feat: rename Array.mkArray to replicate

This PR defines `Vector.flatMap`, changes the order of arguments in
`List.flatMap` for consistency, and aligns the lemmas for
`List`/`Array`/`Vector` `flatMap`.

.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

@@ -161,7 +161,10 @@ def pop (a : Array α) : Array α where
   | ⟨[]⟩ => rfl
   | ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
 
-@[extern "lean_mk_array"]
+def replicate {α : Type u} (n : Nat) (v : α) : Array α where
+  toList := List.replicate n v
+
+@[extern "lean_mk_array", deprecated replicate (since := "2025-01-16")]
 def mkArray {α : Type u} (n : Nat) (v : α) : Array α where
   toList := List.replicate n v
 
@@ -244,8 +247,7 @@ 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 +579,12 @@ 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,
@@ -95,24 +95,29 @@ 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
+theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
   simp [← List.toArray_replicate, List.findSome?_replicate]
 
-@[simp] theorem findSome?_mkArray_of_pos (h : 0 < n) : findSome? f (mkArray n a) = f a := by
-  simp [findSome?_mkArray, Nat.ne_of_gt h]
+@[simp] theorem findSome?_replicate_of_pos (h : 0 < n) : findSome? f (replicate n a) = f a := by
+  simp [findSome?_replicate, Nat.ne_of_gt h]
 
 -- Argument is unused, but used to decide whether `simp` should unfold.
-@[simp] theorem findSome?_mkArray_of_isSome (_ : (f a).isSome) :
-   findSome? f (mkArray n a) = if n = 0 then none else f a := by
-  simp [findSome?_mkArray]
+@[simp] theorem findSome?_replicate_of_isSome (_ : (f a).isSome) :
+   findSome? f (replicate n a) = if n = 0 then none else f a := by
+  simp [findSome?_replicate]
 
-@[simp] theorem findSome?_mkArray_of_isNone (h : (f a).isNone) :
-    findSome? f (mkArray n a) = none := by
+@[simp] theorem findSome?_replicate_of_isNone (h : (f a).isNone) :
+    findSome? f (replicate n a) = none := by
   rw [Option.isNone_iff_eq_none] at h
-  simp [findSome?_mkArray, h]
+  simp [findSome?_replicate, h]
+
+@[deprecated findSome?_replicate (since := "2025-01-16")] abbrev findSome?_mkArray := @findSome?_replicate
+@[deprecated findSome?_replicate_of_pos (since := "2025-01-16")] abbrev findSome?_mkArray_of_pos := @findSome?_replicate_of_pos
+@[deprecated findSome?_replicate_of_isSome (since := "2025-01-16")] abbrev findSome?_mkArray_of_isSome := @findSome?_replicate_of_isSome
+@[deprecated findSome?_replicate_of_isNone (since := "2025-01-16")] abbrev findSome?_mkArray_of_isNone := @findSome?_replicate_of_isNone
 
 /-! ### find? -/
 
@@ -203,7 +208,7 @@ 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 +225,7 @@ 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
@@ -244,34 +249,42 @@ theorem find?_flatMap_eq_none {xs : Array α} {f : α → Array β} {p : β →
     (xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
   simp
 
-theorem find?_mkArray :
-    find? p (mkArray n a) = if n = 0 then none else if p a then some a else none := by
+theorem find?_replicate :
+    find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
   simp [← List.toArray_replicate, List.find?_replicate]
 
-@[simp] theorem find?_mkArray_of_length_pos (h : 0 < n) :
-    find? p (mkArray n a) = if p a then some a else none := by
-  simp [find?_mkArray, Nat.ne_of_gt h]
+@[simp] theorem find?_replicate_of_length_pos (h : 0 < n) :
+    find? p (replicate n a) = if p a then some a else none := by
+  simp [find?_replicate, Nat.ne_of_gt h]
 
-@[simp] theorem find?_mkArray_of_pos (h : p a) :
-    find? p (mkArray n a) = if n = 0 then none else some a := by
-  simp [find?_mkArray, h]
+@[simp] theorem find?_replicate_of_pos (h : p a) :
+    find? p (replicate n a) = if n = 0 then none else some a := by
+  simp [find?_replicate, h]
 
-@[simp] theorem find?_mkArray_of_neg (h : ¬ p a) : find? p (mkArray n a) = none := by
-  simp [find?_mkArray, h]
+@[simp] theorem find?_replicate_of_neg (h : ¬ p a) : find? p (replicate n a) = none := by
+  simp [find?_replicate, h]
 
 -- This isn't a `@[simp]` lemma since there is already a lemma for `l.find? p = none` for any `l`.
-theorem find?_mkArray_eq_none {n : Nat} {a : α} {p : α → Bool} :
-    (mkArray n a).find? p = none ↔ n = 0 ∨ !p a := by
+theorem find?_replicate_eq_none {n : Nat} {a : α} {p : α → Bool} :
+    (replicate n a).find? p = none ↔ n = 0 ∨ !p a := by
   simp [← List.toArray_replicate, List.find?_replicate_eq_none, Classical.or_iff_not_imp_left]
 
-@[simp] theorem find?_mkArray_eq_some {n : Nat} {a b : α} {p : α → Bool} :
-    (mkArray n a).find? p = some b ↔ n ≠ 0 ∧ p a ∧ a = b := by
+@[simp] theorem find?_replicate_eq_some {n : Nat} {a b : α} {p : α → Bool} :
+    (replicate n a).find? p = some b ↔ n ≠ 0 ∧ p a ∧ a = b := by
   simp [← List.toArray_replicate]
 
-@[simp] theorem get_find?_mkArray (n : Nat) (a : α) (p : α → Bool) (h) :
-    ((mkArray n a).find? p).get h = a := by
+@[simp] theorem get_find?_replicate (n : Nat) (a : α) (p : α → Bool) (h) :
+    ((replicate n a).find? p).get h = a := by
   simp [← List.toArray_replicate]
 
+@[deprecated find?_replicate (since := "2025-01-16")] abbrev find?_mkArray := @find?_replicate
+@[deprecated find?_replicate_of_length_pos (since := "2025-01-16")] abbrev find?_mkArray_of_length_pos := @find?_replicate_of_length_pos
+@[deprecated find?_replicate_of_pos (since := "2025-01-16")] abbrev find?_mkArray_of_pos := @find?_replicate_of_pos
+@[deprecated find?_replicate_of_neg (since := "2025-01-16")] abbrev find?_mkArray_of_neg := @find?_replicate_of_neg
+@[deprecated find?_replicate_eq_none (since := "2025-01-16")] abbrev find?_mkArray_eq_none := @find?_replicate_eq_none
+@[deprecated find?_replicate_eq_some (since := "2025-01-16")] abbrev find?_mkArray_eq_some := @find?_replicate_eq_some
+@[deprecated get_find?_mkArray (since := "2025-01-16")] abbrev get_find?_mkArray := @get_find?_replicate
+
 theorem find?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Array α)
     (H : ∀ (a : α), a ∈ xs → P a) (p : β → Bool) :
     (xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by

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]
 
@@ -2551,6 +2395,9 @@ 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 +2546,114 @@ 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 +2667,127 @@ 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/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'
@@ -389,9 +392,29 @@ theorem takeWhile_go_toArray (p : α → Bool) (l : List α) (i : Nat) :
   · simp
   · simp_all [List.set_eq_of_length_le]
 
-@[simp] theorem toArray_replicate (n : Nat) (v : α) : (List.replicate n v).toArray = mkArray n v := rfl
+@[simp] theorem toArray_replicate (n : Nat) (v : α) : (List.replicate n v).toArray = Array.replicate n v := rfl
 
 @[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 =>

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/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

@@ -52,13 +52,15 @@ def elimAsList {motive : Vector α n → Sort u}
 @[inline] def mkEmpty (capacity : Nat) : Vector α 0 := ⟨.mkEmpty capacity, rfl⟩
 
 /-- Makes a vector of size `n` with all cells containing `v`. -/
-@[inline] def mkVector (n) (v : α) : Vector α n := ⟨mkArray n v, by simp⟩
+@[inline] def replicate (n) (v : α) : Vector α n := ⟨Array.replicate n v, by simp⟩
+
+@[deprecated replicate (since := "2025-01-16")] abbrev mkVector := @replicate
 
 /-- Returns a vector of size `1` with element `v`. -/
 @[inline] def singleton (v : α) : Vector α 1 := ⟨#[v], rfl⟩
 
 instance [Inhabited α] : Inhabited (Vector α n) where
-  default := mkVector n default
+  default := replicate n default
 
 /-- Get an element of a vector using a `Fin` index. -/
 @[inline] def get (v : Vector α n) (i : Fin n) : α :=
@@ -103,7 +105,7 @@ 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 +138,18 @@ 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 +172,13 @@ 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) :
@@ -245,7 +269,9 @@ theorem toArray_mk (a : Array α) (h : a.size = n) : (Vector.mk a h).toArray = a
   cases v
   simp
 
-@[simp] theorem toArray_mkVector : (mkVector n a).toArray = mkArray n a := rfl
+@[simp] theorem toArray_replicate : (replicate n a).toArray = Array.replicate n a := rfl
+
+@[deprecated toArray_replicate (since := "2025-01-16")] abbrev toArray_mkVector := @toArray_replicate
 
 @[simp] theorem toArray_inj {v w : Vector α n} : v.toArray = w.toArray ↔ v = w := by
   cases v
@@ -365,7 +391,9 @@ theorem toList_swap (a : Vector α n) (i j) (hi hj) :
   cases v
   simp
 
-@[simp] theorem toList_mkVector : (mkVector n a).toList = List.replicate n a := rfl
+@[simp] theorem toList_replicate : (replicate n a).toList = List.replicate n a := rfl
+
+@[deprecated toList_replicate (since := "2025-01-16")] abbrev toList_mkVector := @toList_replicate
 
 theorem toList_inj {v w : Vector α n} : v.toList = w.toList ↔ v = w := by
   cases v
@@ -444,15 +472,19 @@ theorem exists_push {xs : Vector α (n + 1)} :
 theorem singleton_inj : #v[a] = #v[b] ↔ a = b := by
   simp
 
-/-! ### mkVector -/
+/-! ### replicate -/
 
-@[simp] theorem mkVector_zero : mkVector 0 a = #v[] := rfl
+@[simp] theorem replicate_zero : replicate 0 a = #v[] := rfl
 
-theorem mkVector_succ : mkVector (n + 1) a = (mkVector n a).push a := by
-  simp [mkVector, Array.mkArray_succ]
+theorem replicate_succ : replicate (n + 1) a = (replicate n a).push a := by
+  simp [replicate, Array.replicate_succ]
 
-theorem mkVector_inj : mkVector n a = mkVector n b ↔ n = 0 ∨ a = b := by
-  simp [← toArray_inj, toArray_mkVector, Array.mkArray_inj]
+theorem replicate_inj : replicate n a = replicate n b ↔ n = 0 ∨ a = b := by
+  simp [← toArray_inj, toArray_replicate, Array.replicate_inj]
+
+@[deprecated replicate_zero (since := "2025-01-16")] abbrev mkVector_zero := @replicate_zero
+@[deprecated replicate_succ (since := "2025-01-16")] abbrev mkVector_succ := @replicate_succ
+@[deprecated replicate_inj (since := "2025-01-16")] abbrev mkVector_inj := @replicate_inj
 
 /-! ## L[i] and L[i]? -/
 
@@ -673,6 +705,24 @@ 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] :
@@ -963,15 +1013,17 @@ theorem mem_setIfInBounds (v : Vector α n) (i : Nat) (hi : i < n) (a : α) :
   cases w
   simp
 
-@[simp] theorem mkVector_beq_mkVector [BEq α] {a b : α} {n : Nat} :
-    (mkVector n a == mkVector n b) = (n == 0 || a == b) := by
+@[simp] theorem replicate_beq_replicate [BEq α] {a b : α} {n : Nat} :
+    (replicate n a == replicate n b) = (n == 0 || a == b) := by
   cases n with
   | zero => simp
   | succ n =>
-    rw [mkVector_succ, mkVector_succ, push_beq_push, mkVector_beq_mkVector]
+    rw [replicate_succ, replicate_succ, push_beq_push, replicate_beq_replicate]
     rw [Bool.eq_iff_iff]
     simp +contextual
 
+@[deprecated replicate_beq_replicate (since := "2025-01-16")] abbrev mkVector_beq_mkVector := @replicate_beq_replicate
+
 @[simp] theorem reflBEq_iff [BEq α] [NeZero n] : ReflBEq (Vector α n) ↔ ReflBEq α := by
   match n, NeZero.ne n with
   | n + 1, _ =>
@@ -979,8 +1031,8 @@ theorem mem_setIfInBounds (v : Vector α n) (i : Nat) (hi : i < n) (a : α) :
     · intro h
       constructor
       intro a
-      suffices (mkVector (n + 1) a == mkVector (n + 1) a) = true by
-        rw [mkVector_succ, push_beq_push, Bool.and_eq_true] at this
+      suffices (replicate (n + 1) a == replicate (n + 1) a) = true by
+        rw [replicate_succ, push_beq_push, Bool.and_eq_true] at this
         exact this.2
       simp
     · intro h
@@ -995,15 +1047,15 @@ theorem mem_setIfInBounds (v : Vector α n) (i : Nat) (hi : i < n) (a : α) :
     · intro h
       constructor
       · intro a b h
-        have := mkVector_inj (n := n+1) (a := a) (b := b)
+        have := replicate_inj (n := n+1) (a := a) (b := b)
         simp only [Nat.add_one_ne_zero, false_or] at this
         rw [← this]
         apply eq_of_beq
-        rw [mkVector_beq_mkVector]
+        rw [replicate_beq_replicate]
         simpa
       · intro a
-        suffices (mkVector (n + 1) a == mkVector (n + 1) a) = true by
-          rw [mkVector_beq_mkVector] at this
+        suffices (replicate (n + 1) a == replicate (n + 1) a) = true by
+          rw [replicate_beq_replicate] at this
           simpa
         simp
     · intro h
@@ -1023,11 +1075,12 @@ 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 +1088,528 @@ 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 (replicate n a)
+    simp only [replicate, map_mk, mk.injEq, Array.map_inj_left, Array.mem_replicate, 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 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 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]
+
+/-! 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 +1634,6 @@ 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 +1641,6 @@ 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 +1649,37 @@ 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/Grind.lean

@@ -10,3 +10,5 @@ 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

@@ -8,9 +8,13 @@ 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)
 
@@ -24,6 +28,9 @@ theorem and_eq_of_eq_false_right {a b : Prop} (h : b = False) : (a ∧ b) = Fals
 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]
@@ -34,6 +41,15 @@ theorem or_eq_of_eq_false_right {a b : Prop} (h : b = False) : (a ∨ b) = a :=
 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]
@@ -50,4 +66,44 @@ theorem false_of_not_eq_self {a : Prop} (h : (Not a) = a) : False := by
 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/Tactics.lean

@@ -6,17 +6,49 @@ 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
@@ -27,7 +59,7 @@ namespace Lean.Parser.Tactic
 `grind` tactic and related tactics.
 -/
 
--- TODO: configuration option, parameters
-syntax (name := grind) "grind" : tactic
+-- TODO: parameters
+syntax (name := grind) "grind" optConfig ("on_failure " term)? : tactic
 
 end Lean.Parser.Tactic

src/Init/Grind/Util.lean

@@ -9,11 +9,26 @@ import Init.Core
 namespace Lean.Grind
 
 /-- A helper gadget for annotating nested proofs in goals. -/
-def nestedProof (p : Prop) (h : p) : p := h
+def nestedProof (p : Prop) {h : p} : p := h
 
-set_option pp.proofs true
+/--
+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
 
-theorem nestedProof_congr (p q : Prop) (h : p = q) (hp : p) (hq : q) : HEq (nestedProof p hp) (nestedProof q hq) := by
+/-- 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/IR/ExpandResetReuse.lean

@@ -85,7 +85,7 @@ partial def eraseProjIncForAux (y : VarId) (bs : Array FnBody) (mask : Mask) (ke
 /-- Try to erase `inc` instructions on projections of `y` occurring in the tail of `bs`.
    Return the updated `bs` and a bit mask specifying which `inc`s have been removed. -/
 def eraseProjIncFor (n : Nat) (y : VarId) (bs : Array FnBody) : Array FnBody × Mask :=
-  eraseProjIncForAux y bs (mkArray n none) #[]
+  eraseProjIncForAux y bs (Array.replicate n none) #[]
 
 /-- Replace `reuse x ctor ...` with `ctor ...`, and remove `dec x` -/
 partial def reuseToCtor (x : VarId) : FnBody → FnBody

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/FixedParams.lean

@@ -169,7 +169,7 @@ def mkFixedParamsMap (decls : Array Decl) : NameMap (Array Bool) := Id.run do
   for decl in decls do
     let values := mkInitialValues decl.params.size
     let assignment := mkAssignment decl values
-    let fixed := Array.mkArray decl.params.size true
+    let fixed := Array.replicate decl.params.size true
     match decl.value with
     | .code c =>
       match evalCode c |>.run { main := decl, decls, assignment } |>.run { fixed } with

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/Simp/DiscrM.lean

@@ -98,7 +98,7 @@ where
       return { ctx with discrCtorMap := ctx.discrCtorMap.insert discr ctorInfo, ctorDiscrMap := ctx.ctorDiscrMap.insert ctor.toExpr discr }
     else
       -- For the discrCtor map, the constructor parameters are irrelevant for optimizations that use this information
-      let ctorInfo := .ctor ctorVal (mkArray ctorVal.numParams Arg.erased ++ fieldArgs)
+      let ctorInfo := .ctor ctorVal (Array.replicate ctorVal.numParams Arg.erased ++ fieldArgs)
       return { ctx with discrCtorMap := ctx.discrCtorMap.insert discr ctorInfo }
 
 @[inline, inherit_doc withDiscrCtorImp] def withDiscrCtor [MonadFunctorT DiscrM m] (discr : FVarId) (ctorName : Name) (ctorFields : Array Param) : m α → m α :=

src/Lean/Compiler/LCNF/SpecInfo.lean

@@ -147,7 +147,7 @@ def saveSpecParamInfo (decls : Array Decl) : CompilerM Unit := do
   let mut declsInfo := #[]
   for decl in decls do
     if hasNospecializeAttribute (← getEnv) decl.name then
-      declsInfo := declsInfo.push (mkArray decl.params.size .other)
+      declsInfo := declsInfo.push (Array.replicate decl.params.size .other)
     else
       let specArgs? := getSpecializationArgs? (← getEnv) decl.name
       let contains (i : Nat) : Bool := specArgs?.getD #[] |>.contains i

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/Array.lean

@@ -24,7 +24,7 @@ order, exists in the array.
 -/
 def filterPairsM {m} [Monad m] {α} (a : Array α) (f : α → α → m (Bool × Bool)) :
     m (Array α) := do
-  let mut removed := Array.mkArray a.size false
+  let mut removed := Array.replicate a.size false
   let mut numRemoved := 0
   for h1 : i in [:a.size] do for h2 : j in [i+1:a.size] do
     unless removed[i]! || removed[j]! do

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/FuzzyMatching.lean

@@ -99,11 +99,11 @@ private def fuzzyMatchCore (pattern word : String) (patternRoles wordRoles : Arr
   between the substrings pattern[:i+1] and word[:j+1] assuming that pattern[i] misses at word[j] (k = 0, i.e.
   it was matched earlier), or matches at word[j] (k = 1). A value of `none` corresponds to a score of -∞, and is used
   where no such match/miss is possible or for unneeded parts of the table. -/
-  let mut result : Array (Option Int) := Array.mkArray (pattern.length * word.length * 2) none
-  let mut runLengths : Array Int := Array.mkArray (pattern.length * word.length) 0
+  let mut result : Array (Option Int) := Array.replicate (pattern.length * word.length * 2) none
+  let mut runLengths : Array Int := Array.replicate (pattern.length * word.length) 0
 
   -- penalty for starting a consecutive run at each index
-  let mut startPenalties : Array Int := Array.mkArray word.length 0
+  let mut startPenalties : Array Int := Array.replicate word.length 0
 
   let mut lastSepIdx := 0
   let mut penaltyNs : Int := 0
@@ -124,8 +124,8 @@ private def fuzzyMatchCore (pattern word : String) (patternRoles wordRoles : Arr
    `word.length - pattern.length` at each index (because at the very end, we can only consider fuzzy matches
     of `pattern` with a longer substring of `word`). -/
     for wordIdx in [patternIdx:word.length-(pattern.length - patternIdx - 1)] do
-      let missScore? := 
-        if wordIdx >= 1 then 
+      let missScore? :=
+        if wordIdx >= 1 then
           selectBest
             (getMiss result patternIdx (wordIdx - 1))
             (getMatch result patternIdx (wordIdx - 1))
@@ -134,7 +134,7 @@ private def fuzzyMatchCore (pattern word : String) (patternRoles wordRoles : Arr
       let mut matchScore? := none
 
       if allowMatch (pattern.get ⟨patternIdx⟩) (word.get ⟨wordIdx⟩) (patternRoles.get! patternIdx) (wordRoles.get! wordIdx) then
-        if patternIdx >= 1 then 
+        if patternIdx >= 1 then
           let runLength := runLengths.get! (getIdx (patternIdx - 1) (wordIdx - 1)) + 1
           runLengths := runLengths.set! (getIdx patternIdx wordIdx) runLength
 
@@ -213,7 +213,7 @@ private def fuzzyMatchCore (pattern word : String) (patternRoles wordRoles : Arr
       /- Consecutive character match. -/
       if let some bonus := consecutive then
         /- consecutive run bonus -/
-        score := score + bonus 
+        score := score + bonus
       return score
 
 /-- Match the given pattern with the given word using a fuzzy matching

src/Lean/Data/HashMap.lean

@@ -32,7 +32,7 @@ private def numBucketsForCapacity (capacity : Nat) : Nat :=
 def mkHashMapImp {α : Type u} {β : Type v} (capacity := 8) : HashMapImp α β :=
   { size       := 0
     buckets    :=
-    ⟨mkArray (numBucketsForCapacity capacity).nextPowerOfTwo AssocList.nil,
+    ⟨Array.replicate (numBucketsForCapacity capacity).nextPowerOfTwo AssocList.nil,
      by simp; apply Nat.isPowerOfTwo_nextPowerOfTwo⟩ }
 
 namespace HashMapImp
@@ -101,7 +101,7 @@ decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def expand [Hashable α] (size : Nat) (buckets : HashMapBucket α β) : HashMapImp α β :=
   let bucketsNew : HashMapBucket α β := ⟨
-    mkArray (buckets.val.size * 2) AssocList.nil,
+    Array.replicate (buckets.val.size * 2) AssocList.nil,
     by simp; apply Nat.mul2_isPowerOfTwo_of_isPowerOfTwo buckets.property
   ⟩
   { size    := size,

src/Lean/Data/HashSet.lean

@@ -28,7 +28,7 @@ structure HashSetImp (α : Type u) where
 def mkHashSetImp {α : Type u} (capacity := 8) : HashSetImp α :=
   { size       := 0
     buckets    :=
-    ⟨mkArray ((capacity * 4) / 3).nextPowerOfTwo [],
+    ⟨Array.replicate ((capacity * 4) / 3).nextPowerOfTwo [],
      by simp; apply Nat.isPowerOfTwo_nextPowerOfTwo⟩ }
 
 namespace HashSetImp
@@ -92,7 +92,7 @@ decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def expand [Hashable α] (size : Nat) (buckets : HashSetBucket α) : HashSetImp α :=
   let bucketsNew : HashSetBucket α := ⟨
-    mkArray (buckets.val.size * 2) [],
+    Array.replicate (buckets.val.size * 2) [],
     by simp; apply Nat.mul2_isPowerOfTwo_of_isPowerOfTwo buckets.property
   ⟩
   { size    := size,

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

src/Lean/Data/Lsp/CancelParams.lean

@@ -0,0 +1,25 @@
+/-
+Copyright (c) 2020 Marc Huisinga. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Authors: Marc Huisinga, Wojciech Nawrocki
+-/
+prelude
+import Lean.Data.JsonRpc
+
+/-! # Defines `Lean.Lsp.CancelParams`.
+
+This is separate from `Lean.Data.Lsp.Basic` to reduce transitive dependencies.
+-/
+
+namespace Lean
+namespace Lsp
+
+open Json
+
+structure CancelParams where
+  id : JsonRpc.RequestID
+  deriving Inhabited, BEq, ToJson, FromJson
+
+end Lsp
+end Lean

src/Lean/Data/Lsp/Utf16.lean

@@ -6,7 +6,6 @@ Authors: Marc Huisinga, Wojciech Nawrocki
 -/
 prelude
 import Init.Data.String
-import Init.Data.Array
 import Lean.Data.Lsp.Basic
 import Lean.Data.Position
 import Lean.DeclarationRange

src/Lean/Data/PersistentHashMap.lean

@@ -39,7 +39,7 @@ abbrev maxDepth      : USize  := 7
 abbrev maxCollisions : Nat    := 4
 
 def mkEmptyEntriesArray {α β} : Array (Entry α β (Node α β)) :=
-  (Array.mkArray PersistentHashMap.branching.toNat PersistentHashMap.Entry.null)
+  (Array.replicate PersistentHashMap.branching.toNat PersistentHashMap.Entry.null)
 
 end PersistentHashMap
 

src/Lean/Data/PersistentHashSet.lean

@@ -49,3 +49,8 @@ variable {_ : BEq α} {_ : Hashable α}
 
 @[inline] def fold {β : Type v} (f : β → α → β) (init : β) (s : PersistentHashSet α) : β :=
   Id.run $ s.foldM f init
+
+def toList (s : PersistentHashSet α) : List α :=
+  s.set.toList.map (·.1)
+
+end PersistentHashSet

src/Lean/Elab/App.lean

@@ -1474,7 +1474,7 @@ where
     | field::fields, false => .fieldName field field.getId.getString! none fIdent :: toLVals fields false
 
 /-- Resolve `(.$id:ident)` using the expected type to infer namespace. -/
-private partial def resolveDotName (id : Syntax) (expectedType? : Option Expr) : TermElabM Name := do
+private partial def resolveDotName (id : Syntax) (expectedType? : Option Expr) : TermElabM Expr := do
   tryPostponeIfNoneOrMVar expectedType?
   let some expectedType := expectedType?
     | throwError "invalid dotted identifier notation, expected type must be known"
@@ -1489,7 +1489,7 @@ where
         withForallBody body k
       else
         k body
-  go (resultType : Expr) (expectedType : Expr) (previousExceptions : Array Exception) : TermElabM Name := do
+  go (resultType : Expr) (expectedType : Expr) (previousExceptions : Array Exception) : TermElabM Expr := do
     let resultType ← instantiateMVars resultType
     let resultTypeFn := resultType.cleanupAnnotations.getAppFn
     try
@@ -1497,9 +1497,12 @@ where
       let .const declName .. := resultTypeFn.cleanupAnnotations
         | throwError "invalid dotted identifier notation, expected type is not of the form (... → C ...) where C is a constant{indentExpr expectedType}"
       let idNew := declName ++ id.getId.eraseMacroScopes
-      unless (← getEnv).contains idNew do
+      if (← getEnv).contains idNew then
+        mkConst idNew
+      else if let some (fvar, []) ← resolveLocalName idNew then
+        return fvar
+      else
         throwError "invalid dotted identifier notation, unknown identifier `{idNew}` from expected type{indentExpr expectedType}"
-      return idNew
     catch
       | ex@(.error ..) =>
         match (← unfoldDefinition? resultType) with
@@ -1548,7 +1551,7 @@ private partial def elabAppFn (f : Syntax) (lvals : List LVal) (namedArgs : Arra
     | `(_)       => throwError "placeholders '_' cannot be used where a function is expected"
     | `(.$id:ident) =>
         addCompletionInfo <| CompletionInfo.dotId f id.getId (← getLCtx) expectedType?
-        let fConst ← mkConst (← resolveDotName id expectedType?)
+        let fConst ← resolveDotName id expectedType?
         let s ← observing do
           -- Use (force := true) because we want to record the result of .ident resolution even in patterns
           let fConst ← addTermInfo f fConst expectedType? (force := true)

src/Lean/Elab/Calc.lean

@@ -131,14 +131,18 @@ def throwCalcFailure (steps : Array CalcStepView) (expectedType result : Expr) :
     if ← isDefEqGuarded r er then
       let mut failed := false
       unless ← isDefEqGuarded lhs elhs do
+        let (lhs, elhs) ← addPPExplicitToExposeDiff lhs elhs
+        let (lhsTy, elhsTy) ← addPPExplicitToExposeDiff (← inferType lhs) (← inferType elhs)
         logErrorAt steps[0]!.term m!"\
-          invalid 'calc' step, left-hand side is{indentD m!"{lhs} : {← inferType lhs}"}\n\
-          but is expected to be{indentD m!"{elhs} : {← inferType elhs}"}"
+          invalid 'calc' step, left-hand side is{indentD m!"{lhs} : {lhsTy}"}\n\
+          but is expected to be{indentD m!"{elhs} : {elhsTy}"}"
         failed := true
       unless ← isDefEqGuarded rhs erhs do
+        let (rhs, erhs) ← addPPExplicitToExposeDiff rhs erhs
+        let (rhsTy, erhsTy) ← addPPExplicitToExposeDiff (← inferType rhs) (← inferType erhs)
         logErrorAt steps.back!.term m!"\
-          invalid 'calc' step, right-hand side is{indentD m!"{rhs} : {← inferType rhs}"}\n\
-          but is expected to be{indentD m!"{erhs} : {← inferType erhs}"}"
+          invalid 'calc' step, right-hand side is{indentD m!"{rhs} : {rhsTy}"}\n\
+          but is expected to be{indentD m!"{erhs} : {erhsTy}"}"
         failed := true
       if failed then
         throwAbortTerm

src/Lean/Elab/CheckTactic.lean

@@ -38,6 +38,7 @@ def elabCheckTactic : CommandElab := fun stx => do
       | [next] => do
         let (val, _, _) ← matchCheckGoalType stx (←next.getType)
         if !(← Meta.withReducible <| isDefEq val expTerm) then
+          let (val, expTerm) ← addPPExplicitToExposeDiff val expTerm
           throwErrorAt stx
             m!"Term reduces to{indentExpr val}\nbut is expected to reduce to {indentExpr expTerm}"
       | _ => do

src/Lean/Elab/Deriving.lean

@@ -16,3 +16,4 @@ import Lean.Elab.Deriving.FromToJson
 import Lean.Elab.Deriving.SizeOf
 import Lean.Elab.Deriving.Hashable
 import Lean.Elab.Deriving.Ord
+import Lean.Elab.Deriving.ToExpr

src/Lean/Elab/Deriving/ToExpr.lean

@@ -0,0 +1,237 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kyle Miller
+-/
+prelude
+import Lean.Meta.Transform
+import Lean.Elab.Deriving.Basic
+import Lean.Elab.Deriving.Util
+import Lean.ToLevel
+import Lean.ToExpr
+
+/-!
+# `ToExpr` deriving handler
+
+This module defines a `ToExpr` deriving handler for inductive types.
+It supports mutually inductive types as well.
+
+The `ToExpr` deriving handlers support universe level polymorphism, via the `Lean.ToLevel` class.
+To use `ToExpr` in places where there is universe polymorphism, make sure a `[ToLevel.{u}]` instance is available,
+though be aware that the `ToLevel` mechanism does not support `max` or `imax` expressions.
+
+Implementation note: this deriving handler was initially modeled after the `Repr` deriving handler, but
+1. we need to account for universe levels,
+2. the `ToExpr` class has two fields rather than one, and
+3. we don't handle structures specially.
+-/
+
+namespace Lean.Elab.Deriving.ToExpr
+
+open Lean Elab Parser.Term
+open Meta Command Deriving
+
+/--
+Given `args := #[e₁, e₂, …, eₙ]`, constructs the syntax `Expr.app (… (Expr.app (Expr.app f e₁) e₂) …) eₙ`.
+-/
+def mkAppNTerm (f : Term) (args : Array Term) : MetaM Term :=
+  args.foldlM (fun a b => ``(Expr.app $a $b)) f
+
+/-- Fixes the output of `mkInductiveApp` to explicitly reference universe levels. -/
+def updateIndType (indVal : InductiveVal) (t : Term) : TermElabM Term :=
+  let levels := indVal.levelParams.toArray.map mkIdent
+  match t with
+  | `(@$f $args*) => `(@$f.{$levels,*} $args*)
+  | _ => throwError "(internal error) expecting output of `mkInductiveApp`"
+
+/--
+Creates a term that evaluates to an expression representing the inductive type.
+Uses `toExpr` and `toTypeExpr` for the arguments to the type constructor.
+-/
+def mkToTypeExpr (indVal : InductiveVal) (argNames : Array Name) : TermElabM Term := do
+  let levels ← indVal.levelParams.toArray.mapM (fun u => `(Lean.toLevel.{$(mkIdent u)}))
+  forallTelescopeReducing indVal.type fun xs _ => do
+    let mut args : Array Term := #[]
+    for argName in argNames, x in xs do
+      let a := mkIdent argName
+      if ← Meta.isType x then
+        args := args.push <| ← ``(toTypeExpr $a)
+      else
+        args := args.push <| ← ``(toExpr $a)
+    mkAppNTerm (← ``(Expr.const $(quote indVal.name) [$levels,*])) args
+
+/--
+Creates the body of the `toExpr` function for the `ToExpr` instance, which is a `match` expression
+that calls `toExpr` and `toTypeExpr` to assemble an expression for a given term.
+For recursive inductive types, `auxFunName` refers to the `ToExpr` instance for the current type.
+For mutually recursive types, we rely on the local instances set up by `mkLocalInstanceLetDecls`.
+-/
+def mkToExprBody (header : Header) (indVal : InductiveVal) (auxFunName : Name) (levelInsts : Array Term) :
+    TermElabM Term := do
+  let discrs ← mkDiscrs header indVal
+  let alts ← mkAlts
+  `(match $[$discrs],* with $alts:matchAlt*)
+where
+  /-- Create the `match` cases, one per constructor. -/
+  mkAlts : TermElabM (Array (TSyntax ``matchAlt)) := do
+    let levels ← levelInsts.mapM fun inst => `($(inst).toLevel)
+    let mut alts := #[]
+    for ctorName in indVal.ctors do
+      let ctorInfo ← getConstInfoCtor ctorName
+      let alt ← forallTelescopeReducing ctorInfo.type fun xs _ => do
+        let mut patterns := #[]
+        -- add `_` pattern for indices, before the constructor's pattern
+        for _ in [:indVal.numIndices] do
+          patterns := patterns.push (← `(_))
+        let mut ctorArgs := #[]
+        let mut rhsArgs : Array Term := #[]
+        let mkArg (x : Expr) (a : Term) : TermElabM Term := do
+          if (← inferType x).isAppOf indVal.name then
+            `($(mkIdent auxFunName) $levelInsts* $a)
+          else if ← Meta.isType x then
+            ``(toTypeExpr $a)
+          else
+            ``(toExpr $a)
+        -- add `_` pattern for inductive parameters, which are inaccessible
+        for i in [:ctorInfo.numParams] do
+          let a := mkIdent header.argNames[i]!
+          ctorArgs := ctorArgs.push (← `(_))
+          rhsArgs := rhsArgs.push <| ← mkArg xs[i]! a
+        for i in [:ctorInfo.numFields] do
+          let a := mkIdent (← mkFreshUserName `a)
+          ctorArgs := ctorArgs.push a
+          rhsArgs := rhsArgs.push <| ← mkArg xs[ctorInfo.numParams + i]! a
+        patterns := patterns.push (← `(@$(mkIdent ctorName):ident $ctorArgs:term*))
+        let rhs : Term ← mkAppNTerm (← ``(Expr.const $(quote ctorInfo.name) [$levels,*])) rhsArgs
+        `(matchAltExpr| | $[$patterns:term],* => $rhs)
+      alts := alts.push alt
+    return alts
+
+/--
+For nested and mutually recursive inductive types, we define `partial` instances,
+and the strategy is to have local `ToExpr` instances in scope for the body of each instance.
+This way, each instance can freely use `toExpr` and `toTypeExpr` for each of the types in `ctx`.
+
+This is a modified copy of `Lean.Elab.Deriving.mkLocalInstanceLetDecls`,
+since we need to include the `toTypeExpr` field in the `letDecl`
+Note that, for simplicity, each instance gets its own definition of each others' `toTypeExpr` fields.
+These are very simple fields, so avoiding the duplication is not worth it.
+-/
+def mkLocalInstanceLetDecls (ctx : Deriving.Context) (argNames : Array Name) (levelInsts : Array Term) :
+    TermElabM (Array (TSyntax ``Parser.Term.letDecl)) := do
+  let mut letDecls := #[]
+  for indVal in ctx.typeInfos, auxFunName in ctx.auxFunNames do
+    let currArgNames ← mkInductArgNames indVal
+    let numParams    := indVal.numParams
+    let currIndices  := currArgNames[numParams:]
+    let binders      ← mkImplicitBinders currIndices
+    let argNamesNew  := argNames[:numParams] ++ currIndices
+    let indType      ← mkInductiveApp indVal argNamesNew
+    let instName     ← mkFreshUserName `localinst
+    let toTypeExpr   ← mkToTypeExpr indVal argNames
+    -- Recall that mutually inductive types all use the same universe levels, hence we pass the same ToLevel instances to each aux function.
+    let letDecl      ← `(Parser.Term.letDecl| $(mkIdent instName):ident $binders:implicitBinder* : ToExpr $indType :=
+                          { toExpr     := $(mkIdent auxFunName) $levelInsts*,
+                            toTypeExpr := $toTypeExpr })
+    letDecls := letDecls.push letDecl
+  return letDecls
+
+open TSyntax.Compat in
+/--
+Makes a `toExpr` function for the given inductive type.
+The implementation of each `toExpr` function for a (mutual) inductive type is given as top-level private definitions.
+These are assembled into `ToExpr` instances in `mkInstanceCmds`.
+For mutual/nested inductive types, then each of the types' `ToExpr` instances are provided as local instances,
+to wire together the recursion (necessitating these auxiliary definitions being `partial`).
+-/
+def mkAuxFunction (ctx : Deriving.Context) (i : Nat) : TermElabM Command := do
+  let auxFunName := ctx.auxFunNames[i]!
+  let indVal     := ctx.typeInfos[i]!
+  let header     ← mkHeader ``ToExpr 1 indVal
+  /- We make the `ToLevel` instances be explicit here so that we can pass the instances from the instances to the
+     aux functions. This lets us ensure universe level variables are being lined up,
+     without needing to use `ident.{u₁,…,uₙ}` syntax, which could conditionally be incorrect
+     depending on the ambient CommandElabM scope state.
+     TODO(kmill): deriving handlers should run in a scope with no `universes` or `variables`. -/
+  let (toLevelInsts, levelBinders) := Array.unzip <| ← indVal.levelParams.toArray.mapM fun u => do
+    let inst := mkIdent (← mkFreshUserName `inst)
+    return (inst, ← `(explicitBinderF| ($inst : ToLevel.{$(mkIdent u)})))
+  let mut body ← mkToExprBody header indVal auxFunName toLevelInsts
+  if ctx.usePartial then
+    let letDecls ← mkLocalInstanceLetDecls ctx header.argNames toLevelInsts
+    body ← mkLet letDecls body
+  /- We need to alter the last binder (the one for the "target") to have explicit universe levels
+     so that the `ToLevel` instance arguments can use them. -/
+  let addLevels binder :=
+    match binder with
+    | `(bracketedBinderF| ($a : $ty)) => do `(bracketedBinderF| ($a : $(← updateIndType indVal ty)))
+    | _ => throwError "(internal error) expecting inst binder"
+  let binders := header.binders.pop ++ levelBinders ++ #[← addLevels header.binders.back!]
+  if ctx.usePartial then
+    `(private partial def $(mkIdent auxFunName):ident $binders:bracketedBinder* : Expr := $body:term)
+  else
+    `(private         def $(mkIdent auxFunName):ident $binders:bracketedBinder* : Expr := $body:term)
+
+/--
+Creates all the auxiliary functions (using `mkAuxFunction`) for the (mutual) inductive type(s).
+Wraps the resulting definition commands in `mutual ... end`.
+-/
+def mkAuxFunctions (ctx : Deriving.Context) : TermElabM Syntax := do
+  let mut auxDefs := #[]
+  for i in [:ctx.typeInfos.size] do
+    auxDefs := auxDefs.push (← mkAuxFunction ctx i)
+  `(mutual $auxDefs:command* end)
+
+open TSyntax.Compat in
+/--
+Assuming all of the auxiliary definitions exist,
+creates all the `instance` commands for the `ToExpr` instances for the (mutual) inductive type(s).
+This is a modified copy of `Lean.Elab.Deriving.mkInstanceCmds` to account for `ToLevel` instances.
+-/
+def mkInstanceCmds (ctx : Deriving.Context) (typeNames : Array Name) :
+    TermElabM (Array Command) := do
+  let mut instances := #[]
+  for indVal in ctx.typeInfos, auxFunName in ctx.auxFunNames do
+    if typeNames.contains indVal.name then
+      let argNames     ← mkInductArgNames indVal
+      let binders      ← mkImplicitBinders argNames
+      let binders      := binders ++ (← mkInstImplicitBinders ``ToExpr indVal argNames)
+      let (toLevelInsts, levelBinders) := Array.unzip <| ← indVal.levelParams.toArray.mapM fun u => do
+        let inst := mkIdent (← mkFreshUserName `inst)
+        return (inst, ← `(instBinderF| [$inst : ToLevel.{$(mkIdent u)}]))
+      let binders      := binders ++ levelBinders
+      let indType      ← updateIndType indVal (← mkInductiveApp indVal argNames)
+      let toTypeExpr   ← mkToTypeExpr indVal argNames
+      let instCmd ← `(instance $binders:implicitBinder* : ToExpr $indType where
+                        toExpr := $(mkIdent auxFunName) $toLevelInsts*
+                        toTypeExpr := $toTypeExpr)
+      instances := instances.push instCmd
+  return instances
+
+/--
+Returns all the commands necessary to construct the `ToExpr` instances.
+-/
+def mkToExprInstanceCmds (declNames : Array Name) : TermElabM (Array Syntax) := do
+  let ctx ← mkContext "toExpr" declNames[0]!
+  let cmds := #[← mkAuxFunctions ctx] ++ (← mkInstanceCmds ctx declNames)
+  trace[Elab.Deriving.toExpr] "\n{cmds}"
+  return cmds
+
+/--
+The main entry point to the `ToExpr` deriving handler.
+-/
+def mkToExprInstanceHandler (declNames : Array Name) : CommandElabM Bool := do
+  if (← declNames.allM isInductive) && declNames.size > 0 then
+    let cmds ← withFreshMacroScope <| liftTermElabM <| mkToExprInstanceCmds declNames
+    -- Enable autoimplicits, used for universe levels.
+    withScope (fun scope => { scope with opts := autoImplicit.set scope.opts true }) do
+      elabCommand (mkNullNode cmds)
+    return true
+  else
+    return false
+
+builtin_initialize
+  registerDerivingHandler ``Lean.ToExpr mkToExprInstanceHandler
+  registerTraceClass `Elab.Deriving.toExpr
+
+end Lean.Elab.Deriving.ToExpr

src/Lean/Elab/Import.lean

@@ -5,7 +5,7 @@ Authors: Leonardo de Moura, Sebastian Ullrich
 -/
 prelude
 import Lean.Parser.Module
-import Lean.Data.Json
+import Lean.Util.Paths
 
 namespace Lean.Elab
 
@@ -42,4 +42,12 @@ def printImports (input : String) (fileName : Option String) : IO Unit := do
     let fname ← findOLean dep.module
     IO.println fname
 
+@[export lean_print_import_srcs]
+def printImportSrcs (input : String) (fileName : Option String) : IO Unit := do
+  let sp ← initSrcSearchPath
+  let (deps, _, _) ← parseImports input fileName
+  for dep in deps do
+    let fname ← findLean sp dep.module
+    IO.println fname
+
 end Lean.Elab

src/Lean/Elab/Structure.lean

@@ -691,6 +691,9 @@ private def addProjections (r : ElabHeaderResult) (fieldInfos : Array StructFiel
   let env ← getEnv
   let env ← ofExceptKernelException (mkProjections env r.view.declName projNames.toList r.view.isClass)
   setEnv env
+  for fieldInfo in fieldInfos do
+    if fieldInfo.isSubobject then
+      addDeclarationRangesFromSyntax fieldInfo.declName r.view.ref fieldInfo.ref
 
 private def registerStructure (structName : Name) (infos : Array StructFieldInfo) : TermElabM Unit := do
   let fields ← infos.filterMapM fun info => do
@@ -775,14 +778,14 @@ private def setSourceInstImplicit (type : Expr) : Expr :=
 /--
 Creates a projection function to a non-subobject parent.
 -/
-private partial def mkCoercionToCopiedParent (levelParams : List Name) (params : Array Expr) (view : StructView) (source : Expr) (parentStructName : Name) (parentType : Expr) : MetaM StructureParentInfo := do
+private partial def mkCoercionToCopiedParent (levelParams : List Name) (params : Array Expr) (view : StructView) (source : Expr) (parent : StructParentInfo) (parentType : Expr) : MetaM StructureParentInfo := do
   let isProp ← Meta.isProp parentType
   let env ← getEnv
   let structName := view.declName
   let sourceFieldNames := getStructureFieldsFlattened env structName
-  let binfo := if view.isClass && isClass env parentStructName then BinderInfo.instImplicit else BinderInfo.default
+  let binfo := if view.isClass && isClass env parent.structName then BinderInfo.instImplicit else BinderInfo.default
   let mut declType ← instantiateMVars (← mkForallFVars params (← mkForallFVars #[source] parentType))
-  if view.isClass && isClass env parentStructName then
+  if view.isClass && isClass env parent.structName then
     declType := setSourceInstImplicit declType
   declType := declType.inferImplicit params.size true
   let rec copyFields (parentType : Expr) : MetaM Expr := do
@@ -823,7 +826,8 @@ private partial def mkCoercionToCopiedParent (levelParams : List Name) (params :
   -- (Instances will get instance reducibility in `Lean.Elab.Command.addParentInstances`.)
   if !binfo.isInstImplicit && !(← Meta.isProp parentType) then
     setReducibleAttribute declName
-  return { structName := parentStructName, subobject := false, projFn := declName }
+  addDeclarationRangesFromSyntax declName view.ref parent.ref
+  return { structName := parent.structName, subobject := false, projFn := declName }
 
 private def mkRemainingProjections (levelParams : List Name) (params : Array Expr) (view : StructView)
     (parents : Array StructParentInfo) (fieldInfos : Array StructFieldInfo) : TermElabM (Array StructureParentInfo) := do
@@ -844,7 +848,7 @@ private def mkRemainingProjections (levelParams : List Name) (params : Array Exp
           pure { structName := parent.structName, subobject := true, projFn := info.declName }
         else
           let parent_type := (← instantiateMVars parent.type).replace fun e => parentFVarToConst[e]?
-          mkCoercionToCopiedParent levelParams params view source parent.structName parent_type)
+          mkCoercionToCopiedParent levelParams params view source parent parent_type)
       parentInfos := parentInfos.push parentInfo
       if let some fvar := parent.fvar? then
         parentFVarToConst := parentFVarToConst.insert fvar <|

src/Lean/Elab/Tactic.lean

@@ -45,3 +45,4 @@ import Lean.Elab.Tactic.BVDecide
 import Lean.Elab.Tactic.BoolToPropSimps
 import Lean.Elab.Tactic.Classical
 import Lean.Elab.Tactic.Grind
+import Lean.Elab.Tactic.Monotonicity

src/Lean/Elab/Tactic/BVDecide/Frontend/Attr.lean

@@ -38,6 +38,9 @@ declare_config_elab elabBVDecideConfig Lean.Elab.Tactic.BVDecide.Frontend.BVDeci
 builtin_initialize bvNormalizeExt : Meta.SimpExtension ←
   Meta.registerSimpAttr `bv_normalize "simp theorems used by bv_normalize"
 
+builtin_initialize intToBitVecExt : Meta.SimpExtension ←
+  Meta.registerSimpAttr `int_toBitVec "simp theorems used to convert UIntX/IntX statements into BitVec ones"
+
 /-- Builtin `bv_normalize` simprocs. -/
 builtin_initialize builtinBVNormalizeSimprocsRef : IO.Ref Meta.Simp.Simprocs ← IO.mkRef {}
 

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize.lean

@@ -4,342 +4,28 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Henrik Böving
 -/
 prelude
-import Lean.Meta.AppBuilder
-import Lean.Meta.Tactic.AC.Main
-import Lean.Elab.Tactic.Simp
 import Lean.Elab.Tactic.FalseOrByContra
-import Lean.Elab.Tactic.BVDecide.Frontend.Attr
-import Std.Tactic.BVDecide.Normalize
-import Std.Tactic.BVDecide.Syntax
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Basic
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Simproc
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Rewrite
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.AndFlatten
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.EmbeddedConstraint
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.AC
 
 /-!
-This module contains the implementation of `bv_normalize` which is effectively a custom `bv_normalize`
-simp set that is called like this: `simp only [seval, bv_normalize]`. The rules in `bv_normalize`
-fulfill two goals:
-1. Turn all hypothesis involving `Bool` and `BitVec` into the form `x = true` where `x` only consists
-   of a operations on `Bool` and `BitVec`. In particular no `Prop` should be contained. This makes
-   the reflection procedure further down the pipeline much easier to implement.
-2. Apply simplification rules from the Bitwuzla SMT solver.
+This module contains the implementation of `bv_normalize`, the preprocessing tactic for `bv_decide`.
+It is in essence a (slightly reduced) version of the Bitwuzla preprocessor together with Lean
+specific details.
 -/
 
 namespace Lean.Elab.Tactic.BVDecide
 namespace Frontend.Normalize
 
 open Lean.Meta
-open Std.Tactic.BVDecide.Normalize
 
-builtin_simproc ↓ [bv_normalize] reduceCond (cond _ _ _) := fun e => do
-  let_expr f@cond α c tb eb := e | return .continue
-  let r ← Simp.simp c
-  if r.expr.cleanupAnnotations.isConstOf ``Bool.true then
-    let pr := mkApp (mkApp4 (mkConst ``Bool.cond_pos f.constLevels!) α c tb eb) (← r.getProof)
-    return .visit { expr := tb, proof? := pr }
-  else if r.expr.cleanupAnnotations.isConstOf ``Bool.false then
-    let pr := mkApp (mkApp4 (mkConst ``Bool.cond_neg f.constLevels!) α c tb eb) (← r.getProof)
-    return .visit { expr := eb, proof? := pr }
-  else
-    return .continue
-
-builtin_simproc [bv_normalize] eqToBEq (((_ : Bool) = (_ : Bool))) := fun e => do
-  let_expr Eq _ lhs rhs := e | return .continue
-  match_expr rhs with
-  | Bool.true => return .continue
-  | _ =>
-    let beqApp ← mkAppM ``BEq.beq #[lhs, rhs]
-    let new := mkApp3 (mkConst ``Eq [1]) (mkConst ``Bool) beqApp (mkConst ``Bool.true)
-    let proof := mkApp2 (mkConst ``Bool.eq_to_beq) lhs rhs
-    return .done { expr := new, proof? := some proof }
-
-builtin_simproc [bv_normalize] andOnes ((_ : BitVec _) &&& (_ : BitVec _)) := fun e => do
-  let_expr HAnd.hAnd _ _ _ _ lhs rhs := e | return .continue
-  let some ⟨w, rhsValue⟩ ← getBitVecValue? rhs | return .continue
-  if rhsValue == -1#w then
-    let proof := mkApp2 (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.and_ones) (toExpr w) lhs
-    return .visit { expr := lhs, proof? := some proof }
-  else
-    return .continue
-
-builtin_simproc [bv_normalize] onesAnd ((_ : BitVec _) &&& (_ : BitVec _)) := fun e => do
-  let_expr HAnd.hAnd _ _ _ _ lhs rhs := e | return .continue
-  let some ⟨w, lhsValue⟩ ← getBitVecValue? lhs | return .continue
-  if lhsValue == -1#w then
-    let proof := mkApp2 (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.ones_and) (toExpr w) rhs
-    return .visit { expr := rhs, proof? := some proof }
-  else
-    return .continue
-
-builtin_simproc [bv_normalize] maxUlt (BitVec.ult (_ : BitVec _) (_ : BitVec _)) := fun e => do
-  let_expr BitVec.ult _ lhs rhs := e | return .continue
-  let some ⟨w, lhsValue⟩ ← getBitVecValue? lhs | return .continue
-  if lhsValue == -1#w then
-    let proof := mkApp2 (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.max_ult') (toExpr w) rhs
-    return .visit { expr := toExpr Bool.false, proof? := some proof }
-  else
-    return .continue
-
--- A specialised version of BitVec.neg_eq_not_add so it doesn't trigger on -constant
-builtin_simproc [bv_normalize] neg_eq_not_add (-(_ : BitVec _)) := fun e => do
-  let_expr Neg.neg typ _ val := e | return .continue
-  let_expr BitVec widthExpr := typ | return .continue
-  let some w ← getNatValue? widthExpr | return .continue
-  match ← getBitVecValue? val with
-  | some _ => return .continue
-  | none =>
-    let proof := mkApp2 (mkConst ``BitVec.neg_eq_not_add) (toExpr w) val
-    let expr ← mkAppM ``HAdd.hAdd #[← mkAppM ``Complement.complement #[val], (toExpr 1#w)]
-    return .visit { expr := expr, proof? := some proof }
-
-builtin_simproc [bv_normalize] bv_add_const ((_ : BitVec _) + ((_ : BitVec _) + (_ : BitVec _))) :=
-  fun e => do
-    let_expr HAdd.hAdd _ _ _ _ exp1 rhs := e | return .continue
-    let_expr HAdd.hAdd _ _ _ _ exp2 exp3 := rhs | return .continue
-    let some ⟨w, exp1Val⟩ ←  getBitVecValue? exp1 | return .continue
-    let proofBuilder thm := mkApp4 (mkConst thm) (toExpr w) exp1 exp2 exp3
-    match ← getBitVecValue? exp2 with
-    | some ⟨w', exp2Val⟩ =>
-      if h : w = w' then
-        let newLhs := exp1Val + h ▸ exp2Val
-        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp3]
-        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_left
-        return .visit { expr := expr, proof? := some proof }
-      else
-        return .continue
-    | none =>
-      let some ⟨w', exp3Val⟩ ← getBitVecValue? exp3 | return .continue
-      if h : w = w' then
-        let newLhs := exp1Val + h ▸ exp3Val
-        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp2]
-        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_right
-        return .visit { expr := expr, proof? := some proof }
-      else
-        return .continue
-
-builtin_simproc [bv_normalize] bv_add_const' (((_ : BitVec _) + (_ : BitVec _)) + (_ : BitVec _)) :=
-  fun e => do
-    let_expr HAdd.hAdd _ _ _ _ lhs exp3 := e | return .continue
-    let_expr HAdd.hAdd _ _ _ _ exp1 exp2 := lhs | return .continue
-    let some ⟨w, exp3Val⟩ ←  getBitVecValue? exp3 | return .continue
-    let proofBuilder thm := mkApp4 (mkConst thm) (toExpr w) exp1 exp2 exp3
-    match ← getBitVecValue? exp1 with
-    | some ⟨w', exp1Val⟩ =>
-      if h : w = w' then
-        let newLhs := exp3Val + h ▸ exp1Val
-        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp2]
-        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_left'
-        return .visit { expr := expr, proof? := some proof }
-      else
-        return .continue
-    | none =>
-      let some ⟨w', exp2Val⟩ ← getBitVecValue? exp2 | return .continue
-      if h : w = w' then
-        let newLhs := exp3Val + h ▸ exp2Val
-        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp1]
-        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_right'
-        return .visit { expr := expr, proof? := some proof }
-      else
-        return .continue
-
-/-- Return a number `k` such that `2^k = n`. -/
-private def Nat.log2Exact (n : Nat) : Option Nat := do
-  guard <| n ≠ 0
-  let k := n.log2
-  guard <| Nat.pow 2 k == n
-  return k
-
--- Build an expression for `x ^ y`.
-def mkPow (x y : Expr) : MetaM Expr := mkAppM ``HPow.hPow #[x, y]
-
-builtin_simproc [bv_normalize] bv_udiv_of_two_pow (((_ : BitVec _) / (BitVec.ofNat _ _) : BitVec _)) := fun e => do
-  let_expr HDiv.hDiv _α _β _γ _self x y := e | return .continue
-  let some ⟨w, yVal⟩ ← getBitVecValue? y | return .continue
-  let n := yVal.toNat
-  -- BitVec.ofNat w n, where n =def= 2^k
-  let some k := Nat.log2Exact n | return .continue
-  -- check that k < w.
-  if k ≥ w then return .continue
-  let rhs ← mkAppM ``HShiftRight.hShiftRight #[x, mkNatLit k]
-  -- 2^k = n
-  let hk ← mkDecideProof (← mkEq (← mkPow (mkNatLit 2) (mkNatLit k)) (mkNatLit n))
-  -- k < w
-  let hlt ← mkDecideProof (← mkLt (mkNatLit k) (mkNatLit w))
-  let proof := mkAppN (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.udiv_ofNat_eq_of_lt)
-    #[mkNatLit w, x, mkNatLit n, mkNatLit k, hk, hlt]
-  return .done {
-      expr :=  rhs
-      proof? := some proof
-  }
-
-/--
-A pass in the normalization pipeline. Takes the current goal and produces a refined one or closes
-the goal fully, indicated by returning `none`.
--/
-structure Pass where
-  name : Name
-  run : MVarId → MetaM (Option MVarId)
-
-namespace Pass
-
-/--
-Repeatedly run a list of `Pass` until they either close the goal or an iteration doesn't change
-the goal anymore.
--/
-partial def fixpointPipeline (passes : List Pass) (goal : MVarId) : MetaM (Option MVarId) := do
-  let runPass (goal? : Option MVarId) (pass : Pass) : MetaM (Option MVarId) := do
-    let some goal := goal? | return none
-    withTraceNode `bv (fun _ => return s!"Running pass: {pass.name}") do
-      pass.run goal
-
-  let some newGoal := ← passes.foldlM (init := some goal) runPass | return none
-  if goal != newGoal then
-    trace[Meta.Tactic.bv] m!"Rerunning pipeline on:\n{newGoal}"
-    fixpointPipeline passes newGoal
-  else
-    trace[Meta.Tactic.bv] "Pipeline reached a fixpoint"
-    return newGoal
-
-/--
-Responsible for applying the Bitwuzla style rewrite rules.
--/
-def rewriteRulesPass (maxSteps : Nat) : Pass where
-  name := `rewriteRules
-  run goal := do
-    let bvThms ← bvNormalizeExt.getTheorems
-    let bvSimprocs ← bvNormalizeSimprocExt.getSimprocs
-    let sevalThms ← getSEvalTheorems
-    let sevalSimprocs ← Simp.getSEvalSimprocs
-
-    let simpCtx ← Simp.mkContext
-      (config := { failIfUnchanged := false, zetaDelta := true, maxSteps })
-      (simpTheorems := #[bvThms, sevalThms])
-      (congrTheorems := (← getSimpCongrTheorems))
-
-    let hyps ← goal.getNondepPropHyps
-    let ⟨result?, _⟩ ← simpGoal goal
-      (ctx := simpCtx)
-      (simprocs := #[bvSimprocs, sevalSimprocs])
-      (fvarIdsToSimp := hyps)
-    let some (_, newGoal) := result? | return none
-    return newGoal
-
-/--
-Flatten out ands. That is look for hypotheses of the form `h : (x && y) = true` and replace them
-with `h.left : x = true` and `h.right : y = true`. This can enable more fine grained substitutions
-in embedded constraint substitution.
--/
-partial def andFlatteningPass : Pass where
-  name := `andFlattening
-  run goal := do
-    goal.withContext do
-      let hyps ← goal.getNondepPropHyps
-      let mut newHyps := #[]
-      let mut oldHyps := #[]
-      for fvar in hyps do
-        let hyp : Hypothesis := {
-          userName := (← fvar.getDecl).userName
-          type := ← fvar.getType
-          value := mkFVar fvar
-        }
-        let sizeBefore := newHyps.size
-        newHyps ← splitAnds hyp newHyps
-        if newHyps.size > sizeBefore then
-          oldHyps := oldHyps.push fvar
-      if newHyps.size == 0 then
-        return goal
-      else
-        let (_, goal) ← goal.assertHypotheses newHyps
-        -- Given that we collected the hypotheses in the correct order above the invariant is given
-        let goal ← goal.tryClearMany oldHyps
-        return goal
-where
-  splitAnds (hyp : Hypothesis) (hyps : Array Hypothesis) (first : Bool := true) :
-      MetaM (Array Hypothesis) := do
-    match ← trySplit hyp with
-    | some (left, right) =>
-      let hyps ← splitAnds left hyps false
-      splitAnds right hyps false
-    | none =>
-      if first then
-        return hyps
-      else
-        return hyps.push hyp
-
-  trySplit (hyp : Hypothesis) : MetaM (Option (Hypothesis × Hypothesis)) := do
-    let typ := hyp.type
-    let_expr Eq α eqLhs eqRhs := typ | return none
-    let_expr Bool.and lhs rhs := eqLhs | return none
-    let_expr Bool.true := eqRhs | return none
-    let_expr Bool := α | return none
-    let mkEqTrue (lhs : Expr) : Expr :=
-      mkApp3 (mkConst ``Eq [1]) (mkConst ``Bool) lhs (mkConst ``Bool.true)
-    let leftHyp : Hypothesis := {
-      userName := hyp.userName,
-      type := mkEqTrue lhs,
-      value := mkApp3 (mkConst ``Std.Tactic.BVDecide.Normalize.Bool.and_left) lhs rhs hyp.value
-    }
-    let rightHyp : Hypothesis := {
-      userName := hyp.userName,
-      type := mkEqTrue rhs,
-      value := mkApp3 (mkConst ``Std.Tactic.BVDecide.Normalize.Bool.and_right) lhs rhs hyp.value
-    }
-    return some (leftHyp, rightHyp)
-
-/--
-Substitute embedded constraints. That is look for hypotheses of the form `h : x = true` and use
-them to substitute occurences of `x` within other hypotheses. Additionally this drops all
-redundant top level hypotheses.
--/
-def embeddedConstraintPass (maxSteps : Nat) : Pass where
-  name := `embeddedConstraintSubsitution
-  run goal := do
-    goal.withContext do
-      let hyps ← goal.getNondepPropHyps
-      let mut relevantHyps : SimpTheoremsArray := #[]
-      let mut seen : Std.HashSet Expr := {}
-      let mut duplicates : Array FVarId := #[]
-      for hyp in hyps do
-        let typ ← hyp.getType
-        let_expr Eq α lhs rhs := typ | continue
-        let_expr Bool.true := rhs | continue
-        let_expr Bool := α | continue
-        if seen.contains lhs then
-          -- collect and later remove duplicates on the fly
-          duplicates := duplicates.push hyp
-        else
-          seen := seen.insert lhs
-          let localDecl ← hyp.getDecl
-          let proof  := localDecl.toExpr
-          relevantHyps ← relevantHyps.addTheorem (.fvar hyp) proof
-
-      let goal ← goal.tryClearMany duplicates
-
-      let simpCtx ← Simp.mkContext
-        (config := { failIfUnchanged := false, maxSteps })
-        (simpTheorems := relevantHyps)
-        (congrTheorems := (← getSimpCongrTheorems))
-
-      let ⟨result?, _⟩ ← simpGoal goal (ctx := simpCtx) (fvarIdsToSimp := ← goal.getNondepPropHyps)
-      let some (_, newGoal) := result? | return none
-      return newGoal
-
-/--
-Normalize with respect to Associativity and Commutativity.
--/
-def acNormalizePass : Pass where
-  name := `ac_nf
-  run goal := do
-    let mut newGoal := goal
-    for hyp in (← goal.getNondepPropHyps) do
-      let result ← Lean.Meta.AC.acNfHypMeta newGoal hyp
-
-      if let .some nextGoal := result then
-        newGoal := nextGoal
-      else
-        return none
-
-    return newGoal
-
-def passPipeline (cfg : BVDecideConfig) : List Pass := Id.run do
-  let mut passPipeline := [rewriteRulesPass cfg.maxSteps]
+def passPipeline : PreProcessM (List Pass) := do
+  let mut passPipeline := [rewriteRulesPass]
+  let cfg ← PreProcessM.getConfig
 
   if cfg.acNf then
     passPipeline := passPipeline ++ [acNormalizePass]
@@ -348,18 +34,20 @@ def passPipeline (cfg : BVDecideConfig) : List Pass := Id.run do
     passPipeline := passPipeline ++ [andFlatteningPass]
 
   if cfg.embeddedConstraintSubst then
-    passPipeline := passPipeline ++ [embeddedConstraintPass cfg.maxSteps]
+    passPipeline := passPipeline ++ [embeddedConstraintPass]
 
   return passPipeline
 
-end Pass
-
 def bvNormalize (g : MVarId) (cfg : BVDecideConfig) : MetaM (Option MVarId) := do
-  withTraceNode `bv (fun _ => return "Normalizing goal") do
-    -- Contradiction proof
+  withTraceNode `bv (fun _ => return "Preprocessing goal") do
+    (go g).run cfg g
+where
+  go (g : MVarId) : PreProcessM (Option MVarId) := do
     let some g ← g.falseOrByContra | return none
+
     trace[Meta.Tactic.bv] m!"Running preprocessing pipeline on:\n{g}"
-    Pass.fixpointPipeline (Pass.passPipeline cfg) g
+    let pipeline ← passPipeline
+    Pass.fixpointPipeline pipeline g
 
 @[builtin_tactic Lean.Parser.Tactic.bvNormalize]
 def evalBVNormalize : Tactic := fun

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize/AC.lean

@@ -0,0 +1,39 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+prelude
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Basic
+import Lean.Meta.Tactic.AC.Main
+
+/-!
+This module contains the implementation of the associativity and commutativity normalisation pass
+in the fixpoint pipeline.
+-/
+
+namespace Lean.Elab.Tactic.BVDecide
+namespace Frontend.Normalize
+
+open Lean.Meta
+
+/--
+Normalize with respect to Associativity and Commutativity.
+-/
+def acNormalizePass : Pass where
+  name := `ac_nf
+  run' goal := do
+    let mut newGoal := goal
+    for hyp in (← goal.getNondepPropHyps) do
+      let result ← AC.acNfHypMeta newGoal hyp
+
+      if let .some nextGoal := result then
+        newGoal := nextGoal
+      else
+        return none
+
+    return newGoal
+
+
+end Frontend.Normalize
+end Lean.Elab.Tactic.BVDecide

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize/AndFlatten.lean

@@ -0,0 +1,99 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+prelude
+import Std.Tactic.BVDecide.Normalize.Bool
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Basic
+import Lean.Meta.Tactic.Assert
+
+/-!
+This module contains the implementation of the and flattening pass in the fixpoint pipeline, taking
+hypotheses of the form `h : x && y = true` and splitting them into `h1 : x = true` and
+`h2 : y = true` recursively.
+-/
+
+namespace Lean.Elab.Tactic.BVDecide
+namespace Frontend.Normalize
+
+open Lean.Meta
+
+structure AndFlattenState where
+  hypsToDelete : Array FVarId := #[]
+  hypsToAdd : Array Hypothesis := #[]
+  cache : Std.HashSet Expr := {}
+
+/--
+Flatten out ands. That is look for hypotheses of the form `h : (x && y) = true` and replace them
+with `h.left : x = true` and `h.right : y = true`. This can enable more fine grained substitutions
+in embedded constraint substitution.
+-/
+partial def andFlatteningPass : Pass where
+  name := `andFlattening
+  run' goal := do
+    let (_, { hypsToDelete, hypsToAdd, .. }) ← processGoal goal |>.run {}
+    if hypsToAdd.isEmpty then
+      return goal
+    else
+      let (_, goal) ← goal.assertHypotheses hypsToAdd
+      -- Given that we collected the hypotheses in the correct order above the invariant is given
+      let goal ← goal.tryClearMany hypsToDelete
+      return goal
+where
+  processGoal (goal : MVarId) : StateRefT AndFlattenState MetaM Unit := do
+    goal.withContext do
+      let hyps ← goal.getNondepPropHyps
+      hyps.forM processFVar
+
+  processFVar (fvar : FVarId) : StateRefT AndFlattenState MetaM Unit := do
+    let type ← fvar.getType
+    if (← get).cache.contains type then
+      modify (fun s => { s with hypsToDelete := s.hypsToDelete.push fvar })
+    else
+      let hyp := {
+        userName := (← fvar.getDecl).userName
+        type := type
+        value := mkFVar fvar
+      }
+      let some (lhs, rhs) ← trySplit hyp | return ()
+      modify (fun s => { s with hypsToDelete := s.hypsToDelete.push fvar })
+      splitAnds [lhs, rhs]
+
+  splitAnds (worklist : List Hypothesis) : StateRefT AndFlattenState MetaM Unit := do
+    match worklist with
+    | [] => return ()
+    | hyp :: worklist =>
+      match ← trySplit hyp with
+      | some (left, right) => splitAnds <| left :: right :: worklist
+      | none =>
+        modify (fun s => { s with hypsToAdd := s.hypsToAdd.push hyp })
+        splitAnds worklist
+
+  trySplit (hyp : Hypothesis) :
+      StateRefT AndFlattenState MetaM (Option (Hypothesis × Hypothesis)) := do
+    let typ := hyp.type
+    if (← get).cache.contains typ then
+      return none
+    else
+      modify (fun s => { s with cache := s.cache.insert typ })
+      let_expr Eq _ eqLhs eqRhs := typ | return none
+      let_expr Bool.and lhs rhs := eqLhs | return none
+      let_expr Bool.true := eqRhs | return none
+      let mkEqTrue (lhs : Expr) : Expr :=
+        mkApp3 (mkConst ``Eq [1]) (mkConst ``Bool) lhs (mkConst ``Bool.true)
+      let leftHyp : Hypothesis := {
+        userName := hyp.userName,
+        type := mkEqTrue lhs,
+        value := mkApp3 (mkConst ``Std.Tactic.BVDecide.Normalize.Bool.and_left) lhs rhs hyp.value
+      }
+      let rightHyp : Hypothesis := {
+        userName := hyp.userName,
+        type := mkEqTrue rhs,
+        value := mkApp3 (mkConst ``Std.Tactic.BVDecide.Normalize.Bool.and_right) lhs rhs hyp.value
+      }
+      return some (leftHyp, rightHyp)
+
+
+end Frontend.Normalize
+end Lean.Elab.Tactic.BVDecide

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize/Basic.lean

@@ -0,0 +1,86 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+prelude
+import Lean.Meta.Basic
+import Lean.Elab.Tactic.BVDecide.Frontend.Attr
+
+/-!
+This module contains the basic preprocessing pipeline framework for `bv_normalize`.
+-/
+
+namespace Lean.Elab.Tactic.BVDecide
+namespace Frontend.Normalize
+
+open Lean.Meta
+
+structure PreProcessState where
+  /--
+  Contains `FVarId` that we already know are in `bv_normalize` simp normal form and thus don't
+  need to be processed again when we visit the next time.
+  -/
+  rewriteCache : Std.HashSet FVarId := {}
+
+abbrev PreProcessM : Type → Type := ReaderT BVDecideConfig <| StateRefT PreProcessState MetaM
+
+namespace PreProcessM
+
+def getConfig : PreProcessM BVDecideConfig := read
+
+@[inline]
+def checkRewritten (fvar : FVarId) : PreProcessM Bool := do
+  let val := (← get).rewriteCache.contains fvar
+  trace[Meta.Tactic.bv] m!"{mkFVar fvar} was already rewritten? {val}"
+  return val
+
+@[inline]
+def rewriteFinished (fvar : FVarId) : PreProcessM Unit := do
+  trace[Meta.Tactic.bv] m!"Adding {mkFVar fvar} to the rewritten set"
+  modify (fun s => { s with rewriteCache := s.rewriteCache.insert fvar })
+
+def run (cfg : BVDecideConfig) (goal : MVarId) (x : PreProcessM α) : MetaM α := do
+  let hyps ← goal.getNondepPropHyps
+  ReaderT.run x cfg |>.run' { rewriteCache := Std.HashSet.empty hyps.size }
+
+end PreProcessM
+
+/--
+A pass in the normalization pipeline. Takes the current goal and produces a refined one or closes
+the goal fully, indicated by returning `none`.
+-/
+structure Pass where
+  name : Name
+  run' : MVarId → PreProcessM (Option MVarId)
+
+namespace Pass
+
+def run (pass : Pass) (goal : MVarId) : PreProcessM (Option MVarId) := do
+  withTraceNode `bv (fun _ => return m!"Running pass: {pass.name} on\n{goal}") do
+    pass.run' goal
+
+/--
+Repeatedly run a list of `Pass` until they either close the goal or an iteration doesn't change
+the goal anymore.
+-/
+partial def fixpointPipeline (passes : List Pass) (goal : MVarId) : PreProcessM (Option MVarId) := do
+  let mut newGoal := goal
+  for pass in passes do
+    if let some nextGoal ← pass.run newGoal then
+      newGoal := nextGoal
+    else
+      trace[Meta.Tactic.bv] "Fixpoint iteration solved the goal"
+      return none
+
+  if goal != newGoal then
+    trace[Meta.Tactic.bv] m!"Rerunning pipeline on:\n{newGoal}"
+    fixpointPipeline passes newGoal
+  else
+    trace[Meta.Tactic.bv] "Pipeline reached a fixpoint"
+    return newGoal
+
+end Pass
+
+end Frontend.Normalize
+end Lean.Elab.Tactic.BVDecide

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize/EmbeddedConstraint.lean

@@ -0,0 +1,62 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+prelude
+import Std.Tactic.BVDecide.Normalize.Bool
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Basic
+import Lean.Meta.Tactic.Simp
+
+/-!
+This module contains the implementation of the embedded constraint substitution pass in the fixpoint
+pipeline, substituting hypotheses of the form `h : x = true` in other hypotheses.
+-/
+
+namespace Lean.Elab.Tactic.BVDecide
+namespace Frontend.Normalize
+
+open Lean.Meta
+
+/--
+Substitute embedded constraints. That is look for hypotheses of the form `h : x = true` and use
+them to substitute occurences of `x` within other hypotheses. Additionally this drops all
+redundant top level hypotheses.
+-/
+def embeddedConstraintPass : Pass where
+  name := `embeddedConstraintSubsitution
+  run' goal := do
+    goal.withContext do
+      let hyps ← goal.getNondepPropHyps
+      let mut relevantHyps : SimpTheoremsArray := #[]
+      let mut seen : Std.HashSet Expr := {}
+      let mut duplicates : Array FVarId := #[]
+      for hyp in hyps do
+        let typ ← hyp.getType
+        let_expr Eq _ lhs rhs := typ | continue
+        let_expr Bool.true := rhs | continue
+        if seen.contains lhs then
+          duplicates := duplicates.push hyp
+        else
+          seen := seen.insert lhs
+          let localDecl ← hyp.getDecl
+          let proof := localDecl.toExpr
+          relevantHyps ← relevantHyps.addTheorem (.fvar hyp) proof
+
+      let goal ← goal.tryClearMany duplicates
+
+      if relevantHyps.isEmpty then
+        return goal
+
+      let cfg ← PreProcessM.getConfig
+      let simpCtx ← Simp.mkContext
+        (config := { failIfUnchanged := false, maxSteps := cfg.maxSteps })
+        (simpTheorems := relevantHyps)
+        (congrTheorems := (← getSimpCongrTheorems))
+      let ⟨result?, _⟩ ← simpGoal goal (ctx := simpCtx) (fvarIdsToSimp := ← goal.getNondepPropHyps)
+      let some (_, newGoal) := result? | return none
+      return newGoal
+
+
+end Frontend.Normalize
+end Lean.Elab.Tactic.BVDecide

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize/Rewrite.lean

@@ -0,0 +1,61 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+prelude
+import Lean.Elab.Tactic.Simp
+import Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Basic
+import Lean.Elab.Tactic.BVDecide.Frontend.Attr
+
+/-!
+This module contains the implementation of the rewriting pass in the fixpoint pipeline, applying
+rules from the `bv_normalize` simp set.
+-/
+
+namespace Lean.Elab.Tactic.BVDecide
+namespace Frontend.Normalize
+
+open Lean.Meta
+
+/--
+Responsible for applying the Bitwuzla style rewrite rules.
+-/
+def rewriteRulesPass : Pass where
+  name := `rewriteRules
+  run' goal := do
+    let bvThms ← bvNormalizeExt.getTheorems
+    let bvSimprocs ← bvNormalizeSimprocExt.getSimprocs
+    let sevalThms ← getSEvalTheorems
+    let sevalSimprocs ← Simp.getSEvalSimprocs
+    let cfg ← PreProcessM.getConfig
+
+    let simpCtx ← Simp.mkContext
+      (config := { failIfUnchanged := false, zetaDelta := true, maxSteps := cfg.maxSteps })
+      (simpTheorems := #[bvThms, sevalThms])
+      (congrTheorems := (← getSimpCongrTheorems))
+
+    let hyps ← getHyps goal
+    if hyps.isEmpty then
+      return goal
+    else
+      let ⟨result?, _⟩ ← simpGoal goal
+        (ctx := simpCtx)
+        (simprocs := #[bvSimprocs, sevalSimprocs])
+        (fvarIdsToSimp := hyps)
+
+      let some (_, newGoal) := result? | return none
+      newGoal.withContext do
+        (← newGoal.getNondepPropHyps).forM PreProcessM.rewriteFinished
+      return newGoal
+where
+  getHyps (goal : MVarId) : PreProcessM (Array FVarId) := do
+    goal.withContext do
+      let mut hyps ← goal.getNondepPropHyps
+      let filter hyp := do
+        return !(← PreProcessM.checkRewritten hyp)
+      hyps.filterM filter
+
+
+end Frontend.Normalize
+end Lean.Elab.Tactic.BVDecide

src/Lean/Elab/Tactic/BVDecide/Frontend/Normalize/Simproc.lean

@@ -0,0 +1,164 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+prelude
+import Std.Tactic.BVDecide.Normalize
+import Std.Tactic.BVDecide.Syntax
+import Lean.Elab.Tactic.Simp
+import Lean.Elab.Tactic.BVDecide.Frontend.Attr
+
+/-!
+This module contains implementations of simprocs used in the `bv_normalize` simp set.
+-/
+
+namespace Lean.Elab.Tactic.BVDecide
+namespace Frontend.Normalize
+
+open Lean.Meta
+open Std.Tactic.BVDecide.Normalize
+
+builtin_simproc ↓ [bv_normalize] reduceCond (cond _ _ _) := fun e => do
+  let_expr f@cond α c tb eb := e | return .continue
+  let r ← Simp.simp c
+  if r.expr.cleanupAnnotations.isConstOf ``Bool.true then
+    let pr := mkApp (mkApp4 (mkConst ``Bool.cond_pos f.constLevels!) α c tb eb) (← r.getProof)
+    return .visit { expr := tb, proof? := pr }
+  else if r.expr.cleanupAnnotations.isConstOf ``Bool.false then
+    let pr := mkApp (mkApp4 (mkConst ``Bool.cond_neg f.constLevels!) α c tb eb) (← r.getProof)
+    return .visit { expr := eb, proof? := pr }
+  else
+    return .continue
+
+builtin_simproc [bv_normalize] eqToBEq (((_ : Bool) = (_ : Bool))) := fun e => do
+  let_expr Eq _ lhs rhs := e | return .continue
+  match_expr rhs with
+  | Bool.true => return .continue
+  | _ =>
+    let beqApp ← mkAppM ``BEq.beq #[lhs, rhs]
+    let new := mkApp3 (mkConst ``Eq [1]) (mkConst ``Bool) beqApp (mkConst ``Bool.true)
+    let proof := mkApp2 (mkConst ``Bool.eq_to_beq) lhs rhs
+    return .done { expr := new, proof? := some proof }
+
+builtin_simproc [bv_normalize] andOnes ((_ : BitVec _) &&& (_ : BitVec _)) := fun e => do
+  let_expr HAnd.hAnd _ _ _ _ lhs rhs := e | return .continue
+  let some ⟨w, rhsValue⟩ ← getBitVecValue? rhs | return .continue
+  if rhsValue == -1#w then
+    let proof := mkApp2 (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.and_ones) (toExpr w) lhs
+    return .visit { expr := lhs, proof? := some proof }
+  else
+    return .continue
+
+builtin_simproc [bv_normalize] onesAnd ((_ : BitVec _) &&& (_ : BitVec _)) := fun e => do
+  let_expr HAnd.hAnd _ _ _ _ lhs rhs := e | return .continue
+  let some ⟨w, lhsValue⟩ ← getBitVecValue? lhs | return .continue
+  if lhsValue == -1#w then
+    let proof := mkApp2 (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.ones_and) (toExpr w) rhs
+    return .visit { expr := rhs, proof? := some proof }
+  else
+    return .continue
+
+builtin_simproc [bv_normalize] maxUlt (BitVec.ult (_ : BitVec _) (_ : BitVec _)) := fun e => do
+  let_expr BitVec.ult _ lhs rhs := e | return .continue
+  let some ⟨w, lhsValue⟩ ← getBitVecValue? lhs | return .continue
+  if lhsValue == -1#w then
+    let proof := mkApp2 (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.max_ult') (toExpr w) rhs
+    return .visit { expr := toExpr Bool.false, proof? := some proof }
+  else
+    return .continue
+
+-- A specialised version of BitVec.neg_eq_not_add so it doesn't trigger on -constant
+builtin_simproc [bv_normalize] neg_eq_not_add (-(_ : BitVec _)) := fun e => do
+  let_expr Neg.neg typ _ val := e | return .continue
+  let_expr BitVec widthExpr := typ | return .continue
+  let some w ← getNatValue? widthExpr | return .continue
+  match ← getBitVecValue? val with
+  | some _ => return .continue
+  | none =>
+    let proof := mkApp2 (mkConst ``BitVec.neg_eq_not_add) (toExpr w) val
+    let expr ← mkAppM ``HAdd.hAdd #[← mkAppM ``Complement.complement #[val], (toExpr 1#w)]
+    return .visit { expr := expr, proof? := some proof }
+
+builtin_simproc [bv_normalize] bv_add_const ((_ : BitVec _) + ((_ : BitVec _) + (_ : BitVec _))) :=
+  fun e => do
+    let_expr HAdd.hAdd _ _ _ _ exp1 rhs := e | return .continue
+    let_expr HAdd.hAdd _ _ _ _ exp2 exp3 := rhs | return .continue
+    let some ⟨w, exp1Val⟩ ←  getBitVecValue? exp1 | return .continue
+    let proofBuilder thm := mkApp4 (mkConst thm) (toExpr w) exp1 exp2 exp3
+    match ← getBitVecValue? exp2 with
+    | some ⟨w', exp2Val⟩ =>
+      if h : w = w' then
+        let newLhs := exp1Val + h ▸ exp2Val
+        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp3]
+        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_left
+        return .visit { expr := expr, proof? := some proof }
+      else
+        return .continue
+    | none =>
+      let some ⟨w', exp3Val⟩ ← getBitVecValue? exp3 | return .continue
+      if h : w = w' then
+        let newLhs := exp1Val + h ▸ exp3Val
+        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp2]
+        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_right
+        return .visit { expr := expr, proof? := some proof }
+      else
+        return .continue
+
+builtin_simproc [bv_normalize] bv_add_const' (((_ : BitVec _) + (_ : BitVec _)) + (_ : BitVec _)) :=
+  fun e => do
+    let_expr HAdd.hAdd _ _ _ _ lhs exp3 := e | return .continue
+    let_expr HAdd.hAdd _ _ _ _ exp1 exp2 := lhs | return .continue
+    let some ⟨w, exp3Val⟩ ←  getBitVecValue? exp3 | return .continue
+    let proofBuilder thm := mkApp4 (mkConst thm) (toExpr w) exp1 exp2 exp3
+    match ← getBitVecValue? exp1 with
+    | some ⟨w', exp1Val⟩ =>
+      if h : w = w' then
+        let newLhs := exp3Val + h ▸ exp1Val
+        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp2]
+        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_left'
+        return .visit { expr := expr, proof? := some proof }
+      else
+        return .continue
+    | none =>
+      let some ⟨w', exp2Val⟩ ← getBitVecValue? exp2 | return .continue
+      if h : w = w' then
+        let newLhs := exp3Val + h ▸ exp2Val
+        let expr ← mkAppM ``HAdd.hAdd #[toExpr newLhs, exp1]
+        let proof := proofBuilder ``Std.Tactic.BVDecide.Normalize.BitVec.add_const_right'
+        return .visit { expr := expr, proof? := some proof }
+      else
+        return .continue
+
+/-- Return a number `k` such that `2^k = n`. -/
+private def Nat.log2Exact (n : Nat) : Option Nat := do
+  guard <| n ≠ 0
+  let k := n.log2
+  guard <| Nat.pow 2 k == n
+  return k
+
+-- Build an expression for `x ^ y`.
+def mkPow (x y : Expr) : MetaM Expr := mkAppM ``HPow.hPow #[x, y]
+
+builtin_simproc [bv_normalize] bv_udiv_of_two_pow (((_ : BitVec _) / (BitVec.ofNat _ _) : BitVec _)) := fun e => do
+  let_expr HDiv.hDiv _α _β _γ _self x y := e | return .continue
+  let some ⟨w, yVal⟩ ← getBitVecValue? y | return .continue
+  let n := yVal.toNat
+  -- BitVec.ofNat w n, where n =def= 2^k
+  let some k := Nat.log2Exact n | return .continue
+  -- check that k < w.
+  if k ≥ w then return .continue
+  let rhs ← mkAppM ``HShiftRight.hShiftRight #[x, mkNatLit k]
+  -- 2^k = n
+  let hk ← mkDecideProof (← mkEq (← mkPow (mkNatLit 2) (mkNatLit k)) (mkNatLit n))
+  -- k < w
+  let hlt ← mkDecideProof (← mkLt (mkNatLit k) (mkNatLit w))
+  let proof := mkAppN (mkConst ``Std.Tactic.BVDecide.Normalize.BitVec.udiv_ofNat_eq_of_lt)
+    #[mkNatLit w, x, mkNatLit n, mkNatLit k, hk, hlt]
+  return .done {
+      expr :=  rhs
+      proof? := some proof
+  }
+
+end Frontend.Normalize
+end Lean.Elab.Tactic.BVDecide

src/Lean/Elab/Tactic/BuiltinTactic.lean

@@ -362,9 +362,9 @@ partial def evalChoiceAux (tactics : Array Syntax) (i : Nat) : TacticM Unit :=
   | `(tactic| intro $h:term $hs:term*) => evalTactic (← `(tactic| intro $h:term; intro $hs:term*))
   | _ => throwUnsupportedSyntax
 where
-  introStep (ref : Option Syntax) (n : Name) (typeStx? : Option Syntax := none) : TacticM Unit := do
+  introStep (ref? : Option Syntax) (n : Name) (typeStx? : Option Syntax := none) : TacticM Unit := do
     let fvarId ← liftMetaTacticAux fun mvarId => do
-      let (fvarId, mvarId) ← mvarId.intro n
+      let (fvarId, mvarId) ← withRef? ref? <| mvarId.intro n
       pure (fvarId, [mvarId])
     if let some typeStx := typeStx? then
       withMainContext do
@@ -374,9 +374,9 @@ where
         unless (← isDefEqGuarded type fvarType) do
           throwError "type mismatch at `intro {fvar}`{← mkHasTypeButIsExpectedMsg fvarType type}"
         liftMetaTactic fun mvarId => return [← mvarId.replaceLocalDeclDefEq fvarId type]
-    if let some stx := ref then
+    if let some ref := ref? then
       withMainContext do
-        Term.addLocalVarInfo stx (mkFVar fvarId)
+        Term.addLocalVarInfo ref (mkFVar fvarId)
 
 @[builtin_tactic Lean.Parser.Tactic.introMatch] def evalIntroMatch : Tactic := fun stx => do
   let matchAlts := stx[1]

src/Lean/Elab/Tactic/Classical.lean

@@ -24,11 +24,8 @@ def classical [Monad m] [MonadEnv m] [MonadFinally m] [MonadLiftT MetaM m] (t :
   finally
     modifyEnv Meta.instanceExtension.popScope
 
-@[builtin_tactic Lean.Parser.Tactic.classical]
-def evalClassical : Tactic := fun stx => do
-  match stx with
-  | `(tactic| classical $tacs:tacticSeq) =>
-     classical <| Elab.Tactic.evalTactic tacs
-  | _ => throwUnsupportedSyntax
+@[builtin_tactic Lean.Parser.Tactic.classical, builtin_incremental]
+def evalClassical : Tactic := fun stx =>
+  classical <| Term.withNarrowedArgTacticReuse (argIdx := 1) Elab.Tactic.evalTactic stx
 
 end Lean.Elab.Tactic

src/Lean/Elab/Tactic/Conv/Simp.lean

@@ -7,9 +7,10 @@ prelude
 import Lean.Elab.Tactic.Simp
 import Lean.Elab.Tactic.Split
 import Lean.Elab.Tactic.Conv.Basic
+import Lean.Elab.Tactic.SimpTrace
 
 namespace Lean.Elab.Tactic.Conv
-open Meta
+open Meta Tactic TryThis
 
 def applySimpResult (result : Simp.Result) : TacticM Unit := do
   if result.proof?.isNone then
@@ -23,6 +24,19 @@ def applySimpResult (result : Simp.Result) : TacticM Unit := do
   let (result, _) ← dischargeWrapper.with fun d? => simp lhs ctx (simprocs := simprocs) (discharge? := d?)
   applySimpResult result
 
+@[builtin_tactic Lean.Parser.Tactic.Conv.simpTrace] def evalSimpTrace : Tactic := fun stx => withMainContext do
+  match stx with
+  | `(conv| simp?%$tk $cfg:optConfig $(discharger)? $[only%$o]? $[[$args,*]]?) => do
+    let stx ← `(tactic| simp%$tk $cfg:optConfig $[$discharger]? $[only%$o]? $[[$args,*]]?)
+    let { ctx, simprocs, dischargeWrapper, .. } ← mkSimpContext stx (eraseLocal := false)
+    let lhs ← getLhs
+    let (result, stats) ← dischargeWrapper.with fun d? =>
+      simp lhs ctx (simprocs := simprocs) (discharge? := d?)
+    applySimpResult result
+    let stx ← mkSimpCallStx stx stats.usedTheorems
+    addSuggestion tk stx (origSpan? := ← getRef)
+  | _ => throwUnsupportedSyntax
+
 @[builtin_tactic Lean.Parser.Tactic.Conv.simpMatch] def evalSimpMatch : Tactic := fun _ => withMainContext do
   applySimpResult (← Split.simpMatch (← getLhs))
 
@@ -30,4 +44,15 @@ def applySimpResult (result : Simp.Result) : TacticM Unit := do
   let { ctx, .. } ← mkSimpContext stx (eraseLocal := false) (kind := .dsimp)
   changeLhs (← Lean.Meta.dsimp (← getLhs) ctx).1
 
+@[builtin_tactic Lean.Parser.Tactic.Conv.dsimpTrace] def evalDSimpTrace : Tactic := fun stx => withMainContext do
+  match stx with
+  | `(conv| dsimp?%$tk $cfg:optConfig $[only%$o]? $[[$args,*]]?) =>
+    let stx ← `(tactic| dsimp%$tk $cfg:optConfig $[only%$o]? $[[$args,*]]?)
+    let { ctx, .. } ← mkSimpContext stx (eraseLocal := false) (kind := .dsimp)
+    let (result, stats) ← Lean.Meta.dsimp (← getLhs) ctx
+    changeLhs result
+    let stx ← mkSimpCallStx stx stats.usedTheorems
+    addSuggestion tk stx (origSpan? := ← getRef)
+  | _ => throwUnsupportedSyntax
+
 end Lean.Elab.Tactic.Conv

src/Lean/Elab/Tactic/Grind.lean

@@ -6,22 +6,64 @@ Authors: Leonardo de Moura
 prelude
 import Init.Grind.Tactics
 import Lean.Meta.Tactic.Grind
+import Lean.Elab.Command
 import Lean.Elab.Tactic.Basic
+import Lean.Elab.Tactic.Config
 
 namespace Lean.Elab.Tactic
 open Meta
 
-def grind (mvarId : MVarId) (mainDeclName : Name) : MetaM Unit := do
-  let mvarIds ← Grind.main mvarId mainDeclName
-  unless mvarIds.isEmpty do
-    throwError "`grind` failed\n{goalsToMessageData mvarIds}"
+declare_config_elab elabGrindConfig Grind.Config
+
+open Command Term in
+@[builtin_command_elab Lean.Parser.Command.grindPattern]
+def elabGrindPattern : CommandElab := fun stx => do
+  match stx with
+  | `(grind_pattern $thmName:ident => $terms,*) => do
+    liftTermElabM do
+      let declName ← resolveGlobalConstNoOverload thmName
+      discard <| addTermInfo thmName (← mkConstWithLevelParams declName)
+      let info ← getConstInfo declName
+      forallTelescope info.type fun xs _ => do
+        let patterns ← terms.getElems.mapM fun term => do
+          let pattern ← elabTerm term none
+          synthesizeSyntheticMVarsUsingDefault
+          let pattern ← instantiateMVars pattern
+          let pattern ← Grind.preprocessPattern pattern
+          return pattern.abstract xs
+        Grind.addEMatchTheorem declName xs.size patterns.toList
+  | _ => throwUnsupportedSyntax
+
+def grind (mvarId : MVarId) (config : Grind.Config) (mainDeclName : Name) (fallback : Grind.Fallback) : MetaM Unit := do
+  let goals ← Grind.main mvarId config mainDeclName fallback
+  unless goals.isEmpty do
+    throwError "`grind` failed\n{← Grind.goalsToMessageData goals config}"
+
+private def elabFallback (fallback? : Option Term) : TermElabM (Grind.GoalM Unit) := do
+  let some fallback := fallback? | return (pure ())
+  let type := mkApp (mkConst ``Grind.GoalM) (mkConst ``Unit)
+  let value ← withLCtx {} {} do Term.elabTermAndSynthesize fallback type
+  let auxDeclName ← if let .const declName _ := value then
+    pure declName
+  else
+    let auxDeclName ← Term.mkAuxName `_grind_fallback
+    let decl := Declaration.defnDecl {
+      name := auxDeclName
+      levelParams := []
+      type, value, hints := .opaque, safety := .safe
+    }
+    addAndCompile decl
+    pure auxDeclName
+  unsafe evalConst (Grind.GoalM Unit) auxDeclName
 
 @[builtin_tactic Lean.Parser.Tactic.grind] def evalApplyRfl : Tactic := fun stx => do
   match stx with
-  | `(tactic| grind) =>
+  | `(tactic| grind $config:optConfig $[on_failure $fallback?]?) =>
+    let fallback ← elabFallback fallback?
     logWarningAt stx "The `grind` tactic is experimental and still under development. Avoid using it in production projects"
     let declName := (← Term.getDeclName?).getD `_grind
-    withMainContext do liftMetaFinishingTactic (grind · declName)
+    let config ← elabGrindConfig config
+    withMainContext do liftMetaFinishingTactic (grind · config declName fallback)
   | _ => throwUnsupportedSyntax
 
 end Lean.Elab.Tactic

src/Lean/Elab/Tactic/Induction.lean

@@ -258,11 +258,11 @@ private def saveAltVarsInfo (altMVarId : MVarId) (altStx : Syntax) (fvarIds : Ar
           i := i + 1
 
 open Language in
-def evalAlts (elimInfo : ElimInfo) (alts : Array Alt) (optPreTac : Syntax) (altStxs : Array Syntax)
+def evalAlts (elimInfo : ElimInfo) (alts : Array Alt) (optPreTac : Syntax) (altStxs? : Option (Array Syntax))
     (initialInfo : Info)
     (numEqs : Nat := 0) (numGeneralized : Nat := 0) (toClear : Array FVarId := #[])
     (toTag : Array (Ident × FVarId) := #[]) : TacticM Unit := do
-  let hasAlts := altStxs.size > 0
+  let hasAlts := altStxs?.isSome
   if hasAlts then
     -- default to initial state outside of alts
     -- HACK: because this node has the same span as the original tactic,
@@ -274,9 +274,7 @@ def evalAlts (elimInfo : ElimInfo) (alts : Array Alt) (optPreTac : Syntax) (altS
 where
   -- continuation in the correct info context
   goWithInfo := do
-    let hasAlts := altStxs.size > 0
-
-    if hasAlts then
+    if let some altStxs := altStxs? then
       if let some tacSnap := (← readThe Term.Context).tacSnap? then
         -- incrementality: create a new promise for each alternative, resolve current snapshot to
         -- them, eventually put each of them back in `Context.tacSnap?` in `applyAltStx`
@@ -309,7 +307,8 @@ where
 
   -- continuation in the correct incrementality context
   goWithIncremental (tacSnaps : Array (SnapshotBundle TacticParsedSnapshot)) := do
-    let hasAlts := altStxs.size > 0
+    let hasAlts := altStxs?.isSome
+    let altStxs := altStxs?.getD #[]
     let mut alts := alts
 
     -- initial sanity checks: named cases should be known, wildcards should be last
@@ -343,12 +342,12 @@ where
       let altName := getAltName altStx
       if let some i := alts.findFinIdx? (·.1 == altName) then
         -- cover named alternative
-        applyAltStx tacSnaps altStxIdx altStx alts[i]
+        applyAltStx tacSnaps altStxs altStxIdx altStx alts[i]
         alts := alts.eraseIdx i
       else if !alts.isEmpty && isWildcard altStx then
         -- cover all alternatives
         for alt in alts do
-          applyAltStx tacSnaps altStxIdx altStx alt
+          applyAltStx tacSnaps altStxs altStxIdx altStx alt
         alts := #[]
       else
         throwErrorAt altStx "unused alternative '{altName}'"
@@ -379,7 +378,7 @@ where
         altMVarIds.forM fun mvarId => admitGoal mvarId
 
   /-- Applies syntactic alternative to alternative goal. -/
-  applyAltStx tacSnaps altStxIdx altStx alt := withRef altStx do
+  applyAltStx tacSnaps altStxs altStxIdx altStx alt := withRef altStx do
     let { name := altName, info, mvarId := altMVarId } := alt
     -- also checks for unknown alternatives
     let numFields ← getAltNumFields elimInfo altName
@@ -476,7 +475,7 @@ private def generalizeVars (mvarId : MVarId) (stx : Syntax) (targets : Array Exp
 /--
 Given `inductionAlts` of the form
 ```
-syntax inductionAlts := "with " (tactic)? withPosition( (colGe inductionAlt)+)
+syntax inductionAlts := "with " (tactic)? withPosition( (colGe inductionAlt)*)
 ```
 Return an array containing its alternatives.
 -/
@@ -486,21 +485,30 @@ private def getAltsOfInductionAlts (inductionAlts : Syntax) : Array Syntax :=
 /--
 Given `inductionAlts` of the form
 ```
-syntax inductionAlts := "with " (tactic)? withPosition( (colGe inductionAlt)+)
+syntax inductionAlts := "with " (tactic)? withPosition( (colGe inductionAlt)*)
 ```
-runs `cont alts` where `alts` is an array containing all `inductionAlt`s while disabling incremental
-reuse if any other syntax changed.
+runs `cont (some alts)` where `alts` is an array containing all `inductionAlt`s while disabling incremental
+reuse if any other syntax changed. If there's no `with` clause, then runs `cont none`.
 -/
 private def withAltsOfOptInductionAlts (optInductionAlts : Syntax)
-    (cont : Array Syntax → TacticM α) : TacticM α :=
+    (cont : Option (Array Syntax) → TacticM α) : TacticM α :=
   Term.withNarrowedTacticReuse (stx := optInductionAlts) (fun optInductionAlts =>
     if optInductionAlts.isNone then
       -- if there are no alternatives, what to compare is irrelevant as there will be no reuse
       (mkNullNode #[], mkNullNode #[])
     else
+      -- if there are no alts, then use the `with` token for `inner` for a ref for messages
+      let altStxs := optInductionAlts[0].getArg 2
+      let inner := if altStxs.getNumArgs > 0 then altStxs else optInductionAlts[0][0]
       -- `with` and tactic applied to all branches must be unchanged for reuse
-      (mkNullNode optInductionAlts[0].getArgs[:2], optInductionAlts[0].getArg 2))
-    (fun alts => cont alts.getArgs)
+      (mkNullNode optInductionAlts[0].getArgs[:2], inner))
+    (fun alts? =>
+      if optInductionAlts.isNone then      -- no `with` clause
+        cont none
+      else if alts?.isOfKind nullKind then -- has alts
+        cont (some alts?.getArgs)
+      else                                 -- has `with` clause, but no alts
+        cont (some #[]))
 
 private def getOptPreTacOfOptInductionAlts (optInductionAlts : Syntax) : Syntax :=
   if optInductionAlts.isNone then mkNullNode else optInductionAlts[0][1]
@@ -518,7 +526,7 @@ private def expandMultiAlt? (alt : Syntax) : Option (Array Syntax) := Id.run do
 /--
 Given `inductionAlts` of the form
 ```
-syntax inductionAlts := "with " (tactic)? withPosition( (colGe inductionAlt)+)
+syntax inductionAlts := "with " (tactic)? withPosition( (colGe inductionAlt)*)
 ```
 Return `some inductionAlts'` if one of the alternatives have multiple LHSs, in the new `inductionAlts'`
 all alternatives have a single LHS.
@@ -700,10 +708,10 @@ def evalInduction : Tactic := fun stx =>
         -- unchanged
         -- everything up to the alternatives must be unchanged for reuse
         Term.withNarrowedArgTacticReuse (stx := stx) (argIdx := 4) fun optInductionAlts => do
-        withAltsOfOptInductionAlts optInductionAlts fun alts => do
+        withAltsOfOptInductionAlts optInductionAlts fun alts? => do
           let optPreTac := getOptPreTacOfOptInductionAlts optInductionAlts
           mvarId.assign result.elimApp
-          ElimApp.evalAlts elimInfo result.alts optPreTac alts initInfo (numGeneralized := n) (toClear := targetFVarIds)
+          ElimApp.evalAlts elimInfo result.alts optPreTac alts? initInfo (numGeneralized := n) (toClear := targetFVarIds)
           appendGoals result.others.toList
 where
   checkTargets (targets : Array Expr) : MetaM Unit := do

src/Lean/Elab/Tactic/Monotonicity.lean

@@ -0,0 +1,223 @@
+/-
+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 Lean.Meta.Tactic.Split
+import Lean.Elab.RecAppSyntax
+import Lean.Elab.Tactic.Basic
+import Init.Internal.Order
+
+namespace Lean.Meta.Monotonicity
+
+open Lean Meta
+open Lean.Order
+
+partial def headBetaUnderLambda (f : Expr) : Expr := Id.run do
+  let mut f := f.headBeta
+  if f.isLambda then
+    while f.bindingBody!.isHeadBetaTarget do
+      f := f.updateLambda! f.bindingInfo! f.bindingDomain! f.bindingBody!.headBeta
+  return f
+
+
+/-- Environment extensions for monotonicity lemmas -/
+builtin_initialize monotoneExt :
+    SimpleScopedEnvExtension (Name × Array DiscrTree.Key) (DiscrTree Name) ←
+  registerSimpleScopedEnvExtension {
+    addEntry := fun dt (n, ks) => dt.insertCore ks n
+    initial := {}
+  }
+
+builtin_initialize registerBuiltinAttribute {
+  name := `partial_fixpoint_monotone
+  descr := "monotonicity theorem"
+  add := fun decl _ kind => MetaM.run' do
+    let declTy := (← getConstInfo decl).type
+    let (xs, _, targetTy) ← withReducible <| forallMetaTelescopeReducing declTy
+    let_expr monotone α inst_α β inst_β f := targetTy |
+      throwError "@[partial_fixpoint_monotone] attribute only applies to lemmas proving {.ofConstName ``monotone}"
+    let f := f.headBeta
+    let f ← if f.isLambda then pure f else etaExpand f
+    let f := headBetaUnderLambda f
+    lambdaBoundedTelescope f 1 fun _ e => do
+      let key ← withReducible <| DiscrTree.mkPath e
+      monotoneExt.add (decl, key) kind
+}
+
+/--
+Finds tagged monotonicity theorems of the form `monotone (fun x => e)`.
+-/
+def findMonoThms (e : Expr) : MetaM (Array Name) := do
+  (monotoneExt.getState (← getEnv)).getMatch e
+
+private def defaultFailK (f : Expr) (monoThms : Array Name) : MetaM α :=
+  let extraMsg := if monoThms.isEmpty then m!"" else
+    m!"Tried to apply {.andList (monoThms.toList.map (m!"'{·}'"))}, but failed."
+  throwError "Failed to prove monotonicity of:{indentExpr f}\n{extraMsg}"
+
+private def applyConst (goal : MVarId) (name : Name) : MetaM (List MVarId) := do
+  mapError (f := (m!"Could not apply {.ofConstName name}:{indentD ·}")) do
+    goal.applyConst name (cfg := { synthAssignedInstances := false})
+
+/--
+Base case for solveMonoStep: Handles goals of the form
+```
+monotone (fun f => f.1.2 x y)
+```
+
+It's tricky to solve them compositionally from the outside in, so here we construct the proof
+from the inside out.
+-/
+partial def solveMonoCall (α inst_α : Expr) (e : Expr) : MetaM (Option Expr) := do
+  if e.isApp && !e.appArg!.hasLooseBVars then
+    let some hmono ← solveMonoCall α inst_α e.appFn! | return none
+    let hmonoType ← inferType hmono
+    let_expr monotone _ _ _ inst _ := hmonoType | throwError "solveMonoCall {e}: unexpected type {hmonoType}"
+    let some inst ← whnfUntil inst ``instOrderPi | throwError "solveMonoCall {e}: unexpected instance {inst}"
+    let_expr instOrderPi γ δ inst ← inst | throwError "solveMonoCall {e}: whnfUntil failed?{indentExpr inst}"
+    return ← mkAppOptM ``monotone_apply #[γ, δ, α, inst_α, inst, e.appArg!, none, hmono]
+
+  if e.isProj then
+    let some hmono ← solveMonoCall α inst_α e.projExpr! | return none
+    let hmonoType ← inferType hmono
+    let_expr monotone _ _ _ inst _ := hmonoType | throwError "solveMonoCall {e}: unexpected type {hmonoType}"
+    let some inst ← whnfUntil inst ``instPartialOrderPProd | throwError "solveMonoCall {e}: unexpected instance {inst}"
+    let_expr instPartialOrderPProd β γ inst_β inst_γ ← inst | throwError "solveMonoCall {e}: whnfUntil failed?{indentExpr inst}"
+    let n := if e.projIdx! == 0 then ``monotone_pprod_fst else ``monotone_pprod_snd
+    return ← mkAppOptM n #[β, γ, α, inst_β, inst_γ, inst_α, none, hmono]
+
+  if e == .bvar 0 then
+    let hmono ← mkAppOptM ``monotone_id #[α, inst_α]
+    return some hmono
+
+  return none
+
+
+def solveMonoStep (failK : ∀ {α}, Expr → Array Name → MetaM α := @defaultFailK) (goal : MVarId) : MetaM (List MVarId) :=
+  goal.withContext do
+  trace[Elab.Tactic.monotonicity] "monotonicity at\n{goal}"
+  let type ← goal.getType
+  if type.isForall then
+    let (_, goal) ← goal.intro1P
+    return [goal]
+
+  match_expr type with
+  | monotone α inst_α β inst_β f =>
+    -- Ensure f is not headed by a redex and headed by at least one lambda, and clean some
+    -- redexes left by some of the lemmas we tend to apply
+    let f ← instantiateMVars f
+    let f := f.headBeta
+    let f ← if f.isLambda then pure f else etaExpand f
+    let f := headBetaUnderLambda f
+    let e := f.bindingBody!
+
+    -- No recursive calls left
+    if !e.hasLooseBVars then
+      return ← applyConst goal ``monotone_const
+
+    -- NB: `e` is now an open term.
+
+    -- Look through mdata
+    if e.isMData then
+      let f' := f.updateLambdaE! f.bindingDomain! e.mdataExpr!
+      let goal' ← mkFreshExprSyntheticOpaqueMVar (mkApp type.appFn! f')
+      goal.assign goal'
+      return [goal'.mvarId!]
+
+    -- Float letE to the environment
+    if let .letE n t v b _nonDep := e then
+      if t.hasLooseBVars || v.hasLooseBVars then
+        failK f #[]
+      let goal' ← withLetDecl n t v fun x => do
+        let b' := f.updateLambdaE! f.bindingDomain! (b.instantiate1 x)
+        let goal' ← mkFreshExprSyntheticOpaqueMVar (mkApp type.appFn! b')
+        goal.assign (← mkLetFVars #[x] goal')
+        pure goal'
+      return [goal'.mvarId!]
+
+    -- Float `letFun` to the environment.
+    -- `applyConst` tends to reduce the redex
+    match_expr e with
+    | letFun γ _ v b =>
+      if γ.hasLooseBVars || v.hasLooseBVars then
+        failK f #[]
+      let b' := f.updateLambdaE! f.bindingDomain! b
+      let p  ← mkAppOptM ``monotone_letFun #[α, β, γ, inst_α, inst_β, v, b']
+      let new_goals ← mapError (f := (m!"Could not apply {p}:{indentD ·}")) do
+        goal.apply p
+      let [new_goal] := new_goals
+          | throwError "Unexpected number of goals after {.ofConstName ``monotone_letFun}."
+      let (_, new_goal) ←
+        if b.isLambda then
+          new_goal.intro b.bindingName!
+        else
+          new_goal.intro1
+      return [new_goal]
+    | _ => pure ()
+
+    -- Handle lambdas, preserving the name of the binder
+    if e.isLambda then
+      let [new_goal] ← applyConst goal ``monotone_of_monotone_apply
+        | throwError "Unexpected number of goals after {.ofConstName ``monotone_of_monotone_apply}."
+      let (_, new_goal) ← new_goal.intro e.bindingName!
+      return [new_goal]
+
+    -- A recursive call directly here
+    if e.isBVar then
+      return ← applyConst goal ``monotone_id
+
+    -- A recursive call
+    if let some hmono ← solveMonoCall α inst_α e then
+      trace[Elab.Tactic.monotonicity] "Found recursive call {e}:{indentExpr hmono}"
+      unless ← goal.checkedAssign hmono do
+        trace[Elab.Tactic.monotonicity] "Failed to assign {hmono} : {← inferType hmono} to goal"
+        failK f #[]
+      return []
+
+    let monoThms ← withLocalDeclD `f f.bindingDomain! fun f =>
+      -- The discrimination tree does not like open terms
+      findMonoThms (e.instantiate1 f)
+    trace[Elab.Tactic.monotonicity] "Found monoThms: {monoThms.map MessageData.ofConstName}"
+    for monoThm in monoThms do
+      let new_goals? ← try
+        let new_goals ← applyConst goal monoThm
+        trace[Elab.Tactic.monotonicity] "Succeeded with {.ofConstName monoThm}"
+        pure (some new_goals)
+      catch e =>
+        trace[Elab.Tactic.monotonicity] "{e.toMessageData}"
+        pure none
+      if let some new_goals := new_goals? then
+        return new_goals
+
+    -- Split match-expressions
+    if let some info := isMatcherAppCore? (← getEnv) e then
+      let candidate ← id do
+        let args := e.getAppArgs
+        for i in [info.getFirstDiscrPos : info.getFirstDiscrPos + info.numDiscrs] do
+          if args[i]!.hasLooseBVars then
+            return false
+        return true
+      if candidate then
+        -- We could be even more deliberate here and use the `lifter` lemmas
+        -- for the match statements instead of the `split` tactic.
+        -- For now using `splitMatch` works fine.
+        return ← Split.splitMatch goal e
+
+    failK f monoThms
+  | _ =>
+    throwError "Unexpected goal:{goal}"
+
+partial def solveMono (failK : ∀ {α}, Expr → Array Name → MetaM α := defaultFailK) (goal : MVarId) : MetaM Unit := do
+  let new_goals ← solveMonoStep failK goal
+  new_goals.forM (solveMono failK)
+
+open Elab Tactic in
+@[builtin_tactic Lean.Order.monotonicity]
+def evalMonotonicity : Tactic := fun _stx =>
+  liftMetaTactic Lean.Meta.Monotonicity.solveMonoStep
+
+end Lean.Meta.Monotonicity
+
+builtin_initialize Lean.registerTraceClass `Elab.Tactic.monotonicity

src/Lean/Elab/Tactic/NormCast.lean

@@ -63,7 +63,7 @@ def isNumeral? (e : Expr) : Option (Expr × Nat) :=
   if e.isConstOf ``Nat.zero then
     (mkConst ``Nat, 0)
   else if let Expr.app (Expr.app (Expr.app (Expr.const ``OfNat.ofNat ..) α ..)
-      (Expr.lit (Literal.natVal n) ..) ..) .. := e then
+      (Expr.lit (Literal.natVal n) ..) ..) .. := e.consumeMData then
     some (α, n)
   else
     none

src/Lean/Elab/Tactic/Omega/Frontend.lean

@@ -680,7 +680,7 @@ def omegaTactic (cfg : OmegaConfig) : TacticM Unit := do
 
 /-- The `omega` tactic, for resolving integer and natural linear arithmetic problems. This
 `TacticM Unit` frontend with default configuration can be used as an Aesop rule, for example via
-the tactic call `aesop (add 50% tactic Lean.Omega.omegaDefault)`. -/
+the tactic call `aesop (add 50% tactic Lean.Elab.Tactic.Omega.omegaDefault)`. -/
 def omegaDefault : TacticM Unit := omegaTactic {}
 
 @[builtin_tactic Lean.Parser.Tactic.omega]