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2513be6a09 |
2
.github/ISSUE_TEMPLATE/bug_report.md
vendored
2
.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
|
||||
|
||||
|
||||
6
.github/workflows/ci.yml
vendored
6
.github/workflows/ci.yml
vendored
@@ -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)}`)
|
||||
@@ -316,7 +316,7 @@ jobs:
|
||||
git fetch --depth=1 origin ${{ github.sha }}
|
||||
git checkout FETCH_HEAD flake.nix flake.lock
|
||||
if: github.event_name == 'pull_request'
|
||||
# (needs to be after "Checkout" so files don't get overridden)
|
||||
# (needs to be after "Checkout" so files don't get overriden)
|
||||
- name: Setup emsdk
|
||||
uses: mymindstorm/setup-emsdk@v12
|
||||
with:
|
||||
@@ -452,7 +452,7 @@ jobs:
|
||||
run: ccache -s
|
||||
|
||||
# This job collects results from all the matrix jobs
|
||||
# This can be made the "required" job, instead of listing each
|
||||
# This can be made the “required” job, instead of listing each
|
||||
# matrix job separately
|
||||
all-done:
|
||||
name: Build matrix complete
|
||||
|
||||
20
.github/workflows/pr-release.yml
vendored
20
.github/workflows/pr-release.yml
vendored
@@ -134,7 +134,7 @@ jobs:
|
||||
MESSAGE=""
|
||||
|
||||
if [[ -n "$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"
|
||||
@@ -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
|
||||
|
||||
|
||||
334
RELEASES.md
334
RELEASES.md
@@ -10,317 +10,7 @@ of each version.
|
||||
|
||||
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
|
||||
----------
|
||||
@@ -331,7 +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
|
||||
@@ -691,7 +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.
|
||||
@@ -753,7 +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
|
||||
@@ -789,7 +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.
|
||||
@@ -1250,7 +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
|
||||
@@ -1869,7 +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.
|
||||
@@ -1880,7 +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).
|
||||
@@ -1973,11 +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`.
|
||||
-/
|
||||
@@ -2189,7 +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 \
|
||||
@@ -2212,7 +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
|
||||
|
||||
@@ -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 +82,7 @@ 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. The tactic `rename_i` is used to rename "inaccessible" variables.
|
||||
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`.
|
||||
|
||||
@@ -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:
|
||||
|
||||
@@ -18,14 +18,14 @@ the stdlib.
|
||||
## 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:
|
||||
@@ -61,7 +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
|
||||
@@ -81,6 +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`).
|
||||
|
||||
@@ -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
|
||||
|
||||
14
flake.nix
14
flake.nix
@@ -39,19 +39,7 @@
|
||||
CTEST_OUTPUT_ON_FAILURE = 1;
|
||||
} // pkgs.lib.optionalAttrs pkgs.stdenv.isLinux {
|
||||
GMP = pkgsDist.gmp.override { withStatic = true; };
|
||||
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;
|
||||
});
|
||||
LIBUV = pkgsDist.libuv.overrideAttrs (attrs: { configureFlags = ["--enable-static"]; });
|
||||
GLIBC = pkgsDist.glibc;
|
||||
GLIBC_DEV = pkgsDist.glibc.dev;
|
||||
GCC_LIB = pkgsDist.gcc.cc.lib;
|
||||
|
||||
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 -lpthread -ldl -lrt -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=''"
|
||||
|
||||
@@ -31,15 +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
|
||||
cp /clang64/lib/lib{m,bcrypt,mingw32,moldname,mingwex,msvcrt,pthread,advapi32,shell32,user32,kernel32,ucrtbase,psapi,iphlpapi,userenv,ws2_32,dbghelp,ole32}.* /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'"
|
||||
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 $(pkg-config --libs libuv) -lucrtbase'"
|
||||
echo -n " -DLEAN_EXTRA_LINKER_FLAGS='-lgmp -luv -lucrtbase'"
|
||||
# do not set `LEAN_CC` for tests
|
||||
echo -n " -DAUTO_THREAD_FINALIZATION=OFF -DSTAGE0_AUTO_THREAD_FINALIZATION=OFF"
|
||||
echo -n " -DLEAN_TEST_VARS=''"
|
||||
|
||||
@@ -243,56 +243,11 @@ if("${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 " ${LIBUV_LIBRARIES}")
|
||||
endif()
|
||||
@@ -567,10 +522,6 @@ if(${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} $<TARGET_OBJECTS:util>)
|
||||
@@ -611,10 +562,7 @@ if (${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
|
||||
|
||||
@@ -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
|
||||
|
||||
|
||||
@@ -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
|
||||
|
||||
@@ -25,7 +25,7 @@ theorem ext {x y : ExceptT ε m α} (h : x.run = y.run) : x = y := by
|
||||
@[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]
|
||||
@@ -43,7 +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]
|
||||
|
||||
@@ -62,7 +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 <$> x <*> y := by
|
||||
@@ -84,19 +84,14 @@ instance [Monad m] [LawfulMonad m] : LawfulMonad (ExceptT ε m) where
|
||||
pure_bind := by intros; apply ext; simp [run_bind]
|
||||
bind_assoc := by intros; apply ext; simp [run_bind]; apply bind_congr; intro a; cases a <;> simp
|
||||
|
||||
@[simp] theorem map_throw [Monad m] [LawfulMonad m] {α β : Type _} (f : α → β) (e : ε) :
|
||||
f <$> (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
|
||||
@@ -180,7 +175,7 @@ theorem ext {x y : StateT σ m α} (h : ∀ s, x.run s = y.run s) : x = y :=
|
||||
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
|
||||
|
||||
@@ -215,13 +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]
|
||||
@@ -229,7 +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
|
||||
|
||||
@@ -823,9 +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 :=
|
||||
@@ -838,9 +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
|
||||
@@ -1197,21 +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)
|
||||
|
||||
@@ -1896,8 +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 -/
|
||||
@@ -2060,7 +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
|
||||
|
||||
@@ -2076,7 +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
|
||||
|
||||
@@ -2092,7 +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
|
||||
|
||||
|
||||
@@ -33,6 +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
|
||||
@@ -40,4 +41,3 @@ import Init.Data.ULift
|
||||
import Init.Data.PLift
|
||||
import Init.Data.Zero
|
||||
import Init.Data.NeZero
|
||||
import Init.Data.Function
|
||||
|
||||
@@ -15,4 +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
|
||||
|
||||
@@ -5,7 +5,6 @@ Authors: Joachim Breitner, Mario Carneiro
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Mem
|
||||
import Init.Data.Array.Lemmas
|
||||
import Init.Data.List.Attach
|
||||
|
||||
namespace Array
|
||||
@@ -27,152 +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 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]
|
||||
|
||||
/-! ## 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
|
||||
|
||||
@[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
|
||||
|
||||
/-! ### 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
|
||||
|
||||
@@ -11,78 +11,45 @@ import Init.Data.UInt.Basic
|
||||
import Init.Data.Repr
|
||||
import Init.Data.ToString.Basic
|
||||
import Init.GetElem
|
||||
import Init.Data.List.ToArray
|
||||
universe u v w
|
||||
|
||||
/-! ### Array literal syntax -/
|
||||
|
||||
syntax "#[" withoutPosition(sepBy(term, ", ")) "]" : term
|
||||
|
||||
macro_rules
|
||||
| `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
|
||||
|
||||
variable {α : Type u}
|
||||
|
||||
namespace Array
|
||||
|
||||
/-! ### Preliminary theorems -/
|
||||
|
||||
@[simp] theorem size_set (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
|
||||
List.length_set ..
|
||||
|
||||
@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
|
||||
List.length_concat ..
|
||||
|
||||
theorem ext (a b : Array α)
|
||||
(h₁ : a.size = b.size)
|
||||
(h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
|
||||
: a = b := by
|
||||
let rec extAux (a b : List α)
|
||||
(h₁ : a.length = b.length)
|
||||
(h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
|
||||
: a = b := by
|
||||
induction a generalizing b with
|
||||
| nil =>
|
||||
cases b with
|
||||
| nil => rfl
|
||||
| cons b bs => rw [List.length_cons] at h₁; injection h₁
|
||||
| cons a as ih =>
|
||||
cases b with
|
||||
| nil => rw [List.length_cons] at h₁; injection h₁
|
||||
| cons b bs =>
|
||||
have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
|
||||
have hz₂ : 0 < (b::bs).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
|
||||
have headEq : a = b := h₂ 0 hz₁ hz₂
|
||||
have h₁' : as.length = bs.length := by rw [List.length_cons, List.length_cons] at h₁; injection h₁
|
||||
have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get ⟨i, hi₁⟩ = bs.get ⟨i, hi₂⟩ := by
|
||||
intro i hi₁ hi₂
|
||||
have hi₁' : i+1 < (a::as).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
|
||||
have hi₂' : i+1 < (b::bs).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
|
||||
have : (a::as).get ⟨i+1, hi₁'⟩ = (b::bs).get ⟨i+1, hi₂'⟩ := h₂ (i+1) hi₁' hi₂'
|
||||
apply this
|
||||
have tailEq : as = bs := ih bs h₁' h₂'
|
||||
rw [headEq, tailEq]
|
||||
cases a; cases b
|
||||
apply congrArg
|
||||
apply extAux
|
||||
assumption
|
||||
assumption
|
||||
|
||||
theorem ext' {as bs : Array α} (h : as.toList = bs.toList) : as = bs := by
|
||||
cases as; cases bs; simp at h; rw [h]
|
||||
|
||||
@[simp] theorem toArrayAux_eq (as : List α) (acc : Array α) : (as.toArrayAux acc).toList = acc.toList ++ as := by
|
||||
induction as generalizing acc <;> simp [*, List.toArrayAux, Array.push, List.append_assoc, List.concat_eq_append]
|
||||
|
||||
@[simp] theorem toList_toArray (as : List α) : as.toArray.toList = as := rfl
|
||||
|
||||
@[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [size]
|
||||
|
||||
@[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @toList_toArray
|
||||
variable {α : Type u}
|
||||
|
||||
@[deprecated Array.toList (since := "2024-09-10")] abbrev Array.data := @Array.toList
|
||||
|
||||
/-! ### Externs -/
|
||||
@[extern "lean_mk_array"]
|
||||
def mkArray {α : Type u} (n : Nat) (v : α) : Array α where
|
||||
toList := List.replicate n v
|
||||
|
||||
/--
|
||||
`ofFn f` with `f : Fin n → α` returns the list whose ith element is `f i`.
|
||||
```
|
||||
ofFn f = #[f 0, f 1, ... , f(n - 1)]
|
||||
``` -/
|
||||
def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
|
||||
/-- Auxiliary for `ofFn`. `ofFn.go f i acc = acc ++ #[f i, ..., f(n - 1)]` -/
|
||||
go (i : Nat) (acc : Array α) : Array α :=
|
||||
if h : i < n then go (i+1) (acc.push (f ⟨i, h⟩)) else acc
|
||||
termination_by n - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
/-- The array `#[0, 1, ..., n - 1]`. -/
|
||||
def range (n : Nat) : Array Nat :=
|
||||
n.fold (flip Array.push) (mkEmpty n)
|
||||
|
||||
@[simp] theorem size_mkArray (n : Nat) (v : α) : (mkArray n v).size = n :=
|
||||
List.length_replicate ..
|
||||
|
||||
instance : EmptyCollection (Array α) := ⟨Array.empty⟩
|
||||
instance : Inhabited (Array α) where
|
||||
default := Array.empty
|
||||
|
||||
@[simp] def isEmpty (a : Array α) : Bool :=
|
||||
a.size = 0
|
||||
|
||||
def singleton (v : α) : Array α :=
|
||||
mkArray 1 v
|
||||
|
||||
/-- Low-level version of `size` that directly queries the C array object cached size.
|
||||
While this is not provable, `usize` always returns the exact size of the array since
|
||||
@@ -98,6 +65,29 @@ def usize (a : @& Array α) : USize := a.size.toUSize
|
||||
def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
|
||||
a[i.toNat]
|
||||
|
||||
instance : GetElem (Array α) USize α fun xs i => i.toNat < xs.size where
|
||||
getElem xs i h := xs.uget i h
|
||||
|
||||
def back [Inhabited α] (a : Array α) : α :=
|
||||
a.get! (a.size - 1)
|
||||
|
||||
def get? (a : Array α) (i : Nat) : Option α :=
|
||||
if h : i < a.size then some a[i] else none
|
||||
|
||||
def back? (a : Array α) : Option α :=
|
||||
a.get? (a.size - 1)
|
||||
|
||||
-- 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]
|
||||
|
||||
@[simp] theorem size_set (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
|
||||
List.length_set ..
|
||||
|
||||
@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
|
||||
List.length_concat ..
|
||||
|
||||
/-- Low-level version of `fset` which is as fast as a C array fset.
|
||||
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
|
||||
`fset` may be slightly slower than `uset`. -/
|
||||
@@ -105,19 +95,6 @@ def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
|
||||
def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
|
||||
a.set ⟨i.toNat, h⟩ v
|
||||
|
||||
@[extern "lean_array_pop"]
|
||||
def pop (a : Array α) : Array α where
|
||||
toList := a.toList.dropLast
|
||||
|
||||
@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 := by
|
||||
match a with
|
||||
| ⟨[]⟩ => rfl
|
||||
| ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
|
||||
|
||||
@[extern "lean_mk_array"]
|
||||
def mkArray {α : Type u} (n : Nat) (v : α) : Array α where
|
||||
toList := List.replicate n v
|
||||
|
||||
/--
|
||||
Swaps two entries in an array.
|
||||
|
||||
@@ -131,10 +108,6 @@ def swap (a : Array α) (i j : @& Fin a.size) : Array α :=
|
||||
let a' := a.set i v₂
|
||||
a'.set (size_set a i v₂ ▸ j) v₁
|
||||
|
||||
@[simp] theorem size_swap (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size := by
|
||||
show ((a.set i (a.get j)).set (size_set a i _ ▸ j) (a.get i)).size = a.size
|
||||
rw [size_set, size_set]
|
||||
|
||||
/--
|
||||
Swaps two entries in an array, or returns the array unchanged if either index is out of bounds.
|
||||
|
||||
@@ -148,64 +121,6 @@ def swap! (a : Array α) (i j : @& Nat) : Array α :=
|
||||
else a
|
||||
else a
|
||||
|
||||
/-! ### GetElem instance for `USize`, backed by `uget` -/
|
||||
|
||||
instance : GetElem (Array α) USize α fun xs i => i.toNat < xs.size where
|
||||
getElem xs i h := xs.uget i h
|
||||
|
||||
/-! ### Definitions -/
|
||||
|
||||
instance : EmptyCollection (Array α) := ⟨Array.empty⟩
|
||||
instance : Inhabited (Array α) where
|
||||
default := Array.empty
|
||||
|
||||
@[simp] def isEmpty (a : Array α) : Bool :=
|
||||
a.size = 0
|
||||
|
||||
@[specialize]
|
||||
def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) :
|
||||
∀ (i : Nat) (_ : i ≤ a.size), Bool
|
||||
| 0, _ => true
|
||||
| i+1, h =>
|
||||
p a[i] (b[i]'(hsz ▸ h)) && isEqvAux a b hsz p i (Nat.le_trans (Nat.le_add_right i 1) h)
|
||||
|
||||
@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
|
||||
if h : a.size = b.size then
|
||||
isEqvAux a b h p a.size (Nat.le_refl a.size)
|
||||
else
|
||||
false
|
||||
|
||||
instance [BEq α] : BEq (Array α) :=
|
||||
⟨fun a b => isEqv a b BEq.beq⟩
|
||||
|
||||
/--
|
||||
`ofFn f` with `f : Fin n → α` returns the list whose ith element is `f i`.
|
||||
```
|
||||
ofFn f = #[f 0, f 1, ... , f(n - 1)]
|
||||
``` -/
|
||||
def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
|
||||
/-- Auxiliary for `ofFn`. `ofFn.go f i acc = acc ++ #[f i, ..., f(n - 1)]` -/
|
||||
@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
go (i : Nat) (acc : Array α) : Array α :=
|
||||
if h : i < n then go (i+1) (acc.push (f ⟨i, h⟩)) else acc
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
/-- The array `#[0, 1, ..., n - 1]`. -/
|
||||
def range (n : Nat) : Array Nat :=
|
||||
n.fold (flip Array.push) (mkEmpty n)
|
||||
|
||||
def singleton (v : α) : Array α :=
|
||||
mkArray 1 v
|
||||
|
||||
def back [Inhabited α] (a : Array α) : α :=
|
||||
a.get! (a.size - 1)
|
||||
|
||||
def get? (a : Array α) (i : Nat) : Option α :=
|
||||
if h : i < a.size then some a[i] else none
|
||||
|
||||
def back? (a : Array α) : Option α :=
|
||||
a.get? (a.size - 1)
|
||||
|
||||
@[inline] def swapAt (a : Array α) (i : Fin a.size) (v : α) : α × Array α :=
|
||||
let e := a.get i
|
||||
let a := a.set i v
|
||||
@@ -216,9 +131,13 @@ def swapAt! (a : Array α) (i : Nat) (v : α) : α × Array α :=
|
||||
if h : i < a.size then
|
||||
swapAt a ⟨i, h⟩ v
|
||||
else
|
||||
have : Inhabited (α × Array α) := ⟨(v, a)⟩
|
||||
have : Inhabited α := ⟨v⟩
|
||||
panic! ("index " ++ toString i ++ " out of bounds")
|
||||
|
||||
@[extern "lean_array_pop"]
|
||||
def pop (a : Array α) : Array α where
|
||||
toList := a.toList.dropLast
|
||||
|
||||
def shrink (a : Array α) (n : Nat) : Array α :=
|
||||
let rec loop
|
||||
| 0, a => a
|
||||
@@ -387,12 +306,12 @@ unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad
|
||||
def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
|
||||
-- Note: we cannot use `foldlM` here for the reference implementation because this calls
|
||||
-- `bind` and `pure` too many times. (We are not assuming `m` is a `LawfulMonad`)
|
||||
let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
map (i : Nat) (r : Array β) : m (Array β) := do
|
||||
if hlt : i < as.size then
|
||||
map (i+1) (r.push (← f as[i]))
|
||||
else
|
||||
pure r
|
||||
let rec map (i : Nat) (r : Array β) : m (Array β) := do
|
||||
if hlt : i < as.size then
|
||||
map (i+1) (r.push (← f as[i]))
|
||||
else
|
||||
pure r
|
||||
termination_by as.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
map 0 (mkEmpty as.size)
|
||||
|
||||
@@ -456,8 +375,7 @@ unsafe def anyMUnsafe {α : Type u} {m : Type → Type w} [Monad m] (p : α →
|
||||
@[implemented_by anyMUnsafe]
|
||||
def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
|
||||
let any (stop : Nat) (h : stop ≤ as.size) :=
|
||||
let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
loop (j : Nat) : m Bool := do
|
||||
let rec loop (j : Nat) : m Bool := do
|
||||
if hlt : j < stop then
|
||||
have : j < as.size := Nat.lt_of_lt_of_le hlt h
|
||||
if (← p as[j]) then
|
||||
@@ -466,6 +384,7 @@ def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as :
|
||||
loop (j+1)
|
||||
else
|
||||
pure false
|
||||
termination_by stop - j
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
loop start
|
||||
if h : stop ≤ as.size then
|
||||
@@ -547,28 +466,16 @@ def findRev? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
|
||||
|
||||
@[inline]
|
||||
def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
|
||||
let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
loop (j : Nat) :=
|
||||
let rec loop (j : Nat) :=
|
||||
if h : j < as.size then
|
||||
if p as[j] then some j else loop (j + 1)
|
||||
else none
|
||||
termination_by as.size - j
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
loop 0
|
||||
|
||||
def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
|
||||
a.findIdx? fun a => a == v
|
||||
|
||||
@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
def indexOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
|
||||
if h : i < a.size then
|
||||
let idx : Fin a.size := ⟨i, h⟩;
|
||||
if a.get idx == v then some idx
|
||||
else indexOfAux a v (i+1)
|
||||
else none
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
|
||||
indexOfAux a v 0
|
||||
a.findIdx? fun a => a == v
|
||||
|
||||
@[inline]
|
||||
def any (as : Array α) (p : α → Bool) (start := 0) (stop := as.size) : Bool :=
|
||||
@@ -584,6 +491,13 @@ def contains [BEq α] (as : Array α) (a : α) : Bool :=
|
||||
def elem [BEq α] (a : α) (as : Array α) : Bool :=
|
||||
as.contains a
|
||||
|
||||
@[inline] def getEvenElems (as : Array α) : Array α :=
|
||||
(·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
|
||||
if even then
|
||||
(false, r.push a)
|
||||
else
|
||||
(true, r)
|
||||
|
||||
/-- Convert a `Array α` into an `List α`. This is O(n) in the size of the array. -/
|
||||
-- This function is exported to C, where it is called by `Array.toList`
|
||||
-- (the projection) to implement this functionality.
|
||||
@@ -596,6 +510,17 @@ def toListImpl (as : Array α) : List α :=
|
||||
def toListAppend (as : Array α) (l : List α) : List α :=
|
||||
as.foldr List.cons l
|
||||
|
||||
instance {α : Type u} [Repr α] : Repr (Array α) where
|
||||
reprPrec a _ :=
|
||||
let _ : Std.ToFormat α := ⟨repr⟩
|
||||
if a.size == 0 then
|
||||
"#[]"
|
||||
else
|
||||
Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
|
||||
|
||||
instance [ToString α] : ToString (Array α) where
|
||||
toString a := "#" ++ toString a.toList
|
||||
|
||||
protected def append (as : Array α) (bs : Array α) : Array α :=
|
||||
bs.foldl (init := as) fun r v => r.push v
|
||||
|
||||
@@ -607,31 +532,58 @@ protected def appendList (as : Array α) (bs : List α) : Array α :=
|
||||
instance : HAppend (Array α) (List α) (Array α) := ⟨Array.appendList⟩
|
||||
|
||||
@[inline]
|
||||
def flatMapM [Monad m] (f : α → m (Array β)) (as : Array α) : m (Array β) :=
|
||||
def concatMapM [Monad m] (f : α → m (Array β)) (as : Array α) : m (Array β) :=
|
||||
as.foldlM (init := empty) fun bs a => do return bs ++ (← f a)
|
||||
|
||||
@[deprecated concatMapM (since := "2024-10-16")] abbrev concatMapM := @flatMapM
|
||||
|
||||
@[inline]
|
||||
def flatMap (f : α → Array β) (as : Array α) : Array β :=
|
||||
def concatMap (f : α → Array β) (as : Array α) : Array β :=
|
||||
as.foldl (init := empty) fun bs a => bs ++ f a
|
||||
|
||||
@[deprecated flatMap (since := "2024-10-16")] abbrev concatMap := @flatMap
|
||||
|
||||
/-- Joins array of array into a single array.
|
||||
|
||||
`flatten #[#[a₁, a₂, ⋯], #[b₁, b₂, ⋯], ⋯]` = `#[a₁, a₂, ⋯, b₁, b₂, ⋯]`
|
||||
-/
|
||||
@[inline] def flatten (as : Array (Array α)) : Array α :=
|
||||
def flatten (as : Array (Array α)) : Array α :=
|
||||
as.foldl (init := empty) fun r a => r ++ a
|
||||
|
||||
end Array
|
||||
|
||||
export Array (mkArray)
|
||||
|
||||
syntax "#[" withoutPosition(sepBy(term, ", ")) "]" : term
|
||||
|
||||
macro_rules
|
||||
| `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
|
||||
|
||||
namespace Array
|
||||
|
||||
-- TODO(Leo): cleanup
|
||||
@[specialize]
|
||||
def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
|
||||
if h : i < a.size then
|
||||
have : i < b.size := hsz ▸ h
|
||||
p a[i] b[i] && isEqvAux a b hsz p (i+1)
|
||||
else
|
||||
true
|
||||
termination_by a.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
|
||||
if h : a.size = b.size then
|
||||
isEqvAux a b h p 0
|
||||
else
|
||||
false
|
||||
|
||||
instance [BEq α] : BEq (Array α) :=
|
||||
⟨fun a b => isEqv a b BEq.beq⟩
|
||||
|
||||
@[inline]
|
||||
def filter (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Array α :=
|
||||
as.foldl (init := #[]) (start := start) (stop := stop) fun r a =>
|
||||
if p a then r.push a else r
|
||||
|
||||
@[inline]
|
||||
def filterM {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
|
||||
def filterM [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
|
||||
as.foldlM (init := #[]) (start := start) (stop := stop) fun r a => do
|
||||
if (← p a) then return r.push a else return r
|
||||
|
||||
@@ -666,25 +618,93 @@ def partition (p : α → Bool) (as : Array α) : Array α × Array α := Id.run
|
||||
cs := cs.push a
|
||||
return (bs, cs)
|
||||
|
||||
theorem ext (a b : Array α)
|
||||
(h₁ : a.size = b.size)
|
||||
(h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
|
||||
: a = b := by
|
||||
let rec extAux (a b : List α)
|
||||
(h₁ : a.length = b.length)
|
||||
(h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
|
||||
: a = b := by
|
||||
induction a generalizing b with
|
||||
| nil =>
|
||||
cases b with
|
||||
| nil => rfl
|
||||
| cons b bs => rw [List.length_cons] at h₁; injection h₁
|
||||
| cons a as ih =>
|
||||
cases b with
|
||||
| nil => rw [List.length_cons] at h₁; injection h₁
|
||||
| cons b bs =>
|
||||
have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
|
||||
have hz₂ : 0 < (b::bs).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
|
||||
have headEq : a = b := h₂ 0 hz₁ hz₂
|
||||
have h₁' : as.length = bs.length := by rw [List.length_cons, List.length_cons] at h₁; injection h₁
|
||||
have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get ⟨i, hi₁⟩ = bs.get ⟨i, hi₂⟩ := by
|
||||
intro i hi₁ hi₂
|
||||
have hi₁' : i+1 < (a::as).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
|
||||
have hi₂' : i+1 < (b::bs).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
|
||||
have : (a::as).get ⟨i+1, hi₁'⟩ = (b::bs).get ⟨i+1, hi₂'⟩ := h₂ (i+1) hi₁' hi₂'
|
||||
apply this
|
||||
have tailEq : as = bs := ih bs h₁' h₂'
|
||||
rw [headEq, tailEq]
|
||||
cases a; cases b
|
||||
apply congrArg
|
||||
apply extAux
|
||||
assumption
|
||||
assumption
|
||||
|
||||
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)
|
||||
|
||||
end Array
|
||||
|
||||
-- CLEANUP the following code
|
||||
namespace Array
|
||||
|
||||
def indexOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
|
||||
if h : i < a.size then
|
||||
let idx : Fin a.size := ⟨i, h⟩;
|
||||
if a.get idx == v then some idx
|
||||
else indexOfAux a v (i+1)
|
||||
else none
|
||||
termination_by a.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
|
||||
indexOfAux a v 0
|
||||
|
||||
@[simp] theorem size_swap (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size := by
|
||||
show ((a.set i (a.get j)).set (size_set a i _ ▸ j) (a.get i)).size = a.size
|
||||
rw [size_set, size_set]
|
||||
|
||||
@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 := by
|
||||
match a with
|
||||
| ⟨[]⟩ => rfl
|
||||
| ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
|
||||
|
||||
theorem reverse.termination {i j : Nat} (h : i < j) : j - 1 - (i + 1) < j - i := by
|
||||
rw [Nat.sub_sub, Nat.add_comm]
|
||||
exact Nat.lt_of_le_of_lt (Nat.pred_le _) (Nat.sub_succ_lt_self _ _ h)
|
||||
|
||||
def reverse (as : Array α) : Array α :=
|
||||
if h : as.size ≤ 1 then
|
||||
as
|
||||
else
|
||||
loop as 0 ⟨as.size - 1, Nat.pred_lt (mt (fun h : as.size = 0 => h ▸ by decide) h)⟩
|
||||
where
|
||||
termination {i j : Nat} (h : i < j) : j - 1 - (i + 1) < j - i := by
|
||||
rw [Nat.sub_sub, Nat.add_comm]
|
||||
exact Nat.lt_of_le_of_lt (Nat.pred_le _) (Nat.sub_succ_lt_self _ _ h)
|
||||
loop (as : Array α) (i : Nat) (j : Fin as.size) :=
|
||||
if h : i < j then
|
||||
have := termination h
|
||||
have := reverse.termination h
|
||||
let as := as.swap ⟨i, Nat.lt_trans h j.2⟩ j
|
||||
have : j-1 < as.size := by rw [size_swap]; exact Nat.lt_of_le_of_lt (Nat.pred_le _) j.2
|
||||
loop as (i+1) ⟨j-1, this⟩
|
||||
else
|
||||
as
|
||||
termination_by j - i
|
||||
|
||||
@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
def popWhile (p : α → Bool) (as : Array α) : Array α :=
|
||||
if h : as.size > 0 then
|
||||
if p (as.get ⟨as.size - 1, Nat.sub_lt h (by decide)⟩) then
|
||||
@@ -693,11 +713,11 @@ def popWhile (p : α → Bool) (as : Array α) : Array α :=
|
||||
as
|
||||
else
|
||||
as
|
||||
termination_by as.size
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
def takeWhile (p : α → Bool) (as : Array α) : Array α :=
|
||||
let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
go (i : Nat) (r : Array α) : Array α :=
|
||||
let rec go (i : Nat) (r : Array α) : Array α :=
|
||||
if h : i < as.size then
|
||||
let a := as.get ⟨i, h⟩
|
||||
if p a then
|
||||
@@ -706,6 +726,7 @@ def takeWhile (p : α → Bool) (as : Array α) : Array α :=
|
||||
r
|
||||
else
|
||||
r
|
||||
termination_by as.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
go 0 #[]
|
||||
|
||||
@@ -713,7 +734,6 @@ def takeWhile (p : α → Bool) (as : Array α) : Array α :=
|
||||
|
||||
This function takes worst case O(n) time because
|
||||
it has to backshift all elements at positions greater than `i`.-/
|
||||
@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
|
||||
if h : i.val + 1 < a.size then
|
||||
let a' := a.swap ⟨i.val + 1, h⟩ i
|
||||
@@ -724,8 +744,7 @@ def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
|
||||
termination_by a.size - i.val
|
||||
decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ i.isLt
|
||||
|
||||
-- This is required in `Lean.Data.PersistentHashMap`.
|
||||
@[simp] theorem size_feraseIdx (a : Array α) (i : Fin a.size) : (a.feraseIdx i).size = a.size - 1 := by
|
||||
theorem size_feraseIdx (a : Array α) (i : Fin a.size) : (a.feraseIdx i).size = a.size - 1 := by
|
||||
induction a, i using Array.feraseIdx.induct with
|
||||
| @case1 a i h a' _ ih =>
|
||||
unfold feraseIdx
|
||||
@@ -748,14 +767,14 @@ def erase [BEq α] (as : Array α) (a : α) : Array α :=
|
||||
|
||||
/-- Insert element `a` at position `i`. -/
|
||||
@[inline] def insertAt (as : Array α) (i : Fin (as.size + 1)) (a : α) : Array α :=
|
||||
let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
loop (as : Array α) (j : Fin as.size) :=
|
||||
let rec loop (as : Array α) (j : Fin as.size) :=
|
||||
if i.1 < j then
|
||||
let j' := ⟨j-1, Nat.lt_of_le_of_lt (Nat.pred_le _) j.2⟩
|
||||
let as := as.swap j' j
|
||||
loop as ⟨j', by rw [size_swap]; exact j'.2⟩
|
||||
else
|
||||
as
|
||||
termination_by j.1
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
let j := as.size
|
||||
let as := as.push a
|
||||
@@ -767,7 +786,41 @@ def insertAt! (as : Array α) (i : Nat) (a : α) : Array α :=
|
||||
insertAt as ⟨i, Nat.lt_succ_of_le h⟩ a
|
||||
else panic! "invalid index"
|
||||
|
||||
@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
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 ext' {as bs : Array α} (h : as.toList = bs.toList) : as = bs := by
|
||||
cases as; cases bs; simp at h; rw [h]
|
||||
|
||||
@[simp] theorem toArrayAux_eq (as : List α) (acc : Array α) : (as.toArrayAux acc).toList = acc.toList ++ as := by
|
||||
induction as generalizing acc <;> simp [*, List.toArrayAux, Array.push, List.append_assoc, List.concat_eq_append]
|
||||
|
||||
@[simp] theorem toList_toArray (as : List α) : as.toArray.toList = as := by
|
||||
simp [List.toArray, Array.mkEmpty]
|
||||
|
||||
@[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @toList_toArray
|
||||
|
||||
@[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [size]
|
||||
|
||||
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 [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, *]
|
||||
|
||||
def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) (i : Nat) : Bool :=
|
||||
if h : i < as.size then
|
||||
let a := as[i]
|
||||
@@ -779,6 +832,7 @@ def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) (i : N
|
||||
false
|
||||
else
|
||||
true
|
||||
termination_by as.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
/-- Return true iff `as` is a prefix of `bs`.
|
||||
@@ -789,8 +843,24 @@ def isPrefixOf [BEq α] (as bs : Array α) : Bool :=
|
||||
else
|
||||
false
|
||||
|
||||
@[semireducible, specialize] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) : Array γ :=
|
||||
private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat), i < as.size → Bool
|
||||
| 0, _ => true
|
||||
| i+1, h =>
|
||||
have : i < as.size := Nat.lt_trans (Nat.lt_succ_self _) h;
|
||||
a != as[i] && allDiffAuxAux as a i this
|
||||
|
||||
private def allDiffAux [BEq α] (as : Array α) (i : Nat) : Bool :=
|
||||
if h : i < as.size then
|
||||
allDiffAuxAux as as[i] i h && allDiffAux as (i+1)
|
||||
else
|
||||
true
|
||||
termination_by as.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
def allDiff [BEq α] (as : Array α) : Bool :=
|
||||
allDiffAux as 0
|
||||
|
||||
@[specialize] def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) : Array γ :=
|
||||
if h : i < as.size then
|
||||
let a := as[i]
|
||||
if h : i < bs.size then
|
||||
@@ -800,6 +870,7 @@ def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) (i : Nat)
|
||||
cs
|
||||
else
|
||||
cs
|
||||
termination_by as.size - i
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
@[inline] def zipWith (as : Array α) (bs : Array β) (f : α → β → γ) : Array γ :=
|
||||
@@ -815,66 +886,4 @@ def split (as : Array α) (p : α → Bool) : Array α × Array α :=
|
||||
as.foldl (init := (#[], #[])) fun (as, bs) a =>
|
||||
if p a then (as.push a, bs) else (as, bs.push a)
|
||||
|
||||
/-! ## Auxiliary functions used in metaprogramming.
|
||||
|
||||
We do not intend to provide verification theorems for these functions.
|
||||
-/
|
||||
|
||||
/-! ### eraseReps -/
|
||||
|
||||
/--
|
||||
`O(|l|)`. Erase repeated adjacent elements. Keeps the first occurrence of each run.
|
||||
* `eraseReps #[1, 3, 2, 2, 2, 3, 5] = #[1, 3, 2, 3, 5]`
|
||||
-/
|
||||
def eraseReps {α} [BEq α] (as : Array α) : Array α :=
|
||||
if h : 0 < as.size then
|
||||
let ⟨last, r⟩ := as.foldl (init := (as[0], #[])) fun ⟨last, r⟩ a =>
|
||||
if a == last then ⟨last, r⟩ else ⟨a, r.push last⟩
|
||||
r.push last
|
||||
else
|
||||
#[]
|
||||
|
||||
/-! ### allDiff -/
|
||||
|
||||
private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat), i < as.size → Bool
|
||||
| 0, _ => true
|
||||
| i+1, h =>
|
||||
have : i < as.size := Nat.lt_trans (Nat.lt_succ_self _) h;
|
||||
a != as[i] && allDiffAuxAux as a i this
|
||||
|
||||
@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
|
||||
private def allDiffAux [BEq α] (as : Array α) (i : Nat) : Bool :=
|
||||
if h : i < as.size then
|
||||
allDiffAuxAux as as[i] i h && allDiffAux as (i+1)
|
||||
else
|
||||
true
|
||||
decreasing_by simp_wf; decreasing_trivial_pre_omega
|
||||
|
||||
def allDiff [BEq α] (as : Array α) : Bool :=
|
||||
allDiffAux as 0
|
||||
|
||||
/-! ### getEvenElems -/
|
||||
|
||||
@[inline] def getEvenElems (as : Array α) : Array α :=
|
||||
(·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
|
||||
if even then
|
||||
(false, r.push a)
|
||||
else
|
||||
(true, r)
|
||||
|
||||
/-! ### Repr and ToString -/
|
||||
|
||||
instance {α : Type u} [Repr α] : Repr (Array α) where
|
||||
reprPrec a _ :=
|
||||
let _ : Std.ToFormat α := ⟨repr⟩
|
||||
if a.size == 0 then
|
||||
"#[]"
|
||||
else
|
||||
Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
|
||||
|
||||
instance [ToString α] : ToString (Array α) where
|
||||
toString a := "#" ++ toString a.toList
|
||||
|
||||
end Array
|
||||
|
||||
export Array (mkArray)
|
||||
|
||||
@@ -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 } :=
|
||||
|
||||
@@ -73,7 +73,7 @@ theorem foldr_eq_foldr_toList (f : α → β → β) (init : β) (arr : Array α
|
||||
|
||||
@[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_eq_foldl_toList]
|
||||
@@ -111,8 +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
|
||||
|
||||
@@ -5,49 +5,43 @@ Authors: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.Array.Basic
|
||||
import Init.Data.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 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, rel_of_isEqvAux r a b h a.size (Nat.le_refl ..) h'⟩
|
||||
· intro; contradiction
|
||||
|
||||
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 (fun x y => x = y) a b h
|
||||
exact ext _ _ h (fun i lt _ => by simpa using h' i lt)
|
||||
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 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 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_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) :=
|
||||
|
||||
@@ -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 [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, *]
|
||||
|
||||
end Array
|
||||
File diff suppressed because it is too large
Load Diff
@@ -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
|
||||
@@ -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
|
||||
|
||||
@@ -59,22 +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
|
||||
|
||||
@@ -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
|
||||
|
||||
@@ -1,7 +1,7 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: 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
|
||||
@@ -173,9 +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
|
||||
@@ -269,8 +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`.
|
||||
-/
|
||||
@@ -453,15 +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)`.
|
||||
@@ -474,35 +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
|
||||
@@ -653,7 +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⟩
|
||||
|
||||
@@ -676,13 +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) :=
|
||||
@@ -718,8 +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
|
||||
|
||||
|
||||
@@ -1,7 +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.get idx hidx, 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`.
|
||||
|
||||
@@ -206,18 +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)]
|
||||
@@ -235,26 +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]
|
||||
@@ -391,12 +306,12 @@ 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
|
||||
@@ -407,32 +322,27 @@ theorem setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (x : BitVec w) (i
|
||||
· 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 :
|
||||
@@ -440,23 +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 -/
|
||||
|
||||
/--
|
||||
@@ -499,22 +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]
|
||||
|
||||
/--
|
||||
@@ -527,385 +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 -/
|
||||
|
||||
/--
|
||||
@@ -942,18 +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]
|
||||
|
||||
@@ -1005,20 +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)) :
|
||||
|
||||
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
|
||||
@@ -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_left_inj : ∀ {x y z : Bool}, (x ^^ y) = (x ^^ z) ↔ y = z := bne_left_inj
|
||||
theorem xor_left_inj : ∀ {x y z : Bool}, xor x y = xor x z ↔ y = z := bne_left_inj
|
||||
|
||||
theorem xor_right_inj : ∀ {x y z : Bool}, (x ^^ z) = (y ^^ z) ↔ x = y := bne_right_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 _)
|
||||
|
||||
|
||||
@@ -245,7 +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
|
||||
|
||||
@@ -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 _)
|
||||
|
||||
@@ -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)
|
||||
|
||||
@@ -54,12 +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
|
||||
@@ -73,9 +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 [*]
|
||||
@@ -128,7 +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
|
||||
|
||||
@@ -175,24 +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 _
|
||||
@@ -226,32 +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.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
|
||||
@@ -375,10 +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
|
||||
|
||||
@@ -487,10 +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 :=
|
||||
@@ -520,7 +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) :
|
||||
@@ -558,19 +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)]
|
||||
|
||||
@@ -586,8 +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
|
||||
@@ -636,15 +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) : (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
|
||||
|
||||
@@ -655,7 +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 -/
|
||||
|
||||
@@ -823,12 +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]
|
||||
@@ -838,21 +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)]
|
||||
|
||||
@@ -72,35 +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
|
||||
|
||||
@@ -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
|
||||
@@ -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]
|
||||
|
||||
@@ -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
|
||||
|
||||
@@ -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
|
||||
|
||||
@@ -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
|
||||
|
||||
@@ -23,5 +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
|
||||
|
||||
@@ -73,7 +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]
|
||||
@@ -84,7 +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
|
||||
|
||||
@@ -130,6 +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
|
||||
@@ -294,66 +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.
|
||||
-/
|
||||
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.
|
||||
-/
|
||||
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 [*]
|
||||
|
||||
@@ -528,155 +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]
|
||||
|
||||
/-! ### 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
|
||||
|
||||
@@ -29,10 +29,9 @@ 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`, `bind`, `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`,
|
||||
@@ -44,7 +43,7 @@ The operations are organized as follow:
|
||||
* 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?`.
|
||||
* 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,
|
||||
@@ -219,8 +218,8 @@ 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)]
|
||||
@@ -369,7 +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 -/
|
||||
@@ -543,50 +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|)`. `join 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
|
||||
|
||||
@[deprecated flatten (since := "2024-10-14"), inherit_doc flatten] abbrev join := @flatten
|
||||
@[simp] theorem join_nil : List.join ([] : List (List α)) = [] := rfl
|
||||
@[simp] theorem join_cons : (l :: ls).join = l ++ ls.join := rfl
|
||||
|
||||
/-! ### pure -/
|
||||
|
||||
/-- `pure x = [x]` is the `pure` operation of the list monad. -/
|
||||
@[inline] protected def pure {α : Type u} (a : α) : List α := [a]
|
||||
|
||||
/-! ### 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 -/
|
||||
|
||||
@@ -1474,34 +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,
|
||||
@@ -1537,7 +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 -/
|
||||
|
||||
|
||||
@@ -235,8 +235,8 @@ theorem sizeOf_get [SizeOf α] (as : List α) (i : Fin as.length) : sizeOf (as.g
|
||||
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)
|
||||
|
||||
@@ -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 (config := { contextual := true }) [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 = Nat.sum (l.map (countP p)) := 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 = Nat.sum (l.map (count a)) := 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]
|
||||
|
||||
@@ -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
|
||||
|
||||
@@ -132,14 +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
|
||||
@@ -224,7 +224,7 @@ theorem find?_eq_some : xs.find? p = some b ↔ p b ∧ ∃ as bs, xs = as ++ b
|
||||
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⟩⟩
|
||||
@@ -326,35 +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]
|
||||
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
|
||||
@@ -371,25 +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
|
||||
@@ -624,18 +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
|
||||
@@ -790,15 +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 => 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,
|
||||
@@ -819,7 +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
|
||||
@@ -830,30 +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]
|
||||
@@ -918,7 +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⟩
|
||||
@@ -980,13 +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
|
||||
|
||||
@@ -31,7 +31,7 @@ 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`.
|
||||
@@ -93,29 +93,29 @@ The following operations are given `@[csimp]` replacements below:
|
||||
@[csimp] theorem foldr_eq_foldrTR : @foldr = @foldrTR := by
|
||||
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 -/
|
||||
|
||||
@@ -322,7 +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]
|
||||
|
||||
@@ -55,7 +55,7 @@ See also
|
||||
* `Init.Data.List.Erase` for lemmas about `List.eraseP` and `List.erase`.
|
||||
* `Init.Data.List.Find` for lemmas about `List.find?`, `List.findSome?`, `List.findIdx`,
|
||||
`List.findIdx?`, and `List.indexOf`
|
||||
* `Init.Data.List.MinMax` for lemmas about `List.min?` and `List.max?`.
|
||||
* `Init.Data.List.MinMax` for lemmas about `List.minimum?` and `List.maximum?`.
|
||||
* `Init.Data.List.Pairwise` for lemmas about `List.Pairwise` and `List.Nodup`.
|
||||
* `Init.Data.List.Sublist` for lemmas about `List.Subset`, `List.Sublist`, `List.IsPrefix`,
|
||||
`List.IsSuffix`, and `List.IsInfix`.
|
||||
@@ -191,7 +191,7 @@ theorem get?_eq_some : l.get? n = some a ↔ ∃ h, get l ⟨n, h⟩ = a :=
|
||||
⟨fun e =>
|
||||
have : n < length l := Nat.gt_of_not_le fun hn => by cases get?_len_le hn ▸ e
|
||||
⟨this, by rwa [get?_eq_get this, Option.some.injEq] at e⟩,
|
||||
fun ⟨_, e⟩ => e ▸ get?_eq_get _⟩
|
||||
fun ⟨h, e⟩ => e ▸ get?_eq_get _⟩
|
||||
|
||||
theorem get?_eq_none : l.get? n = none ↔ length l ≤ n :=
|
||||
⟨fun e => Nat.ge_of_not_lt (fun h' => by cases e ▸ get?_eq_some.2 ⟨h', rfl⟩), get?_len_le⟩
|
||||
@@ -203,9 +203,6 @@ theorem get?_eq_none : l.get? n = none ↔ length l ≤ n :=
|
||||
|
||||
@[simp] theorem get_eq_getElem (l : List α) (i : Fin l.length) : l.get i = l[i.1]'i.2 := rfl
|
||||
|
||||
theorem getElem?_eq_some {l : List α} : l[i]? = some a ↔ ∃ h : i < l.length, l[i]'h = a := by
|
||||
simpa using get?_eq_some
|
||||
|
||||
/--
|
||||
If one has `l.get i` in an expression (with `i : Fin l.length`) and `h : l = l'`,
|
||||
`rw [h]` will give a "motive it not type correct" error, as it cannot rewrite the
|
||||
@@ -269,15 +266,9 @@ theorem get!_len_le [Inhabited α] : ∀ {l : List α} {n}, length l ≤ n → l
|
||||
theorem getElem?_eq_some_iff {l : List α} : l[n]? = some a ↔ ∃ h : n < l.length, l[n] = a := by
|
||||
simp only [← get?_eq_getElem?, get?_eq_some, get_eq_getElem]
|
||||
|
||||
theorem some_eq_getElem?_iff {l : List α} : some a = l[n]? ↔ ∃ h : n < l.length, l[n] = a := by
|
||||
rw [eq_comm, getElem?_eq_some_iff]
|
||||
|
||||
@[simp] theorem getElem?_eq_none_iff : l[n]? = none ↔ length l ≤ n := by
|
||||
simp only [← get?_eq_getElem?, get?_eq_none]
|
||||
|
||||
@[simp] theorem none_eq_getElem?_iff {l : List α} {n : Nat} : none = l[n]? ↔ length l ≤ n := by
|
||||
simp [eq_comm (a := none)]
|
||||
|
||||
theorem getElem?_eq_none (h : length l ≤ n) : l[n]? = none := getElem?_eq_none_iff.mpr h
|
||||
|
||||
theorem getElem?_eq (l : List α) (i : Nat) :
|
||||
@@ -492,7 +483,7 @@ theorem getElem?_of_mem {a} {l : List α} (h : a ∈ l) : ∃ n : Nat, l[n]? = s
|
||||
theorem get?_of_mem {a} {l : List α} (h : a ∈ l) : ∃ n, l.get? n = some a :=
|
||||
let ⟨⟨n, _⟩, e⟩ := get_of_mem h; ⟨n, e ▸ get?_eq_get _⟩
|
||||
|
||||
@[simp] theorem getElem_mem : ∀ {l : List α} {n} (h : n < l.length), l[n]'h ∈ l
|
||||
theorem getElem_mem : ∀ {l : List α} {n} (h : n < l.length), l[n]'h ∈ l
|
||||
| _ :: _, 0, _ => .head ..
|
||||
| _ :: l, _+1, _ => .tail _ (getElem_mem (l := l) ..)
|
||||
|
||||
@@ -718,9 +709,9 @@ theorem set_eq_of_length_le {l : List α} {n : Nat} (h : l.length ≤ n) {a : α
|
||||
theorem set_comm (a b : α) : ∀ {n m : Nat} (l : List α), n ≠ m →
|
||||
(l.set n a).set m b = (l.set m b).set n a
|
||||
| _, _, [], _ => by simp
|
||||
| _+1, 0, _ :: _, _ => by simp [set]
|
||||
| 0, _+1, _ :: _, _ => by simp [set]
|
||||
| _+1, _+1, _ :: t, h =>
|
||||
| n+1, 0, _ :: _, _ => by simp [set]
|
||||
| 0, m+1, _ :: _, _ => by simp [set]
|
||||
| n+1, m+1, x :: t, h =>
|
||||
congrArg _ <| set_comm a b t fun h' => h <| Nat.succ_inj'.mpr h'
|
||||
|
||||
@[simp]
|
||||
@@ -881,20 +872,6 @@ theorem foldr_map' {α β : Type u} (g : α → β) (f : α → α → α) (f' :
|
||||
· simp
|
||||
· simp [*, h]
|
||||
|
||||
theorem foldl_assoc {op : α → α → α} [ha : Std.Associative op] :
|
||||
∀ {l : List α} {a₁ a₂}, l.foldl op (op a₁ a₂) = op a₁ (l.foldl op a₂)
|
||||
| [], a₁, a₂ => rfl
|
||||
| a :: l, a₁, a₂ => by
|
||||
simp only [foldl_cons, ha.assoc]
|
||||
rw [foldl_assoc]
|
||||
|
||||
theorem foldr_assoc {op : α → α → α} [ha : Std.Associative op] :
|
||||
∀ {l : List α} {a₁ a₂}, l.foldr op (op a₁ a₂) = op (l.foldr op a₁) a₂
|
||||
| [], a₁, a₂ => rfl
|
||||
| a :: l, a₁, a₂ => by
|
||||
simp only [foldr_cons, ha.assoc]
|
||||
rw [foldr_assoc]
|
||||
|
||||
theorem foldl_hom (f : α₁ → α₂) (g₁ : α₁ → β → α₁) (g₂ : α₂ → β → α₂) (l : List β) (init : α₁)
|
||||
(H : ∀ x y, g₂ (f x) y = f (g₁ x y)) : l.foldl g₂ (f init) = f (l.foldl g₁ init) := by
|
||||
induction l generalizing init <;> simp [*, H]
|
||||
@@ -955,38 +932,6 @@ def foldrRecOn {motive : β → Sort _} : ∀ (l : List α) (op : α → β →
|
||||
x (mem_cons_self x l) :=
|
||||
rfl
|
||||
|
||||
/--
|
||||
We can prove that two folds over the same list are related (by some arbitrary relation)
|
||||
if we know that the initial elements are related and the folding function, for each element of the list,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldl_rel {l : List α} {f g : β → α → β} {a b : β} (r : β → β → Prop)
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f c a) (g c' a)) :
|
||||
r (l.foldl (fun acc a => f acc a) a) (l.foldl (fun acc a => g acc a) b) := by
|
||||
induction l generalizing a b with
|
||||
| nil => simp_all
|
||||
| cons a l ih =>
|
||||
simp only [foldl_cons]
|
||||
apply ih
|
||||
· simp_all
|
||||
· exact fun a m c c' h => h' _ (by simp_all) _ _ h
|
||||
|
||||
/--
|
||||
We can prove that two folds over the same list are related (by some arbitrary relation)
|
||||
if we know that the initial elements are related and the folding function, for each element of the list,
|
||||
preserves the relation.
|
||||
-/
|
||||
theorem foldr_rel {l : List α} {f g : α → β → β} {a b : β} (r : β → β → Prop)
|
||||
(h : r a b) (h' : ∀ (a : α), a ∈ l → ∀ (c c' : β), r c c' → r (f a c) (g a c')) :
|
||||
r (l.foldr (fun a acc => f a acc) a) (l.foldr (fun a acc => g a acc) b) := by
|
||||
induction l generalizing a b with
|
||||
| nil => simp_all
|
||||
| cons a l ih =>
|
||||
simp only [foldr_cons]
|
||||
apply h'
|
||||
· simp
|
||||
· exact ih h fun a m c c' h => h' _ (by simp_all) _ _ h
|
||||
|
||||
/-! ### getLast -/
|
||||
|
||||
theorem getLast_eq_getElem : ∀ (l : List α) (h : l ≠ []),
|
||||
@@ -994,8 +939,8 @@ theorem getLast_eq_getElem : ∀ (l : List α) (h : l ≠ []),
|
||||
match l with
|
||||
| [] => contradiction
|
||||
| a :: l => exact Nat.le_refl _)
|
||||
| [_], _ => rfl
|
||||
| _ :: _ :: _, _ => by
|
||||
| [a], h => rfl
|
||||
| a :: b :: l, h => by
|
||||
simp [getLast, get, Nat.succ_sub_succ, getLast_eq_getElem]
|
||||
|
||||
@[deprecated getLast_eq_getElem (since := "2024-07-15")]
|
||||
@@ -1021,14 +966,14 @@ theorem getLast_eq_getLastD (a l h) : @getLast α (a::l) h = getLastD l a := by
|
||||
theorem getLast!_cons [Inhabited α] : @getLast! α _ (a::l) = getLastD l a := by
|
||||
simp [getLast!, getLast_eq_getLastD]
|
||||
|
||||
@[simp] theorem getLast_mem : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ l
|
||||
theorem getLast_mem : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ l
|
||||
| [], h => absurd rfl h
|
||||
| [_], _ => .head ..
|
||||
| _::a::l, _ => .tail _ <| getLast_mem (cons_ne_nil a l)
|
||||
|
||||
theorem getLast_mem_getLast? : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ getLast? l
|
||||
| [], h => by contradiction
|
||||
| _ :: _, _ => rfl
|
||||
| a :: l, _ => rfl
|
||||
|
||||
theorem getLastD_mem_cons : ∀ (l : List α) (a : α), getLastD l a ∈ a::l
|
||||
| [], _ => .head ..
|
||||
@@ -1100,11 +1045,6 @@ theorem head?_eq_getElem? : ∀ l : List α, head? l = l[0]?
|
||||
| [] => rfl
|
||||
| a :: l => by simp
|
||||
|
||||
theorem head_eq_getElem (l : List α) (h : l ≠ []) : head l h = l[0]'(length_pos.mpr h) := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
| cons _ _ => simp
|
||||
|
||||
theorem head_eq_iff_head?_eq_some {xs : List α} (h) : xs.head h = a ↔ xs.head? = some a := by
|
||||
cases xs with
|
||||
| nil => simp at h
|
||||
@@ -1119,7 +1059,7 @@ theorem head?_eq_some_iff {xs : List α} {a : α} : xs.head? = some a ↔ ∃ ys
|
||||
@[simp] theorem head?_isSome : l.head?.isSome ↔ l ≠ [] := by
|
||||
cases l <;> simp
|
||||
|
||||
@[simp] theorem head_mem : ∀ {l : List α} (h : l ≠ []), head l h ∈ l
|
||||
theorem head_mem : ∀ {l : List α} (h : l ≠ []), head l h ∈ l
|
||||
| [], h => absurd rfl h
|
||||
| _::_, _ => .head ..
|
||||
|
||||
@@ -1134,7 +1074,7 @@ theorem mem_of_mem_head? : ∀ {l : List α} {a : α}, a ∈ l.head? → a ∈ l
|
||||
|
||||
theorem head_mem_head? : ∀ {l : List α} (h : l ≠ []), head l h ∈ head? l
|
||||
| [], h => by contradiction
|
||||
| _ :: _, _ => rfl
|
||||
| a :: l, _ => rfl
|
||||
|
||||
theorem head?_concat {a : α} : (l ++ [a]).head? = l.head?.getD a := by
|
||||
cases l <;> simp
|
||||
@@ -1165,55 +1105,6 @@ theorem tail_eq_tail? (l) : @tail α l = (tail? l).getD [] := by simp [tail_eq_t
|
||||
theorem mem_of_mem_tail {a : α} {l : List α} (h : a ∈ tail l) : a ∈ l := by
|
||||
induction l <;> simp_all
|
||||
|
||||
theorem ne_nil_of_tail_ne_nil {l : List α} : l.tail ≠ [] → l ≠ [] := by
|
||||
cases l <;> simp
|
||||
|
||||
@[simp] theorem getElem_tail (l : List α) (i : Nat) (h : i < l.tail.length) :
|
||||
(tail l)[i] = l[i + 1]'(add_lt_of_lt_sub (by simpa using h)) := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
| cons _ l => simp
|
||||
|
||||
@[simp] theorem getElem?_tail (l : List α) (i : Nat) :
|
||||
(tail l)[i]? = l[i + 1]? := by
|
||||
cases l <;> simp
|
||||
|
||||
@[simp] theorem set_tail (l : List α) (i : Nat) (a : α) :
|
||||
l.tail.set i a = (l.set (i + 1) a).tail := by
|
||||
cases l <;> simp
|
||||
|
||||
theorem one_lt_length_of_tail_ne_nil {l : List α} (h : l.tail ≠ []) : 1 < l.length := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
| cons _ l =>
|
||||
simp only [tail_cons, ne_eq] at h
|
||||
exact Nat.lt_add_of_pos_left (length_pos.mpr h)
|
||||
|
||||
@[simp] theorem head_tail (l : List α) (h : l.tail ≠ []) :
|
||||
(tail l).head h = l[1]'(one_lt_length_of_tail_ne_nil h) := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
| cons _ l => simp [head_eq_getElem]
|
||||
|
||||
@[simp] theorem head?_tail (l : List α) : (tail l).head? = l[1]? := by
|
||||
simp [head?_eq_getElem?]
|
||||
|
||||
@[simp] theorem getLast_tail (l : List α) (h : l.tail ≠ []) :
|
||||
(tail l).getLast h = l.getLast (ne_nil_of_tail_ne_nil h) := by
|
||||
simp only [getLast_eq_getElem, length_tail, getElem_tail]
|
||||
congr
|
||||
match l with
|
||||
| _ :: _ :: l => simp
|
||||
|
||||
theorem getLast?_tail (l : List α) : (tail l).getLast? = if l.length = 1 then none else l.getLast? := by
|
||||
match l with
|
||||
| [] => simp
|
||||
| [a] => simp
|
||||
| _ :: _ :: l =>
|
||||
simp only [tail_cons, length_cons, getLast?_cons_cons]
|
||||
rw [if_neg]
|
||||
rintro ⟨⟩
|
||||
|
||||
/-! ## Basic operations -/
|
||||
|
||||
/-! ### map -/
|
||||
@@ -1331,24 +1222,19 @@ theorem map_eq_iff : map f l = l' ↔ ∀ i : Nat, l'[i]? = l[i]?.map f := by
|
||||
theorem map_eq_foldr (f : α → β) (l : List α) : map f l = foldr (fun a bs => f a :: bs) [] l := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
@[simp] theorem map_set {f : α → β} {l : List α} {i : Nat} {a : α} :
|
||||
(l.set i a).map f = (l.map f).set i (f a) := by
|
||||
induction l generalizing i with
|
||||
| nil => simp
|
||||
| cons b l ih => cases i <;> simp_all
|
||||
|
||||
@[deprecated "Use the reverse direction of `map_set`." (since := "2024-09-20")]
|
||||
theorem set_map {f : α → β} {l : List α} {n : Nat} {a : α} :
|
||||
@[simp] theorem set_map {f : α → β} {l : List α} {n : Nat} {a : α} :
|
||||
(map f l).set n (f a) = map f (l.set n a) := by
|
||||
simp
|
||||
induction l generalizing n with
|
||||
| nil => simp
|
||||
| cons b l ih => cases n <;> simp_all
|
||||
|
||||
@[simp] theorem head_map (f : α → β) (l : List α) (w) :
|
||||
(map f l).head w = f (l.head (by simpa using w)) := by
|
||||
head (map f l) w = f (head l (by simpa using w)) := by
|
||||
cases l
|
||||
· simp at w
|
||||
· simp_all
|
||||
|
||||
@[simp] theorem head?_map (f : α → β) (l : List α) : (map f l).head? = l.head?.map f := by
|
||||
@[simp] theorem head?_map (f : α → β) (l : List α) : head? (map f l) = (head? l).map f := by
|
||||
cases l <;> rfl
|
||||
|
||||
@[simp] theorem map_tail? (f : α → β) (l : List α) : (tail? l).map (map f) = tail? (map f l) := by
|
||||
@@ -1466,7 +1352,7 @@ theorem map_filter_eq_foldr (f : α → β) (p : α → Bool) (as : List α) :
|
||||
|
||||
@[simp] theorem filter_append {p : α → Bool} :
|
||||
∀ (l₁ l₂ : List α), filter p (l₁ ++ l₂) = filter p l₁ ++ filter p l₂
|
||||
| [], _ => rfl
|
||||
| [], l₂ => rfl
|
||||
| a :: l₁, l₂ => by simp [filter]; split <;> simp [filter_append l₁]
|
||||
|
||||
theorem filter_eq_cons_iff {l} {a} {as} :
|
||||
@@ -1671,11 +1557,6 @@ theorem filterMap_eq_cons_iff {l} {b} {bs} :
|
||||
|
||||
/-! ### append -/
|
||||
|
||||
@[simp] theorem nil_append_fun : (([] : List α) ++ ·) = id := rfl
|
||||
|
||||
@[simp] theorem cons_append_fun (a : α) (as : List α) :
|
||||
(fun bs => ((a :: as) ++ bs)) = fun bs => a :: (as ++ bs) := rfl
|
||||
|
||||
theorem getElem_append {l₁ l₂ : List α} (n : Nat) (h) :
|
||||
(l₁ ++ l₂)[n] = if h' : n < l₁.length then l₁[n] else l₂[n - l₁.length]'(by simp at h h'; exact Nat.sub_lt_left_of_lt_add h' h) := by
|
||||
split <;> rename_i h'
|
||||
@@ -1690,7 +1571,7 @@ theorem getElem?_append_left {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.leng
|
||||
|
||||
theorem getElem?_append_right : ∀ {l₁ l₂ : List α} {n : Nat}, l₁.length ≤ n →
|
||||
(l₁ ++ l₂)[n]? = l₂[n - l₁.length]?
|
||||
| [], _, _, _ => rfl
|
||||
| [], _, n, _ => rfl
|
||||
| a :: l, _, n+1, h₁ => by
|
||||
rw [cons_append]
|
||||
simp [Nat.succ_sub_succ_eq_sub, getElem?_append_right (Nat.lt_succ.1 h₁)]
|
||||
@@ -1755,8 +1636,8 @@ theorem append_of_mem {a : α} {l : List α} : a ∈ l → ∃ s t : List α, l
|
||||
|
||||
theorem append_inj :
|
||||
∀ {s₁ s₂ t₁ t₂ : List α}, s₁ ++ t₁ = s₂ ++ t₂ → length s₁ = length s₂ → s₁ = s₂ ∧ t₁ = t₂
|
||||
| [], [], _, _, h, _ => ⟨rfl, h⟩
|
||||
| _ :: _, _ :: _, _, _, h, hl => by
|
||||
| [], [], t₁, t₂, h, _ => ⟨rfl, h⟩
|
||||
| a :: s₁, b :: s₂, t₁, t₂, h, hl => by
|
||||
simp [append_inj (cons.inj h).2 (Nat.succ.inj hl)] at h ⊢; exact h
|
||||
|
||||
theorem append_inj_right (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length s₁ = length s₂) : t₁ = t₂ :=
|
||||
@@ -2011,7 +1892,7 @@ theorem map_eq_append_iff {f : α → β} :
|
||||
map f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
|
||||
rw [← filterMap_eq_map, filterMap_eq_append_iff]
|
||||
|
||||
theorem append_eq_map_iff {f : α → β} :
|
||||
theorem append_eq_map_iff (f : α → β) :
|
||||
L₁ ++ L₂ = map f l ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
|
||||
rw [eq_comm, map_eq_append_iff]
|
||||
|
||||
@@ -2068,98 +1949,106 @@ theorem eq_nil_or_concat : ∀ l : List α, l = [] ∨ ∃ L b, l = concat L b
|
||||
| _, .inl rfl => .inr ⟨[], a, rfl⟩
|
||||
| _, .inr ⟨L, b, rfl⟩ => .inr ⟨a::L, b, rfl⟩
|
||||
|
||||
/-! ### flatten -/
|
||||
/-! ### join -/
|
||||
|
||||
|
||||
@[simp] theorem length_flatten (L : List (List α)) : (flatten L).length = Nat.sum (L.map length) := by
|
||||
@[simp] theorem length_join (L : List (List α)) : (join L).length = Nat.sum (L.map length) := by
|
||||
induction L with
|
||||
| nil => rfl
|
||||
| cons =>
|
||||
simp [flatten, length_append, *]
|
||||
simp [join, length_append, *]
|
||||
|
||||
theorem flatten_singleton (l : List α) : [l].flatten = l := by simp
|
||||
theorem join_singleton (l : List α) : [l].join = l := by simp
|
||||
|
||||
@[simp] theorem mem_flatten : ∀ {L : List (List α)}, a ∈ L.flatten ↔ ∃ l, l ∈ L ∧ a ∈ l
|
||||
@[simp] theorem mem_join : ∀ {L : List (List α)}, a ∈ L.join ↔ ∃ l, l ∈ L ∧ a ∈ l
|
||||
| [] => by simp
|
||||
| b :: l => by simp [mem_flatten, or_and_right, exists_or]
|
||||
| b :: l => by simp [mem_join, or_and_right, exists_or]
|
||||
|
||||
@[simp] theorem flatten_eq_nil_iff {L : List (List α)} : L.flatten = [] ↔ ∀ l ∈ L, l = [] := by
|
||||
@[simp] theorem join_eq_nil_iff {L : List (List α)} : L.join = [] ↔ ∀ l ∈ L, l = [] := by
|
||||
induction L <;> simp_all
|
||||
|
||||
theorem flatten_ne_nil_iff {xs : List (List α)} : xs.flatten ≠ [] ↔ ∃ x, x ∈ xs ∧ x ≠ [] := by
|
||||
@[deprecated join_eq_nil_iff (since := "2024-09-05")] abbrev join_eq_nil := @join_eq_nil_iff
|
||||
|
||||
theorem join_ne_nil_iff {xs : List (List α)} : xs.join ≠ [] ↔ ∃ x, x ∈ xs ∧ x ≠ [] := by
|
||||
simp
|
||||
|
||||
theorem exists_of_mem_flatten : a ∈ flatten L → ∃ l, l ∈ L ∧ a ∈ l := mem_flatten.1
|
||||
@[deprecated join_ne_nil_iff (since := "2024-09-05")] abbrev join_ne_nil := @join_ne_nil_iff
|
||||
|
||||
theorem mem_flatten_of_mem (lL : l ∈ L) (al : a ∈ l) : a ∈ flatten L := mem_flatten.2 ⟨l, lL, al⟩
|
||||
theorem exists_of_mem_join : a ∈ join L → ∃ l, l ∈ L ∧ a ∈ l := mem_join.1
|
||||
|
||||
theorem forall_mem_flatten {p : α → Prop} {L : List (List α)} :
|
||||
(∀ (x) (_ : x ∈ flatten L), p x) ↔ ∀ (l) (_ : l ∈ L) (x) (_ : x ∈ l), p x := by
|
||||
simp only [mem_flatten, forall_exists_index, and_imp]
|
||||
theorem mem_join_of_mem (lL : l ∈ L) (al : a ∈ l) : a ∈ join L := mem_join.2 ⟨l, lL, al⟩
|
||||
|
||||
theorem forall_mem_join {p : α → Prop} {L : List (List α)} :
|
||||
(∀ (x) (_ : x ∈ join L), p x) ↔ ∀ (l) (_ : l ∈ L) (x) (_ : x ∈ l), p x := by
|
||||
simp only [mem_join, forall_exists_index, and_imp]
|
||||
constructor <;> (intros; solve_by_elim)
|
||||
|
||||
theorem flatten_eq_flatMap {L : List (List α)} : flatten L = L.flatMap id := by
|
||||
induction L <;> simp [List.flatMap]
|
||||
theorem join_eq_bind {L : List (List α)} : join L = L.bind id := by
|
||||
induction L <;> simp [List.bind]
|
||||
|
||||
theorem head?_flatten {L : List (List α)} : (flatten L).head? = L.findSome? fun l => l.head? := by
|
||||
theorem head?_join {L : List (List α)} : (join L).head? = L.findSome? fun l => l.head? := by
|
||||
induction L with
|
||||
| nil => rfl
|
||||
| cons =>
|
||||
simp only [findSome?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
-- `getLast?_flatten` is proved later, after the `reverse` section.
|
||||
-- `head_flatten` and `getLast_flatten` are proved in `Init.Data.List.Find`.
|
||||
-- `getLast?_join` is proved later, after the `reverse` section.
|
||||
-- `head_join` and `getLast_join` are proved in `Init.Data.List.Find`.
|
||||
|
||||
theorem foldl_flatten (f : β → α → β) (b : β) (L : List (List α)) :
|
||||
(flatten L).foldl f b = L.foldl (fun b l => l.foldl f b) b := by
|
||||
theorem foldl_join (f : β → α → β) (b : β) (L : List (List α)) :
|
||||
(join L).foldl f b = L.foldl (fun b l => l.foldl f b) b := by
|
||||
induction L generalizing b <;> simp_all
|
||||
|
||||
theorem foldr_flatten (f : α → β → β) (b : β) (L : List (List α)) :
|
||||
(flatten L).foldr f b = L.foldr (fun l b => l.foldr f b) b := by
|
||||
theorem foldr_join (f : α → β → β) (b : β) (L : List (List α)) :
|
||||
(join L).foldr f b = L.foldr (fun l b => l.foldr f b) b := by
|
||||
induction L <;> simp_all
|
||||
|
||||
@[simp] theorem map_flatten (f : α → β) (L : List (List α)) : map f (flatten L) = flatten (map (map f) L) := by
|
||||
@[simp] theorem map_join (f : α → β) (L : List (List α)) : map f (join L) = join (map (map f) L) := by
|
||||
induction L <;> simp_all
|
||||
|
||||
@[simp] theorem filterMap_flatten (f : α → Option β) (L : List (List α)) :
|
||||
filterMap f (flatten L) = flatten (map (filterMap f) L) := by
|
||||
@[simp] theorem filterMap_join (f : α → Option β) (L : List (List α)) :
|
||||
filterMap f (join L) = join (map (filterMap f) L) := by
|
||||
induction L <;> simp [*, filterMap_append]
|
||||
|
||||
@[simp] theorem filter_flatten (p : α → Bool) (L : List (List α)) :
|
||||
filter p (flatten L) = flatten (map (filter p) L) := by
|
||||
@[simp] theorem filter_join (p : α → Bool) (L : List (List α)) :
|
||||
filter p (join L) = join (map (filter p) L) := by
|
||||
induction L <;> simp [*, filter_append]
|
||||
|
||||
theorem flatten_filter_not_isEmpty :
|
||||
∀ {L : List (List α)}, flatten (L.filter fun l => !l.isEmpty) = L.flatten
|
||||
theorem join_filter_not_isEmpty :
|
||||
∀ {L : List (List α)}, join (L.filter fun l => !l.isEmpty) = L.join
|
||||
| [] => rfl
|
||||
| [] :: L
|
||||
| (a :: l) :: L => by
|
||||
simp [flatten_filter_not_isEmpty (L := L)]
|
||||
simp [join_filter_not_isEmpty (L := L)]
|
||||
|
||||
theorem flatten_filter_ne_nil [DecidablePred fun l : List α => l ≠ []] {L : List (List α)} :
|
||||
flatten (L.filter fun l => l ≠ []) = L.flatten := by
|
||||
theorem join_filter_ne_nil [DecidablePred fun l : List α => l ≠ []] {L : List (List α)} :
|
||||
join (L.filter fun l => l ≠ []) = L.join := by
|
||||
simp only [ne_eq, ← isEmpty_iff, Bool.not_eq_true, Bool.decide_eq_false,
|
||||
flatten_filter_not_isEmpty]
|
||||
join_filter_not_isEmpty]
|
||||
|
||||
@[simp] theorem flatten_append (L₁ L₂ : List (List α)) : flatten (L₁ ++ L₂) = flatten L₁ ++ flatten L₂ := by
|
||||
@[deprecated filter_join (since := "2024-08-26")]
|
||||
theorem join_map_filter (p : α → Bool) (l : List (List α)) :
|
||||
(l.map (filter p)).join = (l.join).filter p := by
|
||||
rw [filter_join]
|
||||
|
||||
@[simp] theorem join_append (L₁ L₂ : List (List α)) : join (L₁ ++ L₂) = join L₁ ++ join L₂ := by
|
||||
induction L₁ <;> simp_all
|
||||
|
||||
theorem flatten_concat (L : List (List α)) (l : List α) : flatten (L ++ [l]) = flatten L ++ l := by
|
||||
theorem join_concat (L : List (List α)) (l : List α) : join (L ++ [l]) = join L ++ l := by
|
||||
simp
|
||||
|
||||
theorem flatten_flatten {L : List (List (List α))} : flatten (flatten L) = flatten (map flatten L) := by
|
||||
theorem join_join {L : List (List (List α))} : join (join L) = join (map join L) := by
|
||||
induction L <;> simp_all
|
||||
|
||||
theorem flatten_eq_cons_iff {xs : List (List α)} {y : α} {ys : List α} :
|
||||
xs.flatten = y :: ys ↔
|
||||
∃ as bs cs, xs = as ++ (y :: bs) :: cs ∧ (∀ l, l ∈ as → l = []) ∧ ys = bs ++ cs.flatten := by
|
||||
theorem join_eq_cons_iff {xs : List (List α)} {y : α} {ys : List α} :
|
||||
xs.join = y :: ys ↔
|
||||
∃ as bs cs, xs = as ++ (y :: bs) :: cs ∧ (∀ l, l ∈ as → l = []) ∧ ys = bs ++ cs.join := by
|
||||
constructor
|
||||
· induction xs with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
intro h
|
||||
simp only [flatten_cons] at h
|
||||
simp only [join_cons] at h
|
||||
replace h := h.symm
|
||||
rw [cons_eq_append_iff] at h
|
||||
obtain (⟨rfl, h⟩ | ⟨z⟩) := h
|
||||
@@ -2170,23 +2059,23 @@ theorem flatten_eq_cons_iff {xs : List (List α)} {y : α} {ys : List α} :
|
||||
refine ⟨[], a', xs, ?_⟩
|
||||
simp
|
||||
· rintro ⟨as, bs, cs, rfl, h₁, rfl⟩
|
||||
simp [flatten_eq_nil_iff.mpr h₁]
|
||||
simp [join_eq_nil_iff.mpr h₁]
|
||||
|
||||
theorem flatten_eq_append_iff {xs : List (List α)} {ys zs : List α} :
|
||||
xs.flatten = ys ++ zs ↔
|
||||
(∃ as bs, xs = as ++ bs ∧ ys = as.flatten ∧ zs = bs.flatten) ∨
|
||||
∃ as bs c cs ds, xs = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.flatten ++ bs ∧
|
||||
zs = c :: cs ++ ds.flatten := by
|
||||
theorem join_eq_append_iff {xs : List (List α)} {ys zs : List α} :
|
||||
xs.join = ys ++ zs ↔
|
||||
(∃ as bs, xs = as ++ bs ∧ ys = as.join ∧ zs = bs.join) ∨
|
||||
∃ as bs c cs ds, xs = as ++ (bs ++ c :: cs) :: ds ∧ ys = as.join ++ bs ∧
|
||||
zs = c :: cs ++ ds.join := by
|
||||
constructor
|
||||
· induction xs generalizing ys with
|
||||
| nil =>
|
||||
simp only [flatten_nil, nil_eq, append_eq_nil, and_false, cons_append, false_and, exists_const,
|
||||
simp only [join_nil, nil_eq, append_eq_nil, and_false, cons_append, false_and, exists_const,
|
||||
exists_false, or_false, and_imp, List.cons_ne_nil]
|
||||
rintro rfl rfl
|
||||
exact ⟨[], [], by simp⟩
|
||||
| cons x xs ih =>
|
||||
intro h
|
||||
simp only [flatten_cons] at h
|
||||
simp only [join_cons] at h
|
||||
rw [append_eq_append_iff] at h
|
||||
obtain (⟨ys, rfl, h⟩ | ⟨c', rfl, h⟩) := h
|
||||
· obtain (⟨as, bs, rfl, rfl, rfl⟩ | ⟨as, bs, c, cs, ds, rfl, rfl, rfl⟩) := ih h
|
||||
@@ -2200,15 +2089,18 @@ theorem flatten_eq_append_iff {xs : List (List α)} {ys zs : List α} :
|
||||
· simp
|
||||
· simp
|
||||
|
||||
/-- Two lists of sublists are equal iff their flattens coincide, as well as the lengths of the
|
||||
@[deprecated join_eq_cons_iff (since := "2024-09-05")] abbrev join_eq_cons := @join_eq_cons_iff
|
||||
@[deprecated join_eq_append_iff (since := "2024-09-05")] abbrev join_eq_append := @join_eq_append_iff
|
||||
|
||||
/-- Two lists of sublists are equal iff their joins coincide, as well as the lengths of the
|
||||
sublists. -/
|
||||
theorem eq_iff_flatten_eq : ∀ {L L' : List (List α)},
|
||||
L = L' ↔ L.flatten = L'.flatten ∧ map length L = map length L'
|
||||
theorem eq_iff_join_eq : ∀ {L L' : List (List α)},
|
||||
L = L' ↔ L.join = L'.join ∧ map length L = map length L'
|
||||
| _, [] => by simp_all
|
||||
| [], x' :: L' => by simp_all
|
||||
| x :: L, x' :: L' => by
|
||||
simp
|
||||
rw [eq_iff_flatten_eq]
|
||||
rw [eq_iff_join_eq]
|
||||
constructor
|
||||
· rintro ⟨rfl, h₁, h₂⟩
|
||||
simp_all
|
||||
@@ -2216,86 +2108,86 @@ theorem eq_iff_flatten_eq : ∀ {L L' : List (List α)},
|
||||
obtain ⟨rfl, h⟩ := append_inj h₁ h₂
|
||||
exact ⟨rfl, h, h₃⟩
|
||||
|
||||
/-! ### flatMap -/
|
||||
/-! ### bind -/
|
||||
|
||||
theorem flatMap_def (l : List α) (f : α → List β) : l.flatMap f = flatten (map f l) := by rfl
|
||||
theorem bind_def (l : List α) (f : α → List β) : l.bind f = join (map f l) := by rfl
|
||||
|
||||
@[simp] theorem flatMap_id (l : List (List α)) : List.flatMap l id = l.flatten := by simp [flatMap_def]
|
||||
@[simp] theorem bind_id (l : List (List α)) : List.bind l id = l.join := by simp [bind_def]
|
||||
|
||||
@[simp] theorem mem_flatMap {f : α → List β} {b} {l : List α} : b ∈ l.flatMap f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by
|
||||
simp [flatMap_def, mem_flatten]
|
||||
@[simp] theorem mem_bind {f : α → List β} {b} {l : List α} : b ∈ l.bind f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by
|
||||
simp [bind_def, mem_join]
|
||||
exact ⟨fun ⟨_, ⟨a, h₁, rfl⟩, h₂⟩ => ⟨a, h₁, h₂⟩, fun ⟨a, h₁, h₂⟩ => ⟨_, ⟨a, h₁, rfl⟩, h₂⟩⟩
|
||||
|
||||
theorem exists_of_mem_flatMap {b : β} {l : List α} {f : α → List β} :
|
||||
b ∈ l.flatMap f → ∃ a, a ∈ l ∧ b ∈ f a := mem_flatMap.1
|
||||
theorem exists_of_mem_bind {b : β} {l : List α} {f : α → List β} :
|
||||
b ∈ l.bind f → ∃ a, a ∈ l ∧ b ∈ f a := mem_bind.1
|
||||
|
||||
theorem mem_flatMap_of_mem {b : β} {l : List α} {f : α → List β} {a} (al : a ∈ l) (h : b ∈ f a) :
|
||||
b ∈ l.flatMap f := mem_flatMap.2 ⟨a, al, h⟩
|
||||
theorem mem_bind_of_mem {b : β} {l : List α} {f : α → List β} {a} (al : a ∈ l) (h : b ∈ f a) :
|
||||
b ∈ l.bind f := mem_bind.2 ⟨a, al, h⟩
|
||||
|
||||
@[simp]
|
||||
theorem flatMap_eq_nil_iff {l : List α} {f : α → List β} : List.flatMap l f = [] ↔ ∀ x ∈ l, f x = [] :=
|
||||
flatten_eq_nil_iff.trans <| by
|
||||
theorem bind_eq_nil_iff {l : List α} {f : α → List β} : List.bind l f = [] ↔ ∀ x ∈ l, f x = [] :=
|
||||
join_eq_nil_iff.trans <| by
|
||||
simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
|
||||
|
||||
@[deprecated flatMap_eq_nil_iff (since := "2024-09-05")] abbrev bind_eq_nil := @flatMap_eq_nil_iff
|
||||
@[deprecated bind_eq_nil_iff (since := "2024-09-05")] abbrev bind_eq_nil := @bind_eq_nil_iff
|
||||
|
||||
theorem forall_mem_flatMap {p : β → Prop} {l : List α} {f : α → List β} :
|
||||
(∀ (x) (_ : x ∈ l.flatMap f), p x) ↔ ∀ (a) (_ : a ∈ l) (b) (_ : b ∈ f a), p b := by
|
||||
simp only [mem_flatMap, forall_exists_index, and_imp]
|
||||
theorem forall_mem_bind {p : β → Prop} {l : List α} {f : α → List β} :
|
||||
(∀ (x) (_ : x ∈ l.bind f), p x) ↔ ∀ (a) (_ : a ∈ l) (b) (_ : b ∈ f a), p b := by
|
||||
simp only [mem_bind, forall_exists_index, and_imp]
|
||||
constructor <;> (intros; solve_by_elim)
|
||||
|
||||
theorem flatMap_singleton (f : α → List β) (x : α) : [x].flatMap f = f x :=
|
||||
theorem bind_singleton (f : α → List β) (x : α) : [x].bind f = f x :=
|
||||
append_nil (f x)
|
||||
|
||||
@[simp] theorem flatMap_singleton' (l : List α) : (l.flatMap fun x => [x]) = l := by
|
||||
@[simp] theorem bind_singleton' (l : List α) : (l.bind fun x => [x]) = l := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
theorem head?_flatMap {l : List α} {f : α → List β} :
|
||||
(l.flatMap f).head? = l.findSome? fun a => (f a).head? := by
|
||||
theorem head?_bind {l : List α} {f : α → List β} :
|
||||
(l.bind f).head? = l.findSome? fun a => (f a).head? := by
|
||||
induction l with
|
||||
| nil => rfl
|
||||
| cons =>
|
||||
simp only [findSome?_cons]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem flatMap_append (xs ys : List α) (f : α → List β) :
|
||||
(xs ++ ys).flatMap f = xs.flatMap f ++ ys.flatMap f := by
|
||||
induction xs; {rfl}; simp_all [flatMap_cons, append_assoc]
|
||||
@[simp] theorem bind_append (xs ys : List α) (f : α → List β) :
|
||||
(xs ++ ys).bind f = xs.bind f ++ ys.bind f := by
|
||||
induction xs; {rfl}; simp_all [bind_cons, append_assoc]
|
||||
|
||||
@[deprecated flatMap_append (since := "2024-07-24")] abbrev append_bind := @flatMap_append
|
||||
@[deprecated bind_append (since := "2024-07-24")] abbrev append_bind := @bind_append
|
||||
|
||||
theorem flatMap_assoc {α β} (l : List α) (f : α → List β) (g : β → List γ) :
|
||||
(l.flatMap f).flatMap g = l.flatMap fun x => (f x).flatMap g := by
|
||||
theorem bind_assoc {α β} (l : List α) (f : α → List β) (g : β → List γ) :
|
||||
(l.bind f).bind g = l.bind fun x => (f x).bind g := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
theorem map_flatMap (f : β → γ) (g : α → List β) :
|
||||
∀ l : List α, (l.flatMap g).map f = l.flatMap fun a => (g a).map f
|
||||
theorem map_bind (f : β → γ) (g : α → List β) :
|
||||
∀ l : List α, (l.bind g).map f = l.bind fun a => (g a).map f
|
||||
| [] => rfl
|
||||
| a::l => by simp only [flatMap_cons, map_append, map_flatMap _ _ l]
|
||||
| a::l => by simp only [bind_cons, map_append, map_bind _ _ l]
|
||||
|
||||
theorem flatMap_map (f : α → β) (g : β → List γ) (l : List α) :
|
||||
(map f l).flatMap g = l.flatMap (fun a => g (f a)) := by
|
||||
induction l <;> simp [flatMap_cons, *]
|
||||
theorem bind_map (f : α → β) (g : β → List γ) (l : List α) :
|
||||
(map f l).bind g = l.bind (fun a => g (f a)) := by
|
||||
induction l <;> simp [bind_cons, *]
|
||||
|
||||
theorem map_eq_flatMap {α β} (f : α → β) (l : List α) : map f l = l.flatMap fun x => [f x] := by
|
||||
theorem map_eq_bind {α β} (f : α → β) (l : List α) : map f l = l.bind fun x => [f x] := by
|
||||
simp only [← map_singleton]
|
||||
rw [← flatMap_singleton' l, map_flatMap, flatMap_singleton']
|
||||
rw [← bind_singleton' l, map_bind, bind_singleton']
|
||||
|
||||
theorem filterMap_flatMap {β γ} (l : List α) (g : α → List β) (f : β → Option γ) :
|
||||
(l.flatMap g).filterMap f = l.flatMap fun a => (g a).filterMap f := by
|
||||
theorem filterMap_bind {β γ} (l : List α) (g : α → List β) (f : β → Option γ) :
|
||||
(l.bind g).filterMap f = l.bind fun a => (g a).filterMap f := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
theorem filter_flatMap (l : List α) (g : α → List β) (f : β → Bool) :
|
||||
(l.flatMap g).filter f = l.flatMap fun a => (g a).filter f := by
|
||||
theorem filter_bind (l : List α) (g : α → List β) (f : β → Bool) :
|
||||
(l.bind g).filter f = l.bind fun a => (g a).filter f := by
|
||||
induction l <;> simp [*]
|
||||
|
||||
theorem flatMap_eq_foldl (f : α → List β) (l : List α) :
|
||||
l.flatMap f = l.foldl (fun acc a => acc ++ f a) [] := by
|
||||
suffices ∀ l', l' ++ l.flatMap f = l.foldl (fun acc a => acc ++ f a) l' by simpa using this []
|
||||
theorem bind_eq_foldl (f : α → List β) (l : List α) :
|
||||
l.bind f = l.foldl (fun acc a => acc ++ f a) [] := by
|
||||
suffices ∀ l', l' ++ l.bind f = l.foldl (fun acc a => acc ++ f a) l' by simpa using this []
|
||||
intro l'
|
||||
induction l generalizing l'
|
||||
· simp
|
||||
· next ih => rw [flatMap_cons, ← append_assoc, ih, foldl_cons]
|
||||
· next ih => rw [bind_cons, ← append_assoc, ih, foldl_cons]
|
||||
|
||||
/-! ### replicate -/
|
||||
|
||||
@@ -2398,21 +2290,11 @@ theorem map_eq_replicate_iff {l : List α} {f : α → β} {b : β} :
|
||||
@[simp] theorem map_const (l : List α) (b : β) : map (Function.const α b) l = replicate l.length b :=
|
||||
map_eq_replicate_iff.mpr fun _ _ => rfl
|
||||
|
||||
@[simp] theorem map_const_fun (x : β) : map (Function.const α x) = (replicate ·.length x) := by
|
||||
funext l
|
||||
simp
|
||||
|
||||
/-- Variant of `map_const` using a lambda rather than `Function.const`. -/
|
||||
-- This can not be a `@[simp]` lemma because it would fire on every `List.map`.
|
||||
theorem map_const' (l : List α) (b : β) : map (fun _ => b) l = replicate l.length b :=
|
||||
map_const l b
|
||||
|
||||
@[simp] theorem set_replicate_self : (replicate n a).set i a = replicate n a := by
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· intro i h₁ h₂
|
||||
simp [getElem_set]
|
||||
|
||||
@[simp] theorem append_replicate_replicate : replicate n a ++ replicate m a = replicate (n + m) a := by
|
||||
rw [eq_replicate_iff]
|
||||
constructor
|
||||
@@ -2472,23 +2354,23 @@ theorem filterMap_replicate_of_some {f : α → Option β} (h : f a = some b) :
|
||||
(replicate n a).filterMap f = [] := by
|
||||
simp [filterMap_replicate, h]
|
||||
|
||||
@[simp] theorem flatten_replicate_nil : (replicate n ([] : List α)).flatten = [] := by
|
||||
@[simp] theorem join_replicate_nil : (replicate n ([] : List α)).join = [] := by
|
||||
induction n <;> simp_all [replicate_succ]
|
||||
|
||||
@[simp] theorem flatten_replicate_singleton : (replicate n [a]).flatten = replicate n a := by
|
||||
@[simp] theorem join_replicate_singleton : (replicate n [a]).join = replicate n a := by
|
||||
induction n <;> simp_all [replicate_succ]
|
||||
|
||||
@[simp] theorem flatten_replicate_replicate : (replicate n (replicate m a)).flatten = replicate (n * m) a := by
|
||||
@[simp] theorem join_replicate_replicate : (replicate n (replicate m a)).join = replicate (n * m) a := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
| succ n ih =>
|
||||
simp only [replicate_succ, flatten_cons, ih, append_replicate_replicate, replicate_inj, or_true,
|
||||
simp only [replicate_succ, join_cons, ih, append_replicate_replicate, replicate_inj, or_true,
|
||||
and_true, add_one_mul, Nat.add_comm]
|
||||
|
||||
theorem flatMap_replicate {β} (f : α → List β) : (replicate n a).flatMap f = (replicate n (f a)).flatten := by
|
||||
theorem bind_replicate {β} (f : α → List β) : (replicate n a).bind f = (replicate n (f a)).join := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
| succ n ih => simp only [replicate_succ, flatMap_cons, ih, flatten_cons]
|
||||
| succ n ih => simp only [replicate_succ, bind_cons, ih, join_cons]
|
||||
|
||||
@[simp] theorem isEmpty_replicate : (replicate n a).isEmpty = decide (n = 0) := by
|
||||
cases n <;> simp [replicate_succ]
|
||||
@@ -2663,20 +2545,20 @@ theorem reverse_eq_concat {xs ys : List α} {a : α} :
|
||||
xs.reverse = ys ++ [a] ↔ xs = a :: ys.reverse := by
|
||||
rw [reverse_eq_iff, reverse_concat]
|
||||
|
||||
/-- Reversing a flatten is the same as reversing the order of parts and reversing all parts. -/
|
||||
theorem reverse_flatten (L : List (List α)) :
|
||||
L.flatten.reverse = (L.map reverse).reverse.flatten := by
|
||||
/-- Reversing a join is the same as reversing the order of parts and reversing all parts. -/
|
||||
theorem reverse_join (L : List (List α)) :
|
||||
L.join.reverse = (L.map reverse).reverse.join := by
|
||||
induction L <;> simp_all
|
||||
|
||||
/-- Flattening a reverse is the same as reversing all parts and reversing the flattened result. -/
|
||||
theorem flatten_reverse (L : List (List α)) :
|
||||
L.reverse.flatten = (L.map reverse).flatten.reverse := by
|
||||
/-- Joining a reverse is the same as reversing all parts and reversing the joined result. -/
|
||||
theorem join_reverse (L : List (List α)) :
|
||||
L.reverse.join = (L.map reverse).join.reverse := by
|
||||
induction L <;> simp_all
|
||||
|
||||
theorem reverse_flatMap {β} (l : List α) (f : α → List β) : (l.flatMap f).reverse = l.reverse.flatMap (reverse ∘ f) := by
|
||||
theorem reverse_bind {β} (l : List α) (f : α → List β) : (l.bind f).reverse = l.reverse.bind (reverse ∘ f) := by
|
||||
induction l <;> simp_all
|
||||
|
||||
theorem flatMap_reverse {β} (l : List α) (f : α → List β) : (l.reverse.flatMap f) = (l.flatMap (reverse ∘ f)).reverse := by
|
||||
theorem bind_reverse {β} (l : List α) (f : α → List β) : (l.reverse.bind f) = (l.bind (reverse ∘ f)).reverse := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[simp] theorem reverseAux_eq (as bs : List α) : reverseAux as bs = reverse as ++ bs :=
|
||||
@@ -2696,7 +2578,7 @@ theorem flatMap_reverse {β} (l : List α) (f : α → List β) : (l.reverse.fla
|
||||
@[simp] theorem reverse_replicate (n) (a : α) : reverse (replicate n a) = replicate n a :=
|
||||
eq_replicate_iff.2
|
||||
⟨by rw [length_reverse, length_replicate],
|
||||
fun _ h => eq_of_mem_replicate (mem_reverse.1 h)⟩
|
||||
fun b h => eq_of_mem_replicate (mem_reverse.1 h)⟩
|
||||
|
||||
/-! #### Further results about `getLast` and `getLast?` -/
|
||||
|
||||
@@ -2784,15 +2666,15 @@ theorem getLast_filterMap_of_eq_some {f : α → Option β} {l : List α} {w : l
|
||||
rw [head_filterMap_of_eq_some (by simp_all)]
|
||||
simp_all
|
||||
|
||||
theorem getLast?_flatMap {L : List α} {f : α → List β} :
|
||||
(L.flatMap f).getLast? = L.reverse.findSome? fun a => (f a).getLast? := by
|
||||
simp only [← head?_reverse, reverse_flatMap]
|
||||
rw [head?_flatMap]
|
||||
theorem getLast?_bind {L : List α} {f : α → List β} :
|
||||
(L.bind f).getLast? = L.reverse.findSome? fun a => (f a).getLast? := by
|
||||
simp only [← head?_reverse, reverse_bind]
|
||||
rw [head?_bind]
|
||||
rfl
|
||||
|
||||
theorem getLast?_flatten {L : List (List α)} :
|
||||
(flatten L).getLast? = L.reverse.findSome? fun l => l.getLast? := by
|
||||
simp [← flatMap_id, getLast?_flatMap]
|
||||
theorem getLast?_join {L : List (List α)} :
|
||||
(join L).getLast? = L.reverse.findSome? fun l => l.getLast? := by
|
||||
simp [← bind_id, getLast?_bind]
|
||||
|
||||
theorem getLast?_replicate (a : α) (n : Nat) : (replicate n a).getLast? = if n = 0 then none else some a := by
|
||||
simp only [← head?_reverse, reverse_replicate, head?_replicate]
|
||||
@@ -2901,7 +2783,7 @@ theorem head?_dropLast (xs : List α) : xs.dropLast.head? = if 1 < xs.length the
|
||||
|
||||
theorem getLast_dropLast {xs : List α} (h) :
|
||||
xs.dropLast.getLast h =
|
||||
xs[xs.length - 2]'(match xs, h with | (_ :: _ :: _), _ => Nat.lt_trans (Nat.lt_add_one _) (Nat.lt_add_one _)) := by
|
||||
xs[xs.length - 2]'(match xs, h with | (a :: b :: xs), _ => Nat.lt_trans (Nat.lt_add_one _) (Nat.lt_add_one _)) := by
|
||||
rw [getLast_eq_getElem, getElem_dropLast]
|
||||
congr 1
|
||||
simp; rfl
|
||||
@@ -2925,8 +2807,8 @@ theorem dropLast_cons_of_ne_nil {α : Type u} {x : α}
|
||||
|
||||
theorem dropLast_concat_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l ++ [getLast l h] = l
|
||||
| [], h => absurd rfl h
|
||||
| [_], _ => rfl
|
||||
| _ :: b :: l, _ => by
|
||||
| [a], h => rfl
|
||||
| a :: b :: l, h => by
|
||||
rw [dropLast_cons₂, cons_append, getLast_cons (cons_ne_nil _ _)]
|
||||
congr
|
||||
exact dropLast_concat_getLast (cons_ne_nil b l)
|
||||
@@ -2965,12 +2847,6 @@ theorem dropLast_append_cons : dropLast (l₁ ++ b :: l₂) = l₁ ++ dropLast (
|
||||
dropLast (a :: replicate n a) = replicate n a := by
|
||||
rw [← replicate_succ, dropLast_replicate, Nat.add_sub_cancel]
|
||||
|
||||
@[simp] theorem tail_reverse (l : List α) : l.reverse.tail = l.dropLast.reverse := by
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· intro i h₁ h₂
|
||||
simp [Nat.add_comm i, Nat.sub_add_eq]
|
||||
|
||||
/-!
|
||||
### splitAt
|
||||
|
||||
@@ -3291,22 +3167,18 @@ theorem all_eq_not_any_not (l : List α) (p : α → Bool) : l.all p = !l.any (!
|
||||
| nil => rfl
|
||||
| cons h t ih => simp_all [Bool.and_assoc]
|
||||
|
||||
@[simp] theorem any_flatten {l : List (List α)} : l.flatten.any f = l.any (any · f) := by
|
||||
@[simp] theorem any_join {l : List (List α)} : l.join.any f = l.any (any · f) := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[deprecated any_flatten (since := "2024-10-14")] abbrev any_join := @any_flatten
|
||||
|
||||
@[simp] theorem all_flatten {l : List (List α)} : l.flatten.all f = l.all (all · f) := by
|
||||
@[simp] theorem all_join {l : List (List α)} : l.join.all f = l.all (all · f) := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[deprecated all_flatten (since := "2024-10-14")] abbrev all_join := @all_flatten
|
||||
|
||||
@[simp] theorem any_flatMap {l : List α} {f : α → List β} :
|
||||
(l.flatMap f).any p = l.any fun a => (f a).any p := by
|
||||
@[simp] theorem any_bind {l : List α} {f : α → List β} :
|
||||
(l.bind f).any p = l.any fun a => (f a).any p := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[simp] theorem all_flatMap {l : List α} {f : α → List β} :
|
||||
(l.flatMap f).all p = l.all fun a => (f a).all p := by
|
||||
@[simp] theorem all_bind {l : List α} {f : α → List β} :
|
||||
(l.bind f).all p = l.all fun a => (f a).all p := by
|
||||
induction l <;> simp_all
|
||||
|
||||
@[simp] theorem any_reverse {l : List α} : l.reverse.any f = l.any f := by
|
||||
@@ -3331,72 +3203,4 @@ theorem all_eq_not_any_not (l : List α) (p : α → Bool) : l.all p = !l.any (!
|
||||
(l.insert a).all f = (f a && l.all f) := by
|
||||
simp [all_eq]
|
||||
|
||||
/-! ### Deprecations -/
|
||||
|
||||
|
||||
@[deprecated flatten_nil (since := "2024-10-14")] abbrev join_nil := @flatten_nil
|
||||
@[deprecated flatten_cons (since := "2024-10-14")] abbrev join_cons := @flatten_cons
|
||||
@[deprecated length_flatten (since := "2024-10-14")] abbrev length_join := @length_flatten
|
||||
@[deprecated flatten_singleton (since := "2024-10-14")] abbrev join_singleton := @flatten_singleton
|
||||
@[deprecated mem_flatten (since := "2024-10-14")] abbrev mem_join := @mem_flatten
|
||||
@[deprecated flatten_eq_nil_iff (since := "2024-09-05")] abbrev join_eq_nil := @flatten_eq_nil_iff
|
||||
@[deprecated flatten_eq_nil_iff (since := "2024-10-14")] abbrev join_eq_nil_iff := @flatten_eq_nil_iff
|
||||
@[deprecated flatten_ne_nil_iff (since := "2024-09-05")] abbrev join_ne_nil := @flatten_ne_nil_iff
|
||||
@[deprecated flatten_ne_nil_iff (since := "2024-10-14")] abbrev join_ne_nil_iff := @flatten_ne_nil_iff
|
||||
@[deprecated exists_of_mem_flatten (since := "2024-10-14")] abbrev exists_of_mem_join := @exists_of_mem_flatten
|
||||
@[deprecated mem_flatten_of_mem (since := "2024-10-14")] abbrev mem_join_of_mem := @mem_flatten_of_mem
|
||||
@[deprecated forall_mem_flatten (since := "2024-10-14")] abbrev forall_mem_join := @forall_mem_flatten
|
||||
@[deprecated flatten_eq_flatMap (since := "2024-10-14")] abbrev join_eq_bind := @flatten_eq_flatMap
|
||||
@[deprecated head?_flatten (since := "2024-10-14")] abbrev head?_join := @head?_flatten
|
||||
@[deprecated foldl_flatten (since := "2024-10-14")] abbrev foldl_join := @foldl_flatten
|
||||
@[deprecated foldr_flatten (since := "2024-10-14")] abbrev foldr_join := @foldr_flatten
|
||||
@[deprecated map_flatten (since := "2024-10-14")] abbrev map_join := @map_flatten
|
||||
@[deprecated filterMap_flatten (since := "2024-10-14")] abbrev filterMap_join := @filterMap_flatten
|
||||
@[deprecated filter_flatten (since := "2024-10-14")] abbrev filter_join := @filter_flatten
|
||||
@[deprecated flatten_filter_not_isEmpty (since := "2024-10-14")] abbrev join_filter_not_isEmpty := @flatten_filter_not_isEmpty
|
||||
@[deprecated flatten_filter_ne_nil (since := "2024-10-14")] abbrev join_filter_ne_nil := @flatten_filter_ne_nil
|
||||
@[deprecated filter_flatten (since := "2024-08-26")]
|
||||
theorem join_map_filter (p : α → Bool) (l : List (List α)) :
|
||||
(l.map (filter p)).flatten = (l.flatten).filter p := by
|
||||
rw [filter_flatten]
|
||||
@[deprecated flatten_append (since := "2024-10-14")] abbrev join_append := @flatten_append
|
||||
@[deprecated flatten_concat (since := "2024-10-14")] abbrev join_concat := @flatten_concat
|
||||
@[deprecated flatten_flatten (since := "2024-10-14")] abbrev join_join := @flatten_flatten
|
||||
@[deprecated flatten_eq_cons_iff (since := "2024-09-05")] abbrev join_eq_cons_iff := @flatten_eq_cons_iff
|
||||
@[deprecated flatten_eq_cons_iff (since := "2024-09-05")] abbrev join_eq_cons := @flatten_eq_cons_iff
|
||||
@[deprecated flatten_eq_append_iff (since := "2024-09-05")] abbrev join_eq_append := @flatten_eq_append_iff
|
||||
@[deprecated flatten_eq_append_iff (since := "2024-10-14")] abbrev join_eq_append_iff := @flatten_eq_append_iff
|
||||
@[deprecated eq_iff_flatten_eq (since := "2024-10-14")] abbrev eq_iff_join_eq := @eq_iff_flatten_eq
|
||||
@[deprecated flatten_replicate_nil (since := "2024-10-14")] abbrev join_replicate_nil := @flatten_replicate_nil
|
||||
@[deprecated flatten_replicate_singleton (since := "2024-10-14")] abbrev join_replicate_singleton := @flatten_replicate_singleton
|
||||
@[deprecated flatten_replicate_replicate (since := "2024-10-14")] abbrev join_replicate_replicate := @flatten_replicate_replicate
|
||||
@[deprecated reverse_flatten (since := "2024-10-14")] abbrev reverse_join := @reverse_flatten
|
||||
@[deprecated flatten_reverse (since := "2024-10-14")] abbrev join_reverse := @flatten_reverse
|
||||
@[deprecated getLast?_flatten (since := "2024-10-14")] abbrev getLast?_join := @getLast?_flatten
|
||||
@[deprecated flatten_eq_flatMap (since := "2024-10-16")] abbrev flatten_eq_bind := @flatten_eq_flatMap
|
||||
@[deprecated flatMap_def (since := "2024-10-16")] abbrev bind_def := @flatMap_def
|
||||
@[deprecated flatMap_id (since := "2024-10-16")] abbrev bind_id := @flatMap_id
|
||||
@[deprecated mem_flatMap (since := "2024-10-16")] abbrev mem_bind := @mem_flatMap
|
||||
@[deprecated exists_of_mem_flatMap (since := "2024-10-16")] abbrev exists_of_mem_bind := @exists_of_mem_flatMap
|
||||
@[deprecated mem_flatMap_of_mem (since := "2024-10-16")] abbrev mem_bind_of_mem := @mem_flatMap_of_mem
|
||||
@[deprecated flatMap_eq_nil_iff (since := "2024-10-16")] abbrev bind_eq_nil_iff := @flatMap_eq_nil_iff
|
||||
@[deprecated forall_mem_flatMap (since := "2024-10-16")] abbrev forall_mem_bind := @forall_mem_flatMap
|
||||
@[deprecated flatMap_singleton (since := "2024-10-16")] abbrev bind_singleton := @flatMap_singleton
|
||||
@[deprecated flatMap_singleton' (since := "2024-10-16")] abbrev bind_singleton' := @flatMap_singleton'
|
||||
@[deprecated head?_flatMap (since := "2024-10-16")] abbrev head_bind := @head?_flatMap
|
||||
@[deprecated flatMap_append (since := "2024-10-16")] abbrev bind_append := @flatMap_append
|
||||
@[deprecated flatMap_assoc (since := "2024-10-16")] abbrev bind_assoc := @flatMap_assoc
|
||||
@[deprecated map_flatMap (since := "2024-10-16")] abbrev map_bind := @map_flatMap
|
||||
@[deprecated flatMap_map (since := "2024-10-16")] abbrev bind_map := @flatMap_map
|
||||
@[deprecated map_eq_flatMap (since := "2024-10-16")] abbrev map_eq_bind := @map_eq_flatMap
|
||||
@[deprecated filterMap_flatMap (since := "2024-10-16")] abbrev filterMap_bind := @filterMap_flatMap
|
||||
@[deprecated filter_flatMap (since := "2024-10-16")] abbrev filter_bind := @filter_flatMap
|
||||
@[deprecated flatMap_eq_foldl (since := "2024-10-16")] abbrev bind_eq_foldl := @flatMap_eq_foldl
|
||||
@[deprecated flatMap_replicate (since := "2024-10-16")] abbrev bind_replicate := @flatMap_replicate
|
||||
@[deprecated reverse_flatMap (since := "2024-10-16")] abbrev reverse_bind := @reverse_flatMap
|
||||
@[deprecated flatMap_reverse (since := "2024-10-16")] abbrev bind_reverse := @flatMap_reverse
|
||||
@[deprecated getLast?_flatMap (since := "2024-10-16")] abbrev getLast?_bind := @getLast?_flatMap
|
||||
@[deprecated any_flatMap (since := "2024-10-16")] abbrev any_bind := @any_flatMap
|
||||
@[deprecated all_flatMap (since := "2024-10-16")] abbrev all_bind := @all_flatMap
|
||||
|
||||
end List
|
||||
|
||||
@@ -1,248 +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
|
||||
|
||||
namespace List
|
||||
|
||||
/-! ## Operations using indexes -/
|
||||
|
||||
/-! ### mapIdx -/
|
||||
|
||||
/--
|
||||
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))
|
||||
|
||||
@[simp]
|
||||
theorem mapIdx_nil {f : Nat → α → β} : mapIdx f [] = [] :=
|
||||
rfl
|
||||
|
||||
theorem mapIdx_go_append {l₁ l₂ : List α} {arr : Array β} :
|
||||
mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (List.toArray (mapIdx.go f l₁ arr)) := by
|
||||
generalize h : (l₁ ++ l₂).length = len
|
||||
induction len generalizing l₁ arr with
|
||||
| zero =>
|
||||
have l₁_nil : l₁ = [] := by
|
||||
cases l₁
|
||||
· rfl
|
||||
· contradiction
|
||||
have l₂_nil : l₂ = [] := by
|
||||
cases l₂
|
||||
· rfl
|
||||
· rw [List.length_append] at h; contradiction
|
||||
rw [l₁_nil, l₂_nil]; simp only [mapIdx.go, List.toArray_toList]
|
||||
| succ len ih =>
|
||||
cases l₁ with
|
||||
| nil =>
|
||||
simp only [mapIdx.go, nil_append, List.toArray_toList]
|
||||
| cons head tail =>
|
||||
simp only [mapIdx.go, List.append_eq]
|
||||
rw [ih]
|
||||
· simp only [cons_append, length_cons, length_append, Nat.succ.injEq] at h
|
||||
simp only [length_append, h]
|
||||
|
||||
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]
|
||||
|
||||
@[simp] theorem mapIdx_concat {l : List α} {e : α} :
|
||||
mapIdx f (l ++ [e]) = mapIdx f l ++ [f l.length e] := by
|
||||
unfold mapIdx
|
||||
rw [mapIdx_go_append]
|
||||
simp only [mapIdx.go, Array.size_toArray, mapIdx_go_length, length_nil, Nat.add_zero,
|
||||
Array.push_toList]
|
||||
|
||||
@[simp] theorem mapIdx_singleton {a : α} : mapIdx f [a] = [f 0 a] := by
|
||||
simpa using mapIdx_concat (l := [])
|
||||
|
||||
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
|
||||
|
||||
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_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]
|
||||
|
||||
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_enum_map] at h
|
||||
replace h := exists_of_mem_map h
|
||||
simp only [Prod.exists, mk_mem_enum_iff_getElem?, Function.uncurry_apply_pair] 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_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 : 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 : 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
|
||||
|
||||
@@ -51,27 +51,6 @@ theorem mapM'_eq_mapM [Monad m] [LawfulMonad m] (f : α → m β) (l : List α)
|
||||
@[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 <$> (l.foldlM (fun acc a => return ((← f a) :: acc)) []) := by
|
||||
rw [← mapM'_eq_mapM]
|
||||
induction l with
|
||||
| nil => simp
|
||||
| cons a as ih =>
|
||||
simp only [mapM'_cons, ih, bind_map_left, foldlM_cons, LawfulMonad.bind_assoc, pure_bind,
|
||||
foldlM_cons_eq_append, _root_.map_bind, Functor.map_map, Function.comp_def, reverse_append,
|
||||
reverse_cons, reverse_nil, nil_append, singleton_append]
|
||||
simp [bind_pure_comp]
|
||||
|
||||
/-! ### forM -/
|
||||
|
||||
-- We use `List.forM` as the simp normal form, rather that `ForM.forM`.
|
||||
@@ -87,16 +66,4 @@ theorem mapM_eq_reverse_foldlM_cons [Monad m] [LawfulMonad m] (f : α → m β)
|
||||
(l₁ ++ l₂).forM f = (do l₁.forM f; l₂.forM f) := by
|
||||
induction l₁ <;> simp [*]
|
||||
|
||||
/-! ### allM -/
|
||||
|
||||
theorem allM_eq_not_anyM_not [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
|
||||
allM p as = (! ·) <$> 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
|
||||
|
||||
@@ -10,5 +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
|
||||
|
||||
@@ -5,7 +5,6 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Count
|
||||
import Init.Data.List.Find
|
||||
import Init.Data.List.MinMax
|
||||
import Init.Data.Nat.Lemmas
|
||||
|
||||
@@ -19,26 +18,6 @@ open Nat
|
||||
|
||||
namespace List
|
||||
|
||||
/-! ### dropLast -/
|
||||
|
||||
theorem tail_dropLast (l : List α) : tail (dropLast l) = dropLast (tail l) := by
|
||||
ext1
|
||||
simp only [getElem?_tail, getElem?_dropLast, length_tail]
|
||||
split <;> split
|
||||
· rfl
|
||||
· omega
|
||||
· omega
|
||||
· rfl
|
||||
|
||||
@[simp] theorem dropLast_reverse (l : List α) : l.reverse.dropLast = l.tail.reverse := by
|
||||
apply ext_getElem
|
||||
· simp
|
||||
· intro i h₁ h₂
|
||||
simp only [getElem_dropLast, getElem_reverse, length_tail, getElem_tail]
|
||||
congr
|
||||
simp only [length_dropLast, length_reverse, length_tail] at h₁ h₂
|
||||
omega
|
||||
|
||||
/-! ### filter -/
|
||||
|
||||
theorem length_filter_lt_length_iff_exists {l} :
|
||||
@@ -58,8 +37,7 @@ theorem getElem_eq_getElem_reverse {l : List α} {i} (h : i < l.length) :
|
||||
|
||||
/-- The length of the List returned by `List.leftpad n a l` is equal
|
||||
to the larger of `n` and `l.length` -/
|
||||
-- We don't mark this as a `@[simp]` lemma since we allow `simp` to unfold `leftpad`,
|
||||
-- so the left hand side simplifies directly to `n - l.length + l.length`.
|
||||
@[simp]
|
||||
theorem leftpad_length (n : Nat) (a : α) (l : List α) :
|
||||
(leftpad n a l).length = max n l.length := by
|
||||
simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max]
|
||||
@@ -86,66 +64,70 @@ theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃
|
||||
obtain ⟨h', -⟩ := getElem?_eq_some_iff.1 h
|
||||
exact ⟨h', h⟩
|
||||
|
||||
/-! ### min? -/
|
||||
/-! ### minimum? -/
|
||||
|
||||
-- A specialization of `min?_eq_some_iff` to Nat.
|
||||
theorem min?_eq_some_iff' {xs : List Nat} :
|
||||
xs.min? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
|
||||
min?_eq_some_iff
|
||||
-- A specialization of `minimum?_eq_some_iff` to Nat.
|
||||
theorem minimum?_eq_some_iff' {xs : List Nat} :
|
||||
xs.minimum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
|
||||
minimum?_eq_some_iff
|
||||
(le_refl := Nat.le_refl)
|
||||
(min_eq_or := fun _ _ => Nat.min_def .. ▸ by split <;> simp)
|
||||
(le_min_iff := fun _ _ _ => Nat.le_min)
|
||||
(min_eq_or := fun _ _ => by omega)
|
||||
(le_min_iff := fun _ _ _ => by omega)
|
||||
|
||||
theorem min?_get_le_of_mem {l : List Nat} {a : Nat} (h : a ∈ l) :
|
||||
l.min?.get (isSome_min?_of_mem h) ≤ a := by
|
||||
induction l with
|
||||
| nil => simp at h
|
||||
| cons b t ih =>
|
||||
simp only [min?_cons, Option.get_some] at ih ⊢
|
||||
rcases mem_cons.1 h with (rfl|h)
|
||||
· cases t.min? with
|
||||
| none => simp
|
||||
| some b => simpa using Nat.min_le_left _ _
|
||||
· obtain ⟨q, hq⟩ := Option.isSome_iff_exists.1 (isSome_min?_of_mem h)
|
||||
simp only [hq, Option.elim_some] at ih ⊢
|
||||
exact Nat.le_trans (Nat.min_le_right _ _) (ih h)
|
||||
-- This could be generalized,
|
||||
-- but will first require further work on order typeclasses in the core repository.
|
||||
theorem minimum?_cons' {a : Nat} {l : List Nat} :
|
||||
(a :: l).minimum? = some (match l.minimum? with
|
||||
| none => a
|
||||
| some m => min a m) := by
|
||||
rw [minimum?_eq_some_iff']
|
||||
split <;> rename_i h m
|
||||
· simp_all
|
||||
· rw [minimum?_eq_some_iff'] at m
|
||||
obtain ⟨m, le⟩ := m
|
||||
rw [Nat.min_def]
|
||||
constructor
|
||||
· split
|
||||
· exact mem_cons_self a l
|
||||
· exact mem_cons_of_mem a m
|
||||
· intro b m
|
||||
cases List.mem_cons.1 m with
|
||||
| inl => split <;> omega
|
||||
| inr h =>
|
||||
specialize le b h
|
||||
split <;> omega
|
||||
|
||||
theorem min?_getD_le_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) : l.min?.getD k ≤ a :=
|
||||
Option.get_eq_getD _ ▸ min?_get_le_of_mem h
|
||||
/-! ### maximum? -/
|
||||
|
||||
/-! ### max? -/
|
||||
|
||||
-- A specialization of `max?_eq_some_iff` to Nat.
|
||||
theorem max?_eq_some_iff' {xs : List Nat} :
|
||||
xs.max? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, b ≤ a) :=
|
||||
max?_eq_some_iff
|
||||
-- A specialization of `maximum?_eq_some_iff` to Nat.
|
||||
theorem maximum?_eq_some_iff' {xs : List Nat} :
|
||||
xs.maximum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, b ≤ a) :=
|
||||
maximum?_eq_some_iff
|
||||
(le_refl := Nat.le_refl)
|
||||
(max_eq_or := fun _ _ => Nat.max_def .. ▸ by split <;> simp)
|
||||
(max_le_iff := fun _ _ _ => Nat.max_le)
|
||||
(max_eq_or := fun _ _ => by omega)
|
||||
(max_le_iff := fun _ _ _ => by omega)
|
||||
|
||||
theorem le_max?_get_of_mem {l : List Nat} {a : Nat} (h : a ∈ l) :
|
||||
a ≤ l.max?.get (isSome_max?_of_mem h) := by
|
||||
induction l with
|
||||
| nil => simp at h
|
||||
| cons b t ih =>
|
||||
simp only [max?_cons, Option.get_some] at ih ⊢
|
||||
rcases mem_cons.1 h with (rfl|h)
|
||||
· cases t.max? with
|
||||
| none => simp
|
||||
| some b => simpa using Nat.le_max_left _ _
|
||||
· obtain ⟨q, hq⟩ := Option.isSome_iff_exists.1 (isSome_max?_of_mem h)
|
||||
simp only [hq, Option.elim_some] at ih ⊢
|
||||
exact Nat.le_trans (ih h) (Nat.le_max_right _ _)
|
||||
|
||||
theorem le_max?_getD_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
a ≤ l.max?.getD k :=
|
||||
Option.get_eq_getD _ ▸ le_max?_get_of_mem h
|
||||
|
||||
@[deprecated min?_eq_some_iff' (since := "2024-09-29")] abbrev minimum?_eq_some_iff' := @min?_eq_some_iff'
|
||||
@[deprecated min?_cons' (since := "2024-09-29")] abbrev minimum?_cons' := @min?_cons'
|
||||
@[deprecated min?_getD_le_of_mem (since := "2024-09-29")] abbrev minimum?_getD_le_of_mem := @min?_getD_le_of_mem
|
||||
@[deprecated max?_eq_some_iff' (since := "2024-09-29")] abbrev maximum?_eq_some_iff' := @max?_eq_some_iff'
|
||||
@[deprecated max?_cons' (since := "2024-09-29")] abbrev maximum?_cons' := @max?_cons'
|
||||
@[deprecated le_max?_getD_of_mem (since := "2024-09-29")] abbrev le_maximum?_getD_of_mem := @le_max?_getD_of_mem
|
||||
-- This could be generalized,
|
||||
-- but will first require further work on order typeclasses in the core repository.
|
||||
theorem maximum?_cons' {a : Nat} {l : List Nat} :
|
||||
(a :: l).maximum? = some (match l.maximum? with
|
||||
| none => a
|
||||
| some m => max a m) := by
|
||||
rw [maximum?_eq_some_iff']
|
||||
split <;> rename_i h m
|
||||
· simp_all
|
||||
· rw [maximum?_eq_some_iff'] at m
|
||||
obtain ⟨m, le⟩ := m
|
||||
rw [Nat.max_def]
|
||||
constructor
|
||||
· split
|
||||
· exact mem_cons_of_mem a m
|
||||
· exact mem_cons_self a l
|
||||
· intro b m
|
||||
cases List.mem_cons.1 m with
|
||||
| inl => split <;> omega
|
||||
| inr h =>
|
||||
specialize le b h
|
||||
split <;> omega
|
||||
|
||||
end List
|
||||
|
||||
@@ -28,59 +28,4 @@ theorem count_set [BEq α] (a b : α) (l : List α) (i : Nat) (h : i < l.length)
|
||||
(l.set i a).count b = l.count b - (if l[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
|
||||
simp [count_eq_countP, countP_set, h]
|
||||
|
||||
/--
|
||||
The number of elements satisfying a predicate in a sublist is at least the number of elements satisfying the predicate in the list,
|
||||
minus the difference in the lengths.
|
||||
-/
|
||||
theorem Sublist.le_countP (s : l₁ <+ l₂) (p) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ := by
|
||||
match s with
|
||||
| .slnil => simp
|
||||
| .cons a s =>
|
||||
rename_i l
|
||||
simp only [countP_cons, length_cons]
|
||||
have := s.le_countP p
|
||||
have := s.length_le
|
||||
split <;> omega
|
||||
| .cons₂ a s =>
|
||||
rename_i l₁ l₂
|
||||
simp only [countP_cons, length_cons]
|
||||
have := s.le_countP p
|
||||
have := s.length_le
|
||||
split <;> omega
|
||||
|
||||
theorem IsPrefix.le_countP (s : l₁ <+: l₂) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ :=
|
||||
s.sublist.le_countP _
|
||||
|
||||
theorem IsSuffix.le_countP (s : l₁ <:+ l₂) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ :=
|
||||
s.sublist.le_countP _
|
||||
|
||||
theorem IsInfix.le_countP (s : l₁ <:+: l₂) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ :=
|
||||
s.sublist.le_countP _
|
||||
|
||||
/--
|
||||
The number of elements satisfying a predicate in the tail of a list is
|
||||
at least one less than the number of elements satisfying the predicate in the list.
|
||||
-/
|
||||
theorem le_countP_tail (l) : countP p l - 1 ≤ countP p l.tail := by
|
||||
have := (tail_sublist l).le_countP p
|
||||
simp only [length_tail] at this
|
||||
omega
|
||||
|
||||
variable [BEq α]
|
||||
|
||||
theorem Sublist.le_count (s : l₁ <+ l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
|
||||
s.le_countP _
|
||||
|
||||
theorem IsPrefix.le_count (s : l₁ <+: l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
|
||||
s.sublist.le_count _
|
||||
|
||||
theorem IsSuffix.le_count (s : l₁ <:+ l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
|
||||
s.sublist.le_count _
|
||||
|
||||
theorem IsInfix.le_count (s : l₁ <:+: l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
|
||||
s.sublist.le_count _
|
||||
|
||||
theorem le_count_tail (a : α) (l) : count a l - 1 ≤ count a l.tail :=
|
||||
le_countP_tail _
|
||||
|
||||
end List
|
||||
|
||||
@@ -1,66 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Kim Morrison. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Nat.TakeDrop
|
||||
import Init.Data.List.Erase
|
||||
|
||||
namespace List
|
||||
|
||||
theorem getElem?_eraseIdx (l : List α) (i : Nat) (j : Nat) :
|
||||
(l.eraseIdx i)[j]? = if j < i then l[j]? else l[j + 1]? := by
|
||||
rw [eraseIdx_eq_take_drop_succ, getElem?_append]
|
||||
split <;> rename_i h
|
||||
· rw [getElem?_take]
|
||||
split
|
||||
· rfl
|
||||
· simp_all
|
||||
omega
|
||||
· rw [getElem?_drop]
|
||||
split <;> rename_i h'
|
||||
· simp only [length_take, Nat.min_def, Nat.not_lt] at h
|
||||
split at h
|
||||
· omega
|
||||
· simp_all [getElem?_eq_none]
|
||||
omega
|
||||
· simp only [length_take]
|
||||
simp only [length_take, Nat.min_def, Nat.not_lt] at h
|
||||
split at h
|
||||
· congr 1
|
||||
omega
|
||||
· rw [getElem?_eq_none, getElem?_eq_none] <;> omega
|
||||
|
||||
theorem getElem?_eraseIdx_of_lt (l : List α) (i : Nat) (j : Nat) (h : j < i) :
|
||||
(l.eraseIdx i)[j]? = l[j]? := by
|
||||
rw [getElem?_eraseIdx]
|
||||
simp [h]
|
||||
|
||||
theorem getElem?_eraseIdx_of_ge (l : List α) (i : Nat) (j : Nat) (h : i ≤ j) :
|
||||
(l.eraseIdx i)[j]? = l[j + 1]? := by
|
||||
rw [getElem?_eraseIdx]
|
||||
simp only [dite_eq_ite, ite_eq_right_iff]
|
||||
intro h'
|
||||
omega
|
||||
|
||||
theorem getElem_eraseIdx (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) :
|
||||
(l.eraseIdx i)[j] = if h' : j < i then
|
||||
l[j]'(by have := length_eraseIdx_le l i; omega)
|
||||
else
|
||||
l[j + 1]'(by rw [length_eraseIdx] at h; split at h <;> omega) := by
|
||||
apply Option.some.inj
|
||||
rw [← getElem?_eq_getElem, getElem?_eraseIdx]
|
||||
split <;> simp
|
||||
|
||||
theorem getElem_eraseIdx_of_lt (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) (h' : j < i) :
|
||||
(l.eraseIdx i)[j] = l[j]'(by have := length_eraseIdx_le l i; omega) := by
|
||||
rw [getElem_eraseIdx]
|
||||
simp only [dite_eq_left_iff, Nat.not_lt]
|
||||
intro h'
|
||||
omega
|
||||
|
||||
theorem getElem_eraseIdx_of_ge (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) (h' : i ≤ j) :
|
||||
(l.eraseIdx i)[j] = l[j + 1]'(by rw [length_eraseIdx] at h; split at h <;> omega) := by
|
||||
rw [getElem_eraseIdx, dif_neg]
|
||||
omega
|
||||
@@ -1,32 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Kim Morrison. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Nat.Range
|
||||
import Init.Data.List.Find
|
||||
|
||||
namespace List
|
||||
|
||||
theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : List α} {p q : α → Bool} (w : ∀ x ∈ xs, p x → q x) {i : Nat}
|
||||
(h : xs.findIdx? p = some i) : ∃ j, j ≤ i ∧ xs.findIdx? q = some j := by
|
||||
simp only [findIdx?_eq_findSome?_enum] at h
|
||||
rw [findSome?_eq_some_iff] at h
|
||||
simp only [Option.ite_none_right_eq_some, Option.some.injEq, ite_eq_right_iff, reduceCtorEq,
|
||||
imp_false, Bool.not_eq_true, Prod.forall, exists_and_right, Prod.exists] at h
|
||||
obtain ⟨h, h₁, b, ⟨es, h₂⟩, ⟨hb, rfl⟩, h₃⟩ := h
|
||||
rw [enum_eq_enumFrom, enumFrom_eq_append_iff] at h₂
|
||||
obtain ⟨l₁', l₂', rfl, rfl, h₂⟩ := h₂
|
||||
rw [eq_comm, enumFrom_eq_cons_iff] at h₂
|
||||
obtain ⟨a, as, rfl, h₂, rfl⟩ := h₂
|
||||
simp only [Nat.zero_add, Prod.mk.injEq] at h₂
|
||||
obtain ⟨rfl, rfl⟩ := h₂
|
||||
simp only [findIdx?_append]
|
||||
match h : findIdx? q l₁' with
|
||||
| some j =>
|
||||
refine ⟨j, ?_, by simp⟩
|
||||
rw [findIdx?_eq_some_iff_findIdx_eq] at h
|
||||
omega
|
||||
| none =>
|
||||
refine ⟨l₁'.length, by simp, by simp_all⟩
|
||||
@@ -109,8 +109,7 @@ theorem range'_eq_append_iff : range' s n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs =
|
||||
@[simp] theorem find?_range'_eq_some {s n : Nat} {i : Nat} {p : Nat → Bool} :
|
||||
(range' s n).find? p = some i ↔ p i ∧ i ∈ range' s n ∧ ∀ j, s ≤ j → j < i → !p j := by
|
||||
rw [find?_eq_some]
|
||||
simp only [Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, mem_range'_1,
|
||||
and_congr_right_iff]
|
||||
simp only [Bool.not_eq_true', exists_and_right, mem_range'_1, and_congr_right_iff]
|
||||
simp only [range'_eq_append_iff, eq_comm (a := i :: _), range'_eq_cons_iff]
|
||||
intro h
|
||||
constructor
|
||||
@@ -154,7 +153,7 @@ theorem erase_range' :
|
||||
/-! ### range -/
|
||||
|
||||
theorem reverse_range' : ∀ s n : Nat, reverse (range' s n) = map (s + n - 1 - ·) (range n)
|
||||
| _, 0 => rfl
|
||||
| s, 0 => rfl
|
||||
| s, n + 1 => by
|
||||
rw [range'_1_concat, reverse_append, range_succ_eq_map,
|
||||
show s + (n + 1) - 1 = s + n from rfl, map, map_map]
|
||||
@@ -259,9 +258,6 @@ theorem nodup_iota (n : Nat) : Nodup (iota n) :=
|
||||
| zero => simp at h
|
||||
| succ n => simp
|
||||
|
||||
@[simp] theorem tail_iota (n : Nat) : (iota n).tail = iota (n - 1) := by
|
||||
cases n <;> simp
|
||||
|
||||
@[simp] theorem reverse_iota : reverse (iota n) = range' 1 n := by
|
||||
induction n with
|
||||
| zero => simp
|
||||
@@ -276,15 +272,15 @@ theorem nodup_iota (n : Nat) : Nodup (iota n) :=
|
||||
rw [getLast_eq_head_reverse]
|
||||
simp
|
||||
|
||||
theorem find?_iota_eq_none {n : Nat} {p : Nat → Bool} :
|
||||
theorem find?_iota_eq_none {n : Nat} (p : Nat → Bool) :
|
||||
(iota n).find? p = none ↔ ∀ i, 0 < i → i ≤ n → !p i := by
|
||||
simp
|
||||
|
||||
@[simp] theorem find?_iota_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
|
||||
(iota n).find? p = some i ↔ p i ∧ i ∈ iota n ∧ ∀ j, i < j → j ≤ n → !p j := by
|
||||
rw [find?_eq_some]
|
||||
simp only [iota_eq_reverse_range', reverse_eq_append_iff, reverse_cons, append_assoc, cons_append,
|
||||
nil_append, Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, mem_reverse, mem_range'_1,
|
||||
simp only [iota_eq_reverse_range', reverse_eq_append_iff, reverse_cons, append_assoc,
|
||||
singleton_append, Bool.not_eq_true', exists_and_right, mem_reverse, mem_range'_1,
|
||||
and_congr_right_iff]
|
||||
intro h
|
||||
constructor
|
||||
@@ -358,6 +354,17 @@ theorem map_enumFrom (f : α → β) (n : Nat) (l : List α) :
|
||||
map (Prod.map id f) (enumFrom n l) = enumFrom n (map f l) := by
|
||||
induction l generalizing n <;> simp_all
|
||||
|
||||
@[simp]
|
||||
theorem enumFrom_map_fst (n) :
|
||||
∀ (l : List α), map Prod.fst (enumFrom n l) = range' n l.length
|
||||
| [] => rfl
|
||||
| _ :: _ => congrArg (cons _) (enumFrom_map_fst _ _)
|
||||
|
||||
@[simp]
|
||||
theorem enumFrom_map_snd : ∀ (n) (l : List α), map Prod.snd (enumFrom n l) = l
|
||||
| _, [] => rfl
|
||||
| _, _ :: _ => congrArg (cons _) (enumFrom_map_snd _ _)
|
||||
|
||||
theorem snd_mem_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) : x.2 ∈ l :=
|
||||
enumFrom_map_snd n l ▸ mem_map_of_mem _ h
|
||||
|
||||
@@ -380,6 +387,10 @@ theorem mem_enumFrom {x : α} {i j : Nat} {xs : List α} (h : (i, x) ∈ xs.enum
|
||||
x = xs[i - j]'(by have := le_fst_of_mem_enumFrom h; have := fst_lt_add_of_mem_enumFrom h; omega) :=
|
||||
⟨le_fst_of_mem_enumFrom h, fst_lt_add_of_mem_enumFrom h, snd_eq_of_mem_enumFrom h⟩
|
||||
|
||||
theorem enumFrom_cons' (n : Nat) (x : α) (xs : List α) :
|
||||
enumFrom n (x :: xs) = (n, x) :: (enumFrom n xs).map (Prod.map (· + 1) id) := by
|
||||
rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]
|
||||
|
||||
theorem enumFrom_map (n : Nat) (l : List α) (f : α → β) :
|
||||
enumFrom n (l.map f) = (enumFrom n l).map (Prod.map id f) := by
|
||||
induction l with
|
||||
@@ -396,39 +407,22 @@ theorem enumFrom_append (xs ys : List α) (n : Nat) :
|
||||
rw [cons_append, enumFrom_cons, IH, ← cons_append, ← enumFrom_cons, length, Nat.add_right_comm,
|
||||
Nat.add_assoc]
|
||||
|
||||
theorem enumFrom_eq_cons_iff {l : List α} {n : Nat} :
|
||||
l.enumFrom n = x :: l' ↔ ∃ a as, l = a :: as ∧ x = (n, a) ∧ l' = enumFrom (n + 1) as := by
|
||||
rw [enumFrom_eq_zip_range', zip_eq_cons_iff]
|
||||
constructor
|
||||
· rintro ⟨l₁, l₂, h, rfl, rfl⟩
|
||||
rw [range'_eq_cons_iff] at h
|
||||
obtain ⟨rfl, -, rfl⟩ := h
|
||||
exact ⟨x.2, l₂, by simp [enumFrom_eq_zip_range']⟩
|
||||
· rintro ⟨a, as, rfl, rfl, rfl⟩
|
||||
refine ⟨range' (n+1) as.length, as, ?_⟩
|
||||
simp [enumFrom_eq_zip_range', range'_succ]
|
||||
theorem enumFrom_eq_zip_range' (l : List α) {n : Nat} : l.enumFrom n = (range' n l.length).zip l :=
|
||||
zip_of_prod (enumFrom_map_fst _ _) (enumFrom_map_snd _ _)
|
||||
|
||||
theorem enumFrom_eq_append_iff {l : List α} {n : Nat} :
|
||||
l.enumFrom n = l₁ ++ l₂ ↔
|
||||
∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.enumFrom n ∧ l₂ = l₂'.enumFrom (n + l₁'.length) := by
|
||||
rw [enumFrom_eq_zip_range', zip_eq_append_iff]
|
||||
constructor
|
||||
· rintro ⟨w, x, y, z, h, h', rfl, rfl, rfl⟩
|
||||
rw [range'_eq_append_iff] at h'
|
||||
obtain ⟨k, -, rfl, rfl⟩ := h'
|
||||
simp only [length_range'] at h
|
||||
obtain rfl := h
|
||||
refine ⟨y, z, rfl, ?_⟩
|
||||
simp only [enumFrom_eq_zip_range', length_append, true_and]
|
||||
congr
|
||||
omega
|
||||
· rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
|
||||
simp only [enumFrom_eq_zip_range']
|
||||
refine ⟨range' n l₁'.length, range' (n + l₁'.length) l₂'.length, l₁', l₂', ?_⟩
|
||||
simp [Nat.add_comm]
|
||||
@[simp]
|
||||
theorem unzip_enumFrom_eq_prod (l : List α) {n : Nat} :
|
||||
(l.enumFrom n).unzip = (range' n l.length, l) := by
|
||||
simp only [enumFrom_eq_zip_range', unzip_zip, length_range']
|
||||
|
||||
/-! ### enum -/
|
||||
|
||||
theorem enum_cons : (a::as).enum = (0, a) :: as.enumFrom 1 := rfl
|
||||
|
||||
theorem enum_cons' (x : α) (xs : List α) :
|
||||
enum (x :: xs) = (0, x) :: (enum xs).map (Prod.map (· + 1) id) :=
|
||||
enumFrom_cons' _ _ _
|
||||
|
||||
@[simp]
|
||||
theorem enum_eq_nil {l : List α} : List.enum l = [] ↔ l = [] := enumFrom_eq_nil
|
||||
|
||||
@@ -454,9 +448,6 @@ theorem getElem_enum (l : List α) (i : Nat) (h : i < l.enum.length) :
|
||||
l.enum.getLast? = l.getLast?.map fun a => (l.length - 1, a) := by
|
||||
simp [getLast?_eq_getElem?]
|
||||
|
||||
@[simp] theorem tail_enum (l : List α) : (enum l).tail = enumFrom 1 l.tail := by
|
||||
simp [enum]
|
||||
|
||||
theorem mk_mem_enum_iff_getElem? {i : Nat} {x : α} {l : List α} : (i, x) ∈ enum l ↔ l[i]? = x := by
|
||||
simp [enum, mk_mem_enumFrom_iff_le_and_getElem?_sub]
|
||||
|
||||
@@ -500,13 +491,4 @@ theorem enum_eq_zip_range (l : List α) : l.enum = (range l.length).zip l :=
|
||||
theorem unzip_enum_eq_prod (l : List α) : l.enum.unzip = (range l.length, l) := by
|
||||
simp only [enum_eq_zip_range, unzip_zip, length_range]
|
||||
|
||||
theorem enum_eq_cons_iff {l : List α} :
|
||||
l.enum = x :: l' ↔ ∃ a as, l = a :: as ∧ x = (0, a) ∧ l' = enumFrom 1 as := by
|
||||
rw [enum, enumFrom_eq_cons_iff]
|
||||
|
||||
theorem enum_eq_append_iff {l : List α} :
|
||||
l.enum = l₁ ++ l₂ ↔
|
||||
∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.enum ∧ l₂ = l₂'.enumFrom l₁'.length := by
|
||||
simp [enum, enumFrom_eq_append_iff]
|
||||
|
||||
end List
|
||||
|
||||
@@ -42,7 +42,7 @@ theorem getElem_take' (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j)
|
||||
|
||||
/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
|
||||
length `> i`. Version designed to rewrite from the small list to the big list. -/
|
||||
@[simp] theorem getElem_take (L : List α) {j i : Nat} {h : i < (L.take j).length} :
|
||||
theorem getElem_take (L : List α) {j i : Nat} {h : i < (L.take j).length} :
|
||||
(L.take j)[i] =
|
||||
L[i]'(Nat.lt_of_lt_of_le h (length_take_le' _ _)) := by
|
||||
rw [length_take, Nat.lt_min] at h; rw [getElem_take' L _ h.1]
|
||||
@@ -52,7 +52,7 @@ length `> i`. Version designed to rewrite from the big list to the small list. -
|
||||
@[deprecated getElem_take' (since := "2024-06-12")]
|
||||
theorem get_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
|
||||
get L ⟨i, hi⟩ = get (L.take j) ⟨i, length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩⟩ := by
|
||||
simp
|
||||
simp [getElem_take' _ hi hj]
|
||||
|
||||
/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
|
||||
length `> i`. Version designed to rewrite from the small list to the big list. -/
|
||||
|
||||
@@ -160,25 +160,21 @@ theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x
|
||||
rw [← append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l₁), pairwise_append_comm s]
|
||||
simp only [mem_append, or_comm]
|
||||
|
||||
theorem pairwise_flatten {L : List (List α)} :
|
||||
Pairwise R (flatten L) ↔
|
||||
theorem pairwise_join {L : List (List α)} :
|
||||
Pairwise R (join L) ↔
|
||||
(∀ l ∈ L, Pairwise R l) ∧ Pairwise (fun l₁ l₂ => ∀ x ∈ l₁, ∀ y ∈ l₂, R x y) L := by
|
||||
induction L with
|
||||
| nil => simp
|
||||
| cons l L IH =>
|
||||
simp only [flatten, pairwise_append, IH, mem_flatten, exists_imp, and_imp, forall_mem_cons,
|
||||
simp only [join, pairwise_append, IH, mem_join, exists_imp, and_imp, forall_mem_cons,
|
||||
pairwise_cons, and_assoc, and_congr_right_iff]
|
||||
rw [and_comm, and_congr_left_iff]
|
||||
intros; exact ⟨fun h a b c d e => h c d e a b, fun h c d e a b => h a b c d e⟩
|
||||
|
||||
@[deprecated pairwise_flatten (since := "2024-10-14")] abbrev pairwise_join := @pairwise_flatten
|
||||
|
||||
theorem pairwise_flatMap {R : β → β → Prop} {l : List α} {f : α → List β} :
|
||||
List.Pairwise R (l.flatMap f) ↔
|
||||
theorem pairwise_bind {R : β → β → Prop} {l : List α} {f : α → List β} :
|
||||
List.Pairwise R (l.bind f) ↔
|
||||
(∀ a ∈ l, Pairwise R (f a)) ∧ Pairwise (fun a₁ a₂ => ∀ x ∈ f a₁, ∀ y ∈ f a₂, R x y) l := by
|
||||
simp [List.flatMap, pairwise_flatten, pairwise_map]
|
||||
|
||||
@[deprecated pairwise_flatMap (since := "2024-10-14")] abbrev pairwise_bind := @pairwise_flatMap
|
||||
simp [List.bind, pairwise_join, pairwise_map]
|
||||
|
||||
theorem pairwise_reverse {l : List α} :
|
||||
l.reverse.Pairwise R ↔ l.Pairwise (fun a b => R b a) := by
|
||||
|
||||
@@ -98,8 +98,8 @@ theorem Perm.append_cons (a : α) {h₁ h₂ t₁ t₂ : List α} (p₁ : h₁ ~
|
||||
perm_middle.trans <| by rw [append_nil]
|
||||
|
||||
theorem perm_append_comm : ∀ {l₁ l₂ : List α}, l₁ ++ l₂ ~ l₂ ++ l₁
|
||||
| [], _ => by simp
|
||||
| _ :: _, _ => (perm_append_comm.cons _).trans perm_middle.symm
|
||||
| [], l₂ => by simp
|
||||
| a :: t, l₂ => (perm_append_comm.cons _).trans perm_middle.symm
|
||||
|
||||
theorem perm_append_comm_assoc (l₁ l₂ l₃ : List α) :
|
||||
Perm (l₁ ++ (l₂ ++ l₃)) (l₂ ++ (l₁ ++ l₃)) := by
|
||||
@@ -248,10 +248,6 @@ theorem countP_eq_countP_filter_add (l : List α) (p q : α → Bool) :
|
||||
theorem Perm.count_eq [DecidableEq α] {l₁ l₂ : List α} (p : l₁ ~ l₂) (a) :
|
||||
count a l₁ = count a l₂ := p.countP_eq _
|
||||
|
||||
/-
|
||||
This theorem is a variant of `Perm.foldl_eq` defined in Mathlib which uses typeclasses rather
|
||||
than the explicit `comm` argument.
|
||||
-/
|
||||
theorem Perm.foldl_eq' {f : β → α → β} {l₁ l₂ : List α} (p : l₁ ~ l₂)
|
||||
(comm : ∀ x ∈ l₁, ∀ y ∈ l₁, ∀ (z), f (f z x) y = f (f z y) x)
|
||||
(init) : foldl f init l₁ = foldl f init l₂ := by
|
||||
@@ -268,28 +264,6 @@ theorem Perm.foldl_eq' {f : β → α → β} {l₁ l₂ : List α} (p : l₁ ~
|
||||
refine (IH₁ comm init).trans (IH₂ ?_ _)
|
||||
intros; apply comm <;> apply p₁.symm.subset <;> assumption
|
||||
|
||||
/-
|
||||
This theorem is a variant of `Perm.foldr_eq` defined in Mathlib which uses typeclasses rather
|
||||
than the explicit `comm` argument.
|
||||
-/
|
||||
theorem Perm.foldr_eq' {f : α → β → β} {l₁ l₂ : List α} (p : l₁ ~ l₂)
|
||||
(comm : ∀ x ∈ l₁, ∀ y ∈ l₁, ∀ (z), f y (f x z) = f x (f y z))
|
||||
(init) : foldr f init l₁ = foldr f init l₂ := by
|
||||
induction p using recOnSwap' generalizing init with
|
||||
| nil => simp
|
||||
| cons x _p IH =>
|
||||
simp only [foldr]
|
||||
congr 1
|
||||
apply IH; intros; apply comm <;> exact .tail _ ‹_›
|
||||
| swap' x y _p IH =>
|
||||
simp only [foldr]
|
||||
rw [comm x (.tail _ <| .head _) y (.head _)]
|
||||
congr 2
|
||||
apply IH; intros; apply comm <;> exact .tail _ (.tail _ ‹_›)
|
||||
| trans p₁ _p₂ IH₁ IH₂ =>
|
||||
refine (IH₁ comm init).trans (IH₂ ?_ _)
|
||||
intros; apply comm <;> apply p₁.symm.subset <;> assumption
|
||||
|
||||
theorem Perm.rec_heq {β : List α → Sort _} {f : ∀ a l, β l → β (a :: l)} {b : β []} {l l' : List α}
|
||||
(hl : l ~ l') (f_congr : ∀ {a l l' b b'}, l ~ l' → HEq b b' → HEq (f a l b) (f a l' b'))
|
||||
(f_swap : ∀ {a a' l b}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))) :
|
||||
@@ -461,19 +435,15 @@ theorem Perm.nodup {l l' : List α} (hl : l ~ l') (hR : l.Nodup) : l'.Nodup := h
|
||||
theorem Perm.nodup_iff {l₁ l₂ : List α} : l₁ ~ l₂ → (Nodup l₁ ↔ Nodup l₂) :=
|
||||
Perm.pairwise_iff <| @Ne.symm α
|
||||
|
||||
theorem Perm.flatten {l₁ l₂ : List (List α)} (h : l₁ ~ l₂) : l₁.flatten ~ l₂.flatten := by
|
||||
theorem Perm.join {l₁ l₂ : List (List α)} (h : l₁ ~ l₂) : l₁.join ~ l₂.join := by
|
||||
induction h with
|
||||
| nil => rfl
|
||||
| cons _ _ ih => simp only [flatten_cons, perm_append_left_iff, ih]
|
||||
| swap => simp only [flatten_cons, ← append_assoc, perm_append_right_iff]; exact perm_append_comm ..
|
||||
| cons _ _ ih => simp only [join_cons, perm_append_left_iff, ih]
|
||||
| swap => simp only [join_cons, ← append_assoc, perm_append_right_iff]; exact perm_append_comm ..
|
||||
| trans _ _ ih₁ ih₂ => exact trans ih₁ ih₂
|
||||
|
||||
@[deprecated Perm.flatten (since := "2024-10-14")] abbrev Perm.join := @Perm.flatten
|
||||
|
||||
theorem Perm.flatMap_right {l₁ l₂ : List α} (f : α → List β) (p : l₁ ~ l₂) : l₁.flatMap f ~ l₂.flatMap f :=
|
||||
(p.map _).flatten
|
||||
|
||||
@[deprecated Perm.flatMap_right (since := "2024-10-16")] abbrev Perm.bind_right := @Perm.flatMap_right
|
||||
theorem Perm.bind_right {l₁ l₂ : List α} (f : α → List β) (p : l₁ ~ l₂) : l₁.bind f ~ l₂.bind f :=
|
||||
(p.map _).join
|
||||
|
||||
theorem Perm.eraseP (f : α → Bool) {l₁ l₂ : List α}
|
||||
(H : Pairwise (fun a b => f a → f b → False) l₁) (p : l₁ ~ l₂) : eraseP f l₁ ~ eraseP f l₂ := by
|
||||
|
||||
@@ -5,7 +5,6 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Pairwise
|
||||
import Init.Data.List.Zip
|
||||
|
||||
/-!
|
||||
# Lemmas about `List.range` and `List.enum`
|
||||
@@ -36,16 +35,11 @@ theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step)
|
||||
theorem range'_ne_nil (s : Nat) {n : Nat} : range' s n ≠ [] ↔ n ≠ 0 := by
|
||||
cases n <;> simp
|
||||
|
||||
@[simp] theorem range'_zero : range' s 0 step = [] := by
|
||||
@[simp] theorem range'_zero : range' s 0 = [] := by
|
||||
simp
|
||||
|
||||
@[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
|
||||
|
||||
@[simp] theorem tail_range' (n : Nat) : (range' s n step).tail = range' (s + step) (n - 1) step := by
|
||||
cases n with
|
||||
| zero => simp
|
||||
| succ n => simp [range'_succ]
|
||||
|
||||
@[simp] theorem range'_inj : range' s n = range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s') := by
|
||||
constructor
|
||||
· intro h
|
||||
@@ -92,7 +86,7 @@ theorem map_add_range' (a) : ∀ s n step, map (a + ·) (range' s n step) = rang
|
||||
|
||||
theorem range'_append : ∀ s m n step : Nat,
|
||||
range' s m step ++ range' (s + step * m) n step = range' s (n + m) step
|
||||
| _, 0, _, _ => rfl
|
||||
| s, 0, n, step => rfl
|
||||
| s, m + 1, n, step => by
|
||||
simpa [range', Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
|
||||
using range'_append (s + step) m n step
|
||||
@@ -131,7 +125,7 @@ theorem range'_eq_cons_iff : range' s n = a :: xs ↔ s = a ∧ 0 < n ∧ xs = r
|
||||
/-! ### range -/
|
||||
|
||||
theorem range_loop_range' : ∀ s n : Nat, range.loop s (range' s n) = range' 0 (n + s)
|
||||
| 0, _ => rfl
|
||||
| 0, n => rfl
|
||||
| s + 1, n => by rw [← Nat.add_assoc, Nat.add_right_comm n s 1]; exact range_loop_range' s (n + 1)
|
||||
|
||||
theorem range_eq_range' (n : Nat) : range n = range' 0 n :=
|
||||
@@ -159,9 +153,6 @@ theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) :=
|
||||
theorem range_ne_nil {n : Nat} : range n ≠ [] ↔ n ≠ 0 := by
|
||||
cases n <;> simp
|
||||
|
||||
@[simp] theorem tail_range (n : Nat) : (range n).tail = range' 1 (n - 1) := by
|
||||
rw [range_eq_range', tail_range']
|
||||
|
||||
@[simp]
|
||||
theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by
|
||||
simp only [range_eq_range', range'_sublist_right]
|
||||
@@ -214,9 +205,9 @@ theorem enumFrom_eq_nil {n : Nat} {l : List α} : List.enumFrom n l = [] ↔ l =
|
||||
@[simp]
|
||||
theorem getElem?_enumFrom :
|
||||
∀ n (l : List α) m, (enumFrom n l)[m]? = l[m]?.map fun a => (n + m, a)
|
||||
| _, [], _ => rfl
|
||||
| _, _ :: _, 0 => by simp
|
||||
| n, _ :: l, m + 1 => by
|
||||
| n, [], m => rfl
|
||||
| n, a :: l, 0 => by simp
|
||||
| n, a :: l, m + 1 => by
|
||||
simp only [enumFrom_cons, getElem?_cons_succ]
|
||||
exact (getElem?_enumFrom (n + 1) l m).trans <| by rw [Nat.add_right_comm]; rfl
|
||||
|
||||
@@ -228,12 +219,6 @@ theorem getElem_enumFrom (l : List α) (n) (i : Nat) (h : i < (l.enumFrom n).len
|
||||
simp only [getElem?_enumFrom, getElem?_eq_getElem h]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
theorem tail_enumFrom (l : List α) (n : Nat) : (enumFrom n l).tail = enumFrom (n + 1) l.tail := by
|
||||
induction l generalizing n with
|
||||
| nil => simp
|
||||
| cons _ l ih => simp [ih, enumFrom_cons]
|
||||
|
||||
theorem map_fst_add_enumFrom_eq_enumFrom (l : List α) (n k : Nat) :
|
||||
map (Prod.map (· + n) id) (enumFrom k l) = enumFrom (n + k) l :=
|
||||
ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
|
||||
@@ -242,47 +227,4 @@ theorem map_fst_add_enum_eq_enumFrom (l : List α) (n : Nat) :
|
||||
map (Prod.map (· + n) id) (enum l) = enumFrom n l :=
|
||||
map_fst_add_enumFrom_eq_enumFrom l _ _
|
||||
|
||||
theorem enumFrom_cons' (n : Nat) (x : α) (xs : List α) :
|
||||
enumFrom n (x :: xs) = (n, x) :: (enumFrom n xs).map (Prod.map (· + 1) id) := by
|
||||
rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]
|
||||
|
||||
@[simp]
|
||||
theorem enumFrom_map_fst (n) :
|
||||
∀ (l : List α), map Prod.fst (enumFrom n l) = range' n l.length
|
||||
| [] => rfl
|
||||
| _ :: _ => congrArg (cons _) (enumFrom_map_fst _ _)
|
||||
|
||||
@[simp]
|
||||
theorem enumFrom_map_snd : ∀ (n) (l : List α), map Prod.snd (enumFrom n l) = l
|
||||
| _, [] => rfl
|
||||
| _, _ :: _ => congrArg (cons _) (enumFrom_map_snd _ _)
|
||||
|
||||
theorem enumFrom_eq_zip_range' (l : List α) {n : Nat} : l.enumFrom n = (range' n l.length).zip l :=
|
||||
zip_of_prod (enumFrom_map_fst _ _) (enumFrom_map_snd _ _)
|
||||
|
||||
@[simp]
|
||||
theorem unzip_enumFrom_eq_prod (l : List α) {n : Nat} :
|
||||
(l.enumFrom n).unzip = (range' n l.length, l) := by
|
||||
simp only [enumFrom_eq_zip_range', unzip_zip, length_range']
|
||||
|
||||
/-! ### enum -/
|
||||
|
||||
theorem enum_cons : (a::as).enum = (0, a) :: as.enumFrom 1 := rfl
|
||||
|
||||
theorem enum_cons' (x : α) (xs : List α) :
|
||||
enum (x :: xs) = (0, x) :: (enum xs).map (Prod.map (· + 1) id) :=
|
||||
enumFrom_cons' _ _ _
|
||||
|
||||
theorem enum_eq_enumFrom {l : List α} : l.enum = l.enumFrom 0 := rfl
|
||||
|
||||
theorem enumFrom_eq_map_enum (l : List α) (n : Nat) :
|
||||
enumFrom n l = (enum l).map (Prod.map (· + n) id) := by
|
||||
induction l generalizing n with
|
||||
| nil => simp
|
||||
| cons x xs ih =>
|
||||
simp only [enumFrom_cons, ih, enum_cons, map_cons, Prod.map_apply, Nat.zero_add, id_eq, map_map,
|
||||
cons.injEq, map_inj_left, Function.comp_apply, Prod.forall, Prod.mk.injEq, and_true, true_and]
|
||||
intro a b _
|
||||
exact (succ_add a n).symm
|
||||
|
||||
end List
|
||||
|
||||
@@ -102,7 +102,7 @@ def mergeSortTR (l : List α) (le : α → α → Bool := by exact fun a b => a
|
||||
where run : {n : Nat} → { l : List α // l.length = n } → List α
|
||||
| 0, ⟨[], _⟩ => []
|
||||
| 1, ⟨[a], _⟩ => [a]
|
||||
| _+2, xs =>
|
||||
| n+2, xs =>
|
||||
let (l, r) := splitInTwo xs
|
||||
mergeTR (run l) (run r) le
|
||||
|
||||
@@ -136,13 +136,13 @@ where
|
||||
run : {n : Nat} → { l : List α // l.length = n } → List α
|
||||
| 0, ⟨[], _⟩ => []
|
||||
| 1, ⟨[a], _⟩ => [a]
|
||||
| _+2, xs =>
|
||||
| n+2, xs =>
|
||||
let (l, r) := splitRevInTwo xs
|
||||
mergeTR (run' l) (run r) le
|
||||
run' : {n : Nat} → { l : List α // l.length = n } → List α
|
||||
| 0, ⟨[], _⟩ => []
|
||||
| 1, ⟨[a], _⟩ => [a]
|
||||
| _+2, xs =>
|
||||
| n+2, xs =>
|
||||
let (l, r) := splitRevInTwo' xs
|
||||
mergeTR (run' r) (run l) le
|
||||
|
||||
|
||||
@@ -1,7 +1,7 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Kim Morrison, Eric Wieser, François G. Dorais
|
||||
Authors: Kim Morrison
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Perm
|
||||
@@ -114,40 +114,31 @@ theorem enumLE_trans (trans : ∀ a b c, le a b → le b c → le a c)
|
||||
· simp_all
|
||||
· simp_all
|
||||
|
||||
theorem enumLE_total (total : ∀ a b, le a b || le b a)
|
||||
(a b : Nat × α) : enumLE le a b || enumLE le b a := by
|
||||
theorem enumLE_total (total : ∀ a b, !le a b → le b a)
|
||||
(a b : Nat × α) : !enumLE le a b → enumLE le b a := by
|
||||
simp only [enumLE]
|
||||
split <;> split
|
||||
· simpa using Nat.le_total a.fst b.fst
|
||||
· simpa using Nat.le_of_lt
|
||||
· simp
|
||||
· simp
|
||||
· have := total a.2 b.2
|
||||
simp_all
|
||||
· simp_all [total a.2 b.2]
|
||||
|
||||
/-! ### merge -/
|
||||
|
||||
theorem cons_merge_cons (s : α → α → Bool) (a b l r) :
|
||||
merge (a::l) (b::r) s = if s a b then a :: merge l (b::r) s else b :: merge (a::l) r s := by
|
||||
simp only [merge]
|
||||
|
||||
@[simp] theorem cons_merge_cons_pos (s : α → α → Bool) (l r) (h : s a b) :
|
||||
merge (a::l) (b::r) s = a :: merge l (b::r) s := by
|
||||
rw [cons_merge_cons, if_pos h]
|
||||
|
||||
@[simp] theorem cons_merge_cons_neg (s : α → α → Bool) (l r) (h : ¬ s a b) :
|
||||
merge (a::l) (b::r) s = b :: merge (a::l) r s := by
|
||||
rw [cons_merge_cons, if_neg h]
|
||||
|
||||
@[simp] theorem length_merge (s : α → α → Bool) (l r) :
|
||||
(merge l r s).length = l.length + r.length := by
|
||||
match l, r with
|
||||
| [], r => simp
|
||||
| l, [] => simp
|
||||
| a::l, b::r =>
|
||||
rw [cons_merge_cons]
|
||||
split
|
||||
· simp_arith [length_merge s l (b::r)]
|
||||
· simp_arith [length_merge s (a::l) r]
|
||||
theorem merge_stable : ∀ (xs ys) (_ : ∀ x y, x ∈ xs → y ∈ ys → x.1 ≤ y.1),
|
||||
(merge xs ys (enumLE le)).map (·.2) = merge (xs.map (·.2)) (ys.map (·.2)) le
|
||||
| [], ys, _ => by simp [merge]
|
||||
| xs, [], _ => by simp [merge]
|
||||
| (i, x) :: xs, (j, y) :: ys, h => by
|
||||
simp only [merge, enumLE, map_cons]
|
||||
split <;> rename_i w
|
||||
· rw [if_pos (by simp [h _ _ (mem_cons_self ..) (mem_cons_self ..)])]
|
||||
simp only [map_cons, cons.injEq, true_and]
|
||||
rw [merge_stable, map_cons]
|
||||
exact fun x' y' mx my => h x' y' (mem_cons_of_mem (i, x) mx) my
|
||||
· simp only [↓reduceIte, map_cons, cons.injEq, true_and, reduceCtorEq]
|
||||
rw [merge_stable, map_cons]
|
||||
exact fun x' y' mx my => h x' y' mx (mem_cons_of_mem (j, y) my)
|
||||
|
||||
/--
|
||||
The elements of `merge le xs ys` are exactly the elements of `xs` and `ys`.
|
||||
@@ -167,37 +158,16 @@ theorem mem_merge {a : α} {xs ys : List α} : a ∈ merge xs ys le ↔ a ∈ xs
|
||||
apply or_congr_left
|
||||
simp only [or_comm (a := a = y), or_assoc]
|
||||
|
||||
theorem mem_merge_left (s : α → α → Bool) (h : x ∈ l) : x ∈ merge l r s :=
|
||||
mem_merge.2 <| .inl h
|
||||
|
||||
theorem mem_merge_right (s : α → α → Bool) (h : x ∈ r) : x ∈ merge l r s :=
|
||||
mem_merge.2 <| .inr h
|
||||
|
||||
theorem merge_stable : ∀ (xs ys) (_ : ∀ x y, x ∈ xs → y ∈ ys → x.1 ≤ y.1),
|
||||
(merge xs ys (enumLE le)).map (·.2) = merge (xs.map (·.2)) (ys.map (·.2)) le
|
||||
| [], ys, _ => by simp [merge]
|
||||
| xs, [], _ => by simp [merge]
|
||||
| (i, x) :: xs, (j, y) :: ys, h => by
|
||||
simp only [merge, enumLE, map_cons]
|
||||
split <;> rename_i w
|
||||
· rw [if_pos (by simp [h _ _ (mem_cons_self ..) (mem_cons_self ..)])]
|
||||
simp only [map_cons, cons.injEq, true_and]
|
||||
rw [merge_stable, map_cons]
|
||||
exact fun x' y' mx my => h x' y' (mem_cons_of_mem (i, x) mx) my
|
||||
· simp only [↓reduceIte, map_cons, cons.injEq, true_and, reduceCtorEq]
|
||||
rw [merge_stable, map_cons]
|
||||
exact fun x' y' mx my => h x' y' mx (mem_cons_of_mem (j, y) my)
|
||||
|
||||
-- We enable this instance locally so we can write `Pairwise le` instead of `Pairwise (le · ·)` everywhere.
|
||||
attribute [local instance] boolRelToRel
|
||||
|
||||
/--
|
||||
If the ordering relation `le` is transitive and total (i.e. `le a b || le b a` for all `a, b`)
|
||||
If the ordering relation `le` is transitive and total (i.e. `le a b ∨ le b a` for all `a, b`)
|
||||
then the `merge` of two sorted lists is sorted.
|
||||
-/
|
||||
theorem sorted_merge
|
||||
(trans : ∀ (a b c : α), le a b → le b c → le a c)
|
||||
(total : ∀ (a b : α), le a b || le b a)
|
||||
(total : ∀ (a b : α), !le a b → le b a)
|
||||
(l₁ l₂ : List α) (h₁ : l₁.Pairwise le) (h₂ : l₂.Pairwise le) : (merge l₁ l₂ le).Pairwise le := by
|
||||
induction l₁ generalizing l₂ with
|
||||
| nil => simpa only [merge]
|
||||
@@ -218,10 +188,9 @@ theorem sorted_merge
|
||||
· apply Pairwise.cons
|
||||
· intro z m
|
||||
rw [mem_merge, mem_cons] at m
|
||||
simp only [Bool.not_eq_true] at h
|
||||
rcases m with (⟨rfl|m⟩|m)
|
||||
· simpa [h] using total y z
|
||||
· exact trans _ _ _ (by simpa [h] using total x y) (rel_of_pairwise_cons h₁ m)
|
||||
· exact total _ _ (by simpa using h)
|
||||
· exact trans _ _ _ (total _ _ (by simpa using h)) (rel_of_pairwise_cons h₁ m)
|
||||
· exact rel_of_pairwise_cons h₂ m
|
||||
· exact ih₂ h₂.tail
|
||||
|
||||
@@ -265,7 +234,7 @@ theorem mergeSort_perm : ∀ (l : List α) (le), mergeSort l le ~ l
|
||||
(Perm.of_eq (splitInTwo_fst_append_splitInTwo_snd _)))
|
||||
termination_by l => l.length
|
||||
|
||||
@[simp] theorem length_mergeSort (l : List α) : (mergeSort l le).length = l.length :=
|
||||
@[simp] theorem mergeSort_length (l : List α) : (mergeSort l le).length = l.length :=
|
||||
(mergeSort_perm l le).length_eq
|
||||
|
||||
@[simp] theorem mem_mergeSort {a : α} {l : List α} : a ∈ mergeSort l le ↔ a ∈ l :=
|
||||
@@ -274,13 +243,13 @@ termination_by l => l.length
|
||||
/--
|
||||
The result of `mergeSort` is sorted,
|
||||
as long as the comparison function is transitive (`le a b → le b c → le a c`)
|
||||
and total in the sense that `le a b || le b a`.
|
||||
and total in the sense that `le a b ∨ le b a`.
|
||||
|
||||
The comparison function need not be irreflexive, i.e. `le a b` and `le b a` is allowed even when `a ≠ b`.
|
||||
-/
|
||||
theorem sorted_mergeSort
|
||||
(trans : ∀ (a b c : α), le a b → le b c → le a c)
|
||||
(total : ∀ (a b : α), le a b || le b a) :
|
||||
(total : ∀ (a b : α), !le a b → le b a) :
|
||||
(l : List α) → (mergeSort l le).Pairwise le
|
||||
| [] => by simp [mergeSort]
|
||||
| [a] => by simp [mergeSort]
|
||||
@@ -348,7 +317,7 @@ termination_by _ l => l.length
|
||||
|
||||
theorem mergeSort_cons {le : α → α → Bool}
|
||||
(trans : ∀ (a b c : α), le a b → le b c → le a c)
|
||||
(total : ∀ (a b : α), le a b || le b a)
|
||||
(total : ∀ (a b : α), !le a b → le b a)
|
||||
(a : α) (l : List α) :
|
||||
∃ l₁ l₂, mergeSort (a :: l) le = l₁ ++ a :: l₂ ∧ mergeSort l le = l₁ ++ l₂ ∧
|
||||
∀ b, b ∈ l₁ → !le a b := by
|
||||
@@ -407,7 +376,7 @@ then `c` is still a sublist of `mergeSort le l`.
|
||||
-/
|
||||
theorem sublist_mergeSort
|
||||
(trans : ∀ (a b c : α), le a b → le b c → le a c)
|
||||
(total : ∀ (a b : α), le a b || le b a) :
|
||||
(total : ∀ (a b : α), !le a b → le b a) :
|
||||
∀ {c : List α} (_ : c.Pairwise le) (_ : c <+ l),
|
||||
c <+ mergeSort l le
|
||||
| _, _, .slnil => nil_sublist _
|
||||
@@ -438,45 +407,8 @@ then `[a, b]` is still a sublist of `mergeSort le l`.
|
||||
-/
|
||||
theorem pair_sublist_mergeSort
|
||||
(trans : ∀ (a b c : α), le a b → le b c → le a c)
|
||||
(total : ∀ (a b : α), le a b || le b a)
|
||||
(total : ∀ (a b : α), !le a b → le b a)
|
||||
(hab : le a b) (h : [a, b] <+ l) : [a, b] <+ mergeSort l le :=
|
||||
sublist_mergeSort trans total (pairwise_pair.mpr hab) h
|
||||
|
||||
@[deprecated (since := "2024-09-02")] abbrev mergeSort_stable_pair := @pair_sublist_mergeSort
|
||||
|
||||
theorem map_merge {f : α → β} {r : α → α → Bool} {s : β → β → Bool} {l l' : List α}
|
||||
(hl : ∀ a ∈ l, ∀ b ∈ l', r a b = s (f a) (f b)) :
|
||||
(l.merge l' r).map f = (l.map f).merge (l'.map f) s := by
|
||||
match l, l' with
|
||||
| [], x' => simp
|
||||
| x, [] => simp
|
||||
| x :: xs, x' :: xs' =>
|
||||
simp only [List.forall_mem_cons] at hl
|
||||
simp only [forall_and] at hl
|
||||
simp only [List.map, List.cons_merge_cons]
|
||||
rw [← hl.1.1]
|
||||
split
|
||||
· rw [List.map, map_merge, List.map]
|
||||
simp only [List.forall_mem_cons, forall_and]
|
||||
exact ⟨hl.2.1, hl.2.2⟩
|
||||
· rw [List.map, map_merge, List.map]
|
||||
simp only [List.forall_mem_cons]
|
||||
exact ⟨hl.1.2, hl.2.2⟩
|
||||
|
||||
theorem map_mergeSort {r : α → α → Bool} {s : β → β → Bool} {f : α → β} {l : List α}
|
||||
(hl : ∀ a ∈ l, ∀ b ∈ l, r a b = s (f a) (f b)) :
|
||||
(l.mergeSort r).map f = (l.map f).mergeSort s :=
|
||||
match l with
|
||||
| [] => by simp
|
||||
| [x] => by simp
|
||||
| a :: b :: l => by
|
||||
simp only [mergeSort, splitInTwo_fst, splitInTwo_snd, map_cons]
|
||||
rw [map_merge (fun a am b bm => hl a (mem_of_mem_take (by simpa using am))
|
||||
b (mem_of_mem_drop (by simpa using bm)))]
|
||||
rw [map_mergeSort (s := s) (fun a am b bm => hl a (mem_of_mem_take (by simpa using am))
|
||||
b (mem_of_mem_take (by simpa using bm)))]
|
||||
rw [map_mergeSort (s := s) (fun a am b bm => hl a (mem_of_mem_drop (by simpa using am))
|
||||
b (mem_of_mem_drop (by simpa using bm)))]
|
||||
rw [map_take, map_drop]
|
||||
simp
|
||||
termination_by length l
|
||||
|
||||
@@ -483,30 +483,30 @@ theorem sublist_replicate_iff : l <+ replicate m a ↔ ∃ n, n ≤ m ∧ l = re
|
||||
rw [w]
|
||||
exact (replicate_sublist_replicate a).2 le
|
||||
|
||||
theorem sublist_flatten_of_mem {L : List (List α)} {l} (h : l ∈ L) : l <+ L.flatten := by
|
||||
theorem sublist_join_of_mem {L : List (List α)} {l} (h : l ∈ L) : l <+ L.join := by
|
||||
induction L with
|
||||
| nil => cases h
|
||||
| cons l' L ih =>
|
||||
rcases mem_cons.1 h with (rfl | h)
|
||||
· simp [h]
|
||||
· simp [ih h, flatten_cons, sublist_append_of_sublist_right]
|
||||
· simp [ih h, join_cons, sublist_append_of_sublist_right]
|
||||
|
||||
theorem sublist_flatten_iff {L : List (List α)} {l} :
|
||||
l <+ L.flatten ↔
|
||||
∃ L' : List (List α), l = L'.flatten ∧ ∀ i (_ : i < L'.length), L'[i] <+ L[i]?.getD [] := by
|
||||
theorem sublist_join_iff {L : List (List α)} {l} :
|
||||
l <+ L.join ↔
|
||||
∃ L' : List (List α), l = L'.join ∧ ∀ i (_ : i < L'.length), L'[i] <+ L[i]?.getD [] := by
|
||||
induction L generalizing l with
|
||||
| nil =>
|
||||
constructor
|
||||
· intro w
|
||||
simp only [flatten_nil, sublist_nil] at w
|
||||
simp only [join_nil, sublist_nil] at w
|
||||
subst w
|
||||
exact ⟨[], by simp, fun i x => by cases x⟩
|
||||
· rintro ⟨L', rfl, h⟩
|
||||
simp only [flatten_nil, sublist_nil, flatten_eq_nil_iff]
|
||||
simp only [join_nil, sublist_nil, join_eq_nil_iff]
|
||||
simp only [getElem?_nil, Option.getD_none, sublist_nil] at h
|
||||
exact (forall_getElem (p := (· = []))).1 h
|
||||
| cons l' L ih =>
|
||||
simp only [flatten_cons, sublist_append_iff, ih]
|
||||
simp only [join_cons, sublist_append_iff, ih]
|
||||
constructor
|
||||
· rintro ⟨l₁, l₂, rfl, s, L', rfl, h⟩
|
||||
refine ⟨l₁ :: L', by simp, ?_⟩
|
||||
@@ -517,21 +517,21 @@ theorem sublist_flatten_iff {L : List (List α)} {l} :
|
||||
| nil =>
|
||||
exact ⟨[], [], by simp, by simp, [], by simp, fun i x => by cases x⟩
|
||||
| cons l₁ L' =>
|
||||
exact ⟨l₁, L'.flatten, by simp, by simpa using h 0 (by simp), L', rfl,
|
||||
exact ⟨l₁, L'.join, by simp, by simpa using h 0 (by simp), L', rfl,
|
||||
fun i lt => by simpa using h (i+1) (Nat.add_lt_add_right lt 1)⟩
|
||||
|
||||
theorem flatten_sublist_iff {L : List (List α)} {l} :
|
||||
L.flatten <+ l ↔
|
||||
∃ L' : List (List α), l = L'.flatten ∧ ∀ i (_ : i < L.length), L[i] <+ L'[i]?.getD [] := by
|
||||
theorem join_sublist_iff {L : List (List α)} {l} :
|
||||
L.join <+ l ↔
|
||||
∃ L' : List (List α), l = L'.join ∧ ∀ i (_ : i < L.length), L[i] <+ L'[i]?.getD [] := by
|
||||
induction L generalizing l with
|
||||
| nil =>
|
||||
constructor
|
||||
· intro _
|
||||
exact ⟨[l], by simp, fun i x => by cases x⟩
|
||||
· rintro ⟨L', rfl, _⟩
|
||||
simp only [flatten_nil, nil_sublist]
|
||||
simp only [join_nil, nil_sublist]
|
||||
| cons l' L ih =>
|
||||
simp only [flatten_cons, append_sublist_iff, ih]
|
||||
simp only [join_cons, append_sublist_iff, ih]
|
||||
constructor
|
||||
· rintro ⟨l₁, l₂, rfl, s, L', rfl, h⟩
|
||||
refine ⟨l₁ :: L', by simp, ?_⟩
|
||||
@@ -543,7 +543,7 @@ theorem flatten_sublist_iff {L : List (List α)} {l} :
|
||||
exact ⟨[], [], by simp, by simpa using h 0 (by simp), [], by simp,
|
||||
fun i x => by simpa using h (i+1) (Nat.add_lt_add_right x 1)⟩
|
||||
| cons l₁ L' =>
|
||||
exact ⟨l₁, L'.flatten, by simp, by simpa using h 0 (by simp), L', rfl,
|
||||
exact ⟨l₁, L'.join, by simp, by simpa using h 0 (by simp), L', rfl,
|
||||
fun i lt => by simpa using h (i+1) (Nat.add_lt_add_right lt 1)⟩
|
||||
|
||||
@[simp] theorem isSublist_iff_sublist [BEq α] [LawfulBEq α] {l₁ l₂ : List α} :
|
||||
@@ -725,25 +725,16 @@ theorem infix_iff_suffix_prefix {l₁ l₂ : List α} : l₁ <:+: l₂ ↔ ∃ t
|
||||
theorem IsInfix.eq_of_length (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
|
||||
h.sublist.eq_of_length
|
||||
|
||||
theorem IsInfix.eq_of_length_le (h : l₁ <:+: l₂) : l₂.length ≤ l₁.length → l₁ = l₂ :=
|
||||
h.sublist.eq_of_length_le
|
||||
|
||||
theorem IsPrefix.eq_of_length (h : l₁ <+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
|
||||
h.sublist.eq_of_length
|
||||
|
||||
theorem IsPrefix.eq_of_length_le (h : l₁ <+: l₂) : l₂.length ≤ l₁.length → l₁ = l₂ :=
|
||||
h.sublist.eq_of_length_le
|
||||
|
||||
theorem IsSuffix.eq_of_length (h : l₁ <:+ l₂) : l₁.length = l₂.length → l₁ = l₂ :=
|
||||
h.sublist.eq_of_length
|
||||
|
||||
theorem IsSuffix.eq_of_length_le (h : l₁ <:+ l₂) : l₂.length ≤ l₁.length → l₁ = l₂ :=
|
||||
h.sublist.eq_of_length_le
|
||||
|
||||
theorem prefix_of_prefix_length_le :
|
||||
∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
|
||||
| [], _, _, _, _, _ => nil_prefix
|
||||
| _ :: _, b :: _, _, ⟨_, rfl⟩, ⟨_, e⟩, ll => by
|
||||
| [], l₂, _, _, _, _ => nil_prefix
|
||||
| a :: l₁, b :: l₂, _, ⟨r₁, rfl⟩, ⟨r₂, e⟩, ll => by
|
||||
injection e with _ e'; subst b
|
||||
rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩ (le_of_succ_le_succ ll) with ⟨r₃, rfl⟩
|
||||
exact ⟨r₃, rfl⟩
|
||||
@@ -838,24 +829,6 @@ theorem isPrefix_iff : l₁ <+: l₂ ↔ ∀ i (h : i < l₁.length), l₂[i]? =
|
||||
rw (config := {occs := .pos [2]}) [← Nat.and_forall_add_one]
|
||||
simp [Nat.succ_lt_succ_iff, eq_comm]
|
||||
|
||||
theorem isPrefix_iff_getElem {l₁ l₂ : List α} :
|
||||
l₁ <+: l₂ ↔ ∃ (h : l₁.length ≤ l₂.length), ∀ x (hx : x < l₁.length),
|
||||
l₁[x] = l₂[x]'(Nat.lt_of_lt_of_le hx h) where
|
||||
mp h := ⟨h.length_le, fun _ _ ↦ h.getElem _⟩
|
||||
mpr h := by
|
||||
obtain ⟨hl, h⟩ := h
|
||||
induction l₂ generalizing l₁ with
|
||||
| nil =>
|
||||
simpa using hl
|
||||
| cons _ _ tail_ih =>
|
||||
cases l₁ with
|
||||
| nil =>
|
||||
exact nil_prefix
|
||||
| cons _ _ =>
|
||||
simp only [length_cons, Nat.add_le_add_iff_right, Fin.getElem_fin] at hl h
|
||||
simp only [cons_prefix_cons]
|
||||
exact ⟨h 0 (zero_lt_succ _), tail_ih hl fun a ha ↦ h a.succ (succ_lt_succ ha)⟩
|
||||
|
||||
-- See `Init.Data.List.Nat.Sublist` for `isSuffix_iff` and `ifInfix_iff`.
|
||||
|
||||
theorem isPrefix_filterMap_iff {β} {f : α → Option β} {l₁ : List α} {l₂ : List β} :
|
||||
@@ -938,14 +911,14 @@ theorem isInfix_replicate_iff {n} {a : α} {l : List α} :
|
||||
· simpa using Nat.sub_add_cancel h
|
||||
· simpa using w
|
||||
|
||||
theorem infix_of_mem_flatten : ∀ {L : List (List α)}, l ∈ L → l <:+: flatten L
|
||||
theorem infix_of_mem_join : ∀ {L : List (List α)}, l ∈ L → l <:+: join L
|
||||
| l' :: _, h =>
|
||||
match h with
|
||||
| List.Mem.head .. => infix_append [] _ _
|
||||
| List.Mem.tail _ hlMemL =>
|
||||
IsInfix.trans (infix_of_mem_flatten hlMemL) <| (suffix_append _ _).isInfix
|
||||
IsInfix.trans (infix_of_mem_join hlMemL) <| (suffix_append _ _).isInfix
|
||||
|
||||
@[simp] theorem prefix_append_right_inj (l) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ :=
|
||||
theorem prefix_append_right_inj (l) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ :=
|
||||
exists_congr fun r => by rw [append_assoc, append_right_inj]
|
||||
|
||||
theorem prefix_cons_inj (a) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ :=
|
||||
@@ -1087,11 +1060,4 @@ theorem prefix_iff_eq_take : l₁ <+: l₂ ↔ l₁ = take (length l₁) l₂ :=
|
||||
|
||||
-- See `Init.Data.List.Nat.Sublist` for `suffix_iff_eq_append`, `prefix_take_iff`, and `suffix_iff_eq_drop`.
|
||||
|
||||
/-! ### Deprecations -/
|
||||
|
||||
@[deprecated sublist_flatten_of_mem (since := "2024-10-14")] abbrev sublist_join_of_mem := @sublist_flatten_of_mem
|
||||
@[deprecated sublist_flatten_iff (since := "2024-10-14")] abbrev sublist_join_iff := @sublist_flatten_iff
|
||||
@[deprecated flatten_sublist_iff (since := "2024-10-14")] abbrev flatten_join_iff := @flatten_sublist_iff
|
||||
@[deprecated infix_of_mem_flatten (since := "2024-10-14")] abbrev infix_of_mem_join := @infix_of_mem_flatten
|
||||
|
||||
end List
|
||||
|
||||
@@ -1,23 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Lean FRO. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Henrik Böving
|
||||
-/
|
||||
prelude
|
||||
import Init.Data.List.Basic
|
||||
|
||||
/--
|
||||
Auxiliary definition for `List.toArray`.
|
||||
`List.toArrayAux as r = r ++ as.toArray`
|
||||
-/
|
||||
@[inline_if_reduce]
|
||||
def List.toArrayAux : List α → Array α → Array α
|
||||
| nil, r => r
|
||||
| cons a as, r => toArrayAux as (r.push a)
|
||||
|
||||
/-- Convert a `List α` into an `Array α`. This is O(n) in the length of the list. -/
|
||||
-- This function is exported to C, where it is called by `Array.mk`
|
||||
-- (the constructor) to implement this functionality.
|
||||
@[inline, match_pattern, pp_nodot, export lean_list_to_array]
|
||||
def List.toArrayImpl (as : List α) : Array α :=
|
||||
as.toArrayAux (Array.mkEmpty as.length)
|
||||
@@ -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.Function
|
||||
|
||||
/-!
|
||||
# Lemmas about `List.zip`, `List.zipWith`, `List.zipWithAll`, and `List.unzip`.
|
||||
@@ -17,6 +16,83 @@ open Nat
|
||||
|
||||
/-! ## Zippers -/
|
||||
|
||||
/-! ### zip -/
|
||||
|
||||
theorem zip_map (f : α → γ) (g : β → δ) :
|
||||
∀ (l₁ : List α) (l₂ : List β), zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
|
||||
| [], l₂ => rfl
|
||||
| l₁, [] => by simp only [map, zip_nil_right]
|
||||
| a :: l₁, b :: l₂ => by
|
||||
simp only [map, zip_cons_cons, zip_map, Prod.map]; constructor
|
||||
|
||||
theorem zip_map_left (f : α → γ) (l₁ : List α) (l₂ : List β) :
|
||||
zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]
|
||||
|
||||
theorem zip_map_right (f : β → γ) (l₁ : List α) (l₂ : List β) :
|
||||
zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]
|
||||
|
||||
theorem zip_append :
|
||||
∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
|
||||
zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
|
||||
| [], r₁, l₂, r₂, h => by simp only [eq_nil_of_length_eq_zero h.symm]; rfl
|
||||
| l₁, r₁, [], r₂, h => by simp only [eq_nil_of_length_eq_zero h]; rfl
|
||||
| a :: l₁, r₁, b :: l₂, r₂, h => by
|
||||
simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]
|
||||
|
||||
theorem zip_map' (f : α → β) (g : α → γ) :
|
||||
∀ l : List α, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
|
||||
| [] => rfl
|
||||
| a :: l => by simp only [map, zip_cons_cons, zip_map']
|
||||
|
||||
theorem of_mem_zip {a b} : ∀ {l₁ : List α} {l₂ : List β}, (a, b) ∈ zip l₁ l₂ → a ∈ l₁ ∧ b ∈ l₂
|
||||
| _ :: l₁, _ :: l₂, h => by
|
||||
cases h
|
||||
case head => simp
|
||||
case tail h =>
|
||||
· have := of_mem_zip h
|
||||
exact ⟨Mem.tail _ this.1, Mem.tail _ this.2⟩
|
||||
|
||||
@[deprecated of_mem_zip (since := "2024-07-28")] abbrev mem_zip := @of_mem_zip
|
||||
|
||||
theorem map_fst_zip :
|
||||
∀ (l₁ : List α) (l₂ : List β), l₁.length ≤ l₂.length → map Prod.fst (zip l₁ l₂) = l₁
|
||||
| [], bs, _ => rfl
|
||||
| _ :: as, _ :: bs, h => by
|
||||
simp [Nat.succ_le_succ_iff] at h
|
||||
show _ :: map Prod.fst (zip as bs) = _ :: as
|
||||
rw [map_fst_zip as bs h]
|
||||
| a :: as, [], h => by simp at h
|
||||
|
||||
theorem map_snd_zip :
|
||||
∀ (l₁ : List α) (l₂ : List β), l₂.length ≤ l₁.length → map Prod.snd (zip l₁ l₂) = l₂
|
||||
| _, [], _ => by
|
||||
rw [zip_nil_right]
|
||||
rfl
|
||||
| [], b :: bs, h => by simp at h
|
||||
| a :: as, b :: bs, h => by
|
||||
simp [Nat.succ_le_succ_iff] at h
|
||||
show _ :: map Prod.snd (zip as bs) = _ :: bs
|
||||
rw [map_snd_zip as bs h]
|
||||
|
||||
theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
|
||||
(l.map fun x => (x, f x)) = l.zip (l.map f) := by
|
||||
rw [← zip_map']
|
||||
congr
|
||||
simp
|
||||
|
||||
theorem map_prod_right_eq_zip {l : List α} (f : α → β) :
|
||||
(l.map fun x => (f x, x)) = (l.map f).zip l := by
|
||||
rw [← zip_map']
|
||||
congr
|
||||
simp
|
||||
|
||||
/-- See also `List.zip_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
|
||||
@[simp] theorem zip_replicate' {a : α} {b : β} {n : Nat} :
|
||||
zip (replicate n a) (replicate n b) = replicate n (a, b) := by
|
||||
induction n with
|
||||
| zero => rfl
|
||||
| succ n ih => simp [replicate_succ, ih]
|
||||
|
||||
/-! ### zipWith -/
|
||||
|
||||
theorem zipWith_comm (f : α → β → γ) :
|
||||
@@ -153,7 +229,6 @@ theorem drop_zipWith : (zipWith f l l').drop n = zipWith f (l.drop n) (l'.drop n
|
||||
|
||||
@[deprecated drop_zipWith (since := "2024-07-26")] abbrev zipWith_distrib_drop := @drop_zipWith
|
||||
|
||||
@[simp]
|
||||
theorem tail_zipWith : (zipWith f l l').tail = zipWith f l.tail l'.tail := by
|
||||
rw [← drop_one]; simp [drop_zipWith]
|
||||
|
||||
@@ -173,65 +248,6 @@ theorem zipWith_append (f : α → β → γ) (l la : List α) (l' lb : List β)
|
||||
simp only [length_cons, Nat.succ.injEq] at h
|
||||
simp [ih _ h]
|
||||
|
||||
theorem zipWith_eq_cons_iff {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
|
||||
zipWith f l₁ l₂ = g :: l ↔
|
||||
∃ a l₁' b l₂', l₁ = a :: l₁' ∧ l₂ = b :: l₂' ∧ g = f a b ∧ l = zipWith f l₁' l₂' := by
|
||||
match l₁, l₂ with
|
||||
| [], [] => simp
|
||||
| [], b :: l₂ => simp
|
||||
| a :: l₁, [] => simp
|
||||
| a' :: l₁, b' :: l₂ =>
|
||||
simp only [zip_cons_cons, cons.injEq, Prod.mk.injEq]
|
||||
constructor
|
||||
· rintro ⟨⟨rfl, rfl⟩, rfl⟩
|
||||
refine ⟨a', l₁, b', l₂, by simp⟩
|
||||
· rintro ⟨a, l₁, b, l₂, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl, rfl⟩
|
||||
simp
|
||||
|
||||
theorem zipWith_eq_append_iff {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
|
||||
zipWith f l₁ l₂ = l₁' ++ l₂' ↔
|
||||
∃ w x y z, w.length = y.length ∧ l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zipWith f w y ∧ l₂' = zipWith f x z := by
|
||||
induction l₁ generalizing l₂ l₁' with
|
||||
| nil =>
|
||||
simp
|
||||
constructor
|
||||
· rintro ⟨rfl, rfl⟩
|
||||
exact ⟨[], [], [], by simp⟩
|
||||
· rintro ⟨_, _, _, -, ⟨rfl, rfl⟩, _, rfl, rfl, rfl⟩
|
||||
simp
|
||||
| cons x₁ l₁ ih₁ =>
|
||||
cases l₂ with
|
||||
| nil =>
|
||||
constructor
|
||||
· simp only [zipWith_nil_right, nil_eq, append_eq_nil, exists_and_left, and_imp]
|
||||
rintro rfl rfl
|
||||
exact ⟨[], x₁ :: l₁, [], by simp⟩
|
||||
· rintro ⟨w, x, y, z, h₁, _, h₃, rfl, rfl⟩
|
||||
simp only [nil_eq, append_eq_nil] at h₃
|
||||
obtain ⟨rfl, rfl⟩ := h₃
|
||||
simp
|
||||
| cons x₂ l₂ =>
|
||||
simp only [zipWith_cons_cons]
|
||||
rw [cons_eq_append_iff]
|
||||
constructor
|
||||
· rintro (⟨rfl, rfl⟩ | ⟨l₁'', rfl, h⟩)
|
||||
· exact ⟨[], x₁ :: l₁, [], x₂ :: l₂, by simp⟩
|
||||
· rw [ih₁] at h
|
||||
obtain ⟨w, x, y, z, h, rfl, rfl, h', rfl⟩ := h
|
||||
refine ⟨x₁ :: w, x, x₂ :: y, z, by simp [h, h']⟩
|
||||
· rintro ⟨w, x, y, z, h₁, h₂, h₃, rfl, rfl⟩
|
||||
rw [cons_eq_append_iff] at h₂
|
||||
rw [cons_eq_append_iff] at h₃
|
||||
obtain (⟨rfl, rfl⟩ | ⟨w', rfl, rfl⟩) := h₂
|
||||
· simp only [zipWith_nil_left, true_and, nil_eq, reduceCtorEq, false_and, exists_const,
|
||||
or_false]
|
||||
obtain (⟨rfl, rfl⟩ | ⟨y', rfl, rfl⟩) := h₃
|
||||
· simp
|
||||
· simp_all
|
||||
· obtain (⟨rfl, rfl⟩ | ⟨y', rfl, rfl⟩) := h₃
|
||||
· simp_all
|
||||
· simp_all [zipWith_append, Nat.succ_inj']
|
||||
|
||||
/-- See also `List.zipWith_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
|
||||
@[simp] theorem zipWith_replicate' {a : α} {b : β} {n : Nat} :
|
||||
zipWith f (replicate n a) (replicate n b) = replicate n (f a b) := by
|
||||
@@ -239,121 +255,6 @@ theorem zipWith_eq_append_iff {f : α → β → γ} {l₁ : List α} {l₂ : Li
|
||||
| zero => rfl
|
||||
| succ n ih => simp [replicate_succ, ih]
|
||||
|
||||
theorem map_uncurry_zip_eq_zipWith (f : α → β → γ) (l : List α) (l' : List β) :
|
||||
map (Function.uncurry f) (l.zip l') = zipWith f l l' := by
|
||||
rw [zip]
|
||||
induction l generalizing l' with
|
||||
| nil => simp
|
||||
| cons hl tl ih =>
|
||||
cases l' <;> simp [ih]
|
||||
|
||||
/-! ### zip -/
|
||||
|
||||
theorem zip_eq_zipWith : ∀ (l₁ : List α) (l₂ : List β), zip l₁ l₂ = zipWith Prod.mk l₁ l₂
|
||||
| [], _ => rfl
|
||||
| _, [] => rfl
|
||||
| a :: l₁, b :: l₂ => by simp [zip_cons_cons, zip_eq_zipWith l₁ l₂]
|
||||
|
||||
theorem zip_map (f : α → γ) (g : β → δ) :
|
||||
∀ (l₁ : List α) (l₂ : List β), zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
|
||||
| [], _ => rfl
|
||||
| _, [] => by simp only [map, zip_nil_right]
|
||||
| _ :: _, _ :: _ => by
|
||||
simp only [map, zip_cons_cons, zip_map, Prod.map]; constructor
|
||||
|
||||
theorem zip_map_left (f : α → γ) (l₁ : List α) (l₂ : List β) :
|
||||
zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]
|
||||
|
||||
theorem zip_map_right (f : β → γ) (l₁ : List α) (l₂ : List β) :
|
||||
zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]
|
||||
|
||||
@[simp] theorem tail_zip (l₁ : List α) (l₂ : List β) :
|
||||
(zip l₁ l₂).tail = zip l₁.tail l₂.tail := by
|
||||
cases l₁ <;> cases l₂ <;> simp
|
||||
|
||||
theorem zip_append :
|
||||
∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
|
||||
zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
|
||||
| [], r₁, l₂, r₂, h => by simp only [eq_nil_of_length_eq_zero h.symm]; rfl
|
||||
| l₁, r₁, [], r₂, h => by simp only [eq_nil_of_length_eq_zero h]; rfl
|
||||
| a :: l₁, r₁, b :: l₂, r₂, h => by
|
||||
simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]
|
||||
|
||||
theorem zip_map' (f : α → β) (g : α → γ) :
|
||||
∀ l : List α, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
|
||||
| [] => rfl
|
||||
| a :: l => by simp only [map, zip_cons_cons, zip_map']
|
||||
|
||||
theorem of_mem_zip {a b} : ∀ {l₁ : List α} {l₂ : List β}, (a, b) ∈ zip l₁ l₂ → a ∈ l₁ ∧ b ∈ l₂
|
||||
| _ :: l₁, _ :: l₂, h => by
|
||||
cases h
|
||||
case head => simp
|
||||
case tail h =>
|
||||
· have := of_mem_zip h
|
||||
exact ⟨Mem.tail _ this.1, Mem.tail _ this.2⟩
|
||||
|
||||
@[deprecated of_mem_zip (since := "2024-07-28")] abbrev mem_zip := @of_mem_zip
|
||||
|
||||
theorem map_fst_zip :
|
||||
∀ (l₁ : List α) (l₂ : List β), l₁.length ≤ l₂.length → map Prod.fst (zip l₁ l₂) = l₁
|
||||
| [], _, _ => rfl
|
||||
| _ :: as, _ :: bs, h => by
|
||||
simp [Nat.succ_le_succ_iff] at h
|
||||
show _ :: map Prod.fst (zip as bs) = _ :: as
|
||||
rw [map_fst_zip as bs h]
|
||||
| _ :: _, [], h => by simp at h
|
||||
|
||||
theorem map_snd_zip :
|
||||
∀ (l₁ : List α) (l₂ : List β), l₂.length ≤ l₁.length → map Prod.snd (zip l₁ l₂) = l₂
|
||||
| _, [], _ => by
|
||||
rw [zip_nil_right]
|
||||
rfl
|
||||
| [], b :: bs, h => by simp at h
|
||||
| a :: as, b :: bs, h => by
|
||||
simp [Nat.succ_le_succ_iff] at h
|
||||
show _ :: map Prod.snd (zip as bs) = _ :: bs
|
||||
rw [map_snd_zip as bs h]
|
||||
|
||||
theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
|
||||
(l.map fun x => (x, f x)) = l.zip (l.map f) := by
|
||||
rw [← zip_map']
|
||||
congr
|
||||
simp
|
||||
|
||||
theorem map_prod_right_eq_zip {l : List α} (f : α → β) :
|
||||
(l.map fun x => (f x, x)) = (l.map f).zip l := by
|
||||
rw [← zip_map']
|
||||
congr
|
||||
simp
|
||||
|
||||
@[simp] theorem zip_eq_nil_iff {l₁ : List α} {l₂ : List β} :
|
||||
zip l₁ l₂ = [] ↔ l₁ = [] ∨ l₂ = [] := by
|
||||
simp [zip_eq_zipWith]
|
||||
|
||||
theorem zip_eq_cons_iff {l₁ : List α} {l₂ : List β} :
|
||||
zip l₁ l₂ = (a, b) :: l ↔
|
||||
∃ l₁' l₂', l₁ = a :: l₁' ∧ l₂ = b :: l₂' ∧ l = zip l₁' l₂' := by
|
||||
simp only [zip_eq_zipWith, zipWith_eq_cons_iff]
|
||||
constructor
|
||||
· rintro ⟨a, l₁, b, l₂, rfl, rfl, h, rfl, rfl⟩
|
||||
simp only [Prod.mk.injEq] at h
|
||||
obtain ⟨rfl, rfl⟩ := h
|
||||
simp
|
||||
· rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
|
||||
refine ⟨a, l₁', b, l₂', by simp⟩
|
||||
|
||||
theorem zip_eq_append_iff {l₁ : List α} {l₂ : List β} :
|
||||
zip l₁ l₂ = l₁' ++ l₂' ↔
|
||||
∃ w x y z, w.length = y.length ∧ l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zip w y ∧ l₂' = zip x z := by
|
||||
simp [zip_eq_zipWith, zipWith_eq_append_iff]
|
||||
|
||||
/-- See also `List.zip_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
|
||||
@[simp] theorem zip_replicate' {a : α} {b : β} {n : Nat} :
|
||||
zip (replicate n a) (replicate n b) = replicate n (a, b) := by
|
||||
induction n with
|
||||
| zero => rfl
|
||||
| succ n ih => simp [replicate_succ, ih]
|
||||
|
||||
/-! ### zipWithAll -/
|
||||
|
||||
theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
|
||||
@@ -383,16 +284,12 @@ theorem head?_zipWithAll {f : Option α → Option β → γ} :
|
||||
| none, none => .none | a?, b? => some (f a? b?) := by
|
||||
simp [head?_eq_getElem?, getElem?_zipWithAll]
|
||||
|
||||
@[simp] theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
|
||||
theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
|
||||
(zipWithAll f as bs).head h = f as.head? bs.head? := by
|
||||
apply Option.some.inj
|
||||
rw [← head?_eq_head, head?_zipWithAll]
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem tail_zipWithAll {f : Option α → Option β → γ} :
|
||||
(zipWithAll f as bs).tail = zipWithAll f as.tail bs.tail := by
|
||||
cases as <;> cases bs <;> simp
|
||||
|
||||
theorem zipWithAll_map {μ} (f : Option γ → Option δ → μ) (g : α → γ) (h : β → δ) (l₁ : List α) (l₂ : List β) :
|
||||
zipWithAll f (l₁.map g) (l₂.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) l₁ l₂ := by
|
||||
induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
|
||||
@@ -439,9 +336,9 @@ theorem zip_unzip : ∀ l : List (α × β), zip (unzip l).1 (unzip l).2 = l
|
||||
|
||||
theorem unzip_zip_left :
|
||||
∀ {l₁ : List α} {l₂ : List β}, length l₁ ≤ length l₂ → (unzip (zip l₁ l₂)).1 = l₁
|
||||
| [], _, _ => rfl
|
||||
| _, [], h => by rw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_le_zero h)]; rfl
|
||||
| _ :: _, _ :: _, h => by
|
||||
| [], l₂, _ => rfl
|
||||
| l₁, [], h => by rw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_le_zero h)]; rfl
|
||||
| a :: l₁, b :: l₂, h => by
|
||||
simp only [zip_cons_cons, unzip_cons, unzip_zip_left (le_of_succ_le_succ h)]
|
||||
|
||||
theorem unzip_zip_right :
|
||||
@@ -461,12 +358,6 @@ theorem zip_of_prod {l : List α} {l' : List β} {lp : List (α × β)} (hl : lp
|
||||
(hr : lp.map Prod.snd = l') : lp = l.zip l' := by
|
||||
rw [← hl, ← hr, ← zip_unzip lp, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]
|
||||
|
||||
theorem tail_zip_fst {l : List (α × β)} : l.unzip.1.tail = l.tail.unzip.1 := by
|
||||
simp
|
||||
|
||||
theorem tail_zip_snd {l : List (α × β)} : l.unzip.2.tail = l.tail.unzip.2 := by
|
||||
simp
|
||||
|
||||
@[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
|
||||
unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
|
||||
ext1 <;> simp
|
||||
|
||||
@@ -248,7 +248,7 @@ protected theorem add_mul (n m k : Nat) : (n + m) * k = n * k + m * k :=
|
||||
Nat.right_distrib n m k
|
||||
|
||||
protected theorem mul_assoc : ∀ (n m k : Nat), (n * m) * k = n * (m * k)
|
||||
| _, _, 0 => rfl
|
||||
| n, m, 0 => rfl
|
||||
| n, m, succ k => by simp [mul_succ, Nat.mul_assoc n m k, Nat.left_distrib]
|
||||
instance : Std.Associative (α := Nat) (· * ·) := ⟨Nat.mul_assoc⟩
|
||||
|
||||
@@ -634,8 +634,6 @@ theorem lt_succ_of_lt (h : a < b) : a < succ b := le_succ_of_le h
|
||||
|
||||
theorem lt_add_one_of_lt (h : a < b) : a < b + 1 := le_succ_of_le h
|
||||
|
||||
@[simp] theorem lt_one_iff : n < 1 ↔ n = 0 := Nat.lt_succ_iff.trans <| by rw [le_zero_eq]
|
||||
|
||||
theorem succ_pred_eq_of_ne_zero : ∀ {n}, n ≠ 0 → succ (pred n) = n
|
||||
| _+1, _ => rfl
|
||||
|
||||
|
||||
@@ -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.Nat.Bitwise.Basic
|
||||
|
||||
@@ -36,7 +36,7 @@ private theorem two_mul_sub_one {n : Nat} (n_pos : n > 0) : (2*n - 1) % 2 = 1 :=
|
||||
/-! ### Preliminaries -/
|
||||
|
||||
/--
|
||||
An induction principal that works on division by two.
|
||||
An induction principal that works on divison by two.
|
||||
-/
|
||||
noncomputable def div2Induction {motive : Nat → Sort u}
|
||||
(n : Nat) (ind : ∀(n : Nat), (n > 0 → motive (n/2)) → motive n) : motive n := by
|
||||
@@ -237,7 +237,7 @@ theorem testBit_two_pow_add_eq (x i : Nat) : testBit (2^i + x) i = !(testBit x i
|
||||
| _ p => simp [p]
|
||||
|
||||
theorem testBit_mul_two_pow_add_eq (a b i : Nat) :
|
||||
testBit (2^i*a + b) i = (a%2 = 1 ^^ testBit b i) := by
|
||||
testBit (2^i*a + b) i = Bool.xor (a%2 = 1) (testBit b i) := by
|
||||
match a with
|
||||
| 0 => simp
|
||||
| a+1 =>
|
||||
@@ -570,7 +570,7 @@ theorem or_div_two : (a ||| b) / 2 = a / 2 ||| b / 2 := by
|
||||
/-! ### xor -/
|
||||
|
||||
@[simp] theorem testBit_xor (x y i : Nat) :
|
||||
(x ^^^ y).testBit i = ((x.testBit i) ^^ (y.testBit i)) := by
|
||||
(x ^^^ y).testBit i = Bool.xor (x.testBit i) (y.testBit i) := by
|
||||
simp [HXor.hXor, Xor.xor, xor, testBit_bitwise ]
|
||||
|
||||
@[simp] theorem zero_xor (x : Nat) : 0 ^^^ x = x := by
|
||||
|
||||
@@ -84,7 +84,7 @@ decreasing_by apply div_rec_lemma; assumption
|
||||
protected def mod : @& Nat → @& Nat → Nat
|
||||
/-
|
||||
Nat.modCore is defined by well-founded recursion and thus irreducible. Nevertheless it is
|
||||
desirable if trivial `Nat.mod` calculations, namely
|
||||
desireable if trivial `Nat.mod` calculations, namely
|
||||
* `Nat.mod 0 m` for all `m`
|
||||
* `Nat.mod n (m+n)` for concrete literals `n`
|
||||
reduce definitionally.
|
||||
@@ -134,19 +134,6 @@ theorem mod_eq_of_lt {a b : Nat} (h : a < b) : a % b = a :=
|
||||
if_neg h'
|
||||
(mod_eq a b).symm ▸ this
|
||||
|
||||
@[simp] theorem one_mod_eq_zero_iff {n : Nat} : 1 % n = 0 ↔ n = 1 := by
|
||||
match n with
|
||||
| 0 => simp
|
||||
| 1 => simp
|
||||
| n + 2 =>
|
||||
rw [mod_eq_of_lt (by exact Nat.lt_of_sub_eq_succ rfl)]
|
||||
simp only [add_one_ne_zero, false_iff, ne_eq]
|
||||
exact ne_of_beq_eq_false rfl
|
||||
|
||||
@[simp] theorem Nat.zero_eq_one_mod_iff {n : Nat} : 0 = 1 % n ↔ n = 1 := by
|
||||
rw [eq_comm]
|
||||
simp
|
||||
|
||||
theorem mod_eq_sub_mod {a b : Nat} (h : a ≥ b) : a % b = (a - b) % b :=
|
||||
match eq_zero_or_pos b with
|
||||
| Or.inl h₁ => h₁.symm ▸ (Nat.sub_zero a).symm ▸ rfl
|
||||
@@ -170,13 +157,6 @@ theorem mod_lt (x : Nat) {y : Nat} : y > 0 → x % y < y := by
|
||||
rw [mod_eq_sub_mod h₁]
|
||||
exact h₂ h₃
|
||||
|
||||
@[simp] protected theorem sub_mod_add_mod_cancel (a b : Nat) [NeZero a] : a - b % a + b % a = a := by
|
||||
rw [Nat.sub_add_cancel]
|
||||
cases a with
|
||||
| zero => simp_all
|
||||
| succ a =>
|
||||
exact Nat.le_of_lt (mod_lt b (zero_lt_succ a))
|
||||
|
||||
theorem mod_le (x y : Nat) : x % y ≤ x := by
|
||||
match Nat.lt_or_ge x y with
|
||||
| Or.inl h₁ => rw [mod_eq_of_lt h₁]; apply Nat.le_refl
|
||||
@@ -217,6 +197,7 @@ decreasing_by apply div_rec_lemma; assumption
|
||||
theorem div_eq_sub_div (h₁ : 0 < b) (h₂ : b ≤ a) : a / b = (a - b) / b + 1 := by
|
||||
rw [div_eq a, if_pos]; constructor <;> assumption
|
||||
|
||||
|
||||
theorem mod_add_div (m k : Nat) : m % k + k * (m / k) = m := by
|
||||
induction m, k using mod.inductionOn with rw [div_eq, mod_eq]
|
||||
| base x y h => simp [h]
|
||||
@@ -269,7 +250,7 @@ protected theorem div_div_eq_div_mul (m n k : Nat) : m / n / k = m / (n * k) :=
|
||||
|
||||
theorem div_mul_le_self : ∀ (m n : Nat), m / n * n ≤ m
|
||||
| m, 0 => by simp
|
||||
| _, _+1 => (le_div_iff_mul_le (Nat.succ_pos _)).1 (Nat.le_refl _)
|
||||
| m, n+1 => (le_div_iff_mul_le (Nat.succ_pos _)).1 (Nat.le_refl _)
|
||||
|
||||
theorem div_lt_iff_lt_mul (Hk : 0 < k) : x / k < y ↔ x < y * k := by
|
||||
rw [← Nat.not_le, ← Nat.not_le]; exact not_congr (le_div_iff_mul_le Hk)
|
||||
|
||||
@@ -230,17 +230,6 @@ instance : Std.Associative (α := Nat) min := ⟨Nat.min_assoc⟩
|
||||
@[simp] protected theorem min_self_assoc' {m n : Nat} : min n (min m n) = min n m := by
|
||||
rw [Nat.min_comm m n, ← Nat.min_assoc, Nat.min_self]
|
||||
|
||||
@[simp] theorem min_add_left {a b : Nat} : min a (b + a) = a := by
|
||||
rw [Nat.min_def]
|
||||
simp
|
||||
@[simp] theorem min_add_right {a b : Nat} : min a (a + b) = a := by
|
||||
rw [Nat.min_def]
|
||||
simp
|
||||
@[simp] theorem add_left_min {a b : Nat} : min (b + a) a = a := by
|
||||
rw [Nat.min_comm, min_add_left]
|
||||
@[simp] theorem add_right_min {a b : Nat} : min (a + b) a = a := by
|
||||
rw [Nat.min_comm, min_add_right]
|
||||
|
||||
protected theorem sub_sub_eq_min : ∀ (a b : Nat), a - (a - b) = min a b
|
||||
| 0, _ => by rw [Nat.zero_sub, Nat.zero_min]
|
||||
| _, 0 => by rw [Nat.sub_zero, Nat.sub_self, Nat.min_zero]
|
||||
@@ -295,17 +284,6 @@ protected theorem max_assoc : ∀ (a b c : Nat), max (max a b) c = max a (max b
|
||||
| _+1, _+1, _+1 => by simp only [Nat.succ_max_succ]; exact congrArg succ <| Nat.max_assoc ..
|
||||
instance : Std.Associative (α := Nat) max := ⟨Nat.max_assoc⟩
|
||||
|
||||
@[simp] theorem max_add_left {a b : Nat} : max a (b + a) = b + a := by
|
||||
rw [Nat.max_def]
|
||||
simp
|
||||
@[simp] theorem max_add_right {a b : Nat} : max a (a + b) = a + b := by
|
||||
rw [Nat.max_def]
|
||||
simp
|
||||
@[simp] theorem add_left_max {a b : Nat} : max (b + a) a = b + a := by
|
||||
rw [Nat.max_comm, max_add_left]
|
||||
@[simp] theorem add_right_max {a b : Nat} : max (a + b) a = a + b := by
|
||||
rw [Nat.max_comm, max_add_right]
|
||||
|
||||
protected theorem sub_add_eq_max (a b : Nat) : a - b + b = max a b := by
|
||||
match Nat.le_total a b with
|
||||
| .inl hl => rw [Nat.max_eq_right hl, Nat.sub_eq_zero_iff_le.mpr hl, Nat.zero_add]
|
||||
@@ -605,9 +583,6 @@ theorem add_mod (a b n : Nat) : (a + b) % n = ((a % n) + (b % n)) % n := by
|
||||
| zero => simp_all
|
||||
| succ k => omega
|
||||
|
||||
@[simp] theorem mod_mul_mod {a b c : Nat} : (a % c * b) % c = a * b % c := by
|
||||
rw [mul_mod, mod_mod, ← mul_mod]
|
||||
|
||||
/-! ### pow -/
|
||||
|
||||
theorem pow_succ' {m n : Nat} : m ^ n.succ = m * m ^ n := by
|
||||
@@ -770,16 +745,6 @@ protected theorem two_pow_pred_mod_two_pow (h : 0 < w) :
|
||||
rw [mod_eq_of_lt]
|
||||
apply Nat.pow_pred_lt_pow (by omega) h
|
||||
|
||||
protected theorem pow_lt_pow_iff_pow_mul_le_pow {a n m : Nat} (h : 1 < a) :
|
||||
a ^ n < a ^ m ↔ a ^ n * a ≤ a ^ m := by
|
||||
rw [←Nat.pow_add_one, Nat.pow_le_pow_iff_right (by omega), Nat.pow_lt_pow_iff_right (by omega)]
|
||||
omega
|
||||
|
||||
@[simp]
|
||||
theorem two_pow_pred_mul_two (h : 0 < w) :
|
||||
2 ^ (w - 1) * 2 = 2 ^ w := by
|
||||
simp [← Nat.pow_succ, Nat.sub_add_cancel h]
|
||||
|
||||
/-! ### log2 -/
|
||||
|
||||
@[simp]
|
||||
@@ -874,15 +839,15 @@ theorem shiftLeft_succ_inside (m n : Nat) : m <<< (n+1) = (2*m) <<< n := rfl
|
||||
|
||||
/-- Shiftleft on successor with multiple moved to outside. -/
|
||||
theorem shiftLeft_succ : ∀(m n), m <<< (n + 1) = 2 * (m <<< n)
|
||||
| _, 0 => rfl
|
||||
| _, k + 1 => by
|
||||
| m, 0 => rfl
|
||||
| m, k + 1 => by
|
||||
rw [shiftLeft_succ_inside _ (k+1)]
|
||||
rw [shiftLeft_succ _ k, shiftLeft_succ_inside]
|
||||
|
||||
/-- Shiftright on successor with division moved inside. -/
|
||||
theorem shiftRight_succ_inside : ∀m n, m >>> (n+1) = (m/2) >>> n
|
||||
| _, 0 => rfl
|
||||
| _, k + 1 => by
|
||||
| m, 0 => rfl
|
||||
| m, k + 1 => by
|
||||
rw [shiftRight_succ _ (k+1)]
|
||||
rw [shiftRight_succ_inside _ k, shiftRight_succ]
|
||||
|
||||
|
||||
@@ -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.Omega
|
||||
@@ -15,7 +15,7 @@ in particular
|
||||
and its corollary
|
||||
`Nat.mod_pow_succ : x % b ^ (k + 1) = x % b ^ k + b ^ k * ((x / b ^ k) % b)`.
|
||||
|
||||
It contains the necessary preliminary results relating order and `*` and `/`,
|
||||
It contains the necesssary preliminary results relating order and `*` and `/`,
|
||||
which should probably be moved to their own file.
|
||||
-/
|
||||
|
||||
|
||||
@@ -35,4 +35,4 @@ theorem neZero_iff {n : R} : NeZero n ↔ n ≠ 0 :=
|
||||
⟨fun h ↦ h.out, NeZero.mk⟩
|
||||
|
||||
@[simp] theorem neZero_zero_iff_false {α : Type _} [Zero α] : NeZero (0 : α) ↔ False :=
|
||||
⟨fun _ ↦ NeZero.ne (0 : α) rfl, fun h ↦ h.elim⟩
|
||||
⟨fun h ↦ h.ne rfl, fun h ↦ h.elim⟩
|
||||
|
||||
@@ -8,5 +8,3 @@ import Init.Data.Option.Basic
|
||||
import Init.Data.Option.BasicAux
|
||||
import Init.Data.Option.Instances
|
||||
import Init.Data.Option.Lemmas
|
||||
import Init.Data.Option.Attach
|
||||
import Init.Data.Option.List
|
||||
|
||||
@@ -1,242 +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.Data.Option.Basic
|
||||
import Init.Data.Option.List
|
||||
import Init.Data.List.Attach
|
||||
import Init.BinderPredicates
|
||||
|
||||
namespace Option
|
||||
|
||||
/--
|
||||
Unsafe implementation of `attachWith`, taking advantage of the fact that the representation of
|
||||
`Option {x // P x}` is the same as the input `Option α`.
|
||||
-/
|
||||
@[inline] private unsafe def attachWithImpl
|
||||
(o : Option α) (P : α → Prop) (_ : ∀ x ∈ o, P x) : Option {x // P x} := unsafeCast o
|
||||
|
||||
/-- "Attach" a proof `P x` that holds for the element of `o`, if present,
|
||||
to produce a new option with the same element but in the type `{x // P x}`. -/
|
||||
@[implemented_by attachWithImpl] def attachWith
|
||||
(xs : Option α) (P : α → Prop) (H : ∀ x ∈ xs, P x) : Option {x // P x} :=
|
||||
match xs with
|
||||
| none => none
|
||||
| some x => some ⟨x, H x (mem_some_self x)⟩
|
||||
|
||||
/-- "Attach" the proof that the element of `xs`, if present, is in `xs`
|
||||
to produce a new option with the same elements but in the type `{x // x ∈ xs}`. -/
|
||||
@[inline] def attach (xs : Option α) : Option {x // x ∈ xs} := xs.attachWith _ fun _ => id
|
||||
|
||||
@[simp] theorem attach_none : (none : Option α).attach = none := rfl
|
||||
@[simp] theorem attachWith_none : (none : Option α).attachWith P H = none := rfl
|
||||
|
||||
@[simp] theorem attach_some {x : α} :
|
||||
(some x).attach = some ⟨x, rfl⟩ := rfl
|
||||
@[simp] theorem attachWith_some {x : α} {P : α → Prop} (h : ∀ (b : α), b ∈ some x → P b) :
|
||||
(some x).attachWith P h = some ⟨x, by simpa using h⟩ := rfl
|
||||
|
||||
theorem attach_congr {o₁ o₂ : Option α} (h : o₁ = o₂) :
|
||||
o₁.attach = o₂.attach.map (fun x => ⟨x.1, h ▸ x.2⟩) := by
|
||||
subst h
|
||||
simp
|
||||
|
||||
theorem attachWith_congr {o₁ o₂ : Option α} (w : o₁ = o₂) {P : α → Prop} {H : ∀ x ∈ o₁, P x} :
|
||||
o₁.attachWith P H = o₂.attachWith P fun x h => H _ (w ▸ h) := by
|
||||
subst w
|
||||
simp
|
||||
|
||||
theorem attach_map_coe (o : Option α) (f : α → β) :
|
||||
(o.attach.map fun (i : {i // i ∈ o}) => f i) = o.map f := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem attach_map_val (o : Option α) (f : α → β) :
|
||||
(o.attach.map fun i => f i.val) = o.map f :=
|
||||
attach_map_coe _ _
|
||||
|
||||
@[simp]
|
||||
theorem attach_map_subtype_val (o : Option α) :
|
||||
o.attach.map Subtype.val = o :=
|
||||
(attach_map_coe _ _).trans (congrFun Option.map_id _)
|
||||
|
||||
theorem attachWith_map_coe {p : α → Prop} (f : α → β) (o : Option α) (H : ∀ a ∈ o, p a) :
|
||||
((o.attachWith p H).map fun (i : { i // p i}) => f i.val) = o.map f := by
|
||||
cases o <;> simp [H]
|
||||
|
||||
theorem attachWith_map_val {p : α → Prop} (f : α → β) (o : Option α) (H : ∀ a ∈ o, p a) :
|
||||
((o.attachWith p H).map fun i => f i.val) = o.map f :=
|
||||
attachWith_map_coe _ _ _
|
||||
|
||||
@[simp]
|
||||
theorem attachWith_map_subtype_val {p : α → Prop} (o : Option α) (H : ∀ a ∈ o, p a) :
|
||||
(o.attachWith p H).map Subtype.val = o :=
|
||||
(attachWith_map_coe _ _ _).trans (congrFun Option.map_id _)
|
||||
|
||||
@[simp] theorem mem_attach : ∀ (o : Option α) (x : {x // x ∈ o}), x ∈ o.attach
|
||||
| none, ⟨x, h⟩ => by simp at h
|
||||
| some a, ⟨x, h⟩ => by simpa using h
|
||||
|
||||
@[simp] theorem isNone_attach (o : Option α) : o.attach.isNone = o.isNone := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem isNone_attachWith {p : α → Prop} (o : Option α) (H : ∀ a ∈ o, p a) :
|
||||
(o.attachWith p H).isNone = o.isNone := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem isSome_attach (o : Option α) : o.attach.isSome = o.isSome := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem isSome_attachWith {p : α → Prop} (o : Option α) (H : ∀ a ∈ o, p a) :
|
||||
(o.attachWith p H).isSome = o.isSome := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem attach_eq_none_iff (o : Option α) : o.attach = none ↔ o = none := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem attach_eq_some_iff {o : Option α} {x : {x // x ∈ o}} :
|
||||
o.attach = some x ↔ o = some x.val := by
|
||||
cases o <;> cases x <;> simp
|
||||
|
||||
@[simp] theorem attachWith_eq_none_iff {p : α → Prop} (o : Option α) (H : ∀ a ∈ o, p a) :
|
||||
o.attachWith p H = none ↔ o = none := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem attachWith_eq_some_iff {p : α → Prop} {o : Option α} (H : ∀ a ∈ o, p a) {x : {x // p x}} :
|
||||
o.attachWith p H = some x ↔ o = some x.val := by
|
||||
cases o <;> cases x <;> simp
|
||||
|
||||
@[simp] theorem get_attach {o : Option α} (h : o.attach.isSome = true) :
|
||||
o.attach.get h = ⟨o.get (by simpa using h), by simp⟩ := by
|
||||
cases o
|
||||
· simp at h
|
||||
· simp [get_some]
|
||||
|
||||
@[simp] theorem get_attachWith {p : α → Prop} {o : Option α} (H : ∀ a ∈ o, p a) (h : (o.attachWith p H).isSome) :
|
||||
(o.attachWith p H).get h = ⟨o.get (by simpa using h), H _ (by simp)⟩ := by
|
||||
cases o
|
||||
· simp at h
|
||||
· simp [get_some]
|
||||
|
||||
@[simp] theorem toList_attach (o : Option α) :
|
||||
o.attach.toList = o.toList.attach.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem attach_map {o : Option α} (f : α → β) :
|
||||
(o.map f).attach = o.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem attachWith_map {o : Option α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ o.map f → P b} :
|
||||
(o.map f).attachWith P H = (o.attachWith (P ∘ f) (fun a h => H _ (mem_map_of_mem f h))).map
|
||||
fun ⟨x, h⟩ => ⟨f x, h⟩ := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem map_attach {o : Option α} (f : { x // x ∈ o } → β) :
|
||||
o.attach.map f = o.pmap (fun a (h : a ∈ o) => f ⟨a, h⟩) (fun a h => h) := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem map_attachWith {o : Option α} {P : α → Prop} {H : ∀ (a : α), a ∈ o → P a}
|
||||
(f : { x // P x } → β) :
|
||||
(o.attachWith P H).map f =
|
||||
o.pmap (fun a (h : a ∈ o ∧ P a) => f ⟨a, h.2⟩) (fun a h => ⟨h, H a h⟩) := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem attach_bind {o : Option α} {f : α → Option β} :
|
||||
(o.bind f).attach =
|
||||
o.attach.bind fun ⟨x, h⟩ => (f x).attach.map fun ⟨y, h'⟩ => ⟨y, mem_bind_iff.mpr ⟨x, h, h'⟩⟩ := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem bind_attach {o : Option α} {f : {x // x ∈ o} → Option β} :
|
||||
o.attach.bind f = o.pbind fun a h => f ⟨a, h⟩ := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem pbind_eq_bind_attach {o : Option α} {f : (a : α) → a ∈ o → Option β} :
|
||||
o.pbind f = o.attach.bind fun ⟨x, h⟩ => f x h := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem attach_filter {o : Option α} {p : α → Bool} :
|
||||
(o.filter p).attach =
|
||||
o.attach.bind fun ⟨x, h⟩ => if h' : p x then some ⟨x, by simp_all⟩ else none := by
|
||||
cases o with
|
||||
| none => simp
|
||||
| some a =>
|
||||
simp only [filter_some, attach_some]
|
||||
ext
|
||||
simp only [mem_def, attach_eq_some_iff, ite_none_right_eq_some, some.injEq, some_bind,
|
||||
dite_none_right_eq_some]
|
||||
constructor
|
||||
· rintro ⟨h, w⟩
|
||||
refine ⟨h, by ext; simpa using w⟩
|
||||
· rintro ⟨h, rfl⟩
|
||||
simp [h]
|
||||
|
||||
theorem filter_attach {o : Option α} {p : {x // x ∈ o} → Bool} :
|
||||
o.attach.filter p = o.pbind fun a h => if p ⟨a, h⟩ then some ⟨a, h⟩ else none := by
|
||||
cases o <;> simp [filter_some]
|
||||
|
||||
/-! ## unattach
|
||||
|
||||
`Option.unattach` is the (one-sided) inverse of `Option.attach`. It is a synonym for `Option.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 : Option { 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 [Option.unattach, -Option.map_subtype]` to unfold.
|
||||
-/
|
||||
def unattach {α : Type _} {p : α → Prop} (o : Option { x // p x }) := o.map (·.val)
|
||||
|
||||
@[simp] theorem unattach_none {p : α → Prop} : (none : Option { x // p x }).unattach = none := rfl
|
||||
@[simp] theorem unattach_some {p : α → Prop} {a : { x // p x }} :
|
||||
(some a).unattach = a.val := rfl
|
||||
|
||||
@[simp] theorem isSome_unattach {p : α → Prop} {o : Option { x // p x }} :
|
||||
o.unattach.isSome = o.isSome := by
|
||||
simp [unattach]
|
||||
|
||||
@[simp] theorem isNone_unattach {p : α → Prop} {o : Option { x // p x }} :
|
||||
o.unattach.isNone = o.isNone := by
|
||||
simp [unattach]
|
||||
|
||||
@[simp] theorem unattach_attach (o : Option α) : o.attach.unattach = o := by
|
||||
cases o <;> simp
|
||||
|
||||
@[simp] theorem unattach_attachWith {p : α → Prop} {o : Option α}
|
||||
{H : ∀ a ∈ o, p a} :
|
||||
(o.attachWith p H).unattach = o := by
|
||||
cases o <;> simp
|
||||
|
||||
/-! ### Recognizing higher order functions on subtypes using a function that only depends on the value. -/
|
||||
|
||||
/--
|
||||
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} {o : Option { x // p x }}
|
||||
{f : { x // p x } → β} {g : α → β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
o.map f = o.unattach.map g := by
|
||||
cases o <;> simp [hf]
|
||||
|
||||
@[simp] theorem bind_subtype {p : α → Prop} {o : Option { x // p x }}
|
||||
{f : { x // p x } → Option β} {g : α → Option β} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
(o.bind f) = o.unattach.bind g := by
|
||||
cases o <;> simp [hf]
|
||||
|
||||
@[simp] theorem unattach_filter {p : α → Prop} {o : Option { x // p x }}
|
||||
{f : { x // p x } → Bool} {g : α → Bool} {hf : ∀ x h, f ⟨x, h⟩ = g x} :
|
||||
(o.filter f).unattach = o.unattach.filter g := by
|
||||
cases o
|
||||
· simp
|
||||
· simp only [filter_some, hf, unattach_some]
|
||||
split <;> simp
|
||||
|
||||
end Option
|
||||
@@ -202,7 +202,7 @@ result.
|
||||
instance (α) [BEq α] [LawfulBEq α] : LawfulBEq (Option α) where
|
||||
rfl {x} :=
|
||||
match x with
|
||||
| some _ => LawfulBEq.rfl (α := α)
|
||||
| some x => LawfulBEq.rfl (α := α)
|
||||
| none => rfl
|
||||
eq_of_beq {x y h} := by
|
||||
match x, y with
|
||||
|
||||
@@ -14,7 +14,7 @@ namespace Option
|
||||
|
||||
theorem mem_iff {a : α} {b : Option α} : a ∈ b ↔ b = some a := .rfl
|
||||
|
||||
theorem mem_some {a b : α} : a ∈ some b ↔ b = a := by simp
|
||||
@[simp] theorem mem_some {a b : α} : a ∈ some b ↔ b = a := by simp [mem_iff]
|
||||
|
||||
theorem mem_some_self (a : α) : a ∈ some a := mem_some.2 rfl
|
||||
|
||||
@@ -79,7 +79,7 @@ theorem eq_none_iff_forall_not_mem : o = none ↔ ∀ a, a ∉ o :=
|
||||
|
||||
theorem isSome_iff_exists : isSome x ↔ ∃ a, x = some a := by cases x <;> simp [isSome]
|
||||
|
||||
theorem isSome_eq_isSome : (isSome x = isSome y) ↔ (x = none ↔ y = none) := by
|
||||
@[simp] theorem isSome_eq_isSome : (isSome x = isSome y) ↔ (x = none ↔ y = none) := by
|
||||
cases x <;> cases y <;> simp
|
||||
|
||||
@[simp] theorem isNone_none : @isNone α none = true := rfl
|
||||
@@ -138,10 +138,6 @@ theorem bind_eq_none' {o : Option α} {f : α → Option β} :
|
||||
o.bind f = none ↔ ∀ b a, a ∈ o → b ∉ f a := by
|
||||
simp only [eq_none_iff_forall_not_mem, not_exists, not_and, mem_def, bind_eq_some]
|
||||
|
||||
theorem mem_bind_iff {o : Option α} {f : α → Option β} :
|
||||
b ∈ o.bind f ↔ ∃ a, a ∈ o ∧ b ∈ f a := by
|
||||
cases o <;> simp
|
||||
|
||||
theorem bind_comm {f : α → β → Option γ} (a : Option α) (b : Option β) :
|
||||
(a.bind fun x => b.bind (f x)) = b.bind fun y => a.bind fun x => f x y := by
|
||||
cases a <;> cases b <;> rfl
|
||||
@@ -235,28 +231,10 @@ theorem isSome_filter_of_isSome (p : α → Bool) (o : Option α) (h : (o.filter
|
||||
o.isSome := by
|
||||
cases o <;> simp at h ⊢
|
||||
|
||||
@[simp] theorem filter_eq_none {p : α → Bool} :
|
||||
o.filter p = none ↔ o = none ∨ ∀ a, a ∈ o → ¬ p a := by
|
||||
@[simp] theorem filter_eq_none (p : α → Bool) :
|
||||
Option.filter p o = none ↔ o = none ∨ ∀ a, a ∈ o → ¬ p a := by
|
||||
cases o <;> simp [filter_some]
|
||||
|
||||
@[simp] theorem filter_eq_some {o : Option α} {p : α → Bool} :
|
||||
o.filter p = some a ↔ a ∈ o ∧ p a := by
|
||||
cases o with
|
||||
| none => simp
|
||||
| some a =>
|
||||
simp [filter_some]
|
||||
split <;> rename_i h
|
||||
· simp only [some.injEq, iff_self_and]
|
||||
rintro rfl
|
||||
exact h
|
||||
· simp only [reduceCtorEq, false_iff, not_and, Bool.not_eq_true]
|
||||
rintro rfl
|
||||
simpa using h
|
||||
|
||||
theorem mem_filter_iff {p : α → Bool} {a : α} {o : Option α} :
|
||||
a ∈ o.filter p ↔ a ∈ o ∧ p a := by
|
||||
simp
|
||||
|
||||
@[simp] theorem all_guard (p : α → Prop) [DecidablePred p] (a : α) :
|
||||
Option.all q (guard p a) = (!p a || q a) := by
|
||||
simp only [guard]
|
||||
@@ -330,8 +308,8 @@ theorem guard_comp {p : α → Prop} [DecidablePred p] {f : β → α} :
|
||||
theorem liftOrGet_eq_or_eq {f : α → α → α} (h : ∀ a b, f a b = a ∨ f a b = b) :
|
||||
∀ o₁ o₂, liftOrGet f o₁ o₂ = o₁ ∨ liftOrGet f o₁ o₂ = o₂
|
||||
| none, none => .inl rfl
|
||||
| some _, none => .inl rfl
|
||||
| none, some _ => .inr rfl
|
||||
| some a, none => .inl rfl
|
||||
| none, some b => .inr rfl
|
||||
| some a, some b => by have := h a b; simp [liftOrGet] at this ⊢; exact this
|
||||
|
||||
@[simp] theorem liftOrGet_none_left {f} {b : Option α} : liftOrGet f none b = b := by
|
||||
@@ -372,8 +350,6 @@ end choice
|
||||
|
||||
@[simp] theorem toList_none (α : Type _) : (none : Option α).toList = [] := rfl
|
||||
|
||||
-- See `Init.Data.Option.List` for lemmas about `toList`.
|
||||
|
||||
@[simp] theorem or_some : (some a).or o = some a := rfl
|
||||
@[simp] theorem none_or : none.or o = o := rfl
|
||||
|
||||
@@ -472,6 +448,22 @@ end beq
|
||||
/-! ### ite -/
|
||||
section ite
|
||||
|
||||
@[simp] theorem mem_dite_none_left {x : α} [Decidable p] {l : ¬ p → Option α} :
|
||||
(x ∈ if h : p then none else l h) ↔ ∃ h : ¬ p, x ∈ l h := by
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem mem_dite_none_right {x : α} [Decidable p] {l : p → Option α} :
|
||||
(x ∈ if h : p then l h else none) ↔ ∃ h : p, x ∈ l h := by
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem mem_ite_none_left {x : α} [Decidable p] {l : Option α} :
|
||||
(x ∈ if p then none else l) ↔ ¬ p ∧ x ∈ l := by
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem mem_ite_none_right {x : α} [Decidable p] {l : Option α} :
|
||||
(x ∈ if p then l else none) ↔ p ∧ x ∈ l := by
|
||||
split <;> simp_all
|
||||
|
||||
@[simp] theorem dite_none_left_eq_some {p : Prop} [Decidable p] {b : ¬p → Option β} :
|
||||
(if h : p then none else b h) = some a ↔ ∃ h, b h = some a := by
|
||||
split <;> simp_all
|
||||
@@ -504,22 +496,6 @@ section ite
|
||||
some a = (if p then b else none) ↔ p ∧ some a = b := by
|
||||
split <;> simp_all
|
||||
|
||||
theorem mem_dite_none_left {x : α} [Decidable p] {l : ¬ p → Option α} :
|
||||
(x ∈ if h : p then none else l h) ↔ ∃ h : ¬ p, x ∈ l h := by
|
||||
simp
|
||||
|
||||
theorem mem_dite_none_right {x : α} [Decidable p] {l : p → Option α} :
|
||||
(x ∈ if h : p then l h else none) ↔ ∃ h : p, x ∈ l h := by
|
||||
simp
|
||||
|
||||
theorem mem_ite_none_left {x : α} [Decidable p] {l : Option α} :
|
||||
(x ∈ if p then none else l) ↔ ¬ p ∧ x ∈ l := by
|
||||
simp
|
||||
|
||||
theorem mem_ite_none_right {x : α} [Decidable p] {l : Option α} :
|
||||
(x ∈ if p then l else none) ↔ p ∧ x ∈ l := by
|
||||
simp
|
||||
|
||||
@[simp] theorem isSome_dite {p : Prop} [Decidable p] {b : p → β} :
|
||||
(if h : p then some (b h) else none).isSome = true ↔ p := by
|
||||
split <;> simpa
|
||||
|
||||
@@ -1,14 +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.Data.List.Lemmas
|
||||
|
||||
namespace Option
|
||||
|
||||
@[simp] theorem mem_toList {a : α} {o : Option α} : a ∈ o.toList ↔ a ∈ o := by
|
||||
cases o <;> simp [eq_comm]
|
||||
|
||||
end Option
|
||||
@@ -51,12 +51,6 @@ instance [Repr α] : Repr (id α) :=
|
||||
instance [Repr α] : Repr (Id α) :=
|
||||
inferInstanceAs (Repr α)
|
||||
|
||||
/-
|
||||
This instance allows us to use `Empty` as a type parameter without causing instance synthesis to fail.
|
||||
-/
|
||||
instance : Repr Empty where
|
||||
reprPrec := nofun
|
||||
|
||||
instance : Repr Bool where
|
||||
reprPrec
|
||||
| true, _ => "true"
|
||||
|
||||
@@ -322,7 +322,7 @@ theorem lt_next (s : String) (i : Pos) : i.1 < (s.next i).1 :=
|
||||
|
||||
theorem utf8PrevAux_lt_of_pos : ∀ (cs : List Char) (i p : Pos), p ≠ 0 →
|
||||
(utf8PrevAux cs i p).1 < p.1
|
||||
| [], _, _, h =>
|
||||
| [], i, p, h =>
|
||||
Nat.lt_of_le_of_lt (Nat.zero_le _)
|
||||
(Nat.zero_lt_of_ne_zero (mt (congrArg Pos.mk) h))
|
||||
| c::cs, i, p, h => by
|
||||
|
||||
@@ -117,11 +117,11 @@ def utf8EncodeChar (c : Char) : List UInt8 :=
|
||||
/-- Converts the given `String` to a [UTF-8](https://en.wikipedia.org/wiki/UTF-8) encoded byte array. -/
|
||||
@[extern "lean_string_to_utf8"]
|
||||
def toUTF8 (a : @& String) : ByteArray :=
|
||||
⟨⟨a.data.flatMap utf8EncodeChar⟩⟩
|
||||
⟨⟨a.data.bind utf8EncodeChar⟩⟩
|
||||
|
||||
@[simp] theorem size_toUTF8 (s : String) : s.toUTF8.size = s.utf8ByteSize := by
|
||||
simp [toUTF8, ByteArray.size, Array.size, utf8ByteSize, List.flatMap]
|
||||
induction s.data <;> simp [List.map, List.flatten, utf8ByteSize.go, Nat.add_comm, *]
|
||||
simp [toUTF8, ByteArray.size, Array.size, utf8ByteSize, List.bind]
|
||||
induction s.data <;> simp [List.map, List.join, utf8ByteSize.go, Nat.add_comm, *]
|
||||
|
||||
/-- Accesses a byte in the UTF-8 encoding of the `String`. O(1) -/
|
||||
@[extern "lean_string_get_byte_fast"]
|
||||
|
||||
@@ -156,11 +156,11 @@ theorem getElem?_neg [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem d
|
||||
theorem getElem!_pos [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
|
||||
[Inhabited elem] (c : cont) (i : idx) (h : dom c i) [Decidable (dom c i)] :
|
||||
c[i]! = c[i]'h := by
|
||||
simp [getElem!_def, getElem?_def, h]
|
||||
simp only [getElem!_def, getElem?_def, h]
|
||||
|
||||
theorem getElem!_neg [GetElem? cont idx elem dom] [LawfulGetElem cont idx elem dom]
|
||||
[Inhabited elem] (c : cont) (i : idx) (h : ¬dom c i) [Decidable (dom c i)] : c[i]! = default := by
|
||||
simp [getElem!_def, getElem?_def, h]
|
||||
simp only [getElem!_def, getElem?_def, h]
|
||||
|
||||
namespace Fin
|
||||
|
||||
|
||||
@@ -7,7 +7,7 @@ Additional goodies for writing macros
|
||||
-/
|
||||
prelude
|
||||
import Init.MetaTypes
|
||||
import Init.Data.Array.GetLit
|
||||
import Init.Data.Array.Basic
|
||||
import Init.Data.Option.BasicAux
|
||||
|
||||
namespace Lean
|
||||
@@ -862,7 +862,7 @@ partial def decodeRawStrLitAux (s : String) (i : String.Pos) (num : Nat) : Strin
|
||||
/--
|
||||
Takes the string literal lexical syntax parsed by the parser and interprets it as a string.
|
||||
This is where escape sequences are processed for example.
|
||||
The string `s` is either a plain string literal or a raw string literal.
|
||||
The string `s` is is either a plain string literal or a raw string literal.
|
||||
|
||||
If it returns `none` then the string literal is ill-formed, which indicates a bug in the parser.
|
||||
The function is not required to return `none` if the string literal is ill-formed.
|
||||
|
||||
@@ -535,21 +535,24 @@ syntax (name := includeStr) "include_str " term : term
|
||||
|
||||
/--
|
||||
The `run_cmd doSeq` command executes code in `CommandElabM Unit`.
|
||||
This is the same as `#eval show CommandElabM Unit from discard do doSeq`.
|
||||
This is almost the same as `#eval show CommandElabM Unit from do doSeq`,
|
||||
except that it doesn't print an empty diagnostic.
|
||||
-/
|
||||
syntax (name := runCmd) "run_cmd " doSeq : command
|
||||
|
||||
/--
|
||||
The `run_elab doSeq` command executes code in `TermElabM Unit`.
|
||||
This is the same as `#eval show TermElabM Unit from discard do doSeq`.
|
||||
This is almost the same as `#eval show TermElabM Unit from do doSeq`,
|
||||
except that it doesn't print an empty diagnostic.
|
||||
-/
|
||||
syntax (name := runElab) "run_elab " doSeq : command
|
||||
|
||||
/--
|
||||
The `run_meta doSeq` command executes code in `MetaM Unit`.
|
||||
This is the same as `#eval show MetaM Unit from do discard doSeq`.
|
||||
This is almost the same as `#eval show MetaM Unit from do doSeq`,
|
||||
except that it doesn't print an empty diagnostic.
|
||||
|
||||
(This is effectively a synonym for `run_elab` since `MetaM` lifts to `TermElabM`.)
|
||||
(This is effectively a synonym for `run_elab`.)
|
||||
-/
|
||||
syntax (name := runMeta) "run_meta " doSeq : command
|
||||
|
||||
@@ -672,13 +675,6 @@ Message ordering:
|
||||
|
||||
For example, `#guard_msgs (error, drop all) in cmd` means to check warnings and drop
|
||||
everything else.
|
||||
|
||||
The command elaborator has special support for `#guard_msgs` for linting.
|
||||
The `#guard_msgs` itself wants to capture linter warnings,
|
||||
so it elaborates the command it is attached to as if it were a top-level command.
|
||||
However, the command elaborator runs linters for *all* top-level commands,
|
||||
which would include `#guard_msgs` itself, and would cause duplicate and/or uncaptured linter warnings.
|
||||
The top-level command elaborator only runs the linters if `#guard_msgs` is not present.
|
||||
-/
|
||||
syntax (name := guardMsgsCmd)
|
||||
(docComment)? "#guard_msgs" (ppSpace guardMsgsSpec)? " in" ppLine command : command
|
||||
|
||||
@@ -223,6 +223,38 @@ end Lean
|
||||
| `($_ $array $index) => `($array[$index]?)
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr1] def unexpandMkStr1 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr2] def unexpandMkStr2 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr3] def unexpandMkStr3 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str $a3:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString ++ "." ++ a3.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr4] def unexpandMkStr4 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str $a3:str $a4:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString ++ "." ++ a3.getString ++ "." ++ a4.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr5] def unexpandMkStr5 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str $a3:str $a4:str $a5:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString ++ "." ++ a3.getString ++ "." ++ a4.getString ++ "." ++ a5.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr6] def unexpandMkStr6 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str $a3:str $a4:str $a5:str $a6:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString ++ "." ++ a3.getString ++ "." ++ a4.getString ++ "." ++ a5.getString ++ "." ++ a6.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr7] def unexpandMkStr7 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str $a3:str $a4:str $a5:str $a6:str $a7:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString ++ "." ++ a3.getString ++ "." ++ a4.getString ++ "." ++ a5.getString ++ "." ++ a6.getString ++ "." ++ a7.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Name.mkStr8] def unexpandMkStr8 : Lean.PrettyPrinter.Unexpander
|
||||
| `($(_) $a1:str $a2:str $a3:str $a4:str $a5:str $a6:str $a7:str $a8:str) => return mkNode `Lean.Parser.Term.quotedName #[Syntax.mkNameLit ("`" ++ a1.getString ++ "." ++ a2.getString ++ "." ++ a3.getString ++ "." ++ a4.getString ++ "." ++ a5.getString ++ "." ++ a6.getString ++ "." ++ a7.getString ++ "." ++ a8.getString)]
|
||||
| _ => throw ()
|
||||
|
||||
@[app_unexpander Array.empty] def unexpandArrayEmpty : Lean.PrettyPrinter.Unexpander
|
||||
| _ => `(#[])
|
||||
|
||||
|
||||
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Reference in New Issue
Block a user