fix: case splitting on data

This PR improves the support for `match`-expressions in the `grind`
tactic.

.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()

CMakePresets.json

@@ -26,7 +26,7 @@
       "displayName": "Sanitize build config",
       "cacheVariables": {
         "LEAN_EXTRA_CXX_FLAGS": "-fsanitize=address,undefined",
-        "LEANC_EXTRA_FLAGS": "-fsanitize=address,undefined -fsanitize-link-c++-runtime",
+        "LEANC_EXTRA_CC_FLAGS": "-fsanitize=address,undefined -fsanitize-link-c++-runtime",
         "SMALL_ALLOCATOR": "OFF",
         "BSYMBOLIC": "OFF"
       },

doc/declarations.md

@@ -590,9 +590,9 @@ This table should be read as follows:
  * No other proofs were attempted, either because the parameter has a type without a non-trivial ``WellFounded`` instance (parameter 3), or because it is already clear that no decreasing measure can be found.
 
 
-Lean will print the termination argument it found if ``set_option showInferredTerminationBy true`` is set.
+Lean will print the termination measure it found if ``set_option showInferredTerminationBy true`` is set.
 
-If Lean does not find the termination argument, or if you want to be explicit, you can append a `termination_by` clause to the function definition, after the function's body, but before the `where` clause if present. It is of the form
+If Lean does not find the termination measure, or if you want to be explicit, you can append a `termination_by` clause to the function definition, after the function's body, but before the `where` clause if present. It is of the form
 ```
 termination_by e
 ```
@@ -672,7 +672,7 @@ def num_consts_lst : List Term → Nat
 end
 ```
 
-In a set of mutually recursive function, either all or no functions must have an explicit termination argument (``termination_by``). A change of the default termination tactic (``decreasing_by``) only affects the proofs about the recursive calls of that function, not the other functions in the group.
+In a set of mutually recursive function, either all or no functions must have an explicit termination measure (``termination_by``). A change of the default termination tactic (``decreasing_by``) only affects the proofs about the recursive calls of that function, not the other functions in the group.
 
 ```
 mutual

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/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/lexical_structure.md

@@ -61,7 +61,7 @@ Parts of atomic names can be escaped by enclosing them in pairs of French double
    letterlike_symbols: [℀-⅏]
    escaped_ident_part: "«" [^«»\r\n\t]* "»"
    atomic_ident_rest: atomic_ident_start | [0-9'ⁿ] | subscript
-   subscript: [₀-₉ₐ-ₜᵢ-ᵪ]
+   subscript: [₀-₉ₐ-ₜᵢ-ᵪⱼ]
 ```
 
 String Literals

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" ];

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

@@ -22,6 +22,36 @@ def get_github_token():
         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 {}
@@ -35,11 +65,20 @@ def get_branch_content(repo_url, branch, file_path, github_token):
             return None
     return None
 
-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 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
@@ -64,23 +103,38 @@ def is_merged_into_stable(repo_url, tag_name, stable_branch, github_token):
     stable_commits = [commit['sha'] for commit in commits_response.json()]
     return tag_sha in stable_commits
 
-def parse_version(version_str):
-    # Remove 'v' prefix and split into components
-    # Handle Lean toolchain format (leanprover/lean4:v4.x.y)
-    if ':' in version_str:
-        version_str = version_str.split(':')[1]
-    version = version_str.lstrip('v')
-    # Handle release candidates by removing -rc part for comparison
-    version = version.split('-')[0]
-    return tuple(map(int, version.split('.')))
-
-def is_version_gte(version1, version2):
-    """Check if version1 >= version2"""
-    return parse_version(version1) >= parse_version(version2)
-
 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()
 
@@ -89,6 +143,47 @@ def main():
         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"]
@@ -117,7 +212,7 @@ def main():
         # 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")
+                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")
 

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)
@@ -698,12 +699,12 @@ else()
 endif()
 
 if(NOT ${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
-  add_custom_target(lake_lib ALL
+  add_custom_target(lake_lib
     WORKING_DIRECTORY ${LEAN_SOURCE_DIR}
     DEPENDS leanshared
     COMMAND $(MAKE) -f ${CMAKE_BINARY_DIR}/stdlib.make Lake
     VERBATIM)
-  add_custom_target(lake_shared ALL
+  add_custom_target(lake_shared
     WORKING_DIRECTORY ${LEAN_SOURCE_DIR}
     DEPENDS lake_lib
     COMMAND $(MAKE) -f ${CMAKE_BINARY_DIR}/stdlib.make libLake_shared

src/Init/Core.lean

@@ -516,8 +516,17 @@ The tasks have an overridden representation in the runtime.
 structure Task (α : Type u) : Type u where
   /-- `Task.pure (a : α)` constructs a task that is already resolved with value `a`. -/
   pure ::
-  /-- If `task : Task α` then `task.get : α` blocks the current thread until the
-  value is available, and then returns the result of the task. -/
+  /--
+  Blocks the current thread until the given task has finished execution, and then returns the result
+  of the task. If the current thread is itself executing a (non-dedicated) task, the maximum
+  threadpool size is temporarily increased by one while waiting so as to ensure the process cannot
+  be deadlocked by threadpool starvation. Note that when the current thread is unblocked, more tasks
+  than the configured threadpool size may temporarily be running at the same time until sufficiently
+  many tasks have finished.
+
+  `Task.map` and `Task.bind` should be preferred over `Task.get` for setting up task dependencies
+  where possible as they do not require temporarily growing the threadpool in this way.
+  -/
   get : α
   deriving Inhabited, Nonempty
 

src/Init/Data/Array/Attach.lean

@@ -6,6 +6,7 @@ Authors: Joachim Breitner, Mario Carneiro
 prelude
 import Init.Data.Array.Mem
 import Init.Data.Array.Lemmas
+import Init.Data.Array.Count
 import Init.Data.List.Attach
 
 namespace Array
@@ -142,10 +143,16 @@ theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
   cases l
   simp [List.pmap_eq_map_attach]
 
+@[simp]
+theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l H) :
+    pmap (fun a h => ⟨a, f a h⟩) l H = l.attachWith q (fun x h => f x (H x h)) := by
+  cases l
+  simp [List.pmap_eq_attachWith]
+
 theorem attach_map_coe (l : Array α) (f : α → β) :
     (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
   cases l
-  simp [List.attach_map_coe]
+  simp
 
 theorem attach_map_val (l : Array α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
   attach_map_coe _ _
@@ -172,6 +179,12 @@ theorem mem_attach (l : Array α) : ∀ x, x ∈ l.attach
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
+@[simp]
+theorem mem_attachWith (l : Array α) {q : α → Prop} (H) (x : {x // q x}) :
+    x ∈ l.attachWith q H ↔ x.1 ∈ l := by
+  cases l
+  simp
+
 @[simp]
 theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
     b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
@@ -223,16 +236,16 @@ theorem attachWith_ne_empty_iff {l : Array α} {P : α → Prop} {H : ∀ a ∈
   cases l; simp
 
 @[simp]
-theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) (n : Nat) :
-    (pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (mem_of_getElem? H) := by
+theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) (i : Nat) :
+    (pmap f l h)[i]? = Option.pmap f l[i]? fun x H => h x (mem_of_getElem? H) := by
   cases l; simp
 
 @[simp]
-theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) {n : Nat}
-    (hn : n < (pmap f l h).size) :
-    (pmap f l h)[n] =
-      f (l[n]'(@size_pmap _ _ p f l h ▸ hn))
-        (h _ (getElem_mem (@size_pmap _ _ p f l h ▸ hn))) := by
+theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) {i : Nat}
+    (hi : i < (pmap f l h).size) :
+    (pmap f l h)[i] =
+      f (l[i]'(@size_pmap _ _ p f l h ▸ hi))
+        (h _ (getElem_mem (@size_pmap _ _ p f l h ▸ hi))) := by
   cases l; simp
 
 @[simp]
@@ -256,6 +269,18 @@ theorem getElem_attach {xs : Array α} {i : Nat} (h : i < xs.attach.size) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
+@[simp] theorem pmap_attach (l : Array α) {p : {x // x ∈ l} → Prop} (f : ∀ a, p a → β) (H) :
+    pmap f l.attach H =
+      l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
+  ext <;> simp
+
+@[simp] theorem pmap_attachWith (l : Array α) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
+    pmap f (l.attachWith q H₁) H₂ =
+      l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
+  ext <;> simp
+
 theorem foldl_pmap (l : Array α) {P : α → Prop} (f : (a : α) → P a → β)
   (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
     (l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
@@ -313,11 +338,7 @@ theorem attachWith_map {l : Array α} (f : α → β) {P : β → Prop} {H : ∀
     (l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
   cases l
-  ext
-  · simp
-  · simp only [List.map_toArray, List.attachWith_toArray, List.getElem_toArray,
-      List.getElem_attachWith, List.getElem_map, Function.comp_apply]
-    erw [List.getElem_attachWith] -- Why is `erw` needed here?
+  simp [List.attachWith_map]
 
 theorem map_attachWith {l : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
     (f : { x // P x } → β) :
@@ -347,7 +368,23 @@ theorem attach_filter {l : Array α} (p : α → Bool) :
   simp [List.attach_filter, List.map_filterMap, Function.comp_def]
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
--- Also missing are `filterMap_attach`, `filter_attach`, `filterMap_attachWith` and `filter_attachWith`.
+
+@[simp]
+theorem filterMap_attachWith {q : α → Prop} {l : Array α} {f : {x // q x} → Option β} (H)
+    (w : stop = (l.attachWith q H).size) :
+    (l.attachWith q H).filterMap f 0 stop = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
+  subst w
+  cases l
+  simp [Function.comp_def]
+
+@[simp]
+theorem filter_attachWith {q : α → Prop} {l : Array α} {p : {x // q x} → Bool} (H)
+    (w : stop = (l.attachWith q H).size) :
+    (l.attachWith q H).filter p 0 stop =
+      (l.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
+  subst w
+  cases l
+  simp [Function.comp_def, List.filter_map]
 
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
     pmap f (pmap g l H₁) H₂ =
@@ -427,16 +464,48 @@ theorem reverse_attach (xs : Array α) :
 
 @[simp] theorem back?_attachWith {P : α → Prop} {xs : Array α}
     {H : ∀ (a : α), a ∈ xs → P a} :
-    (xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back?_eq_some h)⟩) := by
+    (xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back? h)⟩) := by
   cases xs
   simp
 
 @[simp]
 theorem back?_attach {xs : Array α} :
-    xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back?_eq_some h⟩ := by
+    xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back? h⟩ := by
   cases xs
   simp
 
+@[simp]
+theorem countP_attach (l : Array α) (p : α → Bool) :
+    l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
+  cases l
+  simp [Function.comp_def]
+
+@[simp]
+theorem countP_attachWith {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) (q : α → Bool) :
+    (l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
+  cases l
+  simp
+
+@[simp]
+theorem count_attach [DecidableEq α] (l : Array α) (a : {x // x ∈ l}) :
+    l.attach.count a = l.count ↑a := by
+  rcases l with ⟨l⟩
+  simp only [List.attach_toArray, List.attachWith_mem_toArray, List.count_toArray]
+  rw [List.map_attach, List.count_eq_countP]
+  simp only [Subtype.beq_iff]
+  rw [List.countP_pmap, List.countP_attach (p := (fun x => x == a.1)), List.count]
+
+@[simp]
+theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
+    (l.attachWith p H).count a = l.count ↑a := by
+  cases l
+  simp
+
+@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (l : Array α) (H₁) :
+    (l.pmap g H₁).countP f =
+      l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
+  simp [pmap_eq_map_attach, countP_map, Function.comp_def]
+
 /-! ## unattach
 
 `Array.unattach` is the (one-sided) inverse of `Array.attach`. It is a synonym for `Array.map Subtype.val`.
@@ -455,7 +524,7 @@ and is ideally subsequently simplified away by `unattach_attach`.
 
 If not, usually the right approach is `simp [Array.unattach, -Array.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) := l.map (·.val)
+def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) : Array α := l.map (·.val)
 
 @[simp] theorem unattach_nil {p : α → Prop} : (#[] : Array { x // p x }).unattach = #[] := rfl
 @[simp] theorem unattach_push {p : α → Prop} {a : { x // p x }} {l : Array { x // p x }} :
@@ -578,4 +647,16 @@ and simplifies these to the function directly taking the value.
   cases l₂
   simp
 
+@[simp] theorem unattach_flatten {p : α → Prop} {l : Array (Array { x // p x })} :
+    l.flatten.unattach = (l.map unattach).flatten := by
+  unfold unattach
+  cases l using array₂_induction
+  simp only [flatten_toArray, List.map_map, Function.comp_def, List.map_id_fun', id_eq,
+    List.map_toArray, List.map_flatten, map_subtype, map_id_fun', List.unattach_toArray, mk.injEq]
+  simp only [List.unattach]
+
+@[simp] theorem unattach_mkArray {p : α → Prop} {n : Nat} {x : { x // p x }} :
+    (Array.mkArray n x).unattach = Array.mkArray n x.1 := by
+  simp [unattach]
+
 end Array

src/Init/Data/Array/Basic.lean

@@ -455,7 +455,7 @@ def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
 /-- Variant of `mapIdxM` which receives the index as a `Fin as.size`. -/
 @[inline]
 def mapFinIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m]
-    (as : Array α) (f : Fin as.size → α → m β) : m (Array β) :=
+    (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → m β) : m (Array β) :=
   let rec @[specialize] map (i : Nat) (j : Nat) (inv : i + j = as.size) (bs : Array β) : m (Array β) := do
     match i, inv with
     | 0,    _  => pure bs
@@ -464,12 +464,12 @@ def mapFinIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m]
         rw [← inv, Nat.add_assoc, Nat.add_comm 1 j, Nat.add_comm]
         apply Nat.le_add_right
       have : i + (j + 1) = as.size := by rw [← inv, Nat.add_comm j 1, Nat.add_assoc]
-      map i (j+1) this (bs.push (← f ⟨j, j_lt⟩ (as.get j j_lt)))
+      map i (j+1) this (bs.push (← f j (as.get j j_lt) j_lt))
   map as.size 0 rfl (mkEmpty as.size)
 
 @[inline]
 def mapIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : Nat → α → m β) (as : Array α) : m (Array β) :=
-  as.mapFinIdxM fun i a => f i a
+  as.mapFinIdxM fun i a _ => f i a
 
 @[inline]
 def findSomeM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m (Option β)) (as : Array α) : m (Option β) := do
@@ -576,13 +576,28 @@ 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))` -/
+@[inline]
+def sum {α} [Add α] [Zero α] : Array α → α :=
+  foldr (· + ·) 0
+
+@[inline]
+def countP {α : Type u} (p : α → Bool) (as : Array α) : Nat :=
+  as.foldr (init := 0) fun a acc => bif p a then acc + 1 else acc
+
+@[inline]
+def count {α : Type u} [BEq α] (a : α) (as : Array α) : Nat :=
+  countP (· == a) as
+
 @[inline]
 def map {α : Type u} {β : Type v} (f : α → β) (as : Array α) : Array β :=
   Id.run <| as.mapM f
 
 /-- Variant of `mapIdx` which receives the index as a `Fin as.size`. -/
 @[inline]
-def mapFinIdx {α : Type u} {β : Type v} (as : Array α) (f : Fin as.size → α → β) : Array β :=
+def mapFinIdx {α : Type u} {β : Type v} (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → β) : Array β :=
   Id.run <| as.mapFinIdxM f
 
 @[inline]

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/Count.lean

@@ -0,0 +1,270 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.Array.Lemmas
+import Init.Data.List.Nat.Count
+
+/-!
+# Lemmas about `Array.countP` and `Array.count`.
+-/
+
+namespace Array
+
+open Nat
+
+/-! ### countP -/
+section countP
+
+variable (p q : α → Bool)
+
+@[simp] theorem countP_empty : countP p #[] = 0 := rfl
+
+@[simp] theorem countP_push_of_pos (l) (pa : p a) : countP p (l.push a) = countP p l + 1 := by
+  rcases l with ⟨l⟩
+  simp_all
+
+@[simp] theorem countP_push_of_neg (l) (pa : ¬p a) : countP p (l.push a) = countP p l := by
+  rcases l with ⟨l⟩
+  simp_all
+
+theorem countP_push (a : α) (l) : countP p (l.push a) = countP p l + if p a then 1 else 0 := by
+  rcases l with ⟨l⟩
+  simp_all
+
+@[simp] theorem countP_singleton (a : α) : countP p #[a] = if p a then 1 else 0 := by
+  simp [countP_push]
+
+theorem size_eq_countP_add_countP (l) : l.size = countP p l + countP (fun a => ¬p a) l := by
+  cases l
+  simp [List.length_eq_countP_add_countP (p := p)]
+
+theorem countP_eq_size_filter (l) : countP p l = (filter p l).size := by
+  cases l
+  simp [List.countP_eq_length_filter]
+
+theorem countP_eq_size_filter' : countP p = size ∘ filter p := by
+  funext l
+  apply countP_eq_size_filter
+
+theorem countP_le_size : countP p l ≤ l.size := by
+  simp only [countP_eq_size_filter]
+  apply size_filter_le
+
+@[simp] theorem countP_append (l₁ l₂) : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
+  cases l₁
+  cases l₂
+  simp
+
+@[simp] theorem countP_pos_iff {p} : 0 < countP p l ↔ ∃ a ∈ l, p a := by
+  cases l
+  simp
+
+@[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p l ↔ ∃ a ∈ l, p a :=
+  countP_pos_iff
+
+@[simp] theorem countP_eq_zero {p} : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
+  cases l
+  simp
+
+@[simp] theorem countP_eq_size {p} : countP p l = l.size ↔ ∀ a ∈ l, p a := by
+  cases l
+  simp
+
+theorem countP_mkArray (p : α → Bool) (a : α) (n : Nat) :
+    countP p (mkArray n a) = if p a then n else 0 := by
+  simp [← List.toArray_replicate, List.countP_replicate]
+
+theorem boole_getElem_le_countP (p : α → Bool) (l : Array α) (i : Nat) (h : i < l.size) :
+    (if p l[i] then 1 else 0) ≤ l.countP p := by
+  cases l
+  simp [List.boole_getElem_le_countP]
+
+theorem countP_set (p : α → Bool) (l : Array α) (i : Nat) (a : α) (h : i < l.size) :
+    (l.set i a).countP p = l.countP p - (if p l[i] then 1 else 0) + (if p a then 1 else 0) := by
+  cases l
+  simp [List.countP_set, h]
+
+theorem countP_filter (l : Array α) :
+    countP p (filter q l) = countP (fun a => p a && q a) l := by
+  cases l
+  simp [List.countP_filter]
+
+@[simp] theorem countP_true : (countP fun (_ : α) => true) = size := by
+  funext l
+  simp
+
+@[simp] theorem countP_false : (countP fun (_ : α) => false) = Function.const _ 0 := by
+  funext l
+  simp
+
+@[simp] theorem countP_map (p : β → Bool) (f : α → β) (l : Array α) :
+    countP p (map f l) = countP (p ∘ f) l := by
+  cases l
+  simp
+
+theorem size_filterMap_eq_countP (f : α → Option β) (l : Array α) :
+    (filterMap f l).size = countP (fun a => (f a).isSome) l := by
+  cases l
+  simp [List.length_filterMap_eq_countP]
+
+theorem countP_filterMap (p : β → Bool) (f : α → Option β) (l : Array α) :
+    countP p (filterMap f l) = countP (fun a => ((f a).map p).getD false) l := by
+  cases l
+  simp [List.countP_filterMap]
+
+@[simp] theorem countP_flatten (l : Array (Array α)) :
+    countP p l.flatten = (l.map (countP p)).sum := by
+  cases l using array₂_induction
+  simp [List.countP_flatten, Function.comp_def]
+
+theorem countP_flatMap (p : β → Bool) (l : Array α) (f : α → Array β) :
+    countP p (l.flatMap f) = sum (map (countP p ∘ f) l) := by
+  cases l
+  simp [List.countP_flatMap, Function.comp_def]
+
+@[simp] theorem countP_reverse (l : Array α) : countP p l.reverse = countP p l := by
+  cases l
+  simp [List.countP_reverse]
+
+variable {p q}
+
+theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
+  cases l
+  simpa using List.countP_mono_left (by simpa using h)
+
+theorem countP_congr (h : ∀ x ∈ l, p x ↔ q x) : countP p l = countP q l :=
+  Nat.le_antisymm
+    (countP_mono_left fun x hx => (h x hx).1)
+    (countP_mono_left fun x hx => (h x hx).2)
+
+end countP
+
+/-! ### count -/
+section count
+
+variable [BEq α]
+
+@[simp] theorem count_empty (a : α) : count a #[] = 0 := rfl
+
+theorem count_push (a b : α) (l : Array α) :
+    count a (l.push b) = count a l + if b == a then 1 else 0 := by
+  simp [count, countP_push]
+
+theorem count_eq_countP (a : α) (l : Array α) : count a l = countP (· == a) l := rfl
+theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
+  funext l
+  apply count_eq_countP
+
+theorem count_le_size (a : α) (l : Array α) : count a l ≤ l.size := countP_le_size _
+
+theorem count_le_count_push (a b : α) (l : Array α) : count a l ≤ count a (l.push b) := by
+  simp [count_push]
+
+@[simp] theorem count_singleton (a b : α) : count a #[b] = if b == a then 1 else 0 := by
+  simp [count_eq_countP]
+
+@[simp] theorem count_append (a : α) : ∀ l₁ l₂, count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
+  countP_append _
+
+@[simp] theorem count_flatten (a : α) (l : Array (Array α)) :
+    count a l.flatten = (l.map (count a)).sum := by
+  cases l using array₂_induction
+  simp [List.count_flatten, Function.comp_def]
+
+@[simp] theorem count_reverse (a : α) (l : Array α) : count a l.reverse = count a l := by
+  cases l
+  simp
+
+theorem boole_getElem_le_count (a : α) (l : Array α) (i : Nat) (h : i < l.size) :
+    (if l[i] == a then 1 else 0) ≤ l.count a := by
+  rw [count_eq_countP]
+  apply boole_getElem_le_countP (· == a)
+
+theorem count_set (a b : α) (l : Array α) (i : Nat) (h : i < l.size) :
+    (l.set i a).count b = l.count b - (if l[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
+  simp [count_eq_countP, countP_set, h]
+
+variable [LawfulBEq α]
+
+@[simp] theorem count_push_self (a : α) (l : Array α) : count a (l.push a) = count a l + 1 := by
+  simp [count_push]
+
+@[simp] theorem count_push_of_ne (h : b ≠ a) (l : Array α) : count a (l.push b) = count a l := by
+  simp_all [count_push, h]
+
+theorem count_singleton_self (a : α) : count a #[a] = 1 := by simp
+
+@[simp]
+theorem count_pos_iff {a : α} {l : Array α} : 0 < count a l ↔ a ∈ l := by
+  simp only [count, countP_pos_iff, beq_iff_eq, exists_eq_right]
+
+@[simp] theorem one_le_count_iff {a : α} {l : Array α} : 1 ≤ count a l ↔ a ∈ l :=
+  count_pos_iff
+
+theorem count_eq_zero_of_not_mem {a : α} {l : Array α} (h : a ∉ l) : count a l = 0 :=
+  Decidable.byContradiction fun h' => h <| count_pos_iff.1 (Nat.pos_of_ne_zero h')
+
+theorem not_mem_of_count_eq_zero {a : α} {l : Array α} (h : count a l = 0) : a ∉ l :=
+  fun h' => Nat.ne_of_lt (count_pos_iff.2 h') h.symm
+
+theorem count_eq_zero {l : Array α} : count a l = 0 ↔ a ∉ l :=
+  ⟨not_mem_of_count_eq_zero, count_eq_zero_of_not_mem⟩
+
+theorem count_eq_size {l : Array α} : count a l = l.size ↔ ∀ b ∈ l, a = b := by
+  rw [count, countP_eq_size]
+  refine ⟨fun h b hb => Eq.symm ?_, fun h b hb => ?_⟩
+  · simpa using h b hb
+  · rw [h b hb, beq_self_eq_true]
+
+@[simp] theorem count_mkArray_self (a : α) (n : Nat) : count a (mkArray n a) = n := by
+  simp [← List.toArray_replicate]
+
+theorem count_mkArray (a b : α) (n : Nat) : count a (mkArray n b) = if b == a then n else 0 := by
+  simp [← List.toArray_replicate, List.count_replicate]
+
+theorem filter_beq (l : Array α) (a : α) : l.filter (· == a) = mkArray (count a l) a := by
+  cases l
+  simp [List.filter_beq]
+
+theorem filter_eq {α} [DecidableEq α] (l : Array α) (a : α) : l.filter (· = a) = mkArray (count a l) a :=
+  filter_beq l a
+
+theorem mkArray_count_eq_of_count_eq_size {l : Array α} (h : count a l = l.size) :
+    mkArray (count a l) a = l := by
+  cases l
+  rw [← toList_inj]
+  simp [List.replicate_count_eq_of_count_eq_length (by simpa using h)]
+
+@[simp] theorem count_filter {l : Array α} (h : p a) : count a (filter p l) = count a l := by
+  cases l
+  simp [List.count_filter, h]
+
+theorem count_le_count_map [DecidableEq β] (l : Array α) (f : α → β) (x : α) :
+    count x l ≤ count (f x) (map f l) := by
+  cases l
+  simp [List.count_le_count_map, countP_map]
+
+theorem count_filterMap {α} [BEq β] (b : β) (f : α → Option β) (l : Array α) :
+    count b (filterMap f l) = countP (fun a => f a == some b) l := by
+  cases l
+  simp [List.count_filterMap, countP_filterMap]
+
+theorem count_flatMap {α} [BEq β] (l : Array α) (f : α → Array β) (x : β) :
+    count x (l.flatMap f) = sum (map (count x ∘ f) l) := by
+  simp [count_eq_countP, countP_flatMap, Function.comp_def]
+
+-- FIXME these theorems can be restored once `List.erase` and `Array.erase` have been related.
+
+-- theorem count_erase (a b : α) (l : Array α) : count a (l.erase b) = count a l - if b == a then 1 else 0 := by
+--   sorry
+
+-- @[simp] theorem count_erase_self (a : α) (l : Array α) :
+--     count a (l.erase a) = count a l - 1 := by rw [count_erase, if_pos (by simp)]
+
+-- @[simp] theorem count_erase_of_ne (ab : a ≠ b) (l : Array α) : count a (l.erase b) = count a l := by
+--   rw [count_erase, if_neg (by simpa using ab.symm), Nat.sub_zero]
+
+end count

src/Init/Data/Array/Find.lean

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

src/Init/Data/Array/MapIdx.lean

@@ -5,6 +5,7 @@ Authors: Mario Carneiro, Kim Morrison
 -/
 prelude
 import Init.Data.Array.Lemmas
+import Init.Data.Array.Attach
 import Init.Data.List.MapIdx
 
 namespace Array
@@ -12,81 +13,82 @@ namespace Array
 /-! ### mapFinIdx -/
 
 -- This could also be proved from `SatisfiesM_mapIdxM` in Batteries.
-theorem mapFinIdx_induction (as : Array α) (f : Fin as.size → α → β)
+theorem mapFinIdx_induction (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → β)
     (motive : Nat → Prop) (h0 : motive 0)
-    (p : Fin as.size → β → Prop)
-    (hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
+    (p : (i : Nat) → β → (h : i < as.size) → Prop)
+    (hs : ∀ i h, motive i → p i (f i as[i] h) h ∧ motive (i + 1)) :
     motive as.size ∧ ∃ eq : (Array.mapFinIdx as f).size = as.size,
-      ∀ i h, p ⟨i, h⟩ ((Array.mapFinIdx as f)[i]) := by
-  let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p ⟨i, h⟩ bs[i]) (hm : motive j) :
+      ∀ i h, p i ((Array.mapFinIdx as f)[i]) h := by
+  let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p i bs[i] h) (hm : motive j) :
     let arr : Array β := Array.mapFinIdxM.map (m := Id) as f i j h bs
-    motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i, h⟩ arr[i] := by
+    motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p i arr[i] h := by
     induction i generalizing j bs with simp [mapFinIdxM.map]
     | zero =>
       have := (Nat.zero_add _).symm.trans h
       exact ⟨this ▸ hm, h₁ ▸ this, fun _ _ => h₂ ..⟩
     | succ i ih =>
-      apply @ih (bs.push (f ⟨j, by omega⟩ as[j])) (j + 1) (by omega) (by simp; omega)
+      apply @ih (bs.push (f j as[j] (by omega))) (j + 1) (by omega) (by simp; omega)
       · intro i i_lt h'
         rw [getElem_push]
         split
         · apply h₂
         · simp only [size_push] at h'
           obtain rfl : i = j := by omega
-          apply (hs ⟨i, by omega⟩ hm).1
-      · exact (hs ⟨j, by omega⟩ hm).2
+          apply (hs i (by omega) hm).1
+      · exact (hs j (by omega) hm).2
   simp [mapFinIdx, mapFinIdxM]; exact go rfl nofun h0
 
-theorem mapFinIdx_spec (as : Array α) (f : Fin as.size → α → β)
-    (p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
+theorem mapFinIdx_spec (as : Array α) (f : (i : Nat) → α → (h : i < as.size) → β)
+    (p : (i : Nat) → β → (h : i < as.size) → Prop) (hs : ∀ i h, p i (f i as[i] h) h) :
     ∃ eq : (Array.mapFinIdx as f).size = as.size,
-      ∀ i h, p ⟨i, h⟩ ((Array.mapFinIdx as f)[i]) :=
-  (mapFinIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
+      ∀ i h, p i ((Array.mapFinIdx as f)[i]) h :=
+  (mapFinIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
 
-@[simp] theorem size_mapFinIdx (a : Array α) (f : Fin a.size → α → β) : (a.mapFinIdx f).size = a.size :=
-  (mapFinIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
+@[simp] theorem size_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) :
+    (a.mapFinIdx f).size = a.size :=
+  (mapFinIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
 
 @[simp] theorem size_zipWithIndex (as : Array α) : as.zipWithIndex.size = as.size :=
   Array.size_mapFinIdx _ _
 
-@[simp] theorem getElem_mapFinIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat)
+@[simp] theorem getElem_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) (i : Nat)
     (h : i < (mapFinIdx a f).size) :
-    (a.mapFinIdx f)[i] = f ⟨i, by simp_all⟩ (a[i]'(by simp_all)) :=
-  (mapFinIdx_spec _ _ (fun i b => b = f i a[i]) fun _ => rfl).2 i _
+    (a.mapFinIdx f)[i] = f i (a[i]'(by simp_all))  (by simp_all) :=
+  (mapFinIdx_spec _ _ (fun i b h => b = f i a[i] h) fun _ _ => rfl).2 i _
 
-@[simp] theorem getElem?_mapFinIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat) :
+@[simp] theorem getElem?_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) (i : Nat) :
     (a.mapFinIdx f)[i]? =
-      a[i]?.pbind fun b h => f ⟨i, (getElem?_eq_some_iff.1 h).1⟩ b := by
+      a[i]?.pbind fun b h => f i b (getElem?_eq_some_iff.1 h).1 := by
   simp only [getElem?_def, size_mapFinIdx, getElem_mapFinIdx]
   split <;> simp_all
 
-@[simp] theorem toList_mapFinIdx (a : Array α) (f : Fin a.size → α → β) :
-    (a.mapFinIdx f).toList = a.toList.mapFinIdx (fun i a => f ⟨i, by simp⟩ a) := by
+@[simp] theorem toList_mapFinIdx (a : Array α) (f : (i : Nat) → α → (h : i < a.size) → β) :
+    (a.mapFinIdx f).toList = a.toList.mapFinIdx (fun i a h => f i a (by simpa)) := by
   apply List.ext_getElem <;> simp
 
 /-! ### mapIdx -/
 
 theorem mapIdx_induction (f : Nat → α → β) (as : Array α)
     (motive : Nat → Prop) (h0 : motive 0)
-    (p : Fin as.size → β → Prop)
-    (hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
+    (p : (i : Nat) → β → (h : i < as.size) → Prop)
+    (hs : ∀ i h, motive i → p i (f i as[i]) h ∧ motive (i + 1)) :
     motive as.size ∧ ∃ eq : (as.mapIdx f).size = as.size,
-      ∀ i h, p ⟨i, h⟩ ((as.mapIdx f)[i]) :=
-  mapFinIdx_induction as (fun i a => f i a) motive h0 p hs
+      ∀ i h, p i ((as.mapIdx f)[i]) h :=
+  mapFinIdx_induction as (fun i a _ => f i a) motive h0 p hs
 
 theorem mapIdx_spec (f : Nat → α → β) (as : Array α)
-    (p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
+    (p : (i : Nat) → β → (h : i < as.size) → Prop) (hs : ∀ i h, p i (f i as[i]) h) :
     ∃ eq : (as.mapIdx f).size = as.size,
-      ∀ i h, p ⟨i, h⟩ ((as.mapIdx f)[i]) :=
-  (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
+      ∀ i h, p i ((as.mapIdx f)[i]) h :=
+  (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
 
 @[simp] theorem size_mapIdx (f : Nat → α → β) (as : Array α) : (as.mapIdx f).size = as.size :=
-  (mapIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
+  (mapIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
 
 @[simp] theorem getElem_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat)
     (h : i < (as.mapIdx f).size) :
     (as.mapIdx f)[i] = f i (as[i]'(by simp_all)) :=
-  (mapIdx_spec _ _ (fun i b => b = f i as[i]) fun _ => rfl).2 i (by simp_all)
+  (mapIdx_spec _ _ (fun i b h => b = f i as[i]) fun _ _ => rfl).2 i (by simp_all)
 
 @[simp] theorem getElem?_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat) :
     (as.mapIdx f)[i]? =
@@ -101,7 +103,7 @@ end Array
 
 namespace List
 
-@[simp] theorem mapFinIdx_toArray (l : List α) (f : Fin l.length → α → β) :
+@[simp] theorem mapFinIdx_toArray (l : List α) (f : (i : Nat) → α → (h : i < l.length) → β) :
     l.toArray.mapFinIdx f = (l.mapFinIdx f).toArray := by
   ext <;> simp
 
@@ -110,3 +112,293 @@ namespace List
   ext <;> simp
 
 end List
+
+namespace Array
+
+/-! ### zipWithIndex -/
+
+@[simp] theorem getElem_zipWithIndex (a : Array α) (i : Nat) (h : i < a.zipWithIndex.size) :
+    (a.zipWithIndex)[i] = (a[i]'(by simp_all), i) := by
+  simp [zipWithIndex]
+
+@[simp] theorem zipWithIndex_toArray {l : List α} :
+    l.toArray.zipWithIndex = (l.enum.map fun (i, x) => (x, i)).toArray := by
+  ext i hi₁ hi₂ <;> simp
+
+@[simp] theorem toList_zipWithIndex (a : Array α) :
+    a.zipWithIndex.toList = a.toList.enum.map (fun (i, a) => (a, i)) := by
+  rcases a with ⟨a⟩
+  simp
+
+theorem mk_mem_zipWithIndex_iff_getElem? {x : α} {i : Nat} {l : Array α} :
+    (x, i) ∈ l.zipWithIndex ↔ l[i]? = x := by
+  rcases l with ⟨l⟩
+  simp only [zipWithIndex_toArray, mem_toArray, List.mem_map, Prod.mk.injEq, Prod.exists,
+    List.mk_mem_enum_iff_getElem?, List.getElem?_toArray]
+  constructor
+  · rintro ⟨a, b, h, rfl, rfl⟩
+    exact h
+  · intro h
+    exact ⟨i, x, by simp [h]⟩
+
+theorem mem_enum_iff_getElem? {x : α × Nat} {l : Array α} : x ∈ l.zipWithIndex ↔ l[x.2]? = some x.1 :=
+  mk_mem_zipWithIndex_iff_getElem?
+
+/-! ### mapFinIdx -/
+
+@[congr] theorem mapFinIdx_congr {xs ys : Array α} (w : xs = ys)
+    (f : (i : Nat) → α → (h : i < xs.size) → β) :
+    mapFinIdx xs f = mapFinIdx ys (fun i a h => f i a (by simp [w]; omega)) := by
+  subst w
+  rfl
+
+@[simp]
+theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #[] f = #[] :=
+  rfl
+
+theorem mapFinIdx_eq_ofFn {as : Array α} {f : (i : Nat) → α → (h : i < as.size) → β} :
+    as.mapFinIdx f = Array.ofFn fun i : Fin as.size => f i as[i] i.2 := by
+  cases as
+  simp [List.mapFinIdx_eq_ofFn]
+
+theorem mapFinIdx_append {K L : Array α} {f : (i : Nat) → α → (h : i < (K ++ L).size) → β} :
+    (K ++ L).mapFinIdx f =
+      K.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
+        L.mapFinIdx (fun i a h => f (i + K.size) a (by simp; omega)) := by
+  cases K
+  cases L
+  simp [List.mapFinIdx_append]
+
+@[simp]
+theorem mapFinIdx_push {l : Array α} {a : α} {f : (i : Nat) → α → (h : i < (l.push a).size) → β} :
+    mapFinIdx (l.push a) f =
+      (mapFinIdx l (fun i a h => f i a (by simp; omega))).push (f l.size a (by simp)) := by
+  simp [← append_singleton, mapFinIdx_append]
+
+theorem mapFinIdx_singleton {a : α} {f : (i : Nat) → α → (h : i < 1) → β} :
+    #[a].mapFinIdx f = #[f 0 a (by simp)] := by
+  simp
+
+theorem mapFinIdx_eq_zipWithIndex_map {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = l.zipWithIndex.attach.map
+      fun ⟨⟨x, i⟩, m⟩ =>
+        f i x (by simp [mk_mem_zipWithIndex_iff_getElem?, getElem?_eq_some_iff] at m; exact m.1) := by
+  ext <;> simp
+
+@[simp]
+theorem mapFinIdx_eq_empty_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = #[] ↔ l = #[] := by
+  cases l
+  simp
+
+theorem mapFinIdx_ne_empty_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f ≠ #[] ↔ l ≠ #[] := by
+  simp
+
+theorem exists_of_mem_mapFinIdx {b : β} {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β}
+    (h : b ∈ l.mapFinIdx f) : ∃ (i : Nat) (h : i < l.size), f i l[i] h = b := by
+  rcases l with ⟨l⟩
+  exact List.exists_of_mem_mapFinIdx (by simpa using h)
+
+@[simp] theorem mem_mapFinIdx {b : β} {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.size), f i l[i] h = b := by
+  rcases l with ⟨l⟩
+  simp
+
+theorem mapFinIdx_eq_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = l' ↔ ∃ h : l'.size = l.size, ∀ (i : Nat) (h : i < l.size), l'[i] = f i l[i] h := by
+  rcases l with ⟨l⟩
+  rcases l' with ⟨l'⟩
+  simpa using List.mapFinIdx_eq_iff
+
+@[simp] theorem mapFinIdx_eq_singleton_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {b : β} :
+    l.mapFinIdx f = #[b] ↔ ∃ (a : α) (w : l = #[a]), f 0 a (by simp [w]) = b := by
+  rcases l with ⟨l⟩
+  simp
+
+theorem mapFinIdx_eq_append_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {l₁ l₂ : Array β} :
+    l.mapFinIdx f = l₁ ++ l₂ ↔
+      ∃ (l₁' : Array α) (l₂' : Array α) (w : l = l₁' ++ l₂'),
+        l₁'.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₁ ∧
+        l₂'.mapFinIdx (fun i a h => f (i + l₁'.size) a (by simp [w]; omega)) = l₂ := by
+  rcases l with ⟨l⟩
+  rcases l₁ with ⟨l₁⟩
+  rcases l₂ with ⟨l₂⟩
+  simp only [List.mapFinIdx_toArray, List.append_toArray, mk.injEq, List.mapFinIdx_eq_append_iff,
+    toArray_eq_append_iff]
+  constructor
+  · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
+    refine ⟨l₁.toArray, l₂.toArray, by simp_all⟩
+  · rintro ⟨⟨l₁⟩, ⟨l₂⟩, rfl, h₁, h₂⟩
+    simp [← toList_inj] at h₁ h₂
+    obtain rfl := h₁
+    obtain rfl := h₂
+    refine ⟨l₁, l₂, by simp_all⟩
+
+theorem mapFinIdx_eq_push_iff {l : Array α} {b : β} {f : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = l₂.push b ↔
+      ∃ (l₁ : Array α) (a : α) (w : l = l₁.push a),
+        l₁.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₂ ∧ b = f (l.size - 1) a (by simp [w]) := by
+  rw [push_eq_append, mapFinIdx_eq_append_iff]
+  constructor
+  · rintro ⟨l₁, l₂, rfl, rfl, h₂⟩
+    simp only [mapFinIdx_eq_singleton_iff, Nat.zero_add] at h₂
+    obtain ⟨a, rfl, rfl⟩ := h₂
+    exact ⟨l₁, a, by simp⟩
+  · rintro ⟨l₁, a, rfl, rfl, rfl⟩
+    exact ⟨l₁, #[a], by simp⟩
+
+theorem mapFinIdx_eq_mapFinIdx_iff {l : Array α} {f g : (i : Nat) → α → (h : i < l.size) → β} :
+    l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Nat) (h : i < l.size), f i l[i] h = g i l[i] h := by
+  rw [eq_comm, mapFinIdx_eq_iff]
+  simp
+
+@[simp] theorem mapFinIdx_mapFinIdx {l : Array α}
+    {f : (i : Nat) → α → (h : i < l.size) → β}
+    {g : (i : Nat) → β → (h : i < (l.mapFinIdx f).size) → γ} :
+    (l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i a h => g i (f i a h) (by simpa using h)) := by
+  simp [mapFinIdx_eq_iff]
+
+theorem mapFinIdx_eq_mkArray_iff {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {b : β} :
+    l.mapFinIdx f = mkArray l.size b ↔ ∀ (i : Nat) (h : i < l.size), f i l[i] h = b := by
+  rcases l with ⟨l⟩
+  rw [← toList_inj]
+  simp [List.mapFinIdx_eq_replicate_iff]
+
+@[simp] theorem mapFinIdx_reverse {l : Array α} {f : (i : Nat) → α → (h : i < l.reverse.size) → β} :
+    l.reverse.mapFinIdx f = (l.mapFinIdx (fun i a h => f (l.size - 1 - i) a (by simp; omega))).reverse := by
+  rcases l with ⟨l⟩
+  simp [List.mapFinIdx_reverse]
+
+/-! ### mapIdx -/
+
+@[simp]
+theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #[] = #[] :=
+  rfl
+
+@[simp] theorem mapFinIdx_eq_mapIdx {l : Array α} {f : (i : Nat) → α → (h : i < l.size) → β} {g : Nat → α → β}
+    (h : ∀ (i : Nat) (h : i < l.size), f i l[i] h = g i l[i]) :
+    l.mapFinIdx f = l.mapIdx g := by
+  simp_all [mapFinIdx_eq_iff]
+
+theorem mapIdx_eq_mapFinIdx {l : Array α} {f : Nat → α → β} :
+    l.mapIdx f = l.mapFinIdx (fun i a _ => f i a) := by
+  simp [mapFinIdx_eq_mapIdx]
+
+theorem mapIdx_eq_zipWithIndex_map {l : Array α} {f : Nat → α → β} :
+    l.mapIdx f = l.zipWithIndex.map fun ⟨a, i⟩ => f i a := by
+  ext <;> simp
+
+theorem mapIdx_append {K L : Array α} :
+    (K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i => f (i + K.size) := by
+  rcases K with ⟨K⟩
+  rcases L with ⟨L⟩
+  simp [List.mapIdx_append]
+
+@[simp]
+theorem mapIdx_push {l : Array α} {a : α} :
+    mapIdx f (l.push a) = (mapIdx f l).push (f l.size a) := by
+  simp [← append_singleton, mapIdx_append]
+
+theorem mapIdx_singleton {a : α} : mapIdx f #[a] = #[f 0 a] := by
+  simp
+
+@[simp]
+theorem mapIdx_eq_empty_iff {l : Array α} : mapIdx f l = #[] ↔ l = #[] := by
+  rcases l with ⟨l⟩
+  simp
+
+theorem mapIdx_ne_empty_iff {l : Array α} :
+    mapIdx f l ≠ #[] ↔ l ≠ #[] := by
+  simp
+
+theorem exists_of_mem_mapIdx {b : β} {l : Array α}
+    (h : b ∈ mapIdx f l) : ∃ (i : Nat) (h : i < l.size), f i l[i] = b := by
+  rw [mapIdx_eq_mapFinIdx] at h
+  simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
+
+@[simp] theorem mem_mapIdx {b : β} {l : Array α} :
+    b ∈ mapIdx f l ↔ ∃ (i : Nat) (h : i < l.size), f i l[i] = b := by
+  constructor
+  · intro h
+    exact exists_of_mem_mapIdx h
+  · rintro ⟨i, h, rfl⟩
+    rw [mem_iff_getElem]
+    exact ⟨i, by simpa using h, by simp⟩
+
+theorem mapIdx_eq_push_iff {l : Array α} {b : β} :
+    mapIdx f l = l₂.push b ↔
+      ∃ (a : α) (l₁ : Array α), l = l₁.push a ∧ mapIdx f l₁ = l₂ ∧ f l₁.size a = b := by
+  rw [mapIdx_eq_mapFinIdx, mapFinIdx_eq_push_iff]
+  simp only [mapFinIdx_eq_mapIdx, exists_and_left, exists_prop]
+  constructor
+  · rintro ⟨l₁, rfl, a, rfl, rfl⟩
+    exact ⟨a, l₁, by simp⟩
+  · rintro ⟨a, l₁, rfl, rfl, rfl⟩
+    exact ⟨l₁, rfl, a, by simp⟩
+
+@[simp] theorem mapIdx_eq_singleton_iff {l : Array α} {f : Nat → α → β} {b : β} :
+    mapIdx f l = #[b] ↔ ∃ (a : α), l = #[a] ∧ f 0 a = b := by
+  rcases l with ⟨l⟩
+  simp [List.mapIdx_eq_singleton_iff]
+
+theorem mapIdx_eq_append_iff {l : Array α} {f : Nat → α → β} {l₁ l₂ : Array β} :
+    mapIdx f l = l₁ ++ l₂ ↔
+      ∃ (l₁' : Array α) (l₂' : Array α), l = l₁' ++ l₂' ∧
+        l₁'.mapIdx f = l₁ ∧
+        l₂'.mapIdx (fun i => f (i + l₁'.size)) = l₂ := by
+  rcases l with ⟨l⟩
+  rcases l₁ with ⟨l₁⟩
+  rcases l₂ with ⟨l₂⟩
+  simp only [List.mapIdx_toArray, List.append_toArray, mk.injEq, List.mapIdx_eq_append_iff,
+    toArray_eq_append_iff]
+  constructor
+  · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
+    exact ⟨l₁.toArray, l₂.toArray, by simp⟩
+  · rintro ⟨⟨l₁⟩, ⟨l₂⟩, rfl, h₁, h₂⟩
+    simp only [List.mapIdx_toArray, mk.injEq, size_toArray] at h₁ h₂
+    obtain rfl := h₁
+    obtain rfl := h₂
+    exact ⟨l₁, l₂, by simp⟩
+
+theorem mapIdx_eq_iff {l : Array α} : mapIdx f l = l' ↔ ∀ i : Nat, l'[i]? = l[i]?.map (f i) := by
+  rcases l with ⟨l⟩
+  rcases l' with ⟨l'⟩
+  simp [List.mapIdx_eq_iff]
+
+theorem mapIdx_eq_mapIdx_iff {l : Array α} :
+    mapIdx f l = mapIdx g l ↔ ∀ i : Nat, (h : i < l.size) → f i l[i] = g i l[i] := by
+  rcases l with ⟨l⟩
+  simp [List.mapIdx_eq_mapIdx_iff]
+
+@[simp] theorem mapIdx_set {l : Array α} {i : Nat} {h : i < l.size} {a : α} :
+    (l.set i a).mapIdx f = (l.mapIdx f).set i (f i a) (by simpa) := by
+  rcases l with ⟨l⟩
+  simp [List.mapIdx_set]
+
+@[simp] theorem mapIdx_setIfInBounds {l : Array α} {i : Nat} {a : α} :
+    (l.setIfInBounds i a).mapIdx f = (l.mapIdx f).setIfInBounds i (f i a) := by
+  rcases l with ⟨l⟩
+  simp [List.mapIdx_set]
+
+@[simp] theorem back?_mapIdx {l : Array α} {f : Nat → α → β} :
+    (mapIdx f l).back? = (l.back?).map (f (l.size - 1)) := by
+  rcases l with ⟨l⟩
+  simp [List.getLast?_mapIdx]
+
+@[simp] theorem mapIdx_mapIdx {l : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
+    (l.mapIdx f).mapIdx g = l.mapIdx (fun i => g i ∘ f i) := by
+  simp [mapIdx_eq_iff]
+
+theorem mapIdx_eq_mkArray_iff {l : Array α} {f : Nat → α → β} {b : β} :
+    mapIdx f l = mkArray l.size b ↔ ∀ (i : Nat) (h : i < l.size), f i l[i] = b := by
+  rcases l with ⟨l⟩
+  rw [← toList_inj]
+  simp [List.mapIdx_eq_replicate_iff]
+
+@[simp] theorem mapIdx_reverse {l : Array α} {f : Nat → α → β} :
+    l.reverse.mapIdx f = (mapIdx (fun i => f (l.size - 1 - i)) l).reverse := by
+  rcases l with ⟨l⟩
+  simp [List.mapIdx_reverse]
+
+end Array

src/Init/Data/BitVec/Basic.lean

@@ -669,4 +669,11 @@ def ofBoolListLE : (bs : List Bool) → BitVec bs.length
 | [] => 0#0
 | b :: bs => concat (ofBoolListLE bs) b
 
+/- ### reverse -/
+
+/-- Reverse the bits in a bitvector. -/
+def reverse : {w : Nat} → BitVec w → BitVec w
+  | 0, x => x
+  | w + 1, x => concat (reverse (x.truncate w)) (x.msb)
+
 end BitVec

src/Init/Data/BitVec/Bitblast.lean

@@ -631,6 +631,13 @@ theorem getLsbD_mul (x y : BitVec w) (i : Nat) :
   · simp
   · omega
 
+theorem getMsbD_mul (x y : BitVec w) (i : Nat) :
+    (x * y).getMsbD i = (mulRec x y w).getMsbD i := by
+  simp only [mulRec_eq_mul_signExtend_setWidth]
+  rw [setWidth_setWidth_of_le]
+  · simp
+  · omega
+
 theorem getElem_mul {x y : BitVec w} {i : Nat} (h : i < w) :
     (x * y)[i] = (mulRec x y w)[i] := by
   simp [mulRec_eq_mul_signExtend_setWidth]
@@ -1084,6 +1091,21 @@ theorem divRec_succ' (m : Nat) (args : DivModArgs w) (qr : DivModState w) :
     divRec m args input := by
   simp [divRec_succ, divSubtractShift]
 
+theorem getElem_udiv (n d : BitVec w) (hy : 0#w < d) (i : Nat) (hi : i < w) :
+    (n / d)[i] = (divRec w {n, d} (DivModState.init w)).q[i] := by
+  rw [udiv_eq_divRec (by assumption)]
+
+theorem getLsbD_udiv (n d : BitVec w) (hy : 0#w < d)  (i : Nat) :
+    (n / d).getLsbD i = (decide (i < w) && (divRec w {n, d} (DivModState.init w)).q.getLsbD i) := by
+  by_cases hi : i < w
+  · simp [udiv_eq_divRec (by assumption)]
+    omega
+  · simp_all
+
+theorem getMsbD_udiv (n d : BitVec w) (hd : 0#w < d)  (i : Nat) :
+    (n / d).getMsbD i = (decide (i < w) && (divRec w {n, d} (DivModState.init w)).q.getMsbD i) := by
+  simp [getMsbD_eq_getLsbD, getLsbD_udiv, udiv_eq_divRec (by assumption)]
+
 /- ### Arithmetic shift right (sshiftRight) recurrence -/
 
 /--

src/Init/Data/BitVec/Lemmas.lean

@@ -905,6 +905,16 @@ instance : Std.LawfulCommIdentity (α := BitVec n) (· ||| · ) (0#n) where
   ext i h
   simp [h]
 
+theorem extractLsb'_or {x y : BitVec w} {start len : Nat} :
+   (x ||| y).extractLsb' start len = (x.extractLsb' start len) ||| (y.extractLsb' start len) := by
+  ext i hi
+  simp [hi]
+
+theorem extractLsb_or {x : BitVec w} {hi lo : Nat} :
+   (x ||| y).extractLsb lo hi = (x.extractLsb lo hi) ||| (y.extractLsb lo hi) := by
+  ext k hk
+  simp [hk, show k ≤ lo - hi by omega]
+
 /-! ### and -/
 
 @[simp] theorem toNat_and (x y : BitVec v) :
@@ -978,6 +988,16 @@ instance : Std.LawfulCommIdentity (α := BitVec n) (· &&& · ) (allOnes n) wher
   ext i h
   simp [h]
 
+theorem extractLsb'_and {x y : BitVec w} {start len : Nat} :
+   (x &&& y).extractLsb' start len = (x.extractLsb' start len) &&& (y.extractLsb' start len) := by
+  ext i hi
+  simp [hi]
+
+theorem extractLsb_and {x : BitVec w} {hi lo : Nat} :
+   (x &&& y).extractLsb lo hi = (x.extractLsb lo hi) &&& (y.extractLsb lo hi) := by
+  ext k hk
+  simp [hk, show k ≤ lo - hi by omega]
+
 /-! ### xor -/
 
 @[simp] theorem toNat_xor (x y : BitVec v) :
@@ -1043,6 +1063,16 @@ instance : Std.LawfulCommIdentity (α := BitVec n) (· ^^^ · ) (0#n) where
   ext i
   simp
 
+theorem extractLsb'_xor {x y : BitVec w} {start len : Nat} :
+   (x ^^^ y).extractLsb' start len = (x.extractLsb' start len) ^^^ (y.extractLsb' start len) := by
+  ext i hi
+  simp [hi]
+
+theorem extractLsb_xor {x : BitVec w} {hi lo : Nat} :
+   (x ^^^ y).extractLsb lo hi = (x.extractLsb lo hi) ^^^ (y.extractLsb lo hi) := by
+  ext k hk
+  simp [hk, show k ≤ lo - hi by omega]
+
 /-! ### not -/
 
 theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
@@ -1149,6 +1179,31 @@ theorem getMsb_not {x : BitVec w} :
 @[simp] theorem msb_not {x : BitVec w} : (~~~x).msb = (decide (0 < w) && !x.msb) := by
   simp [BitVec.msb]
 
+/--
+Negating `x` and then extracting [start..start+len) is the same as extracting and then negating,
+as long as the range [start..start+len) is in bounds.
+See that if the index is out-of-bounds, then `extractLsb` will return `false`,
+which makes the operation not commute.
+-/
+theorem extractLsb'_not_of_lt {x : BitVec w} {start len : Nat} (h : start + len < w) :
+   (~~~ x).extractLsb' start len = ~~~ (x.extractLsb' start len) := by
+  ext i hi
+  simp [hi]
+  omega
+
+/--
+Negating `x` and then extracting [lo:hi] is the same as extracting and then negating.
+For the extraction to be well-behaved,
+we need the range [lo:hi] to be a valid closed interval inside the bitvector:
+1. `lo ≤ hi` for the interval to be a well-formed closed interval.
+2. `hi < w`, for the interval to be contained inside the bitvector.
+-/
+theorem extractLsb_not_of_lt {x : BitVec w} {hi lo : Nat} (hlo : lo ≤ hi) (hhi : hi < w) :
+   (~~~ x).extractLsb hi lo = ~~~ (x.extractLsb hi lo) := by
+  ext k hk
+  simp [hk, show k ≤ hi - lo by omega]
+  omega
+
 /-! ### cast -/
 
 @[simp] theorem not_cast {x : BitVec w} (h : w = w') : ~~~(x.cast h) = (~~~x).cast h := by
@@ -1294,11 +1349,6 @@ theorem allOnes_shiftLeft_or_shiftLeft {x : BitVec w} {n : Nat} :
     BitVec.allOnes w <<< n ||| x <<< n = BitVec.allOnes w <<< n := by
   simp [← shiftLeft_or_distrib]
 
-@[deprecated shiftLeft_add (since := "2024-06-02")]
-theorem shiftLeft_shiftLeft {w : Nat} (x : BitVec w) (n m : Nat) :
-    (x <<< n) <<< m = x <<< (n + m) := by
-  rw [shiftLeft_add]
-
 /-! ### shiftLeft reductions from BitVec to Nat -/
 
 @[simp]
@@ -1318,6 +1368,13 @@ theorem getElem_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {i : Nat} (h : i
     (x <<< y)[i] = (!decide (i < y.toNat) && x.getLsbD (i - y.toNat)) := by
   simp
 
+@[simp] theorem shiftLeft_eq_zero {x : BitVec w} {n : Nat} (hn : w ≤ n) : x <<< n = 0#w := by
+  ext i hi
+  simp [hn, hi]
+  omega
+
+theorem shiftLeft_ofNat_eq {x : BitVec w} {k : Nat} : x <<< (BitVec.ofNat w k) = x <<< (k % 2^w) := rfl
+
 /-! ### ushiftRight -/
 
 @[simp, bv_toNat] theorem toNat_ushiftRight (x : BitVec n) (i : Nat) :
@@ -1449,6 +1506,8 @@ theorem msb_ushiftRight {x : BitVec w} {n : Nat} :
 theorem ushiftRight_eq' (x : BitVec w₁) (y : BitVec w₂) :
     x >>> y = x >>> y.toNat := by rfl
 
+theorem ushiftRight_ofNat_eq {x : BitVec w} {k : Nat} : x >>> (BitVec.ofNat w k) = x >>> (k % 2^w) := rfl
+
 /-! ### sshiftRight -/
 
 theorem sshiftRight_eq {x : BitVec n} {i : Nat} :
@@ -1546,6 +1605,9 @@ theorem sshiftRight_or_distrib (x y : BitVec w) (n : Nat) :
     <;> by_cases w ≤ i
     <;> simp [*]
 
+
+theorem sshiftRight'_ofNat_eq_sshiftRight {x : BitVec w} {k : Nat} : x.sshiftRight' (BitVec.ofNat w k) = x.sshiftRight (k % 2^w) := rfl
+
 /-- The msb after arithmetic shifting right equals the original msb. -/
 @[simp]
 theorem msb_sshiftRight {n : Nat} {x : BitVec w} :
@@ -1946,10 +2008,24 @@ theorem msb_shiftLeft {x : BitVec w} {n : Nat} :
     (x <<< n).msb = x.getMsbD n := by
   simp [BitVec.msb]
 
-@[deprecated shiftRight_add (since := "2024-06-02")]
-theorem shiftRight_shiftRight {w : Nat} (x : BitVec w) (n m : Nat) :
-    (x >>> n) >>> m = x >>> (n + m) := by
-  rw [shiftRight_add]
+theorem ushiftRight_eq_extractLsb'_of_lt {x : BitVec w} {n : Nat} (hn : n < w) :
+    x >>> n = ((0#n) ++ (x.extractLsb' n (w - n))).cast (by omega) := by
+  ext i hi
+  simp only [getLsbD_ushiftRight, getLsbD_cast, getLsbD_append, getLsbD_extractLsb', getLsbD_zero,
+    Bool.if_false_right, Bool.and_self_left, Bool.iff_and_self, decide_eq_true_eq]
+  intros h
+  have := lt_of_getLsbD h
+  omega
+
+theorem shiftLeft_eq_concat_of_lt {x : BitVec w} {n : Nat} (hn : n < w) :
+    x <<< n = (x.extractLsb' 0 (w - n) ++ 0#n).cast (by omega) := by
+  ext i hi
+  simp only [getLsbD_shiftLeft, hi, decide_true, Bool.true_and, getLsbD_cast, getLsbD_append,
+    getLsbD_zero, getLsbD_extractLsb', Nat.zero_add, Bool.if_false_left]
+  by_cases hi' : i < n
+  · simp [hi']
+  · simp [hi']
+    omega
 
 /-! ### rev -/
 
@@ -2063,6 +2139,32 @@ theorem eq_msb_cons_setWidth (x : BitVec (w+1)) : x = (cons x.msb (x.setWidth w)
   ext i
   simp [cons]
 
+
+theorem cons_append (x : BitVec w₁) (y : BitVec w₂) (a : Bool) :
+    (cons a x) ++ y = (cons a (x ++ y)).cast (by omega) := by
+  apply eq_of_toNat_eq
+  simp only [toNat_append, toNat_cons, toNat_cast]
+  rw [Nat.shiftLeft_add, Nat.shiftLeft_or_distrib, Nat.or_assoc]
+
+theorem cons_append_append (x : BitVec w₁) (y : BitVec w₂) (z : BitVec w₃) (a : Bool) :
+    (cons a x) ++ y ++ z = (cons a (x ++ y ++ z)).cast (by omega) := by
+  ext i h
+  simp only [cons, getLsbD_append, getLsbD_cast, getLsbD_ofBool, cast_cast]
+  by_cases h₀ : i < w₁ + w₂ + w₃
+  · simp only [h₀, ↓reduceIte]
+    by_cases h₁ : i < w₃
+    · simp [h₁]
+    · simp only [h₁, ↓reduceIte]
+      by_cases h₂ : i - w₃ < w₂
+      · simp [h₂]
+      · simp [h₂]
+        omega
+  · simp only [show ¬i - w₃ - w₂ < w₁ by omega, ↓reduceIte, show i - w₃ - w₂ - w₁ = 0 by omega,
+      decide_true, Bool.true_and, h₀, show i - (w₁ + w₂ + w₃) = 0 by omega]
+    by_cases h₂ : i < w₃
+    · simp [h₂]; omega
+    · simp [h₂];  omega
+
 /-! ### concat -/
 
 @[simp] theorem toNat_concat (x : BitVec w) (b : Bool) :
@@ -3063,7 +3165,7 @@ theorem getMsbD_rotateLeft_of_lt {n w : Nat} {x : BitVec w} (hi : r < w):
       · simp only [h₁, decide_true, Bool.true_and]
         have h₂ : (r + n) < 2 * (w + 1) := by omega
         congr 1
-        rw [← Nat.sub_mul_eq_mod_of_lt_of_le (n := 1) (by omega) (by omega), Nat.mul_one]
+        rw [← Nat.sub_mul_eq_mod_of_lt_of_le (n := 1) (by omega) (by omega)]
         omega
       · simp [h₁]
 
@@ -3312,6 +3414,11 @@ theorem mul_twoPow_eq_shiftLeft (x : BitVec w) (i : Nat) :
       apply Nat.pow_dvd_pow 2 (by omega)
     simp [Nat.mul_mod, hpow]
 
+theorem twoPow_mul_eq_shiftLeft (x : BitVec w) (i : Nat) :
+    (twoPow w i) * x = x <<< i := by
+  rw [BitVec.mul_comm, mul_twoPow_eq_shiftLeft]
+
+
 theorem twoPow_zero {w : Nat} : twoPow w 0 = 1#w := by
   apply eq_of_toNat_eq
   simp
@@ -3321,6 +3428,12 @@ theorem shiftLeft_eq_mul_twoPow (x : BitVec w) (n : Nat) :
   ext i
   simp [getLsbD_shiftLeft, Fin.is_lt, decide_true, Bool.true_and, mul_twoPow_eq_shiftLeft]
 
+/-- 2^i * 2^j = 2^(i + j) with bitvectors as well -/
+theorem twoPow_mul_twoPow_eq {w : Nat} (i j : Nat) : twoPow w i * twoPow w j = twoPow w (i + j) := by 
+  apply BitVec.eq_of_toNat_eq
+  simp only [toNat_mul, toNat_twoPow]
+  rw [← Nat.mul_mod, Nat.pow_add]
+
 /--
 The unsigned division of `x` by `2^k` equals shifting `x` right by `k`,
 when `k` is less than the bitwidth `w`.
@@ -3383,11 +3496,11 @@ theorem and_one_eq_setWidth_ofBool_getLsbD {x : BitVec w} :
   ext (_ | i) h <;> simp [Bool.and_comm]
 
 @[simp]
-theorem replicate_zero_eq {x : BitVec w} : x.replicate 0 = 0#0 := by
+theorem replicate_zero {x : BitVec w} : x.replicate 0 = 0#0 := by
   simp [replicate]
 
 @[simp]
-theorem replicate_succ_eq {x : BitVec w} :
+theorem replicate_succ {x : BitVec w} :
     x.replicate (n + 1) =
     (x ++ replicate n x).cast (by rw [Nat.mul_succ]; omega) := by
   simp [replicate]
@@ -3399,7 +3512,7 @@ theorem getLsbD_replicate {n w : Nat} (x : BitVec w) :
   induction n generalizing x
   case zero => simp
   case succ n ih =>
-    simp only [replicate_succ_eq, getLsbD_cast, getLsbD_append]
+    simp only [replicate_succ, getLsbD_cast, getLsbD_append]
     by_cases hi : i < w * (n + 1)
     · simp only [hi, decide_true, Bool.true_and]
       by_cases hi' : i < w * n
@@ -3416,6 +3529,33 @@ theorem getElem_replicate {n w : Nat} (x : BitVec w) (h : i < w * n) :
   simp only [← getLsbD_eq_getElem, getLsbD_replicate]
   by_cases h' : w = 0 <;> simp [h'] <;> omega
 
+theorem append_assoc {x₁ : BitVec w₁} {x₂ : BitVec w₂} {x₃ : BitVec w₃} :
+    (x₁ ++ x₂) ++ x₃ = (x₁ ++ (x₂ ++ x₃)).cast (by omega) := by
+  induction w₁ generalizing x₂ x₃
+  case zero => simp
+  case succ n ih =>
+    specialize @ih (setWidth n x₁)
+    rw [← cons_msb_setWidth x₁, cons_append_append, ih, cons_append]
+    ext j h
+    simp [getLsbD_cons, show n + w₂ + w₃ = n + (w₂ + w₃) by omega]
+
+theorem replicate_append_self {x : BitVec w} :
+    x ++ x.replicate n = (x.replicate n ++ x).cast (by omega) := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [replicate_succ]
+    conv => lhs; rw [ih]
+    simp only [cast_cast, cast_eq]
+    rw [← cast_append_left]
+    · rw [append_assoc]; congr
+    · rw [Nat.add_comm, Nat.mul_add, Nat.mul_one]; omega
+
+theorem replicate_succ' {x : BitVec w} :
+    x.replicate (n + 1) =
+    (replicate n x ++ x).cast (by rw [Nat.mul_succ]) := by
+  simp [replicate_append_self]
+
 /-! ### intMin -/
 
 /-- The bitvector of width `w` that has the smallest value when interpreted as an integer. -/
@@ -3539,7 +3679,7 @@ 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]
@@ -3701,6 +3841,57 @@ theorem toInt_abs_eq_natAbs_of_ne_intMin {x : BitVec w} (hx : x ≠ intMin w) :
     x.abs.toInt = x.toInt.natAbs := by
   simp [toInt_abs_eq_natAbs, hx]
 
+/-! ### Reverse -/
+
+theorem getLsbD_reverse {i : Nat} {x : BitVec w} :
+    (x.reverse).getLsbD i = x.getMsbD i := by
+  induction w generalizing i
+  case zero => simp
+  case succ n ih =>
+    simp only [reverse, truncate_eq_setWidth, getLsbD_concat]
+    rcases i with rfl | i
+    · rfl
+    · simp only [Nat.add_one_ne_zero, ↓reduceIte, Nat.add_one_sub_one, ih]
+      rw [getMsbD_setWidth]
+      simp only [show n - (n + 1) = 0 by omega, Nat.zero_le, decide_true, Bool.true_and]
+      congr; omega
+
+theorem getMsbD_reverse {i : Nat} {x : BitVec w} :
+    (x.reverse).getMsbD i = x.getLsbD i := by
+  simp only [getMsbD_eq_getLsbD, getLsbD_reverse]
+  by_cases hi : i < w
+  · simp only [hi, decide_true, show w - 1 - i < w by omega, Bool.true_and]
+    congr; omega
+  · simp [hi, show i ≥ w by omega]
+
+theorem msb_reverse {x : BitVec w} :
+    (x.reverse).msb = x.getLsbD 0 :=
+  by rw [BitVec.msb, getMsbD_reverse]
+
+theorem reverse_append {x : BitVec w} {y : BitVec v} :
+    (x ++ y).reverse = (y.reverse ++ x.reverse).cast (by omega) := by
+  ext i h
+  simp only [getLsbD_append, getLsbD_reverse]
+  by_cases hi : i < v
+  · by_cases hw : w ≤ i
+    · simp [getMsbD_append, getLsbD_cast, getLsbD_append, getLsbD_reverse, hw]
+    · simp [getMsbD_append, getLsbD_cast, getLsbD_append, getLsbD_reverse, hw, show i < w by omega]
+  · by_cases hw : w ≤ i
+    · simp [getMsbD_append, getLsbD_cast, getLsbD_append, hw, show ¬ i < w by omega, getLsbD_reverse]
+    · simp [getMsbD_append, getLsbD_cast, getLsbD_append, hw, show i < w by omega, getLsbD_reverse]
+
+@[simp]
+theorem reverse_cast {w v : Nat} (h : w = v) (x : BitVec w) :
+    (x.cast h).reverse = x.reverse.cast h := by
+  subst h; simp
+
+theorem reverse_replicate {n : Nat} {x : BitVec w} :
+    (x.replicate n).reverse = (x.reverse).replicate n := by
+  induction n with
+  | zero => rfl
+  | succ n ih =>
+    conv => lhs; simp only [replicate_succ']
+    simp [reverse_append, ih]
 
 /-! ### Decidable quantifiers -/
 
@@ -3916,4 +4107,10 @@ abbrev shiftLeft_zero_eq := @shiftLeft_zero
 @[deprecated ushiftRight_zero (since := "2024-10-27")]
 abbrev ushiftRight_zero_eq := @ushiftRight_zero
 
+@[deprecated replicate_zero (since := "2025-01-08")]
+abbrev replicate_zero_eq := @replicate_zero
+
+@[deprecated replicate_succ (since := "2025-01-08")]
+abbrev replicate_succ_eq := @replicate_succ
+
 end BitVec

src/Init/Data/Bool.lean

@@ -620,3 +620,12 @@ but may be used locally.
 -/
 def boolRelToRel : Coe (α → α → Bool) (α → α → Prop) where
   coe r := fun a b => Eq (r a b) true
+
+/-! ### subtypes -/
+
+@[simp] theorem Subtype.beq_iff {α : Type u} [DecidableEq α] {p : α → Prop} {x y : {a : α // p a}} :
+    (x == y) = (x.1 == y.1) := by
+  cases x
+  cases y
+  rw [Bool.eq_iff_iff]
+  simp [beq_iff_eq]

src/Init/Data/Char/Lemmas.lean

@@ -70,5 +70,3 @@ theorem utf8Size_eq (c : Char) : c.utf8Size = 1 ∨ c.utf8Size = 2 ∨ c.utf8Siz
   rfl
 
 end Char
-
-@[deprecated Char.utf8Size (since := "2024-06-04")] abbrev String.csize := Char.utf8Size

src/Init/Data/Int/DivMod.lean

@@ -257,7 +257,7 @@ theorem ofNat_fdiv : ∀ m n : Nat, ↑(m / n) = fdiv ↑m ↑n
 # `bmod` ("balanced" mod)
 
 Balanced mod (and balanced div) are a division and modulus pair such
-that `b * (Int.bdiv a b) + Int.bmod a b = a` and `b/2 ≤ Int.bmod a b <
+that `b * (Int.bdiv a b) + Int.bmod a b = a` and `-b/2 ≤ Int.bmod a b <
 b/2` for all `a : Int` and `b > 0`.
 
 This is used in Omega as well as signed bitvectors.
@@ -266,10 +266,26 @@ This is used in Omega as well as signed bitvectors.
 /--
 Balanced modulus.  This version of Integer modulus uses the
 balanced rounding convention, which guarantees that
-`m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
+`-m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
 to `x` modulo `m`.
 
 If `m = 0`, then `bmod x m = x`.
+
+Examples:
+```
+#eval (7 : Int).bdiv 0 -- 0
+#eval (0 : Int).bdiv 7 -- 0
+
+#eval (12 : Int).bdiv 6 -- 2
+#eval (12 : Int).bdiv 7 -- 2
+#eval (12 : Int).bdiv 8 -- 2
+#eval (12 : Int).bdiv 9 -- 1
+
+#eval (-12 : Int).bdiv 6 -- -2
+#eval (-12 : Int).bdiv 7 -- -2
+#eval (-12 : Int).bdiv 8 -- -1
+#eval (-12 : Int).bdiv 9 -- -1
+```
 -/
 def bmod (x : Int) (m : Nat) : Int :=
   let r := x % m
@@ -281,6 +297,22 @@ def bmod (x : Int) (m : Nat) : Int :=
 /--
 Balanced division.  This returns the unique integer so that
 `b * (Int.bdiv a b) + Int.bmod a b = a`.
+
+Examples:
+```
+#eval (7 : Int).bmod 0 -- 7
+#eval (0 : Int).bmod 7 -- 0
+
+#eval (12 : Int).bmod 6 -- 0
+#eval (12 : Int).bmod 7 -- -2
+#eval (12 : Int).bmod 8 -- -4
+#eval (12 : Int).bmod 9 -- 3
+
+#eval (-12 : Int).bmod 6 -- 0
+#eval (-12 : Int).bmod 7 -- 2
+#eval (-12 : Int).bmod 8 -- -4
+#eval (-12 : Int).bmod 9 -- -3
+```
 -/
 def bdiv (x : Int) (m : Nat) : Int :=
   if m = 0 then

src/Init/Data/List/Attach.lean

@@ -111,6 +111,14 @@ theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
     pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
   rw [attach, attachWith, map_pmap]; exact pmap_congr_left l fun _ _ _ _ => rfl
 
+@[simp]
+theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l H) :
+    pmap (fun a h => ⟨a, f a h⟩) l H = l.attachWith q (fun x h => f x (H x h)) := by
+  induction l with
+  | nil => rfl
+  | cons a l ih =>
+    simp [pmap, attachWith, ih]
+
 theorem attach_map_coe (l : List α) (f : α → β) :
     (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
   rw [attach, attachWith, map_pmap]; exact pmap_eq_map _ _ _ _
@@ -136,10 +144,23 @@ theorem attachWith_map_subtype_val {p : α → Prop} (l : List α) (H : ∀ a
 @[simp]
 theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
   | ⟨a, h⟩ => by
-    have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
+    have := mem_map.1 (by rw [attach_map_subtype_val]; exact h)
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
+@[simp]
+theorem mem_attachWith (l : List α) {q : α → Prop} (H) (x : {x // q x}) :
+    x ∈ l.attachWith q H ↔ x.1 ∈ l := by
+  induction l with
+  | nil => simp
+  | cons a l ih =>
+    simp [ih]
+    constructor
+    · rintro (_ | _) <;> simp_all
+    · rintro (h | h)
+      · simp [← h]
+      · simp_all
+
 @[simp]
 theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
     b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
@@ -266,6 +287,18 @@ theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
+@[simp] theorem pmap_attach (l : List α) {p : {x // x ∈ l} → Prop} (f : ∀ a, p a → β) (H) :
+    pmap f l.attach H =
+      l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
+  apply ext_getElem <;> simp
+
+@[simp] theorem pmap_attachWith (l : List α) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
+    pmap f (l.attachWith q H₁) H₂ =
+      l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
+  apply ext_getElem <;> simp
+
 @[simp] theorem head?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).head? = xs.attach.head?.map fun ⟨a, m⟩ => f a (H a m) := by
@@ -431,7 +464,25 @@ theorem attach_filter {l : List α} (p : α → Bool) :
   split <;> simp
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
--- Also missing are `filterMap_attach`, `filter_attach`, `filterMap_attachWith` and `filter_attachWith`.
+
+@[simp]
+theorem filterMap_attachWith {q : α → Prop} {l : List α} {f : {x // q x} → Option β} (H) :
+    (l.attachWith q H).filterMap f = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [attachWith_cons, filterMap_cons]
+    split <;> simp_all [Function.comp_def]
+
+@[simp]
+theorem filter_attachWith {q : α → Prop} {l : List α} {p : {x // q x} → Bool} (H) :
+    (l.attachWith q H).filter p =
+      (l.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [attachWith_cons, filter_cons]
+    split <;> simp_all [Function.comp_def, filter_map]
 
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
     pmap f (pmap g l H₁) H₂ =
@@ -520,7 +571,7 @@ theorem reverse_attach (xs : List α) :
 
 @[simp] theorem getLast?_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} :
-    (xs.attachWith P H).getLast? = xs.getLast?.pbind (fun a h => some ⟨a, H _ (mem_of_getLast?_eq_some h)⟩) := by
+    (xs.attachWith P H).getLast? = xs.getLast?.pbind (fun a h => some ⟨a, H _ (mem_of_getLast? h)⟩) := by
   rw [getLast?_eq_head?_reverse, reverse_attachWith, head?_attachWith]
   simp
 
@@ -531,7 +582,7 @@ theorem reverse_attach (xs : List α) :
 
 @[simp]
 theorem getLast?_attach {xs : List α} :
-    xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast?_eq_some h⟩ := by
+    xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast? h⟩ := by
   rw [getLast?_eq_head?_reverse, reverse_attach, head?_map, head?_attach]
   simp
 
@@ -560,6 +611,11 @@ theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : List α) (H :
     (l.attachWith p H).count a = l.count ↑a :=
   Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attachWith _ _ _
 
+@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (l : List α) (H₁) :
+    (l.pmap g H₁).countP f =
+      l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
+  simp [pmap_eq_map_attach, countP_map, Function.comp_def]
+
 /-! ## unattach
 
 `List.unattach` is the (one-sided) inverse of `List.attach`. It is a synonym for `List.map Subtype.val`.
@@ -578,7 +634,7 @@ and is ideally subsequently simplified away by `unattach_attach`.
 
 If not, usually the right approach is `simp [List.unattach, -List.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) := l.map (·.val)
+def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) : List α := l.map (·.val)
 
 @[simp] theorem unattach_nil {p : α → Prop} : ([] : List { x // p x }).unattach = [] := rfl
 @[simp] theorem unattach_cons {p : α → Prop} {a : { x // p x }} {l : List { x // p x }} :

src/Init/Data/List/Basic.lean

@@ -258,9 +258,6 @@ theorem ext_get? : ∀ {l₁ l₂ : List α}, (∀ n, l₁.get? n = l₂.get? n)
     have h0 : some a = some a' := h 0
     injection h0 with aa; simp only [aa, ext_get? fun n => h (n+1)]
 
-/-- Deprecated alias for `ext_get?`. The preferred extensionality theorem is now `ext_getElem?`. -/
-@[deprecated ext_get? (since := "2024-06-07")] abbrev ext := @ext_get?
-
 /-! ### getD -/
 
 /--
@@ -606,11 +603,11 @@ 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
@@ -619,11 +616,6 @@ set_option linter.missingDocs false in
 set_option linter.missingDocs false in
 @[deprecated flatMap_cons (since := "2024-10-16")] abbrev cons_flatMap := @flatMap_cons
 
-set_option linter.missingDocs false in
-@[deprecated flatMap_nil (since := "2024-06-15")] abbrev nil_bind := @flatMap_nil
-set_option linter.missingDocs false in
-@[deprecated flatMap_cons (since := "2024-06-15")] abbrev cons_bind := @flatMap_cons
-
 /-! ### replicate -/
 
 /--
@@ -713,11 +705,6 @@ def elem [BEq α] (a : α) : List α → Bool
 theorem elem_cons [BEq α] {a : α} :
     (b::bs).elem a = match a == b with | true => true | false => bs.elem a := rfl
 
-/-- `notElem a l` is `!(elem a l)`. -/
-@[deprecated "Use `!(elem a l)` instead."(since := "2024-06-15")]
-def notElem [BEq α] (a : α) (as : List α) : Bool :=
-  !(as.elem a)
-
 /-! ### contains -/
 
 @[inherit_doc elem] abbrev contains [BEq α] (as : List α) (a : α) : Bool :=
@@ -1533,11 +1520,14 @@ def range' : (start len : Nat) → (step : Nat := 1) → List Nat
 `O(n)`. `iota n` is the numbers from `1` to `n` inclusive, in decreasing order.
 * `iota 5 = [5, 4, 3, 2, 1]`
 -/
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 def iota : Nat → List Nat
   | 0       => []
   | m@(n+1) => m :: iota n
 
+set_option linter.deprecated false in
 @[simp] theorem iota_zero : iota 0 = [] := rfl
+set_option linter.deprecated false in
 @[simp] theorem iota_succ : iota (i+1) = (i+1) :: iota i := rfl
 
 /-! ### enumFrom -/
@@ -1861,12 +1851,14 @@ def unzipTR (l : List (α × β)) : List α × List β :=
 /-! ### iota -/
 
 /-- Tail-recursive version of `List.iota`. -/
+@[deprecated "Use `List.range' 1 n` instead of `iota n`." (since := "2025-01-20")]
 def iotaTR (n : Nat) : List Nat :=
   let rec go : Nat → List Nat → List Nat
     | 0, r => r.reverse
     | m@(n+1), r => go n (m::r)
   go n []
 
+set_option linter.deprecated false in
 @[csimp]
 theorem iota_eq_iotaTR : @iota = @iotaTR :=
   have aux (n : Nat) (r : List Nat) : iotaTR.go n r = r.reverse ++ iota n := by

src/Init/Data/List/Control.lean

@@ -254,6 +254,7 @@ theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as :
     | [], b, _    => pure b
     | a::as', b, h => do
       have : a ∈ as := by
+        clear f
         have ⟨bs, h⟩ := h
         subst h
         exact mem_append_right _ (Mem.head ..)

src/Init/Data/List/Count.lean

@@ -40,7 +40,7 @@ protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 :
 theorem countP_cons (a : α) (l) : countP p (a :: l) = countP p l + if p a then 1 else 0 := by
   by_cases h : p a <;> simp [h]
 
-theorem countP_singleton (a : α) : countP p [a] = if p a then 1 else 0 := by
+@[simp] theorem countP_singleton (a : α) : countP p [a] = if p a then 1 else 0 := by
   simp [countP_cons]
 
 theorem length_eq_countP_add_countP (l) : length l = countP p l + countP (fun a => ¬p a) l := by

src/Init/Data/List/Erase.lean

@@ -6,6 +6,7 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Pairwise
+import Init.Data.List.Find
 
 /-!
 # Lemmas about `List.eraseP` and `List.erase`.
@@ -572,4 +573,19 @@ protected theorem IsPrefix.eraseIdx {l l' : List α} (h : l <+: l') (k : Nat) :
 -- See also `mem_eraseIdx_iff_getElem` and `mem_eraseIdx_iff_getElem?` in
 -- `Init/Data/List/Nat/Basic.lean`.
 
+theorem erase_eq_eraseIdx [BEq α] [LawfulBEq α] (l : List α) (a : α) (i : Nat) (w : l.indexOf a = i) :
+    l.erase a = l.eraseIdx i := by
+  subst w
+  rw [erase_eq_iff]
+  by_cases h : a ∈ l
+  · right
+    obtain ⟨as, bs, rfl, h'⟩ := eq_append_cons_of_mem h
+    refine ⟨as, bs, h', by simp, ?_⟩
+    rw [indexOf_append, if_neg h', indexOf_cons_self, eraseIdx_append_of_length_le] <;>
+      simp
+  · left
+    refine ⟨h, ?_⟩
+    rw [eq_comm, eraseIdx_eq_self]
+    exact Nat.le_of_eq (indexOf_eq_length h).symm
+
 end List

src/Init/Data/List/Find.lean

@@ -884,14 +884,68 @@ theorem IsInfix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l
     List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
   h.sublist.findIdx?_eq_none
 
-/-! ### indexOf -/
+/-! ### indexOf
+
+The verification API for `indexOf` is still incomplete.
+The lemmas below should be made consistent with those for `findIdx` (and proved using them).
+-/
 
 theorem indexOf_cons [BEq α] :
     (x :: xs : List α).indexOf y = bif x == y then 0 else xs.indexOf y + 1 := by
   dsimp [indexOf]
   simp [findIdx_cons]
 
+@[simp] theorem indexOf_cons_self [BEq α] [ReflBEq α] {l : List α} : (a :: l).indexOf a = 0 := by
+  simp [indexOf_cons]
+
+theorem indexOf_append [BEq α] [LawfulBEq α] {l₁ l₂ : List α} {a : α} :
+    (l₁ ++ l₂).indexOf a = if a ∈ l₁ then l₁.indexOf a else l₂.indexOf a + l₁.length := by
+  rw [indexOf, findIdx_append]
+  split <;> rename_i h
+  · rw [if_pos]
+    simpa using h
+  · rw [if_neg]
+    simpa using h
+
+theorem indexOf_eq_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∉ l) : l.indexOf a = l.length := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [mem_cons, not_or] at h
+    simp only [indexOf_cons, cond_eq_if, beq_iff_eq]
+    split <;> simp_all
+
+theorem indexOf_lt_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∈ l) : l.indexOf a < l.length := by
+  induction l with
+  | nil => simp at h
+  | cons x xs ih =>
+    simp only [mem_cons] at h
+    obtain rfl | h := h
+    · simp
+    · simp only [indexOf_cons, cond_eq_if, beq_iff_eq, length_cons]
+      specialize ih h
+      split
+      · exact zero_lt_succ xs.length
+      · exact Nat.add_lt_add_right ih 1
+
+/-! ### indexOf?
+
+The verification API for `indexOf?` is still incomplete.
+The lemmas below should be made consistent with those for `findIdx?` (and proved using them).
+-/
+
+@[simp] theorem indexOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
+    l.indexOf? a = none ↔ a ∉ l := by
+  simp only [indexOf?, findIdx?_eq_none_iff, beq_eq_false_iff_ne, ne_eq]
+  constructor
+  · intro w h
+    specialize w _ h
+    simp at w
+  · rintro w x h rfl
+    contradiction
+
 /-! ### lookup -/
+
 section lookup
 variable [BEq α] [LawfulBEq α]
 

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

@@ -813,11 +813,6 @@ theorem getElem_cons_length (x : α) (xs : List α) (i : Nat) (h : i = xs.length
     (x :: xs)[i]'(by simp [h]) = (x :: xs).getLast (cons_ne_nil x xs) := by
   rw [getLast_eq_getElem]; cases h; rfl
 
-@[deprecated getElem_cons_length (since := "2024-06-12")]
-theorem get_cons_length (x : α) (xs : List α) (n : Nat) (h : n = xs.length) :
-    (x :: xs).get ⟨n, by simp [h]⟩ = (x :: xs).getLast (cons_ne_nil x xs) := by
-  simp [getElem_cons_length, h]
-
 /-! ### getLast? -/
 
 @[simp] theorem getLast?_singleton (a : α) : getLast? [a] = a := rfl
@@ -1026,21 +1021,10 @@ theorem getLast?_tail (l : List α) : (tail l).getLast? = if l.length = 1 then n
   | _ :: _, 0 => by simp
   | _ :: l, i+1 => by simp [getElem?_map f l i]
 
-@[deprecated getElem?_map (since := "2024-06-12")]
-theorem get?_map (f : α → β) : ∀ l i, (map f l).get? i = (l.get? i).map f
-  | [], _ => rfl
-  | _ :: _, 0 => rfl
-  | _ :: l, i+1 => get?_map f l i
-
 @[simp] theorem getElem_map (f : α → β) {l} {i : Nat} {h : i < (map f l).length} :
     (map f l)[i] = f (l[i]'(length_map l f ▸ h)) :=
   Option.some.inj <| by rw [← getElem?_eq_getElem, getElem?_map, getElem?_eq_getElem]; rfl
 
-@[deprecated getElem_map (since := "2024-06-12")]
-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
@@ -1076,9 +1060,31 @@ 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
 
@@ -1087,14 +1093,6 @@ 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
@@ -1272,8 +1270,6 @@ theorem filter_map (f : β → α) (l : List β) : filter p (map f l) = map f (f
   | nil => rfl
   | cons a l IH => by_cases h : p (f a) <;> simp [*]
 
-@[deprecated filter_map (since := "2024-06-15")] abbrev map_filter := @filter_map
-
 theorem map_filter_eq_foldr (f : α → β) (p : α → Bool) (as : List α) :
     map f (filter p as) = foldr (fun a bs => bif p a then f a :: bs else bs) [] as := by
   induction as with
@@ -1318,8 +1314,6 @@ theorem filter_congr {p q : α → Bool} :
     · simp [pa, h.1 ▸ pa, filter_congr h.2]
     · simp [pa, h.1 ▸ pa, filter_congr h.2]
 
-@[deprecated filter_congr (since := "2024-06-20")] abbrev filter_congr' := @filter_congr
-
 theorem head_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p (l.head w)) :
     (filter p l).head ((ne_nil_of_mem (mem_filter.2 ⟨head_mem w, h⟩))) = l.head w := by
   cases l with
@@ -1494,6 +1488,34 @@ 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'
@@ -1519,11 +1541,6 @@ theorem getElem?_append {l₁ l₂ : List α} {i : Nat} :
   · exact getElem?_append_left h
   · exact getElem?_append_right (by simpa using h)
 
-@[deprecated getElem?_append_right (since := "2024-06-12")]
-theorem get?_append_right {l₁ l₂ : List α} {i : Nat} (h : l₁.length ≤ i) :
-    (l₁ ++ l₂).get? i = l₂.get? (i - l₁.length) := by
-  simp [getElem?_append_right, h]
-
 /-- Variant of `getElem_append_left` useful for rewriting from the small list to the big list. -/
 theorem getElem_append_left' (l₂ : List α) {l₁ : List α} {i : Nat} (hi : i < l₁.length) :
     l₁[i] = (l₁ ++ l₂)[i]'(by simpa using Nat.lt_add_right l₂.length hi) := by
@@ -1534,41 +1551,11 @@ theorem getElem_append_right' (l₁ : List α) {l₂ : List α} {i : Nat} (hi :
     l₂[i] = (l₁ ++ l₂)[i + l₁.length]'(by simpa [Nat.add_comm] using Nat.add_lt_add_left hi _) := by
   rw [getElem_append_right] <;> simp [*, le_add_left]
 
-@[deprecated "Deprecated without replacement." (since := "2024-06-12")]
-theorem get_append_right_aux {l₁ l₂ : List α} {i : Nat}
-  (h₁ : l₁.length ≤ i) (h₂ : i < (l₁ ++ l₂).length) : i - l₁.length < l₂.length := by
-  rw [length_append] at h₂
-  exact Nat.sub_lt_left_of_lt_add h₁ h₂
-
-set_option linter.deprecated false in
-@[deprecated getElem_append_right (since := "2024-06-12")]
-theorem get_append_right' {l₁ l₂ : List α} {i : Nat} (h₁ : l₁.length ≤ i) (h₂) :
-    (l₁ ++ l₂).get ⟨i, h₂⟩ = l₂.get ⟨i - l₁.length, get_append_right_aux h₁ h₂⟩ :=
-  Option.some.inj <| by rw [← get?_eq_get, ← get?_eq_get, get?_append_right h₁]
-
 theorem getElem_of_append {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i) :
     l[i]'(eq ▸ h ▸ by simp_arith) = a := Option.some.inj <| by
   rw [← getElem?_eq_getElem, eq, getElem?_append_right (h ▸ Nat.le_refl _), h]
   simp
 
-@[deprecated "Deprecated without replacement." (since := "2024-06-12")]
-theorem get_of_append_proof {l : List α}
-    (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i) : i < length l := eq ▸ h ▸ by simp_arith
-
-set_option linter.deprecated false in
-@[deprecated getElem_of_append (since := "2024-06-12")]
-theorem get_of_append {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i) :
-    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 :
@@ -1585,8 +1572,8 @@ 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 :=
@@ -1614,33 +1601,58 @@ 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
     simp [getLast_cons _, getLast_concat t]
 
-@[deprecated getElem_append (since := "2024-06-12")]
-theorem get_append {l₁ l₂ : List α} (n : Nat) (h : n < l₁.length) :
-    (l₁ ++ l₂).get ⟨n, length_append .. ▸ Nat.lt_add_right _ h⟩ = l₁.get ⟨n, h⟩ := by
-  simp [getElem_append, h]
+@[simp] theorem append_eq_nil_iff : p ++ q = [] ↔ p = [] ∧ q = [] := by
+  cases p <;> simp
 
-@[deprecated getElem_append_left (since := "2024-06-12")]
-theorem get_append_left (as bs : List α) (h : i < as.length) {h'} :
-    (as ++ bs).get ⟨i, h'⟩ = as.get ⟨i, h⟩ := by
-  simp [getElem_append_left, h, h']
+@[deprecated append_eq_nil_iff (since := "2025-01-13")] abbrev append_eq_nil := @append_eq_nil_iff
 
-@[deprecated getElem_append_right (since := "2024-06-12")]
-theorem get_append_right (as bs : List α) (h : as.length ≤ i) {h' h''} :
-    (as ++ bs).get ⟨i, h'⟩ = bs.get ⟨i - as.length, h''⟩ := by
-  simp [getElem_append_right, h, h', h'']
+@[simp] theorem nil_eq_append_iff : [] = a ++ b ↔ a = [] ∧ b = [] := by
+  rw [eq_comm, append_eq_nil_iff]
 
-@[deprecated getElem?_append_left (since := "2024-06-12")]
-theorem get?_append {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.length) :
-    (l₁ ++ l₂).get? n = l₁.get? n := by
-  simp [getElem?_append_left hn]
+@[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
@@ -1691,60 +1703,6 @@ 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
@@ -1873,7 +1831,7 @@ 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 =>
@@ -1888,6 +1846,9 @@ 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
 
@@ -1913,7 +1874,8 @@ 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`.
 
-@[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)) :
@@ -1966,6 +1928,26 @@ 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) ∨
@@ -1974,8 +1956,8 @@ 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 =>
@@ -1994,6 +1976,13 @@ 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)},
@@ -2014,12 +2003,14 @@ 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]
@@ -2032,7 +2023,7 @@ 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]
 
@@ -2141,10 +2132,6 @@ theorem forall_mem_replicate {p : α → Prop} {a : α} {n} :
     (replicate n a)[m] = a :=
   eq_of_mem_replicate (getElem_mem _)
 
-@[deprecated getElem_replicate (since := "2024-06-12")]
-theorem get_replicate (a : α) {n : Nat} (m : Fin _) : (replicate n a).get m = a := by
-  simp
-
 theorem getElem?_replicate : (replicate n a)[m]? = if m < n then some a else none := by
   by_cases h : m < n
   · rw [getElem?_eq_getElem (by simpa), getElem_replicate, if_pos h]
@@ -2216,7 +2203,7 @@ theorem map_const' (l : List α) (b : β) : map (fun _ => b) l = replicate l.len
   · intro i h₁ h₂
     simp [getElem_set]
 
-@[simp] theorem append_replicate_replicate : replicate n a ++ replicate m a = replicate (n + m) a := by
+@[simp] theorem replicate_append_replicate : replicate n a ++ replicate m a = replicate (n + m) a := by
   rw [eq_replicate_iff]
   constructor
   · simp
@@ -2224,6 +2211,9 @@ theorem map_const' (l : List α) (b : β) : map (fun _ => b) l = replicate l.len
     simp only [mem_append, mem_replicate, ne_eq]
     rintro (⟨-, rfl⟩ | ⟨_, rfl⟩) <;> rfl
 
+@[deprecated replicate_append_replicate (since := "2025-01-16")]
+abbrev append_replicate_replicate := @replicate_append_replicate
+
 theorem append_eq_replicate_iff {l₁ l₂ : List α} {a : α} :
     l₁ ++ l₂ = replicate n a ↔
       l₁.length + l₂.length = n ∧ l₁ = replicate l₁.length a ∧ l₂ = replicate l₂.length a := by
@@ -2234,6 +2224,11 @@ theorem append_eq_replicate_iff {l₁ l₂ : List α} {a : α} :
 
 @[deprecated append_eq_replicate_iff (since := "2024-09-05")] abbrev append_eq_replicate := @append_eq_replicate_iff
 
+theorem replicate_eq_append_iff {l₁ l₂ : List α} {a : α} :
+    replicate n a = l₁ ++ l₂ ↔
+      l₁.length + l₂.length = n ∧ l₁ = replicate l₁.length a ∧ l₂ = replicate l₂.length a := by
+  rw [eq_comm, append_eq_replicate_iff]
+
 @[simp] theorem map_replicate : (replicate n a).map f = replicate n (f a) := by
   ext1 n
   simp only [getElem?_map, getElem?_replicate]
@@ -2285,7 +2280,7 @@ theorem filterMap_replicate_of_some {f : α → Option β} (h : f a = some b) :
   induction n with
   | zero => simp
   | succ n ih =>
-    simp only [replicate_succ, flatten_cons, ih, append_replicate_replicate, replicate_inj, or_true,
+    simp only [replicate_succ, flatten_cons, ih, replicate_append_replicate, replicate_inj, or_true,
       and_true, add_one_mul, Nat.add_comm]
 
 theorem flatMap_replicate {β} (f : α → List β) : (replicate n a).flatMap f = (replicate n (f a)).flatten := by
@@ -2337,6 +2332,9 @@ 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
@@ -2369,10 +2367,6 @@ theorem getElem?_reverse' : ∀ {l : List α} (i j), i + j + 1 = length l →
     rw [getElem?_append_left, getElem?_reverse' _ _ this]
     rw [length_reverse, ← this]; apply Nat.lt_add_of_pos_right (Nat.succ_pos _)
 
-@[deprecated getElem?_reverse' (since := "2024-06-12")]
-theorem get?_reverse' {l : List α} (i j) (h : i + j + 1 = length l) : get? l.reverse i = get? l j := by
-  simp [getElem?_reverse' _ _ h]
-
 @[simp]
 theorem getElem?_reverse {l : List α} {i} (h : i < length l) :
     l.reverse[i]? = l[l.length - 1 - i]? :=
@@ -2387,11 +2381,6 @@ theorem getElem_reverse {l : List α} {i} (h : i < l.reverse.length) :
   rw [← getElem?_eq_getElem, ← getElem?_eq_getElem]
   rw [getElem?_reverse (by simpa using h)]
 
-@[deprecated getElem?_reverse (since := "2024-06-12")]
-theorem get?_reverse {l : List α} {i} (h : i < length l) :
-    get? l.reverse i = get? l (l.length - 1 - i) := by
-  simp [getElem?_reverse h]
-
 theorem reverseAux_reverseAux_nil (as bs : List α) : reverseAux (reverseAux as bs) [] = reverseAux bs as := by
   induction as generalizing bs with
   | nil => rfl
@@ -2432,10 +2421,6 @@ theorem mem_of_mem_getLast? {l : List α} {a : α} (h : a ∈ getLast? l) : a
 @[simp] theorem map_reverse (f : α → β) (l : List α) : l.reverse.map f = (l.map f).reverse := by
   induction l <;> simp [*]
 
-@[deprecated map_reverse (since := "2024-06-20")]
-theorem reverse_map (f : α → β) (l : List α) : (l.map f).reverse = l.reverse.map f := by
-  simp
-
 @[simp] theorem filter_reverse (p : α → Bool) (l : List α) : (l.reverse.filter p) = (l.filter p).reverse := by
   induction l with
   | nil => simp
@@ -2561,20 +2546,24 @@ theorem foldr_filterMap (f : α → Option β) (g : β → γ → γ) (l : List
     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)
+theorem foldl_map_hom (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)
+@[deprecated foldl_map_hom (since := "2025-01-20")] abbrev foldl_map' := @foldl_map_hom
+
+theorem foldr_map_hom (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]
 
+@[deprecated foldr_map_hom (since := "2025-01-20")] abbrev foldr_map' := @foldr_map_hom
+
 @[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 [*]
@@ -2761,10 +2750,12 @@ theorem getLast?_eq_some_iff {xs : List α} {a : α} : xs.getLast? = some a ↔
   rw [getLast?_eq_head?_reverse, head?_isSome]
   simp
 
-theorem mem_of_getLast?_eq_some {xs : List α} {a : α} (h : xs.getLast? = some a) : a ∈ xs := by
+theorem mem_of_getLast? {xs : List α} {a : α} (h : xs.getLast? = some a) : a ∈ xs := by
   obtain ⟨ys, rfl⟩ := getLast?_eq_some_iff.1 h
   exact mem_concat_self ys a
 
+@[deprecated mem_of_getLast? (since := "2024-10-21")] abbrev mem_of_getLast?_eq_some := @mem_of_getLast?
+
 @[simp] theorem getLast_reverse {l : List α} (h : l.reverse ≠ []) :
     l.reverse.getLast h = l.head (by simp_all) := by
   simp [getLast_eq_head_reverse]
@@ -2899,11 +2890,6 @@ are often used for theorems about `Array.pop`.
   | _::_::_, 0, _ => rfl
   | _::_::_, i+1, h => getElem_dropLast _ i (Nat.add_one_lt_add_one_iff.mp h)
 
-@[deprecated getElem_dropLast (since := "2024-06-12")]
-theorem get_dropLast (xs : List α) (i : Fin xs.dropLast.length) :
-    xs.dropLast.get i = xs.get ⟨i, Nat.lt_of_lt_of_le i.isLt (length_dropLast .. ▸ Nat.pred_le _)⟩ := by
-  simp
-
 theorem getElem?_dropLast (xs : List α) (i : Nat) :
     xs.dropLast[i]? = if i < xs.length - 1 then xs[i]? else none := by
   split
@@ -3441,29 +3427,6 @@ theorem mem_iff_get? {a} {l : List α} : a ∈ l ↔ ∃ n, l.get? n = some a :=
 
 /-! ### Deprecations -/
 
-@[deprecated getD_eq_getElem?_getD (since := "2024-06-12")]
-theorem getD_eq_get? : ∀ l n (a : α), getD l n a = (get? l n).getD a := by simp
-@[deprecated getElem_singleton (since := "2024-06-12")]
-theorem get_singleton (a : α) (n : Fin 1) : get [a] n = a := by simp
-@[deprecated getElem?_concat_length (since := "2024-06-12")]
-theorem get?_concat_length (l : List α) (a : α) : (l ++ [a]).get? l.length = some a := by simp
-@[deprecated getElem_set_self (since := "2024-06-12")]
-theorem get_set_eq {l : List α} {i : Nat} {a : α} (h : i < (l.set i a).length) :
-    (l.set i a).get ⟨i, h⟩ = a := by
-  simp
-@[deprecated getElem_set_ne (since := "2024-06-12")]
-theorem get_set_ne {l : List α} {i j : Nat} (h : i ≠ j) {a : α}
-    (hj : j < (l.set i a).length) :
-    (l.set i a).get ⟨j, hj⟩ = l.get ⟨j, by simp at hj; exact hj⟩ := by
-  simp [h]
-@[deprecated getElem_set (since := "2024-06-12")]
-theorem get_set {l : List α} {m n} {a : α} (h) :
-    (set l m a).get ⟨n, h⟩ = if m = n then a else l.get ⟨n, length_set .. ▸ h⟩ := by
-  simp [getElem_set]
-@[deprecated cons_inj_right (since := "2024-06-15")] abbrev cons_inj := @cons_inj_right
-@[deprecated ne_nil_of_length_eq_add_one (since := "2024-06-16")]
-abbrev ne_nil_of_length_eq_succ := @ne_nil_of_length_eq_add_one
-
 @[deprecated "Deprecated without replacement." (since := "2024-07-09")]
 theorem get_cons_cons_one : (a₁ :: a₂ :: as).get (1 : Fin (as.length + 2)) = a₂ := rfl
 

src/Init/Data/List/MapIdx.lean

@@ -17,18 +17,19 @@ namespace List
 
 /-! ### mapIdx -/
 
-
 /--
 Given a list `as = [a₀, a₁, ...]` function `f : Fin as.length → α → β`, returns the list
 `[f 0 a₀, f 1 a₁, ...]`.
 -/
-@[inline] def mapFinIdx (as : List α) (f : Fin as.length → α → β) : List β := go as #[] (by simp) where
+@[inline] def mapFinIdx (as : List α) (f : (i : Nat) → α → (h : i < as.length) → β) : List β :=
+  go as #[] (by simp)
+where
   /-- Auxiliary for `mapFinIdx`:
   `mapFinIdx.go [a₀, a₁, ...] acc = acc.toList ++ [f 0 a₀, f 1 a₁, ...]` -/
   @[specialize] go : (bs : List α) → (acc : Array β) → bs.length + acc.size = as.length → List β
   | [], acc, h => acc.toList
   | a :: as, acc, h =>
-    go as (acc.push (f ⟨acc.size, by simp at h; omega⟩ a)) (by simp at h ⊢; omega)
+    go as (acc.push (f acc.size a (by simp at h; omega))) (by simp at h ⊢; omega)
 
 /--
 Given a function `f : Nat → α → β` and `as : List α`, `as = [a₀, a₁, ...]`, returns the list
@@ -43,8 +44,14 @@ Given a function `f : Nat → α → β` and `as : List α`, `as = [a₀, a₁,
 
 /-! ### mapFinIdx -/
 
+@[congr] theorem mapFinIdx_congr {xs ys : List α} (w : xs = ys)
+    (f : (i : Nat) → α → (h : i < xs.length) → β) :
+    mapFinIdx xs f = mapFinIdx ys (fun i a h => f i a (by simp [w]; omega)) := by
+  subst w
+  rfl
+
 @[simp]
-theorem mapFinIdx_nil {f : Fin 0 → α → β} : mapFinIdx [] f = [] :=
+theorem mapFinIdx_nil {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx [] f = [] :=
   rfl
 
 @[simp] theorem length_mapFinIdx_go :
@@ -53,13 +60,16 @@ theorem mapFinIdx_nil {f : Fin 0 → α → β} : mapFinIdx [] f = [] :=
   | nil => simpa using h
   | cons _ _ ih => simp [mapFinIdx.go, ih]
 
-@[simp] theorem length_mapFinIdx {as : List α} {f : Fin as.length → α → β} :
+@[simp] theorem length_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} :
     (as.mapFinIdx f).length = as.length := by
   simp [mapFinIdx, length_mapFinIdx_go]
 
-theorem getElem_mapFinIdx_go {as : List α} {f : Fin as.length → α → β} {i : Nat} {h} {w} :
+theorem getElem_mapFinIdx_go {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} {i : Nat} {h} {w} :
     (mapFinIdx.go as f bs acc h)[i] =
-      if w' : i < acc.size then acc[i] else f ⟨i, by simp at w; omega⟩ (bs[i - acc.size]'(by simp at w; omega)) := by
+      if w' : i < acc.size then
+        acc[i]
+      else
+        f i (bs[i - acc.size]'(by simp at w; omega)) (by simp at w; omega) := by
   induction bs generalizing acc with
   | nil =>
     simp only [length_mapFinIdx_go, length_nil, Nat.zero_add] at w h
@@ -78,29 +88,30 @@ theorem getElem_mapFinIdx_go {as : List α} {f : Fin as.length → α → β} {i
     · have h₃ : i - acc.size = (i - (acc.size + 1)) + 1 := by omega
       simp [h₃]
 
-@[simp] theorem getElem_mapFinIdx {as : List α} {f : Fin as.length → α → β} {i : Nat} {h} :
-    (as.mapFinIdx f)[i] = f ⟨i, by simp at h; omega⟩ (as[i]'(by simp at h; omega)) := by
+@[simp] theorem getElem_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} {i : Nat} {h} :
+    (as.mapFinIdx f)[i] = f i (as[i]'(by simp at h; omega)) (by simp at h; omega) := by
   simp [mapFinIdx, getElem_mapFinIdx_go]
 
-theorem mapFinIdx_eq_ofFn {as : List α} {f : Fin as.length → α → β} :
-    as.mapFinIdx f = List.ofFn fun i : Fin as.length => f i as[i] := by
+theorem mapFinIdx_eq_ofFn {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} :
+    as.mapFinIdx f = List.ofFn fun i : Fin as.length => f i as[i] i.2 := by
   apply ext_getElem <;> simp
 
-@[simp] theorem getElem?_mapFinIdx {l : List α} {f : Fin l.length → α → β} {i : Nat} :
-    (l.mapFinIdx f)[i]? = l[i]?.pbind fun x m => f ⟨i, by simp [getElem?_eq_some_iff] at m; exact m.1⟩ x := by
+@[simp] theorem getElem?_mapFinIdx {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {i : Nat} :
+    (l.mapFinIdx f)[i]? = l[i]?.pbind fun x m => f i x (by simp [getElem?_eq_some_iff] at m; exact m.1) := by
   simp only [getElem?_def, length_mapFinIdx, getElem_mapFinIdx]
   split <;> simp
 
 @[simp]
-theorem mapFinIdx_cons {l : List α} {a : α} {f : Fin (l.length + 1) → α → β} :
-    mapFinIdx (a :: l) f = f 0 a :: mapFinIdx l (fun i => f i.succ) := by
+theorem mapFinIdx_cons {l : List α} {a : α} {f : (i : Nat) → α → (h : i < l.length + 1) → β} :
+    mapFinIdx (a :: l) f = f 0 a (by omega) :: mapFinIdx l (fun i a h => f (i + 1) a (by omega)) := by
   apply ext_getElem
   · simp
   · rintro (_|i) h₁ h₂ <;> simp
 
-theorem mapFinIdx_append {K L : List α} {f : Fin (K ++ L).length → α → β} :
+theorem mapFinIdx_append {K L : List α} {f : (i : Nat) → α → (h : i < (K ++ L).length) → β} :
     (K ++ L).mapFinIdx f =
-      K.mapFinIdx (fun i => f (i.castLE (by simp))) ++ L.mapFinIdx (fun i => f ((i.natAdd K.length).cast (by simp))) := by
+      K.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
+        L.mapFinIdx (fun i a h => f (i + K.length) a (by simp; omega)) := by
   apply ext_getElem
   · simp
   · intro i h₁ h₂
@@ -108,60 +119,57 @@ theorem mapFinIdx_append {K L : List α} {f : Fin (K ++ L).length → α → β}
     simp only [getElem_mapFinIdx, length_mapFinIdx]
     split <;> rename_i h
     · rw [getElem_append_left]
-      congr
     · simp only [Nat.not_lt] at h
       rw [getElem_append_right h]
       congr
-      simp
       omega
 
-@[simp] theorem mapFinIdx_concat {l : List α} {e : α} {f : Fin (l ++ [e]).length → α → β}:
-    (l ++ [e]).mapFinIdx f = l.mapFinIdx (fun i => f (i.castLE (by simp))) ++ [f ⟨l.length, by simp⟩ e] := by
+@[simp] theorem mapFinIdx_concat {l : List α} {e : α} {f : (i : Nat) → α → (h : i < (l ++ [e]).length) → β}:
+    (l ++ [e]).mapFinIdx f = l.mapFinIdx (fun i a h => f i a (by simp; omega)) ++ [f l.length e (by simp)] := by
   simp [mapFinIdx_append]
-  congr
 
-theorem mapFinIdx_singleton {a : α} {f : Fin 1 → α → β} :
-    [a].mapFinIdx f = [f ⟨0, by simp⟩ a] := by
+theorem mapFinIdx_singleton {a : α} {f : (i : Nat) → α → (h : i < 1) → β} :
+    [a].mapFinIdx f = [f 0 a (by simp)] := by
   simp
 
-theorem mapFinIdx_eq_enum_map {l : List α} {f : Fin l.length → α → β} :
+theorem mapFinIdx_eq_enum_map {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
     l.mapFinIdx f = l.enum.attach.map
       fun ⟨⟨i, x⟩, m⟩ =>
-        f ⟨i, by rw [mk_mem_enum_iff_getElem?, getElem?_eq_some_iff] at m; exact m.1⟩ x := by
+        f i x (by rw [mk_mem_enum_iff_getElem?, getElem?_eq_some_iff] at m; exact m.1) := by
   apply ext_getElem <;> simp
 
 @[simp]
-theorem mapFinIdx_eq_nil_iff {l : List α} {f : Fin l.length → α → β} :
+theorem mapFinIdx_eq_nil_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
     l.mapFinIdx f = [] ↔ l = [] := by
   rw [mapFinIdx_eq_enum_map, map_eq_nil_iff, attach_eq_nil_iff, enum_eq_nil_iff]
 
-theorem mapFinIdx_ne_nil_iff {l : List α} {f : Fin l.length → α → β} :
+theorem mapFinIdx_ne_nil_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
     l.mapFinIdx f ≠ [] ↔ l ≠ [] := by
   simp
 
-theorem exists_of_mem_mapFinIdx {b : β} {l : List α} {f : Fin l.length → α → β}
-    (h : b ∈ l.mapFinIdx f) : ∃ (i : Fin l.length), f i l[i] = b := by
+theorem exists_of_mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β}
+    (h : b ∈ l.mapFinIdx f) : ∃ (i : Nat) (h : i < l.length), f i l[i] h = b := by
   rw [mapFinIdx_eq_enum_map] at h
   replace h := exists_of_mem_map h
   simp only [mem_attach, true_and, Subtype.exists, Prod.exists, mk_mem_enum_iff_getElem?] at h
   obtain ⟨i, b, h, rfl⟩ := h
   rw [getElem?_eq_some_iff] at h
   obtain ⟨h', rfl⟩ := h
-  exact ⟨⟨i, h'⟩, rfl⟩
+  exact ⟨i, h', rfl⟩
 
-@[simp] theorem mem_mapFinIdx {b : β} {l : List α} {f : Fin l.length → α → β} :
-    b ∈ l.mapFinIdx f ↔ ∃ (i : Fin l.length), f i l[i] = b := by
+@[simp] theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
+    b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] h = b := by
   constructor
   · intro h
     exact exists_of_mem_mapFinIdx h
   · rintro ⟨i, h, rfl⟩
     rw [mem_iff_getElem]
-    exact ⟨i, by simp⟩
+    exact ⟨i, by simpa using h, by simp⟩
 
-theorem mapFinIdx_eq_cons_iff {l : List α} {b : β} {f : Fin l.length → α → β} :
+theorem mapFinIdx_eq_cons_iff {l : List α} {b : β} {f : (i : Nat) → α → (h : i < l.length) → β} :
     l.mapFinIdx f = b :: l₂ ↔
-      ∃ (a : α) (l₁ : List α) (h : l = a :: l₁),
-        f ⟨0, by simp [h]⟩ a = b ∧ l₁.mapFinIdx (fun i => f (i.succ.cast (by simp [h]))) = l₂ := by
+      ∃ (a : α) (l₁ : List α) (w : l = a :: l₁),
+        f 0 a (by simp [w]) = b ∧ l₁.mapFinIdx (fun i a h => f (i + 1) a (by simp [w]; omega)) = l₂ := by
   cases l with
   | nil => simp
   | cons x l' =>
@@ -169,39 +177,91 @@ theorem mapFinIdx_eq_cons_iff {l : List α} {b : β} {f : Fin l.length → α
       exists_and_left]
     constructor
     · rintro ⟨rfl, rfl⟩
-      refine ⟨x, rfl, l', by simp⟩
-    · rintro ⟨a, ⟨rfl, h⟩, ⟨_, ⟨rfl, rfl⟩, h⟩⟩
-      exact ⟨rfl, h⟩
+      refine ⟨x, l', ⟨rfl, rfl⟩, by simp⟩
+    · rintro ⟨a, l', ⟨rfl, rfl⟩, ⟨rfl, rfl⟩⟩
+      exact ⟨rfl, by simp⟩
 
-theorem mapFinIdx_eq_cons_iff' {l : List α} {b : β} {f : Fin l.length → α → β} :
+theorem mapFinIdx_eq_cons_iff' {l : List α} {b : β} {f : (i : Nat) → α → (h : i < l.length) → β} :
     l.mapFinIdx f = b :: l₂ ↔
-      l.head?.pbind (fun x m => (f ⟨0, by cases l <;> simp_all⟩ x)) = some b ∧
-        l.tail?.attach.map (fun ⟨t, m⟩ => t.mapFinIdx fun i => f (i.succ.cast (by cases l <;> simp_all))) = some l₂ := by
+      l.head?.pbind (fun x m => (f 0 x (by cases l <;> simp_all))) = some b ∧
+        l.tail?.attach.map (fun ⟨t, m⟩ => t.mapFinIdx fun i a h => f (i + 1) a (by cases l <;> simp_all)) = some l₂ := by
   cases l <;> simp
 
-theorem mapFinIdx_eq_iff {l : List α} {f : Fin l.length → α → β} :
-    l.mapFinIdx f = l' ↔ ∃ h : l'.length = l.length, ∀ (i : Nat) (h : i < l.length), l'[i] = f ⟨i, h⟩ l[i] := by
+theorem mapFinIdx_eq_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
+    l.mapFinIdx f = l' ↔ ∃ h : l'.length = l.length, ∀ (i : Nat) (h : i < l.length), l'[i] = f i l[i] h := by
   constructor
   · rintro rfl
     simp
   · rintro ⟨h, w⟩
     apply ext_getElem <;> simp_all
 
-theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : Fin l.length → α → β} :
-    l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Fin l.length), f i l[i] = g i l[i] := by
+@[simp] theorem mapFinIdx_eq_singleton_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {b : β} :
+    l.mapFinIdx f = [b] ↔ ∃ (a : α) (w : l = [a]), f 0 a (by simp [w]) = b := by
+  simp [mapFinIdx_eq_cons_iff]
+
+theorem mapFinIdx_eq_append_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
+    l.mapFinIdx f = l₁ ++ l₂ ↔
+      ∃ (l₁' : List α) (l₂' : List α) (w : l = l₁' ++ l₂'),
+        l₁'.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₁ ∧
+        l₂'.mapFinIdx (fun i a h => f (i + l₁'.length) a (by simp [w]; omega)) = l₂ := by
+  rw [mapFinIdx_eq_iff]
+  constructor
+  · intro ⟨h, w⟩
+    simp only [length_append] at h
+    refine ⟨l.take l₁.length, l.drop l₁.length, by simp, ?_⟩
+    constructor
+    · apply ext_getElem
+      · simp
+        omega
+      · intro i hi₁ hi₂
+        simp only [getElem_mapFinIdx, getElem_take]
+        specialize w i (by omega)
+        rw [getElem_append_left hi₂] at w
+        exact w.symm
+    · apply ext_getElem
+      · simp
+        omega
+      · intro i hi₁ hi₂
+        simp only [getElem_mapFinIdx, getElem_take]
+        simp only [length_take, getElem_drop]
+        have : l₁.length ≤ l.length := by omega
+        simp only [Nat.min_eq_left this, Nat.add_comm]
+        specialize w (i + l₁.length) (by omega)
+        rw [getElem_append_right (by omega)] at w
+        simpa using w.symm
+  · rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
+    refine ⟨by simp, fun i h => ?_⟩
+    rw [getElem_append]
+    split <;> rename_i h'
+    · simp [getElem_append_left (by simpa using h')]
+    · simp only [length_mapFinIdx, Nat.not_lt] at h'
+      have : i - l₁'.length + l₁'.length = i := by omega
+      simp [getElem_append_right h', this]
+
+theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : (i : Nat) → α → (h : i < l.length) → β} :
+    l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Nat) (h : i < l.length), f i l[i] h = g i l[i] h := by
   rw [eq_comm, mapFinIdx_eq_iff]
   simp [Fin.forall_iff]
 
-@[simp] theorem mapFinIdx_mapFinIdx {l : List α} {f : Fin l.length → α → β} {g : Fin _ → β → γ} :
-    (l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i => g (i.cast (by simp)) ∘ f i) := by
+@[simp] theorem mapFinIdx_mapFinIdx {l : List α}
+    {f : (i : Nat) → α → (h : i < l.length) → β}
+    {g : (i : Nat) → β → (h : i < (l.mapFinIdx f).length) → γ} :
+    (l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i a h => g i (f i a h) (by simpa)) := by
   simp [mapFinIdx_eq_iff]
 
-theorem mapFinIdx_eq_replicate_iff {l : List α} {f : Fin l.length → α → β} {b : β} :
-    l.mapFinIdx f = replicate l.length b ↔ ∀ (i : Fin l.length), f i l[i] = b := by
-  simp [eq_replicate_iff, length_mapFinIdx, mem_mapFinIdx, forall_exists_index, true_and]
+theorem mapFinIdx_eq_replicate_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {b : β} :
+    l.mapFinIdx f = replicate l.length b ↔ ∀ (i : Nat) (h : i < l.length), f i l[i] h = b := by
+  rw [eq_replicate_iff, length_mapFinIdx]
+  simp only [mem_mapFinIdx, forall_exists_index, true_and]
+  constructor
+  · intro w i h
+    exact w (f i l[i] h) i h rfl
+  · rintro w b i h rfl
+    exact w i h
 
-@[simp] theorem mapFinIdx_reverse {l : List α} {f : Fin l.reverse.length → α → β} :
-    l.reverse.mapFinIdx f = (l.mapFinIdx (fun i => f ⟨l.length - 1 - i, by simp; omega⟩)).reverse := by
+@[simp] theorem mapFinIdx_reverse {l : List α} {f : (i : Nat) → α → (h : i < l.reverse.length) → β} :
+    l.reverse.mapFinIdx f =
+      (l.mapFinIdx (fun i a h => f (l.length - 1 - i) a (by simp; omega))).reverse := by
   simp [mapFinIdx_eq_iff]
   intro i h
   congr
@@ -262,13 +322,13 @@ theorem getElem?_mapIdx_go : ∀ {l : List α} {arr : Array β} {i : Nat},
   rw [← getElem?_eq_getElem, getElem?_mapIdx, getElem?_eq_getElem (by simpa using h)]
   simp
 
-@[simp] theorem mapFinIdx_eq_mapIdx {l : List α} {f : Fin l.length → α → β} {g : Nat → α → β}
-    (h : ∀ (i : Fin l.length), f i l[i] = g i l[i]) :
+@[simp] theorem mapFinIdx_eq_mapIdx {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {g : Nat → α → β}
+    (h : ∀ (i : Nat) (h : i < l.length), f i l[i] h = g i l[i]) :
     l.mapFinIdx f = l.mapIdx g := by
   simp_all [mapFinIdx_eq_iff]
 
 theorem mapIdx_eq_mapFinIdx {l : List α} {f : Nat → α → β} :
-    l.mapIdx f = l.mapFinIdx (fun i => f i) := by
+    l.mapIdx f = l.mapFinIdx (fun i a _ => f i a) := by
   simp [mapFinIdx_eq_mapIdx]
 
 theorem mapIdx_eq_enum_map {l : List α} :
@@ -328,6 +388,10 @@ theorem mapIdx_eq_cons_iff' {l : List α} {b : β} :
       l.head?.map (f 0) = some b ∧ l.tail?.map (mapIdx fun i => f (i + 1)) = some l₂ := by
   cases l <;> simp
 
+@[simp] theorem mapIdx_eq_singleton_iff {l : List α} {f : Nat → α → β} {b : β} :
+    mapIdx f l = [b] ↔ ∃ (a : α), l = [a] ∧ f 0 a = b := by
+  simp [mapIdx_eq_cons_iff]
+
 theorem mapIdx_eq_iff {l : List α} : mapIdx f l = l' ↔ ∀ i : Nat, l'[i]? = l[i]?.map (f i) := by
   constructor
   · intro w i
@@ -336,6 +400,19 @@ theorem mapIdx_eq_iff {l : List α} : mapIdx f l = l' ↔ ∀ i : Nat, l'[i]? =
     ext1 i
     simp [w]
 
+theorem mapIdx_eq_append_iff {l : List α} :
+    mapIdx f l = l₁ ++ l₂ ↔
+      ∃ (l₁' : List α) (l₂' : List α), l = l₁' ++ l₂' ∧
+        mapIdx f l₁' = l₁ ∧
+        mapIdx (fun i => f (i + l₁'.length)) l₂' = l₂ := by
+  rw [mapIdx_eq_mapFinIdx, mapFinIdx_eq_append_iff]
+  simp only [mapFinIdx_eq_mapIdx, exists_and_left, exists_prop]
+  constructor
+  · rintro ⟨l₁, rfl, l₂, rfl, h⟩
+    refine ⟨l₁, l₂, by simp_all⟩
+  · rintro ⟨l₁, l₂, rfl, rfl, rfl⟩
+    refine ⟨l₁, rfl, l₂, by simp_all⟩
+
 theorem mapIdx_eq_mapIdx_iff {l : List α} :
     mapIdx f l = mapIdx g l ↔ ∀ i : Nat, (h : i < l.length) → f i l[i] = g i l[i] := by
   constructor

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

@@ -195,24 +195,32 @@ theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n -
 
 /-! ### iota -/
 
+section
+set_option linter.deprecated false
+
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem iota_eq_reverse_range' : ∀ n : Nat, iota n = reverse (range' 1 n)
   | 0 => rfl
   | n + 1 => by simp [iota, range'_concat, iota_eq_reverse_range' n, reverse_append, Nat.add_comm]
 
-@[simp] theorem length_iota (n : Nat) : length (iota n) = n := by simp [iota_eq_reverse_range']
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem length_iota (n : Nat) : length (iota n) = n := by simp [iota_eq_reverse_range']
 
-@[simp] theorem iota_eq_nil {n : Nat} : iota n = [] ↔ n = 0 := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem iota_eq_nil {n : Nat} : iota n = [] ↔ n = 0 := by
   cases n <;> simp
 
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem iota_ne_nil {n : Nat} : iota n ≠ [] ↔ n ≠ 0 := by
   cases n <;> simp
 
-@[simp]
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
 theorem mem_iota {m n : Nat} : m ∈ iota n ↔ 0 < m ∧ m ≤ n := by
   simp [iota_eq_reverse_range', Nat.add_comm, Nat.lt_succ]
   omega
 
-@[simp] theorem iota_inj : iota n = iota n' ↔ n = n' := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem iota_inj : iota n = iota n' ↔ n = n' := by
   constructor
   · intro h
     have h' := congrArg List.length h
@@ -221,6 +229,7 @@ theorem mem_iota {m n : Nat} : m ∈ iota n ↔ 0 < m ∧ m ≤ n := by
   · rintro rfl
     simp
 
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem iota_eq_cons_iff : iota n = a :: xs ↔ n = a ∧ 0 < n ∧ xs = iota (n - 1) := by
   simp [iota_eq_reverse_range']
   simp [range'_eq_append_iff, reverse_eq_iff]
@@ -234,6 +243,7 @@ theorem iota_eq_cons_iff : iota n = a :: xs ↔ n = a ∧ 0 < n ∧ xs = iota (n
     rw [eq_comm, range'_eq_singleton]
     omega
 
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem iota_eq_append_iff : iota n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs = (range' (k + 1) (n - k)).reverse ∧ ys = iota k := by
   simp only [iota_eq_reverse_range']
   rw [reverse_eq_append_iff]
@@ -245,42 +255,52 @@ theorem iota_eq_append_iff : iota n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs = (rang
   · rintro ⟨k, h, rfl, rfl⟩
     exact ⟨k, by simp; omega⟩
 
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem pairwise_gt_iota (n : Nat) : Pairwise (· > ·) (iota n) := by
   simpa only [iota_eq_reverse_range', pairwise_reverse] using pairwise_lt_range' 1 n
 
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem nodup_iota (n : Nat) : Nodup (iota n) :=
   (pairwise_gt_iota n).imp Nat.ne_of_gt
 
-@[simp] theorem head?_iota (n : Nat) : (iota n).head? = if n = 0 then none else some n := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem head?_iota (n : Nat) : (iota n).head? = if n = 0 then none else some n := by
   cases n <;> simp
 
-@[simp] theorem head_iota (n : Nat) (h) : (iota n).head h = n := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem head_iota (n : Nat) (h) : (iota n).head h = n := by
   cases n with
   | zero => simp at h
   | succ n => simp
 
-@[simp] theorem tail_iota (n : Nat) : (iota n).tail = iota (n - 1) := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem tail_iota (n : Nat) : (iota n).tail = iota (n - 1) := by
   cases n <;> simp
 
-@[simp] theorem reverse_iota : reverse (iota n) = range' 1 n := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem reverse_iota : reverse (iota n) = range' 1 n := by
   induction n with
   | zero => simp
   | succ n ih =>
     rw [iota_succ, reverse_cons, ih, range'_1_concat, Nat.add_comm]
 
-@[simp] theorem getLast?_iota (n : Nat) : (iota n).getLast? = if n = 0 then none else some 1 := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem getLast?_iota (n : Nat) : (iota n).getLast? = if n = 0 then none else some 1 := by
   rw [getLast?_eq_head?_reverse]
   simp [head?_range']
 
-@[simp] theorem getLast_iota (n : Nat) (h) : (iota n).getLast h = 1 := by
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem getLast_iota (n : Nat) (h) : (iota n).getLast h = 1 := by
   rw [getLast_eq_head_reverse]
   simp
 
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20")]
 theorem find?_iota_eq_none {n : Nat} {p : Nat → Bool} :
     (iota n).find? p = none ↔ ∀ i, 0 < i → i ≤ n → !p i := by
   simp
 
-@[simp] theorem find?_iota_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
+@[deprecated "Use `(List.range' 1 n).reverse` instead of `iota n`." (since := "2025-01-20"), simp]
+theorem find?_iota_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
     (iota n).find? p = some i ↔ p i ∧ i ∈ iota n ∧ ∀ j, i < j → j ≤ n → !p j := by
   rw [find?_eq_some_iff_append]
   simp only [iota_eq_reverse_range', reverse_eq_append_iff, reverse_cons, append_assoc, cons_append,
@@ -317,6 +337,8 @@ theorem find?_iota_eq_none {n : Nat} {p : Nat → Bool} :
       · omega
       · omega
 
+end
+
 /-! ### enumFrom -/
 
 @[simp]

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

@@ -47,41 +47,16 @@ length `> i`. Version designed to rewrite from the small list to the big list. -
     L[i]'(Nat.lt_of_lt_of_le h (length_take_le' _ _)) := by
   rw [length_take, Nat.lt_min] at h; rw [getElem_take' L _ h.1]
 
-/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
-length `> i`. Version designed to rewrite from the big list to the small list. -/
-@[deprecated getElem_take' (since := "2024-06-12")]
-theorem get_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
-    get L ⟨i, hi⟩ = get (L.take j) ⟨i, length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩⟩ := by
-  simp
-
-/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
-length `> i`. Version designed to rewrite from the small list to the big list. -/
-@[deprecated getElem_take (since := "2024-06-12")]
-theorem get_take' (L : List α) {j i} :
-    get (L.take j) i =
-    get L ⟨i.1, Nat.lt_of_lt_of_le i.2 (length_take_le' _ _)⟩ := by
-  simp [getElem_take]
-
 theorem getElem?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
     (l.take n)[m]? = none :=
   getElem?_eq_none <| Nat.le_trans (length_take_le _ _) h
 
-@[deprecated getElem?_take_eq_none (since := "2024-06-12")]
-theorem get?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
-    (l.take n).get? m = none := by
-  simp [getElem?_take_eq_none h]
-
 theorem getElem?_take {l : List α} {n m : Nat} :
     (l.take n)[m]? = if m < n then l[m]? else none := by
   split
   · next h => exact getElem?_take_of_lt h
   · next h => exact getElem?_take_eq_none (Nat.le_of_not_lt h)
 
-@[deprecated getElem?_take (since := "2024-06-12")]
-theorem get?_take_eq_if {l : List α} {n m : Nat} :
-    (l.take n).get? m = if m < n then l.get? m else none := by
-  simp [getElem?_take]
-
 theorem head?_take {l : List α} {n : Nat} :
     (l.take n).head? = if n = 0 then none else l.head? := by
   simp [head?_eq_getElem?, getElem?_take]
@@ -226,13 +201,6 @@ theorem getElem_drop' (L : List α) {i j : Nat} (h : i + j < L.length) :
   · simp [Nat.min_eq_left this, Nat.add_sub_cancel_left]
   · simp [Nat.min_eq_left this, Nat.le_add_right]
 
-/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
-dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
-@[deprecated getElem_drop' (since := "2024-06-12")]
-theorem get_drop (L : List α) {i j : Nat} (h : i + j < L.length) :
-    get L ⟨i + j, h⟩ = get (L.drop i) ⟨j, lt_length_drop L h⟩ := by
-  simp [getElem_drop']
-
 /-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
 dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
 @[simp] theorem getElem_drop (L : List α) {i : Nat} {j : Nat} {h : j < (L.drop i).length} :
@@ -241,15 +209,6 @@ dropping the first `i` elements. Version designed to rewrite from the small list
       exact Nat.add_lt_of_lt_sub (length_drop i L ▸ h)) := by
   rw [getElem_drop']
 
-/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
-dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
-@[deprecated getElem_drop' (since := "2024-06-12")]
-theorem get_drop' (L : List α) {i j} :
-    get (L.drop i) j = get L ⟨i + j, by
-      rw [Nat.add_comm]
-      exact Nat.add_lt_of_lt_sub (length_drop i L ▸ j.2)⟩ := by
-  simp
-
 @[simp]
 theorem getElem?_drop (L : List α) (i j : Nat) : (L.drop i)[j]? = L[i + j]? := by
   ext
@@ -261,10 +220,6 @@ theorem getElem?_drop (L : List α) (i j : Nat) : (L.drop i)[j]? = L[i + j]? :=
     rw [Nat.add_comm] at h
     apply Nat.lt_sub_of_add_lt h
 
-@[deprecated getElem?_drop (since := "2024-06-12")]
-theorem get?_drop (L : List α) (i j : Nat) : get? (L.drop i) j = get? L (i + j) := by
-  simp
-
 theorem mem_take_iff_getElem {l : List α} {a : α} :
     a ∈ l.take n ↔ ∃ (i : Nat) (hm : i < min n l.length), l[i] = a := by
   rw [mem_iff_getElem]

src/Init/Data/List/TakeDrop.lean

@@ -67,17 +67,9 @@ theorem getElem_cons_drop : ∀ (l : List α) (i : Nat) (h : i < l.length),
   | _::_, 0, _ => rfl
   | _::_, i+1, h => getElem_cons_drop _ i (Nat.add_one_lt_add_one_iff.mp h)
 
-@[deprecated getElem_cons_drop (since := "2024-06-12")]
-theorem get_cons_drop (l : List α) (i) : get l i :: drop (i + 1) l = drop i l := by
-  simp
-
 theorem drop_eq_getElem_cons {n} {l : List α} (h : n < l.length) : drop n l = l[n] :: drop (n + 1) l :=
   (getElem_cons_drop _ n h).symm
 
-@[deprecated drop_eq_getElem_cons (since := "2024-06-12")]
-theorem drop_eq_get_cons {n} {l : List α} (h) : drop n l = get l ⟨n, h⟩ :: drop (n + 1) l := by
-  simp [drop_eq_getElem_cons]
-
 @[simp]
 theorem getElem?_take_of_lt {l : List α} {n m : Nat} (h : m < n) : (l.take n)[m]? = l[m]? := by
   induction n generalizing l m with
@@ -91,10 +83,6 @@ theorem getElem?_take_of_lt {l : List α} {n m : Nat} (h : m < n) : (l.take n)[m
       · simp
       · simpa using hn (Nat.lt_of_succ_lt_succ h)
 
-@[deprecated getElem?_take_of_lt (since := "2024-06-12")]
-theorem get?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n).get? m = l.get? m := by
-  simp [getElem?_take_of_lt, h]
-
 theorem getElem?_take_of_succ {l : List α} {n : Nat} : (l.take (n + 1))[n]? = l[n]? := by simp
 
 @[simp] theorem drop_drop (n : Nat) : ∀ (m) (l : List α), drop n (drop m l) = drop (m + n) l
@@ -111,10 +99,6 @@ theorem take_drop : ∀ (m n : Nat) (l : List α), take n (drop m l) = drop m (t
   | _, _, [] => by simp
   | _+1, _, _ :: _ => by simpa [Nat.succ_add, take_succ_cons, drop_succ_cons] using take_drop ..
 
-@[deprecated drop_drop (since := "2024-06-15")]
-theorem drop_add (m n) (l : List α) : drop (m + n) l = drop n (drop m l) := by
-  simp [drop_drop]
-
 @[simp]
 theorem tail_drop (l : List α) (n : Nat) : (l.drop n).tail = l.drop (n + 1) := by
   induction l generalizing n with

src/Init/Data/List/ToArray.lean

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

src/Init/Data/List/Zip.lean

@@ -76,15 +76,6 @@ theorem getElem?_zip_eq_some {l₁ : List α} {l₂ : List β} {z : α × β} {i
   · rintro ⟨h₀, h₁⟩
     exact ⟨_, _, h₀, h₁, rfl⟩
 
-@[deprecated getElem?_zipWith (since := "2024-06-12")]
-theorem get?_zipWith {f : α → β → γ} :
-    (List.zipWith f as bs).get? i = match as.get? i, bs.get? i with
-      | some a, some b => some (f a b) | _, _ => none := by
-  simp [getElem?_zipWith]
-
-set_option linter.deprecated false in
-@[deprecated getElem?_zipWith (since := "2024-06-07")] abbrev zipWith_get? := @get?_zipWith
-
 theorem head?_zipWith {f : α → β → γ} :
     (List.zipWith f as bs).head? = match as.head?, bs.head? with
       | some a, some b => some (f a b) | _, _ => none := by
@@ -203,11 +194,11 @@ 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 =>
@@ -369,15 +360,6 @@ theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
       cases i <;> simp_all
     | cons b bs => cases i <;> simp_all
 
-@[deprecated getElem?_zipWithAll (since := "2024-06-12")]
-theorem get?_zipWithAll {f : Option α → Option β → γ} :
-    (zipWithAll f as bs).get? i = match as.get? i, bs.get? i with
-      | none, none => .none | a?, b? => some (f a? b?) := by
-  simp [getElem?_zipWithAll]
-
-set_option linter.deprecated false in
-@[deprecated getElem?_zipWithAll (since := "2024-06-07")] abbrev zipWithAll_get? := @get?_zipWithAll
-
 theorem head?_zipWithAll {f : Option α → Option β → γ} :
     (zipWithAll f as bs).head? = match as.head?, bs.head? with
       | none, none => .none | a?, b? => some (f a? b?) := by

src/Init/Data/Nat/Basic.lean

@@ -788,9 +788,6 @@ theorem not_eq_zero_of_lt (h : b < a) : a ≠ 0 := by
 theorem pred_lt_of_lt {n m : Nat} (h : m < n) : pred n < n :=
   pred_lt (not_eq_zero_of_lt h)
 
-set_option linter.missingDocs false in
-@[deprecated pred_lt_of_lt (since := "2024-06-01")] abbrev pred_lt' := @pred_lt_of_lt
-
 theorem sub_one_lt_of_lt {n m : Nat} (h : m < n) : n - 1 < n :=
   sub_one_lt (not_eq_zero_of_lt h)
 
@@ -1074,9 +1071,6 @@ theorem pred_mul (n m : Nat) : pred n * m = n * m - m := by
   | zero   => simp
   | succ n => rw [Nat.pred_succ, succ_mul, Nat.add_sub_cancel]
 
-set_option linter.missingDocs false in
-@[deprecated pred_mul (since := "2024-06-01")] abbrev mul_pred_left := @pred_mul
-
 protected theorem sub_one_mul  (n m : Nat) : (n - 1) * m = n * m - m := by
   cases n with
   | zero   => simp
@@ -1086,9 +1080,6 @@ protected theorem sub_one_mul  (n m : Nat) : (n - 1) * m = n * m - m := by
 theorem mul_pred (n m : Nat) : n * pred m = n * m - n := by
   rw [Nat.mul_comm, pred_mul, Nat.mul_comm]
 
-set_option linter.missingDocs false in
-@[deprecated mul_pred (since := "2024-06-01")] abbrev mul_pred_right := @mul_pred
-
 theorem mul_sub_one (n m : Nat) : n * (m - 1) = n * m - n := by
   rw [Nat.mul_comm, Nat.sub_one_mul , Nat.mul_comm]
 

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

@@ -711,6 +711,32 @@ theorem mul_add_lt_is_or {b : Nat} (b_lt : b < 2^i) (a : Nat) : 2^i * a + b = 2^
   rw [mod_two_eq_one_iff_testBit_zero, testBit_shiftLeft]
   simp
 
+theorem testBit_mul_two_pow (x i n : Nat) :
+    (x * 2 ^ n).testBit i = (decide (n ≤ i) && x.testBit (i - n)) := by
+  rw [← testBit_shiftLeft, shiftLeft_eq]
+
+theorem bitwise_mul_two_pow (of_false_false : f false false = false := by rfl) :
+    (bitwise f x y) * 2 ^ n = bitwise f (x * 2 ^ n) (y * 2 ^ n) := by
+  apply Nat.eq_of_testBit_eq
+  simp only [testBit_mul_two_pow, testBit_bitwise of_false_false, Bool.if_false_right]
+  intro i
+  by_cases hn : n ≤ i
+  · simp [hn]
+  · simp [hn, of_false_false]
+
+theorem shiftLeft_bitwise_distrib {a b : Nat} (of_false_false : f false false = false := by rfl) :
+    (bitwise f a b) <<< i = bitwise f (a <<< i) (b <<< i) := by
+  simp [shiftLeft_eq, bitwise_mul_two_pow of_false_false]
+
+theorem shiftLeft_and_distrib {a b : Nat} : (a &&& b) <<< i = a <<< i &&& b <<< i :=
+  shiftLeft_bitwise_distrib
+
+theorem shiftLeft_or_distrib {a b : Nat} : (a ||| b) <<< i = a <<< i ||| b <<< i :=
+  shiftLeft_bitwise_distrib
+
+theorem shiftLeft_xor_distrib {a b : Nat} : (a ^^^ b) <<< i = a <<< i ^^^ b <<< i :=
+  shiftLeft_bitwise_distrib
+
 @[simp] theorem decide_shiftRight_mod_two_eq_one :
     decide (x >>> i % 2 = 1) = x.testBit i := by
   simp only [testBit, one_and_eq_mod_two, mod_two_bne_zero]

src/Init/Data/Nat/Lemmas.lean

@@ -622,6 +622,14 @@ protected theorem pos_of_mul_pos_right {a b : Nat} (h : 0 < a * b) : 0 < a := by
     0 < a * b ↔ 0 < a :=
   ⟨Nat.pos_of_mul_pos_right, fun w => Nat.mul_pos w h⟩
 
+protected theorem pos_of_lt_mul_left {a b c : Nat} (h : a < b * c) : 0 < c := by
+  replace h : 0 < b * c := by omega
+  exact Nat.pos_of_mul_pos_left h
+
+protected theorem pos_of_lt_mul_right {a b c : Nat} (h : a < b * c) : 0 < b := by
+  replace h : 0 < b * c := by omega
+  exact Nat.pos_of_mul_pos_right h
+
 /-! ### div/mod -/
 
 theorem mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 1 :=
@@ -995,11 +1003,6 @@ theorem shiftLeft_add (m n : Nat) : ∀ k, m <<< (n + k) = (m <<< n) <<< k
   | 0 => rfl
   | k + 1 => by simp [← Nat.add_assoc, shiftLeft_add _ _ k, shiftLeft_succ]
 
-@[deprecated shiftLeft_add (since := "2024-06-02")]
-theorem shiftLeft_shiftLeft (m n : Nat) : ∀ k, (m <<< n) <<< k = m <<< (n + k)
-  | 0 => rfl
-  | k + 1 => by simp [← Nat.add_assoc, shiftLeft_shiftLeft _ _ k, shiftLeft_succ]
-
 @[simp] theorem shiftLeft_shiftRight (x n : Nat) : x <<< n >>> n = x := by
   rw [Nat.shiftLeft_eq, Nat.shiftRight_eq_div_pow, Nat.mul_div_cancel _ (Nat.two_pow_pos _)]
 

src/Init/Data/Nat/Linear.lean

@@ -718,8 +718,7 @@ theorem Expr.eq_of_toNormPoly_eq (ctx : Context) (e e' : Expr) (h : e.toNormPoly
 
 end Linear
 
-def elimOffset {α : Sort u} (a b k : Nat) (h₁ : a + k = b + k) (h₂ : a = b → α) : α := by
-  simp_arith at h₁
-  exact h₂ h₁
+def elimOffset {α : Sort u} (a b k : Nat) (h₁ : a + k = b + k) (h₂ : a = b → α) : α :=
+  h₂ (Nat.add_right_cancel h₁)
 
 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/Subtype.lean

@@ -5,6 +5,7 @@ Authors: Johannes Hölzl
 -/
 prelude
 import Init.Ext
+import Init.Core
 
 namespace Subtype
 

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]
@@ -220,23 +234,3 @@ theorem UInt32.toNat_le_of_le {n : UInt32} {m : Nat} (h : m < size) : n ≤ ofNa
 
 theorem UInt32.le_toNat_of_le {n : UInt32} {m : Nat} (h : m < size) : ofNat m ≤ n → m ≤ n.toNat := by
   simp [le_def, BitVec.le_def, UInt32.toNat, toBitVec_eq_of_lt h]
-
-@[deprecated UInt8.toNat_zero (since := "2024-06-23")] protected abbrev UInt8.zero_toNat := @UInt8.toNat_zero
-@[deprecated UInt8.toNat_div (since := "2024-06-23")] protected abbrev UInt8.div_toNat := @UInt8.toNat_div
-@[deprecated UInt8.toNat_mod (since := "2024-06-23")] protected abbrev UInt8.mod_toNat := @UInt8.toNat_mod
-
-@[deprecated UInt16.toNat_zero (since := "2024-06-23")] protected abbrev UInt16.zero_toNat := @UInt16.toNat_zero
-@[deprecated UInt16.toNat_div (since := "2024-06-23")] protected abbrev UInt16.div_toNat := @UInt16.toNat_div
-@[deprecated UInt16.toNat_mod (since := "2024-06-23")] protected abbrev UInt16.mod_toNat := @UInt16.toNat_mod
-
-@[deprecated UInt32.toNat_zero (since := "2024-06-23")] protected abbrev UInt32.zero_toNat := @UInt32.toNat_zero
-@[deprecated UInt32.toNat_div (since := "2024-06-23")] protected abbrev UInt32.div_toNat := @UInt32.toNat_div
-@[deprecated UInt32.toNat_mod (since := "2024-06-23")] protected abbrev UInt32.mod_toNat := @UInt32.toNat_mod
-
-@[deprecated UInt64.toNat_zero (since := "2024-06-23")] protected abbrev UInt64.zero_toNat := @UInt64.toNat_zero
-@[deprecated UInt64.toNat_div (since := "2024-06-23")] protected abbrev UInt64.div_toNat := @UInt64.toNat_div
-@[deprecated UInt64.toNat_mod (since := "2024-06-23")] protected abbrev UInt64.mod_toNat := @UInt64.toNat_mod
-
-@[deprecated USize.toNat_zero (since := "2024-06-23")] protected abbrev USize.zero_toNat := @USize.toNat_zero
-@[deprecated USize.toNat_div (since := "2024-06-23")] protected abbrev USize.div_toNat := @USize.toNat_div
-@[deprecated USize.toNat_mod (since := "2024-06-23")] protected abbrev USize.mod_toNat := @USize.toNat_mod

src/Init/Data/Vector.lean

@@ -5,3 +5,7 @@ Authors: Kim Morrison
 -/
 prelude
 import Init.Data.Vector.Basic
+import Init.Data.Vector.Lemmas
+import Init.Data.Vector.Lex
+import Init.Data.Vector.MapIdx
+import Init.Data.Vector.Count

src/Init/Data/Vector/Attach.lean

@@ -0,0 +1,551 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.Vector.Lemmas
+import Init.Data.Array.Attach
+
+namespace Vector
+
+/--
+`O(n)`. Partial map. If `f : Π a, P a → β` is a partial function defined on
+`a : α` satisfying `P`, then `pmap f l h` is essentially the same as `map f l`
+but is defined only when all members of `l` satisfy `P`, using the proof
+to apply `f`.
+
+We replace this at runtime with a more efficient version via the `csimp` lemma `pmap_eq_pmapImpl`.
+-/
+def pmap {P : α → Prop} (f : ∀ a, P a → β) (l : Vector α n) (H : ∀ a ∈ l, P a) : Vector β n :=
+  Vector.mk (l.toArray.pmap f (fun a m => H a (by simpa using m))) (by simp)
+
+/--
+Unsafe implementation of `attachWith`, taking advantage of the fact that the representation of
+`Vector {x // P x} n` is the same as the input `Vector α n`.
+-/
+@[inline] private unsafe def attachWithImpl
+    (xs : Vector α n) (P : α → Prop) (_ : ∀ x ∈ xs, P x) : Vector {x // P x} n := unsafeCast xs
+
+/-- `O(1)`. "Attach" a proof `P x` that holds for all the elements of `xs` to produce a new array
+  with the same elements but in the type `{x // P x}`. -/
+@[implemented_by attachWithImpl] def attachWith
+    (xs : Vector α n) (P : α → Prop) (H : ∀ x ∈ xs, P x) : Vector {x // P x} n :=
+  Vector.mk (xs.toArray.attachWith P fun x h => H x (by simpa using h)) (by simp)
+
+/-- `O(1)`. "Attach" the proof that the elements of `xs` are in `xs` to produce a new vector
+  with the same elements but in the type `{x // x ∈ xs}`. -/
+@[inline] def attach (xs : Vector α n) : Vector {x // x ∈ xs} n := xs.attachWith _ fun _ => id
+
+@[simp] theorem attachWith_mk {xs : Array α} {h : xs.size = n} {P : α → Prop} {H : ∀ x ∈ mk xs h, P x} :
+    (mk xs h).attachWith P H = mk (xs.attachWith P (by simpa using H)) (by simpa using h) := by
+  simp [attachWith]
+
+@[simp] theorem attach_mk {xs : Array α} {h : xs.size = n} :
+    (mk xs h).attach = mk (xs.attachWith (· ∈ mk xs h) (by simp)) (by simpa using h):= by
+  simp [attach]
+
+@[simp] theorem pmap_mk {xs : Array α} {h : xs.size = n} {P : α → Prop} {f : ∀ a, P a → β}
+    {H : ∀ a ∈ mk xs h, P a} :
+    (mk xs h).pmap f H = mk (xs.pmap f (by simpa using H)) (by simpa using h) := by
+  simp [pmap]
+
+@[simp] theorem toArray_attachWith {l : Vector α n} {P : α → Prop} {H : ∀ x ∈ l, P x} :
+   (l.attachWith P H).toArray = l.toArray.attachWith P (by simpa using H) := by
+  simp [attachWith]
+
+@[simp] theorem toArray_attach {α : Type _} {l : Vector α n} :
+    l.attach.toArray = l.toArray.attachWith (· ∈ l) (by simp) := by
+  simp [attach]
+
+@[simp] theorem toArray_pmap {l : Vector α n} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ l, P a} :
+    (l.pmap f H).toArray = l.toArray.pmap f (fun a m => H a (by simpa using m)) := by
+  simp [pmap]
+
+@[simp] theorem toList_attachWith {l : Vector α n} {P : α → Prop} {H : ∀ x ∈ l, P x} :
+   (l.attachWith P H).toList = l.toList.attachWith P (by simpa using H) := by
+  simp [attachWith]
+
+@[simp] theorem toList_attach {α : Type _} {l : Vector α n} :
+    l.attach.toList = l.toList.attachWith (· ∈ l) (by simp) := by
+  simp [attach]
+
+@[simp] theorem toList_pmap {l : Vector α n} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ l, P a} :
+    (l.pmap f H).toList = l.toList.pmap f (fun a m => H a (by simpa using m)) := by
+  simp [pmap]
+
+/-- Implementation of `pmap` using the zero-copy version of `attach`. -/
+@[inline] private def pmapImpl {P : α → Prop} (f : ∀ a, P a → β) (l : Vector α n) (H : ∀ a ∈ l, P a) :
+    Vector β n := (l.attachWith _ H).map fun ⟨x, h'⟩ => f x h'
+
+@[csimp] private theorem pmap_eq_pmapImpl : @pmap = @pmapImpl := by
+  funext α β n p f L h'
+  rcases L with ⟨L, rfl⟩
+  simp only [pmap, pmapImpl, attachWith_mk, map_mk, Array.map_attachWith, eq_mk]
+  apply Array.pmap_congr_left
+  intro a m h₁ h₂
+  congr
+
+@[simp] theorem pmap_empty {P : α → Prop} (f : ∀ a, P a → β) : pmap f #v[] (by simp) = #v[] := rfl
+
+@[simp] theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (l : Vector α n) (h : ∀ b ∈ l.push a, P b) :
+    pmap f (l.push a) h =
+      (pmap f l (fun a m => by simp at h; exact h a (.inl m))).push (f a (h a (by simp))) := by
+  simp [pmap]
+
+@[simp] theorem attach_empty : (#v[] : Vector α 0).attach = #v[] := rfl
+
+@[simp] theorem attachWith_empty {P : α → Prop} (H : ∀ x ∈ #v[], P x) : (#v[] : Vector α 0).attachWith P H = #v[] := rfl
+
+@[simp]
+theorem pmap_eq_map (p : α → Prop) (f : α → β) (l : Vector α n) (H) :
+    @pmap _ _ _ p (fun a _ => f a) l H = map f l := by
+  cases l; simp
+
+theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (l : Vector α n) {H₁ H₂}
+    (h : ∀ a ∈ l, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f l H₁ = pmap g l H₂ := by
+  rcases l with ⟨l, rfl⟩
+  simp only [pmap_mk, eq_mk]
+  apply Array.pmap_congr_left
+  simpa using h
+
+theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l : Vector α n) (H) :
+    map g (pmap f l H) = pmap (fun a h => g (f a h)) l H := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.map_pmap]
+
+theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (l : Vector α n) (H) :
+    pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun _ h => H _ (mem_map_of_mem _ h) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.pmap_map]
+
+theorem attach_congr {l₁ l₂ : Vector α n} (h : l₁ = l₂) :
+    l₁.attach = l₂.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
+  subst h
+  simp
+
+theorem attachWith_congr {l₁ l₂ : Vector α n} (w : l₁ = l₂) {P : α → Prop} {H : ∀ x ∈ l₁, P x} :
+    l₁.attachWith P H = l₂.attachWith P fun _ h => H _ (w ▸ h) := by
+  subst w
+  simp
+
+@[simp] theorem attach_push {a : α} {l : Vector α n} :
+    (l.push a).attach =
+      (l.attach.map (fun ⟨x, h⟩ => ⟨x, mem_push_of_mem a h⟩)).push ⟨a, by simp⟩ := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.map_attachWith]
+
+@[simp] theorem attachWith_push {a : α} {l : Vector α n} {P : α → Prop} {H : ∀ x ∈ l.push a, P x} :
+    (l.push a).attachWith P H =
+      (l.attachWith P (fun x h => by simp at H; exact H x (.inl h))).push ⟨a, H a (by simp)⟩ := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l : Vector α n) (H) :
+    pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
+  rcases l with ⟨l, rfl⟩
+  simp only [pmap_mk, Array.pmap_eq_map_attach, attach_mk, map_mk, eq_mk]
+  rw [Array.map_attach, Array.map_attachWith]
+  ext i hi₁ hi₂ <;> simp
+
+@[simp]
+theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l : Vector α n) (H) :
+    pmap (fun a h => ⟨a, f a h⟩) l H = l.attachWith q (fun x h => f x (H x h)) := by
+  cases l
+  simp
+
+theorem attach_map_coe (l : Vector α n) (f : α → β) :
+    (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
+  cases l
+  simp
+
+theorem attach_map_val (l : Vector α n) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
+  attach_map_coe _ _
+
+theorem attach_map_subtype_val (l : Vector α n) : l.attach.map Subtype.val = l := by
+  cases l; simp
+
+theorem attachWith_map_coe {p : α → Prop} (f : α → β) (l : Vector α n) (H : ∀ a ∈ l, p a) :
+    ((l.attachWith p H).map fun (i : { i // p i}) => f i) = l.map f := by
+  cases l; simp
+
+theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : Vector α n) (H : ∀ a ∈ l, p a) :
+    ((l.attachWith p H).map fun i => f i.val) = l.map f :=
+  attachWith_map_coe _ _ _
+
+theorem attachWith_map_subtype_val {p : α → Prop} (l : Vector α n) (H : ∀ a ∈ l, p a) :
+    (l.attachWith p H).map Subtype.val = l := by
+  cases l; simp
+
+@[simp]
+theorem mem_attach (l : Vector α n) : ∀ x, x ∈ l.attach
+  | ⟨a, h⟩ => by
+    have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
+    rcases this with ⟨⟨_, _⟩, m, rfl⟩
+    exact m
+
+@[simp]
+theorem mem_attachWith (l : Vector α n) {q : α → Prop} (H) (x : {x // q x}) :
+    x ∈ l.attachWith q H ↔ x.1 ∈ l := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp]
+theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l : Vector α n} {H b} :
+    b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
+  simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, Subtype.exists, eq_comm]
+
+theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {l : Vector α n} {H} {a} (h : a ∈ l) :
+    f a (H a h) ∈ pmap f l H := by
+  rw [mem_pmap]
+  exact ⟨a, h, rfl⟩
+
+theorem pmap_eq_self {l : Vector α n} {p : α → Prop} {hp : ∀ (a : α), a ∈ l → p a}
+    {f : (a : α) → p a → α} : l.pmap f hp = l ↔ ∀ a (h : a ∈ l), f a (hp a h) = a := by
+  cases l; simp [Array.pmap_eq_self]
+
+@[simp]
+theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Vector α n} (h : ∀ a ∈ l, p a) (i : Nat) :
+    (pmap f l h)[i]? = Option.pmap f l[i]? fun x H => h x (mem_of_getElem? H) := by
+  cases l; simp
+
+@[simp]
+theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Vector α n} (h : ∀ a ∈ l, p a) {i : Nat}
+    (hn : i < n) :
+    (pmap f l h)[i] = f (l[i]) (h _ (by simp)) := by
+  cases l; simp
+
+@[simp]
+theorem getElem?_attachWith {xs : Vector α n} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
+    (xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (mem_of_getElem? a)) :=
+  getElem?_pmap ..
+
+@[simp]
+theorem getElem?_attach {xs : Vector α n} {i : Nat} :
+    xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => mem_of_getElem? a) :=
+  getElem?_attachWith
+
+@[simp]
+theorem getElem_attachWith {xs : Vector α n} {P : α → Prop} {H : ∀ a ∈ xs, P a}
+    {i : Nat} (h : i < n) :
+    (xs.attachWith P H)[i] = ⟨xs[i]'(by simpa using h), H _ (getElem_mem (by simpa using h))⟩ :=
+  getElem_pmap _ _ h
+
+@[simp]
+theorem getElem_attach {xs : Vector α n} {i : Nat} (h : i < n) :
+    xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
+  getElem_attachWith h
+
+@[simp] theorem pmap_attach (l : Vector α n) {p : {x // x ∈ l} → Prop} (f : ∀ a, p a → β) (H) :
+    pmap f l.attach H =
+      l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
+  ext <;> simp
+
+@[simp] theorem pmap_attachWith (l : Vector α n) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
+    pmap f (l.attachWith q H₁) H₂ =
+      l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
+  ext <;> simp
+
+theorem foldl_pmap (l : Vector α n) {P : α → Prop} (f : (a : α) → P a → β)
+  (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
+    (l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
+  rw [pmap_eq_map_attach, foldl_map]
+
+theorem foldr_pmap (l : Vector α n) {P : α → Prop} (f : (a : α) → P a → β)
+  (H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) :
+    (l.pmap f H).foldr g x = l.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
+  rw [pmap_eq_map_attach, foldr_map]
+
+/--
+If we fold over `l.attach` with a function that ignores the membership predicate,
+we get the same results as folding over `l` directly.
+
+This is useful when we need to use `attach` to show termination.
+
+Unfortunately this can't be applied by `simp` because of the higher order unification problem,
+and even when rewriting we need to specify the function explicitly.
+See however `foldl_subtype` below.
+-/
+theorem foldl_attach (l : Vector α n) (f : β → α → β) (b : β) :
+    l.attach.foldl (fun acc t => f acc t.1) b = l.foldl f b := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.foldl_attach]
+
+/--
+If we fold over `l.attach` with a function that ignores the membership predicate,
+we get the same results as folding over `l` directly.
+
+This is useful when we need to use `attach` to show termination.
+
+Unfortunately this can't be applied by `simp` because of the higher order unification problem,
+and even when rewriting we need to specify the function explicitly.
+See however `foldr_subtype` below.
+-/
+theorem foldr_attach (l : Vector α n) (f : α → β → β) (b : β) :
+    l.attach.foldr (fun t acc => f t.1 acc) b = l.foldr f b := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.foldr_attach]
+
+theorem attach_map {l : Vector α n} (f : α → β) :
+    (l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
+  cases l
+  ext <;> simp
+
+theorem attachWith_map {l : Vector α n} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ l.map f → P b} :
+    (l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
+      fun ⟨x, h⟩ => ⟨f x, h⟩ := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.attachWith_map]
+
+theorem map_attachWith {l : Vector α n} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
+    (f : { x // P x } → β) :
+    (l.attachWith P H).map f =
+      l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
+  cases l
+  ext <;> simp
+
+/-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
+theorem map_attach {l : Vector α n} (f : { x // x ∈ l } → β) :
+    l.attach.map f = l.pmap (fun a h => f ⟨a, h⟩) (fun _ => id) := by
+  cases l
+  ext <;> simp
+
+theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l : Vector α n) (H₁ H₂) :
+    pmap f (pmap g l H₁) H₂ =
+      pmap (α := { x // x ∈ l }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) l.attach
+        (fun a _ => H₁ a a.2) := by
+  rcases l with ⟨l, rfl⟩
+  ext <;> simp
+
+@[simp] theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (l₁ : Vector ι n) (l₂ : Vector ι m)
+    (h : ∀ a ∈ l₁ ++ l₂, p a) :
+    (l₁ ++ l₂).pmap f h =
+      (l₁.pmap f fun a ha => h a (mem_append_left l₂ ha)) ++
+        l₂.pmap f fun a ha => h a (mem_append_right l₁ ha) := by
+  cases l₁
+  cases l₂
+  simp
+
+theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ : Vector α n) (l₂ : Vector α m)
+    (h₁ : ∀ a ∈ l₁, p a) (h₂ : ∀ a ∈ l₂, p a) :
+    ((l₁ ++ l₂).pmap f fun a ha => (mem_append.1 ha).elim (h₁ a) (h₂ a)) =
+      l₁.pmap f h₁ ++ l₂.pmap f h₂ :=
+  pmap_append f l₁ l₂ _
+
+@[simp] theorem attach_append (xs : Vector α n) (ys : Vector α m) :
+    (xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => (⟨x, mem_append_left ys h⟩ : { x // x ∈ xs ++ ys })) ++
+      ys.attach.map (fun ⟨y, h⟩ => (⟨y, mem_append_right xs h⟩ : { y // y ∈ xs ++ ys })) := by
+  rcases xs with ⟨xs, rfl⟩
+  rcases ys with ⟨ys, rfl⟩
+  simp [Array.map_attachWith]
+
+@[simp] theorem attachWith_append {P : α → Prop} {xs : Vector α n} {ys : Vector α m}
+    {H : ∀ (a : α), a ∈ xs ++ ys → P a} :
+    (xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_left ys h)) ++
+      ys.attachWith P (fun a h => H a (mem_append_right xs h)) := by
+  simp [attachWith, attach_append, map_pmap, pmap_append]
+
+@[simp] theorem pmap_reverse {P : α → Prop} (f : (a : α) → P a → β) (xs : Vector α n)
+    (H : ∀ (a : α), a ∈ xs.reverse → P a) :
+    xs.reverse.pmap f H = (xs.pmap f (fun a h => H a (by simpa using h))).reverse := by
+  induction xs <;> simp_all
+
+theorem reverse_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Vector α n)
+    (H : ∀ (a : α), a ∈ xs → P a) :
+    (xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h)) := by
+  rw [pmap_reverse]
+
+@[simp] theorem attachWith_reverse {P : α → Prop} {xs : Vector α n}
+    {H : ∀ (a : α), a ∈ xs.reverse → P a} :
+    xs.reverse.attachWith P H =
+      (xs.attachWith P (fun a h => H a (by simpa using h))).reverse := by
+  cases xs
+  simp
+
+theorem reverse_attachWith {P : α → Prop} {xs : Vector α n}
+    {H : ∀ (a : α), a ∈ xs → P a} :
+    (xs.attachWith P H).reverse = (xs.reverse.attachWith P (fun a h => H a (by simpa using h))) := by
+  cases xs
+  simp
+
+@[simp] theorem attach_reverse (xs : Vector α n) :
+    xs.reverse.attach = xs.attach.reverse.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
+  cases xs
+  rw [attach_congr (reverse_mk ..)]
+  simp [Array.map_attachWith]
+
+theorem reverse_attach (xs : Vector α n) :
+    xs.attach.reverse = xs.reverse.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
+  cases xs
+  simp [Array.map_attachWith]
+
+@[simp] theorem back?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Vector α n)
+    (H : ∀ (a : α), a ∈ xs → P a) :
+    (xs.pmap f H).back? = xs.attach.back?.map fun ⟨a, m⟩ => f a (H a m) := by
+  cases xs
+  simp
+
+@[simp] theorem back?_attachWith {P : α → Prop} {xs : Vector α n}
+    {H : ∀ (a : α), a ∈ xs → P a} :
+    (xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back? h)⟩) := by
+  cases xs
+  simp
+
+@[simp]
+theorem back?_attach {xs : Vector α n} :
+    xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back? h⟩ := by
+  cases xs
+  simp
+
+@[simp]
+theorem countP_attach (l : Vector α n) (p : α → Bool) :
+    l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
+  cases l
+  simp [Function.comp_def]
+
+@[simp]
+theorem countP_attachWith {p : α → Prop} (l : Vector α n) (H : ∀ a ∈ l, p a) (q : α → Bool) :
+    (l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
+  cases l
+  simp
+
+@[simp]
+theorem count_attach [DecidableEq α] (l : Vector α n) (a : {x // x ∈ l}) :
+    l.attach.count a = l.count ↑a := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp]
+theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : Vector α n) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
+    (l.attachWith p H).count a = l.count ↑a := by
+  cases l
+  simp
+
+@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (l : Vector α n) (H₁) :
+    (l.pmap g H₁).countP f =
+      l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
+  rcases l with ⟨l, rfl⟩
+  simp only [pmap_mk, countP_mk, Array.countP_pmap]
+  simp [Array.countP_eq_size_filter]
+
+/-! ## unattach
+
+`Vector.unattach` is the (one-sided) inverse of `Vector.attach`. It is a synonym for `Vector.map Subtype.val`.
+
+We use it by providing a simp lemma `l.attach.unattach = l`, and simp lemmas which recognize higher order
+functions applied to `l : Vector { x // p x }` which only depend on the value, not the predicate, and rewrite these
+in terms of a simpler function applied to `l.unattach`.
+
+Further, we provide simp lemmas that push `unattach` inwards.
+-/
+
+/--
+A synonym for `l.map (·.val)`. Mostly this should not be needed by users.
+It is introduced as in intermediate step by lemmas such as `map_subtype`,
+and is ideally subsequently simplified away by `unattach_attach`.
+
+If not, usually the right approach is `simp [Vector.unattach, -Vector.map_subtype]` to unfold.
+-/
+def unattach {α : Type _} {p : α → Prop} (l : Vector { x // p x } n) : Vector α n := l.map (·.val)
+
+@[simp] theorem unattach_nil {p : α → Prop} : (#v[] : Vector { x // p x } 0).unattach = #v[] := rfl
+@[simp] theorem unattach_push {p : α → Prop} {a : { x // p x }} {l : Vector { x // p x } n} :
+    (l.push a).unattach = l.unattach.push a.1 := by
+  simp only [unattach, Vector.map_push]
+
+@[simp] theorem unattach_mk {p : α → Prop} {l : Array { x // p x }} {h : l.size = n} :
+    (mk l h).unattach = mk l.unattach (by simpa using h) := by
+  simp [unattach]
+
+@[simp] theorem toArray_unattach {p : α → Prop} {l : Vector { x // p x } n} :
+    l.unattach.toArray = l.toArray.unattach := by
+  simp [unattach]
+
+@[simp] theorem toList_unattach {p : α → Prop} {l : Array { x // p x }} :
+    l.unattach.toList = l.toList.unattach := by
+  simp [unattach]
+
+@[simp] theorem unattach_attach {l : Vector α n} : l.attach.unattach = l := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem unattach_attachWith {p : α → Prop} {l : Vector α n}
+    {H : ∀ a ∈ l, p a} :
+    (l.attachWith p H).unattach = l := by
+  cases l
+  simp
+
+@[simp] theorem getElem?_unattach {p : α → Prop} {l : Vector { x // p x } n} (i : Nat) :
+    l.unattach[i]? = l[i]?.map Subtype.val := by
+  simp [unattach]
+
+@[simp] theorem getElem_unattach
+    {p : α → Prop} {l : Vector { x // p x } n} (i : Nat) (h : i < n) :
+    l.unattach[i] = (l[i]'(by simpa using h)).1 := by
+  simp [unattach]
+
+/-! ### Recognizing higher order functions using a function that only depends on the value. -/
+
+/--
+This lemma identifies folds over arrays of subtypes, where the function only depends on the value, not the proposition,
+and simplifies these to the function directly taking the value.
+-/
+@[simp] theorem foldl_subtype {p : α → Prop} {l : Vector { x // p x } n}
+    {f : β → { x // p x } → β} {g : β → α → β} {x : β}
+    {hf : ∀ b x h, f b ⟨x, h⟩ = g b x} :
+    l.foldl f x = l.unattach.foldl g x := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.foldl_subtype (hf := hf)]
+
+/--
+This lemma identifies folds over arrays of subtypes, where the function only depends on the value, not the proposition,
+and simplifies these to the function directly taking the value.
+-/
+@[simp] theorem foldr_subtype {p : α → Prop} {l : Vector { x // p x } n}
+    {f : { x // p x } → β → β} {g : α → β → β} {x : β}
+    {hf : ∀ x h b, f ⟨x, h⟩ b = g x b} :
+    l.foldr f x = l.unattach.foldr g x := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.foldr_subtype (hf := hf)]
+
+/--
+This lemma identifies maps over arrays of subtypes, where the function only depends on the value, not the proposition,
+and simplifies these to the function directly taking the value.
+-/
+@[simp] theorem map_subtype {p : α → Prop} {l : Vector { x // p x } n}
+    {f : { x // p x } → β} {g : α → β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
+    l.map f = l.unattach.map g := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.map_subtype (hf := hf)]
+
+/-! ### Simp lemmas pushing `unattach` inwards. -/
+
+@[simp] theorem unattach_reverse {p : α → Prop} {l : Vector { x // p x } n} :
+    l.reverse.unattach = l.unattach.reverse := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.unattach_reverse]
+
+
+@[simp] theorem unattach_append {p : α → Prop} {l₁ l₂ : Vector { x // p x } n} :
+    (l₁ ++ l₂).unattach = l₁.unattach ++ l₂.unattach := by
+  rcases l₁
+  rcases l₂
+  simp
+
+@[simp] theorem unattach_flatten {p : α → Prop} {l : Vector (Vector { x // p x } n) n} :
+    l.flatten.unattach = (l.map unattach).flatten := by
+  unfold unattach
+  cases l using vector₂_induction
+  simp only [flatten_mk, Array.map_map, Function.comp_apply, Array.map_subtype,
+    Array.unattach_attach, Array.map_id_fun', id_eq, map_mk, Array.map_flatten, map_subtype,
+    map_id_fun', unattach_mk, eq_mk]
+  unfold Array.unattach
+  rfl
+
+@[simp] theorem unattach_mkVector {p : α → Prop} {n : Nat} {x : { x // p x }} :
+    (mkVector n x).unattach = mkVector n x.1 := by
+  simp [unattach]
+
+end Vector

src/Init/Data/Vector/Basic.lean

@@ -6,6 +6,7 @@ Authors: Shreyas Srinivas, François G. Dorais, Kim Morrison
 
 prelude
 import Init.Data.Array.Lemmas
+import Init.Data.Array.MapIdx
 import Init.Data.Range
 
 /-!
@@ -90,14 +91,12 @@ of bounds.
 /-- The last element of a vector. Panics if the vector is empty. -/
 @[inline] def back! [Inhabited α] (v : Vector α n) : α := v.toArray.back!
 
-/-- The last element of a vector, or `none` if the array is empty. -/
+/-- The last element of a vector, or `none` if the vector is empty. -/
 @[inline] def back? (v : Vector α n) : Option α := v.toArray.back?
 
 /-- The last element of a non-empty vector. -/
 @[inline] def back [NeZero n] (v : Vector α n) : α :=
-  -- TODO: change to just `v[n]`
-  have : Inhabited α := ⟨v[0]'(Nat.pos_of_neZero n)⟩
-  v.back!
+  v[n - 1]'(Nat.sub_one_lt (NeZero.ne n))
 
 /-- The first element of a non-empty vector.  -/
 @[inline] def head [NeZero n] (v : Vector α n) := v[0]'(Nat.pos_of_neZero n)
@@ -170,6 +169,25 @@ 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⟩
 
+/-- Maps elements of a vector using the function `f`, which also receives the index of the element. -/
+@[inline] def mapIdx (f : Nat → α → β) (v : Vector α n) : Vector β n :=
+  ⟨v.toArray.mapIdx f, by simp⟩
+
+/-- Maps elements of a vector using the function `f`,
+which also receives the index of the element, and the fact that the index is less than the size of the vector. -/
+@[inline] def mapFinIdx (v : Vector α n) (f : (i : Nat) → α → (h : i < n) → β) : Vector β n :=
+  ⟨v.toArray.mapFinIdx (fun i a h => f i a (by simpa [v.size_toArray] using h)), 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']⟩
+
+@[inline] def zipWithIndex (v : Vector α n) : Vector (α × Nat) n :=
+  ⟨v.toArray.zipWithIndex, by simp⟩
+
 /-- 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⟩
@@ -294,6 +312,14 @@ no element of the index matches the given value.
 @[inline] def all (v : Vector α n) (p : α → Bool) : Bool :=
   v.toArray.all p
 
+/-- Count the number of elements of a vector that satisfy the predicate `p`. -/
+@[inline] def countP (p : α → Bool) (v : Vector α n) : Nat :=
+  v.toArray.countP p
+
+/-- Count the number of elements of a vector that are equal to `a`. -/
+@[inline] def count [BEq α] (a : α) (v : Vector α n) : Nat :=
+  v.toArray.count a
+
 /-! ### Lexicographic ordering -/
 
 instance instLT [LT α] : LT (Vector α n) := ⟨fun v w => v.toArray < w.toArray⟩

src/Init/Data/Vector/Count.lean

@@ -0,0 +1,233 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.Array.Count
+import Init.Data.Vector.Lemmas
+
+/-!
+# Lemmas about `Vector.countP` and `Vector.count`.
+-/
+
+namespace Vector
+
+open Nat
+
+/-! ### countP -/
+section countP
+
+variable (p q : α → Bool)
+
+@[simp] theorem countP_empty : countP p #v[] = 0 := rfl
+
+@[simp] theorem countP_push_of_pos (l : Vector α n) (pa : p a) : countP p (l.push a) = countP p l + 1 := by
+  rcases l with ⟨l⟩
+  simp_all
+
+@[simp] theorem countP_push_of_neg (l : Vector α n) (pa : ¬p a) : countP p (l.push a) = countP p l := by
+  rcases l with ⟨l, rfl⟩
+  simp_all
+
+theorem countP_push (a : α) (l : Vector α n) : countP p (l.push a) = countP p l + if p a then 1 else 0 := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.countP_push]
+
+@[simp] theorem countP_singleton (a : α) : countP p #v[a] = if p a then 1 else 0 := by
+  simp [countP_push]
+
+theorem size_eq_countP_add_countP (l : Vector α n) : n = countP p l + countP (fun a => ¬p a) l := by
+  rcases l with ⟨l, rfl⟩
+  simp [List.length_eq_countP_add_countP (p := p)]
+
+theorem countP_le_size {l : Vector α n} : countP p l ≤ n := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.countP_le_size (p := p)]
+
+@[simp] theorem countP_append (l₁ : Vector α n) (l₂ : Vector α m) : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
+  cases l₁
+  cases l₂
+  simp
+
+@[simp] theorem countP_pos_iff {p} : 0 < countP p l ↔ ∃ a ∈ l, p a := by
+  cases l
+  simp
+
+@[simp] theorem one_le_countP_iff {p} : 1 ≤ countP p l ↔ ∃ a ∈ l, p a :=
+  countP_pos_iff
+
+@[simp] theorem countP_eq_zero {p} : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
+  cases l
+  simp
+
+@[simp] theorem countP_eq_size {p} : countP p l = l.size ↔ ∀ a ∈ l, p a := by
+  cases l
+  simp
+
+@[simp] theorem countP_cast (p : α → Bool) (l : Vector α n) : countP p (l.cast h) = countP p l := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+theorem countP_mkVector (p : α → Bool) (a : α) (n : Nat) :
+    countP p (mkVector n a) = if p a then n else 0 := by
+  simp only [mkVector_eq_toVector_mkArray, countP_cast, countP_mk]
+  simp [Array.countP_mkArray]
+
+theorem boole_getElem_le_countP (p : α → Bool) (l : Vector α n) (i : Nat) (h : i < n) :
+    (if p l[i] then 1 else 0) ≤ l.countP p := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.boole_getElem_le_countP]
+
+theorem countP_set (p : α → Bool) (l : Vector α n) (i : Nat) (a : α) (h : i < n) :
+    (l.set i a).countP p = l.countP p - (if p l[i] then 1 else 0) + (if p a then 1 else 0) := by
+  cases l
+  simp [Array.countP_set, h]
+
+@[simp] theorem countP_true : (countP fun (_ : α) => true) = (fun (_ : Vector α n) => n) := by
+  funext l
+  rw [countP]
+  simp only [Array.countP_true, l.2]
+
+@[simp] theorem countP_false : (countP fun (_ : α) => false) = (fun (_ : Vector α n) => 0) := by
+  funext l
+  simp
+
+@[simp] theorem countP_map (p : β → Bool) (f : α → β) (l : Vector α n) :
+    countP p (map f l) = countP (p ∘ f) l := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem countP_flatten (l : Vector (Vector α m) n) :
+    countP p l.flatten = (l.map (countP p)).sum := by
+  rcases l with ⟨l, rfl⟩
+  simp [Function.comp_def]
+
+theorem countP_flatMap (p : β → Bool) (l : Vector α n) (f : α → Vector β m) :
+    countP p (l.flatMap f) = (map (countP p ∘ f) l).sum := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.countP_flatMap, Function.comp_def]
+
+@[simp] theorem countP_reverse (l : Vector α n) : countP p l.reverse = countP p l := by
+  cases l
+  simp
+
+variable {p q}
+
+theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
+  cases l
+  simpa using Array.countP_mono_left (by simpa using h)
+
+theorem countP_congr (h : ∀ x ∈ l, p x ↔ q x) : countP p l = countP q l :=
+  Nat.le_antisymm
+    (countP_mono_left fun x hx => (h x hx).1)
+    (countP_mono_left fun x hx => (h x hx).2)
+
+end countP
+
+/-! ### count -/
+section count
+
+variable [BEq α]
+
+@[simp] theorem count_empty (a : α) : count a #v[] = 0 := rfl
+
+theorem count_push (a b : α) (l : Vector α n) :
+    count a (l.push b) = count a l + if b == a then 1 else 0 := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_push]
+
+theorem count_eq_countP (a : α) (l : Vector α n) : count a l = countP (· == a) l := rfl
+
+theorem count_eq_countP' {a : α} : count (n := n) a = countP (· == a) := by
+  funext l
+  apply count_eq_countP
+
+theorem count_le_size (a : α) (l : Vector α n) : count a l ≤ n := countP_le_size _
+
+theorem count_le_count_push (a b : α) (l : Vector α n) : count a l ≤ count a (l.push b) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_push]
+
+@[simp] theorem count_singleton (a b : α) : count a #v[b] = if b == a then 1 else 0 := by
+  simp [count_eq_countP]
+
+@[simp] theorem count_append (a : α) (l₁ : Vector α n) (l₂ : Vector α m) :
+    count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
+  countP_append ..
+
+@[simp] theorem count_flatten (a : α) (l : Vector (Vector α m) n) :
+    count a l.flatten = (l.map (count a)).sum := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_flatten, Function.comp_def]
+
+@[simp] theorem count_reverse (a : α) (l : Vector α n) : count a l.reverse = count a l := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+theorem boole_getElem_le_count (a : α) (l : Vector α n) (i : Nat) (h : i < n) :
+    (if l[i] == a then 1 else 0) ≤ l.count a := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.boole_getElem_le_count, h]
+
+theorem count_set (a b : α) (l : Vector α n) (i : Nat) (h : i < n) :
+    (l.set i a).count b = l.count b - (if l[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_set, h]
+
+@[simp] theorem count_cast (l : Vector α n) : (l.cast h).count a = l.count a := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+variable [LawfulBEq α]
+
+@[simp] theorem count_push_self (a : α) (l : Vector α n) : count a (l.push a) = count a l + 1 := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_push_self]
+
+@[simp] theorem count_push_of_ne (h : b ≠ a) (l : Vector α n) : count a (l.push b) = count a l := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_push_of_ne, h]
+
+theorem count_singleton_self (a : α) : count a #v[a] = 1 := by simp
+
+@[simp]
+theorem count_pos_iff {a : α} {l : Vector α n} : 0 < count a l ↔ a ∈ l := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_pos_iff, beq_iff_eq, exists_eq_right]
+
+@[simp] theorem one_le_count_iff {a : α} {l : Vector α n} : 1 ≤ count a l ↔ a ∈ l :=
+  count_pos_iff
+
+theorem count_eq_zero_of_not_mem {a : α} {l : Vector α n} (h : a ∉ l) : count a l = 0 :=
+  Decidable.byContradiction fun h' => h <| count_pos_iff.1 (Nat.pos_of_ne_zero h')
+
+theorem not_mem_of_count_eq_zero {a : α} {l : Vector α n} (h : count a l = 0) : a ∉ l :=
+  fun h' => Nat.ne_of_lt (count_pos_iff.2 h') h.symm
+
+theorem count_eq_zero {l : Vector α n} : count a l = 0 ↔ a ∉ l :=
+  ⟨not_mem_of_count_eq_zero, count_eq_zero_of_not_mem⟩
+
+theorem count_eq_size {l : Vector α n} : count a l = l.size ↔ ∀ b ∈ l, a = b := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_eq_size]
+
+@[simp] theorem count_mkVector_self (a : α) (n : Nat) : count a (mkVector n a) = n := by
+  simp only [mkVector_eq_toVector_mkArray, count_cast, count_mk]
+  simp
+
+theorem count_mkVector (a b : α) (n : Nat) : count a (mkVector n b) = if b == a then n else 0 := by
+  simp only [mkVector_eq_toVector_mkArray, count_cast, count_mk]
+  simp [Array.count_mkArray]
+
+theorem count_le_count_map [DecidableEq β] (l : Vector α n) (f : α → β) (x : α) :
+    count x l ≤ count (f x) (map f l) := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_le_count_map]
+
+theorem count_flatMap {α} [BEq β] (l : Vector α n) (f : α → Vector β m) (x : β) :
+    count x (l.flatMap f) = (map (count x ∘ f) l).sum := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.count_flatMap, Function.comp_def]
+
+end count

src/Init/Data/Vector/MapIdx.lean

@@ -0,0 +1,333 @@
+/-
+Copyright (c) 2025 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.Array.MapIdx
+import Init.Data.Vector.Lemmas
+
+namespace Vector
+
+/-! ### mapFinIdx -/
+
+@[simp] theorem getElem_mapFinIdx (a : Vector α n) (f : (i : Nat) → α → (h : i < n) → β) (i : Nat)
+    (h : i < n) :
+    (a.mapFinIdx f)[i] = f i a[i] h := by
+  rcases a with ⟨a, rfl⟩
+  simp
+
+@[simp] theorem getElem?_mapFinIdx (a : Vector α n) (f : (i : Nat) → α → (h : i < n) → β) (i : Nat) :
+    (a.mapFinIdx f)[i]? =
+      a[i]?.pbind fun b h => f i b (getElem?_eq_some_iff.1 h).1 := by
+  simp only [getElem?_def, getElem_mapFinIdx]
+  split <;> simp_all
+
+/-! ### mapIdx -/
+
+@[simp] theorem getElem_mapIdx (f : Nat → α → β) (a : Vector α n) (i : Nat) (h : i < n) :
+    (a.mapIdx f)[i] = f i (a[i]'(by simp_all)) := by
+  rcases a with ⟨a, rfl⟩
+  simp
+
+@[simp] theorem getElem?_mapIdx (f : Nat → α → β) (a : Vector α n) (i : Nat) :
+    (a.mapIdx f)[i]? = a[i]?.map (f i) := by
+  rcases a with ⟨a, rfl⟩
+  simp
+
+end Vector
+
+namespace Array
+
+@[simp] theorem mapFinIdx_toVector (l : Array α) (f : (i : Nat) → α → (h : i < l.size) → β) :
+    l.toVector.mapFinIdx f = (l.mapFinIdx f).toVector.cast (by simp) := by
+  ext <;> simp
+
+@[simp] theorem mapIdx_toVector (f : Nat → α → β) (l : Array α) :
+    l.toVector.mapIdx f = (l.mapIdx f).toVector.cast (by simp) := by
+  ext <;> simp
+
+end Array
+
+namespace Vector
+
+/-! ### zipWithIndex -/
+
+@[simp] theorem toList_zipWithIndex (a : Vector α n) :
+    a.zipWithIndex.toList = a.toList.enum.map (fun (i, a) => (a, i)) := by
+  rcases a with ⟨a, rfl⟩
+  simp
+
+@[simp] theorem getElem_zipWithIndex (a : Vector α n) (i : Nat) (h : i < n) :
+    (a.zipWithIndex)[i] = (a[i]'(by simp_all), i) := by
+  rcases a with ⟨a, rfl⟩
+  simp
+
+@[simp] theorem zipWithIndex_toVector {l : Array α} :
+    l.toVector.zipWithIndex = l.zipWithIndex.toVector.cast (by simp) := by
+  ext <;> simp
+
+theorem mk_mem_zipWithIndex_iff_getElem? {x : α} {i : Nat} {l : Vector α n} :
+    (x, i) ∈ l.zipWithIndex ↔ l[i]? = x := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.mk_mem_zipWithIndex_iff_getElem?]
+
+theorem mem_enum_iff_getElem? {x : α × Nat} {l : Vector α n} :
+    x ∈ l.zipWithIndex ↔ l[x.2]? = some x.1 :=
+  mk_mem_zipWithIndex_iff_getElem?
+
+/-! ### mapFinIdx -/
+
+@[congr] theorem mapFinIdx_congr {xs ys : Vector α n} (w : xs = ys)
+    (f : (i : Nat) → α → (h : i < n) → β) :
+    mapFinIdx xs f = mapFinIdx ys f := by
+  subst w
+  rfl
+
+@[simp]
+theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #v[] f = #v[] :=
+  rfl
+
+theorem mapFinIdx_eq_ofFn {as : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+    as.mapFinIdx f = Vector.ofFn fun i : Fin n => f i as[i] i.2 := by
+  rcases as with ⟨as, rfl⟩
+  simp [Array.mapFinIdx_eq_ofFn]
+
+theorem mapFinIdx_append {K : Vector α n} {L : Vector α m} {f : (i : Nat) → α → (h : i < n + m) → β} :
+    (K ++ L).mapFinIdx f =
+      K.mapFinIdx (fun i a h => f i a (by omega)) ++
+        L.mapFinIdx (fun i a h => f (i + n) a (by omega)) := by
+  rcases K with ⟨K, rfl⟩
+  rcases L with ⟨L, rfl⟩
+  simp [Array.mapFinIdx_append]
+
+@[simp]
+theorem mapFinIdx_push {l : Vector α n} {a : α} {f : (i : Nat) → α → (h : i < n + 1) → β} :
+    mapFinIdx (l.push a) f =
+      (mapFinIdx l (fun i a h => f i a (by omega))).push (f l.size a (by simp)) := by
+  simp [← append_singleton, mapFinIdx_append]
+
+theorem mapFinIdx_singleton {a : α} {f : (i : Nat) → α → (h : i < 1) → β} :
+    #v[a].mapFinIdx f = #v[f 0 a (by simp)] := by
+  simp
+
+-- FIXME this lemma can't be stated until we've aligned `List/Array/Vector.attach`:
+-- theorem mapFinIdx_eq_zipWithIndex_map {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+--     l.mapFinIdx f = l.zipWithIndex.attach.map
+--       fun ⟨⟨x, i⟩, m⟩ =>
+--         f i x (by simp [mk_mem_zipWithIndex_iff_getElem?, getElem?_eq_some_iff] at m; exact m.1) := by
+--   ext <;> simp
+
+theorem exists_of_mem_mapFinIdx {b : β} {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β}
+    (h : b ∈ l.mapFinIdx f) : ∃ (i : Nat) (h : i < n), f i l[i] h = b := by
+  rcases l with ⟨l, rfl⟩
+  exact List.exists_of_mem_mapFinIdx (by simpa using h)
+
+@[simp] theorem mem_mapFinIdx {b : β} {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+    b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < n), f i l[i] h = b := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+theorem mapFinIdx_eq_iff {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+    l.mapFinIdx f = l' ↔ ∀ (i : Nat) (h : i < n), l'[i] = f i l[i] h := by
+  rcases l with ⟨l, rfl⟩
+  rcases l' with ⟨l', h⟩
+  simp [mapFinIdx_mk, eq_mk, getElem_mk, Array.mapFinIdx_eq_iff, h]
+
+@[simp] theorem mapFinIdx_eq_singleton_iff {l : Vector α 1} {f : (i : Nat) → α → (h : i < 1) → β} {b : β} :
+    l.mapFinIdx f = #v[b] ↔ ∃ (a : α), l = #v[a] ∧ f 0 a (by omega) = b := by
+  rcases l with ⟨l, h⟩
+  simp only [mapFinIdx_mk, eq_mk, Array.mapFinIdx_eq_singleton_iff]
+  constructor
+  · rintro ⟨a, rfl, rfl⟩
+    exact ⟨a, by simp⟩
+  · rintro ⟨a, rfl, rfl⟩
+    exact ⟨a, by simp⟩
+
+theorem mapFinIdx_eq_append_iff {l : Vector α (n + m)} {f : (i : Nat) → α → (h : i < n + m) → β}
+    {l₁ : Vector β n} {l₂ : Vector β m} :
+    l.mapFinIdx f = l₁ ++ l₂ ↔
+      ∃ (l₁' : Vector α n) (l₂' : Vector α m), l = l₁' ++ l₂' ∧
+        l₁'.mapFinIdx (fun i a h => f i a (by omega)) = l₁ ∧
+        l₂'.mapFinIdx (fun i a h => f (i + n) a (by omega)) = l₂ := by
+  rcases l with ⟨l, h⟩
+  rcases l₁ with ⟨l₁, rfl⟩
+  rcases l₂ with ⟨l₂, rfl⟩
+  simp only [mapFinIdx_mk, mk_append_mk, eq_mk, Array.mapFinIdx_eq_append_iff, toArray_mapFinIdx,
+    mk_eq, toArray_append, exists_and_left, exists_prop]
+  constructor
+  · rintro ⟨l₁', l₂', rfl, h₁, h₂⟩
+    have h₁' := congrArg Array.size h₁
+    have h₂' := congrArg Array.size h₂
+    simp only [Array.size_mapFinIdx] at h₁' h₂'
+    exact ⟨⟨l₁', h₁'⟩, ⟨l₂', h₂'⟩, by simp_all⟩
+  · rintro ⟨⟨l₁, s₁⟩, ⟨l₂, s₂⟩, rfl, h₁, h₂⟩
+    refine ⟨l₁, l₂, by simp_all⟩
+
+theorem mapFinIdx_eq_push_iff {l : Vector α (n + 1)} {b : β} {f : (i : Nat) → α → (h : i < n + 1) → β} {l₂ : Vector β n} :
+    l.mapFinIdx f = l₂.push b ↔
+      ∃ (l₁ : Vector α n) (a : α), l = l₁.push a ∧
+        l₁.mapFinIdx (fun i a h => f i a (by omega)) = l₂ ∧ b = f n a (by omega) := by
+  rcases l with ⟨l, h⟩
+  rcases l₂ with ⟨l₂, rfl⟩
+  simp only [mapFinIdx_mk, push_mk, eq_mk, Array.mapFinIdx_eq_push_iff, mk_eq, toArray_push,
+    toArray_mapFinIdx]
+  constructor
+  · rintro ⟨l₁, a, rfl, h₁, rfl⟩
+    simp only [Array.size_push, Nat.add_right_cancel_iff] at h
+    exact ⟨⟨l₁, h⟩, a, by simp_all⟩
+  · rintro ⟨⟨l₁, h⟩, a, rfl, h₁, rfl⟩
+    exact ⟨l₁, a, by simp_all⟩
+
+theorem mapFinIdx_eq_mapFinIdx_iff {l : Vector α n} {f g : (i : Nat) → α → (h : i < n) → β} :
+    l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Nat) (h : i < n), f i l[i] h = g i l[i] h := by
+  rw [eq_comm, mapFinIdx_eq_iff]
+  simp
+
+@[simp] theorem mapFinIdx_mapFinIdx {l : Vector α n}
+    {f : (i : Nat) → α → (h : i < n) → β}
+    {g : (i : Nat) → β → (h : i < n) → γ} :
+    (l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i a h => g i (f i a h) h) := by
+  simp [mapFinIdx_eq_iff]
+
+theorem mapFinIdx_eq_mkVector_iff {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {b : β} :
+    l.mapFinIdx f = mkVector n b ↔ ∀ (i : Nat) (h : i < n), f i l[i] h = b := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.mapFinIdx_eq_mkArray_iff]
+
+@[simp] theorem mapFinIdx_reverse {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
+    l.reverse.mapFinIdx f = (l.mapFinIdx (fun i a h => f (n - 1 - i) a (by omega))).reverse := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+/-! ### mapIdx -/
+
+@[simp]
+theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #v[] = #v[] :=
+  rfl
+
+@[simp] theorem mapFinIdx_eq_mapIdx {l : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {g : Nat → α → β}
+    (h : ∀ (i : Nat) (h : i < n), f i l[i] h = g i l[i]) :
+    l.mapFinIdx f = l.mapIdx g := by
+  simp_all [mapFinIdx_eq_iff]
+
+theorem mapIdx_eq_mapFinIdx {l : Vector α n} {f : Nat → α → β} :
+    l.mapIdx f = l.mapFinIdx (fun i a _ => f i a) := by
+  simp [mapFinIdx_eq_mapIdx]
+
+theorem mapIdx_eq_zipWithIndex_map {l : Vector α n} {f : Nat → α → β} :
+    l.mapIdx f = l.zipWithIndex.map fun ⟨a, i⟩ => f i a := by
+  ext <;> simp
+
+theorem mapIdx_append {K : Vector α n} {L : Vector α m} :
+    (K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i => f (i + K.size) := by
+  rcases K with ⟨K, rfl⟩
+  rcases L with ⟨L, rfl⟩
+  simp [Array.mapIdx_append]
+
+@[simp]
+theorem mapIdx_push {l : Vector α n} {a : α} :
+    mapIdx f (l.push a) = (mapIdx f l).push (f l.size a) := by
+  simp [← append_singleton, mapIdx_append]
+
+theorem mapIdx_singleton {a : α} : mapIdx f #v[a] = #v[f 0 a] := by
+  simp
+
+theorem exists_of_mem_mapIdx {b : β} {l : Vector α n}
+    (h : b ∈ l.mapIdx f) : ∃ (i : Nat) (h : i < n), f i l[i] = b := by
+  rw [mapIdx_eq_mapFinIdx] at h
+  simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
+
+@[simp] theorem mem_mapIdx {b : β} {l : Vector α n} :
+    b ∈ l.mapIdx f ↔ ∃ (i : Nat) (h : i < n), f i l[i] = b := by
+  constructor
+  · intro h
+    exact exists_of_mem_mapIdx h
+  · rintro ⟨i, h, rfl⟩
+    rw [mem_iff_getElem]
+    exact ⟨i, by simpa using h, by simp⟩
+
+theorem mapIdx_eq_push_iff {l : Vector α (n + 1)} {b : β} :
+    mapIdx f l = l₂.push b ↔
+      ∃ (a : α) (l₁ : Vector α n), l = l₁.push a ∧ mapIdx f l₁ = l₂ ∧ f l₁.size a = b := by
+  rw [mapIdx_eq_mapFinIdx, mapFinIdx_eq_push_iff]
+  simp only [mapFinIdx_eq_mapIdx, exists_and_left, exists_prop]
+  constructor
+  · rintro ⟨l₁, a, rfl, rfl, rfl⟩
+    exact ⟨a, l₁, by simp⟩
+  · rintro ⟨a, l₁, rfl, rfl, rfl⟩
+    exact ⟨l₁, a, rfl, by simp⟩
+
+@[simp] theorem mapIdx_eq_singleton_iff {l : Vector α 1} {f : Nat → α → β} {b : β} :
+    mapIdx f l = #v[b] ↔ ∃ (a : α), l = #v[a] ∧ f 0 a = b := by
+  rcases l with ⟨l⟩
+  simp
+
+theorem mapIdx_eq_append_iff {l : Vector α (n + m)} {f : Nat → α → β} {l₁ : Vector β n} {l₂ : Vector β m} :
+    mapIdx f l = l₁ ++ l₂ ↔
+      ∃ (l₁' : Vector α n) (l₂' : Vector α m), l = l₁' ++ l₂' ∧
+        l₁'.mapIdx f = l₁ ∧
+        l₂'.mapIdx (fun i => f (i + l₁'.size)) = l₂ := by
+  rcases l with ⟨l, h⟩
+  rcases l₁ with ⟨l₁, rfl⟩
+  rcases l₂ with ⟨l₂, rfl⟩
+  rw [mapIdx_eq_mapFinIdx, mapFinIdx_eq_append_iff]
+  simp
+
+theorem mapIdx_eq_iff {l : Vector α n} :
+    mapIdx f l = l' ↔ ∀ (i : Nat) (h : i < n), f i l[i] = l'[i] := by
+  rcases l with ⟨l, rfl⟩
+  rcases l' with ⟨l', h⟩
+  simp only [mapIdx_mk, eq_mk, Array.mapIdx_eq_iff, getElem_mk]
+  constructor
+  · rintro h' i h
+    specialize h' i
+    simp_all
+  · intro h' i
+    specialize h' i
+    by_cases w : i < l.size
+    · specialize h' w
+      simp_all
+    · simp only [Nat.not_lt] at w
+      simp_all [Array.getElem?_eq_none_iff.mpr w]
+
+theorem mapIdx_eq_mapIdx_iff {l : Vector α n} :
+    mapIdx f l = mapIdx g l ↔ ∀ (i : Nat) (h : i < n), f i l[i] = g i l[i] := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.mapIdx_eq_mapIdx_iff]
+
+@[simp] theorem mapIdx_set {l : Vector α n} {i : Nat} {h : i < n} {a : α} :
+    (l.set i a).mapIdx f = (l.mapIdx f).set i (f i a) (by simpa) := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem mapIdx_setIfInBounds {l : Vector α n} {i : Nat} {a : α} :
+    (l.setIfInBounds i a).mapIdx f = (l.mapIdx f).setIfInBounds i (f i a) := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem back?_mapIdx {l : Vector α n} {f : Nat → α → β} :
+    (mapIdx f l).back? = (l.back?).map (f (l.size - 1)) := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem back_mapIdx [NeZero n] {l : Vector α n} {f : Nat → α → β} :
+    (mapIdx f l).back = f (l.size - 1) (l.back) := by
+  rcases l with ⟨l, rfl⟩
+  simp
+
+@[simp] theorem mapIdx_mapIdx {l : Vector α n} {f : Nat → α → β} {g : Nat → β → γ} :
+    (l.mapIdx f).mapIdx g = l.mapIdx (fun i => g i ∘ f i) := by
+  simp [mapIdx_eq_iff]
+
+theorem mapIdx_eq_mkVector_iff {l : Vector α n} {f : Nat → α → β} {b : β} :
+    mapIdx f l = mkVector n b ↔ ∀ (i : Nat) (h : i < n), f i l[i] = b := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.mapIdx_eq_mkArray_iff]
+
+@[simp] theorem mapIdx_reverse {l : Vector α n} {f : Nat → α → β} :
+    l.reverse.mapIdx f = (mapIdx (fun i => f (l.size - 1 - i)) l).reverse := by
+  rcases l with ⟨l, rfl⟩
+  simp [Array.mapIdx_reverse]
+
+end Vector

src/Init/Grind.lean

@@ -11,3 +11,4 @@ import Init.Grind.Cases
 import Init.Grind.Propagator
 import Init.Grind.Util
 import Init.Grind.Offset
+import Init.Grind.PP

src/Init/Grind/Cases.lean

@@ -5,11 +5,7 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.Core
+import Init.Grind.Tactics
 
-attribute [grind_cases] And Prod False Empty True Unit Exists
-
-namespace Lean.Grind.Eager
-
-attribute [scoped grind_cases] Or
-
-end Lean.Grind.Eager
+attribute [grind cases eager] And Prod False Empty True Unit Exists
+attribute [grind cases] Or

src/Init/Grind/Lemmas.lean

@@ -12,6 +12,9 @@ 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)
 
@@ -66,6 +69,12 @@ theorem eq_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a = b) = a := by s
 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

src/Init/Grind/Norm.lean

@@ -12,110 +12,105 @@ import Init.ByCases
 namespace Lean.Grind
 /-!
 Normalization theorems for the `grind` tactic.
-
-We are also going to use simproc's in the future.
 -/
 
--- Not
-attribute [grind_norm] Classical.not_not
-
--- Ne
-attribute [grind_norm] ne_eq
-
--- Iff
-@[grind_norm] theorem iff_eq (p q : Prop) : (p ↔ q) = (p = q) := by
+theorem iff_eq (p q : Prop) : (p ↔ q) = (p = q) := by
   by_cases p <;> by_cases q <;> simp [*]
 
--- Eq
-attribute [grind_norm] eq_self heq_eq_eq
-
--- Prop equality
-@[grind_norm] theorem eq_true_eq (p : Prop) : (p = True) = p := by simp
-@[grind_norm] theorem eq_false_eq (p : Prop) : (p = False) = ¬p := by simp
-@[grind_norm] theorem not_eq_prop (p q : Prop) : (¬(p = q)) = (p = ¬q) := by
+theorem eq_true_eq (p : Prop) : (p = True) = p := by simp
+theorem eq_false_eq (p : Prop) : (p = False) = ¬p := by simp
+theorem not_eq_prop (p q : Prop) : (¬(p = q)) = (p = ¬q) := by
   by_cases p <;> by_cases q <;> simp [*]
 
--- True
-attribute [grind_norm] not_true
-
--- False
-attribute [grind_norm] not_false_eq_true
-
 -- Remark: we disabled the following normalization rule because we want this information when implementing splitting heuristics
 -- Implication as a clause
 theorem imp_eq (p q : Prop) : (p → q) = (¬ p ∨ q) := by
   by_cases p <;> by_cases q <;> simp [*]
 
--- And
-@[grind_norm↓] theorem not_and (p q : Prop) : (¬(p ∧ q)) = (¬p ∨ ¬q) := by
+theorem true_imp_eq (p : Prop) : (True → p) = p := by simp
+theorem false_imp_eq (p : Prop) : (False → p) = True := by simp
+theorem imp_true_eq (p : Prop) : (p → True) = True := by simp
+theorem imp_false_eq (p : Prop) : (p → False) = ¬p := by simp
+theorem imp_self_eq (p : Prop) : (p → p) = True := by simp
+
+theorem not_and (p q : Prop) : (¬(p ∧ q)) = (¬p ∨ ¬q) := by
   by_cases p <;> by_cases q <;> simp [*]
-attribute [grind_norm] and_true true_and and_false false_and and_assoc
 
--- Or
-attribute [grind_norm↓] not_or
-attribute [grind_norm] or_true true_or or_false false_or or_assoc
-
--- ite
-attribute [grind_norm] ite_true ite_false
-@[grind_norm↓] theorem not_ite {_ : Decidable p} (q r : Prop) : (¬ite p q r) = ite p (¬q) (¬r) := by
+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
+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
+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
+theorem not_forall (p : α → Prop) : (¬∀ x, p x) = ∃ x, ¬p x := by simp
 
--- Exists
-@[grind_norm↓] theorem not_exists (p : α → Prop) : (¬∃ x, p x) = ∀ x, ¬p x := by simp
-attribute [grind_norm] exists_const exists_or exists_prop exists_and_left exists_and_right
+theorem not_exists (p : α → Prop) : (¬∃ x, p x) = ∀ x, ¬p x := by simp
 
--- Bool cond
-@[grind_norm] theorem cond_eq_ite (c : Bool) (a b : α) : cond c a b = ite c a b := by
+theorem cond_eq_ite (c : Bool) (a b : α) : cond c a b = ite c a b := by
   cases c <;> simp [*]
 
--- Bool or
-attribute [grind_norm]
-  Bool.or_false Bool.or_true Bool.false_or Bool.true_or Bool.or_eq_true Bool.or_assoc
-
--- Bool and
-attribute [grind_norm]
-  Bool.and_false Bool.and_true Bool.false_and Bool.true_and Bool.and_eq_true Bool.and_assoc
-
--- Bool not
-attribute [grind_norm]
-  Bool.not_not
-
--- beq
-attribute [grind_norm] beq_iff_eq
-
--- bne
-attribute [grind_norm] bne_iff_ne
-
--- Bool not eq true/false
-attribute [grind_norm] Bool.not_eq_true Bool.not_eq_false
-
--- decide
-attribute [grind_norm] decide_eq_true_eq decide_not not_decide_eq_true
-
--- Nat LE
-attribute [grind_norm] Nat.le_zero_eq
-
--- Nat/Int LT
-@[grind_norm] theorem Nat.lt_eq (a b : Nat) : (a < b) = (a + 1 ≤ b) := by
+theorem Nat.lt_eq (a b : Nat) : (a < b) = (a + 1 ≤ b) := by
   simp [Nat.lt, LT.lt]
 
-@[grind_norm] theorem Int.lt_eq (a b : Int) : (a < b) = (a + 1 ≤ b) := by
+theorem Int.lt_eq (a b : Int) : (a < b) = (a + 1 ≤ b) := by
   simp [Int.lt, LT.lt]
 
--- GT GE
-attribute [grind_norm] GT.gt GE.ge
+theorem ge_eq [LE α] (a b : α) : (a ≥ b) = (b ≤ a) := rfl
+theorem gt_eq [LT α] (a b : α) : (a > b) = (b < a) := rfl
 
--- Succ
-attribute [grind_norm] Nat.succ_eq_add_one
+init_grind_norm
+  /- Pre theorems -/
+  not_and not_or not_ite not_forall not_exists
+  |
+  /- Post theorems -/
+  Classical.not_not
+  ne_eq iff_eq eq_self heq_eq_eq
+  -- Prop equality
+  eq_true_eq eq_false_eq not_eq_prop
+  -- True
+  not_true
+  -- False
+  not_false_eq_true
+  -- Implication
+  true_imp_eq false_imp_eq imp_true_eq imp_false_eq imp_self_eq
+  -- And
+  and_true true_and and_false false_and and_assoc
+  -- Or
+  or_true true_or or_false false_or or_assoc
+  -- ite
+  ite_true ite_false ite_true_false ite_false_true
+  -- Forall
+  forall_and
+  -- Exists
+  exists_const exists_or exists_prop exists_and_left exists_and_right
+  -- Bool cond
+  cond_eq_ite
+  -- Bool or
+  Bool.or_false Bool.or_true Bool.false_or Bool.true_or Bool.or_eq_true Bool.or_assoc
+  -- Bool and
+  Bool.and_false Bool.and_true Bool.false_and Bool.true_and Bool.and_eq_true Bool.and_assoc
+  -- Bool not
+  Bool.not_not
+  -- beq
+  beq_iff_eq
+  -- bne
+  bne_iff_ne
+  -- Bool not eq true/false
+  Bool.not_eq_true Bool.not_eq_false
+  -- decide
+  decide_eq_true_eq decide_not not_decide_eq_true
+  -- Nat LE
+  Nat.le_zero_eq
+  -- Nat/Int LT
+  Nat.lt_eq
+  -- Nat.succ
+  Nat.succ_eq_add_one
+  -- Int
+  Int.lt_eq
+  -- GT GE
+  ge_eq gt_eq
 
 end Lean.Grind

src/Init/Grind/Offset.lean

@@ -7,159 +7,86 @@ prelude
 import Init.Core
 import Init.Omega
 
-namespace Lean.Grind.Offset
+namespace Lean.Grind
+abbrev isLt (x y : Nat) : Bool := x < y
+abbrev isLE (x y : Nat) : Bool := x ≤ y
 
-abbrev Var := Nat
-abbrev Context := Lean.RArray Nat
+/-! 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
 
-def fixedVar := 100000000 -- Any big number should work here
+/-! 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
 
-def Var.denote (ctx : Context) (v : Var) : Nat :=
-  bif v == fixedVar then 1 else ctx.get v
+/-! 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
 
-structure Cnstr where
-  x : Var
-  y : Var
-  k : Nat  := 0
-  l : Bool := true
-  deriving Repr, DecidableEq, Inhabited
+/-! 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
 
-def Cnstr.denote (c : Cnstr) (ctx : Context) : Prop :=
-  if c.l then
-    c.x.denote ctx + c.k ≤ c.y.denote ctx
-  else
-    c.x.denote ctx ≤ c.y.denote ctx + c.k
+/-!
+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
 
-def trivialCnstr : Cnstr := { x := 0, y := 0, k := 0, l := true }
+/-!
+Helper theorems for equality propagation
+-/
 
-@[simp] theorem denote_trivial (ctx : Context) : trivialCnstr.denote ctx := by
-  simp [Cnstr.denote, trivialCnstr]
+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
 
-def Cnstr.trans (c₁ c₂ : Cnstr) : Cnstr :=
-  if c₁.y = c₂.x then
-    let { x, k := k₁, l := l₁, .. } := c₁
-    let { y, k := k₂, l := l₂, .. } := c₂
-    match l₁, l₂ with
-    | false, false =>
-      { x, y, k := k₁ + k₂, l := false }
-    | false, true =>
-      if k₁ < k₂ then
-        { x, y, k := k₂ - k₁, l := true }
-      else
-        { x, y, k := k₁ - k₂, l := false }
-    | true, false =>
-      if k₁ < k₂ then
-        { x, y, k := k₂ - k₁, l := false }
-      else
-        { x, y, k := k₁ - k₂, l := true }
-    | true, true =>
-      { x, y, k := k₁ + k₂, l := true }
-  else
-    trivialCnstr
-
-@[simp] theorem Cnstr.denote_trans_easy (ctx : Context) (c₁ c₂ : Cnstr) (h : c₁.y ≠ c₂.x) : (c₁.trans c₂).denote ctx := by
-  simp [*, Cnstr.trans]
-
-@[simp] theorem Cnstr.denote_trans (ctx : Context) (c₁ c₂ : Cnstr) : c₁.denote ctx → c₂.denote ctx → (c₁.trans c₂).denote ctx := by
-  by_cases c₁.y = c₂.x
-  case neg => simp [*]
-  simp [trans, *]
-  let { x, k := k₁, l := l₁, .. } := c₁
-  let { y, k := k₂, l := l₂, .. } := c₂
-  simp_all; split
-  · simp [denote]; omega
-  · split <;> simp [denote] <;> omega
-  · split <;> simp [denote] <;> omega
-  · simp [denote]; omega
-
-def Cnstr.isTrivial (c : Cnstr) : Bool := c.x == c.y && c.k == 0
-
-theorem Cnstr.of_isTrivial (ctx : Context) (c : Cnstr) : c.isTrivial = true → c.denote ctx := by
-  cases c; simp [isTrivial]; intros; simp [*, denote]
-
-def Cnstr.isFalse (c : Cnstr) : Bool := c.x == c.y && c.k != 0 && c.l == true
-
-theorem Cnstr.of_isFalse (ctx : Context) {c : Cnstr} : c.isFalse = true → ¬c.denote ctx := by
-  cases c; simp [isFalse]; intros; simp [*, denote]; omega
-
-def Cnstrs := List Cnstr
-
-def Cnstrs.denoteAnd' (ctx : Context) (c₁ : Cnstr) (c₂ : Cnstrs) : Prop :=
-  match c₂ with
-  | [] => c₁.denote ctx
-  | c::cs => c₁.denote ctx ∧ Cnstrs.denoteAnd' ctx c cs
-
-theorem Cnstrs.denote'_trans (ctx : Context) (c₁ c : Cnstr) (cs : Cnstrs) : c₁.denote ctx → denoteAnd' ctx c cs → denoteAnd' ctx (c₁.trans c) cs := by
-  induction cs
-  next => simp [denoteAnd', *]; apply Cnstr.denote_trans
-  next c cs ih => simp [denoteAnd']; intros; simp [*]
-
-def Cnstrs.trans' (c₁ : Cnstr) (c₂ : Cnstrs) : Cnstr :=
-  match c₂ with
-  | [] => c₁
-  | c::c₂ => trans' (c₁.trans c) c₂
-
-@[simp] theorem Cnstrs.denote'_trans' (ctx : Context) (c₁ : Cnstr) (c₂ : Cnstrs) : denoteAnd' ctx c₁ c₂ → (trans' c₁ c₂).denote ctx := by
-  induction c₂ generalizing c₁
-  next => intros; simp_all [trans', denoteAnd']
-  next c cs ih => simp [denoteAnd']; intros; simp [trans']; apply ih; apply denote'_trans <;> assumption
-
-def Cnstrs.denoteAnd (ctx : Context) (c : Cnstrs) : Prop :=
-  match c with
-  | [] => True
-  | c::cs => denoteAnd' ctx c cs
-
-def Cnstrs.trans (c : Cnstrs) : Cnstr :=
-  match c with
-  | []    => trivialCnstr
-  | c::cs => trans' c cs
-
-theorem Cnstrs.of_denoteAnd_trans {ctx : Context} {c : Cnstrs} : c.denoteAnd ctx → c.trans.denote ctx := by
-  cases c <;> simp [*, trans, denoteAnd] <;> intros <;> simp [*]
-
-def Cnstrs.isFalse (c : Cnstrs) : Bool :=
-  c.trans.isFalse
-
-theorem Cnstrs.unsat' (ctx : Context) (c : Cnstrs) : c.isFalse = true → ¬ c.denoteAnd ctx := by
-  simp [isFalse]; intro h₁ h₂
-  have := of_denoteAnd_trans h₂
-  have := Cnstr.of_isFalse ctx h₁
-  contradiction
-
-/-- `denote ctx [c_1, ..., c_n] C` is `c_1.denote ctx → ... → c_n.denote ctx → C` -/
-def Cnstrs.denote (ctx : Context) (cs : Cnstrs) (C : Prop) : Prop :=
-  match cs with
-  | [] => C
-  | c::cs => c.denote ctx → denote ctx cs C
-
-theorem Cnstrs.not_denoteAnd'_eq (ctx : Context) (c : Cnstr) (cs : Cnstrs) (C : Prop) : (denoteAnd' ctx c cs → C) = denote ctx (c::cs) C := by
-  simp [denote]
-  induction cs generalizing c
-  next => simp [denoteAnd', denote]
-  next c' cs ih =>
-    simp [denoteAnd', denote, *]
-
-theorem Cnstrs.not_denoteAnd_eq (ctx : Context) (cs : Cnstrs) (C : Prop) : (denoteAnd ctx cs → C) = denote ctx cs C := by
-  cases cs
-  next => simp [denoteAnd, denote]
-  next c cs => apply not_denoteAnd'_eq
-
-def Cnstr.isImpliedBy (cs : Cnstrs) (c : Cnstr) : Bool :=
-  cs.trans == c
-
-/-! Main theorems used by `grind`. -/
-
-/-- Auxiliary theorem used by `grind` to prove that a system of offset inequalities is unsatisfiable. -/
-theorem Cnstrs.unsat (ctx : Context) (cs : Cnstrs) : cs.isFalse = true → cs.denote ctx False := by
-  intro h
-  rw [← not_denoteAnd_eq]
-  apply unsat'
-  assumption
-
-/-- Auxiliary theorem used by `grind` to prove an implied offset inequality. -/
-theorem Cnstrs.imp (ctx : Context) (cs : Cnstrs) (c : Cnstr) (h : c.isImpliedBy cs = true)  : cs.denote ctx (c.denote ctx) := by
-  rw [← eq_of_beq h]
-  rw [← not_denoteAnd_eq]
-  apply of_denoteAnd_trans
-
-end Lean.Grind.Offset
+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

@@ -11,10 +11,15 @@ namespace Lean.Parser.Attr
 syntax grindEq     := "="
 syntax grindEqBoth := atomic("_" "=" "_")
 syntax grindEqRhs  := atomic("=" "_")
+syntax grindEqBwd  := atomic("←" "=")
 syntax grindBwd    := "←"
 syntax grindFwd    := "→"
+syntax grindCases  := &"cases"
+syntax grindCasesEager := atomic(&"cases" &"eager")
 
-syntax (name := grind) "grind" (grindEqBoth <|> grindEqRhs <|> grindEq <|> grindBwd <|> grindFwd)? : attr
+syntax grindMod := grindEqBoth <|> grindEqRhs <|> grindEq <|> grindEqBwd <|> grindBwd <|> grindFwd <|> grindCasesEager <|> grindCases
+
+syntax (name := grind) "grind" (grindMod)? : attr
 
 end Lean.Parser.Attr
 
@@ -25,7 +30,7 @@ Passed to `grind` using, for example, the `grind (config := { matchEqs := true }
 -/
 structure Config where
   /-- Maximum number of case-splits in a proof search branch. It does not include splits performed during normalization. -/
-  splits : Nat := 5
+  splits : Nat := 8
   /-- Maximum number of E-matching (aka heuristic theorem instantiation) rounds before each case split. -/
   ematch : Nat := 5
   /--
@@ -43,8 +48,14 @@ structure Config where
   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
+  Otherwise, it performs case-splitting only on types marked with `[grind cases]` attribute. -/
+  splitIndPred : Bool := false
+  /-- 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
+  /-- If `ext` is `true`, `grind` uses extensionality theorems available in the environment. -/
+  ext : Bool := true
   deriving Inhabited, BEq
 
 end Lean.Grind
@@ -55,7 +66,13 @@ namespace Lean.Parser.Tactic
 `grind` tactic and related tactics.
 -/
 
--- TODO: parameters
-syntax (name := grind) "grind" optConfig ("on_failure " term)? : tactic
+syntax grindErase := "-" ident
+syntax grindLemma := (Attr.grindMod)? ident
+syntax grindParam := grindErase <|> grindLemma
+
+syntax (name := grind)
+  "grind" optConfig (&" only")?
+  (" [" withoutPosition(grindParam,*) "]")?
+  ("on_failure " term)? : tactic
 
 end Lean.Parser.Tactic

src/Init/Grind/Util.lean

@@ -9,18 +9,20 @@ 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
 
 /--
-Gadget for marking terms that should not be normalized by `grind`s simplifier.
-`grind` uses a simproc to implement this feature.
+Gadget for marking `match`-expressions that should not be reduced by the `grind` simplifier, but the discriminants should be normalized.
 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
+def simpMatchDiscrsOnly {α : Sort u} (a : α) : α := a
 
 /-- Gadget for representing offsets `t+k` in patterns. -/
 def offset (a b : Nat) : Nat := a + b
 
+/-- Gadget for representing `a = b` in patterns for backward propagation. -/
+def eqBwdPattern (a b : α) : Prop := a = b
+
 /--
 Gadget for annotating the equalities in `match`-equations conclusions.
 `_origin` is the term used to instantiate the `match`-equation using E-matching.
@@ -28,7 +30,15 @@ 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
+/--
+Gadget for annotating conditions of `match` equational lemmas.
+We use this annotation for two different reasons:
+- We don't want to normalize them.
+- We have a propagator for them.
+-/
+def MatchCond (p : Prop) : Prop := p
+
+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/Order.lean

@@ -5,4 +5,5 @@ Authors: Joachim Breitner
 -/
 prelude
 import Init.Internal.Order.Basic
+import Init.Internal.Order.Lemmas
 import Init.Internal.Order.Tactic

src/Init/Internal/Order/Basic.lean

@@ -104,7 +104,7 @@ 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.
+A function is monotone 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.
 -/
@@ -401,6 +401,7 @@ theorem monotone_letFun
     (hmono : ∀ y, monotone (fun x => k x y)) :
   monotone fun (x : α) => letFun v (k x) := hmono v
 
+@[partial_fixpoint_monotone]
 theorem monotone_ite
     {α : Sort u} {β : Sort v} [PartialOrder α] [PartialOrder β]
     (c : Prop) [Decidable c]
@@ -411,6 +412,7 @@ theorem monotone_ite
     · apply hmono₁
     · apply hmono₂
 
+@[partial_fixpoint_monotone]
 theorem monotone_dite
     {α : Sort u} {β : Sort v} [PartialOrder α] [PartialOrder β]
     (c : Prop) [Decidable c]
@@ -440,38 +442,41 @@ instance [PartialOrder α] [PartialOrder β] : PartialOrder (α ×' β) where
     dsimp at *
     rw [rel_antisymm ha.1 hb.1, rel_antisymm ha.2 hb.2]
 
-theorem monotone_pprod [PartialOrder α] [PartialOrder β] [PartialOrder γ]
+@[partial_fixpoint_monotone]
+theorem PProd.monotone_mk [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]
+@[partial_fixpoint_monotone]
+theorem PProd.monotone_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]
+@[partial_fixpoint_monotone]
+theorem PProd.monotone_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⟩
+def PProd.chain.fst [CCPO α] [CCPO β] (c : α ×' β → Prop) : α → Prop := fun a => ∃ b, c ⟨a, b⟩
+def PProd.chain.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
+theorem PProd.chain.chain_fst [CCPO α] [CCPO β] {c : α ×' β → Prop} (hchain : chain c) :
+    chain (chain.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
+theorem PProd.chain.chain_snd [CCPO α] [CCPO β] {c : α ×' β → Prop} (hchain : chain c) :
+    chain (chain.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)⟩
+instance instCCPOPProd [CCPO α] [CCPO β] : CCPO (α ×' β) where
+  csup c := ⟨CCPO.csup (PProd.chain.fst c), CCPO.csup (PProd.chain.snd c)⟩
   csup_spec := by
     intro ⟨a, b⟩ c hchain
     dsimp
@@ -480,32 +485,32 @@ instance [CCPO α] [CCPO β] : CCPO (α ×' β) where
       intro ⟨h₁, h₂⟩ ⟨a', b'⟩ cab
       constructor <;> dsimp at *
       · apply rel_trans ?_ h₁
-        apply le_csup hchain.pprod_fst
+        apply le_csup (PProd.chain.chain_fst hchain)
         exact ⟨b', cab⟩
       · apply rel_trans ?_ h₂
-        apply le_csup hchain.pprod_snd
+        apply le_csup (PProd.chain.chain_snd hchain)
         exact ⟨a', cab⟩
     next =>
       intro h
       constructor <;> dsimp
-      · apply csup_le hchain.pprod_fst
+      · apply csup_le (PProd.chain.chain_fst hchain)
         intro a' ⟨b', hcab⟩
         apply (h _ hcab).1
-      · apply csup_le hchain.pprod_snd
+      · apply csup_le (PProd.chain.chain_snd hchain)
         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
+  apply hadm _ (PProd.chain.chain_fst hchain)
   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
+  apply hadm _ (PProd.chain.chain_snd hchain)
   intro y ⟨x, hxy⟩
   apply h ⟨x,y⟩ hxy
 
@@ -609,6 +614,7 @@ class MonoBind (m : Type u → Type v) [Bind m] [∀ α, PartialOrder (m α)] wh
   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
 
+@[partial_fixpoint_monotone]
 theorem monotone_bind
     (m : Type u → Type v) [Bind m] [∀ α, PartialOrder (m α)] [MonoBind m]
     {α β : Type u}
@@ -634,7 +640,7 @@ noncomputable instance : MonoBind Option where
     · exact FlatOrder.rel.refl
     · exact h _
 
-theorem admissible_eq_some (P : Prop) (y : α) :
+theorem Option.admissible_eq_some (P : Prop) (y : α) :
     admissible (fun (x : Option α) => x = some y → P) := by
   apply admissible_flatOrder; simp
 
@@ -677,7 +683,7 @@ theorem find_spec : ∀ n m, find P n = some m → n ≤ m ∧ P m := by
   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))
+    exact admissible_pi_apply _ (fun n => admissible_pi _ (fun m => Option.admissible_eq_some _ m))
   case hstep =>
     intro f ih n m heq
     simp only [findF] at heq

src/Init/Internal/Order/Lemmas.lean

@@ -0,0 +1,685 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+
+prelude
+
+import Init.Data.List.Control
+import Init.Data.Array.Basic
+import Init.Internal.Order.Basic
+
+/-!
+
+This file contains monotonicity lemmas for higher-order monadic operations (e.g. `mapM`) in the
+standard library. This allows recursive definitions using `partial_fixpoint` to use nested
+recursion.
+
+Ideally, every higher-order monadic funciton in the standard library has a lemma here. At the time
+of writing, this file covers functions from
+
+* Init/Data/Option/Basic.lean
+* Init/Data/List/Control.lean
+* Init/Data/Array/Basic.lean
+
+in the order of their apperance there. No automation to check the exhaustiveness exists yet.
+
+The lemma statements are written manually, but follow a predictable scheme, and could be automated.
+Likewise, the proofs are written very naively. Most of them could be handled by a tactic like
+`monotonicity` (extended to make use of local hypotheses).
+-/
+
+namespace Lean.Order
+
+open Lean.Order
+
+variable {m : Type u → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m]
+variable {α β : Type u}
+variable {γ : Type w} [PartialOrder γ]
+
+@[partial_fixpoint_monotone]
+theorem Functor.monotone_map [LawfulMonad m] (f : γ → m α) (g : α → β) (hmono : monotone f) :
+    monotone (fun x => g <$> f x) := by
+  simp only [← LawfulMonad.bind_pure_comp ]
+  apply monotone_bind _ _ _ hmono
+  apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem Seq.monotone_seq [LawfulMonad m] (f : γ → m α) (g : γ → m (α → β))
+  (hmono₁ : monotone g) (hmono₂ : monotone f) :
+    monotone (fun x => g x <*> f x) := by
+  simp only [← LawfulMonad.bind_map ]
+  apply monotone_bind
+  · assumption
+  · apply monotone_of_monotone_apply
+    intro y
+    apply Functor.monotone_map
+    assumption
+
+@[partial_fixpoint_monotone]
+theorem SeqLeft.monotone_seqLeft [LawfulMonad m] (f : γ → m α) (g : γ → m β)
+  (hmono₁ : monotone g) (hmono₂ : monotone f) :
+    monotone (fun x => g x <* f x) := by
+  simp only [seqLeft_eq]
+  apply Seq.monotone_seq
+  · apply Functor.monotone_map
+    assumption
+  · assumption
+
+@[partial_fixpoint_monotone]
+theorem SeqRight.monotone_seqRight [LawfulMonad m] (f : γ → m α) (g : γ → m β)
+  (hmono₁ : monotone g) (hmono₂ : monotone f) :
+    monotone (fun x => g x *> f x) := by
+  simp only [seqRight_eq]
+  apply Seq.monotone_seq
+  · apply Functor.monotone_map
+    assumption
+  · assumption
+
+namespace Option
+
+@[partial_fixpoint_monotone]
+theorem monotone_bindM (f : γ → α → m (Option β)) (xs : Option α) (hmono : monotone f) :
+    monotone (fun x => xs.bindM (f x)) := by
+  cases xs with
+  | none => apply monotone_const
+  | some x =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_mapM (f : γ → α → m β) (xs : Option α) (hmono : monotone f) :
+    monotone (fun x => xs.mapM (f x)) := by
+  cases xs with
+  | none => apply monotone_const
+  | some x =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_elimM (a : γ → m (Option α)) (n : γ → m β) (s : γ → α → m β)
+    (hmono₁ : monotone a) (hmono₂ : monotone n) (hmono₃ : monotone s) :
+    monotone (fun x => Option.elimM (a x) (n x) (s x)) := by
+  apply monotone_bind
+  · apply hmono₁
+  · apply monotone_of_monotone_apply
+    intro o
+    cases o
+    case none => apply hmono₂
+    case some y =>
+      dsimp only [Option.elim]
+      apply monotone_apply
+      apply hmono₃
+
+omit [MonoBind m] in
+@[partial_fixpoint_monotone]
+theorem monotone_getDM (o : Option α) (y : γ → m α) (hmono : monotone y) :
+    monotone (fun x => o.getDM (y x)) := by
+  cases o
+  · apply hmono
+  · apply monotone_const
+
+end Option
+
+
+namespace List
+
+@[partial_fixpoint_monotone]
+theorem monotone_mapM (f : γ → α → m β) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.mapM (f x)) := by
+  cases xs with
+  | nil => apply monotone_const
+  | cons _ xs =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      dsimp
+      generalize [y] = ys
+      induction xs generalizing ys with
+      | nil => apply monotone_const
+      | cons _ _ ih =>
+        apply monotone_bind
+        · apply monotone_apply
+          apply hmono
+        · apply monotone_of_monotone_apply
+          intro y
+          apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_forM (f : γ → α → m PUnit) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.forM (f x)) := by
+  induction xs with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_filterAuxM
+  {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+  (f : γ → α → m Bool) (xs acc : List α) (hmono : monotone f) :
+    monotone (fun x => xs.filterAuxM (f x) acc) := by
+  induction xs generalizing acc with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_filterM
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.filterM (f x)) := by
+  apply monotone_bind
+  · exact monotone_filterAuxM f xs [] hmono
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_filterRevM
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.filterRevM (f x)) := by
+  exact monotone_filterAuxM f xs.reverse [] hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_foldlM
+    (f : γ → β → α → m β) (init : β) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.foldlM (f x) (init := init)) := by
+  induction xs generalizing init with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_foldrM
+    (f : γ → α → β → m β) (init : β) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.foldrM (f x) (init := init)) := by
+  apply monotone_foldlM
+  apply monotone_of_monotone_apply
+  intro s
+  apply monotone_of_monotone_apply
+  intro a
+  apply monotone_apply (a := s)
+  apply monotone_apply (a := a)
+  apply hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_anyM
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.anyM (f x)) := by
+  induction xs with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y
+      · apply ih
+      · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_allM
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.allM (f x)) := by
+  induction xs with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y
+      · apply monotone_const
+      · apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_findM?
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.findM? (f x)) := by
+  induction xs with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y
+      · apply ih
+      · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_findSomeM?
+    (f : γ → α → m (Option β)) (xs : List α) (hmono : monotone f) :
+    monotone (fun x => xs.findSomeM? (f x)) := by
+  induction xs with
+  | nil => apply monotone_const
+  | cons _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y
+      · apply ih
+      · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_forIn'_loop {α : Type uu}
+    (as : List α) (f : γ → (a : α) → a ∈ as → β → m (ForInStep β)) (as' : List α) (b : β)
+    (p : Exists (fun bs => bs ++ as' = as)) (hmono : monotone f) :
+    monotone (fun x => List.forIn'.loop as (f x) as' b p) := by
+  induction as' generalizing b with
+  | nil => apply monotone_const
+  | cons a as' ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply monotone_apply
+      apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y with
+      | done => apply monotone_const
+      | yield => apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_forIn' {α : Type uu}
+    (as : List α) (init : β) (f : γ → (a : α) → a ∈ as → β → m (ForInStep β)) (hmono : monotone f) :
+    monotone (fun x => forIn' as init (f x)) := by
+  apply monotone_forIn'_loop
+  apply hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_forIn {α : Type uu}
+    (as : List α) (init : β) (f : γ → (a : α) → β → m (ForInStep β)) (hmono : monotone f) :
+    monotone (fun x => forIn as init (f x)) := by
+  apply monotone_forIn' as init _
+  apply monotone_of_monotone_apply
+  intro y
+  apply monotone_of_monotone_apply
+  intro p
+  apply monotone_apply (a := y)
+  apply hmono
+
+end List
+
+namespace Array
+
+@[partial_fixpoint_monotone]
+theorem monotone_modifyM (a : Array α) (i : Nat) (f : γ → α → m α) (hmono : monotone f) :
+    monotone (fun x => a.modifyM i (f x)) := by
+  unfold Array.modifyM
+  split
+  · apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_forIn'_loop {α : Type uu}
+    (as : Array α) (f : γ → (a : α) → a ∈ as → β → m (ForInStep β)) (i : Nat) (h : i ≤ as.size)
+    (b : β) (hmono : monotone f) :
+    monotone (fun x => Array.forIn'.loop as (f x) i h b) := by
+  induction i, h, b using Array.forIn'.loop.induct with
+  | case1 => apply monotone_const
+  | case2 _ _ _ _ _ _ _ ih =>
+    apply monotone_bind
+    · apply monotone_apply
+      apply monotone_apply
+      apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y with
+      | done => apply monotone_const
+      | yield => apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_forIn' {α : Type uu}
+    (as : Array α) (init : β) (f : γ → (a : α) → a ∈ as → β → m (ForInStep β)) (hmono : monotone f) :
+    monotone (fun x => forIn' as init (f x)) := by
+  apply monotone_forIn'_loop
+  apply hmono
+
+
+@[partial_fixpoint_monotone]
+theorem monotone_forIn {α : Type uu}
+    (as : Array α) (init : β) (f : γ → (a : α) → β → m (ForInStep β)) (hmono : monotone f) :
+    monotone (fun x => forIn as init (f x)) := by
+  apply monotone_forIn' as init _
+  apply monotone_of_monotone_apply
+  intro y
+  apply monotone_of_monotone_apply
+  intro p
+  apply monotone_apply (a := y)
+  apply hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_foldlM_loop
+    (f : γ → β → α → m β) (xs : Array α) (stop : Nat) (h : stop ≤ xs.size) (i j : Nat) (b : β)
+    (hmono : monotone f) : monotone (fun x => Array.foldlM.loop (f x) xs stop h i j b) := by
+  induction i, j, b using Array.foldlM.loop.induct (h := h) with
+  | case1 =>
+    simp only [Array.foldlM.loop, ↓reduceDIte, *]
+    apply monotone_const
+  | case2 _ _ _ _ _ ih =>
+    unfold Array.foldlM.loop
+    simp only [↓reduceDIte, *]
+    apply monotone_bind
+    · apply monotone_apply
+      apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      apply ih
+  | case3 =>
+    simp only [Array.foldlM.loop, ↓reduceDIte, *]
+    apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_foldlM
+    (f : γ → β → α → m β) (init : β) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
+    monotone (fun x => xs.foldlM (f x) init start stop) := by
+  unfold Array.foldlM
+  split <;> apply monotone_foldlM_loop (hmono := hmono)
+
+@[partial_fixpoint_monotone]
+theorem monotone_foldrM_fold
+    (f : γ → α → β → m β) (xs : Array α) (stop i : Nat) (h : i ≤ xs.size) (b : β)
+    (hmono : monotone f) : monotone (fun x => Array.foldrM.fold (f x) xs stop i h b) := by
+  induction i, h, b using Array.foldrM.fold.induct (stop := stop) with
+  | case1 =>
+    unfold Array.foldrM.fold
+    simp only [↓reduceIte, *]
+    apply monotone_const
+  | case2  =>
+    unfold Array.foldrM.fold
+    simp only [↓reduceIte, *]
+    apply monotone_const
+  | case3 _ _ _ _ _ _ ih =>
+    unfold Array.foldrM.fold
+    simp only [reduceCtorEq, ↓reduceIte, *]
+    apply monotone_bind
+    · apply monotone_apply
+      apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_foldrM
+    (f : γ → α → β → m β) (init : β) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
+    monotone (fun x => xs.foldrM (f x) init start stop) := by
+  unfold Array.foldrM
+  split
+  · split
+    · apply monotone_foldrM_fold (hmono := hmono)
+    · apply monotone_const
+  · split
+    · apply monotone_foldrM_fold (hmono := hmono)
+    · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_mapM (xs : Array α) (f : γ → α → m β) (hmono : monotone f) :
+    monotone (fun x => xs.mapM (f x)) := by
+  suffices ∀ i r, monotone (fun x => Array.mapM.map (f x) xs i r) by apply this
+  intros i r
+  induction i, r using Array.mapM.map.induct xs
+  case case1 ih =>
+    unfold Array.mapM.map
+    simp only [↓reduceDIte, *]
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · intro y
+      apply monotone_of_monotone_apply
+      apply ih
+  case case2 =>
+    unfold Array.mapM.map
+    simp only [↓reduceDIte, *]
+    apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_mapFinIdxM (xs : Array α) (f : γ → (i : Nat) → α → i < xs.size → m β)
+    (hmono : monotone f) : monotone (fun x => xs.mapFinIdxM (f x)) := by
+  suffices ∀ i j (h : i + j = xs.size) r, monotone (fun x => Array.mapFinIdxM.map xs (f x) i j h r) by apply this
+  intros i j h r
+  induction i, j, h, r using Array.mapFinIdxM.map.induct xs
+  case case1 =>
+    apply monotone_const
+  case case2 ih =>
+    apply monotone_bind
+    · dsimp
+      apply monotone_apply
+      apply monotone_apply
+      apply monotone_apply
+      apply hmono
+    · intro y
+      apply monotone_of_monotone_apply
+      apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_findSomeM?
+    (f : γ → α → m (Option β)) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.findSomeM? (f x)) := by
+  unfold Array.findSomeM?
+  apply monotone_bind
+  · apply monotone_forIn
+    apply monotone_of_monotone_apply
+    intro y
+    apply monotone_of_monotone_apply
+    intro r
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_findM?
+    {m : Type → Type} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.findM? (f x)) := by
+  unfold Array.findM?
+  apply monotone_bind
+  · apply monotone_forIn
+    apply monotone_of_monotone_apply
+    intro y
+    apply monotone_of_monotone_apply
+    intro r
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_findIdxM?
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
+    (f : γ → α → m Bool) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.findIdxM? (f x)) := by
+  unfold Array.findIdxM?
+  apply monotone_bind
+  · apply monotone_forIn
+    apply monotone_of_monotone_apply
+    intro y
+    apply monotone_of_monotone_apply
+    intro r
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_anyM_loop
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
+    (f : γ → α → m Bool) (xs : Array α) (stop : Nat) (h : stop ≤ xs.size) (j : Nat)
+    (hmono : monotone f) : monotone (fun x => Array.anyM.loop (f x) xs stop h j) := by
+  induction j using Array.anyM.loop.induct (h := h) with
+  | case2 =>
+    unfold Array.anyM.loop
+    simp only [↓reduceDIte, *]
+    apply monotone_const
+  | case1 _ _ _ ih =>
+    unfold Array.anyM.loop
+    simp only [↓reduceDIte, *]
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      split
+      · apply monotone_const
+      · apply ih
+
+@[partial_fixpoint_monotone]
+theorem monotone_anyM
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
+    (f : γ → α → m Bool) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
+    monotone (fun x => xs.anyM (f x) start stop) := by
+  unfold Array.anyM
+  split
+  · apply monotone_anyM_loop
+    apply hmono
+  · apply monotone_anyM_loop
+    apply hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_allM
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
+    (f : γ → α → m Bool) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
+    monotone (fun x => xs.allM (f x) start stop) := by
+  unfold Array.allM
+  apply monotone_bind
+  · apply monotone_anyM
+    apply monotone_of_monotone_apply
+    intro y
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_const
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_findSomeRevM?
+    (f : γ → α → m (Option β)) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.findSomeRevM? (f x)) := by
+  unfold Array.findSomeRevM?
+  suffices ∀ i (h : i ≤ xs.size), monotone (fun x => Array.findSomeRevM?.find (f x) xs i h) by apply this
+  intros i h
+  induction i, h using Array.findSomeRevM?.find.induct with
+  | case1 =>
+    unfold Array.findSomeRevM?.find
+    apply monotone_const
+  | case2 _ _ _ _ ih =>
+    unfold Array.findSomeRevM?.find
+    apply monotone_bind
+    · apply monotone_apply
+      apply hmono
+    · apply monotone_of_monotone_apply
+      intro y
+      cases y with
+      | none => apply ih
+      | some y => apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_findRevM?
+    {m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
+    (f : γ → α → m Bool) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.findRevM? (f x)) := by
+  unfold Array.findRevM?
+  apply monotone_findSomeRevM?
+  apply monotone_of_monotone_apply
+  intro y
+  apply monotone_bind
+  · apply monotone_apply
+    apply hmono
+  · apply monotone_const
+
+@[partial_fixpoint_monotone]
+theorem monotone_array_forM
+    (f : γ → α → m PUnit) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
+    monotone (fun x => xs.forM (f x) start stop) := by
+  unfold Array.forM
+  apply monotone_foldlM
+  apply monotone_of_monotone_apply
+  intro y
+  apply hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_array_forRevM
+    (f : γ → α → m PUnit) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
+    monotone (fun x => xs.forRevM (f x) start stop) := by
+  unfold Array.forRevM
+  apply monotone_foldrM
+  apply monotone_of_monotone_apply
+  intro y
+  apply monotone_of_monotone_apply
+  intro z
+  apply monotone_apply
+  apply hmono
+
+@[partial_fixpoint_monotone]
+theorem monotone_flatMapM
+    (f : γ → α → m (Array β)) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.flatMapM (f x)) := by
+  unfold Array.flatMapM
+  apply monotone_foldlM
+  apply monotone_of_monotone_apply
+  intro y
+  apply monotone_of_monotone_apply
+  intro z
+  apply monotone_bind
+  · apply monotone_apply
+    apply hmono
+  · apply monotone_const
+
+
+@[partial_fixpoint_monotone]
+theorem monotone_array_filterMapM
+    (f : γ → α → m (Option β)) (xs : Array α) (hmono : monotone f) :
+    monotone (fun x => xs.filterMapM (f x)) := by
+  unfold Array.filterMapM
+  apply monotone_foldlM
+  apply monotone_of_monotone_apply
+  intro y
+  apply monotone_of_monotone_apply
+  intro z
+  apply monotone_bind
+  · apply monotone_apply
+    apply hmono
+  · apply monotone_const
+
+end Array
+
+end Lean.Order

src/Init/Meta.lean

@@ -93,7 +93,8 @@ def isLetterLike (c : Char) : Bool :=
 def isSubScriptAlnum (c : Char) : Bool :=
   isNumericSubscript c ||
   (0x2090 ≤ c.val && c.val ≤ 0x209c) ||
-  (0x1d62 ≤ c.val && c.val ≤ 0x1d6a)
+  (0x1d62 ≤ c.val && c.val ≤ 0x1d6a) ||
+  c.val == 0x2c7c
 
 @[inline] def isIdFirst (c : Char) : Bool :=
   c.isAlpha || c = '_' || isLetterLike c

src/Init/MetaTypes.lean

@@ -109,6 +109,11 @@ structure Config where
   to find candidate `simp` theorems. It approximates Lean 3 `simp` behavior.
   -/
   index             : Bool := true
+  /--
+  When `true` (default : `true`), then simps will remove unused let-declarations:
+  `let x := v; e` simplifies to `e` when `x` does not occur in `e`.
+  -/
+  zetaUnused : Bool := true
   deriving Inhabited, BEq
 
 end DSimp
@@ -228,6 +233,11 @@ structure Config where
   input and output terms are definitionally equal.
   -/
   implicitDefEqProofs : Bool := true
+  /--
+  When `true` (default : `true`), then simps will remove unused let-declarations:
+  `let x := v; e` simplifies to `e` when `x` does not occur in `e`.
+  -/
+  zetaUnused : Bool := true
   deriving Inhabited, BEq
 
 -- Configuration object for `simp_all`
@@ -248,6 +258,7 @@ def neutralConfig : Simp.Config := {
   autoUnfold        := false
   ground            := false
   zetaDelta         := false
+  zetaUnused        := false
 }
 
 structure NormCastConfig extends Simp.Config where

src/Init/Prelude.lean

@@ -150,6 +150,9 @@ It can also be written as `()`.
 /-- Marker for information that has been erased by the code generator. -/
 unsafe axiom lcErased : Type
 
+/-- Marker for type dependency that has been erased by the code generator. -/
+unsafe axiom lcAny : Type
+
 /--
 Auxiliary unsafe constant used by the Compiler when erasing proofs from code.
 
@@ -3702,8 +3705,7 @@ inductive Syntax where
   /-- Node in the syntax tree.
 
   The `info` field is used by the delaborator to store the position of the
-  subexpression corresponding to this node. The parser sets the `info` field
-  to `none`.
+  subexpression corresponding to this node.
   The parser sets the `info` field to `none`, with position retrieval continuing recursively.
   Nodes created by quotations use the result from `SourceInfo.fromRef` so that they are marked
   as synthetic even when the leading/trailing token is not.

src/Init/Tactics.lean

@@ -1648,17 +1648,6 @@ If there are several with the same priority, it is uses the "most recent one". E
 -/
 syntax (name := simp) "simp" (Tactic.simpPre <|> Tactic.simpPost)? patternIgnore("← " <|> "<- ")? (ppSpace prio)? : attr
 
-/--
-Theorems tagged with the `grind_norm` attribute are used by the `grind` tactic normalizer/pre-processor.
--/
-syntax (name := grind_norm) "grind_norm" (Tactic.simpPre <|> Tactic.simpPost)? (ppSpace prio)? : attr
-
-/--
-Simplification procedures tagged with the `grind_norm_proc` attribute are used by the `grind` tactic normalizer/pre-processor.
--/
-syntax (name := grind_norm_proc) "grind_norm_proc" (Tactic.simpPre <|> Tactic.simpPost)? : attr
-
-
 /-- The possible `norm_cast` kinds: `elim`, `move`, or `squash`. -/
 syntax normCastLabel := &"elim" <|> &"move" <|> &"squash"
 

src/Lean/AddDecl.lean

@@ -14,26 +14,85 @@ register_builtin_option debug.skipKernelTC : Bool := {
   descr    := "skip kernel type checker. WARNING: setting this option to true may compromise soundness because your proofs will not be checked by the Lean kernel"
 }
 
-def Environment.addDecl (env : Environment) (opts : Options) (decl : Declaration)
-    (cancelTk? : Option IO.CancelToken := none) : Except KernelException Environment :=
+/-- Adds given declaration to the environment, respecting `debug.skipKernelTC`. -/
+def Kernel.Environment.addDecl (env : Environment) (opts : Options) (decl : Declaration)
+    (cancelTk? : Option IO.CancelToken := none) : Except Exception Environment :=
   if debug.skipKernelTC.get opts then
     addDeclWithoutChecking env decl
   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
+private def Environment.addDeclAux (env : Environment) (opts : Options) (decl : Declaration)
+    (cancelTk? : Option IO.CancelToken := none) : Except Kernel.Exception Environment :=
+  env.addDeclCore (Core.getMaxHeartbeats opts).toUSize decl cancelTk? (!debug.skipKernelTC.get opts)
+
+@[deprecated "use `Lean.addDecl` instead to ensure new namespaces are registered" (since := "2024-12-03")]
+def Environment.addDecl (env : Environment) (opts : Options) (decl : Declaration)
+    (cancelTk? : Option IO.CancelToken := none) : Except Kernel.Exception Environment :=
+  Environment.addDeclAux env opts decl cancelTk?
+
+private def isNamespaceName : Name → Bool
+  | .str .anonymous _ => true
+  | .str p _          => isNamespaceName p
+  | _                 => false
+
+private def registerNamePrefixes (env : Environment) (name : Name) : Environment :=
+  match name with
+    | .str _ s =>
+      if s.get 0 == '_' then
+        -- Do not register namespaces that only contain internal declarations.
+        env
+      else
+        go env name
+    | _ => env
+where go env
+  | .str p _ => if isNamespaceName p then go (env.registerNamespace p) p else env
+  | _        => env
 
 def addDecl (decl : Declaration) : CoreM Unit := do
+  let mut env ← getEnv
+  -- register namespaces for newly added constants; this used to be done by the kernel itself
+  -- but that is incompatible with moving it to a separate task
+  env := decl.getNames.foldl registerNamePrefixes env
+  if let .inductDecl _ _ types _ := decl then
+    env := types.foldl (registerNamePrefixes · <| ·.name ++ `rec) env
+
+  if !Elab.async.get (← getOptions) then
+    setEnv env
+    return (← doAdd)
+
+  -- convert `Declaration` to `ConstantInfo` to use as a preliminary value in the environment until
+  -- kernel checking has finished; not all cases are supported yet
+  let (name, info, kind) ← match decl with
+    | .thmDecl thm => pure (thm.name, .thmInfo thm, .thm)
+    | .defnDecl defn => pure (defn.name, .defnInfo defn, .defn)
+    | .mutualDefnDecl [defn] => pure (defn.name, .defnInfo defn, .defn)
+    | _ => return (← doAdd)
+
+  -- no environment extension changes to report after kernel checking; ensures we do not
+  -- accidentally wait for this snapshot when querying extension states
+  let async ← env.addConstAsync (reportExts := false) name kind
+  -- report preliminary constant info immediately
+  async.commitConst async.asyncEnv (some info)
+  setEnv async.mainEnv
+  let checkAct ← Core.wrapAsyncAsSnapshot fun _ => do
+    try
+      setEnv async.asyncEnv
+      doAdd
+      async.commitCheckEnv (← getEnv)
+    finally
+      async.commitFailure
+  let t ← BaseIO.mapTask (fun _ => checkAct) env.checked
+  let endRange? := (← getRef).getTailPos?.map fun pos => ⟨pos, pos⟩
+  Core.logSnapshotTask { range? := endRange?, task := t }
+where doAdd := do
   profileitM Exception "type checking" (← getOptions) do
-    withTraceNode `Kernel (fun _ => return m!"typechecking declaration") do
+    withTraceNode `Kernel (fun _ => return m!"typechecking declarations {decl.getNames}") do
       if !(← MonadLog.hasErrors) && decl.hasSorry then
-        logWarning "declaration uses 'sorry'"
-      match (← getEnv).addDecl (← getOptions) decl (← read).cancelTk? with
-      | .ok    env => setEnv env
-      | .error ex  => throwKernelException ex
+        logWarning m!"declaration uses 'sorry'"
+      let env ← (← getEnv).addDeclAux (← getOptions) decl (← read).cancelTk?
+        |> ofExceptKernelException
+      setEnv env
 
 def addAndCompile (decl : Declaration) : CoreM Unit := do
   addDecl decl

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/Main.lean

@@ -33,6 +33,7 @@ def shouldGenerateCode (declName : Name) : CoreM Bool := do
   let some info ← getDeclInfo? declName | return false
   unless info.hasValue (allowOpaque := true) do return false
   if hasMacroInlineAttribute env declName then return false
+  if (getImplementedBy? env declName).isSome then return false
   if (← Meta.isMatcher declName) then return false
   if isCasesOnRecursor env declName then return false
   -- TODO: check if type class instance

src/Lean/Compiler/LCNF/MonoTypes.lean

@@ -72,23 +72,23 @@ The type contains only `→` and constants.
 -/
 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
-    | .const ..        => visitApp type #[]
-    | .app ..          => type.withApp visitApp
-    | .forallE _ d b _ => mkArrow (← toMonoType d) (← toMonoType (b.instantiate1 erasedExpr))
-    | _                => return erasedExpr
+  match type with
+  | .const .. => visitApp type #[]
+  | .app .. => type.withApp visitApp
+  | .forallE _ d b _ =>
+    let monoB ← toMonoType (b.instantiate1 anyExpr)
+    match monoB with
+    | .const ``lcErased _ => return erasedExpr
+    | _ => mkArrow (← toMonoType d) monoB
+  | .sort _ => return erasedExpr
+  | _ => return anyExpr
 where
   visitApp (f : Expr) (args : Array Expr) : CoreM Expr := do
     match f with
+    | .const ``lcErased _ => return erasedExpr
+    | .const ``lcAny _ => return anyExpr
+    | .const ``Decidable _ => return mkConst ``Bool
     | .const declName us =>
-      if declName == ``Decidable then
-        return mkConst ``Bool
       if let some info ← hasTrivialStructure? declName then
         let ctorType ← getOtherDeclBaseType info.ctorName []
         toMonoType (getParamTypes (← instantiateForall ctorType args[:info.numParams]))[info.fieldIdx]!
@@ -98,15 +98,13 @@ where
         for arg in args do
           let .forallE _ d b _ := type.headBeta | unreachable!
           let arg := arg.headBeta
-          if arg.isErased then
-            result := mkApp result arg
-          else if d.isErased || d matches .sort _ then
+          if d matches .const ``lcErased _ | .sort _ then
             result := mkApp result (← toMonoType arg)
           else
             result := mkApp result erasedExpr
           type := b.instantiate1 arg
         return result
-    | _ => return erasedExpr
+    | _ => return anyExpr
 
 /--
 State for the environment extension used to save the LCNF mono phase type for declarations

src/Lean/Compiler/LCNF/PrettyPrinter.lean

@@ -81,7 +81,10 @@ def ppLetDecl (letDecl : LetDecl) : M Format := do
     return f!"let {letDecl.binderName} := {← ppLetValue letDecl.value}"
 
 def getFunType (ps : Array Param) (type : Expr) : CoreM Expr :=
-  instantiateForall type (ps.map (mkFVar ·.fvarId))
+  if type.isErased then
+    pure type
+  else
+    instantiateForall type (ps.map (mkFVar ·.fvarId))
 
 mutual
   partial def ppFunDecl (funDecl : FunDecl) : M Format := do

src/Lean/Compiler/LCNF/ToLCNF.lean

@@ -7,6 +7,7 @@ prelude
 import Lean.ProjFns
 import Lean.Meta.CtorRecognizer
 import Lean.Compiler.BorrowedAnnotation
+import Lean.Compiler.CSimpAttr
 import Lean.Compiler.LCNF.Types
 import Lean.Compiler.LCNF.Bind
 import Lean.Compiler.LCNF.InferType
@@ -472,7 +473,7 @@ where
 
   /-- Giving `f` a constant `.const declName us`, convert `args` into `args'`, and return `.const declName us args'` -/
   visitAppDefaultConst (f : Expr) (args : Array Expr) : M Arg := do
-    let .const declName us := f | unreachable!
+    let .const declName us := CSimp.replaceConstants (← getEnv) f | unreachable!
     let args ← args.mapM visitAppArg
     letValueToArg <| .const declName us args
 
@@ -670,7 +671,7 @@ where
   visitApp (e : Expr) : M Arg := do
     if let some (args, n, t, v, b) := e.letFunAppArgs? then
       visitCore <| mkAppN (.letE n t v b (nonDep := true)) args
-    else if let .const declName _ := e.getAppFn then
+    else if let .const declName _ := CSimp.replaceConstants (← getEnv) e.getAppFn then
       if declName == ``Quot.lift then
         visitQuotLift e
       else if declName == ``Quot.mk then

src/Lean/Compiler/LCNF/ToMono.lean

@@ -150,18 +150,7 @@ where
 
 def toMono : Pass where
   name     := `toMono
-  run      := fun decls => do
-    let decls ← decls.filterM fun decl => do
-      if hasLocalInst decl.type then
-        /-
-        Declaration is a "template" for the code specialization pass.
-        So, we should delete it before going to next phase.
-        -/
-        decl.erase
-        return false
-      else
-        return true
-    decls.mapM (·.toMono)
+  run      := (·.mapM (·.toMono))
   phase    := .base
   phaseOut := .mono
 

src/Lean/Compiler/LCNF/Types.lean

@@ -13,6 +13,7 @@ scoped notation:max "◾" => lcErased
 namespace LCNF
 
 def erasedExpr := mkConst ``lcErased
+def anyExpr := mkConst ``lcAny
 
 def _root_.Lean.Expr.isErased (e : Expr) :=
   e.isAppOf ``lcErased

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

@@ -36,7 +36,7 @@ register_builtin_option Elab.async : Bool := {
   descr := "perform elaboration using multiple threads where possible\
     \n\
     \nThis option defaults to `false` but (when not explicitly set) is overridden to `true` in \
-      `Lean.Language.Lean.process` as used by the cmdline driver and language server. \
+      the language server. \
       Metaprogramming users driving elaboration directly via e.g. \
       `Lean.Elab.Command.elabCommandTopLevel` can opt into asynchronous elaboration by setting \
       this option but then are responsible for processing messages and other data not only in the \
@@ -423,7 +423,11 @@ def wrapAsyncAsSnapshot (act : Unit → CoreM Unit) (desc : String := by exact d
     IO.FS.withIsolatedStreams (isolateStderr := stderrAsMessages.get (← getOptions)) do
       let tid ← IO.getTID
       -- reset trace state and message log so as not to report them twice
-      modify fun st => { st with messages := st.messages.markAllReported, traceState := { tid } }
+      modify fun st => { st with
+        messages := st.messages.markAllReported
+        traceState := { tid }
+        snapshotTasks := #[]
+      }
       try
         withTraceNode `Elab.async (fun _ => return desc) do
           act ()
@@ -514,28 +518,39 @@ 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 Kernel.Exception Environment
+
 def compileDecl (decl : Declaration) : CoreM Unit := do
+  -- don't compile if kernel errored; should be converted into a task dependency when compilation
+  -- is made async as well
+  if !decl.getNames.all (← getEnv).constants.contains then
+    return
   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) =>
+  | Except.error (.other msg) =>
     checkUnsupported decl -- Generate nicer error message for unsupported recursors and axioms
     throwError msg
   | Except.error ex =>
     throwKernelException ex
 
 def compileDecls (decls : List Name) : CoreM Unit := do
+  -- don't compile if kernel errored; should be converted into a task dependency when compilation
+  -- is made async as well
+  if !decls.all (← getEnv).constants.contains then
+    return
   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) =>
+  | Except.error (.other msg) =>
     throwError msg
   | Except.error ex =>
     throwKernelException ex

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/NameTrie.lean

@@ -54,6 +54,10 @@ instance : EmptyCollection (NameTrie β) where
 def NameTrie.find? (t : NameTrie β) (k : Name) : Option β :=
   PrefixTree.find? t (toKey k)
 
+@[inline, inherit_doc PrefixTree.findLongestPrefix?]
+def NameTrie.findLongestPrefix? (t : NameTrie β) (k : Name) : Option β :=
+  PrefixTree.findLongestPrefix? t (toKey k)
+
 @[inline]
 def NameTrie.foldMatchingM [Monad m] (t : NameTrie β) (k : Name) (init : σ) (f : β → σ → m σ) : m σ :=
   PrefixTree.foldMatchingM t (toKey k) init f

src/Lean/Data/PrefixTree.lean

@@ -48,6 +48,17 @@ partial def find? (t : PrefixTreeNode α β) (cmp : α → α → Ordering) (k :
       | some t => loop t ks
   loop t k
 
+/-- Returns the the value of the longest key in `t` that is a prefix of `k`, if any. -/
+@[specialize]
+partial def findLongestPrefix? (t : PrefixTreeNode α β) (cmp : α → α → Ordering) (k : List α) : Option β :=
+  let rec loop acc?
+    | PrefixTreeNode.Node val _, [] => val <|> acc?
+    | PrefixTreeNode.Node val m, k :: ks =>
+      match RBNode.find cmp m k with
+      | none   => val
+      | some t => loop (val <|> acc?) t ks
+  loop none t k
+
 @[specialize]
 partial def foldMatchingM [Monad m] (t : PrefixTreeNode α β) (cmp : α → α → Ordering) (k : List α) (init : σ) (f : β → σ → m σ) : m σ :=
   let rec fold : PrefixTreeNode α β → σ → m σ
@@ -92,6 +103,10 @@ def PrefixTree.insert (t : PrefixTree α β p) (k : List α) (v : β) : PrefixTr
 def PrefixTree.find? (t : PrefixTree α β p) (k : List α) : Option β :=
   t.val.find? p k
 
+@[inline, inherit_doc PrefixTreeNode.findLongestPrefix?]
+def PrefixTree.findLongestPrefix? (t : PrefixTree α β p) (k : List α) : Option β :=
+  t.val.findLongestPrefix? p k
+
 @[inline]
 def PrefixTree.foldMatchingM [Monad m] (t : PrefixTree α β p) (k : List α) (init : σ) (f : β → σ → m σ) : m σ :=
   t.val.foldMatchingM p k init f

src/Lean/Declaration.lean

@@ -193,6 +193,19 @@ def Declaration.definitionVal! : Declaration → DefinitionVal
   | .defnDecl val => val
   | _ => panic! "Expected a `Declaration.defnDecl`."
 
+/--
+Returns all top-level names to be defined by adding this declaration to the environment. This does
+not include auxiliary definitions such as projections.
+-/
+def Declaration.getNames : Declaration → List Name
+  | .axiomDecl val          => [val.name]
+  | .defnDecl val           => [val.name]
+  | .thmDecl val            => [val.name]
+  | .opaqueDecl val         => [val.name]
+  | .quotDecl               => [``Quot, ``Quot.mk, ``Quot.lift, ``Quot.ind]
+  | .mutualDefnDecl defns   => defns.map (·.name)
+  | .inductDecl _ _ types _ => types.map (·.name)
+
 @[specialize] def Declaration.foldExprM {α} {m : Type → Type} [Monad m] (d : Declaration) (f : α → Expr → m α) (a : α) : m α :=
   match d with
   | .quotDecl                                        => pure a
@@ -469,6 +482,10 @@ def isInductive : ConstantInfo → Bool
   | .inductInfo _ => true
   | _             => false
 
+def isDefinition : ConstantInfo → Bool
+  | .defnInfo _ => true
+  | _           => false
+
 def isTheorem : ConstantInfo → Bool
   | .thmInfo _ => true
   | _          => false

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/BuiltinCommand.lean

@@ -124,9 +124,7 @@ private partial def elabChoiceAux (cmds : Array Syntax) (i : Nat) : CommandElabM
   n[1].forArgsM addUnivLevel
 
 @[builtin_command_elab «init_quot»] def elabInitQuot : CommandElab := fun _ => do
-  match (← getEnv).addDecl (← getOptions) Declaration.quotDecl with
-  | Except.ok env   => setEnv env
-  | Except.error ex => throwError (ex.toMessageData (← getOptions))
+  liftCoreM <| addDecl Declaration.quotDecl
 
 @[builtin_command_elab «export»] def elabExport : CommandElab := fun stx => do
   let `(export $ns ($ids*)) := stx | throwUnsupportedSyntax
@@ -294,7 +292,7 @@ def failIfSucceeds (x : CommandElabM Unit) : CommandElabM Unit := do
     modify fun s => { s with messages := {} };
     pure messages
   let restoreMessages (prevMessages : MessageLog) : CommandElabM Unit := do
-    modify fun s => { s with messages := prevMessages ++ s.messages.errorsToWarnings }
+    modify fun s => { s with messages := prevMessages ++ s.messages.errorsToInfos }
   let prevMessages ← resetMessages
   let succeeded ← try
     x

src/Lean/Elab/Command.lean

@@ -313,7 +313,11 @@ def wrapAsyncAsSnapshot (act : Unit → CommandElabM Unit)
     IO.FS.withIsolatedStreams (isolateStderr := Core.stderrAsMessages.get (← getOptions)) do
       let tid ← IO.getTID
       -- reset trace state and message log so as not to report them twice
-      modify fun st => { st with messages := st.messages.markAllReported, traceState := { tid } }
+      modify fun st => { st with
+        messages := st.messages.markAllReported
+        traceState := { tid }
+        snapshotTasks := #[]
+      }
       try
         withTraceNode `Elab.async (fun _ => return desc) do
           act ()

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/PreDefinition/Main.lean

@@ -9,6 +9,7 @@ import Lean.Elab.PreDefinition.Basic
 import Lean.Elab.PreDefinition.Structural
 import Lean.Elab.PreDefinition.WF.Main
 import Lean.Elab.PreDefinition.MkInhabitant
+import Lean.Elab.PreDefinition.PartialFixpoint
 
 namespace Lean.Elab
 open Meta
@@ -162,7 +163,8 @@ def ensureFunIndReservedNamesAvailable (preDefs : Array PreDefinition) : MetaM U
 Checks consistency of a clique of TerminationHints:
 
 * If not all have a hint, the hints are ignored (log error)
-* If one has `structural`, check that all have it, (else throw error)
+* None have both `termination_by` and `nontermination` (throw error)
+* If one has `structural` or `partialFixpoint`, check that all have it (else throw error)
 * A `structural` should not have a `decreasing_by` (else log error)
 
 -/
@@ -171,21 +173,26 @@ def checkTerminationByHints (preDefs : Array PreDefinition) : CoreM Unit := do
   let preDefsWithout := preDefs.filter (·.termination.terminationBy?.isNone)
   let structural :=
     preDefWith.termination.terminationBy? matches some {structural := true, ..}
+  let partialFixpoint := preDefWith.termination.partialFixpoint?.isSome
   for preDef in preDefs do
     if let .some termBy := preDef.termination.terminationBy? then
-      if !structural && !preDefsWithout.isEmpty then
+      if let .some partialFixpointStx := preDef.termination.partialFixpoint? then
+        throwErrorAt partialFixpointStx.ref m!"conflicting annotations: this function cannot \
+          be both terminating and a partial fixpoint"
+
+      if !structural && !partialFixpoint && !preDefsWithout.isEmpty then
         let m := MessageData.andList (preDefsWithout.toList.map (m!"{·.declName}"))
         let doOrDoes := if preDefsWithout.size = 1 then "does" else "do"
         logErrorAt termBy.ref (m!"incomplete set of `termination_by` annotations:\n"++
-          m!"This function is mutually with {m}, which {doOrDoes} not have " ++
+          m!"This function is mutually recursive with {m}, which {doOrDoes} not have " ++
           m!"a `termination_by` clause.\n" ++
           m!"The present clause is ignored.")
 
-      if structural && ! termBy.structural then
+      if structural && !termBy.structural then
         throwErrorAt termBy.ref (m!"Invalid `termination_by`; this function is mutually " ++
           m!"recursive with {preDefWith.declName}, which is marked as `termination_by " ++
           m!"structural` so this one also needs to be marked `structural`.")
-      if ! structural && termBy.structural then
+      if !structural && termBy.structural then
         throwErrorAt termBy.ref (m!"Invalid `termination_by`; this function is mutually " ++
           m!"recursive with {preDefWith.declName}, which is not marked as `structural` " ++
           m!"so this one cannot be `structural` either.")
@@ -194,20 +201,41 @@ def checkTerminationByHints (preDefs : Array PreDefinition) : CoreM Unit := do
           logErrorAt decr.ref (m!"Invalid `decreasing_by`; this function is marked as " ++
             m!"structurally recursive, so no explicit termination proof is needed.")
 
+    if partialFixpoint && preDef.termination.partialFixpoint?.isNone then
+      throwErrorAt preDef.ref (m!"Invalid `termination_by`; this function is mutually " ++
+        m!"recursive with {preDefWith.declName}, which is marked as " ++
+        m!"`nontermination_partialFixpointursive` so this one also needs to be marked " ++
+        m!"`nontermination_partialFixpointursive`.")
+
+    if preDef.termination.partialFixpoint?.isSome then
+        if let .some decr := preDef.termination.decreasingBy? then
+          logErrorAt decr.ref (m!"Invalid `decreasing_by`; this function is marked as " ++
+            m!"nonterminating, so no explicit termination proof is needed.")
+
+    if !partialFixpoint then
+      if let some stx := preDef.termination.partialFixpoint? then
+      throwErrorAt stx.ref (m!"Invalid `termination_by`; this function is mutually " ++
+       m!"recursive with {preDefWith.declName}, which is not also marked as " ++
+        m!"`nontermination_partialFixpointursive`, so this one cannot be either.")
+
 /--
-Elaborates the `TerminationHint` in the clique to `TerminationArguments`
+Elaborates the `TerminationHint` in the clique to `TerminationMeasures`
 -/
-def elabTerminationByHints (preDefs : Array PreDefinition) : TermElabM (Array (Option TerminationArgument)) := do
+def elabTerminationByHints (preDefs : Array PreDefinition) : TermElabM (Array (Option TerminationMeasure)) := do
   preDefs.mapM fun preDef => do
     let arity ← lambdaTelescope preDef.value fun xs _ => pure xs.size
     let hints := preDef.termination
     hints.terminationBy?.mapM
-      (TerminationArgument.elab preDef.declName preDef.type arity hints.extraParams ·)
+      (TerminationMeasure.elab preDef.declName preDef.type arity hints.extraParams ·)
 
 def shouldUseStructural (preDefs : Array PreDefinition) : Bool :=
   preDefs.any fun preDef =>
     preDef.termination.terminationBy? matches some {structural := true, ..}
 
+def shouldUsepartialFixpoint (preDefs : Array PreDefinition) : Bool :=
+  preDefs.any fun preDef =>
+    preDef.termination.partialFixpoint?.isSome
+
 def shouldUseWF (preDefs : Array PreDefinition) : Bool :=
   preDefs.any fun preDef =>
     preDef.termination.terminationBy? matches some {structural := false, ..} ||
@@ -251,16 +279,18 @@ def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLC
           try
             checkCodomainsLevel preDefs
             checkTerminationByHints preDefs
-            let termArg?s ← elabTerminationByHints preDefs
+            let termMeasures?s ← elabTerminationByHints preDefs
             if shouldUseStructural preDefs then
-              structuralRecursion preDefs termArg?s
+              structuralRecursion preDefs termMeasures?s
+            else if shouldUsepartialFixpoint preDefs then
+              partialFixpoint preDefs
             else if shouldUseWF preDefs then
-              wfRecursion preDefs termArg?s
+              wfRecursion preDefs termMeasures?s
             else
               withRef (preDefs[0]!.ref) <| mapError
                 (orelseMergeErrors
-                  (structuralRecursion preDefs termArg?s)
-                  (wfRecursion preDefs termArg?s))
+                  (structuralRecursion preDefs termMeasures?s)
+                  (wfRecursion preDefs termMeasures?s))
                 (fun msg =>
                   let preDefMsgs := preDefs.toList.map (MessageData.ofExpr $ mkConst ·.declName)
                   m!"fail to show termination for{indentD (MessageData.joinSep preDefMsgs Format.line)}\nwith errors\n{msg}")

src/Lean/Elab/PreDefinition/Mutual.lean

@@ -0,0 +1,92 @@
+/-
+Copyright (c) 2021 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura, Joachim Breitner
+-/
+prelude
+import Lean.Elab.PreDefinition.Basic
+
+/-!
+This module contains code common to mutual-via-fixedpoint constructions, i.e.
+well-founded recursion and partial fixed-points, but independent of the details of the mutual packing.
+-/
+
+namespace Lean.Elab.Mutual
+open Meta
+
+partial def withCommonTelescope (preDefs : Array PreDefinition) (k : Array Expr → Array Expr → MetaM α) : MetaM α :=
+  go #[] (preDefs.map (·.value))
+where
+  go (fvars : Array Expr) (vals : Array Expr) : MetaM α := do
+    if !(vals.all fun val => val.isLambda) then
+      k fvars vals
+    else if !(← vals.allM fun val => isDefEq val.bindingDomain! vals[0]!.bindingDomain!) then
+      k fvars vals
+    else
+      withLocalDecl vals[0]!.bindingName! vals[0]!.binderInfo vals[0]!.bindingDomain! fun x =>
+        go (fvars.push x) (vals.map fun val => val.bindingBody!.instantiate1 x)
+
+def getFixedPrefix (preDefs : Array PreDefinition) : MetaM Nat :=
+  withCommonTelescope preDefs fun xs vals => do
+    let resultRef ← IO.mkRef xs.size
+    for val in vals do
+      if (← resultRef.get) == 0 then return 0
+      forEachExpr' val fun e => do
+        if preDefs.any fun preDef => e.isAppOf preDef.declName then
+          let args := e.getAppArgs
+          resultRef.modify (min args.size ·)
+          for arg in args, x in xs do
+            if !(← withoutProofIrrelevance <| withReducible <| isDefEq arg x) then
+              -- We continue searching if e's arguments are not a prefix of `xs`
+              return true
+          return false
+        else
+          return true
+    resultRef.get
+
+def addPreDefsFromUnary (preDefs : Array PreDefinition) (preDefsNonrec : Array PreDefinition)
+    (unaryPreDefNonRec : PreDefinition) : TermElabM Unit := do
+  /-
+  We must remove `implemented_by` attributes from the auxiliary application because
+  this attribute is only relevant for code that is compiled. Moreover, the `[implemented_by <decl>]`
+  attribute would check whether the `unaryPreDef` type matches with `<decl>`'s type, and produce
+  and error. See issue #2899
+  -/
+  let preDefNonRec := unaryPreDefNonRec.filterAttrs fun attr => attr.name != `implemented_by
+  let declNames := preDefs.toList.map (·.declName)
+
+  -- Do not complain if the user sets @[semireducible], which usually is a noop,
+  -- we recognize that below and then do not set @[irreducible]
+  withOptions (allowUnsafeReducibility.set · true) do
+    if unaryPreDefNonRec.declName = preDefs[0]!.declName then
+      addNonRec preDefNonRec (applyAttrAfterCompilation := false)
+    else
+      withEnableInfoTree false do
+        addNonRec preDefNonRec (applyAttrAfterCompilation := false)
+      preDefsNonrec.forM (addNonRec · (applyAttrAfterCompilation := false) (all := declNames))
+
+/--
+Cleans the right-hand-sides of the predefinitions, to prepare for inclusion in the EqnInfos:
+ * Remove RecAppSyntax markers
+ * Abstracts nested proofs (and for that, add the `_unsafe_rec` definitions)
+-/
+def cleanPreDefs (preDefs : Array PreDefinition) : TermElabM (Array PreDefinition) := do
+  addAndCompilePartialRec preDefs
+  let preDefs ← preDefs.mapM (eraseRecAppSyntax ·)
+  let preDefs ← preDefs.mapM (abstractNestedProofs ·)
+  return preDefs
+
+/--
+Assign final attributes to the definitions. Assumes the EqnInfos to be already present.
+-/
+def addPreDefAttributes (preDefs : Array PreDefinition) : TermElabM Unit := do
+  for preDef in preDefs do
+    markAsRecursive preDef.declName
+    generateEagerEqns preDef.declName
+    applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
+    -- Unless the user asks for something else, mark the definition as irreducible
+    unless preDef.modifiers.attrs.any fun a =>
+      a.name = `reducible || a.name = `semireducible do
+      setIrreducibleAttribute preDef.declName
+
+end Lean.Elab.Mutual

src/Lean/Elab/PreDefinition/PartialFixpoint.lean

@@ -0,0 +1,9 @@
+/-
+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.Elab.PreDefinition.PartialFixpoint.Eqns
+import Lean.Elab.PreDefinition.PartialFixpoint.Main
+import Lean.Elab.PreDefinition.PartialFixpoint.Induction

src/Lean/Elab/PreDefinition/PartialFixpoint/Eqns.lean

@@ -0,0 +1,117 @@
+/-
+Copyright (c) 2022 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Lean.Elab.Tactic.Conv
+import Lean.Meta.Tactic.Rewrite
+import Lean.Meta.Tactic.Split
+import Lean.Elab.PreDefinition.Basic
+import Lean.Elab.PreDefinition.Eqns
+import Lean.Meta.ArgsPacker.Basic
+import Init.Data.Array.Basic
+import Init.Internal.Order.Basic
+
+namespace Lean.Elab.PartialFixpoint
+open Meta
+open Eqns
+
+structure EqnInfo extends EqnInfoCore where
+  declNames       : Array Name
+  declNameNonRec  : Name
+  fixedPrefixSize : Nat
+  deriving Inhabited
+
+private def deltaLHSUntilFix (declName declNameNonRec : Name) (mvarId : MVarId) : MetaM MVarId := mvarId.withContext do
+  let target ← mvarId.getType'
+  let some (_, lhs, rhs) := target.eq? | throwTacticEx `deltaLHSUntilFix mvarId "equality expected"
+  let lhs' ← deltaExpand lhs fun n => n == declName || n == declNameNonRec
+  mvarId.replaceTargetDefEq (← mkEq lhs' rhs)
+
+partial def rwFixUnder (lhs : Expr) : MetaM Expr := do
+  if lhs.isAppOfArity ``Order.fix 4 then
+    return mkAppN (mkConst ``Order.fix_eq lhs.getAppFn.constLevels!) lhs.getAppArgs
+  else if lhs.isApp then
+    let h ← rwFixUnder lhs.appFn!
+    mkAppM ``congrFun #[h, lhs.appArg!]
+  else if lhs.isProj then
+    let f := mkLambda `p .default (← inferType lhs.projExpr!) (lhs.updateProj! (.bvar 0))
+    let h ← rwFixUnder lhs.projExpr!
+    mkAppM ``congrArg #[f, h]
+  else
+    throwError "rwFixUnder: unexpected expression {lhs}"
+
+private def rwFixEq (mvarId : MVarId) : MetaM MVarId := mvarId.withContext do
+  let mut mvarId := mvarId
+  let target ← mvarId.getType'
+  let some (_, lhs, rhs) := target.eq? | unreachable!
+  let h ← rwFixUnder lhs
+  let some (_, _, lhsNew) := (← inferType h).eq? | unreachable!
+  let targetNew ← mkEq lhsNew rhs
+  let mvarNew ← mkFreshExprSyntheticOpaqueMVar targetNew
+  mvarId.assign (← mkEqTrans h mvarNew)
+  return mvarNew.mvarId!
+
+private partial def mkProof (declName : Name) (declNameNonRec : Name) (type : Expr) : MetaM Expr := do
+  trace[Elab.definition.partialFixpoint] "proving: {type}"
+  withNewMCtxDepth do
+    let main ← mkFreshExprSyntheticOpaqueMVar type
+    let (_, mvarId) ← main.mvarId!.intros
+    let mvarId ← deltaLHSUntilFix declName declNameNonRec mvarId
+    let mvarId ← rwFixEq mvarId
+    if ← withAtLeastTransparency .all (tryURefl mvarId) then
+      instantiateMVars main
+    else
+      throwError "failed to generate equational theorem for '{declName}'\n{MessageData.ofGoal mvarId}"
+
+def mkEqns (declName : Name) (info : EqnInfo) : MetaM (Array Name) :=
+  withOptions (tactic.hygienic.set · false) do
+  let baseName := declName
+  let eqnTypes ← withNewMCtxDepth <| lambdaTelescope (cleanupAnnotations := true) info.value fun xs body => do
+    let us := info.levelParams.map mkLevelParam
+    let target ← mkEq (mkAppN (Lean.mkConst declName us) xs) body
+    let goal ← mkFreshExprSyntheticOpaqueMVar target
+    withReducible do
+      mkEqnTypes info.declNames goal.mvarId!
+  let mut thmNames := #[]
+  for h : i in [: eqnTypes.size] do
+    let type := eqnTypes[i]
+    trace[Elab.definition.partialFixpoint] "{eqnTypes[i]}"
+    let name := (Name.str baseName eqnThmSuffixBase).appendIndexAfter (i+1)
+    thmNames := thmNames.push name
+    let value ← mkProof declName info.declNameNonRec type
+    let (type, value) ← removeUnusedEqnHypotheses type value
+    addDecl <| Declaration.thmDecl {
+      name, type, value
+      levelParams := info.levelParams
+    }
+  return thmNames
+
+builtin_initialize eqnInfoExt : MapDeclarationExtension EqnInfo ← mkMapDeclarationExtension
+
+def registerEqnsInfo (preDefs : Array PreDefinition) (declNameNonRec : Name) (fixedPrefixSize : Nat) : MetaM Unit := do
+  preDefs.forM fun preDef => ensureEqnReservedNamesAvailable preDef.declName
+  unless preDefs.all fun p => p.kind.isTheorem do
+    unless (← preDefs.allM fun p => isProp p.type) do
+      let declNames := preDefs.map (·.declName)
+      modifyEnv fun env =>
+        preDefs.foldl (init := env) fun env preDef =>
+          eqnInfoExt.insert env preDef.declName { preDef with
+            declNames, declNameNonRec, fixedPrefixSize }
+
+def getEqnsFor? (declName : Name) : MetaM (Option (Array Name)) := do
+  if let some info := eqnInfoExt.find? (← getEnv) declName then
+    mkEqns declName info
+  else
+    return none
+
+def getUnfoldFor? (declName : Name) : MetaM (Option Name) := do
+  let env ← getEnv
+  Eqns.getUnfoldFor? declName fun _ => eqnInfoExt.find? env declName |>.map (·.toEqnInfoCore)
+
+builtin_initialize
+  registerGetEqnsFn getEqnsFor?
+  registerGetUnfoldEqnFn getUnfoldFor?
+
+end Lean.Elab.PartialFixpoint

src/Lean/Elab/PreDefinition/PartialFixpoint/Induction.lean

@@ -0,0 +1,292 @@
+/-
+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.Basic
+import Lean.Meta.Match.MatcherApp.Transform
+import Lean.Meta.Check
+import Lean.Meta.Tactic.Subst
+import Lean.Meta.Injective -- for elimOptParam
+import Lean.Meta.ArgsPacker
+import Lean.Meta.PProdN
+import Lean.Meta.Tactic.Apply
+import Lean.Elab.PreDefinition.PartialFixpoint.Eqns
+import Lean.Elab.Command
+import Lean.Meta.Tactic.ElimInfo
+
+namespace Lean.Elab.PartialFixpoint
+
+open Lean Elab Meta
+
+open Lean.Order
+
+def mkAdmAnd (α instα adm₁ adm₂ : Expr) : MetaM Expr :=
+  mkAppOptM ``admissible_and #[α, instα, none, none, adm₁, adm₂]
+
+partial def mkAdmProj (packedInst : Expr) (i : Nat) (e : Expr) : MetaM Expr := do
+  if let some inst ← whnfUntil packedInst ``instCCPOPProd then
+    let_expr instCCPOPProd α β instα instβ := inst | throwError "mkAdmProj: unexpected instance {inst}"
+    if i == 0 then
+      mkAppOptM ``admissible_pprod_fst #[α, β, instα, instβ, none, e]
+    else
+      let e ← mkAdmProj instβ (i - 1) e
+      mkAppOptM ``admissible_pprod_snd #[α, β, instα, instβ, none, e]
+  else
+    assert! i == 0
+    return e
+
+def CCPOProdProjs (n : Nat) (inst : Expr) : Array Expr := Id.run do
+  let mut insts := #[inst]
+  while insts.size < n do
+    let inst := insts.back!
+    let_expr Lean.Order.instCCPOPProd _ _ inst₁ inst₂ := inst
+      | panic! s!"isOptionFixpoint: unexpected CCPO instance {inst}"
+    insts := insts.pop
+    insts := insts.push inst₁
+    insts := insts.push inst₂
+  return insts
+
+
+/-- `maskArray mask xs` keeps those `x` where the corresponding entry in `mask` is `true` -/
+-- Worth having in the standard libray?
+private def maskArray {α} (mask : Array Bool) (xs : Array α) : Array α := Id.run do
+  let mut ys := #[]
+  for b in mask, x in xs do
+    if b then ys := ys.push x
+  return ys
+
+/-- Appends `_1` etc to `base` unless `n == 1` -/
+private def numberNames (n : Nat) (base : String) : Array Name :=
+  .ofFn (n := n) fun ⟨i, _⟩ =>
+    if n == 1 then .mkSimple base else .mkSimple s!"{base}_{i+1}"
+
+def deriveInduction (name : Name) : MetaM Unit := do
+  mapError (f := (m!"Cannot derive fixpoint induction principle (please report this issue)\n{indentD ·}")) do
+    let some eqnInfo := eqnInfoExt.find? (← getEnv) name |
+      throwError "{name} is not defined by partial_fixpoint"
+
+    let infos ← eqnInfo.declNames.mapM getConstInfoDefn
+    -- First open up the fixed parameters everywhere
+    let e' ← lambdaBoundedTelescope infos[0]!.value eqnInfo.fixedPrefixSize fun xs _ => do
+      -- Now look at the body of an arbitrary of the functions (they are essentially the same
+      -- up to the final projections)
+      let body ← instantiateLambda infos[0]!.value xs
+
+      -- The body should now be of the form of the form (fix … ).2.2.1
+      -- We strip the projections (if present)
+      let body' := PProdN.stripProjs body
+      let some fixApp ← whnfUntil body' ``fix
+        | throwError "Unexpected function body {body}"
+      let_expr fix α instCCPOα F hmono := fixApp
+        | throwError "Unexpected function body {body'}"
+
+      let instCCPOs := CCPOProdProjs infos.size instCCPOα
+      let types ← infos.mapM (instantiateForall ·.type xs)
+      let packedType ← PProdN.pack 0 types
+      let motiveTypes ← types.mapM (mkArrow · (.sort 0))
+      let motiveNames := numberNames motiveTypes.size "motive"
+      withLocalDeclsDND (motiveNames.zip motiveTypes) fun motives => do
+        let packedMotive ←
+          withLocalDeclD (← mkFreshUserName `x) packedType fun x => do
+            mkLambdaFVars #[x] <| ← PProdN.pack 0 <|
+              motives.mapIdx fun idx motive =>
+                mkApp motive (PProdN.proj motives.size idx packedType x)
+
+        let admTypes ← motives.mapIdxM fun i motive => do
+          mkAppOptM ``admissible #[types[i]!, instCCPOs[i]!, some motive]
+        let admNames := numberNames admTypes.size "adm"
+        withLocalDeclsDND (admNames.zip admTypes) fun adms => do
+          let adms' ← adms.mapIdxM fun i adm => mkAdmProj instCCPOα i adm
+          let packedAdm ← PProdN.genMk (mkAdmAnd α instCCPOα) adms'
+          let hNames := numberNames infos.size "h"
+          let hTypes_hmask : Array (Expr × Array Bool) ← infos.mapIdxM fun i _info => do
+            let approxNames := infos.map fun info =>
+              match info.name with
+                | .str _ n => .mkSimple n
+                | _ => `f
+            withLocalDeclsDND (approxNames.zip types) fun approxs => do
+              let ihTypes := approxs.mapIdx fun j approx => mkApp motives[j]! approx
+              withLocalDeclsDND (ihTypes.map (⟨`ih, ·⟩)) fun ihs => do
+                let f ← PProdN.mk 0 approxs
+                let Ff := F.beta #[f]
+                let Ffi := PProdN.proj motives.size i packedType Ff
+                let t := mkApp motives[i]! Ffi
+                let t ← PProdN.reduceProjs t
+                let mask := approxs.map fun approx => t.containsFVar approx.fvarId!
+                let t ← mkForallFVars (maskArray mask approxs ++ maskArray mask ihs) t
+                pure (t, mask)
+          let (hTypes, masks) := hTypes_hmask.unzip
+          withLocalDeclsDND (hNames.zip hTypes) fun hs => do
+            let packedH ←
+              withLocalDeclD `approx packedType fun approx =>
+                let packedIHType := packedMotive.beta #[approx]
+                withLocalDeclD `ih packedIHType fun ih => do
+                  let approxs := PProdN.projs motives.size packedType approx
+                  let ihs := PProdN.projs motives.size packedIHType ih
+                  let e ← PProdN.mk 0 <| hs.mapIdx fun i h =>
+                    let mask := masks[i]!
+                    mkAppN h (maskArray mask approxs ++ maskArray mask ihs)
+                  mkLambdaFVars #[approx, ih] e
+            let e' ← mkAppOptM ``fix_induct #[α, instCCPOα, F, hmono, packedMotive, packedAdm, packedH]
+            -- Should be the type of e', but with the function definitions folded
+            let packedConclusion ← PProdN.pack 0 <| ←
+              motives.mapIdxM fun i motive => do
+                let f ← mkConstWithLevelParams infos[i]!.name
+                return mkApp motive (mkAppN f xs)
+            let e' ← mkExpectedTypeHint e' packedConclusion
+            let e' ← mkLambdaFVars hs e'
+            let e' ← mkLambdaFVars adms e'
+            let e' ← mkLambdaFVars motives e'
+            let e' ← mkLambdaFVars (binderInfoForMVars := .default) (usedOnly := true) xs e'
+            let e' ← instantiateMVars e'
+            trace[Elab.definition.partialFixpoint.induction] "complete body of fixpoint induction principle:{indentExpr e'}"
+            pure e'
+
+    let eTyp ← inferType e'
+    let eTyp ← elimOptParam eTyp
+    -- logInfo m!"eTyp: {eTyp}"
+    let params := (collectLevelParams {} eTyp).params
+    -- Prune unused level parameters, preserving the original order
+    let us := infos[0]!.levelParams.filter (params.contains ·)
+
+    let inductName := name ++ `fixpoint_induct
+    addDecl <| Declaration.thmDecl
+      { name := inductName, levelParams := us, type := eTyp, value := e' }
+
+def isInductName (env : Environment) (name : Name) : Bool := Id.run do
+  let .str p s := name | return false
+  match s with
+  | "fixpoint_induct" =>
+    if let some eqnInfo := eqnInfoExt.find? env p then
+      return p == eqnInfo.declNames[0]!
+    return false
+  | _ => return false
+
+builtin_initialize
+  registerReservedNamePredicate isInductName
+
+  registerReservedNameAction fun name => do
+    if isInductName (← getEnv) name then
+      let .str p _ := name | return false
+      MetaM.run' <| deriveInduction p
+      return true
+    return false
+
+/--
+Returns true if `name` defined by `partial_fixpoint`, the first in its mutual group,
+and all functions are defined using the `CCPO` instance for `Option`.
+-/
+def isOptionFixpoint (env : Environment) (name : Name) : Bool := Option.isSome do
+  let eqnInfo ← eqnInfoExt.find? env name
+  guard <| name == eqnInfo.declNames[0]!
+  let defnInfo ← env.find? eqnInfo.declNameNonRec
+  assert! defnInfo.hasValue
+  let mut value := defnInfo.value!
+  while value.isLambda do value := value.bindingBody!
+  let_expr Lean.Order.fix _ inst _ _ := value | panic! s!"isOptionFixpoint: unexpected value {value}"
+  let insts := CCPOProdProjs eqnInfo.declNames.size inst
+  insts.forM fun inst => do
+    let mut inst := inst
+    while inst.isAppOfArity ``instCCPOPi 3 do
+      guard inst.appArg!.isLambda
+      inst := inst.appArg!.bindingBody!
+    guard <| inst.isAppOfArity ``instCCPOOption 1
+
+def isPartialCorrectnessName (env : Environment) (name : Name) : Bool := Id.run do
+  let .str p s := name | return false
+  unless s == "partial_correctness" do return false
+  return isOptionFixpoint env p
+
+/--
+Given `motive : α → β → γ → Prop`, construct a proof of
+`admissible (fun f => ∀ x y r, f x y = r → motive x y r)`
+-/
+def mkOptionAdm (motive : Expr) : MetaM Expr := do
+  let type ← inferType motive
+  forallTelescope type fun ysr _ => do
+    let P := mkAppN motive ysr
+    let ys := ysr.pop
+    let r := ysr.back!
+    let mut inst ← mkAppM ``Option.admissible_eq_some #[P, r]
+    inst ← mkLambdaFVars #[r] inst
+    inst ← mkAppOptM ``admissible_pi #[none, none, none, none, inst]
+    for y in ys.reverse do
+      inst ← mkLambdaFVars #[y] inst
+      inst ← mkAppOptM ``admissible_pi_apply #[none, none, none, none, inst]
+    pure inst
+
+def derivePartialCorrectness (name : Name) : MetaM Unit := do
+  let fixpointInductThm := name ++ `fixpoint_induct
+  unless (← getEnv).contains fixpointInductThm do
+    deriveInduction name
+
+  mapError (f := (m!"Cannot derive partial correctness theorem (please report this issue)\n{indentD ·}")) do
+    let some eqnInfo := eqnInfoExt.find? (← getEnv) name |
+      throwError "{name} is not defined by partial_fixpoint"
+
+    let infos ← eqnInfo.declNames.mapM getConstInfoDefn
+    -- First open up the fixed parameters everywhere
+    let e' ← lambdaBoundedTelescope infos[0]!.value eqnInfo.fixedPrefixSize fun xs _ => do
+      let types ← infos.mapM (instantiateForall ·.type xs)
+
+      -- for `f : α → β → Option γ`, we expect a `motive : α → β → γ → Prop`
+      let motiveTypes ← types.mapM fun type =>
+        forallTelescopeReducing type fun ys type => do
+          let type ← whnf type
+          let_expr Option γ := type | throwError "Expected `Option`, got:{indentExpr type}"
+          withLocalDeclD (← mkFreshUserName `r) γ fun r =>
+            mkForallFVars (ys.push r) (.sort 0)
+      let motiveDecls ← motiveTypes.mapIdxM fun i motiveType => do
+        let n := if infos.size = 1 then .mkSimple "motive"
+                                   else .mkSimple s!"motive_{i+1}"
+        pure (n, fun _ => pure motiveType)
+      withLocalDeclsD motiveDecls fun motives => do
+        -- the motives, as expected by `f.fixpoint_induct`:
+        -- fun f => ∀ x y r, f x y = some r → motive x y r
+        let motives' ← motives.mapIdxM fun i motive => do
+          withLocalDeclD (← mkFreshUserName `f) types[i]! fun f => do
+            forallTelescope (← inferType motive) fun ysr _ => do
+              let ys := ysr.pop
+              let r := ysr.back!
+              let heq ← mkEq (mkAppN f ys) (← mkAppM ``some #[r])
+              let motive' ← mkArrow heq (mkAppN motive ysr)
+              let motive' ← mkForallFVars ysr motive'
+              mkLambdaFVars #[f] motive'
+
+        let e' ← mkAppOptM fixpointInductThm <| (xs ++ motives').map some
+        let adms ← motives.mapM mkOptionAdm
+        let e' := mkAppN e' adms
+        let e' ← mkLambdaFVars motives e'
+        let e' ← mkLambdaFVars (binderInfoForMVars := .default) (usedOnly := true) xs e'
+        let e' ← instantiateMVars e'
+        trace[Elab.definition.partialFixpoint.induction] "complete body of partial correctness principle:{indentExpr e'}"
+        pure e'
+
+    let eTyp ← inferType e'
+    let eTyp ← elimOptParam eTyp
+    let eTyp ← Core.betaReduce eTyp
+    -- logInfo m!"eTyp: {eTyp}"
+    let params := (collectLevelParams {} eTyp).params
+    -- Prune unused level parameters, preserving the original order
+    let us := infos[0]!.levelParams.filter (params.contains ·)
+
+    let inductName := name ++ `partial_correctness
+    addDecl <| Declaration.thmDecl
+      { name := inductName, levelParams := us, type := eTyp, value := e' }
+
+builtin_initialize
+  registerReservedNamePredicate isPartialCorrectnessName
+
+  registerReservedNameAction fun name => do
+    let .str p s := name | return false
+    unless s == "partial_correctness" do return false
+    unless isOptionFixpoint (← getEnv) p do return false
+    MetaM.run' <| derivePartialCorrectness p
+    return false
+
+end Lean.Elab.PartialFixpoint
+
+builtin_initialize Lean.registerTraceClass `Elab.definition.partialFixpoint.induction

src/Lean/Elab/PreDefinition/PartialFixpoint/Main.lean

@@ -0,0 +1,188 @@
+/-
+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.Elab.PreDefinition.MkInhabitant
+import Lean.Elab.PreDefinition.Mutual
+import Lean.Elab.PreDefinition.PartialFixpoint.Eqns
+import Lean.Elab.Tactic.Monotonicity
+import Init.Internal.Order.Basic
+import Lean.Meta.PProdN
+
+namespace Lean.Elab
+
+open Meta
+open Monotonicity
+
+open Lean.Order
+
+private def replaceRecApps (recFnNames : Array Name) (fixedPrefixSize : Nat) (f : Expr) (e : Expr) : MetaM Expr := do
+  let t ← inferType f
+  return e.replace fun e =>
+    if let some idx := recFnNames.findIdx? (e.isAppOfArity · fixedPrefixSize) then
+      some <| PProdN.proj recFnNames.size idx t f
+    else
+      none
+
+/--
+For pretty error messages:
+Takes `F : (fun f => e)`, where `f` is the packed function, and replaces `f` in `e` with the user-visible
+constants, which are added to the environment temporarily.
+-/
+private def unReplaceRecApps {α} (preDefs : Array PreDefinition) (fixedArgs : Array Expr)
+    (F : Expr) (k : Expr → MetaM α) : MetaM α := do
+  unless F.isLambda do throwError "Expected lambda:{indentExpr F}"
+  withoutModifyingEnv do
+    preDefs.forM addAsAxiom
+    let fns := preDefs.map fun d =>
+      mkAppN (.const d.declName (d.levelParams.map mkLevelParam)) fixedArgs
+    let packedFn ← PProdN.mk 0 fns
+    let e ← lambdaBoundedTelescope F 1 fun f e => do
+      let f := f[0]!
+      -- Replace f with calls to the constants
+      let e := e.replace fun e => do if e == f then return packedFn else none
+      -- And reduce projection redexes
+      let e ← PProdN.reduceProjs e
+      pure e
+    k e
+
+def mkInstCCPOPProd (inst₁ inst₂ : Expr) : MetaM Expr := do
+  mkAppOptM ``instCCPOPProd #[none, none, inst₁, inst₂]
+
+def mkMonoPProd (hmono₁ hmono₂ : Expr) : MetaM Expr := do
+  -- mkAppM does not support the equivalent of (cfg := { synthAssignedInstances := false}),
+  -- so this is a bit more pedestrian
+  let_expr monotone _ inst _ inst₁ _ := (← inferType hmono₁)
+    | throwError "mkMonoPProd: unexpected type of{indentExpr hmono₁}"
+  let_expr monotone _ _ _ inst₂ _ := (← inferType hmono₂)
+    | throwError "mkMonoPProd: unexpected type of{indentExpr hmono₂}"
+  mkAppOptM ``PProd.monotone_mk #[none, none, none, inst₁, inst₂, inst, none, none, hmono₁, hmono₂]
+
+def partialFixpoint (preDefs : Array PreDefinition) : TermElabM Unit := do
+  -- We expect all functions in the clique to have `partial_fixpoint` syntax
+  let hints := preDefs.filterMap (·.termination.partialFixpoint?)
+  assert! preDefs.size = hints.size
+  -- For every function of type `∀ x y, r x y`, an CCPO instance
+  -- ∀ x y, CCPO (r x y), but crucially constructed using `instCCPOPi`
+  let ccpoInsts ← preDefs.mapIdxM fun i preDef => withRef hints[i]!.ref do
+    lambdaTelescope preDef.value fun xs _body => do
+      let type ← instantiateForall preDef.type xs
+      let inst ←
+        try
+          synthInstance (← mkAppM ``CCPO #[type])
+        catch _ =>
+          trace[Elab.definition.partialFixpoint] "No CCPO instance found for {preDef.declName}, trying inhabitation"
+          let msg := m!"failed to compile definition '{preDef.declName}' using `partial_fixpoint`"
+          let w ← mkInhabitantFor msg #[] preDef.type
+          let instNonempty ← mkAppM ``Nonempty.intro #[mkAppN w xs]
+          let classicalWitness ← mkAppOptM ``Classical.ofNonempty #[none, instNonempty]
+          mkAppOptM ``FlatOrder.instCCPO #[none, classicalWitness]
+      mkLambdaFVars xs inst
+
+  let fixedPrefixSize ← Mutual.getFixedPrefix preDefs
+  trace[Elab.definition.partialFixpoint] "fixed prefix size: {fixedPrefixSize}"
+
+  let declNames := preDefs.map (·.declName)
+
+  forallBoundedTelescope preDefs[0]!.type fixedPrefixSize fun fixedArgs _ => do
+    -- ∀ x y, CCPO (rᵢ x y)
+    let ccpoInsts := ccpoInsts.map (·.beta fixedArgs)
+    let types ← preDefs.mapM (instantiateForall ·.type fixedArgs)
+
+    -- (∀ x y, r₁ x y) ×' (∀ x y, r₂ x y)
+    let packedType ← PProdN.pack 0 types
+
+    -- CCPO (∀ x y, rᵢ x y)
+    let ccpoInsts' ← ccpoInsts.mapM fun inst =>
+      lambdaTelescope inst fun xs inst => do
+        let mut inst := inst
+        for x in xs.reverse do
+          inst ← mkAppOptM ``instCCPOPi #[(← inferType x), none, (← mkLambdaFVars #[x] inst)]
+        pure inst
+    -- CCPO ((∀ x y, r₁ x y) ×' (∀ x y, r₂ x y))
+    let packedCCPOInst ← PProdN.genMk mkInstCCPOPProd ccpoInsts'
+    -- Order ((∀ x y, r₁ x y) ×' (∀ x y, r₂ x y))
+    let packedPartialOrderInst ← mkAppOptM ``CCPO.toPartialOrder #[none, packedCCPOInst]
+
+    -- Error reporting hook, presenting monotonicity errors in terms of recursive functions
+    let failK {α} f (monoThms : Array Name) : MetaM α := do
+      unReplaceRecApps preDefs fixedArgs f fun t => do
+        let extraMsg := if monoThms.isEmpty then m!"" else
+          m!"Tried to apply {.andList (monoThms.toList.map (m!"'{.ofConstName ·}'"))}, but failed.\n\
+             Possible cause: A missing `{.ofConstName ``MonoBind}` instance.\n\
+             Use `set_option trace.Elab.Tactic.monotonicity true` to debug."
+        if let some recApp := t.find? hasRecAppSyntax then
+          let some syn := getRecAppSyntax? recApp | panic! "getRecAppSyntax? failed"
+          withRef syn <|
+            throwError "Cannot eliminate recursive call `{syn}` enclosed in{indentExpr t}\n{extraMsg}"
+        else
+          throwError "Cannot eliminate recursive call in{indentExpr t}\n{extraMsg}"
+
+    -- Adjust the body of each function to take the other functions as a
+    -- (packed) parameter
+    let Fs ← preDefs.mapM fun preDef => do
+      let body ← instantiateLambda preDef.value fixedArgs
+      withLocalDeclD (← mkFreshUserName `f) packedType fun f => do
+        let body' ← withoutModifyingEnv do
+          -- replaceRecApps needs the constants in the env to typecheck things
+          preDefs.forM (addAsAxiom ·)
+          replaceRecApps declNames fixedPrefixSize f body
+        mkLambdaFVars #[f] body'
+
+    -- Construct and solve monotonicity goals for each function separately
+    -- This way we preserve the user's parameter names as much as possible
+    -- and can (later) use the user-specified per-function tactic
+    let hmonos ← preDefs.mapIdxM fun i preDef => do
+      let type := types[i]!
+      let F := Fs[i]!
+      let inst ← mkAppOptM ``CCPO.toPartialOrder #[type, ccpoInsts'[i]!]
+      let goal ← mkAppOptM ``monotone #[packedType, packedPartialOrderInst, type, inst, F]
+      if let some term := hints[i]!.term? then
+        let hmono ← Term.withSynthesize <| Term.elabTermEnsuringType term goal
+        let hmono ← instantiateMVars hmono
+        let mvars ← getMVars hmono
+        if mvars.isEmpty then
+          pure hmono
+        else
+          discard <| Term.logUnassignedUsingErrorInfos mvars
+          mkSorry goal (synthetic := true)
+      else
+        let hmono ← mkFreshExprSyntheticOpaqueMVar goal
+        mapError (f := (m!"Could not prove '{preDef.declName}' to be monotone in its recursive calls:{indentD ·}")) do
+          solveMono failK hmono.mvarId!
+        trace[Elab.definition.partialFixpoint] "monotonicity proof for {preDef.declName}: {hmono}"
+        instantiateMVars hmono
+    let hmono ← PProdN.genMk mkMonoPProd hmonos
+
+    let packedValue ← mkAppOptM ``fix #[packedType, packedCCPOInst, none, hmono]
+    trace[Elab.definition.partialFixpoint] "packedValue: {packedValue}"
+
+    let declName :=
+      if preDefs.size = 1 then
+        preDefs[0]!.declName
+      else
+        preDefs[0]!.declName ++ `mutual
+    let packedType' ← mkForallFVars fixedArgs packedType
+    let packedValue' ← mkLambdaFVars fixedArgs packedValue
+    let preDefNonRec := { preDefs[0]! with
+      declName := declName
+      type := packedType'
+      value := packedValue'}
+    let preDefsNonrec ← preDefs.mapIdxM fun fidx preDef => do
+      let us := preDefNonRec.levelParams.map mkLevelParam
+      let value := mkConst preDefNonRec.declName us
+      let value := mkAppN value fixedArgs
+      let value := PProdN.proj preDefs.size fidx packedType value
+      let value ← mkLambdaFVars fixedArgs value
+      pure { preDef with value }
+
+    Mutual.addPreDefsFromUnary preDefs preDefsNonrec preDefNonRec
+    let preDefs ← Mutual.cleanPreDefs preDefs
+    PartialFixpoint.registerEqnsInfo preDefs preDefNonRec.declName fixedPrefixSize
+    Mutual.addPreDefAttributes preDefs
+
+end Lean.Elab
+
+builtin_initialize Lean.registerTraceClass `Elab.definition.partialFixpoint

src/Lean/Elab/PreDefinition/Structural/BRecOn.lean

@@ -294,7 +294,7 @@ def mkBrecOnApp (positions : Positions) (fnIdx : Nat) (brecOnConst : Nat → Exp
     let brecOn := mkAppN brecOn packedFArgs
     let some (size, idx) := positions.findSome? fun pos => (pos.size, ·) <$> pos.indexOf? fnIdx
       | throwError "mkBrecOnApp: Could not find {fnIdx} in {positions}"
-    let brecOn ← PProdN.proj size idx brecOn
+    let brecOn ← PProdN.projM size idx brecOn
     mkLambdaFVars ys (mkAppN brecOn otherArgs)
 
 end Lean.Elab.Structural

src/Lean/Elab/PreDefinition/Structural/FindRecArg.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Joachim Breitner
 -/
 prelude
-import Lean.Elab.PreDefinition.TerminationArgument
+import Lean.Elab.PreDefinition.TerminationMeasure
 import Lean.Elab.PreDefinition.Structural.Basic
 import Lean.Elab.PreDefinition.Structural.RecArgInfo
 
@@ -56,7 +56,7 @@ private def hasBadParamDep? (ys : Array Expr) (indParams : Array Expr) : MetaM (
 
 /--
 Assemble the `RecArgInfo` for the `i`th parameter in the parameter list `xs`. This performs
-various sanity checks on the argument (is it even an inductive type etc).
+various sanity checks on the parameter (is it even of inductive type etc).
 -/
 def getRecArgInfo (fnName : Name) (numFixed : Nat) (xs : Array Expr) (i : Nat) : MetaM RecArgInfo := do
   if h : i < xs.size then
@@ -112,17 +112,17 @@ considered.
 
 The `xs` are the fixed parameters, `value` the body with the fixed prefix instantiated.
 
-Takes the optional user annotations into account (`termArg?`). If this is given and the argument
+Takes the optional user annotation into account (`termMeasure?`). If this is given and the measure
 is unsuitable, throw an error.
 -/
 def getRecArgInfos (fnName : Name) (xs : Array Expr) (value : Expr)
-    (termArg? : Option TerminationArgument) : MetaM (Array RecArgInfo × MessageData) := do
+    (termMeasure? : Option TerminationMeasure) : MetaM (Array RecArgInfo × MessageData) := do
   lambdaTelescope value fun ys _ => do
-    if let .some termArg := termArg? then
-      -- User explicitly asked to use a certain argument, so throw errors eagerly
-      let recArgInfo ← withRef termArg.ref do
-        mapError (f := (m!"cannot use specified parameter for structural recursion:{indentD ·}")) do
-          getRecArgInfo fnName xs.size (xs ++ ys) (← termArg.structuralArg)
+    if let .some termMeasure := termMeasure? then
+      -- User explicitly asked to use a certain measure, so throw errors eagerly
+      let recArgInfo ← withRef termMeasure.ref do
+        mapError (f := (m!"cannot use specified measure for structural recursion:{indentD ·}")) do
+          getRecArgInfo fnName xs.size (xs ++ ys) (← termMeasure.structuralArg)
       return (#[recArgInfo], m!"")
     else
       let mut recArgInfos := #[]
@@ -233,12 +233,12 @@ def allCombinations (xss : Array (Array α)) : Option (Array (Array α)) :=
 
 
 def tryAllArgs (fnNames : Array Name) (xs : Array Expr) (values : Array Expr)
-   (termArg?s : Array (Option TerminationArgument)) (k : Array RecArgInfo → M α) : M α := do
+   (termMeasure?s : Array (Option TerminationMeasure)) (k : Array RecArgInfo → M α) : M α := do
   let mut report := m!""
   -- Gather information on all possible recursive arguments
   let mut recArgInfoss := #[]
-  for fnName in fnNames, value in values, termArg? in termArg?s do
-    let (recArgInfos, thisReport) ← getRecArgInfos fnName xs value termArg?
+  for fnName in fnNames, value in values, termMeasure? in termMeasure?s do
+    let (recArgInfos, thisReport) ← getRecArgInfos fnName xs value termMeasure?
     report := report ++ thisReport
     recArgInfoss := recArgInfoss.push recArgInfos
   -- Put non-indices first

src/Lean/Elab/PreDefinition/Structural/Main.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Joachim Breitner
 -/
 prelude
-import Lean.Elab.PreDefinition.TerminationArgument
+import Lean.Elab.PreDefinition.TerminationMeasure
 import Lean.Elab.PreDefinition.Structural.Basic
 import Lean.Elab.PreDefinition.Structural.FindRecArg
 import Lean.Elab.PreDefinition.Structural.Preprocess
@@ -127,7 +127,7 @@ private def elimMutualRecursion (preDefs : Array PreDefinition) (xs : Array Expr
   let valuesNew ← valuesNew.mapM (mkLambdaFVars xs ·)
   return (Array.zip preDefs valuesNew).map fun ⟨preDef, valueNew⟩ => { preDef with value := valueNew }
 
-private def inferRecArgPos (preDefs : Array PreDefinition) (termArg?s : Array (Option TerminationArgument)) :
+private def inferRecArgPos (preDefs : Array PreDefinition) (termMeasure?s : Array (Option TerminationMeasure)) :
     M (Array Nat × (Array PreDefinition) × Nat) := do
   withoutModifyingEnv do
     preDefs.forM (addAsAxiom ·)
@@ -142,7 +142,7 @@ private def inferRecArgPos (preDefs : Array PreDefinition) (termArg?s : Array (O
       assert! xs.size = maxNumFixed
       let values ← preDefs.mapM (instantiateLambda ·.value xs)
 
-      tryAllArgs fnNames xs values termArg?s fun recArgInfos => do
+      tryAllArgs fnNames xs values termMeasure?s fun recArgInfos => do
         let recArgPoss := recArgInfos.map (·.recArgPos)
         trace[Elab.definition.structural] "Trying argument set {recArgPoss}"
         let numFixed := recArgInfos.foldl (·.min ·.numFixed) maxNumFixed
@@ -156,20 +156,20 @@ private def inferRecArgPos (preDefs : Array PreDefinition) (termArg?s : Array (O
           let preDefs' ← elimMutualRecursion preDefs xs recArgInfos
           return (recArgPoss, preDefs', numFixed)
 
-def reportTermArg (preDef : PreDefinition) (recArgPos : Nat) : MetaM Unit := do
+def reporttermMeasure (preDef : PreDefinition) (recArgPos : Nat) : MetaM Unit := do
   if let some ref := preDef.termination.terminationBy?? then
     let fn ← lambdaTelescope preDef.value fun xs _ => mkLambdaFVars xs xs[recArgPos]!
-    let termArg : TerminationArgument:= {ref := .missing, structural := true, fn}
+    let termMeasure : TerminationMeasure:= {ref := .missing, structural := true, fn}
     let arity ← lambdaTelescope preDef.value fun xs _ => pure xs.size
-    let stx ← termArg.delab arity (extraParams := preDef.termination.extraParams)
+    let stx ← termMeasure.delab arity (extraParams := preDef.termination.extraParams)
     Tactic.TryThis.addSuggestion ref stx
 
 
-def structuralRecursion (preDefs : Array PreDefinition) (termArg?s : Array (Option TerminationArgument)) : TermElabM Unit := do
+def structuralRecursion (preDefs : Array PreDefinition) (termMeasure?s : Array (Option TerminationMeasure)) : TermElabM Unit := do
   let names := preDefs.map (·.declName)
-  let ((recArgPoss, preDefsNonRec, numFixed), state) ← run <| inferRecArgPos preDefs termArg?s
+  let ((recArgPoss, preDefsNonRec, numFixed), state) ← run <| inferRecArgPos preDefs termMeasure?s
   for recArgPos in recArgPoss, preDef in preDefs do
-    reportTermArg preDef recArgPos
+    reporttermMeasure preDef recArgPos
   state.addMatchers.forM liftM
   preDefsNonRec.forM fun preDefNonRec => do
     let preDefNonRec ← eraseRecAppSyntax preDefNonRec

src/Lean/Elab/PreDefinition/TerminationHint.lean

@@ -15,7 +15,7 @@ namespace Lean.Elab
 /-- A single `termination_by` clause -/
 structure TerminationBy where
   ref          : Syntax
-  structural : Bool
+  structural   : Bool
   vars         : TSyntaxArray [`ident, ``Lean.Parser.Term.hole]
   body         : Term
   /--
@@ -33,6 +33,12 @@ structure DecreasingBy where
   tactic    : TSyntax ``Lean.Parser.Tactic.tacticSeq
   deriving Inhabited
 
+/-- A single `partial_fixpoint` clause -/
+structure PartialFixpoint where
+  ref       : Syntax
+  term?     : Option Term
+  deriving Inhabited
+
 /--
 The termination annotations for a single function.
 For `decreasing_by`, we store the whole `decreasing_by tacticSeq` expression, as this
@@ -42,12 +48,13 @@ structure TerminationHints where
   ref : Syntax
   terminationBy?? : Option Syntax
   terminationBy? : Option TerminationBy
+  partialFixpoint? : Option PartialFixpoint
   decreasingBy?  : Option DecreasingBy
   /--
   Here we record the number of parameters past the `:`. It is set by
   `TerminationHints.rememberExtraParams` and used as follows:
 
-  * When we guess the termination argument in `GuessLex` and want to print it in surface-syntax
+  * When we guess the termination measure in `GuessLex` and want to print it in surface-syntax
     compatible form.
   * If there are fewer variables in the `termination_by` annotation than there are extra
     parameters, we know which parameters they should apply to (`TerminationBy.checkVars`).
@@ -55,26 +62,29 @@ structure TerminationHints where
   extraParams : Nat
   deriving Inhabited
 
-def TerminationHints.none : TerminationHints := ⟨.missing, .none, .none, .none, 0⟩
+def TerminationHints.none : TerminationHints := ⟨.missing, .none, .none, .none, .none, 0⟩
 
 /-- Logs warnings when the `TerminationHints` are unexpectedly present.  -/
 def TerminationHints.ensureNone (hints : TerminationHints) (reason : String) : CoreM Unit := do
-  match hints.terminationBy??, hints.terminationBy?, hints.decreasingBy? with
-  | .none, .none, .none => pure ()
-  | .none, .none, .some dec_by =>
+  match hints.terminationBy??, hints.terminationBy?, hints.decreasingBy?, hints.partialFixpoint? with
+  | .none, .none, .none, .none => pure ()
+  | .none, .none, .some dec_by, .none =>
     logWarningAt dec_by.ref m!"unused `decreasing_by`, function is {reason}"
-  | .some term_by?, .none, .none =>
+  | .some term_by?, .none, .none, .none =>
     logWarningAt term_by? m!"unused `termination_by?`, function is {reason}"
-  | .none, .some term_by, .none =>
+  | .none, .some term_by, .none, .none =>
     logWarningAt term_by.ref m!"unused `termination_by`, function is {reason}"
-  | _, _, _ =>
+  | .none, .none, .none, .some partialFixpoint =>
+    logWarningAt partialFixpoint.ref m!"unused `partial_fixpoint`, function is {reason}"
+  | _, _, _, _=>
     logWarningAt hints.ref m!"unused termination hints, function is {reason}"
 
 /-- True if any form of termination hint is present. -/
 def TerminationHints.isNotNone (hints : TerminationHints) : Bool :=
   hints.terminationBy??.isSome ||
   hints.terminationBy?.isSome ||
-  hints.decreasingBy?.isSome
+  hints.decreasingBy?.isSome ||
+  hints.partialFixpoint?.isSome
 
 /--
 Remembers `extraParams` for later use. Needs to happen early enough where we still know
@@ -117,6 +127,8 @@ def elabTerminationHints {m} [Monad m] [MonadError m] (stx : TSyntax ``suffix) :
       | _ => pure none
       else pure none
     let terminationBy? : Option TerminationBy ← if let some t := t? then match t with
+      | `(terminationBy|termination_by partialFixpointursion) =>
+        pure (some {ref := t, structural := false, vars := #[], body := ⟨.missing⟩ : TerminationBy})
       | `(terminationBy|termination_by $[structural%$s]? => $_body) =>
         throwErrorAt t "no extra parameters bounds, please omit the `=>`"
       | `(terminationBy|termination_by $[structural%$s]? $vars* => $body) =>
@@ -124,12 +136,17 @@ def elabTerminationHints {m} [Monad m] [MonadError m] (stx : TSyntax ``suffix) :
       | `(terminationBy|termination_by $[structural%$s]? $body:term) =>
         pure (some {ref := t, structural := s.isSome, vars := #[], body})
       | `(terminationBy?|termination_by?) => pure none
+      | `(partialFixpoint|partial_fixpoint $[monotonicity $_]?) => pure none
       | _ => throwErrorAt t "unexpected `termination_by` syntax"
       else pure none
+    let partialFixpoint? : Option PartialFixpoint ← if let some t := t? then match t with
+      | `(partialFixpoint|partial_fixpoint $[monotonicity $term?]?) => pure (some {ref := t, term?})
+      | _ => pure none
+      else pure none
     let decreasingBy? ← d?.mapM fun d => match d with
       | `(decreasingBy|decreasing_by $tactic) => pure {ref := d, tactic}
       | _ => throwErrorAt d "unexpected `decreasing_by` syntax"
-    return { ref := stx, terminationBy??, terminationBy?, decreasingBy?, extraParams := 0 }
+    return { ref := stx, terminationBy??, terminationBy?, partialFixpoint?, decreasingBy?, extraParams := 0 }
   | _ => throwErrorAt stx s!"Unexpected Termination.suffix syntax: {stx} of kind {stx.raw.getKind}"
 
 end Lean.Elab

src/Lean/Elab/PreDefinition/TerminationMeasure.lean

@@ -14,8 +14,8 @@ import Lean.PrettyPrinter.Delaborator.Basic
 
 /-!
 This module contains
-* the data type `TerminationArgument`, the elaborated form of a `TerminationBy` clause,
-* the `TerminationArguments` type for a clique, and
+* the data type `TerminationMeasure`, the elaborated form of a `TerminationBy` clause,
+* the `TerminationMeasures` type for a clique, and
 * elaboration and deelaboration functions.
 -/
 
@@ -29,28 +29,28 @@ open Lean Meta Elab Term
 Elaborated form for a `termination_by` clause.
 
 The `fn` has the same (value) arity as the recursive functions (stored in
-`arity`), and maps its arguments (including fixed prefix, in unpacked form) to
-the termination argument.
+`arity`), and maps its measures (including fixed prefix, in unpacked form) to
+the termination measure.
 
-If `structural := Bool`, then the `fn` is a lambda picking out exactly one argument.
+If `structural := Bool`, then the `fn` is a lambda picking out exactly one measure.
 -/
-structure TerminationArgument where
+structure TerminationMeasure where
   ref : Syntax
   structural : Bool
   fn : Expr
 deriving Inhabited
 
-/-- A complete set of `TerminationArgument`s, as applicable to a single clique.  -/
-abbrev TerminationArguments := Array TerminationArgument
+/-- A complete set of `TerminationMeasure`s, as applicable to a single clique.  -/
+abbrev TerminationMeasures := Array TerminationMeasure
 
 /--
-Elaborates a `TerminationBy` to an `TerminationArgument`.
+Elaborates a `TerminationBy` to an `TerminationMeasure`.
 
 * `type` is the full type of the original recursive function, including fixed prefix.
 * `hint : TerminationBy` is the syntactic `TerminationBy`.
 -/
-def TerminationArgument.elab (funName : Name) (type : Expr) (arity extraParams : Nat)
-    (hint : TerminationBy) : TermElabM TerminationArgument := withDeclName funName do
+def TerminationMeasure.elab (funName : Name) (type : Expr) (arity extraParams : Nat)
+    (hint : TerminationBy) : TermElabM TerminationMeasure := withDeclName funName do
   assert! extraParams ≤ arity
 
   if h : hint.vars.size > extraParams then
@@ -73,7 +73,7 @@ def TerminationArgument.elab (funName : Name) (type : Expr) (arity extraParams :
         -- Structural recursion: The body has to be a single parameter, whose index we return
         if hint.structural then unless (ys ++ xs).contains body do
           let params := MessageData.andList ((ys ++ xs).toList.map (m!"'{·}'"))
-          throwErrorAt hint.ref m!"The termination argument of a structurally recursive " ++
+          throwErrorAt hint.ref m!"The termination measure of a structurally recursive " ++
             m!"function must be one of the parameters {params}, but{indentExpr body}\nisn't " ++
             m!"one of these."
 
@@ -87,24 +87,24 @@ def TerminationArgument.elab (funName : Name) (type : Expr) (arity extraParams :
     | 1 => "one parameter"
     | n => m!"{n} parameters"
 
-def TerminationArgument.structuralArg (termArg : TerminationArgument) : MetaM Nat := do
-  assert! termArg.structural
-  lambdaTelescope termArg.fn fun ys e => do
+def TerminationMeasure.structuralArg (measure : TerminationMeasure) : MetaM Nat := do
+  assert! measure.structural
+  lambdaTelescope measure.fn fun ys e => do
     let .some idx := ys.indexOf? e
-      | panic! "TerminationArgument.structuralArg: body not one of the parameters"
+      | panic! "TerminationMeasure.structuralArg: body not one of the parameters"
     return idx
 
 
 open PrettyPrinter Delaborator SubExpr Parser.Termination Parser.Term in
 /--
-Delaborates a `TerminationArgument` back to a `TerminationHint`, e.g. for `termination_by?`.
+Delaborates a `TerminationMeasure` back to a `TerminationHint`, e.g. for `termination_by?`.
 
 This needs extra information:
 * `arity` is the value arity of the recursive function
-* `extraParams` indicates how many of the functions arguments are bound “after the colon”.
+* `extraParams` indicates how many of the function's parameters are bound “after the colon”.
 -/
-def TerminationArgument.delab (arity : Nat) (extraParams : Nat) (termArg : TerminationArgument) : MetaM (TSyntax ``terminationBy) := do
-  lambdaBoundedTelescope termArg.fn (arity - extraParams) fun _ys e => do
+def TerminationMeasure.delab (arity : Nat) (extraParams : Nat) (measure : TerminationMeasure) : MetaM (TSyntax ``terminationBy) := do
+  lambdaBoundedTelescope measure.fn (arity - extraParams) fun _ys e => do
     pure (← delabCore e (delab := go extraParams #[])).1
   where
     go : Nat → TSyntaxArray `ident → DelabM (TSyntax ``terminationBy)
@@ -119,7 +119,7 @@ def TerminationArgument.delab (arity : Nat) (extraParams : Nat) (termArg : Termi
       -- drop trailing underscores
       let mut vars := vars
       while ! vars.isEmpty && vars.back!.raw.isOfKind ``hole do vars := vars.pop
-      if termArg.structural then
+      if measure.structural then
         if vars.isEmpty then
           `(terminationBy|termination_by structural $stxBody)
         else