feat: more BitVec lemmas

#3656 used the wrong name in `RefIdent`, which lead to "Find References"
being broken. I really need to set up some tests for this functionality
...

.github/workflows/ci.yml

@@ -98,7 +98,8 @@ jobs:
                 // exclude seriously slow tests
                 "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
               },
-              {
+              // TODO: suddenly started failing in CI
+              /*{
                 "name": "Linux fsanitize",
                 "os": "ubuntu-latest",
                 "quick": false,
@@ -106,7 +107,7 @@ jobs:
                 "CMAKE_OPTIONS": "-DLEAN_EXTRA_CXX_FLAGS=-fsanitize=address,undefined -DLEANC_EXTRA_FLAGS='-fsanitize=address,undefined -fsanitize-link-c++-runtime' -DSMALL_ALLOCATOR=OFF -DBSYMBOLIC=OFF",
                 // exclude seriously slow/problematic tests (laketests crash)
                 "CTEST_OPTIONS": "-E 'interactivetest|leanpkgtest|laketest|benchtest'"
-              },
+              },*/
               {
                 "name": "macOS",
                 "os": "macos-latest",
@@ -445,9 +446,10 @@ jobs:
     name: Build matrix complete
     runs-on: ubuntu-latest
     needs: build
-    if: ${{ always() }}
+    # mark as merely cancelled not failed if builds are cancelled
+    if: ${{ !cancelled() }}
     steps:
-    - if: contains(needs.*.result, 'failure') ||  contains(needs.*.result, 'cancelled')
+    - if: contains(needs.*.result, 'failure')
       uses: actions/github-script@v7
       with:
         script: |

.github/workflows/nix-ci.yml

@@ -6,6 +6,7 @@ on:
     tags:
       - '*'
   pull_request:
+    types: [opened, synchronize, reopened, labeled]
   merge_group:
 
 concurrency:

.github/workflows/pr-release.yml

@@ -126,21 +126,19 @@ jobs:
           if [ "$NIGHTLY_SHA" = "$MERGE_BASE_SHA" ]; then
             echo "The merge base of this PR coincides with the nightly release"
 
-            MATHLIB_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover-community/mathlib4.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
-
-            if [[ -n "$MATHLIB_REMOTE_TAGS" ]]; then
-              echo "... and Mathlib has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
-              MESSAGE=""
-            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."
-            fi
-
             STD_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover/std4.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
+            MATHLIB_REMOTE_TAGS="$(git ls-remote https://github.com/leanprover-community/mathlib4.git nightly-testing-"$MOST_RECENT_NIGHTLY")"
 
             if [[ -n "$STD_REMOTE_TAGS" ]]; then
               echo "... and Std has a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
               MESSAGE=""
+
+              if [[ -n "$MATHLIB_REMOTE_TAGS" ]]; then
+                echo "... and Mathlib has a 'nightly-testing-$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."
+              fi
             else
               echo "... but Std does not yet have a 'nightly-testing-$MOST_RECENT_NIGHTLY' tag."
               MESSAGE="- ❗ Std CI can not be attempted yet, as the \`nightly-testing-$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\`, Std CI should run now."

CODEOWNERS

@@ -13,6 +13,7 @@
 /src/Lean/Data/Lsp/ @mhuisi
 /src/Lean/Elab/Deriving/ @semorrison
 /src/Lean/Elab/Tactic/ @semorrison
+/src/Lean/Language/ @Kha
 /src/Lean/Meta/Tactic/ @leodemoura
 /src/Lean/Parser/ @Kha
 /src/Lean/PrettyPrinter/ @Kha

RELEASES.md

@@ -8,9 +8,88 @@ This file contains work-in-progress notes for the upcoming release, as well as p
 Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
 of each version.
 
-v4.7.0 (development in progress)
+v4.8.0 (development in progress)
 ---------
 
+* **Executables configured with `supportInterpreter := true` on Windows should now be run via `lake exe` to function properly.**
+
+  The way Lean is built on Windows has changed (see PR [#3601](https://github.com/leanprover/lean4/pull/3601)). As a result, Lake now dynamically links executables with `supportInterpreter := true` on Windows to `libleanshared.dll` and `libInit_shared.dll`. Therefore, such executables will not run unless those shared libraries are co-located with the executables or part of `PATH`. Running the executable via `lake exe` will ensure these libraries are part of `PATH`.
+
+  In a related change, the signature of the `nativeFacets` Lake configuration options has changed from a static `Array` to a function `(shouldExport : Bool) → Array`. See its docstring or Lake's [README](src/lake/README.md) for further details on the changed option.
+
+* Lean now generates an error if the type of a theorem is **not** a proposition.
+
+* Importing two different files containing proofs of the same theorem is no longer considered an error. This feature is particularly useful for theorems that are automatically generated on demand (e.g., equational theorems).
+
+* New command `derive_functinal_induction`:
+
+  Derived from the definition of a (possibly mutually) recursive function
+  defined by well-founded recursion, a **functional induction principle** is
+  tailored to proofs about that function. For example from:
+  ```
+  def ackermann : Nat → Nat → Nat
+    | 0, m => m + 1
+    | n+1, 0 => ackermann n 1
+    | n+1, m+1 => ackermann n (ackermann (n + 1) m)
+  derive_functional_induction ackermann
+  ```
+  we get
+  ```
+  ackermann.induct (motive : Nat → Nat → Prop) (case1 : ∀ (m : Nat), motive 0 m)
+    (case2 : ∀ (n : Nat), motive n 1 → motive (Nat.succ n) 0)
+    (case3 : ∀ (n m : Nat), motive (n + 1) m → motive n (ackermann (n + 1) m) → motive (Nat.succ n) (Nat.succ m))
+    (x x : Nat) : motive x x
+  ```
+
+* The termination checker now recognizes more recursion patterns without an
+  explicit `terminatin_by`. In particular the idiom of counting up to an upper
+  bound, as in
+  ```
+  def Array.sum (arr : Array Nat) (i acc : Nat) : Nat :=
+    if _ : i < arr.size then
+      Array.sum arr (i+1) (acc + arr[i])
+    else
+      acc
+  ```
+  is recognized without having to say `termination_by arr.size - i`.
+
+
+Breaking changes:
+
+* Automatically generated equational theorems are now named using suffix `.eq_<idx>` instead of `._eq_<idx>`, and `.def` instead of `._unfold`. Example:
+```
+def fact : Nat → Nat
+  | 0 => 1
+  | n+1 => (n+1) * fact n
+
+theorem ex : fact 0 = 1 := by unfold fact; decide
+
+#check fact.eq_1
+-- fact.eq_1 : fact 0 = 1
+
+#check fact.eq_2
+-- fact.eq_2 (n : Nat) : fact (Nat.succ n) = (n + 1) * fact n
+
+#check fact.def
+/-
+fact.def :
+  ∀ (x : Nat),
+    fact x =
+      match x with
+      | 0 => 1
+      | Nat.succ n => (n + 1) * fact n
+-/
+```
+
+v4.7.0
+---------
+
+* `simp` and `rw` now use instance arguments found by unification,
+  rather than always resynthesizing. For backwards compatibility, the original behaviour is
+  available via `set_option tactic.skipAssignedInstances false`.
+  [#3507](https://github.com/leanprover/lean4/pull/3507) and
+  [#3509](https://github.com/leanprover/lean4/pull/3509).
+
 * When the `pp.proofs` is false, now omitted proofs use `⋯` rather than `_`,
   which gives a more helpful error message when copied from the Infoview.
   The `pp.proofs.threshold` option lets small proofs always be pretty printed.
@@ -94,7 +173,48 @@ v4.7.0 (development in progress)
 * Changed call hierarchy to sort entries and strip private header from names displayed in the call hierarchy. [#3482](https://github.com/leanprover/lean4/pull/3482)
 
 * There is now a low-level error recovery combinator in the parsing framework, primarily intended for DSLs. [#3413](https://github.com/leanprover/lean4/pull/3413)
-* The Library search `exact?` and `apply?` tactics that were originally in
+
+* You can now write `termination_by?` after a declaration to see the automatically inferred
+  termination argument, and turn it into a `termination_by …` clause using the “Try this” widget or a code action. [#3514](https://github.com/leanprover/lean4/pull/3514)
+
+* A large fraction of `Std` has been moved into the Lean repository.
+  This was motivated by:
+  1. Making universally useful tactics such as `ext`, `by_cases`, `change at`,
+    `norm_cast`, `rcases`, `simpa`, `simp?`, `omega`, and `exact?`
+    available to all users of Lean, without imports.
+  2. Minimizing the syntactic changes between plain Lean and Lean with `import Std`.
+  3. Simplifying the development process for the basic data types
+     `Nat`, `Int`, `Fin` (and variants such as `UInt64`), `List`, `Array`,
+     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 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.
+
+* The `/` and `%` notations in `Int` now use `Int.ediv` and `Int.emod`
+  (i.e. the rounding conventions have changed).
+  Previously `Std` overrode these notations, so this is no change for users of `Std`.
+  There is now kernel support for these functions.
+  [#3376](https://github.com/leanprover/lean4/pull/3376).
+
+* `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).
+  * `omega` now has support for `Fin` [#3427](https://github.com/leanprover/lean4/pull/3427),
+    the `<<<` operator [#3433](https://github.com/leanprover/lean4/pull/3433).
+  * During the port `omega` was modified to no longer identify atoms up to definitional equality
+    (so in particular it can no longer prove `id x ≤ x`). [#3525](https://github.com/leanprover/lean4/pull/3525).
+    This may cause some regressions.
+    We plan to provide a general purpose preprocessing tactic later, or an `omega!` mode.
+  * `omega` is now invoked in Lean's automation for termination proofs
+    [#3503](https://github.com/leanprover/lean4/pull/3503) as well as in
+    array indexing proofs [#3515](https://github.com/leanprover/lean4/pull/3515).
+    This automation will be substantially revised in the medium term,
+    and while `omega` does help automate some proofs, we plan to make this much more robust.
+
 * The library search tactics `exact?` and `apply?` that were originally in
   Mathlib are now available in Lean itself.  These use the implementation using
   lazy discrimination trees from `Std`, and thus do not require a disk cache but
@@ -109,10 +229,49 @@ v4.7.0 (development in progress)
   useful for checking tactics (particularly `simp`) behave as expected in test
   suites.
 
+* Previously, app unexpanders would only be applied to entire applications. However, some notations produce
+  functions, and these functions can be given additional arguments. The solution so far has been to write app unexpanders so that they can take an arbitrary number of additional arguments. However this leads to misleading hover information in the Infoview. For example, while `HAdd.hAdd f g 1` pretty prints as `(f + g) 1`, hovering over `f + g` shows `f`. There is no way to fix the situation from within an app unexpander; the expression position for `HAdd.hAdd f g` is absent, and app unexpanders cannot register TermInfo.
+
+  This commit changes the app delaborator to try running app unexpanders on every prefix of an application, from longest to shortest prefix. For efficiency, it is careful to only try this when app delaborators do in fact exist for the head constant, and it also ensures arguments are only delaborated once. Then, in `(f + g) 1`, the `f + g` gets TermInfo registered for that subexpression, making it properly hoverable.
+
+  [#3375](https://github.com/leanprover/lean4/pull/3375)
+
 Breaking changes:
 * `Lean.withTraceNode` and variants got a stronger `MonadAlwaysExcept` assumption to
   fix trace trees not being built on elaboration runtime exceptions. Instances for most elaboration
   monads built on `EIO Exception` should be synthesized automatically.
+* The `match ... with.` and `fun.` notations previously in Std have been replaced by
+  `nomatch ...` and `nofun`. [#3279](https://github.com/leanprover/lean4/pull/3279) and [#3286](https://github.com/leanprover/lean4/pull/3286)
+
+
+Other improvements:
+* several bug fixes for `simp`:
+  * we should not crash when `simp` loops [#3269](https://github.com/leanprover/lean4/pull/3269)
+  * `simp` gets stuck on `autoParam` [#3315](https://github.com/leanprover/lean4/pull/3315)
+  * `simp` fails when custom discharger makes no progress [#3317](https://github.com/leanprover/lean4/pull/3317)
+  * `simp` fails to discharge `autoParam` premises even when it can reduce them to `True` [#3314](https://github.com/leanprover/lean4/pull/3314)
+  * `simp?` suggests generated equations lemma names, fixes [#3547](https://github.com/leanprover/lean4/pull/3547) [#3573](https://github.com/leanprover/lean4/pull/3573)
+* fixes for `match` expressions:
+  * fix regression with builtin literals [#3521](https://github.com/leanprover/lean4/pull/3521)
+  * accept `match` when patterns cover all cases of a `BitVec` finite type [#3538](https://github.com/leanprover/lean4/pull/3538)
+  * fix matching `Int` literals [#3504](https://github.com/leanprover/lean4/pull/3504)
+  * patterns containing int values and constructors [#3496](https://github.com/leanprover/lean4/pull/3496)
+* improve `termination_by` error messages [#3255](https://github.com/leanprover/lean4/pull/3255)
+* fix `rename_i` in macros, fixes [#3553](https://github.com/leanprover/lean4/pull/3553) [#3581](https://github.com/leanprover/lean4/pull/3581)
+* fix excessive resource usage in `generalize`, fixes [#3524](https://github.com/leanprover/lean4/pull/3524) [#3575](https://github.com/leanprover/lean4/pull/3575)
+* an equation lemma with autoParam arguments fails to rewrite, fixing [#2243](https://github.com/leanprover/lean4/pull/2243) [#3316](https://github.com/leanprover/lean4/pull/3316)
+* `add_decl_doc` should check that declarations are local [#3311](https://github.com/leanprover/lean4/pull/3311)
+* instantiate the types of inductives with the right parameters, closing [#3242](https://github.com/leanprover/lean4/pull/3242) [#3246](https://github.com/leanprover/lean4/pull/3246)
+* New simprocs for many basic types. [#3407](https://github.com/leanprover/lean4/pull/3407)
+
+Lake fixes:
+* Warn on fetch cloud release failure [#3401](https://github.com/leanprover/lean4/pull/3401)
+* Cloud release trace & `lake build :release` errors [#3248](https://github.com/leanprover/lean4/pull/3248)
+
+v4.6.1
+---------
+* Backport of [#3552](https://github.com/leanprover/lean4/pull/3552) fixing a performance regression
+  in server startup.
 
 v4.6.0
 ---------

doc/SUMMARY.md

@@ -89,5 +89,6 @@
 - [Testing](./dev/testing.md)
 - [Debugging](./dev/debugging.md)
 - [Commit Convention](./dev/commit_convention.md)
+- [Release checklist](./dev/release_checklist.md)
 - [Building This Manual](./dev/mdbook.md)
 - [Foreign Function Interface](./dev/ffi.md)

doc/dev/ffi.md

@@ -111,6 +111,15 @@ if (lean_io_result_is_ok(res)) {
 lean_io_mark_end_initialization();
 ```
 
+In addition, any other thread not spawned by the Lean runtime itself must be initialized for Lean use by calling
+```c
+void lean_initialize_thread();
+```
+and should be finalized in order to free all thread-local resources by calling
+```c
+void lean_finalize_thread();
+```
+
 ## `@[extern]` in the Interpreter
 
 The interpreter can run Lean declarations for which symbols are available in loaded shared libraries, which includes `@[extern]` declarations.

doc/dev/release_checklist.md

@@ -0,0 +1,201 @@
+# Releasing a stable version
+
+This checklist walks you through releasing a stable version.
+See below for the checklist for release candidates.
+
+We'll use `v4.6.0` as the intended release version as a running example.
+
+- One week before the planned release, ensure that someone has written the first draft of the release blog post
+- `git checkout releases/v4.6.0`
+  (This branch should already exist, from the release candidates.)
+- `git pull`
+- In `src/CMakeLists.txt`, verify you see
+  - `set(LEAN_VERSION_MINOR 6)` (for whichever `6` is appropriate)
+  - `set(LEAN_VERSION_IS_RELEASE 1)`
+  - (both of these should already be in place from the release candidates)
+- It is possible that the `v4.6.0` section of `RELEASES.md` is out of sync between
+  `releases/v4.6.0` and `master`. This should be reconciled:
+  - Run `git diff master RELEASES.md`.
+  - You should expect to see additons on `master` in the `v4.7.0-rc1` section; ignore these.
+    (i.e. the new release notes for the upcoming release candidate).
+  - Reconcile discrepancies in the `v4.6.0` section,
+    usually via copy and paste and a commit to `releases/v4.6.0`.
+- `git tag v4.6.0`
+- `git push origin v4.6.0`
+- Now wait, while CI runs.
+  - You can monitor this at `https://github.com/leanprover/lean4/actions/workflows/ci.yml`,
+    looking for the `v4.6.0` tag.
+  - This step can take up to an hour.
+  - If you are intending to cut the next release candidate on the same day,
+    you may want to start on the release candidate checklist now.
+- Go to https://github.com/leanprover/lean4/releases and verify that the `v4.6.0` release appears.
+  - Edit the release notes on Github to select the "Set as the latest release".
+  - Copy and paste the Github release notes from the previous releases candidate for this version
+    (e.g. `v4.6.0-rc1`), and quickly sanity check.
+- Next, we will move a curated list of downstream repos to the latest stable release.
+  - For each of the repositories listed below:
+    - Make a PR to `master`/`main` changing the toolchain to `v4.6.0`.
+      The PR title should be "chore: bump toolchain to v4.6.0".
+      Since the `v4.6.0` release should be functionally identical to the last release candidate,
+      which the repository should already be on, this PR is a no-op besides changing the toolchain.
+    - Once this is merged, create the tag `v4.6.0` from `master`/`main` and push it.
+    - Merge the tag `v4.6.0` into the stable branch.
+  - We do this for the repositories:
+    - [lean4checker](https://github.com/leanprover/lean4checker)
+      - `lean4checker` uses a different version tagging scheme: use `toolchain/v4.6.0` rather than `v4.6.0`.
+    - [Std](https://github.com/leanprover-community/repl)
+    - [ProofWidgets4](https://github.com/leanprover-community/ProofWidgets4)
+      - `ProofWidgets` uses a sequential version tagging scheme, e.g. `v0.0.29`,
+        which does not refer to the toolchain being used.
+      - Make a new release in this sequence after merging the toolchain bump PR.
+      - `ProofWidgets` does not maintain a `stable` branch.
+    - [Aesop](https://github.com/leanprover-community/aesop)
+    - [Mathlib](https://github.com/leanprover-community/mathlib4)
+      - In addition to updating the `lean-toolchain` and `lakefile.lean`,
+        in `.github/workflows/build.yml.in` in the `lean4checker` section update the line
+        `git checkout toolchain/v4.6.0` to the appropriate tag,
+        and then run `.github/workflows/mk_build_yml.sh`.
+    - [REPL](https://github.com/leanprover-community/repl)
+      - Note that there are two copies of `lean-toolchain`/`lakefile.lean`:
+        in the root, and in `test/Mathlib/`.
+  - Note that there are dependencies between these packages:
+    you should update the lakefile so that you are using the `v4.6.0` tag of upstream repositories
+    (or the sequential tag for `ProofWidgets4`), and run `lake update` before committing.
+  - This means that this process is sequential; each repository must have its bump PR merged,
+    and the new tag pushed, before you can make the PR for the downstream repositories.
+    - `lean4checker` has no dependencies
+    - `Std` has no dependencies
+    - `Aesop` depends on `Std`
+    - `ProofWidgets4` depends on `Std`
+    - `Mathlib` depends on `Aesop`, `ProofWidgets4`, and `lean4checker` (and transitively on `Std`)
+    - `REPL` depends on `Mathlib` (this dependency is only for testing).
+- Merge the release announcement PR for the Lean website - it will be deployed automatically
+- Finally, make an announcement!
+  This should go in https://leanprover.zulipchat.com/#narrow/stream/113486-announce, with topic `v4.6.0`.
+  Please see previous announcements for suggested language.
+  You will want a few bullet points for main topics from the release notes.
+  Link to the blog post from the Zulip announcement.
+  Please also make sure that whoever is handling social media knows the release is out.
+
+## Optimistic(?) time estimates:
+- Initial checks and push the tag: 30 minutes.
+- Note that if `RELEASES.md` has discrepancies this could take longer!
+- Waiting for the release: 60 minutes.
+- Fixing release notes: 10 minutes.
+- Bumping toolchains in downstream repositories, up to creating the Mathlib PR: 30 minutes.
+- Waiting for Mathlib CI and bors: 120 minutes.
+- Finalizing Mathlib tags and stable branch, and updating REPL: 15 minutes.
+- Posting announcement and/or blog post: 20 minutes.
+
+# Creating a release candidate.
+
+This checklist walks you through creating the first release candidate for a version of Lean.
+
+We'll use `v4.7.0-rc1` as the intended release version in this example.
+
+- Decide which nightly release you want to turn into a release candidate.
+  We will use `nightly-2024-02-29` in this example.
+- It is essential that Std and Mathlib already have reviewed branches compatible with this nightly.
+  - Check that both Std and Mathlib's `bump/v4.7.0` branch contain `nightly-2024-02-29`
+    in their `lean-toolchain`.
+  - The steps required to reach that state are beyond the scope of this checklist, but see below!
+- Create the release branch from this nightly tag:
+    ```
+    git remote add nightly https://github.com/leanprover/lean4-nightly.git
+    git fetch nightly tag nightly-2024-02-29
+    git checkout nightly-2024-02-29
+    git checkout -b releases/v4.7.0
+    ```
+- In `RELEASES.md` remove `(development in progress)` from the `v4.7.0` section header.
+- Our current goal is to have written release notes only about major language features or breaking changes,
+  and to rely on automatically generated release notes for bugfixes and minor changes.
+  - Do not wait on `RELEASES.md` being perfect before creating the `release/v4.7.0` branch. It is essential to choose the nightly which will become the release candidate as early as possible, to avoid confusion.
+  - If there are major changes not reflected in `RELEASES.md` already, you may need to solicit help from the authors.
+  - Minor changes and bug fixes do not need to be documented in `RELEASES.md`: they will be added automatically on the Github release page.
+  - Commit your changes to `RELEASES.md`, and push.
+  - Remember that changes to `RELEASES.md` after you have branched `releases/v4.7.0` should also be cherry-picked back to `master`.
+- In `src/CMakeLists.txt`,
+  - verify that you see `set(LEAN_VERSION_MINOR 7)` (for whichever `7` is appropriate); this should already have been updated when the development cycle began.
+  - `set(LEAN_VERSION_IS_RELEASE 1)` (this should be a change; on `master` and nightly releases it is always `0`).
+  - Commit your changes to `src/CMakeLists.txt`, and push.
+- `git tag v4.7.0-rc1`
+- `git push origin v4.7.0-rc1`
+- Now wait, while CI runs.
+  - You can monitor this at `https://github.com/leanprover/lean4/actions/workflows/ci.yml`, looking for the `v4.7.0-rc1` tag.
+  - This step can take up to an hour.
+- Once the release appears at https://github.com/leanprover/lean4/releases/
+  - Edit the release notes on Github to select the "Set as a pre-release box".
+  - Copy the section of `RELEASES.md` for this version into the Github release notes.
+  - Use the title "Changes since v4.6.0 (from RELEASES.md)"
+  - Then in the "previous tag" dropdown, select `v4.6.0`, and click "Generate release notes".
+  - This will add a list of all the commits since the last stable version.
+    - Delete anything already mentioned in the hand-written release notes above.
+    - Delete "update stage0" commits, and anything with a completely inscrutable commit message.
+    - Briefly rearrange the remaining items by category (e.g. `simp`, `lake`, `bug fixes`),
+      but for minor items don't put any work in expanding on commit messages.
+  - (How we want to release notes to look is evolving: please update this section if it looks wrong!)
+- Next, we will move a curated list of downstream repos to the release candidate.
+  - This assumes that there is already a *reviewed* branch `bump/v4.7.0` on each repository
+    containing the required adaptations (or no adaptations are required).
+    The preparation of this branch is beyond the scope of this document.
+  - For each of the target repositories:
+    - Checkout the `bump/v4.7.0` branch.
+    - Verify that the `lean-toolchain` is set to the nightly from which the release candidate was created.
+    - `git merge origin/master`
+    - Change the `lean-toolchain` to `leanprover/lean4:v4.7.0-rc1`
+    - In `lakefile.lean`, change any dependencies which were using `nightly-testing` or `bump/v4.7.0` branches
+      back to `master` or `main`, and run `lake update` for those dependencies.
+    - Run `lake build` to ensure that dependencies are found (but it's okay to stop it after a moment).
+    - `git commit`
+    - `git push`
+    - Open a PR from `bump/v4.7.0` to `master`, and either merge it yourself after CI, if appropriate,
+      or notify the maintainers that it is ready to go.
+    - Once this PR has been merged, tag `master` with `v4.7.0-rc1` and push this tag.
+  - We do this for the same list of repositories as for stable releases, see above.
+    As above, there are dependencies between these, and so the process above is iterative.
+    It greatly helps if you can merge the `bump/v4.7.0` PRs yourself!
+  - For Std/Aesop/Mathlib, which maintain a `nightly-testing` branch, make sure there is a tag
+    `nightly-testing-2024-02-29` with date corresponding to the nightly used for the release
+    (create it if not), and then on the `nightly-testing` branch `git reset --hard master`, and force push.
+- Make an announcement!
+  This should go in https://leanprover.zulipchat.com/#narrow/stream/113486-announce, with topic `v4.7.0-rc1`.
+  Please see previous announcements for suggested language.
+  You will want a few bullet points for main topics from the release notes.
+  Please also make sure that whoever is handling social media knows the release is out.
+- Begin the next development cycle (i.e. for `v4.8.0`) on the Lean repository, by making a PR that:
+  - Updates `src/CMakeLists.txt` to say `set(LEAN_VERSION_MINOR 8)`
+  - Removes `(in development)` from the section heading in `RELEASES.md` for `v4.7.0`,
+    and creates a new `v4.8.0 (in development)` section heading.
+
+## Time estimates:
+Slightly longer than the corresponding steps for a stable release.
+Similar process, but more things go wrong.
+In particular, updating the downstream repositories is significantly more work
+(because we need to merge existing `bump/v4.7.0` branches, not just update a toolchain).
+
+# Preparing `bump/v4.7.0` branches
+
+While not part of the release process per se,
+this is a brief summary of the work that goes into updating Std/Aesop/Mathlib to new versions.
+
+Please read https://leanprover-community.github.io/contribute/tags_and_branches.html
+
+* Each repo has an unreviewed `nightly-testing` branch that
+  receives commits automatically from `master`, and
+  has its toolchain updated automatically for every nightly.
+  (Note: the aesop branch is not automated, and is updated on an as needed basis.)
+  As a consequence this branch is often broken.
+  A bot posts in the (private!) "Mathlib reviewers" stream on Zulip about the status of these branches.
+* We fix the breakages by committing directly to `nightly-testing`: there is no PR process.
+  * This can either be done by the person managing this process directly,
+    or by soliciting assistance from authors of files, or generally helpful people on Zulip!
+* Each repo has a `bump/v4.7.0` which accumulates reviewed changes adapting to new versions.
+* Once `nightly-testing` is working on a given nightly, say `nightly-2024-02-15`, we:
+  * Make sure `bump/v4.7.0` is up to date with `master` (by merging `master`, no PR necessary)
+  * Create from `bump/v4.7.0` a `bump/nightly-2024-02-15` branch.
+  * In that branch, `git merge --squash nightly-testing` to bring across changes from `nightly-testing`.
+  * Sanity check changes, commit, and make a PR to `bump/v4.7.0` from the `bump/nightly-2024-02-15` branch.
+  * Solicit review, merge the PR into `bump/v4,7,0`.
+* It is always okay to merge in the following directions:
+  `master` -> `bump/v4.7.0` -> `bump/nightly-2024-02-15` -> `nightly-testing`.
+  Please remember to push any merges you make to intermediate steps!

doc/examples/bintree.lean

@@ -277,14 +277,13 @@ theorem BinTree.find_insert (b : BinTree β) (k : Nat) (v : β)
     . by_cases' key < k
       cases h; apply ihr; assumption
 
-theorem BinTree.find_insert_of_ne (b : BinTree β) (h : k ≠ k') (v : β)
+theorem BinTree.find_insert_of_ne (b : BinTree β) (ne : k ≠ k') (v : β)
         : (b.insert k v).find? k' = b.find? k' := by
   let ⟨t, h⟩ := b; simp
   induction t with simp
   | leaf =>
-    intros
-    have_eq k k'
-    contradiction
+    intros le
+    exact Nat.lt_of_le_of_ne le ne
   | node left key value right ihl ihr =>
     let .node hl hr bl br := h
     specialize ihl bl

nix/bootstrap.nix

@@ -65,7 +65,7 @@ rec {
     installPhase = ''
       mkdir -p $out/bin $out/lib/lean
       mv bin/lean $out/bin/
-      mv lib/lean/*.so $out/lib/lean
+      mv lib/lean/*.{so,dylib} $out/lib/lean
     '';
     meta.mainProgram = "lean";
   });

nix/buildLeanPackage.nix

@@ -176,7 +176,7 @@ with builtins; let
       # make local "copy" so `drv`'s Nix store path doesn't end up in ccache's hash
       ln -s ${drv.c}/${drv.cPath} src.c
       # on the other hand, a debug build is pretty fast anyway, so preserve the path for gdb
-      leanc -c -o $out/$oPath $leancFlags -fPIC ${if debug then "${drv.c}/${drv.cPath} -g" else "src.c -O3 -DNDEBUG"}
+      leanc -c -o $out/$oPath $leancFlags -fPIC ${if debug then "${drv.c}/${drv.cPath} -g" else "src.c -O3 -DNDEBUG -DLEAN_EXPORTING"}
     '';
   };
   mkMod = mod: deps:

src/CMakeLists.txt

@@ -9,7 +9,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 7)
+set(LEAN_VERSION_MINOR 8)
 set(LEAN_VERSION_PATCH 0)
 set(LEAN_VERSION_IS_RELEASE 0)  # This number is 1 in the release revision, and 0 otherwise.
 set(LEAN_SPECIAL_VERSION_DESC "" CACHE STRING "Additional version description like 'nightly-2018-03-11'")
@@ -503,13 +503,13 @@ file(RELATIVE_PATH LIB ${LEAN_SOURCE_DIR} ${CMAKE_BINARY_DIR}/lib)
 
 # set up libInit_shared only on Windows; see also stdlib.make.in
 if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
-  set(INIT_SHARED_LINKER_FLAGS "-Wl,--whole-archive -lInit ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a -Wl,--no-whole-archive -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libInit_shared.dll.a")
+  set(INIT_SHARED_LINKER_FLAGS "-Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/temp/libInit.a.export ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a -Wl,--no-whole-archive -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libInit_shared.dll.a")
 endif()
 
 if(${CMAKE_SYSTEM_NAME} MATCHES "Darwin")
   set(LEANSHARED_LINKER_FLAGS "-Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libInit.a -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libLean.a -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libleancpp.a ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a ${LEANSHARED_LINKER_FLAGS}")
 elseif(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
-  set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive -lLean -lleancpp -Wl,--no-whole-archive -lInit_shared -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
+  set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive ${CMAKE_BINARY_DIR}/lib/temp/libLean.a.export -lleancpp -Wl,--no-whole-archive -lInit_shared -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
 else()
   set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive -lInit -lLean -lleancpp -Wl,--no-whole-archive ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a ${LEANSHARED_LINKER_FLAGS}")
 endif()

src/Init/ByCases.lean

@@ -37,15 +37,6 @@ theorem apply_ite (f : α → β) (P : Prop) [Decidable P] (x y : α) :
     f (ite P x y) = ite P (f x) (f y) :=
   apply_dite f P (fun _ => x) (fun _ => y)
 
-/-- Negation of the condition `P : Prop` in a `dite` is the same as swapping the branches. -/
-@[simp] theorem dite_not (P : Prop) {_ : Decidable P}  (x : ¬P → α) (y : ¬¬P → α) :
-    dite (¬P) x y = dite P (fun h => y (not_not_intro h)) x := by
-  by_cases h : P <;> simp [h]
-
-/-- Negation of the condition `P : Prop` in a `ite` is the same as swapping the branches. -/
-@[simp] theorem ite_not (P : Prop) {_ : Decidable P} (x y : α) : ite (¬P) x y = ite P y x :=
-  dite_not P (fun _ => x) (fun _ => y)
-
 @[simp] theorem dite_eq_left_iff {P : Prop} [Decidable P] {B : ¬ P → α} :
     dite P (fun _ => a) B = a ↔ ∀ h, B h = a := by
   by_cases P <;> simp [*, forall_prop_of_true, forall_prop_of_false]

src/Init/Classical.lean

@@ -125,16 +125,15 @@ theorem byContradiction {p : Prop} (h : ¬p → False) : p :=
 /-- The Double Negation Theorem: `¬¬P` is equivalent to `P`.
 The left-to-right direction, double negation elimination (DNE),
 is classically true but not constructively. -/
-@[scoped simp] theorem not_not : ¬¬a ↔ a := Decidable.not_not
+@[simp] theorem not_not : ¬¬a ↔ a := Decidable.not_not
 
-@[simp] theorem not_forall {p : α → Prop} : (¬∀ x, p x) ↔ ∃ x, ¬p x := Decidable.not_forall
+@[simp low] theorem not_forall {p : α → Prop} : (¬∀ x, p x) ↔ ∃ x, ¬p x := Decidable.not_forall
 
 theorem not_forall_not {p : α → Prop} : (¬∀ x, ¬p x) ↔ ∃ x, p x := Decidable.not_forall_not
 theorem not_exists_not {p : α → Prop} : (¬∃ x, ¬p x) ↔ ∀ x, p x := Decidable.not_exists_not
 
 theorem forall_or_exists_not (P : α → Prop) : (∀ a, P a) ∨ ∃ a, ¬ P a := by
   rw [← not_forall]; exact em _
-
 theorem exists_or_forall_not (P : α → Prop) : (∃ a, P a) ∨ ∀ a, ¬ P a := by
   rw [← not_exists]; exact em _
 
@@ -147,8 +146,22 @@ theorem not_and_iff_or_not_not : ¬(a ∧ b) ↔ ¬a ∨ ¬b := Decidable.not_an
 
 theorem not_iff : ¬(a ↔ b) ↔ (¬a ↔ b) := Decidable.not_iff
 
+@[simp] theorem imp_iff_left_iff  : (b ↔ a → b) ↔ a ∨ b := Decidable.imp_iff_left_iff
+@[simp] theorem imp_iff_right_iff : (a → b ↔ b) ↔ a ∨ b := Decidable.imp_iff_right_iff
+
+@[simp] theorem and_or_imp : a ∧ b ∨ (a → c) ↔ a → b ∨ c := Decidable.and_or_imp
+
+@[simp] theorem not_imp : ¬(a → b) ↔ a ∧ ¬b := Decidable.not_imp_iff_and_not
+
+@[simp] theorem imp_and_neg_imp_iff (p q : Prop) : (p → q) ∧ (¬p → q) ↔ q :=
+  Iff.intro (fun (a : _ ∧ _) => (Classical.em p).rec a.left a.right)
+            (fun a => And.intro (fun _ => a) (fun _ => a))
+
 end Classical
 
+/- Export for Mathlib compat. -/
+export Classical (imp_iff_right_iff imp_and_neg_imp_iff and_or_imp not_imp)
+
 /-- Extract an element from a existential statement, using `Classical.choose`. -/
 -- This enables projection notation.
 @[reducible] noncomputable def Exists.choose {p : α → Prop} (P : ∃ a, p a) : α := Classical.choose P

src/Init/Control/ExceptCps.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
-import Init.Control.Lawful
+import Init.Control.Lawful.Basic
 
 /-!
 The Exception monad transformer using CPS style.

src/Init/Control/Lawful.lean

@@ -4,373 +4,5 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Ullrich, Leonardo de Moura, Mario Carneiro
 -/
 prelude
-import Init.SimpLemmas
-import Init.Control.Except
-import Init.Control.StateRef
-
-open Function
-
-@[simp] theorem monadLift_self [Monad m] (x : m α) : monadLift x = x :=
-  rfl
-
-class LawfulFunctor (f : Type u → Type v) [Functor f] : Prop where
-  map_const          : (Functor.mapConst : α → f β → f α) = Functor.map ∘ const β
-  id_map   (x : f α) : id <$> x = x
-  comp_map (g : α → β) (h : β → γ) (x : f α) : (h ∘ g) <$> x = h <$> g <$> x
-
-export LawfulFunctor (map_const id_map comp_map)
-
-attribute [simp] id_map
-
-@[simp] theorem id_map' [Functor m] [LawfulFunctor m] (x : m α) : (fun a => a) <$> x = x :=
-  id_map x
-
-class LawfulApplicative (f : Type u → Type v) [Applicative f] extends LawfulFunctor f : Prop where
-  seqLeft_eq  (x : f α) (y : f β)     : x <* y = const β <$> x <*> y
-  seqRight_eq (x : f α) (y : f β)     : x *> y = const α id <$> x <*> y
-  pure_seq    (g : α → β) (x : f α)   : pure g <*> x = g <$> x
-  map_pure    (g : α → β) (x : α)     : g <$> (pure x : f α) = pure (g x)
-  seq_pure    {α β : Type u} (g : f (α → β)) (x : α) : g <*> pure x = (fun h => h x) <$> g
-  seq_assoc   {α β γ : Type u} (x : f α) (g : f (α → β)) (h : f (β → γ)) : h <*> (g <*> x) = ((@comp α β γ) <$> h) <*> g <*> x
-  comp_map g h x := (by
-    repeat rw [← pure_seq]
-    simp [seq_assoc, map_pure, seq_pure])
-
-export LawfulApplicative (seqLeft_eq seqRight_eq pure_seq map_pure seq_pure seq_assoc)
-
-attribute [simp] map_pure seq_pure
-
-@[simp] theorem pure_id_seq [Applicative f] [LawfulApplicative f] (x : f α) : pure id <*> x = x := by
-  simp [pure_seq]
-
-class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m : Prop where
-  bind_pure_comp (f : α → β) (x : m α) : x >>= (fun a => pure (f a)) = f <$> x
-  bind_map       {α β : Type u} (f : m (α → β)) (x : m α) : f >>= (. <$> x) = f <*> x
-  pure_bind      (x : α) (f : α → m β) : pure x >>= f = f x
-  bind_assoc     (x : m α) (f : α → m β) (g : β → m γ) : x >>= f >>= g = x >>= fun x => f x >>= g
-  map_pure g x    := (by rw [← bind_pure_comp, pure_bind])
-  seq_pure g x    := (by rw [← bind_map]; simp [map_pure, bind_pure_comp])
-  seq_assoc x g h := (by simp [← bind_pure_comp, ← bind_map, bind_assoc, pure_bind])
-
-export LawfulMonad (bind_pure_comp bind_map pure_bind bind_assoc)
-attribute [simp] pure_bind bind_assoc
-
-@[simp] theorem bind_pure [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x := by
-  show x >>= (fun a => pure (id a)) = x
-  rw [bind_pure_comp, id_map]
-
-theorem map_eq_pure_bind [Monad m] [LawfulMonad m] (f : α → β) (x : m α) : f <$> x = x >>= fun a => pure (f a) := by
-  rw [← bind_pure_comp]
-
-theorem seq_eq_bind_map {α β : Type u} [Monad m] [LawfulMonad m] (f : m (α → β)) (x : m α) : f <*> x = f >>= (. <$> x) := by
-  rw [← bind_map]
-
-theorem bind_congr [Bind m] {x : m α} {f g : α → m β} (h : ∀ a, f a = g a) : x >>= f = x >>= g := by
-  simp [funext h]
-
-@[simp] theorem bind_pure_unit [Monad m] [LawfulMonad m] {x : m PUnit} : (x >>= fun _ => pure ⟨⟩) = x := by
-  rw [bind_pure]
-
-theorem map_congr [Functor m] {x : m α} {f g : α → β} (h : ∀ a, f a = g a) : (f <$> x : m β) = g <$> x := by
-  simp [funext h]
-
-theorem seq_eq_bind {α β : Type u} [Monad m] [LawfulMonad m] (mf : m (α → β)) (x : m α) : mf <*> x = mf >>= fun f => f <$> x := by
-  rw [bind_map]
-
-theorem seqRight_eq_bind [Monad m] [LawfulMonad m] (x : m α) (y : m β) : x *> y = x >>= fun _ => y := by
-  rw [seqRight_eq]
-  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 [map_eq_pure_bind, seq_eq_bind_map]
-
-/--
-An alternative constructor for `LawfulMonad` which has more
-defaultable fields in the common case.
--/
-theorem LawfulMonad.mk' (m : Type u → Type v) [Monad m]
-    (id_map : ∀ {α} (x : m α), id <$> x = x)
-    (pure_bind : ∀ {α β} (x : α) (f : α → m β), pure x >>= f = f x)
-    (bind_assoc : ∀ {α β γ} (x : m α) (f : α → m β) (g : β → m γ),
-      x >>= f >>= g = x >>= fun x => f x >>= g)
-    (map_const : ∀ {α β} (x : α) (y : m β),
-      Functor.mapConst x y = Function.const β x <$> y := by intros; rfl)
-    (seqLeft_eq : ∀ {α β} (x : m α) (y : m β),
-      x <* y = (x >>= fun a => y >>= fun _ => pure a) := by intros; rfl)
-    (seqRight_eq : ∀ {α β} (x : m α) (y : m β), x *> y = (x >>= fun _ => y) := by intros; rfl)
-    (bind_pure_comp : ∀ {α β} (f : α → β) (x : m α),
-      x >>= (fun y => pure (f y)) = f <$> x := by intros; rfl)
-    (bind_map : ∀ {α β} (f : m (α → β)) (x : m α), f >>= (. <$> x) = f <*> x := by intros; rfl)
-    : LawfulMonad m :=
-  have map_pure {α β} (g : α → β) (x : α) : g <$> (pure x : m α) = pure (g x) := by
-    rw [← bind_pure_comp]; simp [pure_bind]
-  { id_map, bind_pure_comp, bind_map, pure_bind, bind_assoc, map_pure,
-    comp_map := by simp [← bind_pure_comp, bind_assoc, pure_bind]
-    pure_seq := by intros; rw [← bind_map]; simp [pure_bind]
-    seq_pure := by intros; rw [← bind_map]; simp [map_pure, bind_pure_comp]
-    seq_assoc := by simp [← bind_pure_comp, ← bind_map, bind_assoc, pure_bind]
-    map_const := funext fun x => funext (map_const x)
-    seqLeft_eq := by simp [seqLeft_eq, ← bind_map, ← bind_pure_comp, pure_bind, bind_assoc]
-    seqRight_eq := fun x y => by
-      rw [seqRight_eq, ← bind_map, ← bind_pure_comp, bind_assoc]; simp [pure_bind, id_map] }
-
-/-! # Id -/
-
-namespace Id
-
-@[simp] theorem map_eq (x : Id α) (f : α → β) : f <$> x = f x := rfl
-@[simp] theorem bind_eq (x : Id α) (f : α → id β) : x >>= f = f x := rfl
-@[simp] theorem pure_eq (a : α) : (pure a : Id α) = a := rfl
-
-instance : LawfulMonad Id := by
-  refine' { .. } <;> intros <;> rfl
-
-end Id
-
-/-! # ExceptT -/
-
-namespace ExceptT
-
-theorem ext [Monad m] {x y : ExceptT ε m α} (h : x.run = y.run) : x = y := by
-  simp [run] at h
-  assumption
-
-@[simp] theorem run_pure [Monad m] (x : α) : run (pure x : ExceptT ε m α) = pure (Except.ok x) := rfl
-
-@[simp] theorem run_lift  [Monad.{u, v} m] (x : m α) : run (ExceptT.lift x : ExceptT ε m α) = (Except.ok <$> x : m (Except ε α)) := rfl
-
-@[simp] theorem run_throw [Monad m] : run (throw e : ExceptT ε m β) = pure (Except.error e) := rfl
-
-@[simp] theorem run_bind_lift [Monad m] [LawfulMonad m] (x : m α) (f : α → ExceptT ε m β) : run (ExceptT.lift x >>= f : ExceptT ε m β) = x >>= fun a => run (f a) := by
-  simp[ExceptT.run, ExceptT.lift, bind, ExceptT.bind, ExceptT.mk, ExceptT.bindCont, map_eq_pure_bind]
-
-@[simp] theorem bind_throw [Monad m] [LawfulMonad m] (f : α → ExceptT ε m β) : (throw e >>= f) = throw e := by
-  simp [throw, throwThe, MonadExceptOf.throw, bind, ExceptT.bind, ExceptT.bindCont, ExceptT.mk]
-
-theorem run_bind [Monad m] (x : ExceptT ε m α)
-        : run (x >>= f : ExceptT ε m β)
-          =
-          run x >>= fun
-                     | Except.ok x => run (f x)
-                     | Except.error e => pure (Except.error e) :=
-  rfl
-
-@[simp] theorem lift_pure [Monad m] [LawfulMonad m] (a : α) : ExceptT.lift (pure a) = (pure a : ExceptT ε m α) := by
-  simp [ExceptT.lift, pure, ExceptT.pure]
-
-@[simp] theorem run_map [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α)
-    : (f <$> x).run = Except.map f <$> x.run := by
-  simp [Functor.map, ExceptT.map, map_eq_pure_bind]
-  apply bind_congr
-  intro a; cases a <;> simp [Except.map]
-
-protected theorem seq_eq {α β ε : Type u} [Monad m] (mf : ExceptT ε m (α → β)) (x : ExceptT ε m α) : mf <*> x = mf >>= fun f => f <$> x :=
-  rfl
-
-protected theorem bind_pure_comp [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α) : x >>= pure ∘ f = f <$> x := by
-  intros; rfl
-
-protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x <* y = const β <$> x <*> y := by
-  show (x >>= fun a => y >>= fun _ => pure a) = (const (α := α) β <$> x) >>= fun f => f <$> y
-  rw [← ExceptT.bind_pure_comp]
-  apply ext
-  simp [run_bind]
-  apply bind_congr
-  intro
-  | Except.error _ => simp
-  | Except.ok _ =>
-    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 <$> x <*> y := by
-  show (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y
-  rw [← ExceptT.bind_pure_comp]
-  apply ext
-  simp [run_bind]
-  apply bind_congr
-  intro a; cases a <;> simp
-
-instance [Monad m] [LawfulMonad m] : LawfulMonad (ExceptT ε m) where
-  id_map         := by intros; apply ext; simp
-  map_const      := by intros; rfl
-  seqLeft_eq     := ExceptT.seqLeft_eq
-  seqRight_eq    := ExceptT.seqRight_eq
-  pure_seq       := by intros; apply ext; simp [ExceptT.seq_eq, run_bind]
-  bind_pure_comp := ExceptT.bind_pure_comp
-  bind_map       := by intros; rfl
-  pure_bind      := by intros; apply ext; simp [run_bind]
-  bind_assoc     := by intros; apply ext; simp [run_bind]; apply bind_congr; intro a; cases a <;> simp
-
-end ExceptT
-
-/-! # Except -/
-
-instance : LawfulMonad (Except ε) := LawfulMonad.mk'
-  (id_map := fun x => by cases x <;> rfl)
-  (pure_bind := fun a f => rfl)
-  (bind_assoc := fun a f g => by cases a <;> rfl)
-
-instance : LawfulApplicative (Except ε) := inferInstance
-instance : LawfulFunctor (Except ε) := inferInstance
-
-/-! # ReaderT -/
-
-namespace ReaderT
-
-theorem ext {x y : ReaderT ρ m α} (h : ∀ ctx, x.run ctx = y.run ctx) : x = y := by
-  simp [run] at h
-  exact funext h
-
-@[simp] theorem run_pure [Monad m] (a : α) (ctx : ρ) : (pure a : ReaderT ρ m α).run ctx = pure a := rfl
-
-@[simp] theorem run_bind [Monad m] (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) (ctx : ρ)
-    : (x >>= f).run ctx = x.run ctx >>= λ a => (f a).run ctx := rfl
-
-@[simp] theorem run_mapConst [Monad m] (a : α) (x : ReaderT ρ m β) (ctx : ρ)
-    : (Functor.mapConst a x).run ctx = Functor.mapConst a (x.run ctx) := rfl
-
-@[simp] theorem run_map [Monad m] (f : α → β) (x : ReaderT ρ m α) (ctx : ρ)
-    : (f <$> x).run ctx = f <$> x.run ctx := rfl
-
-@[simp] theorem run_monadLift [MonadLiftT n m] (x : n α) (ctx : ρ)
-    : (monadLift x : ReaderT ρ m α).run ctx = (monadLift x : m α) := rfl
-
-@[simp] theorem run_monadMap [MonadFunctor n m] (f : {β : Type u} → n β → n β) (x : ReaderT ρ m α) (ctx : ρ)
-    : (monadMap @f x : ReaderT ρ m α).run ctx = monadMap @f (x.run ctx) := rfl
-
-@[simp] theorem run_read [Monad m] (ctx : ρ) : (ReaderT.read : ReaderT ρ m ρ).run ctx = pure ctx := rfl
-
-@[simp] theorem run_seq {α β : Type u} [Monad m] (f : ReaderT ρ m (α → β)) (x : ReaderT ρ m α) (ctx : ρ)
-    : (f <*> x).run ctx = (f.run ctx <*> x.run ctx) := rfl
-
-@[simp] theorem run_seqRight [Monad m] (x : ReaderT ρ m α) (y : ReaderT ρ m β) (ctx : ρ)
-    : (x *> y).run ctx = (x.run ctx *> y.run ctx) := rfl
-
-@[simp] theorem run_seqLeft [Monad m] (x : ReaderT ρ m α) (y : ReaderT ρ m β) (ctx : ρ)
-    : (x <* y).run ctx = (x.run ctx <* y.run ctx) := rfl
-
-instance [Monad m] [LawfulFunctor m] : LawfulFunctor (ReaderT ρ m) where
-  id_map    := by intros; apply ext; simp
-  map_const := by intros; funext a b; apply ext; intros; simp [map_const]
-  comp_map  := by intros; apply ext; intros; simp [comp_map]
-
-instance [Monad m] [LawfulApplicative m] : LawfulApplicative (ReaderT ρ m) where
-  seqLeft_eq  := by intros; apply ext; intros; simp [seqLeft_eq]
-  seqRight_eq := by intros; apply ext; intros; simp [seqRight_eq]
-  pure_seq    := by intros; apply ext; intros; simp [pure_seq]
-  map_pure    := by intros; apply ext; intros; simp [map_pure]
-  seq_pure    := by intros; apply ext; intros; simp [seq_pure]
-  seq_assoc   := by intros; apply ext; intros; simp [seq_assoc]
-
-instance [Monad m] [LawfulMonad m] : LawfulMonad (ReaderT ρ m) where
-  bind_pure_comp := by intros; apply ext; intros; simp [LawfulMonad.bind_pure_comp]
-  bind_map       := by intros; apply ext; intros; simp [bind_map]
-  pure_bind      := by intros; apply ext; intros; simp
-  bind_assoc     := by intros; apply ext; intros; simp
-
-end ReaderT
-
-/-! # StateRefT -/
-
-instance [Monad m] [LawfulMonad m] : LawfulMonad (StateRefT' ω σ m) :=
-  inferInstanceAs (LawfulMonad (ReaderT (ST.Ref ω σ) m))
-
-/-! # StateT -/
-
-namespace StateT
-
-theorem ext {x y : StateT σ m α} (h : ∀ s, x.run s = y.run s) : x = y :=
-  funext h
-
-@[simp] theorem run'_eq [Monad m] (x : StateT σ m α) (s : σ) : run' x s = (·.1) <$> run x s :=
-  rfl
-
-@[simp] theorem run_pure [Monad m] (a : α) (s : σ) : (pure a : StateT σ m α).run s = pure (a, s) := rfl
-
-@[simp] theorem run_bind [Monad m] (x : StateT σ m α) (f : α → StateT σ m β) (s : σ)
-    : (x >>= f).run s = x.run s >>= λ p => (f p.1).run p.2 := by
-  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, map_eq_pure_bind]
-
-@[simp] theorem run_get [Monad m] (s : σ)    : (get : StateT σ m σ).run s = pure (s, s) := rfl
-
-@[simp] theorem run_set [Monad m] (s s' : σ) : (set s' : StateT σ m PUnit).run s = pure (⟨⟩, s') := rfl
-
-@[simp] theorem run_modify [Monad m] (f : σ → σ) (s : σ) : (modify f : StateT σ m PUnit).run s = pure (⟨⟩, f s) := rfl
-
-@[simp] theorem run_modifyGet [Monad m] (f : σ → α × σ) (s : σ) : (modifyGet f : StateT σ m α).run s = pure ((f s).1, (f s).2) := by
-  simp [modifyGet, MonadStateOf.modifyGet, StateT.modifyGet, run]
-
-@[simp] theorem run_lift {α σ : Type u} [Monad m] (x : m α) (s : σ) : (StateT.lift x : StateT σ m α).run s = x >>= fun a => pure (a, s) := rfl
-
-@[simp] theorem run_bind_lift {α σ : Type u} [Monad m] [LawfulMonad m] (x : m α) (f : α → StateT σ m β) (s : σ) : (StateT.lift x >>= f).run s = x >>= fun a => (f a).run s := by
-  simp [StateT.lift, StateT.run, bind, StateT.bind]
-
-@[simp] theorem run_monadLift {α σ : Type u} [Monad m] [MonadLiftT n m] (x : n α) (s : σ) : (monadLift x : StateT σ m α).run s = (monadLift x : m α) >>= fun a => pure (a, s) := rfl
-
-@[simp] theorem run_monadMap [Monad m] [MonadFunctor n m] (f : {β : Type u} → n β → n β) (x : StateT σ m α) (s : σ)
-    : (monadMap @f x : StateT σ m α).run s = monadMap @f (x.run s) := rfl
-
-@[simp] theorem run_seq {α β σ : Type u} [Monad m] [LawfulMonad m] (f : StateT σ m (α → β)) (x : StateT σ m α) (s : σ) : (f <*> x).run s = (f.run s >>= fun fs => (fun (p : α × σ) => (fs.1 p.1, p.2)) <$> x.run fs.2) := by
-  show (f >>= fun g => g <$> x).run s = _
-  simp
-
-@[simp] theorem run_seqRight [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x *> y).run s = (x.run s >>= fun p => y.run p.2) := by
-  show (x >>= fun _ => y).run s = _
-  simp
-
-@[simp] theorem run_seqLeft {α β σ : Type u} [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x <* y).run s = (x.run s >>= fun p => y.run p.2 >>= fun p' => pure (p.1, p'.2)) := by
-  show (x >>= fun a => y >>= fun _ => pure a).run s = _
-  simp
-
-theorem seqRight_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x *> y = const α id <$> x <*> y := by
-  apply ext; intro s
-  simp [map_eq_pure_bind, const]
-  apply bind_congr; intro p; cases p
-  simp [Prod.eta]
-
-theorem seqLeft_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x <* y = const β <$> x <*> y := by
-  apply ext; intro s
-  simp [map_eq_pure_bind]
-
-instance [Monad m] [LawfulMonad m] : LawfulMonad (StateT σ m) where
-  id_map         := by intros; apply ext; intros; simp[Prod.eta]
-  map_const      := by intros; rfl
-  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; 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
-
-end StateT
-
-/-! # EStateM -/
-
-instance : LawfulMonad (EStateM ε σ) := .mk'
-  (id_map := fun x => funext <| fun s => by
-    dsimp only [EStateM.instMonadEStateM, EStateM.map]
-    match x s with
-    | .ok _ _ => rfl
-    | .error _ _ => rfl)
-  (pure_bind := fun _ _ => rfl)
-  (bind_assoc := fun x _ _ => funext <| fun s => by
-    dsimp only [EStateM.instMonadEStateM, EStateM.bind]
-    match x s with
-    | .ok _ _ => rfl
-    | .error _ _ => rfl)
-  (map_const := fun _ _ => rfl)
-
-/-! # Option -/
-
-instance : LawfulMonad Option := LawfulMonad.mk'
-  (id_map := 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
+import Init.Control.Lawful.Basic
+import Init.Control.Lawful.Instances

src/Init/Control/Lawful/Basic.lean

@@ -0,0 +1,138 @@
+/-
+Copyright (c) 2021 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sebastian Ullrich, Leonardo de Moura, Mario Carneiro
+-/
+prelude
+import Init.SimpLemmas
+import Init.Meta
+
+open Function
+
+@[simp] theorem monadLift_self [Monad m] (x : m α) : monadLift x = x :=
+  rfl
+
+class LawfulFunctor (f : Type u → Type v) [Functor f] : Prop where
+  map_const          : (Functor.mapConst : α → f β → f α) = Functor.map ∘ const β
+  id_map   (x : f α) : id <$> x = x
+  comp_map (g : α → β) (h : β → γ) (x : f α) : (h ∘ g) <$> x = h <$> g <$> x
+
+export LawfulFunctor (map_const id_map comp_map)
+
+attribute [simp] id_map
+
+@[simp] theorem id_map' [Functor m] [LawfulFunctor m] (x : m α) : (fun a => a) <$> x = x :=
+  id_map x
+
+class LawfulApplicative (f : Type u → Type v) [Applicative f] extends LawfulFunctor f : Prop where
+  seqLeft_eq  (x : f α) (y : f β)     : x <* y = const β <$> x <*> y
+  seqRight_eq (x : f α) (y : f β)     : x *> y = const α id <$> x <*> y
+  pure_seq    (g : α → β) (x : f α)   : pure g <*> x = g <$> x
+  map_pure    (g : α → β) (x : α)     : g <$> (pure x : f α) = pure (g x)
+  seq_pure    {α β : Type u} (g : f (α → β)) (x : α) : g <*> pure x = (fun h => h x) <$> g
+  seq_assoc   {α β γ : Type u} (x : f α) (g : f (α → β)) (h : f (β → γ)) : h <*> (g <*> x) = ((@comp α β γ) <$> h) <*> g <*> x
+  comp_map g h x := (by
+    repeat rw [← pure_seq]
+    simp [seq_assoc, map_pure, seq_pure])
+
+export LawfulApplicative (seqLeft_eq seqRight_eq pure_seq map_pure seq_pure seq_assoc)
+
+attribute [simp] map_pure seq_pure
+
+@[simp] theorem pure_id_seq [Applicative f] [LawfulApplicative f] (x : f α) : pure id <*> x = x := by
+  simp [pure_seq]
+
+class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m : Prop where
+  bind_pure_comp (f : α → β) (x : m α) : x >>= (fun a => pure (f a)) = f <$> x
+  bind_map       {α β : Type u} (f : m (α → β)) (x : m α) : f >>= (. <$> x) = f <*> x
+  pure_bind      (x : α) (f : α → m β) : pure x >>= f = f x
+  bind_assoc     (x : m α) (f : α → m β) (g : β → m γ) : x >>= f >>= g = x >>= fun x => f x >>= g
+  map_pure g x    := (by rw [← bind_pure_comp, pure_bind])
+  seq_pure g x    := (by rw [← bind_map]; simp [map_pure, bind_pure_comp])
+  seq_assoc x g h := (by simp [← bind_pure_comp, ← bind_map, bind_assoc, pure_bind])
+
+export LawfulMonad (bind_pure_comp bind_map pure_bind bind_assoc)
+attribute [simp] pure_bind bind_assoc
+
+@[simp] theorem bind_pure [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x := by
+  show x >>= (fun a => pure (id a)) = x
+  rw [bind_pure_comp, id_map]
+
+theorem map_eq_pure_bind [Monad m] [LawfulMonad m] (f : α → β) (x : m α) : f <$> x = x >>= fun a => pure (f a) := by
+  rw [← bind_pure_comp]
+
+theorem seq_eq_bind_map {α β : Type u} [Monad m] [LawfulMonad m] (f : m (α → β)) (x : m α) : f <*> x = f >>= (. <$> x) := by
+  rw [← bind_map]
+
+theorem bind_congr [Bind m] {x : m α} {f g : α → m β} (h : ∀ a, f a = g a) : x >>= f = x >>= g := by
+  simp [funext h]
+
+@[simp] theorem bind_pure_unit [Monad m] [LawfulMonad m] {x : m PUnit} : (x >>= fun _ => pure ⟨⟩) = x := by
+  rw [bind_pure]
+
+theorem map_congr [Functor m] {x : m α} {f g : α → β} (h : ∀ a, f a = g a) : (f <$> x : m β) = g <$> x := by
+  simp [funext h]
+
+theorem seq_eq_bind {α β : Type u} [Monad m] [LawfulMonad m] (mf : m (α → β)) (x : m α) : mf <*> x = mf >>= fun f => f <$> x := by
+  rw [bind_map]
+
+theorem seqRight_eq_bind [Monad m] [LawfulMonad m] (x : m α) (y : m β) : x *> y = x >>= fun _ => y := by
+  rw [seqRight_eq]
+  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 [map_eq_pure_bind, seq_eq_bind_map]
+
+/--
+An alternative constructor for `LawfulMonad` which has more
+defaultable fields in the common case.
+-/
+theorem LawfulMonad.mk' (m : Type u → Type v) [Monad m]
+    (id_map : ∀ {α} (x : m α), id <$> x = x)
+    (pure_bind : ∀ {α β} (x : α) (f : α → m β), pure x >>= f = f x)
+    (bind_assoc : ∀ {α β γ} (x : m α) (f : α → m β) (g : β → m γ),
+      x >>= f >>= g = x >>= fun x => f x >>= g)
+    (map_const : ∀ {α β} (x : α) (y : m β),
+      Functor.mapConst x y = Function.const β x <$> y := by intros; rfl)
+    (seqLeft_eq : ∀ {α β} (x : m α) (y : m β),
+      x <* y = (x >>= fun a => y >>= fun _ => pure a) := by intros; rfl)
+    (seqRight_eq : ∀ {α β} (x : m α) (y : m β), x *> y = (x >>= fun _ => y) := by intros; rfl)
+    (bind_pure_comp : ∀ {α β} (f : α → β) (x : m α),
+      x >>= (fun y => pure (f y)) = f <$> x := by intros; rfl)
+    (bind_map : ∀ {α β} (f : m (α → β)) (x : m α), f >>= (. <$> x) = f <*> x := by intros; rfl)
+    : LawfulMonad m :=
+  have map_pure {α β} (g : α → β) (x : α) : g <$> (pure x : m α) = pure (g x) := by
+    rw [← bind_pure_comp]; simp [pure_bind]
+  { id_map, bind_pure_comp, bind_map, pure_bind, bind_assoc, map_pure,
+    comp_map := by simp [← bind_pure_comp, bind_assoc, pure_bind]
+    pure_seq := by intros; rw [← bind_map]; simp [pure_bind]
+    seq_pure := by intros; rw [← bind_map]; simp [map_pure, bind_pure_comp]
+    seq_assoc := by simp [← bind_pure_comp, ← bind_map, bind_assoc, pure_bind]
+    map_const := funext fun x => funext (map_const x)
+    seqLeft_eq := by simp [seqLeft_eq, ← bind_map, ← bind_pure_comp, pure_bind, bind_assoc]
+    seqRight_eq := fun x y => by
+      rw [seqRight_eq, ← bind_map, ← bind_pure_comp, bind_assoc]; simp [pure_bind, id_map] }
+
+/-! # Id -/
+
+namespace Id
+
+@[simp] theorem map_eq (x : Id α) (f : α → β) : f <$> x = f x := rfl
+@[simp] theorem bind_eq (x : Id α) (f : α → id β) : x >>= f = f x := rfl
+@[simp] theorem pure_eq (a : α) : (pure a : Id α) = a := rfl
+
+instance : LawfulMonad Id := by
+  refine' { .. } <;> intros <;> rfl
+
+end Id
+
+/-! # Option -/
+
+instance : LawfulMonad Option := LawfulMonad.mk'
+  (id_map := 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

src/Init/Control/Lawful/Instances.lean

@@ -0,0 +1,248 @@
+/-
+Copyright (c) 2021 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sebastian Ullrich, Leonardo de Moura, Mario Carneiro
+-/
+prelude
+import Init.Control.Lawful.Basic
+import Init.Control.Except
+import Init.Control.StateRef
+
+open Function
+
+/-! # ExceptT -/
+
+namespace ExceptT
+
+theorem ext [Monad m] {x y : ExceptT ε m α} (h : x.run = y.run) : x = y := by
+  simp [run] at h
+  assumption
+
+@[simp] theorem run_pure [Monad m] (x : α) : run (pure x : ExceptT ε m α) = pure (Except.ok x) := rfl
+
+@[simp] theorem run_lift  [Monad.{u, v} m] (x : m α) : run (ExceptT.lift x : ExceptT ε m α) = (Except.ok <$> x : m (Except ε α)) := rfl
+
+@[simp] theorem run_throw [Monad m] : run (throw e : ExceptT ε m β) = pure (Except.error e) := rfl
+
+@[simp] theorem run_bind_lift [Monad m] [LawfulMonad m] (x : m α) (f : α → ExceptT ε m β) : run (ExceptT.lift x >>= f : ExceptT ε m β) = x >>= fun a => run (f a) := by
+  simp[ExceptT.run, ExceptT.lift, bind, ExceptT.bind, ExceptT.mk, ExceptT.bindCont, map_eq_pure_bind]
+
+@[simp] theorem bind_throw [Monad m] [LawfulMonad m] (f : α → ExceptT ε m β) : (throw e >>= f) = throw e := by
+  simp [throw, throwThe, MonadExceptOf.throw, bind, ExceptT.bind, ExceptT.bindCont, ExceptT.mk]
+
+theorem run_bind [Monad m] (x : ExceptT ε m α)
+        : run (x >>= f : ExceptT ε m β)
+          =
+          run x >>= fun
+                     | Except.ok x => run (f x)
+                     | Except.error e => pure (Except.error e) :=
+  rfl
+
+@[simp] theorem lift_pure [Monad m] [LawfulMonad m] (a : α) : ExceptT.lift (pure a) = (pure a : ExceptT ε m α) := by
+  simp [ExceptT.lift, pure, ExceptT.pure]
+
+@[simp] theorem run_map [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α)
+    : (f <$> x).run = Except.map f <$> x.run := by
+  simp [Functor.map, ExceptT.map, map_eq_pure_bind]
+  apply bind_congr
+  intro a; cases a <;> simp [Except.map]
+
+protected theorem seq_eq {α β ε : Type u} [Monad m] (mf : ExceptT ε m (α → β)) (x : ExceptT ε m α) : mf <*> x = mf >>= fun f => f <$> x :=
+  rfl
+
+protected theorem bind_pure_comp [Monad m] [LawfulMonad m] (f : α → β) (x : ExceptT ε m α) : x >>= pure ∘ f = f <$> x := by
+  intros; rfl
+
+protected theorem seqLeft_eq {α β ε : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] (x : ExceptT ε m α) (y : ExceptT ε m β) : x <* y = const β <$> x <*> y := by
+  show (x >>= fun a => y >>= fun _ => pure a) = (const (α := α) β <$> x) >>= fun f => f <$> y
+  rw [← ExceptT.bind_pure_comp]
+  apply ext
+  simp [run_bind]
+  apply bind_congr
+  intro
+  | Except.error _ => simp
+  | Except.ok _ =>
+    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 <$> x <*> y := by
+  show (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y
+  rw [← ExceptT.bind_pure_comp]
+  apply ext
+  simp [run_bind]
+  apply bind_congr
+  intro a; cases a <;> simp
+
+instance [Monad m] [LawfulMonad m] : LawfulMonad (ExceptT ε m) where
+  id_map         := by intros; apply ext; simp
+  map_const      := by intros; rfl
+  seqLeft_eq     := ExceptT.seqLeft_eq
+  seqRight_eq    := ExceptT.seqRight_eq
+  pure_seq       := by intros; apply ext; simp [ExceptT.seq_eq, run_bind]
+  bind_pure_comp := ExceptT.bind_pure_comp
+  bind_map       := by intros; rfl
+  pure_bind      := by intros; apply ext; simp [run_bind]
+  bind_assoc     := by intros; apply ext; simp [run_bind]; apply bind_congr; intro a; cases a <;> simp
+
+end ExceptT
+
+/-! # Except -/
+
+instance : LawfulMonad (Except ε) := LawfulMonad.mk'
+  (id_map := fun x => by cases x <;> rfl)
+  (pure_bind := fun a f => rfl)
+  (bind_assoc := fun a f g => by cases a <;> rfl)
+
+instance : LawfulApplicative (Except ε) := inferInstance
+instance : LawfulFunctor (Except ε) := inferInstance
+
+/-! # ReaderT -/
+
+namespace ReaderT
+
+theorem ext {x y : ReaderT ρ m α} (h : ∀ ctx, x.run ctx = y.run ctx) : x = y := by
+  simp [run] at h
+  exact funext h
+
+@[simp] theorem run_pure [Monad m] (a : α) (ctx : ρ) : (pure a : ReaderT ρ m α).run ctx = pure a := rfl
+
+@[simp] theorem run_bind [Monad m] (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) (ctx : ρ)
+    : (x >>= f).run ctx = x.run ctx >>= λ a => (f a).run ctx := rfl
+
+@[simp] theorem run_mapConst [Monad m] (a : α) (x : ReaderT ρ m β) (ctx : ρ)
+    : (Functor.mapConst a x).run ctx = Functor.mapConst a (x.run ctx) := rfl
+
+@[simp] theorem run_map [Monad m] (f : α → β) (x : ReaderT ρ m α) (ctx : ρ)
+    : (f <$> x).run ctx = f <$> x.run ctx := rfl
+
+@[simp] theorem run_monadLift [MonadLiftT n m] (x : n α) (ctx : ρ)
+    : (monadLift x : ReaderT ρ m α).run ctx = (monadLift x : m α) := rfl
+
+@[simp] theorem run_monadMap [MonadFunctor n m] (f : {β : Type u} → n β → n β) (x : ReaderT ρ m α) (ctx : ρ)
+    : (monadMap @f x : ReaderT ρ m α).run ctx = monadMap @f (x.run ctx) := rfl
+
+@[simp] theorem run_read [Monad m] (ctx : ρ) : (ReaderT.read : ReaderT ρ m ρ).run ctx = pure ctx := rfl
+
+@[simp] theorem run_seq {α β : Type u} [Monad m] (f : ReaderT ρ m (α → β)) (x : ReaderT ρ m α) (ctx : ρ)
+    : (f <*> x).run ctx = (f.run ctx <*> x.run ctx) := rfl
+
+@[simp] theorem run_seqRight [Monad m] (x : ReaderT ρ m α) (y : ReaderT ρ m β) (ctx : ρ)
+    : (x *> y).run ctx = (x.run ctx *> y.run ctx) := rfl
+
+@[simp] theorem run_seqLeft [Monad m] (x : ReaderT ρ m α) (y : ReaderT ρ m β) (ctx : ρ)
+    : (x <* y).run ctx = (x.run ctx <* y.run ctx) := rfl
+
+instance [Monad m] [LawfulFunctor m] : LawfulFunctor (ReaderT ρ m) where
+  id_map    := by intros; apply ext; simp
+  map_const := by intros; funext a b; apply ext; intros; simp [map_const]
+  comp_map  := by intros; apply ext; intros; simp [comp_map]
+
+instance [Monad m] [LawfulApplicative m] : LawfulApplicative (ReaderT ρ m) where
+  seqLeft_eq  := by intros; apply ext; intros; simp [seqLeft_eq]
+  seqRight_eq := by intros; apply ext; intros; simp [seqRight_eq]
+  pure_seq    := by intros; apply ext; intros; simp [pure_seq]
+  map_pure    := by intros; apply ext; intros; simp [map_pure]
+  seq_pure    := by intros; apply ext; intros; simp [seq_pure]
+  seq_assoc   := by intros; apply ext; intros; simp [seq_assoc]
+
+instance [Monad m] [LawfulMonad m] : LawfulMonad (ReaderT ρ m) where
+  bind_pure_comp := by intros; apply ext; intros; simp [LawfulMonad.bind_pure_comp]
+  bind_map       := by intros; apply ext; intros; simp [bind_map]
+  pure_bind      := by intros; apply ext; intros; simp
+  bind_assoc     := by intros; apply ext; intros; simp
+
+end ReaderT
+
+/-! # StateRefT -/
+
+instance [Monad m] [LawfulMonad m] : LawfulMonad (StateRefT' ω σ m) :=
+  inferInstanceAs (LawfulMonad (ReaderT (ST.Ref ω σ) m))
+
+/-! # StateT -/
+
+namespace StateT
+
+theorem ext {x y : StateT σ m α} (h : ∀ s, x.run s = y.run s) : x = y :=
+  funext h
+
+@[simp] theorem run'_eq [Monad m] (x : StateT σ m α) (s : σ) : run' x s = (·.1) <$> run x s :=
+  rfl
+
+@[simp] theorem run_pure [Monad m] (a : α) (s : σ) : (pure a : StateT σ m α).run s = pure (a, s) := rfl
+
+@[simp] theorem run_bind [Monad m] (x : StateT σ m α) (f : α → StateT σ m β) (s : σ)
+    : (x >>= f).run s = x.run s >>= λ p => (f p.1).run p.2 := by
+  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, map_eq_pure_bind]
+
+@[simp] theorem run_get [Monad m] (s : σ)    : (get : StateT σ m σ).run s = pure (s, s) := rfl
+
+@[simp] theorem run_set [Monad m] (s s' : σ) : (set s' : StateT σ m PUnit).run s = pure (⟨⟩, s') := rfl
+
+@[simp] theorem run_modify [Monad m] (f : σ → σ) (s : σ) : (modify f : StateT σ m PUnit).run s = pure (⟨⟩, f s) := rfl
+
+@[simp] theorem run_modifyGet [Monad m] (f : σ → α × σ) (s : σ) : (modifyGet f : StateT σ m α).run s = pure ((f s).1, (f s).2) := by
+  simp [modifyGet, MonadStateOf.modifyGet, StateT.modifyGet, run]
+
+@[simp] theorem run_lift {α σ : Type u} [Monad m] (x : m α) (s : σ) : (StateT.lift x : StateT σ m α).run s = x >>= fun a => pure (a, s) := rfl
+
+@[simp] theorem run_bind_lift {α σ : Type u} [Monad m] [LawfulMonad m] (x : m α) (f : α → StateT σ m β) (s : σ) : (StateT.lift x >>= f).run s = x >>= fun a => (f a).run s := by
+  simp [StateT.lift, StateT.run, bind, StateT.bind]
+
+@[simp] theorem run_monadLift {α σ : Type u} [Monad m] [MonadLiftT n m] (x : n α) (s : σ) : (monadLift x : StateT σ m α).run s = (monadLift x : m α) >>= fun a => pure (a, s) := rfl
+
+@[simp] theorem run_monadMap [Monad m] [MonadFunctor n m] (f : {β : Type u} → n β → n β) (x : StateT σ m α) (s : σ)
+    : (monadMap @f x : StateT σ m α).run s = monadMap @f (x.run s) := rfl
+
+@[simp] theorem run_seq {α β σ : Type u} [Monad m] [LawfulMonad m] (f : StateT σ m (α → β)) (x : StateT σ m α) (s : σ) : (f <*> x).run s = (f.run s >>= fun fs => (fun (p : α × σ) => (fs.1 p.1, p.2)) <$> x.run fs.2) := by
+  show (f >>= fun g => g <$> x).run s = _
+  simp
+
+@[simp] theorem run_seqRight [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x *> y).run s = (x.run s >>= fun p => y.run p.2) := by
+  show (x >>= fun _ => y).run s = _
+  simp
+
+@[simp] theorem run_seqLeft {α β σ : Type u} [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) (s : σ) : (x <* y).run s = (x.run s >>= fun p => y.run p.2 >>= fun p' => pure (p.1, p'.2)) := by
+  show (x >>= fun a => y >>= fun _ => pure a).run s = _
+  simp
+
+theorem seqRight_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x *> y = const α id <$> x <*> y := by
+  apply ext; intro s
+  simp [map_eq_pure_bind, const]
+  apply bind_congr; intro p; cases p
+  simp [Prod.eta]
+
+theorem seqLeft_eq [Monad m] [LawfulMonad m] (x : StateT σ m α) (y : StateT σ m β) : x <* y = const β <$> x <*> y := by
+  apply ext; intro s
+  simp [map_eq_pure_bind]
+
+instance [Monad m] [LawfulMonad m] : LawfulMonad (StateT σ m) where
+  id_map         := by intros; apply ext; intros; simp[Prod.eta]
+  map_const      := by intros; rfl
+  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; 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
+
+end StateT
+
+/-! # EStateM -/
+
+instance : LawfulMonad (EStateM ε σ) := .mk'
+  (id_map := fun x => funext <| fun s => by
+    dsimp only [EStateM.instMonadEStateM, EStateM.map]
+    match x s with
+    | .ok _ _ => rfl
+    | .error _ _ => rfl)
+  (pure_bind := fun _ _ => rfl)
+  (bind_assoc := fun x _ _ => funext <| fun s => by
+    dsimp only [EStateM.instMonadEStateM, EStateM.bind]
+    match x s with
+    | .ok _ _ => rfl
+    | .error _ _ => rfl)
+  (map_const := fun _ _ => rfl)

src/Init/Control/StateCps.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
-import Init.Control.Lawful
+import Init.Control.Lawful.Basic
 
 /-!
 The State monad transformer using CPS style.

src/Init/Conv.lean

@@ -156,7 +156,6 @@ match [a, b] with
 simplifies to `a`. -/
 syntax (name := simpMatch) "simp_match" : conv
 
-
 /-- Executes the given tactic block without converting `conv` goal into a regular goal. -/
 syntax (name := nestedTacticCore) "tactic'" " => " tacticSeq : conv
 

src/Init/Core.lean

@@ -19,7 +19,7 @@ which applies to all applications of the function).
 -/
 @[simp] def inline {α : Sort u} (a : α) : α := a
 
-theorem id.def {α : Sort u} (a : α) : id a = a := rfl
+theorem id_def {α : Sort u} (a : α) : id a = a := rfl
 
 /--
 `flip f a b` is `f b a`. It is useful for "point-free" programming,
@@ -677,7 +677,7 @@ You can prove theorems about the resulting element by induction on `h`, since
 theorem Eq.substr {α : Sort u} {p : α → Prop} {a b : α} (h₁ : b = a) (h₂ : p a) : p b :=
   h₁ ▸ h₂
 
-theorem cast_eq {α : Sort u} (h : α = α) (a : α) : cast h a = a :=
+@[simp] theorem cast_eq {α : Sort u} (h : α = α) (a : α) : cast h a = a :=
   rfl
 
 /--
@@ -737,13 +737,16 @@ theorem beq_false_of_ne [BEq α] [LawfulBEq α] {a b : α} (h : a ≠ b) : (a ==
 section
 variable {α β φ : Sort u} {a a' : α} {b b' : β} {c : φ}
 
-theorem HEq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : {β : Sort u2} → β → Sort u1} (m : motive a) {β : Sort u2} {b : β} (h : HEq a b) : motive b :=
+/-- Non-dependent recursor for `HEq` -/
+noncomputable def HEq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : {β : Sort u2} → β → Sort u1} (m : motive a) {β : Sort u2} {b : β} (h : HEq a b) : motive b :=
   h.rec m
 
-theorem HEq.ndrecOn.{u1, u2} {α : Sort u2} {a : α} {motive : {β : Sort u2} → β → Sort u1} {β : Sort u2} {b : β} (h : HEq a b) (m : motive a) : motive b :=
+/-- `HEq.ndrec` variant -/
+noncomputable def HEq.ndrecOn.{u1, u2} {α : Sort u2} {a : α} {motive : {β : Sort u2} → β → Sort u1} {β : Sort u2} {b : β} (h : HEq a b) (m : motive a) : motive b :=
   h.rec m
 
-theorem HEq.elim {α : Sort u} {a : α} {p : α → Sort v} {b : α} (h₁ : HEq a b) (h₂ : p a) : p b :=
+/-- `HEq.ndrec` variant -/
+noncomputable def HEq.elim {α : Sort u} {a : α} {p : α → Sort v} {b : α} (h₁ : HEq a b) (h₂ : p a) : p b :=
   eq_of_heq h₁ ▸ h₂
 
 theorem HEq.subst {p : (T : Sort u) → T → Prop} (h₁ : HEq a b) (h₂ : p α a) : p β b :=
@@ -1403,9 +1406,9 @@ theorem false_imp_iff (a : Prop) : (False → a) ↔ True := iff_true_intro Fals
 
 theorem true_imp_iff (α : Prop) : (True → α) ↔ α := imp_iff_right True.intro
 
-@[simp] theorem imp_self : (a → a) ↔ True := iff_true_intro id
+@[simp high] theorem imp_self : (a → a) ↔ True := iff_true_intro id
 
-theorem imp_false : (a → False) ↔ ¬a := Iff.rfl
+@[simp] theorem imp_false : (a → False) ↔ ¬a := Iff.rfl
 
 theorem imp.swap : (a → b → c) ↔ (b → a → c) := Iff.intro flip flip
 

src/Init/Data/AC.lean

@@ -106,7 +106,7 @@ def norm [info : ContextInformation α] (ctx : α) (e : Expr) : List Nat :=
   let xs := if info.isComm ctx then sort xs else xs
   if info.isIdem ctx then mergeIdem xs else xs
 
-theorem List.two_step_induction
+noncomputable def List.two_step_induction
   {motive : List Nat → Sort u}
   (l : List Nat)
   (empty : motive [])

src/Init/Data/Array/Basic.lean

@@ -727,33 +727,38 @@ def takeWhile (p : α → Bool) (as : Array α) : Array α :=
     termination_by as.size - i
   go 0 #[]
 
-def eraseIdxAux (i : Nat) (a : Array α) : Array α :=
-  if h : i < a.size then
-    let idx  : Fin a.size := ⟨i, h⟩;
-    let idx1 : Fin a.size := ⟨i - 1, by exact Nat.lt_of_le_of_lt (Nat.pred_le i) h⟩;
-    let a' := a.swap idx idx1
-    eraseIdxAux (i+1) a'
+/-- Remove the element at a given index from an array without bounds checks, using a `Fin` index.
+
+  This function takes worst case O(n) time because
+  it has to backshift all elements at positions greater than `i`.-/
+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
+    let i' : Fin a'.size := ⟨i.val + 1, by simp [a', h]⟩
+    have : a'.size - i' < a.size - i := by
+      simp [a', Nat.sub_succ_lt_self _ _ i.isLt]
+    a'.feraseIdx i'
   else
     a.pop
-termination_by a.size - i
+termination_by a.size - i.val
 
-def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
-  eraseIdxAux (i.val + 1) a
+derive_functional_induction feraseIdx
 
+theorem size_feraseIdx (a : Array α) (i : Fin a.size) : (a.feraseIdx i).size = a.size - 1 := by
+  induction a, i using feraseIdx.induct with
+  | @case1 a i h a' _ _ ih =>
+    unfold feraseIdx
+    simp [h, a', ih]
+  | case2 a i h =>
+    unfold feraseIdx
+    simp [h]
+
+/-- Remove the element at a given index from an array, or do nothing if the index is out of bounds.
+
+  This function takes worst case O(n) time because
+  it has to backshift all elements at positions greater than `i`.-/
 def eraseIdx (a : Array α) (i : Nat) : Array α :=
-  if i < a.size then eraseIdxAux (i+1) a else a
-
-def eraseIdxSzAux (a : Array α) (i : Nat) (r : Array α) (heq : r.size = a.size) : { r : Array α // r.size = a.size - 1 } :=
-  if h : i < r.size then
-    let idx  : Fin r.size := ⟨i, h⟩;
-    let idx1 : Fin r.size := ⟨i - 1, by exact Nat.lt_of_le_of_lt (Nat.pred_le i) h⟩;
-    eraseIdxSzAux a (i+1) (r.swap idx idx1) ((size_swap r idx idx1).trans heq)
-  else
-    ⟨r.pop, (size_pop r).trans (heq ▸ rfl)⟩
-termination_by r.size - i
-
-def eraseIdx' (a : Array α) (i : Fin a.size) : { r : Array α // r.size = a.size - 1 } :=
-  eraseIdxSzAux a (i.val + 1) a rfl
+  if h : i < a.size then a.feraseIdx ⟨i, h⟩ else a
 
 def erase [BEq α] (as : Array α) (a : α) : Array α :=
   match as.indexOf? a with
@@ -809,7 +814,7 @@ where
     rfl
 
   go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.data.drop i) = as.data := by
-    cases i <;> simp [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, go]
+    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

src/Init/Data/Array/Lemmas.lean

@@ -8,6 +8,7 @@ import Init.Data.Nat.MinMax
 import Init.Data.List.Lemmas
 import Init.Data.Fin.Basic
 import Init.Data.Array.Mem
+import Init.TacticsExtra
 
 /-!
 ## Bootstrapping theorems about arrays

src/Init/Data/Array/QSort.lean

@@ -10,7 +10,7 @@ namespace Array
 -- TODO: remove the [Inhabited α] parameters as soon as we have the tactic framework for automating proof generation and using Array.fget
 
 def qpartition (as : Array α) (lt : α → α → Bool) (lo hi : Nat) : Nat × Array α :=
-  if h : as.size = 0 then (0, as) else have : Inhabited α := ⟨as[0]'(by revert h; cases as.size <;> simp [Nat.zero_lt_succ])⟩ -- TODO: remove
+  if h : as.size = 0 then (0, as) else have : Inhabited α := ⟨as[0]'(by revert h; cases as.size <;> simp)⟩ -- TODO: remove
   let mid := (lo + hi) / 2
   let as  := if lt (as.get! mid) (as.get! lo) then as.swap! lo mid else as
   let as  := if lt (as.get! hi)  (as.get! lo) then as.swap! lo hi  else as

src/Init/Data/BitVec/Basic.lean

@@ -7,6 +7,7 @@ prelude
 import Init.Data.Fin.Basic
 import Init.Data.Nat.Bitwise.Lemmas
 import Init.Data.Nat.Power2
+import Init.Data.Int.Bitwise
 
 /-!
 We define bitvectors. We choose the `Fin` representation over others for its relative efficiency
@@ -72,6 +73,9 @@ protected def toNat (a : BitVec n) : Nat := a.toFin.val
 /-- Return the bound in terms of toNat. -/
 theorem isLt (x : BitVec w) : x.toNat < 2^w := x.toFin.isLt
 
+@[deprecated isLt]
+theorem toNat_lt (x : BitVec n) : x.toNat < 2^n := x.isLt
+
 /-- Theorem for normalizing the bit vector literal representation. -/
 -- TODO: This needs more usage data to assess which direction the simp should go.
 @[simp, bv_toNat] theorem ofNat_eq_ofNat : @OfNat.ofNat (BitVec n) i _ = .ofNat n i := rfl

src/Init/Data/BitVec/Bitblast.lean

@@ -5,6 +5,7 @@ Authors: Harun Khan, Abdalrhman M Mohamed, Joe Hendrix
 -/
 prelude
 import Init.Data.BitVec.Folds
+import Init.Data.Nat.Mod
 
 /-!
 # Bitblasting of bitvectors
@@ -70,24 +71,8 @@ private theorem testBit_limit {x i : Nat} (x_lt_succ : x < 2^(i+1)) :
              _ ≤ x := testBit_implies_ge jp
 
 private theorem mod_two_pow_succ (x i : Nat) :
-  x % 2^(i+1) = 2^i*(x.testBit i).toNat + x % (2 ^ i):= by
-  apply Nat.eq_of_testBit_eq
-  intro j
-  simp only [Nat.mul_add_lt_is_or, testBit_or, testBit_mod_two_pow, testBit_shiftLeft,
-    Nat.testBit_bool_to_nat, Nat.sub_eq_zero_iff_le, Nat.mod_lt, Nat.two_pow_pos,
-    testBit_mul_pow_two]
-  rcases Nat.lt_trichotomy i j with i_lt_j | i_eq_j | j_lt_i
-  · have i_le_j : i ≤ j := Nat.le_of_lt i_lt_j
-    have not_j_le_i : ¬(j ≤ i) := Nat.not_le_of_lt i_lt_j
-    have not_j_lt_i : ¬(j < i) := Nat.not_lt_of_le i_le_j
-    have not_j_lt_i_succ : ¬(j < i + 1) :=
-          Nat.not_le_of_lt (Nat.succ_lt_succ i_lt_j)
-    simp [i_le_j, not_j_le_i, not_j_lt_i, not_j_lt_i_succ]
-  · simp [i_eq_j]
-  · have j_le_i : j ≤ i := Nat.le_of_lt j_lt_i
-    have j_le_i_succ : j < i + 1 := Nat.succ_le_succ j_le_i
-    have not_j_ge_i : ¬(j ≥ i) := Nat.not_le_of_lt j_lt_i
-    simp [j_lt_i, j_le_i, not_j_ge_i, j_le_i_succ]
+    x % 2^(i+1) = 2^i*(x.testBit i).toNat + x % (2 ^ i):= by
+  rw [Nat.mod_pow_succ, Nat.add_comm, Nat.toNat_testBit]
 
 private theorem mod_two_pow_add_mod_two_pow_add_bool_lt_two_pow_succ
      (x y i : Nat) (c : Bool) : x % 2^i + (y % 2^i + c.toNat) < 2^(i+1) := by

src/Init/Data/BitVec/Lemmas.lean

@@ -29,14 +29,12 @@ theorem eq_of_toNat_eq {n} : ∀ {i j : BitVec n}, i.toNat = j.toNat → i = j
 @[bv_toNat] theorem toNat_ne (x y : BitVec n) : x ≠ y ↔ x.toNat ≠ y.toNat := by
   rw [Ne, toNat_eq]
 
-theorem toNat_lt (x : BitVec n) : x.toNat < 2^n := x.toFin.2
-
 theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsb i := rfl
 
 @[simp] theorem getLsb_ofFin (x : Fin (2^n)) (i : Nat) :
   getLsb (BitVec.ofFin x) i = x.val.testBit i := rfl
 
-@[simp] theorem getLsb_ge (x : BitVec w) (i : Nat) (ge : i ≥ w) : getLsb x i = false := by
+@[simp] theorem getLsb_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsb x i = false := by
   let ⟨x, x_lt⟩ := x
   simp
   apply Nat.testBit_lt_two_pow
@@ -72,7 +70,7 @@ theorem eq_of_getMsb_eq {x y : BitVec w}
   else
     have w_pos := Nat.pos_of_ne_zero w_zero
     have r : i ≤ w - 1 := by
-      simp [Nat.le_sub_iff_add_le w_pos, Nat.add_succ]
+      simp [Nat.le_sub_iff_add_le w_pos]
       exact i_lt
     have q_lt : w - 1 - i < w := by
       simp only [Nat.sub_sub]
@@ -89,12 +87,17 @@ theorem eq_of_toFin_eq : ∀ {x y : BitVec w}, x.toFin = y.toFin → x = y
 @[simp] theorem toNat_ofBool (b : Bool) : (ofBool b).toNat = b.toNat := by
   cases b <;> rfl
 
+@[simp] theorem msb_ofBool (b : Bool) : (ofBool b).msb = b := by
+  cases b <;> simp [BitVec.msb]
+
 theorem ofNat_one (n : Nat) : BitVec.ofNat 1 n = BitVec.ofBool (n % 2 = 1) :=  by
   rcases (Nat.mod_two_eq_zero_or_one n) with h | h <;> simp [h, BitVec.ofNat, Fin.ofNat']
 
 theorem ofBool_eq_iff_eq : ∀(b b' : Bool), BitVec.ofBool b = BitVec.ofBool b' ↔ b = b' := by
   decide
 
+@[simp] theorem not_ofBool : ~~~ (ofBool b) = ofBool (!b) := by cases b <;> rfl
+
 @[simp, bv_toNat] theorem toNat_ofFin (x : Fin (2^n)) : (BitVec.ofFin x).toNat = x.val := rfl
 
 @[simp] theorem toNat_ofNatLt (x : Nat) (p : x < 2^w) : (x#'p).toNat = x := rfl
@@ -116,6 +119,8 @@ theorem getLsb_ofNat (n : Nat) (x : Nat) (i : Nat) :
 
 @[simp] theorem getLsb_zero : (0#w).getLsb i = false := by simp [getLsb]
 
+@[simp] theorem getMsb_zero : (0#w).getMsb i = false := by simp [getMsb]
+
 @[simp] theorem toNat_mod_cancel (x : BitVec n) : x.toNat % (2^n) = x.toNat :=
   Nat.mod_eq_of_lt x.isLt
 
@@ -241,6 +246,12 @@ theorem toInt_ofNat {n : Nat} (x : Nat) :
   else
     simp [n_le_i, toNat_ofNat]
 
+theorem zeroExtend'_eq {x : BitVec w} (h : w ≤ v) : x.zeroExtend' h = x.zeroExtend v := by
+  apply eq_of_toNat_eq
+  rw [toNat_zeroExtend, toNat_zeroExtend']
+  rw [Nat.mod_eq_of_lt]
+  exact Nat.lt_of_lt_of_le x.isLt (Nat.pow_le_pow_right (Nat.zero_lt_two) h)
+
 @[simp, bv_toNat] theorem toNat_truncate (x : BitVec n) : (truncate i x).toNat = x.toNat % 2^i :=
   toNat_zeroExtend i x
 
@@ -285,10 +296,25 @@ theorem nat_eq_toNat (x : BitVec w) (y : Nat)
     getLsb (zeroExtend m x) i = (decide (i < m) && getLsb x i) := by
   simp [getLsb, toNat_zeroExtend, Nat.testBit_mod_two_pow]
 
+@[simp] theorem getMsb_zeroExtend_add {x : BitVec w} (h : k ≤ i) :
+    (x.zeroExtend (w + k)).getMsb i = x.getMsb (i - k) := by
+  by_cases h : w = 0
+  · subst h; simp
+  simp only [getMsb, getLsb_zeroExtend]
+  by_cases h₁ : i < w + k <;> by_cases h₂ : i - k < w <;> by_cases h₃ : w + k - 1 - i < w + k
+    <;> simp [h₁, h₂, h₃]
+  · congr 1
+    omega
+  all_goals (first | apply getLsb_ge | apply Eq.symm; apply getLsb_ge)
+    <;> omega
+
 @[simp] theorem getLsb_truncate (m : Nat) (x : BitVec n) (i : Nat) :
     getLsb (truncate m x) i = (decide (i < m) && getLsb x i) :=
   getLsb_zeroExtend m x i
 
+theorem msb_truncate (x : BitVec w) : (x.truncate (k + 1)).msb = x.getLsb k := by
+  simp [BitVec.msb, getMsb]
+
 @[simp] theorem zeroExtend_zeroExtend_of_le (x : BitVec w) (h : k ≤ l) :
     (x.zeroExtend l).zeroExtend k = x.zeroExtend k := by
   ext i
@@ -301,11 +327,18 @@ theorem nat_eq_toNat (x : BitVec w) (y : Nat)
     (x.truncate l).truncate k = x.truncate k :=
   zeroExtend_zeroExtend_of_le x h
 
+@[simp] theorem truncate_cast {h : w = v} : (cast h x).truncate k = x.truncate k := by
+  apply eq_of_getLsb_eq
+  simp
+
 theorem msb_zeroExtend (x : BitVec w) : (x.zeroExtend v).msb = (decide (0 < v) && x.getLsb (v - 1)) := by
   rw [msb_eq_getLsb_last]
   simp only [getLsb_zeroExtend]
   cases getLsb x (v - 1) <;> simp; omega
 
+theorem msb_zeroExtend' (x : BitVec w) (h : w ≤ v) : (x.zeroExtend' h).msb = (decide (0 < v) && x.getLsb (v - 1)) := by
+  rw [zeroExtend'_eq, msb_zeroExtend]
+
 /-! ## extractLsb -/
 
 @[simp]
@@ -353,6 +386,18 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
   rw [← testBit_toNat, getLsb, getLsb]
   simp
 
+@[simp] theorem getMsb_or {x y : BitVec w} : (x ||| y).getMsb i = (x.getMsb i || y.getMsb i) := by
+  simp only [getMsb]
+  by_cases h : i < w <;> simp [h]
+
+@[simp] theorem msb_or {x y : BitVec w} : (x ||| y).msb = (x.msb || y.msb) := by
+  simp [BitVec.msb]
+
+@[simp] theorem truncate_or {x y : BitVec w} :
+    (x ||| y).truncate k = x.truncate k ||| y.truncate k := by
+  ext
+  simp
+
 /-! ### and -/
 
 @[simp] theorem toNat_and (x y : BitVec v) :
@@ -367,6 +412,18 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
   rw [← testBit_toNat, getLsb, getLsb]
   simp
 
+@[simp] theorem getMsb_and {x y : BitVec w} : (x &&& y).getMsb i = (x.getMsb i && y.getMsb i) := by
+  simp only [getMsb]
+  by_cases h : i < w <;> simp [h]
+
+@[simp] theorem msb_and {x y : BitVec w} : (x &&& y).msb = (x.msb && y.msb) := by
+  simp [BitVec.msb]
+
+@[simp] theorem truncate_and {x y : BitVec w} :
+    (x &&& y).truncate k = x.truncate k &&& y.truncate k := by
+  ext
+  simp
+
 /-! ### xor -/
 
 @[simp] theorem toNat_xor (x y : BitVec v) :
@@ -382,6 +439,11 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
   rw [← testBit_toNat, getLsb, getLsb]
   simp
 
+@[simp] theorem truncate_xor {x y : BitVec w} :
+    (x ^^^ y).truncate k = x.truncate k ^^^ y.truncate k := by
+  ext
+  simp
+
 /-! ### not -/
 
 theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
@@ -396,12 +458,12 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
   | y+1 =>
     rw [Nat.succ_eq_add_one] at h
     rw [← h]
-    rw [Nat.testBit_two_pow_sub_succ (toNat_lt _)]
+    rw [Nat.testBit_two_pow_sub_succ (isLt _)]
     · cases w : decide (i < v)
       · simp at w
         simp [w]
         rw [Nat.testBit_lt_two_pow]
-        calc BitVec.toNat x < 2 ^ v := toNat_lt _
+        calc BitVec.toNat x < 2 ^ v := isLt _
           _ ≤ 2 ^ i := Nat.pow_le_pow_of_le_right Nat.zero_lt_two w
       · simp
 
@@ -414,6 +476,30 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
 @[simp] theorem getLsb_not {x : BitVec v} : (~~~x).getLsb i = (decide (i < v) && ! x.getLsb i) := by
   by_cases h' : i < v <;> simp_all [not_def]
 
+@[simp] theorem truncate_not {x : BitVec w} (h : k ≤ w) :
+    (~~~x).truncate k = ~~~(x.truncate k) := by
+  ext
+  simp [h]
+  omega
+
+/-! ### cast -/
+
+@[simp] theorem not_cast {x : BitVec w} (h : w = w') : ~~~(cast h x) = cast h (~~~x) := by
+  ext
+  simp_all [lt_of_getLsb]
+
+@[simp] theorem and_cast {x y : BitVec w} (h : w = w') : cast h x &&& cast h y = cast h (x &&& y) := by
+  ext
+  simp_all [lt_of_getLsb]
+
+@[simp] theorem or_cast {x y : BitVec w} (h : w = w') : cast h x ||| cast h y = cast h (x ||| y) := by
+  ext
+  simp_all [lt_of_getLsb]
+
+@[simp] theorem xor_cast {x y : BitVec w} (h : w = w') : cast h x &&& cast h y = cast h (x &&& y) := by
+  ext
+  simp_all [lt_of_getLsb]
+
 /-! ### shiftLeft -/
 
 @[simp, bv_toNat] theorem toNat_shiftLeft {x : BitVec v} :
@@ -431,6 +517,19 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
   cases h₁ : decide (i < m) <;> cases h₂ : decide (n ≤ i) <;> cases h₃ : decide (i < n)
   all_goals { simp_all <;> omega }
 
+@[simp] theorem getMsb_shiftLeft (x : BitVec w) (i) :
+    (x <<< i).getMsb k = x.getMsb (k + i) := by
+  simp only [getMsb, getLsb_shiftLeft]
+  by_cases h : w = 0
+  · subst h; simp
+  have t : w - 1 - k < w := by omega
+  simp only [t]
+  simp only [decide_True, Nat.sub_sub, Bool.true_and, Nat.add_assoc]
+  by_cases h₁ : k < w <;> by_cases h₂ : w - (1 + k) < i <;> by_cases h₃ : k + i < w
+    <;> simp [h₁, h₂, h₃]
+    <;> (first | apply getLsb_ge | apply Eq.symm; apply getLsb_ge)
+    <;> omega
+
 theorem shiftLeftZeroExtend_eq {x : BitVec w} :
     shiftLeftZeroExtend x n = zeroExtend (w+n) x <<< n := by
   apply eq_of_toNat_eq
@@ -439,7 +538,7 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
   · simp
     rw [Nat.mod_eq_of_lt]
     rw [Nat.shiftLeft_eq, Nat.pow_add]
-    exact Nat.mul_lt_mul_of_pos_right (BitVec.toNat_lt x) (Nat.two_pow_pos _)
+    exact Nat.mul_lt_mul_of_pos_right x.isLt (Nat.two_pow_pos _)
   · omega
 
 @[simp] theorem getLsb_shiftLeftZeroExtend (x : BitVec m) (n : Nat) :
@@ -450,6 +549,10 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
     <;> simp_all
     <;> (rw [getLsb_ge]; omega)
 
+@[simp] theorem msb_shiftLeftZeroExtend (x : BitVec w) (i : Nat) :
+    (shiftLeftZeroExtend x i).msb = x.msb := by
+  simp [shiftLeftZeroExtend_eq, BitVec.msb]
+
 /-! ### ushiftRight -/
 
 @[simp, bv_toNat] theorem toNat_ushiftRight (x : BitVec n) (i : Nat) :
@@ -475,6 +578,59 @@ theorem append_def (x : BitVec v) (y : BitVec w) :
   · simp [h]
   · simp [h]; simp_all
 
+theorem msb_append {x : BitVec w} {y : BitVec v} :
+    (x ++ y).msb = bif (w == 0) then (y.msb) else (x.msb) := by
+  rw [← append_eq, append]
+  simp [msb_zeroExtend']
+  by_cases h : w = 0
+  · subst h
+    simp [BitVec.msb, getMsb]
+  · rw [cond_eq_if]
+    have q : 0 < w + v := by omega
+    have t : y.getLsb (w + v - 1) = false := getLsb_ge _ _ (by omega)
+    simp [h, q, t, BitVec.msb, getMsb]
+
+@[simp] theorem truncate_append {x : BitVec w} {y : BitVec v} :
+    (x ++ y).truncate k = if h : k ≤ v then y.truncate k else (x.truncate (k - v) ++ y).cast (by omega) := by
+  apply eq_of_getLsb_eq
+  intro i
+  simp only [getLsb_zeroExtend, Fin.is_lt, decide_True, getLsb_append, Bool.true_and]
+  split
+  · have t : i < v := by omega
+    simp [t]
+  · by_cases t : i < v
+    · simp [t]
+    · have t' : i - v < k - v := by omega
+      simp [t, t']
+
+@[simp] theorem truncate_cons {x : BitVec w} : (cons a x).truncate w = x := by
+  simp [cons]
+
+@[simp] theorem not_append {x : BitVec w} {y : BitVec v} : ~~~ (x ++ y) = (~~~ x) ++ (~~~ y) := by
+  ext i
+  simp only [getLsb_not, getLsb_append, cond_eq_if]
+  split
+  · simp_all
+  · simp_all; omega
+
+@[simp] theorem and_append {x₁ x₂ : BitVec w} {y₁ y₂ : BitVec v} :
+    (x₁ ++ y₁) &&& (x₂ ++ y₂) = (x₁ &&& x₂) ++ (y₁ &&& y₂) := by
+  ext i
+  simp only [getLsb_append, cond_eq_if]
+  split <;> simp [*]
+
+@[simp] theorem or_append {x₁ x₂ : BitVec w} {y₁ y₂ : BitVec v} :
+    (x₁ ++ y₁) ||| (x₂ ++ y₂) = (x₁ ||| x₂) ++ (y₁ ||| y₂) := by
+  ext i
+  simp only [getLsb_append, cond_eq_if]
+  split <;> simp [*]
+
+@[simp] theorem xor_append {x₁ x₂ : BitVec w} {y₁ y₂ : BitVec v} :
+    (x₁ ++ y₁) ^^^ (x₂ ++ y₂) = (x₁ ^^^ x₂) ++ (y₁ ^^^ y₂) := by
+  ext i
+  simp only [getLsb_append, cond_eq_if]
+  split <;> simp [*]
+
 /-! ### rev -/
 
 theorem getLsb_rev (x : BitVec w) (i : Fin w) :
@@ -497,6 +653,11 @@ theorem getMsb_rev (x : BitVec w) (i : Fin w) :
   let ⟨x, _⟩ := x
   simp [cons, toNat_append, toNat_ofBool]
 
+/-- Variant of `toNat_cons` using `+` instead of `|||`. -/
+theorem toNat_cons' {x : BitVec w} :
+    (cons a x).toNat = (a.toNat <<< w) + x.toNat := by
+  simp [cons, Nat.shiftLeft_eq, Nat.mul_comm _ (2^w), Nat.mul_add_lt_is_or, x.isLt]
+
 @[simp] theorem getLsb_cons (b : Bool) {n} (x : BitVec n) (i : Nat) :
     getLsb (cons b x) i = if i = n then b else getLsb x i := by
   simp only [getLsb, toNat_cons, Nat.testBit_or]
@@ -511,6 +672,9 @@ theorem getMsb_rev (x : BitVec w) (i : Fin w) :
     have p2 : i - n ≠ 0 := by omega
     simp [p1, p2, Nat.testBit_bool_to_nat]
 
+@[simp] theorem msb_cons : (cons a x).msb = a := by
+  simp [cons, msb_cast, msb_append]
+
 theorem truncate_succ (x : BitVec w) :
     truncate (i+1) x = cons (getLsb x i) (truncate i x) := by
   apply eq_of_getLsb_eq
@@ -522,6 +686,30 @@ theorem truncate_succ (x : BitVec w) :
     have j_lt : j.val < i := Nat.lt_of_le_of_ne (Nat.le_of_succ_le_succ j.isLt) j_eq
     simp [j_eq, j_lt]
 
+theorem eq_msb_cons_truncate (x : BitVec (w+1)) : x = (cons x.msb (x.truncate w)) := by
+  ext i
+  simp
+  split <;> rename_i h
+  · simp [BitVec.msb, getMsb, h]
+  · by_cases h' : i < w
+    · simp_all
+    · omega
+
+@[simp] theorem not_cons (x : BitVec w) (b : Bool) : ~~~(cons b x) = cons (!b) (~~~x) := by
+  simp [cons]
+
+@[simp] theorem cons_or_cons (x y : BitVec w) (a b : Bool) :
+    (cons a x) ||| (cons b y) = cons (a || b) (x ||| y) := by
+  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+
+@[simp] theorem cons_and_cons (x y : BitVec w) (a b : Bool) :
+    (cons a x) &&& (cons b y) = cons (a && b) (x &&& y) := by
+  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+
+@[simp] theorem cons_xor_cons (x y : BitVec w) (a b : Bool) :
+    (cons a x) ^^^ (cons b y) = cons (xor a b) (x ^^^ y) := by
+  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+
 /-! ### concat -/
 
 @[simp] theorem toNat_concat (x : BitVec w) (b : Bool) :
@@ -546,6 +734,21 @@ theorem getLsb_concat (x : BitVec w) (b : Bool) (i : Nat) :
 @[simp] theorem getLsb_concat_succ : (concat x b).getLsb (i + 1) = x.getLsb i := by
   simp [getLsb_concat]
 
+@[simp] theorem not_concat (x : BitVec w) (b : Bool) : ~~~(concat x b) = concat (~~~x) !b := by
+  ext i; cases i using Fin.succRecOn <;> simp [*, Nat.succ_lt_succ]
+
+@[simp] theorem concat_or_concat (x y : BitVec w) (a b : Bool) :
+    (concat x a) ||| (concat y b) = concat (x ||| y) (a || b) := by
+  ext i; cases i using Fin.succRecOn <;> simp
+
+@[simp] theorem concat_and_concat (x y : BitVec w) (a b : Bool) :
+    (concat x a) &&& (concat y b) = concat (x &&& y) (a && b) := by
+  ext i; cases i using Fin.succRecOn <;> simp
+
+@[simp] theorem concat_xor_concat (x y : BitVec w) (a b : Bool) :
+    (concat x a) ^^^ (concat y b) = concat (x ^^^ y) (xor a b) := by
+  ext i; cases i using Fin.succRecOn <;> simp
+
 /-! ### add -/
 
 theorem add_def {n} (x y : BitVec n) : x + y = .ofNat n (x.toNat + y.toNat) := rfl
@@ -572,6 +775,10 @@ protected theorem add_comm (x y : BitVec n) : x + y = y + x := by
 
 @[simp] protected theorem zero_add (x : BitVec n) : 0#n + x = x := by simp [add_def]
 
+theorem truncate_add (x y : BitVec w) (h : i ≤ w) :
+    (x + y).truncate i = x.truncate i + y.truncate i := by
+  have dvd : 2^i ∣ 2^w := Nat.pow_dvd_pow _ h
+  simp [bv_toNat, h, Nat.mod_mod_of_dvd _ dvd]
 
 /-! ### sub/neg -/
 

src/Init/Data/Bool.lean

@@ -29,6 +29,8 @@ instance (p : Bool → Prop) [inst : DecidablePred p] : Decidable (∃ x, p x) :
   | _, isTrue hf => isTrue ⟨_, hf⟩
   | isFalse ht, isFalse hf => isFalse fun | ⟨true, h⟩ => absurd h ht | ⟨false, h⟩ => absurd h hf
 
+@[simp] theorem default_bool : default = false := rfl
+
 instance : LE Bool := ⟨(. → .)⟩
 instance : LT Bool := ⟨(!. && .)⟩
 
@@ -48,85 +50,205 @@ theorem ne_false_iff : {b : Bool} → b ≠ false ↔ b = true := by decide
 
 theorem eq_iff_iff {a b : Bool} : a = b ↔ (a ↔ b) := by cases b <;> simp
 
-@[simp] theorem decide_eq_true {b : Bool} : decide (b = true) = b := by cases b <;> simp
-@[simp] theorem decide_eq_false {b : Bool} : decide (b = false) = !b := by cases b <;> simp
-@[simp] theorem decide_true_eq {b : Bool} : decide (true = b) = b := by cases b <;> simp
-@[simp] theorem decide_false_eq {b : Bool} : decide (false = b) = !b := by cases b <;> simp
+@[simp] theorem decide_eq_true  {b : Bool} [Decidable (b = true)]  : decide (b = true)  =  b := by cases b <;> simp
+@[simp] theorem decide_eq_false {b : Bool} [Decidable (b = false)] : decide (b = false) = !b := by cases b <;> simp
+@[simp] theorem decide_true_eq  {b : Bool} [Decidable (true = b)]  : decide (true  = b) =  b := by cases b <;> simp
+@[simp] theorem decide_false_eq {b : Bool} [Decidable (false = b)] : decide (false = b) = !b := by cases b <;> simp
 
 /-! ### and -/
 
-@[simp] theorem not_and_self : ∀ (x : Bool), (!x && x) = false := by decide
+@[simp] theorem and_self_left  : ∀(a b : Bool), (a && (a && b)) = (a && b) := by decide
+@[simp] theorem and_self_right : ∀(a b : Bool), ((a && b) && b) = (a && b) := by decide
 
+@[simp] theorem not_and_self : ∀ (x : Bool), (!x && x) = false := by decide
 @[simp] theorem and_not_self : ∀ (x : Bool), (x && !x) = false := by decide
 
+/-
+Added for confluence with `not_and_self` `and_not_self` on term
+`(b && !b) = true` due to reductions:
+
+1. `(b = true ∨ !b = true)` via `Bool.and_eq_true`
+2. `false = true` via `Bool.and_not_self`
+-/
+@[simp] theorem eq_true_and_eq_false_self : ∀(b : Bool), (b = true ∧ b = false) ↔ False := by decide
+@[simp] theorem eq_false_and_eq_true_self : ∀(b : Bool), (b = false ∧ b = true) ↔ False := by decide
+
 theorem and_comm : ∀ (x y : Bool), (x && y) = (y && x) := by decide
 
 theorem and_left_comm : ∀ (x y z : Bool), (x && (y && z)) = (y && (x && z)) := by decide
-
 theorem and_right_comm : ∀ (x y z : Bool), ((x && y) && z) = ((x && z) && y) := by decide
 
-theorem and_or_distrib_left : ∀ (x y z : Bool), (x && (y || z)) = ((x && y) || (x && z)) := by
-  decide
+/-
+Bool version `and_iff_left_iff_imp`.
 
-theorem and_or_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 -/
-theorem not_and : ∀ (x y : Bool), (!(x && y)) = (!x || !y) := by decide
-
-theorem and_eq_true_iff : ∀ (x y : Bool), (x && y) = true ↔ x = true ∧ y = true := by decide
-
-theorem and_eq_false_iff : ∀ (x y : Bool), (x && y) = false ↔ x = false ∨ y = false := by decide
+Needed for confluence of term `(a && b) ↔ a` which reduces to `(a && b) = a` via
+`Bool.coe_iff_coe` and `a → b` via `Bool.and_eq_true` and
+`and_iff_left_iff_imp`.
+-/
+@[simp] theorem and_iff_left_iff_imp  : ∀(a b : Bool), ((a && b) = a) ↔ (a → b) := by decide
+@[simp] theorem and_iff_right_iff_imp : ∀(a b : Bool), ((a && b) = b) ↔ (b → a) := by decide
+@[simp] theorem iff_self_and : ∀(a b : Bool), (a = (a && b)) ↔ (a → b) := by decide
+@[simp] theorem iff_and_self : ∀(a b : Bool), (b = (a && b)) ↔ (b → a) := by decide
 
 /-! ### or -/
 
-@[simp] theorem not_or_self : ∀ (x : Bool), (!x || x) = true := by decide
+@[simp] theorem or_self_left  : ∀(a b : Bool), (a || (a || b)) = (a || b) := by decide
+@[simp] theorem or_self_right : ∀(a b : Bool), ((a || b) || b) = (a || b) := by decide
 
+@[simp] theorem not_or_self : ∀ (x : Bool), (!x || x) = true := by decide
 @[simp] theorem or_not_self : ∀ (x : Bool), (x || !x) = true := by decide
 
+/-
+Added for confluence with `not_or_self` `or_not_self` on term
+`(b || !b) = true` due to reductions:
+1. `(b = true ∨ !b = true)` via `Bool.or_eq_true`
+2. `true = true` via `Bool.or_not_self`
+-/
+@[simp] theorem eq_true_or_eq_false_self : ∀(b : Bool), (b = true ∨ b = false) ↔ True := by decide
+@[simp] theorem eq_false_or_eq_true_self : ∀(b : Bool), (b = false ∨ b = true) ↔ True := by decide
+
+/-
+Bool version `or_iff_left_iff_imp`.
+
+Needed for confluence of term `(a || b) ↔ a` which reduces to `(a || b) = a` via
+`Bool.coe_iff_coe` and `a → b` via `Bool.or_eq_true` and
+`and_iff_left_iff_imp`.
+-/
+@[simp] theorem or_iff_left_iff_imp  : ∀(a b : Bool), ((a || b) = a) ↔ (b → a) := by decide
+@[simp] theorem or_iff_right_iff_imp : ∀(a b : Bool), ((a || b) = b) ↔ (a → b) := by decide
+@[simp] theorem iff_self_or : ∀(a b : Bool), (a = (a || b)) ↔ (b → a) := by decide
+@[simp] theorem iff_or_self : ∀(a b : Bool), (b = (a || b)) ↔ (a → b) := by decide
+
 theorem or_comm : ∀ (x y : Bool), (x || y) = (y || x) := by decide
 
 theorem or_left_comm : ∀ (x y z : Bool), (x || (y || z)) = (y || (x || z)) := by decide
-
 theorem or_right_comm : ∀ (x y z : Bool), ((x || y) || z) = ((x || z) || y) := by decide
 
-theorem or_and_distrib_left : ∀ (x y z : Bool), (x || (y && z)) = ((x || y) && (x || z)) := by
-  decide
+/-! ### distributivity -/
 
-theorem or_and_distrib_right : ∀ (x y z : Bool), ((x && y) || z) = ((x || z) && (y || z)) := by
-  decide
+theorem and_or_distrib_left  : ∀ (x y z : Bool), (x && (y || z)) = (x && y || x && z) := by decide
+theorem and_or_distrib_right : ∀ (x y z : Bool), ((x || y) && z) = (x && z || y && z) := by decide
+
+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 && 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
 
 /-- De Morgan's law for boolean or -/
-theorem not_or : ∀ (x y : Bool), (!(x || y)) = (!x && !y) := by decide
+@[simp] theorem not_or : ∀ (x y : Bool), (!(x || y)) = (!x && !y) := by decide
 
-theorem or_eq_true_iff : ∀ (x y : Bool), (x || y) = true ↔ x = true ∨ y = true := by decide
+theorem and_eq_true_iff (x y : Bool) : (x && y) = true ↔ x = true ∧ y = true :=
+  Iff.of_eq (and_eq_true x y)
 
-theorem or_eq_false_iff : ∀ (x y : Bool), (x || y) = false ↔ x = false ∧ y = false := by decide
+theorem and_eq_false_iff : ∀ (x y : Bool), (x && y) = false ↔ x = false ∨ y = false := by decide
+
+/-
+New simp rule that replaces `Bool.and_eq_false_eq_eq_false_or_eq_false` in
+Mathlib due to confluence:
+
+Consider the term: `¬((b && c) = true)`:
+
+1. Reduces to `((b && c) = false)` via `Bool.not_eq_true`
+2. Reduces to `¬(b = true ∧ c = true)` via `Bool.and_eq_true`.
+
+
+1. Further reduces to `b = false ∨ c = false` via `Bool.and_eq_false_eq_eq_false_or_eq_false`.
+2. Further reduces to `b = true → c = false` via `not_and` and `Bool.not_eq_true`.
+-/
+@[simp] theorem and_eq_false_imp : ∀ (x y : Bool), (x && y) = false ↔ (x = true → y = false) := by decide
+
+@[simp] theorem or_eq_true_iff : ∀ (x y : Bool), (x || y) = true ↔ x = true ∨ y = true := by decide
+
+@[simp] theorem or_eq_false_iff : ∀ (x y : Bool), (x || y) = false ↔ x = false ∧ y = false := by decide
+
+/-! ### eq/beq/bne -/
+
+/--
+These two rules follow trivially by simp, but are needed to avoid non-termination
+in false_eq and true_eq.
+-/
+@[simp] theorem false_eq_true : (false = true) = False := by simp
+@[simp] theorem true_eq_false : (true = false) = False := by simp
+
+-- The two lemmas below normalize terms with a constant to the
+-- right-hand side but risk non-termination if `false_eq_true` and
+-- `true_eq_false` are disabled.
+@[simp low] theorem false_eq (b : Bool) : (false = b) = (b = false) := by
+  cases b <;> simp
+
+@[simp low] theorem true_eq (b : Bool) : (true = b) = (b = true) := by
+  cases b <;> simp
+
+@[simp] theorem true_beq  : ∀b, (true  == b) =  b := by decide
+@[simp] theorem false_beq : ∀b, (false == b) = !b := by decide
+@[simp] theorem beq_true  : ∀b, (b == true)  =  b := by decide
+@[simp] theorem beq_false : ∀b, (b == false) = !b := by decide
+
+@[simp] theorem true_bne  : ∀(b : Bool), (true  != b) = !b := by decide
+@[simp] theorem false_bne : ∀(b : Bool), (false != b) =  b := by decide
+@[simp] theorem bne_true  : ∀(b : Bool), (b != true)  = !b := by decide
+@[simp] theorem bne_false : ∀(b : Bool), (b != false) =  b := by decide
+
+@[simp] theorem not_beq_self : ∀ (x : Bool), ((!x) == x) = false := by decide
+@[simp] theorem beq_not_self : ∀ (x : Bool), (x   == !x) = false := by decide
+
+@[simp] theorem not_bne_self : ∀ (x : Bool), ((!x) != x) = true := by decide
+@[simp] theorem bne_not_self : ∀ (x : Bool), (x   != !x) = true := by decide
+
+/-
+Added for equivalence with `Bool.not_beq_self` and needed for confluence
+due to `beq_iff_eq`.
+-/
+@[simp] theorem not_eq_self : ∀(b : Bool), ((!b) = b) ↔ False := by decide
+@[simp] theorem eq_not_self : ∀(b : Bool), (b = (!b)) ↔ False := by decide
+
+@[simp] theorem beq_self_left  : ∀(a b : Bool), (a == (a == b)) = b := by decide
+@[simp] theorem beq_self_right : ∀(a b : Bool), ((a == b) == b) = a := by decide
+@[simp] theorem bne_self_left  : ∀(a b : Bool), (a != (a != b)) = b := by decide
+@[simp] theorem bne_self_right : ∀(a b : Bool), ((a != b) != b) = a := by decide
+
+@[simp] theorem not_bne_not : ∀ (x y : Bool), ((!x) != (!y)) = (x != y) := by decide
+
+@[simp] theorem bne_assoc : ∀ (x y z : Bool), ((x != y) != z) = (x != (y != z)) := by decide
+
+@[simp] theorem bne_left_inj  : ∀ (x y z : Bool), (x != y) = (x != z) ↔ y = z := by decide
+@[simp] theorem bne_right_inj : ∀ (x y z : Bool), (x != z) = (y != z) ↔ x = y := by decide
+
+/-! ### coercision related normal forms -/
+
+@[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
+
+@[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
+@[simp] theorem coe_false_iff_true  : ∀(a b : Bool), (a = false ↔ b) ↔ (!a) = b := by decide
+@[simp] theorem coe_false_iff_false : ∀(a b : Bool), (a = false ↔ b = false) ↔ (!a) = (!b) := by decide
 
 /-! ### xor -/
 
-@[simp] theorem false_xor : ∀ (x : Bool), xor false x = x := by decide
+theorem false_xor : ∀ (x : Bool), xor false x = x := false_bne
 
-@[simp] theorem xor_false : ∀ (x : Bool), xor x false = x := by decide
+theorem xor_false : ∀ (x : Bool), xor x false = x := bne_false
 
-@[simp] theorem true_xor : ∀ (x : Bool), xor true x = !x := by decide
+theorem true_xor : ∀ (x : Bool), xor true x = !x := true_bne
 
-@[simp] theorem xor_true : ∀ (x : Bool), xor x true = !x := by decide
+theorem xor_true : ∀ (x : Bool), xor x true = !x := bne_true
 
-@[simp] theorem not_xor_self : ∀ (x : Bool), xor (!x) x = true := by decide
+theorem not_xor_self : ∀ (x : Bool), xor (!x) x = true := not_bne_self
 
-@[simp] theorem xor_not_self : ∀ (x : Bool), xor x (!x) = true := by decide
+theorem xor_not_self : ∀ (x : Bool), xor x (!x) = true := bne_not_self
 
 theorem not_xor : ∀ (x y : Bool), xor (!x) y = !(xor x y) := by decide
 
 theorem xor_not : ∀ (x y : Bool), xor x (!y) = !(xor x y) := by decide
 
-@[simp] theorem not_xor_not : ∀ (x y : Bool), xor (!x) (!y) = (xor x y) := by decide
+theorem not_xor_not : ∀ (x y : Bool), xor (!x) (!y) = (xor x y) := not_bne_not
 
 theorem xor_self : ∀ (x : Bool), xor x x = false := by decide
 
@@ -136,13 +258,11 @@ theorem xor_left_comm : ∀ (x y z : Bool), xor x (xor y z) = xor y (xor x z) :=
 
 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), xor (xor x y) z = xor x (xor y z) := by decide
+theorem xor_assoc : ∀ (x y z : Bool), xor (xor x y) z = xor x (xor y z) := bne_assoc
 
-@[simp]
-theorem xor_left_inj : ∀ (x y z : Bool), xor x y = xor x z ↔ y = z := by decide
+theorem xor_left_inj : ∀ (x y z : Bool), xor x y = xor x z ↔ y = z := bne_left_inj
 
-@[simp]
-theorem xor_right_inj : ∀ (x y z : Bool), xor x z = xor y z ↔ x = y := by decide
+theorem xor_right_inj : ∀ (x y z : Bool), xor x z = xor y z ↔ x = y := bne_right_inj
 
 /-! ### le/lt -/
 
@@ -227,16 +347,152 @@ theorem toNat_lt (b : Bool) : b.toNat < 2 :=
 
 @[simp] theorem toNat_eq_zero (b : Bool) : b.toNat = 0 ↔ b = false := by
   cases b <;> simp
-@[simp] theorem toNat_eq_one (b : Bool) : b.toNat = 1 ↔ b = true := by
+@[simp] theorem toNat_eq_one  (b : Bool) : b.toNat = 1 ↔ b = true := by
   cases b <;> simp
 
-end Bool
+/-! ### ite -/
+
+@[simp] theorem if_true_left  (p : Prop) [h : Decidable p] (f : Bool) :
+    (ite p true f) = (p || f) := by cases h with | _ p => simp [p]
+
+@[simp] theorem if_false_left  (p : Prop) [h : Decidable p] (f : Bool) :
+    (ite p false f) = (!p && f) := by cases h with | _ p => simp [p]
+
+@[simp] theorem if_true_right  (p : Prop) [h : Decidable p] (t : Bool) :
+    (ite p t true) = (!(p : Bool) || t) := by cases h with | _ p => simp [p]
+
+@[simp] theorem if_false_right  (p : Prop) [h : Decidable p] (t : Bool) :
+    (ite p t false) = (p && t) := by cases h with | _ p => simp [p]
+
+@[simp] theorem ite_eq_true_distrib (p : Prop) [h : Decidable p] (t f : Bool) :
+    (ite p t f = true) = ite p (t = true) (f = true) := by
+  cases h with | _ p => simp [p]
+
+@[simp] theorem ite_eq_false_distrib (p : Prop) [h : Decidable p] (t f : Bool) :
+    (ite p t f = false) = ite p (t = false) (f = false) := by
+  cases h with | _ p => simp [p]
+
+/-
+`not_ite_eq_true_eq_true` and related theorems below are added for
+non-confluence.  A motivating example is
+`¬((if u then b else c) = true)`.
+
+This reduces to:
+1. `¬((if u then (b = true) else (c = true))` via `ite_eq_true_distrib`
+2. `(if u then b c) = false)` via `Bool.not_eq_true`.
+
+Similar logic holds for `¬((if u then b else c) = false)` and related
+lemmas.
+-/
+
+@[simp]
+theorem not_ite_eq_true_eq_true (p : Prop) [h : Decidable p] (b c : Bool) :
+  ¬(ite p (b = true) (c = true)) ↔ (ite p (b = false) (c = false)) := by
+  cases h with | _ p => simp [p]
+
+@[simp]
+theorem not_ite_eq_false_eq_false (p : Prop) [h : Decidable p] (b c : Bool) :
+  ¬(ite p (b = false) (c = false)) ↔ (ite p (b = true) (c = true)) := by
+  cases h with | _ p => simp [p]
+
+@[simp]
+theorem not_ite_eq_true_eq_false (p : Prop) [h : Decidable p] (b c : Bool) :
+  ¬(ite p (b = true) (c = false)) ↔ (ite p (b = false) (c = true)) := by
+  cases h with | _ p => simp [p]
+
+@[simp]
+theorem not_ite_eq_false_eq_true (p : Prop) [h : Decidable p] (b c : Bool) :
+  ¬(ite p (b = false) (c = true)) ↔ (ite p (b = true) (c = false)) := by
+  cases h with | _ p => simp [p]
+
+/-
+Added for confluence between `if_true_left` and `ite_false_same` on
+`if b = true then True else b = true`
+-/
+@[simp] theorem eq_false_imp_eq_true : ∀(b:Bool), (b = false → b = true) ↔ (b = true) := by decide
+
+/-
+Added for confluence between `if_true_left` and `ite_false_same` on
+`if b = false then True else b = false`
+-/
+@[simp] theorem eq_true_imp_eq_false : ∀(b:Bool), (b = true → b = false) ↔ (b = false) := by decide
+
 
 /-! ### cond -/
 
-theorem cond_eq_if : (bif b then x else y) = (if b then x else y) := by
+theorem cond_eq_ite {α} (b : Bool) (t e : α) : cond b t e = if b then t else e := by
   cases b <;> simp
 
+theorem cond_eq_if : (bif b then x else y) = (if b then x else y) := cond_eq_ite b x y
+
+@[simp] theorem cond_not (b : Bool) (t e : α) : cond (!b) t e = cond b e t := by
+  cases b <;> rfl
+
+@[simp] theorem cond_self (c : Bool) (t : α) : cond c t t = t := by cases c <;> rfl
+
+/-
+This is a simp rule in Mathlib, but results in non-confluence that is difficult
+to fix as decide distributes over propositions. As an example, observe that
+`cond (decide (p ∧ q)) t f` could simplify to either:
+
+* `if p ∧ q then t else f` via `Bool.cond_decide` or
+* `cond (decide p && decide q) t f` via `Bool.decide_and`.
+
+A possible approach to improve normalization between `cond` and `ite` would be
+to completely simplify away `cond` by making `cond_eq_ite` a `simp` rule, but
+that has not been taken since it could surprise users to migrate pure `Bool`
+operations like `cond` to a mix of `Prop` and `Bool`.
+-/
+theorem cond_decide {α} (p : Prop) [Decidable p] (t e : α) :
+    cond (decide p) t e = if p then t else e := by
+  simp [cond_eq_ite]
+
+@[simp] theorem cond_eq_ite_iff (a : Bool) (p : Prop) [h : Decidable p] (x y u v : α) :
+  (cond a x y = ite p u v) ↔ ite a x y = ite p u v := by
+  simp [Bool.cond_eq_ite]
+
+@[simp] theorem ite_eq_cond_iff (p : Prop) [h : Decidable p] (a : Bool) (x y u v : α) :
+  (ite p x y = cond a u v) ↔ ite p x y = ite a u v := by
+  simp [Bool.cond_eq_ite]
+
+@[simp] theorem cond_eq_true_distrib : ∀(c t f : Bool),
+    (cond c t f = true) = ite (c = true) (t = true) (f = true) := by
+  decide
+
+@[simp] theorem cond_eq_false_distrib : ∀(c t f : Bool),
+    (cond c t f = false) = ite (c = true) (t = false) (f = false) := by decide
+
+protected theorem cond_true  {α : Type u} {a b : α} : cond true  a b = a := cond_true  a b
+protected theorem cond_false {α : Type u} {a b : α} : cond false a b = b := cond_false a b
+
+@[simp] theorem cond_true_left   : ∀(c f : Bool), cond c true f  = ( c || f) := by decide
+@[simp] theorem cond_false_left  : ∀(c f : Bool), cond c false f = (!c && f) := by decide
+@[simp] theorem cond_true_right  : ∀(c t : Bool), cond c t true  = (!c || t) := by decide
+@[simp] theorem cond_false_right : ∀(c t : Bool), cond c t false = ( c && t) := by decide
+
+@[simp] theorem cond_true_same  : ∀(c b : Bool), cond c c b = (c || b) := by decide
+@[simp] theorem cond_false_same : ∀(c b : Bool), cond c b c = (c && b) := by decide
+
+/-# decidability -/
+
+protected theorem decide_coe (b : Bool) [Decidable (b = true)] : decide (b = true) = b := decide_eq_true
+
+@[simp] theorem decide_and (p q : Prop) [dpq : Decidable (p ∧ q)] [dp : Decidable p] [dq : Decidable q] :
+    decide (p ∧ q) = (p && q) := by
+  cases dp with | _ p => simp [p]
+
+@[simp] theorem decide_or (p q : Prop) [dpq : Decidable (p ∨ q)] [dp : Decidable p] [dq : Decidable q] :
+    decide (p ∨ q) = (p || q) := by
+  cases dp with | _ p => simp [p]
+
+@[simp] theorem decide_iff_dist (p q : Prop) [dpq : Decidable (p ↔ q)] [dp : Decidable p] [dq : Decidable q] :
+    decide (p ↔ q) = (decide p == decide q) := by
+  cases dp with | _ p => simp [p]
+
+end Bool
+
+export Bool (cond_eq_if)
+
 /-! ### decide -/
 
 @[simp] theorem false_eq_decide_iff {p : Prop} [h : Decidable p] : false = decide p ↔ ¬p := by

src/Init/Data/Channel.lean

@@ -41,7 +41,7 @@ Sends a message on an `Channel`.
 
 This function does not block.
 -/
-def Channel.send (v : α) (ch : Channel α) : BaseIO Unit :=
+def Channel.send (ch : Channel α) (v : α) : BaseIO Unit :=
   ch.atomically do
     let st ← get
     if st.closed then return

src/Init/Data/Fin/Lemmas.lean

@@ -687,7 +687,7 @@ decreasing_by decreasing_with
 
 @[simp] theorem reverseInduction_last {n : Nat} {motive : Fin (n + 1) → Sort _} {zero succ} :
     (reverseInduction zero succ (Fin.last n) : motive (Fin.last n)) = zero := by
-  rw [reverseInduction]; simp; rfl
+  rw [reverseInduction]; simp
 
 @[simp] theorem reverseInduction_castSucc {n : Nat} {motive : Fin (n + 1) → Sort _} {zero succ}
     (i : Fin n) : reverseInduction (motive := motive) zero succ (castSucc i) =

src/Init/Data/Int.lean

@@ -11,3 +11,4 @@ import Init.Data.Int.DivModLemmas
 import Init.Data.Int.Gcd
 import Init.Data.Int.Lemmas
 import Init.Data.Int.Order
+import Init.Data.Int.Pow

src/Init/Data/Int/DivMod.lean

@@ -158,6 +158,14 @@ instance : Div Int where
 instance : Mod Int where
   mod := Int.emod
 
+@[simp, norm_cast] theorem ofNat_ediv (m n : Nat) : (↑(m / n) : Int) = ↑m / ↑n := rfl
+
+theorem ofNat_div (m n : Nat) : ↑(m / n) = div ↑m ↑n := rfl
+
+theorem ofNat_fdiv : ∀ m n : Nat, ↑(m / n) = fdiv ↑m ↑n
+  | 0, _ => by simp [fdiv]
+  | succ _, _ => rfl
+
 /-!
 # `bmod` ("balanced" mod)
 

src/Init/Data/Int/Gcd.lean

@@ -6,7 +6,12 @@ Authors: Mario Carneiro
 prelude
 import Init.Data.Int.Basic
 import Init.Data.Nat.Gcd
+import Init.Data.Nat.Lcm
+import Init.Data.Int.DivModLemmas
 
+/-!
+Definition and lemmas for gcd and lcm over Int
+-/
 namespace Int
 
 /-! ## gcd -/
@@ -14,4 +19,37 @@ namespace Int
 /-- Computes the greatest common divisor of two integers, as a `Nat`. -/
 def gcd (m n : Int) : Nat := m.natAbs.gcd n.natAbs
 
+theorem gcd_dvd_left {a b : Int} : (gcd a b : Int) ∣ a := by
+  have := Nat.gcd_dvd_left a.natAbs b.natAbs
+  rw [← Int.ofNat_dvd] at this
+  exact Int.dvd_trans this natAbs_dvd_self
+
+theorem gcd_dvd_right {a b : Int} : (gcd a b : Int) ∣ b := by
+  have := Nat.gcd_dvd_right a.natAbs b.natAbs
+  rw [← Int.ofNat_dvd] at this
+  exact Int.dvd_trans this natAbs_dvd_self
+
+@[simp] theorem one_gcd {a : Int} : gcd 1 a = 1 := by simp [gcd]
+@[simp] theorem gcd_one {a : Int} : gcd a 1 = 1 := by simp [gcd]
+
+@[simp] theorem neg_gcd {a b : Int} : gcd (-a) b = gcd a b := by simp [gcd]
+@[simp] theorem gcd_neg {a b : Int} : gcd a (-b) = gcd a b := by simp [gcd]
+
+/-! ## lcm -/
+
+/-- Computes the least common multiple of two integers, as a `Nat`. -/
+def lcm (m n : Int) : Nat := m.natAbs.lcm n.natAbs
+
+theorem lcm_ne_zero (hm : m ≠ 0) (hn : n ≠ 0) : lcm m n ≠ 0 := by
+  simp only [lcm]
+  apply Nat.lcm_ne_zero <;> simpa
+
+theorem dvd_lcm_left {a b : Int} : a ∣ lcm a b :=
+  Int.dvd_trans dvd_natAbs_self (Int.ofNat_dvd.mpr (Nat.dvd_lcm_left a.natAbs b.natAbs))
+
+theorem dvd_lcm_right {a b : Int} : b ∣ lcm a b :=
+  Int.dvd_trans dvd_natAbs_self (Int.ofNat_dvd.mpr (Nat.dvd_lcm_right a.natAbs b.natAbs))
+
+@[simp] theorem lcm_self {a : Int} : lcm a a = a.natAbs := Nat.lcm_self _
+
 end Int

src/Init/Data/Int/Lemmas.lean

@@ -153,7 +153,7 @@ theorem subNatNat_sub (h : n ≤ m) (k : Nat) : subNatNat (m - n) k = subNatNat
 theorem subNatNat_add (m n k : Nat) : subNatNat (m + n) k = m + subNatNat n k := by
   cases n.lt_or_ge k with
   | inl h' =>
-    simp [subNatNat_of_lt h', succ_pred_eq_of_pos (Nat.sub_pos_of_lt h')]
+    simp [subNatNat_of_lt h', sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h')]
     conv => lhs; rw [← Nat.sub_add_cancel (Nat.le_of_lt h')]
     apply subNatNat_add_add
   | inr h' => simp [subNatNat_of_le h',
@@ -169,12 +169,11 @@ theorem subNatNat_add_negSucc (m n k : Nat) :
     rw [subNatNat_sub h', Nat.add_comm]
   | inl h' =>
     have h₂ : m < n + succ k := Nat.lt_of_lt_of_le h' (le_add_right _ _)
-    have h₃ : m ≤ n + k := le_of_succ_le_succ h₂
     rw [subNatNat_of_lt h', subNatNat_of_lt h₂]
-    simp [Nat.add_comm]
-    rw [← add_succ, succ_pred_eq_of_pos (Nat.sub_pos_of_lt h'), add_succ, succ_sub h₃,
-      Nat.pred_succ]
-    rw [Nat.add_comm n, Nat.add_sub_assoc (Nat.le_of_lt h')]
+    simp only [pred_eq_sub_one, negSucc_add_negSucc, succ_eq_add_one, negSucc.injEq]
+    rw [Nat.add_right_comm, sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h'), Nat.sub_sub,
+      ← Nat.add_assoc, succ_sub_succ_eq_sub, Nat.add_comm n,Nat.add_sub_assoc (Nat.le_of_lt h'),
+      Nat.add_comm]
 
 protected theorem add_assoc : ∀ a b c : Int, a + b + c = a + (b + c)
   | (m:Nat), (n:Nat), c => aux1 ..
@@ -188,15 +187,15 @@ protected theorem add_assoc : ∀ a b c : Int, a + b + c = a + (b + c)
   | (m:Nat), -[n+1], -[k+1] => by
     rw [Int.add_comm, Int.add_comm m, Int.add_comm m, ← aux2, Int.add_comm -[k+1]]
   | -[m+1], -[n+1], -[k+1] => by
-    simp [add_succ, Nat.add_comm, Nat.add_left_comm, neg_ofNat_succ]
+    simp [Nat.add_comm, Nat.add_left_comm, Nat.add_assoc]
 where
   aux1 (m n : Nat) : ∀ c : Int, m + n + c = m + (n + c)
     | (k:Nat) => by simp [Nat.add_assoc]
     | -[k+1]  => by simp [subNatNat_add]
   aux2 (m n k : Nat) : -[m+1] + -[n+1] + k = -[m+1] + (-[n+1] + k) := by
-    simp [add_succ]
+    simp
     rw [Int.add_comm, subNatNat_add_negSucc]
-    simp [add_succ, succ_add, Nat.add_comm]
+    simp [Nat.add_comm, Nat.add_left_comm, Nat.add_assoc]
 
 protected theorem add_left_comm (a b c : Int) : a + (b + c) = b + (a + c) := by
   rw [← Int.add_assoc, Int.add_comm a, Int.add_assoc]
@@ -391,7 +390,7 @@ theorem ofNat_mul_subNatNat (m n k : Nat) :
     | inl h =>
       have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)
       simp [subNatNat_of_lt h, subNatNat_of_lt h']
-      rw [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h), ← neg_ofNat_succ, Nat.mul_sub_left_distrib,
+      rw [sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h), ← neg_ofNat_succ, Nat.mul_sub_left_distrib,
         ← succ_pred_eq_of_pos (Nat.sub_pos_of_lt h')]; rfl
     | inr h =>
       have h' : succ m * k ≤ succ m * n := Nat.mul_le_mul_left _ h
@@ -406,7 +405,7 @@ theorem negSucc_mul_subNatNat (m n k : Nat) :
   | inl h =>
     have h' : succ m * n < succ m * k := Nat.mul_lt_mul_of_pos_left h (Nat.succ_pos m)
     rw [subNatNat_of_lt h, subNatNat_of_le (Nat.le_of_lt h')]
-    simp [succ_pred_eq_of_pos (Nat.sub_pos_of_lt h), Nat.mul_sub_left_distrib]
+    simp [sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h), Nat.mul_sub_left_distrib]
   | inr h => cases Nat.lt_or_ge k n with
     | inl h' =>
       have h₁ : succ m * n > succ m * k := Nat.mul_lt_mul_of_pos_left h' (Nat.succ_pos m)
@@ -422,12 +421,12 @@ protected theorem mul_add : ∀ a b c : Int, a * (b + c) = a * b + a * c
     simp [negOfNat_eq_subNatNat_zero]; rw [← subNatNat_add]; rfl
   | (m:Nat), -[n+1],  (k:Nat) => by
     simp [negOfNat_eq_subNatNat_zero]; rw [Int.add_comm, ← subNatNat_add]; rfl
-  | (m:Nat), -[n+1],  -[k+1]  => by simp; rw [← Nat.left_distrib, succ_add]; rfl
+  | (m:Nat), -[n+1],  -[k+1]  => by simp [← Nat.left_distrib, Nat.add_left_comm, Nat.add_assoc]
   | -[m+1],  (n:Nat), (k:Nat) => by simp [Nat.mul_comm]; rw [← Nat.right_distrib, Nat.mul_comm]
   | -[m+1],  (n:Nat), -[k+1]  => by
     simp [negOfNat_eq_subNatNat_zero]; rw [Int.add_comm, ← subNatNat_add]; rfl
   | -[m+1],  -[n+1],  (k:Nat) => by simp [negOfNat_eq_subNatNat_zero]; rw [← subNatNat_add]; rfl
-  | -[m+1],  -[n+1],  -[k+1]  => by simp; rw [← Nat.left_distrib, succ_add]; rfl
+  | -[m+1],  -[n+1],  -[k+1]  => by simp [← Nat.left_distrib, Nat.add_left_comm, Nat.add_assoc]
 
 protected theorem add_mul (a b c : Int) : (a + b) * c = a * c + b * c := by
   simp [Int.mul_comm, Int.mul_add]
@@ -499,33 +498,6 @@ theorem eq_one_of_mul_eq_self_left {a b : Int} (Hpos : a ≠ 0) (H : b * a = a)
 theorem eq_one_of_mul_eq_self_right {a b : Int} (Hpos : b ≠ 0) (H : b * a = b) : a = 1 :=
   Int.eq_of_mul_eq_mul_left Hpos <| by rw [Int.mul_one, H]
 
-/-! # pow -/
-
-protected theorem pow_zero (b : Int) : b^0 = 1 := rfl
-
-protected theorem pow_succ (b : Int) (e : Nat) : b ^ (e+1) = (b ^ e) * b := rfl
-protected theorem pow_succ' (b : Int) (e : Nat) : b ^ (e+1) = b * (b ^ e) := by
-  rw [Int.mul_comm, Int.pow_succ]
-
-theorem pow_le_pow_of_le_left {n m : Nat} (h : n ≤ m) : ∀ (i : Nat), n^i ≤ m^i
-  | 0      => Nat.le_refl _
-  | succ i => Nat.mul_le_mul (pow_le_pow_of_le_left h i) h
-
-theorem pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} : ∀ {j}, i ≤ j → n^i ≤ n^j
-  | 0,      h =>
-    have : i = 0 := eq_zero_of_le_zero h
-    this.symm ▸ Nat.le_refl _
-  | succ j, h =>
-    match le_or_eq_of_le_succ h with
-    | Or.inl h => show n^i ≤ n^j * n from
-      have : n^i * 1 ≤ n^j * n := Nat.mul_le_mul (pow_le_pow_of_le_right hx h) hx
-      Nat.mul_one (n^i) ▸ this
-    | Or.inr h =>
-      h.symm ▸ Nat.le_refl _
-
-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 _)
-
 /-! NatCast lemmas -/
 
 /-!
@@ -545,10 +517,4 @@ theorem natCast_one : ((1 : Nat) : Int) = (1 : Int) := rfl
 @[simp] theorem natCast_mul (a b : Nat) : ((a * b : Nat) : Int) = (a : Int) * (b : Int) := by
   simp
 
-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]
-
 end Int

src/Init/Data/Int/Order.lean

@@ -6,6 +6,7 @@ Authors: Jeremy Avigad, Deniz Aydin, Floris van Doorn, Mario Carneiro
 prelude
 import Init.Data.Int.Lemmas
 import Init.ByCases
+import Init.RCases
 
 /-!
 # Results about the order properties of the integers, and the integers as an ordered ring.
@@ -498,3 +499,524 @@ theorem toNat_add_nat {a : Int} (ha : 0 ≤ a) (n : Nat) : (a + n).toNat = a.toN
 @[simp] theorem toNat_neg_nat : ∀ n : Nat, (-(n : Int)).toNat = 0
   | 0 => rfl
   | _+1 => rfl
+
+/-! ### toNat' -/
+
+theorem mem_toNat' : ∀ (a : Int) (n : Nat), toNat' a = some n ↔ a = n
+  | (m : Nat), n => by simp [toNat', Int.ofNat_inj]
+  | -[m+1], n => by constructor <;> nofun
+
+/-! ## Order properties of the integers -/
+
+protected theorem lt_of_not_ge {a b : Int} : ¬a ≤ b → b < a := Int.not_le.mp
+protected theorem not_le_of_gt {a b : Int} : b < a → ¬a ≤ b := Int.not_le.mpr
+
+protected theorem le_of_not_le {a b : Int} : ¬ a ≤ b → b ≤ a := (Int.le_total a b).resolve_left
+
+@[simp] theorem negSucc_not_pos (n : Nat) : 0 < -[n+1] ↔ False := by
+  simp only [Int.not_lt, iff_false]; constructor
+
+theorem eq_negSucc_of_lt_zero : ∀ {a : Int}, a < 0 → ∃ n : Nat, a = -[n+1]
+  | ofNat _, h => absurd h (Int.not_lt.2 (ofNat_zero_le _))
+  | -[n+1],  _ => ⟨n, rfl⟩
+
+protected theorem lt_of_add_lt_add_left {a b c : Int} (h : a + b < a + c) : b < c := by
+  have : -a + (a + b) < -a + (a + c) := Int.add_lt_add_left h _
+  simp [Int.neg_add_cancel_left] at this
+  assumption
+
+protected theorem lt_of_add_lt_add_right {a b c : Int} (h : a + b < c + b) : a < c :=
+  Int.lt_of_add_lt_add_left (a := b) <| by rwa [Int.add_comm b a, Int.add_comm b c]
+
+protected theorem add_lt_add_iff_left (a : Int) : a + b < a + c ↔ b < c :=
+  ⟨Int.lt_of_add_lt_add_left, (Int.add_lt_add_left · _)⟩
+
+protected theorem add_lt_add_iff_right (c : Int) : a + c < b + c ↔ a < b :=
+  ⟨Int.lt_of_add_lt_add_right, (Int.add_lt_add_right · _)⟩
+
+protected theorem add_lt_add {a b c d : Int} (h₁ : a < b) (h₂ : c < d) : a + c < b + d :=
+  Int.lt_trans (Int.add_lt_add_right h₁ c) (Int.add_lt_add_left h₂ b)
+
+protected theorem add_lt_add_of_le_of_lt {a b c d : Int} (h₁ : a ≤ b) (h₂ : c < d) :
+    a + c < b + d :=
+  Int.lt_of_le_of_lt (Int.add_le_add_right h₁ c) (Int.add_lt_add_left h₂ b)
+
+protected theorem add_lt_add_of_lt_of_le {a b c d : Int} (h₁ : a < b) (h₂ : c ≤ d) :
+    a + c < b + d :=
+  Int.lt_of_lt_of_le (Int.add_lt_add_right h₁ c) (Int.add_le_add_left h₂ b)
+
+protected theorem lt_add_of_pos_right (a : Int) {b : Int} (h : 0 < b) : a < a + b := by
+  have : a + 0 < a + b := Int.add_lt_add_left h a
+  rwa [Int.add_zero] at this
+
+protected theorem lt_add_of_pos_left (a : Int) {b : Int} (h : 0 < b) : a < b + a := by
+  have : 0 + a < b + a := Int.add_lt_add_right h a
+  rwa [Int.zero_add] at this
+
+protected theorem add_nonneg {a b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a + b :=
+  Int.zero_add 0 ▸ Int.add_le_add ha hb
+
+protected theorem add_pos {a b : Int} (ha : 0 < a) (hb : 0 < b) : 0 < a + b :=
+  Int.zero_add 0 ▸ Int.add_lt_add ha hb
+
+protected theorem add_pos_of_pos_of_nonneg {a b : Int} (ha : 0 < a) (hb : 0 ≤ b) : 0 < a + b :=
+  Int.zero_add 0 ▸ Int.add_lt_add_of_lt_of_le ha hb
+
+protected theorem add_pos_of_nonneg_of_pos {a b : Int} (ha : 0 ≤ a) (hb : 0 < b) : 0 < a + b :=
+  Int.zero_add 0 ▸ Int.add_lt_add_of_le_of_lt ha hb
+
+protected theorem add_nonpos {a b : Int} (ha : a ≤ 0) (hb : b ≤ 0) : a + b ≤ 0 :=
+  Int.zero_add 0 ▸ Int.add_le_add ha hb
+
+protected theorem add_neg {a b : Int} (ha : a < 0) (hb : b < 0) : a + b < 0 :=
+  Int.zero_add 0 ▸ Int.add_lt_add ha hb
+
+protected theorem add_neg_of_neg_of_nonpos {a b : Int} (ha : a < 0) (hb : b ≤ 0) : a + b < 0 :=
+  Int.zero_add 0 ▸ Int.add_lt_add_of_lt_of_le ha hb
+
+protected theorem add_neg_of_nonpos_of_neg {a b : Int} (ha : a ≤ 0) (hb : b < 0) : a + b < 0 :=
+  Int.zero_add 0 ▸ Int.add_lt_add_of_le_of_lt ha hb
+
+protected theorem lt_add_of_le_of_pos {a b c : Int} (hbc : b ≤ c) (ha : 0 < a) : b < c + a :=
+  Int.add_zero b ▸ Int.add_lt_add_of_le_of_lt hbc ha
+
+theorem add_one_le_iff {a b : Int} : a + 1 ≤ b ↔ a < b := .rfl
+
+theorem lt_add_one_iff {a b : Int} : a < b + 1 ↔ a ≤ b := Int.add_le_add_iff_right _
+
+@[simp] theorem succ_ofNat_pos (n : Nat) : 0 < (n : Int) + 1 :=
+  lt_add_one_iff.2 (ofNat_zero_le _)
+
+theorem le_add_one {a b : Int} (h : a ≤ b) : a ≤ b + 1 :=
+  Int.le_of_lt (Int.lt_add_one_iff.2 h)
+
+protected theorem nonneg_of_neg_nonpos {a : Int} (h : -a ≤ 0) : 0 ≤ a :=
+  Int.le_of_neg_le_neg <| by rwa [Int.neg_zero]
+
+protected theorem nonpos_of_neg_nonneg {a : Int} (h : 0 ≤ -a) : a ≤ 0 :=
+  Int.le_of_neg_le_neg <| by rwa [Int.neg_zero]
+
+protected theorem lt_of_neg_lt_neg {a b : Int} (h : -b < -a) : a < b :=
+  Int.neg_neg a ▸ Int.neg_neg b ▸ Int.neg_lt_neg h
+
+protected theorem pos_of_neg_neg {a : Int} (h : -a < 0) : 0 < a :=
+  Int.lt_of_neg_lt_neg <| by rwa [Int.neg_zero]
+
+protected theorem neg_of_neg_pos {a : Int} (h : 0 < -a) : a < 0 :=
+  have : -0 < -a := by rwa [Int.neg_zero]
+  Int.lt_of_neg_lt_neg this
+
+protected theorem le_neg_of_le_neg {a b : Int} (h : a ≤ -b) : b ≤ -a := by
+  have h := Int.neg_le_neg h
+  rwa [Int.neg_neg] at h
+
+protected theorem neg_le_of_neg_le {a b : Int} (h : -a ≤ b) : -b ≤ a := by
+  have h := Int.neg_le_neg h
+  rwa [Int.neg_neg] at h
+
+protected theorem lt_neg_of_lt_neg {a b : Int} (h : a < -b) : b < -a := by
+  have h := Int.neg_lt_neg h
+  rwa [Int.neg_neg] at h
+
+protected theorem neg_lt_of_neg_lt {a b : Int} (h : -a < b) : -b < a := by
+  have h := Int.neg_lt_neg h
+  rwa [Int.neg_neg] at h
+
+protected theorem sub_nonpos_of_le {a b : Int} (h : a ≤ b) : a - b ≤ 0 := by
+  have h := Int.add_le_add_right h (-b)
+  rwa [Int.add_right_neg] at h
+
+protected theorem le_of_sub_nonpos {a b : Int} (h : a - b ≤ 0) : a ≤ b := by
+  have h := Int.add_le_add_right h b
+  rwa [Int.sub_add_cancel, Int.zero_add] at h
+
+protected theorem sub_neg_of_lt {a b : Int} (h : a < b) : a - b < 0 := by
+  have h := Int.add_lt_add_right h (-b)
+  rwa [Int.add_right_neg] at h
+
+protected theorem lt_of_sub_neg {a b : Int} (h : a - b < 0) : a < b := by
+  have h := Int.add_lt_add_right h b
+  rwa [Int.sub_add_cancel, Int.zero_add] at h
+
+protected theorem add_le_of_le_neg_add {a b c : Int} (h : b ≤ -a + c) : a + b ≤ c := by
+  have h := Int.add_le_add_left h a
+  rwa [Int.add_neg_cancel_left] at h
+
+protected theorem le_neg_add_of_add_le {a b c : Int} (h : a + b ≤ c) : b ≤ -a + c := by
+  have h := Int.add_le_add_left h (-a)
+  rwa [Int.neg_add_cancel_left] at h
+
+protected theorem add_le_of_le_sub_left {a b c : Int} (h : b ≤ c - a) : a + b ≤ c := by
+  have h := Int.add_le_add_left h a
+  rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h
+
+protected theorem le_sub_left_of_add_le {a b c : Int} (h : a + b ≤ c) : b ≤ c - a := by
+  have h := Int.add_le_add_right h (-a)
+  rwa [Int.add_comm a b, Int.add_neg_cancel_right] at h
+
+protected theorem add_le_of_le_sub_right {a b c : Int} (h : a ≤ c - b) : a + b ≤ c := by
+  have h := Int.add_le_add_right h b
+  rwa [Int.sub_add_cancel] at h
+
+protected theorem le_sub_right_of_add_le {a b c : Int} (h : a + b ≤ c) : a ≤ c - b := by
+  have h := Int.add_le_add_right h (-b)
+  rwa [Int.add_neg_cancel_right] at h
+
+protected theorem le_add_of_neg_add_le {a b c : Int} (h : -b + a ≤ c) : a ≤ b + c := by
+  have h := Int.add_le_add_left h b
+  rwa [Int.add_neg_cancel_left] at h
+
+protected theorem neg_add_le_of_le_add {a b c : Int} (h : a ≤ b + c) : -b + a ≤ c := by
+  have h := Int.add_le_add_left h (-b)
+  rwa [Int.neg_add_cancel_left] at h
+
+protected theorem le_add_of_sub_left_le {a b c : Int} (h : a - b ≤ c) : a ≤ b + c := by
+  have h := Int.add_le_add_right h b
+  rwa [Int.sub_add_cancel, Int.add_comm] at h
+
+protected theorem le_add_of_sub_right_le {a b c : Int} (h : a - c ≤ b) : a ≤ b + c := by
+  have h := Int.add_le_add_right h c
+  rwa [Int.sub_add_cancel] at h
+
+protected theorem sub_right_le_of_le_add {a b c : Int} (h : a ≤ b + c) : a - c ≤ b := by
+  have h := Int.add_le_add_right h (-c)
+  rwa [Int.add_neg_cancel_right] at h
+
+protected theorem le_add_of_neg_add_le_left {a b c : Int} (h : -b + a ≤ c) : a ≤ b + c := by
+  rw [Int.add_comm] at h
+  exact Int.le_add_of_sub_left_le h
+
+protected theorem neg_add_le_left_of_le_add {a b c : Int} (h : a ≤ b + c) : -b + a ≤ c := by
+  rw [Int.add_comm]
+  exact Int.sub_left_le_of_le_add h
+
+protected theorem le_add_of_neg_add_le_right {a b c : Int} (h : -c + a ≤ b) : a ≤ b + c := by
+  rw [Int.add_comm] at h
+  exact Int.le_add_of_sub_right_le h
+
+protected theorem neg_add_le_right_of_le_add {a b c : Int} (h : a ≤ b + c) : -c + a ≤ b := by
+  rw [Int.add_comm] at h
+  exact Int.neg_add_le_left_of_le_add h
+
+protected theorem le_add_of_neg_le_sub_left {a b c : Int} (h : -a ≤ b - c) : c ≤ a + b :=
+  Int.le_add_of_neg_add_le_left (Int.add_le_of_le_sub_right h)
+
+protected theorem neg_le_sub_left_of_le_add {a b c : Int} (h : c ≤ a + b) : -a ≤ b - c := by
+  have h := Int.le_neg_add_of_add_le (Int.sub_left_le_of_le_add h)
+  rwa [Int.add_comm] at h
+
+protected theorem le_add_of_neg_le_sub_right {a b c : Int} (h : -b ≤ a - c) : c ≤ a + b :=
+  Int.le_add_of_sub_right_le (Int.add_le_of_le_sub_left h)
+
+protected theorem neg_le_sub_right_of_le_add {a b c : Int} (h : c ≤ a + b) : -b ≤ a - c :=
+  Int.le_sub_left_of_add_le (Int.sub_right_le_of_le_add h)
+
+protected theorem sub_le_of_sub_le {a b c : Int} (h : a - b ≤ c) : a - c ≤ b :=
+  Int.sub_left_le_of_le_add (Int.le_add_of_sub_right_le h)
+
+protected theorem sub_le_sub_left {a b : Int} (h : a ≤ b) (c : Int) : c - b ≤ c - a :=
+  Int.add_le_add_left (Int.neg_le_neg h) c
+
+protected theorem sub_le_sub_right {a b : Int} (h : a ≤ b) (c : Int) : a - c ≤ b - c :=
+  Int.add_le_add_right h (-c)
+
+protected theorem sub_le_sub {a b c d : Int} (hab : a ≤ b) (hcd : c ≤ d) : a - d ≤ b - c :=
+  Int.add_le_add hab (Int.neg_le_neg hcd)
+
+protected theorem add_lt_of_lt_neg_add {a b c : Int} (h : b < -a + c) : a + b < c := by
+  have h := Int.add_lt_add_left h a
+  rwa [Int.add_neg_cancel_left] at h
+
+protected theorem lt_neg_add_of_add_lt {a b c : Int} (h : a + b < c) : b < -a + c := by
+  have h := Int.add_lt_add_left h (-a)
+  rwa [Int.neg_add_cancel_left] at h
+
+protected theorem add_lt_of_lt_sub_left {a b c : Int} (h : b < c - a) : a + b < c := by
+  have h := Int.add_lt_add_left h a
+  rwa [← Int.add_sub_assoc, Int.add_comm a c, Int.add_sub_cancel] at h
+
+protected theorem lt_sub_left_of_add_lt {a b c : Int} (h : a + b < c) : b < c - a := by
+  have h := Int.add_lt_add_right h (-a)
+  rwa [Int.add_comm a b, Int.add_neg_cancel_right] at h
+
+protected theorem add_lt_of_lt_sub_right {a b c : Int} (h : a < c - b) : a + b < c := by
+  have h := Int.add_lt_add_right h b
+  rwa [Int.sub_add_cancel] at h
+
+protected theorem lt_sub_right_of_add_lt {a b c : Int} (h : a + b < c) : a < c - b := by
+  have h := Int.add_lt_add_right h (-b)
+  rwa [Int.add_neg_cancel_right] at h
+
+protected theorem lt_add_of_neg_add_lt {a b c : Int} (h : -b + a < c) : a < b + c := by
+  have h := Int.add_lt_add_left h b
+  rwa [Int.add_neg_cancel_left] at h
+
+protected theorem neg_add_lt_of_lt_add {a b c : Int} (h : a < b + c) : -b + a < c := by
+  have h := Int.add_lt_add_left h (-b)
+  rwa [Int.neg_add_cancel_left] at h
+
+protected theorem lt_add_of_sub_left_lt {a b c : Int} (h : a - b < c) : a < b + c := by
+  have h := Int.add_lt_add_right h b
+  rwa [Int.sub_add_cancel, Int.add_comm] at h
+
+protected theorem sub_left_lt_of_lt_add {a b c : Int} (h : a < b + c) : a - b < c := by
+  have h := Int.add_lt_add_right h (-b)
+  rwa [Int.add_comm b c, Int.add_neg_cancel_right] at h
+
+protected theorem lt_add_of_sub_right_lt {a b c : Int} (h : a - c < b) : a < b + c := by
+  have h := Int.add_lt_add_right h c
+  rwa [Int.sub_add_cancel] at h
+
+protected theorem sub_right_lt_of_lt_add {a b c : Int} (h : a < b + c) : a - c < b := by
+  have h := Int.add_lt_add_right h (-c)
+  rwa [Int.add_neg_cancel_right] at h
+
+protected theorem lt_add_of_neg_add_lt_left {a b c : Int} (h : -b + a < c) : a < b + c := by
+  rw [Int.add_comm] at h
+  exact Int.lt_add_of_sub_left_lt h
+
+protected theorem neg_add_lt_left_of_lt_add {a b c : Int} (h : a < b + c) : -b + a < c := by
+  rw [Int.add_comm]
+  exact Int.sub_left_lt_of_lt_add h
+
+protected theorem lt_add_of_neg_add_lt_right {a b c : Int} (h : -c + a < b) : a < b + c := by
+  rw [Int.add_comm] at h
+  exact Int.lt_add_of_sub_right_lt h
+
+protected theorem neg_add_lt_right_of_lt_add {a b c : Int} (h : a < b + c) : -c + a < b := by
+  rw [Int.add_comm] at h
+  exact Int.neg_add_lt_left_of_lt_add h
+
+protected theorem lt_add_of_neg_lt_sub_left {a b c : Int} (h : -a < b - c) : c < a + b :=
+  Int.lt_add_of_neg_add_lt_left (Int.add_lt_of_lt_sub_right h)
+
+protected theorem neg_lt_sub_left_of_lt_add {a b c : Int} (h : c < a + b) : -a < b - c := by
+  have h := Int.lt_neg_add_of_add_lt (Int.sub_left_lt_of_lt_add h)
+  rwa [Int.add_comm] at h
+
+protected theorem lt_add_of_neg_lt_sub_right {a b c : Int} (h : -b < a - c) : c < a + b :=
+  Int.lt_add_of_sub_right_lt (Int.add_lt_of_lt_sub_left h)
+
+protected theorem neg_lt_sub_right_of_lt_add {a b c : Int} (h : c < a + b) : -b < a - c :=
+  Int.lt_sub_left_of_add_lt (Int.sub_right_lt_of_lt_add h)
+
+protected theorem sub_lt_of_sub_lt {a b c : Int} (h : a - b < c) : a - c < b :=
+  Int.sub_left_lt_of_lt_add (Int.lt_add_of_sub_right_lt h)
+
+protected theorem sub_lt_sub_left {a b : Int} (h : a < b) (c : Int) : c - b < c - a :=
+  Int.add_lt_add_left (Int.neg_lt_neg h) c
+
+protected theorem sub_lt_sub_right {a b : Int} (h : a < b) (c : Int) : a - c < b - c :=
+  Int.add_lt_add_right h (-c)
+
+protected theorem sub_lt_sub {a b c d : Int} (hab : a < b) (hcd : c < d) : a - d < b - c :=
+  Int.add_lt_add hab (Int.neg_lt_neg hcd)
+
+protected theorem sub_lt_sub_of_le_of_lt {a b c d : Int}
+  (hab : a ≤ b) (hcd : c < d) : a - d < b - c :=
+  Int.add_lt_add_of_le_of_lt hab (Int.neg_lt_neg hcd)
+
+protected theorem sub_lt_sub_of_lt_of_le {a b c d : Int}
+  (hab : a < b) (hcd : c ≤ d) : a - d < b - c :=
+  Int.add_lt_add_of_lt_of_le hab (Int.neg_le_neg hcd)
+
+protected theorem add_le_add_three {a b c d e f : Int}
+    (h₁ : a ≤ d) (h₂ : b ≤ e) (h₃ : c ≤ f) : a + b + c ≤ d + e + f :=
+  Int.add_le_add (Int.add_le_add h₁ h₂) h₃
+
+theorem exists_eq_neg_ofNat {a : Int} (H : a ≤ 0) : ∃ n : Nat, a = -(n : Int) :=
+  let ⟨n, h⟩ := eq_ofNat_of_zero_le (Int.neg_nonneg_of_nonpos H)
+  ⟨n, Int.eq_neg_of_eq_neg h.symm⟩
+
+theorem lt_of_add_one_le {a b : Int} (H : a + 1 ≤ b) : a < b := H
+
+theorem lt_add_one_of_le {a b : Int} (H : a ≤ b) : a < b + 1 := Int.add_le_add_right H 1
+
+theorem le_of_lt_add_one {a b : Int} (H : a < b + 1) : a ≤ b := Int.le_of_add_le_add_right H
+
+theorem sub_one_lt_of_le {a b : Int} (H : a ≤ b) : a - 1 < b :=
+  Int.sub_right_lt_of_lt_add <| lt_add_one_of_le H
+
+theorem le_of_sub_one_lt {a b : Int} (H : a - 1 < b) : a ≤ b :=
+  le_of_lt_add_one <| Int.lt_add_of_sub_right_lt H
+
+theorem le_sub_one_of_lt {a b : Int} (H : a < b) : a ≤ b - 1 := Int.le_sub_right_of_add_le H
+
+theorem lt_of_le_sub_one {a b : Int} (H : a ≤ b - 1) : a < b := Int.add_le_of_le_sub_right H
+
+/- ### Order properties and multiplication -/
+
+protected theorem mul_lt_mul {a b c d : Int}
+    (h₁ : a < c) (h₂ : b ≤ d) (h₃ : 0 < b) (h₄ : 0 ≤ c) : a * b < c * d :=
+  Int.lt_of_lt_of_le (Int.mul_lt_mul_of_pos_right h₁ h₃) (Int.mul_le_mul_of_nonneg_left h₂ h₄)
+
+protected theorem mul_lt_mul' {a b c d : Int}
+    (h₁ : a ≤ c) (h₂ : b < d) (h₃ : 0 ≤ b) (h₄ : 0 < c) : a * b < c * d :=
+  Int.lt_of_le_of_lt (Int.mul_le_mul_of_nonneg_right h₁ h₃) (Int.mul_lt_mul_of_pos_left h₂ h₄)
+
+protected theorem mul_neg_of_pos_of_neg {a b : Int} (ha : 0 < a) (hb : b < 0) : a * b < 0 := by
+  have h : a * b < a * 0 := Int.mul_lt_mul_of_pos_left hb ha
+  rwa [Int.mul_zero] at h
+
+protected theorem mul_neg_of_neg_of_pos {a b : Int} (ha : a < 0) (hb : 0 < b) : a * b < 0 := by
+  have h : a * b < 0 * b := Int.mul_lt_mul_of_pos_right ha hb
+  rwa [Int.zero_mul] at h
+
+protected theorem mul_nonneg_of_nonpos_of_nonpos {a b : Int}
+  (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a * b := by
+  have : 0 * b ≤ a * b := Int.mul_le_mul_of_nonpos_right ha hb
+  rwa [Int.zero_mul] at this
+
+protected theorem mul_lt_mul_of_neg_left {a b c : Int} (h : b < a) (hc : c < 0) : c * a < c * b :=
+  have : -c > 0 := Int.neg_pos_of_neg hc
+  have : -c * b < -c * a := Int.mul_lt_mul_of_pos_left h this
+  have : -(c * b) < -(c * a) := by
+    rwa [← Int.neg_mul_eq_neg_mul, ← Int.neg_mul_eq_neg_mul] at this
+  Int.lt_of_neg_lt_neg this
+
+protected theorem mul_lt_mul_of_neg_right {a b c : Int} (h : b < a) (hc : c < 0) : a * c < b * c :=
+  have : -c > 0 := Int.neg_pos_of_neg hc
+  have : b * -c < a * -c := Int.mul_lt_mul_of_pos_right h this
+  have : -(b * c) < -(a * c) := by
+    rwa [← Int.neg_mul_eq_mul_neg, ← Int.neg_mul_eq_mul_neg] at this
+  Int.lt_of_neg_lt_neg this
+
+protected theorem mul_pos_of_neg_of_neg {a b : Int} (ha : a < 0) (hb : b < 0) : 0 < a * b := by
+  have : 0 * b < a * b := Int.mul_lt_mul_of_neg_right ha hb
+  rwa [Int.zero_mul] at this
+
+protected theorem mul_self_le_mul_self {a b : Int} (h1 : 0 ≤ a) (h2 : a ≤ b) : a * a ≤ b * b :=
+  Int.mul_le_mul h2 h2 h1 (Int.le_trans h1 h2)
+
+protected theorem mul_self_lt_mul_self {a b : Int} (h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b :=
+  Int.mul_lt_mul' (Int.le_of_lt h2) h2 h1 (Int.lt_of_le_of_lt h1 h2)
+
+/- ## sign -/
+
+@[simp] theorem sign_zero : sign 0 = 0 := rfl
+@[simp] theorem sign_one : sign 1 = 1 := rfl
+theorem sign_neg_one : sign (-1) = -1 := rfl
+
+@[simp] theorem sign_of_add_one (x : Nat) : Int.sign (x + 1) = 1 := rfl
+@[simp] theorem sign_negSucc (x : Nat) : Int.sign (Int.negSucc x) = -1 := rfl
+
+theorem natAbs_sign (z : Int) : z.sign.natAbs = if z = 0 then 0 else 1 :=
+  match z with | 0 | succ _ | -[_+1] => rfl
+
+theorem natAbs_sign_of_nonzero {z : Int} (hz : z ≠ 0) : z.sign.natAbs = 1 := by
+  rw [Int.natAbs_sign, if_neg hz]
+
+theorem sign_ofNat_of_nonzero {n : Nat} (hn : n ≠ 0) : Int.sign n = 1 :=
+  match n, Nat.exists_eq_succ_of_ne_zero hn with
+  | _, ⟨n, rfl⟩ => Int.sign_of_add_one n
+
+@[simp] theorem sign_neg (z : Int) : Int.sign (-z) = -Int.sign z := by
+  match z with | 0 | succ _ | -[_+1] => rfl
+
+theorem sign_mul_natAbs : ∀ a : Int, sign a * natAbs a = a
+  | 0      => rfl
+  | succ _ => Int.one_mul _
+  | -[_+1] => (Int.neg_eq_neg_one_mul _).symm
+
+@[simp] theorem sign_mul : ∀ a b, sign (a * b) = sign a * sign b
+  | a, 0 | 0, b => by simp [Int.mul_zero, Int.zero_mul]
+  | succ _, succ _ | succ _, -[_+1] | -[_+1], succ _ | -[_+1], -[_+1] => rfl
+
+theorem sign_eq_one_of_pos {a : Int} (h : 0 < a) : sign a = 1 :=
+  match a, eq_succ_of_zero_lt h with
+  | _, ⟨_, rfl⟩ => rfl
+
+theorem sign_eq_neg_one_of_neg {a : Int} (h : a < 0) : sign a = -1 :=
+  match a, eq_negSucc_of_lt_zero h with
+  | _, ⟨_, rfl⟩ => rfl
+
+theorem eq_zero_of_sign_eq_zero : ∀ {a : Int}, sign a = 0 → a = 0
+  | 0, _ => rfl
+
+theorem pos_of_sign_eq_one : ∀ {a : Int}, sign a = 1 → 0 < a
+  | (_ + 1 : Nat), _ => ofNat_lt.2 (Nat.succ_pos _)
+
+theorem neg_of_sign_eq_neg_one : ∀ {a : Int}, sign a = -1 → a < 0
+  | (_ + 1 : Nat), h => nomatch h
+  | 0, h => nomatch h
+  | -[_+1], _ => negSucc_lt_zero _
+
+theorem sign_eq_one_iff_pos (a : Int) : sign a = 1 ↔ 0 < a :=
+  ⟨pos_of_sign_eq_one, sign_eq_one_of_pos⟩
+
+theorem sign_eq_neg_one_iff_neg (a : Int) : sign a = -1 ↔ a < 0 :=
+  ⟨neg_of_sign_eq_neg_one, sign_eq_neg_one_of_neg⟩
+
+@[simp] theorem sign_eq_zero_iff_zero (a : Int) : sign a = 0 ↔ a = 0 :=
+  ⟨eq_zero_of_sign_eq_zero, fun h => by rw [h, sign_zero]⟩
+
+@[simp] theorem sign_sign : sign (sign x) = sign x := by
+  match x with
+  | 0 => rfl
+  | .ofNat (_ + 1) => rfl
+  | .negSucc _ => rfl
+
+@[simp] theorem sign_nonneg : 0 ≤ sign x ↔ 0 ≤ x := by
+  match x with
+  | 0 => rfl
+  | .ofNat (_ + 1) =>
+    simp (config := { decide := true }) only [sign, true_iff]
+    exact Int.le_add_one (ofNat_nonneg _)
+  | .negSucc _ => simp (config := { decide := true }) [sign]
+
+theorem mul_sign : ∀ i : Int, i * sign i = natAbs i
+  | succ _ => Int.mul_one _
+  | 0 => Int.mul_zero _
+  | -[_+1] => Int.mul_neg_one _
+
+/- ## natAbs -/
+
+theorem natAbs_ne_zero {a : Int} : a.natAbs ≠ 0 ↔ a ≠ 0 := not_congr Int.natAbs_eq_zero
+
+theorem natAbs_mul_self : ∀ {a : Int}, ↑(natAbs a * natAbs a) = a * a
+  | ofNat _ => rfl
+  | -[_+1]  => rfl
+
+theorem eq_nat_or_neg (a : Int) : ∃ n : Nat, a = n ∨ a = -↑n := ⟨_, natAbs_eq a⟩
+
+theorem natAbs_mul_natAbs_eq {a b : Int} {c : Nat}
+    (h : a * b = (c : Int)) : a.natAbs * b.natAbs = c := by rw [← natAbs_mul, h, natAbs]
+
+@[simp] theorem natAbs_mul_self' (a : Int) : (natAbs a * natAbs a : Int) = a * a := by
+  rw [← Int.ofNat_mul, natAbs_mul_self]
+
+theorem natAbs_eq_iff {a : Int} {n : Nat} : a.natAbs = n ↔ a = n ∨ a = -↑n := by
+  rw [← Int.natAbs_eq_natAbs_iff, Int.natAbs_ofNat]
+
+theorem natAbs_add_le (a b : Int) : natAbs (a + b) ≤ natAbs a + natAbs b := by
+  suffices ∀ a b : Nat, natAbs (subNatNat a b.succ) ≤ (a + b).succ by
+    match a, b with
+    | (a:Nat), (b:Nat) => rw [ofNat_add_ofNat, natAbs_ofNat]; apply Nat.le_refl
+    | (a:Nat), -[b+1]  => rw [natAbs_ofNat, natAbs_negSucc]; apply this
+    | -[a+1],  (b:Nat) =>
+      rw [natAbs_negSucc, natAbs_ofNat, Nat.succ_add, Nat.add_comm a b]; apply this
+    | -[a+1],  -[b+1]  => rw [natAbs_negSucc, succ_add]; apply Nat.le_refl
+  refine fun a b => subNatNat_elim a b.succ
+    (fun m n i => n = b.succ → natAbs i ≤ (m + b).succ) ?_
+    (fun i n (e : (n + i).succ = _) => ?_) rfl
+  · rintro i n rfl
+    rw [Nat.add_comm _ i, Nat.add_assoc]
+    exact Nat.le_add_right i (b.succ + b).succ
+  · apply succ_le_succ
+    rw [← succ.inj e, ← Nat.add_assoc, Nat.add_comm]
+    apply Nat.le_add_right
+
+theorem natAbs_sub_le (a b : Int) : natAbs (a - b) ≤ natAbs a + natAbs b := by
+  rw [← Int.natAbs_neg b]; apply natAbs_add_le
+
+theorem negSucc_eq' (m : Nat) : -[m+1] = -m - 1 := by simp only [negSucc_eq, Int.neg_add]; rfl
+
+theorem natAbs_lt_natAbs_of_nonneg_of_lt {a b : Int}
+    (w₁ : 0 ≤ a) (w₂ : a < b) : a.natAbs < b.natAbs :=
+  match a, b, eq_ofNat_of_zero_le w₁, eq_ofNat_of_zero_le (Int.le_trans w₁ (Int.le_of_lt w₂)) with
+  | _, _, ⟨_, rfl⟩, ⟨_, rfl⟩ => ofNat_lt.1 w₂
+
+theorem eq_natAbs_iff_mul_eq_zero : natAbs a = n ↔ (a - n) * (a + n) = 0 := by
+  rw [natAbs_eq_iff, Int.mul_eq_zero, ← Int.sub_neg, Int.sub_eq_zero, Int.sub_eq_zero]
+
+end Int

src/Init/Data/Int/Pow.lean

@@ -0,0 +1,44 @@
+/-
+Copyright (c) 2016 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Deniz Aydin, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.Int.Lemmas
+
+namespace Int
+
+/-! # pow -/
+
+protected theorem pow_zero (b : Int) : b^0 = 1 := rfl
+
+protected theorem pow_succ (b : Int) (e : Nat) : b ^ (e+1) = (b ^ e) * b := rfl
+protected theorem pow_succ' (b : Int) (e : Nat) : b ^ (e+1) = b * (b ^ e) := by
+  rw [Int.mul_comm, Int.pow_succ]
+
+theorem pow_le_pow_of_le_left {n m : Nat} (h : n ≤ m) : ∀ (i : Nat), n^i ≤ m^i
+  | 0      => Nat.le_refl _
+  | i + 1 => Nat.mul_le_mul (pow_le_pow_of_le_left h i) h
+
+theorem pow_le_pow_of_le_right {n : Nat} (hx : n > 0) {i : Nat} : ∀ {j}, i ≤ j → n^i ≤ n^j
+  | 0,      h =>
+    have : i = 0 := Nat.eq_zero_of_le_zero h
+    this.symm ▸ Nat.le_refl _
+  | j + 1, h =>
+    match Nat.le_or_eq_of_le_succ h with
+    | Or.inl h => show n^i ≤ n^j * n from
+      have : n^i * 1 ≤ n^j * n := Nat.mul_le_mul (pow_le_pow_of_le_right hx h) hx
+      Nat.mul_one (n^i) ▸ this
+    | Or.inr h =>
+      h.symm ▸ Nat.le_refl _
+
+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 _)
+
+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]
+
+end Int

src/Init/Data/List/Basic.lean

@@ -458,7 +458,7 @@ contains the longest initial segment for which `p` returns true
 and the second part is everything else.
 
 * `span (· > 5) [6, 8, 9, 5, 2, 9] = ([6, 8, 9], [5, 2, 9])`
-* `span (· > 10) [6, 8, 9, 5, 2, 9] = ([6, 8, 9, 5, 2, 9], [])`
+* `span (· > 10) [6, 8, 9, 5, 2, 9] = ([], [6, 8, 9, 5, 2, 9])`
 -/
 @[inline] def span (p : α → Bool) (as : List α) : List α × List α :=
   loop as []

src/Init/Data/List/BasicAux.lean

@@ -5,9 +5,6 @@ Author: Leonardo de Moura
 -/
 prelude
 import Init.Data.Nat.Linear
-import Init.Data.Array.Basic
-import Init.Data.List.Basic
-import Init.Util
 
 universe u
 

src/Init/Data/List/Lemmas.lean

@@ -6,9 +6,8 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 prelude
 import Init.Data.List.BasicAux
 import Init.Data.List.Control
-import Init.Data.Nat.Lemmas
 import Init.PropLemmas
-import Init.Control.Lawful
+import Init.Control.Lawful.Basic
 import Init.Hints
 
 namespace List
@@ -69,7 +68,7 @@ theorem mem_cons_self (a : α) (l : List α) : a ∈ a :: l := .head ..
 theorem mem_cons_of_mem (y : α) {a : α} {l : List α} : a ∈ l → a ∈ y :: l := .tail _
 
 theorem eq_nil_iff_forall_not_mem {l : List α} : l = [] ↔ ∀ a, a ∉ l := by
-  cases l <;> simp
+  cases l <;> simp [-not_or]
 
 /-! ### append -/
 
@@ -451,9 +450,9 @@ theorem mem_filter : x ∈ filter p as ↔ x ∈ as ∧ p x := by
   induction as with
   | nil => simp [filter]
   | cons a as ih =>
-    by_cases h : p a <;> simp [*, or_and_right]
-    · exact or_congr_left (and_iff_left_of_imp fun | rfl => h).symm
-    · exact (or_iff_right fun ⟨rfl, h'⟩ => h h').symm
+    by_cases h : p a
+    · simp_all [or_and_left]
+    · simp_all [or_and_right]
 
 theorem filter_eq_nil {l} : filter p l = [] ↔ ∀ a, a ∈ l → ¬p a := by
   simp only [eq_nil_iff_forall_not_mem, mem_filter, not_and]

src/Init/Data/Nat.lean

@@ -17,3 +17,5 @@ import Init.Data.Nat.Linear
 import Init.Data.Nat.SOM
 import Init.Data.Nat.Lemmas
 import Init.Data.Nat.Mod
+import Init.Data.Nat.Lcm
+import Init.Data.Nat.Compare

src/Init/Data/Nat/Basic.lean

@@ -10,6 +10,29 @@ universe u
 
 namespace Nat
 
+/-- Compiled version of `Nat.rec` so that we can define `Nat.recAux` to be defeq to `Nat.rec`.
+This is working around the fact that the compiler does not currently support recursors. -/
+private def recCompiled {motive : Nat → Sort u} (zero : motive zero) (succ : (n : Nat) → motive n → motive (Nat.succ n)) : (t : Nat) → motive t
+  | .zero => zero
+  | .succ n => succ n (recCompiled zero succ n)
+
+@[csimp]
+private theorem rec_eq_recCompiled : @Nat.rec = @Nat.recCompiled :=
+  funext fun _ => funext fun _ => funext fun succ => funext fun t =>
+    Nat.recOn t rfl (fun n ih => congrArg (succ n) ih)
+
+/-- Recursor identical to `Nat.rec` but uses notations `0` for `Nat.zero` and `· + 1` for `Nat.succ`.
+Used as the default `Nat` eliminator by the `induction` tactic. -/
+@[elab_as_elim, induction_eliminator]
+protected abbrev recAux {motive : Nat → Sort u} (zero : motive 0) (succ : (n : Nat) → motive n → motive (n + 1)) (t : Nat) : motive t :=
+  Nat.rec zero succ t
+
+/-- Recursor identical to `Nat.casesOn` but uses notations `0` for `Nat.zero` and `· + 1` for `Nat.succ`.
+Used as the default `Nat` eliminator by the `cases` tactic. -/
+@[elab_as_elim, cases_eliminator]
+protected abbrev casesAuxOn {motive : Nat → Sort u} (t : Nat) (zero : motive 0) (succ : (n : Nat) → motive (n + 1)) : motive t :=
+  Nat.casesOn t zero succ
+
 /--
 `Nat.fold` evaluates `f` on the numbers up to `n` exclusive, in increasing order:
 * `Nat.fold f 3 init = init |> f 0 |> f 1 |> f 2`
@@ -125,9 +148,12 @@ theorem add_succ (n m : Nat) : n + succ m = succ (n + m) :=
 theorem add_one (n : Nat) : n + 1 = succ n :=
   rfl
 
-theorem succ_eq_add_one (n : Nat) : succ n = n + 1 :=
+@[simp] theorem succ_eq_add_one (n : Nat) : succ n = n + 1 :=
   rfl
 
+@[simp] theorem add_one_ne_zero (n : Nat) : n + 1 ≠ 0 := nofun
+@[simp] theorem zero_ne_add_one (n : Nat) : 0 ≠ n + 1 := nofun
+
 protected theorem add_comm : ∀ (n m : Nat), n + m = m + n
   | n, 0   => Eq.symm (Nat.zero_add n)
   | n, m+1 => by
@@ -209,6 +235,9 @@ protected theorem mul_assoc : ∀ (n m k : Nat), (n * m) * k = n * (m * k)
 protected theorem mul_left_comm (n m k : Nat) : n * (m * k) = m * (n * k) := by
   rw [← Nat.mul_assoc, Nat.mul_comm n m, Nat.mul_assoc]
 
+protected theorem mul_two (n) : n * 2 = n + n := by rw [Nat.mul_succ, Nat.mul_one]
+protected theorem two_mul (n) : 2 * n = n + n := by rw [Nat.succ_mul, Nat.one_mul]
+
 /-! # Inequalities -/
 
 attribute [simp] Nat.le_refl
@@ -224,7 +253,7 @@ theorem lt_succ_of_le {n m : Nat} : n ≤ m → n < succ m := succ_le_succ
   | zero      => exact rfl
   | succ m ih => apply congrArg pred ih
 
-theorem pred_le : ∀ (n : Nat), pred n ≤ n
+@[simp] theorem pred_le : ∀ (n : Nat), pred n ≤ n
   | zero   => Nat.le.refl
   | succ _ => le_succ _
 
@@ -257,7 +286,7 @@ theorem succ_sub_succ (n m : Nat) : succ n - succ m = n - m :=
 theorem sub_add_eq (a b c : Nat) : a - (b + c) = a - b - c := by
   induction c with
   | zero => simp
-  | succ c ih => simp [Nat.add_succ, Nat.sub_succ, ih]
+  | succ c ih => simp only [Nat.add_succ, Nat.sub_succ, ih]
 
 protected theorem lt_of_lt_of_le {n m k : Nat} : n < m → m ≤ k → n < k :=
   Nat.le_trans
@@ -298,7 +327,8 @@ theorem eq_zero_or_pos : ∀ (n : Nat), n = 0 ∨ n > 0
 protected theorem pos_of_ne_zero {n : Nat} : n ≠ 0 → 0 < n := (eq_zero_or_pos n).resolve_left
 
 theorem lt.base (n : Nat) : n < succ n := Nat.le_refl (succ n)
-theorem lt_succ_self (n : Nat) : n < succ n := lt.base n
+
+@[simp] theorem lt_succ_self (n : Nat) : n < succ n := lt.base n
 
 protected theorem le_total (m n : Nat) : m ≤ n ∨ n ≤ m :=
   match Nat.lt_or_ge m n with
@@ -337,6 +367,12 @@ theorem le_add_right : ∀ (n k : Nat), n ≤ n + k
 theorem le_add_left (n m : Nat): n ≤ m + n :=
   Nat.add_comm n m ▸ le_add_right n m
 
+protected theorem lt_add_left (c : Nat) (h : a < b) : a < c + b :=
+  Nat.lt_of_lt_of_le h (Nat.le_add_left ..)
+
+protected theorem lt_add_right (c : Nat) (h : a < b) : a < b + c :=
+  Nat.lt_of_lt_of_le h (Nat.le_add_right ..)
+
 theorem le.dest : ∀ {n m : Nat}, n ≤ m → Exists (fun k => n + k = m)
   | zero,   zero,   _ => ⟨0, rfl⟩
   | zero,   succ n, _ => ⟨succ n, Nat.add_comm 0 (succ n) ▸ rfl⟩
@@ -426,6 +462,9 @@ protected theorem add_lt_add_left {n m : Nat} (h : n < m) (k : Nat) : k + n < k
 protected theorem add_lt_add_right {n m : Nat} (h : n < m) (k : Nat) : n + k < m + k :=
   Nat.add_comm k m ▸ Nat.add_comm k n ▸ Nat.add_lt_add_left h k
 
+protected theorem lt_add_of_pos_right (h : 0 < k) : n < n + k :=
+  Nat.add_lt_add_left h n
+
 protected theorem zero_lt_one : 0 < (1:Nat) :=
   zero_lt_succ 0
 
@@ -451,6 +490,137 @@ protected theorem le_of_add_le_add_right {a b c : Nat} : a + b ≤ c + b → a
 protected theorem add_le_add_iff_right {n : Nat} : m + n ≤ k + n ↔ m ≤ k :=
   ⟨Nat.le_of_add_le_add_right, fun h => Nat.add_le_add_right h _⟩
 
+/-! ### le/lt -/
+
+protected theorem lt_asymm {a b : Nat} (h : a < b) : ¬ b < a := Nat.not_lt.2 (Nat.le_of_lt h)
+/-- Alias for `Nat.lt_asymm`. -/
+protected abbrev not_lt_of_gt := @Nat.lt_asymm
+/-- Alias for `Nat.lt_asymm`. -/
+protected abbrev not_lt_of_lt := @Nat.lt_asymm
+
+protected theorem lt_iff_le_not_le {m n : Nat} : m < n ↔ m ≤ n ∧ ¬ n ≤ m :=
+  ⟨fun h => ⟨Nat.le_of_lt h, Nat.not_le_of_gt h⟩, fun ⟨_, h⟩ => Nat.lt_of_not_ge h⟩
+/-- Alias for `Nat.lt_iff_le_not_le`. -/
+protected abbrev lt_iff_le_and_not_ge := @Nat.lt_iff_le_not_le
+
+protected theorem lt_iff_le_and_ne {m n : Nat} : m < n ↔ m ≤ n ∧ m ≠ n :=
+  ⟨fun h => ⟨Nat.le_of_lt h, Nat.ne_of_lt h⟩, fun h => Nat.lt_of_le_of_ne h.1 h.2⟩
+
+protected theorem ne_iff_lt_or_gt {a b : Nat} : a ≠ b ↔ a < b ∨ b < a :=
+  ⟨Nat.lt_or_gt_of_ne, fun | .inl h => Nat.ne_of_lt h | .inr h => Nat.ne_of_gt h⟩
+/-- Alias for `Nat.ne_iff_lt_or_gt`. -/
+protected abbrev lt_or_gt := @Nat.ne_iff_lt_or_gt
+
+/-- Alias for `Nat.le_total`. -/
+protected abbrev le_or_ge := @Nat.le_total
+/-- Alias for `Nat.le_total`. -/
+protected abbrev le_or_le := @Nat.le_total
+
+protected theorem eq_or_lt_of_not_lt {a b : Nat} (hnlt : ¬ a < b) : a = b ∨ b < a :=
+  (Nat.lt_trichotomy ..).resolve_left hnlt
+
+protected theorem lt_or_eq_of_le {n m : Nat} (h : n ≤ m) : n < m ∨ n = m :=
+  (Nat.lt_or_ge ..).imp_right (Nat.le_antisymm h)
+
+protected theorem le_iff_lt_or_eq {n m : Nat} : n ≤ m ↔ n < m ∨ n = m :=
+  ⟨Nat.lt_or_eq_of_le, fun | .inl h => Nat.le_of_lt h | .inr rfl => Nat.le_refl _⟩
+
+protected theorem lt_succ_iff : m < succ n ↔ m ≤ n := ⟨le_of_lt_succ, lt_succ_of_le⟩
+
+protected theorem lt_succ_iff_lt_or_eq : m < succ n ↔ m < n ∨ m = n :=
+  Nat.lt_succ_iff.trans Nat.le_iff_lt_or_eq
+
+protected theorem eq_of_lt_succ_of_not_lt (hmn : m < n + 1) (h : ¬ m < n) : m = n :=
+  (Nat.lt_succ_iff_lt_or_eq.1 hmn).resolve_left h
+
+protected theorem eq_of_le_of_lt_succ (h₁ : n ≤ m) (h₂ : m < n + 1) : m = n :=
+  Nat.le_antisymm (le_of_succ_le_succ h₂) h₁
+
+
+/-! ## zero/one/two -/
+
+theorem le_zero : i ≤ 0 ↔ i = 0 := ⟨Nat.eq_zero_of_le_zero, fun | rfl => Nat.le_refl _⟩
+
+/-- Alias for `Nat.zero_lt_one`. -/
+protected abbrev one_pos := @Nat.zero_lt_one
+
+protected theorem two_pos : 0 < 2 := Nat.zero_lt_succ _
+
+protected theorem ne_zero_iff_zero_lt : n ≠ 0 ↔ 0 < n := Nat.pos_iff_ne_zero.symm
+
+protected theorem zero_lt_two : 0 < 2 := Nat.zero_lt_succ _
+
+protected theorem one_lt_two : 1 < 2 := Nat.succ_lt_succ Nat.zero_lt_one
+
+protected theorem eq_zero_of_not_pos (h : ¬0 < n) : n = 0 :=
+  Nat.eq_zero_of_le_zero (Nat.not_lt.1 h)
+
+/-! ## succ/pred -/
+
+attribute [simp] zero_lt_succ
+
+theorem succ_ne_self (n) : succ n ≠ n := Nat.ne_of_gt (lt_succ_self n)
+
+theorem succ_le : succ n ≤ m ↔ n < m := .rfl
+
+theorem lt_succ : m < succ n ↔ m ≤ n := ⟨le_of_lt_succ, lt_succ_of_le⟩
+
+theorem lt_succ_of_lt (h : a < b) : a < succ b := le_succ_of_le h
+
+theorem succ_pred_eq_of_ne_zero : ∀ {n}, n ≠ 0 → succ (pred n) = n
+  | _+1, _ => rfl
+
+theorem eq_zero_or_eq_succ_pred : ∀ n, n = 0 ∨ n = succ (pred n)
+  | 0 => .inl rfl
+  | _+1 => .inr rfl
+
+theorem succ_inj' : succ a = succ b ↔ a = b := ⟨succ.inj, congrArg _⟩
+
+theorem succ_le_succ_iff : succ a ≤ succ b ↔ a ≤ b := ⟨le_of_succ_le_succ, succ_le_succ⟩
+
+theorem succ_lt_succ_iff : succ a < succ b ↔ a < b := ⟨lt_of_succ_lt_succ, succ_lt_succ⟩
+
+theorem pred_inj : ∀ {a b}, 0 < a → 0 < b → pred a = pred b → a = b
+  | _+1, _+1, _, _ => congrArg _
+
+theorem pred_ne_self : ∀ {a}, a ≠ 0 → pred a ≠ a
+  | _+1, _ => (succ_ne_self _).symm
+
+theorem pred_lt_self : ∀ {a}, 0 < a → pred a < a
+  | _+1, _ => lt_succ_self _
+
+theorem pred_lt_pred : ∀ {n m}, n ≠ 0 → n < m → pred n < pred m
+  | _+1, _+1, _, h => lt_of_succ_lt_succ h
+
+theorem pred_le_iff_le_succ : ∀ {n m}, pred n ≤ m ↔ n ≤ succ m
+  | 0, _ => ⟨fun _ => Nat.zero_le _, fun _ => Nat.zero_le _⟩
+  | _+1, _ => Nat.succ_le_succ_iff.symm
+
+theorem le_succ_of_pred_le : pred n ≤ m → n ≤ succ m := pred_le_iff_le_succ.1
+
+theorem pred_le_of_le_succ : n ≤ succ m → pred n ≤ m := pred_le_iff_le_succ.2
+
+theorem lt_pred_iff_succ_lt : ∀ {n m}, n < pred m ↔ succ n < m
+  | _, 0 => ⟨nofun, nofun⟩
+  | _, _+1 => Nat.succ_lt_succ_iff.symm
+
+theorem succ_lt_of_lt_pred : n < pred m → succ n < m := lt_pred_iff_succ_lt.1
+
+theorem lt_pred_of_succ_lt : succ n < m → n < pred m := lt_pred_iff_succ_lt.2
+
+theorem le_pred_iff_lt : ∀ {n m}, 0 < m → (n ≤ pred m ↔ n < m)
+  | 0, _+1, _ => ⟨fun _ => Nat.zero_lt_succ _, fun _ => Nat.zero_le _⟩
+  | _+1, _+1, _ => Nat.lt_pred_iff_succ_lt
+
+theorem le_pred_of_lt (h : n < m) : n ≤ pred m := (le_pred_iff_lt (Nat.zero_lt_of_lt h)).2 h
+
+theorem le_sub_one_of_lt : a < b → a ≤ b - 1 := Nat.le_pred_of_lt
+
+theorem lt_of_le_pred (h : 0 < m) : n ≤ pred m → n < m := (le_pred_iff_lt h).1
+
+theorem exists_eq_succ_of_ne_zero : ∀ {n}, n ≠ 0 → Exists fun k => n = succ k
+  | _+1, _ => ⟨_, rfl⟩
+
 /-! # Basic theorems for comparing numerals -/
 
 theorem ctor_eq_zero : Nat.zero = 0 :=
@@ -462,7 +632,7 @@ protected theorem one_ne_zero : 1 ≠ (0 : Nat) :=
 protected theorem zero_ne_one : 0 ≠ (1 : Nat) :=
   fun h => Nat.noConfusion h
 
-theorem succ_ne_zero (n : Nat) : succ n ≠ 0 :=
+@[simp] theorem succ_ne_zero (n : Nat) : succ n ≠ 0 :=
   fun h => Nat.noConfusion h
 
 /-! # mul + order -/
@@ -573,6 +743,11 @@ theorem succ_pred {a : Nat} (h : a ≠ 0) : a.pred.succ = a := by
 theorem succ_pred_eq_of_pos : ∀ {n}, 0 < n → succ (pred n) = n
   | _+1, _ => rfl
 
+theorem sub_one_add_one_eq_of_pos : ∀ {n}, 0 < n → (n - 1) + 1 = n
+  | _+1, _ => rfl
+
+@[simp] theorem pred_eq_sub_one : pred n = n - 1 := rfl
+
 /-! # sub theorems -/
 
 theorem add_sub_self_left (a b : Nat) : (a + b) - a = b := by
@@ -595,7 +770,7 @@ theorem zero_lt_sub_of_lt (h : i < a) : 0 < a - i := by
   | zero => contradiction
   | succ a ih =>
     match Nat.eq_or_lt_of_le h with
-    | Or.inl h => injection h with h; subst h; rw [←Nat.add_one, Nat.add_sub_self_left]; decide
+    | Or.inl h => injection h with h; subst h; rw [Nat.add_sub_self_left]; decide
     | Or.inr h =>
       have : 0 < a - i := ih (Nat.lt_of_succ_lt_succ h)
       exact Nat.lt_of_lt_of_le this (Nat.sub_le_succ_sub _ _)
@@ -609,7 +784,7 @@ theorem sub_succ_lt_self (a i : Nat) (h : i < a) : a - (i + 1) < a - i := by
 
 theorem sub_ne_zero_of_lt : {a b : Nat} → a < b → b - a ≠ 0
   | 0, 0, h      => absurd h (Nat.lt_irrefl 0)
-  | 0, succ b, _ => by simp
+  | 0, succ b, _ => by simp only [Nat.sub_zero, ne_eq, not_false_eq_true]
   | succ a, 0, h => absurd h (Nat.not_lt_zero a.succ)
   | succ a, succ b, h => by rw [Nat.succ_sub_succ]; exact sub_ne_zero_of_lt (Nat.lt_of_succ_lt_succ h)
 
@@ -627,7 +802,7 @@ theorem add_sub_of_le {a b : Nat} (h : a ≤ b) : a + (b - a) = b := by
 protected theorem add_sub_add_right (n k m : Nat) : (n + k) - (m + k) = n - m := by
   induction k with
   | zero => simp
-  | succ k ih => simp [add_succ, add_succ, succ_sub_succ, ih]
+  | succ k ih => simp [← Nat.add_assoc, ih]
 
 protected theorem add_sub_add_left (k n m : Nat) : (k + n) - (k + m) = n - m := by
   rw [Nat.add_comm k n, Nat.add_comm k m, Nat.add_sub_add_right]
@@ -740,7 +915,7 @@ protected theorem sub_pos_of_lt (h : m < n) : 0 < n - m :=
 protected theorem sub_sub (n m k : Nat) : n - m - k = n - (m + k) := by
   induction k with
   | zero => simp
-  | succ k ih => rw [Nat.add_succ, Nat.sub_succ, Nat.sub_succ, ih]
+  | succ k ih => rw [Nat.add_succ, Nat.sub_succ, Nat.add_succ, Nat.sub_succ, ih]
 
 protected theorem sub_le_sub_left (h : n ≤ m) (k : Nat) : k - m ≤ k - n :=
   match m, le.dest h with

src/Init/Data/Nat/Bitwise/Basic.lean

@@ -51,6 +51,26 @@ instance : Xor Nat := ⟨Nat.xor⟩
 instance : ShiftLeft Nat := ⟨Nat.shiftLeft⟩
 instance : ShiftRight Nat := ⟨Nat.shiftRight⟩
 
+theorem shiftLeft_eq (a b : Nat) : a <<< b = a * 2 ^ b :=
+  match b with
+  | 0 => (Nat.mul_one _).symm
+  | b+1 => (shiftLeft_eq _ b).trans <| by
+    simp [Nat.pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]
+
+@[simp] theorem shiftRight_zero : n >>> 0 = n := rfl
+
+theorem shiftRight_succ (m n) : m >>> (n + 1) = (m >>> n) / 2 := rfl
+
+theorem shiftRight_add (m n : Nat) : ∀ k, m >>> (n + k) = (m >>> n) >>> k
+  | 0 => rfl
+  | k + 1 => by simp [← Nat.add_assoc, shiftRight_add _ _ k, shiftRight_succ]
+
+theorem shiftRight_eq_div_pow (m : Nat) : ∀ n, m >>> n = m / 2 ^ n
+  | 0 => (Nat.div_one _).symm
+  | k + 1 => by
+    rw [shiftRight_add, shiftRight_eq_div_pow m k]
+    simp [Nat.div_div_eq_div_mul, ← Nat.pow_succ, shiftRight_succ]
+
 /-!
 ### testBit
 We define an operation for testing individual bits in the binary representation

src/Init/Data/Nat/Bitwise/Lemmas.lean

@@ -6,6 +6,7 @@ Authors: Joe Hendrix
 
 prelude
 import Init.Data.Bool
+import Init.Data.Int.Pow
 import Init.Data.Nat.Bitwise.Basic
 import Init.Data.Nat.Lemmas
 import Init.TacticsExtra
@@ -23,26 +24,13 @@ namespace Nat
 private theorem one_div_two : 1/2 = 0 := by trivial
 
 private theorem two_pow_succ_sub_succ_div_two : (2 ^ (n+1) - (x + 1)) / 2 = 2^n - (x/2 + 1) := by
-  if h : x + 1 ≤ 2 ^ (n + 1) then
-    apply fun x => (Nat.sub_eq_of_eq_add x).symm
-    apply Eq.trans _
-    apply Nat.add_mul_div_left _ _ Nat.zero_lt_two
-    rw [← Nat.sub_add_comm h]
-    rw [Nat.add_sub_assoc (by omega)]
-    rw [Nat.pow_succ']
-    rw [Nat.mul_add_div Nat.zero_lt_two]
-    simp [show (2 * (x / 2 + 1) - (x + 1)) / 2 = 0 by omega]
-  else
-    rw [Nat.pow_succ'] at *
-    omega
+  omega
 
 private theorem two_pow_succ_sub_one_div_two : (2 ^ (n+1) - 1) / 2 = 2^n - 1 :=
   two_pow_succ_sub_succ_div_two
 
 private theorem two_mul_sub_one {n : Nat} (n_pos : n > 0) : (2*n - 1) % 2 = 1 := by
-  match n with
-  | 0 => contradiction
-  | n + 1 => simp [Nat.mul_succ, Nat.mul_add_mod, mod_eq_of_lt]
+  omega
 
 /-! ### Preliminaries -/
 
@@ -99,6 +87,11 @@ theorem testBit_to_div_mod {x : Nat} : testBit x i = decide (x / 2^i % 2 = 1) :=
   | succ i hyp =>
     simp [hyp, Nat.div_div_eq_div_mul, Nat.pow_succ']
 
+theorem toNat_testBit (x i : Nat) :
+    (x.testBit i).toNat = x / 2 ^ i % 2 := by
+  rw [Nat.testBit_to_div_mod]
+  rcases Nat.mod_two_eq_zero_or_one (x / 2^i) <;> simp_all
+
 theorem ne_zero_implies_bit_true {x : Nat} (xnz : x ≠ 0) : ∃ i, testBit x i := by
   induction x using div2Induction with
   | ind x hyp =>
@@ -269,31 +262,28 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
 
 theorem testBit_one_zero : testBit 1 0 = true := by trivial
 
+theorem not_decide_mod_two_eq_one (x : Nat)
+    : (!decide (x % 2 = 1)) = decide (x % 2 = 0) := by
+  cases Nat.mod_two_eq_zero_or_one x <;> (rename_i p; simp [p])
+
 theorem testBit_two_pow_sub_succ (h₂ : x < 2 ^ n) (i : Nat) :
     testBit (2^n - (x + 1)) i = (decide (i < n) && ! testBit x i) := by
   induction i generalizing n x with
   | zero =>
-    simp only [testBit_zero, zero_eq, Bool.and_eq_true, decide_eq_true_eq,
-      Bool.not_eq_true']
     match n with
     | 0 => simp
     | n+1 =>
-      -- just logic + omega:
-      simp only [zero_lt_succ, decide_True, Bool.true_and]
-      rw [Nat.pow_succ', ← decide_not, decide_eq_decide]
-      rw [Nat.pow_succ'] at h₂
+      simp [not_decide_mod_two_eq_one]
       omega
   | succ i ih =>
     simp only [testBit_succ]
     match n with
     | 0 =>
-      simp only [Nat.pow_zero, succ_sub_succ_eq_sub, Nat.zero_sub, Nat.zero_div, zero_testBit]
-      rw [decide_eq_false] <;> simp
+      simp [decide_eq_false]
     | n+1 =>
       rw [Nat.two_pow_succ_sub_succ_div_two, ih]
       · simp [Nat.succ_lt_succ_iff]
-      · rw [Nat.pow_succ'] at h₂
-        omega
+      · omega
 
 @[simp] theorem testBit_two_pow_sub_one (n i : Nat) : testBit (2^n-1) i = decide (i < n) := by
   rw [testBit_two_pow_sub_succ]
@@ -344,7 +334,7 @@ private theorem eq_0_of_lt_one (x : Nat) : x < 1 ↔ x = 0 :=
       match x with
       | 0 => Eq.refl 0
       | _+1 => False.elim (not_lt_zero _ (Nat.lt_of_succ_lt_succ p)))
-    (fun p => by simp [p, Nat.zero_lt_succ])
+    (fun p => by simp [p])
 
 private theorem eq_0_of_lt (x : Nat) : x < 2^ 0 ↔ x = 0 := eq_0_of_lt_one x
 

src/Init/Data/Nat/Compare.lean

@@ -0,0 +1,57 @@
+/-
+Copyright (c) 2016 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
+-/
+prelude
+import Init.Classical
+import Init.Data.Ord
+
+/-! # Basic lemmas about comparing natural numbers
+
+This file introduce some basic lemmas about compare as applied to natural
+numbers.
+-/
+namespace Nat
+
+theorem compare_def_lt (a b : Nat) :
+    compare a b = if a < b then .lt else if b < a then .gt else .eq := by
+  simp only [compare, compareOfLessAndEq]
+  split
+  · rfl
+  · next h =>
+    match Nat.lt_or_eq_of_le (Nat.not_lt.1 h) with
+    | .inl h => simp [h, Nat.ne_of_gt h]
+    | .inr rfl => simp
+
+theorem compare_def_le (a b : Nat) :
+    compare a b = if a ≤ b then if b ≤ a then .eq else .lt else .gt := by
+  rw [compare_def_lt]
+  split
+  · next hlt => simp [Nat.le_of_lt hlt, Nat.not_le.2 hlt]
+  · next hge =>
+    split
+    · next hgt => simp [Nat.le_of_lt hgt, Nat.not_le.2 hgt]
+    · next hle => simp [Nat.not_lt.1 hge, Nat.not_lt.1 hle]
+
+protected theorem compare_swap (a b : Nat) : (compare a b).swap = compare b a := by
+  simp only [compare_def_le]; (repeat' split) <;> try rfl
+  next h1 h2 => cases h1 (Nat.le_of_not_le h2)
+
+protected theorem compare_eq_eq {a b : Nat} : compare a b = .eq ↔ a = b := by
+  rw [compare_def_lt]; (repeat' split) <;> simp [Nat.ne_of_lt, Nat.ne_of_gt, *]
+  next hlt hgt => exact Nat.le_antisymm (Nat.not_lt.1 hgt) (Nat.not_lt.1 hlt)
+
+protected theorem compare_eq_lt {a b : Nat} : compare a b = .lt ↔ a < b := by
+  rw [compare_def_lt]; (repeat' split) <;> simp [*]
+
+protected theorem compare_eq_gt {a b : Nat} : compare a b = .gt ↔ b < a := by
+  rw [compare_def_lt]; (repeat' split) <;> simp [Nat.le_of_lt, *]
+
+protected theorem compare_ne_gt {a b : Nat} : compare a b ≠ .gt ↔ a ≤ b := by
+  rw [compare_def_le]; (repeat' split) <;> simp [*]
+
+protected theorem compare_ne_lt {a b : Nat} : compare a b ≠ .lt ↔ b ≤ a := by
+  rw [compare_def_le]; (repeat' split) <;> simp [Nat.le_of_not_le, *]
+
+end Nat

src/Init/Data/Nat/Div.lean

@@ -10,6 +10,13 @@ import Init.Data.Nat.Basic
 
 namespace Nat
 
+/--
+Divisibility of natural numbers. `a ∣ b` (typed as `\|`) says that
+there is some `c` such that `b = a * c`.
+-/
+instance : Dvd Nat where
+  dvd a b := Exists (fun c => b = a * c)
+
 theorem div_rec_lemma {x y : Nat} : 0 < y ∧ y ≤ x → x - y < x :=
   fun ⟨ypos, ylex⟩ => sub_lt (Nat.lt_of_lt_of_le ypos ylex) ypos
 
@@ -28,7 +35,7 @@ theorem div_eq (x y : Nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 e
   rw [Nat.div]
   rfl
 
-theorem div.inductionOn.{u}
+def div.inductionOn.{u}
       {motive : Nat → Nat → Sort u}
       (x y : Nat)
       (ind  : ∀ x y, 0 < y ∧ y ≤ x → motive (x - y) y → motive x y)
@@ -95,7 +102,7 @@ protected theorem modCore_eq_mod (x y : Nat) : Nat.modCore x y = x % y := by
 theorem mod_eq (x y : Nat) : x % y = if 0 < y ∧ y ≤ x then (x - y) % y else x := by
   rw [←Nat.modCore_eq_mod, ←Nat.modCore_eq_mod, Nat.modCore]
 
-theorem mod.inductionOn.{u}
+def mod.inductionOn.{u}
       {motive : Nat → Nat → Sort u}
       (x y  : Nat)
       (ind  : ∀ x y, 0 < y ∧ y ≤ x → motive (x - y) y → motive x y)
@@ -198,13 +205,33 @@ theorem le_div_iff_mul_le (k0 : 0 < k) : x ≤ y / k ↔ x * k ≤ y := by
   induction y, k using mod.inductionOn generalizing x with
     (rw [div_eq]; simp [h]; cases x with | zero => simp [zero_le] | succ x => ?_)
   | base y k h =>
-    simp [not_succ_le_zero x, succ_mul, Nat.add_comm]
-    refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_right ..)
+    simp only [add_one, succ_mul, false_iff, Nat.not_le]
+    refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_left ..)
     exact Nat.not_le.1 fun h' => h ⟨k0, h'⟩
   | ind y k h IH =>
-    rw [← add_one, Nat.add_le_add_iff_right, IH k0, succ_mul,
+    rw [Nat.add_le_add_iff_right, IH k0, succ_mul,
         ← Nat.add_sub_cancel (x*k) k, Nat.sub_le_sub_iff_right h.2, Nat.add_sub_cancel]
 
+protected theorem div_div_eq_div_mul (m n k : Nat) : m / n / k = m / (n * k) := by
+  cases eq_zero_or_pos k with
+  | inl k0 => rw [k0, Nat.mul_zero, Nat.div_zero, Nat.div_zero] | inr kpos => ?_
+  cases eq_zero_or_pos n with
+  | inl n0 => rw [n0, Nat.zero_mul, Nat.div_zero, Nat.zero_div] | inr npos => ?_
+
+  apply Nat.le_antisymm
+
+  apply (le_div_iff_mul_le (Nat.mul_pos npos kpos)).2
+  rw [Nat.mul_comm n k, ← Nat.mul_assoc]
+  apply (le_div_iff_mul_le npos).1
+  apply (le_div_iff_mul_le kpos).1
+  (apply Nat.le_refl)
+
+  apply (le_div_iff_mul_le kpos).2
+  apply (le_div_iff_mul_le npos).2
+  rw [Nat.mul_assoc, Nat.mul_comm n k]
+  apply (le_div_iff_mul_le (Nat.mul_pos kpos npos)).1
+  apply Nat.le_refl
+
 theorem div_mul_le_self : ∀ (m n : Nat), m / n * n ≤ m
   | m, 0   => by simp
   | m, n+1 => (le_div_iff_mul_le (Nat.succ_pos _)).1 (Nat.le_refl _)
@@ -266,7 +293,7 @@ theorem sub_mul_div (x n p : Nat) (h₁ : n*p ≤ x) : (x - n*p) / n = x / n - p
         rw [mul_succ] at h₁
         exact h₁
       rw [sub_succ, ← IH h₂, div_eq_sub_div h₀ h₃]
-      simp [add_one, Nat.pred_succ, mul_succ, Nat.sub_sub]
+      simp [Nat.pred_succ, mul_succ, Nat.sub_sub]
 
 theorem mul_sub_div (x n p : Nat) (h₁ : x < n*p) : (n * p - succ x) / n = p - succ (x / n) := by
   have npos : 0 < n := (eq_zero_or_pos _).resolve_left fun n0 => by
@@ -307,4 +334,50 @@ theorem div_eq_of_lt (h₀ : a < b) : a / b = 0 := by
   intro h₁
   apply Nat.not_le_of_gt h₀ h₁.right
 
+protected theorem mul_div_cancel (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m := by
+  let t := add_mul_div_right 0 m H
+  rwa [Nat.zero_add, Nat.zero_div, Nat.zero_add] at t
+
+protected theorem mul_div_cancel_left (m : Nat) {n : Nat} (H : 0 < n) : n * m / n = m := by
+  rw [Nat.mul_comm, Nat.mul_div_cancel _ H]
+
+protected theorem div_le_of_le_mul {m n : Nat} : ∀ {k}, m ≤ k * n → m / k ≤ n
+  | 0, _ => by simp [Nat.div_zero, n.zero_le]
+  | succ k, h => by
+    suffices succ k * (m / succ k) ≤ succ k * n from
+      Nat.le_of_mul_le_mul_left this (zero_lt_succ _)
+    have h1 : succ k * (m / succ k) ≤ m % succ k + succ k * (m / succ k) := Nat.le_add_left _ _
+    have h2 : m % succ k + succ k * (m / succ k) = m := by rw [mod_add_div]
+    have h3 : m ≤ succ k * n := h
+    rw [← h2] at h3
+    exact Nat.le_trans h1 h3
+
+@[simp] theorem mul_div_right (n : Nat) {m : Nat} (H : 0 < m) : m * n / m = n := by
+  induction n <;> simp_all [mul_succ]
+
+@[simp] theorem mul_div_left (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m := by
+  rw [Nat.mul_comm, mul_div_right _ H]
+
+protected theorem div_self (H : 0 < n) : n / n = 1 := by
+  let t := add_div_right 0 H
+  rwa [Nat.zero_add, Nat.zero_div] at t
+
+protected theorem div_eq_of_eq_mul_left (H1 : 0 < n) (H2 : m = k * n) : m / n = k :=
+by rw [H2, Nat.mul_div_cancel _ H1]
+
+protected theorem div_eq_of_eq_mul_right (H1 : 0 < n) (H2 : m = n * k) : m / n = k :=
+by rw [H2, Nat.mul_div_cancel_left _ H1]
+
+protected theorem mul_div_mul_left {m : Nat} (n k : Nat) (H : 0 < m) :
+    m * n / (m * k) = n / k := by rw [← Nat.div_div_eq_div_mul, Nat.mul_div_cancel_left _ H]
+
+protected theorem mul_div_mul_right {m : Nat} (n k : Nat) (H : 0 < m) :
+    n * m / (k * m) = n / k := by rw [Nat.mul_comm, Nat.mul_comm k, Nat.mul_div_mul_left _ _ H]
+
+theorem mul_div_le (m n : Nat) : n * (m / n) ≤ m := by
+  match n, Nat.eq_zero_or_pos n with
+  | _, Or.inl rfl => rw [Nat.zero_mul]; exact m.zero_le
+  | n, Or.inr h => rw [Nat.mul_comm, ← Nat.le_div_iff_mul_le h]; exact Nat.le_refl _
+
+
 end Nat

src/Init/Data/Nat/Dvd.lean

@@ -5,16 +5,10 @@ Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
 -/
 prelude
 import Init.Data.Nat.Div
+import Init.Meta
 
 namespace Nat
 
-/--
-Divisibility of natural numbers. `a ∣ b` (typed as `\|`) says that
-there is some `c` such that `b = a * c`.
--/
-instance : Dvd Nat where
-  dvd a b := Exists (fun c => b = a * c)
-
 protected theorem dvd_refl (a : Nat) : a ∣ a := ⟨1, by simp⟩
 
 protected theorem dvd_zero (a : Nat) : a ∣ 0 := ⟨0, by simp⟩
@@ -97,4 +91,42 @@ protected theorem mul_div_cancel' {n m : Nat} (H : n ∣ m) : n * (m / n) = m :=
 protected theorem div_mul_cancel {n m : Nat} (H : n ∣ m) : m / n * n = m := by
   rw [Nat.mul_comm, Nat.mul_div_cancel' H]
 
+@[simp] theorem mod_mod_of_dvd (a : Nat) (h : c ∣ b) : a % b % c = a % c := by
+  rw (config := {occs := .pos [2]}) [← mod_add_div a b]
+  have ⟨x, h⟩ := h
+  subst h
+  rw [Nat.mul_assoc, add_mul_mod_self_left]
+
+protected theorem dvd_of_mul_dvd_mul_left
+    (kpos : 0 < k) (H : k * m ∣ k * n) : m ∣ n := by
+  let ⟨l, H⟩ := H
+  rw [Nat.mul_assoc] at H
+  exact ⟨_, Nat.eq_of_mul_eq_mul_left kpos H⟩
+
+protected theorem dvd_of_mul_dvd_mul_right (kpos : 0 < k) (H : m * k ∣ n * k) : m ∣ n := by
+  rw [Nat.mul_comm m k, Nat.mul_comm n k] at H; exact Nat.dvd_of_mul_dvd_mul_left kpos H
+
+theorem dvd_sub {k m n : Nat} (H : n ≤ m) (h₁ : k ∣ m) (h₂ : k ∣ n) : k ∣ m - n :=
+  (Nat.dvd_add_iff_left h₂).2 <| by rwa [Nat.sub_add_cancel H]
+
+protected theorem mul_dvd_mul {a b c d : Nat} : a ∣ b → c ∣ d → a * c ∣ b * d
+  | ⟨e, he⟩, ⟨f, hf⟩ =>
+    ⟨e * f, by simp [he, hf, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]⟩
+
+protected theorem mul_dvd_mul_left (a : Nat) (h : b ∣ c) : a * b ∣ a * c :=
+  Nat.mul_dvd_mul (Nat.dvd_refl a) h
+
+protected theorem mul_dvd_mul_right (h: a ∣ b) (c : Nat) : a * c ∣ b * c :=
+  Nat.mul_dvd_mul h (Nat.dvd_refl c)
+
+@[simp] theorem dvd_one {n : Nat} : n ∣ 1 ↔ n = 1 :=
+  ⟨eq_one_of_dvd_one, fun h => h.symm ▸ Nat.dvd_refl _⟩
+
+protected theorem mul_div_assoc (m : Nat) (H : k ∣ n) : m * n / k = m * (n / k) := by
+  match Nat.eq_zero_or_pos k with
+  | .inl h0 => rw [h0, Nat.div_zero, Nat.div_zero, Nat.mul_zero]
+  | .inr hpos =>
+    have h1 : m * n / k = m * (n / k * k) / k := by rw [Nat.div_mul_cancel H]
+    rw [h1, ← Nat.mul_assoc, Nat.mul_div_cancel _ hpos]
+
 end Nat

src/Init/Data/Nat/Gcd.lean

@@ -1,10 +1,12 @@
 /-
 Copyright (c) 2021 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Leonardo de Moura
+Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
 -/
 prelude
 import Init.Data.Nat.Dvd
+import Init.NotationExtra
+import Init.RCases
 
 namespace Nat
 
@@ -14,8 +16,8 @@ def gcd (m n : @& Nat) : Nat :=
     n
   else
     gcd (n % m) m
-termination_by m
-decreasing_by simp_wf; apply mod_lt _ (zero_lt_of_ne_zero _); assumption
+  termination_by m
+  decreasing_by simp_wf; apply mod_lt _ (zero_lt_of_ne_zero _); assumption
 
 @[simp] theorem gcd_zero_left (y : Nat) : gcd 0 y = y :=
   rfl
@@ -69,4 +71,166 @@ theorem dvd_gcd : k ∣ m → k ∣ n → k ∣ gcd m n := by
   | H0 n => rw [gcd_zero_left]; exact kn
   | H1 n m _ IH => rw [gcd_rec]; exact IH ((dvd_mod_iff km).2 kn) km
 
+theorem dvd_gcd_iff : k ∣ gcd m n ↔ k ∣ m ∧ k ∣ n :=
+  ⟨fun h => let ⟨h₁, h₂⟩ := gcd_dvd m n; ⟨Nat.dvd_trans h h₁, Nat.dvd_trans h h₂⟩,
+   fun ⟨h₁, h₂⟩ => dvd_gcd h₁ h₂⟩
+
+theorem gcd_comm (m n : Nat) : gcd m n = gcd n m :=
+  Nat.dvd_antisymm
+    (dvd_gcd (gcd_dvd_right m n) (gcd_dvd_left m n))
+    (dvd_gcd (gcd_dvd_right n m) (gcd_dvd_left n m))
+
+theorem gcd_eq_left_iff_dvd : m ∣ n ↔ gcd m n = m :=
+  ⟨fun h => by rw [gcd_rec, mod_eq_zero_of_dvd h, gcd_zero_left],
+   fun h => h ▸ gcd_dvd_right m n⟩
+
+theorem gcd_eq_right_iff_dvd : m ∣ n ↔ gcd n m = m := by
+  rw [gcd_comm]; exact gcd_eq_left_iff_dvd
+
+theorem gcd_assoc (m n k : Nat) : gcd (gcd m n) k = gcd m (gcd n k) :=
+  Nat.dvd_antisymm
+    (dvd_gcd
+      (Nat.dvd_trans (gcd_dvd_left (gcd m n) k) (gcd_dvd_left m n))
+      (dvd_gcd (Nat.dvd_trans (gcd_dvd_left (gcd m n) k) (gcd_dvd_right m n))
+        (gcd_dvd_right (gcd m n) k)))
+    (dvd_gcd
+      (dvd_gcd (gcd_dvd_left m (gcd n k))
+        (Nat.dvd_trans (gcd_dvd_right m (gcd n k)) (gcd_dvd_left n k)))
+      (Nat.dvd_trans (gcd_dvd_right m (gcd n k)) (gcd_dvd_right n k)))
+
+@[simp] theorem gcd_one_right (n : Nat) : gcd n 1 = 1 := (gcd_comm n 1).trans (gcd_one_left n)
+
+theorem gcd_mul_left (m n k : Nat) : gcd (m * n) (m * k) = m * gcd n k := by
+  induction n, k using gcd.induction with
+  | H0 k => simp
+  | H1 n k _ IH => rwa [← mul_mod_mul_left, ← gcd_rec, ← gcd_rec] at IH
+
+theorem gcd_mul_right (m n k : Nat) : gcd (m * n) (k * n) = gcd m k * n := by
+  rw [Nat.mul_comm m n, Nat.mul_comm k n, Nat.mul_comm (gcd m k) n, gcd_mul_left]
+
+theorem gcd_pos_of_pos_left {m : Nat} (n : Nat) (mpos : 0 < m) : 0 < gcd m n :=
+  pos_of_dvd_of_pos (gcd_dvd_left m n) mpos
+
+theorem gcd_pos_of_pos_right (m : Nat) {n : Nat} (npos : 0 < n) : 0 < gcd m n :=
+  pos_of_dvd_of_pos (gcd_dvd_right m n) npos
+
+theorem div_gcd_pos_of_pos_left (b : Nat) (h : 0 < a) : 0 < a / a.gcd b :=
+  (Nat.le_div_iff_mul_le <| Nat.gcd_pos_of_pos_left _ h).2 (Nat.one_mul _ ▸ Nat.gcd_le_left _ h)
+
+theorem div_gcd_pos_of_pos_right (a : Nat) (h : 0 < b) : 0 < b / a.gcd b :=
+  (Nat.le_div_iff_mul_le <| Nat.gcd_pos_of_pos_right _ h).2 (Nat.one_mul _ ▸ Nat.gcd_le_right _ h)
+
+theorem eq_zero_of_gcd_eq_zero_left {m n : Nat} (H : gcd m n = 0) : m = 0 :=
+  match eq_zero_or_pos m with
+  | .inl H0 => H0
+  | .inr H1 => absurd (Eq.symm H) (ne_of_lt (gcd_pos_of_pos_left _ H1))
+
+theorem eq_zero_of_gcd_eq_zero_right {m n : Nat} (H : gcd m n = 0) : n = 0 := by
+  rw [gcd_comm] at H
+  exact eq_zero_of_gcd_eq_zero_left H
+
+theorem gcd_ne_zero_left : m ≠ 0 → gcd m n ≠ 0 := mt eq_zero_of_gcd_eq_zero_left
+
+theorem gcd_ne_zero_right : n ≠ 0 → gcd m n ≠ 0 := mt eq_zero_of_gcd_eq_zero_right
+
+theorem gcd_div {m n k : Nat} (H1 : k ∣ m) (H2 : k ∣ n) :
+    gcd (m / k) (n / k) = gcd m n / k :=
+  match eq_zero_or_pos k with
+  | .inl H0 => by simp [H0]
+  | .inr H3 => by
+    apply Nat.eq_of_mul_eq_mul_right H3
+    rw [Nat.div_mul_cancel (dvd_gcd H1 H2), ← gcd_mul_right,
+        Nat.div_mul_cancel H1, Nat.div_mul_cancel H2]
+
+theorem gcd_dvd_gcd_of_dvd_left {m k : Nat} (n : Nat) (H : m ∣ k) : gcd m n ∣ gcd k n :=
+  dvd_gcd (Nat.dvd_trans (gcd_dvd_left m n) H) (gcd_dvd_right m n)
+
+theorem gcd_dvd_gcd_of_dvd_right {m k : Nat} (n : Nat) (H : m ∣ k) : gcd n m ∣ gcd n k :=
+  dvd_gcd (gcd_dvd_left n m) (Nat.dvd_trans (gcd_dvd_right n m) H)
+
+theorem gcd_dvd_gcd_mul_left (m n k : Nat) : gcd m n ∣ gcd (k * m) n :=
+  gcd_dvd_gcd_of_dvd_left _ (Nat.dvd_mul_left _ _)
+
+theorem gcd_dvd_gcd_mul_right (m n k : Nat) : gcd m n ∣ gcd (m * k) n :=
+  gcd_dvd_gcd_of_dvd_left _ (Nat.dvd_mul_right _ _)
+
+theorem gcd_dvd_gcd_mul_left_right (m n k : Nat) : gcd m n ∣ gcd m (k * n) :=
+  gcd_dvd_gcd_of_dvd_right _ (Nat.dvd_mul_left _ _)
+
+theorem gcd_dvd_gcd_mul_right_right (m n k : Nat) : gcd m n ∣ gcd m (n * k) :=
+  gcd_dvd_gcd_of_dvd_right _ (Nat.dvd_mul_right _ _)
+
+theorem gcd_eq_left {m n : Nat} (H : m ∣ n) : gcd m n = m :=
+  Nat.dvd_antisymm (gcd_dvd_left _ _) (dvd_gcd (Nat.dvd_refl _) H)
+
+theorem gcd_eq_right {m n : Nat} (H : n ∣ m) : gcd m n = n := by
+  rw [gcd_comm, gcd_eq_left H]
+
+@[simp] theorem gcd_mul_left_left (m n : Nat) : gcd (m * n) n = n :=
+  Nat.dvd_antisymm (gcd_dvd_right _ _) (dvd_gcd (Nat.dvd_mul_left _ _) (Nat.dvd_refl _))
+
+@[simp] theorem gcd_mul_left_right (m n : Nat) : gcd n (m * n) = n := by
+  rw [gcd_comm, gcd_mul_left_left]
+
+@[simp] theorem gcd_mul_right_left (m n : Nat) : gcd (n * m) n = n := by
+  rw [Nat.mul_comm, gcd_mul_left_left]
+
+@[simp] theorem gcd_mul_right_right (m n : Nat) : gcd n (n * m) = n := by
+  rw [gcd_comm, gcd_mul_right_left]
+
+@[simp] theorem gcd_gcd_self_right_left (m n : Nat) : gcd m (gcd m n) = gcd m n :=
+  Nat.dvd_antisymm (gcd_dvd_right _ _) (dvd_gcd (gcd_dvd_left _ _) (Nat.dvd_refl _))
+
+@[simp] theorem gcd_gcd_self_right_right (m n : Nat) : gcd m (gcd n m) = gcd n m := by
+  rw [gcd_comm n m, gcd_gcd_self_right_left]
+
+@[simp] theorem gcd_gcd_self_left_right (m n : Nat) : gcd (gcd n m) m = gcd n m := by
+  rw [gcd_comm, gcd_gcd_self_right_right]
+
+@[simp] theorem gcd_gcd_self_left_left (m n : Nat) : gcd (gcd m n) m = gcd m n := by
+  rw [gcd_comm m n, gcd_gcd_self_left_right]
+
+theorem gcd_add_mul_self (m n k : Nat) : gcd m (n + k * m) = gcd m n := by
+  simp [gcd_rec m (n + k * m), gcd_rec m n]
+
+theorem gcd_eq_zero_iff {i j : Nat} : gcd i j = 0 ↔ i = 0 ∧ j = 0 :=
+  ⟨fun h => ⟨eq_zero_of_gcd_eq_zero_left h, eq_zero_of_gcd_eq_zero_right h⟩,
+   fun h => by simp [h]⟩
+
+/-- Characterization of the value of `Nat.gcd`. -/
+theorem gcd_eq_iff (a b : Nat) :
+    gcd a b = g ↔ g ∣ a ∧ g ∣ b ∧ (∀ c, c ∣ a → c ∣ b → c ∣ g) := by
+  constructor
+  · rintro rfl
+    exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _, fun _ => Nat.dvd_gcd⟩
+  · rintro ⟨ha, hb, hc⟩
+    apply Nat.dvd_antisymm
+    · apply hc
+      · exact gcd_dvd_left a b
+      · exact gcd_dvd_right a b
+    · exact Nat.dvd_gcd ha hb
+
+/-- Represent a divisor of `m * n` as a product of a divisor of `m` and a divisor of `n`. -/
+def prod_dvd_and_dvd_of_dvd_prod {k m n : Nat} (H : k ∣ m * n) :
+    {d : {m' // m' ∣ m} × {n' // n' ∣ n} // k = d.1.val * d.2.val} :=
+  if h0 : gcd k m = 0 then
+    ⟨⟨⟨0, eq_zero_of_gcd_eq_zero_right h0 ▸ Nat.dvd_refl 0⟩,
+      ⟨n, Nat.dvd_refl n⟩⟩,
+      eq_zero_of_gcd_eq_zero_left h0 ▸ (Nat.zero_mul n).symm⟩
+  else by
+    have hd : gcd k m * (k / gcd k m) = k := Nat.mul_div_cancel' (gcd_dvd_left k m)
+    refine ⟨⟨⟨gcd k m, gcd_dvd_right k m⟩, ⟨k / gcd k m, ?_⟩⟩, hd.symm⟩
+    apply Nat.dvd_of_mul_dvd_mul_left (Nat.pos_of_ne_zero h0)
+    rw [hd, ← gcd_mul_right]
+    exact Nat.dvd_gcd (Nat.dvd_mul_right _ _) H
+
+theorem gcd_mul_dvd_mul_gcd (k m n : Nat) : gcd k (m * n) ∣ gcd k m * gcd k n := by
+  let ⟨⟨⟨m', hm'⟩, ⟨n', hn'⟩⟩, (h : gcd k (m * n) = m' * n')⟩ :=
+    prod_dvd_and_dvd_of_dvd_prod <| gcd_dvd_right k (m * n)
+  rw [h]
+  have h' : m' * n' ∣ k := h ▸ gcd_dvd_left ..
+  exact Nat.mul_dvd_mul
+    (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_right m' n') h') hm')
+    (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_left n' m') h') hn')
+
 end Nat

src/Init/Data/Nat/Lcm.lean

@@ -0,0 +1,66 @@
+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
+-/
+prelude
+import Init.Data.Nat.Gcd
+import Init.Data.Nat.Lemmas
+
+namespace Nat
+
+/-- The least common multiple of `m` and `n`, defined using `gcd`. -/
+def lcm (m n : Nat) : Nat := m * n / gcd m n
+
+theorem lcm_comm (m n : Nat) : lcm m n = lcm n m := by
+  rw [lcm, lcm, Nat.mul_comm n m, gcd_comm n m]
+
+@[simp] theorem lcm_zero_left (m : Nat) : lcm 0 m = 0 := by simp [lcm]
+
+@[simp] theorem lcm_zero_right (m : Nat) : lcm m 0 = 0 := by simp [lcm]
+
+@[simp] theorem lcm_one_left (m : Nat) : lcm 1 m = m := by simp [lcm]
+
+@[simp] theorem lcm_one_right (m : Nat) : lcm m 1 = m := by simp [lcm]
+
+@[simp] theorem lcm_self (m : Nat) : lcm m m = m := by
+  match eq_zero_or_pos m with
+  | .inl h => rw [h, lcm_zero_left]
+  | .inr h => simp [lcm, Nat.mul_div_cancel _ h]
+
+theorem dvd_lcm_left (m n : Nat) : m ∣ lcm m n :=
+  ⟨n / gcd m n, by rw [← Nat.mul_div_assoc m (Nat.gcd_dvd_right m n)]; rfl⟩
+
+theorem dvd_lcm_right (m n : Nat) : n ∣ lcm m n := lcm_comm n m ▸ dvd_lcm_left n m
+
+theorem gcd_mul_lcm (m n : Nat) : gcd m n * lcm m n = m * n := by
+  rw [lcm, Nat.mul_div_cancel' (Nat.dvd_trans (gcd_dvd_left m n) (Nat.dvd_mul_right m n))]
+
+theorem lcm_dvd {m n k : Nat} (H1 : m ∣ k) (H2 : n ∣ k) : lcm m n ∣ k := by
+  match eq_zero_or_pos k with
+  | .inl h => rw [h]; exact Nat.dvd_zero _
+  | .inr kpos =>
+    apply Nat.dvd_of_mul_dvd_mul_left (gcd_pos_of_pos_left n (pos_of_dvd_of_pos H1 kpos))
+    rw [gcd_mul_lcm, ← gcd_mul_right, Nat.mul_comm n k]
+    exact dvd_gcd (Nat.mul_dvd_mul_left _ H2) (Nat.mul_dvd_mul_right H1 _)
+
+theorem lcm_assoc (m n k : Nat) : lcm (lcm m n) k = lcm m (lcm n k) :=
+Nat.dvd_antisymm
+  (lcm_dvd
+    (lcm_dvd (dvd_lcm_left m (lcm n k))
+      (Nat.dvd_trans (dvd_lcm_left n k) (dvd_lcm_right m (lcm n k))))
+    (Nat.dvd_trans (dvd_lcm_right n k) (dvd_lcm_right m (lcm n k))))
+  (lcm_dvd
+    (Nat.dvd_trans (dvd_lcm_left m n) (dvd_lcm_left (lcm m n) k))
+    (lcm_dvd (Nat.dvd_trans (dvd_lcm_right m n) (dvd_lcm_left (lcm m n) k))
+      (dvd_lcm_right (lcm m n) k)))
+
+theorem lcm_ne_zero (hm : m ≠ 0) (hn : n ≠ 0) : lcm m n ≠ 0 := by
+  intro h
+  have h1 := gcd_mul_lcm m n
+  rw [h, Nat.mul_zero] at h1
+  match mul_eq_zero.1 h1.symm with
+  | .inl hm1 => exact hm hm1
+  | .inr hn1 => exact hn hn1
+
+end Nat

src/Init/Data/Nat/Lemmas.lean

@@ -4,10 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
 -/
 prelude
-import Init.Data.Nat.Dvd
 import Init.Data.Nat.MinMax
 import Init.Data.Nat.Log2
 import Init.Data.Nat.Power2
+import Init.Omega
 
 /-! # Basic lemmas about natural numbers
 
@@ -19,131 +19,6 @@ and later these lemmas should be organised into other files more systematically.
 -/
 
 namespace Nat
-
-/-! ### le/lt -/
-
-protected theorem lt_asymm {a b : Nat} (h : a < b) : ¬ b < a := Nat.not_lt.2 (Nat.le_of_lt h)
-protected abbrev not_lt_of_gt := @Nat.lt_asymm
-protected abbrev not_lt_of_lt := @Nat.lt_asymm
-
-protected theorem lt_iff_le_not_le {m n : Nat} : m < n ↔ m ≤ n ∧ ¬ n ≤ m :=
-  ⟨fun h => ⟨Nat.le_of_lt h, Nat.not_le_of_gt h⟩, fun ⟨_, h⟩ => Nat.lt_of_not_ge h⟩
-protected abbrev lt_iff_le_and_not_ge := @Nat.lt_iff_le_not_le
-
-protected theorem lt_iff_le_and_ne {m n : Nat} : m < n ↔ m ≤ n ∧ m ≠ n :=
-  ⟨fun h => ⟨Nat.le_of_lt h, Nat.ne_of_lt h⟩, fun h => Nat.lt_of_le_of_ne h.1 h.2⟩
-
-protected theorem ne_iff_lt_or_gt {a b : Nat} : a ≠ b ↔ a < b ∨ b < a :=
-  ⟨Nat.lt_or_gt_of_ne, fun | .inl h => Nat.ne_of_lt h | .inr h => Nat.ne_of_gt h⟩
-protected abbrev lt_or_gt := @Nat.ne_iff_lt_or_gt
-
-protected abbrev le_or_ge := @Nat.le_total
-protected abbrev le_or_le := @Nat.le_total
-
-protected theorem eq_or_lt_of_not_lt {a b : Nat} (hnlt : ¬ a < b) : a = b ∨ b < a :=
-  (Nat.lt_trichotomy ..).resolve_left hnlt
-
-protected theorem lt_or_eq_of_le {n m : Nat} (h : n ≤ m) : n < m ∨ n = m :=
-  (Nat.lt_or_ge ..).imp_right (Nat.le_antisymm h)
-
-protected theorem le_iff_lt_or_eq {n m : Nat} : n ≤ m ↔ n < m ∨ n = m :=
-  ⟨Nat.lt_or_eq_of_le, fun | .inl h => Nat.le_of_lt h | .inr rfl => Nat.le_refl _⟩
-
-protected theorem lt_succ_iff : m < succ n ↔ m ≤ n := ⟨le_of_lt_succ, lt_succ_of_le⟩
-
-protected theorem lt_succ_iff_lt_or_eq : m < succ n ↔ m < n ∨ m = n :=
-  Nat.lt_succ_iff.trans Nat.le_iff_lt_or_eq
-
-protected theorem eq_of_lt_succ_of_not_lt (hmn : m < n + 1) (h : ¬ m < n) : m = n :=
-  (Nat.lt_succ_iff_lt_or_eq.1 hmn).resolve_left h
-
-protected theorem eq_of_le_of_lt_succ (h₁ : n ≤ m) (h₂ : m < n + 1) : m = n :=
-  Nat.le_antisymm (le_of_succ_le_succ h₂) h₁
-
-
-/-! ## zero/one/two -/
-
-theorem le_zero : i ≤ 0 ↔ i = 0 := ⟨Nat.eq_zero_of_le_zero, fun | rfl => Nat.le_refl _⟩
-
-protected abbrev one_pos := @Nat.zero_lt_one
-
-protected theorem two_pos : 0 < 2 := Nat.zero_lt_succ _
-
-theorem add_one_ne_zero (n) : n + 1 ≠ 0 := succ_ne_zero _
-
-protected theorem ne_zero_iff_zero_lt : n ≠ 0 ↔ 0 < n := Nat.pos_iff_ne_zero.symm
-
-protected theorem zero_lt_two : 0 < 2 := Nat.zero_lt_succ _
-
-protected theorem one_lt_two : 1 < 2 := Nat.succ_lt_succ Nat.zero_lt_one
-
-protected theorem eq_zero_of_not_pos (h : ¬0 < n) : n = 0 :=
-  Nat.eq_zero_of_le_zero (Nat.not_lt.1 h)
-
-/-! ## succ/pred -/
-
-attribute [simp] succ_ne_zero zero_lt_succ lt_succ_self Nat.pred_zero Nat.pred_succ Nat.pred_le
-
-theorem succ_ne_self (n) : succ n ≠ n := Nat.ne_of_gt (lt_succ_self n)
-
-theorem succ_le : succ n ≤ m ↔ n < m := .rfl
-
-theorem lt_succ : m < succ n ↔ m ≤ n := ⟨le_of_lt_succ, lt_succ_of_le⟩
-
-theorem lt_succ_of_lt (h : a < b) : a < succ b := le_succ_of_le h
-
-theorem succ_pred_eq_of_ne_zero : ∀ {n}, n ≠ 0 → succ (pred n) = n
-  | _+1, _ => rfl
-
-theorem eq_zero_or_eq_succ_pred : ∀ n, n = 0 ∨ n = succ (pred n)
-  | 0 => .inl rfl
-  | _+1 => .inr rfl
-
-theorem succ_inj' : succ a = succ b ↔ a = b := ⟨succ.inj, congrArg _⟩
-
-theorem succ_le_succ_iff : succ a ≤ succ b ↔ a ≤ b := ⟨le_of_succ_le_succ, succ_le_succ⟩
-
-theorem succ_lt_succ_iff : succ a < succ b ↔ a < b := ⟨lt_of_succ_lt_succ, succ_lt_succ⟩
-
-theorem pred_inj : ∀ {a b}, 0 < a → 0 < b → pred a = pred b → a = b
-  | _+1, _+1, _, _ => congrArg _
-
-theorem pred_ne_self : ∀ {a}, a ≠ 0 → pred a ≠ a
-  | _+1, _ => (succ_ne_self _).symm
-
-theorem pred_lt_self : ∀ {a}, 0 < a → pred a < a
-  | _+1, _ => lt_succ_self _
-
-theorem pred_lt_pred : ∀ {n m}, n ≠ 0 → n < m → pred n < pred m
-  | _+1, _+1, _, h => lt_of_succ_lt_succ h
-
-theorem pred_le_iff_le_succ : ∀ {n m}, pred n ≤ m ↔ n ≤ succ m
-  | 0, _ => ⟨fun _ => Nat.zero_le _, fun _ => Nat.zero_le _⟩
-  | _+1, _ => Nat.succ_le_succ_iff.symm
-
-theorem le_succ_of_pred_le : pred n ≤ m → n ≤ succ m := pred_le_iff_le_succ.1
-
-theorem pred_le_of_le_succ : n ≤ succ m → pred n ≤ m := pred_le_iff_le_succ.2
-
-theorem lt_pred_iff_succ_lt : ∀ {n m}, n < pred m ↔ succ n < m
-  | _, 0 => ⟨nofun, nofun⟩
-  | _, _+1 => Nat.succ_lt_succ_iff.symm
-
-theorem succ_lt_of_lt_pred : n < pred m → succ n < m := lt_pred_iff_succ_lt.1
-
-theorem lt_pred_of_succ_lt : succ n < m → n < pred m := lt_pred_iff_succ_lt.2
-
-theorem le_pred_iff_lt : ∀ {n m}, 0 < m → (n ≤ pred m ↔ n < m)
-  | 0, _+1, _ => ⟨fun _ => Nat.zero_lt_succ _, fun _ => Nat.zero_le _⟩
-  | _+1, _+1, _ => Nat.lt_pred_iff_succ_lt
-
-theorem lt_of_le_pred (h : 0 < m) : n ≤ pred m → n < m := (le_pred_iff_lt h).1
-
-theorem le_pred_of_lt (h : n < m) : n ≤ pred m := (le_pred_iff_lt (Nat.zero_lt_of_lt h)).2 h
-
-theorem exists_eq_succ_of_ne_zero : ∀ {n}, n ≠ 0 → ∃ k, n = succ k
-  | _+1, _ => ⟨_, rfl⟩
-
 /-! ## add -/
 
 protected theorem add_add_add_comm (a b c d : Nat) : (a + b) + (c + d) = (a + c) + (b + d) := by
@@ -191,15 +66,6 @@ protected theorem add_lt_add_of_lt_of_le {a b c d : Nat} (hlt : a < b) (hle : c
     a + c < b + d :=
   Nat.lt_of_le_of_lt (Nat.add_le_add_left hle _) (Nat.add_lt_add_right hlt _)
 
-protected theorem lt_add_left (c : Nat) (h : a < b) : a < c + b :=
-  Nat.lt_of_lt_of_le h (Nat.le_add_left ..)
-
-protected theorem lt_add_right (c : Nat) (h : a < b) : a < b + c :=
-  Nat.lt_of_lt_of_le h (Nat.le_add_right ..)
-
-protected theorem lt_add_of_pos_right (h : 0 < k) : n < n + k :=
-  Nat.add_lt_add_left h n
-
 protected theorem lt_add_of_pos_left : 0 < k → n < k + n := by
   rw [Nat.add_comm]; exact Nat.lt_add_of_pos_right
 
@@ -309,8 +175,6 @@ theorem add_lt_of_lt_sub' {a b c : Nat} : b < c - a → a + b < c := by
 protected theorem sub_add_lt_sub (h₁ : m + k ≤ n) (h₂ : 0 < k) : n - (m + k) < n - m := by
   rw [← Nat.sub_sub]; exact Nat.sub_lt_of_pos_le h₂ (Nat.le_sub_of_add_le' h₁)
 
-theorem le_sub_one_of_lt : a < b → a ≤ b - 1 := Nat.le_pred_of_lt
-
 theorem sub_one_lt_of_le (h₀ : 0 < a) (h₁ : a ≤ b) : a - 1 < b :=
   Nat.lt_of_lt_of_le (Nat.pred_lt' h₀) h₁
 
@@ -470,6 +334,32 @@ protected theorem sub_max_sub_right : ∀ (a b c : Nat), max (a - c) (b - c) = m
   | _, _, 0 => rfl
   | _, _, _+1 => Eq.trans (Nat.pred_max_pred ..) <| congrArg _ (Nat.sub_max_sub_right ..)
 
+protected theorem sub_min_sub_left (a b c : Nat) : min (a - b) (a - c) = a - max b c := by
+  omega
+
+protected theorem sub_max_sub_left (a b c : Nat) : max (a - b) (a - c) = a - min b c := by
+  omega
+
+protected theorem mul_max_mul_right (a b c : Nat) : max (a * c) (b * c) = max a b * c := by
+  induction a generalizing b with
+  | zero => simp
+  | succ i ind =>
+    cases b <;> simp [succ_eq_add_one, Nat.succ_mul, Nat.add_max_add_right, ind]
+
+protected theorem mul_min_mul_right (a b c : Nat) : min (a * c) (b * c) = min a b * c := by
+  induction a generalizing b with
+  | zero => simp
+  | succ i ind =>
+    cases b <;> simp [succ_eq_add_one, Nat.succ_mul, Nat.add_min_add_right, ind]
+
+protected theorem mul_max_mul_left (a b c : Nat) : max (a * b) (a * c) = a * max b c := by
+  repeat rw [Nat.mul_comm a]
+  exact Nat.mul_max_mul_right ..
+
+protected theorem mul_min_mul_left (a b c : Nat) : min (a * b) (a * c) = a * min b c := by
+  repeat rw [Nat.mul_comm a]
+  exact Nat.mul_min_mul_right ..
+
 -- protected theorem sub_min_sub_left (a b c : Nat) : min (a - b) (a - c) = a - max b c := by
 --   induction b, c using Nat.recDiagAux with
 --   | zero_left => rw [Nat.sub_zero, Nat.zero_max]; exact Nat.min_eq_right (Nat.sub_le ..)
@@ -518,10 +408,6 @@ protected theorem mul_right_comm (n m k : Nat) : n * m * k = n * k * m := by
 protected theorem mul_mul_mul_comm (a b c d : Nat) : (a * b) * (c * d) = (a * c) * (b * d) := by
   rw [Nat.mul_assoc, Nat.mul_assoc, Nat.mul_left_comm b]
 
-protected theorem mul_two (n) : n * 2 = n + n := by rw [Nat.mul_succ, Nat.mul_one]
-
-protected theorem two_mul (n) : 2 * n = n + n := by rw [Nat.succ_mul, Nat.one_mul]
-
 theorem mul_eq_zero : ∀ {m n}, n * m = 0 ↔ n = 0 ∨ m = 0
   | 0, _ => ⟨fun _ => .inr rfl, fun _ => rfl⟩
   | _, 0 => ⟨fun _ => .inl rfl, fun _ => Nat.zero_mul ..⟩
@@ -619,68 +505,6 @@ protected theorem pos_of_mul_pos_right {a b : Nat} (h : 0 < a * b) : 0 < a := by
 
 /-! ### div/mod -/
 
-protected theorem div_le_of_le_mul {m n : Nat} : ∀ {k}, m ≤ k * n → m / k ≤ n
-  | 0, _ => by simp [Nat.div_zero, n.zero_le]
-  | succ k, h => by
-    suffices succ k * (m / succ k) ≤ succ k * n from
-      Nat.le_of_mul_le_mul_left this (zero_lt_succ _)
-    have h1 : succ k * (m / succ k) ≤ m % succ k + succ k * (m / succ k) := Nat.le_add_left _ _
-    have h2 : m % succ k + succ k * (m / succ k) = m := by rw [mod_add_div]
-    have h3 : m ≤ succ k * n := h
-    rw [← h2] at h3
-    exact Nat.le_trans h1 h3
-
-@[simp] theorem mul_div_right (n : Nat) {m : Nat} (H : 0 < m) : m * n / m = n := by
-  induction n <;> simp_all [mul_succ]
-
-@[simp] theorem mul_div_left (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m := by
-  rw [Nat.mul_comm, mul_div_right _ H]
-
-protected theorem div_self (H : 0 < n) : n / n = 1 := by
-  let t := add_div_right 0 H
-  rwa [Nat.zero_add, Nat.zero_div] at t
-
-protected theorem mul_div_cancel (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m := by
-  let t := add_mul_div_right 0 m H
-  rwa [Nat.zero_add, Nat.zero_div, Nat.zero_add] at t
-
-protected theorem mul_div_cancel_left (m : Nat) {n : Nat} (H : 0 < n) : n * m / n = m :=
-by rw [Nat.mul_comm, Nat.mul_div_cancel _ H]
-
-protected theorem div_eq_of_eq_mul_left (H1 : 0 < n) (H2 : m = k * n) : m / n = k :=
-by rw [H2, Nat.mul_div_cancel _ H1]
-
-protected theorem div_eq_of_eq_mul_right (H1 : 0 < n) (H2 : m = n * k) : m / n = k :=
-by rw [H2, Nat.mul_div_cancel_left _ H1]
-
-protected theorem div_div_eq_div_mul (m n k : Nat) : m / n / k = m / (n * k) := by
-  cases eq_zero_or_pos k with
-  | inl k0 => rw [k0, Nat.mul_zero, Nat.div_zero, Nat.div_zero] | inr kpos => ?_
-  cases eq_zero_or_pos n with
-  | inl n0 => rw [n0, Nat.zero_mul, Nat.div_zero, Nat.zero_div] | inr npos => ?_
-  apply Nat.le_antisymm
-  · apply (le_div_iff_mul_le (Nat.mul_pos npos kpos)).2
-    rw [Nat.mul_comm n k, ← Nat.mul_assoc]
-    apply (le_div_iff_mul_le npos).1
-    apply (le_div_iff_mul_le kpos).1
-    (apply Nat.le_refl)
-  · apply (le_div_iff_mul_le kpos).2
-    apply (le_div_iff_mul_le npos).2
-    rw [Nat.mul_assoc, Nat.mul_comm n k]
-    apply (le_div_iff_mul_le (Nat.mul_pos kpos npos)).1
-    apply Nat.le_refl
-
-protected theorem mul_div_mul_left {m : Nat} (n k : Nat) (H : 0 < m) :
-    m * n / (m * k) = n / k := by rw [← Nat.div_div_eq_div_mul, Nat.mul_div_cancel_left _ H]
-
-protected theorem mul_div_mul_right {m : Nat} (n k : Nat) (H : 0 < m) :
-    n * m / (k * m) = n / k := by rw [Nat.mul_comm, Nat.mul_comm k, Nat.mul_div_mul_left _ _ H]
-
-theorem mul_div_le (m n : Nat) : n * (m / n) ≤ m := by
-  match n, Nat.eq_zero_or_pos n with
-  | _, Or.inl rfl => rw [Nat.zero_mul]; exact m.zero_le
-  | n, Or.inr h => rw [Nat.mul_comm, ← Nat.le_div_iff_mul_le h]; exact Nat.le_refl _
-
 theorem mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 1 :=
   match n % 2, @Nat.mod_lt n 2 (by decide) with
   | 0, _ => .inl rfl
@@ -692,12 +516,6 @@ theorem le_of_mod_lt {a b : Nat} (h : a % b < a) : b ≤ a :=
 theorem mul_mod_mul_right (z x y : Nat) : (x * z) % (y * z) = (x % y) * z := by
   rw [Nat.mul_comm x z, Nat.mul_comm y z, Nat.mul_comm (x % y) z]; apply mul_mod_mul_left
 
-@[simp] theorem mod_mod_of_dvd (a : Nat) (h : c ∣ b) : a % b % c = a % c := by
-  rw (config := {occs := .pos [2]}) [← mod_add_div a b]
-  have ⟨x, h⟩ := h
-  subst h
-  rw [Nat.mul_assoc, add_mul_mod_self_left]
-
 theorem sub_mul_mod {x k n : Nat} (h₁ : n*k ≤ x) : (x - n*k) % n = x % n := by
   match k with
   | 0 => rw [Nat.mul_zero, Nat.sub_zero]
@@ -738,12 +556,6 @@ theorem pow_succ' {m n : Nat} : m ^ n.succ = m * m ^ n := by
 
 @[simp] theorem pow_eq {m n : Nat} : m.pow n = m ^ n := rfl
 
-theorem shiftLeft_eq (a b : Nat) : a <<< b = a * 2 ^ b :=
-  match b with
-  | 0 => (Nat.mul_one _).symm
-  | b+1 => (shiftLeft_eq _ b).trans <| by
-    simp [Nat.pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]
-
 theorem one_shiftLeft (n : Nat) : 1 <<< n = 2 ^ n := by rw [shiftLeft_eq, Nat.one_mul]
 
 attribute [simp] Nat.pow_zero
@@ -883,37 +695,17 @@ theorem lt_log2_self : n < 2 ^ (n.log2 + 1) :=
 
 /-! ### dvd -/
 
-theorem dvd_sub {k m n : Nat} (H : n ≤ m) (h₁ : k ∣ m) (h₂ : k ∣ n) : k ∣ m - n :=
-  (Nat.dvd_add_iff_left h₂).2 <| by rwa [Nat.sub_add_cancel H]
+protected theorem eq_mul_of_div_eq_right {a b c : Nat} (H1 : b ∣ a) (H2 : a / b = c) :
+    a = b * c := by
+  rw [← H2, Nat.mul_div_cancel' H1]
 
-protected theorem mul_dvd_mul {a b c d : Nat} : a ∣ b → c ∣ d → a * c ∣ b * d
-  | ⟨e, he⟩, ⟨f, hf⟩ =>
-    ⟨e * f, by simp [he, hf, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]⟩
+protected theorem div_eq_iff_eq_mul_right {a b c : Nat} (H : 0 < b) (H' : b ∣ a) :
+    a / b = c ↔ a = b * c :=
+  ⟨Nat.eq_mul_of_div_eq_right H', Nat.div_eq_of_eq_mul_right H⟩
 
-protected theorem mul_dvd_mul_left (a : Nat) (h : b ∣ c) : a * b ∣ a * c :=
-  Nat.mul_dvd_mul (Nat.dvd_refl a) h
-
-protected theorem mul_dvd_mul_right (h: a ∣ b) (c : Nat) : a * c ∣ b * c :=
-  Nat.mul_dvd_mul h (Nat.dvd_refl c)
-
-@[simp] theorem dvd_one {n : Nat} : n ∣ 1 ↔ n = 1 :=
-  ⟨eq_one_of_dvd_one, fun h => h.symm ▸ Nat.dvd_refl _⟩
-
-protected theorem mul_div_assoc (m : Nat) (H : k ∣ n) : m * n / k = m * (n / k) := by
-  match Nat.eq_zero_or_pos k with
-  | .inl h0 => rw [h0, Nat.div_zero, Nat.div_zero, Nat.mul_zero]
-  | .inr hpos =>
-    have h1 : m * n / k = m * (n / k * k) / k := by rw [Nat.div_mul_cancel H]
-    rw [h1, ← Nat.mul_assoc, Nat.mul_div_cancel _ hpos]
-
-protected theorem dvd_of_mul_dvd_mul_left
-    (kpos : 0 < k) (H : k * m ∣ k * n) : m ∣ n := by
-  let ⟨l, H⟩ := H
-  rw [Nat.mul_assoc] at H
-  exact ⟨_, Nat.eq_of_mul_eq_mul_left kpos H⟩
-
-protected theorem dvd_of_mul_dvd_mul_right (kpos : 0 < k) (H : m * k ∣ n * k) : m ∣ n := by
-  rw [Nat.mul_comm m k, Nat.mul_comm n k] at H; exact Nat.dvd_of_mul_dvd_mul_left kpos H
+protected theorem div_eq_iff_eq_mul_left {a b c : Nat} (H : 0 < b) (H' : b ∣ a) :
+    a / b = c ↔ a = c * b := by
+  rw [Nat.mul_comm]; exact Nat.div_eq_iff_eq_mul_right H H'
 
 theorem pow_dvd_pow_iff_pow_le_pow {k l : Nat} :
     ∀ {x : Nat}, 0 < x → (x ^ k ∣ x ^ l ↔ x ^ k ≤ x ^ l)
@@ -937,18 +729,6 @@ theorem pow_dvd_pow_iff_le_right {x k l : Nat} (w : 1 < x) : x ^ k ∣ x ^ l ↔
 theorem pow_dvd_pow_iff_le_right' {b k l : Nat} : (b + 2) ^ k ∣ (b + 2) ^ l ↔ k ≤ l :=
   pow_dvd_pow_iff_le_right (Nat.lt_of_sub_eq_succ rfl)
 
-protected theorem eq_mul_of_div_eq_right {a b c : Nat} (H1 : b ∣ a) (H2 : a / b = c) :
-    a = b * c := by
-  rw [← H2, Nat.mul_div_cancel' H1]
-
-protected theorem div_eq_iff_eq_mul_right {a b c : Nat} (H : 0 < b) (H' : b ∣ a) :
-    a / b = c ↔ a = b * c :=
-  ⟨Nat.eq_mul_of_div_eq_right H', Nat.div_eq_of_eq_mul_right H⟩
-
-protected theorem div_eq_iff_eq_mul_left {a b c : Nat} (H : 0 < b) (H' : b ∣ a) :
-    a / b = c ↔ a = c * b := by
-  rw [Nat.mul_comm]; exact Nat.div_eq_iff_eq_mul_right H H'
-
 protected theorem pow_dvd_pow {m n : Nat} (a : Nat) (h : m ≤ n) : a ^ m ∣ a ^ n := by
   cases Nat.exists_eq_add_of_le h
   case intro k p =>
@@ -983,10 +763,6 @@ theorem shiftLeft_succ : ∀(m n), m <<< (n + 1) = 2 * (m <<< n)
   rw [shiftLeft_succ_inside _ (k+1)]
   rw [shiftLeft_succ _ k, shiftLeft_succ_inside]
 
-@[simp] theorem shiftRight_zero : n >>> 0 = n := rfl
-
-theorem shiftRight_succ (m n) : m >>> (n + 1) = (m >>> n) / 2 := rfl
-
 /-- Shiftright on successor with division moved inside. -/
 theorem shiftRight_succ_inside : ∀m n, m >>> (n+1) = (m/2) >>> n
 | m, 0 => rfl
@@ -1002,19 +778,9 @@ theorem shiftRight_succ_inside : ∀m n, m >>> (n+1) = (m/2) >>> n
   | 0 => by simp [shiftRight]
   | n + 1 => by simp [shiftRight, zero_shiftRight n, shiftRight_succ]
 
-theorem shiftRight_add (m n : Nat) : ∀ k, m >>> (n + k) = (m >>> n) >>> k
-  | 0 => rfl
-  | k + 1 => by simp [add_succ, shiftRight_add, shiftRight_succ]
-
 theorem shiftLeft_shiftLeft (m n : Nat) : ∀ k, (m <<< n) <<< k = m <<< (n + k)
   | 0 => rfl
-  | k + 1 => by simp [add_succ, shiftLeft_shiftLeft _ _ k, shiftLeft_succ]
-
-theorem shiftRight_eq_div_pow (m : Nat) : ∀ n, m >>> n = m / 2 ^ n
-  | 0 => (Nat.div_one _).symm
-  | k + 1 => by
-    rw [shiftRight_add, shiftRight_eq_div_pow m k]
-    simp [Nat.div_div_eq_div_mul, ← Nat.pow_succ, shiftRight_succ]
+  | k + 1 => by simp [← Nat.add_assoc, shiftLeft_shiftLeft _ _ k, shiftLeft_succ]
 
 theorem mul_add_div {m : Nat} (m_pos : m > 0) (x y : Nat) : (m * x + y) / m = x + y / m := by
   match x with

src/Init/Data/Nat/Linear.lean

@@ -4,10 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
-import Init.Coe
 import Init.ByCases
-import Init.Data.Nat.Basic
-import Init.Data.List.Basic
 import Init.Data.Prod
 
 namespace Nat.Linear
@@ -583,7 +580,7 @@ attribute [-simp] Nat.right_distrib Nat.left_distrib
 
 theorem PolyCnstr.denote_mul (ctx : Context) (k : Nat) (c : PolyCnstr) : (c.mul (k+1)).denote ctx = c.denote ctx := by
   cases c; rename_i eq lhs rhs
-  have : k ≠ 0 → k + 1 ≠ 1 := by intro h; match k with | 0 => contradiction | k+1 => simp; apply Nat.succ_ne_zero
+  have : k ≠ 0 → k + 1 ≠ 1 := by intro h; match k with | 0 => contradiction | k+1 => simp
   have : ¬ (k == 0) → (k + 1 == 1) = false := fun h => beq_false_of_ne (this (ne_of_beq_false (Bool.of_not_eq_true h)))
   have : ¬ ((k + 1 == 0) = true)  := fun h => absurd (eq_of_beq h) (Nat.succ_ne_zero k)
   have : (1 == (0 : Nat)) = false := rfl

src/Init/Data/Nat/Log2.lean

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Gabriel Ebner
 -/
 prelude
-import Init.NotationExtra
 import Init.Data.Nat.Linear
 
 namespace Nat

src/Init/Data/Option/Basic.lean

@@ -37,6 +37,13 @@ def toMonad [Monad m] [Alternative m] : Option α → m α
   | none,   _ => none
   | some a, b => b a
 
+/-- Runs `f` on `o`'s value, if any, and returns its result, or else returns `none`.  -/
+@[inline] protected def bindM [Monad m] (f : α → m (Option β)) (o : Option α) : m (Option β) := do
+  if let some a := o then
+    return (← f a)
+  else
+    return none
+
 @[inline] protected def mapM [Monad m] (f : α → m β) (o : Option α) : m (Option β) := do
   if let some a := o then
     return some (← f a)

src/Init/Data/Ord.lean

@@ -5,8 +5,8 @@ Authors: Dany Fabian, Sebastian Ullrich
 -/
 
 prelude
-import Init.Data.Int
 import Init.Data.String
+import Init.Data.Array.Basic
 
 inductive Ordering where
   | lt | eq | gt
@@ -88,11 +88,24 @@ def isGE : Ordering → Bool
 
 end Ordering
 
+/--
+Yields an `Ordering` s.t. `x < y` corresponds to `Ordering.lt` / `Ordering.gt` and
+`x = y` corresponds to `Ordering.eq`.
+-/
 @[inline] def compareOfLessAndEq {α} (x y : α) [LT α] [Decidable (x < y)] [DecidableEq α] : Ordering :=
   if x < y then Ordering.lt
   else if x = y then Ordering.eq
   else Ordering.gt
 
+/--
+Yields an `Ordering` s.t. `x < y` corresponds to `Ordering.lt` / `Ordering.gt` and
+`x == y` corresponds to `Ordering.eq`.
+-/
+@[inline] def compareOfLessAndBEq {α} (x y : α) [LT α] [Decidable (x < y)] [BEq α] : Ordering :=
+  if x < y then .lt
+  else if x == y then .eq
+  else .gt
+
 /--
 Compare `a` and `b` lexicographically by `cmp₁` and `cmp₂`. `a` and `b` are
 first compared by `cmp₁`. If this returns 'equal', `a` and `b` are compared
@@ -148,6 +161,13 @@ instance : Ord USize where
 instance : Ord Char where
   compare x y := compareOfLessAndEq x y
 
+instance [Ord α] : Ord (Option α) where
+  compare
+  | none,   none   => .eq
+  | none,   some _ => .lt
+  | some _, none   => .gt
+  | some x, some y => compare x y
+
 /-- The lexicographic order on pairs. -/
 def lexOrd [Ord α] [Ord β] : Ord (α × β) where
   compare p1 p2 := match compare p1.1 p2.1 with
@@ -211,4 +231,13 @@ returns 'equal', by `ord₂`.
 protected def lex' (ord₁ ord₂ : Ord α) : Ord α where
   compare := compareLex ord₁.compare ord₂.compare
 
+/--
+Creates an order which compares elements of an `Array` in lexicographic order.
+-/
+protected def arrayOrd [a : Ord α] : Ord (Array α) where
+  compare x y :=
+    let _ : LT α := a.toLT
+    let _ : BEq α := a.toBEq
+    compareOfLessAndBEq x.toList y.toList
+
 end Ord

src/Init/Data/Random.lean

@@ -5,7 +5,6 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.System.IO
-import Init.Data.Int
 universe u
 
 /-!

src/Init/NotationExtra.lean

@@ -9,6 +9,7 @@ prelude
 import Init.Meta
 import Init.Data.Array.Subarray
 import Init.Data.ToString
+import Init.Conv
 namespace Lean
 
 macro "Macro.trace[" id:ident "]" s:interpolatedStr(term) : term =>
@@ -123,7 +124,7 @@ calc abc
   _ = xyz := pwxyz
 ```
 
-`calc` has term mode and tactic mode variants. This is the term mode variant.
+`calc` works as a term, as a tactic or as a `conv` tactic.
 
 See [Theorem Proving in Lean 4][tpil4] for more information.
 
@@ -131,45 +132,13 @@ See [Theorem Proving in Lean 4][tpil4] for more information.
 -/
 syntax (name := calc) "calc" calcSteps : term
 
-/-- Step-wise reasoning over transitive relations.
-```
-calc
-  a = b := pab
-  b = c := pbc
-  ...
-  y = z := pyz
-```
-proves `a = z` from the given step-wise proofs. `=` can be replaced with any
-relation implementing the typeclass `Trans`. Instead of repeating the right-
-hand sides, subsequent left-hand sides can be replaced with `_`.
-```
-calc
-  a = b := pab
-  _ = c := pbc
-  ...
-  _ = z := pyz
-```
-It is also possible to write the *first* relation as `<lhs>\n  _ = <rhs> :=
-<proof>`. This is useful for aligning relation symbols:
-```
-calc abc
-  _ = bce := pabce
-  _ = cef := pbcef
-  ...
-  _ = xyz := pwxyz
-```
-
-`calc` has term mode and tactic mode variants. This is the tactic mode variant,
-which supports an additional feature: it works even if the goal is `a = z'`
-for some other `z'`; in this case it will not close the goal but will instead
-leave a subgoal proving `z = z'`.
-
-See [Theorem Proving in Lean 4][tpil4] for more information.
-
-[tpil4]: https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#calculational-proofs
--/
+@[inherit_doc «calc»]
 syntax (name := calcTactic) "calc" calcSteps : tactic
 
+@[inherit_doc «calc»]
+macro tk:"calc" steps:calcSteps : conv =>
+  `(conv| tactic => calc%$tk $steps)
+
 @[app_unexpander Unit.unit] def unexpandUnit : Lean.PrettyPrinter.Unexpander
   | `($(_)) => `(())
 

src/Init/Omega/Int.lean

@@ -4,9 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 -/
 prelude
+import Init.Data.Int.DivMod
 import Init.Data.Int.Order
-import Init.Data.Int.DivModLemmas
-import Init.Data.Nat.Lemmas
 
 /-!
 # Lemmas about `Nat`, `Int`, and `Fin` needed internally by `omega`.
@@ -49,7 +48,7 @@ theorem ofNat_shiftLeft_eq {x y : Nat} : (x <<< y : Int) = (x : Int) * (2 ^ y :
   simp [Nat.shiftLeft_eq]
 
 theorem ofNat_shiftRight_eq_div_pow {x y : Nat} : (x >>> y : Int) = (x : Int) / (2 ^ y : Nat) := by
-  simp [Nat.shiftRight_eq_div_pow]
+  simp only [Nat.shiftRight_eq_div_pow, Int.ofNat_ediv]
 
 -- FIXME these are insane:
 theorem lt_of_not_ge {x y : Int} (h : ¬ (x ≤ y)) : y < x := Int.not_le.mp h

src/Init/Omega/IntList.lean

@@ -5,6 +5,8 @@ Authors: Scott Morrison
 -/
 prelude
 import Init.Data.List.Lemmas
+import Init.Data.Int.DivModLemmas
+import Init.Data.Nat.Gcd
 
 namespace Lean.Omega
 

src/Init/Prelude.lean

@@ -1485,6 +1485,7 @@ instance [ShiftRight α] : HShiftRight α α α where
   hShiftRight a b := ShiftRight.shiftRight a b
 
 open HAdd (hAdd)
+open HSub (hSub)
 open HMul (hMul)
 open HPow (hPow)
 open HAppend (hAppend)
@@ -2035,7 +2036,7 @@ instance : Inhabited UInt64 where
   default := UInt64.ofNatCore 0 (by decide)
 
 /--
-The size of type `UInt16`, that is, `2^System.Platform.numBits`, which may
+The size of type `USize`, that is, `2^System.Platform.numBits`, which may
 be either `2^32` or `2^64` depending on the platform's architecture.
 
 Remark: we define `USize.size` using `(2^numBits - 1) + 1` to ensure the
@@ -2053,7 +2054,7 @@ instance : OfNat (Fin (n+1)) i where
   ofNat := Fin.ofNat i
 ```
 -/
-abbrev USize.size : Nat := Nat.succ (Nat.sub (hPow 2 System.Platform.numBits) 1)
+abbrev USize.size : Nat := hAdd (hSub (hPow 2 System.Platform.numBits) 1) 1
 
 theorem usize_size_eq : Or (Eq USize.size 4294967296) (Eq USize.size 18446744073709551616) :=
   show Or (Eq (Nat.succ (Nat.sub (hPow 2 System.Platform.numBits) 1)) 4294967296) (Eq (Nat.succ (Nat.sub (hPow 2 System.Platform.numBits) 1)) 18446744073709551616) from
@@ -4580,6 +4581,12 @@ def resolveNamespace (n : Name) : MacroM (List Name) := do
 Resolves the given name to an overload list of global definitions.
 The `List String` in each alternative is the deduced list of projections
 (which are ambiguous with name components).
+
+Remark: it will not trigger actions associated with reserved names. Recall that Lean
+has reserved names. For example, a definition `foo` has a reserved name `foo.def` for theorem
+containing stating that `foo` is equal to its definition. The action associated with `foo.def`
+automatically proves the theorem. At the macro level, the name is resolved, but the action is not
+executed. The actions are executed by the elaborator when converting `Syntax` into `Expr`.
 -/
 def resolveGlobalName (n : Name) : MacroM (List (Prod Name (List String))) := do
   (← getMethods).resolveGlobalName n

src/Init/PropLemmas.lean

@@ -11,6 +11,21 @@ import Init.Core
 import Init.NotationExtra
 set_option linter.missingDocs true -- keep it documented
 
+/-! ## cast and equality -/
+
+@[simp] theorem eq_mp_eq_cast  (h : α = β) : Eq.mp  h = cast h := rfl
+@[simp] theorem eq_mpr_eq_cast (h : α = β) : Eq.mpr h = cast h.symm := rfl
+
+@[simp] theorem cast_cast : ∀ (ha : α = β) (hb : β = γ) (a : α),
+    cast hb (cast ha a) = cast (ha.trans hb) a
+  | rfl, rfl, _ => rfl
+
+@[simp] theorem eq_true_eq_id : Eq True = id := by
+  funext _; simp only [true_iff, id_def, eq_iff_iff]
+
+theorem proof_irrel_heq {p q : Prop} (hp : p) (hq : q) : HEq hp hq := by
+  cases propext (iff_of_true hp hq); rfl
+
 /-! ## not -/
 
 theorem not_not_em (a : Prop) : ¬¬(a ∨ ¬a) := fun h => h (.inr (h ∘ .inl))
@@ -104,10 +119,62 @@ theorem and_or_right : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) := by rw [@or
 
 theorem or_imp : (a ∨ b → c) ↔ (a → c) ∧ (b → c) :=
   Iff.intro (fun h => ⟨h ∘ .inl, h ∘ .inr⟩) (fun ⟨ha, hb⟩ => Or.rec ha hb)
-theorem not_or : ¬(p ∨ q) ↔ ¬p ∧ ¬q := or_imp
+
+/-
+`not_or` is made simp for confluence with `¬((b || c) = true)`:
+
+Critical pair:
+1. `¬(b = true ∨ c = true)` via `Bool.or_eq_true`.
+2. `(b || c = false)` via `Bool.not_eq_true` which then
+   reduces to `b = false ∧ c = false` via Mathlib simp lemma
+   `Bool.or_eq_false_eq_eq_false_and_eq_false`.
+
+Both reduce to `b = false ∧ c = false` via `not_or`.
+-/
+@[simp] theorem not_or : ¬(p ∨ q) ↔ ¬p ∧ ¬q := or_imp
 
 theorem not_and_of_not_or_not (h : ¬a ∨ ¬b) : ¬(a ∧ b) := h.elim (mt (·.1)) (mt (·.2))
 
+
+/-! ## Ite -/
+
+@[simp]
+theorem if_false_left [h : Decidable p] :
+    ite p False q ↔ ¬p ∧ q := by cases h <;> (rename_i g; simp [g])
+
+@[simp]
+theorem if_false_right [h : Decidable p] :
+    ite p q False ↔ p ∧ q := by cases h <;> (rename_i g; simp [g])
+
+/-
+`if_true_left` and `if_true_right` are lower priority because
+they introduce disjunctions and we prefer `if_false_left` and
+`if_false_right` if they overlap.
+-/
+
+@[simp low]
+theorem if_true_left [h : Decidable p] :
+    ite p True q ↔ ¬p → q := by cases h <;> (rename_i g; simp [g])
+
+@[simp low]
+theorem if_true_right [h : Decidable p] :
+    ite p q True ↔ p → q := by cases h <;> (rename_i g; simp [g])
+
+/-- Negation of the condition `P : Prop` in a `dite` is the same as swapping the branches. -/
+@[simp] theorem dite_not [hn : Decidable (¬p)] [h : Decidable p]  (x : ¬p → α) (y : ¬¬p → α) :
+    dite (¬p) x y = dite p (fun h => y (not_not_intro h)) x := by
+  cases h <;> (rename_i g; simp [g])
+
+/-- Negation of the condition `P : Prop` in a `ite` is the same as swapping the branches. -/
+@[simp] theorem ite_not (p : Prop) [Decidable p] (x y : α) : ite (¬p) x y = ite p y x :=
+  dite_not (fun _ => x) (fun _ => y)
+
+@[simp] theorem ite_true_same (p q : Prop) [h : Decidable p] : (if p then p else q) = (¬p → q) := by
+  cases h <;> (rename_i g; simp [g])
+
+@[simp] theorem ite_false_same (p q : Prop) [h : Decidable p] : (if p then q else p) = (p ∧ q) := by
+  cases h <;> (rename_i g; simp [g])
+
 /-! ## exists and forall -/
 
 section quantifiers
@@ -268,7 +335,14 @@ end quantifiers
 
 /-! ## decidable -/
 
-theorem Decidable.not_not [Decidable p] : ¬¬p ↔ p := ⟨of_not_not, not_not_intro⟩
+@[simp] theorem Decidable.not_not [Decidable p] : ¬¬p ↔ p := ⟨of_not_not, not_not_intro⟩
+
+/-- Excluded middle.  Added as alias for Decidable.em -/
+abbrev Decidable.or_not_self := em
+
+/-- Excluded middle commuted.  Added as alias for Decidable.em -/
+theorem Decidable.not_or_self (p : Prop) [h : Decidable p] : ¬p ∨ p := by
+  cases h <;> simp [*]
 
 theorem Decidable.by_contra [Decidable p] : (¬p → False) → p := of_not_not
 
@@ -310,7 +384,7 @@ theorem Decidable.not_imp_symm [Decidable a] (h : ¬a → b) (hb : ¬b) : a :=
 theorem Decidable.not_imp_comm [Decidable a] [Decidable b] : (¬a → b) ↔ (¬b → a) :=
   ⟨not_imp_symm, not_imp_symm⟩
 
-theorem Decidable.not_imp_self [Decidable a] : (¬a → a) ↔ a := by
+@[simp] theorem Decidable.not_imp_self [Decidable a] : (¬a → a) ↔ a := by
   have := @imp_not_self (¬a); rwa [not_not] at this
 
 theorem Decidable.or_iff_not_imp_left [Decidable a] : a ∨ b ↔ (¬a → b) :=
@@ -389,8 +463,12 @@ theorem Decidable.and_iff_not_or_not [Decidable a] [Decidable b] : a ∧ b ↔
   rw [← not_and_iff_or_not_not, not_not]
 
 theorem Decidable.imp_iff_right_iff [Decidable a] : (a → b ↔ b) ↔ a ∨ b :=
-  ⟨fun H => (Decidable.em a).imp_right fun ha' => H.1 fun ha => (ha' ha).elim,
-   fun H => H.elim imp_iff_right fun hb => iff_of_true (fun _ => hb) hb⟩
+  Iff.intro
+    (fun h => (Decidable.em a).imp_right fun ha' => h.mp fun ha => (ha' ha).elim)
+    (fun ab => ab.elim imp_iff_right fun hb => iff_of_true (fun _ => hb) hb)
+
+theorem Decidable.imp_iff_left_iff  [Decidable a] : (b ↔ a → b) ↔ a ∨ b :=
+  propext (@Iff.comm (a → b) b) ▸ (@Decidable.imp_iff_right_iff a b _)
 
 theorem Decidable.and_or_imp [Decidable a] : a ∧ b ∨ (a → c) ↔ a → b ∨ c :=
   if ha : a then by simp only [ha, true_and, true_imp_iff]
@@ -435,3 +513,53 @@ protected theorem Decidable.not_forall_not {p : α → Prop} [Decidable (∃ x,
 protected theorem Decidable.not_exists_not {p : α → Prop} [∀ x, Decidable (p x)] :
     (¬∃ x, ¬p x) ↔ ∀ x, p x := by
   simp only [not_exists, Decidable.not_not]
+
+export Decidable (not_imp_self)
+
+/-
+`decide_implies` simp justification.
+
+We have a critical pair from `decide (¬(p ∧ q))`:
+
+ 1. `decide (p → ¬q)` via `not_and`
+ 2. `!decide (p ∧ q)` via `decide_not` This further refines to
+    `!(decide p) || !(decide q)` via `Bool.decide_and` (in Mathlib) and
+    `Bool.not_and` (made simp in Mathlib).
+
+We introduce `decide_implies` below and then both normalize to
+`!(decide p) || !(decide q)`.
+-/
+@[simp]
+theorem decide_implies (u v : Prop)
+    [duv : Decidable (u → v)] [du : Decidable u] {dv : u → Decidable v}
+  : decide (u → v) = dite u (fun h => @decide v (dv h)) (fun _ => true) :=
+  if h : u then by
+    simp [h]
+  else by
+    simp [h]
+
+/-
+`decide_ite` is needed to resolve critical pair with
+
+We have a critical pair from `decide (ite p b c = true)`:
+
+ 1. `ite p b c` via `decide_coe`
+ 2. `decide (ite p (b = true) (c = true))` via `Bool.ite_eq_true_distrib`.
+
+We introduce `decide_ite` so both normalize to `ite p b c`.
+-/
+@[simp]
+theorem decide_ite (u : Prop) [du : Decidable u] (p q : Prop)
+      [dpq : Decidable (ite u p q)] [dp : Decidable p] [dq : Decidable q] :
+    decide (ite u p q) = ite u (decide p) (decide q) := by
+  cases du <;> simp [*]
+
+/- Confluence for `ite_true_same` and `decide_ite`. -/
+@[simp] theorem ite_true_decide_same (p : Prop) [h : Decidable p] (b : Bool) :
+  (if p then decide p else b) = (decide p || b) := by
+  cases h <;> (rename_i pt; simp [pt])
+
+/- Confluence for `ite_false_same` and `decide_ite`. -/
+@[simp] theorem ite_false_decide_same (p : Prop) [h : Decidable p] (b : Bool) :
+  (if p then b else decide p) = (decide p && b) := by
+  cases h <;> (rename_i pt; simp [pt])

src/Init/SimpLemmas.lean

@@ -15,12 +15,15 @@ theorem of_eq_false (h : p = False) : ¬ p := fun hp => False.elim (h.mp hp)
 theorem eq_true (h : p) : p = True :=
   propext ⟨fun _ => trivial, fun _ => h⟩
 
+-- Adding this attribute needs `eq_true`.
+attribute [simp] cast_heq
+
 theorem eq_false (h : ¬ p) : p = False :=
   propext ⟨fun h' => absurd h' h, fun h' => False.elim h'⟩
 
 theorem eq_false' (h : p → False) : p = False := eq_false h
 
-theorem eq_true_of_decide {p : Prop} {_ : Decidable p} (h : decide p = true) : p = True :=
+theorem eq_true_of_decide {p : Prop} [Decidable p] (h : decide p = true) : p = True :=
   eq_true (of_decide_eq_true h)
 
 theorem eq_false_of_decide {p : Prop} {_ : Decidable p} (h : decide p = false) : p = False :=
@@ -124,6 +127,7 @@ end SimprocHelperLemmas
 @[simp] theorem not_true_eq_false : (¬ True) = False := by decide
 
 @[simp] theorem not_iff_self : ¬(¬a ↔ a) | H => iff_not_self H.symm
+attribute [simp] iff_not_self
 
 /-! ## and -/
 
@@ -173,6 +177,11 @@ theorem or_iff_left_of_imp  (hb : b → a) : (a ∨ b) ↔ a  := Iff.intro (Or.r
 @[simp] theorem or_iff_left_iff_imp  : (a ∨ b ↔ a) ↔ (b → a) := Iff.intro (·.mp ∘ Or.inr) or_iff_left_of_imp
 @[simp] theorem or_iff_right_iff_imp : (a ∨ b ↔ b) ↔ (a → b) := by rw [or_comm, or_iff_left_iff_imp]
 
+@[simp] theorem iff_self_or (a b : Prop) : (a ↔ a ∨ b) ↔ (b → a) :=
+  propext (@Iff.comm _ a) ▸ @or_iff_left_iff_imp a b
+@[simp] theorem iff_or_self (a b : Prop) : (b ↔ a ∨ b) ↔ (a → b) :=
+  propext (@Iff.comm _ b) ▸ @or_iff_right_iff_imp a b
+
 /-# Bool -/
 
 @[simp] theorem Bool.or_false (b : Bool) : (b || false) = b  := by cases b <;> rfl
@@ -199,9 +208,9 @@ theorem Bool.or_assoc (a b c : Bool) : (a || b || c) = (a || (b || c)) := by
 @[simp] theorem Bool.not_not (b : Bool) : (!!b) = b := by cases b <;> rfl
 @[simp] theorem Bool.not_true  : (!true) = false := by decide
 @[simp] theorem Bool.not_false : (!false) = true := by decide
-@[simp] theorem Bool.not_beq_true (b : Bool) : (!(b == true)) = (b == false) := by cases b <;> rfl
+@[simp] theorem Bool.not_beq_true  (b : Bool) : (!(b == true)) = (b == false) := by cases b <;> rfl
 @[simp] theorem Bool.not_beq_false (b : Bool) : (!(b == false)) = (b == true) := by cases b <;> rfl
-@[simp] theorem Bool.not_eq_true' (b : Bool) : ((!b) = true) = (b = false) := by cases b <;> simp
+@[simp] theorem Bool.not_eq_true'  (b : Bool) : ((!b) = true) = (b = false) := by cases b <;> simp
 @[simp] theorem Bool.not_eq_false' (b : Bool) : ((!b) = false) = (b = true) := by cases b <;> simp
 
 @[simp] theorem Bool.beq_to_eq (a b : Bool) :
@@ -212,11 +221,14 @@ theorem Bool.or_assoc (a b c : Bool) : (a || b || c) = (a || (b || c)) := by
 @[simp] theorem Bool.not_eq_true (b : Bool) : (¬(b = true)) = (b = false) := by cases b <;> decide
 @[simp] theorem Bool.not_eq_false (b : Bool) : (¬(b = false)) = (b = true) := by cases b <;> decide
 
-@[simp] theorem decide_eq_true_eq {_ : Decidable p} : (decide p = true) = p := propext <| Iff.intro of_decide_eq_true decide_eq_true
-@[simp] theorem decide_not {h : Decidable p} : decide (¬ p) = !decide p := by cases h <;> rfl
-@[simp] theorem not_decide_eq_true {h : Decidable p} : ((!decide p) = true) = ¬ p := by cases h <;> simp [decide, *]
+@[simp] theorem decide_eq_true_eq [Decidable p] : (decide p = true) = p :=
+  propext <| Iff.intro of_decide_eq_true decide_eq_true
+@[simp] theorem decide_not [g : Decidable p] [h : Decidable (Not p)] : decide (Not p) = !(decide p) := by
+  cases g <;> (rename_i gp; simp [gp]; rfl)
+@[simp] theorem not_decide_eq_true [h : Decidable p] : ((!decide p) = true) = ¬ p := by
+  cases h <;> (rename_i hp; simp [decide, hp])
 
-@[simp] theorem heq_eq_eq {α : Sort u} (a b : α) : HEq a b = (a = b) := propext <| Iff.intro eq_of_heq heq_of_eq
+@[simp] theorem heq_eq_eq (a b : α) : HEq a b = (a = b) := propext <| Iff.intro eq_of_heq heq_of_eq
 
 @[simp] theorem cond_true (a b : α) : cond true a b = a := rfl
 @[simp] theorem cond_false (a b : α) : cond false a b = b := rfl
@@ -228,11 +240,29 @@ theorem Bool.or_assoc (a b c : Bool) : (a || b || c) = (a || (b || c)) := by
 @[simp] theorem bne_self_eq_false' [DecidableEq α] (a : α) : (a != a) = false := by simp [bne]
 
 @[simp] theorem decide_False : decide False = false := rfl
-@[simp] theorem decide_True : decide True = true := rfl
+@[simp] theorem decide_True  : decide True  = true := rfl
 
 @[simp] theorem bne_iff_ne [BEq α] [LawfulBEq α] (a b : α) : a != b ↔ a ≠ b := by
   simp [bne]; rw [← beq_iff_eq a b]; simp [-beq_iff_eq]
 
+/-
+Added for critical pair for `¬((a != b) = true)`
+
+1. `(a != b) = false` via `Bool.not_eq_true`
+2. `¬(a ≠ b)` via `bne_iff_ne`
+
+These will both normalize to `a = b` with the first via `bne_eq_false_iff_eq`.
+-/
+@[simp] theorem beq_eq_false_iff_ne [BEq α] [LawfulBEq α]
+    (a b : α) : (a == b) = false ↔ a ≠ b := by
+  rw [ne_eq, ← beq_iff_eq a b]
+  cases a == b <;> decide
+
+@[simp] theorem bne_eq_false_iff_eq [BEq α] [LawfulBEq α] (a b : α) :
+    (a != b) = false ↔ a = b := by
+  rw [bne, ← beq_iff_eq a b]
+  cases a == b <;> decide
+
 /-# Nat -/
 
 @[simp] theorem Nat.le_zero_eq (a : Nat) : (a ≤ 0) = (a = 0) :=

src/Init/Simproc.lean

@@ -31,22 +31,43 @@ Simplification procedures can be also scoped or local.
 -/
 syntax (docComment)? attrKind "simproc " (Tactic.simpPre <|> Tactic.simpPost)? ("[" ident,* "]")? ident " (" term ")" " := " term : command
 
+/--
+Similar to `simproc`, but resulting expression must be definitionally equal to the input one.
+-/
+syntax (docComment)? attrKind "dsimproc " (Tactic.simpPre <|> Tactic.simpPost)? ("[" ident,* "]")? ident " (" term ")" " := " term : command
+
 /--
 A user-defined simplification procedure declaration. To activate this procedure in `simp` tactic,
 we must provide it as an argument, or use the command `attribute` to set its `[simproc]` attribute.
 -/
 syntax (docComment)? "simproc_decl " ident " (" term ")" " := " term : command
 
+/--
+A user-defined defeq simplification procedure declaration. To activate this procedure in `simp` tactic,
+we must provide it as an argument, or use the command `attribute` to set its `[simproc]` attribute.
+-/
+syntax (docComment)? "dsimproc_decl " ident " (" term ")" " := " term : command
+
 /--
 A builtin simplification procedure.
 -/
 syntax (docComment)? attrKind "builtin_simproc " (Tactic.simpPre <|> Tactic.simpPost)? ("[" ident,* "]")? ident " (" term ")" " := " term : command
 
+/--
+A builtin defeq simplification procedure.
+-/
+syntax (docComment)? attrKind "builtin_dsimproc " (Tactic.simpPre <|> Tactic.simpPost)? ("[" ident,* "]")? ident " (" term ")" " := " term : command
+
 /--
 A builtin simplification procedure declaration.
 -/
 syntax (docComment)? "builtin_simproc_decl " ident " (" term ")" " := " term : command
 
+/--
+A builtin defeq simplification procedure declaration.
+-/
+syntax (docComment)? "builtin_dsimproc_decl " ident " (" term ")" " := " term : command
+
 /--
 Auxiliary command for associating a pattern with a simplification procedure.
 -/
@@ -86,33 +107,60 @@ macro_rules
     `($[$doc?:docComment]? def $n:ident : $(mkIdent simprocType) := $body
       simproc_pattern% $pattern => $n)
 
+macro_rules
+  | `($[$doc?:docComment]? dsimproc_decl $n:ident ($pattern:term) := $body) => do
+    let simprocType := `Lean.Meta.Simp.DSimproc
+    `($[$doc?:docComment]? def $n:ident : $(mkIdent simprocType) := $body
+      simproc_pattern% $pattern => $n)
+
 macro_rules
   | `($[$doc?:docComment]? builtin_simproc_decl $n:ident ($pattern:term) := $body) => do
     let simprocType := `Lean.Meta.Simp.Simproc
     `($[$doc?:docComment]? def $n:ident : $(mkIdent simprocType) := $body
       builtin_simproc_pattern% $pattern => $n)
 
+macro_rules
+  | `($[$doc?:docComment]? builtin_dsimproc_decl $n:ident ($pattern:term) := $body) => do
+    let simprocType := `Lean.Meta.Simp.DSimproc
+    `($[$doc?:docComment]? def $n:ident : $(mkIdent simprocType) := $body
+      builtin_simproc_pattern% $pattern => $n)
+
+private def mkAttributeCmds
+    (kind : TSyntax `Lean.Parser.Term.attrKind)
+    (pre? : Option (TSyntax [`Lean.Parser.Tactic.simpPre, `Lean.Parser.Tactic.simpPost]))
+    (ids? : Option (Syntax.TSepArray `ident ","))
+    (n : Ident) : MacroM (Array Syntax) := do
+  let mut cmds := #[]
+  let pushDefault (cmds : Array (TSyntax `command)) : MacroM (Array (TSyntax `command)) := do
+    return cmds.push (← `(attribute [$kind simproc $[$pre?]?] $n))
+  if let some ids := ids? then
+    for id in ids.getElems do
+      let idName := id.getId
+      let (attrName, attrKey) :=
+        if idName == `simp then
+          (`simprocAttr, "simproc")
+        else if idName == `seval then
+          (`sevalprocAttr, "sevalproc")
+        else
+          let idName := idName.appendAfter "_proc"
+          (`Parser.Attr ++ idName, idName.toString)
+      let attrStx : TSyntax `attr := ⟨mkNode attrName #[mkAtom attrKey, mkOptionalNode pre?]⟩
+      cmds := cmds.push (← `(attribute [$kind $attrStx] $n))
+  else
+    cmds ← pushDefault cmds
+  return cmds
+
 macro_rules
   | `($[$doc?:docComment]? $kind:attrKind simproc $[$pre?]? $[ [ $ids?:ident,* ] ]? $n:ident ($pattern:term) := $body) => do
-     let mut cmds := #[(← `($[$doc?:docComment]? simproc_decl $n ($pattern) := $body))]
-     let pushDefault (cmds : Array (TSyntax `command)) : MacroM (Array (TSyntax `command)) := do
-       return cmds.push (← `(attribute [$kind simproc $[$pre?]?] $n))
-     if let some ids := ids? then
-       for id in ids.getElems do
-         let idName := id.getId
-         let (attrName, attrKey) :=
-           if idName == `simp then
-             (`simprocAttr, "simproc")
-           else if idName == `seval then
-             (`sevalprocAttr, "sevalproc")
-           else
-             let idName := idName.appendAfter "_proc"
-             (`Parser.Attr ++ idName, idName.toString)
-         let attrStx : TSyntax `attr := ⟨mkNode attrName #[mkAtom attrKey, mkOptionalNode pre?]⟩
-         cmds := cmds.push (← `(attribute [$kind $attrStx] $n))
-     else
-       cmds ← pushDefault cmds
-     return mkNullNode cmds
+     return mkNullNode <|
+       #[(← `($[$doc?:docComment]? simproc_decl $n ($pattern) := $body))]
+       ++ (← mkAttributeCmds kind pre? ids? n)
+
+macro_rules
+  | `($[$doc?:docComment]? $kind:attrKind dsimproc $[$pre?]? $[ [ $ids?:ident,* ] ]? $n:ident ($pattern:term) := $body) => do
+     return mkNullNode <|
+       #[(← `($[$doc?:docComment]? dsimproc_decl $n ($pattern) := $body))]
+       ++ (← mkAttributeCmds kind pre? ids? n)
 
 macro_rules
   | `($[$doc?:docComment]? $kind:attrKind builtin_simproc $[$pre?]? $n:ident ($pattern:term) := $body) => do
@@ -126,4 +174,16 @@ macro_rules
       attribute [$kind builtin_simproc $[$pre?]?] $n
       attribute [$kind builtin_sevalproc $[$pre?]?] $n)
 
+macro_rules
+  | `($[$doc?:docComment]? $kind:attrKind builtin_dsimproc $[$pre?]? $n:ident ($pattern:term) := $body) => do
+    `($[$doc?:docComment]? builtin_dsimproc_decl $n ($pattern) := $body
+      attribute [$kind builtin_simproc $[$pre?]?] $n)
+  | `($[$doc?:docComment]? $kind:attrKind builtin_dsimproc $[$pre?]? [seval] $n:ident ($pattern:term) := $body) => do
+    `($[$doc?:docComment]? builtin_dsimproc_decl $n ($pattern) := $body
+      attribute [$kind builtin_sevalproc $[$pre?]?] $n)
+  | `($[$doc?:docComment]? $kind:attrKind builtin_dsimproc $[$pre?]? [simp, seval] $n:ident ($pattern:term) := $body) => do
+    `($[$doc?:docComment]? builtin_dsimproc_decl $n ($pattern) := $body
+      attribute [$kind builtin_simproc $[$pre?]?] $n
+      attribute [$kind builtin_sevalproc $[$pre?]?] $n)
+
 end Lean.Parser

src/Init/System/IO.lean

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Nelson, Jared Roesch, Leonardo de Moura, Sebastian Ullrich, Mac Malone
 -/
 prelude
-import Init.Control.EState
 import Init.Control.Reader
 import Init.Data.String
 import Init.Data.ByteArray

src/Init/Tactics.lean

@@ -369,6 +369,14 @@ macro "rfl" : tactic => `(tactic| eq_refl)
 
 macro_rules | `(tactic| rfl) => `(tactic| exact HEq.rfl)
 
+/--
+This tactic applies to a goal whose target has the form `x ~ x`, where `~` is a reflexive
+relation, that is, a relation which has a reflexive lemma tagged with the attribute [refl].
+-/
+syntax (name := applyRfl) "apply_rfl" : tactic
+
+macro_rules | `(tactic| rfl) => `(tactic| apply_rfl)
+
 /--
 `rfl'` is similar to `rfl`, but disables smart unfolding and unfolds all kinds of definitions,
 theorems included (relevant for declarations defined by well-founded recursion).

src/Init/WF.lean

@@ -45,7 +45,7 @@ def apply {α : Sort u} {r : α → α → Prop} (wf : WellFounded r) (a : α) :
 section
 variable {α : Sort u} {r : α → α → Prop} (hwf : WellFounded r)
 
-theorem recursion {C : α → Sort v} (a : α) (h : ∀ x, (∀ y, r y x → C y) → C x) : C a := by
+noncomputable def recursion {C : α → Sort v} (a : α) (h : ∀ x, (∀ y, r y x → C y) → C x) : C a := by
   induction (apply hwf a) with
   | intro x₁ _ ih => exact h x₁ ih
 
@@ -166,13 +166,13 @@ def lt_wfRel : WellFoundedRelation Nat where
       | Or.inl e => subst e; assumption
       | Or.inr e => exact Acc.inv ih e
 
-protected theorem strongInductionOn
+protected noncomputable def strongInductionOn
     {motive : Nat → Sort u}
     (n : Nat)
     (ind : ∀ n, (∀ m, m < n → motive m) → motive n) : motive n :=
   Nat.lt_wfRel.wf.fix ind n
 
-protected theorem caseStrongInductionOn
+protected noncomputable def caseStrongInductionOn
     {motive : Nat → Sort u}
     (a : Nat)
     (zero : motive 0)

src/Lean.lean

@@ -24,6 +24,7 @@ import Lean.Eval
 import Lean.Structure
 import Lean.PrettyPrinter
 import Lean.CoreM
+import Lean.ReservedNameAction
 import Lean.InternalExceptionId
 import Lean.Server
 import Lean.ScopedEnvExtension

src/Lean/Compiler/ImplementedByAttr.lean

@@ -17,7 +17,7 @@ builtin_initialize implementedByAttr : ParametricAttribute Name ← registerPara
   getParam := fun declName stx => do
     let decl ← getConstInfo declName
     let fnNameStx ← Attribute.Builtin.getIdent stx
-    let fnName ← Elab.resolveGlobalConstNoOverloadWithInfo fnNameStx
+    let fnName ← Elab.realizeGlobalConstNoOverloadWithInfo fnNameStx
     let fnDecl ← getConstInfo fnName
     unless decl.levelParams.length == fnDecl.levelParams.length do
       throwError "invalid 'implemented_by' argument '{fnName}', '{fnName}' has {fnDecl.levelParams.length} universe level parameter(s), but '{declName}' has {decl.levelParams.length}"

src/Lean/Compiler/InitAttr.lean

@@ -44,7 +44,7 @@ unsafe def registerInitAttrUnsafe (attrName : Name) (runAfterImport : Bool) (ref
       let decl ← getConstInfo declName
       match (← Attribute.Builtin.getIdent? stx) with
       | some initFnName =>
-        let initFnName ← Elab.resolveGlobalConstNoOverloadWithInfo initFnName
+        let initFnName ← Elab.realizeGlobalConstNoOverloadWithInfo initFnName
         let initDecl ← getConstInfo initFnName
         match getIOTypeArg initDecl.type with
         | none => throwError "initialization function '{initFnName}' must have type of the form `IO <type>`"

src/Lean/CoreM.lean

@@ -289,6 +289,9 @@ def Exception.isMaxHeartbeat (ex : Exception) : Bool :=
 def mkArrow (d b : Expr) : CoreM Expr :=
   return Lean.mkForall (← mkFreshUserName `x) BinderInfo.default d b
 
+/-- Iterated `mkArrow`, creates the expression `a₁ → a₂ → … → aₙ → b`. Also see `arrowDomainsN`. -/
+def mkArrowN (ds : Array Expr) (e : Expr) : CoreM Expr := ds.foldrM mkArrow e
+
 def addDecl (decl : Declaration) : CoreM Unit := do
   profileitM Exception "type checking" (← getOptions) do
     withTraceNode `Kernel (fun _ => return m!"typechecking declaration") do

src/Lean/Data/HashMap.lean

@@ -116,6 +116,22 @@ def expand [Hashable α] (size : Nat) (buckets : HashMapBucket α β) : HashMapI
       else
         (expand size' buckets', false)
 
+@[inline] def insertIfNew [beq : BEq α] [Hashable α] (m : HashMapImp α β) (a : α) (b : β) : HashMapImp α β × Option β :=
+  match m with
+  | ⟨size, buckets⟩ =>
+    let ⟨i, h⟩ := mkIdx (hash a) buckets.property
+    let bkt    := buckets.val[i]
+    if let some b := bkt.find? a then
+      (m, some b)
+    else
+      let size'    := size + 1
+      let buckets' := buckets.update i (AssocList.cons a b bkt) h
+      if numBucketsForCapacity size' ≤ buckets.val.size then
+        ({ size := size', buckets := buckets' }, none)
+      else
+        (expand size' buckets', none)
+
+
 def erase [BEq α] [Hashable α] (m : HashMapImp α β) (a : α) : HashMapImp α β :=
   match m with
   | ⟨ size, buckets ⟩ =>
@@ -125,9 +141,10 @@ def erase [BEq α] [Hashable α] (m : HashMapImp α β) (a : α) : HashMapImp α
     else m
 
 inductive WellFormed [BEq α] [Hashable α] : HashMapImp α β → Prop where
-  | mkWff     : ∀ n,                    WellFormed (mkHashMapImp n)
-  | insertWff : ∀ m a b, WellFormed m → WellFormed (insert m a b |>.1)
-  | eraseWff  : ∀ m a,   WellFormed m → WellFormed (erase m a)
+  | mkWff          : ∀ n,                    WellFormed (mkHashMapImp n)
+  | insertWff      : ∀ m a b, WellFormed m → WellFormed (insert m a b |>.1)
+  | insertIfNewWff : ∀ m a b, WellFormed m → WellFormed (insertIfNew m a b |>.1)
+  | eraseWff       : ∀ m a,   WellFormed m → WellFormed (erase m a)
 
 end HashMapImp
 
@@ -156,13 +173,22 @@ def insert (m : HashMap α β) (a : α) (b : β) : HashMap α β :=
     match h:m.insert a b with
     | (m', _) => ⟨ m', by have aux := WellFormed.insertWff m a b hw; rw [h] at aux; assumption ⟩
 
-/-- Similar to `insert`, but also returns a Boolean flad indicating whether an existing entry has been replaced with `a -> b`. -/
+/-- Similar to `insert`, but also returns a Boolean flag indicating whether an existing entry has been replaced with `a -> b`. -/
 def insert' (m : HashMap α β) (a : α) (b : β) : HashMap α β × Bool :=
   match m with
   | ⟨ m, hw ⟩ =>
     match h:m.insert a b with
     | (m', replaced) => (⟨ m', by have aux := WellFormed.insertWff m a b hw; rw [h] at aux; assumption ⟩, replaced)
 
+/--
+Similar to `insert`, but returns `some old` if the map already had an entry `α → old`.
+If the result is `some old`, the the resulting map is equal to `m`. -/
+def insertIfNew (m : HashMap α β) (a : α) (b : β) : HashMap α β × Option β :=
+  match m with
+  | ⟨ m, hw ⟩ =>
+    match h:m.insertIfNew a b with
+    | (m', old) => (⟨ m', by have aux := WellFormed.insertIfNewWff m a b hw; rw [h] at aux; assumption ⟩, old)
+
 @[inline] def erase (m : HashMap α β) (a : α) : HashMap α β :=
   match m with
   | ⟨ m, hw ⟩ => ⟨ m.erase a, WellFormed.eraseWff m a hw ⟩

src/Lean/Data/JsonRpc.lean

@@ -183,6 +183,9 @@ structure ResponseError (α : Type u) where
 instance [ToJson α] : CoeOut (ResponseError α) Message :=
   ⟨fun r => Message.responseError r.id r.code r.message (r.data?.map toJson)⟩
 
+instance : CoeOut (ResponseError Unit) Message :=
+  ⟨fun r => Message.responseError r.id r.code r.message none⟩
+
 instance : Coe String RequestID := ⟨RequestID.str⟩
 instance : Coe JsonNumber RequestID := ⟨RequestID.num⟩
 

src/Lean/Data/Lsp/Basic.lean

@@ -52,7 +52,7 @@ instance : LE Range := leOfOrd
 structure Location where
   uri : DocumentUri
   range : Range
-  deriving Inhabited, BEq, ToJson, FromJson
+  deriving Inhabited, BEq, ToJson, FromJson, Ord
 
 structure LocationLink where
   originSelectionRange? : Option Range

src/Lean/Data/Lsp/Diagnostics.lean

@@ -25,7 +25,7 @@ open Json
 
 inductive DiagnosticSeverity where
   | error | warning | information | hint
-  deriving Inhabited, BEq
+  deriving Inhabited, BEq, Ord
 
 instance : FromJson DiagnosticSeverity := ⟨fun j =>
   match j.getNat? with
@@ -45,7 +45,7 @@ instance : ToJson DiagnosticSeverity := ⟨fun
 inductive DiagnosticCode where
   | int (i : Int)
   | string (s : String)
-  deriving Inhabited, BEq
+  deriving Inhabited, BEq, Ord
 
 instance : FromJson DiagnosticCode := ⟨fun
   | num (i : Int) => return DiagnosticCode.int i
@@ -62,7 +62,7 @@ inductive DiagnosticTag where
   | unnecessary
   /-- Deprecated or obsolete code. Rendered with a strike-through. -/
   | deprecated
-  deriving Inhabited, BEq
+  deriving Inhabited, BEq, Ord
 
 instance : FromJson DiagnosticTag := ⟨fun j =>
   match j.getNat? with
@@ -80,7 +80,7 @@ instance : ToJson DiagnosticTag := ⟨fun
 structure DiagnosticRelatedInformation where
   location : Location
   message : String
-  deriving Inhabited, BEq, ToJson, FromJson
+  deriving Inhabited, BEq, ToJson, FromJson, Ord
 
 /-- Represents a diagnostic, such as a compiler error or warning. Diagnostic objects are only valid in the scope of a resource.
 
@@ -113,6 +113,29 @@ structure DiagnosticWith (α : Type) where
 def DiagnosticWith.fullRange (d : DiagnosticWith α) : Range :=
   d.fullRange?.getD d.range
 
+local instance [Ord α] : Ord (Array α) := Ord.arrayOrd
+
+/-- Restriction of `DiagnosticWith` to properties that are displayed to users in the InfoView. -/
+private structure DiagnosticWith.UserVisible (α : Type) where
+  range               : Range
+  fullRange?          : Option Range
+  severity?           : Option DiagnosticSeverity
+  code?               : Option DiagnosticCode
+  source?             : Option String
+  message             : α
+  tags?               : Option (Array DiagnosticTag)
+  relatedInformation? : Option (Array DiagnosticRelatedInformation)
+  deriving Ord
+
+/-- Extracts user-visible properties from the given `DiagnosticWith`. -/
+private def DiagnosticWith.UserVisible.ofDiagnostic (d : DiagnosticWith α)
+    : DiagnosticWith.UserVisible α :=
+  { d with }
+
+/-- Compares `DiagnosticWith` instances modulo non-user-facing properties. -/
+def compareByUserVisible [Ord α] (a b : DiagnosticWith α) : Ordering :=
+  compare (DiagnosticWith.UserVisible.ofDiagnostic a) (DiagnosticWith.UserVisible.ofDiagnostic b)
+
 abbrev Diagnostic := DiagnosticWith String
 
 /-- Parameters for the [`textDocument/publishDiagnostics` notification](https://microsoft.github.io/language-server-protocol/specifications/lsp/3.17/specification/#textDocument_publishDiagnostics). -/

src/Lean/Data/Lsp/Internal.lean

@@ -24,44 +24,50 @@ Identifier of a reference.
 -/
 inductive RefIdent where
   /-- Named identifier. These are used in all references that are globally available. -/
-  | const : Name → RefIdent
+  | const (moduleName : Name) (identName : Name) : RefIdent
   /-- Unnamed identifier. These are used for all local references. -/
-  | fvar  : FVarId → RefIdent
+  | fvar (moduleName : Name) (id : FVarId) : RefIdent
   deriving BEq, Hashable, Inhabited
 
 namespace RefIdent
 
-/-- Converts the reference identifier to a string by prefixing it with a symbol. -/
-def toString : RefIdent → String
-  | RefIdent.const n => s!"c:{n}"
-  | RefIdent.fvar id => s!"f:{id.name}"
+instance : ToJson FVarId where
+  toJson id := toJson id.name
 
-/--
-Converts the string representation of a reference identifier back to a reference identifier.
-The string representation must have been created by `RefIdent.toString`.
--/
-def fromString (s : String) : Except String RefIdent := do
-  let sPrefix := s.take 2
-  let sName := s.drop 2
-  -- See `FromJson Name`
-  let name ← match sName with
-    | "[anonymous]" => pure Name.anonymous
-    | _ =>
-      let n := sName.toName
-      if n.isAnonymous then throw s!"expected a Name, got {sName}"
-      else pure n
-  match sPrefix with
-    | "c:" => return RefIdent.const name
-    | "f:" => return RefIdent.fvar <| FVarId.mk name
-    | _ => throw "string must start with 'c:' or 'f:'"
+instance : FromJson FVarId where
+  fromJson? s := return ⟨← fromJson? s⟩
+
+/-- Shortened representation of `RefIdent` for more compact serialization. -/
+inductive RefIdentJsonRepr
+  /-- Shortened representation of `RefIdent.const` for more compact serialization. -/
+  | c (m n : Name)
+  /-- Shortened representation of `RefIdent.fvar` for more compact serialization. -/
+  | f (m : Name) (i : FVarId)
+  deriving FromJson, ToJson
+
+/-- Converts `id` to its compact serialization representation. -/
+def toJsonRepr : (id : RefIdent) → RefIdentJsonRepr
+  | const moduleName identName => .c moduleName identName
+  | fvar moduleName id => .f moduleName id
+
+/-- Converts `repr` to `RefIdent`. -/
+def fromJsonRepr : (repr : RefIdentJsonRepr) → RefIdent
+  | .c m n => const m n
+  | .f m i => fvar m i
+
+/-- Converts `RefIdent` from a JSON for `RefIdentJsonRepr`. -/
+def fromJson? (s : Json) : Except String RefIdent :=
+  return fromJsonRepr (← Lean.FromJson.fromJson? s)
+
+/-- Converts `RefIdent` to a JSON for `RefIdentJsonRepr`. -/
+def toJson (id : RefIdent) : Json :=
+  Lean.ToJson.toJson <| toJsonRepr id
 
 instance : FromJson RefIdent where
-  fromJson?
-    | (s : String) => fromString s
-    | j => Except.error s!"expected a String, got {j}"
+  fromJson? := fromJson?
 
 instance : ToJson RefIdent where
-  toJson ident := toString ident
+  toJson := toJson
 
 end RefIdent
 
@@ -84,6 +90,7 @@ structure RefInfo.Location where
   range       : Lsp.Range
   /-- Parent declaration of the reference. `none` if the reference is itself a declaration. -/
   parentDecl? : Option RefInfo.ParentDecl
+deriving Inhabited
 
 /-- Definition site and usage sites of a reference. Obtained from `Lean.Server.RefInfo`. -/
 structure RefInfo where
@@ -146,17 +153,18 @@ instance : FromJson RefInfo where
 def ModuleRefs := HashMap RefIdent RefInfo
 
 instance : ToJson ModuleRefs where
-  toJson m := Json.mkObj <| m.toList.map fun (ident, info) => (ident.toString, toJson info)
+  toJson m := Json.mkObj <| m.toList.map fun (ident, info) => (ident.toJson.compress, toJson info)
 
 instance : FromJson ModuleRefs where
   fromJson? j := do
     let node ← j.getObj?
     node.foldM (init := HashMap.empty) fun m k v =>
-      return m.insert (← RefIdent.fromString k) (← fromJson? v)
+      return m.insert (← RefIdent.fromJson? (← Json.parse k)) (← fromJson? v)
 
-/-- `$/lean/ileanInfoUpdate` and `$/lean/ileanInfoFinal` watchdog<-worker notifications.
-
-Contains the file's definitions and references. -/
+/--
+Used in the `$/lean/ileanInfoUpdate` and `$/lean/ileanInfoFinal` watchdog <- worker notifications.
+Contains the definitions and references of the file managed by a worker.
+-/
 structure LeanIleanInfoParams where
   /-- Version of the file these references are from. -/
   version    : Nat
@@ -164,4 +172,22 @@ structure LeanIleanInfoParams where
   references : ModuleRefs
   deriving FromJson, ToJson
 
+/--
+Used in the `$/lean/importClosure` watchdog <- worker notification.
+Contains the full import closure of the file managed by a worker.
+-/
+structure LeanImportClosureParams where
+  /-- Full import closure of the file. -/
+  importClosure : Array DocumentUri
+  deriving FromJson, ToJson
+
+/--
+Used in the `$/lean/importClosure` watchdog -> worker notification.
+Informs the worker that one of its dependencies has gone stale and likely needs to be rebuilt.
+-/
+structure LeanStaleDependencyParams where
+  /-- The dependency that is stale. -/
+  staleDependency : DocumentUri
+  deriving FromJson, ToJson
+
 end Lean.Lsp

src/Lean/Data/Lsp/Ipc.lean

@@ -64,9 +64,10 @@ def readRequestAs (expectedMethod : String) (α) [FromJson α] : IpcM (Request
   (←stdout).readLspRequestAs expectedMethod α
 
 /--
-  Reads response, discarding notifications in between. This function is meant
-  purely for testing where we use `collectDiagnostics` explicitly if we do care
-  about such notifications. -/
+Reads response, discarding notifications and server-to-client requests in between.
+This function is meant purely for testing where we use `collectDiagnostics` explicitly
+if we do care about such notifications.
+-/
 partial def readResponseAs (expectedID : RequestID) (α) [FromJson α] :
     IpcM (Response α) := do
   let m ← (←stdout).readLspMessage
@@ -79,20 +80,28 @@ partial def readResponseAs (expectedID : RequestID) (α) [FromJson α] :
     else
       throw $ userError s!"Expected id {expectedID}, got id {id}"
   | .notification .. => readResponseAs expectedID α
-  | _ => throw $ userError s!"Expected JSON-RPC response, got: '{(toJson m).compress}'"
+  | .request .. => readResponseAs expectedID α
+  | .responseError .. => throw $ userError s!"Expected JSON-RPC response, got: '{(toJson m).compress}'"
 
 def waitForExit : IpcM UInt32 := do
   (←read).wait
 
-/-- Waits for the worker to emit all diagnostics for the current document version
-and returns them as a list. -/
+/--
+Waits for the worker to emit all diagnostic notifications for the current document version and
+returns the last notification, if any.
+
+We used to return all notifications but with debouncing in the server, this would not be
+deterministic anymore as what messages are dropped depends on wall-clock timing.
+ -/
 partial def collectDiagnostics (waitForDiagnosticsId : RequestID := 0) (target : DocumentUri) (version : Nat)
-: IpcM (List (Notification PublishDiagnosticsParams)) := do
+: IpcM (Option (Notification PublishDiagnosticsParams)) := do
   writeRequest ⟨waitForDiagnosticsId, "textDocument/waitForDiagnostics", WaitForDiagnosticsParams.mk target version⟩
-  let rec loop : IpcM (List (Notification PublishDiagnosticsParams)) := do
+  loop
+where
+  loop := do
     match (←readMessage) with
     | Message.response id _ =>
-      if id == waitForDiagnosticsId then return []
+      if id == waitForDiagnosticsId then return none
       else loop
     | Message.responseError id _    msg _ =>
       if id == waitForDiagnosticsId then
@@ -100,10 +109,9 @@ partial def collectDiagnostics (waitForDiagnosticsId : RequestID := 0) (target :
       else loop
     | Message.notification "textDocument/publishDiagnostics" (some param) =>
       match fromJson? (toJson param) with
-      | Except.ok diagnosticParam => return ⟨"textDocument/publishDiagnostics", diagnosticParam⟩ :: (←loop)
+      | Except.ok diagnosticParam => return (← loop).getD ⟨"textDocument/publishDiagnostics", diagnosticParam⟩
       | Except.error inner => throw $ userError s!"Cannot decode publishDiagnostics parameters\n{inner}"
     | _ => loop
-  loop
 
 def runWith (lean : System.FilePath) (args : Array String := #[]) (test : IpcM α) : IO α := do
   let proc ← Process.spawn {

src/Lean/Data/Lsp/TextSync.lean

@@ -38,7 +38,7 @@ structure DidOpenTextDocumentParams where
 
 structure TextDocumentChangeRegistrationOptions where
   documentSelector? : Option DocumentSelector := none
-  syncKind : TextDocumentSyncKind
+  syncKind          : TextDocumentSyncKind
   deriving FromJson
 
 inductive TextDocumentContentChangeEvent where
@@ -61,13 +61,18 @@ instance TextDocumentContentChangeEvent.hasToJson : ToJson TextDocumentContentCh
     | TextDocumentContentChangeEvent.fullChange text => [⟨"text", toJson text⟩]⟩
 
 structure DidChangeTextDocumentParams where
-  textDocument : VersionedTextDocumentIdentifier
+  textDocument   : VersionedTextDocumentIdentifier
   contentChanges : Array TextDocumentContentChangeEvent
   deriving ToJson, FromJson
 
+structure DidSaveTextDocumentParams where
+  textDocument : TextDocumentIdentifier
+  text?        : Option String
+  deriving ToJson, FromJson
+
 -- TODO: missing:
 -- WillSaveTextDocumentParams, TextDocumentSaveReason,
--- TextDocumentSaveRegistrationOptions, DidSaveTextDocumentParams
+-- TextDocumentSaveRegistrationOptions
 
 structure SaveOptions where
   includeText : Bool
@@ -81,11 +86,11 @@ structure DidCloseTextDocumentParams where
 
 /-- NOTE: This is defined twice in the spec. The latter version has more fields. -/
 structure TextDocumentSyncOptions where
-  openClose : Bool
-  change : TextDocumentSyncKind
-  willSave : Bool
+  openClose         : Bool
+  change            : TextDocumentSyncKind
+  willSave          : Bool
   willSaveWaitUntil : Bool
-  save? : Option SaveOptions := none
+  save?             : Option SaveOptions
   deriving ToJson, FromJson
 
 end Lsp

src/Lean/Data/Lsp/Window.lean

@@ -0,0 +1,48 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Marc Huisinga
+-/
+prelude
+import Lean.Data.Json
+
+open Lean
+
+inductive MessageType where
+  | error
+  | warning
+  | info
+  | log
+
+instance : FromJson MessageType where
+  fromJson?
+    | (1 : Nat) => .ok .error
+    | (2 : Nat) => .ok .warning
+    | (3 : Nat) => .ok .info
+    | (4 : Nat) => .ok .log
+    | _         => .error "Unknown MessageType ID"
+
+instance : ToJson MessageType where
+  toJson
+    | .error   => 1
+    | .warning => 2
+    | .info    => 3
+    | .log     => 4
+
+structure ShowMessageParams where
+  type    : MessageType
+  message : String
+  deriving FromJson, ToJson
+
+structure MessageActionItem where
+  title : String
+  deriving FromJson, ToJson
+
+structure ShowMessageRequestParams where
+  type     : MessageType
+  message  : String
+  actions? : Option (Array MessageActionItem)
+  deriving FromJson, ToJson
+
+def ShowMessageResponse := Option MessageActionItem
+  deriving FromJson, ToJson

src/Lean/Data/PersistentHashMap.lean

@@ -226,8 +226,10 @@ partial def eraseAux [BEq α] : Node α β → USize → α → Node α β × Bo
   | n@(Node.collision keys vals heq), _, k =>
     match keys.indexOf? k with
     | some idx =>
-      let ⟨keys', keq⟩ := keys.eraseIdx' idx
-      let ⟨vals', veq⟩ := vals.eraseIdx' (Eq.ndrec idx heq)
+      let keys' := keys.feraseIdx idx
+      have keq := keys.size_feraseIdx idx
+      let vals' := vals.feraseIdx (Eq.ndrec idx heq)
+      have veq := vals.size_feraseIdx (Eq.ndrec idx heq)
       have : keys.size - 1 = vals.size - 1 := by rw [heq]
       (Node.collision keys' vals' (keq.trans (this.trans veq.symm)), true)
     | none     => (n, false)

src/Lean/DeclarationRange.lean

@@ -48,14 +48,14 @@ def addDeclarationRanges [MonadEnv m] (declName : Name) (declRanges : Declaratio
 def findDeclarationRangesCore? [Monad m] [MonadEnv m] (declName : Name) : m (Option DeclarationRanges) :=
   return declRangeExt.find? (← getEnv) declName
 
-def findDeclarationRanges? [Monad m] [MonadEnv m] [MonadLiftT IO m] (declName : Name) : m (Option DeclarationRanges) := do
+def findDeclarationRanges? [Monad m] [MonadEnv m] [MonadLiftT BaseIO m] (declName : Name) : m (Option DeclarationRanges) := do
   let env ← getEnv
   let ranges ← if isAuxRecursor env declName || isNoConfusion env declName || (← isRec declName)  then
     findDeclarationRangesCore? declName.getPrefix
   else
     findDeclarationRangesCore? declName
   match ranges with
-  | none => return (← builtinDeclRanges.get (m := IO)).find? declName
+  | none => return (← builtinDeclRanges.get (m := BaseIO)).find? declName
   | some _ => return ranges
 
 end Lean

src/Lean/Elab/BuiltinCommand.lean

@@ -536,7 +536,7 @@ def elabCheckCore (ignoreStuckTC : Bool) : CommandElab
     -- show signature for `#check id`/`#check @id`
     if let `($id:ident) := term then
       try
-        for c in (← resolveGlobalConstWithInfos term) do
+        for c in (← realizeGlobalConstWithInfos term) do
           addCompletionInfo <| .id term id.getId (danglingDot := false) {} none
           logInfoAt tk <| .ofPPFormat { pp := fun
             | some ctx => ctx.runMetaM <| PrettyPrinter.ppSignature c
@@ -760,7 +760,7 @@ def elabRunMeta : CommandElab := fun stx =>
 @[builtin_command_elab Parser.Command.addDocString] def elabAddDeclDoc : CommandElab := fun stx => do
   match stx with
   | `($doc:docComment add_decl_doc $id) =>
-    let declName ← resolveGlobalConstNoOverloadWithInfo id
+    let declName ← liftCoreM <| realizeGlobalConstNoOverloadWithInfo id
     unless ((← getEnv).getModuleIdxFor? declName).isNone do
       throwError "invalid 'add_decl_doc', declaration is in an imported module"
     if let .none ← findDeclarationRangesCore? declName then

src/Lean/Elab/BuiltinTerm.lean

@@ -223,7 +223,7 @@ def elabScientificLit : TermElab := fun stx expectedType? => do
   | none     => throwIllFormedSyntax
 
 @[builtin_term_elab doubleQuotedName] def elabDoubleQuotedName : TermElab := fun stx _ =>
-  return toExpr (← resolveGlobalConstNoOverloadWithInfo stx[2])
+  return toExpr (← realizeGlobalConstNoOverloadWithInfo stx[2])
 
 @[builtin_term_elab declName] def elabDeclName : TermElab := adaptExpander fun _ => do
   let some declName ← getDeclName?

src/Lean/Elab/CheckTactic.lean

@@ -7,6 +7,7 @@ prelude
 import Lean.Elab.Tactic.ElabTerm
 import Lean.Elab.Command
 import Lean.Elab.Tactic.Meta
+import Lean.Meta.CheckTactic
 
 /-!
 Commands to validate tactic results.
@@ -18,15 +19,6 @@ open Lean.Meta CheckTactic
 open Lean.Elab.Tactic
 open Lean.Elab.Command
 
-private def matchCheckGoalType (stx : Syntax) (goalType : Expr) : MetaM (Expr × Expr × Level) := do
-  let u ← mkFreshLevelMVar
-  let type ← mkFreshExprMVar (some (.sort u))
-  let val  ← mkFreshExprMVar (some type)
-  let extType := mkAppN (.const ``CheckGoalType [u]) #[type, val]
-  if !(← isDefEq goalType extType) then
-    throwErrorAt stx "Goal{indentExpr goalType}\nis expected to match {indentExpr extType}"
-  pure (val, type, u)
-
 @[builtin_command_elab Lean.Parser.checkTactic]
 def elabCheckTactic : CommandElab := fun stx => do
   let `(#check_tactic $t ~> $result by $tac) := stx | throwUnsupportedSyntax
@@ -34,11 +26,10 @@ def elabCheckTactic : CommandElab := fun stx => do
     runTermElabM $ fun _vars => do
       let u ← Lean.Elab.Term.elabTerm t none
       let type ← inferType u
-      let lvl ← mkFreshLevelMVar
-      let checkGoalType : Expr := mkApp2 (mkConst ``CheckGoalType [lvl]) type u
+      let checkGoalType ← mkCheckGoalType u type
       let mvar ← mkFreshExprMVar (.some checkGoalType)
-      let (goals, _) ← Lean.Elab.runTactic mvar.mvarId! tac.raw
       let expTerm ← Lean.Elab.Term.elabTerm result (.some type)
+      let (goals, _) ← Lean.Elab.runTactic mvar.mvarId! tac.raw
       match goals with
       | [] =>
         throwErrorAt stx
@@ -51,7 +42,6 @@ def elabCheckTactic : CommandElab := fun stx => do
       | _ => do
         throwErrorAt stx
           m!"{tac} produced multiple goals, but is expected to reduce to {indentExpr expTerm}."
-      pure ()
 
 @[builtin_command_elab Lean.Parser.checkTacticFailure]
 def elabCheckTacticFailure : CommandElab := fun stx => do
@@ -60,8 +50,7 @@ def elabCheckTacticFailure : CommandElab := fun stx => do
     runTermElabM $ fun _vars => do
       let val ← Lean.Elab.Term.elabTerm t none
       let type ← inferType val
-      let lvl ← mkFreshLevelMVar
-      let checkGoalType : Expr := mkApp2 (mkConst ``CheckGoalType [lvl]) type val
+      let checkGoalType ← mkCheckGoalType val type
       let mvar ← mkFreshExprMVar (.some checkGoalType)
       let act := Lean.Elab.runTactic mvar.mvarId! tactic.raw
       match ← try (Term.withoutErrToSorry (some <$> act)) catch _ => pure none with
@@ -73,12 +62,12 @@ def elabCheckTacticFailure : CommandElab := fun stx => do
                 pure m!"{indentExpr val}"
           let msg ←
             match gls with
-              | [] => pure m!"{tactic} expected to fail on {val}, but closed goal."
+              | [] => pure m!"{tactic} expected to fail on {t}, but closed goal."
               | [g] =>
-                pure <| m!"{tactic} expected to fail on {val}, but returned: {←ppGoal g}"
+                pure <| m!"{tactic} expected to fail on {t}, but returned: {←ppGoal g}"
               | gls =>
                 let app m g := do pure <| m ++ (←ppGoal g)
-                let init := m!"{tactic} expected to fail on {val}, but returned goals:"
+                let init := m!"{tactic} expected to fail on {t}, but returned goals:"
                 gls.foldlM (init := init) app
           throwErrorAt stx msg
 

src/Lean/Elab/Command.lean

@@ -141,7 +141,8 @@ private def addTraceAsMessagesCore (ctx : Context) (log : MessageLog) (traceStat
   let mut log := log
   let traces' := traces.toArray.qsort fun ((a, _), _) ((b, _), _) => a < b
   for ((pos, endPos), traceMsg) in traces' do
-    log := log.add <| mkMessageCore ctx.fileName ctx.fileMap (.joinSep traceMsg.toList "\n") .information pos endPos
+    let data := .tagged `_traceMsg <| .joinSep traceMsg.toList "\n"
+    log := log.add <| mkMessageCore ctx.fileName ctx.fileMap data .information pos endPos
   return log
 
 private def addTraceAsMessages : CommandElabM Unit := do
@@ -268,11 +269,6 @@ instance : MonadRecDepth CommandElabM where
   getRecDepth      := return (← read).currRecDepth
   getMaxRecDepth   := return (← get).maxRecDepth
 
-register_builtin_option showPartialSyntaxErrors : Bool := {
-  defValue := false
-  descr    := "show elaboration errors from partial syntax trees (i.e. after parser recovery)"
-}
-
 builtin_initialize registerTraceClass `Elab.command
 
 partial def elabCommand (stx : Syntax) : CommandElabM Unit := do
@@ -321,11 +317,6 @@ def elabCommandTopLevel (stx : Syntax) : CommandElabM Unit := withRef stx do pro
     -- note the order: first process current messages & info trees, then add back old messages & trees,
     -- then convert new traces to messages
     let mut msgs := (← get).messages
-    -- `stx.hasMissing` should imply `initMsgs.hasErrors`, but the latter should be cheaper to check in general
-    if !showPartialSyntaxErrors.get (← getOptions) && initMsgs.hasErrors && stx.hasMissing then
-      -- discard elaboration errors, except for a few important and unlikely misleading ones, on parse error
-      msgs := ⟨msgs.msgs.filter fun msg =>
-        msg.data.hasTag (fun tag => tag == `Elab.synthPlaceholder || tag == `Tactic.unsolvedGoals || (`_traceMsg).isSuffixOf tag)⟩
     for tree in (← getInfoTrees) do
       trace[Elab.info] (← tree.format)
     modify fun st => { st with

src/Lean/Elab/DeclModifiers.lean

@@ -27,6 +27,8 @@ def checkNotAlreadyDeclared {m} [Monad m] [MonadEnv m] [MonadError m] [MonadInfo
     match privateToUserName? declName with
     | none          => throwError "'{declName}' has already been declared"
     | some declName => throwError "private declaration '{declName}' has already been declared"
+  if isReservedName env declName then
+    throwError "'{declName}' is a reserved name"
   if env.contains (mkPrivateName env declName) then
     addInfo (mkPrivateName env declName)
     throwError "a private declaration '{declName}' has already been declared"

src/Lean/Elab/Declaration.lean

@@ -42,7 +42,7 @@ private def isNamedDef (stx : Syntax) : Bool :=
     let decl := stx[1]
     let k := decl.getKind
     k == ``Lean.Parser.Command.abbrev ||
-    k == ``Lean.Parser.Command.def ||
+    k == ``Lean.Parser.Command.definition ||
     k == ``Lean.Parser.Command.theorem ||
     k == ``Lean.Parser.Command.opaque ||
     k == ``Lean.Parser.Command.axiom ||
@@ -166,7 +166,7 @@ private def inductiveSyntaxToView (modifiers : Modifiers) (decl : Syntax) : Comm
     return { ref := ctor, modifiers := ctorModifiers, declName := ctorName, binders := binders, type? := type? : CtorView }
   let computedFields ← (decl[5].getOptional?.map (·[1].getArgs) |>.getD #[]).mapM fun cf => withRef cf do
     return { ref := cf, modifiers := cf[0], fieldId := cf[1].getId, type := ⟨cf[3]⟩, matchAlts := ⟨cf[4]⟩ }
-  let classes ← getOptDerivingClasses decl[6]
+  let classes ← liftCoreM <| getOptDerivingClasses decl[6]
   return {
     ref             := decl
     shortDeclName   := name
@@ -354,7 +354,7 @@ def elabMutual : CommandElab := fun stx => do
     -/
     let declNames ←
        try
-         resolveGlobalConst ident
+         realizeGlobalConst ident
        catch _ =>
          let name := ident.getId.eraseMacroScopes
          if (← Simp.isBuiltinSimproc name) then

src/Lean/Elab/DefView.lean

@@ -142,7 +142,7 @@ def mkDefViewOfExample (modifiers : Modifiers) (stx : Syntax) : DefView :=
 def isDefLike (stx : Syntax) : Bool :=
   let declKind := stx.getKind
   declKind == ``Parser.Command.abbrev ||
-  declKind == ``Parser.Command.def ||
+  declKind == ``Parser.Command.definition ||
   declKind == ``Parser.Command.theorem ||
   declKind == ``Parser.Command.opaque ||
   declKind == ``Parser.Command.instance ||
@@ -152,7 +152,7 @@ def mkDefView (modifiers : Modifiers) (stx : Syntax) : CommandElabM DefView :=
   let declKind := stx.getKind
   if declKind == ``Parser.Command.«abbrev» then
     return mkDefViewOfAbbrev modifiers stx
-  else if declKind == ``Parser.Command.def then
+  else if declKind == ``Parser.Command.definition then
     return mkDefViewOfDef modifiers stx
   else if declKind == ``Parser.Command.theorem then
     return mkDefViewOfTheorem modifiers stx

src/Lean/Elab/Deriving/Basic.lean

@@ -100,10 +100,10 @@ private def tryApplyDefHandler (className : Name) (declName : Name) : CommandEla
 
 @[builtin_command_elab «deriving»] def elabDeriving : CommandElab
   | `(deriving instance $[$classes $[with $argss?]?],* for $[$declNames],*) => do
-     let declNames ← declNames.mapM resolveGlobalConstNoOverloadWithInfo
+     let declNames ← liftCoreM <| declNames.mapM realizeGlobalConstNoOverloadWithInfo
      for cls in classes, args? in argss? do
        try
-         let className ← resolveGlobalConstNoOverloadWithInfo cls
+         let className ← liftCoreM <| realizeGlobalConstNoOverloadWithInfo cls
          withRef cls do
            if declNames.size == 1 && args?.isNone then
              if (← tryApplyDefHandler className declNames[0]!) then
@@ -118,12 +118,12 @@ structure DerivingClassView where
   className : Name
   args? : Option (TSyntax ``Parser.Term.structInst)
 
-def getOptDerivingClasses [Monad m] [MonadEnv m] [MonadResolveName m] [MonadError m] [MonadInfoTree m] (optDeriving : Syntax) : m (Array DerivingClassView) := do
+def getOptDerivingClasses (optDeriving : Syntax) : CoreM (Array DerivingClassView) := do
   match optDeriving with
   | `(Parser.Command.optDeriving| deriving $[$classes $[with $argss?]?],*) =>
     let mut ret := #[]
     for cls in classes, args? in argss? do
-      let className ← resolveGlobalConstNoOverloadWithInfo cls
+      let className ← realizeGlobalConstNoOverloadWithInfo cls
       ret := ret.push { ref := cls, className := className, args? }
     return ret
   | _ => return #[]

src/Lean/Elab/Extra.lean

@@ -488,8 +488,10 @@ def elabBinRelCore (noProp : Bool) (stx : Syntax) (expectedType? : Option Expr)
     ```
     We can improve this failure in the future by applying default instances before reporting a type mismatch.
     -/
-    let lhs ← withRef stx[2] <| toTree stx[2]
-    let rhs ← withRef stx[3] <| toTree stx[3]
+    let lhsStx := stx[2]
+    let rhsStx := stx[3]
+    let lhs ← withRef lhsStx <| toTree lhsStx
+    let rhs ← withRef rhsStx <| toTree rhsStx
     let tree := .binop stx .regular f lhs rhs
     let r ← analyze tree none
     trace[Elab.binrel] "hasUncomparable: {r.hasUncomparable}, maxType: {r.max?}"
@@ -497,10 +499,10 @@ def elabBinRelCore (noProp : Bool) (stx : Syntax) (expectedType? : Option Expr)
       -- Use default elaboration strategy + `toBoolIfNecessary`
       let lhs ← toExprCore lhs
       let rhs ← toExprCore rhs
-      let lhs ← toBoolIfNecessary lhs
-      let rhs ← toBoolIfNecessary rhs
+      let lhs ← withRef lhsStx <| toBoolIfNecessary lhs
+      let rhs ← withRef rhsStx <| toBoolIfNecessary rhs
       let lhsType ← inferType lhs
-      let rhs ← ensureHasType lhsType rhs
+      let rhs ← withRef rhsStx <| ensureHasType lhsType rhs
       elabAppArgs f #[] #[Arg.expr lhs, Arg.expr rhs] expectedType? (explicit := false) (ellipsis := false) (resultIsOutParamSupport := false)
     else
       let mut maxType := r.max?.get!

src/Lean/Elab/Frontend.lean

@@ -4,8 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Sebastian Ullrich
 -/
 prelude
-import Lean.Elab.Import
-import Lean.Elab.Command
+import Lean.Language.Lean
 import Lean.Util.Profile
 import Lean.Server.References
 
@@ -40,7 +39,19 @@ def setCommandState (commandState : Command.State) : FrontendM Unit :=
 
 def elabCommandAtFrontend (stx : Syntax) : FrontendM Unit := do
   runCommandElabM do
+    let initMsgs ← modifyGet fun st => (st.messages, { st with messages := {} })
     Command.elabCommandTopLevel stx
+    let mut msgs := (← get).messages
+    -- `stx.hasMissing` should imply `initMsgs.hasErrors`, but the latter should be cheaper to check
+    -- in general
+    if !Language.Lean.showPartialSyntaxErrors.get (← getOptions) && initMsgs.hasErrors &&
+        stx.hasMissing then
+      -- discard elaboration errors, except for a few important and unlikely misleading ones, on
+      -- parse error
+      msgs := ⟨msgs.msgs.filter fun msg =>
+        msg.data.hasTag (fun tag => tag == `Elab.synthPlaceholder ||
+          tag == `Tactic.unsolvedGoals || (`_traceMsg).isSuffixOf tag)⟩
+    modify ({ · with messages := initMsgs ++ msgs })
 
 def updateCmdPos : FrontendM Unit := do
   modify fun s => { s with cmdPos := s.parserState.pos }
@@ -86,12 +97,8 @@ def process (input : String) (env : Environment) (opts : Options) (fileName : Op
   pure (s.commandState.env, s.commandState.messages)
 
 builtin_initialize
-  registerOption `printMessageEndPos { defValue := false, descr := "print end position of each message in addition to start position" }
   registerTraceClass `Elab.info
 
-def getPrintMessageEndPos (opts : Options) : Bool :=
-  opts.getBool `printMessageEndPos false
-
 @[export lean_run_frontend]
 def runFrontend
     (input : String)
@@ -102,26 +109,50 @@ def runFrontend
     (ileanFileName? : Option String := none)
     : IO (Environment × Bool) := do
   let inputCtx := Parser.mkInputContext input fileName
-  let (header, parserState, messages) ← Parser.parseHeader inputCtx
-  -- allow `env` to be leaked, which would live until the end of the process anyway
-  let (env, messages) ← processHeader (leakEnv := true) header opts messages inputCtx trustLevel
-  let env := env.setMainModule mainModuleName
-  let mut commandState := Command.mkState env messages opts
+  -- TODO: replace with `#lang` processing
+  if /- Lean #lang? -/ true then
+    -- Temporarily keep alive old cmdline driver for the Lean language so that we don't pay the
+    -- overhead of passing the environment between snapshots until we actually make good use of it
+    -- outside the server
+    let (header, parserState, messages) ← Parser.parseHeader inputCtx
+    -- allow `env` to be leaked, which would live until the end of the process anyway
+    let (env, messages) ← processHeader (leakEnv := true) header opts messages inputCtx trustLevel
+    let env := env.setMainModule mainModuleName
+    let mut commandState := Command.mkState env messages opts
 
-  if ileanFileName?.isSome then
-    -- Collect InfoTrees so we can later extract and export their info to the ilean file
-    commandState := { commandState with infoState.enabled := true }
+    if ileanFileName?.isSome then
+      -- Collect InfoTrees so we can later extract and export their info to the ilean file
+      commandState := { commandState with infoState.enabled := true }
 
-  let s ← IO.processCommands inputCtx parserState commandState
-  for msg in s.commandState.messages.toList do
-    IO.print (← msg.toString (includeEndPos := getPrintMessageEndPos opts))
+    let s ← IO.processCommands inputCtx parserState commandState
+    for msg in s.commandState.messages.toList do
+      IO.print (← msg.toString (includeEndPos := Language.printMessageEndPos.get opts))
 
+    if let some ileanFileName := ileanFileName? then
+      let trees := s.commandState.infoState.trees.toArray
+      let references ←
+        Lean.Server.findModuleRefs inputCtx.fileMap trees (localVars := false) |>.toLspModuleRefs
+      let ilean := { module := mainModuleName, references : Lean.Server.Ilean }
+      IO.FS.writeFile ileanFileName $ Json.compress $ toJson ilean
+
+    return (s.commandState.env, !s.commandState.messages.hasErrors)
+
+  let ctx := { inputCtx with mainModuleName, opts, trustLevel }
+  let processor := Language.Lean.process
+  let snap ← processor none ctx
+  let snaps := Language.toSnapshotTree snap
+  snaps.runAndReport opts
   if let some ileanFileName := ileanFileName? then
-    let trees := s.commandState.infoState.trees.toArray
+    let trees := snaps.getAll.concatMap (match ·.infoTree? with | some t => #[t] | _ => #[])
     let references := Lean.Server.findModuleRefs inputCtx.fileMap trees (localVars := false)
     let ilean := { module := mainModuleName, references := ← references.toLspModuleRefs : Lean.Server.Ilean }
     IO.FS.writeFile ileanFileName $ Json.compress $ toJson ilean
 
-  pure (s.commandState.env, !s.commandState.messages.hasErrors)
+  let hasErrors := snaps.getAll.any (·.diagnostics.msgLog.hasErrors)
+  -- TODO: remove default when reworking cmdline interface in Lean; currently the only case
+  -- where we use the environment despite errors in the file is `--stats`
+  let env := Language.Lean.waitForFinalEnv? snap |>.getD (← mkEmptyEnvironment)
+  pure (env, !hasErrors)
+
 
 end Lean.Elab

src/Lean/Elab/GenInjective.lean

@@ -10,8 +10,8 @@ import Lean.Meta.Injective
 namespace Lean.Elab.Command
 
 @[builtin_command_elab genInjectiveTheorems] def elabGenInjectiveTheorems : CommandElab := fun stx => do
-  let declName ← resolveGlobalConstNoOverloadWithInfo stx[1]
   liftTermElabM do
+    let declName ← realizeGlobalConstNoOverloadWithInfo stx[1]
     Meta.mkInjectiveTheorems declName
 
 end Lean.Elab.Command

src/Lean/Elab/GuardMsgs.lean

@@ -73,9 +73,28 @@ structure GuardMsgFailure where
   res : String
 deriving TypeName
 
+/--
+Makes trailing whitespace visible and protectes them against trimming by the editor, by appending
+the symbol ⏎ to such a line (and also to any line that ends with such a symbol, to avoid
+ambiguities in the case the message already had that symbol).
+-/
+def revealTrailingWhitespace (s : String) : String :=
+  s.replace "⏎\n" "⏎⏎\n" |>.replace "\t\n" "\t⏎\n" |>.replace " \n" " ⏎\n"
+
+/- The inverse of `revealTrailingWhitespace` -/
+def removeTrailingWhitespaceMarker (s : String) : String :=
+  s.replace "⏎\n" "\n"
+
+/--
+Strings are compared up to newlines, to allow users to break long lines.
+-/
+def equalUpToNewlines (exp res : String) : Bool :=
+  exp.replace "\n" " " == res.replace "\n" " "
+
 @[builtin_command_elab Lean.guardMsgsCmd] def elabGuardMsgs : CommandElab
   | `(command| $[$dc?:docComment]? #guard_msgs%$tk $(spec?)? in $cmd) => do
-    let expected : String := (← dc?.mapM (getDocStringText ·)).getD "" |>.trim
+    let expected : String := (← dc?.mapM (getDocStringText ·)).getD ""
+        |>.trim |> removeTrailingWhitespaceMarker
     let specFn ← parseGuardMsgsSpec spec?
     let initMsgs ← modifyGet fun st => (st.messages, { st with messages := {} })
     elabCommandTopLevel cmd
@@ -88,8 +107,7 @@ deriving TypeName
       | .drop        => pure ()
       | .passthrough => toPassthrough := toPassthrough.add msg
     let res := "---\n".intercalate (← toCheck.toList.mapM (messageToStringWithoutPos ·)) |>.trim
-    -- We do some whitespace normalization here to allow users to break long lines.
-    if expected.replace "\n" " " == res.replace "\n" " " then
+    if equalUpToNewlines expected res then
       -- Passed. Only put toPassthrough messages back on the message log
       modify fun st => { st with messages := initMsgs ++ toPassthrough }
     else
@@ -119,6 +137,7 @@ def guardMsgsCodeAction : CommandCodeAction := fun _ _ _ node => do
     lazy? := some do
       let some start := stx.getPos? true | return eager
       let some tail := stx.setArg 0 mkNullNode |>.getPos? true | return eager
+      let res := revealTrailingWhitespace res
       let newText := if res.isEmpty then
         ""
       else if res.length ≤ 100-7 && !res.contains '\n' then -- TODO: configurable line length?

src/Lean/Elab/InfoTree/Main.lean

@@ -6,6 +6,7 @@ Authors: Wojciech Nawrocki, Leonardo de Moura, Sebastian Ullrich
 -/
 prelude
 import Lean.Meta.PPGoal
+import Lean.ReservedNameAction
 
 namespace Lean.Elab.CommandContextInfo
 
@@ -94,16 +95,18 @@ partial def InfoTree.substitute (tree : InfoTree) (assignment : PersistentHashMa
     | none      => hole id
     | some tree => substitute tree assignment
 
-def ContextInfo.runMetaM (info : ContextInfo) (lctx : LocalContext) (x : MetaM α) : IO α := do
-  let x := x.run { lctx := lctx } { mctx := info.mctx }
+/-- Embeds a `CoreM` action in `IO` by supplying the information stored in `info`. -/
+def ContextInfo.runCoreM (info : ContextInfo) (x : CoreM α) : IO α := do
   /-
     We must execute `x` using the `ngen` stored in `info`. Otherwise, we may create `MVarId`s and `FVarId`s that
     have been used in `lctx` and `info.mctx`.
   -/
-  let ((a, _), _) ←
+  (·.1) <$>
     x.toIO { options := info.options, currNamespace := info.currNamespace, openDecls := info.openDecls, fileName := "<InfoTree>", fileMap := default }
            { env := info.env, ngen := info.ngen }
-  return a
+
+def ContextInfo.runMetaM (info : ContextInfo) (lctx : LocalContext) (x : MetaM α) : IO α := do
+  (·.1) <$> info.runCoreM (x.run { lctx := lctx } { mctx := info.mctx })
 
 def ContextInfo.toPPContext (info : ContextInfo) (lctx : LocalContext) : PPContext :=
   { env  := info.env, mctx := info.mctx, lctx := lctx,
@@ -279,31 +282,28 @@ def addConstInfo [MonadEnv m] [MonadError m]
     expectedType?
   }
 
-/-- This does the same job as `resolveGlobalConstNoOverload`; resolving an identifier
+/-- This does the same job as `realizeGlobalConstNoOverload`; resolving an identifier
 syntax to a unique fully resolved name or throwing if there are ambiguities.
 But also adds this resolved name to the infotree. This means that when you hover
 over a name in the sourcefile you will see the fully resolved name in the hover info.-/
-def resolveGlobalConstNoOverloadWithInfo [MonadResolveName m] [MonadEnv m] [MonadError m]
-    (id : Syntax) (expectedType? : Option Expr := none) : m Name := do
-  let n ← resolveGlobalConstNoOverload id
+def realizeGlobalConstNoOverloadWithInfo (id : Syntax) (expectedType? : Option Expr := none) : CoreM Name := do
+  let n ← realizeGlobalConstNoOverload id
   if (← getInfoState).enabled then
     -- we do not store a specific elaborator since identifiers are special-cased by the server anyway
     addConstInfo id n expectedType?
   return n
 
-/-- Similar to `resolveGlobalConstNoOverloadWithInfo`, except if there are multiple name resolutions then it returns them as a list. -/
-def resolveGlobalConstWithInfos [MonadResolveName m] [MonadEnv m] [MonadError m]
-    (id : Syntax) (expectedType? : Option Expr := none) : m (List Name) := do
-  let ns ← resolveGlobalConst id
+/-- Similar to `realizeGlobalConstNoOverloadWithInfo`, except if there are multiple name resolutions then it returns them as a list. -/
+def realizeGlobalConstWithInfos (id : Syntax) (expectedType? : Option Expr := none) : CoreM (List Name) := do
+  let ns ← realizeGlobalConst id
   if (← getInfoState).enabled then
     for n in ns do
       addConstInfo id n expectedType?
   return ns
 
-/-- Similar to `resolveGlobalName`, but it also adds the resolved name to the info tree. -/
-def resolveGlobalNameWithInfos [MonadResolveName m] [MonadEnv m] [MonadError m]
-    (ref : Syntax) (id : Name) : m (List (Name × List String)) := do
-  let ns ← resolveGlobalName id
+/-- Similar to `realizeGlobalName`, but it also adds the resolved name to the info tree. -/
+def realizeGlobalNameWithInfos (ref : Syntax) (id : Name) : CoreM (List (Name × List String)) := do
+  let ns ← realizeGlobalName id
   if (← getInfoState).enabled then
     for (n, _) in ns do
       addConstInfo ref n

src/Lean/Elab/InheritDoc.lean

@@ -20,7 +20,7 @@ builtin_initialize
       | `(attr| inherit_doc $[$id?:ident]?) => withRef stx[0] do
         let some id := id?
           | throwError "invalid `[inherit_doc]` attribute, could not infer doc source"
-        let declName ← Elab.resolveGlobalConstNoOverloadWithInfo id
+        let declName ← Elab.realizeGlobalConstNoOverloadWithInfo id
         if (← findDocString? (← getEnv) decl).isSome then
           logWarning m!"{← mkConstWithLevelParams decl} already has a doc string"
         let some doc ← findDocString? (← getEnv) declName

src/Lean/Elab/MutualDef.lean

@@ -68,8 +68,6 @@ private def check (prevHeaders : Array DefViewElabHeader) (newHeader : DefViewEl
     throwError "'partial' theorems are not allowed, 'partial' is a code generation directive"
   if newHeader.kind.isTheorem && newHeader.modifiers.isNoncomputable then
     throwError "'theorem' subsumes 'noncomputable', code is not generated for theorems"
-  if newHeader.modifiers.isNoncomputable && newHeader.modifiers.isUnsafe then
-    throwError "'noncomputable unsafe' is not allowed"
   if newHeader.modifiers.isNoncomputable && newHeader.modifiers.isPartial then
     throwError "'noncomputable partial' is not allowed"
   if newHeader.modifiers.isPartial && newHeader.modifiers.isUnsafe then
@@ -646,6 +644,9 @@ def pushMain (preDefs : Array PreDefinition) (sectionVars : Array Expr) (mainHea
     let termination := termination.rememberExtraParams header.numParams mainVals[i]!
     let value ← mkLambdaFVars sectionVars mainVals[i]!
     let type ← mkForallFVars sectionVars header.type
+    if header.kind.isTheorem then
+      unless (← isProp type) do
+        throwErrorAt header.ref "type of theorem '{header.declName}' is not a proposition{indentExpr type}"
     return preDefs.push {
       ref         := getDeclarationSelectionRef header.ref
       kind        := header.kind
@@ -659,10 +660,14 @@ def pushLetRecs (preDefs : Array PreDefinition) (letRecClosures : List LetRecClo
   letRecClosures.foldlM (init := preDefs) fun preDefs c => do
     let type  := Closure.mkForall c.localDecls c.toLift.type
     let value := Closure.mkLambda c.localDecls c.toLift.val
-    -- Convert any proof let recs inside a `def` to `theorem` kind
     let kind ← if kind.isDefOrAbbrevOrOpaque then
+      -- Convert any proof let recs inside a `def` to `theorem` kind
       withLCtx c.toLift.lctx c.toLift.localInstances do
         return if (← inferType c.toLift.type).isProp then .theorem else kind
+    else if kind.isTheorem then
+      -- Convert any non-proof let recs inside a `theorem` to `def` kind
+      withLCtx c.toLift.lctx c.toLift.localInstances do
+        return if (← inferType c.toLift.type).isProp then .theorem else .def
     else
       pure kind
     return preDefs.push {
@@ -828,7 +833,7 @@ where
     for header in headers, view in views do
       if let some classNamesStx := view.deriving? then
         for classNameStx in classNamesStx do
-          let className ← resolveGlobalConstNoOverload classNameStx
+          let className ← realizeGlobalConstNoOverload classNameStx
           withRef classNameStx do
             unless (← processDefDeriving className header.declName) do
               throwError "failed to synthesize instance '{className}' for '{header.declName}'"

src/Lean/Elab/PreDefinition/Eqns.lean

@@ -317,14 +317,6 @@ def whnfReducibleLHS? (mvarId : MVarId) : MetaM (Option MVarId) := mvarId.withCo
 def tryContradiction (mvarId : MVarId) : MetaM Bool := do
   mvarId.contradictionCore { genDiseq := true }
 
-structure UnfoldEqnExtState where
-  map : PHashMap Name Name := {}
-  deriving Inhabited
-
-/- We generate the unfold equation on demand, and do not save them on .olean files. -/
-builtin_initialize unfoldEqnExt : EnvExtension UnfoldEqnExtState ←
-  registerEnvExtension (pure {})
-
 /--
   Auxiliary method for `mkUnfoldEq`. The structure is based on `mkEqnTypes`.
   `mvarId` is the goal to be proved. It is a goal of the form
@@ -370,9 +362,8 @@ partial def mkUnfoldProof (declName : Name) (mvarId : MVarId) : MetaM Unit := do
 
 /-- Generate the "unfold" lemma for `declName`. -/
 def mkUnfoldEq (declName : Name) (info : EqnInfoCore) : MetaM Name := withLCtx {} {} do
-  let env ← getEnv
   withOptions (tactic.hygienic.set · false) do
-    let baseName := mkPrivateName env declName
+    let baseName := declName
     lambdaTelescope info.value fun xs body => do
       let us := info.levelParams.map mkLevelParam
       let type ← mkEq (mkAppN (Lean.mkConst declName us) xs) body
@@ -380,7 +371,7 @@ def mkUnfoldEq (declName : Name) (info : EqnInfoCore) : MetaM Name := withLCtx {
       mkUnfoldProof declName goal.mvarId!
       let type ← mkForallFVars xs type
       let value ← mkLambdaFVars xs (← instantiateMVars goal)
-      let name := baseName ++ `_unfold
+      let name := baseName ++ `def
       addDecl <| Declaration.thmDecl {
         name, type, value
         levelParams := info.levelParams
@@ -388,13 +379,8 @@ def mkUnfoldEq (declName : Name) (info : EqnInfoCore) : MetaM Name := withLCtx {
       return name
 
 def getUnfoldFor? (declName : Name) (getInfo? : Unit → Option EqnInfoCore) : MetaM (Option Name) := do
-  let env ← getEnv
-  if let some eq := unfoldEqnExt.getState env |>.map.find? declName then
-    return some eq
-  else if let some info := getInfo? () then
-    let eq ← mkUnfoldEq declName info
-    modifyEnv fun env => unfoldEqnExt.modifyState env fun s => { s with map := s.map.insert declName eq }
-    return some eq
+  if let some info := getInfo? () then
+    return some (← mkUnfoldEq declName info)
   else
     return none
 

src/Lean/Elab/PreDefinition/Main.lean

@@ -105,6 +105,7 @@ def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLC
       See issue #2321
       -/
       let preDef ← eraseRecAppSyntax preDefs[0]!
+      ensureEqnReservedNamesAvailable preDef.declName
       if preDef.modifiers.isNoncomputable then
         addNonRec preDef
       else

src/Lean/Elab/PreDefinition/Structural/Eqns.lean

@@ -63,12 +63,12 @@ def mkEqns (info : EqnInfo) : MetaM (Array Name) :=
     let target ← mkEq (mkAppN (Lean.mkConst info.declName us) xs) body
     let goal ← mkFreshExprSyntheticOpaqueMVar target
     mkEqnTypes #[info.declName] goal.mvarId!
-  let baseName := mkPrivateName (← getEnv) info.declName
+  let baseName := info.declName
   let mut thmNames := #[]
   for i in [: eqnTypes.size] do
     let type := eqnTypes[i]!
     trace[Elab.definition.structural.eqns] "{eqnTypes[i]!}"
-    let name := baseName ++ (`_eq).appendIndexAfter (i+1)
+    let name := baseName ++ (`eq).appendIndexAfter (i+1)
     thmNames := thmNames.push name
     let value ← mkProof info.declName type
     let (type, value) ← removeUnusedEqnHypotheses type value
@@ -81,6 +81,7 @@ def mkEqns (info : EqnInfo) : MetaM (Array Name) :=
 builtin_initialize eqnInfoExt : MapDeclarationExtension EqnInfo ← mkMapDeclarationExtension
 
 def registerEqnsInfo (preDef : PreDefinition) (recArgPos : Nat) : CoreM Unit := do
+  ensureEqnReservedNamesAvailable preDef.declName
   modifyEnv fun env => eqnInfoExt.insert env preDef.declName { preDef with recArgPos }
 
 def getEqnsFor? (declName : Name) : MetaM (Option (Array Name)) := do

src/Lean/Elab/PreDefinition/WF/Eqns.lean

@@ -8,6 +8,7 @@ import Lean.Meta.Tactic.Rewrite
 import Lean.Meta.Tactic.Split
 import Lean.Elab.PreDefinition.Basic
 import Lean.Elab.PreDefinition.Eqns
+import Lean.Meta.ArgsPacker.Basic
 
 namespace Lean.Elab.WF
 open Meta
@@ -17,6 +18,7 @@ structure EqnInfo extends EqnInfoCore where
   declNames       : Array Name
   declNameNonRec  : Name
   fixedPrefixSize : Nat
+  argsPacker      : ArgsPacker
   deriving Inhabited
 
 private partial def deltaLHSUntilFix (mvarId : MVarId) : MetaM MVarId := mvarId.withContext do
@@ -107,7 +109,7 @@ private partial def mkProof (declName : Name) (type : Expr) : MetaM Expr := do
 
 def mkEqns (declName : Name) (info : EqnInfo) : MetaM (Array Name) :=
   withOptions (tactic.hygienic.set · false) do
-  let baseName := mkPrivateName (← getEnv) declName
+  let baseName := declName
   let eqnTypes ← withNewMCtxDepth <| lambdaTelescope (cleanupAnnotations := true) info.value fun xs body => do
     let us := info.levelParams.map mkLevelParam
     let target ← mkEq (mkAppN (Lean.mkConst declName us) xs) body
@@ -117,7 +119,7 @@ def mkEqns (declName : Name) (info : EqnInfo) : MetaM (Array Name) :=
   for i in [: eqnTypes.size] do
     let type := eqnTypes[i]!
     trace[Elab.definition.wf.eqns] "{eqnTypes[i]!}"
-    let name := baseName ++ (`_eq).appendIndexAfter (i+1)
+    let name := baseName ++ (`eq).appendIndexAfter (i+1)
     thmNames := thmNames.push name
     let value ← mkProof declName type
     let (type, value) ← removeUnusedEqnHypotheses type value
@@ -129,7 +131,9 @@ def mkEqns (declName : Name) (info : EqnInfo) : MetaM (Array Name) :=
 
 builtin_initialize eqnInfoExt : MapDeclarationExtension EqnInfo ← mkMapDeclarationExtension
 
-def registerEqnsInfo (preDefs : Array PreDefinition) (declNameNonRec : Name) (fixedPrefixSize : Nat) : MetaM Unit := do
+def registerEqnsInfo (preDefs : Array PreDefinition) (declNameNonRec : Name) (fixedPrefixSize : Nat)
+    (argsPacker : ArgsPacker) : MetaM Unit := do
+  preDefs.forM fun preDef => ensureEqnReservedNamesAvailable preDef.declName
   /-
   See issue #2327.
   Remark: we could do better for mutual declarations that mix theorems and definitions. However, this is a rare
@@ -140,7 +144,8 @@ def registerEqnsInfo (preDefs : Array PreDefinition) (declNameNonRec : Name) (fi
       let declNames := preDefs.map (·.declName)
       modifyEnv fun env =>
         preDefs.foldl (init := env) fun env preDef =>
-          eqnInfoExt.insert env preDef.declName { preDef with declNames, declNameNonRec, fixedPrefixSize }
+          eqnInfoExt.insert env preDef.declName { preDef with
+            declNames, declNameNonRec, fixedPrefixSize, argsPacker }
 
 def getEqnsFor? (declName : Name) : MetaM (Option (Array Name)) := do
   if let some info := eqnInfoExt.find? (← getEnv) declName then

src/Lean/Elab/PreDefinition/WF/Fix.lean

@@ -8,12 +8,12 @@ import Lean.Util.HasConstCache
 import Lean.Meta.Match.Match
 import Lean.Meta.Tactic.Simp.Main
 import Lean.Meta.Tactic.Cleanup
+import Lean.Meta.ArgsPacker
 import Lean.Elab.Tactic.Basic
 import Lean.Elab.RecAppSyntax
 import Lean.Elab.PreDefinition.Basic
 import Lean.Elab.PreDefinition.Structural.Basic
 import Lean.Elab.PreDefinition.Structural.BRecOn
-import Lean.Elab.PreDefinition.WF.PackMutual
 import Lean.Data.Array
 
 namespace Lean.Elab.WF
@@ -172,26 +172,28 @@ know which function is making the call.
 The close coupling with how arguments are packed and termination goals look like is not great,
 but it works for now.
 -/
-def groupGoalsByFunction (numFuncs : Nat) (goals : Array MVarId) : MetaM (Array (Array MVarId)) := do
+def groupGoalsByFunction (argsPacker : ArgsPacker) (numFuncs : Nat) (goals : Array MVarId) : MetaM (Array (Array MVarId)) := do
   let mut r := mkArray numFuncs #[]
   for goal in goals do
-    let (.mdata _ (.app _ param)) ← goal.getType
-        | throwError "MVar does not look like like a recursive call"
-    let (funidx, _) ← unpackMutualArg numFuncs param
+    let type ← goal.getType
+    let (.mdata _ (.app _ param)) := type
+        | throwError "MVar does not look like a recursive call:{indentExpr type}"
+    let (funidx, _) ← argsPacker.unpack param
     r := r.modify funidx (·.push goal)
   return r
 
-def solveDecreasingGoals (decrTactics : Array (Option DecreasingBy)) (value : Expr) : MetaM Expr := do
+def solveDecreasingGoals (argsPacker : ArgsPacker) (decrTactics : Array (Option DecreasingBy)) (value : Expr) : MetaM Expr := do
   let goals ← getMVarsNoDelayed value
   let goals ← assignSubsumed goals
-  let goalss ← groupGoalsByFunction decrTactics.size goals
+  let goalss ← groupGoalsByFunction argsPacker decrTactics.size goals
   for goals in goalss, decrTactic? in decrTactics do
     Lean.Elab.Term.TermElabM.run' do
     match decrTactic? with
     | none => do
       for goal in goals do
+        let type ← goal.getType
         let some ref := getRecAppSyntax? (← goal.getType)
-          | throwError "MVar does not look like like a recursive call"
+          | throwError "MVar not annotated as a recursive call:{indentExpr type}"
         withRef ref <| applyDefaultDecrTactic goal
     | some decrTactic => withRef decrTactic.ref do
       unless goals.isEmpty do -- unlikely to be empty
@@ -205,8 +207,8 @@ def solveDecreasingGoals (decrTactics : Array (Option DecreasingBy)) (value : Ex
           Term.reportUnsolvedGoals remainingGoals
   instantiateMVars value
 
-def mkFix (preDef : PreDefinition) (prefixArgs : Array Expr) (wfRel : Expr)
-    (decrTactics : Array (Option DecreasingBy)) : TermElabM Expr := do
+def mkFix (preDef : PreDefinition) (prefixArgs : Array Expr) (argsPacker : ArgsPacker)
+    (wfRel : Expr) (decrTactics : Array (Option DecreasingBy)) : TermElabM Expr := do
   let type ← instantiateForall preDef.type prefixArgs
   let (wfFix, varName) ← forallBoundedTelescope type (some 1) fun x type => do
     let x := x[0]!
@@ -229,7 +231,7 @@ def mkFix (preDef : PreDefinition) (prefixArgs : Array Expr) (wfRel : Expr)
       let val := preDef.value.beta (prefixArgs.push x)
       let val ← processSumCasesOn x F val fun x F val => do
         processPSigmaCasesOn x F val (replaceRecApps preDef.declName prefixArgs.size)
-      let val ← solveDecreasingGoals decrTactics val
+      let val ← solveDecreasingGoals argsPacker decrTactics val
       mkLambdaFVars prefixArgs (mkApp wfFix (← mkLambdaFVars #[x, F] val))
 
 end Lean.Elab.WF