feat: align List.replicate/Array.mkArray/Vector.mkVector lemmas

This PR adds array indexing lemmas for `Vector.flatMap`. (These were not
available for `List` and `Array` due to variable lengths.)

.github/workflows/ci.yml

@@ -238,7 +238,7 @@ jobs:
                 "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 -DCMAKE_LIBRARY_PATH=/usr/lib/i386-linux-gnu/ -DSTAGE0_CMAKE_LIBRARY_PATH=/usr/lib/i386-linux-gnu/",
+                "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/ -DPKG_CONFIG_EXECUTABLE=/usr/bin/i386-linux-gnu-pkg-config",
                 "cmultilib": true,
                 "release": true,
                 "check-level": 2,
@@ -327,7 +327,7 @@ jobs:
         run: |
           sudo dpkg --add-architecture i386
           sudo apt-get update
-          sudo apt-get install -y gcc-multilib g++-multilib ccache libuv1-dev:i386
+          sudo apt-get install -y gcc-multilib g++-multilib ccache libuv1-dev:i386 pkgconf:i386
         if: matrix.cmultilib
       - name: Cache
         uses: actions/cache@v4

CMakeLists.txt

@@ -18,6 +18,9 @@ foreach(var ${vars})
     if("${var}" MATCHES "LLVM*")
       list(APPEND STAGE0_ARGS "-D${var}=${${var}}")
     endif()
+    if("${var}" MATCHES "PKG_CONFIG*")
+      list(APPEND STAGE0_ARGS "-D${var}=${${var}}")
+    endif()
   elseif(("${var}" MATCHES "CMAKE_.*") AND NOT ("${var}" MATCHES "CMAKE_BUILD_TYPE") AND NOT ("${var}" MATCHES "CMAKE_HOME_DIRECTORY"))
     list(APPEND PLATFORM_ARGS "-D${var}=${${var}}")
   endif()

doc/dev/commit_convention.md

@@ -33,6 +33,9 @@ Format of the commit message
  - chore (maintain, ex: travis-ci)
  - perf (performance improvement, optimization, ...)
 
+Every `feat` or `fix` commit must have a `changelog-*` label, and a commit message
+beginning with "This PR " that will be included in the changelog.
+
 ``<subject>`` has the following constraints:
 
  - use imperative, present tense: "change" not "changed" nor "changes"
@@ -44,6 +47,7 @@ Format of the commit message
  - just as in ``<subject>``, use imperative, present tense
  - includes motivation for the change and contrasts with previous
    behavior
+ - If a `changelog-*` label is present, the body must begin with "This PR ".
 
 ``<footer>`` is optional and may contain two items:
 
@@ -60,17 +64,21 @@ Examples
 
 fix: add declarations for operator<<(std::ostream&, expr const&) and operator<<(std::ostream&, context const&) in the kernel
 
+This PR adds declarations `operator<<` for raw printing.
 The actual implementation of these two operators is outside of the
-kernel. They are implemented in the file 'library/printer.cpp'. We
-declare them in the kernel to prevent the following problem. Suppose
-there is a file 'foo.cpp' that does not include 'library/printer.h',
-but contains
+kernel. They are implemented in the file 'library/printer.cpp'.
 
-    expr a;
-    ...
-    std::cout << a << "\n";
-    ...
+We declare them in the kernel to prevent the following problem.
+Suppose there is a file 'foo.cpp' that does not include 'library/printer.h',
+but contains
+```cpp
+expr a;
+...
+std::cout << a << "\n";
+...
+```
 
 The compiler does not generate an error message. It silently uses the
 operator bool() to coerce the expression into a Boolean. This produces
 counter-intuitive behavior, and may confuse developers.
+

doc/dev/ffi.md

@@ -49,8 +49,9 @@ In the case of `@[extern]` all *irrelevant* types are removed first; see next se
   is represented by the representation of that parameter's type.
 
   For example, `{ x : α // p }`, the `Subtype` structure of a value of type `α` and an irrelevant proof, is represented by the representation of `α`.
-* `Nat` is represented by `lean_object *`.
-  Its runtime value is either a pointer to an opaque bignum object or, if the lowest bit of the "pointer" is 1 (`lean_is_scalar`), an encoded unboxed natural number (`lean_box`/`lean_unbox`).
+  Similarly, the signed integer types `Int8`, ..., `Int64`, `ISize` are also represented by the unsigned C types `uint8_t`, ..., `uint64_t`, `size_t`, respectively, because they have a trivial structure.
+* `Nat` and `Int` are represented by `lean_object *`.
+  Their runtime values is either a pointer to an opaque bignum object or, if the lowest bit of the "pointer" is 1 (`lean_is_scalar`), an encoded unboxed natural number or integer (`lean_box`/`lean_unbox`).
 * A universe `Sort u`, type constructor `... → Sort u`, or proposition `p : Prop` is *irrelevant* and is either statically erased (see above) or represented as a `lean_object *` with the runtime value `lean_box(0)`
 * Any other type is represented by `lean_object *`.
   Its runtime value is a pointer to an object of a subtype of `lean_object` (see the "Inductive types" section below) or the unboxed value `lean_box(cidx)` for the `cidx`th constructor of an inductive type if this constructor does not have any relevant parameters.

doc/dev/index.md

@@ -80,3 +80,10 @@ Unlike most Lean projects, all submodules of the `Lean` module begin with the
 `prelude` keyword. This disables the automated import of `Init`, meaning that
 developers need to figure out their own subset of `Init` to import. This is done
 such that changing files in `Init` doesn't force a full rebuild of `Lean`.
+
+### Testing against Mathlib/Batteries
+You can test a Lean PR against Mathlib and Batteries by rebasing your PR
+on to `nightly-with-mathlib` branch. (It is fine to force push after rebasing.)
+CI will generate a branch of Mathlib and Batteries called `lean-pr-testing-NNNN`
+that uses the toolchain for your PR, and will report back to the Lean PR with results from Mathlib CI.
+See https://leanprover-community.github.io/contribute/tags_and_branches.html for more details.

doc/dev/release_checklist.md

@@ -5,11 +5,6 @@ See below for the checklist for release candidates.
 
 We'll use `v4.6.0` as the intended release version as a running example.
 
-- One week before the planned release, ensure that
-  (1) someone has written the release notes and
-  (2) someone has written the first draft of the release blog post.
-  If there is any material in `./releases_drafts/` on the `releases/v4.6.0` branch, then the release notes are not done.
-  (See the section "Writing the release notes".)
 - `git checkout releases/v4.6.0`
   (This branch should already exist, from the release candidates.)
 - `git pull`
@@ -42,16 +37,32 @@ We'll use `v4.6.0` as the intended release version as a running example.
     - Create the tag `v4.6.0` from `master`/`main` and push it.
     - Merge the tag `v4.6.0` into the `stable` branch and push it.
   - We do this for the repositories:
-    - [lean4checker](https://github.com/leanprover/lean4checker)
-      - No dependencies
-      - Toolchain bump PR
-      - Create and push the tag
-      - Merge the tag into `stable`
     - [Batteries](https://github.com/leanprover-community/batteries)
       - No dependencies
       - Toolchain bump PR
       - Create and push the tag
       - Merge the tag into `stable`
+    - [lean4checker](https://github.com/leanprover/lean4checker)
+      - No dependencies
+      - Toolchain bump PR
+      - Create and push the tag
+      - Merge the tag into `stable`
+    - [doc-gen4](https://github.com/leanprover/doc-gen4)
+      - Dependencies: exist, but they're not part of the release workflow
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - There is no `stable` branch; skip this step
+    - [Verso](https://github.com/leanprover/verso)
+      - Dependencies: exist, but they're not part of the release workflow
+      - The `SubVerso` dependency should be compatible with _every_ Lean release simultaneously, rather than following this workflow
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - There is no `stable` branch; skip this step
+    - [Cli](https://github.com/leanprover/lean4-cli)
+      - No dependencies
+      - Toolchain bump PR
+      - Create and push the tag
+      - There is no `stable` branch; skip this step
     - [ProofWidgets4](https://github.com/leanprover-community/ProofWidgets4)
       - Dependencies: `Batteries`
       - Note on versions and branches:
@@ -66,27 +77,20 @@ We'll use `v4.6.0` as the intended release version as a running example.
       - Toolchain bump PR including updated Lake manifest
       - Create and push the tag
       - Merge the tag into `stable`
-    - [doc-gen4](https://github.com/leanprover/doc-gen4)
-      - Dependencies: exist, but they're not part of the release workflow
-      - Toolchain bump PR including updated Lake manifest
-      - Create and push the tag
-      - There is no `stable` branch; skip this step
-    - [Verso](https://github.com/leanprover/verso)
-      - Dependencies: exist, but they're not part of the release workflow
-      - The `SubVerso` dependency should be compatible with _every_ Lean release simultaneously, rather than following this workflow
-      - Toolchain bump PR including updated Lake manifest
-      - Create and push the tag
-      - There is no `stable` branch; skip this step
     - [import-graph](https://github.com/leanprover-community/import-graph)
       - Toolchain bump PR including updated Lake manifest
       - Create and push the tag
       - There is no `stable` branch; skip this step
+    - [plausible](https://github.com/leanprover-community/plausible)
+      - Toolchain bump PR including updated Lake manifest
+      - Create and push the tag
+      - There is no `stable` branch; skip this step
     - [Mathlib](https://github.com/leanprover-community/mathlib4)
       - Dependencies: `Aesop`, `ProofWidgets4`, `lean4checker`, `Batteries`, `doc-gen4`, `import-graph`
       - Toolchain bump PR notes:
         - In addition to updating the `lean-toolchain` and `lakefile.lean`,
           in `.github/workflows/lean4checker.yml` update the line
-          `git checkout v4.6.0` to the appropriate tag. 
+          `git checkout v4.6.0` to the appropriate tag.
         - Push the PR branch to the main Mathlib repository rather than a fork, or CI may not work reliably
         - Create and push the tag
         - Create a new branch from the tag, push it, and open a pull request against `stable`.
@@ -98,6 +102,7 @@ We'll use `v4.6.0` as the intended release version as a running example.
       - Toolchain bump PR including updated Lake manifest
       - Create and push the tag
       - Merge the tag into `stable`
+- Run `scripts/release_checklist.py v4.6.0` to check that everything is in order.
 - The `v4.6.0` section of `RELEASES.md` is out of sync between
   `releases/v4.6.0` and `master`. This should be reconciled:
   - Replace the `v4.6.0` section on `master` with the `v4.6.0` section on `releases/v4.6.0`
@@ -139,16 +144,13 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
     git checkout -b releases/v4.7.0
     ```
 - In `RELEASES.md` replace `Development in progress` in the `v4.7.0` section with `Release notes to be written.`
-- We will rely on automatically generated release notes for release candidates,
-  and the written release notes will be used for stable versions only.
-  It is essential to choose the nightly that will become the release candidate as early as possible, to avoid confusion.
+- It is essential to choose the nightly that will become the release candidate as early as possible, to avoid confusion.
 - In `src/CMakeLists.txt`,
   - verify that you see `set(LEAN_VERSION_MINOR 7)` (for whichever `7` is appropriate); this should already have been updated when the development cycle began.
   - `set(LEAN_VERSION_IS_RELEASE 1)` (this should be a change; on `master` and nightly releases it is always `0`).
   - Commit your changes to `src/CMakeLists.txt`, and push.
 - `git tag v4.7.0-rc1`
 - `git push origin v4.7.0-rc1`
-- Ping the FRO Zulip that release notes need to be written. The release notes do not block completing the rest of this checklist.
 - Now wait, while CI runs.
   - 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.
@@ -248,15 +250,12 @@ Please read https://leanprover-community.github.io/contribute/tags_and_branches.
 
 # Writing the release notes
 
-We are currently trying a system where release notes are compiled all at once from someone looking through the commit history.
-The exact steps are a work in progress.
-Here is the general idea:
+Release notes are automatically generated from the commit history, using `script/release_notes.py`.
 
-* The work is done right on the `releases/v4.6.0` branch sometime after it is created but before the stable release is made.
-  The release notes for `v4.6.0` will later be copied to `master` when we begin a new development cycle.
-* There can be material for release notes entries in commit messages.
-* There can also be pre-written entries in `./releases_drafts`, which should be all incorporated in the release notes and then deleted from the branch.
+Run this as `script/release_notes.py v4.6.0`, where `v4.6.0` is the *previous* release version. This will generate output
+for all commits since that tag. Note that there is output on both stderr, which should be manually reviewed,
+and on stdout, which should be manually copied to `RELEASES.md`.
+
+There can also be pre-written entries in `./releases_drafts`, which should be all incorporated in the release notes and then deleted from the branch.
   See `./releases_drafts/README.md` for more information.
-* The release notes should be written from a downstream expert user's point of view.
 
-This section will be updated when the next release notes are written (for `v4.10.0`).

doc/make/osx-10.9.md

@@ -32,12 +32,13 @@ following to use `g++`.
 cmake -DCMAKE_CXX_COMPILER=g++ ...
 ```
 
-## Required Packages: CMake, GMP, libuv
+## Required Packages: CMake, GMP, libuv, pkgconf
 
 ```bash
 brew install cmake
 brew install gmp
 brew install libuv
+brew install pkgconf
 ```
 
 ## Recommended Packages: 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 libuv1-dev cmake ccache clang
+sudo apt-get install git libgmp-dev libuv1-dev cmake ccache clang pkgconf
 ```

flake.nix

@@ -28,7 +28,7 @@
           stdenv = pkgs.overrideCC pkgs.stdenv lean-packages.llvmPackages.clang;
         } ({
           buildInputs = with pkgs; [
-            cmake gmp libuv ccache cadical
+            cmake gmp libuv ccache cadical pkg-config
             lean-packages.llvmPackages.llvm  # llvm-symbolizer for asan/lsan
             gdb
             tree  # for CI

nix/bootstrap.nix

@@ -1,12 +1,12 @@
 { src, debug ? false, stage0debug ? false, extraCMakeFlags ? [],
-  stdenv, lib, cmake, gmp, libuv, cadical, git, gnumake, bash, buildLeanPackage, writeShellScriptBin, runCommand, symlinkJoin, lndir, perl, gnused, darwin, llvmPackages, linkFarmFromDrvs,
+  stdenv, lib, cmake, pkg-config, gmp, libuv, cadical, git, gnumake, bash, buildLeanPackage, writeShellScriptBin, runCommand, symlinkJoin, lndir, perl, gnused, darwin, llvmPackages, linkFarmFromDrvs,
   ... } @ args:
 with builtins;
 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 ];
+    nativeBuildInputs = [ cmake pkg-config ];
     buildInputs = [ gmp libuv llvmPackages.llvm ];
     # https://github.com/NixOS/nixpkgs/issues/60919
     hardeningDisable = [ "all" ];

releases_drafts/list_lex.md

@@ -1,16 +0,0 @@
-We replace the inductive predicate `List.lt` with an upstreamed version of `List.Lex` from Mathlib.
-(Previously `Lex.lt` was defined in terms of `<`; now it is generalized to take an arbitrary relation.)
-This subtely changes the notion of ordering on `List α`.
-
-`List.lt` was a weaker relation: in particular if `l₁ < l₂`, then
-`a :: l₁ < b :: l₂` may hold according to `List.lt` even if `a` and `b` are merely incomparable
-(either neither `a < b` nor `b < a`), whereas according to `List.Lex` this would require `a = b`.
-
-When `<` is total, in the sense that `¬ · < ·` is antisymmetric, then the two relations coincide.
-
-Mathlib was already overriding the order instances for `List α`,
-so this change should not be noticed by anyone already using Mathlib.
-
-We simultaneously add the boolean valued `List.lex` function, parameterised by a `BEq` typeclass
-and an arbitrary `lt` function. This will support the flexibility previously provided for `List.lt`,
-via a `==` function which is weaker than strict equality.

script/prepare-llvm-linux.sh

@@ -63,8 +63,8 @@ else
 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 ROOT/lib/glibc/libpthread_nonshared.a -Wl,--as-needed -Wl,-Bstatic -lgmp -lunwind -luv -Wl,-Bdynamic -Wl,--no-as-needed -fuse-ld=lld'"
+echo -n " -DLEANC_INTERNAL_FLAGS='--sysroot ROOT -nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang"
+echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='--sysroot ROOT -L ROOT/lib -L ROOT/lib/glibc ROOT/lib/glibc/libc_nonshared.a ROOT/lib/glibc/libpthread_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 -luv -lpthread -ldl -lrt -Wl,--no-as-needed'"
 # do not set `LEAN_CC` for tests

script/prepare-llvm-macos.sh

@@ -48,12 +48,11 @@ if [[ -L llvm-host ]]; then
   echo -n " -DCMAKE_C_COMPILER=$PWD/stage1/bin/clang"
   gcp $GMP/lib/libgmp.a stage1/lib/
   gcp $LIBUV/lib/libuv.a stage1/lib/
-  echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/libc -fuse-ld=lld'"
   echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp -luv'"
 else
   echo -n " -DCMAKE_C_COMPILER=$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'"
 fi
-echo -n " -DLEANC_INTERNAL_FLAGS='-nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang"
+echo -n " -DLEANC_INTERNAL_FLAGS='--sysroot ROOT -nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang"
+echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='--sysroot ROOT -L ROOT/lib -L ROOT/lib/libc -fuse-ld=lld'"
 # do not set `LEAN_CC` for tests
 echo -n " -DLEAN_TEST_VARS=''"

script/prepare-llvm-mingw.sh

@@ -43,7 +43,7 @@ echo -n " -DCMAKE_C_COMPILER=$PWD/stage1/bin/clang.exe -DCMAKE_C_COMPILER_WORKS=
 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 $(pkg-config --static --libs libuv) -lunwind -Wl,-Bdynamic -fuse-ld=lld'"
+echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='--sysroot ROOT -L ROOT/lib -Wl,-Bstatic -lgmp $(pkg-config --static --libs libuv) -lunwind -Wl,-Bdynamic -fuse-ld=lld'"
 # when not using the above flags, link GMP dynamically/as usual. Always link ICU dynamically.
 echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp $(pkg-config --libs libuv) -lucrtbase'"
 # do not set `LEAN_CC` for tests

script/push_repo_release_tag.py

@@ -0,0 +1,69 @@
+#!/usr/bin/env python3
+import sys
+import subprocess
+import requests
+
+def main():
+    if len(sys.argv) != 4:
+        print("Usage: ./push_repo_release_tag.py <repo> <branch> <version_tag>")
+        sys.exit(1)
+
+    repo, branch, version_tag = sys.argv[1], sys.argv[2], sys.argv[3]
+
+    if branch not in {"master", "main"}:
+        print(f"Error: Branch '{branch}' is not 'master' or 'main'.")
+        sys.exit(1)
+
+    # Get the `lean-toolchain` file content
+    lean_toolchain_url = f"https://raw.githubusercontent.com/{repo}/{branch}/lean-toolchain"
+    try:
+        response = requests.get(lean_toolchain_url)
+        response.raise_for_status()
+    except requests.exceptions.RequestException as e:
+        print(f"Error fetching 'lean-toolchain' file: {e}")
+        sys.exit(1)
+
+    lean_toolchain_content = response.text.strip()
+    expected_prefix = "leanprover/lean4:"
+    if not lean_toolchain_content.startswith(expected_prefix) or lean_toolchain_content != f"{expected_prefix}{version_tag}":
+        print(f"Error: 'lean-toolchain' content does not match '{expected_prefix}{version_tag}'.")
+        sys.exit(1)
+
+    # Create and push the tag using `gh`
+    try:
+        # Check if the tag already exists
+        list_tags_cmd = ["gh", "api", f"repos/{repo}/git/matching-refs/tags/v4", "--jq", ".[].ref"]
+        list_tags_output = subprocess.run(list_tags_cmd, capture_output=True, text=True)
+
+        if list_tags_output.returncode == 0:
+            existing_tags = list_tags_output.stdout.strip().splitlines()
+            if f"refs/tags/{version_tag}" in existing_tags:
+                print(f"Error: Tag '{version_tag}' already exists.")
+                print("Existing tags starting with 'v4':")
+                for tag in existing_tags:
+                    print(tag.replace("refs/tags/", ""))
+                sys.exit(1)
+
+        # Get the SHA of the branch
+        get_sha_cmd = [
+            "gh", "api", f"repos/{repo}/git/ref/heads/{branch}", "--jq", ".object.sha"
+        ]
+        sha_result = subprocess.run(get_sha_cmd, capture_output=True, text=True, check=True)
+        sha = sha_result.stdout.strip()
+
+        # Create the tag
+        create_tag_cmd = [
+            "gh", "api", f"repos/{repo}/git/refs",
+            "-X", "POST",
+            "-F", f"ref=refs/tags/{version_tag}",
+            "-F", f"sha={sha}"
+        ]
+        subprocess.run(create_tag_cmd, capture_output=True, text=True, check=True)
+
+        print(f"Successfully created and pushed tag '{version_tag}' to {repo}.")
+    except subprocess.CalledProcessError as e:
+        print(f"Error while creating/pushing tag: {e.stderr.strip() if e.stderr else e}")
+        sys.exit(1)
+
+if __name__ == "__main__":
+    main()

script/release_checklist.py

@@ -0,0 +1,227 @@
+#!/usr/bin/env python3
+
+import argparse
+import yaml
+import requests
+import base64
+import subprocess
+import sys
+import os
+
+def parse_repos_config(file_path):
+    with open(file_path, "r") as f:
+        return yaml.safe_load(f)["repositories"]
+
+def get_github_token():
+    try:
+        import subprocess
+        result = subprocess.run(['gh', 'auth', 'token'], capture_output=True, text=True)
+        if result.returncode == 0:
+            return result.stdout.strip()
+    except FileNotFoundError:
+        print("Warning: 'gh' CLI not found. Some API calls may be rate-limited.")
+    return None
+
+def strip_rc_suffix(toolchain):
+    """Remove -rcX suffix from the toolchain."""
+    return toolchain.split("-")[0]
+
+def branch_exists(repo_url, branch, github_token):
+    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/branches/{branch}"
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+    response = requests.get(api_url, headers=headers)
+    return response.status_code == 200
+
+def tag_exists(repo_url, tag_name, github_token):
+    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/git/refs/tags/{tag_name}"
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+    response = requests.get(api_url, headers=headers)
+    return response.status_code == 200
+
+def release_page_exists(repo_url, tag_name, github_token):
+    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/releases/tags/{tag_name}"
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+    response = requests.get(api_url, headers=headers)
+    return response.status_code == 200
+
+def get_release_notes(repo_url, tag_name, github_token):
+    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/releases/tags/{tag_name}"
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+    response = requests.get(api_url, headers=headers)
+    if response.status_code == 200:
+        return response.json().get("body", "").strip()
+    return None
+
+def get_branch_content(repo_url, branch, file_path, github_token):
+    api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/contents/{file_path}?ref={branch}"
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+    response = requests.get(api_url, headers=headers)
+    if response.status_code == 200:
+        content = response.json().get("content", "")
+        content = content.replace("\n", "")
+        try:
+            return base64.b64decode(content).decode('utf-8').strip()
+        except Exception:
+            return None
+    return None
+
+def parse_version(version_str):
+    # Remove 'v' prefix and extract version and release candidate suffix
+    if ':' in version_str:
+        version_str = version_str.split(':')[1]
+    version = version_str.lstrip('v')
+    parts = version.split('-')
+    base_version = tuple(map(int, parts[0].split('.')))
+    rc_part = parts[1] if len(parts) > 1 and parts[1].startswith('rc') else None
+    rc_number = int(rc_part[2:]) if rc_part else float('inf')  # Treat non-rc as higher than rc
+    return base_version + (rc_number,)
+
+def is_version_gte(version1, version2):
+    """Check if version1 >= version2, including proper handling of release candidates."""
+    return parse_version(version1) >= parse_version(version2)
+
+def is_merged_into_stable(repo_url, tag_name, stable_branch, github_token):
+    # First get the commit SHA for the tag
+    api_base = repo_url.replace("https://github.com/", "https://api.github.com/repos/")
+    headers = {'Authorization': f'token {github_token}'} if github_token else {}
+
+    # Get tag's commit SHA
+    tag_response = requests.get(f"{api_base}/git/refs/tags/{tag_name}", headers=headers)
+    if tag_response.status_code != 200:
+        return False
+    tag_sha = tag_response.json()['object']['sha']
+
+    # Get commits on stable branch containing this SHA
+    commits_response = requests.get(
+        f"{api_base}/commits?sha={stable_branch}&per_page=100",
+        headers=headers
+    )
+    if commits_response.status_code != 200:
+        return False
+
+    # Check if any commit in stable's history matches our tag's SHA
+    stable_commits = [commit['sha'] for commit in commits_response.json()]
+    return tag_sha in stable_commits
+
+def is_release_candidate(version):
+    return "-rc" in version
+
+def check_cmake_version(repo_url, branch, version_major, version_minor, github_token):
+    """Verify the CMake version settings in src/CMakeLists.txt."""
+    cmake_file_path = "src/CMakeLists.txt"
+    content = get_branch_content(repo_url, branch, cmake_file_path, github_token)
+    if content is None:
+        print(f"  ❌ Could not retrieve {cmake_file_path} from {branch}")
+        return False
+
+    expected_lines = [
+        f"set(LEAN_VERSION_MAJOR {version_major})",
+        f"set(LEAN_VERSION_MINOR {version_minor})",
+        f"set(LEAN_VERSION_PATCH 0)",
+        f"set(LEAN_VERSION_IS_RELEASE 1)"
+    ]
+
+    for line in expected_lines:
+        if not any(l.strip().startswith(line) for l in content.splitlines()):
+            print(f"  ❌ Missing or incorrect line in {cmake_file_path}: {line}")
+            return False
+
+    print(f"  ✅ CMake version settings are correct in {cmake_file_path}")
+    return True
+
+def extract_org_repo_from_url(repo_url):
+    """Extract the 'org/repo' part from a GitHub URL."""
+    if repo_url.startswith("https://github.com/"):
+        return repo_url.replace("https://github.com/", "").rstrip("/")
+    return repo_url
+
+def main():
+    github_token = get_github_token()
+
+    if len(sys.argv) != 2:
+        print("Usage: python3 release_checklist.py <toolchain>")
+        sys.exit(1)
+
+    toolchain = sys.argv[1]
+    stripped_toolchain = strip_rc_suffix(toolchain)
+    lean_repo_url = "https://github.com/leanprover/lean4"
+
+    # Preliminary checks
+    print("\nPerforming preliminary checks...")
+
+    # Check for branch releases/v4.Y.0
+    version_major, version_minor, _ = map(int, stripped_toolchain.lstrip('v').split('.'))
+    branch_name = f"releases/v{version_major}.{version_minor}.0"
+    if branch_exists(lean_repo_url, branch_name, github_token):
+        print(f"  ✅ Branch {branch_name} exists")
+
+        # Check CMake version settings
+        check_cmake_version(lean_repo_url, branch_name, version_major, version_minor, github_token)
+    else:
+        print(f"  ❌ Branch {branch_name} does not exist")
+
+    # Check for tag v4.X.Y(-rcZ)
+    if tag_exists(lean_repo_url, toolchain, github_token):
+        print(f"  ✅ Tag {toolchain} exists")
+    else:
+        print(f"  ❌ Tag {toolchain} does not exist.")
+
+    # Check for release page
+    if release_page_exists(lean_repo_url, toolchain, github_token):
+        print(f"  ✅ Release page for {toolchain} exists")
+
+        # Check the first line of the release notes
+        release_notes = get_release_notes(lean_repo_url, toolchain, github_token)
+        if release_notes and release_notes.splitlines()[0].strip() == toolchain:
+            print(f"  ✅ Release notes look good.")
+        else:
+            previous_minor_version = version_minor - 1
+            previous_stable_branch = f"releases/v{version_major}.{previous_minor_version}.0"
+            previous_release = f"v{version_major}.{previous_minor_version}.0"
+            print(f"  ❌ Release notes not published. Please run `script/release_notes.py {previous_release}` on branch `{previous_stable_branch}`.")
+    else:
+        print(f"  ❌ Release page for {toolchain} does not exist")
+
+    # Load repositories and perform further checks
+    print("\nChecking repositories...")
+
+    with open(os.path.join(os.path.dirname(__file__), "release_repos.yml")) as f:
+        repos = yaml.safe_load(f)["repositories"]
+
+    for repo in repos:
+        name = repo["name"]
+        url = repo["url"]
+        branch = repo["branch"]
+        check_stable = repo["stable-branch"]
+        check_tag = repo.get("toolchain-tag", True)
+
+        print(f"\nRepository: {name}")
+
+        # Check if branch is on at least the target toolchain
+        lean_toolchain_content = get_branch_content(url, branch, "lean-toolchain", github_token)
+        if lean_toolchain_content is None:
+            print(f"  ❌ No lean-toolchain file found in {branch} branch")
+            continue
+
+        on_target_toolchain = is_version_gte(lean_toolchain_content.strip(), toolchain)
+        if not on_target_toolchain:
+            print(f"  ❌ Not on target toolchain (needs ≥ {toolchain}, but {branch} is on {lean_toolchain_content.strip()})")
+            continue
+        print(f"  ✅ On compatible toolchain (>= {toolchain})")
+
+        # Only check for tag if toolchain-tag is true
+        if check_tag:
+            if not tag_exists(url, toolchain, github_token):
+                print(f"  ❌ Tag {toolchain} does not exist. Run `script/push_repo_release_tag.py {extract_org_repo_from_url(url)} {branch} {toolchain}`.")
+                continue
+            print(f"  ✅ Tag {toolchain} exists")
+
+        # Only check merging into stable if stable-branch is true and not a release candidate
+        if check_stable and not is_release_candidate(toolchain):
+            if not is_merged_into_stable(url, toolchain, "stable", github_token):
+                print(f"  ❌ Tag {toolchain} is not merged into stable")
+                continue
+            print(f"  ✅ Tag {toolchain} is merged into stable")
+
+if __name__ == "__main__":
+    main()

script/release_notes.py

@@ -0,0 +1,145 @@
+#!/usr/bin/env python3
+
+import sys
+import re
+import json
+import requests
+import subprocess
+from collections import defaultdict
+from git import Repo
+
+def get_commits_since_tag(repo, tag):
+    try:
+        tag_commit = repo.commit(tag)
+        commits = list(repo.iter_commits(f"{tag_commit.hexsha}..HEAD"))
+        return [
+            (commit.hexsha, commit.message.splitlines()[0], commit.message)
+            for commit in commits
+        ]
+    except Exception as e:
+        sys.stderr.write(f"Error retrieving commits: {e}\n")
+        sys.exit(1)
+
+def check_pr_number(first_line):
+    match = re.search(r"\(\#(\d+)\)$", first_line)
+    if match:
+        return int(match.group(1))
+    return None
+
+def fetch_pr_labels(pr_number):
+    try:
+        # Use gh CLI to fetch PR details
+        result = subprocess.run([
+            "gh", "api", f"repos/leanprover/lean4/pulls/{pr_number}"
+        ], capture_output=True, text=True, check=True)
+        pr_data = result.stdout
+        pr_json = json.loads(pr_data)
+        return [label["name"] for label in pr_json.get("labels", [])]
+    except subprocess.CalledProcessError as e:
+        sys.stderr.write(f"Failed to fetch PR #{pr_number} using gh: {e.stderr}\n")
+        return []
+
+def format_section_title(label):
+    title = label.replace("changelog-", "").capitalize()
+    if title == "Doc":
+        return "Documentation"
+    elif title == "Pp":
+        return "Pretty Printing"
+    return title
+
+def sort_sections_order():
+    return [
+        "Language",
+        "Library",
+        "Compiler",
+        "Pretty Printing",
+        "Documentation",
+        "Server",
+        "Lake",
+        "Other",
+        "Uncategorised"
+    ]
+
+def format_markdown_description(pr_number, description):
+    link = f"[#{pr_number}](https://github.com/leanprover/lean4/pull/{pr_number})"
+    return f"{link} {description}"
+
+def main():
+    if len(sys.argv) != 2:
+        sys.stderr.write("Usage: script.py <git-tag>\n")
+        sys.exit(1)
+
+    tag = sys.argv[1]
+    try:
+        repo = Repo(".")
+    except Exception as e:
+        sys.stderr.write(f"Error opening Git repository: {e}\n")
+        sys.exit(1)
+
+    commits = get_commits_since_tag(repo, tag)
+
+    sys.stderr.write(f"Found {len(commits)} commits since tag {tag}:\n")
+    for commit_hash, first_line, _ in commits:
+        sys.stderr.write(f"- {commit_hash}: {first_line}\n")
+
+    changelog = defaultdict(list)
+
+    for commit_hash, first_line, full_message in commits:
+        # Skip commits with the specific first lines
+        if first_line == "chore: update stage0" or first_line.startswith("chore: CI: bump "):
+            continue
+
+        pr_number = check_pr_number(first_line)
+
+        if not pr_number:
+            sys.stderr.write(f"No PR number found in {first_line}\n")
+            continue
+
+        # Remove the first line from the full_message for further processing
+        body = full_message[len(first_line):].strip()
+
+        paragraphs = body.split('\n\n')
+        second_paragraph = paragraphs[0] if len(paragraphs) > 0 else ""
+
+        labels = fetch_pr_labels(pr_number)
+
+        # Skip entries with the "changelog-no" label
+        if "changelog-no" in labels:
+            continue
+
+        report_errors = first_line.startswith("feat:") or first_line.startswith("fix:")
+
+        if not second_paragraph.startswith("This PR "):
+            if report_errors:
+                sys.stderr.write(f"No PR description found in commit:\n{commit_hash}\n{first_line}\n{body}\n\n")
+                fallback_description = re.sub(r":$", "", first_line.split(" ", 1)[1]).rsplit(" (#", 1)[0]
+                markdown_description = format_markdown_description(pr_number, fallback_description)
+            else:
+                continue
+        else:
+            markdown_description = format_markdown_description(pr_number, second_paragraph.replace("This PR ", ""))
+
+        changelog_labels = [label for label in labels if label.startswith("changelog-")]
+        if len(changelog_labels) > 1:
+            sys.stderr.write(f"Warning: Multiple changelog-* labels found for PR #{pr_number}: {changelog_labels}\n")
+
+        if not changelog_labels:
+            if report_errors:
+                sys.stderr.write(f"Warning: No changelog-* label found for PR #{pr_number}\n")
+            else:
+                continue
+
+        for label in changelog_labels:
+            changelog[label].append((pr_number, markdown_description))
+
+    section_order = sort_sections_order()
+    sorted_changelog = sorted(changelog.items(), key=lambda item: section_order.index(format_section_title(item[0])) if format_section_title(item[0]) in section_order else len(section_order))
+
+    for label, entries in sorted_changelog:
+        section_title = format_section_title(label) if label != "Uncategorised" else "Uncategorised"
+        print(f"## {section_title}\n")
+        for _, entry in sorted(entries, key=lambda x: x[0]):
+            print(f"* {entry}\n")
+
+if __name__ == "__main__":
+    main()

script/release_repos.yml

@@ -0,0 +1,86 @@
+repositories:
+  - name: Batteries
+    url: https://github.com/leanprover-community/batteries
+    toolchain-tag: true
+    stable-branch: true
+    branch: main
+    dependencies: []
+
+  - name: lean4checker
+    url: https://github.com/leanprover/lean4checker
+    toolchain-tag: true
+    stable-branch: true
+    branch: master
+    dependencies: []
+
+  - name: doc-gen4
+    url: https://github.com/leanprover/doc-gen4
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
+  - name: Verso
+    url: https://github.com/leanprover/verso
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
+  - name: Cli
+    url: https://github.com/leanprover/lean4-cli
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
+  - name: ProofWidgets4
+    url: https://github.com/leanprover-community/ProofWidgets4
+    toolchain-tag: false
+    stable-branch: false
+    branch: main
+    dependencies:
+      - Batteries
+
+  - name: Aesop
+    url: https://github.com/leanprover-community/aesop
+    toolchain-tag: true
+    stable-branch: true
+    branch: master
+    dependencies:
+      - Batteries
+
+  - name: import-graph
+    url: https://github.com/leanprover-community/import-graph
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
+  - name: plausible
+    url: https://github.com/leanprover-community/plausible
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies: []
+
+  - name: Mathlib
+    url: https://github.com/leanprover-community/mathlib4
+    toolchain-tag: true
+    stable-branch: true
+    branch: master
+    dependencies:
+      - Aesop
+      - ProofWidgets4
+      - lean4checker
+      - Batteries
+      - doc-gen4
+      - import-graph
+
+  - name: REPL
+    url: https://github.com/leanprover-community/repl
+    toolchain-tag: true
+    stable-branch: true
+    branch: master
+    dependencies:
+      - Mathlib

src/CMakeLists.txt

@@ -295,14 +295,15 @@ index 5e8e0166..f3b29134 100644
 	  PATCH_COMMAND git reset --hard HEAD && printf "${LIBUV_PATCH}" > patch.diff && git apply patch.diff
     BUILD_IN_SOURCE ON
     INSTALL_COMMAND "")
-  set(LIBUV_INCLUDE_DIR "${CMAKE_BINARY_DIR}/libuv/src/libuv/include")
-  set(LIBUV_LIBRARIES "${CMAKE_BINARY_DIR}/libuv/src/libuv/libuv.a")
+  set(LIBUV_INCLUDE_DIRS "${CMAKE_BINARY_DIR}/libuv/src/libuv/include")
+  set(LIBUV_LDFLAGS "${CMAKE_BINARY_DIR}/libuv/src/libuv/libuv.a")
 else()
   find_package(LibUV 1.0.0 REQUIRED)
 endif()
-include_directories(${LIBUV_INCLUDE_DIR})
+include_directories(${LIBUV_INCLUDE_DIRS})
 if(NOT LEAN_STANDALONE)
-  string(APPEND LEAN_EXTRA_LINKER_FLAGS " ${LIBUV_LIBRARIES}")
+  string(JOIN " " LIBUV_LDFLAGS ${LIBUV_LDFLAGS})
+  string(APPEND LEAN_EXTRA_LINKER_FLAGS " ${LIBUV_LDFLAGS}")
 endif()
 
 # Windows SDK (for ICU)

src/Init.lean

@@ -37,3 +37,4 @@ import Init.MacroTrace
 import Init.Grind
 import Init.While
 import Init.Syntax
+import Init.Internal

src/Init/Conv.lean

@@ -150,6 +150,10 @@ See the `simp` tactic for more information. -/
 syntax (name := simp) "simp" optConfig (discharger)? (&" only")?
   (" [" withoutPosition((simpStar <|> simpErase <|> simpLemma),*) "]")? : conv
 
+/-- `simp?` takes the same arguments as `simp`, but reports an equivalent call to `simp only`
+that would be sufficient to close the goal. See the `simp?` tactic for more information. -/
+syntax (name := simpTrace) "simp?" optConfig (discharger)? (&" only")? (simpArgs)? : conv
+
 /--
 `dsimp` is the definitional simplifier in `conv`-mode. It differs from `simp` in that it only
 applies theorems that hold by reflexivity.
@@ -167,6 +171,9 @@ example (a : Nat): (0 + 0) = a - a := by
 syntax (name := dsimp) "dsimp" optConfig (discharger)? (&" only")?
   (" [" withoutPosition((simpErase <|> simpLemma),*) "]")? : conv
 
+@[inherit_doc simpTrace]
+syntax (name := dsimpTrace) "dsimp?" optConfig (&" only")? (dsimpArgs)? : conv
+
 /-- `simp_match` simplifies match expressions. For example,
 ```
 match [a, b] with

src/Init/Data/Array/Basic.lean

@@ -244,8 +244,7 @@ def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
 def range (n : Nat) : Array Nat :=
   ofFn fun (i : Fin n) => i
 
-def singleton (v : α) : Array α :=
-  mkArray 1 v
+@[inline] protected def singleton (v : α) : Array α := #[v]
 
 def back! [Inhabited α] (a : Array α) : α :=
   a[a.size - 1]!
@@ -577,6 +576,12 @@ def foldl {α : Type u} {β : Type v} (f : β → α → β) (init : β) (as : A
 def foldr {α : Type u} {β : Type v} (f : α → β → β) (init : β) (as : Array α) (start := as.size) (stop := 0) : β :=
   Id.run <| as.foldrM f init start stop
 
+/-- Sum of an array.
+
+`Array.sum #[a, b, c] = a + (b + (c + 0))` -/
+def sum {α} [Add α] [Zero α] : Array α → α :=
+  foldr (· + ·) 0
+
 @[inline]
 def map {α : Type u} {β : Type v} (f : α → β) (as : Array α) : Array β :=
   Id.run <| as.mapM f

src/Init/Data/Array/Bootstrap.lean

@@ -81,12 +81,18 @@ theorem foldrM_eq_reverse_foldlM_toList [Monad m] (f : α → β → m β) (init
 
 @[simp] theorem toList_empty : (#[] : Array α).toList = [] := rfl
 
-@[simp] theorem append_nil (as : Array α) : as ++ #[] = as := by
+@[simp] theorem append_empty (as : Array α) : as ++ #[] = as := by
   apply ext'; simp only [toList_append, toList_empty, List.append_nil]
 
-@[simp] theorem nil_append (as : Array α) : #[] ++ as = as := by
+@[deprecated append_empty (since := "2025-01-13")]
+abbrev append_nil := @append_empty
+
+@[simp] theorem empty_append (as : Array α) : #[] ++ as = as := by
   apply ext'; simp only [toList_append, toList_empty, List.nil_append]
 
+@[deprecated empty_append (since := "2025-01-13")]
+abbrev nil_append := @empty_append
+
 @[simp] theorem append_assoc (as bs cs : Array α) : as ++ bs ++ cs = as ++ (bs ++ cs) := by
   apply ext'; simp only [toList_append, List.append_assoc]
 

src/Init/Data/Array/Find.lean

@@ -74,12 +74,12 @@ theorem findSome?_append {l₁ l₂ : Array α} : (l₁ ++ l₂).findSome? f = (
 
 theorem getElem?_zero_flatten (L : Array (Array α)) :
     (flatten L)[0]? = L.findSome? fun l => l[0]? := by
-  cases L using array_array_induction
+  cases L using array₂_induction
   simp [← List.head?_eq_getElem?, List.head?_flatten, List.findSome?_map, Function.comp_def]
 
 theorem getElem_zero_flatten.proof {L : Array (Array α)} (h : 0 < L.flatten.size) :
     (L.findSome? fun l => l[0]?).isSome := by
-  cases L using array_array_induction
+  cases L using array₂_induction
   simp only [List.findSome?_toArray, List.findSome?_map, Function.comp_def, List.getElem?_toArray,
     List.findSome?_isSome_iff, isSome_getElem?]
   simp only [flatten_toArray_map_toArray, size_toArray, List.length_flatten,
@@ -95,7 +95,7 @@ theorem getElem_zero_flatten {L : Array (Array α)} (h) :
 
 theorem back?_flatten {L : Array (Array α)} :
     (flatten L).back? = (L.findSomeRev? fun l => l.back?) := by
-  cases L using array_array_induction
+  cases L using array₂_induction
   simp [List.getLast?_flatten, ← List.map_reverse, List.findSome?_map, Function.comp_def]
 
 theorem findSome?_mkArray : findSome? f (mkArray n a) = if n = 0 then none else f a := by
@@ -203,7 +203,7 @@ theorem get_find?_mem {xs : Array α} (h) : (xs.find? p).get h ∈ xs := by
 
 @[simp] theorem find?_flatten (xs : Array (Array α)) (p : α → Bool) :
     xs.flatten.find? p = xs.findSome? (·.find? p) := by
-  cases xs using array_array_induction
+  cases xs using array₂_induction
   simp [List.findSome?_map, Function.comp_def]
 
 theorem find?_flatten_eq_none {xs : Array (Array α)} {p : α → Bool} :
@@ -220,7 +220,7 @@ theorem find?_flatten_eq_some {xs : Array (Array α)} {p : α → Bool} {a : α}
       p a ∧ ∃ (as : Array (Array α)) (ys zs : Array α) (bs : Array (Array α)),
         xs = as.push (ys.push a ++ zs) ++ bs ∧
         (∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
-  cases xs using array_array_induction
+  cases xs using array₂_induction
   simp only [flatten_toArray_map_toArray, List.find?_toArray, List.find?_flatten_eq_some]
   simp only [Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, and_congr_right_iff]
   intro w

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

@@ -17,8 +17,8 @@ namespace Array
 @[simp] theorem lt_toList [LT α] (l₁ l₂ : Array α) : l₁.toList < l₂.toList ↔ l₁ < l₂ := Iff.rfl
 @[simp] theorem le_toList [LT α] (l₁ l₂ : Array α) : l₁.toList ≤ l₂.toList ↔ l₁ ≤ l₂ := Iff.rfl
 
-theorem not_lt_iff_ge [LT α] (l₁ l₂ : List α) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
-theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) :
+protected theorem not_lt_iff_ge [LT α] (l₁ l₂ : List α) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
+protected theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) :
     ¬ l₁ ≤ l₂ ↔ l₂ < l₁ :=
   Decidable.not_not
 
@@ -135,7 +135,7 @@ protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
     {l₁ l₂ : Array α} (h : l₁ < l₂) : l₁ ≤ l₂ :=
   List.le_of_lt h
 
-theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
+protected theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
     [Std.Total (¬ · < · : α → α → Prop)]
@@ -211,7 +211,7 @@ theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α
   cases l₂
   simp_all [List.lex_eq_false_iff_exists]
 
-theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} :
+protected theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} :
     l₁ < l₂ ↔
       (l₁ = l₂.take l₁.size ∧ l₁.size < l₂.size) ∨
         (∃ (i : Nat) (h₁ : i < l₁.size) (h₂ : i < l₂.size),
@@ -221,7 +221,7 @@ theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Arr
   cases l₂
   simp [List.lt_iff_exists]
 
-theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
+protected theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)] {l₁ l₂ : Array α} :
@@ -258,14 +258,14 @@ theorem le_append_left [LT α] [Std.Irrefl (· < · : α → α → Prop)]
   cases l₂
   simpa using List.le_append_left
 
-theorem map_lt [LT α] [LT β]
+protected theorem map_lt [LT α] [LT β]
     {l₁ l₂ : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
     map f l₁ < map f l₂ := by
   cases l₁
   cases l₂
   simpa using List.map_lt w h
 
-theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
+protected theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]

src/Init/Data/BitVec/Lemmas.lean

@@ -9,7 +9,9 @@ import Init.Data.Bool
 import Init.Data.BitVec.Basic
 import Init.Data.Fin.Lemmas
 import Init.Data.Nat.Lemmas
+import Init.Data.Nat.Div.Lemmas
 import Init.Data.Nat.Mod
+import Init.Data.Nat.Div.Lemmas
 import Init.Data.Int.Bitwise.Lemmas
 import Init.Data.Int.Pow
 
@@ -98,6 +100,12 @@ theorem ofFin_eq_ofNat : @BitVec.ofFin w (Fin.mk x lt) = BitVec.ofNat w x := by
 theorem eq_of_toNat_eq {n} : ∀ {x y : BitVec n}, x.toNat = y.toNat → x = y
   | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
 
+/-- Prove nonequality of bitvectors in terms of nat operations. -/
+theorem toNat_ne_iff_ne {n} {x y : BitVec n} : x.toNat ≠ y.toNat ↔ x ≠ y := by
+  constructor
+  · rintro h rfl; apply h rfl
+  · intro h h_eq; apply h <| eq_of_toNat_eq h_eq
+
 @[simp] theorem val_toFin (x : BitVec w) : x.toFin.val = x.toNat := rfl
 
 @[bv_toNat] theorem toNat_eq {x y : BitVec n} : x = y ↔ x.toNat = y.toNat :=
@@ -442,6 +450,10 @@ theorem toInt_eq_toNat_cond (x : BitVec n) :
         (x.toNat : Int) - (2^n : Nat) :=
   rfl
 
+theorem toInt_eq_toNat_of_lt {x : BitVec n} (h : 2 * x.toNat < 2^n) :
+    x.toInt = x.toNat := by
+  simp [toInt_eq_toNat_cond, h]
+
 theorem msb_eq_false_iff_two_mul_lt {x : BitVec w} : x.msb = false ↔ 2 * x.toNat < 2^w := by
   cases w <;> simp [Nat.pow_succ, Nat.mul_comm _ 2, msb_eq_decide, toNat_of_zero_length]
 
@@ -454,6 +466,9 @@ theorem toInt_eq_msb_cond (x : BitVec w) :
   simp only [BitVec.toInt, ← msb_eq_false_iff_two_mul_lt]
   cases x.msb <;> rfl
 
+theorem toInt_eq_toNat_of_msb {x : BitVec w} (h : x.msb = false) :
+    x.toInt = x.toNat := by
+  simp [toInt_eq_msb_cond, h]
 
 theorem toInt_eq_toNat_bmod (x : BitVec n) : x.toInt = Int.bmod x.toNat (2^n) := by
   simp only [toInt_eq_toNat_cond]
@@ -785,6 +800,19 @@ theorem extractLsb'_eq_extractLsb {w : Nat} (x : BitVec w) (start len : Nat) (h
   unfold allOnes
   simp
 
+@[simp] theorem toInt_allOnes : (allOnes w).toInt = if 0 < w then -1 else 0 := by
+  norm_cast
+  by_cases h : w = 0
+  · subst h
+    simp
+  · have : 1 < 2 ^ w := by simp [h]
+    simp [BitVec.toInt]
+    omega
+
+@[simp] theorem toFin_allOnes : (allOnes w).toFin = Fin.ofNat' (2^w) (2^w - 1) := by
+  ext
+  simp
+
 @[simp] theorem getLsbD_allOnes : (allOnes v).getLsbD i = decide (i < v) := by
   simp [allOnes]
 
@@ -1142,11 +1170,16 @@ theorem getMsb_not {x : BitVec w} :
 /-! ### shiftLeft -/
 
 @[simp, bv_toNat] theorem toNat_shiftLeft {x : BitVec v} :
-    BitVec.toNat (x <<< n) = BitVec.toNat x <<< n % 2^v :=
+    (x <<< n).toNat = x.toNat <<< n % 2^v :=
   BitVec.toNat_ofNat _ _
 
+@[simp] theorem toInt_shiftLeft {x : BitVec w} :
+    (x <<< n).toInt = (x.toNat <<< n : Int).bmod (2^w) := by
+  rw [toInt_eq_toNat_bmod, toNat_shiftLeft, Nat.shiftLeft_eq]
+  simp
+
 @[simp] theorem toFin_shiftLeft {n : Nat} (x : BitVec w) :
-    BitVec.toFin (x <<< n) = Fin.ofNat' (2^w) (x.toNat <<< n) := rfl
+    (x <<< n).toFin = Fin.ofNat' (2^w) (x.toNat <<< n) := rfl
 
 @[simp]
 theorem shiftLeft_zero (x : BitVec w) : x <<< 0 = x := by
@@ -2282,6 +2315,12 @@ theorem ofNat_sub_ofNat {n} (x y : Nat) : BitVec.ofNat n x - BitVec.ofNat n y =
 @[simp, bv_toNat] theorem toNat_neg (x : BitVec n) : (- x).toNat = (2^n - x.toNat) % 2^n := by
   simp [Neg.neg, BitVec.neg]
 
+theorem toNat_neg_of_pos {x : BitVec n} (h : 0#n < x) :
+    (- x).toNat = 2^n - x.toNat := by
+  change 0 < x.toNat at h
+  rw [toNat_neg, Nat.mod_eq_of_lt]
+  omega
+
 theorem toInt_neg {x : BitVec w} :
     (-x).toInt = (-x.toInt).bmod (2 ^ w) := by
   rw [← BitVec.zero_sub, toInt_sub]
@@ -2377,6 +2416,54 @@ theorem not_neg (x : BitVec w) : ~~~(-x) = x + -1#w := by
         show (_ - x.toNat) % _ = _ by rw [Nat.mod_eq_of_lt (by omega)]]
       omega
 
+/-! ### fill -/
+
+@[simp]
+theorem getLsbD_fill {w i : Nat} {v : Bool} :
+    (fill w v).getLsbD i = (v && decide (i < w)) := by
+  by_cases h : v
+  <;> simp [h, BitVec.fill, BitVec.negOne_eq_allOnes]
+
+@[simp]
+theorem getMsbD_fill {w i : Nat} {v : Bool} :
+    (fill w v).getMsbD i = (v && decide (i < w)) := by
+  by_cases h : v
+  <;> simp [h, BitVec.fill, BitVec.negOne_eq_allOnes]
+
+@[simp]
+theorem getElem_fill {w i : Nat} {v : Bool} (h : i < w) :
+    (fill w v)[i] = v := by
+  by_cases h : v
+  <;> simp [h, BitVec.fill, BitVec.negOne_eq_allOnes]
+
+@[simp]
+theorem msb_fill {w : Nat} {v : Bool} :
+    (fill w v).msb = (v && decide (0 < w)) := by
+  simp [BitVec.msb]
+
+theorem fill_eq {w : Nat} {v : Bool} : fill w v = if v = true then allOnes w else 0#w := by
+  by_cases h : v <;> (simp only [h] ; ext ; simp)
+
+@[simp]
+theorem fill_true {w : Nat} : fill w true = allOnes w := by
+  simp [fill_eq]
+
+@[simp]
+theorem fill_false {w : Nat} : fill w false = 0#w := by
+  simp [fill_eq]
+
+@[simp] theorem fill_toNat {w : Nat} {v : Bool} :
+    (fill w v).toNat = if v = true then 2^w - 1 else 0 := by
+  by_cases h : v <;> simp [h]
+
+@[simp] theorem fill_toInt {w : Nat} {v : Bool} :
+    (fill w v).toInt = if v = true && 0 < w then -1 else 0 := by
+  by_cases h : v <;> simp [h]
+
+@[simp] theorem fill_toFin {w : Nat} {v : Bool} :
+    (fill w v).toFin = if v = true then (allOnes w).toFin else Fin.ofNat' (2 ^ w) 0 := by
+  by_cases h : v <;> simp [h]
+
 /-! ### mul -/
 
 theorem mul_def {n} {x y : BitVec n} : x * y = (ofFin <| x.toFin * y.toFin) := by rfl
@@ -2520,13 +2607,13 @@ theorem udiv_def {x y : BitVec n} : x / y = BitVec.ofNat n (x.toNat / y.toNat) :
   rw [← udiv_eq]
   simp [udiv, bv_toNat, h, Nat.mod_eq_of_lt]
 
+@[simp]
+theorem toFin_udiv {x y : BitVec n} : (x / y).toFin = x.toFin / y.toFin := by
+  rfl
+
 @[simp, bv_toNat]
 theorem toNat_udiv {x y : BitVec n} : (x / y).toNat = x.toNat / y.toNat := by
-  rw [udiv_def]
-  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)
+  rfl
 
 @[simp]
 theorem zero_udiv {x : BitVec w} : (0#w) / x = 0#w := by
@@ -2562,6 +2649,45 @@ theorem udiv_self {x : BitVec w} :
       ↓reduceIte, toNat_udiv]
     rw [Nat.div_self (by omega), Nat.mod_eq_of_lt (by omega)]
 
+theorem msb_udiv (x y : BitVec w) :
+    (x / y).msb = (x.msb && y == 1#w) := by
+  cases msb_x : x.msb
+  · suffices x.toNat / y.toNat < 2 ^ (w - 1) by simpa [msb_eq_decide]
+    calc
+      x.toNat / y.toNat ≤ x.toNat     := by apply Nat.div_le_self
+                      _ < 2 ^ (w - 1) := by simpa [msb_eq_decide] using msb_x
+  . rcases w with _|w
+    · contradiction
+    · have : (y == 1#_) = decide (y.toNat = 1) := by
+        simp [(· == ·), toNat_eq]
+      simp only [this, Bool.true_and]
+      match hy : y.toNat with
+      | 0 =>
+        obtain rfl : y = 0#_ := eq_of_toNat_eq hy
+        simp
+      | 1 =>
+        obtain rfl : y = 1#_ := eq_of_toNat_eq (by simp [hy])
+        simpa using msb_x
+      | y + 2 =>
+        suffices x.toNat / (y + 2) < 2 ^ w by
+          simp_all [msb_eq_decide, hy]
+        calc
+          x.toNat / (y + 2)
+            ≤ x.toNat / 2 := by apply Nat.div_add_le_right (by omega)
+          _ < 2 ^ w       := by omega
+
+theorem msb_udiv_eq_false_of {x : BitVec w} (h : x.msb = false) (y : BitVec w) :
+    (x / y).msb = false := by
+  simp [msb_udiv, h]
+
+/--
+If `x` is nonnegative (i.e., does not have its msb set),
+then `x / y` is nonnegative, thus `toInt` and `toNat` coincide.
+-/
+theorem toInt_udiv_of_msb {x : BitVec w} (h : x.msb = false) (y : BitVec w) :
+    (x / y).toInt = x.toNat / y.toNat := by
+  simp [toInt_eq_msb_cond, msb_udiv_eq_false_of h]
+
 /-! ### umod -/
 
 theorem umod_def {x y : BitVec n} :
@@ -2574,6 +2700,10 @@ theorem umod_def {x y : BitVec n} :
 theorem toNat_umod {x y : BitVec n} :
     (x % y).toNat = x.toNat % y.toNat := rfl
 
+@[simp]
+theorem toFin_umod {x y : BitVec w} :
+    (x % y).toFin = x.toFin % y.toFin := rfl
+
 @[simp]
 theorem umod_zero {x : BitVec n} : x % 0#n = x := by
   simp [umod_def]
@@ -2601,6 +2731,55 @@ theorem umod_eq_and {x y : BitVec 1} : x % y = x &&& (~~~y) := by
     rcases hy with rfl | rfl <;>
       rfl
 
+theorem umod_eq_of_lt {x y : BitVec w} (h : x < y) :
+    x % y = x := by
+  apply eq_of_toNat_eq
+  simp [Nat.mod_eq_of_lt h]
+
+@[simp]
+theorem msb_umod {x y : BitVec w} :
+    (x % y).msb = (x.msb && (x < y || y == 0#w)) := by
+  rw [msb_eq_decide, toNat_umod]
+  cases msb_x : x.msb
+  · suffices x.toNat % y.toNat < 2 ^ (w - 1) by simpa
+    calc
+      x.toNat % y.toNat ≤ x.toNat     := by apply Nat.mod_le
+                      _ < 2 ^ (w - 1) := by simpa [msb_eq_decide] using msb_x
+  . by_cases hy : y = 0
+    · simp_all [msb_eq_decide]
+    · suffices 2 ^ (w - 1) ≤ x.toNat % y.toNat ↔ x < y by simp_all
+      by_cases x_lt_y : x < y
+      . simp_all [Nat.mod_eq_of_lt x_lt_y, msb_eq_decide]
+      · suffices x.toNat % y.toNat < 2 ^ (w - 1) by
+          simpa [x_lt_y]
+        have y_le_x : y.toNat ≤ x.toNat := by
+          simpa using x_lt_y
+        replace hy : y.toNat ≠ 0 :=
+          toNat_ne_iff_ne.mpr hy
+        by_cases msb_y : y.toNat < 2 ^ (w - 1)
+        · have : x.toNat % y.toNat < y.toNat := Nat.mod_lt _ (by omega)
+          omega
+        · rcases w with _|w
+          · contradiction
+          simp only [Nat.add_one_sub_one]
+          replace msb_y : 2 ^ w ≤ y.toNat := by
+            simpa using msb_y
+          have : y.toNat ≤ y.toNat * (x.toNat / y.toNat) := by
+              apply Nat.le_mul_of_pos_right
+              apply Nat.div_pos y_le_x
+              omega
+          have : x.toNat % y.toNat ≤ x.toNat - y.toNat := by
+            rw [Nat.mod_eq_sub]; omega
+          omega
+
+theorem toInt_umod {x y : BitVec w} :
+    (x % y).toInt = (x.toNat % y.toNat : Int).bmod (2 ^ w) := by
+  simp [toInt_eq_toNat_bmod]
+
+theorem toInt_umod_of_msb {x y : BitVec w} (h : x.msb = false) :
+    (x % y).toInt = x.toInt % y.toNat := by
+  simp [toInt_eq_msb_cond, h]
+
 /-! ### smtUDiv -/
 
 theorem smtUDiv_eq (x y : BitVec w) : smtUDiv x y = if y = 0#w then allOnes w else x / y := by
@@ -2757,7 +2936,12 @@ theorem smod_zero {x : BitVec n} : x.smod 0#n = x := by
 
 /-! # Rotate Left -/
 
-/-- rotateLeft is invariant under `mod` by the bitwidth. -/
+/--`rotateLeft` is defined in terms of left and right shifts. -/
+theorem rotateLeft_def {x : BitVec w} {r : Nat} :
+    x.rotateLeft r = (x <<< (r % w)) ||| (x >>> (w - r % w)) := by
+  simp only [rotateLeft, rotateLeftAux]
+
+/-- `rotateLeft` is invariant under `mod` by the bitwidth. -/
 @[simp]
 theorem rotateLeft_mod_eq_rotateLeft {x : BitVec w} {r : Nat} :
     x.rotateLeft (r % w) = x.rotateLeft r := by
@@ -2901,8 +3085,18 @@ theorem msb_rotateLeft {m w : Nat} {x : BitVec w} :
   · simp
     omega
 
+@[simp]
+theorem toNat_rotateLeft {x : BitVec w} {r : Nat} :
+    (x.rotateLeft r).toNat = (x.toNat <<< (r % w)) % (2^w) ||| x.toNat >>> (w - r % w) := by
+  simp only [rotateLeft_def, toNat_shiftLeft, toNat_ushiftRight, toNat_or]
+
 /-! ## Rotate Right -/
 
+/-- `rotateRight` is defined in terms of left and right shifts. -/
+theorem rotateRight_def {x : BitVec w} {r : Nat} :
+    x.rotateRight r = (x >>> (r % w)) ||| (x <<< (w - r % w)) := by
+  simp only [rotateRight, rotateRightAux]
+
 /--
 Accessing bits in `x.rotateRight r` the range `[0, w-r)` is equal to
 accessing bits `x` in the range `[r, w)`.
@@ -3038,6 +3232,11 @@ theorem msb_rotateRight {r w : Nat} {x : BitVec w} :
     simp [h₁]
   · simp [show w = 0 by omega]
 
+@[simp]
+theorem toNat_rotateRight {x : BitVec w} {r : Nat} :
+    (x.rotateRight r).toNat = (x.toNat >>> (r % w)) ||| x.toNat <<< (w - r % w) % (2^w) := by
+  simp only [rotateRight_def, toNat_shiftLeft, toNat_ushiftRight, toNat_or]
+
 /- ## twoPow -/
 
 theorem twoPow_eq (w : Nat) (i : Nat) : twoPow w i = 1#w <<< i := by
@@ -3340,7 +3539,7 @@ theorem getLsbD_intMax (w : Nat) : (intMax w).getLsbD i = decide (i + 1 < w) :=
 
 /-! ### Non-overflow theorems -/
 
-/-- If `x.toNat * y.toNat < 2^w`, then the multiplication `(x * y)` does not overflow. -/
+/-- If `x.toNat + y.toNat < 2^w`, then the addition `(x + y)` does not overflow. -/
 theorem toNat_add_of_lt {w} {x y : BitVec w} (h : x.toNat + y.toNat < 2^w) :
     (x + y).toNat = x.toNat + y.toNat := by
   rw [BitVec.toNat_add, Nat.mod_eq_of_lt h]

src/Init/Data/Int/DivModLemmas.lean

@@ -534,6 +534,13 @@ theorem mul_emod (a b n : Int) : (a * b) % n = (a % n) * (b % n) % n := by
 @[simp] theorem emod_emod (a b : Int) : (a % b) % b = a % b := by
   conv => rhs; rw [← emod_add_ediv a b, add_mul_emod_self_left]
 
+@[simp] theorem emod_sub_emod (m n k : Int) : (m % n - k) % n = (m - k) % n :=
+  Int.emod_add_emod m n (-k)
+
+@[simp] theorem sub_emod_emod (m n k : Int) : (m - n % k) % k = (m - n) % k := by
+  apply (emod_add_cancel_right (n % k)).mp
+  rw [Int.sub_add_cancel, Int.add_emod_emod, Int.sub_add_cancel]
+
 theorem sub_emod (a b n : Int) : (a - b) % n = (a % n - b % n) % n := by
   apply (emod_add_cancel_right b).mp
   rw [Int.sub_add_cancel, ← Int.add_emod_emod, Int.sub_add_cancel, emod_emod]
@@ -1098,6 +1105,32 @@ theorem bmod_def (x : Int) (m : Nat) : bmod x m =
     (x % m) - m :=
   rfl
 
+theorem bdiv_add_bmod (x : Int) (m : Nat) : m * bdiv x m + bmod x m = x := by
+  unfold bdiv bmod
+  split
+  · simp_all only [Nat.cast_ofNat_Int, Int.mul_zero, emod_zero, Int.zero_add, Int.sub_zero,
+      ite_self]
+  · dsimp only
+    split
+    · exact ediv_add_emod x m
+    · rw [Int.mul_add, Int.mul_one, Int.add_assoc, Int.add_comm m, Int.sub_add_cancel]
+      exact ediv_add_emod x m
+
+theorem bmod_add_bdiv (x : Int) (m : Nat) : bmod x m + m * bdiv x m = x := by
+  rw [Int.add_comm]; exact bdiv_add_bmod x m
+
+theorem bdiv_add_bmod' (x : Int) (m : Nat) : bdiv x m * m + bmod x m = x := by
+  rw [Int.mul_comm]; exact bdiv_add_bmod x m
+
+theorem bmod_add_bdiv' (x : Int) (m : Nat) : bmod x m + bdiv x m * m = x := by
+  rw [Int.add_comm]; exact bdiv_add_bmod' x m
+
+theorem bmod_eq_self_sub_mul_bdiv (x : Int) (m : Nat) : bmod x m = x - m * bdiv x m := by
+  rw [← Int.add_sub_cancel (bmod x m), bmod_add_bdiv]
+
+theorem bmod_eq_self_sub_bdiv_mul (x : Int) (m : Nat) : bmod x m = x - bdiv x m * m := by
+  rw [← Int.add_sub_cancel (bmod x m), bmod_add_bdiv']
+
 theorem bmod_pos (x : Int) (m : Nat) (p : x % m < (m + 1) / 2) : bmod x m = x % m := by
   simp [bmod_def, p]
 

src/Init/Data/List/Basic.lean

@@ -606,11 +606,11 @@ set_option linter.missingDocs false in
 to get a list of lists, and then concatenates them all together.
 * `[2, 3, 2].bind range = [0, 1, 0, 1, 2, 0, 1]`
 -/
-@[inline] def flatMap {α : Type u} {β : Type v} (a : List α) (b : α → List β) : List β := flatten (map b a)
+@[inline] def flatMap {α : Type u} {β : Type v} (b : α → List β) (a : List α) : List β := flatten (map b a)
 
-@[simp] theorem flatMap_nil (f : α → List β) : List.flatMap [] f = [] := by simp [flatten, List.flatMap]
+@[simp] theorem flatMap_nil (f : α → List β) : List.flatMap f [] = [] := by simp [flatten, List.flatMap]
 @[simp] theorem flatMap_cons x xs (f : α → List β) :
-  List.flatMap (x :: xs) f = f x ++ List.flatMap xs f := by simp [flatten, List.flatMap]
+  List.flatMap f (x :: xs) = f x ++ List.flatMap f xs := by simp [flatten, List.flatMap]
 
 set_option linter.missingDocs false in
 @[deprecated flatMap (since := "2024-10-16")] abbrev bind := @flatMap

src/Init/Data/List/Impl.lean

@@ -96,14 +96,14 @@ The following operations are given `@[csimp]` replacements below:
 /-! ### flatMap  -/
 
 /-- Tail recursive version of `List.flatMap`. -/
-@[inline] def flatMapTR (as : List α) (f : α → List β) : List β := go as #[] where
+@[inline] def flatMapTR (f : α → List β) (as : List α) : List β := go as #[] where
   /-- Auxiliary for `flatMap`: `flatMap.go f as = acc.toList ++ bind f as` -/
   @[specialize] go : List α → Array β → List β
   | [], acc => acc.toList
   | x::xs, acc => go xs (acc ++ f x)
 
 @[csimp] theorem flatMap_eq_flatMapTR : @List.flatMap = @flatMapTR := by
-  funext α β as f
+  funext α β f as
   let rec go : ∀ as acc, flatMapTR.go f as acc = acc.toList ++ as.flatMap f
     | [], acc => by simp [flatMapTR.go, flatMap]
     | x::xs, acc => by simp [flatMapTR.go, flatMap, go xs]
@@ -112,7 +112,7 @@ The following operations are given `@[csimp]` replacements below:
 /-! ### flatten -/
 
 /-- Tail recursive version of `List.flatten`. -/
-@[inline] def flattenTR (l : List (List α)) : List α := flatMapTR l id
+@[inline] def flattenTR (l : List (List α)) : List α := l.flatMapTR id
 
 @[csimp] theorem flatten_eq_flattenTR : @flatten = @flattenTR := by
   funext α l; rw [← List.flatMap_id, List.flatMap_eq_flatMapTR]; rfl

src/Init/Data/List/Lemmas.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.Bool
@@ -757,207 +758,6 @@ theorem length_eq_of_beq [BEq α] {l₁ l₂ : List α} (h : l₁ == l₂) : l
     | nil => simp
     | cons b l₂ => simp [isEqv, ih]
 
-/-! ### foldlM and foldrM -/
-
-@[simp] theorem foldlM_reverse [Monad m] (l : List α) (f : β → α → m β) (b) :
-    l.reverse.foldlM f b = l.foldrM (fun x y => f y x) b := rfl
-
-@[simp] theorem foldlM_append [Monad m] [LawfulMonad m] (f : β → α → m β) (b) (l l' : List α) :
-    (l ++ l').foldlM f b = l.foldlM f b >>= l'.foldlM f := by
-  induction l generalizing b <;> simp [*]
-
-@[simp] theorem foldrM_cons [Monad m] [LawfulMonad m] (a : α) (l) (f : α → β → m β) (b) :
-    (a :: l).foldrM f b = l.foldrM f b >>= f a := by
-  simp only [foldrM]
-  induction l <;> simp_all
-
-theorem foldl_eq_foldlM (f : β → α → β) (b) (l : List α) :
-    l.foldl f b = l.foldlM (m := Id) f b := by
-  induction l generalizing b <;> simp [*, foldl]
-
-theorem foldr_eq_foldrM (f : α → β → β) (b) (l : List α) :
-    l.foldr f b = l.foldrM (m := Id) f b := by
-  induction l <;> simp [*, foldr]
-
-@[simp] theorem id_run_foldlM (f : β → α → Id β) (b) (l : List α) :
-    Id.run (l.foldlM f b) = l.foldl f b := (foldl_eq_foldlM f b l).symm
-
-@[simp] theorem id_run_foldrM (f : α → β → Id β) (b) (l : List α) :
-    Id.run (l.foldrM f b) = l.foldr f b := (foldr_eq_foldrM f b l).symm
-
-/-! ### foldl and foldr -/
-
-@[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_cons_nil (l : List α) : l.foldr cons [] = l := by simp
-
-@[deprecated foldr_cons_nil (since := "2024-09-04")] abbrev foldr_self := @foldr_cons_nil
-
-theorem foldl_map (f : β₁ → β₂) (g : α → β₂ → α) (l : List β₁) (init : α) :
-    (l.map f).foldl g init = l.foldl (fun x y => g x (f y)) init := by
-  induction l generalizing init <;> simp [*]
-
-theorem foldr_map (f : α₁ → α₂) (g : α₂ → β → β) (l : List α₁) (init : β) :
-    (l.map f).foldr g init = l.foldr (fun x y => g (f x) y) init := by
-  induction l generalizing init <;> simp [*]
-
-theorem foldl_filterMap (f : α → Option β) (g : γ → β → γ) (l : List α) (init : γ) :
-    (l.filterMap f).foldl g init = l.foldl (fun x y => match f y with | some b => g x b | none => x) init := by
-  induction l generalizing init with
-  | nil => rfl
-  | cons a l ih =>
-    simp only [filterMap_cons, foldl_cons]
-    cases f a <;> simp [ih]
-
-theorem foldr_filterMap (f : α → Option β) (g : β → γ → γ) (l : List α) (init : γ) :
-    (l.filterMap f).foldr g init = l.foldr (fun x y => match f x with | some b => g b y | none => y) init := by
-  induction l generalizing init with
-  | nil => rfl
-  | cons a l ih =>
-    simp only [filterMap_cons, foldr_cons]
-    cases f a <;> simp [ih]
-
-theorem foldl_map' (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α)
-    (h : ∀ x y, f' (g x) (g y) = g (f x y)) :
-    (l.map g).foldl f' (g a) = g (l.foldl f a) := by
-  induction l generalizing a
-  · simp
-  · simp [*, h]
-
-theorem foldr_map' (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α)
-    (h : ∀ x y, f' (g x) (g y) = g (f x y)) :
-    (l.map g).foldr f' (g a) = g (l.foldr f a) := by
-  induction l generalizing a
-  · simp
-  · simp [*, h]
-
-theorem foldl_assoc {op : α → α → α} [ha : Std.Associative op] :
-    ∀ {l : List α} {a₁ a₂}, l.foldl op (op a₁ a₂) = op a₁ (l.foldl op a₂)
-  | [], a₁, a₂ => rfl
-  | a :: l, a₁, a₂ => by
-    simp only [foldl_cons, ha.assoc]
-    rw [foldl_assoc]
-
-theorem foldr_assoc {op : α → α → α} [ha : Std.Associative op] :
-    ∀ {l : List α} {a₁ a₂}, l.foldr op (op a₁ a₂) = op (l.foldr op a₁) a₂
-  | [], a₁, a₂ => rfl
-  | a :: l, a₁, a₂ => by
-    simp only [foldr_cons, ha.assoc]
-    rw [foldr_assoc]
-
-theorem foldl_hom (f : α₁ → α₂) (g₁ : α₁ → β → α₁) (g₂ : α₂ → β → α₂) (l : List β) (init : α₁)
-    (H : ∀ x y, g₂ (f x) y = f (g₁ x y)) : l.foldl g₂ (f init) = f (l.foldl g₁ init) := by
-  induction l generalizing init <;> simp [*, H]
-
-theorem foldr_hom (f : β₁ → β₂) (g₁ : α → β₁ → β₁) (g₂ : α → β₂ → β₂) (l : List α) (init : β₁)
-    (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
-
-/--
-We can prove that two folds over the same list are related (by some arbitrary relation)
-if we know that the initial elements are related and the folding function, for each element of the list,
-preserves the relation.
--/
-theorem foldl_rel {l : List α} {f g : β → α → β} {a b : β} (r : β → β → Prop)
-    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
-    r (l.foldl (fun acc a => f acc a) a) (l.foldl (fun acc a => g acc a) b) := by
-  induction l generalizing a b with
-  | nil => simp_all
-  | cons a l ih =>
-    simp only [foldl_cons]
-    apply ih
-    · simp_all
-    · exact fun a m c c' h => h' _ (by simp_all) _ _ h
-
-/--
-We can prove that two folds over the same list are related (by some arbitrary relation)
-if we know that the initial elements are related and the folding function, for each element of the list,
-preserves the relation.
--/
-theorem foldr_rel {l : List α} {f g : α → β → β} {a b : β} (r : β → β → Prop)
-    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
-    r (l.foldr (fun a acc => f a acc) a) (l.foldr (fun a acc => g a acc) b) := by
-  induction l generalizing a b with
-  | nil => simp_all
-  | cons a l ih =>
-    simp only [foldr_cons]
-    apply h'
-    · simp
-    · exact ih h fun a m c c' h => h' _ (by simp_all) _ _ h
-
-@[simp] theorem foldl_add_const (l : List α) (a b : Nat) :
-    l.foldl (fun x _ => x + a) b = b + a * l.length := by
-  induction l generalizing b with
-  | nil => simp
-  | cons y l ih =>
-    simp only [foldl_cons, ih, length_cons, Nat.mul_add, Nat.mul_one, Nat.add_assoc,
-      Nat.add_comm a]
-
-@[simp] theorem foldr_add_const (l : List α) (a b : Nat) :
-    l.foldr (fun _ x => x + a) b = b + a * l.length := by
-  induction l generalizing b with
-  | nil => simp
-  | cons y l ih =>
-    simp only [foldr_cons, ih, length_cons, Nat.mul_add, Nat.mul_one, Nat.add_assoc]
-
 /-! ### getLast -/
 
 theorem getLast_eq_getElem : ∀ (l : List α) (h : l ≠ []),
@@ -1216,27 +1016,6 @@ theorem getLast?_tail (l : List α) : (tail l).getLast? = if l.length = 1 then n
 
 /-! ### map -/
 
-@[simp] theorem map_id_fun : map (id : α → α) = id := by
-  funext l
-  induction l <;> simp_all
-
-/-- `map_id_fun'` differs from `map_id_fun` by representing the identity function as a lambda, rather than `id`. -/
-@[simp] theorem map_id_fun' : map (fun (a : α) => a) = id := map_id_fun
-
--- This is not a `@[simp]` lemma because `map_id_fun` will apply.
-theorem map_id (l : List α) : map (id : α → α) l = l := by
-  induction l <;> simp_all
-
-/-- `map_id'` differs from `map_id` by representing the identity function as a lambda, rather than `id`. -/
--- This is not a `@[simp]` lemma because `map_id_fun'` will apply.
-theorem map_id' (l : List α) : map (fun (a : α) => a) l = l := map_id l
-
-/-- Variant of `map_id`, with a side condition that the function is pointwise the identity. -/
-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]
-
-theorem map_singleton (f : α → β) (a : α) : map f [a] = [f a] := rfl
-
 @[simp] theorem length_map (as : List α) (f : α → β) : (as.map f).length = as.length := by
   induction as with
   | nil => simp [List.map]
@@ -1262,6 +1041,27 @@ theorem get_map (f : α → β) {l i} :
     get (map f l) i = f (get l ⟨i, length_map l f ▸ i.2⟩) := by
   simp
 
+@[simp] theorem map_id_fun : map (id : α → α) = id := by
+  funext l
+  induction l <;> simp_all
+
+/-- `map_id_fun'` differs from `map_id_fun` by representing the identity function as a lambda, rather than `id`. -/
+@[simp] theorem map_id_fun' : map (fun (a : α) => a) = id := map_id_fun
+
+-- This is not a `@[simp]` lemma because `map_id_fun` will apply.
+theorem map_id (l : List α) : map (id : α → α) l = l := by
+  induction l <;> simp_all
+
+/-- `map_id'` differs from `map_id` by representing the identity function as a lambda, rather than `id`. -/
+-- This is not a `@[simp]` lemma because `map_id_fun'` will apply.
+theorem map_id' (l : List α) : map (fun (a : α) => a) l = l := map_id l
+
+/-- Variant of `map_id`, with a side condition that the function is pointwise the identity. -/
+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]
+
+theorem map_singleton (f : α → β) (a : α) : map f [a] = [f a] := rfl
+
 @[simp] theorem mem_map {f : α → β} : ∀ {l : List α}, b ∈ l.map f ↔ ∃ a, a ∈ l ∧ f a = b
   | [] => by simp
   | _ :: l => by simp [mem_map (l := l), eq_comm (a := b)]
@@ -1276,9 +1076,31 @@ theorem forall_mem_map {f : α → β} {l : List α} {P : β → Prop} :
 
 @[deprecated forall_mem_map (since := "2024-07-25")] abbrev forall_mem_map_iff := @forall_mem_map
 
+@[simp] theorem map_eq_nil_iff {f : α → β} {l : List α} : map f l = [] ↔ l = [] := by
+  constructor <;> exact fun _ => match l with | [] => rfl
+
+@[deprecated map_eq_nil_iff (since := "2024-09-05")] abbrev map_eq_nil := @map_eq_nil_iff
+
+theorem eq_nil_of_map_eq_nil {f : α → β} {l : List α} (h : map f l = []) : l = [] :=
+  map_eq_nil_iff.mp h
+
 @[simp] theorem map_inj_left {f g : α → β} : map f l = map g l ↔ ∀ a ∈ l, f a = g a := by
   induction l <;> simp_all
 
+theorem map_inj_right {f : α → β} (w : ∀ x y, f x = f y → x = y) : map f l = map f l' ↔ l = l' := by
+  induction l generalizing l' with
+  | nil => simp
+  | cons a l ih =>
+    simp only [map_cons]
+    cases l' with
+    | nil => simp
+    | cons a' l' =>
+      simp only [map_cons, cons.injEq, ih, and_congr_left_iff]
+      intro h
+      constructor
+      · apply w
+      · simp +contextual
+
 theorem map_congr_left (h : ∀ a ∈ l, f a = g a) : map f l = map g l :=
   map_inj_left.2 h
 
@@ -1287,14 +1109,6 @@ theorem map_inj : map f = map g ↔ f = g := by
   · intro h; ext a; replace h := congrFun h [a]; simpa using h
   · intro h; subst h; rfl
 
-@[simp] theorem map_eq_nil_iff {f : α → β} {l : List α} : map f l = [] ↔ l = [] := by
-  constructor <;> exact fun _ => match l with | [] => rfl
-
-@[deprecated map_eq_nil_iff (since := "2024-09-05")] abbrev map_eq_nil := @map_eq_nil_iff
-
-theorem eq_nil_of_map_eq_nil {f : α → β} {l : List α} (h : map f l = []) : l = [] :=
-  map_eq_nil_iff.mp h
-
 theorem map_eq_cons_iff {f : α → β} {l : List α} :
     map f l = b :: l₂ ↔ ∃ a l₁, l = a :: l₁ ∧ f a = b ∧ map f l₁ = l₂ := by
   cases l
@@ -1315,6 +1129,10 @@ theorem map_eq_cons_iff' {f : α → β} {l : List α} :
 
 @[deprecated map_eq_cons' (since := "2024-09-05")] abbrev map_eq_cons' := @map_eq_cons_iff'
 
+@[simp] theorem map_eq_singleton_iff {f : α → β} {l : List α} {b : β} :
+    map f l = [b] ↔ ∃ a, l = [a] ∧ f a = b := by
+  simp [map_eq_cons_iff]
+
 theorem map_eq_map_iff : map f l = map g l ↔ ∀ a ∈ l, f a = g a := by
   induction l <;> simp
 
@@ -1481,7 +1299,7 @@ theorem map_filter_eq_foldr (f : α → β) (p : α → Bool) (as : List α) :
 @[simp] theorem filter_append {p : α → Bool} :
     ∀ (l₁ l₂ : List α), filter p (l₁ ++ l₂) = filter p l₁ ++ filter p l₂
   | [], _ => rfl
-  | a :: l₁, l₂ => by simp [filter]; split <;> simp [filter_append l₁]
+  | a :: l₁, l₂ => by simp only [cons_append, filter]; split <;> simp [filter_append l₁]
 
 theorem filter_eq_cons_iff {l} {a} {as} :
     filter p l = a :: as ↔
@@ -1690,6 +1508,34 @@ theorem filterMap_eq_cons_iff {l} {b} {bs} :
 @[simp] theorem cons_append_fun (a : α) (as : List α) :
     (fun bs => ((a :: as) ++ bs)) = fun bs => a :: (as ++ bs) := rfl
 
+@[simp] theorem mem_append {a : α} {s t : List α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
+  induction s <;> simp_all [or_assoc]
+
+theorem not_mem_append {a : α} {s t : List α} (h₁ : a ∉ s) (h₂ : a ∉ t) : a ∉ s ++ t :=
+  mt mem_append.1 $ not_or.mpr ⟨h₁, h₂⟩
+
+@[deprecated mem_append (since := "2025-01-13")]
+theorem mem_append_eq (a : α) (s t : List α) : (a ∈ s ++ t) = (a ∈ s ∨ a ∈ t) :=
+  propext mem_append
+
+@[deprecated mem_append_left (since := "2024-11-20")] abbrev mem_append_of_mem_left := @mem_append_left
+@[deprecated mem_append_right (since := "2024-11-20")] abbrev mem_append_of_mem_right := @mem_append_right
+
+/--
+See also `eq_append_cons_of_mem`, which proves a stronger version
+in which the initial list must not contain the element.
+-/
+theorem append_of_mem {a : α} {l : List α} : a ∈ l → ∃ s t : List α, l = s ++ a :: t
+  | .head l => ⟨[], l, rfl⟩
+  | .tail b h => let ⟨s, t, h'⟩ := append_of_mem h; ⟨b::s, t, by rw [h', cons_append]⟩
+
+theorem mem_iff_append {a : α} {l : List α} : a ∈ l ↔ ∃ s t : List α, l = s ++ a :: t :=
+  ⟨append_of_mem, fun ⟨s, t, e⟩ => e ▸ by simp⟩
+
+theorem forall_mem_append {p : α → Prop} {l₁ l₂ : List α} :
+    (∀ (x) (_ : x ∈ l₁ ++ l₂), p x) ↔ (∀ (x) (_ : x ∈ l₁), p x) ∧ (∀ (x) (_ : x ∈ l₂), p x) := by
+  simp only [mem_append, or_imp, forall_and]
+
 theorem getElem_append {l₁ l₂ : List α} (i : Nat) (h : i < (l₁ ++ l₂).length) :
     (l₁ ++ l₂)[i] = if h' : i < l₁.length then l₁[i] else l₂[i - l₁.length]'(by simp at h h'; exact Nat.sub_lt_left_of_lt_add h' h) := by
   split <;> rename_i h'
@@ -1757,14 +1603,6 @@ theorem get_of_append {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.lengt
     l.get ⟨i, get_of_append_proof eq h⟩ = a := Option.some.inj <| by
   rw [← get?_eq_get, eq, get?_append_right (h ▸ Nat.le_refl _), h, Nat.sub_self]; rfl
 
-/--
-See also `eq_append_cons_of_mem`, which proves a stronger version
-in which the initial list must not contain the element.
--/
-theorem append_of_mem {a : α} {l : List α} : a ∈ l → ∃ s t : List α, l = s ++ a :: t
-  | .head l => ⟨[], l, rfl⟩
-  | .tail b h => let ⟨s, t, h'⟩ := append_of_mem h; ⟨b::s, t, by rw [h', cons_append]⟩
-
 @[simp 1100] theorem singleton_append : [x] ++ l = x :: l := rfl
 
 theorem append_inj :
@@ -1781,8 +1619,8 @@ theorem append_inj_left (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length s₁ = le
 
 /-- Variant of `append_inj` instead requiring equality of the lengths of the second lists. -/
 theorem append_inj' (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length t₁ = length t₂) : s₁ = s₂ ∧ t₁ = t₂ :=
-  append_inj h <| @Nat.add_right_cancel _ (length t₁) _ <| by
-  let hap := congrArg length h; simp only [length_append, ← hl] at hap; exact hap
+  append_inj h <| @Nat.add_right_cancel _ t₁.length _ <| by
+    let hap := congrArg length h; simp only [length_append, ← hl] at hap; exact hap
 
 /-- Variant of `append_inj_right` instead requiring equality of the lengths of the second lists. -/
 theorem append_inj_right' (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length t₁ = length t₂) : t₁ = t₂ :=
@@ -1810,9 +1648,6 @@ theorem append_left_inj {s₁ s₂ : List α} (t) : s₁ ++ t = s₂ ++ t ↔ s
 @[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
-
 theorem getLast_concat {a : α} : ∀ (l : List α), getLast (l ++ [a]) (by simp) = a
   | [] => rfl
   | a::t => by
@@ -1838,6 +1673,54 @@ theorem get?_append {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.length) :
     (l₁ ++ l₂).get? n = l₁.get? n := by
   simp [getElem?_append_left hn]
 
+@[simp] theorem append_eq_nil_iff : p ++ q = [] ↔ p = [] ∧ q = [] := by
+  cases p <;> simp
+
+@[deprecated append_eq_nil_iff (since := "2025-01-13")] abbrev append_eq_nil := @append_eq_nil_iff
+
+@[simp] theorem nil_eq_append_iff : [] = a ++ b ↔ a = [] ∧ b = [] := by
+  rw [eq_comm, append_eq_nil_iff]
+
+@[deprecated nil_eq_append_iff (since := "2024-07-24")] abbrev nil_eq_append := @nil_eq_append_iff
+
+theorem append_ne_nil_of_left_ne_nil {s : List α} (h : s ≠ []) (t : List α) : s ++ t ≠ [] := by simp_all
+theorem append_ne_nil_of_right_ne_nil (s : List α) : t ≠ [] → s ++ t ≠ [] := by simp_all
+
+@[deprecated append_ne_nil_of_left_ne_nil (since := "2024-07-24")]
+theorem append_ne_nil_of_ne_nil_left {s : List α} (h : s ≠ []) (t : List α) : s ++ t ≠ [] := by simp_all
+@[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 append_eq_cons_iff :
+    a ++ b = x :: c ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
+  cases a with simp | cons a as => ?_
+  exact ⟨fun h => ⟨as, by simp [h]⟩, fun ⟨a', ⟨aeq, aseq⟩, h⟩ => ⟨aeq, by rw [aseq, h]⟩⟩
+
+@[deprecated append_eq_cons_iff (since := "2024-07-24")] abbrev append_eq_cons := @append_eq_cons_iff
+
+theorem cons_eq_append_iff :
+    x :: c = a ++ b ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
+  rw [eq_comm, append_eq_cons_iff]
+
+@[deprecated cons_eq_append_iff (since := "2024-07-24")] abbrev cons_eq_append := @cons_eq_append_iff
+
+theorem append_eq_singleton_iff :
+    a ++ b = [x] ↔ (a = [] ∧ b = [x]) ∨ (a = [x] ∧ b = []) := by
+  cases a <;> cases b <;> simp
+
+theorem singleton_eq_append_iff :
+    [x] = a ++ b ↔ (a = [] ∧ b = [x]) ∨ (a = [x] ∧ b = []) := by
+  cases a <;> cases b <;> simp [eq_comm]
+
+theorem append_eq_append_iff {a b c d : List α} :
+    a ++ b = c ++ d ↔ (∃ a', c = a ++ a' ∧ b = a' ++ d) ∨ ∃ c', a = c ++ c' ∧ d = c' ++ b := by
+  induction a generalizing c with
+  | nil => simp_all
+  | cons a as ih => cases c <;> simp [eq_comm, and_assoc, ih, and_or_left]
+
+@[deprecated append_inj (since := "2024-07-24")] abbrev append_inj_of_length_left := @append_inj
+@[deprecated append_inj' (since := "2024-07-24")] abbrev append_inj_of_length_right := @append_inj'
+
 @[simp] theorem head_append_of_ne_nil {l : List α} {w₁} (w₂) :
     head (l ++ l') w₁ = head l w₂ := by
   match l, w₂ with
@@ -1887,60 +1770,6 @@ theorem tail_append {l l' : List α} : (l ++ l').tail = if l.isEmpty then l'.tai
 
 @[deprecated tail_append_of_ne_nil (since := "2024-07-24")] abbrev tail_append_left := @tail_append_of_ne_nil
 
-theorem nil_eq_append_iff : [] = a ++ b ↔ a = [] ∧ b = [] := by
-  rw [eq_comm, append_eq_nil]
-
-@[deprecated nil_eq_append_iff (since := "2024-07-24")] abbrev nil_eq_append := @nil_eq_append_iff
-
-theorem append_ne_nil_of_left_ne_nil {s : List α} (h : s ≠ []) (t : List α) : s ++ t ≠ [] := by simp_all
-theorem append_ne_nil_of_right_ne_nil (s : List α) : t ≠ [] → s ++ t ≠ [] := by simp_all
-
-@[deprecated append_ne_nil_of_left_ne_nil (since := "2024-07-24")]
-theorem append_ne_nil_of_ne_nil_left {s : List α} (h : s ≠ []) (t : List α) : s ++ t ≠ [] := by simp_all
-@[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 append_eq_cons_iff :
-    a ++ b = x :: c ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
-  cases a with simp | cons a as => ?_
-  exact ⟨fun h => ⟨as, by simp [h]⟩, fun ⟨a', ⟨aeq, aseq⟩, h⟩ => ⟨aeq, by rw [aseq, h]⟩⟩
-
-@[deprecated append_eq_cons_iff (since := "2024-07-24")] abbrev append_eq_cons := @append_eq_cons_iff
-
-theorem cons_eq_append_iff :
-    x :: c = a ++ b ↔ (a = [] ∧ b = x :: c) ∨ (∃ a', a = x :: a' ∧ c = a' ++ b) := by
-  rw [eq_comm, append_eq_cons_iff]
-
-@[deprecated cons_eq_append_iff (since := "2024-07-24")] abbrev cons_eq_append := @cons_eq_append_iff
-
-theorem append_eq_append_iff {a b c d : List α} :
-    a ++ b = c ++ d ↔ (∃ a', c = a ++ a' ∧ b = a' ++ d) ∨ ∃ c', a = c ++ c' ∧ d = c' ++ b := by
-  induction a generalizing c with
-  | nil => simp_all
-  | cons a as ih => cases c <;> simp [eq_comm, and_assoc, ih, and_or_left]
-
-@[deprecated append_inj (since := "2024-07-24")] abbrev append_inj_of_length_left := @append_inj
-@[deprecated append_inj' (since := "2024-07-24")] abbrev append_inj_of_length_right := @append_inj'
-
-@[simp] theorem mem_append {a : α} {s t : List α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
-  induction s <;> simp_all [or_assoc]
-
-theorem not_mem_append {a : α} {s t : List α} (h₁ : a ∉ s) (h₂ : a ∉ t) : a ∉ s ++ t :=
-  mt mem_append.1 $ not_or.mpr ⟨h₁, h₂⟩
-
-theorem mem_append_eq (a : α) (s t : List α) : (a ∈ s ++ t) = (a ∈ s ∨ a ∈ t) :=
-  propext mem_append
-
-@[deprecated mem_append_left (since := "2024-11-20")] abbrev mem_append_of_mem_left := @mem_append_left
-@[deprecated mem_append_right (since := "2024-11-20")] abbrev mem_append_of_mem_right := @mem_append_right
-
-theorem mem_iff_append {a : α} {l : List α} : a ∈ l ↔ ∃ s t : List α, l = s ++ a :: t :=
-  ⟨append_of_mem, fun ⟨s, t, e⟩ => e ▸ by simp⟩
-
-theorem forall_mem_append {p : α → Prop} {l₁ l₂ : List α} :
-    (∀ (x) (_ : x ∈ l₁ ++ l₂), p x) ↔ (∀ (x) (_ : x ∈ l₁), p x) ∧ (∀ (x) (_ : x ∈ l₂), p x) := by
-  simp only [mem_append, or_imp, forall_and]
-
 theorem set_append {s t : List α} :
     (s ++ t).set i x = if i < s.length then s.set i x ++ t else s ++ t.set (i - s.length) x := by
   induction s generalizing i with
@@ -1961,16 +1790,6 @@ theorem set_append {s t : List α} :
     (s ++ t).set i x = s ++ t.set (i - s.length) x := by
   rw [set_append, if_neg (by simp_all)]
 
-@[simp] theorem foldrM_append [Monad m] [LawfulMonad m] (f : α → β → m β) (b) (l l' : List α) :
-    (l ++ l').foldrM f b = l'.foldrM f b >>= l.foldrM f := by
-  induction l <;> simp [*]
-
-@[simp] theorem foldl_append {β : Type _} (f : β → α → β) (b) (l l' : List α) :
-    (l ++ l').foldl f b = l'.foldl f (l.foldl f b) := by simp [foldl_eq_foldlM]
-
-@[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_iff {f : α → Option β} :
     filterMap f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂ := by
   constructor
@@ -2079,7 +1898,7 @@ theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ L b, l = concat L b
 
 /-! ### flatten -/
 
-@[simp] theorem length_flatten (L : List (List α)) : (flatten L).length = (L.map length).sum := by
+@[simp] theorem length_flatten (L : List (List α)) : L.flatten.length = (L.map length).sum := by
   induction L with
   | nil => rfl
   | cons =>
@@ -2094,6 +1913,9 @@ theorem flatten_singleton (l : List α) : [l].flatten = l := by simp
 @[simp] theorem flatten_eq_nil_iff {L : List (List α)} : L.flatten = [] ↔ ∀ l ∈ L, l = [] := by
   induction L <;> simp_all
 
+@[simp] theorem nil_eq_flatten_iff {L : List (List α)} : [] = L.flatten ↔ ∀ l ∈ L, l = [] := by
+  rw [eq_comm, flatten_eq_nil_iff]
+
 theorem flatten_ne_nil_iff {xs : List (List α)} : xs.flatten ≠ [] ↔ ∃ x, x ∈ xs ∧ x ≠ [] := by
   simp
 
@@ -2119,15 +1941,8 @@ theorem head?_flatten {L : List (List α)} : (flatten L).head? = L.findSome? fun
 -- `getLast?_flatten` is proved later, after the `reverse` section.
 -- `head_flatten` and `getLast_flatten` are proved in `Init.Data.List.Find`.
 
-theorem foldl_flatten (f : β → α → β) (b : β) (L : List (List α)) :
-    (flatten L).foldl f b = L.foldl (fun b l => l.foldl f b) b := by
-  induction L generalizing b <;> simp_all
-
-theorem foldr_flatten (f : α → β → β) (b : β) (L : List (List α)) :
-    (flatten L).foldr f b = L.foldr (fun l b => l.foldr f b) b := by
-  induction L <;> simp_all
-
-@[simp] theorem map_flatten (f : α → β) (L : List (List α)) : map f (flatten L) = flatten (map (map f) L) := by
+@[simp] theorem map_flatten (f : α → β) (L : List (List α)) :
+    (flatten L).map f = (map (map f) L).flatten := by
   induction L <;> simp_all
 
 @[simp] theorem filterMap_flatten (f : α → Option β) (L : List (List α)) :
@@ -2180,6 +1995,26 @@ theorem flatten_eq_cons_iff {xs : List (List α)} {y : α} {ys : List α} :
   · rintro ⟨as, bs, cs, rfl, h₁, rfl⟩
     simp [flatten_eq_nil_iff.mpr h₁]
 
+theorem cons_eq_flatten_iff {xs : List (List α)} {y : α} {ys : List α} :
+    y :: ys = xs.flatten ↔
+      ∃ as bs cs, xs = as ++ (y :: bs) :: cs ∧ (∀ l, l ∈ as → l = []) ∧ ys = bs ++ cs.flatten := by
+  rw [eq_comm, flatten_eq_cons_iff]
+
+theorem flatten_eq_singleton_iff {xs : List (List α)} {y : α} :
+    xs.flatten = [y] ↔ ∃ as bs, xs = as ++ [y] :: bs ∧ (∀ l, l ∈ as → l = []) ∧ (∀ l, l ∈ bs → l = []) := by
+  rw [flatten_eq_cons_iff]
+  constructor
+  · rintro ⟨as, bs, cs, rfl, h₁, h₂⟩
+    simp at h₂
+    obtain ⟨rfl, h₂⟩ := h₂
+    exact ⟨as, cs, by simp, h₁, h₂⟩
+  · rintro ⟨as, bs, rfl, h₁, h₂⟩
+    exact ⟨as, [], bs, rfl, h₁, by simpa⟩
+
+theorem singleton_eq_flatten_iff {xs : List (List α)} {y : α} :
+    [y] = xs.flatten ↔ ∃ as bs, xs = as ++ [y] :: bs ∧ (∀ l, l ∈ as → l = []) ∧ (∀ l, l ∈ bs → l = []) := by
+  rw [eq_comm, flatten_eq_singleton_iff]
+
 theorem flatten_eq_append_iff {xs : List (List α)} {ys zs : List α} :
     xs.flatten = ys ++ zs ↔
       (∃ as bs, xs = as ++ bs ∧ ys = as.flatten ∧ zs = bs.flatten) ∨
@@ -2188,8 +2023,8 @@ theorem flatten_eq_append_iff {xs : List (List α)} {ys zs : List α} :
   constructor
   · induction xs generalizing ys with
     | nil =>
-      simp only [flatten_nil, nil_eq, append_eq_nil, and_false, cons_append, false_and, exists_const,
-        exists_false, or_false, and_imp, List.cons_ne_nil]
+      simp only [flatten_nil, nil_eq, append_eq_nil_iff, 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 =>
@@ -2208,6 +2043,13 @@ theorem flatten_eq_append_iff {xs : List (List α)} {ys zs : List α} :
     · simp
     · simp
 
+theorem append_eq_flatten_iff {xs : List (List α)} {ys zs : List α} :
+    ys ++ zs = xs.flatten ↔
+      (∃ as bs, xs = as ++ bs ∧ ys = as.flatten ∧ zs = bs.flatten) ∨
+        ∃ as bs c cs ds, xs = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.flatten ++ bs ∧
+          zs = c :: cs ++ ds.flatten := by
+  rw [eq_comm, flatten_eq_append_iff]
+
 /-- Two lists of sublists are equal iff their flattens coincide, as well as the lengths of the
 sublists. -/
 theorem eq_iff_flatten_eq : ∀ {L L' : List (List α)},
@@ -2228,12 +2070,14 @@ theorem eq_iff_flatten_eq : ∀ {L L' : List (List α)},
 
 theorem flatMap_def (l : List α) (f : α → List β) : l.flatMap f = flatten (map f l) := by rfl
 
-@[simp] theorem flatMap_id (l : List (List α)) : List.flatMap l id = l.flatten := by simp [flatMap_def]
+@[simp] theorem flatMap_id (l : List (List α)) : l.flatMap id = l.flatten := by simp [flatMap_def]
+
+@[simp] theorem flatMap_id' (l : List (List α)) : l.flatMap (fun a => a) = l.flatten := by simp [flatMap_def]
 
 @[simp]
 theorem length_flatMap (l : List α) (f : α → List β) :
-    length (l.flatMap f) = sum (map (length ∘ f) l) := by
-  rw [List.flatMap, length_flatten, map_map]
+    length (l.flatMap f) = sum (map (fun a => (f a).length) l) := by
+  rw [List.flatMap, length_flatten, map_map, Function.comp_def]
 
 @[simp] theorem mem_flatMap {f : α → List β} {b} {l : List α} : b ∈ l.flatMap f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by
   simp [flatMap_def, mem_flatten]
@@ -2246,7 +2090,7 @@ theorem mem_flatMap_of_mem {b : β} {l : List α} {f : α → List β} {a} (al :
     b ∈ l.flatMap f := mem_flatMap.2 ⟨a, al, h⟩
 
 @[simp]
-theorem flatMap_eq_nil_iff {l : List α} {f : α → List β} : List.flatMap l f = [] ↔ ∀ x ∈ l, f x = [] :=
+theorem flatMap_eq_nil_iff {l : List α} {f : α → List β} : l.flatMap f = [] ↔ ∀ x ∈ l, f x = [] :=
   flatten_eq_nil_iff.trans <| by
     simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
 
@@ -2430,7 +2274,7 @@ theorem map_const' (l : List α) (b : β) : map (fun _ => b) l = replicate l.len
   · intro i h₁ h₂
     simp [getElem_set]
 
-@[simp] theorem append_replicate_replicate : replicate n a ++ replicate m a = replicate (n + m) a := by
+@[simp] theorem replicate_append_replicate : replicate n a ++ replicate m a = replicate (n + m) a := by
   rw [eq_replicate_iff]
   constructor
   · simp
@@ -2438,6 +2282,9 @@ 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
 
+@[deprecated replicate_append_replicate (since := "2025-01-16")]
+abbrev append_replicate_replicate := @replicate_append_replicate
+
 theorem append_eq_replicate_iff {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
@@ -2448,6 +2295,11 @@ theorem append_eq_replicate_iff {l₁ l₂ : List α} {a : α} :
 
 @[deprecated append_eq_replicate_iff (since := "2024-09-05")] abbrev append_eq_replicate := @append_eq_replicate_iff
 
+theorem replicate_eq_append_iff {l₁ l₂ : List α} {a : α} :
+    replicate n a = l₁ ++ l₂ ↔
+      l₁.length + l₂.length = n ∧ l₁ = replicate l₁.length a ∧ l₂ = replicate l₂.length a := by
+  rw [eq_comm, append_eq_replicate_iff]
+
 @[simp] theorem map_replicate : (replicate n a).map f = replicate n (f a) := by
   ext1 n
   simp only [getElem?_map, getElem?_replicate]
@@ -2499,7 +2351,7 @@ theorem filterMap_replicate_of_some {f : α → Option β} (h : f a = some b) :
   induction n with
   | zero => simp
   | succ n ih =>
-    simp only [replicate_succ, flatten_cons, ih, append_replicate_replicate, replicate_inj, or_true,
+    simp only [replicate_succ, flatten_cons, ih, replicate_append_replicate, replicate_inj, or_true,
       and_true, add_one_mul, Nat.add_comm]
 
 theorem flatMap_replicate {β} (f : α → List β) : (replicate n a).flatMap f = (replicate n (f a)).flatten := by
@@ -2551,6 +2403,9 @@ theorem replicateRecOn {α : Type _} {p : List α → Prop} (m : List α)
     exact hi _ _ _ _ h hn (replicateRecOn (b :: l') h0 hr hi)
 termination_by m.length
 
+@[simp] theorem sum_replicate_nat (n : Nat) (a : Nat) : (replicate n a).sum = n * a := by
+  induction n <;> simp_all [replicate_succ, Nat.add_mul, Nat.add_comm]
+
 /-! ### reverse -/
 
 @[simp] theorem length_reverse (as : List α) : (as.reverse).length = as.length := by
@@ -2699,10 +2554,114 @@ theorem flatMap_reverse {β} (l : List α) (f : α → List β) : (l.reverse.fla
 @[simp] theorem reverseAux_eq (as bs : List α) : reverseAux as bs = reverse as ++ bs :=
   reverseAux_eq_append ..
 
+@[simp] theorem reverse_replicate (n) (a : α) : reverse (replicate n a) = replicate n a :=
+  eq_replicate_iff.2
+    ⟨by rw [length_reverse, length_replicate],
+     fun _ h => eq_of_mem_replicate (mem_reverse.1 h)⟩
+
+
+/-! ### foldlM and foldrM -/
+
+@[simp] theorem foldlM_append [Monad m] [LawfulMonad m] (f : β → α → m β) (b) (l l' : List α) :
+    (l ++ l').foldlM f b = l.foldlM f b >>= l'.foldlM f := by
+  induction l generalizing b <;> simp [*]
+
+@[simp] theorem foldrM_cons [Monad m] [LawfulMonad m] (a : α) (l) (f : α → β → m β) (b) :
+    (a :: l).foldrM f b = l.foldrM f b >>= f a := by
+  simp only [foldrM]
+  induction l <;> simp_all
+
+theorem foldl_eq_foldlM (f : β → α → β) (b) (l : List α) :
+    l.foldl f b = l.foldlM (m := Id) f b := by
+  induction l generalizing b <;> simp [*, foldl]
+
+theorem foldr_eq_foldrM (f : α → β → β) (b) (l : List α) :
+    l.foldr f b = l.foldrM (m := Id) f b := by
+  induction l <;> simp [*, foldr]
+
+@[simp] theorem id_run_foldlM (f : β → α → Id β) (b) (l : List α) :
+    Id.run (l.foldlM f b) = l.foldl f b := (foldl_eq_foldlM f b l).symm
+
+@[simp] theorem id_run_foldrM (f : α → β → Id β) (b) (l : List α) :
+    Id.run (l.foldrM f b) = l.foldr f b := (foldr_eq_foldrM f b l).symm
+
+@[simp] theorem foldlM_reverse [Monad m] (l : List α) (f : β → α → m β) (b) :
+    l.reverse.foldlM f b = l.foldrM (fun x y => f y x) b := rfl
+
 @[simp] theorem foldrM_reverse [Monad m] (l : List α) (f : α → β → m β) (b) :
     l.reverse.foldrM f b = l.foldlM (fun x y => f y x) b :=
   (foldlM_reverse ..).symm.trans <| by simp
 
+/-! ### foldl and foldr -/
+
+@[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_cons_nil (l : List α) : l.foldr cons [] = l := by simp
+
+@[deprecated foldr_cons_nil (since := "2024-09-04")] abbrev foldr_self := @foldr_cons_nil
+
+theorem foldl_map (f : β₁ → β₂) (g : α → β₂ → α) (l : List β₁) (init : α) :
+    (l.map f).foldl g init = l.foldl (fun x y => g x (f y)) init := by
+  induction l generalizing init <;> simp [*]
+
+theorem foldr_map (f : α₁ → α₂) (g : α₂ → β → β) (l : List α₁) (init : β) :
+    (l.map f).foldr g init = l.foldr (fun x y => g (f x) y) init := by
+  induction l generalizing init <;> simp [*]
+
+theorem foldl_filterMap (f : α → Option β) (g : γ → β → γ) (l : List α) (init : γ) :
+    (l.filterMap f).foldl g init = l.foldl (fun x y => match f y with | some b => g x b | none => x) init := by
+  induction l generalizing init with
+  | nil => rfl
+  | cons a l ih =>
+    simp only [filterMap_cons, foldl_cons]
+    cases f a <;> simp [ih]
+
+theorem foldr_filterMap (f : α → Option β) (g : β → γ → γ) (l : List α) (init : γ) :
+    (l.filterMap f).foldr g init = l.foldr (fun x y => match f x with | some b => g b y | none => y) init := by
+  induction l generalizing init with
+  | nil => rfl
+  | cons a l ih =>
+    simp only [filterMap_cons, foldr_cons]
+    cases f a <;> simp [ih]
+
+theorem foldl_map' (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α)
+    (h : ∀ x y, f' (g x) (g y) = g (f x y)) :
+    (l.map g).foldl f' (g a) = g (l.foldl f a) := by
+  induction l generalizing a
+  · simp
+  · simp [*, h]
+
+theorem foldr_map' (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α)
+    (h : ∀ x y, f' (g x) (g y) = g (f x y)) :
+    (l.map g).foldr f' (g a) = g (l.foldr f a) := by
+  induction l generalizing a
+  · simp
+  · simp [*, h]
+
+@[simp] theorem foldrM_append [Monad m] [LawfulMonad m] (f : α → β → m β) (b) (l l' : List α) :
+    (l ++ l').foldrM f b = l'.foldrM f b >>= l.foldrM f := by
+  induction l <;> simp [*]
+
+@[simp] theorem foldl_append {β : Type _} (f : β → α → β) (b) (l l' : List α) :
+    (l ++ l').foldl f b = l'.foldl f (l.foldl f b) := by simp [foldl_eq_foldlM]
+
+@[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 foldl_flatten (f : β → α → β) (b : β) (L : List (List α)) :
+    (flatten L).foldl f b = L.foldl (fun b l => l.foldl f b) b := by
+  induction L generalizing b <;> simp_all
+
+theorem foldr_flatten (f : α → β → β) (b : β) (L : List (List α)) :
+    (flatten L).foldr f b = L.foldr (fun l b => l.foldr f b) b := by
+  induction L <;> simp_all
+
 @[simp] theorem foldl_reverse (l : List α) (f : β → α → β) (b) :
     l.reverse.foldl f b = l.foldr (fun x y => f y x) b := by simp [foldl_eq_foldlM, foldr_eq_foldrM]
 
@@ -2716,10 +2675,127 @@ theorem foldl_eq_foldr_reverse (l : List α) (f : β → α → β) (b) :
 theorem foldr_eq_foldl_reverse (l : List α) (f : α → β → β) (b) :
     l.foldr f b = l.reverse.foldl (fun x y => f y x) b := by simp
 
-@[simp] theorem reverse_replicate (n) (a : α) : reverse (replicate n a) = replicate n a :=
-  eq_replicate_iff.2
-    ⟨by rw [length_reverse, length_replicate],
-     fun _ h => eq_of_mem_replicate (mem_reverse.1 h)⟩
+theorem foldl_assoc {op : α → α → α} [ha : Std.Associative op] :
+    ∀ {l : List α} {a₁ a₂}, l.foldl op (op a₁ a₂) = op a₁ (l.foldl op a₂)
+  | [], a₁, a₂ => rfl
+  | a :: l, a₁, a₂ => by
+    simp only [foldl_cons, ha.assoc]
+    rw [foldl_assoc]
+
+theorem foldr_assoc {op : α → α → α} [ha : Std.Associative op] :
+    ∀ {l : List α} {a₁ a₂}, l.foldr op (op a₁ a₂) = op (l.foldr op a₁) a₂
+  | [], a₁, a₂ => rfl
+  | a :: l, a₁, a₂ => by
+    simp only [foldr_cons, ha.assoc]
+    rw [foldr_assoc]
+
+theorem foldl_hom (f : α₁ → α₂) (g₁ : α₁ → β → α₁) (g₂ : α₂ → β → α₂) (l : List β) (init : α₁)
+    (H : ∀ x y, g₂ (f x) y = f (g₁ x y)) : l.foldl g₂ (f init) = f (l.foldl g₁ init) := by
+  induction l generalizing init <;> simp [*, H]
+
+theorem foldr_hom (f : β₁ → β₂) (g₁ : α → β₁ → β₁) (g₂ : α → β₂ → β₂) (l : List α) (init : β₁)
+    (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
+
+/--
+We can prove that two folds over the same list are related (by some arbitrary relation)
+if we know that the initial elements are related and the folding function, for each element of the list,
+preserves the relation.
+-/
+theorem foldl_rel {l : List α} {f g : β → α → β} {a b : β} (r : β → β → Prop)
+    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
+    r (l.foldl (fun acc a => f acc a) a) (l.foldl (fun acc a => g acc a) b) := by
+  induction l generalizing a b with
+  | nil => simp_all
+  | cons a l ih =>
+    simp only [foldl_cons]
+    apply ih
+    · simp_all
+    · exact fun a m c c' h => h' _ (by simp_all) _ _ h
+
+/--
+We can prove that two folds over the same list are related (by some arbitrary relation)
+if we know that the initial elements are related and the folding function, for each element of the list,
+preserves the relation.
+-/
+theorem foldr_rel {l : List α} {f g : α → β → β} {a b : β} (r : β → β → Prop)
+    (h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
+    r (l.foldr (fun a acc => f a acc) a) (l.foldr (fun a acc => g a acc) b) := by
+  induction l generalizing a b with
+  | nil => simp_all
+  | cons a l ih =>
+    simp only [foldr_cons]
+    apply h'
+    · simp
+    · exact ih h fun a m c c' h => h' _ (by simp_all) _ _ h
+
+@[simp] theorem foldl_add_const (l : List α) (a b : Nat) :
+    l.foldl (fun x _ => x + a) b = b + a * l.length := by
+  induction l generalizing b with
+  | nil => simp
+  | cons y l ih =>
+    simp only [foldl_cons, ih, length_cons, Nat.mul_add, Nat.mul_one, Nat.add_assoc,
+      Nat.add_comm a]
+
+@[simp] theorem foldr_add_const (l : List α) (a b : Nat) :
+    l.foldr (fun _ x => x + a) b = b + a * l.length := by
+  induction l generalizing b with
+  | nil => simp
+  | cons y l ih =>
+    simp only [foldr_cons, ih, length_cons, Nat.mul_add, Nat.mul_one, Nat.add_assoc]
+
 
 /-! #### Further results about `getLast` and `getLast?` -/
 

src/Init/Data/List/Lex.lean

@@ -14,8 +14,8 @@ namespace List
 @[simp] theorem lex_lt [LT α] (l₁ l₂ : List α) : Lex (· < ·) l₁ l₂ ↔ l₁ < l₂ := Iff.rfl
 @[simp] theorem not_lex_lt [LT α] (l₁ l₂ : List α) : ¬ Lex (· < ·) l₁ l₂ ↔ l₂ ≤ l₁ := Iff.rfl
 
-theorem not_lt_iff_ge [LT α] (l₁ l₂ : List α) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
-theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) :
+protected theorem not_lt_iff_ge [LT α] (l₁ l₂ : List α) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
+protected theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) :
     ¬ l₁ ≤ l₂ ↔ l₂ < l₁ :=
   Decidable.not_not
 
@@ -260,7 +260,7 @@ protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
   · exfalso
     exact h' h
 
-theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
+protected theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
     [Std.Total (¬ · < · : α → α → Prop)]
@@ -333,7 +333,7 @@ theorem lex_eq_true_iff_exists [BEq α] (lt : α → α → Bool) :
     cases l₂ with
     | nil => simp [lex]
     | cons b l₂ =>
-      simp only [lex_cons_cons, Bool.or_eq_true, Bool.and_eq_true, ih, isEqv, length_cons]
+      simp [lex_cons_cons, Bool.or_eq_true, Bool.and_eq_true, ih, isEqv, length_cons]
       constructor
       · rintro (hab | ⟨hab, ⟨h₁, h₂⟩ | ⟨i, h₁, h₂, w₁, w₂⟩⟩)
         · exact .inr ⟨0, by simp [hab]⟩
@@ -397,7 +397,7 @@ theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α
     cases l₂ with
     | nil => simp [lex]
     | cons b l₂ =>
-      simp only [lex_cons_cons, Bool.or_eq_false_iff, Bool.and_eq_false_imp, ih, isEqv,
+      simp [lex_cons_cons, Bool.or_eq_false_iff, Bool.and_eq_false_imp, ih, isEqv,
         Bool.and_eq_true, length_cons]
       constructor
       · rintro ⟨hab, h⟩
@@ -435,7 +435,7 @@ theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α
               simpa using w₁ (j + 1) (by simpa)
             · simpa using w₂
 
-theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : List α} :
+protected theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : List α} :
     l₁ < l₂ ↔
       (l₁ = l₂.take l₁.length ∧ l₁.length < l₂.length) ∨
         (∃ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length),
@@ -444,7 +444,7 @@ theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Lis
   rw [← lex_eq_true_iff_lt, lex_eq_true_iff_exists]
   simp
 
-theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
+protected theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)] {l₁ l₂ : List α} :
@@ -489,7 +489,7 @@ theorem IsPrefix.le [LT α] [Std.Irrefl (· < · : α → α → Prop)]
   rcases h with ⟨_, rfl⟩
   apply le_append_left
 
-theorem map_lt [LT α] [LT β]
+protected theorem map_lt [LT α] [LT β]
     {l₁ l₂ : List α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
     map f l₁ < map f l₂ := by
   match l₁, l₂, h with
@@ -497,11 +497,11 @@ theorem map_lt [LT α] [LT β]
   | nil, cons b l₂, h => simp
   | cons a l₁, nil, h => simp at h
   | cons a l₁, cons _ l₂, .cons h =>
-    simp [cons_lt_cons_iff, map_lt w (by simpa using h)]
+    simp [cons_lt_cons_iff, List.map_lt w (by simpa using h)]
   | cons a l₁, cons b l₂, .rel h =>
     simp [cons_lt_cons_iff, w, h]
 
-theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
+protected theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
@@ -510,7 +510,7 @@ theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β
     [Std.Antisymm (¬ · < · : β → β → Prop)]
     {l₁ l₂ : List α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ ≤ l₂) :
     map f l₁ ≤ map f l₂ := by
-  rw [le_iff_exists] at h ⊢
+  rw [List.le_iff_exists] at h ⊢
   obtain (h | ⟨i, h₁, h₂, w₁, w₂⟩) := h
   · left
     rw [h]

src/Init/Data/List/Perm.lean

@@ -510,4 +510,18 @@ theorem Perm.eraseP (f : α → Bool) {l₁ l₂ : List α}
     refine (IH₁ H).trans (IH₂ ((p₁.pairwise_iff ?_).1 H))
     exact fun h h₁ h₂ => h h₂ h₁
 
+theorem perm_insertIdx {α} (x : α) (l : List α) {n} (h : n ≤ l.length) :
+    insertIdx n x l ~ x :: l := by
+  induction l generalizing n with
+  | nil =>
+    cases n with
+    | zero => rfl
+    | succ => cases h
+  | cons _ _ ih =>
+    cases n with
+    | zero => simp [insertIdx]
+    | succ =>
+      simp only [insertIdx, modifyTailIdx]
+      refine .trans (.cons _ (ih (Nat.le_of_succ_le_succ h))) (.swap ..)
+
 end List

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

@@ -253,6 +253,10 @@ theorem merge_perm_append : ∀ {xs ys : List α}, merge xs ys le ~ xs ++ ys
     · exact (merge_perm_append.cons y).trans
         ((Perm.swap x y _).trans (perm_middle.symm.cons x))
 
+theorem Perm.merge (s₁ s₂ : α → α → Bool) (hl : l₁ ~ l₂) (hr : r₁ ~ r₂) :
+    merge l₁ r₁ s₁ ~ merge l₂ r₂ s₂ :=
+  Perm.trans (merge_perm_append ..) <| Perm.trans (Perm.append hl hr) <| Perm.symm (merge_perm_append ..)
+
 /-! ### mergeSort -/
 
 @[simp] theorem mergeSort_nil : [].mergeSort r = [] := by rw [List.mergeSort]

src/Init/Data/List/ToArray.lean

@@ -46,7 +46,7 @@ theorem toArray_cons (a : α) (l : List α) : (a :: l).toArray = #[a] ++ l.toArr
 @[simp] theorem isEmpty_toArray (l : List α) : l.toArray.isEmpty = l.isEmpty := by
   cases l <;> simp [Array.isEmpty]
 
-@[simp] theorem toArray_singleton (a : α) : (List.singleton a).toArray = singleton a := rfl
+@[simp] theorem toArray_singleton (a : α) : (List.singleton a).toArray = Array.singleton a := rfl
 
 @[simp] theorem back!_toArray [Inhabited α] (l : List α) : l.toArray.back! = l.getLast! := by
   simp only [back!, size_toArray, Array.get!_eq_getElem!, getElem!_toArray, getLast!_eq_getElem!]
@@ -143,6 +143,9 @@ theorem forM_toArray [Monad m] (l : List α) (f : α → m PUnit) :
   subst h
   rw [foldl_toList]
 
+@[simp] theorem sum_toArray [Add α] [Zero α] (l : List α) : l.toArray.sum = l.sum := by
+  simp [Array.sum, List.sum]
+
 @[simp] theorem append_toArray (l₁ l₂ : List α) :
     l₁.toArray ++ l₂.toArray = (l₁ ++ l₂).toArray := by
   apply ext'
@@ -394,4 +397,24 @@ theorem takeWhile_go_toArray (p : α → Bool) (l : List α) (i : Nat) :
 @[deprecated toArray_replicate (since := "2024-12-13")]
 abbrev _root_.Array.mkArray_eq_toArray_replicate := @toArray_replicate
 
+@[simp] theorem flatMap_empty {β} (f : α → Array β) : (#[] : Array α).flatMap f = #[] := rfl
+
+theorem flatMap_toArray_cons {β} (f : α → Array β) (a : α) (as : List α) :
+    (a :: as).toArray.flatMap f = f a ++ as.toArray.flatMap f := by
+  simp [Array.flatMap]
+  suffices ∀ cs, List.foldl (fun bs a => bs ++ f a) (f a ++ cs) as =
+      f a ++ List.foldl (fun bs a => bs ++ f a) cs as by
+    erw [empty_append] -- Why doesn't this work via `simp`?
+    simpa using this #[]
+  intro cs
+  induction as generalizing cs <;> simp_all
+
+@[simp] theorem flatMap_toArray {β} (f : α → Array β) (as : List α) :
+    as.toArray.flatMap f = (as.flatMap (fun a => (f a).toList)).toArray := by
+  induction as with
+  | nil => simp
+  | cons a as ih =>
+    apply ext'
+    simp [ih, flatMap_toArray_cons]
+
 end List

src/Init/Data/List/Zip.lean

@@ -203,11 +203,11 @@ theorem zipWith_eq_append_iff {f : α → β → γ} {l₁ : List α} {l₂ : Li
     cases l₂ with
     | nil =>
       constructor
-      · simp only [zipWith_nil_right, nil_eq, append_eq_nil, exists_and_left, and_imp]
+      · simp only [zipWith_nil_right, nil_eq, append_eq_nil_iff, exists_and_left, and_imp]
         rintro rfl  rfl
         exact ⟨[], x₁ :: l₁, [], by simp⟩
       · rintro ⟨w, x, y, z, h₁, _, h₃, rfl, rfl⟩
-        simp only [nil_eq, append_eq_nil] at h₃
+        simp only [nil_eq, append_eq_nil_iff] at h₃
         obtain ⟨rfl, rfl⟩ := h₃
         simp
     | cons x₂ l₂ =>
@@ -259,7 +259,7 @@ theorem zip_map (f : α → γ) (g : β → δ) :
   | [], _ => rfl
   | _, [] => by simp only [map, zip_nil_right]
   | _ :: _, _ :: _ => by
-    simp only [map, zip_cons_cons, zip_map, Prod.map]; constructor
+    simp only [map, zip_cons_cons, zip_map, Prod.map]; try constructor -- TODO: remove try constructor after update stage0
 
 theorem zip_map_left (f : α → γ) (l₁ : List α) (l₂ : List β) :
     zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]

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

@@ -49,4 +49,17 @@ theorem lt_div_mul_self (h : 0 < k) (w : k ≤ x) : x - k < x / k * k := by
   have : x % k < k := mod_lt x h
   omega
 
+theorem div_pos (hba : b ≤ a) (hb : 0 < b) : 0 < a / b := by
+  cases b
+  · contradiction
+  · simp [Nat.pos_iff_ne_zero, div_eq_zero_iff_lt, hba]
+
+theorem div_le_div_left (hcb : c ≤ b) (hc : 0 < c) : a / b ≤ a / c :=
+  (Nat.le_div_iff_mul_le hc).2 <|
+    Nat.le_trans (Nat.mul_le_mul_left _ hcb) (Nat.div_mul_le_self a b)
+
+theorem div_add_le_right {z : Nat} (h : 0 < z) (x y : Nat) :
+    x / (y + z) ≤ x / z :=
+  div_le_div_left (Nat.le_add_left z y) h
+
 end Nat

src/Init/Data/Nat/Lemmas.lean

@@ -622,6 +622,14 @@ protected theorem pos_of_mul_pos_right {a b : Nat} (h : 0 < a * b) : 0 < a := by
     0 < a * b ↔ 0 < a :=
   ⟨Nat.pos_of_mul_pos_right, fun w => Nat.mul_pos w h⟩
 
+protected theorem pos_of_lt_mul_left {a b c : Nat} (h : a < b * c) : 0 < c := by
+  replace h : 0 < b * c := by omega
+  exact Nat.pos_of_mul_pos_left h
+
+protected theorem pos_of_lt_mul_right {a b c : Nat} (h : a < b * c) : 0 < b := by
+  replace h : 0 < b * c := by omega
+  exact Nat.pos_of_mul_pos_right h
+
 /-! ### div/mod -/
 
 theorem mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 1 :=

src/Init/Data/Option/Lemmas.lean

@@ -208,6 +208,15 @@ 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' ..
 
+theorem map_inj_right {f : α → β} {o o' : Option α} (w : ∀ x y, f x = f y → x = y) :
+    o.map f = o'.map f ↔ o = o' := by
+  cases o with
+  | none => cases o' <;> simp
+  | some a =>
+    cases o' with
+    | none => simp
+    | some a' => simpa using ⟨fun h => w _ _ h, fun h => congrArg f h⟩
+
 @[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
@@ -629,6 +638,15 @@ theorem pbind_eq_some_iff {o : Option α} {f : (a : α) → a ∈ o → Option
     · rintro ⟨h, rfl⟩
       rfl
 
+@[simp]
+theorem pmap_eq_map (p : α → Prop) (f : α → β) (o : Option α) (H) :
+    @pmap _ _ p (fun a _ => f a) o H = Option.map f o := by
+  cases o <;> simp
+
+theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (o H) :
+    Option.map g (pmap f o H) = pmap (fun a h => g (f a h)) o H := by
+  cases o <;> simp
+
 /-! ### pelim -/
 
 @[simp] theorem pelim_none : pelim none b f = b := rfl

src/Init/Data/UInt/Basic.lean

@@ -159,6 +159,8 @@ def UInt32.xor (a b : UInt32) : UInt32 := ⟨a.toBitVec ^^^ b.toBitVec⟩
 def UInt32.shiftLeft (a b : UInt32) : UInt32 := ⟨a.toBitVec <<< (mod b 32).toBitVec⟩
 @[extern "lean_uint32_shift_right"]
 def UInt32.shiftRight (a b : UInt32) : UInt32 := ⟨a.toBitVec >>> (mod b 32).toBitVec⟩
+def UInt32.lt (a b : UInt32) : Prop := a.toBitVec < b.toBitVec
+def UInt32.le (a b : UInt32) : Prop := a.toBitVec ≤ b.toBitVec
 
 instance : Add UInt32       := ⟨UInt32.add⟩
 instance : Sub UInt32       := ⟨UInt32.sub⟩
@@ -169,6 +171,8 @@ set_option linter.deprecated false in
 instance : HMod UInt32 Nat UInt32 := ⟨UInt32.modn⟩
 
 instance : Div UInt32       := ⟨UInt32.div⟩
+instance : LT UInt32        := ⟨UInt32.lt⟩
+instance : LE UInt32        := ⟨UInt32.le⟩
 
 @[extern "lean_uint32_complement"]
 def UInt32.complement (a : UInt32) : UInt32 := ⟨~~~a.toBitVec⟩

src/Init/Data/UInt/Bitwise.lean

@@ -13,11 +13,17 @@ macro "declare_bitwise_uint_theorems" typeName:ident bits:term:arg : command =>
 `(
 namespace $typeName
 
-@[simp] protected theorem toBitVec_and (a b : $typeName) : (a &&& b).toBitVec = a.toBitVec &&& b.toBitVec := rfl
-@[simp] protected theorem toBitVec_or (a b : $typeName) : (a ||| b).toBitVec = a.toBitVec ||| b.toBitVec := rfl
-@[simp] protected theorem toBitVec_xor (a b : $typeName) : (a ^^^ b).toBitVec = a.toBitVec ^^^ b.toBitVec := rfl
-@[simp] protected theorem toBitVec_shiftLeft (a b : $typeName) : (a <<< b).toBitVec = a.toBitVec <<< (b.toBitVec % $bits) := rfl
-@[simp] protected theorem toBitVec_shiftRight (a b : $typeName) : (a >>> b).toBitVec = a.toBitVec >>> (b.toBitVec % $bits) := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_add {a b : $typeName} : (a + b).toBitVec = a.toBitVec + b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_sub {a b : $typeName} : (a - b).toBitVec = a.toBitVec - b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_mul {a b : $typeName} : (a * b).toBitVec = a.toBitVec * b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_div {a b : $typeName} : (a / b).toBitVec = a.toBitVec / b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_mod {a b : $typeName} : (a % b).toBitVec = a.toBitVec % b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_not {a : $typeName} : (~~~a).toBitVec = ~~~a.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_and (a b : $typeName) : (a &&& b).toBitVec = a.toBitVec &&& b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_or (a b : $typeName) : (a ||| b).toBitVec = a.toBitVec ||| b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_xor (a b : $typeName) : (a ^^^ b).toBitVec = a.toBitVec ^^^ b.toBitVec := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_shiftLeft (a b : $typeName) : (a <<< b).toBitVec = a.toBitVec <<< (b.toBitVec % $bits) := rfl
+@[simp, int_toBitVec] protected theorem toBitVec_shiftRight (a b : $typeName) : (a >>> b).toBitVec = a.toBitVec >>> (b.toBitVec % $bits) := rfl
 
 @[simp] protected theorem toNat_and (a b : $typeName) : (a &&& b).toNat = a.toNat &&& b.toNat := by simp [toNat]
 @[simp] protected theorem toNat_or (a b : $typeName) : (a ||| b).toNat = a.toNat ||| b.toNat := by simp [toNat]
@@ -37,3 +43,31 @@ declare_bitwise_uint_theorems UInt16 16
 declare_bitwise_uint_theorems UInt32 32
 declare_bitwise_uint_theorems UInt64 64
 declare_bitwise_uint_theorems USize System.Platform.numBits
+
+@[simp, int_toBitVec]
+theorem Bool.toBitVec_toUInt8 {b : Bool} :
+    b.toUInt8.toBitVec = (BitVec.ofBool b).setWidth 8 := by
+  cases b <;> simp [toUInt8]
+
+@[simp, int_toBitVec]
+theorem Bool.toBitVec_toUInt16 {b : Bool} :
+    b.toUInt16.toBitVec = (BitVec.ofBool b).setWidth 16 := by
+  cases b <;> simp [toUInt16]
+
+@[simp, int_toBitVec]
+theorem Bool.toBitVec_toUInt32 {b : Bool} :
+    b.toUInt32.toBitVec = (BitVec.ofBool b).setWidth 32 := by
+  cases b <;> simp [toUInt32]
+
+@[simp, int_toBitVec]
+theorem Bool.toBitVec_toUInt64 {b : Bool} :
+    b.toUInt64.toBitVec = (BitVec.ofBool b).setWidth 64 := by
+  cases b <;> simp [toUInt64]
+
+@[simp, int_toBitVec]
+theorem Bool.toBitVec_toUSize {b : Bool} :
+    b.toUSize.toBitVec = (BitVec.ofBool b).setWidth System.Platform.numBits := by
+  cases b
+  · simp [toUSize]
+  · apply BitVec.eq_of_toNat_eq
+    simp [toUSize]

src/Init/Data/UInt/Lemmas.lean

@@ -41,9 +41,9 @@ macro "declare_uint_theorems" typeName:ident bits:term:arg : command => do
   theorem toNat_ofNat_of_lt {n : Nat} (h : n < size) : (ofNat n).toNat = n := by
     rw [toNat, toBitVec_eq_of_lt h]
 
-  theorem le_def {a b : $typeName} : a ≤ b ↔ a.toBitVec ≤ b.toBitVec := .rfl
+  @[int_toBitVec] theorem le_def {a b : $typeName} : a ≤ b ↔ a.toBitVec ≤ b.toBitVec := .rfl
 
-  theorem lt_def {a b : $typeName} : a < b ↔ a.toBitVec < b.toBitVec := .rfl
+  @[int_toBitVec] theorem lt_def {a b : $typeName} : a < b ↔ a.toBitVec < b.toBitVec := .rfl
 
   theorem le_iff_toNat_le {a b : $typeName} : a ≤ b ↔ a.toNat ≤ b.toNat := .rfl
 
@@ -74,6 +74,11 @@ macro "declare_uint_theorems" typeName:ident bits:term:arg : command => do
   protected theorem toBitVec_inj {a b : $typeName} : a.toBitVec = b.toBitVec ↔ a = b :=
     Iff.intro eq_of_toBitVec_eq toBitVec_eq_of_eq
 
+  open $typeName (eq_of_toBitVec_eq toBitVec_eq_of_eq) in
+  @[int_toBitVec]
+  protected theorem eq_iff_toBitVec_eq {a b : $typeName} : a = b ↔ a.toBitVec = b.toBitVec :=
+    Iff.intro toBitVec_eq_of_eq eq_of_toBitVec_eq
+
   open $typeName (eq_of_toBitVec_eq) in
   protected theorem eq_of_val_eq {a b : $typeName} (h : a.val = b.val) : a = b := by
     rcases a with ⟨⟨_⟩⟩; rcases b with ⟨⟨_⟩⟩; simp_all [val]
@@ -82,10 +87,19 @@ macro "declare_uint_theorems" typeName:ident bits:term:arg : command => do
   protected theorem val_inj {a b : $typeName} : a.val = b.val ↔ a = b :=
     Iff.intro eq_of_val_eq (congrArg val)
 
+  open $typeName (eq_of_toBitVec_eq) in
+  protected theorem toBitVec_ne_of_ne {a b : $typeName} (h : a ≠ b) : a.toBitVec ≠ b.toBitVec :=
+    fun h' => h (eq_of_toBitVec_eq h')
+
   open $typeName (toBitVec_eq_of_eq) in
   protected theorem ne_of_toBitVec_ne {a b : $typeName} (h : a.toBitVec ≠ b.toBitVec) : a ≠ b :=
     fun h' => absurd (toBitVec_eq_of_eq h') h
 
+  open $typeName (ne_of_toBitVec_ne toBitVec_ne_of_ne) in
+  @[int_toBitVec]
+  protected theorem ne_iff_toBitVec_ne {a b : $typeName} : a ≠ b ↔ a.toBitVec ≠ b.toBitVec :=
+    Iff.intro toBitVec_ne_of_ne ne_of_toBitVec_ne
+
   open $typeName (ne_of_toBitVec_ne) in
   protected theorem ne_of_lt {a b : $typeName} (h : a < b) : a ≠ b := by
     apply ne_of_toBitVec_ne
@@ -159,7 +173,7 @@ macro "declare_uint_theorems" typeName:ident bits:term:arg : command => do
   @[simp]
   theorem val_ofNat (n : Nat) : val (no_index (OfNat.ofNat n)) = OfNat.ofNat n := rfl
 
-  @[simp]
+  @[simp, int_toBitVec]
   theorem toBitVec_ofNat (n : Nat) : toBitVec (no_index (OfNat.ofNat n)) = BitVec.ofNat _ n := rfl
 
   @[simp]

src/Init/Data/Vector/Basic.lean

@@ -103,7 +103,7 @@ of bounds.
 @[inline] def head [NeZero n] (v : Vector α n) := v[0]'(Nat.pos_of_neZero n)
 
 /-- Push an element `x` to the end of a vector. -/
-@[inline] def push (x : α) (v : Vector α n)  : Vector α (n + 1) :=
+@[inline] def push (v : Vector α n) (x : α) : Vector α (n + 1) :=
   ⟨v.toArray.push x, by simp⟩
 
 /-- Remove the last element of a vector. -/
@@ -136,6 +136,18 @@ This will perform the update destructively provided that the vector has a refere
 @[inline] def set! (v : Vector α n) (i : Nat) (x : α) : Vector α n :=
   ⟨v.toArray.set! i x, by simp⟩
 
+@[inline] def foldlM [Monad m] (f : β → α → m β) (b : β) (v : Vector α n) : m β :=
+  v.toArray.foldlM f b
+
+@[inline] def foldrM [Monad m] (f : α → β → m β) (b : β) (v : Vector α n) : m β :=
+  v.toArray.foldrM f b
+
+@[inline] def foldl (f : β → α → β) (b : β) (v : Vector α n) : β :=
+  v.toArray.foldl f b
+
+@[inline] def foldr (f : α → β → β) (b : β) (v : Vector α n) : β :=
+  v.toArray.foldr f b
+
 /-- Append two vectors. -/
 @[inline] def append (v : Vector α n) (w : Vector α m) : Vector α (n + m) :=
   ⟨v.toArray ++ w.toArray, by simp⟩
@@ -158,6 +170,13 @@ result is empty. If `stop` is greater than the size of the vector, the size is u
 @[inline] def map (f : α → β) (v : Vector α n) : Vector β n :=
   ⟨v.toArray.map f, by simp⟩
 
+@[inline] def flatten (v : Vector (Vector α n) m) : Vector α (m * n) :=
+  ⟨(v.toArray.map Vector.toArray).flatten,
+    by rcases v; simp_all [Function.comp_def, Array.map_const']⟩
+
+@[inline] def flatMap (v : Vector α n) (f : α → Vector β m) : Vector β (n * m) :=
+  ⟨v.toArray.flatMap fun a => (f a).toArray, by simp [Array.map_const']⟩
+
 /-- Maps corresponding elements of two vectors of equal size using the function `f`. -/
 @[inline] def zipWith (a : Vector α n) (b : Vector β n) (f : α → β → φ) : Vector φ n :=
   ⟨Array.zipWith a.toArray b.toArray f, by simp⟩

src/Init/Data/Vector/Lemmas.lean

@@ -1,10 +1,11 @@
 /-
 Copyright (c) 2024 Shreyas Srinivas. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Shreyas Srinivas, Francois Dorais
+Authors: Shreyas Srinivas, Francois Dorais, Kim Morrison
 -/
 prelude
 import Init.Data.Vector.Basic
+import Init.Data.Array.Attach
 
 /-!
 ## Vectors
@@ -27,6 +28,9 @@ namespace Vector
 
 theorem toArray_mk (a : Array α) (h : a.size = n) : (Vector.mk a h).toArray = a := rfl
 
+@[simp] theorem mk_toArray (v : Vector α n) : mk v.toArray v.2 = v := by
+  rfl
+
 @[simp] theorem getElem_mk {data : Array α} {size : data.size = n} {i : Nat} (h : i < n) :
     (Vector.mk data size)[i] = data[i] := rfl
 
@@ -66,6 +70,18 @@ theorem toArray_mk (a : Array α) (h : a.size = n) : (Vector.mk a h).toArray = a
 @[simp] theorem back?_mk (a : Array α) (h : a.size = n) :
     (Vector.mk a h).back? = a.back? := rfl
 
+@[simp] theorem foldlM_mk [Monad m] (f : β → α → m β) (b : β) (a : Array α) (h : a.size = n) :
+    (Vector.mk a h).foldlM f b = a.foldlM f b := rfl
+
+@[simp] theorem foldrM_mk [Monad m] (f : α → β → m β) (b : β) (a : Array α) (h : a.size = n) :
+    (Vector.mk a h).foldrM f b = a.foldrM f b := rfl
+
+@[simp] theorem foldl_mk (f : β → α → β) (b : β) (a : Array α) (h : a.size = n) :
+    (Vector.mk a h).foldl f b = a.foldl f b := rfl
+
+@[simp] theorem foldr_mk (f : α → β → β) (b : β) (a : Array α) (h : a.size = n) :
+    (Vector.mk a h).foldr f b = a.foldr f b := rfl
+
 @[simp] theorem drop_mk (a : Array α) (h : a.size = n) (m) :
     (Vector.mk a h).drop m = Vector.mk (a.extract m a.size) (by simp [h]) := rfl
 
@@ -141,6 +157,14 @@ theorem toArray_mk (a : Array α) (h : a.size = n) : (Vector.mk a h).toArray = a
 @[simp] theorem all_mk (p : α → Bool) (a : Array α) (h : a.size = n) :
     (Vector.mk a h).all p = a.all p := rfl
 
+@[simp] theorem eq_mk : v = Vector.mk a h ↔ v.toArray = a := by
+  cases v
+  simp
+
+@[simp] theorem mk_eq : Vector.mk a h = v ↔ a = v.toArray := by
+  cases v
+  simp
+
 /-! ### toArray lemmas -/
 
 @[simp] theorem getElem_toArray {α n} (xs : Vector α n) (i : Nat) (h : i < xs.toArray.size) :
@@ -451,7 +475,7 @@ theorem singleton_inj : #v[a] = #v[b] ↔ a = b := by
 theorem mkVector_succ : mkVector (n + 1) a = (mkVector n a).push a := by
   simp [mkVector, Array.mkArray_succ]
 
-theorem mkVector_inj : mkVector n a = mkVector n b ↔ n = 0 ∨ a = b := by
+@[simp] theorem mkVector_inj : mkVector n a = mkVector n b ↔ n = 0 ∨ a = b := by
   simp [← toArray_inj, toArray_mkVector, Array.mkArray_inj]
 
 /-! ## L[i] and L[i]? -/
@@ -673,6 +697,24 @@ theorem forall_getElem {l : Vector α n} {p : α → Prop} :
   rcases l with ⟨l, rfl⟩
   simp [Array.forall_getElem]
 
+
+/-! ### cast -/
+
+@[simp] theorem getElem_cast (a : Vector α n) (h : n = m) (i : Nat) (hi : i < m) :
+    (a.cast h)[i] = a[i] := by
+  cases a
+  simp
+
+@[simp] theorem getElem?_cast {l : Vector α n} {m : Nat} {w : n = m} {i : Nat} :
+    (l.cast w)[i]? = l[i]? := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem mem_cast {a : α} {l : Vector α n} {m : Nat} {w : n = m} :
+    a ∈ l.cast w ↔ a ∈ l := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
 /-! ### Decidability of bounded quantifiers -/
 
 instance {xs : Vector α n} {p : α → Prop} [DecidablePred p] :
@@ -1023,11 +1065,12 @@ theorem mem_setIfInBounds (v : Vector α n) (i : Nat) (hi : i < n) (a : α) :
   cases l₂
   simp
 
-/-! Content below this point has not yet been aligned with `List` and `Array`. -/
+/-! ### map -/
 
-@[simp] theorem getElem_ofFn {α n} (f : Fin n → α) (i : Nat) (h : i < n) :
-    (Vector.ofFn f)[i] = f ⟨i, by simpa using h⟩ := by
-  simp [ofFn]
+@[simp] theorem getElem_map (f : α → β) (a : Vector α n) (i : Nat) (hi : i < n) :
+    (a.map f)[i] = f a[i] := by
+  cases a
+  simp
 
 /-- The empty vector maps to the empty vector. -/
 @[simp]
@@ -1035,6 +1078,680 @@ theorem map_empty (f : α → β) : map f #v[] = #v[] := by
   rw [map, mk.injEq]
   exact Array.map_empty f
 
+@[simp] theorem map_push {f : α → β} {as : Vector α n} {x : α} :
+    (as.push x).map f = (as.map f).push (f x) := by
+  cases as
+  simp
+
+@[simp] theorem map_id_fun : map (n := n) (id : α → α) = id := by
+  funext l
+  induction l <;> simp_all
+
+/-- `map_id_fun'` differs from `map_id_fun` by representing the identity function as a lambda, rather than `id`. -/
+@[simp] theorem map_id_fun' : map (n := n) (fun (a : α) => a) = id := map_id_fun
+
+-- This is not a `@[simp]` lemma because `map_id_fun` will apply.
+theorem map_id (l : Vector α n) : map (id : α → α) l = l := by
+  cases l <;> simp_all
+
+/-- `map_id'` differs from `map_id` by representing the identity function as a lambda, rather than `id`. -/
+-- This is not a `@[simp]` lemma because `map_id_fun'` will apply.
+theorem map_id' (l : Vector α n) : map (fun (a : α) => a) l = l := map_id l
+
+/-- Variant of `map_id`, with a side condition that the function is pointwise the identity. -/
+theorem map_id'' {f : α → α} (h : ∀ x, f x = x) (l : Vector α n) : map f l = l := by
+  simp [show f = id from funext h]
+
+theorem map_singleton (f : α → β) (a : α) : map f #v[a] = #v[f a] := rfl
+
+@[simp] theorem mem_map {f : α → β} {l : Vector α n} : b ∈ l.map f ↔ ∃ a, a ∈ l ∧ f a = b := by
+  cases l
+  simp
+
+theorem exists_of_mem_map (h : b ∈ map f l) : ∃ a, a ∈ l ∧ f a = b := mem_map.1 h
+
+theorem mem_map_of_mem (f : α → β) (h : a ∈ l) : f a ∈ map f l := mem_map.2 ⟨_, h, rfl⟩
+
+theorem forall_mem_map {f : α → β} {l : Vector α n} {P : β → Prop} :
+    (∀ (i) (_ : i ∈ l.map f), P i) ↔ ∀ (j) (_ : j ∈ l), P (f j) := by
+  simp
+
+@[simp] theorem map_inj_left {f g : α → β} : map f l = map g l ↔ ∀ a ∈ l, f a = g a := by
+  cases l <;> simp_all
+
+theorem map_inj_right {f : α → β} (w : ∀ x y, f x = f y → x = y) : map f l = map f l' ↔ l = l' := by
+  cases l
+  cases l'
+  simp [Array.map_inj_right w]
+
+theorem map_congr_left (h : ∀ a ∈ l, f a = g a) : map f l = map g l :=
+  map_inj_left.2 h
+
+theorem map_inj [NeZero n] : map (n := n) f = map g ↔ f = g := by
+  constructor
+  · intro h
+    ext a
+    replace h := congrFun h (mkVector n a)
+    simp only [mkVector, map_mk, mk.injEq, Array.map_inj_left, Array.mem_mkArray,  and_imp,
+      forall_eq_apply_imp_iff] at h
+    exact h (NeZero.ne n)
+  · intro h; subst h; rfl
+
+theorem map_eq_push_iff {f : α → β} {l : Vector α (n + 1)} {l₂ : Vector β n} {b : β} :
+    map f l = l₂.push b ↔ ∃ l₁ a, l = l₁.push a ∧ map f l₁ = l₂ ∧ f a = b := by
+  rcases l with ⟨l, h⟩
+  rcases l₂ with ⟨l₂, rfl⟩
+  simp only [map_mk, push_mk, mk.injEq, Array.map_eq_push_iff]
+  constructor
+  · rintro ⟨l₁, a, rfl, rfl, rfl⟩
+    refine ⟨⟨l₁, by simp⟩, a, by simp⟩
+  · rintro ⟨l₁, a, h₁, h₂, rfl⟩
+    refine ⟨l₁.toArray, a, by simp_all⟩
+
+@[simp] theorem map_eq_singleton_iff {f : α → β} {l : Vector α 1} {b : β} :
+    map f l = #v[b] ↔ ∃ a, l = #v[a] ∧ f a = b := by
+  cases l
+  simp
+
+theorem map_eq_map_iff {f g : α → β} {l : Vector α n} :
+    map f l = map g l ↔ ∀ a ∈ l, f a = g a := by
+  cases l <;> simp_all
+
+theorem map_eq_iff {f : α → β} {l : Vector α n} {l' : Vector β n} :
+    map f l = l' ↔ ∀ i (h : i < n), l'[i] = f l[i] := by
+  rcases l with ⟨l, rfl⟩
+  rcases l' with ⟨l', h'⟩
+  simp only [map_mk, eq_mk, Array.map_eq_iff, getElem_mk]
+  constructor
+  · intro w i h
+    simpa [h, h'] using w i
+  · intro w i
+    if h : i < l.size then
+      simpa [h, h'] using w i h
+    else
+      rw [getElem?_neg, getElem?_neg, Option.map_none'] <;> omega
+
+@[simp] theorem map_set {f : α → β} {l : Vector α n} {i : Nat} {h : i < n} {a : α} :
+    (l.set i a).map f = (l.map f).set i (f a) (by simpa using h) := by
+  cases l
+  simp
+
+@[simp] theorem map_setIfInBounds {f : α → β} {l : Vector α n} {i : Nat} {a : α} :
+    (l.setIfInBounds i a).map f = (l.map f).setIfInBounds i (f a) := by
+  cases l
+  simp
+
+@[simp] theorem map_pop {f : α → β} {l : Vector α n} : l.pop.map f = (l.map f).pop := by
+  cases l
+  simp
+
+@[simp] theorem back?_map {f : α → β} {l : Vector α n} : (l.map f).back? = l.back?.map f := by
+  cases l
+  simp
+
+@[simp] theorem map_map {f : α → β} {g : β → γ} {as : Vector α n} :
+    (as.map f).map g = as.map (g ∘ f) := by
+  cases as
+  simp
+
+/--
+Use this as `induction ass using vector₂_induction` on a hypothesis of the form `ass : Vector (Vector α n) m`.
+The hypothesis `ass` will be replaced with a hypothesis `ass : Array (Array α)`
+along with additional hypotheses `h₁ : ass.size = m` and `h₂ : ∀ xs ∈ ass, xs.size = n`.
+Appearances of the original `ass` in the goal will be replaced with
+`Vector.mk (xss.attach.map (fun ⟨xs, m⟩ => Vector.mk xs ⋯)) ⋯`.
+-/
+-- We can't use `@[cases_eliminator]` here as
+-- `Lean.Meta.getCustomEliminator?` only looks at the top-level constant.
+theorem vector₂_induction (P : Vector (Vector α n) m → Prop)
+    (of : ∀ (xss : Array (Array α)) (h₁ : xss.size = m) (h₂ : ∀ xs ∈ xss, xs.size = n),
+      P (mk (xss.attach.map (fun ⟨xs, m⟩ => mk xs (h₂ xs m))) (by simpa using h₁)))
+    (ass : Vector (Vector α n) m) : P ass := by
+  specialize of (ass.map toArray).toArray (by simp) (by simp)
+  simpa [Array.map_attach, Array.pmap_map] using of
+
+/--
+Use this as `induction ass using vector₃_induction` on a hypothesis of the form `ass : Vector (Vector (Vector α n) m) k`.
+The hypothesis `ass` will be replaced with a hypothesis `ass : Array (Array (Array α))`
+along with additional hypotheses `h₁ : ass.size = k`, `h₂ : ∀ xs ∈ ass, xs.size = m`,
+and `h₃ : ∀ xs ∈ ass, ∀ x ∈ xs, x.size = n`.
+Appearances of the original `ass` in the goal will be replaced with
+`Vector.mk (xss.attach.map (fun ⟨xs, m⟩ => Vector.mk (xs.attach.map (fun ⟨x, m'⟩ => Vector.mk x ⋯)) ⋯)) ⋯`.
+-/
+theorem vector₃_induction (P : Vector (Vector (Vector α n) m) k → Prop)
+    (of : ∀ (xss : Array (Array (Array α))) (h₁ : xss.size = k) (h₂ : ∀ xs ∈ xss, xs.size = m)
+      (h₃ : ∀ xs ∈ xss, ∀ x ∈ xs, x.size = n),
+      P (mk (xss.attach.map (fun ⟨xs, m⟩ =>
+        mk (xs.attach.map (fun ⟨x, m'⟩ =>
+          mk x (h₃ xs m x m'))) (by simpa using h₂ xs m))) (by simpa using h₁)))
+    (ass : Vector (Vector (Vector α n) m) k) : P ass := by
+  specialize of (ass.map (fun as => (as.map toArray).toArray)).toArray (by simp) (by simp) (by simp)
+  simpa [Array.map_attach, Array.pmap_map] using of
+
+/-! ### singleton -/
+
+@[simp] theorem singleton_def (v : α) : Vector.singleton v = #v[v] := rfl
+
+/-! ### append -/
+
+@[simp] theorem append_push {as : Vector α n} {bs : Vector α m} {a : α} :
+    as ++ bs.push a = (as ++ bs).push a := by
+  cases as
+  cases bs
+  simp
+
+theorem singleton_eq_toVector_singleton (a : α) : #v[a] = #[a].toVector := rfl
+
+@[simp] theorem mem_append {a : α} {s : Vector α n} {t : Vector α m} :
+    a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t := by
+  cases s
+  cases t
+  simp
+
+theorem mem_append_left {a : α} {s : Vector α n} {t : Vector α m} (h : a ∈ s) : a ∈ s ++ t :=
+  mem_append.2 (Or.inl h)
+
+theorem mem_append_right {a : α} {s : Vector α n} {t : Vector α m} (h : a ∈ t) : a ∈ s ++ t :=
+  mem_append.2 (Or.inr h)
+
+theorem not_mem_append {a : α} {s : Vector α n} {t : Vector α m} (h₁ : a ∉ s) (h₂ : a ∉ t) :
+    a ∉ s ++ t :=
+  mt mem_append.1 $ not_or.mpr ⟨h₁, h₂⟩
+
+/--
+See also `eq_push_append_of_mem`, which proves a stronger version
+in which the initial array must not contain the element.
+-/
+theorem append_of_mem {a : α} {l : Vector α n} (h : a ∈ l) :
+    ∃ (m k : Nat) (w : m + 1 + k = n) (s : Vector α m) (t : Vector α k),
+      l = (s.push a ++ t).cast w := by
+  rcases l with ⟨l, rfl⟩
+  obtain ⟨s, t, rfl⟩ := Array.append_of_mem (by simpa using h)
+  refine ⟨_, _, by simp, s.toVector, t.toVector, by simp_all⟩
+
+theorem mem_iff_append {a : α} {l : Vector α n} :
+    a ∈ l ↔ ∃ (m k : Nat) (w : m + 1 + k = n) (s : Vector α m) (t : Vector α k),
+      l = (s.push a ++ t).cast w :=
+  ⟨append_of_mem, by rintro ⟨m, k, rfl, s, t, rfl⟩; simp⟩
+
+theorem forall_mem_append {p : α → Prop} {l₁ : Vector α n} {l₂ : Vector α m} :
+    (∀ (x) (_ : x ∈ l₁ ++ l₂), p x) ↔ (∀ (x) (_ : x ∈ l₁), p x) ∧ (∀ (x) (_ : x ∈ l₂), p x) := by
+  simp only [mem_append, or_imp, forall_and]
+
+theorem empty_append (as : Vector α n) : (#v[] : Vector α 0) ++ as = as.cast (by omega) := by
+  rcases as with ⟨as, rfl⟩
+  simp
+
+theorem append_empty (as : Vector α n) : as ++ (#v[] : Vector α 0) = as := by
+  rw [← toArray_inj, toArray_append, Array.append_empty]
+
+theorem getElem_append (a : Vector α n) (b : Vector α m) (i : Nat) (hi : i < n + m) :
+    (a ++ b)[i] = if h : i < n then a[i] else b[i - n] := by
+  rcases a with ⟨a, rfl⟩
+  rcases b with ⟨b, rfl⟩
+  simp [Array.getElem_append, hi]
+
+theorem getElem_append_left {a : Vector α n} {b : Vector α m} {i : Nat} (hi : i < n) :
+    (a ++ b)[i] = a[i] := by simp [getElem_append, hi]
+
+theorem getElem_append_right {a : Vector α n} {b : Vector α m} {i : Nat} (h : i < n + m) (hi : n ≤ i) :
+    (a ++ b)[i] = b[i - n] := by
+  rw [getElem_append, dif_neg (by omega)]
+
+theorem getElem?_append_left {as : Vector α n} {bs : Vector α m} {i : Nat} (hn : i < n) :
+    (as ++ bs)[i]? = as[i]? := by
+  have hn' : i < n + m := by omega
+  simp_all [getElem?_eq_getElem, getElem_append]
+
+theorem getElem?_append_right {as : Vector α n} {bs : Vector α m} {i : Nat} (h : n ≤ i) :
+    (as ++ bs)[i]? = bs[i - n]? := by
+  rcases as with ⟨as, rfl⟩
+  rcases bs with ⟨bs, rfl⟩
+  simp [Array.getElem?_append_right, h]
+
+theorem getElem?_append {as : Vector α n} {bs : Vector α m} {i : Nat} :
+    (as ++ bs)[i]? = if i < n then as[i]? else bs[i - n]? := by
+  split <;> rename_i h
+  · exact getElem?_append_left h
+  · exact getElem?_append_right (by simpa using h)
+
+/-- Variant of `getElem_append_left` useful for rewriting from the small array to the big array. -/
+theorem getElem_append_left' (l₁ : Vector α m) {l₂ : Vector α n} {i : Nat} (hi : i < m) :
+    l₁[i] = (l₁ ++ l₂)[i] := by
+  rw [getElem_append_left] <;> simp
+
+/-- Variant of `getElem_append_right` useful for rewriting from the small array to the big array. -/
+theorem getElem_append_right' (l₁ : Vector α m) {l₂ : Vector α n} {i : Nat} (hi : i < n) :
+    l₂[i] = (l₁ ++ l₂)[i + m] := by
+  rw [getElem_append_right] <;> simp [*, Nat.le_add_left]
+
+theorem getElem_of_append {l : Vector α n} {l₁ : Vector α m} {l₂ : Vector α k}
+    (w : m + 1 + k = n) (eq : l = (l₁.push a ++ l₂).cast w) :
+    l[m] = a := Option.some.inj <| by
+  rw [← getElem?_eq_getElem, eq, getElem?_cast, getElem?_append_left (by simp)]
+  simp
+
+@[simp 1100] theorem append_singleton {a : α} {as : Vector α n} : as ++ #v[a] = as.push a := by
+  cases as
+  simp
+
+theorem append_inj {s₁ s₂ : Vector α n} {t₁ t₂ : Vector α m} (h : s₁ ++ t₁ = s₂ ++ t₂) :
+    s₁ = s₂ ∧ t₁ = t₂ := by
+  rcases s₁ with ⟨s₁, rfl⟩
+  rcases s₂ with ⟨s₂, hs⟩
+  rcases t₁ with ⟨t₁, rfl⟩
+  rcases t₂ with ⟨t₂, ht⟩
+  simpa using Array.append_inj (by simpa using h) (by omega)
+
+theorem append_inj_right {s₁ s₂ : Vector α n} {t₁ t₂ : Vector α m}
+    (h : s₁ ++ t₁ = s₂ ++ t₂) : t₁ = t₂ :=
+  (append_inj h).right
+
+theorem append_inj_left {s₁ s₂ : Vector α n} {t₁ t₂ : Vector α m}
+    (h : s₁ ++ t₁ = s₂ ++ t₂) : s₁ = s₂ :=
+  (append_inj h).left
+
+theorem append_right_inj {t₁ t₂ : Vector α m} (s : Vector α n) : s ++ t₁ = s ++ t₂ ↔ t₁ = t₂ :=
+  ⟨fun h => append_inj_right h, congrArg _⟩
+
+theorem append_left_inj {s₁ s₂ : Vector α n} (t : Vector α m) : s₁ ++ t = s₂ ++ t ↔ s₁ = s₂ :=
+  ⟨fun h => append_inj_left h, congrArg (· ++ _)⟩
+
+theorem append_eq_append_iff {a : Vector α n} {b : Vector α m} {c : Vector α k} {d : Vector α l}
+    (w : k + l = n + m) :
+    a ++ b = (c ++ d).cast w ↔
+      if h : n ≤ k then
+        ∃ a' : Vector α (k - n), c = (a ++ a').cast (by omega) ∧ b = (a' ++ d).cast (by omega)
+      else
+        ∃ c' : Vector α (n - k), a = (c ++ c').cast (by omega) ∧ d = (c' ++ b).cast (by omega) := by
+  rcases a with ⟨a, rfl⟩
+  rcases b with ⟨b, rfl⟩
+  rcases c with ⟨c, rfl⟩
+  rcases d with ⟨d, rfl⟩
+  simp only [mk_append_mk, Array.append_eq_append_iff, mk_eq, toArray_cast]
+  constructor
+  · rintro (⟨a', rfl, rfl⟩ | ⟨c', rfl, rfl⟩)
+    · rw [dif_pos (by simp)]
+      exact ⟨a'.toVector.cast (by simp; omega), by simp⟩
+    · split <;> rename_i h
+      · have hc : c'.size = 0 := by simp at h; omega
+        simp at hc
+        exact ⟨#v[].cast (by simp; omega), by simp_all⟩
+      · exact ⟨c'.toVector.cast (by simp; omega), by simp⟩
+  · split <;> rename_i h
+    · rintro ⟨a', hc, rfl⟩
+      left
+      refine ⟨a'.toArray, hc, rfl⟩
+    · rintro ⟨c', ha, rfl⟩
+      right
+      refine ⟨c'.toArray, ha, rfl⟩
+
+theorem set_append {s : Vector α n} {t : Vector α m} {i : Nat} {x : α} (h : i < n + m) :
+    (s ++ t).set i x =
+      if h' : i < n then
+        s.set i x ++ t
+      else
+        s ++ t.set (i - n) x := by
+  rcases s with ⟨s, rfl⟩
+  rcases t with ⟨t, rfl⟩
+  simp only [mk_append_mk, set_mk, Array.set_append]
+  split <;> simp
+
+@[simp] theorem set_append_left {s : Vector α n} {t : Vector α m} {i : Nat} {x : α} (h : i < n) :
+    (s ++ t).set i x = s.set i x ++ t := by
+  simp [set_append, h]
+
+@[simp] theorem set_append_right {s : Vector α n} {t : Vector α m} {i : Nat} {x : α}
+    (h' : i < n + m) (h : n ≤ i) :
+    (s ++ t).set i x = s ++ t.set (i - n) x := by
+  rw [set_append, dif_neg (by omega)]
+
+theorem setIfInBounds_append {s : Vector α n} {t : Vector α m} {i : Nat} {x : α} :
+    (s ++ t).setIfInBounds i x =
+      if i < n then
+        s.setIfInBounds i x ++ t
+      else
+        s ++ t.setIfInBounds (i - n) x := by
+  rcases s with ⟨s, rfl⟩
+  rcases t with ⟨t, rfl⟩
+  simp only [mk_append_mk, setIfInBounds_mk, Array.setIfInBounds_append]
+  split <;> simp
+
+@[simp] theorem setIfInBounds_append_left {s : Vector α n} {t : Vector α m} {i : Nat} {x : α} (h : i < n) :
+    (s ++ t).setIfInBounds i x = s.setIfInBounds i x ++ t := by
+  simp [setIfInBounds_append, h]
+
+@[simp] theorem setIfInBounds_append_right {s : Vector α n} {t : Vector α m} {i : Nat} {x : α}
+    (h : n ≤ i) :
+    (s ++ t).setIfInBounds i x = s ++ t.setIfInBounds (i - n) x := by
+  rw [setIfInBounds_append, if_neg (by omega)]
+
+@[simp] theorem map_append (f : α → β) (l₁ : Vector α n) (l₂ : Vector α m) :
+    map f (l₁ ++ l₂) = map f l₁ ++ map f l₂ := by
+  rcases l₁ with ⟨l₁, rfl⟩
+  rcases l₂ with ⟨l₂, rfl⟩
+  simp
+
+theorem map_eq_append_iff {f : α → β} :
+    map f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
+  rcases l with ⟨l, h⟩
+  rcases L₁ with ⟨L₁, rfl⟩
+  rcases L₂ with ⟨L₂, rfl⟩
+  simp only [map_mk, mk_append_mk, eq_mk, Array.map_eq_append_iff, mk_eq, toArray_append,
+    toArray_map]
+  constructor
+  · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
+    exact ⟨l₁.toVector.cast (by simp), l₂.toVector.cast (by simp), by simp⟩
+  · rintro ⟨⟨l₁⟩, ⟨l₂⟩, rfl, h₁, h₂⟩
+    exact ⟨l₁, l₂, by simp_all⟩
+
+theorem append_eq_map_iff {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_iff]
+
+/-! ### flatten -/
+
+@[simp] theorem flatten_mk (L : Array (Vector α n)) (h : L.size = m) :
+    (mk L h).flatten =
+      mk (L.map toArray).flatten (by simp [Function.comp_def, Array.map_const', h]) := by
+  simp [flatten]
+
+@[simp] theorem getElem_flatten (l : Vector (Vector β m) n) (i : Nat) (hi : i < n * m) :
+    l.flatten[i] =
+      haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+      haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+      l[i / m][i % m] := by
+  rcases l with ⟨⟨l⟩, rfl⟩
+  simp only [flatten_mk, List.map_toArray, getElem_mk, List.getElem_toArray, Array.flatten_toArray]
+  induction l generalizing i with
+  | nil => simp at hi
+  | cons a l ih =>
+    simp only [List.map_cons, List.map_map, List.flatten_cons]
+    by_cases h : i < m
+    · rw [List.getElem_append_left (by simpa)]
+      have h₁ : i / m = 0 := Nat.div_eq_of_lt h
+      have h₂ : i % m = i := Nat.mod_eq_of_lt h
+      simp [h₁, h₂]
+    · have h₁ : a.toList.length ≤ i := by simp; omega
+      rw [List.getElem_append_right h₁]
+      simp only [Array.length_toList, size_toArray]
+      specialize ih (i - m) (by simp_all [Nat.add_one_mul]; omega)
+      have h₂ : i / m = (i - m) / m + 1 := by
+        conv => lhs; rw [show i = i - m + m by omega]
+        rw [Nat.add_div_right]
+        exact Nat.pos_of_lt_mul_left hi
+      simp only [Array.length_toList, size_toArray] at h₁
+      have h₃ : (i - m) % m = i % m := (Nat.mod_eq_sub_mod h₁).symm
+      simp_all
+
+theorem getElem?_flatten (l : Vector (Vector β m) n) (i : Nat) :
+    l.flatten[i]? =
+      if hi : i < n * m then
+        haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+        haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+        some l[i / m][i % m]
+      else
+        none := by
+  simp [getElem?_def]
+
+@[simp] theorem flatten_singleton (l : Vector α n) : #v[l].flatten = l.cast (by simp) := by
+  simp [flatten]
+
+theorem mem_flatten {L : Vector (Vector α n) m} : a ∈ L.flatten ↔ ∃ l, l ∈ L ∧ a ∈ l := by
+  rcases L with ⟨L, rfl⟩
+  simp [Array.mem_flatten]
+  constructor
+  · rintro ⟨_, ⟨l, h₁, rfl⟩, h₂⟩
+    exact ⟨l, h₁, by simpa using h₂⟩
+  · rintro ⟨l, h₁, h₂⟩
+    exact ⟨l.toArray, ⟨l, h₁, rfl⟩, by simpa using h₂⟩
+
+theorem exists_of_mem_flatten : a ∈ flatten L → ∃ l, l ∈ L ∧ a ∈ l := mem_flatten.1
+
+theorem mem_flatten_of_mem (lL : l ∈ L) (al : a ∈ l) : a ∈ flatten L := mem_flatten.2 ⟨l, lL, al⟩
+
+theorem forall_mem_flatten {p : α → Prop} {L : Vector (Vector α n) m} :
+    (∀ (x) (_ : x ∈ flatten L), p x) ↔ ∀ (l) (_ : l ∈ L) (x) (_ : x ∈ l), p x := by
+  simp only [mem_flatten, forall_exists_index, and_imp]
+  constructor <;> (intros; solve_by_elim)
+
+@[simp] theorem map_flatten (f : α → β) (L : Vector (Vector α n) m) :
+    (flatten L).map f = (map (map f) L).flatten := by
+  induction L using vector₂_induction with
+  | of xss h₁ h₂ => simp
+
+@[simp] theorem flatten_append (L₁ : Vector (Vector α n) m₁) (L₂ : Vector (Vector α n) m₂) :
+    flatten (L₁ ++ L₂) = (flatten L₁ ++ flatten L₂).cast (by simp [Nat.add_mul]) := by
+  induction L₁ using vector₂_induction
+  induction L₂ using vector₂_induction
+  simp
+
+theorem flatten_push (L : Vector (Vector α n) m) (l : Vector α n) :
+    flatten (L.push l) = (flatten L ++ l).cast (by simp [Nat.add_mul]) := by
+  induction L using vector₂_induction
+  rcases l with ⟨l⟩
+  simp [Array.flatten_push]
+
+theorem flatten_flatten {L : Vector (Vector (Vector α n) m) k} :
+    flatten (flatten L) = (flatten (map flatten L)).cast (by simp [Nat.mul_assoc]) := by
+  induction L using vector₃_induction with
+  | of xss h₁ h₂ h₃ =>
+    -- simp [Array.flatten_flatten] -- FIXME: `simp` produces a bad proof here!
+    simp [Array.map_attach, Array.flatten_flatten, Array.map_pmap]
+
+/-- Two vectors of constant length vectors are equal iff their flattens coincide. -/
+theorem eq_iff_flatten_eq {L L' : Vector (Vector α n) m} :
+    L = L' ↔ L.flatten = L'.flatten := by
+  induction L using vector₂_induction with | of L h₁ h₂ =>
+  induction L' using vector₂_induction with | of L' h₁' h₂' =>
+  simp only [eq_mk, flatten_mk, Array.map_map, Function.comp_apply, Array.map_subtype,
+    Array.unattach_attach, Array.map_id_fun', id_eq]
+  constructor
+  · intro h
+    suffices L = L' by simp_all
+    apply Array.ext_getElem?
+    intro i
+    replace h := congrArg (fun x => x[i]?.map (fun x => x.toArray)) h
+    simpa [Option.map_pmap] using h
+  · intro h
+    have w : L.map Array.size = L'.map Array.size := by
+      ext i h h'
+      · simp_all
+      · simp only [Array.getElem_map]
+        rw [h₂ _ (by simp), h₂' _ (by simp)]
+    have := Array.eq_iff_flatten_eq.mpr ⟨h, w⟩
+    subst this
+    rfl
+
+
+/-! ### flatMap -/
+
+@[simp] theorem flatMap_mk (l : Array α) (h : l.size = m) (f : α → Vector β n) :
+    (mk l h).flatMap f =
+      mk (l.flatMap (fun a => (f a).toArray)) (by simp [Array.map_const', h]) := by
+  simp [flatMap]
+
+@[simp] theorem flatMap_toArray (l : Vector α n) (f : α → Vector β m) :
+    l.toArray.flatMap (fun a => (f a).toArray) = (l.flatMap f).toArray := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+theorem flatMap_def (l : Vector α n) (f : α → Vector β m) : l.flatMap f = flatten (map f l) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.flatMap_def, Function.comp_def]
+
+@[simp] theorem getElem_flatMap (l : Vector α n) (f : α → Vector β m) (i : Nat) (hi : i < n * m) :
+    (l.flatMap f)[i] =
+      haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+      haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+      (f (l[i / m]))[i % m] := by
+  rw [flatMap_def, getElem_flatten, getElem_map]
+
+theorem getElem?_flatMap (l : Vector α n) (f : α → Vector β m) (i : Nat) :
+    (l.flatMap f)[i]? =
+      if hi : i < n * m then
+        haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+        haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+        some ((f (l[i / m]))[i % m])
+      else
+        none := by
+  simp [getElem?_def]
+
+@[simp] theorem flatMap_id (l : Vector (Vector α m) n) : l.flatMap id = l.flatten := by simp [flatMap_def]
+
+@[simp] theorem flatMap_id' (l : Vector (Vector α m) n) : l.flatMap (fun a => a) = l.flatten := by simp [flatMap_def]
+
+@[simp] theorem mem_flatMap {f : α → Vector β m} {b} {l : Vector α n} : b ∈ l.flatMap f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by
+  simp [flatMap_def, mem_flatten]
+  exact ⟨fun ⟨_, ⟨a, h₁, rfl⟩, h₂⟩ => ⟨a, h₁, h₂⟩, fun ⟨a, h₁, h₂⟩ => ⟨_, ⟨a, h₁, rfl⟩, h₂⟩⟩
+
+theorem exists_of_mem_flatMap {b : β} {l : Vector α n} {f : α → Vector β m} :
+    b ∈ l.flatMap f → ∃ a, a ∈ l ∧ b ∈ f a := mem_flatMap.1
+
+theorem mem_flatMap_of_mem {b : β} {l : Vector α n} {f : α → Vector β m} {a} (al : a ∈ l) (h : b ∈ f a) :
+    b ∈ l.flatMap f := mem_flatMap.2 ⟨a, al, h⟩
+
+theorem forall_mem_flatMap {p : β → Prop} {l : Vector α n} {f : α → Vector β m} :
+    (∀ (x) (_ : x ∈ l.flatMap f), p x) ↔ ∀ (a) (_ : a ∈ l) (b) (_ : b ∈ f a), p b := by
+  simp only [mem_flatMap, forall_exists_index, and_imp]
+  constructor <;> (intros; solve_by_elim)
+
+theorem flatMap_singleton (f : α → Vector β m) (x : α) : #v[x].flatMap f = (f x).cast (by simp) := by
+  simp [flatMap_def]
+
+@[simp] theorem flatMap_singleton' (l : Vector α n) : (l.flatMap fun x => #v[x]) = l.cast (by simp) := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem flatMap_append (xs ys : Vector α n) (f : α → Vector β m) :
+    (xs ++ ys).flatMap f = (xs.flatMap f ++ ys.flatMap f).cast (by simp [Nat.add_mul]) := by
+  rcases xs with ⟨xs⟩
+  rcases ys with ⟨ys⟩
+  simp [flatMap_def, flatten_append]
+
+theorem flatMap_assoc {α β} (l : Vector α n) (f : α → Vector β m) (g : β → Vector γ k) :
+    (l.flatMap f).flatMap g = (l.flatMap fun x => (f x).flatMap g).cast (by simp [Nat.mul_assoc]) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.flatMap_assoc]
+
+theorem map_flatMap (f : β → γ) (g : α → Vector β m) (l : Vector α n) :
+     (l.flatMap g).map f = l.flatMap fun a => (g a).map f := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.map_flatMap]
+
+theorem flatMap_map (f : α → β) (g : β → Vector γ k) (l : Vector α n) :
+     (map f l).flatMap g = l.flatMap (fun a => g (f a)) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.flatMap_map]
+
+theorem map_eq_flatMap {α β} (f : α → β) (l : Vector α n) :
+    map f l = (l.flatMap fun x => #v[f x]).cast (by simp) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.map_eq_flatMap]
+
+/-! ### mkVector -/
+
+@[simp] theorem mkVector_one : mkVector 1 a = #v[a] := rfl
+
+/-- Variant of `mkVector_succ` that prepends `a` at the beginning of the vector. -/
+theorem mkVector_succ' : mkVector (n + 1) a = (#v[a] ++ mkVector n a).cast (by omega) := by
+  rw [← toArray_inj]
+  simp [Array.mkArray_succ']
+
+@[simp] theorem mem_mkVector {a b : α} {n} : b ∈ mkVector n a ↔ n ≠ 0 ∧ b = a := by
+  unfold mkVector
+  simp
+
+theorem eq_of_mem_mkVector {a b : α} {n} (h : b ∈ mkVector n a) : b = a := (mem_mkVector.1 h).2
+
+theorem forall_mem_mkVector {p : α → Prop} {a : α} {n} :
+    (∀ b, b ∈ mkVector n a → p b) ↔ n = 0 ∨ p a := by
+  cases n <;> simp [mem_mkVector]
+
+@[simp] theorem getElem_mkVector (a : α) (n i : Nat) (h : i < n) : (mkVector n a)[i] = a := by
+  simp [mkVector]
+
+theorem getElem?_mkVector (a : α) (n i : Nat) : (mkVector n a)[i]? = if i < n then some a else none := by
+  simp [getElem?_def]
+
+@[simp] theorem getElem?_mkVector_of_lt {n : Nat} {m : Nat} (h : m < n) : (mkVector n a)[m]? = some a := by
+  simp [getElem?_mkVector, h]
+
+theorem eq_mkVector_of_mem {a : α} {l : Vector α n} (h : ∀ (b) (_ : b ∈ l), b = a) : l = mkVector n a := by
+  rw [← toArray_inj]
+  simpa using Array.eq_mkArray_of_mem (l := l.toArray) (by simpa using h)
+
+theorem eq_mkVector_iff {a : α} {n} {l : Vector α n} :
+    l = mkVector n a ↔ ∀ (b) (_ : b ∈ l), b = a := by
+  rw [← toArray_inj]
+  simpa using Array.eq_mkArray_iff (l := l.toArray) (n := n)
+
+theorem map_eq_mkVector_iff {l : Vector α n} {f : α → β} {b : β} :
+    l.map f = mkVector n b ↔ ∀ x ∈ l, f x = b := by
+  simp [eq_mkVector_iff]
+
+@[simp] theorem map_const (l : Vector α n) (b : β) : map (Function.const α b) l = mkVector n b :=
+  map_eq_mkVector_iff.mpr fun _ _ => rfl
+
+@[simp] theorem map_const_fun (x : β) : map (n := n) (Function.const α x) = fun _ => mkVector n x := by
+  funext l
+  simp
+
+/-- Variant of `map_const` using a lambda rather than `Function.const`. -/
+-- This can not be a `@[simp]` lemma because it would fire on every `List.map`.
+theorem map_const' (l : Vector α n) (b : β) : map (fun _ => b) l = mkVector n b :=
+  map_const l b
+
+@[simp] theorem set_mkVector_self : (mkVector n a).set i a h = mkVector n a := by
+  rw [← toArray_inj]
+  simp
+
+@[simp] theorem setIfInBounds_mkVector_self : (mkVector n a).setIfInBounds i a = mkVector n a := by
+  rw [← toArray_inj]
+  simp
+
+@[simp] theorem mkVector_append_mkVector : mkVector n a ++ mkVector m a = mkVector (n + m) a := by
+  rw [← toArray_inj]
+  simp
+
+theorem append_eq_mkVector_iff {l₁ : Vector α n} {l₂ : Vector α m} {a : α} :
+    l₁ ++ l₂ = mkVector (n + m) a ↔ l₁ = mkVector n a ∧ l₂ = mkVector m a := by
+  simp [← toArray_inj, Array.append_eq_mkArray_iff]
+
+theorem mkVector_eq_append_iff {l₁ : Vector α n} {l₂ : Vector α m} {a : α} :
+    mkVector (n + m) a = l₁ ++ l₂ ↔ l₁ = mkVector n a ∧ l₂ = mkVector m a := by
+  rw [eq_comm, append_eq_mkVector_iff]
+
+@[simp] theorem map_mkVector : (mkVector n a).map f = mkVector n (f a) := by
+  rw [← toArray_inj]
+  simp
+
+
+@[simp] theorem flatten_mkVector_empty : (mkVector n (#v[] : Vector α 0)).flatten = #v[] := by
+  rw [← toArray_inj]
+  simp
+
+@[simp] theorem flatten_mkVector_singleton : (mkVector n #v[a]).flatten = (mkVector n a).cast (by simp) := by
+  ext i h
+  simp [h]
+
+@[simp] theorem flatten_mkVector_mkVector : (mkVector n (mkVector m a)).flatten = mkVector (n * m) a := by
+  ext i h
+  simp [h]
+
+theorem flatMap_mkArray {β} (f : α → Vector β m) : (mkVector n a).flatMap f = (mkVector n (f a)).flatten := by
+  ext i h
+  simp [h]
+
+@[simp] theorem sum_mkArray_nat (n : Nat) (a : Nat) : (mkVector n a).sum = n * a := by
+  simp [toArray_mkVector]
+
+/-! Content below this point has not yet been aligned with `List` and `Array`. -/
+
+@[simp] theorem getElem_ofFn {α n} (f : Fin n → α) (i : Nat) (h : i < n) :
+    (Vector.ofFn f)[i] = f ⟨i, by simpa using h⟩ := by
+  simp [ofFn]
+
 @[simp] theorem getElem_push_last {v : Vector α n} {x : α} : (v.push x)[n] = x := by
   rcases v with ⟨data, rfl⟩
   simp
@@ -1059,28 +1776,6 @@ defeq issues in the implicit size argument.
     subst h
     simp [pop, back, back!, ← Array.eq_push_pop_back!_of_size_ne_zero]
 
-/-! ### append -/
-
-theorem getElem_append (a : Vector α n) (b : Vector α m) (i : Nat) (hi : i < n + m) :
-    (a ++ b)[i] = if h : i < n then a[i] else b[i - n] := by
-  rcases a with ⟨a, rfl⟩
-  rcases b with ⟨b, rfl⟩
-  simp [Array.getElem_append, hi]
-
-theorem getElem_append_left {a : Vector α n} {b : Vector α m} {i : Nat} (hi : i < n) :
-    (a ++ b)[i] = a[i] := by simp [getElem_append, hi]
-
-theorem getElem_append_right {a : Vector α n} {b : Vector α m} {i : Nat} (h : i < n + m) (hi : n ≤ i) :
-    (a ++ b)[i] = b[i - n] := by
-  rw [getElem_append, dif_neg (by omega)]
-
-/-! ### cast -/
-
-@[simp] theorem getElem_cast (a : Vector α n) (h : n = m) (i : Nat) (hi : i < m) :
-    (a.cast h)[i] = a[i] := by
-  cases a
-  simp
-
 /-! ### extract -/
 
 @[simp] theorem getElem_extract (a : Vector α n) (start stop) (i : Nat) (hi : i < min stop n - start) :
@@ -1088,13 +1783,6 @@ theorem getElem_append_right {a : Vector α n} {b : Vector α m} {i : Nat} (h :
   cases a
   simp
 
-/-! ### map -/
-
-@[simp] theorem getElem_map (f : α → β) (a : Vector α n) (i : Nat) (hi : i < n) :
-    (a.map f)[i] = f a[i] := by
-  cases a
-  simp
-
 /-! ### zipWith -/
 
 @[simp] theorem getElem_zipWith (f : α → β → γ) (a : Vector α n) (b : Vector β n) (i : Nat)
@@ -1103,6 +1791,37 @@ theorem getElem_append_right {a : Vector α n} {b : Vector α m} {i : Nat} (h :
   cases b
   simp
 
+/-! ### foldlM and foldrM -/
+
+@[simp] theorem foldlM_append [Monad m] [LawfulMonad m] (f : β → α → m β) (b) (l : Vector α n) (l' : Vector α n') :
+    (l ++ l').foldlM f b = l.foldlM f b >>= l'.foldlM f := by
+  cases l
+  cases l'
+  simp
+
+@[simp] theorem foldrM_push [Monad m] (f : α → β → m β) (init : β) (l : Vector α n) (a : α) :
+    (l.push a).foldrM f init = f a init >>= l.foldrM f := by
+  cases l
+  simp
+
+theorem foldl_eq_foldlM (f : β → α → β) (b) (l : Vector α n) :
+    l.foldl f b = l.foldlM (m := Id) f b := by
+  cases l
+  simp [Array.foldl_eq_foldlM]
+
+theorem foldr_eq_foldrM (f : α → β → β) (b) (l : Vector α n) :
+    l.foldr f b = l.foldrM (m := Id) f b := by
+  cases l
+  simp [Array.foldr_eq_foldrM]
+
+@[simp] theorem id_run_foldlM (f : β → α → Id β) (b) (l : Vector α n) :
+    Id.run (l.foldlM f b) = l.foldl f b := (foldl_eq_foldlM f b l).symm
+
+@[simp] theorem id_run_foldrM (f : α → β → Id β) (b) (l : Vector α n) :
+    Id.run (l.foldrM f b) = l.foldr f b := (foldr_eq_foldrM f b l).symm
+
+/-! ### foldl and foldr -/
+
 /-! ### take -/
 
 @[simp] theorem take_size (a : Vector α n) : a.take n = a.cast (by simp) := by

src/Init/Data/Vector/Lex.lean

@@ -19,6 +19,11 @@ namespace Vector
 @[simp] theorem lt_toList [LT α] (l₁ l₂ : Vector α n) : l₁.toList < l₂.toList ↔ l₁ < l₂ := Iff.rfl
 @[simp] theorem le_toList [LT α] (l₁ l₂ : Vector α n) : l₁.toList ≤ l₂.toList ↔ l₁ ≤ l₂ := Iff.rfl
 
+protected theorem not_lt_iff_ge [LT α] (l₁ l₂ : Vector α n) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
+protected theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : Vector α n) :
+    ¬ l₁ ≤ l₂ ↔ l₂ < l₁ :=
+  Decidable.not_not
+
 @[simp] theorem mk_lt_mk [LT α] :
     Vector.mk (α := α) (n := n) data₁ size₁ < Vector.mk data₂ size₂ ↔ data₁ < data₂ := Iff.rfl
 
@@ -133,7 +138,7 @@ protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
     {l₁ l₂ : Vector α n} (h : l₁ < l₂) : l₁ ≤ l₂ :=
   Array.le_of_lt h
 
-theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
+protected theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]
     [Std.Total (¬ · < · : α → α → Prop)]
@@ -200,14 +205,14 @@ theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α
   rcases l₂ with ⟨l₂, n₂⟩
   simp_all [Array.lex_eq_false_iff_exists, n₂]
 
-theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Vector α n} :
+protected theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Vector α n} :
     l₁ < l₂ ↔
       (∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → l₁[j] = l₂[j]) ∧ l₁[i] < l₂[i]) := by
   cases l₁
   cases l₂
   simp_all [Array.lt_iff_exists]
 
-theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
+protected theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)] {l₁ l₂ : Vector α n} :
@@ -230,12 +235,12 @@ theorem append_left_le [DecidableEq α] [LT α] [DecidableLT α]
     l₁ ++ l₂ ≤ l₁ ++ l₃ := by
   simpa using Array.append_left_le h
 
-theorem map_lt [LT α] [LT β]
+protected theorem map_lt [LT α] [LT β]
     {l₁ l₂ : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
     map f l₁ < map f l₂ := by
   simpa using Array.map_lt w h
 
-theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
+protected theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
     [Std.Irrefl (· < · : α → α → Prop)]
     [Std.Asymm (· < · : α → α → Prop)]
     [Std.Antisymm (¬ · < · : α → α → Prop)]

src/Init/Grind.lean

@@ -8,3 +8,7 @@ import Init.Grind.Norm
 import Init.Grind.Tactics
 import Init.Grind.Lemmas
 import Init.Grind.Cases
+import Init.Grind.Propagator
+import Init.Grind.Util
+import Init.Grind.Offset
+import Init.Grind.PP

src/Init/Grind/Lemmas.lean

@@ -5,10 +5,105 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.Core
+import Init.SimpLemmas
+import Init.Classical
+import Init.ByCases
+import Init.Grind.Util
 
 namespace Lean.Grind
 
+theorem rfl_true : true = true :=
+  rfl
+
 theorem intro_with_eq (p p' q : Prop) (he : p = p') (h : p' → q) : p → q :=
   fun hp => h (he.mp hp)
 
+/-! And -/
+
+theorem and_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a ∧ b) = b := by simp [h]
+theorem and_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a ∧ b) = a := by simp [h]
+theorem and_eq_of_eq_false_left {a b : Prop} (h : a = False) : (a ∧ b) = False := by simp [h]
+theorem and_eq_of_eq_false_right {a b : Prop} (h : b = False) : (a ∧ b) = False := by simp [h]
+
+theorem eq_true_of_and_eq_true_left {a b : Prop} (h : (a ∧ b) = True) : a = True := by simp_all
+theorem eq_true_of_and_eq_true_right {a b : Prop} (h : (a ∧ b) = True) : b = True := by simp_all
+
+theorem or_of_and_eq_false {a b : Prop} (h : (a ∧ b) = False) : (¬a ∨ ¬b) := by
+  by_cases a <;> by_cases b <;> simp_all
+
+/-! Or -/
+
+theorem or_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a ∨ b) = True := by simp [h]
+theorem or_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a ∨ b) = True := by simp [h]
+theorem or_eq_of_eq_false_left {a b : Prop} (h : a = False) : (a ∨ b) = b := by simp [h]
+theorem or_eq_of_eq_false_right {a b : Prop} (h : b = False) : (a ∨ b) = a := by simp [h]
+
+theorem eq_false_of_or_eq_false_left {a b : Prop} (h : (a ∨ b) = False) : a = False := by simp_all
+theorem eq_false_of_or_eq_false_right {a b : Prop} (h : (a ∨ b) = False) : b = False := by simp_all
+
+/-! Implies -/
+
+theorem imp_eq_of_eq_false_left {a b : Prop} (h : a = False) : (a → b) = True := by simp [h]
+theorem imp_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a → b) = True := by simp [h]
+theorem imp_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a → b) = b := by simp [h]
+
+theorem eq_true_of_imp_eq_false {a b : Prop} (h : (a → b) = False) : a = True := by simp_all
+theorem eq_false_of_imp_eq_false {a b : Prop} (h : (a → b) = False) : b = False := by simp_all
+
+/-! Not -/
+
+theorem not_eq_of_eq_true {a : Prop} (h : a = True) : (Not a) = False := by simp [h]
+theorem not_eq_of_eq_false {a : Prop} (h : a = False) : (Not a) = True := by simp [h]
+
+theorem eq_false_of_not_eq_true {a : Prop} (h : (Not a) = True) : a = False := by simp_all
+theorem eq_true_of_not_eq_false {a : Prop} (h : (Not a) = False) : a = True := by simp_all
+
+theorem false_of_not_eq_self {a : Prop} (h : (Not a) = a) : False := by
+  by_cases a <;> simp_all
+
+/-! Eq -/
+
+theorem eq_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a = b) = b := by simp [h]
+theorem eq_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a = b) = a := by simp [h]
+
+theorem eq_congr  {α : Sort u} {a₁ b₁ a₂ b₂ : α} (h₁ : a₁ = a₂) (h₂ : b₁ = b₂) : (a₁ = b₁) = (a₂ = b₂) := by simp [*]
+theorem eq_congr' {α : Sort u} {a₁ b₁ a₂ b₂ : α} (h₁ : a₁ = b₂) (h₂ : b₁ = a₂) : (a₁ = b₁) = (a₂ = b₂) := by rw [h₁, h₂, Eq.comm (a := a₂)]
+
+/- The following two helper theorems are used to case-split `a = b` representing `iff`. -/
+theorem of_eq_eq_true {a b : Prop} (h : (a = b) = True) : (¬a ∨ b) ∧ (¬b ∨ a) := by
+  by_cases a <;> by_cases b <;> simp_all
+theorem of_eq_eq_false {a b : Prop} (h : (a = b) = False) : (¬a ∨ ¬b) ∧ (b ∨ a) := by
+  by_cases a <;> by_cases b <;> simp_all
+
+/-! Forall -/
+
+theorem forall_propagator (p : Prop) (q : p → Prop) (q' : Prop) (h₁ : p = True) (h₂ : q (of_eq_true h₁) = q') : (∀ hp : p, q hp) = q' := by
+  apply propext; apply Iff.intro
+  · intro h'; exact Eq.mp h₂ (h' (of_eq_true h₁))
+  · intro h'; intros; exact Eq.mpr h₂ h'
+
+theorem of_forall_eq_false (α : Sort u) (p : α → Prop) (h : (∀ x : α, p x) = False) : ∃ x : α, ¬ p x := by simp_all
+
+/-! dite -/
+
+theorem dite_cond_eq_true' {α : Sort u} {c : Prop} {_ : Decidable c} {a : c → α} {b : ¬ c → α} {r : α} (h₁ : c = True) (h₂ : a (of_eq_true h₁) = r) : (dite c a b) = r := by simp [h₁, h₂]
+theorem dite_cond_eq_false' {α : Sort u} {c : Prop} {_ : Decidable c} {a : c → α} {b : ¬ c → α} {r : α} (h₁ : c = False) (h₂ : b (of_eq_false h₁) = r) : (dite c a b) = r := by simp [h₁, h₂]
+
+/-! Casts -/
+
+theorem eqRec_heq.{u_1, u_2} {α : Sort u_2} {a : α}
+        {motive : (x : α) → a = x → Sort u_1} (v : motive a (Eq.refl a)) {b : α} (h : a = b)
+        : HEq (@Eq.rec α a motive v b h) v := by
+ subst h; rfl
+
+theorem eqRecOn_heq.{u_1, u_2} {α : Sort u_2} {a : α}
+        {motive : (x : α) → a = x → Sort u_1} {b : α} (h : a = b) (v : motive a (Eq.refl a))
+        : HEq (@Eq.recOn α a motive b h v) v := by
+ subst h; rfl
+
+theorem eqNDRec_heq.{u_1, u_2} {α : Sort u_2} {a : α}
+        {motive : α → Sort u_1} (v : motive a) {b : α} (h : a = b)
+        : HEq (@Eq.ndrec α a motive v b h) v := by
+ subst h; rfl
+
 end Lean.Grind

src/Init/Grind/Norm.lean

@@ -5,6 +5,7 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.SimpLemmas
+import Init.PropLemmas
 import Init.Classical
 import Init.ByCases
 
@@ -40,10 +41,17 @@ attribute [grind_norm] not_true
 -- False
 attribute [grind_norm] not_false_eq_true
 
+-- Remark: we disabled the following normalization rule because we want this information when implementing splitting heuristics
 -- Implication as a clause
-@[grind_norm] theorem imp_eq (p q : Prop) : (p → q) = (¬ p ∨ q) := by
+theorem imp_eq (p q : Prop) : (p → q) = (¬ p ∨ q) := by
   by_cases p <;> by_cases q <;> simp [*]
 
+@[grind_norm] theorem true_imp_eq (p : Prop) : (True → p) = p := by simp
+@[grind_norm] theorem false_imp_eq (p : Prop) : (False → p) = True := by simp
+@[grind_norm] theorem imp_true_eq (p : Prop) : (p → True) = True := by simp
+@[grind_norm] theorem imp_false_eq (p : Prop) : (p → False) = ¬p := by simp
+@[grind_norm] theorem imp_self_eq (p : Prop) : (p → p) = True := by simp
+
 -- And
 @[grind_norm↓] theorem not_and (p q : Prop) : (¬(p ∧ q)) = (¬p ∨ ¬q) := by
   by_cases p <;> by_cases q <;> simp [*]
@@ -58,13 +66,19 @@ attribute [grind_norm] ite_true ite_false
 @[grind_norm↓] theorem not_ite {_ : Decidable p} (q r : Prop) : (¬ite p q r) = ite p (¬q) (¬r) := by
   by_cases p <;> simp [*]
 
+@[grind_norm] theorem ite_true_false {_ : Decidable p} : (ite p True False) = p := by
+  by_cases p <;> simp
+
+@[grind_norm] theorem ite_false_true {_ : Decidable p} : (ite p False True) = ¬p := by
+  by_cases p <;> simp
+
 -- Forall
 @[grind_norm↓] theorem not_forall (p : α → Prop) : (¬∀ x, p x) = ∃ x, ¬p x := by simp
 attribute [grind_norm] forall_and
 
 -- Exists
 @[grind_norm↓] theorem not_exists (p : α → Prop) : (¬∃ x, p x) = ∀ x, ¬p x := by simp
-attribute [grind_norm] exists_const exists_or
+attribute [grind_norm] exists_const exists_or exists_prop exists_and_left exists_and_right
 
 -- Bool cond
 @[grind_norm] theorem cond_eq_ite (c : Bool) (a b : α) : cond c a b = ite c a b := by
@@ -107,4 +121,7 @@ attribute [grind_norm] Nat.le_zero_eq
 -- GT GE
 attribute [grind_norm] GT.gt GE.ge
 
+-- Succ
+attribute [grind_norm] Nat.succ_eq_add_one
+
 end Lean.Grind

src/Init/Grind/Offset.lean

@@ -0,0 +1,92 @@
+/-
+Copyright (c) 2025 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Core
+import Init.Omega
+
+namespace Lean.Grind
+abbrev isLt (x y : Nat) : Bool := x < y
+abbrev isLE (x y : Nat) : Bool := x ≤ y
+
+/-! Theorems for transitivity. -/
+theorem Nat.le_ro (u w v k : Nat) : u ≤ w → w ≤ v + k → u ≤ v + k := by
+  omega
+theorem Nat.le_lo (u w v k : Nat) : u ≤ w → w + k ≤ v → u + k ≤ v := by
+  omega
+theorem Nat.lo_le (u w v k : Nat) : u + k ≤ w → w ≤ v → u + k ≤ v := by
+  omega
+theorem Nat.lo_lo (u w v k₁ k₂ : Nat) : u + k₁ ≤ w → w + k₂ ≤ v → u + (k₁ + k₂) ≤ v := by
+  omega
+theorem Nat.lo_ro_1 (u w v k₁ k₂ : Nat) : isLt k₂ k₁ = true → u + k₁ ≤ w → w ≤ v + k₂ → u + (k₁ - k₂) ≤ v := by
+  simp [isLt]; omega
+theorem Nat.lo_ro_2 (u w v k₁ k₂ : Nat) : u + k₁ ≤ w → w ≤ v + k₂ → u ≤ v + (k₂ - k₁) := by
+  omega
+theorem Nat.ro_le (u w v k : Nat) : u ≤ w + k → w ≤ v → u ≤ v + k := by
+  omega
+theorem Nat.ro_lo_1 (u w v k₁ k₂ : Nat) : u ≤ w + k₁ → w + k₂ ≤ v → u ≤ v + (k₁ - k₂) := by
+  omega
+theorem Nat.ro_lo_2 (u w v k₁ k₂ : Nat) : isLt k₁ k₂ = true → u ≤ w + k₁ → w + k₂ ≤ v → u + (k₂ - k₁) ≤ v := by
+  simp [isLt]; omega
+theorem Nat.ro_ro (u w v k₁ k₂ : Nat) : u ≤ w + k₁ → w ≤ v + k₂ → u ≤ v + (k₁ + k₂) := by
+  omega
+
+/-! Theorems for negating constraints. -/
+theorem Nat.of_le_eq_false (u v : Nat) : ((u ≤ v) = False) → v + 1 ≤ u := by
+  simp; omega
+theorem Nat.of_lo_eq_false_1 (u v : Nat) : ((u + 1 ≤ v) = False) → v ≤ u := by
+  simp; omega
+theorem Nat.of_lo_eq_false (u v k : Nat) : ((u + k ≤ v) = False) → v ≤ u + (k-1) := by
+  simp; omega
+theorem Nat.of_ro_eq_false (u v k : Nat) : ((u ≤ v + k) = False) → v + (k+1) ≤ u := by
+  simp; omega
+
+/-! Theorems for closing a goal. -/
+theorem Nat.unsat_le_lo (u v k : Nat) : isLt 0 k = true → u ≤ v → v + k ≤ u → False := by
+  simp [isLt]; omega
+theorem Nat.unsat_lo_lo (u v k₁ k₂ : Nat) : isLt 0 (k₁+k₂) = true → u + k₁ ≤ v → v + k₂ ≤ u → False := by
+  simp [isLt]; omega
+theorem Nat.unsat_lo_ro (u v k₁ k₂ : Nat) : isLt k₂ k₁ = true → u + k₁ ≤ v → v ≤ u + k₂ → False := by
+  simp [isLt]; omega
+
+/-! Theorems for propagating constraints to `True` -/
+theorem Nat.lo_eq_true_of_lo (u v k₁ k₂ : Nat) : isLE k₂ k₁ = true → u + k₁ ≤ v → (u + k₂ ≤ v) = True :=
+  by simp [isLt]; omega
+theorem Nat.le_eq_true_of_lo (u v k : Nat) : u + k ≤ v → (u ≤ v) = True :=
+  by simp; omega
+theorem Nat.le_eq_true_of_le (u v : Nat) : u ≤ v → (u ≤ v) = True :=
+  by simp
+theorem Nat.ro_eq_true_of_lo (u v k₁ k₂ : Nat) : u + k₁ ≤ v → (u ≤ v + k₂) = True :=
+  by simp; omega
+theorem Nat.ro_eq_true_of_le (u v k : Nat) : u ≤ v → (u ≤ v + k) = True :=
+  by simp; omega
+theorem Nat.ro_eq_true_of_ro (u v k₁ k₂ : Nat) : isLE k₁ k₂ = true → u ≤ v + k₁ → (u ≤ v + k₂) = True :=
+  by simp [isLE]; omega
+
+/-!
+Theorems for propagating constraints to `False`.
+They are variants of the theorems for closing a goal.
+-/
+theorem Nat.lo_eq_false_of_le (u v k : Nat) : isLt 0 k = true → u ≤ v → (v + k ≤ u) = False := by
+ simp [isLt]; omega
+theorem Nat.le_eq_false_of_lo (u v k : Nat) : isLt 0 k = true → u + k ≤ v → (v ≤ u) = False := by
+ simp [isLt]; omega
+theorem Nat.lo_eq_false_of_lo (u v k₁ k₂ : Nat) : isLt 0 (k₁+k₂) = true → u + k₁ ≤ v → (v + k₂ ≤ u) = False := by
+  simp [isLt]; omega
+theorem Nat.ro_eq_false_of_lo (u v k₁ k₂ : Nat) : isLt k₂ k₁ = true → u + k₁ ≤ v → (v ≤ u + k₂) = False := by
+  simp [isLt]; omega
+theorem Nat.lo_eq_false_of_ro (u v k₁ k₂ : Nat) : isLt k₁ k₂ = true → u ≤ v + k₁ → (v + k₂ ≤ u) = False := by
+  simp [isLt]; omega
+
+/-!
+Helper theorems for equality propagation
+-/
+
+theorem Nat.le_of_eq_1 (u v : Nat) : u = v → u ≤ v := by omega
+theorem Nat.le_of_eq_2 (u v : Nat) : u = v → v ≤ u := by omega
+theorem Nat.eq_of_le_of_le (u v : Nat) : u ≤ v → v ≤ u → u = v := by omega
+theorem Nat.le_offset (a k : Nat) : k ≤ a + k := by omega
+
+end Lean.Grind

src/Init/Grind/PP.lean

@@ -0,0 +1,30 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.NotationExtra
+
+namespace Lean.Grind
+/-!
+This is a hackish module for hovering node information in the `grind` tactic state.
+-/
+
+inductive NodeDef where
+  | unit
+
+set_option linter.unusedVariables false in
+def node_def (_ : Nat) {α : Sort u} {a : α} : NodeDef := .unit
+
+@[app_unexpander node_def]
+def nodeDefUnexpander : PrettyPrinter.Unexpander := fun stx => do
+  match stx with
+  | `($_ $id:num) => return mkIdent <| Name.mkSimple $ "#" ++ toString id.getNat
+  | _ => throw ()
+
+@[app_unexpander NodeDef]
+def NodeDefUnexpander : PrettyPrinter.Unexpander := fun _ => do
+  return mkIdent <| Name.mkSimple "NodeDef"
+
+end Lean.Grind

src/Init/Grind/Propagator.lean

@@ -0,0 +1,27 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.NotationExtra
+
+namespace Lean.Parser
+
+/-- A user-defined propagator for the `grind` tactic. -/
+-- TODO: not implemented yet
+syntax (docComment)? "grind_propagator " (Tactic.simpPre <|> Tactic.simpPost) ident " (" ident ")" " := " term : command
+
+/-- A builtin propagator for the `grind` tactic. -/
+syntax (docComment)? "builtin_grind_propagator " ident (Tactic.simpPre <|> Tactic.simpPost) ident " := " term : command
+
+/-- Auxiliary attribute for builtin `grind` propagators. -/
+syntax (name := grindPropagatorBuiltinAttr) "builtin_grind_propagator" (Tactic.simpPre <|> Tactic.simpPost) ident : attr
+
+macro_rules
+  | `($[$doc?:docComment]? builtin_grind_propagator $propagatorName:ident $direction $op:ident := $body) => do
+    let propagatorType := `Lean.Meta.Grind.Propagator
+    `($[$doc?:docComment]? def $propagatorName:ident : $(mkIdent propagatorType) := $body
+      attribute [builtin_grind_propagator $direction $op] $propagatorName)
+
+end Lean.Parser

src/Init/Grind/Tactics.lean

@@ -6,20 +6,60 @@ Authors: Leonardo de Moura
 prelude
 import Init.Tactics
 
+namespace Lean.Parser.Attr
+
+syntax grindEq     := "="
+syntax grindEqBoth := atomic("_" "=" "_")
+syntax grindEqRhs  := atomic("=" "_")
+syntax grindBwd    := "←"
+syntax grindFwd    := "→"
+
+syntax (name := grind) "grind" (grindEqBoth <|> grindEqRhs <|> grindEq <|> grindBwd <|> grindFwd)? : attr
+
+end Lean.Parser.Attr
+
 namespace Lean.Grind
 /--
 The configuration for `grind`.
-Passed to `grind` using, for example, the `grind (config := { eager := true })` syntax.
+Passed to `grind` using, for example, the `grind (config := { matchEqs := true })` syntax.
 -/
 structure Config where
+  /-- Maximum number of case-splits in a proof search branch. It does not include splits performed during normalization. -/
+  splits : Nat := 8
+  /-- Maximum number of E-matching (aka heuristic theorem instantiation) rounds before each case split. -/
+  ematch : Nat := 5
   /--
-  When `eager` is true (default: `false`), `grind` eagerly splits `if-then-else` and `match`
-  expressions.
-  -/
-  eager : Bool := false
+  Maximum term generation.
+  The input goal terms have generation 0. When we instantiate a theorem using a term from generation `n`,
+  the new terms have generation `n+1`. Thus, this parameter limits the length of an instantiation chain. -/
+  gen : Nat := 5
+  /-- Maximum number of theorem instances generated using E-matching in a proof search tree branch. -/
+  instances : Nat := 1000
+  /-- If `matchEqs` is `true`, `grind` uses `match`-equations as E-matching theorems. -/
+  matchEqs : Bool := true
+  /-- If `splitMatch` is `true`, `grind` performs case-splitting on `match`-expressions during the search. -/
+  splitMatch : Bool := true
+  /-- If `splitIte` is `true`, `grind` performs case-splitting on `if-then-else` expressions during the search. -/
+  splitIte : Bool := true
+  /--
+  If `splitIndPred` is `true`, `grind` performs case-splitting on inductive predicates.
+  Otherwise, it performs case-splitting only on types marked with `[grind_split]` attribute. -/
+  splitIndPred : Bool := true
+  /-- By default, `grind` halts as soon as it encounters a sub-goal where no further progress can be made. -/
+  failures : Nat := 1
+  /-- Maximum number of heartbeats (in thousands) the canonicalizer can spend per definitional equality test. -/
+  canonHeartbeats : Nat := 1000
   deriving Inhabited, BEq
 
+end Lean.Grind
+
+namespace Lean.Parser.Tactic
+
 /-!
 `grind` tactic and related tactics.
 -/
-end Lean.Grind
+
+-- TODO: parameters
+syntax (name := grind) "grind" optConfig ("on_failure " term)? : tactic
+
+end Lean.Parser.Tactic

src/Init/Grind/Util.lean

@@ -0,0 +1,34 @@
+/-
+Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Init.Core
+
+namespace Lean.Grind
+
+/-- A helper gadget for annotating nested proofs in goals. -/
+def nestedProof (p : Prop) {h : p} : p := h
+
+/--
+Gadget for marking terms that should not be normalized by `grind`s simplifier.
+`grind` uses a simproc to implement this feature.
+We use it when adding instances of `match`-equations to prevent them from being simplified to true.
+-/
+def doNotSimp {α : Sort u} (a : α) : α := a
+
+/-- Gadget for representing offsets `t+k` in patterns. -/
+def offset (a b : Nat) : Nat := a + b
+
+/--
+Gadget for annotating the equalities in `match`-equations conclusions.
+`_origin` is the term used to instantiate the `match`-equation using E-matching.
+When `EqMatch a b origin` is `True`, we mark `origin` as a resolved case-split.
+-/
+def EqMatch (a b : α) {_origin : α} : Prop := a = b
+
+theorem nestedProof_congr (p q : Prop) (h : p = q) (hp : p) (hq : q) : HEq (@nestedProof p hp) (@nestedProof q hq) := by
+  subst h; apply HEq.refl
+
+end Lean.Grind

src/Init/Internal.lean

@@ -0,0 +1,13 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+prelude
+import Init.Internal.Order
+
+/-!
+This directory is used for components of the standard library that are either considered
+implementation details or not yet ready for public consumption, and that should be available
+without explicit import (in contrast to `Std.Internal`)
+-/

src/Init/Internal/Order.lean

@@ -0,0 +1,8 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+prelude
+import Init.Internal.Order.Basic
+import Init.Internal.Order.Tactic

src/Init/Internal/Order/Basic.lean

@@ -0,0 +1,693 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+prelude
+
+import Init.ByCases
+import Init.RCases
+
+/-!
+This module contains some basic definitions and results from domain theory, intended to be used as
+the underlying construction of the `partial_fixpoint` feature. It is not meant to be used as a
+general purpose library for domain theory, but can be of interest to users who want to extend
+the `partial_fixpoint` machinery (e.g. mark more functions as monotone or register more monads).
+
+This follows the corresponding
+[Isabelle development](https://isabelle.in.tum.de/library/HOL/HOL/Partial_Function.html), as also
+described in [Alexander Krauss: Recursive Definitions of Monadic Functions](https://www21.in.tum.de/~krauss/papers/mrec.pdf).
+-/
+
+universe u v w
+
+namespace Lean.Order
+
+/--
+A partial order is a reflexive, transitive and antisymmetric relation.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+class PartialOrder (α : Sort u) where
+  /--
+  A “less-or-equal-to” or “approximates” relation.
+
+  This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+  -/
+  rel : α → α → Prop
+  rel_refl : ∀ {x}, rel x x
+  rel_trans : ∀ {x y z}, rel x y → rel y z → rel x z
+  rel_antisymm : ∀ {x y}, rel x y → rel y x → x = y
+
+@[inherit_doc] scoped infix:50 " ⊑ " => PartialOrder.rel
+
+section PartialOrder
+
+variable {α  : Sort u} [PartialOrder α]
+
+theorem PartialOrder.rel_of_eq {x y : α} (h : x = y) : x ⊑ y := by cases h; apply rel_refl
+
+/--
+A chain is a totally ordered set (representing a set as a predicate).
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+def chain (c : α → Prop) : Prop := ∀ x y , c x → c y → x ⊑ y ∨ y ⊑ x
+
+end PartialOrder
+
+section CCPO
+
+/--
+A chain-complete partial order (CCPO) is a partial order where every chain a least upper bound.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+class CCPO (α : Sort u) extends PartialOrder α where
+  /--
+  The least upper bound of a chain.
+
+  This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+  -/
+  csup : (α → Prop) → α
+  csup_spec {c : α → Prop} (hc : chain c) : csup c ⊑ x ↔ (∀ y, c y → y ⊑ x)
+
+open PartialOrder CCPO
+
+variable {α  : Sort u} [CCPO α]
+
+theorem csup_le {c : α → Prop} (hchain : chain c) : (∀ y, c y → y ⊑ x) → csup c ⊑ x :=
+  (csup_spec hchain).mpr
+
+theorem le_csup {c : α → Prop} (hchain : chain c) {y : α} (hy : c y) : y ⊑ csup c :=
+  (csup_spec hchain).mp rel_refl y hy
+
+/--
+The bottom element is the least upper bound of the empty chain.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+def bot : α := csup (fun _ => False)
+
+scoped notation "⊥" => bot
+
+theorem bot_le (x : α) : ⊥ ⊑ x := by
+  apply csup_le
+  · intro x y hx hy; contradiction
+  · intro x hx; contradiction
+
+end CCPO
+
+section monotone
+
+variable {α : Sort u} [PartialOrder α]
+variable {β : Sort v} [PartialOrder β]
+
+/--
+A function is monotone if if it maps related elements to releated elements.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+def monotone (f : α → β) : Prop := ∀ x y, x ⊑ y → f x ⊑ f y
+
+theorem monotone_const (c : β) : monotone (fun (_ : α) => c) :=
+  fun _ _ _ => PartialOrder.rel_refl
+
+theorem monotone_id : monotone (fun (x : α) => x) :=
+  fun _ _ hxy => hxy
+
+theorem monotone_compose
+    {γ : Sort w} [PartialOrder γ]
+    {f : α → β} {g : β → γ}
+    (hf : monotone f) (hg : monotone g) :
+   monotone (fun x => g (f x)) := fun _ _ hxy => hg _ _ (hf _ _ hxy)
+
+end monotone
+
+section admissibility
+
+variable {α : Sort u} [CCPO α]
+
+open PartialOrder CCPO
+
+/--
+A predicate is admissable if it can be transferred from the elements of a chain to the chains least
+upper bound. Such predicates can be used in fixpoint induction.
+
+This definition implies `P ⊥`. Sometimes (e.g. in Isabelle) the empty chain is excluded
+from this definition, and `P ⊥` is a separate condition of the induction predicate.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+def admissible (P : α → Prop) :=
+  ∀ (c : α → Prop), chain c → (∀ x, c x → P x) → P (csup c)
+
+theorem admissible_const_true : admissible (fun (_ : α) => True) :=
+  fun _ _ _ => trivial
+
+theorem admissible_and (P Q : α → Prop)
+  (hadm₁ : admissible P) (hadm₂ : admissible Q) : admissible (fun x => P x ∧ Q x) :=
+    fun c hchain h =>
+    ⟨ hadm₁ c hchain fun x hx => (h x hx).1,
+      hadm₂ c hchain fun x hx => (h x hx).2⟩
+
+theorem chain_conj (c P : α → Prop) (hchain : chain c) : chain (fun x => c x ∧ P x) := by
+  intro x y ⟨hcx, _⟩ ⟨hcy, _⟩
+  exact hchain x y hcx hcy
+
+theorem csup_conj (c P : α → Prop) (hchain : chain c) (h : ∀ x, c x → ∃ y, c y ∧ x ⊑ y ∧ P y) :
+    csup c = csup (fun x => c x ∧ P x) := by
+  apply rel_antisymm
+  · apply csup_le hchain
+    intro x hcx
+    obtain ⟨y, hcy, hxy, hPy⟩ := h x hcx
+    apply rel_trans hxy; clear x hcx hxy
+    apply le_csup (chain_conj _ _ hchain) ⟨hcy, hPy⟩
+  · apply csup_le (chain_conj _ _ hchain)
+    intro x ⟨hcx, hPx⟩
+    apply le_csup hchain hcx
+
+theorem admissible_or (P Q : α → Prop)
+  (hadm₁ : admissible P) (hadm₂ : admissible Q) : admissible (fun x => P x ∨ Q x) := by
+  intro c hchain h
+  have : (∀ x, c x → ∃ y, c y ∧ x ⊑ y ∧ P y) ∨ (∀ x, c x → ∃ y, c y ∧ x ⊑ y ∧ Q y) := by
+    open Classical in
+    apply Decidable.or_iff_not_imp_left.mpr
+    intro h'
+    simp only [not_forall, not_imp, not_exists, not_and] at h'
+    obtain ⟨x, hcx, hx⟩ := h'
+    intro y hcy
+    cases hchain x y hcx hcy  with
+    | inl hxy =>
+      refine ⟨y, hcy, rel_refl, ?_⟩
+      cases h y hcy with
+      | inl hPy => exfalso; apply hx y hcy hxy hPy
+      | inr hQy => assumption
+    | inr hyx =>
+      refine ⟨x, hcx, hyx , ?_⟩
+      cases h x hcx with
+      | inl hPx => exfalso; apply hx x hcx rel_refl hPx
+      | inr hQx => assumption
+  cases this with
+  | inl hP =>
+    left
+    rw [csup_conj (h := hP) (hchain := hchain)]
+    apply hadm₁ _ (chain_conj _ _ hchain)
+    intro x ⟨hcx, hPx⟩
+    exact hPx
+  | inr hQ =>
+    right
+    rw [csup_conj (h := hQ) (hchain := hchain)]
+    apply hadm₂ _ (chain_conj _ _ hchain)
+    intro x ⟨hcx, hQx⟩
+    exact hQx
+
+def admissible_pi (P : α → β → Prop)
+  (hadm₁ : ∀ y, admissible (fun x => P x y)) : admissible (fun x => ∀ y, P x y) :=
+    fun c hchain h y => hadm₁ y c hchain fun x hx => h x hx y
+
+end admissibility
+
+section fix
+
+open PartialOrder CCPO
+
+variable {α  : Sort u} [CCPO α]
+
+variable {c : α → Prop} (hchain : chain c)
+
+/--
+The transfinite iteration of a function `f` is a set that is `⊥ ` and is closed under application
+of `f` and `csup`.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+inductive iterates (f : α → α) : α → Prop where
+  | step : iterates f x → iterates f (f x)
+  | sup {c : α → Prop} (hc : chain c) (hi : ∀ x, c x → iterates f x) : iterates f (csup c)
+
+theorem chain_iterates {f : α → α} (hf : monotone f) : chain (iterates f) := by
+  intros x y hx hy
+  induction hx generalizing y
+  case step x hx ih =>
+    induction hy
+    case step y hy _ =>
+      cases ih y hy
+      · left; apply hf; assumption
+      · right; apply hf; assumption
+    case sup c hchain hi ih2 =>
+      show f x ⊑ csup c ∨ csup c ⊑ f x
+      by_cases h : ∃ z, c z ∧ f x ⊑ z
+      · left
+        obtain ⟨z, hz, hfz⟩ := h
+        apply rel_trans hfz
+        apply le_csup hchain hz
+      · right
+        apply csup_le hchain _
+        intro z hz
+        rw [not_exists] at h
+        specialize h z
+        rw [not_and] at h
+        specialize h hz
+        cases ih2 z hz
+        next => contradiction
+        next => assumption
+  case sup c hchain hi ih =>
+    show rel (csup c) y ∨ rel y (csup c)
+    by_cases h : ∃ z, c z ∧ rel y z
+    · right
+      obtain ⟨z, hz, hfz⟩ := h
+      apply rel_trans hfz
+      apply le_csup hchain hz
+    · left
+      apply csup_le hchain _
+      intro z hz
+      rw [not_exists] at h
+      specialize h z
+      rw [not_and] at h
+      specialize h hz
+      cases ih z hz y hy
+      next => assumption
+      next => contradiction
+
+theorem rel_f_of_iterates {f : α → α} (hf : monotone f) {x : α} (hx : iterates f x) : x ⊑ f x := by
+  induction hx
+  case step ih =>
+    apply hf
+    assumption
+  case sup c hchain hi ih =>
+    apply csup_le hchain
+    intro y hy
+    apply rel_trans (ih y hy)
+    apply hf
+    apply le_csup hchain hy
+
+set_option linter.unusedVariables false in
+/--
+The least fixpoint of a monotone function is the least upper bound of its transfinite iteration.
+
+The `monotone f` assumption is not strictly necessarily for the definition, but without this the
+definition is not very meaningful and it simplifies applying theorems like `fix_eq` if every use of
+`fix` already has the monotonicty requirement.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+def fix (f : α → α) (hmono : monotone f) := csup (iterates f)
+
+/--
+The main fixpoint theorem for fixedpoints of monotone functions in chain-complete partial orders.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+theorem fix_eq {f : α → α} (hf : monotone f) : fix f hf = f (fix f hf) := by
+  apply rel_antisymm
+  · apply rel_f_of_iterates hf
+    apply iterates.sup (chain_iterates hf)
+    exact fun _ h => h
+  · apply le_csup (chain_iterates hf)
+    apply iterates.step
+    apply iterates.sup (chain_iterates hf)
+    intro y hy
+    exact hy
+
+/--
+The fixpoint induction theme: An admissible predicate holds for a least fixpoint if it is preserved
+by the fixpoint's function.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+theorem fix_induct {f : α → α} (hf : monotone f)
+    (motive : α → Prop) (hadm: admissible motive)
+    (h : ∀ x, motive x → motive (f x)) : motive (fix f hf) := by
+  apply hadm _ (chain_iterates hf)
+  intro x hiterates
+  induction hiterates with
+  | @step x hiter ih => apply h x ih
+  | @sup c hchain hiter ih => apply hadm c hchain ih
+
+end fix
+
+section fun_order
+
+open PartialOrder
+
+variable {α : Sort u}
+variable {β : α → Sort v}
+variable {γ : Sort w}
+
+instance instOrderPi [∀ x, PartialOrder (β x)] : PartialOrder (∀ x, β x) where
+  rel f g := ∀ x, f x ⊑ g x
+  rel_refl _ := rel_refl
+  rel_trans hf hg x := rel_trans (hf x) (hg x)
+  rel_antisymm hf hg := funext (fun x => rel_antisymm (hf x) (hg x))
+
+theorem monotone_of_monotone_apply [PartialOrder γ] [∀ x, PartialOrder (β x)] (f : γ → (∀ x, β x))
+  (h : ∀ y, monotone (fun x => f x y)) : monotone f :=
+  fun x y hxy z => h z x y hxy
+
+theorem monotone_apply [PartialOrder γ] [∀ x, PartialOrder (β x)] (a : α) (f : γ → ∀ x, β x)
+    (h : monotone f) :
+    monotone (fun x => f x a) := fun _ _ hfg => h _ _ hfg a
+
+theorem chain_apply [∀ x, PartialOrder (β x)] {c : (∀ x, β x) → Prop} (hc : chain c) (x : α) :
+    chain (fun y => ∃ f, c f ∧ f x = y) := by
+  intro _ _ ⟨f, hf, hfeq⟩ ⟨g, hg, hgeq⟩
+  subst hfeq; subst hgeq
+  cases hc f g hf hg
+  next h => left; apply h x
+  next h => right; apply h x
+
+def fun_csup [∀ x, CCPO (β x)] (c : (∀ x, β x) → Prop) (x : α) :=
+  CCPO.csup (fun y => ∃ f, c f ∧ f x = y)
+
+instance instCCPOPi [∀ x, CCPO (β x)] : CCPO (∀ x, β x) where
+  csup := fun_csup
+  csup_spec := by
+    intro f c hc
+    constructor
+    next =>
+      intro hf g hg x
+      apply rel_trans _ (hf x); clear hf
+      apply le_csup (chain_apply hc x)
+      exact ⟨g, hg, rfl⟩
+    next =>
+      intro h x
+      apply csup_le (chain_apply hc x)
+      intro y ⟨z, hz, hyz⟩
+      subst y
+      apply h z hz
+
+def admissible_apply [∀ x, CCPO (β x)] (P : ∀ x, β x → Prop) (x : α)
+  (hadm : admissible (P x)) : admissible (fun (f : ∀ x, β x) => P x (f x)) := by
+  intro c hchain h
+  apply hadm _ (chain_apply hchain x)
+  rintro _ ⟨f, hcf, rfl⟩
+  apply h _ hcf
+
+def admissible_pi_apply [∀ x, CCPO (β x)] (P : ∀ x, β x → Prop) (hadm : ∀ x, admissible (P x)) :
+    admissible (fun (f : ∀ x, β x) => ∀ x, P x (f x)) := by
+  apply admissible_pi
+  intro
+  apply admissible_apply
+  apply hadm
+
+end fun_order
+
+section monotone_lemmas
+
+theorem monotone_letFun
+    {α : Sort u} {β : Sort v} {γ : Sort w} [PartialOrder α] [PartialOrder β]
+    (v : γ) (k : α → γ → β)
+    (hmono : ∀ y, monotone (fun x => k x y)) :
+  monotone fun (x : α) => letFun v (k x) := hmono v
+
+theorem monotone_ite
+    {α : Sort u} {β : Sort v} [PartialOrder α] [PartialOrder β]
+    (c : Prop) [Decidable c]
+    (k₁ : α → β) (k₂ : α → β)
+    (hmono₁ : monotone k₁) (hmono₂ : monotone k₂) :
+  monotone fun x => if c then k₁ x else k₂ x := by
+    split
+    · apply hmono₁
+    · apply hmono₂
+
+theorem monotone_dite
+    {α : Sort u} {β : Sort v} [PartialOrder α] [PartialOrder β]
+    (c : Prop) [Decidable c]
+    (k₁ : α → c → β) (k₂ : α → ¬ c → β)
+    (hmono₁ : monotone k₁) (hmono₂ : monotone k₂) :
+  monotone fun x => dite c (k₁ x) (k₂ x) := by
+    split
+    · apply monotone_apply _ _ hmono₁
+    · apply monotone_apply _ _ hmono₂
+
+end monotone_lemmas
+
+section pprod_order
+
+open PartialOrder
+
+variable {α : Sort u}
+variable {β : Sort v}
+variable {γ : Sort w}
+
+instance [PartialOrder α] [PartialOrder β] : PartialOrder (α ×' β) where
+  rel a b := a.1 ⊑ b.1 ∧ a.2 ⊑ b.2
+  rel_refl := ⟨rel_refl, rel_refl⟩
+  rel_trans ha hb := ⟨rel_trans ha.1 hb.1, rel_trans ha.2 hb.2⟩
+  rel_antisymm := fun {a} {b} ha hb => by
+    cases a; cases b;
+    dsimp at *
+    rw [rel_antisymm ha.1 hb.1, rel_antisymm ha.2 hb.2]
+
+theorem monotone_pprod [PartialOrder α] [PartialOrder β] [PartialOrder γ]
+    {f : γ → α} {g : γ → β} (hf : monotone f) (hg : monotone g) :
+    monotone (fun x => PProd.mk (f x) (g x)) :=
+  fun _ _ h12 => ⟨hf _ _ h12, hg _ _ h12⟩
+
+theorem monotone_pprod_fst [PartialOrder α] [PartialOrder β] [PartialOrder γ]
+    {f : γ → α ×' β} (hf : monotone f) : monotone (fun x => (f x).1) :=
+  fun _ _ h12 => (hf _ _ h12).1
+
+theorem monotone_pprod_snd [PartialOrder α] [PartialOrder β] [PartialOrder γ]
+    {f : γ → α ×' β} (hf : monotone f) : monotone (fun x => (f x).2) :=
+  fun _ _ h12 => (hf _ _ h12).2
+
+def chain_pprod_fst [CCPO α] [CCPO β] (c : α ×' β → Prop) : α → Prop := fun a => ∃ b, c ⟨a, b⟩
+def chain_pprod_snd [CCPO α] [CCPO β] (c : α ×' β → Prop) : β → Prop := fun b => ∃ a, c ⟨a, b⟩
+
+theorem chain.pprod_fst [CCPO α] [CCPO β] (c : α ×' β → Prop) (hchain : chain c) :
+    chain (chain_pprod_fst c) := by
+  intro a₁ a₂ ⟨b₁, h₁⟩ ⟨b₂, h₂⟩
+  cases hchain ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h₁ h₂
+  case inl h => left; exact h.1
+  case inr h => right; exact h.1
+
+theorem chain.pprod_snd [CCPO α] [CCPO β] (c : α ×' β → Prop) (hchain : chain c) :
+    chain (chain_pprod_snd c) := by
+  intro b₁ b₂ ⟨a₁, h₁⟩ ⟨a₂, h₂⟩
+  cases hchain ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h₁ h₂
+  case inl h => left; exact h.2
+  case inr h => right; exact h.2
+
+instance [CCPO α] [CCPO β] : CCPO (α ×' β) where
+  csup c := ⟨CCPO.csup (chain_pprod_fst c), CCPO.csup (chain_pprod_snd c)⟩
+  csup_spec := by
+    intro ⟨a, b⟩ c hchain
+    dsimp
+    constructor
+    next =>
+      intro ⟨h₁, h₂⟩ ⟨a', b'⟩ cab
+      constructor <;> dsimp at *
+      · apply rel_trans ?_ h₁
+        apply le_csup hchain.pprod_fst
+        exact ⟨b', cab⟩
+      · apply rel_trans ?_ h₂
+        apply le_csup hchain.pprod_snd
+        exact ⟨a', cab⟩
+    next =>
+      intro h
+      constructor <;> dsimp
+      · apply csup_le hchain.pprod_fst
+        intro a' ⟨b', hcab⟩
+        apply (h _ hcab).1
+      · apply csup_le hchain.pprod_snd
+        intro b' ⟨a', hcab⟩
+        apply (h _ hcab).2
+
+theorem admissible_pprod_fst {α : Sort u} {β : Sort v} [CCPO α] [CCPO β] (P : α → Prop)
+    (hadm : admissible P) : admissible (fun (x : α ×' β) => P x.1) := by
+  intro c hchain h
+  apply hadm _ hchain.pprod_fst
+  intro x ⟨y, hxy⟩
+  apply h ⟨x,y⟩ hxy
+
+theorem admissible_pprod_snd {α : Sort u} {β : Sort v} [CCPO α] [CCPO β] (P : β → Prop)
+    (hadm : admissible P) : admissible (fun (x : α ×' β) => P x.2) := by
+  intro c hchain h
+  apply hadm _ hchain.pprod_snd
+  intro y ⟨x, hxy⟩
+  apply h ⟨x,y⟩ hxy
+
+end pprod_order
+
+section flat_order
+
+variable {α : Sort u}
+
+set_option linter.unusedVariables false in
+/--
+`FlatOrder b` wraps the type `α` with the flat partial order generated by `∀ x, b ⊑ x`.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+def FlatOrder {α : Sort u} (b : α) := α
+
+variable {b : α}
+
+/--
+The flat partial order generated by `∀ x, b ⊑ x`.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+inductive FlatOrder.rel : (x y : FlatOrder b) → Prop where
+  | bot : rel b x
+  | refl : rel x x
+
+instance FlatOrder.instOrder : PartialOrder (FlatOrder b) where
+  rel := rel
+  rel_refl := .refl
+  rel_trans {x y z : α} (hxy : rel x y) (hyz : rel y z) := by
+    cases hxy <;> cases hyz <;> constructor
+  rel_antisymm {x y : α} (hxy : rel x y) (hyz : rel y x) : x = y := by
+    cases hxy <;> cases hyz <;> constructor
+
+open Classical in
+private theorem Classical.some_spec₂ {α : Sort _} {p : α → Prop} {h : ∃ a, p a} (q : α → Prop)
+    (hpq : ∀ a, p a → q a) : q (choose h) := hpq _ <| choose_spec _
+
+noncomputable def flat_csup (c : FlatOrder b → Prop) : FlatOrder b := by
+  by_cases h : ∃ (x : FlatOrder b), c x ∧ x ≠ b
+  · exact Classical.choose h
+  · exact b
+
+noncomputable instance FlatOrder.instCCPO : CCPO (FlatOrder b) where
+  csup := flat_csup
+  csup_spec := by
+    intro x c hc
+    unfold flat_csup
+    split
+    next hex =>
+      apply Classical.some_spec₂ (q := (· ⊑ x ↔ (∀ y, c y → y ⊑ x)))
+      clear hex
+      intro z ⟨hz, hnb⟩
+      constructor
+      · intro h y hy
+        apply PartialOrder.rel_trans _ h; clear h
+        cases hc y z hy hz
+        next => assumption
+        next h =>
+          cases h
+          · contradiction
+          · constructor
+      · intro h
+        cases h z hz
+        · contradiction
+        · constructor
+    next hnotex =>
+      constructor
+      · intro h y hy; clear h
+        suffices y = b by rw [this]; exact rel.bot
+        rw [not_exists] at hnotex
+        specialize hnotex y
+        rw [not_and] at hnotex
+        specialize hnotex hy
+        rw [@Classical.not_not] at hnotex
+        assumption
+      · intro; exact rel.bot
+
+theorem admissible_flatOrder (P : FlatOrder b → Prop) (hnot : P b) : admissible P := by
+  intro c hchain h
+  by_cases h' : ∃ (x : FlatOrder b), c x ∧ x ≠ b
+  · simp [CCPO.csup, flat_csup, h']
+    apply Classical.some_spec₂ (q := (P ·))
+    intro x ⟨hcx, hneb⟩
+    apply h x hcx
+  · simp [CCPO.csup, flat_csup, h', hnot]
+
+end flat_order
+
+section mono_bind
+
+/--
+The class `MonoBind m` indicates that every `m α` has a `PartialOrder`, and that the bind operation
+on `m` is monotone in both arguments with regard to that order.
+
+This is intended to be used in the construction of `partial_fixpoint`, and not meant to be used otherwise.
+-/
+class MonoBind (m : Type u → Type v) [Bind m] [∀ α, PartialOrder (m α)] where
+  bind_mono_left {a₁ a₂ : m α} {f : α → m b} (h : a₁ ⊑ a₂) : a₁ >>= f ⊑ a₂ >>= f
+  bind_mono_right {a : m α} {f₁ f₂ : α → m b} (h : ∀ x, f₁ x ⊑ f₂ x) : a >>= f₁ ⊑ a >>= f₂
+
+theorem monotone_bind
+    (m : Type u → Type v) [Bind m] [∀ α, PartialOrder (m α)] [MonoBind m]
+    {α β : Type u}
+    {γ : Type w} [PartialOrder γ]
+    (f : γ → m α) (g : γ → α → m β)
+    (hmono₁ : monotone f)
+    (hmono₂ : monotone g) :
+    monotone (fun (x : γ) => f x >>= g x) := by
+  intro x₁ x₂ hx₁₂
+  apply PartialOrder.rel_trans
+  · apply MonoBind.bind_mono_left (hmono₁ _ _ hx₁₂)
+  · apply MonoBind.bind_mono_right (fun y => monotone_apply y _ hmono₂ _ _ hx₁₂)
+
+instance : PartialOrder (Option α) := inferInstanceAs (PartialOrder (FlatOrder none))
+noncomputable instance : CCPO (Option α) := inferInstanceAs (CCPO (FlatOrder none))
+noncomputable instance : MonoBind Option where
+  bind_mono_left h := by
+    cases h
+    · exact FlatOrder.rel.bot
+    · exact FlatOrder.rel.refl
+  bind_mono_right h := by
+    cases ‹Option _›
+    · exact FlatOrder.rel.refl
+    · exact h _
+
+theorem admissible_eq_some (P : Prop) (y : α) :
+    admissible (fun (x : Option α) => x = some y → P) := by
+  apply admissible_flatOrder; simp
+
+instance [Monad m] [inst : ∀ α, PartialOrder (m α)] : PartialOrder (ExceptT ε m α) := inst _
+instance [Monad m] [∀ α, PartialOrder (m α)] [inst : ∀ α, CCPO (m α)] : CCPO (ExceptT ε m α) := inst _
+instance [Monad m] [∀ α, PartialOrder (m α)] [∀ α, CCPO (m α)] [MonoBind m] : MonoBind (ExceptT ε m) where
+  bind_mono_left h₁₂ := by
+    apply MonoBind.bind_mono_left (m := m)
+    exact h₁₂
+  bind_mono_right h₁₂ := by
+    apply MonoBind.bind_mono_right (m := m)
+    intro x
+    cases x
+    · apply PartialOrder.rel_refl
+    · apply h₁₂
+
+end mono_bind
+
+namespace Example
+
+def findF (P : Nat → Bool) (rec : Nat → Option Nat) (x : Nat) : Option Nat :=
+  if P x then
+    some x
+  else
+    rec (x + 1)
+
+noncomputable def find (P : Nat → Bool) : Nat → Option Nat := fix (findF P) <| by
+  unfold findF
+  apply monotone_of_monotone_apply
+  intro n
+  split
+  · apply monotone_const
+  · apply monotone_apply
+    apply monotone_id
+
+theorem find_eq : find P = findF P (find P) := fix_eq ..
+
+theorem find_spec : ∀ n m, find P n = some m → n ≤ m ∧ P m := by
+  unfold find
+  refine fix_induct (motive := fun (f : Nat → Option Nat) => ∀ n m, f n = some m → n ≤ m ∧ P m) _ ?hadm ?hstep
+  case hadm =>
+    -- apply admissible_pi_apply does not work well, hard to infer everything
+    exact admissible_pi_apply _ (fun n => admissible_pi _ (fun m => admissible_eq_some _ m))
+  case hstep =>
+    intro f ih n m heq
+    simp only [findF] at heq
+    split at heq
+    · simp_all
+    · obtain ⟨ih1, ih2⟩ := ih _ _ heq
+      constructor
+      · exact Nat.le_trans (Nat.le_add_right _ _ ) ih1
+      · exact ih2
+
+end Example
+
+end Lean.Order

src/Init/Internal/Order/Tactic.lean

@@ -0,0 +1,20 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+
+prelude
+import Init.Notation
+
+namespace Lean.Order
+/--
+`monotonicity` performs one compositional step solving `monotone` goals,
+using lemma tagged with `@[partial_fixpoint_monotone]`.
+
+This tactic is mostly used internally by lean in `partial_fixpoint` definitions, but
+can be useful on its own for debugging or when proving new `@[partial_fixpoint_monotone]` lemmas.
+-/
+scoped syntax (name := monotonicity) "monotonicity" : tactic
+
+end Lean.Order

src/Init/Prelude.lean

@@ -4170,6 +4170,16 @@ def withRef [Monad m] [MonadRef m] {α} (ref : Syntax) (x : m α) : m α :=
   let ref := replaceRef ref oldRef
   MonadRef.withRef ref x
 
+/--
+If `ref? = some ref`, run `x : m α` with a modified value for the `ref` by calling `withRef`.
+Otherwise, run `x` directly.
+-/
+@[always_inline, inline]
+def withRef? [Monad m] [MonadRef m] {α} (ref? : Option Syntax) (x : m α) : m α :=
+  match ref? with
+  | some ref => withRef ref x
+  | _        => x
+
 /-- A monad that supports syntax quotations. Syntax quotations (in term
     position) are monadic values that when executed retrieve the current "macro
     scope" from the monad and apply it to every identifier they introduce

src/Init/Tactics.lean

@@ -818,7 +818,7 @@ syntax inductionAlt  := ppDedent(ppLine) inductionAltLHS+ " => " (hole <|> synth
 After `with`, there is an optional tactic that runs on all branches, and
 then a list of alternatives.
 -/
-syntax inductionAlts := " with" (ppSpace colGt tactic)? withPosition((colGe inductionAlt)+)
+syntax inductionAlts := " with" (ppSpace colGt tactic)? withPosition((colGe inductionAlt)*)
 
 /--
 Assuming `x` is a variable in the local context with an inductive type,

src/Lean/AddDecl.lean

@@ -21,11 +21,6 @@ def Environment.addDecl (env : Environment) (opts : Options) (decl : Declaration
   else
     addDeclCore env (Core.getMaxHeartbeats opts).toUSize decl cancelTk?
 
-def Environment.addAndCompile (env : Environment) (opts : Options) (decl : Declaration)
-    (cancelTk? : Option IO.CancelToken := none) : Except KernelException Environment := do
-  let env ← addDecl env opts decl cancelTk?
-  compileDecl env opts decl
-
 def addDecl (decl : Declaration) : CoreM Unit := do
   profileitM Exception "type checking" (← getOptions) do
     withTraceNode `Kernel (fun _ => return m!"typechecking declaration") do

src/Lean/Compiler/ExternAttr.lean

@@ -18,6 +18,7 @@ inductive ExternEntry where
   | inline   (backend : Name) (pattern : String)
   | standard (backend : Name) (fn : String)
   | foreign  (backend : Name) (fn : String)
+  deriving BEq, Hashable
 
 /--
 - `@[extern]`
@@ -36,7 +37,7 @@ inductive ExternEntry where
 structure ExternAttrData where
   arity?   : Option Nat := none
   entries  : List ExternEntry
-  deriving Inhabited
+  deriving Inhabited, BEq, Hashable
 
 -- def externEntry := leading_parser optional ident >> optional (nonReservedSymbol "inline ") >> strLit
 -- @[builtin_attr_parser] def extern     := leading_parser nonReservedSymbol "extern " >> optional numLit >> many externEntry

src/Lean/Compiler/InitAttr.lean

@@ -144,11 +144,7 @@ def declareBuiltin (forDecl : Name) (value : Expr) : CoreM Unit := do
   let type := mkApp (mkConst `IO) (mkConst `Unit)
   let decl := Declaration.defnDecl { name, levelParams := [], type, value, hints := ReducibilityHints.opaque,
                                      safety := DefinitionSafety.safe }
-  match (← getEnv).addAndCompile {} decl with
-  -- TODO: pretty print error
-  | Except.error e => do
-    let msg ← (e.toMessageData {}).toString
-    throwError "failed to emit registration code for builtin '{forDecl}': {msg}"
-  | Except.ok env  => IO.ofExcept (setBuiltinInitAttr env name) >>= setEnv
+  addAndCompile decl
+  IO.ofExcept (setBuiltinInitAttr (← getEnv) name) >>= setEnv
 
 end Lean

src/Lean/Compiler/LCNF/Basic.lean

@@ -7,6 +7,7 @@ prelude
 import Init.Data.List.BasicAux
 import Lean.Expr
 import Lean.Meta.Instances
+import Lean.Compiler.ExternAttr
 import Lean.Compiler.InlineAttrs
 import Lean.Compiler.Specialize
 import Lean.Compiler.LCNF.Types
@@ -429,6 +430,80 @@ where
     | .cases c => c.alts.forM fun alt => go alt.getCode
     | .unreach .. | .return .. | .jmp .. => return ()
 
+partial def Code.instantiateValueLevelParams (code : Code) (levelParams : List Name) (us : List Level) : Code :=
+  instCode code
+where
+  instLevel (u : Level) :=
+    u.instantiateParams levelParams us
+
+  instExpr (e : Expr) :=
+    e.instantiateLevelParamsNoCache levelParams us
+
+  instParams (ps : Array Param) :=
+    ps.mapMono fun p => p.updateCore (instExpr p.type)
+
+  instAlt (alt : Alt) :=
+    match alt with
+    | .default k => alt.updateCode (instCode k)
+    | .alt _ ps k => alt.updateAlt! (instParams ps) (instCode k)
+
+  instArg (arg : Arg) : Arg :=
+    match arg with
+    | .type e => arg.updateType! (instExpr e)
+    | .fvar .. | .erased => arg
+
+  instLetValue (e : LetValue) : LetValue :=
+    match e with
+    | .const declName vs args => e.updateConst! declName (vs.mapMono instLevel) (args.mapMono instArg)
+    | .fvar fvarId args => e.updateFVar! fvarId (args.mapMono instArg)
+    | .proj .. | .value .. | .erased => e
+
+  instLetDecl (decl : LetDecl) :=
+    decl.updateCore (instExpr decl.type) (instLetValue decl.value)
+
+  instFunDecl (decl : FunDecl) :=
+    decl.updateCore (instExpr decl.type) (instParams decl.params) (instCode decl.value)
+
+  instCode (code : Code) :=
+    match code with
+    | .let decl k => code.updateLet! (instLetDecl decl) (instCode k)
+    | .jp decl k | .fun decl k => code.updateFun! (instFunDecl decl) (instCode k)
+    | .cases c => code.updateCases! (instExpr c.resultType) c.discr (c.alts.mapMono instAlt)
+    | .jmp fvarId args => code.updateJmp! fvarId (args.mapMono instArg)
+    | .return .. => code
+    | .unreach type => code.updateUnreach! (instExpr type)
+
+inductive DeclValue where
+  | code (code : Code)
+  | extern (externAttrData : ExternAttrData)
+  deriving Inhabited, BEq
+
+partial def DeclValue.size : DeclValue → Nat
+  | .code c => c.size
+  | .extern .. => 0
+
+def DeclValue.mapCode (f : Code → Code) : DeclValue → DeclValue :=
+  fun
+    | .code c => .code (f c)
+    | .extern e => .extern e
+
+def DeclValue.mapCodeM [Monad m] (f : Code → m Code) : DeclValue → m DeclValue :=
+  fun v => do
+    match v with
+    | .code c => return .code (← f c)
+    | .extern .. => return v
+
+def DeclValue.forCodeM [Monad m] (f : Code → m Unit) : DeclValue → m Unit :=
+  fun v => do
+    match v with
+    | .code c => f c
+    | .extern .. => return ()
+
+def DeclValue.isCodeAndM [Monad m] (v : DeclValue) (f : Code → m Bool) : m Bool :=
+  match v with
+  | .code c => f c
+  | .extern .. => pure false
+
 /--
 Declaration being processed by the Lean to Lean compiler passes.
 -/
@@ -455,7 +530,7 @@ structure Decl where
   The body of the declaration, usually changes as it progresses
   through compiler passes.
   -/
-  value : Code
+  value : DeclValue
   /--
   We set this flag to true during LCNF conversion. When we receive
   a block of functions to be compiled, we set this flag to `true`
@@ -536,7 +611,9 @@ We use this function to decide whether we should inline a declaration tagged wit
 `[inline_if_reduce]` or not.
 -/
 def Decl.isCasesOnParam? (decl : Decl) : Option Nat :=
-  go decl.value
+  match decl.value with
+  | .code c => go c
+  | .extern .. => none
 where
   go (code : Code) : Option Nat :=
     match code with
@@ -550,49 +627,6 @@ def Decl.instantiateTypeLevelParams (decl : Decl) (us : List Level) : Expr :=
 def Decl.instantiateParamsLevelParams (decl : Decl) (us : List Level) : Array Param :=
   decl.params.mapMono fun param => param.updateCore (param.type.instantiateLevelParamsNoCache decl.levelParams us)
 
-partial def Decl.instantiateValueLevelParams (decl : Decl) (us : List Level) : Code :=
-  instCode decl.value
-where
-  instLevel (u : Level) :=
-    u.instantiateParams decl.levelParams us
-
-  instExpr (e : Expr) :=
-    e.instantiateLevelParamsNoCache decl.levelParams us
-
-  instParams (ps : Array Param) :=
-    ps.mapMono fun p => p.updateCore (instExpr p.type)
-
-  instAlt (alt : Alt) :=
-    match alt with
-    | .default k => alt.updateCode (instCode k)
-    | .alt _ ps k => alt.updateAlt! (instParams ps) (instCode k)
-
-  instArg (arg : Arg) : Arg :=
-    match arg with
-    | .type e => arg.updateType! (instExpr e)
-    | .fvar .. | .erased => arg
-
-  instLetValue (e : LetValue) : LetValue :=
-    match e with
-    | .const declName vs args => e.updateConst! declName (vs.mapMono instLevel) (args.mapMono instArg)
-    | .fvar fvarId args => e.updateFVar! fvarId (args.mapMono instArg)
-    | .proj .. | .value .. | .erased => e
-
-  instLetDecl (decl : LetDecl) :=
-    decl.updateCore (instExpr decl.type) (instLetValue decl.value)
-
-  instFunDecl (decl : FunDecl) :=
-    decl.updateCore (instExpr decl.type) (instParams decl.params) (instCode decl.value)
-
-  instCode (code : Code) :=
-    match code with
-    | .let decl k => code.updateLet! (instLetDecl decl) (instCode k)
-    | .jp decl k | .fun decl k => code.updateFun! (instFunDecl decl) (instCode k)
-    | .cases c => code.updateCases! (instExpr c.resultType) c.discr (c.alts.mapMono instAlt)
-    | .jmp fvarId args => code.updateJmp! fvarId (args.mapMono instArg)
-    | .return .. => code
-    | .unreach type => code.updateUnreach! (instExpr type)
-
 /--
 Return `true` if the arrow type contains an instance implicit argument.
 -/
@@ -693,7 +727,7 @@ where
       visit k
 
   go : StateM NameSet Unit :=
-    decls.forM fun decl => visit decl.value
+    decls.forM (·.value.forCodeM visit)
 
 def instantiateRangeArgs (e : Expr) (beginIdx endIdx : Nat) (args : Array Arg) : Expr :=
   if !e.hasLooseBVars then

src/Lean/Compiler/LCNF/Bind.lean

@@ -123,7 +123,10 @@ def FunDeclCore.etaExpand (decl : FunDecl) : CompilerM FunDecl := do
   decl.update decl.type params value
 
 def Decl.etaExpand (decl : Decl) : CompilerM Decl := do
-  let some (params, value) ← etaExpandCore? decl.type decl.params decl.value | return decl
-  return { decl with params, value }
+  match decl.value with
+  | .code code =>
+    let some (params, newCode) ← etaExpandCore? decl.type decl.params code | return decl
+    return { decl with params, value := .code newCode}
+  | .extern .. => return decl
 
 end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/CSE.lean

@@ -102,7 +102,7 @@ end CSE
 Common sub-expression elimination
 -/
 def Decl.cse (decl : Decl) : CompilerM Decl := do
-  let value ← decl.value.cse
+  let value ← decl.value.mapCodeM (·.cse)
   return { decl with value }
 
 def cse (phase : Phase := .base) (occurrence := 0) : Pass :=

src/Lean/Compiler/LCNF/Check.lean

@@ -261,7 +261,7 @@ def run (x : CheckM α) : CompilerM α :=
 end Check
 
 def Decl.check (decl : Decl) : CompilerM Unit := do
-  Check.run do Check.checkFunDeclCore decl.name decl.params decl.type decl.value
+  Check.run do decl.value.forCodeM (Check.checkFunDeclCore decl.name decl.params decl.type)
 
 /--
 Check whether every local declaration in the local context is used in one of given `decls`.
@@ -299,7 +299,7 @@ where
 
   visitDecl (decl : Decl) : StateM FVarIdHashSet Unit := do
     visitParams decl.params
-    visitCode decl.value
+    decl.value.forCodeM visitCode
 
   visitDecls (decls : Array Decl) : StateM FVarIdHashSet Unit :=
     decls.forM visitDecl

src/Lean/Compiler/LCNF/CompilerM.lean

@@ -148,7 +148,7 @@ def eraseCodeDecls (decls : Array CodeDecl) : CompilerM Unit := do
 
 def eraseDecl (decl : Decl) : CompilerM Unit := do
   eraseParams decl.params
-  eraseCode decl.value
+  decl.value.forCodeM eraseCode
 
 abbrev Decl.erase (decl : Decl) : CompilerM Unit :=
   eraseDecl decl

src/Lean/Compiler/LCNF/DeclHash.lean

@@ -38,6 +38,7 @@ end
 instance : Hashable Code where
   hash c := hashCode c
 
+deriving instance Hashable for DeclValue
 deriving instance Hashable for Decl
 
-end Lean.Compiler.LCNF
+end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/ElimDead.lean

@@ -95,6 +95,6 @@ def Code.elimDead (code : Code) : CompilerM Code :=
   ElimDead.elimDead code |>.run' {}
 
 def Decl.elimDead (decl : Decl) : CompilerM Decl := do
-  return { decl with value := (← decl.value.elimDead) }
+  return { decl with value := (← decl.value.mapCodeM Code.elimDead) }
 
 end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/ElimDeadBranches.lean

@@ -513,7 +513,7 @@ def inferStep : InterpM Bool := do
     let currentVal ← getFunVal idx
     withReader (fun ctx => { ctx with currFnIdx := idx }) do
       decl.params.forM fun p => updateVarAssignment p.fvarId .top
-      interpCode decl.value
+      decl.value.forCodeM interpCode
     let newVal ← getFunVal idx
     if currentVal != newVal then
       return true
@@ -538,7 +538,7 @@ Use the information produced by the abstract interpreter to:
 -/
 partial def elimDead (assignment : Assignment) (decl : Decl) : CompilerM Decl := do
   trace[Compiler.elimDeadBranches] s!"Eliminating {decl.name} with {repr (← assignment.toArray |>.mapM (fun (name, val) => do return (toString (← getBinderName name), val)))}"
-  return { decl with value := (← go decl.value) }
+  return { decl with value := (← decl.value.mapCodeM go) }
 where
   go (code : Code) : CompilerM Code := do
     match code with

src/Lean/Compiler/LCNF/FixedParams.lean

@@ -141,8 +141,9 @@ partial def evalApp (declName : Name) (args : Array Arg) : FixParamM Unit := do
       let key := (declName, values)
       unless (← get).visited.contains key do
         modify fun s => { s with visited := s.visited.insert key }
-        let assignment := mkAssignment decl values
-        withReader (fun ctx => { ctx with assignment }) <| evalCode decl.value
+        decl.value.forCodeM fun c =>
+          let assignment := mkAssignment decl values
+          withReader (fun ctx => { ctx with assignment }) <| evalCode c
 
 end
 
@@ -169,8 +170,12 @@ def mkFixedParamsMap (decls : Array Decl) : NameMap (Array Bool) := Id.run do
     let values := mkInitialValues decl.params.size
     let assignment := mkAssignment decl values
     let fixed := Array.mkArray decl.params.size true
-    match evalCode decl.value |>.run { main := decl, decls, assignment } |>.run { fixed } with
-    | .ok _ s | .error _ s => result := result.insert decl.name s.fixed
+    match decl.value with
+    | .code c =>
+      match evalCode c |>.run { main := decl, decls, assignment } |>.run { fixed } with
+      | .ok _ s | .error _ s => result := result.insert decl.name s.fixed
+    | .extern .. =>
+      result := result.insert decl.name fixed
   return result
 
 end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/FloatLetIn.lean

@@ -239,7 +239,7 @@ Iterate through `decl`, pushing local declarations that are only used in one
 control flow arm into said arm in order to avoid useless computations.
 -/
 partial def floatLetIn (decl : Decl) : CompilerM Decl := do
-  let newValue ← go decl.value |>.run {}
+  let newValue ← decl.value.mapCodeM go |>.run {}
   return { decl with value := newValue }
 where
   /--

src/Lean/Compiler/LCNF/Internalize.lean

@@ -108,7 +108,7 @@ where
   go (decl : Decl) : InternalizeM Decl := do
     let type ← normExpr decl.type
     let params ← decl.params.mapM internalizeParam
-    let value ← internalizeCode decl.value
+    let value ← decl.value.mapCodeM internalizeCode
     return { decl with type, params, value }
 
 /--

src/Lean/Compiler/LCNF/JoinPoints.lean

@@ -133,7 +133,7 @@ this. This is because otherwise the calls to `myjp` in `f` and `g` would
 produce out of scope join point jumps.
 -/
 partial def find (decl : Decl) : CompilerM FindState := do
-  let (_, candidates) ← go decl.value |>.run none |>.run {} |>.run' {}
+  let (_, candidates) ← decl.value.forCodeM go |>.run none |>.run {} |>.run' {}
   return candidates
 where
   go : Code → FindM Unit
@@ -178,7 +178,7 @@ and all calls to them with `jmp`s.
 partial def replace (decl : Decl) (state : FindState) : CompilerM Decl := do
   let mapper := fun acc cname _ => do return acc.insert cname (← mkFreshJpName)
   let replaceCtx : ReplaceCtx ← state.candidates.foldM (init := .empty) mapper
-  let newValue ← go decl.value |>.run replaceCtx
+  let newValue ← decl.value.mapCodeM go |>.run replaceCtx
   return { decl with value := newValue }
 where
   go (code : Code) : ReplaceM Code := do
@@ -389,7 +389,7 @@ position within the code so we can pull them out as far as possible, hopefully
 enabling new inlining possibilities in the next simplifier run.
 -/
 partial def extend (decl : Decl) : CompilerM Decl := do
-  let newValue ← go decl.value |>.run {} |>.run' {} |>.run' {}
+  let newValue ← decl.value.mapCodeM go |>.run {} |>.run' {} |>.run' {}
   let decl := { decl with value := newValue }
   decl.pullFunDecls
 where
@@ -510,8 +510,8 @@ After we have performed all of these optimizations we can take away the
 code that has as little arguments as possible in the join points.
 -/
 partial def reduce (decl : Decl) : CompilerM Decl := do
-  let (_, analysis) ← goAnalyze decl.value |>.run {} |>.run {} |>.run' {}
-  let newValue ← goReduce decl.value |>.run analysis
+  let (_, analysis) ← decl.value.forCodeM goAnalyze |>.run {} |>.run {} |>.run' {}
+  let newValue ← decl.value.mapCodeM goReduce |>.run analysis
   return { decl with value := newValue }
 where
   goAnalyzeFunDecl (fn : FunDecl) : ReduceAnalysisM Unit := do

src/Lean/Compiler/LCNF/LambdaLifting.lean

@@ -108,9 +108,10 @@ def mkAuxDecl (closure : Array Param) (decl : FunDecl) : LiftM LetDecl := do
 where
   go (nameNew : Name) (safe : Bool) (inlineAttr? : Option InlineAttributeKind) : InternalizeM Decl := do
     let params := (← closure.mapM internalizeParam) ++ (← decl.params.mapM internalizeParam)
-    let value ← internalizeCode decl.value
-    let type ← value.inferType
+    let code ← internalizeCode decl.value
+    let type ← code.inferType
     let type ← mkForallParams params type
+    let value := .code code
     let decl := { name := nameNew, levelParams := [], params, type, value, safe, inlineAttr?, recursive := false : Decl }
     return decl.setLevelParams
 
@@ -149,7 +150,7 @@ mutual
 end
 
 def main (decl : Decl) : LiftM Decl := do
-  let value ← withParams decl.params <| visitCode decl.value
+  let value ← withParams decl.params <| decl.value.mapCodeM visitCode
   return { decl with value }
 
 end LambdaLifting

src/Lean/Compiler/LCNF/Level.lean

@@ -139,6 +139,10 @@ mutual
     | .jmp _ args => visitArgs args
 end
 
+def visitDeclValue : DeclValue → Visitor
+  | .code c => visitCode c
+  | .extern .. => id
+
 end CollectLevelParams
 
 open Lean.CollectLevelParams
@@ -149,7 +153,7 @@ Collect universe level parameters collecting in the type, parameters, and value,
 set `decl.levelParams` with the resulting value.
 -/
 def Decl.setLevelParams (decl : Decl) : Decl :=
-  let levelParams := (visitCode decl.value ∘ visitParams decl.params ∘ visitType decl.type) {} |>.params.toList
+  let levelParams := (visitDeclValue decl.value ∘ visitParams decl.params ∘ visitType decl.type) {} |>.params.toList
   { decl with levelParams }
 
 end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/Main.lean

@@ -28,10 +28,10 @@ and `[specialize]` since they can be partially applied.
 -/
 def shouldGenerateCode (declName : Name) : CoreM Bool := do
   if (← isCompIrrelevant |>.run') then return false
+  let env ← getEnv
+  if isExtern env declName then return true
   let some info ← getDeclInfo? declName | return false
   unless info.hasValue (allowOpaque := true) do return false
-  let env ← getEnv
-  if isExtern env declName then return false
   if hasMacroInlineAttribute env declName then return false
   if (← Meta.isMatcher declName) then return false
   if isCasesOnRecursor env declName then return false

src/Lean/Compiler/LCNF/MonoTypes.lean

@@ -74,8 +74,6 @@ partial def toMonoType (type : Expr) : CoreM Expr := do
   let type := type.headBeta
   if type.isErased then
     return erasedExpr
-  else if type.isErased then
-    return erasedExpr
   else if isTypeFormerType type then
     return erasedExpr
   else match type with

src/Lean/Compiler/LCNF/PrettyPrinter.lean

@@ -105,6 +105,11 @@ mutual
         return f!"⊥ : {← ppExpr type}"
       else
         return "⊥"
+
+  partial def ppDeclValue (b : DeclValue) : M Format := do
+    match b with
+    | .code c => ppCode c
+    | .extern .. => return "extern"
 end
 
 def run (x : M α) : CompilerM α :=
@@ -121,7 +126,7 @@ def ppLetValue (e : LetValue) : CompilerM Format :=
 
 def ppDecl (decl : Decl) : CompilerM Format :=
   PP.run do
-    return f!"def {decl.name}{← PP.ppParams decl.params} : {← PP.ppExpr (← PP.getFunType decl.params decl.type)} :={indentD (← PP.ppCode decl.value)}"
+    return f!"def {decl.name}{← PP.ppParams decl.params} : {← PP.ppExpr (← PP.getFunType decl.params decl.type)} :={indentD (← PP.ppDeclValue decl.value)}"
 
 def ppFunDecl (decl : FunDecl) : CompilerM Format :=
   PP.run do

src/Lean/Compiler/LCNF/Probing.lean

@@ -57,7 +57,7 @@ where
     | .cases (cases : CasesCore Code) => cases.alts.forM (go ·.getCode)
     | .jmp .. | .return .. | .unreach .. => return ()
   start (decls : Array Decl) : StateRefT (Array LetValue) CompilerM Unit :=
-    decls.forM (go ·.value)
+    decls.forM (·.value.forCodeM go)
 
 partial def getJps : Probe Decl FunDecl := fun decls => do
   let (_, res) ← start decls |>.run #[]
@@ -72,10 +72,10 @@ where
     | .jmp .. | .return .. | .unreach .. => return ()
 
   start (decls : Array Decl) : StateRefT (Array FunDecl) CompilerM Unit :=
-    decls.forM fun decl => go decl.value
+    decls.forM (·.value.forCodeM go)
 
 partial def filterByLet (f : LetDecl → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let decl k => do if (← f decl) then return true else go k
@@ -84,7 +84,7 @@ where
   | .jmp .. | .return .. | .unreach .. => return false
 
 partial def filterByFun (f : FunDecl → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k | .jp _ k  => go k
@@ -93,7 +93,7 @@ where
   | .jmp .. | .return .. | .unreach .. => return false
 
 partial def filterByJp (f : FunDecl → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k => go k
@@ -103,7 +103,7 @@ where
   | .jmp .. | .return .. | .unreach .. => return false
 
 partial def filterByFunDecl (f : FunDecl → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k => go k
@@ -112,7 +112,7 @@ where
   | .jmp .. | .return .. | .unreach .. => return false
 
 partial def filterByCases (f : Cases → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k => go k
@@ -121,7 +121,7 @@ where
   | .jmp .. | .return .. | .unreach .. => return false
 
 partial def filterByJmp (f : FVarId → Array Arg → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k => go k
@@ -131,7 +131,7 @@ where
   | .return .. | .unreach .. => return false
 
 partial def filterByReturn (f : FVarId → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k => go k
@@ -141,7 +141,7 @@ where
   | .return var  => f var
 
 partial def filterByUnreach (f : Expr → CompilerM Bool) : Probe Decl Decl :=
-  filter (fun decl => go decl.value)
+  filter (·.value.isCodeAndM go)
 where
   go : Code → CompilerM Bool
   | .let _ k => go k

src/Lean/Compiler/LCNF/PullFunDecls.lean

@@ -172,8 +172,8 @@ open PullFunDecls
 Pull local function declarations and join points in the given declaration.
 -/
 def Decl.pullFunDecls (decl : Decl) : CompilerM Decl := do
-  let (value, ps) ← pull decl.value |>.run []
-  let value := attach ps.toArray value
+  let (value, ps) ← decl.value.mapCodeM pull |>.run []
+  let value := value.mapCode (attach ps.toArray)
   return { decl with value }
 
 def pullFunDecls : Pass :=

src/Lean/Compiler/LCNF/PullLetDecls.lean

@@ -96,8 +96,8 @@ open PullLetDecls
 def Decl.pullLetDecls (decl : Decl) (isCandidateFn : LetDecl → FVarIdSet → CompilerM Bool) : CompilerM Decl := do
   PullM.run (isCandidateFn := isCandidateFn) do
     withParams decl.params do
-      let value ← pullDecls decl.value
-      let value ← attachToPull value
+      let value ← decl.value.mapCodeM pullDecls
+      let value ← value.mapCodeM attachToPull
       return { decl with value }
 
 def Decl.pullInstances (decl : Decl) : CompilerM Decl :=

src/Lean/Compiler/LCNF/ReduceArity.lean

@@ -108,7 +108,7 @@ partial def visit (code : Code) : FindUsedM Unit := do
 
 def collectUsedParams (decl : Decl) : CompilerM FVarIdSet := do
   let params := decl.params.foldl (init := {}) fun s p => s.insert p.fvarId
-  let (_, { used, .. }) ← visit decl.value |>.run { decl, params } |>.run {}
+  let (_, { used, .. }) ← decl.value.forCodeM visit |>.run { decl, params } |>.run {}
   return used
 
 end FindUsed
@@ -146,37 +146,40 @@ end ReduceArity
 open FindUsed ReduceArity Internalize
 
 def Decl.reduceArity (decl : Decl) : CompilerM (Array Decl) := do
-  let used ← collectUsedParams decl
-  if used.size == decl.params.size then
-    return #[decl] -- Declarations uses all parameters
-  else
-    trace[Compiler.reduceArity] "{decl.name}, used params: {used.toList.map mkFVar}"
-    let mask   := decl.params.map fun param => used.contains param.fvarId
-    let auxName   := decl.name ++ `_redArg
-    let mkAuxDecl : CompilerM Decl := do
-      let params := decl.params.filter fun param => used.contains param.fvarId
-      let value  ← reduce decl.value |>.run { declName := decl.name, auxDeclName := auxName, paramMask := mask }
-      let type ← value.inferType
-      let type ← mkForallParams params type
-      let auxDecl := { decl with name := auxName, levelParams := [], type, params, value }
-      auxDecl.saveMono
-      return auxDecl
-    let updateDecl : InternalizeM Decl := do
-      let params ← decl.params.mapM internalizeParam
-      let mut args := #[]
-      for used in mask, param in params do
-        if used then
-          args := args.push param.toArg
-      let letDecl ← mkAuxLetDecl (.const auxName [] args)
-      let value := .let letDecl (.return letDecl.fvarId)
-      let decl := { decl with params, value, inlineAttr? := some .inline, recursive := false }
-      decl.saveMono
-      return decl
-    let unusedParams := decl.params.filter fun param => !used.contains param.fvarId
-    let auxDecl ← mkAuxDecl
-    let decl ← updateDecl |>.run' {}
-    eraseParams unusedParams
-    return #[auxDecl, decl]
+  match decl.value with
+  | .code code =>
+    let used ← collectUsedParams decl
+    if used.size == decl.params.size then
+      return #[decl] -- Declarations uses all parameters
+    else
+      trace[Compiler.reduceArity] "{decl.name}, used params: {used.toList.map mkFVar}"
+      let mask   := decl.params.map fun param => used.contains param.fvarId
+      let auxName   := decl.name ++ `_redArg
+      let mkAuxDecl : CompilerM Decl := do
+        let params := decl.params.filter fun param => used.contains param.fvarId
+        let value  ← decl.value.mapCodeM reduce |>.run { declName := decl.name, auxDeclName := auxName, paramMask := mask }
+        let type ← code.inferType
+        let type ← mkForallParams params type
+        let auxDecl := { decl with name := auxName, levelParams := [], type, params, value }
+        auxDecl.saveMono
+        return auxDecl
+      let updateDecl : InternalizeM Decl := do
+        let params ← decl.params.mapM internalizeParam
+        let mut args := #[]
+        for used in mask, param in params do
+          if used then
+            args := args.push param.toArg
+        let letDecl ← mkAuxLetDecl (.const auxName [] args)
+        let value := .code (.let letDecl (.return letDecl.fvarId))
+        let decl := { decl with params, value, inlineAttr? := some .inline, recursive := false }
+        decl.saveMono
+        return decl
+      let unusedParams := decl.params.filter fun param => !used.contains param.fvarId
+      let auxDecl ← mkAuxDecl
+      let decl ← updateDecl |>.run' {}
+      eraseParams unusedParams
+      return #[auxDecl, decl]
+  | .extern .. => return #[decl]
 
 def reduceArity : Pass where
   phase := .mono
@@ -187,4 +190,4 @@ def reduceArity : Pass where
 builtin_initialize
   registerTraceClass `Compiler.reduceArity (inherited := true)
 
-end Lean.Compiler.LCNF
+end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/ReduceJpArity.lean

@@ -68,7 +68,7 @@ open ReduceJpArity
 Try to reduce arity of join points
 -/
 def Decl.reduceJpArity (decl : Decl) : CompilerM Decl := do
-  let value ← reduce decl.value |>.run {}
+  let value ← decl.value.mapCodeM reduce |>.run {}
   return { decl with value }
 
 def reduceJpArity (phase := Phase.base) : Pass :=

src/Lean/Compiler/LCNF/Renaming.lean

@@ -55,7 +55,7 @@ def Decl.applyRenaming (decl : Decl) (r : Renaming) : CompilerM Decl := do
     return decl
   else
     let params ← decl.params.mapMonoM (·.applyRenaming r)
-    let value ← decl.value.applyRenaming r
+    let value ← decl.value.mapCodeM (·.applyRenaming r)
     return { decl with params, value }
 
-end Lean.Compiler.LCNF
+end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/Simp.lean

@@ -22,19 +22,20 @@ namespace Lean.Compiler.LCNF
 open Simp
 
 def Decl.simp? (decl : Decl) : SimpM (Option Decl) := do
-  updateFunDeclInfo decl.value
+  let .code code := decl.value | return none
+  updateFunDeclInfo code
   traceM `Compiler.simp.inline.info do return m!"{decl.name}:{Format.nest 2 (← (← get).funDeclInfoMap.format)}"
   traceM `Compiler.simp.step do ppDecl decl
-  let value ← simp decl.value
+  let code ← simp code
   let s ← get
-  let value ← value.applyRenaming s.binderRenaming
-  traceM `Compiler.simp.step.new do return m!"{decl.name} :=\n{← ppCode value}"
-  trace[Compiler.simp.stat] "{decl.name}, size: {value.size}, # visited: {s.visited}, # inline: {s.inline}, # inline local: {s.inlineLocal}"
-  if let some value ← simpJpCases? value then
-    let decl := { decl with value }
+  let code ← code.applyRenaming s.binderRenaming
+  traceM `Compiler.simp.step.new do return m!"{decl.name} :=\n{← ppCode code}"
+  trace[Compiler.simp.stat] "{decl.name}, size: {code.size}, # visited: {s.visited}, # inline: {s.inline}, # inline local: {s.inlineLocal}"
+  if let some code ← simpJpCases? code then
+    let decl := { decl with value := .code code }
     decl.reduceJpArity
   else if (← get).simplified then
-    return some { decl with value }
+    return some { decl with value := .code code }
   else
     return none
 

src/Lean/Compiler/LCNF/Simp/InlineCandidate.lean

@@ -43,6 +43,7 @@ def inlineCandidate? (e : LetValue) : SimpM (Option InlineCandidateInfo) := do
     unless (← read).config.inlineDefs do
       return none
     let some decl ← getDecl? declName | return none
+    let .code code := decl.value | return none
     let shouldInline : SimpM Bool := do
       if !decl.inlineIfReduceAttr && decl.recursive then return false
       if mustInline then return true
@@ -63,9 +64,8 @@ def inlineCandidate? (e : LetValue) : SimpM (Option InlineCandidateInfo) := do
       if decl.alwaysInlineAttr then return true
       -- TODO: check inlining quota
       if decl.inlineAttr || decl.inlineIfReduceAttr then return true
-      unless decl.noinlineAttr do
-        if (← isSmall decl.value) then return true
-      return false
+      if decl.noinlineAttr then return false
+      isSmall code
     unless (← shouldInline) do return none
     /- check arity -/
     let arity := decl.getArity
@@ -77,7 +77,7 @@ def inlineCandidate? (e : LetValue) : SimpM (Option InlineCandidateInfo) := do
       let arg := args[paramIdx]!
       unless (← arg.isConstructorApp) do return none
     let params := decl.instantiateParamsLevelParams us
-    let value := decl.instantiateValueLevelParams us
+    let value := code.instantiateValueLevelParams decl.levelParams us
     let type := decl.instantiateTypeLevelParams us
     incInline
     return some {

src/Lean/Compiler/LCNF/Simp/InlineProj.lean

@@ -69,11 +69,14 @@ where
           visit fvarId projs
       else
         let some decl ← getDecl? declName | failure
-        guard (decl.getArity == args.size)
-        let params := decl.instantiateParamsLevelParams us
-        let code := decl.instantiateValueLevelParams us
-        let code ← betaReduce params code args (mustInline := true)
-        visitCode code projs
+        match decl.value with
+        | .code code =>
+          guard (decl.getArity == args.size)
+          let params := decl.instantiateParamsLevelParams us
+          let code := code.instantiateValueLevelParams decl.levelParams us
+          let code ← betaReduce params code args (mustInline := true)
+          visitCode code projs
+        | .extern .. => failure
 
   visitCode (code : Code) (projs : List Nat) : OptionT (StateRefT (Array CodeDecl) SimpM) FVarId := do
     match code with

src/Lean/Compiler/LCNF/Specialize.lean

@@ -222,6 +222,7 @@ def mkSpecDecl (decl : Decl) (us : List Level) (argMask : Array (Option Arg)) (p
     eraseDecl decl
 where
   go (decl : Decl) (nameNew : Name) : InternalizeM Decl := do
+    let .code code := decl.value | panic! "can only specialize decls with code"
     let mut params ← params.mapM internalizeParam
     let decls ← decls.mapM internalizeCodeDecl
     for param in decl.params, arg in argMask do
@@ -235,11 +236,12 @@ where
     for param in decl.params[argMask.size:] do
       let param := { param with type := param.type.instantiateLevelParamsNoCache decl.levelParams us }
       params := params.push (← internalizeParam param)
-    let value := decl.instantiateValueLevelParams us
-    let value ← internalizeCode value
-    let value := attachCodeDecls decls value
-    let type ← value.inferType
+    let code := code.instantiateValueLevelParams decl.levelParams us
+    let code ← internalizeCode code
+    let code := attachCodeDecls decls code
+    let type ← code.inferType
     let type ← mkForallParams params type
+    let value := .code code
     let safe := decl.safe
     let recursive := decl.recursive
     let decl := { name := nameNew, levelParams := levelParamsNew, params, type, value, safe, recursive, inlineAttr? := none : Decl }
@@ -268,6 +270,7 @@ mutual
     let some paramsInfo ← getSpecParamInfo? declName | return none
     unless (← shouldSpecialize paramsInfo args) do return none
     let some decl ← getDecl? declName | return none
+    let .code _ := decl.value | return none
     trace[Compiler.specialize.candidate] "{e.toExpr}, {paramsInfo}"
     let (argMask, params, decls) ← Collector.collect paramsInfo args
     let keyBody := .const declName us (argMask.filterMap id)
@@ -290,7 +293,7 @@ mutual
       let specDecl ← specDecl.simp {}
       let specDecl ← specDecl.simp { etaPoly := true, inlinePartial := true, implementedBy := true }
       let value ← withReader (fun _ => { declName := specDecl.name }) do
-         withParams specDecl.params <| visitCode specDecl.value
+         withParams specDecl.params <| specDecl.value.mapCodeM visitCode
       let specDecl := { specDecl with value }
       modify fun s => { s with decls := s.decls.push specDecl }
       return some (.const specDecl.name usNew argsNew)
@@ -325,7 +328,7 @@ def main (decl : Decl) : SpecializeM Decl := do
   if (← decl.isTemplateLike) then
     return decl
   else
-    let value ← withParams decl.params <| visitCode decl.value
+    let value ← withParams decl.params <| decl.value.mapCodeM visitCode
     return { decl with value }
 
 end Specialize

src/Lean/Compiler/LCNF/Testing.lean

@@ -235,12 +235,17 @@ Assert that the pass under test produces `Decl`s that do not contain
 `Expr.const constName` in their `Code.let` values anymore.
 -/
 def assertDoesNotContainConstAfter (constName : Name) (msg : String) : TestInstaller :=
-  assertForEachDeclAfterEachOccurrence (fun _ decl => !decl.value.containsConst constName) msg
+  assertForEachDeclAfterEachOccurrence
+    fun _ decl =>
+      match decl.value with
+      | .code c => !c.containsConst constName
+      | .extern .. => true
+    msg
 
 def assertNoFun : TestInstaller :=
   assertAfter do
     for decl in (← getDecls) do
-      decl.value.forM fun
+      decl.value.forCodeM fun
         | .fun .. => throwError "declaration `{decl.name}` contains a local function declaration"
         | _ => return ()
 

src/Lean/Compiler/LCNF/ToDecl.lean

@@ -6,6 +6,7 @@ Authors: Leonardo de Moura
 prelude
 import Lean.Meta.Transform
 import Lean.Meta.Match.MatcherInfo
+import Lean.Compiler.ExternAttr
 import Lean.Compiler.ImplementedByAttr
 import Lean.Compiler.LCNF.ToLCNF
 
@@ -96,31 +97,48 @@ The steps for this are roughly:
 def toDecl (declName : Name) : CompilerM Decl := do
   let declName := if let some name := isUnsafeRecName? declName then name else declName
   let some info ← getDeclInfo? declName | throwError "declaration `{declName}` not found"
-  let some value := info.value? (allowOpaque := true) | throwError "declaration `{declName}` does not have a value"
-  let (type, value) ← Meta.MetaM.run' do
-    let type  ← toLCNFType info.type
-    let value ← Meta.lambdaTelescope value fun xs body => do Meta.mkLambdaFVars xs (← Meta.etaExpand body)
-    let value ← replaceUnsafeRecNames value
-    let value ← macroInline value
-    /- Recall that some declarations tagged with `macro_inline` contain matchers. -/
-    let value ← inlineMatchers value
-    /- Recall that `inlineMatchers` may have exposed `ite`s and `dite`s which are tagged as `[macro_inline]`. -/
-    let value ← macroInline value
-    /-
-    Remark: we have disabled the following transformatbion, we will perform it at phase 2, after code specialization.
-    It prevents many optimizations (e.g., "cases-of-ctor").
-    -/
-    -- let value ← applyCasesOnImplementedBy value
-    return (type, value)
-  let value ← toLCNF value
   let safe := !info.isPartial && !info.isUnsafe
   let inlineAttr? := getInlineAttribute? (← getEnv) declName
-  let decl ← if let .fun decl (.return _) := value then
-    eraseFunDecl decl (recursive := false)
-    pure { name := declName, params := decl.params, type, value := decl.value, levelParams := info.levelParams, safe, inlineAttr? : Decl }
+  if let some externAttrData := getExternAttrData? (← getEnv) declName then
+    let paramsFromTypeBinders (expr : Expr) : CompilerM (Array Param) := do
+      let mut params := #[]
+      let mut currentExpr := expr
+      repeat
+        match currentExpr with
+        | .forallE binderName type body _ =>
+          let borrow := isMarkedBorrowed type
+          params := params.push (← mkParam binderName type borrow)
+          currentExpr := body
+        | _ => break
+      return params
+
+    let type ← Meta.MetaM.run' (toLCNFType info.type)
+    let params ← paramsFromTypeBinders type
+    return { name := declName, params, type, value := .extern externAttrData, levelParams := info.levelParams, safe, inlineAttr? }
   else
-    pure { name := declName, params := #[], type, value, levelParams := info.levelParams, safe, inlineAttr? }
-  /- `toLCNF` may eta-reduce simple declarations. -/
-  decl.etaExpand
+    let some value := info.value? (allowOpaque := true) | throwError "declaration `{declName}` does not have a value"
+    let (type, value) ← Meta.MetaM.run' do
+      let type  ← toLCNFType info.type
+      let value ← Meta.lambdaTelescope value fun xs body => do Meta.mkLambdaFVars xs (← Meta.etaExpand body)
+      let value ← replaceUnsafeRecNames value
+      let value ← macroInline value
+      /- Recall that some declarations tagged with `macro_inline` contain matchers. -/
+      let value ← inlineMatchers value
+      /- Recall that `inlineMatchers` may have exposed `ite`s and `dite`s which are tagged as `[macro_inline]`. -/
+      let value ← macroInline value
+      /-
+      Remark: we have disabled the following transformatbion, we will perform it at phase 2, after code specialization.
+      It prevents many optimizations (e.g., "cases-of-ctor").
+      -/
+      -- let value ← applyCasesOnImplementedBy value
+      return (type, value)
+    let code ← toLCNF value
+    let decl ← if let .fun decl (.return _) := code then
+      eraseFunDecl decl (recursive := false)
+      pure { name := declName, params := decl.params, type, value := .code decl.value, levelParams := info.levelParams, safe, inlineAttr? : Decl }
+    else
+      pure { name := declName, params := #[], type, value := .code code, levelParams := info.levelParams, safe, inlineAttr? }
+    /- `toLCNF` may eta-reduce simple declarations. -/
+    decl.etaExpand
 
 end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/ToExpr.lean

@@ -110,7 +110,4 @@ def Code.toExpr (code : Code) (xs : Array FVarId := #[]) : Expr :=
 def FunDeclCore.toExpr (decl : FunDecl) (xs : Array FVarId := #[]) : Expr :=
   run' decl.toExprM xs
 
-def Decl.toExpr (decl : Decl) : Expr :=
-  run do withParams decl.params do mkLambdaM decl.params (← decl.value.toExprM)
-
 end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/ToMono.lean

@@ -143,7 +143,7 @@ where
   go : ToMonoM Decl := do
     let type ← toMonoType decl.type
     let params ← decl.params.mapM (·.toMono)
-    let value ← decl.value.toMono
+    let value ← decl.value.mapCodeM (·.toMono)
     let decl := { decl with type, params, value, levelParams := [] }
     decl.saveMono
     return decl

src/Lean/Compiler/Old.lean

@@ -53,18 +53,3 @@ def isUnsafeRecName? : Name → Option Name
   | _ => none
 
 end Compiler
-
-namespace Environment
-
-/--
-Compile the given block of mutual declarations.
-Assumes the declarations have already been added to the environment using `addDecl`.
--/
-@[extern "lean_compile_decls"]
-opaque compileDecls (env : Environment) (opt : @& Options) (decls : @& List Name) : Except KernelException Environment
-
-/-- Compile the given declaration, it assumes the declaration has already been added to the environment using `addDecl`. -/
-def compileDecl (env : Environment) (opt : @& Options) (decl : @& Declaration) : Except KernelException Environment :=
-  compileDecls env opt (Compiler.getDeclNamesForCodeGen decl)
-
-end Environment

src/Lean/CoreM.lean

@@ -514,13 +514,16 @@ register_builtin_option compiler.enableNew : Bool := {
 @[extern "lean_lcnf_compile_decls"]
 opaque compileDeclsNew (declNames : List Name) : CoreM Unit
 
+@[extern "lean_compile_decls"]
+opaque compileDeclsOld (env : Environment) (opt : @& Options) (decls : @& List Name) : Except KernelException Environment
+
 def compileDecl (decl : Declaration) : CoreM Unit := do
   let opts ← getOptions
   let decls := Compiler.getDeclNamesForCodeGen decl
   if compiler.enableNew.get opts then
     compileDeclsNew decls
   let res ← withTraceNode `compiler (fun _ => return m!"compiling old: {decls}") do
-    return (← getEnv).compileDecl opts decl
+    return compileDeclsOld (← getEnv) opts decls
   match res with
   | Except.ok env => setEnv env
   | Except.error (KernelException.other msg) =>
@@ -533,7 +536,7 @@ def compileDecls (decls : List Name) : CoreM Unit := do
   let opts ← getOptions
   if compiler.enableNew.get opts then
     compileDeclsNew decls
-  match (← getEnv).compileDecls opts decls with
+  match compileDeclsOld (← getEnv) opts decls with
   | Except.ok env   => setEnv env
   | Except.error (KernelException.other msg) =>
     throwError msg

src/Lean/Data/AssocList.lean

@@ -24,7 +24,7 @@ abbrev empty : AssocList α β :=
 
 instance : EmptyCollection (AssocList α β) := ⟨empty⟩
 
-abbrev insert (m : AssocList α β) (k : α) (v : β) : AssocList α β :=
+abbrev insertNew (m : AssocList α β) (k : α) (v : β) : AssocList α β :=
   m.cons k v
 
 def isEmpty : AssocList α β → Bool
@@ -77,6 +77,12 @@ def replace [BEq α] (a : α) (b : β) : AssocList α β → AssocList α β
     | true  => cons a b es
     | false => cons k v (replace a b es)
 
+def insert [BEq α] (m : AssocList α β) (k : α) (v : β) : AssocList α β :=
+  if m.contains k then
+    m.replace k v
+  else
+    m.insertNew k v
+
 def erase [BEq α] (a : α) : AssocList α β → AssocList α β
   | nil         => nil
   | cons k v es => match k == a with

src/Lean/Data/Json/Printer.lean

@@ -11,6 +11,22 @@ import Init.Data.List.Impl
 namespace Lean
 namespace Json
 
+set_option maxRecDepth 1024 in
+/--
+This table contains for each UTF-8 byte whether we need to escape a string that contains it.
+-/
+private def escapeTable : { xs : ByteArray // xs.size = 256 } :=
+  ⟨ByteArray.mk #[
+    1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
+    0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+    0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
+    0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
+    1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
+    1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
+    1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
+    1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
+  ], by rfl⟩
+
 private def escapeAux (acc : String) (c : Char) : String :=
   -- escape ", \, \n and \r, keep all other characters ≥ 0x20 and render characters < 0x20 with \u
   if c = '"' then -- hack to prevent emacs from regarding the rest of the file as a string: "
@@ -39,8 +55,27 @@ private def escapeAux (acc : String) (c : Char) : String :=
     let d4 := Nat.digitChar (n % 16)
     acc ++ "\\u" |>.push d1 |>.push d2 |>.push d3 |>.push d4
 
+private def needEscape (s : String) : Bool :=
+  go s 0
+where
+  go (s : String) (i : Nat) : Bool :=
+    if h : i < s.utf8ByteSize then
+      let byte := s.getUtf8Byte i h
+      have h1 : byte.toNat < 256 := UInt8.toNat_lt_size byte
+      have h2 : escapeTable.val.size = 256 := escapeTable.property
+      if escapeTable.val.get byte.toNat (Nat.lt_of_lt_of_eq h1 h2.symm) == 0 then
+        go s (i + 1)
+      else
+        true
+    else
+      false
+
 def escape (s : String) (acc : String := "") : String :=
-  s.foldl escapeAux acc
+  -- If we don't have any characters that need to be escaped we can just append right away.
+  if needEscape s then
+    s.foldl escapeAux acc
+  else
+    acc ++ s
 
 def renderString (s : String) (acc : String := "") : String :=
   let acc := acc ++ "\""

src/Lean/Data/Lsp.lean

@@ -6,6 +6,7 @@ Authors: Marc Huisinga, Wojciech Nawrocki
 -/
 prelude
 import Lean.Data.Lsp.Basic
+import Lean.Data.Lsp.CancelParams
 import Lean.Data.Lsp.Capabilities
 import Lean.Data.Lsp.Client
 import Lean.Data.Lsp.Communication

src/Lean/Data/Lsp/Basic.lean

@@ -6,7 +6,6 @@ Authors: Marc Huisinga, Wojciech Nawrocki
 -/
 prelude
 import Lean.Data.Json
-import Lean.Data.JsonRpc
 
 /-! Defines most of the 'Basic Structures' in the LSP specification
 (https://microsoft.github.io/language-server-protocol/specifications/specification-current/),
@@ -19,10 +18,6 @@ namespace Lsp
 
 open Json
 
-structure CancelParams where
-  id : JsonRpc.RequestID
-  deriving Inhabited, BEq, ToJson, FromJson
-
 abbrev DocumentUri := String
 
 /-- We adopt the convention that zero-based UTF-16 positions as sent by LSP clients