chore: remove unneeded Batteries reference in repeat doc-string

this fixes a usability paper cut that just annoyed me. When editing a
larger simp proof, I usually want to see the goal state after the simp,
and this is what I see while the `simp` command is complete. But then,
when I start typing, and necessarily type incomplete lemma names, that
error makes `simp` do nothing again and I see the original goal state.
In fact, if a prefix of the simp theorem name I am typing is a valid
identifier, it jumps even more around.

With this PR, using `logException`, I still get the red squiggly lines
for the unknown identifer, but `simp` just ignores that argument and
still shows me the final goal. Much nicer.

I also demoted the message for `[-foo]` when `foo` isn’t `simp` to a
warning and gave it the correct `ref`.

See it in action here: (in the middle, when you suddenly see the
terminal,
I am switching lean versions.)


https://github.com/leanprover/lean4/assets/148037/8cb3c563-1354-4c2d-bcee-26dfa1005ae0

.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
@@ -21,8 +20,10 @@ jobs:
   configure:
     runs-on: ubuntu-latest
     outputs:
-      # Should we run only a quick CI? Yes on a pull request without the full-ci label
-      quick: ${{ steps.set-quick.outputs.quick }}
+      # 0: PRs without special label
+      # 1: PRs with `merge-ci` label, merge queue checks, master commits
+      # 2: PRs with `release-ci` label, releases (incl. nightlies)
+      check-level: ${{ steps.set-level.outputs.check-level }}
       # The build matrix, dynamically generated here
       matrix: ${{ steps.set-matrix.outputs.result }}
       # Should we make a nightly release? If so, this output contains the lean version string, else it is empty
@@ -39,159 +40,6 @@ jobs:
       RELEASE_TAG: ${{ steps.set-release.outputs.RELEASE_TAG }}
 
     steps:
-      - name: Run quick CI?
-        id: set-quick
-        env:
-          quick: ${{
-            github.event_name == 'pull_request' && !contains( github.event.pull_request.labels.*.name, 'full-ci')
-           }}
-        run: |
-          echo "quick=${{env.quick}}" >> "$GITHUB_OUTPUT"
-
-      - name: Configure build matrix
-        id: set-matrix
-        uses: actions/github-script@v7
-        with:
-          script: |
-            const quick = ${{ steps.set-quick.outputs.quick }};
-            console.log(`quick: ${quick}`)
-            let matrix = [
-              {
-                // portable release build: use channel with older glibc (2.27)
-                "name": "Linux LLVM",
-                "os": "ubuntu-latest",
-                "release": false,
-                "quick": false,
-                "shell": "nix-shell --arg pkgsDist \"import (fetchTarball \\\"channel:nixos-19.03\\\") {{}}\" --run \"bash -euxo pipefail {0}\"",
-                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-linux-gnu.tar.zst",
-                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm*",
-                "binary-check": "ldd -v",
-                // foreign code may be linked against more recent glibc
-                // reverse-ffi needs to be updated to link to LLVM libraries
-                "CTEST_OPTIONS": "-E 'foreign|leanlaketest_reverse-ffi'",
-                "CMAKE_OPTIONS": "-DLLVM=ON -DLLVM_CONFIG=${GITHUB_WORKSPACE}/build/llvm-host/bin/llvm-config"
-              },
-              {
-                "name": "Linux release",
-                "os": "ubuntu-latest",
-                "release": true,
-                "quick": true,
-                "shell": "nix-shell --arg pkgsDist \"import (fetchTarball \\\"channel:nixos-19.03\\\") {{}}\" --run \"bash -euxo pipefail {0}\"",
-                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-linux-gnu.tar.zst",
-                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm*",
-                "binary-check": "ldd -v",
-                // foreign code may be linked against more recent glibc
-                "CTEST_OPTIONS": "-E 'foreign'"
-              },
-              {
-                "name": "Linux",
-                "os": "ubuntu-latest",
-                "check-stage3": true,
-                "test-speedcenter": true,
-                "quick": false,
-              },
-              {
-                "name": "Linux Debug",
-                "os": "ubuntu-latest",
-                "quick": false,
-                "CMAKE_OPTIONS": "-DCMAKE_BUILD_TYPE=Debug",
-                // exclude seriously slow tests
-                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
-              },
-              {
-                "name": "Linux fsanitize",
-                "os": "ubuntu-latest",
-                "quick": false,
-                // turn off custom allocator & symbolic functions to make LSAN do its magic
-                "CMAKE_OPTIONS": "-DLEAN_EXTRA_CXX_FLAGS=-fsanitize=address,undefined -DLEANC_EXTRA_FLAGS='-fsanitize=address,undefined -fsanitize-link-c++-runtime' -DSMALL_ALLOCATOR=OFF -DBSYMBOLIC=OFF",
-                // exclude seriously slow/problematic tests (laketests crash)
-                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
-              },
-              {
-                "name": "macOS",
-                "os": "macos-latest",
-                "release": true,
-                "quick": false,
-                "shell": "bash -euxo pipefail {0}",
-                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-apple-darwin.tar.zst",
-                "prepare-llvm": "../script/prepare-llvm-macos.sh lean-llvm*",
-                "binary-check": "otool -L",
-                "tar": "gtar" // https://github.com/actions/runner-images/issues/2619
-              },
-              {
-                "name": "macOS aarch64",
-                "os": "macos-latest",
-                "release": true,
-                "quick": false,
-                "cross": true,
-                "cross_target": "aarch64-apple-darwin",
-                "shell": "bash -euxo pipefail {0}",
-                "CMAKE_OPTIONS": "-DUSE_GMP=OFF -DLEAN_INSTALL_SUFFIX=-darwin_aarch64",
-                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-aarch64-apple-darwin.tar.zst https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-apple-darwin.tar.zst",
-                "prepare-llvm": "../script/prepare-llvm-macos.sh lean-llvm-aarch64-* lean-llvm-x86_64-*",
-                "binary-check": "otool -L",
-                "tar": "gtar" // https://github.com/actions/runner-images/issues/2619
-              },
-              {
-                "name": "Windows",
-                "os": "windows-2022",
-                "release": true,
-                "quick": false,
-                "shell": "msys2 {0}",
-                "CMAKE_OPTIONS": "-G \"Unix Makefiles\" -DUSE_GMP=OFF",
-                // for reasons unknown, interactivetests are flaky on Windows
-                // also, the liasolver test hits “too many exported symbols”
-                "CTEST_OPTIONS": "--repeat until-pass:2 -E 'leanbenchtest_liasolver.lean'",
-                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-w64-windows-gnu.tar.zst",
-                "prepare-llvm": "../script/prepare-llvm-mingw.sh lean-llvm*",
-                // TEMP while compiler tests are deactivated
-                "binary-check": "true"
-              },
-              {
-                "name": "Linux aarch64",
-                "os": "ubuntu-latest",
-                "CMAKE_OPTIONS": "-DUSE_GMP=OFF -DLEAN_INSTALL_SUFFIX=-linux_aarch64",
-                "release": true,
-                "quick": false,
-                "cross": true,
-                "cross_target": "aarch64-unknown-linux-gnu",
-                "shell": "nix-shell --arg pkgsDist \"import (fetchTarball \\\"channel:nixos-19.03\\\") {{ localSystem.config = \\\"aarch64-unknown-linux-gnu\\\"; }}\" --run \"bash -euxo pipefail {0}\"",
-                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-linux-gnu.tar.zst https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-aarch64-linux-gnu.tar.zst",
-                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm-aarch64-* lean-llvm-x86_64-*"
-              },
-              {
-                "name": "Linux 32bit",
-                "os": "ubuntu-latest",
-                // Use 32bit on stage0 and stage1 to keep oleans compatible
-                "CMAKE_OPTIONS": "-DSTAGE0_USE_GMP=OFF -DSTAGE0_LEAN_EXTRA_CXX_FLAGS='-m32' -DSTAGE0_LEANC_OPTS='-m32' -DSTAGE0_MMAP=OFF -DUSE_GMP=OFF -DLEAN_EXTRA_CXX_FLAGS='-m32' -DLEANC_OPTS='-m32' -DMMAP=OFF -DLEAN_INSTALL_SUFFIX=-linux_x86",
-                "cmultilib": true,
-                "release": true,
-                "quick": false,
-                "cross": true,
-                "shell": "bash -euxo pipefail {0}"
-              },
-              {
-                "name": "Web Assembly",
-                "os": "ubuntu-latest",
-                // Build a native 32bit binary in stage0 and use it to compile the oleans and the wasm build
-                "CMAKE_OPTIONS": "-DCMAKE_C_COMPILER_WORKS=1 -DSTAGE0_USE_GMP=OFF -DSTAGE0_LEAN_EXTRA_CXX_FLAGS='-m32' -DSTAGE0_LEANC_OPTS='-m32' -DSTAGE0_CMAKE_CXX_COMPILER=clang++ -DSTAGE0_CMAKE_C_COMPILER=clang -DSTAGE0_CMAKE_EXECUTABLE_SUFFIX=\"\" -DUSE_GMP=OFF -DMMAP=OFF -DSTAGE0_MMAP=OFF -DCMAKE_AR=../emsdk/emsdk-main/upstream/emscripten/emar -DCMAKE_TOOLCHAIN_FILE=../emsdk/emsdk-main/upstream/emscripten/cmake/Modules/Platform/Emscripten.cmake -DLEAN_INSTALL_SUFFIX=-linux_wasm32",
-                "wasm": true,
-                "cmultilib": true,
-                "release": true,
-                "quick": false,
-                "cross": true,
-                "shell": "bash -euxo pipefail {0}",
-                // Just a few selected tests because wasm is slow
-                "CTEST_OPTIONS": "-R \"leantest_1007\\.lean|leantest_Format\\.lean|leanruntest\\_1037.lean|leanruntest_ac_rfl\\.lean\""
-              }
-            ];
-            console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`)
-            if (quick) {
-              return matrix.filter((job) => job.quick)
-            } else {
-              return matrix
-            }
-
       - name: Checkout
         uses: actions/checkout@v3
         # don't schedule nightlies on forks
@@ -242,6 +90,171 @@ jobs:
             echo "Tag ${TAG_NAME} did not match SemVer regex."
           fi
 
+      - name: Set check level
+        id: set-level
+        # 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 setting the `release-ci`/`merge-ci` labels.
+        run: |
+          check_level=0
+
+          if [[ -n "${{ steps.set-nightly.outputs.nightly }}" || -n "${{ steps.set-release.outputs.RELEASE_TAG }}" ]]; then
+            check_level=2
+          elif [[ "${{ github.event_name }}" != "pull_request" ]]; then
+            check_level=1
+          else
+            labels="$(gh api repos/${{ github.repository_owner }}/${{ github.event.repository.name }}/pulls/${{ github.event.pull_request.number }}) --jq '.labels'"
+            if echo "$labels" | grep -q "release-ci"; then
+              check_level=2
+            elif echo "$labels" | grep -q "merge-ci"; then
+              check_level=1
+            fi
+          fi
+
+          echo "check-level=$check_level" >> "$GITHUB_OUTPUT"
+        env:
+          GH_TOKEN: ${{ github.token }}
+
+      - name: Configure build matrix
+        id: set-matrix
+        uses: actions/github-script@v7
+        with:
+          script: |
+            const level = ${{ steps.set-level.outputs.check-level }};
+            console.log(`level: ${level}`);
+            // 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 = ${{ github.event_name != 'pull_request' && github.repository == 'leanprover/lean4' }} ? "-large" : "";
+            let matrix = [
+              {
+                // portable release build: use channel with older glibc (2.27)
+                "name": "Linux LLVM",
+                "os": "ubuntu-latest",
+                "release": false,
+                "check-level": 2,
+                "shell": "nix develop .#oldGlibc -c bash -euxo pipefail {0}",
+                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-linux-gnu.tar.zst",
+                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm*",
+                "binary-check": "ldd -v",
+                // foreign code may be linked against more recent glibc
+                // reverse-ffi needs to be updated to link to LLVM libraries
+                "CTEST_OPTIONS": "-E 'foreign|leanlaketest_reverse-ffi'",
+                "CMAKE_OPTIONS": "-DLLVM=ON -DLLVM_CONFIG=${GITHUB_WORKSPACE}/build/llvm-host/bin/llvm-config"
+              },
+              {
+                "name": "Linux release",
+                "os": "ubuntu-latest",
+                "release": true,
+                "check-level": 0,
+                "shell": "nix develop .#oldGlibc -c bash -euxo pipefail {0}",
+                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-linux-gnu.tar.zst",
+                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm*",
+                "binary-check": "ldd -v",
+                // foreign code may be linked against more recent glibc
+                "CTEST_OPTIONS": "-E 'foreign'"
+              },
+              {
+                "name": "Linux",
+                "os": "ubuntu-latest",
+                "check-stage3": level >= 2,
+                "test-speedcenter": level >= 2,
+                "check-level": 1,
+              },
+              {
+                "name": "Linux Debug",
+                "os": "ubuntu-latest",
+                "check-level": 2,
+                "CMAKE_PRESET": "debug",
+                // exclude seriously slow tests
+                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
+              },
+              // TODO: suddenly started failing in CI
+              /*{
+                "name": "Linux fsanitize",
+                "os": "ubuntu-latest",
+                "check-level": 2,
+                // turn off custom allocator & symbolic functions to make LSAN do its magic
+                "CMAKE_PRESET": "sanitize",
+                // exclude seriously slow/problematic tests (laketests crash)
+                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
+              },*/
+              {
+                "name": "macOS",
+                "os": "macos-13",
+                "release": true,
+                "check-level": 2,
+                "shell": "bash -euxo pipefail {0}",
+                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-apple-darwin.tar.zst",
+                "prepare-llvm": "../script/prepare-llvm-macos.sh lean-llvm*",
+                "binary-check": "otool -L",
+                "tar": "gtar" // https://github.com/actions/runner-images/issues/2619
+              },
+              {
+                "name": "macOS aarch64",
+                "os": "macos-14",
+                "CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-darwin_aarch64",
+                "release": true,
+                "check-level": 1,
+                "shell": "bash -euxo pipefail {0}",
+                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-aarch64-apple-darwin.tar.zst",
+                "prepare-llvm": "../script/prepare-llvm-macos.sh lean-llvm*",
+                "binary-check": "otool -L",
+                "tar": "gtar" // https://github.com/actions/runner-images/issues/2619
+              },
+              {
+                "name": "Windows",
+                "os": "windows-2022",
+                "release": true,
+                "check-level": 2,
+                "shell": "msys2 {0}",
+                "CMAKE_OPTIONS": "-G \"Unix Makefiles\" -DUSE_GMP=OFF",
+                // for reasons unknown, interactivetests are flaky on Windows
+                "CTEST_OPTIONS": "--repeat until-pass:2",
+                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-w64-windows-gnu.tar.zst",
+                "prepare-llvm": "../script/prepare-llvm-mingw.sh lean-llvm*",
+                "binary-check": "ldd"
+              },
+              {
+                "name": "Linux aarch64",
+                "os": "ubuntu-latest",
+                "CMAKE_OPTIONS": "-DUSE_GMP=OFF -DLEAN_INSTALL_SUFFIX=-linux_aarch64",
+                "release": true,
+                "check-level": 2,
+                "cross": true,
+                "cross_target": "aarch64-unknown-linux-gnu",
+                "shell": "nix develop .#oldGlibcAArch -c bash -euxo pipefail {0}",
+                "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-linux-gnu.tar.zst https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-aarch64-linux-gnu.tar.zst",
+                "prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm-aarch64-* lean-llvm-x86_64-*"
+              },
+              {
+                "name": "Linux 32bit",
+                "os": "ubuntu-latest",
+                // Use 32bit on stage0 and stage1 to keep oleans compatible
+                "CMAKE_OPTIONS": "-DSTAGE0_USE_GMP=OFF -DSTAGE0_LEAN_EXTRA_CXX_FLAGS='-m32' -DSTAGE0_LEANC_OPTS='-m32' -DSTAGE0_MMAP=OFF -DUSE_GMP=OFF -DLEAN_EXTRA_CXX_FLAGS='-m32' -DLEANC_OPTS='-m32' -DMMAP=OFF -DLEAN_INSTALL_SUFFIX=-linux_x86",
+                "cmultilib": true,
+                "release": true,
+                "check-level": 2,
+                "cross": true,
+                "shell": "bash -euxo pipefail {0}"
+              },
+              {
+                "name": "Web Assembly",
+                "os": "ubuntu-latest",
+                // Build a native 32bit binary in stage0 and use it to compile the oleans and the wasm build
+                "CMAKE_OPTIONS": "-DCMAKE_C_COMPILER_WORKS=1 -DSTAGE0_USE_GMP=OFF -DSTAGE0_LEAN_EXTRA_CXX_FLAGS='-m32' -DSTAGE0_LEANC_OPTS='-m32' -DSTAGE0_CMAKE_CXX_COMPILER=clang++ -DSTAGE0_CMAKE_C_COMPILER=clang -DSTAGE0_CMAKE_EXECUTABLE_SUFFIX=\"\" -DUSE_GMP=OFF -DMMAP=OFF -DSTAGE0_MMAP=OFF -DCMAKE_AR=../emsdk/emsdk-main/upstream/emscripten/emar -DCMAKE_TOOLCHAIN_FILE=../emsdk/emsdk-main/upstream/emscripten/cmake/Modules/Platform/Emscripten.cmake -DLEAN_INSTALL_SUFFIX=-linux_wasm32",
+                "wasm": true,
+                "cmultilib": true,
+                "release": true,
+                "check-level": 2,
+                "cross": true,
+                "shell": "bash -euxo pipefail {0}",
+                // Just a few selected tests because wasm is slow
+                "CTEST_OPTIONS": "-R \"leantest_1007\\.lean|leantest_Format\\.lean|leanruntest\\_1037.lean|leanruntest_ac_rfl\\.lean\""
+              }
+            ];
+            console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`)
+            return matrix.filter((job) => level >= job["check-level"])
+
   build:
     needs: [configure]
     if: github.event_name != 'schedule' || github.repository == 'leanprover/lean4'
@@ -253,7 +266,7 @@ jobs:
     runs-on: ${{ matrix.os }}
     defaults:
       run:
-        shell: ${{ matrix.shell || 'nix-shell --run "bash -euxo pipefail {0}"' }}
+        shell: ${{ matrix.shell || 'nix develop -c bash -euxo pipefail {0}' }}
     name: ${{ matrix.name }}
     env:
       # must be inside workspace
@@ -278,18 +291,18 @@ 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:
@@ -313,13 +326,16 @@ 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 $PWD/coredumps/%e.%p.%t | sudo tee /proc/sys/kernel/core_pattern
-        if: matrix.os == 'ubuntu-latest'
+        if: runner.os == 'Linux'
+      - name: Set up NPROC
+        run: |
+          echo "NPROC=$(nproc 2>/dev/null || sysctl -n hw.logicalcpu 2>/dev/null || echo 4)" >> $GITHUB_ENV
       - name: Build
         run: |
           mkdir build
@@ -350,8 +366,8 @@ jobs:
             OPTIONS+=(-DLEAN_SPECIAL_VERSION_DESC=${{ needs.configure.outputs.LEAN_SPECIAL_VERSION_DESC }})
           fi
           # contortion to support empty OPTIONS with old macOS bash
-          cmake .. ${{ matrix.CMAKE_OPTIONS }} ${OPTIONS[@]+"${OPTIONS[@]}"} -DLEAN_INSTALL_PREFIX=$PWD/..
-          make -j4
+          cmake .. --preset ${{ matrix.CMAKE_PRESET || 'release' }} -B . ${{ matrix.CMAKE_OPTIONS }} ${OPTIONS[@]+"${OPTIONS[@]}"} -DLEAN_INSTALL_PREFIX=$PWD/..
+          make -j$NPROC
           make install
       - name: Check Binaries
         run: ${{ matrix.binary-check }} lean-*/bin/* || true
@@ -380,26 +396,29 @@ jobs:
           build/stage1/bin/lean --stats src/Lean.lean
         if: ${{ !matrix.cross }}
       - name: Test
+        id: test
         run: |
-          cd build/stage1
           ulimit -c unlimited # coredumps
-          # exclude nonreproducible test
-          ctest -j4 --output-on-failure ${{ matrix.CTEST_OPTIONS }} < /dev/null
-        if: (matrix.wasm || !matrix.cross) && needs.configure.outputs.quick == 'false'
+          ctest --preset ${{ matrix.CMAKE_PRESET || 'release' }} --test-dir build/stage1 -j$NPROC --output-junit test-results.xml ${{ matrix.CTEST_OPTIONS }}
+        if: (matrix.wasm || !matrix.cross) && needs.configure.outputs.check-level >= 1
+      - name: Test Summary
+        uses: test-summary/action@v2
+        with:
+          paths: build/stage1/test-results.xml
+        # prefix `if` above with `always` so it's run even if tests failed
+        if: always() && steps.test.conclusion != 'skipped'
       - name: Check Test Binary
         run: ${{ matrix.binary-check }} tests/compiler/534.lean.out
-        if: ${{ !matrix.cross && needs.configure.outputs.quick == 'false' }}
+        if: (!matrix.cross) && steps.test.conclusion != 'skipped'
       - name: Build Stage 2
         run: |
-          cd build
           ulimit -c unlimited # coredumps
-          make -j4 stage2
+          make -C build -j$NPROC stage2
         if: matrix.test-speedcenter
       - name: Check Stage 3
         run: |
-          cd build
           ulimit -c unlimited # coredumps
-          make -j4 check-stage3
+          make -C build -j$NPROC stage3
         if: matrix.test-speedcenter
       - name: Test Speedcenter Benchmarks
         run: |
@@ -410,15 +429,14 @@ jobs:
         if: matrix.test-speedcenter
       - name: Check rebootstrap
         run: |
-          cd build
           ulimit -c unlimited # coredumps
           # clean rebuild in case of Makefile changes
-          make update-stage0 && rm -rf ./stage* && make -j4
-        if: matrix.name == 'Linux' && needs.configure.outputs.quick == 'false'
+          make -C build update-stage0 && rm -rf build/stage* && make -C build -j$NPROC
+        if: matrix.name == 'Linux' && needs.configure.outputs.check-level >= 1
       - 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/")"
@@ -428,7 +446,7 @@ 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: |
@@ -447,9 +465,10 @@ jobs:
     name: Build matrix complete
     runs-on: ubuntu-latest
     needs: build
-    if: ${{ always() }}
+    # mark as merely cancelled not failed if builds are cancelled
+    if: ${{ !cancelled() }}
     steps:
-    - if: contains(needs.*.result, 'failure') ||  contains(needs.*.result, 'cancelled')
+    - if: contains(needs.*.result, 'failure')
       uses: actions/github-script@v7
       with:
         script: |

.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:
@@ -77,7 +76,13 @@ jobs:
           nix build $NIX_BUILD_ARGS .#cacheRoots -o push-build
       - name: Test
         run: |
-          nix build $NIX_BUILD_ARGS .#test -o push-test
+          nix build --keep-failed $NIX_BUILD_ARGS .#test -o push-test || (ln -s /tmp/nix-build-*/source/src/build/ ./push-test; false)
+      - name: Test Summary
+        uses: test-summary/action@v2
+        with:
+          paths: push-test/test-results.xml
+        if: always()
+        continue-on-error: true
       - name: Build manual
         run: |
           nix build $NIX_BUILD_ARGS --update-input lean --no-write-lock-file ./doc#{lean-mdbook,leanInk,alectryon,test,inked} -o push-doc

.github/workflows/pr-release.yml

@@ -126,24 +126,22 @@ jobs:
           if [ "$NIGHTLY_SHA" = "$MERGE_BASE_SHA" ]; then
             echo "The merge base of this PR coincides with the nightly release"
 
+            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 "$MATHLIB_REMOTE_TAGS" ]]; then
-              echo "... and Mathlib has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
+            if [[ -n "$BATTERIES_REMOTE_TAGS" ]]; then
+              echo "... and Batteries has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
               MESSAGE=""
-            else
-              echo "... but Mathlib does not yet have a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
-              MESSAGE="- ❗ Mathlib CI can not be attempted yet, as the \`nightly-testing-$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
 
-            STD_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover/std4.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
-
-            if [[ -n "$STD_REMOTE_TAGS" ]]; then
-              echo "... and Std has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
-              MESSAGE=""
+              if [[ -n "$MATHLIB_REMOTE_TAGS" ]]; then
+                echo "... and Mathlib has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."    
+              else
+                echo "... but Mathlib does not yet have a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
+                MESSAGE="- ❗ Mathlib CI can not be attempted yet, as the \`nightly-testing-$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
@@ -151,7 +149,9 @@ jobs:
             echo "but 'git merge-base origin/master HEAD' reported: $MERGE_BASE_SHA"
             git -C lean4.git log -10 origin/master
 
-            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 $NIGHTLY_SHA\`."
+            git -C lean4.git fetch origin nightly-with-mathlib 
+            NIGHTLY_WITH_MATHLIB_SHA="$(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
@@ -223,27 +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"
@@ -251,7 +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
 
@@ -268,7 +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/${{ steps.workflow-info.outputs.pullRequestNumber }}"
@@ -321,7 +321,7 @@ jobs:
             git switch -c lean-pr-testing-${{ 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, 'merge-ci') || contains(github.event.label.name, 'release-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)

CMakePresets.json

@@ -0,0 +1,83 @@
+{
+  "version": 2,
+  "cmakeMinimumRequired": {
+    "major": 3,
+    "minor": 10,
+    "patch": 0
+  },
+  "configurePresets": [
+    {
+      "name": "release",
+      "displayName": "Default development optimized build config",
+      "generator": "Unix Makefiles",
+      "binaryDir": "${sourceDir}/build/release"
+    },
+    {
+      "name": "debug",
+      "displayName": "Debug build config",
+      "cacheVariables": {
+        "CMAKE_BUILD_TYPE": "Debug"
+      },
+      "generator": "Unix Makefiles",
+      "binaryDir": "${sourceDir}/build/debug"
+    },
+    {
+      "name": "sanitize",
+      "displayName": "Sanitize build config",
+      "cacheVariables": {
+        "LEAN_EXTRA_CXX_FLAGS": "-fsanitize=address,undefined",
+        "LEANC_EXTRA_FLAGS": "-fsanitize=address,undefined -fsanitize-link-c++-runtime",
+        "SMALL_ALLOCATOR": "OFF",
+        "BSYMBOLIC": "OFF"
+      },
+      "generator": "Unix Makefiles",
+      "binaryDir": "${sourceDir}/build/sanitize"
+    },
+    {
+      "name": "sandebug",
+      "inherits": ["debug", "sanitize"],
+      "displayName": "Sanitize+debug build config",
+      "binaryDir": "${sourceDir}/build/sandebug"
+    }
+  ],
+  "buildPresets": [
+    {
+      "name": "release",
+      "configurePreset": "release"
+    },
+    {
+      "name": "debug",
+      "configurePreset": "debug"
+    },
+    {
+      "name": "sanitize",
+      "configurePreset": "sanitize"
+    },
+    {
+      "name": "sandebug",
+      "configurePreset": "sandebug"
+    }
+  ],
+  "testPresets": [
+    {
+      "name": "release",
+      "configurePreset": "release",
+      "output": {"outputOnFailure": true, "shortProgress": true}
+    },
+    {
+      "name": "debug",
+      "configurePreset": "debug",
+      "inherits": "release"
+    },
+    {
+      "name": "sanitize",
+      "configurePreset": "sanitize",
+      "inherits": "release"
+    },
+    {
+      "name": "sandebug",
+      "configurePreset": "sandebug",
+      "inherits": "release"
+    }
+  ]
+}

CODEOWNERS

@@ -6,17 +6,40 @@
 
 /.github/ @Kha @semorrison
 /RELEASES.md @semorrison
-/src/Init/IO.lean @joehendrix
 /src/kernel/ @leodemoura
 /src/lake/ @tydeu
 /src/Lean/Compiler/ @leodemoura
 /src/Lean/Data/Lsp/ @mhuisi
 /src/Lean/Elab/Deriving/ @semorrison
 /src/Lean/Elab/Tactic/ @semorrison
+/src/Lean/Language/ @Kha
 /src/Lean/Meta/Tactic/ @leodemoura
 /src/Lean/Parser/ @Kha
 /src/Lean/PrettyPrinter/ @Kha
 /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

@@ -1,66 +1,23 @@
 # Lean 4 releases
 
+This file contains release notes for each stable release.
+Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
+of each version.
+During development, drafts of future release notes appear in [`releases_drafts`](https://github.com/leanprover/lean4/tree/master/script).
+
 We intend to provide regular "minor version" releases of the Lean language at approximately monthly intervals.
 There is not yet a strong guarantee of backwards compatibility between versions,
 only an expectation that breaking changes will be documented in this file.
 
-This file contains work-in-progress notes for the upcoming release, as well as previous stable releases.
-Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
-of each version.
-
-v4.8.0 (development in progress)
+v4.9.0
 ---------
 
-* Lean now generates an error if the type of a theorem is **not** a proposition.
+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).
+v4.8.0
+---------
 
-* New command `derive_functinal_induction`:
-
-  Derived from the definition of a (possibly mutually) recursive function
-  defined by well-founded recursion, a **functional induction principle** 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)
-  derive_functional_induction ackermann
-  ```
-  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
-  ```
-
-Breaking changes:
-
-* Automatically generated equational theorems are now named using suffix `.eq_<idx>` instead of `._eq_<idx>`, and `.def` instead of `._unfold`. Example:
-```
-def fact : Nat → Nat
-  | 0 => 1
-  | n+1 => (n+1) * fact n
-
-theorem ex : fact 0 = 1 := by unfold fact; decide
-
-#check fact.eq_1
--- fact.eq_1 : fact 0 = 1
-
-#check fact.eq_2
--- fact.eq_2 (n : Nat) : fact (Nat.succ n) = (n + 1) * fact n
-
-#check fact.def
-/-
-fact.def :
-  ∀ (x : Nat),
-    fact x =
-      match x with
-      | 0 => 1
-      | Nat.succ n => (n + 1) * fact n
--/
-```
+Release candidate, release notes will be copied from branch `releases/v4.8.0` once completed.
 
 v4.7.0
 ---------

default.nix

@@ -1,9 +0,0 @@
-# used for `nix-shell https://github.com/leanprover/lean4/archive/master.tar.gz -A nix`
-{ nix = (import ./shell.nix {}).nix; } //
-(import (
-  fetchTarball {
-    url = "https://github.com/edolstra/flake-compat/archive/c75e76f80c57784a6734356315b306140646ee84.tar.gz";
-    sha256 = "071aal00zp2m9knnhddgr2wqzlx6i6qa1263lv1y7bdn2w20h10h"; }
-) {
-  src =  ./.;
-}).defaultNix

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/flake.nix

@@ -27,7 +27,7 @@
         src = inputs.mdBook;
         cargoDeps = drv.cargoDeps.overrideAttrs (_: {
           inherit src;
-          outputHash = "sha256-1YlPS6cqgxE4fjy9G8pWrpP27YrrbCDnfeyIsX81ZNw=";
+          outputHash = "sha256-CO3A9Kpp4sIvkT9X3p+GTidazk7Fn4jf0AP2PINN44A=";
         });
         doCheck = false;
       });

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/make/index.md

@@ -1,3 +1,7 @@
+These are instructions to set up a working development environment for those who wish to make changes to Lean itself. It is part of the [Development Guide](doc/dev/index.md).
+
+We strongly suggest that new users instead follow the [Quickstart](doc/quickstart.md) to get started using Lean, since this sets up an environment that can automatically manage multiple Lean toolchain versions, which is necessary when working within the Lean ecosystem.
+
 Requirements
 ------------
 
@@ -12,44 +16,32 @@ Platform-Specific Setup
 - [Windows (msys2)](msys2.md)
 - [Windows (WSL)](wsl.md)
 - [macOS (homebrew)](osx-10.9.md)
-- Linux/macOS/WSL via [Nix](https://nixos.org/nix/): Call `nix-shell` in the project root. That's it.
+- Linux/macOS/WSL via [Nix](https://nixos.org/nix/): Call `nix develop` in the project root. That's it.
 
 Generic Build Instructions
 --------------------------
 
-Setting up a basic release build:
+Setting up a basic parallelized release build:
 
 ```bash
-git clone https://github.com/leanprover/lean4 --recurse-submodules
+git clone https://github.com/leanprover/lean4
 cd lean4
-mkdir -p build/release
-cd build/release
-cmake ../..
-make
+cmake --preset release
+make -C build/release -j$(nproc)  # see below for macOS
 ```
-
-For regular development, we recommend running
-```bash
-git config submodule.recurse true
-```
-in the checkout so that `--recurse-submodules` doesn't have to be
-specified with `git pull/checkout/...`.
+You can replace `$(nproc)`, which is not available on macOS and some alternative shells, with the desired parallelism amount.
 
 The above commands will compile the Lean library and binaries into the
-`stage1` subfolder; see below for details. Add `-j N` for an
-appropriate `N` to `make` for a parallel build.
+`stage1` subfolder; see below for details.
 
-For example, on an AMD Ryzen 9 `make` takes 00:04:55, whereas `make -j 10`
-takes 00:01:38.  Your results may vary depending on the speed of your hard
-drive.
-
-You should not usually run `make install` after a successful build.
+You should not usually run `cmake --install` after a successful build.
 See [Dev setup using elan](../dev/index.md#dev-setup-using-elan) on how to properly set up your editor to use the correct stage depending on the source directory.
 
 Useful CMake Configuration Settings
 -----------------------------------
 
-Pass these along with the `cmake ../..` command.
+Pass these along with the `cmake --preset release` command.
+There are also two alternative presets that combine some of these options you can use instead of `release`: `debug` and `sandebug` (sanitize + debug).
 
 * `-D CMAKE_BUILD_TYPE=`\
   Select the build type. Valid values are `RELEASE` (default), `DEBUG`,

doc/make/msvc.md

@@ -1,39 +0,0 @@
-# Compiling Lean with Visual Studio
-
-WARNING: Compiling Lean with Visual Studio doesn't currently work.
-There's an ongoing effort to port Lean to Visual Studio.
-The instructions below are for VS 2017.
-
-In the meantime you can use [MSYS2](msys2.md) or [WSL](wsl.md).
-
-## Installing dependencies
-
-First, install `vcpkg` from https://github.com/Microsoft/vcpkg if you haven't
-done so already.
-Then, open a console in the directory you cloned `vcpkg` to, and type:
-`vcpkg install mpir` for the 32-bit library or
-`vcpkg install mpir:x64-windows` for the x64 one.
-
-In Visual Studio, use the "open folder" feature and open the Lean directory.
-Go to the `CMake->Change CMake Settings` menu. File `CMakeSettings.json` opens.
-In each of the targets, add the following snippet (i.e., after every
-`ctestCommandArgs`):
-
-```json
-    "variables": [
-      {
-        "name": "CMAKE_TOOLCHAIN_FILE",
-        "value": "C:\\path\\to\\vcpkg\\scripts\\buildsystems\\vcpkg.cmake"
-      }
-    ]
-```
-
-## Enable Intellisense
-
-In Visual Studio, press Ctrl+Q and type `CppProperties.json` and press Enter.
-Ensure `includePath` variables include `"${workspaceRoot}\\src"`.
-
-
-## Build Lean
-
-Press F7.

doc/make/msys2.md

@@ -38,10 +38,9 @@ cmake --version
 Then follow the [generic build instructions](index.md) in the MSYS2
 MinGW shell, using:
 ```
-cmake ../.. -G "Unix Makefiles"  -DCMAKE_C_COMPILER=clang -DCMAKE_CXX_COMPILER=clang++
+cmake --preset release -DCMAKE_C_COMPILER=clang -DCMAKE_CXX_COMPILER=clang++
 ```
-instead of `cmake ../..`. This ensures that cmake will call `sh` instead of `cmd.exe`
-for script tasks and it will use the clang compiler instead of gcc, which is required.
+instead of `cmake --preset release`. This will use the clang compiler instead of gcc, which is required with msys2.
 
 ## Install lean
 

doc/make/osx-10.9.md

@@ -1,4 +1,4 @@
-# Install Packages on OS X 10.9
+# Install Packages on OS X 14.5
 
 We assume that you are using [homebrew][homebrew] as a package manager.
 
@@ -22,7 +22,7 @@ brew install gcc
 ```
 To install clang++-3.5 via homebrew, please execute:
 ```bash
-brew install llvm --with-clang --with-asan
+brew install llvm
 ```
 To use compilers other than the default one (Apple's clang++), you
 need to use `-DCMAKE_CXX_COMPILER` option to specify the compiler

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/setup.md

@@ -6,6 +6,7 @@ Platforms built & tested by our CI, available as binary releases via elan (see b
 
 * x86-64 Linux with glibc 2.27+
 * x86-64 macOS 10.15+
+* aarch64 (Apple Silicon) macOS 10.15+
 * x86-64 Windows 10+
 
 ### Tier 2
@@ -16,7 +17,6 @@ Releases may be silently broken due to the lack of automated testing.
 Issue reports and fixes are welcome.
 
 * aarch64 Linux with glibc 2.27+
-* aarch64 (Apple Silicon) macOS
 * x86 (32-bit) Linux
 * Emscripten Web Assembly
 

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.

flake.lock

@@ -1,12 +1,31 @@
 {
   "nodes": {
-    "flake-utils": {
+    "flake-compat": {
+      "flake": false,
       "locked": {
-        "lastModified": 1656928814,
-        "narHash": "sha256-RIFfgBuKz6Hp89yRr7+NR5tzIAbn52h8vT6vXkYjZoM=",
+        "lastModified": 1673956053,
+        "narHash": "sha256-4gtG9iQuiKITOjNQQeQIpoIB6b16fm+504Ch3sNKLd8=",
+        "owner": "edolstra",
+        "repo": "flake-compat",
+        "rev": "35bb57c0c8d8b62bbfd284272c928ceb64ddbde9",
+        "type": "github"
+      },
+      "original": {
+        "owner": "edolstra",
+        "repo": "flake-compat",
+        "type": "github"
+      }
+    },
+    "flake-utils": {
+      "inputs": {
+        "systems": "systems"
+      },
+      "locked": {
+        "lastModified": 1710146030,
+        "narHash": "sha256-SZ5L6eA7HJ/nmkzGG7/ISclqe6oZdOZTNoesiInkXPQ=",
         "owner": "numtide",
         "repo": "flake-utils",
-        "rev": "7e2a3b3dfd9af950a856d66b0a7d01e3c18aa249",
+        "rev": "b1d9ab70662946ef0850d488da1c9019f3a9752a",
         "type": "github"
       },
       "original": {
@@ -18,11 +37,11 @@
     "lean4-mode": {
       "flake": false,
       "locked": {
-        "lastModified": 1676498134,
-        "narHash": "sha256-u3WvyKxOViZG53hkb8wd2/Og6muTecbh+NdflIgVeyk=",
+        "lastModified": 1709737301,
+        "narHash": "sha256-uT9JN2kLNKJK9c/S/WxLjiHmwijq49EgLb+gJUSDpz0=",
         "owner": "leanprover",
         "repo": "lean4-mode",
-        "rev": "2c6ef33f476fdf5eb5e4fa4fa023ba8b11372440",
+        "rev": "f1f24c15134dee3754b82c9d9924866fe6bc6b9f",
         "type": "github"
       },
       "original": {
@@ -31,34 +50,35 @@
         "type": "github"
       }
     },
-    "lowdown-src": {
+    "libgit2": {
       "flake": false,
       "locked": {
-        "lastModified": 1633514407,
-        "narHash": "sha256-Dw32tiMjdK9t3ETl5fzGrutQTzh2rufgZV4A/BbxuD4=",
-        "owner": "kristapsdz",
-        "repo": "lowdown",
-        "rev": "d2c2b44ff6c27b936ec27358a2653caaef8f73b8",
+        "lastModified": 1697646580,
+        "narHash": "sha256-oX4Z3S9WtJlwvj0uH9HlYcWv+x1hqp8mhXl7HsLu2f0=",
+        "owner": "libgit2",
+        "repo": "libgit2",
+        "rev": "45fd9ed7ae1a9b74b957ef4f337bc3c8b3df01b5",
         "type": "github"
       },
       "original": {
-        "owner": "kristapsdz",
-        "repo": "lowdown",
+        "owner": "libgit2",
+        "repo": "libgit2",
         "type": "github"
       }
     },
     "nix": {
       "inputs": {
-        "lowdown-src": "lowdown-src",
+        "flake-compat": "flake-compat",
+        "libgit2": "libgit2",
         "nixpkgs": "nixpkgs",
         "nixpkgs-regression": "nixpkgs-regression"
       },
       "locked": {
-        "lastModified": 1657097207,
-        "narHash": "sha256-SmeGmjWM3fEed3kQjqIAO8VpGmkC2sL1aPE7kKpK650=",
+        "lastModified": 1711102798,
+        "narHash": "sha256-CXOIJr8byjolqG7eqCLa+Wfi7rah62VmLoqSXENaZnw=",
         "owner": "NixOS",
         "repo": "nix",
-        "rev": "f6316b49a0c37172bca87ede6ea8144d7d89832f",
+        "rev": "a22328066416650471c3545b0b138669ea212ab4",
         "type": "github"
       },
       "original": {
@@ -69,16 +89,33 @@
     },
     "nixpkgs": {
       "locked": {
-        "lastModified": 1653988320,
-        "narHash": "sha256-ZaqFFsSDipZ6KVqriwM34T739+KLYJvNmCWzErjAg7c=",
+        "lastModified": 1709083642,
+        "narHash": "sha256-7kkJQd4rZ+vFrzWu8sTRtta5D1kBG0LSRYAfhtmMlSo=",
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "2fa57ed190fd6c7c746319444f34b5917666e5c1",
+        "rev": "b550fe4b4776908ac2a861124307045f8e717c8e",
         "type": "github"
       },
       "original": {
         "owner": "NixOS",
-        "ref": "nixos-22.05-small",
+        "ref": "release-23.11",
+        "repo": "nixpkgs",
+        "type": "github"
+      }
+    },
+    "nixpkgs-old": {
+      "flake": false,
+      "locked": {
+        "lastModified": 1581379743,
+        "narHash": "sha256-i1XCn9rKuLjvCdu2UeXKzGLF6IuQePQKFt4hEKRU5oc=",
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "rev": "34c7eb7545d155cc5b6f499b23a7cb1c96ab4d59",
+        "type": "github"
+      },
+      "original": {
+        "owner": "NixOS",
+        "ref": "nixos-19.03",
         "repo": "nixpkgs",
         "type": "github"
       }
@@ -101,11 +138,11 @@
     },
     "nixpkgs_2": {
       "locked": {
-        "lastModified": 1686089707,
-        "narHash": "sha256-LTNlJcru2qJ0XhlhG9Acp5KyjB774Pza3tRH0pKIb3o=",
+        "lastModified": 1710889954,
+        "narHash": "sha256-Pr6F5Pmd7JnNEMHHmspZ0qVqIBVxyZ13ik1pJtm2QXk=",
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "af21c31b2a1ec5d361ed8050edd0303c31306397",
+        "rev": "7872526e9c5332274ea5932a0c3270d6e4724f3b",
         "type": "github"
       },
       "original": {
@@ -120,7 +157,23 @@
         "flake-utils": "flake-utils",
         "lean4-mode": "lean4-mode",
         "nix": "nix",
-        "nixpkgs": "nixpkgs_2"
+        "nixpkgs": "nixpkgs_2",
+        "nixpkgs-old": "nixpkgs-old"
+      }
+    },
+    "systems": {
+      "locked": {
+        "lastModified": 1681028828,
+        "narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=",
+        "owner": "nix-systems",
+        "repo": "default",
+        "rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e",
+        "type": "github"
+      },
+      "original": {
+        "owner": "nix-systems",
+        "repo": "default",
+        "type": "github"
       }
     }
   },

flake.nix

@@ -2,6 +2,9 @@
   description = "Lean interactive theorem prover";
 
   inputs.nixpkgs.url = "github:NixOS/nixpkgs/nixpkgs-unstable";
+  # old nixpkgs used for portable release with older glibc (2.27)
+  inputs.nixpkgs-old.url = "github:NixOS/nixpkgs/nixos-19.03";
+  inputs.nixpkgs-old.flake = false;
   inputs.flake-utils.url = "github:numtide/flake-utils";
   inputs.nix.url = "github:NixOS/nix";
   inputs.lean4-mode = {
@@ -17,14 +20,41 @@
   #  inputs.lean4-mode.follows = "lean4-mode";
   #};
 
-  outputs = { self, nixpkgs, flake-utils, nix, lean4-mode, ... }@inputs: flake-utils.lib.eachDefaultSystem (system:
+  outputs = { self, nixpkgs, nixpkgs-old, flake-utils, nix, lean4-mode, ... }@inputs: flake-utils.lib.eachDefaultSystem (system:
     let
       pkgs = import nixpkgs {
         inherit system;
         # for `vscode-with-extensions`
         config.allowUnfree = true;
       };
+      # An old nixpkgs for creating releases with an old glibc
+      pkgsDist-old = import nixpkgs-old { inherit system; };
+      # An old nixpkgs for creating releases with an old glibc
+      pkgsDist-old-aarch = import nixpkgs-old { localSystem.config = "aarch64-unknown-linux-gnu"; };
+
       lean-packages = pkgs.callPackage (./nix/packages.nix) { src = ./.; inherit nix lean4-mode; };
+
+      devShellWithDist = pkgsDist: pkgs.mkShell.override {
+	  stdenv = pkgs.overrideCC pkgs.stdenv lean-packages.llvmPackages.clang;
+	} ({
+	  buildInputs = with pkgs; [
+	    cmake gmp ccache
+	    lean-packages.llvmPackages.llvm  # llvm-symbolizer for asan/lsan
+	    # TODO: only add when proven to not affect the flakification
+	    #pkgs.python3
+	  ];
+	  # https://github.com/NixOS/nixpkgs/issues/60919
+	  hardeningDisable = [ "all" ];
+	  # more convenient `ctest` output
+	  CTEST_OUTPUT_ON_FAILURE = 1;
+	} // pkgs.lib.optionalAttrs pkgs.stdenv.isLinux {
+	  GMP = pkgsDist.gmp.override { withStatic = true; };
+	  GLIBC = pkgsDist.glibc;
+	  GLIBC_DEV = pkgsDist.glibc.dev;
+	  GCC_LIB = pkgsDist.gcc.cc.lib;
+	  ZLIB = pkgsDist.zlib;
+	  GDB = pkgsDist.gdb;
+	});
     in {
       packages = lean-packages // rec {
         debug = lean-packages.override { debug = true; };
@@ -49,7 +79,10 @@
       };
       defaultPackage = lean-packages.lean-all;
 
-      inherit (lean-packages) devShell;
+      # The default development shell for working on lean itself
+      devShells.default = devShellWithDist pkgs;
+      devShells.oldGlibc = devShellWithDist pkgsDist-old;
+      devShells.oldGlibcAArch = devShellWithDist pkgsDist-old-aarch;
 
       checks.lean = lean-packages.test;
     }) // rec {

nix/bootstrap.nix

@@ -65,7 +65,7 @@ rec {
     installPhase = ''
       mkdir -p $out/bin $out/lib/lean
       mv bin/lean $out/bin/
-      mv lib/lean/*.so $out/lib/lean
+      mv lib/lean/*.{so,dylib} $out/lib/lean
     '';
     meta.mainProgram = "lean";
   });
@@ -170,16 +170,17 @@ rec {
           ln -sf ${lean-all}/* .
         '';
         buildPhase = ''
-          ctest --output-on-failure -E 'leancomptest_(doc_example|foreign)' -j$NIX_BUILD_CORES
+          ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)|leanlaketest_init' -j$NIX_BUILD_CORES
         '';
         installPhase = ''
-          touch $out
+          mkdir $out
+          mv test-results.xml $out
         '';
       };
       update-stage0 =
         let cTree = symlinkJoin { name = "cs"; paths = [ Init.cTree Lean.cTree ]; }; in
         writeShellScriptBin "update-stage0" ''
-          CSRCS=${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

nix/buildLeanPackage.nix

@@ -176,7 +176,7 @@ with builtins; let
       # make local "copy" so `drv`'s Nix store path doesn't end up in ccache's hash
       ln -s ${drv.c}/${drv.cPath} src.c
       # on the other hand, a debug build is pretty fast anyway, so preserve the path for gdb
-      leanc -c -o $out/$oPath $leancFlags -fPIC ${if debug then "${drv.c}/${drv.cPath} -g" else "src.c -O3 -DNDEBUG"}
+      leanc -c -o $out/$oPath $leancFlags -fPIC ${if debug then "${drv.c}/${drv.cPath} -g" else "src.c -O3 -DNDEBUG -DLEAN_EXPORTING"}
     '';
   };
   mkMod = mod: deps:

releases_drafts/README.md

@@ -0,0 +1,22 @@
+Draft release notes
+-------------------
+
+This folder contains drafts of release notes for inclusion in `RELEASES.md`.
+During the process to create a release candidate, we look through all the commits that make up the release
+to prepare the release notes, and in that process we take these drafts into account.
+
+Guidelines:
+- You should prefer adding release notes to commit messages over adding anything to this folder.
+  A release note should briefly explain the impact of a change from a user's point of view.
+  Please mark these parts out with words such as **release notes** and/or **breaking changes**.
+- It is not necessary to add anything to this folder. It is meant for larger features that span multiple PRs,
+  or for anything that would be helpful when preparing the release notes that might be missed
+  by someone reading through the change log.
+- If the PR that adds a feature simultaneously adds a draft release note, including the PR number is not required
+  since it can be obtained from the git history for the file.
+
+When release notes are prepared, all the draft release notes are deleted from this folder.
+For release candidates beyond the first one, you can either update `RELEASE.md` directly
+or continue to add drafts.
+
+When a release is finalized, we will copy the completed release notes from `RELEASE.md` to the `master` branch.

releases_drafts/messagedata.md

@@ -0,0 +1,13 @@
+* 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`.
+
+part of #3929

releases_drafts/wf.md

@@ -0,0 +1,12 @@
+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.
+
+#4061

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="$REPO_ROOT/script/lib/rebase-editor.sh" git rebase -i "$@"
+

shell.nix

@@ -1,27 +0,0 @@
-let
-  flake = (import ./default.nix);
-  flakePkgs = flake.packages.${builtins.currentSystem};
-in { pkgs ? flakePkgs.nixpkgs, pkgsDist ? pkgs }:
-# use `shell` as default
-(attribs: attribs.shell // attribs) rec {
-  shell = pkgs.mkShell.override {
-    stdenv = pkgs.overrideCC pkgs.stdenv flakePkgs.llvmPackages.clang;
-  } (rec {
-    buildInputs = with pkgs; [
-      cmake gmp ccache
-      flakePkgs.llvmPackages.llvm  # llvm-symbolizer for asan/lsan
-    ];
-    # https://github.com/NixOS/nixpkgs/issues/60919
-    hardeningDisable = [ "all" ];
-    # more convenient `ctest` output
-    CTEST_OUTPUT_ON_FAILURE = 1;
-  } // pkgs.lib.optionalAttrs pkgs.stdenv.isLinux {
-    GMP = pkgsDist.gmp.override { withStatic = true; };
-    GLIBC = pkgsDist.glibc;
-    GLIBC_DEV = pkgsDist.glibc.dev;
-    GCC_LIB = pkgsDist.gcc.cc.lib;
-    ZLIB = pkgsDist.zlib;
-    GDB = pkgsDist.gdb;
-  });
-  nix = flake.devShell.${builtins.currentSystem};
-}

src/CMakeLists.txt

@@ -9,7 +9,7 @@ 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_SPECIAL_VERSION_DESC "" CACHE STRING "Additional version description like 'nightly-2018-03-11'")
@@ -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")
 
@@ -503,13 +509,13 @@ file(RELATIVE_PATH LIB ${LEAN_SOURCE_DIR} ${CMAKE_BINARY_DIR}/lib)
 
 # set up libInit_shared only on Windows; see also stdlib.make.in
 if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
-  set(INIT_SHARED_LINKER_FLAGS "-Wl,--whole-archive -lInit ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a -Wl,--no-whole-archive -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libInit_shared.dll.a")
+  set(INIT_SHARED_LINKER_FLAGS "-Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/temp/libInit.a.export ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a -Wl,--no-whole-archive -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libInit_shared.dll.a")
 endif()
 
 if(${CMAKE_SYSTEM_NAME} MATCHES "Darwin")
   set(LEANSHARED_LINKER_FLAGS "-Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libInit.a -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libLean.a -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libleancpp.a ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a ${LEANSHARED_LINKER_FLAGS}")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
-  set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive -lLean -lleancpp -Wl,--no-whole-archive -lInit_shared -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
+  set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/temp/libLean.a.export -lleancpp -Wl,--no-whole-archive -lInit_shared -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
 else()
   set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive -lInit -lLean -lleancpp -Wl,--no-whole-archive ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a ${LEANSHARED_LINKER_FLAGS}")
 endif()
@@ -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

@@ -33,3 +33,5 @@ import Init.SizeOfLemmas
 import Init.BinderPredicates
 import Init.Ext
 import Init.Omega
+import Init.MacroTrace
+import Init.Grind

src/Init/ByCases.lean

@@ -21,9 +21,9 @@ macro_rules
 
 /-! ## if-then-else -/
 
-@[simp] theorem if_true {h : Decidable True} (t e : α) : ite True t e = t := if_pos trivial
+@[simp] theorem if_true {_ : Decidable True} (t e : α) : ite True t e = t := if_pos trivial
 
-@[simp] theorem if_false {h : Decidable False} (t e : α) : ite False t e = e := if_neg id
+@[simp] theorem if_false {_ : Decidable False} (t e : α) : ite False t e = e := if_neg id
 
 theorem ite_id [Decidable c] {α} (t : α) : (if c then t else t) = t := by split <;> rfl
 
@@ -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/ExceptCps.lean

@@ -18,6 +18,7 @@ namespace ExceptCpsT
 def run {ε α : Type u} [Monad m] (x : ExceptCpsT ε m α) : m (Except ε α) :=
   x _ (fun a => pure (Except.ok a)) (fun e => pure (Except.error e))
 
+set_option linter.unusedVariables false in  -- `s` unused
 @[always_inline, inline]
 def runK {ε α : Type u} (x : ExceptCpsT ε m α) (s : ε) (ok : α → m β) (error : ε → m β) : m β :=
   x _ ok error

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/Conv.lean

@@ -6,7 +6,7 @@ Authors: Leonardo de Moura
 Notation for operators defined at Prelude.lean
 -/
 prelude
-import Init.Meta
+import Init.Tactics
 
 namespace Lean.Parser.Tactic.Conv
 
@@ -201,7 +201,7 @@ macro (name := anyGoals) tk:"any_goals " s:convSeq : conv =>
   with inaccessible names to the given names.
 * `case tag₁ | tag₂ => tac` is equivalent to `(case tag₁ => tac); (case tag₂ => tac)`.
 -/
-macro (name := case) tk:"case " args:sepBy1(caseArg, " | ") arr:" => " s:convSeq : conv =>
+macro (name := case) tk:"case " args:sepBy1(caseArg, "|") arr:" => " s:convSeq : conv =>
   `(conv| tactic' => case%$tk $args|* =>%$arr conv' => ($s); all_goals rfl)
 
 /--
@@ -210,7 +210,7 @@ has been solved after applying `tac`, nor admits the goal if `tac` failed.
 Recall that `case` closes the goal using `sorry` when `tac` fails, and
 the tactic execution is not interrupted.
 -/
-macro (name := case') tk:"case' " args:sepBy1(caseArg, " | ") arr:" => " s:convSeq : conv =>
+macro (name := case') tk:"case' " args:sepBy1(caseArg, "|") arr:" => " s:convSeq : conv =>
   `(conv| tactic' => case'%$tk $args|* =>%$arr conv' => $s)
 
 /--

src/Init/Core.lean

@@ -19,7 +19,7 @@ which applies to all applications of the function).
 -/
 @[simp] def inline {α : Sort u} (a : α) : α := a
 
-theorem id.def {α : Sort u} (a : α) : id a = a := rfl
+theorem id_def {α : Sort u} (a : α) : id a = a := rfl
 
 /--
 `flip f a b` is `f b a`. It is useful for "point-free" programming,
@@ -165,6 +165,7 @@ whose first component is `a : α` and whose second component is `b : β a`
 It is sometimes known as the dependent sum type, since it is the type level version
 of an indexed summation.
 -/
+@[pp_using_anonymous_constructor]
 structure Sigma {α : Type u} (β : α → Type v) where
   /-- Constructor for a dependent pair. If `a : α` and `b : β a` then `⟨a, b⟩ : Sigma β`.
   (This will usually require a type ascription to determine `β`
@@ -190,6 +191,7 @@ which can cause problems for universe level unification,
 because the equation `max 1 u v = ?u + 1` has no solution in level arithmetic.
 `PSigma` is usually only used in automation that constructs pairs of arbitrary types.
 -/
+@[pp_using_anonymous_constructor]
 structure PSigma {α : Sort u} (β : α → Sort v) where
   /-- Constructor for a dependent pair. If `a : α` and `b : β a` then `⟨a, b⟩ : PSigma β`.
   (This will usually require a type ascription to determine `β`
@@ -1112,9 +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)
@@ -1306,7 +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
@@ -1314,6 +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⟩
 
@@ -1594,7 +1598,7 @@ protected def mk' {α : Sort u} [s : Setoid α] (a : α) : Quotient s :=
 The analogue of `Quot.sound`: If `a` and `b` are related by the equivalence relation,
 then they have equal equivalence classes.
 -/
-def sound {α : Sort u} {s : Setoid α} {a b : α} : a ≈ b → Quotient.mk s a = Quotient.mk s b :=
+theorem sound {α : Sort u} {s : Setoid α} {a b : α} : a ≈ b → Quotient.mk s a = Quotient.mk s b :=
   Quot.sound
 
 /--
@@ -2033,4 +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

@@ -10,7 +10,7 @@ import Init.Data.Fin.Basic
 import Init.Data.UInt.Basic
 import Init.Data.Repr
 import Init.Data.ToString.Basic
-import Init.Util
+import Init.GetElem
 universe u v w
 
 namespace Array
@@ -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 :=
@@ -52,13 +53,15 @@ 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]
 
 instance : GetElem (Array α) USize α fun xs i => i.toNat < xs.size where
   getElem xs i h := xs.uget i h
 
+instance : LawfulGetElem (Array α) USize α fun xs i => i.toNat < xs.size where
+
 def back [Inhabited α] (a : Array α) : α :=
   a.get! (a.size - 1)
 
@@ -304,6 +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]
@@ -376,6 +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
@@ -456,24 +461,13 @@ def findRev? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
 
 @[inline]
 def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
-  let rec loop (i : Nat) (j : Nat) (inv : i + j = as.size) : Option Nat :=
-    if hlt : j < as.size then
-      match i, inv with
-      | 0, inv => by
-        apply False.elim
-        rw [Nat.zero_add] at inv
-        rw [inv] at hlt
-        exact absurd hlt (Nat.lt_irrefl _)
-      | i+1, inv =>
-        if p as[j] then
-          some j
-        else
-          have : i + (j+1) = as.size := by
-            rw [← inv, Nat.add_comm j 1, Nat.add_assoc]
-          loop i (j+1) this
-    else
-      none
-  loop as.size 0 rfl
+  let rec loop (j : Nat) :=
+    if h : j < as.size then
+      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 :=
 a.findIdx? fun a => a == v
@@ -567,6 +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
@@ -671,6 +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
@@ -713,6 +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 :=
@@ -725,35 +722,38 @@ 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 #[]
 
-def eraseIdxAux (i : Nat) (a : Array α) : Array α :=
-  if h : i < a.size then
-    let idx  : Fin a.size := ⟨i, h⟩;
-    let idx1 : Fin a.size := ⟨i - 1, by exact Nat.lt_of_le_of_lt (Nat.pred_le i) h⟩;
-    let a' := a.swap idx idx1
-    eraseIdxAux (i+1) a'
+/-- Remove the element at a given index from an array without bounds checks, using a `Fin` index.
+
+  This function takes worst case O(n) time because
+  it has to backshift all elements at positions greater than `i`.-/
+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]⟩
+    a'.feraseIdx i'
   else
     a.pop
-termination_by a.size - i
+termination_by a.size - i.val
+decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ i.isLt
 
-def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
-  eraseIdxAux (i.val + 1) a
+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 =>
+    unfold feraseIdx
+    simp [h, a', ih]
+  | case2 a i h =>
+    unfold feraseIdx
+    simp [h]
 
+/-- Remove the element at a given index from an array, or do nothing if the index is out of bounds.
+
+  This function takes worst case O(n) time because
+  it has to backshift all elements at positions greater than `i`.-/
 def eraseIdx (a : Array α) (i : Nat) : Array α :=
-  if i < a.size then eraseIdxAux (i+1) a else a
-
-def eraseIdxSzAux (a : Array α) (i : Nat) (r : Array α) (heq : r.size = a.size) : { r : Array α // r.size = a.size - 1 } :=
-  if h : i < r.size then
-    let idx  : Fin r.size := ⟨i, h⟩;
-    let idx1 : Fin r.size := ⟨i - 1, by exact Nat.lt_of_le_of_lt (Nat.pred_le i) h⟩;
-    eraseIdxSzAux a (i+1) (r.swap idx idx1) ((size_swap r idx idx1).trans heq)
-  else
-    ⟨r.pop, (size_pop r).trans (heq ▸ rfl)⟩
-termination_by r.size - i
-
-def eraseIdx' (a : Array α) (i : Fin a.size) : { r : Array α // r.size = a.size - 1 } :=
-  eraseIdxSzAux a (i.val + 1) a rfl
+  if h : i < a.size then a.feraseIdx ⟨i, h⟩ else a
 
 def erase [BEq α] (as : Array α) (a : α) : Array α :=
   match as.indexOf? a with
@@ -770,6 +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⟩
@@ -823,6 +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`.-/
@@ -844,6 +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
@@ -859,6 +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

@@ -9,29 +9,46 @@ 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⟩
 
+instance : LawfulGetElem (Subarray α) Nat α fun xs i => i < xs.size where
+
 @[inline] def getD (s : Subarray α) (i : Nat) (v₀ : α) : α :=
   if h : i < s.size then s.get ⟨i, h⟩ else v₀
 
@@ -40,7 +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
 
@@ -48,7 +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
@@ -66,27 +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 :=
@@ -133,15 +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
@@ -618,4 +632,14 @@ section normalization_eqs
 @[simp] theorem zero_eq                                   : BitVec.zero n = 0#n               := rfl
 end normalization_eqs
 
+/-- Converts a list of `Bool`s to a big-endian `BitVec`. -/
+def ofBoolListBE : (bs : List Bool) → BitVec bs.length
+| [] => 0#0
+| b :: bs => cons b (ofBoolListBE bs)
+
+/-- Converts a list of `Bool`s to a little-endian `BitVec`. -/
+def ofBoolListLE : (bs : List Bool) → BitVec bs.length
+| [] => 0#0
+| b :: bs => concat (ofBoolListLE bs) b
+
 end BitVec

src/Init/Data/BitVec/Bitblast.lean

@@ -159,4 +159,80 @@ 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]
+
+/-- If two bitvectors have the same `msb`, then signed and unsigned comparisons coincide -/
+theorem slt_eq_ult_of_msb_eq {x y : BitVec w} (h : x.msb = y.msb) :
+    x.slt y = x.ult y := by
+  simp only [BitVec.slt, toInt_eq_msb_cond, BitVec.ult, decide_eq_decide, h]
+  cases y.msb <;> simp
+
+/-- If two bitvectors have different `msb`s, then unsigned comparison is determined by this bit -/
+theorem ult_eq_msb_of_msb_neq {x y : BitVec w} (h : x.msb ≠ y.msb) :
+    x.ult y = y.msb := by
+  simp only [BitVec.ult, msb_eq_decide, ne_eq, decide_eq_decide] at *
+  omega
+
+/-- If two bitvectors have different `msb`s, then signed and unsigned comparisons are opposites -/
+theorem slt_eq_not_ult_of_msb_neq {x y : BitVec w} (h : x.msb ≠ y.msb) :
+    x.slt y = !x.ult y := by
+  simp only [BitVec.slt, toInt_eq_msb_cond, Bool.eq_not_of_ne h, ult_eq_msb_of_msb_neq h]
+  cases y.msb <;> (simp; omega)
+
+theorem slt_eq_ult (x y : BitVec w) :
+    x.slt y = (x.msb != y.msb).xor (x.ult y) := by
+  by_cases h : x.msb = y.msb
+  · simp [h, slt_eq_ult_of_msb_eq]
+  · have h' : x.msb != y.msb := by simp_all
+    simp [slt_eq_not_ult_of_msb_neq h, h']
+
+theorem slt_eq_not_carry (x y : BitVec w) :
+    x.slt y = (x.msb == y.msb).xor (carry w x (~~~y) true) := by
+  simp only [slt_eq_ult, bne, ult_eq_not_carry]
+  cases x.msb == y.msb <;> simp
+
+theorem sle_eq_not_slt (x y : BitVec w) : x.sle y = !y.slt x := by
+  simp only [BitVec.sle, BitVec.slt, ← decide_not, decide_eq_decide]; omega
+
+theorem sle_eq_carry (x y : BitVec w) :
+    x.sle y = !((x.msb == y.msb).xor (carry w y (~~~x) true)) := by
+  rw [sle_eq_not_slt, slt_eq_not_carry, beq_comm]
+
 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 +1,16 @@
 /-
 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
+import Init.Data.Int.Bitwise.Lemmas
 
 namespace BitVec
 
@@ -41,12 +44,36 @@ theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsb i := rfl
   have p : 2^w ≤ 2^i := Nat.pow_le_pow_of_le_right (by omega) ge
   omega
 
+@[simp] theorem getMsb_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getMsb x i = false := by
+  rw [getMsb]
+  simp only [Bool.and_eq_false_imp, decide_eq_true_eq]
+  omega
+
 theorem lt_of_getLsb (x : BitVec w) (i : Nat) : getLsb x i = true → i < w := by
   if h : i < w then
     simp [h]
   else
     simp [Nat.ge_of_not_lt h]
 
+theorem lt_of_getMsb (x : BitVec w) (i : Nat) : getMsb x i = true → i < w := by
+  if h : i < w then
+    simp [h]
+  else
+    simp [Nat.ge_of_not_lt h]
+
+theorem getMsb_eq_getLsb (x : BitVec w) (i : Nat) : x.getMsb i = (decide (i < w) && x.getLsb (w - 1 - i)) := by
+  rw [getMsb]
+
+theorem getLsb_eq_getMsb (x : BitVec w) (i : Nat) : x.getLsb i = (decide (i < w) && x.getMsb (w - 1 - i)) := by
+  rw [getMsb]
+  by_cases h₁ : i < w <;> by_cases h₂ : w - 1 - i < w <;>
+    simp only [h₁, h₂] <;> simp only [decide_True, decide_False, Bool.false_and, Bool.and_false, Bool.true_and, Bool.and_true]
+  · congr
+    omega
+  all_goals
+    apply getLsb_ge
+    omega
+
 -- We choose `eq_of_getLsb_eq` as the `@[ext]` theorem for `BitVec`
 -- somewhat arbitrarily over `eq_of_getMsg_eq`.
 @[ext] theorem eq_of_getLsb_eq {x y : BitVec w}
@@ -79,7 +106,13 @@ 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
@@ -96,6 +129,8 @@ theorem ofNat_one (n : Nat) : BitVec.ofNat 1 n = BitVec.ofBool (n % 2 = 1) :=  b
 theorem ofBool_eq_iff_eq : ∀(b b' : Bool), BitVec.ofBool b = BitVec.ofBool b' ↔ b = b' := by
   decide
 
+@[simp] theorem not_ofBool : ~~~ (ofBool b) = ofBool (!b) := by cases b <;> rfl
+
 @[simp, bv_toNat] theorem toNat_ofFin (x : Fin (2^n)) : (BitVec.ofFin x).toNat = x.val := rfl
 
 @[simp] theorem toNat_ofNatLt (x : Nat) (p : x < 2^w) : (x#'p).toNat = x := rfl
@@ -107,13 +142,16 @@ theorem ofBool_eq_iff_eq : ∀(b b' : Bool), BitVec.ofBool b = BitVec.ofBool b'
 @[simp, bv_toNat] theorem toNat_ofNat (x w : Nat) : (x#w).toNat = x % 2^w := by
   simp [BitVec.toNat, BitVec.ofNat, Fin.ofNat']
 
+@[simp] theorem toFin_ofNat (x : Nat) : toFin x#w = Fin.ofNat' x (Nat.two_pow_pos w) := rfl
+
 -- Remark: we don't use `[simp]` here because simproc` subsumes it for literals.
 -- If `x` and `n` are not literals, applying this theorem eagerly may not be a good idea.
 theorem 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]
 
@@ -140,8 +178,7 @@ theorem msb_eq_getLsb_last (x : BitVec w) :
     x.getLsb (w-1) = decide (2 ^ (w-1) ≤ x.toNat) := by
   rcases w with rfl | w
   · simp
-  · simp only [Nat.zero_lt_succ, decide_True, getLsb, Nat.testBit, Nat.succ_sub_succ_eq_sub,
-    Nat.sub_zero, Nat.and_one_is_mod, Bool.true_and, Nat.shiftRight_eq_div_pow]
+  · simp only [getLsb, Nat.testBit_to_div_mod, Nat.succ_sub_succ_eq_sub, Nat.sub_zero]
     rcases (Nat.lt_or_ge (BitVec.toNat x) (2 ^ w)) with h | h
     · simp [Nat.div_eq_of_lt h, h]
     · simp only [h]
@@ -188,9 +225,21 @@ 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]
@@ -212,6 +261,12 @@ 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
@@ -227,6 +282,9 @@ theorem toInt_ofNat {n : Nat} (x : Nat) :
   have p : 0 ≤ i % (2^n : Nat) := by omega
   simp [toInt_eq_toNat_bmod, Int.toNat_of_nonneg p]
 
+@[simp] theorem ofInt_natCast (w n : Nat) :
+  BitVec.ofInt w (n : Int) = BitVec.ofNat w n := rfl
+
 /-! ### zeroExtend and truncate -/
 
 @[simp, bv_toNat] theorem toNat_zeroExtend' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
@@ -290,6 +348,19 @@ theorem nat_eq_toNat (x : BitVec w) (y : Nat)
     getLsb (zeroExtend' ge x) i = getLsb x i := by
   simp [getLsb, toNat_zeroExtend']
 
+@[simp] theorem getMsb_zeroExtend' (ge : m ≥ n) (x : BitVec n) (i : Nat) :
+    getMsb (zeroExtend' ge x) i = (decide (i ≥ m - n) && getMsb x (i - (m - n))) := by
+  simp only [getMsb, getLsb_zeroExtend', gt_iff_lt]
+  by_cases h₁ : decide (i < m) <;> by_cases h₂ : decide (i ≥ m - n) <;> by_cases h₃ : decide (i - (m - n) < n) <;>
+    by_cases h₄ : n - 1 - (i - (m - n)) = m - 1 - i
+  all_goals
+    simp only [h₁, h₂, h₃, h₄]
+    simp_all only [ge_iff_le, decide_eq_true_eq, Nat.not_le, Nat.not_lt, Bool.true_and,
+      Bool.false_and, Bool.and_self] <;>
+    (try apply getLsb_ge) <;>
+    (try apply (getLsb_ge _ _ _).symm) <;>
+    omega
+
 @[simp] theorem getLsb_zeroExtend (m : Nat) (x : BitVec n) (i : Nat) :
     getLsb (zeroExtend m x) i = (decide (i < m) && getLsb x i) := by
   simp [getLsb, toNat_zeroExtend, Nat.testBit_mod_two_pow]
@@ -297,7 +368,7 @@ 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]
@@ -396,6 +467,11 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
   ext
   simp
 
+theorem or_assoc (x y z : BitVec w) :
+    x ||| y ||| z = x ||| (y ||| z) := by
+  ext i
+  simp [Bool.or_assoc]
+
 /-! ### and -/
 
 @[simp] theorem toNat_and (x y : BitVec v) :
@@ -422,6 +498,11 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
   ext
   simp
 
+theorem and_assoc (x y z : BitVec w) :
+    x &&& y &&& z = x &&& (y &&& z) := by
+  ext i
+  simp [Bool.and_assoc]
+
 /-! ### xor -/
 
 @[simp] theorem toNat_xor (x y : BitVec v) :
@@ -442,6 +523,11 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
   ext
   simp
 
+theorem xor_assoc (x y z : BitVec w) :
+    x ^^^ y ^^^ z = x ^^^ (y ^^^ z) := by
+  ext i
+  simp [Bool.xor_assoc]
+
 /-! ### not -/
 
 theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
@@ -480,6 +566,24 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
   simp [h]
   omega
 
+/-! ### cast -/
+
+@[simp] theorem not_cast {x : BitVec w} (h : w = w') : ~~~(cast h x) = cast h (~~~x) := by
+  ext
+  simp_all [lt_of_getLsb]
+
+@[simp] theorem and_cast {x y : BitVec w} (h : w = w') : cast h x &&& cast h y = cast h (x &&& y) := by
+  ext
+  simp_all [lt_of_getLsb]
+
+@[simp] theorem or_cast {x y : BitVec w} (h : w = w') : cast h x ||| cast h y = cast h (x ||| y) := by
+  ext
+  simp_all [lt_of_getLsb]
+
+@[simp] theorem xor_cast {x y : BitVec w} (h : w = w') : cast h x &&& cast h y = cast h (x &&& y) := by
+  ext
+  simp_all [lt_of_getLsb]
+
 /-! ### shiftLeft -/
 
 @[simp, bv_toNat] theorem toNat_shiftLeft {x : BitVec v} :
@@ -529,10 +633,26 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
     <;> simp_all
     <;> (rw [getLsb_ge]; omega)
 
+@[simp] theorem getMsb_shiftLeftZeroExtend (x : BitVec m) (n : Nat) :
+    getMsb (shiftLeftZeroExtend x n) i = getMsb x i := by
+  have : n ≤ i + n := by omega
+  simp_all [shiftLeftZeroExtend_eq]
+
 @[simp] theorem msb_shiftLeftZeroExtend (x : BitVec w) (i : Nat) :
     (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) :
@@ -542,6 +662,70 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
     getLsb (x >>> i) j = getLsb x (i+j) := by
   unfold getLsb ; simp
 
+/-! ### sshiftRight -/
+
+theorem sshiftRight_eq {x : BitVec n} {i : Nat} :
+    x.sshiftRight i = BitVec.ofInt n (x.toInt >>> i) := by
+  apply BitVec.eq_of_toInt_eq
+  simp [BitVec.sshiftRight]
+
+/-- if the msb is false, the arithmetic shift right equals logical shift right -/
+theorem sshiftRight_eq_of_msb_false {x : BitVec w} {s : Nat} (h : x.msb = false) :
+    (x.sshiftRight s) = x >>> s := by
+  apply BitVec.eq_of_toNat_eq
+  rw [BitVec.sshiftRight_eq, BitVec.toInt_eq_toNat_cond]
+  have hxbound : 2 * x.toNat < 2 ^ w := (BitVec.msb_eq_false_iff_two_mul_lt x).mp h
+  simp only [hxbound, ↓reduceIte, Int.natCast_shiftRight, Int.ofNat_eq_coe, ofInt_natCast,
+    toNat_ofNat, toNat_ushiftRight]
+  replace hxbound : x.toNat >>> s < 2 ^ w := by
+    rw [Nat.shiftRight_eq_div_pow]
+    exact Nat.lt_of_le_of_lt (Nat.div_le_self ..) x.isLt
+  apply Nat.mod_eq_of_lt hxbound
+
+/--
+If the msb is `true`, the arithmetic shift right equals negating,
+then logical shifting right, then negating again.
+The double negation preserves the lower bits that have been shifted,
+and the outer negation ensures that the high bits are '1'. -/
+theorem sshiftRight_eq_of_msb_true {x : BitVec w} {s : Nat} (h : x.msb = true) :
+    (x.sshiftRight s) = ~~~((~~~x) >>> s) := by
+  apply BitVec.eq_of_toNat_eq
+  rcases w with rfl | w
+  · simp
+  · rw [BitVec.sshiftRight_eq, BitVec.toInt_eq_toNat_cond]
+    have hxbound : (2 * x.toNat ≥ 2 ^ (w + 1)) := (BitVec.msb_eq_true_iff_two_mul_ge x).mp h
+    replace hxbound : ¬ (2 * x.toNat < 2 ^ (w + 1)) := by omega
+    simp only [hxbound, ↓reduceIte, toNat_ofInt, toNat_not, toNat_ushiftRight]
+    rw [← Int.subNatNat_eq_coe, Int.subNatNat_of_lt (by omega),
+        Nat.pred_eq_sub_one, Int.negSucc_shiftRight,
+        Int.emod_negSucc, Int.natAbs_ofNat, Nat.succ_eq_add_one,
+        Int.subNatNat_of_le (by omega), Int.toNat_ofNat, Nat.mod_eq_of_lt,
+        Nat.sub_right_comm]
+    omega
+    · rw [Nat.shiftRight_eq_div_pow]
+      apply Nat.lt_of_le_of_lt (Nat.div_le_self _ _) (by omega)
+
+theorem getLsb_sshiftRight (x : BitVec w) (s i : Nat) :
+    getLsb (x.sshiftRight s) i =
+      (!decide (w ≤ i) && if s + i < w then x.getLsb (s + i) else x.msb) := by
+  rcases hmsb : x.msb with rfl | rfl
+  · simp only [sshiftRight_eq_of_msb_false hmsb, getLsb_ushiftRight, Bool.if_false_right]
+    by_cases hi : i ≥ w
+    · simp only [hi, decide_True, Bool.not_true, Bool.false_and]
+      apply getLsb_ge
+      omega
+    · simp only [hi, decide_False, Bool.not_false, Bool.true_and, Bool.iff_and_self,
+        decide_eq_true_eq]
+      intros hlsb
+      apply BitVec.lt_of_getLsb _ _ hlsb
+  · by_cases hi : i ≥ w
+    · simp [hi]
+    · simp only [sshiftRight_eq_of_msb_true hmsb, getLsb_not, getLsb_ushiftRight, Bool.not_and,
+        Bool.not_not, hi, decide_False, Bool.not_false, Bool.if_true_right, Bool.true_and,
+        Bool.and_iff_right_iff_imp, Bool.or_eq_true, Bool.not_eq_true', decide_eq_false_iff_not,
+        Nat.not_lt, decide_eq_true_eq]
+      omega
+
 /-! ### append -/
 
 theorem append_def (x : BitVec v) (y : BitVec w) :
@@ -553,11 +737,18 @@ theorem append_def (x : BitVec v) (y : BitVec w) :
 
 @[simp] theorem getLsb_append {v : BitVec n} {w : BitVec m} :
     getLsb (v ++ w) i = bif i < m then getLsb w i else getLsb v (i - m) := by
-  simp [append_def]
+  simp only [append_def, getLsb_or, getLsb_shiftLeftZeroExtend, getLsb_zeroExtend']
   by_cases h : i < m
   · simp [h]
   · simp [h]; simp_all
 
+@[simp] theorem getMsb_append {v : BitVec n} {w : BitVec m} :
+    getMsb (v ++ w) i = bif n ≤ i then getMsb w (i - n) else getMsb v i := by
+  simp [append_def]
+  by_cases h : n ≤ i
+  · simp [h]
+  · simp [h]
+
 theorem msb_append {x : BitVec w} {y : BitVec v} :
     (x ++ y).msb = bif (w == 0) then (y.msb) else (x.msb) := by
   rw [← append_eq, append]
@@ -586,6 +777,36 @@ theorem msb_append {x : BitVec w} {y : BitVec v} :
 @[simp] theorem truncate_cons {x : BitVec w} : (cons a x).truncate w = x := by
   simp [cons]
 
+@[simp] theorem not_append {x : BitVec w} {y : BitVec v} : ~~~ (x ++ y) = (~~~ x) ++ (~~~ y) := by
+  ext i
+  simp only [getLsb_not, getLsb_append, cond_eq_if]
+  split
+  · simp_all
+  · simp_all; omega
+
+@[simp] theorem and_append {x₁ x₂ : BitVec w} {y₁ y₂ : BitVec v} :
+    (x₁ ++ y₁) &&& (x₂ ++ y₂) = (x₁ &&& x₂) ++ (y₁ &&& y₂) := by
+  ext i
+  simp only [getLsb_append, cond_eq_if]
+  split <;> simp [*]
+
+@[simp] theorem or_append {x₁ x₂ : BitVec w} {y₁ y₂ : BitVec v} :
+    (x₁ ++ y₁) ||| (x₂ ++ y₂) = (x₁ ||| x₂) ++ (y₁ ||| y₂) := by
+  ext i
+  simp only [getLsb_append, cond_eq_if]
+  split <;> simp [*]
+
+@[simp] theorem xor_append {x₁ x₂ : BitVec w} {y₁ y₂ : BitVec v} :
+    (x₁ ++ y₁) ^^^ (x₂ ++ y₂) = (x₁ ^^^ x₂) ++ (y₁ ^^^ y₂) := by
+  ext i
+  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) :
@@ -630,6 +851,12 @@ theorem toNat_cons' {x : BitVec w} :
 @[simp] theorem msb_cons : (cons a x).msb = a := by
   simp [cons, msb_cast, msb_append]
 
+@[simp] theorem getMsb_cons_zero : (cons a x).getMsb 0 = a := by
+  rw [← BitVec.msb, msb_cons]
+
+@[simp] theorem getMsb_cons_succ : (cons a x).getMsb (i + 1) = x.getMsb i := by
+  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
   apply eq_of_getLsb_eq
@@ -650,6 +877,21 @@ theorem eq_msb_cons_truncate (x : BitVec (w+1)) : x = (cons x.msb (x.truncate w)
     · simp_all
     · omega
 
+@[simp] theorem not_cons (x : BitVec w) (b : Bool) : ~~~(cons b x) = cons (!b) (~~~x) := by
+  simp [cons]
+
+@[simp] theorem cons_or_cons (x y : BitVec w) (a b : Bool) :
+    (cons a x) ||| (cons b y) = cons (a || b) (x ||| y) := by
+  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+
+@[simp] theorem cons_and_cons (x y : BitVec w) (a b : Bool) :
+    (cons a x) &&& (cons b y) = cons (a && b) (x &&& y) := by
+  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+
+@[simp] theorem cons_xor_cons (x y : BitVec w) (a b : Bool) :
+    (cons a x) ^^^ (cons b y) = cons (xor a b) (x ^^^ y) := by
+  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+
 /-! ### concat -/
 
 @[simp] theorem toNat_concat (x : BitVec w) (b : Bool) :
@@ -702,24 +944,42 @@ 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
@@ -749,6 +1009,10 @@ theorem ofNat_sub_ofNat {n} (x y : Nat) : x#n - y#n = .ofNat n (x + (2^n - y % 2
 @[simp, bv_toNat] theorem toNat_neg (x : BitVec n) : (- x).toNat = (2^n - x.toNat) % 2^n := by
   simp [Neg.neg, BitVec.neg]
 
+@[simp] theorem toFin_neg (x : BitVec n) :
+    (-x).toFin = Fin.ofNat' (2^n - x.toNat) (Nat.two_pow_pos _) :=
+  rfl
+
 theorem sub_toAdd {n} (x y : BitVec n) : x - y = x + - y := by
   apply eq_of_toNat_eq
   simp
@@ -757,10 +1021,19 @@ 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
@@ -770,6 +1043,13 @@ 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
@@ -796,6 +1076,15 @@ 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) :
@@ -825,7 +1114,7 @@ protected theorem lt_of_le_ne (x y : BitVec n) (h1 : x <= y) (h2 : ¬ x = y) : x
   simp
   exact Nat.lt_of_le_of_ne
 
-/- ! ### intMax -/
+/-! ### intMax -/
 
 /-- The bitvector of width `w` that has the largest value when interpreted as an integer. -/
 def intMax (w : Nat) : BitVec w  := (2^w - 1)#w
@@ -839,4 +1128,187 @@ theorem toNat_intMax_eq : (intMax w).toNat = 2^w - 1 := by
     omega
   simp [intMax, Nat.shiftLeft_eq, Nat.one_mul, natCast_eq_ofNat, toNat_ofNat, Nat.mod_eq_of_lt h]
 
+/-! ### ofBoolList -/
+
+@[simp] theorem getMsb_ofBoolListBE : (ofBoolListBE bs).getMsb i = bs.getD i false := by
+  induction bs generalizing i <;> cases i <;> simp_all [ofBoolListBE]
+
+@[simp] theorem getLsb_ofBoolListBE :
+    (ofBoolListBE bs).getLsb i = (decide (i < bs.length) && bs.getD (bs.length - 1 - i) false) := by
+  simp [getLsb_eq_getMsb]
+
+@[simp] theorem getLsb_ofBoolListLE : (ofBoolListLE bs).getLsb i = bs.getD i false := by
+  induction bs generalizing i <;> cases i <;> simp_all [ofBoolListLE]
+
+@[simp] theorem getMsb_ofBoolListLE :
+    (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]
+
+
+/--
+Accessing bits in `x.rotateLeft r` the range `[0, r)` is equal to
+accessing bits `x` in the range `[w - r, w)`.
+
+Proof by example:
+Let x := <6 5 4 3 2 1 0> : BitVec 7.
+x.rotateLeft 2 = (<6 5 | 4 3 2 1 0>).rotateLeft 2 = <3 2 1 0 | 6 5>
+
+(x.rotateLeft 2).getLsb ⟨i, i < 2⟩
+= <3 2 1 0 | 6 5>.getLsb ⟨i, i < 2⟩
+= <6 5>[i]
+= <6 5 | 4 3 2 1 0>[i + len(<4 3 2 1 0>)]
+= <6 5 | 4 3 2 1 0>[i + 7 - 2]
+-/
+theorem getLsb_rotateLeftAux_of_le {x : BitVec w} {r : Nat} {i : Nat} (hi : i < r) :
+    (x.rotateLeftAux r).getLsb i = x.getLsb (w - r + i) := by
+  rw [rotateLeftAux, getLsb_or, getLsb_ushiftRight]
+  suffices (x <<< r).getLsb i = false by
+    simp; omega
+  simp only [getLsb_shiftLeft, Bool.and_eq_false_imp, Bool.and_eq_true, decide_eq_true_eq,
+    Bool.not_eq_true', decide_eq_false_iff_not, Nat.not_lt, and_imp]
+  omega
+
+/--
+Accessing bits in `x.rotateLeft r` the range `[r, w)` is equal to
+accessing bits `x` in the range `[0, w - r)`.
+
+Proof by example:
+Let x := <6 5 4 3 2 1 0> : BitVec 7.
+x.rotateLeft 2 = (<6 5 | 4 3 2 1 0>).rotateLeft 2 = <3 2 1 0 | 6 5>
+
+(x.rotateLeft 2).getLsb ⟨i, i ≥ 2⟩
+= <3 2 1 0 | 6 5>.getLsb ⟨i, i ≥ 2⟩
+= <3 2 1 0>[i - 2]
+= <6 5 | 3 2 1 0>[i - 2]
+
+Intuitively, grab the full width (7), then move the marker `|` by `r` to the right `(-2)`
+Then, access the bit at `i` from the right `(+i)`.
+ -/
+theorem getLsb_rotateLeftAux_of_geq {x : BitVec w} {r : Nat} {i : Nat} (hi : i ≥ r) :
+    (x.rotateLeftAux r).getLsb i = (decide (i < w) && x.getLsb (i - r)) := by
+  rw [rotateLeftAux, getLsb_or]
+  suffices (x >>> (w - r)).getLsb i = false by
+    have hiltr : decide (i < r) = false := by
+      simp [hi]
+    simp [getLsb_shiftLeft, Bool.or_false, hi, hiltr, this]
+  simp only [getLsb_ushiftRight]
+  apply getLsb_ge
+  omega
+
+/-- When `r < w`, we give a formula for `(x.rotateRight r).getLsb i`. -/
+theorem getLsb_rotateLeft_of_le {x : BitVec w} {r i : Nat} (hr: r < w) :
+    (x.rotateLeft r).getLsb i =
+      cond (i < r)
+      (x.getLsb (w - r + i))
+      (decide (i < w) && x.getLsb (i - r)) := by
+  · rw [rotateLeft_eq_rotateLeftAux_of_lt hr]
+    by_cases h : i < r
+    · simp [h, getLsb_rotateLeftAux_of_le h]
+    · simp [h, getLsb_rotateLeftAux_of_geq <| Nat.ge_of_not_lt h]
+
+@[simp]
+theorem getLsb_rotateLeft {x : BitVec w} {r i : Nat}  :
+    (x.rotateLeft r).getLsb i =
+      cond (i < r % w)
+      (x.getLsb (w - (r % w) + i))
+      (decide (i < w) && x.getLsb (i - (r % w))) := by
+  rcases w with ⟨rfl, w⟩
+  · simp
+  · rw [← rotateLeft_mod_eq_rotateLeft, getLsb_rotateLeft_of_le (Nat.mod_lt _ (by omega))]
+
+/-! ## Rotate Right -/
+
+/--
+Accessing bits in `x.rotateRight r` the range `[0, w-r)` is equal to
+accessing bits `x` in the range `[r, w)`.
+
+Proof by example:
+Let x := <6 5 4 3 2 1 0> : BitVec 7.
+x.rotateRight 2 = (<6 5 4 3 2 | 1 0>).rotateRight 2 = <1 0 | 6 5 4 3 2>
+
+(x.rotateLeft 2).getLsb ⟨i, i ≤ 7 - 2⟩
+= <1 0 | 6 5 4 3 2>.getLsb ⟨i, i ≤ 7 - 2⟩
+= <6 5 4 3 2>.getLsb i
+= <6 5 4 3 2 | 1 0>[i + 2]
+-/
+theorem getLsb_rotateRightAux_of_le {x : BitVec w} {r : Nat} {i : Nat} (hi : i < w - r) :
+    (x.rotateRightAux r).getLsb i = x.getLsb (r + i) := by
+  rw [rotateRightAux, getLsb_or, getLsb_ushiftRight]
+  suffices (x <<< (w - r)).getLsb i = false by
+    simp only [this, Bool.or_false]
+  simp only [getLsb_shiftLeft, Bool.and_eq_false_imp, Bool.and_eq_true, decide_eq_true_eq,
+    Bool.not_eq_true', decide_eq_false_iff_not, Nat.not_lt, and_imp]
+  omega
+
+/--
+Accessing bits in `x.rotateRight r` the range `[w-r, w)` is equal to
+accessing bits `x` in the range `[0, r)`.
+
+Proof by example:
+Let x := <6 5 4 3 2 1 0> : BitVec 7.
+x.rotateRight 2 = (<6 5 4 3 2 | 1 0>).rotateRight 2 = <1 0 | 6 5 4 3 2>
+
+(x.rotateLeft 2).getLsb ⟨i, i ≥ 7 - 2⟩
+= <1 0 | 6 5 4 3 2>.getLsb ⟨i, i ≤ 7 - 2⟩
+= <1 0>.getLsb (i - len(<6 5 4 3 2>)
+= <6 5 4 3 2 | 1 0> (i - len<6 4 4 3 2>)
+ -/
+theorem getLsb_rotateRightAux_of_geq {x : BitVec w} {r : Nat} {i : Nat} (hi : i ≥ w - r) :
+    (x.rotateRightAux r).getLsb i = (decide (i < w) && x.getLsb (i - (w - r))) := by
+  rw [rotateRightAux, getLsb_or]
+  suffices (x >>> r).getLsb i = false by
+    simp only [this, getLsb_shiftLeft, Bool.false_or]
+    by_cases hiw : i < w
+    <;> simp [hiw, hi]
+  simp only [getLsb_ushiftRight]
+  apply getLsb_ge
+  omega
+
+/-- `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]
+
+/-- When `r < w`, we give a formula for `(x.rotateRight r).getLsb i`. -/
+theorem getLsb_rotateRight_of_le {x : BitVec w} {r i : Nat} (hr: r < w) :
+    (x.rotateRight r).getLsb i =
+      cond (i < w - r)
+      (x.getLsb (r + i))
+      (decide (i < w) && x.getLsb (i - (w - r))) := by
+  · rw [rotateRight_eq_rotateRightAux_of_lt hr]
+    by_cases h : i < w - r
+    · simp [h, getLsb_rotateRightAux_of_le h]
+    · simp [h, getLsb_rotateRightAux_of_geq <| Nat.le_of_not_lt h]
+
+@[simp]
+theorem getLsb_rotateRight {x : BitVec w} {r i : Nat} :
+    (x.rotateRight r).getLsb i =
+      cond (i < w - (r % w))
+      (x.getLsb ((r % w) + i))
+      (decide (i < w) && x.getLsb (i - (w - (r % w)))) := by
+  rcases w with ⟨rfl, w⟩
+  · simp
+  · rw [← rotateRight_mod_eq_rotateRight, getLsb_rotateRight_of_le (Nat.mod_lt _ (by omega))]
+
 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 +222,21 @@ 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
 
+theorem eq_not_of_ne : ∀ {x y : Bool}, x ≠ y → 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 +247,11 @@ 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 +362,8 @@ 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/ByteArray/Basic.lean

@@ -52,9 +52,13 @@ def get : (a : @& ByteArray) → (@& Fin a.size) → UInt8
 instance : GetElem ByteArray Nat UInt8 fun xs i => i < xs.size where
   getElem xs i h := xs.get ⟨i, h⟩
 
+instance : LawfulGetElem ByteArray Nat UInt8 fun xs i => i < xs.size where
+
 instance : GetElem ByteArray USize UInt8 fun xs i => i.val < xs.size where
   getElem xs i h := xs.uget i h
 
+instance : LawfulGetElem ByteArray USize UInt8 fun xs i => i.val < xs.size where
+
 @[extern "lean_byte_array_set"]
 def set! : ByteArray → (@& Nat) → UInt8 → ByteArray
   | ⟨bs⟩, i, b => ⟨bs.set! i b⟩
@@ -195,18 +199,6 @@ instance : ToString ByteArray := ⟨fun bs => bs.toList.toString⟩
 
 /-- Interpret a `ByteArray` of size 8 as a little-endian `UInt64`. -/
 def ByteArray.toUInt64LE! (bs : ByteArray) : UInt64 :=
-  assert! bs.size == 8
-  (bs.get! 0).toUInt64 <<< 0x38 |||
-  (bs.get! 1).toUInt64 <<< 0x30 |||
-  (bs.get! 2).toUInt64 <<< 0x28 |||
-  (bs.get! 3).toUInt64 <<< 0x20 |||
-  (bs.get! 4).toUInt64 <<< 0x18 |||
-  (bs.get! 5).toUInt64 <<< 0x10 |||
-  (bs.get! 6).toUInt64 <<< 0x8  |||
-  (bs.get! 7).toUInt64
-
-/-- Interpret a `ByteArray` of size 8 as a big-endian `UInt64`. -/
-def ByteArray.toUInt64BE! (bs : ByteArray) : UInt64 :=
   assert! bs.size == 8
   (bs.get! 7).toUInt64 <<< 0x38 |||
   (bs.get! 6).toUInt64 <<< 0x30 |||
@@ -216,3 +208,15 @@ def ByteArray.toUInt64BE! (bs : ByteArray) : UInt64 :=
   (bs.get! 2).toUInt64 <<< 0x10 |||
   (bs.get! 1).toUInt64 <<< 0x8  |||
   (bs.get! 0).toUInt64
+
+/-- Interpret a `ByteArray` of size 8 as a big-endian `UInt64`. -/
+def ByteArray.toUInt64BE! (bs : ByteArray) : UInt64 :=
+  assert! bs.size == 8
+  (bs.get! 0).toUInt64 <<< 0x38 |||
+  (bs.get! 1).toUInt64 <<< 0x30 |||
+  (bs.get! 2).toUInt64 <<< 0x28 |||
+  (bs.get! 3).toUInt64 <<< 0x20 |||
+  (bs.get! 4).toUInt64 <<< 0x18 |||
+  (bs.get! 5).toUInt64 <<< 0x10 |||
+  (bs.get! 6).toUInt64 <<< 0x8  |||
+  (bs.get! 7).toUInt64

src/Init/Data/Channel.lean

@@ -41,7 +41,7 @@ Sends a message on an `Channel`.
 
 This function does not block.
 -/
-def Channel.send (v : α) (ch : Channel α) : BaseIO Unit :=
+def Channel.send (ch : Channel α) (v : α) : BaseIO Unit :=
   ch.atomically do
     let st ← get
     if st.closed then return

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

@@ -4,9 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura, Robert Y. Lewis, Keeley Hoek, Mario Carneiro
 -/
 prelude
-import Init.Data.Nat.Div
 import Init.Data.Nat.Bitwise.Basic
-import Init.Coe
 
 open Nat
 
@@ -15,17 +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⟩
 
@@ -35,12 +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⟩
 
@@ -170,9 +194,3 @@ theorem val_add_one_le_of_lt {n : Nat} {a b : Fin n} (h : a < b) : (a : Nat) + 1
 theorem val_add_one_le_of_gt {n : Nat} {a b : Fin n} (h : a > b) : (b : Nat) + 1 ≤ (a : Nat) := h
 
 end Fin
-
-instance [GetElem cont Nat elem dom] : GetElem cont (Fin n) elem fun xs i => dom xs i where
-  getElem xs i h := getElem xs i.1 h
-
-macro_rules
-  | `(tactic| get_elem_tactic_trivial) => `(tactic| apply Fin.val_lt_of_le; get_elem_tactic_trivial; done)

src/Init/Data/Fin/Fold.lean

@@ -6,12 +6,15 @@ Authors: François G. Dorais
 prelude
 import Init.Data.Nat.Linear
 
+namespace Fin
+
 /-- Folds over `Fin n` from the left: `foldl 3 f x = f (f (f x 0) 1) 2`. -/
 @[inline] def foldl (n) (f : α → Fin n → α) (init : α) : α := loop init 0 where
   /-- Inner loop for `Fin.foldl`. `Fin.foldl.loop n f x i = f (f (f x i) ...) (n-1)`  -/
   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
@@ -19,3 +22,5 @@ import Init.Data.Nat.Linear
   loop : {i // i ≤ n} → α → α
   | ⟨0, _⟩, x => x
   | ⟨i+1, h⟩, x => loop ⟨i, Nat.le_of_lt h⟩ (f ⟨i, h⟩ x)
+
+end Fin

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/FloatArray/Basic.lean

@@ -58,9 +58,13 @@ def get? (ds : FloatArray) (i : Nat) : Option Float :=
 instance : GetElem FloatArray Nat Float fun xs i => i < xs.size where
   getElem xs i h := xs.get ⟨i, h⟩
 
+instance : LawfulGetElem FloatArray Nat Float fun xs i => i < xs.size where
+
 instance : GetElem FloatArray USize Float fun xs i => i.val < xs.size where
   getElem xs i h := xs.uget i h
 
+instance : LawfulGetElem FloatArray USize Float fun xs i => i.val < xs.size where
+
 @[extern "lean_float_array_uset"]
 def uset : (a : FloatArray) → (i : USize) → Float → i.toNat < a.size → FloatArray
   | ⟨ds⟩, i, v, h => ⟨ds.uset i v h⟩

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/Bitwise/Lemmas.lean

@@ -0,0 +1,37 @@
+/-
+Copyright (c) 2023 Siddharth Bhat. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Siddharth Bhat, Jeremy Avigad
+-/
+prelude
+import Init.Data.Nat.Bitwise.Lemmas
+import Init.Data.Int.Bitwise
+
+namespace Int
+
+theorem shiftRight_eq (n : Int) (s : Nat) : n >>> s = Int.shiftRight n s := rfl
+@[simp]
+theorem natCast_shiftRight (n s : Nat) : (n : Int) >>> s = n >>> s := rfl
+
+@[simp]
+theorem negSucc_shiftRight (m n : Nat) :
+    -[m+1] >>> n = -[m >>>n +1] := rfl
+
+theorem shiftRight_add (i : Int) (m n : Nat) :
+    i >>> (m + n) = i >>> m >>> n := by
+  simp only [shiftRight_eq, Int.shiftRight]
+  cases i <;> simp [Nat.shiftRight_add]
+
+theorem shiftRight_eq_div_pow (m : Int) (n : Nat) :
+    m >>> n = m / ((2 ^ n) : Nat) := by
+  simp only [shiftRight_eq, Int.shiftRight, Nat.shiftRight_eq_div_pow]
+  split
+  · simp
+  · rw [negSucc_ediv _ (by norm_cast; exact Nat.pow_pos (Nat.zero_lt_two))]
+    rfl
+
+@[simp]
+theorem zero_shiftRight (n : Nat) : (0 : Int) >>> n = 0 := by
+  simp [Int.shiftRight_eq_div_pow]
+
+end Int

src/Init/Data/Int/DivModLemmas.lean

@@ -8,11 +8,14 @@ prelude
 import Init.Data.Int.DivMod
 import Init.Data.Int.Order
 import Init.Data.Nat.Dvd
+import Init.RCases
 
 /-!
 # Lemmas about integer division needed to bootstrap `omega`.
 -/
 
+-- Remove after the next stage0 update
+set_option allowUnsafeReducibility true
 
 open Nat (succ)
 
@@ -141,12 +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
@@ -177,7 +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
 
@@ -224,7 +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]
@@ -413,6 +420,9 @@ theorem negSucc_emod (m : Nat) {b : Int} (bpos : 0 < b) : -[m+1] % b = b - 1 - m
   match b, eq_succ_of_zero_lt bpos with
   | _, ⟨n, rfl⟩ => rfl
 
+theorem emod_negSucc (m : Nat) (n : Int) :
+  (Int.negSucc m) % n = Int.subNatNat (Int.natAbs n) (Nat.succ (m % Int.natAbs n)) := rfl
+
 theorem ofNat_mod_ofNat (m n : Nat) : (m % n : Int) = ↑(m % n) := rfl
 
 theorem emod_nonneg : ∀ (a : Int) {b : Int}, b ≠ 0 → 0 ≤ a % b
@@ -762,11 +772,13 @@ 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
@@ -935,6 +947,7 @@ 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], _, _ => ⟨_⟩
 
@@ -1053,19 +1066,39 @@ 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

@@ -6,7 +6,7 @@ Authors: Jeremy Avigad, Deniz Aydin, Floris van Doorn, Mario Carneiro
 prelude
 import Init.Data.Int.Basic
 import Init.Conv
-import Init.PropLemmas
+import Init.NotationExtra
 
 namespace Int
 
@@ -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

@@ -6,7 +6,6 @@ Authors: Jeremy Avigad, Deniz Aydin, Floris van Doorn, Mario Carneiro
 prelude
 import Init.Data.Int.Lemmas
 import Init.ByCases
-import Init.RCases
 
 /-!
 # Results about the order properties of the integers, and the integers as an ordered ring.
@@ -97,7 +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]
@@ -188,6 +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 [*]
 
@@ -207,6 +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 [*]
 
@@ -812,6 +813,20 @@ protected theorem sub_lt_sub_right {a b : Int} (h : a < b) (c : Int) : a - c < b
 protected theorem sub_lt_sub {a b c d : Int} (hab : a < b) (hcd : c < d) : a - d < b - c :=
   Int.add_lt_add hab (Int.neg_lt_neg hcd)
 
+protected theorem lt_of_sub_lt_sub_left {a b c : Int} (h : c - a < c - b) : b < a :=
+  Int.lt_of_neg_lt_neg <| Int.lt_of_add_lt_add_left h
+
+protected theorem lt_of_sub_lt_sub_right {a b c : Int} (h : a - c < b - c) : a < b :=
+  Int.lt_of_add_lt_add_right h
+
+@[simp] protected theorem sub_lt_sub_left_iff (a b c : Int) :
+    c - a < c - b ↔ b < a :=
+  ⟨Int.lt_of_sub_lt_sub_left, (Int.sub_lt_sub_left · c)⟩
+
+@[simp] protected theorem sub_lt_sub_right_iff (a b c : Int) :
+    a - c < b - c ↔ a < b :=
+  ⟨Int.lt_of_sub_lt_sub_right, (Int.sub_lt_sub_right · c)⟩
+
 protected theorem sub_lt_sub_of_le_of_lt {a b c d : Int}
   (hab : a ≤ b) (hcd : c < d) : a - d < b - c :=
   Int.add_lt_add_of_le_of_lt hab (Int.neg_lt_neg hcd)
@@ -999,7 +1014,8 @@ theorem natAbs_add_le (a b : Int) : natAbs (a + b) ≤ natAbs a + natAbs b := by
   refine fun a b => subNatNat_elim a b.succ
     (fun m n i => n = b.succ → natAbs i ≤ (m + b).succ) ?_
     (fun i n (e : (n + i).succ = _) => ?_) rfl
-  · rintro i n rfl
+  · intro i n h
+    subst h
     rw [Nat.add_comm _ i, Nat.add_assoc]
     exact Nat.le_add_right i (b.succ + b).succ
   · apply succ_le_succ

src/Init/Data/List.lean

@@ -8,3 +8,5 @@ import Init.Data.List.Basic
 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

@@ -7,6 +7,7 @@ prelude
 import Init.SimpLemmas
 import Init.Data.Nat.Basic
 import Init.Data.Nat.Div
+
 set_option linter.missingDocs true -- keep it documented
 open Decidable List
 
@@ -54,15 +55,6 @@ variable {α : Type u} {β : Type v} {γ : Type w}
 
 namespace List
 
-instance : GetElem (List α) Nat α fun as i => i < as.length where
-  getElem as i h := as.get ⟨i, h⟩
-
-@[simp] theorem cons_getElem_zero (a : α) (as : List α) (h : 0 < (a :: as).length) : getElem (a :: as) 0 h = a := by
-  rfl
-
-@[simp] theorem cons_getElem_succ (a : α) (as : List α) (i : Nat) (h : i + 1 < (a :: as).length) : getElem (a :: as) (i+1) h = getElem as i (Nat.lt_of_succ_lt_succ h) := by
-  rfl
-
 theorem length_add_eq_lengthTRAux (as : List α) (n : Nat) : as.length + n = as.lengthTRAux n := by
   induction as generalizing n with
   | nil  => simp [length, lengthTRAux]
@@ -135,6 +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
 
@@ -144,6 +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
@@ -458,7 +454,7 @@ contains the longest initial segment for which `p` returns true
 and the second part is everything else.
 
 * `span (· > 5) [6, 8, 9, 5, 2, 9] = ([6, 8, 9], [5, 2, 9])`
-* `span (· > 10) [6, 8, 9, 5, 2, 9] = ([6, 8, 9, 5, 2, 9], [])`
+* `span (· > 10) [6, 8, 9, 5, 2, 9] = ([], [6, 8, 9, 5, 2, 9])`
 -/
 @[inline] def span (p : α → Bool) (as : List α) : List α × List α :=
   loop as []
@@ -520,11 +516,6 @@ def drop : Nat → List α → List α
 @[simp] theorem drop_nil : ([] : List α).drop i = [] := by
   cases i <;> rfl
 
-theorem get_drop_eq_drop (as : List α) (i : Nat) (h : i < as.length) : as[i] :: as.drop (i+1) = as.drop i :=
-  match as, i with
-  | _::_, 0   => rfl
-  | _::_, i+1 => get_drop_eq_drop _ i _
-
 /--
 `O(min n |xs|)`. Returns the first `n` elements of `xs`, or the whole list if `n` is too large.
 * `take 0 [a, b, c, d, e] = []`

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/Impl.lean

@@ -0,0 +1,261 @@
+/-
+Copyright (c) 2016 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+
+prelude
+import Init.Data.Array.Lemmas
+
+/-!
+## Tail recursive implementations for `List` definitions.
+
+Many of the proofs require theorems about `Array`,
+so these are in a separate file to minimize imports.
+-/
+
+namespace List
+
+/-- Tail recursive version of `erase`. -/
+@[inline] def setTR (l : List α) (n : Nat) (a : α) : List α := go l n #[] where
+  /-- Auxiliary for `setTR`: `setTR.go l a xs n acc = acc.toList ++ set xs a`,
+  unless `n ≥ l.length` in which case it returns `l` -/
+  go : List α → Nat → Array α → List α
+  | [], _, _ => l
+  | _::xs, 0, acc => acc.toListAppend (a::xs)
+  | x::xs, n+1, acc => go xs n (acc.push x)
+
+@[csimp] theorem set_eq_setTR : @set = @setTR := by
+  funext α l n a; simp [setTR]
+  let rec go (acc) : ∀ xs n, l = acc.data ++ xs →
+    setTR.go l a xs n acc = acc.data ++ xs.set n a
+  | [], _ => fun h => by simp [setTR.go, set, h]
+  | x::xs, 0 => by simp [setTR.go, set]
+  | x::xs, n+1 => fun h => by simp [setTR.go, set]; rw [go _ xs]; {simp}; simp [h]
+  exact (go #[] _ _ rfl).symm
+
+/-- Tail recursive version of `erase`. -/
+@[inline] def eraseTR [BEq α] (l : List α) (a : α) : List α := go l #[] where
+  /-- Auxiliary for `eraseTR`: `eraseTR.go l a xs acc = acc.toList ++ erase xs a`,
+  unless `a` is not present in which case it returns `l` -/
+  go : List α → Array α → List α
+  | [], _ => l
+  | x::xs, acc => bif x == a then acc.toListAppend xs else go xs (acc.push x)
+
+@[csimp] theorem erase_eq_eraseTR : @List.erase = @eraseTR := by
+  funext α _ l a; simp [eraseTR]
+  suffices ∀ xs acc, l = acc.data ++ xs → eraseTR.go l a xs acc = acc.data ++ xs.erase a from
+    (this l #[] (by simp)).symm
+  intro xs; induction xs with intro acc h
+  | nil => simp [List.erase, eraseTR.go, h]
+  | cons x xs IH =>
+    simp [List.erase, eraseTR.go]
+    cases x == a <;> simp
+    · rw [IH]; simp; simp; exact h
+
+/-- Tail recursive version of `eraseIdx`. -/
+@[inline] def eraseIdxTR (l : List α) (n : Nat) : List α := go l n #[] where
+  /-- Auxiliary for `eraseIdxTR`: `eraseIdxTR.go l n xs acc = acc.toList ++ eraseIdx xs a`,
+  unless `a` is not present in which case it returns `l` -/
+  go : List α → Nat → Array α → List α
+  | [], _, _ => l
+  | _::as, 0, acc => acc.toListAppend as
+  | a::as, n+1, acc => go as n (acc.push a)
+
+@[csimp] theorem eraseIdx_eq_eraseIdxTR : @eraseIdx = @eraseIdxTR := by
+  funext α l n; simp [eraseIdxTR]
+  suffices ∀ xs acc, l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ xs.eraseIdx n from
+    (this l #[] (by simp)).symm
+  intro xs; induction xs generalizing n with intro acc h
+  | nil => simp [eraseIdx, eraseIdxTR.go, h]
+  | cons x xs IH =>
+    match n with
+    | 0 => simp [eraseIdx, eraseIdxTR.go]
+    | n+1 =>
+      simp [eraseIdx, eraseIdxTR.go]
+      rw [IH]; simp; simp; exact h
+
+/-- Tail recursive version of `bind`. -/
+@[inline] def bindTR (as : List α) (f : α → List β) : List β := go as #[] where
+  /-- Auxiliary for `bind`: `bind.go f as = acc.toList ++ bind f as` -/
+  @[specialize] go : List α → Array β → List β
+  | [], acc => acc.toList
+  | x::xs, acc => go xs (acc ++ f x)
+
+@[csimp] theorem bind_eq_bindTR : @List.bind = @bindTR := by
+  funext α β as f
+  let rec go : ∀ as acc, bindTR.go f as acc = acc.data ++ as.bind f
+    | [], acc => by simp [bindTR.go, bind]
+    | x::xs, acc => by simp [bindTR.go, bind, go xs]
+  exact (go as #[]).symm
+
+/-- Tail recursive version of `join`. -/
+@[inline] def joinTR (l : List (List α)) : List α := bindTR l id
+
+@[csimp] theorem join_eq_joinTR : @join = @joinTR := by
+  funext α l; rw [← List.bind_id, List.bind_eq_bindTR]; rfl
+
+/-- Tail recursive version of `filterMap`. -/
+@[inline] def filterMapTR (f : α → Option β) (l : List α) : List β := go l #[] where
+  /-- Auxiliary for `filterMap`: `filterMap.go f l = acc.toList ++ filterMap f l` -/
+  @[specialize] go : List α → Array β → List β
+  | [], acc => acc.toList
+  | a::as, acc => match f a with
+    | none => go as acc
+    | some b => go as (acc.push b)
+
+@[csimp] theorem filterMap_eq_filterMapTR : @List.filterMap = @filterMapTR := by
+  funext α β f l
+  let rec go : ∀ as acc, filterMapTR.go f as acc = acc.data ++ as.filterMap f
+    | [], acc => by simp [filterMapTR.go, filterMap]
+    | a::as, acc => by simp [filterMapTR.go, filterMap, go as]; split <;> simp [*]
+  exact (go l #[]).symm
+
+/-- Tail recursive version of `replace`. -/
+@[inline] def replaceTR [BEq α] (l : List α) (b c : α) : List α := go l #[] where
+  /-- Auxiliary for `replace`: `replace.go l b c xs acc = acc.toList ++ replace xs b c`,
+  unless `b` is not found in `xs` in which case it returns `l`. -/
+  @[specialize] go : List α → Array α → List α
+  | [], _ => l
+  | a::as, acc => bif a == b then acc.toListAppend (c::as) else go as (acc.push a)
+
+@[csimp] theorem replace_eq_replaceTR : @List.replace = @replaceTR := by
+  funext α _ l b c; simp [replaceTR]
+  suffices ∀ xs acc, l = acc.data ++ xs →
+      replaceTR.go l b c xs acc = acc.data ++ xs.replace b c from
+    (this l #[] (by simp)).symm
+  intro xs; induction xs with intro acc
+  | nil => simp [replace, replaceTR.go]
+  | cons x xs IH =>
+    simp [replace, replaceTR.go]; split <;> simp [*]
+    · intro h; rw [IH]; simp; simp; exact h
+
+/-- Tail recursive version of `take`. -/
+@[inline] def takeTR (n : Nat) (l : List α) : List α := go l n #[] where
+  /-- Auxiliary for `take`: `take.go l xs n acc = acc.toList ++ take n xs`,
+  unless `n ≥ xs.length` in which case it returns `l`. -/
+  @[specialize] go : List α → Nat → Array α → List α
+  | [], _, _ => l
+  | _::_, 0, acc => acc.toList
+  | a::as, n+1, acc => go as n (acc.push a)
+
+@[csimp] theorem take_eq_takeTR : @take = @takeTR := by
+  funext α n l; simp [takeTR]
+  suffices ∀ xs acc, l = acc.data ++ xs → takeTR.go l xs n acc = acc.data ++ xs.take n from
+    (this l #[] (by simp)).symm
+  intro xs; induction xs generalizing n with intro acc
+  | nil => cases n <;> simp [take, takeTR.go]
+  | cons x xs IH =>
+    cases n with simp [take, takeTR.go]
+    | succ n => intro h; rw [IH]; simp; simp; exact h
+
+/-- Tail recursive version of `takeWhile`. -/
+@[inline] def takeWhileTR (p : α → Bool) (l : List α) : List α := go l #[] where
+  /-- Auxiliary for `takeWhile`: `takeWhile.go p l xs acc = acc.toList ++ takeWhile p xs`,
+  unless no element satisfying `p` is found in `xs` in which case it returns `l`. -/
+  @[specialize] go : List α → Array α → List α
+  | [], _ => l
+  | a::as, acc => bif p a then go as (acc.push a) else acc.toList
+
+@[csimp] theorem takeWhile_eq_takeWhileTR : @takeWhile = @takeWhileTR := by
+  funext α p l; simp [takeWhileTR]
+  suffices ∀ xs acc, l = acc.data ++ xs →
+      takeWhileTR.go p l xs acc = acc.data ++ xs.takeWhile p from
+    (this l #[] (by simp)).symm
+  intro xs; induction xs with intro acc
+  | nil => simp [takeWhile, takeWhileTR.go]
+  | cons x xs IH =>
+    simp [takeWhile, takeWhileTR.go]; split <;> simp [*]
+    · intro h; rw [IH]; simp; simp; exact h
+
+/-- Tail recursive version of `foldr`. -/
+@[specialize] def foldrTR (f : α → β → β) (init : β) (l : List α) : β := l.toArray.foldr f init
+
+@[csimp] theorem foldr_eq_foldrTR : @foldr = @foldrTR := by
+  funext α β f init l; simp [foldrTR, Array.foldr_eq_foldr_data, -Array.size_toArray]
+
+/-- Tail recursive version of `zipWith`. -/
+@[inline] def zipWithTR (f : α → β → γ) (as : List α) (bs : List β) : List γ := go as bs #[] where
+  /-- Auxiliary for `zipWith`: `zipWith.go f as bs acc = acc.toList ++ zipWith f as bs` -/
+  go : List α → List β → Array γ → List γ
+  | a::as, b::bs, acc => go as bs (acc.push (f a b))
+  | _, _, acc => acc.toList
+
+@[csimp] theorem zipWith_eq_zipWithTR : @zipWith = @zipWithTR := by
+  funext α β γ f as bs
+  let rec go : ∀ as bs acc, zipWithTR.go f as bs acc = acc.data ++ as.zipWith f bs
+    | [], _, acc | _::_, [], acc => by simp [zipWithTR.go, zipWith]
+    | a::as, b::bs, acc => by simp [zipWithTR.go, zipWith, go as bs]
+  exact (go as bs #[]).symm
+
+/-- Tail recursive version of `unzip`. -/
+def unzipTR (l : List (α × β)) : List α × List β :=
+  l.foldr (fun (a, b) (al, bl) => (a::al, b::bl)) ([], [])
+
+@[csimp] theorem unzip_eq_unzipTR : @unzip = @unzipTR := by
+  funext α β l; simp [unzipTR]; induction l <;> simp [*]
+
+/-- Tail recursive version of `enumFrom`. -/
+def enumFromTR (n : Nat) (l : List α) : List (Nat × α) :=
+  let arr := l.toArray
+  (arr.foldr (fun a (n, acc) => (n-1, (n-1, a) :: acc)) (n + arr.size, [])).2
+
+@[csimp] theorem enumFrom_eq_enumFromTR : @enumFrom = @enumFromTR := by
+  funext α n l; simp [enumFromTR, -Array.size_toArray]
+  let f := fun (a : α) (n, acc) => (n-1, (n-1, a) :: acc)
+  let rec go : ∀ l n, l.foldr f (n + l.length, []) = (n, enumFrom n l)
+    | [], n => rfl
+    | a::as, n => by
+      rw [← show _ + as.length = n + (a::as).length from Nat.succ_add .., foldr, go as]
+      simp [enumFrom, f]
+  rw [Array.foldr_eq_foldr_data]
+  simp [go]
+
+theorem replicateTR_loop_eq : ∀ n, replicateTR.loop a n acc = replicate n a ++ acc
+  | 0 => rfl
+  | n+1 => by rw [← replicateTR_loop_replicate_eq _ 1 n, replicate, replicate,
+    replicateTR.loop, replicateTR_loop_eq n, replicateTR_loop_eq n, append_assoc]; rfl
+
+/-- Tail recursive version of `dropLast`. -/
+@[inline] def dropLastTR (l : List α) : List α := l.toArray.pop.toList
+
+@[csimp] theorem dropLast_eq_dropLastTR : @dropLast = @dropLastTR := by
+  funext α l; simp [dropLastTR]
+
+/-- Tail recursive version of `intersperse`. -/
+def intersperseTR (sep : α) : List α → List α
+  | [] => []
+  | [x] => [x]
+  | x::y::xs => x :: sep :: y :: xs.foldr (fun a r => sep :: a :: r) []
+
+@[csimp] theorem intersperse_eq_intersperseTR : @intersperse = @intersperseTR := by
+  funext α sep l; simp [intersperseTR]
+  match l with
+  | [] | [_] => rfl
+  | x::y::xs => simp [intersperse]; induction xs generalizing y <;> simp [*]
+
+/-- Tail recursive version of `intercalate`. -/
+def intercalateTR (sep : List α) : List (List α) → List α
+  | [] => []
+  | [x] => x
+  | x::xs => go sep.toArray x xs #[]
+where
+  /-- Auxiliary for `intercalateTR`:
+  `intercalateTR.go sep x xs acc = acc.toList ++ intercalate sep.toList (x::xs)` -/
+  go (sep : Array α) : List α → List (List α) → Array α → List α
+  | x, [], acc => acc.toListAppend x
+  | x, y::xs, acc => go sep y xs (acc ++ x ++ sep)
+
+@[csimp] theorem intercalate_eq_intercalateTR : @intercalate = @intercalateTR := by
+  funext α sep l; simp [intercalate, intercalateTR]
+  match l with
+  | [] => rfl
+  | [_] => simp
+  | x::y::xs =>
+    let rec go {acc x} : ∀ xs,
+      intercalateTR.go sep.toArray x xs acc = acc.data ++ join (intersperse sep (x::xs))
+    | [] => by simp [intercalateTR.go]
+    | _::_ => by simp [intercalateTR.go, go]
+    simp [intersperse, go]
+
+end List

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/Basic.lean

@@ -78,6 +78,8 @@ of a number.
 -/
 
 /-- `testBit m n` returns whether the `(n+1)` least significant bit is `1` or `0`-/
-def testBit (m n : Nat) : Bool := (m >>> n) &&& 1 != 0
+def testBit (m n : Nat) : Bool :=
+  -- `1 &&& n` is faster than `n &&& 1` for big `n`.
+  1 &&& (m >>> n) != 0
 
 end Nat

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
 
@@ -49,13 +50,23 @@ 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]
   unfold bitwise
   simp
 
+@[simp] theorem one_and_eq_mod_two (n : Nat) : 1 &&& n = n % 2 := by
+  if n0 : n = 0 then
+    subst n0; decide
+  else
+    simp only [HAnd.hAnd, AndOp.and, land]
+    cases mod_two_eq_zero_or_one n with | _ h => simp [bitwise, n0, h]
+
 @[simp] theorem and_one_is_mod (x : Nat) : x &&& 1 = x % 2 := by
   if xz : x = 0 then
     simp [xz, zero_and]
@@ -70,7 +81,7 @@ noncomputable def div2Induction {motive : Nat → Sort u}
 /-! ### testBit -/
 
 @[simp] theorem zero_testBit (i : Nat) : testBit 0 i = false := by
-  simp only [testBit, zero_shiftRight, zero_and, bne_self_eq_false]
+  simp only [testBit, zero_shiftRight, and_zero, bne_self_eq_false]
 
 @[simp] theorem testBit_zero (x : Nat) : testBit x 0 = decide (x % 2 = 1) := by
   cases mod_two_eq_zero_or_one x with | _ p => simp [testBit, p]
@@ -187,8 +198,6 @@ 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 +241,7 @@ 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 +265,7 @@ 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 +280,7 @@ 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 +288,7 @@ 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 +410,12 @@ 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