perf: cache visited exprs at CheckAssignmentQuick

With this, lean produces the following zoo of rewrite rules:
```
Option.map.eq_1      : Option.map f none = none
Option.map.eq_2      : Option.map f (some x) = some (f x)
Option.map.eq_def    : Option.map f p = match o with | none => none | (some x) => some (f x)
Option.map.eq_unfold : Option.map = fun f p => match o with | none => none | (some x) => some (f x)
```

The `f.eq_unfold` variant is especially useful to rewrite with `rw`
under
binders.

This implements and fixes #5110

.github/workflows/ci.yml

@@ -176,7 +176,7 @@ jobs:
                 "check-level": 2,
                 "CMAKE_PRESET": "debug",
                 // exclude seriously slow tests
-                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
+                "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest|bv_bitblast_stress'"
               },
               // TODO: suddenly started failing in CI
               /*{
@@ -204,7 +204,7 @@ jobs:
                 "os": "macos-14",
                 "CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-darwin_aarch64",
                 "release": true,
-                "check-level": 1,
+                "check-level": 0,
                 "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*",
@@ -226,21 +226,19 @@ jobs:
               },
               {
                 "name": "Linux aarch64",
-                "os": "ubuntu-latest",
+                "os": "nscloud-ubuntu-22.04-arm64-4x8",
                 "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-*"
+                "llvm-url": "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*"
               },
               {
                 "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",
+                "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 -DCMAKE_LIBRARY_PATH=/usr/lib/i386-linux-gnu/ -DSTAGE0_CMAKE_LIBRARY_PATH=/usr/lib/i386-linux-gnu/",
                 "cmultilib": true,
                 "release": true,
                 "check-level": 2,
@@ -251,7 +249,7 @@ jobs:
                 "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",
+                "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 -DSTAGE0_CMAKE_LIBRARY_PATH=/usr/lib/i386-linux-gnu/",
                 "wasm": true,
                 "cmultilib": true,
                 "release": true,
@@ -259,7 +257,7 @@ jobs:
                 "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\""
+                "CTEST_OPTIONS": "-R \"leantest_1007\\.lean|leantest_Format\\.lean|leanruntest\\_1037.lean|leanruntest_ac_rfl\\.lean|leanruntest_libuv\\.lean\""
               }
             ];
             console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`)
@@ -299,11 +297,11 @@ jobs:
         with:
           msystem: clang64
           # `:` means do not prefix with msystem
-          pacboy: "make: python: cmake clang ccache gmp git: zip: unzip: diffutils: binutils: tree: zstd tar:"
+          pacboy: "make: python: cmake clang ccache gmp libuv git: zip: unzip: diffutils: binutils: tree: zstd tar:"
         if: runner.os == 'Windows'
       - name: Install Brew Packages
         run: |
-          brew install ccache tree zstd coreutils gmp
+          brew install ccache tree zstd coreutils gmp libuv
         if: runner.os == 'macOS'
       - name: Checkout
         uses: actions/checkout@v4
@@ -327,17 +325,19 @@ jobs:
         if: matrix.wasm
       - name: Install 32bit c libs
         run: |
+          sudo dpkg --add-architecture i386
           sudo apt-get update
-          sudo apt-get install -y gcc-multilib g++-multilib ccache
+          sudo apt-get install -y gcc-multilib g++-multilib ccache libuv1-dev:i386
         if: matrix.cmultilib
       - name: Cache
-        uses: actions/cache@v3
+        uses: actions/cache@v4
         with:
           path: .ccache
           key: ${{ matrix.name }}-build-v3-${{ github.event.pull_request.head.sha }}
           # fall back to (latest) previous cache
           restore-keys: |
             ${{ matrix.name }}-build-v3
+          save-always: true
       # open nix-shell once for initial setup
       - name: Setup
         run: |
@@ -382,6 +382,12 @@ jobs:
           make -C build install
       - name: Check Binaries
         run: ${{ matrix.binary-check }} lean-*/bin/* || true
+      - name: Count binary symbols
+        run: |
+          for f in lean-*/bin/*; do
+            echo "$f: $(nm $f | grep " T " | wc -l) exported symbols"
+          done
+        if: matrix.name == 'Windows'
       - name: List Install Tree
         run: |
           # omit contents of Init/, ...

.github/workflows/nix-ci.yml

@@ -55,13 +55,14 @@ jobs:
           # the default is to use a virtual merge commit between the PR and master: just use the PR
           ref: ${{ github.event.pull_request.head.sha }}
       - name: Set Up Nix Cache
-        uses: actions/cache@v3
+        uses: actions/cache@v4
         with:
           path: nix-store-cache
           key: ${{ matrix.name }}-nix-store-cache-${{ github.sha }}
           # fall back to (latest) previous cache
           restore-keys: |
             ${{ matrix.name }}-nix-store-cache
+          save-always: true
       - name: Further Set Up Nix Cache
         shell: bash -euxo pipefail {0}
         run: |
@@ -78,13 +79,14 @@ jobs:
           sudo mkdir -m0770 -p /nix/var/cache/ccache
           sudo chown -R $USER /nix/var/cache/ccache
       - name: Setup CCache Cache
-        uses: actions/cache@v3
+        uses: actions/cache@v4
         with:
           path: /nix/var/cache/ccache
           key: ${{ matrix.name }}-nix-ccache-${{ github.sha }}
           # fall back to (latest) previous cache
           restore-keys: |
             ${{ matrix.name }}-nix-ccache
+          save-always: true
       - name: Further Set Up CCache Cache
         run: |
           sudo chown -R root:nixbld /nix/var/cache
@@ -103,7 +105,7 @@ jobs:
         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
+          nix build $NIX_BUILD_ARGS --update-input lean --no-write-lock-file ./doc#{lean-mdbook,leanInk,alectryon,inked} -o push-doc
           nix build $NIX_BUILD_ARGS --update-input lean --no-write-lock-file ./doc
           # https://github.com/netlify/cli/issues/1809
           cp -r --dereference ./result ./dist
@@ -146,5 +148,3 @@ jobs:
       - name: Fixup CCache Cache
         run: |
           sudo chown -R $USER /nix/var/cache
-      - name: CCache stats
-        run: CCACHE_DIR=/nix/var/cache/ccache nix run .#nixpkgs.ccache -- -s

.github/workflows/pr-release.yml

@@ -163,7 +163,8 @@ jobs:
             # so keep in sync
 
             # Use GitHub API to check if a comment already exists
-            existing_comment="$(curl -L -s -H "Authorization: token ${{ secrets.MATHLIB4_BOT }}" \
+            existing_comment="$(curl --retry 3 --location --silent \
+                                    -H "Authorization: token ${{ secrets.MATHLIB4_BOT }}" \
                                     -H "Accept: application/vnd.github.v3+json" \
                                     "https://api.github.com/repos/leanprover/lean4/issues/${{ steps.workflow-info.outputs.pullRequestNumber }}/comments" \
                                     | jq 'first(.[] | select(.body | test("^- . Mathlib") or startswith("Mathlib CI status")) | select(.user.login == "leanprover-community-mathlib4-bot"))')"
@@ -328,7 +329,7 @@ jobs:
             git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
             echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}" > lean-toolchain
             git add lean-toolchain
-            sed -i 's,require "leanprover-community" / "batteries" @ ".\+",require "leanprover-community" / "batteries" @ "git#nightly-testing-'"${MOST_RECENT_NIGHTLY}"'",' lakefile.lean
+            sed -i 's,require "leanprover-community" / "batteries" @ git ".\+",require "leanprover-community" / "batteries" @ git "nightly-testing-'"${MOST_RECENT_NIGHTLY}"'",' lakefile.lean
             lake update batteries
             git add lakefile.lean lake-manifest.json
             git commit -m "Update lean-toolchain for testing https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"

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

@@ -14,8 +14,9 @@ jobs:
         # (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 "Trying to find a run with branch $head_ref and commit $head_sha"
+        run_id="$(gh run list -e pull_request -b "$head_ref" -c "$head_sha" \
+          --workflow 'CI' --limit 1 --json databaseId --jq '.[0].databaseId')"
         echo "Run id: ${run_id}"
         gh run view "$run_id"
         echo "Cancelling (just in case)"
@@ -29,5 +30,6 @@ jobs:
       shell: bash
       env:
         head_ref: ${{ github.head_ref }}
+        head_sha: ${{ github.event.pull_request.head.sha }}
         GH_TOKEN: ${{ github.token }}
         GH_REPO: ${{ github.repository }}

.github/workflows/update-stage0.yml

@@ -47,7 +47,7 @@ jobs:
     #  uses: DeterminateSystems/magic-nix-cache-action@v2
     - if: env.should_update_stage0 == 'yes'
       name: Restore Build Cache
-      uses: actions/cache/restore@v3
+      uses: actions/cache/restore@v4
       with:
         path: nix-store-cache
         key: Nix Linux-nix-store-cache-${{ github.sha }}

CMakeLists.txt

@@ -30,6 +30,35 @@ if(NOT (DEFINED STAGE0_CMAKE_EXECUTABLE_SUFFIX))
     set(STAGE0_CMAKE_EXECUTABLE_SUFFIX "${CMAKE_EXECUTABLE_SUFFIX}")
 endif()
 
+# Don't do anything with cadical on wasm
+if (NOT ${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
+  # On CI Linux, we source cadical from Nix instead; see flake.nix
+  find_program(CADICAL cadical)
+  if(NOT CADICAL)
+    set(CADICAL_CXX c++)
+    find_program(CCACHE ccache)
+    if(CCACHE)
+      set(CADICAL_CXX "${CCACHE} ${CADICAL_CXX}")
+    endif()
+    # missing stdio locking API on Windows
+    if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
+      string(APPEND CADICAL_CXXFLAGS " -DNUNLOCKED")
+    endif()
+    ExternalProject_add(cadical
+      PREFIX cadical
+      GIT_REPOSITORY https://github.com/arminbiere/cadical
+      GIT_TAG rel-1.9.5
+      CONFIGURE_COMMAND ""
+      # https://github.com/arminbiere/cadical/blob/master/BUILD.md#manual-build
+      BUILD_COMMAND $(MAKE) -f ${CMAKE_SOURCE_DIR}/src/cadical.mk CMAKE_EXECUTABLE_SUFFIX=${CMAKE_EXECUTABLE_SUFFIX} CXX=${CADICAL_CXX} CXXFLAGS=${CADICAL_CXXFLAGS}
+      BUILD_IN_SOURCE ON
+      INSTALL_COMMAND "")
+    set(CADICAL ${CMAKE_BINARY_DIR}/cadical/cadical${CMAKE_EXECUTABLE_SUFFIX} CACHE FILEPATH "path to cadical binary" FORCE)
+    set(EXTRA_DEPENDS "cadical")
+  endif()
+  list(APPEND CL_ARGS -DCADICAL=${CADICAL})
+endif()
+
 ExternalProject_add(stage0
   SOURCE_DIR "${LEAN_SOURCE_DIR}/stage0"
   SOURCE_SUBDIR src

CODEOWNERS

@@ -43,3 +43,5 @@
 /src/Init/Guard.lean @digama0
 /src/Lean/Server/CodeActions/ @digama0
 /src/Std/ @TwoFX
+/src/Std/Tactic/BVDecide/ @hargoniX
+/src/Lean/Elab/Tactic/BVDecide/ @hargoniX

LICENSES

@@ -1341,3 +1341,33 @@ whether future versions of the GNU Lesser General Public License shall
 apply, that proxy's public statement of acceptance of any version is
 permanent authorization for you to choose that version for the
 Library.
+==============================================================================
+CaDiCaL is under the MIT License:
+==============================================================================
+MIT License
+
+Copyright (c) 2016-2021 Armin Biere, Johannes Kepler University Linz, Austria
+Copyright (c) 2020-2021 Mathias Fleury, Johannes Kepler University Linz, Austria
+Copyright (c) 2020-2021 Nils Froleyks, Johannes Kepler University Linz, Austria
+Copyright (c) 2022-2024 Katalin Fazekas, Vienna University of Technology, Austria
+Copyright (c) 2021-2024 Armin Biere, University of Freiburg, Germany
+Copyright (c) 2021-2024 Mathias Fleury, University of Freiburg, Germany
+Copyright (c) 2023-2024 Florian Pollitt, University of Freiburg, Germany
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.

RELEASES.md

@@ -8,10 +8,14 @@ This file contains work-in-progress notes for the upcoming release, as well as p
 Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
 of each version.
 
-v4.11.0
+v4.12.0
 ----------
 Development in progress.
 
+v4.11.0
+----------
+Release candidate, release notes will be copied from the branch `releases/v4.11.0` once completed.
+
 v4.10.0
 ----------
 

doc/dev/debugging.md

@@ -5,7 +5,7 @@ Some notes on how to debug Lean, which may also be applicable to debugging Lean
 
 ## Tracing
 
-In `CoreM` and derived monads, we use `trace![traceCls] "msg with {interpolations}"` to fill the structured trace viewable with `set_option trace.traceCls true`.
+In `CoreM` and derived monads, we use `trace[traceCls] "msg with {interpolations}"` to fill the structured trace viewable with `set_option trace.traceCls true`.
 New trace classes have to be registered using `registerTraceClass` first.
 
 Notable trace classes:
@@ -22,7 +22,9 @@ Notable trace classes:
 
 In pure contexts or when execution is aborted before the messages are finally printed, one can instead use the term `dbg_trace "msg with {interpolations}"; val` (`;` can also be replaced by a newline), which will print the message to stderr before evaluating `val`. `dbgTraceVal val` can be used as a shorthand for `dbg_trace "{val}"; val`.
 Note that if the return value is not actually used, the trace code is silently dropped as well.
-In the language server, stderr output is buffered and shown as messages after a command has been elaborated, unless the option `server.stderrAsMessages` is deactivated.
+
+By default, such stderr output is buffered and shown as messages after a command has been elaborated, which is necessary to ensure deterministic ordering of messages under parallelism.
+If Lean aborts the process before it can finish the command or takes too long to do that, using `-DstderrAsMessages=false` avoids this buffering and shows `dbg_trace` output (but not `trace`s or other diagnostics) immediately.
 
 ## Debuggers
 

doc/dev/release_checklist.md

@@ -147,33 +147,31 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
   - You can monitor this at `https://github.com/leanprover/lean4/actions/workflows/ci.yml`, looking for the `v4.7.0-rc1` tag.
   - This step can take up to an hour.
 - (GitHub release notes) Once the release appears at https://github.com/leanprover/lean4/releases/
-  - Edit the release notes on Github to select the "Set as a pre-release box".
-  - If release notes have been written already, copy the section of `RELEASES.md` for this version into the Github release notes
-    and use the title "Changes since v4.6.0 (from RELEASES.md)".
-  - Otherwise, in the "previous tag" dropdown, select `v4.6.0`, and click "Generate release notes".
+  - Verify that the release is marked as a prerelease (this should have been done automatically by the CI release job).
+  - In the "previous tag" dropdown, select `v4.6.0`, and click "Generate release notes".
     This will add a list of all the commits since the last stable version.
-    - Delete anything already mentioned in the hand-written release notes above.
     - Delete "update stage0" commits, and anything with a completely inscrutable commit message.
-    - Briefly rearrange the remaining items by category (e.g. `simp`, `lake`, `bug fixes`),
-      but for minor items don't put any work in expanding on commit messages.
-  - (How we want to release notes to look is evolving: please update this section if it looks wrong!)
 - Next, we will move a curated list of downstream repos to the release candidate.
-  - This assumes that there is already a *reviewed* branch `bump/v4.7.0` on each repository
-    containing the required adaptations (or no adaptations are required).
-    The preparation of this branch is beyond the scope of this document.
+  - This assumes that for each repository either:
+    * There is already a *reviewed* branch `bump/v4.7.0` containing the required adaptations.
+      The preparation of this branch is beyond the scope of this document.
+    * The repository does not need any changes to move to the new version.
   - For each of the target repositories:
-    - Checkout the `bump/v4.7.0` branch.
-    - Verify that the `lean-toolchain` is set to the nightly from which the release candidate was created.
-    - `git merge origin/master`
-    - Change the `lean-toolchain` to `leanprover/lean4:v4.7.0-rc1`
-    - In `lakefile.lean`, change any dependencies which were using `nightly-testing` or `bump/v4.7.0` branches
-      back to `master` or `main`, and run `lake update` for those dependencies.
-    - Run `lake build` to ensure that dependencies are found (but it's okay to stop it after a moment).
-    - `git commit`
-    - `git push`
-    - Open a PR from `bump/v4.7.0` to `master`, and either merge it yourself after CI, if appropriate,
-      or notify the maintainers that it is ready to go.
-    - Once this PR has been merged, tag `master` with `v4.7.0-rc1` and push this tag.
+    - If the repository does not need any changes (i.e. `bump/v4.7.0` does not exist) then create
+      a new PR updating `lean-toolchain` to `leanprover/lean4:v4.7.0-rc1` and running `lake update`.
+    - Otherwise:
+      - Checkout the `bump/v4.7.0` branch.
+      - Verify that the `lean-toolchain` is set to the nightly from which the release candidate was created.
+      - `git merge origin/master`
+      - Change the `lean-toolchain` to `leanprover/lean4:v4.7.0-rc1`
+      - In `lakefile.lean`, change any dependencies which were using `nightly-testing` or `bump/v4.7.0` branches
+        back to `master` or `main`, and run `lake update` for those dependencies.
+      - Run `lake build` to ensure that dependencies are found (but it's okay to stop it after a moment).
+      - `git commit`
+      - `git push`
+      - Open a PR from `bump/v4.7.0` to `master`, and either merge it yourself after CI, if appropriate,
+        or notify the maintainers that it is ready to go.
+    - Once the PR has been merged, tag `master` with `v4.7.0-rc1` and push this tag.
   - 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!
@@ -194,7 +192,7 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
     finalized release notes from the `releases/v4.6.0` branch.
   - Replaces the "development in progress" in the `v4.7.0` section of `RELEASES.md` with
     ```
-    Release candidate, release notes will be copied from `branch releases/v4.7.0` once completed.
+    Release candidate, release notes will be copied from the branch `releases/v4.7.0` once completed.
     ```
     and inserts the following section before that section:
     ```
@@ -203,6 +201,8 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
     Development in progress.
     ```
   - Removes all the entries from the `./releases_drafts/` folder.
+  - Titled "chore: begin development cycle for v4.8.0"
+
 
 ## Time estimates:
 Slightly longer than the corresponding steps for a stable release.

doc/examples/widgets.lean

@@ -4,15 +4,18 @@ open Lean Widget
 /-!
 # The user-widgets system
 
-Proving and programming are inherently interactive tasks. Lots of mathematical objects and data
-structures are visual in nature. *User widgets* let you associate custom interactive UIs with
-sections of a Lean document. User widgets are rendered in the Lean infoview.
+Proving and programming are inherently interactive tasks.
+Lots of mathematical objects and data structures are visual in nature.
+*User widgets* let you associate custom interactive UIs
+with sections of a Lean document.
+User widgets are rendered in the Lean infoview.
 
 ![Rubik's cube](../images/widgets_rubiks.png)
 
 ## Trying it out
 
-To try it out, simply type in the following code and place your cursor over the `#widget` command.
+To try it out, type in the following code and place your cursor over the `#widget` command.
+You can also [view this manual entry in the online editor](https://live.lean-lang.org/#url=https%3A%2F%2Fraw.githubusercontent.com%2Fleanprover%2Flean4%2Fmaster%2Fdoc%2Fexamples%2Fwidgets.lean).
 -/
 
 @[widget_module]
@@ -21,38 +24,37 @@ def helloWidget : Widget.Module where
     import * as React from 'react';
     export default function(props) {
       const name = props.name || 'world'
-      return React.createElement('p', {}, name + '!')
+      return React.createElement('p', {}, 'Hello ' + name + '!')
     }"
 
 #widget helloWidget
 
 /-!
 If you want to dive into a full sample right away, check out
-[`RubiksCube`](https://github.com/leanprover/lean4-samples/blob/main/RubiksCube/).
+[`Rubiks`](https://github.com/leanprover-community/ProofWidgets4/blob/main/ProofWidgets/Demos/Rubiks.lean).
+This sample uses higher-level widget components from the ProofWidgets library.
+
 Below, we'll explain the system piece by piece.
 
 ⚠️ WARNING: All of the user widget APIs are **unstable** and subject to breaking changes.
 
-## Widget sources and instances
+## Widget modules and instances
 
-A *widget source* is a valid JavaScript [ESModule](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules)
-which exports a [React component](https://reactjs.org/docs/components-and-props.html). To access
-React, the module must use `import * as React from 'react'`. Our first example of a widget source
-is of course the value of `helloWidget.javascript`.
+A [widget module](https://leanprover-community.github.io/mathlib4_docs/Lean/Widget/UserWidget.html#Lean.Widget.Module)
+is a valid JavaScript [ESModule](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules)
+that can execute in the Lean infoview.
+Most widget modules export a [React component](https://reactjs.org/docs/components-and-props.html)
+as the piece of user interface to be rendered.
+To access React, the module can use `import * as React from 'react'`.
+Our first example of a widget module is `helloWidget` above.
+Widget modules must be registered with the `@[widget_module]` attribute.
 
-We can register a widget source with the `@[widget]` attribute, giving it a friendlier name
-in the `name` field. This is bundled together in a `UserWidgetDefinition`.
-
-A *widget instance* is then the identifier of a `UserWidgetDefinition` (so `` `helloWidget ``,
-not `"Hello"`) associated with a range of positions in the Lean source code. Widget instances
-are stored in the *infotree* in the same manner as other information about the source file
-such as the type of every expression. In our example, the `#widget` command stores a widget instance
-with the entire line as its range. We can think of a widget instance as an instruction for the
-infoview: "when the user places their cursor here, please render the following widget".
-
-Every widget instance also contains a `props : Json` value. This value is passed as an argument
-to the React component. In our first invocation of `#widget`, we set it to `.null`. Try out what
-happens when you type in:
+A [widget instance](https://leanprover-community.github.io/mathlib4_docs/Lean/Widget/Types.html#Lean.Widget.WidgetInstance)
+is then the identifier of a widget module (e.g. `` `helloWidget ``)
+bundled with a value for its props.
+This value is passed as the argument to the React component.
+In our first invocation of `#widget`, we set it to `.null`.
+Try out what happens when you type in:
 -/
 
 structure HelloWidgetProps where
@@ -62,21 +64,37 @@ structure HelloWidgetProps where
 #widget helloWidget with { name? := "<your name here>" : HelloWidgetProps }
 
 /-!
-💡 NOTE: The RPC system presented below does not depend on JavaScript. However the primary use case
-is the web-based infoview in VSCode.
+Under the hood, widget instances are associated with a range of positions in the source file.
+Widget instances are stored in the *infotree*
+in the same manner as other information about the source file
+such as the type of every expression.
+In our example, the `#widget` command stores a widget instance
+with the entire line as its range.
+One can think of the infotree entry as an instruction for the infoview:
+"when the user places their cursor here, please render the following widget".
+-/
 
+/-!
 ## Querying the Lean server
 
-Besides enabling us to create cool client-side visualizations, user widgets come with the ability
-to communicate with the Lean server. Thanks to this, they have the same metaprogramming capabilities
-as custom elaborators or the tactic framework. To see this in action, let's implement a `#check`
-command as a web input form. This example assumes some familiarity with React.
+💡 NOTE: The RPC system presented below does not depend on JavaScript.
+However, the primary use case is the web-based infoview in VSCode.
 
-The first thing we'll need is to create an *RPC method*. Meaning "Remote Procedure Call", this
-is basically a Lean function callable from widget code (possibly remotely over the internet).
+Besides enabling us to create cool client-side visualizations,
+user widgets have the ability to communicate with the Lean server.
+Thanks to this, they have the same metaprogramming capabilities
+as custom elaborators or the tactic framework.
+To see this in action, let's implement a `#check` command as a web input form.
+This example assumes some familiarity with React.
+
+The first thing we'll need is to create an *RPC method*.
+Meaning "Remote Procedure Call",this is a Lean function callable from widget code
+(possibly remotely over the internet).
 Our method will take in the `name : Name` of a constant in the environment and return its type.
-By convention, we represent the input data as a `structure`. Since it will be sent over from JavaScript,
-we need `FromJson` and `ToJson`. We'll see below why the position field is needed.
+By convention, we represent the input data as a `structure`.
+Since it will be sent over from JavaScript,
+we need `FromJson` and `ToJson` instnace.
+We'll see why the position field is needed later.
 -/
 
 structure GetTypeParams where
@@ -87,25 +105,33 @@ structure GetTypeParams where
   deriving FromJson, ToJson
 
 /-!
-After its arguments, we define the `getType` method. Every RPC method executes in the `RequestM`
-monad and must return a `RequestTask α` where `α` is its "actual" return type. The `Task` is so
-that requests can be handled concurrently. A first guess for `α` might be `Expr`. However,
-expressions in general can be large objects which depend on an `Environment` and `LocalContext`.
-Thus we cannot directly serialize an `Expr` and send it to the widget. Instead, there are two
-options:
-- One is to send a *reference* which points to an object residing on the server. From JavaScript's
-  point of view, references are entirely opaque, but they can be sent back to other RPC methods for
-  further processing.
-- Two is to pretty-print the expression and send its textual representation called `CodeWithInfos`.
-  This representation contains extra data which the infoview uses for interactivity. We take this
-  strategy here.
+After its argument structure, we define the `getType` method.
+RPCs method execute in the `RequestM` monad and must return a `RequestTask α`
+where `α` is the "actual" return type.
+The `Task` is so that requests can be handled concurrently.
+As a first guess, we'd use `Expr` as `α`.
+However, expressions in general can be large objects
+which depend on an `Environment` and `LocalContext`.
+Thus we cannot directly serialize an `Expr` and send it to JavaScript.
+Instead, there are two options:
 
-RPC methods execute in the context of a file, but not any particular `Environment` so they don't
-know about the available `def`initions and `theorem`s. Thus, we need to pass in a position at which
-we want to use the local `Environment`. This is why we store it in `GetTypeParams`. The `withWaitFindSnapAtPos`
-method launches a concurrent computation whose job is to find such an `Environment` and a bit
-more information for us, in the form of a `snap : Snapshot`. With this in hand, we can call
-`MetaM` procedures to find out the type of `name` and pretty-print it.
+- One is to send a *reference* which points to an object residing on the server.
+  From JavaScript's point of view, references are entirely opaque,
+  but they can be sent back to other RPC methods for further processing.
+- The other is to pretty-print the expression and send its textual representation called `CodeWithInfos`.
+  This representation contains extra data which the infoview uses for interactivity.
+  We take this strategy here.
+
+RPC methods execute in the context of a file,
+but not of any particular `Environment`,
+so they don't know about the available `def`initions and `theorem`s.
+Thus, we need to pass in a position at which we want to use the local `Environment`.
+This is why we store it in `GetTypeParams`.
+The `withWaitFindSnapAtPos` method launches a concurrent computation
+whose job is to find such an `Environment` for us,
+in the form of a `snap : Snapshot`.
+With this in hand, we can call `MetaM` procedures
+to find out the type of `name` and pretty-print it.
 -/
 
 open Server RequestM in
@@ -121,18 +147,22 @@ def getType (params : GetTypeParams) : RequestM (RequestTask CodeWithInfos) :=
 /-!
 ## Using infoview components
 
-Now that we have all we need on the server side, let's write the widget source. By importing
-`@leanprover/infoview`, widgets can render UI components used to implement the infoview itself.
-For example, the `<InteractiveCode>` component displays expressions with `term : type` tooltips
-as seen in the goal view. We will use it to implement our custom `#check` display.
+Now that we have all we need on the server side, let's write the widget module.
+By importing `@leanprover/infoview`, widgets can render UI components used to implement the infoview itself.
+For example, the `<InteractiveCode>` component displays expressions
+with `term : type` tooltips as seen in the goal view.
+We will use it to implement our custom `#check` display.
 
 ⚠️ WARNING: Like the other widget APIs, the infoview JS API is **unstable** and subject to breaking changes.
 
-The code below demonstrates useful parts of the API. To make RPC method calls, we use the `RpcContext`.
-The `useAsync` helper packs the results of a call into an `AsyncState` structure which indicates
-whether the call has resolved successfully, has returned an error, or is still in-flight. Based
-on this we either display an `InteractiveCode` with the type, `mapRpcError` the error in order
-to turn it into a readable message, or show a `Loading..` message, respectively.
+The code below demonstrates useful parts of the API.
+To make RPC method calls, we invoke the `useRpcSession` hook.
+The `useAsync` helper packs the results of an RPC call into an `AsyncState` structure
+which indicates whether the call has resolved successfully,
+has returned an error, or is still in-flight.
+Based on this we either display an `InteractiveCode` component with the result,
+`mapRpcError` the error in order to turn it into a readable message,
+or show a `Loading..` message, respectively.
 -/
 
 @[widget_module]
@@ -140,10 +170,10 @@ def checkWidget : Widget.Module where
   javascript := "
 import * as React from 'react';
 const e = React.createElement;
-import { RpcContext, InteractiveCode, useAsync, mapRpcError } from '@leanprover/infoview';
+import { useRpcSession, InteractiveCode, useAsync, mapRpcError } from '@leanprover/infoview';
 
 export default function(props) {
-  const rs = React.useContext(RpcContext)
+  const rs = useRpcSession()
   const [name, setName] = React.useState('getType')
 
   const st = useAsync(() =>
@@ -159,7 +189,7 @@ export default function(props) {
 "
 
 /-!
-Finally we can try out the widget.
+We can now try out the widget.
 -/
 
 #widget checkWidget
@@ -169,30 +199,31 @@ Finally we can try out the widget.
 
 ## Building widget sources
 
-While typing JavaScript inline is fine for a simple example, for real developments we want to use
-packages from NPM, a proper build system, and JSX. Thus, most actual widget sources are built with
-Lake and NPM. They consist of multiple files and may import libraries which don't work as ESModules
-by default. On the other hand a widget source must be a single, self-contained ESModule in the form
-of a string. Readers familiar with web development may already have guessed that to obtain such a
-string, we need a *bundler*. Two popular choices are [`rollup.js`](https://rollupjs.org/guide/en/)
-and [`esbuild`](https://esbuild.github.io/). If we go with `rollup.js`, to make a widget work with
-the infoview we need to:
+While typing JavaScript inline is fine for a simple example,
+for real developments we want to use packages from NPM, a proper build system, and JSX.
+Thus, most actual widget sources are built with Lake and NPM.
+They consist of multiple files and may import libraries which don't work as ESModules by default.
+On the other hand a widget module must be a single, self-contained ESModule in the form of a string.
+Readers familiar with web development may already have guessed that to obtain such a string, we need a *bundler*.
+Two popular choices are [`rollup.js`](https://rollupjs.org/guide/en/)
+and [`esbuild`](https://esbuild.github.io/).
+If we go with `rollup.js`, to make a widget work with the infoview we need to:
 - Set [`output.format`](https://rollupjs.org/guide/en/#outputformat) to `'es'`.
 - [Externalize](https://rollupjs.org/guide/en/#external) `react`, `react-dom`, `@leanprover/infoview`.
   These libraries are already loaded by the infoview so they should not be bundled.
 
-In the RubiksCube sample, we provide a working `rollup.js` build configuration in
-[rollup.config.js](https://github.com/leanprover/lean4-samples/blob/main/RubiksCube/widget/rollup.config.js).
+ProofWidgets provides a working `rollup.js` build configuration in
+[rollup.config.js](https://github.com/leanprover-community/ProofWidgets4/blob/main/widget/rollup.config.js).
 
 ## Inserting text
 
-We can also instruct the editor to insert text, copy text to the clipboard, or
-reveal a certain location in the document.
-To do this, use the `React.useContext(EditorContext)` React context.
-This will return an `EditorConnection` whose `api` field contains a number of methods to
-interact with the text editor.
+Besides making RPC calls, widgets can instruct the editor to carry out certain actions.
+We can insert text, copy text to the clipboard, or highlight a certain location in the document.
+To do this, use the `EditorContext` React context.
+This will return an `EditorConnection`
+whose `api` field contains a number of methods that interact with the editor.
 
-You can see the full API for this [here](https://github.com/leanprover/vscode-lean4/blob/master/lean4-infoview-api/src/infoviewApi.ts#L52)
+The full API can be viewed [here](https://github.com/leanprover/vscode-lean4/blob/master/lean4-infoview-api/src/infoviewApi.ts#L52).
 -/
 
 @[widget_module]
@@ -212,6 +243,4 @@ export default function(props) {
 }
 "
 
-/-! Finally, we can try this out: -/
-
 #widget insertTextWidget

doc/flake.lock

@@ -18,12 +18,15 @@
       }
     },
     "flake-utils": {
+      "inputs": {
+        "systems": "systems"
+      },
       "locked": {
-        "lastModified": 1656928814,
-        "narHash": "sha256-RIFfgBuKz6Hp89yRr7+NR5tzIAbn52h8vT6vXkYjZoM=",
+        "lastModified": 1710146030,
+        "narHash": "sha256-SZ5L6eA7HJ/nmkzGG7/ISclqe6oZdOZTNoesiInkXPQ=",
         "owner": "numtide",
         "repo": "flake-utils",
-        "rev": "7e2a3b3dfd9af950a856d66b0a7d01e3c18aa249",
+        "rev": "b1d9ab70662946ef0850d488da1c9019f3a9752a",
         "type": "github"
       },
       "original": {
@@ -35,13 +38,12 @@
     "lean": {
       "inputs": {
         "flake-utils": "flake-utils",
-        "lean4-mode": "lean4-mode",
-        "nix": "nix",
-        "nixpkgs": "nixpkgs_2"
+        "nixpkgs": "nixpkgs",
+        "nixpkgs-old": "nixpkgs-old"
       },
       "locked": {
         "lastModified": 0,
-        "narHash": "sha256-YnYbmG0oou1Q/GE4JbMNb8/yqUVXBPIvcdQQJHBqtPk=",
+        "narHash": "sha256-saRAtQ6VautVXKDw1XH35qwP0KEBKTKZbg/TRa4N9Vw=",
         "path": "../.",
         "type": "path"
       },
@@ -50,22 +52,6 @@
         "type": "path"
       }
     },
-    "lean4-mode": {
-      "flake": false,
-      "locked": {
-        "lastModified": 1659020985,
-        "narHash": "sha256-+dRaXB7uvN/weSZiKcfSKWhcdJVNg9Vg8k0pJkDNjpc=",
-        "owner": "leanprover",
-        "repo": "lean4-mode",
-        "rev": "37d5c99b7b29c80ab78321edd6773200deb0bca6",
-        "type": "github"
-      },
-      "original": {
-        "owner": "leanprover",
-        "repo": "lean4-mode",
-        "type": "github"
-      }
-    },
     "leanInk": {
       "flake": false,
       "locked": {
@@ -83,22 +69,6 @@
         "type": "github"
       }
     },
-    "lowdown-src": {
-      "flake": false,
-      "locked": {
-        "lastModified": 1633514407,
-        "narHash": "sha256-Dw32tiMjdK9t3ETl5fzGrutQTzh2rufgZV4A/BbxuD4=",
-        "owner": "kristapsdz",
-        "repo": "lowdown",
-        "rev": "d2c2b44ff6c27b936ec27358a2653caaef8f73b8",
-        "type": "github"
-      },
-      "original": {
-        "owner": "kristapsdz",
-        "repo": "lowdown",
-        "type": "github"
-      }
-    },
     "mdBook": {
       "flake": false,
       "locked": {
@@ -115,65 +85,13 @@
         "type": "github"
       }
     },
-    "nix": {
-      "inputs": {
-        "lowdown-src": "lowdown-src",
-        "nixpkgs": "nixpkgs",
-        "nixpkgs-regression": "nixpkgs-regression"
-      },
-      "locked": {
-        "lastModified": 1657097207,
-        "narHash": "sha256-SmeGmjWM3fEed3kQjqIAO8VpGmkC2sL1aPE7kKpK650=",
-        "owner": "NixOS",
-        "repo": "nix",
-        "rev": "f6316b49a0c37172bca87ede6ea8144d7d89832f",
-        "type": "github"
-      },
-      "original": {
-        "owner": "NixOS",
-        "repo": "nix",
-        "type": "github"
-      }
-    },
     "nixpkgs": {
       "locked": {
-        "lastModified": 1653988320,
-        "narHash": "sha256-ZaqFFsSDipZ6KVqriwM34T739+KLYJvNmCWzErjAg7c=",
+        "lastModified": 1710889954,
+        "narHash": "sha256-Pr6F5Pmd7JnNEMHHmspZ0qVqIBVxyZ13ik1pJtm2QXk=",
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "2fa57ed190fd6c7c746319444f34b5917666e5c1",
-        "type": "github"
-      },
-      "original": {
-        "owner": "NixOS",
-        "ref": "nixos-22.05-small",
-        "repo": "nixpkgs",
-        "type": "github"
-      }
-    },
-    "nixpkgs-regression": {
-      "locked": {
-        "lastModified": 1643052045,
-        "narHash": "sha256-uGJ0VXIhWKGXxkeNnq4TvV3CIOkUJ3PAoLZ3HMzNVMw=",
-        "owner": "NixOS",
-        "repo": "nixpkgs",
-        "rev": "215d4d0fd80ca5163643b03a33fde804a29cc1e2",
-        "type": "github"
-      },
-      "original": {
-        "owner": "NixOS",
-        "repo": "nixpkgs",
-        "rev": "215d4d0fd80ca5163643b03a33fde804a29cc1e2",
-        "type": "github"
-      }
-    },
-    "nixpkgs_2": {
-      "locked": {
-        "lastModified": 1657208011,
-        "narHash": "sha256-BlIFwopAykvdy1DYayEkj6ZZdkn+cVgPNX98QVLc0jM=",
-        "owner": "NixOS",
-        "repo": "nixpkgs",
-        "rev": "2770cc0b1e8faa0e20eb2c6aea64c256a706d4f2",
+        "rev": "7872526e9c5332274ea5932a0c3270d6e4724f3b",
         "type": "github"
       },
       "original": {
@@ -183,6 +101,23 @@
         "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"
+      }
+    },
     "root": {
       "inputs": {
         "alectryon": "alectryon",
@@ -194,6 +129,21 @@
         "leanInk": "leanInk",
         "mdBook": "mdBook"
       }
+    },
+    "systems": {
+      "locked": {
+        "lastModified": 1681028828,
+        "narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=",
+        "owner": "nix-systems",
+        "repo": "default",
+        "rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e",
+        "type": "github"
+      },
+      "original": {
+        "owner": "nix-systems",
+        "repo": "default",
+        "type": "github"
+      }
     }
   },
   "root": "root",

doc/flake.nix

@@ -17,7 +17,7 @@
   };
 
   outputs = inputs@{ self, ... }: inputs.flake-utils.lib.eachDefaultSystem (system:
-    with inputs.lean.packages.${system}; with nixpkgs;
+    with inputs.lean.packages.${system}.deprecated; with nixpkgs;
     let
       doc-src = lib.sourceByRegex ../. ["doc.*" "tests(/lean(/beginEndAsMacro.lean)?)?"];
     in {
@@ -44,21 +44,6 @@
           mdbook build -d $out
         '';
       };
-      # We use a separate derivation instead of `checkPhase` so we can push it but not `doc` to the binary cache
-      test = stdenv.mkDerivation {
-        name ="lean-doc-test";
-        src = doc-src;
-        buildInputs = [ lean-mdbook stage1.Lean.lean-package strace ];
-        patchPhase = ''
-          cd doc
-          patchShebangs test
-        '';
-        buildPhase = ''
-          mdbook test
-          touch $out
-        '';
-        dontInstall = true;
-      };
       leanInk = (buildLeanPackage {
         name = "Main";
         src = inputs.leanInk;

doc/make/index.md

@@ -8,6 +8,7 @@ Requirements
 - C++14 compatible compiler
 - [CMake](http://www.cmake.org)
 - [GMP (GNU multiprecision library)](http://gmplib.org/)
+- [LibUV](https://libuv.org/)
 
 Platform-Specific Setup
 -----------------------
@@ -27,9 +28,9 @@ Setting up a basic parallelized release build:
 git clone https://github.com/leanprover/lean4
 cd lean4
 cmake --preset release
-make -C build/release -j$(nproc)  # see below for macOS
+make -C build/release -j$(nproc || sysctl -n hw.logicalcpu)
 ```
-You can replace `$(nproc)`, which is not available on macOS and some alternative shells, with the desired parallelism amount.
+You can replace `$(nproc || sysctl -n hw.logicalcpu)` with the desired parallelism amount.
 
 The above commands will compile the Lean library and binaries into the
 `stage1` subfolder; see below for details.

doc/make/msys2.md

@@ -25,7 +25,7 @@ MSYS2 has a package management system, [pacman][pacman], which is used in Arch L
 Here are the commands to install all dependencies needed to compile Lean on your machine.
 
 ```bash
-pacman -S make python mingw-w64-x86_64-cmake mingw-w64-x86_64-clang mingw-w64-x86_64-ccache git unzip diffutils binutils
+pacman -S make python mingw-w64-x86_64-cmake mingw-w64-x86_64-clang mingw-w64-x86_64-ccache mingw-w64-x86_64-libuv mingw-w64-x86_64-gmp git unzip diffutils binutils
 ```
 
 You should now be able to run these commands:
@@ -64,6 +64,7 @@ they are installed in your MSYS setup:
 - libgcc_s_seh-1.dll
 - libstdc++-6.dll
 - libgmp-10.dll
+- libuv-1.dll
 - libwinpthread-1.dll
 
 The following linux command will do that:

doc/make/osx-10.9.md

@@ -32,15 +32,16 @@ following to use `g++`.
 cmake -DCMAKE_CXX_COMPILER=g++ ...
 ```
 
-## Required Packages: CMake, GMP
+## Required Packages: CMake, GMP, libuv
 
 ```bash
 brew install cmake
 brew install gmp
+brew install libuv
 ```
 
 ## Recommended Packages: CCache
 
 ```bash
 brew install ccache
-```
+```

doc/make/ubuntu.md

@@ -8,5 +8,5 @@ follow the [generic build instructions](index.md).
 ## Basic packages
 
 ```bash
-sudo apt-get install git libgmp-dev cmake ccache clang
+sudo apt-get install git libgmp-dev libuv1-dev cmake ccache clang
 ```

doc/quickstart.md

@@ -5,11 +5,11 @@ See [Setup](./setup.md) for supported platforms and other ways to set up Lean 4.
 
 1. Install [VS Code](https://code.visualstudio.com/).
 
-1. Launch VS Code and install the `lean4` extension by clicking on the "Extensions" sidebar entry and searching for "lean4".
+1. Launch VS Code and install the `Lean 4` extension by clicking on the 'Extensions' sidebar entry and searching for 'Lean 4'.
 
    ![installing the vscode-lean4 extension](images/code-ext.png)
 
-1. Open the Lean 4 setup guide by creating a new text file using "File > New Text File" (`Ctrl+N` / `Cmd+N`), clicking on the ∀-symbol in the top right and selecting "Documentation… > Docs: Show Setup Guide".
+1. Open the Lean 4 setup guide by creating a new text file using 'File > New Text File' (`Ctrl+N` / `Cmd+N`), clicking on the ∀-symbol in the top right and selecting 'Documentation… > Docs: Show Setup Guide'.
 
    ![show setup guide](images/show-setup-guide.png)
 

flake.lock

@@ -1,21 +1,5 @@
 {
   "nodes": {
-    "flake-compat": {
-      "flake": false,
-      "locked": {
-        "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"
@@ -34,75 +18,38 @@
         "type": "github"
       }
     },
-    "lean4-mode": {
-      "flake": false,
-      "locked": {
-        "lastModified": 1709737301,
-        "narHash": "sha256-uT9JN2kLNKJK9c/S/WxLjiHmwijq49EgLb+gJUSDpz0=",
-        "owner": "leanprover",
-        "repo": "lean4-mode",
-        "rev": "f1f24c15134dee3754b82c9d9924866fe6bc6b9f",
-        "type": "github"
-      },
-      "original": {
-        "owner": "leanprover",
-        "repo": "lean4-mode",
-        "type": "github"
-      }
-    },
-    "libgit2": {
-      "flake": false,
-      "locked": {
-        "lastModified": 1697646580,
-        "narHash": "sha256-oX4Z3S9WtJlwvj0uH9HlYcWv+x1hqp8mhXl7HsLu2f0=",
-        "owner": "libgit2",
-        "repo": "libgit2",
-        "rev": "45fd9ed7ae1a9b74b957ef4f337bc3c8b3df01b5",
-        "type": "github"
-      },
-      "original": {
-        "owner": "libgit2",
-        "repo": "libgit2",
-        "type": "github"
-      }
-    },
-    "nix": {
-      "inputs": {
-        "flake-compat": "flake-compat",
-        "libgit2": "libgit2",
-        "nixpkgs": "nixpkgs",
-        "nixpkgs-regression": "nixpkgs-regression"
-      },
-      "locked": {
-        "lastModified": 1711102798,
-        "narHash": "sha256-CXOIJr8byjolqG7eqCLa+Wfi7rah62VmLoqSXENaZnw=",
-        "owner": "NixOS",
-        "repo": "nix",
-        "rev": "a22328066416650471c3545b0b138669ea212ab4",
-        "type": "github"
-      },
-      "original": {
-        "owner": "NixOS",
-        "repo": "nix",
-        "type": "github"
-      }
-    },
     "nixpkgs": {
       "locked": {
-        "lastModified": 1709083642,
-        "narHash": "sha256-7kkJQd4rZ+vFrzWu8sTRtta5D1kBG0LSRYAfhtmMlSo=",
+        "lastModified": 1710889954,
+        "narHash": "sha256-Pr6F5Pmd7JnNEMHHmspZ0qVqIBVxyZ13ik1pJtm2QXk=",
         "owner": "NixOS",
         "repo": "nixpkgs",
-        "rev": "b550fe4b4776908ac2a861124307045f8e717c8e",
+        "rev": "7872526e9c5332274ea5932a0c3270d6e4724f3b",
         "type": "github"
       },
       "original": {
         "owner": "NixOS",
-        "ref": "release-23.11",
+        "ref": "nixpkgs-unstable",
         "repo": "nixpkgs",
         "type": "github"
       }
     },
+    "nixpkgs-cadical": {
+      "locked": {
+        "lastModified": 1722221733,
+        "narHash": "sha256-sga9SrrPb+pQJxG1ttJfMPheZvDOxApFfwXCFO0H9xw=",
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "rev": "12bf09802d77264e441f48e25459c10c93eada2e",
+        "type": "github"
+      },
+      "original": {
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "rev": "12bf09802d77264e441f48e25459c10c93eada2e",
+        "type": "github"
+      }
+    },
     "nixpkgs-old": {
       "flake": false,
       "locked": {
@@ -120,44 +67,11 @@
         "type": "github"
       }
     },
-    "nixpkgs-regression": {
-      "locked": {
-        "lastModified": 1643052045,
-        "narHash": "sha256-uGJ0VXIhWKGXxkeNnq4TvV3CIOkUJ3PAoLZ3HMzNVMw=",
-        "owner": "NixOS",
-        "repo": "nixpkgs",
-        "rev": "215d4d0fd80ca5163643b03a33fde804a29cc1e2",
-        "type": "github"
-      },
-      "original": {
-        "owner": "NixOS",
-        "repo": "nixpkgs",
-        "rev": "215d4d0fd80ca5163643b03a33fde804a29cc1e2",
-        "type": "github"
-      }
-    },
-    "nixpkgs_2": {
-      "locked": {
-        "lastModified": 1710889954,
-        "narHash": "sha256-Pr6F5Pmd7JnNEMHHmspZ0qVqIBVxyZ13ik1pJtm2QXk=",
-        "owner": "NixOS",
-        "repo": "nixpkgs",
-        "rev": "7872526e9c5332274ea5932a0c3270d6e4724f3b",
-        "type": "github"
-      },
-      "original": {
-        "owner": "NixOS",
-        "ref": "nixpkgs-unstable",
-        "repo": "nixpkgs",
-        "type": "github"
-      }
-    },
     "root": {
       "inputs": {
         "flake-utils": "flake-utils",
-        "lean4-mode": "lean4-mode",
-        "nix": "nix",
-        "nixpkgs": "nixpkgs_2",
+        "nixpkgs": "nixpkgs",
+        "nixpkgs-cadical": "nixpkgs-cadical",
         "nixpkgs-old": "nixpkgs-old"
       }
     },

flake.nix

@@ -1,48 +1,36 @@
 {
-  description = "Lean interactive theorem prover";
+  description = "Lean development flake. Not intended for end users.";
 
   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;
+  # for cadical 1.9.5; sync with CMakeLists.txt
+  inputs.nixpkgs-cadical.url = "github:NixOS/nixpkgs/12bf09802d77264e441f48e25459c10c93eada2e";
   inputs.flake-utils.url = "github:numtide/flake-utils";
-  inputs.nix.url = "github:NixOS/nix";
-  inputs.lean4-mode = {
-    url = "github:leanprover/lean4-mode";
-    flake = false;
-  };
-  # used *only* by `stage0-from-input` below
-  #inputs.lean-stage0 = {
-  #  url = github:leanprover/lean4;
-  #  inputs.nixpkgs.follows = "nixpkgs";
-  #  inputs.flake-utils.follows = "flake-utils";
-  #  inputs.nix.follows = "nix";
-  #  inputs.lean4-mode.follows = "lean4-mode";
-  #};
 
-  outputs = { self, nixpkgs, nixpkgs-old, flake-utils, nix, lean4-mode, ... }@inputs: flake-utils.lib.eachDefaultSystem (system:
+  outputs = { self, nixpkgs, nixpkgs-old, flake-utils, ... }@inputs: flake-utils.lib.eachDefaultSystem (system:
     let
-      pkgs = import nixpkgs {
-        inherit system;
-        # for `vscode-with-extensions`
-        config.allowUnfree = true;
-      };
+      pkgs = import nixpkgs { inherit system; };
       # 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"; };
+      pkgsCadical = import inputs.nixpkgs-cadical { inherit system; };
+      cadical = if pkgs.stdenv.isLinux then
+        # use statically-linked cadical on Linux to avoid glibc versioning troubles
+        pkgsCadical.pkgsStatic.cadical.overrideAttrs { doCheck = false; }
+      else pkgsCadical.cadical;
 
-      lean-packages = pkgs.callPackage (./nix/packages.nix) { src = ./.; inherit nix lean4-mode; };
+      lean-packages = pkgs.callPackage (./nix/packages.nix) { src = ./.; };
 
       devShellWithDist = pkgsDist: pkgs.mkShell.override {
           stdenv = pkgs.overrideCC pkgs.stdenv lean-packages.llvmPackages.clang;
         } ({
           buildInputs = with pkgs; [
-            cmake gmp ccache
+            cmake gmp libuv ccache cadical
             lean-packages.llvmPackages.llvm  # llvm-symbolizer for asan/lsan
             gdb
-            # TODO: only add when proven to not affect the flakification
-            #pkgs.python3
             tree  # for CI
           ];
           # https://github.com/NixOS/nixpkgs/issues/60919
@@ -51,6 +39,7 @@
           CTEST_OUTPUT_ON_FAILURE = 1;
         } // pkgs.lib.optionalAttrs pkgs.stdenv.isLinux {
           GMP = pkgsDist.gmp.override { withStatic = true; };
+          LIBUV = pkgsDist.libuv.overrideAttrs (attrs: { configureFlags = ["--enable-static"]; });
           GLIBC = pkgsDist.glibc;
           GLIBC_DEV = pkgsDist.glibc.dev;
           GCC_LIB = pkgsDist.gcc.cc.lib;
@@ -58,41 +47,15 @@
           GDB = pkgsDist.gdb;
         });
     in {
-      packages = lean-packages // rec {
-        debug = lean-packages.override { debug = true; };
-        stage0debug = lean-packages.override { stage0debug = true; };
-        asan = lean-packages.override { extraCMakeFlags = [ "-DLEAN_EXTRA_CXX_FLAGS=-fsanitize=address" "-DLEANC_EXTRA_FLAGS=-fsanitize=address" "-DSMALL_ALLOCATOR=OFF" "-DSYMBOLIC=OFF" ]; };
-        asandebug = asan.override { debug = true; };
-        tsan = lean-packages.override {
-          extraCMakeFlags = [ "-DLEAN_EXTRA_CXX_FLAGS=-fsanitize=thread" "-DLEANC_EXTRA_FLAGS=-fsanitize=thread" "-DCOMPRESSED_OBJECT_HEADER=OFF" ];
-          stage0 = (lean-packages.override {
-            # Compressed headers currently trigger data race reports in tsan.
-            # Turn them off for stage 0 as well so stage 1 can read its own stdlib.
-            extraCMakeFlags = [ "-DCOMPRESSED_OBJECT_HEADER=OFF" ];
-          }).stage1;
-        };
-        tsandebug = tsan.override { debug = true; };
-        stage0-from-input = lean-packages.override {
-          stage0 = pkgs.writeShellScriptBin "lean" ''
-            exec ${inputs.lean-stage0.packages.${system}.lean}/bin/lean -Dinterpreter.prefer_native=false "$@"
-          '';
-        };
-        inherit self;
+      packages = {
+        # to be removed when Nix CI is not needed anymore
+        inherit (lean-packages) cacheRoots test update-stage0-commit ciShell;
+        deprecated = lean-packages;
       };
-      defaultPackage = lean-packages.lean-all;
 
       # 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 {
-      templates.pkg = {
-        path = ./nix/templates/pkg;
-        description = "A custom Lean package";
-      };
-
-      defaultTemplate = templates.pkg;
-    };
+    });
 }

nix/bootstrap.nix

@@ -1,13 +1,13 @@
 { src, debug ? false, stage0debug ? false, extraCMakeFlags ? [],
-  stdenv, lib, cmake, gmp, git, gnumake, bash, buildLeanPackage, writeShellScriptBin, runCommand, symlinkJoin, lndir, perl, gnused, darwin, llvmPackages, linkFarmFromDrvs,
+  stdenv, lib, cmake, gmp, libuv, cadical, git, gnumake, bash, buildLeanPackage, writeShellScriptBin, runCommand, symlinkJoin, lndir, perl, gnused, darwin, llvmPackages, linkFarmFromDrvs,
   ... } @ args:
 with builtins;
-rec {
+lib.warn "The Nix-based build is deprecated" rec {
   inherit stdenv;
   sourceByRegex = p: rs: lib.sourceByRegex p (map (r: "(/src/)?${r}") rs);
   buildCMake = args: stdenv.mkDerivation ({
     nativeBuildInputs = [ cmake ];
-    buildInputs = [ gmp llvmPackages.llvm ];
+    buildInputs = [ gmp libuv llvmPackages.llvm ];
     # https://github.com/NixOS/nixpkgs/issues/60919
     hardeningDisable = [ "all" ];
     dontStrip = (args.debug or debug);
@@ -17,7 +17,7 @@ rec {
     '';
   } // args // {
     src = args.realSrc or (sourceByRegex args.src [ "[a-z].*" "CMakeLists\.txt" ]);
-    cmakeFlags = (args.cmakeFlags or [ "-DSTAGE=1" "-DPREV_STAGE=./faux-prev-stage" "-DUSE_GITHASH=OFF" ]) ++ (args.extraCMakeFlags or extraCMakeFlags) ++ lib.optional (args.debug or debug) [ "-DCMAKE_BUILD_TYPE=Debug" ];
+    cmakeFlags = (args.cmakeFlags or [ "-DSTAGE=1" "-DPREV_STAGE=./faux-prev-stage" "-DUSE_GITHASH=OFF" "-DCADICAL=${cadical}/bin/cadical" ]) ++ (args.extraCMakeFlags or extraCMakeFlags) ++ lib.optional (args.debug or debug) [ "-DCMAKE_BUILD_TYPE=Debug" ];
     preConfigure = args.preConfigure or "" + ''
       # ignore absence of submodule
       sed -i 's!lake/Lake.lean!!' CMakeLists.txt
@@ -26,11 +26,7 @@ rec {
   lean-bin-tools-unwrapped = buildCMake {
     name = "lean-bin-tools";
     outputs = [ "out" "leanc_src" ];
-    realSrc = sourceByRegex (src + "/src") [ "CMakeLists\.txt" "cmake.*" "bin.*" "include.*" ".*\.in" "Leanc\.lean" ];
-    preConfigure = ''
-      touch empty.cpp
-      sed -i 's/add_subdirectory.*//;s/set(LEAN_OBJS.*/set(LEAN_OBJS empty.cpp)/' CMakeLists.txt
-    '';
+    realSrc = sourceByRegex (src + "/src") [ "CMakeLists\.txt" "[a-z].*" ".*\.in" "Leanc\.lean" ];
     dontBuild = true;
     installPhase = ''
       mkdir $out $leanc_src
@@ -45,11 +41,10 @@ rec {
   leancpp = buildCMake {
     name = "leancpp";
     src = src + "/src";
-    buildFlags = [ "leancpp" "leanrt" "leanrt_initial-exec" "shell" ];
+    buildFlags = [ "leancpp" "leanrt" "leanrt_initial-exec" "leanshell" "leanmain" ];
     installPhase = ''
       mkdir -p $out
       mv lib/ $out/
-      mv shell/CMakeFiles/shell.dir/lean.cpp.o $out/lib
       mv runtime/libleanrt_initial-exec.a $out/lib
     '';
   };
@@ -122,12 +117,15 @@ rec {
         touch empty.c
         ${stdenv.cc}/bin/cc -shared -o $out/$libName empty.c
       '';
+      leanshared_1 = runCommand "leanshared_1" { buildInputs = [ stdenv.cc ]; libName = "leanshared_1${stdenv.hostPlatform.extensions.sharedLibrary}"; } ''
+        mkdir $out
+        touch empty.c
+        ${stdenv.cc}/bin/cc -shared -o $out/$libName empty.c
+      '';
       leanshared = runCommand "leanshared" { buildInputs = [ stdenv.cc ]; libName = "libleanshared${stdenv.hostPlatform.extensions.sharedLibrary}"; } ''
         mkdir $out
         LEAN_CC=${stdenv.cc}/bin/cc ${lean-bin-tools-unwrapped}/bin/leanc -shared ${lib.optionalString stdenv.isLinux "-Wl,-Bsymbolic"} \
-          ${if stdenv.isDarwin
-            then "-Wl,-force_load,${Init.staticLib}/libInit.a -Wl,-force_load,${Std.staticLib}/libStd.a -Wl,-force_load,${Lean.staticLib}/libLean.a -Wl,-force_load,${leancpp}/lib/lean/libleancpp.a ${leancpp}/lib/libleanrt_initial-exec.a -lc++"
-            else "-Wl,--whole-archive -lInit -lStd -lLean -lleancpp ${leancpp}/lib/libleanrt_initial-exec.a -Wl,--no-whole-archive -lstdc++"} \
+          -Wl,--whole-archive ${leancpp}/lib/temp/libleanshell.a -lInit -lStd -lLean -lleancpp ${leancpp}/lib/libleanrt_initial-exec.a -Wl,--no-whole-archive -lstdc++ \
           -lm ${stdlibLinkFlags} \
           $(${llvmPackages.libllvm.dev}/bin/llvm-config --ldflags --libs) \
           -o $out/$libName
@@ -135,18 +133,18 @@ rec {
       mods = foldl' (mods: pkg: mods // pkg.mods) {} stdlib;
       print-paths = Lean.makePrintPathsFor [] mods;
       leanc = writeShellScriptBin "leanc" ''
-        LEAN_CC=${stdenv.cc}/bin/cc ${Leanc.executable}/bin/leanc -I${lean-bin-tools-unwrapped}/include ${stdlibLinkFlags} -L${libInit_shared} -L${leanshared} "$@"
+        LEAN_CC=${stdenv.cc}/bin/cc ${Leanc.executable}/bin/leanc -I${lean-bin-tools-unwrapped}/include ${stdlibLinkFlags} -L${libInit_shared} -L${leanshared_1} -L${leanshared} "$@"
       '';
       lean = runCommand "lean" { buildInputs = lib.optional stdenv.isDarwin darwin.cctools; } ''
         mkdir -p $out/bin
-        ${leanc}/bin/leanc ${leancpp}/lib/lean.cpp.o ${libInit_shared}/* ${leanshared}/* -o $out/bin/lean
+        ${leanc}/bin/leanc ${leancpp}/lib/temp/libleanmain.a ${libInit_shared}/* ${leanshared_1}/* ${leanshared}/* -o $out/bin/lean
       '';
       # derivation following the directory layout of the "basic" setup, mostly useful for running tests
       lean-all = stdenv.mkDerivation {
         name = "lean-${desc}";
         buildCommand = ''
           mkdir -p $out/bin $out/lib/lean
-          ln -sf ${leancpp}/lib/lean/* ${lib.concatMapStringsSep " " (l: "${l.modRoot}/* ${l.staticLib}/*") (lib.reverseList stdlib)} ${libInit_shared}/* ${leanshared}/* $out/lib/lean/
+          ln -sf ${leancpp}/lib/lean/* ${lib.concatMapStringsSep " " (l: "${l.modRoot}/* ${l.staticLib}/*") (lib.reverseList stdlib)} ${libInit_shared}/* ${leanshared_1}/* ${leanshared}/* $out/lib/lean/
           # put everything in a single final derivation so `IO.appDir` references work
           cp ${lean}/bin/lean ${leanc}/bin/leanc ${Lake-Main.executable}/bin/lake $out/bin
           # NOTE: `lndir` will not override existing `bin/leanc`
@@ -160,7 +158,7 @@ rec {
       test = buildCMake {
         name = "lean-test-${desc}";
         realSrc = lib.sourceByRegex src [ "src.*" "tests.*" ];
-        buildInputs = [ gmp perl git ];
+        buildInputs = [ gmp libuv perl git cadical ];
         preConfigure = ''
           cd src
         '';
@@ -171,7 +169,7 @@ rec {
           ln -sf ${lean-all}/* .
         '';
         buildPhase = ''
-          ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)' -j$NIX_BUILD_CORES
+          ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)|leanlaketest_reverse-ffi' -j$NIX_BUILD_CORES
         '';
         installPhase = ''
           mkdir $out

nix/buildLeanPackage.nix

@@ -1,5 +1,5 @@
 { lean, lean-leanDeps ? lean, lean-final ? lean, leanc,
-  stdenv, lib, coreutils, gnused, writeShellScriptBin, bash, lean-emacs, lean-vscode, nix, substituteAll, symlinkJoin, linkFarmFromDrvs,
+  stdenv, lib, coreutils, gnused, writeShellScriptBin, bash, substituteAll, symlinkJoin, linkFarmFromDrvs,
   runCommand, darwin, mkShell, ... }:
 let lean-final' = lean-final; in
 lib.makeOverridable (
@@ -197,19 +197,6 @@ with builtins; let
              then map (m: m.module) header.imports
              else abort "errors while parsing imports of ${mod}:\n${lib.concatStringsSep "\n" header.errors}";
     in mkMod mod (map (dep: if modDepsMap ? ${dep} then modCandidates.${dep} else externalModMap.${dep}) deps)) modDepsMap;
-  makeEmacsWrapper = name: emacs: lean: writeShellScriptBin name ''
-    ${emacs} --eval "(progn (setq lean4-rootdir \"${lean}\"))" "$@"
-  '';
-  makeVSCodeWrapper = name: lean: writeShellScriptBin name ''
-    PATH=${lean}/bin:$PATH ${lean-vscode}/bin/code "$@"
-  '';
-  printPaths = deps: writeShellScriptBin "print-paths" ''
-    echo '${toJSON {
-      oleanPath = [(depRoot "print-paths" deps)];
-      srcPath = ["."] ++ map (dep: dep.src) allExternalDeps;
-      loadDynlibPaths = map pathOfSharedLib (loadDynlibsOfDeps deps);
-    }}'
-  '';
   expandGlob = g:
     if typeOf g == "string" then [g]
     else if g.glob == "one" then [g.mod]
@@ -257,48 +244,4 @@ in rec {
         -o $out/bin/${executableName} \
         ${lib.concatStringsSep " " allLinkFlags}
     '') {};
-
-  lean-package = writeShellScriptBin "lean" ''
-    LEAN_PATH=${modRoot}:$LEAN_PATH LEAN_SRC_PATH=$LEAN_SRC_PATH:${src} exec ${lean-final}/bin/lean "$@"
-  '';
-  emacs-package = makeEmacsWrapper "emacs-package" lean-package;
-  vscode-package = makeVSCodeWrapper "vscode-package" lean-package;
-
-  link-ilean = writeShellScriptBin "link-ilean" ''
-    dest=''${1:-.}
-    mkdir -p $dest/build/lib
-    ln -sf ${iTree}/* $dest/build/lib
-  '';
-
-  makePrintPathsFor = deps: mods: printPaths deps // mapAttrs (_: mod: makePrintPathsFor (deps ++ [mod]) mods) mods;
-  print-paths = makePrintPathsFor [] (mods' // externalModMap);
-  # `lean` wrapper that dynamically runs Nix for the actual `lean` executable so the same editor can be
-  # used for multiple projects/after upgrading the `lean` input/for editing both stage 1 and the tests
-  lean-bin-dev = substituteAll {
-    name = "lean";
-    dir = "bin";
-    src = ./lean-dev.in;
-    isExecutable = true;
-    srcRoot = fullSrc;  # use root flake.nix in case of Lean repo
-    inherit bash nix srcTarget srcArgs;
-  };
-  lake-dev = substituteAll {
-    name = "lake";
-    dir = "bin";
-    src = ./lake-dev.in;
-    isExecutable = true;
-    srcRoot = fullSrc;  # use root flake.nix in case of Lean repo
-    inherit bash nix srcTarget srcArgs;
-  };
-  lean-dev = symlinkJoin { name = "lean-dev"; paths = [ lean-bin-dev lake-dev ]; };
-  emacs-dev = makeEmacsWrapper "emacs-dev" "${lean-emacs}/bin/emacs" lean-dev;
-  emacs-path-dev = makeEmacsWrapper "emacs-path-dev" "emacs" lean-dev;
-  vscode-dev = makeVSCodeWrapper "vscode-dev" lean-dev;
-
-  devShell = mkShell {
-    buildInputs = [ nix ];
-    shellHook = ''
-      export LEAN_SRC_PATH="${srcPath}"
-    '';
-  };
 })

nix/packages.nix

@@ -1,9 +1,6 @@
-{ src, pkgs, nix, ... } @ args:
+{ src, pkgs, ... } @ args:
 with pkgs;
 let
-  nix-pinned = writeShellScriptBin "nix" ''
-    ${nix.packages.${system}.default}/bin/nix --experimental-features 'nix-command flakes' --extra-substituters https://lean4.cachix.org/ --option warn-dirty false "$@"
-  '';
   # https://github.com/NixOS/nixpkgs/issues/130963
   llvmPackages = if stdenv.isDarwin then llvmPackages_11 else llvmPackages_15;
   cc = (ccacheWrapper.override rec {
@@ -42,40 +39,9 @@ let
     inherit (lean) stdenv;
     lean = lean.stage1;
     inherit (lean.stage1) leanc;
-    inherit lean-emacs lean-vscode;
-    nix = nix-pinned;
   }));
-  lean4-mode = emacsPackages.melpaBuild {
-    pname = "lean4-mode";
-    version = "1";
-    commit = "1";
-    src = args.lean4-mode;
-    packageRequires = with pkgs.emacsPackages.melpaPackages; [ dash f flycheck magit-section lsp-mode s ];
-    recipe = pkgs.writeText "recipe" ''
-      (lean4-mode
-       :repo "leanprover/lean4-mode"
-       :fetcher github
-       :files ("*.el" "data"))
-    '';
-  };
-  lean-emacs = emacsWithPackages [ lean4-mode ];
-  # updating might be nicer by building from source from a flake input, but this is good enough for now
-  vscode-lean4 = vscode-utils.extensionFromVscodeMarketplace {
-      name = "lean4";
-      publisher = "leanprover";
-      version = "0.0.63";
-      sha256 = "sha256-kjEex7L0F2P4pMdXi4NIZ1y59ywJVubqDqsoYagZNkI=";
-  };
-  lean-vscode = vscode-with-extensions.override {
-    vscodeExtensions = [ vscode-lean4 ];
-  };
 in {
-  inherit cc lean4-mode buildLeanPackage llvmPackages vscode-lean4;
-  lean = lean.stage1;
-  stage0print-paths = lean.stage1.Lean.print-paths;
-  HEAD-as-stage0 = (lean.stage1.Lean.overrideArgs { srcTarget = "..#stage0-from-input.stage0"; srcArgs = "(--override-input lean-stage0 ..\?rev=$(git rev-parse HEAD) -- -Dinterpreter.prefer_native=false \"$@\")"; });
-  HEAD-as-stage1 = (lean.stage1.Lean.overrideArgs { srcTarget = "..\?rev=$(git rev-parse HEAD)#stage0"; });
-  nix = nix-pinned;
+  inherit cc buildLeanPackage llvmPackages;
   nixpkgs = pkgs;
   ciShell = writeShellScriptBin "ciShell" ''
     set -o pipefail
@@ -83,5 +49,4 @@ in {
     # prefix lines with cumulative and individual execution time
     "$@" |& ts -i "(%.S)]" | ts -s "[%M:%S"
   '';
-  vscode = lean-vscode;
-} // lean.stage1.Lean // lean.stage1 // lean
+} // lean.stage1

releases_drafts/hashmap.md

@@ -0,0 +1,3 @@
+* The `Lean` module has switched from `Lean.HashMap` and `Lean.HashSet` to `Std.HashMap` and `Std.HashSet`. `Lean.HashMap` and `Lean.HashSet` are now deprecated and will be removed in a future release. Users of `Lean` APIs that interact with hash maps, for example `Lean.Environment.const2ModIdx`, might encounter minor breakage due to the following breaking changes from `Lean.HashMap` to `Std.HashMap`:
+    * query functions use the term `get` instead of `find`,
+    * the notation `map[key]` no longer returns an optional value but expects a proof that the key is present in the map instead. The previous behavior is available via the `map[key]?` notation.

releases_drafts/libuv.md

@@ -0,0 +1 @@
+* #4963 [LibUV](https://libuv.org/) is now required to build Lean. This change only affects developers who compile Lean themselves instead of obtaining toolchains via `elan`. We have updated the official build instructions with information on how to obtain LibUV on our supported platforms.

releases_drafts/mutualStructural.md

@@ -1,65 +0,0 @@
-* Structural recursion can now be explicitly requested using
-  ```
-  termination_by structural x
-  ```
-  in analogy to the existing `termination_by x` syntax that causes well-founded recursion to be used.
-  (#4542)
-
-* The `termination_by?` syntax no longer forces the use of well-founded recursion, and when structural
-  recursion is inferred, will print the result using the `termination_by` syntax.
-
-* Mutual structural recursion is supported now. This supports both mutual recursion over a non-mutual
-  data type, as well as recursion over mutual or nested data types:
-
-  ```lean
-  mutual
-  def Even : Nat → Prop
-    | 0 => True
-    | n+1 => Odd n
-
-  def Odd : Nat → Prop
-    | 0 => False
-    | n+1 => Even n
-  end
-
-  mutual
-  inductive A
-  | other : B → A
-  | empty
-  inductive B
-  | other : A → B
-  | empty
-  end
-
-  mutual
-  def A.size : A → Nat
-  | .other b => b.size + 1
-  | .empty => 0
-
-  def B.size : B → Nat
-  | .other a => a.size + 1
-  | .empty => 0
-  end
-
-  inductive Tree where | node : List Tree → Tree
-
-  mutual
-  def Tree.size : Tree → Nat
-  | node ts => Tree.list_size ts
-
-  def Tree.list_size : List Tree → Nat
-  | [] => 0
-  | t::ts => Tree.size t + Tree.list_size ts
-  end
-  ```
-
-  Functional induction principles are generated for these functions as well (`A.size.induct`, `A.size.mutual_induct`).
-
-  Nested structural recursion is still not supported.
-
-  PRs #4639, #4715, #4642, #4656, #4684, #4715, #4728, #4575, #4731, #4658, #4734, #4738, #4718,
-  #4733, #4787, #4788, #4789, #4807, #4772
-
-* A bugfix in the structural recursion code may in some cases break existing code, when a parameter
-  of the type of the recursive argument is bound behind indices of that type. This can usually be
-  fixed by reordering the parameters of the function (PR #4672)

releases_drafts/new-variable.md

@@ -0,0 +1,17 @@
+**breaking change**
+
+The effect of the `variable` command on proofs of `theorem`s has been changed. Whether such section variables are accessible in the proof now depends only on the theorem signature and other top-level commands, not on the proof itself.
+This change ensures that
+* the statement of a theorem is independent of its proof. In other words, changes in the proof cannot change the theorem statement.
+* tactics such as `induction` cannot accidentally include a section variable.
+* the proof can be elaborated in parallel to subsequent declarations in a future version of Lean.
+
+The effect of `variable`s on the theorem header as well as on other kinds of declarations is unchanged.
+
+Specifically, section variables are included if they
+* are directly referenced by the theorem header,
+* are included via the new `include` command in the current section and not subsequently mentioned in an `omit` statement,
+* are directly referenced by any variable included by these rules, OR
+* are instance-implicit variables that reference only variables included by these rules.
+
+For porting, a new option `deprecated.oldSectionVars` is included to locally switch back to the old behavior.

script/prepare-llvm-linux.sh

@@ -38,7 +38,7 @@ $CP $GLIBC/lib/*crt* llvm/lib/
 $CP $GLIBC/lib/*crt* stage1/lib/
 # runtime
 (cd llvm; $CP --parents lib/clang/*/lib/*/{clang_rt.*.o,libclang_rt.builtins*} ../stage1)
-$CP llvm/lib/*/lib{c++,c++abi,unwind}.* $GMP/lib/libgmp.a stage1/lib/
+$CP llvm/lib/*/lib{c++,c++abi,unwind}.* $GMP/lib/libgmp.a $LIBUV/lib/libuv.a stage1/lib/
 # LLVM 15 appears to ship the dependencies in 'llvm/lib/<target-triple>/' and 'llvm/include/<target-triple>/'
 # but clang-15 that we use to compile is linked against 'llvm/lib/' and 'llvm/include'
 # https://github.com/llvm/llvm-project/issues/54955
@@ -62,8 +62,8 @@ fi
 # use `-nostdinc` to make sure headers are not visible by default (in particular, not to `#include_next` in the clang headers),
 # but do not change sysroot so users can still link against system libs
 echo -n " -DLEANC_INTERNAL_FLAGS='-nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang"
-echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/glibc ROOT/lib/glibc/libc_nonshared.a -Wl,--as-needed -Wl,-Bstatic -lgmp -lunwind -Wl,-Bdynamic -Wl,--no-as-needed -fuse-ld=lld'"
+echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/glibc ROOT/lib/glibc/libc_nonshared.a -Wl,--as-needed -Wl,-Bstatic -lgmp -lunwind -luv -Wl,-Bdynamic -Wl,--no-as-needed -fuse-ld=lld'"
 # when not using the above flags, link GMP dynamically/as usual
-echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-Wl,--as-needed -lgmp -Wl,--no-as-needed'"
+echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-Wl,--as-needed -lgmp -luv -Wl,--no-as-needed'"
 # do not set `LEAN_CC` for tests
 echo -n " -DLEAN_TEST_VARS=''"

script/prepare-llvm-macos.sh

@@ -9,6 +9,7 @@ set -uxo pipefail
 # use full LLVM release for compiling C++ code, but subset for compiling C code and distribution
 
 GMP=${GMP:-$(brew --prefix)}
+LIBUV=${LIBUV:-$(brew --prefix)}
 
 [[ -d llvm ]] || (mkdir llvm; gtar xf $1 --strip-components 1 --directory llvm)
 [[ -d llvm-host ]] || if [[ "$#" -gt 1 ]]; then
@@ -46,8 +47,9 @@ echo -n " -DLEAN_EXTRA_CXX_FLAGS='${EXTRA_FLAGS:-}'"
 if [[ -L llvm-host ]]; then
   echo -n " -DCMAKE_C_COMPILER=$PWD/stage1/bin/clang"
   gcp $GMP/lib/libgmp.a stage1/lib/
+  gcp $LIBUV/lib/libuv.a stage1/lib/
   echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/libc -fuse-ld=lld'"
-  echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp'"
+  echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp -luv'"
 else
   echo -n " -DCMAKE_C_COMPILER=$PWD/llvm-host/bin/clang -DLEANC_OPTS='--sysroot $PWD/stage1 -resource-dir $PWD/stage1/lib/clang/15.0.1 ${EXTRA_FLAGS:-}'"
   echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/libc -fuse-ld=lld'"

script/prepare-llvm-mingw.sh

@@ -31,15 +31,15 @@ cp /clang64/lib/{crtbegin,crtend,crt2,dllcrt2}.o stage1/lib/
 # runtime
 (cd llvm; cp --parents lib/clang/*/lib/*/libclang_rt.builtins* ../stage1)
 # further dependencies
-cp /clang64/lib/lib{m,bcrypt,mingw32,moldname,mingwex,msvcrt,pthread,advapi32,shell32,user32,kernel32,ucrtbase}.* /clang64/lib/libgmp.a llvm/lib/lib{c++,c++abi,unwind}.a stage1/lib/
+cp /clang64/lib/lib{m,bcrypt,mingw32,moldname,mingwex,msvcrt,pthread,advapi32,shell32,user32,kernel32,ucrtbase}.* /clang64/lib/libgmp.a /clang64/lib/libuv.a llvm/lib/lib{c++,c++abi,unwind}.a stage1/lib/
 echo -n " -DLEAN_STANDALONE=ON"
 echo -n " -DCMAKE_C_COMPILER=$PWD/stage1/bin/clang.exe -DCMAKE_C_COMPILER_WORKS=1 -DCMAKE_CXX_COMPILER=$PWD/llvm/bin/clang++.exe -DCMAKE_CXX_COMPILER_WORKS=1 -DLEAN_CXX_STDLIB='-lc++ -lc++abi'"
 echo -n " -DSTAGE0_CMAKE_C_COMPILER=clang -DSTAGE0_CMAKE_CXX_COMPILER=clang++"
 echo -n " -DLEAN_EXTRA_CXX_FLAGS='--sysroot $PWD/llvm -idirafter /clang64/include/'"
 echo -n " -DLEANC_INTERNAL_FLAGS='--sysroot ROOT -nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang.exe"
-echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -static-libgcc -Wl,-Bstatic -lgmp -lunwind -Wl,-Bdynamic -fuse-ld=lld'"
+echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -static-libgcc -Wl,-Bstatic -lgmp -luv -lunwind -Wl,-Bdynamic -fuse-ld=lld'"
 # when not using the above flags, link GMP dynamically/as usual
-echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp -lucrtbase'"
+echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp -luv -lucrtbase'"
 # do not set `LEAN_CC` for tests
 echo -n " -DAUTO_THREAD_FINALIZATION=OFF -DSTAGE0_AUTO_THREAD_FINALIZATION=OFF"
 echo -n " -DLEAN_TEST_VARS=''"

src/CMakeLists.txt

@@ -10,7 +10,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 11)
+set(LEAN_VERSION_MINOR 12)
 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'")
@@ -243,6 +243,15 @@ if("${USE_GMP}" MATCHES "ON")
   endif()
 endif()
 
+if(NOT "${CMAKE_SYSTEM_NAME}" MATCHES "Emscripten")
+  # LibUV
+  find_package(LibUV 1.0.0 REQUIRED)
+  include_directories(${LIBUV_INCLUDE_DIR})
+endif()
+if(NOT LEAN_STANDALONE)
+  string(APPEND LEAN_EXTRA_LINKER_FLAGS " ${LIBUV_LIBRARIES}")
+endif()
+
 # ccache
 if(CCACHE AND NOT CMAKE_CXX_COMPILER_LAUNCHER AND NOT CMAKE_C_COMPILER_LAUNCHER)
   find_program(CCACHE_PATH ccache)
@@ -373,6 +382,7 @@ if(${CMAKE_SYSTEM_NAME} MATCHES "Linux")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Darwin")
   string(APPEND CMAKE_CXX_FLAGS " -ftls-model=initial-exec")
   string(APPEND INIT_SHARED_LINKER_FLAGS " -install_name @rpath/libInit_shared.dylib")
+  string(APPEND LEANSHARED_1_LINKER_FLAGS " -install_name @rpath/libleanshared_1.dylib")
   string(APPEND LEANSHARED_LINKER_FLAGS " -install_name @rpath/libleanshared.dylib")
   string(APPEND CMAKE_EXE_LINKER_FLAGS " -Wl,-rpath,@executable_path/../lib -Wl,-rpath,@executable_path/../lib/lean")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
@@ -402,8 +412,8 @@ endif()
 # executable or `leanshared`, plugins would try to look them up at load time (even though they
 # are already loaded) and probably fail unless we set up LD_LIBRARY_PATH.
 if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
-  # import library created by the `leanshared` target
-  string(APPEND LEANC_SHARED_LINKER_FLAGS " -lInit_shared -lleanshared")
+  # import libraries created by the stdlib.make targets
+  string(APPEND LEANC_SHARED_LINKER_FLAGS " -lInit_shared -lleanshared_1 -lleanshared")
 elseif("${CMAKE_SYSTEM_NAME}" MATCHES "Darwin")
   string(APPEND LEANC_SHARED_LINKER_FLAGS " -Wl,-undefined,dynamic_lookup")
 endif()
@@ -460,6 +470,22 @@ if(CMAKE_OSX_SYSROOT AND NOT LEAN_STANDALONE)
   string(APPEND LEANC_EXTRA_FLAGS " ${CMAKE_CXX_SYSROOT_FLAG}${CMAKE_OSX_SYSROOT}")
 endif()
 
+add_subdirectory(initialize)
+add_subdirectory(shell)
+# to be included in `leanshared` but not the smaller `leanshared_1` (as it would pull
+# in the world)
+add_library(leaninitialize STATIC $<TARGET_OBJECTS:initialize>)
+set_target_properties(leaninitialize PROPERTIES
+  ARCHIVE_OUTPUT_DIRECTORY ${CMAKE_BINARY_DIR}/lib/temp
+  OUTPUT_NAME leaninitialize)
+add_library(leanshell STATIC util/shell.cpp)
+set_target_properties(leanshell PROPERTIES
+  ARCHIVE_OUTPUT_DIRECTORY ${CMAKE_BINARY_DIR}/lib/temp
+  OUTPUT_NAME leanshell)
+if (${CMAKE_SYSTEM_NAME} MATCHES "Windows")
+  string(APPEND CMAKE_EXE_LINKER_FLAGS " -Wl,--whole-archive -lleanmanifest -Wl,--no-whole-archive")
+endif()
+
 if(${STAGE} GREATER 1)
   # reuse C++ parts, which don't change
   add_library(leanrt_initial-exec STATIC IMPORTED)
@@ -468,13 +494,17 @@ if(${STAGE} GREATER 1)
   add_library(leanrt STATIC IMPORTED)
   set_target_properties(leanrt PROPERTIES
     IMPORTED_LOCATION "${CMAKE_BINARY_DIR}/lib/lean/libleanrt.a")
+  add_library(leancpp_1 STATIC IMPORTED)
+  set_target_properties(leancpp_1 PROPERTIES
+    IMPORTED_LOCATION "${CMAKE_BINARY_DIR}/lib/temp/libleancpp_1.a")
   add_library(leancpp STATIC IMPORTED)
   set_target_properties(leancpp PROPERTIES
     IMPORTED_LOCATION "${CMAKE_BINARY_DIR}/lib/lean/libleancpp.a")
   add_custom_target(copy-leancpp
     COMMAND cmake -E copy_if_different "${PREV_STAGE}/runtime/libleanrt_initial-exec.a" "${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a"
     COMMAND cmake -E copy_if_different "${PREV_STAGE}/lib/lean/libleanrt.a" "${CMAKE_BINARY_DIR}/lib/lean/libleanrt.a"
-    COMMAND cmake -E copy_if_different "${PREV_STAGE}/lib/lean/libleancpp.a" "${CMAKE_BINARY_DIR}/lib/lean/libleancpp.a")
+    COMMAND cmake -E copy_if_different "${PREV_STAGE}/lib/lean/libleancpp.a" "${CMAKE_BINARY_DIR}/lib/lean/libleancpp.a"
+    COMMAND cmake -E copy_if_different "${PREV_STAGE}/lib/temp/libleancpp_1.a" "${CMAKE_BINARY_DIR}/lib/temp/libleancpp_1.a")
   add_dependencies(leancpp copy-leancpp)
   if(LLVM)
       add_custom_target(copy-lean-h-bc
@@ -494,14 +524,23 @@ else()
   set(LEAN_OBJS ${LEAN_OBJS} $<TARGET_OBJECTS:constructions>)
   add_subdirectory(library/compiler)
   set(LEAN_OBJS ${LEAN_OBJS} $<TARGET_OBJECTS:compiler>)
-  add_subdirectory(initialize)
-  set(LEAN_OBJS ${LEAN_OBJS} $<TARGET_OBJECTS:initialize>)
 
-  add_library(leancpp STATIC ${LEAN_OBJS})
+  # leancpp without `initialize` (see `leaninitialize` above)
+  add_library(leancpp_1 STATIC ${LEAN_OBJS})
+  set_target_properties(leancpp_1 PROPERTIES
+    ARCHIVE_OUTPUT_DIRECTORY ${CMAKE_BINARY_DIR}/lib/temp
+    OUTPUT_NAME leancpp_1)
+  add_library(leancpp STATIC ${LEAN_OBJS} $<TARGET_OBJECTS:initialize>)
   set_target_properties(leancpp PROPERTIES
     OUTPUT_NAME leancpp)
 endif()
 
+if((${STAGE} GREATER 0) AND CADICAL)
+  add_custom_target(copy-cadical
+    COMMAND cmake -E copy_if_different "${CADICAL}" "${CMAKE_BINARY_DIR}/bin/cadical${CMAKE_EXECUTABLE_SUFFIX}")
+  add_dependencies(leancpp copy-cadical)
+endif()
+
 # MSYS2 bash usually handles Windows paths relatively well, but not when putting them in the PATH
 string(REGEX REPLACE "^([a-zA-Z]):" "/\\1" LEAN_BIN "${CMAKE_BINARY_DIR}/bin")
 
@@ -509,25 +548,12 @@ string(REGEX REPLACE "^([a-zA-Z]):" "/\\1" LEAN_BIN "${CMAKE_BINARY_DIR}/bin")
 # (also looks nicer in the build log)
 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 ${CMAKE_BINARY_DIR}/lib/temp/libInit.a.export ${CMAKE_BINARY_DIR}/lib/temp/libStd.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,${CMAKE_BINARY_DIR}/lib/lean/libStd.a -Wl,-force_load,${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 ${CMAKE_BINARY_DIR}/lib/temp/libLean.a.export -lleancpp -Wl,--no-whole-archive -lInit_shared -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
-else()
-  set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive -lInit -lStd -lLean -lleancpp -Wl,--no-whole-archive ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a ${LEANSHARED_LINKER_FLAGS}")
-endif()
-
 if (${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
   # We do not use dynamic linking via leanshared for Emscripten to keep things
   # simple. (And we are not interested in `Lake` anyway.) To use dynamic
   # linking, we would probably have to set MAIN_MODULE=2 on `leanshared`,
   # SIDE_MODULE=2 on `lean`, and set CMAKE_SHARED_LIBRARY_SUFFIX to ".js".
-  string(APPEND LEAN_EXE_LINKER_FLAGS " ${TOOLCHAIN_STATIC_LINKER_FLAGS} ${EMSCRIPTEN_SETTINGS} -lnodefs.js -s EXIT_RUNTIME=1 -s MAIN_MODULE=1 -s LINKABLE=1 -s EXPORT_ALL=1")
+  string(APPEND LEAN_EXE_LINKER_FLAGS " ${LIB}/temp/libleanshell.a ${TOOLCHAIN_STATIC_LINKER_FLAGS} ${EMSCRIPTEN_SETTINGS} -lnodefs.js -s EXIT_RUNTIME=1 -s MAIN_MODULE=1 -s LINKABLE=1 -s EXPORT_ALL=1")
 endif()
 
 # Build the compiler using the bootstrapped C sources for stage0, and use
@@ -572,11 +598,11 @@ else()
 
   add_custom_target(leanshared ALL
     WORKING_DIRECTORY ${LEAN_SOURCE_DIR}
-    DEPENDS Init_shared leancpp
+    DEPENDS Init_shared leancpp_1 leancpp leanshell leaninitialize
     COMMAND $(MAKE) -f ${CMAKE_BINARY_DIR}/stdlib.make leanshared
     VERBATIM)
 
-  string(APPEND CMAKE_EXE_LINKER_FLAGS " -lInit_shared -lleanshared")
+  string(APPEND CMAKE_EXE_LINKER_FLAGS " -lInit_shared -lleanshared_1 -lleanshared")
 endif()
 
 if(NOT ${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
@@ -613,7 +639,9 @@ file(COPY ${LEAN_SOURCE_DIR}/bin/leanmake DESTINATION ${CMAKE_BINARY_DIR}/bin)
 
 install(DIRECTORY "${CMAKE_BINARY_DIR}/bin/" USE_SOURCE_PERMISSIONS DESTINATION bin)
 
-add_subdirectory(shell)
+if (${STAGE} GREATER 0 AND CADICAL)
+  install(PROGRAMS "${CADICAL}" DESTINATION bin)
+endif()
 
 add_custom_target(clean-stdlib
   COMMAND rm -rf "${CMAKE_BINARY_DIR}/lib" || true)

src/Init/ByCases.lean

@@ -57,7 +57,7 @@ theorem apply_ite (f : α → β) (P : Prop) [Decidable P] (x y : α) :
 -- We don't mark this as `simp` as it is already handled by `ite_eq_right_iff`.
 theorem ite_some_none_eq_none [Decidable P] :
     (if P then some x else none) = none ↔ ¬ P := by
-  simp only [ite_eq_right_iff]
+  simp only [ite_eq_right_iff, reduceCtorEq]
   rfl
 
 @[simp] theorem ite_some_none_eq_some [Decidable P] :

src/Init/Control/ExceptCps.lean

@@ -34,7 +34,7 @@ instance : Monad (ExceptCpsT ε m) where
   bind x f := fun _ k₁ k₂ => x _ (fun a => f a _ k₁ k₂) k₂
 
 instance : LawfulMonad (ExceptCpsT σ m) := by
-  refine' { .. } <;> intros <;> rfl
+  refine LawfulMonad.mk' _ ?_ ?_ ?_ <;> intros <;> rfl
 
 instance : MonadExceptOf ε (ExceptCpsT ε m) where
   throw e  := fun _ _ k => k e

src/Init/Control/Lawful/Basic.lean

@@ -153,7 +153,7 @@ namespace Id
 @[simp] theorem pure_eq (a : α) : (pure a : Id α) = a := rfl
 
 instance : LawfulMonad Id := by
-  refine' { .. } <;> intros <;> rfl
+  refine LawfulMonad.mk' _ ?_ ?_ ?_ <;> intros <;> rfl
 
 end Id
 

src/Init/Control/StateCps.lean

@@ -35,7 +35,7 @@ instance : Monad (StateCpsT σ m) where
   bind x f := fun δ s k => x δ s fun a s => f a δ s k
 
 instance : LawfulMonad (StateCpsT σ m) := by
-  refine' { .. } <;> intros <;> rfl
+  refine LawfulMonad.mk' _ ?_ ?_ ?_ <;> intros <;> rfl
 
 @[always_inline]
 instance : MonadStateOf σ (StateCpsT σ m) where

src/Init/Core.lean

@@ -36,6 +36,17 @@ and `flip (·<·)` is the greater-than relation.
 
 theorem Function.comp_def {α β δ} (f : β → δ) (g : α → β) : f ∘ g = fun x => f (g x) := rfl
 
+@[simp] theorem Function.const_comp {f : α → β} {c : γ} :
+    (Function.const β c ∘ f) = Function.const α c := by
+  rfl
+@[simp] theorem Function.comp_const {f : β → γ} {b : β} :
+    (f ∘ Function.const α b) = Function.const α (f b) := by
+  rfl
+@[simp] theorem Function.true_comp {f : α → β} : ((fun _ => true) ∘ f) = fun _ => true := by
+  rfl
+@[simp] theorem Function.false_comp {f : α → β} : ((fun _ => false) ∘ f) = fun _ => false := by
+  rfl
+
 attribute [simp] namedPattern
 
 /--
@@ -1104,6 +1115,13 @@ inductive Relation.TransGen {α : Sort u} (r : α → α → Prop) : α → α
 /-- Deprecated synonym for `Relation.TransGen`. -/
 @[deprecated Relation.TransGen (since := "2024-07-16")] abbrev TC := @Relation.TransGen
 
+theorem Relation.TransGen.trans {α : Sort u} {r : α → α → Prop} {a b c} :
+    TransGen r a b → TransGen r b c → TransGen r a c := by
+  intro hab hbc
+  induction hbc with
+  | single h => exact TransGen.tail hab h
+  | tail _ h ih => exact TransGen.tail ih h
+
 /-! # Subtype -/
 
 namespace Subtype
@@ -1546,7 +1564,7 @@ so you should consider the simpler versions if they apply:
 * `Quot.recOnSubsingleton`, when the target type is a `Subsingleton`
 * `Quot.hrecOn`, which uses `HEq (f a) (f b)` instead of a `sound p ▸ f a = f b` assummption
 -/
-protected abbrev rec
+@[elab_as_elim] protected abbrev rec
     (f : (a : α) → motive (Quot.mk r a))
     (h : (a b : α) → (p : r a b) → Eq.ndrec (f a) (sound p) = f b)
     (q : Quot r) : motive q :=
@@ -1632,7 +1650,7 @@ protected theorem ind {α : Sort u} {s : Setoid α} {motive : Quotient s → Pro
 
 /--
 The analogue of `Quot.liftOn`: if `f : α → β` respects the equivalence relation `≈`,
-then it lifts to a function on `Quotient s` such that `lift (mk a) f h = f a`.
+then it lifts to a function on `Quotient s` such that `liftOn (mk a) f h = f a`.
 -/
 protected abbrev liftOn {α : Sort u} {β : Sort v} {s : Setoid α} (q : Quotient s) (f : α → β) (c : (a b : α) → a ≈ b → f a = f b) : β :=
   Quot.liftOn q f c

src/Init/Data.lean

@@ -37,3 +37,5 @@ import Init.Data.Cast
 import Init.Data.Sum
 import Init.Data.BEq
 import Init.Data.Subtype
+import Init.Data.ULift
+import Init.Data.PLift

src/Init/Data/AC.lean

@@ -6,7 +6,7 @@ Authors: Dany Fabian
 
 prelude
 import Init.Classical
-import Init.Data.List
+import Init.ByCases
 
 namespace Lean.Data.AC
 inductive Expr
@@ -260,7 +260,7 @@ theorem Context.evalList_sort (ctx : Context α) (h : ContextInformation.isComm
     simp [ContextInformation.isComm, Option.isSome] at h
     match h₂ : ctx.comm with
     | none =>
-      simp only [h₂] at h
+      simp [h₂] at h
     | some val =>
       simp [h₂] at h
       exact val.down

src/Init/Data/Array/Lemmas.lean

@@ -7,6 +7,7 @@ prelude
 import Init.Data.Nat.MinMax
 import Init.Data.Nat.Lemmas
 import Init.Data.List.Monadic
+import Init.Data.List.Nat.Range
 import Init.Data.Fin.Basic
 import Init.Data.Array.Mem
 import Init.TacticsExtra
@@ -334,8 +335,19 @@ theorem mem_data {a : α} {l : Array α} : a ∈ l.data ↔ a ∈ l := (mem_def
 
 theorem not_mem_nil (a : α) : ¬ a ∈ #[] := nofun
 
+theorem getElem_of_mem {a : α} {as : Array α} :
+    a ∈ as → (∃ (n : Nat) (h : n < as.size), as[n]'h = a) := by
+  intro ha
+  rcases List.getElem_of_mem ha.val with ⟨i, hbound, hi⟩
+  exists i
+  exists hbound
+
 /-- # get lemmas -/
 
+theorem lt_of_getElem {x : α} {a : Array α} {idx : Nat} {hidx : idx < a.size} (_ : a[idx] = x) :
+    idx < a.size :=
+  hidx
+
 theorem getElem?_mem {l : Array α} {i : Fin l.size} : l[i] ∈ l := by
   erw [Array.mem_def, getElem_eq_data_getElem]
   apply List.get_mem
@@ -505,6 +517,13 @@ theorem size_eq_length_data (as : Array α) : as.size = as.data.length := rfl
     simp only [mkEmpty_eq, size_push] at *
     omega
 
+@[simp] theorem data_range (n : Nat) : (range n).data = List.range n := by
+  induction n <;> simp_all [range, Nat.fold, flip, List.range_succ]
+
+@[simp]
+theorem getElem_range {n : Nat} {x : Nat} (h : x < (Array.range n).size) : (Array.range n)[x] = x := by
+  simp [getElem_eq_data_getElem]
+
 set_option linter.deprecated false in
 @[simp] theorem reverse_data (a : Array α) : a.reverse.data = a.data.reverse := by
   let rec go (as : Array α) (i j hj)
@@ -707,12 +726,27 @@ theorem mapIdx_spec (as : Array α) (f : Fin as.size → α → β)
   unfold modify modifyM Id.run
   split <;> simp
 
+theorem getElem_modify {as : Array α} {x i} (h : i < as.size) :
+    (as.modify x f)[i]'(by simp [h]) = if x = i then f as[i] else as[i] := by
+  simp only [modify, modifyM, get_eq_getElem, Id.run, Id.pure_eq]
+  split
+  · simp only [Id.bind_eq, get_set _ _ _ h]; split <;> simp [*]
+  · rw [if_neg (mt (by rintro rfl; exact h) ‹_›)]
+
+theorem getElem_modify_self {as : Array α} {i : Nat} (h : i < as.size) (f : α → α) :
+    (as.modify i f)[i]'(by simp [h]) = f as[i] := by
+  simp [getElem_modify h]
+
+theorem getElem_modify_of_ne {as : Array α} {i : Nat} (hj : j < as.size)
+    (f : α → α) (h : i ≠ j) :
+    (as.modify i f)[j]'(by rwa [size_modify]) = as[j] := by
+  simp [getElem_modify hj, h]
+
+@[deprecated getElem_modify (since := "2024-08-08")]
 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) ‹_›)]
+  simp [getElem_modify h]
 
 /-! ### filter -/
 

src/Init/Data/Array/Mem.lean

@@ -13,11 +13,11 @@ namespace Array
 /-- `a ∈ as` is a predicate which asserts that `a` is in the array `as`. -/
 -- NB: This is defined as a structure rather than a plain def so that a lemma
 -- like `sizeOf_lt_of_mem` will not apply with no actual arrays around.
-structure Mem (a : α) (as : Array α) : Prop where
+structure Mem (as : Array α) (a : α) : Prop where
   val : a ∈ as.data
 
 instance : Membership α (Array α) where
-  mem a as := Mem a as
+  mem := Mem
 
 theorem sizeOf_lt_of_mem [SizeOf α] {as : Array α} (h : a ∈ as) : sizeOf a < sizeOf as := by
   cases as with | _ as =>
@@ -38,8 +38,8 @@ macro "array_get_dec" : tactic =>
     -- 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
+    | (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_get ..)); simp_arith
+    | (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_getElem ..)); simp_arith
     )
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_get_dec)
@@ -52,7 +52,7 @@ macro "array_mem_dec" : tactic =>
   `(tactic| first
     | with_reducible apply Array.sizeOf_lt_of_mem; assumption; done
     | with_reducible
-        apply Nat.lt_trans (Array.sizeOf_lt_of_mem ?h)
+        apply Nat.lt_of_lt_of_le (Array.sizeOf_lt_of_mem ?h)
         case' h => assumption
       simp_arith)
 

src/Init/Data/BitVec/Basic.lean

@@ -20,6 +20,8 @@ We define many of the bitvector operations from the
 of SMT-LIBv2.
 -/
 
+set_option linter.missingDocs true
+
 /--
 A bitvector of the specified width.
 
@@ -34,14 +36,14 @@ structure BitVec (w : Nat) where
   O(1), because we use `Fin` as the internal representation of a bitvector. -/
   toFin : Fin (2^w)
 
-@[deprecated (since := "2024-04-12")]
-protected abbrev Std.BitVec := _root_.BitVec
-
+/--
+Bitvectors have decidable equality. This should be used via the instance `DecidableEq (BitVec n)`.
+-/
 -- We manually derive the `DecidableEq` instances for `BitVec` because
 -- we want to have builtin support for bit-vector literals, and we
 -- need a name for this function to implement `canUnfoldAtMatcher` at `WHNF.lean`.
-def BitVec.decEq (a b : BitVec n) : Decidable (a = b) :=
-  match a, b with
+def BitVec.decEq (x y : BitVec n) : Decidable (x = y) :=
+  match x, y with
   | ⟨n⟩, ⟨m⟩ =>
     if h : n = m then
       isTrue (h ▸ rfl)
@@ -67,9 +69,9 @@ protected def ofNat (n : Nat) (i : Nat) : BitVec n where
 instance instOfNat : OfNat (BitVec n) i where ofNat := .ofNat n i
 instance natCastInst : NatCast (BitVec w) := ⟨BitVec.ofNat w⟩
 
-/-- Given a bitvector `a`, return the underlying `Nat`. This is O(1) because `BitVec` is a
+/-- Given a bitvector `x`, return the underlying `Nat`. This is O(1) because `BitVec` is a
 (zero-cost) wrapper around a `Nat`. -/
-protected def toNat (a : BitVec n) : Nat := a.toFin.val
+protected def toNat (x : BitVec n) : Nat := x.toFin.val
 
 /-- Return the bound in terms of toNat. -/
 theorem isLt (x : BitVec w) : x.toNat < 2^w := x.toFin.isLt
@@ -121,18 +123,18 @@ section getXsb
 @[inline] def getMsb (x : BitVec w) (i : Nat) : Bool := i < w && getLsb x (w-1-i)
 
 /-- Return most-significant bit in bitvector. -/
-@[inline] protected def msb (a : BitVec n) : Bool := getMsb a 0
+@[inline] protected def msb (x : BitVec n) : Bool := getMsb x 0
 
 end getXsb
 
 section Int
 
 /-- Interpret the bitvector as an integer stored in two's complement form. -/
-protected def toInt (a : BitVec n) : Int :=
-  if 2 * a.toNat < 2^n then
-    a.toNat
+protected def toInt (x : BitVec n) : Int :=
+  if 2 * x.toNat < 2^n then
+    x.toNat
   else
-    (a.toNat : Int) - (2^n : Nat)
+    (x.toNat : Int) - (2^n : Nat)
 
 /-- The `BitVec` with value `(2^n + (i mod 2^n)) mod 2^n`.  -/
 protected def ofInt (n : Nat) (i : Int) : BitVec n := .ofNatLt (i % (Int.ofNat (2^n))).toNat (by
@@ -213,7 +215,7 @@ instance : Neg (BitVec n) := ⟨.neg⟩
 /--
 Return the absolute value of a signed bitvector.
 -/
-protected def abs (s : BitVec n) : BitVec n := if s.msb then .neg s else s
+protected def abs (x : BitVec n) : BitVec n := if x.msb then .neg x else x
 
 /--
 Multiplication for bit vectors. This can be interpreted as either signed or unsigned negation
@@ -260,12 +262,12 @@ sdiv 5#4 -2 = -2#4
 sdiv (-7#4) (-2) = 3#4
 ```
 -/
-def sdiv (s t : BitVec n) : BitVec n :=
-  match s.msb, t.msb with
-  | false, false => udiv s t
-  | false, true  => .neg (udiv s (.neg t))
-  | true,  false => .neg (udiv (.neg s) t)
-  | true,  true  => udiv (.neg s) (.neg t)
+def sdiv (x y : BitVec n) : BitVec n :=
+  match x.msb, y.msb with
+  | false, false => udiv x y
+  | false, true  => .neg (udiv x (.neg y))
+  | true,  false => .neg (udiv (.neg x) y)
+  | true,  true  => udiv (.neg x) (.neg y)
 
 /--
 Signed division for bit vectors using SMTLIB rules for division by zero.
@@ -274,40 +276,40 @@ Specifically, `smtSDiv x 0 = if x >= 0 then -1 else 1`
 
 SMT-Lib name: `bvsdiv`.
 -/
-def smtSDiv (s t : BitVec n) : BitVec n :=
-  match s.msb, t.msb with
-  | false, false => smtUDiv s t
-  | false, true  => .neg (smtUDiv s (.neg t))
-  | true,  false => .neg (smtUDiv (.neg s) t)
-  | true,  true  => smtUDiv (.neg s) (.neg t)
+def smtSDiv (x y : BitVec n) : BitVec n :=
+  match x.msb, y.msb with
+  | false, false => smtUDiv x y
+  | false, true  => .neg (smtUDiv x (.neg y))
+  | true,  false => .neg (smtUDiv (.neg x) y)
+  | true,  true  => smtUDiv (.neg x) (.neg y)
 
 /--
 Remainder for signed division rounding to zero.
 
 SMT_Lib name: `bvsrem`.
 -/
-def srem (s t : BitVec n) : BitVec n :=
-  match s.msb, t.msb with
-  | false, false => umod s t
-  | false, true  => umod s (.neg t)
-  | true,  false => .neg (umod (.neg s) t)
-  | true,  true  => .neg (umod (.neg s) (.neg t))
+def srem (x y : BitVec n) : BitVec n :=
+  match x.msb, y.msb with
+  | false, false => umod x y
+  | false, true  => umod x (.neg y)
+  | true,  false => .neg (umod (.neg x) y)
+  | true,  true  => .neg (umod (.neg x) (.neg y))
 
 /--
 Remainder for signed division rounded to negative infinity.
 
 SMT_Lib name: `bvsmod`.
 -/
-def smod (s t : BitVec m) : BitVec m :=
-  match s.msb, t.msb with
-  | false, false => umod s t
+def smod (x y : BitVec m) : BitVec m :=
+  match x.msb, y.msb with
+  | false, false => umod x y
   | false, true =>
-    let u := umod s (.neg t)
-    (if u = .zero m then u else .add u t)
+    let u := umod x (.neg y)
+    (if u = .zero m then u else .add u y)
   | true, false =>
-    let u := umod (.neg s) t
-    (if u = .zero m then u else .sub t u)
-  | true, true => .neg (umod (.neg s) (.neg t))
+    let u := umod (.neg x) y
+    (if u = .zero m then u else .sub y u)
+  | true, true => .neg (umod (.neg x) (.neg y))
 
 end arithmetic
 
@@ -371,8 +373,8 @@ end relations
 
 section cast
 
-/-- `cast eq i` embeds `i` into an equal `BitVec` type. -/
-@[inline] def cast (eq : n = m) (i : BitVec n) : BitVec m := .ofNatLt i.toNat (eq ▸ i.isLt)
+/-- `cast eq x` embeds `x` into an equal `BitVec` type. -/
+@[inline] def cast (eq : n = m) (x : BitVec n) : BitVec m := .ofNatLt x.toNat (eq ▸ x.isLt)
 
 @[simp] theorem cast_ofNat {n m : Nat} (h : n = m) (x : Nat) :
     cast h (BitVec.ofNat n x) = BitVec.ofNat m x := by
@@ -389,7 +391,7 @@ Extraction of bits `start` to `start + len - 1` from a bit vector of size `n` to
 new bitvector of size `len`. If `start + len > n`, then the vector will be zero-padded in the
 high bits.
 -/
-def extractLsb' (start len : Nat) (a : BitVec n) : BitVec len := .ofNat _ (a.toNat >>> start)
+def extractLsb' (start len : Nat) (x : BitVec n) : BitVec len := .ofNat _ (x.toNat >>> start)
 
 /--
 Extraction of bits `hi` (inclusive) down to `lo` (inclusive) from a bit vector of size `n` to
@@ -397,12 +399,12 @@ yield a new bitvector of size `hi - lo + 1`.
 
 SMT-Lib name: `extract`.
 -/
-def extractLsb (hi lo : Nat) (a : BitVec n) : BitVec (hi - lo + 1) := extractLsb' lo _ a
+def extractLsb (hi lo : Nat) (x : BitVec n) : BitVec (hi - lo + 1) := extractLsb' lo _ x
 
 /--
 A version of `zeroExtend` that requires a proof, but is a noop.
 -/
-def zeroExtend' {n w : Nat} (le : n ≤ w) (x : BitVec n)  : BitVec w :=
+def zeroExtend' {n w : Nat} (le : n ≤ w) (x : BitVec n) : BitVec w :=
   x.toNat#'(by
     apply Nat.lt_of_lt_of_le x.isLt
     exact Nat.pow_le_pow_of_le_right (by trivial) le)
@@ -411,8 +413,8 @@ def zeroExtend' {n w : Nat} (le : n ≤ w) (x : BitVec n)  : BitVec w :=
 `shiftLeftZeroExtend x n` returns `zeroExtend (w+n) x <<< n` without
 needing to compute `x % 2^(2+n)`.
 -/
-def shiftLeftZeroExtend (msbs : BitVec w) (m : Nat) : BitVec (w+m) :=
-  let shiftLeftLt {x : Nat} (p : x < 2^w) (m : Nat) : x <<< m < 2^(w+m) := by
+def shiftLeftZeroExtend (msbs : BitVec w) (m : Nat) : BitVec (w + m) :=
+  let shiftLeftLt {x : Nat} (p : x < 2^w) (m : Nat) : x <<< m < 2^(w + m) := by
         simp [Nat.shiftLeft_eq, Nat.pow_add]
         apply Nat.mul_lt_mul_of_pos_right p
         exact (Nat.two_pow_pos m)
@@ -500,24 +502,24 @@ instance : Complement (BitVec w) := ⟨.not⟩
 
 /--
 Left shift for bit vectors. The low bits are filled with zeros. As a numeric operation, this is
-equivalent to `a * 2^s`, modulo `2^n`.
+equivalent to `x * 2^s`, modulo `2^n`.
 
 SMT-Lib name: `bvshl` except this operator uses a `Nat` shift value.
 -/
-protected def shiftLeft (a : BitVec n) (s : Nat) : BitVec n := BitVec.ofNat n (a.toNat <<< s)
+protected def shiftLeft (x : BitVec n) (s : Nat) : BitVec n := BitVec.ofNat n (x.toNat <<< s)
 instance : HShiftLeft (BitVec w) Nat (BitVec w) := ⟨.shiftLeft⟩
 
 /--
 (Logical) right shift for bit vectors. The high bits are filled with zeros.
-As a numeric operation, this is equivalent to `a / 2^s`, rounding down.
+As a numeric operation, this is equivalent to `x / 2^s`, rounding down.
 
 SMT-Lib name: `bvlshr` except this operator uses a `Nat` shift value.
 -/
-def ushiftRight (a : BitVec n) (s : Nat) : BitVec n :=
-  (a.toNat >>> s)#'(by
-  let ⟨a, lt⟩ := a
+def ushiftRight (x : BitVec n) (s : Nat) : BitVec n :=
+  (x.toNat >>> s)#'(by
+  let ⟨x, lt⟩ := x
   simp only [BitVec.toNat, Nat.shiftRight_eq_div_pow, Nat.div_lt_iff_lt_mul (Nat.two_pow_pos s)]
-  rw [←Nat.mul_one a]
+  rw [←Nat.mul_one x]
   exact Nat.mul_lt_mul_of_lt_of_le' lt (Nat.two_pow_pos s) (Nat.le_refl 1))
 
 instance : HShiftRight (BitVec w) Nat (BitVec w) := ⟨.ushiftRight⟩
@@ -525,15 +527,24 @@ instance : HShiftRight (BitVec w) Nat (BitVec w) := ⟨.ushiftRight⟩
 /--
 Arithmetic right shift for bit vectors. The high bits are filled with the
 most-significant bit.
-As a numeric operation, this is equivalent to `a.toInt >>> s`.
+As a numeric operation, this is equivalent to `x.toInt >>> s`.
 
 SMT-Lib name: `bvashr` except this operator uses a `Nat` shift value.
 -/
-def sshiftRight (a : BitVec n) (s : Nat) : BitVec n := .ofInt n (a.toInt >>> s)
+def sshiftRight (x : BitVec n) (s : Nat) : BitVec n := .ofInt n (x.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⟩
 
+/--
+Arithmetic right shift for bit vectors. The high bits are filled with the
+most-significant bit.
+As a numeric operation, this is equivalent to `a.toInt >>> s.toNat`.
+
+SMT-Lib name: `bvashr`.
+-/
+def sshiftRight' (a : BitVec n) (s : BitVec m) : BitVec n := a.sshiftRight s.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 :=
@@ -583,11 +594,9 @@ instance : HAppend (BitVec w) (BitVec v) (BitVec (w + v)) := ⟨.append⟩
 -- TODO: write this using multiplication
 /-- `replicate i x` concatenates `i` copies of `x` into a new vector of length `w*i`. -/
 def replicate : (i : Nat) → BitVec w → BitVec (w*i)
-  | 0,   _ => 0
+  | 0,   _ => 0#0
   | n+1, x =>
-    have hEq : w + w*n = w*(n + 1) := by
-      rw [Nat.mul_add, Nat.add_comm, Nat.mul_one]
-    hEq ▸ (x ++ replicate n x)
+    (x ++ replicate n x).cast (by rw [Nat.mul_succ]; omega)
 
 /-!
 ### Cons and Concat

src/Init/Data/BitVec/Bitblast.lean

@@ -28,6 +28,8 @@ https://github.com/mhk119/lean-smt/blob/bitvec/Smt/Data/Bitwise.lean.
 
 -/
 
+set_option linter.missingDocs true
+
 open Nat Bool
 
 namespace Bool
@@ -287,18 +289,18 @@ theorem sle_eq_carry (x y : BitVec w) :
 A recurrence that describes multiplication as repeated addition.
 Is useful for bitblasting multiplication.
 -/
-def mulRec (l r : BitVec w) (s : Nat) : BitVec w :=
-  let cur := if r.getLsb s then (l <<< s) else 0
+def mulRec (x y : BitVec w) (s : Nat) : BitVec w :=
+  let cur := if y.getLsb s then (x <<< s) else 0
   match s with
   | 0 => cur
-  | s + 1 => mulRec l r s + cur
+  | s + 1 => mulRec x y s + cur
 
-theorem mulRec_zero_eq (l r : BitVec w) :
-    mulRec l r 0 = if r.getLsb 0 then l else 0 := by
+theorem mulRec_zero_eq (x y : BitVec w) :
+    mulRec x y 0 = if y.getLsb 0 then x else 0 := by
   simp [mulRec]
 
-theorem mulRec_succ_eq (l r : BitVec w) (s : Nat) :
-    mulRec l r (s + 1) = mulRec l r s + if r.getLsb (s + 1) then (l <<< (s + 1)) else 0 := rfl
+theorem mulRec_succ_eq (x y : BitVec w) (s : Nat) :
+    mulRec x y (s + 1) = mulRec x y s + if y.getLsb (s + 1) then (x <<< (s + 1)) else 0 := rfl
 
 /--
 Recurrence lemma: truncating to `i+1` bits and then zero extending to `w`
@@ -324,29 +326,29 @@ theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow (x : BitVec w
     by_cases hi : x.getLsb i <;> simp [hi] <;> omega
 
 /--
-Recurrence lemma: multiplying `l` with the first `s` bits of `r` is the
-same as truncating `r` to `s` bits, then zero extending to the original length,
+Recurrence lemma: multiplying `x` with the first `s` bits of `y` is the
+same as truncating `y` to `s` bits, then zero extending to the original length,
 and performing the multplication. -/
-theorem mulRec_eq_mul_signExtend_truncate (l r : BitVec w) (s : Nat) :
-    mulRec l r s = l * ((r.truncate (s + 1)).zeroExtend w) := by
+theorem mulRec_eq_mul_signExtend_truncate (x y : BitVec w) (s : Nat) :
+    mulRec x y s = x * ((y.truncate (s + 1)).zeroExtend w) := by
   induction s
   case zero =>
     simp only [mulRec_zero_eq, ofNat_eq_ofNat, Nat.reduceAdd]
-    by_cases r.getLsb 0
-    case pos hr =>
-      simp only [hr, ↓reduceIte, truncate, zeroExtend_one_eq_ofBool_getLsb_zero,
-        hr, ofBool_true, ofNat_eq_ofNat]
+    by_cases y.getLsb 0
+    case pos hy =>
+      simp only [hy, ↓reduceIte, truncate, zeroExtend_one_eq_ofBool_getLsb_zero,
+        ofBool_true, ofNat_eq_ofNat]
       rw [zeroExtend_ofNat_one_eq_ofNat_one_of_lt (by omega)]
       simp
-    case neg hr =>
-      simp [hr, zeroExtend_one_eq_ofBool_getLsb_zero]
+    case neg hy =>
+      simp [hy, zeroExtend_one_eq_ofBool_getLsb_zero]
   case succ s' hs =>
     rw [mulRec_succ_eq, hs]
     have heq :
-      (if r.getLsb (s' + 1) = true then l <<< (s' + 1) else 0) =
-        (l * (r &&& (BitVec.twoPow w (s' + 1)))) := by
+      (if y.getLsb (s' + 1) = true then x <<< (s' + 1) else 0) =
+        (x * (y &&& (BitVec.twoPow w (s' + 1)))) := by
       simp only [ofNat_eq_ofNat, and_twoPow]
-      by_cases hr : r.getLsb (s' + 1) <;> simp [hr]
+      by_cases hy : y.getLsb (s' + 1) <;> simp [hy]
     rw [heq, ← BitVec.mul_add, ← zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow]
 
 theorem getLsb_mul (x y : BitVec w) (i : Nat) :
@@ -403,12 +405,8 @@ theorem shiftLeftRec_eq {x : BitVec w₁} {y : BitVec w₂} {n : Nat} :
   induction n generalizing x y
   case zero =>
     ext i
-    simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, truncate_one]
-    suffices (y &&& 1#w₂) = zeroExtend w₂ (ofBool (y.getLsb 0)) by simp [this]
-    ext i
-    by_cases h : (↑i : Nat) = 0
-    · simp [h, Bool.and_comm]
-    · simp [h]; omega
+    simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, truncate_one,
+      and_one_eq_zeroExtend_ofBool_getLsb]
   case succ n ih =>
     simp only [shiftLeftRec_succ, and_twoPow]
     rw [ih]
@@ -431,4 +429,128 @@ theorem shiftLeft_eq_shiftLeftRec (x : BitVec w₁) (y : BitVec w₂) :
   · simp [of_length_zero]
   · simp [shiftLeftRec_eq]
 
+/- ### Arithmetic shift right (sshiftRight) recurrence -/
+
+/--
+`sshiftRightRec x y n` shifts `x` arithmetically/signed to the right by the first `n` bits of `y`.
+The theorem `sshiftRight_eq_sshiftRightRec` proves the equivalence of `(x.sshiftRight y)` and `sshiftRightRec`.
+Together with equations `sshiftRightRec_zero`, `sshiftRightRec_succ`,
+this allows us to unfold `sshiftRight` into a circuit for bitblasting.
+-/
+def sshiftRightRec (x : BitVec w₁) (y : BitVec w₂) (n : Nat) : BitVec w₁ :=
+  let shiftAmt := (y &&& (twoPow w₂ n))
+  match n with
+  | 0 => x.sshiftRight' shiftAmt
+  | n + 1 => (sshiftRightRec x y n).sshiftRight' shiftAmt
+
+@[simp]
+theorem sshiftRightRec_zero_eq (x : BitVec w₁) (y : BitVec w₂) :
+    sshiftRightRec x y 0 = x.sshiftRight' (y &&& 1#w₂) := by
+  simp only [sshiftRightRec, twoPow_zero]
+
+@[simp]
+theorem sshiftRightRec_succ_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
+    sshiftRightRec x y (n + 1) = (sshiftRightRec x y n).sshiftRight' (y &&& twoPow w₂ (n + 1)) := by
+  simp [sshiftRightRec]
+
+/--
+If `y &&& z = 0`, `x.sshiftRight (y ||| z) = (x.sshiftRight y).sshiftRight z`.
+This follows as `y &&& z = 0` implies `y ||| z = y + z`,
+and thus `x.sshiftRight (y ||| z) = x.sshiftRight (y + z) = (x.sshiftRight y).sshiftRight z`.
+-/
+theorem sshiftRight'_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
+    (h : y &&& z = 0#w₂) :
+    x.sshiftRight' (y ||| z) = (x.sshiftRight' y).sshiftRight' z := by
+  simp [sshiftRight', ← add_eq_or_of_and_eq_zero _ _ h,
+    toNat_add_of_and_eq_zero h, sshiftRight_add]
+
+theorem sshiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
+    sshiftRightRec x y n = x.sshiftRight' ((y.truncate (n + 1)).zeroExtend w₂) := by
+  induction n generalizing x y
+  case zero =>
+    ext i
+    simp [twoPow_zero, Nat.reduceAdd, and_one_eq_zeroExtend_ofBool_getLsb, truncate_one]
+  case succ n ih =>
+    simp only [sshiftRightRec_succ_eq, and_twoPow, ih]
+    by_cases h : y.getLsb (n + 1)
+    · rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsb_true h,
+        sshiftRight'_or_of_and_eq_zero (by simp), h]
+      simp
+    · rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsb_false (i := n + 1)
+        (by simp [h])]
+      simp [h]
+
+/--
+Show that `x.sshiftRight y` can be written in terms of `sshiftRightRec`.
+This can be unfolded in terms of `sshiftRightRec_zero_eq`, `sshiftRightRec_succ_eq` for bitblasting.
+-/
+theorem sshiftRight_eq_sshiftRightRec (x : BitVec w₁) (y : BitVec w₂) :
+    (x.sshiftRight' y).getLsb i = (sshiftRightRec x y (w₂ - 1)).getLsb i := by
+  rcases w₂ with rfl | w₂
+  · simp [of_length_zero]
+  · simp [sshiftRightRec_eq]
+
+/- ### Logical shift right (ushiftRight) recurrence for bitblasting -/
+
+/--
+`ushiftRightRec x y n` shifts `x` logically to the right by the first `n` bits of `y`.
+
+The theorem `shiftRight_eq_ushiftRightRec` proves the equivalence
+of `(x >>> y)` and `ushiftRightRec`.
+
+Together with equations `ushiftRightRec_zero`, `ushiftRightRec_succ`,
+this allows us to unfold `ushiftRight` into a circuit for bitblasting.
+-/
+def ushiftRightRec (x : BitVec w₁) (y : BitVec w₂) (n : Nat) : BitVec w₁ :=
+  let shiftAmt := (y &&& (twoPow w₂ n))
+  match n with
+  | 0 => x >>> shiftAmt
+  | n + 1 => (ushiftRightRec x y n) >>> shiftAmt
+
+@[simp]
+theorem ushiftRightRec_zero (x : BitVec w₁) (y : BitVec w₂) :
+    ushiftRightRec x y 0 = x >>> (y &&& twoPow w₂ 0) := by
+  simp [ushiftRightRec]
+
+@[simp]
+theorem ushiftRightRec_succ (x : BitVec w₁) (y : BitVec w₂) :
+    ushiftRightRec x y (n + 1) = (ushiftRightRec x y n) >>> (y &&& twoPow w₂ (n + 1)) := by
+  simp [ushiftRightRec]
+
+/--
+If `y &&& z = 0`, `x >>> (y ||| z) = x >>> y >>> z`.
+This follows as `y &&& z = 0` implies `y ||| z = y + z`,
+and thus `x >>> (y ||| z) = x >>> (y + z) = x >>> y >>> z`.
+-/
+theorem ushiftRight'_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
+    (h : y &&& z = 0#w₂) :
+    x >>> (y ||| z) = x >>> y >>> z := by
+  simp [← add_eq_or_of_and_eq_zero _ _ h, toNat_add_of_and_eq_zero h, shiftRight_add]
+
+theorem ushiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
+    ushiftRightRec x y n = x >>> (y.truncate (n + 1)).zeroExtend w₂ := by
+  induction n generalizing x y
+  case zero =>
+    ext i
+    simp only [ushiftRightRec_zero, twoPow_zero, Nat.reduceAdd,
+      and_one_eq_zeroExtend_ofBool_getLsb, truncate_one]
+  case succ n ih =>
+    simp only [ushiftRightRec_succ, and_twoPow]
+    rw [ih]
+    by_cases h : y.getLsb (n + 1) <;> simp only [h, ↓reduceIte]
+    · rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsb_true h,
+        ushiftRight'_or_of_and_eq_zero]
+      simp
+    · simp [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsb_false, h]
+
+/--
+Show that `x >>> y` can be written in terms of `ushiftRightRec`.
+This can be unfolded in terms of `ushiftRightRec_zero`, `ushiftRightRec_succ` for bitblasting.
+-/
+theorem shiftRight_eq_ushiftRightRec (x : BitVec w₁) (y : BitVec w₂) :
+    x >>> y = ushiftRightRec x y (w₂ - 1) := by
+  rcases w₂ with rfl | w₂
+  · simp [of_length_zero]
+  · simp [ushiftRightRec_eq]
+
 end BitVec

src/Init/Data/BitVec/Folds.lean

@@ -8,6 +8,8 @@ import Init.Data.BitVec.Lemmas
 import Init.Data.Nat.Lemmas
 import Init.Data.Fin.Iterate
 
+set_option linter.missingDocs true
+
 namespace BitVec
 
 /--

src/Init/Data/BitVec/Lemmas.lean

@@ -12,6 +12,8 @@ import Init.Data.Nat.Lemmas
 import Init.Data.Nat.Mod
 import Init.Data.Int.Bitwise.Lemmas
 
+set_option linter.missingDocs true
+
 namespace BitVec
 
 /--
@@ -21,7 +23,7 @@ theorem ofFin_eq_ofNat : @BitVec.ofFin w (Fin.mk x lt) = BitVec.ofNat w x := by
   simp only [BitVec.ofNat, Fin.ofNat', lt, Nat.mod_eq_of_lt]
 
 /-- Prove equality of bitvectors in terms of nat operations. -/
-theorem eq_of_toNat_eq {n} : ∀ {i j : BitVec n}, i.toNat = j.toNat → i = j
+theorem eq_of_toNat_eq {n} : ∀ {x y : BitVec n}, x.toNat = y.toNat → x = y
   | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
 
 @[simp] theorem val_toFin (x : BitVec w) : x.toFin.val = x.toNat := rfl
@@ -150,9 +152,6 @@ theorem getLsb_ofNat (n : Nat) (x : Nat) (i : Nat) :
   getLsb (BitVec.ofNat n x) i = (i < n && x.testBit i) := by
   simp [getLsb, BitVec.ofNat, Fin.val_ofNat']
 
-@[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]
 
 @[simp] theorem getMsb_zero : (0#w).getMsb i = false := by simp [getMsb]
@@ -160,6 +159,16 @@ theorem toNat_zero (n : Nat) : (0#n).toNat = 0 := by trivial
 @[simp] theorem toNat_mod_cancel (x : BitVec n) : x.toNat % (2^n) = x.toNat :=
   Nat.mod_eq_of_lt x.isLt
 
+@[simp] theorem sub_toNat_mod_cancel {x : BitVec w} (h : ¬ x = 0#w) :
+    (2 ^ w - x.toNat) % 2 ^ w = 2 ^ w - x.toNat := by
+  simp only [toNat_eq, toNat_ofNat, Nat.zero_mod] at h
+  rw [Nat.mod_eq_of_lt (by omega)]
+
+@[simp] theorem sub_sub_toNat_cancel {x : BitVec w} :
+    2 ^ w - (2 ^ w - x.toNat) = x.toNat := by
+  simp [Nat.sub_sub_eq_min, Nat.min_eq_right]
+  omega
+
 private theorem lt_two_pow_of_le {x m n : Nat} (lt : x < 2 ^ m) (le : m ≤ n) : x < 2 ^ n :=
   Nat.lt_of_lt_of_le lt (Nat.pow_le_pow_of_le_right (by trivial : 0 < 2) le)
 
@@ -226,12 +235,12 @@ theorem toNat_ge_of_msb_true {x : BitVec n} (p : BitVec.msb x = true) : x.toNat
 /-! ### toInt/ofInt -/
 
 /-- Prove equality of bitvectors in terms of nat operations. -/
-theorem toInt_eq_toNat_cond (i : BitVec n) :
-    i.toInt =
-      if 2*i.toNat < 2^n then
-        (i.toNat : Int)
+theorem toInt_eq_toNat_cond (x : BitVec n) :
+    x.toInt =
+      if 2*x.toNat < 2^n then
+        (x.toNat : Int)
       else
-        (i.toNat : Int) - (2^n : Nat) :=
+        (x.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
@@ -258,13 +267,13 @@ theorem toInt_eq_toNat_bmod (x : BitVec n) : x.toInt = Int.bmod x.toNat (2^n) :=
     omega
 
 /-- Prove equality of bitvectors in terms of nat operations. -/
-theorem eq_of_toInt_eq {i j : BitVec n} : i.toInt = j.toInt → i = j := by
+theorem eq_of_toInt_eq {x y : BitVec n} : x.toInt = y.toInt → x = y := by
   intro eq
   simp [toInt_eq_toNat_cond] at eq
   apply eq_of_toNat_eq
   revert eq
-  have _ilt := i.isLt
-  have _jlt := j.isLt
+  have _xlt := x.isLt
+  have _ylt := y.isLt
   split <;> split <;> omega
 
 theorem toInt_inj (x y : BitVec n) : x.toInt = y.toInt ↔ x = y :=
@@ -291,6 +300,17 @@ theorem toInt_ofNat {n : Nat} (x : Nat) :
 @[simp] theorem ofInt_natCast (w n : Nat) :
   BitVec.ofInt w (n : Int) = BitVec.ofNat w n := rfl
 
+@[simp] theorem ofInt_ofNat (w n : Nat) :
+  BitVec.ofInt w (no_index (OfNat.ofNat n)) = BitVec.ofNat w (OfNat.ofNat n) := rfl
+
+theorem toInt_neg_iff {w : Nat} {x : BitVec w} :
+    BitVec.toInt x < 0 ↔ 2 ^ w ≤ 2 * x.toNat := by
+  simp [toInt_eq_toNat_cond]; omega
+
+theorem toInt_pos_iff {w : Nat} {x : BitVec w} :
+    0 ≤ BitVec.toInt x ↔ 2 * x.toNat < 2 ^ w := by
+  simp [toInt_eq_toNat_cond]; omega
+
 /-! ### zeroExtend and truncate -/
 
 theorem truncate_eq_zeroExtend {v : Nat} {x : BitVec w} :
@@ -298,8 +318,7 @@ theorem truncate_eq_zeroExtend {v : Nat} {x : BitVec w} :
 
 @[simp, bv_toNat] theorem toNat_zeroExtend' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
     (zeroExtend' p x).toNat = x.toNat := by
-  unfold zeroExtend'
-  simp [p, x.isLt, Nat.mod_eq_of_lt]
+  simp [zeroExtend']
 
 @[bv_toNat] theorem toNat_zeroExtend (i : Nat) (x : BitVec n) :
     BitVec.toNat (zeroExtend i x) = x.toNat % 2^i := by
@@ -402,11 +421,9 @@ theorem msb_truncate (x : BitVec w) : (x.truncate (k + 1)).msb = x.getLsb k := b
     (x.truncate l).truncate k = x.truncate k :=
   zeroExtend_zeroExtend_of_le x h
 
-/--Truncating by the bitwidth has no effect. -/
-@[simp]
-theorem truncate_eq_self {x : BitVec w} : x.truncate w = x := by
-  ext i
-  simp [getLsb_zeroExtend]
+/-- Truncating by the bitwidth has no effect. -/
+-- This doesn't need to be a `@[simp]` lemma, as `zeroExtend_eq` applies.
+theorem truncate_eq_self {x : BitVec w} : x.truncate w = x := zeroExtend_eq _
 
 @[simp] theorem truncate_cast {h : w = v} : (cast h x).truncate k = x.truncate k := by
   apply eq_of_getLsb_eq
@@ -461,10 +478,18 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
 @[simp] theorem extractLsb_toNat (hi lo : Nat) (x : BitVec n) :
   (extractLsb hi lo x).toNat = (x.toNat >>> lo) % 2^(hi-lo+1) := rfl
 
+@[simp] theorem getLsb_extractLsb' (start len : Nat) (x : BitVec n) (i : Nat) :
+    (extractLsb' start len x).getLsb i = (i < len && x.getLsb (start+i)) := by
+  simp [getLsb, Nat.lt_succ]
+
 @[simp] theorem getLsb_extract (hi lo : Nat) (x : BitVec n) (i : Nat) :
     getLsb (extractLsb hi lo x) i = (i ≤ (hi-lo) && getLsb x (lo+i)) := by
-  unfold getLsb
-  simp [Nat.lt_succ]
+  simp [getLsb, Nat.lt_succ]
+
+theorem extractLsb'_eq_extractLsb {w : Nat} (x : BitVec w) (start len : Nat) (h : len > 0) :
+    x.extractLsb' start len = (x.extractLsb (len - 1 + start) start).cast (by omega) := by
+  apply eq_of_toNat_eq
+  simp [extractLsb, show len - 1 + 1 = len by omega]
 
 /-! ### allOnes -/
 
@@ -505,6 +530,13 @@ theorem or_assoc (x y z : BitVec w) :
     x ||| y ||| z = x ||| (y ||| z) := by
   ext i
   simp [Bool.or_assoc]
+instance : Std.Associative (α := BitVec n) (· ||| ·) := ⟨BitVec.or_assoc⟩
+
+theorem or_comm (x y : BitVec w) :
+    x ||| y = y ||| x := by
+  ext i
+  simp [Bool.or_comm]
+instance : Std.Commutative (fun (x y : BitVec w) => x ||| y) := ⟨BitVec.or_comm⟩
 
 /-! ### and -/
 
@@ -536,11 +568,13 @@ theorem and_assoc (x y z : BitVec w) :
     x &&& y &&& z = x &&& (y &&& z) := by
   ext i
   simp [Bool.and_assoc]
+instance : Std.Associative (α := BitVec n) (· &&& ·) := ⟨BitVec.and_assoc⟩
 
 theorem and_comm (x y : BitVec w) :
     x &&& y = y &&& x := by
   ext i
   simp [Bool.and_comm]
+instance : Std.Commutative (fun (x y : BitVec w) => x &&& y) := ⟨BitVec.and_comm⟩
 
 /-! ### xor -/
 
@@ -557,6 +591,15 @@ theorem and_comm (x y : BitVec w) :
   rw [← testBit_toNat, getLsb, getLsb]
   simp
 
+@[simp] theorem getMsb_xor {x y : BitVec w} :
+    (x ^^^ y).getMsb i = (xor (x.getMsb i) (y.getMsb i)) := by
+  simp only [getMsb]
+  by_cases h : i < w <;> simp [h]
+
+@[simp] theorem msb_xor {x y : BitVec w} :
+    (x ^^^ y).msb = (xor x.msb y.msb) := by
+  simp [BitVec.msb]
+
 @[simp] theorem truncate_xor {x y : BitVec w} :
     (x ^^^ y).truncate k = x.truncate k ^^^ y.truncate k := by
   ext
@@ -566,6 +609,13 @@ theorem xor_assoc (x y z : BitVec w) :
     x ^^^ y ^^^ z = x ^^^ (y ^^^ z) := by
   ext i
   simp [Bool.xor_assoc]
+instance : Std.Associative (fun (x y : BitVec w) => x ^^^ y) := ⟨BitVec.xor_assoc⟩
+
+theorem xor_comm (x y : BitVec w) :
+    x ^^^ y = y ^^^ x := by
+  ext i
+  simp [Bool.xor_comm]
+instance : Std.Commutative (fun (x y : BitVec w) => x ^^^ y) := ⟨BitVec.xor_comm⟩
 
 /-! ### not -/
 
@@ -649,6 +699,27 @@ theorem zero_shiftLeft (n : Nat) : 0#w <<< n = 0#w := by
   cases h₁ : decide (i < m) <;> cases h₂ : decide (n ≤ i) <;> cases h₃ : decide (i < n)
   all_goals { simp_all <;> omega }
 
+theorem shiftLeft_xor_distrib (x y : BitVec w) (n : Nat) :
+    (x ^^^ y) <<< n = (x <<< n) ^^^ (y <<< n) := by
+  ext i
+  simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, getLsb_xor]
+  by_cases h : i < n
+    <;> simp [h]
+
+theorem shiftLeft_and_distrib (x y : BitVec w) (n : Nat) :
+    (x &&& y) <<< n = (x <<< n) &&& (y <<< n) := by
+  ext i
+  simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, getLsb_and]
+  by_cases h : i < n
+    <;> simp [h]
+
+theorem shiftLeft_or_distrib (x y : BitVec w) (n : Nat) :
+    (x ||| y) <<< n = (x <<< n) ||| (y <<< n) := by
+  ext i
+  simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, getLsb_or]
+  by_cases h : i < n
+    <;> simp [h]
+
 @[simp] theorem getMsb_shiftLeft (x : BitVec w) (i) :
     (x <<< i).getMsb k = x.getMsb (k + i) := by
   simp only [getMsb, getLsb_shiftLeft]
@@ -711,7 +782,6 @@ theorem shiftLeft_shiftLeft {w : Nat} (x : BitVec w) (n m : Nat) :
 @[simp]
 theorem shiftLeft_eq' {x : BitVec w₁} {y : BitVec w₂} : x <<< y = x <<< y.toNat := by rfl
 
-@[simp]
 theorem shiftLeft_zero' {x : BitVec w₁} : x <<< 0#w₂ = x := by simp
 
 theorem shiftLeft_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {z : BitVec w₃} :
@@ -731,6 +801,31 @@ theorem getLsb_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {i : Nat} :
     getLsb (x >>> i) j = getLsb x (i+j) := by
   unfold getLsb ; simp
 
+theorem ushiftRight_xor_distrib (x y : BitVec w) (n : Nat) :
+    (x ^^^ y) >>> n = (x >>> n) ^^^ (y >>> n) := by
+  ext
+  simp
+
+theorem ushiftRight_and_distrib (x y : BitVec w) (n : Nat) :
+    (x &&& y) >>> n = (x >>> n) &&& (y >>> n) := by
+  ext
+  simp
+
+theorem ushiftRight_or_distrib (x y : BitVec w)  (n : Nat) :
+    (x ||| y) >>> n = (x >>> n) ||| (y >>> n) := by
+  ext
+  simp
+
+@[simp]
+theorem ushiftRight_zero_eq (x : BitVec w) : x >>> 0 = x := by
+  simp [bv_toNat]
+
+/-! ### ushiftRight reductions from BitVec to Nat -/
+
+@[simp]
+theorem ushiftRight_eq' (x : BitVec w₁) (y : BitVec w₂) :
+    x >>> y = x >>> y.toNat := by rfl
+
 /-! ### sshiftRight -/
 
 theorem sshiftRight_eq {x : BitVec n} {i : Nat} :
@@ -774,7 +869,7 @@ theorem sshiftRight_eq_of_msb_true {x : BitVec w} {s : Nat} (h : x.msb = true) :
     · 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) :
+@[simp] 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
@@ -795,6 +890,90 @@ theorem getLsb_sshiftRight (x : BitVec w) (s i : Nat) :
         Nat.not_lt, decide_eq_true_eq]
       omega
 
+theorem sshiftRight_xor_distrib (x y : BitVec w) (n : Nat) :
+    (x ^^^ y).sshiftRight n = (x.sshiftRight n) ^^^ (y.sshiftRight n) := by
+  ext i
+  simp only [getLsb_sshiftRight, getLsb_xor, msb_xor]
+  split
+    <;> by_cases w ≤ i
+    <;> simp [*]
+
+theorem sshiftRight_and_distrib (x y : BitVec w) (n : Nat) :
+    (x &&& y).sshiftRight n = (x.sshiftRight n) &&& (y.sshiftRight n) := by
+  ext i
+  simp only [getLsb_sshiftRight, getLsb_and, msb_and]
+  split
+    <;> by_cases w ≤ i
+    <;> simp [*]
+
+theorem sshiftRight_or_distrib (x y : BitVec w) (n : Nat) :
+    (x ||| y).sshiftRight n = (x.sshiftRight n) ||| (y.sshiftRight n) := by
+  ext i
+  simp only [getLsb_sshiftRight, getLsb_or, msb_or]
+  split
+    <;> by_cases w ≤ i
+    <;> simp [*]
+
+/-- The msb after arithmetic shifting right equals the original msb. -/
+theorem sshiftRight_msb_eq_msb {n : Nat} {x : BitVec w} :
+    (x.sshiftRight n).msb = x.msb := by
+  rw [msb_eq_getLsb_last, getLsb_sshiftRight, msb_eq_getLsb_last]
+  by_cases hw₀ : w = 0
+  · simp [hw₀]
+  · simp only [show ¬(w ≤ w - 1) by omega, decide_False, Bool.not_false, Bool.true_and,
+      ite_eq_right_iff]
+    intros h
+    simp [show n = 0 by omega]
+
+@[simp] theorem sshiftRight_zero {x : BitVec w} : x.sshiftRight 0 = x := by
+  ext i
+  simp
+
+theorem sshiftRight_add {x : BitVec w} {m n : Nat} :
+    x.sshiftRight (m + n) = (x.sshiftRight m).sshiftRight n := by
+  ext i
+  simp only [getLsb_sshiftRight, Nat.add_assoc]
+  by_cases h₁ : w ≤ (i : Nat)
+  · simp [h₁]
+  · simp only [h₁, decide_False, Bool.not_false, Bool.true_and]
+    by_cases h₂ : n + ↑i < w
+    · simp [h₂]
+    · simp only [h₂, ↓reduceIte]
+      by_cases h₃ : m + (n + ↑i) < w
+      · simp [h₃]
+        omega
+      · simp [h₃, sshiftRight_msb_eq_msb]
+
+/-! ### sshiftRight reductions from BitVec to Nat -/
+
+@[simp]
+theorem sshiftRight_eq' (x : BitVec w) : x.sshiftRight' y = x.sshiftRight y.toNat := rfl
+
+/-! ### udiv -/
+
+theorem udiv_eq {x y : BitVec n} : x.udiv y = BitVec.ofNat n (x.toNat / y.toNat) := by
+  have h : x.toNat / y.toNat < 2 ^ n := Nat.lt_of_le_of_lt (Nat.div_le_self ..) (by omega)
+  simp [udiv, bv_toNat, h, Nat.mod_eq_of_lt]
+
+@[simp, bv_toNat]
+theorem toNat_udiv {x y : BitVec n} : (x.udiv y).toNat = x.toNat / y.toNat := by
+  simp only [udiv_eq]
+  by_cases h : y = 0
+  · simp [h]
+  · rw [toNat_ofNat, Nat.mod_eq_of_lt]
+    exact Nat.lt_of_le_of_lt (Nat.div_le_self ..) (by omega)
+
+/-! ### umod -/
+
+theorem umod_eq {x y : BitVec n} :
+    x.umod y = BitVec.ofNat n (x.toNat % y.toNat) := by
+  have h : x.toNat % y.toNat < 2 ^ n := Nat.lt_of_le_of_lt (Nat.mod_le _ _) x.isLt
+  simp [umod, bv_toNat, Nat.mod_eq_of_lt h]
+
+@[simp, bv_toNat]
+theorem toNat_umod {x y : BitVec n} :
+    (x.umod y).toNat = x.toNat % y.toNat := rfl
+
 /-! ### signExtend -/
 
 /-- Equation theorem for `Int.sub` when both arguments are `Int.ofNat` -/
@@ -812,7 +991,7 @@ theorem signExtend_eq_not_zeroExtend_not_of_msb_false {x : BitVec w} {v : Nat} (
   ext i
   by_cases hv : i < v
   · simp only [signExtend, getLsb, getLsb_zeroExtend, hv, decide_True, Bool.true_and, toNat_ofInt,
-      BitVec.toInt_eq_msb_cond, hmsb, ↓reduceIte]
+      BitVec.toInt_eq_msb_cond, hmsb, ↓reduceIte, reduceCtorEq]
     rw [Int.ofNat_mod_ofNat, Int.toNat_ofNat, Nat.testBit_mod_two_pow]
     simp [BitVec.testBit_toNat]
   · simp only [getLsb_zeroExtend, hv, decide_False, Bool.false_and]
@@ -848,6 +1027,18 @@ theorem signExtend_eq_not_zeroExtend_not_of_msb_true {x : BitVec w} {v : Nat} (h
   · rw [signExtend_eq_not_zeroExtend_not_of_msb_true hmsb]
     by_cases (i < v) <;> by_cases (i < w) <;> simp_all <;> omega
 
+/-- Sign extending to a width smaller than the starting width is a truncation. -/
+theorem signExtend_eq_truncate_of_lt (x : BitVec w) {v : Nat} (hv : v ≤ w):
+  x.signExtend v = x.truncate v := by
+  ext i
+  simp only [getLsb_signExtend, Fin.is_lt, decide_True, Bool.true_and, getLsb_zeroExtend,
+    ite_eq_left_iff, Nat.not_lt]
+  omega
+
+/-- Sign extending to the same bitwidth is a no op. -/
+theorem signExtend_eq (x : BitVec w) : x.signExtend w = x := by
+  rw [signExtend_eq_truncate_of_lt _ (Nat.le_refl _), truncate_eq]
+
 /-! ### append -/
 
 theorem append_def (x : BitVec v) (y : BitVec w) :
@@ -857,15 +1048,15 @@ theorem append_def (x : BitVec v) (y : BitVec w) :
     (x ++ y).toNat = x.toNat <<< n ||| y.toNat :=
   rfl
 
-@[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] theorem getLsb_append {x : BitVec n} {y : BitVec m} :
+    getLsb (x ++ y) i = bif i < m then getLsb y i else getLsb x (i - m) := by
   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] theorem getMsb_append {x : BitVec n} {y : BitVec m} :
+    getMsb (x ++ y) i = bif n ≤ i then getMsb y (i - n) else getMsb x i := by
   simp [append_def]
   by_cases h : n ≤ i
   · simp [h]
@@ -1186,6 +1377,19 @@ theorem neg_eq_not_add (x : BitVec w) : -x = ~~~x + 1 := by
   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)]
 
+@[simp]
+theorem neg_neg {x : BitVec w} : - - x = x := by
+  by_cases h : x = 0#w
+  · simp [h]
+  · simp [bv_toNat, h]
+
+theorem neg_ne_iff_ne_neg {x y : BitVec w} : -x ≠ y ↔ x ≠ -y := by
+  constructor
+  all_goals
+    intro h h'
+    subst h'
+    simp at h
+
 /-! ### mul -/
 
 theorem mul_def {n} {x y : BitVec n} : x * y = (ofFin <| x.toFin * y.toFin) := by rfl
@@ -1262,20 +1466,6 @@ 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 -/
-
-/-- The bitvector of width `w` that has the largest value when interpreted as an integer. -/
-def intMax (w : Nat) : BitVec w := BitVec.ofNat w (2^w - 1)
-
-theorem getLsb_intMax_eq (w : Nat) : (intMax w).getLsb i = decide (i < w) := by
-  simp [intMax, getLsb]
-
-theorem toNat_intMax_eq : (intMax w).toNat = 2^w - 1 := by
-  have h : 2^w - 1 < 2^w := by
-    have pos : 2^w > 0 := Nat.pow_pos (by decide)
-    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
@@ -1549,4 +1739,104 @@ theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsb_true
     simp [hx]
   · by_cases hik' : k < i + 1 <;> simp [hik, hik'] <;> omega
 
+/-- Bitwise and of `(x : BitVec w)` with `1#w` equals zero extending `x.lsb` to `w`. -/
+theorem and_one_eq_zeroExtend_ofBool_getLsb {x : BitVec w} :
+    (x &&& 1#w) = zeroExtend w (ofBool (x.getLsb 0)) := by
+  ext i
+  simp only [getLsb_and, getLsb_one, getLsb_zeroExtend, Fin.is_lt, decide_True, getLsb_ofBool,
+    Bool.true_and]
+  by_cases h : (0 = (i : Nat)) <;> simp [h] <;> omega
+
+@[simp]
+theorem replicate_zero_eq {x : BitVec w} : x.replicate 0 = 0#0 := by
+  simp [replicate]
+
+@[simp]
+theorem replicate_succ_eq {x : BitVec w} :
+    x.replicate (n + 1) =
+    (x ++ replicate n x).cast (by rw [Nat.mul_succ]; omega) := by
+  simp [replicate]
+
+/--
+If a number `w * n ≤ i < w * (n + 1)`, then `i - w * n` equals `i % w`.
+This is true by subtracting `w * n` from the inequality, giving
+`0 ≤ i - w * n < w`, which uniquely identifies `i % w`.
+-/
+private theorem Nat.sub_mul_eq_mod_of_lt_of_le (hlo : w * n ≤ i) (hhi : i < w * (n + 1)) :
+    i - w * n = i % w := by
+  rw [Nat.mod_def]
+  congr
+  symm
+  apply Nat.div_eq_of_lt_le
+    (by rw [Nat.mul_comm]; omega)
+    (by rw [Nat.mul_comm]; omega)
+
+@[simp]
+theorem getLsb_replicate {n w : Nat} (x : BitVec w) :
+    (x.replicate n).getLsb i =
+    (decide (i < w * n) && x.getLsb (i % w)) := by
+  induction n generalizing x
+  case zero => simp
+  case succ n ih =>
+    simp only [replicate_succ_eq, getLsb_cast, getLsb_append]
+    by_cases hi : i < w * (n + 1)
+    · simp only [hi, decide_True, Bool.true_and]
+      by_cases hi' : i < w * n
+      · simp [hi', ih]
+      · simp only [hi', decide_False, cond_false]
+        rw [Nat.sub_mul_eq_mod_of_lt_of_le] <;> omega
+    · rw [Nat.mul_succ] at hi ⊢
+      simp only [show ¬i < w * n by omega, decide_False, cond_false, hi, Bool.false_and]
+      apply BitVec.getLsb_ge (x := x) (i := i - w * n) (ge := by omega)
+
+/-! ### intMin -/
+
+/-- The bitvector of width `w` that has the smallest value when interpreted as an integer. -/
+abbrev intMin (w : Nat) := twoPow w (w - 1)
+
+theorem getLsb_intMin (w : Nat) : (intMin w).getLsb i = decide (i + 1 = w) := by
+  simp only [getLsb_twoPow, boolToPropSimps]
+  omega
+
+@[simp, bv_toNat]
+theorem toNat_intMin : (intMin w).toNat = 2 ^ (w - 1) % 2 ^ w := by
+  simp
+
+@[simp]
+theorem neg_intMin {w : Nat} : -intMin w = intMin w := by
+  by_cases h : 0 < w
+  · simp [bv_toNat, h]
+  · simp only [Nat.not_lt, Nat.le_zero_eq] at h
+    simp [bv_toNat, h]
+
+/-! ### intMax -/
+
+/-- The bitvector of width `w` that has the largest value when interpreted as an integer. -/
+abbrev intMax (w : Nat) := (twoPow w (w - 1)) - 1
+
+@[simp, bv_toNat]
+theorem toNat_intMax : (intMax w).toNat = 2 ^ (w - 1) - 1 := by
+  simp only [intMax]
+  by_cases h : w = 0
+  · simp [h]
+  · have h' : 0 < w := by omega
+    rw [toNat_sub, toNat_twoPow, ← Nat.sub_add_comm (by simpa [h'] using Nat.one_le_two_pow),
+      Nat.add_sub_assoc (by simpa [h'] using Nat.one_le_two_pow),
+      Nat.two_pow_pred_mod_two_pow h', ofNat_eq_ofNat, toNat_ofNat, Nat.one_mod_two_pow h',
+      Nat.add_mod_left, Nat.mod_eq_of_lt]
+    have := Nat.two_pow_pred_lt_two_pow h'
+    have := Nat.two_pow_pos w
+    omega
+
+@[simp]
+theorem getLsb_intMax (w : Nat) : (intMax w).getLsb i = decide (i + 1 < w) := by
+  rw [← testBit_toNat, toNat_intMax, Nat.testBit_two_pow_sub_one, decide_eq_decide]
+  omega
+
+@[simp] theorem intMax_add_one {w : Nat} : intMax w + 1#w = intMin w := by
+  simp only [toNat_eq, toNat_intMax, toNat_add, toNat_intMin, toNat_ofNat, Nat.add_mod_mod]
+  by_cases h : w = 0
+  · simp [h]
+  · rw [Nat.sub_add_cancel (Nat.two_pow_pos (w - 1)), Nat.two_pow_pred_mod_two_pow (by omega)]
+
 end BitVec

src/Init/Data/Bool.lean

@@ -55,6 +55,12 @@ theorem eq_iff_iff {a b : Bool} : a = b ↔ (a ↔ b) := by cases b <;> simp
 theorem decide_true_eq  {b : Bool} [Decidable (true = b)]  : decide (true  = b) =  b := by cases b <;> simp
 theorem decide_false_eq {b : Bool} [Decidable (false = b)] : decide (false = b) = !b := by cases b <;> simp
 
+-- These lemmas assist with confluence.
+@[simp] theorem eq_false_imp_eq_true_iff :
+    ∀(a b : Bool), ((a = false → b = true) ↔ (b = false → a = true)) = True := by decide
+@[simp] theorem eq_true_imp_eq_false_iff :
+    ∀(a b : Bool), ((a = true → b = false) ↔ (b = true → a = false)) = True := by decide
+
 /-! ### and -/
 
 @[simp] theorem and_self_left  : ∀(a b : Bool), (a && (a && b)) = (a && b) := by decide
@@ -91,6 +97,11 @@ Needed for confluence of term `(a && b) ↔ a` which reduces to `(a && b) = a` v
 @[simp] theorem iff_self_and : ∀(a b : Bool), (a = (a && b)) ↔ (a → b) := by decide
 @[simp] theorem iff_and_self : ∀(a b : Bool), (b = (a && b)) ↔ (b → a) := by decide
 
+@[simp] theorem not_and_iff_left_iff_imp  : ∀ (a b : Bool), ((!a && b) = a) ↔ !a ∧ !b := by decide
+@[simp] theorem and_not_iff_right_iff_imp : ∀ (a b : Bool), ((a && !b) = b) ↔ !a ∧ !b := by decide
+@[simp] theorem iff_not_self_and : ∀ (a b : Bool), (a = (!a && b)) ↔ !a ∧ !b := by decide
+@[simp] theorem iff_and_not_self : ∀ (a b : Bool), (b = (a && !b)) ↔ !a ∧ !b := by decide
+
 /-! ### or -/
 
 @[simp] theorem or_self_left  : ∀(a b : Bool), (a || (a || b)) = (a || b) := by decide
@@ -120,6 +131,11 @@ Needed for confluence of term `(a || b) ↔ a` which reduces to `(a || b) = a` v
 @[simp] theorem iff_self_or : ∀(a b : Bool), (a = (a || b)) ↔ (b → a) := by decide
 @[simp] theorem iff_or_self : ∀(a b : Bool), (b = (a || b)) ↔ (a → b) := by decide
 
+@[simp] theorem not_or_iff_left_iff_imp  : ∀ (a b : Bool), ((!a || b) = a) ↔ a ∧ b := by decide
+@[simp] theorem or_not_iff_right_iff_imp : ∀ (a b : Bool), ((a || !b) = b) ↔ a ∧ b := by decide
+@[simp] theorem iff_not_self_or : ∀ (a b : Bool), (a = (!a || b)) ↔ a ∧ b := by decide
+@[simp] theorem iff_or_not_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⟩
 
@@ -134,7 +150,7 @@ theorem and_or_distrib_right : ∀ (x y z : Bool), ((x || y) && z) = (x && z ||
 theorem or_and_distrib_left  : ∀ (x y z : Bool), (x || y && z) = ((x || y) && (x || z)) := by decide
 theorem or_and_distrib_right : ∀ (x y z : Bool), (x && y || z) = ((x || z) && (y || z)) := by decide
 
-theorem and_xor_distrib_left : ∀ (x y z : Bool), (x && xor y z) = xor (x && y) (x && z) := by decide
+theorem and_xor_distrib_left  : ∀ (x y z : Bool), (x && xor y z) = xor (x && y) (x && z) := by decide
 theorem and_xor_distrib_right : ∀ (x y z : Bool), (xor x y && z) = xor (x && z) (y && z) := by decide
 
 /-- De Morgan's law for boolean and -/
@@ -202,8 +218,11 @@ instance : Std.LawfulIdentity (· != ·) false where
 @[simp] theorem not_beq_self : ∀ (x : Bool), ((!x) == x) = false := by decide
 @[simp] theorem beq_not_self : ∀ (x : Bool), (x   == !x) = false := by decide
 
-@[simp] theorem not_bne_self : ∀ (x : Bool), ((!x) != x) = true := by decide
-@[simp] theorem bne_not_self : ∀ (x : Bool), (x   != !x) = true := by decide
+@[simp] theorem not_bne : ∀ (a b : Bool), ((!a) != b) = !(a != b) := by decide
+@[simp] theorem bne_not : ∀ (a b : Bool), (a != !b) = !(a != b) := by decide
+
+theorem not_bne_self : ∀ (x : Bool), ((!x) != x) = true := by decide
+theorem bne_not_self : ∀ (x : Bool), (x   != !x) = true := by decide
 
 /-
 Added for equivalence with `Bool.not_beq_self` and needed for confluence
@@ -235,8 +254,10 @@ theorem beq_eq_decide_eq [BEq α] [LawfulBEq α] [DecidableEq α] (a b : α) :
   · 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
+theorem eq_not : ∀ (a b : Bool), (a = (!b)) ↔ (a ≠ b) := by decide
+theorem not_eq : ∀ (a b : Bool), ((!a) = b) ↔ (a ≠ b) := by decide
 
+@[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
 
 @[simp] theorem coe_iff_coe : ∀(a b : Bool), (a ↔ b) ↔ a = b := by decide
@@ -360,9 +381,6 @@ def toNat (b : Bool) : Nat := cond b 1 0
 theorem toNat_le (c : Bool) : c.toNat ≤ 1 := by
   cases c <;> trivial
 
-@[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 _)
 
@@ -427,17 +445,37 @@ theorem not_ite_eq_false_eq_true (p : Prop) [h : Decidable p] (b c : Bool) :
   cases h with | _ p => simp [p]
 
 /-
-Added for confluence between `if_true_left` and `ite_false_same` on
-`if b = true then True else b = true`
+It would be nice to have this for confluence between `if_true_left` and `ite_false_same` on
+`if b = true then True else b = true`.
+However the discrimination tree key is just `→`, so this is tried too often.
 -/
-@[simp] theorem eq_false_imp_eq_true : ∀(b:Bool), (b = false → b = true) ↔ (b = true) := by decide
+theorem eq_false_imp_eq_true : ∀(b:Bool), (b = false → b = true) ↔ (b = true) := by decide
 
 /-
-Added for confluence between `if_true_left` and `ite_false_same` on
-`if b = false then True else b = false`
+It would be nice to have this for confluence between `if_true_left` and `ite_false_same` on
+`if b = false then True else b = false`.
+However the discrimination tree key is just `→`, so this is tried too often.
 -/
-@[simp] theorem eq_true_imp_eq_false : ∀(b:Bool), (b = true → b = false) ↔ (b = false) := by decide
+theorem eq_true_imp_eq_false : ∀(b:Bool), (b = true → b = false) ↔ (b = false) := by decide
 
+/-! ### forall -/
+
+theorem forall_bool' {p : Bool → Prop} (b : Bool) : (∀ x, p x) ↔ p b ∧ p !b :=
+  ⟨fun h ↦ ⟨h _, h _⟩, fun ⟨h₁, h₂⟩ x ↦ by cases b <;> cases x <;> assumption⟩
+
+@[simp]
+theorem forall_bool {p : Bool → Prop} : (∀ b, p b) ↔ p false ∧ p true :=
+  forall_bool' false
+
+/-! ### exists -/
+
+theorem exists_bool' {p : Bool → Prop} (b : Bool) : (∃ x, p x) ↔ p b ∨ p !b :=
+  ⟨fun ⟨x, hx⟩ ↦ by cases x <;> cases b <;> first | exact .inl ‹_› | exact .inr ‹_›,
+    fun h ↦ by cases h <;> exact ⟨_, ‹_›⟩⟩
+
+@[simp]
+theorem exists_bool {p : Bool → Prop} : (∃ b, p b) ↔ p false ∨ p true :=
+  exists_bool' false
 
 /-! ### cond -/
 
@@ -491,6 +529,10 @@ protected theorem cond_false {α : Type u} {a b : α} : cond false a b = b := co
 @[simp] theorem cond_true_right  : ∀(c t : Bool), cond c t true  = (!c || t) := by decide
 @[simp] theorem cond_false_right : ∀(c t : Bool), cond c t false = ( c && t) := by decide
 
+-- These restore confluence between the above lemmas and `cond_not`.
+@[simp] theorem cond_true_not_same  : ∀ (c b : Bool), cond c (!c) b = (!c && b) := by decide
+@[simp] theorem cond_false_not_same : ∀ (c b : Bool), cond c b (!c) = (!c || b) := by decide
+
 @[simp] theorem cond_true_same  : ∀(c b : Bool), cond c c b = (c || b) := by decide
 @[simp] theorem cond_false_same : ∀(c b : Bool), cond c b c = (c && b) := by decide
 
@@ -504,7 +546,7 @@ theorem apply_cond (f : α → β) {b : Bool} {a a' : α} :
     f (bif b then a else a') = bif b then f a else f a' := by
   cases b <;> simp
 
-/-# decidability -/
+/-! # decidability -/
 
 protected theorem decide_coe (b : Bool) [Decidable (b = true)] : decide (b = true) = b := decide_eq_true
 
@@ -520,6 +562,21 @@ protected theorem decide_coe (b : Bool) [Decidable (b = true)] : decide (b = tru
     decide (p ↔ q) = (decide p == decide q) := by
   cases dp with | _ p => simp [p]
 
+@[boolToPropSimps]
+theorem and_eq_decide (p q : Prop) [dpq : Decidable (p ∧ q)] [dp : Decidable p] [dq : Decidable q] :
+    (p && q) = decide (p ∧ q) := by
+  cases dp with | _ p => simp [p]
+
+@[boolToPropSimps]
+theorem or_eq_decide (p q : Prop) [dpq : Decidable (p ∨ q)] [dp : Decidable p] [dq : Decidable q] :
+    (p || q) = decide (p ∨ q) := by
+  cases dp with | _ p => simp [p]
+
+@[boolToPropSimps]
+theorem decide_beq_decide (p q : Prop) [dpq : Decidable (p ↔ q)] [dp : Decidable p] [dq : Decidable q] :
+    (decide p == decide q) = decide (p ↔ q) := by
+  cases dp with | _ p => simp [p]
+
 end Bool
 
 export Bool (cond_eq_if)
@@ -531,3 +588,19 @@ export Bool (cond_eq_if)
 
 @[simp] theorem true_eq_decide_iff {p : Prop} [h : Decidable p] : true = decide p ↔ p := by
   cases h with | _ q => simp [q]
+
+/-! ### coercions -/
+
+/--
+This should not be turned on globally as an instance because it degrades performance in Mathlib,
+but may be used locally.
+-/
+def boolPredToPred : Coe (α → Bool) (α  → Prop) where
+  coe r := fun a => Eq (r a) true
+
+/--
+This should not be turned on globally as an instance because it degrades performance in Mathlib,
+but may be used locally.
+-/
+def boolRelToRel : Coe (α → α → Bool) (α → α → Prop) where
+  coe r := fun a b => Eq (r a b) true

src/Init/Data/ByteArray/Basic.lean

@@ -191,6 +191,137 @@ def foldlM {β : Type v} {m : Type v → Type w} [Monad m] (f : β → UInt8 →
 def foldl {β : Type v} (f : β → UInt8 → β) (init : β) (as : ByteArray) (start := 0) (stop := as.size) : β :=
   Id.run <| as.foldlM f init start stop
 
+/-- Iterator over the bytes (`UInt8`) of a `ByteArray`.
+
+Typically created by `arr.iter`, where `arr` is a `ByteArray`.
+
+An iterator is *valid* if the position `i` is *valid* for the array `arr`, meaning `0 ≤ i ≤ arr.size`
+
+Most operations on iterators return arbitrary values if the iterator is not valid. The functions in
+the `ByteArray.Iterator` API should rule out the creation of invalid iterators, with two exceptions:
+
+- `Iterator.next iter` is invalid if `iter` is already at the end of the array (`iter.atEnd` is
+  `true`)
+- `Iterator.forward iter n`/`Iterator.nextn iter n` is invalid if `n` is strictly greater than the
+  number of remaining bytes.
+-/
+structure Iterator where
+  /-- The array the iterator is for. -/
+  array : ByteArray
+  /-- The current position.
+
+  This position is not necessarily valid for the array, for instance if one keeps calling
+  `Iterator.next` when `Iterator.atEnd` is true. If the position is not valid, then the
+  current byte is `(default : UInt8)`. -/
+  idx : Nat
+  deriving Inhabited
+
+/-- Creates an iterator at the beginning of an array. -/
+def mkIterator (arr : ByteArray) : Iterator :=
+  ⟨arr, 0⟩
+
+@[inherit_doc mkIterator]
+abbrev iter := mkIterator
+
+/-- The size of an array iterator is the number of bytes remaining. -/
+instance : SizeOf Iterator where
+  sizeOf i := i.array.size - i.idx
+
+theorem Iterator.sizeOf_eq (i : Iterator) : sizeOf i = i.array.size - i.idx :=
+  rfl
+
+namespace Iterator
+
+/-- Number of bytes remaining in the iterator. -/
+def remainingBytes : Iterator → Nat
+  | ⟨arr, i⟩ => arr.size - i
+
+@[inherit_doc Iterator.idx]
+def pos := Iterator.idx
+
+/-- The byte at the current position.
+
+On an invalid position, returns `(default : UInt8)`. -/
+@[inline]
+def curr : Iterator → UInt8
+  | ⟨arr, i⟩ =>
+    if h:i < arr.size then
+      arr[i]'h
+    else
+      default
+
+/-- Moves the iterator's position forward by one byte, unconditionally.
+
+It is only valid to call this function if the iterator is not at the end of the array, *i.e.*
+`Iterator.atEnd` is `false`; otherwise, the resulting iterator will be invalid. -/
+@[inline]
+def next : Iterator → Iterator
+  | ⟨arr, i⟩ => ⟨arr, i + 1⟩
+
+/-- Decreases the iterator's position.
+
+If the position is zero, this function is the identity. -/
+@[inline]
+def prev : Iterator → Iterator
+  | ⟨arr, i⟩ => ⟨arr, i - 1⟩
+
+/-- True if the iterator is past the array's last byte. -/
+@[inline]
+def atEnd : Iterator → Bool
+  | ⟨arr, i⟩ => i ≥ arr.size
+
+/-- True if the iterator is not past the array's last byte. -/
+@[inline]
+def hasNext : Iterator → Bool
+  | ⟨arr, i⟩ => i < arr.size
+
+/-- The byte at the current position. --/
+@[inline]
+def curr' (it : Iterator) (h : it.hasNext) : UInt8 :=
+  match it with
+  | ⟨arr, i⟩ =>
+    have : i < arr.size := by
+      simp only [hasNext, decide_eq_true_eq] at h
+      assumption
+    arr[i]
+
+/-- Moves the iterator's position forward by one byte. --/
+@[inline]
+def next' (it : Iterator) (_h : it.hasNext) : Iterator :=
+  match it with
+  | ⟨arr, i⟩ => ⟨arr, i + 1⟩
+
+/-- True if the position is not zero. -/
+@[inline]
+def hasPrev : Iterator → Bool
+  | ⟨_, i⟩ => i > 0
+
+/-- Moves the iterator's position to the end of the array.
+
+Note that `i.toEnd.atEnd` is always `true`. -/
+@[inline]
+def toEnd : Iterator → Iterator
+  | ⟨arr, _⟩ => ⟨arr, arr.size⟩
+
+/-- Moves the iterator's position several bytes forward.
+
+The resulting iterator is only valid if the number of bytes to skip is less than or equal to
+the number of bytes left in the iterator. -/
+@[inline]
+def forward : Iterator → Nat → Iterator
+  | ⟨arr, i⟩, f => ⟨arr, i + f⟩
+
+@[inherit_doc forward, inline]
+def nextn : Iterator → Nat → Iterator := forward
+
+/-- Moves the iterator's position several bytes back.
+
+If asked to go back more bytes than available, stops at the beginning of the array. -/
+@[inline]
+def prevn : Iterator → Nat → Iterator
+  | ⟨arr, i⟩, f => ⟨arr, i - f⟩
+
+end Iterator
 end ByteArray
 
 def List.toByteArray (bs : List UInt8) : ByteArray :=

src/Init/Data/Char/Basic.lean

@@ -63,27 +63,27 @@ instance : Inhabited Char where
   default := 'A'
 
 /-- Is the character a space (U+0020) a tab (U+0009), a carriage return (U+000D) or a newline (U+000A)? -/
-def isWhitespace (c : Char) : Bool :=
+@[inline] def isWhitespace (c : Char) : Bool :=
   c = ' ' || c = '\t' || c = '\r' || c = '\n'
 
 /-- Is the character in `ABCDEFGHIJKLMNOPQRSTUVWXYZ`? -/
-def isUpper (c : Char) : Bool :=
+@[inline] def isUpper (c : Char) : Bool :=
   c.val ≥ 65 && c.val ≤ 90
 
 /-- Is the character in `abcdefghijklmnopqrstuvwxyz`? -/
-def isLower (c : Char) : Bool :=
+@[inline] def isLower (c : Char) : Bool :=
   c.val ≥ 97 && c.val ≤ 122
 
 /-- Is the character in `ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz`? -/
-def isAlpha (c : Char) : Bool :=
+@[inline] def isAlpha (c : Char) : Bool :=
   c.isUpper || c.isLower
 
 /-- Is the character in `0123456789`? -/
-def isDigit (c : Char) : Bool :=
+@[inline] def isDigit (c : Char) : Bool :=
   c.val ≥ 48 && c.val ≤ 57
 
 /-- Is the character in `ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789`? -/
-def isAlphanum (c : Char) : Bool :=
+@[inline] def isAlphanum (c : Char) : Bool :=
   c.isAlpha || c.isDigit
 
 /-- Convert an upper case character to its lower case character.

src/Init/Data/Fin/Basic.lean

@@ -149,6 +149,9 @@ instance : Inhabited (Fin (no_index (n+1))) where
 
 @[simp] theorem zero_eta : (⟨0, Nat.zero_lt_succ _⟩ : Fin (n + 1)) = 0 := rfl
 
+theorem ne_of_val_ne {i j : Fin n} (h : val i ≠ val j) : i ≠ j :=
+  fun h' => absurd (val_eq_of_eq h') h
+
 theorem val_ne_of_ne {i j : Fin n} (h : i ≠ j) : val i ≠ val j :=
   fun h' => absurd (eq_of_val_eq h') h
 

src/Init/Data/Fin/Lemmas.lean

@@ -11,9 +11,6 @@ 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 -/
@@ -57,9 +54,6 @@ 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 (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
 
@@ -141,6 +135,12 @@ theorem eq_zero_or_eq_succ {n : Nat} : ∀ i : Fin (n + 1), i = 0 ∨ ∃ j : Fi
 theorem eq_succ_of_ne_zero {n : Nat} {i : Fin (n + 1)} (hi : i ≠ 0) : ∃ j : Fin n, i = j.succ :=
   (eq_zero_or_eq_succ i).resolve_left hi
 
+protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
+  Fin.ext_iff.trans Nat.le_antisymm_iff
+
+protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
+  Fin.le_antisymm_iff.2 ⟨h1, h2⟩
+
 @[simp] theorem val_rev (i : Fin n) : rev i = n - (i + 1) := rfl
 
 @[simp] theorem rev_rev (i : Fin n) : rev (rev i) = i := Fin.ext <| by

src/Init/Data/Int.lean

@@ -10,5 +10,6 @@ import Init.Data.Int.DivMod
 import Init.Data.Int.DivModLemmas
 import Init.Data.Int.Gcd
 import Init.Data.Int.Lemmas
+import Init.Data.Int.LemmasAux
 import Init.Data.Int.Order
 import Init.Data.Int.Pow

src/Init/Data/Int/Basic.lean

@@ -322,8 +322,8 @@ protected def pow (m : Int) : Nat → Int
   | 0      => 1
   | succ n => Int.pow m n * m
 
-instance : HPow Int Nat Int where
-  hPow := Int.pow
+instance : NatPow Int where
+  pow := Int.pow
 
 instance : LawfulBEq Int where
   eq_of_beq h := by simp [BEq.beq] at h; assumption

src/Init/Data/Int/DivModLemmas.lean

@@ -14,9 +14,6 @@ import Init.RCases
 # Lemmas about integer division needed to bootstrap `omega`.
 -/
 
--- Remove after the next stage0 update
-set_option allowUnsafeReducibility true
-
 open Nat (succ)
 
 namespace Int
@@ -57,7 +54,7 @@ protected theorem dvd_mul_right (a b : Int) : a ∣ a * b := ⟨_, rfl⟩
 
 protected theorem dvd_mul_left (a b : Int) : b ∣ a * b := ⟨_, Int.mul_comm ..⟩
 
-protected theorem neg_dvd {a b : Int} : -a ∣ b ↔ a ∣ b := by
+@[simp] protected theorem neg_dvd {a b : Int} : -a ∣ b ↔ a ∣ b := by
   constructor <;> exact fun ⟨k, e⟩ =>
     ⟨-k, by simp [e, Int.neg_mul, Int.mul_neg, Int.neg_neg]⟩
 
@@ -357,6 +354,7 @@ theorem add_ediv_of_dvd_left {a b c : Int} (H : c ∣ a) : (a + b) / c = a / c +
 @[simp] theorem mul_ediv_cancel_left (b : Int) (H : a ≠ 0) : (a * b) / a = b :=
   Int.mul_comm .. ▸ Int.mul_ediv_cancel _ H
 
+
 theorem div_nonneg_iff_of_pos {a b : Int} (h : 0 < b) : a / b ≥ 0 ↔ a ≥ 0 := by
   rw [Int.div_def]
   match b, h with
@@ -454,6 +452,12 @@ theorem lt_mul_ediv_self_add {x k : Int} (h : 0 < k) : x < k * (x / k) + k :=
 @[simp] theorem add_mul_emod_self_left (a b c : Int) : (a + b * c) % b = a % b := by
   rw [Int.mul_comm, Int.add_mul_emod_self]
 
+@[simp] theorem add_neg_mul_emod_self {a b c : Int} : (a + -(b * c)) % c = a % c := by
+  rw [Int.neg_mul_eq_neg_mul, add_mul_emod_self]
+
+@[simp] theorem add_neg_mul_emod_self_left {a b c : Int} : (a + -(b * c)) % b = a % b := by
+  rw [Int.neg_mul_eq_mul_neg, add_mul_emod_self_left]
+
 @[simp] theorem add_emod_self {a b : Int} : (a + b) % b = a % b := by
   have := add_mul_emod_self_left a b 1; rwa [Int.mul_one] at this
 
@@ -498,9 +502,12 @@ theorem mul_emod (a b n : Int) : (a * b) % n = (a % n) * (b % n) % n := by
     Int.mul_assoc, Int.mul_assoc, ← Int.mul_add n _ _, add_mul_emod_self_left,
     ← Int.mul_assoc, add_mul_emod_self]
 
-@[local simp] theorem emod_self {a : Int} : a % a = 0 := by
+@[simp] theorem emod_self {a : Int} : a % a = 0 := by
   have := mul_emod_left 1 a; rwa [Int.one_mul] at this
 
+@[simp] theorem neg_emod_self (a : Int) : -a % a = 0 := by
+  rw [neg_emod, Int.sub_self, zero_emod]
+
 @[simp] theorem emod_emod_of_dvd (n : Int) {m k : Int}
     (h : m ∣ k) : (n % k) % m = n % m := by
   conv => rhs; rw [← emod_add_ediv n k]
@@ -596,6 +603,14 @@ theorem emod_eq_zero_of_dvd : ∀ {a b : Int}, a ∣ b → b % a = 0
 theorem dvd_iff_emod_eq_zero (a b : Int) : a ∣ b ↔ b % a = 0 :=
   ⟨emod_eq_zero_of_dvd, dvd_of_emod_eq_zero⟩
 
+@[simp] theorem neg_mul_emod_left (a b : Int) : -(a * b) % b = 0 := by
+  rw [← dvd_iff_emod_eq_zero, Int.dvd_neg]
+  exact Int.dvd_mul_left a b
+
+@[simp] theorem neg_mul_emod_right (a b : Int) : -(a * b) % a = 0 := by
+  rw [← dvd_iff_emod_eq_zero, Int.dvd_neg]
+  exact Int.dvd_mul_right a b
+
 instance decidableDvd : DecidableRel (α := Int) (· ∣ ·) := fun _ _ =>
   decidable_of_decidable_of_iff (dvd_iff_emod_eq_zero ..).symm
 
@@ -620,6 +635,12 @@ theorem neg_ediv_of_dvd : ∀ {a b : Int}, b ∣ a → (-a) / b = -(a / b)
     · simp [bz]
     · rw [Int.neg_mul_eq_mul_neg, Int.mul_ediv_cancel_left _ bz, Int.mul_ediv_cancel_left _ bz]
 
+@[simp] theorem neg_mul_ediv_cancel (a b : Int) (h : b ≠ 0) : -(a * b) / b = -a := by
+  rw [neg_ediv_of_dvd (Int.dvd_mul_left a b), mul_ediv_cancel _ h]
+
+@[simp] theorem neg_mul_ediv_cancel_left (a b : Int) (h : a ≠ 0) : -(a * b) / a = -b := by
+  rw [neg_ediv_of_dvd (Int.dvd_mul_right a b), mul_ediv_cancel_left _ h]
+
 theorem sub_ediv_of_dvd (a : Int) {b c : Int}
     (hcb : c ∣ b) : (a - b) / c = a / c - b / c := by
   rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.add_ediv_of_dvd_right (Int.dvd_neg.2 hcb)]
@@ -635,13 +656,22 @@ theorem sub_ediv_of_dvd (a : Int) {b c : Int}
 @[simp] protected theorem ediv_self {a : Int} (H : a ≠ 0) : a / a = 1 := by
   have := Int.mul_ediv_cancel 1 H; rwa [Int.one_mul] at this
 
+@[simp] protected theorem neg_ediv_self (a : Int) (h : a ≠ 0) : (-a) / a = -1 := by
+  rw [neg_ediv_of_dvd (Int.dvd_refl a), Int.ediv_self h]
+
 @[simp]
-theorem emod_sub_cancel (x y : Int): (x - y)%y = x%y := by
+theorem emod_sub_cancel (x y : Int): (x - y) % y = x % y := by
   by_cases h : y = 0
   · simp [h]
   · simp only [Int.emod_def, Int.sub_ediv_of_dvd, Int.dvd_refl, Int.ediv_self h, Int.mul_sub]
     simp [Int.mul_one, Int.sub_sub, Int.add_comm y]
 
+@[simp] theorem add_neg_emod_self (a b : Int) : (a + -b) % b = a % b := by
+  rw [← Int.sub_eq_add_neg, emod_sub_cancel]
+
+@[simp] theorem neg_add_emod_self (a b : Int) : (-a + b) % a = b % a := by
+  rw [Int.add_comm, add_neg_emod_self]
+
 /-- If `a % b = c` then `b` divides `a - c`. -/
 theorem dvd_sub_of_emod_eq {a b c : Int} (h : a % b = c) : b ∣ a - c := by
   have hx : (a % b) % b = c % b := by
@@ -891,6 +921,14 @@ theorem mod_eq_zero_of_dvd : ∀ {a b : Int}, a ∣ b → mod b a = 0
 theorem dvd_iff_mod_eq_zero (a b : Int) : a ∣ b ↔ mod b a = 0 :=
   ⟨mod_eq_zero_of_dvd, dvd_of_mod_eq_zero⟩
 
+@[simp] theorem neg_mul_mod_right (a b : Int) : (-(a * b)).mod a = 0 := by
+  rw [← dvd_iff_mod_eq_zero, Int.dvd_neg]
+  exact Int.dvd_mul_right a b
+
+@[simp] theorem neg_mul_mod_left (a b : Int) : (-(a * b)).mod b = 0 := by
+  rw [← dvd_iff_mod_eq_zero, Int.dvd_neg]
+  exact Int.dvd_mul_left a b
+
 protected theorem div_mul_cancel {a b : Int} (H : b ∣ a) : a.div b * b = a :=
   div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
 
@@ -903,6 +941,10 @@ protected theorem eq_mul_of_div_eq_right {a b c : Int}
 @[simp] theorem mod_self {a : Int} : a.mod a = 0 := by
   have := mul_mod_left 1 a; rwa [Int.one_mul] at this
 
+@[simp] theorem neg_mod_self (a : Int) : (-a).mod a = 0 := by
+  rw [← dvd_iff_mod_eq_zero, Int.dvd_neg]
+  exact Int.dvd_refl a
+
 theorem lt_div_add_one_mul_self (a : Int) {b : Int} (H : 0 < b) : a < (a.div b + 1) * b := by
   rw [Int.add_mul, Int.one_mul, Int.mul_comm]
   exact Int.lt_add_of_sub_left_lt <| Int.mod_def .. ▸ mod_lt_of_pos _ H
@@ -1091,8 +1133,7 @@ theorem bmod_mul_bmod : Int.bmod (Int.bmod x n * y) n = Int.bmod (x * y) n := by
   next p =>
     simp
   next p =>
-    rw [Int.sub_mul, Int.sub_eq_add_neg, ← Int.mul_neg]
-    simp
+    rw [Int.sub_mul, Int.sub_eq_add_neg, ← Int.mul_neg, bmod_add_mul_cancel, emod_mul_bmod_congr]
 
 @[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]

src/Init/Data/Int/Lemmas.lean

@@ -7,6 +7,7 @@ prelude
 import Init.Data.Int.Basic
 import Init.Conv
 import Init.NotationExtra
+import Init.PropLemmas
 
 namespace Int
 
@@ -288,7 +289,7 @@ protected theorem neg_sub (a b : Int) : -(a - b) = b - a := by
 protected theorem sub_sub_self (a b : Int) : a - (a - b) = b := by
   simp [Int.sub_eq_add_neg, ← Int.add_assoc]
 
-protected theorem sub_neg (a b : Int) : a - -b = a + b := by simp [Int.sub_eq_add_neg]
+@[simp] protected theorem sub_neg (a b : Int) : a - -b = a + b := by simp [Int.sub_eq_add_neg]
 
 @[simp] protected theorem sub_add_cancel (a b : Int) : a - b + b = a :=
   Int.neg_add_cancel_right a b
@@ -444,10 +445,10 @@ protected theorem neg_mul_eq_neg_mul (a b : Int) : -(a * b) = -a * b :=
 protected theorem neg_mul_eq_mul_neg (a b : Int) : -(a * b) = a * -b :=
   Int.neg_eq_of_add_eq_zero <| by rw [← Int.mul_add, Int.add_right_neg, Int.mul_zero]
 
-@[local simp] protected theorem neg_mul (a b : Int) : -a * b = -(a * b) :=
+@[simp] protected theorem neg_mul (a b : Int) : -a * b = -(a * b) :=
   (Int.neg_mul_eq_neg_mul a b).symm
 
-@[local simp] protected theorem mul_neg (a b : Int) : a * -b = -(a * b) :=
+@[simp] protected theorem mul_neg (a b : Int) : a * -b = -(a * b) :=
   (Int.neg_mul_eq_mul_neg a b).symm
 
 protected theorem neg_mul_neg (a b : Int) : -a * -b = a * b := by simp
@@ -486,6 +487,9 @@ protected theorem mul_eq_zero {a b : Int} : a * b = 0 ↔ a = 0 ∨ b = 0 := by
 protected theorem mul_ne_zero {a b : Int} (a0 : a ≠ 0) (b0 : b ≠ 0) : a * b ≠ 0 :=
   Or.rec a0 b0 ∘ Int.mul_eq_zero.mp
 
+@[simp] protected theorem mul_ne_zero_iff (a b : Int) : a * b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 := by
+  rw [ne_eq, Int.mul_eq_zero, not_or, ne_eq]
+
 protected theorem eq_of_mul_eq_mul_right {a b c : Int} (ha : a ≠ 0) (h : b * a = c * a) : b = c :=
   have : (b - c) * a = 0 := by rwa [Int.sub_mul, Int.sub_eq_zero]
   Int.sub_eq_zero.1 <| (Int.mul_eq_zero.mp this).resolve_right ha

src/Init/Data/Int/LemmasAux.lean

@@ -0,0 +1,40 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.Int.Order
+import Init.Omega
+
+
+/-!
+# Further lemmas about `Int` relying on `omega` automation.
+-/
+
+namespace Int
+
+@[simp] theorem toNat_sub' (a : Int) (b : Nat) : a.toNat - b = (a - b).toNat := by
+  simp only [Int.toNat]
+  split <;> rename_i x a
+  · simp only [Int.ofNat_eq_coe]
+    split <;> rename_i y b h
+    · simp at h
+      omega
+    · simp [Int.negSucc_eq] at h
+      omega
+  · simp only [Nat.zero_sub]
+    split <;> rename_i y b h
+    · simp [Int.negSucc_eq] at h
+      omega
+    · rfl
+
+@[simp] theorem toNat_sub_max_self (a : Int) : (a - max a 0).toNat = 0 := by
+  simp [toNat]
+  split <;> simp_all <;> omega
+
+@[simp] theorem toNat_sub_self_max (a : Int) : (a - max 0 a).toNat = 0 := by
+  simp [toNat]
+  split <;> simp_all <;> omega
+
+end Int

src/Init/Data/Int/Order.lean

@@ -240,9 +240,24 @@ theorem le_natAbs {a : Int} : a ≤ natAbs a :=
 theorem negSucc_lt_zero (n : Nat) : -[n+1] < 0 :=
   Int.not_le.1 fun h => let ⟨_, h⟩ := eq_ofNat_of_zero_le h; nomatch h
 
+theorem negSucc_le_zero (n : Nat) : -[n+1] ≤ 0 :=
+  Int.le_of_lt (negSucc_lt_zero n)
+
 @[simp] theorem negSucc_not_nonneg (n : Nat) : 0 ≤ -[n+1] ↔ False := by
   simp only [Int.not_le, iff_false]; exact Int.negSucc_lt_zero n
 
+@[simp] theorem ofNat_max_zero (n : Nat) : (max (n : Int) 0) = n := by
+  rw [Int.max_eq_left (ofNat_zero_le n)]
+
+@[simp] theorem zero_max_ofNat (n : Nat) : (max 0 (n : Int)) = n := by
+  rw [Int.max_eq_right (ofNat_zero_le n)]
+
+@[simp] theorem negSucc_max_zero (n : Nat) : (max (Int.negSucc n) 0) = 0 := by
+  rw [Int.max_eq_right (negSucc_le_zero _)]
+
+@[simp] theorem zero_max_negSucc (n : Nat) : (max 0 (Int.negSucc n)) = 0 := by
+  rw [Int.max_eq_left (negSucc_le_zero _)]
+
 protected theorem add_le_add_left {a b : Int} (h : a ≤ b) (c : Int) : c + a ≤ c + b :=
   let ⟨n, hn⟩ := le.dest h; le.intro n <| by rw [Int.add_assoc, hn]
 
@@ -470,8 +485,16 @@ theorem toNat_eq_max : ∀ a : Int, (toNat a : Int) = max a 0
 
 @[simp] theorem toNat_ofNat (n : Nat) : toNat ↑n = n := rfl
 
+@[simp] theorem toNat_negSucc (n : Nat) : (Int.negSucc n).toNat = 0 := by
+  simp [toNat]
+
 @[simp] theorem toNat_ofNat_add_one {n : Nat} : ((n : Int) + 1).toNat = n + 1 := rfl
 
+@[simp] theorem ofNat_toNat (a : Int) : (a.toNat : Int) = max a 0 := by
+  match a with
+  | Int.ofNat n => simp
+  | Int.negSucc n => simp
+
 theorem self_le_toNat (a : Int) : a ≤ toNat a := by rw [toNat_eq_max]; apply Int.le_max_left
 
 @[simp] theorem le_toNat {n : Nat} {z : Int} (h : 0 ≤ z) : n ≤ z.toNat ↔ (n : Int) ≤ z := by
@@ -1006,7 +1029,7 @@ theorem natAbs_mul_self : ∀ {a : Int}, ↑(natAbs a * natAbs a) = a * a
 theorem eq_nat_or_neg (a : Int) : ∃ n : Nat, a = n ∨ a = -↑n := ⟨_, natAbs_eq a⟩
 
 theorem natAbs_mul_natAbs_eq {a b : Int} {c : Nat}
-    (h : a * b = (c : Int)) : a.natAbs * b.natAbs = c := by rw [← natAbs_mul, h, natAbs]
+    (h : a * b = (c : Int)) : a.natAbs * b.natAbs = c := by rw [← natAbs_mul, h, natAbs.eq_def]
 
 @[simp] theorem natAbs_mul_self' (a : Int) : (natAbs a * natAbs a : Int) = a * a := by
   rw [← Int.ofNat_mul, natAbs_mul_self]

src/Init/Data/List.lean

@@ -21,3 +21,5 @@ import Init.Data.List.Pairwise
 import Init.Data.List.Sublist
 import Init.Data.List.TakeDrop
 import Init.Data.List.Zip
+import Init.Data.List.Perm
+import Init.Data.List.Sort

src/Init/Data/List/Attach.lean

@@ -73,6 +73,13 @@ theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (l H)
   · rfl
   · simp only [*, pmap, map]
 
+@[simp] theorem attach_cons (x : α) (xs : List α) :
+    (x :: xs).attach = ⟨x, mem_cons_self x xs⟩ :: xs.attach.map fun ⟨y, h⟩ => ⟨y, mem_cons_of_mem x h⟩ := by
+  simp only [attach, attachWith, pmap, map_pmap, cons.injEq, true_and]
+  apply pmap_congr
+  intros a _ m' _
+  rfl
+
 theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
     pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
   rw [attach, attachWith, map_pmap]; exact pmap_congr l fun _ _ _ _ => rfl
@@ -86,7 +93,7 @@ theorem attach_map_val (l : List α) (f : α → β) : (l.attach.map fun i => f
 
 @[simp]
 theorem attach_map_subtype_val (l : List α) : l.attach.map Subtype.val = l :=
-  (attach_map_coe _ _).trans l.map_id
+  (attach_map_coe _ _).trans (List.map_id _)
 
 theorem countP_attach (l : List α) (p : α → Bool) : l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
   simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attach_map_subtype_val]
@@ -121,23 +128,14 @@ theorem length_attach (L : List α) : L.attach.length = L.length :=
 theorem pmap_eq_nil {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = [] ↔ l = [] := by
   rw [← length_eq_zero, length_pmap, length_eq_zero]
 
+theorem pmap_ne_nil {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) : xs.pmap f H ≠ [] ↔ xs ≠ [] := by
+  simp
+
 @[simp]
 theorem attach_eq_nil (l : List α) : l.attach = [] ↔ l = [] :=
   pmap_eq_nil
 
-theorem getLast_pmap (p : α → Prop) (f : ∀ a, p a → β) (l : List α)
-    (hl₁ : ∀ a ∈ l, p a) (hl₂ : l ≠ []) :
-    (l.pmap f hl₁).getLast (mt List.pmap_eq_nil.1 hl₂) =
-      f (l.getLast hl₂) (hl₁ _ (List.getLast_mem hl₂)) := by
-  induction l with
-  | nil => apply (hl₂ rfl).elim
-  | cons l_hd l_tl l_ih =>
-    by_cases hl_tl : l_tl = []
-    · simp [hl_tl]
-    · simp only [pmap]
-      rw [getLast_cons, l_ih _ hl_tl]
-      simp only [getLast_cons hl_tl]
-
 theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
     (pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (getElem?_mem H) := by
   induction l generalizing n with
@@ -181,7 +179,22 @@ theorem get_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h :
   simp only [get_eq_getElem]
   simp [getElem_pmap]
 
-theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (l₁ l₂ : List ι)
+@[simp] theorem head?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) : (xs.pmap f H).head? = xs.attach.head?.map fun ⟨a, m⟩ => f a (H a m) := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp at ih
+    simp [head?_pmap, ih]
+
+@[simp] theorem head_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) (h : xs.pmap f H ≠ []) :
+    (xs.pmap f H).head h = f (xs.head (by simpa using h)) (H _ (head_mem _)) := by
+  induction xs with
+  | nil => simp at h
+  | cons x xs ih => simp [head_pmap, ih]
+
+@[simp] theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (l₁ l₂ : List ι)
     (h : ∀ a ∈ l₁ ++ l₂, p a) :
     (l₁ ++ l₂).pmap f h =
       (l₁.pmap f fun a ha => h a (mem_append_left l₂ ha)) ++
@@ -197,3 +210,63 @@ theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ l₂ :
     ((l₁ ++ l₂).pmap f fun a ha => (List.mem_append.1 ha).elim (h₁ a) (h₂ a)) =
       l₁.pmap f h₁ ++ l₂.pmap f h₂ :=
   pmap_append f l₁ l₂ _
+
+@[simp] theorem pmap_reverse {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs.reverse → P a) : xs.reverse.pmap f H = (xs.pmap f (fun a h => H a (by simpa using h))).reverse := by
+  induction xs <;> simp_all
+
+theorem reverse_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) : (xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h)) := by
+  rw [pmap_reverse]
+
+@[simp] theorem attach_append (xs ys : List α) :
+    (xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_of_mem_left ys h⟩) ++
+      ys.attach.map fun ⟨x, h⟩ => ⟨x, mem_append_of_mem_right xs h⟩ := by
+  simp only [attach, attachWith, pmap, map_pmap, pmap_append]
+  congr 1 <;>
+  exact pmap_congr _ fun _ _ _ _ => rfl
+
+@[simp] theorem attach_reverse (xs : List α) : xs.reverse.attach = xs.attach.reverse.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
+  simp only [attach, attachWith, reverse_pmap, map_pmap]
+  apply pmap_congr
+  intros
+  rfl
+
+theorem reverse_attach (xs : List α) : xs.attach.reverse = xs.reverse.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
+  simp only [attach, attachWith, reverse_pmap, map_pmap]
+  apply pmap_congr
+  intros
+  rfl
+
+
+theorem getLast?_attach {xs : List α} :
+    xs.attach.getLast? = match h : xs.getLast? with | none => none | some a => some ⟨a, mem_of_getLast?_eq_some h⟩ := by
+  rw [getLast?_eq_head?_reverse, reverse_attach, head?_map]
+  split <;> rename_i h
+  · simp only [getLast?_eq_none_iff] at h
+    subst h
+    simp
+  · obtain ⟨ys, rfl⟩ := getLast?_eq_some_iff.mp h
+    simp
+
+@[simp] theorem getLast?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) : (xs.pmap f H).getLast? = xs.attach.getLast?.map fun ⟨a, m⟩ => f a (H a m) := by
+  simp only [getLast?_eq_head?_reverse]
+  rw [reverse_pmap, reverse_attach, head?_map, pmap_eq_map_attach, head?_map]
+  simp only [Option.map_map]
+  congr
+
+@[simp] theorem getLast_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) (h : xs.pmap f H ≠ []) :
+    (xs.pmap f H).getLast h = f (xs.getLast (by simpa using h)) (H _ (getLast_mem _)) := by
+  simp only [getLast_eq_iff_getLast_eq_some, getLast?_pmap, Option.map_eq_some', Subtype.exists]
+  refine ⟨xs.getLast (by simpa using h), by simp, ?_⟩
+  simp only [getLast?_attach, and_true]
+  split <;> rename_i h'
+  · simp only [getLast?_eq_none_iff] at h'
+    subst h'
+    simp at h
+  · symm
+    simpa [getLast_eq_iff_getLast_eq_some]
+
+end List

src/Init/Data/List/Basic.lean

@@ -96,7 +96,7 @@ namespace List
 
 /-! ### concat -/
 
-@[simp high] theorem length_concat (as : List α) (a : α) : (concat as a).length = as.length + 1 := by
+theorem length_concat (as : List α) (a : α) : (concat as a).length = as.length + 1 := by
   induction as with
   | nil => rfl
   | cons _ xs ih => simp [concat, ih]
@@ -278,8 +278,9 @@ def getLastD : (as : List α) → (fallback : α) → α
   | [],   a₀ => a₀
   | a::as, _ => getLast (a::as) (fun h => List.noConfusion h)
 
-@[simp] theorem getLastD_nil (a) : @getLastD α [] a = a := rfl
-@[simp] theorem getLastD_cons (a b l) : @getLastD α (b::l) a = getLastD l b := by cases l <;> rfl
+-- These aren't `simp` lemmas since we always simplify `getLastD` in terms of `getLast?`.
+theorem getLastD_nil (a) : @getLastD α [] a = a := rfl
+theorem getLastD_cons (a b l) : @getLastD α (b::l) a = getLastD l b := by cases l <;> rfl
 
 /-! ## Head and tail -/
 
@@ -688,7 +689,7 @@ inductive Mem (a : α) : List α → Prop
   | tail (b : α) {as : List α} : Mem a as → Mem a (b::as)
 
 instance : Membership α (List α) where
-  mem := Mem
+  mem l a := Mem a l
 
 theorem mem_of_elem_eq_true [BEq α] [LawfulBEq α] {a : α} {as : List α} : elem a as = true → a ∈ as := by
   match as with
@@ -962,6 +963,26 @@ def IsInfix (l₁ : List α) (l₂ : List α) : Prop := Exists fun s => Exists f
 
 @[inherit_doc] infixl:50 " <:+: " => IsInfix
 
+/-! ### splitAt -/
+
+/--
+Split a list at an index.
+```
+splitAt 2 [a, b, c] = ([a, b], [c])
+```
+-/
+def splitAt (n : Nat) (l : List α) : List α × List α := go l n [] where
+  /--
+  Auxiliary for `splitAt`:
+  `splitAt.go l xs n acc = (acc.reverse ++ take n xs, drop n xs)` if `n < xs.length`,
+  and `(l, [])` otherwise.
+  -/
+  go : List α → Nat → List α → List α × List α
+  | [], _, _ => (l, []) -- This branch ensures the pointer equality of the result with the input
+                        -- without any runtime branching cost.
+  | x :: xs, n+1, acc => go xs n (x :: acc)
+  | xs, _, acc => (acc.reverse, xs)
+
 /-! ### rotateLeft -/
 
 /--
@@ -1223,6 +1244,36 @@ theorem lookup_cons [BEq α] {k : α} :
     ((k,b)::es).lookup a = match a == k with | true => some b | false => es.lookup a :=
   rfl
 
+/-! ## Permutations -/
+
+/-! ### Perm -/
+
+/--
+`Perm l₁ l₂` or `l₁ ~ l₂` asserts that `l₁` and `l₂` are permutations
+of each other. This is defined by induction using pairwise swaps.
+-/
+inductive Perm : List α → List α → Prop
+  /-- `[] ~ []` -/
+  | nil : Perm [] []
+  /-- `l₁ ~ l₂ → x::l₁ ~ x::l₂` -/
+  | cons (x : α) {l₁ l₂ : List α} : Perm l₁ l₂ → Perm (x :: l₁) (x :: l₂)
+  /-- `x::y::l ~ y::x::l` -/
+  | swap (x y : α) (l : List α) : Perm (y :: x :: l) (x :: y :: l)
+  /-- `Perm` is transitive. -/
+  | trans {l₁ l₂ l₃ : List α} : Perm l₁ l₂ → Perm l₂ l₃ → Perm l₁ l₃
+
+@[inherit_doc] scoped infixl:50 " ~ " => Perm
+
+/-! ### isPerm -/
+
+/--
+`O(|l₁| * |l₂|)`. Computes whether `l₁` is a permutation of `l₂`. See `isPerm_iff` for a
+characterization in terms of `List.Perm`.
+-/
+def isPerm [BEq α] : List α → List α → Bool
+  | [], l₂ => l₂.isEmpty
+  | a :: l₁, l₂ => l₂.contains a && l₁.isPerm (l₂.erase a)
+
 /-! ## Logical operations -/
 
 /-! ### any -/

src/Init/Data/List/BasicAux.lean

@@ -192,7 +192,7 @@ macro "sizeOf_list_dec" : tactic =>
   `(tactic| first
     | with_reducible apply sizeOf_lt_of_mem; assumption; done
     | with_reducible
-        apply Nat.lt_trans (sizeOf_lt_of_mem ?h)
+        apply Nat.lt_of_lt_of_le (sizeOf_lt_of_mem ?h)
         case' h => assumption
       simp_arith)
 
@@ -222,7 +222,7 @@ theorem append_cancel_right {as bs cs : List α} (h : as ++ bs = cs ++ bs) : as
   next => apply append_cancel_right
   next => intro h; simp [h]
 
-@[simp] theorem sizeOf_get [SizeOf α] (as : List α) (i : Fin as.length) : sizeOf (as.get i) < sizeOf as := by
+theorem sizeOf_get [SizeOf α] (as : List α) (i : Fin as.length) : sizeOf (as.get i) < sizeOf as := by
   match as, i with
   | a::as, ⟨0, _⟩  => simp_arith [get]
   | a::as, ⟨i+1, h⟩ =>

src/Init/Data/List/Count.lean

@@ -47,11 +47,11 @@ theorem length_eq_countP_add_countP (l) : length l = countP p l + countP (fun a
     if h : p x then
       rw [countP_cons_of_pos _ _ h, countP_cons_of_neg _ _ _, length, ih]
       · rw [Nat.add_assoc, Nat.add_comm _ 1, Nat.add_assoc]
-      · simp only [h, not_true_eq_false, decide_False, not_false_eq_true]
+      · simp [h]
     else
       rw [countP_cons_of_pos (fun a => ¬p a) _ _, countP_cons_of_neg _ _ h, length, ih]
       · rfl
-      · simp only [h, not_false_eq_true, decide_True]
+      · simp [h]
 
 theorem countP_eq_length_filter (l) : countP p l = length (filter p l) := by
   induction l with
@@ -81,6 +81,10 @@ theorem Sublist.countP_le (s : l₁ <+ l₂) : countP p l₁ ≤ countP p l₂ :
   simp only [countP_eq_length_filter]
   apply s.filter _ |>.length_le
 
+theorem IsPrefix.countP_le (s : l₁ <+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
+theorem IsSuffix.countP_le (s : l₁ <:+ l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
+theorem IsInfix.countP_le (s : l₁ <:+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
+
 theorem countP_filter (l : List α) :
     countP p (filter q l) = countP (fun a => p a ∧ q a) l := by
   simp only [countP_eq_length_filter, filter_filter]
@@ -140,6 +144,10 @@ theorem count_le_length (a : α) (l : List α) : count a l ≤ l.length := count
 
 theorem Sublist.count_le (h : l₁ <+ l₂) (a : α) : count a l₁ ≤ count a l₂ := h.countP_le _
 
+theorem IsPrefix.count_le (h : l₁ <+: l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
+theorem IsSuffix.count_le (h : l₁ <:+ l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
+theorem IsInfix.count_le (h : l₁ <:+: l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
+
 theorem count_le_count_cons (a b : α) (l : List α) : count a l ≤ count a (b :: l) :=
   (sublist_cons_self _ _).count_le _
 
@@ -226,7 +234,7 @@ theorem count_erase (a b : α) :
       rw [if_pos hc_beq, hc, count_cons, Nat.add_sub_cancel]
     else
       have hc_beq := beq_false_of_ne hc
-      simp only [hc_beq, if_false, count_cons, count_cons, count_erase a b l]
+      simp only [hc_beq, if_false, count_cons, count_cons, count_erase a b l, reduceCtorEq]
       if ha : b = a then
         rw [ha, eq_comm] at hc
         rw [if_pos ((beq_iff_eq _ _).2 ha), if_neg (by simpa using Ne.symm hc), Nat.add_zero, Nat.add_zero]

src/Init/Data/List/Erase.lean

@@ -33,6 +33,25 @@ theorem eraseP_of_forall_not {l : List α} (h : ∀ a, a ∈ l → ¬p a) : l.er
   | nil => rfl
   | cons _ _ ih => simp [h _ (.head ..), ih (forall_mem_cons.1 h).2]
 
+@[simp] theorem eraseP_eq_nil (xs : List α) (p : α → Bool) : xs.eraseP p = [] ↔ xs = [] ∨ ∃ x, p x ∧ xs = [x] := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [eraseP_cons, cond_eq_if]
+    split <;> rename_i h
+    · simp only [reduceCtorEq, cons.injEq, false_or]
+      constructor
+      · rintro rfl
+        simpa
+      · rintro ⟨_, _, rfl, rfl⟩
+        rfl
+    · simp only [reduceCtorEq, cons.injEq, false_or, false_iff, not_exists, not_and]
+      rintro x h' rfl
+      simp_all
+
+theorem eraseP_ne_nil (xs : List α) (p : α → Bool) : xs.eraseP p ≠ [] ↔ xs ≠ [] ∧ ∀ x, p x → xs ≠ [x] := by
+  simp
+
 theorem exists_of_eraseP : ∀ {l : List α} {a} (al : a ∈ l) (pa : p a),
     ∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.eraseP p = l₁ ++ l₂
   | b :: l, a, al, pa =>
@@ -159,6 +178,23 @@ theorem eraseP_append (l₁ l₂ : List α) :
     rw [eraseP_append_right _]
     simp_all
 
+theorem eraseP_replicate (n : Nat) (a : α) (p : α → Bool) :
+    (replicate n a).eraseP p = if p a then replicate (n - 1) a else replicate n a := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [replicate_succ, eraseP_cons]
+    split <;> simp [*]
+
+protected theorem IsPrefix.eraseP (h : l₁ <+: l₂) : l₁.eraseP p <+: l₂.eraseP p := by
+  rw [IsPrefix] at h
+  obtain ⟨t, rfl⟩ := h
+  rw [eraseP_append]
+  split
+  · exact prefix_append (eraseP p l₁) t
+  · rw [eraseP_of_forall_not (by simp_all)]
+    exact prefix_append l₁ (eraseP p t)
+
 theorem eraseP_eq_iff {p} {l : List α} :
     l.eraseP p = l' ↔
       ((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
@@ -204,8 +240,11 @@ theorem eraseP_eq_iff {p} {l : List α} :
     (replicate n a).eraseP p = replicate n a := by
   rw [eraseP_of_forall_not (by simp_all)]
 
+theorem Pairwise.eraseP (q) : Pairwise p l → Pairwise p (l.eraseP q) :=
+  Pairwise.sublist <| eraseP_sublist _
+
 theorem Nodup.eraseP (p) : Nodup l → Nodup (l.eraseP p) :=
-  Nodup.sublist <| eraseP_sublist _
+  Pairwise.eraseP p
 
 theorem eraseP_comm {l : List α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
     (l.eraseP p).eraseP q = (l.eraseP q).eraseP p := by
@@ -221,6 +260,12 @@ theorem eraseP_comm {l : List α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
       · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
       · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
 
+theorem head_eraseP_mem (xs : List α) (p : α → Bool) (h) : (xs.eraseP p).head h ∈ xs :=
+  (eraseP_sublist xs).head_mem h
+
+theorem getLast_eraseP_mem (xs : List α) (p : α → Bool) (h) : (xs.eraseP p).getLast h ∈ xs :=
+  (eraseP_sublist xs).getLast_mem h
+
 /-! ### erase -/
 section erase
 variable [BEq α]
@@ -249,6 +294,16 @@ theorem erase_eq_eraseP [LawfulBEq α] (a : α) : ∀ l : List α,  l.erase a =
   | b :: l => by
     if h : a = b then simp [h] else simp [h, Ne.symm h, erase_eq_eraseP a l]
 
+@[simp] theorem erase_eq_nil [LawfulBEq α] (xs : List α) (a : α) :
+    xs.erase a = [] ↔ xs = [] ∨ xs = [a] := by
+  rw [erase_eq_eraseP]
+  simp
+
+theorem erase_ne_nil [LawfulBEq α] (xs : List α) (a : α) :
+    xs.erase a ≠ [] ↔ xs ≠ [] ∧ xs ≠ [a] := by
+  rw [erase_eq_eraseP]
+  simp
+
 theorem exists_erase_eq [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
     ∃ l₁ l₂, a ∉ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l.erase a = l₁ ++ l₂ := by
   let ⟨_, l₁, l₂, h₁, e, h₂, h₃⟩ := exists_of_eraseP h (beq_self_eq_true _)
@@ -271,6 +326,9 @@ theorem erase_subset (a : α) (l : List α) : l.erase a ⊆ l := (erase_sublist
 theorem Sublist.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₁.erase a <+ l₂.erase a := by
   simp only [erase_eq_eraseP']; exact h.eraseP
 
+theorem IsPrefix.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+: l₂) : l₁.erase a <+: l₂.erase a := by
+  simp only [erase_eq_eraseP']; exact h.eraseP
+
 theorem length_erase_le (a : α) (l : List α) : (l.erase a).length ≤ l.length :=
   (erase_sublist a l).length_le
 
@@ -282,7 +340,7 @@ theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈
 
 @[simp] theorem erase_eq_self_iff [LawfulBEq α] {l : List α} : l.erase a = l ↔ a ∉ l := by
   rw [erase_eq_eraseP', eraseP_eq_self_iff]
-  simp
+  simp [forall_mem_ne']
 
 theorem erase_filter [LawfulBEq α] (f : α → Bool) (l : List α) :
     (filter f l).erase a = filter f (l.erase a) := by
@@ -315,6 +373,11 @@ theorem erase_append [LawfulBEq α] {a : α} {l₁ l₂ : List α} :
     (l₁ ++ l₂).erase a = if a ∈ l₁ then l₁.erase a ++ l₂ else l₁ ++ l₂.erase a := by
   simp [erase_eq_eraseP, eraseP_append]
 
+theorem erase_replicate [LawfulBEq α] (n : Nat) (a b : α) :
+    (replicate n a).erase b = if b == a then replicate (n - 1) a else replicate n a := by
+  rw [erase_eq_eraseP]
+  simp [eraseP_replicate]
+
 theorem erase_comm [LawfulBEq α] (a b : α) (l : List α) :
     (l.erase a).erase b = (l.erase b).erase a := by
   if ab : a == b then rw [eq_of_beq ab] else ?_
@@ -354,7 +417,10 @@ theorem erase_eq_iff [LawfulBEq α] {a : α} {l : List α} :
   rw [erase_of_not_mem]
   simp_all
 
-theorem Nodup.erase_eq_filter [BEq α] [LawfulBEq α] {l} (d : Nodup l) (a : α) : l.erase a = l.filter (· != a) := by
+theorem Pairwise.erase [LawfulBEq α] {l : List α} (a) : Pairwise p l → Pairwise p (l.erase a) :=
+  Pairwise.sublist <| erase_sublist _ _
+
+theorem Nodup.erase_eq_filter [LawfulBEq α] {l} (d : Nodup l) (a : α) : l.erase a = l.filter (· != a) := by
   induction d with
   | nil => rfl
   | cons m _n ih =>
@@ -367,14 +433,20 @@ theorem Nodup.erase_eq_filter [BEq α] [LawfulBEq α] {l} (d : Nodup l) (a : α)
       simpa [@eq_comm α] using m
     · simp [beq_false_of_ne h, ih, h]
 
-theorem Nodup.mem_erase_iff [BEq α] [LawfulBEq α] {a : α} (d : Nodup l) : a ∈ l.erase b ↔ a ≠ b ∧ a ∈ l := by
+theorem Nodup.mem_erase_iff [LawfulBEq α] {a : α} (d : Nodup l) : a ∈ l.erase b ↔ a ≠ b ∧ a ∈ l := by
   rw [Nodup.erase_eq_filter d, mem_filter, and_comm, bne_iff_ne]
 
-theorem Nodup.not_mem_erase [BEq α] [LawfulBEq α] {a : α} (h : Nodup l) : a ∉ l.erase a := fun H => by
+theorem Nodup.not_mem_erase [LawfulBEq α] {a : α} (h : Nodup l) : a ∉ l.erase a := fun H => by
   simpa using ((Nodup.mem_erase_iff h).mp H).left
 
-theorem Nodup.erase [BEq α] [LawfulBEq α] (a : α) : Nodup l → Nodup (l.erase a) :=
-  Nodup.sublist <| erase_sublist _ _
+theorem Nodup.erase [LawfulBEq α] (a : α) : Nodup l → Nodup (l.erase a) :=
+  Pairwise.erase a
+
+theorem head_erase_mem (xs : List α) (a : α) (h) : (xs.erase a).head h ∈ xs :=
+  (erase_sublist a xs).head_mem h
+
+theorem getLast_erase_mem (xs : List α) (a : α) (h) : (xs.erase a).getLast h ∈ xs :=
+  (erase_sublist a xs).getLast_mem h
 
 end erase
 
@@ -396,11 +468,26 @@ theorem eraseIdx_eq_take_drop_succ :
   | a::l, 0 => by simp
   | a::l, i + 1 => by simp [eraseIdx_eq_take_drop_succ l i]
 
+@[simp] theorem eraseIdx_eq_nil {l : List α} {i : Nat} : eraseIdx l i = [] ↔ l = [] ∨ (length l = 1 ∧ i = 0) := by
+  match l, i with
+  | [], _
+  | a::l, 0
+  | a::l, i + 1 => simp [Nat.succ_inj']
+
+theorem eraseIdx_ne_nil {l : List α} {i : Nat} : eraseIdx l i ≠ [] ↔ 2 ≤ l.length ∨ (l.length = 1 ∧ i ≠ 0) := by
+  match l with
+  | []
+  | [a]
+  | a::b::l => simp [Nat.succ_inj']
+
 theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
   | [], _ => by simp
   | a::l, 0 => by simp
   | a::l, k + 1 => by simp [eraseIdx_sublist l k]
 
+theorem mem_of_mem_eraseIdx {l : List α} {i : Nat} {a : α} (h : a ∈ l.eraseIdx i) : a ∈ l :=
+  (eraseIdx_sublist _ _).mem h
+
 theorem eraseIdx_subset (l : List α) (k : Nat) : eraseIdx l k ⊆ l := (eraseIdx_sublist l k).subset
 
 @[simp]
@@ -430,6 +517,23 @@ theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤
     | zero => simp_all
     | succ k => simp_all [eraseIdx_cons_succ, Nat.succ_sub_succ]
 
+theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
+    (replicate n a).eraseIdx k = if k < n then replicate (n - 1) a else replicate n a := by
+  split <;> rename_i h
+  · rw [eq_replicate, length_eraseIdx (by simpa using h)]
+    simp only [length_replicate, true_and]
+    intro b m
+    replace m := mem_of_mem_eraseIdx m
+    simp only [mem_replicate] at m
+    exact m.2
+  · rw [eraseIdx_of_length_le (by simpa using h)]
+
+theorem Pairwise.eraseIdx {l : List α} (k) : Pairwise p l → Pairwise p (l.eraseIdx k) :=
+  Pairwise.sublist <| eraseIdx_sublist _ _
+
+theorem Nodup.eraseIdx {l : List α} (k) : Nodup l → Nodup (l.eraseIdx k) :=
+  Pairwise.eraseIdx k
+
 protected theorem IsPrefix.eraseIdx {l l' : List α} (h : l <+: l') (k : Nat) :
     eraseIdx l k <+: eraseIdx l' k := by
   rcases h with ⟨t, rfl⟩

src/Init/Data/List/Find.lean

@@ -1,71 +1,22 @@
 /-
 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
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
+  Kim Morrison, Jannis Limperg
 -/
 prelude
 import Init.Data.List.Lemmas
+import Init.Data.List.Sublist
+import Init.Data.List.Range
 
 /-!
-# Lemmas about `List.find?`, `List.findSome?`, `List.findIdx`, `List.findIdx?`, and `List.indexOf`.
+# Lemmas about `List.findSome?`, `List.find?`, `List.findIdx`, `List.findIdx?`, and `List.indexOf`.
 -/
 
 namespace List
 
 open Nat
 
-/-! ### find? -/
-
-@[simp] theorem find?_cons_of_pos (l) (h : p a) : find? p (a :: l) = some a := by
-  simp [find?, h]
-
-@[simp] theorem find?_cons_of_neg (l) (h : ¬p a) : find? p (a :: l) = find? p l := by
-  simp [find?, h]
-
-@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
-  induction l <;> simp [find?_cons]; split <;> simp [*]
-
-theorem find?_some : ∀ {l}, find? p l = some a → p a
-  | b :: l, H => by
-    by_cases h : p b <;> simp [find?, h] at H
-    · exact H ▸ h
-    · exact find?_some H
-
-@[simp] theorem mem_of_find?_eq_some : ∀ {l}, find? p l = some a → a ∈ l
-  | b :: l, H => by
-    by_cases h : p b <;> simp [find?, h] at H
-    · exact H ▸ .head _
-    · exact .tail _ (mem_of_find?_eq_some H)
-
-@[simp] theorem find?_map (f : β → α) (l : List β) : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
-  induction l with
-  | nil => simp
-  | cons x xs ih =>
-    simp only [map_cons, find?]
-    by_cases h : p (f x) <;> simp [h, ih]
-
-theorem find?_replicate : find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
-  cases n
-  · simp
-  · by_cases p a <;> simp_all [replicate_succ]
-
-@[simp] theorem find?_replicate_of_length_pos (h : 0 < n) : find? p (replicate n a) = if p a then some a else none := by
-  simp [find?_replicate, Nat.ne_of_gt h]
-
-@[simp] theorem find?_replicate_of_pos (h : p a) : find? p (replicate n a) = if n = 0 then none else some a := by
-  simp [find?_replicate, h]
-
-@[simp] theorem find?_replicate_of_neg (h : ¬ p a) : find? p (replicate n a) = none := by
-  simp [find?_replicate, h]
-
-theorem find?_isSome_of_sublist {l₁ l₂ : List α} (h : l₁ <+ l₂) : (l₁.find? p).isSome → (l₂.find? p).isSome := by
-  induction h with
-  | slnil => simp
-  | cons a h ih
-  | cons₂ a h ih =>
-    simp only [find?]
-    split <;> simp_all
-
 /-! ### findSome? -/
 
 @[simp] theorem findSome?_cons_of_isSome (l) (h : (f a).isSome) : findSome? f (a :: l) = f a := by
@@ -84,17 +35,70 @@ theorem exists_of_findSome?_eq_some {l : List α} {f : α → Option β} (w : l.
     simp_all only [findSome?_cons, mem_cons, exists_eq_or_imp]
     split at w <;> simp_all
 
-@[simp] theorem findSome?_map (f : β → γ) (l : List β) : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
+@[simp] theorem findSome?_eq_none : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
+  induction l <;> simp [findSome?_cons]; split <;> simp [*]
+
+@[simp] theorem findSome?_isSome_iff (f : α → Option β) (l : List α) :
+    (l.findSome? f).isSome ↔ ∃ x, x ∈ l ∧ (f x).isSome := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findSome?_cons]
+    split <;> simp_all
+
+@[simp] theorem findSome?_guard (l : List α) : findSome? (Option.guard fun x => p x) l = find? p l := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp [guard, findSome?, find?]
+    split <;> rename_i h
+    · simp only [Option.guard_eq_some] at h
+      obtain ⟨rfl, h⟩ := h
+      simp [h]
+    · simp only [Option.guard_eq_none] at h
+      simp [ih, h]
+
+@[simp] theorem filterMap_head? (f : α → Option β) (l : List α) : (l.filterMap f).head? = l.findSome? f := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [filterMap_cons, findSome?_cons]
+    split <;> simp [*]
+
+@[simp] theorem filterMap_head (f : α → Option β) (l : List α) (h) :
+    (l.filterMap f).head h = (l.findSome? f).get (by simp_all [Option.isSome_iff_ne_none])  := by
+  simp [head_eq_iff_head?_eq_some]
+
+@[simp] theorem filterMap_getLast? (f : α → Option β) (l : List α) : (l.filterMap f).getLast? = l.reverse.findSome? f := by
+  rw [getLast?_eq_head?_reverse]
+  simp [← filterMap_reverse]
+
+@[simp] theorem filterMap_getLast (f : α → Option β) (l : List α) (h) :
+    (l.filterMap f).getLast h = (l.reverse.findSome? f).get (by simp_all [Option.isSome_iff_ne_none]) := by
+  simp [getLast_eq_iff_getLast_eq_some]
+
+@[simp] theorem map_findSome? (f : α → Option β) (g : β → γ) (l : List α) :
+    (l.findSome? f).map g = l.findSome? (Option.map g ∘ f) := by
+  induction l <;> simp [findSome?_cons]; split <;> simp [*]
+
+theorem findSome?_map (f : β → γ) (l : List β) : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
   induction l with
   | nil => simp
   | cons x xs ih =>
     simp only [map_cons, findSome?]
     split <;> simp_all
 
+theorem findSome?_append {l₁ l₂ : List α} : (l₁ ++ l₂).findSome? f = (l₁.findSome? f).or (l₂.findSome? f) := by
+  induction l₁ with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [cons_append, findSome?]
+    split <;> simp_all
+
 theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
-  induction n with
+  cases n with
   | zero => simp
-  | succ n ih =>
+  | succ n =>
     simp only [replicate_succ, findSome?_cons]
     split <;> simp_all
 
@@ -102,22 +106,304 @@ theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none e
   simp [findSome?_replicate, Nat.ne_of_gt h]
 
 -- Argument is unused, but used to decide whether `simp` should unfold.
-@[simp] theorem find?_replicate_of_isSome (_ : (f a).isSome) : findSome? f (replicate n a) = if n = 0 then none else f a := by
+@[simp] theorem findSome?_replicate_of_isSome (_ : (f a).isSome) : findSome? f (replicate n a) = if n = 0 then none else f a := by
   simp [findSome?_replicate]
 
-@[simp] theorem find?_replicate_of_isNone (h : (f a).isNone) : findSome? f (replicate n a) = none := by
+@[simp] theorem findSome?_replicate_of_isNone (h : (f a).isNone) : findSome? f (replicate n a) = none := by
   rw [Option.isNone_iff_eq_none] at h
   simp [findSome?_replicate, h]
 
-theorem findSome?_isSome_of_sublist {l₁ l₂ : List α} (h : l₁ <+ l₂) :
+theorem Sublist.findSome?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
     (l₁.findSome? f).isSome → (l₂.findSome? f).isSome := by
   induction h with
   | slnil => simp
   | cons a h ih
   | cons₂ a h ih =>
     simp only [findSome?]
+    split
+    · simp_all
+    · exact ih
+
+theorem Sublist.findSome?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) :
+    l₂.findSome? f = none → l₁.findSome? f = none := by
+  simp only [List.findSome?_eq_none, Bool.not_eq_true]
+  exact fun w x m => w x (Sublist.mem m h)
+
+theorem IsPrefix.findSome?_eq_some {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <+: l₂) :
+    List.findSome? f l₁ = some b → List.findSome? f l₂ = some b := by
+  rw [IsPrefix] at h
+  obtain ⟨t, rfl⟩ := h
+  simp (config := {contextual := true}) [findSome?_append]
+
+theorem IsPrefix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <+: l₂) :
+    List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
+  h.sublist.findSome?_eq_none
+theorem IsSuffix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <:+ l₂) :
+    List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
+  h.sublist.findSome?_eq_none
+theorem IsInfix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <:+: l₂) :
+    List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
+  h.sublist.findSome?_eq_none
+
+/-! ### find? -/
+
+@[simp] theorem find?_singleton (a : α) (p : α → Bool) : [a].find? p = if p a then some a else none := by
+  simp only [find?]
+  split <;> simp_all
+
+@[simp] theorem find?_cons_of_pos (l) (h : p a) : find? p (a :: l) = some a := by
+  simp [find?, h]
+
+@[simp] theorem find?_cons_of_neg (l) (h : ¬p a) : find? p (a :: l) = find? p l := by
+  simp [find?, h]
+
+@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
+  induction l <;> simp [find?_cons]; split <;> simp [*]
+
+theorem find?_eq_some : xs.find? p = some b ↔ p b ∧ ∃ as bs, xs = as ++ b :: bs ∧ ∀ a ∈ as, !p a := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [find?_cons, exists_and_right]
+    split <;> rename_i h
+    · simp only [Option.some.injEq]
+      constructor
+      · rintro rfl
+        exact ⟨h, [], ⟨xs, rfl⟩, by simp⟩
+      · rintro ⟨-, ⟨as, ⟨⟨bs, h₁⟩, h₂⟩⟩⟩
+        cases as with
+        | nil => simp_all
+        | cons a as =>
+          specialize h₂ a (mem_cons_self _ _)
+          simp only [cons_append] at h₁
+          obtain ⟨rfl, -⟩ := h₁
+          simp_all
+    · simp only [ih, Bool.not_eq_true', exists_and_right, and_congr_right_iff]
+      intro pb
+      constructor
+      · rintro ⟨as, ⟨⟨bs, rfl⟩, h₁⟩⟩
+        refine ⟨x :: as, ⟨⟨bs, rfl⟩, ?_⟩⟩
+        intro a m
+        simp at m
+        obtain (rfl|m) := m
+        · exact h
+        · exact h₁ a m
+      · rintro ⟨as, ⟨bs, h₁⟩, h₂⟩
+        cases as with
+        | nil => simp_all
+        | cons a as =>
+          refine ⟨as, ⟨⟨bs, ?_⟩, fun a m => h₂ a (mem_cons_of_mem _ m)⟩⟩
+          cases h₁
+          simp
+
+@[simp]
+theorem find?_cons_eq_some : (a :: xs).find? p = some b ↔ (p a ∧ a = b) ∨ (!p a ∧ xs.find? p = some b) := by
+  rw [find?_cons]
+  split <;> simp_all
+
+@[simp] theorem find?_isSome (xs : List α) (p : α → Bool) : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [find?_cons, mem_cons, exists_eq_or_imp]
     split <;> simp_all
 
+theorem find?_some : ∀ {l}, find? p l = some a → p a
+  | b :: l, H => by
+    by_cases h : p b <;> simp [find?, h] at H
+    · exact H ▸ h
+    · exact find?_some H
+
+theorem mem_of_find?_eq_some : ∀ {l}, find? p l = some a → a ∈ l
+  | b :: l, H => by
+    by_cases h : p b <;> simp [find?, h] at H
+    · exact H ▸ .head _
+    · exact .tail _ (mem_of_find?_eq_some H)
+
+@[simp] theorem get_find?_mem (xs : List α) (p : α → Bool) (h) : (xs.find? p).get h ∈ xs := by
+  induction xs with
+  | nil => simp at h
+  | cons x xs ih =>
+    simp only [find?_cons]
+    by_cases h : p x
+    · simp [h]
+    · simp only [h]
+      right
+      apply ih
+
+@[simp] theorem find?_filter (xs : List α) (p : α → Bool) (q : α → Bool) :
+    (xs.filter p).find? q = xs.find? (fun a => p a ∧ q a) := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [filter_cons]
+    split <;>
+    · simp only [find?_cons]
+      split <;> simp_all
+
+@[simp] theorem filter_head? (p : α → Bool) (l : List α) : (l.filter p).head? = l.find? p := by
+  rw [← filterMap_eq_filter, filterMap_head?, findSome?_guard]
+
+@[simp] theorem filter_head (p : α → Bool) (l : List α) (h) :
+    (l.filter p).head h = (l.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
+  simp [head_eq_iff_head?_eq_some]
+
+@[simp] theorem filter_getLast? (p : α → Bool) (l : List α) : (l.filter p).getLast? = l.reverse.find? p := by
+  rw [getLast?_eq_head?_reverse]
+  simp [← filter_reverse]
+
+@[simp] theorem filter_getLast (p : α → Bool) (l : List α) (h) :
+    (l.filter p).getLast h = (l.reverse.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
+  simp [getLast_eq_iff_getLast_eq_some]
+
+@[simp] theorem find?_filterMap (xs : List α) (f : α → Option β) (p : β → Bool) :
+    (xs.filterMap f).find? p = (xs.find? (fun a => (f a).any p)).bind f := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [filterMap_cons]
+    split <;>
+    · simp only [find?_cons]
+      split <;> simp_all
+
+@[simp] theorem find?_map (f : β → α) (l : List β) : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [map_cons, find?]
+    by_cases h : p (f x) <;> simp [h, ih]
+
+@[simp] theorem find?_append {l₁ l₂ : List α} : (l₁ ++ l₂).find? p = (l₁.find? p).or (l₂.find? p) := by
+  induction l₁ with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [cons_append, find?]
+    by_cases h : p x <;> simp [h, ih]
+
+@[simp] theorem find?_join (xs : List (List α)) (p : α → Bool) :
+    xs.join.find? p = xs.findSome? (·.find? p) := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [join_cons, find?_append, findSome?_cons, ih]
+    split <;> simp [*]
+
+theorem find?_join_eq_none (xs : List (List α)) (p : α → Bool) :
+    xs.join.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
+  simp
+
+/--
+If `find? p` returns `some a` from `xs.join`, then `p a` holds, and
+some list in `xs` contains `a`, and no earlier element of that list satisfies `p`.
+Moreover, no earlier list in `xs` has an element satisfying `p`.
+-/
+theorem find?_join_eq_some (xs : List (List α)) (p : α → Bool) (a : α) :
+    xs.join.find? p = some a ↔
+      p a ∧ ∃ as ys zs bs, xs = as ++ (ys ++ a :: zs) :: bs ∧
+        (∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
+  rw [find?_eq_some]
+  constructor
+  · rintro ⟨h, ⟨ys, zs, h₁, h₂⟩⟩
+    refine ⟨h, ?_⟩
+    rw [join_eq_append] at h₁
+    obtain (⟨as, bs, rfl, rfl, h₁⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, h₁⟩) := h₁
+    · replace h₁ := h₁.symm
+      rw [join_eq_cons] at h₁
+      obtain ⟨bs, cs, ds, rfl, h₁, rfl⟩ := h₁
+      refine ⟨as ++ bs, [], cs, ds, by simp, ?_⟩
+      simp
+      rintro a (ma | mb) x m
+      · simpa using h₂ x (by simpa using ⟨a, ma, m⟩)
+      · specialize h₁ _ mb
+        simp_all
+    · simp [h₁]
+      refine ⟨as, bs, ?_⟩
+      refine ⟨?_, ?_, ?_⟩
+      · simp_all
+      · intro l ml a m
+        simpa using h₂ a (by simpa using .inl ⟨l, ml, m⟩)
+      · intro x m
+        simpa using h₂ x (by simpa using .inr m)
+  · rintro ⟨h, ⟨as, ys, zs, bs, rfl, h₁, h₂⟩⟩
+    refine ⟨h, as.join ++ ys, zs ++ bs.join, by simp, ?_⟩
+    intro a m
+    simp at m
+    obtain ⟨l, ml, m⟩ | m := m
+    · exact h₁ l ml a m
+    · exact h₂ a m
+
+@[simp] theorem find?_bind (xs : List α) (f : α → List β) (p : β → Bool) :
+    (xs.bind f).find? p = xs.findSome? (fun x => (f x).find? p) := by
+  simp [bind_def, findSome?_map]; rfl
+
+theorem find?_bind_eq_none (xs : List α) (f : α → List β) (p : β → Bool) :
+    (xs.bind f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
+  simp
+
+theorem find?_replicate : find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
+  cases n
+  · simp
+  · by_cases p a <;> simp_all [replicate_succ]
+
+@[simp] theorem find?_replicate_of_length_pos (h : 0 < n) : find? p (replicate n a) = if p a then some a else none := by
+  simp [find?_replicate, Nat.ne_of_gt h]
+
+@[simp] theorem find?_replicate_of_pos (h : p a) : find? p (replicate n a) = if n = 0 then none else some a := by
+  simp [find?_replicate, h]
+
+@[simp] theorem find?_replicate_of_neg (h : ¬ p a) : find? p (replicate n a) = none := by
+  simp [find?_replicate, h]
+
+-- This isn't a `@[simp]` lemma since there is already a lemma for `l.find? p = none` for any `l`.
+theorem find?_replicate_eq_none (n : Nat) (a : α) (p : α → Bool) :
+    (replicate n a).find? p = none ↔ n = 0 ∨ !p a := by
+  simp [Classical.or_iff_not_imp_left]
+
+@[simp] theorem find?_replicate_eq_some (n : Nat) (a b : α) (p : α → Bool) :
+    (replicate n a).find? p = some b ↔ n ≠ 0 ∧ p a ∧ a = b := by
+  cases n <;> simp
+
+@[simp] theorem get_find?_replicate (n : Nat) (a : α) (p : α → Bool) (h) : ((replicate n a).find? p).get h = a := by
+  cases n with
+  | zero => simp at h
+  | succ n => simp
+
+theorem Sublist.find?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) : (l₁.find? p).isSome → (l₂.find? p).isSome := by
+  induction h with
+  | slnil => simp
+  | cons a h ih
+  | cons₂ a h ih =>
+    simp only [find?]
+    split
+    · simp
+    · simpa using ih
+
+theorem Sublist.find?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₂.find? p = none → l₁.find? p = none := by
+  simp only [List.find?_eq_none, Bool.not_eq_true]
+  exact fun w x m => w x (Sublist.mem m h)
+
+theorem IsPrefix.find?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
+    List.find? p l₁ = some b → List.find? p l₂ = some b := by
+  rw [IsPrefix] at h
+  obtain ⟨t, rfl⟩ := h
+  simp (config := {contextual := true}) [find?_append]
+
+theorem IsPrefix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
+    List.find? p l₂ = none → List.find? p l₁ = none :=
+  h.sublist.find?_eq_none
+theorem IsSuffix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+ l₂) :
+    List.find? p l₂ = none → List.find? p l₁ = none :=
+  h.sublist.find?_eq_none
+theorem IsInfix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+: l₂) :
+    List.find? p l₂ = none → List.find? p l₁ = none :=
+  h.sublist.find?_eq_none
+
+theorem find?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) (p : β → Bool) :
+    (xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
+  simp only [pmap_eq_map_attach, find?_map]
+  rfl
+
 /-! ### findIdx -/
 
 theorem findIdx_cons (p : α → Bool) (b : α) (l : List α) :
@@ -135,14 +421,23 @@ where
       cases p head <;> simp only [cond_false, cond_true]
       exact findIdx_go_succ p tail (n + 1)
 
-theorem findIdx_of_get?_eq_some {xs : List α} (w : xs.get? (xs.findIdx p) = some y) : p y := by
+theorem findIdx_of_getElem?_eq_some {xs : List α} (w : xs[xs.findIdx p]? = some y) : p y := by
   induction xs with
   | nil => simp_all
   | cons x xs ih => by_cases h : p x <;> simp_all [findIdx_cons]
 
+theorem findIdx_getElem {xs : List α} {w : xs.findIdx p < xs.length} :
+    p xs[xs.findIdx p] :=
+  xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)
+
+@[deprecated findIdx_of_getElem?_eq_some (since := "2024-08-12")]
+theorem findIdx_of_get?_eq_some {xs : List α} (w : xs.get? (xs.findIdx p) = some y) : p y :=
+  findIdx_of_getElem?_eq_some (by simpa using w)
+
+@[deprecated findIdx_getElem (since := "2024-08-12")]
 theorem findIdx_get {xs : List α} {w : xs.findIdx p < xs.length} :
     p (xs.get ⟨xs.findIdx p, w⟩) :=
-  xs.findIdx_of_get?_eq_some (get?_eq_get w)
+  xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)
 
 theorem findIdx_lt_length_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
     xs.findIdx p < xs.length := by
@@ -153,16 +448,130 @@ theorem findIdx_lt_length_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
     · simp_all only [forall_exists_index, and_imp, mem_cons, exists_eq_or_imp, true_or,
         findIdx_cons, cond_true, length_cons]
       apply Nat.succ_pos
-    · simp_all [findIdx_cons]
-      refine Nat.succ_lt_succ ?_
+    · simp_all [findIdx_cons, Nat.succ_lt_succ_iff]
       obtain ⟨x', m', h'⟩ := h
       exact ih x' m' h'
 
+theorem findIdx_getElem?_eq_getElem_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
+    xs[xs.findIdx p]? = some (xs[xs.findIdx p]'(xs.findIdx_lt_length_of_exists h)) :=
+  getElem?_eq_getElem (findIdx_lt_length_of_exists h)
+
+@[deprecated findIdx_getElem?_eq_getElem_of_exists (since := "2024-08-12")]
 theorem findIdx_get?_eq_get_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
     xs.get? (xs.findIdx p) = some (xs.get ⟨xs.findIdx p, xs.findIdx_lt_length_of_exists h⟩) :=
   get?_eq_get (findIdx_lt_length_of_exists h)
 
-  /-! ### findIdx? -/
+@[simp]
+theorem findIdx_eq_length {p : α → Bool} {xs : List α} :
+    xs.findIdx p = xs.length ↔ ∀ x ∈ xs, p x = false := by
+  induction xs with
+  | nil => simp_all
+  | cons x xs ih =>
+    rw [findIdx_cons, length_cons]
+    simp only [cond_eq_if]
+    split <;> simp_all [Nat.succ.injEq]
+
+theorem findIdx_eq_length_of_false {p : α → Bool} {xs : List α} (h : ∀ x ∈ xs, p x = false) :
+    xs.findIdx p = xs.length := by
+  rw [findIdx_eq_length]
+  exact h
+
+theorem findIdx_le_length (p : α → Bool) {xs : List α} : xs.findIdx p ≤ xs.length := by
+  by_cases e : ∃ x ∈ xs, p x
+  · exact Nat.le_of_lt (findIdx_lt_length_of_exists e)
+  · simp at e
+    exact Nat.le_of_eq (findIdx_eq_length.mpr e)
+
+@[simp]
+theorem findIdx_lt_length {p : α → Bool} {xs : List α} :
+    xs.findIdx p < xs.length ↔ ∃ x ∈ xs, p x := by
+  rw [← Decidable.not_iff_not, Nat.not_lt]
+  have := @Nat.le_antisymm_iff (xs.findIdx p) xs.length
+  simp only [findIdx_le_length, true_and] at this
+  rw [← this, findIdx_eq_length, not_exists]
+  simp only [Bool.not_eq_true, not_and]
+
+/-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/
+theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.findIdx p) :
+    ¬p (xs[i]'(Nat.le_trans h (findIdx_le_length p))) := by
+  revert i
+  induction xs with
+  | nil => intro i h; rw [findIdx_nil] at h; simp at h
+  | cons x xs ih =>
+    intro i h
+    have ho := h
+    rw [findIdx_cons] at h
+    have npx : ¬p x := by intro y; rw [y, cond_true] at h; simp at h
+    simp [npx, cond_false] at h
+    cases i.eq_zero_or_pos with
+    | inl e => simpa only [e, Fin.zero_eta, get_cons_zero]
+    | inr e =>
+      have ipm := Nat.succ_pred_eq_of_pos e
+      have ilt := Nat.le_trans ho (findIdx_le_length p)
+      simp (config := { singlePass := true }) only [← ipm, getElem_cons_succ]
+      rw [← ipm, Nat.succ_lt_succ_iff] at h
+      simpa using ih h
+
+/-- If `¬ p xs[j]` for all `j < i`, then `i ≤ xs.findIdx p`. -/
+theorem le_findIdx_of_not {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length)
+    (h2 : ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji h))) : i ≤ xs.findIdx p := by
+  apply Decidable.byContradiction
+  intro f
+  simp only [Nat.not_le] at f
+  exact absurd (@findIdx_getElem _ p xs (Nat.lt_trans f h)) (h2 (xs.findIdx p) f)
+
+/-- If `¬ p xs[j]` for all `j ≤ i`, then `i < xs.findIdx p`. -/
+theorem lt_findIdx_of_not {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length)
+    (h2 : ∀ j (hji : j ≤ i), ¬p (xs.get ⟨j, Nat.lt_of_le_of_lt hji h⟩)) : i < xs.findIdx p := by
+  apply Decidable.byContradiction
+  intro f
+  simp only [Nat.not_lt] at f
+  exact absurd (@findIdx_getElem _ p xs (Nat.lt_of_le_of_lt f h)) (h2 (xs.findIdx p) f)
+
+/-- `xs.findIdx p = i` iff `p xs[i]` and `¬ p xs [j]` for all `j < i`. -/
+theorem findIdx_eq {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length) :
+    xs.findIdx p = i ↔ p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji h)) := by
+  refine ⟨fun f ↦ ⟨f ▸ (@findIdx_getElem _ p xs (f ▸ h)), fun _ hji ↦ not_of_lt_findIdx (f ▸ hji)⟩,
+    fun ⟨h1, h2⟩ ↦ ?_⟩
+  apply Nat.le_antisymm _ (le_findIdx_of_not h h2)
+  apply Decidable.byContradiction
+  intro h3
+  simp at h3
+  exact not_of_lt_findIdx h3 h1
+
+theorem findIdx_append (p : α → Bool) (l₁ l₂ : List α) :
+    (l₁ ++ l₂).findIdx p =
+      if l₁.findIdx p < l₁.length then l₁.findIdx p else l₂.findIdx p + l₁.length := by
+  simp
+  induction l₁ with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx_cons, length_cons, cons_append]
+    by_cases h : p x
+    · simp [h]
+    · simp only [h, ih, cond_eq_if, Bool.false_eq_true, ↓reduceIte, mem_cons, exists_eq_or_imp,
+        false_or]
+      split <;> simp [Nat.add_assoc]
+
+theorem IsPrefix.findIdx_le {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
+    l₁.findIdx p ≤ l₂.findIdx p := by
+  rw [IsPrefix] at h
+  obtain ⟨t, rfl⟩ := h
+  simp only [findIdx_append, findIdx_lt_length]
+  split
+  · exact Nat.le_refl ..
+  · simp_all [findIdx_eq_length_of_false]
+
+theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂)
+    (lt : l₁.findIdx p < l₁.length) : l₂.findIdx p = l₁.findIdx p := by
+  rw [IsPrefix] at h
+  obtain ⟨t, rfl⟩ := h
+  simp only [findIdx_append, findIdx_lt_length]
+  split
+  · rfl
+  · simp_all
+
+/-! ### findIdx? -/
 
 @[simp] theorem findIdx?_nil : ([] : List α).findIdx? p i = none := rfl
 
@@ -174,16 +583,91 @@ theorem findIdx_get?_eq_get_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
   induction xs generalizing i with simp
   | cons _ _ _ => split <;> simp_all
 
-theorem findIdx?_eq_some_iff (xs : List α) (p : α → Bool) :
-    xs.findIdx? p = some i ↔ (xs.take (i + 1)).map p = replicate i false ++ [true] := by
+@[simp]
+theorem findIdx?_eq_none_iff {xs : List α} {p : α → Bool} :
+    xs.findIdx? p = none ↔ ∀ x, x ∈ xs → p x = false := by
+  induction xs with
+  | nil => simp_all
+  | cons x xs ih =>
+    simp only [findIdx?_cons]
+    split <;> simp_all [cond_eq_if]
+
+theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
+    (xs.findIdx? p).isSome = xs.any p := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx?_cons]
+    split <;> simp_all
+
+theorem findIdx?_isNone {xs : List α} {p : α → Bool} :
+    (xs.findIdx? p).isNone = xs.all (¬p ·) := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx?_cons]
+    split <;> simp_all
+
+theorem findIdx?_eq_some_iff_findIdx_eq {xs : List α} {p : α → Bool} {i : Nat} :
+    xs.findIdx? p = some i ↔ i < xs.length ∧ xs.findIdx p = i := by
+  induction xs generalizing i with
+  | nil => simp_all
+  | cons x xs ih =>
+    simp only [findIdx?_cons, findIdx_cons]
+    split
+    · simp_all [cond_eq_if]
+      rintro rfl
+      exact zero_lt_succ xs.length
+    · simp_all [cond_eq_if, and_assoc]
+      constructor
+      · rintro ⟨a, lt, rfl, rfl⟩
+        simp_all [Nat.succ_lt_succ_iff]
+      · rintro ⟨h, rfl⟩
+        exact ⟨_, by simp_all [Nat.succ_lt_succ_iff], rfl, rfl⟩
+
+theorem findIdx?_eq_some_of_exists {xs : List α} {p : α → Bool} (h : ∃ x, x ∈ xs ∧ p x) :
+    xs.findIdx? p = some (xs.findIdx p) := by
+  rw [findIdx?_eq_some_iff_findIdx_eq]
+  exact ⟨findIdx_lt_length_of_exists h, rfl⟩
+
+theorem findIdx?_eq_none_iff_findIdx_eq {xs : List α} {p : α → Bool} :
+    xs.findIdx? p = none ↔ xs.findIdx p = xs.length := by
+  simp
+
+theorem findIdx?_eq_some_iff_getElem (xs : List α) (p : α → Bool) :
+    xs.findIdx? p = some i ↔
+      ∃ h : i < xs.length, p xs[i] ∧ ∀ j (hji : j < i), ¬p (xs[j]'(Nat.lt_trans hji h)) := by
   induction xs generalizing i with
   | nil => simp
   | cons x xs ih =>
-    simp only [findIdx?_cons, Nat.zero_add, findIdx?_succ, take_succ_cons, map_cons]
-    split <;> cases i <;> simp_all [replicate_succ, succ_inj']
+    simp only [findIdx?_cons, Nat.zero_add, findIdx?_succ]
+    split
+    · simp only [Option.some.injEq, Bool.not_eq_true, length_cons]
+      cases i with
+      | zero => simp_all
+      | succ i =>
+        simp only [Bool.not_eq_true, zero_ne_add_one, getElem_cons_succ, false_iff, not_exists,
+          not_and, Classical.not_forall, Bool.not_eq_false]
+        intros
+        refine ⟨0, zero_lt_succ i, ‹_›⟩
+    · simp only [Option.map_eq_some', ih, Bool.not_eq_true, length_cons]
+      constructor
+      · rintro ⟨a, ⟨⟨h, h₁, h₂⟩, rfl⟩⟩
+        refine ⟨Nat.succ_lt_succ_iff.mpr h, by simpa, fun j hj => ?_⟩
+        cases j with
+        | zero => simp_all
+        | succ j =>
+          apply h₂
+          simp_all [Nat.succ_lt_succ_iff]
+      · rintro ⟨h, h₁, h₂⟩
+        cases i with
+        | zero => simp_all
+        | succ i =>
+          refine ⟨i, ⟨Nat.succ_lt_succ_iff.mp h, by simpa, fun j hj => ?_⟩, rfl⟩
+          simpa using h₂ (j + 1) (Nat.succ_lt_succ_iff.mpr hj)
 
 theorem findIdx?_of_eq_some {xs : List α} {p : α → Bool} (w : xs.findIdx? p = some i) :
-    match xs.get? i with | some a => p a | none => false := by
+    match xs[i]? with | some a => p a | none => false := by
   induction xs generalizing i with
   | nil => simp_all
   | cons x xs ih =>
@@ -191,7 +675,7 @@ theorem findIdx?_of_eq_some {xs : List α} {p : α → Bool} (w : xs.findIdx? p
     split at w <;> cases i <;> simp_all [succ_inj']
 
 theorem findIdx?_of_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p = none) :
-    ∀ i, match xs.get? i with | some a => ¬ p a | none => true := by
+    ∀ i : Nat, match xs[i]? with | some a => ¬ p a | none => true := by
   intro i
   induction xs generalizing i with
   | nil => simp_all
@@ -201,24 +685,90 @@ theorem findIdx?_of_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p
     | zero =>
       split at w <;> simp_all
     | succ i =>
-      simp only [get?_cons_succ]
+      simp only [getElem?_cons_succ]
       apply ih
       split at w <;> simp_all
 
+@[simp] theorem findIdx?_map (f : β → α) (l : List β) : findIdx? p (l.map f) = l.findIdx? (p ∘ f) := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [map_cons, findIdx?]
+    split <;> simp_all
+
 @[simp] theorem findIdx?_append :
     (xs ++ ys : List α).findIdx? p =
-      (xs.findIdx? p <|> (ys.findIdx? p).map fun i => i + xs.length) := by
+      (xs.findIdx? p).or ((ys.findIdx? p).map fun i => i + xs.length) := by
   induction xs with simp
-  | cons _ _ _ => split <;> simp_all [Option.map_orElse, Option.map_map]; rfl
+  | cons _ _ _ => split <;> simp_all [Option.map_or', Option.map_map]; rfl
+
+theorem findIdx?_join {l : List (List α)} {p : α → Bool} :
+    l.join.findIdx? p =
+      (l.findIdx? (·.any p)).map
+        fun i => Nat.sum ((l.take i).map List.length) +
+          (l[i]?.map fun xs => xs.findIdx p).getD 0 := by
+  induction l with
+  | nil => simp
+  | cons xs l ih =>
+    simp only [join, findIdx?_append, map_take, map_cons, findIdx?, any_eq_true, Nat.zero_add,
+      findIdx?_succ]
+    split
+    · simp only [Option.map_some', take_zero, sum_nil, length_cons, zero_lt_succ,
+        getElem?_eq_getElem, getElem_cons_zero, Option.getD_some, Nat.zero_add]
+      rw [Option.or_of_isSome (by simpa [findIdx?_isSome])]
+      rw [findIdx?_eq_some_of_exists ‹_›]
+    · simp_all only [map_take, not_exists, not_and, Bool.not_eq_true, Option.map_map]
+      rw [Option.or_of_isNone (by simpa [findIdx?_isNone])]
+      congr 1
+      ext i
+      simp [Nat.add_comm, Nat.add_assoc]
 
 @[simp] theorem findIdx?_replicate :
     (replicate n a).findIdx? p = if 0 < n ∧ p a then some 0 else none := by
-  induction n with
+  cases n with
   | zero => simp
-  | succ n ih =>
-    simp only [replicate, findIdx?_cons, Nat.zero_add, findIdx?_succ, Nat.zero_lt_succ, true_and]
+  | succ n =>
+    simp only [replicate, findIdx?_cons, Nat.zero_add, findIdx?_succ, zero_lt_succ, true_and]
     split <;> simp_all
 
+theorem findIdx?_eq_enum_findSome? {xs : List α} {p : α → Bool} :
+    xs.findIdx? p = xs.enum.findSome? fun ⟨i, a⟩ => if p a then some i else none := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx?_cons, Nat.zero_add, findIdx?_succ, enum]
+    split
+    · simp_all
+    · simp_all only [enumFrom_cons, ite_false, Option.isNone_none, findSome?_cons_of_isNone, reduceCtorEq]
+      simp [Function.comp_def, ← map_fst_add_enum_eq_enumFrom, findSome?_map]
+
+theorem Sublist.findIdx?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
+    (l₁.findIdx? p).isSome → (l₂.findIdx? p).isSome := by
+  simp only [List.findIdx?_isSome, any_eq_true]
+  rintro ⟨w, m, q⟩
+  exact ⟨w, h.mem m, q⟩
+
+theorem Sublist.findIdx?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) :
+    l₂.findIdx? p = none → l₁.findIdx? p = none := by
+  simp only [findIdx?_eq_none_iff]
+  exact fun w x m => w x (h.mem m)
+
+theorem IsPrefix.findIdx?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
+    List.findIdx? p l₁ = some i → List.findIdx? p l₂ = some i := by
+  rw [IsPrefix] at h
+  obtain ⟨t, rfl⟩ := h
+  intro h
+  simp [findIdx?_append, h]
+theorem IsPrefix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
+    List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
+  h.sublist.findIdx?_eq_none
+theorem IsSuffix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+ l₂) :
+    List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
+  h.sublist.findIdx?_eq_none
+theorem IsInfix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+: l₂) :
+    List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
+  h.sublist.findIdx?_eq_none
+
 /-! ### indexOf -/
 
 theorem indexOf_cons [BEq α] :

src/Init/Data/List/Lemmas.lean

@@ -79,6 +79,11 @@ open Nat
 
 /-! ## Preliminaries -/
 
+/-! ### nil -/
+
+@[simp] theorem nil_eq {α} (xs : List α) : [] = xs ↔ xs = [] := by
+  cases xs <;> simp
+
 /-! ### cons -/
 
 theorem cons_ne_nil (a : α) (l : List α) : a :: l ≠ [] := nofun
@@ -86,6 +91,10 @@ theorem cons_ne_nil (a : α) (l : List α) : a :: l ≠ [] := nofun
 @[simp]
 theorem cons_ne_self (a : α) (l : List α) : a :: l ≠ l := mt (congrArg length) (Nat.succ_ne_self _)
 
+@[simp] theorem ne_cons_self {a : α} {l : List α} : l ≠ a :: l := by
+  rw [ne_eq, eq_comm]
+  simp
+
 theorem head_eq_of_cons_eq (H : h₁ :: t₁ = h₂ :: t₂) : h₁ = h₂ := (cons.inj H).1
 
 theorem tail_eq_of_cons_eq (H : h₁ :: t₁ = h₂ :: t₂) : t₁ = t₂ := (cons.inj H).2
@@ -256,6 +265,18 @@ theorem getElem?_eq_some {l : List α} : l[n]? = some a ↔ ∃ h : n < l.length
 
 theorem getElem?_eq_none (h : length l ≤ n) : l[n]? = none := getElem?_eq_none_iff.mpr h
 
+theorem getElem?_eq (l : List α) (i : Nat) :
+    l[i]? = if h : i < l.length then some l[i] else none := by
+  split <;> simp_all
+
+@[simp] theorem some_getElem_eq_getElem? {α} (xs : List α) (i : Nat) (h : i < xs.length) :
+    (some xs[i] = xs[i]?) ↔ True := by
+  simp [h]
+
+@[simp] theorem getElem?_eq_some_getElem {α} (xs : List α) (i : Nat) (h : i < xs.length) :
+    (xs[i]? = some xs[i]) ↔ True := by
+  simp [h]
+
 theorem getElem_eq_iff {l : List α} {n : Nat} {h : n < l.length} : l[n] = x ↔ l[n]? = some x := by
   simp only [getElem?_eq_some]
   exact ⟨fun w => ⟨h, w⟩, fun h => h.2⟩
@@ -272,6 +293,9 @@ theorem getElem?_cons_zero {l : List α} : (a::l)[0]? = some a := by simp
   simp only [← get?_eq_getElem?]
   rfl
 
+theorem getElem?_cons : (a :: l)[i]? = if i = 0 then some a else l[i-1]? := by
+  cases i <;> simp
+
 theorem getElem?_len_le : ∀ {l : List α} {n}, length l ≤ n → l[n]? = none
   | [], _, _ => rfl
   | _ :: l, _+1, h => by
@@ -340,6 +364,11 @@ theorem get?_concat_length (l : List α) (a : α) : (l ++ [a]).get? l.length = s
 
 theorem mem_cons_self (a : α) (l : List α) : a ∈ a :: l := .head ..
 
+theorem mem_concat_self (xs : List α) (a : α) : a ∈ xs ++ [a] :=
+  mem_append_of_mem_right xs (mem_cons_self a _)
+
+theorem mem_append_cons_self : a ∈ xs ++ a :: ys := mem_append_of_mem_right _ (mem_cons_self _ _)
+
 theorem mem_cons_of_mem (y : α) {a : α} {l : List α} : a ∈ l → a ∈ y :: l := .tail _
 
 theorem exists_mem_of_ne_nil (l : List α) (h : l ≠ []) : ∃ x, x ∈ l :=
@@ -359,27 +388,21 @@ theorem forall_mem_cons {p : α → Prop} {a : α} {l : List α} :
   ⟨fun H => ⟨H _ (.head ..), fun _ h => H _ (.tail _ h)⟩,
    fun ⟨H₁, H₂⟩ _ => fun | .head .. => H₁ | .tail _ h => H₂ _ h⟩
 
-@[simp]
 theorem forall_mem_ne {a : α} {l : List α} : (∀ a' : α, a' ∈ l → ¬a = a') ↔ a ∉ l :=
   ⟨fun h m => h _ m rfl, fun h _ m e => h (e.symm ▸ m)⟩
 
-@[simp]
 theorem forall_mem_ne' {a : α} {l : List α} : (∀ a' : α, a' ∈ l → ¬a' = a) ↔ a ∉ l :=
   ⟨fun h m => h _ m rfl, fun h _ m e => h (e.symm ▸ m)⟩
 
-@[simp]
 theorem any_beq [BEq α] [LawfulBEq α] {l : List α} : (l.any fun x => a == x) ↔ a ∈ l := by
   induction l <;> simp_all
 
-@[simp]
 theorem any_beq' [BEq α] [LawfulBEq α] {l : List α} : (l.any fun x => x == a) ↔ a ∈ l := by
   induction l <;> simp_all [eq_comm (a := a)]
 
-@[simp]
 theorem all_bne [BEq α] [LawfulBEq α] {l : List α} : (l.all fun x => a != x) ↔ a ∉ l := by
   induction l <;> simp_all
 
-@[simp]
 theorem all_bne' [BEq α] [LawfulBEq α] {l : List α} : (l.all fun x => x != a) ↔ a ∉ l := by
   induction l <;> simp_all [eq_comm (a := a)]
 
@@ -503,7 +526,7 @@ theorem isEmpty_iff_length_eq_zero {l : List α} : l.isEmpty ↔ l.length = 0 :=
 @[simp] theorem isEmpty_eq_true {l : List α} : l.isEmpty ↔ l = [] := by
   cases l <;> simp
 
-@[simp] theorem isEmpty_eq_false {l : List α} : ¬ l.isEmpty ↔ l ≠ [] := by
+@[simp] theorem isEmpty_eq_false {l : List α} : l.isEmpty = false ↔ l ≠ [] := by
   cases l <;> simp
 
 /-! ### any / all -/
@@ -542,8 +565,7 @@ theorem get_set_eq {l : List α} {i : Nat} {a : α} (h : i < (l.set i a).length)
     (l.set i a)[i]? = some a := by
   simp_all [getElem?_eq_some]
 
-@[simp]
-theorem getElem?_set_eq' {l : List α} {i : Nat} {a : α} : (set l i a)[i]? = (fun _ => a) <$> l[i]? := by
+theorem getElem?_set_eq' {l : List α} {i : Nat} {a : α} : (set l i a)[i]? = Function.const _ a <$> l[i]? := by
   by_cases h : i < l.length
   · simp [getElem?_set_eq h, getElem?_eq_getElem h]
   · simp only [Nat.not_lt] at h
@@ -600,7 +622,7 @@ theorem getElem?_set {l : List α} {i j : Nat} {a : α} :
 theorem getElem?_set' {l : List α} {i j : Nat} {a : α} :
     (set l i a)[j]? = if i = j then (fun _ => a) <$> l[j]? else l[j]? := by
   by_cases i = j
-  · simp only [getElem?_set_eq', Option.map_eq_map, ↓reduceIte, *]
+  · simp only [getElem?_set_eq', Option.map_eq_map, ↓reduceIte, *]; rfl
   · simp only [ne_eq, not_false_eq_true, getElem?_set_ne, ↓reduceIte, *]
 
 theorem set_eq_of_length_le {l : List α} {n : Nat} (h : l.length ≤ n) {a : α} :
@@ -615,7 +637,7 @@ theorem set_eq_of_length_le {l : List α} {n : Nat} (h : l.length ≤ n) {a : α
       exact Nat.succ_le_succ_iff.mp h
 
 @[simp] theorem set_eq_nil (l : List α) (n : Nat) (a : α) : l.set n a = [] ↔ l = [] := by
-  cases l <;> cases n <;> simp only [set]
+  cases l <;> cases n <;> simp [set]
 
 theorem set_comm (a b : α) : ∀ {n m : Nat} (l : List α), n ≠ m →
     (l.set n a).set m b = (l.set m b).set n a
@@ -710,9 +732,14 @@ theorem foldr_eq_foldrM (f : α → β → β) (b) (l : List α) :
 
 /-! ### foldl and foldr -/
 
-@[simp] theorem foldr_self_append (l : List α) : l.foldr cons l' = l ++ l' := by
+@[simp] theorem foldr_cons_eq_append (l : List α) : l.foldr cons l' = l ++ l' := by
   induction l <;> simp [*]
 
+@[deprecated foldr_cons_eq_append (since := "2024-08-22")] abbrev foldr_self_append := @foldr_cons_eq_append
+
+@[simp] theorem foldl_flip_cons_eq_append (l : List α) : l.foldl (fun x y => y :: x) l' = l.reverse ++ l' := by
+  induction l generalizing l' <;> simp [*]
+
 theorem foldr_self (l : List α) : l.foldr cons [] = l := by simp
 
 theorem foldl_map (f : β₁ → β₂) (g : α → β₂ → α) (l : List β₁) (init : α) :
@@ -745,6 +772,58 @@ theorem foldr_hom (f : β₁ → β₂) (g₁ : α → β₁ → β₁) (g₂ :
     (H : ∀ x y, g₂ x (f y) = f (g₁ x y)) : l.foldr g₂ (f init) = f (l.foldr g₁ init) := by
   induction l <;> simp [*, H]
 
+/--
+Prove a proposition about the result of `List.foldl`,
+by proving it for the initial data,
+and the implication that the operation applied to any element of the list preserves the property.
+
+The motive can take values in `Sort _`, so this may be used to construct data,
+as well as to prove propositions.
+-/
+def foldlRecOn {motive : β → Sort _} : ∀ (l : List α) (op : β → α → β) (b : β) (_ : motive b)
+    (_ : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ l), motive (op b a)), motive (List.foldl op b l)
+  | [], _, _, hb, _ => hb
+  | hd :: tl, op, b, hb, hl =>
+    foldlRecOn tl op (op b hd) (hl b hb hd (mem_cons_self hd tl))
+      fun y hy x hx => hl y hy x (mem_cons_of_mem hd hx)
+
+@[simp] theorem foldlRecOn_nil {motive : β → Sort _} (hb : motive b)
+    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ []), motive (op b a)) :
+    foldlRecOn [] op b hb hl = hb := rfl
+
+@[simp] theorem foldlRecOn_cons {motive : β → Sort _} (hb : motive b)
+    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ x :: l), motive (op b a)) :
+    foldlRecOn (x :: l) op b hb hl =
+      foldlRecOn l op (op b x) (hl b hb x (mem_cons_self x l))
+        (fun b c a m => hl b c a (mem_cons_of_mem x m)) :=
+  rfl
+
+/--
+Prove a proposition about the result of `List.foldr`,
+by proving it for the initial data,
+and the implication that the operation applied to any element of the list preserves the property.
+
+The motive can take values in `Sort _`, so this may be used to construct data,
+as well as to prove propositions.
+-/
+def foldrRecOn {motive : β → Sort _} : ∀ (l : List α) (op : α → β → β) (b : β) (_ : motive b)
+    (_ : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ l), motive (op a b)), motive (List.foldr op b l)
+  | nil, _, _, hb, _ => hb
+  | x :: l, op, b, hb, hl =>
+    hl (foldr op b l)
+      (foldrRecOn l op b hb fun b c a m => hl b c a (mem_cons_of_mem x m)) x (mem_cons_self x l)
+
+@[simp] theorem foldrRecOn_nil {motive : β → Sort _} (hb : motive b)
+    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ []), motive (op a b)) :
+    foldrRecOn [] op b hb hl = hb := rfl
+
+@[simp] theorem foldrRecOn_cons {motive : β → Sort _} (hb : motive b)
+    (hl : ∀ (b : β) (_ : motive b) (a : α) (_ : a ∈ x :: l), motive (op a b)) :
+    foldrRecOn (x :: l) op b hb hl =
+      hl _ (foldrRecOn l op b hb fun b c a m => hl b c a (mem_cons_of_mem x m))
+        x (mem_cons_self x l) :=
+  rfl
+
 /-! ### getLast -/
 
 theorem getLast_eq_getElem : ∀ (l : List α) (h : l ≠ []),
@@ -779,7 +858,7 @@ theorem getLast_eq_getLastD (a l h) : @getLast α (a::l) h = getLastD l a := by
 theorem getLast!_cons [Inhabited α] : @getLast! α _ (a::l) = getLastD l a := by
   simp [getLast!, getLast_eq_getLastD]
 
-theorem getLast_mem : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ l
+@[simp] theorem getLast_mem : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ l
   | [], h => absurd rfl h
   | [_], _ => .head ..
   | _::a::l, _ => .tail _ <| getLast_mem (cons_ne_nil a l)
@@ -799,11 +878,14 @@ theorem get_cons_length (x : α) (xs : List α) (n : Nat) (h : n = xs.length) :
 
 /-! ### getLast? -/
 
-theorem getLast?_cons : @getLast? α (a::l) = getLastD l a := by
-  simp only [getLast?, getLast_eq_getLastD]
-
 @[simp] theorem getLast?_singleton (a : α) : getLast? [a] = a := rfl
 
+theorem getLast?_cons {a : α} : (a::l).getLast? = l.getLast?.getD a := by
+  cases l <;> simp [getLast?, getLast]
+
+@[simp] theorem getLast?_cons_cons : (a :: b :: l).getLast? = (b :: l).getLast? := by
+  simp [getLast?_cons]
+
 theorem getLast?_eq_getLast : ∀ l h, @getLast? α l = some (getLast l h)
   | [], h => nomatch h rfl
   | _::_, _ => rfl
@@ -817,10 +899,10 @@ theorem getLast?_eq_getElem? : ∀ (l : List α), getLast? l = l[l.length - 1]?
 theorem getLast?_eq_get? (l : List α) : getLast? l = l.get? (l.length - 1) := by
   simp [getLast?_eq_getElem?]
 
-@[simp] theorem getLast?_concat (l : List α) : getLast? (l ++ [a]) = some a := by
+theorem getLast?_concat (l : List α) : getLast? (l ++ [a]) = some a := by
   simp [getLast?_eq_getElem?, Nat.succ_sub_succ]
 
-@[simp] theorem getLastD_concat (a b l) : @getLastD α (l ++ [b]) a = b := by
+theorem getLastD_concat (a b l) : @getLastD α (l ++ [b]) a = b := by
   rw [getLastD_eq_getLast?, getLast?_concat]; rfl
 
 /-! ## Head and tail -/
@@ -833,6 +915,11 @@ theorem head!_of_head? [Inhabited α] : ∀ {l : List α}, head? l = some a →
 theorem head?_eq_head : ∀ {l} h, @head? α l = some (head l h)
   | _::_, _ => rfl
 
+theorem head_eq_iff_head?_eq_some {xs : List α} (h) : xs.head h = a ↔ xs.head? = some a := by
+  cases xs with
+  | nil => simp at h
+  | cons x xs => simp
+
 theorem head?_eq_getElem? : ∀ l : List α, head? l = l[0]?
   | [] => rfl
   | a::l => by simp
@@ -840,6 +927,9 @@ theorem head?_eq_getElem? : ∀ l : List α, head? l = l[0]?
 @[simp] theorem head?_eq_none_iff : l.head? = none ↔ l = [] := by
   cases l <;> simp
 
+theorem head?_eq_some_iff {xs : List α} {a : α} : xs.head? = some a ↔ ∃ ys, xs = a :: ys := by
+  cases xs <;> simp_all
+
 @[simp] theorem head_mem : ∀ {l : List α} (h : l ≠ []), head l h ∈ l
   | [], h => absurd rfl h
   | _::_, _ => .head ..
@@ -868,9 +958,16 @@ theorem tail_eq_tail? (l) : @tail α l = (tail? l).getD [] := by simp [tail_eq_t
 
 /-! ### map -/
 
-@[simp] theorem map_id (l : List α) : map id l = l := by induction l <;> simp_all
+@[simp] theorem map_id_fun : map (id : α → α) = id := by
+  funext l
+  induction l <;> simp_all
 
-@[simp] theorem map_id' (l : List α) : map (fun a => a) l = l := by induction l <;> simp_all
+@[simp] theorem map_id_fun' : map (fun (a : α) => a) = id := map_id_fun
+
+theorem map_id (l : List α) : map (id : α → α) l = l := by
+  induction l <;> simp_all
+
+theorem map_id' (l : List α) : map (fun (a : α) => a) l = l := map_id l
 
 theorem map_id'' {f : α → α} (h : ∀ x, f x = x) (l : List α) : map f l = l := by
   simp [show f = id from funext h]
@@ -1057,7 +1154,7 @@ theorem filter_length_eq_length {l} : (filter p l).length = l.length ↔ ∀ a
     · simp_all [or_and_left]
     · simp_all [or_and_right]
 
-theorem filter_eq_nil {l} : filter p l = [] ↔ ∀ a, a ∈ l → ¬p a := by
+@[simp] theorem filter_eq_nil {l} : filter p l = [] ↔ ∀ a, a ∈ l → ¬p a := by
   simp only [eq_nil_iff_forall_not_mem, mem_filter, not_and]
 
 theorem forall_mem_filter {l : List α} {p : α → Bool} {P : α → Prop} :
@@ -1145,8 +1242,13 @@ theorem head_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p
 theorem filterMap_eq_map (f : α → β) : filterMap (some ∘ f) = map f := by
   funext l; induction l <;> simp [*, filterMap_cons]
 
-@[simp] theorem filterMap_some (l : List α) : filterMap some l = l := by
-  erw [filterMap_eq_map, map_id]
+@[simp] theorem filterMap_some_fun : filterMap (some : α → Option α) = id := by
+  funext l
+  erw [filterMap_eq_map]
+  simp
+
+theorem filterMap_some (l : List α) : filterMap some l = l := by
+  rw [filterMap_some_fun, id]
 
 theorem map_filterMap_some_eq_filter_map_isSome (f : α → Option β) (l : List α) :
     (l.filterMap f).map some = (l.map f).filter fun b => b.isSome := by
@@ -1221,7 +1323,7 @@ theorem forall_mem_filterMap {f : α → Option β} {l : List α} {P : β → Pr
   induction l <;> simp [filterMap_cons]; split <;> simp [*]
 
 theorem map_filterMap_of_inv (f : α → Option β) (g : β → α) (H : ∀ x : α, (f x).map g = some x)
-    (l : List α) : map g (filterMap f l) = l := by simp only [map_filterMap, H, filterMap_some]
+    (l : List α) : map g (filterMap f l) = l := by simp only [map_filterMap, H, filterMap_some, id]
 
 theorem head_filterMap_of_eq_some {f : α → Option β} {l : List α} (w : l ≠ []) {b : β} (h : f (l.head w) = some b) :
     (filterMap f l).head ((ne_nil_of_mem (mem_filterMap.2 ⟨_, head_mem w, h⟩))) =
@@ -1244,7 +1346,7 @@ theorem forall_none_of_filterMap_eq_nil (h : filterMap f xs = []) : ∀ x ∈ xs
       | tail _ hmem => exact ih h hmem
     · contradiction
 
-theorem filterMap_eq_nil {l} : filterMap f l = [] ↔ ∀ a ∈ l, f a = none := by
+@[simp] theorem filterMap_eq_nil {l} : filterMap f l = [] ↔ ∀ a ∈ l, f a = none := by
   constructor
   · exact forall_none_of_filterMap_eq_nil
   · intro h
@@ -1363,10 +1465,22 @@ theorem append_right_inj {t₁ t₂ : List α} (s) : s ++ t₁ = s ++ t₂ ↔ t
 theorem append_left_inj {s₁ s₂ : List α} (t) : s₁ ++ t = s₂ ++ t ↔ s₁ = s₂ :=
   ⟨fun h => append_inj_left' h rfl, congrArg (· ++ _)⟩
 
+@[simp] theorem append_left_eq_self {x y : List α} : x ++ y = y ↔ x = [] := by
+  rw [← append_left_inj (s₁ := x), nil_append]
+
+@[simp] theorem self_eq_append_left {x y : List α} : y = x ++ y ↔ x = [] := by
+  rw [eq_comm, append_left_eq_self]
+
+@[simp] theorem append_right_eq_self {x y : List α} : x ++ y = x ↔ y = [] := by
+  rw [← append_right_inj (t₁ := y), append_nil]
+
+@[simp] theorem self_eq_append_right {x y : List α} : x = x ++ y ↔ y = [] := by
+  rw [eq_comm, append_right_eq_self]
+
 @[simp] theorem append_eq_nil : p ++ q = [] ↔ p = [] ∧ q = [] := by
   cases p <;> simp
 
-@[simp] theorem getLast_concat {a : α} : ∀ (l : List α), getLast (l ++ [a]) (by simp) = a
+theorem getLast_concat {a : α} : ∀ (l : List α), getLast (l ++ [a]) (by simp) = a
   | [] => rfl
   | a::t => by
     simp [getLast_cons _, getLast_concat t]
@@ -1391,20 +1505,26 @@ theorem get_append_right (as bs : List α) (h : ¬ i < as.length) {h' h''} :
     (as ++ bs).get ⟨i, h'⟩ = bs.get ⟨i - as.length, h''⟩ := by
   simp [getElem_append_right, h, h', h'']
 
-theorem getElem?_append {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.length) :
+theorem getElem?_append_left {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.length) :
     (l₁ ++ l₂)[n]? = l₁[n]? := by
   have hn' : n < (l₁ ++ l₂).length := Nat.lt_of_lt_of_le hn <|
     length_append .. ▸ Nat.le_add_right ..
   simp_all [getElem?_eq_getElem, getElem_append]
 
-@[deprecated (since := "2024-06-12")]
+@[deprecated getElem?_append_left (since := "2024-06-12")]
 theorem get?_append {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.length) :
     (l₁ ++ l₂).get? n = l₁.get? n := by
-  simp [getElem?_append hn]
+  simp [getElem?_append_left hn]
 
-@[simp] theorem head_append_of_ne_nil {l : List α} (w : l ≠ []) :
-    head (l ++ l') (by simp_all) = head l w := by
-  match l, w with
+theorem getElem?_append {l₁ l₂ : List α} {n : Nat} :
+    (l₁ ++ l₂)[n]? = if n < l₁.length then l₁[n]? else l₂[n - l₁.length]? := by
+  split <;> rename_i h
+  · exact getElem?_append_left h
+  · exact getElem?_append_right (by simpa using h)
+
+@[simp] theorem head_append_of_ne_nil {l : List α} {w₁} (w₂) :
+    head (l ++ l') w₁ = head l w₂ := by
+  match l, w₂ with
   | a :: l, _ => rfl
 
 theorem head_append {l₁ l₂ : List α} (w : l₁ ++ l₂ ≠ []) :
@@ -1427,7 +1547,7 @@ theorem head_append {l₁ l₂ : List α} (w : l₁ ++ l₂ ≠ []) :
 -- `getLast_append_of_ne_nil`, `getLast_append` and `getLast?_append`
 -- are stated and proved later in the `reverse` section.
 
-@[simp] theorem nil_eq_append : [] = a ++ b ↔ a = [] ∧ b = [] := by
+theorem nil_eq_append : [] = a ++ b ↔ a = [] ∧ b = [] := by
   rw [eq_comm, append_eq_nil]
 
 theorem append_ne_nil_of_left_ne_nil {s : List α} (h : s ≠ []) (t : List α) : s ++ t ≠ [] := by simp_all
@@ -1438,6 +1558,14 @@ theorem append_ne_nil_of_ne_nil_left {s : List α} (h : s ≠ []) (t : List α)
 @[deprecated append_ne_nil_of_right_ne_nil (since := "2024-07-24")]
 theorem append_ne_nil_of_ne_nil_right (s : List α) : t ≠ [] → s ++ t ≠ [] := by simp_all
 
+theorem tail_append (xs ys : List α) :
+    (xs ++ ys).tail = if xs.isEmpty then ys.tail else xs.tail ++ ys := by
+  cases xs <;> simp
+
+@[simp] theorem tail_append_of_ne_nil (xs ys : List α) (h : xs ≠ []) :
+    (xs ++ ys).tail = xs.tail ++ ys := by
+  simp_all [tail_append]
+
 theorem append_eq_cons :
     a ++ b = x :: c ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
   cases a with simp | cons a as => ?_
@@ -1453,6 +1581,24 @@ theorem append_eq_append_iff {a b c d : List α} :
   | nil => simp_all
   | cons a as ih => cases c <;> simp [eq_comm, and_assoc, ih, and_or_left]
 
+theorem append_inj_of_length_left {a b c d : List α}
+    (h : a ++ b = c ++ d) (hl : length a = length c) : a = c ∧ b = d := by
+  rcases append_eq_append_iff.mp h with (⟨a', rfl, rfl⟩ | ⟨c', rfl, rfl⟩)
+  · simp only [length_append] at hl
+    have : a'.length = 0 := (Nat.add_left_cancel hl).symm
+    simp_all
+  · simp only [length_append] at hl
+    have : c'.length = 0 := (Nat.add_left_cancel hl.symm).symm
+    simp_all
+
+theorem append_inj_of_length_right {a b c d : List α}
+    (h : a ++ b = c ++ d) (hl : length b = length d) : a = c ∧ b = d := by
+  have : length a = length c :=  by
+    replace h := congrArg length h
+    simp only [length_append, hl] at h
+    exact Nat.add_right_cancel h
+  exact append_inj_of_length_left h this
+
 @[simp] theorem mem_append {a : α} {s t : List α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
   induction s <;> simp_all [or_assoc]
 
@@ -1505,9 +1651,55 @@ theorem set_append {s t : List α} :
 @[simp] theorem foldr_append (f : α → β → β) (b) (l l' : List α) :
     (l ++ l').foldr f b = l.foldr f (l'.foldr f b) := by simp [foldr_eq_foldrM]
 
+theorem filterMap_eq_append (f : α → Option β) :
+    filterMap f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂ := by
+  constructor
+  · induction l generalizing L₁ with
+    | nil =>
+      simp only [filterMap_nil, nil_eq_append, and_imp]
+      rintro rfl rfl
+      exact ⟨[], [], by simp⟩
+    | cons x l ih =>
+      simp only [filterMap_cons]
+      split
+      · intro h
+        obtain ⟨l₁, l₂, rfl, rfl, rfl⟩ := ih h
+        refine ⟨x :: l₁, l₂, ?_⟩
+        simp_all
+      · rename_i b w
+        intro h
+        rcases cons_eq_append.mp h with (⟨rfl, rfl⟩ | ⟨L₁, ⟨rfl, h⟩⟩)
+        · refine ⟨[], x :: l, ?_⟩
+          simp [filterMap_cons, w]
+        · obtain ⟨l₁, l₂, rfl, rfl, rfl⟩ := ih ‹_›
+          refine ⟨x :: l₁, l₂, ?_⟩
+          simp [filterMap_cons, w]
+  · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
+    simp
+
+theorem append_eq_filterMap (f : α → Option β) :
+    L₁ ++ L₂ = filterMap f l ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂ := by
+  rw [eq_comm, filterMap_eq_append]
+
+theorem filter_eq_append (p : α → Bool) :
+    filter p l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filter p l₁ = L₁ ∧ filter p l₂ = L₂ := by
+  rw [← filterMap_eq_filter, filterMap_eq_append]
+
+theorem append_eq_filter (p : α → Bool) :
+    L₁ ++ L₂ = filter p l ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filter p l₁ = L₁ ∧ filter p l₂ = L₂ := by
+  rw [eq_comm, filter_eq_append]
+
 @[simp] theorem map_append (f : α → β) : ∀ l₁ l₂, map f (l₁ ++ l₂) = map f l₁ ++ map f l₂ := by
   intro l₁; induction l₁ <;> intros <;> simp_all
 
+theorem map_eq_append (f : α → β) :
+    map f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
+  rw [← filterMap_eq_map, filterMap_eq_append]
+
+theorem append_eq_map (f : α → β) :
+    L₁ ++ L₂ = map f l ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
+  rw [eq_comm, map_eq_append]
+
 /-! ### concat
 
 Note that `concat_eq_append` is a `@[simp]` lemma, so `concat` should usually not appear in goals.
@@ -1561,13 +1753,21 @@ theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ L b, l = concat L b
   | cons =>
     simp [join, length_append, *]
 
+theorem join_singleton (l : List α) : [l].join = l := by simp
+
 @[simp] theorem mem_join : ∀ {L : List (List α)}, a ∈ L.join ↔ ∃ l, l ∈ L ∧ a ∈ l
   | [] => by simp
   | b :: l => by simp [mem_join, or_and_right, exists_or]
 
-@[simp] theorem join_eq_nil_iff {L : List (List α)} : L.join = [] ↔ ∀ l ∈ L, l = [] := by
+@[simp] theorem join_eq_nil {L : List (List α)} : L.join = [] ↔ ∀ l ∈ L, l = [] := by
   induction L <;> simp_all
 
+@[deprecated join_eq_nil (since := "2024-08-22")]
+theorem join_eq_nil_iff {L : List (List α)} : L.join = [] ↔ ∀ l ∈ L, l = [] := join_eq_nil
+
+theorem join_ne_nil (xs : List (List α)) : xs.join ≠ [] ↔ ∃ x, x ∈ xs ∧ x ≠ [] := by
+  simp
+
 theorem exists_of_mem_join : a ∈ join L → ∃ l, l ∈ L ∧ a ∈ l := mem_join.1
 
 theorem mem_join_of_mem (lL : l ∈ L) (al : a ∈ l) : a ∈ join L := mem_join.2 ⟨l, lL, al⟩
@@ -1608,6 +1808,25 @@ theorem foldr_join (f : α → β → β) (b : β) (L : List (List α)) :
     filter p (join L) = join (map (filter p) L) := by
   induction L <;> simp [*, filter_append]
 
+@[simp]
+theorem join_filter_not_isEmpty  :
+    ∀ {L : List (List α)}, join (L.filter fun l => !l.isEmpty) = L.join
+  | [] => rfl
+  | [] :: L
+  | (a :: l) :: L => by
+      simp [join_filter_not_isEmpty (L := L)]
+
+@[simp]
+theorem join_filter_ne_nil [DecidablePred fun l : List α => l ≠ []] {L : List (List α)} :
+    join (L.filter fun l => l ≠ []) = L.join := by
+  simp only [ne_eq, ← isEmpty_iff, Bool.not_eq_true, Bool.decide_eq_false,
+    join_filter_not_isEmpty]
+
+@[deprecated filter_join (since := "2024-08-26")]
+theorem join_map_filter (p : α → Bool) (l : List (List α)) :
+    (l.map (filter p)).join = (l.join).filter p := by
+  rw [filter_join]
+
 @[simp] theorem join_append (L₁ L₂ : List (List α)) : join (L₁ ++ L₂) = join L₁ ++ join L₂ := by
   induction L₁ <;> simp_all
 
@@ -1617,6 +1836,71 @@ theorem join_concat (L : List (List α)) (l : List α) : join (L ++ [l]) = join
 theorem join_join {L : List (List (List α))} : join (join L) = join (map join L) := by
   induction L <;> simp_all
 
+theorem join_eq_cons (xs : List (List α)) (y : α) (ys : List α) :
+    xs.join = y :: ys ↔
+      ∃ as bs cs, xs = as ++ (y :: bs) :: cs ∧ (∀ l, l ∈ as → l = []) ∧ ys = bs ++ cs.join := by
+  constructor
+  · induction xs with
+    | nil => simp
+    | cons x xs ih =>
+      intro h
+      simp only [join_cons] at h
+      replace h := h.symm
+      rw [cons_eq_append] at h
+      obtain (⟨rfl, h⟩ | ⟨z⟩) := h
+      · obtain ⟨as, bs, cs, rfl, _, rfl⟩ := ih h
+        refine ⟨[] :: as, bs, cs, ?_⟩
+        simpa
+      · obtain ⟨a', rfl, rfl⟩ := z
+        refine ⟨[], a', xs, ?_⟩
+        simp
+  · rintro ⟨as, bs, cs, rfl, h₁, rfl⟩
+    simp [join_eq_nil.mpr h₁]
+
+theorem join_eq_append (xs : List (List α)) (ys zs : List α) :
+    xs.join = ys ++ zs ↔
+      (∃ as bs, xs = as ++ bs ∧ ys = as.join ∧ zs = bs.join) ∨
+        ∃ as bs c cs ds, xs = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.join ++ bs ∧
+          zs = c :: cs ++ ds.join := by
+  constructor
+  · induction xs generalizing ys with
+    | nil =>
+      simp only [join_nil, nil_eq, append_eq_nil, and_false, cons_append, false_and, exists_const,
+        exists_false, or_false, and_imp, List.cons_ne_nil]
+      rintro rfl rfl
+      exact ⟨[], [], by simp⟩
+    | cons x xs ih =>
+      intro h
+      simp only [join_cons] at h
+      rw [append_eq_append_iff] at h
+      obtain (⟨ys, rfl, h⟩ | ⟨c', rfl, h⟩) := h
+      · obtain (⟨as, bs, rfl, rfl, rfl⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, rfl⟩) := ih _ h
+        · exact .inl ⟨x :: as, bs, by simp⟩
+        · exact .inr ⟨x :: as, bs, c, cs, ds, by simp⟩
+      · simp only [h]
+        cases c' with
+        | nil => exact .inl ⟨[ys], xs, by simp⟩
+        | cons x c' => exact .inr ⟨[], ys, x, c', xs, by simp⟩
+  · rintro (⟨as, bs, rfl, rfl, rfl⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, rfl⟩)
+    · simp
+    · simp
+
+/-- Two lists of sublists are equal iff their joins coincide, as well as the lengths of the
+sublists. -/
+theorem eq_iff_join_eq : ∀ (L L' : List (List α)),
+    L = L' ↔ L.join = L'.join ∧ map length L = map length L'
+  | _, [] => by simp_all
+  | [], x' :: L' => by simp_all
+  | x :: L, x' :: L' => by
+    simp
+    rw [eq_iff_join_eq]
+    constructor
+    · rintro ⟨rfl, h₁, h₂⟩
+      simp_all
+    · rintro ⟨h₁, h₂, h₃⟩
+      obtain ⟨rfl, h⟩ := append_inj_of_length_left h₁ h₂
+      exact ⟨rfl, h, h₃⟩
+
 /-! ### bind -/
 
 theorem bind_def (l : List α) (f : α → List β) : l.bind f = join (map f l) := by rfl
@@ -1633,6 +1917,11 @@ theorem exists_of_mem_bind {b : β} {l : List α} {f : α → List β} :
 theorem mem_bind_of_mem {b : β} {l : List α} {f : α → List β} {a} (al : a ∈ l) (h : b ∈ f a) :
     b ∈ l.bind f := mem_bind.2 ⟨a, al, h⟩
 
+@[simp]
+theorem bind_eq_nil {l : List α} {f : α → List β} : List.bind l f = [] ↔ ∀ x ∈ l, f x = [] :=
+  join_eq_nil.trans <| by
+    simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+
 theorem forall_mem_bind {p : β → Prop} {l : List α} {f : α → List β} :
     (∀ (x) (_ : x ∈ l.bind f), p x) ↔ ∀ (a) (_ : a ∈ l) (b) (_ : b ∈ f a), p b := by
   simp only [mem_bind, forall_exists_index, and_imp]
@@ -1720,6 +2009,9 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
 @[simp] theorem replicate_succ_ne_nil (n : Nat) (a : α) : replicate (n+1) a ≠ [] := by
   simp [replicate_succ]
 
+@[simp] theorem replicate_eq_nil (n : Nat) (a : α) : replicate n a = [] ↔ n = 0 := by
+  cases n <;> simp
+
 @[simp] theorem getElem_replicate (a : α) {n : Nat} {m} (h : m < (replicate n a).length) :
     (replicate n a)[m] = a :=
   eq_of_mem_replicate (get_mem _ _ _)
@@ -1744,6 +2036,9 @@ theorem head?_replicate (a : α) (n : Nat) : (replicate n a).head? = if n = 0 th
   · simp at w
   · simp_all [replicate_succ]
 
+@[simp] theorem tail_replicate (n : Nat) (a : α) : (replicate n a).tail = replicate (n - 1) a := by
+  cases n <;> simp [replicate_succ]
+
 @[simp] theorem replicate_inj : replicate n a = replicate m b ↔ n = m ∧ (n = 0 ∨ a = b) :=
   ⟨fun h => have eq : n = m := by simpa using congrArg length h
     ⟨eq, by
@@ -1788,15 +2083,23 @@ theorem map_const' (l : List α) (b : β) : map (fun _ => b) l = replicate l.len
     simp only [mem_append, mem_replicate, ne_eq]
     rintro (⟨-, rfl⟩ | ⟨_, rfl⟩) <;> rfl
 
+theorem append_eq_replicate {l₁ l₂ : List α} {a : α} :
+    l₁ ++ l₂ = replicate n a ↔
+      l₁.length + l₂.length = n ∧ l₁ = replicate l₁.length a ∧ l₂ = replicate l₂.length a := by
+  simp only [eq_replicate, length_append, mem_append, true_and, and_congr_right_iff]
+  exact fun _ =>
+    { mp := fun h => ⟨fun b m => h b (Or.inl m), fun b m => h b (Or.inr m)⟩,
+      mpr := fun h b x => Or.casesOn x (fun m => h.left b m) fun m => h.right b m }
+
 @[simp] theorem map_replicate : (replicate n a).map f = replicate n (f a) := by
   ext1 n
   simp only [getElem?_map, getElem?_replicate]
   split <;> simp
 
 theorem filter_replicate : (replicate n a).filter p = if p a then replicate n a else [] := by
-  induction n with
+  cases n with
   | zero => simp
-  | succ n ih =>
+  | succ n =>
     simp only [replicate_succ, filter_cons]
     split <;> simp_all
 
@@ -1857,36 +2160,48 @@ theorem bind_replicate {β} (f : α → List β) : (replicate n a).bind f = (rep
   | nil => rfl
   | cons a as ih => simp [ih]
 
-@[simp] theorem mem_reverseAux {x : α} : ∀ {as bs}, x ∈ reverseAux as bs ↔ x ∈ as ∨ x ∈ bs
+theorem mem_reverseAux {x : α} : ∀ {as bs}, x ∈ reverseAux as bs ↔ x ∈ as ∨ x ∈ bs
   | [], _ => ⟨.inr, fun | .inr h => h⟩
   | a :: _, _ => by rw [reverseAux, mem_cons, or_assoc, or_left_comm, mem_reverseAux, mem_cons]
 
-@[simp] theorem mem_reverse {x : α} {as : List α} : x ∈ reverse as ↔ x ∈ as := by simp [reverse]
+@[simp] theorem mem_reverse {x : α} {as : List α} : x ∈ reverse as ↔ x ∈ as := by
+  simp [reverse, mem_reverseAux]
 
 @[simp] theorem reverse_eq_nil_iff {xs : List α} : xs.reverse = [] ↔ xs = [] := by
   match xs with
   | [] => simp
   | x :: xs => simp
 
+theorem reverse_ne_nil_iff {xs : List α} : xs.reverse ≠ [] ↔ xs ≠ [] :=
+  not_congr reverse_eq_nil_iff
+
 theorem getElem?_reverse' : ∀ {l : List α} (i j), i + j + 1 = length l →
     l.reverse[i]? = l[j]?
   | [], _, _, _ => rfl
   | a::l, i, 0, h => by simp [Nat.succ.injEq] at h; simp [h, getElem?_append_right, Nat.succ.injEq]
   | a::l, i, j+1, h => by
     have := Nat.succ.inj h; simp at this ⊢
-    rw [getElem?_append, getElem?_reverse' _ _ this]
+    rw [getElem?_append_left, getElem?_reverse' _ _ this]
     rw [length_reverse, ← this]; apply Nat.lt_add_of_pos_right (Nat.succ_pos _)
 
 @[deprecated getElem?_reverse' (since := "2024-06-12")]
 theorem get?_reverse' {l : List α} (i j) (h : i + j + 1 = length l) : get? l.reverse i = get? l j := by
   simp [getElem?_reverse' _ _ h]
 
+@[simp]
 theorem getElem?_reverse {l : List α} {i} (h : i < length l) :
     l.reverse[i]? = l[l.length - 1 - i]? :=
   getElem?_reverse' _ _ <| by
     rw [Nat.add_sub_of_le (Nat.le_sub_one_of_lt h),
       Nat.sub_add_cancel (Nat.lt_of_le_of_lt (Nat.zero_le _) h)]
 
+@[simp]
+theorem getElem_reverse {l : List α} {i} (h : i < l.reverse.length) :
+    l.reverse[i] = l[l.length - 1 - i]'(Nat.sub_one_sub_lt_of_lt (by simpa using h)) := by
+  apply Option.some.inj
+  rw [← getElem?_eq_getElem, ← getElem?_eq_getElem]
+  rw [getElem?_reverse (by simpa using h)]
+
 @[deprecated getElem?_reverse (since := "2024-06-12")]
 theorem get?_reverse {l : List α} {i} (h : i < length l) :
     get? l.reverse i = get? l (l.length - 1 - i) := by
@@ -1903,11 +2218,25 @@ theorem reverseAux_reverseAux_nil (as bs : List α) : reverseAux (reverseAux as
 theorem reverse_eq_iff {as bs : List α} : as.reverse = bs ↔ as = bs.reverse := by
   constructor <;> (rintro rfl; simp)
 
-@[simp] theorem getLast?_reverse (l : List α) : l.reverse.getLast? = l.head? := by cases l <;> simp
+@[simp] theorem reverse_inj {xs ys : List α} : xs.reverse = ys.reverse ↔ xs = ys := by
+  simp [reverse_eq_iff]
+
+@[simp] theorem reverse_eq_cons {xs : List α} {a : α} {ys : List α} :
+    xs.reverse = a :: ys ↔ xs = ys.reverse ++ [a] := by
+  rw [reverse_eq_iff, reverse_cons]
+
+@[simp] theorem getLast?_reverse (l : List α) : l.reverse.getLast? = l.head? := by
+  cases l <;> simp [getLast?_concat]
 
 @[simp] theorem head?_reverse (l : List α) : l.reverse.head? = l.getLast? := by
   rw [← getLast?_reverse, reverse_reverse]
 
+theorem getLast?_eq_head?_reverse {xs : List α} : xs.getLast? = xs.reverse.head? := by
+  simp
+
+theorem head?_eq_getLast?_reverse {xs : List α} : xs.head? = xs.reverse.getLast? := by
+  simp
+
 @[simp] theorem map_reverse (f : α → β) (l : List α) : l.reverse.map f = (l.map f).reverse := by
   induction l <;> simp [*]
 
@@ -1932,8 +2261,16 @@ theorem reverse_map (f : α → β) (l : List α) : (l.map f).reverse = l.revers
 @[simp] theorem reverse_append (as bs : List α) : (as ++ bs).reverse = bs.reverse ++ as.reverse := by
   induction as <;> simp_all
 
-theorem reverse_concat (l : List α) (a : α) : (l.concat a).reverse = a :: l.reverse := by
-  rw [concat_eq_append, reverse_append]; rfl
+@[simp] theorem reverse_eq_append {xs ys zs : List α} :
+    xs.reverse = ys ++ zs ↔ xs = zs.reverse ++ ys.reverse := by
+  rw [reverse_eq_iff, reverse_append]
+
+theorem reverse_concat (l : List α) (a : α) : (l ++ [a]).reverse = a :: l.reverse := by
+  rw [reverse_append]; rfl
+
+theorem reverse_eq_concat {xs ys : List α} {a : α} :
+    xs.reverse = ys ++ [a] ↔ xs = a :: ys.reverse := by
+  rw [reverse_eq_iff, reverse_concat]
 
 /-- Reversing a join is the same as reversing the order of parts and reversing all parts. -/
 theorem reverse_join (L : List (List α)) :
@@ -1951,7 +2288,7 @@ theorem reverse_bind {β} (l : List α) (f : α → List β) : (l.bind f).revers
 theorem bind_reverse {β} (l : List α) (f : α → List β) : (l.reverse.bind f) = (l.bind (reverse ∘ f)).reverse := by
   induction l <;> simp_all
 
-theorem reverseAux_eq (as bs : List α) : reverseAux as bs = reverse as ++ bs :=
+@[simp] theorem reverseAux_eq (as bs : List α) : reverseAux as bs = reverse as ++ bs :=
   reverseAux_eq_append ..
 
 @[simp] theorem foldrM_reverse [Monad m] (l : List α) (f : α → β → m β) (b) :
@@ -1977,16 +2314,32 @@ theorem reverseAux_eq (as bs : List α) : reverseAux as bs = reverse as ++ bs :=
   induction l with
   | nil => contradiction
   | cons a l ih =>
-    simp
+    simp only [reverse_cons]
     by_cases h' : l = []
     · simp_all
-    · rw [getLast_cons, head_append_of_ne_nil, ih]
-      simp_all
+    · simp only [head_eq_iff_head?_eq_some, head?_reverse] at ih
+      simp [ih, h, h', getLast_cons, head_eq_iff_head?_eq_some]
 
 theorem getLast_eq_head_reverse {l : List α} (h : l ≠ []) :
     l.getLast h = l.reverse.head (by simp_all) := by
   rw [← head_reverse]
 
+theorem getLast_eq_iff_getLast_eq_some {xs : List α} (h) : xs.getLast h = a ↔ xs.getLast? = some a := by
+  rw [getLast_eq_head_reverse, head_eq_iff_head?_eq_some]
+  simp
+
+@[simp] theorem getLast?_eq_none_iff {xs : List α} : xs.getLast? = none ↔ xs = [] := by
+  rw [getLast?_eq_head?_reverse, head?_eq_none_iff, reverse_eq_nil_iff]
+
+theorem getLast?_eq_some_iff {xs : List α} {a : α} : xs.getLast? = some a ↔ ∃ ys, xs = ys ++ [a] := by
+  rw [getLast?_eq_head?_reverse, head?_eq_some_iff]
+  simp only [reverse_eq_cons]
+  exact ⟨fun ⟨ys, h⟩ => ⟨ys.reverse, by simpa using h⟩, fun ⟨ys, h⟩ => ⟨ys.reverse, by simpa using h⟩⟩
+
+theorem mem_of_getLast?_eq_some {xs : List α} {a : α} (h : xs.getLast? = some a) : a ∈ xs := by
+  obtain ⟨ys, rfl⟩ := getLast?_eq_some_iff.1 h
+  exact mem_concat_self ys a
+
 @[simp] theorem getLast_reverse {l : List α} (h : l.reverse ≠ []) :
     l.reverse.getLast h = l.head (by simp_all) := by
   simp [getLast_eq_head_reverse]
@@ -1995,8 +2348,8 @@ theorem head_eq_getLast_reverse {l : List α} (h : l ≠ []) :
     l.head h = l.reverse.getLast (by simp_all) := by
   rw [← getLast_reverse]
 
-@[simp] theorem getLast_append_of_ne_nil {l : List α} (h : l' ≠ []) :
-    (l ++ l').getLast (append_ne_nil_of_right_ne_nil l h) = l'.getLast (by simp_all) := by
+@[simp] theorem getLast_append_of_ne_nil {l : List α} {h₁} (h₂ : l' ≠ []) :
+    (l ++ l').getLast h₁ = l'.getLast h₂ := by
   simp only [getLast_eq_head_reverse, reverse_append]
   rw [head_append_of_ne_nil]
 
@@ -2144,6 +2497,14 @@ theorem dropLast_cons_of_ne_nil {α : Type u} {x : α}
     {l : List α} (h : l ≠ []) : (x :: l).dropLast = x :: l.dropLast := by
   simp [dropLast, h]
 
+theorem dropLast_concat_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l ++ [getLast l h] = l
+  | [], h => absurd rfl h
+  | [a], h => rfl
+  | a :: b :: l, h => by
+    rw [dropLast_cons₂, cons_append, getLast_cons (cons_ne_nil _ _)]
+    congr
+    exact dropLast_concat_getLast (cons_ne_nil b l)
+
 @[simp] theorem map_dropLast (f : α → β) (l : List α) : l.dropLast.map f = (l.map f).dropLast := by
   induction l with
   | nil => rfl
@@ -2160,8 +2521,8 @@ theorem dropLast_append {l₁ l₂ : List α} :
     (l₁ ++ l₂).dropLast = if l₂.isEmpty then l₁.dropLast else l₁ ++ l₂.dropLast := by
   split <;> simp_all
 
-@[simp] theorem dropLast_append_cons : dropLast (l₁ ++ b::l₂) = l₁ ++ dropLast (b::l₂) := by
-  simp only [ne_eq, not_false_eq_true, dropLast_append_of_ne_nil]
+theorem dropLast_append_cons : dropLast (l₁ ++ b::l₂) = l₁ ++ dropLast (b::l₂) := by
+  simp
 
 @[simp 1100] theorem dropLast_concat : dropLast (l₁ ++ [b]) = l₁ := by simp
 
@@ -2178,6 +2539,27 @@ theorem dropLast_append {l₁ l₂ : List α} :
     dropLast (a :: replicate n a) = replicate n a := by
   rw [← replicate_succ, dropLast_replicate, Nat.add_sub_cancel]
 
+/-!
+### splitAt
+
+We don't provide any API for `splitAt`, beyond the `@[simp]` lemma
+`splitAt n l = (l.take n, l.drop n)`,
+which is proved in `Init.Data.List.TakeDrop`.
+-/
+
+theorem splitAt_go (n : Nat) (l acc : List α) :
+    splitAt.go l xs n acc =
+      if n < xs.length then (acc.reverse ++ xs.take n, xs.drop n) else (l, []) := by
+  induction xs generalizing n acc with
+  | nil => simp [splitAt.go]
+  | cons x xs ih =>
+    cases n with
+    | zero => simp [splitAt.go]
+    | succ n =>
+      rw [splitAt.go, take_succ_cons, drop_succ_cons, ih n (x :: acc),
+        reverse_cons, append_assoc, singleton_append, length_cons]
+      simp only [Nat.succ_lt_succ_iff]
+
 /-! ## Manipulating elements -/
 
 /-! ### replace -/

src/Init/Data/List/Nat.lean

@@ -7,4 +7,5 @@ prelude
 import Init.Data.List.Nat.Basic
 import Init.Data.List.Nat.Pairwise
 import Init.Data.List.Nat.Range
+import Init.Data.List.Nat.Sublist
 import Init.Data.List.Nat.TakeDrop

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

@@ -25,6 +25,14 @@ theorem length_filter_lt_length_iff_exists (l) :
   simpa [length_eq_countP_add_countP p l, countP_eq_length_filter] using
   countP_pos (fun x => ¬p x) (l := l)
 
+/-! ### reverse -/
+
+theorem getElem_eq_getElem_reverse {l : List α} {i} (h : i < l.length) :
+    l[i] = l.reverse[l.length - 1 - i]'(by simpa using Nat.sub_one_sub_lt_of_lt h) := by
+  rw [getElem_reverse]
+  congr
+  omega
+
 /-! ### leftpad -/
 
 /-- The length of the List returned by `List.leftpad n a l` is equal

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

@@ -5,7 +5,9 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Nat.TakeDrop
+import Init.Data.List.Range
 import Init.Data.List.Pairwise
+import Init.Data.List.Find
 
 /-!
 # Lemmas about `List.range` and `List.enum`
@@ -22,8 +24,6 @@ open Nat
 theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step) n step := by
   simp [range', Nat.add_succ, Nat.mul_succ]
 
-@[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
-
 @[simp] theorem length_range' (s step) : ∀ n : Nat, length (range' s n step) = n
   | 0 => rfl
   | _ + 1 => congrArg succ (length_range' _ _ _)
@@ -31,6 +31,27 @@ theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step)
 @[simp] theorem range'_eq_nil : range' s n step = [] ↔ n = 0 := by
   rw [← length_eq_zero, length_range']
 
+theorem range'_ne_nil (s n : Nat) : range' s n ≠ [] ↔ n ≠ 0 := by
+  cases n <;> simp
+
+@[simp] theorem range'_zero : range' s 0 = [] := by
+  simp
+
+@[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
+
+@[simp] theorem range'_inj : range' s n = range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s') := by
+  constructor
+  · intro h
+    have h' := congrArg List.length h
+    simp at h'
+    subst h'
+    cases n with
+    | zero => simp
+    | succ n =>
+      simp only [range'_succ] at h
+      simp_all
+  · rintro ⟨rfl, rfl | rfl⟩ <;> simp
+
 theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step * i
   | 0 => by simp [range', Nat.not_lt_zero]
   | n + 1 => by
@@ -44,6 +65,29 @@ theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step *
     fun ⟨i, h, e⟩ => e ▸ ⟨Nat.le_add_right .., Nat.add_lt_add_left h _⟩,
     fun ⟨h₁, h₂⟩ => ⟨m - s, Nat.sub_lt_left_of_lt_add h₁ h₂, (Nat.add_sub_cancel' h₁).symm⟩⟩
 
+theorem head?_range' (n : Nat) : (range' s n).head? = if n = 0 then none else some s := by
+  induction n <;> simp_all [range'_succ, head?_append]
+
+@[simp] theorem head_range' (n : Nat) (h) : (range' s n).head h = s := by
+  repeat simp_all [head?_range', head_eq_iff_head?_eq_some]
+
+theorem getLast?_range' (n : Nat) : (range' s n).getLast? = if n = 0 then none else some (s + n - 1) := by
+  induction n generalizing s with
+  | zero => simp
+  | succ n ih =>
+    rw [range'_succ, getLast?_cons, ih]
+    by_cases h : n = 0
+    · rw [if_pos h]
+      simp [h]
+    · rw [if_neg h]
+      simp
+      omega
+
+@[simp] theorem getLast_range' (n : Nat) (h) : (range' s n).getLast h = s + n - 1 := by
+  cases n with
+  | zero => simp at h
+  | succ n => simp [getLast?_range', getLast_eq_iff_getLast_eq_some]
+
 theorem pairwise_lt_range' s n (step := 1) (pos : 0 < step := by simp) :
     Pairwise (· < ·) (range' s n step) :=
   match s, n, step, pos with
@@ -123,6 +167,67 @@ theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ [
 theorem range'_1_concat (s n : Nat) : range' s (n + 1) = range' s n ++ [s + n] := by
   simp [range'_concat]
 
+theorem range'_eq_cons_iff : range' s n = a :: xs ↔ s = a ∧ 0 < n ∧ xs = range' (a + 1) (n - 1) := by
+  induction n generalizing s with
+  | zero => simp
+  | succ n ih =>
+    simp only [range'_succ]
+    simp only [cons.injEq, and_congr_right_iff]
+    rintro rfl
+    simp [eq_comm]
+
+@[simp] theorem range'_eq_singleton {s n a : Nat} : range' s n = [a] ↔ s = a ∧ n = 1 := by
+  rw [range'_eq_cons_iff]
+  simp only [nil_eq, range'_eq_nil, and_congr_right_iff]
+  rintro rfl
+  omega
+
+theorem range'_eq_append_iff : range' s n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs = range' s k ∧ ys = range' (s + k) (n - k) := by
+  induction n generalizing s xs ys with
+  | zero => simp
+  | succ n ih =>
+    simp only [range'_succ]
+    rw [cons_eq_append]
+    constructor
+    · rintro (⟨rfl, rfl⟩ | ⟨a, rfl, h⟩)
+      · exact ⟨0, by simp [range'_succ]⟩
+      · simp only [ih] at h
+        obtain ⟨k, h, rfl, rfl⟩ := h
+        refine ⟨k + 1, ?_⟩
+        simp_all [range'_succ]
+        omega
+    · rintro ⟨k, h, rfl, rfl⟩
+      cases k with
+      | zero => simp [range'_succ]
+      | succ k =>
+        simp only [range'_succ, reduceCtorEq, false_and, cons.injEq, true_and, ih, range'_inj, exists_eq_left', or_true, and_true, false_or]
+        refine ⟨k, ?_⟩
+        simp_all
+        omega
+
+@[simp] theorem find?_range'_eq_some (s n : Nat) (i : Nat) (p : Nat → Bool) :
+    (range' s n).find? p = some i ↔ p i ∧ i ∈ range' s n ∧ ∀ j, s ≤ j → j < i → !p j := by
+  rw [find?_eq_some]
+  simp only [Bool.not_eq_true', exists_and_right, mem_range'_1, and_congr_right_iff]
+  simp only [range'_eq_append_iff, eq_comm (a := i :: _), range'_eq_cons_iff]
+  intro h
+  constructor
+  · rintro ⟨as, ⟨x, k, h₁, rfl, rfl, h₂, rfl⟩, h₃⟩
+    constructor
+    · omega
+    · simpa using h₃
+  · rintro ⟨⟨h₁, h₂⟩, h₃⟩
+    refine ⟨range' s (i - s), ⟨⟨range' (i + 1) (n - (i - s) - 1), i - s, ?_⟩ , ?_⟩⟩
+    · simp; omega
+    · simp only [mem_range'_1, and_imp]
+      intro a a₁ a₂
+      exact h₃ a a₁ (by omega)
+
+@[simp] theorem find?_range'_eq_none (s n : Nat) (p : Nat → Bool) :
+    (range' s n).find? p = none ↔ ∀ i, s ≤ i → i < s + n → !p i := by
+  rw [find?_eq_none]
+  simp
+
 /-! ### range -/
 
 theorem range_loop_range' : ∀ s n : Nat, range.loop s (range' s n) = range' 0 (n + s)
@@ -152,6 +257,9 @@ theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) :=
 @[simp] theorem range_eq_nil {n : Nat} : range n = [] ↔ n = 0 := by
   rw [← length_eq_zero, length_range]
 
+theorem range_ne_nil (n : Nat) : range n ≠ [] ↔ n ≠ 0 := by
+  cases n <;> simp
+
 @[simp]
 theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by
   simp only [range_eq_range', range'_sublist_right]
@@ -187,6 +295,30 @@ theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + ·
   rw [← range'_eq_map_range]
   simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm
 
+theorem head?_range (n : Nat) : (range n).head? = if n = 0 then none else some 0 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [range_succ, head?_append, ih]
+    split <;> simp_all
+
+@[simp] theorem head_range (n : Nat) (h) : (range n).head h = 0 := by
+  cases n with
+  | zero => simp at h
+  | succ n => simp [head?_range, head_eq_iff_head?_eq_some]
+
+theorem getLast?_range (n : Nat) : (range n).getLast? = if n = 0 then none else some (n - 1) := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [range_succ, getLast?_append, ih]
+    split <;> simp_all
+
+@[simp] theorem getLast_range (n : Nat) (h) : (range n).getLast h = n - 1 := by
+  cases n with
+  | zero => simp at h
+  | succ n => simp [getLast?_range, getLast_eq_iff_getLast_eq_some]
+
 theorem take_range (m n : Nat) : take m (range n) = range (min m n) := by
   apply List.ext_getElem
   · simp
@@ -195,6 +327,14 @@ theorem take_range (m n : Nat) : take m (range n) = range (min m n) := by
 theorem nodup_range (n : Nat) : Nodup (range n) := by
   simp (config := {decide := true}) only [range_eq_range', nodup_range']
 
+@[simp] theorem find?_range_eq_some (n : Nat) (i : Nat) (p : Nat → Bool) :
+    (range n).find? p = some i ↔ p i ∧ i ∈ range n ∧ ∀ j, j < i → !p j := by
+  simp [range_eq_range']
+
+@[simp] theorem find?_range_eq_none (n : Nat) (p : Nat → Bool) :
+    (range n).find? p = none ↔ ∀ i, i < n → !p i := by
+  simp [range_eq_range']
+
 /-! ### iota -/
 
 theorem iota_eq_reverse_range' : ∀ n : Nat, iota n = reverse (range' 1 n)
@@ -203,9 +343,49 @@ theorem iota_eq_reverse_range' : ∀ n : Nat, iota n = reverse (range' 1 n)
 
 @[simp] theorem length_iota (n : Nat) : length (iota n) = n := by simp [iota_eq_reverse_range']
 
+@[simp] theorem iota_eq_nil (n : Nat) : iota n = [] ↔ n = 0 := by
+  cases n <;> simp
+
+theorem iota_ne_nil (n : Nat) : iota n ≠ [] ↔ n ≠ 0 := by
+  cases n <;> simp
+
 @[simp]
-theorem mem_iota {m n : Nat} : m ∈ iota n ↔ 1 ≤ m ∧ m ≤ n := by
+theorem mem_iota {m n : Nat} : m ∈ iota n ↔ 0 < m ∧ m ≤ n := by
   simp [iota_eq_reverse_range', Nat.add_comm, Nat.lt_succ]
+  omega
+
+@[simp] theorem iota_inj : iota n = iota n' ↔ n = n' := by
+  constructor
+  · intro h
+    have h' := congrArg List.length h
+    simp at h'
+    exact h'
+  · rintro rfl
+    simp
+
+theorem iota_eq_cons_iff : iota n = a :: xs ↔ n = a ∧ 0 < n ∧ xs = iota (n - 1) := by
+  simp [iota_eq_reverse_range']
+  simp [range'_eq_append_iff, reverse_eq_iff]
+  constructor
+  · rintro ⟨k, h, rfl, h'⟩
+    rw [eq_comm, range'_eq_singleton] at h'
+    simp only [reverse_inj, range'_inj, or_true, and_true]
+    omega
+  · rintro ⟨rfl, h, rfl⟩
+    refine ⟨n - 1, by simp, rfl, ?_⟩
+    rw [eq_comm, range'_eq_singleton]
+    omega
+
+theorem iota_eq_append_iff : iota n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs = (range' (k + 1) (n - k)).reverse ∧ ys = iota k := by
+  simp only [iota_eq_reverse_range']
+  rw [reverse_eq_append]
+  rw [range'_eq_append_iff]
+  simp only [reverse_eq_iff]
+  constructor
+  · rintro ⟨k, h, rfl, rfl⟩
+    simp; omega
+  · rintro ⟨k, h, rfl, rfl⟩
+    exact ⟨k, by simp; omega⟩
 
 theorem pairwise_gt_iota (n : Nat) : Pairwise (· > ·) (iota n) := by
   simpa only [iota_eq_reverse_range', pairwise_reverse] using pairwise_lt_range' 1 n
@@ -213,36 +393,84 @@ theorem pairwise_gt_iota (n : Nat) : Pairwise (· > ·) (iota n) := by
 theorem nodup_iota (n : Nat) : Nodup (iota n) :=
   (pairwise_gt_iota n).imp Nat.ne_of_gt
 
+@[simp] theorem head?_iota (n : Nat) : (iota n).head? = if n = 0 then none else some n := by
+  cases n <;> simp
+
+@[simp] theorem head_iota (n : Nat) (h) : (iota n).head h = n := by
+  cases n with
+  | zero => simp at h
+  | succ n => simp
+
+@[simp] theorem reverse_iota : reverse (iota n) = range' 1 n := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [iota_succ, reverse_cons, ih, range'_1_concat, Nat.add_comm]
+
+@[simp] theorem getLast?_iota (n : Nat) : (iota n).getLast? = if n = 0 then none else some 1 := by
+  rw [getLast?_eq_head?_reverse]
+  simp [head?_range']
+
+@[simp] theorem getLast_iota (n : Nat) (h) : (iota n).getLast h = 1 := by
+  rw [getLast_eq_head_reverse]
+  simp
+
+@[simp] theorem find?_iota_eq_none (n : Nat) (p : Nat → Bool) :
+    (iota n).find? p = none ↔ ∀ i, 0 < i → i ≤ n → !p i := by
+  rw [find?_eq_none]
+  simp
+
+@[simp] theorem find?_iota_eq_some (n : Nat) (i : Nat) (p : Nat → Bool) :
+    (iota n).find? p = some i ↔ p i ∧ i ∈ iota n ∧ ∀ j, i < j → j ≤ n → !p j := by
+  rw [find?_eq_some]
+  simp only [iota_eq_reverse_range', reverse_eq_append, reverse_cons, append_assoc,
+    singleton_append, Bool.not_eq_true', exists_and_right, mem_reverse, mem_range'_1,
+    and_congr_right_iff]
+  intro h
+  constructor
+  · rintro ⟨as, ⟨xs, h⟩, h'⟩
+    constructor
+    · replace h : i ∈ range' 1 n := by
+        rw [h]
+        exact mem_append_cons_self
+      simpa using h
+    · rw [range'_eq_append_iff] at h
+      simp [reverse_eq_iff] at h
+      obtain ⟨k, h₁, rfl, h₂⟩ := h
+      rw [eq_comm, range'_eq_cons_iff, reverse_eq_iff] at h₂
+      obtain ⟨rfl, -, rfl⟩ := h₂
+      intro j j₁ j₂
+      apply h'
+      simp; omega
+  · rintro ⟨⟨i₁, i₂⟩, h⟩
+    refine ⟨(range' (i+1) (n-i)).reverse, ⟨(range' 1 (i-1)).reverse, ?_⟩, ?_⟩
+    · simp [← range'_succ]
+      rw [range'_eq_append_iff]
+      refine ⟨i-1, ?_⟩
+      constructor
+      · omega
+      · simp
+        omega
+    · simp
+      intros a a₁ a₂
+      apply h
+      · omega
+      · omega
+
 /-! ### enumFrom -/
 
 @[simp]
 theorem enumFrom_singleton (x : α) (n : Nat) : enumFrom n [x] = [(n, x)] :=
   rfl
 
-@[simp]
-theorem enumFrom_eq_nil {n : Nat} {l : List α} : List.enumFrom n l = [] ↔ l = [] := by
-  cases l <;> simp
+@[simp] theorem head?_enumFrom (n : Nat) (l : List α) :
+    (enumFrom n l).head? = l.head?.map fun a => (n, a) := by
+  simp [head?_eq_getElem?]
 
-@[simp] theorem enumFrom_length : ∀ {n} {l : List α}, (enumFrom n l).length = l.length
-  | _, [] => rfl
-  | _, _ :: _ => congrArg Nat.succ enumFrom_length
-
-@[simp]
-theorem getElem?_enumFrom :
-    ∀ n (l : List α) m, (enumFrom n l)[m]? = l[m]?.map fun a => (n + m, a)
-  | n, [], m => rfl
-  | n, a :: l, 0 => by simp
-  | n, a :: l, m + 1 => by
-    simp only [enumFrom_cons, getElem?_cons_succ]
-    exact (getElem?_enumFrom (n + 1) l m).trans <| by rw [Nat.add_right_comm]; rfl
-
-@[simp]
-theorem getElem_enumFrom (l : List α) (n) (i : Nat) (h : i < (l.enumFrom n).length) :
-    (l.enumFrom n)[i] = (n + i, l[i]'(by simpa [enumFrom_length] using h)) := by
-  simp only [enumFrom_length] at h
-  rw [getElem_eq_getElem?]
-  simp only [getElem?_enumFrom, getElem?_eq_getElem h]
-  simp
+@[simp] theorem getLast?_enumFrom (n : Nat) (l : List α) :
+    (enumFrom n l).getLast? = l.getLast?.map fun a => (n + l.length - 1, a) := by
+  simp [getLast?_eq_getElem?]
+  cases l <;> simp; omega
 
 theorem mk_add_mem_enumFrom_iff_getElem? {n i : Nat} {x : α} {l : List α} :
     (n + i, x) ∈ enumFrom n l ↔ l[i]? = some x := by
@@ -284,17 +512,24 @@ theorem enumFrom_map_snd : ∀ (n) (l : List α), map Prod.snd (enumFrom n l) =
 theorem snd_mem_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) : x.2 ∈ l :=
   enumFrom_map_snd n l ▸ mem_map_of_mem _ h
 
-theorem mem_enumFrom {x : α} {i j : Nat} (xs : List α) (h : (i, x) ∈ xs.enumFrom j) :
-    j ≤ i ∧ i < j + xs.length ∧ x ∈ xs :=
-  ⟨le_fst_of_mem_enumFrom h, fst_lt_add_of_mem_enumFrom h, snd_mem_of_mem_enumFrom h⟩
+theorem snd_eq_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) :
+    x.2 = l[x.1 - n]'(by have := le_fst_of_mem_enumFrom h; have := fst_lt_add_of_mem_enumFrom h; omega) := by
+  induction l generalizing n with
+  | nil => cases h
+  | cons hd tl ih =>
+    cases h with
+    | head h => simp
+    | tail h m =>
+      specialize ih m
+      have : x.1 - n = x.1 - (n + 1) + 1 := by
+        have := le_fst_of_mem_enumFrom m
+        omega
+      simp [this, ih]
 
-theorem map_fst_add_enumFrom_eq_enumFrom (l : List α) (n k : Nat) :
-    map (Prod.map (· + n) id) (enumFrom k l) = enumFrom (n + k) l :=
-  ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
-
-theorem map_fst_add_enum_eq_enumFrom (l : List α) (n : Nat) :
-    map (Prod.map (· + n) id) (enum l) = enumFrom n l :=
-  map_fst_add_enumFrom_eq_enumFrom l _ _
+theorem mem_enumFrom {x : α} {i j : Nat} {xs : List α} (h : (i, x) ∈ xs.enumFrom j) :
+    j ≤ i ∧ i < j + xs.length ∧
+      x = xs[i - j]'(by have := le_fst_of_mem_enumFrom h; have := fst_lt_add_of_mem_enumFrom h; omega) :=
+  ⟨le_fst_of_mem_enumFrom h, fst_lt_add_of_mem_enumFrom h, snd_eq_of_mem_enumFrom h⟩
 
 theorem enumFrom_cons' (n : Nat) (x : α) (xs : List α) :
     enumFrom n (x :: xs) = (n, x) :: (enumFrom n xs).map (Prod.map (· + 1) id) := by
@@ -349,6 +584,14 @@ theorem getElem_enum (l : List α) (i : Nat) (h : i < l.enum.length) :
     l.enum[i] = (i, l[i]'(by simpa [enum_length] using h)) := by
   simp [enum]
 
+@[simp] theorem head?_enum (l : List α) :
+    l.enum.head? = l.head?.map fun a => (0, a) := by
+  simp [head?_eq_getElem?]
+
+@[simp] theorem getLast?_enum (l : List α) :
+    l.enum.getLast? = l.getLast?.map fun a => (l.length - 1, a) := by
+  simp [getLast?_eq_getElem?]
+
 theorem mk_mem_enum_iff_getElem? {i : Nat} {x : α} {l : List α} : (i, x) ∈ enum l ↔ l[i]? = x := by
   simp [enum, mk_mem_enumFrom_iff_le_and_getElem?_sub]
 
@@ -361,6 +604,14 @@ theorem fst_lt_of_mem_enum {x : Nat × α} {l : List α} (h : x ∈ enum l) : x.
 theorem snd_mem_of_mem_enum {x : Nat × α} {l : List α} (h : x ∈ enum l) : x.2 ∈ l :=
   snd_mem_of_mem_enumFrom h
 
+theorem snd_eq_of_mem_enum {x : Nat × α} {l : List α} (h : x ∈ enum l) :
+    x.2 = l[x.1]'(fst_lt_of_mem_enum h) :=
+  snd_eq_of_mem_enumFrom h
+
+theorem mem_enum {x : α} {i : Nat} {xs : List α} (h : (i, x) ∈ xs.enum) :
+    i < xs.length ∧ x = xs[i]'(fst_lt_of_mem_enum h) :=
+  by simpa using mem_enumFrom h
+
 theorem map_enum (f : α → β) (l : List α) : map (Prod.map id f) (enum l) = enum (map f l) :=
   map_enumFrom f 0 l
 

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

@@ -0,0 +1,174 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Sublist
+import Init.Data.List.Nat.Basic
+import Init.Data.List.Nat.TakeDrop
+import Init.Data.Nat.Lemmas
+
+/-!
+# Further lemmas about `List.IsSuffix` / `List.IsPrefix` / `List.IsInfix`.
+
+These are in a separate file from most of the lemmas about `List.IsSuffix`
+as they required importing more lemmas about natural numbers, and use `omega`.
+-/
+
+namespace List
+
+theorem IsSuffix.getElem {x y : List α} (h : x <:+ y) {n} (hn : n < x.length) :
+    x[n] = y[y.length - x.length + n]'(by have := h.length_le; omega) := by
+  rw [getElem_eq_getElem_reverse, h.reverse.getElem, getElem_reverse]
+  congr
+  have := h.length_le
+  omega
+
+theorem isSuffix_iff : l₁ <:+ l₂ ↔
+    l₁.length ≤ l₂.length ∧ ∀ i (h : i < l₁.length), l₂[i + l₂.length - l₁.length]? = some l₁[i] := by
+  suffices l₁.length ≤ l₂.length ∧ l₁ <:+ l₂ ↔
+      l₁.length ≤ l₂.length ∧ ∀ i (h : i < l₁.length), l₂[i + l₂.length - l₁.length]? = some l₁[i] by
+    constructor
+    · intro h
+      exact this.mp ⟨h.length_le, h⟩
+    · intro h
+      exact (this.mpr h).2
+  simp only [and_congr_right_iff]
+  intro le
+  rw [← reverse_prefix, isPrefix_iff]
+  simp only [length_reverse]
+  constructor
+  · intro w i h
+    specialize w (l₁.length - 1 - i) (by omega)
+    rw [getElem?_reverse (by omega)] at w
+    have p : l₂.length - 1 - (l₁.length - 1 - i) = i + l₂.length - l₁.length := by omega
+    rw [p] at w
+    rw [w, getElem_reverse]
+    congr
+    omega
+  · intro w i h
+    rw [getElem?_reverse]
+    specialize w (l₁.length - 1 - i) (by omega)
+    have p : l₁.length - 1 - i + l₂.length - l₁.length = l₂.length - 1 - i := by omega
+    rw [p] at w
+    rw [w, getElem_reverse]
+    exact Nat.lt_of_lt_of_le h le
+
+theorem isInfix_iff : l₁ <:+: l₂ ↔
+    ∃ k, l₁.length + k ≤ l₂.length ∧ ∀ i (h : i < l₁.length), l₂[i + k]? = some l₁[i] := by
+  constructor
+  · intro h
+    obtain ⟨t, p, s⟩ := infix_iff_suffix_prefix.mp h
+    refine ⟨t.length - l₁.length, by have := p.length_le; have := s.length_le; omega, ?_⟩
+    rw [isSuffix_iff] at p
+    obtain ⟨p', p⟩ := p
+    rw [isPrefix_iff] at s
+    intro i h
+    rw [s _ (by omega)]
+    specialize p i (by omega)
+    rw [Nat.add_sub_assoc (by omega)] at p
+    rw [← getElem?_eq_getElem, p]
+  · rintro ⟨k, le, w⟩
+    refine ⟨l₂.take k, l₂.drop (k + l₁.length), ?_⟩
+    ext1 i
+    rw [getElem?_append]
+    split
+    · rw [getElem?_append]
+      split
+      · rw [getElem?_take]; simp_all; omega
+      · simp_all
+        have p : i = (i - k) + k := by omega
+        rw [p, w _ (by omega), getElem?_eq_getElem]
+        · congr 2
+          omega
+        · omega
+    · rw [getElem?_drop]
+      congr
+      simp_all
+      omega
+
+theorem suffix_iff_eq_append : l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ :=
+  ⟨by rintro ⟨r, rfl⟩; simp only [length_append, Nat.add_sub_cancel_right, take_left], fun e =>
+    ⟨_, e⟩⟩
+
+theorem prefix_take_iff {x y : List α} {n : Nat} : x <+: y.take n ↔ x <+: y ∧ x.length ≤ n := by
+  constructor
+  · intro h
+    constructor
+    · exact List.IsPrefix.trans h <| List.take_prefix n y
+    · replace h := h.length_le
+      rw [length_take, Nat.le_min] at h
+      exact h.left
+  · intro ⟨hp, hl⟩
+    have hl' := hp.length_le
+    rw [List.prefix_iff_eq_take] at *
+    rw [hp, List.take_take]
+    simp [Nat.min_eq_left, hl, hl']
+
+theorem suffix_iff_eq_drop : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length l₁) l₂ :=
+  ⟨fun h => append_cancel_left <| (suffix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,
+    fun e => e.symm ▸ drop_suffix _ _⟩
+
+theorem prefix_take_le_iff {L : List α} (hm : m < L.length) :
+    L.take m <+: L.take n ↔ m ≤ n := by
+  simp only [prefix_iff_eq_take, length_take]
+  induction m generalizing L n with
+  | zero => simp [Nat.min_eq_left, eq_self_iff_true, Nat.zero_le, take]
+  | succ m IH =>
+    cases L with
+    | nil => simp_all
+    | cons l ls =>
+      cases n with
+      | zero =>
+        simp
+      | succ n =>
+        simp only [length_cons, Nat.succ_eq_add_one, Nat.add_lt_add_iff_right] at hm
+        simp [← @IH n ls hm, Nat.min_eq_left, Nat.le_of_lt hm]
+
+@[simp] theorem append_left_sublist_self (xs ys : List α) : xs ++ ys <+ ys ↔ xs = [] := by
+  constructor
+  · intro h
+    replace h := h.length_le
+    simp only [length_append] at h
+    have : xs.length = 0 := by omega
+    simp_all
+  · rintro rfl
+    simp
+@[simp] theorem append_right_sublist_self (xs ys : List α) : xs ++ ys <+ xs ↔ ys = [] := by
+  constructor
+  · intro h
+    replace h := h.length_le
+    simp only [length_append] at h
+    have : ys.length = 0 := by omega
+    simp_all
+  · rintro rfl
+    simp
+
+theorem append_sublist_of_sublist_left (xs ys zs : List α) (h : zs <+ xs) :
+    xs ++ ys <+ zs ↔ ys = [] ∧ xs = zs := by
+  constructor
+  · intro h'
+    have hl := h.length_le
+    have hl' := h'.length_le
+    simp only [length_append] at hl'
+    have : ys.length = 0 := by omega
+    simp_all only [Nat.add_zero, length_eq_zero, true_and, append_nil]
+    exact Sublist.eq_of_length_le h' hl
+  · rintro ⟨rfl, rfl⟩
+    simp
+
+theorem append_sublist_of_sublist_right (xs ys zs : List α) (h : zs <+ ys) :
+    xs ++ ys <+ zs ↔ xs = [] ∧ ys = zs := by
+  constructor
+  · intro h'
+    have hl := h.length_le
+    have hl' := h'.length_le
+    simp only [length_append] at hl'
+    have : xs.length = 0 := by omega
+    simp_all only [Nat.zero_add, length_eq_zero, true_and, append_nil]
+    exact Sublist.eq_of_length_le h' hl
+  · rintro ⟨rfl, rfl⟩
+    simp
+
+end List

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

@@ -5,6 +5,7 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Zip
+import Init.Data.List.Sublist
 import Init.Data.Nat.Lemmas
 
 /-!
@@ -69,20 +70,20 @@ theorem get?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
     (l.take n).get? m = none := by
   simp [getElem?_take_eq_none h]
 
-theorem getElem?_take_eq_if {l : List α} {n m : Nat} :
+theorem getElem?_take {l : List α} {n m : Nat} :
     (l.take n)[m]? = if m < n then l[m]? else none := by
   split
-  · next h => exact getElem?_take h
+  · next h => exact getElem?_take_of_lt h
   · next h => exact getElem?_take_eq_none (Nat.le_of_not_lt h)
 
-@[deprecated getElem?_take_eq_if (since := "2024-06-12")]
+@[deprecated getElem?_take (since := "2024-06-12")]
 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
-  simp [getElem?_take_eq_if]
+  simp [getElem?_take]
 
 theorem head?_take {l : List α} {n : Nat} :
     (l.take n).head? = if n = 0 then none else l.head? := by
-  simp [head?_eq_getElem?, getElem?_take_eq_if]
+  simp [head?_eq_getElem?, getElem?_take]
   split
   · rw [if_neg (by omega)]
   · rw [if_pos (by omega)]
@@ -94,7 +95,7 @@ theorem head_take {l : List α} {n : Nat} (h : l.take n ≠ []) :
   simp_all
 
 theorem getLast?_take {l : List α} : (l.take n).getLast? = if n = 0 then none else l[n - 1]?.or l.getLast? := by
-  rw [getLast?_eq_getElem?, getElem?_take_eq_if, length_take]
+  rw [getLast?_eq_getElem?, getElem?_take, length_take]
   split
   · rw [if_neg (by omega)]
     rw [Nat.min_def]
@@ -127,7 +128,7 @@ theorem take_take : ∀ (n m) (l : List α), take n (take m l) = take (min n m)
 theorem take_set_of_lt (a : α) {n m : Nat} (l : List α) (h : m < n) :
     (l.set n a).take m = l.take m :=
   List.ext_getElem? fun i => by
-    rw [getElem?_take_eq_if, getElem?_take_eq_if]
+    rw [getElem?_take, getElem?_take]
     split
     · next h' => rw [getElem?_set_ne (by omega)]
     · rfl
@@ -199,6 +200,19 @@ theorem map_eq_append_split {f : α → β} {l : List α} {s₁ s₂ : List β}
   rw [← length_map l f, h, length_append]
   apply Nat.le_add_right
 
+theorem take_prefix_take_left (l : List α) {m n : Nat} (h : m ≤ n) : take m l <+: take n l := by
+  rw [isPrefix_iff]
+  intro i w
+  rw [getElem?_take_of_lt, getElem_take', getElem?_eq_getElem]
+  simp only [length_take] at w
+  exact Nat.lt_of_lt_of_le (Nat.lt_of_lt_of_le w (Nat.min_le_left _ _)) h
+
+theorem take_sublist_take_left (l : List α) {m n : Nat} (h : m ≤ n) : take m l <+ take n l :=
+  (take_prefix_take_left l h).sublist
+
+theorem take_subset_take_left (l : List α) {m n : Nat} (h : m ≤ n) : take m l ⊆ take n l :=
+  (take_sublist_take_left l h).subset
+
 /-! ### drop -/
 
 theorem lt_length_drop (L : List α) {i j : Nat} (h : i + j < L.length) : j < (L.drop i).length := by
@@ -261,7 +275,7 @@ theorem head?_drop (l : List α) (n : Nat) :
 theorem head_drop {l : List α} {n : Nat} (h : l.drop n ≠ []) :
     (l.drop n).head h = l[n]'(by simp_all) := by
   have w : n < l.length := length_lt_of_drop_ne_nil h
-  simpa [head?_eq_head, getElem?_eq_getElem, h, w] using head?_drop l n
+  simpa [getElem?_eq_getElem, h, w, head_eq_iff_head?_eq_some] using head?_drop l n
 
 theorem getLast?_drop {l : List α} : (l.drop n).getLast? = if l.length ≤ n then none else l.getLast? := by
   rw [getLast?_eq_getElem?, getElem?_drop]
@@ -319,8 +333,8 @@ theorem set_eq_take_append_cons_drop {l : List α} {n : Nat} {a : α} :
   split <;> rename_i h
   · ext1 m
     by_cases h' : m < n
-    · rw [getElem?_append (by simp [length_take]; omega), getElem?_set_ne (by omega),
-        getElem?_take h']
+    · rw [getElem?_append_left (by simp [length_take]; omega), getElem?_set_ne (by omega),
+        getElem?_take_of_lt h']
     · by_cases h'' : m = n
       · subst h''
         rw [getElem?_set_eq ‹_›, getElem?_append_right, length_take,
@@ -359,40 +373,67 @@ theorem drop_take : ∀ (m n : Nat) (l : List α), drop n (take m l) = take (m -
     congr 1
     omega
 
-theorem take_reverse {α} {xs : List α} {n : Nat} (h : n ≤ xs.length) :
+theorem take_reverse {α} {xs : List α} {n : Nat} :
     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
+  by_cases h : n ≤ xs.length
+  · 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
+  · have w : xs.length - n = 0 := by omega
+    rw [take_of_length_le, w, drop_zero]
+    simp
+    omega
 
-@[deprecated (since := "2024-06-15")] abbrev reverse_take := @take_reverse
-
-theorem drop_reverse {α} {xs : List α} {n : Nat} (h : n ≤ xs.length) :
+theorem drop_reverse {α} {xs : List α} {n : Nat} :
     xs.reverse.drop n = (xs.take (xs.length - n)).reverse := by
-  conv =>
-    rhs
-    rw [← reverse_reverse xs]
-  rw [← reverse_reverse xs] at h
-  generalize xs.reverse = xs' at h ⊢
-  rw [take_reverse]
-  · simp only [length_reverse, reverse_reverse] at *
+  by_cases h : n ≤ xs.length
+  · conv =>
+      rhs
+      rw [← reverse_reverse xs]
+    rw [← reverse_reverse xs] at h
+    generalize xs.reverse = xs' at h ⊢
+    rw [take_reverse]
+    · simp only [length_reverse, reverse_reverse] at *
+      congr
+      omega
+  · have w : xs.length - n = 0 := by omega
+    rw [drop_of_length_le, w, take_zero, reverse_nil]
+    simp
+    omega
+
+theorem reverse_take {l : List α} {n : Nat} :
+    (l.take n).reverse = l.reverse.drop (l.length - n) := by
+  by_cases h : n ≤ l.length
+  · rw [drop_reverse]
     congr
     omega
-  · simp only [length_reverse, sub_le]
+  · have w : l.length - n = 0 := by omega
+    rw [w, drop_zero, take_of_length_le]
+    omega
+
+theorem reverse_drop {l : List α} {n : Nat} :
+    (l.drop n).reverse = l.reverse.take (l.length - n) := by
+  by_cases h : n ≤ l.length
+  · rw [take_reverse]
+    congr
+    omega
+  · have w : l.length - n = 0 := by omega
+    rw [w, take_zero, drop_of_length_le, reverse_nil]
+    omega
 
 /-! ### rotateLeft -/
 

src/Init/Data/List/Pairwise.lean

@@ -5,6 +5,7 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Sublist
+import Init.Data.List.Attach
 
 /-!
 # Lemmas about `List.Pairwise` and `List.Nodup`.
@@ -122,7 +123,7 @@ theorem pairwise_filterMap (f : β → Option α) {l : List β} :
   match e : f a with
   | none =>
     rw [filterMap_cons_none e, pairwise_cons]
-    simp only [e, false_implies, implies_true, true_and, IH]
+    simp only [e, false_implies, implies_true, true_and, IH, reduceCtorEq]
   | some b =>
     rw [filterMap_cons_some e]
     simpa [IH, e] using fun _ =>
@@ -225,6 +226,43 @@ theorem pairwise_iff_forall_sublist : l.Pairwise R ↔ (∀ {a b}, [a,b] <+ l
         intro a b hab
         apply h; exact hab.cons _
 
+theorem Pairwise.rel_of_mem_take_of_mem_drop
+    {l : List α} (h : l.Pairwise R) (hx : x ∈ l.take n) (hy : y ∈ l.drop n) : R x y := by
+  apply pairwise_iff_forall_sublist.mp h
+  rw [← take_append_drop n l, sublist_append_iff]
+  refine ⟨[x], [y], rfl, by simpa, by simpa⟩
+
+theorem Pairwise.rel_of_mem_append
+    {l₁ l₂ : List α} (h : (l₁ ++ l₂).Pairwise R) (hx : x ∈ l₁) (hy : y ∈ l₂) : R x y := by
+  apply pairwise_iff_forall_sublist.mp h
+  rw [sublist_append_iff]
+  exact ⟨[x], [y], rfl, by simpa, by simpa⟩
+
+theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h : ∀ a ∈ l, ∀ b ∈ l, r a b) :
+    l.Pairwise r := by
+  rw [pairwise_iff_forall_sublist]
+  intro a b hab
+  apply h <;> (apply hab.subset; simp)
+
+theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
+    Pairwise R (l.pmap f h) ↔
+      Pairwise (fun b₁ b₂ => ∀ (h₁ : p b₁) (h₂ : p b₂), R (f b₁ h₁) (f b₂ h₂)) l := by
+  induction l with
+  | nil => simp
+  | cons a l ihl =>
+    obtain ⟨_, hl⟩ : p a ∧ ∀ b, b ∈ l → p b := by simpa using h
+    simp only [ihl hl, pairwise_cons, exists₂_imp, pmap, and_congr_left_iff, mem_pmap]
+    refine fun _ => ⟨fun H b hb _ hpb => H _ _ hb rfl, ?_⟩
+    rintro H _ b hb rfl
+    exact H b hb _ _
+
+theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
+    (h : ∀ x ∈ l, p x) {S : β → β → Prop}
+    (hS : ∀ ⦃x⦄ (hx : p x) ⦃y⦄ (hy : p y), R x y → S (f x hx) (f y hy)) :
+    Pairwise S (l.pmap f h) := by
+  refine (pairwise_pmap h).2 (Pairwise.imp_of_mem ?_ hl)
+  intros; apply hS; assumption
+
 /-! ### Nodup -/
 
 @[simp]

src/Init/Data/List/Perm.lean

@@ -0,0 +1,463 @@
+/-
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Pairwise
+import Init.Data.List.Erase
+
+/-!
+# List Permutations
+
+This file introduces the `List.Perm` relation, which is true if two lists are permutations of one
+another.
+
+## Notation
+
+The notation `~` is used for permutation equivalence.
+-/
+
+open Nat
+
+namespace List
+
+open Perm (swap)
+
+@[simp, refl] protected theorem Perm.refl : ∀ l : List α, l ~ l
+  | [] => .nil
+  | x :: xs => (Perm.refl xs).cons x
+
+protected theorem Perm.rfl {l : List α} : l ~ l := .refl _
+
+theorem Perm.of_eq (h : l₁ = l₂) : l₁ ~ l₂ := h ▸ .rfl
+
+protected theorem Perm.symm {l₁ l₂ : List α} (h : l₁ ~ l₂) : l₂ ~ l₁ := by
+  induction h with
+  | nil => exact nil
+  | cons _ _ ih => exact cons _ ih
+  | swap => exact swap ..
+  | trans _ _ ih₁ ih₂ => exact trans ih₂ ih₁
+
+theorem perm_comm {l₁ l₂ : List α} : l₁ ~ l₂ ↔ l₂ ~ l₁ := ⟨Perm.symm, Perm.symm⟩
+
+theorem Perm.swap' (x y : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) : y :: x :: l₁ ~ x :: y :: l₂ :=
+  (swap ..).trans <| p.cons _ |>.cons _
+
+/--
+Similar to `Perm.recOn`, but the `swap` case is generalized to `Perm.swap'`,
+where the tail of the lists are not necessarily the same.
+-/
+@[elab_as_elim] theorem Perm.recOnSwap'
+    {motive : (l₁ : List α) → (l₂ : List α) → l₁ ~ l₂ → Prop} {l₁ l₂ : List α} (p : l₁ ~ l₂)
+    (nil : motive [] [] .nil)
+    (cons : ∀ x {l₁ l₂}, (h : l₁ ~ l₂) → motive l₁ l₂ h → motive (x :: l₁) (x :: l₂) (.cons x h))
+    (swap' : ∀ x y {l₁ l₂}, (h : l₁ ~ l₂) → motive l₁ l₂ h →
+      motive (y :: x :: l₁) (x :: y :: l₂) (.swap' _ _ h))
+    (trans : ∀ {l₁ l₂ l₃}, (h₁ : l₁ ~ l₂) → (h₂ : l₂ ~ l₃) → motive l₁ l₂ h₁ → motive l₂ l₃ h₂ →
+      motive l₁ l₃ (.trans h₁ h₂)) : motive l₁ l₂ p :=
+  have motive_refl l : motive l l (.refl l) :=
+    List.recOn l nil fun x xs ih => cons x (.refl xs) ih
+  Perm.recOn p nil cons (fun x y l => swap' x y (.refl l) (motive_refl l)) trans
+
+theorem Perm.eqv (α) : Equivalence (@Perm α) := ⟨.refl, .symm, .trans⟩
+
+instance isSetoid (α) : Setoid (List α) := .mk Perm (Perm.eqv α)
+
+theorem Perm.mem_iff {a : α} {l₁ l₂ : List α} (p : l₁ ~ l₂) : a ∈ l₁ ↔ a ∈ l₂ := by
+  induction p with
+  | nil => rfl
+  | cons _ _ ih => simp only [mem_cons, ih]
+  | swap => simp only [mem_cons, or_left_comm]
+  | trans _ _ ih₁ ih₂ => simp only [ih₁, ih₂]
+
+theorem Perm.subset {l₁ l₂ : List α} (p : l₁ ~ l₂) : l₁ ⊆ l₂ := fun _ => p.mem_iff.mp
+
+theorem Perm.append_right {l₁ l₂ : List α} (t₁ : List α) (p : l₁ ~ l₂) : l₁ ++ t₁ ~ l₂ ++ t₁ := by
+  induction p with
+  | nil => rfl
+  | cons _ _ ih => exact cons _ ih
+  | swap => exact swap ..
+  | trans _ _ ih₁ ih₂ => exact trans ih₁ ih₂
+
+theorem Perm.append_left {t₁ t₂ : List α} : ∀ l : List α, t₁ ~ t₂ → l ++ t₁ ~ l ++ t₂
+  | [], p => p
+  | x :: xs, p => (p.append_left xs).cons x
+
+theorem Perm.append {l₁ l₂ t₁ t₂ : List α} (p₁ : l₁ ~ l₂) (p₂ : t₁ ~ t₂) : l₁ ++ t₁ ~ l₂ ++ t₂ :=
+  (p₁.append_right t₁).trans (p₂.append_left l₂)
+
+theorem Perm.append_cons (a : α) {h₁ h₂ t₁ t₂ : List α} (p₁ : h₁ ~ h₂) (p₂ : t₁ ~ t₂) :
+    h₁ ++ a :: t₁ ~ h₂ ++ a :: t₂ := p₁.append (p₂.cons a)
+
+@[simp] theorem perm_middle {a : α} : ∀ {l₁ l₂ : List α}, l₁ ++ a :: l₂ ~ a :: (l₁ ++ l₂)
+  | [], _ => .refl _
+  | b :: _, _ => (Perm.cons _ perm_middle).trans (swap a b _)
+
+@[simp] theorem perm_append_singleton (a : α) (l : List α) : l ++ [a] ~ a :: l :=
+  perm_middle.trans <| by rw [append_nil]
+
+theorem perm_append_comm : ∀ {l₁ l₂ : List α}, l₁ ++ l₂ ~ l₂ ++ l₁
+  | [], l₂ => by simp
+  | a :: t, l₂ => (perm_append_comm.cons _).trans perm_middle.symm
+
+theorem perm_append_comm_assoc (l₁ l₂ l₃ : List α) :
+    Perm (l₁ ++ (l₂ ++ l₃)) (l₂ ++ (l₁ ++ l₃)) := by
+  simpa only [List.append_assoc] using perm_append_comm.append_right _
+
+theorem concat_perm (l : List α) (a : α) : concat l a ~ a :: l := by simp
+
+theorem Perm.length_eq {l₁ l₂ : List α} (p : l₁ ~ l₂) : length l₁ = length l₂ := by
+  induction p with
+  | nil => rfl
+  | cons _ _ ih => simp only [length_cons, ih]
+  | swap => rfl
+  | trans _ _ ih₁ ih₂ => simp only [ih₁, ih₂]
+
+theorem Perm.eq_nil {l : List α} (p : l ~ []) : l = [] := eq_nil_of_length_eq_zero p.length_eq
+
+theorem Perm.nil_eq {l : List α} (p : [] ~ l) : [] = l := p.symm.eq_nil.symm
+
+@[simp] theorem perm_nil {l₁ : List α} : l₁ ~ [] ↔ l₁ = [] :=
+  ⟨fun p => p.eq_nil, fun e => e ▸ .rfl⟩
+
+@[simp] theorem nil_perm {l₁ : List α} : [] ~ l₁ ↔ l₁ = [] := perm_comm.trans perm_nil
+
+theorem not_perm_nil_cons (x : α) (l : List α) : ¬[] ~ x :: l := (nomatch ·.symm.eq_nil)
+
+theorem not_perm_cons_nil {l : List α} {a : α} : ¬(Perm (a::l) []) :=
+  fun h => by simpa using h.length_eq
+
+theorem Perm.isEmpty_eq {l l' : List α} (h : Perm l l') : l.isEmpty = l'.isEmpty := by
+  cases l <;> cases l' <;> simp_all
+
+@[simp] theorem reverse_perm : ∀ l : List α, reverse l ~ l
+  | [] => .nil
+  | a :: l => reverse_cons .. ▸ (perm_append_singleton _ _).trans ((reverse_perm l).cons a)
+
+theorem perm_cons_append_cons {l l₁ l₂ : List α} (a : α) (p : l ~ l₁ ++ l₂) :
+    a :: l ~ l₁ ++ a :: l₂ := (p.cons a).trans perm_middle.symm
+
+@[simp] theorem perm_replicate {n : Nat} {a : α} {l : List α} :
+    l ~ replicate n a ↔ l = replicate n a := by
+  refine ⟨fun p => eq_replicate.2 ?_, fun h => h ▸ .rfl⟩
+  exact ⟨p.length_eq.trans <| length_replicate .., fun _b m => eq_of_mem_replicate <| p.subset m⟩
+
+@[simp] theorem replicate_perm {n : Nat} {a : α} {l : List α} :
+    replicate n a ~ l ↔ replicate n a = l := (perm_comm.trans perm_replicate).trans eq_comm
+
+@[simp] theorem perm_singleton {a : α} {l : List α} : l ~ [a] ↔ l = [a] := perm_replicate (n := 1)
+
+@[simp] theorem singleton_perm {a : α} {l : List α} : [a] ~ l ↔ [a] = l := replicate_perm (n := 1)
+
+theorem Perm.eq_singleton (h : l ~ [a]) : l = [a] := perm_singleton.mp h
+theorem Perm.singleton_eq (h : [a] ~ l) : [a] = l := singleton_perm.mp h
+
+theorem singleton_perm_singleton {a b : α} : [a] ~ [b] ↔ a = b := by simp
+
+theorem perm_cons_erase [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : l ~ a :: l.erase a :=
+  let ⟨_l₁, _l₂, _, e₁, e₂⟩ := exists_erase_eq h
+  e₂ ▸ e₁ ▸ perm_middle
+
+theorem Perm.filterMap (f : α → Option β) {l₁ l₂ : List α} (p : l₁ ~ l₂) :
+    filterMap f l₁ ~ filterMap f l₂ := by
+  induction p with
+  | nil => simp
+  | cons x _p IH => cases h : f x <;> simp [h, filterMap_cons, IH, Perm.cons]
+  | swap x y l₂ => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap_cons, swap]
+  | trans _p₁ _p₂ IH₁ IH₂ => exact IH₁.trans IH₂
+
+theorem Perm.map (f : α → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) : map f l₁ ~ map f l₂ :=
+  filterMap_eq_map f ▸ p.filterMap _
+
+theorem Perm.pmap {p : α → Prop} (f : ∀ a, p a → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) {H₁ H₂} :
+    pmap f l₁ H₁ ~ pmap f l₂ H₂ := by
+  induction p with
+  | nil => simp
+  | cons x _p IH => simp [IH, Perm.cons]
+  | swap x y => simp [swap]
+  | trans _p₁ p₂ IH₁ IH₂ => exact IH₁.trans (IH₂ (H₁ := fun a m => H₂ a (p₂.subset m)))
+
+theorem Perm.filter (p : α → Bool) {l₁ l₂ : List α} (s : l₁ ~ l₂) :
+    filter p l₁ ~ filter p l₂ := by rw [← filterMap_eq_filter]; apply s.filterMap
+
+theorem filter_append_perm (p : α → Bool) (l : List α) :
+    filter p l ++ filter (fun x => !p x) l ~ l := by
+  induction l with
+  | nil => rfl
+  | cons x l ih =>
+    by_cases h : p x <;> simp [h]
+    · exact ih.cons x
+    · exact Perm.trans (perm_append_comm.trans (perm_append_comm.cons _)) (ih.cons x)
+
+theorem exists_perm_sublist {l₁ l₂ l₂' : List α} (s : l₁ <+ l₂) (p : l₂ ~ l₂') :
+    ∃ l₁', l₁' ~ l₁ ∧ l₁' <+ l₂' := by
+  induction p generalizing l₁ with
+  | nil => exact ⟨[], sublist_nil.mp s ▸ .rfl, nil_sublist _⟩
+  | cons x _ IH =>
+    match s with
+    | .cons _ s => let ⟨l₁', p', s'⟩ := IH s; exact ⟨l₁', p', s'.cons _⟩
+    | .cons₂ _ s => let ⟨l₁', p', s'⟩ := IH s; exact ⟨x :: l₁', p'.cons x, s'.cons₂ _⟩
+  | swap x y l' =>
+    match s with
+    | .cons _ (.cons _ s) => exact ⟨_, .rfl, (s.cons _).cons _⟩
+    | .cons _ (.cons₂ _ s) => exact ⟨x :: _, .rfl, (s.cons _).cons₂ _⟩
+    | .cons₂ _ (.cons _ s) => exact ⟨y :: _, .rfl, (s.cons₂ _).cons _⟩
+    | .cons₂ _ (.cons₂ _ s) => exact ⟨x :: y :: _, .swap .., (s.cons₂ _).cons₂ _⟩
+  | trans _ _ IH₁ IH₂ =>
+    let ⟨m₁, pm, sm⟩ := IH₁ s
+    let ⟨r₁, pr, sr⟩ := IH₂ sm
+    exact ⟨r₁, pr.trans pm, sr⟩
+
+theorem Perm.sizeOf_eq_sizeOf [SizeOf α] {l₁ l₂ : List α} (h : l₁ ~ l₂) :
+    sizeOf l₁ = sizeOf l₂ := by
+  induction h with
+  | nil => rfl
+  | cons _ _ h_sz₁₂ => simp [h_sz₁₂]
+  | swap => simp [Nat.add_left_comm]
+  | trans _ _ h_sz₁₂ h_sz₂₃ => simp [h_sz₁₂, h_sz₂₃]
+
+theorem Sublist.exists_perm_append {l₁ l₂ : List α} : l₁ <+ l₂ → ∃ l, l₂ ~ l₁ ++ l
+  | Sublist.slnil => ⟨nil, .rfl⟩
+  | Sublist.cons a s =>
+    let ⟨l, p⟩ := Sublist.exists_perm_append s
+    ⟨a :: l, (p.cons a).trans perm_middle.symm⟩
+  | Sublist.cons₂ a s =>
+    let ⟨l, p⟩ := Sublist.exists_perm_append s
+    ⟨l, p.cons a⟩
+
+theorem Perm.countP_eq (p : α → Bool) {l₁ l₂ : List α} (s : l₁ ~ l₂) :
+    countP p l₁ = countP p l₂ := by
+  simp only [countP_eq_length_filter]
+  exact (s.filter _).length_eq
+
+theorem Perm.countP_congr {l₁ l₂ : List α} (s : l₁ ~ l₂) {p p' : α → Bool}
+    (hp : ∀ x ∈ l₁, p x = p' x) : l₁.countP p = l₂.countP p' := by
+  rw [← s.countP_eq p']
+  clear s
+  induction l₁ with
+  | nil => rfl
+  | cons y s hs =>
+    simp only [mem_cons, forall_eq_or_imp] at hp
+    simp only [countP_cons, hs hp.2, hp.1]
+
+theorem countP_eq_countP_filter_add (l : List α) (p q : α → Bool) :
+    l.countP p = (l.filter q).countP p + (l.filter fun a => !q a).countP p :=
+  countP_append .. ▸ Perm.countP_eq _ (filter_append_perm _ _).symm
+
+theorem Perm.count_eq [DecidableEq α] {l₁ l₂ : List α} (p : l₁ ~ l₂) (a) :
+    count a l₁ = count a l₂ := p.countP_eq _
+
+theorem Perm.foldl_eq' {f : β → α → β} {l₁ l₂ : List α} (p : l₁ ~ l₂)
+    (comm : ∀ x ∈ l₁, ∀ y ∈ l₁, ∀ (z), f (f z x) y = f (f z y) x)
+    (init) : foldl f init l₁ = foldl f init l₂ := by
+  induction p using recOnSwap' generalizing init with
+  | nil => simp
+  | cons x _p IH =>
+    simp only [foldl]
+    apply IH; intros; apply comm <;> exact .tail _ ‹_›
+  | swap' x y _p IH =>
+    simp only [foldl]
+    rw [comm x (.tail _ <| .head _) y (.head _)]
+    apply IH; intros; apply comm <;> exact .tail _ (.tail _ ‹_›)
+  | trans p₁ _p₂ IH₁ IH₂ =>
+    refine (IH₁ comm init).trans (IH₂ ?_ _)
+    intros; apply comm <;> apply p₁.symm.subset <;> assumption
+
+theorem Perm.rec_heq {β : List α → Sort _} {f : ∀ a l, β l → β (a :: l)} {b : β []} {l l' : List α}
+    (hl : l ~ l') (f_congr : ∀ {a l l' b b'}, l ~ l' → HEq b b' → HEq (f a l b) (f a l' b'))
+    (f_swap : ∀ {a a' l b}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))) :
+    HEq (@List.rec α β b f l) (@List.rec α β b f l') := by
+  induction hl with
+  | nil => rfl
+  | cons a h ih => exact f_congr h ih
+  | swap a a' l => exact f_swap
+  | trans _h₁ _h₂ ih₁ ih₂ => exact ih₁.trans ih₂
+
+/-- Lemma used to destruct perms element by element. -/
+theorem perm_inv_core {a : α} {l₁ l₂ r₁ r₂ : List α} :
+    l₁ ++ a :: r₁ ~ l₂ ++ a :: r₂ → l₁ ++ r₁ ~ l₂ ++ r₂ := by
+  -- Necessary generalization for `induction`
+  suffices ∀ s₁ s₂ (_ : s₁ ~ s₂) {l₁ l₂ r₁ r₂},
+      l₁ ++ a :: r₁ = s₁ → l₂ ++ a :: r₂ = s₂ → l₁ ++ r₁ ~ l₂ ++ r₂ from (this _ _ · rfl rfl)
+  intro s₁ s₂ p
+  induction p using Perm.recOnSwap' with intro l₁ l₂ r₁ r₂ e₁ e₂
+  | nil =>
+    simp at e₁
+  | cons x p IH =>
+    cases l₁ <;> cases l₂ <;>
+      dsimp at e₁ e₂ <;> injections <;> subst_vars
+    · exact p
+    · exact p.trans perm_middle
+    · exact perm_middle.symm.trans p
+    · exact (IH rfl rfl).cons _
+  | swap' x y p IH =>
+    obtain _ | ⟨y, _ | ⟨z, l₁⟩⟩ := l₁
+      <;> obtain _ | ⟨u, _ | ⟨v, l₂⟩⟩ := l₂
+      <;> dsimp at e₁ e₂ <;> injections <;> subst_vars
+      <;> try exact p.cons _
+    · exact (p.trans perm_middle).cons u
+    · exact ((p.trans perm_middle).cons _).trans (swap _ _ _)
+    · exact (perm_middle.symm.trans p).cons y
+    · exact (swap _ _ _).trans ((perm_middle.symm.trans p).cons u)
+    · exact (IH rfl rfl).swap' _ _
+  | trans p₁ p₂ IH₁ IH₂ =>
+    subst e₁ e₂
+    obtain ⟨l₂, r₂, rfl⟩ := append_of_mem (a := a) (p₁.subset (by simp))
+    exact (IH₁ rfl rfl).trans (IH₂ rfl rfl)
+
+theorem Perm.cons_inv {a : α} {l₁ l₂ : List α} : a :: l₁ ~ a :: l₂ → l₁ ~ l₂ :=
+  perm_inv_core (l₁ := []) (l₂ := [])
+
+@[simp] theorem perm_cons (a : α) {l₁ l₂ : List α} : a :: l₁ ~ a :: l₂ ↔ l₁ ~ l₂ :=
+  ⟨.cons_inv, .cons a⟩
+
+theorem perm_append_left_iff {l₁ l₂ : List α} : ∀ l, l ++ l₁ ~ l ++ l₂ ↔ l₁ ~ l₂
+  | [] => .rfl
+  | a :: l => (perm_cons a).trans (perm_append_left_iff l)
+
+theorem perm_append_right_iff {l₁ l₂ : List α} (l) : l₁ ++ l ~ l₂ ++ l ↔ l₁ ~ l₂ := by
+  refine ⟨fun p => ?_, .append_right _⟩
+  exact (perm_append_left_iff _).1 <| perm_append_comm.trans <| p.trans perm_append_comm
+
+section DecidableEq
+
+variable [DecidableEq α]
+
+theorem Perm.erase (a : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) : l₁.erase a ~ l₂.erase a :=
+  if h₁ : a ∈ l₁ then
+    have h₂ : a ∈ l₂ := p.subset h₁
+    .cons_inv <| (perm_cons_erase h₁).symm.trans <| p.trans (perm_cons_erase h₂)
+  else by
+    have h₂ : a ∉ l₂ := mt p.mem_iff.2 h₁
+    rw [erase_of_not_mem h₁, erase_of_not_mem h₂]; exact p
+
+theorem cons_perm_iff_perm_erase {a : α} {l₁ l₂ : List α} :
+    a :: l₁ ~ l₂ ↔ a ∈ l₂ ∧ l₁ ~ l₂.erase a := by
+  refine ⟨fun h => ?_, fun ⟨m, h⟩ => (h.cons a).trans (perm_cons_erase m).symm⟩
+  have : a ∈ l₂ := h.subset (mem_cons_self a l₁)
+  exact ⟨this, (h.trans <| perm_cons_erase this).cons_inv⟩
+
+theorem perm_iff_count {l₁ l₂ : List α} : l₁ ~ l₂ ↔ ∀ a, count a l₁ = count a l₂ := by
+  refine ⟨Perm.count_eq, fun H => ?_⟩
+  induction l₁ generalizing l₂ with
+  | nil =>
+    match l₂ with
+    | nil => rfl
+    | cons b l₂ =>
+      specialize H b
+      simp at H
+  | cons a l₁ IH =>
+    have : a ∈ l₂ := count_pos_iff_mem.mp (by rw [← H]; simp)
+    refine ((IH fun b => ?_).cons a).trans (perm_cons_erase this).symm
+    specialize H b
+    rw [(perm_cons_erase this).count_eq] at H
+    by_cases h : b = a <;> simpa [h, count_cons, Nat.succ_inj'] using H
+
+theorem isPerm_iff : ∀ {l₁ l₂ : List α}, l₁.isPerm l₂ ↔ l₁ ~ l₂
+  | [], [] => by simp [isPerm, isEmpty]
+  | [], _ :: _ => by simp [isPerm, isEmpty, Perm.nil_eq]
+  | a :: l₁, l₂ => by simp [isPerm, isPerm_iff, cons_perm_iff_perm_erase]
+
+instance decidablePerm (l₁ l₂ : List α) : Decidable (l₁ ~ l₂) := decidable_of_iff _ isPerm_iff
+
+protected theorem Perm.insert (a : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) :
+    l₁.insert a ~ l₂.insert a := by
+  if h : a ∈ l₁ then
+    simp [h, p.subset h, p]
+  else
+    have := p.cons a
+    simpa [h, mt p.mem_iff.2 h] using this
+
+theorem perm_insert_swap (x y : α) (l : List α) :
+    List.insert x (List.insert y l) ~ List.insert y (List.insert x l) := by
+  by_cases xl : x ∈ l <;> by_cases yl : y ∈ l <;> simp [xl, yl]
+  if xy : x = y then simp [xy] else
+  simp [List.insert, xl, yl, xy, Ne.symm xy]
+  constructor
+
+end DecidableEq
+
+theorem Perm.pairwise_iff {R : α → α → Prop} (S : ∀ {x y}, R x y → R y x) :
+    ∀ {l₁ l₂ : List α} (_p : l₁ ~ l₂), Pairwise R l₁ ↔ Pairwise R l₂ :=
+  suffices ∀ {l₁ l₂}, l₁ ~ l₂ → Pairwise R l₁ → Pairwise R l₂
+    from fun p => ⟨this p, this p.symm⟩
+  fun {l₁ l₂} p d => by
+    induction d generalizing l₂ with
+    | nil => rw [← p.nil_eq]; constructor
+    | cons h _ IH =>
+      have : _ ∈ l₂ := p.subset (mem_cons_self _ _)
+      obtain ⟨s₂, t₂, rfl⟩ := append_of_mem this
+      have p' := (p.trans perm_middle).cons_inv
+      refine (pairwise_middle S).2 (pairwise_cons.2 ⟨fun b m => ?_, IH p'⟩)
+      exact h _ (p'.symm.subset m)
+
+theorem Pairwise.perm {R : α → α → Prop} {l l' : List α} (hR : l.Pairwise R) (hl : l ~ l')
+    (hsymm : ∀ {x y}, R x y → R y x) : l'.Pairwise R := (hl.pairwise_iff hsymm).mp hR
+
+theorem Perm.pairwise {R : α → α → Prop} {l l' : List α} (hl : l ~ l') (hR : l.Pairwise R)
+    (hsymm : ∀ {x y}, R x y → R y x) : l'.Pairwise R := hR.perm hl hsymm
+
+/--
+If two lists are sorted by an antisymmetric relation, and permutations of each other,
+they must be equal.
+-/
+theorem Perm.eq_of_sorted : ∀ {l₁ l₂ : List α}
+    (_ : ∀ a b, a ∈ l₁ → b ∈ l₂ → le a b → le b a → a = b)
+    (_ : l₁.Pairwise le) (_ : l₂.Pairwise le) (_ : l₁ ~ l₂), l₁ = l₂
+  | [], [], _, _, _, _ => rfl
+  | [], b :: l₂, _, _, _, h => by simp_all
+  | a :: l₁, [], _, _, _, h => by simp_all
+  | a :: l₁, b :: l₂, w, h₁, h₂, h => by
+    have am : a ∈ b :: l₂ := h.subset (mem_cons_self _ _)
+    have bm : b ∈ a :: l₁ := h.symm.subset (mem_cons_self _ _)
+    have ab : a = b := by
+      simp only [mem_cons] at am
+      rcases am with rfl | am
+      · rfl
+      · simp only [mem_cons] at bm
+        rcases bm with rfl | bm
+        · rfl
+        · exact w _ _ (mem_cons_self _ _) (mem_cons_self _ _)
+            (rel_of_pairwise_cons h₁ bm) (rel_of_pairwise_cons h₂ am)
+    subst ab
+    simp only [perm_cons] at h
+    have := Perm.eq_of_sorted
+      (fun x y hx hy => w x y (mem_cons_of_mem a hx) (mem_cons_of_mem a hy))
+      h₁.tail h₂.tail h
+    simp_all
+
+theorem Nodup.perm {l l' : List α} (hR : l.Nodup) (hl : l ~ l') : l'.Nodup :=
+  Pairwise.perm hR hl (by intro x y h h'; simp_all)
+
+theorem Perm.nodup {l l' : List α} (hl : l ~ l') (hR : l.Nodup) : l'.Nodup := hR.perm hl
+
+theorem Perm.nodup_iff {l₁ l₂ : List α} : l₁ ~ l₂ → (Nodup l₁ ↔ Nodup l₂) :=
+  Perm.pairwise_iff <| @Ne.symm α
+
+theorem Perm.join {l₁ l₂ : List (List α)} (h : l₁ ~ l₂) : l₁.join ~ l₂.join := by
+  induction h with
+  | nil => rfl
+  | cons _ _ ih => simp only [join_cons, perm_append_left_iff, ih]
+  | swap => simp only [join_cons, ← append_assoc, perm_append_right_iff]; exact perm_append_comm ..
+  | trans _ _ ih₁ ih₂ => exact trans ih₁ ih₂
+
+theorem Perm.bind_right {l₁ l₂ : List α} (f : α → List β) (p : l₁ ~ l₂) : l₁.bind f ~ l₂.bind f :=
+  (p.map _).join
+
+theorem Perm.eraseP (f : α → Bool) {l₁ l₂ : List α}
+    (H : Pairwise (fun a b => f a → f b → False) l₁) (p : l₁ ~ l₂) : eraseP f l₁ ~ eraseP f l₂ := by
+  induction p with
+  | nil => simp
+  | cons a p IH =>
+    if h : f a then simp [h, p]
+    else simp [h]; exact IH (pairwise_cons.1 H).2
+  | swap a b l =>
+    by_cases h₁ : f a <;> by_cases h₂ : f b <;> simp [h₁, h₂]
+    · cases (pairwise_cons.1 H).1 _ (mem_cons.2 (Or.inl rfl)) h₂ h₁
+    · apply swap
+  | trans p₁ _ IH₁ IH₂ =>
+    refine (IH₁ H).trans (IH₂ ((p₁.pairwise_iff ?_).1 H))
+    exact fun h h₁ h₂ => h h₂ h₁
+
+end List

src/Init/Data/List/Range.lean

@@ -0,0 +1,57 @@
+/-
+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.Pairwise
+
+/-!
+# Lemmas about `List.range` and `List.enum`
+
+Most of the results are deferred to `Data.Init.List.Nat.Range`, where more results about
+natural arithmetic are available.
+-/
+
+namespace List
+
+open Nat
+
+/-! ## Ranges and enumeration -/
+
+/-! ### enumFrom -/
+
+@[simp]
+theorem enumFrom_eq_nil {n : Nat} {l : List α} : List.enumFrom n l = [] ↔ l = [] := by
+  cases l <;> simp
+
+@[simp] theorem enumFrom_length : ∀ {n} {l : List α}, (enumFrom n l).length = l.length
+  | _, [] => rfl
+  | _, _ :: _ => congrArg Nat.succ enumFrom_length
+
+@[simp]
+theorem getElem?_enumFrom :
+    ∀ n (l : List α) m, (enumFrom n l)[m]? = l[m]?.map fun a => (n + m, a)
+  | n, [], m => rfl
+  | n, a :: l, 0 => by simp
+  | n, a :: l, m + 1 => by
+    simp only [enumFrom_cons, getElem?_cons_succ]
+    exact (getElem?_enumFrom (n + 1) l m).trans <| by rw [Nat.add_right_comm]; rfl
+
+@[simp]
+theorem getElem_enumFrom (l : List α) (n) (i : Nat) (h : i < (l.enumFrom n).length) :
+    (l.enumFrom n)[i] = (n + i, l[i]'(by simpa [enumFrom_length] using h)) := by
+  simp only [enumFrom_length] at h
+  rw [getElem_eq_getElem?]
+  simp only [getElem?_enumFrom, getElem?_eq_getElem h]
+  simp
+
+theorem map_fst_add_enumFrom_eq_enumFrom (l : List α) (n k : Nat) :
+    map (Prod.map (· + n) id) (enumFrom k l) = enumFrom (n + k) l :=
+  ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
+
+theorem map_fst_add_enum_eq_enumFrom (l : List α) (n : Nat) :
+    map (Prod.map (· + n) id) (enum l) = enumFrom n l :=
+  map_fst_add_enumFrom_eq_enumFrom l _ _
+
+end List

src/Init/Data/List/Sort.lean

@@ -0,0 +1,9 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Sort.Basic
+import Init.Data.List.Sort.Impl
+import Init.Data.List.Sort.Lemmas

src/Init/Data/List/Sort/Basic.lean

@@ -0,0 +1,81 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Impl
+
+/-!
+# Definition of `merge` and `mergeSort`.
+
+These definitions are intended for verification purposes,
+and are replaced at runtime by efficient versions in `Init.Data.List.Sort.Impl`.
+-/
+
+namespace List
+
+/--
+`O(min |l| |r|)`. Merge two lists using `le` as a switch.
+
+This version is not tail-recursive,
+but it is replaced at runtime by `mergeTR` using a `@[csimp]` lemma.
+-/
+def merge (le : α → α → Bool) : List α → List α → List α
+  | [], ys => ys
+  | xs, [] => xs
+  | x :: xs, y :: ys =>
+    if le x y then
+      x :: merge le xs (y :: ys)
+    else
+      y :: merge le (x :: xs) ys
+
+@[simp] theorem nil_merge (ys : List α) : merge le [] ys = ys := by simp [merge]
+@[simp] theorem merge_right (xs : List α) : merge le xs [] = xs := by
+  induction xs with
+  | nil => simp [merge]
+  | cons x xs ih => simp [merge, ih]
+
+/--
+Split a list in two equal parts. If the length is odd, the first part will be one element longer.
+-/
+def splitInTwo (l : { l : List α // l.length = n }) :
+    { l : List α // l.length = (n+1)/2 } × { l : List α // l.length = n/2 } :=
+  let r := splitAt ((n+1)/2) l.1
+  (⟨r.1, by simp [r, splitAt_eq, l.2]; omega⟩, ⟨r.2, by simp [r, splitAt_eq, l.2]; omega⟩)
+
+/--
+Simplified implementation of stable merge sort.
+
+This function is designed for reasoning about the algorithm, and is not efficient.
+(It particular it uses the non tail-recursive `merge` function,
+and so can not be run on large lists, but also makes unnecessary traversals of lists.)
+It is replaced at runtime in the compiler by `mergeSortTR₂` using a `@[csimp]` lemma.
+
+Because we want the sort to be stable,
+it is essential that we split the list in two contiguous sublists.
+-/
+def mergeSort (le : α → α → Bool) : List α → List α
+  | [] => []
+  | [a] => [a]
+  | a :: b :: xs =>
+    let lr := splitInTwo ⟨a :: b :: xs, rfl⟩
+    have := by simpa using lr.2.2
+    have := by simpa using lr.1.2
+    merge le (mergeSort le lr.1) (mergeSort le lr.2)
+termination_by l => l.length
+
+
+/--
+Given an ordering relation `le : α → α → Bool`,
+construct the reverse lexicographic ordering on `Nat × α`.
+which first compares the second components using `le`,
+but if these are equivalent (in the sense `le a.2 b.2 && le b.2 a.2`)
+then compares the first components using `≤`.
+
+This function is only used in stating the stability properties of `mergeSort`.
+-/
+def enumLE (le : α → α → Bool) (a b : Nat × α) : Bool :=
+  if le a.2 b.2 then if le b.2 a.2 then a.1 ≤ b.1 else true else false
+
+end List

src/Init/Data/List/Sort/Impl.lean

@@ -0,0 +1,237 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Sort.Lemmas
+
+/-!
+# Replacing `merge` and `mergeSort` at runtime with tail-recursive and faster versions.
+
+We replace `merge` with `mergeTR` using a `@[csimp]` lemma.
+
+We replace `mergeSort` in two steps:
+* first with `mergeSortTR`, which while not tail-recursive itself (it can't be),
+  uses `mergeTR` internally.
+* second with `mergeSortTR₂`, which achieves an ~20% speed-up over `mergeSortTR`
+  by avoiding some unnecessary list reversals.
+
+There is no public API in this file; it solely exists to implement the `@[csimp]` lemmas
+affecting runtime behaviour.
+
+## Future work
+The current runtime implementation could be further improved in a number of ways, e.g.:
+* only walking the list once during splitting,
+* using insertion sort for small chunks rather than splitting all the way down to singletons,
+* identifying already sorted or reverse sorted chunks and skipping them.
+
+Because the theory developed for `mergeSort` is independent of the runtime implementation,
+as long as such improvements are carefully validated by benchmarking,
+they can be done without changing the theory, as long as a `@[csimp]` lemma is provided.
+-/
+
+open List
+
+namespace List.MergeSort.Internal
+
+/--
+`O(min |l| |r|)`. Merge two lists using `le` as a switch.
+-/
+def mergeTR (le : α → α → Bool) (l₁ l₂ : List α) : List α :=
+  go l₁ l₂ []
+where go : List α → List α → List α → List α
+  | [], l₂, acc => reverseAux acc l₂
+  | l₁, [], acc => reverseAux acc l₁
+  | x :: xs, y :: ys, acc =>
+    if le x y then
+      go xs (y :: ys) (x :: acc)
+    else
+      go (x :: xs) ys (y :: acc)
+
+theorem mergeTR_go_eq : mergeTR.go le l₁ l₂ acc = acc.reverse ++ merge le l₁ l₂ := by
+  induction l₁ generalizing l₂ acc with
+  | nil => simp [mergeTR.go, merge, reverseAux_eq]
+  | cons x l₁ ih₁ =>
+    induction l₂ generalizing acc with
+    | nil => simp [mergeTR.go, merge, reverseAux_eq]
+    | cons y l₂ ih₂ =>
+      simp [mergeTR.go, merge]
+      split <;> simp [ih₁, ih₂]
+
+@[csimp] theorem merge_eq_mergeTR : @merge = @mergeTR := by
+  funext
+  simp [mergeTR, mergeTR_go_eq]
+
+/--
+Variant of `splitAt`, that does not reverse the first list, i.e
+`splitRevAt n l = ((l.take n).reverse, l.drop n)`.
+
+This exists solely as an optimization for `mergeSortTR` and `mergeSortTR₂`,
+and should not be used elsewhere.
+-/
+def splitRevAt (n : Nat) (l : List α) : List α × List α := go l n [] where
+  /-- Auxiliary for `splitAtRev`: `splitAtRev.go xs n acc = ((take n xs).reverse ++ acc, drop n xs)`. -/
+  go : List α → Nat → List α → List α × List α
+  | x :: xs, n+1, acc => go xs n (x :: acc)
+  | xs, _, acc => (acc, xs)
+
+theorem splitRevAt_go (xs : List α) (n : Nat) (acc : List α) :
+    splitRevAt.go xs n acc = ((take n xs).reverse ++ acc, drop n xs) := by
+  induction xs generalizing n acc with
+  | nil => simp [splitRevAt.go]
+  | cons x xs ih =>
+    cases n with
+    | zero => simp [splitRevAt.go]
+    | succ n =>
+      rw [splitRevAt.go, ih n (x :: acc), take_succ_cons, reverse_cons, drop_succ_cons,
+        append_assoc, singleton_append]
+
+theorem splitRevAt_eq (n : Nat) (l : List α) : splitRevAt n l = ((l.take n).reverse, l.drop n) := by
+  rw [splitRevAt, splitRevAt_go, append_nil]
+
+/--
+An intermediate speed-up for `mergeSort`.
+This version uses the tail-recurive `mergeTR` function as a subroutine.
+
+This is not the final version we use at runtime, as `mergeSortTR₂` is faster.
+This definition is useful as an intermediate step in proving the `@[csimp]` lemma for `mergeSortTR₂`.
+-/
+def mergeSortTR (le : α → α → Bool) (l : List α) : List α :=
+  run ⟨l, rfl⟩
+where run : {n : Nat} → { l : List α // l.length = n } → List α
+  | 0, ⟨[], _⟩ => []
+  | 1, ⟨[a], _⟩ => [a]
+  | n+2, xs =>
+    let (l, r) := splitInTwo xs
+    mergeTR le (run l) (run r)
+
+/--
+Split a list in two equal parts, reversing the first part.
+If the length is odd, the first part will be one element longer.
+-/
+def splitRevInTwo (l : { l : List α // l.length = n }) :
+    { l : List α // l.length = (n+1)/2 } × { l : List α // l.length = n/2 } :=
+  let r := splitRevAt ((n+1)/2) l.1
+  (⟨r.1, by simp [r, splitRevAt_eq, l.2]; omega⟩, ⟨r.2, by simp [r, splitRevAt_eq, l.2]; omega⟩)
+
+/--
+Split a list in two equal parts, reversing the first part.
+If the length is odd, the second part will be one element longer.
+-/
+def splitRevInTwo' (l : { l : List α // l.length = n }) :
+    { l : List α // l.length = n/2 } × { l : List α // l.length = (n+1)/2 } :=
+  let r := splitRevAt (n/2) l.1
+  (⟨r.1, by simp [r, splitRevAt_eq, l.2]; omega⟩, ⟨r.2, by simp [r, splitRevAt_eq, l.2]; omega⟩)
+
+/--
+Faster version of `mergeSortTR`, which avoids unnecessary list reversals.
+-/
+-- Per the benchmark in `tests/bench/mergeSort/`
+-- (which averages over 4 use cases: already sorted lists, reverse sorted lists, almost sorted lists, and random lists),
+-- for lists of length 10^6, `mergeSortTR₂` is about 20% faster than `mergeSortTR`.
+def mergeSortTR₂ (le : α → α → Bool) (l : List α) : List α :=
+  run ⟨l, rfl⟩
+where
+  run : {n : Nat} → { l : List α // l.length = n } → List α
+  | 0, ⟨[], _⟩ => []
+  | 1, ⟨[a], _⟩ => [a]
+  | n+2, xs =>
+    let (l, r) := splitRevInTwo xs
+    mergeTR le (run' l) (run r)
+  run' : {n : Nat} → { l : List α // l.length = n } → List α
+  | 0, ⟨[], _⟩ => []
+  | 1, ⟨[a], _⟩ => [a]
+  | n+2, xs =>
+    let (l, r) := splitRevInTwo' xs
+    mergeTR le (run' r) (run l)
+
+theorem splitRevInTwo'_fst (l : { l : List α // l.length = n }) :
+    (splitRevInTwo' l).1 = ⟨(splitInTwo ⟨l.1.reverse, by simpa using l.2⟩).2.1, by have := l.2; simp; omega⟩ := by
+  simp only [splitRevInTwo', splitRevAt_eq, reverse_take, splitInTwo_snd]
+  congr
+  have := l.2
+  omega
+theorem splitRevInTwo'_snd (l : { l : List α // l.length = n }) :
+    (splitRevInTwo' l).2 = ⟨(splitInTwo ⟨l.1.reverse, by simpa using l.2⟩).1.1.reverse, by have := l.2; simp; omega⟩ := by
+  simp only [splitRevInTwo', splitRevAt_eq, reverse_take, splitInTwo_fst, reverse_reverse]
+  congr 2
+  have := l.2
+  simp
+  omega
+theorem splitRevInTwo_fst (l : { l : List α // l.length = n }) :
+    (splitRevInTwo l).1 = ⟨(splitInTwo l).1.1.reverse, by have := l.2; simp; omega⟩ := by
+  simp only [splitRevInTwo, splitRevAt_eq, reverse_take, splitInTwo_fst]
+theorem splitRevInTwo_snd (l : { l : List α // l.length = n }) :
+    (splitRevInTwo l).2 = ⟨(splitInTwo l).2.1, by have := l.2; simp; omega⟩ := by
+  simp only [splitRevInTwo, splitRevAt_eq, reverse_take, splitInTwo_snd]
+
+theorem mergeSortTR_run_eq_mergeSort : {n : Nat} → (l : { l : List α // l.length = n }) → mergeSortTR.run le l = mergeSort le l.1
+  | 0, ⟨[], _⟩
+  | 1, ⟨[a], _⟩ => by simp [mergeSortTR.run, mergeSort]
+  | n+2, ⟨a :: b :: l, h⟩ => by
+    cases h
+    simp only [mergeSortTR.run, mergeSortTR.run, mergeSort]
+    rw [merge_eq_mergeTR]
+    rw [mergeSortTR_run_eq_mergeSort, mergeSortTR_run_eq_mergeSort]
+
+-- We don't make this a `@[csimp]` lemma because `mergeSort_eq_mergeSortTR₂` is faster.
+theorem mergeSort_eq_mergeSortTR : @mergeSort = @mergeSortTR := by
+  funext
+  rw [mergeSortTR, mergeSortTR_run_eq_mergeSort]
+
+-- This mutual block is unfortunately quite slow to elaborate.
+set_option maxHeartbeats 400000 in
+mutual
+theorem mergeSortTR₂_run_eq_mergeSort : {n : Nat} → (l : { l : List α // l.length = n }) → mergeSortTR₂.run le l = mergeSort le l.1
+  | 0, ⟨[], _⟩
+  | 1, ⟨[a], _⟩ => by simp [mergeSortTR₂.run, mergeSort]
+  | n+2, ⟨a :: b :: l, h⟩ => by
+    cases h
+    simp only [mergeSortTR₂.run, mergeSort]
+    rw [splitRevInTwo_fst, splitRevInTwo_snd]
+    rw [mergeSortTR₂_run_eq_mergeSort, mergeSortTR₂_run'_eq_mergeSort]
+    rw [merge_eq_mergeTR]
+    rw [reverse_reverse]
+termination_by n => n
+
+theorem mergeSortTR₂_run'_eq_mergeSort : {n : Nat} → (l : { l : List α // l.length = n }) → (w : l' = l.1.reverse) → mergeSortTR₂.run' le l = mergeSort le l'
+  | 0, ⟨[], _⟩, w
+  | 1, ⟨[a], _⟩, w => by simp_all [mergeSortTR₂.run', mergeSort]
+  | n+2, ⟨a :: b :: l, h⟩, w => by
+    cases h
+    simp only [mergeSortTR₂.run', mergeSort]
+    rw [splitRevInTwo'_fst, splitRevInTwo'_snd]
+    rw [mergeSortTR₂_run_eq_mergeSort, mergeSortTR₂_run'_eq_mergeSort _ rfl]
+    rw [← merge_eq_mergeTR]
+    have w' := congrArg length w
+    simp at w'
+    cases l' with
+    | nil => simp at w'
+    | cons x l' =>
+      cases l' with
+      | nil => simp at w'
+      | cons y l' =>
+        rw [mergeSort]
+        congr 2
+        · dsimp at w
+          simp only [w]
+          simp only [splitInTwo_fst, splitInTwo_snd, reverse_take, take_reverse]
+          congr 1
+          rw [w, length_reverse]
+          simp
+        · dsimp at w
+          simp only [w]
+          simp only [reverse_cons, append_assoc, singleton_append, splitInTwo_snd, length_cons]
+          congr 1
+          simp at w'
+          omega
+termination_by n => n
+
+end
+
+@[csimp] theorem mergeSort_eq_mergeSortTR₂ : @mergeSort = @mergeSortTR₂ := by
+  funext
+  rw [mergeSortTR₂, mergeSortTR₂_run_eq_mergeSort]
+
+end List.MergeSort.Internal

src/Init/Data/List/Sort/Lemmas.lean

@@ -0,0 +1,406 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Perm
+import Init.Data.List.Sort.Basic
+import Init.Data.Bool
+
+/-!
+# Basic properties of `mergeSort`.
+
+* `mergeSort_sorted`: `mergeSort` produces a sorted list.
+* `mergeSort_perm`: `mergeSort` is a permutation of the input list.
+* `mergeSort_of_sorted`: `mergeSort` does not change a sorted list.
+* `mergeSort_cons`: proves `mergeSort le (x :: xs) = l₁ ++ x :: l₂` for some `l₁, l₂`
+  so that `mergeSort le xs = l₁ ++ l₂`, and no `a ∈ l₁` satisfies `le a x`.
+* `mergeSort_stable`: if `c` is a sorted sublist of `l`, then `c` is still a sublist of `mergeSort le l`.
+
+-/
+
+namespace List
+
+-- We enable this instance locally so we can write `Sorted le` instead of `Sorted (le · ·)` everywhere.
+attribute [local instance] boolRelToRel
+
+variable {le : α → α → Bool}
+
+/-! ### splitInTwo -/
+
+@[simp] theorem splitInTwo_fst (l : { l : List α // l.length = n }) : (splitInTwo l).1 = ⟨l.1.take ((n+1)/2), by simp [splitInTwo, splitAt_eq, l.2]; omega⟩ := by
+  simp [splitInTwo, splitAt_eq]
+
+@[simp] theorem splitInTwo_snd (l : { l : List α // l.length = n }) : (splitInTwo l).2 = ⟨l.1.drop ((n+1)/2), by simp [splitInTwo, splitAt_eq, l.2]; omega⟩ := by
+  simp [splitInTwo, splitAt_eq]
+
+theorem splitInTwo_fst_append_splitInTwo_snd (l : { l : List α // l.length = n }) : (splitInTwo l).1.1 ++ (splitInTwo l).2.1 = l.1 := by
+  simp
+
+theorem splitInTwo_cons_cons_enumFrom_fst (i : Nat) (l : List α) :
+    (splitInTwo ⟨(i, a) :: (i+1, b) :: l.enumFrom (i+2), rfl⟩).1.1 =
+      (splitInTwo ⟨a :: b :: l, rfl⟩).1.1.enumFrom i := by
+  simp only [length_cons, splitInTwo_fst, enumFrom_length]
+  ext1 j
+  rw [getElem?_take, getElem?_enumFrom, getElem?_take]
+  split
+  · rw [getElem?_cons, getElem?_cons, getElem?_cons, getElem?_cons]
+    split
+    · simp; omega
+    · split
+      · simp; omega
+      · simp only [getElem?_enumFrom]
+        congr
+        ext <;> simp; omega
+  · simp
+
+theorem splitInTwo_cons_cons_enumFrom_snd (i : Nat) (l : List α) :
+    (splitInTwo ⟨(i, a) :: (i+1, b) :: l.enumFrom (i+2), rfl⟩).2.1 =
+      (splitInTwo ⟨a :: b :: l, rfl⟩).2.1.enumFrom (i+(l.length+3)/2) := by
+  simp only [length_cons, splitInTwo_snd, enumFrom_length]
+  ext1 j
+  rw [getElem?_drop, getElem?_enumFrom, getElem?_drop]
+  rw [getElem?_cons, getElem?_cons, getElem?_cons, getElem?_cons]
+  split
+  · simp; omega
+  · split
+    · simp; omega
+    · simp only [getElem?_enumFrom]
+      congr
+      ext <;> simp; omega
+
+theorem splitInTwo_fst_sorted (l : { l : List α // l.length = n }) (h : Pairwise le l.1) : Pairwise le (splitInTwo l).1.1 := by
+  rw [splitInTwo_fst]
+  exact h.take
+
+theorem splitInTwo_snd_sorted (l : { l : List α // l.length = n }) (h : Pairwise le l.1) : Pairwise le (splitInTwo l).2.1 := by
+  rw [splitInTwo_snd]
+  exact h.drop
+
+theorem splitInTwo_fst_le_splitInTwo_snd {l : { l : List α // l.length = n }} (h : Pairwise le l.1) :
+    ∀ a b, a ∈ (splitInTwo l).1.1 → b ∈ (splitInTwo l).2.1 → le a b := by
+  rw [splitInTwo_fst, splitInTwo_snd]
+  intro a b ma mb
+  exact h.rel_of_mem_take_of_mem_drop ma mb
+
+/-! ### enumLE -/
+
+theorem enumLE_trans (trans : ∀ a b c, le a b → le b c → le a c)
+    (a b c : Nat × α) : enumLE le a b → enumLE le b c → enumLE le a c := by
+  simp only [enumLE]
+  split <;> split <;> split <;> rename_i ab₂ ba₂ bc₂
+  · simp_all
+    intro ab₁
+    intro h
+    refine ⟨trans _ _ _ ab₂ bc₂, ?_⟩
+    rcases h with (cd₂ | bc₁)
+    · exact Or.inl (Decidable.byContradiction
+        (fun ca₂ => by simp_all [trans _ _ _ (by simpa using ca₂) ab₂]))
+    · exact Or.inr (Nat.le_trans ab₁ bc₁)
+  · simp_all
+  · simp_all
+    intro h
+    refine ⟨trans _ _ _ ab₂ bc₂, ?_⟩
+    left
+    rcases h with (cb₂ | _)
+    · exact (Decidable.byContradiction
+        (fun ca₂ => by simp_all [trans _ _ _ (by simpa using ca₂) ab₂]))
+    · exact (Decidable.byContradiction
+        (fun ca₂ => by simp_all [trans _ _ _ bc₂ (by simpa using ca₂)]))
+  · simp_all
+  · simp_all
+  · simp_all
+  · simp_all
+  · simp_all
+
+theorem enumLE_total (total : ∀ a b, !le a b → le b a)
+    (a b : Nat × α) : !enumLE le a b → enumLE le b a := by
+  simp only [enumLE]
+  split <;> split
+  · simpa using Nat.le_of_lt
+  · simp
+  · simp
+  · simp_all [total a.2 b.2]
+
+/-! ### merge -/
+
+theorem merge_stable : ∀ (xs ys) (_ : ∀ x y, x ∈ xs → y ∈ ys → x.1 ≤ y.1),
+    (merge (enumLE le) xs ys).map (·.2) = merge le (xs.map (·.2)) (ys.map (·.2))
+  | [], ys, _ => by simp [merge]
+  | xs, [], _ => by simp [merge]
+  | (i, x) :: xs, (j, y) :: ys, h => by
+    simp only [merge, enumLE, map_cons]
+    split <;> rename_i w
+    · rw [if_pos (by simp [h _ _ (mem_cons_self ..) (mem_cons_self ..)])]
+      simp only [map_cons, cons.injEq, true_and]
+      rw [merge_stable, map_cons]
+      exact fun x' y' mx my => h x' y' (mem_cons_of_mem (i, x) mx) my
+    · simp only [↓reduceIte, map_cons, cons.injEq, true_and, reduceCtorEq]
+      rw [merge_stable, map_cons]
+      exact fun x' y' mx my => h x' y' mx (mem_cons_of_mem (j, y) my)
+
+/--
+The elements of `merge le xs ys` are exactly the elements of `xs` and `ys`.
+-/
+-- We subsequently prove that `mergeSort_perm : merge le xs ys ~ xs ++ ys`.
+theorem mem_merge {a : α} {xs ys : List α} : a ∈ merge le xs ys ↔ a ∈ xs ∨ a ∈ ys := by
+  induction xs generalizing ys with
+  | nil => simp [merge]
+  | cons x xs ih =>
+    induction ys with
+    | nil => simp [merge]
+    | cons y ys ih =>
+      simp only [merge]
+      split <;> rename_i h
+      · simp_all [or_assoc]
+      · simp only [mem_cons, or_assoc, Bool.not_eq_true, ih, ← or_assoc]
+        apply or_congr_left
+        simp only [or_comm (a := a = y), or_assoc]
+
+/--
+If the ordering relation `le` is transitive and total (i.e. `le a b ∨ le b a` for all `a, b`)
+then the `merge` of two sorted lists is sorted.
+-/
+theorem merge_sorted
+    (trans : ∀ (a b c : α), le a b → le b c → le a c)
+    (total : ∀ (a b : α), !le a b → le b a)
+    (l₁ l₂ : List α) (h₁ : l₁.Pairwise le) (h₂ : l₂.Pairwise le) : (merge le l₁ l₂).Pairwise le := by
+  induction l₁ generalizing l₂ with
+  | nil => simpa only [merge]
+  | cons x l₁ ih₁ =>
+    induction l₂ with
+    | nil => simpa only [merge]
+    | cons y l₂ ih₂ =>
+      simp only [merge]
+      split <;> rename_i h
+      · apply Pairwise.cons
+        · intro z m
+          rw [mem_merge, mem_cons] at m
+          rcases m with (m|rfl|m)
+          · exact rel_of_pairwise_cons h₁ m
+          · exact h
+          · exact trans _ _ _ h (rel_of_pairwise_cons h₂ m)
+        · exact ih₁ _ h₁.tail h₂
+      · apply Pairwise.cons
+        · intro z m
+          rw [mem_merge, mem_cons] at m
+          rcases m with (⟨rfl|m⟩|m)
+          · exact total _ _ (by simpa using h)
+          · exact trans _ _ _ (total _ _ (by simpa using h)) (rel_of_pairwise_cons h₁ m)
+          · exact rel_of_pairwise_cons h₂ m
+        · exact ih₂ h₂.tail
+
+theorem merge_of_le : ∀ {xs ys : List α} (_ : ∀ a b, a ∈ xs → b ∈ ys → le a b),
+    merge le xs ys = xs ++ ys
+  | [], ys, _
+  | xs, [], _ => by simp [merge]
+  | x :: xs, y :: ys, h => by
+    simp only [merge, cons_append]
+    rw [if_pos, merge_of_le]
+    · intro a b ma mb
+      exact h a b (mem_cons_of_mem _ ma) mb
+    · exact h x y (mem_cons_self _ _) (mem_cons_self _ _)
+
+variable (le) in
+theorem merge_perm_append : ∀ {xs ys : List α}, merge le xs ys ~ xs ++ ys
+  | [], ys => by simp [merge]
+  | xs, [] => by simp [merge]
+  | x :: xs, y :: ys => by
+    simp only [merge]
+    split
+    · exact merge_perm_append.cons x
+    · exact (merge_perm_append.cons y).trans
+        ((Perm.swap x y _).trans (perm_middle.symm.cons x))
+
+/-! ### mergeSort -/
+
+@[simp] theorem mergeSort_nil : [].mergeSort r = [] := by rw [List.mergeSort]
+
+@[simp] theorem mergeSort_singleton (a : α) : [a].mergeSort r = [a] := by rw [List.mergeSort]
+
+variable (le) in
+theorem mergeSort_perm : ∀ (l : List α), mergeSort le l ~ l
+  | [] => by simp [mergeSort]
+  | [a] => by simp [mergeSort]
+  | a :: b :: xs => by
+    simp only [mergeSort]
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).1.1.length < xs.length + 1 + 1 := by simp [splitInTwo_fst]; omega
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).2.1.length < xs.length + 1 + 1 := by simp [splitInTwo_snd]; omega
+    exact (merge_perm_append le).trans
+      (((mergeSort_perm _).append (mergeSort_perm _)).trans
+        (Perm.of_eq (splitInTwo_fst_append_splitInTwo_snd _)))
+termination_by l => l.length
+
+@[simp] theorem mergeSort_length (l : List α) : (mergeSort le l).length = l.length :=
+  (mergeSort_perm le l).length_eq
+
+@[simp] theorem mem_mergeSort {a : α} {l : List α} : a ∈ mergeSort le l ↔ a ∈ l :=
+  (mergeSort_perm le l).mem_iff
+
+/--
+The result of `mergeSort` is sorted,
+as long as the comparison function is transitive (`le a b → le b c → le a c`)
+and total in the sense that `le a b ∨ le b a`.
+
+The comparison function need not be irreflexive, i.e. `le a b` and `le b a` is allowed even when `a ≠ b`.
+-/
+theorem mergeSort_sorted
+    (trans : ∀ (a b c : α), le a b → le b c → le a c)
+    (total : ∀ (a b : α), !le a b → le b a) :
+    (l : List α) → (mergeSort le l).Pairwise le
+  | [] => by simp [mergeSort]
+  | [a] => by simp [mergeSort]
+  | a :: b :: xs => by
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).1.1.length < xs.length + 1 + 1 := by simp [splitInTwo_fst]; omega
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).2.1.length < xs.length + 1 + 1 := by simp [splitInTwo_snd]; omega
+    rw [mergeSort]
+    apply merge_sorted @trans @total
+    apply mergeSort_sorted trans total
+    apply mergeSort_sorted trans total
+termination_by l => l.length
+
+/--
+If the input list is already sorted, then `mergeSort` does not change the list.
+-/
+theorem mergeSort_of_sorted : ∀ {l : List α} (_ : Pairwise le l), mergeSort le l = l
+  | [], _ => by simp [mergeSort]
+  | [a], _ => by simp [mergeSort]
+  | a :: b :: xs, h => by
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).1.1.length < xs.length + 1 + 1 := by simp [splitInTwo_fst]; omega
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).2.1.length < xs.length + 1 + 1 := by simp [splitInTwo_snd]; omega
+    rw [mergeSort]
+    rw [mergeSort_of_sorted (splitInTwo_fst_sorted ⟨a :: b :: xs, rfl⟩ h)]
+    rw [mergeSort_of_sorted (splitInTwo_snd_sorted ⟨a :: b :: xs, rfl⟩ h)]
+    rw [merge_of_le (splitInTwo_fst_le_splitInTwo_snd h)]
+    rw [splitInTwo_fst_append_splitInTwo_snd]
+termination_by l => l.length
+
+/--
+This merge sort algorithm is stable,
+in the sense that breaking ties in the ordering function using the position in the list
+has no effect on the output.
+
+That is, elements which are equal with respect to the ordering function will remain
+in the same order in the output list as they were in the input list.
+
+See also:
+* `mergeSort_stable`: if `c <+ l` and `c.Pairwise le`, then `c <+ mergeSort le l`.
+* `mergeSort_stable_pair`: if `[a, b] <+ l` and `le a b`, then `[a, b] <+ mergeSort le l`)
+-/
+theorem mergeSort_enum {l : List α} :
+    (mergeSort (enumLE le) (l.enum)).map (·.2) = mergeSort le l :=
+  go 0 l
+where go : ∀ (i : Nat) (l : List α),
+    (mergeSort (enumLE le) (l.enumFrom i)).map (·.2) = mergeSort le l
+  | _, []
+  | _, [a] => by simp [mergeSort]
+  | _, a :: b :: xs => by
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).1.1.length < xs.length + 1 + 1 := by simp [splitInTwo_fst]; omega
+    have : (splitInTwo ⟨a :: b :: xs, rfl⟩).2.1.length < xs.length + 1 + 1 := by simp [splitInTwo_snd]; omega
+    simp only [mergeSort, enumFrom]
+    rw [splitInTwo_cons_cons_enumFrom_fst]
+    rw [splitInTwo_cons_cons_enumFrom_snd]
+    rw [merge_stable]
+    · rw [go, go]
+    · simp only [mem_mergeSort, Prod.forall]
+      intros j x k y mx my
+      have := mem_enumFrom mx
+      have := mem_enumFrom my
+      simp_all
+      omega
+termination_by _ l => l.length
+
+theorem mergeSort_cons {le : α → α → Bool}
+    (trans : ∀ (a b c : α), le a b → le b c → le a c)
+    (total : ∀ (a b : α), !le a b → le b a)
+    (a : α) (l : List α) :
+    ∃ l₁ l₂, mergeSort le (a :: l) = l₁ ++ a :: l₂ ∧ mergeSort le l = l₁ ++ l₂ ∧
+      ∀ b, b ∈ l₁ → !le a b := by
+  rw [← mergeSort_enum]
+  rw [enum_cons]
+  have nd : Nodup ((a :: l).enum.map (·.1)) := by rw [enum_map_fst]; exact nodup_range _
+  have m₁ : (0, a) ∈ mergeSort (enumLE le) ((a :: l).enum) :=
+    mem_mergeSort.mpr (mem_cons_self _ _)
+  obtain ⟨l₁, l₂, h⟩ := append_of_mem m₁
+  have s := mergeSort_sorted (enumLE_trans trans) (enumLE_total total) ((a :: l).enum)
+  rw [h] at s
+  have p := mergeSort_perm (enumLE le) ((a :: l).enum)
+  rw [h] at p
+  refine ⟨l₁.map (·.2), l₂.map (·.2), ?_, ?_, ?_⟩
+  · simpa using congrArg (·.map (·.2)) h
+  · rw [← mergeSort_enum.go 1, ← map_append]
+    congr 1
+    have q : mergeSort (enumLE le) (enumFrom 1 l) ~ l₁ ++ l₂ :=
+      (mergeSort_perm (enumLE le) (enumFrom 1 l)).trans
+        (p.symm.trans perm_middle).cons_inv
+    apply Perm.eq_of_sorted (le := enumLE le)
+    · rintro ⟨i, a⟩ ⟨j, b⟩  ha hb
+      simp only [mem_mergeSort] at ha
+      simp only [← q.mem_iff, mem_mergeSort] at hb
+      simp only [enumLE]
+      simp only [Bool.if_false_right, Bool.and_eq_true, Prod.mk.injEq, and_imp]
+      intro ab h ba h'
+      simp only [Bool.decide_eq_true] at ba
+      replace h : i ≤ j := by simpa [ab, ba] using h
+      replace h' : j ≤ i := by simpa [ab, ba] using h'
+      cases Nat.le_antisymm h h'
+      constructor
+      · rfl
+      · have := mem_enumFrom ha
+        have := mem_enumFrom hb
+        simp_all
+    · exact mergeSort_sorted (enumLE_trans trans) (enumLE_total total) ..
+    · exact s.sublist ((sublist_cons_self (0, a) l₂).append_left l₁)
+    · exact q
+  · intro b m
+    simp only [mem_map, Prod.exists, exists_eq_right] at m
+    obtain ⟨j, m⟩ := m
+    replace p := p.map (·.1)
+    have nd' := nd.perm p.symm
+    rw [map_append] at nd'
+    have j0 := nd'.rel_of_mem_append
+      (mem_map_of_mem (·.1) m) (mem_map_of_mem _ (mem_cons_self _ _))
+    simp only [ne_eq] at j0
+    have r := s.rel_of_mem_append m (mem_cons_self _ _)
+    simp_all [enumLE]
+
+/--
+Another statement of stability of merge sort.
+If `c` is a sorted sublist of `l`,
+then `c` is still a sublist of `mergeSort le l`.
+-/
+theorem mergeSort_stable
+    (trans : ∀ (a b c : α), le a b → le b c → le a c)
+    (total : ∀ (a b : α), !le a b → le b a) :
+    ∀ {c : List α} (_ : c.Pairwise le) (_ : c <+ l),
+    c <+ mergeSort le l
+  | _, _, .slnil => nil_sublist _
+  | c, hc, @Sublist.cons _ _ l a h => by
+    obtain ⟨l₁, l₂, h₁, h₂, -⟩ := mergeSort_cons trans total a l
+    rw [h₁]
+    have h' := mergeSort_stable trans total hc h
+    rw [h₂] at h'
+    exact h'.middle a
+  | _, _, @Sublist.cons₂ _ l₁ l₂ a h => by
+    rename_i hc
+    obtain ⟨l₃, l₄, h₁, h₂, h₃⟩ := mergeSort_cons trans total a l₂
+    rw [h₁]
+    have h' := mergeSort_stable trans total hc.tail h
+    rw [h₂] at h'
+    simp only [Bool.not_eq_true', tail_cons] at h₃ h'
+    exact
+      sublist_append_of_sublist_right (Sublist.cons₂ a
+        ((fun w => Sublist.of_sublist_append_right w h') fun b m₁ m₃ =>
+          (Bool.eq_not_self true).mp ((rel_of_pairwise_cons hc m₁).symm.trans (h₃ b m₃))))
+
+/--
+Another statement of stability of merge sort.
+If a pair `[a, b]` is a sublist of `l` and `le a b`,
+then `[a, b]` is still a sublist of `mergeSort le l`.
+-/
+theorem mergeSort_stable_pair
+    (trans : ∀ (a b c : α), le a b → le b c → le a c)
+    (total : ∀ (a b : α), !le a b → le b a)
+    (hab : le a b) (h : [a, b] <+ l) : [a, b] <+ mergeSort le l :=
+  mergeSort_stable trans total (pairwise_pair.mpr hab) h

src/Init/Data/List/Sublist.lean

@@ -1,7 +1,8 @@
 /-
 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
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
+  Kim Morrison
 -/
 prelude
 import Init.Data.List.TakeDrop
@@ -61,18 +62,18 @@ theorem subset_def {l₁ l₂ : List α} : l₁ ⊆ l₂ ↔ ∀ {a : α}, a ∈
 theorem Subset.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=
   fun _ i => h₂ (h₁ i)
 
-instance : Trans (Membership.mem : α → List α → Prop) Subset Membership.mem :=
-  ⟨fun h₁ h₂ => h₂ h₁⟩
+instance : Trans (fun l₁ l₂ => Subset l₂ l₁) (Membership.mem : List α → α → Prop) Membership.mem :=
+  ⟨fun h₁ h₂ => h₁ h₂⟩
 
 instance : Trans (Subset : List α → List α → Prop) Subset Subset :=
   ⟨Subset.trans⟩
 
-@[simp] theorem subset_cons_self (a : α) (l : List α) : l ⊆ a :: l := fun _ => Mem.tail _
+theorem subset_cons_self (a : α) (l : List α) : l ⊆ a :: l := fun _ => Mem.tail _
 
 theorem subset_of_cons_subset {a : α} {l₁ l₂ : List α} : a :: l₁ ⊆ l₂ → l₁ ⊆ l₂ :=
   fun s _ i => s (mem_cons_of_mem _ i)
 
-theorem subset_cons_of_subset (a : α) {l₁ l₂ : List α} : l₁ ⊆ l₂ → l₁ ⊆ a :: l₂ :=
+@[simp] theorem subset_cons_of_subset (a : α) {l₁ l₂ : List α} : l₁ ⊆ l₂ → l₁ ⊆ a :: l₂ :=
   fun s _ i => .tail _ (s i)
 
 theorem cons_subset_cons {l₁ l₂ : List α} (a : α) (s : l₁ ⊆ l₂) : a :: l₁ ⊆ a :: l₂ :=
@@ -84,6 +85,8 @@ theorem cons_subset_cons {l₁ l₂ : List α} (a : α) (s : l₁ ⊆ l₂) : a
 @[simp] theorem subset_nil {l : List α} : l ⊆ [] ↔ l = [] :=
   ⟨fun h => match l with | [] => rfl | _::_ => (nomatch h (.head ..)), fun | rfl => Subset.refl _⟩
 
+theorem eq_nil_of_subset_nil {l : List α} : l ⊆ [] → l = [] := subset_nil.mp
+
 theorem map_subset {l₁ l₂ : List α} (f : α → β) (h : l₁ ⊆ l₂) : map f l₁ ⊆ map f l₂ :=
   fun x => by simp only [mem_map]; exact .imp fun a => .imp_left (@h _)
 
@@ -97,14 +100,14 @@ theorem filterMap_subset {l₁ l₂ : List α} (f : α → Option β) (H : l₁
   rintro ⟨a, h, w⟩
   exact ⟨a, H h, w⟩
 
-@[simp] theorem subset_append_left (l₁ l₂ : List α) : l₁ ⊆ l₁ ++ l₂ := fun _ => mem_append_left _
+theorem subset_append_left (l₁ l₂ : List α) : l₁ ⊆ l₁ ++ l₂ := fun _ => mem_append_left _
 
-@[simp] theorem subset_append_right (l₁ l₂ : List α) : l₂ ⊆ l₁ ++ l₂ := fun _ => mem_append_right _
+theorem subset_append_right (l₁ l₂ : List α) : l₂ ⊆ l₁ ++ l₂ := fun _ => mem_append_right _
 
-theorem subset_append_of_subset_left (l₂ : List α) : l ⊆ l₁ → l ⊆ l₁ ++ l₂ :=
+@[simp] theorem subset_append_of_subset_left (l₂ : List α) : l ⊆ l₁ → l ⊆ l₁ ++ l₂ :=
 fun s => Subset.trans s <| subset_append_left _ _
 
-theorem subset_append_of_subset_right (l₁ : List α) : l ⊆ l₂ → l ⊆ l₁ ++ l₂ :=
+@[simp] theorem subset_append_of_subset_right (l₁ : List α) : l ⊆ l₂ → l ⊆ l₁ ++ l₂ :=
 fun s => Subset.trans s <| subset_append_right _ _
 
 @[simp] theorem append_subset {l₁ l₂ l : List α} :
@@ -152,7 +155,9 @@ theorem Sublist.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ <+ l₂) (h₂ : l
 
 instance : Trans (@Sublist α) Sublist Sublist := ⟨Sublist.trans⟩
 
-@[simp] theorem sublist_cons_self (a : α) (l : List α) : l <+ a :: l := (Sublist.refl l).cons _
+attribute [simp] Sublist.cons
+
+theorem sublist_cons_self (a : α) (l : List α) : l <+ a :: l := (Sublist.refl l).cons _
 
 theorem sublist_of_cons_sublist : a :: l₁ <+ l₂ → l₁ <+ l₂ :=
   (sublist_cons_self a l₁).trans
@@ -177,14 +182,23 @@ theorem Sublist.subset : l₁ <+ l₂ → l₁ ⊆ l₂
   | .cons₂ .., _, .head .. => .head ..
   | .cons₂ _ s, _, .tail _ h => .tail _ (s.subset h)
 
+protected theorem Sublist.mem (hx : a ∈ l₁) (hl : l₁ <+ l₂) : a ∈ l₂ :=
+  hl.subset hx
+
+theorem Sublist.head_mem (s : ys <+ xs) (h) : ys.head h ∈ xs :=
+  s.mem (List.head_mem h)
+
+theorem Sublist.getLast_mem (s : ys <+ xs) (h) : ys.getLast h ∈ xs :=
+  s.mem (List.getLast_mem h)
+
 instance : Trans (@Sublist α) Subset Subset :=
   ⟨fun h₁ h₂ => trans h₁.subset h₂⟩
 
 instance : Trans Subset (@Sublist α) Subset :=
   ⟨fun h₁ h₂ => trans h₁ h₂.subset⟩
 
-instance : Trans (Membership.mem : α → List α → Prop) Sublist Membership.mem :=
-  ⟨fun h₁ h₂ => h₂.subset h₁⟩
+instance : Trans (fun l₁ l₂ => Sublist l₂ l₁) (Membership.mem : List α → α → Prop) Membership.mem :=
+  ⟨fun h₁ h₂ => h₁.subset h₂⟩
 
 theorem mem_of_cons_sublist {a : α} {l₁ l₂ : List α} (s : a :: l₁ <+ l₂) : a ∈ l₂ :=
   (cons_subset.1 s.subset).1
@@ -192,6 +206,9 @@ theorem mem_of_cons_sublist {a : α} {l₁ l₂ : List α} (s : a :: l₁ <+ l
 @[simp] theorem sublist_nil {l : List α} : l <+ [] ↔ l = [] :=
   ⟨fun s => subset_nil.1 s.subset, fun H => H ▸ Sublist.refl _⟩
 
+theorem eq_nil_of_sublist_nil {l : List α} (s : l <+ []) : l = [] :=
+  eq_nil_of_subset_nil <| s.subset
+
 theorem Sublist.length_le : l₁ <+ l₂ → length l₁ ≤ length l₂
   | .slnil => Nat.le_refl 0
   | .cons _l s => le_succ_of_le (length_le s)
@@ -208,6 +225,18 @@ theorem Sublist.eq_of_length_le (s : l₁ <+ l₂) (h : length l₂ ≤ length l
 theorem Sublist.length_eq (s : l₁ <+ l₂) : length l₁ = length l₂ ↔ l₁ = l₂ :=
   ⟨s.eq_of_length, congrArg _⟩
 
+theorem tail_sublist : ∀ l : List α, tail l <+ l
+  | [] => .slnil
+  | a::l => sublist_cons_self a l
+
+protected theorem Sublist.tail : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → tail l₁ <+ tail l₂
+  | _, _, slnil => .slnil
+  | _, _, Sublist.cons _ h => (tail_sublist _).trans h
+  | _, _, Sublist.cons₂ _ h => h
+
+theorem Sublist.of_cons_cons {l₁ l₂ : List α} {a b : α} (h : a :: l₁ <+ b :: l₂) : l₁ <+ l₂ :=
+  h.tail
+
 protected theorem Sublist.map (f : α → β) {l₁ l₂} (s : l₁ <+ l₂) : map f l₁ <+ map f l₂ := by
   induction s with
   | slnil => simp
@@ -223,6 +252,12 @@ protected theorem Sublist.filterMap (f : α → Option β) (s : l₁ <+ l₂) :
 protected theorem Sublist.filter (p : α → Bool) {l₁ l₂} (s : l₁ <+ l₂) : filter p l₁ <+ filter p l₂ := by
   rw [← filterMap_eq_filter]; apply s.filterMap
 
+theorem head_filter_mem (xs : List α) (p : α → Bool) (h) : (xs.filter p).head h ∈ xs :=
+  (filter_sublist xs).head_mem h
+
+theorem getLast_filter_mem (xs : List α) (p : α → Bool) (h) : (xs.filter p).getLast h ∈ xs :=
+  (filter_sublist xs).getLast_mem h
+
 theorem sublist_filterMap_iff {l₁ : List β} {f : α → Option β} :
     l₁ <+ l₂.filterMap f ↔ ∃ l', l' <+ l₂ ∧ l₁ = l'.filterMap f := by
   induction l₂ generalizing l₁ with
@@ -265,11 +300,11 @@ theorem sublist_filter_iff {l₁ : List α} {p : α → Bool} :
     l₁ <+ l₂.filter p ↔ ∃ l', l' <+ l₂ ∧ l₁ = l'.filter p := by
   simp only [← filterMap_eq_filter, sublist_filterMap_iff]
 
-@[simp] theorem sublist_append_left : ∀ l₁ l₂ : List α, l₁ <+ l₁ ++ l₂
+theorem sublist_append_left : ∀ l₁ l₂ : List α, l₁ <+ l₁ ++ l₂
   | [], _ => nil_sublist _
   | _ :: l₁, l₂ => (sublist_append_left l₁ l₂).cons₂ _
 
-@[simp] theorem sublist_append_right : ∀ l₁ l₂ : List α, l₂ <+ l₁ ++ l₂
+theorem sublist_append_right : ∀ l₁ l₂ : List α, l₂ <+ l₁ ++ l₂
   | [], _ => Sublist.refl _
   | _ :: l₁, l₂ => (sublist_append_right l₁ l₂).cons _
 
@@ -278,10 +313,10 @@ theorem sublist_filter_iff {l₁ : List α} {p : α → Bool} :
   obtain ⟨_, _, rfl⟩ := append_of_mem h
   exact ((nil_sublist _).cons₂ _).trans (sublist_append_right ..)
 
-theorem sublist_append_of_sublist_left (s : l <+ l₁) : l <+ l₁ ++ l₂ :=
+@[simp] theorem sublist_append_of_sublist_left (s : l <+ l₁) : l <+ l₁ ++ l₂ :=
   s.trans <| sublist_append_left ..
 
-theorem sublist_append_of_sublist_right (s : l <+ l₂) : l <+ l₁ ++ l₂ :=
+@[simp] theorem sublist_append_of_sublist_right (s : l <+ l₂) : l <+ l₁ ++ l₂ :=
   s.trans <| sublist_append_right ..
 
 @[simp] theorem append_sublist_append_left : ∀ l, l ++ l₁ <+ l ++ l₂ ↔ l₁ <+ l₂
@@ -377,6 +412,27 @@ theorem append_sublist_iff {l₁ l₂ : List α} :
     · rintro ⟨r₁, r₂, rfl, h₁, h₂⟩
       exact Sublist.append h₁ h₂
 
+theorem Sublist.of_sublist_append_left (w : ∀ a, a ∈ l → a ∉ l₂) (h : l <+ l₁ ++ l₂) : l <+ l₁ := by
+  rw [sublist_append_iff] at h
+  obtain ⟨l₁', l₂', rfl, h₁, h₂⟩ := h
+  have : l₂' = [] := by
+    rw [eq_nil_iff_forall_not_mem]
+    exact fun x m => w x (mem_append_of_mem_right l₁' m) (h₂.mem m)
+  simp_all
+
+theorem Sublist.of_sublist_append_right (w : ∀ a, a ∈ l → a ∉ l₁) (h : l <+ l₁ ++ l₂) : l <+ l₂ := by
+  rw [sublist_append_iff] at h
+  obtain ⟨l₁', l₂', rfl, h₁, h₂⟩ := h
+  have : l₁' = [] := by
+    rw [eq_nil_iff_forall_not_mem]
+    exact fun x m => w x (mem_append_of_mem_left l₂' m) (h₁.mem m)
+  simp_all
+
+theorem Sublist.middle {l : List α} (h : l <+ l₁ ++ l₂) (a : α) : l <+ l₁ ++ a :: l₂ := by
+  rw [sublist_append_iff] at h
+  obtain ⟨l₁', l₂', rfl, h₁, h₂⟩ := h
+  exact Sublist.append h₁ (h₂.cons a)
+
 theorem Sublist.reverse : l₁ <+ l₂ → l₁.reverse <+ l₂.reverse
   | .slnil => Sublist.refl _
   | .cons _ h => by rw [reverse_cons]; exact sublist_append_of_sublist_left h.reverse
@@ -446,7 +502,7 @@ theorem sublist_join_iff {L : List (List α)} {l} :
       subst w
       exact ⟨[], by simp, fun i x => by cases x⟩
     · rintro ⟨L', rfl, h⟩
-      simp only [join_nil, sublist_nil, join_eq_nil_iff]
+      simp only [join_nil, sublist_nil, join_eq_nil]
       simp only [getElem?_nil, Option.getD_none, sublist_nil] at h
       exact (forall_getElem L' (· = [])).1 h
   | cons l' L ih =>
@@ -512,6 +568,10 @@ theorem join_sublist_iff {L : List (List α)} {l} :
 instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <+ l₂) :=
   decidable_of_iff (l₁.isSublist l₂) isSublist_iff_sublist
 
+protected theorem Sublist.drop : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → ∀ n, l₁.drop n <+ l₂.drop n
+  | _, _, h, 0 => h
+  | _, _, h, n + 1 => by rw [← drop_tail, ← drop_tail]; exact h.tail.drop n
+
 /-! ### IsPrefix / IsSuffix / IsInfix -/
 
 @[simp] theorem prefix_append (l₁ l₂ : List α) : l₁ <+: l₁ ++ l₂ := ⟨l₂, rfl⟩
@@ -527,17 +587,20 @@ theorem IsPrefix.isInfix : l₁ <+: l₂ → l₁ <:+: l₂ := fun ⟨t, h⟩ =>
 
 theorem IsSuffix.isInfix : l₁ <:+ l₂ → l₁ <:+: l₂ := fun ⟨t, h⟩ => ⟨t, [], by rw [h, append_nil]⟩
 
-@[simp] theorem nil_prefix (l : List α) : [] <+: l := ⟨l, rfl⟩
+@[simp] theorem nil_prefix {l : List α} : [] <+: l := ⟨l, rfl⟩
 
-@[simp] theorem nil_suffix (l : List α) : [] <:+ l := ⟨l, append_nil _⟩
+@[simp] theorem nil_suffix {l : List α} : [] <:+ l := ⟨l, append_nil _⟩
 
-@[simp] theorem nil_infix (l : List α) : [] <:+: l := (nil_prefix _).isInfix
+@[simp] theorem nil_infix {l : List α} : [] <:+: l := nil_prefix.isInfix
 
-@[simp] theorem prefix_refl (l : List α) : l <+: l := ⟨[], append_nil _⟩
+theorem prefix_refl (l : List α) : l <+: l := ⟨[], append_nil _⟩
+@[simp] theorem prefix_rfl {l : List α} : l <+: l := prefix_refl l
 
-@[simp] theorem suffix_refl (l : List α) : l <:+ l := ⟨[], rfl⟩
+theorem suffix_refl (l : List α) : l <:+ l := ⟨[], rfl⟩
+@[simp] theorem suffix_rfl {l : List α} : l <:+ l := suffix_refl l
 
-@[simp] theorem infix_refl (l : List α) : l <:+: l := (prefix_refl l).isInfix
+theorem infix_refl (l : List α) : l <:+: l := prefix_rfl.isInfix
+@[simp] theorem infix_rfl {l : List α} : l <:+: l := infix_refl l
 
 @[simp] theorem suffix_cons (a : α) : ∀ l, l <:+ a :: l := suffix_append [a]
 
@@ -573,6 +636,50 @@ protected theorem IsSuffix.sublist (h : l₁ <:+ l₂) : l₁ <+ l₂ :=
 protected theorem IsSuffix.subset (hl : l₁ <:+ l₂) : l₁ ⊆ l₂ :=
   hl.sublist.subset
 
+@[simp] theorem infix_nil : l <:+: [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ infix_rfl)⟩
+
+@[simp] theorem prefix_nil : l <+: [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ prefix_rfl)⟩
+
+@[simp] theorem suffix_nil : l <:+ [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ suffix_rfl)⟩
+
+theorem eq_nil_of_infix_nil (h : l <:+: []) : l = [] := infix_nil.mp h
+theorem eq_nil_of_prefix_nil (h : l <+: []) : l = [] := prefix_nil.mp h
+theorem eq_nil_of_suffix_nil (h : l <:+ []) : l = [] := suffix_nil.mp h
+
+theorem IsPrefix.ne_nil {x y : List α} (h : x <+: y) (hx : x ≠ []) : y ≠ [] := by
+  rintro rfl; exact hx <| List.prefix_nil.mp h
+
+theorem IsSuffix.ne_nil {x y : List α} (h : x <:+ y) (hx : x ≠ []) : y ≠ [] := by
+  rintro rfl; exact hx <| List.suffix_nil.mp h
+
+theorem IsInfix.ne_nil {x y : List α} (h : x <:+: y) (hx : x ≠ []) : y ≠ [] := by
+  rintro rfl; exact hx <| List.infix_nil.mp h
+
+theorem IsInfix.length_le (h : l₁ <:+: l₂) : l₁.length ≤ l₂.length :=
+  h.sublist.length_le
+
+theorem IsPrefix.length_le (h : l₁ <+: l₂) : l₁.length ≤ l₂.length :=
+  h.sublist.length_le
+
+theorem IsSuffix.length_le (h : l₁ <:+ l₂) : l₁.length ≤ l₂.length :=
+  h.sublist.length_le
+
+theorem IsPrefix.getElem {x y : List α} (h : x <+: y) {n} (hn : n < x.length) :
+    x[n] = y[n]'(Nat.le_trans hn h.length_le) := by
+  obtain ⟨_, rfl⟩ := h
+  exact (List.getElem_append n hn).symm
+
+-- See `Init.Data.List.Nat.Sublist` for `IsSuffix.getElem`.
+
+theorem IsPrefix.mem (hx : a ∈ l₁) (hl : l₁ <+: l₂) : a ∈ l₂ :=
+  hl.subset hx
+
+theorem IsSuffix.mem (hx : a ∈ l₁) (hl : l₁ <:+ l₂) : a ∈ l₂ :=
+  hl.subset hx
+
+theorem IsInfix.mem (hx : a ∈ l₁) (hl : l₁ <:+: l₂) : a ∈ l₂ :=
+  hl.subset hx
+
 @[simp] theorem reverse_suffix : reverse l₁ <:+ reverse l₂ ↔ l₁ <+: l₂ :=
   ⟨fun ⟨r, e⟩ => ⟨reverse r, by rw [← reverse_reverse l₁, ← reverse_append, e, reverse_reverse]⟩,
    fun ⟨r, e⟩ => ⟨reverse r, by rw [← reverse_append, e]⟩⟩
@@ -586,25 +693,35 @@ protected theorem IsSuffix.subset (hl : l₁ <:+ l₂) : l₁ ⊆ l₂ :=
       reverse_reverse]
   · rw [append_assoc, ← reverse_append, ← reverse_append, e]
 
-theorem IsInfix.length_le (h : l₁ <:+: l₂) : l₁.length ≤ l₂.length :=
-  h.sublist.length_le
+theorem IsInfix.reverse : l₁ <:+: l₂ → reverse l₁ <:+: reverse l₂ :=
+  reverse_infix.2
 
-theorem IsPrefix.length_le (h : l₁ <+: l₂) : l₁.length ≤ l₂.length :=
-  h.sublist.length_le
+theorem IsSuffix.reverse : l₁ <:+ l₂ → reverse l₁ <+: reverse l₂ :=
+  reverse_prefix.2
 
-theorem IsSuffix.length_le (h : l₁ <:+ l₂) : l₁.length ≤ l₂.length :=
-  h.sublist.length_le
+theorem IsPrefix.reverse : l₁ <+: l₂ → reverse l₁ <:+ reverse l₂ :=
+  reverse_suffix.2
 
-@[simp] theorem infix_nil : l <:+: [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ infix_refl _)⟩
+theorem IsPrefix.head {x y : List α} (h : x <+: y) (hx : x ≠ []) :
+    x.head hx = y.head (h.ne_nil hx) := by
+  cases x <;> cases y <;> simp only [head_cons, ne_eq, not_true_eq_false] at hx ⊢
+  all_goals (obtain ⟨_, h⟩ := h; injection h)
 
-@[simp] theorem prefix_nil : l <+: [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ prefix_refl _)⟩
+theorem IsSuffix.getLast {x y : List α} (h : x <:+ y) (hx : x ≠ []) :
+    x.getLast hx = y.getLast (h.ne_nil hx) := by
+  rw [← head_reverse (by simpa), h.reverse.head,
+    head_reverse (by rintro h; simp only [reverse_eq_nil_iff] at h; simp_all)]
 
-@[simp] theorem suffix_nil : l <:+ [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ suffix_refl _)⟩
+theorem prefix_concat (a : α) (l) : l <+: concat l a := by simp
 
-theorem infix_iff_prefix_suffix (l₁ l₂ : List α) : l₁ <:+: l₂ ↔ ∃ t, l₁ <+: t ∧ t <:+ l₂ :=
+theorem infix_iff_prefix_suffix {l₁ l₂ : List α} : l₁ <:+: l₂ ↔ ∃ t, l₁ <+: t ∧ t <:+ l₂ :=
   ⟨fun ⟨_, t, e⟩ => ⟨l₁ ++ t, ⟨_, rfl⟩, e ▸ append_assoc .. ▸ ⟨_, rfl⟩⟩,
     fun ⟨_, ⟨t, rfl⟩, s, e⟩ => ⟨s, t, append_assoc .. ▸ e⟩⟩
 
+theorem infix_iff_suffix_prefix {l₁ l₂ : List α} : l₁ <:+: l₂ ↔ ∃ t, l₁ <:+ t ∧ t <+: l₂ :=
+  ⟨fun ⟨s, t, e⟩ => ⟨s ++ l₁, ⟨_, rfl⟩, ⟨t, e⟩⟩,
+    fun ⟨_, ⟨s, rfl⟩, t, e⟩ => ⟨s, t, append_assoc .. ▸ e⟩⟩
+
 theorem IsInfix.eq_of_length (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
   h.sublist.eq_of_length
 
@@ -616,7 +733,7 @@ theorem IsSuffix.eq_of_length (h : l₁ <:+ l₂) : l₁.length = l₂.length
 
 theorem prefix_of_prefix_length_le :
     ∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
-  | [], l₂, _, _, _, _ => nil_prefix _
+  | [], l₂, _, _, _, _ => nil_prefix
   | a :: l₁, b :: l₂, _, ⟨r₁, rfl⟩, ⟨r₂, e⟩, ll => by
     injection e with _ e'; subst b
     rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩ (le_of_succ_le_succ ll) with ⟨r₃, rfl⟩
@@ -650,7 +767,7 @@ theorem prefix_cons_iff : l₁ <+: a :: l₂ ↔ l₁ = [] ∨ ∃ t, l₁ = a :
         refine ⟨s, by simp [h']⟩
 
 @[simp] theorem cons_prefix_cons : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
-  simp only [prefix_cons_iff, cons.injEq, false_or]
+  simp only [prefix_cons_iff, cons.injEq, false_or, List.cons_ne_nil]
   constructor
   · rintro ⟨t, ⟨rfl, rfl⟩, h⟩
     exact ⟨rfl, h⟩
@@ -679,6 +796,121 @@ theorem infix_cons_iff : l₁ <:+: a :: l₂ ↔ l₁ <+: a :: l₂ ∨ l₁ <:+
     · exact h.isInfix
     · exact infix_cons hl₁
 
+theorem prefix_concat_iff {l₁ l₂ : List α} {a : α} :
+    l₁ <+: l₂ ++ [a] ↔ l₁ = l₂ ++ [a] ∨ l₁ <+: l₂ := by
+  simp only [← reverse_suffix, reverse_concat, suffix_cons_iff]
+  simp only [concat_eq_append, ← reverse_concat, reverse_eq_iff, reverse_reverse]
+
+theorem suffix_concat_iff {l₁ l₂ : List α} {a : α} :
+    l₁ <:+ l₂ ++ [a] ↔ l₁ = [] ∨ ∃ t, l₁ = t ++ [a] ∧ t <:+ l₂ := by
+  rw [← reverse_prefix, reverse_concat, prefix_cons_iff]
+  simp only [reverse_eq_nil_iff]
+  apply or_congr_right
+  constructor
+  · rintro ⟨t, w, h⟩
+    exact ⟨t.reverse, by simpa using congrArg reverse w, by simpa using h.reverse⟩
+  · rintro ⟨t, rfl, h⟩
+    exact ⟨t.reverse, by simp, by simpa using h.reverse⟩
+
+theorem infix_concat_iff {l₁ l₂ : List α} {a : α} :
+    l₁ <:+: l₂ ++ [a] ↔ l₁ <:+ l₂ ++ [a] ∨ l₁ <:+: l₂ := by
+  rw [← reverse_infix, reverse_concat, infix_cons_iff, reverse_infix,
+    ← reverse_prefix, reverse_concat]
+
+theorem isPrefix_iff : l₁ <+: l₂ ↔ ∀ i (h : i < l₁.length), l₂[i]? = some l₁[i] := by
+  induction l₁ generalizing l₂ with
+  | nil => simp
+  | cons a l₁ ih =>
+    cases l₂ with
+    | nil =>
+      simpa using ⟨0, by simp⟩
+    | cons b l₂ =>
+      simp only [cons_append, cons_prefix_cons, ih]
+      rw (config := {occs := .pos [2]}) [← Nat.and_forall_add_one]
+      simp [Nat.succ_lt_succ_iff, eq_comm]
+
+-- See `Init.Data.List.Nat.Sublist` for `isSuffix_iff` and `ifInfix_iff`.
+
+theorem isPrefix_filterMap_iff {β} (f : α → Option β) {l₁ : List α} {l₂ : List β} :
+    l₂ <+: filterMap f l₁ ↔ ∃ l, l <+: l₁ ∧ l₂ = filterMap f l := by
+  simp only [IsPrefix, append_eq_filterMap]
+  constructor
+  · rintro ⟨_, l₁, l₂, rfl, rfl, rfl⟩
+    exact ⟨l₁, ⟨l₂, rfl⟩, rfl⟩
+  · rintro ⟨l₁, ⟨l₂, rfl⟩, rfl⟩
+    exact ⟨_, l₁, l₂, rfl, rfl, rfl⟩
+
+theorem isSuffix_filterMap_iff {β} (f : α → Option β) {l₁ : List α} {l₂ : List β} :
+    l₂ <:+ filterMap f l₁ ↔ ∃ l, l <:+ l₁ ∧ l₂ = filterMap f l := by
+  simp only [IsSuffix, append_eq_filterMap]
+  constructor
+  · rintro ⟨_, l₁, l₂, rfl, rfl, rfl⟩
+    exact ⟨l₂, ⟨l₁, rfl⟩, rfl⟩
+  · rintro ⟨l₁, ⟨l₂, rfl⟩, rfl⟩
+    exact ⟨_, l₂, l₁, rfl, rfl, rfl⟩
+
+theorem isInfix_filterMap_iff {β} (f : α → Option β) {l₁ : List α} {l₂ : List β} :
+    l₂ <:+: filterMap f l₁ ↔ ∃ l, l <:+: l₁ ∧ l₂ = filterMap f l := by
+  simp only [IsInfix, append_eq_filterMap, filterMap_eq_append]
+  constructor
+  · rintro ⟨_, _, _, l₁, rfl, ⟨⟨l₂, l₃, rfl, rfl, rfl⟩, rfl⟩⟩
+    exact ⟨l₃, ⟨l₂, l₁, rfl⟩, rfl⟩
+  · rintro ⟨l₃, ⟨l₂, l₁, rfl⟩, rfl⟩
+    exact ⟨_, _, _, l₁, rfl, ⟨⟨l₂, l₃, rfl, rfl, rfl⟩, rfl⟩⟩
+
+theorem isPrefix_filter_iff {p : α → Bool} {l₁ l₂ : List α} :
+    l₂ <+: l₁.filter p ↔ ∃ l, l <+: l₁ ∧ l₂ = l.filter p := by
+  rw [← filterMap_eq_filter, isPrefix_filterMap_iff]
+
+theorem isSuffix_filter_iff {p : α → Bool} {l₁ l₂ : List α} :
+    l₂ <:+ l₁.filter p ↔ ∃ l, l <:+ l₁ ∧ l₂ = l.filter p := by
+  rw [← filterMap_eq_filter, isSuffix_filterMap_iff]
+
+theorem isInfix_filter_iff {p : α → Bool} {l₁ l₂ : List α} :
+    l₂ <:+: l₁.filter p ↔ ∃ l, l <:+: l₁ ∧ l₂ = l.filter p := by
+  rw [← filterMap_eq_filter, isInfix_filterMap_iff]
+
+theorem isPrefix_map_iff {β} (f : α → β) {l₁ : List α} {l₂ : List β} :
+    l₂ <+: l₁.map f ↔ ∃ l, l <+: l₁ ∧ l₂ = l.map f := by
+  rw [← filterMap_eq_map, isPrefix_filterMap_iff]
+
+theorem isSuffix_map_iff {β} (f : α → β) {l₁ : List α} {l₂ : List β} :
+    l₂ <:+ l₁.map f ↔ ∃ l, l <:+ l₁ ∧ l₂ = l.map f := by
+  rw [← filterMap_eq_map, isSuffix_filterMap_iff]
+
+theorem isInfix_map_iff {β} (f : α → β) {l₁ : List α} {l₂ : List β} :
+    l₂ <:+: l₁.map f ↔ ∃ l, l <:+: l₁ ∧ l₂ = l.map f := by
+  rw [← filterMap_eq_map, isInfix_filterMap_iff]
+
+theorem isPrefix_replicate_iff {n} {a : α} {l : List α} :
+    l <+: List.replicate n a ↔ l.length ≤ n ∧ l = List.replicate l.length a := by
+  rw [IsPrefix]
+  simp only [append_eq_replicate]
+  constructor
+  · rintro ⟨_, rfl, _, _⟩
+    exact ⟨le_add_right .., ‹_›⟩
+  · rintro ⟨h, w⟩
+    refine ⟨replicate (n - l.length) a, ?_, ?_⟩
+    · simpa using add_sub_of_le h
+    · simpa using w
+
+theorem isSuffix_replicate_iff {n} {a : α} {l : List α} :
+    l <:+ List.replicate n a ↔ l.length ≤ n ∧ l = List.replicate l.length a := by
+  rw [← reverse_prefix, reverse_replicate, isPrefix_replicate_iff]
+  simp [reverse_eq_iff]
+
+theorem isInfix_replicate_iff {n} {a : α} {l : List α} :
+    l <:+: List.replicate n a ↔ l.length ≤ n ∧ l = List.replicate l.length a := by
+  rw [IsInfix]
+  simp only [append_eq_replicate, length_append]
+  constructor
+  · rintro ⟨_, _, rfl, ⟨-, _, _⟩, _⟩
+    exact ⟨le_add_right_of_le (le_add_left ..), ‹_›⟩
+  · rintro ⟨h, w⟩
+    refine ⟨replicate (n - l.length) a, [], ?_, ?_⟩
+    · simpa using Nat.sub_add_cancel h
+    · simpa using w
+
 theorem infix_of_mem_join : ∀ {L : List (List α)}, l ∈ L → l <:+: join L
   | l' :: _, h =>
     match h with
@@ -689,7 +921,6 @@ theorem infix_of_mem_join : ∀ {L : List (List α)}, l ∈ L → l <:+: join L
 theorem prefix_append_right_inj (l) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ :=
   exists_congr fun r => by rw [append_assoc, append_right_inj]
 
-@[simp]
 theorem prefix_cons_inj (a) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ :=
   prefix_append_right_inj [a]
 
@@ -717,6 +948,60 @@ theorem mem_of_mem_take {l : List α} (h : a ∈ l.take n) : a ∈ l :=
 theorem mem_of_mem_drop {n} {l : List α} (h : a ∈ l.drop n) : a ∈ l :=
   drop_subset _ _ h
 
+theorem drop_suffix_drop_left (l : List α) {m n : Nat} (h : m ≤ n) : drop n l <:+ drop m l := by
+  rw [← Nat.sub_add_cancel h, ← drop_drop]
+  apply drop_suffix
+
+-- See `Init.Data.List.Nat.TakeDrop` for `take_prefix_take_left`.
+
+theorem drop_sublist_drop_left (l : List α) {m n : Nat} (h : m ≤ n) : drop n l <+ drop m l :=
+  (drop_suffix_drop_left l h).sublist
+
+theorem drop_subset_drop_left (l : List α) {m n : Nat} (h : m ≤ n) : drop n l ⊆ drop m l :=
+  (drop_sublist_drop_left l h).subset
+
+theorem takeWhile_prefix (p : α → Bool) : l.takeWhile p <+: l :=
+  ⟨l.dropWhile p, takeWhile_append_dropWhile p l⟩
+
+theorem dropWhile_suffix (p : α → Bool) : l.dropWhile p <:+ l :=
+  ⟨l.takeWhile p, takeWhile_append_dropWhile p l⟩
+
+theorem takeWhile_sublist (p : α → Bool) : l.takeWhile p <+ l :=
+  (takeWhile_prefix p).sublist
+
+theorem dropWhile_sublist (p : α → Bool) : l.dropWhile p <+ l :=
+  (dropWhile_suffix p).sublist
+
+theorem takeWhile_subset {l : List α} (p : α → Bool) : l.takeWhile p ⊆ l :=
+  (takeWhile_sublist p).subset
+
+theorem dropWhile_subset {l : List α} (p : α → Bool) : l.dropWhile p ⊆ l :=
+  (dropWhile_sublist p).subset
+
+theorem dropLast_prefix : ∀ l : List α, l.dropLast <+: l
+  | [] => ⟨nil, by rw [dropLast, List.append_nil]⟩
+  | a :: l => ⟨_, dropLast_concat_getLast (cons_ne_nil a l)⟩
+
+theorem dropLast_sublist (l : List α) : l.dropLast <+ l :=
+  (dropLast_prefix l).sublist
+
+theorem dropLast_subset (l : List α) : l.dropLast ⊆ l :=
+  (dropLast_sublist l).subset
+
+theorem tail_suffix (l : List α) : tail l <:+ l := by rw [← drop_one]; apply drop_suffix
+
+theorem IsPrefix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) : l₁.map f <+: l₂.map f := by
+  obtain ⟨r, rfl⟩ := h
+  rw [map_append]; apply prefix_append
+
+theorem IsSuffix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) : l₁.map f <:+ l₂.map f := by
+  obtain ⟨r, rfl⟩ := h
+  rw [map_append]; apply suffix_append
+
+theorem IsInfix.map {β} (f : α → β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) : l₁.map f <:+: l₂.map f := by
+  obtain ⟨r₁, r₂, rfl⟩ := h
+  rw [map_append, map_append]; apply infix_append
+
 theorem IsPrefix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
     l₁.filter p <+: l₂.filter p := by
   obtain ⟨xs, rfl⟩ := h
@@ -732,6 +1017,21 @@ theorem IsInfix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+
   obtain ⟨xs, ys, rfl⟩ := h
   rw [filter_append, filter_append]; apply infix_append _
 
+theorem IsPrefix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
+    filterMap f l₁ <+: filterMap f l₂ := by
+  obtain ⟨xs, rfl⟩ := h
+  rw [filterMap_append]; apply prefix_append
+
+theorem IsSuffix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) :
+    filterMap f l₁ <:+ filterMap f l₂ := by
+  obtain ⟨xs, rfl⟩ := h
+  rw [filterMap_append]; apply suffix_append
+
+theorem IsInfix.filterMap {β} (f : α → Option β) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) :
+    filterMap f l₁ <:+: filterMap f l₂ := by
+  obtain ⟨xs, ys, rfl⟩ := h
+  rw [filterMap_append, filterMap_append]; apply infix_append
+
 @[simp] theorem isPrefixOf_iff_prefix [BEq α] [LawfulBEq α] {l₁ l₂ : List α} :
     l₁.isPrefixOf l₂ ↔ l₁ <+: l₂ := by
   induction l₁ generalizing l₂ with
@@ -751,4 +1051,13 @@ instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <+: l₂) :=
 instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <:+ l₂) :=
   decidable_of_iff (l₁.isSuffixOf l₂) isSuffixOf_iff_suffix
 
+theorem prefix_iff_eq_append : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ :=
+  ⟨by rintro ⟨r, rfl⟩; rw [drop_left], fun e => ⟨_, e⟩⟩
+
+theorem prefix_iff_eq_take : l₁ <+: l₂ ↔ l₁ = take (length l₁) l₂ :=
+  ⟨fun h => append_cancel_right <| (prefix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,
+    fun e => e.symm ▸ take_prefix _ _⟩
+
+-- See `Init.Data.List.Nat.Sublist` for `suffix_iff_eq_append`, `prefix_take_iff`, and `suffix_iff_eq_drop`.
+
 end List

src/Init/Data/List/TakeDrop.lean

@@ -20,6 +20,11 @@ Further results on `List.take` and `List.drop`, which rely on stronger automatio
 are given in `Init.Data.List.TakeDrop`.
 -/
 
+theorem take_cons {l : List α} (h : 0 < n) : take n (a :: l) = a :: take (n - 1) l := by
+  cases n with
+  | zero => exact absurd h (Nat.lt_irrefl _)
+  | succ n => rfl
+
 @[simp]
 theorem drop_one : ∀ l : List α, drop 1 l = tail l
   | [] | _ :: _ => rfl
@@ -74,7 +79,7 @@ theorem drop_eq_get_cons {n} {l : List α} (h) : drop n l = get l ⟨n, h⟩ ::
   simp [drop_eq_getElem_cons]
 
 @[simp]
-theorem getElem?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n)[m]? = l[m]? := by
+theorem getElem?_take_of_lt {l : List α} {n m : Nat} (h : m < n) : (l.take n)[m]? = l[m]? := by
   induction n generalizing l m with
   | zero =>
     exact absurd h (Nat.not_lt_of_le m.zero_le)
@@ -86,14 +91,31 @@ theorem getElem?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n)[m]? = l
       · simp
       · simpa using hn (Nat.lt_of_succ_lt_succ h)
 
-@[deprecated getElem?_take (since := "2024-06-12")]
+@[deprecated getElem?_take_of_lt (since := "2024-06-12")]
 theorem get?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n).get? m = l.get? m := by
-  simp [getElem?_take, h]
+  simp [getElem?_take_of_lt, h]
+
+theorem getElem?_take_of_succ {l : List α} {n : Nat} : (l.take (n + 1))[n]? = l[n]? := by simp
+
+@[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 ..
+
+@[deprecated drop_drop (since := "2024-06-15")]
+theorem drop_add (m n) (l : List α) : drop (m + n) l = drop m (drop n l) := by
+  simp [drop_drop]
 
 @[simp]
-theorem getElem?_take_of_succ {l : List α} {n : Nat} : (l.take (n + 1))[n]? = l[n]? :=
-  getElem?_take (Nat.lt_succ_self n)
-
 theorem tail_drop (l : List α) (n : Nat) : (l.drop n).tail = l.drop (n + 1) := by
   induction l generalizing n with
   | nil => simp
@@ -102,9 +124,13 @@ theorem tail_drop (l : List α) (n : Nat) : (l.drop n).tail = l.drop (n + 1) :=
     · simp
     · simp [hl]
 
+@[simp]
+theorem drop_tail (l : List α) (n : Nat) : l.tail.drop n = l.drop (n + 1) := by
+  rw [← drop_drop, drop_one]
+
 @[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⟩
+  refine ⟨fun h => ?_, drop_eq_nil_of_le⟩
   induction k generalizing l with
   | zero =>
     simp only [drop] at h
@@ -226,24 +252,6 @@ theorem dropLast_eq_take (l : List α) : l.dropLast = l.take (l.length - 1) := b
     dsimp
     rw [map_drop f t]
 
-@[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 ..
-
-@[deprecated drop_drop (since := "2024-06-15")]
-theorem drop_add (m n) (l : List α) : drop (m + n) l = drop m (drop n l) := by
-  simp [drop_drop]
-
 /-! ### takeWhile and dropWhile -/
 
 theorem takeWhile_cons (p : α → Bool) (a : α) (l : List α) :
@@ -428,6 +436,12 @@ theorem take_takeWhile {l : List α} (p : α → Bool) n :
   | nil => rfl
   | cons h t ih => by_cases p h <;> simp_all
 
+/-! ### splitAt -/
+
+@[simp] theorem splitAt_eq (n : Nat) (l : List α) : splitAt n l = (l.take n, l.drop n) := by
+  rw [splitAt, splitAt_go, reverse_nil, nil_append]
+  split <;> simp_all [take_of_length_le, drop_of_length_le]
+
 /-! ### rotateLeft -/
 
 @[simp] theorem rotateLeft_zero (l : List α) : rotateLeft l 0 = l := by

src/Init/Data/List/Zip.lean

@@ -78,13 +78,13 @@ theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
     (l.map fun x => (x, f x)) = l.zip (l.map f) := by
   rw [← zip_map']
   congr
-  exact map_id _
+  simp
 
 theorem map_prod_right_eq_zip {l : List α} (f : α → β) :
     (l.map fun x => (f x, x)) = (l.map f).zip l := by
   rw [← zip_map']
   congr
-  exact map_id _
+  simp
 
 /-- See also `List.zip_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
 @[simp] theorem zip_replicate' {a : α} {b : β} {n : Nat} :

src/Init/Data/Nat/Basic.lean

@@ -158,7 +158,7 @@ theorem add_one (n : Nat) : n + 1 = succ n :=
   rfl
 
 @[simp] theorem add_one_ne_zero (n : Nat) : n + 1 ≠ 0 := nofun
-@[simp] theorem zero_ne_add_one (n : Nat) : 0 ≠ n + 1 := nofun
+theorem zero_ne_add_one (n : Nat) : 0 ≠ n + 1 := by simp
 
 protected theorem add_comm : ∀ (n m : Nat), n + m = m + n
   | n, 0   => Eq.symm (Nat.zero_add n)
@@ -779,6 +779,11 @@ theorem pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} : ∀ {j}, i ≤
 theorem pos_pow_of_pos {n : Nat} (m : Nat) (h : 0 < n) : 0 < n^m :=
   pow_le_pow_of_le_right h (Nat.zero_le _)
 
+@[simp] theorem zero_pow_of_pos (n : Nat) (h : 0 < n) : 0 ^ n = 0 := by
+  cases n with
+  | zero => cases h
+  | succ n => simp [Nat.pow_succ]
+
 /-! # min/max -/
 
 /--
@@ -887,7 +892,7 @@ theorem sub_succ_lt_self (a i : Nat) (h : i < a) : a - (i + 1) < a - i := by
 
 theorem sub_ne_zero_of_lt : {a b : Nat} → a < b → b - a ≠ 0
   | 0, 0, h      => absurd h (Nat.lt_irrefl 0)
-  | 0, succ b, _ => by simp only [Nat.sub_zero, ne_eq, not_false_eq_true]
+  | 0, succ b, _ => by simp only [Nat.sub_zero, ne_eq, not_false_eq_true, Nat.succ_ne_zero]
   | succ a, 0, h => absurd h (Nat.not_lt_zero a.succ)
   | succ a, succ b, h => by rw [Nat.succ_sub_succ]; exact sub_ne_zero_of_lt (Nat.lt_of_succ_lt_succ h)
 
@@ -1087,6 +1092,10 @@ protected theorem sub_eq_iff_eq_add {c : Nat} (h : b ≤ a) : a - b = c ↔ a =
 protected theorem sub_eq_iff_eq_add' {c : Nat} (h : b ≤ a) : a - b = c ↔ a = b + c := by
   rw [Nat.add_comm, Nat.sub_eq_iff_eq_add h]
 
+protected theorem sub_one_sub_lt_of_lt (h : a < b) : b - 1 - a < b := by
+  rw [← Nat.sub_add_eq]
+  exact sub_lt (zero_lt_of_lt h) (Nat.lt_add_right a Nat.one_pos)
+
 /-! ## Mul sub distrib -/
 
 theorem pred_mul (n m : Nat) : pred n * m = n * m - m := by

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

@@ -40,7 +40,7 @@ An induction principal that works on divison by two.
 -/
 noncomputable def div2Induction {motive : Nat → Sort u}
     (n : Nat) (ind : ∀(n : Nat), (n > 0 → motive (n/2)) → motive n) : motive n := by
-  induction n using Nat.strongInductionOn with
+  induction n using Nat.strongRecOn with
   | ind n hyp =>
     apply ind
     intro n_pos
@@ -86,6 +86,12 @@ noncomputable def div2Induction {motive : Nat → Sort u}
 @[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]
 
+theorem mod_two_eq_one_iff_testBit_zero : (x % 2 = 1) ↔ x.testBit 0 = true := by
+  cases mod_two_eq_zero_or_one x <;> simp_all
+
+theorem mod_two_eq_zero_iff_testBit_zero : (x % 2 = 0) ↔ x.testBit 0 = false := by
+  cases mod_two_eq_zero_or_one x <;> simp_all
+
 theorem testBit_succ (x i : Nat) : testBit x (succ i) = testBit (x/2) i := by
   unfold testBit
   simp [shiftRight_succ_inside]
@@ -94,6 +100,9 @@ theorem testBit_succ (x i : Nat) : testBit x (succ i) = testBit (x/2) i := by
   unfold testBit
   simp [shiftRight_succ_inside]
 
+theorem testBit_div_two (x i : Nat) : testBit (x / 2) i = testBit x (i + 1) := by
+  simp
+
 theorem testBit_to_div_mod {x : Nat} : testBit x i = decide (x / 2^i % 2 = 1) := by
   induction i generalizing x with
   | zero =>
@@ -114,7 +123,7 @@ theorem ne_zero_implies_bit_true {x : Nat} (xnz : x ≠ 0) : ∃ i, testBit x i
     match mod_two_eq_zero_or_one x with
     | Or.inl mod2_eq =>
       rw [←div_add_mod x 2] at xnz
-      simp only [mod2_eq, ne_eq, Nat.mul_eq_zero, Nat.add_zero, false_or] at xnz
+      simp only [mod2_eq, ne_eq, Nat.mul_eq_zero, Nat.add_zero, false_or, reduceCtorEq] at xnz
       have ⟨d, dif⟩   := hyp x_pos xnz
       apply Exists.intro (d+1)
       simp_all
@@ -200,7 +209,7 @@ theorem lt_pow_two_of_testBit (x : Nat) (p : ∀i, i ≥ n → testBit x i = fal
   have x_ge_n := Nat.ge_of_not_lt not_lt
   have ⟨i, ⟨i_ge_n, test_true⟩⟩ := ge_two_pow_implies_high_bit_true x_ge_n
   have test_false := p _ i_ge_n
-  simp only [test_true] at test_false
+  simp [test_true] at test_false
 
 private theorem succ_mod_two : succ x % 2 = 1 - x % 2 := by
   induction x with
@@ -249,7 +258,7 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
 
 @[simp] theorem testBit_mod_two_pow (x j i : Nat) :
     testBit (x % 2^j) i = (decide (i < j) && testBit x i) := by
-  induction x using Nat.strongInductionOn generalizing j i with
+  induction x using Nat.strongRecOn generalizing j i with
   | ind x hyp =>
     rw [mod_eq]
     rcases Nat.lt_or_ge x (2^j) with x_lt_j | x_ge_j
@@ -315,12 +324,44 @@ theorem testBit_one_eq_true_iff_self_eq_zero {i : Nat} :
     Nat.testBit 1 i = true ↔ i = 0 := by
   cases i <;> simp
 
+theorem testBit_two_pow {n m : Nat} : testBit (2 ^ n) m = decide (n = m) := by
+  rw [testBit, shiftRight_eq_div_pow]
+  by_cases h : n = m
+  · simp [h, Nat.div_self (Nat.pow_pos Nat.zero_lt_two)]
+  · simp only [h]
+    cases Nat.lt_or_lt_of_ne h
+    · rw [div_eq_of_lt (Nat.pow_lt_pow_of_lt (by omega) (by omega))]
+      simp
+    · rw [Nat.pow_div _ Nat.two_pos,
+         ← Nat.sub_add_cancel (succ_le_of_lt <| Nat.sub_pos_of_lt (by omega))]
+      simp [Nat.pow_succ, and_one_is_mod, mul_mod_left]
+      omega
+
+@[simp]
+theorem testBit_two_pow_self {n : Nat} : testBit (2 ^ n) n = true := by
+  simp [testBit_two_pow]
+
+@[simp]
+theorem testBit_two_pow_of_ne {n m : Nat} (hm : n ≠ m) : testBit (2 ^ n) m = false := by
+  simp [testBit_two_pow]
+  omega
+
+@[simp] theorem two_pow_sub_one_mod_two : (2 ^ n - 1) % 2 = 1 % 2 ^ n := by
+  cases n with
+  | zero => simp
+  | succ n =>
+    rw [mod_eq_of_lt (a := 1) (Nat.one_lt_two_pow (by omega)), mod_two_eq_one_iff_testBit_zero, testBit_two_pow_sub_one ]
+    simp only [zero_lt_succ, decide_True]
+
+@[simp] theorem mod_two_pos_mod_two_eq_one : x % 2 ^ j % 2 = 1 ↔ (0 < j) ∧ x % 2 = 1 := by
+  rw [mod_two_eq_one_iff_testBit_zero, testBit_mod_two_pow]
+  simp
+
 /-! ### bitwise -/
 
-theorem testBit_bitwise
-  (false_false_axiom : f false false = false) (x y i : Nat)
-: (bitwise f x y).testBit i = f (x.testBit i) (y.testBit i) := by
-  induction i using Nat.strongInductionOn generalizing x y with
+theorem testBit_bitwise (false_false_axiom : f false false = false) (x y i : Nat) :
+    (bitwise f x y).testBit i = f (x.testBit i) (y.testBit i) := by
+  induction i using Nat.strongRecOn generalizing x y with
   | ind i hyp =>
     unfold bitwise
     if x_zero : x = 0 then
@@ -417,6 +458,11 @@ theorem and_pow_two_identity {x : Nat} (lt : x < 2^n) : x &&& 2^n-1 = x := by
   rw [and_pow_two_is_mod]
   apply Nat.mod_eq_of_lt lt
 
+@[simp] theorem and_mod_two_eq_one : (a &&& b) % 2 = 1 ↔ a % 2 = 1 ∧ b % 2 = 1 := by
+  simp only [mod_two_eq_one_iff_testBit_zero]
+  rw [testBit_and]
+  simp
+
 /-! ### lor -/
 
 @[simp] theorem zero_or (x : Nat) : 0 ||| x = x := by
@@ -435,6 +481,11 @@ theorem and_pow_two_identity {x : Nat} (lt : x < 2^n) : x &&& 2^n-1 = x := by
 theorem or_lt_two_pow {x y n : Nat} (left : x < 2^n) (right : y < 2^n) : x ||| y < 2^n :=
   bitwise_lt_two_pow left right
 
+@[simp] theorem or_mod_two_eq_one : (a ||| b) % 2 = 1 ↔ a % 2 = 1 ∨ b % 2 = 1 := by
+  simp only [mod_two_eq_one_iff_testBit_zero]
+  rw [testBit_or]
+  simp
+
 /-! ### xor -/
 
 @[simp] theorem testBit_xor (x y i : Nat) :
@@ -444,6 +495,19 @@ theorem or_lt_two_pow {x y n : Nat} (left : x < 2^n) (right : y < 2^n) : x ||| y
 theorem xor_lt_two_pow {x y n : Nat} (left : x < 2^n) (right : y < 2^n) : x ^^^ y < 2^n :=
   bitwise_lt_two_pow left right
 
+theorem and_xor_distrib_right {a b c : Nat} : (a ^^^ b) &&& c = (a &&& c) ^^^ (b &&& c) := by
+  apply Nat.eq_of_testBit_eq
+  simp [Bool.and_xor_distrib_right]
+
+theorem and_xor_distrib_left {a b c : Nat} : a &&& (b ^^^ c) = (a &&& b) ^^^ (a &&& c) := by
+  apply Nat.eq_of_testBit_eq
+  simp [Bool.and_xor_distrib_left]
+
+@[simp] theorem xor_mod_two_eq_one : ((a ^^^ b) % 2 = 1) ↔ ¬ ((a % 2 = 1) ↔ (b % 2 = 1)) := by
+  simp only [mod_two_eq_one_iff_testBit_zero]
+  rw [testBit_xor]
+  simp
+
 /-! ### Arithmetic -/
 
 theorem testBit_mul_pow_two_add (a : Nat) {b i : Nat} (b_lt : b < 2^i) (j : Nat) :
@@ -505,6 +569,15 @@ theorem mul_add_lt_is_or {b : Nat} (b_lt : b < 2^i) (a : Nat) : 2^i * a + b = 2^
 @[simp] theorem testBit_shiftRight (x : Nat) : testBit (x >>> i) j = testBit x (i+j) := by
   simp [testBit, ←shiftRight_add]
 
+@[simp] theorem shiftLeft_mod_two_eq_one : x <<< i % 2 = 1 ↔ i = 0 ∧ x % 2 = 1 := by
+  rw [mod_two_eq_one_iff_testBit_zero, testBit_shiftLeft]
+  simp
+
+@[simp] theorem decide_shiftRight_mod_two_eq_one :
+    decide (x >>> i % 2 = 1) = x.testBit i := by
+  simp only [testBit, one_and_eq_mod_two, mod_two_bne_zero]
+  exact (Bool.beq_eq_decide_eq _ _).symm
+
 /-! ### le -/
 
 theorem le_of_testBit {n m : Nat} (h : ∀ i, n.testBit i = true → m.testBit i = true) : n ≤ m := by

src/Init/Data/Nat/Div.lean

@@ -48,7 +48,7 @@ def div.inductionOn.{u}
 decreasing_by apply div_rec_lemma; assumption
 
 theorem div_le_self (n k : Nat) : n / k ≤ n := by
-  induction n using Nat.strongInductionOn with
+  induction n using Nat.strongRecOn with
   | ind n ih =>
     rw [div_eq]
     -- Note: manual split to avoid Classical.em which is not yet defined
@@ -203,6 +203,10 @@ theorem mod_add_div (m k : Nat) : m % k + k * (m / k) = m := by
   | base x y h => simp [h]
   | ind x y h IH => simp [h]; rw [Nat.mul_succ, ← Nat.add_assoc, IH, Nat.sub_add_cancel h.2]
 
+theorem mod_def (m k : Nat) : m % k = m - k * (m / k) := by
+  rw [Nat.sub_eq_of_eq_add]
+  apply (Nat.mod_add_div _ _).symm
+
 @[simp] protected theorem div_one (n : Nat) : n / 1 = n := by
   have := mod_add_div n 1
   rwa [mod_one, Nat.zero_add, Nat.one_mul] at this
@@ -217,7 +221,7 @@ theorem le_div_iff_mul_le (k0 : 0 < k) : x ≤ y / k ↔ x * k ≤ y := by
   induction y, k using mod.inductionOn generalizing x with
     (rw [div_eq]; simp [h]; cases x with | zero => simp [zero_le] | succ x => ?_)
   | base y k h =>
-    simp only [add_one, succ_mul, false_iff, Nat.not_le]
+    simp only [add_one, succ_mul, false_iff, Nat.not_le, Nat.succ_ne_zero]
     refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_left ..)
     exact Nat.not_le.1 fun h' => h ⟨k0, h'⟩
   | ind y k h IH =>
@@ -330,7 +334,7 @@ theorem mul_mod_mul_left (z x y : Nat) : (z * x) % (z * y) = z * (x % y) :=
   else if z0 : z = 0 then by
     rw [z0, Nat.zero_mul, Nat.zero_mul, Nat.zero_mul, mod_zero]
   else by
-    induction x using Nat.strongInductionOn with
+    induction x using Nat.strongRecOn with
     | _ n IH =>
       have y0 : y > 0 := Nat.pos_of_ne_zero y0
       have z0 : z > 0 := Nat.pos_of_ne_zero z0

src/Init/Data/Nat/Gcd.lean

@@ -75,7 +75,7 @@ theorem gcd_rec (m n : Nat) : gcd m n = gcd (n % m) m :=
 
 @[elab_as_elim] theorem gcd.induction {P : Nat → Nat → Prop} (m n : Nat)
     (H0 : ∀n, P 0 n) (H1 : ∀ m n, 0 < m → P (n % m) m → P m n) : P m n :=
-  Nat.strongInductionOn (motive := fun m => ∀ n, P m n) m
+  Nat.strongRecOn (motive := fun m => ∀ n, P m n) m
     (fun
     | 0, _ => H0
     | _+1, IH => fun _ => H1 _ _ (succ_pos _) (IH _ (mod_lt _ (succ_pos _)) _) )

src/Init/Data/Nat/Lemmas.lean

@@ -27,6 +27,11 @@ namespace Nat
   ⟨fun ⟨n, h, w⟩ => by cases n with | zero => simp at h | succ n => exact ⟨n, w⟩,
     fun ⟨n, w⟩ => ⟨n + 1, by simp, w⟩⟩
 
+@[simp] theorem exists_eq_add_one : (∃ n, a = n + 1) ↔ 0 < a :=
+  ⟨fun ⟨n, h⟩ => by omega, fun h => ⟨a - 1, by omega⟩⟩
+@[simp] theorem exists_add_one_eq : (∃ n, n + 1 = a) ↔ 0 < a :=
+  ⟨fun ⟨n, h⟩ => by omega, fun h => ⟨a - 1, by omega⟩⟩
+
 /-! ## add -/
 
 protected theorem add_add_add_comm (a b c d : Nat) : (a + b) + (c + d) = (a + c) + (b + d) := by
@@ -50,6 +55,11 @@ protected theorem add_eq_zero_iff : n + m = 0 ↔ n = 0 ∧ m = 0 :=
 @[simp] protected theorem add_right_cancel_iff {n : Nat} : m + n = k + n ↔ m = k :=
   ⟨Nat.add_right_cancel, fun | rfl => rfl⟩
 
+@[simp] protected theorem add_left_eq_self  {a b : Nat} : a + b = b ↔ a = 0 := by omega
+@[simp] protected theorem add_right_eq_self {a b : Nat} : a + b = a ↔ b = 0 := by omega
+@[simp] protected theorem self_eq_add_right {a b : Nat} : a = a + b ↔ b = 0 := by omega
+@[simp] protected theorem self_eq_add_left  {a b : Nat} : a = b + a ↔ b = 0 := by omega
+
 @[simp] protected theorem add_le_add_iff_left {n : Nat} : n + m ≤ n + k ↔ m ≤ k :=
   ⟨Nat.le_of_add_le_add_left, fun h => Nat.add_le_add_left h _⟩
 
@@ -147,17 +157,9 @@ 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 (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' (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
-
 protected theorem le_sub_of_add_le' {n k m : Nat} : m + n ≤ k → n ≤ k - m :=
   Nat.add_comm .. ▸ Nat.le_sub_of_add_le
 
@@ -419,14 +421,6 @@ protected theorem mul_min_mul_left (a b c : Nat) : min (a * b) (a * c) = a * min
 
 /-! ### mul -/
 
-@[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 (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
-
 protected theorem mul_right_comm (n m k : Nat) : n * m * k = n * k * m := by
   rw [Nat.mul_assoc, Nat.mul_comm m, ← Nat.mul_assoc]
 
@@ -539,6 +533,11 @@ theorem mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 1 :=
   | 0, _ => .inl rfl
   | 1, _ => .inr rfl
 
+@[simp] theorem mod_two_bne_zero : ((a % 2) != 0) = (a % 2 == 1) := by
+  cases mod_two_eq_zero_or_one a <;> simp_all
+@[simp] theorem mod_two_bne_one : ((a % 2) != 1) = (a % 2 == 0) := by
+  cases mod_two_eq_zero_or_one a <;> simp_all
+
 theorem le_of_mod_lt {a b : Nat} (h : a % b < a) : b ≤ a :=
   Nat.not_lt.1 fun hf => (ne_of_lt h).elim (Nat.mod_eq_of_lt hf)
 
@@ -649,6 +648,16 @@ protected theorem one_le_two_pow : 1 ≤ 2 ^ n :=
   else
     Nat.le_of_lt (Nat.one_lt_two_pow h)
 
+@[simp] theorem one_mod_two_pow_eq_one : 1 % 2 ^ n = 1 ↔ 0 < n := by
+  cases n with
+  | zero => simp
+  | succ n =>
+    rw [mod_eq_of_lt (a := 1) (Nat.one_lt_two_pow (by omega))]
+    simp
+
+@[simp] theorem one_mod_two_pow (h : 0 < n) : 1 % 2 ^ n = 1 :=
+  one_mod_two_pow_eq_one.mpr h
+
 protected theorem pow_pos (h : 0 < a) : 0 < a^n :=
   match n with
   | 0 => Nat.zero_lt_one
@@ -700,6 +709,36 @@ protected theorem pow_lt_pow_iff_right {a n m : Nat} (h : 1 < a) :
   · intro w
     exact Nat.pow_lt_pow_of_lt h w
 
+@[simp]
+protected theorem pow_pred_mul {x w : Nat} (h : 0 < w) :
+    x ^ (w - 1) * x = x ^ w := by
+  simp [← Nat.pow_succ, succ_eq_add_one, Nat.sub_add_cancel h]
+
+protected theorem pow_pred_lt_pow {x w : Nat} (h₁ : 1 < x) (h₂ : 0 < w) :
+    x ^ (w - 1) < x ^ w :=
+  Nat.pow_lt_pow_of_lt h₁ (by omega)
+
+protected theorem two_pow_pred_lt_two_pow {w : Nat} (h : 0 < w) :
+    2 ^ (w - 1) < 2 ^ w :=
+  Nat.pow_pred_lt_pow (by omega) h
+
+@[simp]
+protected theorem two_pow_pred_add_two_pow_pred (h : 0 < w) :
+    2 ^ (w - 1) + 2 ^ (w - 1) = 2 ^ w := by
+  rw [← Nat.pow_pred_mul h]
+  omega
+
+@[simp]
+protected theorem two_pow_sub_two_pow_pred (h : 0 < w) :
+    2 ^ w - 2 ^ (w - 1) = 2 ^ (w - 1) := by
+  simp [← Nat.two_pow_pred_add_two_pow_pred h]
+
+@[simp]
+protected theorem two_pow_pred_mod_two_pow (h : 0 < w) :
+    2 ^ (w - 1) % 2 ^ w = 2 ^ (w - 1) := by
+  rw [mod_eq_of_lt]
+  apply Nat.pow_pred_lt_pow (by omega) h
+
 /-! ### log2 -/
 
 @[simp]

src/Init/Data/Nat/Mod.lean

@@ -73,4 +73,10 @@ theorem mod_pow_succ {x b k : Nat} :
     x % b ^ (k + 1) = x % b ^ k + b ^ k * ((x / b ^ k) % b) := by
   rw [Nat.pow_succ, Nat.mod_mul]
 
+@[simp] theorem two_pow_mod_two_eq_zero (n : Nat) : 2 ^ n % 2 = 0 ↔ 0 < n := by
+  cases n <;> simp [Nat.pow_succ]
+
+@[simp] theorem two_pow_mod_two_eq_one (n : Nat) : 2 ^ n % 2 = 1 ↔ n = 0 := by
+  cases n <;> simp [Nat.pow_succ]
+
 end Nat

src/Init/Data/Option/Instances.lean

@@ -19,7 +19,7 @@ theorem eq_of_eq_some {α : Type u} : ∀ {x y : Option α}, (∀z, x = some z
 theorem eq_none_of_isNone {α : Type u} : ∀ {o : Option α}, o.isNone → o = none
   | none, _ => rfl
 
-instance : Membership α (Option α) := ⟨fun a b => b = some a⟩
+instance : Membership α (Option α) := ⟨fun b a => b = some a⟩
 
 @[simp] theorem mem_def {a : α} {b : Option α} : a ∈ b ↔ b = some a := .rfl
 

src/Init/Data/Option/Lemmas.lean

@@ -87,6 +87,9 @@ theorem eq_some_iff_get_eq : o = some a ↔ ∃ h : o.isSome, o.get h = a := by
 theorem eq_some_of_isSome : ∀ {o : Option α} (h : o.isSome), o = some (o.get h)
   | some _, _ => rfl
 
+theorem isSome_iff_ne_none : o.isSome ↔ o ≠ none := by
+  cases o <;> simp
+
 theorem not_isSome_iff_eq_none : ¬o.isSome ↔ o = none := by
   cases o <;> simp
 
@@ -159,7 +162,7 @@ theorem map_some : f <$> some a = some (f a) := rfl
 theorem map_eq_some : f <$> x = some b ↔ ∃ a, x = some a ∧ f a = b := map_eq_some'
 
 @[simp] theorem map_eq_none' : x.map f = none ↔ x = none := by
-  cases x <;> simp only [map_none', map_some', eq_self_iff_true]
+  cases x <;> simp [map_none', map_some', eq_self_iff_true]
 
 theorem isSome_map {x : Option α} : (f <$> x).isSome = x.isSome := by
   cases x <;> simp
@@ -167,6 +170,9 @@ theorem isSome_map {x : Option α} : (f <$> x).isSome = x.isSome := by
 @[simp] theorem isSome_map' {x : Option α} : (x.map f).isSome = x.isSome := by
   cases x <;> simp
 
+@[simp] theorem isNone_map' {x : Option α} : (x.map f).isNone = x.isNone := by
+  cases x <;> simp
+
 theorem map_eq_none : f <$> x = none ↔ x = none := map_eq_none'
 
 theorem map_eq_bind {x : Option α} : x.map f = x.bind (some ∘ f) := by
@@ -175,8 +181,19 @@ theorem map_eq_bind {x : Option α} : x.map f = x.bind (some ∘ f) := by
 theorem map_congr {x : Option α} (h : ∀ a, a ∈ x → f a = g a) : x.map f = x.map g := by
   cases x <;> simp only [map_none', map_some', h, mem_def]
 
-@[simp] theorem map_id' : Option.map (@id α) = id := map_id
-@[simp] theorem map_id'' {x : Option α} : (x.map fun a => a) = x := congrFun map_id x
+@[simp] theorem map_id_fun {α : Type u} : Option.map (id : α → α) = id := by
+  funext; simp [map_id]
+
+theorem map_id' {x : Option α} : (x.map fun a => a) = x := congrFun map_id x
+
+@[simp] theorem map_id_fun' {α : Type u} : Option.map (fun (a : α) => a) = id := by
+  funext; simp [map_id']
+
+theorem get_map {f : α → β} {o : Option α} {h : (o.map f).isSome} :
+    (o.map f).get h = f (o.get (by simpa using h)) := by
+  cases o with
+  | none => simp at h
+  | some a => simp
 
 @[simp] theorem map_map (h : β → γ) (g : α → β) (x : Option α) :
     (x.map g).map h = x.map (h ∘ g) := by
@@ -190,6 +207,14 @@ theorem comp_map (h : β → γ) (g : α → β) (x : Option α) : x.map (h ∘
 
 theorem mem_map_of_mem (g : α → β) (h : a ∈ x) : g a ∈ Option.map g x := h.symm ▸ map_some' ..
 
+@[simp] theorem map_if {f : α → β} [Decidable c] :
+     (if c then some a else none).map f = if c then some (f a) else none := by
+  split <;> rfl
+
+@[simp] theorem map_dif {f : α → β} [Decidable c] {a : c → α} :
+     (if h : c then some (a h) else none).map f = if h : c then some (f (a h)) else none := by
+  split <;> rfl
+
 @[simp] theorem filter_none (p : α → Bool) : none.filter p = none := rfl
 theorem filter_some : Option.filter p (some a) = if p a then some a else none := rfl
 
@@ -227,6 +252,15 @@ theorem map_orElse {x y : Option α} : (x <|> y).map f = (x.map f <|> y.map f) :
 @[simp] theorem guard_eq_some [DecidablePred p] : guard p a = some b ↔ a = b ∧ p a :=
   if h : p a then by simp [Option.guard, h] else by simp [Option.guard, h]
 
+@[simp] theorem guard_isSome [DecidablePred p] : (Option.guard p a).isSome ↔ p a :=
+  if h : p a then by simp [Option.guard, h] else by simp [Option.guard, h]
+
+@[simp] theorem guard_eq_none [DecidablePred p] : Option.guard p a = none ↔ ¬ p a :=
+  if h : p a then by simp [Option.guard, h] else by simp [Option.guard, h]
+
+@[simp] theorem guard_pos [DecidablePred p] (h : p a) : Option.guard p a = some a := by
+  simp [Option.guard, h]
+
 theorem liftOrGet_eq_or_eq {f : α → α → α} (h : ∀ a b, f a b = a ∨ f a b = b) :
     ∀ o₁ o₂, liftOrGet f o₁ o₂ = o₁ ∨ liftOrGet f o₁ o₂ = o₂
   | none, none => .inl rfl
@@ -250,7 +284,7 @@ theorem liftOrGet_eq_or_eq {f : α → α → α} (h : ∀ a b, f a b = a ∨ f
 @[simp] theorem getD_map (f : α → β) (x : α) (o : Option α) :
   (o.map f).getD (f x) = f (getD o x) := by cases o <;> rfl
 
-section
+section choice
 
 attribute [local instance] Classical.propDecidable
 
@@ -266,7 +300,7 @@ theorem choice_eq {α : Type _} [Subsingleton α] (a : α) : choice α = some a
 theorem choice_isSome_iff_nonempty {α : Type _} : (choice α).isSome ↔ Nonempty α :=
   ⟨fun h => ⟨(choice α).get h⟩, fun h => by simp only [choice, dif_pos h, isSome_some]⟩
 
-end
+end choice
 
 @[simp] theorem toList_some (a : α) : (a : Option α).toList = [a] := rfl
 
@@ -287,7 +321,7 @@ theorem or_eq_bif : or o o' = bif o.isSome then o else o' := by
 @[simp] theorem or_eq_none : or o o' = none ↔ o = none ∧ o' = none := by
   cases o <;> simp
 
-theorem or_eq_some : or o o' = some a ↔ o = some a ∨ (o = none ∧ o' = some a) := by
+@[simp] theorem or_eq_some : or o o' = some a ↔ o = some a ∨ (o = none ∧ o' = some a) := by
   cases o <;> simp
 
 theorem or_assoc : or (or o₁ o₂) o₃ = or o₁ (or o₂ o₃) := by
@@ -314,3 +348,51 @@ theorem map_or : f <$> or o o' = (f <$> o).or (f <$> o') := by
 
 theorem map_or' : (or o o').map f = (o.map f).or (o'.map f) := by
   cases o <;> rfl
+
+theorem or_of_isSome {o o' : Option α} (h : o.isSome) : o.or o' = o := by
+  match o, h with
+  | some _, _ => simp
+
+theorem or_of_isNone {o o' : Option α} (h : o.isNone) : o.or o' = o' := by
+  match o, h with
+  | none, _ => simp
+
+section ite
+
+@[simp] theorem isSome_dite {p : Prop} [Decidable p] {b : p → β} :
+    (if h : p then some (b h) else none).isSome = true ↔ p := by
+  split <;> simpa
+@[simp] theorem isSome_ite {p : Prop} [Decidable p] :
+    (if p then some b else none).isSome = true ↔ p := by
+  split <;> simpa
+@[simp] theorem isSome_dite' {p : Prop} [Decidable p] {b : ¬ p → β} :
+    (if h : p then none else some (b h)).isSome = true ↔ ¬ p := by
+  split <;> simpa
+@[simp] theorem isSome_ite' {p : Prop} [Decidable p] :
+    (if p then none else some b).isSome = true ↔ ¬ p := by
+  split <;> simpa
+
+@[simp] theorem get_dite {p : Prop} [Decidable p] (b : p → β) (w) :
+    (if h : p then some (b h) else none).get w = b (by simpa using w) := by
+  split
+  · simp
+  · exfalso
+    simp at w
+    contradiction
+@[simp] theorem get_ite {p : Prop} [Decidable p] (h) :
+    (if p then some b else none).get h = b := by
+  simpa using get_dite (p := p) (fun _ => b) (by simpa using h)
+@[simp] theorem get_dite' {p : Prop} [Decidable p] (b : ¬ p → β) (w) :
+    (if h : p then none else some (b h)).get w = b (by simpa using w) := by
+  split
+  · exfalso
+    simp at w
+    contradiction
+  · simp
+@[simp] theorem get_ite' {p : Prop} [Decidable p] (h) :
+    (if p then none else some b).get h = b := by
+  simpa using get_dite' (p := p) (fun _ => b) (by simpa using h)
+
+end ite
+
+end Option

src/Init/Data/PLift.lean

@@ -0,0 +1,12 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Core
+
+deriving instance DecidableEq for PLift
+
+instance [Subsingleton α] : Subsingleton (PLift α) where
+  allEq := fun ⟨a⟩ ⟨b⟩ => congrArg PLift.up (Subsingleton.elim a b)

src/Init/Data/Prod.lean

@@ -5,9 +5,18 @@ Author: Leonardo de Moura
 -/
 prelude
 import Init.SimpLemmas
+import Init.NotationExtra
 
 instance [BEq α] [BEq β] [LawfulBEq α] [LawfulBEq β] : LawfulBEq (α × β) where
   eq_of_beq {a b} (h : a.1 == b.1 && a.2 == b.2) := by
     cases a; cases b
     refine congr (congrArg _ (eq_of_beq ?_)) (eq_of_beq ?_) <;> simp_all
   rfl {a} := by cases a; simp [BEq.beq, LawfulBEq.rfl]
+
+@[simp]
+protected theorem Prod.forall {p : α × β → Prop} : (∀ x, p x) ↔ ∀ a b, p (a, b) :=
+  ⟨fun h a b ↦ h (a, b), fun h ⟨a, b⟩ ↦ h a b⟩
+
+@[simp]
+protected theorem Prod.exists {p : α × β → Prop} : (∃ x, p x) ↔ ∃ a b, p (a, b) :=
+  ⟨fun ⟨⟨a, b⟩, h⟩ ↦ ⟨a, b, h⟩, fun ⟨a, b, h⟩ ↦ ⟨⟨a, b⟩, h⟩⟩

src/Init/Data/Queue.lean

@@ -7,7 +7,7 @@ Simple queue implemented using two lists.
 Note: this is only a temporary placeholder.
 -/
 prelude
-import Init.Data.List
+import Init.Data.Array.Basic
 
 namespace Std
 

src/Init/Data/Range.lean

@@ -15,7 +15,7 @@ structure Range where
   step  : Nat := 1
 
 instance : Membership Nat Range where
-  mem i r := r.start ≤ i ∧ i < r.stop
+  mem r i := r.start ≤ i ∧ i < r.stop
 
 namespace Range
 universe u v

src/Init/Data/String/Basic.lean

@@ -227,7 +227,7 @@ Examples:
 * `"abc".front = 'a'`
 * `"".front = (default : Char)`
 -/
-def front (s : String) : Char :=
+@[inline] def front (s : String) : Char :=
   get s 0
 
 /--
@@ -237,7 +237,7 @@ Examples:
 * `"abc".back = 'c'`
 * `"".back = (default : Char)`
 -/
-def back (s : String) : Char :=
+@[inline] def back (s : String) : Char :=
   get s (prev s s.endPos)
 
 /--
@@ -374,7 +374,7 @@ Examples:
 * `"abba".posOf 'z' = none`
 * `"L∃∀N".posOf '∀' = some ⟨4⟩`
 -/
-def revPosOf (s : String) (c : Char) : Option Pos :=
+@[inline] def revPosOf (s : String) (c : Char) : Option Pos :=
   revPosOfAux s c s.endPos
 
 def findAux (s : String) (p : Char → Bool) (stopPos : Pos) (pos : Pos) : Pos :=
@@ -398,7 +398,7 @@ def revFindAux (s : String) (p : Char → Bool) (pos : Pos) : Option Pos :=
     else revFindAux s p pos
 termination_by pos.1
 
-def revFind (s : String) (p : Char → Bool) : Option Pos :=
+@[inline] def revFind (s : String) (p : Char → Bool) : Option Pos :=
   revFindAux s p s.endPos
 
 abbrev Pos.min (p₁ p₂ : Pos) : Pos :=
@@ -505,7 +505,7 @@ The default separator is `" "`. The separators are not included in the returned
 "ababacabac".splitOn "aba" = ["", "bac", "c"]
 ```
 -/
-def splitOn (s : String) (sep : String := " ") : List String :=
+@[inline] def splitOn (s : String) (sep : String := " ") : List String :=
   if sep == "" then [s] else splitOnAux s sep 0 0 0 []
 
 instance : Inhabited String := ⟨""⟩
@@ -515,16 +515,16 @@ instance : Append String := ⟨String.append⟩
 @[deprecated push (since := "2024-04-06")]
 def str : String → Char → String := push
 
-def pushn (s : String) (c : Char) (n : Nat) : String :=
+@[inline] def pushn (s : String) (c : Char) (n : Nat) : String :=
   n.repeat (fun s => s.push c) s
 
-def isEmpty (s : String) : Bool :=
+@[inline] def isEmpty (s : String) : Bool :=
   s.endPos == 0
 
-def join (l : List String) : String :=
+@[inline] def join (l : List String) : String :=
   l.foldl (fun r s => r ++ s) ""
 
-def singleton (c : Char) : String :=
+@[inline] def singleton (c : Char) : String :=
   "".push c
 
 def intercalate (s : String) : List String → String
@@ -558,10 +558,10 @@ structure Iterator where
   `Iterator.next` when `Iterator.atEnd` is true. If the position is not valid, then the
   current character is `(default : Char)`, similar to `String.get` on an invalid position. -/
   i : Pos
-  deriving DecidableEq
+  deriving DecidableEq, Inhabited
 
 /-- Creates an iterator at the beginning of a string. -/
-def mkIterator (s : String) : Iterator :=
+@[inline] def mkIterator (s : String) : Iterator :=
   ⟨s, 0⟩
 
 @[inherit_doc mkIterator]
@@ -575,66 +575,74 @@ theorem Iterator.sizeOf_eq (i : String.Iterator) : sizeOf i = i.1.utf8ByteSize -
   rfl
 
 namespace Iterator
-@[inherit_doc Iterator.s]
+@[inline, inherit_doc Iterator.s]
 def toString := Iterator.s
 
 /-- Number of bytes remaining in the iterator. -/
-def remainingBytes : Iterator → Nat
+@[inline] def remainingBytes : Iterator → Nat
   | ⟨s, i⟩ => s.endPos.byteIdx - i.byteIdx
 
-@[inherit_doc Iterator.i]
+@[inline, inherit_doc Iterator.i]
 def pos := Iterator.i
 
 /-- The character at the current position.
 
 On an invalid position, returns `(default : Char)`. -/
-def curr : Iterator → Char
+@[inline] def curr : Iterator → Char
   | ⟨s, i⟩ => get s i
 
 /-- Moves the iterator's position forward by one character, unconditionally.
 
 It is only valid to call this function if the iterator is not at the end of the string, *i.e.*
 `Iterator.atEnd` is `false`; otherwise, the resulting iterator will be invalid. -/
-def next : Iterator → Iterator
+@[inline] def next : Iterator → Iterator
   | ⟨s, i⟩ => ⟨s, s.next i⟩
 
 /-- Decreases the iterator's position.
 
 If the position is zero, this function is the identity. -/
-def prev : Iterator → Iterator
+@[inline] def prev : Iterator → Iterator
   | ⟨s, i⟩ => ⟨s, s.prev i⟩
 
 /-- True if the iterator is past the string's last character. -/
-def atEnd : Iterator → Bool
+@[inline] def atEnd : Iterator → Bool
   | ⟨s, i⟩ => i.byteIdx ≥ s.endPos.byteIdx
 
 /-- True if the iterator is not past the string's last character. -/
-def hasNext : Iterator → Bool
+@[inline] def hasNext : Iterator → Bool
   | ⟨s, i⟩ => i.byteIdx < s.endPos.byteIdx
 
 /-- True if the position is not zero. -/
-def hasPrev : Iterator → Bool
+@[inline] def hasPrev : Iterator → Bool
   | ⟨_, i⟩ => i.byteIdx > 0
 
+@[inline] def curr' (it : Iterator) (h : it.hasNext) : Char :=
+  match it with
+  | ⟨s, i⟩ => get' s i (by simpa only [hasNext, endPos, decide_eq_true_eq, String.atEnd, ge_iff_le, Nat.not_le] using h)
+
+@[inline] def next' (it : Iterator) (h : it.hasNext) : Iterator :=
+  match it with
+  | ⟨s, i⟩ => ⟨s, s.next' i (by simpa only [hasNext, endPos, decide_eq_true_eq, String.atEnd, ge_iff_le, Nat.not_le] using h)⟩
+
 /-- Replaces the current character in the string.
 
 Does nothing if the iterator is at the end of the string. If the iterator contains the only
 reference to its string, this function will mutate the string in-place instead of allocating a new
 one. -/
-def setCurr : Iterator → Char → Iterator
+@[inline] def setCurr : Iterator → Char → Iterator
   | ⟨s, i⟩, c => ⟨s.set i c, i⟩
 
 /-- Moves the iterator's position to the end of the string.
 
 Note that `i.toEnd.atEnd` is always `true`. -/
-def toEnd : Iterator → Iterator
+@[inline] def toEnd : Iterator → Iterator
   | ⟨s, _⟩ => ⟨s, s.endPos⟩
 
 /-- Extracts the substring between the positions of two iterators.
 
 Returns the empty string if the iterators are for different strings, or if the position of the first
 iterator is past the position of the second iterator. -/
-def extract : Iterator → Iterator → String
+@[inline] def extract : Iterator → Iterator → String
   | ⟨s₁, b⟩, ⟨s₂, e⟩ =>
     if s₁ ≠ s₂ || b > e then ""
     else s₁.extract b e
@@ -648,7 +656,7 @@ def forward : Iterator → Nat → Iterator
   | it, n+1 => forward it.next n
 
 /-- The remaining characters in an iterator, as a string. -/
-def remainingToString : Iterator → String
+@[inline] def remainingToString : Iterator → String
   | ⟨s, i⟩ => s.extract i s.endPos
 
 @[inherit_doc forward]
@@ -673,7 +681,7 @@ def offsetOfPosAux (s : String) (pos : Pos) (i : Pos) (offset : Nat) : Nat :=
     offsetOfPosAux s pos (s.next i) (offset+1)
 termination_by s.endPos.1 - i.1
 
-def offsetOfPos (s : String) (pos : Pos) : Nat :=
+@[inline] def offsetOfPos (s : String) (pos : Pos) : Nat :=
   offsetOfPosAux s pos 0 0
 
 @[specialize] def foldlAux {α : Type u} (f : α → Char → α) (s : String) (stopPos : Pos) (i : Pos) (a : α) : α :=
@@ -714,7 +722,7 @@ termination_by stopPos.1 - i.1
 @[inline] def all (s : String) (p : Char → Bool) : Bool :=
   !s.any (fun c => !p c)
 
-def contains (s : String) (c : Char) : Bool :=
+@[inline] def contains (s : String) (c : Char) : Bool :=
 s.any (fun a => a == c)
 
 theorem utf8SetAux_of_gt (c' : Char) : ∀ (cs : List Char) {i p : Pos}, i > p → utf8SetAux c' cs i p = cs
@@ -770,7 +778,7 @@ termination_by s.endPos.1 - i.1
 @[inline] def map (f : Char → Char) (s : String) : String :=
   mapAux f 0 s
 
-def isNat (s : String) : Bool :=
+@[inline] def isNat (s : String) : Bool :=
   !s.isEmpty && s.all (·.isDigit)
 
 def toNat? (s : String) : Option Nat :=
@@ -940,7 +948,7 @@ def splitOn (s : Substring) (sep : String := " ") : List Substring :=
 @[inline] def all (s : Substring) (p : Char → Bool) : Bool :=
   !s.any (fun c => !p c)
 
-def contains (s : Substring) (c : Char) : Bool :=
+@[inline] def contains (s : Substring) (c : Char) : Bool :=
   s.any (fun a => a == c)
 
 @[specialize] def takeWhileAux (s : String) (stopPos : String.Pos) (p : Char → Bool) (i : String.Pos) : String.Pos :=
@@ -995,7 +1003,7 @@ termination_by i.1
     let e := takeRightWhileAux s b Char.isWhitespace e
     ⟨s, b, e⟩
 
-def isNat (s : Substring) : Bool :=
+@[inline] def isNat (s : Substring) : Bool :=
   s.all fun c => c.isDigit
 
 def toNat? (s : Substring) : Option Nat :=
@@ -1017,43 +1025,43 @@ end Substring
 
 namespace String
 
-def drop (s : String) (n : Nat) : String :=
+@[inline] def drop (s : String) (n : Nat) : String :=
   (s.toSubstring.drop n).toString
 
-def dropRight (s : String) (n : Nat) : String :=
+@[inline] def dropRight (s : String) (n : Nat) : String :=
   (s.toSubstring.dropRight n).toString
 
-def take (s : String) (n : Nat) : String :=
+@[inline] def take (s : String) (n : Nat) : String :=
   (s.toSubstring.take n).toString
 
-def takeRight (s : String) (n : Nat) : String :=
+@[inline] def takeRight (s : String) (n : Nat) : String :=
   (s.toSubstring.takeRight n).toString
 
-def takeWhile (s : String) (p : Char → Bool) : String :=
+@[inline] def takeWhile (s : String) (p : Char → Bool) : String :=
   (s.toSubstring.takeWhile p).toString
 
-def dropWhile (s : String) (p : Char → Bool) : String :=
+@[inline] def dropWhile (s : String) (p : Char → Bool) : String :=
   (s.toSubstring.dropWhile p).toString
 
-def takeRightWhile (s : String) (p : Char → Bool) : String :=
+@[inline] def takeRightWhile (s : String) (p : Char → Bool) : String :=
   (s.toSubstring.takeRightWhile p).toString
 
-def dropRightWhile (s : String) (p : Char → Bool) : String :=
+@[inline] def dropRightWhile (s : String) (p : Char → Bool) : String :=
   (s.toSubstring.dropRightWhile p).toString
 
-def startsWith (s pre : String) : Bool :=
+@[inline] def startsWith (s pre : String) : Bool :=
   s.toSubstring.take pre.length == pre.toSubstring
 
-def endsWith (s post : String) : Bool :=
+@[inline] def endsWith (s post : String) : Bool :=
   s.toSubstring.takeRight post.length == post.toSubstring
 
-def trimRight (s : String) : String :=
+@[inline] def trimRight (s : String) : String :=
   s.toSubstring.trimRight.toString
 
-def trimLeft (s : String) : String :=
+@[inline] def trimLeft (s : String) : String :=
   s.toSubstring.trimLeft.toString
 
-def trim (s : String) : String :=
+@[inline] def trim (s : String) : String :=
   s.toSubstring.trim.toString
 
 @[inline] def nextWhile (s : String) (p : Char → Bool) (i : String.Pos) : String.Pos :=
@@ -1062,23 +1070,23 @@ def trim (s : String) : String :=
 @[inline] def nextUntil (s : String) (p : Char → Bool) (i : String.Pos) : String.Pos :=
   nextWhile s (fun c => !p c) i
 
-def toUpper (s : String) : String :=
+@[inline] def toUpper (s : String) : String :=
   s.map Char.toUpper
 
-def toLower (s : String) : String :=
+@[inline] def toLower (s : String) : String :=
   s.map Char.toLower
 
-def capitalize (s : String) :=
+@[inline] def capitalize (s : String) :=
   s.set 0 <| s.get 0 |>.toUpper
 
-def decapitalize (s : String) :=
+@[inline] def decapitalize (s : String) :=
   s.set 0 <| s.get 0 |>.toLower
 
 end String
 
 namespace Char
 
-protected def toString (c : Char) : String :=
+@[inline] protected def toString (c : Char) : String :=
   String.singleton c
 
 @[simp] theorem length_toString (c : Char) : c.toString.length = 1 := rfl

src/Init/Data/UInt/Lemmas.lean

@@ -47,11 +47,11 @@ protected theorem ne_of_val_ne {a b : $typeName} (h : a.val ≠ b.val) : a ≠ b
 open $typeName (ne_of_val_ne) in
 protected theorem ne_of_lt {a b : $typeName} (h : a < b) : a ≠ b := ne_of_val_ne (Fin.ne_of_lt h)
 
-@[simp] protected theorem zero_toNat : (0 : $typeName).toNat = 0 := Nat.zero_mod _
-@[simp] protected theorem mod_toNat (a b : $typeName) : (a % b).toNat = a.toNat % b.toNat := Fin.mod_val ..
-@[simp] protected theorem div_toNat (a b : $typeName) : (a / b).toNat = a.toNat / b.toNat := Fin.div_val ..
-@[simp] protected theorem sub_toNat_of_le (a b : $typeName) : b ≤ a → (a - b).toNat = a.toNat - b.toNat := Fin.sub_val_of_le
-@[simp] protected theorem modn_toNat (a : $typeName) (b : Nat) : (a.modn b).toNat = a.toNat % b := Fin.modn_val ..
+@[simp] protected theorem toNat_zero : (0 : $typeName).toNat = 0 := Nat.zero_mod _
+@[simp] protected theorem toNat_mod (a b : $typeName) : (a % b).toNat = a.toNat % b.toNat := Fin.mod_val ..
+@[simp] protected theorem toNat_div (a b : $typeName) : (a / b).toNat = a.toNat / b.toNat := Fin.div_val ..
+@[simp] protected theorem toNat_sub_of_le (a b : $typeName) : b ≤ a → (a - b).toNat = a.toNat - b.toNat := Fin.sub_val_of_le
+@[simp] protected theorem toNat_modn (a : $typeName) (b : Nat) : (a.modn b).toNat = a.toNat % b := Fin.modn_val ..
 protected theorem modn_lt {m : Nat} : ∀ (u : $typeName), m > 0 → toNat (u % m) < m
   | ⟨u⟩, h => Fin.modn_lt u h
 open $typeName (modn_lt) in
@@ -69,3 +69,28 @@ declare_uint_theorems UInt16
 declare_uint_theorems UInt32
 declare_uint_theorems UInt64
 declare_uint_theorems USize
+
+@[deprecated (since := "2024-06-23")] protected abbrev UInt8.zero_toNat := @UInt8.toNat_zero
+@[deprecated (since := "2024-06-23")] protected abbrev UInt8.div_toNat := @UInt8.toNat_div
+@[deprecated (since := "2024-06-23")] protected abbrev UInt8.mod_toNat := @UInt8.toNat_mod
+@[deprecated (since := "2024-06-23")] protected abbrev UInt8.modn_toNat := @UInt8.toNat_modn
+
+@[deprecated (since := "2024-06-23")] protected abbrev UInt16.zero_toNat := @UInt16.toNat_zero
+@[deprecated (since := "2024-06-23")] protected abbrev UInt16.div_toNat := @UInt16.toNat_div
+@[deprecated (since := "2024-06-23")] protected abbrev UInt16.mod_toNat := @UInt16.toNat_mod
+@[deprecated (since := "2024-06-23")] protected abbrev UInt16.modn_toNat := @UInt16.toNat_modn
+
+@[deprecated (since := "2024-06-23")] protected abbrev UInt32.zero_toNat := @UInt32.toNat_zero
+@[deprecated (since := "2024-06-23")] protected abbrev UInt32.div_toNat := @UInt32.toNat_div
+@[deprecated (since := "2024-06-23")] protected abbrev UInt32.mod_toNat := @UInt32.toNat_mod
+@[deprecated (since := "2024-06-23")] protected abbrev UInt32.modn_toNat := @UInt32.toNat_modn
+
+@[deprecated (since := "2024-06-23")] protected abbrev UInt64.zero_toNat := @UInt64.toNat_zero
+@[deprecated (since := "2024-06-23")] protected abbrev UInt64.div_toNat := @UInt64.toNat_div
+@[deprecated (since := "2024-06-23")] protected abbrev UInt64.mod_toNat := @UInt64.toNat_mod
+@[deprecated (since := "2024-06-23")] protected abbrev UInt64.modn_toNat := @UInt64.toNat_modn
+
+@[deprecated (since := "2024-06-23")] protected abbrev USize.zero_toNat := @USize.toNat_zero
+@[deprecated (since := "2024-06-23")] protected abbrev USize.div_toNat := @USize.toNat_div
+@[deprecated (since := "2024-06-23")] protected abbrev USize.mod_toNat := @USize.toNat_mod
+@[deprecated (since := "2024-06-23")] protected abbrev USize.modn_toNat := @USize.toNat_modn

src/Init/Data/ULift.lean

@@ -0,0 +1,12 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Core
+
+deriving instance DecidableEq for ULift
+
+instance [Subsingleton α] : Subsingleton (ULift α) where
+  allEq := fun ⟨a⟩ ⟨b⟩ => congrArg ULift.up (Subsingleton.elim a b)

src/Init/Ext.lean

@@ -78,7 +78,10 @@ end Elab.Tactic.Ext
 end Lean
 
 attribute [ext] Prod PProd Sigma PSigma
-attribute [ext] funext propext Subtype.eq
+attribute [ext] funext propext Subtype.eq Array.ext
 
 @[ext] protected theorem PUnit.ext (x y : PUnit) : x = y := rfl
 protected theorem Unit.ext (x y : Unit) : x = y := rfl
+
+@[ext] protected theorem Thunk.ext : {a b : Thunk α} → a.get = b.get → a = b
+  | {..}, {..}, heq => congrArg _ <| funext fun _ => heq

src/Init/GetElem.lean

@@ -184,7 +184,7 @@ instance [GetElem? cont Nat elem dom] [h : LawfulGetElem cont Nat elem dom] :
 @[simp] theorem getElem!_fin [GetElem? Cont Nat Elem Dom] (a : Cont) (i : Fin n) [Inhabited Elem] : a[i]! = a[i.1]! := rfl
 
 macro_rules
-  | `(tactic| get_elem_tactic_trivial) => `(tactic| apply Fin.val_lt_of_le; get_elem_tactic_trivial; done)
+  | `(tactic| get_elem_tactic_trivial) => `(tactic| (with_reducible apply Fin.val_lt_of_le); get_elem_tactic_trivial; done)
 
 end Fin
 

src/Init/Meta.lean

@@ -75,7 +75,7 @@ See #2572.
 opaque Internal.hasLLVMBackend (u : Unit) : Bool
 
 /-- Valid identifier names -/
-def isGreek (c : Char) : Bool :=
+@[inline] def isGreek (c : Char) : Bool :=
   0x391 ≤ c.val && c.val ≤ 0x3dd
 
 def isLetterLike (c : Char) : Bool :=
@@ -86,7 +86,7 @@ def isLetterLike (c : Char) : Bool :=
   (0x2100 ≤ c.val && c.val ≤ 0x214f) ||                                  -- Letter like block
   (0x1d49c ≤ c.val && c.val ≤ 0x1d59f)                                   -- Latin letters, Script, Double-struck, Fractur
 
-def isNumericSubscript (c : Char) : Bool :=
+@[inline] def isNumericSubscript (c : Char) : Bool :=
   0x2080 ≤ c.val && c.val ≤ 0x2089
 
 def isSubScriptAlnum (c : Char) : Bool :=
@@ -94,16 +94,16 @@ def isSubScriptAlnum (c : Char) : Bool :=
   (0x2090 ≤ c.val && c.val ≤ 0x209c) ||
   (0x1d62 ≤ c.val && c.val ≤ 0x1d6a)
 
-def isIdFirst (c : Char) : Bool :=
+@[inline] def isIdFirst (c : Char) : Bool :=
   c.isAlpha || c = '_' || isLetterLike c
 
-def isIdRest (c : Char) : Bool :=
+@[inline] def isIdRest (c : Char) : Bool :=
   c.isAlphanum || c = '_' || c = '\'' || c == '!' || c == '?' || isLetterLike c || isSubScriptAlnum c
 
 def idBeginEscape := '«'
 def idEndEscape   := '»'
-def isIdBeginEscape (c : Char) : Bool := c = idBeginEscape
-def isIdEndEscape (c : Char) : Bool := c = idEndEscape
+@[inline] def isIdBeginEscape (c : Char) : Bool := c = idBeginEscape
+@[inline] def isIdEndEscape (c : Char) : Bool := c = idEndEscape
 namespace Name
 
 def getRoot : Name → Name
@@ -388,9 +388,9 @@ def getSubstring? (stx : Syntax) (withLeading := true) (withTrailing := true) :
 partial def setTailInfoAux (info : SourceInfo) : Syntax → Option Syntax
   | atom _ val             => some <| atom info val
   | ident _ rawVal val pre => some <| ident info rawVal val pre
-  | node info k args       =>
+  | node info' k args      =>
     match updateLast args (setTailInfoAux info) args.size with
-    | some args => some <| node info k args
+    | some args => some <| node info' k args
     | none      => none
   | _                      => none
 

src/Init/MetaTypes.lean

@@ -104,6 +104,11 @@ structure Config where
   That is, given a local context containing entry `x : t := e`, the free variable `x` reduces to `e`.
   -/
   zetaDelta         : Bool := false
+  /--
+  When `index` (default : `true`) is `false`, `simp` will only use the root symbol
+  to find candidate `simp` theorems. It approximates Lean 3 `simp` behavior.
+  -/
+  index             : Bool := true
   deriving Inhabited, BEq
 
 end DSimp

src/Init/Notation.lean

@@ -336,7 +336,7 @@ macro_rules | `($x == $y) => `(binrel_no_prop% BEq.beq $x $y)
 @[inherit_doc] infixl:30 " || " => or
 @[inherit_doc] notation:max "!" b:40 => not b
 
-@[inherit_doc] infix:50 " ∈ " => Membership.mem
+@[inherit_doc] notation:50 a:50 " ∈ " b:50 => Membership.mem b a
 /-- `a ∉ b` is negated elementhood. It is notation for `¬ (a ∈ b)`. -/
 notation:50 a:50 " ∉ " b:50 => ¬ (a ∈ b)
 
@@ -704,6 +704,17 @@ syntax (name := checkSimp) "#check_simp " term "~>" term : command
 -/
 syntax (name := checkSimpFailure) "#check_simp " term "!~>" : command
 
+/--
+Time the elaboration of a command, and print the result (in milliseconds).
+
+Example usage:
+```
+set_option maxRecDepth 100000 in
+#time example : (List.range 500).length = 500 := rfl
+```
+-/
+syntax (name := timeCmd) "#time " command : command
+
 /--
 `#discr_tree_key  t` prints the discrimination tree keys for a term `t` (or, if it is a single identifier, the type of that constant).
 It uses the default configuration for generating keys.