chore: minimization of variable-induced slowdown

We must erase macro scopes for tags in `case` as we do in `cases .. with
..` and `induction .. with ..`.

.github/ISSUE_TEMPLATE/bug_report.md

@@ -9,9 +9,15 @@ assignees: ''
 
 ### Prerequisites
 
-* [ ] Put an X between the brackets on this line if you have done all of the following:
-    * Check that your issue is not already [filed](https://github.com/leanprover/lean4/issues).
-    * Reduce the issue to a minimal, self-contained, reproducible test case. Avoid dependencies to mathlib4 or std4.
+Please put an X between the brackets as you perform the following steps:
+
+* [ ] Check that your issue is not already filed:
+      https://github.com/leanprover/lean4/issues
+* [ ] Reduce the issue to a minimal, self-contained, reproducible test case.
+      Avoid dependencies to Mathlib or Batteries.
+* [ ] Test your test case against the latest nightly release, for example on
+      https://live.lean-lang.org/#project=lean-nightly
+      (You can also use the settings there to switch to “Lean nightly”)
 
 ### Description
 
@@ -33,8 +39,8 @@ assignees: ''
 
 ### Versions
 
-[Output of `#eval Lean.versionString` or of `lean --version` in the folder that the issue occured in]
-[OS version]
+[Output of `#eval Lean.versionString`]
+[OS version, if not using live.lean-lang.org.]
 
 ### Additional Information
 

.github/workflows/check-stage0.yml

@@ -0,0 +1,57 @@
+name: Check for stage0 changes
+
+on:
+  merge_group:
+  pull_request:
+
+jobs:
+  check-stage0-on-queue:
+    runs-on: ubuntu-latest
+    steps:
+    - uses: actions/checkout@v4
+      with:
+        ref: ${{ github.event.pull_request.head.sha }}
+        filter: blob:none
+        fetch-depth: 0
+
+    - name: Find base commit
+      if: github.event_name == 'pull_request'
+      run: echo "BASE=$(git merge-base origin/${{ github.base_ref }} HEAD)" >> "$GITHUB_ENV"
+
+    - name: Identify stage0 changes
+      run: |
+        git diff "${BASE:-HEAD^}..HEAD" --name-only -- stage0 |
+          grep -v -x -F $'stage0/src/stdlib_flags.h\nstage0/src/lean.mk.in' \
+          > "$RUNNER_TEMP/stage0" || true
+        if test -s "$RUNNER_TEMP/stage0"
+        then
+          echo "CHANGES=yes" >> "$GITHUB_ENV"
+        else
+          echo "CHANGES=no" >> "$GITHUB_ENV"
+        fi
+      shell: bash
+
+    - if: github.event_name == 'pull_request'
+      name: Set label
+      uses: actions/github-script@v7
+      with:
+        script: |
+          const { owner, repo, number: issue_number  } = context.issue;
+          if (process.env.CHANGES == 'yes') {
+            await github.rest.issues.addLabels({ owner, repo, issue_number, labels: ['changes-stage0'] }).catch(() => {});
+          } else {
+            await github.rest.issues.removeLabel({ owner, repo, issue_number, name: 'changes-stage0' }).catch(() => {});
+          }
+
+    - if: env.CHANGES == 'yes'
+      name: Report changes
+      run: |
+        echo "Found changes to stage0/, please do not merge using the merge queue." | tee "$GITHUB_STEP_SUMMARY"
+        # shellcheck disable=SC2129
+        echo '```'                  >> "$GITHUB_STEP_SUMMARY"
+        cat  "$RUNNER_TEMP/stage0"  >> "$GITHUB_STEP_SUMMARY"
+        echo '```'                  >> "$GITHUB_STEP_SUMMARY"
+
+    - if: github.event_name == 'merge_group' && env.CHANGES == 'yes'
+      name: Fail when on the merge queue
+      run: exit 1

.github/workflows/ci.yml

@@ -6,7 +6,6 @@ on:
     tags:
       - '*'
   pull_request:
-    types: [opened, synchronize, reopened, labeled]
   merge_group:
   schedule:
     - cron: '0 7 * * *'  # 8AM CET/11PM PT
@@ -41,12 +40,18 @@ jobs:
     steps:
       - name: Run quick CI?
         id: set-quick
-        env:
-          quick: ${{
-            github.event_name == 'pull_request' && !contains( github.event.pull_request.labels.*.name, 'full-ci')
-           }}
+        # We do not use github.event.pull_request.labels.*.name here because
+        # re-running a run does not update that list, and we do want to be able to
+        # rerun the workflow run after settings the `full-ci` label.
         run: |
-          echo "quick=${{env.quick}}" >> "$GITHUB_OUTPUT"
+          if [ "${{ github.event_name }}" == 'pull_request' ]
+          then
+            echo "quick=$(gh api repos/${{ github.repository_owner }}/${{ github.event.repository.name }}/pulls/${{ github.event.pull_request.number }} --jq '.labels | any(.name == "full-ci") | not')" >> "$GITHUB_OUTPUT"
+          else
+            echo "quick=false" >> "$GITHUB_OUTPUT"
+          fi
+        env:
+          GH_TOKEN: ${{ github.token }}
 
       - name: Configure build matrix
         id: set-matrix
@@ -54,7 +59,10 @@ jobs:
         with:
           script: |
             const quick = ${{ steps.set-quick.outputs.quick }};
-            console.log(`quick: ${quick}`)
+            console.log(`quick: ${quick}`);
+            // use large runners outside PRs where available (original repo)
+            // disabled for now as this mostly just speeds up the test suite which is not a bottleneck
+            // let large = ${{ github.event_name != 'pull_request' && github.repository == 'leanprover/lean4' }} ? "-large" : "";
             let matrix = [
               {
                 // portable release build: use channel with older glibc (2.27)
@@ -110,7 +118,7 @@ jobs:
               },*/
               {
                 "name": "macOS",
-                "os": "macos-latest",
+                "os": "macos-13",
                 "release": true,
                 "quick": false,
                 "shell": "bash -euxo pipefail {0}",
@@ -121,7 +129,7 @@ jobs:
               },
               {
                 "name": "macOS aarch64",
-                "os": "macos-latest",
+                "os": "macos-13",
                 "release": true,
                 "quick": false,
                 "cross": true,
@@ -277,18 +285,18 @@ jobs:
         uses: cachix/install-nix-action@v18
         with:
           install_url: https://releases.nixos.org/nix/nix-2.12.0/install
-        if: matrix.os == 'ubuntu-latest' && !matrix.cmultilib
+        if: runner.os == 'Linux' && !matrix.cmultilib
       - name: Install MSYS2
         uses: msys2/setup-msys2@v2
         with:
           msystem: clang64
           # `:p` means prefix with appropriate msystem prefix
           pacboy: "make python cmake:p clang:p ccache:p gmp:p git zip unzip diffutils binutils tree zstd:p tar"
-        if: matrix.os == 'windows-2022'
+        if: runner.os == 'Windows'
       - name: Install Brew Packages
         run: |
           brew install ccache tree zstd coreutils gmp
-        if: matrix.os == 'macos-latest'
+        if: runner.os == 'macOS'
       - name: Setup emsdk
         uses: mymindstorm/setup-emsdk@v12
         with:
@@ -312,13 +320,13 @@ jobs:
         run: |
           # open nix-shell once for initial setup
           true
-        if: matrix.os == 'ubuntu-latest'
+        if: runner.os == 'Linux'
       - name: Set up core dumps
         run: |
           mkdir -p $PWD/coredumps
           # store in current directory, for easy uploading together with binary
           echo $PWD/coredumps/%e.%p.%t | sudo tee /proc/sys/kernel/core_pattern
-        if: matrix.os == 'ubuntu-latest'
+        if: runner.os == 'Linux'
       - name: Build
         run: |
           mkdir build
@@ -423,7 +431,7 @@ jobs:
       - name: CCache stats
         run: ccache -s
       - name: Show stacktrace for coredumps
-        if: ${{ failure() && matrix.os == 'ubuntu-latest' }}
+        if: ${{ failure() && runner.os == 'Linux' }}
         run: |
           for c in coredumps/*; do
             progbin="$(file $c | sed "s/.*execfn: '\([^']*\)'.*/\1/")"
@@ -433,7 +441,7 @@ jobs:
       # shared libs
       #- name: Upload coredumps
       #  uses: actions/upload-artifact@v3
-      #  if: ${{ failure() && matrix.os == 'ubuntu-latest' }}
+      #  if: ${{ failure() && runner.os == 'Linux' }}
       #  with:
       #    name: coredumps-${{ matrix.name }}
       #    path: |

.github/workflows/copyright-header.yml

@@ -10,7 +10,7 @@ jobs:
 
     - name: Verify .lean files start with a copyright header.
       run: |
-        FILES=$(find . -type d \( -path "./tests" -o -path "./doc" -o -path "./src/lake/examples" -o -path "./src/lake/tests" -o -path "./build" -o -path "./nix" \) -prune -o -type f -name "*.lean" -exec perl -ne 'BEGIN { $/ = undef; } print "$ARGV\n" if !m{\A/-\nCopyright}; exit;' {} \;)
+        FILES=$(find ./src -type d \( -path "./src/lake/examples" -o -path "./src/lake/tests" \) -prune -o -type f -name "*.lean" -exec perl -ne 'BEGIN { $/ = undef; } print "$ARGV\n" if !m{\A/-\nCopyright}; exit;' {} \;)
         if [ -n "$FILES" ]; then
           echo "Found .lean files which do not have a copyright header:"
           echo "$FILES"

.github/workflows/nix-ci.yml

@@ -6,7 +6,6 @@ on:
     tags:
       - '*'
   pull_request:
-    types: [opened, synchronize, reopened, labeled]
   merge_group:
 
 concurrency:

.github/workflows/pr-release.yml

@@ -126,11 +126,11 @@ jobs:
           if [ "$NIGHTLY_SHA" = "$MERGE_BASE_SHA" ]; then
             echo "The merge base of this PR coincides with the nightly release"
 
-            STD_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover/std4.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
+            BATTERIES_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover-community/batteries.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
             MATHLIB_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover-community/mathlib4.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
 
-            if [[ -n "$STD_REMOTE_TAGS" ]]; then
-              echo "... and Std has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
+            if [[ -n "$BATTERIES_REMOTE_TAGS" ]]; then
+              echo "... and Batteries has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
               MESSAGE=""
 
               if [[ -n "$MATHLIB_REMOTE_TAGS" ]]; then
@@ -140,8 +140,8 @@ jobs:
                 MESSAGE="- ❗ Mathlib CI can not be attempted yet, as the \`nightly-testing-$MOST_RECENT_NIGHTLY\` tag does not exist there yet. We will retry when you push more commits. If you rebase your branch onto \`nightly-with-mathlib\`, Mathlib CI should run now."
               fi
             else
-              echo "... but Std does not yet have a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
-              MESSAGE="- ❗ Std CI can not be attempted yet, as the \`nightly-testing-$MOST_RECENT_NIGHTLY\` tag does not exist there yet. We will retry when you push more commits. If you rebase your branch onto \`nightly-with-mathlib\`, Std CI should run now."
+              echo "... but Batteries does not yet have a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
+              MESSAGE="- ❗ Batteries CI can not be attempted yet, as the \`nightly-testing-$MOST_RECENT_NIGHTLY\` tag does not exist there yet. We will retry when you push more commits. If you rebase your branch onto \`nightly-with-mathlib\`, Batteries CI should run now."
             fi
 
           else
@@ -149,8 +149,9 @@ jobs:
             echo "but 'git merge-base origin/master HEAD' reported: $MERGE_BASE_SHA"
             git -C lean4.git log -10 origin/master
 
-            NIGHTLY_WITH_MATHLIB_SHA="$(git -C lean4.git rev-parse "nightly-with-mathlib")"
-            MESSAGE="- ❗ Std/Mathlib CI will not be attempted unless your PR branches off the \`nightly-with-mathlib\` branch. Try \`git rebase $MERGE_BASE_SHA --onto $NIGHTLY_WITH_MATHLIB_SHA\`."
+            git -C lean4.git fetch origin nightly-with-mathlib 
+            NIGHTLY_WITH_MATHLIB_SHA="$(git -C lean4.git rev-parse "origin/nightly-with-mathlib")"
+            MESSAGE="- ❗ Batteries/Mathlib CI will not be attempted unless your PR branches off the \`nightly-with-mathlib\` branch. Try \`git rebase $MERGE_BASE_SHA --onto $NIGHTLY_WITH_MATHLIB_SHA\`."
           fi
 
           if [[ -n "$MESSAGE" ]]; then
@@ -222,27 +223,27 @@ jobs:
               description: description,
             });
 
-      # We next automatically create a Std branch using this toolchain.
-      # Std doesn't itself have a mechanism to report results of CI from this branch back to Lean
-      # Instead this is taken care of by Mathlib CI, which will fail if Std fails.
+      # We next automatically create a Batteries branch using this toolchain.
+      # Batteries doesn't itself have a mechanism to report results of CI from this branch back to Lean
+      # Instead this is taken care of by Mathlib CI, which will fail if Batteries fails.
       - name: Cleanup workspace
         if: steps.workflow-info.outputs.pullRequestNumber != '' && steps.ready.outputs.mathlib_ready == 'true'
         run: |
           sudo rm -rf ./*
 
-      # Checkout the Std repository with all branches
-      - name: Checkout Std repository
+      # 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@v3
         with:
-          repository: leanprover/std4
+          repository: leanprover-community/batteries
           token: ${{ secrets.MATHLIB4_BOT }}
           ref: nightly-testing
           fetch-depth: 0 # This ensures we check out all tags and branches.
 
       - name: Check if tag exists
         if: steps.workflow-info.outputs.pullRequestNumber != '' && steps.ready.outputs.mathlib_ready == 'true'
-        id: check_std_tag
+        id: check_batteries_tag
         run: |
           git config user.name "leanprover-community-mathlib4-bot"
           git config user.email "leanprover-community-mathlib4-bot@users.noreply.github.com"
@@ -250,7 +251,7 @@ jobs:
           if git ls-remote --heads --tags --exit-code origin "nightly-testing-${MOST_RECENT_NIGHTLY}" >/dev/null; then
             BASE="nightly-testing-${MOST_RECENT_NIGHTLY}"
           else
-            echo "This shouldn't be possible: couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' tag at Std. Falling back to 'nightly-testing'."
+            echo "This shouldn't be possible: couldn't find a 'nightly-testing-${MOST_RECENT_NIGHTLY}' tag at Batteries. Falling back to 'nightly-testing'."
             BASE=nightly-testing
           fi
 
@@ -267,7 +268,7 @@ jobs:
           else
             echo "Branch already exists, pushing an empty commit."
             git switch lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}
-            # The Std `nightly-testing` or `nightly-testing-YYYY-MM-DD` branch may have moved since this branch was created, so merge their changes.
+            # The Batteries `nightly-testing` or `nightly-testing-YYYY-MM-DD` branch may have moved since this branch was created, so merge their changes.
             # (This should no longer be possible once `nightly-testing-YYYY-MM-DD` is a tag, but it is still safe to merge.)
             git merge "$BASE" --strategy-option ours --no-commit --allow-unrelated-histories
             git commit --allow-empty -m "Trigger CI for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
@@ -320,7 +321,7 @@ jobs:
             git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
             echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}" > lean-toolchain
             git add lean-toolchain
-            sed -i "s/require std from git \"https:\/\/github.com\/leanprover\/std4\" @ \".\+\"/require std from git \"https:\/\/github.com\/leanprover\/std4\" @ \"nightly-testing-${MOST_RECENT_NIGHTLY}\"/" lakefile.lean
+            sed -i "s/require batteries from git \"https:\/\/github.com\/leanprover-community\/batteries\" @ \".\+\"/require batteries from git \"https:\/\/github.com\/leanprover-community\/batteries\" @ \"nightly-testing-${MOST_RECENT_NIGHTLY}\"/" lakefile.lean
             git add lakefile.lean
             git commit -m "Update lean-toolchain for testing https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
           else

.github/workflows/restart-on-label.yml

@@ -0,0 +1,31 @@
+name: Restart by label
+on:
+  pull_request_target:
+    types:
+      - unlabeled
+      - labeled
+jobs:
+  restart-on-label:
+    runs-on: ubuntu-latest
+    if: contains(github.event.label.name, 'full-ci')
+    steps:
+    - run: |
+        # Finding latest CI workflow run on current pull request
+        # (unfortunately cannot search by PR number, only base branch,
+        # and that is't even unique given PRs from forks, but the risk
+        # of confusion is low and the danger is mild)
+        run_id=$(gh run list -e pull_request -b "$head_ref" --workflow 'CI' --limit 1 \
+          --limit 1 --json databaseId --jq '.[0].databaseId')
+        echo "Run id: ${run_id}"
+        gh run view "$run_id"
+        echo "Cancelling (just in case)"
+        gh run cancel "$run_id" || echo "(failed)"
+        echo "Waiting for 10s"
+        sleep 10
+        echo "Rerunning"
+        gh run rerun "$run_id"
+      shell: bash
+      env:
+        head_ref: ${{ github.head_ref }}
+        GH_TOKEN: ${{ github.token }}
+        GH_REPO: ${{ github.repository }}

CMakeLists.txt

@@ -78,6 +78,10 @@ add_custom_target(update-stage0
   COMMAND $(MAKE) -C stage1 update-stage0
   DEPENDS stage1)
 
+add_custom_target(update-stage0-commit
+  COMMAND $(MAKE) -C stage1 update-stage0-commit
+  DEPENDS stage1)
+
 add_custom_target(test
   COMMAND $(MAKE) -C stage1 test
   DEPENDS stage1)

CODEOWNERS

@@ -6,7 +6,6 @@
 
 /.github/ @Kha @semorrison
 /RELEASES.md @semorrison
-/src/Init/IO.lean @joehendrix
 /src/kernel/ @leodemoura
 /src/lake/ @tydeu
 /src/Lean/Compiler/ @leodemoura
@@ -20,4 +19,27 @@
 /src/Lean/PrettyPrinter/Delaborator/ @kmill
 /src/Lean/Server/ @mhuisi
 /src/Lean/Widget/ @Vtec234
-/src/runtime/io.cpp @joehendrix
+/src/Init/Data/ @semorrison
+/src/Init/Data/Array/Lemmas.lean @digama0
+/src/Init/Data/List/Lemmas.lean @digama0
+/src/Init/Data/List/BasicAux.lean @digama0
+/src/Init/Data/Array/Subarray.lean @david-christiansen
+/src/Lean/Elab/Tactic/RCases.lean @digama0
+/src/Init/RCases.lean @digama0
+/src/Lean/Elab/Tactic/Ext.lean @digama0
+/src/Init/Ext.lean @digama0
+/src/Lean/Elab/Tactic/Simpa.lean @digama0
+/src/Lean/Elab/Tactic/NormCast.lean @digama0
+/src/Lean/Meta/Tactic/NormCast.lean @digama0
+/src/Lean/Meta/Tactic/TryThis.lean @digama0
+/src/Lean/Elab/Tactic/SimpTrace.lean @digama0
+/src/Lean/Elab/Tactic/NoMatch.lean @digama0
+/src/Lean/Elab/Tactic/ShowTerm.lean @digama0
+/src/Lean/Elab/Tactic/Repeat.lean @digama0
+/src/Lean/Meta/Tactic/Repeat.lean @digama0
+/src/Lean/Meta/CoeAttr.lean @digama0
+/src/Lean/Elab/GuardMsgs.lean @digama0
+/src/Lean/Elab/Tactic/Guard.lean @digama0
+/src/Init/Guard.lean @digama0
+/src/Lean/Server/CodeActions/ @digama0
+

README.md

@@ -22,4 +22,4 @@ Please read our [Contribution Guidelines](CONTRIBUTING.md) first.
 
 # Building from Source
 
-See [Building Lean](https://lean-lang.org/lean4/doc/make/index.html).
+See [Building Lean](https://lean-lang.org/lean4/doc/make/index.html) (documentation source: [doc/make/index.md](doc/make/index.md)).

RELEASES.md

@@ -8,7 +8,34 @@ This file contains work-in-progress notes for the upcoming release, as well as p
 Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
 of each version.
 
-v4.8.0 (development in progress)
+v4.8.0
+v4.9.0 (development in progress)
+---------
+
+* Functions defined by well-founded recursion are now marked as
+  `@[irreducible]`, which should prevent expensive and often unfruitful
+  unfolding of such definitions.
+
+  Existing proofs that hold by definitional equality (e.g. `rfl`) can be
+  rewritten to explictly unfold the function definition (using `simp`,
+  `unfold`, `rw`), or the recursive function can be temporariliy made
+  semireducible (using `unseal f in` before the command) or the function
+  definition itself can be marked as `@[semireducible]` to get the previous
+  behavor.
+
+* The `MessageData.ofPPFormat` constructor has been removed.
+  Its functionality has been split into two:
+
+  - for lazy structured messages, please use `MessageData.lazy`;
+  - for embedding `Format` or `FormatWithInfos`, use `MessageData.ofFormatWithInfos`.
+
+  An example migration can be found in [#3929](https://github.com/leanprover/lean4/pull/3929/files#diff-5910592ab7452a0e1b2616c62d22202d2291a9ebb463145f198685aed6299867L109).
+
+* The `MessageData.ofFormat` constructor has been turned into a function.
+  If you need to inspect `MessageData`,
+  you can pattern-match on `MessageData.ofFormatWithInfos`.
+
+v4.8.0
 ---------
 
 * **Executables configured with `supportInterpreter := true` on Windows should now be run via `lake exe` to function properly.**
@@ -21,7 +48,7 @@ v4.8.0 (development in progress)
 
 * Importing two different files containing proofs of the same theorem is no longer considered an error. This feature is particularly useful for theorems that are automatically generated on demand (e.g., equational theorems).
 
-* Funcitonal induction principles.
+* Functional induction principles.
 
   Derived from the definition of a (possibly mutually) recursive function, a **functional induction principle** is created that is tailored to proofs about that function.
 
@@ -57,6 +84,15 @@ v4.8.0 (development in progress)
   ```
   is recognized without having to say `termination_by arr.size - i`.
 
+* Shorter instances names. There is a new algorithm for generating names for anonymous instances.
+  Across Std and Mathlib, the median ratio between lengths of new names and of old names is about 72%.
+  With the old algorithm, the longest name was 1660 characters, and now the longest name is 202 characters.
+  The new algorithm's 95th percentile name length is 67 characters, versus 278 for the old algorithm.
+  While the new algorithm produces names that are 1.2% less unique,
+  it avoids cross-project collisions by adding a module-based suffix
+  when it does not refer to declarations from the same "project" (modules that share the same root).
+  PR [#3089](https://github.com/leanprover/lean4/pull/3089).
+
 * Attribute `@[pp_using_anonymous_constructor]` to make structures pretty print like `⟨x, y, z⟩`
   rather than `{a := x, b := y, c := z}`.
   This attribute is applied to `Sigma`, `PSigma`, `PProd`, `Subtype`, `And`, and `Fin`.
@@ -69,15 +105,489 @@ v4.8.0 (development in progress)
   to enable pretty printing function applications using generalized field notation (defaults to true).
   Field notation can be disabled on a function-by-function basis using the `@[pp_nodot]` attribute.
 
+* Added options `pp.mvars` (default: true) and `pp.mvars.withType` (default: false).
+  When `pp.mvars` is false, expression metavariables pretty print as `?_` and universe metavariables pretty print as `_`.
+  When `pp.mvars.withType` is true, expression metavariables pretty print with a type ascription.
+  These can be set when using `#guard_msgs` to make tests not depend on the particular names of metavariables.
+  [#3798](https://github.com/leanprover/lean4/pull/3798) and
+  [#3978](https://github.com/leanprover/lean4/pull/3978).
+
+* Hovers for terms in `match` expressions in the Infoview now reliably show the correct term.
+
 * Added `@[induction_eliminator]` and `@[cases_eliminator]` attributes to be able to define custom eliminators
   for the `induction` and `cases` tactics, replacing the `@[eliminator]` attribute.
   Gives custom eliminators for `Nat` so that `induction` and `cases` put goal states into terms of `0` and `n + 1`
   rather than `Nat.zero` and `Nat.succ n`.
   Added option `tactic.customEliminators` to control whether to use custom eliminators.
-  [#3629](https://github.com/leanprover/lean4/pull/3629) and
-  [#3655](https://github.com/leanprover/lean4/pull/3655).
+  Added a hack for `rcases`/`rintro`/`obtain` to use the custom eliminator for `Nat`.
+  [#3629](https://github.com/leanprover/lean4/pull/3629),
+  [#3655](https://github.com/leanprover/lean4/pull/3655), and
+  [#3747](https://github.com/leanprover/lean4/pull/3747).
 
-Breaking changes:
+* The `#guard_msgs` command now has options to change whitespace normalization and sensitivity to message ordering.
+  For example, `#guard_msgs (whitespace := lax) in cmd` collapses whitespace before checking messages,
+  and `#guard_msgs (ordering := sorted) in cmd` sorts the messages in lexicographic order before checking.
+  PR [#3883](https://github.com/leanprover/lean4/pull/3883).
+
+* The `#guard_msgs` command now supports showing a diff between the expected and actual outputs. This feature is currently
+  disabled by default, but can be enabled with `set_option guard_msgs.diff true`. Depending on user feedback, this option
+  may default to `true` in a future version of Lean.
+### Language features, tactics, and metaprograms
+
+* **Functional induction principles.**
+  [#3432](https://github.com/leanprover/lean4/pull/3432), [#3620](https://github.com/leanprover/lean4/pull/3620),
+  [#3754](https://github.com/leanprover/lean4/pull/3754), [#3762](https://github.com/leanprover/lean4/pull/3762),
+  [#3738](https://github.com/leanprover/lean4/pull/3738), [#3776](https://github.com/leanprover/lean4/pull/3776),
+  [#3898](https://github.com/leanprover/lean4/pull/3898).
+
+  Derived from the definition of a (possibly mutually) recursive function,
+  a **functional induction principle** is created that is tailored to proofs about that function.
+
+  For example from:
+  ```
+  def ackermann : Nat → Nat → Nat
+    | 0, m => m + 1
+    | n+1, 0 => ackermann n 1
+    | n+1, m+1 => ackermann n (ackermann (n + 1) m)
+  ```
+  we get
+  ```
+  ackermann.induct (motive : Nat → Nat → Prop) (case1 : ∀ (m : Nat), motive 0 m)
+    (case2 : ∀ (n : Nat), motive n 1 → motive (Nat.succ n) 0)
+    (case3 : ∀ (n m : Nat), motive (n + 1) m → motive n (ackermann (n + 1) m) → motive (Nat.succ n) (Nat.succ m))
+    (x x : Nat) : motive x x
+  ```
+
+  It can be used in the `induction` tactic using the `using` syntax:
+  ```
+  induction n, m using ackermann.induct
+  ```
+* The termination checker now recognizes more recursion patterns without an
+  explicit `termination_by`. In particular the idiom of counting up to an upper
+  bound, as in
+  ```
+  def Array.sum (arr : Array Nat) (i acc : Nat) : Nat :=
+    if _ : i < arr.size then
+      Array.sum arr (i+1) (acc + arr[i])
+    else
+      acc
+  ```
+  is recognized without having to say `termination_by arr.size - i`.
+  * [#3630](https://github.com/leanprover/lean4/pull/3630) makes `termination_by?` not use `sizeOf` when not needed
+  * [#3652](https://github.com/leanprover/lean4/pull/3652) improves the `termination_by` syntax.
+  * [#3658](https://github.com/leanprover/lean4/pull/3658) changes how termination arguments are elaborated.
+  * [#3665](https://github.com/leanprover/lean4/pull/3665) refactors GuessLex to allow inferring more complex termination arguments
+  * [#3666](https://github.com/leanprover/lean4/pull/3666) infers termination arguments such as `xs.size - i`
+* [#3629](https://github.com/leanprover/lean4/pull/3629),
+  [#3655](https://github.com/leanprover/lean4/pull/3655),
+  [#3747](https://github.com/leanprover/lean4/pull/3747):
+  Adds `@[induction_eliminator]` and `@[cases_eliminator]` attributes to be able to define custom eliminators
+  for the `induction` and `cases` tactics, replacing the `@[eliminator]` attribute.
+  Gives custom eliminators for `Nat` so that `induction` and `cases` put goal states into terms of `0` and `n + 1`
+  rather than `Nat.zero` and `Nat.succ n`.
+  Added option `tactic.customEliminators` to control whether to use custom eliminators.
+  Added a hack for `rcases`/`rintro`/`obtain` to use the custom eliminator for `Nat`.
+* **Shorter instances names.** There is a new algorithm for generating names for anonymous instances.
+  Across Std and Mathlib, the median ratio between lengths of new names and of old names is about 72%.
+  With the old algorithm, the longest name was 1660 characters, and now the longest name is 202 characters.
+  The new algorithm's 95th percentile name length is 67 characters, versus 278 for the old algorithm.
+  While the new algorithm produces names that are 1.2% less unique,
+  it avoids cross-project collisions by adding a module-based suffix
+  when it does not refer to declarations from the same "project" (modules that share the same root).
+  [#3089](https://github.com/leanprover/lean4/pull/3089)
+  and [#3934](https://github.com/leanprover/lean4/pull/3934).
+* [8d2adf](https://github.com/leanprover/lean4/commit/8d2adf521d2b7636347a5b01bfe473bf0fcfaf31)
+  Importing two different files containing proofs of the same theorem is no longer considered an error.
+  This feature is particularly useful for theorems that are automatically generated on demand (e.g., equational theorems).
+* [84b091](https://github.com/leanprover/lean4/commit/84b0919a116e9be12f933e764474f45d964ce85c)
+  Lean now generates an error if the type of a theorem is **not** a proposition.
+* **Definition transparency.** [47a343](https://github.com/leanprover/lean4/commit/47a34316fc03ce936fddd2d3dce44784c5bcdfa9). `@[reducible]`, `@[semireducible]`, and `@[irreducible]` are now scoped and able to be set for imported declarations.
+* `simp`/`dsimp`
+  * [#3607](https://github.com/leanprover/lean4/pull/3607) enables kernel projection reduction in `dsimp`
+  * [b24fbf](https://github.com/leanprover/lean4/commit/b24fbf44f3aaa112f5d799ef2a341772d1eb222d)
+    and [acdb00](https://github.com/leanprover/lean4/commit/acdb0054d5a0efa724cff596ac26852fad5724c4):
+    `dsimproc` command
+    to define defeq-preserving simplification procedures.
+  * [#3624](https://github.com/leanprover/lean4/pull/3624) makes `dsimp` normalize raw nat literals as `OfNat.ofNat` applications.
+  * [#3628](https://github.com/leanprover/lean4/pull/3628) makes `simp` correctly handle `OfScientific.ofScientific` literals.
+  * [#3654](https://github.com/leanprover/lean4/pull/3654) makes `dsimp?` report used simprocs.
+  * [dee074](https://github.com/leanprover/lean4/commit/dee074dcde03a37b7895a4901df2e4fa490c73c7) fixes equation theorem
+    handling in `simp` for non-recursive definitions.
+  * [#3819](https://github.com/leanprover/lean4/pull/3819) improved performance when simp encounters a loop.
+  * [#3821](https://github.com/leanprover/lean4/pull/3821) fixes discharger/cache interaction.
+  * [#3824](https://github.com/leanprover/lean4/pull/3824) keeps `simp` from breaking `Char` literals.
+  * [#3838](https://github.com/leanprover/lean4/pull/3838) allows `Nat` instances matching to be more lenient.
+  * [#3870](https://github.com/leanprover/lean4/pull/3870) documentation for `simp` configuration options.
+  * [#3972](https://github.com/leanprover/lean4/pull/3972) fixes simp caching.
+  * [#4044](https://github.com/leanprover/lean4/pull/4044) improves cache behavior for "well-behaved" dischargers.
+* `omega`
+  * [#3639](https://github.com/leanprover/lean4/pull/3639), [#3766](https://github.com/leanprover/lean4/pull/3766),
+    [#3853](https://github.com/leanprover/lean4/pull/3853), [#3875](https://github.com/leanprover/lean4/pull/3875):
+    introduces a term canonicalizer.
+  * [#3736](https://github.com/leanprover/lean4/pull/3736) improves handling of positivity for the modulo operator for `Int`.
+  * [#3828](https://github.com/leanprover/lean4/pull/3828) makes it work as a `simp` discharger.
+  * [#3847](https://github.com/leanprover/lean4/pull/3847) adds helpful error messages.
+* `rfl`
+  * [#3671](https://github.com/leanprover/lean4/pull/3671), [#3708](https://github.com/leanprover/lean4/pull/3708): upstreams the `@[refl]` attribute and the `rfl` tactic.
+  * [#3751](https://github.com/leanprover/lean4/pull/3751) makes `apply_rfl` not operate on `Eq` itself.
+  * [#4067](https://github.com/leanprover/lean4/pull/4067) improves error message when there are no goals.
+* [#3719](https://github.com/leanprover/lean4/pull/3719) upstreams the `rw?` tactic, with fixes and improvements in
+  [#3783](https://github.com/leanprover/lean4/pull/3783), [#3794](https://github.com/leanprover/lean4/pull/3794),
+  [#3911](https://github.com/leanprover/lean4/pull/3911).
+* `conv`
+  * [#3659](https://github.com/leanprover/lean4/pull/3659) adds a `conv` version of the `calc` tactic.
+  * [#3763](https://github.com/leanprover/lean4/pull/3763) makes `conv` clean up using `try with_reducible rfl` instead of `try rfl`.
+* `#guard_msgs`
+  * [#3617](https://github.com/leanprover/lean4/pull/3617) introduces whitespace protection using the `⏎` character.
+  * [#3883](https://github.com/leanprover/lean4/pull/3883):
+    The `#guard_msgs` command now has options to change whitespace normalization and sensitivity to message ordering.
+    For example, `#guard_msgs (whitespace := lax) in cmd` collapses whitespace before checking messages,
+    and `#guard_msgs (ordering := sorted) in cmd` sorts the messages in lexicographic order before checking.
+  * [#3931](https://github.com/leanprover/lean4/pull/3931) adds an unused variables ignore function for `#guard_msgs`.
+  * [#3912](https://github.com/leanprover/lean4/pull/3912) adds a diff between the expected and actual outputs. This feature is currently
+    disabled by default, but can be enabled with `set_option guard_msgs.diff true`.
+    Depending on user feedback, this option may default to `true` in a future version of Lean.
+* `do` **notation**
+  * [#3820](https://github.com/leanprover/lean4/pull/3820) makes it an error to lift `(<- ...)` out of a pure `if ... then ... else ...`
+* **Lazy discrimination trees**
+  * [#3610](https://github.com/leanprover/lean4/pull/3610) fixes a name collision for `LazyDiscrTree` that could lead to cache poisoning.
+  * [#3677](https://github.com/leanprover/lean4/pull/3677) simplifies and fixes `LazyDiscrTree` handling for `exact?`/`apply?`.
+  * [#3685](https://github.com/leanprover/lean4/pull/3685) moves general `exact?`/`apply?` functionality into `LazyDiscrTree`.
+  * [#3769](https://github.com/leanprover/lean4/pull/3769) has lemma selection improvements for `rw?` and `LazyDiscrTree`.
+  * [#3818](https://github.com/leanprover/lean4/pull/3818) improves ordering of matches.
+* [#3590](https://github.com/leanprover/lean4/pull/3590) adds `inductive.autoPromoteIndices` option to be able to disable auto promotion of indices in the `inductive` command.
+* **Miscellaneous bug fixes and improvements**
+  * [#3606](https://github.com/leanprover/lean4/pull/3606) preserves `cache` and `dischargeDepth` fields in `Lean.Meta.Simp.Result.mkEqSymm`.
+  * [#3633](https://github.com/leanprover/lean4/pull/3633) makes `elabTermEnsuringType` respect `errToSorry`, improving error recovery of the `have` tactic.
+  * [#3647](https://github.com/leanprover/lean4/pull/3647) enables `noncomputable unsafe` definitions, for deferring implementations until later.
+  * [#3672](https://github.com/leanprover/lean4/pull/3672) adjust namespaces of tactics.
+  * [#3725](https://github.com/leanprover/lean4/pull/3725) fixes `Ord` derive handler for indexed inductive types with unused alternatives.
+  * [#3893](https://github.com/leanprover/lean4/pull/3893) improves performance of derived `Ord` instances.
+  * [#3771](https://github.com/leanprover/lean4/pull/3771) changes error reporting for failing tactic macros. Improves `rfl` error message.
+  * [#3745](https://github.com/leanprover/lean4/pull/3745) fixes elaboration of generalized field notation if the object of the notation is an optional parameter.
+  * [#3799](https://github.com/leanprover/lean4/pull/3799) makes commands such as `universe`, `variable`, `namespace`, etc. require that their argument appear in a later column.
+    Commands that can optionally parse an `ident` or parse any number of `ident`s generally should require
+    that the `ident` use `colGt`. This keeps typos in commands from being interpreted as identifiers.
+  * [#3815](https://github.com/leanprover/lean4/pull/3815) lets the `split` tactic be used for writing code.
+  * [#3822](https://github.com/leanprover/lean4/pull/3822) adds missing info in `induction` tactic for `with` clauses of the form `| cstr a b c => ?_`.
+  * [#3806](https://github.com/leanprover/lean4/pull/3806) fixes `withSetOptionIn` combinator.
+  * [#3844](https://github.com/leanprover/lean4/pull/3844) removes unused `trace.Elab.syntax` option.
+  * [#3896](https://github.com/leanprover/lean4/pull/3896) improves hover and go-to-def for `attribute` command.
+  * [#3989](https://github.com/leanprover/lean4/pull/3989) makes linter options more discoverable.
+  * [#3916](https://github.com/leanprover/lean4/pull/3916) fixes go-to-def for syntax defined with `@[builtin_term_parser]`.
+  * [#3962](https://github.com/leanprover/lean4/pull/3962) fixes how `solveByElim` handles `symm` lemmas, making `exact?`/`apply?` usable again.
+  * [#3968](https://github.com/leanprover/lean4/pull/3968) improves the `@[deprecated]` attribute, adding `(since := "<date>")` field.
+  * [#3768](https://github.com/leanprover/lean4/pull/3768) makes `#print` command show structure fields.
+  * [#3974](https://github.com/leanprover/lean4/pull/3974) makes `exact?%` behave like `by exact?` rather than `by apply?`.
+  * [#3994](https://github.com/leanprover/lean4/pull/3994) makes elaboration of `he ▸ h` notation more predictable.
+  * [#3991](https://github.com/leanprover/lean4/pull/3991) adjusts transparency for `decreasing_trivial` macros.
+  * [#4092](https://github.com/leanprover/lean4/pull/4092) improves performance of `binop%` and `binrel%` expression tree elaborators.
+* **Docs:** [#3748](https://github.com/leanprover/lean4/pull/3748), [#3796](https://github.com/leanprover/lean4/pull/3796),
+[#3800](https://github.com/leanprover/lean4/pull/3800), [#3874](https://github.com/leanprover/lean4/pull/3874),
+[#3863](https://github.com/leanprover/lean4/pull/3863), [#3862](https://github.com/leanprover/lean4/pull/3862),
+[#3891](https://github.com/leanprover/lean4/pull/3891), [#3873](https://github.com/leanprover/lean4/pull/3873),
+[#3908](https://github.com/leanprover/lean4/pull/3908), [#3872](https://github.com/leanprover/lean4/pull/3872).
+
+### Language server and IDE extensions
+
+* [#3432](https://github.com/leanprover/lean4/pull/3432) enables `import` auto-completions.
+* [#3608](https://github.com/leanprover/lean4/pull/3608) fixes issue [leanprover/vscode-lean4#392](https://github.com/leanprover/vscode-lean4/issues/392).
+  Diagnostic ranges had an off-by-one error that would misplace goal states for example.
+* [#3014](https://github.com/leanprover/lean4/pull/3014) introduces snapshot trees, foundational work for incremental tactics and parallelism.
+  [#3849](https://github.com/leanprover/lean4/pull/3849) adds basic incrementality API.
+* [#3271](https://github.com/leanprover/lean4/pull/3271) adds support for server-to-client requests.
+* [#3656](https://github.com/leanprover/lean4/pull/3656) fixes jump to definition when there are conflicting names from different files.
+  Fixes issue [#1170](https://github.com/leanprover/lean4/issues/1170).
+* [#3691](https://github.com/leanprover/lean4/pull/3691), [#3925](https://github.com/leanprover/lean4/pull/3925),
+  [#3932](https://github.com/leanprover/lean4/pull/3932) keep semantic tokens synchronized (used for semantic highlighting), with performance improvements.
+* [#3247](https://github.com/leanprover/lean4/pull/3247) and [#3730](https://github.com/leanprover/lean4/pull/3730)
+  add diagnostics to run "Restart File" when a file dependency is saved.
+* [#3722](https://github.com/leanprover/lean4/pull/3722) uses the correct module names when displaying references.
+* [#3728](https://github.com/leanprover/lean4/pull/3728) makes errors in header reliably appear and makes the "Import out of date" warning be at "hint" severity.
+  [#3739](https://github.com/leanprover/lean4/pull/3739) simplifies the text of this warning.
+* [#3778](https://github.com/leanprover/lean4/pull/3778) fixes [#3462](https://github.com/leanprover/lean4/issues/3462),
+  where info nodes from before the cursor would be used for computing completions.
+* [#3985](https://github.com/leanprover/lean4/pull/3985) makes trace timings appear in Infoview.
+
+### Pretty printing
+
+* [#3797](https://github.com/leanprover/lean4/pull/3797) fixes the hovers over binders so that they show their types.
+* [#3640](https://github.com/leanprover/lean4/pull/3640) and [#3735](https://github.com/leanprover/lean4/pull/3735): Adds attribute `@[pp_using_anonymous_constructor]` to make structures pretty print as `⟨x, y, z⟩`
+  rather than as `{a := x, b := y, c := z}`.
+  This attribute is applied to `Sigma`, `PSigma`, `PProd`, `Subtype`, `And`, and `Fin`.
+* [#3749](https://github.com/leanprover/lean4/pull/3749)
+  Now structure instances pretty print with parent structures' fields inlined.
+  That is, if `B` extends `A`, then `{ toA := { x := 1 }, y := 2 }` now pretty prints as `{ x := 1, y := 2 }`.
+  Setting option `pp.structureInstances.flatten` to false turns this off.
+* [#3737](https://github.com/leanprover/lean4/pull/3737), [#3744](https://github.com/leanprover/lean4/pull/3744)
+  and [#3750](https://github.com/leanprover/lean4/pull/3750):
+  Option `pp.structureProjections` is renamed to `pp.fieldNotation`, and there is now a suboption `pp.fieldNotation.generalized`
+  to enable pretty printing function applications using generalized field notation (defaults to true).
+  Field notation can be disabled on a function-by-function basis using the `@[pp_nodot]` attribute.
+  The notation is not used for theorems.
+* [#4071](https://github.com/leanprover/lean4/pull/4071) fixes interaction between app unexpanders and `pp.fieldNotation.generalized`
+* [#3625](https://github.com/leanprover/lean4/pull/3625) makes `delabConstWithSignature` (used by `#check`) have the ability to put arguments "after the colon"
+  to avoid printing inaccessible names.
+* [#3798](https://github.com/leanprover/lean4/pull/3798),
+  [#3978](https://github.com/leanprover/lean4/pull/3978),
+  [#3798](https://github.com/leanprover/lean4/pull/3980):
+  Adds options `pp.mvars` (default: true) and `pp.mvars.withType` (default: false).
+  When `pp.mvars` is false, expression metavariables pretty print as `?_` and universe metavariables pretty print as `_`.
+  When `pp.mvars.withType` is true, expression metavariables pretty print with a type ascription.
+  These can be set when using `#guard_msgs` to make tests not depend on the particular names of metavariables.
+* [#3917](https://github.com/leanprover/lean4/pull/3917) makes binders hoverable and gives them docstrings.
+* [#4034](https://github.com/leanprover/lean4/pull/4034) makes hovers for RHS terms in `match` expressions in the Infoview reliably show the correct term.
+
+### Library
+
+* `Bool`/`Prop`
+  * [#3508](https://github.com/leanprover/lean4/pull/3508) improves `simp` confluence for `Bool` and `Prop` terms.
+  * Theorems: [#3604](https://github.com/leanprover/lean4/pull/3604)
+* `Nat`
+  * [#3579](https://github.com/leanprover/lean4/pull/3579) makes `Nat.succ_eq_add_one` be a simp lemma, now that `induction`/`cases` uses `n + 1` instead of `Nat.succ n`.
+  * [#3808](https://github.com/leanprover/lean4/pull/3808) replaces `Nat.succ` simp rules with simprocs.
+  * [#3876](https://github.com/leanprover/lean4/pull/3876) adds faster `Nat.repr` implementation in C.
+* `Int`
+  * Theorems: [#3890](https://github.com/leanprover/lean4/pull/3890)
+* `UInt`s
+  * [#3960](https://github.com/leanprover/lean4/pull/3960) improves performance of upcasting.
+* `Array` and `Subarray`
+  * [#3676](https://github.com/leanprover/lean4/pull/3676) removes `Array.eraseIdxAux`, `Array.eraseIdxSzAux`, and `Array.eraseIdx'`.
+  * [#3648](https://github.com/leanprover/lean4/pull/3648) simplifies `Array.findIdx?`.
+  * [#3851](https://github.com/leanprover/lean4/pull/3851) renames fields of `Subarray`.
+* `List`
+  * [#3785](https://github.com/leanprover/lean4/pull/3785) upstreams tail-recursive List operations and `@[csimp]` lemmas.
+* `BitVec`
+  * Theorems: [#3593](https://github.com/leanprover/lean4/pull/3593),
+  [#3593](https://github.com/leanprover/lean4/pull/3593), [#3597](https://github.com/leanprover/lean4/pull/3597),
+  [#3598](https://github.com/leanprover/lean4/pull/3598), [#3721](https://github.com/leanprover/lean4/pull/3721),
+  [#3729](https://github.com/leanprover/lean4/pull/3729), [#3880](https://github.com/leanprover/lean4/pull/3880),
+  [#4039](https://github.com/leanprover/lean4/pull/4039).
+  * [#3884](https://github.com/leanprover/lean4/pull/3884) protects `Std.BitVec`.
+* `String`
+  * [#3832](https://github.com/leanprover/lean4/pull/3832) fixes `String.splitOn`.
+  * [#3959](https://github.com/leanprover/lean4/pull/3959) adds `String.Pos.isValid`.
+  * [#3959](https://github.com/leanprover/lean4/pull/3959) UTF-8 string validation.
+  * [#3961](https://github.com/leanprover/lean4/pull/3961) adds a model implementation for UTF-8 encoding and decoding.
+* `IO`
+  * [#4097](https://github.com/leanprover/lean4/pull/4097) adds `IO.getTaskState` which returns whether a task is finished, actively running, or waiting on other Tasks to finish.
+
+* **Refactors**
+  * [#3605](https://github.com/leanprover/lean4/pull/3605) reduces imports for `Init.Data.Nat` and `Init.Data.Int`.
+  * [#3613](https://github.com/leanprover/lean4/pull/3613) reduces imports for `Init.Omega.Int`.
+  * [#3634](https://github.com/leanprover/lean4/pull/3634) upstreams `Std.Data.Nat`
+    and [#3635](https://github.com/leanprover/lean4/pull/3635) upstreams `Std.Data.Int`.
+  * [#3790](https://github.com/leanprover/lean4/pull/3790) reduces more imports for `omega`.
+  * [#3694](https://github.com/leanprover/lean4/pull/3694) extends `GetElem` interface with `getElem!` and `getElem?` to simplify containers like `RBMap`.
+  * [#3865](https://github.com/leanprover/lean4/pull/3865) renames `Option.toMonad` (see breaking changes below).
+  * [#3882](https://github.com/leanprover/lean4/pull/3882) unifies `lexOrd` with `compareLex`.
+* **Other fixes or improvements**
+  * [#3765](https://github.com/leanprover/lean4/pull/3765) makes `Quotient.sound` be a `theorem`.
+  * [#3645](https://github.com/leanprover/lean4/pull/3645) fixes `System.FilePath.parent` in the case of absolute paths.
+  * [#3660](https://github.com/leanprover/lean4/pull/3660) `ByteArray.toUInt64LE!` and `ByteArray.toUInt64BE!` were swapped.
+  * [#3881](https://github.com/leanprover/lean4/pull/3881), [#3887](https://github.com/leanprover/lean4/pull/3887) fix linearity issues in `HashMap.insertIfNew`, `HashSet.erase`, and `HashMap.erase`.
+    The `HashMap.insertIfNew` fix improves `import` performance.
+  * [#3830](https://github.com/leanprover/lean4/pull/3830) ensures linearity in `Parsec.many*Core`.
+  * [#3930](https://github.com/leanprover/lean4/pull/3930) adds `FS.Stream.isTty` field.
+  * [#3866](https://github.com/leanprover/lean4/pull/3866) deprecates `Option.toBool` in favor of `Option.isSome`.
+  * [#3975](https://github.com/leanprover/lean4/pull/3975) upstreams `Data.List.Init` and `Data.Array.Init` material from Std.
+  * [#3942](https://github.com/leanprover/lean4/pull/3942) adds instances that make `ac_rfl` work without Mathlib.
+  * [#4010](https://github.com/leanprover/lean4/pull/4010) changes `Fin.induction` to use structural induction.
+  * [02753f](https://github.com/leanprover/lean4/commit/02753f6e4c510c385efcbf71fa9a6bec50fce9ab)
+    fixes bug in `reduceLeDiff` simproc.
+  * [#4097](https://github.com/leanprover/lean4/pull/4097)
+    adds `IO.TaskState` and `IO.getTaskState` to get the task from the Lean runtime's task manager.
+* **Docs:** [#3615](https://github.com/leanprover/lean4/pull/3615), [#3664](https://github.com/leanprover/lean4/pull/3664),
+  [#3707](https://github.com/leanprover/lean4/pull/3707), [#3734](https://github.com/leanprover/lean4/pull/3734),
+  [#3868](https://github.com/leanprover/lean4/pull/3868), [#3861](https://github.com/leanprover/lean4/pull/3861),
+  [#3869](https://github.com/leanprover/lean4/pull/3869), [#3858](https://github.com/leanprover/lean4/pull/3858),
+  [#3856](https://github.com/leanprover/lean4/pull/3856), [#3857](https://github.com/leanprover/lean4/pull/3857),
+  [#3867](https://github.com/leanprover/lean4/pull/3867), [#3864](https://github.com/leanprover/lean4/pull/3864),
+  [#3860](https://github.com/leanprover/lean4/pull/3860), [#3859](https://github.com/leanprover/lean4/pull/3859),
+  [#3871](https://github.com/leanprover/lean4/pull/3871), [#3919](https://github.com/leanprover/lean4/pull/3919).
+
+### Lean internals
+
+* **Defeq and WHNF algorithms**
+  * [#3616](https://github.com/leanprover/lean4/pull/3616) gives better support for reducing `Nat.rec` expressions.
+  * [#3774](https://github.com/leanprover/lean4/pull/3774) add tracing for "non-easy" WHNF cases.
+  * [#3807](https://github.com/leanprover/lean4/pull/3807) fixes an `isDefEq` performance issue, now trying structure eta *after* lazy delta reduction.
+  * [#3816](https://github.com/leanprover/lean4/pull/3816) fixes `.yesWithDeltaI` behavior to prevent increasing transparency level when reducing projections.
+  * [#3837](https://github.com/leanprover/lean4/pull/3837) improves heuristic at `isDefEq`.
+  * [#3965](https://github.com/leanprover/lean4/pull/3965) improves `isDefEq` for constraints of the form `t.i =?= s.i`.
+  * [#3977](https://github.com/leanprover/lean4/pull/3977) improves `isDefEqProj`.
+  * [#3981](https://github.com/leanprover/lean4/pull/3981) adds universe constraint approximations to be able to solve `u =?= max u ?v` using `?v = u`.
+    These approximations are only applied when universe constraints cannot be postponed anymore.
+  * [#4004](https://github.com/leanprover/lean4/pull/4004) improves `isDefEqProj` during typeclass resolution.
+  * [#4012](https://github.com/leanprover/lean4/pull/4012) adds `backward.isDefEq.lazyProjDelta` and `backward.isDefEq.lazyWhnfCore` backwards compatibility flags.
+* **Kernel**
+  * [#3966](https://github.com/leanprover/lean4/pull/3966) removes dead code.
+  * [#4035](https://github.com/leanprover/lean4/pull/4035) fixes mismatch for `TheoremVal` between Lean and C++.
+* **Discrimination trees**
+  * [423fed](https://github.com/leanprover/lean4/commit/423fed79a9de75705f34b3e8648db7e076c688d7)
+    and [3218b2](https://github.com/leanprover/lean4/commit/3218b25974d33e92807af3ce42198911c256ff1d):
+    simplify handling of dependent/non-dependent pi types.
+* **Typeclass instance synthesis**
+  * [#3638](https://github.com/leanprover/lean4/pull/3638) eta-reduces synthesized instances
+  * [ce350f](https://github.com/leanprover/lean4/commit/ce350f348161e63fccde6c4a5fe1fd2070e7ce0f) fixes a linearity issue
+  * [917a31](https://github.com/leanprover/lean4/commit/917a31f694f0db44d6907cc2b1485459afe74d49)
+    improves performance by considering at most one answer for subgoals not containing metavariables.
+    [#4008](https://github.com/leanprover/lean4/pull/4008) adds `backward.synthInstance.canonInstances` backward compatibility flag.
+* **Definition processing**
+  * [#3661](https://github.com/leanprover/lean4/pull/3661), [#3767](https://github.com/leanprover/lean4/pull/3767) changes automatically generated equational theorems to be named
+    using suffix `.eq_<idx>` instead of `._eq_<idx>`, and `.eq_def` instead of `._unfold`. (See breaking changes below.)
+    [#3675](https://github.com/leanprover/lean4/pull/3675) adds a mechanism to reserve names.
+    [#3803](https://github.com/leanprover/lean4/pull/3803) fixes reserved name resolution inside namespaces and fixes handling of `match`er declarations and equation lemmas.
+  * [#3662](https://github.com/leanprover/lean4/pull/3662) causes auxiliary definitions nested inside theorems to become `def`s if they are not proofs.
+  * [#4006](https://github.com/leanprover/lean4/pull/4006) makes proposition fields of `structure`s be theorems.
+  * [#4018](https://github.com/leanprover/lean4/pull/4018) makes it an error for a theorem to be `extern`.
+  * [#4047](https://github.com/leanprover/lean4/pull/4047) improves performance making equations for well-founded recursive definitions.
+* **Refactors**
+  * [#3614](https://github.com/leanprover/lean4/pull/3614) avoids unfolding in `Lean.Meta.evalNat`.
+  * [#3621](https://github.com/leanprover/lean4/pull/3621) centralizes functionality for `Fix`/`GuessLex`/`FunInd` in the `ArgsPacker` module.
+  * [#3186](https://github.com/leanprover/lean4/pull/3186) rewrites the UnusedVariable linter to be more performant.
+  * [#3589](https://github.com/leanprover/lean4/pull/3589) removes coercion from `String` to `Name` (see breaking changes below).
+  * [#3237](https://github.com/leanprover/lean4/pull/3237) removes the `lines` field from `FileMap`.
+  * [#3951](https://github.com/leanprover/lean4/pull/3951) makes msg parameter to `throwTacticEx` optional.
+* **Diagnostics**
+  * [#4016](https://github.com/leanprover/lean4/pull/4016), [#4019](https://github.com/leanprover/lean4/pull/4019),
+    [#4020](https://github.com/leanprover/lean4/pull/4020), [#4030](https://github.com/leanprover/lean4/pull/4030),
+    [#4031](https://github.com/leanprover/lean4/pull/4031),
+    [c3714b](https://github.com/leanprover/lean4/commit/c3714bdc6d46845c0428735b283c5b48b23cbcf7),
+    [#4049](https://github.com/leanprover/lean4/pull/4049) adds `set_option diagnostics true` for diagnostic counters.
+    Tracks number of unfolded declarations, instances, reducible declarations, used instances, recursor reductions,
+    `isDefEq` heuristic applications, among others.
+    This option is suggested in exceptional situations, such as at deterministic timeout and maximum recursion depth.
+  * [283587](https://github.com/leanprover/lean4/commit/283587987ab2eb3b56fbc3a19d5f33ab9e04a2ef)
+    adds diagnostic information for `simp`.
+  * [#4043](https://github.com/leanprover/lean4/pull/4043) adds diagnostic information for congruence theorems.
+  * [#4048](https://github.com/leanprover/lean4/pull/4048) display diagnostic information
+    for `set_option diagnostics true in <tactic>` and `set_option diagnostics true in <term>`.
+* **Other features**
+  * [#3800](https://github.com/leanprover/lean4/pull/3800) adds environment extension to record which definitions use structural or well-founded recursion.
+  * [#3801](https://github.com/leanprover/lean4/pull/3801) `trace.profiler` can now export to Firefox Profiler.
+  * [#3918](https://github.com/leanprover/lean4/pull/3918), [#3953](https://github.com/leanprover/lean4/pull/3953) adds `@[builtin_doc]` attribute to make docs and location of a declaration available as a builtin.
+  * [#3939](https://github.com/leanprover/lean4/pull/3939) adds the `lean --json` CLI option to print messages as JSON.
+  * [#3075](https://github.com/leanprover/lean4/pull/3075) improves `test_extern` command.
+  * [#3970](https://github.com/leanprover/lean4/pull/3970) gives monadic generalization of `FindExpr`.
+* **Docs:** [#3743](https://github.com/leanprover/lean4/pull/3743), [#3921](https://github.com/leanprover/lean4/pull/3921),
+  [#3954](https://github.com/leanprover/lean4/pull/3954).
+* **Other fixes:** [#3622](https://github.com/leanprover/lean4/pull/3622),
+  [#3726](https://github.com/leanprover/lean4/pull/3726), [#3823](https://github.com/leanprover/lean4/pull/3823),
+  [#3897](https://github.com/leanprover/lean4/pull/3897), [#3964](https://github.com/leanprover/lean4/pull/3964),
+  [#3946](https://github.com/leanprover/lean4/pull/3946), [#4007](https://github.com/leanprover/lean4/pull/4007),
+  [#4026](https://github.com/leanprover/lean4/pull/4026).
+
+### Compiler, runtime, and FFI
+
+* [#3632](https://github.com/leanprover/lean4/pull/3632) makes it possible to allocate and free thread-local runtime resources for threads not started by Lean itself.
+* [#3627](https://github.com/leanprover/lean4/pull/3627) improves error message about compacting closures.
+* [#3692](https://github.com/leanprover/lean4/pull/3692) fixes deadlock in `IO.Promise.resolve`.
+* [#3753](https://github.com/leanprover/lean4/pull/3753) catches error code from `MoveFileEx` on Windows.
+* [#4028](https://github.com/leanprover/lean4/pull/4028) fixes a double `reset` bug in `ResetReuse` transformation.
+* [6e731b](https://github.com/leanprover/lean4/commit/6e731b4370000a8e7a5cfb675a7f3d7635d21f58)
+  removes `interpreter` copy constructor to avoid potential memory safety issues.
+
+### Lake
+
+* **TOML Lake configurations**. [#3298](https://github.com/leanprover/lean4/pull/3298), [#4104](https://github.com/leanprover/lean4/pull/4104).
+
+  Lake packages can now use TOML as a alternative configuration file format instead of Lean. If the default `lakefile.lean` is missing, Lake will also look for a `lakefile.toml`. The TOML version of the configuration supports a restricted set of the Lake configuration options, only including those which can easily mapped to a TOML data structure. The TOML syntax itself fully compiles with the TOML v1.0.0 specification.
+
+  As part of the introduction of this new feature, we have been helping maintainers of some major packages within the ecosystem switch to this format. For example, the following is Aesop's new `lakefile.toml`:
+
+
+  **[leanprover-community/aesop/lakefile.toml](https://raw.githubusercontent.com/leanprover-community/aesop/de11e0ecf372976e6d627c210573146153090d2d/lakefile.toml)**
+  ```toml
+  name = "aesop"
+  defaultTargets = ["Aesop"]
+  testRunner = "test"
+  precompileModules = false
+
+  [[require]]
+  name = "batteries"
+  git = "https://github.com/leanprover-community/batteries"
+  rev = "main"
+
+  [[lean_lib]]
+  name = "Aesop"
+
+  [[lean_lib]]
+  name = "AesopTest"
+  globs = ["AesopTest.+"]
+  leanOptions = {linter.unusedVariables = false}
+
+  [[lean_exe]]
+  name = "test"
+  srcDir = "scripts"
+  ```
+
+  To assist users who wish to transition their packages between configuration file formats, there is also a new `lake translate-config` command for migrating to/from TOML.
+
+  Running `lake translate-config toml` will produce a `lakefile.toml` version of a package's `lakefile.lean`. Any configuration options unsupported by the TOML format will be discarded during translation, but the original `lakefile.lean` will remain so that you can verify the translation looks good before deleting it.
+
+* **Build progress overhaul.** [#3835](https://github.com/leanprover/lean4/pull/3835), [#4115](https://github.com/leanprover/lean4/pull/4115), [#4127](https://github.com/leanprover/lean4/pull/4127), [#4220](https://github.com/leanprover/lean4/pull/4220), [#4232](https://github.com/leanprover/lean4/pull/4232), [#4236](https://github.com/leanprover/lean4/pull/4236).
+
+  Builds are now managed by a top-level Lake build monitor, this makes the output of Lake builds more standardized and enables producing prettier and more configurable progress reports.
+
+  As part of this change, job isolation has improved. Stray I/O and other build related errors in custom targets are now properly isolated and caught as part of their job. Import errors no longer cause Lake to abort the entire build and are instead localized to the build jobs of the modules in question.
+
+  Lake also now uses ANSI escape sequences to add color and produce progress lines that update in-place; this can be toggled on and off using `--ansi` / `--no-ansi`.
+
+
+  `--wfail` and `--iofail` options have been added that causes a build to fail if any of the jobs log a warning (`--wfail`) or produce any output or log information messages (`--iofail`). Unlike some other build systems, these options do **NOT** convert these logs into errors, and Lake does not abort jobs on such a log (i.e., dependent jobs will still continue unimpeded).
+
+* `lake test`. [#3779](https://github.com/leanprover/lean4/pull/3779).
+
+  Lake now has a built-in `test` command which will run a script or executable labelled `@[test_runner]` (in Lean) or defined as the `testRunner` (in TOML) in the root package.
+
+  Lake also provides a `lake check-test` command which will exit with code `0` if the package has a properly configured test runner or error with `1` otherwise.
+
+* `lake lean`. [#3793](https://github.com/leanprover/lean4/pull/3793).
+
+  The new command `lake lean <file> [-- <args...>]` functions like `lake env lean <file> <args...>`, except that it builds the imports of `file` before running `lean`. This makes it very useful for running test or example code that imports modules that are not guaranteed to have been built beforehand.
+
+* **Miscellaneous bug fixes and improvements**
+  * [#3609](https://github.com/leanprover/lean4/pull/3609) `LEAN_GITHASH` environment variable to override the detected Git hash for Lean when computing traces, useful for testing custom builds of Lean.
+  * [#3795](https://github.com/leanprover/lean4/pull/3795) improves relative package directory path normalization in the pre-rename check.
+  * [#3957](https://github.com/leanprover/lean4/pull/3957) fixes handling of packages that appear multiple times in a dependency tree.
+  * [#3999](https://github.com/leanprover/lean4/pull/3999) makes it an error for there to be a mismatch between a package name and what it is required as. Also adds a special message for the `std`-to-`batteries` rename.
+  * [#4033](https://github.com/leanprover/lean4/pull/4033) fixes quiet mode.
+* **Docs:** [#3704](https://github.com/leanprover/lean4/pull/3704).
+
+### DevOps
+
+* [#3536](https://github.com/leanprover/lean4/pull/3536) and [#3833](https://github.com/leanprover/lean4/pull/3833)
+  add a checklist for the release process.
+* [#3600](https://github.com/leanprover/lean4/pull/3600) runs nix-ci more uniformly.
+* [#3612](https://github.com/leanprover/lean4/pull/3612) avoids argument limits when building on Windows.
+* [#3682](https://github.com/leanprover/lean4/pull/3682) builds Lean's `.o` files in parallel to rest of core.
+* [#3601](https://github.com/leanprover/lean4/pull/3601)
+  changes the way Lean is built on Windows (see breaking changes below).
+  As a result, Lake now dynamically links executables with `supportInterpreter := true` on Windows
+  to `libleanshared.dll` and `libInit_shared.dll`. Therefore, such executables will not run
+  unless those shared libraries are co-located with the executables or part of `PATH`.
+  Running the executable via `lake exe` will ensure these libraries are part of `PATH`.
+
+  In a related change, the signature of the `nativeFacets` Lake configuration options has changed
+  from a static `Array` to a function `(shouldExport : Bool) → Array`.
+  See its docstring or Lake's [README](src/lake/README.md) for further details on the changed option.
+* [#3690](https://github.com/leanprover/lean4/pull/3690) marks "Build matrix complete" as canceled if the build is canceled.
+* [#3700](https://github.com/leanprover/lean4/pull/3700), [#3702](https://github.com/leanprover/lean4/pull/3702),
+  [#3701](https://github.com/leanprover/lean4/pull/3701), [#3834](https://github.com/leanprover/lean4/pull/3834),
+  [#3923](https://github.com/leanprover/lean4/pull/3923): fixes and improvements for std and mathlib CI.
+* [#3712](https://github.com/leanprover/lean4/pull/3712) fixes `nix build .` on macOS.
+* [#3717](https://github.com/leanprover/lean4/pull/3717) replaces `shell.nix` in devShell with `flake.nix`.
+* [#3715](https://github.com/leanprover/lean4/pull/3715) and [#3790](https://github.com/leanprover/lean4/pull/3790) add test result summaries.
+* [#3971](https://github.com/leanprover/lean4/pull/3971) prevents stage0 changes via the merge queue.
+* [#3979](https://github.com/leanprover/lean4/pull/3979) adds handling for `changes-stage0` label.
+* [#3952](https://github.com/leanprover/lean4/pull/3952) adds a script to summarize GitHub issues.
+* [18a699](https://github.com/leanprover/lean4/commit/18a69914da53dbe37c91bc2b9ce65e1dc01752b6)
+  fixes asan linking
+
+### Breaking changes
+
+* Due to the major Lake build refactor, code using the affected parts of the Lake API or relying on the previous output format of Lake builds is likely to have been broken. We have tried to minimize the breakages and, were possible, old definitions have been marked `@[deprecated]` with a reference to the new alternative.
 
 * Automatically generated equational theorems are now named using suffix `.eq_<idx>` instead of `._eq_<idx>`, and `.def` instead of `._unfold`. Example:
 ```
@@ -106,6 +616,12 @@ fact.def :
 
 * The coercion from `String` to `Name` was removed. Previously, it was `Name.mkSimple`, which does not separate strings at dots, but experience showed that this is not always the desired coercion. For the previous behavior, manually insert a call to `Name.mkSimple`.
 
+* The `Subarray` fields `as`, `h₁` and `h₂` have been renamed to `array`, `start_le_stop`, and `stop_le_array_size`, respectively. This more closely follows standard Lean conventions. Deprecated aliases for the field projections were added; these will be removed in a future release.
+
+* The change to the instance name algorithm (described above) can break projects that made use of the auto-generated names.
+
+* `Option.toMonad` has been renamed to `Option.getM` and the unneeded `[Monad m]` instance argument has been removed.
+
 v4.7.0
 ---------
 

doc/BoolExpr.lean

@@ -1,4 +1,4 @@
-open Std
+open Batteries
 open Lean
 
 inductive BoolExpr where

doc/dev/bootstrap.md

@@ -75,26 +75,28 @@ The github repository will automatically update stage0 on `master` once
 
 If you have write access to the lean4 repository, you can also also manually
 trigger that process, for example to be able to use new features in the compiler itself.
-You can do that on <https://github.com/nomeata/lean4/actions/workflows/update-stage0.yml>
+You can do that on <https://github.com/leanprover/lean4/actions/workflows/update-stage0.yml>
 or using Github CLI with
 ```
 gh workflow run update-stage0.yml
 ```
 
-Leaving stage0 updates to the CI automation is preferrable, but should you need
-to do it locally, you can use `make update-stage0` in `build/release`, to
-update `stage0` from `stage1`, `make -C stageN update-stage0` to update from
-another stage, or `nix run .#update-stage0-commit` to update using nix.
+Leaving stage0 updates to the CI automation is preferable, but should you need
+to do it locally, you can use `make update-stage0-commit` in `build/release` to
+update `stage0` from `stage1` or `make -C stageN update-stage0-commit` to
+update from another stage. This command will automatically stage the updated files
+and introduce a commit,so make sure to commit your work before that.
+
+If you rebased the branch (either onto a newer version of `master`, or fixing
+up some commits prior to the stage0 update, recreate the stage0 update commits.
+The script `script/rebase-stage0.sh` can be used for that.
+
+The CI should prevent PRs with changes to stage0 (besides `stdlib_flags.h`)
+from entering `master` through the (squashing!) merge queue, and label such PRs
+with the `changes-stage0` label. Such PRs should have a cleaned up history,
+with separate stage0 update commits; then coordinate with the admins to merge
+your PR using rebase merge, bypassing the merge queue.
 
-Updates to `stage0` should be their own commits in the Git history. So should
-you have to include the stage0 update in your PR (rather than using above
-automation after merging changes), commit your work before running `make
-update-stage0`, commit the updated `stage0` compiler code with the commit
-message:
-```
-chore: update stage0
-```
-and coordinate with the admins to not squash your PR.
 
 ## Further Bootstrapping Complications
 

doc/dev/ffi.md

@@ -53,10 +53,59 @@ In the case of `@[extern]` all *irrelevant* types are removed first; see next se
   Its runtime value is either a pointer to an opaque bignum object or, if the lowest bit of the "pointer" is 1 (`lean_is_scalar`), an encoded unboxed natural number (`lean_box`/`lean_unbox`).
 * A universe `Sort u`, type constructor `... → Sort u`, or proposition `p : Prop` is *irrelevant* and is either statically erased (see above) or represented as a `lean_object *` with the runtime value `lean_box(0)`
 * Any other type is represented by `lean_object *`.
-  Its runtime value is a pointer to an object of a subtype of `lean_object` (see respective declarations in `lean.h`) or the unboxed value `lean_box(cidx)` for the `cidx`th constructor of an inductive type if this constructor does not have any relevant parameters.
+  Its runtime value is a pointer to an object of a subtype of `lean_object` (see the "Inductive types" section below) or the unboxed value `lean_box(cidx)` for the `cidx`th constructor of an inductive type if this constructor does not have any relevant parameters.
 
   Example: the runtime value of `u : Unit` is always `lean_box(0)`.
 
+#### Inductive types
+
+For inductive types which are in the fallback `lean_object *` case above and not trivial constructors, the type is stored as a `lean_ctor_object`, and `lean_is_ctor` will return true. A `lean_ctor_object` stores the constructor index in the header, and the fields are stored in the `m_objs` portion of the object.
+
+The memory order of the fields is derived from the types and order of the fields in the declaration. They are ordered as follows:
+
+* Non-scalar fields stored as `lean_object *`
+* Fields of type `USize`
+* Other scalar fields, in decreasing order by size
+
+Within each group the fields are ordered in declaration order. **Warning**: Trivial wrapper types still count toward a field being treated as non-scalar for this purpose.
+
+* To access fields of the first kind, use `lean_ctor_get(val, i)` to get the `i`th non-scalar field.
+* To access `USize` fields, use `lean_ctor_get_usize(val, n+i)` to get the `i`th usize field and `n` is the total number of fields of the first kind.
+* To access other scalar fields, use `lean_ctor_get_uintN(val, off)` or `lean_ctor_get_usize(val, off)` as appropriate. Here `off` is the byte offset of the field in the structure, starting at `n*sizeof(void*)` where `n` is the number of fields of the first two kinds.
+
+For example, a structure such as
+```lean
+structure S where
+  ptr_1 : Array Nat
+  usize_1 : USize
+  sc64_1 : UInt64
+  ptr_2 : { x : UInt64 // x > 0 } -- wrappers don't count as scalars
+  sc64_2 : Float -- `Float` is 64 bit
+  sc8_1 : Bool
+  sc16_1 : UInt16
+  sc8_2 : UInt8
+  sc64_3 : UInt64
+  usize_2 : USize
+  ptr_3 : Char -- trivial wrapper around `UInt32`
+  sc32_1 : UInt32
+  sc16_2 : UInt16
+```
+would get re-sorted into the following memory order:
+
+* `S.ptr_1` - `lean_ctor_get(val, 0)`
+* `S.ptr_2` - `lean_ctor_get(val, 1)`
+* `S.ptr_3` - `lean_ctor_get(val, 2)`
+* `S.usize_1` - `lean_ctor_get_usize(val, 3)`
+* `S.usize_2` - `lean_ctor_get_usize(val, 4)`
+* `S.sc64_1` - `lean_ctor_get_uint64(val, sizeof(void*)*5)`
+* `S.sc64_2` - `lean_ctor_get_float(val, sizeof(void*)*5 + 8)`
+* `S.sc64_3` - `lean_ctor_get_uint64(val, sizeof(void*)*5 + 16)`
+* `S.sc32_1` - `lean_ctor_get_uint32(val, sizeof(void*)*5 + 24)`
+* `S.sc16_1` - `lean_ctor_get_uint16(val, sizeof(void*)*5 + 28)`
+* `S.sc16_2` - `lean_ctor_get_uint16(val, sizeof(void*)*5 + 30)`
+* `S.sc8_1` - `lean_ctor_get_uint8(val, sizeof(void*)*5 + 32)`
+* `S.sc8_2` - `lean_ctor_get_uint8(val, sizeof(void*)*5 + 33)`
+
 ### Borrowing
 
 By default, all `lean_object *` parameters of an `@[extern]` function are considered *owned*, i.e. the external code is passed a "virtual RC token" and is responsible for passing this token along to another consuming function (exactly once) or freeing it via `lean_dec`.

doc/dev/release_checklist.md

@@ -21,7 +21,7 @@ We'll use `v4.6.0` as the intended release version as a running example.
   - Reconcile discrepancies in the `v4.6.0` section,
     usually via copy and paste and a commit to `releases/v4.6.0`.
 - `git tag v4.6.0`
-- `git push origin v4.6.0`
+- `git push $REMOTE v4.6.0`, where `$REMOTE` is the upstream Lean repository (e.g., `origin`, `upstream`)
 - Now wait, while CI runs.
   - You can monitor this at `https://github.com/leanprover/lean4/actions/workflows/ci.yml`,
     looking for the `v4.6.0` tag.
@@ -34,48 +34,76 @@ We'll use `v4.6.0` as the intended release version as a running example.
     (e.g. `v4.6.0-rc1`), and quickly sanity check.
 - Next, we will move a curated list of downstream repos to the latest stable release.
   - For each of the repositories listed below:
-    - Make a PR to `master`/`main` changing the toolchain to `v4.6.0`.
-      The PR title should be "chore: bump toolchain to v4.6.0".
-      Since the `v4.6.0` release should be functionally identical to the last release candidate,
-      which the repository should already be on, this PR is a no-op besides changing the toolchain.
-    - Once this is merged, create the tag `v4.6.0` from `master`/`main` and push it.
-    - Merge the tag `v4.6.0` into the stable branch.
+    - Make a PR to `master`/`main` changing the toolchain to `v4.6.0`
+      - Update the toolchain file
+      - In the Lakefile, if there are dependencies on specific version tags of dependencies that you've already pushed as part of this process, update them to the new tag.
+        If they depend on `main` or `master`, don't change this; you've just updated the dependency, so it will work and be saved in the manifest
+      - Run `lake update`
+      - The PR title should be "chore: bump toolchain to v4.6.0".
+    - Merge the PR once CI completes.
+    - Create the tag `v4.6.0` from `master`/`main` and push it.
+    - Merge the tag `v4.6.0` into the `stable` branch and push it.
   - We do this for the repositories:
     - [lean4checker](https://github.com/leanprover/lean4checker)
-      - `lean4checker` uses a different version tagging scheme: use `toolchain/v4.6.0` rather than `v4.6.0`.
-    - [Std](https://github.com/leanprover-community/repl)
+      - No dependencies
+      - Note: `lean4checker` uses a different version tagging scheme: use `toolchain/v4.6.0` rather than `v4.6.0`.
+      - Toolchain bump PR
+      - Create and push the tag
+      - Merge the tag into `stable`
+    - [Batteries](https://github.com/leanprover-community/batteries)
+      - No dependencies
+      - Toolchain bump PR
+      - Create and push the tag
+      - Merge the tag into `stable`
     - [ProofWidgets4](https://github.com/leanprover-community/ProofWidgets4)
-      - `ProofWidgets` uses a sequential version tagging scheme, e.g. `v0.0.29`,
-        which does not refer to the toolchain being used.
-      - Make a new release in this sequence after merging the toolchain bump PR.
-      - `ProofWidgets` does not maintain a `stable` branch.
+      - Dependencies: `Batteries`
+      - Note on versions and branches:
+        - `ProofWidgets` uses a sequential version tagging scheme, e.g. `v0.0.29`,
+          which does not refer to the toolchain being used.
+        - Make a new release in this sequence after merging the toolchain bump PR.
+        - `ProofWidgets` does not maintain a `stable` branch.
+      - Toolchain bump PR
+      - Create and push the tag, following the version convention of the repository
     - [Aesop](https://github.com/leanprover-community/aesop)
+      - Dependencies: `Batteries`
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - Merge the tag into `stable`
+    - [doc-gen4](https://github.com/leanprover/doc-gen4)
+      - Dependencies: exist, but they're not part of the release workflow
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - There is no `stable` branch; skip this step
+    - [import-graph](https://github.com/leanprover-community/import-graph)
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - There is no `stable` branch; skip this step
     - [Mathlib](https://github.com/leanprover-community/mathlib4)
-      - In addition to updating the `lean-toolchain` and `lakefile.lean`,
-        in `.github/workflows/build.yml.in` in the `lean4checker` section update the line
-        `git checkout toolchain/v4.6.0` to the appropriate tag,
-        and then run `.github/workflows/mk_build_yml.sh`.
+      - Dependencies: `Aesop`, `ProofWidgets4`, `lean4checker`, `Batteries`, `doc-gen4`, `import-graph`
+      - Toolchain bump PR notes:
+        - In addition to updating the `lean-toolchain` and `lakefile.lean`,
+          in `.github/workflows/build.yml.in` in the `lean4checker` section update the line
+          `git checkout toolchain/v4.6.0` to the appropriate tag,
+          and then run `.github/workflows/mk_build_yml.sh`. Coordinate with
+          a Mathlib maintainer to get this merged.
+        - Push the PR branch to the main Mathlib repository rather than a fork, or CI may not work reliably
+        - Create and push the tag
+        - Create a new branch from the tag, push it, and open a pull request against `stable`.
+          Coordinate with a Mathlib maintainer to get this merged.
     - [REPL](https://github.com/leanprover-community/repl)
+      - Dependencies: `Mathlib` (for test code)
       - Note that there are two copies of `lean-toolchain`/`lakefile.lean`:
-        in the root, and in `test/Mathlib/`.
-  - Note that there are dependencies between these packages:
-    you should update the lakefile so that you are using the `v4.6.0` tag of upstream repositories
-    (or the sequential tag for `ProofWidgets4`), and run `lake update` before committing.
-  - This means that this process is sequential; each repository must have its bump PR merged,
-    and the new tag pushed, before you can make the PR for the downstream repositories.
-    - `lean4checker` has no dependencies
-    - `Std` has no dependencies
-    - `Aesop` depends on `Std`
-    - `ProofWidgets4` depends on `Std`
-    - `Mathlib` depends on `Aesop`, `ProofWidgets4`, and `lean4checker` (and transitively on `Std`)
-    - `REPL` depends on `Mathlib` (this dependency is only for testing).
+        in the root, and in `test/Mathlib/`. Edit both, and run `lake update` in both directories.
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - Merge the tag into `stable`
 - Merge the release announcement PR for the Lean website - it will be deployed automatically
 - Finally, make an announcement!
   This should go in https://leanprover.zulipchat.com/#narrow/stream/113486-announce, with topic `v4.6.0`.
   Please see previous announcements for suggested language.
   You will want a few bullet points for main topics from the release notes.
   Link to the blog post from the Zulip announcement.
-  Please also make sure that whoever is handling social media knows the release is out.
+- Make sure that whoever is handling social media knows the release is out.
 
 ## Optimistic(?) time estimates:
 - Initial checks and push the tag: 30 minutes.
@@ -95,8 +123,8 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
 
 - Decide which nightly release you want to turn into a release candidate.
   We will use `nightly-2024-02-29` in this example.
-- It is essential that Std and Mathlib already have reviewed branches compatible with this nightly.
-  - Check that both Std and Mathlib's `bump/v4.7.0` branch contain `nightly-2024-02-29`
+- It is essential that Batteries and Mathlib already have reviewed branches compatible with this nightly.
+  - Check that both Batteries and Mathlib's `bump/v4.7.0` branch contain `nightly-2024-02-29`
     in their `lean-toolchain`.
   - The steps required to reach that state are beyond the scope of this checklist, but see below!
 - Create the release branch from this nightly tag:
@@ -154,7 +182,7 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
   - We do this for the same list of repositories as for stable releases, see above.
     As above, there are dependencies between these, and so the process above is iterative.
     It greatly helps if you can merge the `bump/v4.7.0` PRs yourself!
-  - For Std/Aesop/Mathlib, which maintain a `nightly-testing` branch, make sure there is a tag
+  - For Batteries/Aesop/Mathlib, which maintain a `nightly-testing` branch, make sure there is a tag
     `nightly-testing-2024-02-29` with date corresponding to the nightly used for the release
     (create it if not), and then on the `nightly-testing` branch `git reset --hard master`, and force push.
 - Make an announcement!
@@ -176,7 +204,7 @@ In particular, updating the downstream repositories is significantly more work
 # Preparing `bump/v4.7.0` branches
 
 While not part of the release process per se,
-this is a brief summary of the work that goes into updating Std/Aesop/Mathlib to new versions.
+this is a brief summary of the work that goes into updating Batteries/Aesop/Mathlib to new versions.
 
 Please read https://leanprover-community.github.io/contribute/tags_and_branches.html
 

doc/examples/Certora2022/ex8.lean

@@ -4,16 +4,16 @@ def ack : Nat → Nat → Nat
   | 0,   y   => y+1
   | x+1, 0   => ack x 1
   | x+1, y+1 => ack x (ack (x+1) y)
-termination_by ack x y => (x, y)
+termination_by x y => (x, y)
 
 def sum (a : Array Int) : Int :=
   let rec go (i : Nat) :=
-     if i < a.size then
+     if _ : i < a.size then
         a[i] + go (i+1)
      else
         0
+  termination_by a.size - i
   go 0
-termination_by go i => a.size - i
 
 set_option pp.proofs true
 #print sum.go

doc/examples/ICERM2022/ctor.lean

@@ -4,43 +4,42 @@ open Lean Meta
 
 def ctor (mvarId : MVarId) (idx : Nat) : MetaM (List MVarId) := do
   /- Set `MetaM` context using `mvarId` -/
-  withMVarContext mvarId do 
+  mvarId.withContext do
     /- Fail if the metavariable is already assigned. -/
-    checkNotAssigned mvarId `ctor
+    mvarId.checkNotAssigned `ctor
     /- Retrieve the target type, instantiateMVars, and use `whnf`. -/
-    let target ← getMVarType' mvarId
+    let target ← mvarId.getType'
     let .const declName us := target.getAppFn
       | throwTacticEx `ctor mvarId "target is not an inductive datatype"
     let .inductInfo { ctors, .. } ← getConstInfo declName
       | throwTacticEx `ctor mvarId "target is not an inductive datatype"
     if idx = 0 then
-      throwTacticEx `ctor mvarId "invalid index, it must be > 0"  
+      throwTacticEx `ctor mvarId "invalid index, it must be > 0"
     else if h : idx - 1 < ctors.length then
-      apply mvarId (.const ctors[idx - 1] us)
+      mvarId.apply (.const ctors[idx - 1] us)
     else
-      throwTacticEx `ctor mvarId "invalid index, inductive datatype has only {ctors.length} contructors"  
+      throwTacticEx `ctor mvarId "invalid index, inductive datatype has only {ctors.length} contructors"
 
 open Elab Tactic
 
-elab "ctor" idx:num : tactic => 
+elab "ctor" idx:num : tactic =>
   liftMetaTactic (ctor · idx.getNat)
 
-example (p : Prop) : p := by 
+example (p : Prop) : p := by
   ctor 1 -- Error
 
-example (h : q) : p ∨ q := by 
+example (h : q) : p ∨ q := by
   ctor 0 -- Error
   exact h
 
-example (h : q) : p ∨ q := by 
+example (h : q) : p ∨ q := by
   ctor 3 -- Error
   exact h
 
-example (h : q) : p ∨ q := by 
+example (h : q) : p ∨ q := by
   ctor 2
   exact h
 
-example (h : q) : p ∨ q := by 
+example (h : q) : p ∨ q := by
   ctor 1
-  exact h -- Error 
-
+  exact h -- Error

doc/examples/ICERM2022/meta.lean

@@ -5,15 +5,15 @@ open Lean Meta
 def ex1 (declName : Name) : MetaM Unit := do
   let info ← getConstInfo declName
   IO.println s!"{declName} : {← ppExpr info.type}"
-  if let some val := info.value? then 
+  if let some val := info.value? then
     IO.println s!"{declName} : {← ppExpr val}"
-    
+
 #eval ex1 ``Nat
 
 def ex2 (declName : Name) : MetaM Unit := do
   let info ← getConstInfo declName
   trace[Meta.debug] "{declName} : {info.type}"
-  if let some val := info.value? then 
+  if let some val := info.value? then
     trace[Meta.debug] "{declName} : {val}"
 
 #eval ex2 ``Add.add
@@ -30,9 +30,9 @@ def ex3 (declName : Name) : MetaM Unit := do
       trace[Meta.debug] "{x} : {← inferType x}"
 
 def myMin [LT α] [DecidableRel (α := α) (·<·)] (a b : α) : α :=
-  if a < b then 
+  if a < b then
     a
-  else 
+  else
     b
 
 set_option trace.Meta.debug true in
@@ -40,7 +40,7 @@ set_option trace.Meta.debug true in
 
 def ex4 : MetaM Unit := do
   let nat := mkConst ``Nat
-  withLocalDeclD `a nat fun a => 
+  withLocalDeclD `a nat fun a =>
   withLocalDeclD `b nat fun b => do
     let e ← mkAppM ``HAdd.hAdd #[a, b]
     trace[Meta.debug] "{e} : {← inferType e}"
@@ -66,15 +66,17 @@ open Elab Term
 
 def ex5 : TermElabM Unit := do
   let nat := Lean.mkConst ``Nat
-  withLocalDeclD `a nat fun a => do 
+  withLocalDeclD `a nat fun a => do
   withLocalDeclD `b nat fun b => do
     let ab ← mkAppM ``HAdd.hAdd #[a, b]
-    let stx ← `(fun x => if x < 10 then $(← exprToSyntax ab) + x else x + $(← exprToSyntax a))
+    let abStx ← exprToSyntax ab
+    let aStx ← exprToSyntax a
+    let stx ← `(fun x => if x < 10 then $abStx + x else x + $aStx)
     let e ← elabTerm stx none
     trace[Meta.debug] "{e} : {← inferType e}"
     let e := mkApp e (mkNatLit 5)
     let e ← whnf e
     trace[Meta.debug] "{e}"
-       
+
 set_option trace.Meta.debug true in
 #eval ex5

doc/examples/NFM2022/nfm8.lean

@@ -4,16 +4,16 @@ def ack : Nat → Nat → Nat
   | 0,   y   => y+1
   | x+1, 0   => ack x 1
   | x+1, y+1 => ack x (ack (x+1) y)
-termination_by ack x y => (x, y)
+termination_by x y => (x, y)
 
 def sum (a : Array Int) : Int :=
   let rec go (i : Nat) :=
-     if i < a.size then
+     if _ : i < a.size then
         a[i] + go (i+1)
      else
         0
+  termination_by a.size - i
   go 0
-termination_by go i => a.size - i
 
 set_option pp.proofs true
 #print sum.go

doc/examples/test_single.sh

@@ -1,4 +1,4 @@
 #!/usr/bin/env bash
 source ../../tests/common.sh
 
-exec_check lean -j 0 -Dlinter.all=false "$f"
+exec_check lean -Dlinter.all=false "$f"

doc/latex/lean4.py

@@ -1,100 +0,0 @@
-# -*- coding: utf-8 -*-
-"""
-    pygments.lexers.theorem
-    ~~~~~~~~~~~~~~~~~~~~~~~
-
-    Lexers for theorem-proving languages.
-
-    :copyright: Copyright 2006-2017 by the Pygments team, see AUTHORS.
-    :license: BSD, see LICENSE for details.
-"""
-
-import re
-
-from pygments.lexer import RegexLexer, default, words
-from pygments.token import Text, Comment, Operator, Keyword, Name, String, \
-    Number, Punctuation, Generic
-
-__all__ = ['Lean4Lexer']
-
-class Lean4Lexer(RegexLexer):
-    """
-    For the `Lean 4 <https://github.com/leanprover/lean4>`_
-    theorem prover.
-
-    .. versionadded:: 2.0
-    """
-    name = 'Lean4'
-    aliases = ['lean4']
-    filenames = ['*.lean']
-    mimetypes = ['text/x-lean']
-
-    flags = re.MULTILINE | re.UNICODE
-
-    keywords1 = (
-        'import', 'abbreviation', 'opaque_hint', 'tactic_hint', 'definition',
-        'renaming', 'inline', 'hiding', 'parameter', 'lemma', 'variable',
-        'theorem', 'axiom', 'inductive', 'structure', 'universe', 'alias',
-        'help', 'options', 'precedence', 'postfix', 'prefix',
-        'infix', 'infixl', 'infixr', 'notation', '#eval',
-        '#check', '#reduce', '#exit', 'coercion', 'end', 'private', 'using', 'namespace',
-        'including', 'instance', 'section', 'context', 'protected', 'expose',
-        'export', 'set_option', 'extends', 'open', 'example',
-        'constant', 'constants', 'print', 'opaque', 'reducible', 'irreducible',
-        'def', 'macro', 'elab', 'syntax', 'macro_rules', 'reduce', 'where',
-        'abbrev', 'noncomputable', 'class', 'attribute', 'synth', 'mutual',
-    )
-
-    keywords2 = (
-        'forall', 'fun', 'Pi', 'obtain', 'from', 'have', 'show', 'assume',
-        'take', 'let', 'if', 'else', 'then', 'by', 'in', 'with', 'begin',
-        'proof', 'qed', 'calc', 'match', 'nomatch', 'do', 'at',
-    )
-
-    keywords3 = (
-        # Sorts
-        'Type', 'Prop', 'Sort',
-    )
-
-    operators = (
-        u'!=', u'#', u'&', u'&&', u'*', u'+', u'-', u'/', u'@', u'!', u'`',
-        u'-.', u'->', u'.', u'..', u'...', u'::', u':>', u';', u';;', u'<',
-        u'<-', u'=', u'==', u'>', u'_', u'|', u'||', u'~', u'=>', u'<=', u'>=',
-        u'/\\', u'\\/', u'∀', u'Π', u'λ', u'↔', u'∧', u'∨', u'≠', u'≤', u'≥',
-        u'¬', u'⁻¹', u'⬝', u'▸', u'→', u'∃', u'ℕ', u'ℤ', u'≈', u'×', u'⌞',
-        u'⌟', u'≡', u'⟨', u'⟩',
-    )
-
-    punctuation = (u'(', u')', u':', u'{', u'}', u'[', u']', u'⦃', u'⦄',
-                   u':=', u',')
-
-    tokens = {
-        'root': [
-            (r'\s+', Text),
-            (r'/-', Comment, 'comment'),
-            (r'--.*?$', Comment.Single),
-            (words(keywords1, prefix=r'\b', suffix=r'\b'), Keyword.Namespace),
-            (words(keywords2, prefix=r'\b', suffix=r'\b'), Keyword),
-            (words(keywords3, prefix=r'\b', suffix=r'\b'), Keyword.Type),
-            (words(operators), Name.Builtin.Pseudo),
-            (words(punctuation), Operator),
-            (u"[A-Za-z_\u03b1-\u03ba\u03bc-\u03fb\u1f00-\u1ffe\u2100-\u214f]"
-             u"[A-Za-z_'\u03b1-\u03ba\u03bc-\u03fb\u1f00-\u1ffe\u2070-\u2079"
-             u"\u207f-\u2089\u2090-\u209c\u2100-\u214f0-9]*", Name),
-            (r'\d+', Number.Integer),
-            (r'"', String.Double, 'string'),
-            (r'[~?][a-z][\w\']*:', Name.Variable)
-        ],
-        'comment': [
-            # Multiline Comments
-            (r'[^/-]', Comment.Multiline),
-            (r'/-', Comment.Multiline, '#push'),
-            (r'-/', Comment.Multiline, '#pop'),
-            (r'[/-]', Comment.Multiline)
-        ],
-        'string': [
-            (r'[^\\"]+', String.Double),
-            (r'\\[n"\\]', String.Escape),
-            ('"', String.Double, '#pop'),
-        ],
-    }

doc/monads/states.lean

@@ -15,7 +15,7 @@ data type containing several important pieces of information. First and foremost
 current player, and it has a random generator.
 -/
 
-open Std (HashMap)
+open Batteries (HashMap)
 abbrev TileIndex := Nat × Nat -- a 2D index
 
 inductive TileState where

doc/syntax_highlight_in_latex.md

@@ -43,7 +43,8 @@ $ pdflatex test.tex
 
 ## Example with `minted`
 
-First [install Pygments](https://pygments.org/download/). Then save [`lean4.py`](https://raw.githubusercontent.com/leanprover/lean4/master/doc/latex/lean4.py), which contains an version of the Lean highlighter updated for Lean 4, and the following sample LaTeX file `test.tex` into the same directory:
+First [install Pygments](https://pygments.org/download/) (version 2.18 or newer).
+Then save the following sample LaTeX file `test.tex` into the same directory:
 
 ```latex
 \documentclass{article}
@@ -51,9 +52,8 @@ First [install Pygments](https://pygments.org/download/). Then save [`lean4.py`]
 % switch to a monospace font supporting more Unicode characters
 \setmonofont{FreeMono}
 \usepackage{minted}
-% instruct minted to use our local theorem.py
-\newmintinline[lean]{lean4.py:Lean4Lexer -x}{bgcolor=white}
-\newminted[leancode]{lean4.py:Lean4Lexer -x}{fontsize=\footnotesize}
+\newmintinline[lean]{lean4}{bgcolor=white}
+\newminted[leancode]{lean4}{fontsize=\footnotesize}
 \usemintedstyle{tango}  % a nice, colorful theme
 
 \begin{document}
@@ -67,9 +67,6 @@ theorem funext {f₁ f₂ : ∀ (x : α), β x} (h : ∀ x, f₁ x = f₂ x) : f
 \end{document}
 ```
 
-If your version of `minted` is v2.7 or newer, but before v3.0,
-you will additionally need to follow the workaround described in https://github.com/gpoore/minted/issues/360.
-
 You can then compile `test.tex` by executing the following command:
 
 ```bash
@@ -81,11 +78,14 @@ Some remarks:
  - either `xelatex` or `lualatex` is required to handle Unicode characters in the code.
  - `--shell-escape` is needed to allow `xelatex` to execute `pygmentize` in a shell.
  - If the chosen monospace font is missing some Unicode symbols, you can direct them to be displayed using a fallback font or other replacement LaTeX code.
-``` latex
-\usepackage{newunicodechar}
-\newfontfamily{\freeserif}{DejaVu Sans}
-\newunicodechar{✝}{\freeserif{✝}}
-\newunicodechar{𝓞}{\ensuremath{\mathcal{O}}}
-```
- - minted has a "helpful" feature that draws red boxes around characters the chosen lexer doesn't recognize.
- Since the Lean lexer cannot encompass all user-defined syntax, it is advisable to [work around](https://tex.stackexchange.com/a/343506/14563) this feature.
+   ``` latex
+   \usepackage{newunicodechar}
+   \newfontfamily{\freeserif}{DejaVu Sans}
+   \newunicodechar{✝}{\freeserif{✝}}
+   \newunicodechar{𝓞}{\ensuremath{\mathcal{O}}}
+   ```
+ - If you are using an old version of Pygments, you can copy 
+   [`lean.py`](https://raw.githubusercontent.com/pygments/pygments/master/pygments/lexers/lean.py) into your working directory,
+   and use `lean4.py:Lean4Lexer -x` instead of `lean4` above.
+   If your version of `minted` is v2.7 or newer, but before v3.0,
+   you will additionally need to follow the workaround described in https://github.com/gpoore/minted/issues/360.

nix/bootstrap.nix

@@ -170,7 +170,7 @@ rec {
           ln -sf ${lean-all}/* .
         '';
         buildPhase = ''
-          ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)' -j$NIX_BUILD_CORES
+          ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)|leanlaketest_init' -j$NIX_BUILD_CORES
         '';
         installPhase = ''
           mkdir $out
@@ -180,7 +180,7 @@ rec {
       update-stage0 =
         let cTree = symlinkJoin { name = "cs"; paths = [ Init.cTree Lean.cTree ]; }; in
         writeShellScriptBin "update-stage0" ''
-          CSRCS=${cTree} CP_C_PARAMS="--dereference --no-preserve=all" ${src + "/script/update-stage0"}
+          CSRCS=${cTree} CP_C_PARAMS="--dereference --no-preserve=all" ${src + "/script/lib/update-stage0"}
         '';
       update-stage0-commit = writeShellScriptBin "update-stage0-commit" ''
         set -euo pipefail

script/collideProfiles.lean

@@ -0,0 +1,28 @@
+import Lean.Util.Profiler
+
+/-!
+
+Usage:
+```sh
+lean --run ./script/collideProfiles.lean **/*.lean.json ... > merged.json
+```
+
+Merges multiple `trace.profiler.output` profiles into a single one while deduplicating samples with
+the same stack. This is useful for building cumulative profiles of medium-to-large projects because
+Firefox Profiler cannot handle hundreds of tracks and the deduplication will also ensure that the
+profile is small enough for uploading.
+
+As ordering of samples is not meaningful after this transformation, only "Call Tree" and "Flame
+Graph" are useful for such profiles.
+-/
+
+open Lean
+
+def main (args : List String) : IO Unit := do
+  let profiles ← args.toArray.mapM fun path => do
+    let json ← IO.FS.readFile ⟨path⟩
+    let profile ← IO.ofExcept $ Json.parse json
+    IO.ofExcept <| fromJson? profile
+  -- NOTE: `collide` should not be interpreted
+  let profile := Firefox.Profile.collide profiles
+  IO.println <| Json.compress <| toJson profile

script/issues_summary.sh

@@ -0,0 +1,39 @@
+#!/bin/bash
+
+# https://chat.openai.com/share/7469c7c3-aceb-4d80-aee5-62982e1f1538
+
+# Output CSV Header
+echo '"Issue URL","Title","Days Since Creation","Days Since Last Update","Total Reactions","Assignee","Labels"'
+
+# Get the current date in YYYY-MM-DD format
+today=$(date +%Y-%m-%d)
+
+# Fetch only open issues (excluding PRs and closed issues) from the repository 'leanprover/lean4'
+issues=$(gh api repos/leanprover/lean4/issues --paginate --jq '.[] | select(.pull_request == null and .state == "open") | {url: .html_url, title: .title, created_at: (.created_at | split("T")[0]), updated_at: (.updated_at | split("T")[0]), number: .number, assignee: (.assignee.login // ""), labels: [.labels[].name] | join(",")}')
+
+# Process each JSON object
+echo "$issues" | while IFS= read -r issue; do
+    # Extract fields from JSON
+    url=$(echo "$issue" | jq -r '.url')
+    title=$(echo "$issue" | jq -r '.title')
+    created_at=$(echo "$issue" | jq -r '.created_at')
+    updated_at=$(echo "$issue" | jq -r '.updated_at')
+    issue_number=$(echo "$issue" | jq -r '.number')
+    assignee=$(echo "$issue" | jq -r '.assignee')
+    labels=$(echo "$issue" | jq -r '.labels')
+
+    # Calculate days since creation and update using macOS compatible date calculation
+    days_since_created=$(( ($(date -jf "%Y-%m-%d" "$today" +%s) - $(date -jf "%Y-%m-%d" "$created_at" +%s)) / 86400 ))
+    days_since_updated=$(( ($(date -jf "%Y-%m-%d" "$today" +%s) - $(date -jf "%Y-%m-%d" "$updated_at" +%s)) / 86400 ))
+
+    # Fetch the total number of reactions for each issue
+    reaction_data=$(gh api repos/leanprover/lean4/issues/$issue_number/reactions --paginate --jq 'length' 2>&1)
+    if [[ $reaction_data == *"Not Found"* ]]; then
+        total_reactions="Error fetching reactions"
+    else
+        total_reactions=$reaction_data
+    fi
+
+    # Format output as CSV by escaping quotes and delimiting with commas
+    echo "\"$url\",\"${title//\"/\"\"}\",\"$days_since_created\",\"$days_since_updated\",\"$total_reactions\",\"$assignee\",\"$labels\""
+done

script/lib/README.md

@@ -0,0 +1,2 @@
+This directory contains various scripts that are *not* meant to be called
+directly, but through other scripts or makefiles.

script/lib/rebase-editor.sh

@@ -0,0 +1,19 @@
+#!/usr/bin/env bash
+
+
+# Script internal to `./script/rebase-stage0.sh`
+
+# Determine OS type for sed in-place editing
+SED_CMD=("sed" "-i")
+if [[ "$OSTYPE" == "darwin"* ]]
+then
+    # macOS requires an empty string argument with -i for in-place editing
+    SED_CMD=("sed" "-i" "")
+fi
+
+if [ "$STAGE0_WITH_NIX" = true ]
+then
+  "${SED_CMD[@]}" '/chore: update stage0/ s,.*,x nix run .#update-stage0-commit,' "$1"
+else
+  "${SED_CMD[@]}" '/chore: update stage0/ s,.*,x make -j32 -C build/release update-stage0 \&\& git commit -m "chore: update stage0",' "$1"
+fi

script/rebase-stage0.sh

@@ -0,0 +1,24 @@
+#!/usr/bin/env bash
+
+# This script rebases onto the given branch/commit, and updates
+# all `chore: update stage0` commits along the way.
+
+# Whether to use nix or make to update stage0
+if [ "$1" = "-nix" ]
+then
+   export STAGE0_WITH_NIX=true
+   shift
+fi
+
+# Check if an argument is provided
+if [ "$#" -eq 0 ]; then
+    echo "Usage: $0 [-nix] <options to git rebase -i>"
+    exit 1
+fi
+
+REPO_ROOT=$(git rev-parse --show-toplevel)
+
+# Run git rebase in interactive mode, but automatically edit the todo list
+# using the defined GIT_SEQUENCE_EDITOR command
+GIT_SEQUENCE_EDITOR="$REPO_ROOT/script/lib/rebase-editor.sh" git rebase -i "$@"
+

src/CMakeLists.txt

@@ -9,9 +9,9 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 8)
+set(LEAN_VERSION_MINOR 9)
 set(LEAN_VERSION_PATCH 0)
-set(LEAN_VERSION_IS_RELEASE 0)  # This number is 1 in the release revision, and 0 otherwise.
+set(LEAN_VERSION_IS_RELEASE 1)  # 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'")
 set(LEAN_VERSION_STRING "${LEAN_VERSION_MAJOR}.${LEAN_VERSION_MINOR}.${LEAN_VERSION_PATCH}")
 if (LEAN_SPECIAL_VERSION_DESC)
@@ -315,6 +315,12 @@ endif()
 string(APPEND TOOLCHAIN_STATIC_LINKER_FLAGS " ${LEAN_CXX_STDLIB}")
 string(APPEND TOOLCHAIN_SHARED_LINKER_FLAGS " ${LEAN_CXX_STDLIB}")
 
+# in local builds, link executables and not just dynlibs against C++ stdlib as well,
+# which is required for e.g. asan
+if(NOT LEAN_STANDALONE)
+  string(APPEND CMAKE_EXE_LINKER_FLAGS " ${LEAN_CXX_STDLIB}")
+endif()
+
 # flags for user binaries = flags for toolchain binaries + Lake
 string(APPEND LEANC_STATIC_LINKER_FLAGS " ${TOOLCHAIN_STATIC_LINKER_FLAGS} -lLake")
 
@@ -585,9 +591,13 @@ endif()
 
 if(PREV_STAGE)
   add_custom_target(update-stage0
-    COMMAND bash -c 'CSRCS=${CMAKE_BINARY_DIR}/lib/temp script/update-stage0'
+    COMMAND bash -c 'CSRCS=${CMAKE_BINARY_DIR}/lib/temp script/lib/update-stage0'
     DEPENDS make_stdlib
     WORKING_DIRECTORY "${LEAN_SOURCE_DIR}/..")
+
+  add_custom_target(update-stage0-commit
+    COMMAND git commit -m "chore: update stage0"
+    DEPENDS update-stage0)
 endif()
 
 # use Bash version for building, use Lean version in bin/ for tests & distribution

src/Init.lean

@@ -34,3 +34,4 @@ import Init.BinderPredicates
 import Init.Ext
 import Init.Omega
 import Init.MacroTrace
+import Init.Grind

src/Init/ByCases.lean

@@ -63,3 +63,16 @@ theorem ite_some_none_eq_none [Decidable P] :
 @[simp] theorem ite_some_none_eq_some [Decidable P] :
     (if P then some x else none) = some y ↔ P ∧ x = y := by
   split <;> simp_all
+
+-- This is not marked as `simp` as it is already handled by `dite_eq_right_iff`.
+theorem dite_some_none_eq_none [Decidable P] {x : P → α} :
+    (if h : P then some (x h) else none) = none ↔ ¬P := by
+  simp only [dite_eq_right_iff]
+  rfl
+
+@[simp] theorem dite_some_none_eq_some [Decidable P] {x : P → α} {y : α} :
+    (if h : P then some (x h) else none) = some y ↔ ∃ h : P, x h = y := by
+  by_cases h : P <;> simp only [h, dite_cond_eq_true, dite_cond_eq_false, Option.some.injEq,
+    false_iff, not_exists]
+  case pos => exact ⟨fun h_eq ↦ Exists.intro h h_eq, fun h_exists => h_exists.2⟩
+  case neg => exact fun h_false _ ↦ h_false

src/Init/Classical.lean

@@ -15,6 +15,13 @@ namespace Classical
 noncomputable def indefiniteDescription {α : Sort u} (p : α → Prop) (h : ∃ x, p x) : {x // p x} :=
   choice <| let ⟨x, px⟩ := h; ⟨⟨x, px⟩⟩
 
+/--
+Given that there exists an element satisfying `p`, returns one such element.
+
+This is a straightforward consequence of, and equivalent to, `Classical.choice`.
+
+See also `choose_spec`, which asserts that the returned value has property `p`.
+-/
 noncomputable def choose {α : Sort u} {p : α → Prop} (h : ∃ x, p x) : α :=
   (indefiniteDescription p h).val
 

src/Init/Control/Basic.lean

@@ -20,8 +20,29 @@ def Functor.discard {f : Type u → Type v} {α : Type u} [Functor f] (x : f α)
 
 export Functor (discard)
 
+/--
+An `Alternative` functor is an `Applicative` functor that can "fail" or be "empty"
+and a binary operation `<|>` that “collects values” or finds the “left-most success”.
+
+Important instances include
+* `Option`, where `failure := none` and `<|>` returns the left-most `some`.
+* Parser combinators typically provide an `Applicative` instance for error-handling and
+  backtracking.
+  
+Error recovery and state can interact subtly. For example, the implementation of `Alternative` for `OptionT (StateT σ Id)` keeps modifications made to the state while recovering from failure, while `StateT σ (OptionT Id)` discards them.
+-/
+-- NB: List instance is in mathlib. Once upstreamed, add
+-- * `List`, where `failure` is the empty list and `<|>` concatenates.
 class Alternative (f : Type u → Type v) extends Applicative f : Type (max (u+1) v) where
+  /--
+  Produces an empty collection or recoverable failure.  The `<|>` operator collects values or recovers
+  from failures. See `Alternative` for more details.
+  -/
   failure : {α : Type u} → f α
+  /--
+  Depending on the `Alternative` instance, collects values or recovers from `failure`s by
+  returning the leftmost success. Can be written using the `<|>` operator syntax.
+  -/
   orElse  : {α : Type u} → f α → (Unit → f α) → f α
 
 instance (f : Type u → Type v) (α : Type u) [Alternative f] : OrElse (f α) := ⟨Alternative.orElse⟩
@@ -30,9 +51,15 @@ variable {f : Type u → Type v} [Alternative f] {α : Type u}
 
 export Alternative (failure)
 
+/--
+If the proposition `p` is true, does nothing, else fails (using `failure`).
+-/
 @[always_inline, inline] def guard {f : Type → Type v} [Alternative f] (p : Prop) [Decidable p] : f Unit :=
   if p then pure () else failure
 
+/--
+Returns `some x` if `f` succeeds with value `x`, else returns `none`.
+-/
 @[always_inline, inline] def optional (x : f α) : f (Option α) :=
   some <$> x <|> pure none
 

src/Init/Control/Lawful/Basic.lean

@@ -12,6 +12,15 @@ open Function
 @[simp] theorem monadLift_self [Monad m] (x : m α) : monadLift x = x :=
   rfl
 
+/--
+The `Functor` typeclass only contains the operations of a functor.
+`LawfulFunctor` further asserts that these operations satisfy the laws of a functor,
+including the preservation of the identity and composition laws:
+```
+id <$> x = x
+(h ∘ g) <$> x = h <$> g <$> x
+```
+-/
 class LawfulFunctor (f : Type u → Type v) [Functor f] : Prop where
   map_const          : (Functor.mapConst : α → f β → f α) = Functor.map ∘ const β
   id_map   (x : f α) : id <$> x = x
@@ -24,6 +33,16 @@ attribute [simp] id_map
 @[simp] theorem id_map' [Functor m] [LawfulFunctor m] (x : m α) : (fun a => a) <$> x = x :=
   id_map x
 
+/--
+The `Applicative` typeclass only contains the operations of an applicative functor.
+`LawfulApplicative` further asserts that these operations satisfy the laws of an applicative functor:
+```
+pure id <*> v = v
+pure (·∘·) <*> u <*> v <*> w = u <*> (v <*> w)
+pure f <*> pure x = pure (f x)
+u <*> pure y = pure (· y) <*> u
+```
+-/
 class LawfulApplicative (f : Type u → Type v) [Applicative f] extends LawfulFunctor f : Prop where
   seqLeft_eq  (x : f α) (y : f β)     : x <* y = const β <$> x <*> y
   seqRight_eq (x : f α) (y : f β)     : x *> y = const α id <$> x <*> y
@@ -42,6 +61,18 @@ attribute [simp] map_pure seq_pure
 @[simp] theorem pure_id_seq [Applicative f] [LawfulApplicative f] (x : f α) : pure id <*> x = x := by
   simp [pure_seq]
 
+/--
+The `Monad` typeclass only contains the operations of a monad.
+`LawfulMonad` further asserts that these operations satisfy the laws of a monad,
+including associativity and identity laws for `bind`:
+```
+pure x >>= f = f x
+x >>= pure = x
+x >>= f >>= g = x >>= (fun x => f x >>= g)
+```
+
+`LawfulMonad.mk'` is an alternative constructor containing useful defaults for many fields.
+-/
 class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m : Prop where
   bind_pure_comp (f : α → β) (x : m α) : x >>= (fun a => pure (f a)) = f <$> x
   bind_map       {α β : Type u} (f : m (α → β)) (x : m α) : f >>= (. <$> x) = f <*> x

src/Init/Control/Lawful/Instances.lean

@@ -235,13 +235,13 @@ end StateT
 
 instance : LawfulMonad (EStateM ε σ) := .mk'
   (id_map := fun x => funext <| fun s => by
-    dsimp only [EStateM.instMonadEStateM, EStateM.map]
+    dsimp only [EStateM.instMonad, EStateM.map]
     match x s with
     | .ok _ _ => rfl
     | .error _ _ => rfl)
   (pure_bind := fun _ _ => rfl)
   (bind_assoc := fun x _ _ => funext <| fun s => by
-    dsimp only [EStateM.instMonadEStateM, EStateM.bind]
+    dsimp only [EStateM.instMonad, EStateM.bind]
     match x s with
     | .ok _ _ => rfl
     | .error _ _ => rfl)

src/Init/Control/Option.lean

@@ -10,7 +10,7 @@ import Init.Control.Except
 
 universe u v
 
-instance : ToBool (Option α) := ⟨Option.toBool⟩
+instance : ToBool (Option α) := ⟨Option.isSome⟩
 
 def OptionT (m : Type u → Type v) (α : Type u) : Type v :=
   m (Option α)

src/Init/Core.lean

@@ -1114,9 +1114,6 @@ theorem eta (a : {x // p x}) (h : p (val a)) : mk (val a) h = a := by
   cases a
   exact rfl
 
-instance {α : Type u} {p : α → Prop} {a : α} (h : p a) : Inhabited {x // p x} where
-  default := ⟨a, h⟩
-
 instance {α : Type u} {p : α → Prop} [DecidableEq α] : DecidableEq {x : α // p x} :=
   fun ⟨a, h₁⟩ ⟨b, h₂⟩ =>
     if h : a = b then isTrue (by subst h; exact rfl)
@@ -1308,7 +1305,6 @@ gen_injective_theorems% Fin
 gen_injective_theorems% Array
 gen_injective_theorems% Sum
 gen_injective_theorems% PSum
-gen_injective_theorems% Nat
 gen_injective_theorems% Option
 gen_injective_theorems% List
 gen_injective_theorems% Except
@@ -1316,6 +1312,12 @@ gen_injective_theorems% EStateM.Result
 gen_injective_theorems% Lean.Name
 gen_injective_theorems% Lean.Syntax
 
+theorem Nat.succ.inj {m n : Nat} : m.succ = n.succ → m = n :=
+  fun x => Nat.noConfusion x id
+
+theorem Nat.succ.injEq (u v : Nat) : (u.succ = v.succ) = (u = v) :=
+  Eq.propIntro Nat.succ.inj (congrArg Nat.succ)
+
 @[simp] theorem beq_iff_eq [BEq α] [LawfulBEq α] (a b : α) : a == b ↔ a = b :=
   ⟨eq_of_beq, by intro h; subst h; exact LawfulBEq.rfl⟩
 
@@ -2035,4 +2037,8 @@ class LawfulCommIdentity (op : α → α → α) (o : outParam α) [hc : Commuta
   left_id a := Eq.trans (hc.comm o a) (right_id a)
   right_id a := Eq.trans (hc.comm a o) (left_id a)
 
+instance : Commutative Or := ⟨fun _ _ => propext or_comm⟩
+instance : Commutative And := ⟨fun _ _ => propext and_comm⟩
+instance : Commutative Iff := ⟨fun _ _ => propext iff_comm⟩
+
 end Std

src/Init/Data.lean

@@ -14,6 +14,7 @@ import Init.Data.String
 import Init.Data.List
 import Init.Data.Int
 import Init.Data.Array
+import Init.Data.Array.Subarray.Split
 import Init.Data.ByteArray
 import Init.Data.FloatArray
 import Init.Data.Fin

src/Init/Data/Array/Basic.lean

@@ -31,6 +31,7 @@ def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
   go (i : Nat) (acc : Array α) : Array α :=
     if h : i < n then go (i+1) (acc.push (f ⟨i, h⟩)) else acc
 termination_by n - i
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 /-- The array `#[0, 1, ..., n - 1]`. -/
 def range (n : Nat) : Array Nat :=
@@ -43,7 +44,7 @@ instance : EmptyCollection (Array α) := ⟨Array.empty⟩
 instance : Inhabited (Array α) where
   default := Array.empty
 
-def isEmpty (a : Array α) : Bool :=
+@[simp] def isEmpty (a : Array α) : Bool :=
   a.size = 0
 
 def singleton (v : α) : Array α :=
@@ -52,7 +53,7 @@ def singleton (v : α) : Array α :=
 /-- Low-level version of `fget` which is as fast as a C array read.
    `Fin` values are represented as tag pointers in the Lean runtime. Thus,
    `fget` may be slightly slower than `uget`. -/
-@[extern "lean_array_uget"]
+@[extern "lean_array_uget", simp]
 def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
   a[i.toNat]
 
@@ -306,6 +307,7 @@ def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
     else
       pure r
   termination_by as.size - i
+  decreasing_by simp_wf; decreasing_trivial_pre_omega
   map 0 (mkEmpty as.size)
 
 @[inline]
@@ -378,6 +380,7 @@ def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as :
       else
         pure false
       termination_by stop - j
+      decreasing_by simp_wf; decreasing_trivial_pre_omega
     loop start
   if h : stop ≤ as.size then
     any stop h
@@ -463,6 +466,7 @@ def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
       if p as[j] then some j else loop (j + 1)
     else none
     termination_by as.size - j
+    decreasing_by simp_wf; decreasing_trivial_pre_omega
   loop 0
 
 def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
@@ -557,6 +561,7 @@ def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (
   else
     true
 termination_by a.size - i
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 @[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
   if h : a.size = b.size then
@@ -661,6 +666,7 @@ def indexOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size)
     else indexOfAux a v (i+1)
   else none
 termination_by a.size - i
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
   indexOfAux a v 0
@@ -703,6 +709,7 @@ def popWhile (p : α → Bool) (as : Array α) : Array α :=
   else
     as
 termination_by as.size
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def takeWhile (p : α → Bool) (as : Array α) : Array α :=
   let rec go (i : Nat) (r : Array α) : Array α :=
@@ -715,6 +722,7 @@ def takeWhile (p : α → Bool) (as : Array α) : Array α :=
     else
       r
     termination_by as.size - i
+    decreasing_by simp_wf; decreasing_trivial_pre_omega
   go 0 #[]
 
 /-- Remove the element at a given index from an array without bounds checks, using a `Fin` index.
@@ -725,16 +733,15 @@ def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
   if h : i.val + 1 < a.size then
     let a' := a.swap ⟨i.val + 1, h⟩ i
     let i' : Fin a'.size := ⟨i.val + 1, by simp [a', h]⟩
-    have : a'.size - i' < a.size - i := by
-      simp [a', Nat.sub_succ_lt_self _ _ i.isLt]
     a'.feraseIdx i'
   else
     a.pop
 termination_by a.size - i.val
+decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ i.isLt
 
 theorem size_feraseIdx (a : Array α) (i : Fin a.size) : (a.feraseIdx i).size = a.size - 1 := by
   induction a, i using Array.feraseIdx.induct with
-  | @case1 a i h a' _ _ ih =>
+  | @case1 a i h a' _ ih =>
     unfold feraseIdx
     simp [h, a', ih]
   | case2 a i h =>
@@ -763,6 +770,7 @@ def erase [BEq α] (as : Array α) (a : α) : Array α :=
     else
       as
     termination_by j.1
+    decreasing_by simp_wf; decreasing_trivial_pre_omega
   let j := as.size
   let as := as.push a
   loop as ⟨j, size_push .. ▸ j.lt_succ_self⟩
@@ -816,6 +824,7 @@ def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) (i : N
   else
     true
 termination_by as.size - i
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 /-- Return true iff `as` is a prefix of `bs`.
 That is, `bs = as ++ t` for some `t : List α`.-/
@@ -837,6 +846,7 @@ private def allDiffAux [BEq α] (as : Array α) (i : Nat) : Bool :=
   else
     true
 termination_by as.size - i
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def allDiff [BEq α] (as : Array α) : Bool :=
   allDiffAux as 0
@@ -852,6 +862,7 @@ def allDiff [BEq α] (as : Array α) : Bool :=
   else
     cs
 termination_by as.size - i
+decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 @[inline] def zipWith (as : Array α) (bs : Array β) (f : α → β → γ) : Array γ :=
   zipWithAux f as bs 0 #[]

src/Init/Data/Array/BasicAux.lean

@@ -48,6 +48,7 @@ where
       let b ← f as[i]
       go (i+1) ⟨acc.val.push b, by simp [acc.property]⟩ hlt
   termination_by as.size - i
+  decreasing_by decreasing_trivial_pre_omega
 
 @[inline] private unsafe def mapMonoMImp [Monad m] (as : Array α) (f : α → m α) : m (Array α) :=
   go 0 as

src/Init/Data/Array/DecidableEq.lean

@@ -21,6 +21,8 @@ theorem eq_of_isEqvAux [DecidableEq α] (a b : Array α) (hsz : a.size = b.size)
     subst heq
     exact absurd (Nat.lt_of_lt_of_le high low) (Nat.lt_irrefl j)
 termination_by a.size - i
+decreasing_by decreasing_trivial_pre_omega
+
 
 theorem eq_of_isEqv [DecidableEq α] (a b : Array α) : Array.isEqv a b (fun x y => x = y) → a = b := by
   simp [Array.isEqv]
@@ -37,6 +39,7 @@ theorem isEqvAux_self [DecidableEq α] (a : Array α) (i : Nat) : Array.isEqvAux
   case inl h => simp [h, isEqvAux_self a (i+1)]
   case inr h => simp [h]
 termination_by a.size - i
+decreasing_by decreasing_trivial_pre_omega
 
 theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (fun x y => x = y) = true := by
   simp [isEqv, isEqvAux_self]

src/Init/Data/Array/Lemmas.lean

@@ -5,6 +5,7 @@ Authors: Mario Carneiro
 -/
 prelude
 import Init.Data.Nat.MinMax
+import Init.Data.Nat.Lemmas
 import Init.Data.List.Lemmas
 import Init.Data.Fin.Basic
 import Init.Data.Array.Mem
@@ -20,6 +21,13 @@ namespace Array
 
 attribute [simp] data_toArray uset
 
+@[simp] theorem singleton_def (v : α) : singleton v = #[v] := rfl
+
+@[simp] theorem toArray_data : (a : Array α) → a.data.toArray = a
+  | ⟨l⟩ => ext' (data_toArray l)
+
+@[simp] theorem data_length {l : Array α} : l.data.length = l.size := rfl
+
 @[simp] theorem mkEmpty_eq (α n) : @mkEmpty α n = #[] := rfl
 
 @[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [size]
@@ -130,6 +138,7 @@ where
       simp [aux (i+1), map_eq_pure_bind]; rfl
     · rw [List.drop_length_le (Nat.ge_of_not_lt ‹_›)]; rfl
   termination_by arr.size - i
+  decreasing_by decreasing_trivial_pre_omega
 
 @[simp] theorem map_data (f : α → β) (arr : Array α) : (arr.map f).data = arr.data.map f := by
   rw [map, mapM_eq_foldlM]
@@ -139,7 +148,8 @@ where
   simp [H]
 
 @[simp] theorem size_map (f : α → β) (arr : Array α) : (arr.map f).size = arr.size := by
-  simp [size]
+  simp only [← data_length]
+  simp
 
 @[simp] theorem pop_data (arr : Array α) : arr.pop.data = arr.data.dropLast := rfl
 
@@ -187,7 +197,8 @@ theorem anyM_stop_le_start [Monad m] (p : α → m Bool) (as : Array α) (start
 theorem mem_def (a : α) (as : Array α) : a ∈ as ↔ a ∈ as.data :=
   ⟨fun | .mk h => h, Array.Mem.mk⟩
 
-/-- # get -/
+/-! # get -/
+
 @[simp] theorem get_eq_getElem (a : Array α) (i : Fin _) : a.get i = a[i.1] := rfl
 
 theorem getElem?_lt
@@ -217,7 +228,7 @@ theorem get!_eq_getD [Inhabited α] (a : Array α) : a.get! n = a.getD n default
 @[simp] theorem get!_eq_getElem? [Inhabited α] (a : Array α) (i : Nat) : a.get! i = (a.get? i).getD default := by
   by_cases p : i < a.size <;> simp [getD_get?, get!_eq_getD, p]
 
-/-- # set -/
+/-! # set -/
 
 @[simp] theorem getElem_set_eq (a : Array α) (i : Fin a.size) (v : α) {j : Nat}
       (eq : i.val = j) (p : j < (a.set i v).size) :
@@ -240,7 +251,7 @@ theorem getElem_set (a : Array α) (i : Fin a.size) (v : α) (j : Nat)
     (ne : i.val ≠ j) : (a.set i v)[j]? = a[j]? := by
   by_cases h : j < a.size <;> simp [getElem?_lt, getElem?_ge, Nat.ge_of_not_lt, ne, h]
 
-/- # setD -/
+/-! # setD -/
 
 @[simp] theorem set!_is_setD : @set! = @setD := rfl
 
@@ -266,4 +277,788 @@ theorem getElem?_setD_eq (a : Array α) {i : Nat} (p : i < a.size) (v : α) : (a
   by_cases h : i < a.size <;>
     simp [setD, Nat.not_lt_of_le, h,  getD_get?]
 
+/-! # ofFn -/
+
+@[simp] theorem size_ofFn_go {n} (f : Fin n → α) (i acc) :
+    (ofFn.go f i acc).size = acc.size + (n - i) := by
+  if hin : i < n then
+    unfold ofFn.go
+    have : 1 + (n - (i + 1)) = n - i :=
+      Nat.sub_sub .. ▸ Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. ▸ hin))
+    rw [dif_pos hin, size_ofFn_go f (i+1), size_push, Nat.add_assoc, this]
+  else
+    have : n - i = 0 := Nat.sub_eq_zero_of_le (Nat.le_of_not_lt hin)
+    unfold ofFn.go
+    simp [hin, this]
+termination_by n - i
+
+@[simp] theorem size_ofFn (f : Fin n → α) : (ofFn f).size = n := by simp [ofFn]
+
+theorem getElem_ofFn_go (f : Fin n → α) (i) {acc k}
+    (hki : k < n) (hin : i ≤ n) (hi : i = acc.size)
+    (hacc : ∀ j, ∀ hj : j < acc.size, acc[j] = f ⟨j, Nat.lt_of_lt_of_le hj (hi ▸ hin)⟩) :
+    haveI : acc.size + (n - acc.size) = n := Nat.add_sub_cancel' (hi ▸ hin)
+    (ofFn.go f i acc)[k]'(by simp [*]) = f ⟨k, hki⟩ := by
+  unfold ofFn.go
+  if hin : i < n then
+    have : 1 + (n - (i + 1)) = n - i :=
+      Nat.sub_sub .. ▸ Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. ▸ hin))
+    simp only [dif_pos hin]
+    rw [getElem_ofFn_go f (i+1) _ hin (by simp [*]) (fun j hj => ?hacc)]
+    cases (Nat.lt_or_eq_of_le <| Nat.le_of_lt_succ (by simpa using hj)) with
+    | inl hj => simp [get_push, hj, hacc j hj]
+    | inr hj => simp [get_push, *]
+  else
+    simp [hin, hacc k (Nat.lt_of_lt_of_le hki (Nat.le_of_not_lt (hi ▸ hin)))]
+termination_by n - i
+
+@[simp] theorem getElem_ofFn (f : Fin n → α) (i : Nat) (h) :
+    (ofFn f)[i] = f ⟨i, size_ofFn f ▸ h⟩ :=
+  getElem_ofFn_go _ _ _ (by simp) (by simp) nofun
+
+/-- # mkArray -/
+
+@[simp] theorem mkArray_data (n : Nat) (v : α) : (mkArray n v).data = List.replicate n v := rfl
+
+@[simp] theorem getElem_mkArray (n : Nat) (v : α) (h : i < (mkArray n v).size) :
+    (mkArray n v)[i] = v := by simp [Array.getElem_eq_data_get]
+
+/-- # mem -/
+
+theorem mem_data {a : α} {l : Array α} : a ∈ l.data ↔ a ∈ l := (mem_def _ _).symm
+
+theorem not_mem_nil (a : α) : ¬ a ∈ #[] := nofun
+
+/-- # get lemmas -/
+
+theorem getElem?_mem {l : Array α} {i : Fin l.size} : l[i] ∈ l := by
+  erw [Array.mem_def, getElem_eq_data_get]
+  apply List.get_mem
+
+theorem getElem_fin_eq_data_get (a : Array α) (i : Fin _) : a[i] = a.data.get i := rfl
+
+@[simp] theorem ugetElem_eq_getElem (a : Array α) {i : USize} (h : i.toNat < a.size) :
+  a[i] = a[i.toNat] := rfl
+
+theorem getElem?_eq_getElem (a : Array α) (i : Nat) (h : i < a.size) : a[i]? = a[i] :=
+  getElem?_pos ..
+
+theorem get?_len_le (a : Array α) (i : Nat) (h : a.size ≤ i) : a[i]? = none := by
+  simp [getElem?_neg, h]
+
+theorem getElem_mem_data (a : Array α) (h : i < a.size) : a[i] ∈ a.data := by
+  simp only [getElem_eq_data_get, List.get_mem]
+
+theorem getElem?_eq_data_get? (a : Array α) (i : Nat) : a[i]? = a.data.get? i := by
+  by_cases i < a.size <;> simp_all [getElem?_pos, getElem?_neg, List.get?_eq_get, eq_comm]; rfl
+
+theorem get?_eq_data_get? (a : Array α) (i : Nat) : a.get? i = a.data.get? i :=
+  getElem?_eq_data_get? ..
+
+theorem get!_eq_get? [Inhabited α] (a : Array α) : a.get! n = (a.get? n).getD default := by
+  simp [get!_eq_getD]
+
+@[simp] theorem back_eq_back? [Inhabited α] (a : Array α) : a.back = a.back?.getD default := by
+  simp [back, back?]
+
+@[simp] theorem back?_push (a : Array α) : (a.push x).back? = some x := by
+  simp [back?, getElem?_eq_data_get?]
+
+theorem back_push [Inhabited α] (a : Array α) : (a.push x).back = x := by simp
+
+theorem get?_push_lt (a : Array α) (x : α) (i : Nat) (h : i < a.size) :
+    (a.push x)[i]? = some a[i] := by
+  rw [getElem?_pos, get_push_lt]
+
+theorem get?_push_eq (a : Array α) (x : α) : (a.push x)[a.size]? = some x := by
+  rw [getElem?_pos, get_push_eq]
+
+theorem get?_push {a : Array α} : (a.push x)[i]? = if i = a.size then some x else a[i]? := by
+  match Nat.lt_trichotomy i a.size with
+  | Or.inl g =>
+    have h1 : i < a.size + 1 := by omega
+    have h2 : i ≠ a.size := by omega
+    simp [getElem?, size_push, g, h1, h2, get_push_lt]
+  | Or.inr (Or.inl heq) =>
+    simp [heq, getElem?_pos, get_push_eq]
+  | Or.inr (Or.inr g) =>
+    simp only [getElem?, size_push]
+    have h1 : ¬ (i < a.size) := by omega
+    have h2 : ¬ (i < a.size + 1) := by omega
+    have h3 : i ≠ a.size := by omega
+    simp [h1, h2, h3]
+
+@[simp] theorem get?_size {a : Array α} : a[a.size]? = none := by
+  simp only [getElem?, Nat.lt_irrefl, dite_false]
+
+@[simp] theorem data_set (a : Array α) (i v) : (a.set i v).data = a.data.set i.1 v := rfl
+
+theorem get_set_eq (a : Array α) (i : Fin a.size) (v : α) :
+    (a.set i v)[i.1] = v := by
+  simp only [set, getElem_eq_data_get, List.get_set_eq]
+
+theorem get?_set_eq (a : Array α) (i : Fin a.size) (v : α) :
+    (a.set i v)[i.1]? = v := by simp [getElem?_pos, i.2]
+
+@[simp] theorem get?_set_ne (a : Array α) (i : Fin a.size) {j : Nat} (v : α)
+    (h : i.1 ≠ j) : (a.set i v)[j]? = a[j]? := by
+  by_cases j < a.size <;> simp [getElem?_pos, getElem?_neg, *]
+
+theorem get?_set (a : Array α) (i : Fin a.size) (j : Nat) (v : α) :
+    (a.set i v)[j]? = if i.1 = j then some v else a[j]? := by
+  if h : i.1 = j then subst j; simp [*] else simp [*]
+
+theorem get_set (a : Array α) (i : Fin a.size) (j : Nat) (hj : j < a.size) (v : α) :
+    (a.set i v)[j]'(by simp [*]) = if i = j then v else a[j] := by
+  if h : i.1 = j then subst j; simp [*] else simp [*]
+
+@[simp] theorem get_set_ne (a : Array α) (i : Fin a.size) {j : Nat} (v : α) (hj : j < a.size)
+    (h : i.1 ≠ j) : (a.set i v)[j]'(by simp [*]) = a[j] := by
+  simp only [set, getElem_eq_data_get, List.get_set_ne _ h]
+
+theorem getElem_setD (a : Array α) (i : Nat) (v : α) (h : i < (setD a i v).size) :
+  (setD a i v)[i] = v := by
+  simp at h
+  simp only [setD, h, dite_true, get_set, ite_true]
+
+theorem set_set (a : Array α) (i : Fin a.size) (v v' : α) :
+    (a.set i v).set ⟨i, by simp [i.2]⟩ v' = a.set i v' := by simp [set, List.set_set]
+
+private theorem fin_cast_val (e : n = n') (i : Fin n) : e ▸ i = ⟨i.1, e ▸ i.2⟩ := by cases e; rfl
+
+theorem swap_def (a : Array α) (i j : Fin a.size) :
+    a.swap i j = (a.set i (a.get j)).set ⟨j.1, by simp [j.2]⟩ (a.get i) := by
+  simp [swap, fin_cast_val]
+
+theorem data_swap (a : Array α) (i j : Fin a.size) :
+    (a.swap i j).data = (a.data.set i (a.get j)).set j (a.get i) := by simp [swap_def]
+
+theorem get?_swap (a : Array α) (i j : Fin a.size) (k : Nat) : (a.swap i j)[k]? =
+    if j = k then some a[i.1] else if i = k then some a[j.1] else a[k]? := by
+  simp [swap_def, get?_set, ← getElem_fin_eq_data_get]
+
+@[simp] theorem swapAt_def (a : Array α) (i : Fin a.size) (v : α) :
+    a.swapAt i v = (a[i.1], a.set i v) := rfl
+
+-- @[simp] -- FIXME: gives a weird linter error
+theorem swapAt!_def (a : Array α) (i : Nat) (v : α) (h : i < a.size) :
+    a.swapAt! i v = (a[i], a.set ⟨i, h⟩ v) := by simp [swapAt!, h]
+
+@[simp] theorem data_pop (a : Array α) : a.pop.data = a.data.dropLast := by simp [pop]
+
+@[simp] theorem pop_empty : (#[] : Array α).pop = #[] := rfl
+
+@[simp] theorem pop_push (a : Array α) : (a.push x).pop = a := by simp [pop]
+
+@[simp] theorem getElem_pop (a : Array α) (i : Nat) (hi : i < a.pop.size) :
+    a.pop[i] = a[i]'(Nat.lt_of_lt_of_le (a.size_pop ▸ hi) (Nat.sub_le _ _)) :=
+  List.get_dropLast ..
+
+theorem eq_empty_of_size_eq_zero {as : Array α} (h : as.size = 0) : as = #[] := by
+  apply ext
+  · simp [h]
+  · intros; contradiction
+
+theorem eq_push_pop_back_of_size_ne_zero [Inhabited α] {as : Array α} (h : as.size ≠ 0) :
+    as = as.pop.push as.back := by
+  apply ext
+  · simp [Nat.sub_add_cancel (Nat.zero_lt_of_ne_zero h)]
+  · intros i h h'
+    if hlt : i < as.pop.size then
+      rw [get_push_lt (h:=hlt), getElem_pop]
+    else
+      have heq : i = as.pop.size :=
+        Nat.le_antisymm (size_pop .. ▸ Nat.le_pred_of_lt h) (Nat.le_of_not_gt hlt)
+      cases heq; rw [get_push_eq, back, ←size_pop, get!_eq_getD, getD, dif_pos h]; rfl
+
+theorem eq_push_of_size_ne_zero {as : Array α} (h : as.size ≠ 0) :
+    ∃ (bs : Array α) (c : α), as = bs.push c :=
+  let _ : Inhabited α := ⟨as[0]⟩
+  ⟨as.pop, as.back, eq_push_pop_back_of_size_ne_zero h⟩
+
+theorem size_eq_length_data (as : Array α) : as.size = as.data.length := rfl
+
+@[simp] theorem size_swap! (a : Array α) (i j) :
+    (a.swap! i j).size = a.size := by unfold swap!; split <;> (try split) <;> simp [size_swap]
+
+@[simp] theorem size_reverse (a : Array α) : a.reverse.size = a.size := by
+  let rec go (as : Array α) (i j) : (reverse.loop as i j).size = as.size := by
+    rw [reverse.loop]
+    if h : i < j then
+      have := reverse.termination h
+      simp [(go · (i+1) ⟨j-1, ·⟩), h]
+    else simp [h]
+    termination_by j - i
+  simp only [reverse]; split <;> simp [go]
+
+@[simp] theorem size_range {n : Nat} : (range n).size = n := by
+  unfold range
+  induction n with
+  | zero      => simp [Nat.fold]
+  | succ k ih =>
+    rw [Nat.fold, flip]
+    simp only [mkEmpty_eq, size_push] at *
+    omega
+
+@[simp] theorem reverse_data (a : Array α) : a.reverse.data = a.data.reverse := by
+  let rec go (as : Array α) (i j hj)
+      (h : i + j + 1 = a.size) (h₂ : as.size = a.size)
+      (H : ∀ k, as.data.get? k = if i ≤ k ∧ k ≤ j then a.data.get? k else a.data.reverse.get? k)
+      (k) : (reverse.loop as i ⟨j, hj⟩).data.get? k = a.data.reverse.get? k := by
+    rw [reverse.loop]; dsimp; split <;> rename_i h₁
+    · have := reverse.termination h₁
+      match j with | j+1 => ?_
+      simp at *
+      rw [(go · (i+1) j)]
+      · rwa [Nat.add_right_comm i]
+      · simp [size_swap, h₂]
+      · intro k
+        rw [← getElem?_eq_data_get?, get?_swap]
+        simp [getElem?_eq_data_get?, getElem_eq_data_get, ← List.get?_eq_get, H, Nat.le_of_lt h₁]
+        split <;> rename_i h₂
+        · simp [← h₂, Nat.not_le.2 (Nat.lt_succ_self _)]
+          exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm
+        split <;> rename_i h₃
+        · simp [← h₃, Nat.not_le.2 (Nat.lt_succ_self _)]
+          exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm
+        simp only [Nat.succ_le, Nat.lt_iff_le_and_ne.trans (and_iff_left h₃),
+          Nat.lt_succ.symm.trans (Nat.lt_iff_le_and_ne.trans (and_iff_left (Ne.symm h₂)))]
+    · rw [H]; split <;> rename_i h₂
+      · cases Nat.le_antisymm (Nat.not_lt.1 h₁) (Nat.le_trans h₂.1 h₂.2)
+        cases Nat.le_antisymm h₂.1 h₂.2
+        exact (List.get?_reverse' _ _ h).symm
+      · rfl
+    termination_by j - i
+  simp only [reverse]; split
+  · match a with | ⟨[]⟩ | ⟨[_]⟩ => rfl
+  · have := Nat.sub_add_cancel (Nat.le_of_not_le ‹_›)
+    refine List.ext <| go _ _ _ _ (by simp [this]) rfl fun k => ?_
+    split; {rfl}; rename_i h
+    simp [← show k < _ + 1 ↔ _ from Nat.lt_succ (n := a.size - 1), this] at h
+    rw [List.get?_eq_none.2 ‹_›, List.get?_eq_none.2 (a.data.length_reverse ▸ ‹_›)]
+
+/-! ### foldl / foldr -/
+
+-- This proof is the pure version of `Array.SatisfiesM_foldlM`,
+-- reproduced to avoid a dependency on `SatisfiesM`.
+theorem foldl_induction
+    {as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → β}
+    (hf : ∀ i : Fin as.size, ∀ b, motive i.1 b → motive (i.1 + 1) (f b as[i])) :
+    motive as.size (as.foldl f init) := by
+  let rec go {i j b} (h₁ : j ≤ as.size) (h₂ : as.size ≤ i + j) (H : motive j b) :
+    (motive as.size) (foldlM.loop (m := Id) f as as.size (Nat.le_refl _) i j b) := by
+    unfold foldlM.loop; split
+    · next hj =>
+      split
+      · cases Nat.not_le_of_gt (by simp [hj]) h₂
+      · exact go hj (by rwa [Nat.succ_add] at h₂) (hf ⟨j, hj⟩ b H)
+    · next hj => exact Nat.le_antisymm h₁ (Nat.ge_of_not_lt hj) ▸ H
+  simpa [foldl, foldlM] using go (Nat.zero_le _) (Nat.le_refl _) h0
+
+-- This proof is the pure version of `Array.SatisfiesM_foldrM`,
+-- reproduced to avoid a dependency on `SatisfiesM`.
+theorem foldr_induction
+    {as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive as.size init) {f : α → β → β}
+    (hf : ∀ i : Fin as.size, ∀ b, motive (i.1 + 1) b → motive i.1 (f as[i] b)) :
+    motive 0 (as.foldr f init) := by
+  let rec go {i b} (hi : i ≤ as.size) (H : motive i b) :
+    (motive 0) (foldrM.fold (m := Id) f as 0 i hi b) := by
+    unfold foldrM.fold; simp; split
+    · next hi => exact (hi ▸ H)
+    · next hi =>
+      split; {simp at hi}
+      · next i hi' =>
+        exact go _ (hf ⟨i, hi'⟩ b H)
+  simp [foldr, foldrM]; split; {exact go _ h0}
+  · next h => exact (Nat.eq_zero_of_not_pos h ▸ h0)
+
+/-! ### map -/
+
+@[simp] theorem mem_map {f : α → β} {l : Array α} : b ∈ l.map f ↔ ∃ a, a ∈ l ∧ f a = b := by
+  simp only [mem_def, map_data, List.mem_map]
+
+theorem mapM_eq_mapM_data [Monad m] [LawfulMonad m] (f : α → m β) (arr : Array α) :
+    arr.mapM f = return mk (← arr.data.mapM f) := by
+  rw [mapM_eq_foldlM, foldlM_eq_foldlM_data, ← List.foldrM_reverse]
+  conv => rhs; rw [← List.reverse_reverse arr.data]
+  induction arr.data.reverse with
+  | nil => simp; rfl
+  | cons a l ih => simp [ih]; simp [map_eq_pure_bind, push]
+
+theorem mapM_map_eq_foldl (as : Array α) (f : α → β) (i) :
+    mapM.map (m := Id) f as i b = as.foldl (start := i) (fun r a => r.push (f a)) b := by
+  unfold mapM.map
+  split <;> rename_i h
+  · simp only [Id.bind_eq]
+    dsimp [foldl, Id.run, foldlM]
+    rw [mapM_map_eq_foldl, dif_pos (by omega), foldlM.loop, dif_pos h]
+    -- Calling `split` here gives a bad goal.
+    have : size as - i = Nat.succ (size as - i - 1) := by omega
+    rw [this]
+    simp [foldl, foldlM, Id.run, Nat.sub_add_eq]
+  · dsimp [foldl, Id.run, foldlM]
+    rw [dif_pos (by omega), foldlM.loop, dif_neg h]
+    rfl
+termination_by as.size - i
+
+theorem map_eq_foldl (as : Array α) (f : α → β) :
+    as.map f = as.foldl (fun r a => r.push (f a)) #[] :=
+  mapM_map_eq_foldl _ _ _
+
+theorem map_induction (as : Array α) (f : α → β) (motive : Nat → Prop) (h0 : motive 0)
+    (p : Fin as.size → β → Prop) (hs : ∀ i, motive i.1 → p i (f as[i]) ∧ motive (i+1)) :
+    motive as.size ∧
+      ∃ eq : (as.map f).size = as.size, ∀ i h, p ⟨i, h⟩ ((as.map f)[i]) := by
+  have t := foldl_induction (as := as) (β := Array β)
+    (motive := fun i arr => motive i ∧ arr.size = i ∧ ∀ i h2, p i arr[i.1])
+    (init := #[]) (f := fun r a => r.push (f a)) ?_ ?_
+  obtain ⟨m, eq, w⟩ := t
+  · refine ⟨m, by simpa [map_eq_foldl] using eq, ?_⟩
+    intro i h
+    simp [eq] at w
+    specialize w ⟨i, h⟩ h
+    simpa [map_eq_foldl] using w
+  · exact ⟨h0, rfl, nofun⟩
+  · intro i b ⟨m, ⟨eq, w⟩⟩
+    refine ⟨?_, ?_, ?_⟩
+    · exact (hs _ m).2
+    · simp_all
+    · intro j h
+      simp at h ⊢
+      by_cases h' : j < size b
+      · rw [get_push]
+        simp_all
+      · rw [get_push, dif_neg h']
+        simp only [show j = i by omega]
+        exact (hs _ m).1
+
+theorem map_spec (as : Array α) (f : α → β) (p : Fin as.size → β → Prop)
+    (hs : ∀ i, p i (f as[i])) :
+    ∃ eq : (as.map f).size = as.size, ∀ i h, p ⟨i, h⟩ ((as.map f)[i]) := by
+  simpa using map_induction as f (fun _ => True) trivial p (by simp_all)
+
+@[simp] theorem getElem_map (f : α → β) (as : Array α) (i : Nat) (h) :
+    ((as.map f)[i]) = f (as[i]'(size_map .. ▸ h)) := by
+  have := map_spec as f (fun i b => b = f (as[i]))
+  simp only [implies_true, true_implies] at this
+  obtain ⟨eq, w⟩ := this
+  apply w
+  simp_all
+
+/-! ### mapIdx -/
+
+-- This could also be prove from `SatisfiesM_mapIdxM`.
+theorem mapIdx_induction (as : Array α) (f : Fin as.size → α → β)
+    (motive : Nat → Prop) (h0 : motive 0)
+    (p : Fin as.size → β → Prop)
+    (hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
+    motive as.size ∧ ∃ eq : (Array.mapIdx as f).size = as.size,
+      ∀ i h, p ⟨i, h⟩ ((Array.mapIdx as f)[i]) := by
+  let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p ⟨i, h⟩ bs[i]) (hm : motive j) :
+    let arr : Array β := Array.mapIdxM.map (m := Id) as f i j h bs
+    motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i, h⟩ arr[i] := by
+    induction i generalizing j bs with simp [mapIdxM.map]
+    | zero =>
+      have := (Nat.zero_add _).symm.trans h
+      exact ⟨this ▸ hm, h₁ ▸ this, fun _ _ => h₂ ..⟩
+    | succ i ih =>
+      apply @ih (bs.push (f ⟨j, by omega⟩ as[j])) (j + 1) (by omega) (by simp; omega)
+      · intro i i_lt h'
+        rw [get_push]
+        split
+        · apply h₂
+        · simp only [size_push] at h'
+          obtain rfl : i = j := by omega
+          apply (hs ⟨i, by omega⟩ hm).1
+      · exact (hs ⟨j, by omega⟩ hm).2
+  simp [mapIdx, mapIdxM]; exact go rfl nofun h0
+
+theorem mapIdx_spec (as : Array α) (f : Fin as.size → α → β)
+    (p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
+    ∃ eq : (Array.mapIdx as f).size = as.size,
+      ∀ i h, p ⟨i, h⟩ ((Array.mapIdx as f)[i]) :=
+  (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
+
+@[simp] theorem size_mapIdx (a : Array α) (f : Fin a.size → α → β) : (a.mapIdx f).size = a.size :=
+  (mapIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
+
+@[simp] theorem size_zipWithIndex (as : Array α) : as.zipWithIndex.size = as.size :=
+  Array.size_mapIdx _ _
+
+@[simp] theorem getElem_mapIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat)
+    (h : i < (mapIdx a f).size) :
+    haveI : i < a.size := by simp_all
+    (a.mapIdx f)[i] = f ⟨i, this⟩ a[i] :=
+  (mapIdx_spec _ _ (fun i b => b = f i a[i]) fun _ => rfl).2 i _
+
+/-! ### modify -/
+
+@[simp] theorem size_modify (a : Array α) (i : Nat) (f : α → α) : (a.modify i f).size = a.size := by
+  unfold modify modifyM Id.run
+  split <;> simp
+
+theorem get_modify {arr : Array α} {x i} (h : i < arr.size) :
+    (arr.modify x f).get ⟨i, by simp [h]⟩ =
+    if x = i then f (arr.get ⟨i, h⟩) else arr.get ⟨i, h⟩ := by
+  simp [modify, modifyM, Id.run]; split
+  · simp [get_set _ _ _ h]; split <;> simp [*]
+  · rw [if_neg (mt (by rintro rfl; exact h) ‹_›)]
+
+/-! ### filter -/
+
+@[simp] theorem filter_data (p : α → Bool) (l : Array α) :
+    (l.filter p).data = l.data.filter p := by
+  dsimp only [filter]
+  rw [foldl_eq_foldl_data]
+  generalize l.data = l
+  suffices ∀ a, (List.foldl (fun r a => if p a = true then push r a else r) a l).data =
+      a.data ++ List.filter p l by
+    simpa using this #[]
+  induction l with simp
+  | cons => split <;> simp [*]
+
+@[simp] theorem filter_filter (q) (l : Array α) :
+    filter p (filter q l) = filter (fun a => p a ∧ q a) l := by
+  apply ext'
+  simp only [filter_data, List.filter_filter]
+
+@[simp] theorem mem_filter : x ∈ filter p as ↔ x ∈ as ∧ p x := by
+  simp only [mem_def, filter_data, List.mem_filter]
+
+theorem mem_of_mem_filter {a : α} {l} (h : a ∈ filter p l) : a ∈ l :=
+  (mem_filter.mp h).1
+
+/-! ### filterMap -/
+
+@[simp] theorem filterMap_data (f : α → Option β) (l : Array α) :
+    (l.filterMap f).data = l.data.filterMap f := by
+  dsimp only [filterMap, filterMapM]
+  rw [foldlM_eq_foldlM_data]
+  generalize l.data = l
+  have this : ∀ a : Array β, (Id.run (List.foldlM (m := Id) ?_ a l)).data =
+    a.data ++ List.filterMap f l := ?_
+  exact this #[]
+  induction l
+  · simp_all [Id.run]
+  · simp_all [Id.run]
+    split <;> simp_all
+
+@[simp] theorem mem_filterMap (f : α → Option β) (l : Array α) {b : β} :
+    b ∈ filterMap f l ↔ ∃ a, a ∈ l ∧ f a = some b := by
+  simp only [mem_def, filterMap_data, List.mem_filterMap]
+
+/-! ### empty -/
+
+theorem size_empty : (#[] : Array α).size = 0 := rfl
+
+theorem empty_data : (#[] : Array α).data = [] := rfl
+
+/-! ### append -/
+
+theorem push_eq_append_singleton (as : Array α) (x) : as.push x = as ++ #[x] := rfl
+
+@[simp] theorem mem_append {a : α} {s t : Array α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
+  simp only [mem_def, append_data, List.mem_append]
+
+theorem size_append (as bs : Array α) : (as ++ bs).size = as.size + bs.size := by
+  simp only [size, append_data, List.length_append]
+
+theorem get_append_left {as bs : Array α} {h : i < (as ++ bs).size} (hlt : i < as.size) :
+    (as ++ bs)[i] = as[i] := by
+  simp only [getElem_eq_data_get]
+  have h' : i < (as.data ++ bs.data).length := by rwa [← data_length, append_data] at h
+  conv => rhs; rw [← List.get_append_left (bs:=bs.data) (h':=h')]
+  apply List.get_of_eq; rw [append_data]
+
+theorem get_append_right {as bs : Array α} {h : i < (as ++ bs).size} (hle : as.size ≤ i)
+    (hlt : i - as.size < bs.size := Nat.sub_lt_left_of_lt_add hle (size_append .. ▸ h)) :
+    (as ++ bs)[i] = bs[i - as.size] := by
+  simp only [getElem_eq_data_get]
+  have h' : i < (as.data ++ bs.data).length := by rwa [← data_length, append_data] at h
+  conv => rhs; rw [← List.get_append_right (h':=h') (h:=Nat.not_lt_of_ge hle)]
+  apply List.get_of_eq; rw [append_data]
+
+@[simp] theorem append_nil (as : Array α) : as ++ #[] = as := by
+  apply ext'; simp only [append_data, empty_data, List.append_nil]
+
+@[simp] theorem nil_append (as : Array α) : #[] ++ as = as := by
+  apply ext'; simp only [append_data, empty_data, List.nil_append]
+
+theorem append_assoc (as bs cs : Array α) : as ++ bs ++ cs = as ++ (bs ++ cs) := by
+  apply ext'; simp only [append_data, List.append_assoc]
+
+/-! ### extract -/
+
+theorem extract_loop_zero (as bs : Array α) (start : Nat) : extract.loop as 0 start bs = bs := by
+  rw [extract.loop]; split <;> rfl
+
+theorem extract_loop_succ (as bs : Array α) (size start : Nat) (h : start < as.size) :
+    extract.loop as (size+1) start bs = extract.loop as size (start+1) (bs.push as[start]) := by
+  rw [extract.loop, dif_pos h]; rfl
+
+theorem extract_loop_of_ge (as bs : Array α) (size start : Nat) (h : start ≥ as.size) :
+    extract.loop as size start bs = bs := by
+  rw [extract.loop, dif_neg (Nat.not_lt_of_ge h)]
+
+theorem extract_loop_eq_aux (as bs : Array α) (size start : Nat) :
+    extract.loop as size start bs = bs ++ extract.loop as size start #[] := by
+  induction size using Nat.recAux generalizing start bs with
+  | zero => rw [extract_loop_zero, extract_loop_zero, append_nil]
+  | succ size ih =>
+    if h : start < as.size then
+      rw [extract_loop_succ (h:=h), ih (bs.push _), push_eq_append_singleton]
+      rw [extract_loop_succ (h:=h), ih (#[].push _), push_eq_append_singleton, nil_append]
+      rw [append_assoc]
+    else
+      rw [extract_loop_of_ge (h:=Nat.le_of_not_lt h)]
+      rw [extract_loop_of_ge (h:=Nat.le_of_not_lt h)]
+      rw [append_nil]
+
+theorem extract_loop_eq (as bs : Array α) (size start : Nat) (h : start + size ≤ as.size) :
+  extract.loop as size start bs = bs ++ as.extract start (start + size) := by
+  simp [extract]; rw [extract_loop_eq_aux, Nat.min_eq_left h, Nat.add_sub_cancel_left]
+
+theorem size_extract_loop (as bs : Array α) (size start : Nat) :
+    (extract.loop as size start bs).size = bs.size + min size (as.size - start) := by
+  induction size using Nat.recAux generalizing start bs with
+  | zero => rw [extract_loop_zero, Nat.zero_min, Nat.add_zero]
+  | succ size ih =>
+    if h : start < as.size then
+      rw [extract_loop_succ (h:=h), ih, size_push, Nat.add_assoc, ←Nat.add_min_add_left,
+        Nat.sub_succ, Nat.one_add, Nat.one_add, Nat.succ_pred_eq_of_pos (Nat.sub_pos_of_lt h)]
+    else
+      have h := Nat.le_of_not_gt h
+      rw [extract_loop_of_ge (h:=h), Nat.sub_eq_zero_of_le h, Nat.min_zero, Nat.add_zero]
+
+@[simp] theorem size_extract (as : Array α) (start stop : Nat) :
+    (as.extract start stop).size = min stop as.size - start := by
+  simp [extract]; rw [size_extract_loop, size_empty, Nat.zero_add, Nat.sub_min_sub_right,
+    Nat.min_assoc, Nat.min_self]
+
+theorem get_extract_loop_lt_aux (as bs : Array α) (size start : Nat) (hlt : i < bs.size) :
+    i < (extract.loop as size start bs).size := by
+  rw [size_extract_loop]
+  apply Nat.lt_of_lt_of_le hlt
+  exact Nat.le_add_right ..
+
+theorem get_extract_loop_lt (as bs : Array α) (size start : Nat) (hlt : i < bs.size)
+    (h := get_extract_loop_lt_aux as bs size start hlt) :
+    (extract.loop as size start bs)[i] = bs[i] := by
+  apply Eq.trans _ (get_append_left (bs:=extract.loop as size start #[]) hlt)
+  · rw [size_append]; exact Nat.lt_of_lt_of_le hlt (Nat.le_add_right ..)
+  · congr; rw [extract_loop_eq_aux]
+
+theorem get_extract_loop_ge_aux (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size)
+    (h : i < (extract.loop as size start bs).size) : start + i - bs.size < as.size := by
+  have h : i < bs.size + (as.size - start) := by
+      apply Nat.lt_of_lt_of_le h
+      rw [size_extract_loop]
+      apply Nat.add_le_add_left
+      exact Nat.min_le_right ..
+  rw [Nat.add_sub_assoc hge]
+  apply Nat.add_lt_of_lt_sub'
+  exact Nat.sub_lt_left_of_lt_add hge h
+
+theorem get_extract_loop_ge (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size)
+    (h : i < (extract.loop as size start bs).size)
+    (h' := get_extract_loop_ge_aux as bs size start hge h) :
+    (extract.loop as size start bs)[i] = as[start + i - bs.size] := by
+  induction size using Nat.recAux generalizing start bs with
+  | zero =>
+    rw [size_extract_loop, Nat.zero_min, Nat.add_zero] at h
+    omega
+  | succ size ih =>
+    have : start < as.size := by
+      apply Nat.lt_of_le_of_lt (Nat.le_add_right start (i - bs.size))
+      rwa [← Nat.add_sub_assoc hge]
+    have : i < (extract.loop as size (start+1) (bs.push as[start])).size := by
+      rwa [← extract_loop_succ]
+    have heq : (extract.loop as (size+1) start bs)[i] =
+        (extract.loop as size (start+1) (bs.push as[start]))[i] := by
+      congr 1; rw [extract_loop_succ]
+    rw [heq]
+    if hi : bs.size = i then
+      cases hi
+      have h₁ : bs.size < (bs.push as[start]).size := by rw [size_push]; exact Nat.lt_succ_self ..
+      have h₂ : bs.size < (extract.loop as size (start+1) (bs.push as[start])).size := by
+        rw [size_extract_loop]; apply Nat.lt_of_lt_of_le h₁; exact Nat.le_add_right ..
+      have h : (extract.loop as size (start + 1) (push bs as[start]))[bs.size] = as[start] := by
+        rw [get_extract_loop_lt as (bs.push as[start]) size (start+1) h₁ h₂, get_push_eq]
+      rw [h]; congr; rw [Nat.add_sub_cancel]
+    else
+      have hge : bs.size + 1 ≤ i := Nat.lt_of_le_of_ne hge hi
+      rw [ih (bs.push as[start]) (start+1) ((size_push ..).symm ▸ hge)]
+      congr 1; rw [size_push, Nat.add_right_comm, Nat.add_sub_add_right]
+
+theorem get_extract_aux {as : Array α} {start stop : Nat} (h : i < (as.extract start stop).size) :
+    start + i < as.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
+
+@[simp] theorem get_extract {as : Array α} {start stop : Nat}
+    (h : i < (as.extract start stop).size) :
+    (as.extract start stop)[i] = as[start + i]'(get_extract_aux h) :=
+  show (extract.loop as (min stop as.size - start) start #[])[i]
+    = as[start + i]'(get_extract_aux h) by rw [get_extract_loop_ge]; rfl; exact Nat.zero_le _
+
+@[simp] theorem extract_all (as : Array α) : as.extract 0 as.size = as := by
+  apply ext
+  · rw [size_extract, Nat.min_self, Nat.sub_zero]
+  · intros; rw [get_extract]; congr; rw [Nat.zero_add]
+
+theorem extract_empty_of_stop_le_start (as : Array α) {start stop : Nat} (h : stop ≤ start) :
+    as.extract start stop = #[] := by
+  simp [extract]; rw [←Nat.sub_min_sub_right, Nat.sub_eq_zero_of_le h, Nat.zero_min,
+    extract_loop_zero]
+
+theorem extract_empty_of_size_le_start (as : Array α) {start stop : Nat} (h : as.size ≤ start) :
+    as.extract start stop = #[] := by
+  simp [extract]; rw [←Nat.sub_min_sub_right, Nat.sub_eq_zero_of_le h, Nat.min_zero,
+    extract_loop_zero]
+
+@[simp] theorem extract_empty (start stop : Nat) : (#[] : Array α).extract start stop = #[] :=
+  extract_empty_of_size_le_start _ (Nat.zero_le _)
+
+/-! ### any -/
+
+-- Auxiliary for `any_iff_exists`.
+theorem anyM_loop_iff_exists (p : α → Bool) (as : Array α) (start stop) (h : stop ≤ as.size) :
+    anyM.loop (m := Id) p as stop h start = true ↔
+      ∃ i : Fin as.size, start ≤ ↑i ∧ ↑i < stop ∧ p as[i] = true := by
+  unfold anyM.loop
+  split <;> rename_i h₁
+  · dsimp
+    split <;> rename_i h₂
+    · simp only [true_iff]
+      refine ⟨⟨start, by omega⟩, by dsimp; omega, by dsimp; omega, h₂⟩
+    · rw [anyM_loop_iff_exists]
+      constructor
+      · rintro ⟨i, ge, lt, h⟩
+        have : start ≠ i := by rintro rfl; omega
+        exact ⟨i, by omega, lt, h⟩
+      · rintro ⟨i, ge, lt, h⟩
+        have : start ≠ i := by rintro rfl; erw [h] at h₂; simp_all
+        exact ⟨i, by omega, lt, h⟩
+  · simp
+    omega
+termination_by stop - start
+
+-- This could also be proved from `SatisfiesM_anyM_iff_exists` in `Batteries.Data.Array.Init.Monadic`
+theorem any_iff_exists (p : α → Bool) (as : Array α) (start stop) :
+    any as p start stop ↔ ∃ i : Fin as.size, start ≤ i.1 ∧ i.1 < stop ∧ p as[i] := by
+  dsimp [any, anyM, Id.run]
+  split
+  · rw [anyM_loop_iff_exists]; rfl
+  · rw [anyM_loop_iff_exists]
+    constructor
+    · rintro ⟨i, ge, _, h⟩
+      exact ⟨i, by omega, by omega, h⟩
+    · rintro ⟨i, ge, _, h⟩
+      exact ⟨i, by omega, by omega, h⟩
+
+theorem any_eq_true (p : α → Bool) (as : Array α) :
+    any as p ↔ ∃ i : Fin as.size, p as[i] := by simp [any_iff_exists, Fin.isLt]
+
+theorem any_def {p : α → Bool} (as : Array α) : as.any p = as.data.any p := by
+  rw [Bool.eq_iff_iff, any_eq_true, List.any_eq_true]; simp only [List.mem_iff_get]
+  exact ⟨fun ⟨i, h⟩ => ⟨_, ⟨i, rfl⟩, h⟩, fun ⟨_, ⟨i, rfl⟩, h⟩ => ⟨i, h⟩⟩
+
+/-! ### all -/
+
+theorem all_eq_not_any_not (p : α → Bool) (as : Array α) (start stop) :
+    all as p start stop = !(any as (!p ·) start stop) := by
+  dsimp [all, allM]
+  rfl
+
+theorem all_iff_forall (p : α → Bool) (as : Array α) (start stop) :
+    all as p start stop ↔ ∀ i : Fin as.size, start ≤ i.1 ∧ i.1 < stop → p as[i] := by
+  rw [all_eq_not_any_not]
+  suffices ¬(any as (!p ·) start stop = true) ↔
+      ∀ i : Fin as.size, start ≤ i.1 ∧ i.1 < stop → p as[i] by
+    simp_all
+  rw [any_iff_exists]
+  simp
+
+theorem all_eq_true (p : α → Bool) (as : Array α) : all as p ↔ ∀ i : Fin as.size, p as[i] := by
+  simp [all_iff_forall, Fin.isLt]
+
+theorem all_def {p : α → Bool} (as : Array α) : as.all p = as.data.all p := by
+  rw [Bool.eq_iff_iff, all_eq_true, List.all_eq_true]; simp only [List.mem_iff_get]
+  constructor
+  · rintro w x ⟨r, rfl⟩
+    rw [← getElem_eq_data_get]
+    apply w
+  · intro w i
+    exact w as[i] ⟨i, (getElem_eq_data_get as i.2).symm⟩
+
+theorem all_eq_true_iff_forall_mem {l : Array α} : l.all p ↔ ∀ x, x ∈ l → p x := by
+  simp only [all_def, List.all_eq_true, mem_def]
+
+/-! ### contains -/
+
+theorem contains_def [DecidableEq α] {a : α} {as : Array α} : as.contains a ↔ a ∈ as := by
+  rw [mem_def, contains, any_def, List.any_eq_true]; simp [and_comm]
+
+instance [DecidableEq α] (a : α) (as : Array α) : Decidable (a ∈ as) :=
+  decidable_of_iff _ contains_def
+
+/-! ### swap -/
+
+open Fin
+
+@[simp] theorem get_swap_right (a : Array α) {i j : Fin a.size} : (a.swap i j)[j.val] = a[i] :=
+  by simp only [swap, fin_cast_val, get_eq_getElem, getElem_set_eq, getElem_fin]
+
+@[simp] theorem get_swap_left (a : Array α) {i j : Fin a.size} : (a.swap i j)[i.val] = a[j] :=
+  if he : ((Array.size_set _ _ _).symm ▸ j).val = i.val then by
+    simp only [←he, fin_cast_val, get_swap_right, getElem_fin]
+  else by
+    apply Eq.trans
+    · apply Array.get_set_ne
+      · simp only [size_set, Fin.isLt]
+      · assumption
+    · simp [get_set_ne]
+
+@[simp] theorem get_swap_of_ne (a : Array α) {i j : Fin a.size} (hp : p < a.size)
+    (hi : p ≠ i) (hj : p ≠ j) : (a.swap i j)[p]'(a.size_swap .. |>.symm ▸ hp) = a[p] := by
+  apply Eq.trans
+  · have : ((a.size_set i (a.get j)).symm ▸ j).val = j.val := by simp only [fin_cast_val]
+    apply Array.get_set_ne
+    · simp only [this]
+      apply Ne.symm
+      · assumption
+  · apply Array.get_set_ne
+    · apply Ne.symm
+      · assumption
+
+theorem get_swap (a : Array α) (i j : Fin a.size) (k : Nat) (hk: k < a.size) :
+    (a.swap i j)[k]'(by simp_all) = if k = i then a[j] else if k = j then a[i]  else a[k] := by
+  split
+  · simp_all only [get_swap_left]
+  · split <;> simp_all
+
+theorem get_swap' (a : Array α) (i j : Fin a.size) (k : Nat) (hk' : k < (a.swap i j).size) :
+    (a.swap i j)[k] = if k = i then a[j] else if k = j then a[i] else a[k]'(by simp_all) := by
+  apply get_swap
+
+@[simp] theorem swap_swap (a : Array α) {i j : Fin a.size} :
+    (a.swap i j).swap ⟨i.1, (a.size_swap ..).symm ▸i.2⟩ ⟨j.1, (a.size_swap ..).symm ▸j.2⟩ = a := by
+  apply ext
+  · simp only [size_swap]
+  · intros
+    simp only [get_swap']
+    split
+    · simp_all
+    · split <;> simp_all
+
+theorem swap_comm (a : Array α) {i j : Fin a.size} : a.swap i j = a.swap j i := by
+  apply ext
+  · simp only [size_swap]
+  · intros
+    simp only [get_swap']
+    split
+    · split <;> simp_all
+    · split <;> simp_all
+
+
 end Array

src/Init/Data/Array/Mem.lean

@@ -27,13 +27,20 @@ theorem sizeOf_lt_of_mem [SizeOf α] {as : Array α} (h : a ∈ as) : sizeOf a <
   cases as with | _ as =>
   exact Nat.lt_trans (List.sizeOf_get ..) (by simp_arith)
 
+@[simp] theorem sizeOf_getElem [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) :
+  sizeOf (as[i]'h) < sizeOf as := sizeOf_get _ _
+
 /-- This tactic, added to the `decreasing_trivial` toolbox, proves that
 `sizeOf arr[i] < sizeOf arr`, which is useful for well founded recursions
 over a nested inductive like `inductive T | mk : Array T → T`. -/
 macro "array_get_dec" : tactic =>
   `(tactic| first
-    | apply sizeOf_get
-    | apply Nat.lt_trans (sizeOf_get ..); simp_arith)
+    -- subsumed by simp
+    -- | with_reducible apply sizeOf_get
+    -- | with_reducible apply sizeOf_getElem
+    | (with_reducible apply Nat.lt_trans (sizeOf_get ..)); simp_arith
+    | (with_reducible apply Nat.lt_trans (sizeOf_getElem ..)); simp_arith
+    )
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_get_dec)
 
@@ -43,9 +50,10 @@ provided that `a ∈ arr` which is useful for well founded recursions over a nes
 -- NB: This is analogue to tactic `sizeOf_list_dec`
 macro "array_mem_dec" : tactic =>
   `(tactic| first
-    | apply Array.sizeOf_lt_of_mem; assumption; done
-    | apply Nat.lt_trans (Array.sizeOf_lt_of_mem ?h)
-      case' h => assumption
+    | with_reducible apply Array.sizeOf_lt_of_mem; assumption; done
+    | with_reducible
+        apply Nat.lt_trans (Array.sizeOf_lt_of_mem ?h)
+        case' h => assumption
       simp_arith)
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_mem_dec)

src/Init/Data/Array/QSort.lean

@@ -27,6 +27,7 @@ def qpartition (as : Array α) (lt : α → α → Bool) (lo hi : Nat) : Nat ×
       let as := as.swap! i hi
       (i, as)
     termination_by hi - j
+    decreasing_by all_goals simp_wf; decreasing_trivial_pre_omega
   loop as lo lo
 
 @[inline] partial def qsort (as : Array α) (lt : α → α → Bool) (low := 0) (high := as.size - 1) : Array α :=

src/Init/Data/Array/Subarray.lean

@@ -9,25 +9,40 @@ import Init.Data.Array.Basic
 universe u v w
 
 structure Subarray (α : Type u)  where
-  as : Array α
+  array : Array α
   start : Nat
   stop : Nat
-  h₁ : start ≤ stop
-  h₂ : stop ≤ as.size
+  start_le_stop : start ≤ stop
+  stop_le_array_size : stop ≤ array.size
+
+@[deprecated Subarray.array (since := "2024-04-13")]
+abbrev Subarray.as (s : Subarray α) : Array α := s.array
+
+@[deprecated Subarray.start_le_stop (since := "2024-04-13")]
+theorem Subarray.h₁ (s : Subarray α) : s.start ≤ s.stop := s.start_le_stop
+
+@[deprecated Subarray.stop_le_array_size (since := "2024-04-13")]
+theorem Subarray.h₂ (s : Subarray α) : s.stop ≤ s.array.size := s.stop_le_array_size
 
 namespace Subarray
 
 def size (s : Subarray α) : Nat :=
   s.stop - s.start
 
+theorem size_le_array_size {s : Subarray α} : s.size ≤ s.array.size := by
+  let {array, start, stop, start_le_stop, stop_le_array_size} := s
+  simp [size]
+  apply Nat.le_trans (Nat.sub_le stop start)
+  assumption
+
 def get (s : Subarray α) (i : Fin s.size) : α :=
-  have : s.start + i.val < s.as.size := by
-   apply Nat.lt_of_lt_of_le _ s.h₂
+  have : s.start + i.val < s.array.size := by
+   apply Nat.lt_of_lt_of_le _ s.stop_le_array_size
    have := i.isLt
    simp [size] at this
    rw [Nat.add_comm]
    exact Nat.add_lt_of_lt_sub this
-  s.as[s.start + i.val]
+  s.array[s.start + i.val]
 
 instance : GetElem (Subarray α) Nat α fun xs i => i < xs.size where
   getElem xs i h := xs.get ⟨i, h⟩
@@ -42,7 +57,7 @@ abbrev get! [Inhabited α] (s : Subarray α) (i : Nat) : α :=
 
 def popFront (s : Subarray α) : Subarray α :=
   if h : s.start < s.stop then
-    { s with start := s.start + 1, h₁ := Nat.le_of_lt_succ (Nat.add_lt_add_right h 1) }
+    { s with start := s.start + 1, start_le_stop := Nat.le_of_lt_succ (Nat.add_lt_add_right h 1) }
   else
     s
 
@@ -50,7 +65,7 @@ def popFront (s : Subarray α) : Subarray α :=
   let sz := USize.ofNat s.stop
   let rec @[specialize] loop (i : USize) (b : β) : m β := do
     if i < sz then
-      let a := s.as.uget i lcProof
+      let a := s.array.uget i lcProof
       match (← f a b) with
       | ForInStep.done  b => pure b
       | ForInStep.yield b => loop (i+1) b
@@ -68,27 +83,27 @@ instance : ForIn m (Subarray α) α where
 
 @[inline]
 def foldlM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Subarray α) : m β :=
-  as.as.foldlM f (init := init) (start := as.start) (stop := as.stop)
+  as.array.foldlM f (init := init) (start := as.start) (stop := as.stop)
 
 @[inline]
 def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Subarray α) : m β :=
-  as.as.foldrM f (init := init) (start := as.stop) (stop := as.start)
+  as.array.foldrM f (init := init) (start := as.stop) (stop := as.start)
 
 @[inline]
 def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Subarray α) : m Bool :=
-  as.as.anyM p (start := as.start) (stop := as.stop)
+  as.array.anyM p (start := as.start) (stop := as.stop)
 
 @[inline]
 def allM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Subarray α) : m Bool :=
-  as.as.allM p (start := as.start) (stop := as.stop)
+  as.array.allM p (start := as.start) (stop := as.stop)
 
 @[inline]
 def forM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Subarray α) : m PUnit :=
-  as.as.forM f (start := as.start) (stop := as.stop)
+  as.array.forM f (start := as.start) (stop := as.stop)
 
 @[inline]
 def forRevM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Subarray α) : m PUnit :=
-  as.as.forRevM f (start := as.stop) (stop := as.start)
+  as.array.forRevM f (start := as.stop) (stop := as.start)
 
 @[inline]
 def foldl {α : Type u} {β : Type v} (f : β → α → β) (init : β) (as : Subarray α) : β :=
@@ -135,15 +150,25 @@ variable {α : Type u}
 
 def toSubarray (as : Array α) (start : Nat := 0) (stop : Nat := as.size) : Subarray α :=
   if h₂ : stop ≤ as.size then
-     if h₁ : start ≤ stop then
-       { as := as, start := start, stop := stop, h₁ := h₁, h₂ := h₂ }
-     else
-       { as := as, start := stop, stop := stop, h₁ := Nat.le_refl _, h₂ := h₂ }
+    if h₁ : start ≤ stop then
+      { array := as, start := start, stop := stop,
+        start_le_stop := h₁, stop_le_array_size := h₂ }
+    else
+      { array := as, start := stop, stop := stop,
+        start_le_stop := Nat.le_refl _, stop_le_array_size := h₂ }
   else
-     if h₁ : start ≤ as.size then
-       { as := as, start := start, stop := as.size, h₁ := h₁, h₂ := Nat.le_refl _ }
-     else
-       { as := as, start := as.size, stop := as.size, h₁ := Nat.le_refl _, h₂ := Nat.le_refl _ }
+    if h₁ : start ≤ as.size then
+      { array := as,
+        start := start,
+        stop := as.size,
+        start_le_stop := h₁,
+        stop_le_array_size := Nat.le_refl _ }
+    else
+      { array := as,
+        start := as.size,
+        stop := as.size,
+        start_le_stop := Nat.le_refl _,
+        stop_le_array_size := Nat.le_refl _ }
 
 @[coe]
 def ofSubarray (s : Subarray α) : Array α := Id.run do

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

@@ -0,0 +1,71 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Thrane Christiansen
+-/
+
+prelude
+import Init.Data.Array.Basic
+import Init.Data.Array.Subarray
+import Init.Omega
+
+/-
+This module contains splitting operations on subarrays that crucially rely on `omega` for proof
+automation. Placing them in another module breaks an import cycle, because `omega` itself uses the
+array library.
+-/
+
+namespace Subarray
+/--
+Splits a subarray into two parts.
+-/
+def split (s : Subarray α) (i : Fin s.size.succ) : (Subarray α × Subarray α) :=
+  let ⟨i', isLt⟩ := i
+  have := s.start_le_stop
+  have := s.stop_le_array_size
+  have : i' ≤ s.stop - s.start := Nat.lt_succ.mp isLt
+  have : s.start + i' ≤ s.stop := by omega
+  have : s.start + i' ≤ s.array.size := by omega
+  have : s.start + i' ≤ s.stop := by
+    simp only [size] at isLt
+    omega
+  let pre := {s with
+    stop := s.start + i',
+    start_le_stop := by omega,
+    stop_le_array_size := by assumption
+  }
+  let post := {s with
+    start := s.start + i'
+    start_le_stop := by assumption
+  }
+  (pre, post)
+
+/--
+Removes the first `i` elements of the subarray. If there are `i` or fewer elements, the resulting
+subarray is empty.
+-/
+def drop (arr : Subarray α) (i : Nat) : Subarray α where
+  array := arr.array
+  start := min (arr.start + i) arr.stop
+  stop := arr.stop
+  start_le_stop := by
+    rw [Nat.min_def]
+    split <;> simp only [Nat.le_refl, *]
+  stop_le_array_size := arr.stop_le_array_size
+
+/--
+Keeps only the first `i` elements of the subarray. If there are `i` or fewer elements, the resulting
+subarray is empty.
+-/
+def take (arr : Subarray α) (i : Nat) : Subarray α where
+  array := arr.array
+  start := arr.start
+  stop := min (arr.start + i) arr.stop
+  start_le_stop := by
+    have := arr.start_le_stop
+    rw [Nat.min_def]
+    split <;> omega
+  stop_le_array_size := by
+    have := arr.stop_le_array_size
+    rw [Nat.min_def]
+    split <;> omega

src/Init/Data/BitVec/Basic.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joe Hendrix, Wojciech Nawrocki, Leonardo de Moura, Mario Carneiro, Alex Keizer
+Authors: Joe Hendrix, Wojciech Nawrocki, Leonardo de Moura, Mario Carneiro, Alex Keizer, Harun Khan, Abdalrhman M Mohamed
 -/
 prelude
 import Init.Data.Fin.Basic
@@ -34,7 +34,8 @@ structure BitVec (w : Nat) where
   O(1), because we use `Fin` as the internal representation of a bitvector. -/
   toFin : Fin (2^w)
 
-@[deprecated] abbrev Std.BitVec := _root_.BitVec
+@[deprecated (since := "2024-04-12")]
+protected abbrev Std.BitVec := _root_.BitVec
 
 -- We manually derive the `DecidableEq` instances for `BitVec` because
 -- we want to have builtin support for bit-vector literals, and we
@@ -73,7 +74,7 @@ protected def toNat (a : BitVec n) : Nat := a.toFin.val
 /-- Return the bound in terms of toNat. -/
 theorem isLt (x : BitVec w) : x.toNat < 2^w := x.toFin.isLt
 
-@[deprecated isLt]
+@[deprecated isLt (since := "2024-03-12")]
 theorem toNat_lt (x : BitVec n) : x.toNat < 2^n := x.isLt
 
 /-- Theorem for normalizing the bit vector literal representation. -/
@@ -533,6 +534,11 @@ def sshiftRight (a : BitVec n) (s : Nat) : BitVec n := .ofInt n (a.toInt >>> s)
 instance {n} : HShiftLeft  (BitVec m) (BitVec n) (BitVec m) := ⟨fun x y => x <<< y.toNat⟩
 instance {n} : HShiftRight (BitVec m) (BitVec n) (BitVec m) := ⟨fun x y => x >>> y.toNat⟩
 
+/-- Auxiliary function for `rotateLeft`, which does not take into account the case where
+the rotation amount is greater than the bitvector width. -/
+def rotateLeftAux (x : BitVec w) (n : Nat) : BitVec w :=
+  x <<< n ||| x >>> (w - n)
+
 /--
 Rotate left for bit vectors. All the bits of `x` are shifted to higher positions, with the top `n`
 bits wrapping around to fill the low bits.
@@ -542,7 +548,15 @@ rotateLeft  0b0011#4 3 = 0b1001
 ```
 SMT-Lib name: `rotate_left` except this operator uses a `Nat` shift amount.
 -/
-def rotateLeft (x : BitVec w) (n : Nat) : BitVec w := x <<< n ||| x >>> (w - n)
+def rotateLeft (x : BitVec w) (n : Nat) : BitVec w := rotateLeftAux x (n % w)
+
+
+/--
+Auxiliary function for `rotateRight`, which does not take into account the case where
+the rotation amount is greater than the bitvector width.
+-/
+def rotateRightAux (x : BitVec w) (n : Nat) : BitVec w :=
+  x >>> n ||| x <<< (w - n)
 
 /--
 Rotate right for bit vectors. All the bits of `x` are shifted to lower positions, with the
@@ -553,7 +567,7 @@ rotateRight 0b01001#5 1 = 0b10100
 ```
 SMT-Lib name: `rotate_right` except this operator uses a `Nat` shift amount.
 -/
-def rotateRight (x : BitVec w) (n : Nat) : BitVec w := x >>> n ||| x <<< (w - n)
+def rotateRight (x : BitVec w) (n : Nat) : BitVec w := rotateRightAux x (n % w)
 
 /--
 Concatenation of bitvectors. This uses the "big endian" convention that the more significant

src/Init/Data/BitVec/Bitblast.lean

@@ -159,4 +159,43 @@ theorem add_eq_adc (w : Nat) (x y : BitVec w) : x + y = (adc x y false).snd := b
 theorem allOnes_sub_eq_not (x : BitVec w) : allOnes w - x = ~~~x := by
   rw [← add_not_self x, BitVec.add_comm, add_sub_cancel]
 
+/-! ### Negation -/
+
+theorem bit_not_testBit (x : BitVec w) (i : Fin w) :
+  getLsb (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsb i)))) ()).snd) i.val = !(getLsb x i.val) := by
+  apply iunfoldr_getLsb (fun _ => ()) i (by simp)
+
+theorem bit_not_add_self (x : BitVec w) :
+  ((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsb i)))) ()).snd + x  = -1 := by
+  simp only [add_eq_adc]
+  apply iunfoldr_replace_snd (fun _ => false) (-1) false rfl
+  intro i; simp only [ BitVec.not, adcb, testBit_toNat]
+  rw [iunfoldr_replace_snd (fun _ => ()) (((iunfoldr (fun i c => (c, !(x.getLsb i)))) ()).snd)]
+  <;> simp [bit_not_testBit, negOne_eq_allOnes, getLsb_allOnes]
+
+theorem bit_not_eq_not (x : BitVec w) :
+  ((iunfoldr (fun i c => (c, !(x.getLsb i)))) ()).snd = ~~~ x := by
+  simp [←allOnes_sub_eq_not, BitVec.eq_sub_iff_add_eq.mpr (bit_not_add_self x), ←negOne_eq_allOnes]
+
+theorem bit_neg_eq_neg (x : BitVec w) : -x = (adc (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsb i)))) ()).snd) (BitVec.ofNat w 1) false).snd:= by
+  simp only [← add_eq_adc]
+  rw [iunfoldr_replace_snd ((fun _ => ())) (((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsb i)))) ()).snd) _ rfl]
+  · rw [BitVec.eq_sub_iff_add_eq.mpr (bit_not_add_self x), sub_toAdd, BitVec.add_comm _ (-x)]
+    simp [← sub_toAdd, BitVec.sub_add_cancel]
+  · simp [bit_not_testBit x _]
+
+/-! ### Inequalities (le / lt) -/
+
+theorem ult_eq_not_carry (x y : BitVec w) : x.ult y = !carry w x (~~~y) true := by
+  simp only [BitVec.ult, carry, toNat_mod_cancel, toNat_not, toNat_true, ge_iff_le, ← decide_not,
+    Nat.not_le, decide_eq_decide]
+  rw [Nat.mod_eq_of_lt (by omega)]
+  omega
+
+theorem ule_eq_not_ult (x y : BitVec w) : x.ule y = !y.ult x := by
+  simp [BitVec.ule, BitVec.ult, ← decide_not]
+
+theorem ule_eq_carry (x y : BitVec w) : x.ule y = carry w y (~~~x) true := by
+  simp [ule_eq_not_ult, ult_eq_not_carry]
+
 end BitVec

src/Init/Data/BitVec/Folds.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joe Hendrix
+Authors: Joe Hendrix, Harun Khan
 -/
 prelude
 import Init.Data.BitVec.Lemmas
@@ -48,6 +48,51 @@ private theorem iunfoldr.eq_test
     intro i
     simp_all [truncate_succ]
 
+theorem iunfoldr_getLsb' {f : Fin w → α → α × Bool} (state : Nat → α)
+    (ind : ∀(i : Fin w), (f i (state i.val)).fst = state (i.val+1)) :
+  (∀ i : Fin w, getLsb (iunfoldr f (state 0)).snd i.val = (f i (state i.val)).snd)
+  ∧ (iunfoldr f (state 0)).fst = state w := by
+  unfold iunfoldr
+  simp
+  apply Fin.hIterate_elim
+        (fun j (p : α × BitVec j) => (hj : j ≤ w) →
+         (∀ i : Fin j,  getLsb p.snd i.val = (f ⟨i.val, Nat.lt_of_lt_of_le i.isLt hj⟩ (state i.val)).snd)
+          ∧ p.fst = state j)
+  case hj => simp
+  case init =>
+    intro
+    apply And.intro
+    · intro i
+      have := Fin.size_pos i
+      contradiction
+    · rfl
+  case step =>
+    intro j ⟨s, v⟩ ih hj
+    apply And.intro
+    case left =>
+      intro i
+      simp only [getLsb_cons]
+      have hj2 : j.val ≤ w := by simp
+      cases (Nat.lt_or_eq_of_le (Nat.lt_succ.mp i.isLt)) with
+      | inl h3 => simp [if_neg, (Nat.ne_of_lt h3)]
+                  exact (ih hj2).1 ⟨i.val, h3⟩
+      | inr h3 => simp [h3, if_pos]
+                  cases (Nat.eq_zero_or_pos j.val) with
+                  | inl hj3 => congr
+                               rw [← (ih hj2).2]
+                  | inr hj3 => congr
+                               exact (ih hj2).2
+    case right =>
+      simp
+      have hj2 : j.val ≤ w := by simp
+      rw [← ind j, ← (ih hj2).2]
+
+
+theorem iunfoldr_getLsb {f : Fin w → α → α × Bool} (state : Nat → α) (i : Fin w)
+    (ind : ∀(i : Fin w), (f i (state i.val)).fst = state (i.val+1)) :
+  getLsb (iunfoldr f (state 0)).snd i.val = (f i (state i.val)).snd := by
+  exact (iunfoldr_getLsb' state ind).1 i
+
 /--
 Correctness theorem for `iunfoldr`.
 -/
@@ -58,4 +103,11 @@ theorem iunfoldr_replace
     iunfoldr f a = (state w, value) := by
   simp [iunfoldr.eq_test state value a init step]
 
+theorem iunfoldr_replace_snd
+  {f : Fin w → α → α × Bool} (state : Nat → α) (value : BitVec w) (a : α)
+    (init : state 0 = a)
+    (step : ∀(i : Fin w), f i (state i.val) = (state (i.val+1), value.getLsb i.val)) :
+    (iunfoldr f a).snd = value := by
+  simp [iunfoldr.eq_test state value a init step]
+
 end BitVec

src/Init/Data/BitVec/Lemmas.lean

@@ -1,13 +1,15 @@
 /-
 Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joe Hendrix
+Authors: Joe Hendrix, Harun Khan, Alex Keizer, Abdalrhman M Mohamed,
+
 -/
 prelude
 import Init.Data.Bool
 import Init.Data.BitVec.Basic
 import Init.Data.Fin.Lemmas
 import Init.Data.Nat.Lemmas
+import Init.Data.Nat.Mod
 
 namespace BitVec
 
@@ -103,7 +105,13 @@ theorem eq_of_getMsb_eq {x y : BitVec w}
     have q := pred ⟨w - 1 - i, q_lt⟩
     simpa [q_lt, Nat.sub_sub_self, r] using q
 
-@[simp] theorem of_length_zero {x : BitVec 0} : x = 0#0 := by ext; simp
+-- This cannot be a `@[simp]` lemma, as it would be tried at every term.
+theorem of_length_zero {x : BitVec 0} : x = 0#0 := by ext; simp
+
+@[simp] theorem toNat_zero_length (x : BitVec 0) : x.toNat = 0 := by simp [of_length_zero]
+@[simp] theorem getLsb_zero_length (x : BitVec 0) : x.getLsb i = false := by simp [of_length_zero]
+@[simp] theorem getMsb_zero_length (x : BitVec 0) : x.getMsb i = false := by simp [of_length_zero]
+@[simp] theorem msb_zero_length (x : BitVec 0) : x.msb = false := by simp [BitVec.msb, of_length_zero]
 
 theorem eq_of_toFin_eq : ∀ {x y : BitVec w}, x.toFin = y.toFin → x = y
   | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
@@ -139,7 +147,8 @@ theorem getLsb_ofNat (n : Nat) (x : Nat) (i : Nat) :
   getLsb (x#n) i = (i < n && x.testBit i) := by
   simp [getLsb, BitVec.ofNat, Fin.val_ofNat']
 
-@[simp, deprecated toNat_ofNat] theorem toNat_zero (n : Nat) : (0#n).toNat = 0 := by trivial
+@[simp, deprecated toNat_ofNat (since := "2024-02-22")]
+theorem toNat_zero (n : Nat) : (0#n).toNat = 0 := by trivial
 
 @[simp] theorem getLsb_zero : (0#w).getLsb i = false := by simp [getLsb]
 
@@ -214,9 +223,21 @@ theorem toInt_eq_toNat_cond (i : BitVec n) :
       if 2*i.toNat < 2^n then
         (i.toNat : Int)
       else
-        (i.toNat : Int) - (2^n : Nat) := by
-  unfold BitVec.toInt
-  split <;> omega
+        (i.toNat : Int) - (2^n : Nat) :=
+  rfl
+
+theorem msb_eq_false_iff_two_mul_lt (x : BitVec w) : x.msb = false ↔ 2 * x.toNat < 2^w := by
+  cases w <;> simp [Nat.pow_succ, Nat.mul_comm _ 2, msb_eq_decide]
+
+theorem msb_eq_true_iff_two_mul_ge (x : BitVec w) : x.msb = true ↔ 2 * x.toNat ≥ 2^w := by
+  simp [← Bool.ne_false_iff, msb_eq_false_iff_two_mul_lt]
+
+/-- Characterize `x.toInt` in terms of `x.msb`. -/
+theorem toInt_eq_msb_cond (x : BitVec w) :
+    x.toInt = if x.msb then (x.toNat : Int) - (2^w : Nat) else (x.toNat : Int) := by
+  simp only [BitVec.toInt, ← msb_eq_false_iff_two_mul_lt]
+  cases x.msb <;> rfl
+
 
 theorem toInt_eq_toNat_bmod (x : BitVec n) : x.toInt = Int.bmod x.toNat (2^n) := by
   simp only [toInt_eq_toNat_cond]
@@ -238,6 +259,12 @@ theorem eq_of_toInt_eq {i j : BitVec n} : i.toInt = j.toInt → i = j := by
   have _jlt := j.isLt
   split <;> split <;> omega
 
+theorem toInt_inj (x y : BitVec n) : x.toInt = y.toInt ↔ x = y :=
+  Iff.intro eq_of_toInt_eq (congrArg BitVec.toInt)
+
+theorem toInt_ne (x y : BitVec n) : x.toInt ≠ y.toInt ↔ x ≠ y  := by
+  rw [Ne, toInt_inj]
+
 @[simp] theorem toNat_ofInt {n : Nat} (i : Int) :
   (BitVec.ofInt n i).toNat = (i % (2^n : Nat)).toNat := by
   unfold BitVec.ofInt
@@ -336,7 +363,7 @@ theorem nat_eq_toNat (x : BitVec w) (y : Nat)
 @[simp] theorem getMsb_zeroExtend_add {x : BitVec w} (h : k ≤ i) :
     (x.zeroExtend (w + k)).getMsb i = x.getMsb (i - k) := by
   by_cases h : w = 0
-  · subst h; simp
+  · subst h; simp [of_length_zero]
   simp only [getMsb, getLsb_zeroExtend]
   by_cases h₁ : i < w + k <;> by_cases h₂ : i - k < w <;> by_cases h₃ : w + k - 1 - i < w + k
     <;> simp [h₁, h₂, h₃]
@@ -595,6 +622,17 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
     (shiftLeftZeroExtend x i).msb = x.msb := by
   simp [shiftLeftZeroExtend_eq, BitVec.msb]
 
+theorem shiftLeft_shiftLeft {w : Nat} (x : BitVec w) (n m : Nat) :
+    (x <<< n) <<< m = x <<< (n + m) := by
+  ext i
+  simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and]
+  rw [show i - (n + m) = (i - m - n) by omega]
+  cases h₂ : decide (i < m) <;>
+  cases h₃ : decide (i - m < w) <;>
+  cases h₄ : decide (i - m < n) <;>
+  cases h₅ : decide (i < n + m) <;>
+    simp at * <;> omega
+
 /-! ### ushiftRight -/
 
 @[simp, bv_toNat] theorem toNat_ushiftRight (x : BitVec n) (i : Nat) :
@@ -680,6 +718,11 @@ theorem msb_append {x : BitVec w} {y : BitVec v} :
   simp only [getLsb_append, cond_eq_if]
   split <;> simp [*]
 
+theorem shiftRight_shiftRight {w : Nat} (x : BitVec w) (n m : Nat) :
+    (x >>> n) >>> m = x >>> (n + m) := by
+  ext i
+  simp [Nat.add_assoc n m i]
+
 /-! ### rev -/
 
 theorem getLsb_rev (x : BitVec w) (i : Fin w) :
@@ -728,8 +771,7 @@ theorem toNat_cons' {x : BitVec w} :
   rw [← BitVec.msb, msb_cons]
 
 @[simp] theorem getMsb_cons_succ : (cons a x).getMsb (i + 1) = x.getMsb i := by
-  simp [cons, cond_eq_if]
-  omega
+  simp [cons, Nat.le_add_left 1 i]
 
 theorem truncate_succ (x : BitVec w) :
     truncate (i+1) x = cons (getLsb x i) (truncate i x) := by
@@ -818,24 +860,42 @@ Definition of bitvector addition as a nat.
   .ofFin x + y = .ofFin (x + y.toFin) := rfl
 @[simp] theorem add_ofFin (x : BitVec n) (y : Fin (2^n)) :
   x + .ofFin y = .ofFin (x.toFin + y) := rfl
-@[simp] theorem ofNat_add_ofNat {n} (x y : Nat) : x#n + y#n = (x + y)#n := by
+
+theorem ofNat_add {n} (x y : Nat) : (x + y)#n = x#n + y#n := by
   apply eq_of_toNat_eq ; simp [BitVec.ofNat]
 
+theorem ofNat_add_ofNat {n} (x y : Nat) : x#n + y#n = (x + y)#n :=
+  (ofNat_add x y).symm
+
 protected theorem add_assoc (x y z : BitVec n) : x + y + z = x + (y + z) := by
   apply eq_of_toNat_eq ; simp [Nat.add_assoc]
+instance : Std.Associative (α := BitVec n) (· + ·) := ⟨BitVec.add_assoc⟩
 
 protected theorem add_comm (x y : BitVec n) : x + y = y + x := by
   simp [add_def, Nat.add_comm]
+instance : Std.Commutative (α := BitVec n) (· + ·) := ⟨BitVec.add_comm⟩
 
 @[simp] protected theorem add_zero (x : BitVec n) : x + 0#n = x := by simp [add_def]
 
 @[simp] protected theorem zero_add (x : BitVec n) : 0#n + x = x := by simp [add_def]
+instance : Std.LawfulIdentity (α := BitVec n) (· + ·) 0#n where
+  left_id := BitVec.zero_add
+  right_id := BitVec.add_zero
 
 theorem truncate_add (x y : BitVec w) (h : i ≤ w) :
     (x + y).truncate i = x.truncate i + y.truncate i := by
   have dvd : 2^i ∣ 2^w := Nat.pow_dvd_pow _ h
   simp [bv_toNat, h, Nat.mod_mod_of_dvd _ dvd]
 
+@[simp, bv_toNat] theorem toInt_add (x y : BitVec w) :
+  (x + y).toInt = (x.toInt + y.toInt).bmod (2^w) := by
+  simp [toInt_eq_toNat_bmod]
+
+theorem ofInt_add {n} (x y : Int) : BitVec.ofInt n (x + y) =
+    BitVec.ofInt n x + BitVec.ofInt n y := by
+  apply eq_of_toInt_eq
+  simp
+
 /-! ### sub/neg -/
 
 theorem sub_def {n} (x y : BitVec n) : x - y = .ofNat n (x.toNat + (2^n - y.toNat)) := by rfl
@@ -873,10 +933,19 @@ theorem sub_toAdd {n} (x y : BitVec n) : x - y = x + - y := by
 
 theorem add_sub_cancel (x y : BitVec w) : x + y - y = x := by
   apply eq_of_toNat_eq
-  have y_toNat_le := Nat.le_of_lt y.toNat_lt
+  have y_toNat_le := Nat.le_of_lt y.isLt
   rw [toNat_sub, toNat_add, Nat.mod_add_mod, Nat.add_assoc, ← Nat.add_sub_assoc y_toNat_le,
     Nat.add_sub_cancel_left, Nat.add_mod_right, toNat_mod_cancel]
 
+theorem sub_add_cancel (x y : BitVec w) : x - y + y = x := by
+  rw [sub_toAdd, BitVec.add_assoc, BitVec.add_comm _ y,
+      ← BitVec.add_assoc, ← sub_toAdd, add_sub_cancel]
+
+theorem eq_sub_iff_add_eq {x y z : BitVec w} : x = z - y ↔ x + y = z := by
+  apply Iff.intro <;> intro h
+  · simp [h, sub_add_cancel]
+  · simp [←h, add_sub_cancel]
+
 theorem negOne_eq_allOnes : -1#w = allOnes w := by
   apply eq_of_toNat_eq
   if g : w = 0 then
@@ -886,6 +955,13 @@ theorem negOne_eq_allOnes : -1#w = allOnes w := by
     have r : (2^w - 1) < 2^w := by omega
     simp [Nat.mod_eq_of_lt q, Nat.mod_eq_of_lt r]
 
+theorem neg_eq_not_add (x : BitVec w) : -x = ~~~x + 1 := by
+  apply eq_of_toNat_eq
+  simp only [toNat_neg, ofNat_eq_ofNat, toNat_add, toNat_not, toNat_ofNat, Nat.add_mod_mod]
+  congr
+  have hx : x.toNat < 2^w := x.isLt
+  rw [Nat.sub_sub, Nat.add_comm 1 x.toNat, ← Nat.sub_sub, Nat.sub_add_cancel (by omega)]
+
 /-! ### mul -/
 
 theorem mul_def {n} {x y : BitVec n} : x * y = (ofFin <| x.toFin * y.toFin) := by rfl
@@ -912,6 +988,15 @@ instance : Std.Associative (fun (x y : BitVec w) => x * y) := ⟨BitVec.mul_asso
 instance : Std.LawfulCommIdentity (fun (x y : BitVec w) => x * y) (1#w) where
   right_id := BitVec.mul_one
 
+@[simp, bv_toNat] theorem toInt_mul (x y : BitVec w) :
+  (x * y).toInt = (x.toInt * y.toInt).bmod (2^w) := by
+  simp [toInt_eq_toNat_bmod]
+
+theorem ofInt_mul {n} (x y : Int) : BitVec.ofInt n (x * y) =
+    BitVec.ofInt n x * BitVec.ofInt n y := by
+  apply eq_of_toInt_eq
+  simp
+
 /-! ### le and lt -/
 
 @[bv_toNat] theorem le_def (x y : BitVec n) :
@@ -971,4 +1056,32 @@ theorem toNat_intMax_eq : (intMax w).toNat = 2^w - 1 := by
     (ofBoolListLE bs).getMsb i = (decide (i < bs.length) && bs.getD (bs.length - 1 - i) false) := by
   simp [getMsb_eq_getLsb]
 
+/-! # Rotate Left -/
+
+/-- rotateLeft is invariant under `mod` by the bitwidth. -/
+@[simp]
+theorem rotateLeft_mod_eq_rotateLeft {x : BitVec w} {r : Nat} :
+    x.rotateLeft (r % w) = x.rotateLeft r := by
+  simp only [rotateLeft, Nat.mod_mod]
+
+/-- `rotateLeft` equals the bit fiddling definition of `rotateLeftAux` when the rotation amount is
+smaller than the bitwidth. -/
+theorem rotateLeft_eq_rotateLeftAux_of_lt {x : BitVec w} {r : Nat} (hr : r < w) :
+    x.rotateLeft r = x.rotateLeftAux r := by
+  simp only [rotateLeft, Nat.mod_eq_of_lt hr]
+
+/-! ## Rotate Right -/
+
+/-- `rotateRight` equals the bit fiddling definition of `rotateRightAux` when the rotation amount is
+smaller than the bitwidth. -/
+theorem rotateRight_eq_rotateRightAux_of_lt {x : BitVec w} {r : Nat} (hr : r < w) :
+    x.rotateRight r = x.rotateRightAux r := by
+  simp only [rotateRight, Nat.mod_eq_of_lt hr]
+
+/-- rotateRight is invariant under `mod` by the bitwidth. -/
+@[simp]
+theorem rotateRight_mod_eq_rotateRight {x : BitVec w} {r : Nat} :
+    x.rotateRight (r % w) = x.rotateRight r := by
+  simp only [rotateRight, Nat.mod_mod]
+
 end BitVec

src/Init/Data/Bool.lean

@@ -74,6 +74,7 @@ Added for confluence with `not_and_self` `and_not_self` on term
 @[simp] theorem eq_false_and_eq_true_self : ∀(b : Bool), (b = false ∧ b = true) ↔ False := by decide
 
 theorem and_comm : ∀ (x y : Bool), (x && y) = (y && x) := by decide
+instance : Std.Commutative (· && ·) := ⟨and_comm⟩
 
 theorem and_left_comm : ∀ (x y z : Bool), (x && (y && z)) = (y && (x && z)) := by decide
 theorem and_right_comm : ∀ (x y z : Bool), ((x && y) && z) = ((x && z) && y) := by decide
@@ -120,6 +121,7 @@ Needed for confluence of term `(a || b) ↔ a` which reduces to `(a || b) = a` v
 @[simp] theorem iff_or_self : ∀(a b : Bool), (b = (a || b)) ↔ (a → b) := by decide
 
 theorem or_comm : ∀ (x y : Bool), (x || y) = (y || x) := by decide
+instance : Std.Commutative (· || ·) := ⟨or_comm⟩
 
 theorem or_left_comm : ∀ (x y z : Bool), (x || (y || z)) = (y || (x || z)) := by decide
 theorem or_right_comm : ∀ (x y z : Bool), ((x || y) || z) = ((x || z) || y) := by decide
@@ -186,12 +188,18 @@ in false_eq and true_eq.
 @[simp] theorem true_beq  : ∀b, (true  == b) =  b := by decide
 @[simp] theorem false_beq : ∀b, (false == b) = !b := by decide
 @[simp] theorem beq_true  : ∀b, (b == true)  =  b := by decide
+instance : Std.LawfulIdentity (· == ·) true where
+  left_id := true_beq
+  right_id := beq_true
 @[simp] theorem beq_false : ∀b, (b == false) = !b := by decide
 
 @[simp] theorem true_bne  : ∀(b : Bool), (true  != b) = !b := by decide
 @[simp] theorem false_bne : ∀(b : Bool), (false != b) =  b := by decide
 @[simp] theorem bne_true  : ∀(b : Bool), (b != true)  = !b := by decide
 @[simp] theorem bne_false : ∀(b : Bool), (b != false) =  b := by decide
+instance : Std.LawfulIdentity (· != ·) false where
+  left_id := false_bne
+  right_id := bne_false
 
 @[simp] theorem not_beq_self : ∀ (x : Bool), ((!x) == x) = false := by decide
 @[simp] theorem beq_not_self : ∀ (x : Bool), (x   == !x) = false := by decide
@@ -214,12 +222,19 @@ due to `beq_iff_eq`.
 @[simp] theorem not_bne_not : ∀ (x y : Bool), ((!x) != (!y)) = (x != y) := by decide
 
 @[simp] theorem bne_assoc : ∀ (x y z : Bool), ((x != y) != z) = (x != (y != z)) := by decide
+instance : Std.Associative (· != ·) := ⟨bne_assoc⟩
 
 @[simp] theorem bne_left_inj  : ∀ (x y z : Bool), (x != y) = (x != z) ↔ y = z := by decide
 @[simp] theorem bne_right_inj : ∀ (x y z : Bool), (x != z) = (y != z) ↔ x = y := by decide
 
 /-! ### coercision related normal forms -/
 
+theorem beq_eq_decide_eq [BEq α] [LawfulBEq α] [DecidableEq α] (a b : α) :
+    (a == b) = decide (a = b) := by
+  cases h : a == b
+  · simp [ne_of_beq_false h]
+  · simp [eq_of_beq h]
+
 @[simp] theorem not_eq_not : ∀ {a b : Bool}, ¬a = !b ↔ a = b := by decide
 
 @[simp] theorem not_not_eq : ∀ {a b : Bool}, ¬(!a) = b ↔ a = b := by decide
@@ -230,6 +245,11 @@ due to `beq_iff_eq`.
 @[simp] theorem coe_false_iff_true  : ∀(a b : Bool), (a = false ↔ b) ↔ (!a) = b := by decide
 @[simp] theorem coe_false_iff_false : ∀(a b : Bool), (a = false ↔ b = false) ↔ (!a) = (!b) := by decide
 
+/-! ### beq properties -/
+
+theorem beq_comm {α} [BEq α] [LawfulBEq α] {a b : α} : (a == b) = (b == a) :=
+  (Bool.coe_iff_coe (a == b) (b == a)).mp (by simp [@eq_comm α])
+
 /-! ### xor -/
 
 theorem false_xor : ∀ (x : Bool), xor false x = x := false_bne
@@ -340,7 +360,8 @@ def toNat (b:Bool) : Nat := cond b 1 0
 theorem toNat_le (c : Bool) : c.toNat ≤ 1 := by
   cases c <;> trivial
 
-@[deprecated toNat_le] abbrev toNat_le_one := toNat_le
+@[deprecated toNat_le (since := "2024-02-23")]
+abbrev toNat_le_one := toNat_le
 
 theorem toNat_lt (b : Bool) : b.toNat < 2 :=
   Nat.lt_succ_of_le (toNat_le _)

src/Init/Data/Char.lean

@@ -5,3 +5,4 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.Data.Char.Basic
+import Init.Data.Char.Lemmas

src/Init/Data/Char/Lemmas.lean

@@ -0,0 +1,25 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Data.Char.Basic
+import Init.Data.UInt.Lemmas
+
+namespace Char
+
+theorem le_def {a b : Char} : a ≤ b ↔ a.1 ≤ b.1 := .rfl
+theorem lt_def {a b : Char} : a < b ↔ a.1 < b.1 := .rfl
+theorem lt_iff_val_lt_val {a b : Char} : a < b ↔ a.val < b.val := Iff.rfl
+@[simp] protected theorem not_le {a b : Char} : ¬ a ≤ b ↔ b < a := UInt32.not_le
+@[simp] protected theorem not_lt {a b : Char} : ¬ a < b ↔ b ≤ a := UInt32.not_lt
+@[simp] protected theorem le_refl (a : Char) : a ≤ a := by simp [le_def]
+@[simp] protected theorem lt_irrefl (a : Char) : ¬ a < a := by simp
+protected theorem le_trans {a b c : Char} : a ≤ b → b ≤ c → a ≤ c := UInt32.le_trans
+protected theorem lt_trans {a b c : Char} : a < b → b < c → a < c := UInt32.lt_trans
+protected theorem le_total (a b : Char) : a ≤ b ∨ b ≤ a := UInt32.le_total a.1 b.1
+protected theorem lt_asymm {a b : Char} (h : a < b) : ¬ b < a := UInt32.lt_asymm h
+protected theorem ne_of_lt {a b : Char} (h : a < b) : a ≠ b := Char.ne_of_val_ne (UInt32.ne_of_lt h)
+
+end Char

src/Init/Data/Fin/Basic.lean

@@ -13,17 +13,40 @@ namespace Fin
 instance coeToNat : CoeOut (Fin n) Nat :=
   ⟨fun v => v.val⟩
 
+/--
+From the empty type `Fin 0`, any desired result `α` can be derived. This is simlar to `Empty.elim`.
+-/
 def elim0.{u} {α : Sort u} : Fin 0 → α
   | ⟨_, h⟩ => absurd h (not_lt_zero _)
 
+/--
+Returns the successor of the argument.
+
+The bound in the result type is increased:
+```
+(2 : Fin 3).succ = (3 : Fin 4)
+```
+This differs from addition, which wraps around:
+```
+(2 : Fin 3) + 1 = (0 : Fin 3)
+```
+-/
 def succ : Fin n → Fin n.succ
   | ⟨i, h⟩ => ⟨i+1, Nat.succ_lt_succ h⟩
 
 variable {n : Nat}
 
+/--
+Returns `a` modulo `n + 1` as a `Fin n.succ`.
+-/
 protected def ofNat {n : Nat} (a : Nat) : Fin n.succ :=
   ⟨a % (n+1), Nat.mod_lt _ (Nat.zero_lt_succ _)⟩
 
+/--
+Returns `a` modulo `n` as a `Fin n`.
+
+The assumption `n > 0` ensures that `Fin n` is nonempty.
+-/
 protected def ofNat' {n : Nat} (a : Nat) (h : n > 0) : Fin n :=
   ⟨a % n, Nat.mod_lt _ h⟩
 
@@ -33,12 +56,15 @@ private theorem mlt {b : Nat} : {a : Nat} → a < n → b % n < n
     have : n > 0 := Nat.lt_trans (Nat.zero_lt_succ _) h;
     Nat.mod_lt _ this
 
+/-- Addition modulo `n` -/
 protected def add : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a + b) % n, mlt h⟩
 
+/-- Multiplication modulo `n` -/
 protected def mul : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a * b) % n, mlt h⟩
 
+/-- Subtraction modulo `n` -/
 protected def sub : Fin n → Fin n → Fin n
   | ⟨a, h⟩, ⟨b, _⟩ => ⟨(a + (n - b)) % n, mlt h⟩
 

src/Init/Data/Fin/Fold.lean

@@ -12,6 +12,7 @@ import Init.Data.Nat.Linear
   loop (x : α) (i : Nat) : α :=
     if h : i < n then loop (f x ⟨i, h⟩) (i+1) else x
   termination_by n - i
+  decreasing_by decreasing_trivial_pre_omega
 
 /-- Folds over `Fin n` from the right: `foldr 3 f x = f 0 (f 1 (f 2 x))`. -/
 @[inline] def foldr (n) (f : Fin n → α → α) (init : α) : α := loop ⟨n, Nat.le_refl n⟩ init where

src/Init/Data/Fin/Iterate.lean

@@ -23,6 +23,7 @@ def hIterateFrom (P : Nat → Sort _) {n} (f : ∀(i : Fin n), P i.val → P (i.
     have p : i = n := (or_iff_left g).mp (Nat.eq_or_lt_of_le ubnd)
     _root_.cast (congrArg P p) a
   termination_by n - i
+  decreasing_by decreasing_trivial_pre_omega
 
 /--
 `hIterate` is a heterogenous iterative operation that applies a

src/Init/Data/Fin/Lemmas.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro
+Authors: Mario Carneiro, Leonardo de Moura
 -/
 prelude
 import Init.Data.Fin.Basic
@@ -11,6 +11,9 @@ import Init.ByCases
 import Init.Conv
 import Init.Omega
 
+-- Remove after the next stage0 update
+set_option allowUnsafeReducibility true
+
 namespace Fin
 
 /-- If you actually have an element of `Fin n`, then the `n` is always positive -/
@@ -59,7 +62,8 @@ theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
 @[simp] theorem val_ofNat' (a : Nat) (is_pos : n > 0) :
   (Fin.ofNat' a is_pos).val = a % n := rfl
 
-@[deprecated ofNat'_zero_val] theorem ofNat'_zero_val : (Fin.ofNat' 0 h).val = 0 := Nat.zero_mod _
+@[deprecated ofNat'_zero_val (since := "2024-02-22")]
+theorem ofNat'_zero_val : (Fin.ofNat' 0 h).val = 0 := Nat.zero_mod _
 
 @[simp] theorem mod_val (a b : Fin n) : (a % b).val = a.val % b.val :=
   rfl
@@ -90,6 +94,18 @@ theorem lt_iff_val_lt_val {a b : Fin n} : a < b ↔ a.val < b.val := Iff.rfl
 
 @[simp] protected theorem not_lt {a b : Fin n} : ¬ a < b ↔ b ≤ a := Nat.not_lt
 
+@[simp] protected theorem le_refl (a : Fin n) : a ≤ a := by simp [le_def]
+
+@[simp] protected theorem lt_irrefl (a : Fin n) : ¬ a < a := by simp
+
+protected theorem le_trans {a b c : Fin n} : a ≤ b → b ≤ c → a ≤ c := Nat.le_trans
+
+protected theorem lt_trans {a b c : Fin n} : a < b → b < c → a < c := Nat.lt_trans
+
+protected theorem le_total (a b : Fin n) : a ≤ b ∨ b ≤ a := Nat.le_total a b
+
+protected theorem lt_asymm {a b : Fin n} (h : a < b) : ¬ b < a := Nat.lt_asymm h
+
 protected theorem ne_of_lt {a b : Fin n} (h : a < b) : a ≠ b := Fin.ne_of_val_ne (Nat.ne_of_lt h)
 
 protected theorem ne_of_gt {a b : Fin n} (h : a < b) : b ≠ a := Fin.ne_of_val_ne (Nat.ne_of_gt h)
@@ -541,7 +557,7 @@ theorem pred_mk {n : Nat} (i : Nat) (h : i < n + 1) (w) : Fin.pred ⟨i, h⟩ w
     ∀ {a b : Fin (n + 1)} {ha : a ≠ 0} {hb : b ≠ 0}, a.pred ha = b.pred hb ↔ a = b
   | ⟨0, _⟩, _, ha, _ => by simp only [mk_zero, ne_eq, not_true] at ha
   | ⟨i + 1, _⟩, ⟨0, _⟩, _, hb => by simp only [mk_zero, ne_eq, not_true] at hb
-  | ⟨i + 1, hi⟩, ⟨j + 1, hj⟩, ha, hb => by simp [ext_iff]
+  | ⟨i + 1, hi⟩, ⟨j + 1, hj⟩, ha, hb => by simp [ext_iff, Nat.succ.injEq]
 
 @[simp] theorem pred_one {n : Nat} :
     Fin.pred (1 : Fin (n + 2)) (Ne.symm (Fin.ne_of_lt one_pos)) = 0 := rfl
@@ -602,6 +618,7 @@ A version of `Fin.succRec` taking `i : Fin n` as the first argument. -/
     @Fin.succRecOn (n + 1) i.succ motive zero succ = succ n i (Fin.succRecOn i zero succ) := by
   cases i; rfl
 
+
 /-- Define `motive i` by induction on `i : Fin (n + 1)` via induction on the underlying `Nat` value.
 This function has two arguments: `zero` handles the base case on `motive 0`,
 and `succ` defines the inductive step using `motive i.castSucc`.
@@ -610,8 +627,12 @@ and `succ` defines the inductive step using `motive i.castSucc`.
 @[elab_as_elim] def induction {motive : Fin (n + 1) → Sort _} (zero : motive 0)
     (succ : ∀ i : Fin n, motive (castSucc i) → motive i.succ) :
     ∀ i : Fin (n + 1), motive i
-  | ⟨0, hi⟩ => by rwa [Fin.mk_zero]
-  | ⟨i+1, hi⟩ => succ ⟨i, Nat.lt_of_succ_lt_succ hi⟩ (induction zero succ ⟨i, Nat.lt_of_succ_lt hi⟩)
+  | ⟨i, hi⟩ => go i hi
+where
+  -- Use a curried function so that this is structurally recursive
+  go : ∀ (i : Nat) (hi : i < n + 1), motive ⟨i, hi⟩
+  | 0, hi => by rwa [Fin.mk_zero]
+  | i+1, hi => succ ⟨i, Nat.lt_of_succ_lt_succ hi⟩ (go i (Nat.lt_of_succ_lt hi))
 
 @[simp] theorem induction_zero {motive : Fin (n + 1) → Sort _} (zero : motive 0)
     (hs : ∀ i : Fin n, motive (castSucc i) → motive i.succ) :
@@ -683,6 +704,7 @@ and `cast` defines the inductive step using `motive i.succ`, inducting downwards
 termination_by n + 1 - i
 decreasing_by decreasing_with
   -- FIXME: we put the proof down here to avoid getting a dummy `have` in the definition
+  try simp only [Nat.succ_sub_succ_eq_sub]
   exact Nat.add_sub_add_right .. ▸ Nat.sub_lt_sub_left i.2 (Nat.lt_succ_self i)
 
 @[simp] theorem reverseInduction_last {n : Nat} {motive : Fin (n + 1) → Sort _} {zero succ} :
@@ -792,15 +814,20 @@ protected theorem mul_one (k : Fin (n + 1)) : k * 1 = k := by
 
 protected theorem mul_comm (a b : Fin n) : a * b = b * a :=
   ext <| by rw [mul_def, mul_def, Nat.mul_comm]
+instance : Std.Commutative (α := Fin n) (· * ·) := ⟨Fin.mul_comm⟩
 
 protected theorem mul_assoc (a b c : Fin n) : a * b * c = a * (b * c) := by
   apply eq_of_val_eq
   simp only [val_mul]
   rw [← Nat.mod_eq_of_lt a.isLt, ← Nat.mod_eq_of_lt b.isLt, ← Nat.mod_eq_of_lt c.isLt]
   simp only [← Nat.mul_mod, Nat.mul_assoc]
+instance : Std.Associative (α := Fin n) (· * ·) := ⟨Fin.mul_assoc⟩
 
 protected theorem one_mul (k : Fin (n + 1)) : (1 : Fin (n + 1)) * k = k := by
   rw [Fin.mul_comm, Fin.mul_one]
+instance : Std.LawfulIdentity (α := Fin (n + 1)) (· * ·) 1 where
+  left_id := Fin.one_mul
+  right_id := Fin.mul_one
 
 protected theorem mul_zero (k : Fin (n + 1)) : k * 0 = 0 := by simp [ext_iff, mul_def]
 
@@ -808,27 +835,3 @@ protected theorem zero_mul (k : Fin (n + 1)) : (0 : Fin (n + 1)) * k = 0 := by
   simp [ext_iff, mul_def]
 
 end Fin
-
-namespace USize
-
-@[simp] theorem lt_def {a b : USize} : a < b ↔ a.toNat < b.toNat := .rfl
-
-@[simp] theorem le_def {a b : USize} : a ≤ b ↔ a.toNat ≤ b.toNat := .rfl
-
-@[simp] theorem zero_toNat : (0 : USize).toNat = 0 := Nat.zero_mod _
-
-@[simp] theorem mod_toNat (a b : USize) : (a % b).toNat = a.toNat % b.toNat :=
-  Fin.mod_val ..
-
-@[simp] theorem div_toNat (a b : USize) : (a / b).toNat = a.toNat / b.toNat :=
-  Fin.div_val ..
-
-@[simp] theorem modn_toNat (a : USize) (b : Nat) : (a.modn b).toNat = a.toNat % b :=
-  Fin.modn_val ..
-
-theorem mod_lt (a b : USize) (h : 0 < b) : a % b < b := USize.modn_lt _ (by simp at h; exact h)
-
-theorem toNat.inj : ∀ {a b : USize}, a.toNat = b.toNat → a = b
-  | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
-
-end USize

src/Init/Data/Int/Basic.lean

@@ -100,7 +100,7 @@ protected def neg (n : @& Int) : Int :=
   ```
 -/
 @[default_instance mid]
-instance : Neg Int where
+instance instNegInt : Neg Int where
   neg := Int.neg
 
 /-- Subtraction of two natural numbers. -/
@@ -173,13 +173,13 @@ inductive NonNeg : Int → Prop where
 /-- Definition of `a ≤ b`, encoded as `b - a ≥ 0`. -/
 protected def le (a b : Int) : Prop := NonNeg (b - a)
 
-instance : LE Int where
+instance instLEInt : LE Int where
   le := Int.le
 
 /-- Definition of `a < b`, encoded as `a + 1 ≤ b`. -/
 protected def lt (a b : Int) : Prop := (a + 1) ≤ b
 
-instance : LT Int where
+instance instLTInt : LT Int where
   lt := Int.lt
 
 set_option bootstrap.genMatcherCode false in

src/Init/Data/Int/DivModLemmas.lean

@@ -14,6 +14,8 @@ import Init.RCases
 # Lemmas about integer division needed to bootstrap `omega`.
 -/
 
+-- Remove after the next stage0 update
+set_option allowUnsafeReducibility true
 
 open Nat (succ)
 
@@ -142,12 +144,14 @@ theorem eq_one_of_mul_eq_one_left {a b : Int} (H : 0 ≤ b) (H' : a * b = 1) : b
   | ofNat _ => show ofNat _ = _ by simp
   | -[_+1] => show -ofNat _ = _ by simp
 
+unseal Nat.div in
 @[simp] protected theorem div_zero : ∀ a : Int, div a 0 = 0
   | ofNat _ => show ofNat _ = _ by simp
   | -[_+1] => rfl
 
 @[simp] theorem zero_fdiv (b : Int) : fdiv 0 b = 0 := by cases b <;> rfl
 
+unseal Nat.div in
 @[simp] protected theorem fdiv_zero : ∀ a : Int, fdiv a 0 = 0
   | 0      => rfl
   | succ _ => rfl
@@ -178,7 +182,7 @@ theorem fdiv_eq_div {a b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : fdiv a b = div a
 
 @[simp] theorem mod_zero : ∀ a : Int, mod a 0 = a
   | ofNat _ => congrArg ofNat <| Nat.mod_zero _
-  | -[_+1] => rfl
+  | -[_+1] => congrArg (fun n => -ofNat n) <| Nat.mod_zero _
 
 @[simp] theorem zero_fmod (b : Int) : fmod 0 b = 0 := by cases b <;> rfl
 
@@ -225,7 +229,9 @@ theorem mod_add_div : ∀ a b : Int, mod a b + b * (a.div b) = a
   | ofNat m, -[n+1] => by
     show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
     rw [Int.neg_mul_neg]; exact congrArg ofNat (Nat.mod_add_div ..)
-  | -[_+1], 0 => rfl
+  | -[m+1], 0 => by
+    show -(↑((succ m) % 0) : Int) + 0 * -↑(succ m / 0) = -↑(succ m)
+    rw [Nat.mod_zero, Int.zero_mul, Int.add_zero]
   | -[m+1], ofNat n => by
     show -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m)
     rw [Int.mul_neg, ← Int.neg_add]
@@ -763,11 +769,13 @@ theorem ediv_eq_ediv_of_mul_eq_mul {a b c d : Int}
   | (n:Nat) => congrArg ofNat (Nat.div_one _)
   | -[n+1] => by simp [Int.div, neg_ofNat_succ]; rfl
 
+unseal Nat.div in
 @[simp] protected theorem div_neg : ∀ a b : Int, a.div (-b) = -(a.div b)
   | ofNat m, 0 => show ofNat (m / 0) = -↑(m / 0) by rw [Nat.div_zero]; rfl
   | ofNat m, -[n+1] | -[m+1], succ n => (Int.neg_neg _).symm
   | ofNat m, succ n | -[m+1], 0 | -[m+1], -[n+1] => rfl
 
+unseal Nat.div in
 @[simp] protected theorem neg_div : ∀ a b : Int, (-a).div b = -(a.div b)
   | 0, n => by simp [Int.neg_zero]
   | succ m, (n:Nat) | -[m+1], 0 | -[m+1], -[n+1] => rfl
@@ -936,6 +944,7 @@ theorem fdiv_nonneg {a b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a.fdiv b :
   match a, b, eq_ofNat_of_zero_le Ha, eq_ofNat_of_zero_le Hb with
   | _, _, ⟨_, rfl⟩, ⟨_, rfl⟩ => ofNat_fdiv .. ▸ ofNat_zero_le _
 
+unseal Nat.div in
 theorem fdiv_nonpos : ∀ {a b : Int}, 0 ≤ a → b ≤ 0 → a.fdiv b ≤ 0
   | 0, 0, _, _ | 0, -[_+1], _, _ | succ _, 0, _, _ | succ _, -[_+1], _, _ => ⟨_⟩
 
@@ -1054,19 +1063,39 @@ theorem emod_add_bmod_congr (x : Int) (n : Nat) : Int.bmod (x%n + y) n = Int.bmo
   simp [Int.emod_def, Int.sub_eq_add_neg]
   rw [←Int.mul_neg, Int.add_right_comm,  Int.bmod_add_mul_cancel]
 
+@[simp]
+theorem emod_mul_bmod_congr (x : Int) (n : Nat) : Int.bmod (x%n * y) n = Int.bmod (x * y) n := by
+  simp [Int.emod_def, Int.sub_eq_add_neg]
+  rw [←Int.mul_neg, Int.add_mul, Int.mul_assoc, Int.bmod_add_mul_cancel]
+
 @[simp]
 theorem bmod_add_bmod_congr : Int.bmod (Int.bmod x n + y) n = Int.bmod (x + y) n := by
+  rw [bmod_def x n]
+  split
+  case inl p =>
+    simp only [emod_add_bmod_congr]
+  case inr p =>
+    rw [Int.sub_eq_add_neg, Int.add_right_comm, ←Int.sub_eq_add_neg]
+    simp
+
+@[simp] theorem add_bmod_bmod : Int.bmod (x + Int.bmod y n) n = Int.bmod (x + y) n := by
+  rw [Int.add_comm x, Int.bmod_add_bmod_congr, Int.add_comm y]
+
+@[simp]
+theorem bmod_mul_bmod : Int.bmod (Int.bmod x n * y) n = Int.bmod (x * y) n := by
   rw [bmod_def x n]
   split
   case inl p =>
     simp
   case inr p =>
-    rw [Int.sub_eq_add_neg, Int.add_right_comm, ←Int.sub_eq_add_neg]
+    rw [Int.sub_mul, Int.sub_eq_add_neg, ← Int.mul_neg]
     simp
 
-@[simp]
-theorem add_bmod_bmod : Int.bmod (x + Int.bmod y n) n = Int.bmod (x + y) n := by
-  rw [Int.add_comm x, Int.bmod_add_bmod_congr, Int.add_comm y]
+@[simp] theorem mul_bmod_bmod : Int.bmod (x * Int.bmod y n) n = Int.bmod (x * y) n := by
+  rw [Int.mul_comm x, bmod_mul_bmod, Int.mul_comm x]
+
+theorem add_bmod (a b : Int) (n : Nat) : (a + b).bmod n = (a.bmod n + b.bmod n).bmod n := by
+  simp
 
 theorem emod_bmod {x : Int} {m : Nat} : bmod (x % m) m = bmod x m := by
   simp [bmod]

src/Init/Data/Int/Lemmas.lean

@@ -137,12 +137,16 @@ protected theorem add_comm : ∀ a b : Int, a + b = b + a
   | ofNat _, -[_+1]  => rfl
   | -[_+1],  ofNat _ => rfl
   | -[_+1],  -[_+1]  => by simp [Nat.add_comm]
+instance : Std.Commutative (α := Int) (· + ·) := ⟨Int.add_comm⟩
 
 @[simp] protected theorem add_zero : ∀ a : Int, a + 0 = a
   | ofNat _ => rfl
   | -[_+1]  => rfl
 
 @[simp] protected theorem zero_add (a : Int) : 0 + a = a := Int.add_comm .. ▸ a.add_zero
+instance : Std.LawfulIdentity (α := Int) (· + ·) 0 where
+  left_id := Int.zero_add
+  right_id := Int.add_zero
 
 theorem ofNat_add_negSucc_of_lt (h : m < n.succ) : ofNat m + -[n+1] = -[n - m+1] :=
   show subNatNat .. = _ by simp [succ_sub (le_of_lt_succ h), subNatNat]
@@ -196,6 +200,7 @@ where
     simp
     rw [Int.add_comm, subNatNat_add_negSucc]
     simp [Nat.add_comm, Nat.add_left_comm, Nat.add_assoc]
+instance : Std.Associative (α := Int) (· + ·) := ⟨Int.add_assoc⟩
 
 protected theorem add_left_comm (a b c : Int) : a + (b + c) = b + (a + c) := by
   rw [← Int.add_assoc, Int.add_comm a, Int.add_assoc]
@@ -351,6 +356,7 @@ protected theorem sub_right_inj (i j k : Int) : (i - k = j - k) ↔ i = j := by
 
 protected theorem mul_comm (a b : Int) : a * b = b * a := by
   cases a <;> cases b <;> simp [Nat.mul_comm]
+instance : Std.Commutative (α := Int) (· * ·) := ⟨Int.mul_comm⟩
 
 theorem ofNat_mul_negOfNat (m n : Nat) : (m : Nat) * negOfNat n = negOfNat (m * n) := by
   cases n <;> rfl
@@ -369,6 +375,7 @@ attribute [local simp] ofNat_mul_negOfNat negOfNat_mul_ofNat
 
 protected theorem mul_assoc (a b c : Int) : a * b * c = a * (b * c) := by
   cases a <;> cases b <;> cases c <;> simp [Nat.mul_assoc]
+instance : Std.Associative (α := Int) (· * ·) := ⟨Int.mul_assoc⟩
 
 protected theorem mul_left_comm (a b c : Int) : a * (b * c) = b * (a * c) := by
   rw [← Int.mul_assoc, ← Int.mul_assoc, Int.mul_comm a]
@@ -458,6 +465,9 @@ protected theorem sub_mul (a b c : Int) : (a - b) * c = a * c - b * c := by
   | -[n+1]  => show -[1 * n +1] = -[n+1] by rw [Nat.one_mul]
 
 @[simp] protected theorem mul_one (a : Int) : a * 1 = a := by rw [Int.mul_comm, Int.one_mul]
+instance : Std.LawfulIdentity (α := Int) (· * ·) 1 where
+  left_id := Int.one_mul
+  right_id := Int.mul_one
 
 protected theorem mul_neg_one (a : Int) : a * -1 = -a := by rw [Int.mul_neg, Int.mul_one]
 

src/Init/Data/Int/Order.lean

@@ -96,7 +96,7 @@ protected theorem le_antisymm {a b : Int} (h₁ : a ≤ b) (h₂ : b ≤ a) : a
   have := Int.ofNat.inj <| Int.add_left_cancel <| this.trans (Int.add_zero _).symm
   rw [← hn, Nat.eq_zero_of_add_eq_zero_left this, ofNat_zero, Int.add_zero a]
 
-protected theorem lt_irrefl (a : Int) : ¬a < a := fun H =>
+@[simp] protected theorem lt_irrefl (a : Int) : ¬a < a := fun H =>
   let ⟨n, hn⟩ := lt.dest H
   have : (a+Nat.succ n) = a+0 := by
     rw [hn, Int.add_zero]
@@ -187,6 +187,7 @@ protected theorem min_comm (a b : Int) : min a b = min b a := by
   by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
   · exact Int.le_antisymm h₁ h₂
   · cases not_or_intro h₁ h₂ <| Int.le_total ..
+instance : Std.Commutative (α := Int) min := ⟨Int.min_comm⟩
 
 protected theorem min_le_right (a b : Int) : min a b ≤ b := by rw [Int.min_def]; split <;> simp [*]
 
@@ -206,6 +207,7 @@ protected theorem max_comm (a b : Int) : max a b = max b a := by
   by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
   · exact Int.le_antisymm h₂ h₁
   · cases not_or_intro h₁ h₂ <| Int.le_total ..
+instance : Std.Commutative (α := Int) max := ⟨Int.max_comm⟩
 
 protected theorem le_max_left (a b : Int) : a ≤ max a b := by rw [Int.max_def]; split <;> simp [*]
 

src/Init/Data/List.lean

@@ -9,3 +9,4 @@ import Init.Data.List.BasicAux
 import Init.Data.List.Control
 import Init.Data.List.Lemmas
 import Init.Data.List.Impl
+import Init.Data.List.TakeDrop

src/Init/Data/List/Basic.lean

@@ -127,6 +127,9 @@ instance : Append (List α) := ⟨List.append⟩
   | nil => rfl
   | cons a as ih =>
     simp_all [HAppend.hAppend, Append.append, List.append]
+instance : Std.LawfulIdentity (α := List α) (· ++ ·) [] where
+  left_id := nil_append
+  right_id := append_nil
 
 @[simp] theorem cons_append (a : α) (as bs : List α) : (a::as) ++ bs = a::(as ++ bs) := rfl
 
@@ -136,6 +139,7 @@ theorem append_assoc (as bs cs : List α) : (as ++ bs) ++ cs = as ++ (bs ++ cs)
   induction as with
   | nil => rfl
   | cons a as ih => simp [ih]
+instance : Std.Associative (α := List α) (· ++ ·) := ⟨append_assoc⟩
 
 theorem append_cons (as : List α) (b : α) (bs : List α) : as ++ b :: bs = as ++ [b] ++ bs := by
   induction as with

src/Init/Data/List/BasicAux.lean

@@ -5,6 +5,7 @@ Author: Leonardo de Moura
 -/
 prelude
 import Init.Data.Nat.Linear
+import Init.Ext
 
 universe u
 
@@ -12,63 +13,157 @@ namespace List
 /-! The following functions can't be defined at `Init.Data.List.Basic`, because they depend on `Init.Util`,
    and `Init.Util` depends on `Init.Data.List.Basic`. -/
 
-def get! [Inhabited α] : List α → Nat → α
+/--
+Returns the `i`-th element in the list (zero-based).
+
+If the index is out of bounds (`i ≥ as.length`), this function panics when executed, and returns
+`default`. See `get?` and `getD` for safer alternatives.
+-/
+def get! [Inhabited α] : (as : List α) → (i : Nat) → α
   | a::_,  0   => a
   | _::as, n+1 => get! as n
   | _,     _   => panic! "invalid index"
 
-def get? : List α → Nat → Option α
+/--
+Returns the `i`-th element in the list (zero-based).
+
+If the index is out of bounds (`i ≥ as.length`), this function returns `none`.
+Also see `get`, `getD` and `get!`.
+-/
+def get? : (as : List α) → (i : Nat) → Option α
   | a::_,  0   => some a
   | _::as, n+1 => get? as n
   | _,     _   => none
 
-def getD (as : List α) (idx : Nat) (a₀ : α) : α :=
-  (as.get? idx).getD a₀
+/--
+Returns the `i`-th element in the list (zero-based).
 
+If the index is out of bounds (`i ≥ as.length`), this function returns `fallback`.
+See also `get?` and `get!`.
+-/
+def getD (as : List α) (i : Nat) (fallback : α) : α :=
+  (as.get? i).getD fallback
+
+@[ext] theorem ext : ∀ {l₁ l₂ : List α}, (∀ n, l₁.get? n = l₂.get? n) → l₁ = l₂
+  | [], [], _ => rfl
+  | a :: l₁, [], h => nomatch h 0
+  | [], a' :: l₂, h => nomatch h 0
+  | a :: l₁, a' :: l₂, h => by
+    have h0 : some a = some a' := h 0
+    injection h0 with aa; simp only [aa, ext fun n => h (n+1)]
+
+/--
+Returns the first element in the list.
+
+If the list is empty, this function panics when executed, and returns `default`.
+See `head` and `headD` for safer alternatives.
+-/
 def head! [Inhabited α] : List α → α
   | []   => panic! "empty list"
   | a::_ => a
 
+/--
+Returns the first element in the list.
+
+If the list is empty, this function returns `none`.
+Also see `headD` and `head!`.
+-/
 def head? : List α → Option α
   | []   => none
   | a::_ => some a
 
-def headD : List α → α → α
-  | [],   a₀ => a₀
+/--
+Returns the first element in the list.
+
+If the list is empty, this function returns `fallback`.
+Also see `head?` and `head!`.
+-/
+def headD : (as : List α) → (fallback : α) → α
+  | [],   fallback => fallback
   | a::_, _  => a
 
+/--
+Returns the first element of a non-empty list.
+-/
 def head : (as : List α) → as ≠ [] → α
   | a::_, _ => a
 
+/--
+Drops the first element of the list.
+
+If the list is empty, this function panics when executed, and returns the empty list.
+See `tail` and `tailD` for safer alternatives.
+-/
 def tail! : List α → List α
   | []    => panic! "empty list"
   | _::as => as
 
+/--
+Drops the first element of the list.
+
+If the list is empty, this function returns `none`.
+Also see `tailD` and `tail!`.
+-/
 def tail? : List α → Option (List α)
   | []    => none
   | _::as => some as
 
-def tailD : List α → List α → List α
-  | [],   as₀ => as₀
-  | _::as, _  => as
+/--
+Drops the first element of the list.
 
+If the list is empty, this function returns `fallback`.
+Also see `head?` and `head!`.
+-/
+def tailD (list fallback : List α) : List α :=
+  match list with
+  | [] => fallback
+  | _ :: tl => tl
+
+/--
+Returns the last element of a non-empty list.
+-/
 def getLast : ∀ (as : List α), as ≠ [] → α
   | [],       h => absurd rfl h
   | [a],      _ => a
   | _::b::as, _ => getLast (b::as) (fun h => List.noConfusion h)
 
+/--
+Returns the last element in the list.
+
+If the list is empty, this function panics when executed, and returns `default`.
+See `getLast` and `getLastD` for safer alternatives.
+-/
 def getLast! [Inhabited α] : List α → α
   | []    => panic! "empty list"
   | a::as => getLast (a::as) (fun h => List.noConfusion h)
 
+/--
+Returns the last element in the list.
+
+If the list is empty, this function returns `none`.
+Also see `getLastD` and `getLast!`.
+-/
 def getLast? : List α → Option α
   | []    => none
   | a::as => some (getLast (a::as) (fun h => List.noConfusion h))
 
-def getLastD : List α → α → α
+/--
+Returns the last element in the list.
+
+If the list is empty, this function returns `fallback`.
+Also see `getLast?` and `getLast!`.
+-/
+def getLastD : (as : List α) → (fallback : α) → α
   | [],   a₀ => a₀
   | a::as, _ => getLast (a::as) (fun h => List.noConfusion h)
 
+/--
+`O(n)`. Rotates the elements of `xs` to the left such that the element at
+`xs[i]` rotates to `xs[(i - n) % l.length]`.
+* `rotateLeft [1, 2, 3, 4, 5] 3 = [4, 5, 1, 2, 3]`
+* `rotateLeft [1, 2, 3, 4, 5] 5 = [1, 2, 3, 4, 5]`
+* `rotateLeft [1, 2, 3, 4, 5] = [2, 3, 4, 5, 1]`
+-/
 def rotateLeft (xs : List α) (n : Nat := 1) : List α :=
   let len := xs.length
   if len ≤ 1 then
@@ -79,6 +174,13 @@ def rotateLeft (xs : List α) (n : Nat := 1) : List α :=
     let e := xs.drop n
     e ++ b
 
+/--
+`O(n)`. Rotates the elements of `xs` to the right such that the element at
+`xs[i]` rotates to `xs[(i + n) % l.length]`.
+* `rotateRight [1, 2, 3, 4, 5] 3 = [3, 4, 5, 1, 2]`
+* `rotateRight [1, 2, 3, 4, 5] 5 = [1, 2, 3, 4, 5]`
+* `rotateRight [1, 2, 3, 4, 5] = [5, 1, 2, 3, 4]`
+-/
 def rotateRight (xs : List α) (n : Nat := 1) : List α :=
   let len := xs.length
   if len ≤ 1 then
@@ -124,9 +226,10 @@ theorem sizeOf_lt_of_mem [SizeOf α] {as : List α} (h : a ∈ as) : sizeOf a <
 over a nested inductive like `inductive T | mk : List T → T`. -/
 macro "sizeOf_list_dec" : tactic =>
   `(tactic| first
-    | apply sizeOf_lt_of_mem; assumption; done
-    | apply Nat.lt_trans (sizeOf_lt_of_mem ?h)
-      case' h => assumption
+    | with_reducible apply sizeOf_lt_of_mem; assumption; done
+    | with_reducible
+        apply Nat.lt_trans (sizeOf_lt_of_mem ?h)
+        case' h => assumption
       simp_arith)
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| sizeOf_list_dec)
@@ -209,6 +312,15 @@ def mapMono (as : List α) (f : α → α) : List α :=
 Monadic generalization of `List.partition`.
 
 This uses `Array.toList` and which isn't imported by `Init.Data.List.Basic`.
+```
+def posOrNeg (x : Int) : Except String Bool :=
+  if x > 0 then pure true
+  else if x < 0 then pure false
+  else throw "Zero is not positive or negative"
+
+partitionM posOrNeg [-1, 2, 3] = Except.ok ([2, 3], [-1])
+partitionM posOrNeg [0, 2, 3] = Except.error "Zero is not positive or negative"
+```
 -/
 @[inline] def partitionM [Monad m] (p : α → m Bool) (l : List α) : m (List α × List α) :=
   go l #[] #[]

src/Init/Data/List/Control.lean

@@ -40,6 +40,13 @@ Finally, we rarely use `mapM` with something that is not a `Monad`.
 Users that want to use `mapM` with `Applicative` should use `mapA` instead.
 -/
 
+/--
+Applies the monadic action `f` on every element in the list, left-to-right, and returns the list of
+results.
+
+See `List.forM` for the variant that discards the results.
+See `List.mapA` for the variant that works with `Applicative`.
+-/
 @[inline]
 def mapM {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f : α → m β) (as : List α) : m (List β) :=
   let rec @[specialize] loop
@@ -47,17 +54,42 @@ def mapM {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f : α
     | a :: as, bs => do loop as ((← f a)::bs)
   loop as []
 
+/--
+Applies the applicative action `f` on every element in the list, left-to-right, and returns the list of
+results.
+
+NB: If `m` is also a `Monad`, then using `mapM` can be more efficient.
+
+See `List.forA` for the variant that discards the results.
+See `List.mapM` for the variant that works with `Monad`.
+
+**Warning**: this function is not tail-recursive, meaning that it may fail with a stack overflow on long lists.
+-/
 @[specialize]
 def mapA {m : Type u → Type v} [Applicative m] {α : Type w} {β : Type u} (f : α → m β) : List α → m (List β)
   | []    => pure []
   | a::as => List.cons <$> f a <*> mapA f as
 
+/--
+Applies the monadic action `f` on every element in the list, left-to-right.
+
+See `List.mapM` for the variant that collects results.
+See `List.forA` for the variant that works with `Applicative`.
+-/
 @[specialize]
 protected def forM {m : Type u → Type v} [Monad m] {α : Type w} (as : List α) (f : α → m PUnit) : m PUnit :=
   match as with
   | []      => pure ⟨⟩
   | a :: as => do f a; List.forM as f
 
+/--
+Applies the applicative action `f` on every element in the list, left-to-right.
+
+NB: If `m` is also a `Monad`, then using `forM` can be more efficient.
+
+See `List.mapA` for the variant that collects results.
+See `List.forM` for the variant that works with `Monad`.
+-/
 @[specialize]
 def forA {m : Type u → Type v} [Applicative m] {α : Type w} (as : List α) (f : α → m PUnit) : m PUnit :=
   match as with
@@ -71,15 +103,27 @@ def filterAuxM {m : Type → Type v} [Monad m] {α : Type} (f : α → m Bool) :
     let b ← f h
     filterAuxM f t (cond b (h :: acc) acc)
 
+/--
+Applies the monadic predicate `p` on every element in the list, left-to-right, and returns those
+elements `x` for which `p x` returns `true`.
+-/
 @[inline]
-def filterM {m : Type → Type v} [Monad m] {α : Type} (f : α → m Bool) (as : List α) : m (List α) := do
-  let as ← filterAuxM f as []
+def filterM {m : Type → Type v} [Monad m] {α : Type} (p : α → m Bool) (as : List α) : m (List α) := do
+  let as ← filterAuxM p as []
   pure as.reverse
 
+/--
+Applies the monadic predicate `p` on every element in the list, right-to-left, and returns those
+elements `x` for which `p x` returns `true`.
+-/
 @[inline]
-def filterRevM {m : Type → Type v} [Monad m] {α : Type} (f : α → m Bool) (as : List α) : m (List α) :=
-  filterAuxM f as.reverse []
+def filterRevM {m : Type → Type v} [Monad m] {α : Type} (p : α → m Bool) (as : List α) : m (List α) :=
+  filterAuxM p as.reverse []
 
+/--
+Applies the monadic function `f` on every element `x` in the list, left-to-right, and returns those
+results `y` for which `f x` returns `some y`.
+-/
 @[inline]
 def filterMapM {m : Type u → Type v} [Monad m] {α β : Type u} (f : α → m (Option β)) (as : List α) : m (List β) :=
   let rec @[specialize] loop
@@ -90,6 +134,16 @@ def filterMapM {m : Type u → Type v} [Monad m] {α β : Type u} (f : α → m
       | some b => loop as (b::bs)
   loop as.reverse []
 
+/--
+Folds a monadic function over a list from left to right:
+```
+foldlM f x₀ [a, b, c] = do
+  let x₁ ← f x₀ a
+  let x₂ ← f x₁ b
+  let x₃ ← f x₂ c
+  pure x₃
+```
+-/
 @[specialize]
 protected def foldlM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w} : (f : s → α → m s) → (init : s) → List α → m s
   | _, s, []      => pure s
@@ -97,10 +151,26 @@ protected def foldlM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w
     let s' ← f s a
     List.foldlM f s' as
 
+/--
+Folds a monadic function over a list from right to left:
+```
+foldrM f x₀ [a, b, c] = do
+  let x₁ ← f c x₀
+  let x₂ ← f b x₁
+  let x₃ ← f a x₂
+  pure x₃
+```
+-/
 @[inline]
 def foldrM {m : Type u → Type v} [Monad m] {s : Type u} {α : Type w} (f : α → s → m s) (init : s) (l : List α) : m s :=
   l.reverse.foldlM (fun s a => f a s) init
 
+/--
+Maps `f` over the list and collects the results with `<|>`.
+```
+firstM f [a, b, c] = f a <|> f b <|> f c <|> failure
+```
+-/
 @[specialize]
 def firstM {m : Type u → Type v} [Alternative m] {α : Type w} {β : Type u} (f : α → m β) : List α → m β
   | []    => failure

src/Init/Data/List/TakeDrop.lean

@@ -0,0 +1,360 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Lemmas
+import Init.Data.Nat.Lemmas
+
+/-!
+# Lemmas about `List.take`, `List.drop`, `List.zip` and `List.zipWith`.
+
+These are in a separate file from most of the list lemmas
+as they required importing more lemmas about natural numbers.
+-/
+
+namespace List
+
+open Nat
+
+/-! ### take -/
+
+abbrev take_succ_cons := @take_cons_succ
+
+@[simp] theorem length_take : ∀ (i : Nat) (l : List α), length (take i l) = min i (length l)
+  | 0, l => by simp [Nat.zero_min]
+  | succ n, [] => by simp [Nat.min_zero]
+  | succ n, _ :: l => by simp [Nat.succ_min_succ, length_take]
+
+theorem length_take_le (n) (l : List α) : length (take n l) ≤ n := by simp [Nat.min_le_left]
+
+theorem length_take_le' (n) (l : List α) : length (take n l) ≤ l.length :=
+  by simp [Nat.min_le_right]
+
+theorem length_take_of_le (h : n ≤ length l) : length (take n l) = n := by simp [Nat.min_eq_left h]
+
+theorem take_all_of_le {n} {l : List α} (h : length l ≤ n) : take n l = l :=
+  take_length_le h
+
+@[simp]
+theorem take_left : ∀ l₁ l₂ : List α, take (length l₁) (l₁ ++ l₂) = l₁
+  | [], _ => rfl
+  | a :: l₁, l₂ => congrArg (cons a) (take_left l₁ l₂)
+
+theorem take_left' {l₁ l₂ : List α} {n} (h : length l₁ = n) : take n (l₁ ++ l₂) = l₁ := by
+  rw [← h]; apply take_left
+
+theorem take_take : ∀ (n m) (l : List α), take n (take m l) = take (min n m) l
+  | n, 0, l => by rw [Nat.min_zero, take_zero, take_nil]
+  | 0, m, l => by rw [Nat.zero_min, take_zero, take_zero]
+  | succ n, succ m, nil => by simp only [take_nil]
+  | succ n, succ m, a :: l => by
+    simp only [take, succ_min_succ, take_take n m l]
+
+theorem take_replicate (a : α) : ∀ n m : Nat, take n (replicate m a) = replicate (min n m) a
+  | n, 0 => by simp [Nat.min_zero]
+  | 0, m => by simp [Nat.zero_min]
+  | succ n, succ m => by simp [succ_min_succ, take_replicate]
+
+theorem map_take (f : α → β) :
+    ∀ (L : List α) (i : Nat), (L.take i).map f = (L.map f).take i
+  | [], i => by simp
+  | _, 0 => by simp
+  | h :: t, n + 1 => by dsimp; rw [map_take f t n]
+
+/-- Taking the first `n` elements in `l₁ ++ l₂` is the same as appending the first `n` elements
+of `l₁` to the first `n - l₁.length` elements of `l₂`. -/
+theorem take_append_eq_append_take {l₁ l₂ : List α} {n : Nat} :
+    take n (l₁ ++ l₂) = take n l₁ ++ take (n - l₁.length) l₂ := by
+  induction l₁ generalizing n
+  · simp
+  · cases n
+    · simp [*]
+    · simp only [cons_append, take_cons_succ, length_cons, succ_eq_add_one, cons.injEq,
+        append_cancel_left_eq, true_and, *]
+      congr 1
+      omega
+
+theorem take_append_of_le_length {l₁ l₂ : List α} {n : Nat} (h : n ≤ l₁.length) :
+    (l₁ ++ l₂).take n = l₁.take n := by
+  simp [take_append_eq_append_take, Nat.sub_eq_zero_of_le h]
+
+/-- Taking the first `l₁.length + i` elements in `l₁ ++ l₂` is the same as appending the first
+`i` elements of `l₂` to `l₁`. -/
+theorem take_append {l₁ l₂ : List α} (i : Nat) :
+    take (l₁.length + i) (l₁ ++ l₂) = l₁ ++ take i l₂ := by
+  rw [take_append_eq_append_take, take_all_of_le (Nat.le_add_right _ _), Nat.add_sub_cancel_left]
+
+/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
+length `> i`. Version designed to rewrite from the big list to the small list. -/
+theorem get_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
+    get L ⟨i, hi⟩ = get (L.take j) ⟨i, length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩⟩ :=
+  get_of_eq (take_append_drop j L).symm _ ▸ get_append ..
+
+/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
+length `> i`. Version designed to rewrite from the small list to the big list. -/
+theorem get_take' (L : List α) {j i} :
+    get (L.take j) i =
+    get L ⟨i.1, Nat.lt_of_lt_of_le i.2 (length_take_le' _ _)⟩ := by
+  let ⟨i, hi⟩ := i; rw [length_take, Nat.lt_min] at hi; rw [get_take L _ hi.1]
+
+theorem get?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n).get? m = l.get? m := by
+  induction n generalizing l m with
+  | zero =>
+    exact absurd h (Nat.not_lt_of_le m.zero_le)
+  | succ _ hn =>
+    cases l with
+    | nil => simp only [take_nil]
+    | cons hd tl =>
+      cases m
+      · simp only [get?, take]
+      · simpa only using hn (Nat.lt_of_succ_lt_succ h)
+
+theorem get?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
+    (l.take n).get? m = none :=
+  get?_eq_none.mpr <| Nat.le_trans (length_take_le _ _) h
+
+theorem get?_take_eq_if {l : List α} {n m : Nat} :
+    (l.take n).get? m = if m < n then l.get? m else none := by
+  split
+  · next h => exact get?_take h
+  · next h => exact get?_take_eq_none (Nat.le_of_not_lt h)
+
+@[simp]
+theorem nth_take_of_succ {l : List α} {n : Nat} : (l.take (n + 1)).get? n = l.get? n :=
+  get?_take (Nat.lt_succ_self n)
+
+theorem take_succ {l : List α} {n : Nat} : l.take (n + 1) = l.take n ++ (l.get? n).toList := by
+  induction l generalizing n with
+  | nil =>
+    simp only [Option.toList, get?, take_nil, append_nil]
+  | cons hd tl hl =>
+    cases n
+    · simp only [Option.toList, get?, eq_self_iff_true, take, nil_append]
+    · simp only [hl, cons_append, get?, eq_self_iff_true, take]
+
+@[simp]
+theorem take_eq_nil_iff {l : List α} {k : Nat} : l.take k = [] ↔ l = [] ∨ k = 0 := by
+  cases l <;> cases k <;> simp [Nat.succ_ne_zero]
+
+@[simp]
+theorem take_eq_take :
+    ∀ {l : List α} {m n : Nat}, l.take m = l.take n ↔ min m l.length = min n l.length
+  | [], m, n => by simp [Nat.min_zero]
+  | _ :: xs, 0, 0 => by simp
+  | x :: xs, m + 1, 0 => by simp [Nat.zero_min, succ_min_succ]
+  | x :: xs, 0, n + 1 => by simp [Nat.zero_min, succ_min_succ]
+  | x :: xs, m + 1, n + 1 => by simp [succ_min_succ, take_eq_take]; omega
+
+theorem take_add (l : List α) (m n : Nat) : l.take (m + n) = l.take m ++ (l.drop m).take n := by
+  suffices take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l) by
+    rw [take_append_drop] at this
+    assumption
+  rw [take_append_eq_append_take, take_all_of_le, append_right_inj]
+  · simp only [take_eq_take, length_take, length_drop]
+    omega
+  apply Nat.le_trans (m := m)
+  · apply length_take_le
+  · apply Nat.le_add_right
+
+theorem take_eq_nil_of_eq_nil : ∀ {as : List α} {i}, as = [] → as.take i = []
+  | _, _, rfl => take_nil
+
+theorem ne_nil_of_take_ne_nil {as : List α} {i : Nat} (h: as.take i ≠ []) : as ≠ [] :=
+  mt take_eq_nil_of_eq_nil h
+
+theorem dropLast_eq_take (l : List α) : l.dropLast = l.take l.length.pred := by
+  cases l with
+  | nil => simp [dropLast]
+  | cons x l =>
+    induction l generalizing x with
+    | nil => simp [dropLast]
+    | cons hd tl hl => simp [dropLast, hl]
+
+theorem dropLast_take {n : Nat} {l : List α} (h : n < l.length) :
+    (l.take n).dropLast = l.take n.pred := by
+  simp only [dropLast_eq_take, length_take, Nat.le_of_lt h, take_take, pred_le, Nat.min_eq_left]
+
+theorem map_eq_append_split {f : α → β} {l : List α} {s₁ s₂ : List β}
+    (h : map f l = s₁ ++ s₂) : ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = s₁ ∧ map f l₂ = s₂ := by
+  have := h
+  rw [← take_append_drop (length s₁) l] at this ⊢
+  rw [map_append] at this
+  refine ⟨_, _, rfl, append_inj this ?_⟩
+  rw [length_map, length_take, Nat.min_eq_left]
+  rw [← length_map l f, h, length_append]
+  apply Nat.le_add_right
+
+/-! ### drop -/
+
+@[simp]
+theorem drop_eq_nil_iff_le {l : List α} {k : Nat} : l.drop k = [] ↔ l.length ≤ k := by
+  refine' ⟨fun h => _, drop_eq_nil_of_le⟩
+  induction k generalizing l with
+  | zero =>
+    simp only [drop] at h
+    simp [h]
+  | succ k hk =>
+    cases l
+    · simp
+    · simp only [drop] at h
+      simpa [Nat.succ_le_succ_iff] using hk h
+
+theorem drop_length_cons {l : List α} (h : l ≠ []) (a : α) :
+    (a :: l).drop l.length = [l.getLast h] := by
+  induction l generalizing a with
+  | nil =>
+    cases h rfl
+  | cons y l ih =>
+    simp only [drop, length]
+    by_cases h₁ : l = []
+    · simp [h₁]
+    rw [getLast_cons' _ h₁]
+    exact ih h₁ y
+
+/-- Dropping the elements up to `n` in `l₁ ++ l₂` is the same as dropping the elements up to `n`
+in `l₁`, dropping the elements up to `n - l₁.length` in `l₂`, and appending them. -/
+theorem drop_append_eq_append_drop {l₁ l₂ : List α} {n : Nat} :
+    drop n (l₁ ++ l₂) = drop n l₁ ++ drop (n - l₁.length) l₂ := by
+  induction l₁ generalizing n
+  · simp
+  · cases n
+    · simp [*]
+    · simp only [cons_append, drop_succ_cons, length_cons, succ_eq_add_one, append_cancel_left_eq, *]
+      congr 1
+      omega
+
+theorem drop_append_of_le_length {l₁ l₂ : List α} {n : Nat} (h : n ≤ l₁.length) :
+    (l₁ ++ l₂).drop n = l₁.drop n ++ l₂ := by
+  simp [drop_append_eq_append_drop, Nat.sub_eq_zero_of_le h]
+
+
+/-- Dropping the elements up to `l₁.length + i` in `l₁ + l₂` is the same as dropping the elements
+up to `i` in `l₂`. -/
+@[simp]
+theorem drop_append {l₁ l₂ : List α} (i : Nat) : drop (l₁.length + i) (l₁ ++ l₂) = drop i l₂ := by
+  rw [drop_append_eq_append_drop, drop_eq_nil_of_le] <;>
+    simp [Nat.add_sub_cancel_left, Nat.le_add_right]
+
+theorem drop_sizeOf_le [SizeOf α] (l : List α) (n : Nat) : sizeOf (l.drop n) ≤ sizeOf l := by
+  induction l generalizing n with
+  | nil => rw [drop_nil]; apply Nat.le_refl
+  | cons _ _ lih =>
+    induction n with
+    | zero => apply Nat.le_refl
+    | succ n =>
+      exact Trans.trans (lih _) (Nat.le_add_left _ _)
+
+theorem lt_length_drop (L : List α) {i j : Nat} (h : i + j < L.length) : j < (L.drop i).length := by
+  have A : i < L.length := Nat.lt_of_le_of_lt (Nat.le.intro rfl) h
+  rw [(take_append_drop i L).symm] at h
+  simpa only [Nat.le_of_lt A, Nat.min_eq_left, Nat.add_lt_add_iff_left, length_take,
+    length_append] using h
+
+/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
+dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
+theorem get_drop (L : List α) {i j : Nat} (h : i + j < L.length) :
+    get L ⟨i + j, h⟩ = get (L.drop i) ⟨j, lt_length_drop L h⟩ := by
+  have : i ≤ L.length := Nat.le_trans (Nat.le_add_right _ _) (Nat.le_of_lt h)
+  rw [get_of_eq (take_append_drop i L).symm ⟨i + j, h⟩, get_append_right'] <;>
+    simp [Nat.min_eq_left this, Nat.add_sub_cancel_left, Nat.le_add_right]
+
+/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
+dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
+theorem get_drop' (L : List α) {i j} :
+    get (L.drop i) j = get L ⟨i + j, by
+      rw [Nat.add_comm]
+      exact Nat.add_lt_of_lt_sub (length_drop i L ▸ j.2)⟩ := by
+  rw [get_drop]
+
+@[simp]
+theorem get?_drop (L : List α) (i j : Nat) : get? (L.drop i) j = get? L (i + j) := by
+  ext
+  simp only [get?_eq_some, get_drop', Option.mem_def]
+  constructor <;> intro ⟨h, ha⟩
+  · exact ⟨_, ha⟩
+  · refine ⟨?_, ha⟩
+    rw [length_drop]
+    rw [Nat.add_comm] at h
+    apply Nat.lt_sub_of_add_lt h
+
+@[simp] theorem drop_drop (n : Nat) : ∀ (m) (l : List α), drop n (drop m l) = drop (n + m) l
+  | m, [] => by simp
+  | 0, l => by simp
+  | m + 1, a :: l =>
+    calc
+      drop n (drop (m + 1) (a :: l)) = drop n (drop m l) := rfl
+      _ = drop (n + m) l := drop_drop n m l
+      _ = drop (n + (m + 1)) (a :: l) := rfl
+
+theorem take_drop : ∀ (m n : Nat) (l : List α), take n (drop m l) = drop m (take (m + n) l)
+  | 0, _, _ => by simp
+  | _, _, [] => by simp
+  | _+1, _, _ :: _ => by simpa [Nat.succ_add, take_succ_cons, drop_succ_cons] using take_drop ..
+
+theorem drop_take : ∀ (m n : Nat) (l : List α), drop n (take m l) = take (m - n) (drop n l)
+  | 0, _, _ => by simp
+  | _, 0, _ => by simp
+  | _, _, [] => by simp
+  | m+1, n+1, h :: t => by
+    simp [take_succ_cons, drop_succ_cons, drop_take m n t]
+    congr 1
+    omega
+
+theorem map_drop (f : α → β) :
+    ∀ (L : List α) (i : Nat), (L.drop i).map f = (L.map f).drop i
+  | [], i => by simp
+  | L, 0 => by simp
+  | h :: t, n + 1 => by
+    dsimp
+    rw [map_drop f t]
+
+theorem reverse_take {α} {xs : List α} (n : Nat) (h : n ≤ xs.length) :
+    xs.reverse.take n = (xs.drop (xs.length - n)).reverse := by
+  induction xs generalizing n <;>
+    simp only [reverse_cons, drop, reverse_nil, Nat.zero_sub, length, take_nil]
+  next xs_hd xs_tl xs_ih =>
+  cases Nat.lt_or_eq_of_le h with
+  | inl h' =>
+    have h' := Nat.le_of_succ_le_succ h'
+    rw [take_append_of_le_length, xs_ih _ h']
+    rw [show xs_tl.length + 1 - n = succ (xs_tl.length - n) from _, drop]
+    · rwa [succ_eq_add_one, Nat.sub_add_comm]
+    · rwa [length_reverse]
+  | inr h' =>
+    subst h'
+    rw [length, Nat.sub_self, drop]
+    suffices xs_tl.length + 1 = (xs_tl.reverse ++ [xs_hd]).length by
+      rw [this, take_length, reverse_cons]
+    rw [length_append, length_reverse]
+    rfl
+
+@[simp]
+theorem get_cons_drop : ∀ (l : List α) i, get l i :: drop (i + 1) l = drop i l
+  | _::_, ⟨0, _⟩ => rfl
+  | _::_, ⟨i+1, _⟩ => get_cons_drop _ ⟨i, _⟩
+
+theorem drop_eq_get_cons {n} {l : List α} (h) : drop n l = get l ⟨n, h⟩ :: drop (n + 1) l :=
+  (get_cons_drop _ ⟨n, h⟩).symm
+
+theorem drop_eq_nil_of_eq_nil : ∀ {as : List α} {i}, as = [] → as.drop i = []
+  | _, _, rfl => drop_nil
+
+theorem ne_nil_of_drop_ne_nil {as : List α} {i : Nat} (h: as.drop i ≠ []) : as ≠ [] :=
+  mt drop_eq_nil_of_eq_nil h
+
+/-! ### zipWith -/
+
+@[simp] theorem length_zipWith (f : α → β → γ) (l₁ l₂) :
+    length (zipWith f l₁ l₂) = min (length l₁) (length l₂) := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;>
+    simp_all [succ_min_succ, Nat.zero_min, Nat.min_zero]
+
+/-! ### zip -/
+
+@[simp] theorem length_zip (l₁ : List α) (l₂ : List β) :
+    length (zip l₁ l₂) = min (length l₁) (length l₂) := by
+  simp [zip]
+
+end List

src/Init/Data/Nat.lean

@@ -19,3 +19,4 @@ import Init.Data.Nat.Lemmas
 import Init.Data.Nat.Mod
 import Init.Data.Nat.Lcm
 import Init.Data.Nat.Compare
+import Init.Data.Nat.Simproc

src/Init/Data/Nat/Basic.lean

@@ -137,6 +137,9 @@ instance : LawfulBEq Nat where
 @[simp] protected theorem zero_add : ∀ (n : Nat), 0 + n = n
   | 0   => rfl
   | n+1 => congrArg succ (Nat.zero_add n)
+instance : Std.LawfulIdentity (α := Nat) (· + ·) 0 where
+  left_id := Nat.zero_add
+  right_id := Nat.add_zero
 
 theorem succ_add : ∀ (n m : Nat), (succ n) + m = succ (n + m)
   | _, 0   => rfl
@@ -160,10 +163,12 @@ protected theorem add_comm : ∀ (n m : Nat), n + m = m + n
     have : succ (n + m) = succ (m + n) := by apply congrArg; apply Nat.add_comm
     rw [succ_add m n]
     apply this
+instance : Std.Commutative (α := Nat) (· + ·) := ⟨Nat.add_comm⟩
 
 protected theorem add_assoc : ∀ (n m k : Nat), (n + m) + k = n + (m + k)
   | _, _, 0      => rfl
   | n, m, succ k => congrArg succ (Nat.add_assoc n m k)
+instance : Std.Associative (α := Nat) (· + ·) := ⟨Nat.add_assoc⟩
 
 protected theorem add_left_comm (n m k : Nat) : n + (m + k) = m + (n + k) := by
   rw [← Nat.add_assoc, Nat.add_comm n m, Nat.add_assoc]
@@ -174,7 +179,7 @@ protected theorem add_right_comm (n m k : Nat) : (n + m) + k = (n + k) + m := by
 protected theorem add_left_cancel {n m k : Nat} : n + m = n + k → m = k := by
   induction n with
   | zero => simp
-  | succ n ih => simp [succ_add]; intro h; apply ih h
+  | succ n ih => simp [succ_add, succ.injEq]; intro h; apply ih h
 
 protected theorem add_right_cancel {n m k : Nat} (h : n + m = k + m) : n = k := by
   rw [Nat.add_comm n m, Nat.add_comm k m] at h
@@ -207,12 +212,16 @@ theorem succ_mul (n m : Nat) : (succ n) * m = (n * m) + m := by
 protected theorem mul_comm : ∀ (n m : Nat), n * m = m * n
   | n, 0      => (Nat.zero_mul n).symm ▸ (Nat.mul_zero n).symm ▸ rfl
   | n, succ m => (mul_succ n m).symm ▸ (succ_mul m n).symm ▸ (Nat.mul_comm n m).symm ▸ rfl
+instance : Std.Commutative (α := Nat) (· * ·) := ⟨Nat.mul_comm⟩
 
 @[simp] protected theorem mul_one : ∀ (n : Nat), n * 1 = n :=
   Nat.zero_add
 
 @[simp] protected theorem one_mul (n : Nat) : 1 * n = n :=
   Nat.mul_comm n 1 ▸ Nat.mul_one n
+instance : Std.LawfulIdentity (α := Nat) (· * ·) 1 where
+  left_id := Nat.one_mul
+  right_id := Nat.mul_one
 
 protected theorem left_distrib (n m k : Nat) : n * (m + k) = n * m + n * k := by
   induction n with
@@ -231,6 +240,7 @@ protected theorem add_mul (n m k : Nat) : (n + m) * k = n * k + m * k :=
 protected theorem mul_assoc : ∀ (n m k : Nat), (n * m) * k = n * (m * k)
   | n, m, 0      => rfl
   | n, m, succ k => by simp [mul_succ, Nat.mul_assoc n m k, Nat.left_distrib]
+instance : Std.Associative (α := Nat) (· * ·) := ⟨Nat.mul_assoc⟩
 
 protected theorem mul_left_comm (n m k : Nat) : n * (m * k) = m * (n * k) := by
   rw [← Nat.mul_assoc, Nat.mul_comm n m, Nat.mul_assoc]
@@ -248,7 +258,7 @@ theorem lt_succ_of_le {n m : Nat} : n ≤ m → n < succ m := succ_le_succ
 
 @[simp] protected theorem sub_zero (n : Nat) : n - 0 = n := rfl
 
-@[simp] theorem succ_sub_succ_eq_sub (n m : Nat) : succ n - succ m = n - m := by
+theorem succ_sub_succ_eq_sub (n m : Nat) : succ n - succ m = n - m := by
   induction m with
   | zero      => exact rfl
   | succ m ih => apply congrArg pred ih
@@ -574,7 +584,7 @@ theorem eq_zero_or_eq_succ_pred : ∀ n, n = 0 ∨ n = succ (pred n)
   | 0 => .inl rfl
   | _+1 => .inr rfl
 
-theorem succ_inj' : succ a = succ b ↔ a = b := ⟨succ.inj, congrArg _⟩
+theorem succ_inj' : succ a = succ b ↔ a = b := (Nat.succ.injEq a b).to_iff
 
 theorem succ_le_succ_iff : succ a ≤ succ b ↔ a ≤ b := ⟨le_of_succ_le_succ, succ_le_succ⟩
 
@@ -802,7 +812,7 @@ theorem add_sub_of_le {a b : Nat} (h : a ≤ b) : a + (b - a) = b := by
 protected theorem add_sub_add_right (n k m : Nat) : (n + k) - (m + k) = n - m := by
   induction k with
   | zero => simp
-  | succ k ih => simp [← Nat.add_assoc, ih]
+  | succ k ih => simp [← Nat.add_assoc, succ_sub_succ_eq_sub, ih]
 
 protected theorem add_sub_add_left (k n m : Nat) : (k + n) - (k + m) = n - m := by
   rw [Nat.add_comm k n, Nat.add_comm k m, Nat.add_sub_add_right]

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

@@ -9,6 +9,7 @@ import Init.Data.Bool
 import Init.Data.Int.Pow
 import Init.Data.Nat.Bitwise.Basic
 import Init.Data.Nat.Lemmas
+import Init.Data.Nat.Simproc
 import Init.TacticsExtra
 import Init.Omega
 
@@ -49,7 +50,10 @@ noncomputable def div2Induction {motive : Nat → Sort u}
       apply hyp
       exact Nat.div_lt_self n_pos (Nat.le_refl _)
 
-@[simp] theorem zero_and (x : Nat) : 0 &&& x = 0 := by rfl
+@[simp] theorem zero_and (x : Nat) : 0 &&& x = 0 := by
+  simp only [HAnd.hAnd, AndOp.and, land]
+  unfold bitwise
+  simp
 
 @[simp] theorem and_zero (x : Nat) : x &&& 0 = 0 := by
   simp only [HAnd.hAnd, AndOp.and, land]
@@ -187,8 +191,6 @@ theorem lt_pow_two_of_testBit (x : Nat) (p : ∀i, i ≥ n → testBit x i = fal
   have test_false := p _ i_ge_n
   simp only [test_true] at test_false
 
-/-! ### testBit -/
-
 private theorem succ_mod_two : succ x % 2 = 1 - x % 2 := by
   induction x with
   | zero =>
@@ -232,7 +234,7 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
     rw [Nat.sub_eq_zero_iff_le] at i_sub_j_eq
     exact Nat.not_le_of_gt j_lt_i i_sub_j_eq
   | d+1 =>
-    simp [Nat.pow_succ, Nat.mul_comm _ 2,  Nat.mul_add_mod]
+    simp [Nat.pow_succ, Nat.mul_comm _ 2, Nat.mul_add_mod]
 
 @[simp] theorem testBit_mod_two_pow (x j i : Nat) :
     testBit (x % 2^j) i = (decide (i < j) && testBit x i) := by
@@ -256,7 +258,7 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
             exact Nat.lt_add_of_pos_right (Nat.two_pow_pos j)
       simp only [hyp y y_lt_x]
       if i_lt_j : i < j then
-        rw [ Nat.add_comm _ (2^_), testBit_two_pow_add_gt i_lt_j]
+        rw [Nat.add_comm _ (2^_), testBit_two_pow_add_gt i_lt_j]
       else
         simp [i_lt_j]
 
@@ -271,7 +273,7 @@ theorem testBit_two_pow_sub_succ (h₂ : x < 2 ^ n) (i : Nat) :
   induction i generalizing n x with
   | zero =>
     match n with
-    | 0 => simp
+    | 0 => simp [succ_sub_succ_eq_sub]
     | n+1 =>
       simp [not_decide_mod_two_eq_one]
       omega
@@ -279,7 +281,7 @@ theorem testBit_two_pow_sub_succ (h₂ : x < 2 ^ n) (i : Nat) :
     simp only [testBit_succ]
     match n with
     | 0 =>
-      simp [decide_eq_false]
+      simp [decide_eq_false, succ_sub_succ_eq_sub]
     | n+1 =>
       rw [Nat.two_pow_succ_sub_succ_div_two, ih]
       · simp [Nat.succ_lt_succ_iff]
@@ -401,12 +403,12 @@ theorem and_pow_two_identity {x : Nat} (lt : x < 2^n) : x &&& 2^n-1 = x := by
 
 /-! ### lor -/
 
-@[simp] theorem or_zero (x : Nat) : 0 ||| x = x := by
+@[simp] theorem zero_or (x : Nat) : 0 ||| x = x := by
   simp only [HOr.hOr, OrOp.or, lor]
   unfold bitwise
   simp [@eq_comm _ 0]
 
-@[simp] theorem zero_or (x : Nat) : x ||| 0 = x := by
+@[simp] theorem or_zero (x : Nat) : x ||| 0 = x := by
   simp only [HOr.hOr, OrOp.or, lor]
   unfold bitwise
   simp [@eq_comm _ 0]

src/Init/Data/Nat/Div.lean

@@ -28,7 +28,7 @@ protected def div (x y : @& Nat) : Nat :=
     0
 decreasing_by apply div_rec_lemma; assumption
 
-instance : Div Nat := ⟨Nat.div⟩
+instance instDiv : Div Nat := ⟨Nat.div⟩
 
 theorem div_eq (x y : Nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 else 0 := by
   show Nat.div x y = _
@@ -82,22 +82,34 @@ decreasing_by apply div_rec_lemma; assumption
 
 @[extern "lean_nat_mod"]
 protected def mod : @& Nat → @& Nat → Nat
-  /- This case is not needed mathematically as the case below is equal to it; however, it makes
-  `0 % n = 0` true definitionally rather than just propositionally.
-  This property is desirable for `Fin n`, as it means `(ofNat 0 : Fin n).val = 0` by definition.
-  Primarily, this is valuable because mathlib in Lean3 assumed this was true definitionally, and so
-  keeping this definitional equality makes mathlib easier to port to mathlib4. -/
+  /-
+  Nat.modCore is defined by well-founded recursion and thus irreducible. Nevertheless it is
+  desireable if trivial `Nat.mod` calculations, namely
+  * `Nat.mod 0 m` for all `m`
+  * `Nat.mod n (m+n)` for concrete literals `n`
+  reduce definitionally.
+  This property is desirable for `Fin n` literals, as it means `(ofNat 0 : Fin n).val = 0` by
+  definition.
+   -/
   | 0, _ => 0
-  | x@(_ + 1), y => Nat.modCore x y
+  | n@(_ + 1), m =>
+    if m ≤ n -- NB: if n < m does not reduce as well as `m ≤ n`!
+    then Nat.modCore n m
+    else n
 
-instance : Mod Nat := ⟨Nat.mod⟩
+instance instMod : Mod Nat := ⟨Nat.mod⟩
 
-protected theorem modCore_eq_mod (x y : Nat) : Nat.modCore x y = x % y := by
-  cases x with
-  | zero =>
+protected theorem modCore_eq_mod (n m : Nat) : Nat.modCore n m = n % m := by
+  show Nat.modCore n m = Nat.mod n m
+  match n, m with
+  | 0, _ =>
     rw [Nat.modCore]
     exact if_neg fun ⟨hlt, hle⟩ => Nat.lt_irrefl _ (Nat.lt_of_lt_of_le hlt hle)
-  | succ x => rfl
+  | (_ + 1), _ =>
+    rw [Nat.mod]; dsimp
+    refine iteInduction (fun _ => rfl) (fun h => ?false) -- cannot use `split` this early yet
+    rw [Nat.modCore]
+    exact if_neg fun ⟨_hlt, hle⟩ => h hle
 
 theorem mod_eq (x y : Nat) : x % y = if 0 < y ∧ y ≤ x then (x - y) % y else x := by
   rw [←Nat.modCore_eq_mod, ←Nat.modCore_eq_mod, Nat.modCore]

src/Init/Data/Nat/Gcd.lean

@@ -10,6 +10,24 @@ import Init.RCases
 
 namespace Nat
 
+/--
+Computes the greatest common divisor of two natural numbers.
+
+This reference implementation via the Euclidean algorithm
+is overridden in both the kernel and the compiler to efficiently
+evaluate using the "bignum" representation (see `Nat`).
+The definition provided here is the logical model
+(and it is soundness-critical that they coincide).
+
+The GCD of two natural numbers is the largest natural number
+that divides both arguments.
+In particular, the GCD of a number and `0` is the number itself:
+```
+example : Nat.gcd 10 15 = 5 := rfl
+example : Nat.gcd 0 5 = 5 := rfl
+example : Nat.gcd 7 0 = 7 := rfl
+```
+-/
 @[extern "lean_nat_gcd"]
 def gcd (m n : @& Nat) : Nat :=
   if m = 0 then
@@ -19,11 +37,11 @@ def gcd (m n : @& Nat) : Nat :=
   termination_by m
   decreasing_by simp_wf; apply mod_lt _ (zero_lt_of_ne_zero _); assumption
 
-@[simp] theorem gcd_zero_left (y : Nat) : gcd 0 y = y :=
-  rfl
+@[simp] theorem gcd_zero_left (y : Nat) : gcd 0 y = y := by
+  rw [gcd]; rfl
 
-theorem gcd_succ (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x) :=
-  rfl
+theorem gcd_succ (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x) := by
+  rw [gcd]; rfl
 
 @[simp] theorem gcd_one_left (n : Nat) : gcd 1 n = 1 := by
   rw [gcd_succ, mod_one]
@@ -36,13 +54,17 @@ theorem gcd_succ (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x) :=
     -- `simp [gcd_succ]` produces an invalid term unless `gcd_succ` is proved with `id rfl` instead
     rw [gcd_succ]
     exact gcd_zero_left _
+instance : Std.LawfulIdentity gcd 0 where
+  left_id := gcd_zero_left
+  right_id := gcd_zero_right
 
 @[simp] theorem gcd_self (n : Nat) : gcd n n = n := by
   cases n <;> simp [gcd_succ]
+instance : Std.IdempotentOp gcd := ⟨gcd_self⟩
 
 theorem gcd_rec (m n : Nat) : gcd m n = gcd (n % m) m :=
   match m with
-  | 0 => by have := (mod_zero n).symm; rwa [gcd_zero_right]
+  | 0 => by have := (mod_zero n).symm; rwa [gcd, gcd_zero_right]
   | _ + 1 => by simp [gcd_succ]
 
 @[elab_as_elim] theorem gcd.induction {P : Nat → Nat → Prop} (m n : Nat)
@@ -79,6 +101,7 @@ theorem gcd_comm (m n : Nat) : gcd m n = gcd n m :=
   Nat.dvd_antisymm
     (dvd_gcd (gcd_dvd_right m n) (gcd_dvd_left m n))
     (dvd_gcd (gcd_dvd_right n m) (gcd_dvd_left n m))
+instance : Std.Commutative gcd := ⟨gcd_comm⟩
 
 theorem gcd_eq_left_iff_dvd : m ∣ n ↔ gcd m n = m :=
   ⟨fun h => by rw [gcd_rec, mod_eq_zero_of_dvd h, gcd_zero_left],

src/Init/Data/Nat/Lcm.lean

@@ -14,6 +14,7 @@ def lcm (m n : Nat) : Nat := m * n / gcd m n
 
 theorem lcm_comm (m n : Nat) : lcm m n = lcm n m := by
   rw [lcm, lcm, Nat.mul_comm n m, gcd_comm n m]
+instance : Std.Commutative lcm := ⟨lcm_comm⟩
 
 @[simp] theorem lcm_zero_left (m : Nat) : lcm 0 m = 0 := by simp [lcm]
 
@@ -22,11 +23,15 @@ theorem lcm_comm (m n : Nat) : lcm m n = lcm n m := by
 @[simp] theorem lcm_one_left (m : Nat) : lcm 1 m = m := by simp [lcm]
 
 @[simp] theorem lcm_one_right (m : Nat) : lcm m 1 = m := by simp [lcm]
+instance : Std.LawfulIdentity lcm 1 where
+  left_id := lcm_one_left
+  right_id := lcm_one_right
 
 @[simp] theorem lcm_self (m : Nat) : lcm m m = m := by
   match eq_zero_or_pos m with
   | .inl h => rw [h, lcm_zero_left]
   | .inr h => simp [lcm, Nat.mul_div_cancel _ h]
+instance : Std.IdempotentOp lcm := ⟨lcm_self⟩
 
 theorem dvd_lcm_left (m n : Nat) : m ∣ lcm m n :=
   ⟨n / gcd m n, by rw [← Nat.mul_div_assoc m (Nat.gcd_dvd_right m n)]; rfl⟩
@@ -54,6 +59,7 @@ Nat.dvd_antisymm
     (Nat.dvd_trans (dvd_lcm_left m n) (dvd_lcm_left (lcm m n) k))
     (lcm_dvd (Nat.dvd_trans (dvd_lcm_right m n) (dvd_lcm_left (lcm m n) k))
       (dvd_lcm_right (lcm m n) k)))
+instance : Std.Associative lcm := ⟨lcm_assoc⟩
 
 theorem lcm_ne_zero (hm : m ≠ 0) (hn : n ≠ 0) : lcm m n ≠ 0 := by
   intro h

src/Init/Data/Nat/Lemmas.lean

@@ -88,7 +88,7 @@ protected theorem add_pos_right (m) (h : 0 < n) : 0 < m + n :=
   Nat.lt_of_lt_of_le h (Nat.le_add_left ..)
 
 protected theorem add_self_ne_one : ∀ n, n + n ≠ 1
-  | n+1, h => by rw [Nat.succ_add, Nat.succ_inj'] at h; contradiction
+  | n+1, h => by rw [Nat.succ_add, Nat.succ.injEq] at h; contradiction
 
 /-! ## sub -/
 
@@ -137,14 +137,14 @@ protected theorem sub_le_iff_le_add' {a b c : Nat} : a - b ≤ c ↔ a ≤ b + c
 protected theorem le_sub_iff_add_le {n : Nat} (h : k ≤ m) : n ≤ m - k ↔ n + k ≤ m :=
   ⟨Nat.add_le_of_le_sub h, Nat.le_sub_of_add_le⟩
 
-@[deprecated Nat.le_sub_iff_add_le]
+@[deprecated Nat.le_sub_iff_add_le (since := "2024-02-19")]
 protected theorem add_le_to_le_sub (n : Nat) (h : m ≤ k) : n + m ≤ k ↔ n ≤ k - m :=
   (Nat.le_sub_iff_add_le h).symm
 
 protected theorem add_le_of_le_sub' {n k m : Nat} (h : m ≤ k) : n ≤ k - m → m + n ≤ k :=
   Nat.add_comm .. ▸ Nat.add_le_of_le_sub h
 
-@[deprecated Nat.add_le_of_le_sub']
+@[deprecated Nat.add_le_of_le_sub' (since := "2024-02-19")]
 protected theorem add_le_of_le_sub_left {n k m : Nat} (h : m ≤ k) : n ≤ k - m → m + n ≤ k :=
   Nat.add_le_of_le_sub' h
 
@@ -200,6 +200,7 @@ theorem succ_min_succ (x y) : min (succ x) (succ y) = succ (min x y) := by
   | inr h => rw [Nat.min_eq_right h, Nat.min_eq_right (Nat.succ_le_succ h)]
 
 @[simp] protected theorem min_self (a : Nat) : min a a = a := Nat.min_eq_left (Nat.le_refl _)
+instance : Std.IdempotentOp (α := Nat) min := ⟨Nat.min_self⟩
 
 @[simp] protected theorem zero_min (a) : min 0 a = 0 := Nat.min_eq_left (Nat.zero_le _)
 
@@ -210,6 +211,7 @@ protected theorem min_assoc : ∀ (a b c : Nat), min (min a b) c = min a (min b
   | _, 0, _ => by rw [Nat.zero_min, Nat.min_zero, Nat.zero_min]
   | _, _, 0 => by rw [Nat.min_zero, Nat.min_zero, Nat.min_zero]
   | _+1, _+1, _+1 => by simp only [Nat.succ_min_succ]; exact congrArg succ <| Nat.min_assoc ..
+instance : Std.Associative (α := Nat) min := ⟨Nat.min_assoc⟩
 
 protected theorem sub_sub_eq_min : ∀ (a b : Nat), a - (a - b) = min a b
   | 0, _ => by rw [Nat.zero_sub, Nat.zero_min]
@@ -249,16 +251,21 @@ protected theorem max_lt {a b c : Nat} : max a b < c ↔ a < c ∧ b < c := by
   rw [← Nat.succ_le, ← Nat.succ_max_succ a b]; exact Nat.max_le
 
 @[simp] protected theorem max_self (a : Nat) : max a a = a := Nat.max_eq_right (Nat.le_refl _)
+instance : Std.IdempotentOp (α := Nat) max := ⟨Nat.max_self⟩
 
 @[simp] protected theorem zero_max (a) : max 0 a = a := Nat.max_eq_right (Nat.zero_le _)
 
 @[simp] protected theorem max_zero (a) : max a 0 = a := Nat.max_eq_left (Nat.zero_le _)
+instance : Std.LawfulIdentity (α := Nat) max 0 where
+  left_id := Nat.zero_max
+  right_id := Nat.max_zero
 
 protected theorem max_assoc : ∀ (a b c : Nat), max (max a b) c = max a (max b c)
   | 0, _, _ => by rw [Nat.zero_max, Nat.zero_max]
   | _, 0, _ => by rw [Nat.zero_max, Nat.max_zero]
   | _, _, 0 => by rw [Nat.max_zero, Nat.max_zero]
   | _+1, _+1, _+1 => by simp only [Nat.succ_max_succ]; exact congrArg succ <| Nat.max_assoc ..
+instance : Std.Associative (α := Nat) max := ⟨Nat.max_assoc⟩
 
 protected theorem sub_add_eq_max (a b : Nat) : a - b + b = max a b := by
   match Nat.le_total a b with
@@ -394,11 +401,11 @@ protected theorem mul_min_mul_left (a b c : Nat) : min (a * b) (a * c) = a * min
 
 /-! ### mul -/
 
-@[deprecated Nat.mul_le_mul_left]
+@[deprecated Nat.mul_le_mul_left (since := "2024-02-19")]
 protected theorem mul_le_mul_of_nonneg_left {a b c : Nat} : a ≤ b → c * a ≤ c * b :=
   Nat.mul_le_mul_left c
 
-@[deprecated Nat.mul_le_mul_right]
+@[deprecated Nat.mul_le_mul_right (since := "2024-02-19")]
 protected theorem mul_le_mul_of_nonneg_right {a b c : Nat} : a ≤ b → a * c ≤ b * c :=
   Nat.mul_le_mul_right c
 
@@ -471,6 +478,7 @@ protected theorem mul_lt_mul_of_lt_of_lt {a b c d : Nat} (hac : a < c) (hbd : b
 
 theorem succ_mul_succ (a b) : succ a * succ b = a * b + a + b + 1 := by
   rw [succ_mul, mul_succ]; rfl
+
 theorem mul_le_add_right (m k n : Nat) : k * m ≤ m + n ↔ (k-1) * m ≤ n := by
   match k with
   | 0 =>
@@ -670,6 +678,10 @@ protected theorem pow_lt_pow_iff_right {a n m : Nat} (h : 1 < a) :
 
 /-! ### log2 -/
 
+@[simp]
+theorem log2_zero : Nat.log2 0 = 0 := by
+  simp [Nat.log2]
+
 theorem le_log2 (h : n ≠ 0) : k ≤ n.log2 ↔ 2 ^ k ≤ n := by
   match k with
   | 0 => simp [show 1 ≤ n from Nat.pos_of_ne_zero h]
@@ -690,7 +702,7 @@ theorem log2_self_le (h : n ≠ 0) : 2 ^ n.log2 ≤ n := (le_log2 h).1 (Nat.le_r
 
 theorem lt_log2_self : n < 2 ^ (n.log2 + 1) :=
   match n with
-  | 0 => Nat.zero_lt_two
+  | 0 => by simp
   | n+1 => (log2_lt n.succ_ne_zero).1 (Nat.le_refl _)
 
 /-! ### dvd -/
@@ -782,6 +794,9 @@ theorem shiftLeft_shiftLeft (m n : Nat) : ∀ k, (m <<< n) <<< k = m <<< (n + k)
   | 0 => rfl
   | k + 1 => by simp [← Nat.add_assoc, shiftLeft_shiftLeft _ _ k, shiftLeft_succ]
 
+@[simp] theorem shiftLeft_shiftRight (x n : Nat) : x <<< n >>> n = x := by
+  rw [Nat.shiftLeft_eq, Nat.shiftRight_eq_div_pow, Nat.mul_div_cancel _ (Nat.two_pow_pos _)]
+
 theorem mul_add_div {m : Nat} (m_pos : m > 0) (x y : Nat) : (m * x + y) / m = x + y / m := by
   match x with
   | 0 => simp

src/Init/Data/Nat/Linear.lean

@@ -580,7 +580,7 @@ attribute [-simp] Nat.right_distrib Nat.left_distrib
 
 theorem PolyCnstr.denote_mul (ctx : Context) (k : Nat) (c : PolyCnstr) : (c.mul (k+1)).denote ctx = c.denote ctx := by
   cases c; rename_i eq lhs rhs
-  have : k ≠ 0 → k + 1 ≠ 1 := by intro h; match k with | 0 => contradiction | k+1 => simp
+  have : k ≠ 0 → k + 1 ≠ 1 := by intro h; match k with | 0 => contradiction | k+1 => simp [Nat.succ.injEq]
   have : ¬ (k == 0) → (k + 1 == 1) = false := fun h => beq_false_of_ne (this (ne_of_beq_false (Bool.of_not_eq_true h)))
   have : ¬ ((k + 1 == 0) = true)  := fun h => absurd (eq_of_beq h) (Nat.succ_ne_zero k)
   have : (1 == (0 : Nat)) = false := rfl
@@ -714,4 +714,10 @@ theorem Expr.eq_of_toNormPoly_eq (ctx : Context) (e e' : Expr) (h : e.toNormPoly
   simp [Expr.toNormPoly, Poly.norm] at h
   assumption
 
-end Nat.Linear
+end Linear
+
+def elimOffset {α : Sort u} (a b k : Nat) (h₁ : a + k = b + k) (h₂ : a = b → α) : α := by
+  simp_arith at h₁
+  exact h₂ h₁
+
+end Nat

src/Init/Data/Nat/MinMax.lean

@@ -17,6 +17,7 @@ protected theorem min_comm (a b : Nat) : min a b = min b a := by
   | .inl h => simp [Nat.min_def, h, Nat.le_of_lt, Nat.not_le_of_lt]
   | .inr (.inl h) => simp [Nat.min_def, h]
   | .inr (.inr h) => simp [Nat.min_def, h, Nat.le_of_lt, Nat.not_le_of_lt]
+instance : Std.Commutative (α := Nat) min := ⟨Nat.min_comm⟩
 
 protected theorem min_le_right (a b : Nat) : min a b ≤ b := by
   by_cases (a <= b) <;> simp [Nat.min_def, *]
@@ -47,6 +48,7 @@ protected theorem max_comm (a b : Nat) : max a b = max b a := by
   by_cases h₁ : a ≤ b <;> by_cases h₂ : b ≤ a <;> simp [h₁, h₂]
   · exact Nat.le_antisymm h₂ h₁
   · cases not_or_intro h₁ h₂ <| Nat.le_total ..
+instance : Std.Commutative (α := Nat) max := ⟨Nat.max_comm⟩
 
 protected theorem le_max_left ( a b : Nat) : a ≤ max a b := by
   by_cases (a <= b) <;> simp [Nat.max_def, *]

src/Init/Data/Nat/Simproc.lean

@@ -0,0 +1,108 @@
+/-
+Copyright (c) 2023 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joe Hendrix
+-/
+prelude
+import Init.Data.Bool
+import Init.Data.Nat.Basic
+import Init.Data.Nat.Lemmas
+
+/-!
+This contains lemmas used by the Nat simprocs for simplifying arithmetic
+addition offsets.
+-/
+namespace Nat.Simproc
+
+/- Sub proofs -/
+
+theorem sub_add_eq_comm (a b c : Nat) : a - (b + c) = a - c - b := by
+  rw [Nat.add_comm b c]
+  exact Nat.sub_add_eq a c b
+
+theorem add_sub_add_le (a c : Nat) {b d : Nat} (h : b ≤ d) : a + b - (c + d) = a - (c + (d-b)) := by
+  induction b generalizing a c d with
+  | zero =>
+    simp
+  | succ b ind =>
+    match d with
+    | 0 =>
+      contradiction
+    | d + 1 =>
+      have g := Nat.le_of_succ_le_succ h
+      rw [Nat.add_succ a, Nat.add_succ c, Nat.succ_sub_succ, Nat.succ_sub_succ,
+          ind _ _ g]
+
+theorem add_sub_add_ge (a c : Nat) {b d : Nat} (h : b ≥ d) : a + b - (c + d) = a + (b - d) - c := by
+  rw [Nat.add_comm c d, Nat.sub_add_eq, Nat.add_sub_assoc h a]
+
+theorem add_sub_le (a : Nat) {b c : Nat} (h : b ≤ c) : a + b - c = a - (c - b) := by
+  have p := add_sub_add_le a 0 h
+  simp only [Nat.zero_add] at p
+  exact p
+
+/- Eq proofs -/
+
+theorem add_eq_gt (a : Nat) {b c : Nat} (h : b > c) : (a + b = c) = False :=
+  eq_false (Nat.ne_of_gt (Nat.lt_of_lt_of_le h (le_add_left b a)))
+
+theorem eq_add_gt (a : Nat) {b c : Nat} (h : c > a) : (a = b + c) = False := by
+  rw [@Eq.comm Nat a (b + c)]
+  exact add_eq_gt b h
+
+theorem add_eq_add_le (a c : Nat) {b d : Nat} (h : b ≤ d) : (a + b = c + d) = (a = c + (d - b)) := by
+  have g : b ≤ c + d := Nat.le_trans h (le_add_left d c)
+  rw [← Nat.add_sub_assoc h, @Eq.comm _ a, Nat.sub_eq_iff_eq_add g, @Eq.comm _ (a + b)]
+
+theorem add_eq_add_ge (a c : Nat) {b d : Nat} (h : b ≥ d) : (a + b = c + d) = (a + (b - d) = c) := by
+  rw [@Eq.comm _ (a + b) _, add_eq_add_le c a h, @Eq.comm _ _ c]
+
+theorem add_eq_le (a : Nat) {b c : Nat} (h : b ≤ c) : (a + b = c) = (a = c - b) := by
+  have r := add_eq_add_le a 0 h
+  simp only [Nat.zero_add] at r
+  exact r
+
+theorem eq_add_le {a : Nat} (b : Nat) {c : Nat} (h : c ≤ a) : (a = b + c) = (b = a - c) := by
+  rw [@Eq.comm Nat a (b + c)]
+  exact add_eq_le b h
+
+/- Lemmas for lifting Eq proofs to beq -/
+
+theorem beqEqOfEqEq {a b c d : Nat} (p : (a = b) = (c = d)) : (a == b) = (c == d) := by
+  simp only [Bool.beq_eq_decide_eq, p]
+
+theorem beqFalseOfEqFalse {a b : Nat} (p : (a = b) = False) : (a == b) = false := by
+  simp [Bool.beq_eq_decide_eq, p]
+
+theorem bneEqOfEqEq {a b c d : Nat} (p : (a = b) = (c = d)) : (a != b) = (c != d) := by
+  simp only [bne, beqEqOfEqEq p]
+
+theorem bneTrueOfEqFalse {a b : Nat} (p : (a = b) = False) : (a != b) = true := by
+  simp [bne, beqFalseOfEqFalse p]
+
+/- le proofs -/
+
+theorem add_le_add_le (a c : Nat) {b d : Nat} (h : b ≤ d) : (a + b ≤ c + d) = (a ≤ c + (d - b)) := by
+  rw [← Nat.add_sub_assoc h, Nat.le_sub_iff_add_le]
+  exact Nat.le_trans h (le_add_left d c)
+
+theorem add_le_add_ge (a c : Nat) {b d : Nat} (h : b ≥ d) : (a + b ≤ c + d) = (a + (b - d) ≤ c) := by
+  rw [← Nat.add_sub_assoc h, Nat.sub_le_iff_le_add]
+
+theorem add_le_le (a : Nat) {b c : Nat} (h : b ≤ c) : (a + b ≤ c) = (a ≤ c - b) := by
+  have r := add_le_add_le a 0 h
+  simp only [Nat.zero_add] at r
+  exact r
+
+theorem add_le_gt (a : Nat) {b c : Nat} (h : b > c) : (a + b ≤ c) = False :=
+  eq_false (Nat.not_le_of_gt (Nat.lt_of_lt_of_le h (le_add_left b a)))
+
+theorem le_add_le (a : Nat) {b c : Nat} (h : a ≤ c) : (a ≤ b + c) = True :=
+  eq_true (Nat.le_trans h (le_add_left c b))
+
+theorem le_add_ge (a : Nat) {b c : Nat} (h : a ≥ c) : (a ≤ b + c) = (a - c ≤ b) := by
+  have r := add_le_add_ge 0 b h
+  simp only [Nat.zero_add] at r
+  exact r
+
+end Nat.Simproc

src/Init/Data/Option/Basic.lean

@@ -13,29 +13,36 @@ namespace Option
 deriving instance DecidableEq for Option
 deriving instance BEq for Option
 
-def toMonad [Monad m] [Alternative m] : Option α → m α
+/-- 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
 
-@[inline] def toBool : Option α → Bool
-  | some _ => true
-  | none   => false
+@[deprecated getM (since := "2024-04-17")]
+def toMonad [Monad m] [Alternative m] : Option α → m α := getM
 
+/-- Returns `true` on `some x` and `false` on `none`. -/
 @[inline] def isSome : Option α → Bool
   | some _ => true
   | none   => false
 
+@[deprecated isSome, inline] def toBool : Option α → Bool := isSome
+
+/-- Returns `true` on `none` and `false` on `some x`. -/
 @[inline] def isNone : Option α → Bool
   | some _ => false
   | none   => true
 
+/--
+`x?.isEqSome y` is equivalent to `x? == some y`, but avoids an allocation.
+-/
 @[inline] def isEqSome [BEq α] : Option α → α → Bool
   | some a, b => a == b
   | none,   _ => false
 
 @[inline] protected def bind : Option α → (α → Option β) → Option β
   | none,   _ => none
-  | some a, b => b a
+  | some a, f => f a
 
 /-- Runs `f` on `o`'s value, if any, and returns its result, or else returns `none`.  -/
 @[inline] protected def bindM [Monad m] (f : α → m (Option β)) (o : Option α) : m (Option β) := do
@@ -44,6 +51,10 @@ def toMonad [Monad m] [Alternative m] : Option α → m α
   else
     return none
 
+/--
+Runs a monadic function `f` on an optional value.
+If the optional value is `none` the function is not called.
+-/
 @[inline] protected def mapM [Monad m] (f : α → m β) (o : Option α) : m (Option β) := do
   if let some a := o then
     return some (← f a)
@@ -53,18 +64,24 @@ def toMonad [Monad m] [Alternative m] : Option α → m α
 theorem map_id : (Option.map id : Option α → Option α) = id :=
   funext (fun o => match o with | none => rfl | some _ => rfl)
 
+/-- Keeps an optional value only if it satisfies the predicate `p`. -/
 @[always_inline, inline] protected def filter (p : α → Bool) : Option α → Option α
   | some a => if p a then some a else none
   | none   => none
 
+/-- Checks that an optional value satisfies a predicate `p` or is `none`. -/
 @[always_inline, inline] protected def all (p : α → Bool) : Option α → Bool
   | some a => p a
   | none   => true
 
+/-- Checks that an optional value is not `none` and the value satisfies a predicate `p`. -/
 @[always_inline, inline] protected def any (p : α → Bool) : Option α → Bool
   | some a => p a
   | none   => false
 
+/--
+Implementation of `OrElse`'s `<|>` syntax for `Option`.
+-/
 @[always_inline, macro_inline] protected def orElse : Option α → (Unit → Option α) → Option α
   | some a, _ => some a
   | none,   b => b ()

src/Init/Data/Ord.lean

@@ -114,7 +114,18 @@ by `cmp₂` to break the tie.
 @[inline] def compareLex (cmp₁ cmp₂ : α → β → Ordering) (a : α) (b : β) : Ordering :=
   (cmp₁ a b).then (cmp₂ a b)
 
+/--
+`Ord α` provides a computable total order on `α`, in terms of the
+`compare : α → α → Ordering` function.
+
+Typically instances will be transitive, reflexive, and antisymmetric,
+but this is not enforced by the typeclass.
+
+There is a derive handler, so appending `deriving Ord` to an inductive type or structure
+will attempt to create an `Ord` instance.
+-/
 class Ord (α : Type u) where
+  /-- Compare two elements in `α` using the comparator contained in an `[Ord α]` instance. -/
   compare : α → α → Ordering
 
 export Ord (compare)
@@ -171,15 +182,13 @@ instance [Ord α] : Ord (Option α) where
 
 /-- The lexicographic order on pairs. -/
 def lexOrd [Ord α] [Ord β] : Ord (α × β) where
-  compare p1 p2 := match compare p1.1 p2.1 with
-    | .eq => compare p1.2 p2.2
-    | o   => o
+  compare := compareLex (compareOn (·.1)) (compareOn (·.2))
 
 def ltOfOrd [Ord α] : LT α where
-  lt a b := compare a b == Ordering.lt
+  lt a b := compare a b = Ordering.lt
 
 instance [Ord α] : DecidableRel (@LT.lt α ltOfOrd) :=
-  inferInstanceAs (DecidableRel (fun a b => compare a b == Ordering.lt))
+  inferInstanceAs (DecidableRel (fun a b => compare a b = Ordering.lt))
 
 def leOfOrd [Ord α] : LE α where
   le a b := (compare a b).isLE

src/Init/Data/Repr.lean

@@ -13,11 +13,24 @@ open Sum Subtype Nat
 
 open Std
 
+/--
+A typeclass that specifies the standard way of turning values of some type into `Format`.
+
+When rendered this `Format` should be as close as possible to something that can be parsed as the
+input value.
+-/
 class Repr (α : Type u) where
+  /--
+  Turn a value of type `α` into `Format` at a given precedence. The precedence value can be used
+  to avoid parentheses if they are not necessary.
+  -/
   reprPrec : α → Nat → Format
 
 export Repr (reprPrec)
 
+/--
+Turn `a` into `Format` using its `Repr` instance. The precedence level is initially set to 0.
+-/
 abbrev repr [Repr α] (a : α) : Format :=
   reprPrec a 0
 
@@ -103,6 +116,11 @@ instance {p : α → Prop} [Repr α] : Repr (Subtype p) where
 
 namespace Nat
 
+/-
+We have pure functions for calculating the decimal representation of a `Nat` (`toDigits`), but also
+a fast variant that handles small numbers (`USize`) via C code (`lean_string_of_usize`).
+-/
+
 def digitChar (n : Nat) : Char :=
   if n = 0 then '0' else
   if n = 1 then '1' else
@@ -133,6 +151,20 @@ def toDigitsCore (base : Nat) : Nat → Nat → List Char → List Char
 def toDigits (base : Nat) (n : Nat) : List Char :=
   toDigitsCore base (n+1) n []
 
+@[extern "lean_string_of_usize"]
+protected def _root_.USize.repr (n : @& USize) : String :=
+  (toDigits 10 n.toNat).asString
+
+/-- We statically allocate and memoize reprs for small natural numbers. -/
+private def reprArray : Array String := Id.run do
+  List.range 128 |>.map (·.toUSize.repr) |> Array.mk
+
+private def reprFast (n : Nat) : String :=
+  if h : n < 128 then Nat.reprArray.get ⟨n, h⟩ else
+  if h : n < USize.size then (USize.ofNatCore n h).repr
+  else (toDigits 10 n).asString
+
+@[implemented_by reprFast]
 protected def repr (n : Nat) : String :=
   (toDigits 10 n).asString
 
@@ -162,6 +194,32 @@ def toSuperDigits (n : Nat) : List Char :=
 def toSuperscriptString (n : Nat) : String :=
   (toSuperDigits n).asString
 
+def subDigitChar (n : Nat) : Char :=
+  if n = 0 then '₀' else
+  if n = 1 then '₁' else
+  if n = 2 then '₂' else
+  if n = 3 then '₃' else
+  if n = 4 then '₄' else
+  if n = 5 then '₅' else
+  if n = 6 then '₆' else
+  if n = 7 then '₇' else
+  if n = 8 then '₈' else
+  if n = 9 then '₉' else
+  '*'
+
+partial def toSubDigitsAux : Nat → List Char → List Char
+  | n, ds =>
+    let d  := subDigitChar <| n % 10;
+    let n' := n / 10;
+    if n' = 0 then d::ds
+    else toSubDigitsAux n' (d::ds)
+
+def toSubDigits (n : Nat) : List Char :=
+  toSubDigitsAux n []
+
+def toSubscriptString (n : Nat) : String :=
+  (toSubDigits n).asString
+
 end Nat
 
 instance : Repr Nat where

src/Init/Data/Stream.lean

@@ -94,7 +94,8 @@ instance : Stream (Subarray α) α where
   next? s :=
     if h : s.start < s.stop then
       have : s.start + 1 ≤ s.stop := Nat.succ_le_of_lt h
-      some (s.as.get ⟨s.start, Nat.lt_of_lt_of_le h s.h₂⟩, { s with start := s.start + 1, h₁ := this })
+      some (s.array.get ⟨s.start, Nat.lt_of_lt_of_le h s.stop_le_array_size⟩,
+        { s with start := s.start + 1, start_le_stop := this })
     else
       none
 

src/Init/Data/String.lean

@@ -6,3 +6,4 @@ Authors: Leonardo de Moura
 prelude
 import Init.Data.String.Basic
 import Init.Data.String.Extra
+import Init.Data.String.Lemmas

src/Init/Data/String/Basic.lean

@@ -24,34 +24,88 @@ instance : LT String :=
 instance decLt (s₁ s₂ : @& String) : Decidable (s₁ < s₂) :=
   List.hasDecidableLt s₁.data s₂.data
 
+@[reducible] protected def le (a b : String) : Prop := ¬ b < a
+
+instance : LE String :=
+  ⟨String.le⟩
+
+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"]
 def length : (@& String) → Nat
   | ⟨s⟩ => s.length
 
-/-- The internal implementation uses dynamic arrays and will perform destructive updates
-   if the String is not shared. -/
+/--
+Pushes a character onto the end of a string.
+
+The internal implementation uses dynamic arrays and will perform destructive updates
+if the string is not shared.
+
+Example: `"abc".push 'd' = "abcd"`
+-/
 @[extern "lean_string_push"]
 def push : String → Char → String
   | ⟨s⟩, c => ⟨s ++ [c]⟩
 
-/-- The internal implementation uses dynamic arrays and will perform destructive updates
-   if the String is not shared. -/
+/--
+Appends two strings.
+
+The internal implementation uses dynamic arrays and will perform destructive updates
+if the string is not shared.
+
+Example: `"abc".append "def" = "abcdef"`
+-/
 @[extern "lean_string_append"]
 def append : String → (@& String) → String
   | ⟨a⟩, ⟨b⟩ => ⟨a ++ b⟩
 
-/-- O(n) in the runtime, where n is the length of the String -/
+/--
+Converts a string to a list of characters.
+
+Even though the logical model of strings is as a structure that wraps a list of characters,
+this operation takes time and space linear in the length of the string, because the compiler
+uses an optimized representation as dynamic arrays.
+
+Example: `"abc".toList = ['a', 'b', 'c']`
+-/
 def toList (s : String) : List Char :=
   s.data
 
+/-- Returns true if `p` is a valid UTF-8 position in the string `s`, meaning that `p ≤ s.endPos`
+and `p` lies on a UTF-8 character boundary. This has an O(1) implementation in the runtime. -/
+@[extern "lean_string_is_valid_pos"]
+def Pos.isValid (s : @&String) (p : @& Pos) : Bool :=
+  go s.data 0
+where
+  go : List Char → Pos → Bool
+  | [],    i => i = p
+  | c::cs, i => if i = p then true else go cs (i + c)
+
 def utf8GetAux : List Char → Pos → Pos → Char
   | [],    _, _ => default
   | c::cs, i, p => if i = p then c else utf8GetAux cs (i + c) p
 
 /--
-  Return character at position `p`. If `p` is not a valid position
-  returns `(default : Char)`.
-  See `utf8GetAux` for the reference implementation.
+Returns the character at position `p` of a string. If `p` is not a valid position,
+returns `(default : Char)`.
+
+See `utf8GetAux` for the reference implementation.
+
+Examples:
+* `"abc".get ⟨1⟩ = 'b'`
+* `"abc".get ⟨3⟩ = (default : Char) = 'A'`
+
+Positions can also be invalid if a byte index points into the middle of a multi-byte UTF-8
+character. For example,`"L∃∀N".get ⟨2⟩ = (default : Char) = 'A'`.
 -/
 @[extern "lean_string_utf8_get"]
 def get (s : @& String) (p : @& Pos) : Char :=
@@ -62,12 +116,30 @@ def utf8GetAux? : List Char → Pos → Pos → Option Char
   | [],    _, _ => none
   | c::cs, i, p => if i = p then c else utf8GetAux? cs (i + c) p
 
+/--
+Returns the character at position `p`. If `p` is not a valid position, returns `none`.
+
+Examples:
+* `"abc".get? ⟨1⟩ = some 'b'`
+* `"abc".get? ⟨3⟩ = none`
+
+Positions can also be invalid if a byte index points into the middle of a multi-byte UTF-8
+character. For example, `"L∃∀N".get? ⟨2⟩ = none`
+-/
 @[extern "lean_string_utf8_get_opt"]
 def get? : (@& String) → (@& Pos) → Option Char
   | ⟨s⟩, p => utf8GetAux? s 0 p
 
 /--
-  Similar to `get`, but produces a panic error message if `p` is not a valid `String.Pos`.
+Returns the character at position `p` of a string. If `p` is not a valid position,
+returns `(default : Char)` and produces a panic error message.
+
+Examples:
+* `"abc".get! ⟨1⟩ = 'b'`
+* `"abc".get! ⟨3⟩` panics
+
+Positions can also be invalid if a byte index points into the middle of a multi-byte UTF-8 character. For example,
+`"L∃∀N".get! ⟨2⟩` panics.
 -/
 @[extern "lean_string_utf8_get_bang"]
 def get! (s : @& String) (p : @& Pos) : Char :=
@@ -79,13 +151,49 @@ def utf8SetAux (c' : Char) : List Char → Pos → Pos → List Char
   | c::cs, i, p =>
     if i = p then (c'::cs) else c::(utf8SetAux c' cs (i + c) p)
 
+/--
+Replaces the character at a specified position in a string with a new character. If the position
+is invalid, the string is returned unchanged.
+
+If both the replacement character and the replaced character are ASCII characters and the string
+is not shared, destructive updates are used.
+
+Examples:
+* `"abc".set ⟨1⟩ 'B' = "aBc"`
+* `"abc".set ⟨3⟩ 'D' = "abc"`
+* `"L∃∀N".set ⟨4⟩ 'X' = "L∃XN"`
+
+Because `'∃'` is a multi-byte character, the byte index `2` in `L∃∀N` is an invalid position,
+so `"L∃∀N".set ⟨2⟩ 'X' = "L∃∀N"`.
+-/
 @[extern "lean_string_utf8_set"]
 def set : String → (@& Pos) → Char → String
   | ⟨s⟩, i, c => ⟨utf8SetAux c s 0 i⟩
 
+/--
+Replaces the character at position `p` in the string `s` with the result of applying `f` to that character.
+If `p` is an invalid position, the string is returned unchanged.
+
+Examples:
+* `abc.modify ⟨1⟩ Char.toUpper = "aBc"`
+* `abc.modify ⟨3⟩ Char.toUpper = "abc"`
+-/
 def modify (s : String) (i : Pos) (f : Char → Char) : String :=
   s.set i <| f <| s.get i
 
+/--
+Returns the next position in a string after position `p`. If `p` is not a valid position or `p = s.endPos`,
+the result is unspecified.
+
+Examples:
+Given `def abc := "abc"` and `def lean := "L∃∀N"`,
+* `abc.get (0 |> abc.next) = 'b'`
+* `lean.get (0 |> lean.next |> lean.next) = '∀'`
+
+Cases where the result is unspecified:
+* `"abc".next ⟨3⟩`, since `3 = s.endPos`
+* `"L∃∀N".next ⟨2⟩`, since `2` points into the middle of a multi-byte UTF-8 character
+-/
 @[extern "lean_string_utf8_next"]
 def next (s : @& String) (p : @& Pos) : Pos :=
   let c := get s p
@@ -97,16 +205,52 @@ def utf8PrevAux : List Char → Pos → Pos → Pos
     let i' := i + c
     if i' = p then i else utf8PrevAux cs i' p
 
+/--
+Returns the position in a string before a specified position, `p`. If `p = ⟨0⟩`, returns `0`.
+If `p` is not a valid position, the result is unspecified.
+
+Examples:
+Given `def abc := "abc"` and `def lean := "L∃∀N"`,
+* `abc.get (abc.endPos |> abc.prev) = 'c'`
+* `lean.get (lean.endPos |> lean.prev |> lean.prev |> lean.prev) = '∃'`
+* `"L∃∀N".prev ⟨3⟩` is unspecified, since byte 3 occurs in the middle of the multi-byte character `'∃'`.
+-/
 @[extern "lean_string_utf8_prev"]
 def prev : (@& String) → (@& Pos) → Pos
   | ⟨s⟩, p => if p = 0 then 0 else utf8PrevAux s 0 p
 
+/--
+Returns the first character in `s`. If `s = ""`, returns `(default : Char)`.
+
+Examples:
+* `"abc".front = 'a'`
+* `"".front = (default : Char)`
+-/
 def front (s : String) : Char :=
   get s 0
 
+/--
+Returns the last character in `s`. If `s = ""`, returns `(default : Char)`.
+
+Examples:
+* `"abc".back = 'c'`
+* `"".back = (default : Char)`
+-/
 def back (s : String) : Char :=
   get s (prev s s.endPos)
 
+/--
+Returns `true` if a specified position is greater than or equal to the position which
+points to the end of a string. Otherwise, returns `false`.
+
+Examples:
+Given `def abc := "abc"` and `def lean := "L∃∀N"`,
+* `(0 |> abc.next |> abc.next |> abc.atEnd) = false`
+* `(0 |> abc.next |> abc.next |> abc.next |> abc.next |> abc.atEnd) = true`
+* `(0 |> lean.next |> lean.next |> lean.next |> lean.next |> lean.atEnd) = true`
+
+Because `"L∃∀N"` contains multi-byte characters, `lean.next (lean.next 0)` is not equal to `abc.next (abc.next 0)`.
+-/
 @[extern "lean_string_utf8_at_end"]
 def atEnd : (@& String) → (@& Pos) → Bool
   | s, p => p.byteIdx ≥ utf8ByteSize s
@@ -245,12 +389,21 @@ termination_by s.endPos.1 - i.1
 @[specialize] def split (s : String) (p : Char → Bool) : List String :=
   splitAux s p 0 0 []
 
+/--
+Auxiliary for `splitOn`. Preconditions:
+* `sep` is not empty
+* `b <= i` are indexes into `s`
+* `j` is an index into `sep`, and not at the end
+
+It represents the state where we have currently parsed some split parts into `r` (in reverse order),
+`b` is the beginning of the string / the end of the previous match of `sep`, and the first `j` bytes
+of `sep` match the bytes `i-j .. i` of `s`.
+-/
 def splitOnAux (s sep : String) (b : Pos) (i : Pos) (j : Pos) (r : List String) : List String :=
-  if h : s.atEnd i then
+  if s.atEnd i then
     let r := (s.extract b i)::r
     r.reverse
   else
-    have := Nat.sub_lt_sub_left (Nat.gt_of_not_le (mt decide_eq_true h)) (lt_next s _)
     if s.get i == sep.get j then
       let i := s.next i
       let j := sep.next j
@@ -259,9 +412,42 @@ def splitOnAux (s sep : String) (b : Pos) (i : Pos) (j : Pos) (r : List String)
       else
         splitOnAux s sep b i j r
     else
-      splitOnAux s sep b (s.next i) 0 r
-termination_by s.endPos.1 - i.1
+      splitOnAux s sep b (s.next (i - j)) 0 r
+termination_by (s.endPos.1 - (i - j).1, sep.endPos.1 - j.1)
+decreasing_by
+  all_goals simp_wf
+  focus
+    rename_i h _ _
+    left; exact Nat.sub_lt_sub_left
+      (Nat.lt_of_le_of_lt (Nat.sub_le ..) (Nat.gt_of_not_le (mt decide_eq_true h)))
+      (Nat.lt_of_le_of_lt (Nat.sub_le ..) (lt_next s _))
+  focus
+    rename_i i₀ j₀ _ eq h'
+    rw [show (s.next i₀ - sep.next j₀).1 = (i₀ - j₀).1 by
+      show (_ + csize _) - (_ + csize _) = _
+      rw [(beq_iff_eq ..).1 eq, Nat.add_sub_add_right]; rfl]
+    right; exact Nat.sub_lt_sub_left
+      (Nat.lt_of_le_of_lt (Nat.le_add_right ..) (Nat.gt_of_not_le (mt decide_eq_true h')))
+      (lt_next sep _)
+  focus
+    rename_i h _
+    left; exact Nat.sub_lt_sub_left
+      (Nat.lt_of_le_of_lt (Nat.sub_le ..) (Nat.gt_of_not_le (mt decide_eq_true h)))
+      (lt_next s _)
 
+/--
+Splits a string `s` on occurrences of the separator `sep`. When `sep` is empty, it returns `[s]`;
+when `sep` occurs in overlapping patterns, the first match is taken. There will always be exactly
+`n+1` elements in the returned list if there were `n` nonoverlapping matches of `sep` in the string.
+The default separator is `" "`. The separators are not included in the returned substrings.
+
+```
+"here is some text ".splitOn = ["here", "is", "some", "text", ""]
+"here is some text ".splitOn "some" = ["here is ", " text "]
+"here is some text ".splitOn "" = ["here is some text "]
+"ababacabac".splitOn "aba" = ["", "bac", "c"]
+```
+-/
 def splitOn (s : String) (sep : String := " ") : List String :=
   if sep == "" then [s] else splitOnAux s sep 0 0 0 []
 
@@ -542,13 +728,15 @@ def substrEq (s1 : String) (off1 : String.Pos) (s2 : String) (off2 : String.Pos)
   off1.byteIdx + sz ≤ s1.endPos.byteIdx && off2.byteIdx + sz ≤ s2.endPos.byteIdx && loop off1 off2 { byteIdx := off1.byteIdx + sz }
 where
   loop (off1 off2 stop1 : Pos) :=
-    if h : off1.byteIdx < stop1.byteIdx then
+    if _h : off1.byteIdx < stop1.byteIdx then
       let c₁ := s1.get off1
       let c₂ := s2.get off2
-      have := Nat.sub_lt_sub_left h (Nat.add_lt_add_left (one_le_csize c₁) off1.1)
       c₁ == c₂ && loop (off1 + c₁) (off2 + c₂) stop1
     else true
   termination_by stop1.1 - off1.1
+  decreasing_by
+    have := Nat.sub_lt_sub_left _h (Nat.add_lt_add_left (one_le_csize c₁) off1.1)
+    decreasing_tactic
 
 /-- Return true iff `p` is a prefix of `s` -/
 def isPrefixOf (p : String) (s : String) : Bool :=
@@ -763,6 +951,10 @@ def beq (ss1 ss2 : Substring) : Bool :=
 
 instance hasBeq : BEq Substring := ⟨beq⟩
 
+/-- Checks whether two substrings have the same position and content. -/
+def sameAs (ss1 ss2 : Substring) : Bool :=
+  ss1.startPos == ss2.startPos && ss1 == ss2
+
 end Substring
 
 namespace String

src/Init/Data/String/Extra.lean

@@ -17,32 +17,132 @@ def toNat! (s : String) : Nat :=
   else
     panic! "Nat expected"
 
-/--
-  Convert a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded `ByteArray` string to `String`.
-  The result is unspecified if `a` is not properly UTF-8 encoded.
--/
-@[extern "lean_string_from_utf8_unchecked"]
-opaque fromUTF8Unchecked (a : @& ByteArray) : String
+def utf8DecodeChar? (a : ByteArray) (i : Nat) : Option Char := do
+  let c ← a[i]?
+  if c &&& 0x80 == 0 then
+    some ⟨c.toUInt32, .inl (Nat.lt_trans c.1.2 (by decide))⟩
+  else if c &&& 0xe0 == 0xc0 then
+    let c1 ← a[i+1]?
+    guard (c1 &&& 0xc0 == 0x80)
+    let r := ((c &&& 0x1f).toUInt32 <<< 6) ||| (c1 &&& 0x3f).toUInt32
+    guard (0x80 ≤ r)
+    -- TODO: Prove h from the definition of r once we have the necessary lemmas
+    if h : r < 0xd800 then some ⟨r, .inl h⟩ else none
+  else if c &&& 0xf0 == 0xe0 then
+    let c1 ← a[i+1]?
+    let c2 ← a[i+2]?
+    guard (c1 &&& 0xc0 == 0x80 && c2 &&& 0xc0 == 0x80)
+    let r :=
+      ((c &&& 0x0f).toUInt32 <<< 12) |||
+      ((c1 &&& 0x3f).toUInt32 <<< 6) |||
+      (c2 &&& 0x3f).toUInt32
+    guard (0x800 ≤ r)
+    -- TODO: Prove `r < 0x110000` from the definition of r once we have the necessary lemmas
+    if h : r < 0xd800 ∨ 0xdfff < r ∧ r < 0x110000 then some ⟨r, h⟩ else none
+  else if c &&& 0xf8 == 0xf0 then
+    let c1 ← a[i+1]?
+    let c2 ← a[i+2]?
+    let c3 ← a[i+3]?
+    guard (c1 &&& 0xc0 == 0x80 && c2 &&& 0xc0 == 0x80 && c3 &&& 0xc0 == 0x80)
+    let r :=
+      ((c &&& 0x07).toUInt32 <<< 18) |||
+      ((c1 &&& 0x3f).toUInt32 <<< 12) |||
+      ((c2 &&& 0x3f).toUInt32 <<< 6) |||
+      (c3 &&& 0x3f).toUInt32
+    if h : 0x10000 ≤ r ∧ r < 0x110000 then
+      some ⟨r, .inr ⟨Nat.lt_of_lt_of_le (by decide) h.1, h.2⟩⟩
+    else none
+  else
+    none
 
-/-- Convert the given `String` to a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded byte array. -/
+/-- Returns true if the given byte array consists of valid UTF-8. -/
+@[extern "lean_string_validate_utf8"]
+def validateUTF8 (a : @& ByteArray) : Bool :=
+  (loop 0).isSome
+where
+  loop (i : Nat) : Option Unit := do
+    if i < a.size then
+      let c ← utf8DecodeChar? a i
+      loop (i + csize c)
+    else pure ()
+  termination_by a.size - i
+  decreasing_by exact Nat.sub_lt_sub_left ‹_› (Nat.lt_add_of_pos_right (one_le_csize c))
+
+/-- Converts a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded `ByteArray` string to `String`. -/
+@[extern "lean_string_from_utf8"]
+def fromUTF8 (a : @& ByteArray) (h : validateUTF8 a) : String :=
+  loop 0 ""
+where
+  loop (i : Nat) (acc : String) : String :=
+    if i < a.size then
+      let c := (utf8DecodeChar? a i).getD default
+      loop (i + csize c) (acc.push c)
+    else acc
+  termination_by a.size - i
+  decreasing_by exact Nat.sub_lt_sub_left ‹_› (Nat.lt_add_of_pos_right (one_le_csize c))
+
+/-- Converts a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded `ByteArray` string to `String`,
+or returns `none` if `a` is not properly UTF-8 encoded. -/
+@[inline] def fromUTF8? (a : ByteArray) : Option String :=
+  if h : validateUTF8 a then fromUTF8 a h else none
+
+/-- Converts a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded `ByteArray` string to `String`,
+or panics if `a` is not properly UTF-8 encoded. -/
+@[inline] def fromUTF8! (a : ByteArray) : String :=
+  if h : validateUTF8 a then fromUTF8 a h else panic! "invalid UTF-8 string"
+
+def utf8EncodeChar (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]
+
+@[simp] theorem length_utf8EncodeChar (c : Char) : (utf8EncodeChar c).length = csize c := by
+  simp [csize, utf8EncodeChar, Char.utf8Size]
+  cases Decidable.em (c.val ≤ 0x7f) <;> simp [*]
+  cases Decidable.em (c.val ≤ 0x7ff) <;> simp [*]
+  cases Decidable.em (c.val ≤ 0xffff) <;> simp [*]
+
+/-- Converts the given `String` to a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded byte array. -/
 @[extern "lean_string_to_utf8"]
-opaque toUTF8 (a : @& String) : ByteArray
+def toUTF8 (a : @& String) : ByteArray :=
+  ⟨⟨a.data.bind utf8EncodeChar⟩⟩
+
+@[simp] theorem size_toUTF8 (s : String) : s.toUTF8.size = s.utf8ByteSize := by
+  simp [toUTF8, ByteArray.size, Array.size, utf8ByteSize, List.bind]
+  induction s.data <;> simp [List.map, List.join, utf8ByteSize.go, Nat.add_comm, *]
 
 /-- Accesses a byte in the UTF-8 encoding of the `String`. O(1) -/
 @[extern "lean_string_get_byte_fast"]
-opaque getUtf8Byte (s : @& String) (n : Nat) (h : n < s.utf8ByteSize) : UInt8
+def getUtf8Byte (s : @& String) (n : Nat) (h : n < s.utf8ByteSize) : UInt8 :=
+  (toUTF8 s).get ⟨n, size_toUTF8 _ ▸ h⟩
 
 theorem Iterator.sizeOf_next_lt_of_hasNext (i : String.Iterator) (h : i.hasNext) : sizeOf i.next < sizeOf i := by
   cases i; rename_i s pos; simp [Iterator.next, Iterator.sizeOf_eq]; simp [Iterator.hasNext] at h
   exact Nat.sub_lt_sub_left h (String.lt_next s pos)
 
-macro_rules | `(tactic| decreasing_trivial) => `(tactic| apply String.Iterator.sizeOf_next_lt_of_hasNext; assumption)
+macro_rules
+| `(tactic| decreasing_trivial) =>
+  `(tactic| with_reducible apply String.Iterator.sizeOf_next_lt_of_hasNext; assumption)
 
 theorem Iterator.sizeOf_next_lt_of_atEnd (i : String.Iterator) (h : ¬ i.atEnd = true) : sizeOf i.next < sizeOf i :=
   have h : i.hasNext := decide_eq_true <| Nat.gt_of_not_le <| mt decide_eq_true h
   sizeOf_next_lt_of_hasNext i h
 
-macro_rules | `(tactic| decreasing_trivial) => `(tactic| apply String.Iterator.sizeOf_next_lt_of_atEnd; assumption)
+macro_rules
+| `(tactic| decreasing_trivial) =>
+  `(tactic| with_reducible apply String.Iterator.sizeOf_next_lt_of_atEnd; assumption)
 
 namespace Iterator
 
@@ -98,4 +198,35 @@ def removeLeadingSpaces (s : String) : String :=
   let n := findLeadingSpacesSize s
   if n == 0 then s else removeNumLeadingSpaces n s
 
+/--
+Replaces each `\r\n` with `\n` to normalize line endings,
+but does not validate that there are no isolated `\r` characters.
+It is an optimized version of `String.replace text "\r\n" "\n"`.
+-/
+def crlfToLf (text : String) : String :=
+  go "" 0 0
+where
+  go (acc : String) (accStop pos : String.Pos) : String :=
+    if h : text.atEnd pos then
+      -- note: if accStop = 0 then acc is empty
+      if accStop = 0 then text else acc ++ text.extract accStop pos
+    else
+      let c := text.get' pos h
+      let pos' := text.next' pos h
+      if h' : ¬ text.atEnd pos' ∧ c == '\r' ∧ text.get pos' == '\n' then
+        let acc := acc ++ text.extract accStop pos
+        go acc pos' (text.next' pos' h'.1)
+      else
+        go acc accStop pos'
+  termination_by text.utf8ByteSize - pos.byteIdx
+  decreasing_by
+    decreasing_with
+      show text.utf8ByteSize - (text.next' (text.next' pos _) _).byteIdx < text.utf8ByteSize - pos.byteIdx
+      have k := Nat.gt_of_not_le <| mt decide_eq_true h
+      exact Nat.sub_lt_sub_left k (Nat.lt_trans (String.lt_next text pos) (String.lt_next _ _))
+    decreasing_with
+      show text.utf8ByteSize - (text.next' pos _).byteIdx < text.utf8ByteSize - pos.byteIdx
+      have k := Nat.gt_of_not_le <| mt decide_eq_true h
+      exact Nat.sub_lt_sub_left k (String.lt_next _ _)
+
 end String

src/Init/Data/String/Lemmas.lean

@@ -0,0 +1,21 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Data.Char.Lemmas
+
+namespace String
+
+protected theorem data_eq_of_eq {a b : String} (h : a = b) : a.data = b.data :=
+  h ▸ rfl
+protected theorem ne_of_data_ne {a b : String} (h : a.data ≠ b.data) : a ≠ b :=
+  fun h' => absurd (String.data_eq_of_eq h') h
+@[simp] protected theorem lt_irrefl (s : String) : ¬ s < s :=
+  List.lt_irrefl' Char.lt_irrefl s.data
+protected theorem ne_of_lt {a b : String} (h : a < b) : a ≠ b := by
+  have := String.lt_irrefl a
+  intro h; subst h; contradiction
+
+end String

src/Init/Data/UInt.lean

@@ -6,3 +6,4 @@ Authors: Henrik Böving
 prelude
 import Init.Data.UInt.Basic
 import Init.Data.UInt.Log2
+import Init.Data.UInt.Lemmas

src/Init/Data/UInt/Basic.lean

@@ -103,7 +103,7 @@ def UInt16.shiftLeft (a b : UInt16) : UInt16 := ⟨a.val <<< (modn b 16).val⟩
 @[extern "lean_uint16_to_uint8"]
 def UInt16.toUInt8 (a : UInt16) : UInt8 := a.toNat.toUInt8
 @[extern "lean_uint8_to_uint16"]
-def UInt8.toUInt16 (a : UInt8) : UInt16 := a.toNat.toUInt16
+def UInt8.toUInt16 (a : UInt8) : UInt16 := ⟨a.val, Nat.lt_trans a.1.2 (by decide)⟩
 @[extern "lean_uint16_shift_right"]
 def UInt16.shiftRight (a b : UInt16) : UInt16 := ⟨a.val >>> (modn b 16).val⟩
 def UInt16.lt (a b : UInt16) : Prop := a.val < b.val
@@ -186,9 +186,9 @@ def UInt32.toUInt8 (a : UInt32) : UInt8 := a.toNat.toUInt8
 @[extern "lean_uint32_to_uint16"]
 def UInt32.toUInt16 (a : UInt32) : UInt16 := a.toNat.toUInt16
 @[extern "lean_uint8_to_uint32"]
-def UInt8.toUInt32 (a : UInt8) : UInt32 := a.toNat.toUInt32
+def UInt8.toUInt32 (a : UInt8) : UInt32 := ⟨a.val, Nat.lt_trans a.1.2 (by decide)⟩
 @[extern "lean_uint16_to_uint32"]
-def UInt16.toUInt32 (a : UInt16) : UInt32 := a.toNat.toUInt32
+def UInt16.toUInt32 (a : UInt16) : UInt32 := ⟨a.val, Nat.lt_trans a.1.2 (by decide)⟩
 
 instance UInt32.instOfNat : OfNat UInt32 n := ⟨UInt32.ofNat n⟩
 instance : Add UInt32       := ⟨UInt32.add⟩
@@ -244,11 +244,11 @@ def UInt64.toUInt16 (a : UInt64) : UInt16 := a.toNat.toUInt16
 @[extern "lean_uint64_to_uint32"]
 def UInt64.toUInt32 (a : UInt64) : UInt32 := a.toNat.toUInt32
 @[extern "lean_uint8_to_uint64"]
-def UInt8.toUInt64 (a : UInt8) : UInt64 := a.toNat.toUInt64
+def UInt8.toUInt64 (a : UInt8) : UInt64 := ⟨a.val, Nat.lt_trans a.1.2 (by decide)⟩
 @[extern "lean_uint16_to_uint64"]
-def UInt16.toUInt64 (a : UInt16) : UInt64 := a.toNat.toUInt64
+def UInt16.toUInt64 (a : UInt16) : UInt64 := ⟨a.val, Nat.lt_trans a.1.2 (by decide)⟩
 @[extern "lean_uint32_to_uint64"]
-def UInt32.toUInt64 (a : UInt32) : UInt64 := a.toNat.toUInt64
+def UInt32.toUInt64 (a : UInt32) : UInt64 := ⟨a.val, Nat.lt_trans a.1.2 (by decide)⟩
 
 instance UInt64.instOfNat : OfNat UInt64 n := ⟨UInt64.ofNat n⟩
 instance : Add UInt64       := ⟨UInt64.add⟩
@@ -321,7 +321,7 @@ def USize.shiftLeft (a b : USize) : USize := ⟨a.val <<< (modn b System.Platfor
 @[extern "lean_usize_shift_right"]
 def USize.shiftRight (a b : USize) : USize := ⟨a.val >>> (modn b System.Platform.numBits).val⟩
 @[extern "lean_uint32_to_usize"]
-def UInt32.toUSize (a : UInt32) : USize := a.toNat.toUSize
+def UInt32.toUSize (a : UInt32) : USize := USize.ofNat32 a.val a.1.2
 @[extern "lean_usize_to_uint32"]
 def USize.toUInt32 (a : USize) : UInt32 := a.toNat.toUInt32
 
@@ -364,6 +364,3 @@ instance (a b : USize) : Decidable (a < b) := USize.decLt a b
 instance (a b : USize) : Decidable (a ≤ b) := USize.decLe a b
 instance : Max USize := maxOfLe
 instance : Min USize := minOfLe
-
-theorem USize.modn_lt {m : Nat} : ∀ (u : USize), m > 0 → USize.toNat (u % m) < m
-  | ⟨u⟩, h => Fin.modn_lt u h

src/Init/Data/UInt/Lemmas.lean

@@ -0,0 +1,66 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Data.UInt.Basic
+import Init.Data.Fin.Lemmas
+
+set_option hygiene false in
+macro "declare_uint_theorems" typeName:ident : command =>
+`(
+namespace $typeName
+
+instance : Inhabited $typeName where
+  default := 0
+
+theorem zero_def : (0 : $typeName) = ⟨0⟩ := rfl
+theorem one_def : (1 : $typeName) = ⟨1⟩ := rfl
+theorem sub_def (a b : $typeName) : a - b = ⟨a.val - b.val⟩ := rfl
+theorem mul_def (a b : $typeName) : a * b = ⟨a.val * b.val⟩ := rfl
+theorem mod_def (a b : $typeName) : a % b = ⟨a.val % b.val⟩ := rfl
+theorem add_def (a b : $typeName) : a + b = ⟨a.val + b.val⟩ := rfl
+
+@[simp] theorem mk_val_eq : ∀ (a : $typeName), mk a.val = a
+  | ⟨_, _⟩ => rfl
+theorem val_eq_of_lt {a : Nat} : a < size → ((ofNat a).val : Nat) = a :=
+  Nat.mod_eq_of_lt
+
+theorem le_def {a b : $typeName} : a ≤ b ↔ a.1 ≤ b.1 := .rfl
+theorem lt_def {a b : $typeName} : a < b ↔ a.1 < b.1 := .rfl
+theorem lt_iff_val_lt_val {a b : $typeName} : a < b ↔ a.val < b.val := .rfl
+@[simp] protected theorem not_le {a b : $typeName} : ¬ a ≤ b ↔ b < a := Fin.not_le
+@[simp] protected theorem not_lt {a b : $typeName} : ¬ a < b ↔ b ≤ a := Fin.not_lt
+@[simp] protected theorem le_refl (a : $typeName) : a ≤ a := by simp [le_def]
+@[simp] protected theorem lt_irrefl (a : $typeName) : ¬ a < a := by simp
+protected theorem le_trans {a b c : $typeName} : a ≤ b → b ≤ c → a ≤ c := Fin.le_trans
+protected theorem lt_trans {a b c : $typeName} : a < b → b < c → a < c := Fin.lt_trans
+protected theorem le_total (a b : $typeName) : a ≤ b ∨ b ≤ a := Fin.le_total a.1 b.1
+protected theorem lt_asymm {a b : $typeName} (h : a < b) : ¬ b < a := Fin.lt_asymm h
+protected theorem val_eq_of_eq {a b : $typeName} (h : a = b) : a.val = b.val := h ▸ rfl
+protected theorem eq_of_val_eq {a b : $typeName} (h : a.val = b.val) : a = b := by cases a; cases b; simp at h; simp [h]
+open $typeName (val_eq_of_eq) in
+protected theorem ne_of_val_ne {a b : $typeName} (h : a.val ≠ b.val) : a ≠ b := fun h' => absurd (val_eq_of_eq h') h
+open $typeName (ne_of_val_ne) in
+protected theorem ne_of_lt {a b : $typeName} (h : a < b) : a ≠ b := ne_of_val_ne (Fin.ne_of_lt h)
+
+@[simp] protected theorem zero_toNat : (0 : $typeName).toNat = 0 := Nat.zero_mod _
+@[simp] protected theorem mod_toNat (a b : $typeName) : (a % b).toNat = a.toNat % b.toNat := Fin.mod_val ..
+@[simp] protected theorem div_toNat (a b : $typeName) : (a / b).toNat = a.toNat / b.toNat := Fin.div_val ..
+@[simp] protected theorem modn_toNat (a : $typeName) (b : Nat) : (a.modn b).toNat = a.toNat % b := Fin.modn_val ..
+protected theorem modn_lt {m : Nat} : ∀ (u : $typeName), m > 0 → toNat (u % m) < m
+  | ⟨u⟩, h => Fin.modn_lt u h
+open $typeName (modn_lt) in
+protected theorem mod_lt (a b : $typeName) (h : 0 < b) : a % b < b := modn_lt _ (by simp [lt_def] at h; exact h)
+protected theorem toNat.inj : ∀ {a b : $typeName}, a.toNat = b.toNat → a = b
+  | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
+
+end $typeName
+)
+
+declare_uint_theorems UInt8
+declare_uint_theorems UInt16
+declare_uint_theorems UInt32
+declare_uint_theorems UInt64
+declare_uint_theorems USize

src/Init/Grind.lean

@@ -0,0 +1,10 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Grind.Norm
+import Init.Grind.Tactics
+import Init.Grind.Lemmas
+import Init.Grind.Cases

src/Init/Grind/Cases.lean

@@ -0,0 +1,15 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Core
+
+attribute [grind_cases] And Prod False Empty True Unit Exists
+
+namespace Lean.Grind.Eager
+
+attribute [scoped grind_cases] Or
+
+end Lean.Grind.Eager

src/Init/Grind/Lemmas.lean

@@ -0,0 +1,14 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Core
+
+namespace Lean.Grind
+
+theorem intro_with_eq (p p' q : Prop) (he : p = p') (h : p' → q) : p → q :=
+  fun hp => h (he.mp hp)
+
+end Lean.Grind

src/Init/Grind/Norm.lean

@@ -0,0 +1,110 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.SimpLemmas
+import Init.Classical
+import Init.ByCases
+
+namespace Lean.Grind
+/-!
+Normalization theorems for the `grind` tactic.
+
+We are also going to use simproc's in the future.
+-/
+
+-- Not
+attribute [grind_norm] Classical.not_not
+
+-- Ne
+attribute [grind_norm] ne_eq
+
+-- Iff
+@[grind_norm] theorem iff_eq (p q : Prop) : (p ↔ q) = (p = q) := by
+  by_cases p <;> by_cases q <;> simp [*]
+
+-- Eq
+attribute [grind_norm] eq_self heq_eq_eq
+
+-- Prop equality
+@[grind_norm] theorem eq_true_eq (p : Prop) : (p = True) = p := by simp
+@[grind_norm] theorem eq_false_eq (p : Prop) : (p = False) = ¬p := by simp
+@[grind_norm] theorem not_eq_prop (p q : Prop) : (¬(p = q)) = (p = ¬q) := by
+  by_cases p <;> by_cases q <;> simp [*]
+
+-- True
+attribute [grind_norm] not_true
+
+-- False
+attribute [grind_norm] not_false_eq_true
+
+-- Implication as a clause
+@[grind_norm] theorem imp_eq (p q : Prop) : (p → q) = (¬ p ∨ q) := by
+  by_cases p <;> by_cases q <;> simp [*]
+
+-- And
+@[grind_norm↓] theorem not_and (p q : Prop) : (¬(p ∧ q)) = (¬p ∨ ¬q) := by
+  by_cases p <;> by_cases q <;> simp [*]
+attribute [grind_norm] and_true true_and and_false false_and and_assoc
+
+-- Or
+attribute [grind_norm↓] not_or
+attribute [grind_norm] or_true true_or or_false false_or or_assoc
+
+-- ite
+attribute [grind_norm] ite_true ite_false
+@[grind_norm↓] theorem not_ite {_ : Decidable p} (q r : Prop) : (¬ite p q r) = ite p (¬q) (¬r) := by
+  by_cases p <;> simp [*]
+
+-- Forall
+@[grind_norm↓] theorem not_forall (p : α → Prop) : (¬∀ x, p x) = ∃ x, ¬p x := by simp
+attribute [grind_norm] forall_and
+
+-- Exists
+@[grind_norm↓] theorem not_exists (p : α → Prop) : (¬∃ x, p x) = ∀ x, ¬p x := by simp
+attribute [grind_norm] exists_const exists_or
+
+-- Bool cond
+@[grind_norm] theorem cond_eq_ite (c : Bool) (a b : α) : cond c a b = ite c a b := by
+  cases c <;> simp [*]
+
+-- Bool or
+attribute [grind_norm]
+  Bool.or_false Bool.or_true Bool.false_or Bool.true_or Bool.or_eq_true Bool.or_assoc
+
+-- Bool and
+attribute [grind_norm]
+  Bool.and_false Bool.and_true Bool.false_and Bool.true_and Bool.and_eq_true Bool.and_assoc
+
+-- Bool not
+attribute [grind_norm]
+  Bool.not_not
+
+-- beq
+attribute [grind_norm] beq_iff_eq
+
+-- bne
+attribute [grind_norm] bne_iff_ne
+
+-- Bool not eq true/false
+attribute [grind_norm] Bool.not_eq_true Bool.not_eq_false
+
+-- decide
+attribute [grind_norm] decide_eq_true_eq decide_not not_decide_eq_true
+
+-- Nat LE
+attribute [grind_norm] Nat.le_zero_eq
+
+-- Nat/Int LT
+@[grind_norm] theorem Nat.lt_eq (a b : Nat) : (a < b) = (a + 1 ≤ b) := by
+  simp [Nat.lt, LT.lt]
+
+@[grind_norm] theorem Int.lt_eq (a b : Int) : (a < b) = (a + 1 ≤ b) := by
+  simp [Int.lt, LT.lt]
+
+-- GT GE
+attribute [grind_norm] GT.gt GE.ge
+
+end Lean.Grind

src/Init/Grind/Tactics.lean

@@ -0,0 +1,25 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Tactics
+
+namespace Lean.Grind
+/--
+The configuration for `grind`.
+Passed to `grind` using, for example, the `grind (config := { eager := true })` syntax.
+-/
+structure Config where
+  /--
+  When `eager` is true (default: `false`), `grind` eagerly splits `if-then-else` and `match`
+  expressions.
+  -/
+  eager : Bool := false
+  deriving Inhabited, BEq
+
+/-!
+`grind` tactic and related tactics.
+-/
+end Lean.Grind

src/Init/Meta.lean

@@ -1057,6 +1057,7 @@ where
     else
       Syntax.mkCApp (Name.mkStr2 "Array" ("mkArray" ++ toString xs.size)) args
   termination_by xs.size - i
+  decreasing_by decreasing_trivial_pre_omega
 
 instance [Quote α `term] : Quote (Array α) `term where
   quote := quoteArray

src/Init/MetaTypes.lean

@@ -68,45 +68,121 @@ namespace Simp
 
 def defaultMaxSteps := 100000
 
+/--
+The configuration for `simp`.
+Passed to `simp` using, for example, the `simp (config := {contextual := true})` syntax.
+
+See also `Lean.Meta.Simp.neutralConfig`.
+-/
 structure Config where
+  /--
+  The maximum number of subexpressions to visit when performing simplification.
+  The default is 100000.
+  -/
   maxSteps          : Nat  := defaultMaxSteps
+  /--
+  When simp discharges side conditions for conditional lemmas, it can recursively apply simplification.
+  The `maxDischargeDepth` (default: 2) is the maximum recursion depth when recursively applying simplification to side conditions.
+  -/
   maxDischargeDepth : Nat  := 2
+  /--
+  When `contextual` is true (default: `false`) and simplification encounters an implication `p → q`
+  it includes `p` as an additional simp lemma when simplifying `q`.
+  -/
   contextual        : Bool := false
+  /--
+  When true (default: `true`) then the simplifier caches the result of simplifying each subexpression, if possible.
+  -/
   memoize           : Bool := true
+  /--
+  When `singlePass` is `true` (default: `false`), the simplifier runs through a single round of simplification,
+  which consists of running pre-methods, recursing using congruence lemmas, and then running post-methods.
+  Otherwise, when it is `false`, it iteratively applies this simplification procedure.
+  -/
   singlePass        : Bool := false
-  /-- `let x := v; e[x]` reduces to `e[v]`. -/
+  /--
+  When `true` (default: `true`), performs zeta reduction of let expressions.
+  That is, `let x := v; e[x]` reduces to `e[v]`.
+  See also `zetaDelta`.
+  -/
   zeta              : Bool := true
+  /--
+  When `true` (default: `true`), performs beta reduction of applications of `fun` expressions.
+  That is, `(fun x => e[x]) v` reduces to `e[v]`.
+  -/
   beta              : Bool := true
+  /--
+  TODO (currently unimplemented). When `true` (default: `true`), performs eta reduction for `fun` expressions.
+  That is, `(fun x => f x)` reduces to `f`.
+  -/
   eta               : Bool := true
+  /--
+  Configures how to determine definitional equality between two structure instances.
+  See documentation for `Lean.Meta.EtaStructMode`.
+  -/
   etaStruct         : EtaStructMode := .all
+  /--
+  When `true` (default: `true`), reduces `match` expressions applied to constructors.
+  -/
   iota              : Bool := true
+  /--
+  When `true` (default: `true`), reduces projections of structure constructors.
+  -/
   proj              : Bool := true
+  /--
+  When `true` (default: `false`), rewrites a proposition `p` to `True` or `False` by inferring
+  a `Decidable p` instance and reducing it.
+  -/
   decide            : Bool := false
+  /--  When `true` (default: `false`), simplifies simple arithmetic expressions. -/
   arith             : Bool := false
+  /--
+  When `true` (default: `false`), unfolds definitions.
+  This can be enabled using the `simp!` syntax.
+  -/
   autoUnfold        : Bool := false
   /--
-    If `dsimp := true`, then switches to `dsimp` on dependent arguments where there is no congruence theorem that allows
-    `simp` to visit them. If `dsimp := false`, then argument is not visited.
+  When `true` (default: `true`) then switches to `dsimp` on dependent arguments
+  if there is no congruence theorem that would allow `simp` to visit them.
+  When `dsimp` is `false`, then the argument is not visited.
   -/
   dsimp             : Bool := true
-  /-- If `failIfUnchanged := true`, then calls to `simp`, `dsimp`, or `simp_all`
-  will fail if they do not make progress. -/
+  /--
+  If `failIfUnchanged` is `true` (default: `true`), then calls to `simp`, `dsimp`, or `simp_all`
+  will fail if they do not make progress.
+  -/
   failIfUnchanged   : Bool := true
-  /-- If `ground := true`, then ground terms are reduced. A term is ground when
-  it does not contain free or meta variables. Reduction is interrupted at a function application `f ...`
-  if `f` is marked to not be unfolded. -/
+  /--
+  If `ground` is `true` (default: `false`), then ground terms are reduced.
+  A term is ground when it does not contain free or meta variables.
+  Reduction is interrupted at a function application `f ...` if `f` is marked to not be unfolded.
+  Ground term reduction applies `@[seval]` lemmas.
+  -/
   ground            : Bool := false
-  /-- If `unfoldPartialApp := true`, then calls to `simp`, `dsimp`, or `simp_all`
-  will unfold even partial applications of `f` when we request `f` to be unfolded. -/
+  /--
+  If `unfoldPartialApp` is `true` (default: `false`), then calls to `simp`, `dsimp`, or `simp_all`
+  will unfold even partial applications of `f` when we request `f` to be unfolded.
+  -/
   unfoldPartialApp  : Bool := false
-  /-- Given a local context containing entry `x : t := e`, free variable `x` reduces to `e`. -/
+  /--
+  When `true` (default: `false`), local definitions are unfolded.
+  That is, given a local context containing entry `x : t := e`, the free variable `x` reduces to `e`.
+  -/
   zetaDelta         : Bool := false
+  /--
+  When `index` (default : `true`) is `false`, `simp` will only use the root symbol
+  to find candidate `simp` theorems. It approximates Lean 3 `simp` behavior.
+  -/
+  index             : Bool := true
   deriving Inhabited, BEq
 
 -- Configuration object for `simp_all`
 structure ConfigCtx extends Config where
   contextual := true
 
+/--
+A neutral configuration for `simp`, turning off all reductions and other built-in simplifications.
+-/
 def neutralConfig : Simp.Config := {
   zeta              := false
   beta              := false

src/Init/Notation.lean

@@ -296,7 +296,7 @@ macro_rules | `($x - $y)   => `(binop% HSub.hSub $x $y)
 macro_rules | `($x * $y)   => `(binop% HMul.hMul $x $y)
 macro_rules | `($x / $y)   => `(binop% HDiv.hDiv $x $y)
 macro_rules | `($x % $y)   => `(binop% HMod.hMod $x $y)
--- exponentiation should be considered a right action (#2220)
+-- exponentiation should be considered a right action (#2854)
 macro_rules | `($x ^ $y)   => `(rightact% HPow.hPow $x $y)
 macro_rules | `($x ++ $y)  => `(binop% HAppend.hAppend $x $y)
 macro_rules | `(- $x)      => `(unop% Neg.neg $x)
@@ -492,9 +492,12 @@ The attribute `@[deprecated]` on a declaration indicates that the declaration
 is discouraged for use in new code, and/or should be migrated away from in
 existing code. It may be removed in a future version of the library.
 
-`@[deprecated myBetterDef]` means that `myBetterDef` is the suggested replacement.
+* `@[deprecated myBetterDef]` means that `myBetterDef` is the suggested replacement.
+* `@[deprecated myBetterDef "use myBetterDef instead"]` allows customizing the deprecation message.
+* `@[deprecated (since := "2024-04-21")]` records when the deprecation was first applied.
 -/
-syntax (name := deprecated) "deprecated" (ppSpace ident)? : attr
+syntax (name := deprecated) "deprecated" (ppSpace ident)? (ppSpace str)?
+    (" (" &"since" " := " str ")")? : attr
 
 /--
 The `@[coe]` attribute on a function (which should also appear in a
@@ -552,15 +555,52 @@ except that it doesn't print an empty diagnostic.
 -/
 syntax (name := runMeta) "run_meta " doSeq : command
 
-/-- Element that can be part of a `#guard_msgs` specification. -/
-syntax guardMsgsSpecElt := &"drop"? (&"info" <|> &"warning" <|> &"error" <|> &"all")
+set_option linter.missingDocs false in
+syntax guardMsgsFilterSeverity := &"info" <|> &"warning" <|> &"error" <|> &"all"
 
-/-- Specification for `#guard_msgs` command. -/
+/--
+A message filter specification for `#guard_msgs`.
+- `info`, `warning`, `error`: capture messages with the given severity level.
+- `all`: capture all messages (the default).
+- `drop info`, `drop warning`, `drop error`: drop messages with the given severity level.
+- `drop all`: drop every message.
+These filters are processed in left-to-right order.
+-/
+syntax guardMsgsFilter := &"drop"? guardMsgsFilterSeverity
+
+set_option linter.missingDocs false in
+syntax guardMsgsWhitespaceArg := &"exact" <|> &"normalized" <|> &"lax"
+
+/--
+Whitespace handling for `#guard_msgs`:
+- `whitespace := exact` requires an exact whitespace match.
+- `whitespace := normalized` converts all newline characters to a space before matching
+  (the default). This allows breaking long lines.
+- `whitespace := lax` collapses whitespace to a single space before matching.
+In all cases, leading and trailing whitespace is trimmed before matching.
+-/
+syntax guardMsgsWhitespace := &"whitespace" " := " guardMsgsWhitespaceArg
+
+set_option linter.missingDocs false in
+syntax guardMsgsOrderingArg := &"exact" <|> &"sorted"
+
+/--
+Message ordering for `#guard_msgs`:
+- `ordering := exact` uses the exact ordering of the messages (the default).
+- `ordering := sorted` sorts the messages in lexicographic order.
+  This helps with testing commands that are non-deterministic in their ordering.
+-/
+syntax guardMsgsOrdering := &"ordering" " := " guardMsgsOrderingArg
+
+set_option linter.missingDocs false in
+syntax guardMsgsSpecElt := guardMsgsFilter <|> guardMsgsWhitespace <|> guardMsgsOrdering
+
+set_option linter.missingDocs false in
 syntax guardMsgsSpec := "(" guardMsgsSpecElt,* ")"
 
 /--
-`#guard_msgs` captures the messages generated by another command and checks that they
-match the contents of the docstring attached to the `#guard_msgs` command.
+`/-- ... -/ #guard_msgs in cmd` captures the messages generated by the command `cmd`
+and checks that they match the contents of the docstring.
 
 Basic example:
 ```lean
@@ -570,10 +610,10 @@ error: unknown identifier 'x'
 #guard_msgs in
 example : α := x
 ```
-This checks that there is such an error and then consumes the message entirely.
+This checks that there is such an error and then consumes the message.
 
-By default, the command intercepts all messages, but there is a way to specify which types
-of messages to consider. For example, we can select only warnings:
+By default, the command captures all messages, but the filter condition can be adjusted.
+For example, we can select only warnings:
 ```lean
 /--
 warning: declaration uses 'sorry'
@@ -586,29 +626,37 @@ or only errors
 #guard_msgs(error) in
 example : α := sorry
 ```
-In this last example, since the message is not intercepted there is a warning on `sorry`.
+In the previous example, since warnings are not captured there is a warning on `sorry`.
 We can drop the warning completely with
 ```lean
 #guard_msgs(error, drop warning) in
 example : α := sorry
 ```
 
-Syntax description:
+In general, `#guard_msgs` accepts a comma-separated list of configuration clauses in parentheses:
 ```
-#guard_msgs (drop? info|warning|error|all,*)? in cmd
+#guard_msgs (configElt,*) in cmd
 ```
+By default, the configuration list is `(all, whitespace := normalized, ordering := exact)`.
 
-If there is no specification, `#guard_msgs` intercepts all messages.
-Otherwise, if there is one, the specification is considered in left-to-right order, and the first
-that applies chooses the outcome of the message:
-- `info`, `warning`, `error`: intercept a message with the given severity level.
-- `all`: intercept any message (so `#guard_msgs in cmd` and `#guard_msgs (all) in cmd`
-  are equivalent).
-- `drop info`, `drop warning`, `drop error`: intercept a message with the given severity
-  level and then drop it. These messages are not checked.
-- `drop all`: intercept a message and drop it.
+Message filters (processed in left-to-right order):
+- `info`, `warning`, `error`: capture messages with the given severity level.
+- `all`: capture all messages (the default).
+- `drop info`, `drop warning`, `drop error`: drop messages with the given severity level.
+- `drop all`: drop every message.
 
-For example, `#guard_msgs (error, drop all) in cmd` means to check warnings and then drop
+Whitespace handling (after trimming leading and trailing whitespace):
+- `whitespace := exact` requires an exact whitespace match.
+- `whitespace := normalized` converts all newline characters to a space before matching
+  (the default). This allows breaking long lines.
+- `whitespace := lax` collapses whitespace to a single space before matching.
+
+Message ordering:
+- `ordering := exact` uses the exact ordering of the messages (the default).
+- `ordering := sorted` sorts the messages in lexicographic order.
+  This helps with testing commands that are non-deterministic in their ordering.
+
+For example, `#guard_msgs (error, drop all) in cmd` means to check warnings and drop
 everything else.
 -/
 syntax (name := guardMsgsCmd)
@@ -639,4 +687,27 @@ syntax (name := checkSimp) "#check_simp " term "~>" term : command
 -/
 syntax (name := checkSimpFailure) "#check_simp " term "!~>" : command
 
+/--
+The `seal foo` command ensures that the definition of `foo` is sealed, meaning it is marked as `[irreducible]`.
+This command is particularly useful in contexts where you want to prevent the reduction of `foo` in proofs.
+
+In terms of functionality, `seal foo` is equivalent to `attribute [local irreducible] foo`.
+This attribute specifies that `foo` should be treated as irreducible only within the local scope,
+which helps in maintaining the desired abstraction level without affecting global settings.
+-/
+syntax "seal " (ppSpace ident)+ : command
+
+/--
+The `unseal foo` command ensures that the definition of `foo` is unsealed, meaning it is marked as `[semireducible]`, the
+default reducibility setting. This command is useful when you need to allow some level of reduction of `foo` in proofs.
+
+Functionally, `unseal foo` is equivalent to `attribute [local semireducible] foo`.
+Applying this attribute makes `foo` semireducible only within the local scope.
+-/
+syntax "unseal " (ppSpace ident)+ : command
+
+macro_rules
+  | `(seal $fs:ident*) => `(attribute [local irreducible] $fs:ident*)
+  | `(unseal $fs:ident*) => `(attribute [local semireducible] $fs:ident*)
+
 end Parser

src/Init/NotationExtra.lean

@@ -87,6 +87,7 @@ macro:35 xs:bracketedExplicitBinders " × " b:term:35  : term => expandBrackedBi
 macro:35 xs:bracketedExplicitBinders " ×' " b:term:35 : term => expandBrackedBinders ``PSigma xs b
 end
 
+namespace Lean
 -- first step of a `calc` block
 syntax calcFirstStep := ppIndent(colGe term (" := " term)?)
 -- enforce indentation of calc steps so we know when to stop parsing them
@@ -136,6 +137,7 @@ syntax (name := calcTactic) "calc" calcSteps : tactic
 @[inherit_doc «calc»]
 macro tk:"calc" steps:calcSteps : conv =>
   `(conv| tactic => calc%$tk $steps)
+end Lean
 
 @[app_unexpander Unit.unit] def unexpandUnit : Lean.PrettyPrinter.Unexpander
   | `($(_)) => `(())
@@ -361,6 +363,7 @@ macro_rules
   | `(letI $_:ident $_* : $_ := $_; $_) => Lean.Macro.throwUnsupported -- handled by elab
 
 
+namespace Lean
 syntax cdotTk := patternIgnore("· " <|> ". ")
 /-- `· tac` focuses on the main goal and tries to solve it using `tac`, or else fails. -/
 syntax (name := cdot) cdotTk tacticSeqIndentGt : tactic
@@ -368,12 +371,11 @@ syntax (name := cdot) cdotTk tacticSeqIndentGt : tactic
 /--
   Similar to `first`, but succeeds only if one the given tactics solves the current goal.
 -/
-syntax (name := solve) "solve" withPosition((ppDedent(ppLine) colGe "| " tacticSeq)+) : tactic
+syntax (name := solveTactic) "solve" withPosition((ppDedent(ppLine) colGe "| " tacticSeq)+) : tactic
 
 macro_rules
   | `(tactic| solve $[| $ts]* ) => `(tactic| focus first $[| ($ts); done]*)
 
-namespace Lean
 /-! # `repeat` and `while` notation -/
 
 inductive Loop where

src/Init/Omega/Coeffs.lean

@@ -68,7 +68,7 @@ abbrev map (f : Int → Int) (xs : Coeffs) : Coeffs := List.map f xs
 /-- Shim for `.enum.find?`. -/
 abbrev findIdx? (f : Int → Bool) (xs : Coeffs) : Option Nat :=
   -- List.findIdx? f xs
-  -- We could avoid `Std.Data.List.Basic` by using the less efficient:
+  -- We could avoid `Batteries.Data.List.Basic` by using the less efficient:
   xs.enum.find? (f ·.2) |>.map (·.1)
 /-- Shim for `IntList.bmod`. -/
 abbrev bmod (x : Coeffs) (m : Nat) : Coeffs := IntList.bmod x m