fix: lake: report bad imports from a library build (#13340)

This PR fixes a Lake issue where library builds would not produce
informative errors about bad imports (unlike module builds).

.claude/CLAUDE.md

@@ -7,6 +7,11 @@ To build Lean you should use `make -j$(nproc) -C build/release`.
 The build uses `ccache`, and in a sandbox `ccache` may complain about read-only file systems.
 Use `CCACHE_READONLY` and `CCACHE_TEMPDIR` instead of disabling ccache completely.
 
+To rebuild individual modules without a full build, use Lake directly:
+```
+cd src && lake build Init.Prelude
+```
+
 ## Running Tests
 
 See `tests/README.md` for full documentation. Quick reference:
@@ -56,6 +61,11 @@ make -C build/release/stage2 clean-stdlib
 ```
 must be run manually before building.
 
+To rebuild individual stage 2 modules without a full `make stage2`, use Lake directly:
+```
+cd build/release/stage2 && lake build Init.Prelude
+```
+
 ## New features
 
 When asked to implement new features:

.claude/commands/release.md

@@ -157,6 +157,16 @@ Note: `gh pr checks --watch` exits as soon as ALL checks complete (pass or fail)
 fail while others are still running, `--watch` will continue until everything settles, then exit
 with a non-zero code. So a background `--watch` finishing = all checks done; check which failed.
 
+## Mathlib Bump Branches
+
+Mathlib `bump/v4.X.0` branches live on the **fork** `leanprover-community/mathlib4-nightly-testing`,
+NOT on `leanprover-community/mathlib4`.
+
+## Never Force-Update Remote Refs Without Confirmation
+
+Never force-update an existing remote branch or tag via `git push --force` or the GitHub API
+without explicit user confirmation.
+
 ## Error Handling
 
 **CRITICAL**: If something goes wrong or a command fails:

.github/workflows/build-template.yml

@@ -131,7 +131,7 @@ jobs:
           [ -d build ] || mkdir build
           cd build
           # arguments passed to `cmake`
-          OPTIONS=(-DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true)
+          OPTIONS=(-DWFAIL=ON)
           if [[ -n '${{ matrix.release }}' ]]; then
             # this also enables githash embedding into stage 1 library, which prohibits reusing
             # `.olean`s across commits, so we don't do it in the fast non-release CI
@@ -276,10 +276,10 @@ jobs:
       - name: Check rebootstrap
         run: |
           set -e
-          # clean rebuild in case of Makefile changes/Lake does not detect uncommited stage 0
-          # changes yet
+          git config user.email "stage0@lean-fro.org"
+          git config user.name "update-stage0"
           make -C build update-stage0
-          make -C build/stage1 clean-stdlib
+          git commit --allow-empty -m "chore: update-stage0"
           time make -C build -j$NPROC
           time ctest --preset ${{ matrix.CMAKE_PRESET || 'release' }} --test-dir build/stage1 -j$NPROC
         if: matrix.check-rebootstrap

.github/workflows/ci.yml

@@ -143,7 +143,7 @@ jobs:
           CMAKE_MAJOR=$(grep -E "^set\(LEAN_VERSION_MAJOR " src/CMakeLists.txt | grep -oE '[0-9]+')
           CMAKE_MINOR=$(grep -E "^set\(LEAN_VERSION_MINOR " src/CMakeLists.txt | grep -oE '[0-9]+')
           CMAKE_PATCH=$(grep -E "^set\(LEAN_VERSION_PATCH " src/CMakeLists.txt | grep -oE '[0-9]+')
-          CMAKE_IS_RELEASE=$(grep -m 1 -E "^set\(LEAN_VERSION_IS_RELEASE " src/CMakeLists.txt | sed -nE 's/^set\(LEAN_VERSION_IS_RELEASE ([0-9]+)\).*/\1/p')
+          CMAKE_IS_RELEASE=$(grep -m 1 -E "^set\(LEAN_VERSION_IS_RELEASE " src/CMakeLists.txt | grep -oE '[0-9]+' | head -1)
 
           # Expected values from tag parsing
           TAG_MAJOR="${{ steps.set-release.outputs.LEAN_VERSION_MAJOR }}"

.github/workflows/update-stage0.yml

@@ -77,7 +77,7 @@ jobs:
       # sync options with `Linux Lake` to ensure cache reuse
       run: |
         mkdir -p build
-        cmake --preset release -B build -DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true
+        cmake --preset release -B build -DWFAIL=ON
       shell: 'nix develop -c bash -euxo pipefail {0}'
     - if: env.should_update_stage0 == 'yes'
       run: |

.gitignore

@@ -34,3 +34,4 @@ wdErr.txt
 wdIn.txt
 wdOut.txt
 downstream_releases/
+.claude/worktrees/

.gitpod.yml

@@ -6,6 +6,6 @@ vscode:
     - leanprover.lean4
 
 tasks:
-  - name: Release build
-    init: cmake --preset release
+  - name: Build
+    init: cmake --preset dev
     command: make -C build/release -j$(nproc || sysctl -n hw.logicalcpu)

.vscode/tasks.json

@@ -11,6 +11,15 @@
 				"isDefault": true
 			}
 		},
+		{
+			"label": "build stage2",
+			"type": "shell",
+			"command": "make -C build/release stage2 -j$(nproc 2>/dev/null || sysctl -n hw.logicalcpu 2>/dev/null || echo 4)",
+			"problemMatcher": [],
+			"group": {
+				"kind": "build"
+			}
+		},
 		{
 			"label": "build-old",
 			"type": "shell",

CMakeLists.txt

@@ -1,4 +1,6 @@
 cmake_minimum_required(VERSION 3.21)
+include(ExternalProject)
+include(FetchContent)
 
 if(NOT CMAKE_GENERATOR MATCHES "Makefiles")
   message(FATAL_ERROR "Only makefile generators are supported")
@@ -34,7 +36,6 @@ foreach(var ${vars})
   endif()
 endforeach()
 
-include(ExternalProject)
 project(LEAN CXX C)
 
 if(NOT (DEFINED STAGE0_CMAKE_EXECUTABLE_SUFFIX))
@@ -119,17 +120,17 @@ if(NOT CMAKE_SYSTEM_NAME MATCHES "Emscripten")
 endif()
 
 if(USE_MIMALLOC)
-  ExternalProject_Add(
+  FetchContent_Declare(
     mimalloc
-    PREFIX mimalloc
     GIT_REPOSITORY https://github.com/microsoft/mimalloc
     GIT_TAG v2.2.3
-    # just download, we compile it as part of each stage as it is small
-    CONFIGURE_COMMAND ""
-    BUILD_COMMAND ""
-    INSTALL_COMMAND ""
+    # Unnecessarily deep directory structure, but it saves us from a complicated
+    # stage0 update for now. If we ever update the other dependencies like
+    # cadical, it might be worth reorganizing the directory structure.
+    SOURCE_DIR
+    "${CMAKE_BINARY_DIR}/mimalloc/src/mimalloc"
   )
-  list(APPEND EXTRA_DEPENDS mimalloc)
+  FetchContent_MakeAvailable(mimalloc)
 endif()
 
 if(NOT STAGE1_PREV_STAGE)

CMakePresets.json

@@ -8,16 +8,26 @@
   "configurePresets": [
     {
       "name": "release",
-      "displayName": "Default development optimized build config",
+      "displayName": "Release build config",
       "generator": "Unix Makefiles",
       "binaryDir": "${sourceDir}/build/release"
     },
+    {
+      "name": "dev",
+      "displayName": "Default development optimized build config",
+      "cacheVariables": {
+        "STRIP_BINARIES": "OFF"
+      },
+      "generator": "Unix Makefiles",
+      "binaryDir": "${sourceDir}/build/dev"
+    },
     {
       "name": "debug",
       "displayName": "Debug build config",
       "cacheVariables": {
+        "CMAKE_BUILD_TYPE": "Debug",
         "LEAN_EXTRA_CXX_FLAGS": "-DLEAN_DEFAULT_THREAD_STACK_SIZE=16*1024*1024",
-        "CMAKE_BUILD_TYPE": "Debug"
+        "STRIP_BINARIES": "OFF"
       },
       "generator": "Unix Makefiles",
       "binaryDir": "${sourceDir}/build/debug"
@@ -26,7 +36,8 @@
       "name": "reldebug",
       "displayName": "Release with assertions enabled",
       "cacheVariables": {
-        "CMAKE_BUILD_TYPE": "RelWithAssert"
+        "CMAKE_BUILD_TYPE": "RelWithAssert",
+        "STRIP_BINARIES": "OFF"
       },
       "generator": "Unix Makefiles",
       "binaryDir": "${sourceDir}/build/reldebug"
@@ -38,6 +49,7 @@
         "LEAN_EXTRA_CXX_FLAGS": "-fsanitize=address,undefined -DLEAN_DEFAULT_THREAD_STACK_SIZE=16*1024*1024",
         "LEANC_EXTRA_CC_FLAGS": "-fsanitize=address,undefined",
         "LEAN_EXTRA_LINKER_FLAGS": "-fsanitize=address,undefined -fsanitize-link-c++-runtime",
+        "STRIP_BINARIES": "OFF",
         "SMALL_ALLOCATOR": "OFF",
         "USE_MIMALLOC": "OFF",
         "BSYMBOLIC": "OFF",
@@ -58,6 +70,10 @@
       "name": "release",
       "configurePreset": "release"
     },
+    {
+      "name": "dev",
+      "configurePreset": "dev"
+    },
     {
       "name": "debug",
       "configurePreset": "debug"
@@ -81,6 +97,11 @@
       "configurePreset": "release",
       "output": {"outputOnFailure": true, "shortProgress": true}
     },
+    {
+      "name": "dev",
+      "configurePreset": "dev",
+      "output": {"outputOnFailure": true, "shortProgress": true}
+    },
     {
       "name": "debug",
       "configurePreset": "debug",

doc/make/index.md

@@ -30,6 +30,9 @@ cd lean4
 cmake --preset release
 make -C build/release -j$(nproc || sysctl -n hw.logicalcpu)
 ```
+
+For development, `cmake --preset dev` is recommended instead.
+
 You can replace `$(nproc || sysctl -n hw.logicalcpu)` with the desired parallelism amount.
 
 The above commands will compile the Lean library and binaries into the

script/release_checklist.py

@@ -311,16 +311,16 @@ def check_cmake_version(repo_url, branch, version_major, version_minor, github_t
         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)"
+    expected_patterns = [
+        (f"LEAN_VERSION_MAJOR", rf"^set\(LEAN_VERSION_MAJOR\s+{version_major}[\s)]", f"set(LEAN_VERSION_MAJOR {version_major} ...)"),
+        (f"LEAN_VERSION_MINOR", rf"^set\(LEAN_VERSION_MINOR\s+{version_minor}[\s)]", f"set(LEAN_VERSION_MINOR {version_minor} ...)"),
+        (f"LEAN_VERSION_PATCH", rf"^set\(LEAN_VERSION_PATCH\s+0[\s)]", f"set(LEAN_VERSION_PATCH 0 ...)"),
+        (f"LEAN_VERSION_IS_RELEASE", rf"^set\(LEAN_VERSION_IS_RELEASE\s+1[\s)]", 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}")
+    for name, pattern, display in expected_patterns:
+        if not any(re.match(pattern, l.strip()) for l in content.splitlines()):
+            print(f"  ❌ Missing or incorrect line in {cmake_file_path}: {display}")
             return False
 
     print(f"  ✅ CMake version settings are correct in {cmake_file_path}")
@@ -343,11 +343,11 @@ def check_stage0_version(repo_url, branch, version_major, version_minor, github_
     for line in content.splitlines():
         stripped = line.strip()
         if stripped.startswith("set(LEAN_VERSION_MAJOR "):
-            actual = stripped.split()[-1].rstrip(")")
+            actual = stripped.split()[1].rstrip(")")
             if actual != str(version_major):
                 errors.append(f"LEAN_VERSION_MAJOR: expected {version_major}, found {actual}")
         elif stripped.startswith("set(LEAN_VERSION_MINOR "):
-            actual = stripped.split()[-1].rstrip(")")
+            actual = stripped.split()[1].rstrip(")")
             if actual != str(version_minor):
                 errors.append(f"LEAN_VERSION_MINOR: expected {version_minor}, found {actual}")
 

script/release_repos.yml

@@ -28,6 +28,14 @@ repositories:
     branch: main
     dependencies: []
 
+  - name: leansqlite
+    url: https://github.com/leanprover/leansqlite
+    toolchain-tag: true
+    stable-branch: false
+    branch: main
+    dependencies:
+      - plausible
+
   - name: verso
     url: https://github.com/leanprover/verso
     toolchain-tag: true
@@ -100,7 +108,7 @@ repositories:
     toolchain-tag: true
     stable-branch: false
     branch: main
-    dependencies: [lean4-cli, BibtexQuery, mathlib4]
+    dependencies: [lean4-cli, BibtexQuery, mathlib4, leansqlite]
 
   - name: cslib
     url: https://github.com/leanprover/cslib

script/release_steps.py

@@ -481,11 +481,9 @@ def execute_release_steps(repo, version, config):
         run_command("lake update", cwd=repo_path, stream_output=True)
     elif repo_name == "verso":
         # verso has nested Lake projects in test-projects/ that each have their own
-        # lake-manifest.json with a subverso pin. After updating the root manifest via
-        # `lake update`, sync the de-modulized subverso rev into all sub-manifests.
-        # The sub-projects use an old toolchain (v4.21.0) that doesn't support module/prelude
-        # syntax, so they need the de-modulized version (tagged no-modules/<root-rev>).
-        # The "SubVerso version consistency" CI check accepts either the root or de-modulized rev.
+        # lake-manifest.json with a subverso pin and their own lean-toolchain.
+        # After updating the root manifest via `lake update`, sync the de-modulized
+        # subverso rev into all sub-manifests, and update their lean-toolchain files.
         run_command("lake update", cwd=repo_path, stream_output=True)
         print(blue("Syncing de-modulized subverso rev to test-project sub-manifests..."))
         sync_script = (
@@ -498,6 +496,15 @@ def execute_release_steps(repo, version, config):
         )
         run_command(sync_script, cwd=repo_path)
         print(green("Synced de-modulized subverso rev to all test-project sub-manifests"))
+        # Update all lean-toolchain files in test-projects/ to match the root
+        print(blue("Updating lean-toolchain files in test-projects/..."))
+        find_result = run_command("find test-projects -name lean-toolchain", cwd=repo_path)
+        for tc_path in find_result.stdout.strip().splitlines():
+            if tc_path:
+                tc_file = repo_path / tc_path
+                with open(tc_file, "w") as f:
+                    f.write(f"leanprover/lean4:{version}\n")
+                print(green(f"  Updated {tc_path}"))
     elif dependencies:
         run_command(f'perl -pi -e \'s/"v4\\.[0-9]+(\\.[0-9]+)?(-rc[0-9]+)?"/"' + version + '"/g\' lakefile.*', cwd=repo_path)
         run_command("lake update", cwd=repo_path, stream_output=True)
@@ -659,56 +666,61 @@ def execute_release_steps(repo, version, config):
         # Fetch latest changes to ensure we have the most up-to-date nightly-testing branch
         print(blue("Fetching latest changes from origin..."))
         run_command("git fetch origin", cwd=repo_path)
-        
-        try:
-            print(blue("Merging origin/nightly-testing..."))
-            run_command("git merge origin/nightly-testing", cwd=repo_path)
-            print(green("Merge completed successfully"))
-        except subprocess.CalledProcessError:
-            # Merge failed due to conflicts - check which files are conflicted
-            print(blue("Merge conflicts detected, checking which files are affected..."))
-            
-            # Get conflicted files using git status
-            status_result = run_command("git status --porcelain", cwd=repo_path)
-            conflicted_files = []
-            
-            for line in status_result.stdout.splitlines():
-                if len(line) >= 2 and line[:2] in ['UU', 'AA', 'DD', 'AU', 'UA', 'DU', 'UD']:
-                    # Extract filename (skip the first 3 characters which are status codes)
-                    conflicted_files.append(line[3:])
-            
-            # Filter out allowed files
-            allowed_patterns = ['lean-toolchain', 'lake-manifest.json']
-            problematic_files = []
-            
-            for file in conflicted_files:
-                is_allowed = any(pattern in file for pattern in allowed_patterns)
-                if not is_allowed:
-                    problematic_files.append(file)
-            
-            if problematic_files:
-                # There are conflicts in non-allowed files - fail
-                print(red("❌ Merge failed!"))
-                print(red(f"Merging nightly-testing resulted in conflicts in:"))
-                for file in problematic_files:
-                    print(red(f"  - {file}"))
-                print(red("Please resolve these conflicts manually."))
-                return
-            else:
-                # Only allowed files are conflicted - resolve them automatically
-                print(green(f"✅ Only allowed files conflicted: {', '.join(conflicted_files)}"))
-                print(blue("Resolving conflicts automatically..."))
-                
-                # For lean-toolchain and lake-manifest.json, keep our versions
+
+        # Check if nightly-testing branch exists on origin (use local ref after fetch for exact match)
+        nightly_check = run_command("git show-ref --verify --quiet refs/remotes/origin/nightly-testing", cwd=repo_path, check=False)
+        if nightly_check.returncode != 0:
+            print(yellow("No nightly-testing branch found on origin, skipping merge"))
+        else:
+            try:
+                print(blue("Merging origin/nightly-testing..."))
+                run_command("git merge origin/nightly-testing", cwd=repo_path)
+                print(green("Merge completed successfully"))
+            except subprocess.CalledProcessError:
+                # Merge failed due to conflicts - check which files are conflicted
+                print(blue("Merge conflicts detected, checking which files are affected..."))
+
+                # Get conflicted files using git status
+                status_result = run_command("git status --porcelain", cwd=repo_path)
+                conflicted_files = []
+
+                for line in status_result.stdout.splitlines():
+                    if len(line) >= 2 and line[:2] in ['UU', 'AA', 'DD', 'AU', 'UA', 'DU', 'UD']:
+                        # Extract filename (skip the first 3 characters which are status codes)
+                        conflicted_files.append(line[3:])
+
+                # Filter out allowed files
+                allowed_patterns = ['lean-toolchain', 'lake-manifest.json']
+                problematic_files = []
+
                 for file in conflicted_files:
-                    print(blue(f"Keeping our version of {file}"))
-                    run_command(f"git checkout --ours {file}", cwd=repo_path)
-                
-                # Complete the merge
-                run_command("git add .", cwd=repo_path)
-                run_command("git commit --no-edit", cwd=repo_path)
-                
-                print(green("✅ Merge completed successfully with automatic conflict resolution"))
+                    is_allowed = any(pattern in file for pattern in allowed_patterns)
+                    if not is_allowed:
+                        problematic_files.append(file)
+
+                if problematic_files:
+                    # There are conflicts in non-allowed files - fail
+                    print(red("❌ Merge failed!"))
+                    print(red(f"Merging nightly-testing resulted in conflicts in:"))
+                    for file in problematic_files:
+                        print(red(f"  - {file}"))
+                    print(red("Please resolve these conflicts manually."))
+                    return
+                else:
+                    # Only allowed files are conflicted - resolve them automatically
+                    print(green(f"✅ Only allowed files conflicted: {', '.join(conflicted_files)}"))
+                    print(blue("Resolving conflicts automatically..."))
+
+                    # For lean-toolchain and lake-manifest.json, keep our versions
+                    for file in conflicted_files:
+                        print(blue(f"Keeping our version of {file}"))
+                        run_command(f"git checkout --ours {file}", cwd=repo_path)
+
+                    # Complete the merge
+                    run_command("git add .", cwd=repo_path)
+                    run_command("git commit --no-edit", cwd=repo_path)
+
+                    print(green("✅ Merge completed successfully with automatic conflict resolution"))
 
     # Build and test (skip for Mathlib)
     if repo_name not in ["mathlib4"]:

src/CMakeLists.txt

@@ -8,7 +8,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4 CACHE STRING "")
-set(LEAN_VERSION_MINOR 30 CACHE STRING "")
+set(LEAN_VERSION_MINOR 31 CACHE STRING "")
 set(LEAN_VERSION_PATCH 0 CACHE STRING "")
 set(LEAN_VERSION_IS_RELEASE 0 CACHE STRING "") # This number is 1 in the release revision, and 0 otherwise.
 set(LEAN_SPECIAL_VERSION_DESC "" CACHE STRING "Additional version description like 'nightly-2018-03-11'")
@@ -80,6 +80,7 @@ option(CCACHE "use ccache" ON)
 option(SPLIT_STACK "SPLIT_STACK" OFF)
 # When OFF we disable LLVM support
 option(LLVM "LLVM" OFF)
+option(STRIP_BINARIES "Strip produced binaries" ON)
 
 # When ON we include githash in the version string
 option(USE_GITHASH "GIT_HASH" ON)
@@ -115,11 +116,19 @@ option(CHECK_OLEAN_VERSION "Only load .olean files compiled with the current ver
 option(USE_LAKE "Use Lake instead of lean.mk for building core libs from language server" ON)
 option(USE_LAKE_CACHE "Use the Lake artifact cache for stage 1 builds (requires USE_LAKE)" OFF)
 
-set(LEAN_EXTRA_MAKE_OPTS "" CACHE STRING "extra options to lean --make")
+set(LEAN_EXTRA_OPTS "" CACHE STRING "extra options to lean (via lake or make)")
+set(LEAN_EXTRA_MAKE_OPTS "" CACHE STRING "extra options to leanmake")
 set(LEANC_CC ${CMAKE_C_COMPILER} CACHE STRING "C compiler to use in `leanc`")
 
 # Temporary, core-only flags. Must be synced with stdlib_flags.h.
-string(APPEND LEAN_EXTRA_MAKE_OPTS " -Dbackward.do.legacy=false")
+string(APPEND LEAN_EXTRA_OPTS " -Dbackward.do.legacy=false")
+
+# option used by the CI to fail on warnings
+option(WFAIL "Fail build if warnings are emitted by Lean" ON)
+if(WFAIL MATCHES "ON")
+  string(APPEND LAKE_EXTRA_ARGS " --wfail")
+  string(APPEND LEAN_EXTRA_MAKE_OPTS " -DwarningAsError=true")
+endif()
 
 if(LAZY_RC MATCHES "ON")
   set(LEAN_LAZY_RC "#define LEAN_LAZY_RC")
@@ -197,7 +206,7 @@ set(CMAKE_ARCHIVE_OUTPUT_DIRECTORY "${CMAKE_BINARY_DIR}/lib/lean")
 
 # OSX default thread stack size is very small. Moreover, in Debug mode, each new stack frame consumes a lot of extra memory.
 if((MULTI_THREAD MATCHES "ON") AND (CMAKE_SYSTEM_NAME MATCHES "Darwin"))
-  string(APPEND LEAN_EXTRA_MAKE_OPTS " -s40000")
+  string(APPEND LEAN_EXTRA_OPTS " -s40000")
 endif()
 
 # We want explicit stack probes in huge Lean stack frames for robust stack overflow detection
@@ -614,6 +623,38 @@ else()
       OUTPUT_VARIABLE GIT_SHA1
       OUTPUT_STRIP_TRAILING_WHITESPACE
     )
+    # Fallback for jj workspaces where git cannot find .git directly.
+    # Use `jj git root` to find the backing git repo, then `jj log` to
+    # resolve the current workspace's commit (git HEAD points to the root
+    # workspace, not the current one).
+    if("${GIT_SHA1}" STREQUAL "")
+      find_program(JJ_EXECUTABLE jj)
+      if(JJ_EXECUTABLE)
+        execute_process(
+          COMMAND "${JJ_EXECUTABLE}" git root
+          WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}"
+          OUTPUT_VARIABLE _jj_git_dir
+          OUTPUT_STRIP_TRAILING_WHITESPACE
+          ERROR_QUIET
+          RESULT_VARIABLE _jj_git_root_result
+        )
+        execute_process(
+          COMMAND "${JJ_EXECUTABLE}" log -r @ --no-graph -T "commit_id"
+          WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}"
+          OUTPUT_VARIABLE _jj_commit
+          OUTPUT_STRIP_TRAILING_WHITESPACE
+          ERROR_QUIET
+          RESULT_VARIABLE _jj_rev_result
+        )
+        if(_jj_git_root_result EQUAL 0 AND _jj_rev_result EQUAL 0)
+          execute_process(
+            COMMAND git --git-dir "${_jj_git_dir}" ls-tree "${_jj_commit}" stage0 --object-only
+            OUTPUT_VARIABLE GIT_SHA1
+            OUTPUT_STRIP_TRAILING_WHITESPACE
+          )
+        endif()
+      endif()
+    endif()
     message(STATUS "stage0 sha1: ${GIT_SHA1}")
     # Now that we've prepared the information for the next stage, we can forget that we will use
     # Lake in the future as we won't use it in this stage
@@ -637,6 +678,9 @@ else()
   set(LEAN_PATH_SEPARATOR ":")
 endif()
 
+# inherit genral options for lean.mk.in and stdlib.make.in
+string(APPEND LEAN_EXTRA_MAKE_OPTS " ${LEAN_EXTRA_OPTS}")
+
 # Version
 configure_file("${LEAN_SOURCE_DIR}/version.h.in" "${LEAN_BINARY_DIR}/include/lean/version.h")
 if(STAGE EQUAL 0)
@@ -797,7 +841,14 @@ if(LLVM AND STAGE GREATER 0)
   set(EXTRA_LEANMAKE_OPTS "LLVM=1")
 endif()
 
-set(STDLIBS Init Std Lean Leanc LeanIR)
+set(
+  STDLIBS
+  Init
+  Std
+  Lean
+  Leanc
+  LeanIR
+)
 if(NOT CMAKE_SYSTEM_NAME MATCHES "Emscripten")
   list(APPEND STDLIBS Lake LeanChecker)
 endif()
@@ -905,10 +956,7 @@ if(PREV_STAGE)
 endif()
 
 if(NOT CMAKE_SYSTEM_NAME MATCHES "Emscripten")
-  add_custom_target(leanir ALL
-    DEPENDS leanshared
-    COMMAND $(MAKE) -f ${CMAKE_BINARY_DIR}/stdlib.make leanir
-    VERBATIM)
+  add_custom_target(leanir ALL DEPENDS leanshared COMMAND $(MAKE) -f ${CMAKE_BINARY_DIR}/stdlib.make leanir VERBATIM)
 endif()
 
 # use Bash version for building, use Lean version in bin/ for tests & distribution
@@ -1017,7 +1065,7 @@ string(REPLACE "ROOT" "${CMAKE_BINARY_DIR}" LEANC_CC "${LEANC_CC}")
 string(REPLACE "ROOT" "${CMAKE_BINARY_DIR}" LEANC_INTERNAL_FLAGS "${LEANC_INTERNAL_FLAGS}")
 string(REPLACE "ROOT" "${CMAKE_BINARY_DIR}" LEANC_INTERNAL_LINKER_FLAGS "${LEANC_INTERNAL_LINKER_FLAGS}")
 
-toml_escape("${LEAN_EXTRA_MAKE_OPTS}" LEAN_EXTRA_OPTS_TOML)
+toml_escape("${LEAN_EXTRA_OPTS}" LEAN_EXTRA_OPTS_TOML)
 
 if(CMAKE_BUILD_TYPE MATCHES "Debug|Release|RelWithDebInfo|MinSizeRel")
   set(CMAKE_BUILD_TYPE_TOML "${CMAKE_BUILD_TYPE}")

src/Init/Data/Array/Basic.lean

@@ -1085,6 +1085,17 @@ Examples:
 def sum {α} [Add α] [Zero α] : Array α → α :=
   foldr (· + ·) 0
 
+/--
+Computes the product of the elements of an array.
+
+Examples:
+ * `#[a, b, c].prod = a * (b * (c * 1))`
+ * `#[1, 2, 5].prod = 10`
+-/
+@[inline, expose]
+def prod {α} [Mul α] [One α] : Array α → α :=
+  foldr (· * ·) 1
+
 /--
 Counts the number of elements in the array `as` that satisfy the Boolean predicate `p`.
 

src/Init/Data/Array/Int.lean

@@ -7,6 +7,7 @@ module
 
 prelude
 public import Init.Data.List.Int.Sum
+public import Init.Data.List.Int.Prod
 public import Init.Data.Array.MinMax
 import Init.Data.Int.Lemmas
 
@@ -74,4 +75,17 @@ theorem sum_div_length_le_max_of_max?_eq_some_int {xs : Array Int} (h : xs.max?
   simpa [List.max?_toArray, List.sum_toArray] using
     List.sum_div_length_le_max_of_max?_eq_some_int (by simpa using h)
 
+@[simp] theorem prod_replicate_int {n : Nat} {a : Int} : (replicate n a).prod = a ^ n := by
+  rw [← List.toArray_replicate, List.prod_toArray]
+  simp
+
+theorem prod_append_int {as₁ as₂ : Array Int} : (as₁ ++ as₂).prod = as₁.prod * as₂.prod := by
+  simp [prod_append]
+
+theorem prod_reverse_int (xs : Array Int) : xs.reverse.prod = xs.prod := by
+  simp [prod_reverse]
+
+theorem prod_eq_foldl_int {xs : Array Int} : xs.prod = xs.foldl (init := 1) (· * ·) := by
+  simp only [foldl_eq_foldr_reverse, Int.mul_comm, ← prod_eq_foldr, prod_reverse_int]
+
 end Array

src/Init/Data/Array/Lemmas.lean

@@ -4380,6 +4380,47 @@ theorem sum_eq_foldl [Zero α] [Add α] [Std.Associative (α := α) (· + ·)]
     xs.sum = xs.foldl (init := 0) (· + ·) := by
   simp [← sum_toList, List.sum_eq_foldl]
 
+/-! ### prod -/
+
+@[simp, grind =] theorem prod_empty [Mul α] [One α] : (#[] : Array α).prod = 1 := rfl
+theorem prod_eq_foldr [Mul α] [One α] {xs : Array α} :
+    xs.prod = xs.foldr (init := 1) (· * ·) :=
+  rfl
+
+@[simp, grind =]
+theorem prod_toList [Mul α] [One α] {as : Array α} : as.toList.prod = as.prod := by
+  cases as
+  simp [Array.prod, List.prod]
+
+@[simp, grind =]
+theorem prod_append [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.LawfulLeftIdentity (α := α) (· * ·) 1]
+    {as₁ as₂ : Array α} : (as₁ ++ as₂).prod = as₁.prod * as₂.prod := by
+  simp [← prod_toList, List.prod_append]
+
+@[simp, grind =]
+theorem prod_singleton [Mul α] [One α] [Std.LawfulRightIdentity (· * ·) (1 : α)] {x : α} :
+    #[x].prod = x := by
+  simp [Array.prod_eq_foldr, Std.LawfulRightIdentity.right_id x]
+
+@[simp, grind =]
+theorem prod_push [Mul α] [One α] [Std.Associative (α := α) (· * ·)]
+    [Std.LawfulIdentity (· * ·) (1 : α)] {xs : Array α} {x : α} :
+    (xs.push x).prod = xs.prod * x := by
+  simp [Array.prod_eq_foldr, Std.LawfulRightIdentity.right_id, Std.LawfulLeftIdentity.left_id,
+    ← Array.foldr_assoc]
+
+@[simp, grind =]
+theorem prod_reverse [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.Commutative (α := α) (· * ·)]
+    [Std.LawfulLeftIdentity (α := α) (· * ·) 1] (xs : Array α) : xs.reverse.prod = xs.prod := by
+  simp [← prod_toList, List.prod_reverse]
+
+theorem prod_eq_foldl [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.LawfulIdentity (· * ·) (1 : α)] {xs : Array α} :
+    xs.prod = xs.foldl (init := 1) (· * ·) := by
+  simp [← prod_toList, List.prod_eq_foldl]
+
 theorem foldl_toList_eq_flatMap {l : List α} {acc : Array β}
     {F : Array β → α → Array β} {G : α → List β}
     (H : ∀ acc a, (F acc a).toList = acc.toList ++ G a) :

src/Init/Data/Array/MinMax.lean

@@ -113,7 +113,7 @@ public theorem _root_.List.min?_toArray [Min α] {l : List α} :
   · simp [List.min_toArray, List.min_eq_get_min?, - List.get_min?]
   · simp_all
 
-@[simp, grind =]
+@[simp, grind =, cbv_eval ←]
 public theorem min?_toList [Min α] {xs : Array α} :
     xs.toList.min? = xs.min? := by
   cases xs; simp
@@ -153,7 +153,7 @@ public theorem _root_.List.max?_toArray [Max α] {l : List α} :
   · simp [List.max_toArray, List.max_eq_get_max?, - List.get_max?]
   · simp_all
 
-@[simp, grind =]
+@[simp, grind =, cbv_eval ←]
 public theorem max?_toList [Max α] {xs : Array α} :
     xs.toList.max? = xs.max? := by
   cases xs; simp

src/Init/Data/Array/Nat.lean

@@ -8,6 +8,7 @@ module
 prelude
 public import Init.Data.Array.MinMax
 import Init.Data.List.Nat.Sum
+import Init.Data.List.Nat.Prod
 import Init.Data.Array.Lemmas
 
 public section
@@ -81,4 +82,24 @@ theorem sum_div_length_le_max_of_max?_eq_some_nat {xs : Array Nat} (h : xs.max?
   simpa [List.max?_toArray, List.sum_toArray] using
     List.sum_div_length_le_max_of_max?_eq_some_nat (by simpa using h)
 
+protected theorem prod_pos_iff_forall_pos_nat {xs : Array Nat} : 0 < xs.prod ↔ ∀ x ∈ xs, 0 < x := by
+  simp [← prod_toList, List.prod_pos_iff_forall_pos_nat]
+
+protected theorem prod_eq_zero_iff_exists_zero_nat {xs : Array Nat} :
+    xs.prod = 0 ↔ ∃ x ∈ xs, x = 0 := by
+  simp [← prod_toList, List.prod_eq_zero_iff_exists_zero_nat]
+
+@[simp] theorem prod_replicate_nat {n : Nat} {a : Nat} : (replicate n a).prod = a ^ n := by
+  rw [← List.toArray_replicate, List.prod_toArray]
+  simp
+
+theorem prod_append_nat {as₁ as₂ : Array Nat} : (as₁ ++ as₂).prod = as₁.prod * as₂.prod := by
+  simp [prod_append]
+
+theorem prod_reverse_nat (xs : Array Nat) : xs.reverse.prod = xs.prod := by
+  simp [prod_reverse]
+
+theorem prod_eq_foldl_nat {xs : Array Nat} : xs.prod = xs.foldl (init := 1) (· * ·) := by
+  simp only [foldl_eq_foldr_reverse, Nat.mul_comm, ← prod_eq_foldr, prod_reverse_nat]
+
 end Array

src/Init/Data/ByteArray/Basic.lean

@@ -20,12 +20,20 @@ universe u
 
 namespace ByteArray
 
-deriving instance BEq for ByteArray
+@[extern "lean_sarray_dec_eq"]
+def beq (lhs rhs : @& ByteArray) : Bool :=
+  lhs.data == rhs.data
+
+instance : BEq ByteArray where
+  beq := beq
 
 attribute [ext] ByteArray
 
-instance : DecidableEq ByteArray :=
-  fun _ _ => decidable_of_decidable_of_iff ByteArray.ext_iff.symm
+@[extern "lean_sarray_dec_eq"]
+def decEq (lhs rhs : @& ByteArray) : Decidable (lhs = rhs) :=
+  decidable_of_decidable_of_iff ByteArray.ext_iff.symm
+
+instance : DecidableEq ByteArray := decEq
 
 instance : Inhabited ByteArray where
   default := empty

src/Init/Data/Fin/Lemmas.lean

@@ -527,6 +527,14 @@ theorem castLE_of_eq {m n : Nat} (h : m = n) {h' : m ≤ n} : castLE h' = Fin.ca
 
 @[simp, grind =] theorem val_castAdd (m : Nat) (i : Fin n) : (castAdd m i : Nat) = i := rfl
 
+/-
+**Note**
+The current pattern inference heuristic includes the implicit term `n + m` as pattern of the pattern,
+but arithmetic is problematic in patterns because it is an interpreted symbol. For example,
+we will fail to match `@val n (castNat 0 i)`. Thus, we mark the implicit subterm with `no_index`
+-/
+grind_pattern val_castAdd => @val (no_index _) (castAdd m i)
+
 @[deprecated val_castAdd (since := "2025-11-21")]
 theorem coe_castAdd (m : Nat) (i : Fin n) : (castAdd m i : Nat) = i := rfl
 
@@ -637,7 +645,15 @@ theorem exists_castSucc_eq {n : Nat} {i : Fin (n + 1)} : (∃ j, castSucc j = i)
 
 theorem succ_castSucc {n : Nat} (i : Fin n) : i.castSucc.succ = i.succ.castSucc := rfl
 
-@[simp, grind =] theorem val_addNat (m : Nat) (i : Fin n) : (addNat i m : Nat) = i + m := rfl
+@[simp] theorem val_addNat (m : Nat) (i : Fin n) : (addNat i m : Nat) = i + m := rfl
+
+/-
+**Note**
+The current pattern inference heuristic includes the implicit term `n + m` as pattern of the pattern,
+but arithmetic is problematic in patterns because it is an interpreted symbol. For example,
+we will fail to match `@val n (addNat i 0)`. Thus, we mark the implicit subterm with `no_index`
+-/
+grind_pattern val_addNat => @val (no_index _) (addNat i m)
 
 @[deprecated val_addNat (since := "2025-11-21")]
 theorem coe_addNat (m : Nat) (i : Fin n) : (addNat i m : Nat) = i + m := rfl

src/Init/Data/Iterators/Consumers/Collect.lean

@@ -66,7 +66,7 @@ lists are prepend-only, this `toListRev` is usually more efficient that `toList`
 If the iterator is not finite, this function might run forever. The variant
 `it.ensureTermination.toListRev` always terminates after finitely many steps.
 -/
-@[always_inline, inline]
+@[always_inline, inline, cbv_opaque]
 def Iter.toListRev {α : Type w} {β : Type w}
     [Iterator α Id β] (it : Iter (α := α) β) : List β :=
   it.toIterM.toListRev.run

src/Init/Data/Iterators/Consumers/Loop.lean

@@ -226,7 +226,7 @@ any element emitted by the iterator {name}`it`.
 {lit}`O(|xs|)`. Short-circuits upon encountering the first match. The elements in {name}`it` are
 examined in order of iteration.
 -/
-@[inline]
+@[inline, cbv_opaque]
 def Iter.any {α β : Type w}
     [Iterator α Id β] [IteratorLoop α Id Id]
     (p : β → Bool) (it : Iter (α := α) β) : Bool :=
@@ -292,7 +292,7 @@ all element emitted by the iterator {name}`it`.
 {lit}`O(|xs|)`. Short-circuits upon encountering the first match. The elements in {name}`it` are
 examined in order of iteration.
 -/
-@[inline]
+@[inline, cbv_opaque]
 def Iter.all {α β : Type w}
     [Iterator α Id β] [IteratorLoop α Id Id]
     (p : β → Bool) (it : Iter (α := α) β) : Bool :=
@@ -644,7 +644,7 @@ Examples:
 * `[7, 6].iter.first? = some 7`
 * `[].iter.first? = none`
 -/
-@[inline]
+@[inline, cbv_opaque]
 def Iter.first? {α β : Type w} [Iterator α Id β] [IteratorLoop α Id Id]
     (it : Iter (α := α) β) : Option β :=
   it.toIterM.first?.run

src/Init/Data/Iterators/Lemmas/Combinators/Monadic/FilterMap.lean

@@ -271,7 +271,7 @@ private def optionPelim' {α : Type u_1} (t : Option α) {β :  Sort u_2}
 
 /--
 Inserts an `Option` case distinction after the first computation of a call to `MonadAttach.pbind`.
-This lemma is useful for simplifying the second computation, which often involes `match` expressions
+This lemma is useful for simplifying the second computation, which often involves `match` expressions
 that use `pbind`'s proof term.
 -/
 private theorem pbind_eq_pbind_if_isSome [Monad m] [MonadAttach m] (x : m (Option α)) (f : (_ : _) → _ → m β) :

src/Init/Data/Iterators/Lemmas/Consumers/Collect.lean

@@ -110,6 +110,7 @@ theorem Iter.reverse_toListRev_ensureTermination [Iterator α Id β] [Finite α
     it.ensureTermination.toListRev.reverse = it.toList := by
   simp
 
+@[cbv_eval]
 theorem Iter.toListRev_eq {α β} [Iterator α Id β] [Finite α Id]
     {it : Iter (α := α) β} :
     it.toListRev = it.toList.reverse := by

src/Init/Data/Iterators/Lemmas/Consumers/Loop.lean

@@ -637,6 +637,7 @@ theorem Iter.any_eq_forIn {α β : Type w} [Iterator α Id β]
           return .yield false)).run := by
   simp [any_eq_anyM, anyM_eq_forIn]
 
+@[cbv_eval ←]
 theorem Iter.any_toList {α β : Type w} [Iterator α Id β]
     [Finite α Id] [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
     {it : Iter (α := α) β} {p : β → Bool} :
@@ -727,6 +728,7 @@ theorem Iter.all_eq_forIn {α β : Type w} [Iterator α Id β]
           return .done false)).run := by
   simp [all_eq_allM, allM_eq_forIn]
 
+@[cbv_eval ←]
 theorem Iter.all_toList {α β : Type w} [Iterator α Id β]
     [Finite α Id] [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
     {it : Iter (α := α) β} {p : β → Bool} :
@@ -954,7 +956,7 @@ theorem Iter.first?_eq_match_step {α β : Type w} [Iterator α Id β] [Iterator
   generalize it.toIterM.step.run.inflate = s
   rcases s with ⟨_|_|_, _⟩ <;> simp [Iter.first?_eq_first?_toIterM]
 
-@[simp, grind =]
+@[simp, grind =, cbv_eval ←]
 theorem Iter.head?_toList {α β : Type w} [Iterator α Id β] [IteratorLoop α Id Id]
     [Finite α Id] [LawfulIteratorLoop α Id Id] {it : Iter (α := α) β} :
     it.toList.head? = it.first? := by

src/Init/Data/List/Basic.lean

@@ -2056,6 +2056,20 @@ def sum {α} [Add α] [Zero α] : List α → α :=
 @[simp, grind =] theorem sum_cons [Add α] [Zero α] {a : α} {l : List α} : (a::l).sum = a + l.sum := rfl
 theorem sum_eq_foldr [Add α] [Zero α] {l : List α} : l.sum = l.foldr (· + ·) 0 := rfl
 
+/--
+Computes the product of the elements of a list.
+
+Examples:
+ * `[a, b, c].prod = a * (b * (c * 1))`
+ * `[1, 2, 5].prod = 10`
+-/
+def prod {α} [Mul α] [One α] : List α → α :=
+  foldr (· * ·) 1
+
+@[simp, grind =] theorem prod_nil [Mul α] [One α] : ([] : List α).prod = 1 := rfl
+@[simp, grind =] theorem prod_cons [Mul α] [One α] {a : α} {l : List α} : (a::l).prod = a * l.prod := rfl
+theorem prod_eq_foldr [Mul α] [One α] {l : List α} : l.prod = l.foldr (· * ·) 1 := rfl
+
 /-! ### range -/
 
 /--

src/Init/Data/List/Int.lean

@@ -7,3 +7,4 @@ module
 
 prelude
 public import Init.Data.List.Int.Sum
+public import Init.Data.List.Int.Prod

src/Init/Data/List/Int/Prod.lean

@@ -0,0 +1,31 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+module
+
+prelude
+import Init.Data.List.Lemmas
+import Init.Data.Int.Lemmas
+public import Init.Data.Int.Pow
+public import Init.Data.List.Basic
+
+public section
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
+namespace List
+
+@[simp]
+theorem prod_replicate_int {n : Nat} {a : Int} : (replicate n a).prod = a ^ n := by
+  induction n <;> simp_all [replicate_succ, Int.pow_succ, Int.mul_comm]
+
+theorem prod_append_int {l₁ l₂ : List Int} : (l₁ ++ l₂).prod = l₁.prod * l₂.prod := by
+  simp [prod_append]
+
+theorem prod_reverse_int (xs : List Int) : xs.reverse.prod = xs.prod := by
+  simp [prod_reverse]
+
+end List

src/Init/Data/List/Lemmas.lean

@@ -1878,6 +1878,24 @@ theorem sum_reverse [Zero α] [Add α] [Std.Associative (α := α) (· + ·)]
     simp_all [sum_append, Std.Commutative.comm (α := α) _ 0,
       Std.LawfulLeftIdentity.left_id, Std.Commutative.comm]
 
+@[simp, grind =]
+theorem prod_append [Mul α] [One α] [Std.LawfulLeftIdentity (α := α) (· * ·) 1]
+    [Std.Associative (α := α) (· * ·)] {l₁ l₂ : List α} : (l₁ ++ l₂).prod = l₁.prod * l₂.prod := by
+  induction l₁ generalizing l₂ <;> simp_all [Std.Associative.assoc, Std.LawfulLeftIdentity.left_id]
+
+@[simp, grind =]
+theorem prod_singleton [Mul α] [One α] [Std.LawfulRightIdentity (· * ·) (1 : α)] {x : α} :
+    [x].prod = x := by
+  simp [List.prod_eq_foldr, Std.LawfulRightIdentity.right_id x]
+
+@[simp, grind =]
+theorem prod_reverse [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.Commutative (α := α) (· * ·)]
+    [Std.LawfulLeftIdentity (α := α) (· * ·) 1] (xs : List α) : xs.reverse.prod = xs.prod := by
+  induction xs <;>
+    simp_all [prod_append, Std.Commutative.comm (α := α) _ 1,
+      Std.LawfulLeftIdentity.left_id, Std.Commutative.comm]
+
 /-! ### concat
 
 Note that `concat_eq_append` is a `@[simp]` lemma, so `concat` should usually not appear in goals.
@@ -2784,6 +2802,11 @@ theorem sum_eq_foldl [Zero α] [Add α] [Std.Associative (α := α) (· + ·)]
     xs.sum = xs.foldl (init := 0) (· + ·) := by
   simp [sum_eq_foldr, foldl_eq_apply_foldr, Std.LawfulLeftIdentity.left_id]
 
+theorem prod_eq_foldl [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.LawfulIdentity (· * ·) (1 : α)] {xs : List α} :
+    xs.prod = xs.foldl (init := 1) (· * ·) := by
+  simp [prod_eq_foldr, foldl_eq_apply_foldr, Std.LawfulLeftIdentity.left_id]
+
 -- The argument `f : α₁ → α₂` is intentionally explicit, as it is sometimes not found by unification.
 theorem foldl_hom (f : α₁ → α₂) {g₁ : α₁ → β → α₁} {g₂ : α₂ → β → α₂} {l : List β} {init : α₁}
     (H : ∀ x y, g₂ (f x) y = f (g₁ x y)) : l.foldl g₂ (f init) = f (l.foldl g₁ init) := by

src/Init/Data/List/Nat.lean

@@ -13,6 +13,7 @@ public import Init.Data.List.Nat.Sublist
 public import Init.Data.List.Nat.TakeDrop
 public import Init.Data.List.Nat.Count
 public import Init.Data.List.Nat.Sum
+public import Init.Data.List.Nat.Prod
 public import Init.Data.List.Nat.Erase
 public import Init.Data.List.Nat.Find
 public import Init.Data.List.Nat.BEq

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

@@ -0,0 +1,50 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+module
+
+prelude
+import Init.Data.List.Lemmas
+public import Init.BinderPredicates
+public import Init.NotationExtra
+import Init.Data.Nat.Lemmas
+
+public section
+
+set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
+set_option linter.indexVariables true -- Enforce naming conventions for index variables.
+
+namespace List
+
+protected theorem prod_eq_zero_iff_exists_zero_nat {l : List Nat} : l.prod = 0 ↔ ∃ x ∈ l, x = 0 := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp [Nat.mul_eq_zero, ih, eq_comm (a := (0 : Nat))]
+
+protected theorem prod_pos_iff_forall_pos_nat {l : List Nat} : 0 < l.prod ↔ ∀ x ∈ l, 0 < x := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [prod_cons, mem_cons, forall_eq_or_imp, ← ih]
+    constructor
+    · intro h
+      exact ⟨Nat.pos_of_mul_pos_right h, Nat.pos_of_mul_pos_left h⟩
+    · exact fun ⟨hx, hxs⟩ => Nat.mul_pos hx hxs
+
+@[simp]
+theorem prod_replicate_nat {n : Nat} {a : Nat} : (replicate n a).prod = a ^ n := by
+  induction n <;> simp_all [replicate_succ, Nat.pow_succ, Nat.mul_comm]
+
+theorem prod_append_nat {l₁ l₂ : List Nat} : (l₁ ++ l₂).prod = l₁.prod * l₂.prod := by
+  simp [prod_append]
+
+theorem prod_reverse_nat (xs : List Nat) : xs.reverse.prod = xs.prod := by
+  simp [prod_reverse]
+
+theorem prod_eq_foldl_nat {xs : List Nat} : xs.prod = xs.foldl (init := 1) (· * ·) := by
+  simp only [foldl_eq_foldr_reverse, Nat.mul_comm, ← prod_eq_foldr, prod_reverse_nat]
+
+end List

src/Init/Data/List/Perm.lean

@@ -606,6 +606,13 @@ theorem sum_nat {l₁ l₂ : List Nat} (h : l₁ ~ l₂) : l₁.sum = l₂.sum :
   | swap => simpa [List.sum_cons] using Nat.add_left_comm ..
   | trans _ _ ih₁ ih₂ => simp [ih₁, ih₂]
 
+theorem prod_nat {l₁ l₂ : List Nat} (h : l₁ ~ l₂) : l₁.prod = l₂.prod := by
+  induction h with
+  | nil => simp
+  | cons _ _ ih => simp [ih]
+  | swap => simpa [List.prod_cons] using Nat.mul_left_comm ..
+  | trans _ _ ih₁ ih₂ => simp [ih₁, ih₂]
+
 theorem all_eq {l₁ l₂ : List α} {f : α → Bool} (hp : l₁.Perm l₂) : l₁.all f = l₂.all f := by
   rw [Bool.eq_iff_iff]; simp [hp.mem_iff]
 
@@ -615,6 +622,9 @@ theorem any_eq {l₁ l₂ : List α} {f : α → Bool} (hp : l₁.Perm l₂) : l
 grind_pattern Perm.sum_nat => l₁ ~ l₂, l₁.sum
 grind_pattern Perm.sum_nat => l₁ ~ l₂, l₂.sum
 
+grind_pattern Perm.prod_nat => l₁ ~ l₂, l₁.prod
+grind_pattern Perm.prod_nat => l₁ ~ l₂, l₂.prod
+
 end Perm
 
 end List

src/Init/Data/List/ToArray.lean

@@ -213,6 +213,9 @@ theorem forM_toArray [Monad m] (l : List α) (f : α → m PUnit) :
 @[simp, grind =] theorem sum_toArray [Add α] [Zero α] (l : List α) : l.toArray.sum = l.sum := by
   simp [Array.sum, List.sum]
 
+@[simp, grind =] theorem prod_toArray [Mul α] [One α] (l : List α) : l.toArray.prod = l.prod := by
+  simp [Array.prod, List.prod]
+
 @[simp, grind =] theorem append_toArray (l₁ l₂ : List α) :
     l₁.toArray ++ l₂.toArray = (l₁ ++ l₂).toArray := by
   apply ext'

src/Init/Data/Nat/Gcd.lean

@@ -60,7 +60,7 @@ theorem gcd_def (x y : Nat) : gcd x y = if x = 0 then y else gcd (y % x) x := by
   cases n with
   | zero => simp
   | succ n =>
-    -- `simp [gcd_succ]` produces an invalid term unless `gcd_succ` is proved with `id rfl` instead
+    -- `simp [gcd_succ]` produces an invalid term unless `gcd_succ` is proved with `(rfl)` instead
     rw [gcd_succ]
     exact gcd_zero_left _
 instance : Std.LawfulIdentity gcd 0 where

src/Init/Data/Ord/String.lean

@@ -9,7 +9,7 @@ prelude
 public import Init.Data.Order.Ord
 public import Init.Data.String.Basic
 import Init.Data.Char.Lemmas
-import Init.Data.String.Lemmas
+import Init.Data.String.Lemmas.StringOrder
 
 public section
 

src/Init/Data/Order/Lemmas.lean

@@ -243,6 +243,10 @@ public theorem lt_iff_le_and_ne [LE α] [LT α] [LawfulOrderLT α] [IsPartialOrd
     a < b ↔ a ≤ b ∧ a ≠ b := by
   simpa [le_iff_lt_or_eq, or_and_right] using Std.ne_of_lt
 
+public theorem lt_trichotomy [LT α] [Std.Trichotomous (α := α) (· < ·)] (a b : α) :
+    a < b ∨ a = b ∨ b < a :=
+  Trichotomous.rel_or_eq_or_rel_swap
+
 end LT
 end Std
 

src/Init/Data/Range/Polymorphic/NatLemmas.lean

@@ -1718,7 +1718,7 @@ theorem toArray_roc_append_toArray_roc {l m n : Nat} (h : l ≤ m) (h' : m ≤ n
 @[simp]
 theorem getElem_toArray_roc {m n i : Nat} (_h : i < (m<...=n).toArray.size) :
     (m<...=n).toArray[i]'_h = m + 1 + i := by
-simp [toArray_roc_eq_toArray_rco]
+  simp [toArray_roc_eq_toArray_rco]
 
 theorem getElem?_toArray_roc {m n i : Nat} :
     (m<...=n).toArray[i]? = if i < n - m then some (m + 1 + i) else none := by

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

@@ -193,6 +193,7 @@ public theorem Array.toSubarray_eq_toSubarray_of_min_eq_min {xs : Array α}
         simp [*]; omega
       · simp
 
+@[cbv_eval]
 public theorem Array.toSubarray_eq_min {xs : Array α} {lo hi : Nat} :
     xs.toSubarray lo hi = ⟨⟨xs, min lo (min hi xs.size), min hi xs.size, Nat.min_le_right _ _,
       Nat.min_le_right _ _⟩⟩ := by

src/Init/Data/String/Basic.lean

@@ -1386,6 +1386,11 @@ theorem Slice.copy_eq_copy_sliceTo {s : Slice} {pos : s.Pos} :
   rw [Nat.max_eq_right]
   exact pos.offset_str_le_offset_endExclusive
 
+@[simp]
+theorem Slice.sliceTo_append_sliceFrom {s : Slice} {pos : s.Pos} :
+    (s.sliceTo pos).copy ++ (s.sliceFrom pos).copy = s.copy :=
+  copy_eq_copy_sliceTo.symm
+
 /-- Given a slice `s` and a position on `s.copy`, obtain the corresponding position on `s`. -/
 @[inline]
 def Pos.ofCopy {s : Slice} (pos : s.copy.Pos) : s.Pos where
@@ -1745,6 +1750,31 @@ theorem Slice.Pos.offset_cast {s t : Slice} {pos : s.Pos} {h : s.copy = t.copy}
 theorem Slice.Pos.cast_rfl {s : Slice} {pos : s.Pos} : pos.cast rfl = pos :=
   Slice.Pos.ext (by simp)
 
+@[simp]
+theorem Slice.Pos.cast_cast {s t u : Slice} {hst : s.copy = t.copy} {htu : t.copy = u.copy}
+    {pos : s.Pos} : (pos.cast hst).cast htu = pos.cast (hst.trans htu) :=
+  Slice.Pos.ext (by simp)
+
+@[simp]
+theorem Slice.Pos.cast_inj {s t : Slice} {hst : s.copy = t.copy} {p q : s.Pos} : p.cast hst = q.cast hst ↔ p = q := by
+  simp [Slice.Pos.ext_iff]
+
+@[simp]
+theorem Slice.Pos.cast_startPos {s t : Slice} {hst : s.copy = t.copy} : s.startPos.cast hst = t.startPos :=
+  Slice.Pos.ext (by simp)
+
+@[simp]
+theorem Slice.Pos.cast_eq_startPos {s t : Slice} {p : s.Pos} {hst : s.copy = t.copy} : p.cast hst = t.startPos ↔ p = s.startPos := by
+  rw [← cast_startPos (hst := hst), Pos.cast_inj]
+
+@[simp]
+theorem Slice.Pos.cast_endPos {s t : Slice} {hst : s.copy = t.copy} : s.endPos.cast hst = t.endPos :=
+  Slice.Pos.ext (by simp [← rawEndPos_copy, hst])
+
+@[simp]
+theorem Slice.Pos.cast_eq_endPos {s t : Slice} {p : s.Pos} {hst : s.copy = t.copy} : p.cast hst = t.endPos ↔ p = s.endPos := by
+  rw [← cast_endPos (hst := hst), Pos.cast_inj]
+
 @[simp]
 theorem Slice.Pos.cast_le_cast_iff {s t : Slice} {pos pos' : s.Pos} {h : s.copy = t.copy} :
     pos.cast h ≤ pos'.cast h ↔ pos ≤ pos' := by
@@ -1755,6 +1785,22 @@ theorem Slice.Pos.cast_lt_cast_iff {s t : Slice} {pos pos' : s.Pos} {h : s.copy
     pos.cast h < pos'.cast h ↔ pos < pos' := by
   simp [Slice.Pos.lt_iff]
 
+theorem Slice.Pos.cast_le_iff {s t : Slice} {pos : s.Pos} {pos' : t.Pos} {h : s.copy = t.copy} :
+    pos.cast h ≤ pos' ↔ pos ≤ pos'.cast h.symm := by
+  simp [Slice.Pos.le_iff]
+
+theorem Slice.Pos.le_cast_iff {s t : Slice} {pos : t.Pos} {pos' : s.Pos} {h : s.copy = t.copy} :
+    pos ≤ pos'.cast h ↔ pos.cast h.symm ≤ pos' := by
+  simp [Slice.Pos.le_iff]
+
+theorem Slice.Pos.cast_lt_iff {s t : Slice} {pos : s.Pos} {pos' : t.Pos} {h : s.copy = t.copy} :
+    pos.cast h < pos' ↔ pos < pos'.cast h.symm := by
+  simp [Slice.Pos.lt_iff]
+
+theorem Slice.Pos.lt_cast_iff {s t : Slice} {pos : t.Pos} {pos' : s.Pos} {h : s.copy = t.copy} :
+    pos < pos'.cast h ↔ pos.cast h.symm < pos' := by
+  simp [Slice.Pos.lt_iff]
+
 /-- Constructs a valid position on `t` from a valid position on `s` and a proof that `s = t`. -/
 @[inline]
 def Pos.cast {s t : String} (pos : s.Pos) (h : s = t) : t.Pos where
@@ -1769,6 +1815,31 @@ theorem Pos.offset_cast {s t : String} {pos : s.Pos} {h : s = t} :
 theorem Pos.cast_rfl {s : String} {pos : s.Pos} : pos.cast rfl = pos :=
   Pos.ext (by simp)
 
+@[simp]
+theorem Pos.cast_cast {s t u : String} {hst : s = t} {htu : t = u}
+    {pos : s.Pos} : (pos.cast hst).cast htu = pos.cast (hst.trans htu) :=
+  Pos.ext (by simp)
+
+@[simp]
+theorem Pos.cast_inj {s t : String} {hst : s = t} {p q : s.Pos} : p.cast hst = q.cast hst ↔ p = q := by
+  simp [Pos.ext_iff]
+
+@[simp]
+theorem Pos.cast_startPos {s t : String} {hst : s = t} : s.startPos.cast hst = t.startPos := by
+  subst hst; simp
+
+@[simp]
+theorem Pos.cast_eq_startPos {s t : String} {hst : s = t} {p : s.Pos} : p.cast hst = t.startPos ↔ p = s.startPos := by
+  rw [← Pos.cast_startPos (hst := hst), Pos.cast_inj]
+
+@[simp]
+theorem Pos.cast_endPos {s t : String} {hst : s = t} : s.endPos.cast hst = t.endPos := by
+  subst hst; simp
+
+@[simp]
+theorem Pos.cast_eq_endPos {s t : String} {hst : s = t} {p : s.Pos} : p.cast hst = t.endPos ↔ p = s.endPos := by
+  rw [← Pos.cast_endPos (hst := hst), Pos.cast_inj]
+
 @[simp]
 theorem Pos.cast_le_cast_iff {s t : String} {pos pos' : s.Pos} {h : s = t} :
     pos.cast h ≤ pos'.cast h ↔ pos ≤ pos' := by
@@ -1779,6 +1850,22 @@ theorem Pos.cast_lt_cast_iff {s t : String} {pos pos' : s.Pos} {h : s = t} :
     pos.cast h < pos'.cast h ↔ pos < pos' := by
   cases h; simp
 
+theorem Pos.cast_le_iff {s t : String} {pos : s.Pos} {pos' : t.Pos} {h : s = t} :
+    pos.cast h ≤ pos' ↔ pos ≤ pos'.cast h.symm := by
+  simp [Pos.le_iff]
+
+theorem Pos.le_cast_iff {s t : String} {pos : t.Pos} {pos' : s.Pos} {h : s = t} :
+    pos ≤ pos'.cast h ↔ pos.cast h.symm ≤ pos' := by
+  simp [Pos.le_iff]
+
+theorem Pos.cast_lt_iff {s t : String} {pos : s.Pos} {pos' : t.Pos} {h : s = t} :
+    pos.cast h < pos' ↔ pos < pos'.cast h.symm := by
+  simp [Pos.lt_iff]
+
+theorem Pos.lt_cast_iff {s t : String} {pos : t.Pos} {pos' : s.Pos} {h : s = t} :
+    pos < pos'.cast h ↔ pos.cast h.symm < pos' := by
+  simp [Pos.lt_iff]
+
 theorem Pos.copy_toSlice_eq_cast {s : String} (p : s.Pos) :
     p.toSlice.copy = p.cast copy_toSlice.symm :=
   Pos.ext (by simp)
@@ -2054,6 +2141,10 @@ theorem Pos.le_ofToSlice_iff {s : String} {p : s.Pos} {q : s.toSlice.Pos} :
 theorem Pos.toSlice_lt_toSlice_iff {s : String} {p q : s.Pos} :
     p.toSlice < q.toSlice ↔ p < q := Iff.rfl
 
+@[simp]
+theorem Pos.toSlice_le_toSlice_iff {s : String} {p q : s.Pos} :
+    p.toSlice ≤ q.toSlice ↔ p ≤ q := Iff.rfl
+
 theorem Pos.next_le_of_lt {s : String} {p q : s.Pos} {h} : p < q → p.next h ≤ q := by
   rw [next, Pos.ofToSlice_le_iff, ← Pos.toSlice_lt_toSlice_iff]
   exact Slice.Pos.next_le_of_lt

src/Init/Data/String/Decode.lean

@@ -363,7 +363,7 @@ theorem toBitVec_eq_of_parseFirstByte_eq_threeMore {b : UInt8} (h : parseFirstBy
 public def isInvalidContinuationByte (b : UInt8) : Bool :=
   b &&& 0xc0 != 0x80
 
-theorem isInvalidContinutationByte_eq_false_iff {b : UInt8} :
+theorem isInvalidContinuationByte_eq_false_iff {b : UInt8} :
     isInvalidContinuationByte b = false ↔ b &&& 0xc0 = 0x80 := by
   simp [isInvalidContinuationByte]
 

src/Init/Data/String/Iter/Intercalate.lean

@@ -17,6 +17,7 @@ namespace Std
 /--
 Appends all the elements in the iterator, in order.
 -/
+@[inline]
 public def Iter.joinString {α β : Type} [Iterator α Id β] [ToString β]
     (it : Std.Iter (α := α) β) : String :=
   (it.map toString).fold (init := "") (· ++ ·)

src/Init/Data/String/Lemmas.lean

@@ -20,49 +20,4 @@ public import Init.Data.String.Lemmas.Intercalate
 public import Init.Data.String.Lemmas.Iter
 public import Init.Data.String.Lemmas.Hashable
 public import Init.Data.String.Lemmas.TakeDrop
-import Init.Data.Order.Lemmas
-public import Init.Data.String.Basic
-import Init.Data.Char.Lemmas
-import Init.Data.Char.Order
-import Init.Data.List.Lex
-
-public section
-
-open Std
-
-namespace String
-
-@[deprecated toList_inj (since := "2025-10-30")]
-protected theorem data_eq_of_eq {a b : String} (h : a = b) : a.toList = b.toList :=
-  h ▸ rfl
-@[deprecated toList_inj (since := "2025-10-30")]
-protected theorem ne_of_data_ne {a b : String} (h : a.toList ≠ b.toList) : a ≠ b := by
-  simpa [← toList_inj]
-
-@[simp] protected theorem not_le {a b : String} : ¬ a ≤ b ↔ b < a := Decidable.not_not
-@[simp] protected theorem not_lt {a b : String} : ¬ a < b ↔ b ≤ a := Iff.rfl
-@[simp] protected theorem le_refl (a : String) : a ≤ a := List.le_refl _
-@[simp] protected theorem lt_irrefl (a : String) : ¬ a < a := List.lt_irrefl _
-
-attribute [local instance] Char.notLTTrans Char.ltTrichotomous Char.ltAsymm
-
-protected theorem le_trans {a b c : String} : a ≤ b → b ≤ c → a ≤ c := List.le_trans
-protected theorem lt_trans {a b c : String} : a < b → b < c → a < c := List.lt_trans
-protected theorem le_total (a b : String) : a ≤ b ∨ b ≤ a := List.le_total _ _
-protected theorem le_antisymm {a b : String} : a ≤ b → b ≤ a → a = b := fun h₁ h₂ => String.ext (List.le_antisymm (as := a.toList) (bs := b.toList) h₁ h₂)
-protected theorem lt_asymm {a b : String} (h : a < b) : ¬ b < a := List.lt_asymm h
-protected theorem ne_of_lt {a b : String} (h : a < b) : a ≠ b := by
-  have := String.lt_irrefl a
-  intro h; subst h; contradiction
-
-instance instIsLinearOrder : IsLinearOrder String := by
-  apply IsLinearOrder.of_le
-  case le_antisymm => constructor; apply String.le_antisymm
-  case le_trans => constructor; apply String.le_trans
-  case le_total => constructor; apply String.le_total
-
-instance : LawfulOrderLT String where
-  lt_iff a b := by
-    simp [← String.not_le, Decidable.imp_iff_not_or, Std.Total.total]
-
-end String
+public import Init.Data.String.Lemmas.StringOrder

src/Init/Data/String/Lemmas/Basic.lean

@@ -22,6 +22,10 @@ public section
 
 namespace String
 
+@[simp]
+theorem singleton_inj {c d : Char} : singleton c = singleton d ↔ c = d := by
+  simp [← toList_inj]
+
 @[simp]
 theorem singleton_append_inj : singleton c ++ s = singleton d ++ t ↔ c = d ∧ s = t := by
   simp [← toList_inj]
@@ -191,18 +195,74 @@ theorem sliceTo_slice {s : String} {p₁ p₂ h p} :
 theorem Slice.sliceFrom_startPos {s : Slice} : s.sliceFrom s.startPos = s := by
   ext <;> simp
 
+@[simp]
+theorem Slice.sliceFrom_eq_self_iff {s : Slice} {p : s.Pos} : s.sliceFrom p = s ↔ p = s.startPos := by
+  refine ⟨?_, by rintro rfl; simp⟩
+  rcases s with ⟨str, startInclusive, endExclusive, h⟩
+  simp [sliceFrom, Slice.startPos, String.Pos.ext_iff, Pos.Raw.ext_iff, Slice.Pos.ext_iff]
+
 @[simp]
 theorem Slice.sliceTo_endPos {s : Slice} : s.sliceTo s.endPos = s := by
   ext <;> simp
 
+@[simp]
+theorem Slice.sliceTo_eq_self_iff {s : Slice} {p : s.Pos} : s.sliceTo p = s ↔ p = s.endPos := by
+  refine ⟨?_, by rintro rfl; simp⟩
+  rcases s with ⟨str, startInclusive, endExclusive, h⟩
+  simp [sliceTo, Slice.endPos, String.Pos.ext_iff, Pos.Raw.ext_iff, Slice.Pos.ext_iff,
+    utf8ByteSize_eq]
+  omega
+
+@[simp]
+theorem Slice.slice_startPos {s : Slice} {p : s.Pos} :
+    s.slice s.startPos p (Pos.startPos_le _) = s.sliceTo p := by
+  ext <;> simp
+
+@[simp]
+theorem Slice.slice_eq_self_iff {s : Slice} {p₁ p₂ : s.Pos} {h} :
+    s.slice p₁ p₂ h = s ↔ p₁ = s.startPos ∧ p₂ = s.endPos := by
+  refine ⟨?_, by rintro ⟨rfl, rfl⟩; simp⟩
+  rcases s with ⟨str, startInclusive, endExclusive, h⟩
+  simp [slice, Slice.endPos, String.Pos.ext_iff, Pos.Raw.ext_iff, Slice.Pos.ext_iff,
+    utf8ByteSize_eq]
+  omega
+
+@[simp]
+theorem Slice.slice_endPos {s : Slice} {p : s.Pos} :
+    s.slice p s.endPos (Pos.le_endPos _) = s.sliceFrom p := by
+  ext <;> simp
+
 @[simp]
 theorem sliceFrom_startPos {s : String} : s.sliceFrom s.startPos = s := by
   ext <;> simp
 
+@[simp]
+theorem sliceFrom_eq_toSlice_iff {s : String} {p : s.Pos} : s.sliceFrom p = s.toSlice ↔ p = s.startPos := by
+  simp [← sliceFrom_toSlice]
+
 @[simp]
 theorem sliceTo_endPos {s : String} : s.sliceTo s.endPos = s := by
   ext <;> simp
 
+@[simp]
+theorem sliceTo_eq_toSlice_iff {s : String} {p : s.Pos} : s.sliceTo p = s.toSlice ↔ p = s.endPos := by
+  simp [← sliceTo_toSlice]
+
+@[simp]
+theorem slice_startPos {s : String} {p : s.Pos} :
+    s.slice s.startPos p (Pos.startPos_le _) = s.sliceTo p := by
+  ext <;> simp
+
+@[simp]
+theorem slice_endPos {s : String} {p : s.Pos} :
+    s.slice p s.endPos (Pos.le_endPos _) = s.sliceFrom p := by
+  ext <;> simp
+
+@[simp]
+theorem slice_eq_toSlice_iff {s : String} {p₁ p₂ : s.Pos} {h} :
+    s.slice p₁ p₂ h = s.toSlice ↔ p₁ = s.startPos ∧ p₂ = s.endPos := by
+  simp [← slice_toSlice]
+
 end Iterate
 
 theorem Slice.copy_eq_copy_slice {s : Slice} {pos₁ pos₂ : s.Pos} {h} :
@@ -292,4 +352,39 @@ theorem nextn_endPos {s : String} : s.endPos.nextn n = s.endPos := by
 
 end Pos
 
+@[simp]
+theorem Slice.Pos.cast_toSlice_copy {s : Slice} {pos : s.Pos} :
+    pos.copy.toSlice.cast (by simp) = pos := by
+  ext; simp
+
+@[simp]
+theorem Slice.Pos.sliceFrom_eq_startPos {s : Slice} {p : s.Pos} :
+    (Pos.sliceFrom p p (Pos.le_refl _)) = Slice.startPos _ := by
+  simp [← Pos.ofSliceFrom_inj]
+
+@[simp]
+theorem Slice.Pos.sliceFrom_endPos {s : Slice} {p : s.Pos} :
+    (Pos.sliceFrom p s.endPos (Pos.le_endPos _)) = Slice.endPos _ := by
+  simp [← Pos.ofSliceFrom_inj]
+
+@[simp]
+theorem Slice.Pos.sliceTo_startPos {s : Slice} {p : s.Pos} :
+    (Pos.sliceTo p s.startPos (Pos.startPos_le _)) = Slice.startPos _ := by
+  simp [← Pos.ofSliceTo_inj]
+
+@[simp]
+theorem Slice.Pos.sliceTo_eq_endPos {s : Slice} {p : s.Pos} :
+    (Pos.sliceTo p p (Pos.le_refl _)) = Slice.endPos _ := by
+  simp [← Pos.ofSliceTo_inj]
+
+@[simp]
+theorem Slice.Pos.slice_eq_startPos {s : Slice} {p₀ p₁ : s.Pos} {h} :
+    (Pos.slice p₀ p₀ p₁ (Pos.le_refl _) h) = Slice.startPos _ := by
+  simp [← Pos.ofSlice_inj]
+
+@[simp]
+theorem Slice.Pos.slice_eq_endPos {s : Slice} {p₀ p₁ : s.Pos} {h} :
+    (Pos.slice p₁ p₀ p₁ h (Pos.le_refl _)) = Slice.endPos _ := by
+  simp [← Pos.ofSlice_inj]
+
 end String

src/Init/Data/String/Lemmas/Intercalate.lean

@@ -77,6 +77,15 @@ theorem join_cons : join (s :: l) = s ++ join l := by
 theorem toList_join {l : List String} : (String.join l).toList = l.flatMap String.toList := by
   induction l <;> simp_all
 
+@[simp]
+theorem join_append {l m : List String} : String.join (l ++ m) = String.join l ++ String.join m := by
+  simp [← toList_inj]
+
+@[simp]
+theorem length_join {l : List String} : (String.join l).length = (l.map String.length).sum := by
+  simp only [← length_toList, toList_join, List.length_flatMap]
+  simp
+
 namespace Slice
 
 @[simp]

src/Init/Data/String/Lemmas/Iterate.lean

@@ -31,7 +31,7 @@ namespace Slice
 /--
 A list of all positions starting at {name}`p`.
 
-This function is not meant to be used in actual progams. Actual programs should use
+This function is not meant to be used in actual programs. Actual programs should use
 {name}`Slice.positionsFrom` or {name}`Slice.positions`.
 -/
 protected def Model.positionsFrom {s : Slice} (p : s.Pos) : List { p : s.Pos // p ≠ s.endPos } :=
@@ -206,7 +206,7 @@ end Slice
 /--
 A list of all positions starting at {name}`p`.
 
-This function is not meant to be used in actual progams. Actual programs should use
+This function is not meant to be used in actual programs. Actual programs should use
 {name}`Slice.positionsFrom` or {name}`Slice.positions`.
 -/
 protected def Model.positionsFrom {s : String} (p : s.Pos) : List { p : s.Pos // p ≠ s.endPos } :=

src/Init/Data/String/Lemmas/Order.lean

@@ -368,21 +368,41 @@ theorem Slice.Pos.ofSliceTo_ne_endPos {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₀))
   simpa [← lt_endPos_iff, ← ofSliceTo_lt_ofSliceTo_iff] using h
 
+theorem Slice.Pos.ne_endPos_of_sliceTo_ne_endPos {s : Slice} {p p₀ : s.Pos} {h₀}
+    (h : Pos.sliceTo p₀ p h₀ ≠ Slice.endPos _) : p ≠ s.endPos := by
+  rw [← Pos.ofSliceTo_sliceTo (h := h₀)]
+  apply Pos.ofSliceTo_ne_endPos h
+
 theorem Slice.Pos.ofSliceFrom_ne_startPos {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos}
     (h : p ≠ (s.sliceFrom p₀).startPos) : Pos.ofSliceFrom p ≠ s.startPos := by
   refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
   simpa [← startPos_lt_iff, ← ofSliceFrom_lt_ofSliceFrom_iff] using h
 
+theorem Slice.Pos.ne_startPos_of_sliceFrom_ne_startPos {s : Slice} {p p₀ : s.Pos} {h₀}
+    (h : Pos.sliceFrom p₀ p h₀ ≠ Slice.startPos _) : p ≠ s.startPos := by
+  rw [← Pos.ofSliceFrom_sliceFrom (h := h₀)]
+  apply Pos.ofSliceFrom_ne_startPos h
+
 theorem Pos.ofSliceTo_ne_endPos {s : String} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos}
     (h : p ≠ (s.sliceTo p₀).endPos) : Pos.ofSliceTo p ≠ s.endPos := by
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₀))
   simpa [← Slice.Pos.lt_endPos_iff, ← ofSliceTo_lt_ofSliceTo_iff] using h
 
+theorem Pos.ne_endPos_of_sliceTo_ne_endPos {s : String} {p p₀ : s.Pos} {h₀}
+    (h : Pos.sliceTo p₀ p h₀ ≠ Slice.endPos _) : p ≠ s.endPos := by
+  rw [← Pos.ofSliceTo_sliceTo (h := h₀)]
+  apply Pos.ofSliceTo_ne_endPos h
+
 theorem Pos.ofSliceFrom_ne_startPos {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos}
     (h : p ≠ (s.sliceFrom p₀).startPos) : Pos.ofSliceFrom p ≠ s.startPos := by
   refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
   simpa [← Slice.Pos.startPos_lt_iff, ← ofSliceFrom_lt_ofSliceFrom_iff] using h
 
+theorem Pos.ne_startPos_of_sliceFrom_ne_startPos {s : String} {p p₀ : s.Pos} {h₀}
+    (h : Pos.sliceFrom p₀ p h₀ ≠ Slice.startPos _) : p ≠ s.startPos := by
+  rw [← Pos.ofSliceFrom_sliceFrom (h := h₀)]
+  apply Pos.ofSliceFrom_ne_startPos h
+
 theorem Slice.Pos.ofSliceTo_next {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
     Pos.ofSliceTo (p.next h) = (Pos.ofSliceTo p).next (ofSliceTo_ne_endPos h) := by
   rw [eq_comm, Pos.next_eq_iff]
@@ -514,21 +534,41 @@ theorem Slice.Pos.ofSlice_ne_endPos {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₁))
   simpa [← lt_endPos_iff, ← ofSlice_lt_ofSlice_iff] using h
 
+theorem Slice.Pos.ne_endPos_of_slice_ne_endPos {s : Slice} {p p₀ p₁ : s.Pos} {h₁ h₂}
+    (h : Pos.slice p p₀ p₁ h₁ h₂ ≠ Slice.endPos _) : p ≠ s.endPos := by
+  rw [← Pos.ofSlice_slice (h₁ := h₁) (h₂ := h₂)]
+  apply Pos.ofSlice_ne_endPos h
+
 theorem Slice.Pos.ofSlice_ne_startPos {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
     (h : p ≠ (s.slice p₀ p₁ h).startPos) : Pos.ofSlice p ≠ s.startPos := by
   refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
   simpa [← startPos_lt_iff, ← ofSlice_lt_ofSlice_iff] using h
 
+theorem Slice.Pos.ne_startPos_of_slice_ne_startPos {s : Slice} {p p₀ p₁ : s.Pos} {h₁ h₂}
+    (h : Pos.slice p p₀ p₁ h₁ h₂ ≠ Slice.startPos _) : p ≠ s.startPos := by
+  rw [← Pos.ofSlice_slice (h₁ := h₁) (h₂ := h₂)]
+  apply Pos.ofSlice_ne_startPos h
+
 theorem Pos.ofSlice_ne_endPos {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
     (h : p ≠ (s.slice p₀ p₁ h).endPos) : Pos.ofSlice p ≠ s.endPos := by
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₁))
   simpa [← Slice.Pos.lt_endPos_iff, ← ofSlice_lt_ofSlice_iff] using h
 
+theorem Pos.ne_endPos_of_slice_ne_endPos {s : String} {p p₀ p₁ : s.Pos} {h₁ h₂}
+    (h : Pos.slice p p₀ p₁ h₁ h₂ ≠ Slice.endPos _) : p ≠ s.endPos := by
+  rw [← Pos.ofSlice_slice (h₁ := h₁) (h₂ := h₂)]
+  apply Pos.ofSlice_ne_endPos h
+
 theorem Pos.ofSlice_ne_startPos {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
     (h : p ≠ (s.slice p₀ p₁ h).startPos) : Pos.ofSlice p ≠ s.startPos := by
   refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
   simpa [← Slice.Pos.startPos_lt_iff, ← ofSlice_lt_ofSlice_iff] using h
 
+theorem Pos.ne_startPos_of_slice_ne_startPos {s : String} {p p₀ p₁ : s.Pos} {h₁ h₂}
+    (h : Pos.slice p p₀ p₁ h₁ h₂ ≠ Slice.startPos _) : p ≠ s.startPos := by
+  rw [← Pos.ofSlice_slice (h₁ := h₁) (h₂ := h₂)]
+  apply Pos.ofSlice_ne_startPos h
+
 @[simp]
 theorem Slice.Pos.offset_le_rawEndPos {s : Slice} {p : s.Pos} :
     p.offset ≤ s.rawEndPos :=
@@ -581,21 +621,37 @@ theorem Slice.Pos.get_eq_get_ofSliceTo {s : Slice} {p₀ : s.Pos} {pos : (s.slic
     pos.get h = (ofSliceTo pos).get (ofSliceTo_ne_endPos h) := by
   simp [Slice.Pos.get]
 
+theorem Slice.Pos.get_sliceTo {s : Slice} {p₀ p : s.Pos} {h h'} :
+    (Pos.sliceTo p₀ p h).get h' = p.get (ne_endPos_of_sliceTo_ne_endPos h') := by
+  simp [get_eq_get_ofSliceTo]
+
 theorem Pos.get_eq_get_ofSliceTo {s : String} {p₀ : s.Pos}
     {pos : (s.sliceTo p₀).Pos} {h} :
     pos.get h = (ofSliceTo pos).get (ofSliceTo_ne_endPos h) := by
   simp [Pos.get, Slice.Pos.get]
 
+theorem Pos.get_sliceTo {s : String} {p₀ p : s.Pos} {h h'} :
+    (Pos.sliceTo p₀ p h).get h' = p.get (ne_endPos_of_sliceTo_ne_endPos h') := by
+  simp [get_eq_get_ofSliceTo]
+
 theorem Slice.Pos.get_eq_get_ofSlice {s : Slice} {p₀ p₁ : s.Pos} {h}
     {pos : (s.slice p₀ p₁ h).Pos} {h'} :
     pos.get h' = (ofSlice pos).get (ofSlice_ne_endPos h') := by
   simp [Slice.Pos.get, Nat.add_assoc]
 
+theorem Slice.Pos.get_slice {s : Slice} {p p₀ p₁ : s.Pos} {h₁ h₂ h} :
+    (Pos.slice p p₀ p₁ h₁ h₂).get h = p.get (ne_endPos_of_slice_ne_endPos h) := by
+  simp [get_eq_get_ofSlice]
+
 theorem Pos.get_eq_get_ofSlice {s : String} {p₀ p₁ : s.Pos} {h}
     {pos : (s.slice p₀ p₁ h).Pos} {h'} :
     pos.get h' = (ofSlice pos).get (ofSlice_ne_endPos h') := by
   simp [Pos.get, Slice.Pos.get]
 
+theorem Pos.get_slice {s : String} {p p₀ p₁ : s.Pos} {h₁ h₂ h} :
+    (Pos.slice p p₀ p₁ h₁ h₂).get h = p.get (ne_endPos_of_slice_ne_endPos h) := by
+  simp [get_eq_get_ofSlice]
+
 theorem Slice.Pos.ofSlice_next {s : Slice} {p₀ p₁ : s.Pos} {h}
     {p : (s.slice p₀ p₁ h).Pos} {h'} :
     Pos.ofSlice (p.next h') = (Pos.ofSlice p).next (ofSlice_ne_endPos h') := by

src/Init/Data/String/Lemmas/Pattern/Basic.lean

@@ -12,7 +12,7 @@ public import Init.Data.Iterators.Consumers.Collect
 import all Init.Data.String.Pattern.Basic
 import Init.Data.String.OrderInstances
 import Init.Data.String.Lemmas.IsEmpty
-import Init.Data.String.Lemmas.Basic
+public import Init.Data.String.Lemmas.Basic
 import Init.Data.String.Lemmas.Order
 import Init.Data.String.Termination
 import Init.Data.Order.Lemmas
@@ -40,7 +40,7 @@ framework.
 /--
 This data-carrying typeclass is used to give semantics to a pattern type that implements
 {name}`ForwardPattern` and/or {name}`ToForwardSearcher` by providing an abstract, not necessarily
-decidable {name}`PatternModel.Matches` predicate that implementates of {name}`ForwardPattern`
+decidable {name}`PatternModel.Matches` predicate that implementations of {name}`ForwardPattern`
 and {name}`ToForwardSearcher` can be validated against.
 
 Correctness results for generic functions relying on the pattern infrastructure, for example the
@@ -52,19 +52,23 @@ The corresponding compatibility typeclasses are
 {name (scope := "Init.Data.String.Lemmas.Pattern.Basic")}`String.Slice.Pattern.Model.LawfulForwardPatternModel`
 and
 {name (scope := "Init.Data.String.Lemmas.Pattern.Basic")}`String.Slice.Pattern.Model.LawfulToForwardSearcherModel`.
-
-We include the condition that the empty string is not a match. This is necessary for the theory to
-work out as there is just no reasonable notion of searching that works for the empty string that is
-still specific enough to yield reasonably strong correctness results for operations based on
-searching.
-
-This means that pattern types that allow searching for the empty string will have to special-case
-the empty string in their correctness statements.
 -/
 class PatternModel {ρ : Type} (pat : ρ) : Type where
   /-- The predicate that says which strings match the pattern. -/
   Matches : String → Prop
-  not_matches_empty : ¬ Matches ""
+
+/--
+Type class for the condition that the empty string is not a match. This is necessary for the theory to
+work out as there is just no reasonable notion of searching that works for the empty string that is
+still specific enough to yield reasonably strong correctness results for operations based on
+searching.
+-/
+class StrictPatternModel {ρ : Type} (pat : ρ) [PatternModel pat] : Prop where
+  not_matches_empty : ¬ PatternModel.Matches pat ""
+
+theorem not_matches_empty {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat] :
+    ¬ PatternModel.Matches pat "" :=
+  StrictPatternModel.not_matches_empty
 
 /--
 Predicate stating that the region between the start of the slice {name}`s` and the position
@@ -74,10 +78,10 @@ Predicate stating that the region between the start of the slice {name}`s` and t
 structure IsMatch (pat : ρ) [PatternModel pat] {s : Slice} (endPos : s.Pos) : Prop where
   matches_copy : PatternModel.Matches pat (s.sliceTo endPos).copy
 
-theorem IsMatch.ne_startPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+theorem IsMatch.ne_startPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {pos : s.Pos}
     (h : IsMatch pat pos) : pos ≠ s.startPos := by
   intro hc
-  apply PatternModel.not_matches_empty (pat := pat)
+  apply not_matches_empty (pat := pat)
   simpa [hc] using h.matches_copy
 
 theorem isMatch_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
@@ -90,6 +94,21 @@ theorem isMatch_iff_exists_splits {pat : ρ} [PatternModel pat] {s : Slice} {pos
   refine ⟨fun h => ⟨_, _, pos.splits, h⟩, fun ⟨t₁, t₂, h₁, h₂⟩ => ?_⟩
   rwa [h₁.eq_left pos.splits] at h₂
 
+@[simp]
+theorem isMatch_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (h : s.copy = t.copy) {pos : s.Pos} :
+    IsMatch pat (pos.cast h) ↔ IsMatch pat pos := by
+  simp [isMatch_iff]
+
+@[simp]
+theorem isMatch_sliceTo_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos p : s.Pos} {h} :
+    IsMatch pat (Pos.sliceTo p pos h) ↔ IsMatch pat pos := by
+  simp [isMatch_iff]
+
+@[simp]
+theorem isMatch_ofSliceTo_iff {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos} {pos : (s.sliceTo p).Pos} :
+    IsMatch pat (Pos.ofSliceTo pos) ↔ IsMatch pat pos := by
+  rw [← isMatch_sliceTo_iff (p := p) (h := Pos.ofSliceTo_le), Pos.sliceTo_ofSliceTo]
+
 /--
 Predicate stating that the region between the position {name}`startPos` and the end of the slice
 {name}`s` matches the pattern {name}`pat`. Note that there might be a longer match.
@@ -97,10 +116,10 @@ Predicate stating that the region between the position {name}`startPos` and the
 structure IsRevMatch (pat : ρ) [PatternModel pat] {s : Slice} (startPos : s.Pos) : Prop where
   matches_copy : PatternModel.Matches pat (s.sliceFrom startPos).copy
 
-theorem IsRevMatch.ne_endPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+theorem IsRevMatch.ne_endPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {pos : s.Pos}
     (h : IsRevMatch pat pos) : pos ≠ s.endPos := by
   intro hc
-  apply PatternModel.not_matches_empty (pat := pat)
+  apply not_matches_empty (pat := pat)
   simpa [hc] using h.matches_copy
 
 theorem isRevMatch_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
@@ -113,6 +132,21 @@ theorem isRevMatch_iff_exists_splits {pat : ρ} [PatternModel pat] {s : Slice} {
   refine ⟨fun h => ⟨_, _, pos.splits, h⟩, fun ⟨t₁, t₂, h₁, h₂⟩ => ?_⟩
   rwa [h₁.eq_right pos.splits] at h₂
 
+@[simp]
+theorem isRevMatch_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (h : s.copy = t.copy) {pos : s.Pos} :
+    IsRevMatch pat (pos.cast h) ↔ IsRevMatch pat pos := by
+  simp [isRevMatch_iff]
+
+@[simp]
+theorem isRevMatch_sliceFrom_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos p : s.Pos} {h} :
+    IsRevMatch pat (Pos.sliceFrom p pos h) ↔ IsRevMatch pat pos := by
+  simp [isRevMatch_iff]
+
+@[simp]
+theorem isRevMatch_ofSliceFrom_iff {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos} {pos : (s.sliceFrom p).Pos} :
+    IsRevMatch pat (Pos.ofSliceFrom pos) ↔ IsRevMatch pat pos := by
+  rw [← isRevMatch_sliceFrom_iff (p := p) (h := Pos.le_ofSliceFrom), Pos.sliceFrom_ofSliceFrom]
+
 /--
 Predicate stating that the region between the start of the slice {name}`s` and the position
 {name}`pos` matches the pattern {name}`pat`, and that there is no longer match starting at the
@@ -125,10 +159,19 @@ structure IsLongestMatch (pat : ρ) [PatternModel pat] {s : Slice} (pos : s.Pos)
   isMatch : IsMatch pat pos
   not_isMatch : ∀ pos', pos < pos' → ¬ IsMatch pat pos'
 
-theorem IsLongestMatch.ne_startPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+theorem isLongestMatch_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsLongestMatch pat pos ↔ IsMatch pat pos ∧ ∀ pos', pos < pos' → ¬ IsMatch pat pos' :=
+  ⟨fun ⟨h, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'⟩⟩
+
+theorem IsLongestMatch.ne_startPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {pos : s.Pos}
     (h : IsLongestMatch pat pos) : pos ≠ s.startPos :=
   h.isMatch.ne_startPos
 
+@[simp]
+theorem not_isLongestMatch_startPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} :
+    ¬IsLongestMatch pat s.startPos :=
+  fun h => h.ne_startPos rfl
+
 theorem IsLongestMatch.eq {pat : ρ} [PatternModel pat] {s : Slice} {pos pos' : s.Pos}
     (h : IsLongestMatch pat pos) (h' : IsLongestMatch pat pos') : pos = pos' := by
   apply Std.le_antisymm
@@ -149,9 +192,37 @@ theorem IsLongestMatch.le_of_isMatch {pat : ρ} [PatternModel pat] {s : Slice} {
     (h : IsLongestMatch pat pos) (h' : IsMatch pat pos') : pos' ≤ pos :=
   Std.not_lt.1 (fun hlt => h.not_isMatch _ hlt h')
 
+@[simp]
+theorem isLongestMatch_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice}
+    (hst : s.copy = t.copy) {pos : s.Pos} :
+    IsLongestMatch pat (pos.cast hst) ↔ IsLongestMatch pat pos := by
+  simp only [isLongestMatch_iff, isMatch_cast_iff, and_congr_right_iff]
+  refine fun _ => ⟨fun h p hp => ?_, fun h p hp => ?_⟩
+  · rw [← isMatch_cast_iff hst]
+    exact h _ (by simpa)
+  · have : p = (p.cast hst.symm).cast hst := by simp
+    rw [this, isMatch_cast_iff hst]
+    exact h _ (by rwa [this, Pos.cast_lt_cast_iff] at hp)
+
+theorem IsLongestMatch.of_eq {pat : ρ} [PatternModel pat] {s t : Slice} {pos : s.Pos} {pos' : t.Pos}
+    (h : IsLongestMatch pat pos) (h₁ : s.copy = t.copy) (h₂ : pos.cast h₁ = pos') :
+    IsLongestMatch pat pos' := by
+  subst h₂; simpa
+
+theorem IsLongestMatch.sliceTo {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+    (h : IsLongestMatch pat pos) (p : s.Pos) (hp : pos ≤ p) : IsLongestMatch pat (Pos.sliceTo p pos hp) := by
+  simp [isLongestMatch_iff] at ⊢ h
+  refine ⟨h.1, fun p hp => ?_⟩
+  rw [← isMatch_ofSliceTo_iff]
+  exact h.2 _ (by simpa [Pos.sliceTo_lt_iff] using hp)
+
+theorem isLongestMatch_of_ofSliceTo {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos} {pos : (s.sliceTo p).Pos}
+    (h : IsLongestMatch pat (Pos.ofSliceTo pos)) : IsLongestMatch pat pos := by
+  simpa using h.sliceTo p
+
 /--
 Predicate stating that the region between the start of the slice {name}`s` and the position
-{name}`pos` matches the patten {name}`pat`, and that there is no longer match starting at the
+{name}`pos` matches the pattern {name}`pat`, and that there is no longer match starting at the
 beginning of the slice. This is what a correct matcher should match.
 
 In some cases, being a match and being a longest match will coincide, see
@@ -161,10 +232,19 @@ structure IsLongestRevMatch (pat : ρ) [PatternModel pat] {s : Slice} (pos : s.P
   isRevMatch : IsRevMatch pat pos
   not_isRevMatch : ∀ pos', pos' < pos → ¬ IsRevMatch pat pos'
 
-theorem IsLongestRevMatch.ne_endPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+theorem isLongestRevMatch_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch pat pos ↔ IsRevMatch pat pos ∧ ∀ pos', pos' < pos → ¬ IsRevMatch pat pos' :=
+  ⟨fun ⟨h, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'⟩⟩
+
+theorem IsLongestRevMatch.ne_endPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {pos : s.Pos}
     (h : IsLongestRevMatch pat pos) : pos ≠ s.endPos :=
   h.isRevMatch.ne_endPos
 
+@[simp]
+theorem not_isLongestRevMatch_endPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} :
+    ¬IsLongestRevMatch pat s.endPos :=
+  fun h => h.ne_endPos rfl
+
 theorem IsLongestRevMatch.eq {pat : ρ} [PatternModel pat] {s : Slice} {pos pos' : s.Pos}
     (h : IsLongestRevMatch pat pos) (h' : IsLongestRevMatch pat pos') : pos = pos' := by
   apply Std.le_antisymm
@@ -185,6 +265,34 @@ theorem IsLongestRevMatch.le_of_isRevMatch {pat : ρ} [PatternModel pat] {s : Sl
     (h : IsLongestRevMatch pat pos) (h' : IsRevMatch pat pos') : pos ≤ pos' :=
   Std.not_lt.1 (fun hlt => h.not_isRevMatch _ hlt h')
 
+@[simp]
+theorem isLongestRevMatch_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice}
+    (hst : s.copy = t.copy) {pos : s.Pos} :
+    IsLongestRevMatch pat (pos.cast hst) ↔ IsLongestRevMatch pat pos := by
+  simp only [isLongestRevMatch_iff, isRevMatch_cast_iff, and_congr_right_iff]
+  refine fun _ => ⟨fun h p hp => ?_, fun h p hp => ?_⟩
+  · rw [← isRevMatch_cast_iff hst]
+    exact h _ (by simpa)
+  · have : p = (p.cast hst.symm).cast hst := by simp
+    rw [this, isRevMatch_cast_iff hst]
+    exact h _ (by rwa [this, Pos.cast_lt_cast_iff] at hp)
+
+theorem IsLongestRevMatch.of_eq {pat : ρ} [PatternModel pat] {s t : Slice} {pos : s.Pos} {pos' : t.Pos}
+    (h : IsLongestRevMatch pat pos) (h₁ : s.copy = t.copy) (h₂ : pos.cast h₁ = pos') :
+    IsLongestRevMatch pat pos' := by
+  subst h₂; simpa
+
+theorem IsLongestRevMatch.sliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+    (h : IsLongestRevMatch pat pos) (p : s.Pos) (hp : p ≤ pos) : IsLongestRevMatch pat (Pos.sliceFrom p pos hp) := by
+  simp [isLongestRevMatch_iff] at ⊢ h
+  refine ⟨h.1, fun p' hp' => ?_⟩
+  rw [← isRevMatch_ofSliceFrom_iff]
+  exact h.2 _ (by simpa [Pos.lt_sliceFrom_iff] using hp')
+
+theorem isLongestRevMatch_of_ofSliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos} {pos : (s.sliceFrom p).Pos}
+    (h : IsLongestRevMatch pat (Pos.ofSliceFrom pos)) : IsLongestRevMatch pat pos := by
+  simpa using h.sliceFrom p
+
 /--
 Predicate stating that a match for a given pattern is never a proper prefix of another match.
 
@@ -228,7 +336,7 @@ theorem isLongestRevMatch_iff_isRevMatch {ρ : Type} (pat : ρ) [PatternModel pa
   exact ht₅ (NoSuffixPatternModel.eq_empty _ _ ht₂ (ht₅'' ▸ ht₂'))
 
 /--
-Predicate stating that the slice formed by {name}`startPos` and {name}`endPos` contains is a match
+Predicate stating that the slice formed by {name}`startPos` and {name}`endPos` contains a match
 of {name}`pat` in {name}`s` and it is longest among matches starting at {name}`startPos`.
 -/
 structure IsLongestMatchAt (pat : ρ) [PatternModel pat] {s : Slice} (startPos endPos : s.Pos) : Prop where
@@ -240,12 +348,21 @@ theorem isLongestMatchAt_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos₁ p
       ∃ (h : pos₁ ≤ pos₂), IsLongestMatch pat (Slice.Pos.sliceFrom _ _ h) :=
   ⟨fun ⟨h, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'⟩⟩
 
-theorem IsLongestMatchAt.lt {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+theorem IsLongestMatchAt.lt {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {startPos endPos : s.Pos}
     (h : IsLongestMatchAt pat startPos endPos) : startPos < endPos := by
   have := h.isLongestMatch_sliceFrom.ne_startPos
   rw [← Pos.startPos_lt_iff, ← Slice.Pos.ofSliceFrom_lt_ofSliceFrom_iff] at this
   simpa
 
+theorem IsLongestMatchAt.ne {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestMatchAt pat startPos endPos) : startPos ≠ endPos :=
+  Std.ne_of_lt h.lt
+
+@[simp]
+theorem not_isLongestMatchAt_self {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {startPos : s.Pos} :
+    ¬IsLongestMatchAt pat startPos startPos :=
+  fun h => h.ne rfl
+
 theorem IsLongestMatchAt.eq {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos endPos' : s.Pos}
     (h : IsLongestMatchAt pat startPos endPos) (h' : IsLongestMatchAt pat startPos endPos') :
     endPos = endPos' := by
@@ -282,6 +399,77 @@ theorem isLongestMatchAt_startPos_iff {pat : ρ} [PatternModel pat] {s : Slice}
     ⟨fun h => isLongestMatch_of_eq (by simp) (by simp) h,
      fun h => isLongestMatch_of_eq (by simp) (by simp) h⟩
 
+theorem isLongestMatch_iff_isLongestMatchAt_ofSliceFrom {pat : ρ} [PatternModel pat]
+    {s : Slice} {base : s.Pos} (endPos : (s.sliceFrom base).Pos) :
+    IsLongestMatch pat endPos ↔ IsLongestMatchAt pat base (Pos.ofSliceFrom endPos) := by
+  simp [← isLongestMatchAt_startPos_iff, isLongestMatchAt_iff_isLongestMatchAt_ofSliceFrom]
+
+theorem IsLongestMatchAt.matches_slice {pat : ρ} [PatternModel pat] {s : Slice}
+    {startPos endPos : s.Pos} (h : IsLongestMatchAt pat startPos endPos) :
+    PatternModel.Matches pat (s.slice startPos endPos h.le).copy := by
+  simpa using h.isLongestMatch_sliceFrom.isMatch.matches_copy
+
+@[simp]
+theorem isLongestMatchAt_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {startPos endPos : s.Pos} :
+    IsLongestMatchAt pat (startPos.cast hst) (endPos.cast hst) ↔ IsLongestMatchAt pat startPos endPos := by
+  simp [isLongestMatchAt_iff, Pos.sliceFrom_cast]
+
+theorem IsLongestMatchAt.of_eq {pat : ρ} [PatternModel pat] {s t : Slice} {s₁ e₁ : s.Pos} {s₂ e₂ : t.Pos}
+    (h : IsLongestMatchAt pat s₁ e₁) (h₁ : s.copy = t.copy) (h₂ : s₁.cast h₁ = s₂) (h₃ : e₁.cast h₁ = e₂) :
+    IsLongestMatchAt pat s₂ e₂ := by
+  subst h₂ h₃; simpa
+
+theorem IsLongestMatchAt.sliceTo {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestMatchAt pat startPos endPos) (p : s.Pos) (hp : endPos ≤ p) :
+    IsLongestMatchAt pat (Pos.sliceTo p startPos (by exact Std.le_trans h.le hp)) (Pos.sliceTo p endPos hp) := by
+  simp only [isLongestMatchAt_iff, Pos.sliceTo_le_sliceTo_iff] at ⊢ h
+  obtain ⟨h, hp'⟩ := h
+  exact ⟨h, (hp'.sliceTo (Pos.sliceFrom startPos p (Std.le_trans h hp)) (by simpa)).of_eq (by simp) (by ext; simp)⟩
+
+theorem isLongestMatchAt_of_ofSliceTo {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos} {startPos endPos : (s.sliceTo p).Pos}
+    (h : IsLongestMatchAt pat (Pos.ofSliceTo startPos) (Pos.ofSliceTo endPos)) :
+    IsLongestMatchAt pat startPos endPos := by
+  simpa using h.sliceTo p Pos.ofSliceTo_le
+
+/--
+Predicate stating that the range between two positions of {name}`s` can be covered by longest
+matches of the pattern within {name}`s`.
+-/
+inductive IsLongestMatchAtChain (pat : ρ) [PatternModel pat] {s : Slice} : s.Pos → s.Pos → Prop where
+  | nil (p : s.Pos) : IsLongestMatchAtChain pat p p
+  | cons (startPos middlePos endPos : s.Pos) : IsLongestMatchAt pat startPos middlePos →
+      IsLongestMatchAtChain pat middlePos endPos → IsLongestMatchAtChain pat startPos endPos
+
+attribute [simp] IsLongestMatchAtChain.nil
+
+theorem IsLongestMatchAtChain.eq_of_isLongestMatchAt_self {pat : ρ} [PatternModel pat] {s : Slice}
+    {startPos endPos : s.Pos} (h : IsLongestMatchAtChain pat startPos endPos) (h' : IsLongestMatchAt pat startPos startPos) :
+    startPos = endPos := by
+  induction h with
+  | nil => rfl
+  | cons p₁ p₂ p₃ h₁ h₂ ih =>
+    obtain rfl : p₁ = p₂ := h'.eq h₁
+    exact ih h₁
+
+theorem IsLongestMatchAtChain.le {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestMatchAtChain pat startPos endPos) : startPos ≤ endPos := by
+  induction h with
+  | nil => exact Std.le_refl _
+  | cons p₁ p₂ p₃ h₁ h₂ ih => exact Std.le_trans h₁.le ih
+
+theorem IsLongestMatchAtChain.sliceTo {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestMatchAtChain pat startPos endPos) (p : s.Pos) (hp : endPos ≤ p) :
+    IsLongestMatchAtChain pat (Pos.sliceTo p startPos (by exact Std.le_trans h.le hp)) (Pos.sliceTo p endPos hp) := by
+  induction h with
+  | nil => simp
+  | cons p₁ p₂ p₃ h₁ h₂ ih => exact .cons _ _ _ (h₁.sliceTo p (Std.le_trans h₂.le hp)) (ih hp)
+
+theorem isLongestMatchAtChain_of_ofSliceTo {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos}
+    {startPos endPos : (s.sliceTo p).Pos} (h : IsLongestMatchAtChain pat (Pos.ofSliceTo startPos) (Pos.ofSliceTo endPos)) :
+    IsLongestMatchAtChain pat startPos endPos := by
+  simpa using h.sliceTo p Pos.ofSliceTo_le
+
 /--
 Predicate stating that the slice formed by {name}`startPos` and {name}`endPos` contains is a match
 of {name}`pat` in {name}`s` and it is longest among matches ending at {name}`endPos`.
@@ -295,12 +483,21 @@ theorem isLongestRevMatchAt_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos
       ∃ (h : pos₁ ≤ pos₂), IsLongestRevMatch pat (Slice.Pos.sliceTo _ _ h) :=
   ⟨fun ⟨h, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'⟩⟩
 
-theorem IsLongestRevMatchAt.lt {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+theorem IsLongestRevMatchAt.lt {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {startPos endPos : s.Pos}
     (h : IsLongestRevMatchAt pat startPos endPos) : startPos < endPos := by
   have := h.isLongestRevMatch_sliceTo.ne_endPos
   rw [← Pos.lt_endPos_iff, ← Slice.Pos.ofSliceTo_lt_ofSliceTo_iff] at this
   simpa
 
+theorem IsLongestRevMatchAt.ne {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestRevMatchAt pat startPos endPos) : startPos ≠ endPos :=
+  Std.ne_of_lt h.lt
+
+@[simp]
+theorem not_isLongestRevMatchAt_self {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} {endPos : s.Pos} :
+    ¬IsLongestRevMatchAt pat endPos endPos :=
+  fun h => h.ne rfl
+
 theorem IsLongestRevMatchAt.eq {pat : ρ} [PatternModel pat] {s : Slice} {startPos startPos' endPos : s.Pos}
     (h : IsLongestRevMatchAt pat startPos endPos) (h' : IsLongestRevMatchAt pat startPos' endPos) :
     startPos = startPos' := by
@@ -335,6 +532,77 @@ theorem isLongestRevMatchAt_endPos_iff {pat : ρ} [PatternModel pat] {s : Slice}
     ⟨fun h => isLongestRevMatch_of_eq (by simp) (by simp) h,
      fun h => isLongestRevMatch_of_eq (by simp) (by simp) h⟩
 
+theorem isLongestRevMatch_iff_isLongestRevMatchAt_ofSliceTo {pat : ρ} [PatternModel pat]
+    {s : Slice} {base : s.Pos} (startPos : (s.sliceTo base).Pos) :
+    IsLongestRevMatch pat startPos ↔ IsLongestRevMatchAt pat (Pos.ofSliceTo startPos) base := by
+  simp [← isLongestRevMatchAt_endPos_iff, isLongestRevMatchAt_iff_isLongestRevMatchAt_ofSliceTo]
+
+theorem IsLongestRevMatchAt.matches_slice {pat : ρ} [PatternModel pat] {s : Slice}
+    {startPos endPos : s.Pos} (h : IsLongestRevMatchAt pat startPos endPos) :
+    PatternModel.Matches pat (s.slice startPos endPos h.le).copy := by
+  simpa using h.isLongestRevMatch_sliceTo.isRevMatch.matches_copy
+
+@[simp]
+theorem isLongestRevMatchAt_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {startPos endPos : s.Pos} :
+    IsLongestRevMatchAt pat (startPos.cast hst) (endPos.cast hst) ↔ IsLongestRevMatchAt pat startPos endPos := by
+  simp [isLongestRevMatchAt_iff, Pos.sliceTo_cast]
+
+theorem IsLongestRevMatchAt.of_eq {pat : ρ} [PatternModel pat] {s t : Slice} {s₁ e₁ : s.Pos} {s₂ e₂ : t.Pos}
+    (h : IsLongestRevMatchAt pat s₁ e₁) (h₁ : s.copy = t.copy) (h₂ : s₁.cast h₁ = s₂) (h₃ : e₁.cast h₁ = e₂) :
+    IsLongestRevMatchAt pat s₂ e₂ := by
+  subst h₂ h₃; simpa
+
+theorem IsLongestRevMatchAt.sliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestRevMatchAt pat startPos endPos) (p : s.Pos) (hp : p ≤ startPos) :
+    IsLongestRevMatchAt pat (Pos.sliceFrom p startPos hp) (Pos.sliceFrom p endPos (by exact Std.le_trans hp h.le)) := by
+  simp only [isLongestRevMatchAt_iff, Pos.sliceFrom_le_sliceFrom_iff] at ⊢ h
+  obtain ⟨h, hp'⟩ := h
+  exact ⟨h, (hp'.sliceFrom (Pos.sliceTo endPos p (Std.le_trans hp h)) (by simpa)).of_eq (by simp) (by ext; simp)⟩
+
+theorem isLongestRevMatchAt_of_ofSliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos} {startPos endPos : (s.sliceFrom p).Pos}
+    (h : IsLongestRevMatchAt pat (Pos.ofSliceFrom startPos) (Pos.ofSliceFrom endPos)) :
+    IsLongestRevMatchAt pat startPos endPos := by
+  simpa using h.sliceFrom p Pos.le_ofSliceFrom
+
+/--
+Predicate stating that the range between two positions of {name}`s` can be covered by longest
+reverse matches of the pattern within {name}`s`.
+-/
+inductive IsLongestRevMatchAtChain (pat : ρ) [PatternModel pat] {s : Slice} : s.Pos → s.Pos → Prop where
+  | nil (p : s.Pos) : IsLongestRevMatchAtChain pat p p
+  | cons (startPos middlePos endPos : s.Pos) : IsLongestRevMatchAtChain pat startPos middlePos →
+      IsLongestRevMatchAt pat middlePos endPos → IsLongestRevMatchAtChain pat startPos endPos
+
+attribute [simp] IsLongestRevMatchAtChain.nil
+
+theorem IsLongestRevMatchAtChain.eq_of_isLongestRevMatchAt_self {pat : ρ} [PatternModel pat] {s : Slice}
+    {startPos endPos : s.Pos} (h : IsLongestRevMatchAtChain pat startPos endPos) (h' : IsLongestRevMatchAt pat endPos endPos) :
+    startPos = endPos := by
+  induction h with
+  | nil => rfl
+  | cons mid endP hchain hmatch ih =>
+    obtain rfl := hmatch.eq h'
+    exact ih hmatch
+
+theorem IsLongestRevMatchAtChain.le {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestRevMatchAtChain pat startPos endPos) : startPos ≤ endPos := by
+  induction h with
+  | nil => exact Std.le_refl _
+  | cons mid endP hchain hmatch ih => exact Std.le_trans ih hmatch.le
+
+theorem IsLongestRevMatchAtChain.sliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestRevMatchAtChain pat startPos endPos) (p : s.Pos) (hp : p ≤ startPos) :
+    IsLongestRevMatchAtChain pat (Pos.sliceFrom p startPos hp) (Pos.sliceFrom p endPos (by exact Std.le_trans hp h.le)) := by
+  induction h with
+  | nil => simp
+  | cons mid endP hchain hmatch ih => exact .cons _ _ _ ih (hmatch.sliceFrom p (Std.le_trans hp hchain.le))
+
+theorem isLongestRevMatchAtChain_of_ofSliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {p : s.Pos}
+    {startPos endPos : (s.sliceFrom p).Pos} (h : IsLongestRevMatchAtChain pat (Pos.ofSliceFrom startPos) (Pos.ofSliceFrom endPos)) :
+    IsLongestRevMatchAtChain pat startPos endPos := by
+  simpa using h.sliceFrom p Pos.le_ofSliceFrom
+
 /--
 Predicate stating that there is a (longest) match starting at the given position.
 -/
@@ -360,7 +628,7 @@ theorem matchesAt_iff_exists_isMatch {pat : ρ} [PatternModel pat] {s : Slice}
      by simpa using hq⟩⟩
 
 @[simp]
-theorem not_matchesAt_endPos {pat : ρ} [PatternModel pat] {s : Slice} :
+theorem not_matchesAt_endPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} :
     ¬ MatchesAt pat s.endPos := by
   simp only [matchesAt_iff_exists_isMatch, Pos.endPos_le, exists_prop_eq]
   intro h
@@ -380,6 +648,14 @@ theorem IsLongestMatchAt.matchesAt {pat : ρ} [PatternModel pat] {s : Slice} {st
     (h : IsLongestMatchAt pat startPos endPos) : MatchesAt pat startPos where
   exists_isLongestMatchAt := ⟨_, h⟩
 
+@[simp]
+theorem matchesAt_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {pos : s.Pos} : MatchesAt pat (pos.cast hst) ↔ MatchesAt pat pos := by
+  simp only [matchesAt_iff_exists_isLongestMatchAt]
+  refine ⟨fun ⟨endPos, h⟩ => ?_, fun ⟨endPos, h⟩ => ?_⟩
+  · exact ⟨endPos.cast hst.symm, by simpa [← isLongestMatchAt_cast_iff hst]⟩
+  · exact ⟨endPos.cast hst, by simpa⟩
+
 /--
 Predicate stating that there is a (longest) match ending at the given position.
 -/
@@ -405,13 +681,13 @@ theorem revMatchesAt_iff_exists_isRevMatch {pat : ρ} [PatternModel pat] {s : Sl
      by simpa using hq⟩⟩
 
 @[simp]
-theorem not_revMatchesAt_startPos {pat : ρ} [PatternModel pat] {s : Slice} :
+theorem not_revMatchesAt_startPos {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice} :
     ¬ RevMatchesAt pat s.startPos := by
   simp only [revMatchesAt_iff_exists_isRevMatch, Pos.le_startPos, exists_prop_eq]
   intro h
   simpa [← Pos.ofSliceTo_inj] using h.ne_endPos
 
-theorem revMatchesAt_iff_revMatchesAt_ofSliceto {pat : ρ} [PatternModel pat] {s : Slice} {base : s.Pos}
+theorem revMatchesAt_iff_revMatchesAt_ofSliceTo {pat : ρ} [PatternModel pat] {s : Slice} {base : s.Pos}
     {pos : (s.sliceTo base).Pos} : RevMatchesAt pat pos ↔ RevMatchesAt pat (Pos.ofSliceTo pos) := by
   simp only [revMatchesAt_iff_exists_isLongestRevMatchAt]
   constructor
@@ -425,6 +701,14 @@ theorem IsLongestRevMatchAt.revMatchesAt {pat : ρ} [PatternModel pat] {s : Slic
     (h : IsLongestRevMatchAt pat startPos endPos) : RevMatchesAt pat endPos where
   exists_isLongestRevMatchAt := ⟨_, h⟩
 
+@[simp]
+theorem revMatchesAt_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {pos : s.Pos} : RevMatchesAt pat (pos.cast hst) ↔ RevMatchesAt pat pos := by
+  simp only [revMatchesAt_iff_exists_isLongestRevMatchAt]
+  refine ⟨fun ⟨endPos, h⟩ => ?_, fun ⟨endPos, h⟩ => ?_⟩
+  · exact ⟨endPos.cast hst.symm, by simpa [← isLongestRevMatchAt_cast_iff hst]⟩
+  · exact ⟨endPos.cast hst, by simpa⟩
+
 open Classical in
 /--
 Noncomputable model function returning the end point of the longest match starting at the given
@@ -450,6 +734,21 @@ theorem matchAt?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat]
   | case1 h => simpa using ⟨h⟩
   | case2 h => simpa using fun ⟨h'⟩ => h h'
 
+theorem lt_of_matchAt?_eq_some {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat]
+    {s : Slice} {startPos endPos : s.Pos} (h : matchAt? pat startPos = some endPos) :
+    startPos < endPos :=
+  (matchAt?_eq_some_iff.1 h).lt
+
+@[simp]
+theorem matchAt?_cast {ρ : Type} (pat : ρ) [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {startPos : s.Pos} :
+    matchAt? pat (startPos.cast hst) = (matchAt? pat startPos).map (Slice.Pos.cast · hst) := by
+  refine Option.ext (fun endPos => ?_)
+  have : endPos = (endPos.cast hst.symm).cast hst := by simp
+  conv => lhs; rw [this, matchAt?_eq_some_iff, isLongestMatchAt_cast_iff]
+  simp only [Option.map_eq_some_iff, matchAt?_eq_some_iff]
+  exact ⟨fun h => ⟨_, ⟨h, by simp⟩⟩, by rintro ⟨pos, h, rfl⟩; simpa⟩
+
 open Classical in
 /--
 Noncomputable model function returning the start point of the longest match ending at the given
@@ -475,6 +774,21 @@ theorem revMatchAt?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat]
   | case1 h => simpa using ⟨h⟩
   | case2 h => simpa using fun ⟨h'⟩ => h h'
 
+theorem lt_of_revMatchAt?_eq_some {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat]
+    {s : Slice} {startPos endPos : s.Pos} (h : revMatchAt? pat endPos = some startPos) :
+    startPos < endPos :=
+  (revMatchAt?_eq_some_iff.1 h).lt
+
+@[simp]
+theorem revMatchAt?_cast {ρ : Type} (pat : ρ) [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {startPos : s.Pos} :
+    revMatchAt? pat (startPos.cast hst) = (revMatchAt? pat startPos).map (Slice.Pos.cast · hst) := by
+  refine Option.ext (fun endPos => ?_)
+  have : endPos = (endPos.cast hst.symm).cast hst := by simp
+  conv => lhs; rw [this, revMatchAt?_eq_some_iff, isLongestRevMatchAt_cast_iff]
+  simp only [Option.map_eq_some_iff, revMatchAt?_eq_some_iff]
+  exact ⟨fun h => ⟨_, ⟨h, by simp⟩⟩, by rintro ⟨pos, h, rfl⟩; simpa⟩
+
 /--
 Predicate stating compatibility between {name}`PatternModel` and {name}`ForwardPattern`.
 
@@ -505,8 +819,8 @@ theorem LawfulForwardPatternModel.skipPrefix?_eq_none_iff {ρ : Type} {pat : ρ}
 /--
 Predicate stating compatibility between {name}`PatternModel` and {name}`BackwardPattern`.
 
-This extends {name}`LawfulForwardPattern`, but it is much stronger because it forces the
-{name}`ForwardPattern` to match the longest prefix of the given slice that matches the property
+This extends {name}`LawfulBackwardPattern`, but it is much stronger because it forces the
+{name}`BackwardPattern` to match the longest prefix of the given slice that matches the property
 supplied by the {name}`PatternModel` instance.
 -/
 class LawfulBackwardPatternModel {ρ : Type} (pat : ρ) [BackwardPattern pat]
@@ -570,6 +884,24 @@ theorem IsValidSearchFrom.endPos_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
   cases hl
   exact IsValidSearchFrom.endPos
 
+theorem isValidSearchFrom_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {pos : s.Pos} {l : List (SearchStep t)} :
+    IsValidSearchFrom pat (pos.cast hst) l ↔ IsValidSearchFrom pat pos (l.map (·.cast hst.symm)) := by
+  suffices ∀ (s t : Slice) (hst : s.copy = t.copy) (pos : s.Pos) (l : List (SearchStep s)),
+      IsValidSearchFrom pat pos l → IsValidSearchFrom pat (pos.cast hst) (l.map (·.cast hst)) from
+    ⟨fun h => by simpa using this _ _ hst.symm _ _ h, fun h => by
+      have hcomp : (SearchStep.cast hst) ∘ (SearchStep.cast hst.symm) = id := by ext; simp
+      simpa [hcomp] using this _ _ hst _ _ h⟩
+  intro s t hst pos l hl
+  induction hl with
+  | endPos => simpa using IsValidSearchFrom.endPos
+  | matched h₁ h₂ ih =>
+    simpa only [List.map_cons, SearchStep.cast_matched] using IsValidSearchFrom.matched (by simpa) ih
+  | mismatched h₁ h₂ h₃ ih =>
+    simp only [List.map_cons, SearchStep.cast_rejected]
+    refine IsValidSearchFrom.mismatched (by simpa) (fun p hp₁ hp₂ hp₃ => ?_) ih
+    exact h₂ (p.cast hst.symm) (by simpa [Pos.le_cast_iff]) (by simpa [Pos.cast_lt_iff]) (by simpa)
+
 /--
 Predicate stating compatibility between {name}`PatternModel` and {name}`ToForwardSearcher`.
 
@@ -663,6 +995,24 @@ theorem IsValidRevSearchFrom.startPos_of_eq {pat : ρ} [PatternModel pat] {s : S
   cases hl
   exact IsValidRevSearchFrom.startPos
 
+theorem isValidRevSearchFrom_cast_iff {pat : ρ} [PatternModel pat] {s t : Slice} (hst : s.copy = t.copy)
+    {pos : s.Pos} {l : List (SearchStep t)} :
+    IsValidRevSearchFrom pat (pos.cast hst) l ↔ IsValidRevSearchFrom pat pos (l.map (·.cast hst.symm)) := by
+  suffices ∀ (s t : Slice) (hst : s.copy = t.copy) (pos : s.Pos) (l : List (SearchStep s)),
+      IsValidRevSearchFrom pat pos l → IsValidRevSearchFrom pat (pos.cast hst) (l.map (·.cast hst)) from
+    ⟨fun h => by simpa using this _ _ hst.symm _ _ h, fun h => by
+      have hcomp : (SearchStep.cast hst) ∘ (SearchStep.cast hst.symm) = id := by ext; simp
+      simpa [hcomp] using this _ _ hst _ _ h⟩
+  intro s t hst pos l hl
+  induction hl with
+  | startPos => simpa using IsValidRevSearchFrom.startPos
+  | matched h₁ h₂ ih =>
+    simpa only [List.map_cons, SearchStep.cast_matched] using IsValidRevSearchFrom.matched (by simpa) ih
+  | mismatched h₁ h₂ h₃ ih =>
+    simp only [List.map_cons, SearchStep.cast_rejected]
+    refine IsValidRevSearchFrom.mismatched (by simpa) (fun p hp₁ hp₂ hp₃ => ?_) ih
+    exact h₂ (p.cast hst.symm) (by simpa [Pos.lt_cast_iff]) (by simpa [Pos.cast_le_iff]) (by simpa)
+
 /--
 Predicate stating compatibility between {name}`PatternModel` and {name}`ToBackwardSearcher`.
 

src/Init/Data/String/Lemmas/Pattern/Char.lean

@@ -28,7 +28,9 @@ namespace String.Slice.Pattern.Model.Char
 
 instance {c : Char} : PatternModel c where
   Matches s := s = String.singleton c
-  not_matches_empty := by simp
+
+instance {c : Char} : StrictPatternModel c where
+  not_matches_empty := by simp [PatternModel.Matches]
 
 instance {c : Char} : NoPrefixPatternModel c :=
   .of_length_eq (by simp +contextual [PatternModel.Matches])
@@ -168,11 +170,61 @@ theorem isLongestMatchAt_iff_isLongestMatchAt_beq {c : Char} {s : Slice}
     IsLongestMatchAt c pos pos' ↔ IsLongestMatchAt (· == c) pos pos' := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff_isLongestMatch_beq]
 
+theorem isLongestMatchAtChain_iff_isLongestMatchAtChain_beq {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestMatchAtChain c pos pos' ↔ IsLongestMatchAtChain (· == c) pos pos' := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | nil => simp
+    | cons p₁ p₂ p₃ h₁ h₂ ih => exact .cons _ _ _ (isLongestMatchAt_iff_isLongestMatchAt_beq.1 h₁) ih
+  · induction h with
+    | nil => simp
+    | cons p₁ p₂ p₃ h₁ h₂ ih => exact .cons _ _ _ (isLongestMatchAt_iff_isLongestMatchAt_beq.2 h₁) ih
+
+theorem isLongestMatchAtChain_iff {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestMatchAtChain c pos pos' ↔ pos ≤ pos' ∧ ∀ pos'', pos ≤ pos'' → (h : pos'' < pos') → pos''.get (Pos.ne_endPos_of_lt h) = c := by
+  simp [isLongestMatchAtChain_iff_isLongestMatchAtChain_beq, CharPred.isLongestMatchAtChain_iff]
+
+theorem isLongestMatchAtChain_iff_toList {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestMatchAtChain c pos pos' ↔
+      ∃ (h : pos ≤ pos'), (s.slice pos pos' h).copy.toList = List.replicate (s.slice pos pos' h).copy.length c := by
+  simp [isLongestMatchAtChain_iff_isLongestMatchAtChain_beq, CharPred.isLongestMatchAtChain_iff_toList,
+    List.eq_replicate_iff]
+
+theorem isLongestMatchAtChain_startPos_endPos_iff_toList {c : Char} {s : Slice} :
+    IsLongestMatchAtChain c s.startPos s.endPos ↔ s.copy.toList = List.replicate s.copy.length c := by
+  simp [isLongestMatchAtChain_iff_isLongestMatchAtChain_beq,
+    CharPred.isLongestMatchAtChain_startPos_endPos_iff_toList, List.eq_replicate_iff]
+
 theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_beq {c : Char} {s : Slice}
     {pos pos' : s.Pos} :
     IsLongestRevMatchAt c pos pos' ↔ IsLongestRevMatchAt (· == c) pos pos' := by
   simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff_isLongestRevMatch_beq]
 
+theorem isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_beq {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAtChain c pos pos' ↔ IsLongestRevMatchAtChain (· == c) pos pos' := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | nil => simp
+    | cons p₂ p₃ _ hmatch ih => exact .cons _ _ _ ih (isLongestRevMatchAt_iff_isLongestRevMatchAt_beq.1 hmatch)
+  · induction h with
+    | nil => simp
+    | cons p₂ p₃ _ hmatch ih => exact .cons _ _ _ ih (isLongestRevMatchAt_iff_isLongestRevMatchAt_beq.2 hmatch)
+
+theorem isLongestRevMatchAtChain_iff {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAtChain c pos pos' ↔ pos ≤ pos' ∧ ∀ pos'', pos ≤ pos'' → (h : pos'' < pos') → pos''.get (Pos.ne_endPos_of_lt h) = c := by
+  simp [isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_beq, CharPred.isLongestRevMatchAtChain_iff]
+
+theorem isLongestRevMatchAtChain_iff_toList {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAtChain c pos pos' ↔
+      ∃ (h : pos ≤ pos'), (s.slice pos pos' h).copy.toList = List.replicate (s.slice pos pos' h).copy.length c := by
+  simp [isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_beq, CharPred.isLongestRevMatchAtChain_iff_toList,
+    List.eq_replicate_iff]
+
+theorem isLongestRevMatchAtChain_startPos_endPos_iff_toList {c : Char} {s : Slice} :
+    IsLongestRevMatchAtChain c s.startPos s.endPos ↔ s.copy.toList = List.replicate s.copy.length c := by
+  simp [isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_beq,
+    CharPred.isLongestRevMatchAtChain_startPos_endPos_iff_toList, List.eq_replicate_iff]
+
 theorem matchesAt_iff_matchesAt_beq {c : Char} {s : Slice} {pos : s.Pos} :
     MatchesAt c pos ↔ MatchesAt (· == c) pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_beq]
@@ -242,18 +294,21 @@ theorem skipPrefix?_char_eq_skipPrefix?_beq {c : Char} {s : Slice} :
 theorem Pattern.ForwardPattern.skipPrefix?_char_eq_skipPrefix?_beq {c : Char} {s : Slice} :
     skipPrefix? c s = skipPrefix? (· == c) s := (rfl)
 
+theorem Pos.skip?_char_eq_skip?_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    pos.skip? c = pos.skip? (· == c) := (rfl)
+
 theorem Pos.skipWhile_char_eq_skipWhile_beq {c : Char} {s : Slice} (curr : s.Pos) :
     Pos.skipWhile curr c = Pos.skipWhile curr (· == c) := by
   fun_induction Pos.skipWhile curr c with
   | case1 pos nextCurr h₁ h₂ ih =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_char_eq_skipPrefix?_beq, h₁, h₂, ih]
+    simp [← Pos.skip?_char_eq_skip?_beq, h₁, h₂, ih]
   | case2 pos nextCurr h ih =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_char_eq_skipPrefix?_beq, h, ih]
+    simp [← Pos.skip?_char_eq_skip?_beq, h, ih]
   | case3 pos h =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_char_eq_skipPrefix?_beq]
+    simp [← Pos.skip?_char_eq_skip?_beq, h]
 
 theorem skipPrefixWhile_char_eq_skipPrefixWhile_beq {c : Char} {s : Slice} :
     s.skipPrefixWhile c = s.skipPrefixWhile (· == c) :=
@@ -269,7 +324,7 @@ theorem takeWhile_char_eq_takeWhile_beq {c : Char} {s : Slice} :
 
 theorem all_char_eq_all_beq {c : Char} {s : Slice} :
     s.all c = s.all (· == c) := by
-  simp only [all, dropWhile_char_eq_dropWhile_beq]
+  simp only [all, skipPrefixWhile_char_eq_skipPrefixWhile_beq]
 
 theorem find?_char_eq_find?_beq {c : Char} {s : Slice} :
     s.find? c = s.find? (· == c) :=
@@ -298,18 +353,21 @@ theorem dropSuffix_char_eq_dropSuffix_beq {c : Char} {s : Slice} :
 theorem Pattern.BackwardPattern.skipSuffix?_char_eq_skipSuffix?_beq {c : Char} {s : Slice} :
     skipSuffix? c s = skipSuffix? (· == c) s := (rfl)
 
+theorem Pos.revSkip?_char_eq_revSkip?_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    pos.revSkip? c = pos.revSkip? (· == c) := (rfl)
+
 theorem Pos.revSkipWhile_char_eq_revSkipWhile_beq {c : Char} {s : Slice} (curr : s.Pos) :
     Pos.revSkipWhile curr c = Pos.revSkipWhile curr (· == c) := by
   fun_induction Pos.revSkipWhile curr c with
   | case1 pos nextCurr h₁ h₂ ih =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_char_eq_skipSuffix?_beq, h₁, h₂, ih]
+    simp [← Pos.revSkip?_char_eq_revSkip?_beq, h₁, h₂, ih]
   | case2 pos nextCurr h ih =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_char_eq_skipSuffix?_beq, h, ih]
+    simp [← Pos.revSkip?_char_eq_revSkip?_beq, h, ih]
   | case3 pos h =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_char_eq_skipSuffix?_beq]
+    simp [← Pos.revSkip?_char_eq_revSkip?_beq, h]
 
 theorem skipSuffixWhile_char_eq_skipSuffixWhile_beq {c : Char} {s : Slice} :
     s.skipSuffixWhile c = s.skipSuffixWhile (· == c) :=
@@ -323,4 +381,16 @@ theorem takeEndWhile_char_eq_takeEndWhile_beq {c : Char} {s : Slice} :
     s.takeEndWhile c = s.takeEndWhile (· == c) := by
   simp only [takeEndWhile]; exact congrArg _ skipSuffixWhile_char_eq_skipSuffixWhile_beq
 
+theorem revFind?_char_eq_revFind?_beq {c : Char} {s : Slice} :
+    s.revFind? c = s.revFind? (· == c) :=
+  (rfl)
+
+theorem Pos.revFind?_char_eq_revFind?_beq {c : Char} {s : Slice} {p : s.Pos} :
+    p.revFind? c = p.revFind? (· == c) :=
+  (rfl)
+
+theorem revAll_char_eq_revAll_beq {c : Char} {s : Slice} :
+    s.revAll c = s.revAll (· == c) := by
+  simp [revAll, skipSuffixWhile_char_eq_skipSuffixWhile_beq]
+
 end String.Slice

src/Init/Data/String/Lemmas/Pattern/Find/Basic.lean

@@ -23,8 +23,8 @@ open Std String.Slice Pattern Pattern.Model
 
 namespace String.Slice
 
-theorem Pattern.Model.find?_eq_some_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
-    [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
+theorem Pattern.Model.find?_eq_some_iff {ρ : Type} (pat : ρ) [PatternModel pat] [StrictPatternModel pat]
+    {σ : Slice → Type} [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} {pos : s.Pos} :
     s.find? pat = some pos ↔ MatchesAt pat pos ∧ (∀ pos', pos' < pos → ¬ MatchesAt pat pos') := by
@@ -40,8 +40,8 @@ theorem Pattern.Model.find?_eq_some_iff {ρ : Type} (pat : ρ) [PatternModel pat
   | matched h₁ _ _ => have := h₁.matchesAt; grind
   | mismatched => grind
 
-theorem Pattern.Model.find?_eq_none_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
-    [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
+theorem Pattern.Model.find?_eq_none_iff {ρ : Type} (pat : ρ) [PatternModel pat] [StrictPatternModel pat]
+    {σ : Slice → Type} [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} :
     s.find? pat = none ↔ ∀ (pos : s.Pos), ¬ MatchesAt pat pos := by
@@ -65,15 +65,15 @@ theorem find?_eq_none_iff {ρ : Type} (pat : ρ) {σ : Slice → Type}
     [ToForwardSearcher pat σ] {s : Slice} : s.find? pat = none ↔ s.contains pat = false := by
   rw [← Option.isNone_iff_eq_none, ← Option.isSome_eq_false_iff, isSome_find?]
 
-theorem Pattern.Model.contains_eq_false_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
-    [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
+theorem Pattern.Model.contains_eq_false_iff {ρ : Type} (pat : ρ) [PatternModel pat] [StrictPatternModel pat]
+    {σ : Slice → Type} [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} :
     s.contains pat = false ↔ ∀ (pos : s.Pos), ¬ MatchesAt pat pos := by
   rw [← find?_eq_none_iff, Slice.find?_eq_none_iff]
 
-theorem Pattern.Model.contains_eq_true_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
-    [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
+theorem Pattern.Model.contains_eq_true_iff {ρ : Type} (pat : ρ) [PatternModel pat] [StrictPatternModel pat]
+    {σ : Slice → Type} [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} :
     s.contains pat ↔ ∃ (pos : s.Pos), MatchesAt pat pos := by
@@ -85,7 +85,7 @@ theorem Pos.find?_eq_find?_sliceFrom {ρ : Type} {pat : ρ} {σ : Slice → Type
     p.find? pat = ((s.sliceFrom p).find? pat).map Pos.ofSliceFrom :=
   (rfl)
 
-theorem Pattern.Model.posFind?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.posFind?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} {pos pos' : s.Pos} :
@@ -100,8 +100,8 @@ theorem Pattern.Model.posFind?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel
     refine ⟨Pos.sliceFrom _ _ h₁, ⟨by simpa using h₂, fun p hp₁ hp₂ => ?_⟩, by simp⟩
     exact h₃ (Pos.ofSliceFrom p) Slice.Pos.le_ofSliceFrom (Pos.lt_sliceFrom_iff.1 hp₁) hp₂
 
-theorem Pattern.Model.posFind?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat] {σ : Slice → Type}
-    [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
+theorem Pattern.Model.posFind?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat]
+    {σ : Slice → Type} [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} {pos : s.Pos} :
     pos.find? pat = none ↔ ∀ pos', pos ≤ pos' → ¬ MatchesAt pat pos' := by

src/Init/Data/String/Lemmas/Pattern/Find/String.lean

@@ -49,9 +49,10 @@ theorem contains_slice_iff {t s : Slice} :
   by_cases ht : t.isEmpty
   · simp [contains_eq_true_of_isEmpty ht s, copy_eq_empty_iff.mpr ht, String.toList_empty]
   · simp only [Bool.not_eq_true] at ht
+    have := Pattern.Model.ForwardSliceSearcher.strictPatternModel ht
     have := Pattern.Model.ForwardSliceSearcher.lawfulToForwardSearcherModel ht
     simp only [Pattern.Model.contains_eq_true_iff,
-      Pattern.Model.ForwardSliceSearcher.exists_matchesAt_iff_eq_append ht, isInfix_toList_iff]
+      Pattern.Model.ForwardSliceSearcher.exists_matchesAt_iff_eq_append, isInfix_toList_iff]
 
 @[simp]
 theorem contains_string_iff {t : String} {s : Slice} :

src/Init/Data/String/Lemmas/Pattern/Pred.lean

@@ -18,6 +18,7 @@ import Init.Data.String.Lemmas.Basic
 import Init.Data.String.Lemmas.Order
 import Init.Data.Order.Lemmas
 import Init.Data.String.OrderInstances
+import Init.Data.String.Lemmas.Iterate
 import Init.Omega
 import Init.Data.String.Lemmas.FindPos
 
@@ -27,8 +28,9 @@ namespace String.Slice.Pattern.Model.CharPred
 
 instance {p : Char → Bool} : PatternModel p where
   Matches s := ∃ c, s = singleton c ∧ p c
-  not_matches_empty := by
-    simp
+
+instance {p : Char → Bool} : StrictPatternModel p where
+  not_matches_empty := by simp [PatternModel.Matches]
 
 instance {p : Char → Bool} : NoPrefixPatternModel p :=
   .of_length_eq (by simp +contextual [PatternModel.Matches])
@@ -71,6 +73,39 @@ theorem isLongestMatchAt_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos}
   simp +contextual [Model.isLongestMatchAt_iff, isLongestMatch_iff, ← Pos.ofSliceFrom_inj,
     Pos.get_eq_get_ofSliceFrom, Pos.ofSliceFrom_next]
 
+theorem isLongestMatchAtChain_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestMatchAtChain p pos pos' ↔ pos ≤ pos' ∧ ∀ pos'', pos ≤ pos'' → (h : pos'' < pos') → p (pos''.get (Pos.ne_endPos_of_lt h)) := by
+  induction pos using WellFounded.induction Pos.wellFounded_gt with | h pos ih
+  obtain (h|rfl|h) := Std.lt_trichotomy pos pos'
+  · refine ⟨fun h => ?_, fun ⟨h₁, h₂⟩ => ?_⟩
+    · cases h with
+      | nil => exact (Std.lt_irrefl h).elim
+      | cons _ mid _ h₁ h₂ =>
+        obtain ⟨h₀, rfl, h₁'⟩ := isLongestMatchAt_iff.1 h₁
+        refine ⟨Std.le_of_lt h, fun pos'' hp₁ hp₂ => ?_⟩
+        obtain (hh|rfl) := Std.le_iff_lt_or_eq.1 hp₁
+        · exact ((ih (pos.next (Pos.ne_endPos_of_lt h)) Pos.lt_next).1 h₂).2 _ (by simpa) hp₂
+        · exact h₁'
+    · refine .cons _ (pos.next (Pos.ne_endPos_of_lt h)) _ ?_ ((ih _ Pos.lt_next).2 ?_)
+      · exact isLongestMatchAt_iff.2 ⟨Pos.ne_endPos_of_lt h, rfl, h₂ _ (by simp) h⟩
+      · exact ⟨by simpa, fun pos'' hp₁ hp₂ => h₂ _ (Std.le_trans Pos.le_next hp₁) hp₂⟩
+  · simpa using fun _ h₁ h₂ => (Std.lt_irrefl (Std.lt_of_le_of_lt h₁ h₂)).elim
+  · simpa [Std.not_le.2 h] using fun h' => (Std.not_le.2 h h'.le).elim
+
+theorem isLongestMatchAtChain_iff_toList {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestMatchAtChain p pos pos' ↔ ∃ (h : pos ≤ pos'), ∀ c, c ∈ (s.slice pos pos' h).copy.toList → p c := by
+  simp only [isLongestMatchAtChain_iff, mem_toList_copy_iff_exists_get, Pos.get_eq_get_ofSlice,
+    forall_exists_index]
+  refine ⟨fun ⟨h₁, h₂⟩ => ⟨h₁, fun c p' hp => ?_⟩, fun ⟨h₁, h₂⟩ => ⟨h₁, fun p' hp₁ hp₂ => ?_⟩⟩
+  · rintro rfl
+    exact h₂ _ Pos.le_ofSlice (by simp [Pos.ofSlice_lt_iff, h₁, hp])
+  · refine h₂ _ (Pos.slice p' _ _ hp₁ (Std.le_of_lt hp₂)) ?_ (by simp)
+    rwa [← Pos.lt_endPos_iff, ← Pos.slice_eq_endPos (h := h₁), Pos.slice_lt_slice_iff]
+
+theorem isLongestMatchAtChain_startPos_endPos_iff_toList {p : Char → Bool} {s : Slice} :
+    IsLongestMatchAtChain p s.startPos s.endPos ↔ ∀ c, c ∈ s.copy.toList → p c := by
+  simp [isLongestMatchAtChain_iff_toList]
+
 theorem isLongestRevMatchAt_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
     IsLongestRevMatchAt p pos pos' ↔ ∃ h, pos = pos'.prev h ∧ p ((pos'.prev h).get (by simp)) := by
   simp +contextual [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff, ← Pos.ofSliceTo_inj,
@@ -84,6 +119,35 @@ theorem isLongestRevMatchAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos}
     (hc : p ((pos.prev h).get (by simp))) : IsLongestRevMatchAt p (pos.prev h) pos :=
   isLongestRevMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAtChain_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAtChain p pos pos' ↔ pos ≤ pos' ∧ ∀ pos'', pos ≤ pos'' → (h : pos'' < pos') → p (pos''.get (Pos.ne_endPos_of_lt h)) := by
+  induction pos' using WellFounded.induction Pos.wellFounded_lt with | h pos' ih
+  obtain (h|rfl|h) := Std.lt_trichotomy pos pos'
+  · refine ⟨fun h => ?_, fun ⟨h₁, h₂⟩ => ?_⟩
+    · cases h with
+      | nil => exact (Std.lt_irrefl h).elim
+      | cons _ _ hchain hmatch =>
+        obtain ⟨hne, hmid, hp⟩ := isLongestRevMatchAt_iff.1 hmatch
+        refine ⟨Std.le_of_lt h, fun pos'' hp₁ hp₂ => ?_⟩
+        rcases Std.le_iff_lt_or_eq.1 (Pos.le_prev_iff_lt.2 hp₂) with hh | heq
+        · exact ((ih _ Pos.prev_lt).1 (hmid ▸ hchain)).2 _ hp₁ hh
+        · exact heq ▸ hp
+    · have hne : pos' ≠ s.startPos := Slice.Pos.ne_startPos_of_lt h
+      refine .cons _ (pos'.prev hne) _ ((ih _ Pos.prev_lt).2 ?_)
+        (isLongestRevMatchAt_of_get (h₂ _ (Pos.le_prev_iff_lt.2 h) Pos.prev_lt))
+      exact ⟨Pos.le_prev_iff_lt.2 h, fun pos'' hp₁ hp₂ =>
+        h₂ _ hp₁ (Std.lt_trans hp₂ Pos.prev_lt)⟩
+  · simpa using fun _ h₁ h₂ => (Std.lt_irrefl (Std.lt_of_le_of_lt h₁ h₂)).elim
+  · simpa [Std.not_le.2 h] using fun h' => (Std.not_le.2 h h'.le).elim
+
+theorem isLongestRevMatchAtChain_iff_toList {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAtChain p pos pos' ↔ ∃ (h : pos ≤ pos'), ∀ c, c ∈ (s.slice pos pos' h).copy.toList → p c :=
+  isLongestRevMatchAtChain_iff.trans (isLongestMatchAtChain_iff.symm.trans isLongestMatchAtChain_iff_toList)
+
+theorem isLongestRevMatchAtChain_startPos_endPos_iff_toList {p : Char → Bool} {s : Slice} :
+    IsLongestRevMatchAtChain p s.startPos s.endPos ↔ ∀ c, c ∈ s.copy.toList → p c := by
+  simp [isLongestRevMatchAtChain_iff_toList]
+
 instance {p : Char → Bool} : LawfulForwardPatternModel p where
   skipPrefix?_eq_some_iff {s} pos := by
     simp [isLongestMatch_iff, ForwardPattern.skipPrefix?, and_comm, eq_comm (b := pos)]
@@ -128,7 +192,9 @@ namespace Decidable
 
 instance {p : Char → Prop} [DecidablePred p] : PatternModel p where
   Matches := PatternModel.Matches (decide <| p ·)
-  not_matches_empty := PatternModel.not_matches_empty (pat := (decide <| p ·))
+
+instance {p : Char → Prop} [DecidablePred p] : StrictPatternModel p where
+  not_matches_empty := StrictPatternModel.not_matches_empty (pat := (decide <| p ·))
 
 instance {p : Char → Prop} [DecidablePred p] : NoPrefixPatternModel p where
   eq_empty := NoPrefixPatternModel.eq_empty (pat := (decide <| p ·))
@@ -182,6 +248,32 @@ theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_decide {p : Char → Prop} [
     IsLongestRevMatchAt p pos pos' ↔ IsLongestRevMatchAt (decide <| p ·) pos pos' := by
   simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff_isLongestRevMatch_decide]
 
+theorem isLongestMatchAtChain_iff_isLongestMatchAtChain_decide {p : Char → Prop} [DecidablePred p]
+    {s : Slice} {pos pos' : s.Pos} :
+    IsLongestMatchAtChain p pos pos' ↔ IsLongestMatchAtChain (decide <| p ·) pos pos' := by
+  constructor
+  · intro h; induction h with
+    | nil => exact .nil _
+    | cons _ mid _ hmatch hchain ih =>
+      exact .cons _ mid _ (isLongestMatchAt_iff_isLongestMatchAt_decide.1 hmatch) ih
+  · intro h; induction h with
+    | nil => exact .nil _
+    | cons _ mid _ hmatch hchain ih =>
+      exact .cons _ mid _ (isLongestMatchAt_iff_isLongestMatchAt_decide.2 hmatch) ih
+
+theorem isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_decide {p : Char → Prop} [DecidablePred p]
+    {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAtChain p pos pos' ↔ IsLongestRevMatchAtChain (decide <| p ·) pos pos' := by
+  constructor
+  · intro h; induction h with
+    | nil => exact .nil _
+    | cons _ _ hchain hmatch ih =>
+      exact .cons _ _ _ ih (isLongestRevMatchAt_iff_isLongestRevMatchAt_decide.1 hmatch)
+  · intro h; induction h with
+    | nil => exact .nil _
+    | cons _ _ hchain hmatch ih =>
+      exact .cons _ _ _ ih (isLongestRevMatchAt_iff_isLongestRevMatchAt_decide.2 hmatch)
+
 theorem isLongestMatchAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos pos' : s.Pos} :
     IsLongestMatchAt p pos pos' ↔ ∃ h, pos' = pos.next h ∧ p (pos.get h) := by
@@ -319,6 +411,9 @@ theorem dropPrefix_prop_eq_dropPrefix_decide {p : Char → Prop} [DecidablePred
 theorem skipPrefix?_prop_eq_skipPrefix?_decide {p : Char → Prop} [DecidablePred p] {s : Slice} :
     s.skipPrefix? p = s.skipPrefix? (decide <| p ·) := (rfl)
 
+theorem Pos.skip?_prop_eq_skip?_decide {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    pos.skip? p = pos.skip? (decide <| p ·) := (rfl)
+
 theorem Pattern.ForwardPattern.skipPrefix?_prop_eq_skipPrefix?_decide
     {p : Char → Prop} [DecidablePred p] {s : Slice} :
     skipPrefix? p s = skipPrefix? (decide <| p ·) s := (rfl)
@@ -329,13 +424,13 @@ theorem Pos.skipWhile_prop_eq_skipWhile_decide {p : Char → Prop} [DecidablePre
   fun_induction Pos.skipWhile curr p with
   | case1 pos nextCurr h₁ h₂ ih =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_prop_eq_skipPrefix?_decide, h₁, h₂, ih]
+    simp [← Pos.skip?_prop_eq_skip?_decide, h₁, h₂, ih]
   | case2 pos nextCurr h ih =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_prop_eq_skipPrefix?_decide, h, ih]
+    simp [← Pos.skip?_prop_eq_skip?_decide, h, ih]
   | case3 pos h =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_prop_eq_skipPrefix?_decide]
+    simp [← Pos.skip?_prop_eq_skip?_decide, h]
 
 theorem skipPrefixWhile_prop_eq_skipPrefixWhile_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} :
@@ -352,7 +447,7 @@ theorem takeWhile_prop_eq_takeWhile_decide {p : Char → Prop} [DecidablePred p]
 
 theorem all_prop_eq_all_decide {p : Char → Prop} [DecidablePred p] {s : Slice} :
     s.all p = s.all (decide <| p ·) := by
-  simp only [all, dropWhile_prop_eq_dropWhile_decide]
+  simp only [all, skipPrefixWhile_prop_eq_skipPrefixWhile_decide]
 
 theorem find?_prop_eq_find?_decide {p : Char → Prop} [DecidablePred p] {s : Slice} :
     s.find? p = s.find? (decide <| p ·) :=
@@ -383,19 +478,22 @@ theorem Pattern.BackwardPattern.skipSuffix?_prop_eq_skipSuffix?_decide
     {p : Char → Prop} [DecidablePred p] {s : Slice} :
     skipSuffix? p s = skipSuffix? (decide <| p ·) s := (rfl)
 
+theorem Pos.revSkip?_prop_eq_revSkip?_decide {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    pos.revSkip? p = pos.revSkip? (decide <| p ·) := (rfl)
+
 theorem Pos.revSkipWhile_prop_eq_revSkipWhile_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} (curr : s.Pos) :
     Pos.revSkipWhile curr p = Pos.revSkipWhile curr (decide <| p ·) := by
   fun_induction Pos.revSkipWhile curr p with
   | case1 pos nextCurr h₁ h₂ ih =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_prop_eq_skipSuffix?_decide, h₁, h₂, ih]
+    simp [← Pos.revSkip?_prop_eq_revSkip?_decide, h₁, h₂, ih]
   | case2 pos nextCurr h ih =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_prop_eq_skipSuffix?_decide, h, ih]
+    simp [← Pos.revSkip?_prop_eq_revSkip?_decide, h, ih]
   | case3 pos h =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_prop_eq_skipSuffix?_decide]
+    simp [← Pos.revSkip?_prop_eq_revSkip?_decide, h]
 
 theorem skipSuffixWhile_prop_eq_skipSuffixWhile_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} :
@@ -412,4 +510,8 @@ theorem takeEndWhile_prop_eq_takeEndWhile_decide {p : Char → Prop} [DecidableP
     s.takeEndWhile p = s.takeEndWhile (decide <| p ·) := by
   simp only [takeEndWhile]; exact congrArg _ skipSuffixWhile_prop_eq_skipSuffixWhile_decide
 
+theorem revAll_prop_eq_revAll_decide {p : Char → Prop} [DecidablePred p] {s : Slice} :
+    s.revAll p = s.revAll (decide <| p ·) := by
+  simp only [revAll, skipSuffixWhile_prop_eq_skipSuffixWhile_decide]
+
 end String.Slice

src/Init/Data/String/Lemmas/Pattern/Split/Basic.lean

@@ -36,7 +36,7 @@ This gives a low-level correctness proof from which higher-level API lemmas can
 namespace String.Slice.Pattern.Model
 
 @[cbv_opaque]
-public protected noncomputable def split {ρ : Type} (pat : ρ) [PatternModel pat] {s : Slice}
+public protected noncomputable def split {ρ : Type} (pat : ρ) [PatternModel pat] [StrictPatternModel pat] {s : Slice}
     (firstRejected curr : s.Pos) (hle : firstRejected ≤ curr) : List s.Subslice :=
   if h : curr = s.endPos then
     [s.subslice _ _ hle]
@@ -49,12 +49,12 @@ public protected noncomputable def split {ρ : Type} (pat : ρ) [PatternModel pa
 termination_by curr
 
 @[simp]
-public theorem split_endPos {ρ : Type} {pat : ρ} [PatternModel pat] {s : Slice}
+public theorem split_endPos {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat] {s : Slice}
     {firstRejected : s.Pos} :
     Model.split (s := s) pat firstRejected s.endPos (by simp) = [s.subslice firstRejected s.endPos (by simp)] := by
   simp [Model.split]
 
-public theorem split_eq_of_isLongestMatchAt {ρ : Type} {pat : ρ} [PatternModel pat]
+public theorem split_eq_of_isLongestMatchAt {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat]
     {s : Slice} {firstRejected start stop : s.Pos} {hle} (h : IsLongestMatchAt pat start stop) :
     Model.split pat firstRejected start hle =
       s.subslice _ _ hle :: Model.split pat stop stop (by exact Std.le_refl _) := by
@@ -63,7 +63,7 @@ public theorem split_eq_of_isLongestMatchAt {ρ : Type} {pat : ρ} [PatternModel
   · congr <;> exact (matchAt?_eq_some_iff.1 ‹_›).eq h
   · simp [matchAt?_eq_some_iff.2 ‹_›] at *
 
-public theorem split_eq_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pat]
+public theorem split_eq_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat]
     {s : Slice} {firstRejected start} (stop : s.Pos) (h₀ : start ≤ stop) {hle}
     (h : ∀ p, start ≤ p → p < stop → ¬ MatchesAt pat p) :
     Model.split pat firstRejected start hle =
@@ -80,7 +80,7 @@ public theorem split_eq_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pa
   · obtain rfl : start = stop := Std.le_antisymm h₀ (Std.not_lt.1 h')
     simp
 
-public theorem split_eq_next_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pat]
+public theorem split_eq_next_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pat] [StrictPatternModel pat]
     {s : Slice} {firstRejected start} {hle} (hs : start ≠ s.endPos) (h : ¬ MatchesAt pat start) :
     Model.split pat firstRejected start hle =
       Model.split pat firstRejected (start.next hs) (by exact Std.le_trans hle (by simp)) := by
@@ -103,7 +103,7 @@ def splitFromSteps {s : Slice} (currPos : s.Pos) (l : List (SearchStep s)) : Lis
   | .matched p q :: l => s.subslice! currPos p :: splitFromSteps q l
 
 theorem IsValidSearchFrom.splitFromSteps_eq_extend_split {ρ : Type} (pat : ρ)
-    [PatternModel pat] (l : List (SearchStep s)) (pos pos' : s.Pos) (h₀ : pos ≤ pos')
+    [PatternModel pat] [StrictPatternModel pat] (l : List (SearchStep s)) (pos pos' : s.Pos) (h₀ : pos ≤ pos')
     (h' : ∀ p, pos ≤ p → p < pos' → ¬ MatchesAt pat p)
     (h : IsValidSearchFrom pat pos' l) :
     splitFromSteps pos l = Model.split pat pos pos' h₀ := by
@@ -155,7 +155,7 @@ end Model
 open Model
 
 @[cbv_eval]
-public theorem toList_splitToSubslice_eq_modelSplit {ρ : Type} (pat : ρ) [PatternModel pat]
+public theorem toList_splitToSubslice_eq_modelSplit {ρ : Type} (pat : ρ) [PatternModel pat] [StrictPatternModel pat]
     {σ : Slice → Type} [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [LawfulToForwardSearcherModel pat] (s : Slice) :
     (s.splitToSubslice pat).toList = Model.split pat s.startPos s.startPos (by exact Std.le_refl _) := by
@@ -168,7 +168,7 @@ end Pattern
 open Pattern
 
 public theorem toList_splitToSubslice_of_isEmpty {ρ : Type} (pat : ρ)
-    [Model.PatternModel pat] {σ : Slice → Type}
+    [Model.PatternModel pat] [Model.StrictPatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [Model.LawfulToForwardSearcherModel pat] {s : Slice}
     (h : s.isEmpty = true) :
@@ -182,7 +182,7 @@ public theorem toList_split_eq_splitToSubslice {ρ : Type} (pat : ρ) {σ : Slic
   simp [split, Std.Iter.toList_map]
 
 public theorem toList_split_of_isEmpty {ρ : Type} (pat : ρ)
-    [Model.PatternModel pat] {σ : Slice → Type}
+    [Model.PatternModel pat] [Model.StrictPatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [Model.LawfulToForwardSearcherModel pat] {s : Slice}
     (h : s.isEmpty = true) :
@@ -200,7 +200,7 @@ public theorem split_eq_split_toSlice {ρ : Type} {pat : ρ} {σ : Slice → Typ
 
 @[simp]
 public theorem toList_split_empty {ρ : Type} (pat : ρ)
-    [Model.PatternModel pat] {σ : Slice → Type}
+    [Model.PatternModel pat] [Model.StrictPatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [Model.LawfulToForwardSearcherModel pat] :
     ("".split pat).toList.map Slice.copy = [""] := by

src/Init/Data/String/Lemmas/Pattern/String/Basic.lean

@@ -10,6 +10,9 @@ public import Init.Data.String.Pattern.String
 public import Init.Data.String.Lemmas.Pattern.Basic
 import Init.Data.String.Lemmas.IsEmpty
 import Init.Data.String.Lemmas.Basic
+import Init.Data.String.Lemmas.Intercalate
+import Init.Data.String.OrderInstances
+import Init.Data.String.Lemmas.Splits
 import Init.Data.ByteArray.Lemmas
 import Init.Omega
 
@@ -20,17 +23,10 @@ namespace String.Slice.Pattern.Model
 namespace ForwardSliceSearcher
 
 instance {pat : Slice} : PatternModel pat where
-  /-
-  See the docstring of `PatternModel` for an explanation about why we disallow matching the
-  empty string.
+  Matches s := s = pat.copy
 
-  Requiring `s ≠ ""` is a trick that allows us to give a `PatternModel` instance
-  unconditionally, without forcing `pat.copy` to be non-empty (which would make it very awkward
-  to state theorems about the instance). It does not change anything about the fact that all lemmas
-  about this instance require `pat.isEmpty = false`.
-  -/
-  Matches s := s ≠ "" ∧ s = pat.copy
-  not_matches_empty := by simp
+theorem strictPatternModel {pat : Slice} (hpat : pat.isEmpty = false) : StrictPatternModel pat where
+  not_matches_empty := by simpa [PatternModel.Matches]
 
 instance {pat : Slice} : NoPrefixPatternModel pat :=
   .of_length_eq (by simp +contextual [PatternModel.Matches])
@@ -38,59 +34,111 @@ instance {pat : Slice} : NoPrefixPatternModel pat :=
 instance {pat : Slice} : NoSuffixPatternModel pat :=
   .of_length_eq (by simp +contextual [PatternModel.Matches])
 
-theorem isMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem isMatch_iff {pat s : Slice} {pos : s.Pos} :
     IsMatch pat pos ↔ (s.sliceTo pos).copy = pat.copy := by
-  simp only [Model.isMatch_iff, PatternModel.Matches, ne_eq, copy_eq_empty_iff,
-    Bool.not_eq_true, and_iff_right_iff_imp]
-  intro h'
-  rw [← isEmpty_copy (s := s.sliceTo pos), h', isEmpty_copy, h]
+  simp [Model.isMatch_iff, PatternModel.Matches]
 
-theorem isRevMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem isRevMatch_iff {pat s : Slice} {pos : s.Pos} :
     IsRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat.copy := by
-  simp only [Model.isRevMatch_iff, PatternModel.Matches, ne_eq, copy_eq_empty_iff,
-    Bool.not_eq_true, and_iff_right_iff_imp]
-  intro h'
-  rw [← isEmpty_copy (s := s.sliceFrom pos), h', isEmpty_copy, h]
+  simp [Model.isRevMatch_iff, PatternModel.Matches]
 
-theorem isLongestMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestMatch_iff {pat s : Slice} {pos : s.Pos} :
     IsLongestMatch pat pos ↔ (s.sliceTo pos).copy = pat.copy := by
-  rw [isLongestMatch_iff_isMatch, isMatch_iff h]
+  rw [isLongestMatch_iff_isMatch, isMatch_iff]
 
-theorem isLongestRevMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestRevMatch_iff {pat s : Slice} {pos : s.Pos} :
     IsLongestRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat.copy := by
-  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff h]
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
 
-theorem isLongestMatchAt_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestMatchAt_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat.copy := by
-  simp [Model.isLongestMatchAt_iff, isLongestMatch_iff h]
+  simp [Model.isLongestMatchAt_iff, isLongestMatch_iff]
 
-theorem isLongestRevMatchAt_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestMatchAtChain_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestMatchAtChain pat pos₁ pos₂ ↔
+      ∃ h n, (s.slice pos₁ pos₂ h).copy = String.join (List.replicate n pat.copy) := by
+  refine ⟨fun h => ⟨h.le, ?_⟩, fun ⟨h, n, h'⟩ => ?_⟩
+  · induction h with
+    | nil => simpa using ⟨0, by simp⟩
+    | cons p₁ p₂ p₃ h₁ h₂ ih =>
+      rw [isLongestMatchAt_iff] at h₁
+      obtain ⟨n, ih⟩ := ih
+      obtain ⟨h₀, h₁⟩ := h₁
+      have : (s.slice p₁ p₃ (Std.le_trans h₀ h₂.le)).copy = (s.slice p₁ p₂ h₀).copy ++ (s.slice p₂ p₃ h₂.le).copy := by
+        simp [(Slice.Pos.slice p₂ _ _ h₀ h₂.le).splits.eq_append]
+      refine ⟨n + 1, ?_⟩
+      rw [this, h₁, ih]
+      simp [← String.join_cons, ← List.replicate_succ]
+  · induction n generalizing pos₁ pos₂ with
+    | zero => simp_all
+    | succ n ih =>
+      rw [List.replicate_succ, String.join_cons] at h'
+      refine .cons _ (Pos.ofSlice (Pos.ofEqAppend h')) _ ?_ (ih ?_ Pos.ofSlice_le ?_)
+      · simpa [isLongestMatchAt_iff] using (Pos.splits_ofEqAppend h').copy_sliceTo_eq
+      · simpa [sliceFrom_slice ▸ (Pos.splits_ofEqAppend h').copy_sliceFrom_eq] using ⟨n, rfl⟩
+      · simpa using (Pos.splits_ofEqAppend h').copy_sliceFrom_eq
+
+theorem isLongestMatchAtChain_startPos_endPos_iff {pat s : Slice} :
+    IsLongestMatchAtChain pat s.startPos s.endPos ↔
+      ∃ n, s.copy = String.join (List.replicate n pat.copy) := by
+  simp [isLongestMatchAtChain_iff]
+
+theorem isLongestRevMatchAt_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestRevMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat.copy := by
-  simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff h]
+  simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff]
 
-theorem isLongestMatchAt_iff_splits {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestRevMatchAtChain_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestRevMatchAtChain pat pos₁ pos₂ ↔
+      ∃ h n, (s.slice pos₁ pos₂ h).copy = String.join (List.replicate n pat.copy) := by
+  refine ⟨fun h => ⟨h.le, ?_⟩, fun ⟨h, n, h'⟩ => ?_⟩
+  · induction h with
+    | nil => simpa using ⟨0, by simp⟩
+    | cons p₂ p₃ h₁ h₂ ih =>
+      rw [isLongestRevMatchAt_iff] at h₂
+      obtain ⟨n, ih⟩ := ih
+      obtain ⟨h₀, h₂⟩ := h₂
+      have : (s.slice pos₁ p₃ (Std.le_trans h₁.le h₀)).copy = (s.slice pos₁ p₂ h₁.le).copy ++ (s.slice p₂ p₃ h₀).copy := by
+        simp [(Slice.Pos.slice p₂ _ _ (IsLongestRevMatchAtChain.le ‹_›) h₀).splits.eq_append]
+      refine ⟨n + 1, ?_⟩
+      rw [this, h₂, ih]
+      simp [← List.replicate_append_replicate]
+  · induction n generalizing pos₁ pos₂ with
+    | zero => simp_all
+    | succ n ih =>
+      have h'' : (s.slice pos₁ pos₂ h).copy = String.join (List.replicate n pat.copy) ++ pat.copy := by
+        rw [h', List.replicate_succ', String.join_append]; simp
+      refine .cons _ (Pos.ofSlice (Pos.ofEqAppend h'')) _ (ih ?_ Pos.le_ofSlice ?_) ?_
+      · simpa [sliceTo_slice ▸ (Pos.splits_ofEqAppend h'').copy_sliceTo_eq] using ⟨n, rfl⟩
+      · simpa using (Pos.splits_ofEqAppend h'').copy_sliceTo_eq
+      · simpa [isLongestRevMatchAt_iff] using (Pos.splits_ofEqAppend h'').copy_sliceFrom_eq
+
+theorem isLongestRevMatchAtChain_startPos_endPos_iff {pat s : Slice} :
+    IsLongestRevMatchAtChain pat s.startPos s.endPos ↔
+      ∃ n, s.copy = String.join (List.replicate n pat.copy) := by
+  simp [isLongestRevMatchAtChain_iff]
+
+theorem isLongestMatchAt_iff_splits {pat s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt pat pos₁ pos₂ ↔ ∃ t₁ t₂, pos₁.Splits t₁ (pat.copy ++ t₂) ∧
       pos₂.Splits (t₁ ++ pat.copy) t₂ := by
-  simp only [isLongestMatchAt_iff h, copy_slice_eq_iff_splits]
+  simp only [isLongestMatchAt_iff, copy_slice_eq_iff_splits]
 
-theorem isLongestRevMatchAt_iff_splits {pat s : Slice} {pos₁ pos₂ : s.Pos}
-    (h : pat.isEmpty = false) :
+theorem isLongestRevMatchAt_iff_splits {pat s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestRevMatchAt pat pos₁ pos₂ ↔ ∃ t₁ t₂, pos₁.Splits t₁ (pat.copy ++ t₂) ∧
       pos₂.Splits (t₁ ++ pat.copy) t₂ := by
-  simp only [isLongestRevMatchAt_iff h, copy_slice_eq_iff_splits]
+  simp only [isLongestRevMatchAt_iff, copy_slice_eq_iff_splits]
 
-theorem isLongestMatch_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestMatch_iff_splits {pat s : Slice} {pos : s.Pos} :
     IsLongestMatch pat pos ↔ ∃ t, pos.Splits pat.copy t := by
-  rw [isLongestMatch_iff h, copy_sliceTo_eq_iff_exists_splits]
+  rw [isLongestMatch_iff, copy_sliceTo_eq_iff_exists_splits]
 
-theorem isLongestRevMatch_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem isLongestRevMatch_iff_splits {pat s : Slice} {pos : s.Pos} :
     IsLongestRevMatch pat pos ↔ ∃ t, pos.Splits t pat.copy := by
-  rw [isLongestRevMatch_iff h, copy_sliceFrom_eq_iff_exists_splits]
+  rw [isLongestRevMatch_iff, copy_sliceFrom_eq_iff_exists_splits]
 
 theorem isLongestMatchAt_iff_extract {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
     IsLongestMatchAt pat pos₁ pos₂ ↔
       s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx = pat.copy.toByteArray := by
-  rw [isLongestMatchAt_iff h]
+  rw [isLongestMatchAt_iff]
   refine ⟨fun ⟨h, h'⟩ => ?_, fun h' => ?_⟩
   · simp [← h', toByteArray_copy_slice]
   · rw [← Slice.toByteArray_copy_ne_empty_iff, ← h', ne_eq, ByteArray.extract_eq_empty_iff] at h
@@ -102,7 +150,7 @@ theorem isLongestRevMatchAt_iff_extract {pat s : Slice} {pos₁ pos₂ : s.Pos}
     IsLongestRevMatchAt pat pos₁ pos₂ ↔
       s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx =
         pat.copy.toByteArray := by
-  rw [isLongestRevMatchAt_iff h]
+  rw [isLongestRevMatchAt_iff]
   refine ⟨fun ⟨h, h'⟩ => ?_, fun h' => ?_⟩
   · simp [← h', toByteArray_copy_slice]
   · rw [← Slice.toByteArray_copy_ne_empty_iff, ← h', ne_eq, ByteArray.extract_eq_empty_iff] at h
@@ -130,21 +178,21 @@ theorem offset_of_isLongestRevMatchAt {pat s : Slice} {pos₁ pos₂ : s.Pos}
   suffices pos₂.offset.byteIdx ≤ s.utf8ByteSize by omega
   simpa [Pos.le_iff, Pos.Raw.le_iff] using pos₂.le_endPos
 
-theorem matchesAt_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem matchesAt_iff_splits {pat s : Slice} {pos : s.Pos} :
     MatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ (pat.copy ++ t₂) := by
-  simp only [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_splits h]
+  simp only [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_splits]
   exact ⟨fun ⟨e, t₁, t₂, ht₁, ht₂⟩ => ⟨t₁, t₂, ht₁⟩,
     fun ⟨t₁, t₂, ht⟩ => ⟨ht.rotateRight, t₁, t₂, ht, ht.splits_rotateRight⟩⟩
 
-theorem revMatchesAt_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+theorem revMatchesAt_iff_splits {pat s : Slice} {pos : s.Pos} :
     RevMatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits (t₁ ++ pat.copy) t₂ := by
-  simp only [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_splits h]
+  simp only [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_splits]
   exact ⟨fun ⟨e, t₁, t₂, ht₁, ht₂⟩ => ⟨t₁, t₂, ht₂⟩,
     fun ⟨t₁, t₂, ht⟩ => ⟨ht.rotateLeft, t₁, t₂, ht.splits_rotateLeft, ht⟩⟩
 
-theorem exists_matchesAt_iff_eq_append {pat s : Slice} (h : pat.isEmpty = false) :
+theorem exists_matchesAt_iff_eq_append {pat s : Slice} :
     (∃ (pos : s.Pos), MatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat.copy ++ t₂ := by
-  simp only [matchesAt_iff_splits h]
+  simp only [matchesAt_iff_splits]
   constructor
   · rintro ⟨pos, t₁, t₂, hsplit⟩
     exact ⟨t₁, t₂, by rw [hsplit.eq_append, append_assoc]⟩
@@ -154,9 +202,9 @@ theorem exists_matchesAt_iff_eq_append {pat s : Slice} (h : pat.isEmpty = false)
         ⟨t₁, pat.copy ++ t₂, by rw [← append_assoc]; exact heq, rfl⟩
     exact ⟨s.pos _ hvalid, t₁, t₂, ⟨by rw [← append_assoc]; exact heq, by simp⟩⟩
 
-theorem exists_revMatchesAt_iff_eq_append {pat s : Slice} (h : pat.isEmpty = false) :
+theorem exists_revMatchesAt_iff_eq_append {pat s : Slice} :
     (∃ (pos : s.Pos), RevMatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat.copy ++ t₂ := by
-  simp only [revMatchesAt_iff_splits h]
+  simp only [revMatchesAt_iff_splits]
   constructor
   · rintro ⟨pos, t₁, t₂, hsplit⟩
     exact ⟨t₁, t₂, by rw [hsplit.eq_append, append_assoc]⟩
@@ -233,8 +281,10 @@ end ForwardSliceSearcher
 namespace ForwardStringSearcher
 
 instance {pat : String} : PatternModel pat where
-  Matches s := s ≠ "" ∧ s = pat
-  not_matches_empty := by simp
+  Matches s := s = pat
+
+theorem strictPatternModel {pat : String} (h : pat ≠ "") : StrictPatternModel pat where
+  not_matches_empty := by simpa [PatternModel.Matches]
 
 instance {pat : String} : NoPrefixPatternModel pat :=
   .of_length_eq (by simp +contextual [PatternModel.Matches])
@@ -267,12 +317,60 @@ theorem isLongestMatchAt_iff_isLongestMatchAt_toSlice {pat : String} {s : Slice}
       IsLongestMatchAt (ρ := Slice) pat.toSlice pos₁ pos₂ := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff_isLongestMatch_toSlice]
 
+theorem isLongestMatchAtChain_iff_isLongestMatchAtChain_toSlice {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestMatchAtChain pat pos₁ pos₂ ↔
+      IsLongestMatchAtChain pat.toSlice pos₁ pos₂ := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | nil => simp
+    | cons p₁ p₂ p₃ h₁ h₂ ih =>
+      exact .cons _ _ _ (isLongestMatchAt_iff_isLongestMatchAt_toSlice.1 h₁) ih
+  · induction h with
+    | nil => simp
+    | cons p₁ p₂ p₃ h₁ h₂ ih =>
+      exact .cons _ _ _ (isLongestMatchAt_iff_isLongestMatchAt_toSlice.2 h₁) ih
+
+theorem isLongestMatchAtChain_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestMatchAtChain pat pos₁ pos₂ ↔
+      ∃ h n, (s.slice pos₁ pos₂ h).copy = String.join (List.replicate n pat) := by
+  simp [isLongestMatchAtChain_iff_isLongestMatchAtChain_toSlice,
+    ForwardSliceSearcher.isLongestMatchAtChain_iff]
+
+theorem isLongestMatchAtChain_startPos_endPos_iff {pat : String} {s : Slice} :
+    IsLongestMatchAtChain pat s.startPos s.endPos ↔
+      ∃ n, s.copy = String.join (List.replicate n pat) := by
+  simp [isLongestMatchAtChain_iff]
+
 theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice {pat : String} {s : Slice}
     {pos₁ pos₂ : s.Pos} :
     IsLongestRevMatchAt (ρ := String) pat pos₁ pos₂ ↔
       IsLongestRevMatchAt (ρ := Slice) pat.toSlice pos₁ pos₂ := by
   simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff_isLongestRevMatch_toSlice]
 
+theorem isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_toSlice {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestRevMatchAtChain pat pos₁ pos₂ ↔
+      IsLongestRevMatchAtChain pat.toSlice pos₁ pos₂ := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | nil => simp
+    | cons p₂ p₃ _ hmatch ih =>
+      exact .cons _ _ _ ih (isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice.1 hmatch)
+  · induction h with
+    | nil => simp
+    | cons p₂ p₃ _ hmatch ih =>
+      exact .cons _ _ _ ih (isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice.2 hmatch)
+
+theorem isLongestRevMatchAtChain_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestRevMatchAtChain pat pos₁ pos₂ ↔
+      ∃ h n, (s.slice pos₁ pos₂ h).copy = String.join (List.replicate n pat) := by
+  simp [isLongestRevMatchAtChain_iff_isLongestRevMatchAtChain_toSlice,
+    ForwardSliceSearcher.isLongestRevMatchAtChain_iff]
+
+theorem isLongestRevMatchAtChain_startPos_endPos_iff {pat : String} {s : Slice} :
+    IsLongestRevMatchAtChain pat s.startPos s.endPos ↔
+      ∃ n, s.copy = String.join (List.replicate n pat) := by
+  simp [isLongestRevMatchAtChain_iff]
+
 theorem matchesAt_iff_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
     MatchesAt (ρ := String) pat pos ↔ MatchesAt (ρ := Slice) pat.toSlice pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_toSlice]
@@ -282,61 +380,55 @@ theorem revMatchesAt_iff_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
   simp [revMatchesAt_iff_exists_isLongestRevMatchAt,
     isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
 
-private theorem toSlice_isEmpty (h : pat ≠ "") : pat.toSlice.isEmpty = false := by
-  rwa [isEmpty_toSlice, isEmpty_eq_false_iff]
-
-theorem isMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+theorem isMatch_iff {pat : String} {s : Slice} {pos : s.Pos} :
     IsMatch pat pos ↔ (s.sliceTo pos).copy = pat := by
-  rw [isMatch_iff_slice, ForwardSliceSearcher.isMatch_iff (toSlice_isEmpty h)]
+  rw [isMatch_iff_slice, ForwardSliceSearcher.isMatch_iff]
   simp
 
-theorem isRevMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+theorem isRevMatch_iff {pat : String} {s : Slice} {pos : s.Pos} :
     IsRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat := by
-  rw [isRevMatch_iff_slice, ForwardSliceSearcher.isRevMatch_iff (toSlice_isEmpty h)]
+  rw [isRevMatch_iff_slice, ForwardSliceSearcher.isRevMatch_iff]
   simp
 
-theorem isLongestMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+theorem isLongestMatch_iff {pat : String} {s : Slice} {pos : s.Pos} :
     IsLongestMatch pat pos ↔ (s.sliceTo pos).copy = pat := by
-  rw [isLongestMatch_iff_isMatch, isMatch_iff h]
+  rw [isLongestMatch_iff_isMatch, isMatch_iff]
 
-theorem isLongestRevMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+theorem isLongestRevMatch_iff {pat : String} {s : Slice} {pos : s.Pos} :
     IsLongestRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat := by
-  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff h]
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
 
-theorem isLongestMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} (h : pat ≠ "") :
+theorem isLongestMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat := by
   rw [isLongestMatchAt_iff_isLongestMatchAt_toSlice,
-    ForwardSliceSearcher.isLongestMatchAt_iff (toSlice_isEmpty h)]
+    ForwardSliceSearcher.isLongestMatchAt_iff]
   simp
 
-theorem isLongestRevMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} (h : pat ≠ "") :
+theorem isLongestRevMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestRevMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat := by
   rw [isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice,
-    ForwardSliceSearcher.isLongestRevMatchAt_iff (toSlice_isEmpty h)]
+    ForwardSliceSearcher.isLongestRevMatchAt_iff]
   simp
 
-theorem isLongestMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
-    (h : pat ≠ "") :
+theorem isLongestMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt pat pos₁ pos₂ ↔
       ∃ t₁ t₂, pos₁.Splits t₁ (pat ++ t₂) ∧ pos₂.Splits (t₁ ++ pat) t₂ := by
   rw [isLongestMatchAt_iff_isLongestMatchAt_toSlice,
-    ForwardSliceSearcher.isLongestMatchAt_iff_splits (toSlice_isEmpty h)]
+    ForwardSliceSearcher.isLongestMatchAt_iff_splits]
   simp
 
-theorem isLongestRevMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
-    (h : pat ≠ "") :
+theorem isLongestRevMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestRevMatchAt pat pos₁ pos₂ ↔
       ∃ t₁ t₂, pos₁.Splits t₁ (pat ++ t₂) ∧ pos₂.Splits (t₁ ++ pat) t₂ := by
   rw [isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice,
-    ForwardSliceSearcher.isLongestRevMatchAt_iff_splits (toSlice_isEmpty h)]
+    ForwardSliceSearcher.isLongestRevMatchAt_iff_splits]
   simp
 
-theorem isLongestMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
-    (h : pat ≠ "") :
+theorem isLongestMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} (h : pat ≠ "") :
     IsLongestMatchAt pat pos₁ pos₂ ↔
       s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx = pat.toByteArray := by
   rw [isLongestMatchAt_iff_isLongestMatchAt_toSlice,
-    ForwardSliceSearcher.isLongestMatchAt_iff_extract (toSlice_isEmpty h)]
+    ForwardSliceSearcher.isLongestMatchAt_iff_extract (by simpa)]
   simp
 
 theorem isLongestRevMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
@@ -344,38 +436,38 @@ theorem isLongestRevMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos
     IsLongestRevMatchAt pat pos₁ pos₂ ↔
       s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx = pat.toByteArray := by
   rw [isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice,
-    ForwardSliceSearcher.isLongestRevMatchAt_iff_extract (toSlice_isEmpty h)]
+    ForwardSliceSearcher.isLongestRevMatchAt_iff_extract (by simpa)]
   simp
 
 theorem offset_of_isLongestMatchAt {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
     (h : pat ≠ "") (h' : IsLongestMatchAt pat pos₁ pos₂) :
     pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
   rw [show pat.utf8ByteSize = pat.toSlice.utf8ByteSize from utf8ByteSize_toSlice.symm]
-  exact ForwardSliceSearcher.offset_of_isLongestMatchAt (toSlice_isEmpty h)
+  exact ForwardSliceSearcher.offset_of_isLongestMatchAt (by simpa)
     (isLongestMatchAt_iff_isLongestMatchAt_toSlice.1 h')
 
 theorem offset_of_isLongestRevMatchAt {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
     (h : pat ≠ "") (h' : IsLongestRevMatchAt pat pos₁ pos₂) :
     pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
   rw [show pat.utf8ByteSize = pat.toSlice.utf8ByteSize from utf8ByteSize_toSlice.symm]
-  exact ForwardSliceSearcher.offset_of_isLongestRevMatchAt (toSlice_isEmpty h)
+  exact ForwardSliceSearcher.offset_of_isLongestRevMatchAt (by simpa)
     (isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice.1 h')
 
-theorem matchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+theorem matchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} :
     MatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ (pat ++ t₂) := by
   rw [matchesAt_iff_toSlice,
-    ForwardSliceSearcher.matchesAt_iff_splits (toSlice_isEmpty h)]
+    ForwardSliceSearcher.matchesAt_iff_splits]
   simp
 
-theorem revMatchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+theorem revMatchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} :
     RevMatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits (t₁ ++ pat) t₂ := by
   rw [revMatchesAt_iff_toSlice,
-    ForwardSliceSearcher.revMatchesAt_iff_splits (toSlice_isEmpty h)]
+    ForwardSliceSearcher.revMatchesAt_iff_splits]
   simp
 
-theorem exists_matchesAt_iff_eq_append {pat : String} {s : Slice} (h : pat ≠ "") :
+theorem exists_matchesAt_iff_eq_append {pat : String} {s : Slice} :
     (∃ (pos : s.Pos), MatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat ++ t₂ := by
-  simp only [matchesAt_iff_splits h]
+  simp only [matchesAt_iff_splits]
   constructor
   · rintro ⟨pos, t₁, t₂, hsplit⟩
     exact ⟨t₁, t₂, by rw [hsplit.eq_append, append_assoc]⟩
@@ -385,12 +477,12 @@ theorem exists_matchesAt_iff_eq_append {pat : String} {s : Slice} (h : pat ≠ "
         ⟨t₁, pat ++ t₂, by rw [← append_assoc]; exact heq, rfl⟩
     exact ⟨s.pos _ hvalid, t₁, t₂, ⟨by rw [← append_assoc]; exact heq, by simp⟩⟩
 
-theorem exists_revMatchesAt_iff_eq_append {pat : String} {s : Slice} (h : pat ≠ "") :
+theorem exists_revMatchesAt_iff_eq_append {pat : String} {s : Slice} :
     (∃ (pos : s.Pos), RevMatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat ++ t₂ := by
   rw [show (∃ (pos : s.Pos), RevMatchesAt (ρ := String) pat pos) ↔
       (∃ (pos : s.Pos), RevMatchesAt (ρ := Slice) pat.toSlice pos) from by
     simp [revMatchesAt_iff_toSlice],
-    ForwardSliceSearcher.exists_revMatchesAt_iff_eq_append (toSlice_isEmpty h)]
+    ForwardSliceSearcher.exists_revMatchesAt_iff_eq_append]
   simp
 
 theorem matchesAt_iff_isLongestMatchAt {pat : String} {s : Slice} {pos : s.Pos}
@@ -398,7 +490,7 @@ theorem matchesAt_iff_isLongestMatchAt {pat : String} {s : Slice} {pos : s.Pos}
     MatchesAt pat pos ↔ ∃ (h : (pos.offset.increaseBy pat.utf8ByteSize).IsValidForSlice s),
       IsLongestMatchAt pat pos (s.pos _ h) := by
   have key := ForwardSliceSearcher.matchesAt_iff_isLongestMatchAt (pat := pat.toSlice)
-    (toSlice_isEmpty h) (pos := pos)
+    (by simpa) (pos := pos)
   simp only [utf8ByteSize_toSlice, ← isLongestMatchAt_iff_isLongestMatchAt_toSlice] at key
   rwa [matchesAt_iff_toSlice]
 
@@ -408,7 +500,7 @@ theorem revMatchesAt_iff_isLongestRevMatchAt {pat : String} {s : Slice} {pos : s
       ∃ (h : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s),
         IsLongestRevMatchAt pat (s.pos _ h) pos := by
   have key := ForwardSliceSearcher.revMatchesAt_iff_isLongestRevMatchAt (pat := pat.toSlice)
-    (toSlice_isEmpty h) (pos := pos)
+    (by simpa) (pos := pos)
   simp only [utf8ByteSize_toSlice, ← isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice] at key
   rwa [revMatchesAt_iff_toSlice]
 
@@ -418,14 +510,14 @@ theorem matchesAt_iff_getElem {pat : String} {s : Slice} {pos : s.Pos} (h : pat
         ∀ (j), (hj : j < pat.toByteArray.size) →
           pat.toByteArray[j] = s.copy.toByteArray[pos.offset.byteIdx + j] := by
   have key := ForwardSliceSearcher.matchesAt_iff_getElem (pat := pat.toSlice)
-    (toSlice_isEmpty h) (pos := pos)
+    (by simpa) (pos := pos)
   simp only [copy_toSlice] at key
   rwa [matchesAt_iff_toSlice]
 
 theorem le_of_matchesAt {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "")
     (h' : MatchesAt pat pos) : pos.offset.increaseBy pat.utf8ByteSize ≤ s.rawEndPos := by
   rw [show pat.utf8ByteSize = pat.toSlice.utf8ByteSize from utf8ByteSize_toSlice.symm]
-  exact ForwardSliceSearcher.le_of_matchesAt (toSlice_isEmpty h)
+  exact ForwardSliceSearcher.le_of_matchesAt (by simpa)
     (matchesAt_iff_toSlice.1 h')
 
 theorem matchesAt_iff_matchesAt_toSlice {pat : String} {s : Slice}

src/Init/Data/String/Lemmas/Pattern/String/ForwardPattern.lean

@@ -56,7 +56,7 @@ theorem skipPrefix?_eq_some_iff {pat s : Slice} {pos : s.Pos} :
     simp only [reduceCtorEq, false_iff]
     intro heq
     have := h (s.sliceFrom pos).copy
-    simp [← heq, pos.splits.eq_append] at this
+    simp [← heq, -sliceTo_append_sliceFrom, pos.splits.eq_append] at this
 
 theorem isSome_skipPrefix? {pat s : Slice} : (skipPrefix? pat s).isSome = startsWith pat s := by
   fun_cases skipPrefix? <;> simp_all
@@ -76,14 +76,11 @@ namespace Model.ForwardSliceSearcher
 
 open Pattern.ForwardSliceSearcher
 
-public instance {pat : Slice} : LawfulForwardPattern pat where
+public instance {pat : Slice} : LawfulForwardPatternModel pat where
   skipPrefixOfNonempty?_eq _ := rfl
   startsWith_eq _ := isSome_skipPrefix?.symm
-
-public theorem lawfulForwardPatternModel {pat : Slice} (hpat : pat.isEmpty = false) :
-    LawfulForwardPatternModel pat where
   skipPrefix?_eq_some_iff pos := by
-    simp [ForwardPattern.skipPrefix?, skipPrefix?_eq_some_iff, isLongestMatch_iff hpat]
+    simp [ForwardPattern.skipPrefix?, skipPrefix?_eq_some_iff, isLongestMatch_iff]
 
 end Model.ForwardSliceSearcher
 
@@ -91,14 +88,11 @@ namespace Model.ForwardStringSearcher
 
 open Pattern.ForwardSliceSearcher
 
-public instance {pat : String} : LawfulForwardPattern pat where
+public instance {pat : String} : LawfulForwardPatternModel pat where
   skipPrefixOfNonempty?_eq _ := rfl
   startsWith_eq _ := isSome_skipPrefix?.symm
-
-public theorem lawfulForwardPatternModel {pat : String} (hpat : pat ≠ "") :
-    LawfulForwardPatternModel pat where
   skipPrefix?_eq_some_iff pos := by
-    simp [ForwardPattern.skipPrefix?, skipPrefix?_eq_some_iff, isLongestMatch_iff hpat]
+    simp [ForwardPattern.skipPrefix?, skipPrefix?_eq_some_iff, isLongestMatch_iff]
 
 end Model.ForwardStringSearcher
 
@@ -153,7 +147,7 @@ theorem skipSuffix?_eq_some_iff {pat s : Slice} {pos : s.Pos} :
     simp only [reduceCtorEq, false_iff]
     intro heq
     have := h (s.sliceTo pos).copy
-    simp [← heq, pos.splits.eq_append] at this
+    simp [← heq, -sliceTo_append_sliceFrom, pos.splits.eq_append] at this
 
 theorem isSome_skipSuffix? {pat s : Slice} : (skipSuffix? pat s).isSome = endsWith pat s := by
   fun_cases skipSuffix? <;> simp_all
@@ -173,15 +167,12 @@ namespace Model.BackwardSliceSearcher
 
 open Pattern.BackwardSliceSearcher
 
-public instance {pat : Slice} : LawfulBackwardPattern pat where
+public instance {pat : Slice} : LawfulBackwardPatternModel pat where
   skipSuffixOfNonempty?_eq _ := rfl
   endsWith_eq _ := isSome_skipSuffix?.symm
-
-public theorem lawfulBackwardPatternModel {pat : Slice} (hpat : pat.isEmpty = false) :
-    LawfulBackwardPatternModel pat where
   skipSuffix?_eq_some_iff pos := by
     simp [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff,
-      ForwardSliceSearcher.isLongestRevMatch_iff hpat]
+      ForwardSliceSearcher.isLongestRevMatch_iff]
 
 end Model.BackwardSliceSearcher
 
@@ -189,15 +180,12 @@ namespace Model.BackwardStringSearcher
 
 open Pattern.BackwardSliceSearcher
 
-public instance {pat : String} : LawfulBackwardPattern pat where
+public instance {pat : String} : LawfulBackwardPatternModel pat where
   skipSuffixOfNonempty?_eq _ := rfl
   endsWith_eq _ := isSome_skipSuffix?.symm
-
-public theorem lawfulBackwardPatternModel {pat : String} (hpat : pat ≠ "") :
-    LawfulBackwardPatternModel pat where
   skipSuffix?_eq_some_iff pos := by
     simp [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff,
-      ForwardStringSearcher.isLongestRevMatch_iff hpat]
+      ForwardStringSearcher.isLongestRevMatch_iff]
 
 end Model.BackwardStringSearcher
 
@@ -219,19 +207,22 @@ public theorem Pattern.ForwardPattern.skipPrefix?_string_eq_skipPrefix?_toSlice
     {pat : String} {s : Slice} :
     skipPrefix? pat s = skipPrefix? pat.toSlice s := (rfl)
 
+public theorem Pos.skip?_string_eq_skip?_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
+    pos.skip? pat = pos.skip? pat.toSlice := (rfl)
+
 public theorem Pos.skipWhile_string_eq_skipWhile_toSlice {pat : String} {s : Slice}
     (curr : s.Pos) :
     Pos.skipWhile curr pat = Pos.skipWhile curr pat.toSlice := by
   fun_induction Pos.skipWhile curr pat with
   | case1 pos nextCurr h₁ h₂ ih =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_string_eq_skipPrefix?_toSlice, h₁, h₂, ih]
+    simp [← Pos.skip?_string_eq_skip?_toSlice, h₁, h₂, ih]
   | case2 pos nextCurr h ih =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_string_eq_skipPrefix?_toSlice, h, ih]
+    simp [← Pos.skip?_string_eq_skip?_toSlice, h, ih]
   | case3 pos h =>
     conv => rhs; rw [Pos.skipWhile]
-    simp [← Pattern.ForwardPattern.skipPrefix?_string_eq_skipPrefix?_toSlice]
+    simp [← Pos.skip?_string_eq_skip?_toSlice, h]
 
 public theorem skipPrefixWhile_string_eq_skipPrefixWhile_toSlice {pat : String} {s : Slice} :
     s.skipPrefixWhile pat = s.skipPrefixWhile pat.toSlice :=
@@ -247,7 +238,7 @@ public theorem takeWhile_string_eq_takeWhile_toSlice {pat : String} {s : Slice}
 
 public theorem all_string_eq_all_toSlice {pat : String} {s : Slice} :
     s.all pat = s.all pat.toSlice := by
-  simp only [all, dropWhile_string_eq_dropWhile_toSlice]
+  simp only [all, skipPrefixWhile_string_eq_skipPrefixWhile_toSlice]
 
 public theorem endsWith_string_eq_endsWith_toSlice {pat : String} {s : Slice} :
     s.endsWith pat = s.endsWith pat.toSlice := (rfl)
@@ -265,19 +256,22 @@ public theorem Pattern.BackwardPattern.skipSuffix?_string_eq_skipSuffix?_toSlice
     {pat : String} {s : Slice} :
     skipSuffix? pat s = skipSuffix? pat.toSlice s := (rfl)
 
+public theorem Pos.revSkip?_string_eq_revSkip?_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
+    pos.revSkip? pat = pos.revSkip? pat.toSlice := (rfl)
+
 public theorem Pos.revSkipWhile_string_eq_revSkipWhile_toSlice {pat : String} {s : Slice}
     (curr : s.Pos) :
     Pos.revSkipWhile curr pat = Pos.revSkipWhile curr pat.toSlice := by
   fun_induction Pos.revSkipWhile curr pat with
   | case1 pos nextCurr h₁ h₂ ih =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_string_eq_skipSuffix?_toSlice, h₁, h₂, ih]
+    simp [← Pos.revSkip?_string_eq_revSkip?_toSlice, h₁, h₂, ih]
   | case2 pos nextCurr h ih =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_string_eq_skipSuffix?_toSlice, h, ih]
+    simp [← Pos.revSkip?_string_eq_revSkip?_toSlice, h, ih]
   | case3 pos h =>
     conv => rhs; rw [Pos.revSkipWhile]
-    simp [← Pattern.BackwardPattern.skipSuffix?_string_eq_skipSuffix?_toSlice]
+    simp [← Pos.revSkip?_string_eq_revSkip?_toSlice, h]
 
 public theorem skipSuffixWhile_string_eq_skipSuffixWhile_toSlice {pat : String} {s : Slice} :
     s.skipSuffixWhile pat = s.skipSuffixWhile pat.toSlice :=
@@ -291,4 +285,8 @@ public theorem takeEndWhile_string_eq_takeEndWhile_toSlice {pat : String} {s : S
     s.takeEndWhile pat = s.takeEndWhile pat.toSlice := by
   simp only [takeEndWhile]; exact congrArg _ skipSuffixWhile_string_eq_skipSuffixWhile_toSlice
 
+public theorem revAll_string_eq_revAll_toSlice {pat : String} {s : Slice} :
+    s.revAll pat = s.revAll pat.toSlice := by
+  simp [revAll, skipSuffixWhile_string_eq_skipSuffixWhile_toSlice]
+
 end String.Slice

src/Init/Data/String/Lemmas/Pattern/TakeDrop/Char.lean

@@ -56,6 +56,77 @@ theorem eq_append_of_dropPrefix?_char_eq_some {c : Char} {s res : Slice} (h : s.
     s.copy = singleton c ++ res.copy := by
   simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
 
+theorem Pos.skip?_char_eq_some_iff {c : Char} {s : Slice} {pos res : s.Pos} :
+    pos.skip? c = some res ↔ ∃ h, res = pos.next h ∧ pos.get h = c := by
+  simp [Pattern.Model.Pos.skip?_eq_some_iff, Char.isLongestMatchAt_iff]
+
+@[simp]
+theorem Pos.skip?_char_eq_none_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    pos.skip? c = none ↔ ∀ h, pos.get h ≠ c := by
+  simp [Pattern.Model.Pos.skip?_eq_none_iff, Char.matchesAt_iff]
+
+theorem Pos.get_skipWhile_char_ne {c : Char} {s : Slice} {pos : s.Pos} {h} :
+    (pos.skipWhile c).get h ≠ c := by
+  have := Pattern.Model.Pos.not_matchesAt_skipWhile c pos
+  simp_all [Char.matchesAt_iff]
+
+theorem Pos.skipWhile_char_eq_self_iff_get {c : Char} {s : Slice} {pos : s.Pos} :
+    pos.skipWhile c = pos ↔ ∀ h, pos.get h ≠ c := by
+  simp [Pattern.Model.Pos.skipWhile_eq_self_iff, Char.matchesAt_iff]
+
+theorem Pos.get_eq_of_lt_skipWhile_char {c : Char} {s : Slice} {pos pos' : s.Pos}
+    (h₁ : pos ≤ pos') (h₂ : pos' < pos.skipWhile c) : pos'.get (ne_endPos_of_lt h₂) = c :=
+  (Char.isLongestMatchAtChain_iff.1 (Pattern.Model.Pos.isLongestMatchAtChain_skipWhile c pos)).2 _ h₁ h₂
+
+theorem get_skipPrefixWhile_char_ne {c : Char} {s : Slice} {h} :
+    (s.skipPrefixWhile c).get h ≠ c := by
+  simp [skipPrefixWhile_eq_skipWhile_startPos, Pos.get_skipWhile_char_ne]
+
+theorem get_eq_of_lt_skipPrefixWhile_char {c : Char} {s : Slice} {pos : s.Pos} (h : pos < s.skipPrefixWhile c) :
+    pos.get (Pos.ne_endPos_of_lt h) = c :=
+  Pos.get_eq_of_lt_skipWhile_char (Pos.startPos_le _) (by rwa [skipPrefixWhile_eq_skipWhile_startPos] at h)
+
+@[simp]
+theorem all_char_iff {c : Char} {s : Slice} : s.all c ↔ s.copy.toList = List.replicate s.copy.length c := by
+  rw [Bool.eq_iff_iff]
+  simp [Pattern.Model.all_eq_true_iff, Char.isLongestMatchAtChain_startPos_endPos_iff_toList]
+
+theorem Pos.revSkip?_char_eq_some_iff {c : Char} {s : Slice} {pos res : s.Pos} :
+    pos.revSkip? c = some res ↔ ∃ h, res = pos.prev h ∧ (pos.prev h).get (by simp) = c := by
+  simp [Pattern.Model.Pos.revSkip?_eq_some_iff, Char.isLongestRevMatchAt_iff]
+
+@[simp]
+theorem Pos.revSkip?_char_eq_none_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    pos.revSkip? c = none ↔ ∀ h, (pos.prev h).get (by simp) ≠ c := by
+  simp [Pattern.Model.Pos.revSkip?_eq_none_iff, Char.revMatchesAt_iff]
+
+theorem Pos.get_revSkipWhile_char_ne {c : Char} {s : Slice} {pos : s.Pos} {h} :
+    ((pos.revSkipWhile c).prev h).get (by simp) ≠ c := by
+  have := Pattern.Model.Pos.not_revMatchesAt_revSkipWhile c pos
+  simp_all [Char.revMatchesAt_iff]
+
+theorem Pos.revSkipWhile_char_eq_self_iff_get {c : Char} {s : Slice} {pos : s.Pos} :
+    pos.revSkipWhile c = pos ↔ ∀ h, (pos.prev h).get (by simp) ≠ c := by
+  simp [Pattern.Model.Pos.revSkipWhile_eq_self_iff, Char.revMatchesAt_iff]
+
+theorem Pos.get_eq_of_revSkipWhile_le_char {c : Char} {s : Slice} {pos pos' : s.Pos}
+    (h₁ : pos' < pos) (h₂ : pos.revSkipWhile c ≤ pos') : pos'.get (Pos.ne_endPos_of_lt h₁) = c :=
+  (Char.isLongestRevMatchAtChain_iff.1 (Pattern.Model.Pos.isLongestRevMatchAtChain_revSkipWhile c pos)).2 _ h₂ h₁
+
+theorem get_skipSuffixWhile_char_ne {c : Char} {s : Slice} {h} :
+    ((s.skipSuffixWhile c).prev h).get (by simp) ≠ c := by
+  simp [skipSuffixWhile_eq_revSkipWhile_endPos, Pos.get_revSkipWhile_char_ne]
+
+theorem get_eq_of_skipSuffixWhile_le_char {c : Char} {s : Slice} {pos : s.Pos}
+    (h : s.skipSuffixWhile c ≤ pos) (h' : pos < s.endPos) :
+    pos.get (Pos.ne_endPos_of_lt h') = c :=
+  Pos.get_eq_of_revSkipWhile_le_char h' (by rwa [skipSuffixWhile_eq_revSkipWhile_endPos] at h)
+
+@[simp]
+theorem revAll_char_iff {c : Char} {s : Slice} : s.revAll c ↔ s.copy.toList = List.replicate s.copy.length c := by
+  rw [Bool.eq_iff_iff]
+  simp [Pattern.Model.revAll_eq_true_iff, Char.isLongestRevMatchAtChain_startPos_endPos_iff_toList]
+
 theorem skipSuffix?_char_eq_some_iff {c : Char} {s : Slice} {pos : s.Pos} :
     s.skipSuffix? c = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
   rw [Pattern.Model.skipSuffix?_eq_some_iff, Char.isLongestRevMatch_iff]
@@ -100,19 +171,19 @@ theorem skipPrefix?_char_eq_some_iff {c : Char} {s : String} {pos : s.Pos} :
 
 theorem startsWith_char_iff_get {c : Char} {s : String} :
     s.startsWith c ↔ ∃ h, s.startPos.get h = c := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_char_iff_get]
+  simp [← startsWith_toSlice, Slice.startsWith_char_iff_get]
 
 theorem startsWith_char_eq_false_iff_get {c : Char} {s : String} :
     s.startsWith c = false ↔ ∀ h, s.startPos.get h ≠ c := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_char_eq_false_iff_get]
+  simp [← startsWith_toSlice, Slice.startsWith_char_eq_false_iff_get]
 
 theorem startsWith_char_eq_head? {c : Char} {s : String} :
     s.startsWith c = (s.toList.head? == some c) := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_char_eq_head?]
+  simp [← startsWith_toSlice, Slice.startsWith_char_eq_head?]
 
 theorem startsWith_char_iff_exists_append {c : Char} {s : String} :
     s.startsWith c ↔ ∃ t, s = singleton c ++ t := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_char_iff_exists_append]
+  simp [← startsWith_toSlice, Slice.startsWith_char_iff_exists_append]
 
 theorem startsWith_char_eq_false_iff_forall_append {c : Char} {s : String} :
     s.startsWith c = false ↔ ∀ t, s ≠ singleton c ++ t := by
@@ -130,19 +201,19 @@ theorem skipSuffix?_char_eq_some_iff {c : Char} {s : String} {pos : s.Pos} :
 
 theorem endsWith_char_iff_get {c : Char} {s : String} :
     s.endsWith c ↔ ∃ h, (s.endPos.prev h).get (by simp) = c := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_iff_get, Pos.prev_toSlice]
+  simp [← endsWith_toSlice, Slice.endsWith_char_iff_get, Pos.prev_toSlice]
 
 theorem endsWith_char_eq_false_iff_get {c : Char} {s : String} :
     s.endsWith c = false ↔ ∀ h, (s.endPos.prev h).get (by simp) ≠ c := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_eq_false_iff_get, Pos.prev_toSlice]
+  simp [← endsWith_toSlice, Slice.endsWith_char_eq_false_iff_get, Pos.prev_toSlice]
 
 theorem endsWith_char_eq_getLast? {c : Char} {s : String} :
     s.endsWith c = (s.toList.getLast? == some c) := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_eq_getLast?]
+  simp [← endsWith_toSlice, Slice.endsWith_char_eq_getLast?]
 
 theorem endsWith_char_iff_exists_append {c : Char} {s : String} :
     s.endsWith c ↔ ∃ t, s = t ++ singleton c := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_iff_exists_append]
+  simp [← endsWith_toSlice, Slice.endsWith_char_iff_exists_append]
 
 theorem endsWith_char_eq_false_iff_forall_append {c : Char} {s : String} :
     s.endsWith c = false ↔ ∀ t, s ≠ t ++ singleton c := by

src/Init/Data/String/Lemmas/Pattern/TakeDrop/Pred.lean

@@ -8,11 +8,16 @@ module
 prelude
 public import Init.Data.String.Slice
 public import Init.Data.String.TakeDrop
+public import Init.Data.String.Lemmas.Order
 import Init.Data.String.Lemmas.Pattern.TakeDrop.Basic
 import Init.Data.String.Lemmas.Pattern.Pred
 import Init.Data.Option.Lemmas
 import Init.Data.String.Lemmas.FindPos
+import Init.Data.String.Lemmas.Intercalate
 import Init.ByCases
+import Init.Data.Order.Lemmas
+import Init.Data.String.OrderInstances
+import Init.Data.String.Lemmas.Basic
 
 public section
 
@@ -49,6 +54,80 @@ theorem eq_append_of_dropPrefix?_bool_eq_some {p : Char → Bool} {s res : Slice
   obtain ⟨_, ⟨c, ⟨rfl, h₁⟩⟩, h₂⟩ := by simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
   exact ⟨_, h₂, h₁⟩
 
+@[simp]
+theorem Pos.skip?_bool_eq_some_iff {p : Char → Bool} {s : Slice} {pos res : s.Pos} :
+    pos.skip? p = some res ↔ ∃ h, res = pos.next h ∧ p (pos.get h) := by
+  simp [Pattern.Model.Pos.skip?_eq_some_iff, CharPred.isLongestMatchAt_iff]
+
+@[simp]
+theorem Pos.skip?_bool_eq_none_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    pos.skip? p = none ↔ ∀ h, p (pos.get h) = false := by
+  simp [Pattern.Model.Pos.skip?_eq_none_iff, CharPred.matchesAt_iff]
+
+theorem Pos.apply_skipWhile_bool_eq_false {p : Char → Bool} {s : Slice} {pos : s.Pos} {h} :
+    p ((pos.skipWhile p).get h) = false := by
+  have := Pattern.Model.Pos.not_matchesAt_skipWhile p pos
+  simp_all [CharPred.matchesAt_iff]
+
+theorem Pos.skipWhile_bool_eq_self_iff_get {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    pos.skipWhile p = pos ↔ ∀ h, p (pos.get h) = false := by
+  simp [Pattern.Model.Pos.skipWhile_eq_self_iff, CharPred.matchesAt_iff]
+
+theorem Pos.apply_eq_true_of_lt_skipWhile_bool {p : Char → Bool} {s : Slice} {pos pos' : s.Pos}
+    (h₁ : pos ≤ pos') (h₂ : pos' < pos.skipWhile p) : p (pos'.get (ne_endPos_of_lt h₂)) = true :=
+  (CharPred.isLongestMatchAtChain_iff.1 (Pattern.Model.Pos.isLongestMatchAtChain_skipWhile p pos)).2 _ h₁ h₂
+
+theorem apply_skipPrefixWhile_bool_eq_false {p : Char → Bool} {s : Slice} {h} :
+    p ((s.skipPrefixWhile p).get h) = false := by
+  simp [skipPrefixWhile_eq_skipWhile_startPos, Pos.apply_skipWhile_bool_eq_false]
+
+theorem apply_eq_true_of_lt_skipPrefixWhile_bool {p : Char → Bool} {s : Slice} {pos : s.Pos} (h : pos < s.skipPrefixWhile p) :
+    p (pos.get (Pos.ne_endPos_of_lt h)) = true :=
+  Pos.apply_eq_true_of_lt_skipWhile_bool (Pos.startPos_le _) (skipPrefixWhile_eq_skipWhile_startPos ▸ h)
+
+@[simp]
+theorem all_bool_eq {p : Char → Bool} {s : Slice} : s.all p = s.copy.toList.all p := by
+  rw [Bool.eq_iff_iff, Pattern.Model.all_eq_true_iff,
+    CharPred.isLongestMatchAtChain_startPos_endPos_iff_toList, List.all_eq_true]
+
+@[simp]
+theorem Pos.skip?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos res : s.Pos} :
+    pos.skip? P = some res ↔ ∃ h, res = pos.next h ∧ P (pos.get h) := by
+  simp [Pos.skip?_prop_eq_skip?_decide, skip?_bool_eq_some_iff]
+
+@[simp]
+theorem Pos.skip?_prop_eq_none_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
+    pos.skip? P = none ↔ ∀ h, ¬ P (pos.get h) := by
+  simp [Pos.skip?_prop_eq_skip?_decide, skip?_bool_eq_none_iff]
+
+theorem Pos.apply_skipWhile_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} {h} :
+    ¬ P ((pos.skipWhile P).get h) := by
+  have := Pattern.Model.Pos.not_matchesAt_skipWhile P pos
+  simp_all [CharPred.Decidable.matchesAt_iff]
+
+theorem Pos.skipWhile_prop_eq_self_iff_get {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
+    pos.skipWhile P = pos ↔ ∀ h, ¬ P (pos.get h) := by
+  simp [Pos.skipWhile_prop_eq_skipWhile_decide, skipWhile_bool_eq_self_iff_get]
+
+theorem Pos.apply_of_lt_skipWhile_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {pos pos' : s.Pos}
+    (h₁ : pos ≤ pos') (h₂ : pos' < pos.skipWhile P) : P (pos'.get (ne_endPos_of_lt h₂)) := by
+  simp [Pos.skipWhile_prop_eq_skipWhile_decide] at h₂
+  simpa using apply_eq_true_of_lt_skipWhile_bool h₁ h₂
+
+theorem apply_skipPrefixWhile_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {h} :
+    ¬ P ((s.skipPrefixWhile P).get h) := by
+  simp [skipPrefixWhile_eq_skipWhile_startPos, Pos.apply_skipWhile_prop]
+
+theorem apply_of_lt_skipPrefixWhile_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos}
+    (h : pos < s.skipPrefixWhile P) : P (pos.get (Pos.ne_endPos_of_lt h)) := by
+  simp [skipPrefixWhile_prop_eq_skipPrefixWhile_decide] at h
+  simpa using apply_eq_true_of_lt_skipPrefixWhile_bool h
+
+@[simp]
+theorem all_prop_eq {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.all P = s.copy.toList.all (decide <| P ·) := by
+  simp [all_prop_eq_all_decide]
+
 theorem skipPrefix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
     s.skipPrefix? P = some pos ↔ ∃ h, pos = s.startPos.next h ∧ P (s.startPos.get h) := by
   simp [skipPrefix?_prop_eq_skipPrefix?_decide, skipPrefix?_bool_eq_some_iff]
@@ -65,7 +144,7 @@ theorem startsWith_prop_eq_head? {P : Char → Prop} [DecidablePred P] {s : Slic
     s.startsWith P = s.copy.toList.head?.any (decide <| P ·) := by
   simp [startsWith_prop_eq_startsWith_decide, startsWith_bool_eq_head?]
 
-theorem eq_append_of_dropPrefix_prop_eq_some {P : Char → Prop} [DecidablePred P] {s res : Slice} (h : s.dropPrefix? P = some res) :
+theorem eq_append_of_dropPrefix?_prop_eq_some {P : Char → Prop} [DecidablePred P] {s res : Slice} (h : s.dropPrefix? P = some res) :
     ∃ c, s.copy = singleton c ++ res.copy ∧ P c := by
   rw [dropPrefix?_prop_eq_dropPrefix?_decide] at h
   simpa using eq_append_of_dropPrefix?_bool_eq_some h
@@ -118,6 +197,83 @@ theorem eq_append_of_dropSuffix?_prop_eq_some {P : Char → Prop} [DecidablePred
   rw [dropSuffix?_prop_eq_dropSuffix?_decide] at h
   simpa using eq_append_of_dropSuffix?_bool_eq_some h
 
+@[simp]
+theorem Pos.revSkip?_bool_eq_some_iff {p : Char → Bool} {s : Slice} {pos res : s.Pos} :
+    pos.revSkip? p = some res ↔ ∃ h, res = pos.prev h ∧ p ((pos.prev h).get (by simp)) := by
+  simp [Pattern.Model.Pos.revSkip?_eq_some_iff, CharPred.isLongestRevMatchAt_iff]
+
+@[simp]
+theorem Pos.revSkip?_bool_eq_none_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    pos.revSkip? p = none ↔ ∀ h, p ((pos.prev h).get (by simp)) = false := by
+  simp [Pattern.Model.Pos.revSkip?_eq_none_iff, CharPred.revMatchesAt_iff]
+
+theorem Pos.apply_revSkipWhile_bool_eq_false {p : Char → Bool} {s : Slice} {pos : s.Pos} {h} :
+    p (((pos.revSkipWhile p).prev h).get (by simp)) = false := by
+  have := Pattern.Model.Pos.not_revMatchesAt_revSkipWhile p pos
+  simp_all [CharPred.revMatchesAt_iff]
+
+theorem Pos.revSkipWhile_bool_eq_self_iff_get {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    pos.revSkipWhile p = pos ↔ ∀ h, p ((pos.prev h).get (by simp)) = false := by
+  simp [Pattern.Model.Pos.revSkipWhile_eq_self_iff, CharPred.revMatchesAt_iff]
+
+theorem Pos.apply_eq_true_of_revSkipWhile_le_bool {p : Char → Bool} {s : Slice} {pos pos' : s.Pos}
+    (h₁ : pos' < pos) (h₂ : pos.revSkipWhile p ≤ pos') : p (pos'.get (Pos.ne_endPos_of_lt h₁)) = true :=
+  (CharPred.isLongestRevMatchAtChain_iff.1 (Pattern.Model.Pos.isLongestRevMatchAtChain_revSkipWhile p pos)).2 _ h₂ h₁
+
+theorem apply_skipSuffixWhile_bool_eq_false {p : Char → Bool} {s : Slice} {h} :
+    p (((s.skipSuffixWhile p).prev h).get (by simp)) = false := by
+  simp [skipSuffixWhile_eq_revSkipWhile_endPos, Pos.apply_revSkipWhile_bool_eq_false]
+
+theorem apply_eq_true_of_skipSuffixWhile_le_bool {p : Char → Bool} {s : Slice} {pos : s.Pos}
+    (h : s.skipSuffixWhile p ≤ pos) (h' : pos < s.endPos) :
+    p (pos.get (Pos.ne_endPos_of_lt h')) = true :=
+  Pos.apply_eq_true_of_revSkipWhile_le_bool h' (skipSuffixWhile_eq_revSkipWhile_endPos ▸ h)
+
+@[simp]
+theorem revAll_bool_eq {p : Char → Bool} {s : Slice} : s.revAll p = s.copy.toList.all p := by
+  rw [Bool.eq_iff_iff, Pattern.Model.revAll_eq_true_iff,
+    CharPred.isLongestRevMatchAtChain_startPos_endPos_iff_toList, List.all_eq_true]
+
+@[simp]
+theorem Pos.revSkip?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos res : s.Pos} :
+    pos.revSkip? P = some res ↔ ∃ h, res = pos.prev h ∧ P ((pos.prev h).get (by simp)) := by
+  simp [Pos.revSkip?_prop_eq_revSkip?_decide, revSkip?_bool_eq_some_iff]
+
+@[simp]
+theorem Pos.revSkip?_prop_eq_none_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
+    pos.revSkip? P = none ↔ ∀ h, ¬ P ((pos.prev h).get (by simp)) := by
+  simp [Pos.revSkip?_prop_eq_revSkip?_decide, revSkip?_bool_eq_none_iff]
+
+theorem Pos.apply_revSkipWhile_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} {h} :
+    ¬ P (((pos.revSkipWhile P).prev h).get (by simp)) := by
+  have := Pattern.Model.Pos.not_revMatchesAt_revSkipWhile P pos
+  simp_all [CharPred.Decidable.revMatchesAt_iff]
+
+theorem Pos.revSkipWhile_prop_eq_self_iff_get {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
+    pos.revSkipWhile P = pos ↔ ∀ h, ¬ P ((pos.prev h).get (by simp)) := by
+  simp [Pos.revSkipWhile_prop_eq_revSkipWhile_decide, revSkipWhile_bool_eq_self_iff_get]
+
+theorem Pos.apply_of_revSkipWhile_le_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {pos pos' : s.Pos}
+    (h₁ : pos' < pos) (h₂ : pos.revSkipWhile P ≤ pos') : P (pos'.get (Pos.ne_endPos_of_lt h₁)) := by
+  have h₂' : pos.revSkipWhile (decide <| P ·) ≤ pos' :=
+    Pos.revSkipWhile_prop_eq_revSkipWhile_decide (p := P) pos ▸ h₂
+  simpa using Pos.apply_eq_true_of_revSkipWhile_le_bool h₁ h₂'
+
+theorem apply_skipSuffixWhile_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {h} :
+    ¬ P (((s.skipSuffixWhile P).prev h).get (by simp)) := by
+  have := Pattern.Model.Pos.not_revMatchesAt_revSkipWhile P s.endPos
+  simp_all [CharPred.Decidable.revMatchesAt_iff, skipSuffixWhile_eq_revSkipWhile_endPos]
+
+theorem apply_of_skipSuffixWhile_le_prop {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos}
+    (h : s.skipSuffixWhile P ≤ pos) (h' : pos < s.endPos) :
+    P (pos.get (Pos.ne_endPos_of_lt h')) :=
+  Pos.apply_of_revSkipWhile_le_prop h' (skipSuffixWhile_eq_revSkipWhile_endPos (pat := P) ▸ h)
+
+@[simp]
+theorem revAll_prop_eq {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.revAll P = s.copy.toList.all (decide <| P ·) := by
+  simp [revAll_prop_eq_revAll_decide, revAll_bool_eq]
+
 end Slice
 
 theorem skipPrefix?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos : s.Pos} :
@@ -127,21 +283,58 @@ theorem skipPrefix?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos : s.P
 
 theorem startsWith_bool_iff_get {p : Char → Bool} {s : String} :
     s.startsWith p ↔ ∃ h, p (s.startPos.get h) = true := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_bool_iff_get]
+  simp [← startsWith_toSlice, Slice.startsWith_bool_iff_get]
 
 theorem startsWith_bool_eq_false_iff_get {p : Char → Bool} {s : String} :
     s.startsWith p = false ↔ ∀ h, p (s.startPos.get h) = false := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_bool_eq_false_iff_get]
+  simp [← startsWith_toSlice, Slice.startsWith_bool_eq_false_iff_get]
 
 theorem startsWith_bool_eq_head? {p : Char → Bool} {s : String} :
     s.startsWith p = s.toList.head?.any p := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_bool_eq_head?]
+  simp [← startsWith_toSlice, Slice.startsWith_bool_eq_head?]
 
 theorem eq_append_of_dropPrefix?_bool_eq_some {p : Char → Bool} {s : String} {res : Slice} (h : s.dropPrefix? p = some res) :
     ∃ c, s = singleton c ++ res.copy ∧ p c = true := by
   rw [dropPrefix?_eq_dropPrefix?_toSlice] at h
   simpa using Slice.eq_append_of_dropPrefix?_bool_eq_some h
 
+@[simp]
+theorem Pos.skip?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos res : s.Pos} :
+    pos.skip? p = some res ↔ ∃ h, res = pos.next h ∧ p (pos.get h) := by
+  simp [skip?_eq_skip?_toSlice, ← toSlice_inj, toSlice_next]
+
+@[simp]
+theorem Pos.skip?_bool_eq_none_iff {p : Char → Bool} {s : String} {pos : s.Pos} :
+    pos.skip? p = none ↔ ∀ h, p (pos.get h) = false := by
+  simp [skip?_eq_skip?_toSlice]
+
+theorem Pos.apply_skipWhile_bool_eq_false {p : Char → Bool} {s : String} {pos : s.Pos} {h} :
+    p ((pos.skipWhile p).get h) = false := by
+  simp [skipWhile_eq_skipWhile_toSlice, Slice.Pos.apply_skipWhile_bool_eq_false]
+
+theorem Pos.skipWhile_bool_eq_self_iff_get {p : Char → Bool} {s : String} {pos : s.Pos} :
+    pos.skipWhile p = pos ↔ ∀ h, p (pos.get h) = false := by
+  simp [skipWhile_eq_skipWhile_toSlice, ← toSlice_inj, Slice.Pos.skipWhile_bool_eq_self_iff_get]
+
+theorem Pos.apply_eq_true_of_lt_skipWhile_bool {p : Char → Bool} {s : String} {pos pos' : s.Pos}
+    (h₁ : pos ≤ pos') (h₂ : pos' < pos.skipWhile p) : p (pos'.get (ne_endPos_of_lt h₂)) = true := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.Pos.apply_eq_true_of_lt_skipWhile_bool (toSlice_le_toSlice_iff.2 h₁)
+    (by simpa [skipWhile_eq_skipWhile_toSlice] using h₂)
+
+theorem apply_skipPrefixWhile_bool_eq_false {p : Char → Bool} {s : String} {h} :
+    p ((s.skipPrefixWhile p).get h) = false := by
+  simp [skipPrefixWhile_eq_skipPrefixWhile_toSlice, Slice.apply_skipPrefixWhile_bool_eq_false]
+
+theorem apply_eq_true_of_lt_skipPrefixWhile_bool {p : Char → Bool} {s : String} {pos : s.Pos} (h : pos < s.skipPrefixWhile p) :
+    p (pos.get (Pos.ne_endPos_of_lt h)) = true := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.apply_eq_true_of_lt_skipPrefixWhile_bool (by simpa [skipPrefixWhile_eq_skipPrefixWhile_toSlice] using h)
+
+@[simp]
+theorem all_bool_eq {p : Char → Bool} {s : String} : s.all p = s.toList.all p := by
+  simp [← all_toSlice]
+
 theorem skipPrefix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
     s.skipPrefix? P = some pos ↔ ∃ h, pos = s.startPos.next h ∧ P (s.startPos.get h) := by
   simp [skipPrefix?_eq_skipPrefix?_toSlice, Slice.skipPrefix?_prop_eq_some_iff, ← Pos.toSlice_inj,
@@ -149,20 +342,20 @@ theorem skipPrefix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s :
 
 theorem startsWith_prop_iff_get {P : Char → Prop} [DecidablePred P] {s : String} :
     s.startsWith P ↔ ∃ h, P (s.startPos.get h) := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_prop_iff_get]
+  simp [← startsWith_toSlice, Slice.startsWith_prop_iff_get]
 
 theorem startsWith_prop_eq_false_iff_get {P : Char → Prop} [DecidablePred P] {s : String} :
     s.startsWith P = false ↔ ∀ h, ¬ P (s.startPos.get h) := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_prop_eq_false_iff_get]
+  simp [← startsWith_toSlice, Slice.startsWith_prop_eq_false_iff_get]
 
 theorem startsWith_prop_eq_head? {P : Char → Prop} [DecidablePred P] {s : String} :
     s.startsWith P = s.toList.head?.any (decide <| P ·) := by
-  simp [startsWith_eq_startsWith_toSlice, Slice.startsWith_prop_eq_head?]
+  simp [← startsWith_toSlice, Slice.startsWith_prop_eq_head?]
 
 theorem eq_append_of_dropPrefix?_prop_eq_some {P : Char → Prop} [DecidablePred P] {s : String} {res : Slice}
     (h : s.dropPrefix? P = some res) : ∃ c, s = singleton c ++ res.copy ∧ P c := by
   rw [dropPrefix?_eq_dropPrefix?_toSlice] at h
-  simpa using Slice.eq_append_of_dropPrefix_prop_eq_some h
+  simpa using Slice.eq_append_of_dropPrefix?_prop_eq_some h
 
 theorem skipSuffix?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos : s.Pos} :
     s.skipSuffix? p = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) = true := by
@@ -171,15 +364,15 @@ theorem skipSuffix?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos : s.P
 
 theorem endsWith_bool_iff_get {p : Char → Bool} {s : String} :
     s.endsWith p ↔ ∃ h, p ((s.endPos.prev h).get (by simp)) = true := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_bool_iff_get, Pos.prev_toSlice]
+  simp [← endsWith_toSlice, Slice.endsWith_bool_iff_get, Pos.prev_toSlice]
 
 theorem endsWith_bool_eq_false_iff_get {p : Char → Bool} {s : String} :
     s.endsWith p = false ↔ ∀ h, p ((s.endPos.prev h).get (by simp)) = false := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_bool_eq_false_iff_get, Pos.prev_toSlice]
+  simp [← endsWith_toSlice, Slice.endsWith_bool_eq_false_iff_get, Pos.prev_toSlice]
 
 theorem endsWith_bool_eq_getLast? {p : Char → Bool} {s : String} :
     s.endsWith p = s.toList.getLast?.any p := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_bool_eq_getLast?]
+  simp [← endsWith_toSlice, Slice.endsWith_bool_eq_getLast?]
 
 theorem eq_append_of_dropSuffix?_bool_eq_some {p : Char → Bool} {s : String} {res : Slice} (h : s.dropSuffix? p = some res) :
     ∃ c, s = res.copy ++ singleton c ∧ p c = true := by
@@ -193,19 +386,154 @@ theorem skipSuffix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s :
 
 theorem endsWith_prop_iff_get {P : Char → Prop} [DecidablePred P] {s : String} :
     s.endsWith P ↔ ∃ h, P ((s.endPos.prev h).get (by simp)) := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_prop_iff_get, Pos.prev_toSlice]
+  simp [← endsWith_toSlice, Slice.endsWith_prop_iff_get, Pos.prev_toSlice]
 
 theorem endsWith_prop_eq_false_iff_get {P : Char → Prop} [DecidablePred P] {s : String} :
     s.endsWith P = false ↔ ∀ h, ¬ P ((s.endPos.prev h).get (by simp)) := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_prop_eq_false_iff_get, Pos.prev_toSlice]
+  simp [← endsWith_toSlice, Slice.endsWith_prop_eq_false_iff_get, Pos.prev_toSlice]
 
 theorem endsWith_prop_eq_getLast? {P : Char → Prop} [DecidablePred P] {s : String} :
     s.endsWith P = s.toList.getLast?.any (decide <| P ·) := by
-  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_prop_eq_getLast?]
+  simp [← endsWith_toSlice, Slice.endsWith_prop_eq_getLast?]
 
 theorem eq_append_of_dropSuffix?_prop_eq_some {P : Char → Prop} [DecidablePred P] {s : String} {res : Slice}
     (h : s.dropSuffix? P = some res) : ∃ c, s = res.copy ++ singleton c ∧ P c := by
   rw [dropSuffix?_eq_dropSuffix?_toSlice] at h
   simpa using Slice.eq_append_of_dropSuffix?_prop_eq_some h
 
+@[simp]
+theorem Pos.skip?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos res : s.Pos} :
+    pos.skip? P = some res ↔ ∃ h, res = pos.next h ∧ P (pos.get h) := by
+  simp [skip?_eq_skip?_toSlice, ← toSlice_inj, toSlice_next]
+
+@[simp]
+theorem Pos.skip?_prop_eq_none_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
+    pos.skip? P = none ↔ ∀ h, ¬ P (pos.get h) := by
+  simp [skip?_eq_skip?_toSlice]
+
+theorem Pos.apply_skipWhile_prop {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} {h} :
+    ¬ P ((pos.skipWhile P).get h) := by
+  simp [skipWhile_eq_skipWhile_toSlice, Slice.Pos.apply_skipWhile_prop]
+
+theorem Pos.skipWhile_prop_eq_self_iff_get {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
+    pos.skipWhile P = pos ↔ ∀ h, ¬ P (pos.get h) := by
+  simp [skipWhile_eq_skipWhile_toSlice, ← toSlice_inj, Slice.Pos.skipWhile_prop_eq_self_iff_get]
+
+theorem Pos.apply_of_lt_skipWhile_prop {P : Char → Prop} [DecidablePred P] {s : String} {pos pos' : s.Pos}
+    (h₁ : pos ≤ pos') (h₂ : pos' < pos.skipWhile P) : P (pos'.get (ne_endPos_of_lt h₂)) := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.Pos.apply_of_lt_skipWhile_prop (toSlice_le_toSlice_iff.2 h₁)
+    (by simpa [skipWhile_eq_skipWhile_toSlice] using h₂)
+
+theorem apply_skipPrefixWhile_prop {P : Char → Prop} [DecidablePred P] {s : String} {h} :
+    ¬ P ((s.skipPrefixWhile P).get h) := by
+  simp [skipPrefixWhile_eq_skipPrefixWhile_toSlice, Slice.apply_skipPrefixWhile_prop]
+
+theorem apply_of_lt_skipPrefixWhile_prop {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos}
+    (h : pos < s.skipPrefixWhile P) : P (pos.get (Pos.ne_endPos_of_lt h)) := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.apply_of_lt_skipPrefixWhile_prop (by simpa [skipPrefixWhile_eq_skipPrefixWhile_toSlice] using h)
+
+@[simp]
+theorem all_prop_eq {P : Char → Prop} [DecidablePred P] {s : String} :
+    s.all P = s.toList.all (decide <| P ·) := by
+  simp [← all_toSlice]
+
+@[simp]
+theorem Pos.revSkip?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos res : s.Pos} :
+    pos.revSkip? p = some res ↔ ∃ h, res = pos.prev h ∧ p ((pos.prev h).get (by simp)) := by
+  simp [revSkip?_eq_revSkip?_toSlice, ← toSlice_inj, toSlice_prev, get_eq_get_toSlice]
+
+@[simp]
+theorem Pos.revSkip?_bool_eq_none_iff {p : Char → Bool} {s : String} {pos : s.Pos} :
+    pos.revSkip? p = none ↔ ∀ h, p ((pos.prev h).get (by simp)) = false := by
+  simp [revSkip?_eq_revSkip?_toSlice, Pos.prev_toSlice]
+
+theorem Pos.apply_revSkipWhile_bool_eq_false {p : Char → Bool} {s : String} {pos : s.Pos} {h} :
+    p (((pos.revSkipWhile p).prev h).get (by simp)) = false := by
+  have h' : pos.toSlice.revSkipWhile p ≠ s.toSlice.startPos := by
+    simpa [Pos.revSkipWhile_eq_revSkipWhile_toSlice, ← toSlice_inj] using h
+  have := Slice.Pos.apply_revSkipWhile_bool_eq_false (pos := pos.toSlice) (h := h')
+  simpa [Pos.revSkipWhile_eq_revSkipWhile_toSlice, Pos.prev_ofToSlice]
+
+theorem Pos.revSkipWhile_bool_eq_self_iff_get {p : Char → Bool} {s : String} {pos : s.Pos} :
+    pos.revSkipWhile p = pos ↔ ∀ h, p ((pos.prev h).get (by simp)) = false := by
+  simp [Pos.revSkipWhile_eq_revSkipWhile_toSlice, ← toSlice_inj, Slice.Pos.revSkipWhile_bool_eq_self_iff_get,
+    Pos.prev_toSlice]
+
+theorem Pos.apply_eq_true_of_revSkipWhile_le_bool {p : Char → Bool} {s : String} {pos pos' : s.Pos}
+    (h₁ : pos' < pos) (h₂ : pos.revSkipWhile p ≤ pos') : p (pos'.get (ne_endPos_of_lt h₁)) = true := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.Pos.apply_eq_true_of_revSkipWhile_le_bool
+    (Pos.toSlice_lt_toSlice_iff.2 h₁)
+    (by simpa [Pos.revSkipWhile_eq_revSkipWhile_toSlice, Pos.ofToSlice_le_iff] using h₂)
+
+theorem apply_skipSuffixWhile_bool_eq_false {p : Char → Bool} {s : String} {h} :
+    p (((s.skipSuffixWhile p).prev h).get (by simp)) = false := by
+  have h' : s.toSlice.skipSuffixWhile p ≠ s.toSlice.startPos := by
+    simpa [skipSuffixWhile_eq_skipSuffixWhile_toSlice, ← Pos.toSlice_inj] using h
+  have := Slice.apply_skipSuffixWhile_bool_eq_false (s := s.toSlice) (h := h')
+  simpa [skipSuffixWhile_eq_skipSuffixWhile_toSlice, Pos.prev_ofToSlice]
+
+theorem apply_eq_true_of_skipSuffixWhile_le_bool {p : Char → Bool} {s : String} {pos : s.Pos}
+    (h : s.skipSuffixWhile p ≤ pos) (h' : pos < s.endPos) :
+    p (pos.get (Pos.ne_endPos_of_lt h')) = true := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.apply_eq_true_of_skipSuffixWhile_le_bool
+    (by simpa [skipSuffixWhile_eq_skipSuffixWhile_toSlice, Pos.ofToSlice_le_iff] using h)
+    (by simpa [Pos.toSlice_lt_toSlice_iff] using h')
+
+@[simp]
+theorem revAll_bool_eq {p : Char → Bool} {s : String} : s.revAll p = s.toList.all p := by
+  simp [← revAll_toSlice]
+
+@[simp]
+theorem Pos.revSkip?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos res : s.Pos} :
+    pos.revSkip? P = some res ↔ ∃ h, res = pos.prev h ∧ P ((pos.prev h).get (by simp)) := by
+  simp [revSkip?_eq_revSkip?_toSlice, ← toSlice_inj, toSlice_prev, get_eq_get_toSlice]
+
+@[simp]
+theorem Pos.revSkip?_prop_eq_none_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
+    pos.revSkip? P = none ↔ ∀ h, ¬ P ((pos.prev h).get (by simp)) := by
+  simp [revSkip?_eq_revSkip?_toSlice, Pos.prev_toSlice]
+
+theorem Pos.apply_revSkipWhile_prop {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} {h} :
+    ¬ P (((pos.revSkipWhile P).prev h).get (by simp)) := by
+  have h' : pos.toSlice.revSkipWhile P ≠ s.toSlice.startPos := by
+    simpa [Pos.revSkipWhile_eq_revSkipWhile_toSlice, ← toSlice_inj] using h
+  have := Slice.Pos.apply_revSkipWhile_prop (pos := pos.toSlice) (h := h')
+  simpa [Pos.revSkipWhile_eq_revSkipWhile_toSlice, Pos.prev_ofToSlice]
+
+theorem Pos.revSkipWhile_prop_eq_self_iff_get {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
+    pos.revSkipWhile P = pos ↔ ∀ h, ¬ P ((pos.prev h).get (by simp)) := by
+  simp [Pos.revSkipWhile_eq_revSkipWhile_toSlice, ← toSlice_inj,
+    Slice.Pos.revSkipWhile_prop_eq_self_iff_get, Pos.prev_toSlice]
+
+theorem Pos.apply_of_revSkipWhile_le_prop {P : Char → Prop} [DecidablePred P] {s : String} {pos pos' : s.Pos}
+    (h₁ : pos' < pos) (h₂ : pos.revSkipWhile P ≤ pos') : P (pos'.get (ne_endPos_of_lt h₁)) := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.Pos.apply_of_revSkipWhile_le_prop
+    (Pos.toSlice_lt_toSlice_iff.2 h₁)
+    (by simpa [Pos.revSkipWhile_eq_revSkipWhile_toSlice, Pos.ofToSlice_le_iff] using h₂)
+
+theorem apply_skipSuffixWhile_prop {P : Char → Prop} [DecidablePred P] {s : String} {h} :
+    ¬ P (((s.skipSuffixWhile P).prev h).get (by simp)) := by
+  have h' : s.toSlice.skipSuffixWhile P ≠ s.toSlice.startPos := by
+    simpa [skipSuffixWhile_eq_skipSuffixWhile_toSlice, ← Pos.toSlice_inj] using h
+  have := Slice.apply_skipSuffixWhile_prop (s := s.toSlice) (h := h')
+  simpa [skipSuffixWhile_eq_skipSuffixWhile_toSlice, Pos.prev_ofToSlice]
+
+theorem apply_of_skipSuffixWhile_le_prop {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos}
+    (h : s.skipSuffixWhile P ≤ pos) (h' : pos < s.endPos) :
+    P (pos.get (Pos.ne_endPos_of_lt h')) := by
+  rw [Pos.get_eq_get_toSlice]
+  exact Slice.apply_of_skipSuffixWhile_le_prop
+    (by simpa [skipSuffixWhile_eq_skipSuffixWhile_toSlice, Pos.ofToSlice_le_iff] using h)
+    (by simpa [Pos.toSlice_lt_toSlice_iff] using h')
+
+@[simp]
+theorem revAll_prop_eq {P : Char → Prop} [DecidablePred P] {s : String} :
+    s.revAll P = s.toList.all (decide <| P ·) := by
+  simp [← revAll_toSlice]
+
 end String

src/Init/Data/String/Lemmas/Pattern/TakeDrop/String.lean

@@ -30,11 +30,7 @@ theorem skipPrefix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true)
 @[simp]
 theorem skipPrefix?_slice_eq_some_iff {pat s : Slice} {pos : s.Pos} :
     s.skipPrefix? pat = some pos ↔ ∃ t, pos.Splits pat.copy t := by
-  match h : pat.isEmpty with
-  | false =>
-    have := ForwardSliceSearcher.lawfulForwardPatternModel h
-    rw [Pattern.Model.skipPrefix?_eq_some_iff, ForwardSliceSearcher.isLongestMatch_iff_splits h]
-  | true => simp [skipPrefix?_slice_of_isEmpty h, (show pat.copy = "" by simpa), eq_comm]
+  rw [Pattern.Model.skipPrefix?_eq_some_iff, ForwardSliceSearcher.isLongestMatch_iff_splits]
 
 theorem startsWith_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
     s.startsWith pat = true := by
@@ -43,14 +39,10 @@ theorem startsWith_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true)
 @[simp]
 theorem startsWith_slice_iff {pat s : Slice} :
     s.startsWith pat ↔ pat.copy.toList <+: s.copy.toList := by
-  match h : pat.isEmpty with
-  | false =>
-    have := ForwardSliceSearcher.lawfulForwardPatternModel h
-    simp only [Model.startsWith_iff, ForwardSliceSearcher.matchesAt_iff_splits h,
-      splits_startPos_iff, exists_and_left, exists_eq_left]
-    simp only [← toList_inj, toList_append, List.prefix_iff_exists_append_eq]
-    exact ⟨fun ⟨t, ht⟩ => ⟨t.toList, by simp [ht]⟩, fun ⟨t, ht⟩ => ⟨String.ofList t, by simp [← ht]⟩⟩
-  | true => simp [startsWith_slice_of_isEmpty h, (show pat.copy = "" by simpa)]
+  simp only [Model.startsWith_iff, ForwardSliceSearcher.matchesAt_iff_splits,
+    splits_startPos_iff, exists_and_left, exists_eq_left]
+  simp only [← toList_inj, toList_append, List.prefix_iff_exists_append_eq]
+  exact ⟨fun ⟨t, ht⟩ => ⟨t.toList, by simp [ht]⟩, fun ⟨t, ht⟩ => ⟨String.ofList t, by simp [← ht]⟩⟩
 
 @[simp]
 theorem startsWith_slice_eq_false_iff {pat s : Slice} :
@@ -63,14 +55,18 @@ theorem dropPrefix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true)
 
 theorem eq_append_of_dropPrefix?_slice_eq_some {pat s res : Slice} (h : s.dropPrefix? pat = some res) :
     s.copy = pat.copy ++ res.copy := by
-  match hpat : pat.isEmpty with
-  | false =>
-    have := ForwardSliceSearcher.lawfulForwardPatternModel hpat
-    have := Pattern.Model.eq_append_of_dropPrefix?_eq_some h
-    simp only [PatternModel.Matches] at this
-    obtain ⟨_, ⟨-, rfl⟩, h⟩ := this
-    exact h
-  | true => simp [Option.some.inj (h ▸ dropPrefix?_slice_of_isEmpty hpat), (show pat.copy = "" by simpa)]
+  have := Pattern.Model.eq_append_of_dropPrefix?_eq_some h
+  simp only [PatternModel.Matches] at this
+  obtain ⟨_, ⟨-, rfl⟩, h⟩ := this
+  exact h
+
+@[simp]
+theorem all_slice_iff {pat s : Slice} : s.all pat ↔ ∃ n, s.copy = String.join (List.replicate n pat.copy) := by
+  simp [Pattern.Model.all_eq_true_iff, ForwardSliceSearcher.isLongestMatchAtChain_startPos_endPos_iff]
+
+@[simp]
+theorem revAll_slice_iff {pat s : Slice} : s.revAll pat ↔ ∃ n, s.copy = String.join (List.replicate n pat.copy) := by
+  simp [Pattern.Model.revAll_eq_true_iff, ForwardSliceSearcher.isLongestRevMatchAtChain_startPos_endPos_iff]
 
 @[simp]
 theorem skipPrefix?_string_eq_some_iff {pat : String} {s : Slice} {pos : s.Pos} :
@@ -104,6 +100,7 @@ theorem eq_append_of_dropPrefix?_string_eq_some {pat : String} {s res : Slice} (
   rw [dropPrefix?_string_eq_dropPrefix?_toSlice] at h
   simpa using eq_append_of_dropPrefix?_slice_eq_some h
 
+
 theorem skipSuffix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
     s.skipSuffix? pat = some s.endPos := by
   rw [skipSuffix?_eq_backwardPatternSkipSuffix?, BackwardSliceSearcher.skipSuffix?_of_isEmpty hpat]
@@ -111,11 +108,7 @@ theorem skipSuffix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true)
 @[simp]
 theorem skipSuffix?_slice_eq_some_iff {pat s : Slice} {pos : s.Pos} :
     s.skipSuffix? pat = some pos ↔ ∃ t, pos.Splits t pat.copy := by
-  match h : pat.isEmpty with
-  | false =>
-    have := BackwardSliceSearcher.lawfulBackwardPatternModel h
-    rw [Pattern.Model.skipSuffix?_eq_some_iff, ForwardSliceSearcher.isLongestRevMatch_iff_splits h]
-  | true => simp [skipSuffix?_slice_of_isEmpty h, (show pat.copy = "" by simpa), eq_comm]
+  rw [Pattern.Model.skipSuffix?_eq_some_iff, ForwardSliceSearcher.isLongestRevMatch_iff_splits]
 
 theorem endsWith_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
     s.endsWith pat = true := by
@@ -124,14 +117,10 @@ theorem endsWith_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
 @[simp]
 theorem endsWith_slice_iff {pat s : Slice} :
     s.endsWith pat ↔ pat.copy.toList <:+ s.copy.toList := by
-  match h : pat.isEmpty with
-  | false =>
-    have := BackwardSliceSearcher.lawfulBackwardPatternModel h
-    simp only [Model.endsWith_iff, ForwardSliceSearcher.revMatchesAt_iff_splits h,
-      splits_endPos_iff, exists_eq_right]
-    simp only [← toList_inj, toList_append, List.suffix_iff_exists_append_eq]
-    exact ⟨fun ⟨t, ht⟩ => ⟨t.toList, by simp [ht]⟩, fun ⟨t, ht⟩ => ⟨String.ofList t, by simp [← ht]⟩⟩
-  | true => simp [endsWith_slice_of_isEmpty h, (show pat.copy = "" by simpa)]
+  simp only [Model.endsWith_iff, ForwardSliceSearcher.revMatchesAt_iff_splits,
+    splits_endPos_iff, exists_eq_right]
+  simp only [← toList_inj, toList_append, List.suffix_iff_exists_append_eq]
+  exact ⟨fun ⟨t, ht⟩ => ⟨t.toList, by simp [ht]⟩, fun ⟨t, ht⟩ => ⟨String.ofList t, by simp [← ht]⟩⟩
 
 @[simp]
 theorem endsWith_slice_eq_false_iff {pat s : Slice} :
@@ -144,14 +133,10 @@ theorem dropSuffix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true)
 
 theorem eq_append_of_dropSuffix?_slice_eq_some {pat s res : Slice} (h : s.dropSuffix? pat = some res) :
     s.copy = res.copy ++ pat.copy := by
-  match hpat : pat.isEmpty with
-  | false =>
-    have := BackwardSliceSearcher.lawfulBackwardPatternModel hpat
-    have := Pattern.Model.eq_append_of_dropSuffix?_eq_some h
-    simp only [PatternModel.Matches] at this
-    obtain ⟨_, ⟨-, rfl⟩, h⟩ := this
-    exact h
-  | true => simp [Option.some.inj (h ▸ dropSuffix?_slice_of_isEmpty hpat), (show pat.copy = "" by simpa)]
+  have := Pattern.Model.eq_append_of_dropSuffix?_eq_some h
+  simp only [PatternModel.Matches] at this
+  obtain ⟨_, ⟨-, rfl⟩, h⟩ := this
+  exact h
 
 @[simp]
 theorem skipSuffix?_string_eq_some_iff' {pat : String} {s : Slice} {pos : s.Pos} :
@@ -208,12 +193,12 @@ theorem startsWith_slice_of_isEmpty {pat : Slice} {s : String} (hpat : pat.isEmp
 @[simp]
 theorem startsWith_slice_iff {pat : Slice} {s : String} :
     s.startsWith pat ↔ pat.copy.toList <+: s.toList := by
-  simp [startsWith_eq_startsWith_toSlice]
+  simp [← startsWith_toSlice]
 
 @[simp]
 theorem startsWith_slice_eq_false_iff {pat : Slice} {s : String} :
     s.startsWith pat = false ↔ ¬ (pat.copy.toList <+: s.toList) := by
-  simp [startsWith_eq_startsWith_toSlice]
+  simp [← startsWith_toSlice]
 
 theorem dropPrefix?_slice_of_isEmpty {pat : Slice} {s : String} (hpat : pat.isEmpty = true) :
     s.dropPrefix? pat = some s.toSlice := by
@@ -239,21 +224,21 @@ theorem skipPrefix?_string_eq_some_iff {pat s : String} {pos : s.Pos} :
 
 @[simp]
 theorem startsWith_string_empty {s : String} : s.startsWith "" = true := by
-  simp [startsWith_eq_startsWith_toSlice]
+  simp [← startsWith_toSlice]
 
 @[simp]
 theorem startsWith_string_iff {pat s : String} :
     s.startsWith pat ↔ pat.toList <+: s.toList := by
-  simp [startsWith_eq_startsWith_toSlice]
+  simp [← startsWith_toSlice]
 
 @[simp]
 theorem startsWith_string_eq_false_iff {pat s : String} :
     s.startsWith pat = false ↔ ¬ (pat.toList <+: s.toList) := by
-  simp [startsWith_eq_startsWith_toSlice]
+  simp [← startsWith_toSlice]
 
 @[simp]
 theorem dropPrefix?_string_empty {s : String} : s.dropPrefix? "" = some s.toSlice := by
-  simp [dropPrefix?_eq_dropPrefix?_toSlice]
+  simp [← dropPrefix?_toSlice]
 
 theorem eq_append_of_dropPrefix?_string_eq_some {s pat : String} {res : Slice} (h : s.dropPrefix? pat = some res) :
     s = pat ++ res.copy := by

src/Init/Data/String/Lemmas/Splits.lean

@@ -99,6 +99,11 @@ theorem Pos.splits {s : String} (p : s.Pos) :
   eq_append := by simp [← toByteArray_inj, Slice.toByteArray_copy, ← size_toByteArray]
   offset_eq_rawEndPos := by simp
 
+@[simp]
+theorem sliceTo_append_sliceFrom {s : String} {pos : s.Pos} :
+    (s.sliceTo pos).copy ++ (s.sliceFrom pos).copy = s :=
+  pos.splits.eq_append.symm
+
 theorem Slice.Pos.splits {s : Slice} (p : s.Pos) :
     p.Splits (s.sliceTo p).copy (s.sliceFrom p).copy where
   eq_append := copy_eq_copy_sliceTo
@@ -375,6 +380,10 @@ theorem Slice.copy_sliceTo_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ :
   · rintro ⟨t₂, h⟩
     exact p.splits.eq_left h
 
+theorem Slice.copy_sliceTo_eq_iff_splits {s : Slice} {p : s.Pos} {t₁ : String} :
+    (s.sliceTo p).copy = t₁ ↔ p.Splits t₁ (s.sliceFrom p).copy :=
+  ⟨fun h => h ▸ p.splits, p.splits.eq_left⟩
+
 theorem Slice.copy_sliceFrom_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₂ : String} :
     (s.sliceFrom p).copy = t₂ ↔ ∃ t₁, p.Splits t₁ t₂ := by
   refine ⟨?_, ?_⟩
@@ -383,14 +392,26 @@ theorem Slice.copy_sliceFrom_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₂
   · rintro ⟨t₂, h⟩
     exact p.splits.eq_right h
 
+theorem Slice.copy_sliceFrom_eq_iff_splits {s : Slice} {p : s.Pos} {t₂ : String} :
+    (s.sliceFrom p).copy = t₂ ↔ p.Splits (s.sliceTo p).copy t₂ :=
+  ⟨fun h => h ▸ p.splits, p.splits.eq_right⟩
+
 theorem copy_sliceTo_eq_iff_exists_splits {s : String} {p : s.Pos} {t₁ : String} :
     (s.sliceTo p).copy = t₁ ↔ ∃ t₂, p.Splits t₁ t₂ := by
   simp [← Pos.splits_toSlice_iff, ← Slice.copy_sliceTo_eq_iff_exists_splits]
 
+theorem copy_sliceTo_eq_iff_splits {s : String} {p : s.Pos} {t₁ : String} :
+    (s.sliceTo p).copy = t₁ ↔ p.Splits t₁ (s.sliceFrom p).copy :=
+  ⟨fun h => h ▸ p.splits, p.splits.eq_left⟩
+
 theorem copy_sliceFrom_eq_iff_exists_splits {s : String} {p : s.Pos} {t₂ : String} :
     (s.sliceFrom p).copy = t₂ ↔ ∃ t₁, p.Splits t₁ t₂ := by
   simp [← Pos.splits_toSlice_iff, ← Slice.copy_sliceFrom_eq_iff_exists_splits]
 
+theorem copy_sliceFrom_eq_iff_splits {s : String} {p : s.Pos} {t₂ : String} :
+    (s.sliceFrom p).copy = t₂ ↔ p.Splits (s.sliceTo p).copy t₂ :=
+  ⟨fun h => h ▸ p.splits, p.splits.eq_right⟩
+
 theorem Pos.Splits.offset_eq_decreaseBy {s : String} {p : s.Pos} (h : p.Splits t₁ t₂) :
     p.offset = s.rawEndPos.decreaseBy t₂.utf8ByteSize := by
   simp [h.offset_eq_rawEndPos, h.eq_append, Pos.Raw.ext_iff]
@@ -641,6 +662,28 @@ theorem Pos.splits_append_rawEndPos {s t : String} :
   eq_append := rfl
   offset_eq_rawEndPos := rfl
 
+/--
+Given a slice `s` such that `s.copy = t₁ ++ t₂`, obtain the position sitting between `t₁` and `t₂`.
+-/
+def Slice.Pos.ofEqAppend {s : Slice} {t₁ t₂ : String} (h : s.copy = t₁ ++ t₂) : s.Pos :=
+  s.pos t₁.rawEndPos
+    (by simpa [← Pos.Raw.isValid_copy_iff, h] using ((Pos.Raw.isValid_rawEndPos).append_right t₂))
+
+theorem Slice.Pos.splits_ofEqAppend {s : Slice} {t₁ t₂ : String} (h : s.copy = t₁ ++ t₂) :
+    (ofEqAppend h).Splits t₁ t₂ where
+  eq_append := h
+  offset_eq_rawEndPos := by simp [ofEqAppend]
+
+/--
+Given a string `s` such that `s = t₁ ++ t₂`, obtain the position sitting between `t₁` and `t₂`.
+-/
+def Pos.ofEqAppend {s t₁ t₂ : String} (h : s = t₁ ++ t₂) : s.Pos :=
+  ((t₁ ++ t₂).pos t₁.rawEndPos ((Pos.Raw.isValid_rawEndPos).append_right t₂)).cast h.symm
+
+theorem Pos.splits_ofEqAppend {s t₁ t₂ : String} (h : s = t₁ ++ t₂) : (ofEqAppend h).Splits t₁ t₂ where
+  eq_append := h
+  offset_eq_rawEndPos := by simp [ofEqAppend]
+
 theorem Pos.Splits.copy_sliceTo_eq {s : String} {p : s.Pos} (h : p.Splits t₁ t₂) :
     (s.sliceTo p).copy = t₁ :=
   p.splits.eq_left h
@@ -740,4 +783,44 @@ theorem splits_prevn_endPos (s : String) (n : Nat) :
     (s.endPos.prevn n).Splits (String.ofList (s.toList.take (s.length - n))) (String.ofList (s.toList.drop (s.length - n))) := by
   simpa using s.splits_endPos.prevn n
 
+@[simp]
+theorem Slice.copy_sliceFrom_cast {s t : Slice} (hst : s.copy = t.copy) {pos : s.Pos} :
+    (t.sliceFrom (pos.cast hst)).copy = (s.sliceFrom pos).copy := by
+  simpa [copy_sliceFrom_eq_iff_exists_splits] using ⟨_, pos.splits⟩
+
+@[simp]
+theorem Slice.copy_sliceTo_cast {s t : Slice} (hst : s.copy = t.copy) {pos : s.Pos} :
+    (t.sliceTo (pos.cast hst)).copy = (s.sliceTo pos).copy := by
+  simpa [copy_sliceTo_eq_iff_exists_splits] using ⟨_, pos.splits⟩
+
+@[simp]
+theorem copy_sliceFrom_cast {s t : String} (hst : s = t) {pos : s.Pos} :
+    (t.sliceFrom (pos.cast hst)).copy = (s.sliceFrom pos).copy := by
+  simpa [copy_sliceFrom_eq_iff_exists_splits] using ⟨_, pos.splits⟩
+
+@[simp]
+theorem copy_sliceTo_cast {s t : String} (hst : s = t) {pos : s.Pos} :
+    (t.sliceTo (pos.cast hst)).copy = (s.sliceTo pos).copy := by
+  simpa [copy_sliceTo_eq_iff_exists_splits] using ⟨_, pos.splits⟩
+
+theorem Slice.Pos.sliceFrom_cast {s t : Slice} {hst : s.copy = t.copy} (p q : s.Pos) {h} :
+    Slice.Pos.sliceFrom (p.cast hst) (q.cast hst) h =
+      (Slice.Pos.sliceFrom p q (by simpa using h)).cast (by simp) := by
+  ext1; simp
+
+theorem Slice.Pos.sliceTo_cast {s t : Slice} {hst : s.copy = t.copy} (p q : s.Pos) {h} :
+    Slice.Pos.sliceTo (p.cast hst) (q.cast hst) h =
+      (Slice.Pos.sliceTo p q (by simpa using h)).cast (by simp) := by
+  ext1; simp
+
+theorem Pos.sliceFrom_cast {s t : String} {hst : s = t} (p q : s.Pos) {h} :
+    Pos.sliceFrom (p.cast hst) (q.cast hst) h =
+      (Pos.sliceFrom p q (by simpa using h)).cast (by simp) := by
+  ext1; simp
+
+theorem Pos.sliceTo_cast {s t : String} {hst : s = t} (p q : s.Pos) {h} :
+    Pos.sliceTo (p.cast hst) (q.cast hst) h =
+      (Pos.sliceTo p q (by simpa using h)).cast (by simp) := by
+  ext1; simp
+
 end String

src/Init/Data/String/Lemmas/StringOrder.lean

@@ -0,0 +1,49 @@
+/-
+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
+-/
+module
+
+prelude
+public import Init.Data.String.Basic
+public import Init.Data.Order.Classes
+import Init.Data.List.Lex
+import Init.Data.Char.Lemmas
+import Init.Data.Char.Order
+import Init.Data.Order.Factories
+import Init.Data.Order.Lemmas
+
+public section
+
+open Std
+
+namespace String
+
+@[simp] protected theorem not_le {a b : String} : ¬ a ≤ b ↔ b < a := Decidable.not_not
+@[simp] protected theorem not_lt {a b : String} : ¬ a < b ↔ b ≤ a := Iff.rfl
+@[simp] protected theorem le_refl (a : String) : a ≤ a := List.le_refl _
+@[simp] protected theorem lt_irrefl (a : String) : ¬ a < a := List.lt_irrefl _
+
+attribute [local instance] Char.notLTTrans Char.ltTrichotomous Char.ltAsymm
+
+protected theorem le_trans {a b c : String} : a ≤ b → b ≤ c → a ≤ c := List.le_trans
+protected theorem lt_trans {a b c : String} : a < b → b < c → a < c := List.lt_trans
+protected theorem le_total (a b : String) : a ≤ b ∨ b ≤ a := List.le_total _ _
+protected theorem le_antisymm {a b : String} : a ≤ b → b ≤ a → a = b := fun h₁ h₂ => String.ext (List.le_antisymm (as := a.toList) (bs := b.toList) h₁ h₂)
+protected theorem lt_asymm {a b : String} (h : a < b) : ¬ b < a := List.lt_asymm h
+protected theorem ne_of_lt {a b : String} (h : a < b) : a ≠ b := by
+  have := String.lt_irrefl a
+  intro h; subst h; contradiction
+
+instance instIsLinearOrder : IsLinearOrder String := by
+  apply IsLinearOrder.of_le
+  case le_antisymm => constructor; apply String.le_antisymm
+  case le_trans => constructor; apply String.le_trans
+  case le_total => constructor; apply String.le_total
+
+instance : LawfulOrderLT String where
+  lt_iff a b := by
+    simp [← String.not_le, Decidable.imp_iff_not_or, Std.Total.total]
+
+end String

src/Init/Data/String/Pattern/Basic.lean

@@ -96,6 +96,44 @@ theorem endPos_ofSliceFrom {s : Slice} {p : s.Pos} {st : SearchStep (s.sliceFrom
     st.ofSliceFrom.endPos = Slice.Pos.ofSliceFrom st.endPos := by
   cases st <;> simp [ofSliceFrom]
 
+/--
+Converts a {lean}`SearchStep s` into a {lean}`SearchStep t` by applying {name}`Slice.Pos.cast` to the
+start and end position.
+-/
+@[inline]
+def cast {s t : Slice} (hst : s.copy = t.copy) : SearchStep s → SearchStep t
+  | .rejected startPos endPos => .rejected (startPos.cast hst) (endPos.cast hst)
+  | .matched startPos endPos => .matched (startPos.cast hst) (endPos.cast hst)
+
+@[simp]
+theorem cast_rejected {s t : Slice} {hst : s.copy = t.copy} {startPos endPos : s.Pos} :
+    (SearchStep.rejected startPos endPos).cast hst = .rejected (startPos.cast hst) (endPos.cast hst) :=
+  (rfl)
+
+@[simp]
+theorem cast_matched {s t : Slice} {hst : s.copy = t.copy} {startPos endPos : s.Pos} :
+    (SearchStep.matched startPos endPos).cast hst = .matched (startPos.cast hst) (endPos.cast hst) :=
+  (rfl)
+
+@[simp]
+theorem startPos_cast {s t : Slice} (hst : s.copy = t.copy) {st : SearchStep s} :
+    (st.cast hst).startPos = st.startPos.cast hst := by
+  cases st <;> simp
+
+@[simp]
+theorem endPos_cast {s t : Slice} (hst : s.copy = t.copy) {st : SearchStep s} :
+    (st.cast hst).endPos = st.endPos.cast hst := by
+  cases st <;> simp
+
+@[simp]
+theorem cast_rfl {s : Slice} {st : SearchStep s} : st.cast rfl = st := by
+  cases st <;> simp
+
+@[simp]
+theorem cast_cast {s t u : Slice} {hst : s.copy = t.copy} {htu : t.copy = u.copy} {st : SearchStep s} :
+    (st.cast hst).cast htu = st.cast (hst.trans htu) := by
+  cases st <;> simp
+
 end SearchStep
 
 /--

src/Init/Data/String/Search.lean

@@ -311,23 +311,6 @@ def Internal.containsImpl (s : String) (c : Char) : Bool :=
 def Internal.anyImpl (s : String) (p : Char → Bool) :=
   String.any s p
 
-/--
-Checks whether a slice only consists of matches of the pattern {name}`pat`.
-
-Short-circuits at the first pattern mis-match.
-
-This function is generic over all currently supported patterns.
-
-Examples:
- * {lean}`"brown".all Char.isLower = true`
- * {lean}`"brown and orange".all Char.isLower = false`
- * {lean}`"aaaaaa".all 'a' = true`
- * {lean}`"aaaaaa".all "aa" = true`
- * {lean}`"aaaaaaa".all "aa" = false`
--/
-@[inline, suggest_for String.every] def all (s : String) (pat : ρ) [ForwardPattern pat] : Bool :=
-  s.toSlice.all pat
-
 /--
 Checks whether the string can be interpreted as the decimal representation of a natural number.
 

src/Init/Data/String/Slice.lean

@@ -426,13 +426,13 @@ Advances {name}`pos` as long as {name}`pat` matches.
 -/
 @[specialize pat]
 def Pos.skipWhile {s : Slice} (pos : s.Pos) (pat : ρ) [ForwardPattern pat] : s.Pos :=
-  if let some nextCurr := ForwardPattern.skipPrefix? pat (s.sliceFrom pos) then
-    if pos < Pos.ofSliceFrom nextCurr then
-      skipWhile (Pos.ofSliceFrom nextCurr) pat
+  match pos.skip? pat with
+  | some nextCurr =>
+    if pos < nextCurr then
+      skipWhile nextCurr pat
     else
       pos
-  else
-    pos
+  | none => pos
 termination_by pos
 
 /--
@@ -572,7 +572,7 @@ Examples:
 -/
 @[inline]
 def all (s : Slice) (pat : ρ) [ForwardPattern pat] : Bool :=
-  s.dropWhile pat |>.isEmpty
+  s.skipPrefixWhile pat == s.endPos
 
 end ForwardPatternUsers
 
@@ -706,14 +706,14 @@ Returns {name}`none` otherwise.
 This function is generic over all currently supported patterns.
 -/
 @[inline]
-def Pos.revSkip? {s : Slice} (pos : s.Pos) (pat : ρ) [ForwardPattern pat] : Option s.Pos :=
-  ((s.sliceFrom pos).skipPrefix? pat).map Pos.ofSliceFrom
+def Pos.revSkip? {s : Slice} (pos : s.Pos) (pat : ρ) [BackwardPattern pat] : Option s.Pos :=
+  ((s.sliceTo pos).skipSuffix? pat).map Pos.ofSliceTo
 
 /--
 If {name}`pat` matches a suffix of {name}`s`, returns the remainder. Returns {name}`none` otherwise.
 
 Use {name (scope := "Init.Data.String.Slice")}`String.Slice.dropSuffix` to return the slice
-unchanged when {name}`pat` does not match a prefix.
+unchanged when {name}`pat` does not match a suffix.
 
 This function is generic over all currently supported patterns.
 
@@ -765,23 +765,53 @@ Rewinds {name}`pos` as long as {name}`pat` matches.
 -/
 @[specialize pat]
 def Pos.revSkipWhile {s : Slice} (pos : s.Pos) (pat : ρ) [BackwardPattern pat] : s.Pos :=
-  if let some nextCurr := BackwardPattern.skipSuffix? pat (s.sliceTo pos) then
-    if Pos.ofSliceTo nextCurr < pos then
-      revSkipWhile (Pos.ofSliceTo nextCurr) pat
+  match pos.revSkip? pat with
+  | some nextCurr =>
+    if nextCurr < pos then
+      revSkipWhile nextCurr pat
     else
       pos
-  else
-    pos
+  | none => pos
 termination_by pos.down
 
 /--
-Returns the position a the start of the longest suffix of {name}`s` for which {name}`pat` matches
+Returns the position at the start of the longest suffix of {name}`s` for which {name}`pat` matches
 (potentially repeatedly).
 -/
 @[inline]
 def skipSuffixWhile (s : Slice) (pat : ρ) [BackwardPattern pat] : s.Pos :=
   s.endPos.revSkipWhile pat
 
+/--
+Checks whether a slice only consists of matches of the pattern {name}`pat`, starting from the back
+of the string.
+
+Short-circuits at the first pattern mis-match.
+
+This function is generic over all currently supported patterns.
+
+For many types of patterns, this function can be expected to return the same result as
+{name}`Slice.all`. If mismatches are expected to occur close to the end of the string, this function
+might be more efficient.
+
+For some types of patterns, this function will return a different result than {name}`Slice.all`.
+Consider, for example, a pattern that matches the longest string at the given position that matches
+the regular expression {lean}`"a|aa|ab"`. Then, given the input string {lean}`"aab"`, performing
+{name}`Slice.all` will greedily match the prefix {lean}`"aa"` and then get stuck on the remainder
+{lean}`"b"`, causing it to return {lean}`false`. On the other hand, {name}`Slice.revAll` will match
+the suffix {lean}`"ab"` and then match the remainder {lean}`"a"`, so it will return {lean}`true`.
+
+Examples:
+ * {lean}`"brown".toSlice.revAll Char.isLower = true`
+ * {lean}`"brown and orange".toSlice.revAll Char.isLower = false`
+ * {lean}`"aaaaaa".toSlice.revAll 'a' = true`
+ * {lean}`"aaaaaa".toSlice.revAll "aa" = true`
+ * {lean}`"aaaaaaa".toSlice.revAll "aa" = false`
+-/
+@[inline]
+def revAll (s : Slice) (pat : ρ) [BackwardPattern pat] : Bool :=
+  s.skipSuffixWhile pat == s.startPos
+
 /--
 Creates a new slice that contains the longest suffix of {name}`s` for which {name}`pat` matched
 (potentially repeatedly).

src/Init/Data/String/Subslice.lean

@@ -23,7 +23,7 @@ Given a {name}`Slice` {name}`s`, the type {lean}`s.Subslice` is the type of half
 in {name}`s` delineated by a valid position on both sides.
 
 This type is useful to track regions of interest within some larger slice that is also of interest.
-In contrast, {name}`Slice` is used to track regions of interest whithin some larger string that is
+In contrast, {name}`Slice` is used to track regions of interest within some larger string that is
 not or no longer relevant.
 
 Equality on {name}`Subslice` is somewhat better behaved than on {name}`Slice`, but note that there

src/Init/Data/String/TakeDrop.lean

@@ -224,6 +224,53 @@ Returns the position after the longest prefix of {name}`s` for which {name}`pat`
 @[inline] def skipPrefixWhile (s : String) (pat : ρ) [ForwardPattern pat] : s.Pos :=
   Pos.ofToSlice (s.toSlice.skipPrefixWhile pat)
 
+/--
+Checks whether a string only consists of matches of the pattern {name}`pat`.
+
+Short-circuits at the first pattern mis-match.
+
+This function is generic over all currently supported patterns.
+
+Examples:
+ * {lean}`"brown".all Char.isLower = true`
+ * {lean}`"brown and orange".all Char.isLower = false`
+ * {lean}`"aaaaaa".all 'a' = true`
+ * {lean}`"aaaaaa".all "aa" = true`
+ * {lean}`"aaaaaaa".all "aa" = false`
+-/
+@[inline, suggest_for String.every] def all (s : String) (pat : ρ) [ForwardPattern pat] : Bool :=
+  s.toSlice.all pat
+
+/--
+Checks whether a string only consists of matches of the pattern {name}`pat`, starting from the back
+of the string.
+
+Short-circuits at the first pattern mis-match.
+
+This function is generic over all currently supported patterns.
+
+For many types of patterns, this function can be expected to return the same result as
+{name}`String.all`. If mismatches are expected to occur close to the end of the string, this function
+might be more efficient.
+
+For some types of patterns, this function will return a different result than {name}`String.all`.
+Consider, for example, a pattern that matches the longest string at the given position that matches
+the regular expression {lean}`"a|aa|ab"`. Then, given the input string {lean}`"aab"`, performing
+{name}`String.all` will greedily match the prefix {lean}`"aa"` and then get stuck on the remainder
+{lean}`"b"`, causing it to return {lean}`false`. On the other hand, {name}`String.revAll` will match
+the suffix {lean}`"ab"` and then match the remainder {lean}`"a"`, so it will return {lean}`true`.
+
+Examples:
+ * {lean}`"brown".revAll Char.isLower = true`
+ * {lean}`"brown and orange".revAll Char.isLower = false`
+ * {lean}`"aaaaaa".revAll 'a' = true`
+ * {lean}`"aaaaaa".revAll "aa" = true`
+ * {lean}`"aaaaaaa".revAll "aa" = false`
+-/
+@[inline]
+def revAll (s : String) (pat : ρ) [BackwardPattern pat] : Bool :=
+  s.toSlice.revAll pat
+
 /--
 If {name}`pat` matches at {name}`pos`, returns the position after the end of the match.
 Returns {name}`none` otherwise.
@@ -314,7 +361,7 @@ Returns {name}`none` otherwise.
 This function is generic over all currently supported patterns.
 -/
 @[inline]
-def Pos.revSkip? {s : String} (pos : s.Pos) (pat : ρ) [ForwardPattern pat] : Option s.Pos :=
+def Pos.revSkip? {s : String} (pos : s.Pos) (pat : ρ) [BackwardPattern pat] : Option s.Pos :=
   (pos.toSlice.revSkip? pat).map Pos.ofToSlice
 
 /--
@@ -461,7 +508,7 @@ def dropPrefix? (s : String) (pat : ρ) [ForwardPattern pat] : Option String.Sli
 If {name}`pat` matches a suffix of {name}`s`, returns the remainder. Returns {name}`none` otherwise.
 
 Use {name (scope := "Init.Data.String.TakeDrop")}`String.dropSuffix` to return the slice
-unchanged when {name}`pat` does not match a prefix.
+unchanged when {name}`pat` does not match a suffix.
 
 This is a cheap operation because it does not allocate a new string to hold the result.
 To convert the result into a string, use {name}`String.Slice.copy`.

src/Init/Data/Vector/Basic.lean

@@ -506,6 +506,16 @@ Examples:
 @[inline, expose] def sum [Add α] [Zero α] (xs : Vector α n) : α :=
   xs.toArray.sum
 
+/--
+Computes the product of the elements of a vector.
+
+Examples:
+ * `#v[a, b, c].prod = a * (b * (c * 1))`
+ * `#v[1, 2, 5].prod = 10`
+-/
+@[inline, expose] def prod [Mul α] [One α] (xs : Vector α n) : α :=
+  xs.toArray.prod
+
 /--
 Pad a vector on the left with a given element.
 

src/Init/Data/Vector/Int.lean

@@ -30,4 +30,16 @@ theorem sum_reverse_int (xs : Vector Int n) : xs.reverse.sum = xs.sum := by
 theorem sum_eq_foldl_int {xs : Vector Int n} : xs.sum = xs.foldl (b := 0) (· + ·) := by
   simp only [foldl_eq_foldr_reverse, Int.add_comm, ← sum_eq_foldr, sum_reverse_int]
 
+@[simp] theorem prod_replicate_int {n : Nat} {a : Int} : (replicate n a).prod = a ^ n := by
+  simp [← prod_toArray, Array.prod_replicate_int]
+
+theorem prod_append_int {as₁ as₂ : Vector Int n} : (as₁ ++ as₂).prod = as₁.prod * as₂.prod := by
+  simp [← prod_toArray]
+
+theorem prod_reverse_int (xs : Vector Int n) : xs.reverse.prod = xs.prod := by
+  simp [prod_reverse]
+
+theorem prod_eq_foldl_int {xs : Vector Int n} : xs.prod = xs.foldl (b := 1) (· * ·) := by
+  simp only [foldl_eq_foldr_reverse, Int.mul_comm, ← prod_eq_foldr, prod_reverse_int]
+
 end Vector

src/Init/Data/Vector/Lemmas.lean

@@ -278,6 +278,12 @@ theorem toArray_mk {xs : Array α} (h : xs.size = n) : (Vector.mk xs h).toArray
 @[simp, grind =] theorem sum_toArray [Add α] [Zero α] {xs : Vector α n} :
     xs.toArray.sum = xs.sum := rfl
 
+@[simp] theorem prod_mk [Mul α] [One α] {xs : Array α} (h : xs.size = n) :
+    (Vector.mk xs h).prod = xs.prod := rfl
+
+@[simp, grind =] theorem prod_toArray [Mul α] [One α] {xs : Vector α n} :
+    xs.toArray.prod = xs.prod := rfl
+
 @[simp] theorem eq_mk : xs = Vector.mk as h ↔ xs.toArray = as := by
   cases xs
   simp
@@ -551,6 +557,10 @@ theorem toArray_toList {xs : Vector α n} : xs.toList.toArray = xs.toArray := rf
     xs.toList.sum = xs.sum := by
   rw [← toList_toArray, Array.sum_toList, sum_toArray]
 
+@[simp, grind =] theorem prod_toList [Mul α] [One α] {xs : Vector α n} :
+    xs.toList.prod = xs.prod := by
+  rw [← toList_toArray, Array.prod_toList, prod_toArray]
+
 @[simp] theorem getElem_toList {xs : Vector α n} {i : Nat} (h : i < xs.toList.length) :
     xs.toList[i] = xs[i]'(by simpa using h) := by
   cases xs
@@ -3134,3 +3144,39 @@ theorem sum_eq_foldl [Zero α] [Add α]
     {xs : Vector α n} :
     xs.sum = xs.foldl (b := 0) (· + ·) := by
   simp [← sum_toList, List.sum_eq_foldl]
+
+/-! ### prod -/
+
+@[simp, grind =] theorem prod_empty [Mul α] [One α] : (#v[] : Vector α 0).prod = 1 := rfl
+theorem prod_eq_foldr [Mul α] [One α] {xs : Vector α n} :
+    xs.prod = xs.foldr (b := 1) (· * ·) :=
+  rfl
+
+@[simp, grind =]
+theorem prod_append [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.LeftIdentity (α := α) (· * ·) 1] [Std.LawfulLeftIdentity (α := α) (· * ·) 1]
+    {as₁ as₂ : Vector α n} : (as₁ ++ as₂).prod = as₁.prod * as₂.prod := by
+  simp [← prod_toList, List.prod_append]
+
+@[simp, grind =]
+theorem prod_singleton [Mul α] [One α] [Std.LawfulRightIdentity (· * ·) (1 : α)] {x : α} :
+    #v[x].prod = x := by
+  simp [← prod_toList, Std.LawfulRightIdentity.right_id x]
+
+@[simp, grind =]
+theorem prod_push [Mul α] [One α] [Std.Associative (α := α) (· * ·)]
+    [Std.LawfulIdentity (· * ·) (1 : α)] {xs : Vector α n} {x : α} :
+    (xs.push x).prod = xs.prod * x := by
+  simp [← prod_toArray]
+
+@[simp, grind =]
+theorem prod_reverse [One α] [Mul α] [Std.Associative (α := α) (· * ·)]
+    [Std.Commutative (α := α) (· * ·)]
+    [Std.LawfulLeftIdentity (α := α) (· * ·) 1] (xs : Vector α n) : xs.reverse.prod = xs.prod := by
+  simp [← prod_toList, List.prod_reverse]
+
+theorem prod_eq_foldl [One α] [Mul α]
+    [Std.Associative (α := α) (· * ·)] [Std.LawfulIdentity (· * ·) (1 : α)]
+    {xs : Vector α n} :
+    xs.prod = xs.foldl (b := 1) (· * ·) := by
+  simp [← prod_toList, List.prod_eq_foldl]

src/Init/Data/Vector/Nat.lean

@@ -37,4 +37,23 @@ theorem sum_reverse_nat (xs : Vector Nat n) : xs.reverse.sum = xs.sum := by
 theorem sum_eq_foldl_nat {xs : Vector Nat n} : xs.sum = xs.foldl (b := 0) (· + ·) := by
   simp only [foldl_eq_foldr_reverse, Nat.add_comm, ← sum_eq_foldr, sum_reverse_nat]
 
+protected theorem prod_pos_iff_forall_pos_nat {xs : Vector Nat n} : 0 < xs.prod ↔ ∀ x ∈ xs, 0 < x := by
+  simp [← prod_toArray, Array.prod_pos_iff_forall_pos_nat]
+
+protected theorem prod_eq_zero_iff_exists_zero_nat {xs : Vector Nat n} :
+    xs.prod = 0 ↔ ∃ x ∈ xs, x = 0 := by
+  simp [← prod_toArray, Array.prod_eq_zero_iff_exists_zero_nat]
+
+@[simp] theorem prod_replicate_nat {n : Nat} {a : Nat} : (replicate n a).prod = a ^ n := by
+  simp [← prod_toArray, Array.prod_replicate_nat]
+
+theorem prod_append_nat {as₁ as₂ : Vector Nat n} : (as₁ ++ as₂).prod = as₁.prod * as₂.prod := by
+  simp [← prod_toArray]
+
+theorem prod_reverse_nat (xs : Vector Nat n) : xs.reverse.prod = xs.prod := by
+  simp [prod_reverse]
+
+theorem prod_eq_foldl_nat {xs : Vector Nat n} : xs.prod = xs.foldl (b := 1) (· * ·) := by
+  simp only [foldl_eq_foldr_reverse, Nat.mul_comm, ← prod_eq_foldr, prod_reverse_nat]
+
 end Vector

src/Init/Grind/Ring/Basic.lean

@@ -564,6 +564,28 @@ end Ring
 
 end IsCharP
 
+/--
+`PowIdentity α p` states that `x ^ p = x` holds for all elements of `α`.
+
+The primary source of instances is Fermat's little theorem: for a finite field with `q` elements,
+`x ^ q = x` for every `x`. For `Fin p` or `ZMod p` with prime `p`, this gives `x ^ p = x`.
+
+The `grind` ring solver uses this typeclass to add the relation `x ^ p - x = 0` to the
+Groebner basis, which allows it to reduce high-degree polynomials. Mathlib can provide
+instances for general finite fields via `FiniteField.pow_card`.
+-/
+class PowIdentity (α : Type u) [CommSemiring α] (p : outParam Nat) : Prop where
+  /-- Every element satisfies `x ^ p = x`. -/
+  pow_eq (x : α) : x ^ p = x
+
+namespace PowIdentity
+
+variable [CommSemiring α] [PowIdentity α p]
+
+theorem pow (x : α) : x ^ p = x := pow_eq x
+
+end PowIdentity
+
 open AddCommGroup
 
 theorem no_int_zero_divisors {α : Type u} [IntModule α] [NoNatZeroDivisors α] {k : Int} {a : α}

src/Init/Grind/Ring/Envelope.lean

@@ -193,7 +193,7 @@ theorem mul_assoc (a b c : Q α) : mul (mul a b) c = mul a (mul b c) := by
   simp [Semiring.left_distrib, Semiring.right_distrib]; refine ⟨0, ?_⟩; ac_rfl
 
 theorem mul_one (a : Q α) : mul a (natCast 1) = a := by
-obtain ⟨⟨_, _⟩⟩ := a; simp
+  obtain ⟨⟨_, _⟩⟩ := a; simp
 
 theorem one_mul (a : Q α) : mul (natCast 1) a = a := by
   obtain ⟨⟨_, _⟩⟩ := a; simp

src/Init/Grind/Util.lean

@@ -30,13 +30,13 @@ simpMatchDiscrsOnly (match 0 with | 0 => true | _ => false) = true
 ```
 using `eq_self`.
 -/
-def simpMatchDiscrsOnly {α : Sort u} (a : α) : α := a
+@[expose] def simpMatchDiscrsOnly {α : Sort u} (a : α) : α := a
 
 /--
 Gadget for protecting lambda abstractions created by `abstractGroundMismatches?`
 from beta reduction during preprocessing. See `ProveEq.lean` for details.
 -/
-def abstractFn {α : Sort u} (a : α) : α := a
+@[expose] def abstractFn {α : Sort u} (a : α) : α := a
 
 /-- Gadget for representing offsets `t+k` in patterns. -/
 def offset (a b : Nat) : Nat := a + b

src/Init/GrindInstances/Ring/Fin.lean

@@ -156,6 +156,12 @@ instance [i : NeZero n] : ToInt.Pow (Fin n) (.co 0 n) where
       rw [pow_succ, ToInt.Mul.toInt_mul, ih, ← ToInt.wrap_toInt,
         ← IntInterval.wrap_mul (by simp), Int.pow_succ, ToInt.wrap_toInt]
 
+instance : PowIdentity (Fin 2) 2 where
+  pow_eq x := by
+    match x with
+    | ⟨0, _⟩ => rfl
+    | ⟨1, _⟩ => rfl
+
 end Fin
 
 end Lean.Grind

src/Init/Notation.lean

@@ -624,6 +624,23 @@ existing code. It may be removed in a future version of the library.
 syntax (name := deprecated) "deprecated" (ppSpace ident)? (ppSpace str)?
     (" (" &"since" " := " str ")")? : attr
 
+/--
+The attribute `@[deprecated_arg old new]` marks a named parameter as deprecated.
+
+When a caller uses the old name with a replacement available, a deprecation warning is emitted
+and the argument is silently forwarded to the new parameter. When no replacement is provided,
+the parameter is treated as removed and using it produces an error.
+
+* `@[deprecated_arg old new (since := "2026-03-18")]` marks `old` as a deprecated alias for `new`.
+* `@[deprecated_arg old new "use foo instead" (since := "2026-03-18")]` adds a custom message.
+* `@[deprecated_arg old (since := "2026-03-18")]` marks `old` as a removed parameter (no replacement).
+* `@[deprecated_arg old "no longer needed" (since := "2026-03-18")]` removed with a custom message.
+
+A warning is emitted if `(since := "...")` is omitted.
+-/
+syntax (name := deprecated_arg) "deprecated_arg" ppSpace ident (ppSpace ident)? (ppSpace str)?
+    (" (" &"since" " := " str ")")? : attr
+
 /--
 The attribute `@[suggest_for ..]` on a declaration suggests likely ways in which
 someone might **incorrectly** refer to a definition.

src/Init/Omega/LinearCombo.lean

@@ -36,9 +36,6 @@ private local instance : ToString Int where
 private local instance : Repr Int where
   reprPrec i prec := if i < 0 then Repr.addAppParen (toString i) prec else toString i
 
-private local instance : Append String where
-  append := String.Internal.append
-
 /-- Internal representation of a linear combination of atoms, and a constant term. -/
 structure LinearCombo where
   /-- Constant term. -/

src/Init/Prelude.lean

@@ -145,7 +145,7 @@ Examples:
 The constant function that ignores its argument.
 
 If `a : α`, then `Function.const β a : β → α` is the “constant function with value `a`”. For all
-arguments `b : β`, `Function.const β a b = a`.
+arguments `b : β`, `Function.const β a b = a`. It is often written directly as `fun _ => a`.
 
 Examples:
  * `Function.const Bool 10 true = 10`
@@ -185,13 +185,9 @@ example : foo.default = (default, default) :=
 abbrev inferInstance {α : Sort u} [i : α] : α := i
 
 set_option checkBinderAnnotations false in
-/-- `inferInstanceAs α` synthesizes an instance of type `α`, transporting it from a
-definitionally equal type if necessary. This is useful when `α` is definitionally equal to
-some `α'` for which instances are registered, as it prevents leaking the definition's RHS
-at lower transparencies.
-
-`inferInstanceAs` requires an expected type from context. If you just need to synthesize an
-instance without transporting between types, use `inferInstance` instead.
+/--
+`inferInstanceAs α` synthesizes an instance of type `α` and then adjusts it to conform to the
+expected type `β`, which must be inferable from context.
 
 Example:
 ```
@@ -199,7 +195,26 @@ def D := Nat
 instance : Inhabited D := inferInstanceAs (Inhabited Nat)
 ```
 
-See `Lean.Meta.WrapInstance` for details.
+The adjustment will make sure that when the resulting instance will not "leak" the RHS `Nat` when
+reduced at transparency levels below `semireducible`, i.e. where `D` would not be unfolded either,
+preventing "defeq abuse".
+
+More specifically, given the "source type" (the argument) and "target type" (the expected type),
+`inferInstanceAs` synthesizes an instance for the source type and then unfolds and rewraps its
+components (fields, nested instances) as necessary to make them compatible with the target type. The
+individual steps are represented by the following options, which all default to enabled and can be
+disabled to help with porting:
+
+* `backward.inferInstanceAs.wrap`: master switch for instance adjustment in both `inferInstanceAs`
+  and the default deriving handler
+* `backward.inferInstanceAs.wrap.reuseSubInstances`: reuse existing instances for the target type
+  for sub-instance fields to avoid non-defeq instance diamonds
+* `backward.inferInstanceAs.wrap.instances`: wrap non-reducible instances in auxiliary definitions
+* `backward.inferInstanceAs.wrap.data`: wrap data fields in auxiliary definitions (proof fields are
+  always wrapped)
+
+If you just need to synthesize an instance without transporting between types, use `inferInstance`
+instead, potentially with a type annotation for the expected type.
 -/
 abbrev «inferInstanceAs» (α : Sort u) [i : α] : α := i
 
@@ -3673,7 +3688,7 @@ def panic {α : Sort u} [Inhabited α] (msg : String) : α :=
   panicCore msg
 
 -- TODO: this be applied directly to `Inhabited`'s definition when we remove the above workaround
-attribute [nospecialize] Inhabited
+attribute [weak_specialize] Inhabited
 
 /--
 The `>>=` operator is overloaded via instances of `bind`.
@@ -3739,7 +3754,7 @@ class Functor (f : Type u → Type v) : Type (max (u+1) v) where
   /--
   Mapping a constant function.
 
-  Given `a : α` and `v : f α`, `mapConst a v` is equivalent to `Function.const _ a <$> v`. For some
+  Given `a : α` and `v : f β`, `mapConst a v` is equivalent to `(fun _ => a) <$> v`. For some
   functors, this can be implemented more efficiently; for all other functors, the default
   implementation may be used.
   -/

src/Init/System/IO.lean

@@ -1880,3 +1880,12 @@ lead to undefined behavior.
 -/
 @[extern "lean_runtime_forget"]
 def Runtime.forget (a : α) : BaseIO Unit := return
+
+set_option linter.unusedVariables false in
+/--
+Ensures `a` remains at least alive until the call site by holding a reference to `a`. This can be useful
+for unsafe code (such as an FFI) that relies on a Lean object not being freed until after some point
+in the program. At runtime, this will be a no-op as the C compiler will optimize away this call.
+-/
+@[extern "lean_runtime_hold"]
+def Runtime.hold (a : @& α) : BaseIO Unit := return

src/Lean/AddDecl.lean

@@ -9,6 +9,7 @@ prelude
 public import Lean.Meta.Sorry
 public import Lean.Util.CollectAxioms
 public import Lean.OriginalConstKind
+import Lean.Compiler.MetaAttr
 import all Lean.OriginalConstKind  -- for accessing `privateConstKindsExt`
 
 public section
@@ -208,8 +209,12 @@ where
       catch _ => pure ()
 
 
-def addAndCompile (decl : Declaration) (logCompileErrors : Bool := true) : CoreM Unit := do
+def addAndCompile (decl : Declaration) (logCompileErrors : Bool := true)
+    (markMeta : Bool := false) : CoreM Unit := do
   addDecl decl
+  if markMeta then
+    for n in decl.getNames do
+      modifyEnv (Lean.markMeta · n)
   compileDecl decl (logErrors := logCompileErrors)
 
 end Lean

src/Lean/Attributes.lean

@@ -186,11 +186,11 @@ def registerTagAttribute (name : Name) (descr : String)
     mkInitial       := pure {}
     addImportedFn   := fun _ _ => pure {}
     addEntryFn      := fun (s : NameSet) n => s.insert n
-    exportEntriesFnEx := fun env es _ =>
-      let r : Array Name := es.foldl (fun a e => a.push e) #[]
-      -- Do not export info for private defs
-      let r := r.filter (env.contains (skipRealize := false))
-      r.qsort Name.quickLt
+    exportEntriesFnEx := fun env es =>
+      let all : Array Name := es.foldl (fun a e => a.push e) #[] |>.qsort Name.quickLt
+      -- Do not export info for private defs at exported/server levels
+      let exported := all.filter ((env.setExporting true).contains (skipRealize := false))
+      { exported, server := exported, «private» := all }
     statsFn         := fun s => "tag attribute" ++ Format.line ++ "number of local entries: " ++ format s.size
     asyncMode       := asyncMode
     replay?         := some fun _ newState newConsts s =>
@@ -266,15 +266,14 @@ def registerParametricAttribute (impl : ParametricAttributeImpl α) : IO (Parame
     mkInitial       := pure ([], {})
     addImportedFn   := fun _ => pure ([], {})
     addEntryFn      := fun (decls, m) (p : Name × α) => (p.1 :: decls, m.insert p.1 p.2)
-    exportEntriesFnEx := fun env (decls, m) lvl => Id.run do
-      let mut r := if impl.preserveOrder then
+    exportEntriesFnEx := fun env (decls, m) => Id.run do
+      let all := if impl.preserveOrder then
         decls.toArray.reverse.filterMap (fun n => return (n, ← m.find? n))
       else
         let r := m.foldl (fun a n p => a.push (n, p)) #[]
         r.qsort (fun a b => Name.quickLt a.1 b.1)
-      if lvl != .private then
-        r := r.filter (fun ⟨n, a⟩ => impl.filterExport env n a)
-      r
+      let exported := all.filter (fun ⟨n, a⟩ => impl.filterExport env n a)
+      { exported, server := exported, «private» := all }
     statsFn         := fun (_, m) => "parametric attribute" ++ Format.line ++ "number of local entries: " ++ format m.size
   }
   let attrImpl : AttributeImpl := {
@@ -333,11 +332,11 @@ def registerEnumAttributes (attrDescrs : List (Name × String × α))
     mkInitial       := pure {}
     addImportedFn   := fun _ _ => pure {}
     addEntryFn      := fun (s : NameMap α) (p : Name × α) => s.insert p.1 p.2
-    exportEntriesFnEx := fun env m _ =>
-      let r : Array (Name × α) := m.foldl (fun a n p => a.push (n, p)) #[]
-      -- Do not export info for private defs
-      let r := r.filter (env.contains (skipRealize := false) ·.1)
-      r.qsort (fun a b => Name.quickLt a.1 b.1)
+    exportEntriesFnEx := fun env m =>
+      let all : Array (Name × α) := m.foldl (fun a n p => a.push (n, p)) #[] |>.qsort (fun a b => Name.quickLt a.1 b.1)
+      -- Do not export info for private defs at exported/server levels
+      let exported := all.filter ((env.setExporting true).contains (skipRealize := false) ·.1)
+      { exported, server := exported, «private» := all }
     statsFn         := fun s => "enumeration attribute extension" ++ Format.line ++ "number of local entries: " ++ format s.size
     -- We assume (and check in `modifyState`) that, if used asynchronously, enum attributes are set
     -- only in the same context in which the tagged declaration was created

src/Lean/Compiler/ExternAttr.lean

@@ -55,11 +55,6 @@ private def syntaxToExternAttrData (stx : Syntax) : AttrM ExternAttrData := do
       entries := entries.push <| ExternEntry.inline backend str
   return { entries := entries.toList }
 
--- Forward declaration
-set_option compiler.ignoreBorrowAnnotation true in
-@[extern "lean_add_extern"]
-opaque addExtern (declName : Name) (externAttrData : ExternAttrData) : CoreM Unit
-
 builtin_initialize externAttr : ParametricAttribute ExternAttrData ←
   registerParametricAttribute {
     name := `extern
@@ -71,7 +66,7 @@ builtin_initialize externAttr : ParametricAttribute ExternAttrData ←
         if let some (.thmInfo ..) := env.find? declName then
           -- We should not mark theorems as extern
           return ()
-        addExtern declName externAttrData
+        compileDecls #[declName]
   }
 
 def getExternAttrData? (env : Environment) (n : Name) : Option ExternAttrData :=

src/Lean/Compiler/IR.lean

@@ -6,7 +6,6 @@ Authors: Leonardo de Moura
 module
 
 prelude
-public import Lean.Compiler.IR.AddExtern
 public import Lean.Compiler.IR.Basic
 public import Lean.Compiler.IR.Format
 public import Lean.Compiler.IR.CompilerM

src/Lean/Compiler/IR/AddExtern.lean

@@ -1,86 +0,0 @@
-/-
-Copyright (c) 2025 Lean FRO LLC. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Cameron Zwarich
--/
-
-module
-
-prelude
-import Init.While
-import Lean.Compiler.IR.ToIR
-import Lean.Compiler.LCNF.ToImpureType
-import Lean.Compiler.LCNF.ToImpure
-import Lean.Compiler.LCNF.ExplicitBoxing
-import Lean.Compiler.LCNF.Internalize
-public import Lean.Compiler.ExternAttr
-import Lean.Compiler.LCNF.ExplicitRC
-import Lean.Compiler.Options
-
-public section
-
-namespace Lean.IR
-
-set_option compiler.ignoreBorrowAnnotation true in
-@[export lean_add_extern]
-def addExtern (declName : Name) (externAttrData : ExternAttrData) : CoreM Unit := do
-  if !isPrivateName declName then
-    modifyEnv (Compiler.LCNF.setDeclPublic · declName)
-  let monoDecl ← addMono declName
-  let impureDecls ← addImpure monoDecl
-  addIr impureDecls
-where
-  addMono (declName : Name) : CoreM (Compiler.LCNF.Decl .pure) := do
-    let type ← Compiler.LCNF.getOtherDeclMonoType declName
-    let mut typeIter := type
-    let mut params := #[]
-    let ignoreBorrow := Compiler.compiler.ignoreBorrowAnnotation.get (← getOptions)
-    repeat
-      let .forallE binderName ty b _ := typeIter | break
-      let borrow := !ignoreBorrow && isMarkedBorrowed ty
-      params := params.push {
-        fvarId := (← mkFreshFVarId)
-        type := ty,
-        binderName,
-        borrow
-      }
-      typeIter := b
-    let decl := {
-      name := declName,
-      levelParams := [],
-      value := .extern externAttrData,
-      inlineAttr? := some .noinline,
-      type,
-      params,
-    }
-    decl.saveMono
-    return decl
-
-  addImpure (decl : Compiler.LCNF.Decl .pure) : CoreM (Array (Compiler.LCNF.Decl .impure)) := do
-    let type ← Compiler.LCNF.lowerResultType decl.type decl.params.size
-    let params ← decl.params.mapM fun param =>
-      return { param with type := ← Compiler.LCNF.toImpureType param.type }
-    let decl : Compiler.LCNF.Decl .impure := {
-      name := decl.name,
-      levelParams := decl.levelParams,
-      value := .extern externAttrData
-      inlineAttr? := some .noinline,
-      type,
-      params
-    }
-    Compiler.LCNF.CompilerM.run (phase := .impure) do
-      let decl ← decl.internalize
-      decl.saveImpure
-      let decls ← Compiler.LCNF.addBoxedVersions #[decl]
-      let decls ← Compiler.LCNF.runExplicitRc decls
-      for decl in decls do
-        decl.saveImpure
-        modifyEnv fun env => Compiler.LCNF.recordFinalImpureDecl env decl.name
-      return decls
-
-  addIr (decls : Array (Compiler.LCNF.Decl .impure)) : CoreM Unit := do
-    let decls ← toIR decls
-    logDecls `result decls
-    addDecls decls
-
-end Lean.IR

src/Lean/Compiler/IR/CompilerM.lean

@@ -86,11 +86,11 @@ builtin_initialize declMapExt : SimplePersistentEnvExtension Decl DeclMap ←
     addEntryFn    := fun s d => s.insert d.name d
     -- Store `meta` closure only in `.olean`, turn all other decls into opaque externs.
     -- Leave storing the remainder for `meta import` and server `#eval` to `exportIREntries` below.
-    exportEntriesFnEx? := some fun env s entries _ =>
+    exportEntriesFnEx? := some fun env s entries =>
       let decls := entries.foldl (init := #[]) fun decls decl => decls.push decl
       let entries := sortDecls decls
       -- Do not save all IR even in .olean.private as it will be in .ir anyway
-      if env.header.isModule then
+      .uniform <| if env.header.isModule then
         entries.filterMap fun d => do
           if isDeclMeta env d.name then
             return d
@@ -126,12 +126,12 @@ private def exportIREntries (env : Environment) : Array (Name × Array EnvExtens
   -- save all initializers independent of meta/private. Non-meta initializers will only be used when
   -- .ir is actually loaded, and private ones iff visible.
   let initDecls : Array (Name × Name) :=
-    regularInitAttr.ext.exportEntriesFn env (regularInitAttr.ext.getState env) .private
+    (regularInitAttr.ext.exportEntriesFn env (regularInitAttr.ext.getState env)).private
   -- safety: cast to erased type
   let initDecls : Array EnvExtensionEntry := unsafe unsafeCast initDecls
 
   -- needed during initialization via interpreter
-  let modPkg : Array (Option PkgId) := modPkgExt.exportEntriesFn env (modPkgExt.getState env) .private
+  let modPkg : Array (Option PkgId) := (modPkgExt.exportEntriesFn env (modPkgExt.getState env)).private
   -- safety: cast to erased type
   let modPkg : Array EnvExtensionEntry := unsafe unsafeCast modPkg
 

src/Lean/Compiler/LCNF/Basic.lean

@@ -1230,7 +1230,14 @@ def instantiateRevRangeArgs (e : Expr) (beginIdx endIdx : Nat) (args : Array (Ar
   else
     e.instantiateRevRange beginIdx endIdx (args.map (·.toExpr))
 
-/-- Lookup function for compiler extensions with sorted persisted state that works in both `lean` and `leanir`. -/
+/--
+Lookup function for compiler extensions with sorted persisted state that works in both `lean` and
+`leanir`.
+
+`preferImported` defaults to false because in `leanir`, we do not want to mix information from
+`meta` compilation in `lean` with our own state. But in `lean`, setting `preferImported` can help
+with avoiding unnecessary task blocks.
+-/
 @[inline] def findExtEntry? [Inhabited σ] (env : Environment) (ext : PersistentEnvExtension α β σ) (declName : Name)
     (findAtSorted? : Array α → Name → Option α')
     (findInState? : σ → Name → Option α') : Option α' :=

src/Lean/Compiler/LCNF/Check.lean

@@ -232,6 +232,7 @@ partial def checkCases (c : Cases .pure) : CheckM Unit := do
       withParams params do check k
 
 partial def check (code : Code .pure) : CheckM Unit := do
+  checkSystem "LCNF check"
   match code with
   | .let decl k => checkLetDecl decl; withFVarId decl.fvarId do check k
   | .fun decl k =>

src/Lean/Compiler/LCNF/CoalesceRC.lean

@@ -0,0 +1,104 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Henrik Böving
+-/
+module
+
+prelude
+public import Lean.Compiler.LCNF.CompilerM
+public import Lean.Compiler.LCNF.PassManager
+
+namespace Lean.Compiler.LCNF
+
+/-!
+# Coalesce Reference Counting Operations
+
+This pass coalesces multiple `inc`/`dec` operations on the same variable within a basic block.
+Within a basic block, it is always safe to:
+- Move all increments on a variable to the first `inc` location (summing the counts). Because if
+  there are later `inc`s no intermediate operation can observe RC=1 (as the value must stay alive
+  until the later inc) and thus doing all relevant `inc` in the beginning doesn't change
+  semantics.
+- Move all decrements on a variable to the last `dec` location (summing the counts). Because the
+  value is guaranteed to stay alive until at least the last `dec` anyway so a similar argument to
+  `inc` holds.
+
+Crucially this pass must be placed after `expandResetReuse` as that one relies on `inc`s still being
+present in their original location for optimization purposes.
+-/
+
+private structure State where
+  /-- Total inc count per variable in the current basic block (accumulated going forward). -/
+  incTotal : Std.HashMap FVarId Nat := {}
+  /-- Total dec count per variable in the current basic block (accumulated going forward). -/
+  decTotal : Std.HashMap FVarId Nat := {}
+  /--
+  Inc count seen so far per variable going backward. When this equals `incTotal`, we've
+  reached the first inc and should emit the coalesced operation.
+  -/
+  incAccum : Std.HashMap FVarId Nat := {}
+  /--
+  Whether we've already emitted the coalesced dec for a variable (going backward, the first
+  dec encountered is the last in the block).
+  -/
+  decPlaced : Std.HashSet FVarId := {}
+
+private abbrev M := StateRefT State CompilerM
+
+/--
+Coalesce inc/dec operations within individual basic blocks.
+-/
+partial def Code.coalesceRC (code : Code .impure) : CompilerM (Code .impure) := do
+  go code |>.run' {}
+where
+  go (code : Code .impure) : M (Code .impure) := do
+    match code with
+    | .inc fvarId n check persistent k _ =>
+      modify fun s => { s with incTotal := s.incTotal.alter fvarId (fun v? => some ((v?.getD 0) + n)) }
+      let k ← go k
+      modify fun s => { s with incAccum := s.incAccum.alter fvarId (fun v? => some ((v?.getD 0) + n)) }
+      let s ← get
+      if s.incAccum[fvarId]! == s.incTotal[fvarId]! then
+        return .inc fvarId s.incTotal[fvarId]! check persistent k
+      else
+        return k
+    | .dec fvarId n check persistent k _ =>
+      modify fun s => { s with decTotal := s.decTotal.alter fvarId (fun v? => some ((v?.getD 0) + n)) }
+      let k ← go k
+      let s ← get
+      if !s.decPlaced.contains fvarId then
+        modify fun s => { s with decPlaced := s.decPlaced.insert fvarId }
+        return .dec fvarId s.decTotal[fvarId]! check persistent k
+      else
+        return k
+    | .let _ k =>
+      let k ← go k
+      return code.updateCont! k
+    | .jp decl k =>
+      let value ← decl.value.coalesceRC
+      let decl ← decl.updateValue value
+      let k ← go k
+      return code.updateFun! decl k
+    | .cases c =>
+      let alts ← c.alts.mapMonoM (·.mapCodeM (·.coalesceRC))
+      return code.updateAlts! alts
+    | .del _ k _ =>
+      let k ← go k
+      return code.updateCont! k
+    | .oset (k := k) .. | .uset (k := k) .. | .sset (k := k) .. | .setTag (k := k) .. =>
+      let k ← go k
+      return code.updateCont! k
+    | .return .. | .jmp .. | .unreach .. => return code
+
+def Decl.coalesceRC (decl : Decl .impure) : CompilerM (Decl .impure) := do
+  let value ← decl.value.mapCodeM Code.coalesceRC
+  return { decl with value }
+
+public def coalesceRC : Pass :=
+  .mkPerDeclaration `coalesceRc .impure Decl.coalesceRC
+
+builtin_initialize
+  registerTraceClass `Compiler.coalesceRc (inherited := true)
+
+end Lean.Compiler.LCNF

src/Lean/Compiler/LCNF/ElimDeadBranches.lean

@@ -291,10 +291,9 @@ builtin_initialize functionSummariesExt : SimplePersistentEnvExtension (Name ×
   registerSimplePersistentEnvExtension {
     addImportedFn := fun _ => {}
     addEntryFn := fun s ⟨e, n⟩ => s.insert e n
-    exportEntriesFnEx? := some fun _ s _ => fun
+    exportEntriesFnEx? := some fun _ s _ =>
       -- preserved for non-modules, make non-persistent at some point?
-      | .private => s.toArray.qsort decLt
-      | _ => #[]
+      { exported := #[], server := #[], «private» := s.toArray.qsort decLt }
     asyncMode := .sync  -- compilation is non-parallel anyway
     replay? := some <| SimplePersistentEnvExtension.replayOfFilter (!·.contains ·.1) (fun s ⟨e, n⟩ => s.insert e n)
   }

src/Lean/Compiler/LCNF/ExpandResetReuse.lean

@@ -69,8 +69,8 @@ open ImpureType
 abbrev Mask := Array (Option FVarId)
 
 /--
-Try to erase `inc` instructions on projections of `targetId` occuring in the tail of `ds`.
-Return the updated `ds` and mask contianing the `FVarId`s whose `inc` was removed.
+Try to erase `inc` instructions on projections of `targetId` occurring in the tail of `ds`.
+Return the updated `ds` and mask containing the `FVarId`s whose `inc` was removed.
 -/
 partial def eraseProjIncFor (nFields : Nat) (targetId : FVarId) (ds : Array (CodeDecl .impure)) :
     CompilerM (Array (CodeDecl .impure) × Mask) := do

src/Lean/Compiler/LCNF/ExplicitRC.lean

@@ -31,9 +31,12 @@ namespace Lean.Compiler.LCNF
 open ImpureType
 
 /-!
-The following section contains the derived value analysis. It figures out a dependency tree of
+The following section contains the derived value analysis. It figures out a dependency graph of
 values that were derived from other values through projections or `Array` accesses. This information
 is later used in the derived borrow analysis to reduce reference counting pressure.
+
+When a derived value has more than one parent, it is derived from one of the parent values but we
+cannot statically determine which one.
 -/
 
 /--
@@ -41,10 +44,10 @@ Contains information about values derived through various forms of projection fr
 -/
 structure DerivedValInfo where
   /--
-  The variable this value was derived from. This is always set except for parameters as they have no
-  value to be derived from.
+  The set of variables this value may derive from. This is always set except for parameters as they
+  have no value to be derived from.
   -/
-  parent? : Option FVarId
+  parents : Array FVarId
   /--
   The set of variables that were derived from this value.
   -/
@@ -56,59 +59,85 @@ abbrev DerivedValMap := Std.HashMap FVarId DerivedValInfo
 namespace CollectDerivedValInfo
 
 structure State where
+  /--
+  The dependency graph of values.
+  -/
   varMap : DerivedValMap := {}
-  borrowedParams : FVarIdHashSet := {}
+  /--
+  The set of values that are to be interpreted as being borrowed by nature. This currently includes:
+  - borrowed parameters
+  - variables that are initialized from constants
+  -/
+  borrowedValues : FVarIdHashSet := {}
 
 abbrev M := StateRefT State CompilerM
 
 @[inline]
-def visitParam (p : Param .impure) : M Unit :=
+def addDerivedValue (parents : Array FVarId) (child : FVarId) : M Unit := do
   modify fun s => { s with
-    varMap := s.varMap.insert p.fvarId {
-      parent? := none
-      children := {}
-    }
-    borrowedParams :=
-      if p.borrow && p.type.isPossibleRef then
-        s.borrowedParams.insert p.fvarId
-      else
-        s.borrowedParams
+    varMap :=
+      let varMap := parents.foldl (init := s.varMap)
+        (·.modify · (fun info => { info with children := info.children.insert child }))
+      varMap.insert child { parents := parents, children := {} }
   }
 
 @[inline]
-def addDerivedValue (parent : FVarId) (child : FVarId) : M Unit := do
-  modify fun s => { s with
-    varMap :=
-      s.varMap
-        |>.modify parent (fun info => { info with children := info.children.insert child })
-        |>.insert child { parent? := some parent, children := {} }
-  }
+def addBorrowedValue (fvarId : FVarId) : M Unit := do
+  modify fun s => { s with borrowedValues := s.borrowedValues.insert fvarId }
 
-def removeFromParent (child : FVarId) : M Unit := do
-  if let some parent := (← get).varMap.get? child |>.bind (·.parent?) then
-    modify fun s => { s with
-      varMap := s.varMap.modify parent fun info =>
-        { info with children := info.children.erase child }
-    }
+def addDerivedLetValue (parents : Array FVarId) (child : FVarId) : M Unit := do
+  let type ← getType child
+  if !type.isPossibleRef then
+    return ()
+  let parents ← parents.filterM fun fvarId => do
+    let type ← getType fvarId
+    return type.isPossibleRef
+  addDerivedValue parents child
+  if parents.isEmpty then
+      addBorrowedValue child
+
+@[inline]
+def visitParam (p : Param .impure) : M Unit := do
+  addDerivedValue #[] p.fvarId
+  if p.borrow && p.type.isPossibleRef then
+    addBorrowedValue p.fvarId
+
+def removeFromParents (child : FVarId) : M Unit := do
+  if let some entry := (← get).varMap.get? child then
+    for parent in entry.parents do
+      modify fun s => { s with
+        varMap := s.varMap.modify parent fun info =>
+          { info with children := info.children.erase child }
+      }
 
 partial def collectCode (code : Code .impure) : M Unit := do
   match code with
   | .let decl k =>
     match decl.value with
     | .oproj _ parent =>
-      addDerivedValue parent decl.fvarId
+      addDerivedLetValue #[parent] decl.fvarId
     -- Keep in sync with PropagateBorrow, InferBorrow
     | .fap ``Array.getInternal args =>
       if let .fvar parent := args[1]! then
-        addDerivedValue parent decl.fvarId
+        addDerivedLetValue #[parent] decl.fvarId
     | .fap ``Array.get!Internal args =>
+      let mut parents := #[]
+      /-
+      Because execution may continue after a panic, the value resulting from a get!InternalBorrowed
+      may be derived from either the `Inhabited` instance or the `Array` argument.
+      -/
+      if let .fvar parent := args[1]! then
+        parents := parents.push parent
       if let .fvar parent := args[2]! then
-        addDerivedValue parent decl.fvarId
+        parents := parents.push parent
+      addDerivedLetValue parents decl.fvarId
     | .fap ``Array.uget args =>
       if let .fvar parent := args[1]! then
-        addDerivedValue parent decl.fvarId
+        addDerivedLetValue #[parent] decl.fvarId
+    | .fap _ #[] =>
+      addDerivedLetValue #[] decl.fvarId
     | .reset _ target =>
-      removeFromParent target
+      removeFromParents target
     | _ => pure ()
     collectCode k
   | .jp decl k =>
@@ -125,8 +154,8 @@ Collect the derived value tree as well as the set of parameters that take object
 -/
 def collect (ps : Array (Param .impure)) (code : Code .impure) :
     CompilerM (DerivedValMap × FVarIdHashSet) := do
-  let ⟨_, { varMap, borrowedParams }⟩ ← go |>.run {}
-  return ⟨varMap, borrowedParams⟩
+  let ⟨_, { varMap, borrowedValues }⟩ ← go |>.run {}
+  return ⟨varMap, borrowedValues⟩
 where
   go : M Unit := do
     ps.forM visitParam
@@ -170,13 +199,21 @@ def LiveVars.erase (liveVars : LiveVars) (fvarId : FVarId) : LiveVars :=
   let borrows := liveVars.borrows.erase fvarId
   { vars, borrows }
 
+@[inline]
+def LiveVars.insertBorrow (liveVars : LiveVars) (fvarId : FVarId) : LiveVars :=
+  { liveVars with borrows := liveVars.borrows.insert fvarId }
+
+@[inline]
+def LiveVars.insertLive (liveVars : LiveVars) (fvarId : FVarId) : LiveVars :=
+  { liveVars with vars := liveVars.vars.insert fvarId }
+
 abbrev JPLiveVarMap := FVarIdMap LiveVars
 
 structure Context where
   /--
-  The set of all parameters that are borrowed and take potential objects as arguments.
+  The set of all values that are borrowed and potentially objects
   -/
-  borrowedParams : FVarIdHashSet
+  borrowedValues : FVarIdHashSet
   /--
   The derived value tree.
   -/
@@ -277,18 +314,21 @@ def withCollectLiveVars (x : RcM α) : RcM (α × LiveVars) := do
   return (ret, collected)
 
 /--
-Traverse the transitive closure of values derived from `fvarId` and add them to `s` if they pass
-`shouldAdd`.
+Traverse the transitive closure of values derived from `fvarId` and add them to `s` if:
+- they pass `shouldAdd`.
+- all their parents are accessible
 -/
 @[specialize]
-partial def addDescendants (fvarId : FVarId) (derivedValMap : DerivedValMap) (s : FVarIdHashSet)
-    (shouldAdd : FVarId → Bool := fun _ => true) : FVarIdHashSet :=
+partial def addDescendants (fvarId : FVarId) (derivedValMap : DerivedValMap) (liveVars : LiveVars)
+    (shouldAdd : FVarId → Bool := fun _ => true) : LiveVars :=
   if let some info := derivedValMap.get? fvarId then
-    info.children.fold (init := s) fun s child =>
-      let s := if shouldAdd child then s.insert child else s
-      addDescendants child derivedValMap s shouldAdd
+    info.children.fold (init := liveVars) fun liveVars child =>
+      let cinfo := derivedValMap.get! child
+      let parentsOk := cinfo.parents.all fun fvarId => (liveVars.vars.contains fvarId || liveVars.borrows.contains fvarId)
+      let liveVars := if parentsOk && shouldAdd child then liveVars.insertBorrow child else liveVars
+      addDescendants child derivedValMap liveVars shouldAdd
   else
-    s
+    liveVars
 
 /--
 Mark `fvarId` as live from here on out and if there are any derived values that are not live anymore
@@ -299,20 +339,21 @@ alive after all).
 def useVar (fvarId : FVarId) (shouldBorrow : FVarId → Bool := fun _ => true) : RcM Unit := do
   if !(← isLive fvarId) then
     let derivedValMap := (← read).derivedValMap
+    modifyLive fun liveVars => { liveVars with vars := liveVars.vars.insert fvarId }
     modifyLive fun liveVars =>
-      { liveVars with
-          borrows := addDescendants fvarId derivedValMap liveVars.borrows fun y =>
-            !liveVars.vars.contains y && shouldBorrow y
-          vars := liveVars.vars.insert fvarId
-      }
+      addDescendants fvarId derivedValMap liveVars fun y =>
+        !liveVars.vars.contains y && shouldBorrow y
 
 def useArgs (args : Array (Arg .impure)) : RcM Unit := do
   args.forM fun arg =>
     match arg with
     | .fvar fvarId =>
       useVar fvarId fun y =>
-        -- If a value is used as an argument we are going to mark it live anyways so don't mark it
-        -- as borrowed.
+        /-
+        If we are in a situation like `f x y` where `x` would imply that `y` remains borrowed we are
+        going to mark `y` as being live instead of borrowed later on anyways. Instead we skip this
+        intermediate state and don't even begin to consider it as borrowed.
+        -/
         args.all fun arg =>
           match arg with
           | .fvar z => y != z
@@ -341,9 +382,9 @@ def setRetLiveVars : RcM Unit := do
   let derivedValMap := (← read).derivedValMap
   -- At the end of a function no values are live and all borrows derived from parameters will still
   -- be around.
-  let borrows := (← read).borrowedParams.fold (init := {}) fun borrows x =>
-    addDescendants x derivedValMap (borrows.insert x)
-  modifyLive fun _ => { vars := {}, borrows }
+  let liveVars := (← read).borrowedValues.fold (init := {}) fun liveVars x =>
+    addDescendants x derivedValMap (liveVars.insertBorrow x)
+  modifyLive (fun _ => liveVars)
 
 @[inline]
 def addInc (fvarId : FVarId) (k : Code .impure) (n : Nat := 1) : RcM (Code .impure) := do
@@ -625,9 +666,9 @@ partial def Code.explicitRc (code : Code .impure) : RcM (Code .impure) := do
 def Decl.explicitRc (decl : Decl .impure) :
     CompilerM (Decl .impure) := do
   let value ← decl.value.mapCodeM fun code => do
-    let ⟨derivedValMap, borrowedParams⟩ ← CollectDerivedValInfo.collect decl.params code
+    let ⟨derivedValMap, borrowedValues⟩ ← CollectDerivedValInfo.collect decl.params code
     go code |>.run {
-      borrowedParams,
+      borrowedValues,
       derivedValMap,
     } |>.run' {}
   return { decl with value }

src/Lean/Compiler/LCNF/InferBorrow.lean

@@ -67,7 +67,7 @@ structure ParamMap where
   The set of fvars that were already annotated as borrowed before arriving at this pass. We try to
   preserve the annotations here if possible.
   -/
-  annoatedBorrows : Std.HashSet FVarId := {}
+  annotatedBorrows : Std.HashSet FVarId := {}
 
 namespace ParamMap
 
@@ -95,7 +95,7 @@ where
         modify fun m =>
           { m with
             map := m.map.insert (.decl decl.name) (initParamsIfNotExported exported decl.params),
-            annoatedBorrows := decl.params.foldl (init := m.annoatedBorrows) fun acc p =>
+            annotatedBorrows := decl.params.foldl (init := m.annotatedBorrows) fun acc p =>
               if p.borrow then acc.insert p.fvarId else acc
           }
         goCode decl.name code
@@ -116,7 +116,7 @@ where
       modify fun m =>
         { m with
           map := m.map.insert (.jp declName decl.fvarId) (initParams decl.params),
-          annoatedBorrows := decl.params.foldl (init := m.annoatedBorrows) fun acc p =>
+          annotatedBorrows := decl.params.foldl (init := m.annotatedBorrows) fun acc p =>
             if p.borrow then acc.insert p.fvarId else acc
         }
       goCode declName decl.value
@@ -286,7 +286,7 @@ where
 
   ownFVar (fvarId : FVarId) (reason : OwnReason) : InferM Unit := do
     unless (← get).owned.contains fvarId do
-      if !reason.isForced && (← get).paramMap.annoatedBorrows.contains fvarId then
+      if !reason.isForced && (← get).paramMap.annotatedBorrows.contains fvarId then
         trace[Compiler.inferBorrow] "user annotation blocked owning {← PP.run <| PP.ppFVar fvarId}: {← reason.toString}"
       else
         trace[Compiler.inferBorrow] "own {← PP.run <| PP.ppFVar fvarId}: {← reason.toString}"
@@ -375,7 +375,6 @@ where
     match v with
     | .reset _ x => ownFVar z (.resetReuse z); ownFVar x (.resetReuse z)
     | .reuse x _ _ args => ownFVar z (.resetReuse z); ownFVar x (.resetReuse z); ownArgsIfParam z args
-    | .ctor _ args => ownFVar z (.constructorResult z); ownArgsIfParam z args
     | .oproj _ x _ =>
       if ← isOwned x then ownFVar z (.forwardProjectionProp z)
       if ← isOwned z then ownFVar x (.backwardProjectionProp z)
@@ -384,6 +383,8 @@ where
       if let .fvar parent := args[1]! then
         if ← isOwned parent then ownFVar z (.forwardProjectionProp z)
     | .fap ``Array.get!Internal args =>
+      if let .fvar parent := args[1]! then
+        if ← isOwned parent then ownFVar z (.forwardProjectionProp z)
       if let .fvar parent := args[2]! then
         if ← isOwned parent then ownFVar z (.forwardProjectionProp z)
     | .fap ``Array.uget args =>
@@ -396,6 +397,9 @@ where
         let ps ← getParamInfo (.decl f)
         ownFVar z (.functionCallResult z)
         ownArgsUsingParams args ps (.functionCallArg z)
+    | .ctor i args =>
+      if !i.isScalar then
+        ownFVar z (.constructorResult z); ownArgsIfParam z args
     | .fvar x args =>
       ownFVar z (.functionCallResult z); ownFVar x (.fvarCall z); ownArgs (.fvarCall z) args
     | .pap _ args => ownFVar z (.functionCallResult z); ownArgs (.partialApplication z) args

src/Lean/Compiler/LCNF/Main.lean

@@ -78,9 +78,13 @@ def isValidMainType (type : Expr) : Bool :=
     isValidResultName resultName
   | _ => false
 
+/-- A postponed call of `compileDecls`. -/
 structure PostponedCompileDecls where
+  /-- Declaration names of this mutual group. -/
   declNames : Array Name
-deriving BEq, Hashable
+  /-- Options at time of original call, to be restored for tracing etc. -/
+  options : Options
+deriving BEq
 
 /--
 Saves postponed `compileDecls` calls.
@@ -96,21 +100,25 @@ builtin_initialize postponedCompileDeclsExt : SimplePersistentEnvExtension Postp
     asyncMode     := .sync
     replay?       := some <| SimplePersistentEnvExtension.replayOfFilter
       (fun s e => !e.declNames.any s.contains) (fun s e => e.declNames.foldl (·.insert · e) s)
-    exportEntriesFnEx? := some fun _ _ es lvl =>
+    exportEntriesFnEx? := some fun _ _ es =>
       -- `leanir` imports the target module privately
-      if lvl == .private then es.toArray else #[]
+      { exported := #[], server := #[], «private» := es.toArray }
   }
 
-def resumeCompilation (declName : Name) : CoreM Unit := do
+def resumeCompilation (declName : Name) (baseOpts : Options) : CoreM Unit := do
   let some decls := postponedCompileDeclsExt.getState (← getEnv) |>.find? declName | return
+  let opts := baseOpts.mergeBy (fun _ base _ => base) decls.options
+  let opts := compiler.postponeCompile.set opts false
   modifyEnv (postponedCompileDeclsExt.modifyState · fun s => decls.declNames.foldl (·.erase) s)
-  withOptions (compiler.postponeCompile.set · false) do
+  -- NOTE: we *must* throw away the current options as they could depend on the specific recursion
+  -- we did to get here.
+  withOptions (fun _ => opts) do
   Core.prependError m!"Failed to compile `{declName}`" do
-    (← compileDeclsRef.get) decls.declNames
+    (← compileDeclsRef.get) decls.declNames baseOpts
 
 namespace PassManager
 
-partial def run (declNames : Array Name) : CompilerM Unit := withAtLeastMaxRecDepth 8192 do
+partial def run (declNames : Array Name) (baseOpts : Options) : CompilerM Unit := withAtLeastMaxRecDepth 8192 do
   /-
   Note: we need to increase the recursion depth because we currently do to save phase1
   declarations in .olean files. Then, we have to recursively compile all dependencies,
@@ -141,11 +149,14 @@ partial def run (declNames : Array Name) : CompilerM Unit := withAtLeastMaxRecDe
 
   -- Now that we have done all input checks, check for postponement
   if (← getEnv).header.isModule && (← compiler.postponeCompile.getM) then
-    modifyEnv (postponedCompileDeclsExt.addEntry · { declNames := decls.map (·.name) })
+    modifyEnv (postponedCompileDeclsExt.addEntry · { declNames := decls.map (·.name), options := ← getOptions })
     -- meta defs are compiled locally so they are available for execution/compilation without
     -- importing `.ir` but still marked for `leanir` compilation so that we do not have to persist
     -- module-local compilation information between the two processes
-    if !decls.any (isMarkedMeta (← getEnv) ·.name) then
+    if decls.any (isMarkedMeta (← getEnv) ·.name) then
+      -- avoid re-compiling the meta defs in this process; the entry for `leanir` is not affected
+      modifyEnv (postponedCompileDeclsExt.modifyState · fun s => decls.foldl (·.erase ·.name) s)
+    else
       trace[Compiler] "postponing compilation of {decls.map (·.name)}"
       return
 
@@ -157,7 +168,7 @@ partial def run (declNames : Array Name) : CompilerM Unit := withAtLeastMaxRecDe
       let .let { value := .const c .., .. } .. := c | return
       -- Need to do some lookups to get the actual name passed to `compileDecls`
       let c := Compiler.getImplementedBy? (← getEnv) c |>.getD c
-      resumeCompilation c
+      resumeCompilation c baseOpts
 
   let decls := markRecDecls decls
   let manager ← getPassManager
@@ -188,6 +199,7 @@ where
     profileitM Exception profilerName (← getOptions) do
       let mut state : (pu : Purity) × Array (Decl pu) := ⟨inPhase, decls⟩
       for pass in passes do
+        checkSystem "LCNF compiler"
         state ← withTraceNode `Compiler (fun _ => return m!"compiler phase: {pass.phase}, pass: {pass.name}") do
           let decls ← withPhase pass.phase do
             state.fst.withAssertPurity pass.phase.toPurity fun h => do
@@ -199,9 +211,9 @@ where
 
 end PassManager
 
-def main (declNames : Array Name) : CoreM Unit := do
+def main (declNames : Array Name) (baseOpts : Options) : CoreM Unit := do
   withTraceNode `Compiler (fun _ => return m!"compiling: {declNames}") do
-    CompilerM.run <| PassManager.run declNames
+    CompilerM.run <| PassManager.run declNames baseOpts
 
 builtin_initialize
   compileDeclsRef.set main

src/Lean/Compiler/LCNF/PassManager.lean

@@ -121,7 +121,7 @@ def mkPerDeclaration (name : Name) (phase : Phase)
   occurrence := occurrence
   phase := phase
   name := name
-  run := fun xs => xs.mapM run
+  run := fun xs => xs.mapM fun decl => do checkSystem "LCNF compiler"; run decl
 
 end Pass
 

src/Lean/Compiler/LCNF/Passes.lean

@@ -26,6 +26,7 @@ public import Lean.Compiler.LCNF.SimpCase
 public import Lean.Compiler.LCNF.InferBorrow
 public import Lean.Compiler.LCNF.ExplicitBoxing
 public import Lean.Compiler.LCNF.ExplicitRC
+public import Lean.Compiler.LCNF.CoalesceRC
 public import Lean.Compiler.LCNF.Toposort
 public import Lean.Compiler.LCNF.ExpandResetReuse
 public import Lean.Compiler.LCNF.SimpleGroundExpr
@@ -149,6 +150,7 @@ def builtinPassManager : PassManager := {
     explicitBoxing,
     explicitRc,
     expandResetReuse,
+    coalesceRC,
     pushProj (occurrence := 1),
     detectSimpleGround,
     inferVisibility (phase := .impure),

src/Lean/Compiler/LCNF/PhaseExt.lean

@@ -93,16 +93,15 @@ def mkDeclExt (phase : Phase) (name : Name := by exact decl_name%) :
     mkInitial := pure {},
     addImportedFn := fun _ => pure {},
     addEntryFn := fun s decl => s.insert decl.name decl
-    exportEntriesFnEx env s level := Id.run do
-      let mut entries := sortedEntries s declLt
-      if level != .private then
-        entries := entries.filterMap fun decl => do
-          guard <| isDeclPublic env decl.name
-          if isDeclTransparent env phase decl.name then
-            some decl
-          else
-            some { decl with value := .extern { entries := [.opaque] } }
-      return entries
+    exportEntriesFnEx env s := Id.run do
+      let all := sortedEntries s declLt
+      let exported := all.filterMap fun decl => do
+        guard <| isDeclPublic env decl.name
+        if isDeclTransparent env phase decl.name then
+          some decl
+        else
+          some { decl with value := .extern { entries := [.opaque] } }
+      return { exported, server := exported, «private» := all }
     statsFn := statsFn,
     asyncMode := .sync,
     replay? := some (replayFn phase)
@@ -138,13 +137,12 @@ def mkSigDeclExt (phase : Phase) (name : Name := by exact decl_name%) :
     mkInitial := pure {},
     addImportedFn := fun _ => pure {},
     addEntryFn := fun s sig => s.insert sig.name sig
-    exportEntriesFnEx env s level := Id.run do
-      let mut entries := sortedEntries s sigLt
-      if level != .private then
-        entries := entries.filterMap fun sig => do
-          guard <| isDeclPublic env sig.name
-          some sig
-      return entries
+    exportEntriesFnEx env s := Id.run do
+      let all := sortedEntries s sigLt
+      let exported := all.filterMap fun sig => do
+        guard <| isDeclPublic env sig.name
+        some sig
+      return { exported, server := exported, «private» := all }
     statsFn := statsFn,
     asyncMode := .sync,
     replay? := some (replayFn phase)

src/Lean/Compiler/LCNF/PropagateBorrow.lean

@@ -114,6 +114,9 @@ where
         let parentVal ← getOwnedness parent
         join z parentVal
     | .fap ``Array.get!Internal args =>
+      if let .fvar parent := args[1]! then
+        let parentVal ← getOwnedness parent
+        join z parentVal
       if let .fvar parent := args[2]! then
         let parentVal ← getOwnedness parent
         join z parentVal
@@ -124,7 +127,10 @@ where
     | .fap _ args =>
       let value := if args.isEmpty then .borrow else .own
       join z value
-    | .ctor .. | .fvar .. | .pap .. | .sproj .. | .uproj .. | .erased .. | .lit .. =>
+    | .ctor i _ =>
+      let value := if i.isScalar then .borrow else .own
+      join z value
+    | .fvar .. | .pap .. | .sproj .. | .uproj .. | .erased .. | .lit .. =>
       join z .own
     | _ => unreachable!
 

src/Lean/Compiler/LCNF/ResetReuse.lean

@@ -28,7 +28,7 @@ inserts addition instructions to attempt to reuse the memory right away instead
 allocator.
 
 For this the paper defines three functions:
-- `R` (called `Decl.insertResetReuse` here) which looks for candidates that might be elligible for
+- `R` (called `Decl.insertResetReuse` here) which looks for candidates that might be eligible for
   reuse. For these variables it invokes `D`.
 - `D` which looks for code regions in which the target variable is dead (i.e. no longer read from),
   it then invokes `S`. If `S` succeeds it inserts a `reset` instruction to match the `reuse`

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

@@ -217,6 +217,8 @@ Simplify `code`
 -/
 partial def simp (code : Code .pure) : SimpM (Code .pure) := withIncRecDepth do
   incVisited
+  if (← get).visited % 128 == 0 then
+    checkSystem "LCNF simp"
   match code with
   | .let decl k =>
     let baseDecl := decl

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

@@ -146,7 +146,7 @@ Similar to the default `Lean.withIncRecDepth`, but include the `inlineStack` in
 @[inline] def withIncRecDepth (x : SimpM α) : SimpM α := do
   let curr ← MonadRecDepth.getRecDepth
   let max  ← MonadRecDepth.getMaxRecDepth
-  if curr == max then
+  if max != 0 && curr == max then
     throwMaxRecDepth
   else
     MonadRecDepth.withRecDepth (curr+1) x