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https://github.com/leanprover/lean4.git
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| Author | SHA1 | Date | |
|---|---|---|---|
|
2513be6a09 |
4
.github/ISSUE_TEMPLATE/bug_report.md
vendored
4
.github/ISSUE_TEMPLATE/bug_report.md
vendored
@@ -25,7 +25,7 @@ Please put an X between the brackets as you perform the following steps:
|
||||
|
||||
### Context
|
||||
|
||||
[Broader context that the issue occurred in. If there was any prior discussion on [the Lean Zulip](https://leanprover.zulipchat.com), link it here as well.]
|
||||
[Broader context that the issue occured in. If there was any prior discussion on [the Lean Zulip](https://leanprover.zulipchat.com), link it here as well.]
|
||||
|
||||
### Steps to Reproduce
|
||||
|
||||
@@ -39,7 +39,7 @@ Please put an X between the brackets as you perform the following steps:
|
||||
|
||||
### Versions
|
||||
|
||||
[Output of `#version` or `#eval Lean.versionString`]
|
||||
[Output of `#eval Lean.versionString`]
|
||||
[OS version, if not using live.lean-lang.org.]
|
||||
|
||||
### Additional Information
|
||||
|
||||
8
.github/PULL_REQUEST_TEMPLATE.md
vendored
8
.github/PULL_REQUEST_TEMPLATE.md
vendored
@@ -5,10 +5,6 @@
|
||||
* Include the link to your `RFC` or `bug` issue in the description.
|
||||
* If the issue does not already have approval from a developer, submit the PR as draft.
|
||||
* The PR title/description will become the commit message. Keep it up-to-date as the PR evolves.
|
||||
* For `feat/fix` PRs, the first paragraph starting with "This PR" must be present and will become a
|
||||
changelog entry unless the PR is labeled with `no-changelog`. If the PR does not have this label,
|
||||
it must instead be categorized with one of the `changelog-*` labels (which will be done by a
|
||||
reviewer for external PRs).
|
||||
* A toolchain of the form `leanprover/lean4-pr-releases:pr-release-NNNN` for Linux and M-series Macs will be generated upon build. To generate binaries for Windows and Intel-based Macs as well, write a comment containing `release-ci` on its own line.
|
||||
* If you rebase your PR onto `nightly-with-mathlib` then CI will test Mathlib against your PR.
|
||||
* You can manage the `awaiting-review`, `awaiting-author`, and `WIP` labels yourself, by writing a comment containing one of these labels on its own line.
|
||||
@@ -16,6 +12,4 @@
|
||||
|
||||
---
|
||||
|
||||
This PR <short changelog summary for feat/fix, see above>.
|
||||
|
||||
Closes <`RFC` or `bug` issue number fixed by this PR, if any>
|
||||
Closes #0000 (`RFC` or `bug` issue number fixed by this PR, if any)
|
||||
|
||||
8
.github/dependabot.yml
vendored
8
.github/dependabot.yml
vendored
@@ -1,8 +0,0 @@
|
||||
version: 2
|
||||
updates:
|
||||
- package-ecosystem: "github-actions"
|
||||
directory: "/"
|
||||
schedule:
|
||||
interval: "monthly"
|
||||
commit-message:
|
||||
prefix: "chore: CI"
|
||||
2
.github/workflows/actionlint.yml
vendored
2
.github/workflows/actionlint.yml
vendored
@@ -17,6 +17,6 @@ jobs:
|
||||
- name: Checkout
|
||||
uses: actions/checkout@v4
|
||||
- name: actionlint
|
||||
uses: raven-actions/actionlint@v2
|
||||
uses: raven-actions/actionlint@v1
|
||||
with:
|
||||
pyflakes: false # we do not use python scripts
|
||||
|
||||
8
.github/workflows/check-prelude.yml
vendored
8
.github/workflows/check-prelude.yml
vendored
@@ -11,9 +11,7 @@ jobs:
|
||||
with:
|
||||
# the default is to use a virtual merge commit between the PR and master: just use the PR
|
||||
ref: ${{ github.event.pull_request.head.sha }}
|
||||
sparse-checkout: |
|
||||
src/Lean
|
||||
src/Std
|
||||
sparse-checkout: src/Lean
|
||||
- name: Check Prelude
|
||||
run: |
|
||||
failed_files=""
|
||||
@@ -21,8 +19,8 @@ jobs:
|
||||
if ! grep -q "^prelude$" "$file"; then
|
||||
failed_files="$failed_files$file\n"
|
||||
fi
|
||||
done < <(find src/Lean src/Std -name '*.lean' -print0)
|
||||
done < <(find src/Lean -name '*.lean' -print0)
|
||||
if [ -n "$failed_files" ]; then
|
||||
echo -e "The following files should use 'prelude':\n$failed_files"
|
||||
exit 1
|
||||
fi
|
||||
fi
|
||||
16
.github/workflows/ci.yml
vendored
16
.github/workflows/ci.yml
vendored
@@ -217,7 +217,7 @@ jobs:
|
||||
"release": true,
|
||||
"check-level": 2,
|
||||
"shell": "msys2 {0}",
|
||||
"CMAKE_OPTIONS": "-G \"Unix Makefiles\"",
|
||||
"CMAKE_OPTIONS": "-G \"Unix Makefiles\" -DUSE_GMP=OFF",
|
||||
// for reasons unknown, interactivetests are flaky on Windows
|
||||
"CTEST_OPTIONS": "--repeat until-pass:2",
|
||||
"llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-w64-windows-gnu.tar.zst",
|
||||
@@ -227,7 +227,7 @@ jobs:
|
||||
{
|
||||
"name": "Linux aarch64",
|
||||
"os": "nscloud-ubuntu-22.04-arm64-4x8",
|
||||
"CMAKE_OPTIONS": "-DLEAN_INSTALL_SUFFIX=-linux_aarch64",
|
||||
"CMAKE_OPTIONS": "-DUSE_GMP=OFF -DLEAN_INSTALL_SUFFIX=-linux_aarch64",
|
||||
"release": true,
|
||||
"check-level": 2,
|
||||
"shell": "nix develop .#oldGlibcAArch -c bash -euxo pipefail {0}",
|
||||
@@ -257,7 +257,7 @@ jobs:
|
||||
"cross": true,
|
||||
"shell": "bash -euxo pipefail {0}",
|
||||
// Just a few selected tests because wasm is slow
|
||||
"CTEST_OPTIONS": "-R \"leantest_1007\\.lean|leantest_Format\\.lean|leanruntest\\_1037.lean|leanruntest_ac_rfl\\.lean|leanruntest_tempfile.lean\\.|leanruntest_libuv\\.lean\""
|
||||
"CTEST_OPTIONS": "-R \"leantest_1007\\.lean|leantest_Format\\.lean|leanruntest\\_1037.lean|leanruntest_ac_rfl\\.lean|leanruntest_libuv\\.lean\""
|
||||
}
|
||||
];
|
||||
console.log(`matrix:\n${JSON.stringify(matrix, null, 2)}`)
|
||||
@@ -316,9 +316,9 @@ jobs:
|
||||
git fetch --depth=1 origin ${{ github.sha }}
|
||||
git checkout FETCH_HEAD flake.nix flake.lock
|
||||
if: github.event_name == 'pull_request'
|
||||
# (needs to be after "Checkout" so files don't get overridden)
|
||||
# (needs to be after "Checkout" so files don't get overriden)
|
||||
- name: Setup emsdk
|
||||
uses: mymindstorm/setup-emsdk@v14
|
||||
uses: mymindstorm/setup-emsdk@v12
|
||||
with:
|
||||
version: 3.1.44
|
||||
actions-cache-folder: emsdk
|
||||
@@ -452,7 +452,7 @@ jobs:
|
||||
run: ccache -s
|
||||
|
||||
# This job collects results from all the matrix jobs
|
||||
# This can be made the "required" job, instead of listing each
|
||||
# This can be made the “required” job, instead of listing each
|
||||
# matrix job separately
|
||||
all-done:
|
||||
name: Build matrix complete
|
||||
@@ -492,7 +492,7 @@ jobs:
|
||||
with:
|
||||
path: artifacts
|
||||
- name: Release
|
||||
uses: softprops/action-gh-release@v2
|
||||
uses: softprops/action-gh-release@v1
|
||||
with:
|
||||
files: artifacts/*/*
|
||||
fail_on_unmatched_files: true
|
||||
@@ -536,7 +536,7 @@ jobs:
|
||||
echo -e "\n*Full commit log*\n" >> diff.md
|
||||
git log --oneline "$last_tag"..HEAD | sed 's/^/* /' >> diff.md
|
||||
- name: Release Nightly
|
||||
uses: softprops/action-gh-release@v2
|
||||
uses: softprops/action-gh-release@v1
|
||||
with:
|
||||
body_path: diff.md
|
||||
prerelease: true
|
||||
|
||||
24
.github/workflows/labels-from-comments.yml
vendored
24
.github/workflows/labels-from-comments.yml
vendored
@@ -1,8 +1,7 @@
|
||||
# This workflow allows any user to add one of the `awaiting-review`, `awaiting-author`, `WIP`,
|
||||
# `release-ci`, or a `changelog-XXX` label by commenting on the PR or issue.
|
||||
# or `release-ci` labels by commenting on the PR or issue.
|
||||
# If any labels from the set {`awaiting-review`, `awaiting-author`, `WIP`} are added, other labels
|
||||
# from that set are removed automatically at the same time.
|
||||
# Similarly, if any `changelog-XXX` label is added, other `changelog-YYY` labels are removed.
|
||||
|
||||
name: Label PR based on Comment
|
||||
|
||||
@@ -12,7 +11,7 @@ on:
|
||||
|
||||
jobs:
|
||||
update-label:
|
||||
if: github.event.issue.pull_request != null && (contains(github.event.comment.body, 'awaiting-review') || contains(github.event.comment.body, 'awaiting-author') || contains(github.event.comment.body, 'WIP') || contains(github.event.comment.body, 'release-ci') || contains(github.event.comment.body, 'changelog-'))
|
||||
if: github.event.issue.pull_request != null && (contains(github.event.comment.body, 'awaiting-review') || contains(github.event.comment.body, 'awaiting-author') || contains(github.event.comment.body, 'WIP') || contains(github.event.comment.body, 'release-ci'))
|
||||
runs-on: ubuntu-latest
|
||||
|
||||
steps:
|
||||
@@ -21,14 +20,13 @@ jobs:
|
||||
with:
|
||||
github-token: ${{ secrets.GITHUB_TOKEN }}
|
||||
script: |
|
||||
const { owner, repo, number: issue_number } = context.issue;
|
||||
const { owner, repo, number: issue_number } = context.issue;
|
||||
const commentLines = context.payload.comment.body.split('\r\n');
|
||||
|
||||
const awaitingReview = commentLines.includes('awaiting-review');
|
||||
const awaitingAuthor = commentLines.includes('awaiting-author');
|
||||
const wip = commentLines.includes('WIP');
|
||||
const releaseCI = commentLines.includes('release-ci');
|
||||
const changelogMatch = commentLines.find(line => line.startsWith('changelog-'));
|
||||
|
||||
if (awaitingReview || awaitingAuthor || wip) {
|
||||
await github.rest.issues.removeLabel({ owner, repo, issue_number, name: 'awaiting-review' }).catch(() => {});
|
||||
@@ -49,19 +47,3 @@ jobs:
|
||||
if (releaseCI) {
|
||||
await github.rest.issues.addLabels({ owner, repo, issue_number, labels: ['release-ci'] });
|
||||
}
|
||||
|
||||
if (changelogMatch) {
|
||||
const changelogLabel = changelogMatch.trim();
|
||||
const { data: existingLabels } = await github.rest.issues.listLabelsOnIssue({ owner, repo, issue_number });
|
||||
const changelogLabels = existingLabels.filter(label => label.name.startsWith('changelog-'));
|
||||
|
||||
// Remove all other changelog labels
|
||||
for (const label of changelogLabels) {
|
||||
if (label.name !== changelogLabel) {
|
||||
await github.rest.issues.removeLabel({ owner, repo, issue_number, name: label.name }).catch(() => {});
|
||||
}
|
||||
}
|
||||
|
||||
// Add the new changelog label
|
||||
await github.rest.issues.addLabels({ owner, repo, issue_number, labels: [changelogLabel] });
|
||||
}
|
||||
|
||||
12
.github/workflows/nix-ci.yml
vendored
12
.github/workflows/nix-ci.yml
vendored
@@ -96,7 +96,7 @@ jobs:
|
||||
nix build $NIX_BUILD_ARGS .#cacheRoots -o push-build
|
||||
- name: Test
|
||||
run: |
|
||||
nix build --keep-failed $NIX_BUILD_ARGS .#test -o push-test || (ln -s /tmp/nix-build-*/build/source/src/build ./push-test; false)
|
||||
nix build --keep-failed $NIX_BUILD_ARGS .#test -o push-test || (ln -s /tmp/nix-build-*/source/src/build/ ./push-test; false)
|
||||
- name: Test Summary
|
||||
uses: test-summary/action@v2
|
||||
with:
|
||||
@@ -110,6 +110,14 @@ jobs:
|
||||
# https://github.com/netlify/cli/issues/1809
|
||||
cp -r --dereference ./result ./dist
|
||||
if: matrix.name == 'Nix Linux'
|
||||
- name: Check manual for broken links
|
||||
id: lychee
|
||||
uses: lycheeverse/lychee-action@v1.9.0
|
||||
with:
|
||||
fail: false # report errors but do not block CI on temporary failures
|
||||
# gmplib.org consistently times out from GH actions
|
||||
# the GitHub token is to avoid rate limiting
|
||||
args: --base './dist' --no-progress --github-token ${{ secrets.GITHUB_TOKEN }} --exclude 'gmplib.org' './dist/**/*.html'
|
||||
- name: Rebuild Nix Store Cache
|
||||
run: |
|
||||
rm -rf nix-store-cache || true
|
||||
@@ -121,7 +129,7 @@ jobs:
|
||||
python3 -c 'import base64; print("alias="+base64.urlsafe_b64encode(bytes.fromhex("${{github.sha}}")).decode("utf-8").rstrip("="))' >> "$GITHUB_OUTPUT"
|
||||
echo "message=`git log -1 --pretty=format:"%s"`" >> "$GITHUB_OUTPUT"
|
||||
- name: Publish manual to Netlify
|
||||
uses: nwtgck/actions-netlify@v3.0
|
||||
uses: nwtgck/actions-netlify@v2.0
|
||||
id: publish-manual
|
||||
with:
|
||||
publish-dir: ./dist
|
||||
|
||||
25
.github/workflows/pr-body.yml
vendored
25
.github/workflows/pr-body.yml
vendored
@@ -1,25 +0,0 @@
|
||||
name: Check PR body for changelog convention
|
||||
|
||||
on:
|
||||
merge_group:
|
||||
pull_request:
|
||||
types: [opened, synchronize, reopened, edited, labeled, converted_to_draft, ready_for_review]
|
||||
|
||||
jobs:
|
||||
check-pr-body:
|
||||
runs-on: ubuntu-latest
|
||||
steps:
|
||||
- name: Check PR body
|
||||
if: github.event_name == 'pull_request'
|
||||
uses: actions/github-script@v7
|
||||
with:
|
||||
script: |
|
||||
const { title, body, labels, draft } = context.payload.pull_request;
|
||||
if (!draft && /^(feat|fix):/.test(title) && !labels.some(label => label.name == "changelog-no")) {
|
||||
if (!labels.some(label => label.name.startsWith("changelog-"))) {
|
||||
core.setFailed('feat/fix PR must have a `changelog-*` label');
|
||||
}
|
||||
if (!/^This PR [^<]/.test(body)) {
|
||||
core.setFailed('feat/fix PR must have changelog summary starting with "This PR ..." as first line.');
|
||||
}
|
||||
}
|
||||
30
.github/workflows/pr-release.yml
vendored
30
.github/workflows/pr-release.yml
vendored
@@ -34,7 +34,7 @@ jobs:
|
||||
- name: Download artifact from the previous workflow.
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
id: download-artifact
|
||||
uses: dawidd6/action-download-artifact@v6 # https://github.com/marketplace/actions/download-workflow-artifact
|
||||
uses: dawidd6/action-download-artifact@v2 # https://github.com/marketplace/actions/download-workflow-artifact
|
||||
with:
|
||||
run_id: ${{ github.event.workflow_run.id }}
|
||||
path: artifacts
|
||||
@@ -60,7 +60,7 @@ jobs:
|
||||
GH_TOKEN: ${{ secrets.PR_RELEASES_TOKEN }}
|
||||
- name: Release
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: softprops/action-gh-release@v2
|
||||
uses: softprops/action-gh-release@v1
|
||||
with:
|
||||
name: Release for PR ${{ steps.workflow-info.outputs.pullRequestNumber }}
|
||||
# There are coredumps files here as well, but all in deeper subdirectories.
|
||||
@@ -75,7 +75,7 @@ jobs:
|
||||
|
||||
- name: Report release status
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: actions/github-script@v7
|
||||
uses: actions/github-script@v6
|
||||
with:
|
||||
script: |
|
||||
await github.rest.repos.createCommitStatus({
|
||||
@@ -111,7 +111,7 @@ jobs:
|
||||
|
||||
- name: 'Setup jq'
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' }}
|
||||
uses: dcarbone/install-jq-action@v2.1.0
|
||||
uses: dcarbone/install-jq-action@v1.0.1
|
||||
|
||||
# Check that the most recently nightly coincides with 'git merge-base HEAD master'
|
||||
- name: Check merge-base and nightly-testing-YYYY-MM-DD
|
||||
@@ -134,7 +134,7 @@ jobs:
|
||||
MESSAGE=""
|
||||
|
||||
if [[ -n "$MATHLIB_REMOTE_TAGS" ]]; then
|
||||
echo "... and Mathlib has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
|
||||
echo "... and Mathlib has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
|
||||
else
|
||||
echo "... but Mathlib does not yet have a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
|
||||
MESSAGE="- ❗ Mathlib CI can not be attempted yet, as the \`nightly-testing-$MOST_RECENT_NIGHTLY\` tag does not exist there yet. We will retry when you push more commits. If you rebase your branch onto \`nightly-with-mathlib\`, Mathlib CI should run now."
|
||||
@@ -149,7 +149,7 @@ jobs:
|
||||
echo "but 'git merge-base origin/master HEAD' reported: $MERGE_BASE_SHA"
|
||||
git -C lean4.git log -10 origin/master
|
||||
|
||||
git -C lean4.git fetch origin nightly-with-mathlib
|
||||
git -C lean4.git fetch origin nightly-with-mathlib
|
||||
NIGHTLY_WITH_MATHLIB_SHA="$(git -C lean4.git rev-parse "origin/nightly-with-mathlib")"
|
||||
MESSAGE="- ❗ Batteries/Mathlib CI will not be attempted unless your PR branches off the \`nightly-with-mathlib\` branch. Try \`git rebase $MERGE_BASE_SHA --onto $NIGHTLY_WITH_MATHLIB_SHA\`."
|
||||
fi
|
||||
@@ -164,10 +164,10 @@ jobs:
|
||||
|
||||
# Use GitHub API to check if a comment already exists
|
||||
existing_comment="$(curl --retry 3 --location --silent \
|
||||
-H "Authorization: token ${{ secrets.MATHLIB4_COMMENT_BOT }}" \
|
||||
-H "Authorization: token ${{ secrets.MATHLIB4_BOT }}" \
|
||||
-H "Accept: application/vnd.github.v3+json" \
|
||||
"https://api.github.com/repos/leanprover/lean4/issues/${{ steps.workflow-info.outputs.pullRequestNumber }}/comments" \
|
||||
| jq 'first(.[] | select(.body | test("^- . Mathlib") or startswith("Mathlib CI status")) | select(.user.login == "leanprover-community-bot"))')"
|
||||
| jq 'first(.[] | select(.body | test("^- . Mathlib") or startswith("Mathlib CI status")) | select(.user.login == "leanprover-community-mathlib4-bot"))')"
|
||||
existing_comment_id="$(echo "$existing_comment" | jq -r .id)"
|
||||
existing_comment_body="$(echo "$existing_comment" | jq -r .body)"
|
||||
|
||||
@@ -177,14 +177,14 @@ jobs:
|
||||
echo "Posting message to the comments: $MESSAGE"
|
||||
|
||||
# Append new result to the existing comment or post a new comment
|
||||
# It's essential we use the MATHLIB4_COMMENT_BOT token here, so that Mathlib CI can subsequently edit the comment.
|
||||
# It's essential we use the MATHLIB4_BOT token here, so that Mathlib CI can subsequently edit the comment.
|
||||
if [ -z "$existing_comment_id" ]; then
|
||||
INTRO="Mathlib CI status ([docs](https://leanprover-community.github.io/contribute/tags_and_branches.html)):"
|
||||
# Post new comment with a bullet point
|
||||
echo "Posting as new comment at leanprover/lean4/issues/${{ steps.workflow-info.outputs.pullRequestNumber }}/comments"
|
||||
curl -L -s \
|
||||
-X POST \
|
||||
-H "Authorization: token ${{ secrets.MATHLIB4_COMMENT_BOT }}" \
|
||||
-H "Authorization: token ${{ secrets.MATHLIB4_BOT }}" \
|
||||
-H "Accept: application/vnd.github.v3+json" \
|
||||
-d "$(jq --null-input --arg intro "$INTRO" --arg val "$MESSAGE" '{"body":($intro + "\n" + $val)}')" \
|
||||
"https://api.github.com/repos/leanprover/lean4/issues/${{ steps.workflow-info.outputs.pullRequestNumber }}/comments"
|
||||
@@ -193,7 +193,7 @@ jobs:
|
||||
echo "Appending to existing comment at leanprover/lean4/issues/${{ steps.workflow-info.outputs.pullRequestNumber }}/comments"
|
||||
curl -L -s \
|
||||
-X PATCH \
|
||||
-H "Authorization: token ${{ secrets.MATHLIB4_COMMENT_BOT }}" \
|
||||
-H "Authorization: token ${{ secrets.MATHLIB4_BOT }}" \
|
||||
-H "Accept: application/vnd.github.v3+json" \
|
||||
-d "$(jq --null-input --arg existing "$existing_comment_body" --arg message "$MESSAGE" '{"body":($existing + "\n" + $message)}')" \
|
||||
"https://api.github.com/repos/leanprover/lean4/issues/comments/$existing_comment_id"
|
||||
@@ -208,7 +208,7 @@ jobs:
|
||||
|
||||
- name: Report mathlib base
|
||||
if: ${{ steps.workflow-info.outputs.pullRequestNumber != '' && steps.ready.outputs.mathlib_ready == 'true' }}
|
||||
uses: actions/github-script@v7
|
||||
uses: actions/github-script@v6
|
||||
with:
|
||||
script: |
|
||||
const description =
|
||||
@@ -329,18 +329,16 @@ jobs:
|
||||
git switch -c lean-pr-testing-${{ steps.workflow-info.outputs.pullRequestNumber }} "$BASE"
|
||||
echo "leanprover/lean4-pr-releases:pr-release-${{ 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-${{ steps.workflow-info.outputs.pullRequestNumber }}",' lakefile.lean
|
||||
sed -i 's,require "leanprover-community" / "batteries" @ git ".\+",require "leanprover-community" / "batteries" @ git "nightly-testing-'"${MOST_RECENT_NIGHTLY}"'",' 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, merging $BASE and bumping Batteries."
|
||||
echo "Branch already exists, pushing an empty commit."
|
||||
git switch lean-pr-testing-${{ 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 "$BASE" --strategy-option ours --no-commit --allow-unrelated-histories
|
||||
lake update batteries
|
||||
git add lake-manifest.json
|
||||
git commit --allow-empty -m "Trigger CI for https://github.com/leanprover/lean4/pull/${{ steps.workflow-info.outputs.pullRequestNumber }}"
|
||||
fi
|
||||
|
||||
|
||||
2
.github/workflows/stale.yml
vendored
2
.github/workflows/stale.yml
vendored
@@ -11,7 +11,7 @@ jobs:
|
||||
stale:
|
||||
runs-on: ubuntu-latest
|
||||
steps:
|
||||
- uses: actions/stale@v9
|
||||
- uses: actions/stale@v8
|
||||
with:
|
||||
days-before-stale: -1
|
||||
days-before-pr-stale: 30
|
||||
|
||||
11
CODEOWNERS
11
CODEOWNERS
@@ -4,14 +4,14 @@
|
||||
# Listed persons will automatically be asked by GitHub to review a PR touching these paths.
|
||||
# If multiple names are listed, a review by any of them is considered sufficient by default.
|
||||
|
||||
/.github/ @Kha @kim-em
|
||||
/RELEASES.md @kim-em
|
||||
/.github/ @Kha @semorrison
|
||||
/RELEASES.md @semorrison
|
||||
/src/kernel/ @leodemoura
|
||||
/src/lake/ @tydeu
|
||||
/src/Lean/Compiler/ @leodemoura
|
||||
/src/Lean/Data/Lsp/ @mhuisi
|
||||
/src/Lean/Elab/Deriving/ @kim-em
|
||||
/src/Lean/Elab/Tactic/ @kim-em
|
||||
/src/Lean/Elab/Deriving/ @semorrison
|
||||
/src/Lean/Elab/Tactic/ @semorrison
|
||||
/src/Lean/Language/ @Kha
|
||||
/src/Lean/Meta/Tactic/ @leodemoura
|
||||
/src/Lean/Parser/ @Kha
|
||||
@@ -19,7 +19,7 @@
|
||||
/src/Lean/PrettyPrinter/Delaborator/ @kmill
|
||||
/src/Lean/Server/ @mhuisi
|
||||
/src/Lean/Widget/ @Vtec234
|
||||
/src/Init/Data/ @kim-em
|
||||
/src/Init/Data/ @semorrison
|
||||
/src/Init/Data/Array/Lemmas.lean @digama0
|
||||
/src/Init/Data/List/Lemmas.lean @digama0
|
||||
/src/Init/Data/List/BasicAux.lean @digama0
|
||||
@@ -45,4 +45,3 @@
|
||||
/src/Std/ @TwoFX
|
||||
/src/Std/Tactic/BVDecide/ @hargoniX
|
||||
/src/Lean/Elab/Tactic/BVDecide/ @hargoniX
|
||||
/src/Std/Sat/ @hargoniX
|
||||
|
||||
657
RELEASES.md
657
RELEASES.md
@@ -8,642 +8,9 @@ This file contains work-in-progress notes for the upcoming release, as well as p
|
||||
Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
|
||||
of each version.
|
||||
|
||||
v4.15.0
|
||||
----------
|
||||
|
||||
Development in progress.
|
||||
|
||||
v4.14.0
|
||||
----------
|
||||
|
||||
Release candidate, release notes will be copied from the branch `releases/v4.14.0` once completed.
|
||||
|
||||
v4.13.0
|
||||
----------
|
||||
|
||||
**Full Changelog**: https://github.com/leanprover/lean4/compare/v4.12.0...v4.13.0
|
||||
|
||||
### Language features, tactics, and metaprograms
|
||||
|
||||
* `structure` command
|
||||
* [#5511](https://github.com/leanprover/lean4/pull/5511) allows structure parents to be type synonyms.
|
||||
* [#5531](https://github.com/leanprover/lean4/pull/5531) allows default values for structure fields to be noncomputable.
|
||||
|
||||
* `rfl` and `apply_rfl` tactics
|
||||
* [#3714](https://github.com/leanprover/lean4/pull/3714), [#3718](https://github.com/leanprover/lean4/pull/3718) improve the `rfl` tactic and give better error messages.
|
||||
* [#3772](https://github.com/leanprover/lean4/pull/3772) makes `rfl` no longer use kernel defeq for ground terms.
|
||||
* [#5329](https://github.com/leanprover/lean4/pull/5329) tags `Iff.refl` with `@[refl]` (@Parcly-Taxel)
|
||||
* [#5359](https://github.com/leanprover/lean4/pull/5359) ensures that the `rfl` tactic tries `Iff.rfl` (@Parcly-Taxel)
|
||||
|
||||
* `unfold` tactic
|
||||
* [#4834](https://github.com/leanprover/lean4/pull/4834) let `unfold` do zeta-delta reduction of local definitions, incorporating functionality of the Mathlib `unfold_let` tactic.
|
||||
|
||||
* `omega` tactic
|
||||
* [#5382](https://github.com/leanprover/lean4/pull/5382) fixes spurious error in [#5315](https://github.com/leanprover/lean4/issues/5315)
|
||||
* [#5523](https://github.com/leanprover/lean4/pull/5523) supports `Int.toNat`
|
||||
|
||||
* `simp` tactic
|
||||
* [#5479](https://github.com/leanprover/lean4/pull/5479) lets `simp` apply rules with higher-order patterns.
|
||||
|
||||
* `induction` tactic
|
||||
* [#5494](https://github.com/leanprover/lean4/pull/5494) fixes `induction`’s "pre-tactic" block to always be indented, avoiding unintended uses of it.
|
||||
|
||||
* `ac_nf` tactic
|
||||
* [#5524](https://github.com/leanprover/lean4/pull/5524) adds `ac_nf`, a counterpart to `ac_rfl`, for normalizing expressions with respect to associativity and commutativity. Tests it with `BitVec` expressions.
|
||||
|
||||
* `bv_decide`
|
||||
* [#5211](https://github.com/leanprover/lean4/pull/5211) makes `extractLsb'` the primitive `bv_decide` understands, rather than `extractLsb` (@alexkeizer)
|
||||
* [#5365](https://github.com/leanprover/lean4/pull/5365) adds `bv_decide` diagnoses.
|
||||
* [#5375](https://github.com/leanprover/lean4/pull/5375) adds `bv_decide` normalization rules for `ofBool (a.getLsbD i)` and `ofBool a[i]` (@alexkeizer)
|
||||
* [#5423](https://github.com/leanprover/lean4/pull/5423) enhances the rewriting rules of `bv_decide`
|
||||
* [#5433](https://github.com/leanprover/lean4/pull/5433) presents the `bv_decide` counterexample at the API
|
||||
* [#5484](https://github.com/leanprover/lean4/pull/5484) handles `BitVec.ofNat` with `Nat` fvars in `bv_decide`
|
||||
* [#5506](https://github.com/leanprover/lean4/pull/5506), [#5507](https://github.com/leanprover/lean4/pull/5507) add `bv_normalize` rules.
|
||||
* [#5568](https://github.com/leanprover/lean4/pull/5568) generalize the `bv_normalize` pipeline to support more general preprocessing passes
|
||||
* [#5573](https://github.com/leanprover/lean4/pull/5573) gets `bv_normalize` up-to-date with the current `BitVec` rewrites
|
||||
* Cleanups: [#5408](https://github.com/leanprover/lean4/pull/5408), [#5493](https://github.com/leanprover/lean4/pull/5493), [#5578](https://github.com/leanprover/lean4/pull/5578)
|
||||
|
||||
|
||||
* Elaboration improvements
|
||||
* [#5266](https://github.com/leanprover/lean4/pull/5266) preserve order of overapplied arguments in `elab_as_elim` procedure.
|
||||
* [#5510](https://github.com/leanprover/lean4/pull/5510) generalizes `elab_as_elim` to allow arbitrary motive applications.
|
||||
* [#5283](https://github.com/leanprover/lean4/pull/5283), [#5512](https://github.com/leanprover/lean4/pull/5512) refine how named arguments suppress explicit arguments. Breaking change: some previously omitted explicit arguments may need explicit `_` arguments now.
|
||||
* [#5376](https://github.com/leanprover/lean4/pull/5376) modifies projection instance binder info for instances, making parameters that are instance implicit in the type be implicit.
|
||||
* [#5402](https://github.com/leanprover/lean4/pull/5402) localizes universe metavariable errors to `let` bindings and `fun` binders if possible. Makes "cannot synthesize metavariable" errors take precedence over unsolved universe level errors.
|
||||
* [#5419](https://github.com/leanprover/lean4/pull/5419) must not reduce `ite` in the discriminant of `match`-expression when reducibility setting is `.reducible`
|
||||
* [#5474](https://github.com/leanprover/lean4/pull/5474) have autoparams report parameter/field on failure
|
||||
* [#5530](https://github.com/leanprover/lean4/pull/5530) makes automatic instance names about types with hygienic names be hygienic.
|
||||
|
||||
* Deriving handlers
|
||||
* [#5432](https://github.com/leanprover/lean4/pull/5432) makes `Repr` deriving instance handle explicit type parameters
|
||||
|
||||
* Functional induction
|
||||
* [#5364](https://github.com/leanprover/lean4/pull/5364) adds more equalities in context, more careful cleanup.
|
||||
|
||||
* Linters
|
||||
* [#5335](https://github.com/leanprover/lean4/pull/5335) fixes the unused variables linter complaining about match/tactic combinations
|
||||
* [#5337](https://github.com/leanprover/lean4/pull/5337) fixes the unused variables linter complaining about some wildcard patterns
|
||||
|
||||
* Other fixes
|
||||
* [#4768](https://github.com/leanprover/lean4/pull/4768) fixes a parse error when `..` appears with a `.` on the next line
|
||||
|
||||
* Metaprogramming
|
||||
* [#3090](https://github.com/leanprover/lean4/pull/3090) handles level parameters in `Meta.evalExpr` (@eric-wieser)
|
||||
* [#5401](https://github.com/leanprover/lean4/pull/5401) instance for `Inhabited (TacticM α)` (@alexkeizer)
|
||||
* [#5412](https://github.com/leanprover/lean4/pull/5412) expose Kernel.check for debugging purposes
|
||||
* [#5556](https://github.com/leanprover/lean4/pull/5556) improves the "invalid projection" type inference error in `inferType`.
|
||||
* [#5587](https://github.com/leanprover/lean4/pull/5587) allows `MVarId.assertHypotheses` to set `BinderInfo` and `LocalDeclKind`.
|
||||
* [#5588](https://github.com/leanprover/lean4/pull/5588) adds `MVarId.tryClearMany'`, a variant of `MVarId.tryClearMany`.
|
||||
|
||||
|
||||
|
||||
### Language server, widgets, and IDE extensions
|
||||
|
||||
* [#5205](https://github.com/leanprover/lean4/pull/5205) decreases the latency of auto-completion in tactic blocks.
|
||||
* [#5237](https://github.com/leanprover/lean4/pull/5237) fixes symbol occurrence highlighting in VS Code not highlighting occurrences when moving the text cursor into the identifier from the right.
|
||||
* [#5257](https://github.com/leanprover/lean4/pull/5257) fixes several instances of incorrect auto-completions being reported.
|
||||
* [#5299](https://github.com/leanprover/lean4/pull/5299) allows auto-completion to report completions for global identifiers when the elaborator fails to provide context-specific auto-completions.
|
||||
* [#5312](https://github.com/leanprover/lean4/pull/5312) fixes the server breaking when changing whitespace after the module header.
|
||||
* [#5322](https://github.com/leanprover/lean4/pull/5322) fixes several instances of auto-completion reporting non-existent namespaces.
|
||||
* [#5428](https://github.com/leanprover/lean4/pull/5428) makes sure to always report some recent file range as progress when waiting for elaboration.
|
||||
|
||||
|
||||
### Pretty printing
|
||||
|
||||
* [#4979](https://github.com/leanprover/lean4/pull/4979) make pretty printer escape identifiers that are tokens.
|
||||
* [#5389](https://github.com/leanprover/lean4/pull/5389) makes formatter use the current token table.
|
||||
* [#5513](https://github.com/leanprover/lean4/pull/5513) use breakable instead of unbreakable whitespace when formatting tokens.
|
||||
|
||||
|
||||
### Library
|
||||
|
||||
* [#5222](https://github.com/leanprover/lean4/pull/5222) reduces allocations in `Json.compress`.
|
||||
* [#5231](https://github.com/leanprover/lean4/pull/5231) upstreams `Zero` and `NeZero`
|
||||
* [#5292](https://github.com/leanprover/lean4/pull/5292) refactors `Lean.Elab.Deriving.FromToJson` (@arthur-adjedj)
|
||||
* [#5415](https://github.com/leanprover/lean4/pull/5415) implements `Repr Empty` (@TomasPuverle)
|
||||
* [#5421](https://github.com/leanprover/lean4/pull/5421) implements `To/FromJSON Empty` (@TomasPuverle)
|
||||
|
||||
* Logic
|
||||
* [#5263](https://github.com/leanprover/lean4/pull/5263) allows simplifying `dite_not`/`decide_not` with only `Decidable (¬p)`.
|
||||
* [#5268](https://github.com/leanprover/lean4/pull/5268) fixes binders on `ite_eq_left_iff`
|
||||
* [#5284](https://github.com/leanprover/lean4/pull/5284) turns off `Inhabited (Sum α β)` instances
|
||||
* [#5355](https://github.com/leanprover/lean4/pull/5355) adds simp lemmas for `LawfulBEq`
|
||||
* [#5374](https://github.com/leanprover/lean4/pull/5374) add `Nonempty` instances for products, allowing more `partial` functions to elaborate successfully
|
||||
* [#5447](https://github.com/leanprover/lean4/pull/5447) updates Pi instance names
|
||||
* [#5454](https://github.com/leanprover/lean4/pull/5454) makes some instance arguments implicit
|
||||
* [#5456](https://github.com/leanprover/lean4/pull/5456) adds `heq_comm`
|
||||
* [#5529](https://github.com/leanprover/lean4/pull/5529) moves `@[simp]` from `exists_prop'` to `exists_prop`
|
||||
|
||||
* `Bool`
|
||||
* [#5228](https://github.com/leanprover/lean4/pull/5228) fills gaps in Bool lemmas
|
||||
* [#5332](https://github.com/leanprover/lean4/pull/5332) adds notation `^^` for Bool.xor
|
||||
* [#5351](https://github.com/leanprover/lean4/pull/5351) removes `_root_.and` (and or/not/xor) and instead exports/uses `Bool.and` (etc.).
|
||||
|
||||
* `BitVec`
|
||||
* [#5240](https://github.com/leanprover/lean4/pull/5240) removes BitVec simps with complicated RHS
|
||||
* [#5247](https://github.com/leanprover/lean4/pull/5247) `BitVec.getElem_zeroExtend`
|
||||
* [#5248](https://github.com/leanprover/lean4/pull/5248) simp lemmas for BitVec, improving confluence
|
||||
* [#5249](https://github.com/leanprover/lean4/pull/5249) removes `@[simp]` from some BitVec lemmas
|
||||
* [#5252](https://github.com/leanprover/lean4/pull/5252) changes `BitVec.intMin/Max` from abbrev to def
|
||||
* [#5278](https://github.com/leanprover/lean4/pull/5278) adds `BitVec.getElem_truncate` (@tobiasgrosser)
|
||||
* [#5281](https://github.com/leanprover/lean4/pull/5281) adds udiv/umod bitblasting for `bv_decide` (@bollu)
|
||||
* [#5297](https://github.com/leanprover/lean4/pull/5297) `BitVec` unsigned order theoretic results
|
||||
* [#5313](https://github.com/leanprover/lean4/pull/5313) adds more basic BitVec ordering theory for UInt
|
||||
* [#5314](https://github.com/leanprover/lean4/pull/5314) adds `toNat_sub_of_le` (@bollu)
|
||||
* [#5357](https://github.com/leanprover/lean4/pull/5357) adds `BitVec.truncate` lemmas
|
||||
* [#5358](https://github.com/leanprover/lean4/pull/5358) introduces `BitVec.setWidth` to unify zeroExtend and truncate (@tobiasgrosser)
|
||||
* [#5361](https://github.com/leanprover/lean4/pull/5361) some BitVec GetElem lemmas
|
||||
* [#5385](https://github.com/leanprover/lean4/pull/5385) adds `BitVec.ofBool_[and|or|xor]_ofBool` theorems (@tobiasgrosser)
|
||||
* [#5404](https://github.com/leanprover/lean4/pull/5404) more of `BitVec.getElem_*` (@tobiasgrosser)
|
||||
* [#5410](https://github.com/leanprover/lean4/pull/5410) BitVec analogues of `Nat.{mul_two, two_mul, mul_succ, succ_mul}` (@bollu)
|
||||
* [#5411](https://github.com/leanprover/lean4/pull/5411) `BitVec.toNat_{add,sub,mul_of_lt}` for BitVector non-overflow reasoning (@bollu)
|
||||
* [#5413](https://github.com/leanprover/lean4/pull/5413) adds `_self`, `_zero`, and `_allOnes` for `BitVec.[and|or|xor]` (@tobiasgrosser)
|
||||
* [#5416](https://github.com/leanprover/lean4/pull/5416) adds LawCommIdentity + IdempotentOp for `BitVec.[and|or|xor]` (@tobiasgrosser)
|
||||
* [#5418](https://github.com/leanprover/lean4/pull/5418) decidable quantifers for BitVec
|
||||
* [#5450](https://github.com/leanprover/lean4/pull/5450) adds `BitVec.toInt_[intMin|neg|neg_of_ne_intMin]` (@tobiasgrosser)
|
||||
* [#5459](https://github.com/leanprover/lean4/pull/5459) missing BitVec lemmas
|
||||
* [#5469](https://github.com/leanprover/lean4/pull/5469) adds `BitVec.[not_not, allOnes_shiftLeft_or_shiftLeft, allOnes_shiftLeft_and_shiftLeft]` (@luisacicolini)
|
||||
* [#5478](https://github.com/leanprover/lean4/pull/5478) adds `BitVec.(shiftLeft_add_distrib, shiftLeft_ushiftRight)` (@luisacicolini)
|
||||
* [#5487](https://github.com/leanprover/lean4/pull/5487) adds `sdiv_eq`, `smod_eq` to allow `sdiv`/`smod` bitblasting (@bollu)
|
||||
* [#5491](https://github.com/leanprover/lean4/pull/5491) adds `BitVec.toNat_[abs|sdiv|smod]` (@tobiasgrosser)
|
||||
* [#5492](https://github.com/leanprover/lean4/pull/5492) `BitVec.(not_sshiftRight, not_sshiftRight_not, getMsb_not, msb_not)` (@luisacicolini)
|
||||
* [#5499](https://github.com/leanprover/lean4/pull/5499) `BitVec.Lemmas` - drop non-terminal simps (@tobiasgrosser)
|
||||
* [#5505](https://github.com/leanprover/lean4/pull/5505) unsimps `BitVec.divRec_succ'`
|
||||
* [#5508](https://github.com/leanprover/lean4/pull/5508) adds `BitVec.getElem_[add|add_add_bool|mul|rotateLeft|rotateRight…` (@tobiasgrosser)
|
||||
* [#5554](https://github.com/leanprover/lean4/pull/5554) adds `Bitvec.[add, sub, mul]_eq_xor` and `width_one_cases` (@luisacicolini)
|
||||
|
||||
* `List`
|
||||
* [#5242](https://github.com/leanprover/lean4/pull/5242) improve naming for `List.mergeSort` lemmas
|
||||
* [#5302](https://github.com/leanprover/lean4/pull/5302) provide `mergeSort` comparator autoParam
|
||||
* [#5373](https://github.com/leanprover/lean4/pull/5373) fix name of `List.length_mergeSort`
|
||||
* [#5377](https://github.com/leanprover/lean4/pull/5377) upstream `map_mergeSort`
|
||||
* [#5378](https://github.com/leanprover/lean4/pull/5378) modify signature of lemmas about `mergeSort`
|
||||
* [#5245](https://github.com/leanprover/lean4/pull/5245) avoid importing `List.Basic` without List.Impl
|
||||
* [#5260](https://github.com/leanprover/lean4/pull/5260) review of List API
|
||||
* [#5264](https://github.com/leanprover/lean4/pull/5264) review of List API
|
||||
* [#5269](https://github.com/leanprover/lean4/pull/5269) remove HashMap's duplicated Pairwise and Sublist
|
||||
* [#5271](https://github.com/leanprover/lean4/pull/5271) remove @[simp] from `List.head_mem` and similar
|
||||
* [#5273](https://github.com/leanprover/lean4/pull/5273) lemmas about `List.attach`
|
||||
* [#5275](https://github.com/leanprover/lean4/pull/5275) reverse direction of `List.tail_map`
|
||||
* [#5277](https://github.com/leanprover/lean4/pull/5277) more `List.attach` lemmas
|
||||
* [#5285](https://github.com/leanprover/lean4/pull/5285) `List.count` lemmas
|
||||
* [#5287](https://github.com/leanprover/lean4/pull/5287) use boolean predicates in `List.filter`
|
||||
* [#5289](https://github.com/leanprover/lean4/pull/5289) `List.mem_ite_nil_left` and analogues
|
||||
* [#5293](https://github.com/leanprover/lean4/pull/5293) cleanup of `List.findIdx` / `List.take` lemmas
|
||||
* [#5294](https://github.com/leanprover/lean4/pull/5294) switch primes on `List.getElem_take`
|
||||
* [#5300](https://github.com/leanprover/lean4/pull/5300) more `List.findIdx` theorems
|
||||
* [#5310](https://github.com/leanprover/lean4/pull/5310) fix `List.all/any` lemmas
|
||||
* [#5311](https://github.com/leanprover/lean4/pull/5311) fix `List.countP` lemmas
|
||||
* [#5316](https://github.com/leanprover/lean4/pull/5316) `List.tail` lemma
|
||||
* [#5331](https://github.com/leanprover/lean4/pull/5331) fix implicitness of `List.getElem_mem`
|
||||
* [#5350](https://github.com/leanprover/lean4/pull/5350) `List.replicate` lemmas
|
||||
* [#5352](https://github.com/leanprover/lean4/pull/5352) `List.attachWith` lemmas
|
||||
* [#5353](https://github.com/leanprover/lean4/pull/5353) `List.head_mem_head?`
|
||||
* [#5360](https://github.com/leanprover/lean4/pull/5360) lemmas about `List.tail`
|
||||
* [#5391](https://github.com/leanprover/lean4/pull/5391) review of `List.erase` / `List.find` lemmas
|
||||
* [#5392](https://github.com/leanprover/lean4/pull/5392) `List.fold` / `attach` lemmas
|
||||
* [#5393](https://github.com/leanprover/lean4/pull/5393) `List.fold` relators
|
||||
* [#5394](https://github.com/leanprover/lean4/pull/5394) lemmas about `List.maximum?`
|
||||
* [#5403](https://github.com/leanprover/lean4/pull/5403) theorems about `List.toArray`
|
||||
* [#5405](https://github.com/leanprover/lean4/pull/5405) reverse direction of `List.set_map`
|
||||
* [#5448](https://github.com/leanprover/lean4/pull/5448) add lemmas about `List.IsPrefix` (@Command-Master)
|
||||
* [#5460](https://github.com/leanprover/lean4/pull/5460) missing `List.set_replicate_self`
|
||||
* [#5518](https://github.com/leanprover/lean4/pull/5518) rename `List.maximum?` to `max?`
|
||||
* [#5519](https://github.com/leanprover/lean4/pull/5519) upstream `List.fold` lemmas
|
||||
* [#5520](https://github.com/leanprover/lean4/pull/5520) restore `@[simp]` on `List.getElem_mem` etc.
|
||||
* [#5521](https://github.com/leanprover/lean4/pull/5521) List simp fixes
|
||||
* [#5550](https://github.com/leanprover/lean4/pull/5550) `List.unattach` and simp lemmas
|
||||
* [#5594](https://github.com/leanprover/lean4/pull/5594) induction-friendly `List.min?_cons`
|
||||
|
||||
* `Array`
|
||||
* [#5246](https://github.com/leanprover/lean4/pull/5246) cleanup imports of Array.Lemmas
|
||||
* [#5255](https://github.com/leanprover/lean4/pull/5255) split Init.Data.Array.Lemmas for better bootstrapping
|
||||
* [#5288](https://github.com/leanprover/lean4/pull/5288) rename `Array.data` to `Array.toList`
|
||||
* [#5303](https://github.com/leanprover/lean4/pull/5303) cleanup of `List.getElem_append` variants
|
||||
* [#5304](https://github.com/leanprover/lean4/pull/5304) `Array.not_mem_empty`
|
||||
* [#5400](https://github.com/leanprover/lean4/pull/5400) reorganization in Array/Basic
|
||||
* [#5420](https://github.com/leanprover/lean4/pull/5420) make `Array` functions either semireducible or use structural recursion
|
||||
* [#5422](https://github.com/leanprover/lean4/pull/5422) refactor `DecidableEq (Array α)`
|
||||
* [#5452](https://github.com/leanprover/lean4/pull/5452) refactor of Array
|
||||
* [#5458](https://github.com/leanprover/lean4/pull/5458) cleanup of Array docstrings after refactor
|
||||
* [#5461](https://github.com/leanprover/lean4/pull/5461) restore `@[simp]` on `Array.swapAt!_def`
|
||||
* [#5465](https://github.com/leanprover/lean4/pull/5465) improve Array GetElem lemmas
|
||||
* [#5466](https://github.com/leanprover/lean4/pull/5466) `Array.foldX` lemmas
|
||||
* [#5472](https://github.com/leanprover/lean4/pull/5472) @[simp] lemmas about `List.toArray`
|
||||
* [#5485](https://github.com/leanprover/lean4/pull/5485) reverse simp direction for `toArray_concat`
|
||||
* [#5514](https://github.com/leanprover/lean4/pull/5514) `Array.eraseReps`
|
||||
* [#5515](https://github.com/leanprover/lean4/pull/5515) upstream `Array.qsortOrd`
|
||||
* [#5516](https://github.com/leanprover/lean4/pull/5516) upstream `Subarray.empty`
|
||||
* [#5526](https://github.com/leanprover/lean4/pull/5526) fix name of `Array.length_toList`
|
||||
* [#5527](https://github.com/leanprover/lean4/pull/5527) reduce use of deprecated lemmas in Array
|
||||
* [#5534](https://github.com/leanprover/lean4/pull/5534) cleanup of Array GetElem lemmas
|
||||
* [#5536](https://github.com/leanprover/lean4/pull/5536) fix `Array.modify` lemmas
|
||||
* [#5551](https://github.com/leanprover/lean4/pull/5551) upstream `Array.flatten` lemmas
|
||||
* [#5552](https://github.com/leanprover/lean4/pull/5552) switch obvious cases of array "bang"`[]!` indexing to rely on hypothesis (@TomasPuverle)
|
||||
* [#5577](https://github.com/leanprover/lean4/pull/5577) add missing simp to `Array.size_feraseIdx`
|
||||
* [#5586](https://github.com/leanprover/lean4/pull/5586) `Array/Option.unattach`
|
||||
|
||||
* `Option`
|
||||
* [#5272](https://github.com/leanprover/lean4/pull/5272) remove @[simp] from `Option.pmap/pbind` and add simp lemmas
|
||||
* [#5307](https://github.com/leanprover/lean4/pull/5307) restoring Option simp confluence
|
||||
* [#5354](https://github.com/leanprover/lean4/pull/5354) remove @[simp] from `Option.bind_map`
|
||||
* [#5532](https://github.com/leanprover/lean4/pull/5532) `Option.attach`
|
||||
* [#5539](https://github.com/leanprover/lean4/pull/5539) fix explicitness of `Option.mem_toList`
|
||||
|
||||
* `Nat`
|
||||
* [#5241](https://github.com/leanprover/lean4/pull/5241) add @[simp] to `Nat.add_eq_zero_iff`
|
||||
* [#5261](https://github.com/leanprover/lean4/pull/5261) Nat bitwise lemmas
|
||||
* [#5262](https://github.com/leanprover/lean4/pull/5262) `Nat.testBit_add_one` should not be a global simp lemma
|
||||
* [#5267](https://github.com/leanprover/lean4/pull/5267) protect some Nat bitwise theorems
|
||||
* [#5305](https://github.com/leanprover/lean4/pull/5305) rename Nat bitwise lemmas
|
||||
* [#5306](https://github.com/leanprover/lean4/pull/5306) add `Nat.self_sub_mod` lemma
|
||||
* [#5503](https://github.com/leanprover/lean4/pull/5503) restore @[simp] to upstreamed `Nat.lt_off_iff`
|
||||
|
||||
* `Int`
|
||||
* [#5301](https://github.com/leanprover/lean4/pull/5301) rename `Int.div/mod` to `Int.tdiv/tmod`
|
||||
* [#5320](https://github.com/leanprover/lean4/pull/5320) add `ediv_nonneg_of_nonpos_of_nonpos` to DivModLemmas (@sakehl)
|
||||
|
||||
* `Fin`
|
||||
* [#5250](https://github.com/leanprover/lean4/pull/5250) missing lemma about `Fin.ofNat'`
|
||||
* [#5356](https://github.com/leanprover/lean4/pull/5356) `Fin.ofNat'` uses `NeZero`
|
||||
* [#5379](https://github.com/leanprover/lean4/pull/5379) remove some @[simp]s from Fin lemmas
|
||||
* [#5380](https://github.com/leanprover/lean4/pull/5380) missing Fin @[simp] lemmas
|
||||
|
||||
* `HashMap`
|
||||
* [#5244](https://github.com/leanprover/lean4/pull/5244) (`DHashMap`|`HashMap`|`HashSet`).(`getKey?`|`getKey`|`getKey!`|`getKeyD`)
|
||||
* [#5362](https://github.com/leanprover/lean4/pull/5362) remove the last use of `Lean.(HashSet|HashMap)`
|
||||
* [#5369](https://github.com/leanprover/lean4/pull/5369) `HashSet.ofArray`
|
||||
* [#5370](https://github.com/leanprover/lean4/pull/5370) `HashSet.partition`
|
||||
* [#5581](https://github.com/leanprover/lean4/pull/5581) `Singleton`/`Insert`/`Union` instances for `HashMap`/`Set`
|
||||
* [#5582](https://github.com/leanprover/lean4/pull/5582) `HashSet.all`/`any`
|
||||
* [#5590](https://github.com/leanprover/lean4/pull/5590) adding `Insert`/`Singleton`/`Union` instances for `HashMap`/`Set.Raw`
|
||||
* [#5591](https://github.com/leanprover/lean4/pull/5591) `HashSet.Raw.all/any`
|
||||
|
||||
* `Monads`
|
||||
* [#5463](https://github.com/leanprover/lean4/pull/5463) upstream some monad lemmas
|
||||
* [#5464](https://github.com/leanprover/lean4/pull/5464) adjust simp attributes on monad lemmas
|
||||
* [#5522](https://github.com/leanprover/lean4/pull/5522) more monadic simp lemmas
|
||||
|
||||
* Simp lemma cleanup
|
||||
* [#5251](https://github.com/leanprover/lean4/pull/5251) remove redundant simp annotations
|
||||
* [#5253](https://github.com/leanprover/lean4/pull/5253) remove Int simp lemmas that can't fire
|
||||
* [#5254](https://github.com/leanprover/lean4/pull/5254) variables appearing on both sides of an iff should be implicit
|
||||
* [#5381](https://github.com/leanprover/lean4/pull/5381) cleaning up redundant simp lemmas
|
||||
|
||||
|
||||
### Compiler, runtime, and FFI
|
||||
|
||||
* [#4685](https://github.com/leanprover/lean4/pull/4685) fixes a typo in the C `run_new_frontend` signature
|
||||
* [#4729](https://github.com/leanprover/lean4/pull/4729) has IR checker suggest using `noncomputable`
|
||||
* [#5143](https://github.com/leanprover/lean4/pull/5143) adds a shared library for Lake
|
||||
* [#5437](https://github.com/leanprover/lean4/pull/5437) removes (syntactically) duplicate imports (@euprunin)
|
||||
* [#5462](https://github.com/leanprover/lean4/pull/5462) updates `src/lake/lakefile.toml` to the adjusted Lake build process
|
||||
* [#5541](https://github.com/leanprover/lean4/pull/5541) removes new shared libs before build to better support Windows
|
||||
* [#5558](https://github.com/leanprover/lean4/pull/5558) make `lean.h` compile with MSVC (@kant2002)
|
||||
* [#5564](https://github.com/leanprover/lean4/pull/5564) removes non-conforming size-0 arrays (@eric-wieser)
|
||||
|
||||
|
||||
### Lake
|
||||
* Reservoir build cache. Lake will now attempt to fetch a pre-built copy of the package from Reservoir before building it. This is only enabled for packages in the leanprover or leanprover-community organizations on versions indexed by Reservoir. Users can force Lake to build packages from the source by passing --no-cache on the CLI or by setting the LAKE_NO_CACHE environment variable to true. [#5486](https://github.com/leanprover/lean4/pull/5486), [#5572](https://github.com/leanprover/lean4/pull/5572), [#5583](https://github.com/leanprover/lean4/pull/5583), [#5600](https://github.com/leanprover/lean4/pull/5600), [#5641](https://github.com/leanprover/lean4/pull/5641), [#5642](https://github.com/leanprover/lean4/pull/5642).
|
||||
* [#5504](https://github.com/leanprover/lean4/pull/5504) lake new and lake init now produce TOML configurations by default.
|
||||
* [#5878](https://github.com/leanprover/lean4/pull/5878) fixes a serious issue where Lake would delete path dependencies when attempting to cleanup a dependency required with an incorrect name.
|
||||
|
||||
* **Breaking changes**
|
||||
* [#5641](https://github.com/leanprover/lean4/pull/5641) A Lake build of target within a package will no longer build a package's dependencies package-level extra target dependencies. At the technical level, a package's extraDep facet no longer transitively builds its dependencies’ extraDep facets (which include their extraDepTargets).
|
||||
|
||||
### Documentation fixes
|
||||
|
||||
* [#3918](https://github.com/leanprover/lean4/pull/3918) `@[builtin_doc]` attribute (@digama0)
|
||||
* [#4305](https://github.com/leanprover/lean4/pull/4305) explains the borrow syntax (@eric-wieser)
|
||||
* [#5349](https://github.com/leanprover/lean4/pull/5349) adds documentation for `groupBy.loop` (@vihdzp)
|
||||
* [#5473](https://github.com/leanprover/lean4/pull/5473) fixes typo in `BitVec.mul` docstring (@llllvvuu)
|
||||
* [#5476](https://github.com/leanprover/lean4/pull/5476) fixes typos in `Lean.MetavarContext`
|
||||
* [#5481](https://github.com/leanprover/lean4/pull/5481) removes mention of `Lean.withSeconds` (@alexkeizer)
|
||||
* [#5497](https://github.com/leanprover/lean4/pull/5497) updates documentation and tests for `toUIntX` functions (@TomasPuverle)
|
||||
* [#5087](https://github.com/leanprover/lean4/pull/5087) mentions that `inferType` does not ensure type correctness
|
||||
* Many fixes to spelling across the doc-strings, (@euprunin): [#5425](https://github.com/leanprover/lean4/pull/5425) [#5426](https://github.com/leanprover/lean4/pull/5426) [#5427](https://github.com/leanprover/lean4/pull/5427) [#5430](https://github.com/leanprover/lean4/pull/5430) [#5431](https://github.com/leanprover/lean4/pull/5431) [#5434](https://github.com/leanprover/lean4/pull/5434) [#5435](https://github.com/leanprover/lean4/pull/5435) [#5436](https://github.com/leanprover/lean4/pull/5436) [#5438](https://github.com/leanprover/lean4/pull/5438) [#5439](https://github.com/leanprover/lean4/pull/5439) [#5440](https://github.com/leanprover/lean4/pull/5440) [#5599](https://github.com/leanprover/lean4/pull/5599)
|
||||
|
||||
### Changes to CI
|
||||
|
||||
* [#5343](https://github.com/leanprover/lean4/pull/5343) allows addition of `release-ci` label via comment (@thorimur)
|
||||
* [#5344](https://github.com/leanprover/lean4/pull/5344) sets check level correctly during workflow (@thorimur)
|
||||
* [#5444](https://github.com/leanprover/lean4/pull/5444) Mathlib's `lean-pr-testing-NNNN` branches should use Batteries' `lean-pr-testing-NNNN` branches
|
||||
* [#5489](https://github.com/leanprover/lean4/pull/5489) commit `lake-manifest.json` when updating `lean-pr-testing` branches
|
||||
* [#5490](https://github.com/leanprover/lean4/pull/5490) use separate secrets for commenting and branching in `pr-release.yml`
|
||||
|
||||
v4.12.0
|
||||
----------
|
||||
|
||||
### Language features, tactics, and metaprograms
|
||||
|
||||
* `bv_decide` tactic. This release introduces a new tactic for proving goals involving `BitVec` and `Bool`. It reduces the goal to a SAT instance that is refuted by an external solver, and the resulting LRAT proof is checked in Lean. This is used to synthesize a proof of the goal by reflection. As this process uses verified algorithms, proofs generated by this tactic use `Lean.ofReduceBool`, so this tactic includes the Lean compiler as part of the trusted code base. The external solver CaDiCaL is included with Lean and does not need to be installed separately to make use of `bv_decide`.
|
||||
|
||||
For example, we can use `bv_decide` to verify that a bit twiddling formula leaves at most one bit set:
|
||||
```lean
|
||||
def popcount (x : BitVec 64) : BitVec 64 :=
|
||||
let rec go (x pop : BitVec 64) : Nat → BitVec 64
|
||||
| 0 => pop
|
||||
| n + 1 => go (x >>> 2) (pop + (x &&& 1)) n
|
||||
go x 0 64
|
||||
|
||||
example (x : BitVec 64) : popcount ((x &&& (x - 1)) ^^^ x) ≤ 1 := by
|
||||
simp only [popcount, popcount.go]
|
||||
bv_decide
|
||||
```
|
||||
When the external solver fails to refute the SAT instance generated by `bv_decide`, it can report a counterexample:
|
||||
```lean
|
||||
/--
|
||||
error: The prover found a counterexample, consider the following assignment:
|
||||
x = 0xffffffffffffffff#64
|
||||
-/
|
||||
#guard_msgs in
|
||||
example (x : BitVec 64) : x < x + 1 := by
|
||||
bv_decide
|
||||
```
|
||||
|
||||
See `Lean.Elab.Tactic.BVDecide` for a more detailed overview, and look in `tests/lean/run/bv_*` for examples.
|
||||
|
||||
[#5013](https://github.com/leanprover/lean4/pull/5013), [#5074](https://github.com/leanprover/lean4/pull/5074), [#5100](https://github.com/leanprover/lean4/pull/5100), [#5113](https://github.com/leanprover/lean4/pull/5113), [#5137](https://github.com/leanprover/lean4/pull/5137), [#5203](https://github.com/leanprover/lean4/pull/5203), [#5212](https://github.com/leanprover/lean4/pull/5212), [#5220](https://github.com/leanprover/lean4/pull/5220).
|
||||
|
||||
* `simp` tactic
|
||||
* [#4988](https://github.com/leanprover/lean4/pull/4988) fixes a panic in the `reducePow` simproc.
|
||||
* [#5071](https://github.com/leanprover/lean4/pull/5071) exposes the `index` option to the `dsimp` tactic, introduced to `simp` in [#4202](https://github.com/leanprover/lean4/pull/4202).
|
||||
* [#5159](https://github.com/leanprover/lean4/pull/5159) fixes a panic at `Fin.isValue` simproc.
|
||||
* [#5167](https://github.com/leanprover/lean4/pull/5167) and [#5175](https://github.com/leanprover/lean4/pull/5175) rename the `simpCtorEq` simproc to `reduceCtorEq` and makes it optional. (See breaking changes.)
|
||||
* [#5187](https://github.com/leanprover/lean4/pull/5187) ensures `reduceCtorEq` is enabled in the `norm_cast` tactic.
|
||||
* [#5073](https://github.com/leanprover/lean4/pull/5073) modifies the simp debug trace messages to tag with "dpre" and "dpost" instead of "pre" and "post" when in definitional rewrite mode. [#5054](https://github.com/leanprover/lean4/pull/5054) explains the `reduce` steps for `trace.Debug.Meta.Tactic.simp` trace messages.
|
||||
* `ext` tactic
|
||||
* [#4996](https://github.com/leanprover/lean4/pull/4996) reduces default maximum iteration depth from 1000000 to 100.
|
||||
* `induction` tactic
|
||||
* [#5117](https://github.com/leanprover/lean4/pull/5117) fixes a bug where `let` bindings in minor premises wouldn't be counted correctly.
|
||||
|
||||
* `omega` tactic
|
||||
* [#5157](https://github.com/leanprover/lean4/pull/5157) fixes a panic.
|
||||
|
||||
* `conv` tactic
|
||||
* [#5149](https://github.com/leanprover/lean4/pull/5149) improves `arg n` to handle subsingleton instance arguments.
|
||||
|
||||
* [#5044](https://github.com/leanprover/lean4/pull/5044) upstreams the `#time` command.
|
||||
* [#5079](https://github.com/leanprover/lean4/pull/5079) makes `#check` and `#reduce` typecheck the elaborated terms.
|
||||
|
||||
* **Incrementality**
|
||||
* [#4974](https://github.com/leanprover/lean4/pull/4974) fixes regression where we would not interrupt elaboration of previous document versions.
|
||||
* [#5004](https://github.com/leanprover/lean4/pull/5004) fixes a performance regression.
|
||||
* [#5001](https://github.com/leanprover/lean4/pull/5001) disables incremental body elaboration in presence of `where` clauses in declarations.
|
||||
* [#5018](https://github.com/leanprover/lean4/pull/5018) enables infotrees on the command line for ilean generation.
|
||||
* [#5040](https://github.com/leanprover/lean4/pull/5040) and [#5056](https://github.com/leanprover/lean4/pull/5056) improve performance of info trees.
|
||||
* [#5090](https://github.com/leanprover/lean4/pull/5090) disables incrementality in the `case .. | ..` tactic.
|
||||
* [#5312](https://github.com/leanprover/lean4/pull/5312) fixes a bug where changing whitespace after the module header could break subsequent commands.
|
||||
|
||||
* **Definitions**
|
||||
* [#5016](https://github.com/leanprover/lean4/pull/5016) and [#5066](https://github.com/leanprover/lean4/pull/5066) add `clean_wf` tactic to clean up tactic state in `decreasing_by`. This can be disabled with `set_option debug.rawDecreasingByGoal false`.
|
||||
* [#5055](https://github.com/leanprover/lean4/pull/5055) unifies equational theorems between structural and well-founded recursion.
|
||||
* [#5041](https://github.com/leanprover/lean4/pull/5041) allows mutually recursive functions to use different parameter names among the “fixed parameter prefix”
|
||||
* [#4154](https://github.com/leanprover/lean4/pull/4154) and [#5109](https://github.com/leanprover/lean4/pull/5109) add fine-grained equational lemmas for non-recursive functions. See breaking changes.
|
||||
* [#5129](https://github.com/leanprover/lean4/pull/5129) unifies equation lemmas for recursive and non-recursive definitions. The `backward.eqns.deepRecursiveSplit` option can be set to `false` to get the old behavior. See breaking changes.
|
||||
* [#5141](https://github.com/leanprover/lean4/pull/5141) adds `f.eq_unfold` lemmas. Now Lean produces the following zoo of rewrite rules:
|
||||
```
|
||||
Option.map.eq_1 : Option.map f none = none
|
||||
Option.map.eq_2 : Option.map f (some x) = some (f x)
|
||||
Option.map.eq_def : Option.map f p = match o with | none => none | (some x) => some (f x)
|
||||
Option.map.eq_unfold : Option.map = fun f p => match o with | none => none | (some x) => some (f x)
|
||||
```
|
||||
The `f.eq_unfold` variant is especially useful to rewrite with `rw` under binders.
|
||||
* [#5136](https://github.com/leanprover/lean4/pull/5136) fixes bugs in recursion over predicates.
|
||||
|
||||
* **Variable inclusion**
|
||||
* [#5206](https://github.com/leanprover/lean4/pull/5206) documents that `include` currently only applies to theorems.
|
||||
|
||||
* **Elaboration**
|
||||
* [#4926](https://github.com/leanprover/lean4/pull/4926) fixes a bug where autoparam errors were associated to an incorrect source position.
|
||||
* [#4833](https://github.com/leanprover/lean4/pull/4833) fixes an issue where cdot anonymous functions (e.g. `(· + ·)`) would not handle ambiguous notation correctly. Numbers the parameters, making this example expand as `fun x1 x2 => x1 + x2` rather than `fun x x_1 => x + x_1`.
|
||||
* [#5037](https://github.com/leanprover/lean4/pull/5037) improves strength of the tactic that proves array indexing is in bounds.
|
||||
* [#5119](https://github.com/leanprover/lean4/pull/5119) fixes a bug in the tactic that proves indexing is in bounds where it could loop in the presence of mvars.
|
||||
* [#5072](https://github.com/leanprover/lean4/pull/5072) makes the structure type clickable in "not a field of structure" errors for structure instance notation.
|
||||
* [#4717](https://github.com/leanprover/lean4/pull/4717) fixes a bug where mutual `inductive` commands could create terms that the kernel rejects.
|
||||
* [#5142](https://github.com/leanprover/lean4/pull/5142) fixes a bug where `variable` could fail when mixing binder updates and declarations.
|
||||
|
||||
* **Other fixes or improvements**
|
||||
* [#5118](https://github.com/leanprover/lean4/pull/5118) changes the definition of the `syntheticHole` parser so that hovering over `_` in `?_` gives the docstring for synthetic holes.
|
||||
* [#5173](https://github.com/leanprover/lean4/pull/5173) uses the emoji variant selector for ✅️,❌️,💥️ in messages, improving fonts selection.
|
||||
* [#5183](https://github.com/leanprover/lean4/pull/5183) fixes a bug in `rename_i` where implementation detail hypotheses could be renamed.
|
||||
|
||||
### Language server, widgets, and IDE extensions
|
||||
|
||||
* [#4821](https://github.com/leanprover/lean4/pull/4821) resolves two language server bugs that especially affect Windows users. (1) Editing the header could result in the watchdog not correctly restarting the file worker, which would lead to the file seemingly being processed forever. (2) On an especially slow Windows machine, we found that starting the language server would sometimes not succeed at all. This PR also resolves an issue where we would not correctly emit messages that we received while the file worker is being restarted to the corresponding file worker after the restart.
|
||||
* [#5006](https://github.com/leanprover/lean4/pull/5006) updates the user widget manual.
|
||||
* [#5193](https://github.com/leanprover/lean4/pull/5193) updates the quickstart guide with the new display name for the Lean 4 extension ("Lean 4").
|
||||
* [#5185](https://github.com/leanprover/lean4/pull/5185) fixes a bug where over time "import out of date" messages would accumulate.
|
||||
* [#4900](https://github.com/leanprover/lean4/pull/4900) improves ilean loading performance by about a factor of two. Optimizes the JSON parser and the conversion from JSON to Lean data structures; see PR description for details.
|
||||
* **Other fixes or improvements**
|
||||
* [#5031](https://github.com/leanprover/lean4/pull/5031) localizes an instance in `Lsp.Diagnostics`.
|
||||
|
||||
### Pretty printing
|
||||
|
||||
* [#4976](https://github.com/leanprover/lean4/pull/4976) introduces `@[app_delab]`, a macro for creating delaborators for particular constants. The `@[app_delab ident]` syntax resolves `ident` to its constant name `name` and then expands to `@[delab app.name]`.
|
||||
* [#4982](https://github.com/leanprover/lean4/pull/4982) fixes a bug where the pretty printer assumed structure projections were type correct (such terms can appear in type mismatch errors). Improves hoverability of `#print` output for structures.
|
||||
* [#5218](https://github.com/leanprover/lean4/pull/5218) and [#5239](https://github.com/leanprover/lean4/pull/5239) add `pp.exprSizes` debugging option. When true, each pretty printed expression is prefixed with `[size a/b/c]`, where `a` is the size without sharing, `b` is the actual size, and `c` is the size with the maximum possible sharing.
|
||||
|
||||
### Library
|
||||
|
||||
* [#5020](https://github.com/leanprover/lean4/pull/5020) swaps the parameters to `Membership.mem`. A purpose of this change is to make set-like `CoeSort` coercions to refer to the eta-expanded function `fun x => Membership.mem s x`, which can reduce in many computations. Another is that having the `s` argument first leads to better discrimination tree keys. (See breaking changes.)
|
||||
* `Array`
|
||||
* [#4970](https://github.com/leanprover/lean4/pull/4970) adds `@[ext]` attribute to `Array.ext`.
|
||||
* [#4957](https://github.com/leanprover/lean4/pull/4957) deprecates `Array.get_modify`.
|
||||
* `List`
|
||||
* [#4995](https://github.com/leanprover/lean4/pull/4995) upstreams `List.findIdx` lemmas.
|
||||
* [#5029](https://github.com/leanprover/lean4/pull/5029), [#5048](https://github.com/leanprover/lean4/pull/5048) and [#5132](https://github.com/leanprover/lean4/pull/5132) add `List.Sublist` lemmas, some upstreamed. [#5077](https://github.com/leanprover/lean4/pull/5077) fixes implicitness in refl/rfl lemma binders. add `List.Sublist` theorems.
|
||||
* [#5047](https://github.com/leanprover/lean4/pull/5047) upstreams `List.Pairwise` lemmas.
|
||||
* [#5053](https://github.com/leanprover/lean4/pull/5053), [#5124](https://github.com/leanprover/lean4/pull/5124), and [#5161](https://github.com/leanprover/lean4/pull/5161) add `List.find?/findSome?/findIdx?` theorems.
|
||||
* [#5039](https://github.com/leanprover/lean4/pull/5039) adds `List.foldlRecOn` and `List.foldrRecOn` recursion principles to prove things about `List.foldl` and `List.foldr`.
|
||||
* [#5069](https://github.com/leanprover/lean4/pull/5069) upstreams `List.Perm`.
|
||||
* [#5092](https://github.com/leanprover/lean4/pull/5092) and [#5107](https://github.com/leanprover/lean4/pull/5107) add `List.mergeSort` and a fast `@[csimp]` implementation.
|
||||
* [#5103](https://github.com/leanprover/lean4/pull/5103) makes the simp lemmas for `List.subset` more aggressive.
|
||||
* [#5106](https://github.com/leanprover/lean4/pull/5106) changes the statement of `List.getLast?_cons`.
|
||||
* [#5123](https://github.com/leanprover/lean4/pull/5123) and [#5158](https://github.com/leanprover/lean4/pull/5158) add `List.range` and `List.iota` lemmas.
|
||||
* [#5130](https://github.com/leanprover/lean4/pull/5130) adds `List.join` lemmas.
|
||||
* [#5131](https://github.com/leanprover/lean4/pull/5131) adds `List.append` lemmas.
|
||||
* [#5152](https://github.com/leanprover/lean4/pull/5152) adds `List.erase(|P|Idx)` lemmas.
|
||||
* [#5127](https://github.com/leanprover/lean4/pull/5127) makes miscellaneous lemma updates.
|
||||
* [#5153](https://github.com/leanprover/lean4/pull/5153) and [#5160](https://github.com/leanprover/lean4/pull/5160) add lemmas about `List.attach` and `List.pmap`.
|
||||
* [#5164](https://github.com/leanprover/lean4/pull/5164), [#5177](https://github.com/leanprover/lean4/pull/5177), and [#5215](https://github.com/leanprover/lean4/pull/5215) add `List.find?` and `List.range'/range/iota` lemmas.
|
||||
* [#5196](https://github.com/leanprover/lean4/pull/5196) adds `List.Pairwise_erase` and related lemmas.
|
||||
* [#5151](https://github.com/leanprover/lean4/pull/5151) and [#5163](https://github.com/leanprover/lean4/pull/5163) improve confluence of `List` simp lemmas. [#5105](https://github.com/leanprover/lean4/pull/5105) and [#5102](https://github.com/leanprover/lean4/pull/5102) adjust `List` simp lemmas.
|
||||
* [#5178](https://github.com/leanprover/lean4/pull/5178) removes `List.getLast_eq_iff_getLast_eq_some` as a simp lemma.
|
||||
* [#5210](https://github.com/leanprover/lean4/pull/5210) reverses the meaning of `List.getElem_drop` and `List.getElem_drop'`.
|
||||
* [#5214](https://github.com/leanprover/lean4/pull/5214) moves `@[csimp]` lemmas earlier where possible.
|
||||
* `Nat` and `Int`
|
||||
* [#5104](https://github.com/leanprover/lean4/pull/5104) adds `Nat.add_left_eq_self` and relatives.
|
||||
* [#5146](https://github.com/leanprover/lean4/pull/5146) adds missing `Nat.and_xor_distrib_(left|right)`.
|
||||
* [#5148](https://github.com/leanprover/lean4/pull/5148) and [#5190](https://github.com/leanprover/lean4/pull/5190) improve `Nat` and `Int` simp lemma confluence.
|
||||
* [#5165](https://github.com/leanprover/lean4/pull/5165) adjusts `Int` simp lemmas.
|
||||
* [#5166](https://github.com/leanprover/lean4/pull/5166) adds `Int` lemmas relating `neg` and `emod`/`mod`.
|
||||
* [#5208](https://github.com/leanprover/lean4/pull/5208) reverses the direction of the `Int.toNat_sub` simp lemma.
|
||||
* [#5209](https://github.com/leanprover/lean4/pull/5209) adds `Nat.bitwise` lemmas.
|
||||
* [#5230](https://github.com/leanprover/lean4/pull/5230) corrects the docstrings for integer division and modulus.
|
||||
* `Option`
|
||||
* [#5128](https://github.com/leanprover/lean4/pull/5128) and [#5154](https://github.com/leanprover/lean4/pull/5154) add `Option` lemmas.
|
||||
* `BitVec`
|
||||
* [#4889](https://github.com/leanprover/lean4/pull/4889) adds `sshiftRight` bitblasting.
|
||||
* [#4981](https://github.com/leanprover/lean4/pull/4981) adds `Std.Associative` and `Std.Commutative` instances for `BitVec.[and|or|xor]`.
|
||||
* [#4913](https://github.com/leanprover/lean4/pull/4913) enables `missingDocs` error for `BitVec` modules.
|
||||
* [#4930](https://github.com/leanprover/lean4/pull/4930) makes parameter names for `BitVec` more consistent.
|
||||
* [#5098](https://github.com/leanprover/lean4/pull/5098) adds `BitVec.intMin`. Introduces `boolToPropSimps` simp set for converting from boolean to propositional expressions.
|
||||
* [#5200](https://github.com/leanprover/lean4/pull/5200) and [#5217](https://github.com/leanprover/lean4/pull/5217) rename `BitVec.getLsb` to `BitVec.getLsbD`, etc., to bring naming in line with `List`/`Array`/etc.
|
||||
* **Theorems:** [#4977](https://github.com/leanprover/lean4/pull/4977), [#4951](https://github.com/leanprover/lean4/pull/4951), [#4667](https://github.com/leanprover/lean4/pull/4667), [#5007](https://github.com/leanprover/lean4/pull/5007), [#4997](https://github.com/leanprover/lean4/pull/4997), [#5083](https://github.com/leanprover/lean4/pull/5083), [#5081](https://github.com/leanprover/lean4/pull/5081), [#4392](https://github.com/leanprover/lean4/pull/4392)
|
||||
* `UInt`
|
||||
* [#4514](https://github.com/leanprover/lean4/pull/4514) fixes naming convention for `UInt` lemmas.
|
||||
* `Std.HashMap` and `Std.HashSet`
|
||||
* [#4943](https://github.com/leanprover/lean4/pull/4943) deprecates variants of hash map query methods. (See breaking changes.)
|
||||
* [#4917](https://github.com/leanprover/lean4/pull/4917) switches the library and Lean to `Std.HashMap` and `Std.HashSet` almost everywhere.
|
||||
* [#4954](https://github.com/leanprover/lean4/pull/4954) deprecates `Lean.HashMap` and `Lean.HashSet`.
|
||||
* [#5023](https://github.com/leanprover/lean4/pull/5023) cleans up lemma parameters.
|
||||
|
||||
* `Std.Sat` (for `bv_decide`)
|
||||
* [#4933](https://github.com/leanprover/lean4/pull/4933) adds definitions of SAT and CNF.
|
||||
* [#4953](https://github.com/leanprover/lean4/pull/4953) defines "and-inverter graphs" (AIGs) as described in section 3 of [Davis-Swords 2013](https://arxiv.org/pdf/1304.7861.pdf).
|
||||
|
||||
* **Parsec**
|
||||
* [#4774](https://github.com/leanprover/lean4/pull/4774) generalizes the `Parsec` library, allowing parsing of iterable data beyond `String` such as `ByteArray`. (See breaking changes.)
|
||||
* [#5115](https://github.com/leanprover/lean4/pull/5115) moves `Lean.Data.Parsec` to `Std.Internal.Parsec` for bootstrappng reasons.
|
||||
|
||||
* `Thunk`
|
||||
* [#4969](https://github.com/leanprover/lean4/pull/4969) upstreams `Thunk.ext`.
|
||||
|
||||
* **IO**
|
||||
* [#4973](https://github.com/leanprover/lean4/pull/4973) modifies `IO.FS.lines` to handle `\r\n` on all operating systems instead of just on Windows.
|
||||
* [#5125](https://github.com/leanprover/lean4/pull/5125) adds `createTempFile` and `withTempFile` for creating temporary files that can only be read and written by the current user.
|
||||
|
||||
* **Other fixes or improvements**
|
||||
* [#4945](https://github.com/leanprover/lean4/pull/4945) adds `Array`, `Bool` and `Prod` utilities from LeanSAT.
|
||||
* [#4960](https://github.com/leanprover/lean4/pull/4960) adds `Relation.TransGen.trans`.
|
||||
* [#5012](https://github.com/leanprover/lean4/pull/5012) states `WellFoundedRelation Nat` using `<`, not `Nat.lt`.
|
||||
* [#5011](https://github.com/leanprover/lean4/pull/5011) uses `≠` instead of `Not (Eq ...)` in `Fin.ne_of_val_ne`.
|
||||
* [#5197](https://github.com/leanprover/lean4/pull/5197) upstreams `Fin.le_antisymm`.
|
||||
* [#5042](https://github.com/leanprover/lean4/pull/5042) reduces usage of `refine'`.
|
||||
* [#5101](https://github.com/leanprover/lean4/pull/5101) adds about `if-then-else` and `Option`.
|
||||
* [#5112](https://github.com/leanprover/lean4/pull/5112) adds basic instances for `ULift` and `PLift`.
|
||||
* [#5133](https://github.com/leanprover/lean4/pull/5133) and [#5168](https://github.com/leanprover/lean4/pull/5168) make fixes from running the simpNF linter over Lean.
|
||||
* [#5156](https://github.com/leanprover/lean4/pull/5156) removes a bad simp lemma in `omega` theory.
|
||||
* [#5155](https://github.com/leanprover/lean4/pull/5155) improves confluence of `Bool` simp lemmas.
|
||||
* [#5162](https://github.com/leanprover/lean4/pull/5162) improves confluence of `Function.comp` simp lemmas.
|
||||
* [#5191](https://github.com/leanprover/lean4/pull/5191) improves confluence of `if-then-else` simp lemmas.
|
||||
* [#5147](https://github.com/leanprover/lean4/pull/5147) adds `@[elab_as_elim]` to `Quot.rec`, `Nat.strongInductionOn` and `Nat.casesStrongInductionOn`, and also renames the latter two to `Nat.strongRecOn` and `Nat.casesStrongRecOn` (deprecated in [#5179](https://github.com/leanprover/lean4/pull/5179)).
|
||||
* [#5180](https://github.com/leanprover/lean4/pull/5180) disables some simp lemmas with bad discrimination tree keys.
|
||||
* [#5189](https://github.com/leanprover/lean4/pull/5189) cleans up internal simp lemmas that had leaked.
|
||||
* [#5198](https://github.com/leanprover/lean4/pull/5198) cleans up `allowUnsafeReducibility`.
|
||||
* [#5229](https://github.com/leanprover/lean4/pull/5229) removes unused lemmas from some `simp` tactics.
|
||||
* [#5199](https://github.com/leanprover/lean4/pull/5199) removes >6 month deprecations.
|
||||
|
||||
### Lean internals
|
||||
|
||||
* **Performance**
|
||||
* Some core algorithms have been rewritten in C++ for performance.
|
||||
* [#4910](https://github.com/leanprover/lean4/pull/4910) and [#4912](https://github.com/leanprover/lean4/pull/4912) reimplement `instantiateLevelMVars`.
|
||||
* [#4915](https://github.com/leanprover/lean4/pull/4915), [#4922](https://github.com/leanprover/lean4/pull/4922), and [#4931](https://github.com/leanprover/lean4/pull/4931) reimplement `instantiateExprMVars`, 30% faster on a benchmark.
|
||||
* [#4934](https://github.com/leanprover/lean4/pull/4934) has optimizations for the kernel's `Expr` equality test.
|
||||
* [#4990](https://github.com/leanprover/lean4/pull/4990) fixes bug in hashing for the kernel's `Expr` equality test.
|
||||
* [#4935](https://github.com/leanprover/lean4/pull/4935) and [#4936](https://github.com/leanprover/lean4/pull/4936) skip some `PreDefinition` transformations if they are not needed.
|
||||
* [#5225](https://github.com/leanprover/lean4/pull/5225) adds caching for visited exprs at `CheckAssignmentQuick` in `ExprDefEq`.
|
||||
* [#5226](https://github.com/leanprover/lean4/pull/5226) maximizes term sharing at `instantiateMVarDeclMVars`, used by `runTactic`.
|
||||
* **Diagnostics and profiling**
|
||||
* [#4923](https://github.com/leanprover/lean4/pull/4923) adds profiling for `instantiateMVars` in `Lean.Elab.MutualDef`, which can be a bottleneck there.
|
||||
* [#4924](https://github.com/leanprover/lean4/pull/4924) adds diagnostics for large theorems, controlled by the `diagnostics.threshold.proofSize` option.
|
||||
* [#4897](https://github.com/leanprover/lean4/pull/4897) improves display of diagnostic results.
|
||||
* **Other fixes or improvements**
|
||||
* [#4921](https://github.com/leanprover/lean4/pull/4921) cleans up `Expr.betaRev`.
|
||||
* [#4940](https://github.com/leanprover/lean4/pull/4940) fixes tests by not writing directly to stdout, which is unreliable now that elaboration and reporting are executed in separate threads.
|
||||
* [#4955](https://github.com/leanprover/lean4/pull/4955) documents that `stderrAsMessages` is now the default on the command line as well.
|
||||
* [#4647](https://github.com/leanprover/lean4/pull/4647) adjusts documentation for building on macOS.
|
||||
* [#4987](https://github.com/leanprover/lean4/pull/4987) makes regular mvar assignments take precedence over delayed ones in `instantiateMVars`. Normally delayed assignment metavariables are never directly assigned, but on errors Lean assigns `sorry` to unassigned metavariables.
|
||||
* [#4967](https://github.com/leanprover/lean4/pull/4967) adds linter name to errors when a linter crashes.
|
||||
* [#5043](https://github.com/leanprover/lean4/pull/5043) cleans up command line snapshots logic.
|
||||
* [#5067](https://github.com/leanprover/lean4/pull/5067) minimizes some imports.
|
||||
* [#5068](https://github.com/leanprover/lean4/pull/5068) generalizes the monad for `addMatcherInfo`.
|
||||
* [f71a1f](https://github.com/leanprover/lean4/commit/f71a1fb4ae958fccb3ad4d48786a8f47ced05c15) adds missing test for [#5126](https://github.com/leanprover/lean4/issues/5126).
|
||||
* [#5201](https://github.com/leanprover/lean4/pull/5201) restores a test.
|
||||
* [#3698](https://github.com/leanprover/lean4/pull/3698) fixes a bug where label attributes did not pass on the attribute kind.
|
||||
* Typos: [#5080](https://github.com/leanprover/lean4/pull/5080), [#5150](https://github.com/leanprover/lean4/pull/5150), [#5202](https://github.com/leanprover/lean4/pull/5202)
|
||||
|
||||
### Compiler, runtime, and FFI
|
||||
|
||||
* [#3106](https://github.com/leanprover/lean4/pull/3106) moves frontend to new snapshot architecture. Note that `Frontend.processCommand` and `FrontendM` are no longer used by Lean core, but they will be preserved.
|
||||
* [#4919](https://github.com/leanprover/lean4/pull/4919) adds missing include in runtime for `AUTO_THREAD_FINALIZATION` feature on Windows.
|
||||
* [#4941](https://github.com/leanprover/lean4/pull/4941) adds more `LEAN_EXPORT`s for Windows.
|
||||
* [#4911](https://github.com/leanprover/lean4/pull/4911) improves formatting of CLI help text for the frontend.
|
||||
* [#4950](https://github.com/leanprover/lean4/pull/4950) improves file reading and writing.
|
||||
* `readBinFile` and `readFile` now only require two system calls (`stat` + `read`) instead of one `read` per 1024 byte chunk.
|
||||
* `Handle.getLine` and `Handle.putStr` no longer get tripped up by NUL characters.
|
||||
* [#4971](https://github.com/leanprover/lean4/pull/4971) handles the SIGBUS signal when detecting stack overflows.
|
||||
* [#5062](https://github.com/leanprover/lean4/pull/5062) avoids overwriting existing signal handlers, like in [rust-lang/rust#69685](https://github.com/rust-lang/rust/pull/69685).
|
||||
* [#4860](https://github.com/leanprover/lean4/pull/4860) improves workarounds for building on Windows. Splits `libleanshared` on Windows to avoid symbol limit, removes the `LEAN_EXPORT` denylist workaround, adds missing `LEAN_EXPORT`s.
|
||||
* [#4952](https://github.com/leanprover/lean4/pull/4952) output panics into Lean's redirected stderr, ensuring panics ARE visible as regular messages in the language server and properly ordered in relation to other messages on the command line.
|
||||
* [#4963](https://github.com/leanprover/lean4/pull/4963) links LibUV.
|
||||
|
||||
### Lake
|
||||
|
||||
* [#5030](https://github.com/leanprover/lean4/pull/5030) removes dead code.
|
||||
* [#4770](https://github.com/leanprover/lean4/pull/4770) adds additional fields to the package configuration which will be used by Reservoir. See the PR description for details.
|
||||
|
||||
|
||||
### DevOps/CI
|
||||
* [#4914](https://github.com/leanprover/lean4/pull/4914) and [#4937](https://github.com/leanprover/lean4/pull/4937) improve the release checklist.
|
||||
* [#4925](https://github.com/leanprover/lean4/pull/4925) ignores stale leanpkg tests.
|
||||
* [#5003](https://github.com/leanprover/lean4/pull/5003) upgrades `actions/cache` in CI.
|
||||
* [#5010](https://github.com/leanprover/lean4/pull/5010) sets `save-always` in cache actions in CI.
|
||||
* [#5008](https://github.com/leanprover/lean4/pull/5008) adds more libuv search patterns for the speedcenter.
|
||||
* [#5009](https://github.com/leanprover/lean4/pull/5009) reduce number of runs in the speedcenter for "fast" benchmarks from 10 to 3.
|
||||
* [#5014](https://github.com/leanprover/lean4/pull/5014) adjusts lakefile editing to use new `git` syntax in `pr-release` workflow.
|
||||
* [#5025](https://github.com/leanprover/lean4/pull/5025) has `pr-release` workflow pass `--retry` to `curl`.
|
||||
* [#5022](https://github.com/leanprover/lean4/pull/5022) builds MacOS Aarch64 release for PRs by default.
|
||||
* [#5045](https://github.com/leanprover/lean4/pull/5045) adds libuv to the required packages heading in macos docs.
|
||||
* [#5034](https://github.com/leanprover/lean4/pull/5034) fixes the install name of `libleanshared_1` on macOS.
|
||||
* [#5051](https://github.com/leanprover/lean4/pull/5051) fixes Windows stage 0.
|
||||
* [#5052](https://github.com/leanprover/lean4/pull/5052) fixes 32bit stage 0 builds in CI.
|
||||
* [#5057](https://github.com/leanprover/lean4/pull/5057) avoids rebuilding `leanmanifest` in each build.
|
||||
* [#5099](https://github.com/leanprover/lean4/pull/5099) makes `restart-on-label` workflow also filter by commit SHA.
|
||||
* [#4325](https://github.com/leanprover/lean4/pull/4325) adds CaDiCaL.
|
||||
|
||||
### Breaking changes
|
||||
|
||||
* [LibUV](https://libuv.org/) is now required to build Lean. This change only affects developers who compile Lean themselves instead of obtaining toolchains via `elan`. We have updated the official build instructions with information on how to obtain LibUV on our supported platforms. ([#4963](https://github.com/leanprover/lean4/pull/4963))
|
||||
|
||||
* Recursive definitions with a `decreasing_by` clause that begins with `simp_wf` may break. Try removing `simp_wf` or replacing it with `simp`. ([#5016](https://github.com/leanprover/lean4/pull/5016))
|
||||
|
||||
* The behavior of `rw [f]` where `f` is a non-recursive function defined by pattern matching changed.
|
||||
|
||||
For example, preciously, `rw [Option.map]` would rewrite `Option.map f o` to `match o with … `. Now this rewrite fails because it will use the equational lemmas, and these require constructors – just like for `List.map`.
|
||||
|
||||
Remedies:
|
||||
* Split on `o` before rewriting.
|
||||
* Use `rw [Option.map.eq_def]`, which rewrites any (saturated) application of `Option.map`.
|
||||
* Use `set_option backward.eqns.nonrecursive false` when *defining* the function in question.
|
||||
([#4154](https://github.com/leanprover/lean4/pull/4154))
|
||||
|
||||
* The unified handling of equation lemmas for recursive and non-recursive functions can break existing code, as there now can be extra equational lemmas:
|
||||
|
||||
* Explicit uses of `f.eq_2` might have to be adjusted if the numbering changed.
|
||||
|
||||
* Uses of `rw [f]` or `simp [f]` may no longer apply if they previously matched (and introduced a `match` statement), when the equational lemmas got more fine-grained.
|
||||
|
||||
In this case either case analysis on the parameters before rewriting helps, or setting the option `backward.eqns.deepRecursiveSplit false` while *defining* the function.
|
||||
|
||||
([#5129](https://github.com/leanprover/lean4/pull/5129), [#5207](https://github.com/leanprover/lean4/pull/5207))
|
||||
|
||||
* The `reduceCtorEq` simproc is now optional, and it might need to be included in lists of simp lemmas, like `simp only [reduceCtorEq]`. This simproc is responsible for reducing equalities of constructors. ([#5167](https://github.com/leanprover/lean4/pull/5167))
|
||||
|
||||
* `Nat.strongInductionOn` is now `Nat.strongRecOn` and `Nat.caseStrongInductionOn` to `Nat.caseStrongRecOn`. ([#5147](https://github.com/leanprover/lean4/pull/5147))
|
||||
|
||||
* The parameters to `Membership.mem` have been swapped, which affects all `Membership` instances. ([#5020](https://github.com/leanprover/lean4/pull/5020))
|
||||
|
||||
* The meanings of `List.getElem_drop` and `List.getElem_drop'` have been reversed and the first is now a simp lemma. ([#5210](https://github.com/leanprover/lean4/pull/5210))
|
||||
|
||||
* The `Parsec` library has moved from `Lean.Data.Parsec` to `Std.Internal.Parsec`. The `Parsec` type is now more general with a parameter for an iterable. Users parsing strings can migrate to `Parser` in the `Std.Internal.Parsec.String` namespace, which also includes string-focused parsing combinators. ([#4774](https://github.com/leanprover/lean4/pull/4774))
|
||||
|
||||
* The `Lean` module has switched from `Lean.HashMap` and `Lean.HashSet` to `Std.HashMap` and `Std.HashSet` ([#4943](https://github.com/leanprover/lean4/pull/4943)). `Lean.HashMap` and `Lean.HashSet` are now deprecated ([#4954](https://github.com/leanprover/lean4/pull/4954)) and will be removed in a future release. Users of `Lean` APIs that interact with hash maps, for example `Lean.Environment.const2ModIdx`, might encounter minor breakage due to the following changes from `Lean.HashMap` to `Std.HashMap`:
|
||||
* query functions use the term `get` instead of `find`, ([#4943](https://github.com/leanprover/lean4/pull/4943))
|
||||
* the notation `map[key]` no longer returns an optional value but instead expects a proof that the key is present in the map. The previous behavior is available via the `map[key]?` notation.
|
||||
|
||||
Development in progress.
|
||||
|
||||
v4.11.0
|
||||
----------
|
||||
@@ -654,7 +21,7 @@ v4.11.0
|
||||
|
||||
See breaking changes below.
|
||||
|
||||
PRs: [#4883](https://github.com/leanprover/lean4/pull/4883), [#4814](https://github.com/leanprover/lean4/pull/4814), [#5000](https://github.com/leanprover/lean4/pull/5000), [#5036](https://github.com/leanprover/lean4/pull/5036), [#5138](https://github.com/leanprover/lean4/pull/5138), [0edf1b](https://github.com/leanprover/lean4/commit/0edf1bac392f7e2fe0266b28b51c498306363a84).
|
||||
PRs: [#4883](https://github.com/leanprover/lean4/pull/4883), [1242ff](https://github.com/leanprover/lean4/commit/1242ffbfb5a79296041683682268e770fc3cf820), [#5000](https://github.com/leanprover/lean4/pull/5000), [#5036](https://github.com/leanprover/lean4/pull/5036), [#5138](https://github.com/leanprover/lean4/pull/5138), [0edf1b](https://github.com/leanprover/lean4/commit/0edf1bac392f7e2fe0266b28b51c498306363a84).
|
||||
|
||||
* **Recursive definitions**
|
||||
* Structural recursion can now be explicitly requested using
|
||||
@@ -1014,7 +381,7 @@ v4.10.0
|
||||
|
||||
* **Commands**
|
||||
* [#4370](https://github.com/leanprover/lean4/pull/4370) makes the `variable` command fully elaborate binders during validation, fixing an issue where some errors would be reported only at the next declaration.
|
||||
* [#4408](https://github.com/leanprover/lean4/pull/4408) fixes a discrepancy in universe parameter order between `theorem` and `def` declarations.
|
||||
* [#4408](https://github.com/leanprover/lean4/pull/4408) fixes a discrepency in universe parameter order between `theorem` and `def` declarations.
|
||||
* [#4493](https://github.com/leanprover/lean4/pull/4493) and
|
||||
[#4482](https://github.com/leanprover/lean4/pull/4482) fix a discrepancy in the elaborators for `theorem`, `def`, and `example`,
|
||||
making `Prop`-valued `example`s and other definition commands elaborate like `theorem`s.
|
||||
@@ -1076,7 +443,7 @@ v4.10.0
|
||||
* [#4454](https://github.com/leanprover/lean4/pull/4454) adds public `Name.isInternalDetail` function for filtering declarations using naming conventions for internal names.
|
||||
|
||||
* **Other fixes or improvements**
|
||||
* [#4416](https://github.com/leanprover/lean4/pull/4416) sorts the output of `#print axioms` for determinism.
|
||||
* [#4416](https://github.com/leanprover/lean4/pull/4416) sorts the ouput of `#print axioms` for determinism.
|
||||
* [#4528](https://github.com/leanprover/lean4/pull/4528) fixes error message range for the cdot focusing tactic.
|
||||
|
||||
### Language server, widgets, and IDE extensions
|
||||
@@ -1112,7 +479,7 @@ v4.10.0
|
||||
* [#4372](https://github.com/leanprover/lean4/pull/4372) fixes linearity in `HashMap.insert` and `HashMap.erase`, leading to a 40% speedup in a replace-heavy workload.
|
||||
* `Option`
|
||||
* [#4403](https://github.com/leanprover/lean4/pull/4403) generalizes type of `Option.forM` from `Unit` to `PUnit`.
|
||||
* [#4504](https://github.com/leanprover/lean4/pull/4504) remove simp attribute from `Option.elim` and instead adds it to individual reduction lemmas, making unfolding less aggressive.
|
||||
* [#4504](https://github.com/leanprover/lean4/pull/4504) remove simp attribute from `Option.elim` and instead adds it to individal reduction lemmas, making unfolding less aggressive.
|
||||
* `Nat`
|
||||
* [#4242](https://github.com/leanprover/lean4/pull/4242) adds missing theorems for `n + 1` and `n - 1` normal forms.
|
||||
* [#4486](https://github.com/leanprover/lean4/pull/4486) makes `Nat.min_assoc` be a simp lemma.
|
||||
@@ -1573,7 +940,7 @@ While most changes could be considered to be a breaking change, this section mak
|
||||
In particular, tactics embedded in the type will no longer make use of the type of `value` in expressions such as `let x : type := value; body`.
|
||||
* Now functions defined by well-founded recursion are marked with `@[irreducible]` by default ([#4061](https://github.com/leanprover/lean4/pull/4061)).
|
||||
Existing proofs that hold by definitional equality (e.g. `rfl`) can be
|
||||
rewritten to explicitly unfold the function definition (using `simp`,
|
||||
rewritten to explictly unfold the function definition (using `simp`,
|
||||
`unfold`, `rw`), or the recursive function can be temporarily made
|
||||
semireducible (using `unseal f in` before the command), or the function
|
||||
definition itself can be marked as `@[semireducible]` to get the previous
|
||||
@@ -2192,7 +1559,7 @@ v4.7.0
|
||||
and `BitVec` as we begin making the APIs and simp normal forms for these types
|
||||
more complete and consistent.
|
||||
4. Laying the groundwork for the Std roadmap, as a library focused on
|
||||
essential datatypes not provided by the core language (e.g. `RBMap`)
|
||||
essential datatypes not provided by the core langauge (e.g. `RBMap`)
|
||||
and utilities such as basic IO.
|
||||
While we have achieved most of our initial aims in `v4.7.0-rc1`,
|
||||
some upstreaming will continue over the coming months.
|
||||
@@ -2203,7 +1570,7 @@ v4.7.0
|
||||
There is now kernel support for these functions.
|
||||
[#3376](https://github.com/leanprover/lean4/pull/3376).
|
||||
|
||||
* `omega`, our integer linear arithmetic tactic, is now available in the core language.
|
||||
* `omega`, our integer linear arithmetic tactic, is now availabe in the core langauge.
|
||||
* It is supplemented by a preprocessing tactic `bv_omega` which can solve goals about `BitVec`
|
||||
which naturally translate into linear arithmetic problems.
|
||||
[#3435](https://github.com/leanprover/lean4/pull/3435).
|
||||
@@ -2296,11 +1663,11 @@ v4.6.0
|
||||
/-
|
||||
The `Step` type has three constructors: `.done`, `.visit`, `.continue`.
|
||||
* The constructor `.done` instructs `simp` that the result does
|
||||
not need to be simplified further.
|
||||
not need to be simplied further.
|
||||
* The constructor `.visit` instructs `simp` to visit the resulting expression.
|
||||
* The constructor `.continue` instructs `simp` to try other simplification procedures.
|
||||
|
||||
All three constructors take a `Result`. The `.continue` constructor may also take `none`.
|
||||
All three constructors take a `Result`. The `.continue` contructor may also take `none`.
|
||||
`Result` has two fields `expr` (the new expression), and `proof?` (an optional proof).
|
||||
If the new expression is definitionally equal to the input one, then `proof?` can be omitted or set to `none`.
|
||||
-/
|
||||
@@ -2512,7 +1879,7 @@ v4.5.0
|
||||
---------
|
||||
|
||||
* Modify the lexical syntax of string literals to have string gaps, which are escape sequences of the form `"\" newline whitespace*`.
|
||||
These have the interpretation of an empty string and allow a string to flow across multiple lines without introducing additional whitespace.
|
||||
These have the interpetation of an empty string and allow a string to flow across multiple lines without introducing additional whitespace.
|
||||
The following is equivalent to `"this is a string"`.
|
||||
```lean
|
||||
"this is \
|
||||
@@ -2535,7 +1902,7 @@ v4.5.0
|
||||
|
||||
If the well-founded relation you want to use is not the one that the
|
||||
`WellFoundedRelation` type class would infer for your termination argument,
|
||||
you can use `WellFounded.wrap` from the std library to explicitly give one:
|
||||
you can use `WellFounded.wrap` from the std libarary to explicitly give one:
|
||||
```diff
|
||||
-termination_by' ⟨r, hwf⟩
|
||||
+termination_by x => hwf.wrap x
|
||||
|
||||
@@ -73,7 +73,7 @@ update the archived C source code of the stage 0 compiler in `stage0/src`.
|
||||
The github repository will automatically update stage0 on `master` once
|
||||
`src/stdlib_flags.h` and `stage0/src/stdlib_flags.h` are out of sync.
|
||||
|
||||
If you have write access to the lean4 repository, you can also manually
|
||||
If you have write access to the lean4 repository, you can also also manually
|
||||
trigger that process, for example to be able to use new features in the compiler itself.
|
||||
You can do that on <https://github.com/leanprover/lean4/actions/workflows/update-stage0.yml>
|
||||
or using Github CLI with
|
||||
|
||||
@@ -71,12 +71,6 @@ We'll use `v4.6.0` as the intended release version as a running example.
|
||||
- Toolchain bump PR including updated Lake manifest
|
||||
- Create and push the tag
|
||||
- There is no `stable` branch; skip this step
|
||||
- [Verso](https://github.com/leanprover/verso)
|
||||
- Dependencies: exist, but they're not part of the release workflow
|
||||
- The `SubVerso` dependency should be compatible with _every_ Lean release simultaneously, rather than following this workflow
|
||||
- Toolchain bump PR including updated Lake manifest
|
||||
- Create and push the tag
|
||||
- There is no `stable` branch; skip this step
|
||||
- [import-graph](https://github.com/leanprover-community/import-graph)
|
||||
- Toolchain bump PR including updated Lake manifest
|
||||
- Create and push the tag
|
||||
|
||||
@@ -18,7 +18,7 @@ def ctor (mvarId : MVarId) (idx : Nat) : MetaM (List MVarId) := do
|
||||
else if h : idx - 1 < ctors.length then
|
||||
mvarId.apply (.const ctors[idx - 1] us)
|
||||
else
|
||||
throwTacticEx `ctor mvarId "invalid index, inductive datatype has only {ctors.length} constructors"
|
||||
throwTacticEx `ctor mvarId "invalid index, inductive datatype has only {ctors.length} contructors"
|
||||
|
||||
open Elab Tactic
|
||||
|
||||
|
||||
@@ -149,7 +149,7 @@ We now define the constant folding optimization that traverses a term if replace
|
||||
/-!
|
||||
The correctness of the `Term.constFold` is proved using induction, case-analysis, and the term simplifier.
|
||||
We prove all cases but the one for `plus` using `simp [*]`. This tactic instructs the term simplifier to
|
||||
use hypotheses such as `a = b` as rewriting/simplifications rules.
|
||||
use hypotheses such as `a = b` as rewriting/simplications rules.
|
||||
We use the `split` to break the nested `match` expression in the `plus` case into two cases.
|
||||
The local variables `iha` and `ihb` are the induction hypotheses for `a` and `b`.
|
||||
The modifier `←` in a term simplifier argument instructs the term simplifier to use the equation as a rewriting rule in
|
||||
|
||||
@@ -12,17 +12,17 @@ Remark: this example is based on an example found in the Idris manual.
|
||||
Vectors
|
||||
--------
|
||||
|
||||
A `Vec` is a list of size `n` whose elements belong to a type `α`.
|
||||
A `Vector` is a list of size `n` whose elements belong to a type `α`.
|
||||
-/
|
||||
|
||||
inductive Vec (α : Type u) : Nat → Type u
|
||||
| nil : Vec α 0
|
||||
| cons : α → Vec α n → Vec α (n+1)
|
||||
inductive Vector (α : Type u) : Nat → Type u
|
||||
| nil : Vector α 0
|
||||
| cons : α → Vector α n → Vector α (n+1)
|
||||
|
||||
/-!
|
||||
We can overload the `List.cons` notation `::` and use it to create `Vec`s.
|
||||
We can overload the `List.cons` notation `::` and use it to create `Vector`s.
|
||||
-/
|
||||
infix:67 " :: " => Vec.cons
|
||||
infix:67 " :: " => Vector.cons
|
||||
|
||||
/-!
|
||||
Now, we define the types of our simple functional language.
|
||||
@@ -50,11 +50,11 @@ the builtin instance for `Add Int` as the solution.
|
||||
/-!
|
||||
Expressions are indexed by the types of the local variables, and the type of the expression itself.
|
||||
-/
|
||||
inductive HasType : Fin n → Vec Ty n → Ty → Type where
|
||||
inductive HasType : Fin n → Vector Ty n → Ty → Type where
|
||||
| stop : HasType 0 (ty :: ctx) ty
|
||||
| pop : HasType k ctx ty → HasType k.succ (u :: ctx) ty
|
||||
|
||||
inductive Expr : Vec Ty n → Ty → Type where
|
||||
inductive Expr : Vector Ty n → Ty → Type where
|
||||
| var : HasType i ctx ty → Expr ctx ty
|
||||
| val : Int → Expr ctx Ty.int
|
||||
| lam : Expr (a :: ctx) ty → Expr ctx (Ty.fn a ty)
|
||||
@@ -102,8 +102,8 @@ indexed over the types in scope. Since an environment is just another form of li
|
||||
to the vector of local variable types, we overload again the notation `::` so that we can use the usual list syntax.
|
||||
Given a proof that a variable is defined in the context, we can then produce a value from the environment.
|
||||
-/
|
||||
inductive Env : Vec Ty n → Type where
|
||||
| nil : Env Vec.nil
|
||||
inductive Env : Vector Ty n → Type where
|
||||
| nil : Env Vector.nil
|
||||
| cons : Ty.interp a → Env ctx → Env (a :: ctx)
|
||||
|
||||
infix:67 " :: " => Env.cons
|
||||
|
||||
@@ -225,7 +225,7 @@ We now define the constant folding optimization that traverses a term if replace
|
||||
/-!
|
||||
The correctness of the `constFold` is proved using induction, case-analysis, and the term simplifier.
|
||||
We prove all cases but the one for `plus` using `simp [*]`. This tactic instructs the term simplifier to
|
||||
use hypotheses such as `a = b` as rewriting/simplifications rules.
|
||||
use hypotheses such as `a = b` as rewriting/simplications rules.
|
||||
We use the `split` to break the nested `match` expression in the `plus` case into two cases.
|
||||
The local variables `iha` and `ihb` are the induction hypotheses for `a` and `b`.
|
||||
The modifier `←` in a term simplifier argument instructs the term simplifier to use the equation as a rewriting rule in
|
||||
|
||||
@@ -29,7 +29,7 @@ inductive HasType : Expr → Ty → Prop
|
||||
|
||||
/-!
|
||||
We can easily show that if `e` has type `t₁` and type `t₂`, then `t₁` and `t₂` must be equal
|
||||
by using the `cases` tactic. This tactic creates a new subgoal for every constructor,
|
||||
by using the the `cases` tactic. This tactic creates a new subgoal for every constructor,
|
||||
and automatically discharges unreachable cases. The tactic combinator `tac₁ <;> tac₂` applies
|
||||
`tac₂` to each subgoal produced by `tac₁`. Then, the tactic `rfl` is used to close all produced
|
||||
goals using reflexivity.
|
||||
@@ -82,7 +82,9 @@ theorem Expr.typeCheck_correct (h₁ : HasType e ty) (h₂ : e.typeCheck ≠ .un
|
||||
/-!
|
||||
Now, we prove that if `Expr.typeCheck e` returns `Maybe.unknown`, then forall `ty`, `HasType e ty` does not hold.
|
||||
The notation `e.typeCheck` is sugar for `Expr.typeCheck e`. Lean can infer this because we explicitly said that `e` has type `Expr`.
|
||||
The proof is by induction on `e` and case analysis. Note that the tactic `simp [typeCheck]` is applied to all goal generated by the `induction` tactic, and closes
|
||||
The proof is by induction on `e` and case analysis. The tactic `rename_i` is used to to rename "inaccessible" variables.
|
||||
We say a variable is inaccessible if it is introduced by a tactic (e.g., `cases`) or has been shadowed by another variable introduced
|
||||
by the user. Note that the tactic `simp [typeCheck]` is applied to all goal generated by the `induction` tactic, and closes
|
||||
the cases corresponding to the constructors `Expr.nat` and `Expr.bool`.
|
||||
-/
|
||||
theorem Expr.typeCheck_complete {e : Expr} : e.typeCheck = .unknown → ¬ HasType e ty := by
|
||||
|
||||
@@ -93,7 +93,7 @@ Meaning "Remote Procedure Call",this is a Lean function callable from widget cod
|
||||
Our method will take in the `name : Name` of a constant in the environment and return its type.
|
||||
By convention, we represent the input data as a `structure`.
|
||||
Since it will be sent over from JavaScript,
|
||||
we need `FromJson` and `ToJson` instance.
|
||||
we need `FromJson` and `ToJson` instnace.
|
||||
We'll see why the position field is needed later.
|
||||
-/
|
||||
|
||||
|
||||
@@ -396,7 +396,7 @@ Every expression in Lean has a natural computational interpretation, unless it i
|
||||
|
||||
* *β-reduction* : An expression ``(λ x, t) s`` β-reduces to ``t[s/x]``, that is, the result of replacing ``x`` by ``s`` in ``t``.
|
||||
* *ζ-reduction* : An expression ``let x := s in t`` ζ-reduces to ``t[s/x]``.
|
||||
* *δ-reduction* : If ``c`` is a defined constant with definition ``t``, then ``c`` δ-reduces to ``t``.
|
||||
* *δ-reduction* : If ``c`` is a defined constant with definition ``t``, then ``c`` δ-reduces to to ``t``.
|
||||
* *ι-reduction* : When a function defined by recursion on an inductive type is applied to an element given by an explicit constructor, the result ι-reduces to the specified function value, as described in [Inductive Types](inductive.md).
|
||||
|
||||
The reduction relation is transitive, which is to say, is ``s`` reduces to ``s'`` and ``t`` reduces to ``t'``, then ``s t`` reduces to ``s' t'``, ``λ x, s`` reduces to ``λ x, s'``, and so on. If ``s`` and ``t`` reduce to a common term, they are said to be *definitionally equal*. Definitional equality is defined to be the smallest equivalence relation that satisfies all these properties and also includes α-equivalence and the following two relations:
|
||||
|
||||
@@ -1,6 +1,6 @@
|
||||
These are instructions to set up a working development environment for those who wish to make changes to Lean itself. It is part of the [Development Guide](../dev/index.md).
|
||||
These are instructions to set up a working development environment for those who wish to make changes to Lean itself. It is part of the [Development Guide](doc/dev/index.md).
|
||||
|
||||
We strongly suggest that new users instead follow the [Quickstart](../quickstart.md) to get started using Lean, since this sets up an environment that can automatically manage multiple Lean toolchain versions, which is necessary when working within the Lean ecosystem.
|
||||
We strongly suggest that new users instead follow the [Quickstart](doc/quickstart.md) to get started using Lean, since this sets up an environment that can automatically manage multiple Lean toolchain versions, which is necessary when working within the Lean ecosystem.
|
||||
|
||||
Requirements
|
||||
------------
|
||||
|
||||
@@ -15,24 +15,17 @@ Mode](https://docs.microsoft.com/en-us/windows/apps/get-started/enable-your-devi
|
||||
which will allow Lean to create symlinks that e.g. enable go-to-definition in
|
||||
the stdlib.
|
||||
|
||||
## Installing the Windows SDK
|
||||
|
||||
Install the Windows SDK from [Microsoft](https://developer.microsoft.com/en-us/windows/downloads/windows-sdk/).
|
||||
The oldest supported version is 10.0.18362.0. If you installed the Windows SDK to the default location,
|
||||
then there should be a directory with the version number at `C:\Program Files (x86)\Windows Kits\10\Include`.
|
||||
If there are multiple directories, only the highest version number matters.
|
||||
|
||||
## Installing dependencies
|
||||
|
||||
[The official webpage of MSYS2][msys2] provides one-click installers.
|
||||
Once installed, you should run the "MSYS2 CLANG64" shell from the start menu (the one that runs `clang64.exe`).
|
||||
Do not run "MSYS2 MSYS" or "MSYS2 MINGW64" instead!
|
||||
MSYS2 has a package management system, [pacman][pacman].
|
||||
Once installed, you should run the "MSYS2 MinGW 64-bit shell" from the start menu (the one that runs `mingw64.exe`).
|
||||
Do not run "MSYS2 MSYS" instead!
|
||||
MSYS2 has a package management system, [pacman][pacman], which is used in Arch Linux.
|
||||
|
||||
Here are the commands to install all dependencies needed to compile Lean on your machine.
|
||||
|
||||
```bash
|
||||
pacman -S make python mingw-w64-clang-x86_64-cmake mingw-w64-clang-x86_64-clang mingw-w64-clang-x86_64-ccache mingw-w64-clang-x86_64-libuv mingw-w64-clang-x86_64-gmp git unzip diffutils binutils
|
||||
pacman -S make python mingw-w64-x86_64-cmake mingw-w64-x86_64-clang mingw-w64-x86_64-ccache mingw-w64-x86_64-libuv mingw-w64-x86_64-gmp git unzip diffutils binutils
|
||||
```
|
||||
|
||||
You should now be able to run these commands:
|
||||
@@ -68,7 +61,8 @@ If you want a version that can run independently of your MSYS install
|
||||
then you need to copy the following dependent DLL's from where ever
|
||||
they are installed in your MSYS setup:
|
||||
|
||||
- libc++.dll
|
||||
- libgcc_s_seh-1.dll
|
||||
- libstdc++-6.dll
|
||||
- libgmp-10.dll
|
||||
- libuv-1.dll
|
||||
- libwinpthread-1.dll
|
||||
@@ -88,6 +82,6 @@ version clang to your path.
|
||||
|
||||
**-bash: gcc: command not found**
|
||||
|
||||
Make sure `/clang64/bin` is in your PATH environment. If it is not then
|
||||
check you launched the MSYS2 CLANG64 shell from the start menu.
|
||||
(The one that runs `clang64.exe`).
|
||||
Make sure `/mingw64/bin` is in your PATH environment. If it is not then
|
||||
check you launched the MSYS2 MinGW 64-bit shell from the start menu.
|
||||
(The one that runs `mingw64.exe`).
|
||||
|
||||
@@ -138,8 +138,8 @@ definition:
|
||||
|
||||
-/
|
||||
instance : Applicative List where
|
||||
pure := List.singleton
|
||||
seq f x := List.flatMap f fun y => Functor.map y (x ())
|
||||
pure := List.pure
|
||||
seq f x := List.bind f fun y => Functor.map y (x ())
|
||||
/-!
|
||||
|
||||
Notice you can now sequence a _list_ of functions and a _list_ of items.
|
||||
|
||||
@@ -128,8 +128,8 @@ Applying the identity function through an applicative structure should not chang
|
||||
values or structure. For example:
|
||||
-/
|
||||
instance : Applicative List where
|
||||
pure := List.singleton
|
||||
seq f x := List.flatMap f fun y => Functor.map y (x ())
|
||||
pure := List.pure
|
||||
seq f x := List.bind f fun y => Functor.map y (x ())
|
||||
|
||||
#eval pure id <*> [1, 2, 3] -- [1, 2, 3]
|
||||
/-!
|
||||
@@ -171,7 +171,7 @@ of data contained in the container resulting in a new container that has the sam
|
||||
|
||||
`u <*> pure y = pure (. y) <*> u`.
|
||||
|
||||
This law is a little more complicated, so don't sweat it too much. It states that the order that
|
||||
This law is is a little more complicated, so don't sweat it too much. It states that the order that
|
||||
you wrap things shouldn't matter. One the left, you apply any applicative `u` over a pure wrapped
|
||||
object. On the right, you first wrap a function applying the object as an argument. Note that `(·
|
||||
y)` is short hand for: `fun f => f y`. Then you apply this to the first applicative `u`. These
|
||||
@@ -235,8 +235,8 @@ structure or its values.
|
||||
Left identity is `x >>= pure = x` and is demonstrated by the following examples on a monadic `List`:
|
||||
-/
|
||||
instance : Monad List where
|
||||
pure := List.singleton
|
||||
bind := List.flatMap
|
||||
pure := List.pure
|
||||
bind := List.bind
|
||||
|
||||
def a := ["apple", "orange"]
|
||||
|
||||
|
||||
@@ -192,8 +192,8 @@ implementation of `pure` and `bind`.
|
||||
|
||||
-/
|
||||
instance : Monad List where
|
||||
pure := List.singleton
|
||||
bind := List.flatMap
|
||||
pure := List.pure
|
||||
bind := List.bind
|
||||
/-!
|
||||
|
||||
Like you saw with the applicative `seq` operator, the `bind` operator applies the given function
|
||||
|
||||
@@ -7,7 +7,7 @@ Platforms built & tested by our CI, available as binary releases via elan (see b
|
||||
* x86-64 Linux with glibc 2.27+
|
||||
* x86-64 macOS 10.15+
|
||||
* aarch64 (Apple Silicon) macOS 10.15+
|
||||
* x86-64 Windows 11 (any version), Windows 10 (version 1903 or higher), Windows Server 2022
|
||||
* x86-64 Windows 10+
|
||||
|
||||
### Tier 2
|
||||
|
||||
|
||||
20
flake.nix
20
flake.nix
@@ -38,24 +38,8 @@
|
||||
# more convenient `ctest` output
|
||||
CTEST_OUTPUT_ON_FAILURE = 1;
|
||||
} // pkgs.lib.optionalAttrs pkgs.stdenv.isLinux {
|
||||
GMP = (pkgsDist.gmp.override { withStatic = true; }).overrideAttrs (attrs:
|
||||
pkgs.lib.optionalAttrs (pkgs.stdenv.system == "aarch64-linux") {
|
||||
# would need additional linking setup on Linux aarch64, we don't use it anywhere else either
|
||||
hardeningDisable = [ "stackprotector" ];
|
||||
});
|
||||
LIBUV = pkgsDist.libuv.overrideAttrs (attrs: {
|
||||
configureFlags = ["--enable-static"];
|
||||
hardeningDisable = [ "stackprotector" ];
|
||||
# Sync version with CMakeLists.txt
|
||||
version = "1.48.0";
|
||||
src = pkgs.fetchFromGitHub {
|
||||
owner = "libuv";
|
||||
repo = "libuv";
|
||||
rev = "v1.48.0";
|
||||
sha256 = "100nj16fg8922qg4m2hdjh62zv4p32wyrllsvqr659hdhjc03bsk";
|
||||
};
|
||||
doCheck = false;
|
||||
});
|
||||
GMP = pkgsDist.gmp.override { withStatic = true; };
|
||||
LIBUV = pkgsDist.libuv.overrideAttrs (attrs: { configureFlags = ["--enable-static"]; });
|
||||
GLIBC = pkgsDist.glibc;
|
||||
GLIBC_DEV = pkgsDist.glibc.dev;
|
||||
GCC_LIB = pkgsDist.gcc.cc.lib;
|
||||
|
||||
@@ -170,7 +170,7 @@ lib.warn "The Nix-based build is deprecated" rec {
|
||||
ln -sf ${lean-all}/* .
|
||||
'';
|
||||
buildPhase = ''
|
||||
ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)|leanlaketest_reverse-ffi|leanruntest_timeIO' -j$NIX_BUILD_CORES
|
||||
ctest --output-junit test-results.xml --output-on-failure -E 'leancomptest_(doc_example|foreign)|leanlaketest_reverse-ffi' -j$NIX_BUILD_CORES
|
||||
'';
|
||||
installPhase = ''
|
||||
mkdir $out
|
||||
|
||||
3
releases_drafts/hashmap.md
Normal file
3
releases_drafts/hashmap.md
Normal file
@@ -0,0 +1,3 @@
|
||||
* The `Lean` module has switched from `Lean.HashMap` and `Lean.HashSet` to `Std.HashMap` and `Std.HashSet`. `Lean.HashMap` and `Lean.HashSet` are now deprecated and will be removed in a future release. Users of `Lean` APIs that interact with hash maps, for example `Lean.Environment.const2ModIdx`, might encounter minor breakage due to the following breaking changes from `Lean.HashMap` to `Std.HashMap`:
|
||||
* query functions use the term `get` instead of `find`,
|
||||
* the notation `map[key]` no longer returns an optional value but expects a proof that the key is present in the map instead. The previous behavior is available via the `map[key]?` notation.
|
||||
1
releases_drafts/libuv.md
Normal file
1
releases_drafts/libuv.md
Normal file
@@ -0,0 +1 @@
|
||||
* #4963 [LibUV](https://libuv.org/) is now required to build Lean. This change only affects developers who compile Lean themselves instead of obtaining toolchains via `elan`. We have updated the official build instructions with information on how to obtain LibUV on our supported platforms.
|
||||
@@ -17,7 +17,7 @@ for f in $(git ls-files src ':!:src/lake/*' ':!:src/Leanc.lean'); do
|
||||
done
|
||||
|
||||
# special handling for Lake files due to its nested directory
|
||||
# copy the README to ensure the `stage0/src/lake` directory is committed
|
||||
# copy the README to ensure the `stage0/src/lake` directory is comitted
|
||||
for f in $(git ls-files 'src/lake/Lake/*' src/lake/Lake.lean src/lake/LakeMain.lean src/lake/README.md ':!:src/lakefile.toml'); do
|
||||
if [[ $f == *.lean ]]; then
|
||||
f=${f#src/lake}
|
||||
|
||||
@@ -48,8 +48,6 @@ $CP llvm-host/lib/*/lib{c++,c++abi,unwind}.* llvm-host/lib/
|
||||
$CP -r llvm/include/*-*-* llvm-host/include/
|
||||
# glibc: use for linking (so Lean programs don't embed newer symbol versions), but not for running (because libc.so, librt.so, and ld.so must be compatible)!
|
||||
$CP $GLIBC/lib/libc_nonshared.a stage1/lib/glibc
|
||||
# libpthread_nonshared.a must be linked in order to be able to use `pthread_atfork(3)`. LibUV uses this function.
|
||||
$CP $GLIBC/lib/libpthread_nonshared.a stage1/lib/glibc
|
||||
for f in $GLIBC/lib/lib{c,dl,m,rt,pthread}-*; do b=$(basename $f); cp $f stage1/lib/glibc/${b%-*}.so; done
|
||||
OPTIONS=()
|
||||
echo -n " -DLEAN_STANDALONE=ON"
|
||||
@@ -64,8 +62,8 @@ fi
|
||||
# use `-nostdinc` to make sure headers are not visible by default (in particular, not to `#include_next` in the clang headers),
|
||||
# but do not change sysroot so users can still link against system libs
|
||||
echo -n " -DLEANC_INTERNAL_FLAGS='-nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang"
|
||||
echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/glibc ROOT/lib/glibc/libc_nonshared.a ROOT/lib/glibc/libpthread_nonshared.a -Wl,--as-needed -Wl,-Bstatic -lgmp -lunwind -luv -Wl,-Bdynamic -Wl,--no-as-needed -fuse-ld=lld'"
|
||||
echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -L ROOT/lib/glibc ROOT/lib/glibc/libc_nonshared.a -Wl,--as-needed -Wl,-Bstatic -lgmp -lunwind -luv -Wl,-Bdynamic -Wl,--no-as-needed -fuse-ld=lld'"
|
||||
# when not using the above flags, link GMP dynamically/as usual
|
||||
echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-Wl,--as-needed -lgmp -luv -lpthread -ldl -lrt -Wl,--no-as-needed'"
|
||||
echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-Wl,--as-needed -lgmp -luv -Wl,--no-as-needed'"
|
||||
# do not set `LEAN_CC` for tests
|
||||
echo -n " -DLEAN_TEST_VARS=''"
|
||||
|
||||
@@ -31,21 +31,15 @@ cp /clang64/lib/{crtbegin,crtend,crt2,dllcrt2}.o stage1/lib/
|
||||
# runtime
|
||||
(cd llvm; cp --parents lib/clang/*/lib/*/libclang_rt.builtins* ../stage1)
|
||||
# further dependencies
|
||||
# Note: even though we're linking against libraries like `libbcrypt.a` which appear to be static libraries from the file name,
|
||||
# we're not actually linking statically against the code.
|
||||
# Rather, `libbcrypt.a` is an import library (see https://en.wikipedia.org/wiki/Dynamic-link_library#Import_libraries) that just
|
||||
# tells the compiler how to dynamically link against `bcrypt.dll` (which is located in the System32 folder).
|
||||
# This distinction is relevant specifically for `libicu.a`/`icu.dll` because there we want updates to the time zone database to
|
||||
# be delivered to users via Windows Update without having to recompile Lean or Lean programs.
|
||||
cp /clang64/lib/lib{m,bcrypt,mingw32,moldname,mingwex,msvcrt,pthread,advapi32,shell32,user32,kernel32,ucrtbase,psapi,iphlpapi,userenv,ws2_32,dbghelp,ole32,icu}.* /clang64/lib/libgmp.a /clang64/lib/libuv.a llvm/lib/lib{c++,c++abi,unwind}.a stage1/lib/
|
||||
cp /clang64/lib/lib{m,bcrypt,mingw32,moldname,mingwex,msvcrt,pthread,advapi32,shell32,user32,kernel32,ucrtbase}.* /clang64/lib/libgmp.a /clang64/lib/libuv.a llvm/lib/lib{c++,c++abi,unwind}.a stage1/lib/
|
||||
echo -n " -DLEAN_STANDALONE=ON"
|
||||
echo -n " -DCMAKE_C_COMPILER=$PWD/stage1/bin/clang.exe -DCMAKE_C_COMPILER_WORKS=1 -DCMAKE_CXX_COMPILER=$PWD/llvm/bin/clang++.exe -DCMAKE_CXX_COMPILER_WORKS=1 -DLEAN_CXX_STDLIB='-lc++ -lc++abi'"
|
||||
echo -n " -DSTAGE0_CMAKE_C_COMPILER=clang -DSTAGE0_CMAKE_CXX_COMPILER=clang++"
|
||||
echo -n " -DLEAN_EXTRA_CXX_FLAGS='--sysroot $PWD/llvm -idirafter /clang64/include/'"
|
||||
echo -n " -DLEANC_INTERNAL_FLAGS='--sysroot ROOT -nostdinc -isystem ROOT/include/clang' -DLEANC_CC=ROOT/bin/clang.exe"
|
||||
echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -static-libgcc -Wl,-Bstatic -lgmp $(pkg-config --static --libs libuv) -lunwind -Wl,-Bdynamic -fuse-ld=lld'"
|
||||
# when not using the above flags, link GMP dynamically/as usual. Always link ICU dynamically.
|
||||
echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp $(pkg-config --libs libuv) -lucrtbase'"
|
||||
echo -n " -DLEANC_INTERNAL_LINKER_FLAGS='-L ROOT/lib -static-libgcc -Wl,-Bstatic -lgmp -luv -lunwind -Wl,-Bdynamic -fuse-ld=lld'"
|
||||
# when not using the above flags, link GMP dynamically/as usual
|
||||
echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp -luv -lucrtbase'"
|
||||
# do not set `LEAN_CC` for tests
|
||||
echo -n " -DAUTO_THREAD_FINALIZATION=OFF -DSTAGE0_AUTO_THREAD_FINALIZATION=OFF"
|
||||
echo -n " -DLEAN_TEST_VARS=''"
|
||||
|
||||
@@ -10,15 +10,13 @@ endif()
|
||||
include(ExternalProject)
|
||||
project(LEAN CXX C)
|
||||
set(LEAN_VERSION_MAJOR 4)
|
||||
set(LEAN_VERSION_MINOR 15)
|
||||
set(LEAN_VERSION_MINOR 12)
|
||||
set(LEAN_VERSION_PATCH 0)
|
||||
set(LEAN_VERSION_IS_RELEASE 0) # This number is 1 in the release revision, and 0 otherwise.
|
||||
set(LEAN_SPECIAL_VERSION_DESC "" CACHE STRING "Additional version description like 'nightly-2018-03-11'")
|
||||
set(LEAN_VERSION_STRING "${LEAN_VERSION_MAJOR}.${LEAN_VERSION_MINOR}.${LEAN_VERSION_PATCH}")
|
||||
if (LEAN_SPECIAL_VERSION_DESC)
|
||||
string(APPEND LEAN_VERSION_STRING "-${LEAN_SPECIAL_VERSION_DESC}")
|
||||
elseif (NOT LEAN_VERSION_IS_RELEASE)
|
||||
string(APPEND LEAN_VERSION_STRING "-pre")
|
||||
endif()
|
||||
|
||||
set(LEAN_PLATFORM_TARGET "" CACHE STRING "LLVM triple of the target platform")
|
||||
@@ -157,10 +155,6 @@ endif ()
|
||||
# We want explicit stack probes in huge Lean stack frames for robust stack overflow detection
|
||||
string(APPEND LEANC_EXTRA_FLAGS " -fstack-clash-protection")
|
||||
|
||||
# This makes signed integer overflow guaranteed to match 2's complement.
|
||||
string(APPEND CMAKE_CXX_FLAGS " -fwrapv")
|
||||
string(APPEND LEANC_EXTRA_FLAGS " -fwrapv")
|
||||
|
||||
if(NOT MULTI_THREAD)
|
||||
message(STATUS "Disabled multi-thread support, it will not be safe to run multiple threads in parallel")
|
||||
set(AUTO_THREAD_FINALIZATION OFF)
|
||||
@@ -249,77 +243,15 @@ if("${USE_GMP}" MATCHES "ON")
|
||||
endif()
|
||||
endif()
|
||||
|
||||
# LibUV
|
||||
if("${CMAKE_SYSTEM_NAME}" MATCHES "Emscripten")
|
||||
# Only on WebAssembly we compile LibUV ourselves
|
||||
set(LIBUV_EMSCRIPTEN_FLAGS "${EMSCRIPTEN_SETTINGS}")
|
||||
|
||||
# LibUV does not compile on WebAssembly without modifications because
|
||||
# building LibUV on a platform requires including stub implementations
|
||||
# for features not present on the target platform. This patch includes
|
||||
# the minimum amount of stub implementations needed for successfully
|
||||
# running Lean on WebAssembly and using LibUV's temporary file support.
|
||||
# It still leaves several symbols completely undefined: uv__fs_event_close,
|
||||
# uv__hrtime, uv__io_check_fd, uv__io_fork, uv__io_poll, uv__platform_invalidate_fd
|
||||
# uv__platform_loop_delete, uv__platform_loop_init. Making additional
|
||||
# LibUV features available on WebAssembly might require adapting the
|
||||
# patch to include additional LibUV source files.
|
||||
set(LIBUV_PATCH_IN "
|
||||
diff --git a/CMakeLists.txt b/CMakeLists.txt
|
||||
index 5e8e0166..f3b29134 100644
|
||||
--- a/CMakeLists.txt
|
||||
+++ b/CMakeLists.txt
|
||||
@@ -317,6 +317,11 @@ if(CMAKE_SYSTEM_NAME STREQUAL \"GNU\")
|
||||
src/unix/hurd.c)
|
||||
endif()
|
||||
|
||||
+if(CMAKE_SYSTEM_NAME STREQUAL \"Emscripten\")
|
||||
+ list(APPEND uv_sources
|
||||
+ src/unix/no-proctitle.c)
|
||||
+endif()
|
||||
+
|
||||
if(CMAKE_SYSTEM_NAME STREQUAL \"Linux\")
|
||||
list(APPEND uv_defines _GNU_SOURCE _POSIX_C_SOURCE=200112)
|
||||
list(APPEND uv_libraries dl rt)
|
||||
")
|
||||
string(REPLACE "\n" "\\n" LIBUV_PATCH ${LIBUV_PATCH_IN})
|
||||
|
||||
ExternalProject_add(libuv
|
||||
PREFIX libuv
|
||||
GIT_REPOSITORY https://github.com/libuv/libuv
|
||||
# Sync version with flake.nix
|
||||
GIT_TAG v1.48.0
|
||||
CMAKE_ARGS -DCMAKE_BUILD_TYPE=Release -DLIBUV_BUILD_TESTS=OFF -DLIBUV_BUILD_SHARED=OFF -DCMAKE_AR=${CMAKE_AR} -DCMAKE_TOOLCHAIN_FILE=${CMAKE_TOOLCHAIN_FILE} -DCMAKE_POSITION_INDEPENDENT_CODE=ON -DCMAKE_C_FLAGS=${LIBUV_EMSCRIPTEN_FLAGS}
|
||||
PATCH_COMMAND git reset --hard HEAD && printf "${LIBUV_PATCH}" > patch.diff && git apply patch.diff
|
||||
BUILD_IN_SOURCE ON
|
||||
INSTALL_COMMAND "")
|
||||
set(LIBUV_INCLUDE_DIR "${CMAKE_BINARY_DIR}/libuv/src/libuv/include")
|
||||
set(LIBUV_LIBRARIES "${CMAKE_BINARY_DIR}/libuv/src/libuv/libuv.a")
|
||||
else()
|
||||
if(NOT "${CMAKE_SYSTEM_NAME}" MATCHES "Emscripten")
|
||||
# LibUV
|
||||
find_package(LibUV 1.0.0 REQUIRED)
|
||||
include_directories(${LIBUV_INCLUDE_DIR})
|
||||
endif()
|
||||
include_directories(${LIBUV_INCLUDE_DIR})
|
||||
if(NOT LEAN_STANDALONE)
|
||||
string(APPEND LEAN_EXTRA_LINKER_FLAGS " ${LIBUV_LIBRARIES}")
|
||||
endif()
|
||||
|
||||
# Windows SDK (for ICU)
|
||||
if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
|
||||
# Pass 'tools' to skip MSVC version check (as MSVC/Visual Studio is not necessarily installed)
|
||||
find_package(WindowsSDK REQUIRED COMPONENTS tools)
|
||||
|
||||
# This will give a semicolon-separated list of include directories
|
||||
get_windowssdk_include_dirs(${WINDOWSSDK_LATEST_DIR} WINDOWSSDK_INCLUDE_DIRS)
|
||||
|
||||
# To successfully build against Windows SDK headers, the Windows SDK headers must have lower
|
||||
# priority than other system headers, so use `-idirafter`. Unfortunately, CMake does not
|
||||
# support this using `include_directories`.
|
||||
string(REPLACE ";" "\" -idirafter \"" WINDOWSSDK_INCLUDE_DIRS "${WINDOWSSDK_INCLUDE_DIRS}")
|
||||
string(APPEND CMAKE_CXX_FLAGS " -idirafter \"${WINDOWSSDK_INCLUDE_DIRS}\"")
|
||||
|
||||
string(APPEND LEAN_EXTRA_LINKER_FLAGS " -licu")
|
||||
endif()
|
||||
|
||||
# ccache
|
||||
if(CCACHE AND NOT CMAKE_CXX_COMPILER_LAUNCHER AND NOT CMAKE_C_COMPILER_LAUNCHER)
|
||||
find_program(CCACHE_PATH ccache)
|
||||
@@ -503,7 +435,7 @@ endif()
|
||||
# Git HASH
|
||||
if(USE_GITHASH)
|
||||
include(GetGitRevisionDescription)
|
||||
get_git_head_revision(GIT_REFSPEC GIT_SHA1 ALLOW_LOOKING_ABOVE_CMAKE_SOURCE_DIR)
|
||||
get_git_head_revision(GIT_REFSPEC GIT_SHA1)
|
||||
if(${GIT_SHA1} MATCHES "GITDIR-NOTFOUND")
|
||||
message(STATUS "Failed to read git_sha1")
|
||||
set(GIT_SHA1 "")
|
||||
@@ -590,10 +522,6 @@ if(${STAGE} GREATER 1)
|
||||
endif()
|
||||
else()
|
||||
add_subdirectory(runtime)
|
||||
if("${CMAKE_SYSTEM_NAME}" MATCHES "Emscripten")
|
||||
add_dependencies(leanrt libuv)
|
||||
add_dependencies(leanrt_initial-exec libuv)
|
||||
endif()
|
||||
|
||||
add_subdirectory(util)
|
||||
set(LEAN_OBJS ${LEAN_OBJS} $<TARGET_OBJECTS:util>)
|
||||
@@ -634,10 +562,7 @@ if (${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
|
||||
# simple. (And we are not interested in `Lake` anyway.) To use dynamic
|
||||
# linking, we would probably have to set MAIN_MODULE=2 on `leanshared`,
|
||||
# SIDE_MODULE=2 on `lean`, and set CMAKE_SHARED_LIBRARY_SUFFIX to ".js".
|
||||
# We set `ERROR_ON_UNDEFINED_SYMBOLS=0` because our build of LibUV does not
|
||||
# define all symbols, see the comment about LibUV on WebAssembly further up
|
||||
# in this file.
|
||||
string(APPEND LEAN_EXE_LINKER_FLAGS " ${LIB}/temp/libleanshell.a ${TOOLCHAIN_STATIC_LINKER_FLAGS} ${EMSCRIPTEN_SETTINGS} -lnodefs.js -s EXIT_RUNTIME=1 -s MAIN_MODULE=1 -s LINKABLE=1 -s EXPORT_ALL=1 -s ERROR_ON_UNDEFINED_SYMBOLS=0")
|
||||
string(APPEND LEAN_EXE_LINKER_FLAGS " ${LIB}/temp/libleanshell.a ${TOOLCHAIN_STATIC_LINKER_FLAGS} ${EMSCRIPTEN_SETTINGS} -lnodefs.js -s EXIT_RUNTIME=1 -s MAIN_MODULE=1 -s LINKABLE=1 -s EXPORT_ALL=1")
|
||||
endif()
|
||||
|
||||
# Build the compiler using the bootstrapped C sources for stage0, and use
|
||||
|
||||
@@ -35,5 +35,3 @@ import Init.Ext
|
||||
import Init.Omega
|
||||
import Init.MacroTrace
|
||||
import Init.Grind
|
||||
import Init.While
|
||||
import Init.Syntax
|
||||
|
||||
@@ -40,23 +40,21 @@ theorem apply_ite (f : α → β) (P : Prop) [Decidable P] (x y : α) :
|
||||
/-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/
|
||||
@[simp] theorem dite_eq_ite [Decidable P] : (dite P (fun _ => a) fun _ => b) = ite P a b := rfl
|
||||
|
||||
@[deprecated "Use `ite_eq_right_iff`" (since := "2024-09-18")]
|
||||
-- We don't mark this as `simp` as it is already handled by `ite_eq_right_iff`.
|
||||
theorem ite_some_none_eq_none [Decidable P] :
|
||||
(if P then some x else none) = none ↔ ¬ P := by
|
||||
simp only [ite_eq_right_iff, reduceCtorEq]
|
||||
rfl
|
||||
|
||||
@[deprecated "Use `Option.ite_none_right_eq_some`" (since := "2024-09-18")]
|
||||
theorem ite_some_none_eq_some [Decidable P] :
|
||||
@[simp] theorem ite_some_none_eq_some [Decidable P] :
|
||||
(if P then some x else none) = some y ↔ P ∧ x = y := by
|
||||
split <;> simp_all
|
||||
|
||||
@[deprecated "Use `dite_eq_right_iff" (since := "2024-09-18")]
|
||||
-- This is not marked as `simp` as it is already handled by `dite_eq_right_iff`.
|
||||
theorem dite_some_none_eq_none [Decidable P] {x : P → α} :
|
||||
(if h : P then some (x h) else none) = none ↔ ¬P := by
|
||||
simp
|
||||
|
||||
@[deprecated "Use `Option.dite_none_right_eq_some`" (since := "2024-09-18")]
|
||||
theorem dite_some_none_eq_some [Decidable P] {x : P → α} {y : α} :
|
||||
@[simp] theorem dite_some_none_eq_some [Decidable P] {x : P → α} {y : α} :
|
||||
(if h : P then some (x h) else none) = some y ↔ ∃ h : P, x h = y := by
|
||||
by_cases h : P <;> simp [h]
|
||||
|
||||
@@ -80,8 +80,6 @@ noncomputable scoped instance (priority := low) propDecidable (a : Prop) : Decid
|
||||
noncomputable def decidableInhabited (a : Prop) : Inhabited (Decidable a) where
|
||||
default := inferInstance
|
||||
|
||||
instance (a : Prop) : Nonempty (Decidable a) := ⟨propDecidable a⟩
|
||||
|
||||
noncomputable def typeDecidableEq (α : Sort u) : DecidableEq α :=
|
||||
fun _ _ => inferInstance
|
||||
|
||||
@@ -123,11 +121,11 @@ theorem propComplete (a : Prop) : a = True ∨ a = False :=
|
||||
| Or.inl ha => Or.inl (eq_true ha)
|
||||
| Or.inr hn => Or.inr (eq_false hn)
|
||||
|
||||
-- this supersedes byCases in Decidable
|
||||
-- this supercedes byCases in Decidable
|
||||
theorem byCases {p q : Prop} (hpq : p → q) (hnpq : ¬p → q) : q :=
|
||||
Decidable.byCases (dec := propDecidable _) hpq hnpq
|
||||
|
||||
-- this supersedes byContradiction in Decidable
|
||||
-- this supercedes byContradiction in Decidable
|
||||
theorem byContradiction {p : Prop} (h : ¬p → False) : p :=
|
||||
Decidable.byContradiction (dec := propDecidable _) h
|
||||
|
||||
|
||||
@@ -8,42 +8,6 @@ import Init.Core
|
||||
|
||||
universe u v w
|
||||
|
||||
/--
|
||||
A `ForIn'` instance, which handles `for h : x in c do`,
|
||||
can also handle `for x in x do` by ignoring `h`, and so provides a `ForIn` instance.
|
||||
|
||||
Note that this instance will cause a potentially non-defeq duplication if both `ForIn` and `ForIn'`
|
||||
instances are provided for the same type.
|
||||
-/
|
||||
-- We set the priority to 500 so it is below the default,
|
||||
-- but still above the low priority instance from `Stream`.
|
||||
instance (priority := 500) instForInOfForIn' [ForIn' m ρ α d] : ForIn m ρ α where
|
||||
forIn x b f := forIn' x b fun a _ => f a
|
||||
|
||||
@[simp] theorem forIn'_eq_forIn [d : Membership α ρ] [ForIn' m ρ α d] {β} [Monad m] (x : ρ) (b : β)
|
||||
(f : (a : α) → a ∈ x → β → m (ForInStep β)) (g : (a : α) → β → m (ForInStep β))
|
||||
(h : ∀ a m b, f a m b = g a b) :
|
||||
forIn' x b f = forIn x b g := by
|
||||
simp [instForInOfForIn']
|
||||
congr
|
||||
apply funext
|
||||
intro a
|
||||
apply funext
|
||||
intro m
|
||||
apply funext
|
||||
intro b
|
||||
simp [h]
|
||||
rfl
|
||||
|
||||
/-- Extract the value from a `ForInStep`, ignoring whether it is `done` or `yield`. -/
|
||||
def ForInStep.value (x : ForInStep α) : α :=
|
||||
match x with
|
||||
| ForInStep.done b => b
|
||||
| ForInStep.yield b => b
|
||||
|
||||
@[simp] theorem ForInStep.value_done (b : β) : (ForInStep.done b).value = b := rfl
|
||||
@[simp] theorem ForInStep.value_yield (b : β) : (ForInStep.yield b).value = b := rfl
|
||||
|
||||
@[reducible]
|
||||
def Functor.mapRev {f : Type u → Type v} [Functor f] {α β : Type u} : f α → (α → β) → f β :=
|
||||
fun a f => f <$> a
|
||||
|
||||
@@ -33,10 +33,6 @@ attribute [simp] id_map
|
||||
@[simp] theorem id_map' [Functor m] [LawfulFunctor m] (x : m α) : (fun a => a) <$> x = x :=
|
||||
id_map x
|
||||
|
||||
@[simp] theorem Functor.map_map [Functor f] [LawfulFunctor f] (m : α → β) (g : β → γ) (x : f α) :
|
||||
g <$> m <$> x = (fun a => g (m a)) <$> x :=
|
||||
(comp_map _ _ _).symm
|
||||
|
||||
/--
|
||||
The `Applicative` typeclass only contains the operations of an applicative functor.
|
||||
`LawfulApplicative` further asserts that these operations satisfy the laws of an applicative functor:
|
||||
@@ -87,16 +83,12 @@ class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m
|
||||
seq_assoc x g h := (by simp [← bind_pure_comp, ← bind_map, bind_assoc, pure_bind])
|
||||
|
||||
export LawfulMonad (bind_pure_comp bind_map pure_bind bind_assoc)
|
||||
attribute [simp] pure_bind bind_assoc bind_pure_comp
|
||||
attribute [simp] pure_bind bind_assoc
|
||||
|
||||
@[simp] theorem bind_pure [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x := by
|
||||
show x >>= (fun a => pure (id a)) = x
|
||||
rw [bind_pure_comp, id_map]
|
||||
|
||||
/--
|
||||
Use `simp [← bind_pure_comp]` rather than `simp [map_eq_pure_bind]`,
|
||||
as `bind_pure_comp` is in the default simp set, so also using `map_eq_pure_bind` would cause a loop.
|
||||
-/
|
||||
theorem map_eq_pure_bind [Monad m] [LawfulMonad m] (f : α → β) (x : m α) : f <$> x = x >>= fun a => pure (f a) := by
|
||||
rw [← bind_pure_comp]
|
||||
|
||||
@@ -117,24 +109,10 @@ theorem seq_eq_bind {α β : Type u} [Monad m] [LawfulMonad m] (mf : m (α →
|
||||
|
||||
theorem seqRight_eq_bind [Monad m] [LawfulMonad m] (x : m α) (y : m β) : x *> y = x >>= fun _ => y := by
|
||||
rw [seqRight_eq]
|
||||
simp only [map_eq_pure_bind, const, seq_eq_bind_map, bind_assoc, pure_bind, id_eq, bind_pure]
|
||||
simp [map_eq_pure_bind, seq_eq_bind_map, const]
|
||||
|
||||
theorem seqLeft_eq_bind [Monad m] [LawfulMonad m] (x : m α) (y : m β) : x <* y = x >>= fun a => y >>= fun _ => pure a := by
|
||||
rw [seqLeft_eq]
|
||||
simp only [map_eq_pure_bind, seq_eq_bind_map, bind_assoc, pure_bind, const_apply]
|
||||
|
||||
@[simp] theorem map_bind [Monad m] [LawfulMonad m] (f : β → γ) (x : m α) (g : α → m β) :
|
||||
f <$> (x >>= g) = x >>= fun a => f <$> g a := by
|
||||
rw [← bind_pure_comp, LawfulMonad.bind_assoc]
|
||||
simp [bind_pure_comp]
|
||||
|
||||
@[simp] theorem bind_map_left [Monad m] [LawfulMonad m] (f : α → β) (x : m α) (g : β → m γ) :
|
||||
((f <$> x) >>= fun b => g b) = (x >>= fun a => g (f a)) := by
|
||||
rw [← bind_pure_comp]
|
||||
simp only [bind_assoc, pure_bind]
|
||||
|
||||
@[simp] theorem Functor.map_unit [Monad m] [LawfulMonad m] {a : m PUnit} : (fun _ => PUnit.unit) <$> a = a := by
|
||||
simp [map]
|
||||
rw [seqLeft_eq]; simp [map_eq_pure_bind, seq_eq_bind_map]
|
||||
|
||||
/--
|
||||
An alternative constructor for `LawfulMonad` which has more
|
||||
@@ -183,9 +161,9 @@ end Id
|
||||
|
||||
instance : LawfulMonad Option := LawfulMonad.mk'
|
||||
(id_map := fun x => by cases x <;> rfl)
|
||||
(pure_bind := fun _ _ => rfl)
|
||||
(bind_assoc := fun x _ _ => by cases x <;> rfl)
|
||||
(bind_pure_comp := fun _ x => by cases x <;> rfl)
|
||||
(pure_bind := fun x f => rfl)
|
||||
(bind_assoc := fun x f g => by cases x <;> rfl)
|
||||
(bind_pure_comp := fun f x => by cases x <;> rfl)
|
||||
|
||||
instance : LawfulApplicative Option := inferInstance
|
||||
instance : LawfulFunctor Option := inferInstance
|
||||
|
||||
@@ -7,7 +7,6 @@ prelude
|
||||
import Init.Control.Lawful.Basic
|
||||
import Init.Control.Except
|
||||
import Init.Control.StateRef
|
||||
import Init.Ext
|
||||
|
||||
open Function
|
||||
|
||||
@@ -15,7 +14,7 @@ open Function
|
||||
|
||||
namespace ExceptT
|
||||
|
||||
@[ext] theorem ext {x y : ExceptT ε m α} (h : x.run = y.run) : x = y := by
|
||||
theorem ext {x y : ExceptT ε m α} (h : x.run = y.run) : x = y := by
|
||||
simp [run] at h
|
||||
assumption
|
||||
|
||||
@@ -26,7 +25,7 @@ namespace ExceptT
|
||||
@[simp] theorem run_throw [Monad m] : run (throw e : ExceptT ε m β) = pure (Except.error e) := rfl
|
||||
|
||||
@[simp] theorem run_bind_lift [Monad m] [LawfulMonad m] (x : m α) (f : α → ExceptT ε m β) : run (ExceptT.lift x >>= f : ExceptT ε m β) = x >>= fun a => run (f a) := by
|
||||
simp [ExceptT.run, ExceptT.lift, bind, ExceptT.bind, ExceptT.mk, ExceptT.bindCont]
|
||||
simp[ExceptT.run, ExceptT.lift, bind, ExceptT.bind, ExceptT.mk, ExceptT.bindCont, map_eq_pure_bind]
|
||||
|
||||
@[simp] theorem bind_throw [Monad m] [LawfulMonad m] (f : α → ExceptT ε m β) : (throw e >>= f) = throw e := by
|
||||
simp [throw, throwThe, MonadExceptOf.throw, bind, ExceptT.bind, ExceptT.bindCont, ExceptT.mk]
|
||||
@@ -44,7 +43,7 @@ theorem run_bind [Monad m] (x : ExceptT ε m α)
|
||||
|
||||
@[simp] theorem run_map [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α)
|
||||
: (f <$> x).run = Except.map f <$> x.run := by
|
||||
simp [Functor.map, ExceptT.map, ←bind_pure_comp]
|
||||
simp [Functor.map, ExceptT.map, map_eq_pure_bind]
|
||||
apply bind_congr
|
||||
intro a; cases a <;> simp [Except.map]
|
||||
|
||||
@@ -63,7 +62,7 @@ protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad
|
||||
intro
|
||||
| Except.error _ => simp
|
||||
| Except.ok _ =>
|
||||
simp [←bind_pure_comp]; apply bind_congr; intro b;
|
||||
simp [map_eq_pure_bind]; apply bind_congr; intro b;
|
||||
cases b <;> simp [comp, Except.map, const]
|
||||
|
||||
protected theorem seqRight_eq [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x *> y = const α id <$> x <*> y := by
|
||||
@@ -85,19 +84,14 @@ instance [Monad m] [LawfulMonad m] : LawfulMonad (ExceptT ε m) where
|
||||
pure_bind := by intros; apply ext; simp [run_bind]
|
||||
bind_assoc := by intros; apply ext; simp [run_bind]; apply bind_congr; intro a; cases a <;> simp
|
||||
|
||||
@[simp] theorem map_throw [Monad m] [LawfulMonad m] {α β : Type _} (f : α → β) (e : ε) :
|
||||
f <$> (throw e : ExceptT ε m α) = (throw e : ExceptT ε m β) := by
|
||||
simp only [ExceptT.instMonad, ExceptT.map, ExceptT.mk, throw, throwThe, MonadExceptOf.throw,
|
||||
pure_bind]
|
||||
|
||||
end ExceptT
|
||||
|
||||
/-! # Except -/
|
||||
|
||||
instance : LawfulMonad (Except ε) := LawfulMonad.mk'
|
||||
(id_map := fun x => by cases x <;> rfl)
|
||||
(pure_bind := fun _ _ => rfl)
|
||||
(bind_assoc := fun a _ _ => by cases a <;> rfl)
|
||||
(pure_bind := fun a f => rfl)
|
||||
(bind_assoc := fun a f g => by cases a <;> rfl)
|
||||
|
||||
instance : LawfulApplicative (Except ε) := inferInstance
|
||||
instance : LawfulFunctor (Except ε) := inferInstance
|
||||
@@ -106,7 +100,7 @@ instance : LawfulFunctor (Except ε) := inferInstance
|
||||
|
||||
namespace ReaderT
|
||||
|
||||
@[ext] theorem ext {x y : ReaderT ρ m α} (h : ∀ ctx, x.run ctx = y.run ctx) : x = y := by
|
||||
theorem ext {x y : ReaderT ρ m α} (h : ∀ ctx, x.run ctx = y.run ctx) : x = y := by
|
||||
simp [run] at h
|
||||
exact funext h
|
||||
|
||||
@@ -168,7 +162,7 @@ instance [Monad m] [LawfulMonad m] : LawfulMonad (StateRefT' ω σ m) :=
|
||||
|
||||
namespace StateT
|
||||
|
||||
@[ext] theorem ext {x y : StateT σ m α} (h : ∀ s, x.run s = y.run s) : x = y :=
|
||||
theorem ext {x y : StateT σ m α} (h : ∀ s, x.run s = y.run s) : x = y :=
|
||||
funext h
|
||||
|
||||
@[simp] theorem run'_eq [Monad m] (x : StateT σ m α) (s : σ) : run' x s = (·.1) <$> run x s :=
|
||||
@@ -181,7 +175,7 @@ namespace StateT
|
||||
simp [bind, StateT.bind, run]
|
||||
|
||||
@[simp] theorem run_map {α β σ : Type u} [Monad m] [LawfulMonad m] (f : α → β) (x : StateT σ m α) (s : σ) : (f <$> x).run s = (fun (p : α × σ) => (f p.1, p.2)) <$> x.run s := by
|
||||
simp [Functor.map, StateT.map, run, ←bind_pure_comp]
|
||||
simp [Functor.map, StateT.map, run, map_eq_pure_bind]
|
||||
|
||||
@[simp] theorem run_get [Monad m] (s : σ) : (get : StateT σ m σ).run s = pure (s, s) := rfl
|
||||
|
||||
@@ -216,13 +210,13 @@ theorem run_bind_lift {α σ : Type u} [Monad m] [LawfulMonad m] (x : m α) (f :
|
||||
|
||||
theorem seqRight_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x *> y = const α id <$> x <*> y := by
|
||||
apply ext; intro s
|
||||
simp [←bind_pure_comp, const]
|
||||
simp [map_eq_pure_bind, const]
|
||||
apply bind_congr; intro p; cases p
|
||||
simp [Prod.eta]
|
||||
|
||||
theorem seqLeft_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x <* y = const β <$> x <*> y := by
|
||||
apply ext; intro s
|
||||
simp [←bind_pure_comp]
|
||||
simp [map_eq_pure_bind]
|
||||
|
||||
instance [Monad m] [LawfulMonad m] : LawfulMonad (StateT σ m) where
|
||||
id_map := by intros; apply ext; intros; simp[Prod.eta]
|
||||
@@ -230,7 +224,7 @@ instance [Monad m] [LawfulMonad m] : LawfulMonad (StateT σ m) where
|
||||
seqLeft_eq := seqLeft_eq
|
||||
seqRight_eq := seqRight_eq
|
||||
pure_seq := by intros; apply ext; intros; simp
|
||||
bind_pure_comp := by intros; apply ext; intros; simp
|
||||
bind_pure_comp := by intros; apply ext; intros; simp; apply LawfulMonad.bind_pure_comp
|
||||
bind_map := by intros; rfl
|
||||
pure_bind := by intros; apply ext; intros; simp
|
||||
bind_assoc := by intros; apply ext; intros; simp
|
||||
|
||||
@@ -6,7 +6,8 @@ Authors: Leonardo de Moura, Sebastian Ullrich
|
||||
The State monad transformer using IO references.
|
||||
-/
|
||||
prelude
|
||||
import Init.System.ST
|
||||
import Init.System.IO
|
||||
import Init.Control.State
|
||||
|
||||
def StateRefT' (ω : Type) (σ : Type) (m : Type → Type) (α : Type) : Type := ReaderT (ST.Ref ω σ) m α
|
||||
|
||||
|
||||
@@ -7,7 +7,6 @@ Notation for operators defined at Prelude.lean
|
||||
-/
|
||||
prelude
|
||||
import Init.Tactics
|
||||
import Init.Meta
|
||||
|
||||
namespace Lean.Parser.Tactic.Conv
|
||||
|
||||
@@ -47,20 +46,12 @@ scoped syntax (name := withAnnotateState)
|
||||
/-- `skip` does nothing. -/
|
||||
syntax (name := skip) "skip" : conv
|
||||
|
||||
/--
|
||||
Traverses into the left subterm of a binary operator.
|
||||
|
||||
In general, for an `n`-ary operator, it traverses into the second to last argument.
|
||||
It is a synonym for `arg -2`.
|
||||
-/
|
||||
/-- Traverses into the left subterm of a binary operator.
|
||||
(In general, for an `n`-ary operator, it traverses into the second to last argument.) -/
|
||||
syntax (name := lhs) "lhs" : conv
|
||||
|
||||
/--
|
||||
Traverses into the right subterm of a binary operator.
|
||||
|
||||
In general, for an `n`-ary operator, it traverses into the last argument.
|
||||
It is a synonym for `arg -1`.
|
||||
-/
|
||||
/-- Traverses into the right subterm of a binary operator.
|
||||
(In general, for an `n`-ary operator, it traverses into the last argument.) -/
|
||||
syntax (name := rhs) "rhs" : conv
|
||||
|
||||
/-- Traverses into the function of a (unary) function application.
|
||||
@@ -83,17 +74,13 @@ subgoals for all the function arguments. For example, if the target is `f x y` t
|
||||
`congr` produces two subgoals, one for `x` and one for `y`. -/
|
||||
syntax (name := congr) "congr" : conv
|
||||
|
||||
syntax argArg := "@"? "-"? num
|
||||
|
||||
/--
|
||||
* `arg i` traverses into the `i`'th argument of the target. For example if the
|
||||
target is `f a b c d` then `arg 1` traverses to `a` and `arg 3` traverses to `c`.
|
||||
The index may be negative; `arg -1` traverses into the last argument,
|
||||
`arg -2` into the second-to-last argument, and so on.
|
||||
* `arg @i` is the same as `arg i` but it counts all arguments instead of just the
|
||||
explicit arguments.
|
||||
* `arg 0` traverses into the function. If the target is `f a b c d`, `arg 0` traverses into `f`. -/
|
||||
syntax (name := arg) "arg " argArg : conv
|
||||
syntax (name := arg) "arg " "@"? num : conv
|
||||
|
||||
/-- `ext x` traverses into a binder (a `fun x => e` or `∀ x, e` expression)
|
||||
to target `e`, introducing name `x` in the process. -/
|
||||
@@ -143,11 +130,11 @@ For example, if we are searching for `f _` in `f (f a) = f b`:
|
||||
syntax (name := pattern) "pattern " (occs)? term : conv
|
||||
|
||||
/-- `rw [thm]` rewrites the target using `thm`. See the `rw` tactic for more information. -/
|
||||
syntax (name := rewrite) "rewrite" optConfig rwRuleSeq : conv
|
||||
syntax (name := rewrite) "rewrite" (config)? rwRuleSeq : conv
|
||||
|
||||
/-- `simp [thm]` performs simplification using `thm` and marked `@[simp]` lemmas.
|
||||
See the `simp` tactic for more information. -/
|
||||
syntax (name := simp) "simp" optConfig (discharger)? (&" only")?
|
||||
syntax (name := simp) "simp" (config)? (discharger)? (&" only")?
|
||||
(" [" withoutPosition((simpStar <|> simpErase <|> simpLemma),*) "]")? : conv
|
||||
|
||||
/--
|
||||
@@ -164,7 +151,7 @@ example (a : Nat): (0 + 0) = a - a := by
|
||||
rw [← Nat.sub_self a]
|
||||
```
|
||||
-/
|
||||
syntax (name := dsimp) "dsimp" optConfig (discharger)? (&" only")?
|
||||
syntax (name := dsimp) "dsimp" (config)? (discharger)? (&" only")?
|
||||
(" [" withoutPosition((simpErase <|> simpLemma),*) "]")? : conv
|
||||
|
||||
/-- `simp_match` simplifies match expressions. For example,
|
||||
@@ -260,12 +247,12 @@ macro (name := failIfSuccess) tk:"fail_if_success " s:convSeq : conv =>
|
||||
|
||||
/-- `rw [rules]` applies the given list of rewrite rules to the target.
|
||||
See the `rw` tactic for more information. -/
|
||||
macro "rw" c:optConfig s:rwRuleSeq : conv => `(conv| rewrite $c:optConfig $s)
|
||||
macro "rw" c:(config)? s:rwRuleSeq : conv => `(conv| rewrite $[$c]? $s)
|
||||
|
||||
/-- `erw [rules]` is a shorthand for `rw (transparency := .default) [rules]`.
|
||||
/-- `erw [rules]` is a shorthand for `rw (config := { transparency := .default }) [rules]`.
|
||||
This does rewriting up to unfolding of regular definitions (by comparison to regular `rw`
|
||||
which only unfolds `@[reducible]` definitions). -/
|
||||
macro "erw" c:optConfig s:rwRuleSeq : conv => `(conv| rw $[$(getConfigItems c)]* (transparency := .default) $s:rwRuleSeq)
|
||||
macro "erw" s:rwRuleSeq : conv => `(conv| rw (config := { transparency := .default }) $s)
|
||||
|
||||
/-- `args` traverses into all arguments. Synonym for `congr`. -/
|
||||
macro "args" : conv => `(conv| congr)
|
||||
@@ -276,7 +263,7 @@ macro "right" : conv => `(conv| rhs)
|
||||
/-- `intro` traverses into binders. Synonym for `ext`. -/
|
||||
macro "intro" xs:(ppSpace colGt ident)* : conv => `(conv| ext $xs*)
|
||||
|
||||
syntax enterArg := ident <|> argArg
|
||||
syntax enterArg := ident <|> ("@"? num)
|
||||
|
||||
/-- `enter [arg, ...]` is a compact way to describe a path to a subterm.
|
||||
It is a shorthand for other conv tactics as follows:
|
||||
@@ -285,7 +272,12 @@ It is a shorthand for other conv tactics as follows:
|
||||
* `enter [x]` (where `x` is an identifier) is equivalent to `ext x`.
|
||||
For example, given the target `f (g a (fun x => x b))`, `enter [1, 2, x, 1]`
|
||||
will traverse to the subterm `b`. -/
|
||||
syntax (name := enter) "enter" " [" withoutPosition(enterArg,+) "]" : conv
|
||||
syntax "enter" " [" withoutPosition(enterArg,+) "]" : conv
|
||||
macro_rules
|
||||
| `(conv| enter [$i:num]) => `(conv| arg $i)
|
||||
| `(conv| enter [@$i]) => `(conv| arg @$i)
|
||||
| `(conv| enter [$id:ident]) => `(conv| ext $id)
|
||||
| `(conv| enter [$arg, $args,*]) => `(conv| (enter [$arg]; enter [$args,*]))
|
||||
|
||||
/-- The `apply thm` conv tactic is the same as `apply thm` the tactic.
|
||||
There are no restrictions on `thm`, but strange results may occur if `thm`
|
||||
|
||||
@@ -324,6 +324,7 @@ class ForIn' (m : Type u₁ → Type u₂) (ρ : Type u) (α : outParam (Type v)
|
||||
|
||||
export ForIn' (forIn')
|
||||
|
||||
|
||||
/--
|
||||
Auxiliary type used to compile `do` notation. It is used when compiling a do block
|
||||
nested inside a combinator like `tryCatch`. It encodes the possible ways the
|
||||
@@ -822,9 +823,6 @@ theorem iff_iff_implies_and_implies {a b : Prop} : (a ↔ b) ↔ (a → b) ∧ (
|
||||
protected theorem Iff.rfl {a : Prop} : a ↔ a :=
|
||||
Iff.refl a
|
||||
|
||||
-- And, also for backward compatibility, we try `Iff.rfl.` using `exact` (see #5366)
|
||||
macro_rules | `(tactic| rfl) => `(tactic| exact Iff.rfl)
|
||||
|
||||
theorem Iff.of_eq (h : a = b) : a ↔ b := h ▸ Iff.rfl
|
||||
|
||||
theorem Iff.trans (h₁ : a ↔ b) (h₂ : b ↔ c) : a ↔ c :=
|
||||
@@ -837,9 +835,6 @@ instance : Trans Iff Iff Iff where
|
||||
theorem Eq.comm {a b : α} : a = b ↔ b = a := Iff.intro Eq.symm Eq.symm
|
||||
theorem eq_comm {a b : α} : a = b ↔ b = a := Eq.comm
|
||||
|
||||
theorem HEq.comm {a : α} {b : β} : HEq a b ↔ HEq b a := Iff.intro HEq.symm HEq.symm
|
||||
theorem heq_comm {a : α} {b : β} : HEq a b ↔ HEq b a := HEq.comm
|
||||
|
||||
@[symm] theorem Iff.symm (h : a ↔ b) : b ↔ a := Iff.intro h.mpr h.mp
|
||||
theorem Iff.comm: (a ↔ b) ↔ (b ↔ a) := Iff.intro Iff.symm Iff.symm
|
||||
theorem iff_comm : (a ↔ b) ↔ (b ↔ a) := Iff.comm
|
||||
@@ -861,21 +856,16 @@ theorem Exists.elim {α : Sort u} {p : α → Prop} {b : Prop}
|
||||
|
||||
/-! # Decidable -/
|
||||
|
||||
@[simp] theorem decide_true (h : Decidable True) : @decide True h = true :=
|
||||
theorem decide_true_eq_true (h : Decidable True) : @decide True h = true :=
|
||||
match h with
|
||||
| isTrue _ => rfl
|
||||
| isFalse h => False.elim <| h ⟨⟩
|
||||
|
||||
@[simp] theorem decide_false (h : Decidable False) : @decide False h = false :=
|
||||
theorem decide_false_eq_false (h : Decidable False) : @decide False h = false :=
|
||||
match h with
|
||||
| isFalse _ => rfl
|
||||
| isTrue h => False.elim h
|
||||
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated decide_true (since := "2024-11-05")] abbrev decide_true_eq_true := decide_true
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated decide_false (since := "2024-11-05")] abbrev decide_false_eq_false := decide_false
|
||||
|
||||
/-- Similar to `decide`, but uses an explicit instance -/
|
||||
@[inline] def toBoolUsing {p : Prop} (d : Decidable p) : Bool :=
|
||||
decide (h := d)
|
||||
@@ -1201,21 +1191,6 @@ end
|
||||
|
||||
/-! # Product -/
|
||||
|
||||
instance [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (α × β) :=
|
||||
Nonempty.elim h1 fun x =>
|
||||
Nonempty.elim h2 fun y =>
|
||||
⟨(x, y)⟩
|
||||
|
||||
instance [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (MProd α β) :=
|
||||
Nonempty.elim h1 fun x =>
|
||||
Nonempty.elim h2 fun y =>
|
||||
⟨⟨x, y⟩⟩
|
||||
|
||||
instance [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (PProd α β) :=
|
||||
Nonempty.elim h1 fun x =>
|
||||
Nonempty.elim h2 fun y =>
|
||||
⟨⟨x, y⟩⟩
|
||||
|
||||
instance [Inhabited α] [Inhabited β] : Inhabited (α × β) where
|
||||
default := (default, default)
|
||||
|
||||
@@ -1389,7 +1364,6 @@ gen_injective_theorems% Except
|
||||
gen_injective_theorems% EStateM.Result
|
||||
gen_injective_theorems% Lean.Name
|
||||
gen_injective_theorems% Lean.Syntax
|
||||
gen_injective_theorems% BitVec
|
||||
|
||||
theorem Nat.succ.inj {m n : Nat} : m.succ = n.succ → m = n :=
|
||||
fun x => Nat.noConfusion x id
|
||||
@@ -1869,8 +1843,7 @@ section
|
||||
variable {α : Type u}
|
||||
variable (r : α → α → Prop)
|
||||
|
||||
instance Quotient.decidableEq {α : Sort u} {s : Setoid α} [d : ∀ (a b : α), Decidable (a ≈ b)]
|
||||
: DecidableEq (Quotient s) :=
|
||||
instance {α : Sort u} {s : Setoid α} [d : ∀ (a b : α), Decidable (a ≈ b)] : DecidableEq (Quotient s) :=
|
||||
fun (q₁ q₂ : Quotient s) =>
|
||||
Quotient.recOnSubsingleton₂ q₁ q₂
|
||||
fun a₁ a₂ =>
|
||||
@@ -1902,8 +1875,7 @@ theorem funext {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x}
|
||||
show extfunApp (Quot.mk eqv f) = extfunApp (Quot.mk eqv g)
|
||||
exact congrArg extfunApp (Quot.sound h)
|
||||
|
||||
instance Pi.instSubsingleton {α : Sort u} {β : α → Sort v} [∀ a, Subsingleton (β a)] :
|
||||
Subsingleton (∀ a, β a) where
|
||||
instance {α : Sort u} {β : α → Sort v} [∀ a, Subsingleton (β a)] : Subsingleton (∀ a, β a) where
|
||||
allEq f g := funext fun a => Subsingleton.elim (f a) (g a)
|
||||
|
||||
/-! # Squash -/
|
||||
@@ -1922,12 +1894,12 @@ represents an element of `Squash α` the same as `α` itself
|
||||
`Squash.lift` will extract a value in any subsingleton `β` from a function on `α`,
|
||||
while `Nonempty.rec` can only do the same when `β` is a proposition.
|
||||
-/
|
||||
def Squash (α : Sort u) := Quot (fun (_ _ : α) => True)
|
||||
def Squash (α : Type u) := Quot (fun (_ _ : α) => True)
|
||||
|
||||
/-- The canonical quotient map into `Squash α`. -/
|
||||
def Squash.mk {α : Sort u} (x : α) : Squash α := Quot.mk _ x
|
||||
def Squash.mk {α : Type u} (x : α) : Squash α := Quot.mk _ x
|
||||
|
||||
theorem Squash.ind {α : Sort u} {motive : Squash α → Prop} (h : ∀ (a : α), motive (Squash.mk a)) : ∀ (q : Squash α), motive q :=
|
||||
theorem Squash.ind {α : Type u} {motive : Squash α → Prop} (h : ∀ (a : α), motive (Squash.mk a)) : ∀ (q : Squash α), motive q :=
|
||||
Quot.ind h
|
||||
|
||||
/-- If `β` is a subsingleton, then a function `α → β` lifts to `Squash α → β`. -/
|
||||
@@ -1941,6 +1913,15 @@ instance : Subsingleton (Squash α) where
|
||||
apply Quot.sound
|
||||
trivial
|
||||
|
||||
/-! # Relations -/
|
||||
|
||||
/--
|
||||
`Antisymm (·≤·)` says that `(·≤·)` is antisymmetric, that is, `a ≤ b → b ≤ a → a = b`.
|
||||
-/
|
||||
class Antisymm {α : Sort u} (r : α → α → Prop) : Prop where
|
||||
/-- An antisymmetric relation `(·≤·)` satisfies `a ≤ b → b ≤ a → a = b`. -/
|
||||
antisymm {a b : α} : r a b → r b a → a = b
|
||||
|
||||
namespace Lean
|
||||
/-! # Kernel reduction hints -/
|
||||
|
||||
@@ -2057,7 +2038,7 @@ class IdempotentOp (op : α → α → α) : Prop where
|
||||
`LeftIdentify op o` indicates `o` is a left identity of `op`.
|
||||
|
||||
This class does not require a proof that `o` is an identity, and
|
||||
is used primarily for inferring the identity using class resolution.
|
||||
is used primarily for infering the identity using class resoluton.
|
||||
-/
|
||||
class LeftIdentity (op : α → β → β) (o : outParam α) : Prop
|
||||
|
||||
@@ -2073,7 +2054,7 @@ class LawfulLeftIdentity (op : α → β → β) (o : outParam α) extends LeftI
|
||||
`RightIdentify op o` indicates `o` is a right identity `o` of `op`.
|
||||
|
||||
This class does not require a proof that `o` is an identity, and is used
|
||||
primarily for inferring the identity using class resolution.
|
||||
primarily for infering the identity using class resoluton.
|
||||
-/
|
||||
class RightIdentity (op : α → β → α) (o : outParam β) : Prop
|
||||
|
||||
@@ -2089,7 +2070,7 @@ class LawfulRightIdentity (op : α → β → α) (o : outParam β) extends Righ
|
||||
`Identity op o` indicates `o` is a left and right identity of `op`.
|
||||
|
||||
This class does not require a proof that `o` is an identity, and is used
|
||||
primarily for inferring the identity using class resolution.
|
||||
primarily for infering the identity using class resoluton.
|
||||
-/
|
||||
class Identity (op : α → α → α) (o : outParam α) extends LeftIdentity op o, RightIdentity op o : Prop
|
||||
|
||||
@@ -2116,14 +2097,4 @@ instance : Commutative Or := ⟨fun _ _ => propext or_comm⟩
|
||||
instance : Commutative And := ⟨fun _ _ => propext and_comm⟩
|
||||
instance : Commutative Iff := ⟨fun _ _ => propext iff_comm⟩
|
||||
|
||||
/--
|
||||
`Antisymm (·≤·)` says that `(·≤·)` is antisymmetric, that is, `a ≤ b → b ≤ a → a = b`.
|
||||
-/
|
||||
class Antisymm (r : α → α → Prop) : Prop where
|
||||
/-- An antisymmetric relation `(·≤·)` satisfies `a ≤ b → b ≤ a → a = b`. -/
|
||||
antisymm {a b : α} : r a b → r b a → a = b
|
||||
|
||||
@[deprecated Antisymm (since := "2024-10-16"), inherit_doc Antisymm]
|
||||
abbrev _root_.Antisymm (r : α → α → Prop) : Prop := Std.Antisymm r
|
||||
|
||||
end Std
|
||||
|
||||
@@ -19,7 +19,6 @@ import Init.Data.ByteArray
|
||||
import Init.Data.FloatArray
|
||||
import Init.Data.Fin
|
||||
import Init.Data.UInt
|
||||
import Init.Data.SInt
|
||||
import Init.Data.Float
|
||||
import Init.Data.Option
|
||||
import Init.Data.Ord
|
||||
@@ -34,6 +33,7 @@ import Init.Data.Prod
|
||||
import Init.Data.AC
|
||||
import Init.Data.Queue
|
||||
import Init.Data.Channel
|
||||
import Init.Data.Cast
|
||||
import Init.Data.Sum
|
||||
import Init.Data.BEq
|
||||
import Init.Data.Subtype
|
||||
@@ -41,6 +41,3 @@ import Init.Data.ULift
|
||||
import Init.Data.PLift
|
||||
import Init.Data.Zero
|
||||
import Init.Data.NeZero
|
||||
import Init.Data.Function
|
||||
import Init.Data.RArray
|
||||
import Init.Data.Vector
|
||||
|
||||
@@ -15,7 +15,3 @@ import Init.Data.Array.BasicAux
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.Array.TakeDrop
|
||||
import Init.Data.Array.Bootstrap
|
||||
import Init.Data.Array.GetLit
|
||||
import Init.Data.Array.MapIdx
|
||||
import Init.Data.Array.Set
|
||||
import Init.Data.Array.Monadic
|
||||
|
||||
@@ -5,22 +5,10 @@ Authors: Joachim Breitner, Mario Carneiro
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Mem
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.List.Attach
|
||||
|
||||
namespace Array
|
||||
|
||||
/--
|
||||
`O(n)`. Partial map. If `f : Π a, P a → β` is a partial function defined on
|
||||
`a : α` satisfying `P`, then `pmap f l h` is essentially the same as `map f l`
|
||||
but is defined only when all members of `l` satisfy `P`, using the proof
|
||||
to apply `f`.
|
||||
|
||||
We replace this at runtime with a more efficient version via the `csimp` lemma `pmap_eq_pmapImpl`.
|
||||
-/
|
||||
def pmap {P : α → Prop} (f : ∀ a, P a → β) (l : Array α) (H : ∀ a ∈ l, P a) : Array β :=
|
||||
(l.toList.pmap f (fun a m => H a (mem_def.mpr m))).toArray
|
||||
|
||||
/--
|
||||
Unsafe implementation of `attachWith`, taking advantage of the fact that the representation of
|
||||
`Array {x // P x}` is the same as the input `Array α`.
|
||||
@@ -38,546 +26,4 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
|
||||
with the same elements but in the type `{x // x ∈ xs}`. -/
|
||||
@[inline] def attach (xs : Array α) : Array {x // x ∈ xs} := xs.attachWith _ fun _ => id
|
||||
|
||||
@[simp] theorem _root_.List.attachWith_toArray {l : List α} {P : α → Prop} {H : ∀ x ∈ l.toArray, P x} :
|
||||
l.toArray.attachWith P H = (l.attachWith P (by simpa using H)).toArray := by
|
||||
simp [attachWith]
|
||||
|
||||
@[simp] theorem _root_.List.attach_toArray {l : List α} :
|
||||
l.toArray.attach = (l.attachWith (· ∈ l.toArray) (by simp)).toArray := by
|
||||
simp [attach]
|
||||
|
||||
@[simp] theorem _root_.List.pmap_toArray {l : List α} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ l.toArray, P a} :
|
||||
l.toArray.pmap f H = (l.pmap f (by simpa using H)).toArray := by
|
||||
simp [pmap]
|
||||
|
||||
@[simp] theorem toList_attachWith {l : Array α} {P : α → Prop} {H : ∀ x ∈ l, P x} :
|
||||
(l.attachWith P H).toList = l.toList.attachWith P (by simpa [mem_toList] using H) := by
|
||||
simp [attachWith]
|
||||
|
||||
@[simp] theorem toList_attach {α : Type _} {l : Array α} :
|
||||
l.attach.toList = l.toList.attachWith (· ∈ l) (by simp [mem_toList]) := by
|
||||
simp [attach]
|
||||
|
||||
@[simp] theorem toList_pmap {l : Array α} {P : α → Prop} {f : ∀ a, P a → β} {H : ∀ a ∈ l, P a} :
|
||||
(l.pmap f H).toList = l.toList.pmap f (fun a m => H a (mem_def.mpr m)) := by
|
||||
simp [pmap]
|
||||
|
||||
/-- Implementation of `pmap` using the zero-copy version of `attach`. -/
|
||||
@[inline] private def pmapImpl {P : α → Prop} (f : ∀ a, P a → β) (l : Array α) (H : ∀ a ∈ l, P a) :
|
||||
Array β := (l.attachWith _ H).map fun ⟨x, h'⟩ => f x h'
|
||||
|
||||
@[csimp] private theorem pmap_eq_pmapImpl : @pmap = @pmapImpl := by
|
||||
funext α β p f L h'
|
||||
cases L
|
||||
simp only [pmap, pmapImpl, List.attachWith_toArray, List.map_toArray, mk.injEq, List.map_attachWith]
|
||||
apply List.pmap_congr_left
|
||||
intro a m h₁ h₂
|
||||
congr
|
||||
|
||||
@[simp] theorem pmap_empty {P : α → Prop} (f : ∀ a, P a → β) : pmap f #[] (by simp) = #[] := rfl
|
||||
|
||||
@[simp] theorem pmap_push {P : α → Prop} (f : ∀ a, P a → β) (a : α) (l : Array α) (h : ∀ b ∈ l.push a, P b) :
|
||||
pmap f (l.push a) h =
|
||||
(pmap f l (fun a m => by simp at h; exact h a (.inl m))).push (f a (h a (by simp))) := by
|
||||
simp [pmap]
|
||||
|
||||
@[simp] theorem attach_empty : (#[] : Array α).attach = #[] := rfl
|
||||
|
||||
@[simp] theorem attachWith_empty {P : α → Prop} (H : ∀ x ∈ #[], P x) : (#[] : Array α).attachWith P H = #[] := rfl
|
||||
|
||||
@[simp] theorem _root_.List.attachWith_mem_toArray {l : List α} :
|
||||
l.attachWith (fun x => x ∈ l.toArray) (fun x h => by simpa using h) =
|
||||
l.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
|
||||
simp only [List.attachWith, List.attach, List.map_pmap]
|
||||
apply List.pmap_congr_left
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
theorem pmap_eq_map (p : α → Prop) (f : α → β) (l : Array α) (H) :
|
||||
@pmap _ _ p (fun a _ => f a) l H = map f l := by
|
||||
cases l; simp
|
||||
|
||||
theorem pmap_congr_left {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (l : Array α) {H₁ H₂}
|
||||
(h : ∀ a ∈ l, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f l H₁ = pmap g l H₂ := by
|
||||
cases l
|
||||
simp only [mem_toArray] at h
|
||||
simp only [List.pmap_toArray, mk.injEq]
|
||||
rw [List.pmap_congr_left _ h]
|
||||
|
||||
theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l H) :
|
||||
map g (pmap f l H) = pmap (fun a h => g (f a h)) l H := by
|
||||
cases l
|
||||
simp [List.map_pmap]
|
||||
|
||||
theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (l H) :
|
||||
pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun _ h => H _ (mem_map_of_mem _ h) := by
|
||||
cases l
|
||||
simp [List.pmap_map]
|
||||
|
||||
theorem attach_congr {l₁ l₂ : Array α} (h : l₁ = l₂) :
|
||||
l₁.attach = l₂.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
|
||||
subst h
|
||||
simp
|
||||
|
||||
theorem attachWith_congr {l₁ l₂ : Array α} (w : l₁ = l₂) {P : α → Prop} {H : ∀ x ∈ l₁, P x} :
|
||||
l₁.attachWith P H = l₂.attachWith P fun _ h => H _ (w ▸ h) := by
|
||||
subst w
|
||||
simp
|
||||
|
||||
@[simp] theorem attach_push {a : α} {l : Array α} :
|
||||
(l.push a).attach =
|
||||
(l.attach.map (fun ⟨x, h⟩ => ⟨x, mem_push_of_mem a h⟩)).push ⟨a, by simp⟩ := by
|
||||
cases l
|
||||
rw [attach_congr (List.push_toArray _ _)]
|
||||
simp [Function.comp_def]
|
||||
|
||||
@[simp] theorem attachWith_push {a : α} {l : Array α} {P : α → Prop} {H : ∀ x ∈ l.push a, P x} :
|
||||
(l.push a).attachWith P H =
|
||||
(l.attachWith P (fun x h => by simp at H; exact H x (.inl h))).push ⟨a, H a (by simp)⟩ := by
|
||||
cases l
|
||||
simp [attachWith_congr (List.push_toArray _ _)]
|
||||
|
||||
theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
|
||||
pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
|
||||
cases l
|
||||
simp [List.pmap_eq_map_attach]
|
||||
|
||||
theorem attach_map_coe (l : Array α) (f : α → β) :
|
||||
(l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
|
||||
cases l
|
||||
simp [List.attach_map_coe]
|
||||
|
||||
theorem attach_map_val (l : Array α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
|
||||
attach_map_coe _ _
|
||||
|
||||
@[simp]
|
||||
theorem attach_map_subtype_val (l : Array α) : l.attach.map Subtype.val = l := by
|
||||
cases l; simp
|
||||
|
||||
theorem attachWith_map_coe {p : α → Prop} (f : α → β) (l : Array α) (H : ∀ a ∈ l, p a) :
|
||||
((l.attachWith p H).map fun (i : { i // p i}) => f i) = l.map f := by
|
||||
cases l; simp
|
||||
|
||||
theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : Array α) (H : ∀ a ∈ l, p a) :
|
||||
((l.attachWith p H).map fun i => f i.val) = l.map f :=
|
||||
attachWith_map_coe _ _ _
|
||||
|
||||
@[simp]
|
||||
theorem attachWith_map_subtype_val {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) :
|
||||
(l.attachWith p H).map Subtype.val = l := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem mem_attach (l : Array α) : ∀ x, x ∈ l.attach
|
||||
| ⟨a, h⟩ => by
|
||||
have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
|
||||
rcases this with ⟨⟨_, _⟩, m, rfl⟩
|
||||
exact m
|
||||
|
||||
@[simp]
|
||||
theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
|
||||
b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
|
||||
simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, Subtype.exists, eq_comm]
|
||||
|
||||
theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {l H} {a} (h : a ∈ l) :
|
||||
f a (H a h) ∈ pmap f l H := by
|
||||
rw [mem_pmap]
|
||||
exact ⟨a, h, rfl⟩
|
||||
|
||||
@[simp]
|
||||
theorem size_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : (pmap f l H).size = l.size := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem size_attach {L : Array α} : L.attach.size = L.size := by
|
||||
cases L; simp
|
||||
|
||||
@[simp]
|
||||
theorem size_attachWith {p : α → Prop} {l : Array α} {H} : (l.attachWith p H).size = l.size := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem pmap_eq_empty_iff {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = #[] ↔ l = #[] := by
|
||||
cases l; simp
|
||||
|
||||
theorem pmap_ne_empty_iff {P : α → Prop} (f : (a : α) → P a → β) {xs : Array α}
|
||||
(H : ∀ (a : α), a ∈ xs → P a) : xs.pmap f H ≠ #[] ↔ xs ≠ #[] := by
|
||||
cases xs; simp
|
||||
|
||||
theorem pmap_eq_self {l : Array α} {p : α → Prop} (hp : ∀ (a : α), a ∈ l → p a)
|
||||
(f : (a : α) → p a → α) : l.pmap f hp = l ↔ ∀ a (h : a ∈ l), f a (hp a h) = a := by
|
||||
cases l; simp [List.pmap_eq_self]
|
||||
|
||||
@[simp]
|
||||
theorem attach_eq_empty_iff {l : Array α} : l.attach = #[] ↔ l = #[] := by
|
||||
cases l; simp
|
||||
|
||||
theorem attach_ne_empty_iff {l : Array α} : l.attach ≠ #[] ↔ l ≠ #[] := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem attachWith_eq_empty_iff {l : Array α} {P : α → Prop} {H : ∀ a ∈ l, P a} :
|
||||
l.attachWith P H = #[] ↔ l = #[] := by
|
||||
cases l; simp
|
||||
|
||||
theorem attachWith_ne_empty_iff {l : Array α} {P : α → Prop} {H : ∀ a ∈ l, P a} :
|
||||
l.attachWith P H ≠ #[] ↔ l ≠ #[] := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) (n : Nat) :
|
||||
(pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (mem_of_getElem? H) := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) {n : Nat}
|
||||
(hn : n < (pmap f l h).size) :
|
||||
(pmap f l h)[n] =
|
||||
f (l[n]'(@size_pmap _ _ p f l h ▸ hn))
|
||||
(h _ (getElem_mem (@size_pmap _ _ p f l h ▸ hn))) := by
|
||||
cases l; simp
|
||||
|
||||
@[simp]
|
||||
theorem getElem?_attachWith {xs : Array α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
|
||||
(xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (mem_of_getElem? a)) :=
|
||||
getElem?_pmap ..
|
||||
|
||||
@[simp]
|
||||
theorem getElem?_attach {xs : Array α} {i : Nat} :
|
||||
xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => mem_of_getElem? a) :=
|
||||
getElem?_attachWith
|
||||
|
||||
@[simp]
|
||||
theorem getElem_attachWith {xs : Array α} {P : α → Prop} {H : ∀ a ∈ xs, P a}
|
||||
{i : Nat} (h : i < (xs.attachWith P H).size) :
|
||||
(xs.attachWith P H)[i] = ⟨xs[i]'(by simpa using h), H _ (getElem_mem (by simpa using h))⟩ :=
|
||||
getElem_pmap ..
|
||||
|
||||
@[simp]
|
||||
theorem getElem_attach {xs : Array α} {i : Nat} (h : i < xs.attach.size) :
|
||||
xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
|
||||
getElem_attachWith h
|
||||
|
||||
theorem foldl_pmap (l : Array α) {P : α → Prop} (f : (a : α) → P a → β)
|
||||
(H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
|
||||
(l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
|
||||
rw [pmap_eq_map_attach, foldl_map]
|
||||
|
||||
theorem foldr_pmap (l : Array α) {P : α → Prop} (f : (a : α) → P a → β)
|
||||
(H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) :
|
||||
(l.pmap f H).foldr g x = l.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
|
||||
rw [pmap_eq_map_attach, foldr_map]
|
||||
|
||||
/--
|
||||
If we fold over `l.attach` with a function that ignores the membership predicate,
|
||||
we get the same results as folding over `l` directly.
|
||||
|
||||
This is useful when we need to use `attach` to show termination.
|
||||
|
||||
Unfortunately this can't be applied by `simp` because of the higher order unification problem,
|
||||
and even when rewriting we need to specify the function explicitly.
|
||||
See however `foldl_subtype` below.
|
||||
-/
|
||||
theorem foldl_attach (l : Array α) (f : β → α → β) (b : β) :
|
||||
l.attach.foldl (fun acc t => f acc t.1) b = l.foldl f b := by
|
||||
rcases l with ⟨l⟩
|
||||
simp only [List.attach_toArray, List.attachWith_mem_toArray, List.map_attach, size_toArray,
|
||||
List.length_pmap, List.foldl_toArray', mem_toArray, List.foldl_subtype]
|
||||
congr
|
||||
ext
|
||||
simpa using fun a => List.mem_of_getElem? a
|
||||
|
||||
/--
|
||||
If we fold over `l.attach` with a function that ignores the membership predicate,
|
||||
we get the same results as folding over `l` directly.
|
||||
|
||||
This is useful when we need to use `attach` to show termination.
|
||||
|
||||
Unfortunately this can't be applied by `simp` because of the higher order unification problem,
|
||||
and even when rewriting we need to specify the function explicitly.
|
||||
See however `foldr_subtype` below.
|
||||
-/
|
||||
theorem foldr_attach (l : Array α) (f : α → β → β) (b : β) :
|
||||
l.attach.foldr (fun t acc => f t.1 acc) b = l.foldr f b := by
|
||||
rcases l with ⟨l⟩
|
||||
simp only [List.attach_toArray, List.attachWith_mem_toArray, List.map_attach, size_toArray,
|
||||
List.length_pmap, List.foldr_toArray', mem_toArray, List.foldr_subtype]
|
||||
congr
|
||||
ext
|
||||
simpa using fun a => List.mem_of_getElem? a
|
||||
|
||||
theorem attach_map {l : Array α} (f : α → β) :
|
||||
(l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
|
||||
cases l
|
||||
ext <;> simp
|
||||
|
||||
theorem attachWith_map {l : Array α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ l.map f → P b} :
|
||||
(l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
|
||||
fun ⟨x, h⟩ => ⟨f x, h⟩ := by
|
||||
cases l
|
||||
ext
|
||||
· simp
|
||||
· simp only [List.map_toArray, List.attachWith_toArray, List.getElem_toArray,
|
||||
List.getElem_attachWith, List.getElem_map, Function.comp_apply]
|
||||
erw [List.getElem_attachWith] -- Why is `erw` needed here?
|
||||
|
||||
theorem map_attachWith {l : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
|
||||
(f : { x // P x } → β) :
|
||||
(l.attachWith P H).map f =
|
||||
l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
|
||||
cases l
|
||||
ext <;> simp
|
||||
|
||||
/-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
|
||||
theorem map_attach {l : Array α} (f : { x // x ∈ l } → β) :
|
||||
l.attach.map f = l.pmap (fun a h => f ⟨a, h⟩) (fun _ => id) := by
|
||||
cases l
|
||||
ext <;> simp
|
||||
|
||||
theorem attach_filterMap {l : Array α} {f : α → Option β} :
|
||||
(l.filterMap f).attach = l.attach.filterMap
|
||||
fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩) := by
|
||||
cases l
|
||||
rw [attach_congr (List.filterMap_toArray f _)]
|
||||
simp [List.attach_filterMap, List.map_filterMap, Function.comp_def]
|
||||
|
||||
theorem attach_filter {l : Array α} (p : α → Bool) :
|
||||
(l.filter p).attach = l.attach.filterMap
|
||||
fun x => if w : p x.1 then some ⟨x.1, mem_filter.mpr ⟨x.2, w⟩⟩ else none := by
|
||||
cases l
|
||||
rw [attach_congr (List.filter_toArray p _)]
|
||||
simp [List.attach_filter, List.map_filterMap, Function.comp_def]
|
||||
|
||||
-- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
|
||||
-- Also missing are `filterMap_attach`, `filter_attach`, `filterMap_attachWith` and `filter_attachWith`.
|
||||
|
||||
theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
|
||||
pmap f (pmap g l H₁) H₂ =
|
||||
pmap (α := { x // x ∈ l }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) l.attach
|
||||
(fun a _ => H₁ a a.2) := by
|
||||
cases l
|
||||
simp [List.pmap_pmap, List.pmap_map]
|
||||
|
||||
@[simp] theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (l₁ l₂ : Array ι)
|
||||
(h : ∀ a ∈ l₁ ++ l₂, p a) :
|
||||
(l₁ ++ l₂).pmap f h =
|
||||
(l₁.pmap f fun a ha => h a (mem_append_left l₂ ha)) ++
|
||||
l₂.pmap f fun a ha => h a (mem_append_right l₁ ha) := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
simp
|
||||
|
||||
theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ l₂ : Array α)
|
||||
(h₁ : ∀ a ∈ l₁, p a) (h₂ : ∀ a ∈ l₂, p a) :
|
||||
((l₁ ++ l₂).pmap f fun a ha => (mem_append.1 ha).elim (h₁ a) (h₂ a)) =
|
||||
l₁.pmap f h₁ ++ l₂.pmap f h₂ :=
|
||||
pmap_append f l₁ l₂ _
|
||||
|
||||
@[simp] theorem attach_append (xs ys : Array α) :
|
||||
(xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_left ys h⟩) ++
|
||||
ys.attach.map fun ⟨x, h⟩ => ⟨x, mem_append_right xs h⟩ := by
|
||||
cases xs
|
||||
cases ys
|
||||
rw [attach_congr (List.append_toArray _ _)]
|
||||
simp [List.attach_append, Function.comp_def]
|
||||
|
||||
@[simp] theorem attachWith_append {P : α → Prop} {xs ys : Array α}
|
||||
{H : ∀ (a : α), a ∈ xs ++ ys → P a} :
|
||||
(xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_left ys h)) ++
|
||||
ys.attachWith P (fun a h => H a (mem_append_right xs h)) := by
|
||||
simp [attachWith, attach_append, map_pmap, pmap_append]
|
||||
|
||||
@[simp] theorem pmap_reverse {P : α → Prop} (f : (a : α) → P a → β) (xs : Array α)
|
||||
(H : ∀ (a : α), a ∈ xs.reverse → P a) :
|
||||
xs.reverse.pmap f H = (xs.pmap f (fun a h => H a (by simpa using h))).reverse := by
|
||||
induction xs <;> simp_all
|
||||
|
||||
theorem reverse_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Array α)
|
||||
(H : ∀ (a : α), a ∈ xs → P a) :
|
||||
(xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h)) := by
|
||||
rw [pmap_reverse]
|
||||
|
||||
@[simp] theorem attachWith_reverse {P : α → Prop} {xs : Array α}
|
||||
{H : ∀ (a : α), a ∈ xs.reverse → P a} :
|
||||
xs.reverse.attachWith P H =
|
||||
(xs.attachWith P (fun a h => H a (by simpa using h))).reverse := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
theorem reverse_attachWith {P : α → Prop} {xs : Array α}
|
||||
{H : ∀ (a : α), a ∈ xs → P a} :
|
||||
(xs.attachWith P H).reverse = (xs.reverse.attachWith P (fun a h => H a (by simpa using h))) := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
@[simp] theorem attach_reverse (xs : Array α) :
|
||||
xs.reverse.attach = xs.attach.reverse.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
|
||||
cases xs
|
||||
rw [attach_congr (List.reverse_toArray _)]
|
||||
simp
|
||||
|
||||
theorem reverse_attach (xs : Array α) :
|
||||
xs.attach.reverse = xs.reverse.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
@[simp] theorem back?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Array α)
|
||||
(H : ∀ (a : α), a ∈ xs → P a) :
|
||||
(xs.pmap f H).back? = xs.attach.back?.map fun ⟨a, m⟩ => f a (H a m) := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
@[simp] theorem back?_attachWith {P : α → Prop} {xs : Array α}
|
||||
{H : ∀ (a : α), a ∈ xs → P a} :
|
||||
(xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back?_eq_some h)⟩) := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
theorem back?_attach {xs : Array α} :
|
||||
xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back?_eq_some h⟩ := by
|
||||
cases xs
|
||||
simp
|
||||
|
||||
/-! ## unattach
|
||||
|
||||
`Array.unattach` is the (one-sided) inverse of `Array.attach`. It is a synonym for `Array.map Subtype.val`.
|
||||
|
||||
We use it by providing a simp lemma `l.attach.unattach = l`, and simp lemmas which recognize higher order
|
||||
functions applied to `l : Array { x // p x }` which only depend on the value, not the predicate, and rewrite these
|
||||
in terms of a simpler function applied to `l.unattach`.
|
||||
|
||||
Further, we provide simp lemmas that push `unattach` inwards.
|
||||
-/
|
||||
|
||||
/--
|
||||
A synonym for `l.map (·.val)`. Mostly this should not be needed by users.
|
||||
It is introduced as in intermediate step by lemmas such as `map_subtype`,
|
||||
and is ideally subsequently simplified away by `unattach_attach`.
|
||||
|
||||
If not, usually the right approach is `simp [Array.unattach, -Array.map_subtype]` to unfold.
|
||||
-/
|
||||
def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) := l.map (·.val)
|
||||
|
||||
@[simp] theorem unattach_nil {p : α → Prop} : (#[] : Array { x // p x }).unattach = #[] := rfl
|
||||
@[simp] theorem unattach_push {p : α → Prop} {a : { x // p x }} {l : Array { x // p x }} :
|
||||
(l.push a).unattach = l.unattach.push a.1 := by
|
||||
simp only [unattach, Array.map_push]
|
||||
|
||||
@[simp] theorem size_unattach {p : α → Prop} {l : Array { x // p x }} :
|
||||
l.unattach.size = l.size := by
|
||||
unfold unattach
|
||||
simp
|
||||
|
||||
@[simp] theorem _root_.List.unattach_toArray {p : α → Prop} {l : List { x // p x }} :
|
||||
l.toArray.unattach = l.unattach.toArray := by
|
||||
simp only [unattach, List.map_toArray, List.unattach]
|
||||
|
||||
@[simp] theorem toList_unattach {p : α → Prop} {l : Array { x // p x }} :
|
||||
l.unattach.toList = l.toList.unattach := by
|
||||
simp only [unattach, toList_map, List.unattach]
|
||||
|
||||
@[simp] theorem unattach_attach {l : Array α} : l.attach.unattach = l := by
|
||||
cases l
|
||||
simp only [List.attach_toArray, List.unattach_toArray, List.unattach_attachWith]
|
||||
|
||||
@[simp] theorem unattach_attachWith {p : α → Prop} {l : Array α}
|
||||
{H : ∀ a ∈ l, p a} :
|
||||
(l.attachWith p H).unattach = l := by
|
||||
cases l
|
||||
simp
|
||||
|
||||
@[simp] theorem getElem?_unattach {p : α → Prop} {l : Array { x // p x }} (i : Nat) :
|
||||
l.unattach[i]? = l[i]?.map Subtype.val := by
|
||||
simp [unattach]
|
||||
|
||||
@[simp] theorem getElem_unattach
|
||||
{p : α → Prop} {l : Array { x // p x }} (i : Nat) (h : i < l.unattach.size) :
|
||||
l.unattach[i] = (l[i]'(by simpa using h)).1 := by
|
||||
simp [unattach]
|
||||
|
||||
/-! ### Recognizing higher order functions using a function that only depends on the value. -/
|
||||
|
||||
/--
|
||||
This lemma identifies folds over arrays of subtypes, where the function only depends on the value, not the proposition,
|
||||
and simplifies these to the function directly taking the value.
|
||||
-/
|
||||
theorem foldl_subtype {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : β → { x // p x } → β} {g : β → α → β} {x : β}
|
||||
{hf : ∀ b x h, f b ⟨x, h⟩ = g b x} :
|
||||
l.foldl f x = l.unattach.foldl g x := by
|
||||
cases l
|
||||
simp only [List.foldl_toArray', List.unattach_toArray]
|
||||
rw [List.foldl_subtype] -- Why can't simp do this?
|
||||
simp [hf]
|
||||
|
||||
/-- Variant of `foldl_subtype` with side condition to check `stop = l.size`. -/
|
||||
@[simp] theorem foldl_subtype' {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : β → { x // p x } → β} {g : β → α → β} {x : β}
|
||||
{hf : ∀ b x h, f b ⟨x, h⟩ = g b x} (h : stop = l.size) :
|
||||
l.foldl f x 0 stop = l.unattach.foldl g x := by
|
||||
subst h
|
||||
rwa [foldl_subtype]
|
||||
|
||||
/--
|
||||
This lemma identifies folds over arrays of subtypes, where the function only depends on the value, not the proposition,
|
||||
and simplifies these to the function directly taking the value.
|
||||
-/
|
||||
theorem foldr_subtype {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : { x // p x } → β → β} {g : α → β → β} {x : β}
|
||||
{hf : ∀ x h b, f ⟨x, h⟩ b = g x b} :
|
||||
l.foldr f x = l.unattach.foldr g x := by
|
||||
cases l
|
||||
simp only [List.foldr_toArray', List.unattach_toArray]
|
||||
rw [List.foldr_subtype]
|
||||
simp [hf]
|
||||
|
||||
/-- Variant of `foldr_subtype` with side condition to check `stop = l.size`. -/
|
||||
@[simp] theorem foldr_subtype' {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : { x // p x } → β → β} {g : α → β → β} {x : β}
|
||||
{hf : ∀ x h b, f ⟨x, h⟩ b = g x b} (h : start = l.size) :
|
||||
l.foldr f x start 0 = l.unattach.foldr g x := by
|
||||
subst h
|
||||
rwa [foldr_subtype]
|
||||
|
||||
/--
|
||||
This lemma identifies maps over arrays of subtypes, where the function only depends on the value, not the proposition,
|
||||
and simplifies these to the function directly taking the value.
|
||||
-/
|
||||
@[simp] theorem map_subtype {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : { x // p x } → β} {g : α → β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
l.map f = l.unattach.map g := by
|
||||
cases l
|
||||
simp only [List.map_toArray, List.unattach_toArray]
|
||||
rw [List.map_subtype]
|
||||
simp [hf]
|
||||
|
||||
@[simp] theorem filterMap_subtype {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : { x // p x } → Option β} {g : α → Option β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
l.filterMap f = l.unattach.filterMap g := by
|
||||
cases l
|
||||
simp only [size_toArray, List.filterMap_toArray', List.unattach_toArray, List.length_unattach,
|
||||
mk.injEq]
|
||||
rw [List.filterMap_subtype]
|
||||
simp [hf]
|
||||
|
||||
@[simp] theorem unattach_filter {p : α → Prop} {l : Array { x // p x }}
|
||||
{f : { x // p x } → Bool} {g : α → Bool} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
(l.filter f).unattach = l.unattach.filter g := by
|
||||
cases l
|
||||
simp [hf]
|
||||
|
||||
/-! ### Simp lemmas pushing `unattach` inwards. -/
|
||||
|
||||
@[simp] theorem unattach_reverse {p : α → Prop} {l : Array { x // p x }} :
|
||||
l.reverse.unattach = l.unattach.reverse := by
|
||||
cases l
|
||||
simp
|
||||
|
||||
@[simp] theorem unattach_append {p : α → Prop} {l₁ l₂ : Array { x // p x }} :
|
||||
(l₁ ++ l₂).unattach = l₁.unattach ++ l₂.unattach := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
simp
|
||||
|
||||
end Array
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -34,7 +34,7 @@ private theorem List.of_toArrayAux_eq_toArrayAux {as bs : List α} {cs ds : Arra
|
||||
|
||||
@[simp] theorem List.toArray_eq_toArray_eq (as bs : List α) : (as.toArray = bs.toArray) = (as = bs) := by
|
||||
apply propext; apply Iff.intro
|
||||
· intro h; simpa [toArray] using h
|
||||
· intro h; simp [toArray] at h; have := of_toArrayAux_eq_toArrayAux h rfl; exact this.1
|
||||
· intro h; rw [h]
|
||||
|
||||
def Array.mapM' [Monad m] (f : α → m β) (as : Array α) : m { bs : Array β // bs.size = as.size } :=
|
||||
@@ -60,7 +60,7 @@ where
|
||||
if ptrEq a b then
|
||||
go (i+1) as
|
||||
else
|
||||
go (i+1) (as.set i b h)
|
||||
go (i+1) (as.set ⟨i, h⟩ b)
|
||||
else
|
||||
return as
|
||||
|
||||
|
||||
@@ -5,64 +5,59 @@ Authors: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Basic
|
||||
import Init.Omega
|
||||
universe u v
|
||||
|
||||
namespace Array
|
||||
-- TODO: CLEANUP
|
||||
|
||||
@[specialize] def binSearchAux {α : Type u} {β : Type v} (lt : α → α → Bool) (found : Option α → β) (as : Array α) (k : α) :
|
||||
(lo : Fin (as.size + 1)) → (hi : Fin as.size) → (lo.1 ≤ hi.1) → β
|
||||
| lo, hi, h =>
|
||||
let m := (lo.1 + hi.1)/2
|
||||
let a := as[m]
|
||||
if lt a k then
|
||||
if h' : m + 1 ≤ hi.1 then
|
||||
binSearchAux lt found as k ⟨m+1, by omega⟩ hi h'
|
||||
else found none
|
||||
else if lt k a then
|
||||
if h' : m = 0 ∨ m - 1 < lo.1 then found none
|
||||
else binSearchAux lt found as k lo ⟨m-1, by omega⟩ (by simp; omega)
|
||||
else found (some a)
|
||||
termination_by lo hi => hi.1 - lo.1
|
||||
namespace Array
|
||||
-- TODO: remove the [Inhabited α] parameters as soon as we have the tactic framework for automating proof generation and using Array.fget
|
||||
-- TODO: remove `partial` using well-founded recursion
|
||||
|
||||
@[specialize] partial def binSearchAux {α : Type u} {β : Type v} [Inhabited β] (lt : α → α → Bool) (found : Option α → β) (as : Array α) (k : α) : Nat → Nat → β
|
||||
| lo, hi =>
|
||||
if lo <= hi then
|
||||
let _ := Inhabited.mk k
|
||||
let m := (lo + hi)/2
|
||||
let a := as.get! m
|
||||
if lt a k then binSearchAux lt found as k (m+1) hi
|
||||
else if lt k a then
|
||||
if m == 0 then found none
|
||||
else binSearchAux lt found as k lo (m-1)
|
||||
else found (some a)
|
||||
else found none
|
||||
|
||||
@[inline] def binSearch {α : Type} (as : Array α) (k : α) (lt : α → α → Bool) (lo := 0) (hi := as.size - 1) : Option α :=
|
||||
if h : lo < as.size then
|
||||
if lo < as.size then
|
||||
let hi := if hi < as.size then hi else as.size - 1
|
||||
if w : lo ≤ hi then
|
||||
binSearchAux lt id as k ⟨lo, by omega⟩ ⟨hi, by simp [hi]; split <;> omega⟩ (by simp [hi]; omega)
|
||||
else
|
||||
none
|
||||
binSearchAux lt id as k lo hi
|
||||
else
|
||||
none
|
||||
|
||||
@[inline] def binSearchContains {α : Type} (as : Array α) (k : α) (lt : α → α → Bool) (lo := 0) (hi := as.size - 1) : Bool :=
|
||||
if h : lo < as.size then
|
||||
if lo < as.size then
|
||||
let hi := if hi < as.size then hi else as.size - 1
|
||||
if w : lo ≤ hi then
|
||||
binSearchAux lt Option.isSome as k ⟨lo, by omega⟩ ⟨hi, by simp [hi]; split <;> omega⟩ (by simp [hi]; omega)
|
||||
else
|
||||
false
|
||||
binSearchAux lt Option.isSome as k lo hi
|
||||
else
|
||||
false
|
||||
|
||||
@[specialize] private def binInsertAux {α : Type u} {m : Type u → Type v} [Monad m]
|
||||
@[specialize] private partial def binInsertAux {α : Type u} {m : Type u → Type v} [Monad m]
|
||||
(lt : α → α → Bool)
|
||||
(merge : α → m α)
|
||||
(add : Unit → m α)
|
||||
(as : Array α)
|
||||
(k : α) : (lo : Fin as.size) → (hi : Fin as.size) → (lo.1 ≤ hi.1) → (lt as[lo] k) → m (Array α)
|
||||
| lo, hi, h, w =>
|
||||
let mid := (lo.1 + hi.1)/2
|
||||
let midVal := as[mid]
|
||||
if w₁ : lt midVal k then
|
||||
if h' : mid = lo then do let v ← add (); pure <| as.insertIdx (lo+1) v
|
||||
else binInsertAux lt merge add as k ⟨mid, by omega⟩ hi (by simp; omega) w₁
|
||||
else if w₂ : lt k midVal then
|
||||
have : mid ≠ lo := fun z => by simp [midVal, z] at w₁; simp_all
|
||||
binInsertAux lt merge add as k lo ⟨mid, by omega⟩ (by simp; omega) w
|
||||
(k : α) : Nat → Nat → m (Array α)
|
||||
| lo, hi =>
|
||||
let _ := Inhabited.mk k
|
||||
-- as[lo] < k < as[hi]
|
||||
let mid := (lo + hi)/2
|
||||
let midVal := as.get! mid
|
||||
if lt midVal k then
|
||||
if mid == lo then do let v ← add (); pure <| as.insertAt! (lo+1) v
|
||||
else binInsertAux lt merge add as k mid hi
|
||||
else if lt k midVal then
|
||||
binInsertAux lt merge add as k lo mid
|
||||
else do
|
||||
as.modifyM mid <| fun v => merge v
|
||||
termination_by lo hi => hi.1 - lo.1
|
||||
|
||||
@[specialize] def binInsertM {α : Type u} {m : Type u → Type v} [Monad m]
|
||||
(lt : α → α → Bool)
|
||||
@@ -70,12 +65,13 @@ termination_by lo hi => hi.1 - lo.1
|
||||
(add : Unit → m α)
|
||||
(as : Array α)
|
||||
(k : α) : m (Array α) :=
|
||||
if h : as.size = 0 then do let v ← add (); pure <| as.push v
|
||||
else if lt k as[0] then do let v ← add (); pure <| as.insertIdx 0 v
|
||||
else if h' : !lt as[0] k then as.modifyM 0 <| merge
|
||||
else if lt as[as.size - 1] k then do let v ← add (); pure <| as.push v
|
||||
else if !lt k as[as.size - 1] then as.modifyM (as.size - 1) <| merge
|
||||
else binInsertAux lt merge add as k ⟨0, by omega⟩ ⟨as.size - 1, by omega⟩ (by simp) (by simpa using h')
|
||||
let _ := Inhabited.mk k
|
||||
if as.isEmpty then do let v ← add (); pure <| as.push v
|
||||
else if lt k (as.get! 0) then do let v ← add (); pure <| as.insertAt! 0 v
|
||||
else if !lt (as.get! 0) k then as.modifyM 0 <| merge
|
||||
else if lt as.back k then do let v ← add (); pure <| as.push v
|
||||
else if !lt k as.back then as.modifyM (as.size - 1) <| merge
|
||||
else binInsertAux lt merge add as k 0 (as.size - 1)
|
||||
|
||||
@[inline] def binInsert {α : Type u} (lt : α → α → Bool) (as : Array α) (k : α) : Array α :=
|
||||
Id.run <| binInsertM lt (fun _ => k) (fun _ => k) as k
|
||||
|
||||
@@ -15,26 +15,26 @@ This file contains some theorems about `Array` and `List` needed for `Init.Data.
|
||||
|
||||
namespace Array
|
||||
|
||||
theorem foldlM_toList.aux [Monad m]
|
||||
theorem foldlM_eq_foldlM_toList.aux [Monad m]
|
||||
(f : β → α → m β) (arr : Array α) (i j) (H : arr.size ≤ i + j) (b) :
|
||||
foldlM.loop f arr arr.size (Nat.le_refl _) i j b = (arr.toList.drop j).foldlM f b := by
|
||||
unfold foldlM.loop
|
||||
split; split
|
||||
· cases Nat.not_le_of_gt ‹_› (Nat.zero_add _ ▸ H)
|
||||
· rename_i i; rw [Nat.succ_add] at H
|
||||
simp [foldlM_toList.aux f arr i (j+1) H]
|
||||
rw (occs := [2]) [← List.getElem_cons_drop_succ_eq_drop ‹_›]
|
||||
simp [foldlM_eq_foldlM_toList.aux f arr i (j+1) H]
|
||||
rw (config := {occs := .pos [2]}) [← List.get_drop_eq_drop _ _ ‹_›]
|
||||
rfl
|
||||
· rw [List.drop_of_length_le (Nat.ge_of_not_lt ‹_›)]; rfl
|
||||
|
||||
@[simp] theorem foldlM_toList [Monad m]
|
||||
theorem foldlM_eq_foldlM_toList [Monad m]
|
||||
(f : β → α → m β) (init : β) (arr : Array α) :
|
||||
arr.toList.foldlM f init = arr.foldlM f init := by
|
||||
simp [foldlM, foldlM_toList.aux]
|
||||
arr.foldlM f init = arr.toList.foldlM f init := by
|
||||
simp [foldlM, foldlM_eq_foldlM_toList.aux]
|
||||
|
||||
@[simp] theorem foldl_toList (f : β → α → β) (init : β) (arr : Array α) :
|
||||
arr.toList.foldl f init = arr.foldl f init :=
|
||||
List.foldl_eq_foldlM .. ▸ foldlM_toList ..
|
||||
theorem foldl_eq_foldl_toList (f : β → α → β) (init : β) (arr : Array α) :
|
||||
arr.foldl f init = arr.toList.foldl f init :=
|
||||
List.foldl_eq_foldlM .. ▸ foldlM_eq_foldlM_toList ..
|
||||
|
||||
theorem foldrM_eq_reverse_foldlM_toList.aux [Monad m]
|
||||
(f : α → β → m β) (arr : Array α) (init : β) (i h) :
|
||||
@@ -42,7 +42,7 @@ theorem foldrM_eq_reverse_foldlM_toList.aux [Monad m]
|
||||
unfold foldrM.fold
|
||||
match i with
|
||||
| 0 => simp [List.foldlM, List.take]
|
||||
| i+1 => rw [← List.take_concat_get _ _ h]; simp [← (aux f arr · i)]
|
||||
| i+1 => rw [← List.take_concat_get _ _ h]; simp [← (aux f arr · i)]; rfl
|
||||
|
||||
theorem foldrM_eq_reverse_foldlM_toList [Monad m] (f : α → β → m β) (init : β) (arr : Array α) :
|
||||
arr.foldrM f init = arr.toList.reverse.foldlM (fun x y => f y x) init := by
|
||||
@@ -51,45 +51,34 @@ theorem foldrM_eq_reverse_foldlM_toList [Monad m] (f : α → β → m β) (init
|
||||
match arr, this with | _, .inl rfl => rfl | arr, .inr h => ?_
|
||||
simp [foldrM, h, ← foldrM_eq_reverse_foldlM_toList.aux, List.take_length]
|
||||
|
||||
@[simp] theorem foldrM_toList [Monad m]
|
||||
theorem foldrM_eq_foldrM_toList [Monad m]
|
||||
(f : α → β → m β) (init : β) (arr : Array α) :
|
||||
arr.toList.foldrM f init = arr.foldrM f init := by
|
||||
arr.foldrM f init = arr.toList.foldrM f init := by
|
||||
rw [foldrM_eq_reverse_foldlM_toList, List.foldlM_reverse]
|
||||
|
||||
@[simp] theorem foldr_toList (f : α → β → β) (init : β) (arr : Array α) :
|
||||
arr.toList.foldr f init = arr.foldr f init :=
|
||||
List.foldr_eq_foldrM .. ▸ foldrM_toList ..
|
||||
theorem foldr_eq_foldr_toList (f : α → β → β) (init : β) (arr : Array α) :
|
||||
arr.foldr f init = arr.toList.foldr f init :=
|
||||
List.foldr_eq_foldrM .. ▸ foldrM_eq_foldrM_toList ..
|
||||
|
||||
@[simp] theorem push_toList (arr : Array α) (a : α) : (arr.push a).toList = arr.toList ++ [a] := by
|
||||
simp [push, List.concat_eq_append]
|
||||
|
||||
@[simp] theorem toListAppend_eq (arr : Array α) (l) : arr.toListAppend l = arr.toList ++ l := by
|
||||
simp [toListAppend, ← foldr_toList]
|
||||
simp [toListAppend, foldr_eq_foldr_toList]
|
||||
|
||||
@[simp] theorem toListImpl_eq (arr : Array α) : arr.toListImpl = arr.toList := by
|
||||
simp [toListImpl, ← foldr_toList]
|
||||
simp [toListImpl, foldr_eq_foldr_toList]
|
||||
|
||||
@[simp] theorem pop_toList (arr : Array α) : arr.pop.toList = arr.toList.dropLast := rfl
|
||||
|
||||
@[simp] theorem append_eq_append (arr arr' : Array α) : arr.append arr' = arr ++ arr' := rfl
|
||||
|
||||
@[simp] theorem toList_append (arr arr' : Array α) :
|
||||
@[simp] theorem append_toList (arr arr' : Array α) :
|
||||
(arr ++ arr').toList = arr.toList ++ arr'.toList := by
|
||||
rw [← append_eq_append]; unfold Array.append
|
||||
rw [← foldl_toList]
|
||||
rw [foldl_eq_foldl_toList]
|
||||
induction arr'.toList generalizing arr <;> simp [*]
|
||||
|
||||
@[simp] theorem toList_empty : (#[] : Array α).toList = [] := rfl
|
||||
|
||||
@[simp] theorem append_nil (as : Array α) : as ++ #[] = as := by
|
||||
apply ext'; simp only [toList_append, toList_empty, List.append_nil]
|
||||
|
||||
@[simp] theorem nil_append (as : Array α) : #[] ++ as = as := by
|
||||
apply ext'; simp only [toList_append, toList_empty, List.nil_append]
|
||||
|
||||
@[simp] theorem append_assoc (as bs cs : Array α) : as ++ bs ++ cs = as ++ (bs ++ cs) := by
|
||||
apply ext'; simp only [toList_append, List.append_assoc]
|
||||
|
||||
@[simp] theorem appendList_eq_append
|
||||
(arr : Array α) (l : List α) : arr.appendList l = arr ++ l := rfl
|
||||
|
||||
@@ -98,44 +87,20 @@ theorem foldrM_eq_reverse_foldlM_toList [Monad m] (f : α → β → m β) (init
|
||||
rw [← appendList_eq_append]; unfold Array.appendList
|
||||
induction l generalizing arr <;> simp [*]
|
||||
|
||||
@[deprecated "Use the reverse direction of `foldrM_toList`." (since := "2024-11-13")]
|
||||
theorem foldrM_eq_foldrM_toList [Monad m]
|
||||
(f : α → β → m β) (init : β) (arr : Array α) :
|
||||
arr.foldrM f init = arr.toList.foldrM f init := by
|
||||
simp
|
||||
@[deprecated foldlM_eq_foldlM_toList (since := "2024-09-09")]
|
||||
abbrev foldlM_eq_foldlM_data := @foldlM_eq_foldlM_toList
|
||||
|
||||
@[deprecated "Use the reverse direction of `foldlM_toList`." (since := "2024-11-13")]
|
||||
theorem foldlM_eq_foldlM_toList [Monad m]
|
||||
(f : β → α → m β) (init : β) (arr : Array α) :
|
||||
arr.foldlM f init = arr.toList.foldlM f init:= by
|
||||
simp
|
||||
|
||||
@[deprecated "Use the reverse direction of `foldr_toList`." (since := "2024-11-13")]
|
||||
theorem foldr_eq_foldr_toList
|
||||
(f : α → β → β) (init : β) (arr : Array α) :
|
||||
arr.foldr f init = arr.toList.foldr f init := by
|
||||
simp
|
||||
|
||||
@[deprecated "Use the reverse direction of `foldl_toList`." (since := "2024-11-13")]
|
||||
theorem foldl_eq_foldl_toList
|
||||
(f : β → α → β) (init : β) (arr : Array α) :
|
||||
arr.foldl f init = arr.toList.foldl f init:= by
|
||||
simp
|
||||
|
||||
@[deprecated foldlM_toList (since := "2024-09-09")]
|
||||
abbrev foldlM_eq_foldlM_data := @foldlM_toList
|
||||
|
||||
@[deprecated foldl_toList (since := "2024-09-09")]
|
||||
abbrev foldl_eq_foldl_data := @foldl_toList
|
||||
@[deprecated foldl_eq_foldl_toList (since := "2024-09-09")]
|
||||
abbrev foldl_eq_foldl_data := @foldl_eq_foldl_toList
|
||||
|
||||
@[deprecated foldrM_eq_reverse_foldlM_toList (since := "2024-09-09")]
|
||||
abbrev foldrM_eq_reverse_foldlM_data := @foldrM_eq_reverse_foldlM_toList
|
||||
|
||||
@[deprecated foldrM_toList (since := "2024-09-09")]
|
||||
abbrev foldrM_eq_foldrM_data := @foldrM_toList
|
||||
@[deprecated foldrM_eq_foldrM_toList (since := "2024-09-09")]
|
||||
abbrev foldrM_eq_foldrM_data := @foldrM_eq_foldrM_toList
|
||||
|
||||
@[deprecated foldr_toList (since := "2024-09-09")]
|
||||
abbrev foldr_eq_foldr_data := @foldr_toList
|
||||
@[deprecated foldr_eq_foldr_toList (since := "2024-09-09")]
|
||||
abbrev foldr_eq_foldr_data := @foldr_eq_foldr_toList
|
||||
|
||||
@[deprecated push_toList (since := "2024-09-09")]
|
||||
abbrev push_data := @push_toList
|
||||
@@ -146,8 +111,8 @@ abbrev toList_eq := @toListImpl_eq
|
||||
@[deprecated pop_toList (since := "2024-09-09")]
|
||||
abbrev pop_data := @pop_toList
|
||||
|
||||
@[deprecated toList_append (since := "2024-09-09")]
|
||||
abbrev append_data := @toList_append
|
||||
@[deprecated append_toList (since := "2024-09-09")]
|
||||
abbrev append_data := @append_toList
|
||||
|
||||
@[deprecated appendList_toList (since := "2024-09-09")]
|
||||
abbrev appendList_data := @appendList_toList
|
||||
|
||||
@@ -5,81 +5,43 @@ Authors: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Basic
|
||||
import Init.Data.BEq
|
||||
import Init.Data.List.Nat.BEq
|
||||
import Init.ByCases
|
||||
|
||||
namespace Array
|
||||
|
||||
theorem rel_of_isEqvAux
|
||||
{r : α → α → Bool} {a b : Array α} (hsz : a.size = b.size) {i : Nat} (hi : i ≤ a.size)
|
||||
(heqv : Array.isEqvAux a b hsz r i hi)
|
||||
{j : Nat} (hj : j < i) : r (a[j]'(Nat.lt_of_lt_of_le hj hi)) (b[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi))) := by
|
||||
induction i with
|
||||
| zero => contradiction
|
||||
| succ i ih =>
|
||||
simp only [Array.isEqvAux, Bool.and_eq_true, decide_eq_true_eq] at heqv
|
||||
by_cases hj' : j < i
|
||||
next =>
|
||||
exact ih _ heqv.right hj'
|
||||
next =>
|
||||
replace hj' : j = i := Nat.eq_of_le_of_lt_succ (Nat.not_lt.mp hj') hj
|
||||
subst hj'
|
||||
exact heqv.left
|
||||
theorem eq_of_isEqvAux [DecidableEq α] (a b : Array α) (hsz : a.size = b.size) (i : Nat) (hi : i ≤ a.size) (heqv : Array.isEqvAux a b hsz (fun x y => x = y) i) (j : Nat) (low : i ≤ j) (high : j < a.size) : a[j] = b[j]'(hsz ▸ high) := by
|
||||
by_cases h : i < a.size
|
||||
· unfold Array.isEqvAux at heqv
|
||||
simp [h] at heqv
|
||||
have hind := eq_of_isEqvAux a b hsz (i+1) (Nat.succ_le_of_lt h) heqv.2
|
||||
by_cases heq : i = j
|
||||
· subst heq; exact heqv.1
|
||||
· exact hind j (Nat.succ_le_of_lt (Nat.lt_of_le_of_ne low heq)) high
|
||||
· have heq : i = a.size := Nat.le_antisymm hi (Nat.ge_of_not_lt h)
|
||||
subst heq
|
||||
exact absurd (Nat.lt_of_lt_of_le high low) (Nat.lt_irrefl j)
|
||||
termination_by a.size - i
|
||||
decreasing_by decreasing_trivial_pre_omega
|
||||
|
||||
theorem isEqvAux_of_rel {r : α → α → Bool} {a b : Array α} (hsz : a.size = b.size) {i : Nat} (hi : i ≤ a.size)
|
||||
(w : ∀ j, (hj : j < i) → r (a[j]'(Nat.lt_of_lt_of_le hj hi)) (b[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi)))) : Array.isEqvAux a b hsz r i hi := by
|
||||
induction i with
|
||||
| zero => simp [Array.isEqvAux]
|
||||
| succ i ih =>
|
||||
simp only [isEqvAux, Bool.and_eq_true]
|
||||
exact ⟨w i (Nat.lt_add_one i), ih _ fun j hj => w j (Nat.lt_add_right 1 hj)⟩
|
||||
|
||||
theorem rel_of_isEqv {r : α → α → Bool} {a b : Array α} :
|
||||
Array.isEqv a b r → ∃ h : a.size = b.size, ∀ (i : Nat) (h' : i < a.size), r (a[i]) (b[i]'(h ▸ h')) := by
|
||||
simp only [isEqv]
|
||||
split <;> rename_i h
|
||||
· exact fun h' => ⟨h, fun i => rel_of_isEqvAux h (Nat.le_refl ..) h'⟩
|
||||
· intro; contradiction
|
||||
theorem eq_of_isEqv [DecidableEq α] (a b : Array α) : Array.isEqv a b (fun x y => x = y) → a = b := by
|
||||
simp [Array.isEqv]
|
||||
split
|
||||
next hsz =>
|
||||
intro h
|
||||
have aux := eq_of_isEqvAux a b hsz 0 (Nat.zero_le ..) h
|
||||
exact ext a b hsz fun i h _ => aux i (Nat.zero_le ..) _
|
||||
next => intro; contradiction
|
||||
|
||||
theorem isEqv_iff_rel (a b : Array α) (r) :
|
||||
Array.isEqv a b r ↔ ∃ h : a.size = b.size, ∀ (i : Nat) (h' : i < a.size), r (a[i]) (b[i]'(h ▸ h')) :=
|
||||
⟨rel_of_isEqv, fun ⟨h, w⟩ => by
|
||||
simp only [isEqv, ← h, ↓reduceDIte]
|
||||
exact isEqvAux_of_rel h (by simp [h]) w⟩
|
||||
theorem isEqvAux_self [DecidableEq α] (a : Array α) (i : Nat) : Array.isEqvAux a a rfl (fun x y => x = y) i = true := by
|
||||
unfold Array.isEqvAux
|
||||
split
|
||||
next h => simp [h, isEqvAux_self a (i+1)]
|
||||
next h => simp [h]
|
||||
termination_by a.size - i
|
||||
decreasing_by decreasing_trivial_pre_omega
|
||||
|
||||
theorem isEqv_eq_decide (a b : Array α) (r) :
|
||||
Array.isEqv a b r =
|
||||
if h : a.size = b.size then decide (∀ (i : Nat) (h' : i < a.size), r (a[i]) (b[i]'(h ▸ h'))) else false := by
|
||||
by_cases h : Array.isEqv a b r
|
||||
· simp only [h, Bool.true_eq]
|
||||
simp only [isEqv_iff_rel] at h
|
||||
obtain ⟨h, w⟩ := h
|
||||
simp [h, w]
|
||||
· let h' := h
|
||||
simp only [Bool.not_eq_true] at h
|
||||
simp only [h, Bool.false_eq, dite_eq_right_iff, decide_eq_false_iff_not, Classical.not_forall,
|
||||
Bool.not_eq_true]
|
||||
simpa [isEqv_iff_rel] using h'
|
||||
|
||||
@[simp] theorem isEqv_toList [BEq α] (a b : Array α) : (a.toList.isEqv b.toList r) = (a.isEqv b r) := by
|
||||
simp [isEqv_eq_decide, List.isEqv_eq_decide]
|
||||
|
||||
theorem eq_of_isEqv [DecidableEq α] (a b : Array α) (h : Array.isEqv a b (fun x y => x = y)) : a = b := by
|
||||
have ⟨h, h'⟩ := rel_of_isEqv h
|
||||
exact ext _ _ h (fun i lt _ => by simpa using h' i lt)
|
||||
|
||||
theorem isEqvAux_self (r : α → α → Bool) (hr : ∀ a, r a a) (a : Array α) (i : Nat) (h : i ≤ a.size) :
|
||||
Array.isEqvAux a a rfl r i h = true := by
|
||||
induction i with
|
||||
| zero => simp [Array.isEqvAux]
|
||||
| succ i ih =>
|
||||
simp_all only [isEqvAux, Bool.and_self]
|
||||
|
||||
theorem isEqv_self_beq [BEq α] [ReflBEq α] (a : Array α) : Array.isEqv a a (· == ·) = true := by
|
||||
simp [isEqv, isEqvAux_self]
|
||||
|
||||
theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (· = ·) = true := by
|
||||
theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (fun x y => x = y) = true := by
|
||||
simp [isEqv, isEqvAux_self]
|
||||
|
||||
instance [DecidableEq α] : DecidableEq (Array α) :=
|
||||
@@ -88,22 +50,4 @@ instance [DecidableEq α] : DecidableEq (Array α) :=
|
||||
| true => isTrue (eq_of_isEqv a b h)
|
||||
| false => isFalse fun h' => by subst h'; rw [isEqv_self] at h; contradiction
|
||||
|
||||
theorem beq_eq_decide [BEq α] (a b : Array α) :
|
||||
(a == b) = if h : a.size = b.size then
|
||||
decide (∀ (i : Nat) (h' : i < a.size), a[i] == b[i]'(h ▸ h')) else false := by
|
||||
simp [BEq.beq, isEqv_eq_decide]
|
||||
|
||||
@[simp] theorem beq_toList [BEq α] (a b : Array α) : (a.toList == b.toList) = (a == b) := by
|
||||
simp [beq_eq_decide, List.beq_eq_decide]
|
||||
|
||||
end Array
|
||||
|
||||
namespace List
|
||||
|
||||
@[simp] theorem isEqv_toArray [BEq α] (a b : List α) : (a.toArray.isEqv b.toArray r) = (a.isEqv b r) := by
|
||||
simp [isEqv_eq_decide, Array.isEqv_eq_decide]
|
||||
|
||||
@[simp] theorem beq_toArray [BEq α] (a b : List α) : (a.toArray == b.toArray) = (a == b) := by
|
||||
simp [beq_eq_decide, Array.beq_eq_decide]
|
||||
|
||||
end List
|
||||
|
||||
@@ -1,281 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Find
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.Array.Attach
|
||||
|
||||
/-!
|
||||
# Lemmas about `Array.findSome?`, `Array.find?`.
|
||||
-/
|
||||
|
||||
namespace Array
|
||||
|
||||
open Nat
|
||||
|
||||
/-! ### findSome? -/
|
||||
|
||||
@[simp] theorem findSomeRev?_push_of_isSome (l : Array α) (h : (f a).isSome) : (l.push a).findSomeRev? f = f a := by
|
||||
cases l; simp_all
|
||||
|
||||
@[simp] theorem findSomeRev?_push_of_isNone (l : Array α) (h : (f a).isNone) : (l.push a).findSomeRev? f = l.findSomeRev? f := by
|
||||
cases l; simp_all
|
||||
|
||||
theorem exists_of_findSome?_eq_some {f : α → Option β} {l : Array α} (w : l.findSome? f = some b) :
|
||||
∃ a, a ∈ l ∧ f a = b := by
|
||||
cases l; simp_all [List.exists_of_findSome?_eq_some]
|
||||
|
||||
@[simp] theorem findSome?_eq_none_iff : findSome? p l = none ↔ ∀ x ∈ l, p x = none := by
|
||||
cases l; simp
|
||||
|
||||
@[simp] theorem findSome?_isSome_iff {f : α → Option β} {l : Array α} :
|
||||
(l.findSome? f).isSome ↔ ∃ x, x ∈ l ∧ (f x).isSome := by
|
||||
cases l; simp
|
||||
|
||||
theorem findSome?_eq_some_iff {f : α → Option β} {l : Array α} {b : β} :
|
||||
l.findSome? f = some b ↔ ∃ (l₁ : Array α) (a : α) (l₂ : Array α), l = l₁.push a ++ l₂ ∧ f a = some b ∧ ∀ x ∈ l₁, f x = none := by
|
||||
cases l
|
||||
simp only [List.findSome?_toArray, List.findSome?_eq_some_iff]
|
||||
constructor
|
||||
· rintro ⟨l₁, a, l₂, rfl, h₁, h₂⟩
|
||||
exact ⟨l₁.toArray, a, l₂.toArray, by simp_all⟩
|
||||
· rintro ⟨l₁, a, l₂, h₀, h₁, h₂⟩
|
||||
exact ⟨l₁.toList, a, l₂.toList, by simpa using congrArg toList h₀, h₁, by simpa⟩
|
||||
|
||||
@[simp] theorem findSome?_guard (l : Array α) : findSome? (Option.guard fun x => p x) l = find? p l := by
|
||||
cases l; simp
|
||||
|
||||
@[simp] theorem getElem?_zero_filterMap (f : α → Option β) (l : Array α) : (l.filterMap f)[0]? = l.findSome? f := by
|
||||
cases l; simp [← List.head?_eq_getElem?]
|
||||
|
||||
@[simp] theorem getElem_zero_filterMap (f : α → Option β) (l : Array α) (h) :
|
||||
(l.filterMap f)[0] = (l.findSome? f).get (by cases l; simpa [List.length_filterMap_eq_countP] using h) := by
|
||||
cases l; simp [← List.head_eq_getElem, ← getElem?_zero_filterMap]
|
||||
|
||||
@[simp] theorem back?_filterMap (f : α → Option β) (l : Array α) : (l.filterMap f).back? = l.findSomeRev? f := by
|
||||
cases l; simp
|
||||
|
||||
@[simp] theorem back!_filterMap [Inhabited β] (f : α → Option β) (l : Array α) :
|
||||
(l.filterMap f).back! = (l.findSomeRev? f).getD default := by
|
||||
cases l; simp
|
||||
|
||||
@[simp] theorem map_findSome? (f : α → Option β) (g : β → γ) (l : Array α) :
|
||||
(l.findSome? f).map g = l.findSome? (Option.map g ∘ f) := by
|
||||
cases l; simp
|
||||
|
||||
theorem findSome?_map (f : β → γ) (l : Array β) : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
|
||||
cases l; simp [List.findSome?_map]
|
||||
|
||||
theorem findSome?_append {l₁ l₂ : Array α} : (l₁ ++ l₂).findSome? f = (l₁.findSome? f).or (l₂.findSome? f) := by
|
||||
cases l₁; cases l₂; simp [List.findSome?_append]
|
||||
|
||||
theorem getElem?_zero_flatten (L : Array (Array α)) :
|
||||
(flatten L)[0]? = L.findSome? fun l => l[0]? := by
|
||||
cases L using array_array_induction
|
||||
simp [← List.head?_eq_getElem?, List.head?_flatten, List.findSome?_map, Function.comp_def]
|
||||
|
||||
theorem getElem_zero_flatten.proof {L : Array (Array α)} (h : 0 < L.flatten.size) :
|
||||
(L.findSome? fun l => l[0]?).isSome := by
|
||||
cases L using array_array_induction
|
||||
simp only [List.findSome?_toArray, List.findSome?_map, Function.comp_def, List.getElem?_toArray,
|
||||
List.findSome?_isSome_iff, List.isSome_getElem?]
|
||||
simp only [flatten_toArray_map_toArray, size_toArray, List.length_flatten,
|
||||
Nat.sum_pos_iff_exists_pos, List.mem_map] at h
|
||||
obtain ⟨_, ⟨xs, m, rfl⟩, h⟩ := h
|
||||
exact ⟨xs, m, by simpa using h⟩
|
||||
|
||||
theorem getElem_zero_flatten {L : Array (Array α)} (h) :
|
||||
(flatten L)[0] = (L.findSome? fun l => l[0]?).get (getElem_zero_flatten.proof h) := by
|
||||
have t := getElem?_zero_flatten L
|
||||
simp [getElem?_eq_getElem, h] at t
|
||||
simp [← t]
|
||||
|
||||
theorem back?_flatten {L : Array (Array α)} :
|
||||
(flatten L).back? = (L.findSomeRev? fun l => l.back?) := by
|
||||
cases L using array_array_induction
|
||||
simp [List.getLast?_flatten, ← List.map_reverse, List.findSome?_map, Function.comp_def]
|
||||
|
||||
theorem findSome?_mkArray : findSome? f (mkArray n a) = if n = 0 then none else f a := by
|
||||
simp [mkArray_eq_toArray_replicate, List.findSome?_replicate]
|
||||
|
||||
@[simp] theorem findSome?_mkArray_of_pos (h : 0 < n) : findSome? f (mkArray n a) = f a := by
|
||||
simp [findSome?_mkArray, Nat.ne_of_gt h]
|
||||
|
||||
-- Argument is unused, but used to decide whether `simp` should unfold.
|
||||
@[simp] theorem findSome?_mkArray_of_isSome (_ : (f a).isSome) :
|
||||
findSome? f (mkArray n a) = if n = 0 then none else f a := by
|
||||
simp [findSome?_mkArray]
|
||||
|
||||
@[simp] theorem findSome?_mkArray_of_isNone (h : (f a).isNone) :
|
||||
findSome? f (mkArray n a) = none := by
|
||||
rw [Option.isNone_iff_eq_none] at h
|
||||
simp [findSome?_mkArray, h]
|
||||
|
||||
/-! ### find? -/
|
||||
|
||||
@[simp] theorem find?_singleton (a : α) (p : α → Bool) :
|
||||
#[a].find? p = if p a then some a else none := by
|
||||
simp [singleton_eq_toArray_singleton]
|
||||
|
||||
@[simp] theorem findRev?_push_of_pos (l : Array α) (h : p a) :
|
||||
findRev? p (l.push a) = some a := by
|
||||
cases l; simp [h]
|
||||
|
||||
@[simp] theorem findRev?_cons_of_neg (l : Array α) (h : ¬p a) :
|
||||
findRev? p (l.push a) = findRev? p l := by
|
||||
cases l; simp [h]
|
||||
|
||||
@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
|
||||
cases l; simp
|
||||
|
||||
theorem find?_eq_some_iff_append {xs : Array α} :
|
||||
xs.find? p = some b ↔ p b ∧ ∃ (as bs : Array α), xs = as.push b ++ bs ∧ ∀ a ∈ as, !p a := by
|
||||
rcases xs with ⟨xs⟩
|
||||
simp only [List.find?_toArray, List.find?_eq_some_iff_append, Bool.not_eq_eq_eq_not,
|
||||
Bool.not_true, exists_and_right, and_congr_right_iff]
|
||||
intro w
|
||||
constructor
|
||||
· rintro ⟨as, ⟨⟨x, rfl⟩, h⟩⟩
|
||||
exact ⟨as.toArray, ⟨x.toArray, by simp⟩ , by simpa using h⟩
|
||||
· rintro ⟨as, ⟨⟨x, h'⟩, h⟩⟩
|
||||
exact ⟨as.toList, ⟨x.toList, by simpa using congrArg Array.toList h'⟩,
|
||||
by simpa using h⟩
|
||||
|
||||
@[simp]
|
||||
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] theorem find?_isSome {xs : Array α} {p : α → Bool} : (xs.find? p).isSome ↔ ∃ x, x ∈ xs ∧ p x := by
|
||||
cases xs; simp
|
||||
|
||||
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
|
||||
|
||||
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
|
||||
|
||||
theorem get_find?_mem {xs : Array α} (h) : (xs.find? p).get h ∈ xs := by
|
||||
cases xs
|
||||
simp [List.get_find?_mem]
|
||||
|
||||
@[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] theorem getElem?_zero_filter (p : α → Bool) (l : Array α) :
|
||||
(l.filter p)[0]? = l.find? p := by
|
||||
cases l; simp [← List.head?_eq_getElem?]
|
||||
|
||||
@[simp] theorem getElem_zero_filter (p : α → Bool) (l : Array α) (h) :
|
||||
(l.filter p)[0] =
|
||||
(l.find? p).get (by cases l; simpa [← List.countP_eq_length_filter] using h) := by
|
||||
cases l
|
||||
simp [List.getElem_zero_eq_head]
|
||||
|
||||
@[simp] theorem back?_filter (p : α → Bool) (l : Array α) : (l.filter p).back? = l.findRev? p := by
|
||||
cases l; simp
|
||||
|
||||
@[simp] theorem back!_filter [Inhabited α] (p : α → Bool) (l : Array α) :
|
||||
(l.filter p).back! = (l.findRev? p).get! := by
|
||||
cases l; simp [Option.get!_eq_getD]
|
||||
|
||||
@[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] 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] theorem find?_append {l₁ l₂ : Array α} :
|
||||
(l₁ ++ l₂).find? p = (l₁.find? p).or (l₂.find? p) := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
simp
|
||||
|
||||
@[simp] theorem find?_flatten (xs : Array (Array α)) (p : α → Bool) :
|
||||
xs.flatten.find? p = xs.findSome? (·.find? p) := by
|
||||
cases xs using array_array_induction
|
||||
simp [List.findSome?_map, Function.comp_def]
|
||||
|
||||
theorem find?_flatten_eq_none {xs : Array (Array α)} {p : α → Bool} :
|
||||
xs.flatten.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
|
||||
simp
|
||||
|
||||
/--
|
||||
If `find? p` returns `some a` from `xs.flatten`, then `p a` holds, and
|
||||
some array in `xs` contains `a`, and no earlier element of that array satisfies `p`.
|
||||
Moreover, no earlier array in `xs` has an element satisfying `p`.
|
||||
-/
|
||||
theorem find?_flatten_eq_some {xs : Array (Array α)} {p : α → Bool} {a : α} :
|
||||
xs.flatten.find? p = some a ↔
|
||||
p a ∧ ∃ (as : Array (Array α)) (ys zs : Array α) (bs : Array (Array α)),
|
||||
xs = as.push (ys.push a ++ zs) ++ bs ∧
|
||||
(∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
|
||||
cases xs using array_array_induction
|
||||
simp only [flatten_toArray_map_toArray, List.find?_toArray, List.find?_flatten_eq_some]
|
||||
simp only [Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, and_congr_right_iff]
|
||||
intro w
|
||||
constructor
|
||||
· rintro ⟨as, ys, ⟨⟨zs, bs, rfl⟩, h₁, h₂⟩⟩
|
||||
exact ⟨as.toArray.map List.toArray, ys.toArray,
|
||||
⟨zs.toArray, bs.toArray.map List.toArray, by simp⟩, by simpa using h₁, by simpa using h₂⟩
|
||||
· rintro ⟨as, ys, ⟨⟨zs, bs, h⟩, h₁, h₂⟩⟩
|
||||
replace h := congrArg (·.map Array.toList) (congrArg Array.toList h)
|
||||
simp [Function.comp_def] at h
|
||||
exact ⟨as.toList.map Array.toList, ys.toList,
|
||||
⟨zs.toList, bs.toList.map Array.toList, by simpa using h⟩,
|
||||
by simpa using h₁, by simpa using h₂⟩
|
||||
|
||||
@[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]
|
||||
|
||||
theorem find?_flatMap_eq_none {xs : Array α} {f : α → Array β} {p : β → Bool} :
|
||||
(xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
|
||||
simp
|
||||
|
||||
theorem find?_mkArray :
|
||||
find? p (mkArray n a) = if n = 0 then none else if p a then some a else none := by
|
||||
simp [mkArray_eq_toArray_replicate, List.find?_replicate]
|
||||
|
||||
@[simp] theorem find?_mkArray_of_length_pos (h : 0 < n) :
|
||||
find? p (mkArray n a) = if p a then some a else none := by
|
||||
simp [find?_mkArray, Nat.ne_of_gt h]
|
||||
|
||||
@[simp] theorem find?_mkArray_of_pos (h : p a) :
|
||||
find? p (mkArray n a) = if n = 0 then none else some a := by
|
||||
simp [find?_mkArray, h]
|
||||
|
||||
@[simp] theorem find?_mkArray_of_neg (h : ¬ p a) : find? p (mkArray n a) = none := by
|
||||
simp [find?_mkArray, h]
|
||||
|
||||
-- This isn't a `@[simp]` lemma since there is already a lemma for `l.find? p = none` for any `l`.
|
||||
theorem find?_mkArray_eq_none {n : Nat} {a : α} {p : α → Bool} :
|
||||
(mkArray n a).find? p = none ↔ n = 0 ∨ !p a := by
|
||||
simp [mkArray_eq_toArray_replicate, List.find?_replicate_eq_none, Classical.or_iff_not_imp_left]
|
||||
|
||||
@[simp] theorem find?_mkArray_eq_some {n : Nat} {a b : α} {p : α → Bool} :
|
||||
(mkArray n a).find? p = some b ↔ n ≠ 0 ∧ p a ∧ a = b := by
|
||||
simp [mkArray_eq_toArray_replicate]
|
||||
|
||||
@[simp] theorem get_find?_mkArray (n : Nat) (a : α) (p : α → Bool) (h) :
|
||||
((mkArray n a).find? p).get h = a := by
|
||||
simp [mkArray_eq_toArray_replicate]
|
||||
|
||||
theorem find?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Array α)
|
||||
(H : ∀ (a : α), a ∈ xs → P a) (p : β → Bool) :
|
||||
(xs.pmap f H).find? p = (xs.attach.find? (fun ⟨a, m⟩ => p (f a (H a m)))).map fun ⟨a, m⟩ => f a (H a m) := by
|
||||
simp only [pmap_eq_map_attach, find?_map]
|
||||
rfl
|
||||
|
||||
end Array
|
||||
@@ -1,46 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Leonardo de Moura
|
||||
-/
|
||||
|
||||
prelude
|
||||
import Init.Data.Array.Basic
|
||||
|
||||
namespace Array
|
||||
|
||||
/-! ### getLit -/
|
||||
|
||||
-- auxiliary declaration used in the equation compiler when pattern matching array literals.
|
||||
abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
|
||||
have := h₁.symm ▸ h₂
|
||||
a[i]
|
||||
|
||||
theorem extLit {n : Nat}
|
||||
(a b : Array α)
|
||||
(hsz₁ : a.size = n) (hsz₂ : b.size = n)
|
||||
(h : (i : Nat) → (hi : i < n) → a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
|
||||
Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ _ => h i (hsz₁ ▸ hi₁)
|
||||
|
||||
def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
|
||||
| 0, _, acc => acc
|
||||
| (i+1), hi, acc => toListLitAux a n hsz i (Nat.le_of_succ_le hi) (a.getLit i hsz (Nat.lt_of_lt_of_eq (Nat.lt_of_lt_of_le (Nat.lt_succ_self i) hi) hsz) :: acc)
|
||||
|
||||
def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
|
||||
List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.le_refl _) []
|
||||
|
||||
theorem toArrayLit_eq (as : Array α) (n : Nat) (hsz : as.size = n) : as = toArrayLit as n hsz := by
|
||||
apply ext'
|
||||
simp [toArrayLit, toList_toArray]
|
||||
have hle : n ≤ as.size := hsz ▸ Nat.le_refl _
|
||||
have hge : as.size ≤ n := hsz ▸ Nat.le_refl _
|
||||
have := go n hle
|
||||
rw [List.drop_eq_nil_of_le hge] at this
|
||||
rw [this]
|
||||
where
|
||||
getLit_eq (as : Array α) (i : Nat) (h₁ : as.size = n) (h₂ : i < n) : as.getLit i h₁ h₂ = getElem as.toList i ((id (α := as.toList.length = n) h₁) ▸ h₂) :=
|
||||
rfl
|
||||
go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.toList.drop i) = as.toList := by
|
||||
induction i <;> simp only [List.drop, toListLitAux, getLit_eq, List.getElem_cons_drop_succ_eq_drop, *]
|
||||
|
||||
end Array
|
||||
@@ -6,7 +6,7 @@ Authors: Leonardo de Moura
|
||||
prelude
|
||||
import Init.Data.Array.Basic
|
||||
|
||||
@[inline] def Array.insertionSort (a : Array α) (lt : α → α → Bool := by exact (· < ·)) : Array α :=
|
||||
@[inline] def Array.insertionSort (a : Array α) (lt : α → α → Bool) : Array α :=
|
||||
traverse a 0 a.size
|
||||
where
|
||||
@[specialize] traverse (a : Array α) (i : Nat) (fuel : Nat) : Array α :=
|
||||
@@ -23,6 +23,6 @@ where
|
||||
| j'+1 =>
|
||||
have h' : j' < a.size := by subst j; exact Nat.lt_trans (Nat.lt_succ_self _) h
|
||||
if lt a[j] a[j'] then
|
||||
swapLoop (a.swap j j') j' (by rw [size_swap]; assumption; done)
|
||||
swapLoop (a.swap ⟨j, h⟩ ⟨j', h'⟩) j' (by rw [size_swap]; assumption; done)
|
||||
else
|
||||
a
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -1,112 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2022 Mario Carneiro. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Mario Carneiro, Kim Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.List.MapIdx
|
||||
|
||||
namespace Array
|
||||
|
||||
/-! ### mapFinIdx -/
|
||||
|
||||
-- This could also be proved from `SatisfiesM_mapIdxM` in Batteries.
|
||||
theorem mapFinIdx_induction (as : Array α) (f : Fin as.size → α → β)
|
||||
(motive : Nat → Prop) (h0 : motive 0)
|
||||
(p : Fin as.size → β → Prop)
|
||||
(hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
|
||||
motive as.size ∧ ∃ eq : (Array.mapFinIdx as f).size = as.size,
|
||||
∀ i h, p ⟨i, h⟩ ((Array.mapFinIdx as f)[i]) := by
|
||||
let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p ⟨i, h⟩ bs[i]) (hm : motive j) :
|
||||
let arr : Array β := Array.mapFinIdxM.map (m := Id) as f i j h bs
|
||||
motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i, h⟩ arr[i] := by
|
||||
induction i generalizing j bs with simp [mapFinIdxM.map]
|
||||
| zero =>
|
||||
have := (Nat.zero_add _).symm.trans h
|
||||
exact ⟨this ▸ hm, h₁ ▸ this, fun _ _ => h₂ ..⟩
|
||||
| succ i ih =>
|
||||
apply @ih (bs.push (f ⟨j, by omega⟩ as[j])) (j + 1) (by omega) (by simp; omega)
|
||||
· intro i i_lt h'
|
||||
rw [getElem_push]
|
||||
split
|
||||
· apply h₂
|
||||
· simp only [size_push] at h'
|
||||
obtain rfl : i = j := by omega
|
||||
apply (hs ⟨i, by omega⟩ hm).1
|
||||
· exact (hs ⟨j, by omega⟩ hm).2
|
||||
simp [mapFinIdx, mapFinIdxM]; exact go rfl nofun h0
|
||||
|
||||
theorem mapFinIdx_spec (as : Array α) (f : Fin as.size → α → β)
|
||||
(p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
|
||||
∃ eq : (Array.mapFinIdx as f).size = as.size,
|
||||
∀ i h, p ⟨i, h⟩ ((Array.mapFinIdx as f)[i]) :=
|
||||
(mapFinIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
|
||||
|
||||
@[simp] theorem size_mapFinIdx (a : Array α) (f : Fin a.size → α → β) : (a.mapFinIdx f).size = a.size :=
|
||||
(mapFinIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
|
||||
|
||||
@[simp] theorem size_zipWithIndex (as : Array α) : as.zipWithIndex.size = as.size :=
|
||||
Array.size_mapFinIdx _ _
|
||||
|
||||
@[simp] theorem getElem_mapFinIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat)
|
||||
(h : i < (mapFinIdx a f).size) :
|
||||
(a.mapFinIdx f)[i] = f ⟨i, by simp_all⟩ (a[i]'(by simp_all)) :=
|
||||
(mapFinIdx_spec _ _ (fun i b => b = f i a[i]) fun _ => rfl).2 i _
|
||||
|
||||
@[simp] theorem getElem?_mapFinIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat) :
|
||||
(a.mapFinIdx f)[i]? =
|
||||
a[i]?.pbind fun b h => f ⟨i, (getElem?_eq_some_iff.1 h).1⟩ b := by
|
||||
simp only [getElem?_def, size_mapFinIdx, getElem_mapFinIdx]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem toList_mapFinIdx (a : Array α) (f : Fin a.size → α → β) :
|
||||
(a.mapFinIdx f).toList = a.toList.mapFinIdx (fun i a => f ⟨i, by simp⟩ a) := by
|
||||
apply List.ext_getElem <;> simp
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
theorem mapIdx_induction (f : Nat → α → β) (as : Array α)
|
||||
(motive : Nat → Prop) (h0 : motive 0)
|
||||
(p : Fin as.size → β → Prop)
|
||||
(hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
|
||||
motive as.size ∧ ∃ eq : (as.mapIdx f).size = as.size,
|
||||
∀ i h, p ⟨i, h⟩ ((as.mapIdx f)[i]) :=
|
||||
mapFinIdx_induction as (fun i a => f i a) motive h0 p hs
|
||||
|
||||
theorem mapIdx_spec (f : Nat → α → β) (as : Array α)
|
||||
(p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
|
||||
∃ eq : (as.mapIdx f).size = as.size,
|
||||
∀ i h, p ⟨i, h⟩ ((as.mapIdx f)[i]) :=
|
||||
(mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
|
||||
|
||||
@[simp] theorem size_mapIdx (f : Nat → α → β) (as : Array α) : (as.mapIdx f).size = as.size :=
|
||||
(mapIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
|
||||
|
||||
@[simp] theorem getElem_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat)
|
||||
(h : i < (as.mapIdx f).size) :
|
||||
(as.mapIdx f)[i] = f i (as[i]'(by simp_all)) :=
|
||||
(mapIdx_spec _ _ (fun i b => b = f i as[i]) fun _ => rfl).2 i (by simp_all)
|
||||
|
||||
@[simp] theorem getElem?_mapIdx (f : Nat → α → β) (as : Array α) (i : Nat) :
|
||||
(as.mapIdx f)[i]? =
|
||||
as[i]?.map (f i) := by
|
||||
simp [getElem?_def, size_mapIdx, getElem_mapIdx]
|
||||
|
||||
@[simp] theorem toList_mapIdx (f : Nat → α → β) (as : Array α) :
|
||||
(as.mapIdx f).toList = as.toList.mapIdx (fun i a => f i a) := by
|
||||
apply List.ext_getElem <;> simp
|
||||
|
||||
end Array
|
||||
|
||||
namespace List
|
||||
|
||||
@[simp] theorem mapFinIdx_toArray (l : List α) (f : Fin l.length → α → β) :
|
||||
l.toArray.mapFinIdx f = (l.mapFinIdx f).toArray := by
|
||||
ext <;> simp
|
||||
|
||||
@[simp] theorem mapIdx_toArray (f : Nat → α → β) (l : List α) :
|
||||
l.toArray.mapIdx f = (l.mapIdx f).toArray := by
|
||||
ext <;> simp
|
||||
|
||||
end List
|
||||
@@ -10,16 +10,25 @@ import Init.Data.List.BasicAux
|
||||
|
||||
namespace Array
|
||||
|
||||
/-- `a ∈ as` is a predicate which asserts that `a` is in the array `as`. -/
|
||||
-- NB: This is defined as a structure rather than a plain def so that a lemma
|
||||
-- like `sizeOf_lt_of_mem` will not apply with no actual arrays around.
|
||||
structure Mem (as : Array α) (a : α) : Prop where
|
||||
val : a ∈ as.toList
|
||||
|
||||
instance : Membership α (Array α) where
|
||||
mem := Mem
|
||||
|
||||
theorem sizeOf_lt_of_mem [SizeOf α] {as : Array α} (h : a ∈ as) : sizeOf a < sizeOf as := by
|
||||
cases as with | _ as =>
|
||||
exact Nat.lt_trans (List.sizeOf_lt_of_mem h.val) (by simp_arith)
|
||||
|
||||
theorem sizeOf_get [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) : sizeOf (as.get i h) < sizeOf as := by
|
||||
theorem sizeOf_get [SizeOf α] (as : Array α) (i : Fin as.size) : sizeOf (as.get i) < sizeOf as := by
|
||||
cases as with | _ as =>
|
||||
simpa using Nat.lt_trans (List.sizeOf_get _ ⟨i, h⟩) (by simp_arith)
|
||||
exact Nat.lt_trans (List.sizeOf_get ..) (by simp_arith)
|
||||
|
||||
@[simp] theorem sizeOf_getElem [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) :
|
||||
sizeOf (as[i]'h) < sizeOf as := sizeOf_get _ _ h
|
||||
sizeOf (as[i]'h) < sizeOf as := sizeOf_get _ _
|
||||
|
||||
/-- This tactic, added to the `decreasing_trivial` toolbox, proves that
|
||||
`sizeOf arr[i] < sizeOf arr`, which is useful for well founded recursions
|
||||
|
||||
@@ -1,159 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.Array.Attach
|
||||
import Init.Data.List.Monadic
|
||||
|
||||
/-!
|
||||
# Lemmas about `Array.forIn'` and `Array.forIn`.
|
||||
-/
|
||||
|
||||
namespace Array
|
||||
|
||||
open Nat
|
||||
|
||||
/-! ## Monadic operations -/
|
||||
|
||||
/-! ### mapM -/
|
||||
|
||||
theorem mapM_eq_foldlM_push [Monad m] [LawfulMonad m] (f : α → m β) (l : Array α) :
|
||||
mapM f l = l.foldlM (fun acc a => return (acc.push (← f a))) #[] := by
|
||||
rcases l with ⟨l⟩
|
||||
simp only [List.mapM_toArray, bind_pure_comp, size_toArray, List.foldlM_toArray']
|
||||
rw [List.mapM_eq_reverse_foldlM_cons]
|
||||
simp only [bind_pure_comp, Functor.map_map]
|
||||
suffices ∀ (k), (fun a => a.reverse.toArray) <$> List.foldlM (fun acc a => (fun a => a :: acc) <$> f a) k l =
|
||||
List.foldlM (fun acc a => acc.push <$> f a) k.reverse.toArray l by
|
||||
exact this []
|
||||
intro k
|
||||
induction l generalizing k with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp [ih, List.foldlM_cons]
|
||||
|
||||
/-! ### foldlM and foldrM -/
|
||||
|
||||
theorem foldlM_map [Monad m] (f : β₁ → β₂) (g : α → β₂ → m α) (l : Array β₁) (init : α) :
|
||||
(l.map f).foldlM g init = l.foldlM (fun x y => g x (f y)) init := by
|
||||
cases l
|
||||
rw [List.map_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldlM_map]
|
||||
|
||||
theorem foldrM_map [Monad m] [LawfulMonad m] (f : β₁ → β₂) (g : β₂ → α → m α) (l : Array β₁)
|
||||
(init : α) : (l.map f).foldrM g init = l.foldrM (fun x y => g (f x) y) init := by
|
||||
cases l
|
||||
rw [List.map_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldrM_map]
|
||||
|
||||
theorem foldlM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : γ → β → m γ) (l : Array α) (init : γ) :
|
||||
(l.filterMap f).foldlM g init =
|
||||
l.foldlM (fun x y => match f y with | some b => g x b | none => pure x) init := by
|
||||
cases l
|
||||
rw [List.filterMap_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldlM_filterMap]
|
||||
rfl
|
||||
|
||||
theorem foldrM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : β → γ → m γ) (l : Array α) (init : γ) :
|
||||
(l.filterMap f).foldrM g init =
|
||||
l.foldrM (fun x y => match f x with | some b => g b y | none => pure y) init := by
|
||||
cases l
|
||||
rw [List.filterMap_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldrM_filterMap]
|
||||
rfl
|
||||
|
||||
theorem foldlM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : β → α → m β) (l : Array α) (init : β) :
|
||||
(l.filter p).foldlM g init =
|
||||
l.foldlM (fun x y => if p y then g x y else pure x) init := by
|
||||
cases l
|
||||
rw [List.filter_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldlM_filter]
|
||||
|
||||
theorem foldrM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : α → β → m β) (l : Array α) (init : β) :
|
||||
(l.filter p).foldrM g init =
|
||||
l.foldrM (fun x y => if p x then g x y else pure y) init := by
|
||||
cases l
|
||||
rw [List.filter_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldrM_filter]
|
||||
|
||||
/-! ### forIn' -/
|
||||
|
||||
/--
|
||||
We can express a for loop over an array as a fold,
|
||||
in which whenever we reach `.done b` we keep that value through the rest of the fold.
|
||||
-/
|
||||
theorem forIn'_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(l : Array α) (f : (a : α) → a ∈ l → β → m (ForInStep β)) (init : β) :
|
||||
forIn' l init f = ForInStep.value <$>
|
||||
l.attach.foldlM (fun b ⟨a, m⟩ => match b with
|
||||
| .yield b => f a m b
|
||||
| .done b => pure (.done b)) (ForInStep.yield init) := by
|
||||
cases l
|
||||
rw [List.attach_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp only [List.forIn'_toArray, List.forIn'_eq_foldlM, List.attachWith_mem_toArray, size_toArray,
|
||||
List.length_map, List.length_attach, List.foldlM_toArray', List.foldlM_map]
|
||||
congr
|
||||
|
||||
/-- We can express a for loop over an array which always yields as a fold. -/
|
||||
@[simp] theorem forIn'_yield_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(l : Array α) (f : (a : α) → a ∈ l → β → m γ) (g : (a : α) → a ∈ l → β → γ → β) (init : β) :
|
||||
forIn' l init (fun a m b => (fun c => .yield (g a m b c)) <$> f a m b) =
|
||||
l.attach.foldlM (fun b ⟨a, m⟩ => g a m b <$> f a m b) init := by
|
||||
cases l
|
||||
rw [List.attach_toArray] -- Why doesn't this fire via `simp`?
|
||||
simp [List.foldlM_map]
|
||||
|
||||
theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
|
||||
(l : Array α) (f : (a : α) → a ∈ l → β → β) (init : β) :
|
||||
forIn' l init (fun a m b => pure (.yield (f a m b))) =
|
||||
pure (f := m) (l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init) := by
|
||||
cases l
|
||||
simp [List.forIn'_pure_yield_eq_foldl, List.foldl_map]
|
||||
|
||||
@[simp] theorem forIn'_yield_eq_foldl
|
||||
(l : Array α) (f : (a : α) → a ∈ l → β → β) (init : β) :
|
||||
forIn' (m := Id) l init (fun a m b => .yield (f a m b)) =
|
||||
l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init := by
|
||||
cases l
|
||||
simp [List.foldl_map]
|
||||
|
||||
/--
|
||||
We can express a for loop over an array as a fold,
|
||||
in which whenever we reach `.done b` we keep that value through the rest of the fold.
|
||||
-/
|
||||
theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(f : α → β → m (ForInStep β)) (init : β) (l : Array α) :
|
||||
forIn l init f = ForInStep.value <$>
|
||||
l.foldlM (fun b a => match b with
|
||||
| .yield b => f a b
|
||||
| .done b => pure (.done b)) (ForInStep.yield init) := by
|
||||
cases l
|
||||
simp only [List.forIn_toArray, List.forIn_eq_foldlM, size_toArray, List.foldlM_toArray']
|
||||
congr
|
||||
|
||||
/-- We can express a for loop over an array which always yields as a fold. -/
|
||||
@[simp] theorem forIn_yield_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(l : Array α) (f : α → β → m γ) (g : α → β → γ → β) (init : β) :
|
||||
forIn l init (fun a b => (fun c => .yield (g a b c)) <$> f a b) =
|
||||
l.foldlM (fun b a => g a b <$> f a b) init := by
|
||||
cases l
|
||||
simp [List.foldlM_map]
|
||||
|
||||
theorem forIn_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
|
||||
(l : Array α) (f : α → β → β) (init : β) :
|
||||
forIn l init (fun a b => pure (.yield (f a b))) =
|
||||
pure (f := m) (l.foldl (fun b a => f a b) init) := by
|
||||
cases l
|
||||
simp [List.forIn_pure_yield_eq_foldl, List.foldl_map]
|
||||
|
||||
@[simp] theorem forIn_yield_eq_foldl
|
||||
(l : Array α) (f : α → β → β) (init : β) :
|
||||
forIn (m := Id) l init (fun a b => .yield (f a b)) =
|
||||
l.foldl (fun b a => f a b) init := by
|
||||
cases l
|
||||
simp [List.foldl_map]
|
||||
|
||||
end Array
|
||||
@@ -5,7 +5,6 @@ Authors: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Basic
|
||||
import Init.Data.Ord
|
||||
|
||||
namespace Array
|
||||
-- TODO: remove the [Inhabited α] parameters as soon as we have the tactic framework for automating proof generation and using Array.fget
|
||||
@@ -13,19 +12,19 @@ namespace Array
|
||||
def qpartition (as : Array α) (lt : α → α → Bool) (lo hi : Nat) : Nat × Array α :=
|
||||
if h : as.size = 0 then (0, as) else have : Inhabited α := ⟨as[0]'(by revert h; cases as.size <;> simp)⟩ -- TODO: remove
|
||||
let mid := (lo + hi) / 2
|
||||
let as := if lt (as.get! mid) (as.get! lo) then as.swapIfInBounds lo mid else as
|
||||
let as := if lt (as.get! hi) (as.get! lo) then as.swapIfInBounds lo hi else as
|
||||
let as := if lt (as.get! mid) (as.get! hi) then as.swapIfInBounds mid hi else as
|
||||
let as := if lt (as.get! mid) (as.get! lo) then as.swap! lo mid else as
|
||||
let as := if lt (as.get! hi) (as.get! lo) then as.swap! lo hi else as
|
||||
let as := if lt (as.get! mid) (as.get! hi) then as.swap! mid hi else as
|
||||
let pivot := as.get! hi
|
||||
let rec loop (as : Array α) (i j : Nat) :=
|
||||
if h : j < hi then
|
||||
if lt (as.get! j) pivot then
|
||||
let as := as.swapIfInBounds i j
|
||||
let as := as.swap! i j
|
||||
loop as (i+1) (j+1)
|
||||
else
|
||||
loop as i (j+1)
|
||||
else
|
||||
let as := as.swapIfInBounds i hi
|
||||
let as := as.swap! i hi
|
||||
(i, as)
|
||||
termination_by hi - j
|
||||
decreasing_by all_goals simp_wf; decreasing_trivial_pre_omega
|
||||
@@ -45,11 +44,4 @@ def qpartition (as : Array α) (lt : α → α → Bool) (lo hi : Nat) : Nat ×
|
||||
else as
|
||||
sort as low high
|
||||
|
||||
set_option linter.unusedVariables.funArgs false in
|
||||
/--
|
||||
Sort an array using `compare` to compare elements.
|
||||
-/
|
||||
def qsortOrd [ord : Ord α] (xs : Array α) : Array α :=
|
||||
xs.qsort fun x y => compare x y |>.isLT
|
||||
|
||||
end Array
|
||||
|
||||
@@ -1,41 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Leonardo de Moura, Mario Carneiro
|
||||
-/
|
||||
prelude
|
||||
import Init.Tactics
|
||||
|
||||
|
||||
/--
|
||||
Set an element in an array, using a proof that the index is in bounds.
|
||||
(This proof can usually be omitted, and will be synthesized automatically.)
|
||||
|
||||
This will perform the update destructively provided that `a` has a reference
|
||||
count of 1 when called.
|
||||
-/
|
||||
@[extern "lean_array_fset"]
|
||||
def Array.set (a : Array α) (i : @& Nat) (v : α) (h : i < a.size := by get_elem_tactic) :
|
||||
Array α where
|
||||
toList := a.toList.set i v
|
||||
|
||||
/--
|
||||
Set an element in an array, or do nothing if the index is out of bounds.
|
||||
|
||||
This will perform the update destructively provided that `a` has a reference
|
||||
count of 1 when called.
|
||||
-/
|
||||
@[inline] def Array.setIfInBounds (a : Array α) (i : Nat) (v : α) : Array α :=
|
||||
dite (LT.lt i a.size) (fun h => a.set i v h) (fun _ => a)
|
||||
|
||||
@[deprecated Array.setIfInBounds (since := "2024-11-24")] abbrev Array.setD := @Array.setIfInBounds
|
||||
|
||||
/--
|
||||
Set an element in an array, or panic if the index is out of bounds.
|
||||
|
||||
This will perform the update destructively provided that `a` has a reference
|
||||
count of 1 when called.
|
||||
-/
|
||||
@[extern "lean_array_set"]
|
||||
def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
|
||||
Array.setIfInBounds a i v
|
||||
@@ -15,6 +15,15 @@ structure Subarray (α : Type u) where
|
||||
start_le_stop : start ≤ stop
|
||||
stop_le_array_size : stop ≤ array.size
|
||||
|
||||
@[deprecated Subarray.array (since := "2024-04-13")]
|
||||
abbrev Subarray.as (s : Subarray α) : Array α := s.array
|
||||
|
||||
@[deprecated Subarray.start_le_stop (since := "2024-04-13")]
|
||||
theorem Subarray.h₁ (s : Subarray α) : s.start ≤ s.stop := s.start_le_stop
|
||||
|
||||
@[deprecated Subarray.stop_le_array_size (since := "2024-04-13")]
|
||||
theorem Subarray.h₂ (s : Subarray α) : s.stop ≤ s.array.size := s.stop_le_array_size
|
||||
|
||||
namespace Subarray
|
||||
|
||||
def size (s : Subarray α) : Nat :=
|
||||
@@ -39,7 +48,7 @@ instance : GetElem (Subarray α) Nat α fun xs i => i < xs.size where
|
||||
getElem xs i h := xs.get ⟨i, h⟩
|
||||
|
||||
@[inline] def getD (s : Subarray α) (i : Nat) (v₀ : α) : α :=
|
||||
if h : i < s.size then s[i] else v₀
|
||||
if h : i < s.size then s.get ⟨i, h⟩ else v₀
|
||||
|
||||
abbrev get! [Inhabited α] (s : Subarray α) (i : Nat) : α :=
|
||||
getD s i default
|
||||
@@ -50,22 +59,6 @@ def popFront (s : Subarray α) : Subarray α :=
|
||||
else
|
||||
s
|
||||
|
||||
/--
|
||||
The empty subarray.
|
||||
-/
|
||||
protected def empty : Subarray α where
|
||||
array := #[]
|
||||
start := 0
|
||||
stop := 0
|
||||
start_le_stop := Nat.le_refl 0
|
||||
stop_le_array_size := Nat.le_refl 0
|
||||
|
||||
instance : EmptyCollection (Subarray α) :=
|
||||
⟨Subarray.empty⟩
|
||||
|
||||
instance : Inhabited (Subarray α) :=
|
||||
⟨{}⟩
|
||||
|
||||
@[inline] unsafe def forInUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (s : Subarray α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
|
||||
let sz := USize.ofNat s.stop
|
||||
let rec @[specialize] loop (i : USize) (b : β) : m β := do
|
||||
|
||||
@@ -23,13 +23,16 @@ def split (s : Subarray α) (i : Fin s.size.succ) : (Subarray α × Subarray α)
|
||||
let ⟨i', isLt⟩ := i
|
||||
have := s.start_le_stop
|
||||
have := s.stop_le_array_size
|
||||
have : i' ≤ s.stop - s.start := Nat.lt_succ.mp isLt
|
||||
have : s.start + i' ≤ s.stop := by omega
|
||||
have : s.start + i' ≤ s.array.size := by omega
|
||||
have : s.start + i' ≤ s.stop := by
|
||||
simp only [size] at isLt
|
||||
omega
|
||||
let pre := {s with
|
||||
stop := s.start + i',
|
||||
start_le_stop := by omega,
|
||||
stop_le_array_size := by omega
|
||||
stop_le_array_size := by assumption
|
||||
}
|
||||
let post := {s with
|
||||
start := s.start + i'
|
||||
@@ -45,7 +48,9 @@ def drop (arr : Subarray α) (i : Nat) : Subarray α where
|
||||
array := arr.array
|
||||
start := min (arr.start + i) arr.stop
|
||||
stop := arr.stop
|
||||
start_le_stop := by omega
|
||||
start_le_stop := by
|
||||
rw [Nat.min_def]
|
||||
split <;> simp only [Nat.le_refl, *]
|
||||
stop_le_array_size := arr.stop_le_array_size
|
||||
|
||||
/--
|
||||
@@ -58,7 +63,9 @@ def take (arr : Subarray α) (i : Nat) : Subarray α where
|
||||
stop := min (arr.start + i) arr.stop
|
||||
start_le_stop := by
|
||||
have := arr.start_le_stop
|
||||
omega
|
||||
rw [Nat.min_def]
|
||||
split <;> omega
|
||||
stop_le_array_size := by
|
||||
have := arr.stop_le_array_size
|
||||
omega
|
||||
rw [Nat.min_def]
|
||||
split <;> omega
|
||||
|
||||
@@ -12,7 +12,6 @@ namespace Array
|
||||
theorem exists_of_uset (self : Array α) (i d h) :
|
||||
∃ l₁ l₂, self.toList = l₁ ++ self[i] :: l₂ ∧ List.length l₁ = i.toNat ∧
|
||||
(self.uset i d h).toList = l₁ ++ d :: l₂ := by
|
||||
simpa only [ugetElem_eq_getElem, getElem_eq_getElem_toList, uset, toList_set] using
|
||||
List.exists_of_set _
|
||||
simpa [Array.getElem_eq_toList_getElem] using List.exists_of_set _
|
||||
|
||||
end Array
|
||||
|
||||
@@ -1,7 +1,7 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison
|
||||
Authors: Scott Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.BitVec.Basic
|
||||
|
||||
@@ -1,20 +1,19 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joe Hendrix, Wojciech Nawrocki, Leonardo de Moura, Mario Carneiro, Alex Keizer, Harun Khan, Abdalrhman M Mohamed, Siddharth Bhat
|
||||
Authors: Joe Hendrix, Wojciech Nawrocki, Leonardo de Moura, Mario Carneiro, Alex Keizer, Harun Khan, Abdalrhman M Mohamed
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Fin.Basic
|
||||
import Init.Data.Nat.Bitwise.Lemmas
|
||||
import Init.Data.Nat.Power2
|
||||
import Init.Data.Int.Bitwise
|
||||
import Init.Data.BitVec.BasicAux
|
||||
|
||||
/-!
|
||||
We define the basic algebraic structure of bitvectors. We choose the `Fin` representation over
|
||||
others for its relative efficiency (Lean has special support for `Nat`), and the fact that bitwise
|
||||
operations on `Fin` are already defined. Some other possible representations are `List Bool`,
|
||||
`{ l : List Bool // l.length = w }`, `Fin w → Bool`.
|
||||
We define bitvectors. We choose the `Fin` representation over others for its relative efficiency
|
||||
(Lean has special support for `Nat`), alignment with `UIntXY` types which are also represented
|
||||
with `Fin`, and the fact that bitwise operations on `Fin` are already defined. Some other possible
|
||||
representations are `List Bool`, `{ l : List Bool // l.length = w }`, `Fin w → Bool`.
|
||||
|
||||
We define many of the bitvector operations from the
|
||||
[`QF_BV` logic](https://smtlib.cs.uiowa.edu/logics-all.shtml#QF_BV).
|
||||
@@ -23,12 +22,63 @@ of SMT-LIBv2.
|
||||
|
||||
set_option linter.missingDocs true
|
||||
|
||||
/--
|
||||
A bitvector of the specified width.
|
||||
|
||||
This is represented as the underlying `Nat` number in both the runtime
|
||||
and the kernel, inheriting all the special support for `Nat`.
|
||||
-/
|
||||
structure BitVec (w : Nat) where
|
||||
/-- Construct a `BitVec w` from a number less than `2^w`.
|
||||
O(1), because we use `Fin` as the internal representation of a bitvector. -/
|
||||
ofFin ::
|
||||
/-- Interpret a bitvector as a number less than `2^w`.
|
||||
O(1), because we use `Fin` as the internal representation of a bitvector. -/
|
||||
toFin : Fin (2^w)
|
||||
|
||||
/--
|
||||
Bitvectors have decidable equality. This should be used via the instance `DecidableEq (BitVec n)`.
|
||||
-/
|
||||
-- We manually derive the `DecidableEq` instances for `BitVec` because
|
||||
-- we want to have builtin support for bit-vector literals, and we
|
||||
-- need a name for this function to implement `canUnfoldAtMatcher` at `WHNF.lean`.
|
||||
def BitVec.decEq (x y : BitVec n) : Decidable (x = y) :=
|
||||
match x, y with
|
||||
| ⟨n⟩, ⟨m⟩ =>
|
||||
if h : n = m then
|
||||
isTrue (h ▸ rfl)
|
||||
else
|
||||
isFalse (fun h' => BitVec.noConfusion h' (fun h' => absurd h' h))
|
||||
|
||||
instance : DecidableEq (BitVec n) := BitVec.decEq
|
||||
|
||||
namespace BitVec
|
||||
|
||||
section Nat
|
||||
|
||||
/-- The `BitVec` with value `i`, given a proof that `i < 2^n`. -/
|
||||
@[match_pattern]
|
||||
protected def ofNatLt {n : Nat} (i : Nat) (p : i < 2^n) : BitVec n where
|
||||
toFin := ⟨i, p⟩
|
||||
|
||||
/-- The `BitVec` with value `i mod 2^n`. -/
|
||||
@[match_pattern]
|
||||
protected def ofNat (n : Nat) (i : Nat) : BitVec n where
|
||||
toFin := Fin.ofNat' (2^n) i
|
||||
|
||||
instance instOfNat : OfNat (BitVec n) i where ofNat := .ofNat n i
|
||||
instance natCastInst : NatCast (BitVec w) := ⟨BitVec.ofNat w⟩
|
||||
|
||||
/-- Given a bitvector `x`, return the underlying `Nat`. This is O(1) because `BitVec` is a
|
||||
(zero-cost) wrapper around a `Nat`. -/
|
||||
protected def toNat (x : BitVec n) : Nat := x.toFin.val
|
||||
|
||||
/-- Return the bound in terms of toNat. -/
|
||||
theorem isLt (x : BitVec w) : x.toNat < 2^w := x.toFin.isLt
|
||||
|
||||
@[deprecated isLt (since := "2024-03-12")]
|
||||
theorem toNat_lt (x : BitVec n) : x.toNat < 2^n := x.isLt
|
||||
|
||||
/-- Theorem for normalizing the bit vector literal representation. -/
|
||||
-- TODO: This needs more usage data to assess which direction the simp should go.
|
||||
@[simp, bv_toNat] theorem ofNat_eq_ofNat : @OfNat.ofNat (BitVec n) i _ = .ofNat n i := rfl
|
||||
@@ -123,9 +173,6 @@ instance : GetElem (BitVec w) Nat Bool fun _ i => i < w where
|
||||
theorem getElem_eq_testBit_toNat (x : BitVec w) (i : Nat) (h : i < w) :
|
||||
x[i] = x.toNat.testBit i := rfl
|
||||
|
||||
theorem getLsbD_eq_getElem {x : BitVec w} {i : Nat} (h : i < w) :
|
||||
x.getLsbD i = x[i] := rfl
|
||||
|
||||
end getElem
|
||||
|
||||
section Int
|
||||
@@ -188,6 +235,22 @@ end repr_toString
|
||||
|
||||
section arithmetic
|
||||
|
||||
/--
|
||||
Addition for bit vectors. This can be interpreted as either signed or unsigned addition
|
||||
modulo `2^n`.
|
||||
|
||||
SMT-Lib name: `bvadd`.
|
||||
-/
|
||||
protected def add (x y : BitVec n) : BitVec n := .ofNat n (x.toNat + y.toNat)
|
||||
instance : Add (BitVec n) := ⟨BitVec.add⟩
|
||||
|
||||
/--
|
||||
Subtraction for bit vectors. This can be interpreted as either signed or unsigned subtraction
|
||||
modulo `2^n`.
|
||||
-/
|
||||
protected def sub (x y : BitVec n) : BitVec n := .ofNat n ((2^n - y.toNat) + x.toNat)
|
||||
instance : Sub (BitVec n) := ⟨BitVec.sub⟩
|
||||
|
||||
/--
|
||||
Negation for bit vectors. This can be interpreted as either signed or unsigned negation
|
||||
modulo `2^n`.
|
||||
@@ -203,8 +266,8 @@ Return the absolute value of a signed bitvector.
|
||||
protected def abs (x : BitVec n) : BitVec n := if x.msb then .neg x else x
|
||||
|
||||
/--
|
||||
Multiplication for bit vectors. This can be interpreted as either signed or unsigned
|
||||
multiplication modulo `2^n`.
|
||||
Multiplication for bit vectors. This can be interpreted as either signed or unsigned negation
|
||||
modulo `2^n`.
|
||||
|
||||
SMT-Lib name: `bvmul`.
|
||||
-/
|
||||
@@ -321,6 +384,10 @@ SMT-Lib name: `bvult`.
|
||||
-/
|
||||
protected def ult (x y : BitVec n) : Bool := x.toNat < y.toNat
|
||||
|
||||
instance : LT (BitVec n) where lt := (·.toNat < ·.toNat)
|
||||
instance (x y : BitVec n) : Decidable (x < y) :=
|
||||
inferInstanceAs (Decidable (x.toNat < y.toNat))
|
||||
|
||||
/--
|
||||
Unsigned less-than-or-equal-to for bit vectors.
|
||||
|
||||
@@ -328,6 +395,10 @@ SMT-Lib name: `bvule`.
|
||||
-/
|
||||
protected def ule (x y : BitVec n) : Bool := x.toNat ≤ y.toNat
|
||||
|
||||
instance : LE (BitVec n) where le := (·.toNat ≤ ·.toNat)
|
||||
instance (x y : BitVec n) : Decidable (x ≤ y) :=
|
||||
inferInstanceAs (Decidable (x.toNat ≤ y.toNat))
|
||||
|
||||
/--
|
||||
Signed less-than for bit vectors.
|
||||
|
||||
@@ -379,15 +450,13 @@ SMT-Lib name: `extract`.
|
||||
def extractLsb (hi lo : Nat) (x : BitVec n) : BitVec (hi - lo + 1) := extractLsb' lo _ x
|
||||
|
||||
/--
|
||||
A version of `setWidth` that requires a proof, but is a noop.
|
||||
A version of `zeroExtend` that requires a proof, but is a noop.
|
||||
-/
|
||||
def setWidth' {n w : Nat} (le : n ≤ w) (x : BitVec n) : BitVec w :=
|
||||
def zeroExtend' {n w : Nat} (le : n ≤ w) (x : BitVec n) : BitVec w :=
|
||||
x.toNat#'(by
|
||||
apply Nat.lt_of_lt_of_le x.isLt
|
||||
exact Nat.pow_le_pow_of_le_right (by trivial) le)
|
||||
|
||||
@[deprecated setWidth' (since := "2024-09-18"), inherit_doc setWidth'] abbrev zeroExtend' := @setWidth'
|
||||
|
||||
/--
|
||||
`shiftLeftZeroExtend x n` returns `zeroExtend (w+n) x <<< n` without
|
||||
needing to compute `x % 2^(2+n)`.
|
||||
@@ -400,35 +469,22 @@ def shiftLeftZeroExtend (msbs : BitVec w) (m : Nat) : BitVec (w + m) :=
|
||||
(msbs.toNat <<< m)#'(shiftLeftLt msbs.isLt m)
|
||||
|
||||
/--
|
||||
Transform `x` of length `w` into a bitvector of length `v`, by either:
|
||||
- zero extending, that is, adding zeros in the high bits until it has length `v`, if `v > w`, or
|
||||
- truncating the high bits, if `v < w`.
|
||||
Zero extend vector `x` of length `w` by adding zeros in the high bits until it has length `v`.
|
||||
If `v < w` then it truncates the high bits instead.
|
||||
|
||||
SMT-Lib name: `zero_extend`.
|
||||
-/
|
||||
def setWidth (v : Nat) (x : BitVec w) : BitVec v :=
|
||||
def zeroExtend (v : Nat) (x : BitVec w) : BitVec v :=
|
||||
if h : w ≤ v then
|
||||
setWidth' h x
|
||||
zeroExtend' h x
|
||||
else
|
||||
.ofNat v x.toNat
|
||||
|
||||
/--
|
||||
Transform `x` of length `w` into a bitvector of length `v`, by either:
|
||||
- zero extending, that is, adding zeros in the high bits until it has length `v`, if `v > w`, or
|
||||
- truncating the high bits, if `v < w`.
|
||||
|
||||
SMT-Lib name: `zero_extend`.
|
||||
Truncate the high bits of bitvector `x` of length `w`, resulting in a vector of length `v`.
|
||||
If `v > w` then it zero-extends the vector instead.
|
||||
-/
|
||||
abbrev zeroExtend := @setWidth
|
||||
|
||||
/--
|
||||
Transform `x` of length `w` into a bitvector of length `v`, by either:
|
||||
- zero extending, that is, adding zeros in the high bits until it has length `v`, if `v > w`, or
|
||||
- truncating the high bits, if `v < w`.
|
||||
|
||||
SMT-Lib name: `zero_extend`.
|
||||
-/
|
||||
abbrev truncate := @setWidth
|
||||
abbrev truncate := @zeroExtend
|
||||
|
||||
/--
|
||||
Sign extend a vector of length `w`, extending with `i` additional copies of the most significant
|
||||
@@ -579,7 +635,7 @@ input is on the left, so `0xAB#8 ++ 0xCD#8 = 0xABCD#16`.
|
||||
SMT-Lib name: `concat`.
|
||||
-/
|
||||
def append (msbs : BitVec n) (lsbs : BitVec m) : BitVec (n+m) :=
|
||||
shiftLeftZeroExtend msbs m ||| setWidth' (Nat.le_add_left m n) lsbs
|
||||
shiftLeftZeroExtend msbs m ||| zeroExtend' (Nat.le_add_left m n) lsbs
|
||||
|
||||
instance : HAppend (BitVec w) (BitVec v) (BitVec (w + v)) := ⟨.append⟩
|
||||
|
||||
@@ -602,13 +658,6 @@ result of appending a single bit to the front in the naive implementation).
|
||||
That is, the new bit is the least significant bit. -/
|
||||
def concat {n} (msbs : BitVec n) (lsb : Bool) : BitVec (n+1) := msbs ++ (ofBool lsb)
|
||||
|
||||
/--
|
||||
`x.shiftConcat b` shifts all bits of `x` to the left by `1` and sets the least significant bit to `b`.
|
||||
It is a non-dependent version of `concat` that does not change the total bitwidth.
|
||||
-/
|
||||
def shiftConcat (x : BitVec n) (b : Bool) : BitVec n :=
|
||||
(x.concat b).truncate n
|
||||
|
||||
/-- Prepend a single bit to the front of a bitvector, using big endian order (see `append`).
|
||||
That is, the new bit is the most significant bit. -/
|
||||
def cons {n} (msb : Bool) (lsbs : BitVec n) : BitVec (n+1) :=
|
||||
@@ -631,16 +680,6 @@ def twoPow (w : Nat) (i : Nat) : BitVec w := 1#w <<< i
|
||||
|
||||
end bitwise
|
||||
|
||||
/-- Compute a hash of a bitvector, combining 64-bit words using `mixHash`. -/
|
||||
def hash (bv : BitVec n) : UInt64 :=
|
||||
if n ≤ 64 then
|
||||
bv.toFin.val.toUInt64
|
||||
else
|
||||
mixHash (bv.toFin.val.toUInt64) (hash ((bv >>> 64).setWidth (n - 64)))
|
||||
|
||||
instance : Hashable (BitVec n) where
|
||||
hash := hash
|
||||
|
||||
section normalization_eqs
|
||||
/-! We add simp-lemmas that rewrite bitvector operations into the equivalent notation -/
|
||||
@[simp] theorem append_eq (x : BitVec w) (y : BitVec v) : BitVec.append x y = x ++ y := rfl
|
||||
@@ -654,8 +693,6 @@ section normalization_eqs
|
||||
@[simp] theorem add_eq (x y : BitVec w) : BitVec.add x y = x + y := rfl
|
||||
@[simp] theorem sub_eq (x y : BitVec w) : BitVec.sub x y = x - y := rfl
|
||||
@[simp] theorem mul_eq (x y : BitVec w) : BitVec.mul x y = x * y := rfl
|
||||
@[simp] theorem udiv_eq (x y : BitVec w) : BitVec.udiv x y = x / y := rfl
|
||||
@[simp] theorem umod_eq (x y : BitVec w) : BitVec.umod x y = x % y := rfl
|
||||
@[simp] theorem zero_eq : BitVec.zero n = 0#n := rfl
|
||||
end normalization_eqs
|
||||
|
||||
|
||||
@@ -1,52 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joe Hendrix, Wojciech Nawrocki, Leonardo de Moura, Mario Carneiro, Alex Keizer, Harun Khan, Abdalrhman M Mohamed
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Fin.Basic
|
||||
|
||||
set_option linter.missingDocs true
|
||||
|
||||
/-!
|
||||
This module exists to provide the very basic `BitVec` definitions required for
|
||||
`Init.Data.UInt.BasicAux`.
|
||||
-/
|
||||
|
||||
namespace BitVec
|
||||
|
||||
section Nat
|
||||
|
||||
/-- The `BitVec` with value `i mod 2^n`. -/
|
||||
@[match_pattern]
|
||||
protected def ofNat (n : Nat) (i : Nat) : BitVec n where
|
||||
toFin := Fin.ofNat' (2^n) i
|
||||
|
||||
instance instOfNat : OfNat (BitVec n) i where ofNat := .ofNat n i
|
||||
|
||||
/-- Return the bound in terms of toNat. -/
|
||||
theorem isLt (x : BitVec w) : x.toNat < 2^w := x.toFin.isLt
|
||||
|
||||
end Nat
|
||||
|
||||
section arithmetic
|
||||
|
||||
/--
|
||||
Addition for bit vectors. This can be interpreted as either signed or unsigned addition
|
||||
modulo `2^n`.
|
||||
|
||||
SMT-Lib name: `bvadd`.
|
||||
-/
|
||||
protected def add (x y : BitVec n) : BitVec n := .ofNat n (x.toNat + y.toNat)
|
||||
instance : Add (BitVec n) := ⟨BitVec.add⟩
|
||||
|
||||
/--
|
||||
Subtraction for bit vectors. This can be interpreted as either signed or unsigned subtraction
|
||||
modulo `2^n`.
|
||||
-/
|
||||
protected def sub (x y : BitVec n) : BitVec n := .ofNat n ((2^n - y.toNat) + x.toNat)
|
||||
instance : Sub (BitVec n) := ⟨BitVec.sub⟩
|
||||
|
||||
end arithmetic
|
||||
|
||||
end BitVec
|
||||
@@ -1,7 +1,7 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Harun Khan, Abdalrhman M Mohamed, Joe Hendrix, Siddharth Bhat
|
||||
Authors: Harun Khan, Abdalrhman M Mohamed, Joe Hendrix
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.BitVec.Folds
|
||||
@@ -18,80 +18,6 @@ as vectors of bits into proofs about Lean `BitVec` values.
|
||||
The module is named for the bit-blasting operation in an SMT solver that converts bitvector
|
||||
expressions into expressions about individual bits in each vector.
|
||||
|
||||
### Example: How bitblasting works for multiplication
|
||||
|
||||
We explain how the lemmas here are used for bitblasting,
|
||||
by using multiplication as a prototypical example.
|
||||
Other bitblasters for other operations follow the same pattern.
|
||||
To bitblast a multiplication of the form `x * y`,
|
||||
we must unfold the above into a form that the SAT solver understands.
|
||||
|
||||
We assume that the solver already knows how to bitblast addition.
|
||||
This is known to `bv_decide`, by exploiting the lemma `add_eq_adc`,
|
||||
which says that `x + y : BitVec w` equals `(adc x y false).2`,
|
||||
where `adc` builds an add-carry circuit in terms of the primitive operations
|
||||
(bitwise and, bitwise or, bitwise xor) that bv_decide already understands.
|
||||
In this way, we layer bitblasters on top of each other,
|
||||
by reducing the multiplication bitblaster to an addition operation.
|
||||
|
||||
The core lemma is given by `getLsbD_mul`:
|
||||
|
||||
```lean
|
||||
x y : BitVec w ⊢ (x * y).getLsbD i = (mulRec x y w).getLsbD i
|
||||
```
|
||||
|
||||
Which says that the `i`th bit of `x * y` can be obtained by
|
||||
evaluating the `i`th bit of `(mulRec x y w)`.
|
||||
Once again, we assume that `bv_decide` knows how to implement `getLsbD`,
|
||||
given that `mulRec` can be understood by `bv_decide`.
|
||||
|
||||
We write two lemmas to enable `bv_decide` to unfold `(mulRec x y w)`
|
||||
into a complete circuit, **when `w` is a known constant**`.
|
||||
This is given by two recurrence lemmas, `mulRec_zero_eq` and `mulRec_succ_eq`,
|
||||
which are applied repeatedly when the width is `0` and when the width is `w' + 1`:
|
||||
|
||||
```lean
|
||||
mulRec_zero_eq :
|
||||
mulRec x y 0 =
|
||||
if y.getLsbD 0 then x else 0
|
||||
|
||||
mulRec_succ_eq
|
||||
mulRec x y (s + 1) =
|
||||
mulRec x y s +
|
||||
if y.getLsbD (s + 1) then (x <<< (s + 1)) else 0 := rfl
|
||||
```
|
||||
|
||||
By repeatedly applying the lemmas `mulRec_zero_eq` and `mulRec_succ_eq`,
|
||||
one obtains a circuit for multiplication.
|
||||
Note that this circuit uses `BitVec.add`, `BitVec.getLsbD`, `BitVec.shiftLeft`.
|
||||
Here, `BitVec.add` and `BitVec.shiftLeft` are (recursively) bitblasted by `bv_decide`,
|
||||
using the lemmas `add_eq_adc` and `shiftLeft_eq_shiftLeftRec`,
|
||||
and `BitVec.getLsbD` is a primitive that `bv_decide` knows how to reduce to SAT.
|
||||
|
||||
The two lemmas, `mulRec_zero_eq`, and `mulRec_succ_eq`,
|
||||
are used in `Std.Tactic.BVDecide.BVExpr.bitblast.blastMul`
|
||||
to prove the correctness of the circuit that is built by `bv_decide`.
|
||||
|
||||
```lean
|
||||
def blastMul (aig : AIG BVBit) (input : AIG.BinaryRefVec aig w) : AIG.RefVecEntry BVBit w
|
||||
theorem denote_blastMul (aig : AIG BVBit) (lhs rhs : BitVec w) (assign : Assignment) :
|
||||
...
|
||||
⟦(blastMul aig input).aig, (blastMul aig input).vec[idx], assign.toAIGAssignment⟧
|
||||
=
|
||||
(lhs * rhs).getLsbD idx
|
||||
```
|
||||
|
||||
The definition and theorem above are internal to `bv_decide`,
|
||||
and use `mulRec_{zero,succ}_eq` to prove that the circuit built by `bv_decide`
|
||||
computes the correct value for multiplication.
|
||||
|
||||
To zoom out, therefore, we follow two steps:
|
||||
First, we prove bitvector lemmas to unfold a high-level operation (such as multiplication)
|
||||
into already bitblastable operations (such as addition and left shift).
|
||||
We then use these lemmas to prove the correctness of the circuit that `bv_decide` builds.
|
||||
|
||||
We use this workflow to implement bitblasting for all SMT-LIB2 operations.
|
||||
|
||||
## Main results
|
||||
* `x + y : BitVec w` is `(adc x y false).2`.
|
||||
|
||||
@@ -174,30 +100,6 @@ theorem carry_succ (i : Nat) (x y : BitVec w) (c : Bool) :
|
||||
exact mod_two_pow_add_mod_two_pow_add_bool_lt_two_pow_succ ..
|
||||
cases x.toNat.testBit i <;> cases y.toNat.testBit i <;> (simp; omega)
|
||||
|
||||
theorem carry_succ_one (i : Nat) (x : BitVec w) (h : 0 < w) :
|
||||
carry (i+1) x (1#w) false = decide (∀ j ≤ i, x.getLsbD j = true) := by
|
||||
induction i with
|
||||
| zero => simp [carry_succ, h]
|
||||
| succ i ih =>
|
||||
rw [carry_succ, ih]
|
||||
simp only [getLsbD_one, add_one_ne_zero, decide_false, Bool.and_false, atLeastTwo_false_mid]
|
||||
cases hx : x.getLsbD (i+1)
|
||||
case false =>
|
||||
have : ∃ j ≤ i + 1, x.getLsbD j = false :=
|
||||
⟨i+1, by omega, hx⟩
|
||||
simpa
|
||||
case true =>
|
||||
suffices
|
||||
(∀ (j : Nat), j ≤ i → x.getLsbD j = true)
|
||||
↔ (∀ (j : Nat), j ≤ i + 1 → x.getLsbD j = true) by
|
||||
simpa
|
||||
constructor
|
||||
· intro h j hj
|
||||
rcases Nat.le_or_eq_of_le_succ hj with (hj' | rfl)
|
||||
· apply h; assumption
|
||||
· exact hx
|
||||
· intro h j hj; apply h; omega
|
||||
|
||||
/--
|
||||
If `x &&& y = 0`, then the carry bit `(x + y + 0)` is always `false` for any index `i`.
|
||||
Intuitively, this is because a carry is only produced when at least two of `x`, `y`, and the
|
||||
@@ -230,18 +132,18 @@ theorem toNat_add_of_and_eq_zero {x y : BitVec w} (h : x &&& y = 0#w) :
|
||||
simp [not_eq_true, carry_of_and_eq_zero h]
|
||||
|
||||
/-- Carry function for bitwise addition. -/
|
||||
def adcb (x y c : Bool) : Bool × Bool := (atLeastTwo x y c, x ^^ (y ^^ c))
|
||||
def adcb (x y c : Bool) : Bool × Bool := (atLeastTwo x y c, Bool.xor x (Bool.xor y c))
|
||||
|
||||
/-- Bitwise addition implemented via a ripple carry adder. -/
|
||||
def adc (x y : BitVec w) : Bool → Bool × BitVec w :=
|
||||
iunfoldr fun (i : Fin w) c => adcb (x.getLsbD i) (y.getLsbD i) c
|
||||
|
||||
theorem getLsbD_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool) :
|
||||
getLsbD (x + y + setWidth w (ofBool c)) i =
|
||||
(getLsbD x i ^^ (getLsbD y i ^^ carry i x y c)) := by
|
||||
getLsbD (x + y + zeroExtend w (ofBool c)) i =
|
||||
Bool.xor (getLsbD x i) (Bool.xor (getLsbD y i) (carry i x y c)) := by
|
||||
let ⟨x, x_lt⟩ := x
|
||||
let ⟨y, y_lt⟩ := y
|
||||
simp only [getLsbD, toNat_add, toNat_setWidth, i_lt, toNat_ofFin, toNat_ofBool,
|
||||
simp only [getLsbD, toNat_add, toNat_zeroExtend, i_lt, toNat_ofFin, toNat_ofBool,
|
||||
Nat.mod_add_mod, Nat.add_mod_mod]
|
||||
apply Eq.trans
|
||||
rw [← Nat.div_add_mod x (2^i), ← Nat.div_add_mod y (2^i)]
|
||||
@@ -249,7 +151,7 @@ theorem getLsbD_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool
|
||||
[ Nat.testBit_mod_two_pow,
|
||||
Nat.testBit_mul_two_pow_add_eq,
|
||||
i_lt,
|
||||
decide_true,
|
||||
decide_True,
|
||||
Bool.true_and,
|
||||
Nat.add_assoc,
|
||||
Nat.add_left_comm (_%_) (_ * _) _,
|
||||
@@ -259,26 +161,15 @@ theorem getLsbD_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool
|
||||
|
||||
theorem getLsbD_add {i : Nat} (i_lt : i < w) (x y : BitVec w) :
|
||||
getLsbD (x + y) i =
|
||||
(getLsbD x i ^^ (getLsbD y i ^^ carry i x y false)) := by
|
||||
Bool.xor (getLsbD x i) (Bool.xor (getLsbD y i) (carry i x y false)) := by
|
||||
simpa using getLsbD_add_add_bool i_lt x y false
|
||||
|
||||
theorem getElem_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool) :
|
||||
(x + y + setWidth w (ofBool c))[i] =
|
||||
(x[i] ^^ (y[i] ^^ carry i x y c)) := by
|
||||
simp only [← getLsbD_eq_getElem]
|
||||
rw [getLsbD_add_add_bool]
|
||||
omega
|
||||
|
||||
theorem getElem_add {i : Nat} (i_lt : i < w) (x y : BitVec w) :
|
||||
(x + y)[i] = (x[i] ^^ (y[i] ^^ carry i x y false)) := by
|
||||
simpa using getElem_add_add_bool i_lt x y false
|
||||
|
||||
theorem adc_spec (x y : BitVec w) (c : Bool) :
|
||||
adc x y c = (carry w x y c, x + y + setWidth w (ofBool c)) := by
|
||||
adc x y c = (carry w x y c, x + y + zeroExtend w (ofBool c)) := by
|
||||
simp only [adc]
|
||||
apply iunfoldr_replace
|
||||
(fun i => carry i x y c)
|
||||
(x + y + setWidth w (ofBool c))
|
||||
(x + y + zeroExtend w (ofBool c))
|
||||
c
|
||||
case init =>
|
||||
simp [carry, Nat.mod_one]
|
||||
@@ -291,21 +182,6 @@ theorem add_eq_adc (w : Nat) (x y : BitVec w) : x + y = (adc x y false).snd := b
|
||||
|
||||
/-! ### add -/
|
||||
|
||||
theorem getMsbD_add {i : Nat} {i_lt : i < w} {x y : BitVec w} :
|
||||
getMsbD (x + y) i =
|
||||
Bool.xor (getMsbD x i) (Bool.xor (getMsbD y i) (carry (w - 1 - i) x y false)) := by
|
||||
simp [getMsbD, getLsbD_add, i_lt, show w - 1 - i < w by omega]
|
||||
|
||||
theorem msb_add {w : Nat} {x y: BitVec w} :
|
||||
(x + y).msb =
|
||||
Bool.xor x.msb (Bool.xor y.msb (carry (w - 1) x y false)) := by
|
||||
simp only [BitVec.msb, BitVec.getMsbD]
|
||||
by_cases h : w ≤ 0
|
||||
· simp [h, show w = 0 by omega]
|
||||
· rw [getLsbD_add (x := x)]
|
||||
simp [show w > 0 by omega]
|
||||
omega
|
||||
|
||||
/-- Adding a bitvector to its own complement yields the all ones bitpattern -/
|
||||
@[simp] theorem add_not_self (x : BitVec w) : x + ~~~x = allOnes w := by
|
||||
rw [add_eq_adc, adc, iunfoldr_replace (fun _ => false) (allOnes w)]
|
||||
@@ -331,30 +207,6 @@ theorem add_eq_or_of_and_eq_zero {w : Nat} (x y : BitVec w)
|
||||
simp_all [hx]
|
||||
· by_cases hx : x.getLsbD i <;> simp_all [hx]
|
||||
|
||||
/-! ### Sub-/
|
||||
|
||||
theorem getLsbD_sub {i : Nat} {i_lt : i < w} {x y : BitVec w} :
|
||||
(x - y).getLsbD i
|
||||
= (x.getLsbD i ^^ ((~~~y + 1#w).getLsbD i ^^ carry i x (~~~y + 1#w) false)) := by
|
||||
rw [sub_toAdd, BitVec.neg_eq_not_add, getLsbD_add]
|
||||
omega
|
||||
|
||||
theorem getMsbD_sub {i : Nat} {i_lt : i < w} {x y : BitVec w} :
|
||||
(x - y).getMsbD i =
|
||||
(x.getMsbD i ^^ ((~~~y + 1).getMsbD i ^^ carry (w - 1 - i) x (~~~y + 1) false)) := by
|
||||
rw [sub_toAdd, neg_eq_not_add, getMsbD_add]
|
||||
· rfl
|
||||
· omega
|
||||
|
||||
theorem getElem_sub {i : Nat} {x y : BitVec w} (h : i < w) :
|
||||
(x - y)[i] = (x[i] ^^ ((~~~y + 1#w)[i] ^^ carry i x (~~~y + 1#w) false)) := by
|
||||
simp [← getLsbD_eq_getElem, getLsbD_sub, h]
|
||||
|
||||
theorem msb_sub {x y: BitVec w} :
|
||||
(x - y).msb
|
||||
= (x.msb ^^ ((~~~y + 1#w).msb ^^ carry (w - 1 - 0) x (~~~y + 1#w) false)) := by
|
||||
simp [sub_toAdd, BitVec.neg_eq_not_add, msb_add]
|
||||
|
||||
/-! ### Negation -/
|
||||
|
||||
theorem bit_not_testBit (x : BitVec w) (i : Fin w) :
|
||||
@@ -380,121 +232,6 @@ theorem bit_neg_eq_neg (x : BitVec w) : -x = (adc (((iunfoldr (fun (i : Fin w) c
|
||||
simp [← sub_toAdd, BitVec.sub_add_cancel]
|
||||
· simp [bit_not_testBit x _]
|
||||
|
||||
/--
|
||||
Remember that negating a bitvector is equal to incrementing the complement
|
||||
by one, i.e., `-x = ~~~x + 1`. See also `neg_eq_not_add`.
|
||||
|
||||
This computation has two crucial properties:
|
||||
- The least significant bit of `-x` is the same as the least significant bit of `x`, and
|
||||
- The `i+1`-th least significant bit of `-x` is the complement of the `i+1`-th bit of `x`, unless
|
||||
all of the preceding bits are `false`, in which case the bit is equal to the `i+1`-th bit of `x`
|
||||
-/
|
||||
theorem getLsbD_neg {i : Nat} {x : BitVec w} :
|
||||
getLsbD (-x) i =
|
||||
(getLsbD x i ^^ decide (i < w) && decide (∃ j < i, getLsbD x j = true)) := by
|
||||
rw [neg_eq_not_add]
|
||||
by_cases hi : i < w
|
||||
· rw [getLsbD_add hi]
|
||||
have : 0 < w := by omega
|
||||
simp only [getLsbD_not, hi, decide_true, Bool.true_and, getLsbD_one, this, not_bne,
|
||||
_root_.true_and, not_eq_eq_eq_not]
|
||||
cases i with
|
||||
| zero =>
|
||||
have carry_zero : carry 0 ?x ?y false = false := by
|
||||
simp [carry]; omega
|
||||
simp [hi, carry_zero]
|
||||
| succ =>
|
||||
rw [carry_succ_one _ _ (by omega), ← Bool.xor_not, ← decide_not]
|
||||
simp only [add_one_ne_zero, decide_false, getLsbD_not, and_eq_true, decide_eq_true_eq,
|
||||
not_eq_eq_eq_not, Bool.not_true, false_bne, not_exists, _root_.not_and, not_eq_true,
|
||||
bne_right_inj, decide_eq_decide]
|
||||
constructor
|
||||
· rintro h j hj; exact And.right <| h j (by omega)
|
||||
· rintro h j hj; exact ⟨by omega, h j (by omega)⟩
|
||||
· have h_ge : w ≤ i := by omega
|
||||
simp [getLsbD_ge _ _ h_ge, h_ge, hi]
|
||||
|
||||
theorem getElem_neg {i : Nat} {x : BitVec w} (h : i < w) :
|
||||
(-x)[i] = (x[i] ^^ decide (∃ j < i, x.getLsbD j = true)) := by
|
||||
simp [← getLsbD_eq_getElem, getLsbD_neg, h]
|
||||
|
||||
theorem getMsbD_neg {i : Nat} {x : BitVec w} :
|
||||
getMsbD (-x) i =
|
||||
(getMsbD x i ^^ decide (∃ j < w, i < j ∧ getMsbD x j = true)) := by
|
||||
simp only [getMsbD, getLsbD_neg, Bool.decide_and, Bool.and_eq_true, decide_eq_true_eq]
|
||||
by_cases hi : i < w
|
||||
case neg =>
|
||||
simp [hi]; omega
|
||||
case pos =>
|
||||
have h₁ : w - 1 - i < w := by omega
|
||||
simp only [hi, decide_true, h₁, Bool.true_and, Bool.bne_right_inj, decide_eq_decide]
|
||||
constructor
|
||||
· rintro ⟨j, hj, h⟩
|
||||
refine ⟨w - 1 - j, by omega, by omega, by omega, _root_.cast ?_ h⟩
|
||||
congr; omega
|
||||
· rintro ⟨j, hj₁, hj₂, -, h⟩
|
||||
exact ⟨w - 1 - j, by omega, h⟩
|
||||
|
||||
theorem msb_neg {w : Nat} {x : BitVec w} :
|
||||
(-x).msb = ((x != 0#w && x != intMin w) ^^ x.msb) := by
|
||||
simp only [BitVec.msb, getMsbD_neg]
|
||||
by_cases hmin : x = intMin _
|
||||
case pos =>
|
||||
have : (∃ j, j < w ∧ 0 < j ∧ 0 < w ∧ j = 0) ↔ False := by
|
||||
simp; omega
|
||||
simp [hmin, getMsbD_intMin, this]
|
||||
case neg =>
|
||||
by_cases hzero : x = 0#w
|
||||
case pos => simp [hzero]
|
||||
case neg =>
|
||||
have w_pos : 0 < w := by
|
||||
cases w
|
||||
· rw [@of_length_zero x] at hzero
|
||||
contradiction
|
||||
· omega
|
||||
suffices ∃ j, j < w ∧ 0 < j ∧ x.getMsbD j = true
|
||||
by simp [show x != 0#w by simpa, show x != intMin w by simpa, this]
|
||||
false_or_by_contra
|
||||
rename_i getMsbD_x
|
||||
simp only [not_exists, _root_.not_and, not_eq_true] at getMsbD_x
|
||||
/- `getMsbD` says that all bits except the msb are `false` -/
|
||||
cases hmsb : x.msb
|
||||
case true =>
|
||||
apply hmin
|
||||
apply eq_of_getMsbD_eq
|
||||
rintro ⟨i, hi⟩
|
||||
simp only [getMsbD_intMin, w_pos, decide_true, Bool.true_and]
|
||||
cases i
|
||||
case zero => exact hmsb
|
||||
case succ => exact getMsbD_x _ hi (by omega)
|
||||
case false =>
|
||||
apply hzero
|
||||
apply eq_of_getMsbD_eq
|
||||
rintro ⟨i, hi⟩
|
||||
simp only [getMsbD_zero]
|
||||
cases i
|
||||
case zero => exact hmsb
|
||||
case succ => exact getMsbD_x _ hi (by omega)
|
||||
|
||||
/-! ### abs -/
|
||||
|
||||
theorem msb_abs {w : Nat} {x : BitVec w} :
|
||||
x.abs.msb = (decide (x = intMin w) && decide (0 < w)) := by
|
||||
simp only [BitVec.abs, getMsbD_neg, ne_eq, decide_not, Bool.not_bne]
|
||||
by_cases h₀ : 0 < w
|
||||
· by_cases h₁ : x = intMin w
|
||||
· simp [h₁, msb_intMin]
|
||||
· simp only [neg_eq, h₁, decide_false]
|
||||
by_cases h₂ : x.msb
|
||||
· simp [h₂, msb_neg]
|
||||
and_intros
|
||||
· by_cases h₃ : x = 0#w
|
||||
· simp [h₃] at h₂
|
||||
· simp [h₃]
|
||||
· simp [h₁]
|
||||
· simp [h₂]
|
||||
· simp [BitVec.msb, show w = 0 by omega]
|
||||
|
||||
/-! ### Inequalities (le / lt) -/
|
||||
|
||||
theorem ult_eq_not_carry (x y : BitVec w) : x.ult y = !carry w x (~~~y) true := by
|
||||
@@ -569,48 +306,43 @@ theorem mulRec_succ_eq (x y : BitVec w) (s : Nat) :
|
||||
Recurrence lemma: truncating to `i+1` bits and then zero extending to `w`
|
||||
equals truncating upto `i` bits `[0..i-1]`, and then adding the `i`th bit of `x`.
|
||||
-/
|
||||
theorem setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (x : BitVec w) (i : Nat) :
|
||||
setWidth w (x.setWidth (i + 1)) =
|
||||
setWidth w (x.setWidth i) + (x &&& twoPow w i) := by
|
||||
theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow (x : BitVec w) (i : Nat) :
|
||||
zeroExtend w (x.truncate (i + 1)) =
|
||||
zeroExtend w (x.truncate i) + (x &&& twoPow w i) := by
|
||||
rw [add_eq_or_of_and_eq_zero]
|
||||
· ext k
|
||||
simp only [getLsbD_setWidth, Fin.is_lt, decide_true, Bool.true_and, getLsbD_or, getLsbD_and]
|
||||
simp only [getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
|
||||
by_cases hik : i = k
|
||||
· subst hik
|
||||
simp
|
||||
· simp only [getLsbD_twoPow, hik, decide_false, Bool.and_false, Bool.or_false]
|
||||
· simp only [getLsbD_twoPow, hik, decide_False, Bool.and_false, Bool.or_false]
|
||||
by_cases hik' : k < (i + 1)
|
||||
· have hik'' : k < i := by omega
|
||||
simp [hik', hik'']
|
||||
· have hik'' : ¬ (k < i) := by omega
|
||||
simp [hik', hik'']
|
||||
· ext k
|
||||
simp only [and_twoPow, getLsbD_and, getLsbD_setWidth, Fin.is_lt, decide_true, Bool.true_and,
|
||||
simp only [and_twoPow, getLsbD_and, getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and,
|
||||
getLsbD_zero, and_eq_false_imp, and_eq_true, decide_eq_true_eq, and_imp]
|
||||
by_cases hi : x.getLsbD i <;> simp [hi] <;> omega
|
||||
|
||||
@[deprecated setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (since := "2024-09-18"),
|
||||
inherit_doc setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow]
|
||||
abbrev zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow :=
|
||||
@setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow
|
||||
|
||||
/--
|
||||
Recurrence lemma: multiplying `x` with the first `s` bits of `y` is the
|
||||
same as truncating `y` to `s` bits, then zero extending to the original length,
|
||||
and performing the multplication. -/
|
||||
theorem mulRec_eq_mul_signExtend_setWidth (x y : BitVec w) (s : Nat) :
|
||||
mulRec x y s = x * ((y.setWidth (s + 1)).setWidth w) := by
|
||||
theorem mulRec_eq_mul_signExtend_truncate (x y : BitVec w) (s : Nat) :
|
||||
mulRec x y s = x * ((y.truncate (s + 1)).zeroExtend w) := by
|
||||
induction s
|
||||
case zero =>
|
||||
simp only [mulRec_zero_eq, ofNat_eq_ofNat, Nat.reduceAdd]
|
||||
by_cases y.getLsbD 0
|
||||
case pos hy =>
|
||||
simp only [hy, ↓reduceIte, setWidth_one_eq_ofBool_getLsb_zero,
|
||||
simp only [hy, ↓reduceIte, truncate, zeroExtend_one_eq_ofBool_getLsb_zero,
|
||||
ofBool_true, ofNat_eq_ofNat]
|
||||
rw [setWidth_ofNat_one_eq_ofNat_one_of_lt (by omega)]
|
||||
rw [zeroExtend_ofNat_one_eq_ofNat_one_of_lt (by omega)]
|
||||
simp
|
||||
case neg hy =>
|
||||
simp [hy, setWidth_one_eq_ofBool_getLsb_zero]
|
||||
simp [hy, zeroExtend_one_eq_ofBool_getLsb_zero]
|
||||
case succ s' hs =>
|
||||
rw [mulRec_succ_eq, hs]
|
||||
have heq :
|
||||
@@ -618,23 +350,16 @@ theorem mulRec_eq_mul_signExtend_setWidth (x y : BitVec w) (s : Nat) :
|
||||
(x * (y &&& (BitVec.twoPow w (s' + 1)))) := by
|
||||
simp only [ofNat_eq_ofNat, and_twoPow]
|
||||
by_cases hy : y.getLsbD (s' + 1) <;> simp [hy]
|
||||
rw [heq, ← BitVec.mul_add, ← setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow]
|
||||
|
||||
@[deprecated mulRec_eq_mul_signExtend_setWidth (since := "2024-09-18"),
|
||||
inherit_doc mulRec_eq_mul_signExtend_setWidth]
|
||||
abbrev mulRec_eq_mul_signExtend_truncate := @mulRec_eq_mul_signExtend_setWidth
|
||||
rw [heq, ← BitVec.mul_add, ← zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow]
|
||||
|
||||
theorem getLsbD_mul (x y : BitVec w) (i : Nat) :
|
||||
(x * y).getLsbD i = (mulRec x y w).getLsbD i := by
|
||||
simp only [mulRec_eq_mul_signExtend_setWidth]
|
||||
rw [setWidth_setWidth_of_le]
|
||||
simp only [mulRec_eq_mul_signExtend_truncate]
|
||||
rw [truncate, ← truncate_eq_zeroExtend, ← truncate_eq_zeroExtend,
|
||||
truncate_truncate_of_le]
|
||||
· simp
|
||||
· omega
|
||||
|
||||
theorem getElem_mul {x y : BitVec w} {i : Nat} (h : i < w) :
|
||||
(x * y)[i] = (mulRec x y w)[i] := by
|
||||
simp [mulRec_eq_mul_signExtend_setWidth]
|
||||
|
||||
/-! ## shiftLeft recurrence for bitblasting -/
|
||||
|
||||
/--
|
||||
@@ -677,22 +402,22 @@ theorem shiftLeft_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
|
||||
`shiftLeftRec x y n` shifts `x` to the left by the first `n` bits of `y`.
|
||||
-/
|
||||
theorem shiftLeftRec_eq {x : BitVec w₁} {y : BitVec w₂} {n : Nat} :
|
||||
shiftLeftRec x y n = x <<< (y.setWidth (n + 1)).setWidth w₂ := by
|
||||
shiftLeftRec x y n = x <<< (y.truncate (n + 1)).zeroExtend w₂ := by
|
||||
induction n generalizing x y
|
||||
case zero =>
|
||||
ext i
|
||||
simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, setWidth_one,
|
||||
and_one_eq_setWidth_ofBool_getLsbD]
|
||||
simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, truncate_one,
|
||||
and_one_eq_zeroExtend_ofBool_getLsbD]
|
||||
case succ n ih =>
|
||||
simp only [shiftLeftRec_succ, and_twoPow]
|
||||
rw [ih]
|
||||
by_cases h : y.getLsbD (n + 1)
|
||||
· simp only [h, ↓reduceIte]
|
||||
rw [setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true h,
|
||||
rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true h,
|
||||
shiftLeft_or_of_and_eq_zero]
|
||||
simp [and_twoPow]
|
||||
· simp only [h, false_eq_true, ↓reduceIte, shiftLeft_zero']
|
||||
rw [setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false (i := n + 1)]
|
||||
rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false (i := n + 1)]
|
||||
simp [h]
|
||||
|
||||
/--
|
||||
@@ -705,385 +430,6 @@ theorem shiftLeft_eq_shiftLeftRec (x : BitVec w₁) (y : BitVec w₂) :
|
||||
· simp [of_length_zero]
|
||||
· simp [shiftLeftRec_eq]
|
||||
|
||||
/-! # udiv/urem recurrence for bitblasting
|
||||
|
||||
In order to prove the correctness of the division algorithm on the integers,
|
||||
one shows that `n.div d = q` and `n.mod d = r` iff `n = d * q + r` and `0 ≤ r < d`.
|
||||
Mnemonic: `n` is the numerator, `d` is the denominator, `q` is the quotient, and `r` the remainder.
|
||||
|
||||
This *uniqueness of decomposition* is not true for bitvectors.
|
||||
For `n = 0, d = 3, w = 3`, we can write:
|
||||
- `0 = 0 * 3 + 0` (`q = 0`, `r = 0 < 3`.)
|
||||
- `0 = 2 * 3 + 2 = 6 + 2 ≃ 0 (mod 8)` (`q = 2`, `r = 2 < 3`).
|
||||
|
||||
Such examples can be created by choosing different `(q, r)` for a fixed `(d, n)`
|
||||
such that `(d * q + r)` overflows and wraps around to equal `n`.
|
||||
|
||||
This tells us that the division algorithm must have more restrictions than just the ones
|
||||
we have for integers. These restrictions are captured in `DivModState.Lawful`.
|
||||
The key idea is to state the relationship in terms of the toNat values of {n, d, q, r}.
|
||||
If the division equation `d.toNat * q.toNat + r.toNat = n.toNat` holds,
|
||||
then `n.udiv d = q` and `n.umod d = r`.
|
||||
|
||||
Following this, we implement the division algorithm by repeated shift-subtract.
|
||||
|
||||
References:
|
||||
- Fast 32-bit Division on the DSP56800E: Minimized nonrestoring division algorithm by David Baca
|
||||
- Bitwuzla sources for bitblasting.h
|
||||
-/
|
||||
|
||||
private theorem Nat.div_add_eq_left_of_lt {x y z : Nat} (hx : z ∣ x) (hy : y < z) (hz : 0 < z) :
|
||||
(x + y) / z = x / z := by
|
||||
refine Nat.div_eq_of_lt_le ?lo ?hi
|
||||
· apply Nat.le_trans
|
||||
· exact div_mul_le_self x z
|
||||
· omega
|
||||
· simp only [succ_eq_add_one, Nat.add_mul, Nat.one_mul]
|
||||
apply Nat.add_lt_add_of_le_of_lt
|
||||
· apply Nat.le_of_eq
|
||||
exact (Nat.div_eq_iff_eq_mul_left hz hx).mp rfl
|
||||
· exact hy
|
||||
|
||||
/-- If the division equation `d.toNat * q.toNat + r.toNat = n.toNat` holds,
|
||||
then `n.udiv d = q`. -/
|
||||
theorem udiv_eq_of_mul_add_toNat {d n q r : BitVec w} (hd : 0 < d)
|
||||
(hrd : r < d)
|
||||
(hdqnr : d.toNat * q.toNat + r.toNat = n.toNat) :
|
||||
n / d = q := by
|
||||
apply BitVec.eq_of_toNat_eq
|
||||
rw [toNat_udiv]
|
||||
replace hdqnr : (d.toNat * q.toNat + r.toNat) / d.toNat = n.toNat / d.toNat := by
|
||||
simp [hdqnr]
|
||||
rw [Nat.div_add_eq_left_of_lt] at hdqnr
|
||||
· rw [← hdqnr]
|
||||
exact mul_div_right q.toNat hd
|
||||
· exact Nat.dvd_mul_right d.toNat q.toNat
|
||||
· exact hrd
|
||||
· exact hd
|
||||
|
||||
/-- If the division equation `d.toNat * q.toNat + r.toNat = n.toNat` holds,
|
||||
then `n.umod d = r`. -/
|
||||
theorem umod_eq_of_mul_add_toNat {d n q r : BitVec w} (hrd : r < d)
|
||||
(hdqnr : d.toNat * q.toNat + r.toNat = n.toNat) :
|
||||
n % d = r := by
|
||||
apply BitVec.eq_of_toNat_eq
|
||||
rw [toNat_umod]
|
||||
replace hdqnr : (d.toNat * q.toNat + r.toNat) % d.toNat = n.toNat % d.toNat := by
|
||||
simp [hdqnr]
|
||||
rw [Nat.add_mod, Nat.mul_mod_right] at hdqnr
|
||||
simp only [Nat.zero_add, mod_mod] at hdqnr
|
||||
replace hrd : r.toNat < d.toNat := by
|
||||
simpa [BitVec.lt_def] using hrd
|
||||
rw [Nat.mod_eq_of_lt hrd] at hdqnr
|
||||
simp [hdqnr]
|
||||
|
||||
/-! ### DivModState -/
|
||||
|
||||
/-- `DivModState` is a structure that maintains the state of recursive `divrem` calls. -/
|
||||
structure DivModState (w : Nat) : Type where
|
||||
/-- The number of bits in the numerator that are not yet processed -/
|
||||
wn : Nat
|
||||
/-- The number of bits in the remainder (and quotient) -/
|
||||
wr : Nat
|
||||
/-- The current quotient. -/
|
||||
q : BitVec w
|
||||
/-- The current remainder. -/
|
||||
r : BitVec w
|
||||
|
||||
|
||||
/-- `DivModArgs` contains the arguments to a `divrem` call which remain constant throughout
|
||||
execution. -/
|
||||
structure DivModArgs (w : Nat) where
|
||||
/-- the numerator (aka, dividend) -/
|
||||
n : BitVec w
|
||||
/-- the denumerator (aka, divisor)-/
|
||||
d : BitVec w
|
||||
|
||||
/-- A `DivModState` is lawful if the remainder width `wr` plus the numerator width `wn` equals `w`,
|
||||
and the bitvectors `r` and `n` have values in the bounds given by bitwidths `wr`, resp. `wn`.
|
||||
|
||||
This is a proof engineering choice: an alternative world could have been
|
||||
`r : BitVec wr` and `n : BitVec wn`, but this required much more dependent typing coercions.
|
||||
|
||||
Instead, we choose to declare all involved bitvectors as length `w`, and then prove that
|
||||
the values are within their respective bounds.
|
||||
|
||||
We start with `wn = w` and `wr = 0`, and then in each step, we decrement `wn` and increment `wr`.
|
||||
In this way, we grow a legal remainder in each loop iteration.
|
||||
-/
|
||||
structure DivModState.Lawful {w : Nat} (args : DivModArgs w) (qr : DivModState w) : Prop where
|
||||
/-- The sum of widths of the dividend and remainder is `w`. -/
|
||||
hwrn : qr.wr + qr.wn = w
|
||||
/-- The denominator is positive. -/
|
||||
hdPos : 0 < args.d
|
||||
/-- The remainder is strictly less than the denominator. -/
|
||||
hrLtDivisor : qr.r.toNat < args.d.toNat
|
||||
/-- The remainder is morally a `Bitvec wr`, and so has value less than `2^wr`. -/
|
||||
hrWidth : qr.r.toNat < 2^qr.wr
|
||||
/-- The quotient is morally a `Bitvec wr`, and so has value less than `2^wr`. -/
|
||||
hqWidth : qr.q.toNat < 2^qr.wr
|
||||
/-- The low `(w - wn)` bits of `n` obey the invariant for division. -/
|
||||
hdiv : args.n.toNat >>> qr.wn = args.d.toNat * qr.q.toNat + qr.r.toNat
|
||||
|
||||
/-- A lawful DivModState implies `w > 0`. -/
|
||||
def DivModState.Lawful.hw {args : DivModArgs w} {qr : DivModState w}
|
||||
{h : DivModState.Lawful args qr} : 0 < w := by
|
||||
have hd := h.hdPos
|
||||
rcases w with rfl | w
|
||||
· have hcontra : args.d = 0#0 := by apply Subsingleton.elim
|
||||
rw [hcontra] at hd
|
||||
simp at hd
|
||||
· omega
|
||||
|
||||
/-- An initial value with both `q, r = 0`. -/
|
||||
def DivModState.init (w : Nat) : DivModState w := {
|
||||
wn := w
|
||||
wr := 0
|
||||
q := 0#w
|
||||
r := 0#w
|
||||
}
|
||||
|
||||
/-- The initial state is lawful. -/
|
||||
def DivModState.lawful_init {w : Nat} (args : DivModArgs w) (hd : 0#w < args.d) :
|
||||
DivModState.Lawful args (DivModState.init w) := by
|
||||
simp only [BitVec.DivModState.init]
|
||||
exact {
|
||||
hwrn := by simp only; omega,
|
||||
hdPos := by assumption
|
||||
hrLtDivisor := by simp [BitVec.lt_def] at hd ⊢; assumption
|
||||
hrWidth := by simp [DivModState.init],
|
||||
hqWidth := by simp [DivModState.init],
|
||||
hdiv := by
|
||||
simp only [DivModState.init, toNat_ofNat, zero_mod, Nat.mul_zero, Nat.add_zero];
|
||||
rw [Nat.shiftRight_eq_div_pow]
|
||||
apply Nat.div_eq_of_lt args.n.isLt
|
||||
}
|
||||
|
||||
/--
|
||||
A lawful DivModState with a fully consumed dividend (`wn = 0`) witnesses that the
|
||||
quotient has been correctly computed.
|
||||
-/
|
||||
theorem DivModState.udiv_eq_of_lawful {n d : BitVec w} {qr : DivModState w}
|
||||
(h_lawful : DivModState.Lawful {n, d} qr)
|
||||
(h_final : qr.wn = 0) :
|
||||
n / d = qr.q := by
|
||||
apply udiv_eq_of_mul_add_toNat h_lawful.hdPos h_lawful.hrLtDivisor
|
||||
have hdiv := h_lawful.hdiv
|
||||
simp only [h_final] at *
|
||||
omega
|
||||
|
||||
/--
|
||||
A lawful DivModState with a fully consumed dividend (`wn = 0`) witnesses that the
|
||||
remainder has been correctly computed.
|
||||
-/
|
||||
theorem DivModState.umod_eq_of_lawful {qr : DivModState w}
|
||||
(h : DivModState.Lawful {n, d} qr)
|
||||
(h_final : qr.wn = 0) :
|
||||
n % d = qr.r := by
|
||||
apply umod_eq_of_mul_add_toNat h.hrLtDivisor
|
||||
have hdiv := h.hdiv
|
||||
simp only [shiftRight_zero] at hdiv
|
||||
simp only [h_final] at *
|
||||
exact hdiv.symm
|
||||
|
||||
/-! ### DivModState.Poised -/
|
||||
|
||||
/--
|
||||
A `Poised` DivModState is a state which is `Lawful` and furthermore, has at least
|
||||
one numerator bit left to process `(0 < wn)`
|
||||
|
||||
The input to the shift subtractor is a legal input to `divrem`, and we also need to have an
|
||||
input bit to perform shift subtraction on, and thus we need `0 < wn`.
|
||||
-/
|
||||
structure DivModState.Poised {w : Nat} (args : DivModArgs w) (qr : DivModState w)
|
||||
extends DivModState.Lawful args qr : Type where
|
||||
/-- Only perform a round of shift-subtract if we have dividend bits. -/
|
||||
hwn_lt : 0 < qr.wn
|
||||
|
||||
/--
|
||||
In the shift subtract input, the dividend is at least one bit long (`wn > 0`), so
|
||||
the remainder has bits to be computed (`wr < w`).
|
||||
-/
|
||||
def DivModState.wr_lt_w {qr : DivModState w} (h : qr.Poised args) : qr.wr < w := by
|
||||
have hwrn := h.hwrn
|
||||
have hwn_lt := h.hwn_lt
|
||||
omega
|
||||
|
||||
/-! ### Division shift subtractor -/
|
||||
|
||||
/--
|
||||
One round of the division algorithm, that tries to perform a subtract shift.
|
||||
Note that this should only be called when `r.msb = false`, so we will not overflow.
|
||||
-/
|
||||
def divSubtractShift (args : DivModArgs w) (qr : DivModState w) : DivModState w :=
|
||||
let {n, d} := args
|
||||
let wn := qr.wn - 1
|
||||
let wr := qr.wr + 1
|
||||
let r' := shiftConcat qr.r (n.getLsbD wn)
|
||||
if r' < d then {
|
||||
q := qr.q.shiftConcat false, -- If `r' < d`, then we do not have a quotient bit.
|
||||
r := r'
|
||||
wn, wr
|
||||
} else {
|
||||
q := qr.q.shiftConcat true, -- Otherwise, `r' ≥ d`, and we have a quotient bit.
|
||||
r := r' - d -- we subtract to maintain the invariant that `r < d`.
|
||||
wn, wr
|
||||
}
|
||||
|
||||
/-- The value of shifting right by `wn - 1` equals shifting by `wn` and grabbing the lsb at `(wn - 1)`. -/
|
||||
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
|
||||
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)]
|
||||
· simp
|
||||
· omega
|
||||
· apply BitVec.toNat_ushiftRight_lt
|
||||
omega
|
||||
|
||||
/--
|
||||
This is used when proving the correctness of the division algorithm,
|
||||
where we know that `r < d`.
|
||||
We then want to show that `((r.shiftConcat b) - d) < d` as the loop invariant.
|
||||
In arithmetic, this is the same as showing that
|
||||
`r * 2 + 1 - d < d`, which this theorem establishes.
|
||||
-/
|
||||
private theorem two_mul_add_sub_lt_of_lt_of_lt_two (h : a < x) (hy : y < 2) :
|
||||
2 * a + y - x < x := by omega
|
||||
|
||||
/-- We show that the output of `divSubtractShift` is lawful, which tells us that it
|
||||
obeys the division equation. -/
|
||||
theorem lawful_divSubtractShift (qr : DivModState w) (h : qr.Poised args) :
|
||||
DivModState.Lawful args (divSubtractShift args qr) := by
|
||||
rcases args with ⟨n, d⟩
|
||||
simp only [divSubtractShift, decide_eq_true_eq]
|
||||
-- We add these hypotheses for `omega` to find them later.
|
||||
have ⟨⟨hrwn, hd, hrd, hr, hn, hrnd⟩, hwn_lt⟩ := h
|
||||
have : d.toNat * (qr.q.toNat * 2) = d.toNat * qr.q.toNat * 2 := by rw [Nat.mul_assoc]
|
||||
by_cases rltd : shiftConcat qr.r (n.getLsbD (qr.wn - 1)) < d
|
||||
· simp only [rltd, ↓reduceIte]
|
||||
constructor <;> try bv_omega
|
||||
case pos.hrWidth => apply toNat_shiftConcat_lt_of_lt <;> omega
|
||||
case pos.hqWidth => apply toNat_shiftConcat_lt_of_lt <;> omega
|
||||
case pos.hdiv =>
|
||||
simp [qr.toNat_shiftRight_sub_one_eq h, h.hdiv, this,
|
||||
toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hrWidth,
|
||||
toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hqWidth]
|
||||
omega
|
||||
· simp only [rltd, ↓reduceIte]
|
||||
constructor <;> try bv_omega
|
||||
case neg.hrLtDivisor =>
|
||||
simp only [lt_def, Nat.not_lt] at rltd
|
||||
rw [BitVec.toNat_sub_of_le rltd,
|
||||
toNat_shiftConcat_eq_of_lt (hk := qr.wr_lt_w h) (hx := h.hrWidth),
|
||||
Nat.mul_comm]
|
||||
apply two_mul_add_sub_lt_of_lt_of_lt_two <;> bv_omega
|
||||
case neg.hrWidth =>
|
||||
simp only
|
||||
have hdr' : d ≤ (qr.r.shiftConcat (n.getLsbD (qr.wn - 1))) :=
|
||||
BitVec.not_lt_iff_le.mp rltd
|
||||
have hr' : ((qr.r.shiftConcat (n.getLsbD (qr.wn - 1)))).toNat < 2 ^ (qr.wr + 1) := by
|
||||
apply toNat_shiftConcat_lt_of_lt <;> bv_omega
|
||||
rw [BitVec.toNat_sub_of_le hdr']
|
||||
omega
|
||||
case neg.hqWidth =>
|
||||
apply toNat_shiftConcat_lt_of_lt <;> omega
|
||||
case neg.hdiv =>
|
||||
have rltd' := (BitVec.not_lt_iff_le.mp rltd)
|
||||
simp only [qr.toNat_shiftRight_sub_one_eq h,
|
||||
BitVec.toNat_sub_of_le rltd',
|
||||
toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hrWidth]
|
||||
simp only [BitVec.le_def,
|
||||
toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hrWidth] at rltd'
|
||||
simp only [toNat_shiftConcat_eq_of_lt (qr.wr_lt_w h) h.hqWidth, h.hdiv, Nat.mul_add]
|
||||
bv_omega
|
||||
|
||||
/-! ### Core division algorithm circuit -/
|
||||
|
||||
/-- A recursive definition of division for bitblasting, in terms of a shift-subtraction circuit. -/
|
||||
def divRec {w : Nat} (m : Nat) (args : DivModArgs w) (qr : DivModState w) :
|
||||
DivModState w :=
|
||||
match m with
|
||||
| 0 => qr
|
||||
| m + 1 => divRec m args <| divSubtractShift args qr
|
||||
|
||||
@[simp]
|
||||
theorem divRec_zero (qr : DivModState w) :
|
||||
divRec 0 args qr = qr := rfl
|
||||
|
||||
@[simp]
|
||||
theorem divRec_succ (m : Nat) (args : DivModArgs w) (qr : DivModState w) :
|
||||
divRec (m + 1) args qr =
|
||||
divRec m args (divSubtractShift args qr) := rfl
|
||||
|
||||
/-- The output of `divRec` is a lawful state -/
|
||||
theorem lawful_divRec {args : DivModArgs w} {qr : DivModState w}
|
||||
(h : DivModState.Lawful args qr) :
|
||||
DivModState.Lawful args (divRec qr.wn args qr) := by
|
||||
generalize hm : qr.wn = m
|
||||
induction m generalizing qr
|
||||
case zero =>
|
||||
exact h
|
||||
case succ wn' ih =>
|
||||
simp only [divRec_succ]
|
||||
apply ih
|
||||
· apply lawful_divSubtractShift
|
||||
constructor
|
||||
· assumption
|
||||
· omega
|
||||
· simp only [divSubtractShift, hm]
|
||||
split <;> rfl
|
||||
|
||||
/-- The output of `divRec` has no more bits left to process (i.e., `wn = 0`) -/
|
||||
@[simp]
|
||||
theorem wn_divRec (args : DivModArgs w) (qr : DivModState w) :
|
||||
(divRec qr.wn args qr).wn = 0 := by
|
||||
generalize hm : qr.wn = m
|
||||
induction m generalizing qr
|
||||
case zero =>
|
||||
assumption
|
||||
case succ wn' ih =>
|
||||
apply ih
|
||||
simp only [divSubtractShift, hm]
|
||||
split <;> rfl
|
||||
|
||||
/-- The result of `udiv` agrees with the result of the division recurrence. -/
|
||||
theorem udiv_eq_divRec (hd : 0#w < d) :
|
||||
let out := divRec w {n, d} (DivModState.init w)
|
||||
n / d = out.q := by
|
||||
have := DivModState.lawful_init {n, d} hd
|
||||
have := lawful_divRec this
|
||||
apply DivModState.udiv_eq_of_lawful this (wn_divRec ..)
|
||||
|
||||
/-- The result of `umod` agrees with the result of the division recurrence. -/
|
||||
theorem umod_eq_divRec (hd : 0#w < d) :
|
||||
let out := divRec w {n, d} (DivModState.init w)
|
||||
n % d = out.r := by
|
||||
have := DivModState.lawful_init {n, d} hd
|
||||
have := lawful_divRec this
|
||||
apply DivModState.umod_eq_of_lawful this (wn_divRec ..)
|
||||
|
||||
theorem divRec_succ' (m : Nat) (args : DivModArgs w) (qr : DivModState w) :
|
||||
divRec (m+1) args qr =
|
||||
let wn := qr.wn - 1
|
||||
let wr := qr.wr + 1
|
||||
let r' := shiftConcat qr.r (args.n.getLsbD wn)
|
||||
let input : DivModState _ :=
|
||||
if r' < args.d then {
|
||||
q := qr.q.shiftConcat false,
|
||||
r := r'
|
||||
wn, wr
|
||||
} else {
|
||||
q := qr.q.shiftConcat true,
|
||||
r := r' - args.d
|
||||
wn, wr
|
||||
}
|
||||
divRec m args input := by
|
||||
simp [divRec_succ, divSubtractShift]
|
||||
|
||||
/- ### Arithmetic shift right (sshiftRight) recurrence -/
|
||||
|
||||
/--
|
||||
@@ -1100,8 +446,8 @@ def sshiftRightRec (x : BitVec w₁) (y : BitVec w₂) (n : Nat) : BitVec w₁ :
|
||||
|
||||
@[simp]
|
||||
theorem sshiftRightRec_zero_eq (x : BitVec w₁) (y : BitVec w₂) :
|
||||
sshiftRightRec x y 0 = x.sshiftRight' (y &&& twoPow w₂ 0) := by
|
||||
simp only [sshiftRightRec]
|
||||
sshiftRightRec x y 0 = x.sshiftRight' (y &&& 1#w₂) := by
|
||||
simp only [sshiftRightRec, twoPow_zero]
|
||||
|
||||
@[simp]
|
||||
theorem sshiftRightRec_succ_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
|
||||
@@ -1120,18 +466,18 @@ theorem sshiftRight'_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
|
||||
toNat_add_of_and_eq_zero h, sshiftRight_add]
|
||||
|
||||
theorem sshiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
|
||||
sshiftRightRec x y n = x.sshiftRight' ((y.setWidth (n + 1)).setWidth w₂) := by
|
||||
sshiftRightRec x y n = x.sshiftRight' ((y.truncate (n + 1)).zeroExtend w₂) := by
|
||||
induction n generalizing x y
|
||||
case zero =>
|
||||
ext i
|
||||
simp [twoPow_zero, Nat.reduceAdd, and_one_eq_setWidth_ofBool_getLsbD, setWidth_one]
|
||||
simp [twoPow_zero, Nat.reduceAdd, and_one_eq_zeroExtend_ofBool_getLsbD, truncate_one]
|
||||
case succ n ih =>
|
||||
simp only [sshiftRightRec_succ_eq, and_twoPow, ih]
|
||||
by_cases h : y.getLsbD (n + 1)
|
||||
· rw [setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true h,
|
||||
· rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true h,
|
||||
sshiftRight'_or_of_and_eq_zero (by simp [and_twoPow]), h]
|
||||
simp
|
||||
· rw [setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false (i := n + 1)
|
||||
· rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false (i := n + 1)
|
||||
(by simp [h])]
|
||||
simp [h]
|
||||
|
||||
@@ -1183,20 +529,20 @@ theorem ushiftRight'_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
|
||||
simp [← add_eq_or_of_and_eq_zero _ _ h, toNat_add_of_and_eq_zero h, shiftRight_add]
|
||||
|
||||
theorem ushiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
|
||||
ushiftRightRec x y n = x >>> (y.setWidth (n + 1)).setWidth w₂ := by
|
||||
ushiftRightRec x y n = x >>> (y.truncate (n + 1)).zeroExtend w₂ := by
|
||||
induction n generalizing x y
|
||||
case zero =>
|
||||
ext i
|
||||
simp only [ushiftRightRec_zero, twoPow_zero, Nat.reduceAdd,
|
||||
and_one_eq_setWidth_ofBool_getLsbD, setWidth_one]
|
||||
and_one_eq_zeroExtend_ofBool_getLsbD, truncate_one]
|
||||
case succ n ih =>
|
||||
simp only [ushiftRightRec_succ, and_twoPow]
|
||||
rw [ih]
|
||||
by_cases h : y.getLsbD (n + 1) <;> simp only [h, ↓reduceIte]
|
||||
· rw [setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true h,
|
||||
· rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true h,
|
||||
ushiftRight'_or_of_and_eq_zero]
|
||||
simp [and_twoPow]
|
||||
· simp [setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false, h]
|
||||
· simp [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false, h]
|
||||
|
||||
/--
|
||||
Show that `x >>> y` can be written in terms of `ushiftRightRec`.
|
||||
|
||||
@@ -48,7 +48,7 @@ private theorem iunfoldr.eq_test
|
||||
simp only [init, eq_nil]
|
||||
case step =>
|
||||
intro i
|
||||
simp_all [setWidth_succ]
|
||||
simp_all [truncate_succ]
|
||||
|
||||
theorem iunfoldr_getLsbD' {f : Fin w → α → α × Bool} (state : Nat → α)
|
||||
(ind : ∀(i : Fin w), (f i (state i.val)).fst = state (i.val+1)) :
|
||||
@@ -65,7 +65,7 @@ theorem iunfoldr_getLsbD' {f : Fin w → α → α × Bool} (state : Nat → α)
|
||||
intro
|
||||
apply And.intro
|
||||
· intro i
|
||||
have := Fin.pos i
|
||||
have := Fin.size_pos i
|
||||
contradiction
|
||||
· rfl
|
||||
case step =>
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -12,8 +12,6 @@ namespace Bool
|
||||
/-- Boolean exclusive or -/
|
||||
abbrev xor : Bool → Bool → Bool := bne
|
||||
|
||||
@[inherit_doc] infixl:33 " ^^ " => xor
|
||||
|
||||
instance (p : Bool → Prop) [inst : DecidablePred p] : Decidable (∀ x, p x) :=
|
||||
match inst true, inst false with
|
||||
| isFalse ht, _ => isFalse fun h => absurd (h _) ht
|
||||
@@ -147,8 +145,8 @@ theorem and_or_distrib_right : ∀ (x y z : Bool), ((x || y) && z) = (x && z ||
|
||||
theorem or_and_distrib_left : ∀ (x y z : Bool), (x || y && z) = ((x || y) && (x || z)) := by decide
|
||||
theorem or_and_distrib_right : ∀ (x y z : Bool), (x && y || z) = ((x || z) && (y || z)) := by decide
|
||||
|
||||
theorem and_xor_distrib_left : ∀ (x y z : Bool), (x && (y ^^ z)) = ((x && y) ^^ (x && z)) := by decide
|
||||
theorem and_xor_distrib_right : ∀ (x y z : Bool), ((x ^^ y) && z) = ((x && z) ^^ (y && z)) := by decide
|
||||
theorem and_xor_distrib_left : ∀ (x y z : Bool), (x && xor y z) = xor (x && y) (x && z) := by decide
|
||||
theorem and_xor_distrib_right : ∀ (x y z : Bool), (xor x y && z) = xor (x && z) (y && z) := by decide
|
||||
|
||||
/-- De Morgan's law for boolean and -/
|
||||
@[simp] theorem not_and : ∀ (x y : Bool), (!(x && y)) = (!x || !y) := by decide
|
||||
@@ -238,8 +236,8 @@ theorem not_bne_not : ∀ (x y : Bool), ((!x) != (!y)) = (x != y) := by simp
|
||||
@[simp] theorem bne_assoc : ∀ (x y z : Bool), ((x != y) != z) = (x != (y != z)) := by decide
|
||||
instance : Std.Associative (· != ·) := ⟨bne_assoc⟩
|
||||
|
||||
@[simp] theorem bne_right_inj : ∀ {x y z : Bool}, (x != y) = (x != z) ↔ y = z := by decide
|
||||
@[simp] theorem bne_left_inj : ∀ {x y z : Bool}, (x != z) = (y != z) ↔ x = y := by decide
|
||||
@[simp] theorem bne_left_inj : ∀ {x y z : Bool}, (x != y) = (x != z) ↔ y = z := by decide
|
||||
@[simp] theorem bne_right_inj : ∀ {x y z : Bool}, (x != z) = (y != z) ↔ x = y := by decide
|
||||
|
||||
theorem eq_not_of_ne : ∀ {x y : Bool}, x ≠ y → x = !y := by decide
|
||||
|
||||
@@ -254,6 +252,15 @@ theorem beq_eq_decide_eq [BEq α] [LawfulBEq α] [DecidableEq α] (a b : α) :
|
||||
theorem eq_not : ∀ {a b : Bool}, (a = (!b)) ↔ (a ≠ b) := by decide
|
||||
theorem not_eq : ∀ {a b : Bool}, ((!a) = b) ↔ (a ≠ b) := by decide
|
||||
|
||||
@[simp] theorem not_eq_not : ∀ {a b : Bool}, ¬a = !b ↔ a = b := by decide
|
||||
@[simp] theorem not_not_eq : ∀ {a b : Bool}, ¬(!a) = b ↔ a = b := by decide
|
||||
|
||||
/--
|
||||
We move `!` from the left hand side of an equality to the right hand side.
|
||||
This helps confluence, and also helps combining pairs of `!`s.
|
||||
-/
|
||||
@[simp] theorem not_eq_eq_eq_not : ∀ {a b : Bool}, ((!a) = b) ↔ (a = !b) := by decide
|
||||
|
||||
@[simp] theorem coe_iff_coe : ∀{a b : Bool}, (a ↔ b) ↔ a = b := by decide
|
||||
|
||||
@[simp] theorem coe_true_iff_false : ∀{a b : Bool}, (a ↔ b = false) ↔ a = (!b) := by decide
|
||||
@@ -267,37 +274,37 @@ theorem beq_comm {α} [BEq α] [LawfulBEq α] {a b : α} : (a == b) = (b == a) :
|
||||
|
||||
/-! ### xor -/
|
||||
|
||||
theorem false_xor : ∀ (x : Bool), (false ^^ x) = x := false_bne
|
||||
theorem false_xor : ∀ (x : Bool), xor false x = x := false_bne
|
||||
|
||||
theorem xor_false : ∀ (x : Bool), (x ^^ false) = x := bne_false
|
||||
theorem xor_false : ∀ (x : Bool), xor x false = x := bne_false
|
||||
|
||||
theorem true_xor : ∀ (x : Bool), (true ^^ x) = !x := true_bne
|
||||
theorem true_xor : ∀ (x : Bool), xor true x = !x := true_bne
|
||||
|
||||
theorem xor_true : ∀ (x : Bool), (x ^^ true) = !x := bne_true
|
||||
theorem xor_true : ∀ (x : Bool), xor x true = !x := bne_true
|
||||
|
||||
theorem not_xor_self : ∀ (x : Bool), (!x ^^ x) = true := not_bne_self
|
||||
theorem not_xor_self : ∀ (x : Bool), xor (!x) x = true := not_bne_self
|
||||
|
||||
theorem xor_not_self : ∀ (x : Bool), (x ^^ !x) = true := bne_not_self
|
||||
theorem xor_not_self : ∀ (x : Bool), xor x (!x) = true := bne_not_self
|
||||
|
||||
theorem not_xor : ∀ (x y : Bool), (!x ^^ y) = !(x ^^ y) := by decide
|
||||
theorem not_xor : ∀ (x y : Bool), xor (!x) y = !(xor x y) := by decide
|
||||
|
||||
theorem xor_not : ∀ (x y : Bool), (x ^^ !y) = !(x ^^ y) := by decide
|
||||
theorem xor_not : ∀ (x y : Bool), xor x (!y) = !(xor x y) := by decide
|
||||
|
||||
theorem not_xor_not : ∀ (x y : Bool), (!x ^^ !y) = (x ^^ y) := not_bne_not
|
||||
theorem not_xor_not : ∀ (x y : Bool), xor (!x) (!y) = (xor x y) := not_bne_not
|
||||
|
||||
theorem xor_self : ∀ (x : Bool), (x ^^ x) = false := by decide
|
||||
theorem xor_self : ∀ (x : Bool), xor x x = false := by decide
|
||||
|
||||
theorem xor_comm : ∀ (x y : Bool), (x ^^ y) = (y ^^ x) := by decide
|
||||
theorem xor_comm : ∀ (x y : Bool), xor x y = xor y x := by decide
|
||||
|
||||
theorem xor_left_comm : ∀ (x y z : Bool), (x ^^ (y ^^ z)) = (y ^^ (x ^^ z)) := by decide
|
||||
theorem xor_left_comm : ∀ (x y z : Bool), xor x (xor y z) = xor y (xor x z) := by decide
|
||||
|
||||
theorem xor_right_comm : ∀ (x y z : Bool), ((x ^^ y) ^^ z) = ((x ^^ z) ^^ y) := by decide
|
||||
theorem xor_right_comm : ∀ (x y z : Bool), xor (xor x y) z = xor (xor x z) y := by decide
|
||||
|
||||
theorem xor_assoc : ∀ (x y z : Bool), ((x ^^ y) ^^ z) = (x ^^ (y ^^ z)) := bne_assoc
|
||||
theorem xor_assoc : ∀ (x y z : Bool), xor (xor x y) z = xor x (xor y z) := bne_assoc
|
||||
|
||||
theorem xor_right_inj : ∀ {x y z : Bool}, (x ^^ y) = (x ^^ z) ↔ y = z := bne_right_inj
|
||||
theorem xor_left_inj : ∀ {x y z : Bool}, xor x y = xor x z ↔ y = z := bne_left_inj
|
||||
|
||||
theorem xor_left_inj : ∀ {x y z : Bool}, (x ^^ z) = (y ^^ z) ↔ x = y := bne_left_inj
|
||||
theorem xor_right_inj : ∀ {x y z : Bool}, xor x z = xor y z ↔ x = y := bne_right_inj
|
||||
|
||||
/-! ### le/lt -/
|
||||
|
||||
@@ -368,14 +375,13 @@ theorem and_or_inj_left_iff :
|
||||
/-- convert a `Bool` to a `Nat`, `false -> 0`, `true -> 1` -/
|
||||
def toNat (b : Bool) : Nat := cond b 1 0
|
||||
|
||||
@[simp, bv_toNat] theorem toNat_false : false.toNat = 0 := rfl
|
||||
@[simp] theorem toNat_false : false.toNat = 0 := rfl
|
||||
|
||||
@[simp, bv_toNat] theorem toNat_true : true.toNat = 1 := rfl
|
||||
@[simp] theorem toNat_true : true.toNat = 1 := rfl
|
||||
|
||||
theorem toNat_le (c : Bool) : c.toNat ≤ 1 := by
|
||||
cases c <;> trivial
|
||||
|
||||
@[bv_toNat]
|
||||
theorem toNat_lt (b : Bool) : b.toNat < 2 :=
|
||||
Nat.lt_succ_of_le (toNat_le _)
|
||||
|
||||
|
||||
@@ -42,7 +42,7 @@ def usize (a : @& ByteArray) : USize :=
|
||||
a.size.toUSize
|
||||
|
||||
@[extern "lean_byte_array_uget"]
|
||||
def uget : (a : @& ByteArray) → (i : USize) → (h : i.toNat < a.size := by get_elem_tactic) → UInt8
|
||||
def uget : (a : @& ByteArray) → (i : USize) → i.toNat < a.size → UInt8
|
||||
| ⟨bs⟩, i, h => bs[i]
|
||||
|
||||
@[extern "lean_byte_array_get"]
|
||||
@@ -50,11 +50,11 @@ def get! : (@& ByteArray) → (@& Nat) → UInt8
|
||||
| ⟨bs⟩, i => bs.get! i
|
||||
|
||||
@[extern "lean_byte_array_fget"]
|
||||
def get : (a : @& ByteArray) → (i : @& Nat) → (h : i < a.size := by get_elem_tactic) → UInt8
|
||||
| ⟨bs⟩, i, _ => bs[i]
|
||||
def get : (a : @& ByteArray) → (@& Fin a.size) → UInt8
|
||||
| ⟨bs⟩, i => bs.get i
|
||||
|
||||
instance : GetElem ByteArray Nat UInt8 fun xs i => i < xs.size where
|
||||
getElem xs i h := xs.get i
|
||||
getElem xs i h := xs.get ⟨i, h⟩
|
||||
|
||||
instance : GetElem ByteArray USize UInt8 fun xs i => i.val < xs.size where
|
||||
getElem xs i h := xs.uget i h
|
||||
@@ -64,11 +64,11 @@ def set! : ByteArray → (@& Nat) → UInt8 → ByteArray
|
||||
| ⟨bs⟩, i, b => ⟨bs.set! i b⟩
|
||||
|
||||
@[extern "lean_byte_array_fset"]
|
||||
def set : (a : ByteArray) → (i : @& Nat) → UInt8 → (h : i < a.size := by get_elem_tactic) → ByteArray
|
||||
| ⟨bs⟩, i, b, h => ⟨bs.set i b h⟩
|
||||
def set : (a : ByteArray) → (@& Fin a.size) → UInt8 → ByteArray
|
||||
| ⟨bs⟩, i, b => ⟨bs.set i b⟩
|
||||
|
||||
@[extern "lean_byte_array_uset"]
|
||||
def uset : (a : ByteArray) → (i : USize) → UInt8 → (h : i.toNat < a.size := by get_elem_tactic) → ByteArray
|
||||
def uset : (a : ByteArray) → (i : USize) → UInt8 → i.toNat < a.size → ByteArray
|
||||
| ⟨bs⟩, i, v, h => ⟨bs.uset i v h⟩
|
||||
|
||||
@[extern "lean_byte_array_hash"]
|
||||
@@ -108,18 +108,8 @@ def toList (bs : ByteArray) : List UInt8 :=
|
||||
|
||||
@[inline] def findIdx? (a : ByteArray) (p : UInt8 → Bool) (start := 0) : Option Nat :=
|
||||
let rec @[specialize] loop (i : Nat) :=
|
||||
if h : i < a.size then
|
||||
if p a[i] then some i else loop (i+1)
|
||||
else
|
||||
none
|
||||
termination_by a.size - i
|
||||
decreasing_by decreasing_trivial_pre_omega
|
||||
loop start
|
||||
|
||||
@[inline] def findFinIdx? (a : ByteArray) (p : UInt8 → Bool) (start := 0) : Option (Fin a.size) :=
|
||||
let rec @[specialize] loop (i : Nat) :=
|
||||
if h : i < a.size then
|
||||
if p a[i] then some ⟨i, h⟩ else loop (i+1)
|
||||
if i < a.size then
|
||||
if p (a.get! i) then some i else loop (i+1)
|
||||
else
|
||||
none
|
||||
termination_by a.size - i
|
||||
@@ -154,7 +144,7 @@ protected def forIn {β : Type v} {m : Type v → Type w} [Monad m] (as : ByteAr
|
||||
have h' : i < as.size := Nat.lt_of_lt_of_le (Nat.lt_succ_self i) h
|
||||
have : as.size - 1 < as.size := Nat.sub_lt (Nat.zero_lt_of_lt h') (by decide)
|
||||
have : as.size - 1 - i < as.size := Nat.lt_of_le_of_lt (Nat.sub_le (as.size - 1) i) this
|
||||
match (← f as[as.size - 1 - i] b) with
|
||||
match (← f (as.get ⟨as.size - 1 - i, this⟩) b) with
|
||||
| ForInStep.done b => pure b
|
||||
| ForInStep.yield b => loop i (Nat.le_of_lt h') b
|
||||
loop as.size (Nat.le_refl _) b
|
||||
@@ -188,7 +178,7 @@ def foldlM {β : Type v} {m : Type v → Type w} [Monad m] (f : β → UInt8 →
|
||||
match i with
|
||||
| 0 => pure b
|
||||
| i'+1 =>
|
||||
loop i' (j+1) (← f b as[j])
|
||||
loop i' (j+1) (← f b (as.get ⟨j, Nat.lt_of_lt_of_le hlt h⟩))
|
||||
else
|
||||
pure b
|
||||
loop (stop - start) start init
|
||||
@@ -255,7 +245,7 @@ On an invalid position, returns `(default : UInt8)`. -/
|
||||
@[inline]
|
||||
def curr : Iterator → UInt8
|
||||
| ⟨arr, i⟩ =>
|
||||
if h : i < arr.size then
|
||||
if h:i < arr.size then
|
||||
arr[i]'h
|
||||
else
|
||||
default
|
||||
|
||||
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.UInt.BasicAux
|
||||
import Init.Data.UInt.Basic
|
||||
|
||||
/-- Determines if the given integer is a valid [Unicode scalar value](https://www.unicode.org/glossary/#unicode_scalar_value).
|
||||
|
||||
@@ -42,10 +42,8 @@ theorem isValidUInt32 (n : Nat) (h : isValidCharNat n) : n < UInt32.size := by
|
||||
|
||||
theorem isValidChar_of_isValidCharNat (n : Nat) (h : isValidCharNat n) : isValidChar (UInt32.ofNat' n (isValidUInt32 n h)) :=
|
||||
match h with
|
||||
| Or.inl h =>
|
||||
Or.inl (UInt32.ofNat'_lt_of_lt _ (by decide) h)
|
||||
| Or.inr ⟨h₁, h₂⟩ =>
|
||||
Or.inr ⟨UInt32.lt_ofNat'_of_lt _ (by decide) h₁, UInt32.ofNat'_lt_of_lt _ (by decide) h₂⟩
|
||||
| Or.inl h => Or.inl h
|
||||
| Or.inr ⟨h₁, h₂⟩ => Or.inr ⟨h₁, h₂⟩
|
||||
|
||||
theorem isValidChar_zero : isValidChar 0 :=
|
||||
Or.inl (by decide)
|
||||
@@ -59,7 +57,7 @@ theorem isValidChar_zero : isValidChar 0 :=
|
||||
c.val.toUInt8
|
||||
|
||||
/-- The numbers from 0 to 256 are all valid UTF-8 characters, so we can embed one in the other. -/
|
||||
def ofUInt8 (n : UInt8) : Char := ⟨n.toUInt32, .inl (Nat.lt_trans n.toBitVec.isLt (by decide))⟩
|
||||
def ofUInt8 (n : UInt8) : Char := ⟨n.toUInt32, .inl (Nat.lt_trans n.1.2 (by decide))⟩
|
||||
|
||||
instance : Inhabited Char where
|
||||
default := 'A'
|
||||
|
||||
@@ -14,7 +14,7 @@ instance coeToNat : CoeOut (Fin n) Nat :=
|
||||
⟨fun v => v.val⟩
|
||||
|
||||
/--
|
||||
From the empty type `Fin 0`, any desired result `α` can be derived. This is similar to `Empty.elim`.
|
||||
From the empty type `Fin 0`, any desired result `α` can be derived. This is simlar to `Empty.elim`.
|
||||
-/
|
||||
def elim0.{u} {α : Sort u} : Fin 0 → α
|
||||
| ⟨_, h⟩ => absurd h (not_lt_zero _)
|
||||
@@ -165,7 +165,6 @@ theorem modn_lt : ∀ {m : Nat} (i : Fin n), m > 0 → (modn i m).val < m
|
||||
theorem val_lt_of_le (i : Fin b) (h : b ≤ n) : i.val < n :=
|
||||
Nat.lt_of_lt_of_le i.isLt h
|
||||
|
||||
/-- If you actually have an element of `Fin n`, then the `n` is always positive -/
|
||||
protected theorem pos (i : Fin n) : 0 < n :=
|
||||
Nat.lt_of_le_of_lt (Nat.zero_le _) i.2
|
||||
|
||||
|
||||
@@ -5,217 +5,22 @@ Authors: François G. Dorais
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Nat.Linear
|
||||
import Init.Control.Lawful.Basic
|
||||
import Init.Data.Fin.Lemmas
|
||||
|
||||
namespace Fin
|
||||
|
||||
/-- Folds over `Fin n` from the left: `foldl 3 f x = f (f (f x 0) 1) 2`. -/
|
||||
@[inline] def foldl (n) (f : α → Fin n → α) (init : α) : α := loop init 0 where
|
||||
/-- Inner loop for `Fin.foldl`. `Fin.foldl.loop n f x i = f (f (f x i) ...) (n-1)` -/
|
||||
@[semireducible] loop (x : α) (i : Nat) : α :=
|
||||
loop (x : α) (i : Nat) : α :=
|
||||
if h : i < n then loop (f x ⟨i, h⟩) (i+1) else x
|
||||
termination_by n - i
|
||||
|
||||
/-- Folds over `Fin n` from the right: `foldr 3 f x = f 0 (f 1 (f 2 x))`. -/
|
||||
@[inline] def foldr (n) (f : Fin n → α → α) (init : α) : α := loop n (Nat.le_refl n) init where
|
||||
/-- Inner loop for `Fin.foldr`. `Fin.foldr.loop n f i x = f 0 (f ... (f (i-1) x))` -/
|
||||
loop : (i : _) → i ≤ n → α → α
|
||||
| 0, _, x => x
|
||||
| i+1, h, x => loop i (Nat.le_of_lt h) (f ⟨i, h⟩ x)
|
||||
termination_by structural i => i
|
||||
|
||||
/--
|
||||
Folds a monadic function over `Fin n` from left to right:
|
||||
```
|
||||
Fin.foldlM n f x₀ = do
|
||||
let x₁ ← f x₀ 0
|
||||
let x₂ ← f x₁ 1
|
||||
...
|
||||
let xₙ ← f xₙ₋₁ (n-1)
|
||||
pure xₙ
|
||||
```
|
||||
-/
|
||||
@[inline] def foldlM [Monad m] (n) (f : α → Fin n → m α) (init : α) : m α := loop init 0 where
|
||||
/--
|
||||
Inner loop for `Fin.foldlM`.
|
||||
```
|
||||
Fin.foldlM.loop n f xᵢ i = do
|
||||
let xᵢ₊₁ ← f xᵢ i
|
||||
...
|
||||
let xₙ ← f xₙ₋₁ (n-1)
|
||||
pure xₙ
|
||||
```
|
||||
-/
|
||||
loop (x : α) (i : Nat) : m α := do
|
||||
if h : i < n then f x ⟨i, h⟩ >>= (loop · (i+1)) else pure x
|
||||
termination_by n - i
|
||||
decreasing_by decreasing_trivial_pre_omega
|
||||
|
||||
/--
|
||||
Folds a monadic function over `Fin n` from right to left:
|
||||
```
|
||||
Fin.foldrM n f xₙ = do
|
||||
let xₙ₋₁ ← f (n-1) xₙ
|
||||
let xₙ₋₂ ← f (n-2) xₙ₋₁
|
||||
...
|
||||
let x₀ ← f 0 x₁
|
||||
pure x₀
|
||||
```
|
||||
-/
|
||||
@[inline] def foldrM [Monad m] (n) (f : Fin n → α → m α) (init : α) : m α :=
|
||||
loop ⟨n, Nat.le_refl n⟩ init where
|
||||
/--
|
||||
Inner loop for `Fin.foldrM`.
|
||||
```
|
||||
Fin.foldrM.loop n f i xᵢ = do
|
||||
let xᵢ₋₁ ← f (i-1) xᵢ
|
||||
...
|
||||
let x₁ ← f 1 x₂
|
||||
let x₀ ← f 0 x₁
|
||||
pure x₀
|
||||
```
|
||||
-/
|
||||
loop : {i // i ≤ n} → α → m α
|
||||
| ⟨0, _⟩, x => pure x
|
||||
| ⟨i+1, h⟩, x => f ⟨i, h⟩ x >>= loop ⟨i, Nat.le_of_lt h⟩
|
||||
|
||||
/-! ### foldlM -/
|
||||
|
||||
theorem foldlM_loop_lt [Monad m] (f : α → Fin n → m α) (x) (h : i < n) :
|
||||
foldlM.loop n f x i = f x ⟨i, h⟩ >>= (foldlM.loop n f . (i+1)) := by
|
||||
rw [foldlM.loop, dif_pos h]
|
||||
|
||||
theorem foldlM_loop_eq [Monad m] (f : α → Fin n → m α) (x) : foldlM.loop n f x n = pure x := by
|
||||
rw [foldlM.loop, dif_neg (Nat.lt_irrefl _)]
|
||||
|
||||
theorem foldlM_loop [Monad m] (f : α → Fin (n+1) → m α) (x) (h : i < n+1) :
|
||||
foldlM.loop (n+1) f x i = f x ⟨i, h⟩ >>= (foldlM.loop n (fun x j => f x j.succ) . i) := by
|
||||
if h' : i < n then
|
||||
rw [foldlM_loop_lt _ _ h]
|
||||
congr; funext
|
||||
rw [foldlM_loop_lt _ _ h', foldlM_loop]; rfl
|
||||
else
|
||||
cases Nat.le_antisymm (Nat.le_of_lt_succ h) (Nat.not_lt.1 h')
|
||||
rw [foldlM_loop_lt]
|
||||
congr; funext
|
||||
rw [foldlM_loop_eq, foldlM_loop_eq]
|
||||
termination_by n - i
|
||||
|
||||
@[simp] theorem foldlM_zero [Monad m] (f : α → Fin 0 → m α) (x) : foldlM 0 f x = pure x :=
|
||||
foldlM_loop_eq ..
|
||||
|
||||
theorem foldlM_succ [Monad m] (f : α → Fin (n+1) → m α) (x) :
|
||||
foldlM (n+1) f x = f x 0 >>= foldlM n (fun x j => f x j.succ) := foldlM_loop ..
|
||||
|
||||
/-! ### foldrM -/
|
||||
|
||||
theorem foldrM_loop_zero [Monad m] (f : Fin n → α → m α) (x) :
|
||||
foldrM.loop n f ⟨0, Nat.zero_le _⟩ x = pure x := by
|
||||
rw [foldrM.loop]
|
||||
|
||||
theorem foldrM_loop_succ [Monad m] (f : Fin n → α → m α) (x) (h : i < n) :
|
||||
foldrM.loop n f ⟨i+1, h⟩ x = f ⟨i, h⟩ x >>= foldrM.loop n f ⟨i, Nat.le_of_lt h⟩ := by
|
||||
rw [foldrM.loop]
|
||||
|
||||
theorem foldrM_loop [Monad m] [LawfulMonad m] (f : Fin (n+1) → α → m α) (x) (h : i+1 ≤ n+1) :
|
||||
foldrM.loop (n+1) f ⟨i+1, h⟩ x =
|
||||
foldrM.loop n (fun j => f j.succ) ⟨i, Nat.le_of_succ_le_succ h⟩ x >>= f 0 := by
|
||||
induction i generalizing x with
|
||||
| zero =>
|
||||
rw [foldrM_loop_zero, foldrM_loop_succ, pure_bind]
|
||||
conv => rhs; rw [←bind_pure (f 0 x)]
|
||||
congr; funext; exact foldrM_loop_zero ..
|
||||
| succ i ih =>
|
||||
rw [foldrM_loop_succ, foldrM_loop_succ, bind_assoc]
|
||||
congr; funext; exact ih ..
|
||||
|
||||
@[simp] theorem foldrM_zero [Monad m] (f : Fin 0 → α → m α) (x) : foldrM 0 f x = pure x :=
|
||||
foldrM_loop_zero ..
|
||||
|
||||
theorem foldrM_succ [Monad m] [LawfulMonad m] (f : Fin (n+1) → α → m α) (x) :
|
||||
foldrM (n+1) f x = foldrM n (fun i => f i.succ) x >>= f 0 := foldrM_loop ..
|
||||
|
||||
/-! ### foldl -/
|
||||
|
||||
theorem foldl_loop_lt (f : α → Fin n → α) (x) (h : i < n) :
|
||||
foldl.loop n f x i = foldl.loop n f (f x ⟨i, h⟩) (i+1) := by
|
||||
rw [foldl.loop, dif_pos h]
|
||||
|
||||
theorem foldl_loop_eq (f : α → Fin n → α) (x) : foldl.loop n f x n = x := by
|
||||
rw [foldl.loop, dif_neg (Nat.lt_irrefl _)]
|
||||
|
||||
theorem foldl_loop (f : α → Fin (n+1) → α) (x) (h : i < n+1) :
|
||||
foldl.loop (n+1) f x i = foldl.loop n (fun x j => f x j.succ) (f x ⟨i, h⟩) i := by
|
||||
if h' : i < n then
|
||||
rw [foldl_loop_lt _ _ h]
|
||||
rw [foldl_loop_lt _ _ h', foldl_loop]; rfl
|
||||
else
|
||||
cases Nat.le_antisymm (Nat.le_of_lt_succ h) (Nat.not_lt.1 h')
|
||||
rw [foldl_loop_lt]
|
||||
rw [foldl_loop_eq, foldl_loop_eq]
|
||||
|
||||
@[simp] theorem foldl_zero (f : α → Fin 0 → α) (x) : foldl 0 f x = x :=
|
||||
foldl_loop_eq ..
|
||||
|
||||
theorem foldl_succ (f : α → Fin (n+1) → α) (x) :
|
||||
foldl (n+1) f x = foldl n (fun x i => f x i.succ) (f x 0) :=
|
||||
foldl_loop ..
|
||||
|
||||
theorem foldl_succ_last (f : α → Fin (n+1) → α) (x) :
|
||||
foldl (n+1) f x = f (foldl n (f · ·.castSucc) x) (last n) := by
|
||||
rw [foldl_succ]
|
||||
induction n generalizing x with
|
||||
| zero => simp [foldl_succ, Fin.last]
|
||||
| succ n ih => rw [foldl_succ, ih (f · ·.succ), foldl_succ]; simp [succ_castSucc]
|
||||
|
||||
theorem foldl_eq_foldlM (f : α → Fin n → α) (x) :
|
||||
foldl n f x = foldlM (m:=Id) n f x := by
|
||||
induction n generalizing x <;> simp [foldl_succ, foldlM_succ, *]
|
||||
|
||||
/-! ### foldr -/
|
||||
|
||||
theorem foldr_loop_zero (f : Fin n → α → α) (x) :
|
||||
foldr.loop n f 0 (Nat.zero_le _) x = x := by
|
||||
rw [foldr.loop]
|
||||
|
||||
theorem foldr_loop_succ (f : Fin n → α → α) (x) (h : i < n) :
|
||||
foldr.loop n f (i+1) h x = foldr.loop n f i (Nat.le_of_lt h) (f ⟨i, h⟩ x) := by
|
||||
rw [foldr.loop]
|
||||
|
||||
theorem foldr_loop (f : Fin (n+1) → α → α) (x) (h : i+1 ≤ n+1) :
|
||||
foldr.loop (n+1) f (i+1) h x =
|
||||
f 0 (foldr.loop n (fun j => f j.succ) i (Nat.le_of_succ_le_succ h) x) := by
|
||||
induction i generalizing x with
|
||||
| zero => simp [foldr_loop_succ, foldr_loop_zero]
|
||||
| succ i ih => rw [foldr_loop_succ, ih]; rfl
|
||||
|
||||
@[simp] theorem foldr_zero (f : Fin 0 → α → α) (x) : foldr 0 f x = x :=
|
||||
foldr_loop_zero ..
|
||||
|
||||
theorem foldr_succ (f : Fin (n+1) → α → α) (x) :
|
||||
foldr (n+1) f x = f 0 (foldr n (fun i => f i.succ) x) := foldr_loop ..
|
||||
|
||||
theorem foldr_succ_last (f : Fin (n+1) → α → α) (x) :
|
||||
foldr (n+1) f x = foldr n (f ·.castSucc) (f (last n) x) := by
|
||||
induction n generalizing x with
|
||||
| zero => simp [foldr_succ, Fin.last]
|
||||
| succ n ih => rw [foldr_succ, ih (f ·.succ), foldr_succ]; simp [succ_castSucc]
|
||||
|
||||
theorem foldr_eq_foldrM (f : Fin n → α → α) (x) :
|
||||
foldr n f x = foldrM (m:=Id) n f x := by
|
||||
induction n <;> simp [foldr_succ, foldrM_succ, *]
|
||||
|
||||
theorem foldl_rev (f : Fin n → α → α) (x) :
|
||||
foldl n (fun x i => f i.rev x) x = foldr n f x := by
|
||||
induction n generalizing x with
|
||||
| zero => simp
|
||||
| succ n ih => rw [foldl_succ, foldr_succ_last, ← ih]; simp [rev_succ]
|
||||
|
||||
theorem foldr_rev (f : α → Fin n → α) (x) :
|
||||
foldr n (fun i x => f x i.rev) x = foldl n f x := by
|
||||
induction n generalizing x with
|
||||
| zero => simp
|
||||
| succ n ih => rw [foldl_succ_last, foldr_succ, ← ih]; simp [rev_succ]
|
||||
/-- Folds over `Fin n` from the right: `foldr 3 f x = f 0 (f 1 (f 2 x))`. -/
|
||||
@[inline] def foldr (n) (f : Fin n → α → α) (init : α) : α := loop ⟨n, Nat.le_refl n⟩ init where
|
||||
/-- Inner loop for `Fin.foldr`. `Fin.foldr.loop n f i x = f 0 (f ... (f (i-1) x))` -/
|
||||
loop : {i // i ≤ n} → α → α
|
||||
| ⟨0, _⟩, x => x
|
||||
| ⟨i+1, h⟩, x => loop ⟨i, Nat.le_of_lt h⟩ (f ⟨i, h⟩ x)
|
||||
|
||||
end Fin
|
||||
|
||||
@@ -26,7 +26,7 @@ def hIterateFrom (P : Nat → Sort _) {n} (f : ∀(i : Fin n), P i.val → P (i.
|
||||
decreasing_by decreasing_trivial_pre_omega
|
||||
|
||||
/--
|
||||
`hIterate` is a heterogeneous iterative operation that applies a
|
||||
`hIterate` is a heterogenous iterative operation that applies a
|
||||
index-dependent function `f` to a value `init : P start` a total of
|
||||
`stop - start` times to produce a value of type `P stop`.
|
||||
|
||||
@@ -35,7 +35,7 @@ Concretely, `hIterate start stop f init` is equal to
|
||||
init |> f start _ |> f (start+1) _ ... |> f (end-1) _
|
||||
```
|
||||
|
||||
Because it is heterogeneous and must return a value of type `P stop`,
|
||||
Because it is heterogenous and must return a value of type `P stop`,
|
||||
`hIterate` requires proof that `start ≤ stop`.
|
||||
|
||||
One can prove properties of `hIterate` using the general theorem
|
||||
@@ -70,7 +70,7 @@ private theorem hIterateFrom_elim {P : Nat → Sort _}(Q : ∀(i : Nat), P i →
|
||||
|
||||
/-
|
||||
`hIterate_elim` provides a mechanism for showing that the result of
|
||||
`hIterate` satisfies a property `Q stop` by showing that the states
|
||||
`hIterate` satisifies a property `Q stop` by showing that the states
|
||||
at the intermediate indices `i : start ≤ i < stop` satisfy `Q i`.
|
||||
-/
|
||||
theorem hIterate_elim {P : Nat → Sort _} (Q : ∀(i : Nat), P i → Prop)
|
||||
|
||||
@@ -13,19 +13,17 @@ import Init.Omega
|
||||
|
||||
namespace Fin
|
||||
|
||||
@[deprecated Fin.pos (since := "2024-11-11")]
|
||||
theorem size_pos (i : Fin n) : 0 < n := i.pos
|
||||
/-- If you actually have an element of `Fin n`, then the `n` is always positive -/
|
||||
theorem size_pos (i : Fin n) : 0 < n := Nat.lt_of_le_of_lt (Nat.zero_le _) i.2
|
||||
|
||||
theorem mod_def (a m : Fin n) : a % m = Fin.mk (a % m) (Nat.lt_of_le_of_lt (Nat.mod_le _ _) a.2) :=
|
||||
rfl
|
||||
|
||||
theorem mul_def (a b : Fin n) : a * b = Fin.mk ((a * b) % n) (Nat.mod_lt _ a.pos) := rfl
|
||||
theorem mul_def (a b : Fin n) : a * b = Fin.mk ((a * b) % n) (Nat.mod_lt _ a.size_pos) := rfl
|
||||
|
||||
theorem sub_def (a b : Fin n) : a - b = Fin.mk (((n - b) + a) % n) (Nat.mod_lt _ a.pos) := rfl
|
||||
theorem sub_def (a b : Fin n) : a - b = Fin.mk (((n - b) + a) % n) (Nat.mod_lt _ a.size_pos) := rfl
|
||||
|
||||
theorem pos' : ∀ [Nonempty (Fin n)], 0 < n | ⟨i⟩ => i.pos
|
||||
|
||||
@[deprecated pos' (since := "2024-11-11")] abbrev size_pos' := @pos'
|
||||
theorem size_pos' : ∀ [Nonempty (Fin n)], 0 < n | ⟨i⟩ => i.size_pos
|
||||
|
||||
@[simp] theorem is_lt (a : Fin n) : (a : Nat) < n := a.2
|
||||
|
||||
@@ -56,12 +54,7 @@ theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
|
||||
@[simp] theorem val_ofNat' (n : Nat) [NeZero n] (a : Nat) :
|
||||
(Fin.ofNat' n a).val = a % n := rfl
|
||||
|
||||
@[simp] theorem ofNat'_self {n : Nat} [NeZero n] : Fin.ofNat' n n = 0 := by
|
||||
ext
|
||||
simp
|
||||
congr
|
||||
|
||||
@[simp] theorem ofNat'_val_eq_self [NeZero n] (x : Fin n) : (Fin.ofNat' n x) = x := by
|
||||
@[simp] theorem ofNat'_val_eq_self [NeZero n](x : Fin n) : (Fin.ofNat' n x) = x := by
|
||||
ext
|
||||
rw [val_ofNat', Nat.mod_eq_of_lt]
|
||||
exact x.2
|
||||
@@ -75,9 +68,6 @@ theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
|
||||
@[simp] theorem modn_val (a : Fin n) (b : Nat) : (a.modn b).val = a.val % b :=
|
||||
rfl
|
||||
|
||||
@[simp] theorem val_eq_zero (a : Fin 1) : a.val = 0 :=
|
||||
Nat.eq_zero_of_le_zero <| Nat.le_of_lt_succ a.isLt
|
||||
|
||||
theorem ite_val {n : Nat} {c : Prop} [Decidable c] {x : c → Fin n} (y : ¬c → Fin n) :
|
||||
(if h : c then x h else y h).val = if h : c then (x h).val else (y h).val := by
|
||||
by_cases c <;> simp [*]
|
||||
@@ -130,7 +120,7 @@ theorem mk_le_of_le_val {b : Fin n} {a : Nat} (h : a ≤ b) :
|
||||
|
||||
@[simp] theorem mk_lt_mk {x y : Nat} {hx hy} : (⟨x, hx⟩ : Fin n) < ⟨y, hy⟩ ↔ x < y := .rfl
|
||||
|
||||
@[simp] theorem val_zero (n : Nat) [NeZero n] : ((0 : Fin n) : Nat) = 0 := rfl
|
||||
@[simp] theorem val_zero (n : Nat) : (0 : Fin (n + 1)).1 = 0 := rfl
|
||||
|
||||
@[simp] theorem mk_zero : (⟨0, Nat.succ_pos n⟩ : Fin (n + 1)) = 0 := rfl
|
||||
|
||||
@@ -177,24 +167,8 @@ theorem rev_eq {n a : Nat} (i : Fin (n + 1)) (h : n = a + i) :
|
||||
@[simp] theorem rev_lt_rev {i j : Fin n} : rev i < rev j ↔ j < i := by
|
||||
rw [← Fin.not_le, ← Fin.not_le, rev_le_rev]
|
||||
|
||||
/-! ### last -/
|
||||
|
||||
@[simp] theorem val_last (n : Nat) : last n = n := rfl
|
||||
|
||||
@[simp] theorem last_zero : (Fin.last 0 : Fin 1) = 0 := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem zero_eq_last_iff {n : Nat} : (0 : Fin (n + 1)) = last n ↔ n = 0 := by
|
||||
constructor
|
||||
· intro h
|
||||
simp_all [Fin.ext_iff]
|
||||
· rintro rfl
|
||||
simp
|
||||
|
||||
@[simp] theorem last_eq_zero_iff {n : Nat} : Fin.last n = 0 ↔ n = 0 := by
|
||||
simp [eq_comm (a := Fin.last n)]
|
||||
|
||||
theorem le_last (i : Fin (n + 1)) : i ≤ last n := Nat.le_of_lt_succ i.is_lt
|
||||
|
||||
theorem last_pos : (0 : Fin (n + 2)) < last (n + 1) := Nat.succ_pos _
|
||||
@@ -228,32 +202,10 @@ instance subsingleton_one : Subsingleton (Fin 1) := subsingleton_iff_le_one.2 (b
|
||||
|
||||
theorem fin_one_eq_zero (a : Fin 1) : a = 0 := Subsingleton.elim a 0
|
||||
|
||||
@[simp] theorem zero_eq_one_iff {n : Nat} [NeZero n] : (0 : Fin n) = 1 ↔ n = 1 := by
|
||||
constructor
|
||||
· intro h
|
||||
simp [Fin.ext_iff] at h
|
||||
change 0 % n = 1 % n at h
|
||||
rw [eq_comm] at h
|
||||
simpa using h
|
||||
· rintro rfl
|
||||
simp
|
||||
|
||||
@[simp] theorem one_eq_zero_iff {n : Nat} [NeZero n] : (1 : Fin n) = 0 ↔ n = 1 := by
|
||||
rw [eq_comm]
|
||||
simp
|
||||
|
||||
theorem add_def (a b : Fin n) : a + b = Fin.mk ((a + b) % n) (Nat.mod_lt _ a.pos) := rfl
|
||||
theorem add_def (a b : Fin n) : a + b = Fin.mk ((a + b) % n) (Nat.mod_lt _ a.size_pos) := rfl
|
||||
|
||||
theorem val_add (a b : Fin n) : (a + b).val = (a.val + b.val) % n := rfl
|
||||
|
||||
@[simp] protected theorem zero_add [NeZero n] (k : Fin n) : (0 : Fin n) + k = k := by
|
||||
ext
|
||||
simp [Fin.add_def, Nat.mod_eq_of_lt k.2]
|
||||
|
||||
@[simp] protected theorem add_zero [NeZero n] (k : Fin n) : k + 0 = k := by
|
||||
ext
|
||||
simp [add_def, Nat.mod_eq_of_lt k.2]
|
||||
|
||||
theorem val_add_one_of_lt {n : Nat} {i : Fin n.succ} (h : i < last _) : (i + 1).1 = i + 1 := by
|
||||
match n with
|
||||
| 0 => cases h
|
||||
@@ -377,10 +329,6 @@ theorem succ_succ_ne_one (a : Fin n) : Fin.succ (Fin.succ a) ≠ 1 :=
|
||||
|
||||
@[simp] theorem cast_mk (h : n = m) (i : Nat) (hn : i < n) : cast h ⟨i, hn⟩ = ⟨i, h ▸ hn⟩ := rfl
|
||||
|
||||
@[simp] theorem cast_refl (n : Nat) (h : n = n) : cast h = id := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem cast_trans {k : Nat} (h : n = m) (h' : m = k) {i : Fin n} :
|
||||
cast h' (cast h i) = cast (Eq.trans h h') i := rfl
|
||||
|
||||
@@ -489,10 +437,6 @@ theorem succ_castSucc {n : Nat} (i : Fin n) : i.castSucc.succ = castSucc i.succ
|
||||
|
||||
@[simp] theorem coe_addNat (m : Nat) (i : Fin n) : (addNat i m : Nat) = i + m := rfl
|
||||
|
||||
@[simp] theorem addNat_zero (n : Nat) (i : Fin n) : addNat i 0 = i := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem addNat_one {i : Fin n} : addNat i 1 = i.succ := rfl
|
||||
|
||||
theorem le_coe_addNat (m : Nat) (i : Fin n) : m ≤ addNat i m :=
|
||||
@@ -522,7 +466,7 @@ theorem cast_addNat_left {n n' m : Nat} (i : Fin n') (h : n' + m = n + m) :
|
||||
|
||||
theorem le_coe_natAdd (m : Nat) (i : Fin n) : m ≤ natAdd m i := Nat.le_add_right ..
|
||||
|
||||
@[simp] theorem natAdd_zero {n : Nat} : natAdd 0 = cast (Nat.zero_add n).symm := by ext; simp
|
||||
theorem natAdd_zero {n : Nat} : natAdd 0 = cast (Nat.zero_add n).symm := by ext; simp
|
||||
|
||||
/-- For rewriting in the reverse direction, see `Fin.cast_natAdd_right`. -/
|
||||
theorem natAdd_cast {n n' : Nat} (m : Nat) (i : Fin n') (h : n' = n) :
|
||||
@@ -560,19 +504,9 @@ theorem cast_addNat {n : Nat} (m : Nat) (i : Fin n) :
|
||||
|
||||
@[simp] theorem natAdd_last {m n : Nat} : natAdd n (last m) = last (n + m) := rfl
|
||||
|
||||
@[simp] theorem addNat_last (n : Nat) :
|
||||
addNat (last n) m = cast (by omega) (last (n + m)) := by
|
||||
ext
|
||||
simp
|
||||
|
||||
theorem natAdd_castSucc {m n : Nat} {i : Fin m} : natAdd n (castSucc i) = castSucc (natAdd n i) :=
|
||||
rfl
|
||||
|
||||
@[simp] theorem natAdd_eq_addNat (n : Nat) (i : Fin n) : Fin.natAdd n i = i.addNat n := by
|
||||
ext
|
||||
simp
|
||||
omega
|
||||
|
||||
theorem rev_castAdd (k : Fin n) (m : Nat) : rev (castAdd m k) = addNat (rev k) m := Fin.ext <| by
|
||||
rw [val_rev, coe_castAdd, coe_addNat, val_rev, Nat.sub_add_comm (Nat.succ_le_of_lt k.is_lt)]
|
||||
|
||||
@@ -588,8 +522,8 @@ theorem rev_succ (k : Fin n) : rev (succ k) = castSucc (rev k) := k.rev_addNat 1
|
||||
@[simp] theorem coe_pred (j : Fin (n + 1)) (h : j ≠ 0) : (j.pred h : Nat) = j - 1 := rfl
|
||||
|
||||
@[simp] theorem succ_pred : ∀ (i : Fin (n + 1)) (h : i ≠ 0), (i.pred h).succ = i
|
||||
| ⟨0, _⟩, hi => by simp only [mk_zero, ne_eq, not_true] at hi
|
||||
| ⟨_ + 1, _⟩, _ => rfl
|
||||
| ⟨0, h⟩, hi => by simp only [mk_zero, ne_eq, not_true] at hi
|
||||
| ⟨n + 1, h⟩, hi => rfl
|
||||
|
||||
@[simp]
|
||||
theorem pred_succ (i : Fin n) {h : i.succ ≠ 0} : i.succ.pred h = i := by
|
||||
@@ -638,15 +572,6 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
|
||||
@[simp] theorem subNat_mk {i : Nat} (h₁ : i < n + m) (h₂ : m ≤ i) :
|
||||
subNat m ⟨i, h₁⟩ h₂ = ⟨i - m, Nat.sub_lt_right_of_lt_add h₂ h₁⟩ := rfl
|
||||
|
||||
@[simp] theorem subNat_zero (i : Fin n) (h : 0 ≤ (i : Nat)): Fin.subNat 0 i h = i := by
|
||||
ext
|
||||
simp
|
||||
|
||||
@[simp] theorem subNat_one_succ (i : Fin (n + 1)) (h : 1 ≤ (i : Nat)) : (subNat 1 i h).succ = i := by
|
||||
ext
|
||||
simp
|
||||
omega
|
||||
|
||||
@[simp] theorem pred_castSucc_succ (i : Fin n) :
|
||||
pred (castSucc i.succ) (Fin.ne_of_gt (castSucc_pos i.succ_pos)) = castSucc i := rfl
|
||||
|
||||
@@ -657,7 +582,7 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
|
||||
subNat m (addNat i m) h = i := Fin.ext <| Nat.add_sub_cancel i m
|
||||
|
||||
@[simp] theorem natAdd_subNat_cast {i : Fin (n + m)} (h : n ≤ i) :
|
||||
natAdd n (subNat n (cast (Nat.add_comm ..) i) h) = i := by simp [← cast_addNat]
|
||||
natAdd n (subNat n (cast (Nat.add_comm ..) i) h) = i := by simp [← cast_addNat]; rfl
|
||||
|
||||
/-! ### recursion and induction principles -/
|
||||
|
||||
@@ -825,12 +750,12 @@ theorem addCases_right {m n : Nat} {motive : Fin (m + n) → Sort _} {left right
|
||||
|
||||
/-! ### add -/
|
||||
|
||||
theorem ofNat'_add [NeZero n] (x : Nat) (y : Fin n) :
|
||||
@[simp] theorem ofNat'_add [NeZero n] (x : Nat) (y : Fin n) :
|
||||
Fin.ofNat' n x + y = Fin.ofNat' n (x + y.val) := by
|
||||
apply Fin.eq_of_val_eq
|
||||
simp [Fin.ofNat', Fin.add_def]
|
||||
|
||||
theorem add_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
|
||||
@[simp] theorem add_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
|
||||
x + Fin.ofNat' n y = Fin.ofNat' n (x.val + y) := by
|
||||
apply Fin.eq_of_val_eq
|
||||
simp [Fin.ofNat', Fin.add_def]
|
||||
@@ -840,21 +765,16 @@ theorem add_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
|
||||
protected theorem coe_sub (a b : Fin n) : ((a - b : Fin n) : Nat) = ((n - b) + a) % n := by
|
||||
cases a; cases b; rfl
|
||||
|
||||
theorem ofNat'_sub [NeZero n] (x : Nat) (y : Fin n) :
|
||||
@[simp] theorem ofNat'_sub [NeZero n] (x : Nat) (y : Fin n) :
|
||||
Fin.ofNat' n x - y = Fin.ofNat' n ((n - y.val) + x) := by
|
||||
apply Fin.eq_of_val_eq
|
||||
simp [Fin.ofNat', Fin.sub_def]
|
||||
|
||||
theorem sub_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
|
||||
@[simp] theorem sub_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
|
||||
x - Fin.ofNat' n y = Fin.ofNat' n ((n - y % n) + x.val) := by
|
||||
apply Fin.eq_of_val_eq
|
||||
simp [Fin.ofNat', Fin.sub_def]
|
||||
|
||||
@[simp] protected theorem sub_self [NeZero n] {x : Fin n} : x - x = 0 := by
|
||||
ext
|
||||
rw [Fin.sub_def]
|
||||
simp
|
||||
|
||||
private theorem _root_.Nat.mod_eq_sub_of_lt_two_mul {x n} (h₁ : n ≤ x) (h₂ : x < 2 * n) :
|
||||
x % n = x - n := by
|
||||
rw [Nat.mod_eq, if_pos (by omega), Nat.mod_eq_of_lt (by omega)]
|
||||
|
||||
@@ -31,7 +31,7 @@ opaque floatSpec : FloatSpec := {
|
||||
structure Float where
|
||||
val : floatSpec.float
|
||||
|
||||
instance : Nonempty Float := ⟨{ val := floatSpec.val }⟩
|
||||
instance : Inhabited Float := ⟨{ val := floatSpec.val }⟩
|
||||
|
||||
@[extern "lean_float_add"] opaque Float.add : Float → Float → Float
|
||||
@[extern "lean_float_sub"] opaque Float.sub : Float → Float → Float
|
||||
@@ -47,25 +47,6 @@ def Float.lt : Float → Float → Prop := fun a b =>
|
||||
def Float.le : Float → Float → Prop := fun a b =>
|
||||
floatSpec.le a.val b.val
|
||||
|
||||
/--
|
||||
Raw transmutation from `UInt64`.
|
||||
|
||||
Floats and UInts have the same endianness on all supported platforms.
|
||||
IEEE 754 very precisely specifies the bit layout of floats.
|
||||
-/
|
||||
@[extern "lean_float_of_bits"] opaque Float.ofBits : UInt64 → Float
|
||||
|
||||
/--
|
||||
Raw transmutation to `UInt64`.
|
||||
|
||||
Floats and UInts have the same endianness on all supported platforms.
|
||||
IEEE 754 very precisely specifies the bit layout of floats.
|
||||
|
||||
Note that this function is distinct from `Float.toUInt64`, which attempts
|
||||
to preserve the numeric value, and not the bitwise value.
|
||||
-/
|
||||
@[extern "lean_float_to_bits"] opaque Float.toBits : Float → UInt64
|
||||
|
||||
instance : Add Float := ⟨Float.add⟩
|
||||
instance : Sub Float := ⟨Float.sub⟩
|
||||
instance : Mul Float := ⟨Float.mul⟩
|
||||
@@ -91,35 +72,21 @@ instance floatDecLt (a b : Float) : Decidable (a < b) := Float.decLt a b
|
||||
instance floatDecLe (a b : Float) : Decidable (a ≤ b) := Float.decLe a b
|
||||
|
||||
@[extern "lean_float_to_string"] opaque Float.toString : Float → String
|
||||
/-- If the given float is non-negative, truncates the value to the nearest non-negative integer.
|
||||
If negative or NaN, returns `0`.
|
||||
If larger than the maximum value for `UInt8` (including Inf), returns the maximum value of `UInt8`
|
||||
(i.e. `UInt8.size - 1`).
|
||||
-/
|
||||
|
||||
/-- If the given float is positive, truncates the value to the nearest positive integer.
|
||||
If negative or larger than the maximum value for UInt8, returns 0. -/
|
||||
@[extern "lean_float_to_uint8"] opaque Float.toUInt8 : Float → UInt8
|
||||
/-- If the given float is non-negative, truncates the value to the nearest non-negative integer.
|
||||
If negative or NaN, returns `0`.
|
||||
If larger than the maximum value for `UInt16` (including Inf), returns the maximum value of `UInt16`
|
||||
(i.e. `UInt16.size - 1`).
|
||||
-/
|
||||
/-- If the given float is positive, truncates the value to the nearest positive integer.
|
||||
If negative or larger than the maximum value for UInt16, returns 0. -/
|
||||
@[extern "lean_float_to_uint16"] opaque Float.toUInt16 : Float → UInt16
|
||||
/-- If the given float is non-negative, truncates the value to the nearest non-negative integer.
|
||||
If negative or NaN, returns `0`.
|
||||
If larger than the maximum value for `UInt32` (including Inf), returns the maximum value of `UInt32`
|
||||
(i.e. `UInt32.size - 1`).
|
||||
-/
|
||||
/-- If the given float is positive, truncates the value to the nearest positive integer.
|
||||
If negative or larger than the maximum value for UInt32, returns 0. -/
|
||||
@[extern "lean_float_to_uint32"] opaque Float.toUInt32 : Float → UInt32
|
||||
/-- If the given float is non-negative, truncates the value to the nearest non-negative integer.
|
||||
If negative or NaN, returns `0`.
|
||||
If larger than the maximum value for `UInt64` (including Inf), returns the maximum value of `UInt64`
|
||||
(i.e. `UInt64.size - 1`).
|
||||
-/
|
||||
/-- If the given float is positive, truncates the value to the nearest positive integer.
|
||||
If negative or larger than the maximum value for UInt64, returns 0. -/
|
||||
@[extern "lean_float_to_uint64"] opaque Float.toUInt64 : Float → UInt64
|
||||
/-- If the given float is non-negative, truncates the value to the nearest non-negative integer.
|
||||
If negative or NaN, returns `0`.
|
||||
If larger than the maximum value for `USize` (including Inf), returns the maximum value of `USize`
|
||||
(i.e. `USize.size - 1`). This value is platform dependent).
|
||||
-/
|
||||
/-- If the given float is positive, truncates the value to the nearest positive integer.
|
||||
If negative or larger than the maximum value for USize, returns 0. -/
|
||||
@[extern "lean_float_to_usize"] opaque Float.toUSize : Float → USize
|
||||
|
||||
@[extern "lean_float_isnan"] opaque Float.isNaN : Float → Bool
|
||||
@@ -136,9 +103,6 @@ instance : ToString Float where
|
||||
|
||||
@[extern "lean_uint64_to_float"] opaque UInt64.toFloat (n : UInt64) : Float
|
||||
|
||||
instance : Inhabited Float where
|
||||
default := UInt64.toFloat 0
|
||||
|
||||
instance : Repr Float where
|
||||
reprPrec n prec := if n < UInt64.toFloat 0 then Repr.addAppParen (toString n) prec else toString n
|
||||
|
||||
|
||||
@@ -46,8 +46,8 @@ def uget : (a : @& FloatArray) → (i : USize) → i.toNat < a.size → Float
|
||||
| ⟨ds⟩, i, h => ds[i]
|
||||
|
||||
@[extern "lean_float_array_fget"]
|
||||
def get : (ds : @& FloatArray) → (i : @& Nat) → (h : i < ds.size := by get_elem_tactic) → Float
|
||||
| ⟨ds⟩, i, h => ds.get i h
|
||||
def get : (ds : @& FloatArray) → (@& Fin ds.size) → Float
|
||||
| ⟨ds⟩, i => ds.get i
|
||||
|
||||
@[extern "lean_float_array_get"]
|
||||
def get! : (@& FloatArray) → (@& Nat) → Float
|
||||
@@ -55,23 +55,23 @@ def get! : (@& FloatArray) → (@& Nat) → Float
|
||||
|
||||
def get? (ds : FloatArray) (i : Nat) : Option Float :=
|
||||
if h : i < ds.size then
|
||||
some (ds.get i h)
|
||||
ds.get ⟨i, h⟩
|
||||
else
|
||||
none
|
||||
|
||||
instance : GetElem FloatArray Nat Float fun xs i => i < xs.size where
|
||||
getElem xs i h := xs.get i h
|
||||
getElem xs i h := xs.get ⟨i, h⟩
|
||||
|
||||
instance : GetElem FloatArray USize Float fun xs i => i.val < xs.size where
|
||||
getElem xs i h := xs.uget i h
|
||||
|
||||
@[extern "lean_float_array_uset"]
|
||||
def uset : (a : FloatArray) → (i : USize) → Float → (h : i.toNat < a.size := by get_elem_tactic) → FloatArray
|
||||
def uset : (a : FloatArray) → (i : USize) → Float → i.toNat < a.size → FloatArray
|
||||
| ⟨ds⟩, i, v, h => ⟨ds.uset i v h⟩
|
||||
|
||||
@[extern "lean_float_array_fset"]
|
||||
def set : (ds : FloatArray) → (i : @& Nat) → Float → (h : i < ds.size := by get_elem_tactic) → FloatArray
|
||||
| ⟨ds⟩, i, d, h => ⟨ds.set i d h⟩
|
||||
def set : (ds : FloatArray) → (@& Fin ds.size) → Float → FloatArray
|
||||
| ⟨ds⟩, i, d => ⟨ds.set i d⟩
|
||||
|
||||
@[extern "lean_float_array_set"]
|
||||
def set! : FloatArray → (@& Nat) → Float → FloatArray
|
||||
@@ -83,7 +83,7 @@ def isEmpty (s : FloatArray) : Bool :=
|
||||
partial def toList (ds : FloatArray) : List Float :=
|
||||
let rec loop (i r) :=
|
||||
if h : i < ds.size then
|
||||
loop (i+1) (ds[i] :: r)
|
||||
loop (i+1) (ds.get ⟨i, h⟩ :: r)
|
||||
else
|
||||
r.reverse
|
||||
loop 0 []
|
||||
@@ -115,7 +115,7 @@ protected def forIn {β : Type v} {m : Type v → Type w} [Monad m] (as : FloatA
|
||||
have h' : i < as.size := Nat.lt_of_lt_of_le (Nat.lt_succ_self i) h
|
||||
have : as.size - 1 < as.size := Nat.sub_lt (Nat.zero_lt_of_lt h') (by decide)
|
||||
have : as.size - 1 - i < as.size := Nat.lt_of_le_of_lt (Nat.sub_le (as.size - 1) i) this
|
||||
match (← f as[as.size - 1 - i] b) with
|
||||
match (← f (as.get ⟨as.size - 1 - i, this⟩) b) with
|
||||
| ForInStep.done b => pure b
|
||||
| ForInStep.yield b => loop i (Nat.le_of_lt h') b
|
||||
loop as.size (Nat.le_refl _) b
|
||||
@@ -149,7 +149,7 @@ def foldlM {β : Type v} {m : Type v → Type w} [Monad m] (f : β → Float →
|
||||
match i with
|
||||
| 0 => pure b
|
||||
| i'+1 =>
|
||||
loop i' (j+1) (← f b (as[j]'(Nat.lt_of_lt_of_le hlt h)))
|
||||
loop i' (j+1) (← f b (as.get ⟨j, Nat.lt_of_lt_of_le hlt h⟩))
|
||||
else
|
||||
pure b
|
||||
loop (stop - start) start init
|
||||
|
||||
@@ -1,35 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison
|
||||
-/
|
||||
|
||||
prelude
|
||||
import Init.Core
|
||||
|
||||
namespace Function
|
||||
|
||||
@[inline]
|
||||
def curry : (α × β → φ) → α → β → φ := fun f a b => f (a, b)
|
||||
|
||||
/-- Interpret a function with two arguments as a function on `α × β` -/
|
||||
@[inline]
|
||||
def uncurry : (α → β → φ) → α × β → φ := fun f a => f a.1 a.2
|
||||
|
||||
@[simp]
|
||||
theorem curry_uncurry (f : α → β → φ) : curry (uncurry f) = f :=
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
theorem uncurry_curry (f : α × β → φ) : uncurry (curry f) = f :=
|
||||
funext fun ⟨_a, _b⟩ => rfl
|
||||
|
||||
@[simp]
|
||||
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
|
||||
rfl
|
||||
|
||||
end Function
|
||||
@@ -48,15 +48,9 @@ instance : Hashable UInt64 where
|
||||
instance : Hashable USize where
|
||||
hash n := n.toUInt64
|
||||
|
||||
instance : Hashable ByteArray where
|
||||
hash as := as.foldl (fun r a => mixHash r (hash a)) 7
|
||||
|
||||
instance : Hashable (Fin n) where
|
||||
hash v := v.val.toUInt64
|
||||
|
||||
instance : Hashable Char where
|
||||
hash c := c.val.toUInt64
|
||||
|
||||
instance : Hashable Int where
|
||||
hash
|
||||
| Int.ofNat n => UInt64.ofNat (2 * n)
|
||||
|
||||
@@ -194,7 +194,7 @@ theorem fdiv_eq_tdiv {a b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : fdiv a b = tdiv
|
||||
@[simp, norm_cast] theorem ofNat_emod (m n : Nat) : (↑(m % n) : Int) = m % n := rfl
|
||||
|
||||
|
||||
/-! ### mod definitions -/
|
||||
/-! ### mod definitiions -/
|
||||
|
||||
theorem emod_add_ediv : ∀ a b : Int, a % b + b * (a / b) = a
|
||||
| ofNat _, ofNat _ => congrArg ofNat <| Nat.mod_add_div ..
|
||||
@@ -253,7 +253,7 @@ theorem tmod_def (a b : Int) : tmod a b = a - b * a.tdiv b := by
|
||||
|
||||
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, ofNat n => congrArg ofNat <| Nat.mod_add_div ..
|
||||
| succ m, -[n+1] => by
|
||||
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,
|
||||
@@ -289,8 +289,8 @@ theorem fmod_eq_tmod {a b : Int} (Ha : 0 ≤ a) (Hb : 0 ≤ b) : fmod a b = tmod
|
||||
|
||||
@[simp] protected theorem ediv_neg : ∀ a b : Int, a / (-b) = -(a / b)
|
||||
| ofNat m, 0 => show ofNat (m / 0) = -↑(m / 0) by rw [Nat.div_zero]; rfl
|
||||
| ofNat _, -[_+1] => (Int.neg_neg _).symm
|
||||
| ofNat _, succ _ | -[_+1], 0 | -[_+1], succ _ | -[_+1], -[_+1] => rfl
|
||||
| ofNat m, -[n+1] => (Int.neg_neg _).symm
|
||||
| ofNat m, succ n | -[m+1], 0 | -[m+1], succ n | -[m+1], -[n+1] => rfl
|
||||
|
||||
theorem ediv_neg' {a b : Int} (Ha : a < 0) (Hb : 0 < b) : a / b < 0 :=
|
||||
match a, b, eq_negSucc_of_lt_zero Ha, eq_succ_of_zero_lt Hb with
|
||||
@@ -339,7 +339,7 @@ theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c
|
||||
| _, ⟨k, rfl⟩, -[n+1] => show (a - n.succ * k.succ).ediv k.succ = a.ediv k.succ - n.succ by
|
||||
rw [← Int.add_sub_cancel (ediv ..), ← this, Int.sub_add_cancel]
|
||||
fun {k n} => @fun
|
||||
| ofNat _ => congrArg ofNat <| Nat.add_mul_div_right _ _ k.succ_pos
|
||||
| ofNat m => congrArg ofNat <| Nat.add_mul_div_right _ _ k.succ_pos
|
||||
| -[m+1] => by
|
||||
show ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat)
|
||||
by_cases h : m < n * k.succ
|
||||
@@ -396,7 +396,7 @@ theorem add_mul_ediv_left (a : Int) {b : Int}
|
||||
rw [Int.mul_neg, Int.ediv_neg, Int.ediv_neg]; apply congrArg Neg.neg; apply this
|
||||
fun m k b =>
|
||||
match b, k with
|
||||
| ofNat _, _ => congrArg ofNat (Nat.mul_div_mul_left _ _ m.succ_pos)
|
||||
| ofNat n, k => congrArg ofNat (Nat.mul_div_mul_left _ _ m.succ_pos)
|
||||
| -[n+1], 0 => by
|
||||
rw [Int.ofNat_zero, Int.mul_zero, Int.ediv_zero, Int.ediv_zero]
|
||||
| -[n+1], succ k => congrArg negSucc <|
|
||||
@@ -822,14 +822,14 @@ theorem ediv_eq_ediv_of_mul_eq_mul {a b c d : Int}
|
||||
unseal Nat.div in
|
||||
@[simp] protected theorem tdiv_neg : ∀ a b : Int, a.tdiv (-b) = -(a.tdiv b)
|
||||
| ofNat m, 0 => show ofNat (m / 0) = -↑(m / 0) by rw [Nat.div_zero]; rfl
|
||||
| ofNat _, -[_+1] | -[_+1], succ _ => (Int.neg_neg _).symm
|
||||
| ofNat _, succ _ | -[_+1], 0 | -[_+1], -[_+1] => rfl
|
||||
| ofNat m, -[n+1] | -[m+1], succ n => (Int.neg_neg _).symm
|
||||
| ofNat m, succ n | -[m+1], 0 | -[m+1], -[n+1] => rfl
|
||||
|
||||
unseal Nat.div in
|
||||
@[simp] protected theorem neg_tdiv : ∀ a b : Int, (-a).tdiv b = -(a.tdiv b)
|
||||
| 0, n => by simp [Int.neg_zero]
|
||||
| succ _, (n:Nat) | -[_+1], 0 | -[_+1], -[_+1] => rfl
|
||||
| succ _, -[_+1] | -[_+1], succ _ => (Int.neg_neg _).symm
|
||||
| succ m, (n:Nat) | -[m+1], 0 | -[m+1], -[n+1] => rfl
|
||||
| succ m, -[n+1] | -[m+1], succ n => (Int.neg_neg _).symm
|
||||
|
||||
protected theorem neg_tdiv_neg (a b : Int) : (-a).tdiv (-b) = a.tdiv b := by
|
||||
simp [Int.tdiv_neg, Int.neg_tdiv, Int.neg_neg]
|
||||
@@ -1125,17 +1125,6 @@ theorem emod_add_bmod_congr (x : Int) (n : Nat) : Int.bmod (x%n + y) n = Int.bmo
|
||||
simp [Int.emod_def, Int.sub_eq_add_neg]
|
||||
rw [←Int.mul_neg, Int.add_right_comm, Int.bmod_add_mul_cancel]
|
||||
|
||||
@[simp]
|
||||
theorem emod_sub_bmod_congr (x : Int) (n : Nat) : Int.bmod (x%n - y) n = Int.bmod (x - y) n := by
|
||||
simp only [emod_def, Int.sub_eq_add_neg]
|
||||
rw [←Int.mul_neg, Int.add_right_comm, Int.bmod_add_mul_cancel]
|
||||
|
||||
@[simp]
|
||||
theorem sub_emod_bmod_congr (x : Int) (n : Nat) : Int.bmod (x - y%n) n = Int.bmod (x - y) n := by
|
||||
simp only [emod_def]
|
||||
rw [Int.sub_eq_add_neg, Int.neg_sub, Int.sub_eq_add_neg, ← Int.add_assoc, Int.add_right_comm,
|
||||
Int.bmod_add_mul_cancel, Int.sub_eq_add_neg]
|
||||
|
||||
@[simp]
|
||||
theorem emod_mul_bmod_congr (x : Int) (n : Nat) : Int.bmod (x%n * y) n = Int.bmod (x * y) n := by
|
||||
simp [Int.emod_def, Int.sub_eq_add_neg]
|
||||
@@ -1151,28 +1140,9 @@ theorem bmod_add_bmod_congr : Int.bmod (Int.bmod x n + y) n = Int.bmod (x + y) n
|
||||
rw [Int.sub_eq_add_neg, Int.add_right_comm, ←Int.sub_eq_add_neg]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
theorem bmod_sub_bmod_congr : Int.bmod (Int.bmod x n - y) n = Int.bmod (x - y) n := by
|
||||
rw [Int.bmod_def x n]
|
||||
split
|
||||
next p =>
|
||||
simp only [emod_sub_bmod_congr]
|
||||
next p =>
|
||||
rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.add_right_comm, ←Int.sub_eq_add_neg, ← Int.sub_eq_add_neg]
|
||||
simp [emod_sub_bmod_congr]
|
||||
|
||||
@[simp] theorem add_bmod_bmod : Int.bmod (x + Int.bmod y n) n = Int.bmod (x + y) n := by
|
||||
rw [Int.add_comm x, Int.bmod_add_bmod_congr, Int.add_comm y]
|
||||
|
||||
@[simp] theorem sub_bmod_bmod : Int.bmod (x - Int.bmod y n) n = Int.bmod (x - y) n := by
|
||||
rw [Int.bmod_def y n]
|
||||
split
|
||||
next p =>
|
||||
simp [sub_emod_bmod_congr]
|
||||
next p =>
|
||||
rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, ← Int.add_assoc, ← Int.sub_eq_add_neg]
|
||||
simp [sub_emod_bmod_congr]
|
||||
|
||||
@[simp]
|
||||
theorem bmod_mul_bmod : Int.bmod (Int.bmod x n * y) n = Int.bmod (x * y) n := by
|
||||
rw [bmod_def x n]
|
||||
@@ -1267,7 +1237,7 @@ theorem bmod_le {x : Int} {m : Nat} (h : 0 < m) : bmod x m ≤ (m - 1) / 2 := by
|
||||
_ = ((m + 1 - 2) + 2)/2 := by simp
|
||||
_ = (m - 1) / 2 + 1 := by
|
||||
rw [add_ediv_of_dvd_right]
|
||||
· simp +decide only [Int.ediv_self]
|
||||
· simp (config := {decide := true}) only [Int.ediv_self]
|
||||
congr 2
|
||||
rw [Int.add_sub_assoc, ← Int.sub_neg]
|
||||
congr
|
||||
@@ -1285,7 +1255,7 @@ theorem bmod_natAbs_plus_one (x : Int) (w : 1 < x.natAbs) : bmod x (x.natAbs + 1
|
||||
simp only [bmod, ofNat_eq_coe, natAbs_ofNat, natCast_add, ofNat_one,
|
||||
emod_self_add_one (ofNat_nonneg x)]
|
||||
match x with
|
||||
| 0 => rw [if_pos] <;> simp +decide
|
||||
| 0 => rw [if_pos] <;> simp (config := {decide := true})
|
||||
| (x+1) =>
|
||||
rw [if_neg]
|
||||
· simp [← Int.sub_sub]
|
||||
|
||||
@@ -181,12 +181,12 @@ theorem subNatNat_add_negSucc (m n k : Nat) :
|
||||
Nat.add_comm]
|
||||
|
||||
protected theorem add_assoc : ∀ a b c : Int, a + b + c = a + (b + c)
|
||||
| (m:Nat), (n:Nat), _ => aux1 ..
|
||||
| (m:Nat), (n:Nat), c => aux1 ..
|
||||
| Nat.cast m, b, Nat.cast k => by
|
||||
rw [Int.add_comm, ← aux1, Int.add_comm k, aux1, Int.add_comm b]
|
||||
| a, (n:Nat), (k:Nat) => by
|
||||
rw [Int.add_comm, Int.add_comm a, ← aux1, Int.add_comm a, Int.add_comm k]
|
||||
| -[_+1], -[_+1], (k:Nat) => aux2 ..
|
||||
| -[m+1], -[n+1], (k:Nat) => aux2 ..
|
||||
| -[m+1], (n:Nat), -[k+1] => by
|
||||
rw [Int.add_comm, ← aux2, Int.add_comm n, ← aux2, Int.add_comm -[m+1]]
|
||||
| (m:Nat), -[n+1], -[k+1] => by
|
||||
@@ -329,22 +329,22 @@ theorem toNat_sub (m n : Nat) : toNat (m - n) = m - n := by
|
||||
/- ## add/sub injectivity -/
|
||||
|
||||
@[simp]
|
||||
protected theorem add_left_inj {i j : Int} (k : Int) : (i + k = j + k) ↔ i = j := by
|
||||
protected theorem add_right_inj {i j : Int} (k : Int) : (i + k = j + k) ↔ i = j := by
|
||||
apply Iff.intro
|
||||
· intro p
|
||||
rw [←Int.add_sub_cancel i k, ←Int.add_sub_cancel j k, p]
|
||||
· exact congrArg (· + k)
|
||||
|
||||
@[simp]
|
||||
protected theorem add_right_inj {i j : Int} (k : Int) : (k + i = k + j) ↔ i = j := by
|
||||
protected theorem add_left_inj {i j : Int} (k : Int) : (k + i = k + j) ↔ i = j := by
|
||||
simp [Int.add_comm k]
|
||||
|
||||
@[simp]
|
||||
protected theorem sub_right_inj {i j : Int} (k : Int) : (k - i = k - j) ↔ i = j := by
|
||||
protected theorem sub_left_inj {i j : Int} (k : Int) : (k - i = k - j) ↔ i = j := by
|
||||
simp [Int.sub_eq_add_neg, Int.neg_inj]
|
||||
|
||||
@[simp]
|
||||
protected theorem sub_left_inj {i j : Int} (k : Int) : (i - k = j - k) ↔ i = j := by
|
||||
protected theorem sub_right_inj {i j : Int} (k : Int) : (i - k = j - k) ↔ i = j := by
|
||||
simp [Int.sub_eq_add_neg]
|
||||
|
||||
/- ## Ring properties -/
|
||||
|
||||
@@ -512,8 +512,8 @@ theorem toNat_add_nat {a : Int} (ha : 0 ≤ a) (n : Nat) : (a + n).toNat = a.toN
|
||||
|
||||
@[simp] theorem pred_toNat : ∀ i : Int, (i - 1).toNat = i.toNat - 1
|
||||
| 0 => rfl
|
||||
| (_+1:Nat) => by simp [ofNat_add]
|
||||
| -[_+1] => rfl
|
||||
| (n+1:Nat) => by simp [ofNat_add]
|
||||
| -[n+1] => rfl
|
||||
|
||||
theorem toNat_sub_toNat_neg : ∀ n : Int, ↑n.toNat - ↑(-n).toNat = n
|
||||
| 0 => rfl
|
||||
@@ -1007,9 +1007,9 @@ theorem sign_eq_neg_one_iff_neg {a : Int} : sign a = -1 ↔ a < 0 :=
|
||||
match x with
|
||||
| 0 => rfl
|
||||
| .ofNat (_ + 1) =>
|
||||
simp +decide only [sign, true_iff]
|
||||
simp (config := { decide := true }) only [sign, true_iff]
|
||||
exact Int.le_add_one (ofNat_nonneg _)
|
||||
| .negSucc _ => simp +decide [sign]
|
||||
| .negSucc _ => simp (config := { decide := true }) [sign]
|
||||
|
||||
theorem mul_sign : ∀ i : Int, i * sign i = natAbs i
|
||||
| succ _ => Int.mul_one _
|
||||
|
||||
@@ -5,7 +5,6 @@ Authors: Jeremy Avigad, Deniz Aydin, Floris van Doorn, Mario Carneiro
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Int.Lemmas
|
||||
import Init.Data.Nat.Lemmas
|
||||
|
||||
namespace Int
|
||||
|
||||
@@ -36,24 +35,10 @@ theorem pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} : ∀ {j}, i ≤
|
||||
theorem pos_pow_of_pos {n : Nat} (m : Nat) (h : 0 < n) : 0 < n^m :=
|
||||
pow_le_pow_of_le_right h (Nat.zero_le _)
|
||||
|
||||
@[norm_cast]
|
||||
theorem natCast_pow (b n : Nat) : ((b^n : Nat) : Int) = (b : Int) ^ n := by
|
||||
match n with
|
||||
| 0 => rfl
|
||||
| n + 1 =>
|
||||
simp only [Nat.pow_succ, Int.pow_succ, natCast_mul, natCast_pow _ n]
|
||||
|
||||
@[simp]
|
||||
protected theorem two_pow_pred_sub_two_pow {w : Nat} (h : 0 < w) :
|
||||
((2 ^ (w - 1) : Nat) - (2 ^ w : Nat) : Int) = - ((2 ^ (w - 1) : Nat) : Int) := by
|
||||
rw [← Nat.two_pow_pred_add_two_pow_pred h]
|
||||
omega
|
||||
|
||||
@[simp]
|
||||
protected theorem two_pow_pred_sub_two_pow' {w : Nat} (h : 0 < w) :
|
||||
(2 : Int) ^ (w - 1) - (2 : Int) ^ w = - (2 : Int) ^ (w - 1) := by
|
||||
norm_cast
|
||||
rw [← Nat.two_pow_pred_add_two_pow_pred h]
|
||||
simp [h]
|
||||
|
||||
end Int
|
||||
|
||||
@@ -23,6 +23,3 @@ import Init.Data.List.TakeDrop
|
||||
import Init.Data.List.Zip
|
||||
import Init.Data.List.Perm
|
||||
import Init.Data.List.Sort
|
||||
import Init.Data.List.ToArray
|
||||
import Init.Data.List.MapIdx
|
||||
import Init.Data.List.OfFn
|
||||
|
||||
@@ -13,7 +13,7 @@ namespace List
|
||||
`a : α` satisfying `P`, then `pmap f l h` is essentially the same as `map f l`
|
||||
but is defined only when all members of `l` satisfy `P`, using the proof
|
||||
to apply `f`. -/
|
||||
def pmap {P : α → Prop} (f : ∀ a, P a → β) : ∀ l : List α, (H : ∀ a ∈ l, P a) → List β
|
||||
@[simp] def pmap {P : α → Prop} (f : ∀ a, P a → β) : ∀ l : List α, (H : ∀ a ∈ l, P a) → List β
|
||||
| [], _ => []
|
||||
| a :: l, H => f a (forall_mem_cons.1 H).1 :: pmap f l (forall_mem_cons.1 H).2
|
||||
|
||||
@@ -46,11 +46,6 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
|
||||
| cons _ L', hL' => congrArg _ <| go L' fun _ hx => hL' (.tail _ hx)
|
||||
exact go L h'
|
||||
|
||||
@[simp] theorem pmap_nil {P : α → Prop} (f : ∀ a, P a → β) : pmap f [] (by simp) = [] := rfl
|
||||
|
||||
@[simp] theorem pmap_cons {P : α → Prop} (f : ∀ a, P a → β) (a : α) (l : List α) (h : ∀ b ∈ a :: l, P b) :
|
||||
pmap f (a :: l) h = f a (forall_mem_cons.1 h).1 :: pmap f l (forall_mem_cons.1 h).2 := rfl
|
||||
|
||||
@[simp] theorem attach_nil : ([] : List α).attach = [] := rfl
|
||||
|
||||
@[simp] theorem attachWith_nil : ([] : List α).attachWith P H = [] := rfl
|
||||
@@ -78,7 +73,7 @@ theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l H)
|
||||
· simp only [*, pmap, map]
|
||||
|
||||
theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (l H) :
|
||||
pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun _ h => H _ (mem_map_of_mem _ h) := by
|
||||
pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun a h => H _ (mem_map_of_mem _ h) := by
|
||||
induction l
|
||||
· rfl
|
||||
· simp only [*, pmap, map]
|
||||
@@ -89,7 +84,7 @@ theorem attach_congr {l₁ l₂ : List α} (h : l₁ = l₂) :
|
||||
simp
|
||||
|
||||
theorem attachWith_congr {l₁ l₂ : List α} (w : l₁ = l₂) {P : α → Prop} {H : ∀ x ∈ l₁, P x} :
|
||||
l₁.attachWith P H = l₂.attachWith P fun _ h => H _ (w ▸ h) := by
|
||||
l₁.attachWith P H = l₂.attachWith P fun x h => H _ (w ▸ h) := by
|
||||
subst w
|
||||
simp
|
||||
|
||||
@@ -135,6 +130,24 @@ theorem attachWith_map_subtype_val {p : α → Prop} (l : List α) (H : ∀ a
|
||||
(l.attachWith p H).map Subtype.val = l :=
|
||||
(attachWith_map_coe _ _ _).trans (List.map_id _)
|
||||
|
||||
theorem countP_attach (l : List α) (p : α → Bool) :
|
||||
l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
|
||||
simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attach_map_subtype_val]
|
||||
|
||||
theorem countP_attachWith {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) (q : α → Bool) :
|
||||
(l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
|
||||
simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attachWith_map_subtype_val]
|
||||
|
||||
@[simp]
|
||||
theorem count_attach [DecidableEq α] (l : List α) (a : {x // x ∈ l}) :
|
||||
l.attach.count a = l.count ↑a :=
|
||||
Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attach _ _
|
||||
|
||||
@[simp]
|
||||
theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
|
||||
(l.attachWith p H).count a = l.count ↑a :=
|
||||
Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attachWith _ _ _
|
||||
|
||||
@[simp]
|
||||
theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
|
||||
| ⟨a, h⟩ => by
|
||||
@@ -153,7 +166,7 @@ theorem mem_pmap_of_mem {p : α → Prop} {f : ∀ a, p a → β} {l H} {a} (h :
|
||||
exact ⟨a, h, rfl⟩
|
||||
|
||||
@[simp]
|
||||
theorem length_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : (pmap f l H).length = l.length := by
|
||||
theorem length_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : length (pmap f l H) = length l := by
|
||||
induction l
|
||||
· rfl
|
||||
· simp only [*, pmap, length]
|
||||
@@ -174,13 +187,6 @@ theorem pmap_ne_nil_iff {P : α → Prop} (f : (a : α) → P a → β) {xs : Li
|
||||
(H : ∀ (a : α), a ∈ xs → P a) : xs.pmap f H ≠ [] ↔ xs ≠ [] := by
|
||||
simp
|
||||
|
||||
theorem pmap_eq_self {l : List α} {p : α → Prop} (hp : ∀ (a : α), a ∈ l → p a)
|
||||
(f : (a : α) → p a → α) : l.pmap f hp = l ↔ ∀ a (h : a ∈ l), f a (hp a h) = a := by
|
||||
rw [pmap_eq_map_attach]
|
||||
conv => lhs; rhs; rw [← attach_map_subtype_val l]
|
||||
rw [map_inj_left]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
theorem attach_eq_nil_iff {l : List α} : l.attach = [] ↔ l = [] :=
|
||||
pmap_eq_nil_iff
|
||||
@@ -204,7 +210,7 @@ theorem attachWith_ne_nil_iff {l : List α} {P : α → Prop} {H : ∀ a ∈ l,
|
||||
|
||||
@[simp]
|
||||
theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
|
||||
(pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (mem_of_getElem? H) := by
|
||||
(pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (getElem?_mem H) := by
|
||||
induction l generalizing n with
|
||||
| nil => simp
|
||||
| cons hd tl hl =>
|
||||
@@ -220,7 +226,7 @@ theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h
|
||||
· simp_all
|
||||
|
||||
theorem get?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
|
||||
get? (pmap f l h) n = Option.pmap f (get? l n) fun x H => h x (mem_of_get? H) := by
|
||||
get? (pmap f l h) n = Option.pmap f (get? l n) fun x H => h x (get?_mem H) := by
|
||||
simp only [get?_eq_getElem?]
|
||||
simp [getElem?_pmap, h]
|
||||
|
||||
@@ -243,18 +249,18 @@ theorem get_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h :
|
||||
(hn : n < (pmap f l h).length) :
|
||||
get (pmap f l h) ⟨n, hn⟩ =
|
||||
f (get l ⟨n, @length_pmap _ _ p f l h ▸ hn⟩)
|
||||
(h _ (getElem_mem (@length_pmap _ _ p f l h ▸ hn))) := by
|
||||
(h _ (get_mem l n (@length_pmap _ _ p f l h ▸ hn))) := by
|
||||
simp only [get_eq_getElem]
|
||||
simp [getElem_pmap]
|
||||
|
||||
@[simp]
|
||||
theorem getElem?_attachWith {xs : List α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
|
||||
(xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (mem_of_getElem? a)) :=
|
||||
(xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (getElem?_mem a)) :=
|
||||
getElem?_pmap ..
|
||||
|
||||
@[simp]
|
||||
theorem getElem?_attach {xs : List α} {i : Nat} :
|
||||
xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => mem_of_getElem? a) :=
|
||||
xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => getElem?_mem a) :=
|
||||
getElem?_attachWith
|
||||
|
||||
@[simp]
|
||||
@@ -306,68 +312,12 @@ theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
|
||||
| nil => simp at h
|
||||
| cons x xs => simp [head_attach, h]
|
||||
|
||||
@[simp] theorem tail_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
|
||||
(H : ∀ (a : α), a ∈ xs → P a) :
|
||||
(xs.pmap f H).tail = xs.tail.pmap f (fun a h => H a (mem_of_mem_tail h)) := by
|
||||
cases xs <;> simp
|
||||
|
||||
@[simp] theorem tail_attachWith {P : α → Prop} {xs : List α}
|
||||
{H : ∀ (a : α), a ∈ xs → P a} :
|
||||
(xs.attachWith P H).tail = xs.tail.attachWith P (fun a h => H a (mem_of_mem_tail h)) := by
|
||||
cases xs <;> simp
|
||||
|
||||
@[simp] theorem tail_attach (xs : List α) :
|
||||
xs.attach.tail = xs.tail.attach.map (fun ⟨x, h⟩ => ⟨x, mem_of_mem_tail h⟩) := by
|
||||
cases xs <;> simp
|
||||
|
||||
theorem foldl_pmap (l : List α) {P : α → Prop} (f : (a : α) → P a → β)
|
||||
(H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
|
||||
(l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
|
||||
rw [pmap_eq_map_attach, foldl_map]
|
||||
|
||||
theorem foldr_pmap (l : List α) {P : α → Prop} (f : (a : α) → P a → β)
|
||||
(H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) :
|
||||
(l.pmap f H).foldr g x = l.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
|
||||
rw [pmap_eq_map_attach, foldr_map]
|
||||
|
||||
/--
|
||||
If we fold over `l.attach` with a function that ignores the membership predicate,
|
||||
we get the same results as folding over `l` directly.
|
||||
|
||||
This is useful when we need to use `attach` to show termination.
|
||||
|
||||
Unfortunately this can't be applied by `simp` because of the higher order unification problem,
|
||||
and even when rewriting we need to specify the function explicitly.
|
||||
See however `foldl_subtype` below.
|
||||
-/
|
||||
theorem foldl_attach (l : List α) (f : β → α → β) (b : β) :
|
||||
l.attach.foldl (fun acc t => f acc t.1) b = l.foldl f b := by
|
||||
induction l generalizing b with
|
||||
| nil => simp
|
||||
| cons a l ih => rw [foldl_cons, attach_cons, foldl_cons, foldl_map, ih]
|
||||
|
||||
/--
|
||||
If we fold over `l.attach` with a function that ignores the membership predicate,
|
||||
we get the same results as folding over `l` directly.
|
||||
|
||||
This is useful when we need to use `attach` to show termination.
|
||||
|
||||
Unfortunately this can't be applied by `simp` because of the higher order unification problem,
|
||||
and even when rewriting we need to specify the function explicitly.
|
||||
See however `foldr_subtype` below.
|
||||
-/
|
||||
theorem foldr_attach (l : List α) (f : α → β → β) (b : β) :
|
||||
l.attach.foldr (fun t acc => f t.1 acc) b = l.foldr f b := by
|
||||
induction l generalizing b with
|
||||
| nil => simp
|
||||
| cons a l ih => rw [foldr_cons, attach_cons, foldr_cons, foldr_map, ih]
|
||||
|
||||
theorem attach_map {l : List α} (f : α → β) :
|
||||
(l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
theorem attachWith_map {l : List α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ l.map f → P b} :
|
||||
(l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
|
||||
(l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun a h => H _ (mem_map_of_mem f h))).map
|
||||
fun ⟨x, h⟩ => ⟨f x, h⟩ := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
@@ -459,16 +409,16 @@ theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ l₂ :
|
||||
pmap_append f l₁ l₂ _
|
||||
|
||||
@[simp] theorem attach_append (xs ys : List α) :
|
||||
(xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_left ys h⟩) ++
|
||||
ys.attach.map fun ⟨x, h⟩ => ⟨x, mem_append_right xs h⟩ := by
|
||||
(xs ++ ys).attach = xs.attach.map (fun ⟨x, h⟩ => ⟨x, mem_append_of_mem_left ys h⟩) ++
|
||||
ys.attach.map fun ⟨x, h⟩ => ⟨x, mem_append_of_mem_right xs h⟩ := by
|
||||
simp only [attach, attachWith, pmap, map_pmap, pmap_append]
|
||||
congr 1 <;>
|
||||
exact pmap_congr_left _ fun _ _ _ _ => rfl
|
||||
|
||||
@[simp] theorem attachWith_append {P : α → Prop} {xs ys : List α}
|
||||
{H : ∀ (a : α), a ∈ xs ++ ys → P a} :
|
||||
(xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_left ys h)) ++
|
||||
ys.attachWith P (fun a h => H a (mem_append_right xs h)) := by
|
||||
(xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_of_mem_left ys h)) ++
|
||||
ys.attachWith P (fun a h => H a (mem_append_of_mem_right xs h)) := by
|
||||
simp only [attachWith, attach_append, map_pmap, pmap_append]
|
||||
|
||||
@[simp] theorem pmap_reverse {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
|
||||
@@ -542,164 +492,4 @@ theorem getLast_attach {xs : List α} (h : xs.attach ≠ []) :
|
||||
xs.attach.getLast h = ⟨xs.getLast (by simpa using h), getLast_mem (by simpa using h)⟩ := by
|
||||
simp only [getLast_eq_head_reverse, reverse_attach, head_map, head_attach]
|
||||
|
||||
@[simp]
|
||||
theorem countP_attach (l : List α) (p : α → Bool) :
|
||||
l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
|
||||
simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attach_map_subtype_val]
|
||||
|
||||
@[simp]
|
||||
theorem countP_attachWith {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) (q : α → Bool) :
|
||||
(l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
|
||||
simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attachWith_map_subtype_val]
|
||||
|
||||
@[simp]
|
||||
theorem count_attach [DecidableEq α] (l : List α) (a : {x // x ∈ l}) :
|
||||
l.attach.count a = l.count ↑a :=
|
||||
Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attach _ _
|
||||
|
||||
@[simp]
|
||||
theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
|
||||
(l.attachWith p H).count a = l.count ↑a :=
|
||||
Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attachWith _ _ _
|
||||
|
||||
/-! ## unattach
|
||||
|
||||
`List.unattach` is the (one-sided) inverse of `List.attach`. It is a synonym for `List.map Subtype.val`.
|
||||
|
||||
We use it by providing a simp lemma `l.attach.unattach = l`, and simp lemmas which recognize higher order
|
||||
functions applied to `l : List { x // p x }` which only depend on the value, not the predicate, and rewrite these
|
||||
in terms of a simpler function applied to `l.unattach`.
|
||||
|
||||
Further, we provide simp lemmas that push `unattach` inwards.
|
||||
-/
|
||||
|
||||
/--
|
||||
A synonym for `l.map (·.val)`. Mostly this should not be needed by users.
|
||||
It is introduced as an intermediate step by lemmas such as `map_subtype`,
|
||||
and is ideally subsequently simplified away by `unattach_attach`.
|
||||
|
||||
If not, usually the right approach is `simp [List.unattach, -List.map_subtype]` to unfold.
|
||||
-/
|
||||
def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) := l.map (·.val)
|
||||
|
||||
@[simp] theorem unattach_nil {p : α → Prop} : ([] : List { x // p x }).unattach = [] := rfl
|
||||
@[simp] theorem unattach_cons {p : α → Prop} {a : { x // p x }} {l : List { x // p x }} :
|
||||
(a :: l).unattach = a.val :: l.unattach := rfl
|
||||
|
||||
@[simp] theorem length_unattach {p : α → Prop} {l : List { x // p x }} :
|
||||
l.unattach.length = l.length := by
|
||||
unfold unattach
|
||||
simp
|
||||
|
||||
@[simp] theorem unattach_attach {l : List α} : l.attach.unattach = l := by
|
||||
unfold unattach
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, Function.comp_def]
|
||||
|
||||
@[simp] theorem unattach_attachWith {p : α → Prop} {l : List α}
|
||||
{H : ∀ a ∈ l, p a} :
|
||||
(l.attachWith p H).unattach = l := by
|
||||
unfold unattach
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, Function.comp_def]
|
||||
|
||||
@[simp] theorem getElem?_unattach {p : α → Prop} {l : List { x // p x }} (i : Nat) :
|
||||
l.unattach[i]? = l[i]?.map Subtype.val := by
|
||||
simp [unattach]
|
||||
|
||||
@[simp] theorem getElem_unattach
|
||||
{p : α → Prop} {l : List { x // p x }} (i : Nat) (h : i < l.unattach.length) :
|
||||
l.unattach[i] = (l[i]'(by simpa using h)).1 := by
|
||||
simp [unattach]
|
||||
|
||||
/-! ### Recognizing higher order functions on subtypes using a function that only depends on the value. -/
|
||||
|
||||
/--
|
||||
This lemma identifies folds over lists of subtypes, where the function only depends on the value, not the proposition,
|
||||
and simplifies these to the function directly taking the value.
|
||||
-/
|
||||
@[simp] theorem foldl_subtype {p : α → Prop} {l : List { x // p x }}
|
||||
{f : β → { x // p x } → β} {g : β → α → β} {x : β}
|
||||
{hf : ∀ b x h, f b ⟨x, h⟩ = g b x} :
|
||||
l.foldl f x = l.unattach.foldl g x := by
|
||||
unfold unattach
|
||||
induction l generalizing x with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, hf]
|
||||
|
||||
/--
|
||||
This lemma identifies folds over lists of subtypes, where the function only depends on the value, not the proposition,
|
||||
and simplifies these to the function directly taking the value.
|
||||
-/
|
||||
@[simp] theorem foldr_subtype {p : α → Prop} {l : List { x // p x }}
|
||||
{f : { x // p x } → β → β} {g : α → β → β} {x : β}
|
||||
{hf : ∀ x h b, f ⟨x, h⟩ b = g x b} :
|
||||
l.foldr f x = l.unattach.foldr g x := by
|
||||
unfold unattach
|
||||
induction l generalizing x with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, hf]
|
||||
|
||||
/--
|
||||
This lemma identifies maps over lists of subtypes, where the function only depends on the value, not the proposition,
|
||||
and simplifies these to the function directly taking the value.
|
||||
-/
|
||||
@[simp] theorem map_subtype {p : α → Prop} {l : List { x // p x }}
|
||||
{f : { x // p x } → β} {g : α → β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
l.map f = l.unattach.map g := by
|
||||
unfold unattach
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, hf]
|
||||
|
||||
@[simp] theorem filterMap_subtype {p : α → Prop} {l : List { x // p x }}
|
||||
{f : { x // p x } → Option β} {g : α → Option β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
l.filterMap f = l.unattach.filterMap g := by
|
||||
unfold unattach
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, hf, filterMap_cons]
|
||||
|
||||
@[simp] theorem flatMap_subtype {p : α → Prop} {l : List { x // p x }}
|
||||
{f : { x // p x } → List β} {g : α → List β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
(l.flatMap f) = l.unattach.flatMap g := by
|
||||
unfold unattach
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a l ih => simp [ih, hf]
|
||||
|
||||
@[deprecated flatMap_subtype (since := "2024-10-16")] abbrev bind_subtype := @flatMap_subtype
|
||||
|
||||
@[simp] theorem unattach_filter {p : α → Prop} {l : List { x // p x }}
|
||||
{f : { x // p x } → Bool} {g : α → Bool} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
(l.filter f).unattach = l.unattach.filter g := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a l ih =>
|
||||
simp only [filter_cons, hf, unattach_cons]
|
||||
split <;> simp [ih]
|
||||
|
||||
/-! ### Simp lemmas pushing `unattach` inwards. -/
|
||||
|
||||
@[simp] theorem unattach_reverse {p : α → Prop} {l : List { x // p x }} :
|
||||
l.reverse.unattach = l.unattach.reverse := by
|
||||
simp [unattach, -map_subtype]
|
||||
|
||||
@[simp] theorem unattach_append {p : α → Prop} {l₁ l₂ : List { x // p x }} :
|
||||
(l₁ ++ l₂).unattach = l₁.unattach ++ l₂.unattach := by
|
||||
simp [unattach, -map_subtype]
|
||||
|
||||
@[simp] theorem unattach_flatten {p : α → Prop} {l : List (List { x // p x })} :
|
||||
l.flatten.unattach = (l.map unattach).flatten := by
|
||||
unfold unattach
|
||||
induction l <;> simp_all
|
||||
|
||||
@[deprecated unattach_flatten (since := "2024-10-14")] abbrev unattach_join := @unattach_flatten
|
||||
|
||||
@[simp] theorem unattach_replicate {p : α → Prop} {n : Nat} {x : { x // p x }} :
|
||||
(List.replicate n x).unattach = List.replicate n x.1 := by
|
||||
simp [unattach, -map_subtype]
|
||||
|
||||
end List
|
||||
|
||||
@@ -29,23 +29,22 @@ The operations are organized as follow:
|
||||
* Lexicographic ordering: `lt`, `le`, and instances.
|
||||
* Head and tail operators: `head`, `head?`, `headD?`, `tail`, `tail?`, `tailD`.
|
||||
* Basic operations:
|
||||
`map`, `filter`, `filterMap`, `foldr`, `append`, `flatten`, `pure`, `flatMap`, `replicate`, and
|
||||
`map`, `filter`, `filterMap`, `foldr`, `append`, `join`, `pure`, `bind`, `replicate`, and
|
||||
`reverse`.
|
||||
* Additional functions defined in terms of these: `leftpad`, `rightPad`, and `reduceOption`.
|
||||
* Operations using indexes: `mapIdx`.
|
||||
* List membership: `isEmpty`, `elem`, `contains`, `mem` (and the `∈` notation),
|
||||
and decidability for predicates quantifying over membership in a `List`.
|
||||
* Sublists: `take`, `drop`, `takeWhile`, `dropWhile`, `partition`, `dropLast`,
|
||||
`isPrefixOf`, `isPrefixOf?`, `isSuffixOf`, `isSuffixOf?`, `Subset`, `Sublist`,
|
||||
`rotateLeft` and `rotateRight`.
|
||||
* Manipulating elements: `replace`, `modify`, `insert`, `insertIdx`, `erase`, `eraseP`, `eraseIdx`.
|
||||
* Manipulating elements: `replace`, `insert`, `erase`, `eraseP`, `eraseIdx`.
|
||||
* Finding elements: `find?`, `findSome?`, `findIdx`, `indexOf`, `findIdx?`, `indexOf?`,
|
||||
`countP`, `count`, and `lookup`.
|
||||
* Logic: `any`, `all`, `or`, and `and`.
|
||||
* Zippers: `zipWith`, `zip`, `zipWithAll`, and `unzip`.
|
||||
* Ranges and enumeration: `range`, `iota`, `enumFrom`, and `enum`.
|
||||
* Minima and maxima: `min?` and `max?`.
|
||||
* Other functions: `intersperse`, `intercalate`, `eraseDups`, `eraseReps`, `span`, `splitBy`,
|
||||
* Minima and maxima: `minimum?` and `maximum?`.
|
||||
* Other functions: `intersperse`, `intercalate`, `eraseDups`, `eraseReps`, `span`, `groupBy`,
|
||||
`removeAll`
|
||||
(currently these functions are mostly only used in meta code,
|
||||
and do not have API suitable for verification).
|
||||
@@ -122,11 +121,6 @@ protected def beq [BEq α] : List α → List α → Bool
|
||||
| a::as, b::bs => a == b && List.beq as bs
|
||||
| _, _ => false
|
||||
|
||||
@[simp] theorem beq_nil_nil [BEq α] : List.beq ([] : List α) ([] : List α) = true := rfl
|
||||
@[simp] theorem beq_cons_nil [BEq α] (a : α) (as : List α) : List.beq (a::as) [] = false := rfl
|
||||
@[simp] theorem beq_nil_cons [BEq α] (a : α) (as : List α) : List.beq [] (a::as) = false := rfl
|
||||
theorem beq_cons₂ [BEq α] (a b : α) (as bs : List α) : List.beq (a::as) (b::bs) = (a == b && List.beq as bs) := rfl
|
||||
|
||||
instance [BEq α] : BEq (List α) := ⟨List.beq⟩
|
||||
|
||||
instance [BEq α] [LawfulBEq α] : LawfulBEq (List α) where
|
||||
@@ -224,14 +218,14 @@ def get? : (as : List α) → (i : Nat) → Option α
|
||||
|
||||
theorem ext_get? : ∀ {l₁ l₂ : List α}, (∀ n, l₁.get? n = l₂.get? n) → l₁ = l₂
|
||||
| [], [], _ => rfl
|
||||
| _ :: _, [], h => nomatch h 0
|
||||
| [], _ :: _, h => nomatch h 0
|
||||
| a :: l₁, [], h => nomatch h 0
|
||||
| [], a' :: l₂, h => nomatch h 0
|
||||
| a :: l₁, a' :: l₂, h => by
|
||||
have h0 : some a = some a' := h 0
|
||||
injection h0 with aa; simp only [aa, ext_get? fun n => h (n+1)]
|
||||
|
||||
/-- Deprecated alias for `ext_get?`. The preferred extensionality theorem is now `ext_getElem?`. -/
|
||||
@[deprecated ext_get? (since := "2024-06-07")] abbrev ext := @ext_get?
|
||||
@[deprecated (since := "2024-06-07")] abbrev ext := @ext_get?
|
||||
|
||||
/-! ### getD -/
|
||||
|
||||
@@ -374,7 +368,7 @@ def tailD (list fallback : List α) : List α :=
|
||||
/-! ## Basic `List` operations.
|
||||
|
||||
We define the basic functional programming operations on `List`:
|
||||
`map`, `filter`, `filterMap`, `foldr`, `append`, `flatten`, `pure`, `bind`, `replicate`, and `reverse`.
|
||||
`map`, `filter`, `filterMap`, `foldr`, `append`, `join`, `pure`, `bind`, `replicate`, and `reverse`.
|
||||
-/
|
||||
|
||||
/-! ### map -/
|
||||
@@ -548,53 +542,41 @@ theorem reverseAux_eq_append (as bs : List α) : reverseAux as bs = reverseAux a
|
||||
simp [reverse, reverseAux]
|
||||
rw [← reverseAux_eq_append]
|
||||
|
||||
/-! ### flatten -/
|
||||
/-! ### join -/
|
||||
|
||||
/--
|
||||
`O(|flatten L|)`. `flatten L` concatenates all the lists in `L` into one list.
|
||||
* `flatten [[a], [], [b, c], [d, e, f]] = [a, b, c, d, e, f]`
|
||||
`O(|join L|)`. `join L` concatenates all the lists in `L` into one list.
|
||||
* `join [[a], [], [b, c], [d, e, f]] = [a, b, c, d, e, f]`
|
||||
-/
|
||||
def flatten : List (List α) → List α
|
||||
def join : List (List α) → List α
|
||||
| [] => []
|
||||
| a :: as => a ++ flatten as
|
||||
| a :: as => a ++ join as
|
||||
|
||||
@[simp] theorem flatten_nil : List.flatten ([] : List (List α)) = [] := rfl
|
||||
@[simp] theorem flatten_cons : (l :: ls).flatten = l ++ ls.flatten := rfl
|
||||
@[simp] theorem join_nil : List.join ([] : List (List α)) = [] := rfl
|
||||
@[simp] theorem join_cons : (l :: ls).join = l ++ ls.join := rfl
|
||||
|
||||
@[deprecated flatten (since := "2024-10-14"), inherit_doc flatten] abbrev join := @flatten
|
||||
/-! ### pure -/
|
||||
|
||||
/-! ### singleton -/
|
||||
/-- `pure x = [x]` is the `pure` operation of the list monad. -/
|
||||
@[inline] protected def pure {α : Type u} (a : α) : List α := [a]
|
||||
|
||||
/-- `singleton x = [x]`. -/
|
||||
@[inline] protected def singleton {α : Type u} (a : α) : List α := [a]
|
||||
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated singleton (since := "2024-10-16")] protected abbrev pure := @singleton
|
||||
|
||||
/-! ### flatMap -/
|
||||
/-! ### bind -/
|
||||
|
||||
/--
|
||||
`flatMap xs f` applies `f` to each element of `xs`
|
||||
`bind xs f` is the bind operation of the list monad. It applies `f` to each element of `xs`
|
||||
to get a list of lists, and then concatenates them all together.
|
||||
* `[2, 3, 2].bind range = [0, 1, 0, 1, 2, 0, 1]`
|
||||
-/
|
||||
@[inline] def flatMap {α : Type u} {β : Type v} (a : List α) (b : α → List β) : List β := flatten (map b a)
|
||||
@[inline] protected def bind {α : Type u} {β : Type v} (a : List α) (b : α → List β) : List β := join (map b a)
|
||||
|
||||
@[simp] theorem flatMap_nil (f : α → List β) : List.flatMap [] f = [] := by simp [flatten, List.flatMap]
|
||||
@[simp] theorem flatMap_cons x xs (f : α → List β) :
|
||||
List.flatMap (x :: xs) f = f x ++ List.flatMap xs f := by simp [flatten, List.flatMap]
|
||||
@[simp] theorem bind_nil (f : α → List β) : List.bind [] f = [] := by simp [join, List.bind]
|
||||
@[simp] theorem bind_cons x xs (f : α → List β) :
|
||||
List.bind (x :: xs) f = f x ++ List.bind xs f := by simp [join, List.bind]
|
||||
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated flatMap (since := "2024-10-16")] abbrev bind := @flatMap
|
||||
@[deprecated bind_nil (since := "2024-06-15")] abbrev nil_bind := @bind_nil
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated flatMap_nil (since := "2024-10-16")] abbrev nil_flatMap := @flatMap_nil
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated flatMap_cons (since := "2024-10-16")] abbrev cons_flatMap := @flatMap_cons
|
||||
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated flatMap_nil (since := "2024-06-15")] abbrev nil_bind := @flatMap_nil
|
||||
set_option linter.missingDocs false in
|
||||
@[deprecated flatMap_cons (since := "2024-06-15")] abbrev cons_bind := @flatMap_cons
|
||||
@[deprecated bind_cons (since := "2024-06-15")] abbrev cons_bind := @bind_cons
|
||||
|
||||
/-! ### replicate -/
|
||||
|
||||
@@ -682,7 +664,7 @@ theorem elem_cons [BEq α] {a : α} :
|
||||
(b::bs).elem a = match a == b with | true => true | false => bs.elem a := rfl
|
||||
|
||||
/-- `notElem a l` is `!(elem a l)`. -/
|
||||
@[deprecated "Use `!(elem a l)` instead."(since := "2024-06-15")]
|
||||
@[deprecated (since := "2024-06-15")]
|
||||
def notElem [BEq α] (a : α) (as : List α) : Bool :=
|
||||
!(as.elem a)
|
||||
|
||||
@@ -726,13 +708,13 @@ theorem elem_eq_true_of_mem [BEq α] [LawfulBEq α] {a : α} {as : List α} (h :
|
||||
instance [BEq α] [LawfulBEq α] (a : α) (as : List α) : Decidable (a ∈ as) :=
|
||||
decidable_of_decidable_of_iff (Iff.intro mem_of_elem_eq_true elem_eq_true_of_mem)
|
||||
|
||||
theorem mem_append_left {a : α} {as : List α} (bs : List α) : a ∈ as → a ∈ as ++ bs := by
|
||||
theorem mem_append_of_mem_left {a : α} {as : List α} (bs : List α) : a ∈ as → a ∈ as ++ bs := by
|
||||
intro h
|
||||
induction h with
|
||||
| head => apply Mem.head
|
||||
| tail => apply Mem.tail; assumption
|
||||
|
||||
theorem mem_append_right {b : α} {bs : List α} (as : List α) : b ∈ bs → b ∈ as ++ bs := by
|
||||
theorem mem_append_of_mem_right {b : α} {bs : List α} (as : List α) : b ∈ bs → b ∈ as ++ bs := by
|
||||
intro h
|
||||
induction as with
|
||||
| nil => simp [h]
|
||||
@@ -1113,50 +1095,12 @@ theorem replace_cons [BEq α] {a : α} :
|
||||
(a::as).replace b c = match b == a with | true => c::as | false => a :: replace as b c :=
|
||||
rfl
|
||||
|
||||
/-! ### modify -/
|
||||
|
||||
/--
|
||||
Apply a function to the nth tail of `l`. Returns the input without
|
||||
using `f` if the index is larger than the length of the List.
|
||||
```
|
||||
modifyTailIdx f 2 [a, b, c] = [a, b] ++ f [c]
|
||||
```
|
||||
-/
|
||||
@[simp] def modifyTailIdx (f : List α → List α) : Nat → List α → List α
|
||||
| 0, l => f l
|
||||
| _+1, [] => []
|
||||
| n+1, a :: l => a :: modifyTailIdx f n l
|
||||
|
||||
/-- Apply `f` to the head of the list, if it exists. -/
|
||||
@[inline] def modifyHead (f : α → α) : List α → List α
|
||||
| [] => []
|
||||
| a :: l => f a :: l
|
||||
|
||||
@[simp] theorem modifyHead_nil (f : α → α) : [].modifyHead f = [] := by rw [modifyHead]
|
||||
@[simp] theorem modifyHead_cons (a : α) (l : List α) (f : α → α) :
|
||||
(a :: l).modifyHead f = f a :: l := by rw [modifyHead]
|
||||
|
||||
/--
|
||||
Apply `f` to the nth element of the list, if it exists, replacing that element with the result.
|
||||
-/
|
||||
def modify (f : α → α) : Nat → List α → List α :=
|
||||
modifyTailIdx (modifyHead f)
|
||||
|
||||
/-! ### insert -/
|
||||
|
||||
/-- Inserts an element into a list without duplication. -/
|
||||
@[inline] protected def insert [BEq α] (a : α) (l : List α) : List α :=
|
||||
if l.elem a then l else a :: l
|
||||
|
||||
/--
|
||||
`insertIdx n a l` inserts `a` into the list `l` after the first `n` elements of `l`
|
||||
```
|
||||
insertIdx 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]
|
||||
```
|
||||
-/
|
||||
def insertIdx (n : Nat) (a : α) : List α → List α :=
|
||||
modifyTailIdx (cons a) n
|
||||
|
||||
/-! ### erase -/
|
||||
|
||||
/--
|
||||
@@ -1427,10 +1371,10 @@ def zipWithAll (f : Option α → Option β → γ) : List α → List β → Li
|
||||
| a :: as, [] => (a :: as).map fun a => f (some a) none
|
||||
| a :: as, b :: bs => f a b :: zipWithAll f as bs
|
||||
|
||||
@[simp] theorem zipWithAll_nil :
|
||||
@[simp] theorem zipWithAll_nil_right :
|
||||
zipWithAll f as [] = as.map fun a => f (some a) none := by
|
||||
cases as <;> rfl
|
||||
@[simp] theorem nil_zipWithAll :
|
||||
@[simp] theorem zipWithAll_nil_left :
|
||||
zipWithAll f [] bs = bs.map fun b => f none (some b) := rfl
|
||||
@[simp] theorem zipWithAll_cons_cons :
|
||||
zipWithAll f (a :: as) (b :: bs) = f (some a) (some b) :: zipWithAll f as bs := rfl
|
||||
@@ -1451,25 +1395,12 @@ def unzip : List (α × β) → List α × List β
|
||||
|
||||
/-! ## Ranges and enumeration -/
|
||||
|
||||
/-- Sum of a list.
|
||||
|
||||
`List.sum [a, b, c] = a + (b + (c + 0))` -/
|
||||
def sum {α} [Add α] [Zero α] : List α → α :=
|
||||
foldr (· + ·) 0
|
||||
|
||||
@[simp] theorem sum_nil [Add α] [Zero α] : ([] : List α).sum = 0 := rfl
|
||||
@[simp] theorem sum_cons [Add α] [Zero α] {a : α} {l : List α} : (a::l).sum = a + l.sum := rfl
|
||||
|
||||
/-- Sum of a list of natural numbers. -/
|
||||
@[deprecated List.sum (since := "2024-10-17")]
|
||||
-- This is not in the `List` namespace as later `List.sum` will be defined polymorphically.
|
||||
protected def _root_.Nat.sum (l : List Nat) : Nat := l.foldr (·+·) 0
|
||||
|
||||
set_option linter.deprecated false in
|
||||
@[simp, deprecated sum_nil (since := "2024-10-17")]
|
||||
theorem _root_.Nat.sum_nil : Nat.sum ([] : List Nat) = 0 := rfl
|
||||
set_option linter.deprecated false in
|
||||
@[simp, deprecated sum_cons (since := "2024-10-17")]
|
||||
theorem _root_.Nat.sum_cons (a : Nat) (l : List Nat) :
|
||||
@[simp] theorem _root_.Nat.sum_nil : Nat.sum ([] : List Nat) = 0 := rfl
|
||||
@[simp] theorem _root_.Nat.sum_cons (a : Nat) (l : List Nat) :
|
||||
Nat.sum (a::l) = a + Nat.sum l := rfl
|
||||
|
||||
/-! ### range -/
|
||||
@@ -1533,34 +1464,30 @@ def enum : List α → List (Nat × α) := enumFrom 0
|
||||
|
||||
/-! ## Minima and maxima -/
|
||||
|
||||
/-! ### min? -/
|
||||
/-! ### minimum? -/
|
||||
|
||||
/--
|
||||
Returns the smallest element of the list, if it is not empty.
|
||||
* `[].min? = none`
|
||||
* `[4].min? = some 4`
|
||||
* `[1, 4, 2, 10, 6].min? = some 1`
|
||||
* `[].minimum? = none`
|
||||
* `[4].minimum? = some 4`
|
||||
* `[1, 4, 2, 10, 6].minimum? = some 1`
|
||||
-/
|
||||
def min? [Min α] : List α → Option α
|
||||
def minimum? [Min α] : List α → Option α
|
||||
| [] => none
|
||||
| a::as => some <| as.foldl min a
|
||||
|
||||
@[inherit_doc min?, deprecated min? (since := "2024-09-29")] abbrev minimum? := @min?
|
||||
|
||||
/-! ### max? -/
|
||||
/-! ### maximum? -/
|
||||
|
||||
/--
|
||||
Returns the largest element of the list, if it is not empty.
|
||||
* `[].max? = none`
|
||||
* `[4].max? = some 4`
|
||||
* `[1, 4, 2, 10, 6].max? = some 10`
|
||||
* `[].maximum? = none`
|
||||
* `[4].maximum? = some 4`
|
||||
* `[1, 4, 2, 10, 6].maximum? = some 10`
|
||||
-/
|
||||
def max? [Max α] : List α → Option α
|
||||
def maximum? [Max α] : List α → Option α
|
||||
| [] => none
|
||||
| a::as => some <| as.foldl max a
|
||||
|
||||
@[inherit_doc max?, deprecated max? (since := "2024-09-29")] abbrev maximum? := @max?
|
||||
|
||||
/-! ## Other list operations
|
||||
|
||||
The functions are currently mostly used in meta code,
|
||||
@@ -1596,7 +1523,7 @@ def intersperse (sep : α) : List α → List α
|
||||
* `intercalate sep [a, b, c] = a ++ sep ++ b ++ sep ++ c`
|
||||
-/
|
||||
def intercalate (sep : List α) (xs : List (List α)) : List α :=
|
||||
(intersperse sep xs).flatten
|
||||
join (intersperse sep xs)
|
||||
|
||||
/-! ### eraseDups -/
|
||||
|
||||
@@ -1648,23 +1575,23 @@ where
|
||||
| true => loop as (a::rs)
|
||||
| false => (rs.reverse, a::as)
|
||||
|
||||
/-! ### splitBy -/
|
||||
/-! ### groupBy -/
|
||||
|
||||
/--
|
||||
`O(|l|)`. `splitBy R l` splits `l` into chains of elements
|
||||
`O(|l|)`. `groupBy R l` splits `l` into chains of elements
|
||||
such that adjacent elements are related by `R`.
|
||||
|
||||
* `splitBy (·==·) [1, 1, 2, 2, 2, 3, 2] = [[1, 1], [2, 2, 2], [3], [2]]`
|
||||
* `splitBy (·<·) [1, 2, 5, 4, 5, 1, 4] = [[1, 2, 5], [4, 5], [1, 4]]`
|
||||
* `groupBy (·==·) [1, 1, 2, 2, 2, 3, 2] = [[1, 1], [2, 2, 2], [3], [2]]`
|
||||
* `groupBy (·<·) [1, 2, 5, 4, 5, 1, 4] = [[1, 2, 5], [4, 5], [1, 4]]`
|
||||
-/
|
||||
@[specialize] def splitBy (R : α → α → Bool) : List α → List (List α)
|
||||
@[specialize] def groupBy (R : α → α → Bool) : List α → List (List α)
|
||||
| [] => []
|
||||
| a::as => loop as a [] []
|
||||
where
|
||||
/--
|
||||
The arguments of `splitBy.loop l ag g gs` represent the following:
|
||||
The arguments of `groupBy.loop l ag g gs` represent the following:
|
||||
|
||||
- `l : List α` are the elements which we still need to split.
|
||||
- `l : List α` are the elements which we still need to group.
|
||||
- `ag : α` is the previous element for which a comparison was performed.
|
||||
- `g : List α` is the group currently being assembled, in **reverse order**.
|
||||
- `gs : List (List α)` is all of the groups that have been completed, in **reverse order**.
|
||||
@@ -1675,8 +1602,6 @@ where
|
||||
| false => loop as a [] ((ag::g).reverse::gs)
|
||||
| [], ag, g, gs => ((ag::g).reverse::gs).reverse
|
||||
|
||||
@[deprecated splitBy (since := "2024-10-30"), inherit_doc splitBy] abbrev groupBy := @splitBy
|
||||
|
||||
/-! ### removeAll -/
|
||||
|
||||
/-- `O(|xs|)`. Computes the "set difference" of lists,
|
||||
|
||||
@@ -232,12 +232,11 @@ theorem sizeOf_get [SizeOf α] (as : List α) (i : Fin as.length) : sizeOf (as.g
|
||||
apply Nat.lt_trans ih
|
||||
simp_arith
|
||||
|
||||
theorem le_antisymm [LT α] [s : Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
{as bs : List α} (h₁ : as ≤ bs) (h₂ : bs ≤ as) : as = bs :=
|
||||
theorem le_antisymm [LT α] [s : Antisymm (¬ · < · : α → α → Prop)] {as bs : List α} (h₁ : as ≤ bs) (h₂ : bs ≤ as) : as = bs :=
|
||||
match as, bs with
|
||||
| [], [] => rfl
|
||||
| [], _::_ => False.elim <| h₂ (List.lt.nil ..)
|
||||
| _::_, [] => False.elim <| h₁ (List.lt.nil ..)
|
||||
| [], b::bs => False.elim <| h₂ (List.lt.nil ..)
|
||||
| a::as, [] => False.elim <| h₁ (List.lt.nil ..)
|
||||
| a::as, b::bs => by
|
||||
by_cases hab : a < b
|
||||
· exact False.elim <| h₂ (List.lt.head _ _ hab)
|
||||
@@ -249,8 +248,7 @@ theorem le_antisymm [LT α] [s : Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
have : a = b := s.antisymm hab hba
|
||||
simp [this, ih]
|
||||
|
||||
instance [LT α] [Std.Antisymm (¬ · < · : α → α → Prop)] :
|
||||
Std.Antisymm (· ≤ · : List α → List α → Prop) where
|
||||
instance [LT α] [Antisymm (¬ · < · : α → α → Prop)] : Antisymm (· ≤ · : List α → List α → Prop) where
|
||||
antisymm h₁ h₂ := le_antisymm h₁ h₂
|
||||
|
||||
end List
|
||||
|
||||
@@ -5,8 +5,6 @@ Author: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import Init.Control.Basic
|
||||
import Init.Control.Id
|
||||
import Init.Control.Lawful
|
||||
import Init.Data.List.Basic
|
||||
|
||||
namespace List
|
||||
@@ -209,16 +207,6 @@ def findM? {m : Type → Type u} [Monad m] {α : Type} (p : α → m Bool) : Lis
|
||||
| true => pure (some a)
|
||||
| false => findM? p as
|
||||
|
||||
@[simp]
|
||||
theorem findM?_id (p : α → Bool) (as : List α) : findM? (m := Id) p as = as.find? p := by
|
||||
induction as with
|
||||
| nil => rfl
|
||||
| cons a as ih =>
|
||||
simp only [findM?, find?]
|
||||
cases p a with
|
||||
| true => rfl
|
||||
| false => rw [ih]; rfl
|
||||
|
||||
@[specialize]
|
||||
def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f : α → m (Option β)) : List α → m (Option β)
|
||||
| [] => pure none
|
||||
@@ -227,27 +215,26 @@ def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f
|
||||
| some b => pure (some b)
|
||||
| none => findSomeM? f as
|
||||
|
||||
@[simp]
|
||||
theorem findSomeM?_id (f : α → Option β) (as : List α) : findSomeM? (m := Id) f as = as.findSome? f := by
|
||||
induction as with
|
||||
| nil => rfl
|
||||
| cons a as ih =>
|
||||
simp only [findSomeM?, findSome?]
|
||||
cases f a with
|
||||
| some b => rfl
|
||||
| none => rw [ih]; rfl
|
||||
@[inline] protected def forIn {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : List α) (init : β) (f : α → β → m (ForInStep β)) : m β :=
|
||||
let rec @[specialize] loop
|
||||
| [], b => pure b
|
||||
| a::as, b => do
|
||||
match (← f a b) with
|
||||
| ForInStep.done b => pure b
|
||||
| ForInStep.yield b => loop as b
|
||||
loop as init
|
||||
|
||||
theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
|
||||
as.findM? p = as.findSomeM? fun a => return if (← p a) then some a else none := by
|
||||
induction as with
|
||||
| nil => rfl
|
||||
| cons a as ih =>
|
||||
simp only [findM?, findSomeM?]
|
||||
simp [ih]
|
||||
congr
|
||||
apply funext
|
||||
intro b
|
||||
cases b <;> simp
|
||||
instance : ForIn m (List α) α where
|
||||
forIn := List.forIn
|
||||
|
||||
@[simp] theorem forIn_eq_forIn [Monad m] : @List.forIn α β m _ = forIn := rfl
|
||||
|
||||
@[simp] theorem forIn_nil [Monad m] (f : α → β → m (ForInStep β)) (b : β) : forIn [] b f = pure b :=
|
||||
rfl
|
||||
|
||||
@[simp] theorem forIn_cons [Monad m] (f : α → β → m (ForInStep β)) (a : α) (as : List α) (b : β)
|
||||
: forIn (a::as) b f = f a b >>= fun | ForInStep.done b => pure b | ForInStep.yield b => forIn as b f :=
|
||||
rfl
|
||||
|
||||
@[inline] protected def forIn' {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : List α) (init : β) (f : (a : α) → a ∈ as → β → m (ForInStep β)) : m β :=
|
||||
let rec @[specialize] loop : (as' : List α) → (b : β) → Exists (fun bs => bs ++ as' = as) → m β
|
||||
@@ -256,7 +243,7 @@ theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as :
|
||||
have : a ∈ as := by
|
||||
have ⟨bs, h⟩ := h
|
||||
subst h
|
||||
exact mem_append_right _ (Mem.head ..)
|
||||
exact mem_append_of_mem_right _ (Mem.head ..)
|
||||
match (← f a this b) with
|
||||
| ForInStep.done b => pure b
|
||||
| ForInStep.yield b =>
|
||||
@@ -267,15 +254,14 @@ theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as :
|
||||
instance : ForIn' m (List α) α inferInstance where
|
||||
forIn' := List.forIn'
|
||||
|
||||
-- No separate `ForIn` instance is required because it can be derived from `ForIn'`.
|
||||
|
||||
@[simp] theorem forIn'_eq_forIn' [Monad m] : @List.forIn' α β m _ = forIn' := rfl
|
||||
|
||||
@[simp] theorem forIn'_nil [Monad m] (f : (a : α) → a ∈ [] → β → m (ForInStep β)) (b : β) : forIn' [] b f = pure b :=
|
||||
rfl
|
||||
|
||||
@[simp] theorem forIn_nil [Monad m] (f : α → β → m (ForInStep β)) (b : β) : forIn [] b f = pure b :=
|
||||
rfl
|
||||
@[simp] theorem forIn'_eq_forIn {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : List α) (init : β) (f : α → β → m (ForInStep β)) : forIn' as init (fun a _ b => f a b) = forIn as init f := by
|
||||
simp [forIn', forIn, List.forIn, List.forIn']
|
||||
have : ∀ cs h, List.forIn'.loop cs (fun a _ b => f a b) as init h = List.forIn.loop f as init := by
|
||||
intro cs h
|
||||
induction as generalizing cs init with
|
||||
| nil => intros; rfl
|
||||
| cons a as ih => intros; simp [List.forIn.loop, List.forIn'.loop, ih]
|
||||
apply this
|
||||
|
||||
instance : ForM m (List α) α where
|
||||
forM := List.forM
|
||||
|
||||
@@ -115,13 +115,6 @@ theorem IsPrefix.countP_le (s : l₁ <+: l₂) : countP p l₁ ≤ countP p l₂
|
||||
theorem IsSuffix.countP_le (s : l₁ <:+ l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
|
||||
theorem IsInfix.countP_le (s : l₁ <:+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
|
||||
|
||||
-- See `Init.Data.List.Nat.Count` for `Sublist.le_countP : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁`.
|
||||
|
||||
theorem countP_tail_le (l) : countP p l.tail ≤ countP p l :=
|
||||
(tail_sublist l).countP_le _
|
||||
|
||||
-- See `Init.Data.List.Nat.Count` for `le_countP_tail : countP p l - 1 ≤ countP p l.tail`.
|
||||
|
||||
theorem countP_filter (l : List α) :
|
||||
countP p (filter q l) = countP (fun a => p a && q a) l := by
|
||||
simp only [countP_eq_length_filter, filter_filter]
|
||||
@@ -153,15 +146,13 @@ theorem countP_filterMap (p : β → Bool) (f : α → Option β) (l : List α)
|
||||
simp only [length_filterMap_eq_countP]
|
||||
congr
|
||||
ext a
|
||||
simp +contextual [Option.getD_eq_iff, Option.isSome_eq_isSome]
|
||||
simp (config := { contextual := true }) [Option.getD_eq_iff]
|
||||
|
||||
@[simp] theorem countP_flatten (l : List (List α)) :
|
||||
countP p l.flatten = (l.map (countP p)).sum := by
|
||||
simp only [countP_eq_length_filter, filter_flatten]
|
||||
@[simp] theorem countP_join (l : List (List α)) :
|
||||
countP p l.join = Nat.sum (l.map (countP p)) := by
|
||||
simp only [countP_eq_length_filter, filter_join]
|
||||
simp [countP_eq_length_filter']
|
||||
|
||||
@[deprecated countP_flatten (since := "2024-10-14")] abbrev countP_join := @countP_flatten
|
||||
|
||||
@[simp] theorem countP_reverse (l : List α) : countP p l.reverse = countP p l := by
|
||||
simp [countP_eq_length_filter, filter_reverse]
|
||||
|
||||
@@ -216,13 +207,6 @@ theorem IsPrefix.count_le (h : l₁ <+: l₂) (a : α) : count a l₁ ≤ count
|
||||
theorem IsSuffix.count_le (h : l₁ <:+ l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
|
||||
theorem IsInfix.count_le (h : l₁ <:+: l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
|
||||
|
||||
-- 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) : count a l.tail ≤ count a l :=
|
||||
(tail_sublist l).count_le _
|
||||
|
||||
-- See `Init.Data.List.Nat.Count` for `le_count_tail : count a l - 1 ≤ count a l.tail`.
|
||||
|
||||
theorem count_le_count_cons (a b : α) (l : List α) : count a l ≤ count a (b :: l) :=
|
||||
(sublist_cons_self _ _).count_le _
|
||||
|
||||
@@ -232,10 +216,8 @@ theorem count_singleton (a b : α) : count a [b] = if b == a then 1 else 0 := by
|
||||
@[simp] theorem count_append (a : α) : ∀ l₁ l₂, count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
|
||||
countP_append _
|
||||
|
||||
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']
|
||||
|
||||
@[deprecated count_flatten (since := "2024-10-14")] abbrev count_join := @count_flatten
|
||||
theorem count_join (a : α) (l : List (List α)) : count a l.join = Nat.sum (l.map (count a)) := by
|
||||
simp only [count_eq_countP, countP_join, count_eq_countP']
|
||||
|
||||
@[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]
|
||||
@@ -315,7 +297,7 @@ theorem replicate_count_eq_of_count_eq_length {l : List α} (h : count a l = len
|
||||
theorem count_le_count_map [DecidableEq β] (l : List α) (f : α → β) (x : α) :
|
||||
count x l ≤ count (f x) (map f l) := by
|
||||
rw [count, count, countP_map]
|
||||
apply countP_mono_left; simp +contextual
|
||||
apply countP_mono_left; simp (config := { contextual := true })
|
||||
|
||||
theorem count_filterMap {α} [BEq β] (b : β) (f : α → Option β) (l : List α) :
|
||||
count b (filterMap f l) = countP (fun a => f a == some b) l := by
|
||||
|
||||
@@ -52,9 +52,9 @@ theorem eraseP_of_forall_not {l : List α} (h : ∀ a, a ∈ l → ¬p a) : l.er
|
||||
theorem eraseP_ne_nil {xs : List α} {p : α → Bool} : xs.eraseP p ≠ [] ↔ xs ≠ [] ∧ ∀ x, p x → xs ≠ [x] := by
|
||||
simp
|
||||
|
||||
theorem exists_of_eraseP : ∀ {l : List α} {a} (_ : a ∈ l) (_ : p a),
|
||||
theorem exists_of_eraseP : ∀ {l : List α} {a} (al : a ∈ l) (pa : p a),
|
||||
∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.eraseP p = l₁ ++ l₂
|
||||
| b :: l, _, al, pa =>
|
||||
| b :: l, a, al, pa =>
|
||||
if pb : p b then
|
||||
⟨b, [], l, forall_mem_nil _, pb, by simp [pb]⟩
|
||||
else
|
||||
@@ -109,10 +109,6 @@ protected theorem Sublist.eraseP : l₁ <+ l₂ → l₁.eraseP p <+ l₂.eraseP
|
||||
theorem length_eraseP_le (l : List α) : (l.eraseP p).length ≤ l.length :=
|
||||
l.eraseP_sublist.length_le
|
||||
|
||||
theorem le_length_eraseP (l : List α) : l.length - 1 ≤ (l.eraseP p).length := by
|
||||
rw [length_eraseP]
|
||||
split <;> simp
|
||||
|
||||
theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (eraseP_subset _ ·)
|
||||
|
||||
@[simp] theorem mem_eraseP_of_neg {l : List α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
|
||||
@@ -168,8 +164,8 @@ theorem eraseP_append_left {a : α} (pa : p a) :
|
||||
|
||||
theorem eraseP_append_right :
|
||||
∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁++l₂) = l₁ ++ l₂.eraseP p
|
||||
| [], _, _ => rfl
|
||||
| _ :: _, _, h => by
|
||||
| [], l₂, _ => rfl
|
||||
| x :: xs, l₂, h => by
|
||||
simp [(forall_mem_cons.1 h).1, eraseP_append_right _ (forall_mem_cons.1 h).2]
|
||||
|
||||
theorem eraseP_append (l₁ l₂ : List α) :
|
||||
@@ -336,10 +332,6 @@ theorem IsPrefix.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+: l₂) : l₁
|
||||
theorem length_erase_le (a : α) (l : List α) : (l.erase a).length ≤ l.length :=
|
||||
(erase_sublist a l).length_le
|
||||
|
||||
theorem le_length_erase [LawfulBEq α] (a : α) (l : List α) : l.length - 1 ≤ (l.erase a).length := by
|
||||
rw [length_erase]
|
||||
split <;> simp
|
||||
|
||||
theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈ l := erase_subset _ _ h
|
||||
|
||||
@[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {l : List α} (ab : a ≠ b) :
|
||||
@@ -460,22 +452,13 @@ end erase
|
||||
|
||||
/-! ### eraseIdx -/
|
||||
|
||||
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
|
||||
| nil => simp
|
||||
| cons x l ih =>
|
||||
cases i with
|
||||
| zero => simp
|
||||
| succ i =>
|
||||
simp only [eraseIdx, length_cons, ih, add_one_lt_add_one_iff, Nat.add_one_sub_one]
|
||||
split
|
||||
· cases l <;> simp_all
|
||||
· rfl
|
||||
|
||||
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]
|
||||
theorem length_eraseIdx : ∀ {l i}, i < length l → length (@eraseIdx α l i) = length l - 1
|
||||
| [], _, _ => rfl
|
||||
| _::_, 0, _ => by simp [eraseIdx]
|
||||
| x::xs, i+1, h => by
|
||||
have : i < length xs := Nat.lt_of_succ_lt_succ h
|
||||
simp [eraseIdx, ← Nat.add_one]
|
||||
rw [length_eraseIdx this, Nat.sub_add_cancel (Nat.lt_of_le_of_lt (Nat.zero_le _) this)]
|
||||
|
||||
@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
|
||||
|
||||
@@ -485,8 +468,6 @@ theorem eraseIdx_eq_take_drop_succ :
|
||||
| a::l, 0 => by simp
|
||||
| a::l, i + 1 => by simp [eraseIdx_eq_take_drop_succ l i]
|
||||
|
||||
-- See `Init.Data.List.Nat.Erase` for `getElem?_eraseIdx` and `getElem_eraseIdx`.
|
||||
|
||||
@[simp] theorem eraseIdx_eq_nil {l : List α} {i : Nat} : eraseIdx l i = [] ↔ l = [] ∨ (length l = 1 ∧ i = 0) := by
|
||||
match l, i with
|
||||
| [], _
|
||||
@@ -518,13 +499,6 @@ theorem eraseIdx_eq_self : ∀ {l : List α} {k : Nat}, eraseIdx l k = l ↔ len
|
||||
theorem eraseIdx_of_length_le {l : List α} {k : Nat} (h : length l ≤ k) : eraseIdx l k = l := by
|
||||
rw [eraseIdx_eq_self.2 h]
|
||||
|
||||
theorem length_eraseIdx_le (l : List α) (i : Nat) : length (l.eraseIdx i) ≤ length l :=
|
||||
(eraseIdx_sublist l i).length_le
|
||||
|
||||
theorem le_length_eraseIdx (l : List α) (i : Nat) : length l - 1 ≤ length (l.eraseIdx i) := by
|
||||
rw [length_eraseIdx]
|
||||
split <;> simp
|
||||
|
||||
theorem eraseIdx_append_of_lt_length {l : List α} {k : Nat} (hk : k < length l) (l' : List α) :
|
||||
eraseIdx (l ++ l') k = eraseIdx l k ++ l' := by
|
||||
induction l generalizing k with
|
||||
@@ -546,7 +520,7 @@ theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤
|
||||
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
|
||||
· rw [eq_replicate_iff, length_eraseIdx_of_lt (by simpa using h)]
|
||||
· rw [eq_replicate_iff, length_eraseIdx (by simpa using h)]
|
||||
simp only [length_replicate, true_and]
|
||||
intro b m
|
||||
replace m := mem_of_mem_eraseIdx m
|
||||
|
||||
@@ -10,8 +10,7 @@ import Init.Data.List.Sublist
|
||||
import Init.Data.List.Range
|
||||
|
||||
/-!
|
||||
Lemmas about `List.findSome?`, `List.find?`, `List.findIdx`, `List.findIdx?`, `List.indexOf`,
|
||||
and `List.lookup`.
|
||||
# Lemmas about `List.findSome?`, `List.find?`, `List.findIdx`, `List.findIdx?`, and `List.indexOf`.
|
||||
-/
|
||||
|
||||
namespace List
|
||||
@@ -96,22 +95,22 @@ theorem findSome?_eq_some_iff {f : α → Option β} {l : List α} {b : β} :
|
||||
· simp only [Option.guard_eq_none] at h
|
||||
simp [ih, h]
|
||||
|
||||
@[simp] theorem head?_filterMap (f : α → Option β) (l : List α) : (l.filterMap f).head? = l.findSome? f := by
|
||||
@[simp] theorem filterMap_head? (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] 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] theorem filterMap_head (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] theorem getLast?_filterMap (f : α → Option β) (l : List α) : (l.filterMap f).getLast? = l.reverse.findSome? f := by
|
||||
@[simp] theorem filterMap_getLast? (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] theorem getLast_filterMap (f : α → Option β) (l : List α) (h) :
|
||||
@[simp] theorem filterMap_getLast (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]
|
||||
|
||||
@@ -133,14 +132,14 @@ theorem findSome?_append {l₁ l₂ : List α} : (l₁ ++ l₂).findSome? f = (l
|
||||
simp only [cons_append, findSome?]
|
||||
split <;> simp_all
|
||||
|
||||
theorem head_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
|
||||
(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]
|
||||
theorem head_join {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
|
||||
(join 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?_join]
|
||||
|
||||
theorem getLast_flatten {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
|
||||
(flatten L).getLast (by simpa using h) =
|
||||
theorem getLast_join {L : List (List α)} (h : ∃ l, l ∈ L ∧ l ≠ []) :
|
||||
(join L).getLast (by simpa using h) =
|
||||
(L.reverse.findSome? fun l => l.getLast?).get (by simpa using h) := by
|
||||
simp [getLast_eq_iff_getLast_eq_some, getLast?_flatten]
|
||||
simp [getLast_eq_iff_getLast_eq_some, getLast?_join]
|
||||
|
||||
theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
|
||||
cases n with
|
||||
@@ -180,7 +179,7 @@ theorem IsPrefix.findSome?_eq_some {l₁ l₂ : List α} {f : α → Option β}
|
||||
List.findSome? f l₁ = some b → List.findSome? f l₂ = some b := by
|
||||
rw [IsPrefix] at h
|
||||
obtain ⟨t, rfl⟩ := h
|
||||
simp +contextual [findSome?_append]
|
||||
simp (config := {contextual := true}) [findSome?_append]
|
||||
|
||||
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 :=
|
||||
@@ -207,8 +206,7 @@ theorem IsInfix.findSome?_eq_none {l₁ l₂ : List α} {f : α → Option β} (
|
||||
@[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 :
|
||||
xs.find? p = some b ↔ p b ∧ ∃ as bs, xs = as ++ b :: bs ∧ ∀ a ∈ as, !p a := by
|
||||
theorem find?_eq_some : xs.find? p = some b ↔ p b ∧ ∃ as bs, xs = as ++ b :: bs ∧ ∀ a ∈ as, !p a := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
@@ -226,7 +224,7 @@ theorem find?_eq_some_iff_append :
|
||||
simp only [cons_append] at h₁
|
||||
obtain ⟨rfl, -⟩ := h₁
|
||||
simp_all
|
||||
· simp only [ih, Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, and_congr_right_iff]
|
||||
· simp only [ih, Bool.not_eq_true', exists_and_right, and_congr_right_iff]
|
||||
intro pb
|
||||
constructor
|
||||
· rintro ⟨as, ⟨⟨bs, rfl⟩, h₁⟩⟩
|
||||
@@ -244,9 +242,6 @@ theorem find?_eq_some_iff_append :
|
||||
cases h₁
|
||||
simp
|
||||
|
||||
@[deprecated find?_eq_some_iff_append (since := "2024-11-06")]
|
||||
abbrev find?_eq_some := @find?_eq_some_iff_append
|
||||
|
||||
@[simp]
|
||||
theorem find?_cons_eq_some : (a :: xs).find? p = some b ↔ (p a ∧ a = b) ∨ (!p a ∧ xs.find? p = some b) := by
|
||||
rw [find?_cons]
|
||||
@@ -292,18 +287,18 @@ theorem get_find?_mem (xs : List α) (p : α → Bool) (h) : (xs.find? p).get h
|
||||
· simp only [find?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[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] theorem filter_head? (p : α → Bool) (l : List α) : (l.filter p).head? = l.find? p := by
|
||||
rw [← filterMap_eq_filter, filterMap_head?, findSome?_guard]
|
||||
|
||||
@[simp] theorem head_filter (p : α → Bool) (l : List α) (h) :
|
||||
@[simp] theorem filter_head (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] theorem getLast?_filter (p : α → Bool) (l : List α) : (l.filter p).getLast? = l.reverse.find? p := by
|
||||
@[simp] theorem filter_getLast? (p : α → Bool) (l : List α) : (l.filter p).getLast? = l.reverse.find? p := by
|
||||
rw [getLast?_eq_head?_reverse]
|
||||
simp [← filter_reverse]
|
||||
|
||||
@[simp] theorem getLast_filter (p : α → Bool) (l : List α) (h) :
|
||||
@[simp] theorem filter_getLast (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]
|
||||
|
||||
@@ -331,35 +326,35 @@ theorem get_find?_mem (xs : List α) (p : α → Bool) (h) : (xs.find? p).get h
|
||||
simp only [cons_append, find?]
|
||||
by_cases h : p x <;> simp [h, ih]
|
||||
|
||||
@[simp] theorem find?_flatten (xs : List (List α)) (p : α → Bool) :
|
||||
xs.flatten.find? p = xs.findSome? (·.find? p) := by
|
||||
@[simp] theorem find?_join (xs : List (List α)) (p : α → Bool) :
|
||||
xs.join.find? p = xs.findSome? (·.find? p) := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [flatten_cons, find?_append, findSome?_cons, ih]
|
||||
simp only [join_cons, find?_append, findSome?_cons, ih]
|
||||
split <;> simp [*]
|
||||
|
||||
theorem find?_flatten_eq_none {xs : List (List α)} {p : α → Bool} :
|
||||
xs.flatten.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
|
||||
theorem find?_join_eq_none {xs : List (List α)} {p : α → Bool} :
|
||||
xs.join.find? p = none ↔ ∀ ys ∈ xs, ∀ x ∈ ys, !p x := by
|
||||
simp
|
||||
|
||||
/--
|
||||
If `find? p` returns `some a` from `xs.flatten`, then `p a` holds, and
|
||||
If `find? p` returns `some a` from `xs.join`, then `p a` holds, and
|
||||
some list in `xs` contains `a`, and no earlier element of that list satisfies `p`.
|
||||
Moreover, no earlier list in `xs` has an element satisfying `p`.
|
||||
-/
|
||||
theorem find?_flatten_eq_some {xs : List (List α)} {p : α → Bool} {a : α} :
|
||||
xs.flatten.find? p = some a ↔
|
||||
theorem find?_join_eq_some {xs : List (List α)} {p : α → Bool} {a : α} :
|
||||
xs.join.find? p = some a ↔
|
||||
p a ∧ ∃ as ys zs bs, xs = as ++ (ys ++ a :: zs) :: bs ∧
|
||||
(∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) := by
|
||||
rw [find?_eq_some_iff_append]
|
||||
rw [find?_eq_some]
|
||||
constructor
|
||||
· rintro ⟨h, ⟨ys, zs, h₁, h₂⟩⟩
|
||||
refine ⟨h, ?_⟩
|
||||
rw [flatten_eq_append_iff] at h₁
|
||||
rw [join_eq_append_iff] at h₁
|
||||
obtain (⟨as, bs, rfl, rfl, h₁⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, h₁⟩) := h₁
|
||||
· replace h₁ := h₁.symm
|
||||
rw [flatten_eq_cons_iff] at h₁
|
||||
rw [join_eq_cons_iff] at h₁
|
||||
obtain ⟨bs, cs, ds, rfl, h₁, rfl⟩ := h₁
|
||||
refine ⟨as ++ bs, [], cs, ds, by simp, ?_⟩
|
||||
simp
|
||||
@@ -376,25 +371,21 @@ theorem find?_flatten_eq_some {xs : List (List α)} {p : α → Bool} {a : α} :
|
||||
· intro x m
|
||||
simpa using h₂ x (by simpa using .inr m)
|
||||
· rintro ⟨h, ⟨as, ys, zs, bs, rfl, h₁, h₂⟩⟩
|
||||
refine ⟨h, as.flatten ++ ys, zs ++ bs.flatten, by simp, ?_⟩
|
||||
refine ⟨h, as.join ++ ys, zs ++ bs.join, by simp, ?_⟩
|
||||
intro a m
|
||||
simp at m
|
||||
obtain ⟨l, ml, m⟩ | m := m
|
||||
· exact h₁ l ml a m
|
||||
· exact h₂ a m
|
||||
|
||||
@[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
|
||||
@[simp] theorem find?_bind (xs : List α) (f : α → List β) (p : β → Bool) :
|
||||
(xs.bind f).find? p = xs.findSome? (fun x => (f x).find? p) := by
|
||||
simp [bind_def, findSome?_map]; rfl
|
||||
|
||||
@[deprecated find?_flatMap (since := "2024-10-16")] abbrev find?_bind := @find?_flatMap
|
||||
|
||||
theorem find?_flatMap_eq_none {xs : List α} {f : α → List β} {p : β → Bool} :
|
||||
(xs.flatMap f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
|
||||
theorem find?_bind_eq_none {xs : List α} {f : α → List β} {p : β → Bool} :
|
||||
(xs.bind f).find? p = none ↔ ∀ x ∈ xs, ∀ y ∈ f x, !p y := by
|
||||
simp
|
||||
|
||||
@[deprecated find?_flatMap_eq_none (since := "2024-10-16")] abbrev find?_bind_eq_none := @find?_flatMap_eq_none
|
||||
|
||||
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
|
||||
@@ -441,7 +432,7 @@ theorem IsPrefix.find?_eq_some {l₁ l₂ : List α} {p : α → Bool} (h : l₁
|
||||
List.find? p l₁ = some b → List.find? p l₂ = some b := by
|
||||
rw [IsPrefix] at h
|
||||
obtain ⟨t, rfl⟩ := h
|
||||
simp +contextual [find?_append]
|
||||
simp (config := {contextual := true}) [find?_append]
|
||||
|
||||
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 :=
|
||||
@@ -567,7 +558,7 @@ theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs
|
||||
| inr e =>
|
||||
have ipm := Nat.succ_pred_eq_of_pos e
|
||||
have ilt := Nat.le_trans ho (findIdx_le_length p)
|
||||
simp +singlePass only [← ipm, getElem_cons_succ]
|
||||
simp (config := { singlePass := true }) only [← ipm, getElem_cons_succ]
|
||||
rw [← ipm, Nat.succ_lt_succ_iff] at h
|
||||
simpa using ih h
|
||||
|
||||
@@ -600,14 +591,15 @@ theorem findIdx_eq {p : α → Bool} {xs : List α} {i : Nat} (h : i < xs.length
|
||||
|
||||
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
|
||||
if ∃ x, x ∈ l₁ ∧ p x = true then l₁.findIdx p else l₂.findIdx p + l₁.length := by
|
||||
induction l₁ with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [findIdx_cons, length_cons, cons_append]
|
||||
by_cases h : p x
|
||||
· simp [h]
|
||||
· simp only [h, ih, cond_eq_if, Bool.false_eq_true, ↓reduceIte, add_one_lt_add_one_iff]
|
||||
· simp only [h, ih, cond_eq_if, Bool.false_eq_true, ↓reduceIte, mem_cons, exists_eq_or_imp,
|
||||
false_or]
|
||||
split <;> simp [Nat.add_assoc]
|
||||
|
||||
theorem IsPrefix.findIdx_le {l₁ l₂ : List α} {p : α → Bool} (h : l₁ <+: l₂) :
|
||||
@@ -628,18 +620,6 @@ theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α
|
||||
· rfl
|
||||
· simp_all
|
||||
|
||||
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
|
||||
| cons x xs ih =>
|
||||
simp only [findIdx_cons, cond_eq_if]
|
||||
split
|
||||
· simp
|
||||
· split
|
||||
· simp_all
|
||||
· simp only [Nat.add_le_add_iff_right]
|
||||
exact ih fun _ m w => h _ (mem_cons_of_mem x m) w
|
||||
|
||||
/-! ### findIdx? -/
|
||||
|
||||
@[simp] theorem findIdx?_nil : ([] : List α).findIdx? p i = none := rfl
|
||||
@@ -794,15 +774,15 @@ theorem findIdx?_of_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p
|
||||
induction xs with simp
|
||||
| cons _ _ _ => split <;> simp_all [Option.map_or', Option.map_map]; rfl
|
||||
|
||||
theorem findIdx?_flatten {l : List (List α)} {p : α → Bool} :
|
||||
l.flatten.findIdx? p =
|
||||
theorem findIdx?_join {l : List (List α)} {p : α → Bool} :
|
||||
l.join.findIdx? p =
|
||||
(l.findIdx? (·.any p)).map
|
||||
fun i => ((l.take i).map List.length).sum +
|
||||
fun i => Nat.sum ((l.take i).map List.length) +
|
||||
(l[i]?.map fun xs => xs.findIdx p).getD 0 := by
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons xs l ih =>
|
||||
simp only [flatten, findIdx?_append, map_take, map_cons, findIdx?, any_eq_true, Nat.zero_add,
|
||||
simp only [join, findIdx?_append, map_take, map_cons, findIdx?, any_eq_true, Nat.zero_add,
|
||||
findIdx?_succ]
|
||||
split
|
||||
· simp only [Option.map_some', take_zero, sum_nil, length_cons, zero_lt_succ,
|
||||
@@ -823,7 +803,7 @@ theorem findIdx?_flatten {l : List (List α)} {p : α → Bool} :
|
||||
simp only [replicate, findIdx?_cons, Nat.zero_add, findIdx?_succ, zero_lt_succ, true_and]
|
||||
split <;> simp_all
|
||||
|
||||
theorem findIdx?_eq_findSome?_enum {xs : List α} {p : α → Bool} :
|
||||
theorem findIdx?_eq_enum_findSome? {xs : List α} {p : α → Bool} :
|
||||
xs.findIdx? p = xs.enum.findSome? fun ⟨i, a⟩ => if p a then some i else none := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
@@ -834,30 +814,6 @@ theorem findIdx?_eq_findSome?_enum {xs : List α} {p : α → Bool} :
|
||||
· simp_all only [enumFrom_cons, ite_false, Option.isNone_none, findSome?_cons_of_isNone, reduceCtorEq]
|
||||
simp [Function.comp_def, ← map_fst_add_enum_eq_enumFrom, findSome?_map]
|
||||
|
||||
theorem findIdx?_eq_fst_find?_enum {xs : List α} {p : α → Bool} :
|
||||
xs.findIdx? p = (xs.enum.find? fun ⟨_, x⟩ => p x).map (·.1) := by
|
||||
induction xs with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [findIdx?_cons, Nat.zero_add, findIdx?_start_succ, enum_cons]
|
||||
split
|
||||
· simp_all
|
||||
· simp only [Option.map_map, enumFrom_eq_map_enum, Bool.false_eq_true, not_false_eq_true,
|
||||
find?_cons_of_neg, find?_map, *]
|
||||
congr
|
||||
|
||||
-- See also `findIdx_le_findIdx`.
|
||||
theorem findIdx?_eq_none_of_findIdx?_eq_none {xs : List α} {p q : α → Bool} (w : ∀ x ∈ xs, p x → q x) :
|
||||
xs.findIdx? q = none → xs.findIdx? p = none := by
|
||||
simp only [findIdx?_eq_none_iff]
|
||||
intro h x m
|
||||
cases z : p x
|
||||
· rfl
|
||||
· exfalso
|
||||
specialize w x m z
|
||||
specialize h x m
|
||||
simp_all
|
||||
|
||||
theorem Sublist.findIdx?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
|
||||
(l₁.findIdx? p).isSome → (l₂.findIdx? p).isSome := by
|
||||
simp only [List.findIdx?_isSome, any_eq_true]
|
||||
@@ -922,7 +878,7 @@ theorem lookup_eq_some_iff {l : List (α × β)} {k : α} {b : β} :
|
||||
simp only [lookup_eq_findSome?, findSome?_eq_some_iff]
|
||||
constructor
|
||||
· rintro ⟨l₁, a, l₂, rfl, h₁, h₂⟩
|
||||
simp only [beq_iff_eq, Option.ite_none_right_eq_some, Option.some.injEq] at h₁
|
||||
simp only [beq_iff_eq, ite_some_none_eq_some] at h₁
|
||||
obtain ⟨rfl, rfl⟩ := h₁
|
||||
simp at h₂
|
||||
exact ⟨l₁, l₂, rfl, by simpa using h₂⟩
|
||||
@@ -984,13 +940,4 @@ theorem IsInfix.lookup_eq_none {l₁ l₂ : List (α × β)} (h : l₁ <:+: l₂
|
||||
|
||||
end lookup
|
||||
|
||||
/-! ### Deprecations -/
|
||||
|
||||
@[deprecated head_flatten (since := "2024-10-14")] abbrev head_join := @head_flatten
|
||||
@[deprecated getLast_flatten (since := "2024-10-14")] abbrev getLast_join := @getLast_flatten
|
||||
@[deprecated find?_flatten (since := "2024-10-14")] abbrev find?_join := @find?_flatten
|
||||
@[deprecated find?_flatten_eq_none (since := "2024-10-14")] abbrev find?_join_eq_none := @find?_flatten_eq_none
|
||||
@[deprecated find?_flatten_eq_some (since := "2024-10-14")] abbrev find?_join_eq_some := @find?_flatten_eq_some
|
||||
@[deprecated findIdx?_flatten (since := "2024-10-14")] abbrev findIdx?_join := @findIdx?_flatten
|
||||
|
||||
end List
|
||||
|
||||
@@ -23,7 +23,7 @@ namespace List
|
||||
The following operations are already tail-recursive, and do not need `@[csimp]` replacements:
|
||||
`get`, `foldl`, `beq`, `isEqv`, `reverse`, `elem` (and hence `contains`), `drop`, `dropWhile`,
|
||||
`partition`, `isPrefixOf`, `isPrefixOf?`, `find?`, `findSome?`, `lookup`, `any` (and hence `or`),
|
||||
`all` (and hence `and`) , `range`, `eraseDups`, `eraseReps`, `span`, `splitBy`.
|
||||
`all` (and hence `and`) , `range`, `eraseDups`, `eraseReps`, `span`, `groupBy`.
|
||||
|
||||
The following operations are still missing `@[csimp]` replacements:
|
||||
`concat`, `zipWithAll`.
|
||||
@@ -31,14 +31,14 @@ The following operations are still missing `@[csimp]` replacements:
|
||||
The following operations are not recursive to begin with
|
||||
(or are defined in terms of recursive primitives):
|
||||
`isEmpty`, `isSuffixOf`, `isSuffixOf?`, `rotateLeft`, `rotateRight`, `insert`, `zip`, `enum`,
|
||||
`min?`, `max?`, and `removeAll`.
|
||||
`minimum?`, `maximum?`, and `removeAll`.
|
||||
|
||||
The following operations were already given `@[csimp]` replacements in `Init/Data/List/Basic.lean`:
|
||||
`length`, `map`, `filter`, `replicate`, `leftPad`, `unzip`, `range'`, `iota`, `intersperse`.
|
||||
|
||||
The following operations are given `@[csimp]` replacements below:
|
||||
`set`, `filterMap`, `foldr`, `append`, `bind`, `join`,
|
||||
`take`, `takeWhile`, `dropLast`, `replace`, `modify`, `insertIdx`, `erase`, `eraseIdx`, `zipWith`,
|
||||
`take`, `takeWhile`, `dropLast`, `replace`, `erase`, `eraseIdx`, `zipWith`,
|
||||
`enumFrom`, and `intercalate`.
|
||||
|
||||
-/
|
||||
@@ -91,31 +91,31 @@ The following operations are given `@[csimp]` replacements below:
|
||||
@[specialize] def foldrTR (f : α → β → β) (init : β) (l : List α) : β := l.toArray.foldr f init
|
||||
|
||||
@[csimp] theorem foldr_eq_foldrTR : @foldr = @foldrTR := by
|
||||
funext α β f init l; simp [foldrTR, ← Array.foldr_toList, -Array.size_toArray]
|
||||
funext α β f init l; simp [foldrTR, Array.foldr_eq_foldr_toList, -Array.size_toArray]
|
||||
|
||||
/-! ### flatMap -/
|
||||
/-! ### bind -/
|
||||
|
||||
/-- Tail recursive version of `List.flatMap`. -/
|
||||
@[inline] def flatMapTR (as : List α) (f : α → List β) : List β := go as #[] where
|
||||
/-- Auxiliary for `flatMap`: `flatMap.go f as = acc.toList ++ bind f as` -/
|
||||
/-- Tail recursive version of `List.bind`. -/
|
||||
@[inline] def bindTR (as : List α) (f : α → List β) : List β := go as #[] where
|
||||
/-- Auxiliary for `bind`: `bind.go f as = acc.toList ++ bind f as` -/
|
||||
@[specialize] go : List α → Array β → List β
|
||||
| [], acc => acc.toList
|
||||
| x::xs, acc => go xs (acc ++ f x)
|
||||
|
||||
@[csimp] theorem flatMap_eq_flatMapTR : @List.flatMap = @flatMapTR := by
|
||||
@[csimp] theorem bind_eq_bindTR : @List.bind = @bindTR := by
|
||||
funext α β as f
|
||||
let rec go : ∀ as acc, flatMapTR.go f as acc = acc.toList ++ as.flatMap f
|
||||
| [], acc => by simp [flatMapTR.go, flatMap]
|
||||
| x::xs, acc => by simp [flatMapTR.go, flatMap, go xs]
|
||||
let rec go : ∀ as acc, bindTR.go f as acc = acc.toList ++ as.bind f
|
||||
| [], acc => by simp [bindTR.go, bind]
|
||||
| x::xs, acc => by simp [bindTR.go, bind, go xs]
|
||||
exact (go as #[]).symm
|
||||
|
||||
/-! ### flatten -/
|
||||
/-! ### join -/
|
||||
|
||||
/-- Tail recursive version of `List.flatten`. -/
|
||||
@[inline] def flattenTR (l : List (List α)) : List α := flatMapTR l id
|
||||
/-- Tail recursive version of `List.join`. -/
|
||||
@[inline] def joinTR (l : List (List α)) : List α := bindTR l id
|
||||
|
||||
@[csimp] theorem flatten_eq_flattenTR : @flatten = @flattenTR := by
|
||||
funext α l; rw [← List.flatMap_id, List.flatMap_eq_flatMapTR]; rfl
|
||||
@[csimp] theorem join_eq_joinTR : @join = @joinTR := by
|
||||
funext α l; rw [← List.bind_id, List.bind_eq_bindTR]; rfl
|
||||
|
||||
/-! ## Sublists -/
|
||||
|
||||
@@ -197,41 +197,6 @@ The following operations are given `@[csimp]` replacements below:
|
||||
· simp [*]
|
||||
· intro h; rw [IH] <;> simp_all
|
||||
|
||||
/-! ### modify -/
|
||||
|
||||
/-- Tail-recursive version of `modify`. -/
|
||||
def modifyTR (f : α → α) (n : Nat) (l : List α) : List α := go l n #[] where
|
||||
/-- Auxiliary for `modifyTR`: `modifyTR.go f l n acc = acc.toList ++ modify f n l`. -/
|
||||
go : List α → Nat → Array α → List α
|
||||
| [], _, acc => acc.toList
|
||||
| a :: l, 0, acc => acc.toListAppend (f a :: l)
|
||||
| a :: l, n+1, acc => go l n (acc.push a)
|
||||
|
||||
theorem modifyTR_go_eq : ∀ l n, modifyTR.go f l n acc = acc.toList ++ modify f n l
|
||||
| [], n => by cases n <;> simp [modifyTR.go, modify]
|
||||
| a :: l, 0 => by simp [modifyTR.go, modify]
|
||||
| a :: l, n+1 => by simp [modifyTR.go, modify, modifyTR_go_eq l]
|
||||
|
||||
@[csimp] theorem modify_eq_modifyTR : @modify = @modifyTR := by
|
||||
funext α f n l; simp [modifyTR, modifyTR_go_eq]
|
||||
|
||||
/-! ### insertIdx -/
|
||||
|
||||
/-- Tail-recursive version of `insertIdx`. -/
|
||||
@[inline] def insertIdxTR (n : Nat) (a : α) (l : List α) : List α := go n l #[] where
|
||||
/-- Auxiliary for `insertIdxTR`: `insertIdxTR.go a n l acc = acc.toList ++ insertIdx n a l`. -/
|
||||
go : Nat → List α → Array α → List α
|
||||
| 0, l, acc => acc.toListAppend (a :: l)
|
||||
| _, [], acc => acc.toList
|
||||
| n+1, a :: l, acc => go n l (acc.push a)
|
||||
|
||||
theorem insertIdxTR_go_eq : ∀ n l, insertIdxTR.go a n l acc = acc.toList ++ insertIdx n a l
|
||||
| 0, l | _+1, [] => by simp [insertIdxTR.go, insertIdx]
|
||||
| n+1, a :: l => by simp [insertIdxTR.go, insertIdx, insertIdxTR_go_eq n l]
|
||||
|
||||
@[csimp] theorem insertIdx_eq_insertIdxTR : @insertIdx = @insertIdxTR := by
|
||||
funext α f n l; simp [insertIdxTR, insertIdxTR_go_eq]
|
||||
|
||||
/-! ### erase -/
|
||||
|
||||
/-- Tail recursive version of `List.erase`. -/
|
||||
@@ -331,7 +296,7 @@ def enumFromTR (n : Nat) (l : List α) : List (Nat × α) :=
|
||||
| a::as, n => by
|
||||
rw [← show _ + as.length = n + (a::as).length from Nat.succ_add .., foldr, go as]
|
||||
simp [enumFrom, f]
|
||||
rw [← Array.foldr_toList]
|
||||
rw [Array.foldr_eq_foldr_toList]
|
||||
simp [go]
|
||||
|
||||
/-! ## Other list operations -/
|
||||
@@ -357,7 +322,7 @@ where
|
||||
| [_] => simp
|
||||
| x::y::xs =>
|
||||
let rec go {acc x} : ∀ xs,
|
||||
intercalateTR.go sep.toArray x xs acc = acc.toList ++ flatten (intersperse sep (x::xs))
|
||||
intercalateTR.go sep.toArray x xs acc = acc.toList ++ join (intersperse sep (x::xs))
|
||||
| [] => by simp [intercalateTR.go]
|
||||
| _::_ => by simp [intercalateTR.go, go]
|
||||
simp [intersperse, go]
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -1,408 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison, Mario Carneiro
|
||||
-/
|
||||
|
||||
prelude
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.List.Nat.Range
|
||||
import Init.Data.List.OfFn
|
||||
import Init.Data.Fin.Lemmas
|
||||
import Init.Data.Option.Attach
|
||||
|
||||
namespace List
|
||||
|
||||
/-! ## Operations using indexes -/
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
|
||||
/--
|
||||
Given a list `as = [a₀, a₁, ...]` function `f : Fin as.length → α → β`, returns the list
|
||||
`[f 0 a₀, f 1 a₁, ...]`.
|
||||
-/
|
||||
@[inline] def mapFinIdx (as : List α) (f : Fin as.length → α → β) : List β := go as #[] (by simp) where
|
||||
/-- Auxiliary for `mapFinIdx`:
|
||||
`mapFinIdx.go [a₀, a₁, ...] acc = acc.toList ++ [f 0 a₀, f 1 a₁, ...]` -/
|
||||
@[specialize] go : (bs : List α) → (acc : Array β) → bs.length + acc.size = as.length → List β
|
||||
| [], acc, h => acc.toList
|
||||
| a :: as, acc, h =>
|
||||
go as (acc.push (f ⟨acc.size, by simp at h; omega⟩ a)) (by simp at h ⊢; omega)
|
||||
|
||||
/--
|
||||
Given a function `f : Nat → α → β` and `as : List α`, `as = [a₀, a₁, ...]`, returns the list
|
||||
`[f 0 a₀, f 1 a₁, ...]`.
|
||||
-/
|
||||
@[inline] def mapIdx (f : Nat → α → β) (as : List α) : List β := go as #[] where
|
||||
/-- Auxiliary for `mapIdx`:
|
||||
`mapIdx.go [a₀, a₁, ...] acc = acc.toList ++ [f acc.size a₀, f (acc.size + 1) a₁, ...]` -/
|
||||
@[specialize] go : List α → Array β → List β
|
||||
| [], acc => acc.toList
|
||||
| a :: as, acc => go as (acc.push (f acc.size a))
|
||||
|
||||
/-! ### mapFinIdx -/
|
||||
|
||||
@[simp]
|
||||
theorem mapFinIdx_nil {f : Fin 0 → α → β} : mapFinIdx [] f = [] :=
|
||||
rfl
|
||||
|
||||
@[simp] theorem length_mapFinIdx_go :
|
||||
(mapFinIdx.go as f bs acc h).length = as.length := by
|
||||
induction bs generalizing acc with
|
||||
| nil => simpa using h
|
||||
| cons _ _ ih => simp [mapFinIdx.go, ih]
|
||||
|
||||
@[simp] theorem length_mapFinIdx {as : List α} {f : Fin as.length → α → β} :
|
||||
(as.mapFinIdx f).length = as.length := by
|
||||
simp [mapFinIdx, length_mapFinIdx_go]
|
||||
|
||||
theorem getElem_mapFinIdx_go {as : List α} {f : Fin as.length → α → β} {i : Nat} {h} {w} :
|
||||
(mapFinIdx.go as f bs acc h)[i] =
|
||||
if w' : i < acc.size then acc[i] else f ⟨i, by simp at w; omega⟩ (bs[i - acc.size]'(by simp at w; omega)) := by
|
||||
induction bs generalizing acc with
|
||||
| nil =>
|
||||
simp only [length_mapFinIdx_go, length_nil, Nat.zero_add] at w h
|
||||
simp only [mapFinIdx.go, Array.getElem_toList]
|
||||
rw [dif_pos]
|
||||
| cons _ _ ih =>
|
||||
simp [mapFinIdx.go]
|
||||
rw [ih]
|
||||
simp
|
||||
split <;> rename_i h₁ <;> split <;> rename_i h₂
|
||||
· rw [Array.getElem_push_lt]
|
||||
· have h₃ : i = acc.size := by omega
|
||||
subst h₃
|
||||
simp
|
||||
· omega
|
||||
· have h₃ : i - acc.size = (i - (acc.size + 1)) + 1 := by omega
|
||||
simp [h₃]
|
||||
|
||||
@[simp] theorem getElem_mapFinIdx {as : List α} {f : Fin as.length → α → β} {i : Nat} {h} :
|
||||
(as.mapFinIdx f)[i] = f ⟨i, by simp at h; omega⟩ (as[i]'(by simp at h; omega)) := by
|
||||
simp [mapFinIdx, getElem_mapFinIdx_go]
|
||||
|
||||
theorem mapFinIdx_eq_ofFn {as : List α} {f : Fin as.length → α → β} :
|
||||
as.mapFinIdx f = List.ofFn fun i : Fin as.length => f i as[i] := by
|
||||
apply ext_getElem <;> simp
|
||||
|
||||
@[simp] theorem getElem?_mapFinIdx {l : List α} {f : Fin l.length → α → β} {i : Nat} :
|
||||
(l.mapFinIdx f)[i]? = l[i]?.pbind fun x m => f ⟨i, by simp [getElem?_eq_some] at m; exact m.1⟩ x := by
|
||||
simp only [getElem?_eq, length_mapFinIdx, getElem_mapFinIdx]
|
||||
split <;> simp
|
||||
|
||||
@[simp]
|
||||
theorem mapFinIdx_cons {l : List α} {a : α} {f : Fin (l.length + 1) → α → β} :
|
||||
mapFinIdx (a :: l) f = f 0 a :: mapFinIdx l (fun i => f i.succ) := by
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· rintro (_|i) h₁ h₂ <;> simp
|
||||
|
||||
theorem mapFinIdx_append {K L : List α} {f : Fin (K ++ L).length → α → β} :
|
||||
(K ++ L).mapFinIdx f =
|
||||
K.mapFinIdx (fun i => f (i.castLE (by simp))) ++ L.mapFinIdx (fun i => f ((i.natAdd K.length).cast (by simp))) := by
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· intro i h₁ h₂
|
||||
rw [getElem_append]
|
||||
simp only [getElem_mapFinIdx, length_mapFinIdx]
|
||||
split <;> rename_i h
|
||||
· rw [getElem_append_left]
|
||||
congr
|
||||
· simp only [Nat.not_lt] at h
|
||||
rw [getElem_append_right h]
|
||||
congr
|
||||
simp
|
||||
omega
|
||||
|
||||
@[simp] theorem mapFinIdx_concat {l : List α} {e : α} {f : Fin (l ++ [e]).length → α → β}:
|
||||
(l ++ [e]).mapFinIdx f = l.mapFinIdx (fun i => f (i.castLE (by simp))) ++ [f ⟨l.length, by simp⟩ e] := by
|
||||
simp [mapFinIdx_append]
|
||||
congr
|
||||
|
||||
theorem mapFinIdx_singleton {a : α} {f : Fin 1 → α → β} :
|
||||
[a].mapFinIdx f = [f ⟨0, by simp⟩ a] := by
|
||||
simp
|
||||
|
||||
theorem mapFinIdx_eq_enum_map {l : List α} {f : Fin l.length → α → β} :
|
||||
l.mapFinIdx f = l.enum.attach.map
|
||||
fun ⟨⟨i, x⟩, m⟩ => f ⟨i, by rw [mk_mem_enum_iff_getElem?, getElem?_eq_some] at m; exact m.1⟩ x := by
|
||||
apply ext_getElem <;> simp
|
||||
|
||||
@[simp]
|
||||
theorem mapFinIdx_eq_nil_iff {l : List α} {f : Fin l.length → α → β} :
|
||||
l.mapFinIdx f = [] ↔ l = [] := by
|
||||
rw [mapFinIdx_eq_enum_map, map_eq_nil_iff, attach_eq_nil_iff, enum_eq_nil_iff]
|
||||
|
||||
theorem mapFinIdx_ne_nil_iff {l : List α} {f : Fin l.length → α → β} :
|
||||
l.mapFinIdx f ≠ [] ↔ l ≠ [] := by
|
||||
simp
|
||||
|
||||
theorem exists_of_mem_mapFinIdx {b : β} {l : List α} {f : Fin l.length → α → β}
|
||||
(h : b ∈ l.mapFinIdx f) : ∃ (i : Fin l.length), f i l[i] = b := by
|
||||
rw [mapFinIdx_eq_enum_map] at h
|
||||
replace h := exists_of_mem_map h
|
||||
simp only [mem_attach, true_and, Subtype.exists, Prod.exists, mk_mem_enum_iff_getElem?] at h
|
||||
obtain ⟨i, b, h, rfl⟩ := h
|
||||
rw [getElem?_eq_some_iff] at h
|
||||
obtain ⟨h', rfl⟩ := h
|
||||
exact ⟨⟨i, h'⟩, rfl⟩
|
||||
|
||||
@[simp] theorem mem_mapFinIdx {b : β} {l : List α} {f : Fin l.length → α → β} :
|
||||
b ∈ l.mapFinIdx f ↔ ∃ (i : Fin l.length), f i l[i] = b := by
|
||||
constructor
|
||||
· intro h
|
||||
exact exists_of_mem_mapFinIdx h
|
||||
· rintro ⟨i, h, rfl⟩
|
||||
rw [mem_iff_getElem]
|
||||
exact ⟨i, by simp⟩
|
||||
|
||||
theorem mapFinIdx_eq_cons_iff {l : List α} {b : β} {f : Fin l.length → α → β} :
|
||||
l.mapFinIdx f = b :: l₂ ↔
|
||||
∃ (a : α) (l₁ : List α) (h : l = a :: l₁),
|
||||
f ⟨0, by simp [h]⟩ a = b ∧ l₁.mapFinIdx (fun i => f (i.succ.cast (by simp [h]))) = l₂ := by
|
||||
cases l with
|
||||
| nil => simp
|
||||
| cons x l' =>
|
||||
simp only [mapFinIdx_cons, cons.injEq, length_cons, Fin.zero_eta, Fin.cast_succ_eq,
|
||||
exists_and_left]
|
||||
constructor
|
||||
· rintro ⟨rfl, rfl⟩
|
||||
refine ⟨x, rfl, l', by simp⟩
|
||||
· rintro ⟨a, ⟨rfl, h⟩, ⟨_, ⟨rfl, rfl⟩, h⟩⟩
|
||||
exact ⟨rfl, h⟩
|
||||
|
||||
theorem mapFinIdx_eq_cons_iff' {l : List α} {b : β} {f : Fin l.length → α → β} :
|
||||
l.mapFinIdx f = b :: l₂ ↔
|
||||
l.head?.pbind (fun x m => (f ⟨0, by cases l <;> simp_all⟩ x)) = some b ∧
|
||||
l.tail?.attach.map (fun ⟨t, m⟩ => t.mapFinIdx fun i => f (i.succ.cast (by cases l <;> simp_all))) = some l₂ := by
|
||||
cases l <;> simp
|
||||
|
||||
theorem mapFinIdx_eq_iff {l : List α} {f : Fin l.length → α → β} :
|
||||
l.mapFinIdx f = l' ↔ ∃ h : l'.length = l.length, ∀ (i : Nat) (h : i < l.length), l'[i] = f ⟨i, h⟩ l[i] := by
|
||||
constructor
|
||||
· rintro rfl
|
||||
simp
|
||||
· rintro ⟨h, w⟩
|
||||
apply ext_getElem <;> simp_all
|
||||
|
||||
theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : Fin l.length → α → β} :
|
||||
l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Fin l.length), f i l[i] = g i l[i] := by
|
||||
rw [eq_comm, mapFinIdx_eq_iff]
|
||||
simp [Fin.forall_iff]
|
||||
|
||||
@[simp] theorem mapFinIdx_mapFinIdx {l : List α} {f : Fin l.length → α → β} {g : Fin _ → β → γ} :
|
||||
(l.mapFinIdx f).mapFinIdx g = l.mapFinIdx (fun i => g (i.cast (by simp)) ∘ f i) := by
|
||||
simp [mapFinIdx_eq_iff]
|
||||
|
||||
theorem mapFinIdx_eq_replicate_iff {l : List α} {f : Fin l.length → α → β} {b : β} :
|
||||
l.mapFinIdx f = replicate l.length b ↔ ∀ (i : Fin l.length), f i l[i] = b := by
|
||||
simp [eq_replicate_iff, length_mapFinIdx, mem_mapFinIdx, forall_exists_index, true_and]
|
||||
|
||||
@[simp] theorem mapFinIdx_reverse {l : List α} {f : Fin l.reverse.length → α → β} :
|
||||
l.reverse.mapFinIdx f = (l.mapFinIdx (fun i => f ⟨l.length - 1 - i, by simp; omega⟩)).reverse := by
|
||||
simp [mapFinIdx_eq_iff]
|
||||
intro i h
|
||||
congr
|
||||
omega
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
@[simp]
|
||||
theorem mapIdx_nil {f : Nat → α → β} : mapIdx f [] = [] :=
|
||||
rfl
|
||||
|
||||
theorem mapIdx_go_length {arr : Array β} :
|
||||
length (mapIdx.go f l arr) = length l + arr.size := by
|
||||
induction l generalizing arr with
|
||||
| nil => simp only [mapIdx.go, length_nil, Nat.zero_add]
|
||||
| cons _ _ ih =>
|
||||
simp only [mapIdx.go, ih, Array.size_push, Nat.add_succ, length_cons, Nat.add_comm]
|
||||
|
||||
theorem length_mapIdx_go : ∀ {l : List α} {arr : Array β},
|
||||
(mapIdx.go f l arr).length = l.length + arr.size
|
||||
| [], _ => by simp [mapIdx.go]
|
||||
| a :: l, _ => by
|
||||
simp only [mapIdx.go, length_cons]
|
||||
rw [length_mapIdx_go]
|
||||
simp
|
||||
omega
|
||||
|
||||
@[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} {arr : Array β} {i : Nat},
|
||||
(mapIdx.go f l arr)[i]? =
|
||||
if h : i < arr.size then some arr[i] else Option.map (f i) l[i - arr.size]?
|
||||
| [], arr, i => by
|
||||
simp only [mapIdx.go, Array.toListImpl_eq, getElem?_eq, Array.length_toList,
|
||||
Array.getElem_eq_getElem_toList, length_nil, Nat.not_lt_zero, ↓reduceDIte, Option.map_none']
|
||||
| a :: l, arr, i => by
|
||||
rw [mapIdx.go, getElem?_mapIdx_go]
|
||||
simp only [Array.size_push]
|
||||
split <;> split
|
||||
· simp only [Option.some.injEq]
|
||||
rw [Array.getElem_eq_getElem_toList]
|
||||
simp only [Array.push_toList]
|
||||
rw [getElem_append_left, Array.getElem_eq_getElem_toList]
|
||||
· have : i = arr.size := by omega
|
||||
simp_all
|
||||
· omega
|
||||
· have : i - arr.size = i - (arr.size + 1) + 1 := by omega
|
||||
simp_all
|
||||
|
||||
@[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] 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)]
|
||||
simp
|
||||
|
||||
@[simp] theorem mapFinIdx_eq_mapIdx {l : List α} {f : Fin l.length → α → β} {g : Nat → α → β}
|
||||
(h : ∀ (i : Fin l.length), f i l[i] = g i l[i]) :
|
||||
l.mapFinIdx f = l.mapIdx g := by
|
||||
simp_all [mapFinIdx_eq_iff]
|
||||
|
||||
theorem mapIdx_eq_mapFinIdx {l : List α} {f : Nat → α → β} :
|
||||
l.mapIdx f = l.mapFinIdx (fun i => f i) := by
|
||||
simp [mapFinIdx_eq_mapIdx]
|
||||
|
||||
theorem mapIdx_eq_enum_map {l : List α} :
|
||||
l.mapIdx f = l.enum.map (Function.uncurry f) := by
|
||||
ext1 i
|
||||
simp only [getElem?_mapIdx, Option.map, getElem?_map, getElem?_enum]
|
||||
split <;> simp
|
||||
|
||||
@[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_enum_map, enum_eq_zip_range, map_uncurry_zip_eq_zipWith,
|
||||
range_succ_eq_map, zipWith_map_left]
|
||||
|
||||
theorem mapIdx_append {K L : List α} :
|
||||
(K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i => f (i + K.length) := by
|
||||
induction K generalizing f with
|
||||
| nil => rfl
|
||||
| cons _ _ ih => simp [ih (f := fun i => f (i + 1)), Nat.add_assoc]
|
||||
|
||||
@[simp] theorem mapIdx_concat {l : List α} {e : α} :
|
||||
mapIdx f (l ++ [e]) = mapIdx f l ++ [f l.length e] := by
|
||||
simp [mapIdx_append]
|
||||
|
||||
theorem mapIdx_singleton {a : α} : mapIdx f [a] = [f 0 a] := by
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
theorem mapIdx_eq_nil_iff {l : List α} : List.mapIdx f l = [] ↔ l = [] := by
|
||||
rw [List.mapIdx_eq_enum_map, List.map_eq_nil_iff, List.enum_eq_nil_iff]
|
||||
|
||||
theorem mapIdx_ne_nil_iff {l : List α} :
|
||||
List.mapIdx f l ≠ [] ↔ l ≠ [] := by
|
||||
simp
|
||||
|
||||
theorem exists_of_mem_mapIdx {b : β} {l : List α}
|
||||
(h : b ∈ mapIdx f l) : ∃ (i : Nat) (h : i < l.length), f i l[i] = b := by
|
||||
rw [mapIdx_eq_mapFinIdx] at h
|
||||
simpa [Fin.exists_iff] using exists_of_mem_mapFinIdx h
|
||||
|
||||
@[simp] theorem mem_mapIdx {b : β} {l : List α} :
|
||||
b ∈ mapIdx f l ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] = b := by
|
||||
constructor
|
||||
· intro h
|
||||
exact exists_of_mem_mapIdx h
|
||||
· rintro ⟨i, h, rfl⟩
|
||||
rw [mem_iff_getElem]
|
||||
exact ⟨i, by simpa using h, by simp⟩
|
||||
|
||||
theorem mapIdx_eq_cons_iff {l : List α} {b : β} :
|
||||
mapIdx f l = b :: l₂ ↔
|
||||
∃ (a : α) (l₁ : List α), l = a :: l₁ ∧ f 0 a = b ∧ mapIdx (fun i => f (i + 1)) l₁ = l₂ := by
|
||||
cases l <;> simp [and_assoc]
|
||||
|
||||
theorem mapIdx_eq_cons_iff' {l : List α} {b : β} :
|
||||
mapIdx f l = b :: l₂ ↔
|
||||
l.head?.map (f 0) = some b ∧ l.tail?.map (mapIdx fun i => f (i + 1)) = some l₂ := by
|
||||
cases l <;> simp
|
||||
|
||||
theorem mapIdx_eq_iff {l : List α} : mapIdx f l = l' ↔ ∀ i : Nat, l'[i]? = l[i]?.map (f i) := by
|
||||
constructor
|
||||
· intro w i
|
||||
simpa using congrArg (fun l => l[i]?) w.symm
|
||||
· intro w
|
||||
ext1 i
|
||||
simp [w]
|
||||
|
||||
theorem mapIdx_eq_mapIdx_iff {l : List α} :
|
||||
mapIdx f l = mapIdx g l ↔ ∀ i : Nat, (h : i < l.length) → f i l[i] = g i l[i] := by
|
||||
constructor
|
||||
· intro w i h
|
||||
simpa [h] using congrArg (fun l => l[i]?) w
|
||||
· intro w
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· intro i h₁ h₂
|
||||
simp [w]
|
||||
|
||||
@[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
|
||||
split
|
||||
· split <;> simp_all
|
||||
· rfl
|
||||
|
||||
@[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] theorem head?_mapIdx {l : List α} {f : Nat → α → β} : (mapIdx f l).head? = l.head?.map (f 0) := by
|
||||
cases l <;> simp
|
||||
|
||||
@[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
|
||||
| cons _ _ =>
|
||||
simp only [← getElem_cons_length _ _ _ rfl]
|
||||
simp only [mapIdx_cons]
|
||||
simp only [← getElem_cons_length _ _ _ rfl]
|
||||
simp only [← mapIdx_cons, getElem_mapIdx]
|
||||
simp
|
||||
|
||||
@[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] 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]
|
||||
|
||||
theorem mapIdx_eq_replicate_iff {l : List α} {f : Nat → α → β} {b : β} :
|
||||
mapIdx f l = replicate l.length b ↔ ∀ (i : Nat) (h : i < l.length), f i l[i] = b := by
|
||||
simp only [eq_replicate_iff, length_mapIdx, mem_mapIdx, forall_exists_index, true_and]
|
||||
constructor
|
||||
· intro w i h
|
||||
apply w _ _ _ rfl
|
||||
· rintro w _ i h rfl
|
||||
exact w i h
|
||||
|
||||
@[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
|
||||
by_cases h : i < l.length
|
||||
· simp [getElem?_reverse, h]
|
||||
congr
|
||||
omega
|
||||
· simp at h
|
||||
rw [getElem?_eq_none (by simp [h]), getElem?_eq_none (by simp [h])]
|
||||
simp
|
||||
|
||||
end List
|
||||
@@ -7,7 +7,7 @@ prelude
|
||||
import Init.Data.List.Lemmas
|
||||
|
||||
/-!
|
||||
# Lemmas about `List.min?` and `List.max?.
|
||||
# Lemmas about `List.minimum?` and `List.maximum?.
|
||||
-/
|
||||
|
||||
namespace List
|
||||
@@ -16,32 +16,24 @@ open Nat
|
||||
|
||||
/-! ## Minima and maxima -/
|
||||
|
||||
/-! ### min? -/
|
||||
/-! ### minimum? -/
|
||||
|
||||
@[simp] theorem min?_nil [Min α] : ([] : List α).min? = none := rfl
|
||||
@[simp] theorem minimum?_nil [Min α] : ([] : List α).minimum? = none := rfl
|
||||
|
||||
-- We don't put `@[simp]` on `min?_cons'`,
|
||||
-- We don't put `@[simp]` on `minimum?_cons`,
|
||||
-- because the definition in terms of `foldl` is not useful for proofs.
|
||||
theorem min?_cons' [Min α] {xs : List α} : (x :: xs).min? = foldl min x xs := rfl
|
||||
theorem minimum?_cons [Min α] {xs : List α} : (x :: xs).minimum? = foldl min x xs := rfl
|
||||
|
||||
@[simp] theorem min?_cons [Min α] [Std.Associative (min : α → α → α)] {xs : List α} :
|
||||
(x :: xs).min? = some (xs.min?.elim x (min x)) := by
|
||||
cases xs <;> simp [min?_cons', foldl_assoc]
|
||||
@[simp] theorem minimum?_eq_none_iff {xs : List α} [Min α] : xs.minimum? = none ↔ xs = [] := by
|
||||
cases xs <;> simp [minimum?]
|
||||
|
||||
@[simp] theorem min?_eq_none_iff {xs : List α} [Min α] : xs.min? = none ↔ xs = [] := by
|
||||
cases xs <;> simp [min?]
|
||||
|
||||
theorem isSome_min?_of_mem {l : List α} [Min α] {a : α} (h : a ∈ l) :
|
||||
l.min?.isSome := by
|
||||
cases l <;> simp_all [List.min?_cons']
|
||||
|
||||
theorem min?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
|
||||
{xs : List α} → xs.min? = some a → a ∈ xs := by
|
||||
theorem minimum?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
|
||||
{xs : List α} → xs.minimum? = some a → a ∈ xs := by
|
||||
intro xs
|
||||
match xs with
|
||||
| nil => simp
|
||||
| x :: xs =>
|
||||
simp only [min?_cons', Option.some.injEq, List.mem_cons]
|
||||
simp only [minimum?_cons, Option.some.injEq, List.mem_cons]
|
||||
intro eq
|
||||
induction xs generalizing x with
|
||||
| nil =>
|
||||
@@ -57,12 +49,12 @@ theorem min?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b
|
||||
|
||||
-- See also `Init.Data.List.Nat.Basic` for specialisations of the next two results to `Nat`.
|
||||
|
||||
theorem le_min?_iff [Min α] [LE α]
|
||||
theorem le_minimum?_iff [Min α] [LE α]
|
||||
(le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) :
|
||||
{xs : List α} → xs.min? = some a → ∀ {x}, x ≤ a ↔ ∀ b, b ∈ xs → x ≤ b
|
||||
{xs : List α} → xs.minimum? = some a → ∀ {x}, x ≤ a ↔ ∀ b, b ∈ xs → x ≤ b
|
||||
| nil => by simp
|
||||
| cons x xs => by
|
||||
rw [min?]
|
||||
rw [minimum?]
|
||||
intro eq y
|
||||
simp only [Option.some.injEq] at eq
|
||||
induction xs generalizing x with
|
||||
@@ -75,58 +67,46 @@ theorem le_min?_iff [Min α] [LE α]
|
||||
|
||||
-- This could be refactored by designing appropriate typeclasses to replace `le_refl`, `min_eq_or`,
|
||||
-- and `le_min_iff`.
|
||||
theorem min?_eq_some_iff [Min α] [LE α] [anti : Std.Antisymm ((· : α) ≤ ·)]
|
||||
theorem minimum?_eq_some_iff [Min α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
|
||||
(le_refl : ∀ a : α, a ≤ a)
|
||||
(min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b)
|
||||
(le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) {xs : List α} :
|
||||
xs.min? = some a ↔ a ∈ xs ∧ ∀ b, b ∈ xs → a ≤ b := by
|
||||
refine ⟨fun h => ⟨min?_mem min_eq_or h, (le_min?_iff le_min_iff h).1 (le_refl _)⟩, ?_⟩
|
||||
xs.minimum? = some a ↔ a ∈ xs ∧ ∀ b, b ∈ xs → a ≤ b := by
|
||||
refine ⟨fun h => ⟨minimum?_mem min_eq_or h, (le_minimum?_iff le_min_iff h).1 (le_refl _)⟩, ?_⟩
|
||||
intro ⟨h₁, h₂⟩
|
||||
cases xs with
|
||||
| nil => simp at h₁
|
||||
| cons x xs =>
|
||||
exact congrArg some <| anti.1
|
||||
((le_min?_iff le_min_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
|
||||
(h₂ _ (min?_mem min_eq_or (xs := x::xs) rfl))
|
||||
((le_minimum?_iff le_min_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
|
||||
(h₂ _ (minimum?_mem min_eq_or (xs := x::xs) rfl))
|
||||
|
||||
theorem min?_replicate [Min α] {n : Nat} {a : α} (w : min a a = a) :
|
||||
(replicate n a).min? = if n = 0 then none else some a := by
|
||||
theorem minimum?_replicate [Min α] {n : Nat} {a : α} (w : min a a = a) :
|
||||
(replicate n a).minimum? = if n = 0 then none else some a := by
|
||||
induction n with
|
||||
| zero => rfl
|
||||
| succ n ih => cases n <;> simp_all [replicate_succ, min?_cons']
|
||||
| succ n ih => cases n <;> simp_all [replicate_succ, minimum?_cons]
|
||||
|
||||
@[simp] theorem min?_replicate_of_pos [Min α] {n : Nat} {a : α} (w : min a a = a) (h : 0 < n) :
|
||||
(replicate n a).min? = some a := by
|
||||
simp [min?_replicate, Nat.ne_of_gt h, w]
|
||||
@[simp] theorem minimum?_replicate_of_pos [Min α] {n : Nat} {a : α} (w : min a a = a) (h : 0 < n) :
|
||||
(replicate n a).minimum? = some a := by
|
||||
simp [minimum?_replicate, Nat.ne_of_gt h, w]
|
||||
|
||||
theorem foldl_min [Min α] [Std.IdempotentOp (min : α → α → α)] [Std.Associative (min : α → α → α)]
|
||||
{l : List α} {a : α} : l.foldl (init := a) min = min a (l.min?.getD a) := by
|
||||
cases l <;> simp [min?, foldl_assoc, Std.IdempotentOp.idempotent]
|
||||
/-! ### maximum? -/
|
||||
|
||||
/-! ### max? -/
|
||||
@[simp] theorem maximum?_nil [Max α] : ([] : List α).maximum? = none := rfl
|
||||
|
||||
@[simp] theorem max?_nil [Max α] : ([] : List α).max? = none := rfl
|
||||
|
||||
-- We don't put `@[simp]` on `max?_cons'`,
|
||||
-- We don't put `@[simp]` on `maximum?_cons`,
|
||||
-- because the definition in terms of `foldl` is not useful for proofs.
|
||||
theorem max?_cons' [Max α] {xs : List α} : (x :: xs).max? = foldl max x xs := rfl
|
||||
theorem maximum?_cons [Max α] {xs : List α} : (x :: xs).maximum? = foldl max x xs := rfl
|
||||
|
||||
@[simp] theorem max?_cons [Max α] [Std.Associative (max : α → α → α)] {xs : List α} :
|
||||
(x :: xs).max? = some (xs.max?.elim x (max x)) := by
|
||||
cases xs <;> simp [max?_cons', foldl_assoc]
|
||||
@[simp] theorem maximum?_eq_none_iff {xs : List α} [Max α] : xs.maximum? = none ↔ xs = [] := by
|
||||
cases xs <;> simp [maximum?]
|
||||
|
||||
@[simp] theorem max?_eq_none_iff {xs : List α} [Max α] : xs.max? = none ↔ xs = [] := by
|
||||
cases xs <;> simp [max?]
|
||||
|
||||
theorem isSome_max?_of_mem {l : List α} [Max α] {a : α} (h : a ∈ l) :
|
||||
l.max?.isSome := by
|
||||
cases l <;> simp_all [List.max?_cons']
|
||||
|
||||
theorem max?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
|
||||
{xs : List α} → xs.max? = some a → a ∈ xs
|
||||
theorem maximum?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
|
||||
{xs : List α} → xs.maximum? = some a → a ∈ xs
|
||||
| nil => by simp
|
||||
| cons x xs => by
|
||||
rw [max?]; rintro ⟨⟩
|
||||
rw [maximum?]; rintro ⟨⟩
|
||||
induction xs generalizing x with simp at *
|
||||
| cons y xs ih =>
|
||||
rcases ih (max x y) with h | h <;> simp [h]
|
||||
@@ -134,61 +114,40 @@ theorem max?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b
|
||||
|
||||
-- See also `Init.Data.List.Nat.Basic` for specialisations of the next two results to `Nat`.
|
||||
|
||||
theorem max?_le_iff [Max α] [LE α]
|
||||
theorem maximum?_le_iff [Max α] [LE α]
|
||||
(max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) :
|
||||
{xs : List α} → xs.max? = some a → ∀ {x}, a ≤ x ↔ ∀ b ∈ xs, b ≤ x
|
||||
{xs : List α} → xs.maximum? = some a → ∀ {x}, a ≤ x ↔ ∀ b ∈ xs, b ≤ x
|
||||
| nil => by simp
|
||||
| cons x xs => by
|
||||
rw [max?]; rintro ⟨⟩ y
|
||||
rw [maximum?]; rintro ⟨⟩ y
|
||||
induction xs generalizing x with
|
||||
| nil => simp
|
||||
| cons y xs ih => simp [ih, max_le_iff, and_assoc]
|
||||
|
||||
-- This could be refactored by designing appropriate typeclasses to replace `le_refl`, `max_eq_or`,
|
||||
-- and `le_min_iff`.
|
||||
theorem max?_eq_some_iff [Max α] [LE α] [anti : Std.Antisymm ((· : α) ≤ ·)]
|
||||
theorem maximum?_eq_some_iff [Max α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
|
||||
(le_refl : ∀ a : α, a ≤ a)
|
||||
(max_eq_or : ∀ a b : α, max a b = a ∨ max a b = b)
|
||||
(max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) {xs : List α} :
|
||||
xs.max? = some a ↔ a ∈ xs ∧ ∀ b ∈ xs, b ≤ a := by
|
||||
refine ⟨fun h => ⟨max?_mem max_eq_or h, (max?_le_iff max_le_iff h).1 (le_refl _)⟩, ?_⟩
|
||||
xs.maximum? = some a ↔ a ∈ xs ∧ ∀ b ∈ xs, b ≤ a := by
|
||||
refine ⟨fun h => ⟨maximum?_mem max_eq_or h, (maximum?_le_iff max_le_iff h).1 (le_refl _)⟩, ?_⟩
|
||||
intro ⟨h₁, h₂⟩
|
||||
cases xs with
|
||||
| nil => simp at h₁
|
||||
| cons x xs =>
|
||||
exact congrArg some <| anti.1
|
||||
(h₂ _ (max?_mem max_eq_or (xs := x::xs) rfl))
|
||||
((max?_le_iff max_le_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
|
||||
(h₂ _ (maximum?_mem max_eq_or (xs := x::xs) rfl))
|
||||
((maximum?_le_iff max_le_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
|
||||
|
||||
theorem max?_replicate [Max α] {n : Nat} {a : α} (w : max a a = a) :
|
||||
(replicate n a).max? = if n = 0 then none else some a := by
|
||||
theorem maximum?_replicate [Max α] {n : Nat} {a : α} (w : max a a = a) :
|
||||
(replicate n a).maximum? = if n = 0 then none else some a := by
|
||||
induction n with
|
||||
| zero => rfl
|
||||
| succ n ih => cases n <;> simp_all [replicate_succ, max?_cons']
|
||||
| succ n ih => cases n <;> simp_all [replicate_succ, maximum?_cons]
|
||||
|
||||
@[simp] theorem max?_replicate_of_pos [Max α] {n : Nat} {a : α} (w : max a a = a) (h : 0 < n) :
|
||||
(replicate n a).max? = some a := by
|
||||
simp [max?_replicate, Nat.ne_of_gt h, w]
|
||||
|
||||
theorem foldl_max [Max α] [Std.IdempotentOp (max : α → α → α)] [Std.Associative (max : α → α → α)]
|
||||
{l : List α} {a : α} : l.foldl (init := a) max = max a (l.max?.getD a) := by
|
||||
cases l <;> simp [max?, foldl_assoc, Std.IdempotentOp.idempotent]
|
||||
|
||||
@[deprecated min?_nil (since := "2024-09-29")] abbrev minimum?_nil := @min?_nil
|
||||
@[deprecated min?_cons (since := "2024-09-29")] abbrev minimum?_cons := @min?_cons
|
||||
@[deprecated min?_eq_none_iff (since := "2024-09-29")] abbrev mininmum?_eq_none_iff := @min?_eq_none_iff
|
||||
@[deprecated min?_mem (since := "2024-09-29")] abbrev minimum?_mem := @min?_mem
|
||||
@[deprecated le_min?_iff (since := "2024-09-29")] abbrev le_minimum?_iff := @le_min?_iff
|
||||
@[deprecated min?_eq_some_iff (since := "2024-09-29")] abbrev minimum?_eq_some_iff := @min?_eq_some_iff
|
||||
@[deprecated min?_replicate (since := "2024-09-29")] abbrev minimum?_replicate := @min?_replicate
|
||||
@[deprecated min?_replicate_of_pos (since := "2024-09-29")] abbrev minimum?_replicate_of_pos := @min?_replicate_of_pos
|
||||
@[deprecated max?_nil (since := "2024-09-29")] abbrev maximum?_nil := @max?_nil
|
||||
@[deprecated max?_cons (since := "2024-09-29")] abbrev maximum?_cons := @max?_cons
|
||||
@[deprecated max?_eq_none_iff (since := "2024-09-29")] abbrev maximum?_eq_none_iff := @max?_eq_none_iff
|
||||
@[deprecated max?_mem (since := "2024-09-29")] abbrev maximum?_mem := @max?_mem
|
||||
@[deprecated max?_le_iff (since := "2024-09-29")] abbrev maximum?_le_iff := @max?_le_iff
|
||||
@[deprecated max?_eq_some_iff (since := "2024-09-29")] abbrev maximum?_eq_some_iff := @max?_eq_some_iff
|
||||
@[deprecated max?_replicate (since := "2024-09-29")] abbrev maximum?_replicate := @max?_replicate
|
||||
@[deprecated max?_replicate_of_pos (since := "2024-09-29")] abbrev maximum?_replicate_of_pos := @max?_replicate_of_pos
|
||||
@[simp] theorem maximum?_replicate_of_pos [Max α] {n : Nat} {a : α} (w : max a a = a) (h : 0 < n) :
|
||||
(replicate n a).maximum? = some a := by
|
||||
simp [maximum?_replicate, Nat.ne_of_gt h, w]
|
||||
|
||||
end List
|
||||
|
||||
@@ -5,7 +5,6 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.TakeDrop
|
||||
import Init.Data.List.Attach
|
||||
|
||||
/-!
|
||||
# Lemmas about `List.mapM` and `List.forM`.
|
||||
@@ -49,79 +48,9 @@ theorem mapM'_eq_mapM [Monad m] [LawfulMonad m] (f : α → m β) (l : List α)
|
||||
@[simp] theorem mapM_cons [Monad m] [LawfulMonad m] (f : α → m β) :
|
||||
(a :: l).mapM f = (return (← f a) :: (← l.mapM f)) := by simp [← mapM'_eq_mapM, mapM']
|
||||
|
||||
@[simp] theorem mapM_id {l : List α} {f : α → Id β} : l.mapM f = l.map f := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[simp] theorem mapM_append [Monad m] [LawfulMonad m] (f : α → m β) {l₁ l₂ : List α} :
|
||||
(l₁ ++ l₂).mapM f = (return (← l₁.mapM f) ++ (← l₂.mapM f)) := by induction l₁ <;> simp [*]
|
||||
|
||||
/-- Auxiliary lemma for `mapM_eq_reverse_foldlM_cons`. -/
|
||||
theorem foldlM_cons_eq_append [Monad m] [LawfulMonad m] (f : α → m β) (as : List α) (b : β) (bs : List β) :
|
||||
(as.foldlM (init := b :: bs) fun acc a => return ((← f a) :: acc)) =
|
||||
(· ++ b :: bs) <$> as.foldlM (init := []) fun acc a => return ((← f a) :: acc) := by
|
||||
induction as generalizing b bs with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [bind_pure_comp] at ih
|
||||
simp [ih, _root_.map_bind, Functor.map_map, Function.comp_def]
|
||||
|
||||
theorem mapM_eq_reverse_foldlM_cons [Monad m] [LawfulMonad m] (f : α → m β) (l : List α) :
|
||||
mapM f l = reverse <$> (l.foldlM (fun acc a => return ((← f a) :: acc)) []) := by
|
||||
rw [← mapM'_eq_mapM]
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [mapM'_cons, ih, bind_map_left, foldlM_cons, LawfulMonad.bind_assoc, pure_bind,
|
||||
foldlM_cons_eq_append, _root_.map_bind, Functor.map_map, Function.comp_def, reverse_append,
|
||||
reverse_cons, reverse_nil, nil_append, singleton_append]
|
||||
simp [bind_pure_comp]
|
||||
|
||||
/-! ### foldlM and foldrM -/
|
||||
|
||||
theorem foldlM_map [Monad m] (f : β₁ → β₂) (g : α → β₂ → m α) (l : List β₁) (init : α) :
|
||||
(l.map f).foldlM g init = l.foldlM (fun x y => g x (f y)) init := by
|
||||
induction l generalizing g init <;> simp [*]
|
||||
|
||||
theorem foldrM_map [Monad m] [LawfulMonad m] (f : β₁ → β₂) (g : β₂ → α → m α) (l : List β₁)
|
||||
(init : α) : (l.map f).foldrM g init = l.foldrM (fun x y => g (f x) y) init := by
|
||||
induction l generalizing g init <;> simp [*]
|
||||
|
||||
theorem foldlM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : γ → β → m γ) (l : List α) (init : γ) :
|
||||
(l.filterMap f).foldlM g init =
|
||||
l.foldlM (fun x y => match f y with | some b => g x b | none => pure x) init := by
|
||||
induction l generalizing init with
|
||||
| nil => rfl
|
||||
| cons a l ih =>
|
||||
simp only [filterMap_cons, foldlM_cons]
|
||||
cases f a <;> simp [ih]
|
||||
|
||||
theorem foldrM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : β → γ → m γ) (l : List α) (init : γ) :
|
||||
(l.filterMap f).foldrM g init =
|
||||
l.foldrM (fun x y => match f x with | some b => g b y | none => pure y) init := by
|
||||
induction l generalizing init with
|
||||
| nil => rfl
|
||||
| cons a l ih =>
|
||||
simp only [filterMap_cons, foldrM_cons]
|
||||
cases f a <;> simp [ih]
|
||||
|
||||
theorem foldlM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : β → α → m β) (l : List α) (init : β) :
|
||||
(l.filter p).foldlM g init =
|
||||
l.foldlM (fun x y => if p y then g x y else pure x) init := by
|
||||
induction l generalizing init with
|
||||
| nil => rfl
|
||||
| cons a l ih =>
|
||||
simp only [filter_cons, foldlM_cons]
|
||||
split <;> simp [ih]
|
||||
|
||||
theorem foldrM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : α → β → m β) (l : List α) (init : β) :
|
||||
(l.filter p).foldrM g init =
|
||||
l.foldrM (fun x y => if p x then g x y else pure y) init := by
|
||||
induction l generalizing init with
|
||||
| nil => rfl
|
||||
| cons a l ih =>
|
||||
simp only [filter_cons, foldrM_cons]
|
||||
split <;> simp [ih]
|
||||
|
||||
/-! ### forM -/
|
||||
|
||||
-- We use `List.forM` as the simp normal form, rather that `ForM.forM`.
|
||||
@@ -137,186 +66,4 @@ theorem foldrM_filter [Monad m] [LawfulMonad m] (p : α → Bool) (g : α → β
|
||||
(l₁ ++ l₂).forM f = (do l₁.forM f; l₂.forM f) := by
|
||||
induction l₁ <;> simp [*]
|
||||
|
||||
/-! ### forIn' -/
|
||||
|
||||
theorem forIn'_loop_congr [Monad m] {as bs : List α}
|
||||
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
|
||||
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
|
||||
{b : β} (ha : ∃ ys, ys ++ xs = as) (hb : ∃ ys, ys ++ xs = bs)
|
||||
(h : ∀ a m m' b, f a m b = g a m' b) : forIn'.loop as f xs b ha = forIn'.loop bs g xs b hb := by
|
||||
induction xs generalizing b with
|
||||
| nil => simp [forIn'.loop]
|
||||
| cons a xs ih =>
|
||||
simp only [forIn'.loop] at *
|
||||
congr 1
|
||||
· rw [h]
|
||||
· funext s
|
||||
obtain b | b := s
|
||||
· rfl
|
||||
· simp
|
||||
rw [ih]
|
||||
|
||||
@[simp] theorem forIn'_cons [Monad m] {a : α} {as : List α}
|
||||
(f : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)) (b : β) :
|
||||
forIn' (a::as) b f = f a (mem_cons_self a as) b >>=
|
||||
fun | ForInStep.done b => pure b | ForInStep.yield b => forIn' as b fun a' m b => f a' (mem_cons_of_mem a m) b := by
|
||||
simp only [forIn', List.forIn', forIn'.loop]
|
||||
congr 1
|
||||
funext s
|
||||
obtain b | b := s
|
||||
· rfl
|
||||
· apply forIn'_loop_congr
|
||||
intros
|
||||
rfl
|
||||
|
||||
@[simp] theorem forIn_cons [Monad m] (f : α → β → m (ForInStep β)) (a : α) (as : List α) (b : β) :
|
||||
forIn (a::as) b f = f a b >>= fun | ForInStep.done b => pure b | ForInStep.yield b => forIn as b f := by
|
||||
have := forIn'_cons (a := a) (as := as) (fun a' _ b => f a' b) b
|
||||
simpa only [forIn'_eq_forIn]
|
||||
|
||||
@[congr] theorem forIn'_congr [Monad m] {as bs : List α} (w : as = bs)
|
||||
{b b' : β} (hb : b = b')
|
||||
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
|
||||
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
|
||||
(h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
|
||||
forIn' as b f = forIn' bs b' g := by
|
||||
induction bs generalizing as b b' with
|
||||
| nil =>
|
||||
subst w
|
||||
simp [hb, forIn'_nil]
|
||||
| cons b bs ih =>
|
||||
cases as with
|
||||
| nil => simp at w
|
||||
| cons a as =>
|
||||
simp only [cons.injEq] at w
|
||||
obtain ⟨rfl, rfl⟩ := w
|
||||
simp only [forIn'_cons]
|
||||
congr 1
|
||||
· simp [h, hb]
|
||||
· funext s
|
||||
obtain b | b := s
|
||||
· rfl
|
||||
· simp
|
||||
rw [ih rfl rfl]
|
||||
intro a m b
|
||||
exact h a (mem_cons_of_mem _ m) b
|
||||
|
||||
/--
|
||||
We can express a for loop over a list as a fold,
|
||||
in which whenever we reach `.done b` we keep that value through the rest of the fold.
|
||||
-/
|
||||
theorem forIn'_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(l : List α) (f : (a : α) → a ∈ l → β → m (ForInStep β)) (init : β) :
|
||||
forIn' l init f = ForInStep.value <$>
|
||||
l.attach.foldlM (fun b ⟨a, m⟩ => match b with
|
||||
| .yield b => f a m b
|
||||
| .done b => pure (.done b)) (ForInStep.yield init) := by
|
||||
induction l generalizing init with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [forIn'_cons, attach_cons, foldlM_cons, _root_.map_bind]
|
||||
congr 1
|
||||
funext x
|
||||
match x with
|
||||
| .done b =>
|
||||
clear ih
|
||||
dsimp
|
||||
induction as with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [attach_cons, map_cons, map_map, Function.comp_def, foldlM_cons, pure_bind]
|
||||
specialize ih (fun a m b => f a (by
|
||||
simp only [mem_cons] at m
|
||||
rcases m with rfl|m
|
||||
· apply mem_cons_self
|
||||
· exact mem_cons_of_mem _ (mem_cons_of_mem _ m)) b)
|
||||
simp [ih, List.foldlM_map]
|
||||
| .yield b =>
|
||||
simp [ih, List.foldlM_map]
|
||||
|
||||
/-- We can express a for loop over a list which always yields as a fold. -/
|
||||
@[simp] theorem forIn'_yield_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(l : List α) (f : (a : α) → a ∈ l → β → m γ) (g : (a : α) → a ∈ l → β → γ → β) (init : β) :
|
||||
forIn' l init (fun a m b => (fun c => .yield (g a m b c)) <$> f a m b) =
|
||||
l.attach.foldlM (fun b ⟨a, m⟩ => g a m b <$> f a m b) init := by
|
||||
simp only [forIn'_eq_foldlM]
|
||||
generalize l.attach = l'
|
||||
induction l' generalizing init <;> simp_all
|
||||
|
||||
theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
|
||||
(l : List α) (f : (a : α) → a ∈ l → β → β) (init : β) :
|
||||
forIn' l init (fun a m b => pure (.yield (f a m b))) =
|
||||
pure (f := m) (l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init) := by
|
||||
simp only [forIn'_eq_foldlM]
|
||||
generalize l.attach = l'
|
||||
induction l' generalizing init <;> simp_all
|
||||
|
||||
@[simp] theorem forIn'_yield_eq_foldl
|
||||
(l : List α) (f : (a : α) → a ∈ l → β → β) (init : β) :
|
||||
forIn' (m := Id) l init (fun a m b => .yield (f a m b)) =
|
||||
l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init := by
|
||||
simp only [forIn'_eq_foldlM]
|
||||
generalize l.attach = l'
|
||||
induction l' generalizing init <;> simp_all
|
||||
|
||||
/--
|
||||
We can express a for loop over a list as a fold,
|
||||
in which whenever we reach `.done b` we keep that value through the rest of the fold.
|
||||
-/
|
||||
theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(f : α → β → m (ForInStep β)) (init : β) (l : List α) :
|
||||
forIn l init f = ForInStep.value <$>
|
||||
l.foldlM (fun b a => match b with
|
||||
| .yield b => f a b
|
||||
| .done b => pure (.done b)) (ForInStep.yield init) := by
|
||||
induction l generalizing init with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [foldlM_cons, bind_pure_comp, forIn_cons, _root_.map_bind]
|
||||
congr 1
|
||||
funext x
|
||||
match x with
|
||||
| .done b =>
|
||||
clear ih
|
||||
dsimp
|
||||
induction as with
|
||||
| nil => simp
|
||||
| cons a as ih => simp [ih]
|
||||
| .yield b =>
|
||||
simp [ih]
|
||||
|
||||
/-- We can express a for loop over a list which always yields as a fold. -/
|
||||
@[simp] theorem forIn_yield_eq_foldlM [Monad m] [LawfulMonad m]
|
||||
(l : List α) (f : α → β → m γ) (g : α → β → γ → β) (init : β) :
|
||||
forIn l init (fun a b => (fun c => .yield (g a b c)) <$> f a b) =
|
||||
l.foldlM (fun b a => g a b <$> f a b) init := by
|
||||
simp only [forIn_eq_foldlM]
|
||||
induction l generalizing init <;> simp_all
|
||||
|
||||
theorem forIn_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
|
||||
(l : List α) (f : α → β → β) (init : β) :
|
||||
forIn l init (fun a b => pure (.yield (f a b))) =
|
||||
pure (f := m) (l.foldl (fun b a => f a b) init) := by
|
||||
simp only [forIn_eq_foldlM]
|
||||
induction l generalizing init <;> simp_all
|
||||
|
||||
@[simp] theorem forIn_yield_eq_foldl
|
||||
(l : List α) (f : α → β → β) (init : β) :
|
||||
forIn (m := Id) l init (fun a b => .yield (f a b)) =
|
||||
l.foldl (fun b a => f a b) init := by
|
||||
simp only [forIn_eq_foldlM]
|
||||
induction l generalizing init <;> simp_all
|
||||
|
||||
/-! ### allM -/
|
||||
|
||||
theorem allM_eq_not_anyM_not [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
|
||||
allM p as = (! ·) <$> anyM ((! ·) <$> p ·) as := by
|
||||
induction as with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [allM, anyM, bind_map_left, _root_.map_bind]
|
||||
congr
|
||||
funext b
|
||||
split <;> simp_all
|
||||
|
||||
end List
|
||||
|
||||
@@ -10,8 +10,3 @@ import Init.Data.List.Nat.Range
|
||||
import Init.Data.List.Nat.Sublist
|
||||
import Init.Data.List.Nat.TakeDrop
|
||||
import Init.Data.List.Nat.Count
|
||||
import Init.Data.List.Nat.Erase
|
||||
import Init.Data.List.Nat.Find
|
||||
import Init.Data.List.Nat.BEq
|
||||
import Init.Data.List.Nat.Modify
|
||||
import Init.Data.List.Nat.InsertIdx
|
||||
|
||||
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Reference in New Issue
Block a user