+    obtain rfl : c = d := by simpa [List.utf8DecodeChar?_utf8Encode_singleton_append] using h
+    rw [utf8Decode?go_eq_utf8Decode?go_extract, utf8Decode?]
+    simp only [List.push_toArray, List.nil_append, Nat.zero_add]
+    congr 1
+    apply ByteArray.utf8Decode?.go.congr _ rfl rfl
+    apply extract_append_eq_right
+    simp [List.utf8Encode_singleton]
+
+@[simp]
+theorem List.utf8Decode?_utf8Encode {l : List Char} :
+    l.utf8Encode.utf8Decode? = some l.toArray := by
+  induction l with
+  | nil => simp
+  | cons c l ih =>
+    rw [← List.singleton_append, List.utf8Encode_append]
+    simp only [ByteArray.utf8Decode?_utf8Encode_singleton_append, cons_append, nil_append,
+      Option.map_eq_some_iff, Array.append_eq_toArray_iff, cons.injEq, true_and]
+    refine ⟨l.toArray, ih, by simp⟩
+
+@[simp]
+theorem ByteArray.utf8Encode_get_utf8Decode? {b : ByteArray} {h} :
+    (b.utf8Decode?.get h).toList.utf8Encode = b := by
+  obtain ⟨l, rfl⟩ := isSome_utf8Decode?_iff.1 h
+  simp
+
+@[simp]
+theorem List.data_asString {l : List Char} : l.asString.data = l := by
+  simp [String.data, String.Internal.toArray]
+
+@[simp]
+theorem String.asString_data {b : String} : b.data.asString = b := by
+  obtain ⟨l, rfl⟩ := String.exists_eq_asString b
+  rw [List.data_asString]
+
+theorem List.asString_injective {l₁ l₂ : List Char} (h : l₁.asString = l₂.asString) : l₁ = l₂ := by
+  simpa using congrArg String.data h
+
+theorem List.asString_inj {l₁ l₂ : List Char} : l₁.asString = l₂.asString ↔ l₁ = l₂ :=
+  ⟨asString_injective, (· ▸ rfl)⟩
+
+theorem String.data_injective {s₁ s₂ : String} (h : s₁.data = s₂.data) : s₁ = s₂ := by
+  simpa using congrArg List.asString h
+
+theorem String.data_inj {s₁ s₂ : String} : s₁.data = s₂.data ↔ s₁ = s₂ :=
+  ⟨data_injective, (· ▸ rfl)⟩
+
+@[simp]
+theorem String.data_append {l₁ l₂ : String} : (l₁ ++ l₂).data = l₁.data ++ l₂.data := by
+  apply List.asString_injective
+  simp
+
+@[simp]
+theorem String.utf8encode_data {b : String} : b.data.utf8Encode = b.bytes := by
+  have := congrArg String.bytes (String.asString_data (b := b))
+  rwa [← List.bytes_asString]
+
+@[simp]
+theorem String.utf8ByteSize_empty : "".utf8ByteSize = 0 := (rfl)
+
+@[simp]
+theorem String.utf8ByteSize_append {s t : String} :
+    (s ++ t).utf8ByteSize = s.utf8ByteSize + t.utf8ByteSize := by
+  simp [utf8ByteSize]
+
+@[simp]
+theorem String.size_bytes {s : String} : s.bytes.size = s.utf8ByteSize := rfl
+
+@[simp]
+theorem String.bytes_push {s : String} {c : Char} : (s.push c).bytes = s.bytes ++ [c].utf8Encode := by
+  simp [push]
+
+-- This is just to keep the proof of `set_next_add` below from breaking; if that lemma goes away
+-- or the proof is rewritten, it can be removed.
+private noncomputable def String.utf8ByteSize' : String → Nat
+  | s => go s.data
+where
+  go : List Char → Nat
+  | []    => 0
+  | c::cs => go cs + c.utf8Size
+
+private theorem String.utf8ByteSize'_eq (s : String) : s.utf8ByteSize' = s.utf8ByteSize := by
+  suffices ∀ l, utf8ByteSize'.go l = l.asString.utf8ByteSize by
+    obtain ⟨m, rfl⟩ := s.exists_eq_asString
+    rw [utf8ByteSize', this, asString_data]
+  intro l
+  induction l with
+  | nil => simp [utf8ByteSize'.go]
+  | cons c cs ih =>
+    rw [utf8ByteSize'.go, ih, ← List.singleton_append, List.asString_append,
+      utf8ByteSize_append, Nat.add_comm]
+    congr
+    rw [← size_bytes, List.bytes_asString, List.utf8Encode_singleton,
+      List.size_toByteArray, length_utf8EncodeChar]
+
+end
+
 namespace String
 
 instance : HAdd String.Pos String.Pos String.Pos where
@@ -54,8 +413,6 @@ instance : LT String :=
 instance decidableLT (s₁ s₂ : @& String) : Decidable (s₁ < s₂) :=
   List.decidableLT s₁.data s₂.data
 
-
-
 /--
 Non-strict inequality on strings, typically used via the `≤` operator.
 
@@ -69,32 +426,6 @@ instance : LE String :=
 instance decLE (s₁ s₂ : String) : Decidable (s₁ ≤ s₂) :=
   inferInstanceAs (Decidable (Not _))
 
-/--
-Returns the length of a string in Unicode code points.
-
-Examples:
-* `"".length = 0`
-* `"abc".length = 3`
-* `"L∃∀N".length = 4`
--/
-@[extern "lean_string_length", expose]
-def length : (@& String) → Nat
-  | ⟨s⟩ => s.length
-
-/--
-Appends two strings. Usually accessed via the `++` operator.
-
-The internal implementation will perform destructive updates if the string is not shared.
-
-Examples:
- * `"abc".append "def" = "abcdef"`
- * `"abc" ++ "def" = "abcdef"`
- * `"" ++ "" = ""`
--/
-@[extern "lean_string_append", expose]
-def append : String → (@& String) → String
-  | ⟨a⟩, ⟨b⟩ => ⟨a ++ b⟩
-
 /--
 Converts a string to a list of characters.
 
@@ -153,8 +484,7 @@ Examples:
 -/
 @[extern "lean_string_utf8_get", expose]
 def get (s : @& String) (p : @& Pos) : Char :=
-  match s with
-  | ⟨s⟩ => utf8GetAux s 0 p
+  utf8GetAux s.data 0 p
 
 def utf8GetAux? : List Char → Pos → Pos → Option Char
   | [],    _, _ => none
@@ -175,7 +505,7 @@ Examples:
 -/
 @[extern "lean_string_utf8_get_opt"]
 def get? : (@& String) → (@& Pos) → Option Char
-  | ⟨s⟩, p => utf8GetAux? s 0 p
+  | s, p => utf8GetAux? s.data 0 p
 
 /--
 Returns the character at position `p` of a string. Panics if `p` is not a valid position.
@@ -191,7 +521,7 @@ Examples
 @[extern "lean_string_utf8_get_bang", expose]
 def get! (s : @& String) (p : @& Pos) : Char :=
   match s with
-  | ⟨s⟩ => utf8GetAux s 0 p
+  | s => utf8GetAux s.data 0 p
 
 def utf8SetAux (c' : Char) : List Char → Pos → Pos → List Char
   | [],    _, _ => []
@@ -214,7 +544,7 @@ Examples:
 -/
 @[extern "lean_string_utf8_set"]
 def set : String → (@& Pos) → Char → String
-  | ⟨s⟩, i, c => ⟨utf8SetAux c s 0 i⟩
+  | s, i, c => (utf8SetAux c s.data 0 i).asString
 
 /--
 Replaces the character at position `p` in the string `s` with the result of applying `f` to that
@@ -270,7 +600,7 @@ Examples:
 -/
 @[extern "lean_string_utf8_prev", expose]
 def prev : (@& String) → (@& Pos) → Pos
-  | ⟨s⟩, p => utf8PrevAux s 0 p
+  | s, p => utf8PrevAux s.data 0 p
 
 /--
 Returns the first character in `s`. If `s = ""`, returns `(default : Char)`.
@@ -336,7 +666,7 @@ Examples:
 @[extern "lean_string_utf8_get_fast", expose]
 def get' (s : @& String) (p : @& Pos) (h : ¬ s.atEnd p) : Char :=
   match s with
-  | ⟨s⟩ => utf8GetAux s 0 p
+  | s => utf8GetAux s.data 0 p
 
 /--
 Returns the next position in a string after position `p`. The result is unspecified if `p` is not a
@@ -360,16 +690,6 @@ def next' (s : @& String) (p : @& Pos) (h : ¬ s.atEnd p) : Pos :=
   let c := get s p
   p + c
 
-theorem _root_.Char.utf8Size_pos (c : Char) : 0 < c.utf8Size := by
-  repeat first | apply iteInduction (motive := (0 < ·)) <;> intros | decide
-
-theorem _root_.Char.utf8Size_le_four (c : Char) : c.utf8Size ≤ 4 := by
-  repeat first | apply iteInduction (motive := (· ≤ 4)) <;> intros | decide
-
-theorem _root_.Char.utf8Size_eq (c : Char) : c.utf8Size = 1 ∨ c.utf8Size = 2 ∨ c.utf8Size = 3 ∨ c.utf8Size = 4 := by
-  match c.utf8Size, c.utf8Size_pos, c.utf8Size_le_four with
-  | 1, _, _ | 2, _, _ | 3, _, _ | 4, _, _ => simp
-
 @[deprecated Char.utf8Size_pos (since := "2026-06-04")] abbrev one_le_csize := Char.utf8Size_pos
 
 @[simp] theorem pos_lt_eq (p₁ p₂ : Pos) : (p₁ < p₂) = (p₁.1 < p₂.1) := rfl
@@ -534,7 +854,7 @@ Examples:
 -/
 @[extern "lean_string_utf8_extract", expose]
 def extract : (@& String) → (@& Pos) → (@& Pos) → String
-  | ⟨s⟩, b, e => if b.byteIdx ≥ e.byteIdx then "" else ⟨go₁ s 0 b e⟩
+  | s, b, e => if b.byteIdx ≥ e.byteIdx then "" else (go₁ s.data 0 b e).asString
 where
   go₁ : List Char → Pos → Pos → Pos → List Char
     | [],        _, _, _ => []
@@ -1030,37 +1350,31 @@ theorem utf8SetAux_of_gt (c' : Char) : ∀ (cs : List Char) {i p : Pos}, i > p
 theorem set_next_add (s : String) (i : Pos) (c : Char) (b₁ b₂)
     (h : (s.next i).1 + b₁ = s.endPos.1 + b₂) :
     ((s.set i c).next i).1 + b₁ = (s.set i c).endPos.1 + b₂ := by
-  simp [next, get, set, endPos, utf8ByteSize] at h ⊢
+  simp [next, get, set, endPos, ← utf8ByteSize'_eq, utf8ByteSize'] at h ⊢
   rw [Nat.add_comm i.1, Nat.add_assoc] at h ⊢
   let rec foo : ∀ cs a b₁ b₂,
-    (utf8GetAux cs a i).utf8Size + b₁ = utf8ByteSize.go cs + b₂ →
-    (utf8GetAux (utf8SetAux c cs a i) a i).utf8Size + b₁ = utf8ByteSize.go (utf8SetAux c cs a i) + b₂
+    (utf8GetAux cs a i).utf8Size + b₁ = utf8ByteSize'.go cs + b₂ →
+    (utf8GetAux (utf8SetAux c cs a i) a i).utf8Size + b₁ = utf8ByteSize'.go (utf8SetAux c cs a i) + b₂
   | [], _, _, _, h => h
   | c'::cs, a, b₁, b₂, h => by
     unfold utf8SetAux
-    apply iteInduction (motive := fun p => (utf8GetAux p a i).utf8Size + b₁ = utf8ByteSize.go p + b₂) <;>
-      intro h' <;> simp [utf8GetAux, h', utf8ByteSize.go] at h ⊢
+    apply iteInduction (motive := fun p => (utf8GetAux p a i).utf8Size + b₁ = utf8ByteSize'.go p + b₂) <;>
+      intro h' <;> simp [utf8GetAux, h', utf8ByteSize'.go] at h ⊢
     next =>
       rw [Nat.add_assoc, Nat.add_left_comm] at h ⊢; rw [Nat.add_left_cancel h]
     next =>
       rw [Nat.add_assoc] at h ⊢
       refine foo cs (a + c') b₁ (c'.utf8Size + b₂) h
-  exact foo s.1 0 _ _ h
+  exact foo s.data 0 _ _ h
 
 theorem mapAux_lemma (s : String) (i : Pos) (c : Char) (h : ¬s.atEnd i) :
-    (s.set i c).endPos.1 - ((s.set i c).next i).1 < s.endPos.1 - i.1 :=
+    (s.set i c).endPos.1 - ((s.set i c).next i).1 < s.endPos.1 - i.1 := by
   suffices (s.set i c).endPos.1 - ((s.set i c).next i).1 = s.endPos.1 - (s.next i).1 by
     rw [this]
     apply Nat.sub_lt_sub_left (Nat.gt_of_not_le (mt decide_eq_true h)) (lt_next ..)
-  Nat.sub.elim (motive := (_ = ·)) _ _
-    (fun _ k e =>
-      have := set_next_add _ _ _ k 0 e.symm
-      Nat.sub_eq_of_eq_add <| this.symm.trans <| Nat.add_comm ..)
-    (fun h => by
-      have ⟨k, e⟩ := Nat.le.dest h
-      rw [Nat.succ_add] at e
-      have : ((s.set i c).next i).1 = _ := set_next_add _ _ c 0 k.succ e.symm
-      exact Nat.sub_eq_zero_of_le (this ▸ Nat.le_add_right ..))
+  have := set_next_add s i c (s.endPos.byteIdx - (s.next i).byteIdx) 0
+  have := set_next_add s i c 0 ((s.next i).byteIdx - s.endPos.byteIdx)
+  omega
 
 @[specialize] def mapAux (f : Char → Char) (i : Pos) (s : String) : String :=
   if h : s.atEnd i then s
@@ -2044,40 +2358,51 @@ def stripSuffix (s : String) (suff : String) : String :=
 
 end String
 
-namespace Char
-
-@[simp] theorem length_toString (c : Char) : c.toString.length = 1 := rfl
-
-end Char
-
 namespace String
 
+@[ext]
 theorem ext {s₁ s₂ : String} (h : s₁.data = s₂.data) : s₁ = s₂ :=
-  show ⟨s₁.data⟩ = (⟨s₂.data⟩ : String) from h ▸ rfl
-
-theorem ext_iff {s₁ s₂ : String} : s₁ = s₂ ↔ s₁.data = s₂.data := ⟨fun h => h ▸ rfl, ext⟩
+  data_injective h
 
 @[simp] theorem default_eq : default = "" := rfl
 
-@[simp] theorem length_mk (s : List Char) : (String.mk s).length = s.length := rfl
+@[simp]
+theorem String.mk_eq_asString (s : List Char) : String.mk s = List.asString s := rfl
 
-@[simp] theorem length_empty : "".length = 0 := rfl
+@[simp]
+theorem _root_.List.length_asString (s : List Char) : (List.asString s).length = s.length := by
+  rw [← length_data, List.data_asString]
 
-@[simp] theorem length_singleton (c : Char) : (String.singleton c).length = 1 := rfl
+@[simp] theorem length_empty : "".length = 0 := by simp [← length_data, data_empty]
+
+@[simp]
+theorem bytes_singleton {c : Char} : (String.singleton c).bytes = [c].utf8Encode := by
+  simp [singleton]
+
+theorem singleton_eq {c : Char} : String.singleton c = [c].asString := by
+  simp [← bytes_inj]
+
+@[simp] theorem data_singleton (c : Char) : (String.singleton c).data = [c] := by
+  simp [singleton_eq]
+
+@[simp]
+theorem length_singleton {c : Char} : (String.singleton c).length = 1 := by
+  simp [← length_data]
+
+theorem push_eq_append (c : Char) : String.push s c = s ++ singleton c := by
+  simp [← bytes_inj]
+
+@[simp] theorem data_push (c : Char) : (String.push s c).data = s.data ++ [c] := by
+  simp [push_eq_append]
 
 @[simp] theorem length_push (c : Char) : (String.push s c).length = s.length + 1 := by
-  rw [push, length_mk, List.length_append, List.length_singleton, Nat.succ.injEq]
-  rfl
+  simp [← length_data]
 
 @[simp] theorem length_pushn (c : Char) (n : Nat) : (pushn s c n).length = s.length + n := by
   unfold pushn; induction n <;> simp [Nat.repeat, Nat.add_assoc, *]
 
 @[simp] theorem length_append (s t : String) : (s ++ t).length = s.length + t.length := by
-  simp only [length, append, List.length_append]
-
-@[simp] theorem data_push (s : String) (c : Char) : (s.push c).data = s.data ++ [c] := rfl
-
-@[simp] theorem data_append (s t : String) : (s ++ t).data = s.data ++ t.data := rfl
+  simp [← length_data]
 
 attribute [simp] toList -- prefer `String.data` over `String.toList` in lemmas
 
@@ -2161,10 +2486,8 @@ end Pos
 theorem lt_next' (s : String) (p : Pos) : p < next s p := lt_next ..
 
 @[simp] theorem prev_zero (s : String) : prev s 0 = 0 := by
-  cases s with | mk cs
-  cases cs
-  next => rfl
-  next => simp [prev, utf8PrevAux, Pos.le_iff]
+  rw [prev]
+  cases s.data <;> simp [utf8PrevAux, Pos.le_iff]
 
 @[simp] theorem get'_eq (s : String) (p : Pos) (h) : get' s p h = get s p := rfl
 
@@ -2174,19 +2497,20 @@ theorem lt_next' (s : String) (p : Pos) : p < next s p := lt_next ..
 -- so for proving can be unfolded.
 attribute [simp] toSubstring'
 
-theorem singleton_eq (c : Char) : singleton c = ⟨[c]⟩ := rfl
-
-@[simp] theorem data_singleton (c : Char) : (singleton c).data = [c] := rfl
-
-@[simp] theorem append_empty (s : String) : s ++ "" = s := ext (List.append_nil _)
-
-@[simp] theorem empty_append (s : String) : "" ++ s = s := rfl
-
 theorem append_assoc (s₁ s₂ s₃ : String) : (s₁ ++ s₂) ++ s₃ = s₁ ++ (s₂ ++ s₃) :=
-  ext (List.append_assoc ..)
+  ext (by simp [data_append])
 
 end String
 
+namespace Char
+
+theorem toString_eq_singleton {c : Char} : c.toString = String.singleton c := rfl
+
+@[simp] theorem length_toString (c : Char) : c.toString.length = 1 := by
+  simp [toString_eq_singleton]
+
+end Char
+
 open String
 
 namespace Substring

src/Init/Data/String/Bootstrap.lean

@@ -8,6 +8,7 @@ module
 prelude
 public import Init.Data.List.Basic
 public import Init.Data.Char.Basic
+public import Init.Data.ByteArray.Bootstrap
 
 public section
 
@@ -31,7 +32,11 @@ Examples:
 -/
 @[extern "lean_string_push", expose]
 def push : String → Char → String
-  | ⟨s⟩, c => ⟨s ++ [c]⟩
+  | ⟨b, h⟩, c => ⟨b.append (List.utf8Encode [c]), ?pf⟩
+where finally
+  have ⟨m, hm⟩ := h
+  cases hm
+  exact .intro (m ++ [c]) (by simp [List.utf8Encode, List.toByteArray_append'])
 
 /--
 Returns a new string that contains only the character `c`.
@@ -124,8 +129,21 @@ Examples:
  * `[].asString = ""`
  * `['a', 'a', 'a'].asString = "aaa"`
 -/
+@[extern "lean_string_mk", expose]
+def String.mk (data : List Char) : String :=
+  ⟨List.utf8Encode data,.intro data rfl⟩
+
+/--
+Creates a string that contains the characters in a list, in order.
+
+Examples:
+ * `['L', '∃', '∀', 'N'].asString = "L∃∀N"`
+ * `[].asString = ""`
+ * `['a', 'a', 'a'].asString = "aaa"`
+-/
+@[expose]
 def List.asString (s : List Char) : String :=
-  ⟨s⟩
+  String.mk s
 
 namespace Substring.Internal
 

src/Init/Data/String/Extra.lean

@@ -133,92 +133,15 @@ the corresponding string, or panics if the array is not a valid UTF-8 encoding o
 @[inline] def fromUTF8! (a : ByteArray) : String :=
   if h : validateUTF8 a then fromUTF8 a h else panic! "invalid UTF-8 string"
 
-/--
-Returns the sequence of bytes in a character's UTF-8 encoding.
--/
-def utf8EncodeCharFast (c : Char) : List UInt8 :=
-  let v := c.val
-  if v ≤ 0x7f then
-    [v.toUInt8]
-  else if v ≤ 0x7ff then
-    [(v >>>  6).toUInt8 &&& 0x1f ||| 0xc0,
-              v.toUInt8 &&& 0x3f ||| 0x80]
-  else if v ≤ 0xffff then
-    [(v >>> 12).toUInt8 &&& 0x0f ||| 0xe0,
-     (v >>>  6).toUInt8 &&& 0x3f ||| 0x80,
-              v.toUInt8 &&& 0x3f ||| 0x80]
-  else
-    [(v >>> 18).toUInt8 &&& 0x07 ||| 0xf0,
-     (v >>> 12).toUInt8 &&& 0x3f ||| 0x80,
-     (v >>>  6).toUInt8 &&& 0x3f ||| 0x80,
-              v.toUInt8 &&& 0x3f ||| 0x80]
-
-private theorem Nat.add_two_pow_eq_or_of_lt {b : Nat} (i : Nat) (b_lt : b < 2 ^ i) (a : Nat) :
-    b + 2 ^ i * a = b ||| 2 ^ i * a := by
-  rw [Nat.add_comm, Nat.or_comm, Nat.two_pow_add_eq_or_of_lt b_lt]
-
-@[csimp]
-theorem utf8EncodeChar_eq_utf8EncodeCharFast : @utf8EncodeChar = @utf8EncodeCharFast := by
-  funext c
-  simp only [utf8EncodeChar, utf8EncodeCharFast, UInt8.ofNat_uInt32ToNat, UInt8.ofNat_add,
-    UInt8.reduceOfNat, UInt32.le_iff_toNat_le, UInt32.reduceToNat]
-  split
-  · rfl
-  · split
-    · simp only [List.cons.injEq, ← UInt8.toNat_inj, UInt8.toNat_add, UInt8.toNat_ofNat',
-        Nat.reducePow, UInt8.reduceToNat, Nat.mod_add_mod, UInt8.toNat_or, UInt8.toNat_and,
-        UInt32.toNat_toUInt8, UInt32.toNat_shiftRight, UInt32.reduceToNat, Nat.reduceMod, and_true]
-      refine ⟨?_, ?_⟩
-      · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 5 (by omega) 6,
-          Nat.and_two_pow_sub_one_eq_mod _ 5, Nat.shiftRight_eq_div_pow,
-          Nat.mod_eq_of_lt (b := 256) (by omega)]
-      · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 6 (by omega) 2,
-          Nat.and_two_pow_sub_one_eq_mod _ 6, Nat.mod_mod_of_dvd _ (by decide)]
-    · split
-      · simp only [List.cons.injEq, ← UInt8.toNat_inj, UInt8.toNat_add, UInt8.toNat_ofNat',
-          Nat.reducePow, UInt8.reduceToNat, Nat.mod_add_mod, UInt8.toNat_or, UInt8.toNat_and,
-          UInt32.toNat_toUInt8, UInt32.toNat_shiftRight, UInt32.reduceToNat, Nat.reduceMod, and_true]
-        refine ⟨?_, ?_, ?_⟩
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 4 (by omega) 14,
-            Nat.and_two_pow_sub_one_eq_mod _ 4, Nat.shiftRight_eq_div_pow,
-            Nat.mod_eq_of_lt (b := 256) (by omega)]
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 6 (by omega) 2,
-            Nat.and_two_pow_sub_one_eq_mod _ 6, Nat.shiftRight_eq_div_pow,
-            Nat.mod_mod_of_dvd (c.val.toNat / 2 ^ 6) (by decide)]
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 6 (by omega) 2,
-            Nat.and_two_pow_sub_one_eq_mod _ 6, Nat.mod_mod_of_dvd c.val.toNat (by decide)]
-      · simp only [List.cons.injEq, ← UInt8.toNat_inj, UInt8.toNat_add, UInt8.toNat_ofNat',
-          Nat.reducePow, UInt8.reduceToNat, Nat.mod_add_mod, UInt8.toNat_or, UInt8.toNat_and,
-          UInt32.toNat_toUInt8, UInt32.toNat_shiftRight, UInt32.reduceToNat, Nat.reduceMod, and_true]
-        refine ⟨?_, ?_, ?_, ?_⟩
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 3 (by omega) 30,
-            Nat.and_two_pow_sub_one_eq_mod _ 3, Nat.shiftRight_eq_div_pow,
-            Nat.mod_mod_of_dvd _ (by decide)]
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 6 (by omega) 2,
-            Nat.and_two_pow_sub_one_eq_mod _ 6, Nat.shiftRight_eq_div_pow,
-            Nat.mod_mod_of_dvd (c.val.toNat / 2 ^ 12) (by decide)]
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 6 (by omega) 2,
-            Nat.and_two_pow_sub_one_eq_mod _ 6, Nat.shiftRight_eq_div_pow,
-            Nat.mod_mod_of_dvd (c.val.toNat / 2 ^ 6) (by decide)]
-        · rw [Nat.mod_eq_of_lt (by omega), Nat.add_two_pow_eq_or_of_lt 6 (by omega) 2,
-            Nat.and_two_pow_sub_one_eq_mod _ 6, Nat.mod_mod_of_dvd c.val.toNat (by decide)]
-
-@[simp] theorem length_utf8EncodeChar (c : Char) : (utf8EncodeChar c).length = c.utf8Size := by
-  simp [Char.utf8Size, utf8EncodeChar_eq_utf8EncodeCharFast, utf8EncodeCharFast]
-  cases Decidable.em (c.val ≤ 0x7f) <;> simp [*]
-  cases Decidable.em (c.val ≤ 0x7ff) <;> simp [*]
-  cases Decidable.em (c.val ≤ 0xffff) <;> simp [*]
-
 /--
 Encodes a string in UTF-8 as an array of bytes.
 -/
 @[extern "lean_string_to_utf8"]
 def toUTF8 (a : @& String) : ByteArray :=
-  ⟨⟨a.data.flatMap utf8EncodeChar⟩⟩
+  a.bytes
 
 @[simp] theorem size_toUTF8 (s : String) : s.toUTF8.size = s.utf8ByteSize := by
-  simp [toUTF8, ByteArray.size, Array.size, utf8ByteSize, List.flatMap]
-  induction s.data <;> simp [List.map, utf8ByteSize.go, Nat.add_comm, *]
+  rfl
 
 /--
 Accesses the indicated byte in the UTF-8 encoding of a string.

src/Init/Data/Sum/Basic.lean

@@ -142,9 +142,9 @@ inductive LiftRel (r : α → γ → Prop) (s : β → δ → Prop) : α ⊕ β
 @[simp, grind =] theorem liftRel_inl_inl : LiftRel r s (inl a) (inl c) ↔ r a c :=
   ⟨fun h => by cases h; assumption, LiftRel.inl⟩
 
-@[simp, grind] theorem not_liftRel_inl_inr : ¬LiftRel r s (inl a) (inr d) := nofun
+@[simp, grind ←] theorem not_liftRel_inl_inr : ¬LiftRel r s (inl a) (inr d) := nofun
 
-@[simp, grind] theorem not_liftRel_inr_inl : ¬LiftRel r s (inr b) (inl c) := nofun
+@[simp, grind ←] theorem not_liftRel_inr_inl : ¬LiftRel r s (inr b) (inl c) := nofun
 
 @[simp, grind =] theorem liftRel_inr_inr : LiftRel r s (inr b) (inr d) ↔ s b d :=
   ⟨fun h => by cases h; assumption, LiftRel.inr⟩
@@ -179,7 +179,7 @@ attribute [simp] Lex.sep
 @[simp, grind =] theorem lex_inr_inr : Lex r s (inr b₁) (inr b₂) ↔ s b₁ b₂ :=
   ⟨fun h => by cases h; assumption, Lex.inr⟩
 
-@[simp, grind] theorem lex_inr_inl : ¬Lex r s (inr b) (inl a) := nofun
+@[simp, grind ←] theorem lex_inr_inl : ¬Lex r s (inr b) (inl a) := nofun
 
 instance instDecidableRelSumLex [DecidableRel r] [DecidableRel s] : DecidableRel (Lex r s)
   | inl _, inl _ => decidable_of_iff' _ lex_inl_inl

src/Init/Data/UInt/Bitwise.lean

@@ -1327,3 +1327,14 @@ theorem UInt64.right_le_or {a b : UInt64} : b ≤ a ||| b := by
   simpa [UInt64.le_iff_toNat_le] using Nat.right_le_or
 theorem USize.right_le_or {a b : USize} : b ≤ a ||| b := by
   simpa [USize.le_iff_toNat_le] using Nat.right_le_or
+
+theorem UInt8.and_lt_add_one {b c : UInt8} (h : c ≠ -1) : b &&& c < c + 1 :=
+  UInt8.lt_of_le_of_lt UInt8.and_le_right (UInt8.lt_add_one h)
+theorem UInt16.and_lt_add_one {b c : UInt16} (h : c ≠ -1) : b &&& c < c + 1 :=
+  UInt16.lt_of_le_of_lt UInt16.and_le_right (UInt16.lt_add_one h)
+theorem UInt32.and_lt_add_one {b c : UInt32} (h : c ≠ -1) : b &&& c < c + 1 :=
+  UInt32.lt_of_le_of_lt UInt32.and_le_right (UInt32.lt_add_one h)
+theorem UInt64.and_lt_add_one {b c : UInt64} (h : c ≠ -1) : b &&& c < c + 1 :=
+  UInt64.lt_of_le_of_lt UInt64.and_le_right (UInt64.lt_add_one h)
+theorem USize.and_lt_add_one {b c : USize} (h : c ≠ -1) : b &&& c < c + 1 :=
+  USize.lt_of_le_of_lt USize.and_le_right (USize.lt_add_one h)

src/Init/Data/UInt/Lemmas.lean

@@ -311,6 +311,13 @@ theorem UInt64.ofNat_mod_size : ofNat (x % 2 ^ 64) = ofNat x := by
 theorem USize.ofNat_mod_size : ofNat (x % 2 ^ System.Platform.numBits) = ofNat x := by
   simp [ofNat, BitVec.ofNat, Fin.ofNat]
 
+theorem UInt8.ofNat_size : ofNat size = 0 := by decide
+theorem UInt16.ofNat_size : ofNat size = 0 := by decide
+theorem UInt32.ofNat_size : ofNat size = 0 := by decide
+theorem UInt64.ofNat_size : ofNat size = 0 := by decide
+theorem USize.ofNat_size : ofNat size = 0 := by
+  simp [ofNat, BitVec.ofNat, USize.eq_iff_toBitVec_eq]
+
 theorem UInt8.lt_ofNat_iff {n : UInt8} {m : Nat} (h : m < size) : n < ofNat m ↔ n.toNat < m := by
   rw [lt_iff_toNat_lt, toNat_ofNat_of_lt' h]
 theorem UInt8.ofNat_lt_iff {n : UInt8} {m : Nat} (h : m < size) : ofNat m < n ↔ m < n.toNat := by
@@ -3156,3 +3163,15 @@ protected theorem USize.sub_lt {a b : USize} (hb : 0 < b) (hab : b ≤ a) : a -
   rw [lt_iff_toNat_lt, USize.toNat_sub_of_le _ _ hab]
   refine Nat.sub_lt ?_ (USize.lt_iff_toNat_lt.1 hb)
   exact USize.lt_iff_toNat_lt.1 (USize.lt_of_lt_of_le hb hab)
+
+theorem UInt8.lt_add_one {c : UInt8} (h : c ≠ -1) : c < c + 1 :=
+  UInt8.lt_iff_toBitVec_lt.2 (BitVec.lt_add_one (by simpa [← UInt8.toBitVec_inj] using h))
+theorem UInt16.lt_add_one {c : UInt16} (h : c ≠ -1) : c < c + 1 :=
+  UInt16.lt_iff_toBitVec_lt.2 (BitVec.lt_add_one (by simpa [← UInt16.toBitVec_inj] using h))
+theorem UInt32.lt_add_one {c : UInt32} (h : c ≠ -1) : c < c + 1 :=
+  UInt32.lt_iff_toBitVec_lt.2 (BitVec.lt_add_one (by simpa [← UInt32.toBitVec_inj] using h))
+theorem UInt64.lt_add_one {c : UInt64} (h : c ≠ -1) : c < c + 1 :=
+  UInt64.lt_iff_toBitVec_lt.2 (BitVec.lt_add_one (by simpa [← UInt64.toBitVec_inj] using h))
+theorem USize.lt_add_one {c : USize} (h : c ≠ -1) : c < c + 1 :=
+  USize.lt_iff_toBitVec_lt.2 (BitVec.lt_add_one
+    (by simpa [← USize.toBitVec_inj, BitVec.neg_one_eq_allOnes] using h))

src/Init/Data/Vector/Attach.lean

@@ -193,14 +193,14 @@ theorem attachWith_map_subtype_val {p : α → Prop} {xs : Vector α n} (H : ∀
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind ←]
 theorem mem_attach (xs : Vector α n) : ∀ 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, grind]
+@[simp, grind =]
 theorem mem_attachWith {xs : Vector α n} {q : α → Prop} (H) (x : {x // q x}) :
     x ∈ xs.attachWith q H ↔ x.1 ∈ xs := by
   rcases xs with ⟨xs, rfl⟩
@@ -211,12 +211,13 @@ theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {xs : Vector α n} {H
     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 : Vector α n} {H} {a} (h : a ∈ xs) :
     f a (H a h) ∈ pmap f xs H := by
   rw [mem_pmap]
   exact ⟨a, h, rfl⟩
 
+grind_pattern mem_pmap_of_mem => _ ∈ pmap f xs H, a ∈ xs
+
 theorem pmap_eq_self {xs : Vector α n} {p : α → Prop} {hp : ∀ (a : α), a ∈ xs → p a}
     {f : (a : α) → p a → α} : xs.pmap f hp = xs ↔ ∀ a (h : a ∈ xs), f a (hp a h) = a := by
   rcases xs with ⟨xs, rfl⟩

src/Init/Data/Vector/Basic.lean

@@ -36,7 +36,7 @@ structure Vector (α : Type u) (n : Nat) where
   size_toArray : toArray.size = n
 deriving Repr, DecidableEq
 
-attribute [simp, grind] Vector.size_toArray
+attribute [simp, grind =] Vector.size_toArray
 
 /--
 Converts an array to a vector. The resulting vector's size is the array's size.

src/Init/Data/Vector/Find.lean

@@ -32,11 +32,11 @@ open Nat
 
 /-! ### findSome? -/
 
-@[simp, grind] theorem findSome?_empty : (#v[] : Vector α 0).findSome? f = none := rfl
-@[simp, grind] theorem findSome?_push {xs : Vector α n} : (xs.push a).findSome? f = (xs.findSome? f).or (f a) := by
+@[simp, grind =] theorem findSome?_empty : (#v[] : Vector α 0).findSome? f = none := rfl
+@[simp, grind =] theorem findSome?_push {xs : Vector α n} : (xs.push a).findSome? f = (xs.findSome? f).or (f a) := by
   cases xs; simp
 
-@[grind]
+@[grind =]
 theorem findSome?_singleton {a : α} {f : α → Option β} : #v[a].findSome? f = f a := by
   simp
 
@@ -228,11 +228,12 @@ theorem mem_of_find?_eq_some {xs : Vector α n} (h : find? p xs = some a) : a
   simp at h
   simpa using Array.mem_of_find?_eq_some h
 
-@[grind]
 theorem get_find?_mem {xs : Vector α n} (h) : (xs.find? p).get h ∈ xs := by
   cases xs
   simp [Array.get_find?_mem]
 
+grind_pattern get_find?_mem => (xs.find? p).get h
+
 @[simp, grind =] theorem find?_map {f : β → α} {xs : Vector β n} :
     find? p (xs.map f) = (xs.find? (p ∘ f)).map f := by
   cases xs; simp

src/Init/Data/Vector/Lemmas.lean

@@ -266,12 +266,12 @@ theorem toArray_mk {xs : Array α} (h : xs.size = n) : (Vector.mk xs h).toArray
 
 /-! ### toArray lemmas -/
 
-@[simp, grind] theorem getElem_toArray {α n} {xs : Vector α n} {i : Nat} (h : i < xs.toArray.size) :
+@[simp, grind =] theorem getElem_toArray {α n} {xs : Vector α n} {i : Nat} (h : i < xs.toArray.size) :
     xs.toArray[i] = xs[i]'(by simpa using h) := by
   cases xs
   simp
 
-@[simp, grind] theorem getElem?_toArray {α n} {xs : Vector α n} {i : Nat} :
+@[simp, grind =] theorem getElem?_toArray {α n} {xs : Vector α n} {i : Nat} :
     xs.toArray[i]? = xs[i]? := by
   cases xs
   simp
@@ -280,45 +280,45 @@ theorem toArray_mk {xs : Array α} (h : xs.size = n) : (Vector.mk xs h).toArray
     (xs ++ ys).toArray = xs.toArray ++ ys.toArray := rfl
 
 set_option linter.indexVariables false in
-@[simp, grind] theorem toArray_drop {xs : Vector α n} {i} :
+@[simp, grind =] theorem toArray_drop {xs : Vector α n} {i} :
     (xs.drop i).toArray = xs.toArray.extract i n := by
   simp [drop]
 
 @[simp, grind =] theorem toArray_empty : (#v[] : Vector α 0).toArray = #[] := rfl
 
-@[simp, grind] theorem toArray_emptyWithCapacity {cap} :
+@[simp, grind =] theorem toArray_emptyWithCapacity {cap} :
     (Vector.emptyWithCapacity (α := α) cap).toArray = Array.emptyWithCapacity cap := rfl
 
 @[deprecated toArray_emptyWithCapacity (since := "2025-03-12")]
 abbrev toArray_mkEmpty := @toArray_emptyWithCapacity
 
-@[simp, grind] theorem toArray_eraseIdx {xs : Vector α n} {i} (h) :
+@[simp, grind =] theorem toArray_eraseIdx {xs : Vector α n} {i} (h) :
     (xs.eraseIdx i h).toArray = xs.toArray.eraseIdx i (by simp [h]) := rfl
 
-@[simp, grind] theorem toArray_eraseIdx! {xs : Vector α n} {i} (hi : i < n) :
+@[simp, grind =] theorem toArray_eraseIdx! {xs : Vector α n} {i} (hi : i < n) :
     (xs.eraseIdx! i).toArray = xs.toArray.eraseIdx! i := by
   cases xs; simp_all [Array.eraseIdx!]
 
-@[simp, grind] theorem toArray_insertIdx {xs : Vector α n} {i x} (h) :
+@[simp, grind =] theorem toArray_insertIdx {xs : Vector α n} {i x} (h) :
     (xs.insertIdx i x h).toArray = xs.toArray.insertIdx i x (by simp [h]) := rfl
 
-@[simp, grind] theorem toArray_insertIdx! {xs : Vector α n} {i x} (hi : i ≤ n) :
+@[simp, grind =] theorem toArray_insertIdx! {xs : Vector α n} {i x} (hi : i ≤ n) :
     (xs.insertIdx! i x).toArray = xs.toArray.insertIdx! i x := by
   cases xs; simp_all [Array.insertIdx!]
 
-@[simp, grind] theorem toArray_cast {xs : Vector α n} (h : n = m) :
+@[simp, grind =] theorem toArray_cast {xs : Vector α n} (h : n = m) :
     (xs.cast h).toArray = xs.toArray := rfl
 
-@[simp, grind] theorem toArray_extract {xs : Vector α n} {start stop} :
+@[simp, grind =] theorem toArray_extract {xs : Vector α n} {start stop} :
     (xs.extract start stop).toArray = xs.toArray.extract start stop := rfl
 
-@[simp, grind] theorem toArray_map {f : α → β} {xs : Vector α n} :
+@[simp, grind =] theorem toArray_map {f : α → β} {xs : Vector α n} :
     (xs.map f).toArray = xs.toArray.map f := rfl
 
-@[simp, grind] theorem toArray_mapIdx {f : Nat → α → β} {xs : Vector α n} :
+@[simp, grind =] theorem toArray_mapIdx {f : Nat → α → β} {xs : Vector α n} :
     (xs.mapIdx f).toArray = xs.toArray.mapIdx f := rfl
 
-@[simp, grind] theorem toArray_mapFinIdx {f : (i : Nat) → α → (h : i < n) → β} {xs : Vector α n} :
+@[simp, grind =] theorem toArray_mapFinIdx {f : (i : Nat) → α → (h : i < n) → β} {xs : Vector α n} :
     (xs.mapFinIdx f).toArray =
       xs.toArray.mapFinIdx (fun i a h => f i a (by simpa [xs.size_toArray] using h)) :=
   rfl
@@ -336,42 +336,42 @@ private theorem toArray_mapM_go [Monad m] [LawfulMonad m] {f : α → m β} {xs
     rfl
   · simp
 
-@[simp, grind] theorem toArray_mapM [Monad m] [LawfulMonad m] {f : α → m β} {xs : Vector α n} :
+@[simp, grind =] theorem toArray_mapM [Monad m] [LawfulMonad m] {f : α → m β} {xs : Vector α n} :
     toArray <$> xs.mapM f = xs.toArray.mapM f := by
   rcases xs with ⟨xs, rfl⟩
   unfold mapM
   rw [toArray_mapM_go]
   rfl
 
-@[simp, grind] theorem toArray_ofFn {f : Fin n → α} : (Vector.ofFn f).toArray = Array.ofFn f := rfl
+@[simp, grind =] theorem toArray_ofFn {f : Fin n → α} : (Vector.ofFn f).toArray = Array.ofFn f := rfl
 
-@[simp, grind] theorem toArray_pop {xs : Vector α n} : xs.pop.toArray = xs.toArray.pop := rfl
+@[simp, grind =] theorem toArray_pop {xs : Vector α n} : xs.pop.toArray = xs.toArray.pop := rfl
 
-@[simp, grind] theorem toArray_push {xs : Vector α n} {x} : (xs.push x).toArray = xs.toArray.push x := rfl
+@[simp, grind =] theorem toArray_push {xs : Vector α n} {x} : (xs.push x).toArray = xs.toArray.push x := rfl
 
-@[simp, grind] theorem toArray_beq_toArray [BEq α] {xs : Vector α n} {ys : Vector α n} :
+@[simp, grind =] theorem toArray_beq_toArray [BEq α] {xs : Vector α n} {ys : Vector α n} :
     (xs.toArray == ys.toArray) = (xs == ys) := by
   simp [instBEq, isEqv, Array.instBEq, Array.isEqv, xs.2, ys.2]
 
-@[simp, grind] theorem toArray_range : (Vector.range n).toArray = Array.range n := rfl
+@[simp, grind =] theorem toArray_range : (Vector.range n).toArray = Array.range n := rfl
 
-@[simp, grind] theorem toArray_reverse (xs : Vector α n) : xs.reverse.toArray = xs.toArray.reverse := rfl
+@[simp, grind =] theorem toArray_reverse (xs : Vector α n) : xs.reverse.toArray = xs.toArray.reverse := rfl
 
-@[simp, grind] theorem toArray_set {xs : Vector α n} {i x} (h) :
+@[simp, grind =] theorem toArray_set {xs : Vector α n} {i x} (h) :
     (xs.set i x).toArray = xs.toArray.set i x (by simpa using h):= rfl
 
-@[simp, grind] theorem toArray_set! {xs : Vector α n} {i x} :
+@[simp, grind =] theorem toArray_set! {xs : Vector α n} {i x} :
     (xs.set! i x).toArray = xs.toArray.set! i x := rfl
 
-@[simp, grind] theorem toArray_setIfInBounds {xs : Vector α n} {i x} :
+@[simp, grind =] theorem toArray_setIfInBounds {xs : Vector α n} {i x} :
     (xs.setIfInBounds i x).toArray = xs.toArray.setIfInBounds i x := rfl
 
-@[simp, grind] theorem toArray_singleton {x : α} : (Vector.singleton x).toArray = #[x] := rfl
+@[simp, grind =] theorem toArray_singleton {x : α} : (Vector.singleton x).toArray = #[x] := rfl
 
-@[simp, grind] theorem toArray_swap {xs : Vector α n} {i j} (hi hj) : (xs.swap i j).toArray =
+@[simp, grind =] theorem toArray_swap {xs : Vector α n} {i j} (hi hj) : (xs.swap i j).toArray =
     xs.toArray.swap i j (by simp [hj]) (by simp [hi]) := rfl
 
-@[simp, grind] theorem toArray_swapIfInBounds {xs : Vector α n} {i j} :
+@[simp, grind =] theorem toArray_swapIfInBounds {xs : Vector α n} {i j} :
     (xs.swapIfInBounds i j).toArray = xs.toArray.swapIfInBounds i j := rfl
 
 theorem toArray_swapAt {xs : Vector α n} {i x} (h) :
@@ -383,98 +383,98 @@ theorem toArray_swapAt! {xs : Vector α n} {i x} :
     ((xs.swapAt! i x).fst, (xs.swapAt! i x).snd.toArray) =
       ((xs.toArray.swapAt! i x).fst, (xs.toArray.swapAt! i x).snd) := rfl
 
-@[simp, grind] theorem toArray_take {xs : Vector α n} {i} : (xs.take i).toArray = xs.toArray.take i := rfl
+@[simp, grind =] theorem toArray_take {xs : Vector α n} {i} : (xs.take i).toArray = xs.toArray.take i := rfl
 
-@[simp, grind] theorem toArray_zipIdx {xs : Vector α n} (k : Nat := 0) :
+@[simp, grind =] theorem toArray_zipIdx {xs : Vector α n} (k : Nat := 0) :
     (xs.zipIdx k).toArray = xs.toArray.zipIdx k := rfl
 
-@[simp, grind] theorem toArray_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} :
+@[simp, grind =] theorem toArray_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} :
     (Vector.zipWith f as bs).toArray = Array.zipWith f as.toArray bs.toArray := rfl
 
-@[simp, grind] theorem anyM_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
+@[simp, grind =] theorem anyM_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
     xs.toArray.anyM p = xs.anyM p := by
   cases xs
   simp
 
-@[simp, grind] theorem allM_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
+@[simp, grind =] theorem allM_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
     xs.toArray.allM p = xs.allM p := by
   cases xs
   simp
 
-@[simp, grind] theorem any_toArray {p : α → Bool} {xs : Vector α n} :
+@[simp, grind =] theorem any_toArray {p : α → Bool} {xs : Vector α n} :
     xs.toArray.any p = xs.any p := by
   cases xs
   simp
 
-@[simp, grind] theorem all_toArray {p : α → Bool} {xs : Vector α n} :
+@[simp, grind =] theorem all_toArray {p : α → Bool} {xs : Vector α n} :
     xs.toArray.all p = xs.all p := by
   cases xs
   simp
 
-@[simp, grind] theorem countP_toArray {p : α → Bool} {xs : Vector α n} :
+@[simp, grind =] theorem countP_toArray {p : α → Bool} {xs : Vector α n} :
     xs.toArray.countP p = xs.countP p := by
   cases xs
   simp
 
-@[simp, grind] theorem count_toArray [BEq α] {a : α} {xs : Vector α n} :
+@[simp, grind =] theorem count_toArray [BEq α] {a : α} {xs : Vector α n} :
     xs.toArray.count a = xs.count a := by
   cases xs
   simp
 
-@[simp, grind] theorem replace_toArray [BEq α] {xs : Vector α n} {a b} :
+@[simp, grind =] theorem replace_toArray [BEq α] {xs : Vector α n} {a b} :
     xs.toArray.replace a b = (xs.replace a b).toArray := rfl
 
-@[simp, grind] theorem find?_toArray {p : α → Bool} {xs : Vector α n} :
+@[simp, grind =] theorem find?_toArray {p : α → Bool} {xs : Vector α n} :
     xs.toArray.find? p = xs.find? p := by
   cases xs
   simp
 
-@[simp, grind] theorem findSome?_toArray {f : α → Option β} {xs : Vector α n} :
+@[simp, grind =] theorem findSome?_toArray {f : α → Option β} {xs : Vector α n} :
     xs.toArray.findSome? f = xs.findSome? f := by
   cases xs
   simp
 
-@[simp, grind] theorem findRev?_toArray {p : α → Bool} {xs : Vector α n} :
+@[simp, grind =] theorem findRev?_toArray {p : α → Bool} {xs : Vector α n} :
     xs.toArray.findRev? p = xs.findRev? p := by
   cases xs
   simp
 
-@[simp, grind] theorem findSomeRev?_toArray {f : α → Option β} {xs : Vector α n} :
+@[simp, grind =] theorem findSomeRev?_toArray {f : α → Option β} {xs : Vector α n} :
     xs.toArray.findSomeRev? f = xs.findSomeRev? f := by
   cases xs
   simp
 
-@[simp, grind] theorem findM?_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
+@[simp, grind =] theorem findM?_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
     xs.toArray.findM? p = xs.findM? p := by
   cases xs
   simp
 
-@[simp, grind] theorem findSomeM?_toArray [Monad m] {f : α → m (Option β)} {xs : Vector α n} :
+@[simp, grind =] theorem findSomeM?_toArray [Monad m] {f : α → m (Option β)} {xs : Vector α n} :
     xs.toArray.findSomeM? f = xs.findSomeM? f := by
   cases xs
   simp
 
-@[simp, grind] theorem findRevM?_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
+@[simp, grind =] theorem findRevM?_toArray [Monad m] {p : α → m Bool} {xs : Vector α n} :
     xs.toArray.findRevM? p = xs.findRevM? p := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem findSomeRevM?_toArray [Monad m] {f : α → m (Option β)} {xs : Vector α n} :
+@[simp, grind =] theorem findSomeRevM?_toArray [Monad m] {f : α → m (Option β)} {xs : Vector α n} :
     xs.toArray.findSomeRevM? f = xs.findSomeRevM? f := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem finIdxOf?_toArray [BEq α] {a : α} {xs : Vector α n} :
+@[simp, grind =] theorem finIdxOf?_toArray [BEq α] {a : α} {xs : Vector α n} :
     xs.toArray.finIdxOf? a = (xs.finIdxOf? a).map (Fin.cast xs.size_toArray.symm) := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem findFinIdx?_toArray {p : α → Bool} {xs : Vector α n} :
+@[simp, grind =] theorem findFinIdx?_toArray {p : α → Bool} {xs : Vector α n} :
     xs.toArray.findFinIdx? p = (xs.findFinIdx? p).map (Fin.cast xs.size_toArray.symm) := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem toArray_replicate : (replicate n a).toArray = Array.replicate n a := rfl
+@[simp, grind =] theorem toArray_replicate : (replicate n a).toArray = Array.replicate n a := rfl
 
 @[deprecated toArray_replicate (since := "2025-03-18")]
 abbrev toArray_mkVector := @toArray_replicate
@@ -503,13 +503,13 @@ protected theorem ext {xs ys : Vector α n} (h : (i : Nat) → (_ : i < n) → x
 
 /-! ### toList -/
 
-@[simp, grind] theorem length_toList {xs : Vector α n} : xs.toList.length = n := by
+@[simp, grind =] theorem length_toList {xs : Vector α n} : xs.toList.length = n := by
   rcases xs with ⟨xs, rfl⟩
   simp [toList]
 
 @[grind =_] theorem toList_toArray {xs : Vector α n} : xs.toArray.toList = xs.toList := rfl
 
-@[simp, grind] theorem toList_mk : (Vector.mk xs h).toList = xs.toList := rfl
+@[simp, grind =] theorem toList_mk : (Vector.mk xs h).toList = xs.toList := rfl
 
 @[simp] theorem getElem_toList {xs : Vector α n} {i : Nat} (h : i < xs.toList.length) :
     xs.toList[i] = xs[i]'(by simpa using h) := by
@@ -784,12 +784,12 @@ theorem singleton_inj : #v[a] = #v[b] ↔ a = b := by
 
 /-! ### replicate -/
 
-@[simp, grind] theorem replicate_zero : replicate 0 a = #v[] := rfl
+@[simp, grind =] theorem replicate_zero : replicate 0 a = #v[] := rfl
 
 @[deprecated replicate_zero (since := "2025-03-18")]
 abbrev replicate_mkVector := @replicate_zero
 
-@[grind]
+@[grind =]
 theorem replicate_succ : replicate (n + 1) a = (replicate n a).push a := by
   simp [replicate, Array.replicate_succ]
 
@@ -895,26 +895,35 @@ theorem getElem?_push_size {xs : Vector α n} {x : α} : (xs.push x)[n]? = some
 theorem getElem_singleton {a : α} (h : i < 1) : #v[a][i] = a := by
   simp
 
-@[grind]
+@[grind =]
 theorem getElem?_singleton {a : α} {i : Nat} : #v[a][i]? = if i = 0 then some a else none := by
   simp [List.getElem?_singleton]
 
 /-! ### mem -/
 
-@[simp, grind] theorem getElem_mem {xs : Vector α n} {i : Nat} (h : i < n) : xs[i] ∈ xs := by
+@[simp] theorem getElem_mem {xs : Vector α n} {i : Nat} (h : i < n) : xs[i] ∈ xs := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
+grind_pattern getElem_mem => xs[i] ∈ xs
+
 theorem not_mem_empty (a : α) : ¬ a ∈ #v[] := nofun
 
-@[simp] theorem mem_push {xs : Vector α n} {x y : α} : x ∈ xs.push y ↔ x ∈ xs ∨ x = y := by
+@[simp, grind =] theorem mem_push {xs : Vector α n} {x y : α} : x ∈ xs.push y ↔ x ∈ xs ∨ x = y := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[grind] theorem mem_or_eq_of_mem_push {a b : α} {xs : Vector α n} :
+theorem mem_or_eq_of_mem_push {a b : α} {xs : Vector α n} :
     a ∈ xs.push b → a ∈ xs ∨ a = b := Vector.mem_push.mp
 
-@[grind] theorem mem_push_self {xs : Vector α n} {x : α} : x ∈ xs.push x :=
+-- This pattern may be excessively general:
+-- it fires anytime we ae thinking about membership of vectors,
+-- and constructing a list via `push`, even if the elements are unrelated.
+-- Nevertheless in practice it is quite helpful!
+grind_pattern mem_or_eq_of_mem_push => xs.push b, a ∈ xs
+
+
+theorem mem_push_self {xs : Vector α n} {x : α} : x ∈ xs.push x :=
   mem_push.2 (Or.inr rfl)
 
 theorem eq_push_append_of_mem {xs : Vector α n} {x : α} (h : x ∈ xs) :
@@ -926,7 +935,7 @@ theorem eq_push_append_of_mem {xs : Vector α n} {x : α} (h : x ∈ xs) :
   obtain rfl := h
   exact ⟨_, _, as.toVector, bs.toVector, by simp, by simp, by simpa using w⟩
 
-@[grind] theorem mem_push_of_mem {xs : Vector α n} {x : α} (y : α) (h : x ∈ xs) : x ∈ xs.push y :=
+theorem mem_push_of_mem {xs : Vector α n} {x : α} (y : α) (h : x ∈ xs) : x ∈ xs.push y :=
   mem_push.2 (Or.inl h)
 
 theorem exists_mem_of_size_pos {xs : Vector α n} (h : 0 < n) : ∃ x, x ∈ xs := by
@@ -1213,9 +1222,9 @@ theorem contains_iff [BEq α] [LawfulBEq α] {a : α} {as : Vector α n} :
 instance [BEq α] [LawfulBEq α] (a : α) (as : Vector α n) : Decidable (a ∈ as) :=
   decidable_of_decidable_of_iff contains_iff
 
-@[grind] theorem contains_empty [BEq α] : (#v[] : Vector α 0).contains a = false := by simp
+@[grind =] theorem contains_empty [BEq α] : (#v[] : Vector α 0).contains a = false := by simp
 
-@[simp, grind] theorem contains_eq_mem [BEq α] [LawfulBEq α] {a : α} {as : Vector α n} :
+@[simp, grind =] theorem contains_eq_mem [BEq α] [LawfulBEq α] {a : α} {as : Vector α n} :
     as.contains a = decide (a ∈ as) := by
   rw [Bool.eq_iff_iff, contains_iff, decide_eq_true_iff]
 
@@ -1236,7 +1245,7 @@ instance [BEq α] [LawfulBEq α] (a : α) (as : Vector α n) : Decidable (a ∈
 
 /-! ### set -/
 
-@[grind] theorem getElem_set {xs : Vector α n} {i : Nat} {x : α} (hi : i < n) {j : Nat} (hj : j < n) :
+@[grind =] theorem getElem_set {xs : Vector α n} {i : Nat} {x : α} (hi : i < n) {j : Nat} (hj : j < n) :
     (xs.set i x hi)[j] = if i = j then x else xs[j] := by
   cases xs
   split <;> simp_all
@@ -1249,7 +1258,7 @@ instance [BEq α] [LawfulBEq α] (a : α) (as : Vector α n) : Decidable (a ∈
 @[simp] theorem getElem_set_ne {xs : Vector α n} {x : α} (hi : i < n) (hj : j < n) (h : i ≠ j) :
     (xs.set i x hi)[j] = xs[j] := by simp [getElem_set, h]
 
-@[grind] theorem getElem?_set {xs : Vector α n} {x : α} (hi : i < n) :
+@[grind =] theorem getElem?_set {xs : Vector α n} {x : α} (hi : i < n) :
     (xs.set i x hi)[j]? = if i = j then some x else xs[j]? := by
   cases xs
   split <;> simp_all
@@ -1294,10 +1303,10 @@ grind_pattern mem_or_eq_of_mem_set => a ∈ xs.set i b
 
 /-! ### setIfInBounds -/
 
-@[simp, grind] theorem setIfInBounds_empty {i : Nat} {a : α} :
+@[simp, grind =] theorem setIfInBounds_empty {i : Nat} {a : α} :
     #v[].setIfInBounds i a = #v[] := rfl
 
-@[grind] theorem getElem_setIfInBounds {xs : Vector α n} {x : α} (hj : j < n) :
+@[grind =] theorem getElem_setIfInBounds {xs : Vector α n} {x : α} (hj : j < n) :
     (xs.setIfInBounds i x)[j] = if i = j then x else xs[j] := by
   cases xs
   split <;> simp_all
@@ -1310,7 +1319,7 @@ grind_pattern mem_or_eq_of_mem_set => a ∈ xs.set i b
 @[simp] theorem getElem_setIfInBounds_ne {xs : Vector α n} {x : α} (hj : j < n) (h : i ≠ j) :
     (xs.setIfInBounds i x)[j] = xs[j] := by simp [getElem_setIfInBounds, h]
 
-@[grind] theorem getElem?_setIfInBounds {xs : Vector α n} {x : α} :
+@[grind =] theorem getElem?_setIfInBounds {xs : Vector α n} {x : α} :
     (xs.setIfInBounds i x)[j]? = if i = j then if i < n then some x else none else xs[j]? := by
   rcases xs with ⟨xs, rfl⟩
   simp [Array.getElem?_setIfInBounds]
@@ -1347,7 +1356,7 @@ theorem mem_setIfInBounds {xs : Vector α n} {a : α} (hi : i < n) :
 
 /-! ### BEq -/
 
-@[simp, grind] theorem push_beq_push [BEq α] {a b : α} {n : Nat} {xs : Vector α n} {ys : Vector α n} :
+@[simp, grind =] theorem push_beq_push [BEq α] {a b : α} {n : Nat} {xs : Vector α n} {ys : Vector α n} :
     (xs.push a == ys.push b) = (xs == ys && a == b) := by
   cases xs
   cases ys
@@ -1410,16 +1419,16 @@ abbrev mkVector_beq_mkVector := @replicate_beq_replicate
 
 /-! ### back -/
 
-@[grind] theorem back_singleton {a : α} : #v[a].back = a := by simp
+@[grind =] theorem back_singleton {a : α} : #v[a].back = a := by simp
 
-@[grind]
+@[grind =]
 theorem back_eq_getElem [NeZero n] {xs : Vector α n} : xs.back = xs[n - 1]'(by have := NeZero.ne n; omega) := by
   rcases xs with ⟨xs, rfl⟩
   simp [Array.back_eq_getElem]
 
-@[grind] theorem back?_empty : (#v[] : Vector α 0).back? = none := by simp
+@[grind =] theorem back?_empty : (#v[] : Vector α 0).back? = none := by simp
 
-@[grind] theorem back?_eq_getElem? {xs : Vector α n} : xs.back? = xs[n - 1]? := by
+@[grind =] theorem back?_eq_getElem? {xs : Vector α n} : xs.back? = xs[n - 1]? := by
   rcases xs with ⟨xs, rfl⟩
   simp [Array.back?_eq_getElem?]
 
@@ -1430,22 +1439,22 @@ theorem back_eq_getElem [NeZero n] {xs : Vector α n} : xs.back = xs[n - 1]'(by
 /-! ### map -/
 
 -- The argument `f : α → β` is explicit, to facilitate rewriting from right to left.
-@[simp, grind] theorem getElem_map (f : α → β) {xs : Vector α n} (hi : i < n) :
+@[simp, grind =] theorem getElem_map (f : α → β) {xs : Vector α n} (hi : i < n) :
     (xs.map f)[i] = f xs[i] := by
   cases xs
   simp
 
-@[simp, grind] theorem getElem?_map {f : α → β} {xs : Vector α n} {i : Nat}:
+@[simp, grind =] theorem getElem?_map {f : α → β} {xs : Vector α n} {i : Nat}:
     (xs.map f)[i]? = xs[i]?.map f := by
   cases xs
   simp
 
 /-- The empty vector maps to the empty vector. -/
-@[grind]
+@[grind =]
 theorem map_empty {f : α → β} : map f #v[] = #v[] := by
   simp
 
-@[simp, grind] theorem map_push {f : α → β} {as : Vector α n} {x : α} :
+@[simp, grind =] theorem map_push {f : α → β} {as : Vector α n} {x : α} :
     (as.push x).map f = (as.map f).push (f x) := by
   cases as
   simp
@@ -1620,7 +1629,7 @@ theorem append_push {as : Vector α n} {bs : Vector α m} {a : α} :
 
 theorem singleton_eq_toVector_singleton {a : α} : #v[a] = #[a].toVector := rfl
 
-@[simp, grind] theorem mem_append {a : α} {xs : Vector α n} {ys : Vector α m} :
+@[simp, grind =] theorem mem_append {a : α} {xs : Vector α n} {ys : Vector α m} :
     a ∈ xs ++ ys ↔ a ∈ xs ∨ a ∈ ys := by
   cases xs
   cases ys
@@ -1656,16 +1665,16 @@ theorem forall_mem_append {p : α → Prop} {xs : Vector α n} {ys : Vector α m
     (∀ (x) (_ : x ∈ xs ++ ys), p x) ↔ (∀ (x) (_ : x ∈ xs), p x) ∧ (∀ (x) (_ : x ∈ ys), p x) := by
   simp only [mem_append, or_imp, forall_and]
 
-@[simp, grind]
+@[simp, grind =]
 theorem empty_append {xs : Vector α n} : (#v[] : Vector α 0) ++ xs = xs.cast (by omega) := by
   rcases xs with ⟨as, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem append_empty {xs : Vector α n} : xs ++ (#v[] : Vector α 0) = xs := by
   rw [← toArray_inj, toArray_append, Array.append_empty]
 
-@[grind]
+@[grind =]
 theorem getElem_append {xs : Vector α n} {ys : Vector α m} (hi : i < n + m) :
     (xs ++ ys)[i] = if h : i < n then xs[i] else ys[i - n] := by
   rcases xs with ⟨xs, rfl⟩
@@ -1692,7 +1701,7 @@ theorem getElem?_append_right {xs : Vector α n} {ys : Vector α m} (h : n ≤ i
   rcases ys with ⟨ys, rfl⟩
   simp [Array.getElem?_append_right, h]
 
-@[grind]
+@[grind =]
 theorem getElem?_append {xs : Vector α n} {ys : Vector α m} {i : Nat} :
     (xs ++ ys)[i]? = if i < n then xs[i]? else ys[i - n]? := by
   split <;> rename_i h
@@ -1771,7 +1780,7 @@ theorem append_eq_append_iff {ws : Vector α n} {xs : Vector α m} {ys : Vector
       right
       refine ⟨cs.toArray, ha, rfl⟩
 
-@[simp, grind] theorem append_assoc {xs : Vector α n} {ys : Vector α m} {zs : Vector α k} :
+@[simp, grind =] theorem append_assoc {xs : Vector α n} {ys : Vector α m} {zs : Vector α k} :
     (xs ++ ys) ++ zs = (xs ++ (ys ++ zs)).cast (by omega) := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
@@ -1779,14 +1788,14 @@ theorem append_eq_append_iff {ws : Vector α n} {xs : Vector α m} {ys : Vector
   simp [Array.append_assoc]
 
 -- Variant for rewriting the other direction: we can't use `append_assoc` as it has a `cast` on the right-hand side.
-@[grind] theorem append_assoc_symm {xs : Vector α n} {ys : Vector α m} {zs : Vector α k} :
+@[grind =] theorem append_assoc_symm {xs : Vector α n} {ys : Vector α m} {zs : Vector α k} :
     xs ++ (ys ++ zs) = ((xs ++ ys) ++ zs).cast (by omega) := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
   rcases zs with ⟨zs, rfl⟩
   simp [Array.append_assoc]
 
-@[grind] theorem set_append {xs : Vector α n} {ys : Vector α m} {i : Nat} {x : α} (h : i < n + m) :
+@[grind =] theorem set_append {xs : Vector α n} {ys : Vector α m} {i : Nat} {x : α} (h : i < n + m) :
     (xs ++ ys).set i x =
       if h' : i < n then
         xs.set i x ++ ys
@@ -1806,7 +1815,7 @@ theorem append_eq_append_iff {ws : Vector α n} {xs : Vector α m} {ys : Vector
     (xs ++ ys).set i x = xs ++ ys.set (i - n) x := by
   rw [set_append, dif_neg (by omega)]
 
-@[grind] theorem setIfInBounds_append {xs : Vector α n} {ys : Vector α m} {i : Nat} {x : α} :
+@[grind =] theorem setIfInBounds_append {xs : Vector α n} {ys : Vector α m} {i : Nat} {x : α} :
     (xs ++ ys).setIfInBounds i x =
       if i < n then
         xs.setIfInBounds i x ++ ys
@@ -1826,7 +1835,7 @@ theorem append_eq_append_iff {ws : Vector α n} {xs : Vector α m} {ys : Vector
     (xs ++ ys).setIfInBounds i x = xs ++ ys.setIfInBounds (i - n) x := by
   rw [setIfInBounds_append, if_neg (by omega)]
 
-@[simp, grind] theorem map_append {f : α → β} {xs : Vector α n} {ys : Vector α m} :
+@[simp, grind =] theorem map_append {f : α → β} {xs : Vector α n} {ys : Vector α m} :
     map f (xs ++ ys) = map f xs ++ map f ys := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
@@ -1895,7 +1904,7 @@ theorem getElem?_flatten {xss : Vector (Vector β m) n} {i : Nat} :
         none := by
   simp [getElem?_def]
 
-@[simp, grind] theorem flatten_singleton {xs : Vector α n} : #v[xs].flatten = xs.cast (by simp) := by
+@[simp, grind =] theorem flatten_singleton {xs : Vector α n} : #v[xs].flatten = xs.cast (by simp) := by
   simp [flatten]
 
 set_option linter.listVariables false in
@@ -1922,17 +1931,17 @@ theorem forall_mem_flatten {p : α → Prop} {xss : Vector (Vector α n) m} :
   induction xss using vector₂_induction with
   | of xss h₁ h₂ => simp
 
-@[simp, grind] theorem flatten_append {xss₁ : Vector (Vector α n) m₁} {xss₂ : Vector (Vector α n) m₂} :
+@[simp, grind =] theorem flatten_append {xss₁ : Vector (Vector α n) m₁} {xss₂ : Vector (Vector α n) m₂} :
     flatten (xss₁ ++ xss₂) = (flatten xss₁ ++ flatten xss₂).cast (by simp [Nat.add_mul]) := by
   induction xss₁ using vector₂_induction
   induction xss₂ using vector₂_induction
   simp
 
-@[grind] theorem append_flatten {xss₁ : Vector (Vector α n) m₁} {xss₂ : Vector (Vector α n) m₂} :
+@[grind =] theorem append_flatten {xss₁ : Vector (Vector α n) m₁} {xss₂ : Vector (Vector α n) m₂} :
     flatten xss₁ ++ flatten xss₂ = (flatten (xss₁ ++ xss₂)).cast (by simp [Nat.add_mul]) := by
   simp
 
-@[grind] theorem flatten_push {xss : Vector (Vector α n) m} {xs : Vector α n} :
+@[grind =] theorem flatten_push {xss : Vector (Vector α n) m} {xs : Vector α n} :
     flatten (xss.push xs) = (flatten xss ++ xs).cast (by simp [Nat.add_mul]) := by
   induction xss using vector₂_induction
   rcases xs with ⟨xs⟩
@@ -1982,10 +1991,10 @@ theorem flatMap_def {xs : Vector α n} {f : α → Vector β m} : xs.flatMap f =
   rcases xs with ⟨xs, rfl⟩
   simp [Array.flatMap_def, Function.comp_def]
 
-@[simp, grind] theorem flatMap_empty {f : α → Vector β m} :
+@[simp, grind =] theorem flatMap_empty {f : α → Vector β m} :
     (#v[] : Vector α 0).flatMap f = #v[].cast (by simp) := rfl
 
-@[simp, grind] theorem flatMap_push {xs : Vector α n} {x : α} {f : α → Vector β m} :
+@[simp, grind =] theorem flatMap_push {xs : Vector α n} {x : α} {f : α → Vector β m} :
     (xs.push x).flatMap f = (xs.flatMap f ++ f x).cast (by simp [Nat.add_mul]) := by
   rcases xs with ⟨xs, rfl⟩
   simp
@@ -2011,7 +2020,7 @@ theorem getElem?_flatMap {xs : Vector α n} {f : α → Vector β m} {i : Nat} :
 
 @[simp] theorem flatMap_id' {xss : Vector (Vector α m) n} : xss.flatMap (fun xs => xs) = xss.flatten := by simp [flatMap_def]
 
-@[simp, grind] theorem mem_flatMap {f : α → Vector β m} {b} {xs : Vector α n} : b ∈ xs.flatMap f ↔ ∃ a, a ∈ xs ∧ b ∈ f a := by
+@[simp, grind =] theorem mem_flatMap {f : α → Vector β m} {b} {xs : Vector α n} : b ∈ xs.flatMap f ↔ ∃ a, a ∈ xs ∧ b ∈ f a := by
   simp [flatMap_def, mem_flatten]
   exact ⟨fun ⟨_, ⟨a, h₁, rfl⟩, h₂⟩ => ⟨a, h₁, h₂⟩, fun ⟨a, h₁, h₂⟩ => ⟨_, ⟨a, h₁, rfl⟩, h₂⟩⟩
 
@@ -2074,7 +2083,7 @@ theorem replicate_succ' : replicate (n + 1) a = (#v[a] ++ replicate n a).cast (b
 @[deprecated replicate_succ' (since := "2025-03-18")]
 abbrev mkVector_succ' := @replicate_succ'
 
-@[simp, grind] theorem mem_replicate {a b : α} {n} : b ∈ replicate n a ↔ n ≠ 0 ∧ b = a := by
+@[simp, grind =] theorem mem_replicate {a b : α} {n} : b ∈ replicate n a ↔ n ≠ 0 ∧ b = a := by
   unfold replicate
   simp only [mem_mk]
   simp
@@ -2094,14 +2103,14 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
 @[deprecated forall_mem_replicate (since := "2025-03-18")]
 abbrev forall_mem_mkVector := @forall_mem_replicate
 
-@[simp, grind] theorem getElem_replicate {a : α} (h : i < n) : (replicate n a)[i] = a := by
+@[simp, grind =] theorem getElem_replicate {a : α} (h : i < n) : (replicate n a)[i] = a := by
   rw [replicate_eq_mk_replicate, getElem_mk]
   simp
 
 @[deprecated getElem_replicate (since := "2025-03-18")]
 abbrev getElem_mkVector := @getElem_replicate
 
-@[grind] theorem getElem?_replicate {a : α} {n i : Nat} : (replicate n a)[i]? = if i < n then some a else none := by
+@[grind =] theorem getElem?_replicate {a : α} {n i : Nat} : (replicate n a)[i]? = if i < n then some a else none := by
   simp [getElem?_def]
 
 @[deprecated getElem?_replicate (since := "2025-03-18")]
@@ -2227,16 +2236,16 @@ abbrev sum_mkVector := @sum_replicate_nat
 
 theorem reverse_empty : reverse (#v[] : Vector α 0) = #v[] := rfl
 
-@[simp, grind] theorem reverse_push {as : Vector α n} {a : α} :
+@[simp, grind =] theorem reverse_push {as : Vector α n} {a : α} :
     (as.push a).reverse = (#v[a] ++ as.reverse).cast (by omega) := by
   rcases as with ⟨as, rfl⟩
   simp [Array.reverse_push]
 
-@[simp, grind] theorem mem_reverse {x : α} {as : Vector α n} : x ∈ as.reverse ↔ x ∈ as := by
+@[simp, grind =] theorem mem_reverse {x : α} {as : Vector α n} : x ∈ as.reverse ↔ x ∈ as := by
   cases as
   simp
 
-@[simp, grind] theorem getElem_reverse {xs : Vector α n} {i : Nat} (hi : i < n) :
+@[simp, grind =] theorem getElem_reverse {xs : Vector α n} {i : Nat} (hi : i < n) :
     (xs.reverse)[i] = xs[n - 1 - i] := by
   rcases xs with ⟨xs, rfl⟩
   simp
@@ -2252,14 +2261,14 @@ theorem getElem?_reverse' {xs : Vector α n} {i j : Nat} (h : i + j + 1 = n) : x
   rcases xs with ⟨xs, rfl⟩
   simpa using Array.getElem?_reverse' h
 
-@[simp, grind]
+@[simp, grind =]
 theorem getElem?_reverse {xs : Vector α n} {i} (h : i < n) :
     xs.reverse[i]? = xs[n - 1 - i]? := by
   cases xs
   simp_all
 
 -- The argument `xs : Vector α n` is explicit so we can rewrite from right to left.
-@[simp, grind] theorem reverse_reverse (xs : Vector α n) : xs.reverse.reverse = xs := by
+@[simp, grind =] theorem reverse_reverse (xs : Vector α n) : xs.reverse.reverse = xs := by
   rcases xs with ⟨xs, rfl⟩
   simp [Array.reverse_reverse]
 
@@ -2279,13 +2288,13 @@ theorem reverse_eq_iff {xs ys : Vector α n} : xs.reverse = ys ↔ xs = ys.rever
   rcases xs with ⟨xs, rfl⟩
   simp [Array.map_reverse]
 
-@[simp, grind] theorem reverse_append {xs : Vector α n} {ys : Vector α m} :
+@[simp, grind =] theorem reverse_append {xs : Vector α n} {ys : Vector α m} :
     (xs ++ ys).reverse = (ys.reverse ++ xs.reverse).cast (by omega) := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
   simp [Array.reverse_append]
 
-@[grind] theorem append_reverse {xs : Vector α n} {ys : Vector α m} :
+@[grind =] theorem append_reverse {xs : Vector α n} {ys : Vector α m} :
     ys.reverse ++ xs.reverse = (xs ++ ys).reverse.cast (by omega) := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
@@ -2320,7 +2329,7 @@ theorem flatMap_reverse {xs : Vector α n} {f : α → Vector β m} :
   rcases xs with ⟨xs, rfl⟩
   simp [Array.flatMap_reverse, Function.comp_def]
 
-@[simp, grind] theorem reverse_replicate {n : Nat} {a : α} : reverse (replicate n a) = replicate n a := by
+@[simp, grind =] theorem reverse_replicate {n : Nat} {a : α} : reverse (replicate n a) = replicate n a := by
   rw [← toArray_inj]
   simp
 
@@ -2345,7 +2354,7 @@ set_option linter.indexVariables false in
   rcases as with ⟨as, rfl⟩
   simp
 
-@[grind] theorem extract_empty {start stop : Nat} :
+@[grind =] theorem extract_empty {start stop : Nat} :
     (#v[] : Vector α 0).extract start stop = #v[].cast (by simp) := by
   simp
 
@@ -2361,11 +2370,11 @@ theorem foldlM_empty [Monad m] {f : β → α → m β} {init : β} :
     foldlM f init #v[] = return init := by
   simp
 
-@[grind] theorem foldrM_empty [Monad m] {f : α → β → m β} {init : β} :
+@[grind =] theorem foldrM_empty [Monad m] {f : α → β → m β} {init : β} :
     foldrM f init #v[] = return init := by
   simp
 
-@[simp, grind] theorem foldlM_push [Monad m] [LawfulMonad m] {xs : Vector α n} {a : α} {f : β → α → m β} {b} :
+@[simp, grind =] theorem foldlM_push [Monad m] [LawfulMonad m] {xs : Vector α n} {a : α} {f : β → α → m β} {b} :
     (xs.push a).foldlM f b = xs.foldlM f b >>= fun b => f b a := by
   rcases xs with ⟨xs, rfl⟩
   simp
@@ -2410,16 +2419,16 @@ theorem id_run_foldrM {f : α → β → Id β} {b} {xs : Vector α n} :
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem foldrM_push [Monad m] {f : α → β → m β} {init : β} {xs : Vector α n} {a : α} :
+@[simp, grind =] theorem foldrM_push [Monad m] {f : α → β → m β} {init : β} {xs : Vector α n} {a : α} :
     (xs.push a).foldrM f init = f a init >>= xs.foldrM f := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
 /-! ### foldl / foldr -/
 
-@[grind] theorem foldl_empty {f : β → α → β} {init : β} : (#v[].foldl f init) = init := rfl
+@[grind =] theorem foldl_empty {f : β → α → β} {init : β} : (#v[].foldl f init) = init := rfl
 
-@[grind] theorem foldr_empty {f : α → β → β} {init : β} : (#v[].foldr f init) = init := rfl
+@[grind =] theorem foldr_empty {f : α → β → β} {init : β} : (#v[].foldr f init) = init := rfl
 
 @[congr]
 theorem foldl_congr {xs ys : Vector α n} (h₀ : xs = ys) {f g : β → α → β} (h₁ : f = g)
@@ -2433,12 +2442,12 @@ theorem foldr_congr {xs ys : Vector α n} (h₀ : xs = ys) {f g : α → β →
     xs.foldr f a = ys.foldr g b := by
   congr
 
-@[simp, grind] theorem foldl_push {f : β → α → β} {init : β} {xs : Vector α n} {a : α} :
+@[simp, grind =] theorem foldl_push {f : β → α → β} {init : β} {xs : Vector α n} {a : α} :
     (xs.push a).foldl f init = f (xs.foldl f init) a := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem foldr_push {f : α → β → β} {init : β} {xs : Vector α n} {a : α} :
+@[simp, grind =] theorem foldr_push {f : α → β → β} {init : β} {xs : Vector α n} {a : α} :
     (xs.push a).foldr f init = xs.foldr f (f a init) := by
   rcases xs with ⟨xs, rfl⟩
   simp
@@ -2490,21 +2499,21 @@ theorem foldr_map_hom {g : α → β} {f : α → α → α} {f' : β → β →
 @[simp, grind _=_] theorem foldr_append {f : α → β → β} {b} {xs : Vector α n} {ys : Vector α k} :
     (xs ++ ys).foldr f b = xs.foldr f (ys.foldr f b) := foldrM_append
 
-@[simp, grind] theorem foldl_flatten {f : β → α → β} {b} {xss : Vector (Vector α m) n} :
+@[simp, grind =] theorem foldl_flatten {f : β → α → β} {b} {xss : Vector (Vector α m) n} :
     (flatten xss).foldl f b = xss.foldl (fun b xs => xs.foldl f b) b := by
   cases xss using vector₂_induction
   simp [Array.foldl_flatten', Array.foldl_map']
 
-@[simp, grind] theorem foldr_flatten {f : α → β → β} {b} {xss : Vector (Vector α m) n} :
+@[simp, grind =] theorem foldr_flatten {f : α → β → β} {b} {xss : Vector (Vector α m) n} :
     (flatten xss).foldr f b = xss.foldr (fun xs b => xs.foldr f b) b := by
   cases xss using vector₂_induction
   simp [Array.foldr_flatten', Array.foldr_map']
 
-@[simp, grind] theorem foldl_reverse {xs : Vector α n} {f : β → α → β} {b} :
+@[simp, grind =] theorem foldl_reverse {xs : Vector α n} {f : β → α → β} {b} :
     xs.reverse.foldl f b = xs.foldr (fun x y => f y x) b :=
   foldlM_reverse
 
-@[simp, grind] theorem foldr_reverse {xs : Vector α n} {f : α → β → β} {b} :
+@[simp, grind =] theorem foldr_reverse {xs : Vector α n} {f : α → β → β} {b} :
     xs.reverse.foldr f b = xs.foldl (fun x y => f y x) b :=
   (foldl_reverse ..).symm.trans <| by simp
 
@@ -2598,7 +2607,7 @@ theorem back?_eq_some_iff {xs : Vector α n} {a : α} :
   simp only [mk_append_mk, back_mk]
   rw [Array.back_append_of_size_pos]
 
-@[grind] theorem back_append {xs : Vector α n} {ys : Vector α m} [NeZero (n + m)] :
+@[grind =] theorem back_append {xs : Vector α n} {ys : Vector α m} [NeZero (n + m)] :
     (xs ++ ys).back =
       if h' : m = 0 then
         have : NeZero n := by subst h'; simp_all
@@ -2629,7 +2638,7 @@ theorem back_append_left {xs : Vector α n} {ys : Vector α 0} [NeZero n] :
   simp only [mk_append_mk, back_mk]
   rw [Array.back_append_left _ h]
 
-@[simp, grind] theorem back?_append {xs : Vector α n} {ys : Vector α m} : (xs ++ ys).back? = ys.back?.or xs.back? := by
+@[simp, grind =] theorem back?_append {xs : Vector α n} {ys : Vector α m} : (xs ++ ys).back? = ys.back?.or xs.back? := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
   simp
@@ -2681,24 +2690,28 @@ theorem contains_iff_exists_mem_beq [BEq α] {xs : Vector α n} {a : α} :
   rcases xs with ⟨xs, rfl⟩
   simp [Array.contains_iff_exists_mem_beq]
 
+-- We add this as a `grind` lemma because it is useful without `LawfulBEq α`.
+-- With `LawfulBEq α`, it would be better to use `contains_iff_mem` directly.
+grind_pattern contains_iff_exists_mem_beq => xs.contains a
+
 @[grind _=_]
 theorem contains_iff_mem [BEq α] [LawfulBEq α] {xs : Vector α n} {a : α} :
     xs.contains a ↔ a ∈ xs := by
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_toList [BEq α] {xs : Vector α n} {x : α} :
     xs.toList.contains x = xs.contains x := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_toArray [BEq α] {xs : Vector α n} {x : α} :
     xs.toArray.contains x = xs.contains x := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_map [BEq β] {xs : Vector α n} {x : β} {f : α → β} :
     (xs.map f).contains x = xs.any (fun a => x == f a) := by
   rcases xs with ⟨xs⟩
@@ -2723,19 +2736,19 @@ theorem contains_append [BEq α] {xs : Vector α n} {ys : Vector α m} {x : α}
   rcases ys with ⟨ys, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_flatten [BEq α] {xs : Vector (Vector α n) m} {x : α} :
     (xs.flatten).contains x = xs.any fun xs => xs.contains x := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_reverse [BEq α] {xs : Vector α n} {x : α} :
     (xs.reverse).contains x = xs.contains x := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind]
+@[simp, grind =]
 theorem contains_flatMap [BEq β] {xs : Vector α n} {f : α → Vector β m} {x : β} :
     (xs.flatMap f).contains x = xs.any fun a => (f a).contains x := by
   rcases xs with ⟨xs, rfl⟩
@@ -2747,7 +2760,7 @@ theorem contains_flatMap [BEq β] {xs : Vector α n} {f : α → Vector β m} {x
 
 @[simp] theorem pop_push {xs : Vector α n} {x : α} : (xs.push x).pop = xs := by simp [pop]
 
-@[simp, grind] theorem getElem_pop {xs : Vector α n} {i : Nat} (h : i < n - 1) :
+@[simp, grind =] theorem getElem_pop {xs : Vector α n} {i : Nat} (h : i < n - 1) :
     xs.pop[i] = xs[i] := by
   rcases xs with ⟨xs, rfl⟩
   simp
@@ -2760,7 +2773,7 @@ defeq issues in the implicit size argument.
     @getElem (Vector α n) Nat α (fun _ i => i < n) instGetElemNatLt xs.pop i h = xs[i] :=
   getElem_pop h
 
-@[grind] theorem getElem?_pop {xs : Vector α n} {i : Nat} :
+@[grind =] theorem getElem?_pop {xs : Vector α n} {i : Nat} :
     xs.pop[i]? = if i < n - 1 then xs[i]? else none := by
   rcases xs with ⟨xs, rfl⟩
   simp [Array.getElem?_pop]
@@ -2908,15 +2921,15 @@ theorem all_filterMap {xs : Vector α n} {f : α → Option β} {p : β → Bool
   unfold all
   apply allM_congr w h
 
-@[simp, grind] theorem any_flatten {xss : Vector (Vector α n) m} : xss.flatten.any f = xss.any (any · f) := by
+@[simp, grind =] theorem any_flatten {xss : Vector (Vector α n) m} : xss.flatten.any f = xss.any (any · f) := by
   cases xss using vector₂_induction
   simp
 
-@[simp, grind] theorem all_flatten {xss : Vector (Vector α n) m} : xss.flatten.all f = xss.all (all · f) := by
+@[simp, grind =] theorem all_flatten {xss : Vector (Vector α n) m} : xss.flatten.all f = xss.all (all · f) := by
   cases xss using vector₂_induction
   simp
 
-@[simp, grind] theorem any_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
+@[simp, grind =] theorem any_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
     (xs.flatMap f).any p = xs.any fun a => (f a).any p := by
   rcases xs with ⟨xs⟩
   simp only [flatMap_mk, any_mk, Array.size_flatMap, size_toArray, Array.any_flatMap']
@@ -2925,7 +2938,7 @@ theorem all_filterMap {xs : Vector α n} {f : α → Option β} {p : β → Bool
   congr
   simp [Vector.size_toArray]
 
-@[simp, grind] theorem all_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
+@[simp, grind =] theorem all_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
     (xs.flatMap f).all p = xs.all fun a => (f a).all p := by
   rcases xs with ⟨xs⟩
   simp only [flatMap_mk, all_mk, Array.size_flatMap, size_toArray, Array.all_flatMap']
@@ -2934,11 +2947,11 @@ theorem all_filterMap {xs : Vector α n} {f : α → Option β} {p : β → Bool
   congr
   simp [Vector.size_toArray]
 
-@[simp, grind] theorem any_reverse {xs : Vector α n} : xs.reverse.any f  = xs.any f := by
+@[simp, grind =] theorem any_reverse {xs : Vector α n} : xs.reverse.any f  = xs.any f := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem all_reverse {xs : Vector α n} : xs.reverse.all f = xs.all f := by
+@[simp, grind =] theorem all_reverse {xs : Vector α n} : xs.reverse.all f = xs.all f := by
   rcases xs with ⟨xs, rfl⟩
   simp
 
@@ -2974,9 +2987,9 @@ variable [BEq α]
   rcases xs with ⟨xs, rfl⟩
   simp
 
-@[simp, grind] theorem replace_empty : (#v[] : Vector α 0).replace a b = #v[] := by simp
+@[simp, grind =] theorem replace_empty : (#v[] : Vector α 0).replace a b = #v[] := by simp
 
-@[grind] theorem replace_singleton {a b c : α} : #v[a].replace b c = #v[if a == b then c else a] := by
+@[grind =] theorem replace_singleton {a b c : α} : #v[a].replace b c = #v[if a == b then c else a] := by
   simp
 
 -- This hypothesis could probably be dropped from some of the lemmas below,
@@ -2987,7 +3000,7 @@ variable [LawfulBEq α]
   rcases xs with ⟨xs, rfl⟩
   simp_all
 
-@[grind] theorem getElem?_replace {xs : Vector α n} {i : Nat} :
+@[grind =] theorem getElem?_replace {xs : Vector α n} {i : Nat} :
     (xs.replace a b)[i]? = if xs[i]? == some a then if a ∈ xs.take i then some a else some b else xs[i]? := by
   rcases xs with ⟨xs, rfl⟩
   simp [Array.getElem?_replace, -beq_iff_eq]
@@ -2996,7 +3009,7 @@ theorem getElem?_replace_of_ne {xs : Vector α n} {i : Nat} (h : xs[i]? ≠ some
     (xs.replace a b)[i]? = xs[i]? := by
   simp_all [getElem?_replace]
 
-@[grind] theorem getElem_replace {xs : Vector α n} {i : Nat} (h : i < n) :
+@[grind =] theorem getElem_replace {xs : Vector α n} {i : Nat} (h : i < n) :
     (xs.replace a b)[i] = if xs[i] == a then if a ∈ xs.take i then a else b else xs[i] := by
   apply Option.some.inj
   rw [← getElem?_eq_getElem, getElem?_replace]
@@ -3007,7 +3020,7 @@ theorem getElem_replace_of_ne {xs : Vector α n} {i : Nat} {h : i < n} (h' : xs[
   rw [getElem_replace h]
   simp [h']
 
-@[grind] theorem replace_append {xs : Vector α n} {ys : Vector α m} :
+@[grind =] theorem replace_append {xs : Vector α n} {ys : Vector α m} :
     (xs ++ ys).replace a b = if a ∈ xs then xs.replace a b ++ ys else xs ++ ys.replace a b := by
   rcases xs with ⟨xs, rfl⟩
   rcases ys with ⟨ys, rfl⟩
@@ -3022,7 +3035,7 @@ theorem replace_append_right {xs : Vector α n} {ys : Vector α m} (h : ¬ a ∈
     (xs ++ ys).replace a b = xs ++ ys.replace a b := by
   simp [replace_append, h]
 
-@[grind] theorem replace_push {xs : Vector α n} {a b c : α} :
+@[grind =] theorem replace_push {xs : Vector α n} {a b c : α} :
     (xs.push a).replace b c = if b ∈ xs then (xs.replace b c).push a else xs.push (if b == a then c else a) := by
   rcases xs with ⟨xs, rfl⟩
   simp only [push_mk, replace_mk, Array.replace_push, mem_mk]
@@ -3091,7 +3104,7 @@ theorem take_size {as : Vector α n} : as.take n = as.cast (by simp) := by
 
 /-! ### swap -/
 
-@[grind] theorem getElem_swap {xs : Vector α n} {i j : Nat} (hi hj) {k : Nat} (hk : k < n) :
+@[grind =] theorem getElem_swap {xs : Vector α n} {i j : Nat} (hi hj) {k : Nat} (hk : k < n) :
     (xs.swap i j hi hj)[k] = if k = i then xs[j] else if k = j then xs[i] else xs[k] := by
   cases xs
   simp_all [Array.getElem_swap]
@@ -3108,7 +3121,7 @@ theorem take_size {as : Vector α n} : as.take n = as.cast (by simp) := by
       (hi' : k ≠ i) (hj' : k ≠ j) : (xs.swap i j hi hj)[k] = xs[k] := by
   simp_all [getElem_swap]
 
-@[grind]
+@[grind =]
 theorem getElem?_swap {xs : Vector α n} {i j : Nat} (hi hj) {k : Nat} : (xs.swap i j hi hj)[k]? =
     if j = k then some xs[i] else if i = k then some xs[j] else xs[k]? := by
   rcases xs with ⟨xs, rfl⟩

src/Init/GetElem.lean

@@ -166,25 +166,25 @@ export LawfulGetElem (getElem?_def getElem!_def)
 instance (priority := low) [GetElem coll idx elem valid] [∀ xs i, Decidable (valid xs i)] :
     LawfulGetElem coll idx elem valid where
 
-@[simp, grind] theorem getElem?_pos [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
+@[simp, grind =] theorem getElem?_pos [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
     (c : cont) (i : idx) (h : dom c i) : c[i]? = some (c[i]'h) := by
   have : Decidable (dom c i) := .isTrue h
   rw [getElem?_def]
   exact dif_pos h
 
-@[simp, grind] theorem getElem?_neg [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
+@[simp, grind =] theorem getElem?_neg [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
     (c : cont) (i : idx) (h : ¬dom c i) : c[i]? = none := by
   have : Decidable (dom c i) := .isFalse h
   rw [getElem?_def]
   exact dif_neg h
 
-@[simp, grind] theorem getElem!_pos [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
+@[simp, grind =] theorem getElem!_pos [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
     [Inhabited elem] (c : cont) (i : idx) (h : dom c i) :
     c[i]! = c[i]'h := by
   have : Decidable (dom c i) := .isTrue h
   simp [getElem!_def, h]
 
-@[simp, grind] theorem getElem!_neg [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
+@[simp, grind =] theorem getElem!_neg [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
     [Inhabited elem] (c : cont) (i : idx) (h : ¬dom c i) : c[i]! = default := by
   have : Decidable (dom c i) := .isFalse h
   simp [getElem!_def, h]
@@ -291,18 +291,20 @@ namespace List
 instance : GetElem (List α) Nat α fun as i => i < as.length where
   getElem as i h := as.get ⟨i, h⟩
 
-@[simp, grind]
+@[simp, grind =]
 theorem getElem_cons_zero (a : α) (as : List α) (h : 0 < (a :: as).length) :
     getElem (a :: as) 0 h = a := rfl
 
-@[simp, grind]
+@[simp, grind =]
 theorem getElem_cons_succ (a : α) (as : List α) (i : Nat) (h : i + 1 < (a :: as).length) : getElem (a :: as) (i+1) h = getElem as i (Nat.lt_of_succ_lt_succ h) :=
     rfl
 
-@[simp, grind] theorem getElem_mem : ∀ {l : List α} {n} (h : n < l.length), l[n]'h ∈ l
+@[simp] theorem getElem_mem : ∀ {l : List α} {n} (h : n < l.length), l[n]'h ∈ l
   | _ :: _, 0, _ => .head ..
   | _ :: l, _+1, _ => .tail _ (getElem_mem (l := l) ..)
 
+grind_pattern getElem_mem => l[n]'h ∈ l
+
 theorem getElem_cons_drop_succ_eq_drop {as : List α} {i : Nat} (h : i < as.length) :
     as[i] :: as.drop (i+1) = as.drop i :=
   match as, i with

src/Init/Grind/AC.lean

@@ -10,6 +10,8 @@ public import Init.Core
 public import Init.Data.Nat.Lemmas
 public import Init.Data.RArray
 public import Init.Data.Bool
+import Init.LawfulBEqTactics
+
 @[expose] public section
 
 namespace Lean.Grind.AC
@@ -38,7 +40,7 @@ attribute [local simp] Expr.denote_var Expr.denote_op
 inductive Seq where
   | var (x : Var)
   | cons (x : Var) (s : Seq)
-  deriving Inhabited, Repr, BEq
+  deriving Inhabited, Repr, BEq, ReflBEq, LawfulBEq
 
 -- Kernel version for Seq.beq
 noncomputable def Seq.beq' (s₁ : Seq) : Seq → Bool :=
@@ -55,12 +57,6 @@ theorem Seq.beq'_eq (s₁ s₂ : Seq) : s₁.beq' s₂ = (s₁ = s₂) := by
 
 attribute [local simp] Seq.beq'_eq
 
-instance : LawfulBEq Seq where
-  eq_of_beq {a} := by
-    induction a <;> intro b <;> cases b <;> simp! [BEq.beq]
-    next x₁ s₁ ih x₂ s₂ => intro h₁ h₂; simp [h₁, ih h₂]
-  rfl := by intro a; induction a <;> simp! [BEq.beq]; assumption
-
 noncomputable def Seq.denote {α} (ctx : Context α) (s : Seq) : α :=
   Seq.rec (fun x => x.denote ctx) (fun x _ ih => ctx.op (x.denote ctx) ih) s
 

src/Init/Grind/Attr.lean

@@ -98,28 +98,34 @@ syntax grindEqBwd  := patternIgnore(atomic("←" "=") <|> atomic("<-" "="))
 The `←` modifier instructs `grind` to select a multi-pattern from the conclusion of theorem.
 In other words, `grind` will use the theorem for backwards reasoning.
 This may fail if not all of the arguments to the theorem appear in the conclusion.
+Each time it encounters a subexpression which covers an argument which was not
+previously covered, it adds that subexpression as a pattern, until all arguments have been covered.
+If `grind!` is used, then only minimal indexable subexpressions are considered.
 -/
 syntax grindBwd    := patternIgnore("←" <|> "<-") (grindGen)?
 /--
 The `→` modifier instructs `grind` to select a multi-pattern from the hypotheses of the theorem.
 In other words, `grind` will use the theorem for forwards reasoning.
 To generate a pattern, it traverses the hypotheses of the theorem from left to right.
-Each time it encounters a minimal indexable subexpression which covers an argument which was not
+Each time it encounters a subexpression which covers an argument which was not
 previously covered, it adds that subexpression as a pattern, until all arguments have been covered.
+If `grind!` is used, then only minimal indexable subexpressions are considered.
 -/
 syntax grindFwd    := patternIgnore("→" <|> "->")
 /--
 The `⇐` modifier instructs `grind` to select a multi-pattern by traversing the conclusion, and then
 all the hypotheses from right to left.
-Each time it encounters a minimal indexable subexpression which covers an argument which was not
+Each time it encounters a subexpression which covers an argument which was not
 previously covered, it adds that subexpression as a pattern, until all arguments have been covered.
+If `grind!` is used, then only minimal indexable subexpressions are considered.
 -/
 syntax grindRL     := patternIgnore("⇐" <|> "<=")
 /--
 The `⇒` modifier instructs `grind` to select a multi-pattern by traversing all the hypotheses from
 left to right, followed by the conclusion.
-Each time it encounters a minimal indexable subexpression which covers an argument which was not
+Each time it encounters a subexpression which covers an argument which was not
 previously covered, it adds that subexpression as a pattern, until all arguments have been covered.
+If `grind!` is used, then only minimal indexable subexpressions are considered.
 -/
 syntax grindLR     := patternIgnore("⇒" <|> "=>")
 /--
@@ -195,6 +201,8 @@ syntax grindMod :=
     <|> grindFwd <|> grindRL <|> grindLR <|> grindUsr <|> grindCasesEager
     <|> grindCases <|> grindIntro <|> grindExt <|> grindGen <|> grindSym
 syntax (name := grind) "grind" (ppSpace grindMod)? : attr
+syntax (name := grind!) "grind!" (ppSpace grindMod)? : attr
 syntax (name := grind?) "grind?" (ppSpace grindMod)? : attr
+syntax (name := grind!?) "grind!?" (ppSpace grindMod)? : attr
 end Attr
 end Lean.Parser

src/Init/Grind/Ordered/Linarith.lean

@@ -13,6 +13,7 @@ import all Init.Data.Ord.Basic
 public import Init.Data.AC
 import all Init.Data.AC
 public import Init.Data.RArray
+import Init.LawfulBEqTactics
 
 @[expose] public section
 
@@ -55,7 +56,7 @@ def Expr.denote {α} [IntModule α] (ctx : Context α) : Expr → α
 inductive Poly where
   | nil
   | add (k : Int) (v : Var) (p : Poly)
-  deriving BEq, Repr
+  deriving BEq, ReflBEq, LawfulBEq, Repr
 
 def Poly.denote {α} [IntModule α] (ctx : Context α) (p : Poly) : α :=
   match p with
@@ -233,18 +234,6 @@ theorem Expr.denote_norm {α} [IntModule α] (ctx : Context α) (e : Expr) : e.n
   simp [norm, toPoly', Expr.denote_toPoly'_go, Poly.denote]
 
 attribute [local simp] Expr.denote_norm
-
-instance : LawfulBEq Poly where
-  eq_of_beq {a} := by
-    induction a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
-    next ih =>
-      intro _ _ h
-      exact ih h
-  rfl := by
-    intro a
-    induction a <;> simp! [BEq.beq]
-    assumption
-
 attribute [local simp] Poly.denote'_eq_denote
 
 def Poly.leadCoeff (p : Poly) : Int :=

src/Init/Grind/PP.lean

@@ -7,7 +7,7 @@ module
 
 prelude
 public import Init.NotationExtra
-meta import Init.Data.String.Basic
+public meta import Init.Data.String.Basic
 
 public section
 

src/Init/Grind/Ring.lean

@@ -4,13 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kim Morrison
 -/
 module
-
 prelude
 public import Init.Grind.Ring.Basic
-public import Init.Grind.Ring.Poly
 public import Init.Grind.Ring.Field
 public import Init.Grind.Ring.Envelope
-public import Init.Grind.Ring.OfSemiring
+public import Init.Grind.Ring.CommSolver
+public import Init.Grind.Ring.CommSemiringAdapter
 public import Init.Grind.Ring.ToInt
-
-public section

src/Init/Grind/Ring/Basic.lean

@@ -179,6 +179,20 @@ theorem ofNat_mul (a b : Nat) : OfNat.ofNat (α := α) (a * b) = OfNat.ofNat a *
 theorem natCast_mul (a b : Nat) : ((a * b : Nat) : α) = ((a : α) * (b : α)) := by
   rw [← ofNat_eq_natCast, ofNat_mul, ofNat_eq_natCast, ofNat_eq_natCast]
 
+theorem natCast_mul_comm (a : Nat) (b : α) : a * b = b * a := by
+  induction a
+  next => simp [Semiring.natCast_zero, mul_zero, zero_mul]
+  next ih =>
+    rw [Semiring.natCast_succ, Semiring.left_distrib, Semiring.right_distrib, ih]
+    simp [Semiring.mul_one, Semiring.one_mul]
+
+theorem natCast_mul_left_comm (a : α) (b : Nat) (c : α) : a * (b * c) = b * (a * c) := by
+  induction b
+  next => simp [Semiring.natCast_zero, mul_zero, zero_mul]
+  next ih =>
+    rw [Semiring.natCast_succ, Semiring.right_distrib, Semiring.left_distrib, ih,
+        Semiring.right_distrib, Semiring.one_mul, Semiring.one_mul]
+
 theorem pow_one (a : α) : a ^ 1 = a := by
   rw [pow_succ, pow_zero, one_mul]
 
@@ -331,6 +345,18 @@ theorem intCast_mul (x y : Int) : ((x * y : Int) : α) = ((x : α) * (y : α)) :
     rw [Int.neg_mul_neg, intCast_neg, intCast_neg, neg_mul, mul_neg, neg_neg, intCast_mul_aux,
       intCast_natCast, intCast_natCast]
 
+theorem intCast_mul_comm (a : Int) (b : α) : a * b = b * a := by
+  have : a = a.natAbs ∨ a = -a.natAbs := by exact Int.natAbs_eq a
+  cases this
+  next h => rw [h, Ring.intCast_natCast, Semiring.natCast_mul_comm]
+  next h => rw [h, Ring.intCast_neg, Ring.intCast_natCast, Ring.mul_neg, Ring.neg_mul, Semiring.natCast_mul_comm]
+
+theorem intCast_mul_left_comm (a : α) (b : Int) (c : α) : a * (b * c) = b * (a * c) := by
+  have : b = b.natAbs ∨ b = -b.natAbs := by exact Int.natAbs_eq b
+  cases this
+  next h => rw [h, Ring.intCast_natCast, Semiring.natCast_mul_left_comm]
+  next h => rw [h, Ring.intCast_neg, Ring.intCast_natCast, Ring.neg_mul, Ring.neg_mul, Ring.mul_neg, Semiring.natCast_mul_left_comm]
+
 theorem intCast_pow (x : Int) (k : Nat) : ((x ^ k : Int) : α) = (x : α) ^ k := by
   induction k
   next => simp [pow_zero, Int.pow_zero, intCast_one]

src/Init/Grind/Ring/CommSemiringAdapter.lean

@@ -4,79 +4,65 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 module
-
 prelude
 public import Init.Grind.Ring.Envelope
 public import Init.Data.Hashable
 public import Init.Data.RArray
-public import Init.Grind.Ring.Poly
-
+public import Init.Grind.Ring.CommSolver
 @[expose] public section
+namespace Lean.Grind
+namespace CommRing
 
-namespace Lean.Grind.Ring.OfSemiring
-/-!
-Helper definitions and theorems for converting `Semiring` expressions into `Ring` ones.
-We use them to implement `grind`
--/
-abbrev Var := Nat
-inductive Expr where
-  | num  (v : Nat)
-  | var  (i : Var)
-  | add  (a b : Expr)
-  | mul (a b : Expr)
-  | pow (a : Expr) (k : Nat)
-  deriving Inhabited, BEq, Hashable
+def Expr.denoteS {α} [Semiring α] (ctx : Context α) : Expr → α
+  | .num k     => OfNat.ofNat (α := α) k.natAbs
+  | .natCast k => OfNat.ofNat (α := α) k
+  | .var v     => v.denote ctx
+  | .add a b   => denoteS ctx a + denoteS ctx b
+  | .mul a b   => denoteS ctx a * denoteS ctx b
+  | .pow a k   => denoteS ctx a ^ k
+  | .sub .. | .neg .. | .intCast .. => 0
 
-abbrev Context (α : Type u) := RArray α
+def Expr.denoteSAsRing {α} [Semiring α] (ctx : Context α) : Expr → Ring.OfSemiring.Q α
+  | .num k     => OfNat.ofNat (α := Ring.OfSemiring.Q α) k.natAbs
+  | .natCast k => OfNat.ofNat (α := Ring.OfSemiring.Q α) k
+  | .var v     => Ring.OfSemiring.toQ (v.denote ctx)
+  | .add a b   => denoteSAsRing ctx a + denoteSAsRing ctx b
+  | .mul a b   => denoteSAsRing ctx a * denoteSAsRing ctx b
+  | .pow a k   => denoteSAsRing ctx a ^ k
+  | .sub .. | .neg .. | .intCast .. => 0
 
-def Var.denote {α} (ctx : Context α) (v : Var) : α :=
-  ctx.get v
+attribute [local simp] Ring.OfSemiring.toQ_add Ring.OfSemiring.toQ_mul Ring.OfSemiring.toQ_ofNat
+  Ring.OfSemiring.toQ_pow Ring.OfSemiring.toQ_zero in
+theorem Expr.denoteAsRing_eq {α} [Semiring α] (ctx : Context α) (e : Expr) : e.denoteSAsRing ctx = Ring.OfSemiring.toQ (e.denoteS ctx) := by
+  induction e <;> simp [denoteS, denoteSAsRing, *]
 
-def Expr.denote {α} [Semiring α] (ctx : Context α) : Expr → α
-  | .num k    => OfNat.ofNat (α := α) k
-  | .var v    => v.denote ctx
-  | .add a b  => denote ctx a + denote ctx b
-  | .mul a b  => denote ctx a * denote ctx b
-  | .pow a k  => denote ctx a ^ k
-
-attribute [local instance] ofSemiring
-
-def Expr.denoteAsRing {α} [Semiring α] (ctx : Context α) : Expr → Q α
-  | .num k    => OfNat.ofNat (α := Q α) k
-  | .var v    => toQ (v.denote ctx)
-  | .add a b  => denoteAsRing ctx a + denoteAsRing ctx b
-  | .mul a b  => denoteAsRing ctx a * denoteAsRing ctx b
-  | .pow a k  => denoteAsRing ctx a ^ k
-
-attribute [local simp] toQ_add toQ_mul toQ_ofNat toQ_pow
-
-theorem Expr.denoteAsRing_eq {α} [Semiring α] (ctx : Context α) (e : Expr) : e.denoteAsRing ctx = toQ (e.denote ctx) := by
-  induction e <;> simp [denote, denoteAsRing, *]
-
-theorem of_eq {α} [Semiring α] (ctx : Context α) (lhs rhs : Expr)
-    : lhs.denote ctx = rhs.denote ctx → lhs.denoteAsRing ctx = rhs.denoteAsRing ctx := by
-  intro h; replace h := congrArg toQ h
-  simpa [← Expr.denoteAsRing_eq] using h
-
-theorem of_diseq {α} [Semiring α] [AddRightCancel α] (ctx : Context α) (lhs rhs : Expr)
-    : lhs.denote ctx ≠ rhs.denote ctx → lhs.denoteAsRing ctx ≠ rhs.denoteAsRing ctx := by
-  intro h₁ h₂
-  simp [Expr.denoteAsRing_eq] at h₂
-  replace h₂ := toQ_inj h₂
-  contradiction
-
-def Expr.toPoly : Expr → CommRing.Poly
-  | .num n   => .num n
+def Expr.toPolyS : Expr → CommRing.Poly
+  | .num n   => .num n.natAbs
   | .var x   => CommRing.Poly.ofVar x
-  | .add a b => a.toPoly.combine b.toPoly
-  | .mul a b => a.toPoly.mul b.toPoly
+  | .add a b => a.toPolyS.combine b.toPolyS
+  | .mul a b => a.toPolyS.mul b.toPolyS
   | .pow a k =>
     match a with
-    | .num n => .num (n^k)
+    | .num n => .num (n.natAbs ^ k)
     | .var x => CommRing.Poly.ofMon (.mult {x, k} .unit)
-    | _ => a.toPoly.pow k
+    | _ => a.toPolyS.pow k
+  | .natCast n => .num n
+  | .sub .. | .neg  .. | .intCast .. => .num 0
 
-end Ring.OfSemiring
+def Expr.toPolyS_nc : Expr → CommRing.Poly
+  | .num n   => .num n.natAbs
+  | .var x   => CommRing.Poly.ofVar x
+  | .add a b => a.toPolyS_nc.combine b.toPolyS_nc
+  | .mul a b => a.toPolyS_nc.mul_nc b.toPolyS_nc
+  | .pow a k =>
+    match a with
+    | .num n => .num (n.natAbs ^ k)
+    | .var x => CommRing.Poly.ofMon (.mult {x, k} .unit)
+    | _ => a.toPolyS_nc.pow_nc k
+  | .natCast n => .num n
+  | .sub .. | .neg  .. | .intCast .. => .num 0
+
+end CommRing
 
 namespace CommRing
 attribute [local instance] Semiring.natCast Ring.intCast
@@ -109,15 +95,15 @@ def Poly.denoteS [Semiring α] (ctx : Context α) (p : Poly) : α :=
 
 attribute [local simp] natCast_one natCast_zero zero_mul mul_zero one_mul mul_one add_zero zero_add denoteSInt_eq
 
-theorem Poly.denoteS_ofMon {α} [CommSemiring α] (ctx : Context α) (m : Mon)
+theorem Poly.denoteS_ofMon {α} [Semiring α] (ctx : Context α) (m : Mon)
     : denoteS ctx (ofMon m) = m.denote ctx := by
   simp [ofMon, denoteS]
 
-theorem Poly.denoteS_ofVar {α} [CommSemiring α] (ctx : Context α) (x : Var)
+theorem Poly.denoteS_ofVar {α} [Semiring α] (ctx : Context α) (x : Var)
     : denoteS ctx (ofVar x) = x.denote ctx := by
   simp [ofVar, denoteS_ofMon, Mon.denote_ofVar]
 
-theorem Poly.denoteS_addConst {α} [CommSemiring α] (ctx : Context α) (p : Poly) (k : Int)
+theorem Poly.denoteS_addConst {α} [Semiring α] (ctx : Context α) (p : Poly) (k : Int)
     : k ≥ 0 → p.NonnegCoeffs → (addConst p k).denoteS ctx = p.denoteS ctx + k.toNat := by
   simp [addConst, cond_eq_if]; split
   next => subst k; simp
@@ -130,7 +116,7 @@ theorem Poly.denoteS_addConst {α} [CommSemiring α] (ctx : Context α) (p : Pol
       intro _ h; cases h
       next h₁ h₂ => simp [*, add_assoc]
 
-theorem Poly.denoteS_insert {α} [CommSemiring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+theorem Poly.denoteS_insert {α} [Semiring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
     : k ≥ 0 → p.NonnegCoeffs → (insert k m p).denoteS ctx = k.toNat * m.denote ctx + p.denoteS ctx := by
   simp [insert, cond_eq_if] <;> split
   next => simp [*]
@@ -157,13 +143,13 @@ theorem Poly.denoteS_insert {α} [CommSemiring α] (ctx : Context α) (k : Int)
         intro hk hn; cases hn; rename_i hn₁ hn₂
         rw [ih hk hn₂, add_left_comm]
 
-theorem Poly.denoteS_concat {α} [CommSemiring α] (ctx : Context α) (p₁ p₂ : Poly)
+theorem Poly.denoteS_concat {α} [Semiring α] (ctx : Context α) (p₁ p₂ : Poly)
     : p₁.NonnegCoeffs → p₂.NonnegCoeffs → (concat p₁ p₂).denoteS ctx = p₁.denoteS ctx + p₂.denoteS ctx := by
   fun_induction concat <;> intro h₁ h₂; simp [*, denoteS]
   next => cases h₁; rw [add_comm, denoteS_addConst] <;> assumption
   next ih => cases h₁; next hn₁ hn₂ => rw [denoteS, denoteS, ih hn₂ h₂, add_assoc]
 
-theorem Poly.denoteS_mulConst {α} [CommSemiring α] (ctx : Context α) (k : Int) (p : Poly)
+theorem Poly.denoteS_mulConst {α} [Semiring α] (ctx : Context α) (k : Int) (p : Poly)
     : k ≥ 0 → p.NonnegCoeffs → (mulConst k p).denoteS ctx = k.toNat * p.denoteS ctx := by
   simp [mulConst, cond_eq_if] <;> split
   next => simp [denoteS, *, zero_mul]
@@ -177,30 +163,7 @@ theorem Poly.denoteS_mulConst {α} [CommSemiring α] (ctx : Context α) (k : Int
       intro h₁ h₂; cases h₂; rename_i h₂ h₃
       rw [Int.toNat_mul, natCast_mul, left_distrib, mul_assoc, ih h₁ h₃] <;> assumption
 
-theorem Poly.denoteS_mulMon {α} [CommSemiring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
-    : k ≥ 0 → p.NonnegCoeffs → (mulMon k m p).denoteS ctx = k.toNat * m.denote ctx * p.denoteS ctx := by
-  simp [mulMon, cond_eq_if] <;> split
-  next => simp [denoteS, *]
-  next =>
-    split
-    next h =>
-      intro h₁ h₂
-      simp at h; simp [*, Mon.denote, denoteS_mulConst _ _ _ h₁ h₂]
-    next =>
-      fun_induction mulMon.go <;> simp [denoteS, *]
-      next h => simp +zetaDelta at h; simp [*]
-      next =>
-        intro h₁ h₂; cases h₂
-        rw [Int.toNat_mul]
-        simp [natCast_mul, CommSemiring.mul_comm, CommSemiring.mul_left_comm, mul_assoc]
-        assumption; assumption
-      next ih =>
-        intro h₁ h₂; cases h₂; rename_i h₂ h₃
-        rw [Int.toNat_mul]
-        simp [Mon.denote_mul, natCast_mul, left_distrib, CommSemiring.mul_left_comm, mul_assoc, ih h₁ h₃]
-        assumption; assumption
-
-theorem Poly.denoteS_combine {α} [CommSemiring α] (ctx : Context α) (p₁ p₂ : Poly)
+theorem Poly.denoteS_combine {α} [Semiring α] (ctx : Context α) (p₁ p₂ : Poly)
     : p₁.NonnegCoeffs → p₂.NonnegCoeffs → (combine p₁ p₂).denoteS ctx = p₁.denoteS ctx + p₂.denoteS ctx := by
   unfold combine; generalize hugeFuel = fuel
   fun_induction combine.go
@@ -233,6 +196,93 @@ theorem Poly.denoteS_combine {α} [CommSemiring α] (ctx : Context α) (p₁ p
     rename_i h₂
     simp [denoteS, ih h₁ h₂, add_left_comm, add_assoc]
 
+theorem Poly.denoteS_mulMon {α} [CommSemiring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+    : k ≥ 0 → p.NonnegCoeffs → (mulMon k m p).denoteS ctx = k.toNat * m.denote ctx * p.denoteS ctx := by
+  simp [mulMon, cond_eq_if] <;> split
+  next => simp [denoteS, *]
+  next =>
+    split
+    next h =>
+      intro h₁ h₂
+      simp at h; simp [*, Mon.denote, denoteS_mulConst _ _ _ h₁ h₂]
+    next =>
+      fun_induction mulMon.go <;> simp [denoteS, *]
+      next h => simp +zetaDelta at h; simp [*]
+      next =>
+        intro h₁ h₂; cases h₂
+        rw [Int.toNat_mul]
+        simp [natCast_mul, CommSemiring.mul_comm, CommSemiring.mul_left_comm, mul_assoc]
+        assumption; assumption
+      next ih =>
+        intro h₁ h₂; cases h₂; rename_i h₂ h₃
+        rw [Int.toNat_mul]
+        simp [Mon.denote_mul, natCast_mul, left_distrib, CommSemiring.mul_left_comm, mul_assoc, ih h₁ h₃]
+        assumption; assumption
+
+theorem Poly.addConst_NonnegCoeffs {p : Poly} {k : Int} : k ≥ 0 → p.NonnegCoeffs → (p.addConst k).NonnegCoeffs := by
+  simp [addConst, cond_eq_if]; split
+  next => intros; assumption
+  fun_induction addConst.go
+  next h _ => intro _ h; cases h; constructor; apply Int.add_nonneg <;> assumption
+  next ih => intro h₁ h₂; cases h₂; constructor; assumption; apply ih <;> assumption
+
+theorem Poly.insert_Nonneg (k : Int) (m : Mon) (p : Poly) : k ≥ 0 → p.NonnegCoeffs → (p.insert k m).NonnegCoeffs := by
+  intro h₁ h₂
+  fun_cases Poly.insert
+  next => assumption
+  next => apply Poly.addConst_NonnegCoeffs <;> assumption
+  next =>
+    fun_induction Poly.insert.go
+    next => constructor <;> assumption
+    next => cases h₂; assumption
+    next => simp +zetaDelta; cases h₂; constructor; omega; assumption
+    next => constructor <;> assumption
+    next ih =>
+      cases h₂; constructor
+      next => assumption
+      next => apply ih; assumption
+
+theorem Poly.denoteS_mulMon_nc_go {α} [Semiring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly) (acc : Poly)
+    : k ≥ 0 → p.NonnegCoeffs → acc.NonnegCoeffs
+      → (mulMon_nc.go k m p acc).denoteS ctx = k.toNat * m.denote ctx * p.denoteS ctx + acc.denoteS ctx := by
+  fun_induction mulMon_nc.go with simp [*]
+  | case1 acc k' =>
+    intro h₁ h₂ h₃; cases h₂
+    have : k * k' ≥ 0 := by apply Int.mul_nonneg <;> assumption
+    simp [denoteS_insert, denoteS, Int.toNat_mul, Semiring.natCast_mul, Semiring.mul_assoc, *]
+    rw [← Semiring.natCast_mul_comm]
+  | case2 acc k' m' p ih =>
+    intro h₁ h₂ h₃; rcases h₂
+    next _ h₂ =>
+    have : k * k' ≥ 0 := by apply Int.mul_nonneg <;> assumption
+    have : (insert (k * k') (m.mul_nc m') acc).NonnegCoeffs := by apply Poly.insert_Nonneg <;> assumption
+    rw [ih h₁ h₂ this]
+    simp [denoteS_insert, Int.toNat_mul, Semiring.natCast_mul, denoteS, left_distrib, Mon.denote_mul_nc, *]
+    simp only [← Semiring.add_assoc]
+    congr 1
+    rw [Semiring.add_comm]
+    congr 1
+    rw [Semiring.natCast_mul_left_comm]
+    conv => enter [1, 1]; rw [Semiring.natCast_mul_comm]
+    simp [Semiring.mul_assoc]
+
+theorem Poly.num_zero_NonnegCoeffs : (num 0).NonnegCoeffs := by
+  apply NonnegCoeffs.num; simp
+
+theorem Poly.denoteS_mulMon_nc {α} [Semiring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+    : k ≥ 0 → p.NonnegCoeffs → (mulMon_nc k m p).denoteS ctx = k.toNat * m.denote ctx * p.denoteS ctx := by
+  simp [mulMon_nc, cond_eq_if] <;> split
+  next => simp [denoteS, *]
+  next =>
+    split
+    next h =>
+      intro h₁ h₂
+      simp at h; simp [*, Mon.denote, denoteS_mulConst _ _ _ h₁ h₂]
+    next =>
+      intro h₁ h₂
+      have := Poly.denoteS_mulMon_nc_go ctx k m p (.num 0) h₁ h₂ Poly.num_zero_NonnegCoeffs
+      simp [this, denoteS]
+
 theorem Poly.mulConst_NonnegCoeffs {p : Poly} {k : Int} : k ≥ 0 → p.NonnegCoeffs → (p.mulConst k).NonnegCoeffs := by
   simp [mulConst, cond_eq_if]; split
   next => intros; constructor; decide
@@ -259,12 +309,29 @@ theorem Poly.mulMon_NonnegCoeffs {p : Poly} {k : Int} (m : Mon) : k ≥ 0 → p.
     apply Int.mul_nonneg <;> assumption
     apply ih <;> assumption
 
-theorem Poly.addConst_NonnegCoeffs {p : Poly} {k : Int} : k ≥ 0 → p.NonnegCoeffs → (p.addConst k).NonnegCoeffs := by
-  simp [addConst, cond_eq_if]; split
-  next => intros; assumption
-  fun_induction addConst.go
-  next h _ => intro _ h; cases h; constructor; apply Int.add_nonneg <;> assumption
-  next ih => intro h₁ h₂; cases h₂; constructor; assumption; apply ih <;> assumption
+theorem Poly.mulMon_nc_go_NonnegCoeffs {p : Poly} {k : Int} (m : Mon) {acc : Poly}
+    : k ≥ 0 → p.NonnegCoeffs → acc.NonnegCoeffs → (Poly.mulMon_nc.go k m p acc).NonnegCoeffs := by
+  intro h₁ h₂ h₃
+  fun_induction Poly.mulMon_nc.go
+  next k' =>
+    cases h₂
+    have : k*k' ≥ 0 := by apply Int.mul_nonneg <;> assumption
+    apply Poly.insert_Nonneg <;> assumption
+  next ih =>
+    cases h₂; next h₂ =>
+    apply ih; assumption
+    apply insert_Nonneg
+    next => apply Int.mul_nonneg <;> assumption
+    next => assumption
+
+theorem Poly.mulMon_nc_NonnegCoeffs {p : Poly} {k : Int} (m : Mon) : k ≥ 0 → p.NonnegCoeffs → (p.mulMon_nc k m).NonnegCoeffs := by
+  simp [mulMon_nc, cond_eq_if]; split
+  next => intros; constructor; decide
+  split
+  next => intros; apply mulConst_NonnegCoeffs <;> assumption
+  intro h₁ h₂
+  apply Poly.mulMon_nc_go_NonnegCoeffs; assumption; assumption
+  exact Poly.num_zero_NonnegCoeffs
 
 theorem Poly.concat_NonnegCoeffs {p₁ p₂ : Poly} : p₁.NonnegCoeffs → p₂.NonnegCoeffs → (p₁.concat p₂).NonnegCoeffs := by
   fun_induction Poly.concat
@@ -312,14 +379,40 @@ theorem Poly.mul_NonnegCoeffs {p₁ p₂ : Poly} : p₁.NonnegCoeffs → p₂.No
   unfold mul; intros; apply mul_go_NonnegCoeffs
   assumption; assumption; constructor; decide
 
+theorem Poly.mul_nc_go_NonnegCoeffs (p₁ p₂ acc : Poly)
+    : p₁.NonnegCoeffs → p₂.NonnegCoeffs → acc.NonnegCoeffs → (mul_nc.go p₂ p₁ acc).NonnegCoeffs := by
+  fun_induction mul_nc.go
+  next =>
+    intro h₁ h₂ h₃
+    cases h₁; rename_i h₁
+    have := mulConst_NonnegCoeffs h₁ h₂
+    apply combine_NonnegCoeffs <;> assumption
+  next ih =>
+    intro h₁ h₂ h₃
+    cases h₁
+    apply ih
+    assumption; assumption
+    apply Poly.combine_NonnegCoeffs; assumption
+    apply Poly.mulMon_nc_NonnegCoeffs <;> assumption
+
+theorem Poly.mul_nc_NonnegCoeffs {p₁ p₂ : Poly} : p₁.NonnegCoeffs → p₂.NonnegCoeffs → (p₁.mul_nc p₂).NonnegCoeffs := by
+  unfold mul_nc; intros; apply mul_nc_go_NonnegCoeffs
+  assumption; assumption; constructor; decide
+
 theorem Poly.pow_NonnegCoeffs {p : Poly} (k : Nat) : p.NonnegCoeffs → (p.pow k).NonnegCoeffs := by
   fun_induction Poly.pow
   next => intros; constructor; decide
   next => intros; assumption
   next ih => intro h; apply mul_NonnegCoeffs; assumption; apply ih; assumption
 
-theorem Poly.num_zero_NonnegCoeffs : (num 0).NonnegCoeffs := by
-  apply NonnegCoeffs.num; simp
+theorem Poly.pow_nc_NonnegCoeffs {p : Poly} (k : Nat) : p.NonnegCoeffs → (p.pow_nc k).NonnegCoeffs := by
+  fun_induction Poly.pow_nc
+  next => intros; constructor; decide
+  next => intros; assumption
+  next ih =>
+    intro h; apply mul_nc_NonnegCoeffs
+    next => apply ih; assumption
+    next => assumption
 
 theorem Poly.denoteS_mul_go {α} [CommSemiring α] (ctx : Context α) (p₁ p₂ acc : Poly)
     : p₁.NonnegCoeffs → p₂.NonnegCoeffs → acc.NonnegCoeffs → (mul.go p₂ p₁ acc).denoteS ctx = acc.denoteS ctx + p₁.denoteS ctx * p₂.denoteS ctx := by
@@ -342,6 +435,27 @@ theorem Poly.denoteS_mul {α} [CommSemiring α] (ctx : Context α) (p₁ p₂ :
   intro h₁ h₂
   simp [mul, denoteS_mul_go, denoteS, Poly.num_zero_NonnegCoeffs, *]
 
+theorem Poly.denoteS_mul_nc_go {α} [Semiring α] (ctx : Context α) (p₁ p₂ acc : Poly)
+    : p₁.NonnegCoeffs → p₂.NonnegCoeffs → acc.NonnegCoeffs → (mul_nc.go p₂ p₁ acc).denoteS ctx = acc.denoteS ctx + p₁.denoteS ctx * p₂.denoteS ctx := by
+  fun_induction mul_nc.go <;> intro h₁ h₂ h₃
+  next k =>
+    cases h₁; rename_i h₁
+    have := p₂.mulConst_NonnegCoeffs h₁ h₂
+    simp [denoteS, denoteS_combine, denoteS_mulConst, *]
+  next acc a m p ih =>
+    cases h₁; rename_i h₁ h₁'
+    have := p₂.mulMon_nc_NonnegCoeffs m h₁ h₂
+    have := acc.combine_NonnegCoeffs h₃ this
+    replace ih := ih h₁' h₂ this
+    rw [ih, denoteS_combine, denoteS_mulMon_nc]
+    simp [denoteS, add_assoc, right_distrib]
+    all_goals assumption
+
+theorem Poly.denoteS_mul_nc {α} [Semiring α] (ctx : Context α) (p₁ p₂ : Poly)
+    : p₁.NonnegCoeffs → p₂.NonnegCoeffs → (mul_nc p₁ p₂).denoteS ctx = p₁.denoteS ctx * p₂.denoteS ctx := by
+  intro h₁ h₂
+  simp [mul_nc, denoteS_mul_nc_go, denoteS, Poly.num_zero_NonnegCoeffs, *]
+
 theorem Poly.denoteS_pow {α} [CommSemiring α] (ctx : Context α) (p : Poly) (k : Nat)
    : p.NonnegCoeffs → (pow p k).denoteS ctx = p.denoteS ctx ^ k := by
  fun_induction pow <;> intro h₁
@@ -353,13 +467,19 @@ theorem Poly.denoteS_pow {α} [CommSemiring α] (ctx : Context α) (p : Poly) (k
   assumption
   apply Poly.pow_NonnegCoeffs; assumption
 
-end CommRing
+theorem Poly.denoteS_pow_nc {α} [Semiring α] (ctx : Context α) (p : Poly) (k : Nat)
+   : p.NonnegCoeffs → (pow_nc p k).denoteS ctx = p.denoteS ctx ^ k := by
+ fun_induction pow_nc <;> intro h₁
+ next => simp [denoteS, pow_zero]
+ next => simp [pow_succ, pow_zero]
+ next ih =>
+  replace ih := ih h₁
+  rw [denoteS_mul_nc, ih, pow_succ]
+  apply Poly.pow_nc_NonnegCoeffs; assumption
+  assumption
 
-namespace Ring.OfSemiring
-open CommRing
-
-theorem Expr.toPoly_NonnegCoeffs {e : Expr} : e.toPoly.NonnegCoeffs := by
-  fun_induction toPoly
+theorem Expr.toPolyS_NonnegCoeffs {e : Expr} : e.toPolyS.NonnegCoeffs := by
+  fun_induction toPolyS
   next => constructor; apply Int.natCast_nonneg
   next => simp [Poly.ofVar, Poly.ofMon]; constructor; decide; constructor; decide
   next => apply Poly.combine_NonnegCoeffs <;> assumption
@@ -367,29 +487,89 @@ theorem Expr.toPoly_NonnegCoeffs {e : Expr} : e.toPoly.NonnegCoeffs := by
   next => constructor; apply Int.pow_nonneg; apply Int.natCast_nonneg
   next => constructor; decide; constructor; decide
   next => apply Poly.pow_NonnegCoeffs; assumption
+  next => constructor; apply Int.ofNat_zero_le
+  all_goals exact Poly.num_zero_NonnegCoeffs
 
-theorem Expr.denoteS_toPoly {α} [CommSemiring α] (ctx : Context α) (e : Expr)
-   : e.toPoly.denoteS ctx = e.denote ctx := by
-  fun_induction toPoly
-    <;> simp [denote, Poly.denoteS, Poly.denoteS_ofVar, denoteSInt_eq, Semiring.ofNat_eq_natCast]
-  next => simp [CommRing.Var.denote, Var.denote]
-  next ih₁ ih₂ => rw [Poly.denoteS_combine, ih₁, ih₂] <;> apply toPoly_NonnegCoeffs
-  next ih₁ ih₂ => rw [Poly.denoteS_mul, ih₁, ih₂] <;> apply toPoly_NonnegCoeffs
-  next => rw [Int.toNat_pow_of_nonneg, Semiring.natCast_pow, Int.toNat_natCast]; apply Int.natCast_nonneg
-  next =>
-    simp [Poly.ofMon, Poly.denoteS, denoteSInt_eq, Power.denote_eq, Mon.denote,
-          Semiring.natCast_zero, Semiring.natCast_one, Semiring.one_mul, Semiring.add_zero,
-          CommRing.Var.denote, Var.denote, Semiring.mul_one]
-  next ih => rw [Poly.denoteS_pow, ih]; apply toPoly_NonnegCoeffs
+attribute [local simp] Expr.toPolyS_NonnegCoeffs
+
+theorem Expr.toPolyS_nc_NonnegCoeffs {e : Expr} : e.toPolyS_nc.NonnegCoeffs := by
+  fun_induction toPolyS_nc
+  next => constructor; apply Int.natCast_nonneg
+  next => simp [Poly.ofVar, Poly.ofMon]; constructor; decide; constructor; decide
+  next => apply Poly.combine_NonnegCoeffs <;> assumption
+  next => apply Poly.mul_nc_NonnegCoeffs <;> assumption
+  next => constructor; apply Int.pow_nonneg; apply Int.natCast_nonneg
+  next => constructor; decide; constructor; decide
+  next => apply Poly.pow_nc_NonnegCoeffs; assumption
+  next => constructor; apply Int.ofNat_zero_le
+  all_goals exact Poly.num_zero_NonnegCoeffs
+
+attribute [local simp] Expr.toPolyS_nc_NonnegCoeffs
+
+theorem Expr.denoteS_toPolyS {α} [CommSemiring α] (ctx : Context α) (e : Expr)
+   : e.toPolyS.denoteS ctx = e.denoteS ctx := by
+  fun_induction toPolyS <;> simp [denoteS, Poly.denoteS, Poly.denoteS_ofVar, denoteSInt_eq]
+  next => simp [Semiring.ofNat_eq_natCast]
+  next => simp [Poly.denoteS_combine] <;> simp [*]
+  next => simp [Poly.denoteS_mul] <;> simp [*]
+  next => rw [Int.toNat_pow_of_nonneg, Semiring.natCast_pow, Int.toNat_natCast, ← Semiring.ofNat_eq_natCast]
+          apply Int.natCast_nonneg
+  next => simp [Poly.ofMon, Poly.denoteS, denoteSInt_eq, Power.denote_eq, Mon.denote,
+                Semiring.natCast_zero, Semiring.natCast_one, Semiring.one_mul,
+                CommRing.Var.denote, Var.denote, Semiring.mul_one]
+  next ih => rw [Poly.denoteS_pow, ih]; apply toPolyS_NonnegCoeffs
+  next => simp [Semiring.natCast_eq_ofNat]
+
+theorem Expr.denoteS_toPolyS_nc {α} [Semiring α] (ctx : Context α) (e : Expr)
+   : e.toPolyS_nc.denoteS ctx = e.denoteS ctx := by
+  fun_induction Expr.toPolyS_nc <;> simp [denoteS, Poly.denoteS, Poly.denoteS_ofVar, denoteSInt_eq]
+  next => simp [Semiring.ofNat_eq_natCast]
+  next => simp [Poly.denoteS_combine] <;> simp [*]
+  next => simp [Poly.denoteS_mul_nc] <;> simp [*]
+  next => rw [Int.toNat_pow_of_nonneg, Semiring.natCast_pow, Int.toNat_natCast, ← Semiring.ofNat_eq_natCast]
+          apply Int.natCast_nonneg
+  next => simp [Poly.ofMon, Poly.denoteS, denoteSInt_eq, Power.denote_eq, Mon.denote,
+                Semiring.natCast_zero, Semiring.natCast_one, Semiring.one_mul,
+                CommRing.Var.denote, Var.denote, Semiring.mul_one]
+  next ih => rw [Poly.denoteS_pow_nc, ih]; apply toPolyS_nc_NonnegCoeffs
+  next => simp [Semiring.natCast_eq_ofNat]
 
 def eq_normS_cert (lhs rhs : Expr) : Bool :=
-  lhs.toPoly == rhs.toPoly
+  lhs.toPolyS == rhs.toPolyS
 
 theorem eq_normS {α} [CommSemiring α] (ctx : Context α) (lhs rhs : Expr)
-    : eq_normS_cert lhs rhs → lhs.denote ctx = rhs.denote ctx := by
+    : eq_normS_cert lhs rhs → lhs.denoteS ctx = rhs.denoteS ctx := by
   simp [eq_normS_cert]; intro h
   replace h := congrArg (Poly.denoteS ctx) h
-  simp [Expr.denoteS_toPoly, *] at h
+  simp [Expr.denoteS_toPolyS, *] at h
   assumption
 
-end Lean.Grind.Ring.OfSemiring
+def eq_normS_nc_cert (lhs rhs : Expr) : Bool :=
+  lhs.toPolyS_nc == rhs.toPolyS_nc
+
+theorem eq_normS_nc {α} [Semiring α] (ctx : Context α) (lhs rhs : Expr)
+    : eq_normS_nc_cert lhs rhs → lhs.denoteS ctx = rhs.denoteS ctx := by
+  simp [eq_normS_nc_cert]; intro h
+  replace h := congrArg (Poly.denoteS ctx) h
+  simp [Expr.denoteS_toPolyS_nc, *] at h
+  assumption
+
+end CommRing
+
+namespace Ring.OfSemiring
+open CommRing
+
+theorem of_eq {α} [Semiring α] (ctx : Context α) (lhs rhs : Expr)
+    : lhs.denoteS ctx = rhs.denoteS ctx → lhs.denoteSAsRing ctx = rhs.denoteSAsRing ctx := by
+  intro h; replace h := congrArg toQ h
+  simpa [← Expr.denoteAsRing_eq] using h
+
+theorem of_diseq {α} [Semiring α] [AddRightCancel α] (ctx : Context α) (lhs rhs : Expr)
+    : lhs.denoteS ctx ≠ rhs.denoteS ctx → lhs.denoteSAsRing ctx ≠ rhs.denoteSAsRing ctx := by
+  intro h₁ h₂
+  simp [Expr.denoteAsRing_eq] at h₂
+  replace h₂ := toQ_inj h₂
+  contradiction
+
+end Ring.OfSemiring
+end Lean.Grind

src/Init/Grind/Ring/CommSolver.lean

@@ -4,27 +4,32 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 module
-
 prelude
 public import Init.Data.Nat.Lemmas
 public import Init.Data.Int.LemmasAux
 public import Init.Data.Hashable
 public import Init.Data.Ord.Basic
-import all Init.Data.Ord.Basic
 public import Init.Data.RArray
 public import Init.Grind.Ring.Basic
 public import Init.Grind.Ring.Field
 public import Init.Grind.Ordered.Ring
 public import Init.GrindInstances.Ring.Int
+import all Init.Data.Ord.Basic
+import Init.LawfulBEqTactics
 
 @[expose] public section
 
-open Std
+namespace Lean.Grind.CommRing
+/-!
+Data-structures, definitions and theorems for implementing the
+`grind` solver and normalizer for commutative rings and its extensions (e.g., fields,
+commutative semirings, etc.)
 
-namespace Lean.Grind
+The solver uses proof-by-reflection.
+-/
+open Std
 -- These are no longer global instances, so we need to turn them on here.
 attribute [local instance] Semiring.natCast Ring.intCast
-namespace CommRing
 abbrev Var := Nat
 
 inductive Expr where
@@ -41,18 +46,15 @@ inductive Expr where
 
 abbrev Context (α : Type u) := RArray α
 
-@[expose]
 def Var.denote {α} (ctx : Context α) (v : Var) : α :=
   ctx.get v
 
-@[expose]
 noncomputable def denoteInt {α} [Ring α] (k : Int) : α :=
   Bool.rec
     (OfNat.ofNat (α := α) k.natAbs)
     (- OfNat.ofNat (α := α) k.natAbs)
     (Int.blt' k 0)
 
-@[expose]
 noncomputable def Expr.denote {α} [Ring α] (ctx : Context α) (e : Expr) : α :=
   Expr.rec
     (fun k => denoteInt k)
@@ -69,11 +71,7 @@ noncomputable def Expr.denote {α} [Ring α] (ctx : Context α) (e : Expr) : α
 structure Power where
   x : Var
   k : Nat
-  deriving BEq, Repr, Inhabited, Hashable
-
-instance : LawfulBEq Power where
-  eq_of_beq {a} := by cases a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
-  rfl := by intro a; cases a <;> simp! [BEq.beq]
+  deriving BEq, ReflBEq, LawfulBEq, Repr, Inhabited, Hashable
 
 protected noncomputable def Power.beq' (pw₁ pw₂ : Power) : Bool :=
   Power.rec (fun x₁ k₁ => Power.rec (fun x₂ k₂ => Nat.beq x₁ x₂ && Nat.beq k₁ k₂) pw₂) pw₁
@@ -81,11 +79,9 @@ protected noncomputable def Power.beq' (pw₁ pw₂ : Power) : Bool :=
 @[simp] theorem Power.beq'_eq (pw₁ pw₂ : Power) : pw₁.beq' pw₂ = (pw₁ = pw₂) := by
   cases pw₁; cases pw₂; simp [Power.beq']
 
-@[expose]
 def Power.varLt (p₁ p₂ : Power) : Bool :=
   p₁.x.blt p₂.x
 
-@[expose]
 def Power.denote {α} [Semiring α] (ctx : Context α) : Power → α
   | {x, k} =>
     match k with
@@ -96,18 +92,7 @@ def Power.denote {α} [Semiring α] (ctx : Context α) : Power → α
 inductive Mon where
   | unit
   | mult (p : Power) (m : Mon)
-  deriving BEq, Repr, Inhabited, Hashable
-
-instance : LawfulBEq Mon where
-  eq_of_beq {a} := by
-    induction a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
-    next p₁ m₁ p₂ m₂ ih =>
-      cases p₁ <;> cases p₂ <;> simp <;> intros <;> simp [*]
-      next h => exact ih h
-  rfl := by
-    intro a
-    induction a <;> simp! [BEq.beq]
-    assumption
+  deriving BEq, ReflBEq, LawfulBEq, Repr, Inhabited, Hashable
 
 protected noncomputable def Mon.beq' (m₁ : Mon) : Mon → Bool :=
   Mon.rec
@@ -121,12 +106,10 @@ protected noncomputable def Mon.beq' (m₁ : Mon) : Mon → Bool :=
   simp [← ih m₂, ← Bool.and'_eq_and]
   rfl
 
-@[expose]
 def Mon.denote {α} [Semiring α] (ctx : Context α) : Mon → α
   | unit => 1
   | .mult p m => p.denote ctx * denote ctx m
 
-@[expose]
 def Mon.denote' {α} [Semiring α] (ctx : Context α) (m : Mon) : α :=
   match m with
   | .unit => 1
@@ -137,17 +120,14 @@ where
     | .unit => acc
     | .mult pw m => go m (acc * (pw.denote ctx))
 
-@[expose]
 def Mon.ofVar (x : Var) : Mon :=
   .mult { x, k := 1 } .unit
 
-@[expose]
 def Mon.concat (m₁ m₂ : Mon) : Mon :=
   match m₁ with
   | .unit => m₂
   | .mult pw m₁ => .mult pw (concat m₁ m₂)
 
-@[expose]
 def Mon.mulPow (pw : Power) (m : Mon) : Mon :=
   match m with
   | .unit =>
@@ -160,15 +140,23 @@ def Mon.mulPow (pw : Power) (m : Mon) : Mon :=
     else
       .mult { x := pw.x, k := pw.k + pw'.k } m
 
-@[expose]
+-- **Note**: We use the `_nc` suffix for functions for the non-commutative case
+
+def Mon.mulPow_nc (pw : Power) (m : Mon) : Mon :=
+  match m with
+  | .unit      => .mult pw .unit
+  | .mult pw' m =>
+    bif pw.x == pw'.x then
+      .mult { x := pw.x, k := pw.k + pw'.k } m
+    else
+      .mult pw (.mult pw' m)
+
 def Mon.length : Mon → Nat
   | .unit => 0
   | .mult _ m => 1 + length m
 
-@[expose]
 def hugeFuel := 1000000
 
-@[expose]
 def Mon.mul (m₁ m₂ : Mon) : Mon :=
   -- We could use `m₁.length + m₂.length` to avoid hugeFuel
   go hugeFuel m₁ m₂
@@ -188,18 +176,21 @@ where
         else
           .mult { x := pw₁.x, k := pw₁.k + pw₂.k } (go fuel m₁ m₂)
 
-@[expose]
+def Mon.mul_nc (m₁ m₂ : Mon) : Mon :=
+  match m₁ with
+  | .unit          => m₂
+  | .mult pw .unit => m₂.mulPow_nc pw
+  | .mult pw m₁    => .mult pw (mul_nc m₁ m₂)
+
 def Mon.degree : Mon → Nat
   | .unit => 0
   | .mult pw m => pw.k + degree m
 
-@[expose]
 def Var.revlex (x y : Var) : Ordering :=
   bif x.blt y then .gt
   else bif y.blt x then .lt
   else .eq
 
-@[expose]
 def powerRevlex (k₁ k₂ : Nat) : Ordering :=
   bif k₁.blt k₂ then .gt
   else bif k₂.blt k₁ then .lt
@@ -212,11 +203,9 @@ theorem powerRevlex_k_eq_powerRevlex (k₁ k₂ : Nat) : powerRevlex_k k₁ k₂
   simp [powerRevlex_k, powerRevlex, cond] <;> split <;> simp [*]
   split <;> simp [*]
 
-@[expose]
 def Power.revlex (p₁ p₂ : Power) : Ordering :=
   p₁.x.revlex p₂.x |>.then (powerRevlex p₁.k p₂.k)
 
-@[expose]
 def Mon.revlexWF (m₁ m₂ : Mon) : Ordering :=
   match m₁, m₂ with
   | .unit, .unit => .eq
@@ -230,7 +219,6 @@ def Mon.revlexWF (m₁ m₂ : Mon) : Ordering :=
     else
       revlexWF (.mult pw₁ m₁) m₂ |>.then .gt
 
-@[expose]
 def Mon.revlexFuel (fuel : Nat) (m₁ m₂ : Mon) : Ordering :=
   match fuel with
   | 0 =>
@@ -250,11 +238,9 @@ def Mon.revlexFuel (fuel : Nat) (m₁ m₂ : Mon) : Ordering :=
       else
         revlexFuel fuel (.mult pw₁ m₁) m₂ |>.then .gt
 
-@[expose]
 def Mon.revlex (m₁ m₂ : Mon) : Ordering :=
   revlexFuel hugeFuel m₁ m₂
 
-@[expose]
 def Mon.grevlex (m₁ m₂ : Mon) : Ordering :=
   compare m₁.degree m₂.degree |>.then (revlex m₁ m₂)
 
@@ -328,7 +314,7 @@ theorem Mon.grevlex_k_eq_grevlex (m₁ m₂ : Mon) : m₁.grevlex_k m₂ = m₁.
 inductive Poly where
   | num (k : Int)
   | add (k : Int) (v : Mon) (p : Poly)
-  deriving BEq, Repr, Inhabited, Hashable
+  deriving BEq, ReflBEq, LawfulBEq, Repr, Inhabited, Hashable
 
 protected noncomputable def Poly.beq' (p₁ : Poly) : Poly → Bool :=
   Poly.rec
@@ -346,27 +332,11 @@ protected noncomputable def Poly.beq' (p₁ : Poly) : Poly → Bool :=
   intro _ _; subst k₁ m₁
   simp [← ih p₂, ← Bool.and'_eq_and]; rfl
 
-instance : LawfulBEq Poly where
-  eq_of_beq {a} := by
-    induction a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
-    intro h₁ h₂ h₃
-    rename_i m₁ p₁ _ m₂ p₂ ih
-    replace h₂ : m₁ == m₂ := h₂
-    simp [ih h₃, eq_of_beq h₂]
-  rfl := by
-    intro a
-    induction a <;> simp! [BEq.beq]
-    rename_i k m p ih
-    change m == m ∧ p == p
-    simp [ih]
-
-@[expose]
 def Poly.denote [Ring α] (ctx : Context α) (p : Poly) : α :=
   match p with
   | .num k => Int.cast k
   | .add k m p => k • (m.denote ctx) + denote ctx p
 
-@[expose]
 def Poly.denote' [Ring α] (ctx : Context α) (p : Poly) : α :=
   match p with
   | .num k => Int.cast k
@@ -384,21 +354,17 @@ where
     | .num k => acc + Int.cast k
     | .add k m p => go p (acc + denoteTerm k m)
 
-@[expose]
 def Poly.ofMon (m : Mon) : Poly :=
   .add 1 m (.num 0)
 
-@[expose]
 def Poly.ofVar (x : Var) : Poly :=
   ofMon (Mon.ofVar x)
 
-@[expose]
 def Poly.isSorted : Poly → Bool
   | .num _ => true
   | .add _ _ (.num _) => true
   | .add _ m₁ (.add k m₂ p) => m₁.grevlex m₂ == .gt && (Poly.add k m₂ p).isSorted
 
-@[expose]
 def Poly.addConst (p : Poly) (k : Int) : Poly :=
   bif k == 0 then
     p
@@ -424,7 +390,6 @@ theorem Poly.addConst_k_eq_addConst (p : Poly) (k : Int) : addConst_k p k = addC
     induction p <;> simp [addConst.go]
     next ih => rw [← ih]
 
-@[expose]
 def Poly.insert (k : Int) (m : Mon) (p : Poly) : Poly :=
   bif k == 0 then
     p
@@ -446,13 +411,11 @@ where
       | .gt => .add k m (.add k' m' p)
       | .lt => .add k' m' (go p)
 
-@[expose]
 def Poly.concat (p₁ p₂ : Poly) : Poly :=
   match p₁ with
   | .num k₁ => p₂.addConst k₁
   | .add k m p₁ => .add k m (concat p₁ p₂)
 
-@[expose]
 def Poly.mulConst (k : Int) (p : Poly) : Poly :=
   bif k == 0 then
     .num 0
@@ -491,7 +454,6 @@ noncomputable def Poly.mulConst_k (k : Int) (p : Poly) : Poly :=
       next => rfl
       next k m p ih => simp [mulConst.go, ← ih]
 
-@[expose]
 def Poly.mulMon (k : Int) (m : Mon) (p : Poly) : Poly :=
   bif k == 0 then
     .num 0
@@ -545,7 +507,19 @@ noncomputable def Poly.mulMon_k (k : Int) (m : Mon) (p : Poly) : Poly :=
           simp [h]
       next ih => simp [← ih]
 
-@[expose]
+def Poly.mulMon_nc (k : Int) (m : Mon) (p : Poly) : Poly :=
+  bif k == 0 then
+    .num 0
+  else bif m == .unit then
+    p.mulConst k
+  else
+    go p (.num 0)
+where
+  go (p : Poly) (acc : Poly) : Poly :=
+    match p with
+    | .num k' => acc.insert (k*k') m
+    | .add k' m' p => go p (acc.insert (k*k') (m.mul_nc m'))
+
 def Poly.combine (p₁ p₂ : Poly) : Poly :=
   go hugeFuel p₁ p₂
 where
@@ -609,7 +583,6 @@ noncomputable def Poly.combine_k : Poly → Poly → Poly :=
     next h => simp [h]; rw [← ih p₁ (add k₂ m₂ p₂)]; rfl
     next h => simp [h]; rw [← ih (add k₁ m₁ p₁) p₂]; rfl
 
-@[expose]
 def Poly.mul (p₁ : Poly) (p₂ : Poly) : Poly :=
   go p₁ (.num 0)
 where
@@ -618,14 +591,26 @@ where
     | .num k => acc.combine (p₂.mulConst k)
     | .add k m p₁ => go p₁ (acc.combine (p₂.mulMon k m))
 
-@[expose]
+def Poly.mul_nc (p₁ : Poly) (p₂ : Poly) : Poly :=
+  go p₁ (.num 0)
+where
+  go (p₁ : Poly) (acc : Poly) : Poly :=
+    match p₁ with
+    | .num k => acc.combine (p₂.mulConst k)
+    | .add k m p₁ => go p₁ (acc.combine (p₂.mulMon_nc k m))
+
 def Poly.pow (p : Poly) (k : Nat) : Poly :=
   match k with
   | 0 => .num 1
   | 1 => p
   | k+1 => p.mul (pow p k)
 
-@[expose]
+def Poly.pow_nc (p : Poly) (k : Nat) : Poly :=
+  match k with
+  | 0 => .num 1
+  | 1 => p
+  | k+1 => (pow_nc p k).mul_nc p
+
 def Expr.toPoly : Expr → Poly
   | .num k     => .num k
   | .intCast k => .num k
@@ -645,7 +630,7 @@ def Expr.toPoly : Expr → Poly
     | .var x => Poly.ofMon (.mult {x, k} .unit)
     | _ => a.toPoly.pow k
 
-@[expose] noncomputable def Expr.toPoly_k (e : Expr) : Poly :=
+noncomputable def Expr.toPoly_k (e : Expr) : Poly :=
   Expr.rec
     (fun k => .num k) (fun k => .num k) (fun k => .num k)
     (fun x => .ofVar x)
@@ -691,6 +676,25 @@ def Expr.toPoly : Expr → Poly
             | x => a.toPoly.pow k
       cases a <;> try simp [*]
 
+def Expr.toPoly_nc : Expr → Poly
+  | .num k     => .num k
+  | .intCast k => .num k
+  | .natCast k => .num k
+  | .var x     => Poly.ofVar x
+  | .add a b   => a.toPoly_nc.combine b.toPoly_nc
+  | .mul a b   => a.toPoly_nc.mul_nc b.toPoly_nc
+  | .neg a     => a.toPoly_nc.mulConst (-1)
+  | .sub a b   => a.toPoly_nc.combine (b.toPoly_nc.mulConst (-1))
+  | .pow a k   =>
+    bif k == 0 then
+      .num 1
+    else  match a with
+    | .num n => .num (n^k)
+    | .intCast n => .num (n^k)
+    | .natCast n => .num (n^k)
+    | .var x => Poly.ofMon (.mult {x, k} .unit)
+    | _ => a.toPoly_nc.pow_nc k
+
 def Poly.normEq0 (p : Poly) (c : Nat) : Poly :=
   match p with
   | .num a =>
@@ -707,13 +711,11 @@ Once we can specialize definitions before they reach the kernel,
 we can merge the two versions. Until then, the `IsCharP` definitions will carry the `C` suffix.
 We use them whenever we can infer the characteristic using type class instance synthesis.
 -/
-@[expose]
 def Poly.addConstC (p : Poly) (k : Int) (c : Nat) : Poly :=
   match p with
   | .num k' => .num ((k' + k) % c)
   | .add k' m p => .add k' m (addConstC p k c)
 
-@[expose]
 def Poly.insertC (k : Int) (m : Mon) (p : Poly) (c : Nat) : Poly :=
   let k := k % c
   bif k == 0 then
@@ -734,7 +736,6 @@ where
       | .gt => .add k m (.add k' m' p)
       | .lt => .add k' m' (go k p)
 
-@[expose]
 def Poly.mulConstC (k : Int) (p : Poly) (c : Nat) : Poly :=
   let k := k % c
   bif k == 0 then
@@ -753,7 +754,6 @@ where
     else
       .add k m (go p)
 
-@[expose]
 def Poly.mulMonC (k : Int) (m : Mon) (p : Poly) (c : Nat) : Poly :=
   let k := k % c
   bif k == 0 then
@@ -777,7 +777,20 @@ where
      else
        .add k (m.mul m') (go p)
 
-@[expose]
+def Poly.mulMonC_nc (k : Int) (m : Mon) (p : Poly) (c : Nat) : Poly :=
+  let k := k % c
+  bif k == 0 then
+    .num 0
+  else bif m == .unit then
+    p.mulConstC k c
+  else
+    go p (.num 0)
+where
+  go (p : Poly) (acc : Poly) : Poly :=
+    match p with
+    | .num k' => acc.insert (k*k' % c) m
+    | .add k' m' p => go p (acc.insert (k*k' % c) (m.mul_nc m'))
+
 def Poly.combineC (p₁ p₂ : Poly) (c : Nat) : Poly :=
   go hugeFuel p₁ p₂
 where
@@ -799,7 +812,6 @@ where
         | .gt => .add k₁ m₁ (go fuel p₁ (.add k₂ m₂ p₂))
         | .lt => .add k₂ m₂ (go fuel (.add k₁ m₁ p₁) p₂)
 
-@[expose]
 def Poly.mulC (p₁ : Poly) (p₂ : Poly) (c : Nat) : Poly :=
   go p₁ (.num 0)
 where
@@ -808,14 +820,26 @@ where
     | .num k => acc.combineC (p₂.mulConstC k c) c
     | .add k m p₁ => go p₁ (acc.combineC (p₂.mulMonC k m c) c)
 
-@[expose]
+def Poly.mulC_nc (p₁ : Poly) (p₂ : Poly) (c : Nat) : Poly :=
+  go p₁ (.num 0)
+where
+  go (p₁ : Poly) (acc : Poly) : Poly :=
+    match p₁ with
+    | .num k => acc.combineC (p₂.mulConstC k c) c
+    | .add k m p₁ => go p₁ (acc.combineC (p₂.mulMonC_nc k m c) c)
+
 def Poly.powC (p : Poly) (k : Nat) (c : Nat) : Poly :=
   match k with
   | 0 => .num 1
   | 1 => p
   | k+1 => p.mulC (powC p k c) c
 
-@[expose]
+def Poly.powC_nc (p : Poly) (k : Nat) (c : Nat) : Poly :=
+  match k with
+  | 0 => .num 1
+  | 1 => p
+  | k+1 => (powC_nc p k c).mulC_nc p c
+
 def Expr.toPolyC (e : Expr) (c : Nat) : Poly :=
   go e
 where
@@ -836,6 +860,26 @@ where
       | .var x => Poly.ofMon (.mult {x, k} .unit)
       | _ => (go a).powC k c
 
+def Expr.toPolyC_nc (e : Expr) (c : Nat) : Poly :=
+  go e
+where
+  go : Expr → Poly
+    | .num k     => .num (k % c)
+    | .natCast k => .num (k % c)
+    | .intCast k => .num (k % c)
+    | .var x     => Poly.ofVar x
+    | .add a b   => (go a).combineC (go b) c
+    | .mul a b   => (go a).mulC_nc (go b) c
+    | .neg a     => (go a).mulConstC (-1) c
+    | .sub a b   => (go a).combineC ((go b).mulConstC (-1) c) c
+    | .pow a k   =>
+      bif k == 0 then
+        .num 1
+      else match a with
+      | .num n => .num ((n^k) % c)
+      | .var x => Poly.ofMon (.mult {x, k} .unit)
+      | _ => (go a).powC_nc k c
+
 /-!
 Theorems for justifying the procedure for commutative rings in `grind`.
 -/
@@ -845,7 +889,7 @@ open Semiring hiding add_zero add_comm add_assoc
 open Ring hiding sub_eq_add_neg
 open CommSemiring
 
-theorem denoteInt_eq {α} [CommRing α] (k : Int) : denoteInt (α := α) k = k := by
+theorem denoteInt_eq {α} [Ring α] (k : Int) : denoteInt (α := α) k = k := by
   simp [denoteInt] <;> cases h : k.blt' 0 <;> simp <;> simp at h
   next h => rw [ofNat_eq_natCast, ← intCast_natCast, ← Int.eq_natAbs_of_nonneg h]
   next h => rw [ofNat_eq_natCast, ← intCast_natCast, ← Ring.intCast_neg, ← Int.eq_neg_natAbs_of_nonpos (Int.le_of_lt h)]
@@ -888,6 +932,13 @@ theorem Mon.denote_mulPow {α} [CommSemiring α] (ctx : Context α) (p : Power)
     have := eq_of_blt_false h₁ h₂
     simp [Power.denote_eq, pow_add, mul_assoc, this]
 
+theorem Mon.denote_mulPow_nc {α} [Semiring α] (ctx : Context α) (p : Power) (m : Mon)
+    : denote ctx (mulPow_nc p m) = p.denote ctx * m.denote ctx := by
+  fun_cases mulPow_nc <;> simp [denote, *]
+  next h =>
+    simp at h
+    simp [Power.denote_eq, pow_add, mul_assoc, h]
+
 theorem Mon.denote_mul {α} [CommSemiring α] (ctx : Context α) (m₁ m₂ : Mon)
     : denote ctx (mul m₁ m₂) = m₁.denote ctx * m₂.denote ctx := by
   unfold mul
@@ -899,6 +950,10 @@ theorem Mon.denote_mul {α} [CommSemiring α] (ctx : Context α) (m₁ m₂ : Mo
     have := eq_of_blt_false h₁ h₂
     simp [Power.denote_eq, pow_add, this]
 
+theorem Mon.denote_mul_nc {α} [Semiring α] (ctx : Context α) (m₁ m₂ : Mon)
+    : denote ctx (mul_nc m₁ m₂) = m₁.denote ctx * m₂.denote ctx := by
+  fun_induction mul_nc <;> simp [denote, Semiring.one_mul, Semiring.mul_one, denote_mulPow_nc, Semiring.mul_assoc, *]
+
 theorem Var.eq_of_revlex {x₁ x₂ : Var} : x₁.revlex x₂ = .eq → x₁ = x₂ := by
   simp [revlex, cond_eq_if] <;> split <;> simp
   next h₁ => intro h₂; exact Nat.le_antisymm h₂ (Nat.ge_of_not_lt h₁)
@@ -954,15 +1009,15 @@ theorem Poly.denote'_eq_denote {α} [Ring α] (ctx : Context α) (p : Poly) : p.
     fun_induction denote'.go <;> simp [denote, *, Ring.intCast_zero, Semiring.add_zero, denoteTerm_eq]
     next ih => simp [denoteTerm_eq] at ih; simp [ih, Semiring.add_assoc, zsmul_eq_intCast_mul]
 
-theorem Poly.denote_ofMon {α} [CommRing α] (ctx : Context α) (m : Mon)
+theorem Poly.denote_ofMon {α} [Ring α] (ctx : Context α) (m : Mon)
     : denote ctx (ofMon m) = m.denote ctx := by
   simp [ofMon, denote, intCast_one, intCast_zero, one_mul, add_zero, zsmul_eq_intCast_mul]
 
-theorem Poly.denote_ofVar {α} [CommRing α] (ctx : Context α) (x : Var)
+theorem Poly.denote_ofVar {α} [Ring α] (ctx : Context α) (x : Var)
     : denote ctx (ofVar x) = x.denote ctx := by
   simp [ofVar, denote_ofMon, Mon.denote_ofVar]
 
-theorem Poly.denote_addConst {α} [CommRing α] (ctx : Context α) (p : Poly) (k : Int) : (addConst p k).denote ctx = p.denote ctx + k := by
+theorem Poly.denote_addConst {α} [Ring α] (ctx : Context α) (p : Poly) (k : Int) : (addConst p k).denote ctx = p.denote ctx + k := by
   simp [addConst, cond_eq_if]; split
   next => simp [*, intCast_zero, add_zero]
   next =>
@@ -970,7 +1025,7 @@ theorem Poly.denote_addConst {α} [CommRing α] (ctx : Context α) (p : Poly) (k
     next => rw [intCast_add]
     next => simp [add_comm, add_left_comm]
 
-theorem Poly.denote_insert {α} [CommRing α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+theorem Poly.denote_insert {α} [Ring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
     : (insert k m p).denote ctx = k * m.denote ctx + p.denote ctx := by
   simp [insert, cond_eq_if] <;> split
   next => simp [*, intCast_zero, zero_mul, zero_add]
@@ -987,13 +1042,13 @@ theorem Poly.denote_insert {α} [CommRing α] (ctx : Context α) (k : Int) (m :
       next =>
         rw [add_left_comm]
 
-theorem Poly.denote_concat {α} [CommRing α] (ctx : Context α) (p₁ p₂ : Poly)
+theorem Poly.denote_concat {α} [Ring α] (ctx : Context α) (p₁ p₂ : Poly)
     : (concat p₁ p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
   fun_induction concat <;> simp [*, denote_addConst, denote]
   next => rw [add_comm]
   next => rw [add_assoc]
 
-theorem Poly.denote_mulConst {α} [CommRing α] (ctx : Context α) (k : Int) (p : Poly)
+theorem Poly.denote_mulConst {α} [Ring α] (ctx : Context α) (k : Int) (p : Poly)
     : (mulConst k p).denote ctx = k * p.denote ctx := by
   simp [mulConst, cond_eq_if] <;> split
   next => simp [denote, *, intCast_zero, zero_mul]
@@ -1017,7 +1072,28 @@ theorem Poly.denote_mulMon {α} [CommRing α] (ctx : Context α) (k : Int) (m :
       next => simp [intCast_mul, intCast_zero, add_zero, mul_comm, mul_left_comm, mul_assoc]
       next => simp [Mon.denote_mul, intCast_mul, left_distrib, mul_left_comm, mul_assoc]
 
-theorem Poly.denote_combine {α} [CommRing α] (ctx : Context α) (p₁ p₂ : Poly)
+theorem Poly.denote_mulMon_nc_go {α} [Ring α] (ctx : Context α) (k : Int) (m : Mon) (p acc : Poly)
+    : (mulMon_nc.go k m p acc).denote ctx = k * m.denote ctx * p.denote ctx + acc.denote ctx := by
+  fun_induction mulMon_nc.go <;> simp [denote, denote_insert, zsmul_eq_intCast_mul]
+  next => rw [Ring.intCast_mul, Semiring.mul_assoc, Semiring.mul_assoc, ← Ring.intCast_mul_comm]
+  next ih =>
+    rw [ih, denote_insert, Mon.denote_mul_nc, Semiring.left_distrib, Ring.intCast_mul]
+    rw [Ring.intCast_mul_left_comm]; simp [← Semiring.mul_assoc]
+    conv => enter [1, 2, 1, 1, 1]; rw [Ring.intCast_mul_comm]
+    simp [Semiring.add_assoc, Semiring.add_comm, add_left_comm]
+
+theorem Poly.denote_mulMon_nc {α} [Ring α] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+    : (mulMon_nc k m p).denote ctx = k * m.denote ctx * p.denote ctx := by
+  simp [mulMon_nc, cond_eq_if] <;> split
+  next => simp [denote, *, intCast_zero, zero_mul]
+  next =>
+    split
+    next h =>
+      simp at h; simp [*, Mon.denote, mul_one, denote_mulConst]
+    next =>
+      rw [denote_mulMon_nc_go]; simp [denote, Ring.intCast_zero, add_zero]
+
+theorem Poly.denote_combine {α} [Ring α] (ctx : Context α) (p₁ p₂ : Poly)
     : (combine p₁ p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
   unfold combine; generalize hugeFuel = fuel
   fun_induction combine.go
@@ -1038,6 +1114,15 @@ theorem Poly.denote_mul {α} [CommRing α] (ctx : Context α) (p₁ p₂ : Poly)
     : (mul p₁ p₂).denote ctx = p₁.denote ctx * p₂.denote ctx := by
   simp [mul, denote_mul_go, denote, intCast_zero, zero_add]
 
+theorem Poly.denote_mul_nc_go {α} [Ring α] (ctx : Context α) (p₁ p₂ acc : Poly)
+    : (mul_nc.go p₂ p₁ acc).denote ctx = acc.denote ctx + p₁.denote ctx * p₂.denote ctx := by
+  fun_induction mul_nc.go
+    <;> simp [denote_combine, denote_mulConst, denote, *, right_distrib, denote_mulMon_nc, add_assoc, zsmul_eq_intCast_mul]
+
+theorem Poly.denote_mul_nc {α} [Ring α] (ctx : Context α) (p₁ p₂ : Poly)
+    : (mul_nc p₁ p₂).denote ctx = p₁.denote ctx * p₂.denote ctx := by
+  simp [mul_nc, denote_mul_nc_go, denote, intCast_zero, zero_add]
+
 theorem Poly.denote_pow {α} [CommRing α] (ctx : Context α) (p : Poly) (k : Nat)
    : (pow p k).denote ctx = p.denote ctx ^ k := by
  fun_induction pow
@@ -1045,6 +1130,13 @@ theorem Poly.denote_pow {α} [CommRing α] (ctx : Context α) (p : Poly) (k : Na
  next => simp [pow_succ, pow_zero, one_mul]
  next => simp [denote_mul, *, pow_succ, mul_comm]
 
+theorem Poly.denote_pow_nc {α} [Ring α] (ctx : Context α) (p : Poly) (k : Nat)
+   : (pow_nc p k).denote ctx = p.denote ctx ^ k := by
+ fun_induction pow_nc
+ next => simp [denote, intCast_one, pow_zero]
+ next => simp [pow_succ, pow_zero, one_mul]
+ next => simp [denote_mul_nc, *, pow_succ]
+
 theorem Expr.denote_toPoly {α} [CommRing α] (ctx : Context α) (e : Expr)
    : e.toPoly.denote ctx = e.denote ctx := by
   fun_induction toPoly
@@ -1056,21 +1148,37 @@ theorem Expr.denote_toPoly {α} [CommRing α] (ctx : Context α) (e : Expr)
   next => rw [Ring.intCast_natCast]
   next => simp [Poly.denote_ofMon, Mon.denote, Power.denote_eq, mul_one]
 
+theorem Expr.denote_toPoly_nc {α} [Ring α] (ctx : Context α) (e : Expr)
+   : e.toPoly_nc.denote ctx = e.denote ctx := by
+  fun_induction toPoly_nc
+    <;> simp [denote, Poly.denote, Poly.denote_ofVar, Poly.denote_combine,
+          Poly.denote_mul_nc, Poly.denote_mulConst, Poly.denote_pow_nc, intCast_pow, intCast_neg, intCast_one,
+          neg_mul, one_mul, sub_eq_add_neg, denoteInt_eq, *]
+  next => rw [Ring.intCast_natCast]
+  next a k h => simp at h; simp [h, Semiring.pow_zero]
+  next => rw [Ring.intCast_natCast]
+  next => simp [Poly.denote_ofMon, Mon.denote, Power.denote_eq, mul_one]
+
 theorem Expr.eq_of_toPoly_eq {α} [CommRing α] (ctx : Context α) (a b : Expr) (h : a.toPoly == b.toPoly) : a.denote ctx = b.denote ctx := by
   have h := congrArg (Poly.denote ctx) (eq_of_beq h)
   simp [denote_toPoly] at h
   assumption
 
+theorem Expr.eq_of_toPoly_nc_eq {α} [Ring α] (ctx : Context α) (a b : Expr) (h : a.toPoly_nc == b.toPoly_nc) : a.denote ctx = b.denote ctx := by
+  have h := congrArg (Poly.denote ctx) (eq_of_beq h)
+  simp [denote_toPoly_nc] at h
+  assumption
+
 /-!
 Theorems for justifying the procedure for commutative rings with a characteristic in `grind`.
 -/
 
-theorem Poly.denote_addConstC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (p : Poly) (k : Int) : (addConstC p k c).denote ctx = p.denote ctx + k := by
+theorem Poly.denote_addConstC {α c} [Ring α] [IsCharP α c] (ctx : Context α) (p : Poly) (k : Int) : (addConstC p k c).denote ctx = p.denote ctx + k := by
   fun_induction addConstC <;> simp [denote, *]
   next => rw [IsCharP.intCast_emod, intCast_add]
   next => simp [add_comm, add_left_comm]
 
-theorem Poly.denote_insertC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+theorem Poly.denote_insertC {α c} [Ring α] [IsCharP α c] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
     : (insertC k m p c).denote ctx = k * m.denote ctx + p.denote ctx := by
   simp [insertC, cond_eq_if] <;> split
   next =>
@@ -1087,7 +1195,7 @@ theorem Poly.denote_insertC {α c} [CommRing α] [IsCharP α c] (ctx : Context
     next => rw [IsCharP.intCast_emod]
     next => rw [add_left_comm]
 
-theorem Poly.denote_mulConstC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (k : Int) (p : Poly)
+theorem Poly.denote_mulConstC {α c} [Ring α] [IsCharP α c] (ctx : Context α) (k : Int) (p : Poly)
     : (mulConstC k p c).denote ctx = k * p.denote ctx := by
   simp [mulConstC, cond_eq_if] <;> split
   next =>
@@ -1136,7 +1244,29 @@ theorem Poly.denote_mulMonC {α c} [CommRing α] [IsCharP α c] (ctx : Context
         simp +zetaDelta [*, IsCharP.intCast_emod, Mon.denote_mul, intCast_mul, left_distrib,
           mul_left_comm, mul_assoc, zsmul_eq_intCast_mul]
 
-theorem Poly.denote_combineC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (p₁ p₂ : Poly)
+theorem Poly.denote_mulMonC_nc_go {α c} [Ring α] [IsCharP α c] (ctx : Context α) (k : Int) (m : Mon) (p acc : Poly)
+    : (mulMonC_nc.go k m c p acc).denote ctx = k * m.denote ctx * p.denote ctx + acc.denote ctx := by
+  fun_induction mulMonC_nc.go <;> simp [denote, denote_insert, zsmul_eq_intCast_mul]
+  next => rw [IsCharP.intCast_emod (x := k * _) (p := c), Ring.intCast_mul, Semiring.mul_assoc, Semiring.mul_assoc, ← Ring.intCast_mul_comm]
+  next ih =>
+    rw [ih, denote_insert, Mon.denote_mul_nc, IsCharP.intCast_emod (x := k * _) (p := c),
+        Semiring.left_distrib, Ring.intCast_mul]
+    rw [Ring.intCast_mul_left_comm]; simp [← Semiring.mul_assoc]
+    conv => enter [1, 2, 1, 1, 1]; rw [Ring.intCast_mul_comm]
+    simp [Semiring.add_assoc, Semiring.add_comm, add_left_comm]
+
+theorem Poly.denote_mulMonC_nc {α c} [Ring α] [IsCharP α c] (ctx : Context α) (k : Int) (m : Mon) (p : Poly)
+    : (mulMonC_nc k m p c).denote ctx = k * m.denote ctx * p.denote ctx := by
+  simp [mulMonC_nc, cond_eq_if] <;> split
+  next =>
+    rw [← IsCharP.intCast_emod (p := c)]
+    simp [denote, *, intCast_zero, zero_mul]
+  next =>
+    split
+    next h => simp at h; simp [*, Mon.denote, mul_one, denote_mulConstC, IsCharP.intCast_emod]
+    next => rw [Poly.denote_mulMonC_nc_go, denote, Ring.intCast_zero, add_zero]
+
+theorem Poly.denote_combineC {α c} [Ring α] [IsCharP α c] (ctx : Context α) (p₁ p₂ : Poly)
     : (combineC p₁ p₂ c).denote ctx = p₁.denote ctx + p₂.denote ctx := by
   unfold combineC; generalize hugeFuel = fuel
   fun_induction combineC.go
@@ -1160,6 +1290,15 @@ theorem Poly.denote_mulC {α c} [CommRing α] [IsCharP α c] (ctx : Context α)
     : (mulC p₁ p₂ c).denote ctx = p₁.denote ctx * p₂.denote ctx := by
   simp [mulC, denote_mulC_go, denote, intCast_zero, zero_add]
 
+theorem Poly.denote_mulC_nc_go {α c} [Ring α] [IsCharP α c] (ctx : Context α) (p₁ p₂ acc : Poly)
+    : (mulC_nc.go p₂ c p₁ acc).denote ctx = acc.denote ctx + p₁.denote ctx * p₂.denote ctx := by
+  fun_induction mulC_nc.go
+    <;> simp [denote_combineC, denote_mulConstC, denote, *, right_distrib, denote_mulMonC_nc, add_assoc, zsmul_eq_intCast_mul]
+
+theorem Poly.denote_mulC_nc {α c} [Ring α] [IsCharP α c] (ctx : Context α) (p₁ p₂ : Poly)
+    : (mulC_nc p₁ p₂ c).denote ctx = p₁.denote ctx * p₂.denote ctx := by
+  simp [mulC_nc, denote_mulC_nc_go, denote, intCast_zero, zero_add]
+
 theorem Poly.denote_powC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (p : Poly) (k : Nat)
    : (powC p k c).denote ctx = p.denote ctx ^ k := by
  fun_induction powC
@@ -1167,6 +1306,13 @@ theorem Poly.denote_powC {α c} [CommRing α] [IsCharP α c] (ctx : Context α)
  next => simp [pow_succ, pow_zero, one_mul]
  next => simp [denote_mulC, *, pow_succ, mul_comm]
 
+theorem Poly.denote_powC_nc {α c} [Ring α] [IsCharP α c] (ctx : Context α) (p : Poly) (k : Nat)
+   : (powC_nc p k c).denote ctx = p.denote ctx ^ k := by
+ fun_induction powC_nc
+ next => simp [denote, intCast_one, pow_zero]
+ next => simp [pow_succ, pow_zero, one_mul]
+ next => simp [denote_mulC_nc, *, pow_succ]
+
 theorem Expr.denote_toPolyC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (e : Expr)
    : (e.toPolyC c).denote ctx = e.denote ctx := by
   unfold toPolyC
@@ -1182,17 +1328,37 @@ theorem Expr.denote_toPolyC {α c} [CommRing α] [IsCharP α c] (ctx : Context
   next => rw [IsCharP.intCast_emod, intCast_pow]
   next => simp [Poly.denote_ofMon, Mon.denote, Power.denote_eq, mul_one]
 
+theorem Expr.denote_toPolyC_nc {α c} [Ring α] [IsCharP α c] (ctx : Context α) (e : Expr)
+   : (e.toPolyC_nc c).denote ctx = e.denote ctx := by
+  unfold toPolyC_nc
+  fun_induction toPolyC_nc.go
+    <;> simp [denote, Poly.denote, Poly.denote_ofVar, Poly.denote_combineC,
+          Poly.denote_mulC_nc, Poly.denote_mulConstC, Poly.denote_powC_nc, denoteInt_eq, *]
+  next => rw [IsCharP.intCast_emod]
+  next => rw [IsCharP.intCast_emod, Ring.intCast_natCast]
+  next => rw [IsCharP.intCast_emod]
+  next => rw [intCast_neg, neg_mul, intCast_one, one_mul]
+  next => rw [intCast_neg, neg_mul, intCast_one, one_mul, sub_eq_add_neg]
+  next a k h => simp at h; simp [h, Semiring.pow_zero, Ring.intCast_one]
+  next => rw [IsCharP.intCast_emod, intCast_pow]
+  next => simp [Poly.denote_ofMon, Mon.denote, Power.denote_eq, mul_one]
+
 theorem Expr.eq_of_toPolyC_eq {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (a b : Expr)
     (h : a.toPolyC c == b.toPolyC c) : a.denote ctx = b.denote ctx := by
   have h := congrArg (Poly.denote ctx) (eq_of_beq h)
   simp [denote_toPolyC] at h
   assumption
 
+theorem Expr.eq_of_toPolyC_nc_eq {α c} [Ring α] [IsCharP α c] (ctx : Context α) (a b : Expr)
+    (h : a.toPolyC_nc c == b.toPolyC_nc c) : a.denote ctx = b.denote ctx := by
+  have h := congrArg (Poly.denote ctx) (eq_of_beq h)
+  simp [denote_toPolyC_nc] at h
+  assumption
+
 namespace Stepwise
 /-!
 Theorems for stepwise proof-term construction
 -/
-@[expose]
 noncomputable def core_cert (lhs rhs : Expr) (p : Poly) : Bool :=
   (lhs.sub rhs).toPoly_k.beq' p
 
@@ -1202,7 +1368,6 @@ theorem core {α} [CommRing α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
   simp [Expr.denote_toPoly, Expr.denote]
   simp [sub_eq_zero_iff]
 
-@[expose]
 noncomputable def superpose_cert (k₁ : Int) (m₁ : Mon) (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) : Bool :=
   (p₁.mulMon_k k₁ m₁).combine_k (p₂.mulMon_k k₂ m₂) |>.beq' p
 
@@ -1211,7 +1376,6 @@ theorem superpose {α} [CommRing α] (ctx : Context α) (k₁ : Int) (m₁ : Mon
   simp [superpose_cert]; intro _ h₁ h₂; subst p
   simp [Poly.denote_combine, Poly.denote_mulMon, h₁, h₂, mul_zero, add_zero]
 
-@[expose]
 noncomputable def simp_cert (k₁ : Int) (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) : Bool :=
   (p₁.mulConst_k k₁).combine_k (p₂.mulMon_k k₂ m₂) |>.beq' p
 
@@ -1220,32 +1384,26 @@ theorem simp {α} [CommRing α] (ctx : Context α) (k₁ : Int) (p₁ : Poly) (k
   simp [simp_cert]; intro _ h₁ h₂; subst p
   simp [Poly.denote_combine, Poly.denote_mulMon, Poly.denote_mulConst, h₁, h₂, mul_zero, add_zero]
 
-@[expose]
 noncomputable def mul_cert (p₁ : Poly) (k : Int) (p : Poly) : Bool :=
   p₁.mulConst_k k |>.beq' p
 
-@[expose]
 def mul {α} [CommRing α] (ctx : Context α) (p₁ : Poly) (k : Int) (p : Poly)
     : mul_cert p₁ k p → p₁.denote ctx = 0 → p.denote ctx = 0 := by
   simp [mul_cert]; intro _ h; subst p
   simp [Poly.denote_mulConst, *, mul_zero]
 
-@[expose]
 noncomputable def div_cert (p₁ : Poly) (k : Int) (p : Poly) : Bool :=
   !Int.beq' k 0 |>.and' (p.mulConst_k k |>.beq' p₁)
 
-@[expose]
 def div {α} [CommRing α] (ctx : Context α) [NoNatZeroDivisors α] (p₁ : Poly) (k : Int) (p : Poly)
     : div_cert p₁ k p → p₁.denote ctx = 0 → p.denote ctx = 0 := by
   simp [div_cert]; intro hnz _ h; subst p₁
   simp [Poly.denote_mulConst, ← zsmul_eq_intCast_mul] at h
   exact no_int_zero_divisors hnz h
 
-@[expose]
 noncomputable def unsat_eq_cert (p : Poly) (k : Int) : Bool :=
   !Int.beq' k 0 |>.and' (p.beq' (.num k))
 
-@[expose]
 def unsat_eq {α} [CommRing α] (ctx : Context α) [IsCharP α 0] (p : Poly) (k : Int)
     : unsat_eq_cert p k → p.denote ctx = 0 → False := by
   simp [unsat_eq_cert]; intro h _; subst p; simp [Poly.denote]
@@ -1256,7 +1414,6 @@ def unsat_eq {α} [CommRing α] (ctx : Context α) [IsCharP α 0] (p : Poly) (k
 theorem d_init {α} [CommRing α] (ctx : Context α) (p : Poly) : (1:Int) * p.denote ctx = p.denote ctx := by
   rw [intCast_one, one_mul]
 
-@[expose]
 noncomputable def d_step1_cert (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) : Bool :=
   p.beq' (p₁.combine_k (p₂.mulMon_k k₂ m₂))
 
@@ -1265,7 +1422,6 @@ theorem d_step1 {α} [CommRing α] (ctx : Context α) (k : Int) (init : Poly) (p
   simp [d_step1_cert]; intro _ h₁ h₂; subst p
   simp [Poly.denote_combine, Poly.denote_mulMon, h₂, mul_zero, add_zero, h₁]
 
-@[expose]
 noncomputable def d_stepk_cert (k₁ : Int) (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) : Bool :=
   p.beq' ((p₁.mulConst_k k₁).combine_k (p₂.mulMon_k k₂ m₂))
 
@@ -1275,7 +1431,6 @@ theorem d_stepk {α} [CommRing α] (ctx : Context α) (k₁ : Int) (k : Int) (in
   simp [Poly.denote_combine, Poly.denote_mulMon, Poly.denote_mulConst, h₂, mul_zero, add_zero]
   rw [intCast_mul, mul_assoc, h₁]
 
-@[expose]
 noncomputable def imp_1eq_cert (lhs rhs : Expr) (p₁ p₂ : Poly) : Bool :=
   (lhs.sub rhs).toPoly_k.beq' p₁ |>.and' (p₂.beq' (.num 0))
 
@@ -1284,7 +1439,6 @@ theorem imp_1eq {α} [CommRing α] (ctx : Context α) (lhs rhs : Expr) (p₁ p
   simp [imp_1eq_cert, intCast_one, one_mul]; intro _ _; subst p₁ p₂
   simp [Expr.denote_toPoly, Expr.denote, sub_eq_zero_iff, Poly.denote, intCast_zero]
 
-@[expose]
 noncomputable def imp_keq_cert (lhs rhs : Expr) (k : Int) (p₁ p₂ : Poly) : Bool :=
   !Int.beq' k 0 |>.and' ((lhs.sub rhs).toPoly_k.beq' p₁ |>.and' (p₂.beq' (.num 0)))
 
@@ -1295,7 +1449,6 @@ theorem imp_keq  {α} [CommRing α] (ctx : Context α) [NoNatZeroDivisors α] (k
   intro h; replace h := no_int_zero_divisors hnz h
   rw [← sub_eq_zero_iff, h]
 
-@[expose]
 noncomputable def core_certC (lhs rhs : Expr) (p : Poly) (c : Nat) : Bool :=
   (lhs.sub rhs).toPolyC c |>.beq' p
 
@@ -1305,7 +1458,6 @@ theorem coreC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (lhs rhs :
   simp [Expr.denote_toPolyC, Expr.denote]
   simp [sub_eq_zero_iff]
 
-@[expose]
 noncomputable def superpose_certC (k₁ : Int) (m₁ : Mon) (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) (c : Nat) : Bool :=
   (p₁.mulMonC k₁ m₁ c).combineC (p₂.mulMonC k₂ m₂ c) c |>.beq' p
 
@@ -1314,28 +1466,23 @@ theorem superposeC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (k₁
   simp [superpose_certC]; intro _ h₁ h₂; subst p
   simp [Poly.denote_combineC, Poly.denote_mulMonC, h₁, h₂, mul_zero, add_zero]
 
-@[expose]
 noncomputable def mul_certC (p₁ : Poly) (k : Int) (p : Poly) (c : Nat) : Bool :=
   p₁.mulConstC k c |>.beq' p
 
-@[expose]
 def mulC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (p₁ : Poly) (k : Int) (p : Poly)
     : mul_certC p₁ k p c → p₁.denote ctx = 0 → p.denote ctx = 0 := by
   simp [mul_certC]; intro _ h; subst p
   simp [Poly.denote_mulConstC, *, mul_zero]
 
-@[expose]
 noncomputable def div_certC (p₁ : Poly) (k : Int) (p : Poly) (c : Nat) : Bool :=
   !Int.beq' k 0 |>.and' ((p.mulConstC k c).beq' p₁)
 
-@[expose]
 def divC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) [NoNatZeroDivisors α] (p₁ : Poly) (k : Int) (p : Poly)
     : div_certC p₁ k p c → p₁.denote ctx = 0 → p.denote ctx = 0 := by
   simp [div_certC]; intro hnz _ h; subst p₁
   simp [Poly.denote_mulConstC, ← zsmul_eq_intCast_mul] at h
   exact no_int_zero_divisors hnz h
 
-@[expose]
 noncomputable def simp_certC (k₁ : Int) (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) (c : Nat) : Bool :=
   (p₁.mulConstC k₁ c).combineC (p₂.mulMonC k₂ m₂ c) c |>.beq' p
 
@@ -1344,11 +1491,9 @@ theorem simpC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (k₁ : Int
   simp [simp_certC]; intro _ h₁ h₂; subst p
   simp [Poly.denote_combineC, Poly.denote_mulMonC, Poly.denote_mulConstC, h₁, h₂, mul_zero, add_zero]
 
-@[expose]
 noncomputable def unsat_eq_certC (p : Poly) (k : Int) (c : Nat) : Bool :=
   !Int.beq' (k % c) 0 |>.and' (p.beq' (.num k))
 
-@[expose]
 def unsat_eqC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (p : Poly) (k : Int)
     : unsat_eq_certC p k c → p.denote ctx = 0 → False := by
   simp [unsat_eq_certC]; intro h _; subst p; simp [Poly.denote]
@@ -1356,7 +1501,6 @@ def unsat_eqC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (p : Poly)
   simp [h] at this
   assumption
 
-@[expose]
 noncomputable def d_step1_certC (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) (c : Nat) : Bool :=
   p.beq' (p₁.combineC (p₂.mulMonC k₂ m₂ c) c)
 
@@ -1365,7 +1509,6 @@ theorem d_step1C {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (k : Int
   simp [d_step1_certC]; intro _ h₁ h₂; subst p
   simp [Poly.denote_combineC, Poly.denote_mulMonC, h₂, mul_zero, add_zero, h₁]
 
-@[expose]
 noncomputable def d_stepk_certC (k₁ : Int) (p₁ : Poly) (k₂ : Int) (m₂ : Mon) (p₂ : Poly) (p : Poly) (c : Nat) : Bool :=
   p.beq' ((p₁.mulConstC k₁ c).combineC (p₂.mulMonC k₂ m₂ c) c)
 
@@ -1375,7 +1518,6 @@ theorem d_stepkC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (k₁ :
   simp [Poly.denote_combineC, Poly.denote_mulMonC, Poly.denote_mulConstC, h₂, mul_zero, add_zero]
   rw [intCast_mul, mul_assoc, h₁]
 
-@[expose]
 noncomputable def imp_1eq_certC (lhs rhs : Expr) (p₁ p₂ : Poly) (c : Nat) : Bool :=
   ((lhs.sub rhs).toPolyC c).beq' p₁ |>.and' (p₂.beq' (.num 0))
 
@@ -1384,7 +1526,6 @@ theorem imp_1eqC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) (lhs rhs
   simp [imp_1eq_certC, intCast_one, one_mul]; intro _ _; subst p₁ p₂
   simp [Expr.denote_toPolyC, Expr.denote, sub_eq_zero_iff, Poly.denote, intCast_zero]
 
-@[expose]
 noncomputable def imp_keq_certC (lhs rhs : Expr) (k : Int) (p₁ p₂ : Poly) (c : Nat) : Bool :=
   !Int.beq' k 0 |>.and' (((lhs.sub rhs).toPolyC c).beq' p₁ |>.and' (p₂.beq' (.num 0)))
 
@@ -1399,7 +1540,6 @@ end Stepwise
 
 /-! IntModule interface -/
 
-@[expose]
 def Mon.denoteAsIntModule [CommRing α] (ctx : Context α) (m : Mon) : α :=
   match m with
   | .unit => One.one
@@ -1410,7 +1550,6 @@ where
     | .unit => acc
     | .mult pw m => go m (acc * pw.denote ctx)
 
-@[expose]
 def Poly.denoteAsIntModule [CommRing α] (ctx : Context α) (p : Poly) : α :=
   match p with
   | .num k => k • (One.one : α)
@@ -1511,7 +1650,6 @@ theorem inv_split {α} [Field α] (a : α) : if a = 0 then a⁻¹ = 0 else a * a
   next h => simp [h, Field.inv_zero]
   next h => rw [Field.mul_inv_cancel h]
 
-@[expose]
 noncomputable def one_eq_zero_unsat_cert (p : Poly) :=
   p.beq' (.num 1) || p.beq' (.num (-1))
 
@@ -1551,7 +1689,6 @@ theorem Poly.normEq0_eq {α} [CommRing α] (ctx : Context α) (p : Poly) (c : Na
     simp [denote, normEq0, cond_eq_if]; split <;> simp [denote, zsmul_eq_intCast_mul, *]
     next h' => rw [of_mod_eq_0 h h', Semiring.zero_mul, zero_add]
 
-@[expose]
 noncomputable def eq_normEq0_cert (c : Nat) (p₁ p₂ p : Poly) : Bool :=
   p₁.beq' (.num c) && (p.beq' (p₂.normEq0 c))
 
@@ -1571,7 +1708,6 @@ theorem gcd_eq_0 [CommRing α] (g n m a b : Int) (h : g = a * n + b * m)
   rw [← Ring.intCast_add, h₂, zero_add, ← h] at h₁
   rw [Ring.intCast_zero, h₁]
 
-@[expose]
 def eq_gcd_cert (a b : Int) (p₁ p₂ p : Poly) : Bool :=
   match p₁ with
   | .add .. => false
@@ -1589,7 +1725,6 @@ theorem eq_gcd {α} [CommRing α] (ctx : Context α) (a b : Int) (p₁ p₂ p :
   rename_i n m g
   apply gcd_eq_0 g n m a b
 
-@[expose]
 noncomputable def d_normEq0_cert (c : Nat) (p₁ p₂ p : Poly) : Bool :=
   p₂.beq' (.num c) |>.and' (p.beq' (p₁.normEq0 c))
 
@@ -1598,11 +1733,10 @@ theorem d_normEq0 {α} [CommRing α] (ctx : Context α) (k : Int) (c : Nat) (ini
   simp [d_normEq0_cert]; intro _ h₁ h₂; subst p p₂; simp [Poly.denote]
   intro h; rw [p₁.normEq0_eq] <;> assumption
 
-@[expose] noncomputable def norm_int_cert (e : Expr) (p : Poly) : Bool :=
+noncomputable def norm_int_cert (e : Expr) (p : Poly) : Bool :=
   e.toPoly_k.beq' p
 
 theorem norm_int (ctx : Context Int) (e : Expr) (p : Poly) : norm_int_cert e p → e.denote ctx = p.denote' ctx := by
   simp [norm_int_cert, Poly.denote'_eq_denote]; intro; subst p; simp [Expr.denote_toPoly]
 
-end CommRing
-end Lean.Grind
+end Lean.Grind.CommRing

src/Init/Grind/Ring/Envelope.lean

@@ -277,6 +277,9 @@ attribute [-simp] Q.mk
 
 /-! Embedding theorems -/
 
+theorem toQ_zero : toQ (0 : α) = (0 : Q α) := by
+  simp; apply Quot.sound; simp
+
 theorem toQ_add (a b : α) : toQ (a + b) = toQ a + toQ b := by
   simp
 

src/Init/Grind/Ring/Field.lean

@@ -169,7 +169,8 @@ theorem zpow_add {a : α} (h : a ≠ 0) (m n : Int) : a ^ (m + n) = a ^ m * a ^
     | zero => simp [Int.add_neg_one, zpow_sub_one h, zpow_neg_one]
     | succ n ih => rw [Int.natCast_add_one, Int.neg_add, Int.add_neg_one, ← Int.add_sub_assoc, zpow_sub_one h, zpow_sub_one h, ih, Semiring.mul_assoc]
 
-instance [IsCharP α 0] : NoNatZeroDivisors α := NoNatZeroDivisors.mk' <| by
+-- This is expensive as an instance. Let's see what breaks without it.
+def noNatZeroDivisors.ofIsCharPZero [IsCharP α 0] : NoNatZeroDivisors α := NoNatZeroDivisors.mk' <| by
   intro a b h w
   have := IsCharP.natCast_eq_zero_iff (α := α) 0 a
   simp only [Nat.mod_zero, h, iff_false] at this

src/Init/Grind/Tactics.lean

@@ -126,6 +126,52 @@ structure Config where
   abstractProof := true
   deriving Inhabited, BEq
 
+/--
+A minimal configuration, with ematching and splitting disabled, and all solver modules turned off.
+`grind` will not do anything in this configuration,
+which can be used a starting point for minimal configurations.
+-/
+-- This is a `structure` rather than `def` so we can use `declare_config_elab`.
+structure NoopConfig extends Config where
+  -- Disable splitting
+  splits := 0
+  -- We don't override the various `splitMatch` / `splitIte` settings separately.
+
+  -- Disable e-matching
+  ematch := 0
+  -- We don't override `matchEqs` separately.
+
+  -- Disable extensionality
+  ext       := false
+  extAll    := false
+  etaStruct := false
+  funext    := false
+
+  -- Disable all solver modules
+  ring      := false
+  linarith  := false
+  cutsat    := false
+  ac        := false
+
+/--
+A `grind` configuration that only uses `cutsat` and splitting.
+
+Note: `cutsat` benefits from some amount of instantiation, e.g. `Nat.max_def`.
+We don't currently have a mechanism to enable only a small set of lemmas.
+-/
+-- This is a `structure` rather than `def` so we can use `declare_config_elab`.
+structure CutsatConfig extends NoopConfig where
+  cutsat := true
+  -- Allow the default number of splits.
+  splits := ({} : Config).splits
+
+/--
+A `grind` configuration that only uses `ring`.
+-/
+-- This is a `structure` rather than `def` so we can use `declare_config_elab`.
+structure GrobnerConfig extends NoopConfig where
+  ring := true
+
 end Lean.Grind
 
 namespace Lean.Parser.Tactic
@@ -134,9 +180,11 @@ namespace Lean.Parser.Tactic
 `grind` tactic and related tactics.
 -/
 
-syntax grindErase := "-" ident
-syntax grindLemma := ppGroup((Attr.grindMod ppSpace)? ident)
-syntax grindParam := grindErase <|> grindLemma
+syntax grindErase    := "-" ident
+syntax grindLemma    := ppGroup((Attr.grindMod ppSpace)? ident)
+-- `!` for enabling minimal indexable subexpression restriction
+syntax grindLemmaMin := ppGroup("!" (Attr.grindMod ppSpace)? ident)
+syntax grindParam    := grindErase <|> grindLemma <|> grindLemmaMin
 
 /--
 `grind` is a tactic inspired by modern SMT solvers. **Picture a virtual whiteboard**:
@@ -420,6 +468,23 @@ syntax (name := grindTrace)
   (" [" withoutPosition(grindParam,*) "]")?
   (&" on_failure " term)? : tactic
 
+/--
+`cutsat` solves linear integer arithmetic goals.
+
+It is a implemented as a thin wrapper around the `grind` tactic, enabling only the `cutsat` solver.
+Please use `grind` instead if you need additional capabilities.
+-/
+syntax (name := cutsat) "cutsat" optConfig : tactic
+
+/--
+`grobner` solves goals that can be phrased as polynomial equations (with further polynomial equations as hypotheses)
+over commutative (semi)rings, using the Grobner basis algorithm.
+
+It is a implemented as a thin wrapper around the `grind` tactic, enabling only the `grobner` solver.
+Please use `grind` instead if you need additional capabilities.
+-/
+syntax (name := grobner) "grobner" optConfig : tactic
+
 /-!
 Sets symbol priorities for the E-matching pattern inference procedure used in `grind`
 -/

src/Init/Grind/ToInt.lean

@@ -7,6 +7,7 @@ module
 
 prelude
 public import Init.Data.Int.DivMod.Lemmas
+import Init.LawfulBEqTactics
 
 public section
 
@@ -37,11 +38,7 @@ inductive IntInterval : Type where
     io (hi : Int)
   | /-- The infinite interval `(-∞, ∞)`. -/
     ii
-  deriving BEq, DecidableEq, Inhabited
-
-instance : LawfulBEq IntInterval where
-   rfl := by intro a; cases a <;> simp_all! [BEq.beq]
-   eq_of_beq := by intro a b; cases a <;> cases b <;> simp_all! [BEq.beq]
+  deriving BEq, ReflBEq, LawfulBEq, DecidableEq, Inhabited
 
 namespace IntInterval
 

src/Init/LawfulBEqTactics.lean

@@ -0,0 +1,104 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+
+module
+prelude
+public import Init.Prelude
+public import Init.Notation
+public import Init.Tactics
+public import Init.Core
+import Init.Data.Bool
+import Init.ByCases
+
+public section
+
+namespace DerivingHelpers
+
+macro "deriving_ReflEq_tactic" : tactic => `(tactic|(
+  intro x
+  induction x
+  all_goals
+    simp only [ BEq.refl, ↓reduceDIte, Bool.and_true, *, reduceBEq ]
+))
+
+theorem and_true_curry {a b : Bool} {P : Prop}
+    (h : a → b → P) : (a && b) → P := by
+  rw [Bool.and_eq_true_iff]
+  intro h'
+  apply h h'.1 h'.2
+
+
+theorem deriving_lawful_beq_helper_dep {x y : α} [BEq α] [ReflBEq α]
+  {t : (x == y) = true → Bool} {P : Prop}
+    (inst : (x == y) = true → x = y)
+    (k : (h : x = y) → t (h ▸ ReflBEq.rfl) = true → P) :
+    (if h : (x == y) then t h else false) = true → P := by
+  intro h
+  by_cases hxy : x = y
+  · subst hxy
+    apply k rfl
+    rw [dif_pos (BEq.refl x)] at h
+    exact h
+  · by_cases hxy' : x == y
+    · exact False.elim <| hxy (inst hxy')
+    · rw [dif_neg hxy'] at h
+      contradiction
+
+theorem deriving_lawful_beq_helper_nd {x y : α} [BEq α] [ReflBEq α]
+    {P : Prop}
+    (inst : (x == y) = true → x = y)
+    (k : x = y → P) :
+    (x == y) = true → P := by
+  intro h
+  by_cases hxy : x = y
+  · subst hxy
+    apply k rfl
+  · exact False.elim <| hxy (inst h)
+
+end DerivingHelpers
+
+syntax "deriving_LawfulEq_tactic_step" : tactic
+macro_rules
+| `(tactic| deriving_LawfulEq_tactic_step) =>
+  `(tactic| fail "deriving_LawfulEq_tactic_step failed")
+macro_rules
+| `(tactic| deriving_LawfulEq_tactic_step) =>
+  `(tactic| ( with_reducible change dite (_ == _) _ _ = true → _
+              refine DerivingHelpers.deriving_lawful_beq_helper_dep ?_ ?_
+              · solve | apply_assumption | simp | fail "could not discharge eq_of_beq assumption"
+              intro h
+              cases h
+              dsimp only
+    ))
+macro_rules
+| `(tactic| deriving_LawfulEq_tactic_step) =>
+  `(tactic| ( with_reducible change (_ == _) = true → _
+              refine DerivingHelpers.deriving_lawful_beq_helper_nd ?_ ?_
+              · solve | apply_assumption | simp | fail "could not discharge eq_of_beq assumption"
+              intro h
+              subst h
+    ))
+macro_rules
+| `(tactic| deriving_LawfulEq_tactic_step) =>
+  `(tactic| ( with_reducible change (_ == _ && _) = true → _
+              refine DerivingHelpers.and_true_curry ?_))
+macro_rules
+| `(tactic| deriving_LawfulEq_tactic_step) =>
+  `(tactic| rfl)
+macro_rules
+| `(tactic| deriving_LawfulEq_tactic_step) =>
+  `(tactic| intro _; trivial)
+
+macro "deriving_LawfulEq_tactic" : tactic => `(tactic|(
+  intro x
+  induction x
+  all_goals
+    intro y
+    cases y
+    all_goals
+      simp only [reduceBEq]
+      repeat deriving_LawfulEq_tactic_step
+))

src/Init/MetaTypes.lean

@@ -94,7 +94,7 @@ structure Config where
   -/
   decide            : Bool := false
   /--
-  When `true` (default: `false`), unfolds definitions.
+  When `true` (default: `false`), unfolds applications of functions defined by pattern matching, when one of the patterns applies.
   This can be enabled using the `simp!` syntax.
   -/
   autoUnfold        : Bool := false
@@ -208,7 +208,7 @@ structure Config where
   /--  When `true` (default: `false`), simplifies simple arithmetic expressions. -/
   arith             : Bool := false
   /--
-  When `true` (default: `false`), unfolds definitions.
+  When `true` (default: `false`), unfolds applications of functions defined by pattern matching, when one of the patterns applies.
   This can be enabled using the `simp!` syntax.
   -/
   autoUnfold        : Bool := false