feat: implicit public meta import Init in non-prelude files (#13323)

Ensure metaprograms have implicit access to `Init` like everyone else.
Closes #13310.

.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

@@ -33,7 +33,7 @@ jobs:
         include: ${{fromJson(inputs.config)}}
       # complete all jobs
       fail-fast: false
-    runs-on: ${{ endsWith(matrix.os, '-with-cache') && fromJSON(format('["{0}", "nscloud-git-mirror-1gb"]', matrix.os)) || matrix.os }}
+    runs-on: ${{ endsWith(matrix.os, '-with-cache') && fromJSON(format('["{0}", "nscloud-git-mirror-5gb"]', matrix.os)) || matrix.os }}
     defaults:
       run:
         shell: ${{ matrix.shell || 'nix develop -c bash -euxo pipefail {0}' }}
@@ -78,7 +78,7 @@ jobs:
       # (needs to be after "Install *" to use the right shell)
       - name: CI Merge Checkout
         run: |
-          git fetch --depth=1 origin ${{ github.sha }}
+          git fetch --depth=${{ matrix.name == 'Linux Lake (Cached)' && '10' || '1' }} origin ${{ github.sha }}
           git checkout FETCH_HEAD flake.nix flake.lock script/prepare-* tests/elab/importStructure.lean
         if: github.event_name == 'pull_request'
       # (needs to be after "Checkout" so files don't get overridden)
@@ -125,13 +125,13 @@ jobs:
           else
             echo "TARGET_STAGE=stage1" >> $GITHUB_ENV
           fi
-      - name: Build
+      - name: Configure Build
         run: |
           ulimit -c unlimited  # coredumps
           [ -d build ] || mkdir build
           cd build
           # arguments passed to `cmake`
-          OPTIONS=(-DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true)
+          OPTIONS=(-DWFAIL=ON)
           if [[ -n '${{ 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
@@ -162,7 +162,21 @@ jobs:
           fi
           # contortion to support empty OPTIONS with old macOS bash
           cmake .. --preset ${{ matrix.CMAKE_PRESET || 'release' }} -B . ${{ matrix.CMAKE_OPTIONS }} ${OPTIONS[@]+"${OPTIONS[@]}"} -DLEAN_INSTALL_PREFIX=$PWD/..
-          time make $TARGET_STAGE -j$NPROC
+      - name: Build Stage 0 & Configure Stage 1
+        run: |
+          ulimit -c unlimited  # coredumps
+          time make -C build stage1-configure -j$NPROC
+      - name: Download Lake Cache
+        if: matrix.name == 'Linux Lake (Cached)'
+        run: |
+          cd src
+          ../build/stage0/bin/lake cache get --repo=${{ github.repository }}
+        timeout-minutes: 20 # prevent excessive hanging from network issues
+        continue-on-error: true
+      - name: Build Target Stage
+        run: |
+          ulimit -c unlimited  # coredumps
+          time make -C build $TARGET_STAGE -j$NPROC
       # Should be done as early as possible and in particular *before* "Check rebootstrap" which
       # changes the state of stage1/
       - name: Save Cache
@@ -181,6 +195,21 @@ jobs:
             build/stage1/**/*.c
             build/stage1/**/*.c.o*' || '' }}
           key: ${{ steps.restore-cache.outputs.cache-primary-key }}
+      - name: Upload Lake Cache
+        # Caching on cancellation created some mysterious issues perhaps related to improper build
+        # shutdown. Also, since this needs access to secrets, it cannot be run on forks.
+        if: matrix.name == 'Linux Lake' && !cancelled() && (github.event_name != 'pull_request' || github.event.pull_request.head.repo.full_name == github.repository)
+        run: |
+          curl --version
+          cd src
+          time ../build/stage0/bin/lake build -o ../build/lake-mappings.jsonl
+          time ../build/stage0/bin/lake cache put ../build/lake-mappings.jsonl --repo=${{ github.repository }}
+        env:
+          LAKE_CACHE_KEY: ${{ secrets.LAKE_CACHE_KEY }}
+          LAKE_CACHE_ARTIFACT_ENDPOINT: ${{ vars.LAKE_CACHE_ENDPOINT }}/a1
+          LAKE_CACHE_REVISION_ENDPOINT: ${{ vars.LAKE_CACHE_ENDPOINT }}/r1
+        timeout-minutes: 20 # prevent excessive hanging from network issues
+        continue-on-error: true
       - name: Install
         run: |
           make -C build/$TARGET_STAGE install
@@ -247,10 +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/check-empty-pr.yml

@@ -0,0 +1,29 @@
+name: Check for empty PR
+
+on:
+  merge_group:
+  pull_request:
+
+jobs:
+  check-empty-pr:
+    runs-on: ubuntu-latest
+    steps:
+    - uses: actions/checkout@v6
+      with:
+        ref: ${{ github.event_name == 'pull_request' && github.event.pull_request.head.sha || github.sha }}
+        fetch-depth: 0
+        filter: tree:0
+
+    - name: Check for empty diff
+      run: |
+        if [[ "${{ github.event_name }}" == "pull_request" ]]; then
+          base=$(git merge-base "origin/${{ github.base_ref }}" HEAD)
+        else
+          base=$(git rev-parse HEAD^1)
+        fi
+        if git diff --quiet "$base" HEAD --; then
+          echo "This PR introduces no changes compared to its base branch." | tee "$GITHUB_STEP_SUMMARY"
+          echo "It may be a duplicate of an already-merged PR." | tee -a "$GITHUB_STEP_SUMMARY"
+          exit 1
+        fi
+      shell: bash

.github/workflows/ci.yml

@@ -76,9 +76,20 @@ jobs:
               fi
               echo "nightly=$LEAN_VERSION_STRING" >> "$GITHUB_OUTPUT"
             else
-              # Scheduled: do nothing if commit already has a different tag
+              # Scheduled: do nothing if commit already has a different tag (e.g. a release tag)
               LEAN_VERSION_STRING="nightly-$(date -u +%F)"
-              if [[ "$(git name-rev --name-only --tags --no-undefined HEAD 2> /dev/null || echo "$LEAN_VERSION_STRING")" == "$LEAN_VERSION_STRING" ]]; then
+              HEAD_TAG="$(git name-rev --name-only --tags --no-undefined HEAD 2> /dev/null || true)"
+              if [[ -n "$HEAD_TAG" && "$HEAD_TAG" != "$LEAN_VERSION_STRING" ]]; then
+                echo "HEAD already tagged as ${HEAD_TAG}, skipping nightly"
+              elif git rev-parse "refs/tags/${LEAN_VERSION_STRING}" >/dev/null 2>&1; then
+                # Today's nightly already exists (e.g. from a manual release), create a revision
+                REV=1
+                while git rev-parse "refs/tags/${LEAN_VERSION_STRING}-rev${REV}" >/dev/null 2>&1; do
+                  REV=$((REV + 1))
+                done
+                LEAN_VERSION_STRING="${LEAN_VERSION_STRING}-rev${REV}"
+                echo "nightly=$LEAN_VERSION_STRING" >> "$GITHUB_OUTPUT"
+              else
                 echo "nightly=$LEAN_VERSION_STRING" >> "$GITHUB_OUTPUT"
               fi
             fi
@@ -132,7 +143,7 @@ jobs:
           CMAKE_MAJOR=$(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 }}"
@@ -244,7 +255,7 @@ jobs:
                 // portable release build: use channel with older glibc (2.26)
                 "name": "Linux release",
                 // usually not a bottleneck so make exclusive to `fast-ci`
-                "os": large && fast ? "nscloud-ubuntu-22.04-amd64-8x16-with-cache" : "ubuntu-latest",
+                "os": large && fast ? "nscloud-ubuntu-24.04-amd64-8x16-with-cache" : "ubuntu-latest",
                 "release": true,
                 // Special handling for release jobs. We want:
                 // 1. To run it in PRs so developers get PR toolchains (so secondary without tests is sufficient)
@@ -265,7 +276,7 @@ jobs:
               },
               {
                 "name": "Linux Lake",
-                "os": large ? "nscloud-ubuntu-22.04-amd64-8x16-with-cache" : "ubuntu-latest",
+                "os": large ? "nscloud-ubuntu-24.04-amd64-8x16-with-cache" : "ubuntu-latest",
                 "enabled": true,
                 "check-rebootstrap": level >= 1,
                 "check-stage3": level >= 2,
@@ -273,7 +284,19 @@ jobs:
                 // NOTE: `test-bench` currently seems to be broken on `ubuntu-latest`
                 "test-bench": large && level >= 2,
                 // We are not warning-free yet on all platforms, start here
-                "CMAKE_OPTIONS": "-DLEAN_EXTRA_CXX_FLAGS=-Werror",
+                "CMAKE_OPTIONS": "-DLEAN_EXTRA_CXX_FLAGS=-Werror -DUSE_LAKE_CACHE=ON",
+              },
+              {
+                "name": "Linux Lake (Cached)",
+                "os": large ? "nscloud-ubuntu-24.04-amd64-8x16-with-cache" : "ubuntu-latest",
+                "enabled": true,
+                "check-rebootstrap": level >= 1,
+                "check-stage3": level >= 2,
+                "test": true,
+                "secondary": true,
+                // NOTE: `test-bench` currently seems to be broken on `ubuntu-latest`
+                "test-bench": large && level >= 2,
+                "CMAKE_OPTIONS": "-DLEAN_EXTRA_CXX_FLAGS=-Werror -DUSE_LAKE_CACHE=ON",
               },
               {
                 "name": "Linux Reldebug",
@@ -287,7 +310,7 @@ jobs:
               {
                 "name": "Linux fsanitize",
                 // Always run on large if available, more reliable regarding timeouts
-                "os": large ? "nscloud-ubuntu-22.04-amd64-16x32-with-cache" : "ubuntu-latest",
+                "os": large ? "nscloud-ubuntu-24.04-amd64-16x32-with-cache" : "ubuntu-latest",
                 "enabled": level >= 2,
                 // do not fail nightlies on this for now
                 "secondary": level <= 2,

.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

@@ -236,7 +236,7 @@ def parse_version(version_str):
 def is_version_gte(version1, version2):
     """Check if version1 >= version2, including proper handling of release candidates."""
     # Check if version1 is a nightly toolchain
-    if version1.startswith("leanprover/lean4:nightly-"):
+    if version1.startswith("leanprover/lean4:nightly-") or version1.startswith("leanprover/lean4-nightly:"):
         return False
     return parse_version(version1) >= parse_version(version2)
 
@@ -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

@@ -14,13 +14,6 @@ repositories:
     bump-branch: true
     dependencies: []
 
-  - name: lean4checker
-    url: https://github.com/leanprover/lean4checker
-    toolchain-tag: true
-    stable-branch: true
-    branch: master
-    dependencies: []
-
   - name: quote4
     url: https://github.com/leanprover-community/quote4
     toolchain-tag: true
@@ -35,6 +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
@@ -107,7 +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/Control/ExceptCps.lean

@@ -37,7 +37,7 @@ set_option linter.unusedVariables false in  -- `s` unused
 Use a monadic action that may throw an exception by providing explicit success and failure
 continuations.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def runK {ε α : Type u} (x : ExceptCpsT ε m α) (s : ε) (ok : α → m β) (error : ε → m β) : m β :=
   x _ ok error
 
@@ -83,6 +83,8 @@ of `True`.
 -/
 instance : MonadAttach (ExceptCpsT ε m) := .trivial
 
+@[simp] theorem throw_bind [Monad m] (e : ε) (f : α → ExceptCpsT ε m β) : (throw e >>= f : ExceptCpsT ε m β) = throw e := rfl
+
 @[simp] theorem run_pure [Monad m] : run (pure x : ExceptCpsT ε m α) = pure (Except.ok x) := rfl
 
 @[simp] theorem run_lift {α ε : Type u} [Monad m] (x : m α) : run (ExceptCpsT.lift x : ExceptCpsT ε m α) = (x >>= fun a => pure (Except.ok a) : m (Except ε α)) := rfl
@@ -91,7 +93,20 @@ instance : MonadAttach (ExceptCpsT ε m) := .trivial
 
 @[simp] theorem run_bind_lift [Monad m] (x : m α) (f : α → ExceptCpsT ε m β) : run (ExceptCpsT.lift x >>= f : ExceptCpsT ε m β) = x >>= fun a => run (f a) := rfl
 
-@[simp] theorem run_bind_throw [Monad m] (e : ε) (f : α → ExceptCpsT ε m β) : run (throw e >>= f : ExceptCpsT ε m β) = run (throw e) := rfl
+@[deprecated throw_bind (since := "2026-03-13")]
+theorem run_bind_throw [Monad m] (e : ε) (f : α → ExceptCpsT ε m β) : run (throw e >>= f : ExceptCpsT ε m β) = run (throw e) := rfl
+
+@[simp] theorem runK_pure :
+    runK (pure x : ExceptCpsT ε m α) s ok error = ok x := rfl
+
+@[simp] theorem runK_lift {α ε : Type u} [Monad m] (x : m α) (s : ε) (ok : α → m β) (error : ε → m β) :
+    runK (ExceptCpsT.lift x : ExceptCpsT ε m α) s ok error = x >>= ok := rfl
+
+@[simp] theorem runK_throw [Monad m] :
+    runK (throw e : ExceptCpsT ε m β) s ok error = error e := rfl
+
+@[simp] theorem runK_bind_lift [Monad m] (x : m α) (f : α → ExceptCpsT ε m β) :
+    runK (ExceptCpsT.lift x >>= f : ExceptCpsT ε m β) s ok error = x >>= fun a => runK (f a) s ok error := rfl
 
 @[simp] theorem runCatch_pure [Monad m] : runCatch (pure x : ExceptCpsT α m α) = pure x := rfl
 
@@ -102,6 +117,7 @@ instance : MonadAttach (ExceptCpsT ε m) := .trivial
 
 @[simp] theorem runCatch_bind_lift [Monad m] (x : m α) (f : α → ExceptCpsT β m β) : runCatch (ExceptCpsT.lift x >>= f : ExceptCpsT β m β) = x >>= fun a => runCatch (f a) := rfl
 
-@[simp] theorem runCatch_bind_throw [Monad m] (e : β) (f : α → ExceptCpsT β m β) : runCatch (throw e >>= f : ExceptCpsT β m β) = pure e := rfl
+@[deprecated throw_bind (since := "2026-03-13")]
+theorem runCatch_bind_throw [Monad m] (e : β) (f : α → ExceptCpsT β m β) : runCatch (throw e >>= f : ExceptCpsT β m β) = pure e := rfl
 
 end ExceptCpsT

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/BEq.lean

@@ -66,3 +66,8 @@ theorem BEq.neq_of_beq_of_neq [BEq α] [PartialEquivBEq α] {a b c : α} :
 instance (priority := low) [BEq α] [LawfulBEq α] : EquivBEq α where
   symm h := beq_iff_eq.2 <| Eq.symm <| beq_iff_eq.1 h
   trans hab hbc := beq_iff_eq.2 <| (beq_iff_eq.1 hab).trans <| beq_iff_eq.1 hbc
+
+theorem equivBEq_of_iff_apply_eq [BEq α] (f : α → β) (hf : ∀ a b, a == b ↔ f a = f b) : EquivBEq α where
+  rfl := by simp [hf]
+  symm := by simp [hf, eq_comm]
+  trans hab hbc := (hf _ _).2 (Eq.trans ((hf _ _).1 hab) ((hf _ _).1 hbc))

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/Char/Lemmas.lean

@@ -98,4 +98,8 @@ theorem toNat_inj {c d : Char} : c.toNat = d.toNat ↔ c = d := by
 theorem isDigit_iff_toNat {c : Char} : c.isDigit ↔ '0'.toNat ≤ c.toNat ∧ c.toNat ≤ '9'.toNat := by
   simp [isDigit, UInt32.le_iff_toNat_le]
 
+@[simp]
+theorem toNat_mk {val : UInt32} {h} : (Char.mk val h).toNat = val.toNat := by
+  simp [← toNat_val]
+
 end Char

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/Nat/ToString.lean

@@ -298,7 +298,7 @@ theorem ofDigitChars_cons {c : Char} {cs : List Char} {init : Nat} :
   simp [ofDigitChars]
 
 theorem ofDigitChars_cons_digitChar_of_lt_ten {n : Nat} (hn : n < 10) {cs : List Char} {init : Nat} :
-    ofDigitChars 10 (n.digitChar :: cs) init = ofDigitChars 10 cs (10 * init + n) := by
+    ofDigitChars b (n.digitChar :: cs) init = ofDigitChars b cs (b * init + n) := by
   simp [ofDigitChars_cons, Nat.toNat_digitChar_sub_48_of_lt_ten hn]
 
 theorem ofDigitChars_eq_ofDigitChars_zero {l : List Char} {init : Nat} :
@@ -320,15 +320,17 @@ theorem ofDigitChars_replicate_zero {n : Nat} : ofDigitChars b (List.replicate n
   | zero => simp
   | succ n ih => simp [List.replicate_succ, ofDigitChars_cons, ih, Nat.pow_succ, Nat.mul_assoc]
 
-@[simp]
-theorem ofDigitChars_toDigits {n : Nat} : ofDigitChars 10 (toDigits 10 n) 0 = n := by
-  have : 1 < 10 := by decide
-  induction n using base_induction 10 this with
+theorem ofDigitChars_toDigits {b n : Nat} (hb' : 1 < b) (hb : b ≤ 10) : ofDigitChars b (toDigits b n) 0 = n := by
+  induction n using base_induction b hb' with
   | single m hm =>
-    simp [Nat.toDigits_of_lt_base hm, ofDigitChars_cons_digitChar_of_lt_ten hm]
+    simp [Nat.toDigits_of_lt_base hm, ofDigitChars_cons_digitChar_of_lt_ten (by omega : m < 10)]
   | digit m k hk hm ih =>
-    rw [← Nat.toDigits_append_toDigits this hm hk,
+    rw [← Nat.toDigits_append_toDigits hb' hm hk,
       ofDigitChars_append, ih, Nat.toDigits_of_lt_base hk,
-      ofDigitChars_cons_digitChar_of_lt_ten hk, ofDigitChars_nil]
+      ofDigitChars_cons_digitChar_of_lt_ten (Nat.lt_of_lt_of_le hk hb), ofDigitChars_nil]
+
+@[simp]
+theorem ofDigitChars_ten_toDigits {n : Nat} : ofDigitChars 10 (toDigits 10 n) 0 = n :=
+  ofDigitChars_toDigits (by decide) (by decide)
 
 end Nat

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/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

@@ -852,6 +852,10 @@ theorem Slice.rawEndPos_copy {s : Slice} : s.copy.rawEndPos = s.rawEndPos := by
 theorem copy_toSlice {s : String} : s.toSlice.copy = s := by
   simp [← toByteArray_inj, Slice.toByteArray_copy, ← size_toByteArray]
 
+@[simp]
+theorem copy_comp_toSlice : String.Slice.copy ∘ String.toSlice = id := by
+  ext; simp
+
 theorem Slice.getUTF8Byte_eq_getUTF8Byte_copy {s : Slice} {p : Pos.Raw} {h : p < s.rawEndPos} :
     s.getUTF8Byte p h = s.copy.getUTF8Byte p (by simpa) := by
   simp [getUTF8Byte, String.getUTF8Byte, toByteArray_copy, ByteArray.getElem_extract]
@@ -1382,6 +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
@@ -1741,6 +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
@@ -1751,6 +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
@@ -1765,6 +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
@@ -1775,6 +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)
@@ -2050,6 +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/Defs.lean

@@ -187,6 +187,9 @@ theorem append_right_inj (s : String) {t₁ t₂ : String} :
 theorem append_assoc {s₁ s₂ s₃ : String} : s₁ ++ s₂ ++ s₃ = s₁ ++ (s₂ ++ s₃) := by
   simp [← toByteArray_inj, ByteArray.append_assoc]
 
+instance : Std.Associative (α := String) (· ++ ·) where
+  assoc _ _ _ := append_assoc
+
 @[simp]
 theorem utf8ByteSize_eq_zero_iff {s : String} : s.utf8ByteSize = 0 ↔ s = "" := by
   refine ⟨fun h => ?_, fun h => h ▸ utf8ByteSize_empty⟩

src/Init/Data/String/Iter.lean

@@ -6,29 +6,5 @@ Authors: Markus Himmel
 module
 
 prelude
-public import Init.Data.Iterators.Combinators.FilterMap
-public import Init.Data.Iterators.Consumers.Collect
-
-set_option doc.verso true
-
-namespace Std
-
-/--
-Convenience function for turning an iterator into a list of strings, provided the output of the
-iterator implements {name}`ToString`.
--/
-@[inline]
-public abbrev Iter.toStringList {α β : Type} [Iterator α Id β] [ToString β]
-    (it : Iter (α := α) β) : List String :=
-  it.map toString |>.toList
-
-/--
-Convenience function for turning an iterator into an array of strings, provided the output of the
-iterator implements {name}`ToString`.
--/
-@[inline]
-public abbrev Iter.toStringArray {α β : Type} [Iterator α Id β] [ToString β]
-    (it : Iter (α := α) β) : Array String :=
-  it.map toString |>.toArray
-
-end Std
+public import Init.Data.String.Iter.Basic
+public import Init.Data.String.Iter.Intercalate

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

@@ -0,0 +1,34 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Markus Himmel
+-/
+module
+
+prelude
+public import Init.Data.Iterators.Combinators.FilterMap
+public import Init.Data.Iterators.Consumers.Collect
+
+set_option doc.verso true
+
+namespace Std
+
+/--
+Convenience function for turning an iterator into a list of strings, provided the output of the
+iterator implements {name}`ToString`.
+-/
+@[inline]
+public abbrev Iter.toStringList {α β : Type} [Iterator α Id β] [ToString β]
+    (it : Iter (α := α) β) : List String :=
+  it.map toString |>.toList
+
+/--
+Convenience function for turning an iterator into an array of strings, provided the output of the
+iterator implements {name}`ToString`.
+-/
+@[inline]
+public abbrev Iter.toStringArray {α β : Type} [Iterator α Id β] [ToString β]
+    (it : Iter (α := α) β) : Array String :=
+  it.map toString |>.toArray
+
+end Std

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

@@ -0,0 +1,37 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Julia Markus Himmel
+-/
+module
+
+prelude
+public import Init.Data.Iterators.Combinators.Monadic.FilterMap
+public import Init.Data.String.Basic
+import Init.Data.String.Slice
+
+set_option doc.verso true
+
+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 := "") (· ++ ·)
+
+/--
+Appends the elements of the iterator into a string, placing the separator {name}`s` between them.
+-/
+@[inline]
+public def Iter.intercalateString {α β : Type} [Iterator α Id β] [ToString β]
+    (s : String.Slice) (it : Std.Iter (α := α) β) : String :=
+  it.map toString
+    |>.fold (init := none) (fun
+      | none, sl => some sl
+      | some str, sl => some (str ++ s ++ sl))
+    |>.getD ""
+
+end Std

src/Init/Data/String/Lemmas.lean

@@ -17,49 +17,7 @@ public import Init.Data.String.Lemmas.Pattern
 public import Init.Data.String.Lemmas.Slice
 public import Init.Data.String.Lemmas.Iterate
 public import Init.Data.String.Lemmas.Intercalate
-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.Iter
+public import Init.Data.String.Lemmas.Hashable
+public import Init.Data.String.Lemmas.TakeDrop
+public import Init.Data.String.Lemmas.StringOrder

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

@@ -7,6 +7,7 @@ module
 
 prelude
 public import Init.Data.String.Basic
+import all Init.Data.String.Basic
 import Init.Data.ByteArray.Lemmas
 import Init.Data.Nat.MinMax
 
@@ -21,6 +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]
@@ -56,6 +61,11 @@ theorem singleton_ne_empty {c : Char} : singleton c ≠ "" := by
 theorem empty_ne_singleton {c : Char} : "" ≠ singleton c := by
   simp
 
+@[simp]
+theorem ofList_cons {c : Char} {l : List Char} :
+    String.ofList (c :: l) = String.singleton c ++ String.ofList l := by
+  simp [← toList_inj]
+
 @[simp]
 theorem Slice.Pos.copy_inj {s : Slice} {p₁ p₂ : s.Pos} : p₁.copy = p₂.copy ↔ p₁ = p₂ := by
   simp [String.Pos.ext_iff, Pos.ext_iff]
@@ -185,18 +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} :
@@ -244,4 +310,81 @@ theorem Pos.get_ofToSlice {s : String} {p : (s.toSlice).Pos} {h} :
 @[simp]
 theorem push_empty {c : Char} : "".push c = singleton c := rfl
 
+namespace Slice.Pos
+
+@[simp]
+theorem nextn_zero {s : Slice} {p : s.Pos} : p.nextn 0 = p := by
+  simp [nextn]
+
+theorem nextn_add_one {s : Slice} {p : s.Pos} :
+    p.nextn (n + 1) = if h : p = s.endPos then p else (p.next h).nextn n := by
+  simp [nextn]
+
+@[simp]
+theorem nextn_endPos {s : Slice} : s.endPos.nextn n = s.endPos := by
+  cases n <;> simp [nextn_add_one]
+
+end Slice.Pos
+
+namespace Pos
+
+theorem nextn_eq_nextn_toSlice {s : String} {p : s.Pos} : p.nextn n = Pos.ofToSlice (p.toSlice.nextn n) :=
+  (rfl)
+
+@[simp]
+theorem nextn_zero {s : String} {p : s.Pos} : p.nextn 0 = p := by
+  simp [nextn_eq_nextn_toSlice]
+
+theorem nextn_add_one {s : String} {p : s.Pos} :
+    p.nextn (n + 1) = if h : p = s.endPos then p else (p.next h).nextn n := by
+  simp only [nextn_eq_nextn_toSlice, Slice.Pos.nextn_add_one, endPos_toSlice, toSlice_inj]
+  split <;> simp [Pos.next_toSlice]
+
+theorem nextn_toSlice {s : String} {p : s.Pos} : p.toSlice.nextn n = (p.nextn n).toSlice := by
+  induction n generalizing p with simp_all [nextn_add_one, Slice.Pos.nextn_add_one, apply_dite Pos.toSlice, next_toSlice]
+
+theorem toSlice_nextn {s : String} {p : s.Pos} : (p.nextn n).toSlice = p.toSlice.nextn n :=
+  nextn_toSlice.symm
+
+@[simp]
+theorem nextn_endPos {s : String} : s.endPos.nextn n = s.endPos := by
+  cases n <;> simp [nextn_add_one]
+
+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/FindPos.lean

@@ -11,6 +11,8 @@ import all Init.Data.String.FindPos
 import Init.Data.String.OrderInstances
 import Init.Data.String.Lemmas.Order
 import Init.Data.Order.Lemmas
+import Init.Data.Option.Lemmas
+import Init.ByCases
 
 public section
 
@@ -199,6 +201,10 @@ theorem Pos.prev_eq_iff {s : Slice} {p q : s.Pos} {h} :
 theorem Pos.prev_lt {s : Slice} {p : s.Pos} {h} : p.prev h < p := by
   simp
 
+@[simp]
+theorem Pos.prev_le {s : Slice} {p : s.Pos} {h} : p.prev h ≤ p :=
+  Std.le_of_lt (by simp)
+
 @[simp]
 theorem Pos.prev_ne_endPos {s : Slice} {p : s.Pos} {h} : p.prev h ≠ s.endPos :=
   ne_endPos_of_lt prev_lt
@@ -209,6 +215,29 @@ theorem Pos.prevn_le {s : Slice} {p : s.Pos} {n : Nat} : p.prevn n ≤ p := by
   | case2 p n h ih => exact Std.le_of_lt (by simpa using ih)
   | case3 => simp
 
+theorem Pos.ofSliceTo_prev {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    Pos.ofSliceTo (p.prev h) = (Pos.ofSliceTo p).prev (by simpa [← Pos.ofSliceTo_inj] using h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceTo_lt_ofSliceTo_iff, Pos.le_ofSliceTo_iff]
+  simp [Pos.lt_ofSliceTo_iff]
+
+theorem Pos.prev_ofSliceTo {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    (Pos.ofSliceTo p).prev h = Pos.ofSliceTo (p.prev (by simpa [← Pos.ofSliceTo_inj])) := by
+  simp [ofSliceTo_prev]
+
+theorem Pos.ofSliceFrom_prev {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    Pos.ofSliceFrom (p.prev h) = (Pos.ofSliceFrom p).prev (by exact ofSliceFrom_ne_startPos h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceFrom_lt_ofSliceFrom_iff,  Pos.le_ofSliceFrom_iff]
+  simp [Pos.lt_ofSliceFrom_iff]
+
+theorem Pos.ofSlice_prev {s : Slice} {p₀ p₁ : s.Pos} {h}
+    {p : (s.slice p₀ p₁ h).Pos} {h'} :
+    Pos.ofSlice (p.prev h') = (Pos.ofSlice p).prev (by exact ofSlice_ne_startPos h') := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [ofSlice_lt_ofSlice_iff,  le_ofSlice_iff]
+  simpa +contextual [← ofSlice_lt_ofSlice_iff] using fun q hq => Std.le_of_lt (Std.lt_of_lt_of_le hq ofSlice_le)
+
 @[simp]
 theorem Pos.prev_next {s : Slice} {p : s.Pos} {h} : (p.next h).prev (by simp) = p :=
   prev_eq_iff.2 (by simp)
@@ -217,6 +246,23 @@ theorem Pos.prev_next {s : Slice} {p : s.Pos} {h} : (p.next h).prev (by simp) =
 theorem Pos.next_prev {s : Slice} {p : s.Pos} {h} : (p.prev h).next (by simp) = p :=
   next_eq_iff.2 (by simp)
 
+theorem Pos.prev?_eq_dif {s : Slice} {p : s.Pos} : p.prev? = if h : p = s.startPos then none else some (p.prev h) :=
+  (rfl)
+
+theorem Pos.prev?_eq_some_prev {s : Slice} {p : s.Pos} (h : p ≠ s.startPos) : p.prev? = some (p.prev h) := by
+  simp [Pos.prev?, h]
+
+@[simp]
+theorem Pos.prev?_eq_none_iff {s : Slice} {p : s.Pos} : p.prev? = none ↔ p = s.startPos := by
+  simp [Pos.prev?]
+
+theorem Pos.prev?_eq_none {s : Slice} {p : s.Pos} (h : p = s.startPos) : p.prev? = none :=
+  prev?_eq_none_iff.2 h
+
+@[simp]
+theorem Pos.prev?_startPos {s : Slice} : s.startPos.prev? = none := by
+  simp
+
 end Slice
 
 @[simp]
@@ -420,6 +466,10 @@ theorem Pos.prev_eq_iff {s : String} {p q : s.Pos} {h} :
 theorem Pos.prev_lt {s : String} {p : s.Pos} {h} : p.prev h < p := by
   simp
 
+@[simp]
+theorem Pos.prev_le {s : String} {p : s.Pos} {h} : p.prev h ≤ p :=
+  Std.le_of_lt (by simp)
+
 @[simp]
 theorem Pos.prev_ne_endPos {s : String} {p : s.Pos} {h} : p.prev h ≠ s.endPos :=
   ne_endPos_of_lt prev_lt
@@ -428,14 +478,45 @@ theorem Pos.toSlice_prev {s : String} {p : s.Pos} {h} :
     (p.prev h).toSlice = p.toSlice.prev (by simpa [toSlice_inj]) := by
   simp [prev]
 
+theorem Pos.ofToSlice_prev {s : String} {p : s.toSlice.Pos} {h} :
+    Pos.ofToSlice (p.prev h) = (Pos.ofToSlice p).prev (by simpa [← toSlice_inj]) := by
+  simp [prev]
+
 theorem Pos.prev_toSlice {s : String} {p : s.Pos} {h} :
     p.toSlice.prev h = (p.prev (by simpa [← toSlice_inj])).toSlice := by
   simp [prev]
 
+theorem Pos.prev_ofToSlice {s : String} {p : s.toSlice.Pos} {h} :
+    (Pos.ofToSlice p).prev h = Pos.ofToSlice (p.prev (by simpa [← ofToSlice_inj])) := by
+  simp [prev]
+
 theorem Pos.prevn_le {s : String} {p : s.Pos} {n : Nat} :
     p.prevn n ≤ p := by
   simpa [Pos.le_iff, ← offset_toSlice] using Slice.Pos.prevn_le
 
+theorem Pos.ofSliceTo_prev {s : String} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    Pos.ofSliceTo (p.prev h) = (Pos.ofSliceTo p).prev (by simpa [← Pos.ofSliceTo_inj] using h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceTo_lt_ofSliceTo_iff, Pos.le_ofSliceTo_iff]
+  simp [Pos.lt_ofSliceTo_iff]
+
+theorem Pos.prev_ofSliceTo {s : String} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    (Pos.ofSliceTo p).prev h = Pos.ofSliceTo (p.prev (by simpa [← Pos.ofSliceTo_inj])) := by
+  simp [ofSliceTo_prev]
+
+theorem Pos.ofSliceFrom_prev {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    Pos.ofSliceFrom (p.prev h) = (Pos.ofSliceFrom p).prev (by exact ofSliceFrom_ne_startPos h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceFrom_lt_ofSliceFrom_iff, Pos.le_ofSliceFrom_iff]
+  simp [Pos.lt_ofSliceFrom_iff]
+
+theorem Pos.ofSlice_prev {s : String} {p₀ p₁ : s.Pos} {h}
+    {p : (s.slice p₀ p₁ h).Pos} {h'} :
+    Pos.ofSlice (p.prev h') = (Pos.ofSlice p).prev (by exact ofSlice_ne_startPos h') := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [ofSlice_lt_ofSlice_iff, le_ofSlice_iff]
+  simpa +contextual [← ofSlice_lt_ofSlice_iff] using fun q hq => Std.le_of_lt (Std.lt_of_lt_of_le hq ofSlice_le)
+
 @[simp]
 theorem Pos.prev_next {s : String} {p : s.Pos} {h} : (p.next h).prev (by simp) = p :=
   prev_eq_iff.2 (by simp)
@@ -444,4 +525,71 @@ theorem Pos.prev_next {s : String} {p : s.Pos} {h} : (p.next h).prev (by simp) =
 theorem Pos.next_prev {s : String} {p : s.Pos} {h} : (p.prev h).next (by simp) = p :=
   next_eq_iff.2 (by simp)
 
+theorem Pos.prev?_eq_prev?_toSlice {s : String} {p : s.Pos} : p.prev? = p.toSlice.prev?.map Pos.ofToSlice :=
+  (rfl)
+
+theorem Pos.prev?_toSlice {s : String} {p : s.Pos} : p.toSlice.prev? = p.prev?.map Pos.toSlice := by
+  simp [prev?_eq_prev?_toSlice]
+
+theorem Pos.prev?_eq_dif {s : String} {p : s.Pos} : p.prev? = if h : p = s.startPos then none else some (p.prev h) := by
+  simp [prev?_eq_prev?_toSlice, Slice.Pos.prev?_eq_dif, apply_dite (Option.map Pos.ofToSlice),
+    ofToSlice_prev]
+
+theorem Pos.prev?_eq_some_prev {s : String} {p : s.Pos} (h : p ≠ s.startPos) : p.prev? = some (p.prev h) := by
+  simp [prev?_eq_prev?_toSlice, Slice.Pos.prev?_eq_some_prev (by simpa : p.toSlice ≠ s.toSlice.startPos),
+    ofToSlice_prev]
+
+@[simp]
+theorem Pos.prev?_eq_none_iff {s : String} {p : s.Pos} : p.prev? = none ↔ p = s.startPos := by
+  simp [prev?_eq_prev?_toSlice]
+
+theorem Pos.prev?_eq_none {s : String} {p : s.Pos} (h : p = s.startPos) : p.prev? = none :=
+  prev?_eq_none_iff.2 h
+
+@[simp]
+theorem Pos.prev?_startPos {s : String} : s.startPos.prev? = none := by
+  simp
+
+namespace Slice.Pos
+
+@[simp]
+theorem prevn_zero {s : Slice} {p : s.Pos} : p.prevn 0 = p := by
+  simp [prevn]
+
+theorem prevn_add_one {s : Slice} {p : s.Pos} :
+    p.prevn (n + 1) = if h : p = s.startPos then p else (p.prev h).prevn n := by
+  simp [prevn]
+
+@[simp]
+theorem prevn_startPos {s : Slice} : s.startPos.prevn n = s.startPos := by
+  cases n <;> simp [prevn_add_one]
+
+end Slice.Pos
+
+namespace Pos
+
+theorem prevn_eq_prevn_toSlice {s : String} {p : s.Pos} : p.prevn n = Pos.ofToSlice (p.toSlice.prevn n) :=
+  (rfl)
+
+@[simp]
+theorem prevn_zero {s : String} {p : s.Pos} : p.prevn 0 = p := by
+  simp [prevn_eq_prevn_toSlice]
+
+theorem prevn_add_one {s : String} {p : s.Pos} :
+    p.prevn (n + 1) = if h : p = s.startPos then p else (p.prev h).prevn n := by
+  simp only [prevn_eq_prevn_toSlice, Slice.Pos.prevn_add_one, startPos_toSlice, toSlice_inj]
+  split <;> simp [Pos.prev_toSlice]
+
+theorem prevn_toSlice {s : String} {p : s.Pos} : p.toSlice.prevn n = (p.prevn n).toSlice := by
+  induction n generalizing p with simp_all [prevn_add_one, Slice.Pos.prevn_add_one, apply_dite Pos.toSlice, prev_toSlice]
+
+theorem toSlice_prevn {s : String} {p : s.Pos} : (p.prevn n).toSlice = p.toSlice.prevn n :=
+  prevn_toSlice.symm
+
+@[simp]
+theorem prevn_startPos {s : String} : s.startPos.prevn n = s.startPos := by
+  cases n <;> simp [prevn_add_one]
+
+end Pos
+
 end String

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

@@ -0,0 +1,25 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Julia Markus Himmel
+-/
+module
+
+prelude
+public import Init.Data.String.Slice
+public import Init.Data.LawfulHashable
+import all Init.Data.String.Slice
+import Init.Data.String.Lemmas.Slice
+
+namespace String
+
+public theorem hash_eq {s : String} : hash s = String.hash s := rfl
+
+namespace Slice
+
+public theorem hash_eq {s : String.Slice} : hash s = String.hash s.copy := (rfl)
+
+public instance : LawfulHashable String.Slice where
+  hash_eq a b hab := by simp [hash_eq, beq_eq_true_iff.1 hab]
+
+end String.Slice

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

@@ -10,6 +10,7 @@ public import Init.Data.String.Defs
 import all Init.Data.String.Defs
 public import Init.Data.String.Slice
 import all Init.Data.String.Slice
+import Init.ByCases
 
 public section
 
@@ -42,6 +43,16 @@ theorem intercalate_cons_of_ne_nil {s t : String} {l : List String} (h : l ≠ [
   match l, h with
   | u::l, _ => by simp
 
+theorem intercalate_append_of_ne_nil {l m : List String} {s : String} (hl : l ≠ []) (hm : m ≠ []) :
+    s.intercalate (l ++ m) = s.intercalate l ++ s ++ s.intercalate m := by
+  induction l with
+  | nil => simp_all
+  | cons hd tl ih =>
+    rw [List.cons_append, intercalate_cons_of_ne_nil (by simp_all)]
+    by_cases ht : tl = []
+    · simp_all
+    · simp [ih ht, intercalate_cons_of_ne_nil ht, String.append_assoc]
+
 @[simp]
 theorem toList_intercalate {s : String} {l : List String} :
     (s.intercalate l).toList = s.toList.intercalate (l.map String.toList) := by
@@ -49,6 +60,32 @@ theorem toList_intercalate {s : String} {l : List String} :
   | nil => simp
   | cons hd tl ih => cases tl <;> simp_all
 
+theorem join_eq_foldl : join l = l.foldl (fun r s => r ++ s) "" :=
+  (rfl)
+
+@[simp]
+theorem join_nil : join [] = "" := by
+  simp [join]
+
+@[simp]
+theorem join_cons : join (s :: l) = s ++ join l := by
+  simp only [join, List.foldl_cons, empty_append]
+  conv => lhs; rw [← String.append_empty (s := s)]
+  rw [List.foldl_assoc]
+
+@[simp]
+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]
@@ -65,6 +102,10 @@ theorem intercalate_eq {s : Slice} {l : List Slice} :
   | nil => simp [intercalate]
   | cons hd tl ih => cases tl <;> simp_all [intercalate, intercalate.go, intercalateGo_append]
 
+@[simp]
+theorem join_eq {l : List Slice} : join l = String.join (l.map copy) := by
+  simp [join, String.join, List.foldl_map]
+
 end Slice
 
 end String

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

@@ -204,7 +204,7 @@ theorem Slice.copy_sliceTo_startPos {s : Slice} : (s.sliceTo s.startPos).copy =
   simp
 
 @[simp]
-theorem Slice.copy_sliceFrom_startPos {s : Slice} : (s.sliceFrom s.endPos).copy = "" := by
+theorem Slice.copy_sliceFrom_endPos {s : Slice} : (s.sliceFrom s.endPos).copy = "" := by
   simp
 
 end CopyEqEmpty

src/Init/Data/String/Lemmas/Iter.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: Julia Markus Himmel
+-/
+module
+
+prelude
+public import Init.Data.String.Iter.Intercalate
+public import Init.Data.String.Slice
+import all Init.Data.String.Iter.Intercalate
+import all Init.Data.String.Defs
+import Init.Data.String.Lemmas.Intercalate
+import Init.Data.Iterators.Lemmas.Consumers.Loop
+import Init.Data.Iterators.Lemmas.Combinators.FilterMap
+
+namespace Std.Iter
+
+@[simp]
+public theorem joinString_eq {α β : Type} [Std.Iterator α Id β] [Std.Iterators.Finite α Id]
+    [ToString β] {it : Std.Iter (α := α) β} :
+    it.joinString = String.join (it.toList.map toString) := by
+  rw [joinString, String.join, ← foldl_toList, toList_map]
+
+@[simp]
+public theorem intercalateString_eq {α β : Type} [Std.Iterator α Id β] [Std.Iterators.Finite α Id]
+    [ToString β] {s : String.Slice} {it : Std.Iter (α := α) β} :
+    it.intercalateString s = s.copy.intercalate (it.toList.map toString) := by
+  simp only [intercalateString, String.appendSlice_eq, ← foldl_toList, toList_map]
+  generalize s.copy = s
+  suffices ∀ (l m : List String),
+      (l.foldl (init := if m = [] then none else some (s.intercalate m))
+        (fun | none, sl => some sl | some str, sl => some (str ++ s ++ sl))).getD ""
+        = s.intercalate (m ++ l) by
+    simpa [-foldl_toList] using this (it.toList.map toString) []
+  intro l m
+  induction l generalizing m with
+  | nil => cases m <;> simp
+  | cons hd tl ih =>
+    rw [List.append_cons, ← ih, List.foldl_cons]
+    congr
+    simp only [List.append_eq_nil_iff, List.cons_ne_self, and_false, ↓reduceIte]
+    match m with
+    | [] => simp
+    | x::xs =>
+      simp only [reduceCtorEq, ↓reduceIte, List.cons_append, Option.some.injEq]
+      rw [← List.cons_append, String.intercalate_append_of_ne_nil (by simp) (by simp),
+        String.intercalate_singleton]
+
+end Std.Iter

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

@@ -11,6 +11,7 @@ import Init.Data.String.OrderInstances
 import Init.Data.String.Lemmas.Basic
 import Init.Data.Order.Lemmas
 import Init.Omega
+import Init.ByCases
 
 public section
 
@@ -70,7 +71,7 @@ theorem Pos.le_startPos {s : String} (p : s.Pos) : p ≤ s.startPos ↔ p = s.st
   ⟨fun h => Std.le_antisymm h (startPos_le _), by simp +contextual⟩
 
 @[simp]
-theorem Pos.startPos_lt_iff {s : String} {p : s.Pos} : s.startPos < p ↔ p ≠ s.startPos := by
+theorem Pos.startPos_lt_iff {s : String} (p : s.Pos) : s.startPos < p ↔ p ≠ s.startPos := by
   simp [← le_startPos, Std.not_le]
 
 @[simp]
@@ -235,6 +236,10 @@ theorem Slice.Pos.ofSliceFrom_next {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom
     Pos.next_le_iff_lt, true_and]
   simp [Pos.ofSliceFrom_lt_iff]
 
+theorem Slice.Pos.next_ofSliceFrom {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    (Pos.ofSliceFrom p).next h = Pos.ofSliceFrom (p.next (by simpa [← Pos.ofSliceFrom_inj])) := by
+  simp [ofSliceFrom_next]
+
 theorem Pos.ofSliceFrom_next {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
     Pos.ofSliceFrom (p.next h) = (Pos.ofSliceFrom p).next (by simpa [← Pos.ofSliceFrom_inj] using h) := by
   rw [eq_comm, Pos.next_eq_iff]
@@ -242,6 +247,10 @@ theorem Pos.ofSliceFrom_next {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀)
     Slice.Pos.next_le_iff_lt, true_and]
   simp [Pos.ofSliceFrom_lt_iff]
 
+theorem Pos.next_ofSliceFrom {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    (Pos.ofSliceFrom p).next h = Pos.ofSliceFrom (p.next (by simpa [← Pos.ofSliceFrom_inj])) := by
+  simp [Pos.ofSliceFrom_next]
+
 theorem Slice.Pos.le_ofSliceTo_iff {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {q : s.Pos} :
     q ≤ Pos.ofSliceTo p ↔ ∃ h, Slice.Pos.sliceTo p₀ q h ≤ p := by
   refine ⟨fun h => ⟨Slice.Pos.le_trans h Pos.ofSliceTo_le, ?_⟩, fun ⟨h, h'⟩ => ?_⟩
@@ -359,11 +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]
@@ -406,16 +445,130 @@ theorem Pos.slice_le_slice_iff {s : String} {p₀ p₁ : s.Pos} {q r : s.Pos}
   simp [Slice.Pos.le_iff, Pos.le_iff, Pos.Raw.le_iff] at h₁ h₁' ⊢
   omega
 
+theorem Slice.Pos.le_ofSlice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q ≤ Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Slice.Pos.slice q p₀ p₁ h₀ h₁ ≤ p := by
+  refine ⟨fun h => ⟨Std.le_trans h ofSlice_le, fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Slice.Pos.slice_ofSlice (pos := p), slice_le_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Slice.Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_le_ofSlice_iff] using h h₀
+    · exact Std.le_of_lt (Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice)
+
+theorem Slice.Pos.ofSlice_lt_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p < q ↔ ∀ h₁, ∃ h₀, p < Slice.Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_le, le_ofSlice_iff]
+
+theorem Slice.Pos.lt_ofSlice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q < Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Slice.Pos.slice q p₀ p₁ h₀ h₁ < p := by
+  refine ⟨fun h => ⟨Std.le_of_lt (Std.lt_of_lt_of_le h ofSlice_le), fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Slice.Pos.slice_ofSlice (pos := p), slice_lt_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Slice.Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_lt_ofSlice_iff] using h h₀
+    · exact Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice
+
+theorem Slice.Pos.ofSlice_le_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p ≤ q ↔ ∀ h₁, ∃ h₀, p ≤ Slice.Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_lt, lt_ofSlice_iff]
+
+theorem Pos.le_ofSlice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q ≤ Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Pos.slice q p₀ p₁ h₀ h₁ ≤ p := by
+  refine ⟨fun h => ⟨Std.le_trans h ofSlice_le, fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Pos.slice_ofSlice (pos := p), slice_le_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_le_ofSlice_iff] using h h₀
+    · exact Std.le_of_lt (Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice)
+
+theorem Pos.ofSlice_lt_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p < q ↔ ∀ h₁, ∃ h₀, p < Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_le, le_ofSlice_iff]
+
+theorem Pos.lt_ofSlice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q < Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Pos.slice q p₀ p₁ h₀ h₁ < p := by
+  refine ⟨fun h => ⟨Std.le_of_lt (Std.lt_of_lt_of_le h ofSlice_le), fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Pos.slice_ofSlice (pos := p), slice_lt_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_lt_ofSlice_iff] using h h₀
+    · exact Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice
+
+theorem Pos.ofSlice_le_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p ≤ q ↔ ∀ h₁, ∃ h₀, p ≤ Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_lt, lt_ofSlice_iff]
+
+theorem Slice.Pos.slice_le_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Slice.Pos.slice q p₀ p₁ h₀ h₁ ≤ p ↔ q ≤ Pos.ofSlice p := by
+  simp [le_ofSlice_iff, h₀, h₁]
+
+theorem Slice.Pos.lt_slice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p < Slice.Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p < q := by
+  simp [ofSlice_lt_iff, h₀, h₁]
+
+theorem Slice.Pos.slice_lt_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Slice.Pos.slice q p₀ p₁ h₀ h₁ < p ↔ q < Pos.ofSlice p := by
+  simp [lt_ofSlice_iff, h₀, h₁]
+
+theorem Slice.Pos.le_slice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p ≤ Slice.Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p ≤ q := by
+  simp [ofSlice_le_iff, h₀, h₁]
+
+theorem Pos.slice_le_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Pos.slice q p₀ p₁ h₀ h₁ ≤ p ↔ q ≤ Pos.ofSlice p := by
+  simp [le_ofSlice_iff, h₀, h₁]
+
+theorem Pos.lt_slice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p < Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p < q := by
+  simp [ofSlice_lt_iff, h₀, h₁]
+
+theorem Pos.slice_lt_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Pos.slice q p₀ p₁ h₀ h₁ < p ↔ q < Pos.ofSlice p := by
+  simp [lt_ofSlice_iff, h₀, h₁]
+
+theorem Pos.le_slice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p ≤ Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p ≤ q := by
+  simp [ofSlice_le_iff, h₀, h₁]
+
 theorem Slice.Pos.ofSlice_ne_endPos {s : Slice} {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 [← 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 :=
@@ -468,21 +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/Char.lean

@@ -20,28 +20,44 @@ import Init.Data.String.Lemmas.Order
 import Init.Data.Order.Lemmas
 import Init.Data.String.OrderInstances
 import Init.Omega
+import Init.Data.String.Lemmas.FindPos
 
 public section
 
 namespace String.Slice.Pattern.Model.Char
 
-instance {c : Char} : ForwardPatternModel c where
+instance {c : Char} : PatternModel c where
   Matches s := s = String.singleton c
-  not_matches_empty := by simp
 
-instance {c : Char} : NoPrefixForwardPatternModel c :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+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])
+
+instance {c : Char} : NoSuffixPatternModel c :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
     IsMatch c pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ s.startPos.get h = c := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, sliceTo_copy_eq_iff_exists_splits]
+  simp only [Model.isMatch_iff, PatternModel.Matches, copy_sliceTo_eq_iff_exists_splits]
   refine ⟨?_, ?_⟩
   · simp only [splits_singleton_iff]
     exact fun ⟨t₂, h, h₁, h₂, h₃⟩ => ⟨h, h₁, h₂⟩
   · rintro ⟨h, rfl, rfl⟩
     exact ⟨_, Slice.splits_next_startPos⟩
 
+theorem isRevMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    IsRevMatch c pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, copy_sliceFrom_eq_iff_exists_splits]
+  refine ⟨?_, ?_⟩
+  · simp only [splits_singleton_right_iff]
+    exact fun ⟨t₂, h, h₁, h₂, h₃⟩ => ⟨h, h₁, h₂⟩
+  · rintro ⟨h, rfl, rfl⟩
+    exact ⟨_, Slice.splits_prev_endPos⟩
+
 theorem isLongestMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
     IsLongestMatch c pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ s.startPos.get h = c := by
@@ -52,21 +68,46 @@ theorem isLongestMatchAt_iff {c : Char} {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 isLongestRevMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch c pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
+
+theorem isLongestRevMatchAt_iff {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAt c pos pos' ↔ ∃ h, pos = pos'.prev h ∧ (pos'.prev h).get (by simp) = c := by
+  simp +contextual [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff, ← Pos.ofSliceTo_inj,
+    Pos.get_eq_get_ofSliceTo, Pos.ofSliceTo_prev]
+
 theorem isLongestMatchAt_of_get_eq {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : pos.get h = c) : IsLongestMatchAt c pos (pos.next h) :=
   isLongestMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAt_of_get_eq {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : (pos.prev h).get (by simp) = c) : IsLongestRevMatchAt c (pos.prev h) pos :=
+  isLongestRevMatchAt_iff.2 ⟨h, by simp [hc]⟩
+
 instance {c : Char} : LawfulForwardPatternModel c where
   skipPrefix?_eq_some_iff {s} pos := by
     simp [isLongestMatch_iff, ForwardPattern.skipPrefix?, and_comm, eq_comm (b := pos)]
 
+instance {c : Char} : LawfulBackwardPatternModel c where
+  skipSuffix?_eq_some_iff {s} pos := by
+    simp [isLongestRevMatch_iff, BackwardPattern.skipSuffix?, and_comm, eq_comm (b := pos)]
+
 theorem toSearcher_eq {c : Char} {s : Slice} :
   ToForwardSearcher.toSearcher c s = ToForwardSearcher.toSearcher (· == c) s := (rfl)
 
+theorem toBackwardSearcher_eq {c : Char} {s : Slice} :
+  ToBackwardSearcher.toSearcher c s = ToBackwardSearcher.toSearcher (· == c) s := (rfl)
+
 theorem matchesAt_iff {c : Char} {s : Slice} {pos : s.Pos} :
     MatchesAt c pos ↔ ∃ (h : pos ≠ s.endPos), pos.get h = c := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff, exists_comm]
 
+theorem revMatchesAt_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt c pos ↔ ∃ (h : pos ≠ s.startPos), (pos.prev h).get (by simp) = c := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff, exists_comm]
+
 theorem matchesAt_iff_splits {c : Char} {s : Slice} {pos : s.Pos} :
     MatchesAt c pos ↔ ∃ t₁ t₂, pos.Splits t₁ (singleton c ++ t₂) := by
   rw [matchesAt_iff]
@@ -77,37 +118,131 @@ theorem matchesAt_iff_splits {c : Char} {s : Slice} {pos : s.Pos} :
     have hne := hs.ne_endPos_of_singleton
     exact ⟨hne, (singleton_append_inj.mp (hs.eq_right (pos.splits_next_right hne))).1.symm⟩
 
+theorem revMatchesAt_iff_splits {c : Char} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt c pos ↔ ∃ t₁ t₂, pos.Splits (t₁ ++ singleton c) t₂ := by
+  rw [revMatchesAt_iff]
+  refine ⟨?_, ?_⟩
+  · rintro ⟨h, rfl⟩
+    exact ⟨_, _, pos.splits_prev_right h⟩
+  · rintro ⟨t₁, t₂, hs⟩
+    have hne := hs.ne_startPos_of_singleton
+    refine ⟨hne, ?_⟩
+    have := hs.eq_left (pos.splits_prev_right hne)
+    simp only [append_singleton, push_inj] at this
+    exact this.2.symm
+
 theorem not_matchesAt_of_get_ne {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : pos.get h ≠ c) : ¬ MatchesAt c pos := by
   simp [matchesAt_iff, hc]
 
+theorem not_revMatchesAt_of_get_ne {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : (pos.prev h).get (by simp) ≠ c) : ¬ RevMatchesAt c pos := by
+  simp [revMatchesAt_iff, hc]
+
 theorem matchAt?_eq {s : Slice} {pos : s.Pos} {c : Char} :
     matchAt? c pos =
       if h₀ : ∃ (h : pos ≠ s.endPos), pos.get h = c then some (pos.next h₀.1) else none := by
   split <;> simp_all [isLongestMatchAt_iff, matchesAt_iff]
 
+theorem revMatchAt?_eq {s : Slice} {pos : s.Pos} {c : Char} :
+    revMatchAt? c pos =
+      if h₀ : ∃ (h : pos ≠ s.startPos), (pos.prev h).get (by simp) = c then some (pos.prev h₀.1) else none := by
+  split <;> simp_all [isLongestRevMatchAt_iff, revMatchesAt_iff]
+
 theorem isMatch_iff_isMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
     IsMatch c pos ↔ IsMatch (· == c) pos := by
   simp [isMatch_iff, CharPred.isMatch_iff, beq_iff_eq]
 
+theorem isRevMatch_iff_isRevMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    IsRevMatch c pos ↔ IsRevMatch (· == c) pos := by
+  simp [isRevMatch_iff, CharPred.isRevMatch_iff, beq_iff_eq]
+
 theorem isLongestMatch_iff_isLongestMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
     IsLongestMatch c pos ↔ IsLongestMatch (· == c) pos := by
   simp [isLongestMatch_iff_isMatch, isMatch_iff_isMatch_beq]
 
+theorem isLongestRevMatch_iff_isLongestRevMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch c pos ↔ IsLongestRevMatch (· == c) pos := by
+  simp [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff_isRevMatch_beq]
+
 theorem isLongestMatchAt_iff_isLongestMatchAt_beq {c : Char} {s : Slice}
     {pos pos' : s.Pos} :
     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]
 
+theorem revMatchesAt_iff_revMatchesAt_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt c pos ↔ RevMatchesAt (· == c) pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+
 theorem matchAt?_eq_matchAt?_beq {c : Char} {s : Slice} {pos : s.Pos} :
     matchAt? c pos = matchAt? (· == c) pos := by
   refine Option.ext (fun pos' => ?_)
   simp [matchAt?_eq_some_iff, isLongestMatchAt_iff_isLongestMatchAt_beq]
 
+theorem revMatchAt?_eq_revMatchAt?_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    revMatchAt? c pos = revMatchAt? (· == c) pos := by
+  refine Option.ext (fun pos' => ?_)
+  simp [revMatchAt?_eq_some_iff, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+
 theorem isValidSearchFrom_iff_isValidSearchFrom_beq {c : Char} {s : Slice} {p : s.Pos}
     {l : List (SearchStep s)} : IsValidSearchFrom c p l ↔ IsValidSearchFrom (· == c) p l := by
   refine ⟨fun h => ?_, fun h => ?_⟩
@@ -120,11 +255,28 @@ theorem isValidSearchFrom_iff_isValidSearchFrom_beq {c : Char} {s : Slice} {p :
     | matched => simp_all [IsValidSearchFrom.matched, isLongestMatchAt_iff_isLongestMatchAt_beq]
     | mismatched => simp_all [IsValidSearchFrom.mismatched, matchesAt_iff_matchesAt_beq]
 
+theorem isValidRevSearchFrom_iff_isValidRevSearchFrom_beq {c : Char} {s : Slice} {p : s.Pos}
+    {l : List (SearchStep s)} : IsValidRevSearchFrom c p l ↔ IsValidRevSearchFrom (· == c) p l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_beq]
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_beq]
+
 instance {c : Char} : LawfulToForwardSearcherModel c where
   isValidSearchFrom_toList s := by
     simpa [toSearcher_eq, isValidSearchFrom_iff_isValidSearchFrom_beq] using
       LawfulToForwardSearcherModel.isValidSearchFrom_toList (pat := (· == c)) (s := s)
 
+instance {c : Char} : LawfulToBackwardSearcherModel c where
+  isValidRevSearchFrom_toList s := by
+    simpa [toBackwardSearcher_eq, isValidRevSearchFrom_iff_isValidRevSearchFrom_beq] using
+      LawfulToBackwardSearcherModel.isValidRevSearchFrom_toList (pat := (· == c)) (s := s)
+
 end Pattern.Model.Char
 
 theorem startsWith_char_eq_startsWith_beq {c : Char} {s : Slice} :
@@ -142,18 +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) :=
@@ -169,7 +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) :=
@@ -198,18 +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) :=
@@ -223,4 +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) [ForwardPatternModel 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) [ForwardPatternMo
   | matched h₁ _ _ => have := h₁.matchesAt; grind
   | mismatched => grind
 
-theorem Pattern.Model.find?_eq_none_iff {ρ : Type} (pat : ρ) [ForwardPatternModel 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) [ForwardPatternModel 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) [ForwardPatternModel 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} [ForwardPatternModel 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} [ForwardPatter
     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} [ForwardPatternModel 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

@@ -18,125 +18,321 @@ 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
 
 public section
 
 namespace String.Slice.Pattern.Model.CharPred
 
-instance {p : Char → Bool} : ForwardPatternModel p where
+instance {p : Char → Bool} : PatternModel p where
   Matches s := ∃ c, s = singleton c ∧ p c
-  not_matches_empty := by
-    simp
 
-instance {p : Char → Bool} : NoPrefixForwardPatternModel p :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+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])
+
+instance {p : Char → Bool} : NoSuffixPatternModel p :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
     IsMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, sliceTo_copy_eq_iff_exists_splits]
+  simp only [Model.isMatch_iff, PatternModel.Matches, copy_sliceTo_eq_iff_exists_splits]
   refine ⟨?_, ?_⟩
   · simp only [splits_singleton_iff]
     refine fun ⟨c, ⟨t₂, h, h₁, h₂, h₃⟩, hc⟩ => ⟨h, h₁, h₂ ▸ hc⟩
   · rintro ⟨h, rfl, h'⟩
     exact ⟨s.startPos.get h, ⟨_, Slice.splits_next_startPos⟩, h'⟩
 
+theorem isRevMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    IsRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, copy_sliceFrom_eq_iff_exists_splits]
+  refine ⟨?_, ?_⟩
+  · simp only [splits_singleton_right_iff]
+    refine fun ⟨c, ⟨t₂, h, h₁, h₂, h₃⟩, hc⟩ => ⟨h, h₁, h₂ ▸ hc⟩
+  · rintro ⟨h, rfl, h'⟩
+    exact ⟨(s.endPos.prev h).get (by simp), ⟨_, Slice.splits_prev_endPos⟩, h'⟩
+
 theorem isLongestMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
     IsLongestMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
   rw [isLongestMatch_iff_isMatch, isMatch_iff]
 
+theorem isLongestRevMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
+
 theorem isLongestMatchAt_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
     IsLongestMatchAt p pos pos' ↔ ∃ h, pos' = pos.next h ∧ p (pos.get h) := by
   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,
+    Pos.get_eq_get_ofSliceTo, Pos.ofSliceTo_prev]
+
 theorem isLongestMatchAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : p (pos.get h)) : IsLongestMatchAt p pos (pos.next h) :=
   isLongestMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (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)]
 
+instance {p : Char → Bool} : LawfulBackwardPatternModel p where
+  skipSuffix?_eq_some_iff {s} pos := by
+    simp [isLongestRevMatch_iff, BackwardPattern.skipSuffix?, and_comm, eq_comm (b := pos)]
+
 instance {p : Char → Bool} : LawfulToForwardSearcherModel p :=
   .defaultImplementation
 
+instance {p : Char → Bool} : LawfulToBackwardSearcherModel p :=
+  .defaultImplementation
+
 theorem matchesAt_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
     MatchesAt p pos ↔ ∃ (h : pos ≠ s.endPos), p (pos.get h) := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff, exists_comm]
 
+theorem revMatchesAt_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt p pos ↔ ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff, exists_comm]
+
 theorem not_matchesAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : p (pos.get h) = false) : ¬ MatchesAt p pos := by
   simp [matchesAt_iff, hc]
 
+theorem not_revMatchesAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : p ((pos.prev h).get (by simp)) = false) : ¬ RevMatchesAt p pos := by
+  simp [revMatchesAt_iff, hc]
+
 theorem matchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Bool} :
     matchAt? p pos =
       if h₀ : ∃ (h : pos ≠ s.endPos), p (pos.get h) then some (pos.next h₀.1) else none := by
   split <;> simp_all [isLongestMatchAt_iff, matchesAt_iff]
 
+theorem revMatchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Bool} :
+    revMatchAt? p pos =
+      if h₀ : ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) then some (pos.prev h₀.1) else none := by
+  split <;> simp_all [isLongestRevMatchAt_iff, revMatchesAt_iff]
+
 namespace Decidable
 
-instance {p : Char → Prop} [DecidablePred p] : ForwardPatternModel p where
-  Matches := ForwardPatternModel.Matches (decide <| p ·)
-  not_matches_empty := ForwardPatternModel.not_matches_empty (pat := (decide <| p ·))
+instance {p : Char → Prop} [DecidablePred p] : PatternModel p where
+  Matches := PatternModel.Matches (decide <| p ·)
 
-instance {p : Char → Prop} [DecidablePred p] : NoPrefixForwardPatternModel p where
-  eq_empty := NoPrefixForwardPatternModel.eq_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 ·))
+
+instance {p : Char → Prop} [DecidablePred p] : NoSuffixPatternModel p where
+  eq_empty := NoSuffixPatternModel.eq_empty (pat := (decide <| p ·))
 
 theorem isMatch_iff_isMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     IsMatch p pos ↔ IsMatch (decide <| p ·) pos :=
   ⟨fun ⟨h⟩ => ⟨h⟩, fun ⟨h⟩ => ⟨h⟩⟩
 
+theorem isRevMatch_iff_isRevMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    IsRevMatch p pos ↔ IsRevMatch (decide <| p ·) pos :=
+  ⟨fun ⟨h⟩ => ⟨h⟩, fun ⟨h⟩ => ⟨h⟩⟩
+
 theorem isMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     IsMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
   simp [isMatch_iff_isMatch_decide, CharPred.isMatch_iff]
 
+theorem isRevMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    IsRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  simp [isRevMatch_iff_isRevMatch_decide, CharPred.isRevMatch_iff]
+
 theorem isLongestMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     IsLongestMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
   rw [isLongestMatch_iff_isMatch, isMatch_iff]
 
+theorem isLongestRevMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  simp [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
+
 theorem isLongestMatch_iff_isLongestMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos : s.Pos} : IsLongestMatch p pos ↔ IsLongestMatch (decide <| p ·) pos := by
   simp [isLongestMatch_iff_isMatch, isMatch_iff_isMatch_decide]
 
+theorem isLongestRevMatch_iff_isLongestRevMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos : s.Pos} : IsLongestRevMatch p pos ↔ IsLongestRevMatch (decide <| p ·) pos := by
+  simp [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff_isRevMatch_decide]
+
 theorem isLongestMatchAt_iff_isLongestMatchAt_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} {pos pos' : s.Pos} :
     IsLongestMatchAt p pos pos' ↔ IsLongestMatchAt (decide <| p ·) pos pos' := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff_isLongestMatch_decide]
 
+theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_decide {p : Char → Prop} [DecidablePred p]
+    {s : Slice} {pos pos' : s.Pos} :
+    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
   simp [isLongestMatchAt_iff_isLongestMatchAt_decide, CharPred.isLongestMatchAt_iff]
 
+theorem isLongestRevMatchAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos pos' : s.Pos} :
+    IsLongestRevMatchAt p pos pos' ↔ ∃ h, pos = pos'.prev h ∧ p ((pos'.prev h).get (by simp)) := by
+  simp [isLongestRevMatchAt_iff_isLongestRevMatchAt_decide, CharPred.isLongestRevMatchAt_iff]
+
 theorem isLongestMatchAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
     {h : pos ≠ s.endPos} (hc : p (pos.get h)) : IsLongestMatchAt p pos (pos.next h) :=
   isLongestMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
+    {h : pos ≠ s.startPos} (hc : p ((pos.prev h).get (by simp))) :
+    IsLongestRevMatchAt p (pos.prev h) pos :=
+  isLongestRevMatchAt_iff.2 ⟨h, by simp [hc]⟩
+
 theorem matchesAt_iff_matchesAt_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos : s.Pos} : MatchesAt p pos ↔ MatchesAt (decide <| p ·) pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_decide]
 
+theorem revMatchesAt_iff_revMatchesAt_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos : s.Pos} : RevMatchesAt p pos ↔ RevMatchesAt (decide <| p ·) pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+
 theorem matchAt?_eq_matchAt?_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos : s.Pos} : matchAt? p pos = matchAt? (decide <| p ·) pos := by
   ext endPos
   simp [isLongestMatchAt_iff_isLongestMatchAt_decide]
 
+theorem revMatchAt?_eq_revMatchAt?_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos : s.Pos} : revMatchAt? p pos = revMatchAt? (decide <| p ·) pos := by
+  ext startPos
+  simp [isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+
 theorem skipPrefix?_eq_skipPrefix?_decide {p : Char → Prop} [DecidablePred p] :
     ForwardPattern.skipPrefix? p = ForwardPattern.skipPrefix? (decide <| p ·) := rfl
 
+theorem skipSuffix?_eq_skipSuffix?_decide {p : Char → Prop} [DecidablePred p] :
+    BackwardPattern.skipSuffix? p = BackwardPattern.skipSuffix? (decide <| p ·) := rfl
+
 instance {p : Char → Prop} [DecidablePred p] : LawfulForwardPatternModel p where
   skipPrefix?_eq_some_iff {s} pos := by
     rw [skipPrefix?_eq_skipPrefix?_decide, isLongestMatch_iff_isLongestMatch_decide]
     exact LawfulForwardPatternModel.skipPrefix?_eq_some_iff ..
 
+instance {p : Char → Prop} [DecidablePred p] : LawfulBackwardPatternModel p where
+  skipSuffix?_eq_some_iff {s} pos := by
+    rw [skipSuffix?_eq_skipSuffix?_decide, isLongestRevMatch_iff_isLongestRevMatch_decide]
+    exact LawfulBackwardPatternModel.skipSuffix?_eq_some_iff ..
+
 theorem toSearcher_eq {p : Char → Prop} [DecidablePred p] {s : Slice} :
     ToForwardSearcher.toSearcher p s = ToForwardSearcher.toSearcher (decide <| p ·) s := (rfl)
 
+theorem toBackwardSearcher_eq {p : Char → Prop} [DecidablePred p] {s : Slice} :
+    ToBackwardSearcher.toSearcher p s = ToBackwardSearcher.toSearcher (decide <| p ·) s := (rfl)
+
 theorem isValidSearchFrom_iff_isValidSearchFrom_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} {pos : s.Pos} {l : List (SearchStep s)} :
     IsValidSearchFrom p pos l ↔ IsValidSearchFrom (decide <| p ·) pos l := by
@@ -150,24 +346,55 @@ theorem isValidSearchFrom_iff_isValidSearchFrom_decide {p : Char → Prop} [Deci
     | matched => simp_all [IsValidSearchFrom.matched, isLongestMatchAt_iff_isLongestMatchAt_decide]
     | mismatched => simp_all [IsValidSearchFrom.mismatched, matchesAt_iff_matchesAt_decide]
 
+theorem isValidRevSearchFrom_iff_isValidRevSearchFrom_decide {p : Char → Prop} [DecidablePred p]
+    {s : Slice} {pos : s.Pos} {l : List (SearchStep s)} :
+    IsValidRevSearchFrom p pos l ↔ IsValidRevSearchFrom (decide <| p ·) pos l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_decide]
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_decide]
+
 instance {p : Char → Prop} [DecidablePred p] : LawfulToForwardSearcherModel p where
   isValidSearchFrom_toList s := by
     simpa [toSearcher_eq, isValidSearchFrom_iff_isValidSearchFrom_decide] using
       LawfulToForwardSearcherModel.isValidSearchFrom_toList (pat := (decide <| p ·)) (s := s)
 
+instance {p : Char → Prop} [DecidablePred p] : LawfulToBackwardSearcherModel p where
+  isValidRevSearchFrom_toList s := by
+    simpa [toBackwardSearcher_eq, isValidRevSearchFrom_iff_isValidRevSearchFrom_decide] using
+      LawfulToBackwardSearcherModel.isValidRevSearchFrom_toList (pat := (decide <| p ·)) (s := s)
+
 theorem matchesAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     MatchesAt p pos ↔ ∃ (h : pos ≠ s.endPos), p (pos.get h) := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff, exists_comm]
 
+theorem revMatchesAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    RevMatchesAt p pos ↔ ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff, exists_comm]
+
 theorem not_matchesAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
     {h : pos ≠ s.endPos} (hc : ¬ p (pos.get h)) : ¬ MatchesAt p pos := by
   simp [matchesAt_iff, hc]
 
+theorem not_revMatchesAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
+    {h : pos ≠ s.startPos} (hc : ¬ p ((pos.prev h).get (by simp))) : ¬ RevMatchesAt p pos := by
+  simp [revMatchesAt_iff, hc]
+
 theorem matchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Prop} [DecidablePred p] :
     matchAt? p pos =
       if h₀ : ∃ (h : pos ≠ s.endPos), p (pos.get h) then some (pos.next h₀.1) else none := by
   split <;> simp_all [isLongestMatchAt_iff, matchesAt_iff]
 
+theorem revMatchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Prop} [DecidablePred p] :
+    revMatchAt? p pos =
+      if h₀ : ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) then some (pos.prev h₀.1) else none := by
+  split <;> simp_all [isLongestRevMatchAt_iff, revMatchesAt_iff]
+
 end Decidable
 
 end Pattern.Model.CharPred
@@ -184,6 +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)
@@ -194,13 +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} :
@@ -217,7 +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 ·) :=
@@ -248,19 +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} :
@@ -277,4 +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

@@ -28,7 +28,7 @@ set_option doc.verso true
 # Verification of {name}`String.Slice.splitToSubslice`
 
 This PR verifies the {name}`String.Slice.splitToSubslice` function by relating it to a model
-implementation based on the {name}`String.Slice.Pattern.Model.ForwardPatternModel` class.
+implementation based on the {name}`String.Slice.Pattern.Model.PatternModel` class.
 
 This gives a low-level correctness proof from which higher-level API lemmas can be derived.
 -/
@@ -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) [ForwardPatternModel 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) [ForwardPatternM
 termination_by curr
 
 @[simp]
-public theorem split_endPos {ρ : Type} {pat : ρ} [ForwardPatternModel 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} [ForwardPatternModel 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} [ForwardPatte
   · 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} [ForwardPatternModel 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} [ForwardPatternM
   · 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} [ForwardPatternModel 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)
-    [ForwardPatternModel 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) [ForwardPatternModel 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.ForwardPatternModel 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.ForwardPatternModel 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.ForwardPatternModel 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/Split/Char.lean

@@ -23,6 +23,7 @@ import Init.Data.String.OrderInstances
 import Init.Data.String.Lemmas.Order
 import Init.Data.String.Lemmas.Intercalate
 import Init.Data.List.SplitOn.Lemmas
+import Init.Data.String.Lemmas.Slice
 
 public section
 
@@ -70,6 +71,11 @@ theorem Slice.toList_split_intercalate {c : Char} {l : List Slice} (hl : ∀ s
   · simp_all
   · rw [List.splitOn_intercalate] <;> simp_all
 
+theorem Slice.toList_split_intercalate_beq {c : Char} {l : List Slice} (hl : ∀ s ∈ l, c ∉ s.copy.toList) :
+    ((Slice.intercalate (String.singleton c) l).split c).toList ==
+      if l = [] then ["".toSlice] else l := by
+  split <;> simp_all [toList_split_intercalate hl, beq_list_iff]
+
 theorem toList_split_intercalate {c : Char} {l : List String} (hl : ∀ s ∈ l, c ∉ s.toList) :
     ((String.intercalate (String.singleton c) l).split c).toList.map (·.copy) =
       if l = [] then [""] else l := by
@@ -78,4 +84,9 @@ theorem toList_split_intercalate {c : Char} {l : List String} (hl : ∀ s ∈ l,
   · simp_all
   · rw [List.splitOn_intercalate] <;> simp_all
 
+theorem toList_split_intercalate_beq {c : Char} {l : List String} (hl : ∀ s ∈ l, c ∉ s.toList) :
+    ((String.intercalate (String.singleton c) l).split c).toList ==
+      if l = [] then ["".toSlice] else l.map String.toSlice := by
+  split <;> simp_all [toList_split_intercalate hl, Slice.beq_list_iff]
+
 end String

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
 
@@ -19,52 +22,135 @@ namespace String.Slice.Pattern.Model
 
 namespace ForwardSliceSearcher
 
-instance {pat : Slice} : ForwardPatternModel pat where
-  /-
-  See the docstring of `ForwardPatternModel` for an explanation about why we disallow matching the
-  empty string.
+instance {pat : Slice} : PatternModel pat where
+  Matches s := s = pat.copy
 
-  Requiring `s ≠ ""` is a trick that allows us to give a `ForwardPatternModel` 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} : NoPrefixForwardPatternModel pat :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+instance {pat : Slice} : NoPrefixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
-theorem isMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+instance {pat : Slice} : NoSuffixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
+
+theorem isMatch_iff {pat s : Slice} {pos : s.Pos} :
     IsMatch pat pos ↔ (s.sliceTo pos).copy = pat.copy := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.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 isLongestMatch_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 [Model.isRevMatch_iff, PatternModel.Matches]
+
+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 isLongestMatchAt_iff {pat s : Slice} {pos₁ pos₂ : 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]
+
+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 isLongestMatchAt_iff_splits {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]
+
+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 isLongestMatch_iff_splits {pat s : Slice} {pos : 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, copy_slice_eq_iff_splits]
+
+theorem isLongestMatch_iff_splits {pat s : Slice} {pos : s.Pos} :
     IsLongestMatch pat pos ↔ ∃ t, pos.Splits pat.copy t := by
-  simp only [← isLongestMatchAt_startPos_iff, isLongestMatchAt_iff_splits h, splits_startPos_iff,
-    and_assoc, exists_and_left, exists_eq_left, empty_append]
-  exact ⟨fun ⟨h, _, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'.eq_append.symm, h'⟩⟩
+  rw [isLongestMatch_iff, copy_sliceTo_eq_iff_exists_splits]
+
+theorem isLongestRevMatch_iff_splits {pat s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch pat pos ↔ ∃ t, pos.Splits t pat.copy := by
+  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
+    exact ⟨by simp [Pos.le_iff, Pos.Raw.le_iff]; omega,
+      by simp [← h', ← toByteArray_inj, toByteArray_copy_slice]⟩
+
+theorem isLongestRevMatchAt_iff_extract {pat s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat.isEmpty = false) :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔
+      s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx =
+        pat.copy.toByteArray := by
+  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
@@ -81,15 +167,32 @@ theorem offset_of_isLongestMatchAt {pat s : Slice} {pos₁ pos₂ : s.Pos} (h :
   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 offset_of_isLongestRevMatchAt {pat s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat.isEmpty = false) (h' : IsLongestRevMatchAt pat pos₁ pos₂) :
+    pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
+  simp only [Pos.Raw.ext_iff, Pos.Raw.byteIdx_increaseBy]
+  rw [isLongestRevMatchAt_iff_extract h] at h'
+  rw [← Slice.toByteArray_copy_ne_empty_iff, ← h', ne_eq, ByteArray.extract_eq_empty_iff] at h
+  replace h' := congrArg ByteArray.size h'
+  simp only [ByteArray.size_extract, size_toByteArray, utf8ByteSize_copy] at h'
+  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} :
     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 exists_matchesAt_iff_eq_append {pat s : Slice} (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]
+  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} :
     (∃ (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]⟩
@@ -99,6 +202,18 @@ 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} :
+    (∃ (pos : s.Pos), RevMatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat.copy ++ t₂ := by
+  simp only [revMatchesAt_iff_splits]
+  constructor
+  · rintro ⟨pos, t₁, t₂, hsplit⟩
+    exact ⟨t₁, t₂, by rw [hsplit.eq_append, append_assoc]⟩
+  · rintro ⟨t₁, t₂, heq⟩
+    have hvalid : (t₁ ++ pat.copy).rawEndPos.IsValidForSlice s :=
+      Pos.Raw.isValidForSlice_iff_exists_append.mpr
+        ⟨t₁ ++ pat.copy, t₂, heq, rfl⟩
+    exact ⟨s.pos _ hvalid, t₁, t₂, ⟨heq, by simp⟩⟩
+
 theorem matchesAt_iff_isLongestMatchAt {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     MatchesAt pat pos ↔ ∃ (h : (pos.offset.increaseBy pat.utf8ByteSize).IsValidForSlice s),
       IsLongestMatchAt pat pos (s.pos _ h) := by
@@ -108,6 +223,25 @@ theorem matchesAt_iff_isLongestMatchAt {pat s : Slice} {pos : s.Pos} (h : pat.is
   obtain rfl : p = s.pos _ this := by simpa [Pos.ext_iff] using offset_of_isLongestMatchAt h h'
   exact h'
 
+theorem revMatchesAt_iff_isLongestRevMatchAt {pat s : Slice} {pos : s.Pos}
+    (h : pat.isEmpty = false) :
+    RevMatchesAt pat pos ↔
+      ∃ (h : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s),
+        IsLongestRevMatchAt pat (s.pos _ h) pos := by
+  refine ⟨fun ⟨⟨p, h'⟩⟩ => ?_, fun ⟨_, h⟩ => ⟨⟨_, h⟩⟩⟩
+  have hoff := offset_of_isLongestRevMatchAt h h'
+  have hvalid : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s := by
+    rw [show pos.offset.decreaseBy pat.utf8ByteSize = p.offset from by
+      simp [Pos.Raw.ext_iff, Pos.Raw.byteIdx_decreaseBy, Pos.Raw.byteIdx_increaseBy] at hoff ⊢
+      omega]
+    exact p.isValidForSlice
+  refine ⟨hvalid, ?_⟩
+  obtain rfl : p = s.pos _ hvalid := by
+    simp only [Pos.ext_iff, offset_pos]
+    simp [Pos.Raw.ext_iff, Pos.Raw.byteIdx_decreaseBy, Pos.Raw.byteIdx_increaseBy] at hoff ⊢
+    omega
+  exact h'
+
 theorem matchesAt_iff_getElem {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     MatchesAt pat pos ↔
       ∃ (h : pos.offset.byteIdx + pat.copy.toByteArray.size ≤ s.copy.toByteArray.size),
@@ -146,81 +280,194 @@ end ForwardSliceSearcher
 
 namespace ForwardStringSearcher
 
-instance {pat : String} : ForwardPatternModel pat where
-  Matches s := s ≠ "" ∧ s = pat
-  not_matches_empty := by simp
+instance {pat : String} : PatternModel pat where
+  Matches s := s = pat
 
-instance {pat : String} : NoPrefixForwardPatternModel pat :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+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])
+
+instance {pat : String} : NoSuffixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff_slice {pat : String} {s : Slice} {pos : s.Pos} :
     IsMatch (ρ := String) pat pos ↔ IsMatch (ρ := Slice) pat.toSlice pos := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, copy_toSlice]
+  simp only [Model.isMatch_iff, PatternModel.Matches, copy_toSlice]
+
+theorem isRevMatch_iff_slice {pat : String} {s : Slice} {pos : s.Pos} :
+    IsRevMatch (ρ := String) pat pos ↔ IsRevMatch (ρ := Slice) pat.toSlice pos := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, copy_toSlice]
 
 theorem isLongestMatch_iff_isLongestMatch_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
     IsLongestMatch (ρ := String) pat pos ↔ IsLongestMatch (ρ := Slice) pat.toSlice pos where
   mp h := ⟨isMatch_iff_slice.1 h.isMatch, fun p hp hm => h.not_isMatch p hp (isMatch_iff_slice.2 hm)⟩
   mpr h := ⟨isMatch_iff_slice.2 h.isMatch, fun p hp hm => h.not_isMatch p hp (isMatch_iff_slice.1 hm)⟩
 
+theorem isLongestRevMatch_iff_isLongestRevMatch_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch (ρ := String) pat pos ↔ IsLongestRevMatch (ρ := Slice) pat.toSlice pos where
+  mp h := ⟨isRevMatch_iff_slice.1 h.isRevMatch,
+    fun p hp hm => h.not_isRevMatch p hp (isRevMatch_iff_slice.2 hm)⟩
+  mpr h := ⟨isRevMatch_iff_slice.2 h.isRevMatch,
+    fun p hp hm => h.not_isRevMatch p hp (isRevMatch_iff_slice.1 hm)⟩
+
 theorem isLongestMatchAt_iff_isLongestMatchAt_toSlice {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt (ρ := String) pat pos₁ pos₂ ↔
       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]
 
-private theorem toSlice_isEmpty (h : pat ≠ "") : pat.toSlice.isEmpty = false := by
-  rwa [isEmpty_toSlice, isEmpty_eq_false_iff]
+theorem revMatchesAt_iff_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt (ρ := String) pat pos ↔ RevMatchesAt (ρ := Slice) pat.toSlice pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt,
+    isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
 
-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 isLongestMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
-    IsLongestMatch pat pos ↔ (s.sliceTo pos).copy = pat := by
-  rw [isLongestMatch_iff_isMatch, isMatch_iff h]
+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]
+  simp
 
-theorem isLongestMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : 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]
+
+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]
+
+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 isLongestMatchAt_iff_splits {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]
+  simp
+
+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 isLongestMatchAt_iff_extract {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]
+  simp
+
+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}
+    (h : pat ≠ "") :
+    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 (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 matchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+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 (by simpa)
+    (isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice.1 h')
+
+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 exists_matchesAt_iff_eq_append {pat : String} {s : Slice} (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]
+  simp
+
+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]⟩
@@ -230,35 +477,58 @@ 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} :
+    (∃ (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]
+  simp
+
 theorem matchesAt_iff_isLongestMatchAt {pat : String} {s : Slice} {pos : s.Pos}
     (h : pat ≠ "") :
     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]
 
+theorem revMatchesAt_iff_isLongestRevMatchAt {pat : String} {s : Slice} {pos : s.Pos}
+    (h : pat ≠ "") :
+    RevMatchesAt pat pos ↔
+      ∃ (h : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s),
+        IsLongestRevMatchAt pat (s.pos _ h) pos := by
+  have key := ForwardSliceSearcher.revMatchesAt_iff_isLongestRevMatchAt (pat := pat.toSlice)
+    (by simpa) (pos := pos)
+  simp only [utf8ByteSize_toSlice, ← isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice] at key
+  rwa [revMatchesAt_iff_toSlice]
+
 theorem matchesAt_iff_getElem {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
     MatchesAt pat pos ↔
       ∃ (h : pos.offset.byteIdx + pat.toByteArray.size ≤ s.copy.toByteArray.size),
         ∀ (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}
     {pos : s.Pos} : MatchesAt pat pos ↔ MatchesAt pat.toSlice pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_toSlice]
 
+theorem revMatchesAt_iff_revMatchesAt_toSlice {pat : String} {s : Slice}
+    {pos : s.Pos} : RevMatchesAt pat pos ↔ RevMatchesAt pat.toSlice pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt,
+    isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+
 theorem toSearcher_eq {pat : String} {s : Slice} :
     ToForwardSearcher.toSearcher pat s = ToForwardSearcher.toSearcher pat.toSlice s := (rfl)
 
@@ -275,6 +545,21 @@ theorem isValidSearchFrom_iff_isValidSearchFrom_toSlice {pat : String}
     | matched => simp_all [IsValidSearchFrom.matched, isLongestMatchAt_iff_isLongestMatchAt_toSlice]
     | mismatched => simp_all [IsValidSearchFrom.mismatched, matchesAt_iff_matchesAt_toSlice]
 
+theorem isValidRevSearchFrom_iff_isValidRevSearchFrom_toSlice {pat : String}
+    {s : Slice} {pos : s.Pos} {l : List (SearchStep s)} :
+    IsValidRevSearchFrom pat pos l ↔ IsValidRevSearchFrom pat.toSlice pos l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched,
+        isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_toSlice]
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched,
+        isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_toSlice]
+
 end ForwardStringSearcher
 
 end String.Slice.Pattern.Model

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
@@ -76,12 +76,11 @@ namespace Model.ForwardSliceSearcher
 
 open Pattern.ForwardSliceSearcher
 
-public theorem lawfulForwardPatternModel {pat : Slice} (hpat : pat.isEmpty = false) :
-    LawfulForwardPatternModel pat where
-  skipPrefixOfNonempty?_eq h := rfl
-  startsWith_eq s := isSome_skipPrefix?.symm
+public instance {pat : Slice} : LawfulForwardPatternModel pat where
+  skipPrefixOfNonempty?_eq _ := rfl
+  startsWith_eq _ := isSome_skipPrefix?.symm
   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
 
@@ -89,15 +88,107 @@ namespace Model.ForwardStringSearcher
 
 open Pattern.ForwardSliceSearcher
 
-public theorem lawfulForwardPatternModel {pat : String} (hpat : pat ≠ "") :
-    LawfulForwardPatternModel pat where
-  skipPrefixOfNonempty?_eq h := rfl
-  startsWith_eq s := isSome_skipPrefix?.symm
+public instance {pat : String} : LawfulForwardPatternModel pat where
+  skipPrefixOfNonempty?_eq _ := rfl
+  startsWith_eq _ := isSome_skipPrefix?.symm
   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
 
+namespace BackwardSliceSearcher
+
+theorem endsWith_iff {pat s : Slice} : endsWith pat s ↔ ∃ t, s.copy = t ++ pat.copy := by
+  rw [endsWith]
+  simp [Internal.memcmpSlice_eq_true_iff, utf8ByteSize_eq_size_toByteArray_copy, -size_toByteArray]
+  generalize pat.copy = pat
+  generalize s.copy = s
+  refine ⟨fun ⟨h₁, h₂⟩ => ?_, ?_⟩
+  · rw [Nat.sub_add_cancel h₁] at h₂
+    suffices (s.rawEndPos.unoffsetBy pat.rawEndPos).IsValid s by
+      have h₃ : (s.sliceFrom (s.pos _ this)).copy = pat := by
+        rw [← toByteArray_inj, (s.pos _ this).splits.toByteArray_right_eq]
+        simpa [offset_pos, Pos.Raw.byteIdx_unoffsetBy, byteIdx_rawEndPos]
+      have := (s.pos _ this).splits
+      rw [h₃] at this
+      exact ⟨_, this.eq_append⟩
+    rw [Pos.Raw.isValid_iff_isValidUTF8_extract_utf8ByteSize]
+    refine ⟨by simp [Pos.Raw.le_iff, Pos.Raw.byteIdx_unoffsetBy], ?_⟩
+    simp only [size_toByteArray] at h₂
+    simpa [Pos.Raw.byteIdx_unoffsetBy, byteIdx_rawEndPos, h₂] using pat.isValidUTF8
+  · rintro ⟨t, rfl⟩
+    exact ⟨by simp, by rw [Nat.sub_add_cancel (by simp)]; exact
+      ByteArray.extract_append_eq_right (by simp) (by simp)⟩
+
+theorem skipSuffix?_eq_some_iff {pat s : Slice} {pos : s.Pos} :
+    skipSuffix? pat s = some pos ↔ (s.sliceFrom pos).copy = pat.copy := by
+  fun_cases skipSuffix? with
+  | case1 h =>
+    simp only [Option.some.injEq]
+    obtain ⟨t, ht⟩ := endsWith_iff.1 h
+    have hpc : pat.copy.utf8ByteSize = pat.utf8ByteSize := Slice.utf8ByteSize_copy
+    have hsz : s.utf8ByteSize = t.utf8ByteSize + pat.utf8ByteSize := by
+      have := congrArg String.utf8ByteSize ht
+      simp only [utf8ByteSize_append, Slice.utf8ByteSize_copy] at this
+      exact this
+    have hoff : (s.endPos.offset.unoffsetBy pat.rawEndPos) = t.rawEndPos := by
+      ext
+      simp only [offset_endPos, Pos.Raw.byteIdx_unoffsetBy, byteIdx_rawEndPos,
+        String.byteIdx_rawEndPos]
+      omega
+    have hval : (s.endPos.offset.unoffsetBy pat.rawEndPos).IsValidForSlice s :=
+      Pos.Raw.isValidForSlice_iff_exists_append.mpr ⟨t, pat.copy, ht, hoff⟩
+    have hsp : (s.pos _ hval).Splits t pat.copy := ⟨ht, hoff⟩
+    rw [Slice.pos!_eq_pos hval]
+    exact ⟨(· ▸ hsp.copy_sliceFrom_eq),
+      fun h => hsp.pos_eq_of_eq_right (h ▸ pos.splits)⟩
+  | case2 h =>
+    simp only [endsWith_iff, not_exists] at h
+    simp only [reduceCtorEq, false_iff]
+    intro heq
+    have := h (s.sliceTo pos).copy
+    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
+
+public theorem endsWith_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    BackwardPattern.endsWith pat s = true := by
+  suffices pat.copy = "" by simp [BackwardPattern.endsWith, endsWith_iff, this]
+  simpa
+
+public theorem skipSuffix?_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    BackwardPattern.skipSuffix? pat s = some s.endPos := by
+  simpa [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff]
+
+end BackwardSliceSearcher
+
+namespace Model.BackwardSliceSearcher
+
+open Pattern.BackwardSliceSearcher
+
+public instance {pat : Slice} : LawfulBackwardPatternModel pat where
+  skipSuffixOfNonempty?_eq _ := rfl
+  endsWith_eq _ := isSome_skipSuffix?.symm
+  skipSuffix?_eq_some_iff pos := by
+    simp [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff,
+      ForwardSliceSearcher.isLongestRevMatch_iff]
+
+end Model.BackwardSliceSearcher
+
+namespace Model.BackwardStringSearcher
+
+open Pattern.BackwardSliceSearcher
+
+public instance {pat : String} : LawfulBackwardPatternModel pat where
+  skipSuffixOfNonempty?_eq _ := rfl
+  endsWith_eq _ := isSome_skipSuffix?.symm
+  skipSuffix?_eq_some_iff pos := by
+    simp [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff,
+      ForwardStringSearcher.isLongestRevMatch_iff]
+
+end Model.BackwardStringSearcher
+
 end Pattern
 
 public theorem startsWith_string_eq_startsWith_toSlice {pat : String} {s : Slice} :
@@ -116,19 +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 :=
@@ -144,7 +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)
@@ -162,19 +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 :=
@@ -188,4 +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

@@ -11,6 +11,8 @@ public import Init.Data.String.TakeDrop
 import Init.Data.String.Lemmas.Pattern.TakeDrop.Basic
 import Init.Data.String.Lemmas.Pattern.Char
 import Init.Data.Option.Lemmas
+import Init.Data.String.Lemmas.FindPos
+import Init.Data.List.Sublist
 
 public section
 
@@ -52,7 +54,113 @@ theorem startsWith_char_eq_false_iff_forall_append {c : Char} {s : Slice} :
 
 theorem eq_append_of_dropPrefix?_char_eq_some {c : Char} {s res : Slice} (h : s.dropPrefix? c = some res) :
     s.copy = singleton c ++ res.copy := by
-  simpa [ForwardPatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
+  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]
+
+theorem endsWith_char_iff_get {c : Char} {s : Slice} :
+    s.endsWith c ↔ ∃ h, (s.endPos.prev h).get (by simp) = c := by
+  simp [Pattern.Model.endsWith_iff, Char.revMatchesAt_iff]
+
+theorem endsWith_char_eq_false_iff_get {c : Char} {s : Slice} :
+    s.endsWith c = false ↔ ∀ h, (s.endPos.prev h).get (by simp) ≠ c := by
+  simp [Pattern.Model.endsWith_eq_false_iff, Char.revMatchesAt_iff]
+
+theorem endsWith_char_iff_exists_append {c : Char} {s : Slice} :
+    s.endsWith c ↔ ∃ t, s.copy = t ++ singleton c := by
+  rw [Pattern.Model.endsWith_iff, Char.revMatchesAt_iff_splits]
+  simp only [splits_endPos_iff, exists_eq_right, eq_comm (a := s.copy)]
+
+theorem endsWith_char_eq_getLast? {c : Char} {s : Slice} :
+    s.endsWith c = (s.copy.toList.getLast? == some c) := by
+  rw [Bool.eq_iff_iff, endsWith_char_iff_exists_append, beq_iff_eq,
+    ← List.singleton_suffix_iff_getLast?_eq_some, List.suffix_iff_exists_eq_append]
+  constructor
+  · rintro ⟨t, ht⟩
+    exact ⟨t.toList, by rw [ht, toList_append, toList_singleton]⟩
+  · rintro ⟨l, hl⟩
+    exact ⟨ofList l, by rw [← toList_inj, toList_append, toList_singleton, toList_ofList]; exact hl⟩
+
+theorem endsWith_char_eq_false_iff_forall_append {c : Char} {s : Slice} :
+    s.endsWith c = false ↔ ∀ t, s.copy ≠ t ++ singleton c := by
+  simp [← Bool.not_eq_true, endsWith_char_iff_exists_append]
+
+theorem eq_append_of_dropSuffix?_char_eq_some {c : Char} {s res : Slice} (h : s.dropSuffix? c = some res) :
+    s.copy = res.copy ++ singleton c := by
+  simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropSuffix?_eq_some h
 
 end Slice
 
@@ -63,19 +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
@@ -86,4 +194,34 @@ theorem eq_append_of_dropPrefix?_char_eq_some {c : Char} {s : String} {res : Sli
   rw [dropPrefix?_eq_dropPrefix?_toSlice] at h
   simpa using Slice.eq_append_of_dropPrefix?_char_eq_some h
 
+theorem skipSuffix?_char_eq_some_iff {c : Char} {s : String} {pos : s.Pos} :
+    s.skipSuffix? c = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  simp [skipSuffix?_eq_skipSuffix?_toSlice, Slice.skipSuffix?_char_eq_some_iff, ← Pos.toSlice_inj,
+    Pos.prev_toSlice]
+
+theorem endsWith_char_iff_get {c : Char} {s : String} :
+    s.endsWith c ↔ ∃ h, (s.endPos.prev h).get (by simp) = c := by
+  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_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_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_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
+  simp [← Bool.not_eq_true, endsWith_char_iff_exists_append]
+
+theorem eq_append_of_dropSuffix?_char_eq_some {c : Char} {s : String} {res : Slice} (h : s.dropSuffix? c = some res) :
+    s = res.copy ++ singleton c := by
+  rw [dropSuffix?_eq_dropSuffix?_toSlice] at h
+  simpa using Slice.eq_append_of_dropSuffix?_char_eq_some h
+
 end String

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

@@ -8,10 +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
 
@@ -45,9 +51,83 @@ theorem startsWith_bool_eq_head? {p : Char → Bool} {s : Slice} :
 
 theorem eq_append_of_dropPrefix?_bool_eq_some {p : Char → Bool} {s res : Slice} (h : s.dropPrefix? p = some res) :
     ∃ c, s.copy = singleton c ++ res.copy ∧ p c = true := by
-  obtain ⟨_, ⟨c, ⟨rfl, h₁⟩⟩, h₂⟩ := by simpa [ForwardPatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
+  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]
@@ -64,11 +144,136 @@ 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
 
+theorem skipSuffix?_bool_eq_some_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? p = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) = true := by
+  rw [Pattern.Model.skipSuffix?_eq_some_iff, CharPred.isLongestRevMatch_iff]
+
+theorem endsWith_bool_iff_get {p : Char → Bool} {s : Slice} :
+    s.endsWith p ↔ ∃ h, p ((s.endPos.prev h).get (by simp)) = true := by
+  simp [Pattern.Model.endsWith_iff, CharPred.revMatchesAt_iff]
+
+theorem endsWith_bool_eq_false_iff_get {p : Char → Bool} {s : Slice} :
+    s.endsWith p = false ↔ ∀ h, p ((s.endPos.prev h).get (by simp)) = false := by
+  simp [Pattern.Model.endsWith_eq_false_iff, CharPred.revMatchesAt_iff]
+
+theorem endsWith_bool_eq_getLast? {p : Char → Bool} {s : Slice} :
+    s.endsWith p = s.copy.toList.getLast?.any p := by
+  rw [Bool.eq_iff_iff, Pattern.Model.endsWith_iff, CharPred.revMatchesAt_iff]
+  by_cases h : s.endPos = s.startPos
+  · refine ⟨fun ⟨h', _⟩ => by simp_all, ?_⟩
+    have : s.copy = "" := by simp_all [Slice.startPos_eq_endPos_iff.mp h.symm]
+    simp [this]
+  · obtain ⟨t, ht⟩ := s.splits_endPos.exists_eq_append_singleton_of_ne_startPos h
+    simp [h, ht]
+
+theorem eq_append_of_dropSuffix?_bool_eq_some {p : Char → Bool} {s res : Slice} (h : s.dropSuffix? p = some res) :
+    ∃ c, s.copy = res.copy ++ singleton c ∧ p c = true := by
+  obtain ⟨_, ⟨c, ⟨rfl, h₁⟩⟩, h₂⟩ := by simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropSuffix?_eq_some h
+  exact ⟨_, h₂, h₁⟩
+
+theorem skipSuffix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? P = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ P ((s.endPos.prev h).get (by simp)) := by
+  simp [skipSuffix?_prop_eq_skipSuffix?_decide, skipSuffix?_bool_eq_some_iff]
+
+theorem endsWith_prop_iff_get {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.endsWith P ↔ ∃ h, P ((s.endPos.prev h).get (by simp)) := by
+  simp [endsWith_prop_eq_endsWith_decide, endsWith_bool_iff_get]
+
+theorem endsWith_prop_eq_false_iff_get {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.endsWith P = false ↔ ∀ h, ¬ P ((s.endPos.prev h).get (by simp)) := by
+  simp [endsWith_prop_eq_endsWith_decide, endsWith_bool_eq_false_iff_get]
+
+theorem endsWith_prop_eq_getLast? {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.endsWith P = s.copy.toList.getLast?.any (decide <| P ·) := by
+  simp [endsWith_prop_eq_endsWith_decide, endsWith_bool_eq_getLast?]
+
+theorem eq_append_of_dropSuffix?_prop_eq_some {P : Char → Prop} [DecidablePred P] {s res : Slice} (h : s.dropSuffix? P = some res) :
+    ∃ c, s.copy = res.copy ++ singleton c ∧ P c := by
+  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} :
@@ -78,21 +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,
@@ -100,19 +342,198 @@ 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
+  simp [skipSuffix?_eq_skipSuffix?_toSlice, Slice.skipSuffix?_bool_eq_some_iff, ← Pos.toSlice_inj,
+    Pos.prev_toSlice]
+
+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_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_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_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
+  rw [dropSuffix?_eq_dropSuffix?_toSlice] at h
+  simpa using Slice.eq_append_of_dropSuffix?_bool_eq_some h
+
+theorem skipSuffix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
+    s.skipSuffix? P = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ P ((s.endPos.prev h).get (by simp)) := by
+  simp [skipSuffix?_eq_skipSuffix?_toSlice, Slice.skipSuffix?_prop_eq_some_iff, ← Pos.toSlice_inj,
+    Pos.prev_toSlice]
+
+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_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_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_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 [ForwardPatternModel.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 +100,76 @@ 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]
+
+@[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
+  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
+  rw [endsWith_eq_backwardPatternEndsWith, BackwardSliceSearcher.endsWith_of_isEmpty hpat]
+
+@[simp]
+theorem endsWith_slice_iff {pat s : Slice} :
+    s.endsWith pat ↔ pat.copy.toList <:+ s.copy.toList := by
+  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} :
+    s.endsWith pat = false ↔ ¬ (pat.copy.toList <:+ s.copy.toList) := by
+  simp [← Bool.not_eq_true, endsWith_slice_iff]
+
+theorem dropSuffix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    s.dropSuffix? pat = some s := by
+  simp [dropSuffix?_eq_map_skipSuffix?, skipSuffix?_slice_of_isEmpty hpat]
+
+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
+  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} :
+    s.skipSuffix? pat = some pos ↔ ∃ t, pos.Splits t pat := by
+  simp [skipSuffix?_string_eq_skipSuffix?_toSlice]
+
+@[simp]
+theorem skipSuffix?_string_empty {s : Slice} : s.skipSuffix? "" = some s.endPos := by
+  simp
+
+@[simp]
+theorem endsWith_string_iff {pat : String} {s : Slice} :
+    s.endsWith pat ↔ pat.toList <:+ s.copy.toList := by
+  simp [endsWith_string_eq_endsWith_toSlice]
+
+@[simp]
+theorem endsWith_string_empty {s : Slice} : s.endsWith "" = true := by
+  simp
+
+@[simp]
+theorem endsWith_string_eq_false_iff {pat : String} {s : Slice} :
+    s.endsWith pat = false ↔ ¬ (pat.toList <:+ s.copy.toList) := by
+  simp [endsWith_string_eq_endsWith_toSlice]
+
+@[simp]
+theorem dropSuffix?_string_empty {s : Slice} : s.dropSuffix? "" = some s := by
+  simpa [dropSuffix?_string_eq_dropSuffix?_toSlice] using dropSuffix?_slice_of_isEmpty (by simp)
+
+theorem eq_append_of_dropSuffix?_string_eq_some {pat : String} {s res : Slice} (h : s.dropSuffix? pat = some res) :
+    s.copy = res.copy ++ pat := by
+  rw [dropSuffix?_string_eq_dropSuffix?_toSlice] at h
+  simpa using eq_append_of_dropSuffix?_slice_eq_some h
+
 end Slice
 
 theorem skipPrefix?_slice_of_isEmpty {pat : Slice} {s : String} (hpat : pat.isEmpty = true) :
@@ -127,12 +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
@@ -158,21 +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/Search.lean

@@ -8,6 +8,8 @@ module
 prelude
 public import Init.Data.String.Search
 import all Init.Data.String.Search
+import Init.Data.String.Lemmas.Slice
+import Init.Data.String.Lemmas.FindPos
 
 public section
 
@@ -28,4 +30,42 @@ theorem Pos.le_find {s : String} (pos : s.Pos) (pattern : ρ) [ToForwardSearcher
     pos ≤ pos.find pattern := by
   simp [Pos.find, ← toSlice_le]
 
+@[simp]
+theorem front?_toSlice {s : String} : s.toSlice.front? = s.front? :=
+  (rfl)
+
+theorem front?_eq_get? {s : String} : s.front? = s.startPos.get? := by
+  simp [← front?_toSlice, ← Pos.get?_toSlice, Slice.front?_eq_get?]
+
+theorem front?_eq {s : String} : s.front? = s.toList.head? := by
+  simp [← front?_toSlice, Slice.front?_eq]
+
+@[simp]
+theorem front_toSlice {s : String} : s.toSlice.front = s.front :=
+  (rfl)
+
+@[simp]
+theorem front_eq {s : String} : s.front = s.front?.getD default := by
+  simp [← front_toSlice, Slice.front_eq]
+
+@[simp]
+theorem back?_toSlice {s : String} : s.toSlice.back? = s.back? :=
+  (rfl)
+
+theorem back?_eq_get? {s : String} : s.back? = s.endPos.prev?.bind Pos.get? := by
+  simp only [← back?_toSlice, Slice.back?_eq_get?, endPos_toSlice, Slice.Pos.prev?_eq_dif,
+    startPos_toSlice, Pos.toSlice_inj, Pos.prev?_eq_dif]
+  split <;> simp [← Pos.get?_toSlice, Pos.toSlice_prev]
+
+theorem back?_eq {s : String} : s.back? = s.toList.getLast? := by
+  simp [← back?_toSlice, Slice.back?_eq]
+
+@[simp]
+theorem back_toSlice {s : String} : s.toSlice.back = s.back :=
+  (rfl)
+
+@[simp]
+theorem back_eq {s : String} : s.back = s.back?.getD default := by
+  simp [← back_toSlice, Slice.back_eq]
+
 end String

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

@@ -11,6 +11,8 @@ import all Init.Data.String.Slice
 import Init.Data.String.Lemmas.Pattern.Memcmp
 import Init.Data.String.Lemmas.Basic
 import Init.Data.ByteArray.Lemmas
+import Init.Data.String.Lemmas.IsEmpty
+import Init.Data.String.Lemmas.FindPos
 
 public section
 
@@ -33,9 +35,104 @@ theorem beq_eq_true_iff {s t : Slice} : s == t ↔ s.copy = t.copy := by
 theorem beq_eq_false_iff {s t : Slice} : (s == t) = false ↔ s.copy ≠ t.copy := by
   simp [← Bool.not_eq_true]
 
-theorem beq_eq_decide {s t : Slice} : (s == t) = decide (s.copy = t.copy) := by
-  cases h : s == t <;> simp_all
+theorem beq_eq_decide {s t : Slice} : (s == t) = decide (s.copy = t.copy) :=
+  Bool.eq_iff_iff.2 (by simp)
+
+instance : EquivBEq String.Slice :=
+  equivBEq_of_iff_apply_eq copy (by simp)
+
+theorem beq_list_iff {l l' : List String.Slice} : l == l' ↔ l.map copy = l'.map copy := by
+  induction l generalizing l' <;> cases l' <;> simp_all
+
+theorem beq_list_eq_false_iff {l l' : List String.Slice} :
+    (l == l') = false ↔ l.map copy ≠ l'.map copy := by
+  simp [← Bool.not_eq_true, beq_list_iff]
+
+theorem beq_list_eq_decide {l l' : List String.Slice} :
+    (l == l') = decide (l.map copy = l'.map copy) :=
+  Bool.eq_iff_iff.2 (by simp [beq_list_iff])
 
 end BEq
 
-end String.Slice
+namespace Pos
+
+theorem get?_eq_dif {s : Slice} {p : s.Pos} : p.get? = if h : p = s.endPos then none else some (p.get h) :=
+  (rfl)
+
+theorem get?_eq_some_get {s : Slice} {p : s.Pos} (h : p ≠ s.endPos) : p.get? = some (p.get h) := by
+  simp [Pos.get?, h]
+
+@[simp]
+theorem get?_eq_none_iff {s : Slice} {p : s.Pos} : p.get? = none ↔ p = s.endPos := by
+  simp [Pos.get?]
+
+theorem get?_eq_none {s : Slice} {p : s.Pos} (h : p = s.endPos) : p.get? = none :=
+  get?_eq_none_iff.2 h
+
+@[simp]
+theorem get?_endPos {s : Slice} : s.endPos.get? = none := by
+  simp
+
+end Pos
+
+end Slice
+
+namespace Pos
+
+theorem get?_toSlice {s : String} {p : s.Pos} : p.toSlice.get? = p.get? :=
+  (rfl)
+
+theorem get?_eq_dif {s : String} {p : s.Pos} : p.get? = if h : p = s.endPos then none else some (p.get h) := by
+  simp [← get?_toSlice, Slice.Pos.get?_eq_dif]
+
+theorem get?_eq_some_get {s : String} {p : s.Pos} (h : p ≠ s.endPos) : p.get? = some (p.get h) := by
+  simpa [← get?_toSlice] using Slice.Pos.get?_eq_some_get (by simpa)
+
+@[simp]
+theorem get?_eq_none_iff {s : String} {p : s.Pos} : p.get? = none ↔ p = s.endPos := by
+  simp [← get?_toSlice]
+
+theorem get?_eq_none {s : String} {p : s.Pos} (h : p = s.endPos) : p.get? = none :=
+  get?_eq_none_iff.2 h
+
+@[simp]
+theorem get?_endPos {s : String} : s.endPos.get? = none := by
+  simp
+
+end Pos
+
+namespace Slice
+
+theorem front?_eq_get? {s : Slice} : s.front? = s.startPos.get? :=
+  (rfl)
+
+theorem front?_eq {s : Slice} : s.front? = s.copy.toList.head? := by
+  simp only [front?_eq_get?, Pos.get?_eq_dif]
+  split
+  · simp_all [startPos_eq_endPos_iff, eq_comm (a := none)]
+  · rename_i h
+    obtain ⟨t, ht⟩ := s.splits_startPos.exists_eq_singleton_append h
+    simp [ht]
+
+@[simp]
+theorem front_eq {s : Slice} : s.front = s.front?.getD default := by
+  simp [front]
+
+theorem back?_eq_get? {s : Slice} : s.back? = s.endPos.prev?.bind Pos.get? :=
+  (rfl)
+
+theorem back?_eq {s : Slice} : s.back? = s.copy.toList.getLast? := by
+  simp [back?_eq_get?, Pos.prev?_eq_dif]
+  split
+  · simp_all [startPos_eq_endPos_iff, eq_comm (a := s.endPos), eq_comm (a := none)]
+  · rename_i h
+    obtain ⟨t, ht⟩ := s.splits_endPos.exists_eq_append_singleton_of_ne_startPos h
+    simp [ht, Pos.get?_eq_some_get]
+
+@[simp]
+theorem back_eq {s : Slice} : s.back = s.back?.getD default := by
+  simp [back]
+
+end Slice
+
+end String

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

@@ -17,6 +17,8 @@ import Init.Data.String.OrderInstances
 import Init.Data.Nat.Order
 import Init.Omega
 import Init.Data.String.Lemmas.FindPos
+import Init.Data.List.TakeDrop
+import Init.Data.List.Nat.TakeDrop
 
 /-!
 # `Splits` predicates on `String.Pos` and `String.Slice.Pos`.
@@ -97,6 +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
@@ -365,7 +372,7 @@ theorem Slice.Pos.Splits.of_prev {s : Slice} {p : s.Pos} {hp}
   obtain ⟨rfl, rfl, rfl⟩ := by simpa using h.eq (splits_prev p hp)
   exact splits_prev_right p hp
 
-theorem Slice.sliceTo_copy_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ : String} :
+theorem Slice.copy_sliceTo_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ : String} :
     (s.sliceTo p).copy = t₁ ↔ ∃ t₂, p.Splits t₁ t₂ := by
   refine ⟨?_, ?_⟩
   · rintro rfl
@@ -373,13 +380,37 @@ theorem Slice.sliceTo_copy_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ :
   · rintro ⟨t₂, h⟩
     exact p.splits.eq_left h
 
-theorem sliceTo_copy_eq_iff_exists_splits {s : String} {p : s.Pos} {t₁ : String} :
-    (s.sliceTo p).copy = t₁ ↔ ∃ t₂, p.Splits t₁ t₂ := by
+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 ⟨?_, ?_⟩
   · rintro rfl
     exact ⟨_, p.splits⟩
   · rintro ⟨t₂, h⟩
-    exact p.splits.eq_left 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
@@ -425,8 +456,7 @@ theorem Slice.splits_singleton_iff {s : Slice} {p : s.Pos} {c : Char} {t : Strin
       simp [startPos_ne_endPos_iff, ← copy_ne_empty_iff, h.eq_append]
     have spl : (s.startPos.next this).Splits (singleton c) t := by
       rw [← empty_append (s := singleton c)]
-      apply Pos.Splits.next
-      simp [h.eq_append]
+      exact Pos.Splits.next (by simp [h.eq_append])
     refine ⟨this, ⟨h.pos_eq spl, ?_, h.eq_append⟩⟩
     rw [← empty_append (s := singleton c)] at spl
     exact spl.get_eq_of_singleton
@@ -440,6 +470,27 @@ theorem splits_singleton_iff {s : String} {p : s.Pos} {c : Char} {t : String} :
   rw [← Pos.splits_toSlice_iff, Slice.splits_singleton_iff]
   simp [← Pos.ofToSlice_inj]
 
+theorem Slice.splits_singleton_right_iff {s : Slice} {p : s.Pos} {c : Char} {t : String} :
+    p.Splits t (singleton c) ↔
+      ∃ h, p = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c ∧ s.copy = t ++ singleton c := by
+  refine ⟨fun h => ?_, ?_⟩
+  · have : s.endPos ≠ s.startPos := by
+      simp [ne_comm (a := s.endPos), startPos_ne_endPos_iff, ← copy_ne_empty_iff, h.eq_append]
+    have spl : (s.endPos.prev this).Splits t (singleton c) := by
+      rw [← append_empty (s := singleton c)]
+      exact Pos.Splits.prev (by simp [h.eq_append])
+    refine ⟨this, ⟨h.pos_eq spl, ?_, h.eq_append⟩⟩
+    exact (h.eq_append ▸ Pos.next_prev (h := this) ▸ s.splits_endPos).get_eq_of_singleton
+  · rintro ⟨h, ⟨rfl, rfl, h'⟩⟩
+    rw [← String.append_empty (s := singleton _)]
+    exact Pos.Splits.prev (by simp [h'])
+
+theorem splits_singleton_right_iff {s : String} {p : s.Pos} {c : Char} {t : String} :
+    p.Splits t (singleton c) ↔
+      ∃ h, p = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c ∧ s = t ++ singleton c := by
+  rw [← Pos.splits_toSlice_iff, Slice.splits_singleton_right_iff]
+  simp [← Pos.ofToSlice_inj, Pos.prev_toSlice]
+
 theorem Slice.splits_next_startPos {s : Slice} {h : s.startPos ≠ s.endPos} :
     (s.startPos.next h).Splits
       (singleton (s.startPos.get h)) (s.sliceFrom (s.startPos.next h)).copy := by
@@ -454,6 +505,20 @@ theorem splits_next_startPos {s : String} {h : s.startPos ≠ s.endPos} :
   rw [← Pos.splits_toSlice_iff]
   apply (Slice.splits_next_startPos).of_eq <;> simp [String.Pos.next_toSlice]
 
+theorem Slice.splits_prev_endPos {s : Slice} {h : s.endPos ≠ s.startPos} :
+    (s.endPos.prev h).Splits
+      (s.sliceTo (s.endPos.prev h)).copy (singleton ((s.endPos.prev h).get (by simp))) := by
+  rw [← String.append_empty (s := singleton _)]
+  apply Slice.Pos.Splits.prev
+  have := Slice.Pos.splits_prev_right s.endPos h
+  rwa [copy_sliceFrom_endPos] at this
+
+theorem splits_prev_endPos {s : String} {h : s.endPos ≠ s.startPos} :
+    (s.endPos.prev h).Splits
+      (s.sliceTo (s.endPos.prev h)).copy (singleton ((s.endPos.prev h).get (by simp))) := by
+  rw [← Pos.splits_toSlice_iff]
+  apply (Slice.splits_prev_endPos).of_eq <;> simp [String.Pos.prev_toSlice, h]
+
 theorem Slice.Pos.Splits.toByteArray_eq_left {s : Slice} {p : s.Pos} {t₁ t₂ : String} (h : p.Splits t₁ t₂) :
     t₁.toByteArray = s.copy.toByteArray.extract 0 p.offset.byteIdx := by
   rw [h.eq_left p.splits]
@@ -597,6 +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
@@ -649,4 +736,91 @@ theorem Slice.splits_slice {s : Slice} {p₀ p₁ : s.Pos} (h) (p : (s.slice p
     p.Splits (s.slice p₀ (Pos.ofSlice p) Pos.le_ofSlice).copy (s.slice (Pos.ofSlice p) p₁ Pos.ofSlice_le).copy := by
   simpa using p.splits
 
+theorem Slice.Pos.Splits.nextn {s : Slice} {t₁ t₂ : String} {p : s.Pos} (h : p.Splits t₁ t₂) (n : Nat) :
+    (p.nextn n).Splits (t₁ ++ String.ofList (t₂.toList.take n)) (String.ofList (t₂.toList.drop n)) := by
+  induction n generalizing p t₁ t₂ with
+  | zero => simpa
+  | succ n ih =>
+    rw [Pos.nextn_add_one]
+    split
+    · simp_all
+    · obtain ⟨t₂, rfl⟩ := h.exists_eq_singleton_append ‹_›
+      simpa [← append_assoc] using ih h.next
+
+theorem Slice.splits_nextn_startPos (s : Slice) (n : Nat) :
+    (s.startPos.nextn n).Splits (String.ofList (s.copy.toList.take n)) (String.ofList (s.copy.toList.drop n)) := by
+  simpa using s.splits_startPos.nextn n
+
+theorem Pos.Splits.nextn {s t₁ t₂ : String} {p : s.Pos} (h : p.Splits t₁ t₂) (i : Nat) :
+    (p.nextn i).Splits (t₁ ++ String.ofList (t₂.toList.take i)) (String.ofList (t₂.toList.drop i)) := by
+  simpa [← splits_toSlice_iff, toSlice_nextn] using h.toSlice.nextn i
+
+theorem splits_nextn_startPos (s : String) (n : Nat) :
+    (s.startPos.nextn n).Splits (String.ofList (s.toList.take n)) (String.ofList (s.toList.drop n)) := by
+  simpa using s.splits_startPos.nextn n
+
+theorem Slice.Pos.Splits.prevn {s : Slice} {t₁ t₂ : String} {p : s.Pos} (h : p.Splits t₁ t₂) (n : Nat) :
+    (p.prevn n).Splits (String.ofList (t₁.toList.take (t₁.length - n))) (String.ofList (t₁.toList.drop (t₁.length - n)) ++ t₂) := by
+  induction n generalizing p t₁ t₂ with
+  | zero => simpa [← String.length_toList]
+  | succ n ih =>
+    rw [Pos.prevn_add_one]
+    split
+    · simp_all
+    · obtain ⟨t₂, rfl⟩ := h.exists_eq_append_singleton_of_ne_startPos ‹_›
+      simpa [Nat.add_sub_add_right, List.take_append, List.drop_append, ← append_assoc] using ih h.prev
+
+theorem Slice.splits_prevn_endPos (s : Slice) (n : Nat) :
+    (s.endPos.prevn n).Splits (String.ofList (s.copy.toList.take (s.copy.length - n)))
+      (String.ofList (s.copy.toList.drop (s.copy.length - n))) := by
+  simpa using s.splits_endPos.prevn n
+
+theorem Pos.Splits.prevn {s t₁ t₂ : String} {p : s.Pos} (h : p.Splits t₁ t₂) (n : Nat) :
+    (p.prevn n).Splits (String.ofList (t₁.toList.take (t₁.length - n))) (String.ofList (t₁.toList.drop (t₁.length - n)) ++ t₂) := by
+  simpa [← splits_toSlice_iff, toSlice_prevn] using h.toSlice.prevn n
+
+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/Lemmas/TakeDrop.lean

@@ -0,0 +1,86 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Julia Markus Himmel
+-/
+module
+
+prelude
+public import Init.Data.String.TakeDrop
+import all Init.Data.String.Slice
+import all Init.Data.String.TakeDrop
+import Init.Data.String.Lemmas.Splits
+
+public section
+
+namespace String
+
+namespace Slice
+
+theorem drop_eq_sliceFrom {s : Slice} {n : Nat} : s.drop n = s.sliceFrom (s.startPos.nextn n) :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_drop {s : Slice} {n : Nat} : (s.drop n).copy.toList = s.copy.toList.drop n := by
+  simp [drop_eq_sliceFrom, (s.splits_nextn_startPos n).copy_sliceFrom_eq]
+
+theorem dropEnd_eq_sliceTo {s : Slice} {n : Nat} : s.dropEnd n = s.sliceTo (s.endPos.prevn n) :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_dropEnd {s : Slice} {n : Nat} :
+    (s.dropEnd n).copy.toList = s.copy.toList.take (s.copy.length - n) := by
+  simp [dropEnd_eq_sliceTo, (s.splits_prevn_endPos n).copy_sliceTo_eq]
+
+theorem take_eq_sliceTo {s : Slice} {n : Nat} : s.take n = s.sliceTo (s.startPos.nextn n) :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_take {s : Slice} {n : Nat} : (s.take n).copy.toList = s.copy.toList.take n := by
+  simp [take_eq_sliceTo, (s.splits_nextn_startPos n).copy_sliceTo_eq]
+
+theorem takeEnd_eq_sliceFrom {s : Slice} {n : Nat} : s.takeEnd n = s.sliceFrom (s.endPos.prevn n) :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_takeEnd {s : Slice} {n : Nat} :
+    (s.takeEnd n).copy.toList = s.copy.toList.drop (s.copy.length - n) := by
+  simp [takeEnd_eq_sliceFrom, (s.splits_prevn_endPos n).copy_sliceFrom_eq]
+
+end Slice
+
+@[simp]
+theorem drop_toSlice {s : String} {n : Nat} : s.toSlice.drop n = s.drop n :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_drop {s : String} {n : Nat} : (s.drop n).copy.toList = s.toList.drop n := by
+  simp [← drop_toSlice]
+
+@[simp]
+theorem dropEnd_toSlice {s : String} {n : Nat} : s.toSlice.dropEnd n = s.dropEnd n :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_dropEnd {s : String} {n : Nat} :
+    (s.dropEnd n).copy.toList = s.toList.take (s.length - n) := by
+  simp [← dropEnd_toSlice]
+
+@[simp]
+theorem take_toSlice {s : String} {n : Nat} : s.toSlice.take n = s.take n :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_take {s : String} {n : Nat} : (s.take n).copy.toList = s.toList.take n := by
+  simp [← take_toSlice]
+
+@[simp]
+theorem takeEnd_toSlice {s : String} {n : Nat} : s.toSlice.takeEnd n = s.takeEnd n :=
+  (rfl)
+
+@[simp]
+theorem toList_copy_takeEnd {s : String} {n : Nat} :
+    (s.takeEnd n).copy.toList = s.toList.drop (s.length - n) := by
+  simp [← takeEnd_toSlice]
+
+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
 
 /--
@@ -117,7 +155,7 @@ class ForwardPattern {ρ : Type} (pat : ρ) where
   -/
   startsWith : (s : Slice) → Bool := fun s => (skipPrefix? s).isSome
 
-@[deprecated ForwardPattern.dropPrefix? (since := "2026-03-19")]
+@[deprecated ForwardPattern.skipPrefix? (since := "2026-03-19")]
 def ForwardPattern.dropPrefix? {ρ : Type} (pat : ρ) [ForwardPattern pat] (s : Slice) : Option s.Pos :=
   ForwardPattern.skipPrefix? pat s
 

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

@@ -47,8 +47,8 @@ instance {c : Char} : LawfulBackwardPattern c where
   skipSuffixOfNonempty?_eq h := LawfulBackwardPattern.skipSuffixOfNonempty?_eq (pat := (· == c)) h
   endsWith_eq s := LawfulBackwardPattern.endsWith_eq (pat := (· == c)) s
 
-instance {c : Char} : ToBackwardSearcher c (ToBackwardSearcher.DefaultBackwardSearcher c) :=
-  .defaultImplementation
+instance {c : Char} : ToBackwardSearcher c (ToBackwardSearcher.DefaultBackwardSearcher (· == c)) where
+  toSearcher s := ToBackwardSearcher.toSearcher (· == c) s
 
 end Char
 

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

@@ -139,8 +139,9 @@ instance {p : Char → Prop} [DecidablePred p] : LawfulBackwardPattern p where
   skipSuffixOfNonempty?_eq h := LawfulBackwardPattern.skipSuffixOfNonempty?_eq (pat := (decide <| p ·)) h
   endsWith_eq s := LawfulBackwardPattern.endsWith_eq (pat := (decide <| p ·)) s
 
-instance {p : Char → Prop} [DecidablePred p] : ToBackwardSearcher p (ToBackwardSearcher.DefaultBackwardSearcher p) :=
-  .defaultImplementation
+instance {p : Char → Prop} [DecidablePred p] :
+    ToBackwardSearcher p (ToBackwardSearcher.DefaultBackwardSearcher (decide <| p ·)) where
+  toSearcher s := ToBackwardSearcher.toSearcher (decide <| p ·) s
 
 end Decidable
 

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

@@ -11,7 +11,7 @@ public import Init.Data.Ord.Basic
 public import Init.Data.Iterators.Combinators.FilterMap
 public import Init.Data.String.ToSlice
 public import Init.Data.String.Subslice
-public import Init.Data.String.Iter
+public import Init.Data.String.Iter.Basic
 public import Init.Data.String.Iterate
 import Init.Data.Iterators.Consumers.Collect
 import Init.Data.Iterators.Consumers.Loop
@@ -84,10 +84,11 @@ instance : ToString String.Slice where
 theorem toStringToString_eq : ToString.toString = String.Slice.copy := (rfl)
 
 @[extern "lean_slice_hash"]
-opaque hash (s : @& Slice) : UInt64
+protected def hash (s : @& Slice) : UInt64 :=
+  String.hash s.copy
 
 instance : Hashable Slice where
-  hash := hash
+  hash := Slice.hash
 
 instance : LT Slice where
   lt x y := x.copy < y.copy
@@ -425,13 +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
 
 /--
@@ -571,7 +572,7 @@ Examples:
 -/
 @[inline]
 def all (s : Slice) (pat : ρ) [ForwardPattern pat] : Bool :=
-  s.dropWhile pat |>.isEmpty
+  s.skipPrefixWhile pat == s.endPos
 
 end ForwardPatternUsers
 
@@ -705,14 +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.
 
@@ -764,23 +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).
@@ -1151,6 +1182,19 @@ where go (acc : String) (s : Slice) : List Slice → String
   | a :: as => go (acc ++ s ++ a) s as
   | []      => acc
 
+/--
+Appends all the slices in a list of slices, in order.
+
+Use {name}`String.Slice.intercalate` to place a separator string between the strings in a list.
+
+Examples:
+ * {lean}`String.Slice.join ["gr", "ee", "n"] = "green"`
+ * {lean}`String.Slice.join ["b", "", "l", "", "ue"] = "blue"`
+ * {lean}`String.Slice.join [] = ""`
+-/
+def join (l : List String.Slice) : String :=
+  l.foldl (fun (r : String) (s : String.Slice) => r ++ s) ""
+
 /--
 Converts a string to the Lean compiler's representation of names. The resulting name is
 hierarchical, and the string is split at the dots ({lean}`'.'`).

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/Interactive.lean

@@ -107,6 +107,9 @@ syntax (name := showLocalThms) "show_local_thms" : grind
 -/
 syntax (name := showTerm) "show_term " grindSeq : grind
 
+/-- Shows the pending goals. -/
+syntax (name := showGoals) "show_goals" : grind
+
 declare_syntax_cat grind_ref (behavior := both)
 
 syntax:max anchor : grind_ref
@@ -315,5 +318,8 @@ Only available in `sym =>` mode.
 -/
 syntax (name := symSimp) "simp" (ppSpace colGt ident)? (" [" ident,* "]")? : grind
 
+/-- `exact e` closes the main goal if its target type matches that of `e`. -/
+macro "exact " e:term : grind => `(grind| tactic => exact $e:term)
+
 end Grind
 end Lean.Parser.Tactic

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/Util.lean

@@ -30,7 +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.
+-/
+@[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`
@@ -185,15 +185,36 @@ 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` and normalizes it to
-"instance normal form": the result is a constructor application whose sub-instance fields
-are canonical instances and whose types match `α` exactly. 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. See `Lean.Meta.InstanceNormalForm`
-for details. Example:
+/--
+`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:
 ```
-#check inferInstanceAs (Inhabited Nat) -- Inhabited Nat
+def D := Nat
+instance : Inhabited D := inferInstanceAs (Inhabited Nat)
 ```
+
+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
 
@@ -3261,7 +3282,7 @@ Version of `Array.get!Internal` that does not increment the reference count of i
 This is only intended for direct use by the compiler.
 -/
 @[extern "lean_array_get_borrowed"]
-unsafe opaque Array.get!InternalBorrowed {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α
+unsafe opaque Array.get!InternalBorrowed {α : Type u} [@&Inhabited α] (a : @& Array α) (i : @& Nat) : α
 
 /--
 Use the indexing notation `a[i]!` instead.
@@ -3269,7 +3290,7 @@ Use the indexing notation `a[i]!` instead.
 Access an element from an array, or panic if the index is out of bounds.
 -/
 @[extern "lean_array_get"]
-def Array.get!Internal {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
+def Array.get!Internal {α : Type u} [@&Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
   Array.getD a i default
 
 /--
@@ -3648,8 +3669,8 @@ will prevent the actual monad from being "copied" to the code being specialized.
 When we reimplement the specializer, we may consider copying `inst` if it also
 occurs outside binders or if it is an instance.
 -/
-@[never_extract, extern "lean_panic_fn"]
-def panicCore {α : Sort u} [Inhabited α] (msg : String) : α := default
+@[never_extract, extern "lean_panic_fn_borrowed"]
+def panicCore {α : Sort u} [@&Inhabited α] (msg : String) : α := default
 
 /--
 `(panic "msg" : α)` has a built-in implementation which prints `msg` to
@@ -3667,7 +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`.
@@ -3733,7 +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.
   -/
@@ -4082,7 +4103,7 @@ Actions in the resulting monad are functions that take the local value as a para
 ordinary actions in `m`.
 -/
 def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
-  ρ → m α
+  (a : @&ρ) → m α
 
 /--
 Interpret `ρ → m α` as an element of `ReaderT ρ m α`.

src/Init/Sym/Simp/SimprocDSL.lean

@@ -49,6 +49,14 @@ syntax (name := ground) "ground" : sym_simproc
 /-- Simplify telescope binders but not the final body. -/
 syntax (name := telescope) "telescope" : sym_simproc
 
+/-- Simplify control-flow expressions (`if-then-else`, `match`, `cond`, `dite`).
+Visits only conditions and discriminants. Intended as a `pre` simproc. -/
+syntax (name := control) "control" : sym_simproc
+
+/-- Simplify arrow telescopes (`p₁ → p₂ → ... → q`) without entering binders.
+Simplifies each `pᵢ` and `q` individually. Intended as a `pre` simproc. -/
+syntax (name := arrowTelescope) "arrow_telescope" : sym_simproc
+
 /-- Rewrite using a named theorem set. Optionally specify a discharger for conditional rewrites. -/
 syntax (name := rewriteSet) "rewrite" ident (" with " sym_discharger)? : sym_simproc
 

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/Init/Tactics.lean

@@ -2259,42 +2259,6 @@ with grind
 ```
 This is more convenient than the equivalent `· by rename_i _ acc _; exact I1 acc`.
 
-### Witnesses
-
-When a specification has a parameter whose type is tagged with `@[mvcgen_witness_type]`, `mvcgen`
-classifies the corresponding goal as a *witness* rather than a verification condition.
-Witnesses are concrete values that the user must provide (inspired by zero-knowledge proofs),
-as opposed to invariants (predicates maintained across loop iterations) or verification conditions
-(propositions to prove).
-
-Witness goals are labelled `witness1`, `witness2`, etc. and can be provided in a `witnesses` section
-that appears before the `invariants` section:
-```
-mvcgen [...] witnesses
-· W1
-· W2
-invariants
-· I1
-with grind
-```
-Like invariants, witnesses support case label syntax:
-```
-mvcgen [...] witnesses
-| witness1 => W1
-```
-
-See the `@[mvcgen_witness_type]` attribute for how to register custom witness types.
-
-### Invariant and witness type attributes
-
-The `@[mvcgen_invariant_type]` and `@[mvcgen_witness_type]` tag attributes control how `mvcgen`
-classifies subgoals:
-* A goal whose type is an application of a type tagged with `@[mvcgen_invariant_type]` is classified
-  as an invariant (`inv<n>`).
-* A goal whose type is an application of a type tagged with `@[mvcgen_witness_type]` is classified
-  as a witness (`witness<n>`).
-* All other goals are classified as verification conditions (`vc<n>`).
-
 ### Invariant suggestions
 
 `mvcgen` will suggest invariants for you if you use the `invariants?` keyword.

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