mirror of
https://github.com/leanprover/lean4.git
synced 2026-03-21 20:34:07 +00:00
Compare commits
3 Commits
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grind_spli
| Author | SHA1 | Date | |
|---|---|---|---|
|
|
762eca7832 | ||
|
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51f34c8425 | ||
|
|
960ed43dae |
17
.github/workflows/build-template.yml
vendored
17
.github/workflows/build-template.yml
vendored
@@ -82,7 +82,7 @@ jobs:
|
||||
- name: CI Merge Checkout
|
||||
run: |
|
||||
git fetch --depth=1 origin ${{ github.sha }}
|
||||
git checkout FETCH_HEAD flake.nix flake.lock script/prepare-* tests/lean/run/importStructure.lean
|
||||
git checkout FETCH_HEAD flake.nix flake.lock script/prepare-*
|
||||
if: github.event_name == 'pull_request'
|
||||
# (needs to be after "Checkout" so files don't get overridden)
|
||||
- name: Setup emsdk
|
||||
@@ -127,12 +127,9 @@ jobs:
|
||||
[ -d build ] || mkdir build
|
||||
cd build
|
||||
# arguments passed to `cmake`
|
||||
OPTIONS=(-DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true)
|
||||
if [[ -n '${{ 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
|
||||
OPTIONS+=(-DCHECK_OLEAN_VERSION=ON)
|
||||
fi
|
||||
# this also enables githash embedding into stage 1 library
|
||||
OPTIONS=(-DCHECK_OLEAN_VERSION=ON)
|
||||
OPTIONS+=(-DLEAN_EXTRA_MAKE_OPTS=-DwarningAsError=true)
|
||||
if [[ -n '${{ matrix.cross_target }}' ]]; then
|
||||
# used by `prepare-llvm`
|
||||
export EXTRA_FLAGS=--target=${{ matrix.cross_target }}
|
||||
@@ -196,7 +193,7 @@ jobs:
|
||||
run: |
|
||||
ulimit -c unlimited # coredumps
|
||||
time ctest --preset ${{ matrix.CMAKE_PRESET || 'release' }} --test-dir build/stage1 -j$NPROC --output-junit test-results.xml ${{ matrix.CTEST_OPTIONS }}
|
||||
if: (matrix.wasm || !matrix.cross) && (inputs.check-level >= 1 || matrix.test)
|
||||
if: (matrix.wasm || !matrix.cross) && (inputs.check-level >= 1 || matrix.name == 'Linux release')
|
||||
- name: Test Summary
|
||||
uses: test-summary/action@v2
|
||||
with:
|
||||
@@ -213,7 +210,7 @@ jobs:
|
||||
- name: Check Stage 3
|
||||
run: |
|
||||
make -C build -j$NPROC check-stage3
|
||||
if: matrix.check-stage3
|
||||
if: matrix.test-speedcenter
|
||||
- name: Test Speedcenter Benchmarks
|
||||
run: |
|
||||
# Necessary for some timing metrics but does not work on Namespace runners
|
||||
@@ -227,7 +224,7 @@ jobs:
|
||||
run: |
|
||||
# clean rebuild in case of Makefile changes
|
||||
make -C build update-stage0 && rm -rf build/stage* && make -C build -j$NPROC
|
||||
if: matrix.check-rebootstrap
|
||||
if: matrix.name == 'Linux' && inputs.check-level >= 1
|
||||
- name: CCache stats
|
||||
if: always()
|
||||
run: ccache -s
|
||||
|
||||
22
.github/workflows/ci.yml
vendored
22
.github/workflows/ci.yml
vendored
@@ -166,7 +166,7 @@ jobs:
|
||||
"name": "Linux release",
|
||||
"os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
|
||||
"release": true,
|
||||
"check-level": 2,
|
||||
"check-level": 0,
|
||||
"shell": "nix develop .#oldGlibc -c bash -euxo pipefail {0}",
|
||||
"llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/19.1.2/lean-llvm-x86_64-linux-gnu.tar.zst",
|
||||
"prepare-llvm": "../script/prepare-llvm-linux.sh lean-llvm*",
|
||||
@@ -176,13 +176,21 @@ jobs:
|
||||
},
|
||||
{
|
||||
"name": "Linux Lake",
|
||||
"os": large ? "nscloud-ubuntu-22.04-amd64-8x8" : "ubuntu-latest",
|
||||
"os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
|
||||
"check-level": 0,
|
||||
"test": true,
|
||||
"check-rebootstrap": level >= 1,
|
||||
// just a secondary build job for now until false positives can be excluded
|
||||
"secondary": true,
|
||||
"CMAKE_OPTIONS": "-DUSE_LAKE=ON",
|
||||
// TODO: importStructure is not compatible with .olean caching
|
||||
// TODO: why does scopedMacros fail?
|
||||
"CTEST_OPTIONS": "-E 'scopedMacros|importStructure'"
|
||||
},
|
||||
{
|
||||
"name": "Linux",
|
||||
"os": large ? "nscloud-ubuntu-22.04-amd64-4x8" : "ubuntu-latest",
|
||||
"check-stage3": level >= 2,
|
||||
"test-speedcenter": level >= 2,
|
||||
"CMAKE_OPTIONS": "-DUSE_LAKE=ON",
|
||||
"check-level": 1,
|
||||
},
|
||||
{
|
||||
"name": "Linux Reldebug",
|
||||
@@ -356,7 +364,7 @@ jobs:
|
||||
with:
|
||||
path: artifacts
|
||||
- name: Release
|
||||
uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
|
||||
uses: softprops/action-gh-release@v2
|
||||
with:
|
||||
files: artifacts/*/*
|
||||
fail_on_unmatched_files: true
|
||||
@@ -400,7 +408,7 @@ jobs:
|
||||
echo -e "\n*Full commit log*\n" >> diff.md
|
||||
git log --oneline "$last_tag"..HEAD | sed 's/^/* /' >> diff.md
|
||||
- name: Release Nightly
|
||||
uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
|
||||
uses: softprops/action-gh-release@v2
|
||||
with:
|
||||
body_path: diff.md
|
||||
prerelease: true
|
||||
|
||||
64
.github/workflows/pr-release.yml
vendored
64
.github/workflows/pr-release.yml
vendored
@@ -48,30 +48,19 @@ jobs:
|
||||
git -C lean4.git remote add origin https://github.com/${{ github.repository_owner }}/lean4.git
|
||||
git -C lean4.git fetch -n origin master
|
||||
git -C lean4.git fetch -n origin "${{ steps.workflow-info.outputs.sourceHeadSha }}"
|
||||
|
||||
# Create both the original tag and the SHA-suffixed tag
|
||||
SHORT_SHA="${{ steps.workflow-info.outputs.sourceHeadSha }}"
|
||||
SHORT_SHA="${SHORT_SHA:0:7}"
|
||||
|
||||
# Export the short SHA for use in subsequent steps
|
||||
echo "SHORT_SHA=${SHORT_SHA}" >> "$GITHUB_ENV"
|
||||
|
||||
git -C lean4.git tag -f pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }} "${{ steps.workflow-info.outputs.sourceHeadSha }}"
|
||||
git -C lean4.git tag -f pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-"${SHORT_SHA}" "${{ steps.workflow-info.outputs.sourceHeadSha }}"
|
||||
|
||||
git -C lean4.git remote add pr-releases https://foo:'${{ secrets.PR_RELEASES_TOKEN }}'@github.com/${{ github.repository_owner }}/lean4-pr-releases.git
|
||||
git -C lean4.git push -f pr-releases pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}
|
||||
git -C lean4.git push -f pr-releases pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-"${SHORT_SHA}"
|
||||
- name: Delete existing release if present
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
run: |
|
||||
# Try to delete any existing release for the current PR (just the version without the SHA suffix).
|
||||
# Try to delete any existing release for the current PR.
|
||||
gh release delete --repo ${{ github.repository_owner }}/lean4-pr-releases pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }} -y || true
|
||||
env:
|
||||
GH_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}
|
||||
- name: Release (short format)
|
||||
- name: Release
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
|
||||
uses: softprops/action-gh-release@v2
|
||||
with:
|
||||
name: Release for PR ${{ steps.workflow-info.outputs.pullRequestNumber }}
|
||||
# There are coredumps files here as well, but all in deeper subdirectories.
|
||||
@@ -84,22 +73,7 @@ jobs:
|
||||
# The token used here must have `workflow` privileges.
|
||||
GITHUB_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}
|
||||
|
||||
- name: Release (SHA-suffixed format)
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: softprops/action-gh-release@da05d552573ad5aba039eaac05058a918a7bf631
|
||||
with:
|
||||
name: Release for PR ${{ steps.workflow-info.outputs.pullRequestNumber }} (${{ steps.workflow-info.outputs.sourceHeadSha }})
|
||||
# There are coredumps files here as well, but all in deeper subdirectories.
|
||||
files: artifacts/*/*
|
||||
fail_on_unmatched_files: true
|
||||
draft: false
|
||||
tag_name: pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}
|
||||
repository: ${{ github.repository_owner }}/lean4-pr-releases
|
||||
env:
|
||||
# The token used here must have `workflow` privileges.
|
||||
GITHUB_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}
|
||||
|
||||
- name: Report release status (short format)
|
||||
- name: Report release status
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: actions/github-script@v7
|
||||
with:
|
||||
@@ -113,20 +87,6 @@ jobs:
|
||||
description: "${{ github.repository_owner }}/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}",
|
||||
});
|
||||
|
||||
- name: Report release status (SHA-suffixed format)
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: actions/github-script@v7
|
||||
with:
|
||||
script: |
|
||||
await github.rest.repos.createCommitStatus({
|
||||
owner: context.repo.owner,
|
||||
repo: context.repo.repo,
|
||||
sha: "${{ steps.workflow-info.outputs.sourceHeadSha }}",
|
||||
state: "success",
|
||||
context: "PR toolchain (SHA-suffixed)",
|
||||
description: "${{ github.repository_owner }}/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}",
|
||||
});
|
||||
|
||||
- name: Add label
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: actions/github-script@v7
|
||||
@@ -322,18 +282,16 @@ jobs:
|
||||
if [ "$EXISTS" = "0" ]; then
|
||||
echo "Branch does not exist, creating it."
|
||||
git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}" > lean-toolchain
|
||||
git add lean-toolchain
|
||||
git commit -m "Update lean-toolchain for testing https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
else
|
||||
echo "Branch already exists, updating lean-toolchain."
|
||||
echo "Branch already exists, pushing an empty commit."
|
||||
git switch lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}
|
||||
# The Batteries `nightly-testing` or `nightly-testing-YYYY-MM-DD` branch may have moved since this branch was created, so merge their changes.
|
||||
# (This should no longer be possible once `nightly-testing-YYYY-MM-DD` is a tag, but it is still safe to merge.)
|
||||
git merge "$BASE" --strategy-option ours --no-commit --allow-unrelated-histories
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
|
||||
git add lean-toolchain
|
||||
git commit -m "Update lean-toolchain for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
git commit --allow-empty -m "Trigger CI for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
fi
|
||||
|
||||
- name: Push changes
|
||||
@@ -388,23 +346,21 @@ jobs:
|
||||
if [ "$EXISTS" = "0" ]; then
|
||||
echo "Branch does not exist, creating it."
|
||||
git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}" > lean-toolchain
|
||||
git add lean-toolchain
|
||||
sed -i 's,require "leanprover-community" / "batteries" @ git ".\+",require "leanprover-community" / "batteries" @ git "lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}",' lakefile.lean
|
||||
lake update batteries
|
||||
git add lakefile.lean lake-manifest.json
|
||||
git commit -m "Update lean-toolchain for testing https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
else
|
||||
echo "Branch already exists, updating lean-toolchain and bumping Batteries."
|
||||
echo "Branch already exists, merging $BASE and bumping Batteries."
|
||||
git switch lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }}
|
||||
# The Mathlib `nightly-testing` branch or `nightly-testing-YYYY-MM-DD` tag may have moved since this branch was created, so merge their changes.
|
||||
# (This should no longer be possible once `nightly-testing-YYYY-MM-DD` is a tag, but it is still safe to merge.)
|
||||
git merge "$BASE" --strategy-option ours --no-commit --allow-unrelated-histories
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ steps.workflow-info.outputs.pullRequestNumber }}-${{ env.SHORT_SHA }}" > lean-toolchain
|
||||
git add lean-toolchain
|
||||
lake update batteries
|
||||
git add lake-manifest.json
|
||||
git commit -m "Update lean-toolchain for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
git commit --allow-empty -m "Trigger CI for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
fi
|
||||
|
||||
- name: Push changes
|
||||
|
||||
@@ -50,7 +50,7 @@ We'll use `v4.6.0` as the intended release version as a running example.
|
||||
- Re-running `script/release_checklist.py` will then create the tag `v4.6.0` from `master`/`main` and push it (unless `toolchain-tag: false` in the `release_repos.yml` file)
|
||||
- `script/release_checklist.py` will then merge the tag `v4.6.0` into the `stable` branch and push it (unless `stable-branch: false` in the `release_repos.yml` file).
|
||||
- Special notes on repositories with exceptional requirements:
|
||||
- `doc-gen4` has additional dependencies which we do not update at each toolchain release, although occasionally these break and need to be updated manually.
|
||||
- `doc-gen4` has addition dependencies which we do not update at each toolchain release, although occasionally these break and need to be updated manually.
|
||||
- `verso`:
|
||||
- The `subverso` dependency is unusual in that it needs to be compatible with _every_ Lean release simultaneously.
|
||||
Usually you don't need to do anything.
|
||||
@@ -94,8 +94,6 @@ We'll use `v4.6.0` as the intended release version as a running example.
|
||||
|
||||
This checklist walks you through creating the first release candidate for a version of Lean.
|
||||
|
||||
For subsequent release candidates, the process is essentially the same, but we start out with the `releases/v4.7.0` branch already created.
|
||||
|
||||
We'll use `v4.7.0-rc1` as the intended release version in this example.
|
||||
|
||||
- Decide which nightly release you want to turn into a release candidate.
|
||||
@@ -114,7 +112,7 @@ We'll use `v4.7.0-rc1` as the intended release version in this example.
|
||||
git fetch nightly tag nightly-2024-02-29
|
||||
git checkout nightly-2024-02-29
|
||||
git checkout -b releases/v4.7.0
|
||||
git push --set-upstream origin releases/v4.7.0
|
||||
git push --set-upstream origin releases/v4.18.0
|
||||
```
|
||||
- In `src/CMakeLists.txt`,
|
||||
- verify that you see `set(LEAN_VERSION_MINOR 7)` (for whichever `7` is appropriate); this should already have been updated when the development cycle began.
|
||||
|
||||
@@ -1,9 +0,0 @@
|
||||
#!/usr/bin/env bash
|
||||
set -euo pipefail
|
||||
|
||||
# We benchmark against stage 2 to test new optimizations.
|
||||
timeout -s KILL 1h time bash -c 'mkdir -p build/release; cd build/release; cmake ../.. && make -j$(nproc) stage2' 1>&2
|
||||
export PATH=$PWD/build/release/stage2/bin:$PATH
|
||||
cd tests/bench
|
||||
timeout -s KILL 1h time temci exec --config speedcenter.yaml --in speedcenter.exec.velcom.yaml 1>&2
|
||||
temci report run_output.yaml --reporter codespeed2
|
||||
@@ -53,23 +53,6 @@ def tag_exists(repo_url, tag_name, github_token):
|
||||
matching_tags = response.json()
|
||||
return any(tag["ref"] == f"refs/tags/{tag_name}" for tag in matching_tags)
|
||||
|
||||
def commit_hash_for_tag(repo_url, tag_name, github_token):
|
||||
# Use /git/matching-refs/tags/ to get all matching tags
|
||||
api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/git/matching-refs/tags/{tag_name}"
|
||||
headers = {'Authorization': f'token {github_token}'} if github_token else {}
|
||||
response = requests.get(api_url, headers=headers)
|
||||
|
||||
if response.status_code != 200:
|
||||
return False
|
||||
|
||||
# Check if any of the returned refs exactly match our tag
|
||||
matching_tags = response.json()
|
||||
matching_commits = [tag["object"]["sha"] for tag in matching_tags if tag["ref"] == f"refs/tags/{tag_name}"]
|
||||
if len(matching_commits) != 1:
|
||||
return None
|
||||
else:
|
||||
return matching_commits[0]
|
||||
|
||||
def release_page_exists(repo_url, tag_name, github_token):
|
||||
api_url = repo_url.replace("https://github.com/", "https://api.github.com/repos/") + f"/releases/tags/{tag_name}"
|
||||
headers = {'Authorization': f'token {github_token}'} if github_token else {}
|
||||
@@ -303,14 +286,6 @@ def main():
|
||||
lean4_success = False
|
||||
else:
|
||||
print(f" ✅ Tag {toolchain} exists")
|
||||
commit_hash = commit_hash_for_tag(lean_repo_url, toolchain, github_token)
|
||||
SHORT_HASH_LENGTH = 7 # Lake abbreviates the Lean commit to 7 characters.
|
||||
if commit_hash is None:
|
||||
print(f" ❌ Could not resolve tag {toolchain} to a commit.")
|
||||
lean4_success = False
|
||||
elif commit_hash[0] == '0' and commit_hash[:SHORT_HASH_LENGTH].isnumeric():
|
||||
print(f" ❌ Short commit hash {commit_hash[:SHORT_HASH_LENGTH]} is numeric and starts with 0, causing issues for version parsing. Try regenerating the last commit to get a new hash.")
|
||||
lean4_success = False
|
||||
|
||||
if not release_page_exists(lean_repo_url, toolchain, github_token):
|
||||
print(f" ❌ Release page for {toolchain} does not exist")
|
||||
|
||||
@@ -94,7 +94,6 @@ def generate_script(repo, version, config):
|
||||
"echo 'This repo has nightly-testing infrastructure'",
|
||||
f"git merge origin/bump/{version.split('-rc')[0]}",
|
||||
"echo 'Please resolve any conflicts.'",
|
||||
"grep nightly-testing lakefile.* && echo 'Please ensure the lakefile does not include nightly-testing versions.'",
|
||||
""
|
||||
])
|
||||
if re.search(r'rc\d+$', version) and repo_name in ["verso", "reference-manual"]:
|
||||
|
||||
@@ -10,7 +10,7 @@ endif()
|
||||
include(ExternalProject)
|
||||
project(LEAN CXX C)
|
||||
set(LEAN_VERSION_MAJOR 4)
|
||||
set(LEAN_VERSION_MINOR 22)
|
||||
set(LEAN_VERSION_MINOR 21)
|
||||
set(LEAN_VERSION_PATCH 0)
|
||||
set(LEAN_VERSION_IS_RELEASE 0) # This number is 1 in the release revision, and 0 otherwise.
|
||||
set(LEAN_SPECIAL_VERSION_DESC "" CACHE STRING "Additional version description like 'nightly-2018-03-11'")
|
||||
|
||||
@@ -7,7 +7,6 @@ module
|
||||
|
||||
prelude
|
||||
import Init.Prelude
|
||||
meta import Init.Prelude
|
||||
set_option linter.missingDocs true -- keep it documented
|
||||
|
||||
/-!
|
||||
|
||||
@@ -148,7 +148,7 @@ attribute [simp] pure_bind bind_assoc bind_pure_comp
|
||||
attribute [grind] pure_bind
|
||||
|
||||
@[simp] theorem bind_pure [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x := by
|
||||
change x >>= (fun a => pure (id a)) = x
|
||||
show x >>= (fun a => pure (id a)) = x
|
||||
rw [bind_pure_comp, id_map]
|
||||
|
||||
/--
|
||||
|
||||
@@ -58,7 +58,7 @@ protected theorem bind_pure_comp [Monad m] (f : α → β) (x : ExceptT ε m α)
|
||||
intros; rfl
|
||||
|
||||
protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x <* y = const β <$> x <*> y := by
|
||||
change (x >>= fun a => y >>= fun _ => pure a) = (const (α := α) β <$> x) >>= fun f => f <$> y
|
||||
show (x >>= fun a => y >>= fun _ => pure a) = (const (α := α) β <$> x) >>= fun f => f <$> y
|
||||
rw [← ExceptT.bind_pure_comp]
|
||||
apply ext
|
||||
simp [run_bind]
|
||||
@@ -70,7 +70,7 @@ protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad
|
||||
cases b <;> simp [comp, Except.map, const]
|
||||
|
||||
protected theorem seqRight_eq [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x *> y = const α id <$> x <*> y := by
|
||||
change (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y
|
||||
show (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y
|
||||
rw [← ExceptT.bind_pure_comp]
|
||||
apply ext
|
||||
simp [run_bind]
|
||||
@@ -206,15 +206,15 @@ theorem run_bind_lift {α σ : Type u} [Monad m] [LawfulMonad m] (x : m α) (f :
|
||||
(monadMap @f x : StateT σ m α).run s = monadMap @f (x.run s) := rfl
|
||||
|
||||
@[simp] theorem run_seq {α β σ : Type u} [Monad m] [LawfulMonad m] (f : StateT σ m (α → β)) (x : StateT σ m α) (s : σ) : (f <*> x).run s = (f.run s >>= fun fs => (fun (p : α × σ) => (fs.1 p.1, p.2)) <$> x.run fs.2) := by
|
||||
change (f >>= fun g => g <$> x).run s = _
|
||||
show (f >>= fun g => g <$> x).run s = _
|
||||
simp
|
||||
|
||||
@[simp] theorem run_seqRight [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x *> y).run s = (x.run s >>= fun p => y.run p.2) := by
|
||||
change (x >>= fun _ => y).run s = _
|
||||
show (x >>= fun _ => y).run s = _
|
||||
simp
|
||||
|
||||
@[simp] theorem run_seqLeft {α β σ : Type u} [Monad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x <* y).run s = (x.run s >>= fun p => y.run p.2 >>= fun p' => pure (p.1, p'.2)) := by
|
||||
change (x >>= fun a => y >>= fun _ => pure a).run s = _
|
||||
show (x >>= fun a => y >>= fun _ => pure a).run s = _
|
||||
simp
|
||||
|
||||
theorem seqRight_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x *> y = const α id <$> x <*> y := by
|
||||
|
||||
@@ -9,7 +9,7 @@ module
|
||||
|
||||
prelude
|
||||
import Init.Tactics
|
||||
meta import Init.Meta
|
||||
import Init.Meta
|
||||
|
||||
namespace Lean.Parser.Tactic.Conv
|
||||
|
||||
|
||||
@@ -8,7 +8,7 @@ notation, basic datatypes and type classes
|
||||
module
|
||||
|
||||
prelude
|
||||
meta import Init.Prelude
|
||||
import Init.Prelude
|
||||
import Init.SizeOf
|
||||
set_option linter.missingDocs true -- keep it documented
|
||||
|
||||
@@ -2252,7 +2252,7 @@ theorem funext {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x}
|
||||
Quot.liftOn f
|
||||
(fun (f : ∀ (x : α), β x) => f x)
|
||||
(fun _ _ h => h x)
|
||||
change extfunApp (Quot.mk eqv f) = extfunApp (Quot.mk eqv g)
|
||||
show extfunApp (Quot.mk eqv f) = extfunApp (Quot.mk eqv g)
|
||||
exact congrArg extfunApp (Quot.sound h)
|
||||
|
||||
/--
|
||||
|
||||
@@ -246,7 +246,7 @@ def swap (xs : Array α) (i j : @& Nat) (hi : i < xs.size := by get_elem_tactic)
|
||||
xs'.set j v₁ (Nat.lt_of_lt_of_eq hj (size_set _).symm)
|
||||
|
||||
@[simp] theorem size_swap {xs : Array α} {i j : Nat} {hi hj} : (xs.swap i j hi hj).size = xs.size := by
|
||||
change ((xs.set i xs[j]).set j xs[i]
|
||||
show ((xs.set i xs[j]).set j xs[i]
|
||||
(Nat.lt_of_lt_of_eq hj (size_set _).symm)).size = xs.size
|
||||
rw [size_set, size_set]
|
||||
|
||||
|
||||
@@ -52,7 +52,6 @@ theorem countP_push {a : α} {xs : Array α} : countP p (xs.push a) = countP p x
|
||||
rcases xs with ⟨xs⟩
|
||||
simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem countP_singleton {a : α} : countP p #[a] = if p a then 1 else 0 := by
|
||||
simp
|
||||
|
||||
@@ -60,12 +59,10 @@ theorem size_eq_countP_add_countP {xs : Array α} : xs.size = countP p xs + coun
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.length_eq_countP_add_countP (p := p)]
|
||||
|
||||
@[grind _=_]
|
||||
theorem countP_eq_size_filter {xs : Array α} : countP p xs = (filter p xs).size := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.countP_eq_length_filter]
|
||||
|
||||
@[grind =]
|
||||
theorem countP_eq_size_filter' : countP p = size ∘ filter p := by
|
||||
funext xs
|
||||
apply countP_eq_size_filter
|
||||
@@ -74,7 +71,7 @@ theorem countP_le_size : countP p xs ≤ xs.size := by
|
||||
simp only [countP_eq_size_filter]
|
||||
apply size_filter_le
|
||||
|
||||
@[simp, grind =] theorem countP_append {xs ys : Array α} : countP p (xs ++ ys) = countP p xs + countP p ys := by
|
||||
@[simp] theorem countP_append {xs ys : Array α} : countP p (xs ++ ys) = countP p xs + countP p ys := by
|
||||
rcases xs with ⟨xs⟩
|
||||
rcases ys with ⟨ys⟩
|
||||
simp
|
||||
@@ -149,7 +146,7 @@ theorem countP_flatMap {p : β → Bool} {xs : Array α} {f : α → Array β} :
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.countP_flatMap, Function.comp_def]
|
||||
|
||||
@[simp, grind =] theorem countP_reverse {xs : Array α} : countP p xs.reverse = countP p xs := by
|
||||
@[simp] theorem countP_reverse {xs : Array α} : countP p xs.reverse = countP p xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.countP_reverse]
|
||||
|
||||
@@ -176,7 +173,7 @@ variable [BEq α]
|
||||
cases xs
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem count_empty {a : α} : count a #[] = 0 := rfl
|
||||
@[simp] theorem count_empty {a : α} : count a #[] = 0 := rfl
|
||||
|
||||
theorem count_push {a b : α} {xs : Array α} :
|
||||
count a (xs.push b) = count a xs + if b == a then 1 else 0 := by
|
||||
@@ -189,28 +186,21 @@ theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
|
||||
|
||||
theorem count_le_size {a : α} {xs : Array α} : count a xs ≤ xs.size := countP_le_size
|
||||
|
||||
grind_pattern count_le_size => count a xs
|
||||
|
||||
@[grind =]
|
||||
theorem count_eq_size_filter {a : α} {xs : Array α} : count a xs = (filter (· == a) xs).size := by
|
||||
simp [count, countP_eq_size_filter]
|
||||
|
||||
theorem count_le_count_push {a b : α} {xs : Array α} : count a xs ≤ count a (xs.push b) := by
|
||||
simp [count_push]
|
||||
|
||||
@[grind =]
|
||||
theorem count_singleton {a b : α} : count a #[b] = if b == a then 1 else 0 := by
|
||||
simp [count_eq_countP]
|
||||
|
||||
@[simp, grind =] theorem count_append {a : α} {xs ys : Array α} : count a (xs ++ ys) = count a xs + count a ys :=
|
||||
@[simp] theorem count_append {a : α} {xs ys : Array α} : count a (xs ++ ys) = count a xs + count a ys :=
|
||||
countP_append
|
||||
|
||||
@[simp, grind =] theorem count_flatten {a : α} {xss : Array (Array α)} :
|
||||
@[simp] theorem count_flatten {a : α} {xss : Array (Array α)} :
|
||||
count a xss.flatten = (xss.map (count a)).sum := by
|
||||
cases xss using array₂_induction
|
||||
simp [List.count_flatten, Function.comp_def]
|
||||
|
||||
@[simp, grind =] theorem count_reverse {a : α} {xs : Array α} : count a xs.reverse = count a xs := by
|
||||
@[simp] theorem count_reverse {a : α} {xs : Array α} : count a xs.reverse = count a xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
|
||||
|
||||
@@ -24,7 +24,6 @@ open Nat
|
||||
|
||||
/-! ### eraseP -/
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_empty : #[].eraseP p = #[] := by simp
|
||||
|
||||
theorem eraseP_of_forall_mem_not {xs : Array α} (h : ∀ a, a ∈ xs → ¬p a) : xs.eraseP p = xs := by
|
||||
@@ -65,7 +64,6 @@ theorem exists_or_eq_self_of_eraseP (p) (xs : Array α) :
|
||||
let ⟨_, ys, zs, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
|
||||
rw [e₂]; simp [size_append, e₁]
|
||||
|
||||
@[grind =]
|
||||
theorem size_eraseP {xs : Array α} : (xs.eraseP p).size = if xs.any p then xs.size - 1 else xs.size := by
|
||||
split <;> rename_i h
|
||||
· simp only [any_eq_true] at h
|
||||
@@ -83,12 +81,11 @@ theorem le_size_eraseP {xs : Array α} : xs.size - 1 ≤ (xs.eraseP p).size := b
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.le_length_eraseP
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_mem_eraseP {xs : Array α} : a ∈ xs.eraseP p → a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.mem_of_mem_eraseP
|
||||
|
||||
@[simp, grind] theorem mem_eraseP_of_neg {xs : Array α} (pa : ¬p a) : a ∈ xs.eraseP p ↔ a ∈ xs := by
|
||||
@[simp] theorem mem_eraseP_of_neg {xs : Array α} (pa : ¬p a) : a ∈ xs.eraseP p ↔ a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.mem_eraseP_of_neg pa
|
||||
|
||||
@@ -96,18 +93,15 @@ theorem mem_of_mem_eraseP {xs : Array α} : a ∈ xs.eraseP p → a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
|
||||
@[grind _=_]
|
||||
theorem eraseP_map {f : β → α} {xs : Array β} : (xs.map f).eraseP p = (xs.eraseP (p ∘ f)).map f := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.eraseP_map
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_filterMap {f : α → Option β} {xs : Array α} :
|
||||
(filterMap f xs).eraseP p = filterMap f (xs.eraseP (fun x => match f x with | some y => p y | none => false)) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.eraseP_filterMap
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_filter {f : α → Bool} {xs : Array α} :
|
||||
(filter f xs).eraseP p = filter f (xs.eraseP (fun x => p x && f x)) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -125,7 +119,6 @@ theorem eraseP_append_right {xs : Array α} ys (h : ∀ b ∈ xs, ¬p b) :
|
||||
rcases ys with ⟨ys⟩
|
||||
simpa using List.eraseP_append_right ys (by simpa using h)
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_append {xs : Array α} {ys : Array α} :
|
||||
(xs ++ ys).eraseP p = if xs.any p then xs.eraseP p ++ ys else xs ++ ys.eraseP p := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -133,7 +126,6 @@ theorem eraseP_append {xs : Array α} {ys : Array α} :
|
||||
simp only [List.append_toArray, List.eraseP_toArray, List.eraseP_append, List.any_toArray]
|
||||
split <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_replicate {n : Nat} {a : α} {p : α → Bool} :
|
||||
(replicate n a).eraseP p = if p a then replicate (n - 1) a else replicate n a := by
|
||||
simp only [← List.toArray_replicate, List.eraseP_toArray, List.eraseP_replicate]
|
||||
@@ -173,7 +165,6 @@ theorem eraseP_eq_iff {p} {xs : Array α} :
|
||||
· exact Or.inl h
|
||||
· exact Or.inr ⟨a, l₁, by simpa using h₁, h₂, ⟨l, by simp⟩⟩
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_comm {xs : Array α} (h : ∀ a ∈ xs, ¬ p a ∨ ¬ q a) :
|
||||
(xs.eraseP p).eraseP q = (xs.eraseP q).eraseP p := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -217,7 +208,6 @@ theorem exists_erase_eq [LawfulBEq α] {a : α} {xs : Array α} (h : a ∈ xs) :
|
||||
(xs.erase a).size = xs.size - 1 := by
|
||||
rw [erase_eq_eraseP]; exact size_eraseP_of_mem h (beq_self_eq_true a)
|
||||
|
||||
@[grind =]
|
||||
theorem size_erase [LawfulBEq α] {a : α} {xs : Array α} :
|
||||
(xs.erase a).size = if a ∈ xs then xs.size - 1 else xs.size := by
|
||||
rw [erase_eq_eraseP, size_eraseP]
|
||||
@@ -232,12 +222,11 @@ theorem le_size_erase [LawfulBEq α] {a : α} {xs : Array α} : xs.size - 1 ≤
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.le_length_erase
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.mem_of_mem_erase (by simpa using h)
|
||||
|
||||
@[simp, grind] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {xs : Array α} (ab : a ≠ b) :
|
||||
@[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {xs : Array α} (ab : a ≠ b) :
|
||||
a ∈ xs.erase b ↔ a ∈ xs :=
|
||||
erase_eq_eraseP b xs ▸ mem_eraseP_of_neg (mt eq_of_beq ab.symm)
|
||||
|
||||
@@ -245,7 +234,6 @@ theorem mem_of_mem_erase {a b : α} {xs : Array α} (h : a ∈ xs.erase b) : a
|
||||
rw [erase_eq_eraseP', eraseP_eq_self_iff]
|
||||
simp [forall_mem_ne']
|
||||
|
||||
@[grind _=_]
|
||||
theorem erase_filter [LawfulBEq α] {f : α → Bool} {xs : Array α} :
|
||||
(filter f xs).erase a = filter f (xs.erase a) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -263,7 +251,6 @@ theorem erase_append_right [LawfulBEq α] {a : α} {xs : Array α} (ys : Array
|
||||
rcases ys with ⟨ys⟩
|
||||
simpa using List.erase_append_right ys (by simpa using h)
|
||||
|
||||
@[grind =]
|
||||
theorem erase_append [LawfulBEq α] {a : α} {xs ys : Array α} :
|
||||
(xs ++ ys).erase a = if a ∈ xs then xs.erase a ++ ys else xs ++ ys.erase a := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -271,7 +258,6 @@ theorem erase_append [LawfulBEq α] {a : α} {xs ys : Array α} :
|
||||
simp only [List.append_toArray, List.erase_toArray, List.erase_append, mem_toArray]
|
||||
split <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem erase_replicate [LawfulBEq α] {n : Nat} {a b : α} :
|
||||
(replicate n a).erase b = if b == a then replicate (n - 1) a else replicate n a := by
|
||||
simp only [← List.toArray_replicate, List.erase_toArray]
|
||||
@@ -283,7 +269,6 @@ abbrev erase_mkArray := @erase_replicate
|
||||
|
||||
-- The arguments `a b` are explicit,
|
||||
-- so they can be specified to prevent `simp` repeatedly applying the lemma.
|
||||
@[grind =]
|
||||
theorem erase_comm [LawfulBEq α] (a b : α) {xs : Array α} :
|
||||
(xs.erase a).erase b = (xs.erase b).erase a := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -327,7 +312,6 @@ theorem eraseIdx_eq_eraseIdxIfInBounds {xs : Array α} {i : Nat} (h : i < xs.siz
|
||||
xs.eraseIdx i h = xs.eraseIdxIfInBounds i := by
|
||||
simp [eraseIdxIfInBounds, h]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_eq_take_drop_succ {xs : Array α} {i : Nat} (h) :
|
||||
xs.eraseIdx i h = xs.take i ++ xs.drop (i + 1) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -338,7 +322,6 @@ theorem eraseIdx_eq_take_drop_succ {xs : Array α} {i : Nat} (h) :
|
||||
rw [List.take_of_length_le]
|
||||
simp
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_eraseIdx {xs : Array α} {i : Nat} (h : i < xs.size) {j : Nat} :
|
||||
(xs.eraseIdx i)[j]? = if j < i then xs[j]? else xs[j + 1]? := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -356,7 +339,6 @@ theorem getElem?_eraseIdx_of_ge {xs : Array α} {i : Nat} (h : i < xs.size) {j :
|
||||
intro h'
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem getElem_eraseIdx {xs : Array α} {i : Nat} (h : i < xs.size) {j : Nat} (h' : j < (xs.eraseIdx i).size) :
|
||||
(xs.eraseIdx i)[j] = if h'' : j < i then
|
||||
xs[j]
|
||||
@@ -380,7 +362,6 @@ theorem eraseIdx_ne_empty_iff {xs : Array α} {i : Nat} {h} : xs.eraseIdx i ≠
|
||||
simp [h]
|
||||
· simp
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_mem_eraseIdx {xs : Array α} {i : Nat} {h} {a : α} (h : a ∈ xs.eraseIdx i) : a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.mem_of_mem_eraseIdx (by simpa using h)
|
||||
@@ -392,29 +373,13 @@ theorem eraseIdx_append_of_lt_size {xs : Array α} {k : Nat} (hk : k < xs.size)
|
||||
simp at hk
|
||||
simp [List.eraseIdx_append_of_lt_length, *]
|
||||
|
||||
theorem eraseIdx_append_of_size_le {xs : Array α} {k : Nat} (hk : xs.size ≤ k) (ys : Array α) (h) :
|
||||
theorem eraseIdx_append_of_length_le {xs : Array α} {k : Nat} (hk : xs.size ≤ k) (ys : Array α) (h) :
|
||||
eraseIdx (xs ++ ys) k = xs ++ eraseIdx ys (k - xs.size) (by simp at h; omega) := by
|
||||
rcases xs with ⟨l⟩
|
||||
rcases ys with ⟨l'⟩
|
||||
simp at hk
|
||||
simp [List.eraseIdx_append_of_length_le, *]
|
||||
|
||||
@[deprecated eraseIdx_append_of_size_le (since := "2025-06-11")]
|
||||
abbrev eraseIdx_append_of_length_le := @eraseIdx_append_of_size_le
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_append {xs ys : Array α} (h : k < (xs ++ ys).size) :
|
||||
eraseIdx (xs ++ ys) k =
|
||||
if h' : k < xs.size then
|
||||
eraseIdx xs k ++ ys
|
||||
else
|
||||
xs ++ eraseIdx ys (k - xs.size) (by simp at h; omega) := by
|
||||
split <;> rename_i h
|
||||
· simp [eraseIdx_append_of_lt_size h]
|
||||
· rw [eraseIdx_append_of_size_le]
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} {h} :
|
||||
(replicate n a).eraseIdx k = replicate (n - 1) a := by
|
||||
simp at h
|
||||
@@ -463,48 +428,6 @@ theorem eraseIdx_set_gt {xs : Array α} {i : Nat} {j : Nat} {a : α} (h : i < j)
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.eraseIdx_set_gt, *]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_set {xs : Array α} {i : Nat} {a : α} {hi : i < xs.size} {j : Nat} {hj : j < (xs.set i a).size} :
|
||||
(xs.set i a).eraseIdx j =
|
||||
if h' : j < i then
|
||||
(xs.eraseIdx j).set (i - 1) a (by simp; omega)
|
||||
else if h'' : j = i then
|
||||
xs.eraseIdx i
|
||||
else
|
||||
(xs.eraseIdx j (by simp at hj; omega)).set i a (by simp at hj ⊢; omega) := by
|
||||
split <;> rename_i h'
|
||||
· rw [eraseIdx_set_lt]
|
||||
omega
|
||||
· split <;> rename_i h''
|
||||
· subst h''
|
||||
rw [eraseIdx_set_eq]
|
||||
· rw [eraseIdx_set_gt]
|
||||
omega
|
||||
|
||||
theorem set_eraseIdx_le {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (h : i ≤ j) (hj : j < (xs.eraseIdx i).size) :
|
||||
(xs.eraseIdx i).set j a = (xs.set (j + 1) a (by simp at hj; omega)).eraseIdx i (by simp at ⊢; omega) := by
|
||||
rw [eraseIdx_set_lt]
|
||||
· simp
|
||||
· omega
|
||||
|
||||
theorem set_eraseIdx_gt {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (h : j < i) (hj : j < (xs.eraseIdx i).size) :
|
||||
(xs.eraseIdx i).set j a = (xs.set j a).eraseIdx i (by simp at ⊢; omega) := by
|
||||
rw [eraseIdx_set_gt]
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem set_eraseIdx {xs : Array α} {i : Nat} {w : i < xs.size} {j : Nat} {a : α} (hj : j < (xs.eraseIdx i).size) :
|
||||
(xs.eraseIdx i).set j a =
|
||||
if h' : i ≤ j then
|
||||
(xs.set (j + 1) a (by simp at hj; omega)).eraseIdx i (by simp at ⊢; omega)
|
||||
else
|
||||
(xs.set j a).eraseIdx i (by simp at ⊢; omega) := by
|
||||
split <;> rename_i h'
|
||||
· rw [set_eraseIdx_le]
|
||||
omega
|
||||
· rw [set_eraseIdx_gt]
|
||||
omega
|
||||
|
||||
@[simp] theorem set_getElem_succ_eraseIdx_succ
|
||||
{xs : Array α} {i : Nat} (h : i + 1 < xs.size) :
|
||||
(xs.eraseIdx (i + 1)).set i xs[i + 1] (by simp; omega) = xs.eraseIdx i := by
|
||||
|
||||
@@ -23,10 +23,10 @@ Examples:
|
||||
-/
|
||||
protected def finRange (n : Nat) : Array (Fin n) := ofFn fun i => i
|
||||
|
||||
@[simp, grind =] theorem size_finRange {n} : (Array.finRange n).size = n := by
|
||||
@[simp] theorem size_finRange {n} : (Array.finRange n).size = n := by
|
||||
simp [Array.finRange]
|
||||
|
||||
@[simp, grind =] theorem getElem_finRange {i : Nat} (h : i < (Array.finRange n).size) :
|
||||
@[simp] theorem getElem_finRange {i : Nat} (h : i < (Array.finRange n).size) :
|
||||
(Array.finRange n)[i] = Fin.cast size_finRange ⟨i, h⟩ := by
|
||||
simp [Array.finRange]
|
||||
|
||||
@@ -49,7 +49,6 @@ theorem finRange_succ_last {n} :
|
||||
· simp_all
|
||||
omega
|
||||
|
||||
@[grind _=_]
|
||||
theorem finRange_reverse {n} : (Array.finRange n).reverse = (Array.finRange n).map Fin.rev := by
|
||||
ext i h
|
||||
· simp
|
||||
|
||||
@@ -38,22 +38,11 @@ theorem findSome?_singleton {a : α} {f : α → Option β} : #[a].findSome? f =
|
||||
@[simp] theorem findSomeRev?_push_of_isNone {xs : Array α} (h : (f a).isNone) : (xs.push a).findSomeRev? f = xs.findSomeRev? f := by
|
||||
cases xs; simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem findSomeRev?_push {xs : Array α} {a : α} {f : α → Option β} :
|
||||
(xs.push a).findSomeRev? f = (f a).or (xs.findSomeRev? f) := by
|
||||
match h : f a with
|
||||
| some b =>
|
||||
rw [findSomeRev?_push_of_isSome]
|
||||
all_goals simp_all
|
||||
| none =>
|
||||
rw [findSomeRev?_push_of_isNone]
|
||||
all_goals simp_all
|
||||
|
||||
theorem exists_of_findSome?_eq_some {f : α → Option β} {xs : Array α} (w : xs.findSome? f = some b) :
|
||||
∃ a, a ∈ xs ∧ f a = some b := by
|
||||
cases xs; simp_all [List.exists_of_findSome?_eq_some]
|
||||
|
||||
@[simp, grind =] theorem findSome?_eq_none_iff : findSome? p xs = none ↔ ∀ x ∈ xs, p x = none := by
|
||||
@[simp] theorem findSome?_eq_none_iff : findSome? p xs = none ↔ ∀ x ∈ xs, p x = none := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp] theorem findSome?_isSome_iff {f : α → Option β} {xs : Array α} :
|
||||
@@ -70,39 +59,36 @@ theorem findSome?_eq_some_iff {f : α → Option β} {xs : Array α} {b : β} :
|
||||
· rintro ⟨xs, a, ys, h₀, h₁, h₂⟩
|
||||
exact ⟨xs.toList, a, ys.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩
|
||||
|
||||
@[simp, grind =] theorem findSome?_guard {xs : Array α} : findSome? (Option.guard p) xs = find? p xs := by
|
||||
@[simp] theorem findSome?_guard {xs : Array α} : findSome? (Option.guard fun x => p x) xs = find? p xs := by
|
||||
cases xs; simp
|
||||
|
||||
theorem find?_eq_findSome?_guard {xs : Array α} : find? p xs = findSome? (Option.guard p) xs :=
|
||||
theorem find?_eq_findSome?_guard {xs : Array α} : find? p xs = findSome? (Option.guard fun x => p x) xs :=
|
||||
findSome?_guard.symm
|
||||
|
||||
@[simp, grind =] theorem getElem?_zero_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f)[0]? = xs.findSome? f := by
|
||||
@[simp] theorem getElem?_zero_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f)[0]? = xs.findSome? f := by
|
||||
cases xs; simp [← List.head?_eq_getElem?]
|
||||
|
||||
@[simp, grind =] theorem getElem_zero_filterMap {f : α → Option β} {xs : Array α} (h) :
|
||||
@[simp] theorem getElem_zero_filterMap {f : α → Option β} {xs : Array α} (h) :
|
||||
(xs.filterMap f)[0] = (xs.findSome? f).get (by cases xs; simpa [List.length_filterMap_eq_countP] using h) := by
|
||||
cases xs; simp [← List.head_eq_getElem, ← getElem?_zero_filterMap]
|
||||
|
||||
@[simp, grind =] theorem back?_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f).back? = xs.findSomeRev? f := by
|
||||
@[simp] theorem back?_filterMap {f : α → Option β} {xs : Array α} : (xs.filterMap f).back? = xs.findSomeRev? f := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem back!_filterMap [Inhabited β] {f : α → Option β} {xs : Array α} :
|
||||
@[simp] theorem back!_filterMap [Inhabited β] {f : α → Option β} {xs : Array α} :
|
||||
(xs.filterMap f).back! = (xs.findSomeRev? f).getD default := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind _=_] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Array α} :
|
||||
@[simp] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Array α} :
|
||||
(xs.findSome? f).map g = xs.findSome? (Option.map g ∘ f) := by
|
||||
cases xs; simp
|
||||
|
||||
@[grind _=_]
|
||||
theorem findSome?_map {f : β → γ} {xs : Array β} : findSome? p (xs.map f) = xs.findSome? (p ∘ f) := by
|
||||
cases xs; simp [List.findSome?_map]
|
||||
|
||||
@[grind =]
|
||||
theorem findSome?_append {xs ys : Array α} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
|
||||
cases xs; cases ys; simp [List.findSome?_append]
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_zero_flatten (xss : Array (Array α)) :
|
||||
(flatten xss)[0]? = xss.findSome? fun xs => xs[0]? := by
|
||||
cases xss using array₂_induction
|
||||
@@ -118,14 +104,12 @@ theorem getElem_zero_flatten.proof {xss : Array (Array α)} (h : 0 < xss.flatten
|
||||
obtain ⟨_, ⟨xs, m, rfl⟩, h⟩ := h
|
||||
exact ⟨xs, m, by simpa using h⟩
|
||||
|
||||
@[grind =]
|
||||
theorem getElem_zero_flatten {xss : Array (Array α)} (h) :
|
||||
(flatten xss)[0] = (xss.findSome? fun xs => xs[0]?).get (getElem_zero_flatten.proof h) := by
|
||||
have t := getElem?_zero_flatten xss
|
||||
simp [getElem?_eq_getElem, h] at t
|
||||
simp [← t]
|
||||
|
||||
@[grind =]
|
||||
theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
|
||||
simp [← List.toArray_replicate, List.findSome?_replicate]
|
||||
|
||||
@@ -156,9 +140,8 @@ abbrev findSome?_mkArray_of_isNone := @findSome?_replicate_of_isNone
|
||||
|
||||
/-! ### find? -/
|
||||
|
||||
@[simp, grind =] theorem find?_empty : find? p #[] = none := rfl
|
||||
@[simp] theorem find?_empty : find? p #[] = none := rfl
|
||||
|
||||
@[grind =]
|
||||
theorem find?_singleton {a : α} {p : α → Bool} :
|
||||
#[a].find? p = if p a then some a else none := by
|
||||
simp
|
||||
@@ -167,26 +150,11 @@ theorem find?_singleton {a : α} {p : α → Bool} :
|
||||
findRev? p (xs.push a) = some a := by
|
||||
cases xs; simp [h]
|
||||
|
||||
@[simp] theorem findRev?_push_of_neg {xs : Array α} (h : ¬p a) :
|
||||
@[simp] theorem findRev?_cons_of_neg {xs : Array α} (h : ¬p a) :
|
||||
findRev? p (xs.push a) = findRev? p xs := by
|
||||
cases xs; simp [h]
|
||||
|
||||
@[deprecated findRev?_push_of_neg (since := "2025-06-12")]
|
||||
abbrev findRev?_cons_of_neg := @findRev?_push_of_neg
|
||||
|
||||
@[grind =]
|
||||
theorem finRev?_push {xs : Array α} :
|
||||
findRev? p (xs.push a) = (Option.guard p a).or (xs.findRev? p) := by
|
||||
cases h : p a
|
||||
· rw [findRev?_push_of_neg, Option.guard_eq_none_iff.mpr h]
|
||||
all_goals simp [h]
|
||||
· rw [findRev?_push_of_pos, Option.guard_eq_some_iff.mpr ⟨rfl, h⟩]
|
||||
all_goals simp [h]
|
||||
|
||||
@[deprecated finRev?_push (since := "2025-06-12")]
|
||||
abbrev findRev?_cons := @finRev?_push
|
||||
|
||||
@[simp, grind =] theorem find?_eq_none : find? p xs = none ↔ ∀ x ∈ xs, ¬ p x := by
|
||||
@[simp] theorem find?_eq_none : find? p xs = none ↔ ∀ x ∈ xs, ¬ p x := by
|
||||
cases xs; simp
|
||||
|
||||
theorem find?_eq_some_iff_append {xs : Array α} :
|
||||
@@ -210,63 +178,60 @@ theorem find?_push_eq_some {xs : Array α} :
|
||||
(xs.push a).find? p = some b ↔ xs.find? p = some b ∨ (xs.find? p = none ∧ (p a ∧ a = b)) := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem find?_isSome {xs : Array α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
@[simp] theorem find?_isSome {xs : Array α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
cases xs; simp
|
||||
|
||||
@[grind →]
|
||||
theorem find?_some {xs : Array α} (h : find? p xs = some a) : p a := by
|
||||
cases xs
|
||||
simp at h
|
||||
exact List.find?_some h
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_find?_eq_some {xs : Array α} (h : find? p xs = some a) : a ∈ xs := by
|
||||
cases xs
|
||||
simp at h
|
||||
simpa using List.mem_of_find?_eq_some h
|
||||
|
||||
@[grind]
|
||||
theorem get_find?_mem {xs : Array α} (h) : (xs.find? p).get h ∈ xs := by
|
||||
cases xs
|
||||
simp [List.get_find?_mem]
|
||||
|
||||
@[simp, grind =] theorem find?_filter {xs : Array α} (p q : α → Bool) :
|
||||
@[simp] theorem find?_filter {xs : Array α} (p q : α → Bool) :
|
||||
(xs.filter p).find? q = xs.find? (fun a => p a ∧ q a) := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem getElem?_zero_filter {p : α → Bool} {xs : Array α} :
|
||||
@[simp] theorem getElem?_zero_filter {p : α → Bool} {xs : Array α} :
|
||||
(xs.filter p)[0]? = xs.find? p := by
|
||||
cases xs; simp [← List.head?_eq_getElem?]
|
||||
|
||||
@[simp, grind =] theorem getElem_zero_filter {p : α → Bool} {xs : Array α} (h) :
|
||||
@[simp] theorem getElem_zero_filter {p : α → Bool} {xs : Array α} (h) :
|
||||
(xs.filter p)[0] =
|
||||
(xs.find? p).get (by cases xs; simpa [← List.countP_eq_length_filter] using h) := by
|
||||
cases xs
|
||||
simp [List.getElem_zero_eq_head]
|
||||
|
||||
@[simp, grind =] theorem back?_filter {p : α → Bool} {xs : Array α} : (xs.filter p).back? = xs.findRev? p := by
|
||||
@[simp] theorem back?_filter {p : α → Bool} {xs : Array α} : (xs.filter p).back? = xs.findRev? p := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem back!_filter [Inhabited α] {p : α → Bool} {xs : Array α} :
|
||||
@[simp] theorem back!_filter [Inhabited α] {p : α → Bool} {xs : Array α} :
|
||||
(xs.filter p).back! = (xs.findRev? p).get! := by
|
||||
cases xs; simp [Option.get!_eq_getD]
|
||||
|
||||
@[simp, grind =] theorem find?_filterMap {xs : Array α} {f : α → Option β} {p : β → Bool} :
|
||||
@[simp] theorem find?_filterMap {xs : Array α} {f : α → Option β} {p : β → Bool} :
|
||||
(xs.filterMap f).find? p = (xs.find? (fun a => (f a).any p)).bind f := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem find?_map {f : β → α} {xs : Array β} :
|
||||
@[simp] theorem find?_map {f : β → α} {xs : Array β} :
|
||||
find? p (xs.map f) = (xs.find? (p ∘ f)).map f := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem find?_append {xs ys : Array α} :
|
||||
@[simp] theorem find?_append {xs ys : Array α} :
|
||||
(xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
|
||||
cases xs
|
||||
cases ys
|
||||
simp
|
||||
|
||||
@[simp, grind _=_] theorem find?_flatten {xss : Array (Array α)} {p : α → Bool} :
|
||||
xss.flatten.find? p = xss.findSome? (find? p) := by
|
||||
@[simp] theorem find?_flatten {xss : Array (Array α)} {p : α → Bool} :
|
||||
xss.flatten.find? p = xss.findSome? (·.find? p) := by
|
||||
cases xss using array₂_induction
|
||||
simp [List.findSome?_map, Function.comp_def]
|
||||
|
||||
@@ -305,7 +270,7 @@ theorem find?_flatten_eq_some_iff {xss : Array (Array α)} {p : α → Bool} {a
|
||||
@[deprecated find?_flatten_eq_some_iff (since := "2025-02-03")]
|
||||
abbrev find?_flatten_eq_some := @find?_flatten_eq_some_iff
|
||||
|
||||
@[simp, grind =] theorem find?_flatMap {xs : Array α} {f : α → Array β} {p : β → Bool} :
|
||||
@[simp] theorem find?_flatMap {xs : Array α} {f : α → Array β} {p : β → Bool} :
|
||||
(xs.flatMap f).find? p = xs.findSome? (fun x => (f x).find? p) := by
|
||||
cases xs
|
||||
simp [List.find?_flatMap, Array.flatMap_toArray]
|
||||
@@ -317,7 +282,6 @@ theorem find?_flatMap_eq_none_iff {xs : Array α} {f : α → Array β} {p : β
|
||||
@[deprecated find?_flatMap_eq_none_iff (since := "2025-02-03")]
|
||||
abbrev find?_flatMap_eq_none := @find?_flatMap_eq_none_iff
|
||||
|
||||
@[grind =]
|
||||
theorem find?_replicate :
|
||||
find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
|
||||
simp [← List.toArray_replicate, List.find?_replicate]
|
||||
@@ -370,7 +334,6 @@ abbrev find?_mkArray_eq_some := @find?_replicate_eq_some_iff
|
||||
@[deprecated get_find?_replicate (since := "2025-03-18")]
|
||||
abbrev get_find?_mkArray := @get_find?_replicate
|
||||
|
||||
@[grind =]
|
||||
theorem find?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Array α}
|
||||
(H : ∀ (a : α), a ∈ xs → P a) {p : β → Bool} :
|
||||
(xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
|
||||
@@ -384,15 +347,12 @@ theorem find?_eq_some_iff_getElem {xs : Array α} {p : α → Bool} {b : α} :
|
||||
|
||||
/-! ### findIdx -/
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_empty : findIdx p #[] = 0 := rfl
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_singleton {a : α} {p : α → Bool} :
|
||||
#[a].findIdx p = if p a then 0 else 1 := by
|
||||
simp
|
||||
|
||||
@[grind →]
|
||||
theorem findIdx_of_getElem?_eq_some {xs : Array α} (w : xs[xs.findIdx p]? = some y) : p y := by
|
||||
rcases xs with ⟨xs⟩
|
||||
exact List.findIdx_of_getElem?_eq_some (by simpa using w)
|
||||
@@ -401,8 +361,6 @@ theorem findIdx_getElem {xs : Array α} {w : xs.findIdx p < xs.size} :
|
||||
p xs[xs.findIdx p] :=
|
||||
xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)
|
||||
|
||||
grind_pattern findIdx_getElem => xs[xs.findIdx p]
|
||||
|
||||
theorem findIdx_lt_size_of_exists {xs : Array α} (h : ∃ x ∈ xs, p x) :
|
||||
xs.findIdx p < xs.size := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -429,24 +387,18 @@ theorem findIdx_le_size {p : α → Bool} {xs : Array α} : xs.findIdx p ≤ xs.
|
||||
· simp at e
|
||||
exact Nat.le_of_eq (findIdx_eq_size.mpr e)
|
||||
|
||||
grind_pattern findIdx_le_size => xs.findIdx p, xs.size
|
||||
|
||||
@[simp]
|
||||
theorem findIdx_lt_size {p : α → Bool} {xs : Array α} :
|
||||
xs.findIdx p < xs.size ↔ ∃ x ∈ xs, p x := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
|
||||
grind_pattern findIdx_lt_size => xs.findIdx p, xs.size
|
||||
|
||||
/-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/
|
||||
theorem not_of_lt_findIdx {p : α → Bool} {xs : Array α} {i : Nat} (h : i < xs.findIdx p) :
|
||||
p (xs[i]'(Nat.le_trans h findIdx_le_size)) = false := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.not_of_lt_findIdx (by simpa using h)
|
||||
|
||||
grind_pattern not_of_lt_findIdx => xs.findIdx p, xs[i]
|
||||
|
||||
/-- If `¬ p xs[j]` for all `j < i`, then `i ≤ xs.findIdx p`. -/
|
||||
theorem le_findIdx_of_not {p : α → Bool} {xs : Array α} {i : Nat} (h : i < xs.size)
|
||||
(h2 : ∀ j (hji : j < i), p (xs[j]'(Nat.lt_trans hji h)) = false) : i ≤ xs.findIdx p := by
|
||||
@@ -474,7 +426,6 @@ theorem findIdx_eq {p : α → Bool} {xs : Array α} {i : Nat} (h : i < xs.size)
|
||||
simp at h3
|
||||
simp_all [not_of_lt_findIdx h3]
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_append {p : α → Bool} {xs ys : Array α} :
|
||||
(xs ++ ys).findIdx p =
|
||||
if xs.findIdx p < xs.size then xs.findIdx p else ys.findIdx p + xs.size := by
|
||||
@@ -482,7 +433,6 @@ theorem findIdx_append {p : α → Bool} {xs ys : Array α} :
|
||||
rcases ys with ⟨ys⟩
|
||||
simp [List.findIdx_append]
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_push {xs : Array α} {a : α} {p : α → Bool} :
|
||||
(xs.push a).findIdx p = if xs.findIdx p < xs.size then xs.findIdx p else xs.size + if p a then 0 else 1 := by
|
||||
simp only [push_eq_append, findIdx_append]
|
||||
@@ -505,7 +455,7 @@ theorem false_of_mem_extract_findIdx {xs : Array α} {p : α → Bool} (h : x
|
||||
rcases xs with ⟨xs⟩
|
||||
exact List.false_of_mem_take_findIdx (by simpa using h)
|
||||
|
||||
@[simp, grind =] theorem findIdx_extract {xs : Array α} {i : Nat} {p : α → Bool} :
|
||||
@[simp] theorem findIdx_extract {xs : Array α} {i : Nat} {p : α → Bool} :
|
||||
(xs.extract 0 i).findIdx p = min i (xs.findIdx p) := by
|
||||
cases xs
|
||||
simp
|
||||
@@ -517,24 +467,24 @@ theorem false_of_mem_extract_findIdx {xs : Array α} {p : α → Bool} (h : x
|
||||
|
||||
/-! ### findIdx? -/
|
||||
|
||||
@[simp, grind =] theorem findIdx?_empty : (#[] : Array α).findIdx? p = none := by simp
|
||||
@[grind =] theorem findIdx?_singleton {a : α} {p : α → Bool} :
|
||||
@[simp] theorem findIdx?_empty : (#[] : Array α).findIdx? p = none := by simp
|
||||
theorem findIdx?_singleton {a : α} {p : α → Bool} :
|
||||
#[a].findIdx? p = if p a then some 0 else none := by
|
||||
simp
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem findIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
|
||||
xs.findIdx? p = none ↔ ∀ x, x ∈ xs → p x = false := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem findIdx?_isSome {xs : Array α} {p : α → Bool} :
|
||||
(xs.findIdx? p).isSome = xs.any p := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.findIdx?_isSome]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem findIdx?_isNone {xs : Array α} {p : α → Bool} :
|
||||
(xs.findIdx? p).isNone = xs.all (¬p ·) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -576,19 +526,18 @@ theorem of_findIdx?_eq_none {xs : Array α} {p : α → Bool} (w : xs.findIdx? p
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.of_findIdx?_eq_none (by simpa using w)
|
||||
|
||||
@[simp, grind =] theorem findIdx?_map {f : β → α} {xs : Array β} {p : α → Bool} :
|
||||
@[simp] theorem findIdx?_map {f : β → α} {xs : Array β} {p : α → Bool} :
|
||||
findIdx? p (xs.map f) = xs.findIdx? (p ∘ f) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.findIdx?_map]
|
||||
|
||||
@[simp, grind =] theorem findIdx?_append :
|
||||
@[simp] theorem findIdx?_append :
|
||||
(xs ++ ys : Array α).findIdx? p =
|
||||
(xs.findIdx? p).or ((ys.findIdx? p).map fun i => i + xs.size) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
rcases ys with ⟨ys⟩
|
||||
simp [List.findIdx?_append]
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx?_push {xs : Array α} {a : α} {p : α → Bool} :
|
||||
(xs.push a).findIdx? p = (xs.findIdx? p).or (if p a then some xs.size else none) := by
|
||||
simp only [push_eq_append, findIdx?_append]
|
||||
@@ -604,7 +553,7 @@ theorem findIdx?_flatten {xss : Array (Array α)} {p : α → Bool} :
|
||||
cases xss using array₂_induction
|
||||
simp [List.findIdx?_flatten, Function.comp_def]
|
||||
|
||||
@[simp, grind =] theorem findIdx?_replicate :
|
||||
@[simp] theorem findIdx?_replicate :
|
||||
(replicate n a).findIdx? p = if 0 < n ∧ p a then some 0 else none := by
|
||||
rw [← List.toArray_replicate]
|
||||
simp only [List.findIdx?_toArray]
|
||||
@@ -629,7 +578,6 @@ theorem findIdx?_eq_none_of_findIdx?_eq_none {xs : Array α} {p q : α → Bool}
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.findIdx?_eq_none_of_findIdx?_eq_none (by simpa using w)
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_eq_getD_findIdx? {xs : Array α} {p : α → Bool} :
|
||||
xs.findIdx p = (xs.findIdx? p).getD xs.size := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -646,17 +594,15 @@ theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : Array α} {p q : α → Bo
|
||||
cases xs
|
||||
simp [hf]
|
||||
|
||||
@[simp, grind =] theorem findIdx?_take {xs : Array α} {i : Nat} {p : α → Bool} :
|
||||
@[simp] theorem findIdx?_take {xs : Array α} {i : Nat} {p : α → Bool} :
|
||||
(xs.take i).findIdx? p = (xs.findIdx? p).bind (Option.guard (fun j => j < i)) := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
/-! ### findFinIdx? -/
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p #[] = none := by simp
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
|
||||
#[a].findFinIdx? p = if p a then some ⟨0, by simp⟩ else none := by
|
||||
simp
|
||||
@@ -674,7 +620,7 @@ theorem findFinIdx?_eq_pmap_findIdx? {xs : Array α} {p : α → Bool} :
|
||||
(fun i h => h) := by
|
||||
simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]
|
||||
|
||||
@[simp, grind =] theorem findFinIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
|
||||
@[simp] theorem findFinIdx?_eq_none_iff {xs : Array α} {p : α → Bool} :
|
||||
xs.findFinIdx? p = none ↔ ∀ x, x ∈ xs → ¬ p x := by
|
||||
simp [findFinIdx?_eq_pmap_findIdx?]
|
||||
|
||||
@@ -690,14 +636,12 @@ theorem findFinIdx?_eq_some_iff {xs : Array α} {p : α → Bool} {i : Fin xs.si
|
||||
· rintro ⟨h, w⟩
|
||||
exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_push {xs : Array α} {a : α} {p : α → Bool} :
|
||||
(xs.push a).findFinIdx? p =
|
||||
((xs.findFinIdx? p).map (Fin.castLE (by simp))).or (if p a then some ⟨xs.size, by simp⟩ else none) := by
|
||||
simp only [findFinIdx?_eq_pmap_findIdx?, findIdx?_push, Option.pmap_or]
|
||||
split <;> rename_i h _ <;> split <;> simp [h]
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_append {xs ys : Array α} {p : α → Bool} :
|
||||
(xs ++ ys).findFinIdx? p =
|
||||
((xs.findFinIdx? p).map (Fin.castLE (by simp))).or
|
||||
@@ -707,13 +651,13 @@ theorem findFinIdx?_append {xs ys : Array α} {p : α → Bool} :
|
||||
· simp [h, Option.pmap_map, Option.map_pmap, Nat.add_comm]
|
||||
· simp [h]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isSome_findFinIdx? {xs : Array α} {p : α → Bool} :
|
||||
(xs.findFinIdx? p).isSome = xs.any p := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [Array.size]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isNone_findFinIdx? {xs : Array α} {p : α → Bool} :
|
||||
(xs.findFinIdx? p).isNone = xs.all (fun x => ¬ p x) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
@@ -734,7 +678,6 @@ The verification API for `idxOf` is still incomplete.
|
||||
The lemmas below should be made consistent with those for `findIdx` (and proved using them).
|
||||
-/
|
||||
|
||||
@[grind =]
|
||||
theorem idxOf_append [BEq α] [LawfulBEq α] {xs ys : Array α} {a : α} :
|
||||
(xs ++ ys).idxOf a = if a ∈ xs then xs.idxOf a else ys.idxOf a + xs.size := by
|
||||
rw [idxOf, findIdx_append]
|
||||
@@ -748,23 +691,10 @@ theorem idxOf_eq_size [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∉ xs) : x
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.idxOf_eq_length (by simpa using h)]
|
||||
|
||||
theorem idxOf_lt_length_of_mem [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∈ xs) : xs.idxOf a < xs.size := by
|
||||
theorem idxOf_lt_length [BEq α] [LawfulBEq α] {xs : Array α} (h : a ∈ xs) : xs.idxOf a < xs.size := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.idxOf_lt_length_of_mem (by simpa using h)]
|
||||
simp [List.idxOf_lt_length (by simpa using h)]
|
||||
|
||||
theorem idxOf_le_size [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
xs.idxOf a ≤ xs.size := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.idxOf_le_length]
|
||||
|
||||
grind_pattern idxOf_le_size => xs.idxOf a, xs.size
|
||||
|
||||
theorem idxOf_lt_size_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
xs.idxOf a < xs.size ↔ a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.idxOf_lt_length_iff]
|
||||
|
||||
grind_pattern idxOf_lt_size_iff => xs.idxOf a, xs.size
|
||||
|
||||
/-! ### idxOf?
|
||||
|
||||
@@ -772,20 +702,19 @@ The verification API for `idxOf?` is still incomplete.
|
||||
The lemmas below should be made consistent with those for `findIdx?` (and proved using them).
|
||||
-/
|
||||
|
||||
@[grind =] theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := by simp
|
||||
theorem idxOf?_empty [BEq α] : (#[] : Array α).idxOf? a = none := by simp
|
||||
|
||||
@[simp, grind =] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
xs.idxOf? a = none ↔ a ∉ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.idxOf?_eq_none_iff]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isSome_idxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
(xs.idxOf? a).isSome ↔ a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
|
||||
@[grind =]
|
||||
theorem isNone_idxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
(xs.idxOf? a).isNone = ¬ a ∈ xs := by
|
||||
simp
|
||||
@@ -800,9 +729,9 @@ theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : Array α} {a : α} :
|
||||
xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
|
||||
simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
|
||||
|
||||
@[grind =] theorem finIdxOf?_empty [BEq α] : (#[] : Array α).finIdxOf? a = none := by simp
|
||||
theorem finIdxOf?_empty [BEq α] : (#[] : Array α).finIdxOf? a = none := by simp
|
||||
|
||||
@[simp, grind =] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
xs.finIdxOf? a = none ↔ a ∉ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.finIdxOf?_eq_none_iff, Array.size]
|
||||
@@ -813,16 +742,14 @@ theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : Array α} {a : α} :
|
||||
unfold Array.size at i ⊢
|
||||
simp [List.finIdxOf?_eq_some_iff]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem isSome_finIdxOf? [BEq α] [PartialEquivBEq α] {xs : Array α} {a : α} :
|
||||
(xs.finIdxOf? a).isSome = xs.contains a := by
|
||||
@[simp]
|
||||
theorem isSome_finIdxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
(xs.finIdxOf? a).isSome ↔ a ∈ xs := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [Array.size]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem isNone_finIdxOf? [BEq α] [PartialEquivBEq α] {xs : Array α} {a : α} :
|
||||
(xs.finIdxOf? a).isNone = !xs.contains a := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [Array.size]
|
||||
theorem isNone_finIdxOf? [BEq α] [LawfulBEq α] {xs : Array α} {a : α} :
|
||||
(xs.finIdxOf? a).isNone = ¬ a ∈ xs := by
|
||||
simp
|
||||
|
||||
end Array
|
||||
|
||||
@@ -89,8 +89,6 @@ theorem size_pos_of_mem {a : α} {xs : Array α} (h : a ∈ xs) : 0 < xs.size :=
|
||||
simp only [mem_toArray] at h
|
||||
simpa using List.length_pos_of_mem h
|
||||
|
||||
grind_pattern size_pos_of_mem => a ∈ xs, xs.size
|
||||
|
||||
theorem exists_mem_of_size_pos {xs : Array α} (h : 0 < xs.size) : ∃ a, a ∈ xs := by
|
||||
cases xs
|
||||
simpa using List.exists_mem_of_length_pos h
|
||||
@@ -1500,19 +1498,6 @@ theorem forall_mem_filter {p : α → Bool} {xs : Array α} {P : α → Prop} :
|
||||
(∀ (i) (_ : i ∈ xs.filter p), P i) ↔ ∀ (j) (_ : j ∈ xs), p j → P j := by
|
||||
simp
|
||||
|
||||
@[grind] theorem getElem_filter {xs : Array α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).size) :
|
||||
p (xs.filter p)[i] :=
|
||||
(mem_filter.mp (getElem_mem h)).2
|
||||
|
||||
theorem getElem?_filter {xs : Array α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).size)
|
||||
(w : (xs.filter p)[i]? = some a) : p a := by
|
||||
rw [getElem?_eq_getElem] at w
|
||||
simp only [Option.some.injEq] at w
|
||||
rw [← w]
|
||||
apply getElem_filter h
|
||||
|
||||
grind_pattern getElem?_filter => (xs.filter p)[i]?, some a
|
||||
|
||||
@[simp] theorem filter_filter {p q : α → Bool} {xs : Array α} :
|
||||
filter p (filter q xs) = filter (fun a => p a && q a) xs := by
|
||||
apply ext'
|
||||
@@ -3636,8 +3621,8 @@ We can prove that two folds over the same array are related (by some arbitrary r
|
||||
if we know that the initial elements are related and the folding function, for each element of the array,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldl_rel {xs : Array α} {f : β → α → β} {g : γ → α → γ} {a : β} {b : γ} {r : β → γ → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f c a) (g c' a)) :
|
||||
theorem foldl_rel {xs : Array α} {f g : β → α → β} {a b : β} {r : β → β → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
|
||||
r (xs.foldl (fun acc a => f acc a) a) (xs.foldl (fun acc a => g acc a) b) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.foldl_rel h (by simpa using h')
|
||||
@@ -3647,8 +3632,8 @@ We can prove that two folds over the same array are related (by some arbitrary r
|
||||
if we know that the initial elements are related and the folding function, for each element of the array,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldr_rel {xs : Array α} {f : α → β → β} {g : α → γ → γ} {a : β} {b : γ} {r : β → γ → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f a c) (g a c')) :
|
||||
theorem foldr_rel {xs : Array α} {f g : α → β → β} {a b : β} {r : β → β → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
|
||||
r (xs.foldr (fun a acc => f a acc) a) (xs.foldr (fun a acc => g a acc) b) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simpa using List.foldr_rel h (by simpa using h')
|
||||
@@ -4541,7 +4526,7 @@ abbrev contains_def [DecidableEq α] {a : α} {xs : Array α} : xs.contains a
|
||||
(zip xs ys).size = min xs.size ys.size :=
|
||||
size_zipWith
|
||||
|
||||
@[simp, grind =] theorem getElem_zipWith {xs : Array α} {ys : Array β} {f : α → β → γ} {i : Nat}
|
||||
@[simp] theorem getElem_zipWith {xs : Array α} {ys : Array β} {f : α → β → γ} {i : Nat}
|
||||
(hi : i < (zipWith f xs ys).size) :
|
||||
(zipWith f xs ys)[i] = f (xs[i]'(by simp at hi; omega)) (ys[i]'(by simp at hi; omega)) := by
|
||||
cases xs
|
||||
|
||||
@@ -51,27 +51,27 @@ theorem mapFinIdx_spec {xs : Array α} {f : (i : Nat) → α → (h : i < xs.siz
|
||||
∀ i h, p i ((Array.mapFinIdx xs f)[i]) h :=
|
||||
(mapFinIdx_induction _ _ (fun _ => True) trivial p fun _ _ _ => ⟨hs .., trivial⟩).2
|
||||
|
||||
@[simp, grind =] theorem size_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
@[simp] theorem size_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
(xs.mapFinIdx f).size = xs.size :=
|
||||
(mapFinIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
|
||||
|
||||
@[simp, grind =] theorem size_zipIdx {xs : Array α} {k : Nat} : (xs.zipIdx k).size = xs.size :=
|
||||
@[simp] theorem size_zipIdx {xs : Array α} {k : Nat} : (xs.zipIdx k).size = xs.size :=
|
||||
Array.size_mapFinIdx
|
||||
|
||||
@[deprecated size_zipIdx (since := "2025-01-21")] abbrev size_zipWithIndex := @size_zipIdx
|
||||
|
||||
@[simp, grind =] theorem getElem_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {i : Nat}
|
||||
@[simp] theorem getElem_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {i : Nat}
|
||||
(h : i < (xs.mapFinIdx f).size) :
|
||||
(xs.mapFinIdx f)[i] = f i (xs[i]'(by simp_all)) (by simp_all) :=
|
||||
(mapFinIdx_spec (p := fun i b h => b = f i xs[i] h) fun _ _ => rfl).2 i _
|
||||
|
||||
@[simp, grind =] theorem getElem?_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {i : Nat} :
|
||||
@[simp] theorem getElem?_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} {i : Nat} :
|
||||
(xs.mapFinIdx f)[i]? =
|
||||
xs[i]?.pbind fun b h => some <| f i b (getElem?_eq_some_iff.1 h).1 := by
|
||||
simp only [getElem?_def, size_mapFinIdx, getElem_mapFinIdx]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem toList_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
@[simp] theorem toList_mapFinIdx {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
(xs.mapFinIdx f).toList = xs.toList.mapFinIdx (fun i a h => f i a (by simpa)) := by
|
||||
apply List.ext_getElem <;> simp
|
||||
|
||||
@@ -91,20 +91,20 @@ theorem mapIdx_spec {f : Nat → α → β} {xs : Array α}
|
||||
∀ i h, p i ((xs.mapIdx f)[i]) h :=
|
||||
(mapIdx_induction (motive := fun _ => True) trivial fun _ _ _ => ⟨hs .., trivial⟩).2
|
||||
|
||||
@[simp, grind =] theorem size_mapIdx {f : Nat → α → β} {xs : Array α} : (xs.mapIdx f).size = xs.size :=
|
||||
@[simp] theorem size_mapIdx {f : Nat → α → β} {xs : Array α} : (xs.mapIdx f).size = xs.size :=
|
||||
(mapIdx_spec (p := fun _ _ _ => True) (hs := fun _ _ => trivial)).1
|
||||
|
||||
@[simp, grind =] theorem getElem_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat}
|
||||
@[simp] theorem getElem_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat}
|
||||
(h : i < (xs.mapIdx f).size) :
|
||||
(xs.mapIdx f)[i] = f i (xs[i]'(by simp_all)) :=
|
||||
(mapIdx_spec (p := fun i b h => b = f i xs[i]) fun _ _ => rfl).2 i (by simp_all)
|
||||
|
||||
@[simp, grind =] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat} :
|
||||
@[simp] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Array α} {i : Nat} :
|
||||
(xs.mapIdx f)[i]? =
|
||||
xs[i]?.map (f i) := by
|
||||
simp [getElem?_def, size_mapIdx, getElem_mapIdx]
|
||||
|
||||
@[simp, grind =] theorem toList_mapIdx {f : Nat → α → β} {xs : Array α} :
|
||||
@[simp] theorem toList_mapIdx {f : Nat → α → β} {xs : Array α} :
|
||||
(xs.mapIdx f).toList = xs.toList.mapIdx (fun i a => f i a) := by
|
||||
apply List.ext_getElem <;> simp
|
||||
|
||||
@@ -126,7 +126,7 @@ namespace Array
|
||||
|
||||
/-! ### zipIdx -/
|
||||
|
||||
@[simp, grind =] theorem getElem_zipIdx {xs : Array α} {k : Nat} {i : Nat} (h : i < (xs.zipIdx k).size) :
|
||||
@[simp] theorem getElem_zipIdx {xs : Array α} {k : Nat} {i : Nat} (h : i < (xs.zipIdx k).size) :
|
||||
(xs.zipIdx k)[i] = (xs[i]'(by simp_all), k + i) := by
|
||||
simp [zipIdx]
|
||||
|
||||
@@ -140,7 +140,7 @@ abbrev getElem_zipWithIndex := @getElem_zipIdx
|
||||
@[deprecated zipIdx_toArray (since := "2025-01-21")]
|
||||
abbrev zipWithIndex_toArray := @zipIdx_toArray
|
||||
|
||||
@[simp, grind =] theorem toList_zipIdx {xs : Array α} {k : Nat} :
|
||||
@[simp] theorem toList_zipIdx {xs : Array α} {k : Nat} :
|
||||
(xs.zipIdx k).toList = xs.toList.zipIdx k := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
@@ -185,7 +185,7 @@ abbrev mem_zipWithIndex_iff_getElem? := @mem_zipIdx_iff_getElem?
|
||||
subst w
|
||||
rfl
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #[] f = #[] :=
|
||||
rfl
|
||||
|
||||
@@ -195,7 +195,6 @@ theorem mapFinIdx_eq_ofFn {xs : Array α} {f : (i : Nat) → α → (h : i < xs.
|
||||
simp only [List.mapFinIdx_toArray, List.mapFinIdx_eq_ofFn, Fin.getElem_fin, List.getElem_toArray]
|
||||
simp [Array.size]
|
||||
|
||||
@[grind =]
|
||||
theorem mapFinIdx_append {xs ys : Array α} {f : (i : Nat) → α → (h : i < (xs ++ ys).size) → β} :
|
||||
(xs ++ ys).mapFinIdx f =
|
||||
xs.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
|
||||
@@ -204,7 +203,7 @@ theorem mapFinIdx_append {xs ys : Array α} {f : (i : Nat) → α → (h : i < (
|
||||
cases ys
|
||||
simp [List.mapFinIdx_append, Array.size]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapFinIdx_push {xs : Array α} {a : α} {f : (i : Nat) → α → (h : i < (xs.push a).size) → β} :
|
||||
mapFinIdx (xs.push a) f =
|
||||
(mapFinIdx xs (fun i a h => f i a (by simp; omega))).push (f xs.size a (by simp)) := by
|
||||
@@ -238,7 +237,7 @@ theorem exists_of_mem_mapFinIdx {b : β} {xs : Array α} {f : (i : Nat) → α
|
||||
rcases xs with ⟨xs⟩
|
||||
exact List.exists_of_mem_mapFinIdx (by simpa using h)
|
||||
|
||||
@[simp, grind =] theorem mem_mapFinIdx {b : β} {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
@[simp] theorem mem_mapFinIdx {b : β} {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
b ∈ xs.mapFinIdx f ↔ ∃ (i : Nat) (h : i < xs.size), f i xs[i] h = b := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
@@ -291,7 +290,7 @@ theorem mapFinIdx_eq_mapFinIdx_iff {xs : Array α} {f g : (i : Nat) → α → (
|
||||
rw [eq_comm, mapFinIdx_eq_iff]
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_mapFinIdx {xs : Array α}
|
||||
@[simp] theorem mapFinIdx_mapFinIdx {xs : Array α}
|
||||
{f : (i : Nat) → α → (h : i < xs.size) → β}
|
||||
{g : (i : Nat) → β → (h : i < (xs.mapFinIdx f).size) → γ} :
|
||||
(xs.mapFinIdx f).mapFinIdx g = xs.mapFinIdx (fun i a h => g i (f i a h) (by simpa using h)) := by
|
||||
@@ -306,14 +305,14 @@ theorem mapFinIdx_eq_replicate_iff {xs : Array α} {f : (i : Nat) → α → (h
|
||||
@[deprecated mapFinIdx_eq_replicate_iff (since := "2025-03-18")]
|
||||
abbrev mapFinIdx_eq_mkArray_iff := @mapFinIdx_eq_replicate_iff
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_reverse {xs : Array α} {f : (i : Nat) → α → (h : i < xs.reverse.size) → β} :
|
||||
@[simp] theorem mapFinIdx_reverse {xs : Array α} {f : (i : Nat) → α → (h : i < xs.reverse.size) → β} :
|
||||
xs.reverse.mapFinIdx f = (xs.mapFinIdx (fun i a h => f (xs.size - 1 - i) a (by simp; omega))).reverse := by
|
||||
rcases xs with ⟨l⟩
|
||||
simp [List.mapFinIdx_reverse, Array.size]
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #[] = #[] :=
|
||||
rfl
|
||||
|
||||
@@ -333,14 +332,13 @@ theorem mapIdx_eq_zipIdx_map {xs : Array α} {f : Nat → α → β} :
|
||||
@[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-21")]
|
||||
abbrev mapIdx_eq_zipWithIndex_map := @mapIdx_eq_zipIdx_map
|
||||
|
||||
@[grind =]
|
||||
theorem mapIdx_append {xs ys : Array α} :
|
||||
(xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx (fun i => f (i + xs.size)) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
rcases ys with ⟨ys⟩
|
||||
simp [List.mapIdx_append]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapIdx_push {xs : Array α} {a : α} :
|
||||
mapIdx f (xs.push a) = (mapIdx f xs).push (f xs.size a) := by
|
||||
simp [← append_singleton, mapIdx_append]
|
||||
@@ -362,7 +360,7 @@ theorem exists_of_mem_mapIdx {b : β} {xs : Array α}
|
||||
rw [mapIdx_eq_mapFinIdx] at h
|
||||
simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
|
||||
|
||||
@[simp, grind =] theorem mem_mapIdx {b : β} {xs : Array α} :
|
||||
@[simp] theorem mem_mapIdx {b : β} {xs : Array α} :
|
||||
b ∈ mapIdx f xs ↔ ∃ (i : Nat) (h : i < xs.size), f i xs[i] = b := by
|
||||
constructor
|
||||
· intro h
|
||||
@@ -416,7 +414,7 @@ theorem mapIdx_eq_mapIdx_iff {xs : Array α} :
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.mapIdx_eq_mapIdx_iff]
|
||||
|
||||
@[simp, grind =] theorem mapIdx_set {f : Nat → α → β} {xs : Array α} {i : Nat} {h : i < xs.size} {a : α} :
|
||||
@[simp] theorem mapIdx_set {f : Nat → α → β} {xs : Array α} {i : Nat} {h : i < xs.size} {a : α} :
|
||||
(xs.set i a).mapIdx f = (xs.mapIdx f).set i (f i a) (by simpa) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.mapIdx_set]
|
||||
@@ -426,17 +424,17 @@ theorem mapIdx_eq_mapIdx_iff {xs : Array α} :
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.mapIdx_set]
|
||||
|
||||
@[simp, grind =] theorem back?_mapIdx {xs : Array α} {f : Nat → α → β} :
|
||||
@[simp] theorem back?_mapIdx {xs : Array α} {f : Nat → α → β} :
|
||||
(mapIdx f xs).back? = (xs.back?).map (f (xs.size - 1)) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.getLast?_mapIdx]
|
||||
|
||||
@[simp, grind =] theorem back_mapIdx {xs : Array α} {f : Nat → α → β} (h) :
|
||||
@[simp] theorem back_mapIdx {xs : Array α} {f : Nat → α → β} (h) :
|
||||
(xs.mapIdx f).back h = f (xs.size - 1) (xs.back (by simpa using h)) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.getLast_mapIdx]
|
||||
|
||||
@[simp, grind =] theorem mapIdx_mapIdx {xs : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
|
||||
@[simp] theorem mapIdx_mapIdx {xs : Array α} {f : Nat → α → β} {g : Nat → β → γ} :
|
||||
(xs.mapIdx f).mapIdx g = xs.mapIdx (fun i => g i ∘ f i) := by
|
||||
simp [mapIdx_eq_iff]
|
||||
|
||||
@@ -449,7 +447,7 @@ theorem mapIdx_eq_replicate_iff {xs : Array α} {f : Nat → α → β} {b : β}
|
||||
@[deprecated mapIdx_eq_replicate_iff (since := "2025-03-18")]
|
||||
abbrev mapIdx_eq_mkArray_iff := @mapIdx_eq_replicate_iff
|
||||
|
||||
@[simp, grind =] theorem mapIdx_reverse {xs : Array α} {f : Nat → α → β} :
|
||||
@[simp] theorem mapIdx_reverse {xs : Array α} {f : Nat → α → β} :
|
||||
xs.reverse.mapIdx f = (mapIdx (fun i => f (xs.size - 1 - i)) xs).reverse := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [List.mapIdx_reverse]
|
||||
@@ -458,7 +456,7 @@ end Array
|
||||
|
||||
namespace List
|
||||
|
||||
@[grind =] theorem mapFinIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
|
||||
@[grind] theorem mapFinIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
|
||||
{f : (i : Nat) → α → (h : i < l.length) → m β} :
|
||||
l.toArray.mapFinIdxM f = toArray <$> l.mapFinIdxM f := by
|
||||
let rec go (i : Nat) (acc : Array β) (inv : i + acc.size = l.length) :
|
||||
@@ -479,7 +477,7 @@ namespace List
|
||||
simp only [Array.mapFinIdxM, mapFinIdxM]
|
||||
exact go _ #[] _
|
||||
|
||||
@[grind =] theorem mapIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
|
||||
@[grind] theorem mapIdxM_toArray [Monad m] [LawfulMonad m] {l : List α}
|
||||
{f : Nat → α → m β} :
|
||||
l.toArray.mapIdxM f = toArray <$> l.mapIdxM f := by
|
||||
let rec go (bs : List α) (acc : Array β) (inv : bs.length + acc.size = l.length) :
|
||||
|
||||
@@ -23,7 +23,7 @@ namespace Array
|
||||
|
||||
/-! ### ofFn -/
|
||||
|
||||
@[simp, grind =] theorem ofFn_zero {f : Fin 0 → α} : ofFn f = #[] := by
|
||||
@[simp] theorem ofFn_zero {f : Fin 0 → α} : ofFn f = #[] := by
|
||||
simp [ofFn, ofFn.go]
|
||||
|
||||
theorem ofFn_succ {f : Fin (n+1) → α} :
|
||||
@@ -42,10 +42,10 @@ theorem ofFn_add {n m} {f : Fin (n + m) → α} :
|
||||
| zero => simp
|
||||
| succ m ih => simp [ofFn_succ, ih]
|
||||
|
||||
@[simp, grind =] theorem _root_.List.toArray_ofFn {f : Fin n → α} : (List.ofFn f).toArray = Array.ofFn f := by
|
||||
@[simp] theorem _root_.List.toArray_ofFn {f : Fin n → α} : (List.ofFn f).toArray = Array.ofFn f := by
|
||||
ext <;> simp
|
||||
|
||||
@[simp, grind =] theorem toList_ofFn {f : Fin n → α} : (Array.ofFn f).toList = List.ofFn f := by
|
||||
@[simp] theorem toList_ofFn {f : Fin n → α} : (Array.ofFn f).toList = List.ofFn f := by
|
||||
apply List.ext_getElem <;> simp
|
||||
|
||||
theorem ofFn_succ' {f : Fin (n+1) → α} :
|
||||
@@ -58,7 +58,7 @@ theorem ofFn_eq_empty_iff {f : Fin n → α} : ofFn f = #[] ↔ n = 0 := by
|
||||
rw [← Array.toList_inj]
|
||||
simp
|
||||
|
||||
@[simp 500, grind =]
|
||||
@[simp 500]
|
||||
theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
|
||||
constructor
|
||||
· intro w
|
||||
@@ -73,7 +73,7 @@ theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i =
|
||||
def ofFnM {n} [Monad m] (f : Fin n → m α) : m (Array α) :=
|
||||
Fin.foldlM n (fun xs i => xs.push <$> f i) (Array.emptyWithCapacity n)
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem ofFnM_zero [Monad m] {f : Fin 0 → m α} : ofFnM f = pure #[] := by
|
||||
simp [ofFnM]
|
||||
|
||||
@@ -109,7 +109,7 @@ theorem ofFnM_add {n m} [Monad m] [LawfulMonad m] {f : Fin (n + k) → m α} :
|
||||
funext x
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
|
||||
@[simp] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
|
||||
toList <$> ofFnM f = List.ofFnM f := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
|
||||
@@ -91,26 +91,17 @@ theorem Perm.mem_iff {a : α} {xs ys : Array α} (p : xs ~ ys) : a ∈ xs ↔ a
|
||||
simp only [perm_iff_toList_perm] at p
|
||||
simpa using p.mem_iff
|
||||
|
||||
grind_pattern Perm.mem_iff => xs ~ ys, a ∈ xs
|
||||
grind_pattern Perm.mem_iff => xs ~ ys, a ∈ ys
|
||||
|
||||
theorem Perm.append {xs ys as bs : Array α} (p₁ : xs ~ ys) (p₂ : as ~ bs) :
|
||||
xs ++ as ~ ys ++ bs := by
|
||||
cases xs; cases ys; cases as; cases bs
|
||||
simp only [append_toArray, perm_iff_toList_perm] at p₁ p₂ ⊢
|
||||
exact p₁.append p₂
|
||||
|
||||
grind_pattern Perm.append => xs ~ ys, as ~ bs, xs ++ as
|
||||
grind_pattern Perm.append => xs ~ ys, as ~ bs, ys ++ bs
|
||||
|
||||
theorem Perm.push (x : α) {xs ys : Array α} (p : xs ~ ys) :
|
||||
xs.push x ~ ys.push x := by
|
||||
rw [push_eq_append_singleton]
|
||||
exact p.append .rfl
|
||||
|
||||
grind_pattern Perm.push => xs ~ ys, xs.push x
|
||||
grind_pattern Perm.push => xs ~ ys, ys.push x
|
||||
|
||||
theorem Perm.push_comm (x y : α) {xs ys : Array α} (p : xs ~ ys) :
|
||||
(xs.push x).push y ~ (ys.push y).push x := by
|
||||
cases xs; cases ys
|
||||
|
||||
@@ -128,16 +128,6 @@ theorem erase_range' :
|
||||
simp only [← List.toArray_range', List.erase_toArray]
|
||||
simp [List.erase_range']
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range' {a s n step} (h : 0 < step := by simp) :
|
||||
count a (range' s n step) = if ∃ i, i < n ∧ a = s + step * i then 1 else 0 := by
|
||||
rw [← List.toArray_range', List.count_toArray, ← List.count_range' h]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range_1' {a s n} :
|
||||
count a (range' s n) = if s ≤ a ∧ a < s + n then 1 else 0 := by
|
||||
rw [← List.toArray_range', List.count_toArray, ← List.count_range_1']
|
||||
|
||||
/-! ### range -/
|
||||
|
||||
@[grind _=_]
|
||||
@@ -201,10 +191,6 @@ theorem self_mem_range_succ {n : Nat} : n ∈ range (n + 1) := by simp
|
||||
theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n - (i + 1)) := by
|
||||
simp [range_eq_range', erase_range']
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range {a n} :
|
||||
count a (range n) = if a < n then 1 else 0 := by
|
||||
rw [← List.toArray_range, List.count_toArray, ← List.count_range]
|
||||
|
||||
/-! ### zipIdx -/
|
||||
|
||||
|
||||
@@ -45,7 +45,6 @@ theorem zipWith_self {f : α → α → δ} {xs : Array α} : zipWith f xs xs =
|
||||
See also `getElem?_zipWith'` for a variant
|
||||
using `Option.map` and `Option.bind` rather than a `match`.
|
||||
-/
|
||||
@[grind =]
|
||||
theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
|
||||
(zipWith f as bs)[i]? = match as[i]?, bs[i]? with
|
||||
| some a, some b => some (f a b) | _, _ => none := by
|
||||
@@ -77,35 +76,31 @@ theorem getElem?_zip_eq_some {as : Array α} {bs : Array β} {z : α × β} {i :
|
||||
· rintro ⟨h₀, h₁⟩
|
||||
exact ⟨_, _, h₀, h₁, rfl⟩
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem zipWith_map {μ} {f : γ → δ → μ} {g : α → γ} {h : β → δ} {as : Array α} {bs : Array β} :
|
||||
zipWith f (as.map g) (bs.map h) = zipWith (fun a b => f (g a) (h b)) as bs := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWith_map]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_map_left {as : Array α} {bs : Array β} {f : α → α'} {g : α' → β → γ} :
|
||||
zipWith g (as.map f) bs = zipWith (fun a b => g (f a) b) as bs := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWith_map_left]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_map_right {as : Array α} {bs : Array β} {f : β → β'} {g : α → β' → γ} :
|
||||
zipWith g as (bs.map f) = zipWith (fun a b => g a (f b)) as bs := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWith_map_right]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} {i : δ} :
|
||||
(zipWith f as bs).foldr g i = (zip as bs).foldr (fun p r => g (f p.1 p.2) r) i := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWith_foldr_eq_zip_foldr]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} :
|
||||
(zipWith f as bs).foldl g i = (zip as bs).foldl (fun r p => g r (f p.1 p.2)) i := by
|
||||
cases as
|
||||
@@ -116,26 +111,22 @@ theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} :
|
||||
theorem zipWith_eq_empty_iff {f : α → β → γ} {as : Array α} {bs : Array β} : zipWith f as bs = #[] ↔ as = #[] ∨ bs = #[] := by
|
||||
cases as <;> cases bs <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {cs : Array γ} {ds : Array δ} :
|
||||
map f (zipWith g cs ds) = zipWith (fun x y => f (g x y)) cs ds := by
|
||||
cases cs
|
||||
cases ds
|
||||
simp [List.map_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem take_zipWith : (zipWith f as bs).take i = zipWith f (as.take i) (bs.take i) := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.take_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem extract_zipWith : (zipWith f as bs).extract i j = zipWith f (as.extract i j) (bs.extract i j) := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.drop_zipWith, List.take_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_append {f : α → β → γ} {as as' : Array α} {bs bs' : Array β}
|
||||
(h : as.size = bs.size) :
|
||||
zipWith f (as ++ as') (bs ++ bs') = zipWith f as bs ++ zipWith f as' bs' := by
|
||||
@@ -161,7 +152,7 @@ theorem zipWith_eq_append_iff {f : α → β → γ} {as : Array α} {bs : Array
|
||||
· rintro ⟨⟨ws⟩, ⟨xs⟩, ⟨ys⟩, ⟨zs⟩, h, rfl, rfl, h₁, h₂⟩
|
||||
exact ⟨ws, xs, ys, zs, by simp_all⟩
|
||||
|
||||
@[simp, grind =] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
|
||||
@[simp] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
|
||||
zipWith f (replicate m a) (replicate n b) = replicate (min m n) (f a b) := by
|
||||
simp [← List.toArray_replicate]
|
||||
|
||||
@@ -193,7 +184,6 @@ theorem zipWith_eq_zipWith_take_min (as : Array α) (bs : Array β) :
|
||||
simp
|
||||
rw [List.zipWith_eq_zipWith_take_min]
|
||||
|
||||
@[grind =]
|
||||
theorem reverse_zipWith (h : as.size = bs.size) :
|
||||
(zipWith f as bs).reverse = zipWith f as.reverse bs.reverse := by
|
||||
cases as
|
||||
@@ -210,7 +200,7 @@ theorem lt_size_right_of_zip {i : Nat} {as : Array α} {bs : Array β} (h : i <
|
||||
i < bs.size :=
|
||||
lt_size_right_of_zipWith h
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem getElem_zip {as : Array α} {bs : Array β} {i : Nat} {h : i < (zip as bs).size} :
|
||||
(zip as bs)[i] =
|
||||
(as[i]'(lt_size_left_of_zip h), bs[i]'(lt_size_right_of_zip h)) :=
|
||||
@@ -221,22 +211,18 @@ theorem zip_eq_zipWith {as : Array α} {bs : Array β} : zip as bs = zipWith Pro
|
||||
cases bs
|
||||
simp [List.zip_eq_zipWith]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map {f : α → γ} {g : β → δ} {as : Array α} {bs : Array β} :
|
||||
zip (as.map f) (bs.map g) = (zip as bs).map (Prod.map f g) := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zip_map]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map_left {f : α → γ} {as : Array α} {bs : Array β} :
|
||||
zip (as.map f) bs = (zip as bs).map (Prod.map f id) := by rw [← zip_map, map_id]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map_right {f : β → γ} {as : Array α} {bs : Array β} :
|
||||
zip as (bs.map f) = (zip as bs).map (Prod.map id f) := by rw [← zip_map, map_id]
|
||||
|
||||
@[grind =]
|
||||
theorem zip_append {as bs : Array α} {cs ds : Array β} (_h : as.size = cs.size) :
|
||||
zip (as ++ bs) (cs ++ ds) = zip as cs ++ zip bs ds := by
|
||||
cases as
|
||||
@@ -245,7 +231,6 @@ theorem zip_append {as bs : Array α} {cs ds : Array β} (_h : as.size = cs.size
|
||||
cases ds
|
||||
simp_all [List.zip_append]
|
||||
|
||||
@[grind =]
|
||||
theorem zip_map' {f : α → β} {g : α → γ} {xs : Array α} :
|
||||
zip (xs.map f) (xs.map g) = xs.map fun a => (f a, g a) := by
|
||||
cases xs
|
||||
@@ -291,7 +276,7 @@ theorem zip_eq_append_iff {as : Array α} {bs : Array β} :
|
||||
∃ as₁ as₂ bs₁ bs₂, as₁.size = bs₁.size ∧ as = as₁ ++ as₂ ∧ bs = bs₁ ++ bs₂ ∧ xs = zip as₁ bs₁ ∧ ys = zip as₂ bs₂ := by
|
||||
simp [zip_eq_zipWith, zipWith_eq_append_iff]
|
||||
|
||||
@[simp, grind =] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
|
||||
@[simp] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
|
||||
zip (replicate m a) (replicate n b) = replicate (min m n) (a, b) := by
|
||||
simp [← List.toArray_replicate]
|
||||
|
||||
@@ -308,7 +293,6 @@ theorem zip_eq_zip_take_min {as : Array α} {bs : Array β} :
|
||||
|
||||
/-! ### zipWithAll -/
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
|
||||
(zipWithAll f as bs)[i]? = match as[i]?, bs[i]? with
|
||||
| none, none => .none | a?, b? => some (f a? b?) := by
|
||||
@@ -317,35 +301,31 @@ theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
|
||||
simp [List.getElem?_zipWithAll]
|
||||
rfl
|
||||
|
||||
@[grind =]
|
||||
theorem zipWithAll_map {μ} {f : Option γ → Option δ → μ} {g : α → γ} {h : β → δ} {as : Array α} {bs : Array β} :
|
||||
zipWithAll f (as.map g) (bs.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) as bs := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWithAll_map]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWithAll_map_left {as : Array α} {bs : Array β} {f : α → α'} {g : Option α' → Option β → γ} :
|
||||
zipWithAll g (as.map f) bs = zipWithAll (fun a b => g (f <$> a) b) as bs := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWithAll_map_left]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWithAll_map_right {as : Array α} {bs : Array β} {f : β → β'} {g : Option α → Option β' → γ} :
|
||||
zipWithAll g as (bs.map f) = zipWithAll (fun a b => g a (f <$> b)) as bs := by
|
||||
cases as
|
||||
cases bs
|
||||
simp [List.zipWithAll_map_right]
|
||||
|
||||
@[grind =]
|
||||
theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option δ → α} {cs : Array γ} {ds : Array δ} :
|
||||
map f (zipWithAll g cs ds) = zipWithAll (fun x y => f (g x y)) cs ds := by
|
||||
cases cs
|
||||
cases ds
|
||||
simp [List.map_zipWithAll]
|
||||
|
||||
@[simp, grind =] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
|
||||
@[simp] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
|
||||
zipWithAll f (replicate n a) (replicate n b) = replicate n (f (some a) (some b)) := by
|
||||
simp [← List.toArray_replicate]
|
||||
|
||||
@@ -362,7 +342,6 @@ theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
|
||||
theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
|
||||
simp
|
||||
|
||||
@[grind =]
|
||||
theorem unzip_eq_map {xs : Array (α × β)} : unzip xs = (xs.map Prod.fst, xs.map Prod.snd) := by
|
||||
cases xs
|
||||
simp [List.unzip_eq_map]
|
||||
@@ -396,11 +375,9 @@ theorem zip_of_prod {as : Array α} {bs : Array β} {xs : Array (α × β)} (hl
|
||||
(hr : xs.map Prod.snd = bs) : xs = as.zip bs := by
|
||||
rw [← hl, ← hr, ← zip_unzip xs, ← fst_unzip, ← snd_unzip, zip_unzip, zip_unzip]
|
||||
|
||||
@[simp, grind =] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
|
||||
@[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
|
||||
unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
|
||||
ext1 <;> simp
|
||||
|
||||
@[deprecated unzip_replicate (since := "2025-03-18")]
|
||||
abbrev unzip_mkArray := @unzip_replicate
|
||||
|
||||
end Array
|
||||
|
||||
@@ -174,7 +174,7 @@ recommended_spelling "zero" for "0#n" in [BitVec.ofNat, «term__#__»]
|
||||
recommended_spelling "one" for "1#n" in [BitVec.ofNat, «term__#__»]
|
||||
|
||||
/-- Unexpander for bitvector literals. -/
|
||||
@[app_unexpander BitVec.ofNat] meta def unexpandBitVecOfNat : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander BitVec.ofNat] def unexpandBitVecOfNat : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $n $i:num) => `($i:num#$n)
|
||||
| _ => throw ()
|
||||
|
||||
@@ -183,7 +183,7 @@ scoped syntax:max term:max noWs "#'" noWs term:max : term
|
||||
macro_rules | `($i#'$p) => `(BitVec.ofNatLT $i $p)
|
||||
|
||||
/-- Unexpander for bitvector literals without truncation. -/
|
||||
@[app_unexpander BitVec.ofNatLT] meta def unexpandBitVecOfNatLt : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander BitVec.ofNatLT] def unexpandBitVecOfNatLt : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $i $p) => `($i#'$p)
|
||||
| _ => throw ()
|
||||
|
||||
|
||||
@@ -1023,7 +1023,7 @@ theorem DivModState.toNat_shiftRight_sub_one_eq
|
||||
{args : DivModArgs w} {qr : DivModState w} (h : qr.Poised args) :
|
||||
args.n.toNat >>> (qr.wn - 1)
|
||||
= (args.n.toNat >>> qr.wn) * 2 + (args.n.getLsbD (qr.wn - 1)).toNat := by
|
||||
change BitVec.toNat (args.n >>> (qr.wn - 1)) = _
|
||||
show BitVec.toNat (args.n >>> (qr.wn - 1)) = _
|
||||
have {..} := h -- break the structure down for `omega`
|
||||
rw [shiftRight_sub_one_eq_shiftConcat args.n h.hwn_lt]
|
||||
rw [toNat_shiftConcat_eq_of_lt (k := w - qr.wn)]
|
||||
|
||||
@@ -882,19 +882,6 @@ theorem slt_eq_sle_and_ne {x y : BitVec w} : x.slt y = (x.sle y && x != y) := by
|
||||
apply Bool.eq_iff_iff.2
|
||||
simp [BitVec.slt, BitVec.sle, Int.lt_iff_le_and_ne, BitVec.toInt_inj]
|
||||
|
||||
/-- For all bitvectors `x, y`, either `x` is signed less than `y`,
|
||||
or is equal to `y`, or is signed greater than `y`. -/
|
||||
theorem slt_trichotomy (x y : BitVec w) : x.slt y ∨ x = y ∨ y.slt x := by
|
||||
simpa [slt_iff_toInt_lt, ← toInt_inj]
|
||||
using Int.lt_trichotomy x.toInt y.toInt
|
||||
|
||||
/-- For all bitvectors `x, y`, either `x` is unsigned less than `y`,
|
||||
or is equal to `y`, or is unsigned greater than `y`. -/
|
||||
theorem lt_trichotomy (x y : BitVec w) :
|
||||
x < y ∨ x = y ∨ y < x := by
|
||||
simpa [← ult_iff_lt, ult_eq_decide, decide_eq_true_eq, ← toNat_inj]
|
||||
using Nat.lt_trichotomy x.toNat y.toNat
|
||||
|
||||
/-! ### setWidth, zeroExtend and truncate -/
|
||||
|
||||
@[simp]
|
||||
@@ -3354,17 +3341,6 @@ theorem toNat_add_of_not_uaddOverflow {x y : BitVec w} (h : ¬ uaddOverflow x y)
|
||||
· simp only [uaddOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h
|
||||
rw [toNat_add, Nat.mod_eq_of_lt h]
|
||||
|
||||
/--
|
||||
Unsigned addition overflow reassociation.
|
||||
If `(x + y)` and `(y + z)` do not overflow, then `(x + y) + z` overflows iff `x + (y + z)` overflows.
|
||||
-/
|
||||
theorem uaddOverflow_assoc {x y z : BitVec w} (h : ¬ x.uaddOverflow y) (h' : ¬ y.uaddOverflow z) :
|
||||
(x + y).uaddOverflow z = x.uaddOverflow (y + z) := by
|
||||
simp only [uaddOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h h'
|
||||
simp only [uaddOverflow, toNat_add, ge_iff_le, decide_eq_decide]
|
||||
repeat rw [Nat.mod_eq_of_lt (by omega)]
|
||||
omega
|
||||
|
||||
protected theorem add_assoc (x y z : BitVec n) : x + y + z = x + (y + z) := by
|
||||
apply eq_of_toNat_eq ; simp [Nat.add_assoc]
|
||||
instance : Std.Associative (α := BitVec n) (· + ·) := ⟨BitVec.add_assoc⟩
|
||||
@@ -3403,20 +3379,6 @@ theorem toInt_add_of_not_saddOverflow {x y : BitVec w} (h : ¬ saddOverflow x y)
|
||||
_root_.not_or, Int.not_le, Int.not_lt] at h
|
||||
rw [toInt_add, Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]
|
||||
|
||||
/--
|
||||
Signed addition overflow reassociation.
|
||||
If `(x + y)` and `(y + z)` do not overflow, then `(x + y) + z` overflows iff `x + (y + z)` overflows.
|
||||
-/
|
||||
theorem saddOverflow_assoc {x y z : BitVec w} (h : ¬ x.saddOverflow y) (h' : ¬ y.saddOverflow z) :
|
||||
(x + y).saddOverflow z = x.saddOverflow (y + z) := by
|
||||
rcases w with _|w
|
||||
· simp [of_length_zero]
|
||||
· simp only [saddOverflow, Nat.add_one_sub_one, ge_iff_le, Bool.or_eq_true, decide_eq_true_eq,
|
||||
_root_.not_or, Int.not_le, Int.not_lt] at h h'
|
||||
simp only [bool_to_prop, saddOverflow, toInt_add, ge_iff_le, Nat.add_one_sub_one]
|
||||
repeat rw [Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]
|
||||
omega
|
||||
|
||||
@[simp]
|
||||
theorem shiftLeft_add_distrib {x y : BitVec w} {n : Nat} :
|
||||
(x + y) <<< n = x <<< n + y <<< n := by
|
||||
@@ -3845,18 +3807,6 @@ theorem toNat_mul_of_not_umulOverflow {x y : BitVec w} (h : ¬ umulOverflow x y)
|
||||
· simp only [umulOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h
|
||||
rw [toNat_mul, Nat.mod_eq_of_lt h]
|
||||
|
||||
/--
|
||||
Unsigned multiplication overflow reassociation.
|
||||
If `(x * y)` and `(y * z)` do not overflow, then `(x * y) * z` overflows iff `x * (y * z)` overflows.
|
||||
-/
|
||||
theorem umulOverflow_assoc {x y z : BitVec w} (h : ¬ x.umulOverflow y) (h' : ¬ y.umulOverflow z) :
|
||||
|
||||
(x * y).umulOverflow z = x.umulOverflow (y * z) := by
|
||||
simp only [umulOverflow, ge_iff_le, decide_eq_true_eq, Nat.not_le] at h h'
|
||||
simp only [umulOverflow, toNat_mul, ge_iff_le, decide_eq_decide]
|
||||
repeat rw [Nat.mod_eq_of_lt (by omega)]
|
||||
rw [Nat.mul_assoc]
|
||||
|
||||
@[simp]
|
||||
theorem toInt_mul_of_not_smulOverflow {x y : BitVec w} (h : ¬ smulOverflow x y) :
|
||||
(x * y).toInt = x.toInt * y.toInt := by
|
||||
@@ -3866,20 +3816,6 @@ theorem toInt_mul_of_not_smulOverflow {x y : BitVec w} (h : ¬ smulOverflow x y)
|
||||
_root_.not_or, Int.not_le, Int.not_lt] at h
|
||||
rw [toInt_mul, Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]
|
||||
|
||||
/--
|
||||
Signed multiplication overflow reassociation.
|
||||
If `(x * y)` and `(y * z)` do not overflow, then `(x * y) * z` overflows iff `x * (y * z)` overflows.
|
||||
-/
|
||||
theorem smulOverflow_assoc {x y z : BitVec w} (h : ¬ x.smulOverflow y) (h' : ¬ y.smulOverflow z) :
|
||||
(x * y).smulOverflow z = x.smulOverflow (y * z) := by
|
||||
rcases w with _|w
|
||||
· simp [of_length_zero]
|
||||
· simp only [smulOverflow, Nat.add_one_sub_one, ge_iff_le, Bool.or_eq_true, decide_eq_true_eq,
|
||||
_root_.not_or, Int.not_le, Int.not_lt] at h h'
|
||||
simp only [smulOverflow, toInt_mul, Nat.add_one_sub_one, ge_iff_le, bool_to_prop]
|
||||
repeat rw [Int.bmod_eq_of_le (by push_cast; omega) (by push_cast; omega)]
|
||||
rw [Int.mul_assoc]
|
||||
|
||||
theorem ofInt_mul {n} (x y : Int) : BitVec.ofInt n (x * y) =
|
||||
BitVec.ofInt n x * BitVec.ofInt n y := by
|
||||
apply eq_of_toInt_eq
|
||||
|
||||
@@ -183,7 +183,10 @@ theorem foldrM_loop [Monad m] [LawfulMonad m] (f : Fin (n+1) → α → m α) (x
|
||||
| zero =>
|
||||
rw [foldrM_loop_zero, foldrM_loop_succ, pure_bind]
|
||||
conv => rhs; rw [←bind_pure (f 0 x)]
|
||||
rfl
|
||||
congr
|
||||
try -- TODO: block can be deleted after bootstrapping
|
||||
funext
|
||||
simp [foldrM_loop_zero]
|
||||
| succ i ih =>
|
||||
rw [foldrM_loop_succ, foldrM_loop_succ, bind_assoc]
|
||||
congr; funext; exact ih ..
|
||||
|
||||
@@ -1079,17 +1079,6 @@ theorem val_neg {n : Nat} [NeZero n] (x : Fin n) :
|
||||
have := Fin.val_ne_zero_iff.mpr h
|
||||
omega
|
||||
|
||||
protected theorem sub_eq_add_neg {n : Nat} (x y : Fin n) : x - y = x + -y := by
|
||||
by_cases h : n = 0
|
||||
· subst h
|
||||
apply elim0 x
|
||||
· replace h : NeZero n := ⟨h⟩
|
||||
ext
|
||||
rw [Fin.coe_sub, Fin.val_add, val_neg]
|
||||
split
|
||||
· simp_all
|
||||
· simp [Nat.add_comm]
|
||||
|
||||
/-! ### mul -/
|
||||
|
||||
theorem ofNat_mul [NeZero n] (x : Nat) (y : Fin n) :
|
||||
|
||||
@@ -3,6 +3,7 @@ Copyright (c) 2016 Jeremy Avigad. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Jeremy Avigad, Mario Carneiro
|
||||
-/
|
||||
|
||||
module
|
||||
|
||||
prelude
|
||||
@@ -98,7 +99,7 @@ theorem ofNat_emod (m n : Nat) : (↑(m % n) : Int) = m % n := natCast_emod m n
|
||||
theorem emod_add_ediv : ∀ a b : Int, a % b + b * (a / b) = a
|
||||
| ofNat _, ofNat _ => congrArg ofNat <| Nat.mod_add_div ..
|
||||
| ofNat m, -[n+1] => by
|
||||
change (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
|
||||
show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
|
||||
rw [Int.neg_mul_neg]; exact congrArg ofNat <| Nat.mod_add_div ..
|
||||
| -[_+1], 0 => by rw [emod_zero]; rfl
|
||||
| -[m+1], succ n => aux m n.succ
|
||||
@@ -148,7 +149,7 @@ theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c
|
||||
fun {k n} => @fun
|
||||
| ofNat _ => congrArg ofNat <| Nat.add_mul_div_right _ _ k.succ_pos
|
||||
| -[m+1] => by
|
||||
change ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat)
|
||||
show ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat)
|
||||
by_cases h : m < n * k.succ
|
||||
· rw [← Int.ofNat_sub h, ← Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)]
|
||||
apply congrArg ofNat
|
||||
@@ -157,7 +158,7 @@ theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c
|
||||
have H {a b : Nat} (h : a ≤ b) : (a : Int) + -((b : Int) + 1) = -[b - a +1] := by
|
||||
rw [negSucc_eq, Int.ofNat_sub h]
|
||||
simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc]
|
||||
change ediv (↑(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(↑(m / succ k) + 1 : Int)
|
||||
show ediv (↑(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(↑(m / succ k) + 1 : Int)
|
||||
rw [H h, H ((Nat.le_div_iff_mul_le k.succ_pos).2 h)]
|
||||
apply congrArg negSucc
|
||||
rw [Nat.mul_comm, Nat.sub_mul_div_of_le]; rwa [Nat.mul_comm]
|
||||
|
||||
@@ -3,6 +3,7 @@ Copyright (c) 2016 Jeremy Avigad. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Jeremy Avigad, Mario Carneiro, Kim Morrison, Markus Himmel
|
||||
-/
|
||||
|
||||
module
|
||||
|
||||
prelude
|
||||
@@ -202,9 +203,6 @@ theorem tdiv_eq_ediv_of_nonneg : ∀ {a b : Int}, 0 ≤ a → a.tdiv b = a / b
|
||||
| succ _, succ _, _ => rfl
|
||||
| succ _, -[_+1], _ => rfl
|
||||
|
||||
@[simp] theorem natCast_tdiv_eq_ediv {a : Nat} {b : Int} : (a : Int).tdiv b = a / b :=
|
||||
tdiv_eq_ediv_of_nonneg (by simp)
|
||||
|
||||
theorem tdiv_eq_ediv {a b : Int} :
|
||||
a.tdiv b = a / b + if 0 ≤ a ∨ b ∣ a then 0 else sign b := by
|
||||
simp only [dvd_iff_emod_eq_zero]
|
||||
@@ -331,17 +329,17 @@ theorem fdiv_eq_ediv_of_dvd {a b : Int} (h : b ∣ a) : a.fdiv b = a / b := by
|
||||
theorem tmod_add_tdiv : ∀ a b : Int, tmod a b + b * (a.tdiv b) = a
|
||||
| ofNat _, ofNat _ => congrArg ofNat (Nat.mod_add_div ..)
|
||||
| ofNat m, -[n+1] => by
|
||||
change (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
|
||||
show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m
|
||||
rw [Int.neg_mul_neg]; exact congrArg ofNat (Nat.mod_add_div ..)
|
||||
| -[m+1], 0 => by
|
||||
change -(↑((succ m) % 0) : Int) + 0 * -↑(succ m / 0) = -↑(succ m)
|
||||
show -(↑((succ m) % 0) : Int) + 0 * -↑(succ m / 0) = -↑(succ m)
|
||||
rw [Nat.mod_zero, Int.zero_mul, Int.add_zero]
|
||||
| -[m+1], ofNat n => by
|
||||
change -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m)
|
||||
show -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m)
|
||||
rw [Int.mul_neg, ← Int.neg_add]
|
||||
exact congrArg (-ofNat ·) (Nat.mod_add_div ..)
|
||||
| -[m+1], -[n+1] => by
|
||||
change -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)
|
||||
show -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)
|
||||
rw [Int.neg_mul, ← Int.neg_add]
|
||||
exact congrArg (-ofNat ·) (Nat.mod_add_div ..)
|
||||
|
||||
@@ -363,17 +361,17 @@ theorem fmod_add_fdiv : ∀ a b : Int, a.fmod b + b * a.fdiv b = a
|
||||
| 0, ofNat _ | 0, -[_+1] => congrArg ofNat <| by simp
|
||||
| succ _, ofNat _ => congrArg ofNat <| Nat.mod_add_div ..
|
||||
| succ m, -[n+1] => by
|
||||
change subNatNat (m % succ n) n + (↑(succ n * (m / succ n)) + n + 1) = (m + 1)
|
||||
show subNatNat (m % succ n) n + (↑(succ n * (m / succ n)) + n + 1) = (m + 1)
|
||||
rw [Int.add_comm _ n, ← Int.add_assoc, ← Int.add_assoc,
|
||||
Int.subNatNat_eq_coe, Int.sub_add_cancel]
|
||||
exact congrArg (ofNat · + 1) <| Nat.mod_add_div ..
|
||||
| -[_+1], 0 => by rw [fmod_zero]; rfl
|
||||
| -[m+1], succ n => by
|
||||
change subNatNat .. - (↑(succ n * (m / succ n)) + ↑(succ n)) = -↑(succ m)
|
||||
show subNatNat .. - (↑(succ n * (m / succ n)) + ↑(succ n)) = -↑(succ m)
|
||||
rw [Int.subNatNat_eq_coe, ← Int.sub_sub, ← Int.neg_sub, Int.sub_sub, Int.sub_sub_self]
|
||||
exact congrArg (-ofNat ·) <| Nat.succ_add .. ▸ Nat.mod_add_div .. ▸ rfl
|
||||
| -[m+1], -[n+1] => by
|
||||
change -(↑(succ m % succ n) : Int) + -↑(succ n * (succ m / succ n)) = -↑(succ m)
|
||||
show -(↑(succ m % succ n) : Int) + -↑(succ n * (succ m / succ n)) = -↑(succ m)
|
||||
rw [← Int.neg_add]; exact congrArg (-ofNat ·) <| Nat.mod_add_div ..
|
||||
|
||||
/-- Variant of `fmod_add_fdiv` with the multiplication written the other way around. -/
|
||||
@@ -574,7 +572,7 @@ theorem neg_one_ediv (b : Int) : -1 / b = -b.sign :=
|
||||
· refine Nat.le_trans ?_ (Nat.le_add_right _ _)
|
||||
rw [← Nat.mul_div_mul_left _ _ m.succ_pos]
|
||||
apply Nat.div_mul_le_self
|
||||
· change m.succ * n.succ ≤ _
|
||||
· show m.succ * n.succ ≤ _
|
||||
rw [Nat.mul_left_comm]
|
||||
apply Nat.mul_le_mul_left
|
||||
apply (Nat.div_lt_iff_lt_mul k.succ_pos).1
|
||||
@@ -2747,7 +2745,7 @@ theorem bmod_lt {x : Int} {m : Nat} (h : 0 < m) : bmod x m < (m + 1) / 2 := by
|
||||
split
|
||||
· assumption
|
||||
· apply Int.lt_of_lt_of_le
|
||||
· change _ < 0
|
||||
· show _ < 0
|
||||
have : x % m < m := emod_lt_of_pos x (natCast_pos.mpr h)
|
||||
exact Int.sub_neg_of_lt this
|
||||
· exact Int.le.intro_sub _ rfl
|
||||
|
||||
@@ -339,7 +339,7 @@ protected theorem add_sub_assoc (a b c : Int) : a + b - c = a + (b - c) := by
|
||||
match m with
|
||||
| 0 => rfl
|
||||
| succ m =>
|
||||
change ofNat (n - succ m) = subNatNat n (succ m)
|
||||
show ofNat (n - succ m) = subNatNat n (succ m)
|
||||
rw [subNatNat, Nat.sub_eq_zero_of_le h]
|
||||
|
||||
@[deprecated negSucc_eq (since := "2025-03-11")]
|
||||
|
||||
@@ -1665,7 +1665,7 @@ theorem natCast_sub (x y : Nat)
|
||||
(NatCast.natCast x : Int) + -1*NatCast.natCast y
|
||||
else
|
||||
(0 : Int) := by
|
||||
change (↑(x - y) : Int) = if (↑y : Int) + (-1)*↑x ≤ 0 then (↑x : Int) + (-1)*↑y else 0
|
||||
show (↑(x - y) : Int) = if (↑y : Int) + (-1)*↑x ≤ 0 then ↑x + (-1)*↑y else 0
|
||||
rw [Int.neg_mul, ← Int.sub_eq_add_neg, Int.one_mul]
|
||||
rw [Int.neg_mul, ← Int.sub_eq_add_neg, Int.one_mul]
|
||||
split
|
||||
|
||||
@@ -19,13 +19,6 @@ protected theorem pow_succ (b : Int) (e : Nat) : b ^ (e+1) = (b ^ e) * b := rfl
|
||||
protected theorem pow_succ' (b : Int) (e : Nat) : b ^ (e+1) = b * (b ^ e) := by
|
||||
rw [Int.mul_comm, Int.pow_succ]
|
||||
|
||||
protected theorem zero_pow {n : Nat} (h : n ≠ 0) : (0 : Int) ^ n = 0 := by
|
||||
match n, h with
|
||||
| n + 1, _ => simp [Int.pow_succ]
|
||||
|
||||
protected theorem one_pow {n : Nat} : (1 : Int) ^ n = 1 := by
|
||||
induction n with simp_all [Int.pow_succ]
|
||||
|
||||
protected theorem pow_pos {n : Int} {m : Nat} : 0 < n → 0 < n ^ m := by
|
||||
induction m with
|
||||
| zero => simp
|
||||
|
||||
@@ -9,7 +9,6 @@ prelude
|
||||
import Init.SimpLemmas
|
||||
import Init.Data.Nat.Basic
|
||||
import Init.Data.List.Notation
|
||||
import Init.Data.Nat.Div.Basic
|
||||
|
||||
@[expose] section
|
||||
|
||||
@@ -1625,8 +1624,8 @@ def find? (p : α → Bool) : List α → Option α
|
||||
| true => some a
|
||||
| false => find? p as
|
||||
|
||||
@[simp, grind =] theorem find?_nil : ([] : List α).find? p = none := rfl
|
||||
@[grind =]theorem find?_cons : (a::as).find? p = match p a with | true => some a | false => as.find? p :=
|
||||
@[simp] theorem find?_nil : ([] : List α).find? p = none := rfl
|
||||
theorem find?_cons : (a::as).find? p = match p a with | true => some a | false => as.find? p :=
|
||||
rfl
|
||||
|
||||
/-! ### findSome? -/
|
||||
@@ -1846,8 +1845,8 @@ def lookup [BEq α] : α → List (α × β) → Option β
|
||||
| true => some b
|
||||
| false => lookup a as
|
||||
|
||||
@[simp, grind =] theorem lookup_nil [BEq α] : ([] : List (α × β)).lookup a = none := rfl
|
||||
@[grind =] theorem lookup_cons [BEq α] {k : α} :
|
||||
@[simp] theorem lookup_nil [BEq α] : ([] : List (α × β)).lookup a = none := rfl
|
||||
theorem lookup_cons [BEq α] {k : α} :
|
||||
((k, b)::as).lookup a = match a == k with | true => some b | false => as.lookup a :=
|
||||
rfl
|
||||
|
||||
|
||||
@@ -64,8 +64,8 @@ theorem length_eq_countP_add_countP (p : α → Bool) {l : List α} : length l =
|
||||
· rfl
|
||||
· simp [h]
|
||||
|
||||
@[grind _=_] -- This to quite aggressive, as it introduces `filter` based reasoning whenever we see `countP`.
|
||||
theorem countP_eq_length_filter {l : List α} : countP p l = (filter p l).length := by
|
||||
@[grind =]
|
||||
theorem countP_eq_length_filter {l : List α} : countP p l = length (filter p l) := by
|
||||
induction l with
|
||||
| nil => rfl
|
||||
| cons x l ih =>
|
||||
@@ -82,7 +82,7 @@ theorem countP_le_length : countP p l ≤ l.length := by
|
||||
simp only [countP_eq_length_filter]
|
||||
apply length_filter_le
|
||||
|
||||
@[simp, grind =] theorem countP_append {l₁ l₂ : List α} : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
|
||||
@[simp] theorem countP_append {l₁ l₂ : List α} : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
|
||||
simp only [countP_eq_length_filter, filter_append, length_append]
|
||||
|
||||
@[simp] theorem countP_pos_iff {p} : 0 < countP p l ↔ ∃ a ∈ l, p a := by
|
||||
@@ -120,24 +120,10 @@ theorem Sublist.countP_le (s : l₁ <+ l₂) : countP p l₁ ≤ countP p l₂ :
|
||||
simp only [countP_eq_length_filter]
|
||||
apply s.filter _ |>.length_le
|
||||
|
||||
grind_pattern Sublist.countP_le => l₁ <+ l₂, countP p l₁
|
||||
grind_pattern Sublist.countP_le => l₁ <+ l₂, countP p l₂
|
||||
|
||||
theorem IsPrefix.countP_le (s : l₁ <+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le
|
||||
|
||||
grind_pattern IsPrefix.countP_le => l₁ <+: l₂, countP p l₁
|
||||
grind_pattern IsPrefix.countP_le => l₁ <+: l₂, countP p l₂
|
||||
|
||||
theorem IsSuffix.countP_le (s : l₁ <:+ l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le
|
||||
|
||||
grind_pattern IsSuffix.countP_le => l₁ <:+ l₂, countP p l₁
|
||||
grind_pattern IsSuffix.countP_le => l₁ <:+ l₂, countP p l₂
|
||||
|
||||
theorem IsInfix.countP_le (s : l₁ <:+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le
|
||||
|
||||
grind_pattern IsInfix.countP_le => l₁ <:+: l₂, countP p l₁
|
||||
grind_pattern IsInfix.countP_le => l₁ <:+: l₂, countP p l₂
|
||||
|
||||
-- See `Init.Data.List.Nat.Count` for `Sublist.le_countP : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁`.
|
||||
|
||||
@[grind]
|
||||
@@ -188,7 +174,7 @@ theorem countP_flatMap {p : β → Bool} {l : List α} {f : α → List β} :
|
||||
countP p (l.flatMap f) = sum (map (countP p ∘ f) l) := by
|
||||
rw [List.flatMap, countP_flatten, map_map]
|
||||
|
||||
@[simp, grind =] theorem countP_reverse {l : List α} : countP p l.reverse = countP p l := by
|
||||
@[simp] theorem countP_reverse {l : List α} : countP p l.reverse = countP p l := by
|
||||
simp [countP_eq_length_filter, filter_reverse]
|
||||
|
||||
theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
|
||||
@@ -217,22 +203,18 @@ section count
|
||||
|
||||
variable [BEq α]
|
||||
|
||||
@[simp, grind =] theorem count_nil {a : α} : count a [] = 0 := rfl
|
||||
@[simp] theorem count_nil {a : α} : count a [] = 0 := rfl
|
||||
|
||||
@[grind]
|
||||
theorem count_cons {a b : α} {l : List α} :
|
||||
count a (b :: l) = count a l + if b == a then 1 else 0 := by
|
||||
simp [count, countP_cons]
|
||||
|
||||
theorem count_eq_countP {a : α} {l : List α} : count a l = countP (· == a) l := rfl
|
||||
@[grind =] theorem count_eq_countP {a : α} {l : List α} : count a l = countP (· == a) l := rfl
|
||||
theorem count_eq_countP' {a : α} : count a = countP (· == a) := by
|
||||
funext l
|
||||
apply count_eq_countP
|
||||
|
||||
@[grind =]
|
||||
theorem count_eq_length_filter {a : α} {l : List α} : count a l = (filter (· == a) l).length := by
|
||||
simp [count, countP_eq_length_filter]
|
||||
|
||||
@[grind]
|
||||
theorem count_tail : ∀ {l : List α} {a : α},
|
||||
l.tail.count a = l.count a - if l.head? == some a then 1 else 0
|
||||
@@ -241,28 +223,12 @@ theorem count_tail : ∀ {l : List α} {a : α},
|
||||
|
||||
theorem count_le_length {a : α} {l : List α} : count a l ≤ l.length := countP_le_length
|
||||
|
||||
grind_pattern count_le_length => count a l
|
||||
|
||||
theorem Sublist.count_le (a : α) (h : l₁ <+ l₂) : count a l₁ ≤ count a l₂ := h.countP_le
|
||||
|
||||
grind_pattern Sublist.count_le => l₁ <+ l₂, count a l₁
|
||||
grind_pattern Sublist.count_le => l₁ <+ l₂, count a l₂
|
||||
|
||||
theorem IsPrefix.count_le (a : α) (h : l₁ <+: l₂) : count a l₁ ≤ count a l₂ := h.sublist.count_le a
|
||||
|
||||
grind_pattern IsPrefix.count_le => l₁ <+: l₂, count a l₁
|
||||
grind_pattern IsPrefix.count_le => l₁ <+: l₂, count a l₂
|
||||
|
||||
theorem IsSuffix.count_le (a : α) (h : l₁ <:+ l₂) : count a l₁ ≤ count a l₂ := h.sublist.count_le a
|
||||
|
||||
grind_pattern IsSuffix.count_le => l₁ <:+ l₂, count a l₁
|
||||
grind_pattern IsSuffix.count_le => l₁ <:+ l₂, count a l₂
|
||||
|
||||
theorem IsInfix.count_le (a : α) (h : l₁ <:+: l₂) : count a l₁ ≤ count a l₂ := h.sublist.count_le a
|
||||
|
||||
grind_pattern IsInfix.count_le => l₁ <:+: l₂, count a l₁
|
||||
grind_pattern IsInfix.count_le => l₁ <:+: l₂, count a l₂
|
||||
|
||||
-- See `Init.Data.List.Nat.Count` for `Sublist.le_count : count a l₂ - (l₂.length - l₁.length) ≤ countP a l₁`.
|
||||
|
||||
theorem count_tail_le {a : α} {l : List α} : count a l.tail ≤ count a l :=
|
||||
@@ -279,11 +245,10 @@ theorem count_singleton {a b : α} : count a [b] = if b == a then 1 else 0 := by
|
||||
@[simp, grind =] theorem count_append {a : α} {l₁ l₂ : List α} : count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
|
||||
countP_append
|
||||
|
||||
@[grind =]
|
||||
theorem count_flatten {a : α} {l : List (List α)} : count a l.flatten = (l.map (count a)).sum := by
|
||||
simp only [count_eq_countP, countP_flatten, count_eq_countP']
|
||||
|
||||
@[simp, grind =] theorem count_reverse {a : α} {l : List α} : count a l.reverse = count a l := by
|
||||
@[simp] theorem count_reverse {a : α} {l : List α} : count a l.reverse = count a l := by
|
||||
simp only [count_eq_countP, countP_eq_length_filter, filter_reverse, length_reverse]
|
||||
|
||||
@[grind]
|
||||
|
||||
@@ -23,9 +23,9 @@ open Nat
|
||||
|
||||
/-! ### eraseP -/
|
||||
|
||||
@[simp, grind =] theorem eraseP_nil : [].eraseP p = [] := rfl
|
||||
@[simp] theorem eraseP_nil : [].eraseP p = [] := rfl
|
||||
|
||||
@[grind =] theorem eraseP_cons {a : α} {l : List α} :
|
||||
theorem eraseP_cons {a : α} {l : List α} :
|
||||
(a :: l).eraseP p = bif p a then l else a :: l.eraseP p := rfl
|
||||
|
||||
@[simp] theorem eraseP_cons_of_pos {l : List α} {p} (h : p a) : (a :: l).eraseP p = l := by
|
||||
@@ -92,7 +92,7 @@ theorem exists_or_eq_self_of_eraseP (p) (l : List α) :
|
||||
let ⟨_, l₁, l₂, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
|
||||
rw [e₂]; simp [length_append, e₁]
|
||||
|
||||
@[grind =] theorem length_eraseP {l : List α} : (l.eraseP p).length = if l.any p then l.length - 1 else l.length := by
|
||||
theorem length_eraseP {l : List α} : (l.eraseP p).length = if l.any p then l.length - 1 else l.length := by
|
||||
split <;> rename_i h
|
||||
· simp only [any_eq_true] at h
|
||||
obtain ⟨x, m, h⟩ := h
|
||||
@@ -106,13 +106,8 @@ theorem eraseP_sublist {l : List α} : l.eraseP p <+ l := by
|
||||
| .inl h => rw [h]; apply Sublist.refl
|
||||
| .inr ⟨c, l₁, l₂, _, _, h₃, h₄⟩ => rw [h₄, h₃]; simp
|
||||
|
||||
grind_pattern eraseP_sublist => l.eraseP p, List.Sublist
|
||||
|
||||
theorem eraseP_subset {l : List α} : l.eraseP p ⊆ l := eraseP_sublist.subset
|
||||
|
||||
grind_pattern eraseP_subset => l.eraseP p, List.Subset
|
||||
|
||||
@[grind ←]
|
||||
protected theorem Sublist.eraseP : l₁ <+ l₂ → l₁.eraseP p <+ l₂.eraseP p
|
||||
| .slnil => Sublist.refl _
|
||||
| .cons a s => by
|
||||
@@ -152,12 +147,10 @@ theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (erase
|
||||
· intro; obtain ⟨x, m, h⟩ := h; simp_all
|
||||
· simp_all
|
||||
|
||||
@[grind _=_]
|
||||
theorem eraseP_map {f : β → α} : ∀ {l : List β}, (map f l).eraseP p = map f (l.eraseP (p ∘ f))
|
||||
| [] => rfl
|
||||
| b::l => by by_cases h : p (f b) <;> simp [h, eraseP_map, eraseP_cons_of_pos]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_filterMap {f : α → Option β} : ∀ {l : List α},
|
||||
(filterMap f l).eraseP p = filterMap f (l.eraseP (fun x => match f x with | some y => p y | none => false))
|
||||
| [] => rfl
|
||||
@@ -172,7 +165,6 @@ theorem eraseP_filterMap {f : α → Option β} : ∀ {l : List α},
|
||||
· simp only [w, cond_false]
|
||||
rw [filterMap_cons_some h, eraseP_filterMap]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_filter {f : α → Bool} {l : List α} :
|
||||
(filter f l).eraseP p = filter f (l.eraseP (fun x => p x && f x)) := by
|
||||
rw [← filterMap_eq_filter, eraseP_filterMap]
|
||||
@@ -182,19 +174,18 @@ theorem eraseP_filter {f : α → Bool} {l : List α} :
|
||||
split <;> split at * <;> simp_all
|
||||
|
||||
theorem eraseP_append_left {a : α} (pa : p a) :
|
||||
∀ {l₁ : List α} l₂, a ∈ l₁ → (l₁ ++ l₂).eraseP p = l₁.eraseP p ++ l₂
|
||||
∀ {l₁ : List α} l₂, a ∈ l₁ → (l₁++l₂).eraseP p = l₁.eraseP p ++ l₂
|
||||
| x :: xs, l₂, h => by
|
||||
by_cases h' : p x <;> simp [h']
|
||||
rw [eraseP_append_left pa l₂ ((mem_cons.1 h).resolve_left (mt _ h'))]
|
||||
intro | rfl => exact pa
|
||||
|
||||
theorem eraseP_append_right :
|
||||
∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁ ++ l₂) = l₁ ++ l₂.eraseP p
|
||||
∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁++l₂) = l₁ ++ l₂.eraseP p
|
||||
| [], _, _ => rfl
|
||||
| _ :: _, _, h => by
|
||||
simp [(forall_mem_cons.1 h).1, eraseP_append_right _ (forall_mem_cons.1 h).2]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_append {l₁ l₂ : List α} :
|
||||
(l₁ ++ l₂).eraseP p = if l₁.any p then l₁.eraseP p ++ l₂ else l₁ ++ l₂.eraseP p := by
|
||||
split <;> rename_i h
|
||||
@@ -205,7 +196,6 @@ theorem eraseP_append {l₁ l₂ : List α} :
|
||||
rw [eraseP_append_right _]
|
||||
simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_replicate {n : Nat} {a : α} {p : α → Bool} :
|
||||
(replicate n a).eraseP p = if p a then replicate (n - 1) a else replicate n a := by
|
||||
induction n with
|
||||
@@ -222,7 +212,6 @@ theorem eraseP_replicate {n : Nat} {a : α} {p : α → Bool} :
|
||||
(replicate n a).eraseP p = replicate n a := by
|
||||
rw [eraseP_of_forall_not (by simp_all)]
|
||||
|
||||
@[grind ←]
|
||||
protected theorem IsPrefix.eraseP (h : l₁ <+: l₂) : l₁.eraseP p <+: l₂.eraseP p := by
|
||||
rw [IsPrefix] at h
|
||||
obtain ⟨t, rfl⟩ := h
|
||||
@@ -269,15 +258,13 @@ theorem eraseP_eq_iff {p} {l : List α} :
|
||||
subst p
|
||||
simp_all
|
||||
|
||||
@[grind ←]
|
||||
theorem Pairwise.eraseP (q) : Pairwise p l → Pairwise p (l.eraseP q) :=
|
||||
Pairwise.sublist <| eraseP_sublist
|
||||
|
||||
@[grind ←]
|
||||
@[grind]
|
||||
theorem Nodup.eraseP (p) : Nodup l → Nodup (l.eraseP p) :=
|
||||
Pairwise.eraseP p
|
||||
|
||||
@[grind =]
|
||||
theorem eraseP_comm {l : List α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
|
||||
(l.eraseP p).eraseP q = (l.eraseP q).eraseP p := by
|
||||
induction l with
|
||||
@@ -370,7 +357,6 @@ theorem exists_erase_eq [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
|
||||
length (l.erase a) = length l - 1 := by
|
||||
rw [erase_eq_eraseP]; exact length_eraseP_of_mem h (beq_self_eq_true a)
|
||||
|
||||
@[grind =]
|
||||
theorem length_erase [LawfulBEq α] {a : α} {l : List α} :
|
||||
length (l.erase a) = if a ∈ l then length l - 1 else length l := by
|
||||
rw [erase_eq_eraseP, length_eraseP]
|
||||
@@ -379,17 +365,11 @@ theorem length_erase [LawfulBEq α] {a : α} {l : List α} :
|
||||
theorem erase_sublist {a : α} {l : List α} : l.erase a <+ l :=
|
||||
erase_eq_eraseP' a l ▸ eraseP_sublist ..
|
||||
|
||||
grind_pattern length_erase => l.erase a, List.Sublist
|
||||
|
||||
theorem erase_subset {a : α} {l : List α} : l.erase a ⊆ l := erase_sublist.subset
|
||||
|
||||
grind_pattern erase_subset => l.erase a, List.Subset
|
||||
|
||||
@[grind ←]
|
||||
theorem Sublist.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₁.erase a <+ l₂.erase a := by
|
||||
simp only [erase_eq_eraseP']; exact h.eraseP
|
||||
|
||||
@[grind ←]
|
||||
theorem IsPrefix.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+: l₂) : l₁.erase a <+: l₂.erase a := by
|
||||
simp only [erase_eq_eraseP']; exact h.eraseP
|
||||
|
||||
@@ -411,7 +391,6 @@ theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈
|
||||
rw [erase_eq_eraseP', eraseP_eq_self_iff]
|
||||
simp [forall_mem_ne']
|
||||
|
||||
@[grind _=_]
|
||||
theorem erase_filter [LawfulBEq α] {f : α → Bool} {l : List α} :
|
||||
(filter f l).erase a = filter f (l.erase a) := by
|
||||
induction l with
|
||||
@@ -439,12 +418,10 @@ theorem erase_append_right [LawfulBEq α] {a : α} {l₁ : List α} (l₂ : List
|
||||
rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right]
|
||||
intros b h' h''; rw [eq_of_beq h''] at h; exact h h'
|
||||
|
||||
@[grind =]
|
||||
theorem erase_append [LawfulBEq α] {a : α} {l₁ l₂ : List α} :
|
||||
(l₁ ++ l₂).erase a = if a ∈ l₁ then l₁.erase a ++ l₂ else l₁ ++ l₂.erase a := by
|
||||
simp [erase_eq_eraseP, eraseP_append]
|
||||
|
||||
@[grind =]
|
||||
theorem erase_replicate [LawfulBEq α] {n : Nat} {a b : α} :
|
||||
(replicate n a).erase b = if b == a then replicate (n - 1) a else replicate n a := by
|
||||
rw [erase_eq_eraseP]
|
||||
@@ -452,7 +429,6 @@ theorem erase_replicate [LawfulBEq α] {n : Nat} {a b : α} :
|
||||
|
||||
-- The arguments `a b` are explicit,
|
||||
-- so they can be specified to prevent `simp` repeatedly applying the lemma.
|
||||
@[grind =]
|
||||
theorem erase_comm [LawfulBEq α] (a b : α) {l : List α} :
|
||||
(l.erase a).erase b = (l.erase b).erase a := by
|
||||
if ab : a == b then rw [eq_of_beq ab] else ?_
|
||||
@@ -492,7 +468,6 @@ theorem erase_eq_iff [LawfulBEq α] {a : α} {l : List α} :
|
||||
rw [erase_of_not_mem]
|
||||
simp_all
|
||||
|
||||
@[grind ←]
|
||||
theorem Pairwise.erase [LawfulBEq α] {l : List α} (a) : Pairwise p l → Pairwise p (l.erase a) :=
|
||||
Pairwise.sublist <| erase_sublist
|
||||
|
||||
@@ -545,7 +520,6 @@ end erase
|
||||
|
||||
/-! ### eraseIdx -/
|
||||
|
||||
@[grind =]
|
||||
theorem length_eraseIdx {l : List α} {i : Nat} :
|
||||
(l.eraseIdx i).length = if i < l.length then l.length - 1 else l.length := by
|
||||
induction l generalizing i with
|
||||
@@ -563,9 +537,8 @@ theorem length_eraseIdx_of_lt {l : List α} {i} (h : i < length l) :
|
||||
(l.eraseIdx i).length = length l - 1 := by
|
||||
simp [length_eraseIdx, h]
|
||||
|
||||
@[simp, grind =] theorem eraseIdx_zero {l : List α} : eraseIdx l 0 = l.tail := by cases l <;> rfl
|
||||
@[simp] theorem eraseIdx_zero {l : List α} : eraseIdx l 0 = l.tail := by cases l <;> rfl
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_eq_take_drop_succ :
|
||||
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
|
||||
| nil, _ => by simp
|
||||
@@ -592,7 +565,6 @@ theorem eraseIdx_ne_nil_iff {l : List α} {i : Nat} : eraseIdx l i ≠ [] ↔ 2
|
||||
@[deprecated eraseIdx_ne_nil_iff (since := "2025-01-30")]
|
||||
abbrev eraseIdx_ne_nil := @eraseIdx_ne_nil_iff
|
||||
|
||||
@[grind]
|
||||
theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
|
||||
| [], _ => by simp
|
||||
| a::l, 0 => by simp
|
||||
@@ -601,7 +573,6 @@ theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
|
||||
theorem mem_of_mem_eraseIdx {l : List α} {i : Nat} {a : α} (h : a ∈ l.eraseIdx i) : a ∈ l :=
|
||||
(eraseIdx_sublist _ _).mem h
|
||||
|
||||
@[grind]
|
||||
theorem eraseIdx_subset {l : List α} {k : Nat} : eraseIdx l k ⊆ l :=
|
||||
(eraseIdx_sublist _ _).subset
|
||||
|
||||
@@ -641,15 +612,6 @@ theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤
|
||||
| zero => simp_all
|
||||
| succ k => simp_all [eraseIdx_cons_succ, Nat.succ_sub_succ]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_append :
|
||||
eraseIdx (l ++ l') k = if k < length l then eraseIdx l k ++ l' else l ++ eraseIdx l' (k - length l) := by
|
||||
split <;> rename_i h
|
||||
· simp [eraseIdx_append_of_lt_length h]
|
||||
· rw [eraseIdx_append_of_length_le]
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
|
||||
(replicate n a).eraseIdx k = if k < n then replicate (n - 1) a else replicate n a := by
|
||||
split <;> rename_i h
|
||||
@@ -661,15 +623,12 @@ theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
|
||||
exact m.2
|
||||
· rw [eraseIdx_of_length_le (by simpa using h)]
|
||||
|
||||
@[grind ←]
|
||||
theorem Pairwise.eraseIdx {l : List α} (k) : Pairwise p l → Pairwise p (l.eraseIdx k) :=
|
||||
Pairwise.sublist <| eraseIdx_sublist _ _
|
||||
|
||||
@[grind ←]
|
||||
theorem Nodup.eraseIdx {l : List α} (k) : Nodup l → Nodup (l.eraseIdx k) :=
|
||||
Pairwise.eraseIdx k
|
||||
|
||||
@[grind ←]
|
||||
protected theorem IsPrefix.eraseIdx {l l' : List α} (h : l <+: l') (k : Nat) :
|
||||
eraseIdx l k <+: eraseIdx l' k := by
|
||||
rcases h with ⟨t, rfl⟩
|
||||
|
||||
@@ -23,14 +23,14 @@ Examples:
|
||||
-/
|
||||
def finRange (n : Nat) : List (Fin n) := ofFn fun i => i
|
||||
|
||||
@[simp, grind =] theorem length_finRange {n : Nat} : (List.finRange n).length = n := by
|
||||
@[simp] theorem length_finRange {n : Nat} : (List.finRange n).length = n := by
|
||||
simp [List.finRange]
|
||||
|
||||
@[simp, grind =] theorem getElem_finRange {i : Nat} (h : i < (List.finRange n).length) :
|
||||
@[simp] theorem getElem_finRange {i : Nat} (h : i < (List.finRange n).length) :
|
||||
(finRange n)[i] = Fin.cast length_finRange ⟨i, h⟩ := by
|
||||
simp [List.finRange]
|
||||
|
||||
@[simp, grind =] theorem finRange_zero : finRange 0 = [] := by simp [finRange]
|
||||
@[simp] theorem finRange_zero : finRange 0 = [] := by simp [finRange]
|
||||
|
||||
theorem finRange_succ {n} : finRange (n+1) = 0 :: (finRange n).map Fin.succ := by
|
||||
apply List.ext_getElem; simp; intro i; cases i <;> simp
|
||||
@@ -46,7 +46,6 @@ theorem finRange_succ_last {n} :
|
||||
· rfl
|
||||
· next h => exact Fin.eq_last_of_not_lt h
|
||||
|
||||
@[grind _=_]
|
||||
theorem finRange_reverse {n} : (finRange n).reverse = (finRange n).map Fin.rev := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
|
||||
@@ -45,7 +45,7 @@ theorem exists_of_findSome?_eq_some {l : List α} {f : α → Option β} (w : l.
|
||||
simp_all only [findSome?_cons, mem_cons, exists_eq_or_imp]
|
||||
split at w <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
|
||||
@[simp] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
|
||||
induction l <;> simp [findSome?_cons]; split <;> simp [*]
|
||||
|
||||
@[simp] theorem findSome?_isSome_iff {f : α → Option β} {l : List α} :
|
||||
@@ -91,7 +91,7 @@ theorem findSome?_eq_some_iff {f : α → Option β} {l : List α} {b : β} :
|
||||
obtain ⟨⟨rfl, rfl⟩, rfl⟩ := h₁
|
||||
exact ⟨l₁, a, l₂, rfl, h₂, fun a' w => h₃ a' (mem_cons_of_mem p w)⟩
|
||||
|
||||
@[simp, grind =] theorem findSome?_guard {l : List α} : findSome? (Option.guard p) l = find? p l := by
|
||||
@[simp] theorem findSome?_guard {l : List α} : findSome? (Option.guard fun x => p x) l = find? p l := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
@@ -103,33 +103,32 @@ theorem findSome?_eq_some_iff {f : α → Option β} {l : List α} {b : β} :
|
||||
· simp only [Option.guard_eq_none_iff] at h
|
||||
simp [ih, h]
|
||||
|
||||
theorem find?_eq_findSome?_guard {l : List α} : find? p l = findSome? (Option.guard p) l :=
|
||||
theorem find?_eq_findSome?_guard {l : List α} : find? p l = findSome? (Option.guard fun x => p x) l :=
|
||||
findSome?_guard.symm
|
||||
|
||||
@[simp, grind =] theorem head?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).head? = l.findSome? f := by
|
||||
@[simp] theorem head?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).head? = l.findSome? f := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [filterMap_cons, findSome?_cons]
|
||||
split <;> simp [*]
|
||||
|
||||
@[simp, grind =] theorem head_filterMap {f : α → Option β} {l : List α} (h) :
|
||||
@[simp] theorem head_filterMap {f : α → Option β} {l : List α} (h) :
|
||||
(l.filterMap f).head h = (l.findSome? f).get (by simp_all [Option.isSome_iff_ne_none]) := by
|
||||
simp [head_eq_iff_head?_eq_some]
|
||||
|
||||
@[simp, grind =] theorem getLast?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).getLast? = l.reverse.findSome? f := by
|
||||
@[simp] theorem getLast?_filterMap {f : α → Option β} {l : List α} : (l.filterMap f).getLast? = l.reverse.findSome? f := by
|
||||
rw [getLast?_eq_head?_reverse]
|
||||
simp [← filterMap_reverse]
|
||||
|
||||
@[simp, grind =] theorem getLast_filterMap {f : α → Option β} {l : List α} (h) :
|
||||
@[simp] theorem getLast_filterMap {f : α → Option β} {l : List α} (h) :
|
||||
(l.filterMap f).getLast h = (l.reverse.findSome? f).get (by simp_all [Option.isSome_iff_ne_none]) := by
|
||||
simp [getLast_eq_iff_getLast?_eq_some]
|
||||
|
||||
@[simp, grind _=_] theorem map_findSome? {f : α → Option β} {g : β → γ} {l : List α} :
|
||||
@[simp] theorem map_findSome? {f : α → Option β} {g : β → γ} {l : List α} :
|
||||
(l.findSome? f).map g = l.findSome? (Option.map g ∘ f) := by
|
||||
induction l <;> simp [findSome?_cons]; split <;> simp [*]
|
||||
|
||||
@[grind _=_]
|
||||
theorem findSome?_map {f : β → γ} {l : List β} : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
@@ -137,18 +136,15 @@ theorem findSome?_map {f : β → γ} {l : List β} : findSome? p (l.map f) = l.
|
||||
simp only [map_cons, findSome?]
|
||||
split <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem head_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
|
||||
(flatten L).head (by simpa using h) = (L.findSome? head?).get (by simpa using h) := by
|
||||
(flatten L).head (by simpa using h) = (L.findSome? fun l => l.head?).get (by simpa using h) := by
|
||||
simp [head_eq_iff_head?_eq_some, head?_flatten]
|
||||
|
||||
@[grind =]
|
||||
theorem getLast_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
|
||||
(flatten L).getLast (by simpa using h) =
|
||||
(L.reverse.findSome? getLast?).get (by simpa using h) := by
|
||||
(L.reverse.findSome? fun l => l.getLast?).get (by simpa using h) := by
|
||||
simp [getLast_eq_iff_getLast?_eq_some, getLast?_flatten]
|
||||
|
||||
@[grind =]
|
||||
theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
|
||||
cases n with
|
||||
| zero => simp
|
||||
@@ -178,9 +174,6 @@ theorem Sublist.findSome?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
|
||||
· simp_all
|
||||
· exact ih
|
||||
|
||||
grind_pattern Sublist.findSome?_isSome => l₁ <+ l₂, l₁.findSome? f
|
||||
grind_pattern Sublist.findSome?_isSome => l₁ <+ l₂, l₂.findSome? f
|
||||
|
||||
theorem Sublist.findSome?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) :
|
||||
l₂.findSome? f = none → l₁.findSome? f = none := by
|
||||
simp only [List.findSome?_eq_none_iff, Bool.not_eq_true]
|
||||
@@ -192,30 +185,16 @@ theorem IsPrefix.findSome?_eq_some {l₁ l₂ : List α} {f : α → Option β}
|
||||
obtain ⟨t, rfl⟩ := h
|
||||
simp +contextual [findSome?_append]
|
||||
|
||||
grind_pattern IsPrefix.findSome?_eq_some => l₁ <+: l₂, l₁.findSome? f, some b
|
||||
grind_pattern IsPrefix.findSome?_eq_some => l₁ <+: l₂, l₂.findSome? f, some b
|
||||
|
||||
theorem IsPrefix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <+: l₂) :
|
||||
List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
|
||||
h.sublist.findSome?_eq_none
|
||||
|
||||
grind_pattern IsPrefix.findSome?_eq_none => l₁ <+: l₂, l₂.findSome? f
|
||||
grind_pattern IsPrefix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
|
||||
|
||||
theorem IsSuffix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <:+ l₂) :
|
||||
List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
|
||||
h.sublist.findSome?_eq_none
|
||||
|
||||
grind_pattern IsSuffix.findSome?_eq_none => l₁ <+: l₂, l₂.findSome? f
|
||||
grind_pattern IsSuffix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
|
||||
|
||||
theorem IsInfix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (h : l₁ <:+: l₂) :
|
||||
List.findSome? f l₂ = none → List.findSome? f l₁ = none :=
|
||||
h.sublist.findSome?_eq_none
|
||||
|
||||
grind_pattern IsInfix.findSome?_eq_none => l₁ <+: l₂, l₂.findSome? f
|
||||
grind_pattern IsInfix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
|
||||
|
||||
/-! ### find? -/
|
||||
|
||||
@[simp] theorem find?_cons_of_pos {l} (h : p a) : find? p (a :: l) = some a := by
|
||||
@@ -224,7 +203,7 @@ grind_pattern IsInfix.findSome?_eq_none => l₁ <+: l₂, l₁.findSome? f
|
||||
@[simp] theorem find?_cons_of_neg {l} (h : ¬p a) : find? p (a :: l) = find? p l := by
|
||||
simp [find?, h]
|
||||
|
||||
@[simp, grind =] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
|
||||
@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
|
||||
induction l <;> simp [find?_cons]; split <;> simp [*]
|
||||
|
||||
theorem find?_eq_some_iff_append :
|
||||
@@ -269,28 +248,25 @@ theorem find?_cons_eq_some : (a :: xs).find? p = some b ↔ (p a ∧ a = b) ∨
|
||||
rw [find?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem find?_isSome {xs : List α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
@[simp] theorem find?_isSome {xs : List α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [find?_cons, mem_cons, exists_eq_or_imp]
|
||||
split <;> simp_all
|
||||
|
||||
@[grind →]
|
||||
theorem find?_some : ∀ {l}, find? p l = some a → p a
|
||||
| b :: l, H => by
|
||||
by_cases h : p b <;> simp [find?, h] at H
|
||||
· exact H ▸ h
|
||||
· exact find?_some H
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_find?_eq_some : ∀ {l}, find? p l = some a → a ∈ l
|
||||
| b :: l, H => by
|
||||
by_cases h : p b <;> simp [find?, h] at H
|
||||
· exact H ▸ .head _
|
||||
· exact .tail _ (mem_of_find?_eq_some H)
|
||||
|
||||
@[grind]
|
||||
theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h ∈ xs := by
|
||||
induction xs with
|
||||
| nil => simp at h
|
||||
@@ -302,7 +278,7 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
|
||||
right
|
||||
apply ih
|
||||
|
||||
@[simp, grind =] theorem find?_filter {xs : List α} {p : α → Bool} {q : α → Bool} :
|
||||
@[simp] theorem find?_filter {xs : List α} {p : α → Bool} {q : α → Bool} :
|
||||
(xs.filter p).find? q = xs.find? (fun a => p a ∧ q a) := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
@@ -312,22 +288,22 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
|
||||
· simp only [find?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem head?_filter {p : α → Bool} {l : List α} : (l.filter p).head? = l.find? p := by
|
||||
@[simp] theorem head?_filter {p : α → Bool} {l : List α} : (l.filter p).head? = l.find? p := by
|
||||
rw [← filterMap_eq_filter, head?_filterMap, findSome?_guard]
|
||||
|
||||
@[simp, grind =] theorem head_filter {p : α → Bool} {l : List α} (h) :
|
||||
@[simp] theorem head_filter {p : α → Bool} {l : List α} (h) :
|
||||
(l.filter p).head h = (l.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
|
||||
simp [head_eq_iff_head?_eq_some]
|
||||
|
||||
@[simp, grind =] theorem getLast?_filter {p : α → Bool} {l : List α} : (l.filter p).getLast? = l.reverse.find? p := by
|
||||
@[simp] theorem getLast?_filter {p : α → Bool} {l : List α} : (l.filter p).getLast? = l.reverse.find? p := by
|
||||
rw [getLast?_eq_head?_reverse]
|
||||
simp [← filter_reverse]
|
||||
|
||||
@[simp, grind =] theorem getLast_filter {p : α → Bool} {l : List α} (h) :
|
||||
@[simp] theorem getLast_filter {p : α → Bool} {l : List α} (h) :
|
||||
(l.filter p).getLast h = (l.reverse.find? p).get (by simp_all [Option.isSome_iff_ne_none]) := by
|
||||
simp [getLast_eq_iff_getLast?_eq_some]
|
||||
|
||||
@[simp, grind =] theorem find?_filterMap {xs : List α} {f : α → Option β} {p : β → Bool} :
|
||||
@[simp] theorem find?_filterMap {xs : List α} {f : α → Option β} {p : β → Bool} :
|
||||
(xs.filterMap f).find? p = (xs.find? (fun a => (f a).any p)).bind f := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
@@ -337,15 +313,15 @@ theorem get_find?_mem {xs : List α} {p : α → Bool} (h) : (xs.find? p).get h
|
||||
· simp only [find?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem find?_map {f : β → α} {l : List β} : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
|
||||
@[simp] theorem find?_map {f : β → α} {l : List β} : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [map_cons, find?]
|
||||
by_cases h : p (f x) <;> simp [h, ih]
|
||||
|
||||
@[simp, grind _=_] theorem find?_flatten {xss : List (List α)} {p : α → Bool} :
|
||||
xss.flatten.find? p = xss.findSome? (find? p) := by
|
||||
@[simp] theorem find?_flatten {xss : List (List α)} {p : α → Bool} :
|
||||
xss.flatten.find? p = xss.findSome? (·.find? p) := by
|
||||
induction xss with
|
||||
| nil => simp
|
||||
| cons _ _ ih =>
|
||||
@@ -402,7 +378,7 @@ theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a :
|
||||
@[deprecated find?_flatten_eq_some_iff (since := "2025-02-03")]
|
||||
abbrev find?_flatten_eq_some := @find?_flatten_eq_some_iff
|
||||
|
||||
@[simp, grind =] theorem find?_flatMap {xs : List α} {f : α → List β} {p : β → Bool} :
|
||||
@[simp] theorem find?_flatMap {xs : List α} {f : α → List β} {p : β → Bool} :
|
||||
(xs.flatMap f).find? p = xs.findSome? (fun x => (f x).find? p) := by
|
||||
simp [flatMap_def, findSome?_map]; rfl
|
||||
|
||||
@@ -410,7 +386,6 @@ theorem find?_flatMap_eq_none_iff {xs : List α} {f : α → List β} {p : β
|
||||
(xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
|
||||
simp
|
||||
|
||||
@[grind =]
|
||||
theorem find?_replicate : find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
|
||||
cases n
|
||||
· simp
|
||||
@@ -455,9 +430,6 @@ theorem Sublist.find?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) : (l₁.fi
|
||||
· simp
|
||||
· simpa using ih
|
||||
|
||||
grind_pattern Sublist.find?_isSome => l₁ <+ l₂, l₁.find? p
|
||||
grind_pattern Sublist.find?_isSome => l₁ <+ l₂, l₂.find? p
|
||||
|
||||
theorem Sublist.find?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₂.find? p = none → l₁.find? p = none := by
|
||||
simp only [List.find?_eq_none, Bool.not_eq_true]
|
||||
exact fun w x m => w x (Sublist.mem m h)
|
||||
@@ -468,31 +440,16 @@ theorem IsPrefix.find?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁
|
||||
obtain ⟨t, rfl⟩ := h
|
||||
simp +contextual [find?_append]
|
||||
|
||||
grind_pattern IsPrefix.find?_eq_some => l₁ <+: l₂, l₁.find? p, some b
|
||||
grind_pattern IsPrefix.find?_eq_some => l₁ <+: l₂, l₂.find? p, some b
|
||||
|
||||
theorem IsPrefix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
|
||||
List.find? p l₂ = none → List.find? p l₁ = none :=
|
||||
h.sublist.find?_eq_none
|
||||
|
||||
grind_pattern Sublist.find?_eq_none => l₁ <+ l₂, l₂.find? p
|
||||
grind_pattern Sublist.find?_eq_none => l₁ <+ l₂, l₁.find? p
|
||||
|
||||
theorem IsSuffix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+ l₂) :
|
||||
List.find? p l₂ = none → List.find? p l₁ = none :=
|
||||
h.sublist.find?_eq_none
|
||||
|
||||
grind_pattern IsPrefix.find?_eq_none => l₁ <+: l₂, l₂.find? p
|
||||
grind_pattern IsPrefix.find?_eq_none => l₁ <+: l₂, l₁.find? p
|
||||
|
||||
theorem IsInfix.find?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+: l₂) :
|
||||
List.find? p l₂ = none → List.find? p l₁ = none :=
|
||||
h.sublist.find?_eq_none
|
||||
|
||||
grind_pattern IsSuffix.find?_eq_none => l₁ <:+ l₂, l₂.find? p
|
||||
grind_pattern IsSuffix.find?_eq_none => l₁ <:+ l₂, l₁.find? p
|
||||
|
||||
@[grind =]
|
||||
theorem find?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : List α}
|
||||
(H : ∀ (a : α), a ∈ xs → P a) {p : β → Bool} :
|
||||
(xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
|
||||
@@ -525,9 +482,9 @@ private theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
|
||||
ext
|
||||
simp only [Nat.add_comm i, Function.comp_apply, Nat.add_assoc]
|
||||
|
||||
@[simp, grind =] theorem findIdx?_nil : ([] : List α).findIdx? p = none := rfl
|
||||
@[simp] theorem findIdx?_nil : ([] : List α).findIdx? p = none := rfl
|
||||
|
||||
@[grind =] theorem findIdx?_cons :
|
||||
theorem findIdx?_cons :
|
||||
(x :: xs).findIdx? p = if p x then some 0 else (xs.findIdx? p).map fun i => i + 1 := by
|
||||
simp [findIdx?, findIdx?_go_eq]
|
||||
|
||||
@@ -536,7 +493,6 @@ private theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
|
||||
|
||||
/-! ### findIdx -/
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_cons {p : α → Bool} {b : α} {l : List α} :
|
||||
(b :: l).findIdx p = bif p b then 0 else (l.findIdx p) + 1 := by
|
||||
cases H : p b with
|
||||
@@ -555,7 +511,6 @@ where
|
||||
@[simp] theorem findIdx_singleton {a : α} {p : α → Bool} : [a].findIdx p = if p a then 0 else 1 := by
|
||||
simp [findIdx_cons, findIdx_nil]
|
||||
|
||||
@[grind →]
|
||||
theorem findIdx_of_getElem?_eq_some {xs : List α} (w : xs[xs.findIdx p]? = some y) : p y := by
|
||||
induction xs with
|
||||
| nil => simp_all
|
||||
@@ -565,8 +520,6 @@ theorem findIdx_getElem {xs : List α} {w : xs.findIdx p < xs.length} :
|
||||
p xs[xs.findIdx p] :=
|
||||
xs.findIdx_of_getElem?_eq_some (getElem?_eq_getElem w)
|
||||
|
||||
grind_pattern findIdx_getElem => xs[xs.findIdx p]
|
||||
|
||||
theorem findIdx_lt_length_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
|
||||
xs.findIdx p < xs.length := by
|
||||
induction xs with
|
||||
@@ -605,8 +558,6 @@ theorem findIdx_le_length {p : α → Bool} {xs : List α} : xs.findIdx p ≤ xs
|
||||
· simp at e
|
||||
exact Nat.le_of_eq (findIdx_eq_length.mpr e)
|
||||
|
||||
grind_pattern findIdx_le_length => xs.findIdx p, xs.length
|
||||
|
||||
@[simp]
|
||||
theorem findIdx_lt_length {p : α → Bool} {xs : List α} :
|
||||
xs.findIdx p < xs.length ↔ ∃ x ∈ xs, p x := by
|
||||
@@ -616,8 +567,6 @@ theorem findIdx_lt_length {p : α → Bool} {xs : List α} :
|
||||
rw [← this, findIdx_eq_length, not_exists]
|
||||
simp only [Bool.not_eq_true, not_and]
|
||||
|
||||
grind_pattern findIdx_lt_length => xs.findIdx p, xs.length
|
||||
|
||||
/-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/
|
||||
theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.findIdx p) :
|
||||
p (xs[i]'(Nat.le_trans h findIdx_le_length)) = false := by
|
||||
@@ -642,8 +591,6 @@ theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs
|
||||
rw [← ipm, Nat.succ_lt_succ_iff] at h
|
||||
simpa using ih h
|
||||
|
||||
grind_pattern not_of_lt_findIdx => xs.findIdx p, xs[i]
|
||||
|
||||
/-- If `¬ p xs[j]` for all `j < i`, then `i ≤ xs.findIdx p`. -/
|
||||
theorem le_findIdx_of_not {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length)
|
||||
(h2 : ∀ j (hji : j < i), p (xs[j]'(Nat.lt_trans hji h)) = false) : i ≤ xs.findIdx p := by
|
||||
@@ -671,7 +618,6 @@ theorem findIdx_eq {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length
|
||||
simp at h3
|
||||
simp_all [not_of_lt_findIdx h3]
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_append {p : α → Bool} {l₁ l₂ : List α} :
|
||||
(l₁ ++ l₂).findIdx p =
|
||||
if l₁.findIdx p < l₁.length then l₁.findIdx p else l₂.findIdx p + l₁.length := by
|
||||
@@ -693,9 +639,6 @@ theorem IsPrefix.findIdx_le {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+
|
||||
· exact Nat.le_refl ..
|
||||
· simp_all [findIdx_eq_length_of_false]
|
||||
|
||||
grind_pattern IsPrefix.findIdx_le => l₁ <:+ l₂, l₁.findIdx p
|
||||
grind_pattern IsPrefix.findIdx_le => l₁ <:+ l₂, l₂.findIdx p
|
||||
|
||||
theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂)
|
||||
(lt : l₁.findIdx p < l₁.length) : l₂.findIdx p = l₁.findIdx p := by
|
||||
rw [IsPrefix] at h
|
||||
@@ -705,8 +648,6 @@ theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α
|
||||
· rfl
|
||||
· simp_all
|
||||
|
||||
grind_pattern IsPrefix.findIdx_eq_of_findIdx_lt_length => l₁ <:+ l₂, l₁.findIdx p, l₂.findIdx p
|
||||
|
||||
theorem findIdx_le_findIdx {l : List α} {p q : α → Bool} (h : ∀ x ∈ l, p x → q x) : l.findIdx q ≤ l.findIdx p := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
@@ -730,7 +671,7 @@ theorem findIdx_le_findIdx {l : List α} {p q : α → Bool} (h : ∀ x ∈ l, p
|
||||
|
||||
/-! ### findIdx? -/
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem findIdx?_eq_none_iff {xs : List α} {p : α → Bool} :
|
||||
xs.findIdx? p = none ↔ ∀ x, x ∈ xs → p x = false := by
|
||||
induction xs with
|
||||
@@ -739,7 +680,7 @@ theorem findIdx?_eq_none_iff {xs : List α} {p : α → Bool} :
|
||||
simp only [findIdx?_cons]
|
||||
split <;> simp_all [cond_eq_if]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
|
||||
(xs.findIdx? p).isSome = xs.any p := by
|
||||
induction xs with
|
||||
@@ -748,7 +689,7 @@ theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
|
||||
simp only [findIdx?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem findIdx?_isNone {xs : List α} {p : α → Bool} :
|
||||
(xs.findIdx? p).isNone = xs.all (¬p ·) := by
|
||||
induction xs with
|
||||
@@ -854,14 +795,14 @@ theorem of_findIdx?_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p
|
||||
@[deprecated of_findIdx?_eq_none (since := "2025-02-02")]
|
||||
abbrev findIdx?_of_eq_none := @of_findIdx?_eq_none
|
||||
|
||||
@[simp, grind _=_] theorem findIdx?_map {f : β → α} {l : List β} : findIdx? p (l.map f) = l.findIdx? (p ∘ f) := by
|
||||
@[simp] theorem findIdx?_map {f : β → α} {l : List β} : findIdx? p (l.map f) = l.findIdx? (p ∘ f) := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [map_cons, findIdx?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem findIdx?_append :
|
||||
@[simp] theorem findIdx?_append :
|
||||
(xs ++ ys : List α).findIdx? p =
|
||||
(xs.findIdx? p).or ((ys.findIdx? p).map fun i => i + xs.length) := by
|
||||
induction xs with simp [findIdx?_cons]
|
||||
@@ -883,7 +824,7 @@ theorem findIdx?_flatten {l : List (List α)} {p : α → Bool} :
|
||||
· rw [Option.or_of_isNone (by simp_all [findIdx?_isNone])]
|
||||
simp [Function.comp_def, Nat.add_comm, Nat.add_assoc]
|
||||
|
||||
@[simp, grind =] theorem findIdx?_replicate :
|
||||
@[simp] theorem findIdx?_replicate :
|
||||
(replicate n a).findIdx? p = if 0 < n ∧ p a then some 0 else none := by
|
||||
cases n with
|
||||
| zero => simp
|
||||
@@ -937,38 +878,22 @@ theorem Sublist.findIdx?_eq_none {l₁ l₂ : List α} (h : l₁ <+ l₂) :
|
||||
simp only [findIdx?_eq_none_iff]
|
||||
exact fun w x m => w x (h.mem m)
|
||||
|
||||
grind_pattern Sublist.findIdx?_eq_none => l₁ <+ l₂, l₁.findIdx? p
|
||||
grind_pattern Sublist.findIdx?_eq_none => l₁ <+ l₂, l₂.findIdx? p
|
||||
|
||||
theorem IsPrefix.findIdx?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
|
||||
List.findIdx? p l₁ = some i → List.findIdx? p l₂ = some i := by
|
||||
rw [IsPrefix] at h
|
||||
obtain ⟨t, rfl⟩ := h
|
||||
intro h
|
||||
simp [findIdx?_append, h]
|
||||
|
||||
theorem IsPrefix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
|
||||
List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
|
||||
h.sublist.findIdx?_eq_none
|
||||
|
||||
grind_pattern IsPrefix.findIdx?_eq_none => l₁ <+: l₂, l₁.findIdx? p
|
||||
grind_pattern IsPrefix.findIdx?_eq_none => l₁ <+: l₂, l₂.findIdx? p
|
||||
|
||||
theorem IsSuffix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+ l₂) :
|
||||
List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
|
||||
h.sublist.findIdx?_eq_none
|
||||
|
||||
grind_pattern IsSuffix.findIdx?_eq_none => l₁ <:+ l₂, l₁.findIdx? p
|
||||
grind_pattern IsSuffix.findIdx?_eq_none => l₁ <:+ l₂, l₂.findIdx? p
|
||||
|
||||
theorem IsInfix.findIdx?_eq_none {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <:+: l₂) :
|
||||
List.findIdx? p l₂ = none → List.findIdx? p l₁ = none :=
|
||||
h.sublist.findIdx?_eq_none
|
||||
|
||||
grind_pattern IsInfix.findIdx?_eq_none => l₁ <:+: l₂, l₁.findIdx? p
|
||||
grind_pattern IsInfix.findIdx?_eq_none => l₁ <:+: l₂, l₂.findIdx? p
|
||||
|
||||
@[grind =]
|
||||
theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :
|
||||
xs.findIdx p = (xs.findIdx? p).getD xs.length := by
|
||||
induction xs with
|
||||
@@ -989,7 +914,7 @@ theorem findIdx_eq_getD_findIdx? {xs : List α} {p : α → Bool} :
|
||||
|
||||
/-! ### findFinIdx? -/
|
||||
|
||||
@[simp, grind =] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
|
||||
@[simp] theorem findFinIdx?_nil {p : α → Bool} : findFinIdx? p [] = none := rfl
|
||||
|
||||
theorem findIdx?_go_eq_map_findFinIdx?_go_val {xs : List α} {p : α → Bool} {i : Nat} {h} :
|
||||
List.findIdx?.go p xs i =
|
||||
@@ -1015,7 +940,6 @@ theorem findFinIdx?_eq_pmap_findIdx? {xs : List α} {p : α → Bool} :
|
||||
(fun i h => h) := by
|
||||
simp [findIdx?_eq_map_findFinIdx?_val, Option.pmap_map]
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
|
||||
findFinIdx? p (x :: xs) = if p x then some 0 else (findFinIdx? p xs).map Fin.succ := by
|
||||
rw [← Option.map_inj_right (f := Fin.val) (fun a b => Fin.eq_of_val_eq)]
|
||||
@@ -1026,7 +950,6 @@ theorem findFinIdx?_cons {p : α → Bool} {x : α} {xs : List α} :
|
||||
· rw [findIdx?_eq_map_findFinIdx?_val]
|
||||
simp [Function.comp_def]
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_append {xs ys : List α} {p : α → Bool} :
|
||||
(xs ++ ys).findFinIdx? p =
|
||||
((xs.findFinIdx? p).map (Fin.castLE (by simp))).or
|
||||
@@ -1036,11 +959,11 @@ theorem findFinIdx?_append {xs ys : List α} {p : α → Bool} :
|
||||
· simp [h, Option.pmap_map, Option.map_pmap, Nat.add_comm]
|
||||
· simp [h]
|
||||
|
||||
@[simp, grind =] theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
|
||||
@[simp] theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
|
||||
[a].findFinIdx? p = if p a then some ⟨0, by simp⟩ else none := by
|
||||
simp [findFinIdx?_cons, findFinIdx?_nil]
|
||||
|
||||
@[simp, grind =] theorem findFinIdx?_eq_none_iff {l : List α} {p : α → Bool} :
|
||||
@[simp] theorem findFinIdx?_eq_none_iff {l : List α} {p : α → Bool} :
|
||||
l.findFinIdx? p = none ↔ ∀ x ∈ l, ¬ p x := by
|
||||
simp [findFinIdx?_eq_pmap_findIdx?]
|
||||
|
||||
@@ -1056,7 +979,7 @@ theorem findFinIdx?_eq_some_iff {xs : List α} {p : α → Bool} {i : Fin xs.len
|
||||
· rintro ⟨h, w⟩
|
||||
exact ⟨i, ⟨i.2, h, fun j hji => w ⟨j, by omega⟩ hji⟩, rfl⟩
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isSome_findFinIdx? {l : List α} {p : α → Bool} :
|
||||
(l.findFinIdx? p).isSome = l.any p := by
|
||||
induction l with
|
||||
@@ -1065,7 +988,7 @@ theorem isSome_findFinIdx? {l : List α} {p : α → Bool} :
|
||||
simp only [findFinIdx?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isNone_findFinIdx? {l : List α} {p : α → Bool} :
|
||||
(l.findFinIdx? p).isNone = l.all (fun x => ¬ p x) := by
|
||||
induction l with
|
||||
@@ -1090,7 +1013,6 @@ The verification API for `idxOf` is still incomplete.
|
||||
The lemmas below should be made consistent with those for `findIdx` (and proved using them).
|
||||
-/
|
||||
|
||||
@[grind =]
|
||||
theorem idxOf_cons [BEq α] :
|
||||
(x :: xs : List α).idxOf y = bif x == y then 0 else xs.idxOf y + 1 := by
|
||||
dsimp [idxOf]
|
||||
@@ -1105,7 +1027,6 @@ abbrev indexOf_cons := @idxOf_cons
|
||||
@[deprecated idxOf_cons_self (since := "2025-01-29")]
|
||||
abbrev indexOf_cons_self := @idxOf_cons_self
|
||||
|
||||
@[grind =]
|
||||
theorem idxOf_append [BEq α] [LawfulBEq α] {l₁ l₂ : List α} {a : α} :
|
||||
(l₁ ++ l₂).idxOf a = if a ∈ l₁ then l₁.idxOf a else l₂.idxOf a + l₁.length := by
|
||||
rw [idxOf, findIdx_append]
|
||||
@@ -1129,7 +1050,7 @@ theorem idxOf_eq_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∉ l) : l.
|
||||
@[deprecated idxOf_eq_length (since := "2025-01-29")]
|
||||
abbrev indexOf_eq_length := @idxOf_eq_length
|
||||
|
||||
theorem idxOf_lt_length_of_mem [BEq α] [EquivBEq α] {l : List α} (h : a ∈ l) : l.idxOf a < l.length := by
|
||||
theorem idxOf_lt_length [BEq α] [EquivBEq α] {l : List α} (h : a ∈ l) : l.idxOf a < l.length := by
|
||||
induction l with
|
||||
| nil => simp at h
|
||||
| cons x xs ih =>
|
||||
@@ -1142,23 +1063,8 @@ theorem idxOf_lt_length_of_mem [BEq α] [EquivBEq α] {l : List α} (h : a ∈ l
|
||||
· exact zero_lt_succ xs.length
|
||||
· exact Nat.add_lt_add_right ih 1
|
||||
|
||||
theorem idxOf_le_length [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
l.idxOf a ≤ l.length := by
|
||||
simpa [idxOf] using findIdx_le_length
|
||||
|
||||
grind_pattern idxOf_le_length => l.idxOf a, l.length
|
||||
|
||||
theorem idxOf_lt_length_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
l.idxOf a < l.length ↔ a ∈ l := by
|
||||
constructor
|
||||
· intro h
|
||||
simpa [idxOf] using h
|
||||
· exact idxOf_lt_length_of_mem
|
||||
|
||||
grind_pattern idxOf_lt_length_iff => l.idxOf a, l.length
|
||||
|
||||
@[deprecated idxOf_lt_length_of_mem (since := "2025-01-29")]
|
||||
abbrev indexOf_lt_length := @idxOf_lt_length_of_mem
|
||||
@[deprecated idxOf_lt_length (since := "2025-01-29")]
|
||||
abbrev indexOf_lt_length := @idxOf_lt_length
|
||||
|
||||
/-! ### finIdxOf?
|
||||
|
||||
@@ -1170,14 +1076,14 @@ theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : List α} {a : α} :
|
||||
xs.idxOf? a = (xs.finIdxOf? a).map (·.val) := by
|
||||
simp [idxOf?, finIdxOf?, findIdx?_eq_map_findFinIdx?_val]
|
||||
|
||||
@[simp, grind =] theorem finIdxOf?_nil [BEq α] : ([] : List α).finIdxOf? a = none := rfl
|
||||
@[simp] theorem finIdxOf?_nil [BEq α] : ([] : List α).finIdxOf? a = none := rfl
|
||||
|
||||
@[grind =] theorem finIdxOf?_cons [BEq α] {a : α} {xs : List α} :
|
||||
theorem finIdxOf?_cons [BEq α] {a : α} {xs : List α} :
|
||||
(a :: xs).finIdxOf? b =
|
||||
if a == b then some ⟨0, by simp⟩ else (xs.finIdxOf? b).map (·.succ) := by
|
||||
simp [finIdxOf?, findFinIdx?_cons]
|
||||
|
||||
@[simp, grind =] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
@[simp] theorem finIdxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
l.finIdxOf? a = none ↔ a ∉ l := by
|
||||
simp only [finIdxOf?, findFinIdx?_eq_none_iff, beq_iff_eq]
|
||||
constructor
|
||||
@@ -1190,19 +1096,18 @@ theorem idxOf?_eq_map_finIdxOf?_val [BEq α] {xs : List α} {a : α} :
|
||||
l.finIdxOf? a = some i ↔ l[i] = a ∧ ∀ j (_ : j < i), ¬l[j] = a := by
|
||||
simp only [finIdxOf?, findFinIdx?_eq_some_iff, beq_iff_eq]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem isSome_finIdxOf? [BEq α] [PartialEquivBEq α] {l : List α} {a : α} :
|
||||
(l.finIdxOf? a).isSome = l.contains a := by
|
||||
@[simp]
|
||||
theorem isSome_finIdxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
(l.finIdxOf? a).isSome ↔ a ∈ l := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [finIdxOf?_cons]
|
||||
split <;> simp_all [BEq.comm]
|
||||
split <;> simp_all [@eq_comm _ x a]
|
||||
|
||||
@[simp]
|
||||
theorem isNone_finIdxOf? [BEq α] [PartialEquivBEq α] {l : List α} {a : α} :
|
||||
(l.finIdxOf? a).isNone = !l.contains a := by
|
||||
rw [← isSome_finIdxOf?, Option.not_isSome]
|
||||
theorem isNone_finIdxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
(l.finIdxOf? a).isNone = ¬ a ∈ l := by
|
||||
simp
|
||||
|
||||
/-! ### idxOf?
|
||||
|
||||
@@ -1210,16 +1115,16 @@ The verification API for `idxOf?` is still incomplete.
|
||||
The lemmas below should be made consistent with those for `findIdx?` (and proved using them).
|
||||
-/
|
||||
|
||||
@[simp, grind =] theorem idxOf?_nil [BEq α] : ([] : List α).idxOf? a = none := rfl
|
||||
@[simp] theorem idxOf?_nil [BEq α] : ([] : List α).idxOf? a = none := rfl
|
||||
|
||||
@[grind =] theorem idxOf?_cons [BEq α] {a : α} {xs : List α} {b : α} :
|
||||
theorem idxOf?_cons [BEq α] {a : α} {xs : List α} {b : α} :
|
||||
(a :: xs).idxOf? b = if a == b then some 0 else (xs.idxOf? b).map (· + 1) := by
|
||||
simp [idxOf?, findIdx?_cons]
|
||||
|
||||
@[simp] theorem idxOf?_singleton [BEq α] {a b : α} : [a].idxOf? b = if a == b then some 0 else none := by
|
||||
simp [idxOf?_cons, idxOf?_nil]
|
||||
|
||||
@[simp, grind =] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
@[simp] theorem idxOf?_eq_none_iff [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
l.idxOf? a = none ↔ a ∉ l := by
|
||||
simp only [idxOf?, findIdx?_eq_none_iff, beq_eq_false_iff_ne, ne_eq]
|
||||
constructor
|
||||
@@ -1232,7 +1137,7 @@ The lemmas below should be made consistent with those for `findIdx?` (and proved
|
||||
@[deprecated idxOf?_eq_none_iff (since := "2025-01-29")]
|
||||
abbrev indexOf?_eq_none_iff := @idxOf?_eq_none_iff
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isSome_idxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
(l.idxOf? a).isSome ↔ a ∈ l := by
|
||||
induction l with
|
||||
@@ -1241,7 +1146,6 @@ theorem isSome_idxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
simp only [idxOf?_cons]
|
||||
split <;> simp_all [@eq_comm _ x a]
|
||||
|
||||
@[grind =]
|
||||
theorem isNone_idxOf? [BEq α] [LawfulBEq α] {l : List α} {a : α} :
|
||||
(l.idxOf? a).isNone = ¬ a ∈ l := by
|
||||
simp
|
||||
@@ -1268,7 +1172,7 @@ theorem lookup_eq_findSome? {l : List (α × β)} {k : α} :
|
||||
simp only [lookup_cons, findSome?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem lookup_eq_none_iff {l : List (α × β)} {k : α} :
|
||||
@[simp] theorem lookup_eq_none_iff {l : List (α × β)} {k : α} :
|
||||
l.lookup k = none ↔ ∀ p ∈ l, k != p.1 := by
|
||||
simp [lookup_eq_findSome?]
|
||||
|
||||
@@ -1288,12 +1192,10 @@ theorem lookup_eq_some_iff {l : List (α × β)} {k : α} {b : β} :
|
||||
· rintro ⟨l₁, l₂, rfl, h⟩
|
||||
exact ⟨l₁, (k, b), l₂, rfl, by simp, by simpa using h⟩
|
||||
|
||||
@[grind =]
|
||||
theorem lookup_append {l₁ l₂ : List (α × β)} {k : α} :
|
||||
(l₁ ++ l₂).lookup k = (l₁.lookup k).or (l₂.lookup k) := by
|
||||
simp [lookup_eq_findSome?, findSome?_append]
|
||||
|
||||
@[grind =]
|
||||
theorem lookup_replicate {k : α} :
|
||||
(replicate n (a,b)).lookup k = if n = 0 then none else if k == a then some b else none := by
|
||||
induction n with
|
||||
@@ -1328,9 +1230,6 @@ theorem Sublist.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <+ l₂)
|
||||
simp only [lookup_eq_findSome?]
|
||||
exact h.findSome?_eq_none
|
||||
|
||||
grind_pattern Sublist.lookup_isSome => l₁ <+ l₂, l₁.lookup k
|
||||
grind_pattern Sublist.lookup_isSome => l₁ <+ l₂, l₂.lookup k
|
||||
|
||||
theorem IsPrefix.lookup_eq_some {l₁ l₂ : List (α × β)} (h : l₁ <+: l₂) :
|
||||
List.lookup k l₁ = some b → List.lookup k l₂ = some b := by
|
||||
simp only [lookup_eq_findSome?]
|
||||
@@ -1339,24 +1238,13 @@ theorem IsPrefix.lookup_eq_some {l₁ l₂ : List (α × β)} (h : l₁ <+: l₂
|
||||
theorem IsPrefix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <+: l₂) :
|
||||
List.lookup k l₂ = none → List.lookup k l₁ = none :=
|
||||
h.sublist.lookup_eq_none
|
||||
|
||||
grind_pattern IsPrefix.lookup_eq_none => l₁ <+: l₂, l₁.lookup k
|
||||
grind_pattern IsPrefix.lookup_eq_none => l₁ <+: l₂, l₂.lookup k
|
||||
|
||||
theorem IsSuffix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <:+ l₂) :
|
||||
List.lookup k l₂ = none → List.lookup k l₁ = none :=
|
||||
h.sublist.lookup_eq_none
|
||||
|
||||
grind_pattern IsSuffix.lookup_eq_none => l₁ <:+ l₂, l₁.lookup k
|
||||
grind_pattern IsSuffix.lookup_eq_none => l₁ <:+ l₂, l₂.lookup k
|
||||
|
||||
theorem IsInfix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <:+: l₂) :
|
||||
List.lookup k l₂ = none → List.lookup k l₁ = none :=
|
||||
h.sublist.lookup_eq_none
|
||||
|
||||
grind_pattern IsInfix.lookup_eq_none => l₁ <:+: l₂, l₁.lookup k
|
||||
grind_pattern IsInfix.lookup_eq_none => l₁ <:+: l₂, l₂.lookup k
|
||||
|
||||
end lookup
|
||||
|
||||
end List
|
||||
|
||||
@@ -261,11 +261,11 @@ Examples:
|
||||
/-- Tail recursive implementation of `findRev?`. This is only used at runtime. -/
|
||||
def findRev?TR (p : α → Bool) (l : List α) : Option α := l.reverse.find? p
|
||||
|
||||
@[simp, grind =] theorem find?_singleton {a : α} : [a].find? p = if p a then some a else none := by
|
||||
@[simp] theorem find?_singleton {a : α} : [a].find? p = if p a then some a else none := by
|
||||
simp only [find?]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem find?_append {xs ys : List α} : (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
|
||||
@[simp] theorem find?_append {xs ys : List α} : (xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
|
||||
induction xs with
|
||||
| nil => simp [find?]
|
||||
| cons x xs ih =>
|
||||
@@ -287,12 +287,12 @@ def findRev?TR (p : α → Bool) (l : List α) : Option α := l.reverse.find? p
|
||||
/-- Tail recursive implementation of `finSomedRev?`. This is only used at runtime. -/
|
||||
def findSomeRev?TR (f : α → Option β) (l : List α) : Option β := l.reverse.findSome? f
|
||||
|
||||
@[simp, grind =] theorem findSome?_singleton {a : α} :
|
||||
@[simp] theorem findSome?_singleton {a : α} :
|
||||
[a].findSome? f = f a := by
|
||||
simp only [findSome?_cons, findSome?_nil]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem findSome?_append {xs ys : List α} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
|
||||
@[simp] theorem findSome?_append {xs ys : List α} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
|
||||
induction xs with
|
||||
| nil => simp [findSome?]
|
||||
| cons x xs ih =>
|
||||
|
||||
@@ -109,11 +109,9 @@ abbrev length_eq_zero := @length_eq_zero_iff
|
||||
theorem eq_nil_iff_length_eq_zero : l = [] ↔ length l = 0 :=
|
||||
length_eq_zero_iff.symm
|
||||
|
||||
theorem length_pos_of_mem {a : α} : ∀ {l : List α}, a ∈ l → 0 < length l
|
||||
@[grind →] theorem length_pos_of_mem {a : α} : ∀ {l : List α}, a ∈ l → 0 < length l
|
||||
| _::_, _ => Nat.zero_lt_succ _
|
||||
|
||||
grind_pattern length_pos_of_mem => a ∈ l, length l
|
||||
|
||||
theorem exists_mem_of_length_pos : ∀ {l : List α}, 0 < length l → ∃ a, a ∈ l
|
||||
| _::_, _ => ⟨_, .head ..⟩
|
||||
|
||||
@@ -1320,19 +1318,6 @@ theorem forall_mem_filter {l : List α} {p : α → Bool} {P : α → Prop} :
|
||||
(∀ (i) (_ : i ∈ l.filter p), P i) ↔ ∀ (j) (_ : j ∈ l), p j → P j := by
|
||||
simp
|
||||
|
||||
@[grind] theorem getElem_filter {xs : List α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).length) :
|
||||
p (xs.filter p)[i] :=
|
||||
(mem_filter.mp (getElem_mem h)).2
|
||||
|
||||
theorem getElem?_filter {xs : List α} {p : α → Bool} {i : Nat} (h : i < (xs.filter p).length)
|
||||
(w : (xs.filter p)[i]? = some a) : p a := by
|
||||
rw [getElem?_eq_getElem] at w
|
||||
simp only [Option.some.injEq] at w
|
||||
rw [← w]
|
||||
apply getElem_filter h
|
||||
|
||||
grind_pattern getElem?_filter => (xs.filter p)[i]?, some a
|
||||
|
||||
@[simp] theorem filter_filter : ∀ {l}, filter p (filter q l) = filter (fun a => p a && q a) l
|
||||
| [] => rfl
|
||||
| a :: l => by by_cases hp : p a <;> by_cases hq : q a <;> simp [hp, hq, filter_filter]
|
||||
@@ -2793,8 +2778,8 @@ We can prove that two folds over the same list are related (by some arbitrary re
|
||||
if we know that the initial elements are related and the folding function, for each element of the list,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldl_rel {l : List α} {f : β → α → β} {g : γ → α → γ} {a : β} {b : γ} {r : β → γ → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c : β) (c' : γ), r c c' → r (f c a) (g c' a)) :
|
||||
theorem foldl_rel {l : List α} {f g : β → α → β} {a b : β} {r : β → β → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
|
||||
r (l.foldl (fun acc a => f acc a) a) (l.foldl (fun acc a => g acc a) b) := by
|
||||
induction l generalizing a b with
|
||||
| nil => simp_all
|
||||
@@ -2809,8 +2794,8 @@ We can prove that two folds over the same list are related (by some arbitrary re
|
||||
if we know that the initial elements are related and the folding function, for each element of the list,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldr_rel {l : List α} {f : α → β → β} {g : α → γ → γ} {a : β} {b : γ} {r : β → γ → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c : β) (c' : γ), r c c' → r (f a c) (g a c')) :
|
||||
theorem foldr_rel {l : List α} {f g : α → β → β} {a b : β} {r : β → β → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
|
||||
r (l.foldr (fun a acc => f a acc) a) (l.foldr (fun a acc => g a acc) b) := by
|
||||
induction l generalizing a b with
|
||||
| nil => simp_all
|
||||
|
||||
@@ -91,7 +91,7 @@ is valid.
|
||||
subst w
|
||||
rfl
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapFinIdx_nil {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx [] f = [] :=
|
||||
rfl
|
||||
|
||||
@@ -101,7 +101,7 @@ theorem mapFinIdx_nil {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx
|
||||
| nil => simpa using h
|
||||
| cons _ _ ih => simp [mapFinIdx.go, ih]
|
||||
|
||||
@[simp, grind =] theorem length_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} :
|
||||
@[simp] theorem length_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} :
|
||||
(as.mapFinIdx f).length = as.length := by
|
||||
simp [mapFinIdx, length_mapFinIdx_go]
|
||||
|
||||
@@ -129,7 +129,7 @@ theorem getElem_mapFinIdx_go {as : List α} {f : (i : Nat) → α → (h : i < a
|
||||
· have h₃ : i - acc.size = (i - (acc.size + 1)) + 1 := by omega
|
||||
simp [h₃]
|
||||
|
||||
@[simp, grind =] theorem getElem_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} {i : Nat} {h} :
|
||||
@[simp] theorem getElem_mapFinIdx {as : List α} {f : (i : Nat) → α → (h : i < as.length) → β} {i : Nat} {h} :
|
||||
(as.mapFinIdx f)[i] = f i (as[i]'(by simp at h; omega)) (by simp at h; omega) := by
|
||||
simp [mapFinIdx, getElem_mapFinIdx_go]
|
||||
|
||||
@@ -137,19 +137,18 @@ theorem mapFinIdx_eq_ofFn {as : List α} {f : (i : Nat) → α → (h : i < as.l
|
||||
as.mapFinIdx f = List.ofFn fun i : Fin as.length => f i as[i] i.2 := by
|
||||
apply ext_getElem <;> simp
|
||||
|
||||
@[simp, grind =] theorem getElem?_mapFinIdx {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {i : Nat} :
|
||||
@[simp] theorem getElem?_mapFinIdx {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} {i : Nat} :
|
||||
(l.mapFinIdx f)[i]? = l[i]?.pbind fun x m => some <| f i x (by simp [getElem?_eq_some_iff] at m; exact m.1) := by
|
||||
simp only [getElem?_def, length_mapFinIdx, getElem_mapFinIdx]
|
||||
split <;> simp
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapFinIdx_cons {l : List α} {a : α} {f : (i : Nat) → α → (h : i < l.length + 1) → β} :
|
||||
mapFinIdx (a :: l) f = f 0 a (by omega) :: mapFinIdx l (fun i a h => f (i + 1) a (by omega)) := by
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· rintro (_|i) h₁ h₂ <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem mapFinIdx_append {xs ys : List α} {f : (i : Nat) → α → (h : i < (xs ++ ys).length) → β} :
|
||||
(xs ++ ys).mapFinIdx f =
|
||||
xs.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
|
||||
@@ -166,7 +165,7 @@ theorem mapFinIdx_append {xs ys : List α} {f : (i : Nat) → α → (h : i < (x
|
||||
congr
|
||||
omega
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_concat {l : List α} {e : α} {f : (i : Nat) → α → (h : i < (l ++ [e]).length) → β}:
|
||||
@[simp] theorem mapFinIdx_concat {l : List α} {e : α} {f : (i : Nat) → α → (h : i < (l ++ [e]).length) → β}:
|
||||
(l ++ [e]).mapFinIdx f = l.mapFinIdx (fun i a h => f i a (by simp; omega)) ++ [f l.length e (by simp)] := by
|
||||
simp [mapFinIdx_append]
|
||||
|
||||
@@ -202,7 +201,7 @@ theorem exists_of_mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α →
|
||||
obtain ⟨h', rfl⟩ := h
|
||||
exact ⟨i, h', rfl⟩
|
||||
|
||||
@[simp, grind =] theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
|
||||
@[simp] theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
|
||||
b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] h = b := by
|
||||
constructor
|
||||
· intro h
|
||||
@@ -288,7 +287,7 @@ theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : (i : Nat) → α → (h
|
||||
rw [eq_comm, mapFinIdx_eq_iff]
|
||||
simp [Fin.forall_iff]
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_mapFinIdx {l : List α}
|
||||
@[simp] theorem mapFinIdx_mapFinIdx {l : List α}
|
||||
{f : (i : Nat) → α → (h : i < l.length) → β}
|
||||
{g : (i : Nat) → β → (h : i < (l.mapFinIdx f).length) → γ} :
|
||||
(l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i a h => g i (f i a h) (by simpa)) := by
|
||||
@@ -304,7 +303,7 @@ theorem mapFinIdx_eq_replicate_iff {l : List α} {f : (i : Nat) → α → (h :
|
||||
· rintro w b i h rfl
|
||||
exact w i h
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_reverse {l : List α} {f : (i : Nat) → α → (h : i < l.reverse.length) → β} :
|
||||
@[simp] theorem mapFinIdx_reverse {l : List α} {f : (i : Nat) → α → (h : i < l.reverse.length) → β} :
|
||||
l.reverse.mapFinIdx f =
|
||||
(l.mapFinIdx (fun i a h => f (l.length - 1 - i) a (by simp; omega))).reverse := by
|
||||
simp [mapFinIdx_eq_iff]
|
||||
@@ -314,7 +313,7 @@ theorem mapFinIdx_eq_replicate_iff {l : List α} {f : (i : Nat) → α → (h :
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapIdx_nil {f : Nat → α → β} : mapIdx f [] = [] :=
|
||||
rfl
|
||||
|
||||
@@ -334,7 +333,7 @@ theorem length_mapIdx_go : ∀ {l : List α} {acc : Array β},
|
||||
simp
|
||||
omega
|
||||
|
||||
@[simp, grind =] theorem length_mapIdx {l : List α} : (l.mapIdx f).length = l.length := by
|
||||
@[simp] theorem length_mapIdx {l : List α} : (l.mapIdx f).length = l.length := by
|
||||
simp [mapIdx, length_mapIdx_go]
|
||||
|
||||
theorem getElem?_mapIdx_go : ∀ {l : List α} {acc : Array β} {i : Nat},
|
||||
@@ -357,11 +356,11 @@ theorem getElem?_mapIdx_go : ∀ {l : List α} {acc : Array β} {i : Nat},
|
||||
· have : i - acc.size = i - (acc.size + 1) + 1 := by omega
|
||||
simp_all
|
||||
|
||||
@[simp, grind =] theorem getElem?_mapIdx {l : List α} {i : Nat} :
|
||||
@[simp] theorem getElem?_mapIdx {l : List α} {i : Nat} :
|
||||
(l.mapIdx f)[i]? = Option.map (f i) l[i]? := by
|
||||
simp [mapIdx, getElem?_mapIdx_go]
|
||||
|
||||
@[simp, grind =] theorem getElem_mapIdx {l : List α} {f : Nat → α → β} {i : Nat} {h : i < (l.mapIdx f).length} :
|
||||
@[simp] theorem getElem_mapIdx {l : List α} {f : Nat → α → β} {i : Nat} {h : i < (l.mapIdx f).length} :
|
||||
(l.mapIdx f)[i] = f i (l[i]'(by simpa using h)) := by
|
||||
apply Option.some_inj.mp
|
||||
rw [← getElem?_eq_getElem, getElem?_mapIdx, getElem?_eq_getElem (by simpa using h)]
|
||||
@@ -385,19 +384,18 @@ theorem mapIdx_eq_zipIdx_map {l : List α} {f : Nat → α → β} :
|
||||
@[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-21")]
|
||||
abbrev mapIdx_eq_enum_map := @mapIdx_eq_zipIdx_map
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapIdx_cons {l : List α} {a : α} :
|
||||
mapIdx f (a :: l) = f 0 a :: mapIdx (fun i => f (i + 1)) l := by
|
||||
simp [mapIdx_eq_zipIdx_map, List.zipIdx_succ]
|
||||
|
||||
@[grind =]
|
||||
theorem mapIdx_append {xs ys : List α} :
|
||||
(xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx fun i => f (i + xs.length) := by
|
||||
induction xs generalizing f with
|
||||
| nil => rfl
|
||||
| cons _ _ ih => simp [ih (f := fun i => f (i + 1)), Nat.add_assoc]
|
||||
|
||||
@[simp, grind =] theorem mapIdx_concat {l : List α} {e : α} :
|
||||
@[simp] theorem mapIdx_concat {l : List α} {e : α} :
|
||||
mapIdx f (l ++ [e]) = mapIdx f l ++ [f l.length e] := by
|
||||
simp [mapIdx_append]
|
||||
|
||||
@@ -417,7 +415,7 @@ theorem exists_of_mem_mapIdx {b : β} {l : List α}
|
||||
rw [mapIdx_eq_mapFinIdx] at h
|
||||
simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
|
||||
|
||||
@[simp, grind =] theorem mem_mapIdx {b : β} {l : List α} :
|
||||
@[simp] theorem mem_mapIdx {b : β} {l : List α} :
|
||||
b ∈ mapIdx f l ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] = b := by
|
||||
constructor
|
||||
· intro h
|
||||
@@ -472,7 +470,7 @@ theorem mapIdx_eq_mapIdx_iff {l : List α} :
|
||||
· intro i h₁ h₂
|
||||
simp [w]
|
||||
|
||||
@[simp, grind =] theorem mapIdx_set {l : List α} {i : Nat} {a : α} :
|
||||
@[simp] theorem mapIdx_set {l : List α} {i : Nat} {a : α} :
|
||||
(l.set i a).mapIdx f = (l.mapIdx f).set i (f i a) := by
|
||||
simp only [mapIdx_eq_iff, getElem?_set, length_mapIdx, getElem?_mapIdx]
|
||||
intro i
|
||||
@@ -480,16 +478,16 @@ theorem mapIdx_eq_mapIdx_iff {l : List α} :
|
||||
· split <;> simp_all
|
||||
· rfl
|
||||
|
||||
@[simp, grind =] theorem head_mapIdx {l : List α} {f : Nat → α → β} {w : mapIdx f l ≠ []} :
|
||||
@[simp] theorem head_mapIdx {l : List α} {f : Nat → α → β} {w : mapIdx f l ≠ []} :
|
||||
(mapIdx f l).head w = f 0 (l.head (by simpa using w)) := by
|
||||
cases l with
|
||||
| nil => simp at w
|
||||
| cons _ _ => simp
|
||||
|
||||
@[simp, grind =] theorem head?_mapIdx {l : List α} {f : Nat → α → β} : (mapIdx f l).head? = l.head?.map (f 0) := by
|
||||
@[simp] theorem head?_mapIdx {l : List α} {f : Nat → α → β} : (mapIdx f l).head? = l.head?.map (f 0) := by
|
||||
cases l <;> simp
|
||||
|
||||
@[simp, grind =] theorem getLast_mapIdx {l : List α} {f : Nat → α → β} {h} :
|
||||
@[simp] theorem getLast_mapIdx {l : List α} {f : Nat → α → β} {h} :
|
||||
(mapIdx f l).getLast h = f (l.length - 1) (l.getLast (by simpa using h)) := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
@@ -500,13 +498,13 @@ theorem mapIdx_eq_mapIdx_iff {l : List α} :
|
||||
simp only [← mapIdx_cons, getElem_mapIdx]
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem getLast?_mapIdx {l : List α} {f : Nat → α → β} :
|
||||
@[simp] theorem getLast?_mapIdx {l : List α} {f : Nat → α → β} :
|
||||
(mapIdx f l).getLast? = (getLast? l).map (f (l.length - 1)) := by
|
||||
cases l
|
||||
· simp
|
||||
· rw [getLast?_eq_getLast, getLast?_eq_getLast, getLast_mapIdx] <;> simp
|
||||
|
||||
@[simp, grind =] theorem mapIdx_mapIdx {l : List α} {f : Nat → α → β} {g : Nat → β → γ} :
|
||||
@[simp] theorem mapIdx_mapIdx {l : List α} {f : Nat → α → β} {g : Nat → β → γ} :
|
||||
(l.mapIdx f).mapIdx g = l.mapIdx (fun i => g i ∘ f i) := by
|
||||
simp [mapIdx_eq_iff]
|
||||
|
||||
@@ -519,7 +517,7 @@ theorem mapIdx_eq_replicate_iff {l : List α} {f : Nat → α → β} {b : β} :
|
||||
· rintro w _ i h rfl
|
||||
exact w i h
|
||||
|
||||
@[simp, grind =] theorem mapIdx_reverse {l : List α} {f : Nat → α → β} :
|
||||
@[simp] theorem mapIdx_reverse {l : List α} {f : Nat → α → β} :
|
||||
l.reverse.mapIdx f = (mapIdx (fun i => f (l.length - 1 - i)) l).reverse := by
|
||||
simp [mapIdx_eq_iff]
|
||||
intro i
|
||||
|
||||
@@ -60,25 +60,6 @@ theorem count_replace [BEq α] [LawfulBEq α] {a b c : α} {l : List α} :
|
||||
if l.contains a then l.count c + (if b == c then 1 else 0) - (if a == c then 1 else 0) else l.count c := by
|
||||
simp [count_eq_countP, countP_replace]
|
||||
|
||||
@[grind =] theorem count_insert [BEq α] [LawfulBEq α] {a b : α} {l : List α} :
|
||||
count a (List.insert b l) = max (count a l) (if b == a then 1 else 0) := by
|
||||
simp only [List.insert, contains_eq_mem, decide_eq_true_eq, beq_iff_eq]
|
||||
split <;> rename_i h
|
||||
· split <;> rename_i h'
|
||||
· rw [Nat.max_def]
|
||||
simp only [beq_iff_eq] at h'
|
||||
split
|
||||
· have := List.count_pos_iff.mpr (h' ▸ h)
|
||||
omega
|
||||
· rfl
|
||||
· simp [h']
|
||||
· rw [count_cons]
|
||||
split <;> rename_i h'
|
||||
· simp only [beq_iff_eq] at h'
|
||||
rw [count_eq_zero.mpr (h' ▸ h)]
|
||||
simp [h']
|
||||
· simp
|
||||
|
||||
/--
|
||||
The number of elements satisfying a predicate in a sublist is at least the number of elements satisfying the predicate in the list,
|
||||
minus the difference in the lengths.
|
||||
|
||||
@@ -14,7 +14,6 @@ set_option linter.indexVariables true -- Enforce naming conventions for index va
|
||||
|
||||
namespace List
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_eraseIdx {l : List α} {i : Nat} {j : Nat} :
|
||||
(l.eraseIdx i)[j]? = if j < i then l[j]? else l[j + 1]? := by
|
||||
rw [eraseIdx_eq_take_drop_succ, getElem?_append]
|
||||
@@ -50,7 +49,6 @@ theorem getElem?_eraseIdx_of_ge {l : List α} {i : Nat} {j : Nat} (h : i ≤ j)
|
||||
intro h'
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem getElem_eraseIdx {l : List α} {i : Nat} {j : Nat} (h : j < (l.eraseIdx i).length) :
|
||||
(l.eraseIdx i)[j] = if h' : j < i then
|
||||
l[j]'(by have := length_eraseIdx_le l i; omega)
|
||||
@@ -125,48 +123,6 @@ theorem eraseIdx_set_gt {l : List α} {i : Nat} {j : Nat} {a : α} (h : i < j) :
|
||||
· have t : i ≠ n := by omega
|
||||
simp [t]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_set {xs : List α} {i : Nat} {a : α} {j : Nat} :
|
||||
(xs.set i a).eraseIdx j =
|
||||
if j < i then
|
||||
(xs.eraseIdx j).set (i - 1) a
|
||||
else if j = i then
|
||||
xs.eraseIdx i
|
||||
else
|
||||
(xs.eraseIdx j).set i a := by
|
||||
split <;> rename_i h'
|
||||
· rw [eraseIdx_set_lt]
|
||||
omega
|
||||
· split <;> rename_i h''
|
||||
· subst h''
|
||||
rw [eraseIdx_set_eq]
|
||||
· rw [eraseIdx_set_gt]
|
||||
omega
|
||||
|
||||
theorem set_eraseIdx_le {xs : List α} {i : Nat} {j : Nat} {a : α} (h : i ≤ j) :
|
||||
(xs.eraseIdx i).set j a = (xs.set (j + 1) a).eraseIdx i := by
|
||||
rw [eraseIdx_set_lt]
|
||||
· simp
|
||||
· omega
|
||||
|
||||
theorem set_eraseIdx_gt {xs : List α} {i : Nat} {j : Nat} {a : α} (h : j < i) :
|
||||
(xs.eraseIdx i).set j a = (xs.set j a).eraseIdx i := by
|
||||
rw [eraseIdx_set_gt]
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem set_eraseIdx {xs : List α} {i : Nat} {j : Nat} {a : α} :
|
||||
(xs.eraseIdx i).set j a =
|
||||
if i ≤ j then
|
||||
(xs.set (j + 1) a).eraseIdx i
|
||||
else
|
||||
(xs.set j a).eraseIdx i := by
|
||||
split <;> rename_i h'
|
||||
· rw [set_eraseIdx_le]
|
||||
omega
|
||||
· rw [set_eraseIdx_gt]
|
||||
omega
|
||||
|
||||
@[simp] theorem set_getElem_succ_eraseIdx_succ
|
||||
{l : List α} {i : Nat} (h : i + 1 < l.length) :
|
||||
(l.eraseIdx (i + 1)).set i l[i + 1] = l.eraseIdx i := by
|
||||
|
||||
@@ -158,26 +158,6 @@ theorem erase_range' :
|
||||
simp [p]
|
||||
omega
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range' {a s n step} (h : 0 < step := by simp) :
|
||||
count a (range' s n step) = if ∃ i, i < n ∧ a = s + step * i then 1 else 0 := by
|
||||
rw [(nodup_range' step h).count]
|
||||
simp only [mem_range']
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range_1' {a s n} :
|
||||
count a (range' s n) = if s ≤ a ∧ a < s + n then 1 else 0 := by
|
||||
rw [count_range' (by simp)]
|
||||
split <;> rename_i h
|
||||
· obtain ⟨i, h, rfl⟩ := h
|
||||
simp [h]
|
||||
· simp at h
|
||||
rw [if_neg]
|
||||
simp only [not_and, Nat.not_lt]
|
||||
intro w
|
||||
specialize h (a - s)
|
||||
omega
|
||||
|
||||
/-! ### range -/
|
||||
|
||||
theorem reverse_range' : ∀ {s n : Nat}, reverse (range' s n) = map (s + n - 1 - ·) (range n)
|
||||
@@ -220,12 +200,6 @@ theorem find?_range_eq_none {n : Nat} {p : Nat → Bool} :
|
||||
theorem erase_range : (range n).erase i = range (min n i) ++ range' (i + 1) (n - (i + 1)) := by
|
||||
simp [range_eq_range', erase_range']
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range {a n} :
|
||||
count a (range n) = if a < n then 1 else 0 := by
|
||||
rw [range_eq_range', count_range_1']
|
||||
simp
|
||||
|
||||
/-! ### iota -/
|
||||
|
||||
section
|
||||
|
||||
@@ -538,7 +538,7 @@ theorem dropWhile_eq_drop_findIdx_not {xs : List α} {p : α → Bool} :
|
||||
|
||||
/-! ### zipWith -/
|
||||
|
||||
@[simp, grind =] theorem length_zipWith {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
|
||||
@[simp] theorem length_zipWith {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
|
||||
length (zipWith f l₁ l₂) = min (length l₁) (length l₂) := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;>
|
||||
simp_all [succ_min_succ, Nat.zero_min, Nat.min_zero]
|
||||
@@ -549,7 +549,7 @@ theorem lt_length_left_of_zipWith {f : α → β → γ} {i : Nat} {l : List α}
|
||||
theorem lt_length_right_of_zipWith {f : α → β → γ} {i : Nat} {l : List α} {l' : List β}
|
||||
(h : i < (zipWith f l l').length) : i < l'.length := by rw [length_zipWith] at h; omega
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem getElem_zipWith {f : α → β → γ} {l : List α} {l' : List β}
|
||||
{i : Nat} {h : i < (zipWith f l l').length} :
|
||||
(zipWith f l l')[i] =
|
||||
@@ -566,7 +566,6 @@ theorem zipWith_eq_zipWith_take_min : ∀ {l₁ : List α} {l₂ : List β},
|
||||
| _, [] => by simp
|
||||
| a :: l₁, b :: l₂ => by simp [succ_min_succ, zipWith_eq_zipWith_take_min (l₁ := l₁) (l₂ := l₂)]
|
||||
|
||||
@[grind =]
|
||||
theorem reverse_zipWith (h : l.length = l'.length) :
|
||||
(zipWith f l l').reverse = zipWith f l.reverse l'.reverse := by
|
||||
induction l generalizing l' with
|
||||
@@ -579,14 +578,14 @@ theorem reverse_zipWith (h : l.length = l'.length) :
|
||||
have : tl.reverse.length = tl'.reverse.length := by simp [h]
|
||||
simp [hl h, zipWith_append this]
|
||||
|
||||
@[simp, grind =] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
|
||||
@[simp] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
|
||||
zipWith f (replicate m a) (replicate n b) = replicate (min m n) (f a b) := by
|
||||
rw [zipWith_eq_zipWith_take_min]
|
||||
simp
|
||||
|
||||
/-! ### zip -/
|
||||
|
||||
@[simp, grind =] theorem length_zip {l₁ : List α} {l₂ : List β} :
|
||||
@[simp] theorem length_zip {l₁ : List α} {l₂ : List β} :
|
||||
length (zip l₁ l₂) = min (length l₁) (length l₂) := by
|
||||
simp [zip]
|
||||
|
||||
@@ -598,7 +597,7 @@ theorem lt_length_right_of_zip {i : Nat} {l : List α} {l' : List β} (h : i < (
|
||||
i < l'.length :=
|
||||
lt_length_right_of_zipWith h
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem getElem_zip {l : List α} {l' : List β} {i : Nat} {h : i < (zip l l').length} :
|
||||
(zip l l')[i] =
|
||||
(l[i]'(lt_length_left_of_zip h), l'[i]'(lt_length_right_of_zip h)) :=
|
||||
@@ -610,7 +609,7 @@ theorem zip_eq_zip_take_min : ∀ {l₁ : List α} {l₂ : List β},
|
||||
| _, [] => by simp
|
||||
| a :: l₁, b :: l₂ => by simp [succ_min_succ, zip_eq_zip_take_min (l₁ := l₁) (l₂ := l₂)]
|
||||
|
||||
@[simp, grind =] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
|
||||
@[simp] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
|
||||
zip (replicate m a) (replicate n b) = replicate (min m n) (a, b) := by
|
||||
rw [zip_eq_zip_take_min]
|
||||
simp
|
||||
|
||||
@@ -6,7 +6,7 @@ Author: Leonardo de Moura
|
||||
module
|
||||
|
||||
prelude
|
||||
meta import Init.Data.Nat.Div.Basic
|
||||
import Init.Data.Nat.Div.Basic
|
||||
|
||||
/-!
|
||||
# Notation for `List` literals.
|
||||
|
||||
@@ -34,14 +34,14 @@ to each potential index in order, starting at `0`.
|
||||
def ofFnM {n} [Monad m] (f : Fin n → m α) : m (List α) :=
|
||||
List.reverse <$> Fin.foldlM n (fun xs i => (· :: xs) <$> f i) []
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem length_ofFn {f : Fin n → α} : (ofFn f).length = n := by
|
||||
simp only [ofFn]
|
||||
induction n with
|
||||
| zero => simp
|
||||
| succ n ih => simp [Fin.foldr_succ, ih]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
protected theorem getElem_ofFn {f : Fin n → α} (h : i < (ofFn f).length) :
|
||||
(ofFn f)[i] = f ⟨i, by simp_all⟩ := by
|
||||
simp only [ofFn]
|
||||
@@ -55,7 +55,7 @@ protected theorem getElem_ofFn {f : Fin n → α} (h : i < (ofFn f).length) :
|
||||
apply ih
|
||||
simp_all
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
protected theorem getElem?_ofFn {f : Fin n → α} :
|
||||
(ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none :=
|
||||
if h : i < (ofFn f).length
|
||||
@@ -67,7 +67,7 @@ protected theorem getElem?_ofFn {f : Fin n → α} :
|
||||
simpa using h
|
||||
|
||||
/-- `ofFn` on an empty domain is the empty list. -/
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem ofFn_zero {f : Fin 0 → α} : ofFn f = [] := by
|
||||
rw [ofFn, Fin.foldr_zero]
|
||||
|
||||
@@ -98,7 +98,7 @@ theorem ofFn_add {n m} {f : Fin (n + m) → α} :
|
||||
theorem ofFn_eq_nil_iff {f : Fin n → α} : ofFn f = [] ↔ n = 0 := by
|
||||
cases n <;> simp only [ofFn_zero, ofFn_succ, eq_self_iff_true, Nat.succ_ne_zero, reduceCtorEq]
|
||||
|
||||
@[simp 500, grind =]
|
||||
@[simp 500]
|
||||
theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
|
||||
constructor
|
||||
· intro w
|
||||
@@ -107,17 +107,17 @@ theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i =
|
||||
· rintro ⟨i, rfl⟩
|
||||
apply mem_of_getElem (i := i) <;> simp
|
||||
|
||||
@[grind =] theorem head_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
|
||||
theorem head_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
|
||||
(ofFn f).head h = f ⟨0, Nat.pos_of_ne_zero (mt ofFn_eq_nil_iff.2 h)⟩ := by
|
||||
rw [← getElem_zero (length_ofFn ▸ Nat.pos_of_ne_zero (mt ofFn_eq_nil_iff.2 h)),
|
||||
List.getElem_ofFn]
|
||||
|
||||
@[grind =]theorem getLast_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
|
||||
theorem getLast_ofFn {n} {f : Fin n → α} (h : ofFn f ≠ []) :
|
||||
(ofFn f).getLast h = f ⟨n - 1, Nat.sub_one_lt (mt ofFn_eq_nil_iff.2 h)⟩ := by
|
||||
simp [getLast_eq_getElem, length_ofFn, List.getElem_ofFn]
|
||||
|
||||
/-- `ofFnM` on an empty domain is the empty list. -/
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem ofFnM_zero [Monad m] [LawfulMonad m] {f : Fin 0 → m α} : ofFnM f = pure [] := by
|
||||
simp [ofFnM]
|
||||
|
||||
|
||||
@@ -159,7 +159,7 @@ theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y →
|
||||
|
||||
@[grind =] theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {a : α} {l₁ l₂ : List α} :
|
||||
Pairwise R (l₁ ++ a :: l₂) ↔ Pairwise R (a :: (l₁ ++ l₂)) := by
|
||||
change Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
|
||||
show Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
|
||||
rw [← append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l₁), pairwise_append_comm s]
|
||||
simp only [mem_append, or_comm]
|
||||
|
||||
@@ -279,11 +279,7 @@ theorem nodup_nil : @Nodup α [] :=
|
||||
theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by
|
||||
simp only [Nodup, pairwise_cons, forall_mem_ne]
|
||||
|
||||
@[grind =] theorem nodup_append {l₁ l₂ : List α} :
|
||||
(l₁ ++ l₂).Nodup ↔ l₁.Nodup ∧ l₂.Nodup ∧ ∀ a ∈ l₁, ∀ b ∈ l₂, a ≠ b :=
|
||||
pairwise_append
|
||||
|
||||
theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
|
||||
@[grind →] theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
|
||||
Pairwise.sublist
|
||||
|
||||
grind_pattern Nodup.sublist => l₁ <+ l₂, Nodup l₁
|
||||
@@ -316,48 +312,4 @@ theorem getElem?_inj {xs : List α}
|
||||
@[simp, grind =] theorem nodup_replicate {n : Nat} {a : α} :
|
||||
(replicate n a).Nodup ↔ n ≤ 1 := by simp [Nodup]
|
||||
|
||||
theorem Nodup.count [BEq α] [LawfulBEq α] {a : α} {l : List α} (h : Nodup l) : count a l = if a ∈ l then 1 else 0 := by
|
||||
split <;> rename_i h'
|
||||
· obtain ⟨s, t, rfl⟩ := List.append_of_mem h'
|
||||
rw [nodup_append] at h
|
||||
simp_all
|
||||
rw [count_eq_zero.mpr ?_, count_eq_zero.mpr ?_]
|
||||
· exact h.2.1.1
|
||||
· intro w
|
||||
simpa using h.2.2 _ w
|
||||
· rw [count_eq_zero_of_not_mem h']
|
||||
|
||||
grind_pattern Nodup.count => count a l, Nodup l
|
||||
|
||||
@[grind =]
|
||||
theorem nodup_iff_count [BEq α] [LawfulBEq α] {l : List α} : l.Nodup ↔ ∀ a, count a l ≤ 1 := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons x l ih =>
|
||||
constructor
|
||||
· intro h a
|
||||
simp at h
|
||||
rw [count_cons]
|
||||
split <;> rename_i h'
|
||||
· simp at h'
|
||||
rw [count_eq_zero.mpr ?_]
|
||||
· exact Nat.le_refl _
|
||||
· exact h' ▸ h.1
|
||||
· simp at h'
|
||||
refine ih.mp h.2 a
|
||||
· intro h
|
||||
simp only [count_cons] at h
|
||||
simp only [nodup_cons]
|
||||
constructor
|
||||
· intro w
|
||||
specialize h x
|
||||
simp at h
|
||||
have := count_pos_iff.mpr w
|
||||
replace h := le_of_lt_succ h
|
||||
apply Nat.lt_irrefl _ (Nat.lt_of_lt_of_le this h)
|
||||
· rw [ih]
|
||||
intro a
|
||||
specialize h a
|
||||
exact le_of_add_right_le h
|
||||
|
||||
end List
|
||||
|
||||
@@ -23,7 +23,8 @@ The notation `~` is used for permutation equivalence.
|
||||
-/
|
||||
|
||||
set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
|
||||
set_option linter.indexVariables true -- Enforce naming conventions for index variables.
|
||||
-- TODO: restore after an update-stage0
|
||||
-- set_option linter.indexVariables true -- Enforce naming conventions for index variables.
|
||||
|
||||
open Nat
|
||||
|
||||
@@ -89,9 +90,6 @@ theorem Perm.mem_iff {a : α} {l₁ l₂ : List α} (p : l₁ ~ l₂) : a ∈ l
|
||||
| swap => simp only [mem_cons, or_left_comm]
|
||||
| trans _ _ ih₁ ih₂ => simp only [ih₁, ih₂]
|
||||
|
||||
grind_pattern Perm.mem_iff => l₁ ~ l₂, a ∈ l₁
|
||||
grind_pattern Perm.mem_iff => l₁ ~ l₂, a ∈ l₂
|
||||
|
||||
theorem Perm.subset {l₁ l₂ : List α} (p : l₁ ~ l₂) : l₁ ⊆ l₂ := fun _ => p.mem_iff.mp
|
||||
|
||||
theorem Perm.append_right {l₁ l₂ : List α} (t₁ : List α) (p : l₁ ~ l₂) : l₁ ++ t₁ ~ l₂ ++ t₁ := by
|
||||
@@ -108,15 +106,9 @@ theorem Perm.append_left {t₁ t₂ : List α} : ∀ l : List α, t₁ ~ t₂
|
||||
theorem Perm.append {l₁ l₂ t₁ t₂ : List α} (p₁ : l₁ ~ l₂) (p₂ : t₁ ~ t₂) : l₁ ++ t₁ ~ l₂ ++ t₂ :=
|
||||
(p₁.append_right t₁).trans (p₂.append_left l₂)
|
||||
|
||||
grind_pattern Perm.append => l₁ ~ l₂, t₁ ~ t₂, l₁ ++ t₁
|
||||
grind_pattern Perm.append => l₁ ~ l₂, t₁ ~ t₂, l₂ ++ t₂
|
||||
|
||||
theorem Perm.append_cons (a : α) {l₁ l₂ r₁ r₂ : List α} (p₁ : l₁ ~ l₂) (p₂ : r₁ ~ r₂) :
|
||||
l₁ ++ a :: r₁ ~ l₂ ++ a :: r₂ := p₁.append (p₂.cons a)
|
||||
|
||||
grind_pattern Perm.append_cons => l₁ ~ l₂, r₁ ~ r₂, l₁ ++ a :: r₁
|
||||
grind_pattern Perm.append_cons => l₁ ~ l₂, r₁ ~ r₂, l₂ ++ a :: r₂
|
||||
|
||||
@[simp] theorem perm_middle {a : α} : ∀ {l₁ l₂ : List α}, l₁ ++ a :: l₂ ~ a :: (l₁ ++ l₂)
|
||||
| [], _ => .refl _
|
||||
| b :: _, _ => (Perm.cons _ perm_middle).trans (swap a b _)
|
||||
@@ -202,15 +194,9 @@ theorem Perm.filterMap (f : α → Option β) {l₁ l₂ : List α} (p : l₁ ~
|
||||
| swap x y l₂ => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap_cons, swap]
|
||||
| trans _p₁ _p₂ IH₁ IH₂ => exact IH₁.trans IH₂
|
||||
|
||||
grind_pattern Perm.filterMap => l₁ ~ l₂, filterMap f l₁
|
||||
grind_pattern Perm.filterMap => l₁ ~ l₂, filterMap f l₂
|
||||
|
||||
theorem Perm.map (f : α → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) : map f l₁ ~ map f l₂ :=
|
||||
filterMap_eq_map ▸ p.filterMap _
|
||||
|
||||
grind_pattern Perm.map => l₁ ~ l₂, map f l₁
|
||||
grind_pattern Perm.map => l₁ ~ l₂, map f l₂
|
||||
|
||||
theorem Perm.pmap {p : α → Prop} (f : ∀ a, p a → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) {H₁ H₂} :
|
||||
pmap f l₁ H₁ ~ pmap f l₂ H₂ := by
|
||||
induction p with
|
||||
@@ -219,18 +205,12 @@ theorem Perm.pmap {p : α → Prop} (f : ∀ a, p a → β) {l₁ l₂ : List α
|
||||
| swap x y => simp [swap]
|
||||
| trans _p₁ p₂ IH₁ IH₂ => exact IH₁.trans (IH₂ (H₁ := fun a m => H₂ a (p₂.subset m)))
|
||||
|
||||
grind_pattern Perm.pmap => l₁ ~ l₂, pmap f l₁ H₁
|
||||
grind_pattern Perm.pmap => l₁ ~ l₂, pmap f l₂ H₂
|
||||
|
||||
theorem Perm.unattach {α : Type u} {p : α → Prop} {l₁ l₂ : List { x // p x }} (h : l₁ ~ l₂) :
|
||||
l₁.unattach.Perm l₂.unattach := h.map _
|
||||
|
||||
theorem Perm.filter (p : α → Bool) {l₁ l₂ : List α} (s : l₁ ~ l₂) :
|
||||
filter p l₁ ~ filter p l₂ := by rw [← filterMap_eq_filter]; apply s.filterMap
|
||||
|
||||
grind_pattern Perm.filter => l₁ ~ l₂, filter p l₁
|
||||
grind_pattern Perm.filter => l₁ ~ l₂, filter p l₂
|
||||
|
||||
theorem filter_append_perm (p : α → Bool) (l : List α) :
|
||||
filter p l ++ filter (fun x => !p x) l ~ l := by
|
||||
induction l with
|
||||
@@ -408,16 +388,12 @@ theorem Perm.erase (a : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) : l₁.erase
|
||||
have h₂ : a ∉ l₂ := mt p.mem_iff.2 h₁
|
||||
rw [erase_of_not_mem h₁, erase_of_not_mem h₂]; exact p
|
||||
|
||||
grind_pattern Perm.erase => l₁ ~ l₂, l₁.erase a
|
||||
grind_pattern Perm.erase => l₁ ~ l₂, l₂.erase a
|
||||
|
||||
theorem cons_perm_iff_perm_erase {a : α} {l₁ l₂ : List α} :
|
||||
a :: l₁ ~ l₂ ↔ a ∈ l₂ ∧ l₁ ~ l₂.erase a := by
|
||||
refine ⟨fun h => ?_, fun ⟨m, h⟩ => (h.cons a).trans (perm_cons_erase m).symm⟩
|
||||
have : a ∈ l₂ := h.subset mem_cons_self
|
||||
exact ⟨this, (h.trans <| perm_cons_erase this).cons_inv⟩
|
||||
|
||||
@[grind =]
|
||||
theorem perm_iff_count {l₁ l₂ : List α} : l₁ ~ l₂ ↔ ∀ a, count a l₁ = count a l₂ := by
|
||||
refine ⟨Perm.count_eq, fun H => ?_⟩
|
||||
induction l₁ generalizing l₂ with
|
||||
@@ -434,12 +410,6 @@ theorem perm_iff_count {l₁ l₂ : List α} : l₁ ~ l₂ ↔ ∀ a, count a l
|
||||
rw [(perm_cons_erase this).count_eq] at H
|
||||
by_cases h : b = a <;> simpa [h, count_cons, Nat.succ_inj] using H
|
||||
|
||||
theorem Perm.count (h : l₁ ~ l₂) (a : α) : count a l₁ = count a l₂ := by
|
||||
rw [perm_iff_count.mp h]
|
||||
|
||||
grind_pattern Perm.count => l₁ ~ l₂, count a l₁
|
||||
grind_pattern Perm.count => l₁ ~ l₂, count a l₂
|
||||
|
||||
theorem isPerm_iff : ∀ {l₁ l₂ : List α}, l₁.isPerm l₂ ↔ l₁ ~ l₂
|
||||
| [], [] => by simp [isPerm, isEmpty]
|
||||
| [], _ :: _ => by simp [isPerm, isEmpty, Perm.nil_eq]
|
||||
@@ -455,9 +425,6 @@ protected theorem Perm.insert (a : α) {l₁ l₂ : List α} (p : l₁ ~ l₂) :
|
||||
have := p.cons a
|
||||
simpa [h, mt p.mem_iff.2 h] using this
|
||||
|
||||
grind_pattern Perm.insert => l₁ ~ l₂, l₁.insert a
|
||||
grind_pattern Perm.insert => l₁ ~ l₂, l₂.insert a
|
||||
|
||||
theorem perm_insert_swap (x y : α) (l : List α) :
|
||||
List.insert x (List.insert y l) ~ List.insert y (List.insert x l) := by
|
||||
by_cases xl : x ∈ l <;> by_cases yl : y ∈ l <;> simp [xl, yl]
|
||||
@@ -524,9 +491,6 @@ theorem Perm.nodup {l l' : List α} (hl : l ~ l') (hR : l.Nodup) : l'.Nodup := h
|
||||
theorem Perm.nodup_iff {l₁ l₂ : List α} : l₁ ~ l₂ → (Nodup l₁ ↔ Nodup l₂) :=
|
||||
Perm.pairwise_iff <| @Ne.symm α
|
||||
|
||||
grind_pattern Perm.nodup_iff => l₁ ~ l₂, Nodup l₁
|
||||
grind_pattern Perm.nodup_iff => l₁ ~ l₂, Nodup l₂
|
||||
|
||||
theorem Perm.flatten {l₁ l₂ : List (List α)} (h : l₁ ~ l₂) : l₁.flatten ~ l₂.flatten := by
|
||||
induction h with
|
||||
| nil => rfl
|
||||
@@ -577,30 +541,20 @@ theorem perm_insertIdx {α} (x : α) (l : List α) {i} (h : i ≤ l.length) :
|
||||
|
||||
namespace Perm
|
||||
|
||||
theorem take {l₁ l₂ : List α} (h : l₁ ~ l₂) {i : Nat} (w : l₁.drop i ~ l₂.drop i) :
|
||||
l₁.take i ~ l₂.take i := by
|
||||
theorem take {l₁ l₂ : List α} (h : l₁ ~ l₂) {n : Nat} (w : l₁.drop n ~ l₂.drop n) :
|
||||
l₁.take n ~ l₂.take n := by
|
||||
classical
|
||||
rw [perm_iff_count] at h w ⊢
|
||||
rw [← take_append_drop i l₁, ← take_append_drop i l₂] at h
|
||||
rw [← take_append_drop n l₁, ← take_append_drop n l₂] at h
|
||||
simpa only [count_append, w, Nat.add_right_cancel_iff] using h
|
||||
|
||||
theorem drop {l₁ l₂ : List α} (h : l₁ ~ l₂) {i : Nat} (w : l₁.take i ~ l₂.take i) :
|
||||
l₁.drop i ~ l₂.drop i := by
|
||||
theorem drop {l₁ l₂ : List α} (h : l₁ ~ l₂) {n : Nat} (w : l₁.take n ~ l₂.take n) :
|
||||
l₁.drop n ~ l₂.drop n := by
|
||||
classical
|
||||
rw [perm_iff_count] at h w ⊢
|
||||
rw [← take_append_drop i l₁, ← take_append_drop i l₂] at h
|
||||
rw [← take_append_drop n l₁, ← take_append_drop n l₂] at h
|
||||
simpa only [count_append, w, Nat.add_left_cancel_iff] using h
|
||||
|
||||
theorem sum_nat {l₁ l₂ : List Nat} (h : l₁ ~ l₂) : l₁.sum = l₂.sum := by
|
||||
induction h with
|
||||
| nil => simp
|
||||
| cons _ _ ih => simp [ih]
|
||||
| swap => simpa [List.sum_cons] using Nat.add_left_comm ..
|
||||
| trans _ _ ih₁ ih₂ => simp [ih₁, ih₂]
|
||||
|
||||
grind_pattern Perm.sum_nat => l₁ ~ l₂, l₁.sum
|
||||
grind_pattern Perm.sum_nat => l₁ ~ l₂, l₂.sum
|
||||
|
||||
end Perm
|
||||
|
||||
end List
|
||||
|
||||
@@ -225,7 +225,7 @@ theorem zipIdx_eq_nil_iff {l : List α} {i : Nat} : List.zipIdx l i = [] ↔ l =
|
||||
| [], _ => rfl
|
||||
| _ :: _, _ => congrArg Nat.succ length_zipIdx
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem getElem?_zipIdx :
|
||||
∀ {l : List α} {i j}, (zipIdx l i)[j]? = l[j]?.map fun a => (a, i + j)
|
||||
| [], _, _ => rfl
|
||||
@@ -234,7 +234,7 @@ theorem getElem?_zipIdx :
|
||||
simp only [zipIdx_cons, getElem?_cons_succ]
|
||||
exact getElem?_zipIdx.trans <| by rw [Nat.add_right_comm]; rfl
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem getElem_zipIdx {l : List α} (h : i < (l.zipIdx j).length) :
|
||||
(l.zipIdx j)[i] = (l[i]'(by simpa [length_zipIdx] using h), j + i) := by
|
||||
simp only [length_zipIdx] at h
|
||||
@@ -242,7 +242,7 @@ theorem getElem_zipIdx {l : List α} (h : i < (l.zipIdx j).length) :
|
||||
simp only [getElem?_zipIdx, getElem?_eq_getElem h]
|
||||
simp
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem tail_zipIdx {l : List α} {i : Nat} : (zipIdx l i).tail = zipIdx l.tail (i + 1) := by
|
||||
induction l generalizing i with
|
||||
| nil => simp
|
||||
|
||||
@@ -685,7 +685,7 @@ theorem replace_toArray [BEq α] [LawfulBEq α] (l : List α) (a b : α) :
|
||||
· rw [if_pos (by omega), if_pos, if_neg]
|
||||
· simp only [mem_take_iff_getElem, not_exists]
|
||||
intro k hk
|
||||
simpa using h.2 ⟨k, by omega⟩ (by change k < i.1; omega)
|
||||
simpa using h.2 ⟨k, by omega⟩ (by show k < i.1; omega)
|
||||
· subst h₃
|
||||
simpa using h.1
|
||||
· rw [if_neg (by omega)]
|
||||
|
||||
@@ -46,7 +46,6 @@ theorem zipWith_self {f : α → α → δ} : ∀ {l : List α}, zipWith f l l =
|
||||
See also `getElem?_zipWith'` for a variant
|
||||
using `Option.map` and `Option.bind` rather than a `match`.
|
||||
-/
|
||||
@[grind =]
|
||||
theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
|
||||
(zipWith f as bs)[i]? = match as[i]?, bs[i]? with
|
||||
| some a, some b => some (f a b) | _, _ => none := by
|
||||
@@ -84,39 +83,33 @@ theorem getElem?_zip_eq_some {l₁ : List α} {l₂ : List β} {z : α × β} {i
|
||||
· rintro ⟨h₀, h₁⟩
|
||||
exact ⟨_, _, h₀, h₁, rfl⟩
|
||||
|
||||
@[grind =]
|
||||
theorem head?_zipWith {f : α → β → γ} :
|
||||
(List.zipWith f as bs).head? = match as.head?, bs.head? with
|
||||
| some a, some b => some (f a b) | _, _ => none := by
|
||||
simp [head?_eq_getElem?, getElem?_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem head_zipWith {f : α → β → γ} (h):
|
||||
(List.zipWith f as bs).head h = f (as.head (by rintro rfl; simp_all)) (bs.head (by rintro rfl; simp_all)) := by
|
||||
apply Option.some.inj
|
||||
rw [← head?_eq_head, head?_zipWith, head?_eq_head, head?_eq_head]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem zipWith_map {μ} {f : γ → δ → μ} {g : α → γ} {h : β → δ} {l₁ : List α} {l₂ : List β} :
|
||||
zipWith f (l₁.map g) (l₂.map h) = zipWith (fun a b => f (g a) (h b)) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_map_left {l₁ : List α} {l₂ : List β} {f : α → α'} {g : α' → β → γ} :
|
||||
zipWith g (l₁.map f) l₂ = zipWith (fun a b => g (f a) b) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_map_right {l₁ : List α} {l₂ : List β} {f : β → β'} {g : α → β' → γ} :
|
||||
zipWith g l₁ (l₂.map f) = zipWith (fun a b => g a (f b)) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} {i : δ} {g : γ → δ → δ} :
|
||||
(zipWith f l₁ l₂).foldr g i = (zip l₁ l₂).foldr (fun p r => g (f p.1 p.2) r) i := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} {g : δ → γ → δ} :
|
||||
(zipWith f l₁ l₂).foldl g i = (zip l₁ l₂).foldl (fun r p => g r (f p.1 p.2)) i := by
|
||||
induction l₁ generalizing i l₂ <;> cases l₂ <;> simp_all
|
||||
@@ -125,7 +118,6 @@ theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} {g : δ →
|
||||
theorem zipWith_eq_nil_iff {f : α → β → γ} {l l'} : zipWith f l l' = [] ↔ l = [] ∨ l' = [] := by
|
||||
cases l <;> cases l' <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {l : List γ} {l' : List δ} :
|
||||
map f (zipWith g l l') = zipWith (fun x y => f (g x y)) l l' := by
|
||||
induction l generalizing l' with
|
||||
@@ -135,7 +127,6 @@ theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {l : Li
|
||||
· simp
|
||||
· simp [hl]
|
||||
|
||||
@[grind =]
|
||||
theorem take_zipWith : (zipWith f l l').take i = zipWith f (l.take i) (l'.take i) := by
|
||||
induction l generalizing l' i with
|
||||
| nil => simp
|
||||
@@ -146,7 +137,6 @@ theorem take_zipWith : (zipWith f l l').take i = zipWith f (l.take i) (l'.take i
|
||||
· simp
|
||||
· simp [hl]
|
||||
|
||||
@[grind =]
|
||||
theorem drop_zipWith : (zipWith f l l').drop i = zipWith f (l.drop i) (l'.drop i) := by
|
||||
induction l generalizing l' i with
|
||||
| nil => simp
|
||||
@@ -157,11 +147,10 @@ theorem drop_zipWith : (zipWith f l l').drop i = zipWith f (l.drop i) (l'.drop i
|
||||
· simp
|
||||
· simp [hl]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem tail_zipWith : (zipWith f l l').tail = zipWith f l.tail l'.tail := by
|
||||
rw [← drop_one]; simp [drop_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_append {f : α → β → γ} {l₁ l₁' : List α} {l₂ l₂' : List β}
|
||||
(h : l₁.length = l₂.length) :
|
||||
zipWith f (l₁ ++ l₁') (l₂ ++ l₂') = zipWith f l₁ l₂ ++ zipWith f l₁' l₂' := by
|
||||
@@ -265,26 +254,22 @@ theorem zip_eq_zipWith : ∀ {l₁ : List α} {l₂ : List β}, zip l₁ l₂ =
|
||||
| _, [] => rfl
|
||||
| a :: l₁, b :: l₂ => by simp [zip_cons_cons, zip_eq_zipWith (l₁ := l₁)]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map {f : α → γ} {g : β → δ} :
|
||||
∀ {l₁ : List α} {l₂ : List β}, zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
|
||||
| [], _ => rfl
|
||||
| _, [] => by simp only [map, zip_nil_right]
|
||||
| _ :: _, _ :: _ => by simp only [map, zip_cons_cons, zip_map, Prod.map]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map_left {f : α → γ} {l₁ : List α} {l₂ : List β} :
|
||||
zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map_right {f : β → γ} {l₁ : List α} {l₂ : List β} :
|
||||
zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]
|
||||
|
||||
@[simp, grind =] theorem tail_zip {l₁ : List α} {l₂ : List β} :
|
||||
@[simp] theorem tail_zip {l₁ : List α} {l₂ : List β} :
|
||||
(zip l₁ l₂).tail = zip l₁.tail l₂.tail := by
|
||||
cases l₁ <;> cases l₂ <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem zip_append :
|
||||
∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
|
||||
zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
|
||||
@@ -293,7 +278,6 @@ theorem zip_append :
|
||||
| _ :: _, _, _ :: _, _, h => by
|
||||
simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]
|
||||
|
||||
@[grind =]
|
||||
theorem zip_map' {f : α → β} {g : α → γ} :
|
||||
∀ {l : List α}, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
|
||||
| [] => rfl
|
||||
@@ -312,7 +296,7 @@ theorem map_fst_zip :
|
||||
| [], _, _ => rfl
|
||||
| _ :: as, _ :: bs, h => by
|
||||
simp [Nat.succ_le_succ_iff] at h
|
||||
change _ :: map Prod.fst (zip as bs) = _ :: as
|
||||
show _ :: map Prod.fst (zip as bs) = _ :: as
|
||||
rw [map_fst_zip (l₁ := as) h]
|
||||
| _ :: _, [], h => by simp at h
|
||||
|
||||
@@ -324,7 +308,7 @@ theorem map_snd_zip :
|
||||
| [], b :: bs, h => by simp at h
|
||||
| a :: as, b :: bs, h => by
|
||||
simp [Nat.succ_le_succ_iff] at h
|
||||
change _ :: map Prod.snd (zip as bs) = _ :: bs
|
||||
show _ :: map Prod.snd (zip as bs) = _ :: bs
|
||||
rw [map_snd_zip (l₂ := bs) h]
|
||||
|
||||
theorem map_prod_left_eq_zip {l : List α} {f : α → β} :
|
||||
@@ -369,7 +353,6 @@ theorem zip_eq_append_iff {l₁ : List α} {l₂ : List β} :
|
||||
|
||||
/-! ### zipWithAll -/
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
|
||||
(zipWithAll f as bs)[i]? = match as[i]?, bs[i]? with
|
||||
| none, none => .none | a?, b? => some (f a? b?) := by
|
||||
@@ -383,38 +366,33 @@ theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
|
||||
cases i <;> simp_all
|
||||
| cons b bs => cases i <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem head?_zipWithAll {f : Option α → Option β → γ} :
|
||||
(zipWithAll f as bs).head? = match as.head?, bs.head? with
|
||||
| none, none => .none | a?, b? => some (f a? b?) := by
|
||||
simp [head?_eq_getElem?, getElem?_zipWithAll]
|
||||
|
||||
@[simp, grind =] theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
|
||||
@[simp] theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
|
||||
(zipWithAll f as bs).head h = f as.head? bs.head? := by
|
||||
apply Option.some.inj
|
||||
rw [← head?_eq_head, head?_zipWithAll]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem tail_zipWithAll {f : Option α → Option β → γ} :
|
||||
@[simp] theorem tail_zipWithAll {f : Option α → Option β → γ} :
|
||||
(zipWithAll f as bs).tail = zipWithAll f as.tail bs.tail := by
|
||||
cases as <;> cases bs <;> simp
|
||||
|
||||
@[grind =]
|
||||
theorem zipWithAll_map {μ} {f : Option γ → Option δ → μ} {g : α → γ} {h : β → δ} {l₁ : List α} {l₂ : List β} :
|
||||
zipWithAll f (l₁.map g) (l₂.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem zipWithAll_map_left {l₁ : List α} {l₂ : List β} {f : α → α'} {g : Option α' → Option β → γ} :
|
||||
zipWithAll g (l₁.map f) l₂ = zipWithAll (fun a b => g (f <$> a) b) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem zipWithAll_map_right {l₁ : List α} {l₂ : List β} {f : β → β'} {g : Option α → Option β' → γ} :
|
||||
zipWithAll g l₁ (l₂.map f) = zipWithAll (fun a b => g a (f <$> b)) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option δ → α} {l : List γ} {l' : List δ} :
|
||||
map f (zipWithAll g l l') = zipWithAll (fun x y => f (g x y)) l l' := by
|
||||
induction l generalizing l' with
|
||||
@@ -422,7 +400,7 @@ theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option
|
||||
| cons hd tl hl =>
|
||||
cases l' <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
|
||||
@[simp] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
|
||||
zipWithAll f (replicate n a) (replicate n b) = replicate n (f (some a) (some b)) := by
|
||||
induction n with
|
||||
| zero => rfl
|
||||
@@ -430,13 +408,12 @@ theorem map_zipWithAll {δ : Type _} {f : α → β} {g : Option γ → Option
|
||||
|
||||
/-! ### unzip -/
|
||||
|
||||
@[simp, grind =] theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
|
||||
@[simp] theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[simp, grind =] theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
|
||||
@[simp] theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem unzip_eq_map : ∀ {l : List (α × β)}, unzip l = (l.map Prod.fst, l.map Prod.snd)
|
||||
| [] => rfl
|
||||
| (a, b) :: l => by simp only [unzip_cons, map_cons, unzip_eq_map (l := l)]
|
||||
@@ -476,6 +453,6 @@ theorem tail_zip_fst {l : List (α × β)} : l.unzip.1.tail = l.tail.unzip.1 :=
|
||||
theorem tail_zip_snd {l : List (α × β)} : l.unzip.2.tail = l.tail.unzip.2 := by
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
|
||||
@[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
|
||||
unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
|
||||
ext1 <;> simp
|
||||
|
||||
@@ -1069,7 +1069,7 @@ protected theorem sub_lt_sub_right : ∀ {a b c : Nat}, c ≤ a → a < b → a
|
||||
exact Nat.sub_lt_sub_right (le_of_succ_le_succ hle) (lt_of_succ_lt_succ h)
|
||||
|
||||
protected theorem sub_self_add (n m : Nat) : n - (n + m) = 0 := by
|
||||
change (n + 0) - (n + m) = 0
|
||||
show (n + 0) - (n + m) = 0
|
||||
rw [Nat.add_sub_add_left, Nat.zero_sub]
|
||||
|
||||
@[simp] protected theorem sub_eq_zero_of_le {n m : Nat} (h : n ≤ m) : n - m = 0 := by
|
||||
|
||||
@@ -9,7 +9,6 @@ prelude
|
||||
import Init.WF
|
||||
import Init.WFTactics
|
||||
import Init.Data.Nat.Basic
|
||||
meta import Init.MetaTypes
|
||||
|
||||
@[expose] section
|
||||
|
||||
@@ -76,7 +75,7 @@ private theorem div.go.fuel_congr (x y fuel1 fuel2 : Nat) (hy : 0 < y) (h1 : x <
|
||||
termination_by structural fuel1
|
||||
|
||||
theorem div_eq (x y : Nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 else 0 := by
|
||||
change Nat.div _ _ = ite _ (Nat.div _ _ + 1) _
|
||||
show Nat.div _ _ = ite _ (Nat.div _ _ + 1) _
|
||||
unfold Nat.div
|
||||
split
|
||||
next =>
|
||||
@@ -258,7 +257,7 @@ protected def mod : @& Nat → @& Nat → Nat
|
||||
instance instMod : Mod Nat := ⟨Nat.mod⟩
|
||||
|
||||
protected theorem modCore_eq_mod (n m : Nat) : Nat.modCore n m = n % m := by
|
||||
change Nat.modCore n m = Nat.mod n m
|
||||
show Nat.modCore n m = Nat.mod n m
|
||||
match n, m with
|
||||
| 0, _ =>
|
||||
rw [Nat.modCore_eq]
|
||||
@@ -522,7 +521,7 @@ theorem mul_sub_div (x n p : Nat) (h₁ : x < n*p) : (n * p - (x + 1)) / n = p -
|
||||
rw [Nat.mul_sub_right_distrib, Nat.mul_comm]
|
||||
exact Nat.sub_le_sub_left ((div_lt_iff_lt_mul npos).1 (lt_succ_self _)) _
|
||||
focus
|
||||
change succ (pred (n * p - x)) ≤ (succ (pred (p - x / n))) * n
|
||||
show succ (pred (n * p - x)) ≤ (succ (pred (p - x / n))) * n
|
||||
rw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h₁),
|
||||
fun h => succ_pred_eq_of_pos (Nat.sub_pos_of_lt h)] -- TODO: why is the function needed?
|
||||
focus
|
||||
|
||||
@@ -1772,12 +1772,6 @@ instance decidableExistsLE' {p : (m : Nat) → m ≤ k → Prop} [I : ∀ m h, D
|
||||
intro
|
||||
exact ⟨fun ⟨h, w⟩ => ⟨le_of_lt_succ h, w⟩, fun ⟨h, w⟩ => ⟨lt_add_one_of_le h, w⟩⟩
|
||||
|
||||
instance decidableExistsFin (P : Fin n → Prop) [DecidablePred P] : Decidable (∃ i, P i) :=
|
||||
decidable_of_iff (∃ k, k < n ∧ ((h: k < n) → P ⟨k, h⟩))
|
||||
⟨fun ⟨k, a⟩ => Exists.intro ⟨k, a.left⟩ (a.right a.left),
|
||||
fun ⟨i, e⟩ => Exists.intro i.val ⟨i.isLt, fun _ => e⟩⟩
|
||||
|
||||
|
||||
/-! ### Results about `List.sum` specialized to `Nat` -/
|
||||
|
||||
protected theorem sum_pos_iff_exists_pos {l : List Nat} : 0 < l.sum ↔ ∃ x ∈ l, 0 < x := by
|
||||
|
||||
@@ -149,7 +149,7 @@ instance : LawfulBEq PolyCnstr where
|
||||
rw [h₁, h₂, h₃]
|
||||
rfl {a} := by
|
||||
cases a; rename_i eq lhs rhs
|
||||
change (eq == eq && (lhs == lhs && rhs == rhs)) = true
|
||||
show (eq == eq && (lhs == lhs && rhs == rhs)) = true
|
||||
simp
|
||||
|
||||
structure ExprCnstr where
|
||||
|
||||
@@ -85,4 +85,4 @@ theorem Membership.get_elem_helper {i n : Nat} {r : Std.Range} (h₁ : i ∈ r)
|
||||
i < n := h₂ ▸ h₁.2.1
|
||||
|
||||
macro_rules
|
||||
| `(tactic| get_elem_tactic_extensible) => `(tactic| apply Membership.get_elem_helper; assumption; rfl)
|
||||
| `(tactic| get_elem_tactic_trivial) => `(tactic| apply Membership.get_elem_helper; assumption; rfl)
|
||||
|
||||
@@ -342,12 +342,11 @@ instance : Repr Int where
|
||||
def hexDigitRepr (n : Nat) : String :=
|
||||
String.singleton <| Nat.digitChar n
|
||||
|
||||
def Char.quoteCore (c : Char) (inString : Bool := false) : String :=
|
||||
def Char.quoteCore (c : Char) : String :=
|
||||
if c = '\n' then "\\n"
|
||||
else if c = '\t' then "\\t"
|
||||
else if c = '\\' then "\\\\"
|
||||
else if c = '\"' then "\\\""
|
||||
else if !inString && c = '\'' then "\\\'"
|
||||
else if c.toNat <= 31 ∨ c = '\x7f' then "\\x" ++ smallCharToHex c
|
||||
else String.singleton c
|
||||
where
|
||||
@@ -384,7 +383,7 @@ Examples:
|
||||
-/
|
||||
def String.quote (s : String) : String :=
|
||||
if s.isEmpty then "\"\""
|
||||
else s.foldl (fun s c => s ++ c.quoteCore (inString := true)) "\"" ++ "\""
|
||||
else s.foldl (fun s c => s ++ c.quoteCore) "\"" ++ "\""
|
||||
|
||||
instance : Repr String where
|
||||
reprPrec s _ := s.quote
|
||||
|
||||
@@ -586,7 +586,7 @@ decreasing_by
|
||||
focus
|
||||
rename_i i₀ j₀ _ eq h'
|
||||
rw [show (s.next i₀ - sep.next j₀).1 = (i₀ - j₀).1 by
|
||||
change (_ + Char.utf8Size _) - (_ + Char.utf8Size _) = _
|
||||
show (_ + Char.utf8Size _) - (_ + Char.utf8Size _) = _
|
||||
rw [(beq_iff_eq ..).1 eq, Nat.add_sub_add_right]; rfl]
|
||||
right; exact Nat.sub_lt_sub_left
|
||||
(Nat.lt_of_le_of_lt (Nat.le_add_right ..) (Nat.gt_of_not_le (mt decide_eq_true h')))
|
||||
|
||||
@@ -295,11 +295,11 @@ where
|
||||
termination_by text.utf8ByteSize - pos.byteIdx
|
||||
decreasing_by
|
||||
decreasing_with
|
||||
change text.utf8ByteSize - (text.next (text.next pos)).byteIdx < text.utf8ByteSize - pos.byteIdx
|
||||
show text.utf8ByteSize - (text.next (text.next pos)).byteIdx < text.utf8ByteSize - pos.byteIdx
|
||||
have k := Nat.gt_of_not_le <| mt decide_eq_true h
|
||||
exact Nat.sub_lt_sub_left k (Nat.lt_trans (String.lt_next text pos) (String.lt_next _ _))
|
||||
decreasing_with
|
||||
change text.utf8ByteSize - (text.next pos).byteIdx < text.utf8ByteSize - pos.byteIdx
|
||||
show text.utf8ByteSize - (text.next pos).byteIdx < text.utf8ByteSize - pos.byteIdx
|
||||
have k := Nat.gt_of_not_le <| mt decide_eq_true h
|
||||
exact Nat.sub_lt_sub_left k (String.lt_next _ _)
|
||||
|
||||
|
||||
@@ -6,7 +6,8 @@ Author: Leonardo de Moura
|
||||
module
|
||||
|
||||
prelude
|
||||
meta import Init.Meta
|
||||
import Init.Meta
|
||||
import Init.Data.ToString.Basic
|
||||
|
||||
syntax:max "s!" interpolatedStr(term) : term
|
||||
|
||||
|
||||
@@ -7,7 +7,6 @@ Authors: Shreyas Srinivas, François G. Dorais, Kim Morrison
|
||||
module
|
||||
|
||||
prelude
|
||||
meta import Init.Coe
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.Array.MapIdx
|
||||
import Init.Data.Array.InsertIdx
|
||||
|
||||
@@ -40,7 +40,6 @@ theorem countP_push {a : α} {xs : Vector α n} : countP p (xs.push a) = countP
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.countP_push]
|
||||
|
||||
@[grind =]
|
||||
theorem countP_singleton {a : α} : countP p #v[a] = if p a then 1 else 0 := by
|
||||
simp
|
||||
|
||||
@@ -52,7 +51,7 @@ theorem countP_le_size {xs : Vector α n} : countP p xs ≤ n := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.countP_le_size (p := p)]
|
||||
|
||||
@[simp, grind =] theorem countP_append {xs : Vector α n} {ys : Vector α m} : countP p (xs ++ ys) = countP p xs + countP p ys := by
|
||||
@[simp] theorem countP_append {xs : Vector α n} {ys : Vector α m} : countP p (xs ++ ys) = countP p xs + countP p ys := by
|
||||
cases xs
|
||||
cases ys
|
||||
simp
|
||||
@@ -117,7 +116,7 @@ theorem countP_flatMap {p : β → Bool} {xs : Vector α n} {f : α → Vector
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.countP_flatMap, Function.comp_def]
|
||||
|
||||
@[simp, grind =] theorem countP_reverse {xs : Vector α n} : countP p xs.reverse = countP p xs := by
|
||||
@[simp] theorem countP_reverse {xs : Vector α n} : countP p xs.reverse = countP p xs := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@@ -137,7 +136,7 @@ section count
|
||||
|
||||
variable [BEq α]
|
||||
|
||||
@[simp, grind =] theorem count_empty {a : α} : count a #v[] = 0 := rfl
|
||||
@[simp] theorem count_empty {a : α} : count a #v[] = 0 := rfl
|
||||
|
||||
theorem count_push {a b : α} {xs : Vector α n} :
|
||||
count a (xs.push b) = count a xs + if b == a then 1 else 0 := by
|
||||
@@ -152,25 +151,23 @@ theorem count_eq_countP' {a : α} : count (n := n) a = countP (· == a) := by
|
||||
|
||||
theorem count_le_size {a : α} {xs : Vector α n} : count a xs ≤ n := countP_le_size
|
||||
|
||||
grind_pattern count_le_size => count a xs
|
||||
|
||||
theorem count_le_count_push {a b : α} {xs : Vector α n} : count a xs ≤ count a (xs.push b) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.count_push]
|
||||
|
||||
@[simp, grind =] theorem count_singleton {a b : α} : count a #v[b] = if b == a then 1 else 0 := by
|
||||
@[simp] theorem count_singleton {a b : α} : count a #v[b] = if b == a then 1 else 0 := by
|
||||
simp [count_eq_countP]
|
||||
|
||||
@[simp, grind =] theorem count_append {a : α} {xs : Vector α n} {ys : Vector α m} :
|
||||
@[simp] theorem count_append {a : α} {xs : Vector α n} {ys : Vector α m} :
|
||||
count a (xs ++ ys) = count a xs + count a ys :=
|
||||
countP_append ..
|
||||
|
||||
@[simp, grind =] theorem count_flatten {a : α} {xss : Vector (Vector α m) n} :
|
||||
@[simp] theorem count_flatten {a : α} {xss : Vector (Vector α m) n} :
|
||||
count a xss.flatten = (xss.map (count a)).sum := by
|
||||
rcases xss with ⟨xss, rfl⟩
|
||||
simp [Array.count_flatten, Function.comp_def]
|
||||
|
||||
@[simp, grind =] theorem count_reverse {a : α} {xs : Vector α n} : count a xs.reverse = count a xs := by
|
||||
@[simp] theorem count_reverse {a : α} {xs : Vector α n} : count a xs.reverse = count a xs := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
|
||||
@@ -22,13 +22,11 @@ open Nat
|
||||
|
||||
/-! ### eraseIdx -/
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_eq_take_drop_succ {xs : Vector α n} {i : Nat} (h) :
|
||||
xs.eraseIdx i = (xs.take i ++ xs.drop (i + 1)).cast (by omega) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.eraseIdx_eq_take_drop_succ, *]
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_eraseIdx {xs : Vector α n} {i : Nat} (h : i < n) {j : Nat} :
|
||||
(xs.eraseIdx i)[j]? = if j < i then xs[j]? else xs[j + 1]? := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
@@ -46,14 +44,12 @@ theorem getElem?_eraseIdx_of_ge {xs : Vector α n} {i : Nat} (h : i < n) {j : Na
|
||||
intro h'
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem getElem_eraseIdx {xs : Vector α n} {i : Nat} (h : i < n) {j : Nat} (h' : j < n - 1) :
|
||||
(xs.eraseIdx i)[j] = if h'' : j < i then xs[j] else xs[j + 1] := by
|
||||
apply Option.some.inj
|
||||
rw [← getElem?_eq_getElem, getElem?_eraseIdx]
|
||||
split <;> simp
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_mem_eraseIdx {xs : Vector α n} {i : Nat} {h} {a : α} (h : a ∈ xs.eraseIdx i) : a ∈ xs := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simpa using Array.mem_of_mem_eraseIdx (by simpa using h)
|
||||
@@ -68,23 +64,13 @@ theorem eraseIdx_append_of_length_le {xs : Vector α n} {k : Nat} (hk : n ≤ k)
|
||||
eraseIdx (xs ++ xs') k = (xs ++ eraseIdx xs' (k - n)).cast (by omega) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
rcases xs' with ⟨xs'⟩
|
||||
simp [Array.eraseIdx_append_of_size_le, *]
|
||||
simp [Array.eraseIdx_append_of_length_le, *]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_append {xs : Vector α n} {ys : Vector α m} {k : Nat} (h : k < n + m) :
|
||||
eraseIdx (xs ++ ys) k = if h' : k < n then (eraseIdx xs k ++ ys).cast (by omega) else (xs ++ eraseIdx ys (k - n) (by omega)).cast (by omega) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
rcases ys with ⟨ys, rfl⟩
|
||||
simp only [mk_append_mk, eraseIdx_mk, Array.eraseIdx_append]
|
||||
split <;> simp
|
||||
|
||||
@[grind = ]
|
||||
theorem eraseIdx_cast {xs : Vector α n} {k : Nat} (h : k < m) :
|
||||
eraseIdx (xs.cast w) k h = (eraseIdx xs k).cast (by omega) := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} {h} :
|
||||
(replicate n a).eraseIdx k = replicate (n - 1) a := by
|
||||
rw [replicate_eq_mk_replicate, eraseIdx_mk]
|
||||
@@ -126,45 +112,6 @@ theorem eraseIdx_set_gt {xs : Vector α n} {i : Nat} {j : Nat} {a : α} (h : i <
|
||||
rcases xs with ⟨xs⟩
|
||||
simp [Array.eraseIdx_set_gt, *]
|
||||
|
||||
@[grind =]
|
||||
theorem eraseIdx_set {xs : Vector α n} {i : Nat} {a : α} {hi : i < n} {j : Nat} {hj : j < n} :
|
||||
(xs.set i a).eraseIdx j =
|
||||
if h' : j < i then
|
||||
(xs.eraseIdx j).set (i - 1) a
|
||||
else if h'' : j = i then
|
||||
xs.eraseIdx i
|
||||
else
|
||||
(xs.eraseIdx j).set i a := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp only [set_mk, eraseIdx_mk, Array.eraseIdx_set]
|
||||
split
|
||||
· simp
|
||||
· split <;> simp
|
||||
|
||||
theorem set_eraseIdx_le {xs : Vector α n} {i : Nat} {w : i < n} {j : Nat} {a : α} (h : i ≤ j) (hj : j < n - 1) :
|
||||
(xs.eraseIdx i).set j a = (xs.set (j + 1) a).eraseIdx i := by
|
||||
rw [eraseIdx_set_lt]
|
||||
· simp
|
||||
· omega
|
||||
|
||||
theorem set_eraseIdx_gt {xs : Vector α n} {i : Nat} {w : i < n} {j : Nat} {a : α} (h : j < i) (hj : j < n - 1) :
|
||||
(xs.eraseIdx i).set j a = (xs.set j a).eraseIdx i := by
|
||||
rw [eraseIdx_set_gt]
|
||||
omega
|
||||
|
||||
@[grind =]
|
||||
theorem set_eraseIdx {xs : Vector α n} {i : Nat} {w : i < n} {j : Nat} {a : α} (hj : j < n - 1) :
|
||||
(xs.eraseIdx i).set j a =
|
||||
if h' : i ≤ j then
|
||||
(xs.set (j + 1) a).eraseIdx i
|
||||
else
|
||||
(xs.set j a).eraseIdx i := by
|
||||
split <;> rename_i h'
|
||||
· rw [set_eraseIdx_le]
|
||||
omega
|
||||
· rw [set_eraseIdx_gt]
|
||||
omega
|
||||
|
||||
@[simp] theorem set_getElem_succ_eraseIdx_succ
|
||||
{xs : Vector α n} {i : Nat} (h : i + 1 < n) :
|
||||
(xs.eraseIdx (i + 1)).set i xs[i + 1] = xs.eraseIdx i := by
|
||||
|
||||
@@ -17,7 +17,7 @@ namespace Vector
|
||||
/-- `finRange n` is the vector of all elements of `Fin n` in order. -/
|
||||
protected def finRange (n : Nat) : Vector (Fin n) n := ofFn fun i => i
|
||||
|
||||
@[simp, grind =] theorem getElem_finRange {i : Nat} (h : i < n) :
|
||||
@[simp] theorem getElem_finRange {i : Nat} (h : i < n) :
|
||||
(Vector.finRange n)[i] = ⟨i, h⟩ := by
|
||||
simp [Vector.finRange]
|
||||
|
||||
@@ -39,7 +39,6 @@ theorem finRange_succ_last {n} :
|
||||
· simp_all
|
||||
omega
|
||||
|
||||
@[grind _=_]
|
||||
theorem finRange_reverse {n} : (Vector.finRange n).reverse = (Vector.finRange n).map Fin.rev := by
|
||||
ext i h
|
||||
simp
|
||||
|
||||
@@ -44,23 +44,12 @@ theorem findSome?_singleton {a : α} {f : α → Option β} : #v[a].findSome? f
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp only [push_mk, findSomeRev?_mk, Array.findSomeRev?_push_of_isNone, h]
|
||||
|
||||
@[grind =]
|
||||
theorem findSomeRev?_push {xs : Vector α n} {a : α} {f : α → Option β} :
|
||||
(xs.push a).findSomeRev? f = (f a).or (xs.findSomeRev? f) := by
|
||||
match h : f a with
|
||||
| some b =>
|
||||
rw [findSomeRev?_push_of_isSome]
|
||||
all_goals simp_all
|
||||
| none =>
|
||||
rw [findSomeRev?_push_of_isNone]
|
||||
all_goals simp_all
|
||||
|
||||
theorem exists_of_findSome?_eq_some {f : α → Option β} {xs : Vector α n} (w : xs.findSome? f = some b) :
|
||||
∃ a, a ∈ xs ∧ f a = some b := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simpa using Array.exists_of_findSome?_eq_some (by simpa using w)
|
||||
|
||||
@[simp, grind =] theorem findSome?_eq_none_iff {f : α → Option β} {xs : Vector α n} :
|
||||
@[simp] theorem findSome?_eq_none_iff {f : α → Option β} {xs : Vector α n} :
|
||||
xs.findSome? f = none ↔ ∀ x ∈ xs, f x = none := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
@@ -83,27 +72,24 @@ theorem findSome?_eq_some_iff {f : α → Option β} {xs : Vector α n} {b : β}
|
||||
· rintro ⟨k₁, k₂, h, ys, a, zs, w, h₁, h₂⟩
|
||||
exact ⟨ys.toArray, a, zs.toArray, by simp [w], h₁, by simpa using h₂⟩
|
||||
|
||||
@[simp, grind =] theorem findSome?_guard {xs : Vector α n} : findSome? (Option.guard p) xs = find? p xs := by
|
||||
@[simp] theorem findSome?_guard {xs : Vector α n} : findSome? (Option.guard fun x => p x) xs = find? p xs := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
theorem find?_eq_findSome?_guard {xs : Vector α n} : find? p xs = findSome? (Option.guard p) xs :=
|
||||
theorem find?_eq_findSome?_guard {xs : Vector α n} : find? p xs = findSome? (Option.guard fun x => p x) xs :=
|
||||
findSome?_guard.symm
|
||||
|
||||
@[simp, grind =] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Vector α n} :
|
||||
@[simp] theorem map_findSome? {f : α → Option β} {g : β → γ} {xs : Vector α n} :
|
||||
(xs.findSome? f).map g = xs.findSome? (Option.map g ∘ f) := by
|
||||
cases xs; simp
|
||||
|
||||
@[grind _=_]
|
||||
theorem findSome?_map {f : β → γ} {xs : Vector β n} : findSome? p (xs.map f) = xs.findSome? (p ∘ f) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.findSome?_map]
|
||||
|
||||
@[grind =]
|
||||
theorem findSome?_append {xs : Vector α n₁} {ys : Vector α n₂} : (xs ++ ys).findSome? f = (xs.findSome? f).or (ys.findSome? f) := by
|
||||
cases xs; cases ys; simp [Array.findSome?_append]
|
||||
|
||||
@[grind =]
|
||||
theorem getElem?_zero_flatten {xss : Vector (Vector α m) n} :
|
||||
(flatten xss)[0]? = xss.findSome? fun xs => xs[0]? := by
|
||||
cases xss using vector₂_induction
|
||||
@@ -120,14 +106,12 @@ theorem getElem_zero_flatten.proof {xss : Vector (Vector α m) n} (h : 0 < n * m
|
||||
Option.isSome_some, and_true]
|
||||
exact ⟨⟨xss[0], h₂ _ (by simp)⟩, by simp⟩
|
||||
|
||||
@[grind =]
|
||||
theorem getElem_zero_flatten {xss : Vector (Vector α m) n} (h : 0 < n * m) :
|
||||
(flatten xss)[0] = (xss.findSome? fun xs => xs[0]?).get (getElem_zero_flatten.proof h) := by
|
||||
have t := getElem?_zero_flatten (xss := xss)
|
||||
simp [getElem?_eq_getElem, h] at t
|
||||
simp [← t]
|
||||
|
||||
@[grind =]
|
||||
theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
|
||||
rw [replicate_eq_mk_replicate, findSome?_mk, Array.findSome?_replicate]
|
||||
|
||||
@@ -158,9 +142,9 @@ abbrev findSome?_mkVector_of_isNone := @findSome?_replicate_of_isNone
|
||||
|
||||
/-! ### find? -/
|
||||
|
||||
@[simp, grind =] theorem find?_empty : find? p #v[] = none := rfl
|
||||
@[simp] theorem find?_empty : find? p #v[] = none := rfl
|
||||
|
||||
@[grind =]theorem find?_singleton {a : α} {p : α → Bool} :
|
||||
theorem find?_singleton {a : α} {p : α → Bool} :
|
||||
#v[a].find? p = if p a then some a else none := by
|
||||
simp
|
||||
|
||||
@@ -168,23 +152,11 @@ abbrev findSome?_mkVector_of_isNone := @findSome?_replicate_of_isNone
|
||||
findRev? p (xs.push a) = some a := by
|
||||
cases xs; simp [h]
|
||||
|
||||
@[simp] theorem findRev?_push_of_neg {xs : Vector α n} (h : ¬p a) :
|
||||
@[simp] theorem findRev?_cons_of_neg {xs : Vector α n} (h : ¬p a) :
|
||||
findRev? p (xs.push a) = findRev? p xs := by
|
||||
cases xs; simp [h]
|
||||
|
||||
@[deprecated findRev?_push_of_neg (since := "2025-06-12")]
|
||||
abbrev findRev?_cons_of_neg := @findRev?_push_of_neg
|
||||
|
||||
@[grind =]
|
||||
theorem finRev?_push {xs : Vector α n} :
|
||||
findRev? p (xs.push a) = (Option.guard p a).or (xs.findRev? p) := by
|
||||
cases h : p a
|
||||
· rw [findRev?_push_of_neg, Option.guard_eq_none_iff.mpr h]
|
||||
all_goals simp [h]
|
||||
· rw [findRev?_push_of_pos, Option.guard_eq_some_iff.mpr ⟨rfl, h⟩]
|
||||
all_goals simp [h]
|
||||
|
||||
@[simp, grind =] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
|
||||
@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
|
||||
cases l; simp
|
||||
|
||||
theorem find?_eq_some_iff_append {xs : Vector α n} :
|
||||
@@ -209,38 +181,35 @@ theorem find?_push_eq_some {xs : Vector α n} :
|
||||
(xs.push a).find? p = some b ↔ xs.find? p = some b ∨ (xs.find? p = none ∧ (p a ∧ a = b)) := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem find?_isSome {xs : Vector α n} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
@[simp] theorem find?_isSome {xs : Vector α n} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
cases xs; simp
|
||||
|
||||
@[grind →]
|
||||
theorem find?_some {xs : Vector α n} (h : find? p xs = some a) : p a := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp at h
|
||||
exact Array.find?_some h
|
||||
|
||||
@[grind →]
|
||||
theorem mem_of_find?_eq_some {xs : Vector α n} (h : find? p xs = some a) : a ∈ xs := by
|
||||
cases xs
|
||||
simp at h
|
||||
simpa using Array.mem_of_find?_eq_some h
|
||||
|
||||
@[grind]
|
||||
theorem get_find?_mem {xs : Vector α n} (h) : (xs.find? p).get h ∈ xs := by
|
||||
cases xs
|
||||
simp [Array.get_find?_mem]
|
||||
|
||||
@[simp, grind =] theorem find?_map {f : β → α} {xs : Vector β n} :
|
||||
@[simp] theorem find?_map {f : β → α} {xs : Vector β n} :
|
||||
find? p (xs.map f) = (xs.find? (p ∘ f)).map f := by
|
||||
cases xs; simp
|
||||
|
||||
@[simp, grind =] theorem find?_append {xs : Vector α n₁} {ys : Vector α n₂} :
|
||||
@[simp] theorem find?_append {xs : Vector α n₁} {ys : Vector α n₂} :
|
||||
(xs ++ ys).find? p = (xs.find? p).or (ys.find? p) := by
|
||||
cases xs
|
||||
cases ys
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem find?_flatten {xs : Vector (Vector α m) n} {p : α → Bool} :
|
||||
xs.flatten.find? p = xs.findSome? (find? p) := by
|
||||
@[simp] theorem find?_flatten {xs : Vector (Vector α m) n} {p : α → Bool} :
|
||||
xs.flatten.find? p = xs.findSome? (·.find? p) := by
|
||||
cases xs using vector₂_induction
|
||||
simp [Array.findSome?_map, Function.comp_def]
|
||||
|
||||
@@ -248,7 +217,7 @@ theorem find?_flatten_eq_none_iff {xs : Vector (Vector α m) n} {p : α → Bool
|
||||
xs.flatten.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem find?_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
|
||||
@[simp] theorem find?_flatMap {xs : Vector α n} {f : α → Vector β m} {p : β → Bool} :
|
||||
(xs.flatMap f).find? p = xs.findSome? (fun x => (f x).find? p) := by
|
||||
cases xs
|
||||
simp [Array.find?_flatMap, Array.flatMap_toArray]
|
||||
@@ -258,7 +227,6 @@ theorem find?_flatMap_eq_none_iff {xs : Vector α n} {f : α → Vector β m} {p
|
||||
(xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
|
||||
simp
|
||||
|
||||
@[grind =]
|
||||
theorem find?_replicate :
|
||||
find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
|
||||
rw [replicate_eq_mk_replicate, find?_mk, Array.find?_replicate]
|
||||
@@ -310,7 +278,6 @@ abbrev find?_mkVector_eq_some_iff := @find?_replicate_eq_some_iff
|
||||
@[deprecated get_find?_replicate (since := "2025-03-18")]
|
||||
abbrev get_find?_mkVector := @get_find?_replicate
|
||||
|
||||
@[grind =]
|
||||
theorem find?_pmap {P : α → Prop} {f : (a : α) → P a → β} {xs : Vector α n}
|
||||
(H : ∀ (a : α), a ∈ xs → P a) {p : β → Bool} :
|
||||
(xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
|
||||
@@ -324,10 +291,8 @@ theorem find?_eq_some_iff_getElem {xs : Vector α n} {p : α → Bool} {b : α}
|
||||
|
||||
/-! ### findFinIdx? -/
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_empty {p : α → Bool} : findFinIdx? p (#v[] : Vector α 0) = none := by simp
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
|
||||
#[a].findFinIdx? p = if p a then some ⟨0, by simp⟩ else none := by
|
||||
simp
|
||||
@@ -337,12 +302,6 @@ theorem findFinIdx?_singleton {a : α} {p : α → Bool} :
|
||||
subst w
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem findFinIdx?_eq_none_iff {xs : Vector α n} {p : α → Bool} :
|
||||
xs.findFinIdx? p = none ↔ ∀ x, x ∈ xs → ¬ p x := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.findFinIdx?_eq_none_iff]
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_push {xs : Vector α n} {a : α} {p : α → Bool} :
|
||||
(xs.push a).findFinIdx? p =
|
||||
((xs.findFinIdx? p).map Fin.castSucc).or (if p a then some ⟨n, by simp⟩ else none) := by
|
||||
@@ -350,7 +309,6 @@ theorem findFinIdx?_push {xs : Vector α n} {a : α} {p : α → Bool} :
|
||||
simp [Array.findFinIdx?_push, Option.map_or, Function.comp_def]
|
||||
congr
|
||||
|
||||
@[grind =]
|
||||
theorem findFinIdx?_append {xs : Vector α n₁} {ys : Vector α n₂} {p : α → Bool} :
|
||||
(xs ++ ys).findFinIdx? p =
|
||||
((xs.findFinIdx? p).map (Fin.castLE (by simp))).or
|
||||
@@ -359,13 +317,13 @@ theorem findFinIdx?_append {xs : Vector α n₁} {ys : Vector α n₂} {p : α
|
||||
rcases ys with ⟨ys, rfl⟩
|
||||
simp [Array.findFinIdx?_append, Option.map_or, Function.comp_def]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isSome_findFinIdx? {xs : Vector α n} {p : α → Bool} :
|
||||
(xs.findFinIdx? p).isSome = xs.any p := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem isNone_findFinIdx? {xs : Vector α n} {p : α → Bool} :
|
||||
(xs.findFinIdx? p).isNone = xs.all (fun x => ¬ p x) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
|
||||
@@ -2538,23 +2538,23 @@ theorem foldr_hom (f : β₁ → β₂) {g₁ : α → β₁ → β₁} {g₂ :
|
||||
rw [Array.foldr_hom _ H]
|
||||
|
||||
/--
|
||||
We can prove that two folds over the same vector are related (by some arbitrary relation)
|
||||
if we know that the initial elements are related and the folding function, for each element of the
|
||||
vector, preserves the relation.
|
||||
We can prove that two folds over the same array are related (by some arbitrary relation)
|
||||
if we know that the initial elements are related and the folding function, for each element of the array,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldl_rel {xs : Vector α n} {f : β → α → β} {g : γ → α → γ} {a : β} {b : γ} {r : β → γ → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f c a) (g c' a)) :
|
||||
theorem foldl_rel {xs : Vector α n} {f g : β → α → β} {a b : β} {r : β → β → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
|
||||
r (xs.foldl (fun acc a => f acc a) a) (xs.foldl (fun acc a => g acc a) b) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simpa using Array.foldl_rel h (by simpa using h')
|
||||
|
||||
/--
|
||||
We can prove that two folds over the same vector are related (by some arbitrary relation)
|
||||
if we know that the initial elements are related and the folding function, for each element of the
|
||||
vector, preserves the relation.
|
||||
We can prove that two folds over the same array are related (by some arbitrary relation)
|
||||
if we know that the initial elements are related and the folding function, for each element of the array,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldr_rel {xs : Vector α n} {f : α → β → β} {g : α → γ → γ} {a : β} {b : γ} {r : β → γ → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c : β) (c' : γ), r c c' → r (f a c) (g a c')) :
|
||||
theorem foldr_rel {xs : Vector α n} {f g : α → β → β} {a b : β} {r : β → β → Prop}
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ xs → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
|
||||
r (xs.foldr (fun a acc => f a acc) a) (xs.foldr (fun a acc => g a acc) b) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simpa using Array.foldr_rel h (by simpa using h')
|
||||
@@ -3076,7 +3076,7 @@ theorem getElem_push_last {xs : Vector α n} {x : α} : (xs.push x)[n] = x := by
|
||||
|
||||
/-! ### zipWith -/
|
||||
|
||||
@[simp, grind =] theorem getElem_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} {i : Nat}
|
||||
@[simp] theorem getElem_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} {i : Nat}
|
||||
(hi : i < n) : (zipWith f as bs)[i] = f as[i] bs[i] := by
|
||||
cases as
|
||||
cases bs
|
||||
|
||||
@@ -19,13 +19,13 @@ namespace Vector
|
||||
|
||||
/-! ### mapFinIdx -/
|
||||
|
||||
@[simp, grind =] theorem getElem_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat}
|
||||
@[simp] theorem getElem_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat}
|
||||
(h : i < n) :
|
||||
(xs.mapFinIdx f)[i] = f i xs[i] h := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem getElem?_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat} :
|
||||
@[simp] theorem getElem?_mapFinIdx {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} {i : Nat} :
|
||||
(xs.mapFinIdx f)[i]? =
|
||||
xs[i]?.pbind fun b h => some <| f i b (getElem?_eq_some_iff.1 h).1 := by
|
||||
simp only [getElem?_def, getElem_mapFinIdx]
|
||||
@@ -33,12 +33,12 @@ namespace Vector
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
@[simp, grind =] theorem getElem_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} (h : i < n) :
|
||||
@[simp] theorem getElem_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} (h : i < n) :
|
||||
(xs.mapIdx f)[i] = f i (xs[i]'(by simp_all)) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} :
|
||||
@[simp] theorem getElem?_mapIdx {f : Nat → α → β} {xs : Vector α n} {i : Nat} :
|
||||
(xs.mapIdx f)[i]? = xs[i]?.map (f i) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
@@ -47,11 +47,11 @@ end Vector
|
||||
|
||||
namespace Array
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_toVector {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
@[simp] theorem mapFinIdx_toVector {xs : Array α} {f : (i : Nat) → α → (h : i < xs.size) → β} :
|
||||
xs.toVector.mapFinIdx f = (xs.mapFinIdx f).toVector.cast (by simp) := by
|
||||
ext <;> simp
|
||||
|
||||
@[simp, grind =] theorem mapIdx_toVector {f : Nat → α → β} {xs : Array α} :
|
||||
@[simp] theorem mapIdx_toVector {f : Nat → α → β} {xs : Array α} :
|
||||
xs.toVector.mapIdx f = (xs.mapIdx f).toVector.cast (by simp) := by
|
||||
ext <;> simp
|
||||
|
||||
@@ -61,12 +61,12 @@ namespace Vector
|
||||
|
||||
/-! ### zipIdx -/
|
||||
|
||||
@[simp, grind =] theorem toList_zipIdx {xs : Vector α n} (k : Nat := 0) :
|
||||
@[simp] theorem toList_zipIdx {xs : Vector α n} (k : Nat := 0) :
|
||||
(xs.zipIdx k).toList = xs.toList.zipIdx k := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem getElem_zipIdx {xs : Vector α n} {i : Nat} {h : i < n} :
|
||||
@[simp] theorem getElem_zipIdx {xs : Vector α n} {i : Nat} {h : i < n} :
|
||||
(xs.zipIdx k)[i] = (xs[i]'(by simp_all), k + i) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
@@ -116,7 +116,7 @@ abbrev mem_zipWithIndex_iff_getElem? := @mem_zipIdx_iff_getElem?
|
||||
subst w
|
||||
rfl
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapFinIdx_empty {f : (i : Nat) → α → (h : i < 0) → β} : mapFinIdx #v[] f = #v[] :=
|
||||
rfl
|
||||
|
||||
@@ -125,7 +125,6 @@ theorem mapFinIdx_eq_ofFn {as : Vector α n} {f : (i : Nat) → α → (h : i <
|
||||
rcases as with ⟨as, rfl⟩
|
||||
simp [Array.mapFinIdx_eq_ofFn]
|
||||
|
||||
@[grind =]
|
||||
theorem mapFinIdx_append {xs : Vector α n} {ys : Vector α m} {f : (i : Nat) → α → (h : i < n + m) → β} :
|
||||
(xs ++ ys).mapFinIdx f =
|
||||
xs.mapFinIdx (fun i a h => f i a (by omega)) ++
|
||||
@@ -134,7 +133,7 @@ theorem mapFinIdx_append {xs : Vector α n} {ys : Vector α m} {f : (i : Nat)
|
||||
rcases ys with ⟨ys, rfl⟩
|
||||
simp [Array.mapFinIdx_append]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapFinIdx_push {xs : Vector α n} {a : α} {f : (i : Nat) → α → (h : i < n + 1) → β} :
|
||||
mapFinIdx (xs.push a) f =
|
||||
(mapFinIdx xs (fun i a h => f i a (by omega))).push (f n a (by simp)) := by
|
||||
@@ -155,7 +154,7 @@ theorem exists_of_mem_mapFinIdx {b : β} {xs : Vector α n} {f : (i : Nat) →
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
exact List.exists_of_mem_mapFinIdx (by simpa using h)
|
||||
|
||||
@[simp, grind =] theorem mem_mapFinIdx {b : β} {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
|
||||
@[simp] theorem mem_mapFinIdx {b : β} {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
|
||||
b ∈ xs.mapFinIdx f ↔ ∃ (i : Nat) (h : i < n), f i xs[i] h = b := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
@@ -216,7 +215,7 @@ theorem mapFinIdx_eq_mapFinIdx_iff {xs : Vector α n} {f g : (i : Nat) → α
|
||||
rw [eq_comm, mapFinIdx_eq_iff]
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_mapFinIdx {xs : Vector α n}
|
||||
@[simp] theorem mapFinIdx_mapFinIdx {xs : Vector α n}
|
||||
{f : (i : Nat) → α → (h : i < n) → β}
|
||||
{g : (i : Nat) → β → (h : i < n) → γ} :
|
||||
(xs.mapFinIdx f).mapFinIdx g = xs.mapFinIdx (fun i a h => g i (f i a h) h) := by
|
||||
@@ -230,14 +229,14 @@ theorem mapFinIdx_eq_replicate_iff {xs : Vector α n} {f : (i : Nat) → α →
|
||||
@[deprecated mapFinIdx_eq_replicate_iff (since := "2025-03-18")]
|
||||
abbrev mapFinIdx_eq_mkVector_iff := @mapFinIdx_eq_replicate_iff
|
||||
|
||||
@[simp, grind =] theorem mapFinIdx_reverse {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
|
||||
@[simp] theorem mapFinIdx_reverse {xs : Vector α n} {f : (i : Nat) → α → (h : i < n) → β} :
|
||||
xs.reverse.mapFinIdx f = (xs.mapFinIdx (fun i a h => f (n - 1 - i) a (by omega))).reverse := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapIdx_empty {f : Nat → α → β} : mapIdx f #v[] = #v[] :=
|
||||
rfl
|
||||
|
||||
@@ -257,14 +256,13 @@ theorem mapIdx_eq_zipIdx_map {xs : Vector α n} {f : Nat → α → β} :
|
||||
@[deprecated mapIdx_eq_zipIdx_map (since := "2025-01-27")]
|
||||
abbrev mapIdx_eq_zipWithIndex_map := @mapIdx_eq_zipIdx_map
|
||||
|
||||
@[grind =]
|
||||
theorem mapIdx_append {xs : Vector α n} {ys : Vector α m} :
|
||||
(xs ++ ys).mapIdx f = xs.mapIdx f ++ ys.mapIdx fun i => f (i + n) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
rcases ys with ⟨ys, rfl⟩
|
||||
simp [Array.mapIdx_append]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem mapIdx_push {xs : Vector α n} {a : α} :
|
||||
mapIdx f (xs.push a) = (mapIdx f xs).push (f n a) := by
|
||||
simp [← append_singleton, mapIdx_append]
|
||||
@@ -277,7 +275,7 @@ theorem exists_of_mem_mapIdx {b : β} {xs : Vector α n}
|
||||
rw [mapIdx_eq_mapFinIdx] at h
|
||||
simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
|
||||
|
||||
@[simp, grind =] theorem mem_mapIdx {b : β} {xs : Vector α n} :
|
||||
@[simp] theorem mem_mapIdx {b : β} {xs : Vector α n} :
|
||||
b ∈ xs.mapIdx f ↔ ∃ (i : Nat) (h : i < n), f i xs[i] = b := by
|
||||
constructor
|
||||
· intro h
|
||||
@@ -334,7 +332,7 @@ theorem mapIdx_eq_mapIdx_iff {xs : Vector α n} :
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.mapIdx_eq_mapIdx_iff]
|
||||
|
||||
@[simp, grind =] theorem mapIdx_set {xs : Vector α n} {i : Nat} {h : i < n} {a : α} :
|
||||
@[simp] theorem mapIdx_set {xs : Vector α n} {i : Nat} {h : i < n} {a : α} :
|
||||
(xs.set i a).mapIdx f = (xs.mapIdx f).set i (f i a) (by simpa) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
@@ -344,17 +342,17 @@ theorem mapIdx_eq_mapIdx_iff {xs : Vector α n} :
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem back?_mapIdx {xs : Vector α n} {f : Nat → α → β} :
|
||||
@[simp] theorem back?_mapIdx {xs : Vector α n} {f : Nat → α → β} :
|
||||
(mapIdx f xs).back? = (xs.back?).map (f (n - 1)) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem back_mapIdx [NeZero n] {xs : Vector α n} {f : Nat → α → β} :
|
||||
@[simp] theorem back_mapIdx [NeZero n] {xs : Vector α n} {f : Nat → α → β} :
|
||||
(mapIdx f xs).back = f (n - 1) (xs.back) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp
|
||||
|
||||
@[simp, grind =] theorem mapIdx_mapIdx {xs : Vector α n} {f : Nat → α → β} {g : Nat → β → γ} :
|
||||
@[simp] theorem mapIdx_mapIdx {xs : Vector α n} {f : Nat → α → β} {g : Nat → β → γ} :
|
||||
(xs.mapIdx f).mapIdx g = xs.mapIdx (fun i => g i ∘ f i) := by
|
||||
simp [mapIdx_eq_iff]
|
||||
|
||||
@@ -366,7 +364,7 @@ theorem mapIdx_eq_replicate_iff {xs : Vector α n} {f : Nat → α → β} {b :
|
||||
@[deprecated mapIdx_eq_replicate_iff (since := "2025-03-18")]
|
||||
abbrev mapIdx_eq_mkVector_iff := @mapIdx_eq_replicate_iff
|
||||
|
||||
@[simp, grind =] theorem mapIdx_reverse {xs : Vector α n} {f : Nat → α → β} :
|
||||
@[simp] theorem mapIdx_reverse {xs : Vector α n} {f : Nat → α → β} :
|
||||
xs.reverse.mapIdx f = (mapIdx (fun i => f (n - 1 - i)) xs).reverse := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
simp [Array.mapIdx_reverse]
|
||||
|
||||
@@ -20,15 +20,15 @@ set_option linter.indexVariables true -- Enforce naming conventions for index va
|
||||
|
||||
namespace Vector
|
||||
|
||||
@[simp, grind =] theorem getElem_ofFn {α n} {f : Fin n → α} (h : i < n) :
|
||||
@[simp] theorem getElem_ofFn {α n} {f : Fin n → α} (h : i < n) :
|
||||
(Vector.ofFn f)[i] = f ⟨i, by simpa using h⟩ := by
|
||||
simp [ofFn]
|
||||
|
||||
@[simp, grind =] theorem getElem?_ofFn {α n} {f : Fin n → α} :
|
||||
theorem getElem?_ofFn {α n} {f : Fin n → α} :
|
||||
(ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none := by
|
||||
simp [getElem?_def]
|
||||
|
||||
@[simp 500, grind =]
|
||||
@[simp 500]
|
||||
theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
|
||||
constructor
|
||||
· intro w
|
||||
@@ -37,7 +37,7 @@ theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i =
|
||||
· rintro ⟨i, rfl⟩
|
||||
apply mem_of_getElem (i := i) <;> simp
|
||||
|
||||
@[grind =] theorem back_ofFn {n} [NeZero n] {f : Fin n → α} :
|
||||
theorem back_ofFn {n} [NeZero n] {f : Fin n → α} :
|
||||
(ofFn f).back = f ⟨n - 1, by have := NeZero.ne n; omega⟩ := by
|
||||
simp [back]
|
||||
|
||||
@@ -71,7 +71,7 @@ def ofFnM {n} [Monad m] (f : Fin n → m α) : m (Vector α n) :=
|
||||
else
|
||||
pure ⟨acc, by omega⟩
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem ofFnM_zero [Monad m] {f : Fin 0 → m α} : Vector.ofFnM f = pure #v[] := by
|
||||
simp [ofFnM, ofFnM.go]
|
||||
|
||||
@@ -114,13 +114,13 @@ theorem ofFnM_add {n m} [Monad m] [LawfulMonad m] {f : Fin (n + k) → m α} :
|
||||
| zero => simp
|
||||
| succ k ih => simp [ofFnM_succ, ih, ← push_append]
|
||||
|
||||
@[simp, grind =] theorem toArray_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
|
||||
@[simp, grind] theorem toArray_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
|
||||
toArray <$> ofFnM f = Array.ofFnM f := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
| succ n ih => simp [ofFnM_succ, Array.ofFnM_succ, ← ih]
|
||||
|
||||
@[simp, grind =] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
|
||||
@[simp, grind] theorem toList_ofFnM [Monad m] [LawfulMonad m] {f : Fin n → m α} :
|
||||
toList <$> Vector.ofFnM f = List.ofFnM f := by
|
||||
unfold toList
|
||||
suffices Array.toList <$> (toArray <$> ofFnM f) = List.ofFnM f by
|
||||
|
||||
@@ -120,17 +120,6 @@ theorem range'_eq_append_iff : range' s (n + m) = xs ++ ys ↔ xs = range' s n
|
||||
(range' s n).find? p = none ↔ ∀ i, s ≤ i → i < s + n → !p i := by
|
||||
simp [range'_eq_mk_range']
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range' {a s n step} (h : 0 < step := by simp) :
|
||||
count a (range' s n step) = if ∃ i, i < n ∧ a = s + step * i then 1 else 0 := by
|
||||
rw [range'_eq_mk_range', count_mk, ← Array.count_range' h]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range_1' {a s n} :
|
||||
count a (range' s n) = if s ≤ a ∧ a < s + n then 1 else 0 := by
|
||||
rw [range'_eq_mk_range', count_mk, ← Array.count_range_1']
|
||||
|
||||
|
||||
/-! ### range -/
|
||||
|
||||
@[simp, grind =] theorem getElem_range {i : Nat} (hi : i < n) : (Vector.range n)[i] = i := by
|
||||
@@ -182,12 +171,6 @@ theorem self_mem_range_succ {n : Nat} : n ∈ range (n + 1) := by simp
|
||||
(range n).find? p = none ↔ ∀ i, i < n → !p i := by
|
||||
simp [range_eq_range']
|
||||
|
||||
@[simp, grind =]
|
||||
theorem count_range {a n} :
|
||||
count a (range n) = if a < n then 1 else 0 := by
|
||||
rw [range_eq_range', count_range_1']
|
||||
simp
|
||||
|
||||
/-! ### zipIdx -/
|
||||
|
||||
@[simp]
|
||||
|
||||
@@ -46,7 +46,6 @@ theorem zipWith_self {f : α → α → δ} {xs : Vector α n} : zipWith f xs xs
|
||||
See also `getElem?_zipWith'` for a variant
|
||||
using `Option.map` and `Option.bind` rather than a `match`.
|
||||
-/
|
||||
@[grind =]
|
||||
theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
|
||||
(zipWith f as bs)[i]? = match as[i]?, bs[i]? with
|
||||
| some a, some b => some (f a b) | _, _ => none := by
|
||||
@@ -75,61 +74,53 @@ theorem getElem?_zip_eq_some {as : Vector α n} {bs : Vector β n} {z : α × β
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.getElem?_zip_eq_some]
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem zipWith_map {μ} {f : γ → δ → μ} {g : α → γ} {h : β → δ} {as : Vector α n} {bs : Vector β n} :
|
||||
zipWith f (as.map g) (bs.map h) = zipWith (fun a b => f (g a) (h b)) as bs := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.zipWith_map]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_map_left {as : Vector α n} {bs : Vector β n} {f : α → α'} {g : α' → β → γ} :
|
||||
zipWith g (as.map f) bs = zipWith (fun a b => g (f a) b) as bs := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.zipWith_map_left]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_map_right {as : Vector α n} {bs : Vector β n} {f : β → β'} {g : α → β' → γ} :
|
||||
zipWith g as (bs.map f) = zipWith (fun a b => g a (f b)) as bs := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.zipWith_map_right]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} {i : δ} :
|
||||
(zipWith f as bs).foldr g i = (zip as bs).foldr (fun p r => g (f p.1 p.2) r) i := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simpa using Array.zipWith_foldr_eq_zip_foldr
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} {i : δ} :
|
||||
(zipWith f as bs).foldl g i = (zip as bs).foldl (fun r p => g r (f p.1 p.2)) i := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simpa using Array.zipWith_foldl_eq_zip_foldl
|
||||
|
||||
@[grind =]
|
||||
theorem map_zipWith {δ : Type _} {f : α → β} {g : γ → δ → α} {as : Vector γ n} {bs : Vector δ n} :
|
||||
map f (zipWith g as bs) = zipWith (fun x y => f (g x y)) as bs := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.map_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem take_zipWith : (zipWith f as bs).take i = zipWith f (as.take i) (bs.take i) := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.take_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem extract_zipWith : (zipWith f as bs).extract i j = zipWith f (as.extract i j) (bs.extract i j) := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.extract_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem zipWith_append {f : α → β → γ}
|
||||
{as : Vector α n} {as' : Vector α m} {bs : Vector β n} {bs' : Vector β m} :
|
||||
zipWith f (as ++ as') (bs ++ bs') = zipWith f as bs ++ zipWith f as' bs' := by
|
||||
@@ -156,8 +147,7 @@ theorem zipWith_eq_append_iff {f : α → β → γ} {as : Vector α (n + m)} {b
|
||||
simp only at w₁ w₂
|
||||
exact ⟨as₁, as₂, bs₁, bs₂, by simpa [hw, hy] using ⟨w₁, w₂⟩⟩
|
||||
|
||||
@[simp, grind =]
|
||||
theorem zipWith_replicate {a : α} {b : β} {n : Nat} :
|
||||
@[simp] theorem zipWith_replicate {a : α} {b : β} {n : Nat} :
|
||||
zipWith f (replicate n a) (replicate n b) = replicate n (f a b) := by
|
||||
ext
|
||||
simp
|
||||
@@ -177,7 +167,6 @@ theorem map_zip_eq_zipWith {f : α × β → γ} {as : Vector α n} {bs : Vector
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.map_zip_eq_zipWith]
|
||||
|
||||
@[grind =]
|
||||
theorem reverse_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector β n} :
|
||||
(zipWith f as bs).reverse = zipWith f as.reverse bs.reverse := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
@@ -186,7 +175,7 @@ theorem reverse_zipWith {f : α → β → γ} {as : Vector α n} {bs : Vector
|
||||
|
||||
/-! ### zip -/
|
||||
|
||||
@[simp, grind =]
|
||||
@[simp]
|
||||
theorem getElem_zip {as : Vector α n} {bs : Vector β n} {i : Nat} {h : i < n} :
|
||||
(zip as bs)[i] = (as[i], bs[i]) :=
|
||||
getElem_zipWith ..
|
||||
@@ -196,22 +185,18 @@ theorem zip_eq_zipWith {as : Vector α n} {bs : Vector β n} : zip as bs = zipWi
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.zip_eq_zipWith, h]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map {f : α → γ} {g : β → δ} {as : Vector α n} {bs : Vector β n} :
|
||||
zip (as.map f) (bs.map g) = (zip as bs).map (Prod.map f g) := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
rcases bs with ⟨bs, h⟩
|
||||
simp [Array.zip_map, h]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map_left {f : α → γ} {as : Vector α n} {bs : Vector β n} :
|
||||
zip (as.map f) bs = (zip as bs).map (Prod.map f id) := by rw [← zip_map, map_id]
|
||||
|
||||
@[grind _=_]
|
||||
theorem zip_map_right {f : β → γ} {as : Vector α n} {bs : Vector β n} :
|
||||
zip as (bs.map f) = (zip as bs).map (Prod.map id f) := by rw [← zip_map, map_id]
|
||||
|
||||
@[grind =]
|
||||
theorem zip_append {as : Vector α n} {bs : Vector β n} {as' : Vector α m} {bs' : Vector β m} :
|
||||
zip (as ++ as') (bs ++ bs') = zip as bs ++ zip as' bs' := by
|
||||
rcases as with ⟨as, rfl⟩
|
||||
@@ -220,7 +205,6 @@ theorem zip_append {as : Vector α n} {bs : Vector β n} {as' : Vector α m} {bs
|
||||
rcases bs' with ⟨bs', h'⟩
|
||||
simp [Array.zip_append, h, h']
|
||||
|
||||
@[grind =]
|
||||
theorem zip_map' {f : α → β} {g : α → γ} {xs : Vector α n} :
|
||||
zip (xs.map f) (xs.map g) = xs.map fun a => (f a, g a) := by
|
||||
rcases xs with ⟨xs, rfl⟩
|
||||
@@ -264,8 +248,7 @@ theorem zip_eq_append_iff {as : Vector α (n + m)} {bs : Vector β (n + m)} {xs
|
||||
∃ as₁ as₂ bs₁ bs₂, as = as₁ ++ as₂ ∧ bs = bs₁ ++ bs₂ ∧ xs = zip as₁ bs₁ ∧ ys = zip as₂ bs₂ := by
|
||||
simp [zip_eq_zipWith, zipWith_eq_append_iff]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem zip_replicate {a : α} {b : β} {n : Nat} :
|
||||
@[simp] theorem zip_replicate {a : α} {b : β} {n : Nat} :
|
||||
zip (replicate n a) (replicate n b) = replicate n (a, b) := by
|
||||
ext <;> simp
|
||||
|
||||
@@ -274,17 +257,14 @@ abbrev zip_mkVector := @zip_replicate
|
||||
|
||||
/-! ### unzip -/
|
||||
|
||||
@[simp, grind =]
|
||||
theorem unzip_fst : (unzip xs).fst = xs.map Prod.fst := by
|
||||
@[simp] theorem unzip_fst : (unzip xs).fst = xs.map Prod.fst := by
|
||||
cases xs
|
||||
simp_all
|
||||
|
||||
@[simp, grind =]
|
||||
theorem unzip_snd : (unzip xs).snd = xs.map Prod.snd := by
|
||||
@[simp] theorem unzip_snd : (unzip xs).snd = xs.map Prod.snd := by
|
||||
cases xs
|
||||
simp_all
|
||||
|
||||
@[grind =]
|
||||
theorem unzip_eq_map {xs : Vector (α × β) n} : unzip xs = (xs.map Prod.fst, xs.map Prod.snd) := by
|
||||
cases xs
|
||||
simp [List.unzip_eq_map]
|
||||
@@ -316,8 +296,7 @@ theorem zip_of_prod {as : Vector α n} {bs : Vector β n} {xs : Vector (α × β
|
||||
(hr : xs.map Prod.snd = bs) : xs = as.zip bs := by
|
||||
rw [← hl, ← hr, ← zip_unzip xs, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]
|
||||
|
||||
@[simp, grind =]
|
||||
theorem unzip_replicate {a : α} {b : β} {n : Nat} :
|
||||
@[simp] theorem unzip_replicate {a : α} {b : β} {n : Nat} :
|
||||
unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
|
||||
ext1 <;> simp
|
||||
|
||||
|
||||
@@ -47,7 +47,7 @@ proof in the context using `have`, because `get_elem_tactic` tries
|
||||
|
||||
The proof side-condition `valid xs i` is automatically dispatched by the
|
||||
`get_elem_tactic` tactic; this tactic can be extended by adding more clauses to
|
||||
`get_elem_tactic_extensible` using `macro_rules`.
|
||||
`get_elem_tactic_trivial` using `macro_rules`.
|
||||
|
||||
`xs[i]?` and `xs[i]!` do not impose a proof obligation; the former returns
|
||||
an `Option elem`, with `none` signalling that the value isn't present, and
|
||||
@@ -281,7 +281,7 @@ instance [GetElem? cont Nat elem dom] [h : LawfulGetElem cont Nat elem dom] :
|
||||
@[simp, grind =] theorem getElem!_fin [GetElem? Cont Nat Elem Dom] (a : Cont) (i : Fin n) [Inhabited Elem] : a[i]! = a[i.1]! := rfl
|
||||
|
||||
macro_rules
|
||||
| `(tactic| get_elem_tactic_extensible) => `(tactic| (with_reducible apply Fin.val_lt_of_le); get_elem_tactic_extensible; done)
|
||||
| `(tactic| get_elem_tactic_trivial) => `(tactic| (with_reducible apply Fin.val_lt_of_le); get_elem_tactic_trivial; done)
|
||||
|
||||
end Fin
|
||||
|
||||
|
||||
@@ -120,6 +120,7 @@ theorem pow_add (a : α) (k₁ k₂ : Nat) : a ^ (k₁ + k₂) = a^k₁ * a^k₂
|
||||
instance : NatModule α where
|
||||
hMul a x := a * x
|
||||
add_zero := by simp [add_zero]
|
||||
zero_add := by simp [zero_add]
|
||||
add_assoc := by simp [add_assoc]
|
||||
add_comm := by simp [add_comm]
|
||||
zero_hmul := by simp [natCast_zero, zero_mul]
|
||||
@@ -272,6 +273,7 @@ theorem intCast_pow (x : Int) (k : Nat) : ((x ^ k : Int) : α) = (x : α) ^ k :=
|
||||
instance : IntModule α where
|
||||
hMul a x := a * x
|
||||
add_zero := by simp [add_zero]
|
||||
zero_add := by simp [zero_add]
|
||||
add_assoc := by simp [add_assoc]
|
||||
add_comm := by simp [add_comm]
|
||||
zero_hmul := by simp [intCast_zero, zero_mul]
|
||||
|
||||
@@ -34,9 +34,4 @@ instance : CommRing (BitVec w) where
|
||||
instance : IsCharP (BitVec w) (2 ^ w) where
|
||||
ofNat_eq_zero_iff {x} := by simp [BitVec.ofInt, BitVec.toNat_eq]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add (BitVec w) (some 0) (some (2^w)) := inferInstance
|
||||
example : ToInt.Neg (BitVec w) (some 0) (some (2^w)) := inferInstance
|
||||
example : ToInt.Sub (BitVec w) (some 0) (some (2^w)) := inferInstance
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -100,9 +100,6 @@ instance (n : Nat) [NeZero n] : IsCharP (Fin n) n where
|
||||
simp only [Nat.zero_mod]
|
||||
simp only [Fin.mk.injEq]
|
||||
|
||||
example [NeZero n] : ToInt.Neg (Fin n) (some 0) (some n) := inferInstance
|
||||
example [NeZero n] : ToInt.Sub (Fin n) (some 0) (some n) := inferInstance
|
||||
|
||||
end Fin
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -11,13 +11,13 @@ import Init.Data.Hashable
|
||||
import all Init.Data.Ord
|
||||
import Init.Data.RArray
|
||||
import Init.Grind.CommRing.Basic
|
||||
import Init.Grind.CommRing.Field
|
||||
import Init.Grind.Ordered.Ring
|
||||
|
||||
namespace Lean.Grind
|
||||
namespace CommRing
|
||||
|
||||
-- These are no longer global instances, so we need to turn them on here.
|
||||
attribute [local instance] Semiring.natCast Ring.intCast
|
||||
namespace CommRing
|
||||
|
||||
abbrev Var := Nat
|
||||
|
||||
inductive Expr where
|
||||
@@ -35,13 +35,13 @@ abbrev Context (α : Type u) := RArray α
|
||||
def Var.denote {α} (ctx : Context α) (v : Var) : α :=
|
||||
ctx.get v
|
||||
|
||||
def denoteInt {α} [Ring α] (k : Int) : α :=
|
||||
def denoteInt {α} [CommRing α] (k : Int) : α :=
|
||||
bif k < 0 then
|
||||
- OfNat.ofNat (α := α) k.natAbs
|
||||
else
|
||||
OfNat.ofNat (α := α) k.natAbs
|
||||
|
||||
def Expr.denote {α} [Ring α] (ctx : Context α) : Expr → α
|
||||
def Expr.denote {α} [CommRing α] (ctx : Context α) : Expr → α
|
||||
| .add a b => denote ctx a + denote ctx b
|
||||
| .sub a b => denote ctx a - denote ctx b
|
||||
| .mul a b => denote ctx a * denote ctx b
|
||||
@@ -205,7 +205,7 @@ instance : LawfulBEq Poly where
|
||||
intro a
|
||||
induction a <;> simp! [BEq.beq]
|
||||
next k m p ih =>
|
||||
change m == m ∧ p == p
|
||||
show m == m ∧ p == p
|
||||
simp [ih]
|
||||
|
||||
def Poly.denote [CommRing α] (ctx : Context α) (p : Poly) : α :=
|
||||
@@ -532,7 +532,7 @@ theorem Mon.denote_concat {α} [CommRing α] (ctx : Context α) (m₁ m₂ : Mon
|
||||
next p₁ m₁ ih => rw [mul_assoc]
|
||||
|
||||
private theorem le_of_blt_false {a b : Nat} : a.blt b = false → b ≤ a := by
|
||||
intro h; apply Nat.le_of_not_gt; change ¬a < b
|
||||
intro h; apply Nat.le_of_not_gt; show ¬a < b
|
||||
rw [← Nat.blt_eq, h]; simp
|
||||
|
||||
private theorem eq_of_blt_false {a b : Nat} : a.blt b = false → b.blt a = false → a = b := by
|
||||
@@ -1139,136 +1139,5 @@ theorem imp_keqC {α c} [CommRing α] [IsCharP α c] (ctx : Context α) [NoNatZe
|
||||
|
||||
end Stepwise
|
||||
|
||||
/-! IntModule interface -/
|
||||
|
||||
def Mon.denoteAsIntModule [CommRing α] (ctx : Context α) (m : Mon) : α :=
|
||||
match m with
|
||||
| .unit => One.one
|
||||
| .mult pw m => go m (pw.denote ctx)
|
||||
where
|
||||
go (m : Mon) (acc : α) : α :=
|
||||
match m with
|
||||
| .unit => acc
|
||||
| .mult pw m => go m (acc * pw.denote ctx)
|
||||
|
||||
def Poly.denoteAsIntModule [CommRing α] (ctx : Context α) (p : Poly) : α :=
|
||||
match p with
|
||||
| .num k => Int.cast k * One.one
|
||||
| .add k m p => Int.cast k * m.denoteAsIntModule ctx + denoteAsIntModule ctx p
|
||||
|
||||
theorem Mon.denoteAsIntModule_go_eq_denote {α} [CommRing α] (ctx : Context α) (m : Mon) (acc : α)
|
||||
: denoteAsIntModule.go ctx m acc = acc * m.denote ctx := by
|
||||
induction m generalizing acc <;> simp [*, denoteAsIntModule.go, denote, mul_one, One.one, *, mul_assoc]
|
||||
|
||||
theorem Mon.denoteAsIntModule_eq_denote {α} [CommRing α] (ctx : Context α) (m : Mon)
|
||||
: m.denoteAsIntModule ctx = m.denote ctx := by
|
||||
cases m <;> simp [denoteAsIntModule, denote, denoteAsIntModule_go_eq_denote]; rfl
|
||||
|
||||
theorem Poly.denoteAsIntModule_eq_denote {α} [CommRing α] (ctx : Context α) (p : Poly) : p.denoteAsIntModule ctx = p.denote ctx := by
|
||||
induction p <;> simp [*, denoteAsIntModule, denote, mul_one, One.one, Mon.denoteAsIntModule_eq_denote]
|
||||
|
||||
open Stepwise
|
||||
|
||||
theorem eq_norm {α} [CommRing α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert lhs rhs p → lhs.denote ctx = rhs.denote ctx → p.denoteAsIntModule ctx = 0 := by
|
||||
rw [Poly.denoteAsIntModule_eq_denote]; apply core
|
||||
|
||||
theorem diseq_norm {α} [CommRing α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert lhs rhs p → lhs.denote ctx ≠ rhs.denote ctx → p.denoteAsIntModule ctx ≠ 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h; subst p; simp [Expr.denote_toPoly, Expr.denote]
|
||||
intro h; rw [sub_eq_zero_iff] at h; contradiction
|
||||
|
||||
open IntModule.IsOrdered
|
||||
|
||||
theorem le_norm {α} [CommRing α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert lhs rhs p → lhs.denote ctx ≤ rhs.denote ctx → p.denoteAsIntModule ctx ≤ 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h; subst p; simp [Expr.denote_toPoly, Expr.denote]
|
||||
replace h := add_le_left h ((-1) * rhs.denote ctx)
|
||||
rw [neg_mul, ← sub_eq_add_neg, one_mul, ← sub_eq_add_neg, sub_self] at h
|
||||
assumption
|
||||
|
||||
theorem lt_norm {α} [CommRing α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert lhs rhs p → lhs.denote ctx < rhs.denote ctx → p.denoteAsIntModule ctx < 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h; subst p; simp [Expr.denote_toPoly, Expr.denote]
|
||||
replace h := add_lt_left h ((-1) * rhs.denote ctx)
|
||||
rw [neg_mul, ← sub_eq_add_neg, one_mul, ← sub_eq_add_neg, sub_self] at h
|
||||
assumption
|
||||
|
||||
theorem not_le_norm {α} [CommRing α] [LinearOrder α] [Ring.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert rhs lhs p → ¬ lhs.denote ctx ≤ rhs.denote ctx → p.denoteAsIntModule ctx < 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h₁; subst p; simp [Expr.denote_toPoly, Expr.denote]
|
||||
replace h₁ := LinearOrder.lt_of_not_le h₁
|
||||
replace h₁ := add_lt_left h₁ (-lhs.denote ctx)
|
||||
simp [← sub_eq_add_neg, sub_self] at h₁
|
||||
assumption
|
||||
|
||||
theorem not_lt_norm {α} [CommRing α] [LinearOrder α] [Ring.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert rhs lhs p → ¬ lhs.denote ctx < rhs.denote ctx → p.denoteAsIntModule ctx ≤ 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h₁; subst p; simp [Expr.denote_toPoly, Expr.denote]
|
||||
replace h₁ := LinearOrder.le_of_not_lt h₁
|
||||
replace h₁ := add_le_left h₁ (-lhs.denote ctx)
|
||||
simp [← sub_eq_add_neg, sub_self] at h₁
|
||||
assumption
|
||||
|
||||
theorem not_le_norm' {α} [CommRing α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert lhs rhs p → ¬ lhs.denote ctx ≤ rhs.denote ctx → ¬ p.denoteAsIntModule ctx ≤ 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h₁; subst p; simp [Expr.denote_toPoly, Expr.denote]; intro h
|
||||
replace h := add_le_right (rhs.denote ctx) h
|
||||
rw [sub_eq_add_neg, add_left_comm, ← sub_eq_add_neg, sub_self] at h; simp [add_zero] at h
|
||||
contradiction
|
||||
|
||||
theorem not_lt_norm' {α} [CommRing α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: core_cert lhs rhs p → ¬ lhs.denote ctx < rhs.denote ctx → ¬ p.denoteAsIntModule ctx < 0 := by
|
||||
simp [core_cert, Poly.denoteAsIntModule_eq_denote]; intro _ h₁; subst p; simp [Expr.denote_toPoly, Expr.denote]; intro h
|
||||
replace h := add_lt_right (rhs.denote ctx) h
|
||||
rw [sub_eq_add_neg, add_left_comm, ← sub_eq_add_neg, sub_self] at h; simp [add_zero] at h
|
||||
contradiction
|
||||
|
||||
theorem inv_int_eq [Field α] [IsCharP α 0] (b : Int) : b != 0 → (denoteInt b : α) * (denoteInt b)⁻¹ = 1 := by
|
||||
simp; intro h
|
||||
have : (denoteInt b : α) ≠ 0 := by
|
||||
simp [denoteInt_eq]; intro h
|
||||
have := IsCharP.intCast_eq_zero_iff (α := α) 0 b; simp [*] at this
|
||||
rw [Field.mul_inv_cancel this]
|
||||
|
||||
theorem inv_int_eqC {α c} [Field α] [IsCharP α c] (b : Int) : b % c != 0 → (denoteInt b : α) * (denoteInt b)⁻¹ = 1 := by
|
||||
simp; intro h
|
||||
have : (denoteInt b : α) ≠ 0 := by
|
||||
simp [denoteInt_eq]; intro h
|
||||
have := IsCharP.intCast_eq_zero_iff (α := α) c b; simp [*] at this
|
||||
rw [Field.mul_inv_cancel this]
|
||||
|
||||
theorem inv_zero_eqC {α c} [Field α] [IsCharP α c] (b : Int) : b % c == 0 → (denoteInt b : α)⁻¹ = 0 := by
|
||||
simp [denoteInt_eq]; intro h
|
||||
have : (b : α) = 0 := by
|
||||
have := IsCharP.intCast_eq_zero_iff (α := α) c b
|
||||
simp [*]
|
||||
simp [this, Field.inv_zero]
|
||||
|
||||
open Classical in
|
||||
theorem inv_split {α} [Field α] (a : α) : if a = 0 then a⁻¹ = 0 else a * a⁻¹ = 1 := by
|
||||
split
|
||||
next h => simp [h, Field.inv_zero]
|
||||
next h => rw [Field.mul_inv_cancel h]
|
||||
|
||||
def one_eq_zero_unsat_cert (p : Poly) :=
|
||||
p == .num 1 || p == .num (-1)
|
||||
|
||||
theorem one_eq_zero_unsat {α} [Field α] (ctx : Context α) (p : Poly) : one_eq_zero_unsat_cert p → p.denote ctx = 0 → False := by
|
||||
simp [one_eq_zero_unsat_cert]; intro h; cases h <;> simp [*, Poly.denote, intCast_one, intCast_neg]
|
||||
next => rw [Eq.comm]; apply Field.zero_ne_one
|
||||
next => rw [← neg_eq_zero, neg_neg, Eq.comm]; apply Field.zero_ne_one
|
||||
|
||||
theorem diseq_to_eq {α} [Field α] (a b : α) : a ≠ b → (a - b)*(a - b)⁻¹ = 1 := by
|
||||
intro h
|
||||
have : a - b ≠ 0 := by
|
||||
intro h'; rw [Ring.sub_eq_zero_iff.mp h'] at h
|
||||
contradiction
|
||||
exact Field.mul_inv_cancel this
|
||||
|
||||
theorem diseq0_to_eq {α} [Field α] (a : α) : a ≠ 0 → a*a⁻¹ = 1 := by
|
||||
exact Field.mul_inv_cancel
|
||||
|
||||
end CommRing
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -45,11 +45,6 @@ instance : IsCharP Int8 (2 ^ 8) where
|
||||
simp [Int8.ofInt_eq_iff_bmod_eq_toInt,
|
||||
← Int.dvd_iff_bmod_eq_zero, ← Nat.dvd_iff_mod_eq_zero, Int.ofNat_dvd_right]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add Int8 (some (-(2^7))) (some (2^7)) := inferInstance
|
||||
example : ToInt.Neg Int8 (some (-(2^7))) (some (2^7)) := inferInstance
|
||||
example : ToInt.Sub Int8 (some (-(2^7))) (some (2^7)) := inferInstance
|
||||
|
||||
instance : NatCast Int16 where
|
||||
natCast x := Int16.ofNat x
|
||||
|
||||
@@ -81,11 +76,6 @@ instance : IsCharP Int16 (2 ^ 16) where
|
||||
simp [Int16.ofInt_eq_iff_bmod_eq_toInt,
|
||||
← Int.dvd_iff_bmod_eq_zero, ← Nat.dvd_iff_mod_eq_zero, Int.ofNat_dvd_right]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add Int16 (some (-(2^15))) (some (2^15)) := inferInstance
|
||||
example : ToInt.Neg Int16 (some (-(2^15))) (some (2^15)) := inferInstance
|
||||
example : ToInt.Sub Int16 (some (-(2^15))) (some (2^15)) := inferInstance
|
||||
|
||||
instance : NatCast Int32 where
|
||||
natCast x := Int32.ofNat x
|
||||
|
||||
@@ -117,11 +107,6 @@ instance : IsCharP Int32 (2 ^ 32) where
|
||||
simp [Int32.ofInt_eq_iff_bmod_eq_toInt,
|
||||
← Int.dvd_iff_bmod_eq_zero, ← Nat.dvd_iff_mod_eq_zero, Int.ofNat_dvd_right]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add Int32 (some (-(2^31))) (some (2^31)) := inferInstance
|
||||
example : ToInt.Neg Int32 (some (-(2^31))) (some (2^31)) := inferInstance
|
||||
example : ToInt.Sub Int32 (some (-(2^31))) (some (2^31)) := inferInstance
|
||||
|
||||
instance : NatCast Int64 where
|
||||
natCast x := Int64.ofNat x
|
||||
|
||||
@@ -153,11 +138,6 @@ instance : IsCharP Int64 (2 ^ 64) where
|
||||
simp [Int64.ofInt_eq_iff_bmod_eq_toInt,
|
||||
← Int.dvd_iff_bmod_eq_zero, ← Nat.dvd_iff_mod_eq_zero, Int.ofNat_dvd_right]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add Int64 (some (-(2^63))) (some (2^63)) := inferInstance
|
||||
example : ToInt.Neg Int64 (some (-(2^63))) (some (2^63)) := inferInstance
|
||||
example : ToInt.Sub Int64 (some (-(2^63))) (some (2^63)) := inferInstance
|
||||
|
||||
instance : NatCast ISize where
|
||||
natCast x := ISize.ofNat x
|
||||
|
||||
@@ -191,9 +171,4 @@ instance : IsCharP ISize (2 ^ numBits) where
|
||||
simp [ISize.ofInt_eq_iff_bmod_eq_toInt,
|
||||
← Int.dvd_iff_bmod_eq_zero, ← Nat.dvd_iff_mod_eq_zero, Int.ofNat_dvd_right]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add ISize (some (-(2^(numBits-1)))) (some (2^(numBits-1))) := inferInstance
|
||||
example : ToInt.Neg ISize (some (-(2^(numBits-1)))) (some (2^(numBits-1))) := inferInstance
|
||||
example : ToInt.Sub ISize (some (-(2^(numBits-1)))) (some (2^(numBits-1))) := inferInstance
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -149,11 +149,6 @@ instance : IsCharP UInt8 256 where
|
||||
have : OfNat.ofNat x = UInt8.ofNat x := rfl
|
||||
simp [this, UInt8.ofNat_eq_iff_mod_eq_toNat]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add UInt8 (some 0) (some (2^8)) := inferInstance
|
||||
example : ToInt.Neg UInt8 (some 0) (some (2^8)) := inferInstance
|
||||
example : ToInt.Sub UInt8 (some 0) (some (2^8)) := inferInstance
|
||||
|
||||
instance : CommRing UInt16 where
|
||||
add_assoc := UInt16.add_assoc
|
||||
add_comm := UInt16.add_comm
|
||||
@@ -179,11 +174,6 @@ instance : IsCharP UInt16 65536 where
|
||||
have : OfNat.ofNat x = UInt16.ofNat x := rfl
|
||||
simp [this, UInt16.ofNat_eq_iff_mod_eq_toNat]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add UInt16 (some 0) (some (2^16)) := inferInstance
|
||||
example : ToInt.Neg UInt16 (some 0) (some (2^16)) := inferInstance
|
||||
example : ToInt.Sub UInt16 (some 0) (some (2^16)) := inferInstance
|
||||
|
||||
instance : CommRing UInt32 where
|
||||
add_assoc := UInt32.add_assoc
|
||||
add_comm := UInt32.add_comm
|
||||
@@ -209,11 +199,6 @@ instance : IsCharP UInt32 4294967296 where
|
||||
have : OfNat.ofNat x = UInt32.ofNat x := rfl
|
||||
simp [this, UInt32.ofNat_eq_iff_mod_eq_toNat]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add UInt32 (some 0) (some (2^32)) := inferInstance
|
||||
example : ToInt.Neg UInt32 (some 0) (some (2^32)) := inferInstance
|
||||
example : ToInt.Sub UInt32 (some 0) (some (2^32)) := inferInstance
|
||||
|
||||
instance : CommRing UInt64 where
|
||||
add_assoc := UInt64.add_assoc
|
||||
add_comm := UInt64.add_comm
|
||||
@@ -239,11 +224,6 @@ instance : IsCharP UInt64 18446744073709551616 where
|
||||
have : OfNat.ofNat x = UInt64.ofNat x := rfl
|
||||
simp [this, UInt64.ofNat_eq_iff_mod_eq_toNat]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add UInt64 (some 0) (some (2^64)) := inferInstance
|
||||
example : ToInt.Neg UInt64 (some 0) (some (2^64)) := inferInstance
|
||||
example : ToInt.Sub UInt64 (some 0) (some (2^64)) := inferInstance
|
||||
|
||||
instance : CommRing USize where
|
||||
add_assoc := USize.add_assoc
|
||||
add_comm := USize.add_comm
|
||||
@@ -271,9 +251,4 @@ instance : IsCharP USize (2 ^ numBits) where
|
||||
have : OfNat.ofNat x = USize.ofNat x := rfl
|
||||
simp [this, USize.ofNat_eq_iff_mod_eq_toNat]
|
||||
|
||||
-- Verify we can derive the instances showing how `toInt` interacts with operations:
|
||||
example : ToInt.Add USize (some 0) (some (2^numBits)) := inferInstance
|
||||
example : ToInt.Neg USize (some 0) (some (2^numBits)) := inferInstance
|
||||
example : ToInt.Sub USize (some 0) (some (2^numBits)) := inferInstance
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -7,12 +7,12 @@ module
|
||||
|
||||
prelude
|
||||
import Init.Data.Int.Order
|
||||
import Init.Grind.ToInt
|
||||
|
||||
namespace Lean.Grind
|
||||
|
||||
class NatModule (M : Type u) extends Zero M, Add M, HMul Nat M M where
|
||||
add_zero : ∀ a : M, a + 0 = a
|
||||
zero_add : ∀ a : M, 0 + a = a
|
||||
add_comm : ∀ a b : M, a + b = b + a
|
||||
add_assoc : ∀ a b c : M, a + b + c = a + (b + c)
|
||||
zero_hmul : ∀ a : M, 0 * a = 0
|
||||
@@ -26,6 +26,7 @@ attribute [instance 100] NatModule.toZero NatModule.toAdd NatModule.toHMul
|
||||
|
||||
class IntModule (M : Type u) extends Zero M, Add M, Neg M, Sub M, HMul Int M M where
|
||||
add_zero : ∀ a : M, a + 0 = a
|
||||
zero_add : ∀ a : M, 0 + a = a
|
||||
add_comm : ∀ a b : M, a + b = b + a
|
||||
add_assoc : ∀ a b c : M, a + b + c = a + (b + c)
|
||||
zero_hmul : ∀ a : M, (0 : Int) * a = 0
|
||||
@@ -37,15 +38,6 @@ class IntModule (M : Type u) extends Zero M, Add M, Neg M, Sub M, HMul Int M M w
|
||||
neg_add_cancel : ∀ a : M, -a + a = 0
|
||||
sub_eq_add_neg : ∀ a b : M, a - b = a + -b
|
||||
|
||||
namespace NatModule
|
||||
|
||||
variable {M : Type u} [NatModule M]
|
||||
|
||||
theorem zero_add (a : M) : 0 + a = a := by
|
||||
rw [add_comm, add_zero]
|
||||
|
||||
end NatModule
|
||||
|
||||
namespace IntModule
|
||||
|
||||
attribute [instance 100] IntModule.toZero IntModule.toAdd IntModule.toNeg IntModule.toSub IntModule.toHMul
|
||||
@@ -60,21 +52,9 @@ instance toNatModule (M : Type u) [i : IntModule M] : NatModule M :=
|
||||
|
||||
variable {M : Type u} [IntModule M]
|
||||
|
||||
instance (priority := 100) (M : Type u) [IntModule M] : SMul Nat M where
|
||||
smul a x := (a : Int) * x
|
||||
|
||||
instance (priority := 100) (M : Type u) [IntModule M] : SMul Int M where
|
||||
smul a x := a * x
|
||||
|
||||
theorem zero_add (a : M) : 0 + a = a := by
|
||||
rw [add_comm, add_zero]
|
||||
|
||||
theorem add_neg_cancel (a : M) : a + -a = 0 := by
|
||||
rw [add_comm, neg_add_cancel]
|
||||
|
||||
theorem add_left_comm (a b c : M) : a + (b + c) = b + (a + c) := by
|
||||
rw [← add_assoc, ← add_assoc, add_comm a]
|
||||
|
||||
theorem add_left_inj {a b : M} (c : M) : a + c = b + c ↔ a = b :=
|
||||
⟨fun h => by simpa [add_assoc, add_neg_cancel, add_zero] using (congrArg (· + -c) h),
|
||||
fun g => congrArg (· + c) g⟩
|
||||
@@ -131,17 +111,4 @@ class NoNatZeroDivisors (α : Type u) [Zero α] [HMul Nat α α] where
|
||||
|
||||
export NoNatZeroDivisors (no_nat_zero_divisors)
|
||||
|
||||
instance [ToInt α (some lo) (some hi)] [IntModule α] [ToInt.Zero α (some lo) (some hi)] [ToInt.Add α (some lo) (some hi)] : ToInt.Neg α (some lo) (some hi) where
|
||||
toInt_neg x := by
|
||||
have := (ToInt.Add.toInt_add (-x) x).symm
|
||||
rw [IntModule.neg_add_cancel, ToInt.Zero.toInt_zero] at this
|
||||
rw [ToInt.wrap_eq_wrap_iff] at this
|
||||
simp at this
|
||||
rw [← ToInt.wrap_toInt]
|
||||
rw [ToInt.wrap_eq_wrap_iff]
|
||||
simpa
|
||||
|
||||
instance [ToInt α (some lo) (some hi)] [IntModule α] [ToInt.Add α (some lo) (some hi)] [ToInt.Neg α (some lo) (some hi)] : ToInt.Sub α (some lo) (some hi) :=
|
||||
ToInt.Sub.of_sub_eq_add_neg IntModule.sub_eq_add_neg
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -12,7 +12,6 @@ import Init.Classical
|
||||
import Init.ByCases
|
||||
import Init.Data.Int.Linear
|
||||
import Init.Data.Int.Pow
|
||||
import Init.Grind.CommRing.Field
|
||||
|
||||
namespace Lean.Grind
|
||||
/-!
|
||||
@@ -178,16 +177,15 @@ init_grind_norm
|
||||
Nat.add_eq Nat.sub_eq Nat.mul_eq Nat.zero_eq Nat.le_eq
|
||||
Nat.div_zero Nat.mod_zero Nat.div_one Nat.mod_one
|
||||
Nat.sub_sub Nat.pow_zero Nat.pow_one Nat.sub_self
|
||||
Nat.one_pow
|
||||
-- Int
|
||||
Int.lt_eq
|
||||
Int.emod_neg Int.ediv_neg
|
||||
Int.ediv_zero Int.emod_zero
|
||||
Int.ediv_one Int.emod_one
|
||||
Int.negSucc_eq
|
||||
|
||||
natCast_eq natCast_div natCast_mod
|
||||
natCast_add natCast_mul
|
||||
Int.one_pow
|
||||
|
||||
Int.pow_zero Int.pow_one
|
||||
-- GT GE
|
||||
ge_eq gt_eq
|
||||
@@ -199,7 +197,5 @@ init_grind_norm
|
||||
-- Function composition
|
||||
Function.const_apply Function.comp_apply Function.const_comp
|
||||
Function.comp_const Function.true_comp Function.false_comp
|
||||
-- Field
|
||||
Field.div_eq_mul_inv Field.inv_zero Field.inv_inv Field.inv_one Field.inv_neg
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -11,4 +11,3 @@ import Init.Grind.Ordered.Module
|
||||
import Init.Grind.Ordered.Ring
|
||||
import Init.Grind.Ordered.Field
|
||||
import Init.Grind.Ordered.Int
|
||||
import Init.Grind.Ordered.Linarith
|
||||
|
||||
@@ -185,54 +185,4 @@ theorem mul_le_mul_iff_of_neg_left {a b c : R} (h : c < 0) : c * a ≤ c * b ↔
|
||||
|
||||
end Field.IsOrdered
|
||||
|
||||
theorem Field.zero_lt_one [Field α] [LinearOrder α] [Ring.IsOrdered α] : (0 : α) < 1 := by
|
||||
cases LinearOrder.trichotomy (0:α) 1
|
||||
next => assumption
|
||||
next h =>
|
||||
cases h
|
||||
next => have := Field.zero_ne_one (α := α); contradiction
|
||||
next h =>
|
||||
have := Ring.IsOrdered.mul_pos_of_neg_of_neg h h
|
||||
simp [Semiring.one_mul] at this
|
||||
assumption
|
||||
|
||||
theorem Field.minus_one_lt_zero [Field α] [LinearOrder α] [Ring.IsOrdered α] : (-1 : α) < 0 := by
|
||||
have h := Field.zero_lt_one (α := α)
|
||||
have := IntModule.IsOrdered.add_lt_left h (-1)
|
||||
rw [Semiring.zero_add, Ring.add_neg_cancel] at this
|
||||
assumption
|
||||
|
||||
theorem ofNat_nonneg [Field α] [LinearOrder α] [Ring.IsOrdered α] (x : Nat) : (OfNat.ofNat x : α) ≥ 0 := by
|
||||
induction x
|
||||
next => simp [OfNat.ofNat, Zero.zero]; apply Preorder.le_refl
|
||||
next n ih =>
|
||||
have := Field.zero_lt_one (α := α)
|
||||
rw [Semiring.ofNat_succ]
|
||||
replace ih := IntModule.IsOrdered.add_le_left ih 1
|
||||
rw [Semiring.zero_add] at ih
|
||||
have := Preorder.lt_of_lt_of_le this ih
|
||||
exact Preorder.le_of_lt this
|
||||
|
||||
instance [Field α] [LinearOrder α] [Ring.IsOrdered α] : IsCharP α 0 where
|
||||
ofNat_eq_zero_iff := by
|
||||
intro x
|
||||
simp only [Nat.mod_zero]; constructor
|
||||
next =>
|
||||
intro h
|
||||
cases x
|
||||
next => rfl
|
||||
next x =>
|
||||
rw [Semiring.ofNat_succ] at h
|
||||
replace h := congrArg (· - 1) h; simp at h
|
||||
rw [Ring.sub_eq_add_neg, Semiring.add_assoc, Ring.add_neg_cancel,
|
||||
Ring.sub_eq_add_neg, Semiring.zero_add, Semiring.add_zero] at h
|
||||
have h₁ : (OfNat.ofNat x : α) < 0 := by
|
||||
have := Field.minus_one_lt_zero (α := α)
|
||||
rw [h]; assumption
|
||||
have h₂ := ofNat_nonneg (α := α) x
|
||||
have : (0 : α) < 0 := Preorder.lt_of_le_of_lt h₂ h₁
|
||||
simp
|
||||
exact (Preorder.lt_irrefl 0) this
|
||||
next => intro h; rw [OfNat.ofNat, h]; rfl
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -1,502 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2025 Amazon.com, Inc. or its affiliates. All Rights Reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Leonardo de Moura
|
||||
-/
|
||||
module
|
||||
prelude
|
||||
import Init.Grind.Ordered.Module
|
||||
import Init.Grind.Ordered.Ring
|
||||
import all Init.Data.Ord
|
||||
import all Init.Data.AC
|
||||
import Init.Data.RArray
|
||||
|
||||
/-!
|
||||
Support for the linear arithmetic module for `IntModule` in `grind`
|
||||
-/
|
||||
|
||||
namespace Lean.Grind.Linarith
|
||||
abbrev Var := Nat
|
||||
open IntModule
|
||||
|
||||
attribute [local simp] add_zero zero_add zero_hmul hmul_zero one_hmul
|
||||
|
||||
inductive Expr where
|
||||
| zero
|
||||
| var (i : Var)
|
||||
| add (a b : Expr)
|
||||
| sub (a b : Expr)
|
||||
| neg (a : Expr)
|
||||
| mul (k : Int) (a : Expr)
|
||||
deriving Inhabited, BEq
|
||||
|
||||
abbrev Context (α : Type u) := RArray α
|
||||
|
||||
def Var.denote {α} (ctx : Context α) (v : Var) : α :=
|
||||
ctx.get v
|
||||
|
||||
def Expr.denote {α} [IntModule α] (ctx : Context α) : Expr → α
|
||||
| zero => 0
|
||||
| .var v => v.denote ctx
|
||||
| .add a b => denote ctx a + denote ctx b
|
||||
| .sub a b => denote ctx a - denote ctx b
|
||||
| .mul k a => k * denote ctx a
|
||||
| .neg a => -denote ctx a
|
||||
|
||||
inductive Poly where
|
||||
| nil
|
||||
| add (k : Int) (v : Var) (p : Poly)
|
||||
deriving BEq
|
||||
|
||||
def Poly.denote {α} [IntModule α] (ctx : Context α) (p : Poly) : α :=
|
||||
match p with
|
||||
| .nil => 0
|
||||
| .add k v p => k * v.denote ctx + denote ctx p
|
||||
|
||||
/--
|
||||
Similar to `Poly.denote`, but produces a denotation better for normalization.
|
||||
-/
|
||||
def Poly.denote' {α} [IntModule α] (ctx : Context α) (p : Poly) : α :=
|
||||
match p with
|
||||
| .nil => 0
|
||||
| .add 1 v p => go (v.denote ctx) p
|
||||
| .add k v p => go (k * v.denote ctx) p
|
||||
where
|
||||
go (r : α) (p : Poly) : α :=
|
||||
match p with
|
||||
| .nil => r
|
||||
| .add 1 v p => go (r + v.denote ctx) p
|
||||
| .add k v p => go (r + k * v.denote ctx) p
|
||||
|
||||
-- Helper instance for `ac_rfl`
|
||||
local instance {α} [IntModule α] : Std.Associative (· + · : α → α → α) where
|
||||
assoc := IntModule.add_assoc
|
||||
-- Helper instance for `ac_rfl`
|
||||
local instance {α} [IntModule α] : Std.Commutative (· + · : α → α → α) where
|
||||
comm := IntModule.add_comm
|
||||
|
||||
theorem Poly.denote'_go_eq_denote {α} [IntModule α] (ctx : Context α) (p : Poly) (r : α) : denote'.go ctx r p = p.denote ctx + r := by
|
||||
induction r, p using denote'.go.induct ctx <;> simp [denote'.go, denote]
|
||||
next ih => rw [ih]; ac_rfl
|
||||
next ih => rw [ih]; ac_rfl
|
||||
|
||||
theorem Poly.denote'_eq_denote {α} [IntModule α] (ctx : Context α) (p : Poly) : p.denote' ctx = p.denote ctx := by
|
||||
unfold denote' <;> split <;> simp [denote, denote'_go_eq_denote] <;> ac_rfl
|
||||
|
||||
def Poly.insert (k : Int) (v : Var) (p : Poly) : Poly :=
|
||||
match p with
|
||||
| .nil => .add k v .nil
|
||||
| .add k' v' p =>
|
||||
bif Nat.blt v' v then
|
||||
.add k v <| .add k' v' p
|
||||
else bif Nat.beq v v' then
|
||||
if Int.add k k' == 0 then
|
||||
p
|
||||
else
|
||||
.add (Int.add k k') v' p
|
||||
else
|
||||
.add k' v' (insert k v p)
|
||||
|
||||
/-- Normalizes the given polynomial by fusing monomial and constants. -/
|
||||
def Poly.norm (p : Poly) : Poly :=
|
||||
match p with
|
||||
| .nil => .nil
|
||||
| .add k v p => (norm p).insert k v
|
||||
|
||||
def Poly.append (p₁ p₂ : Poly) : Poly :=
|
||||
match p₁ with
|
||||
| .nil => p₂
|
||||
| .add k x p₁ => .add k x (append p₁ p₂)
|
||||
|
||||
def Poly.combine' (fuel : Nat) (p₁ p₂ : Poly) : Poly :=
|
||||
match fuel with
|
||||
| 0 => p₁.append p₂
|
||||
| fuel + 1 => match p₁, p₂ with
|
||||
| .nil, p₂ => p₂
|
||||
| p₁, .nil => p₁
|
||||
| .add a₁ x₁ p₁, .add a₂ x₂ p₂ =>
|
||||
bif Nat.beq x₁ x₂ then
|
||||
let a := a₁ + a₂
|
||||
bif a == 0 then
|
||||
combine' fuel p₁ p₂
|
||||
else
|
||||
.add a x₁ (combine' fuel p₁ p₂)
|
||||
else bif Nat.blt x₂ x₁ then
|
||||
.add a₁ x₁ (combine' fuel p₁ (.add a₂ x₂ p₂))
|
||||
else
|
||||
.add a₂ x₂ (combine' fuel (.add a₁ x₁ p₁) p₂)
|
||||
|
||||
def Poly.combine (p₁ p₂ : Poly) : Poly :=
|
||||
combine' 100000000 p₁ p₂
|
||||
|
||||
/-- Converts the given expression into a polynomial. -/
|
||||
def Expr.toPoly' (e : Expr) : Poly :=
|
||||
go 1 e .nil
|
||||
where
|
||||
go (coeff : Int) : Expr → (Poly → Poly)
|
||||
| .zero => id
|
||||
| .var v => (.add coeff v ·)
|
||||
| .add a b => go coeff a ∘ go coeff b
|
||||
| .sub a b => go coeff a ∘ go (-coeff) b
|
||||
| .mul k a => bif k == 0 then id else go (Int.mul coeff k) a
|
||||
| .neg a => go (-coeff) a
|
||||
|
||||
/-- Converts the given expression into a polynomial, and then normalizes it. -/
|
||||
def Expr.norm (e : Expr) : Poly :=
|
||||
e.toPoly'.norm
|
||||
|
||||
/--
|
||||
`p.mul k` multiplies all coefficients and constant of the polynomial `p` by `k`.
|
||||
-/
|
||||
def Poly.mul' (p : Poly) (k : Int) : Poly :=
|
||||
match p with
|
||||
| .nil => .nil
|
||||
| .add k' v p => .add (k*k') v (mul' p k)
|
||||
|
||||
def Poly.mul (p : Poly) (k : Int) : Poly :=
|
||||
if k == 0 then
|
||||
.nil
|
||||
else
|
||||
p.mul' k
|
||||
|
||||
@[simp] theorem Poly.denote_mul {α} [IntModule α] (ctx : Context α) (p : Poly) (k : Int) : (p.mul k).denote ctx = k * p.denote ctx := by
|
||||
simp [mul]
|
||||
split
|
||||
next => simp [*, denote]
|
||||
next =>
|
||||
induction p <;> simp [mul', denote, *]
|
||||
rw [mul_hmul, hmul_add]
|
||||
|
||||
theorem Poly.denote_insert {α} [IntModule α] (ctx : Context α) (k : Int) (v : Var) (p : Poly) :
|
||||
(p.insert k v).denote ctx = p.denote ctx + k * v.denote ctx := by
|
||||
fun_induction p.insert k v <;> simp [denote]
|
||||
next => ac_rfl
|
||||
next h₁ h₂ h₃ =>
|
||||
simp at h₃; simp at h₂; subst h₂
|
||||
rw [add_comm, ← add_assoc, ← add_hmul, h₃, zero_hmul, zero_add]
|
||||
next h _ => simp at h; subst h; rw [add_hmul]; ac_rfl
|
||||
next ih => rw [ih]; ac_rfl
|
||||
|
||||
attribute [local simp] Poly.denote_insert
|
||||
|
||||
theorem Poly.denote_norm {α} [IntModule α] (ctx : Context α) (p : Poly) : p.norm.denote ctx = p.denote ctx := by
|
||||
induction p <;> simp [denote, norm, add_comm, *]
|
||||
|
||||
attribute [local simp] Poly.denote_norm
|
||||
|
||||
theorem Poly.denote_append {α} [IntModule α] (ctx : Context α) (p₁ p₂ : Poly) : (p₁.append p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
|
||||
induction p₁ <;> simp [append, denote, *]; ac_rfl
|
||||
|
||||
attribute [local simp] Poly.denote_append
|
||||
|
||||
theorem Poly.denote_combine' {α} [IntModule α] (ctx : Context α) (fuel : Nat) (p₁ p₂ : Poly) : (p₁.combine' fuel p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
|
||||
fun_induction p₁.combine' fuel p₂ <;>
|
||||
simp_all +zetaDelta [denote, ← Int.add_mul]
|
||||
next h _ =>
|
||||
rw [Int.add_comm] at h
|
||||
rw [add_left_comm, add_assoc, ← add_assoc, ← add_hmul, h, zero_hmul, zero_add]
|
||||
next => rw [add_hmul]; ac_rfl
|
||||
all_goals ac_rfl
|
||||
|
||||
theorem Poly.denote_combine {α} [IntModule α] (ctx : Context α) (p₁ p₂ : Poly) : (p₁.combine p₂).denote ctx = p₁.denote ctx + p₂.denote ctx := by
|
||||
simp [combine, denote_combine']
|
||||
|
||||
attribute [local simp] Poly.denote_combine
|
||||
|
||||
theorem Expr.denote_toPoly'_go {α} [IntModule α] {k p} (ctx : Context α) (e : Expr)
|
||||
: (toPoly'.go k e p).denote ctx = k * e.denote ctx + p.denote ctx := by
|
||||
induction k, e using Expr.toPoly'.go.induct generalizing p <;> simp [toPoly'.go, denote, Poly.denote, *, hmul_add]
|
||||
next => ac_rfl
|
||||
next => rw [sub_eq_add_neg, neg_hmul, hmul_add, hmul_neg]; ac_rfl
|
||||
next h => simp at h; subst h; simp
|
||||
next ih => simp at ih; rw [ih, mul_hmul]
|
||||
next => rw [hmul_neg, neg_hmul]
|
||||
|
||||
theorem Expr.denote_norm {α} [IntModule α] (ctx : Context α) (e : Expr) : e.norm.denote ctx = e.denote ctx := by
|
||||
simp [norm, toPoly', Expr.denote_toPoly'_go, Poly.denote]
|
||||
|
||||
attribute [local simp] Expr.denote_norm
|
||||
|
||||
instance : LawfulBEq Poly where
|
||||
eq_of_beq {a} := by
|
||||
induction a <;> intro b <;> cases b <;> simp_all! [BEq.beq]
|
||||
next ih =>
|
||||
intro _ _ h
|
||||
exact ih h
|
||||
rfl := by
|
||||
intro a
|
||||
induction a <;> simp! [BEq.beq]
|
||||
assumption
|
||||
|
||||
attribute [local simp] Poly.denote'_eq_denote
|
||||
|
||||
def Poly.leadCoeff (p : Poly) : Int :=
|
||||
match p with
|
||||
| .add a _ _ => a
|
||||
| _ => 1
|
||||
|
||||
open IntModule.IsOrdered
|
||||
|
||||
/-!
|
||||
Helper theorems for conflict resolution during model construction.
|
||||
-/
|
||||
|
||||
private theorem le_add_le {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] {a b : α}
|
||||
(h₁ : a ≤ 0) (h₂ : b ≤ 0) : a + b ≤ 0 := by
|
||||
replace h₁ := add_le_left h₁ b; simp at h₁
|
||||
exact Preorder.le_trans h₁ h₂
|
||||
|
||||
private theorem le_add_lt {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] {a b : α}
|
||||
(h₁ : a ≤ 0) (h₂ : b < 0) : a + b < 0 := by
|
||||
replace h₁ := add_le_left h₁ b; simp at h₁
|
||||
exact Preorder.lt_of_le_of_lt h₁ h₂
|
||||
|
||||
private theorem lt_add_lt {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] {a b : α}
|
||||
(h₁ : a < 0) (h₂ : b < 0) : a + b < 0 := by
|
||||
replace h₁ := add_lt_left h₁ b; simp at h₁
|
||||
exact Preorder.lt_trans h₁ h₂
|
||||
|
||||
private theorem coe_natAbs_nonneg (a : Int) : (a.natAbs : Int) ≥ 0 := by
|
||||
exact Int.natCast_nonneg a.natAbs
|
||||
|
||||
def le_le_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
|
||||
let a₁ := p₁.leadCoeff.natAbs
|
||||
let a₂ := p₂.leadCoeff.natAbs
|
||||
p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
|
||||
|
||||
theorem le_le_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: le_le_combine_cert p₁ p₂ p₃ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 → p₃.denote' ctx ≤ 0 := by
|
||||
simp [le_le_combine_cert]; intro _ h₁ h₂; subst p₃; simp
|
||||
replace h₁ := hmul_nonpos (coe_natAbs_nonneg p₂.leadCoeff) h₁
|
||||
replace h₂ := hmul_nonpos (coe_natAbs_nonneg p₁.leadCoeff) h₂
|
||||
exact le_add_le h₁ h₂
|
||||
|
||||
def le_lt_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
|
||||
let a₁ := p₁.leadCoeff.natAbs
|
||||
let a₂ := p₂.leadCoeff.natAbs
|
||||
a₁ > (0 : Int) && p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
|
||||
|
||||
theorem le_lt_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: le_lt_combine_cert p₁ p₂ p₃ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx < 0 → p₃.denote' ctx < 0 := by
|
||||
simp [-Int.natAbs_pos, -Int.ofNat_pos, le_lt_combine_cert]; intro hp _ h₁ h₂; subst p₃; simp
|
||||
replace h₁ := hmul_nonpos (coe_natAbs_nonneg p₂.leadCoeff) h₁
|
||||
replace h₂ := hmul_neg (↑p₁.leadCoeff.natAbs) h₂ |>.mp hp
|
||||
exact le_add_lt h₁ h₂
|
||||
|
||||
theorem le_eq_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: le_le_combine_cert p₁ p₂ p₃ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx = 0 → p₃.denote' ctx ≤ 0 := by
|
||||
simp [le_le_combine_cert]; intro _ h₁ h₂; subst p₃; simp [h₂]
|
||||
replace h₁ := hmul_nonpos (coe_natAbs_nonneg p₂.leadCoeff) h₁
|
||||
assumption
|
||||
|
||||
def lt_lt_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
|
||||
let a₁ := p₁.leadCoeff.natAbs
|
||||
let a₂ := p₂.leadCoeff.natAbs
|
||||
a₂ > (0 : Int) && a₁ > (0 : Int) && p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
|
||||
|
||||
theorem lt_lt_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: lt_lt_combine_cert p₁ p₂ p₃ → p₁.denote' ctx < 0 → p₂.denote' ctx < 0 → p₃.denote' ctx < 0 := by
|
||||
simp [-Int.natAbs_pos, -Int.ofNat_pos, lt_lt_combine_cert]; intro hp₁ hp₂ _ h₁ h₂; subst p₃; simp
|
||||
replace h₁ := hmul_neg (↑p₂.leadCoeff.natAbs) h₁ |>.mp hp₁
|
||||
replace h₂ := hmul_neg (↑p₁.leadCoeff.natAbs) h₂ |>.mp hp₂
|
||||
exact lt_add_lt h₁ h₂
|
||||
|
||||
def lt_eq_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
|
||||
let a₁ := p₁.leadCoeff.natAbs
|
||||
let a₂ := p₂.leadCoeff.natAbs
|
||||
a₂ > (0 : Int) && p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
|
||||
|
||||
theorem lt_eq_combine {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: lt_eq_combine_cert p₁ p₂ p₃ → p₁.denote' ctx < 0 → p₂.denote' ctx = 0 → p₃.denote' ctx < 0 := by
|
||||
simp [-Int.natAbs_pos, -Int.ofNat_pos, lt_eq_combine_cert]; intro hp₁ _ h₁ h₂; subst p₃; simp [h₂]
|
||||
replace h₁ := hmul_neg (↑p₂.leadCoeff.natAbs) h₁ |>.mp hp₁
|
||||
assumption
|
||||
|
||||
theorem eq_eq_combine {α} [IntModule α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: le_le_combine_cert p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx = 0 → p₃.denote' ctx = 0 := by
|
||||
simp [le_le_combine_cert]; intro _ h₁ h₂; subst p₃; simp [h₁, h₂]
|
||||
|
||||
def diseq_split_cert (p₁ p₂ : Poly) : Bool :=
|
||||
p₂ == p₁.mul (-1)
|
||||
|
||||
-- We need `LinearOrder` to use `trichotomy`
|
||||
theorem diseq_split {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly)
|
||||
: diseq_split_cert p₁ p₂ → p₁.denote' ctx ≠ 0 → p₁.denote' ctx < 0 ∨ p₂.denote' ctx < 0 := by
|
||||
simp [diseq_split_cert]; intro _ h₁; subst p₂; simp
|
||||
cases LinearOrder.trichotomy (p₁.denote ctx) 0
|
||||
next h => exact Or.inl h
|
||||
next h =>
|
||||
apply Or.inr
|
||||
simp [h₁] at h
|
||||
rw [← neg_pos_iff, neg_hmul, neg_neg, one_hmul]; assumption
|
||||
|
||||
theorem diseq_split_resolve {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly)
|
||||
: diseq_split_cert p₁ p₂ → p₁.denote' ctx ≠ 0 → ¬p₁.denote' ctx < 0 → p₂.denote' ctx < 0 := by
|
||||
intro h₁ h₂ h₃
|
||||
exact (diseq_split ctx p₁ p₂ h₁ h₂).resolve_left h₃
|
||||
|
||||
def eq_diseq_combine_cert (p₁ p₂ p₃ : Poly) : Bool :=
|
||||
let a₁ := p₁.leadCoeff.natAbs
|
||||
let a₂ := p₂.leadCoeff.natAbs
|
||||
a₁ ≠ 0 && p₃ == (p₁.mul a₂ |>.combine (p₂.mul a₁))
|
||||
|
||||
theorem eq_diseq_combine {α} [IntModule α] [NoNatZeroDivisors α] (ctx : Context α) (p₁ p₂ p₃ : Poly)
|
||||
: eq_diseq_combine_cert p₁ p₂ p₃ → p₁.denote' ctx = 0 → p₂.denote' ctx ≠ 0 → p₃.denote' ctx ≠ 0 := by
|
||||
simp [- Int.natAbs_eq_zero, -Int.natCast_eq_zero, eq_diseq_combine_cert]; intro hne _ h₁ h₂; subst p₃
|
||||
simp [h₁, h₂]; intro h
|
||||
have := no_nat_zero_divisors (p₁.leadCoeff.natAbs) (p₂.denote ctx) hne h
|
||||
contradiction
|
||||
|
||||
def eq_diseq_combine_cert' (p₁ p₂ p₃ : Poly) (k : Int) : Bool :=
|
||||
p₃ == (p₁.mul k |>.combine p₂)
|
||||
|
||||
/-
|
||||
Special case of `eq_diseq_combine` where leading coefficient `c₁` of `p₁` is `-k*c₂`, where
|
||||
`c₂` is the leading coefficient of `p₂`.
|
||||
-/
|
||||
theorem eq_diseq_combine' {α} [IntModule α] (ctx : Context α) (p₁ p₂ p₃ : Poly) (k : Int)
|
||||
: eq_diseq_combine_cert' p₁ p₂ p₃ k → p₁.denote' ctx = 0 → p₂.denote' ctx ≠ 0 → p₃.denote' ctx ≠ 0 := by
|
||||
simp [eq_diseq_combine_cert']; intro _ h₁ h₂; subst p₃
|
||||
simp [h₁, h₂]
|
||||
|
||||
/-!
|
||||
Helper theorems for internalizing facts into the linear arithmetic procedure
|
||||
-/
|
||||
|
||||
def norm_cert (lhs rhs : Expr) (p : Poly) :=
|
||||
p == (lhs.sub rhs).norm
|
||||
|
||||
theorem eq_norm {α} [IntModule α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → lhs.denote ctx = rhs.denote ctx → p.denote' ctx = 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
|
||||
theorem le_of_eq {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → lhs.denote ctx = rhs.denote ctx → p.denote' ctx ≤ 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
apply Preorder.le_refl
|
||||
|
||||
theorem diseq_norm {α} [IntModule α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → lhs.denote ctx ≠ rhs.denote ctx → p.denote' ctx ≠ 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
intro h
|
||||
replace h := congrArg (rhs.denote ctx + ·) h; simp [sub_eq_add_neg] at h
|
||||
rw [add_left_comm, ← sub_eq_add_neg, sub_self, add_zero] at h
|
||||
contradiction
|
||||
|
||||
theorem le_norm {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → lhs.denote ctx ≤ rhs.denote ctx → p.denote' ctx ≤ 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
replace h₁ := add_le_left h₁ (-rhs.denote ctx)
|
||||
simp [← sub_eq_add_neg, sub_self] at h₁
|
||||
assumption
|
||||
|
||||
theorem lt_norm {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → lhs.denote ctx < rhs.denote ctx → p.denote' ctx < 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
replace h₁ := add_lt_left h₁ (-rhs.denote ctx)
|
||||
simp [← sub_eq_add_neg, sub_self] at h₁
|
||||
assumption
|
||||
|
||||
theorem not_le_norm {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert rhs lhs p → ¬ lhs.denote ctx ≤ rhs.denote ctx → p.denote' ctx < 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
replace h₁ := LinearOrder.lt_of_not_le h₁
|
||||
replace h₁ := add_lt_left h₁ (-lhs.denote ctx)
|
||||
simp [← sub_eq_add_neg, sub_self] at h₁
|
||||
assumption
|
||||
|
||||
theorem not_lt_norm {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert rhs lhs p → ¬ lhs.denote ctx < rhs.denote ctx → p.denote' ctx ≤ 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]
|
||||
replace h₁ := LinearOrder.le_of_not_lt h₁
|
||||
replace h₁ := add_le_left h₁ (-lhs.denote ctx)
|
||||
simp [← sub_eq_add_neg, sub_self] at h₁
|
||||
assumption
|
||||
|
||||
-- If the module does not have a linear order, we can still put the expressions in polynomial forms
|
||||
|
||||
theorem not_le_norm' {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → ¬ lhs.denote ctx ≤ rhs.denote ctx → ¬ p.denote' ctx ≤ 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]; intro h
|
||||
replace h := add_le_right (rhs.denote ctx) h
|
||||
rw [sub_eq_add_neg, add_left_comm, ← sub_eq_add_neg, sub_self] at h; simp at h
|
||||
contradiction
|
||||
|
||||
theorem not_lt_norm' {α} [IntModule α] [Preorder α] [IntModule.IsOrdered α] (ctx : Context α) (lhs rhs : Expr) (p : Poly)
|
||||
: norm_cert lhs rhs p → ¬ lhs.denote ctx < rhs.denote ctx → ¬ p.denote' ctx < 0 := by
|
||||
simp [norm_cert]; intro _ h₁; subst p; simp [Expr.denote, h₁, sub_self]; intro h
|
||||
replace h := add_lt_right (rhs.denote ctx) h
|
||||
rw [sub_eq_add_neg, add_left_comm, ← sub_eq_add_neg, sub_self] at h; simp at h
|
||||
contradiction
|
||||
|
||||
/-!
|
||||
Equality detection
|
||||
-/
|
||||
def eq_of_le_ge_cert (p₁ p₂ : Poly) : Bool :=
|
||||
p₂ == p₁.mul (-1)
|
||||
|
||||
theorem eq_of_le_ge {α} [IntModule α] [PartialOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ : Poly) (p₂ : Poly)
|
||||
: eq_of_le_ge_cert p₁ p₂ → p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 → p₁.denote' ctx = 0 := by
|
||||
simp [eq_of_le_ge_cert]
|
||||
intro; subst p₂; simp
|
||||
intro h₁ h₂
|
||||
replace h₂ := add_le_left h₂ (p₁.denote ctx)
|
||||
rw [add_comm, neg_hmul, one_hmul, ← sub_eq_add_neg, sub_self, zero_add] at h₂
|
||||
exact PartialOrder.le_antisymm h₁ h₂
|
||||
|
||||
/-!
|
||||
Helper theorems for closing the goal
|
||||
-/
|
||||
|
||||
theorem diseq_unsat {α} [IntModule α] (ctx : Context α) : (Poly.nil).denote ctx ≠ 0 → False := by
|
||||
simp [Poly.denote]
|
||||
|
||||
theorem lt_unsat {α} [IntModule α] [Preorder α] (ctx : Context α) : (Poly.nil).denote ctx < 0 → False := by
|
||||
simp [Poly.denote]; intro h
|
||||
have := Preorder.lt_iff_le_not_le.mp h
|
||||
simp at this
|
||||
|
||||
def zero_lt_one_cert (p : Poly) : Bool :=
|
||||
p == .add (-1) 0 .nil
|
||||
|
||||
theorem zero_lt_one {α} [Ring α] [Preorder α] [Ring.IsOrdered α] (ctx : Context α) (p : Poly)
|
||||
: zero_lt_one_cert p → (0 : Var).denote ctx = One.one → p.denote' ctx < 0 := by
|
||||
simp [zero_lt_one_cert]; intro _ h; subst p; simp [Poly.denote, h, One.one, neg_hmul]
|
||||
rw [neg_lt_iff, neg_zero]; apply Ring.IsOrdered.zero_lt_one
|
||||
|
||||
/-!
|
||||
Coefficient normalization
|
||||
-/
|
||||
|
||||
def coeff_cert (p₁ p₂ : Poly) (k : Nat) :=
|
||||
k > 0 && p₁ == p₂.mul k
|
||||
|
||||
theorem eq_coeff {α} [IntModule α] [NoNatZeroDivisors α] (ctx : Context α) (p₁ p₂ : Poly) (k : Nat)
|
||||
: coeff_cert p₁ p₂ k → p₁.denote' ctx = 0 → p₂.denote' ctx = 0 := by
|
||||
simp [coeff_cert]; intro h _; subst p₁; simp
|
||||
exact no_nat_zero_divisors k (p₂.denote ctx) (Nat.ne_zero_of_lt h)
|
||||
|
||||
theorem le_coeff {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly) (k : Nat)
|
||||
: coeff_cert p₁ p₂ k → p₁.denote' ctx ≤ 0 → p₂.denote' ctx ≤ 0 := by
|
||||
simp [coeff_cert]; intro h _; subst p₁; simp
|
||||
have : ↑k > (0 : Int) := Int.natCast_pos.mpr h
|
||||
intro h₁; apply Classical.byContradiction
|
||||
intro h₂; replace h₂ := LinearOrder.lt_of_not_le h₂
|
||||
replace h₂ := IsOrdered.hmul_pos (↑k) h₂ |>.mp this
|
||||
exact Preorder.lt_irrefl 0 (Preorder.lt_of_lt_of_le h₂ h₁)
|
||||
|
||||
theorem lt_coeff {α} [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (ctx : Context α) (p₁ p₂ : Poly) (k : Nat)
|
||||
: coeff_cert p₁ p₂ k → p₁.denote' ctx < 0 → p₂.denote' ctx < 0 := by
|
||||
simp [coeff_cert]; intro h _; subst p₁; simp
|
||||
have : ↑k > (0 : Int) := Int.natCast_pos.mpr h
|
||||
intro h₁; apply Classical.byContradiction
|
||||
intro h₂; replace h₂ := LinearOrder.le_of_not_lt h₂
|
||||
replace h₂ := IsOrdered.hmul_nonneg (Int.le_of_lt this) h₂
|
||||
exact Preorder.lt_irrefl 0 (Preorder.lt_of_le_of_lt h₂ h₁)
|
||||
|
||||
theorem diseq_neg {α} [IntModule α] (ctx : Context α) (p p' : Poly) : p' == p.mul (-1) → p.denote' ctx ≠ 0 → p'.denote' ctx ≠ 0 := by
|
||||
simp; intro _ _; subst p'; simp [neg_hmul]
|
||||
intro h; replace h := congrArg (- ·) h; simp [neg_neg, neg_zero] at h
|
||||
contradiction
|
||||
|
||||
end Lean.Grind.Linarith
|
||||
@@ -12,57 +12,12 @@ import Init.Grind.Ordered.Order
|
||||
|
||||
namespace Lean.Grind
|
||||
|
||||
class NatModule.IsOrdered (M : Type u) [Preorder M] [NatModule M] where
|
||||
add_le_left_iff : ∀ {a b : M} (c : M), a ≤ b ↔ a + c ≤ b + c
|
||||
hmul_pos : ∀ (k : Nat) {a : M}, 0 < a → (0 < k ↔ 0 < k * a)
|
||||
hmul_nonneg : ∀ {k : Nat} {a : M}, 0 ≤ a → 0 ≤ k * a
|
||||
|
||||
class IntModule.IsOrdered (M : Type u) [Preorder M] [IntModule M] where
|
||||
neg_le_iff : ∀ a b : M, -a ≤ b ↔ -b ≤ a
|
||||
add_le_left : ∀ {a b : M}, a ≤ b → (c : M) → a + c ≤ b + c
|
||||
hmul_pos : ∀ (k : Int) {a : M}, 0 < a → (0 < k ↔ 0 < k * a)
|
||||
hmul_nonneg : ∀ {k : Int} {a : M}, 0 ≤ k → 0 ≤ a → 0 ≤ k * a
|
||||
|
||||
namespace NatModule.IsOrdered
|
||||
|
||||
variable {M : Type u} [Preorder M] [NatModule M] [NatModule.IsOrdered M]
|
||||
|
||||
theorem add_le_right_iff {a b : M} (c : M) : a ≤ b ↔ c + a ≤ c + b := by
|
||||
rw [add_comm c a, add_comm c b,add_le_left_iff]
|
||||
|
||||
theorem add_le_left {a b : M} (h : a ≤ b) (c : M) : a + c ≤ b + c :=
|
||||
(add_le_left_iff c).mp h
|
||||
|
||||
theorem add_le_right {a b : M} (h : a ≤ b) (c : M) : c + a ≤ c + b :=
|
||||
(add_le_right_iff c).mp h
|
||||
|
||||
theorem add_lt_left {a b : M} (h : a < b) (c : M) : a + c < b + c := by
|
||||
simp only [Preorder.lt_iff_le_not_le] at h ⊢
|
||||
constructor
|
||||
· exact add_le_left h.1 _
|
||||
· intro w
|
||||
apply h.2
|
||||
exact (add_le_left_iff c).mpr w
|
||||
|
||||
theorem add_lt_right {a b : M} (h : a < b) (c : M) : c + a < c + b := by
|
||||
rw [add_comm c a, add_comm c b]
|
||||
exact add_lt_left h c
|
||||
|
||||
theorem add_lt_left_iff {a b : M} (c : M) : a < b ↔ a + c < b + c := by
|
||||
constructor
|
||||
· exact fun h => add_lt_left h c
|
||||
· intro w
|
||||
simp only [Preorder.lt_iff_le_not_le] at w ⊢
|
||||
constructor
|
||||
· exact (add_le_left_iff c).mpr w.1
|
||||
· intro h
|
||||
exact w.2 ((add_le_left_iff c).mp h)
|
||||
|
||||
theorem add_lt_right_iff {a b : M} (c : M) : a < b ↔ c + a < c + b := by
|
||||
rw [add_comm c a, add_comm c b, add_lt_left_iff]
|
||||
|
||||
end NatModule.IsOrdered
|
||||
|
||||
namespace IntModule.IsOrdered
|
||||
|
||||
variable {M : Type u} [Preorder M] [IntModule M] [IntModule.IsOrdered M]
|
||||
@@ -86,7 +41,7 @@ theorem neg_pos_iff {a : M} : 0 < -a ↔ a < 0 := by
|
||||
rw [lt_neg_iff, neg_zero]
|
||||
|
||||
theorem add_lt_left {a b : M} (h : a < b) (c : M) : a + c < b + c := by
|
||||
simp only [Preorder.lt_iff_le_not_le] at h ⊢
|
||||
simp [Preorder.lt_iff_le_not_le] at h ⊢
|
||||
constructor
|
||||
· exact add_le_left h.1 _
|
||||
· intro w
|
||||
|
||||
@@ -91,19 +91,6 @@ theorem trichotomy (a b : α) : a < b ∨ a = b ∨ b < a := by
|
||||
| inl h => right; right; exact h
|
||||
| inr h => right; left; exact h.symm
|
||||
|
||||
theorem le_of_not_lt {α} [LinearOrder α] {a b : α} (h : ¬ a < b) : b ≤ a := by
|
||||
cases LinearOrder.trichotomy a b
|
||||
next => contradiction
|
||||
next h => apply PartialOrder.le_iff_lt_or_eq.mpr; cases h <;> simp [*]
|
||||
|
||||
theorem lt_of_not_le {α} [LinearOrder α] {a b : α} (h : ¬ a ≤ b) : b < a := by
|
||||
cases LinearOrder.trichotomy a b
|
||||
next h₁ h₂ => have := Preorder.lt_iff_le_not_le.mp h₂; simp [h] at this
|
||||
next h =>
|
||||
cases h
|
||||
next h => subst a; exact False.elim <| h (Preorder.le_refl b)
|
||||
next => assumption
|
||||
|
||||
end LinearOrder
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -20,13 +20,13 @@ set_option linter.unusedVariables false in
|
||||
def node_def (_ : Nat) {α : Sort u} {a : α} : NodeDef := .unit
|
||||
|
||||
@[app_unexpander node_def]
|
||||
meta def nodeDefUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
def nodeDefUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
match stx with
|
||||
| `($_ $id:num) => return mkIdent <| Name.mkSimple $ "#" ++ toString id.getNat
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander NodeDef]
|
||||
meta def NodeDefUnexpander : PrettyPrinter.Unexpander := fun _ => do
|
||||
def NodeDefUnexpander : PrettyPrinter.Unexpander := fun _ => do
|
||||
return mkIdent <| Name.mkSimple "NodeDef"
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -55,7 +55,7 @@ and stores `c` in `x`, `b` in `a`, and the proof that `c = some b` in `h`.
|
||||
-/
|
||||
def genPattern {α : Sort u} (_h : Prop) (x : α) (_val : α) : α := x
|
||||
|
||||
/-- Similar to `genPattern` but for the heterogeneous case -/
|
||||
/-- Similar to `genPattern` but for the heterogenous case -/
|
||||
def genHEqPattern {α β : Sort u} (_h : Prop) (x : α) (_val : β) : α := x
|
||||
end Lean.Grind
|
||||
|
||||
@@ -175,7 +175,7 @@ structure Config where
|
||||
-/
|
||||
zeta := true
|
||||
/--
|
||||
When `true` (default: `true`), uses procedure for handling equalities over commutative rings.
|
||||
When `true` (default: `false`), uses procedure for handling equalities over commutative rings.
|
||||
-/
|
||||
ring := true
|
||||
ringSteps := 10000
|
||||
@@ -184,15 +184,6 @@ structure Config where
|
||||
proof terms, instead of a single-step Nullstellensatz certificate
|
||||
-/
|
||||
ringNull := false
|
||||
/--
|
||||
When `true` (default: `true`), uses procedure for handling linear arithmetic for `IntModule`, and
|
||||
`CommRing`.
|
||||
-/
|
||||
linarith := true
|
||||
/--
|
||||
When `true` (default: `true`), uses procedure for handling linear integer arithmetic for `Int` and `Nat`.
|
||||
-/
|
||||
cutsat := true
|
||||
deriving Inhabited, BEq
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -27,6 +27,9 @@ The typeclass `ToInt.Add α lo? hi?` then asserts that `toInt (x + y) = wrap lo?
|
||||
There are many variants for other operations.
|
||||
|
||||
These typeclasses are used solely in the `grind` tactic to lift linear inequalities into `Int`.
|
||||
|
||||
-- TODO: instances for `ToInt.Mod` (only exists for `Fin n` so far)
|
||||
-- TODO: typeclasses for LT, and other algebraic operations.
|
||||
-/
|
||||
|
||||
namespace Lean.Grind
|
||||
@@ -37,73 +40,21 @@ class ToInt (α : Type u) (lo? hi? : outParam (Option Int)) where
|
||||
le_toInt : lo? = some lo → lo ≤ toInt x
|
||||
toInt_lt : hi? = some hi → toInt x < hi
|
||||
|
||||
@[simp]
|
||||
@[simp 500]
|
||||
def ToInt.wrap (lo? hi? : Option Int) (x : Int) : Int :=
|
||||
match lo?, hi? with
|
||||
| some lo, some hi => (x - lo) % (hi - lo) + lo
|
||||
| _, _ => x
|
||||
|
||||
class ToInt.Zero (α : Type u) [Zero α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
toInt_zero : toInt (0 : α) = wrap lo? hi? 0
|
||||
|
||||
class ToInt.Add (α : Type u) [Add α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
toInt_add : ∀ x y : α, toInt (x + y) = wrap lo? hi? (toInt x + toInt y)
|
||||
|
||||
class ToInt.Neg (α : Type u) [Neg α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
toInt_neg : ∀ x : α, toInt (-x) = wrap lo? hi? (-toInt x)
|
||||
|
||||
class ToInt.Sub (α : Type u) [Sub α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
toInt_sub : ∀ x y : α, toInt (x - y) = wrap lo? hi? (toInt x - toInt y)
|
||||
|
||||
class ToInt.Mod (α : Type u) [Mod α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
/-- One might expect a `wrap` on the right hand side,
|
||||
but in practice this stronger statement is usually true. -/
|
||||
toInt_mod : ∀ x y : α, toInt (x % y) = toInt x % toInt y
|
||||
|
||||
class ToInt.LE (α : Type u) [LE α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
le_iff : ∀ x y : α, x ≤ y ↔ toInt x ≤ toInt y
|
||||
|
||||
class ToInt.LT (α : Type u) [LT α] (lo? hi? : outParam (Option Int)) [ToInt α lo? hi?] where
|
||||
lt_iff : ∀ x y : α, x < y ↔ toInt x < toInt y
|
||||
|
||||
/-! ## Helper theorems -/
|
||||
|
||||
theorem ToInt.wrap_add (lo? hi? : Option Int) (x y : Int) :
|
||||
ToInt.wrap lo? hi? (x + y) = ToInt.wrap lo? hi? (ToInt.wrap lo? hi? x + ToInt.wrap lo? hi? y) := by
|
||||
simp only [wrap]
|
||||
split <;> rename_i lo hi
|
||||
· dsimp
|
||||
rw [Int.add_left_inj, Int.emod_eq_emod_iff_emod_sub_eq_zero, Int.emod_def (x - lo), Int.emod_def (y - lo)]
|
||||
have : (x + y - lo -
|
||||
(x - lo - (hi - lo) * ((x - lo) / (hi - lo)) + lo +
|
||||
(y - lo - (hi - lo) * ((y - lo) / (hi - lo)) + lo) - lo)) =
|
||||
(hi - lo) * ((x - lo) / (hi - lo) + (y - lo) / (hi - lo)) := by
|
||||
simp only [Int.mul_add]
|
||||
omega
|
||||
rw [this]
|
||||
exact Int.mul_emod_right ..
|
||||
· simp
|
||||
|
||||
@[simp]
|
||||
theorem ToInt.wrap_toInt (lo? hi? : Option Int) [ToInt α lo? hi?] (x : α) :
|
||||
ToInt.wrap lo? hi? (ToInt.toInt x) = ToInt.toInt x := by
|
||||
simp only [wrap]
|
||||
split
|
||||
· have := ToInt.le_toInt (x := x) rfl
|
||||
have := ToInt.toInt_lt (x := x) rfl
|
||||
rw [Int.emod_eq_of_lt (by omega) (by omega)]
|
||||
omega
|
||||
· rfl
|
||||
|
||||
theorem ToInt.wrap_eq_bmod {i : Int} (h : 0 ≤ i) :
|
||||
ToInt.wrap (some (-i)) (some i) x = x.bmod ((2 * i).toNat) := by
|
||||
match i, h with
|
||||
| (i : Nat), _ =>
|
||||
have : (2 * (i : Int)).toNat = 2 * i := by omega
|
||||
rw [this]
|
||||
simp only [this]
|
||||
simp [Int.bmod_eq_emod, ← Int.two_mul]
|
||||
have : (2 * (i : Int) + 1) / 2 = i := by omega
|
||||
rw [this]
|
||||
simp only [this]
|
||||
by_cases h : i = 0
|
||||
· simp [h]
|
||||
split
|
||||
@@ -118,7 +69,7 @@ theorem ToInt.wrap_eq_bmod {i : Int} (h : 0 ≤ i) :
|
||||
have : x - 2 * ↑i * (x / (2 * ↑i)) - ↑i - (x + ↑i) = (2 * (i : Int)) * (- (x / (2 * i)) - 1) := by
|
||||
simp only [Int.mul_sub, Int.mul_neg]
|
||||
omega
|
||||
rw [this]
|
||||
simp only [this]
|
||||
exact Int.dvd_mul_right ..
|
||||
· rw [← Int.sub_eq_add_neg, Int.sub_eq_iff_eq_add, Int.natCast_zero, Int.sub_zero]
|
||||
rw [Int.emod_eq_iff (by omega)]
|
||||
@@ -130,27 +81,17 @@ theorem ToInt.wrap_eq_bmod {i : Int} (h : 0 ≤ i) :
|
||||
have : x - 2 * ↑i * (x / (2 * ↑i)) + ↑i - (x + ↑i) = (2 * (i : Int)) * (- (x / (2 * i))) := by
|
||||
simp only [Int.mul_neg]
|
||||
omega
|
||||
rw [this]
|
||||
simp only [this]
|
||||
exact Int.dvd_mul_right ..
|
||||
|
||||
theorem ToInt.wrap_eq_wrap_iff :
|
||||
ToInt.wrap (some lo) (some hi) x = ToInt.wrap (some lo) (some hi) y ↔ (x - y) % (hi - lo) = 0 := by
|
||||
simp only [wrap]
|
||||
rw [Int.add_left_inj]
|
||||
rw [Int.emod_eq_emod_iff_emod_sub_eq_zero]
|
||||
have : x - lo - (y - lo) = x - y := by omega
|
||||
rw [this]
|
||||
class ToInt.Add (α : Type u) [Add α] (lo? hi? : Option Int) [ToInt α lo? hi?] where
|
||||
toInt_add : ∀ x y : α, toInt (x + y) = wrap lo? hi? (toInt x + toInt y)
|
||||
|
||||
/-- Construct a `ToInt.Sub` instance from a `ToInt.Add` and `ToInt.Neg` instance and
|
||||
a `sub_eq_add_neg` assumption. -/
|
||||
def ToInt.Sub.of_sub_eq_add_neg {α : Type u} [_root_.Add α] [_root_.Neg α] [_root_.Sub α]
|
||||
(sub_eq_add_neg : ∀ x y : α, x - y = x + -y)
|
||||
{lo? hi? : Option Int} [ToInt α lo? hi?] [Add α lo? hi?] [Neg α lo? hi?] : ToInt.Sub α lo? hi? where
|
||||
toInt_sub x y := by
|
||||
rw [sub_eq_add_neg, ToInt.Add.toInt_add, ToInt.Neg.toInt_neg, Int.sub_eq_add_neg]
|
||||
conv => rhs; rw [ToInt.wrap_add, ToInt.wrap_toInt]
|
||||
class ToInt.Mod (α : Type u) [Mod α] (lo? hi? : Option Int) [ToInt α lo? hi?] where
|
||||
toInt_mod : ∀ x y : α, toInt (x % y) = wrap lo? hi? (toInt x % toInt y)
|
||||
|
||||
/-! ## Instances for concrete types-/
|
||||
class ToInt.LE (α : Type u) [LE α] (lo? hi? : Option Int) [ToInt α lo? hi?] where
|
||||
le_iff : ∀ x y : α, x ≤ y ↔ toInt x ≤ toInt y
|
||||
|
||||
instance : ToInt Int none none where
|
||||
toInt := id
|
||||
@@ -163,21 +104,9 @@ instance : ToInt Int none none where
|
||||
instance : ToInt.Add Int none none where
|
||||
toInt_add := by simp
|
||||
|
||||
instance : ToInt.Neg Int none none where
|
||||
toInt_neg x := by simp
|
||||
|
||||
instance : ToInt.Sub Int none none where
|
||||
toInt_sub x y := by simp
|
||||
|
||||
instance : ToInt.Mod Int none none where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE Int none none where
|
||||
le_iff x y := by simp
|
||||
|
||||
instance : ToInt.LT Int none none where
|
||||
lt_iff x y := by simp
|
||||
|
||||
instance : ToInt Nat (some 0) none where
|
||||
toInt := Nat.cast
|
||||
toInt_inj x y := Int.ofNat_inj.mp
|
||||
@@ -189,15 +118,9 @@ instance : ToInt Nat (some 0) none where
|
||||
instance : ToInt.Add Nat (some 0) none where
|
||||
toInt_add := by simp
|
||||
|
||||
instance : ToInt.Mod Nat (some 0) none where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE Nat (some 0) none where
|
||||
le_iff x y := by simp
|
||||
|
||||
instance : ToInt.LT Nat (some 0) none where
|
||||
lt_iff x y := by simp
|
||||
|
||||
-- Mathlib will add a `ToInt ℕ+ (some 1) none` instance.
|
||||
|
||||
instance : ToInt (Fin n) (some 0) (some n) where
|
||||
@@ -211,21 +134,17 @@ instance : ToInt (Fin n) (some 0) (some n) where
|
||||
instance : ToInt.Add (Fin n) (some 0) (some n) where
|
||||
toInt_add x y := by rfl
|
||||
|
||||
instance [NeZero n] : ToInt.Zero (Fin n) (some 0) (some n) where
|
||||
toInt_zero := by rfl
|
||||
|
||||
-- The `ToInt.Neg` and `ToInt.Sub` instances are generated automatically from the `IntModule (Fin n)` instance.
|
||||
|
||||
instance : ToInt.Mod (Fin n) (some 0) (some n) where
|
||||
toInt_mod x y := by
|
||||
simp only [toInt_fin, Fin.mod_val, Int.natCast_emod]
|
||||
simp only [toInt_fin, Fin.mod_val, Int.natCast_emod, ToInt.wrap, Int.sub_zero, Int.add_zero]
|
||||
rw [Int.emod_eq_of_lt (b := n)]
|
||||
· omega
|
||||
· rw [Int.ofNat_mod_ofNat, ← Fin.mod_val]
|
||||
exact Int.ofNat_lt.mpr (x % y).isLt
|
||||
|
||||
instance : ToInt.LE (Fin n) (some 0) (some n) where
|
||||
le_iff x y := by simpa using Fin.le_def
|
||||
|
||||
instance : ToInt.LT (Fin n) (some 0) (some n) where
|
||||
lt_iff x y := by simpa using Fin.lt_def
|
||||
|
||||
instance : ToInt UInt8 (some 0) (some (2^8)) where
|
||||
toInt x := (x.toNat : Int)
|
||||
toInt_inj x y w := UInt8.toNat_inj.mp (Int.ofNat_inj.mp w)
|
||||
@@ -237,18 +156,9 @@ instance : ToInt UInt8 (some 0) (some (2^8)) where
|
||||
instance : ToInt.Add UInt8 (some 0) (some (2^8)) where
|
||||
toInt_add x y := by simp
|
||||
|
||||
instance : ToInt.Zero UInt8 (some 0) (some (2^8)) where
|
||||
toInt_zero := by simp
|
||||
|
||||
instance : ToInt.Mod UInt8 (some 0) (some (2^8)) where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE UInt8 (some 0) (some (2^8)) where
|
||||
le_iff x y := by simpa using UInt8.le_iff_toBitVec_le
|
||||
|
||||
instance : ToInt.LT UInt8 (some 0) (some (2^8)) where
|
||||
lt_iff x y := by simpa using UInt8.lt_iff_toBitVec_lt
|
||||
|
||||
instance : ToInt UInt16 (some 0) (some (2^16)) where
|
||||
toInt x := (x.toNat : Int)
|
||||
toInt_inj x y w := UInt16.toNat_inj.mp (Int.ofNat_inj.mp w)
|
||||
@@ -260,18 +170,9 @@ instance : ToInt UInt16 (some 0) (some (2^16)) where
|
||||
instance : ToInt.Add UInt16 (some 0) (some (2^16)) where
|
||||
toInt_add x y := by simp
|
||||
|
||||
instance : ToInt.Zero UInt16 (some 0) (some (2^16)) where
|
||||
toInt_zero := by simp
|
||||
|
||||
instance : ToInt.Mod UInt16 (some 0) (some (2^16)) where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE UInt16 (some 0) (some (2^16)) where
|
||||
le_iff x y := by simpa using UInt16.le_iff_toBitVec_le
|
||||
|
||||
instance : ToInt.LT UInt16 (some 0) (some (2^16)) where
|
||||
lt_iff x y := by simpa using UInt16.lt_iff_toBitVec_lt
|
||||
|
||||
instance : ToInt UInt32 (some 0) (some (2^32)) where
|
||||
toInt x := (x.toNat : Int)
|
||||
toInt_inj x y w := UInt32.toNat_inj.mp (Int.ofNat_inj.mp w)
|
||||
@@ -283,18 +184,9 @@ instance : ToInt UInt32 (some 0) (some (2^32)) where
|
||||
instance : ToInt.Add UInt32 (some 0) (some (2^32)) where
|
||||
toInt_add x y := by simp
|
||||
|
||||
instance : ToInt.Zero UInt32 (some 0) (some (2^32)) where
|
||||
toInt_zero := by simp
|
||||
|
||||
instance : ToInt.Mod UInt32 (some 0) (some (2^32)) where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE UInt32 (some 0) (some (2^32)) where
|
||||
le_iff x y := by simpa using UInt32.le_iff_toBitVec_le
|
||||
|
||||
instance : ToInt.LT UInt32 (some 0) (some (2^32)) where
|
||||
lt_iff x y := by simpa using UInt32.lt_iff_toBitVec_lt
|
||||
|
||||
instance : ToInt UInt64 (some 0) (some (2^64)) where
|
||||
toInt x := (x.toNat : Int)
|
||||
toInt_inj x y w := UInt64.toNat_inj.mp (Int.ofNat_inj.mp w)
|
||||
@@ -306,18 +198,9 @@ instance : ToInt UInt64 (some 0) (some (2^64)) where
|
||||
instance : ToInt.Add UInt64 (some 0) (some (2^64)) where
|
||||
toInt_add x y := by simp
|
||||
|
||||
instance : ToInt.Zero UInt64 (some 0) (some (2^64)) where
|
||||
toInt_zero := by simp
|
||||
|
||||
instance : ToInt.Mod UInt64 (some 0) (some (2^64)) where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE UInt64 (some 0) (some (2^64)) where
|
||||
le_iff x y := by simpa using UInt64.le_iff_toBitVec_le
|
||||
|
||||
instance : ToInt.LT UInt64 (some 0) (some (2^64)) where
|
||||
lt_iff x y := by simpa using UInt64.lt_iff_toBitVec_lt
|
||||
|
||||
instance : ToInt USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
toInt x := (x.toNat : Int)
|
||||
toInt_inj x y w := USize.toNat_inj.mp (Int.ofNat_inj.mp w)
|
||||
@@ -333,18 +216,9 @@ instance : ToInt USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
instance : ToInt.Add USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
toInt_add x y := by simp
|
||||
|
||||
instance : ToInt.Zero USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
toInt_zero := by simp
|
||||
|
||||
instance : ToInt.Mod USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
le_iff x y := by simpa using USize.le_iff_toBitVec_le
|
||||
|
||||
instance : ToInt.LT USize (some 0) (some (2^System.Platform.numBits)) where
|
||||
lt_iff x y := by simpa using USize.lt_iff_toBitVec_lt
|
||||
|
||||
instance : ToInt Int8 (some (-2^7)) (some (2^7)) where
|
||||
toInt x := x.toInt
|
||||
toInt_inj x y w := Int8.toInt_inj.mp w
|
||||
@@ -358,22 +232,9 @@ instance : ToInt.Add Int8 (some (-2^7)) (some (2^7)) where
|
||||
simp [Int.bmod_eq_emod]
|
||||
split <;> · simp; omega
|
||||
|
||||
instance : ToInt.Zero Int8 (some (-2^7)) (some (2^7)) where
|
||||
toInt_zero := by
|
||||
-- simp -- FIXME: succeeds, but generates a `(kernel) application type mismatch` error!
|
||||
change (0 : Int8).toInt = _
|
||||
rw [Int8.toInt_zero]
|
||||
decide
|
||||
|
||||
-- Note that we can not define `ToInt.Mod` instances for `Int8`,
|
||||
-- because the condition does not hold unless `0 ≤ x.toInt ∨ y.toInt ∣ x.toInt ∨ y = 0`.
|
||||
|
||||
instance : ToInt.LE Int8 (some (-2^7)) (some (2^7)) where
|
||||
le_iff x y := by simpa using Int8.le_iff_toInt_le
|
||||
|
||||
instance : ToInt.LT Int8 (some (-2^7)) (some (2^7)) where
|
||||
lt_iff x y := by simpa using Int8.lt_iff_toInt_lt
|
||||
|
||||
instance : ToInt Int16 (some (-2^15)) (some (2^15)) where
|
||||
toInt x := x.toInt
|
||||
toInt_inj x y w := Int16.toInt_inj.mp w
|
||||
@@ -387,19 +248,9 @@ instance : ToInt.Add Int16 (some (-2^15)) (some (2^15)) where
|
||||
simp [Int.bmod_eq_emod]
|
||||
split <;> · simp; omega
|
||||
|
||||
instance : ToInt.Zero Int16 (some (-2^15)) (some (2^15)) where
|
||||
toInt_zero := by
|
||||
-- simp -- FIXME: succeeds, but generates a `(kernel) application type mismatch` error!
|
||||
change (0 : Int16).toInt = _
|
||||
rw [Int16.toInt_zero]
|
||||
decide
|
||||
|
||||
instance : ToInt.LE Int16 (some (-2^15)) (some (2^15)) where
|
||||
le_iff x y := by simpa using Int16.le_iff_toInt_le
|
||||
|
||||
instance : ToInt.LT Int16 (some (-2^15)) (some (2^15)) where
|
||||
lt_iff x y := by simpa using Int16.lt_iff_toInt_lt
|
||||
|
||||
instance : ToInt Int32 (some (-2^31)) (some (2^31)) where
|
||||
toInt x := x.toInt
|
||||
toInt_inj x y w := Int32.toInt_inj.mp w
|
||||
@@ -413,19 +264,9 @@ instance : ToInt.Add Int32 (some (-2^31)) (some (2^31)) where
|
||||
simp [Int.bmod_eq_emod]
|
||||
split <;> · simp; omega
|
||||
|
||||
instance : ToInt.Zero Int32 (some (-2^31)) (some (2^31)) where
|
||||
toInt_zero := by
|
||||
-- simp -- FIXME: succeeds, but generates a `(kernel) application type mismatch` error!
|
||||
change (0 : Int32).toInt = _
|
||||
rw [Int32.toInt_zero]
|
||||
decide
|
||||
|
||||
instance : ToInt.LE Int32 (some (-2^31)) (some (2^31)) where
|
||||
le_iff x y := by simpa using Int32.le_iff_toInt_le
|
||||
|
||||
instance : ToInt.LT Int32 (some (-2^31)) (some (2^31)) where
|
||||
lt_iff x y := by simpa using Int32.lt_iff_toInt_lt
|
||||
|
||||
instance : ToInt Int64 (some (-2^63)) (some (2^63)) where
|
||||
toInt x := x.toInt
|
||||
toInt_inj x y w := Int64.toInt_inj.mp w
|
||||
@@ -439,44 +280,31 @@ instance : ToInt.Add Int64 (some (-2^63)) (some (2^63)) where
|
||||
simp [Int.bmod_eq_emod]
|
||||
split <;> · simp; omega
|
||||
|
||||
instance : ToInt.Zero Int64 (some (-2^63)) (some (2^63)) where
|
||||
toInt_zero := by
|
||||
-- simp -- FIXME: succeeds, but generates a `(kernel) application type mismatch` error!
|
||||
change (0 : Int64).toInt = _
|
||||
rw [Int64.toInt_zero]
|
||||
decide
|
||||
|
||||
instance : ToInt.LE Int64 (some (-2^63)) (some (2^63)) where
|
||||
le_iff x y := by simpa using Int64.le_iff_toInt_le
|
||||
|
||||
instance : ToInt.LT Int64 (some (-2^63)) (some (2^63)) where
|
||||
lt_iff x y := by simpa using Int64.lt_iff_toInt_lt
|
||||
instance : ToInt (BitVec 0) (some 0) (some 1) where
|
||||
toInt x := 0
|
||||
toInt_inj x y w := by simp at w; exact BitVec.eq_of_zero_length rfl
|
||||
le_toInt {lo x} w := by simp at w; subst w; exact Int.zero_le_ofNat 0
|
||||
toInt_lt {hi x} w := by simp at w; subst w; exact Int.one_pos
|
||||
|
||||
instance : ToInt (BitVec v) (some 0) (some (2^v)) where
|
||||
toInt x := (x.toNat : Int)
|
||||
toInt_inj x y w :=
|
||||
BitVec.eq_of_toNat_eq (Int.ofNat_inj.mp w)
|
||||
le_toInt {lo x} w := by simp at w; subst w; exact Int.natCast_nonneg x.toNat
|
||||
toInt_lt {hi x} w := by
|
||||
simp at w; subst w;
|
||||
simpa using Int.ofNat_lt.mpr (BitVec.isLt x)
|
||||
@[simp] theorem toInt_bitVec_0 (x : BitVec 0) : ToInt.toInt x = 0 := rfl
|
||||
|
||||
@[simp] theorem toInt_bitVec (x : BitVec v) : ToInt.toInt x = (x.toNat : Int) := rfl
|
||||
instance [NeZero v] : ToInt (BitVec v) (some (-2^(v-1))) (some (2^(v-1))) where
|
||||
toInt x := x.toInt
|
||||
toInt_inj x y w := BitVec.toInt_inj.mp w
|
||||
le_toInt {lo x} w := by simp at w; subst w; exact BitVec.le_toInt x
|
||||
toInt_lt {hi x} w := by simp at w; subst w; exact BitVec.toInt_lt
|
||||
|
||||
instance : ToInt.Add (BitVec v) (some 0) (some (2^v)) where
|
||||
toInt_add x y := by simp
|
||||
@[simp] theorem toInt_bitVec [NeZero v] (x : BitVec v) : ToInt.toInt x = x.toInt := rfl
|
||||
|
||||
instance : ToInt.Zero (BitVec v) (some 0) (some (2^v)) where
|
||||
toInt_zero := by simp
|
||||
|
||||
instance : ToInt.Mod (BitVec v) (some 0) (some (2^v)) where
|
||||
toInt_mod x y := by simp
|
||||
|
||||
instance : ToInt.LE (BitVec v) (some 0) (some (2^v)) where
|
||||
le_iff x y := by simpa using BitVec.le_def
|
||||
|
||||
instance : ToInt.LT (BitVec v) (some 0) (some (2^v)) where
|
||||
lt_iff x y := by simpa using BitVec.lt_def
|
||||
instance [i : NeZero v] : ToInt.Add (BitVec v) (some (-2^(v-1))) (some (2^(v-1))) where
|
||||
toInt_add x y := by
|
||||
rw [toInt_bitVec, BitVec.toInt_add, ToInt.wrap_eq_bmod (Int.pow_nonneg (by decide))]
|
||||
have : ((2 : Int) * 2 ^ (v - 1)).toNat = 2 ^ v := by
|
||||
match v, i with | v + 1, _ => simp [← Int.pow_succ', Int.toNat_pow_of_nonneg]
|
||||
simp [this]
|
||||
|
||||
instance : ToInt ISize (some (-2^(System.Platform.numBits-1))) (some (2^(System.Platform.numBits-1))) where
|
||||
toInt x := x.toInt
|
||||
@@ -498,16 +326,4 @@ instance : ToInt.Add ISize (some (-2^(System.Platform.numBits-1))) (some (2^(Sys
|
||||
simp
|
||||
simp [p₁, p₂]
|
||||
|
||||
instance : ToInt.Zero ISize (some (-2^(System.Platform.numBits-1))) (some (2^(System.Platform.numBits-1))) where
|
||||
toInt_zero := by
|
||||
rw [toInt_isize]
|
||||
rw [ISize.toInt_zero, ToInt.wrap_eq_bmod (Int.pow_nonneg (by decide))]
|
||||
simp
|
||||
|
||||
instance instToIntLEISize : ToInt.LE ISize (some (-2^(System.Platform.numBits-1))) (some (2^(System.Platform.numBits-1))) where
|
||||
le_iff x y := by simpa using ISize.le_iff_toInt_le
|
||||
|
||||
instance instToIntLTISize : ToInt.LT ISize (some (-2^(System.Platform.numBits-1))) (some (2^(System.Platform.numBits-1))) where
|
||||
lt_iff x y := by simpa using ISize.lt_iff_toInt_lt
|
||||
|
||||
end Lean.Grind
|
||||
|
||||
@@ -57,25 +57,25 @@ theorem nestedProof_congr (p q : Prop) (h : p = q) (hp : p) (hq : q) : HEq (@nes
|
||||
subst h; apply HEq.refl
|
||||
|
||||
@[app_unexpander nestedProof]
|
||||
meta def nestedProofUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
def nestedProofUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
match stx with
|
||||
| `($_ $p:term) => `(‹$p›)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander MatchCond]
|
||||
meta def matchCondUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
def matchCondUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
match stx with
|
||||
| `($_ $p:term) => `($p)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander EqMatch]
|
||||
meta def eqMatchUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
def eqMatchUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
match stx with
|
||||
| `($_ $lhs:term $rhs:term) => `($lhs = $rhs)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander offset]
|
||||
meta def offsetUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
def offsetUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
match stx with
|
||||
| `($_ $lhs:term $rhs:term) => `($lhs + $rhs)
|
||||
| _ => throw ()
|
||||
@@ -88,7 +88,7 @@ many `NatCast.natCast` applications which is too verbose. We add the following
|
||||
unexpander to cope with this issue.
|
||||
-/
|
||||
@[app_unexpander NatCast.natCast]
|
||||
meta def natCastUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
def natCastUnexpander : PrettyPrinter.Unexpander := fun stx => do
|
||||
match stx with
|
||||
| `($_ $a:term) => `(↑$a)
|
||||
| _ => throw ()
|
||||
|
||||
@@ -363,7 +363,7 @@ theorem chain_iterates {f : α → α} (hf : monotone f) : chain (iterates f) :=
|
||||
· left; apply hf; assumption
|
||||
· right; apply hf; assumption
|
||||
case sup c hchain hi ih2 =>
|
||||
change f x ⊑ csup c ∨ csup c ⊑ f x
|
||||
show f x ⊑ csup c ∨ csup c ⊑ f x
|
||||
by_cases h : ∃ z, c z ∧ f x ⊑ z
|
||||
· left
|
||||
obtain ⟨z, hz, hfz⟩ := h
|
||||
@@ -380,7 +380,7 @@ theorem chain_iterates {f : α → α} (hf : monotone f) : chain (iterates f) :=
|
||||
next => contradiction
|
||||
next => assumption
|
||||
case sup c hchain hi ih =>
|
||||
change rel (csup c) y ∨ rel y (csup c)
|
||||
show rel (csup c) y ∨ rel y (csup c)
|
||||
by_cases h : ∃ z, c z ∧ rel y z
|
||||
· right
|
||||
obtain ⟨z, hz, hfz⟩ := h
|
||||
|
||||
@@ -13,8 +13,6 @@ import Init.MetaTypes
|
||||
import Init.Syntax
|
||||
import Init.Data.Array.GetLit
|
||||
import Init.Data.Option.BasicAux
|
||||
meta import Init.Data.Array.Basic
|
||||
meta import Init.Syntax
|
||||
|
||||
namespace Lean
|
||||
|
||||
@@ -248,7 +246,7 @@ def appendBefore (n : Name) (pre : String) : Name :=
|
||||
| num p n => Name.mkNum (Name.mkStr p pre) n
|
||||
|
||||
protected theorem beq_iff_eq {m n : Name} : m == n ↔ m = n := by
|
||||
change m.beq n ↔ _
|
||||
show m.beq n ↔ _
|
||||
induction m generalizing n <;> cases n <;> simp_all [Name.beq, And.comm]
|
||||
|
||||
instance : LawfulBEq Name where
|
||||
@@ -1612,7 +1610,7 @@ macro (name := declareSimpLikeTactic) doc?:(docComment)?
|
||||
else
|
||||
pure (← `(``dsimp), ← `("dsimp"), ← `($[$doc?:docComment]? syntax (name := $tacName) $tacToken:str optConfig (discharger)? (&" only")? (" [" (simpErase <|> simpLemma),* "]")? (location)? : tactic))
|
||||
`($stx:command
|
||||
@[macro $tacName] meta def expandSimp : Macro := fun s => do
|
||||
@[macro $tacName] def expandSimp : Macro := fun s => do
|
||||
let cfg ← `(optConfig| $cfg)
|
||||
let s := s.setKind $kind
|
||||
let s := s.setArg 0 (mkAtomFrom s[0] $tkn (canonical := true))
|
||||
|
||||
@@ -57,9 +57,8 @@ It is immediately converted to `Lean.Meta.Simp.Config` by `Lean.Elab.Tactic.elab
|
||||
-/
|
||||
structure Config where
|
||||
/--
|
||||
When `true` (default: `true`), performs zeta reduction of `let` and `have` expressions.
|
||||
When `true` (default: `true`), performs zeta reduction of let expressions.
|
||||
That is, `let x := v; e[x]` reduces to `e[v]`.
|
||||
If `zetaHave` is `false` then `have` expressions are not zeta reduced.
|
||||
See also `zetaDelta`.
|
||||
-/
|
||||
zeta : Bool := true
|
||||
@@ -108,8 +107,7 @@ structure Config where
|
||||
unfoldPartialApp : Bool := false
|
||||
/--
|
||||
When `true` (default: `false`), local definitions are unfolded.
|
||||
That is, given a local context containing `x : t := e`, then the free variable `x` reduces to `e`.
|
||||
Otherwise, `x` must be provided as a `simp` argument.
|
||||
That is, given a local context containing entry `x : t := e`, the free variable `x` reduces to `e`.
|
||||
-/
|
||||
zetaDelta : Bool := false
|
||||
/--
|
||||
@@ -118,17 +116,10 @@ structure Config where
|
||||
-/
|
||||
index : Bool := true
|
||||
/--
|
||||
When `true` (default : `true`), then `simp` will remove unused `let` and `have` expressions:
|
||||
When `true` (default : `true`), then simps will remove unused let-declarations:
|
||||
`let x := v; e` simplifies to `e` when `x` does not occur in `e`.
|
||||
-/
|
||||
zetaUnused : Bool := true
|
||||
/--
|
||||
(Unimplemented)
|
||||
When `false` (default: `true`), then disables zeta reduction of `have` expressions.
|
||||
If `zeta` is `false`, then this option has no effect.
|
||||
Unused `have`s are still removed if `zeta` or `zetaUnused` are true.
|
||||
-/
|
||||
zetaHave : Bool := true
|
||||
deriving Inhabited, BEq
|
||||
|
||||
end DSimp
|
||||
@@ -170,9 +161,8 @@ structure Config where
|
||||
-/
|
||||
singlePass : Bool := false
|
||||
/--
|
||||
When `true` (default: `true`), performs zeta reduction of `let` and `have` expressions.
|
||||
When `true` (default: `true`), performs zeta reduction of let expressions.
|
||||
That is, `let x := v; e[x]` reduces to `e[v]`.
|
||||
If `zetaHave` is `false` then `have` expressions are not zeta reduced.
|
||||
See also `zetaDelta`.
|
||||
-/
|
||||
zeta : Bool := true
|
||||
@@ -236,8 +226,7 @@ structure Config where
|
||||
unfoldPartialApp : Bool := false
|
||||
/--
|
||||
When `true` (default: `false`), local definitions are unfolded.
|
||||
That is, given a local context containing `x : t := e`, then the free variable `x` reduces to `e`.
|
||||
Otherwise, `x` must be provided as a `simp` argument.
|
||||
That is, given a local context containing entry `x : t := e`, the free variable `x` reduces to `e`.
|
||||
-/
|
||||
zetaDelta : Bool := false
|
||||
/--
|
||||
@@ -251,7 +240,7 @@ structure Config where
|
||||
-/
|
||||
implicitDefEqProofs : Bool := true
|
||||
/--
|
||||
When `true` (default : `true`), then `simp` will remove unused `let` and `have` expressions:
|
||||
When `true` (default : `true`), then simps will remove unused let-declarations:
|
||||
`let x := v; e` simplifies to `e` when `x` does not occur in `e`.
|
||||
-/
|
||||
zetaUnused : Bool := true
|
||||
@@ -260,19 +249,6 @@ structure Config where
|
||||
convert them into `simp` exceptions.
|
||||
-/
|
||||
catchRuntime : Bool := true
|
||||
/--
|
||||
(Unimplemented)
|
||||
When `false` (default: `true`), then disables zeta reduction of `have` expressions.
|
||||
If `zeta` is `false`, then this option has no effect.
|
||||
Unused `have`s are still removed if `zeta` or `zetaUnused` are true.
|
||||
-/
|
||||
zetaHave : Bool := true
|
||||
/--
|
||||
(Unimplemented)
|
||||
When `true` (default : `true`), then `simp` will attempt to transform `let`s into `have`s
|
||||
if they are non-dependent. This only applies when `zeta := false`.
|
||||
-/
|
||||
letToHave : Bool := true
|
||||
deriving Inhabited, BEq
|
||||
|
||||
-- Configuration object for `simp_all`
|
||||
@@ -294,7 +270,6 @@ def neutralConfig : Simp.Config := {
|
||||
ground := false
|
||||
zetaDelta := false
|
||||
zetaUnused := false
|
||||
letToHave := false
|
||||
}
|
||||
|
||||
structure NormCastConfig extends Simp.Config where
|
||||
|
||||
@@ -8,6 +8,7 @@ Notation for operators defined at Prelude.lean
|
||||
module
|
||||
|
||||
prelude
|
||||
import Init.Prelude
|
||||
import Init.Coe
|
||||
set_option linter.missingDocs true -- keep it documented
|
||||
|
||||
@@ -26,14 +27,14 @@ of a lean file. For example, `def foo := 1` is a `command`, as is
|
||||
`namespace Foo` and `end Foo`. Commands generally have an effect on the state of
|
||||
adding something to the environment (like a new definition), as well as
|
||||
commands like `variable` which modify future commands within a scope. -/
|
||||
meta def command : Category := {}
|
||||
def command : Category := {}
|
||||
|
||||
/-- `term` is the builtin syntax category for terms. A term denotes an expression
|
||||
in lean's type theory, for example `2 + 2` is a term. The difference between
|
||||
`Term` and `Expr` is that the former is a kind of syntax, while the latter is
|
||||
the result of elaboration. For example `by simp` is also a `Term`, but it elaborates
|
||||
to different `Expr`s depending on the context. -/
|
||||
meta def term : Category := {}
|
||||
def term : Category := {}
|
||||
|
||||
/-- `tactic` is the builtin syntax category for tactics. These appear after
|
||||
`by` in proofs, and they are programs that take in the proof context
|
||||
@@ -43,28 +44,28 @@ a term of the expected type. For example, `simp` is a tactic, used in:
|
||||
example : 2 + 2 = 4 := by simp
|
||||
```
|
||||
-/
|
||||
meta def tactic : Category := {}
|
||||
def tactic : Category := {}
|
||||
|
||||
/-- `doElem` is a builtin syntax category for elements that can appear in the `do` notation.
|
||||
For example, `let x ← e` is a `doElem`, and a `do` block consists of a list of `doElem`s. -/
|
||||
meta def doElem : Category := {}
|
||||
def doElem : Category := {}
|
||||
|
||||
/-- `structInstFieldDecl` is the syntax category for value declarations for fields in structure instance notation.
|
||||
For example, the `:= 1` and `| 0 => 0 | n + 1 => n` in `{ x := 1, f | 0 => 0 | n + 1 => n }` are in the `structInstFieldDecl` class. -/
|
||||
meta def structInstFieldDecl : Category := {}
|
||||
def structInstFieldDecl : Category := {}
|
||||
|
||||
/-- `level` is a builtin syntax category for universe levels.
|
||||
This is the `u` in `Sort u`: it can contain `max` and `imax`, addition with
|
||||
constants, and variables. -/
|
||||
meta def level : Category := {}
|
||||
def level : Category := {}
|
||||
|
||||
/-- `attr` is a builtin syntax category for attributes.
|
||||
Declarations can be annotated with attributes using the `@[...]` notation. -/
|
||||
meta def attr : Category := {}
|
||||
def attr : Category := {}
|
||||
|
||||
/-- `stx` is a builtin syntax category for syntax. This is the abbreviated
|
||||
parser notation used inside `syntax` and `macro` declarations. -/
|
||||
meta def stx : Category := {}
|
||||
def stx : Category := {}
|
||||
|
||||
/-- `prio` is a builtin syntax category for priorities.
|
||||
Priorities are used in many different attributes.
|
||||
@@ -72,7 +73,7 @@ Higher numbers denote higher priority, and for example typeclass search will
|
||||
try high priority instances before low priority.
|
||||
In addition to literals like `37`, you can also use `low`, `mid`, `high`, as well as
|
||||
add and subtract priorities. -/
|
||||
meta def prio : Category := {}
|
||||
def prio : Category := {}
|
||||
|
||||
/-- `prec` is a builtin syntax category for precedences. A precedence is a value
|
||||
that expresses how tightly a piece of syntax binds: for example `1 + 2 * 3` is
|
||||
@@ -83,7 +84,7 @@ In addition to literals like `37`, there are some special named precedence level
|
||||
* `max` for the highest precedence used in term parsers (not actually the maximum possible value)
|
||||
* `lead` for the precedence of terms not supposed to be used as arguments
|
||||
and you can also add and subtract precedences. -/
|
||||
meta def prec : Category := {}
|
||||
def prec : Category := {}
|
||||
|
||||
end Parser.Category
|
||||
|
||||
|
||||
@@ -9,11 +9,10 @@ module
|
||||
|
||||
prelude
|
||||
import Init.Data.ToString.Basic
|
||||
import Init.Data.Array.Subarray
|
||||
import Init.Conv
|
||||
import Init.Meta
|
||||
import Init.While
|
||||
meta import Init.Data.Option.Basic
|
||||
meta import Init.Data.Array.Subarray
|
||||
|
||||
namespace Lean
|
||||
|
||||
@@ -143,13 +142,13 @@ macro tk:"calc" steps:calcSteps : conv =>
|
||||
`(conv| tactic => calc%$tk $steps)
|
||||
end Lean
|
||||
|
||||
@[app_unexpander Unit.unit] meta def unexpandUnit : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Unit.unit] def unexpandUnit : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_)) => `(())
|
||||
|
||||
@[app_unexpander List.nil] meta def unexpandListNil : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander List.nil] def unexpandListNil : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_)) => `([])
|
||||
|
||||
@[app_unexpander List.cons] meta def unexpandListCons : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander List.cons] def unexpandListCons : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $x $tail) =>
|
||||
match tail with
|
||||
| `([]) => `([$x])
|
||||
@@ -158,105 +157,105 @@ end Lean
|
||||
| _ => throw ()
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander List.toArray] meta def unexpandListToArray : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander List.toArray] def unexpandListToArray : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) [$xs,*]) => `(#[$xs,*])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Prod.mk] meta def unexpandProdMk : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Prod.mk] def unexpandProdMk : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $x ($y, $ys,*)) => `(($x, $y, $ys,*))
|
||||
| `($(_) $x $y) => `(($x, $y))
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander ite] meta def unexpandIte : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander ite] def unexpandIte : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $c $t $e) => `(if $c then $t else $e)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Eq.ndrec] meta def unexpandEqNDRec : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Eq.ndrec] def unexpandEqNDRec : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $m $h) => `($h ▸ $m)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Eq.rec] meta def unexpandEqRec : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Eq.rec] def unexpandEqRec : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $m $h) => `($h ▸ $m)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Exists] meta def unexpandExists : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Exists] def unexpandExists : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) fun $x:ident => ∃ $xs:binderIdent*, $b) => `(∃ $x:ident $xs:binderIdent*, $b)
|
||||
| `($(_) fun $x:ident => $b) => `(∃ $x:ident, $b)
|
||||
| `($(_) fun ($x:ident : $t) => $b) => `(∃ ($x:ident : $t), $b)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Sigma] meta def unexpandSigma : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Sigma] def unexpandSigma : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) fun ($x:ident : $t) => $b) => `(($x:ident : $t) × $b)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander PSigma] meta def unexpandPSigma : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander PSigma] def unexpandPSigma : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) fun ($x:ident : $t) => $b) => `(($x:ident : $t) ×' $b)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Subtype] meta def unexpandSubtype : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Subtype] def unexpandSubtype : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) fun ($x:ident : $type) => $p) => `({ $x : $type // $p })
|
||||
| `($(_) fun $x:ident => $p) => `({ $x // $p })
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander TSyntax] meta def unexpandTSyntax : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander TSyntax] def unexpandTSyntax : Lean.PrettyPrinter.Unexpander
|
||||
| `($f [$k]) => `($f $k)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander TSyntaxArray] meta def unexpandTSyntaxArray : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander TSyntaxArray] def unexpandTSyntaxArray : Lean.PrettyPrinter.Unexpander
|
||||
| `($f [$k]) => `($f $k)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Syntax.TSepArray] meta def unexpandTSepArray : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Syntax.TSepArray] def unexpandTSepArray : Lean.PrettyPrinter.Unexpander
|
||||
| `($f [$k] $sep) => `($f $k $sep)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander GetElem.getElem] meta def unexpandGetElem : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander GetElem.getElem] def unexpandGetElem : Lean.PrettyPrinter.Unexpander
|
||||
| `($_ $array $index $_) => `($array[$index])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander getElem!] meta def unexpandGetElem! : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander getElem!] def unexpandGetElem! : Lean.PrettyPrinter.Unexpander
|
||||
| `($_ $array $index) => `($array[$index]!)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander getElem?] meta def unexpandGetElem? : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander getElem?] def unexpandGetElem? : Lean.PrettyPrinter.Unexpander
|
||||
| `($_ $array $index) => `($array[$index]?)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.empty] meta def unexpandArrayEmpty : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.empty] def unexpandArrayEmpty : Lean.PrettyPrinter.Unexpander
|
||||
| _ => `(#[])
|
||||
|
||||
@[app_unexpander Array.mkArray0] meta def unexpandMkArray0 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray0] def unexpandMkArray0 : Lean.PrettyPrinter.Unexpander
|
||||
| _ => `(#[])
|
||||
|
||||
@[app_unexpander Array.mkArray1] meta def unexpandMkArray1 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray1] def unexpandMkArray1 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1) => `(#[$a1])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray2] meta def unexpandMkArray2 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray2] def unexpandMkArray2 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2) => `(#[$a1, $a2])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray3] meta def unexpandMkArray3 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray3] def unexpandMkArray3 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2 $a3) => `(#[$a1, $a2, $a3])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray4] meta def unexpandMkArray4 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray4] def unexpandMkArray4 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2 $a3 $a4) => `(#[$a1, $a2, $a3, $a4])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray5] meta def unexpandMkArray5 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray5] def unexpandMkArray5 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2 $a3 $a4 $a5) => `(#[$a1, $a2, $a3, $a4, $a5])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray6] meta def unexpandMkArray6 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray6] def unexpandMkArray6 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2 $a3 $a4 $a5 $a6) => `(#[$a1, $a2, $a3, $a4, $a5, $a6])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray7] meta def unexpandMkArray7 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray7] def unexpandMkArray7 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2 $a3 $a4 $a5 $a6 $a7) => `(#[$a1, $a2, $a3, $a4, $a5, $a6, $a7])
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.mkArray8] meta def unexpandMkArray8 : Lean.PrettyPrinter.Unexpander
|
||||
@[app_unexpander Array.mkArray8] def unexpandMkArray8 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1 $a2 $a3 $a4 $a5 $a6 $a7 $a8) => `(#[$a1, $a2, $a3, $a4, $a5, $a6, $a7, $a8])
|
||||
| _ => throw ()
|
||||
|
||||
@@ -361,13 +360,13 @@ namespace Lean
|
||||
|
||||
/-- Unexpander for the `{ x }` notation. -/
|
||||
@[app_unexpander singleton]
|
||||
meta def singletonUnexpander : Lean.PrettyPrinter.Unexpander
|
||||
def singletonUnexpander : Lean.PrettyPrinter.Unexpander
|
||||
| `($_ $a) => `({ $a:term })
|
||||
| _ => throw ()
|
||||
|
||||
/-- Unexpander for the `{ x, y, ... }` notation. -/
|
||||
@[app_unexpander insert]
|
||||
meta def insertUnexpander : Lean.PrettyPrinter.Unexpander
|
||||
def insertUnexpander : Lean.PrettyPrinter.Unexpander
|
||||
| `($_ $a { $ts:term,* }) => `({$a:term, $ts,*})
|
||||
| _ => throw ()
|
||||
|
||||
|
||||
@@ -74,28 +74,6 @@ theorem let_body_congr {α : Sort u} {β : α → Sort v} {b b' : (a : α) →
|
||||
(a : α) (h : ∀ x, b x = b' x) : (let x := a; b x) = (let x := a; b' x) :=
|
||||
(funext h : b = b') ▸ rfl
|
||||
|
||||
theorem have_unused {α : Sort u} {β : Sort v} (a : α) {b b' : β}
|
||||
(h : b = b') : (have _ := a; b) = b' := h
|
||||
|
||||
theorem have_unused_dep {α : Sort u} {β : Sort v} (a : α) {b : α → β} {b' : β}
|
||||
(h : ∀ x, b x = b') : (have x := a; b x) = b' := h a
|
||||
|
||||
theorem have_congr {α : Sort u} {β : Sort v} {a a' : α} {f f' : α → β}
|
||||
(h₁ : a = a') (h₂ : ∀ x, f x = f' x) : (have x := a; f x) = (have x := a'; f' x) :=
|
||||
@congr α β f f' a a' (funext h₂) h₁
|
||||
|
||||
theorem have_val_congr {α : Sort u} {β : Sort v} {a a' : α} {f : α → β}
|
||||
(h : a = a') : (have x := a; f x) = (have x := a'; f x) :=
|
||||
@congrArg α β a a' f h
|
||||
|
||||
theorem have_body_congr_dep {α : Sort u} {β : α → Sort v} (a : α) {f f' : (x : α) → β x}
|
||||
(h : ∀ x, f x = f' x) : (have x := a; f x) = (have x := a; f' x) :=
|
||||
h a
|
||||
|
||||
theorem have_body_congr {α : Sort u} {β : Sort v} (a : α) {f f' : α → β}
|
||||
(h : ∀ x, f x = f' x) : (have x := a; f x) = (have x := a; f' x) :=
|
||||
h a
|
||||
|
||||
theorem letFun_unused {α : Sort u} {β : Sort v} (a : α) {b b' : β} (h : b = b') : @letFun α (fun _ => β) a (fun _ => b) = b' :=
|
||||
h
|
||||
|
||||
|
||||
@@ -149,7 +149,7 @@ Because the resulting value is treated as a side-effect-free term, the compiler
|
||||
duplicate, or delete calls to this function. The side effect may even be hoisted into a constant,
|
||||
causing the side effect to occur at initialization time, even if it would otherwise never be called.
|
||||
-/
|
||||
@[noinline] unsafe def unsafeBaseIO (fn : BaseIO α) : α :=
|
||||
@[inline] unsafe def unsafeBaseIO (fn : BaseIO α) : α :=
|
||||
match fn.run () with
|
||||
| EStateM.Result.ok a _ => a
|
||||
|
||||
@@ -567,7 +567,7 @@ def addHeartbeats (count : Nat) : BaseIO Unit := do
|
||||
/--
|
||||
Whether a file should be opened for reading, writing, creation and writing, or appending.
|
||||
|
||||
At the operating system level, this translates to the mode of a file handle (i.e., a set of `open`
|
||||
A the operating system level, this translates to the mode of a file handle (i.e., a set of `open`
|
||||
flags and an `fdopen` mode).
|
||||
|
||||
None of the modes represented by this datatype translate line endings (i.e. `O_BINARY` on Windows).
|
||||
|
||||
@@ -535,12 +535,6 @@ syntax (name := change) "change " term (location)? : tactic
|
||||
-/
|
||||
syntax (name := changeWith) "change " term " with " term (location)? : tactic
|
||||
|
||||
/--
|
||||
`show t` finds the first goal whose target unifies with `t`. It makes that the main goal,
|
||||
performs the unification, and replaces the target with the unified version of `t`.
|
||||
-/
|
||||
syntax (name := «show») "show " term : tactic
|
||||
|
||||
/--
|
||||
Extracts `let` and `let_fun` expressions from within the target or a local hypothesis,
|
||||
introducing new local definitions.
|
||||
@@ -701,7 +695,7 @@ syntax (name := dsimp) "dsimp" optConfig (discharger)? (&" only")?
|
||||
A `simpArg` is either a `*`, `-lemma` or a simp lemma specification
|
||||
(which includes the `↑` `↓` `←` specifications for pre, post, reverse rewriting).
|
||||
-/
|
||||
meta def simpArg := simpStar.binary `orelse (simpErase.binary `orelse simpLemma)
|
||||
def simpArg := simpStar.binary `orelse (simpErase.binary `orelse simpLemma)
|
||||
|
||||
/-- A simp args list is a list of `simpArg`. This is the main argument to `simp`. -/
|
||||
syntax simpArgs := " [" simpArg,* "]"
|
||||
@@ -710,7 +704,7 @@ syntax simpArgs := " [" simpArg,* "]"
|
||||
A `dsimpArg` is similar to `simpArg`, but it does not have the `simpStar` form
|
||||
because it does not make sense to use hypotheses in `dsimp`.
|
||||
-/
|
||||
meta def dsimpArg := simpErase.binary `orelse simpLemma
|
||||
def dsimpArg := simpErase.binary `orelse simpLemma
|
||||
|
||||
/-- A dsimp args list is a list of `dsimpArg`. This is the main argument to `dsimp`. -/
|
||||
syntax dsimpArgs := " [" dsimpArg,* "]"
|
||||
@@ -870,6 +864,11 @@ The `let` tactic is for adding definitions to the local context of the main goal
|
||||
local variables `x : α`, `y : β`, and `z : γ`.
|
||||
-/
|
||||
macro "let " d:letDecl : tactic => `(tactic| refine_lift let $d:letDecl; ?_)
|
||||
/--
|
||||
`show t` finds the first goal whose target unifies with `t`. It makes that the main goal,
|
||||
performs the unification, and replaces the target with the unified version of `t`.
|
||||
-/
|
||||
macro "show " e:term : tactic => `(tactic| refine_lift show $e from ?_) -- TODO: fix, see comment
|
||||
/-- `let rec f : t := e` adds a recursive definition `f` to the current goal.
|
||||
The syntax is the same as term-mode `let rec`. -/
|
||||
syntax (name := letrec) withPosition(atomic("let " &"rec ") letRecDecls) : tactic
|
||||
@@ -1918,35 +1917,30 @@ syntax "‹" withoutPosition(term) "›" : term
|
||||
macro_rules | `(‹$type›) => `((by assumption : $type))
|
||||
|
||||
/--
|
||||
`get_elem_tactic_extensible` is an extensible tactic automatically called
|
||||
`get_elem_tactic_trivial` is an extensible tactic automatically called
|
||||
by the notation `arr[i]` to prove any side conditions that arise when
|
||||
constructing the term (e.g. the index is in bounds of the array).
|
||||
The default behavior is to try `simp +arith` and `omega`
|
||||
The default behavior is to just try `trivial` (which handles the case
|
||||
where `i < arr.size` is in the context) and `simp +arith` and `omega`
|
||||
(for doing linear arithmetic in the index).
|
||||
|
||||
(Note that the core tactic `get_elem_tactic` has already tried
|
||||
`done` and `assumption` before the extensible tactic is called.)
|
||||
-/
|
||||
syntax "get_elem_tactic_extensible" : tactic
|
||||
|
||||
/-- `get_elem_tactic_trivial` has been deprecated in favour of `get_elem_tactic_extensible`. -/
|
||||
syntax "get_elem_tactic_trivial" : tactic
|
||||
|
||||
macro_rules | `(tactic| get_elem_tactic_extensible) => `(tactic| omega)
|
||||
macro_rules | `(tactic| get_elem_tactic_extensible) => `(tactic| simp +arith; done)
|
||||
macro_rules | `(tactic| get_elem_tactic_extensible) => `(tactic| trivial)
|
||||
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| omega)
|
||||
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| simp +arith; done)
|
||||
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| trivial)
|
||||
|
||||
/--
|
||||
`get_elem_tactic` is the tactic automatically called by the notation `arr[i]`
|
||||
to prove any side conditions that arise when constructing the term
|
||||
(e.g. the index is in bounds of the array). It just delegates to
|
||||
`get_elem_tactic_extensible` and gives a diagnostic error message otherwise;
|
||||
users are encouraged to extend `get_elem_tactic_extensible` instead of this tactic.
|
||||
`get_elem_tactic_trivial` and gives a diagnostic error message otherwise;
|
||||
users are encouraged to extend `get_elem_tactic_trivial` instead of this tactic.
|
||||
-/
|
||||
macro "get_elem_tactic" : tactic =>
|
||||
`(tactic| first
|
||||
/-
|
||||
Recall that `macro_rules` (namely, for `get_elem_tactic_extensible`) are tried in reverse order.
|
||||
Recall that `macro_rules` (namely, for `get_elem_tactic_trivial`) are tried in reverse order.
|
||||
We first, however, try `done`, since the necessary proof may already have been
|
||||
found during unification, in which case there is no goal to solve (see #6999).
|
||||
If a goal is present, we want `assumption` to be tried first.
|
||||
@@ -1959,15 +1953,15 @@ macro "get_elem_tactic" : tactic =>
|
||||
If `omega` is used to "fill" this proof, we will have a more complex proof term that
|
||||
cannot be inferred by unification.
|
||||
We hardcoded `assumption` here to ensure users cannot accidentally break this IF
|
||||
they add new `macro_rules` for `get_elem_tactic_extensible`.
|
||||
they add new `macro_rules` for `get_elem_tactic_trivial`.
|
||||
|
||||
TODO: Implement priorities for `macro_rules`.
|
||||
TODO: Ensure we have **high-priority** macro_rules for `get_elem_tactic_extensible` which are
|
||||
TODO: Ensure we have **high-priority** macro_rules for `get_elem_tactic_trivial` which are
|
||||
just `done` and `assumption`.
|
||||
-/
|
||||
| done
|
||||
| assumption
|
||||
| get_elem_tactic_extensible
|
||||
| get_elem_tactic_trivial
|
||||
| fail "failed to prove index is valid, possible solutions:
|
||||
- Use `have`-expressions to prove the index is valid
|
||||
- Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid
|
||||
|
||||
@@ -41,6 +41,3 @@ import Lean.PrivateName
|
||||
import Lean.PremiseSelection
|
||||
import Lean.Namespace
|
||||
import Lean.EnvExtension
|
||||
import Lean.ErrorExplanation
|
||||
import Lean.DefEqAttrib
|
||||
import Lean.Shell
|
||||
|
||||
@@ -64,6 +64,13 @@ Checks whether the declaration was originally declared as a theorem; see also
|
||||
def wasOriginallyTheorem (env : Environment) (declName : Name) : Bool :=
|
||||
getOriginalConstKind? env declName |>.map (· matches .thm) |>.getD false
|
||||
|
||||
-- HACK: remove together with MutualDef HACK when `[dsimp]` is introduced
|
||||
private def isSimpleRflProof (proof : Expr) : Bool :=
|
||||
if let .lam _ _ proof _ := proof then
|
||||
isSimpleRflProof proof
|
||||
else
|
||||
proof.isAppOfArity ``rfl 2
|
||||
|
||||
private def looksLikeRelevantTheoremProofType (type : Expr) : Bool :=
|
||||
if let .forallE _ _ type _ := type then
|
||||
looksLikeRelevantTheoremProofType type
|
||||
@@ -83,6 +90,9 @@ def addDecl (decl : Declaration) : CoreM Unit := do
|
||||
let (name, info, kind) ← match decl with
|
||||
| .thmDecl thm =>
|
||||
let exportProof := !(← getEnv).header.isModule ||
|
||||
-- We should preserve rfl theorems but also we should not override a decision to hide by the
|
||||
-- MutualDef elaborator via `withoutExporting`
|
||||
(← getEnv).isExporting && isSimpleRflProof thm.value ||
|
||||
-- TODO: this is horrible...
|
||||
looksLikeRelevantTheoremProofType thm.type
|
||||
if !exportProof then
|
||||
|
||||
Some files were not shown because too many files have changed in this diff Show More
Reference in New Issue
Block a user