chore: upstream orphaned tests from Std

.github/workflows/ci.yml

@@ -140,12 +140,10 @@ jobs:
                 "shell": "msys2 {0}",
                 "CMAKE_OPTIONS": "-G \"Unix Makefiles\" -DUSE_GMP=OFF",
                 // for reasons unknown, interactivetests are flaky on Windows
-                // also, the liasolver test hits “too many exported symbols”
-                "CTEST_OPTIONS": "--repeat until-pass:2 -E 'leanbenchtest_liasolver.lean'",
+                "CTEST_OPTIONS": "--repeat until-pass:2",
                 "llvm-url": "https://github.com/leanprover/lean-llvm/releases/download/15.0.1/lean-llvm-x86_64-w64-windows-gnu.tar.zst",
                 "prepare-llvm": "../script/prepare-llvm-mingw.sh lean-llvm*",
-                // TEMP while compiler tests are deactivated
-                "binary-check": "true"
+                "binary-check": "ldd"
               },
               {
                 "name": "Linux aarch64",

.github/workflows/nix-ci.yml

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

RELEASES.md

@@ -8,69 +8,9 @@ This file contains work-in-progress notes for the upcoming release, as well as p
 Please check the [releases](https://github.com/leanprover/lean4/releases) page for the current status
 of each version.
 
-v4.8.0 (development in progress)
+v4.7.0 (development in progress)
 ---------
 
-* 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
-  ```
-
-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.
@@ -78,10 +18,6 @@ v4.7.0
 
 * `pp.proofs.withType` is now set to false by default to reduce noise in the info view.
 
-* The pretty printer for applications now handles the case of over-application itself when applying app unexpanders.
-  In particular, the ``| `($_ $a $b $xs*) => `(($a + $b) $xs*)`` case of an `app_unexpander` is no longer necessary.
-  [#3495](https://github.com/leanprover/lean4/pull/3495).
-
 * New `simp` (and `dsimp`) configuration option: `zetaDelta`. It is `false` by default.
   The `zeta` option is still `true` by default, but their meaning has changed.
   - When `zeta := true`, `simp` and `dsimp` reduce terms of the form
@@ -90,7 +26,7 @@ v4.7.0
     the context. For example, suppose the context contains `x := val`. Then,
     any occurrence of `x` is replaced with `val`.
 
-  See [issue #2682](https://github.com/leanprover/lean4/pull/2682) for additional details. Here are some examples:
+  See issue [#2682](https://github.com/leanprover/lean4/pull/2682) for additional details. Here are some examples:
   ```
   example (h : z = 9) : let x := 5; let y := 4; x + y = z := by
     intro x
@@ -131,7 +67,7 @@ v4.7.0
   ```
 
 * When adding new local theorems to `simp`, the system assumes that the function application arguments
-  have been annotated with `no_index`. This modification, which addresses [issue #2670](https://github.com/leanprover/lean4/issues/2670),
+  have been annotated with `no_index`. This modification, which addresses issue [#2670](https://github.com/leanprover/lean4/issues/2670),
   restores the Lean 3 behavior that users expect. With this modification, the following examples are now operational:
   ```
   example {α β : Type} {f : α × β → β → β} (h : ∀ p : α × β, f p p.2 = p.2)
@@ -147,178 +83,76 @@ v4.7.0
 
 * Improved the error messages produced by the `decide` tactic. [#3422](https://github.com/leanprover/lean4/pull/3422)
 
-* Improved auto-completion performance. [#3460](https://github.com/leanprover/lean4/pull/3460)
-
-* Improved initial language server startup performance. [#3552](https://github.com/leanprover/lean4/pull/3552)
-
-* 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)
-
-* 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
-  have a slightly longer startup time.  The order used for selection lemmas has
-  changed as well to favor goals purely based on how many terms in the head
-  pattern match the current goal.
-
-* The `solve_by_elim` tactic has been ported from `Std` to Lean so that library
-  search can use it.
-
-* New `#check_tactic` and `#check_simp` commands have been added.  These are
-  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
 ---------
 
 * Add custom simplification procedures (aka `simproc`s) to `simp`. Simprocs can be triggered by the simplifier on a specified term-pattern. Here is an small example:
-  ```lean
-  import Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat
+```lean
+import Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat
 
-  def foo (x : Nat) : Nat :=
-    x + 10
+def foo (x : Nat) : Nat :=
+  x + 10
 
-  /--
-  The `simproc` `reduceFoo` is invoked on terms that match the pattern `foo _`.
-  -/
-  simproc reduceFoo (foo _) :=
-    /- A term of type `Expr → SimpM Step -/
-    fun e => do
-      /-
-      The `Step` type has three constructors: `.done`, `.visit`, `.continue`.
-      * The constructor `.done` instructs `simp` that the result does
-        not need to be simplied further.
-      * The constructor `.visit` instructs `simp` to visit the resulting expression.
-      * The constructor `.continue` instructs `simp` to try other simplification procedures.
-
-      All three constructors take a `Result`. The `.continue` contructor may also take `none`.
-      `Result` has two fields `expr` (the new expression), and `proof?` (an optional proof).
-       If the new expression is definitionally equal to the input one, then `proof?` can be omitted or set to `none`.
-      -/
-      /- `simp` uses matching modulo reducibility. So, we ensure the term is a `foo`-application. -/
-      unless e.isAppOfArity ``foo 1 do
-        return .continue
-      /- `Nat.fromExpr?` tries to convert an expression into a `Nat` value -/
-      let some n ← Nat.fromExpr? e.appArg!
-        | return .continue
-      return .done { expr := Lean.mkNatLit (n+10) }
-  ```
-  We disable simprocs support by using the command `set_option simprocs false`. This command is particularly useful when porting files to v4.6.0.
-  Simprocs can be scoped, manually added to `simp` commands, and suppressed using `-`. They are also supported by `simp?`. `simp only` does not execute any `simproc`. Here are some examples for the `simproc` defined above.
-  ```lean
-  example : x + foo 2 = 12 + x := by
-    set_option simprocs false in
-      /- This `simp` command does not make progress since `simproc`s are disabled. -/
-      fail_if_success simp
-    simp_arith
-
-  example : x + foo 2 = 12 + x := by
-    /- `simp only` must not use the default simproc set. -/
-    fail_if_success simp only
-    simp_arith
-
-  example : x + foo 2 = 12 + x := by
+/--
+The `simproc` `reduceFoo` is invoked on terms that match the pattern `foo _`.
+-/
+simproc reduceFoo (foo _) :=
+  /- A term of type `Expr → SimpM Step -/
+  fun e => do
     /-
-    `simp only` does not use the default simproc set,
-    but we can provide simprocs as arguments. -/
-    simp only [reduceFoo]
-    simp_arith
+    The `Step` type has three constructors: `.done`, `.visit`, `.continue`.
+    * The constructor `.done` instructs `simp` that the result does
+      not need to be simplied further.
+    * The constructor `.visit` instructs `simp` to visit the resulting expression.
+    * The constructor `.continue` instructs `simp` to try other simplification procedures.
 
-  example : x + foo 2 = 12 + x := by
-    /- We can use `-` to disable `simproc`s. -/
-    fail_if_success simp [-reduceFoo]
-    simp_arith
-  ```
-  The command `register_simp_attr <id>` now creates a `simp` **and** a `simproc` set with the name `<id>`. The following command instructs Lean to insert the `reduceFoo` simplification procedure into the set `my_simp`. If no set is specified, Lean uses the default `simp` set.
-  ```lean
-  simproc [my_simp] reduceFoo (foo _) := ...
-  ```
+    All three constructors take a `Result`. The `.continue` contructor may also take `none`.
+    `Result` has two fields `expr` (the new expression), and `proof?` (an optional proof).
+     If the new expression is definitionally equal to the input one, then `proof?` can be omitted or set to `none`.
+    -/
+    /- `simp` uses matching modulo reducibility. So, we ensure the term is a `foo`-application. -/
+    unless e.isAppOfArity ``foo 1 do
+      return .continue
+    /- `Nat.fromExpr?` tries to convert an expression into a `Nat` value -/
+    let some n ← Nat.fromExpr? e.appArg!
+      | return .continue
+    return .done { expr := Lean.mkNatLit (n+10) }
+```
+We disable simprocs support by using the command `set_option simprocs false`. This command is particularly useful when porting files to v4.6.0.
+Simprocs can be scoped, manually added to `simp` commands, and suppressed using `-`. They are also supported by `simp?`. `simp only` does not execute any `simproc`. Here are some examples for the `simproc` defined above.
+```lean
+example : x + foo 2 = 12 + x := by
+  set_option simprocs false in
+    /- This `simp` command does not make progress since `simproc`s are disabled. -/
+    fail_if_success simp
+  simp_arith
+
+example : x + foo 2 = 12 + x := by
+  /- `simp only` must not use the default simproc set. -/
+  fail_if_success simp only
+  simp_arith
+
+example : x + foo 2 = 12 + x := by
+  /-
+  `simp only` does not use the default simproc set,
+  but we can provide simprocs as arguments. -/
+  simp only [reduceFoo]
+  simp_arith
+
+example : x + foo 2 = 12 + x := by
+  /- We can use `-` to disable `simproc`s. -/
+  fail_if_success simp [-reduceFoo]
+  simp_arith
+```
+The command `register_simp_attr <id>` now creates a `simp` **and** a `simproc` set with the name `<id>`. The following command instructs Lean to insert the `reduceFoo` simplification procedure into the set `my_simp`. If no set is specified, Lean uses the default `simp` set.
+```lean
+simproc [my_simp] reduceFoo (foo _) := ...
+```
 
 * The syntax of the `termination_by` and `decreasing_by` termination hints is overhauled:
 
@@ -457,7 +291,7 @@ v4.6.0
   and hence greatly reduces the reliance on costly structure eta reduction. This has a large impact on mathlib,
   reducing total CPU instructions by 3% and enabling impactful refactors like leanprover-community/mathlib4#8386
   which reduces the build time by almost 20%.
-  See [PR #2478](https://github.com/leanprover/lean4/pull/2478) and [RFC #2451](https://github.com/leanprover/lean4/issues/2451).
+  See PR [#2478](https://github.com/leanprover/lean4/pull/2478) and RFC [#2451](https://github.com/leanprover/lean4/issues/2451).
 
 * Add pretty printer settings to omit deeply nested terms (`pp.deepTerms false` and `pp.deepTerms.threshold`) ([PR #3201](https://github.com/leanprover/lean4/pull/3201))
 
@@ -476,7 +310,7 @@ Other improvements:
 * produce simpler proof terms in `rw` [#3121](https://github.com/leanprover/lean4/pull/3121)
 * fuse nested `mkCongrArg` calls in proofs generated by `simp` [#3203](https://github.com/leanprover/lean4/pull/3203)
 * `induction using` followed by a general term [#3188](https://github.com/leanprover/lean4/pull/3188)
-* allow generalization in `let` [#3060](https://github.com/leanprover/lean4/pull/3060), fixing [#3065](https://github.com/leanprover/lean4/issues/3065)
+* allow generalization in `let` [#3060](https://github.com/leanprover/lean4/pull/3060, fixing [#3065](https://github.com/leanprover/lean4/issues/3065)
 * reducing out-of-bounds `swap!` should return `a`, not `default`` [#3197](https://github.com/leanprover/lean4/pull/3197), fixing [#3196](https://github.com/leanprover/lean4/issues/3196)
 * derive `BEq` on structure with `Prop`-fields [#3191](https://github.com/leanprover/lean4/pull/3191), fixing [#3140](https://github.com/leanprover/lean4/issues/3140)
 * refine through more `casesOnApp`/`matcherApp` [#3176](https://github.com/leanprover/lean4/pull/3176), fixing [#3175](https://github.com/leanprover/lean4/pull/3175)

doc/SUMMARY.md

@@ -89,6 +89,5 @@
 - [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

@@ -111,15 +111,6 @@ 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

@@ -1,201 +0,0 @@
-# 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

@@ -277,13 +277,14 @@ 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) (ne : k ≠ k') (v : β)
+theorem BinTree.find_insert_of_ne (b : BinTree β) (h : 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 le
-    exact Nat.lt_of_le_of_ne le ne
+    intros
+    have_eq k k'
+    contradiction
   | node left key value right ihl ihr =>
     let .node hl hr bl br := h
     specialize ihl bl

src/CMakeLists.txt

@@ -9,7 +9,7 @@ endif()
 include(ExternalProject)
 project(LEAN CXX C)
 set(LEAN_VERSION_MAJOR 4)
-set(LEAN_VERSION_MINOR 8)
+set(LEAN_VERSION_MINOR 7)
 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'")
@@ -501,18 +501,24 @@ string(REGEX REPLACE "^([a-zA-Z]):" "/\\1" LEAN_BIN "${CMAKE_BINARY_DIR}/bin")
 # (also looks nicer in the build log)
 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")
+if(${CMAKE_SYSTEM_NAME} MATCHES "Darwin")
+  string(APPEND INIT_SHARED_LINKER_FLAGS " -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libInit.a -Wl,-force_load,${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a")
+else()
+  string(APPEND INIT_SHARED_LINKER_FLAGS " -Wl,--whole-archive -lInit ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a -Wl,--no-whole-archive")
+  if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
+    string(APPEND INIT_SHARED_LINKER_FLAGS " -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libInit_shared.dll.a")
+  endif()
 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,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
+  string(APPEND LEANSHARED_LINKER_FLAGS " -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libLean.a -Wl,-force_load,${CMAKE_BINARY_DIR}/lib/lean/libleancpp.a")
 else()
-  set(LEANSHARED_LINKER_FLAGS "-Wl,--whole-archive -lInit -lLean -lleancpp -Wl,--no-whole-archive ${CMAKE_BINARY_DIR}/runtime/libleanrt_initial-exec.a ${LEANSHARED_LINKER_FLAGS}")
+  string(APPEND LEANSHARED_LINKER_FLAGS " -Wl,--whole-archive -lLean -lleancpp -Wl,--no-whole-archive")
+  if(${CMAKE_SYSTEM_NAME} MATCHES "Windows")
+    string(APPEND LEANSHARED_LINKER_FLAGS " -Wl,--out-implib,${CMAKE_BINARY_DIR}/lib/lean/libleanshared.dll.a")
+  endif()
 endif()
+string(APPEND LEANSHARED_LINKER_FLAGS " -lInit_shared")
 
 if (${CMAKE_SYSTEM_NAME} MATCHES "Emscripten")
   # We do not use dynamic linking via leanshared for Emscripten to keep things

src/Init/ByCases.lean

@@ -37,6 +37,15 @@ 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

@@ -125,15 +125,16 @@ 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. -/
-@[simp] theorem not_not : ¬¬a ↔ a := Decidable.not_not
+@[scoped simp] theorem not_not : ¬¬a ↔ a := Decidable.not_not
 
-@[simp low] theorem not_forall {p : α → Prop} : (¬∀ x, p x) ↔ ∃ x, ¬p x := Decidable.not_forall
+@[simp] 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 _
 
@@ -146,22 +147,8 @@ 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.Basic
+import Init.Control.Lawful
 
 /-!
 The Exception monad transformer using CPS style.

src/Init/Control/Lawful.lean

@@ -4,5 +4,373 @@ 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.Lawful.Instances
+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

src/Init/Control/Lawful/Basic.lean

@@ -1,138 +0,0 @@
-/-
-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

@@ -1,248 +0,0 @@
-/-
-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.Basic
+import Init.Control.Lawful
 
 /-!
 The State monad transformer using CPS style.

src/Init/Conv.lean

@@ -156,6 +156,7 @@ 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

@@ -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
 
-@[simp] theorem cast_eq {α : Sort u} (h : α = α) (a : α) : cast h a = a :=
+theorem cast_eq {α : Sort u} (h : α = α) (a : α) : cast h a = a :=
   rfl
 
 /--
@@ -737,16 +737,13 @@ 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}
 
-/-- 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 :=
+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 :=
   h.rec m
 
-/-- `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 :=
+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 :=
   h.rec m
 
-/-- `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 :=
+theorem 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 :=
@@ -1406,9 +1403,9 @@ 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 high] theorem imp_self : (a → a) ↔ True := iff_true_intro id
+@[simp] theorem imp_self : (a → a) ↔ True := iff_true_intro id
 
-@[simp] theorem imp_false : (a → False) ↔ ¬a := Iff.rfl
+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
 
-noncomputable def List.two_step_induction
+theorem List.two_step_induction
   {motive : List Nat → Sort u}
   (l : List Nat)
   (empty : motive [])

src/Init/Data/Array/Basic.lean

@@ -809,7 +809,7 @@ where
     rfl
 
   go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.data.drop i) = as.data := by
-    induction i <;> simp [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, *]
+    cases i <;> simp [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, go]
 
 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

@@ -8,7 +8,6 @@ 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)⟩ -- TODO: remove
+  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
   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

@@ -7,7 +7,6 @@ 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
@@ -73,9 +72,6 @@ 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
@@ -128,20 +124,13 @@ section Int
 
 /-- Interpret the bitvector as an integer stored in two's complement form. -/
 protected def toInt (a : BitVec n) : Int :=
-  if 2 * a.toNat < 2^n then
-    a.toNat
-  else
-    (a.toNat : Int) - (2^n : Nat)
+  if a.msb then Int.ofNat a.toNat - Int.ofNat (2^n) else a.toNat
 
 /-- The `BitVec` with value `(2^n + (i mod 2^n)) mod 2^n`.  -/
-protected def ofInt (n : Nat) (i : Int) : BitVec n := .ofNatLt (i % (Int.ofNat (2^n))).toNat (by
-  apply (Int.toNat_lt _).mpr
-  · apply Int.emod_lt_of_pos
-    exact Int.ofNat_pos.mpr (Nat.two_pow_pos _)
-  · apply Int.emod_nonneg
-    intro eq
-    apply Nat.ne_of_gt (Nat.two_pow_pos n)
-    exact Int.ofNat_inj.mp eq)
+protected def ofInt (n : Nat) (i : Int) : BitVec n :=
+  match i with
+  | Int.ofNat x => .ofNat n x
+  | Int.negSucc x => BitVec.ofNatLt (2^n - x % 2^n - 1) (by omega)
 
 instance : IntCast (BitVec w) := ⟨BitVec.ofInt w⟩
 

src/Init/Data/BitVec/Bitblast.lean

@@ -5,7 +5,6 @@ Authors: Harun Khan, Abdalrhman M Mohamed, Joe Hendrix
 -/
 prelude
 import Init.Data.BitVec.Folds
-import Init.Data.Nat.Mod
 
 /-!
 # Bitblasting of bitvectors
@@ -71,8 +70,24 @@ 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
-  rw [Nat.mod_pow_succ, Nat.add_comm, Nat.toNat_testBit]
+  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]
 
 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

@@ -29,12 +29,14 @@ 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 : w ≤ i) : getLsb x i = false := by
+@[simp] theorem getLsb_ge (x : BitVec w) (i : Nat) (ge : i ≥ w) : getLsb x i = false := by
   let ⟨x, x_lt⟩ := x
   simp
   apply Nat.testBit_lt_two_pow
@@ -70,7 +72,7 @@ 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]
+      simp [Nat.le_sub_iff_add_le w_pos, Nat.add_succ]
       exact i_lt
     have q_lt : w - 1 - i < w := by
       simp only [Nat.sub_sub]
@@ -87,9 +89,6 @@ 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']
 
@@ -117,8 +116,6 @@ 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
 
@@ -136,35 +133,21 @@ theorem msb_eq_getLsb_last (x : BitVec w) :
   · simp [BitVec.eq_nil x]
   · simp
 
-@[bv_toNat] theorem getLsb_last (x : BitVec w) :
-    x.getLsb (w-1) = decide (2 ^ (w-1) ≤ x.toNat) := by
-  rcases w with rfl | w
-  · simp
-  · simp only [Nat.zero_lt_succ, decide_True, getLsb, Nat.testBit, Nat.succ_sub_succ_eq_sub,
+@[bv_toNat] theorem getLsb_last (x : BitVec (w + 1)) :
+    x.getLsb w = decide (2 ^ w ≤ x.toNat) := by
+  simp only [Nat.zero_lt_succ, decide_True, getLsb, Nat.testBit, Nat.succ_sub_succ_eq_sub,
     Nat.sub_zero, Nat.and_one_is_mod, Bool.true_and, Nat.shiftRight_eq_div_pow]
-    rcases (Nat.lt_or_ge (BitVec.toNat x) (2 ^ w)) with h | h
-    · simp [Nat.div_eq_of_lt h, h]
-    · simp only [h]
-      rw [Nat.div_eq_sub_div (Nat.two_pow_pos w) h, Nat.div_eq_of_lt]
-      · decide
-      · have : BitVec.toNat x < 2^w + 2^w := by simpa [Nat.pow_succ, Nat.mul_two] using x.isLt
-        omega
+  rcases (Nat.lt_or_ge (BitVec.toNat x) (2 ^ w)) with h | h
+  · simp [Nat.div_eq_of_lt h, h]
+  · simp only [h]
+    rw [Nat.div_eq_sub_div (Nat.two_pow_pos w) h, Nat.div_eq_of_lt]
+    · decide
+    · have : BitVec.toNat x < 2^w + 2^w := by simpa [Nat.pow_succ, Nat.mul_two] using x.isLt
+      omega
 
-@[bv_toNat] theorem getLsb_succ_last (x : BitVec (w + 1)) :
-    x.getLsb w = decide (2 ^ w ≤ x.toNat) := getLsb_last x
-
-@[bv_toNat] theorem msb_eq_decide (x : BitVec w) : BitVec.msb x = decide (2 ^ (w-1) ≤ x.toNat) := by
+@[bv_toNat] theorem msb_eq_decide (x : BitVec (w + 1)) : BitVec.msb x = decide (2 ^ w ≤ x.toNat) := by
   simp [msb_eq_getLsb_last, getLsb_last]
 
-theorem toNat_ge_of_msb_true {x : BitVec n} (p : BitVec.msb x = true) : x.toNat ≥ 2^(n-1) := by
-  match n with
-  | 0 =>
-    simp [BitVec.msb, BitVec.getMsb] at p
-  | n + 1 =>
-    simp [BitVec.msb_eq_decide] at p
-    simp only [Nat.add_sub_cancel]
-    exact p
-
 /-! ### cast -/
 
 @[simp, bv_toNat] theorem toNat_cast (h : w = v) (x : BitVec w) : (cast h x).toNat = x.toNat := rfl
@@ -180,53 +163,6 @@ theorem toNat_ge_of_msb_true {x : BitVec n} (p : BitVec.msb x = true) : x.toNat
 @[simp] theorem msb_cast (h : w = v) (x : BitVec w) : (cast h x).msb = x.msb := by
   simp [BitVec.msb]
 
-/-! ### toInt/ofInt -/
-
-/-- Prove equality of bitvectors in terms of nat operations. -/
-theorem toInt_eq_toNat_cond (i : BitVec n) :
-    i.toInt =
-      if 2*i.toNat < 2^n then
-        (i.toNat : Int)
-      else
-        (i.toNat : Int) - (2^n : Nat) := by
-  unfold BitVec.toInt
-  split <;> omega
-
-theorem toInt_eq_toNat_bmod (x : BitVec n) : x.toInt = Int.bmod x.toNat (2^n) := by
-  simp only [toInt_eq_toNat_cond]
-  split
-  case inl g =>
-    rw [Int.bmod_pos] <;> simp only [←Int.ofNat_emod, toNat_mod_cancel]
-    omega
-  case inr g =>
-    rw [Int.bmod_neg] <;> simp only [←Int.ofNat_emod, toNat_mod_cancel]
-    omega
-
-/-- Prove equality of bitvectors in terms of nat operations. -/
-theorem eq_of_toInt_eq {i j : BitVec n} : i.toInt = j.toInt → i = j := by
-  intro eq
-  simp [toInt_eq_toNat_cond] at eq
-  apply eq_of_toNat_eq
-  revert eq
-  have _ilt := i.isLt
-  have _jlt := j.isLt
-  split <;> split <;> omega
-
-@[simp] theorem toNat_ofInt {n : Nat} (i : Int) :
-  (BitVec.ofInt n i).toNat = (i % (2^n : Nat)).toNat := by
-  unfold BitVec.ofInt
-  simp
-
-theorem toInt_ofNat {n : Nat} (x : Nat) :
-  (BitVec.ofNat n x).toInt = (x : Int).bmod (2^n) := by
-  simp [toInt_eq_toNat_bmod]
-
-@[simp] theorem toInt_ofInt {n : Nat} (i : Int) :
-  (BitVec.ofInt n i).toInt = i.bmod (2^n) := by
-  have _ := Nat.two_pow_pos n
-  have p : 0 ≤ i % (2^n : Nat) := by omega
-  simp [toInt_eq_toNat_bmod, Int.toNat_of_nonneg p]
-
 /-! ### zeroExtend and truncate -/
 
 @[simp, bv_toNat] theorem toNat_zeroExtend' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
@@ -244,12 +180,6 @@ 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
 
@@ -294,25 +224,10 @@ 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
@@ -325,18 +240,11 @@ theorem msb_truncate (x : BitVec w) : (x.truncate (k + 1)).msb = x.getLsb k := b
     (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]
@@ -384,18 +292,6 @@ 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) :
@@ -410,18 +306,6 @@ 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) :
@@ -437,11 +321,6 @@ 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
@@ -456,12 +335,12 @@ 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 (isLt _)]
+    rw [Nat.testBit_two_pow_sub_succ (toNat_lt _)]
     · cases w : decide (i < v)
       · simp at w
         simp [w]
         rw [Nat.testBit_lt_two_pow]
-        calc BitVec.toNat x < 2 ^ v := isLt _
+        calc BitVec.toNat x < 2 ^ v := toNat_lt _
           _ ≤ 2 ^ i := Nat.pow_le_pow_of_le_right Nat.zero_lt_two w
       · simp
 
@@ -474,12 +353,6 @@ 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
-
 /-! ### shiftLeft -/
 
 @[simp, bv_toNat] theorem toNat_shiftLeft {x : BitVec v} :
@@ -497,19 +370,6 @@ 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
@@ -518,7 +378,7 @@ 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 x.isLt (Nat.two_pow_pos _)
+    exact Nat.mul_lt_mul_of_pos_right (BitVec.toNat_lt x) (Nat.two_pow_pos _)
   · omega
 
 @[simp] theorem getLsb_shiftLeftZeroExtend (x : BitVec m) (n : Nat) :
@@ -529,10 +389,6 @@ 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) :
@@ -558,34 +414,6 @@ 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]
-
 /-! ### rev -/
 
 theorem getLsb_rev (x : BitVec w) (i : Fin w) :
@@ -608,11 +436,6 @@ 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]
@@ -627,9 +450,6 @@ theorem toNat_cons' {x : BitVec 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
@@ -641,15 +461,6 @@ 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
-
 /-! ### concat -/
 
 @[simp] theorem toNat_concat (x : BitVec w) (b : Bool) :
@@ -674,21 +485,6 @@ 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
@@ -715,10 +511,6 @@ 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

@@ -29,8 +29,6 @@ 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 := ⟨(!. && .)⟩
 
@@ -50,205 +48,85 @@ 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} [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
+@[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
 
 /-! ### and -/
 
-@[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
 
-/-
-Bool version `and_iff_left_iff_imp`.
+theorem and_or_distrib_left : ∀ (x y z : Bool), (x && (y || z)) = ((x && y) || (x && z)) := 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
+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
 
 /-! ### or -/
 
-@[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
 
-/-! ### distributivity -/
+theorem or_and_distrib_left : ∀ (x y z : Bool), (x || (y && z)) = ((x || y) && (x || 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
+theorem or_and_distrib_right : ∀ (x y z : Bool), ((x && y) || z) = ((x || z) && (y || z)) := by
+  decide
 
 /-- De Morgan's law for boolean or -/
-@[simp] theorem not_or : ∀ (x y : Bool), (!(x || y)) = (!x && !y) := by decide
+theorem not_or : ∀ (x y : Bool), (!(x || y)) = (!x && !y) := 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_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
-
-/-
-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
+theorem or_eq_false_iff : ∀ (x y : Bool), (x || y) = false ↔ x = false ∧ y = false := by decide
 
 /-! ### xor -/
 
-theorem false_xor : ∀ (x : Bool), xor false x = x := false_bne
+@[simp] theorem false_xor : ∀ (x : Bool), xor false x = x := by decide
 
-theorem xor_false : ∀ (x : Bool), xor x false = x := bne_false
+@[simp] theorem xor_false : ∀ (x : Bool), xor x false = x := by decide
 
-theorem true_xor : ∀ (x : Bool), xor true x = !x := true_bne
+@[simp] theorem true_xor : ∀ (x : Bool), xor true x = !x := by decide
 
-theorem xor_true : ∀ (x : Bool), xor x true = !x := bne_true
+@[simp] theorem xor_true : ∀ (x : Bool), xor x true = !x := by decide
 
-theorem not_xor_self : ∀ (x : Bool), xor (!x) x = true := not_bne_self
+@[simp] theorem not_xor_self : ∀ (x : Bool), xor (!x) x = true := by decide
 
-theorem xor_not_self : ∀ (x : Bool), xor x (!x) = true := bne_not_self
+@[simp] theorem xor_not_self : ∀ (x : Bool), xor x (!x) = true := by decide
 
 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
 
-theorem not_xor_not : ∀ (x y : Bool), xor (!x) (!y) = (xor x y) := not_bne_not
+@[simp] theorem not_xor_not : ∀ (x y : Bool), xor (!x) (!y) = (xor x y) := by decide
 
 theorem xor_self : ∀ (x : Bool), xor x x = false := by decide
 
@@ -258,11 +136,13 @@ 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) := bne_assoc
+theorem xor_assoc : ∀ (x y z : Bool), xor (xor x y) z = xor x (xor 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_left_inj : ∀ (x y z : Bool), xor x y = xor x z ↔ y = z := by decide
 
-theorem xor_right_inj : ∀ (x y z : Bool), xor x z = xor y z ↔ x = y := bne_right_inj
+@[simp]
+theorem xor_right_inj : ∀ (x y z : Bool), xor x z = xor y z ↔ x = y := by decide
 
 /-! ### le/lt -/
 
@@ -347,151 +227,15 @@ 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
 
-/-! ### 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_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)
+/-! ### cond -/
+
+theorem cond_eq_if : (bif b then x else y) = (if b then x else y) := by
+  cases b <;> simp
 
 /-! ### decide -/
 

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
+  rw [reverseInduction]; simp; rfl
 
 @[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

@@ -11,4 +11,3 @@ 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

@@ -158,52 +158,4 @@ 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)
-
-Balanced mod (and balanced div) are a division and modulus pair such
-that `b * (Int.bdiv a b) + Int.bmod a b = a` and `b/2 ≤ Int.bmod a b <
-b/2` for all `a : Int` and `b > 0`.
-
-This is used in Omega as well as signed bitvectors.
--/
-
-/--
-Balanced modulus.  This version of Integer modulus uses the
-balanced rounding convention, which guarantees that
-`m/2 ≤ bmod x m < m/2` for `m ≠ 0` and `bmod x m` is congruent
-to `x` modulo `m`.
-
-If `m = 0`, then `bmod x m = x`.
--/
-def bmod (x : Int) (m : Nat) : Int :=
-  let r := x % m
-  if r < (m + 1) / 2 then
-    r
-  else
-    r - m
-
-/--
-Balanced division.  This returns the unique integer so that
-`b * (Int.bdiv a b) + Int.bmod a b = a`.
--/
-def bdiv (x : Int) (m : Nat) : Int :=
-  if m = 0 then
-    0
-  else
-    let q := x / m
-    let r := x % m
-    if r < (m + 1) / 2 then
-      q
-    else
-      q + 1
-
 end Int

src/Init/Data/Int/Gcd.lean

@@ -6,12 +6,7 @@ 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 -/
@@ -19,37 +14,4 @@ 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', sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h')]
+    simp [subNatNat_of_lt h', succ_pred_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',
@@ -169,11 +169,12 @@ 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 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]
+    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')]
 
 protected theorem add_assoc : ∀ a b c : Int, a + b + c = a + (b + c)
   | (m:Nat), (n:Nat), c => aux1 ..
@@ -187,15 +188,15 @@ 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 [Nat.add_comm, Nat.add_left_comm, Nat.add_assoc]
+    simp [add_succ, Nat.add_comm, Nat.add_left_comm, neg_ofNat_succ]
 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
+    simp [add_succ]
     rw [Int.add_comm, subNatNat_add_negSucc]
-    simp [Nat.add_comm, Nat.add_left_comm, Nat.add_assoc]
+    simp [add_succ, succ_add, Nat.add_comm]
 
 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]
@@ -320,27 +321,6 @@ theorem toNat_sub (m n : Nat) : toNat (m - n) = m - n := by
   · exact (Nat.add_sub_cancel_left ..).symm
   · dsimp; rw [Nat.add_assoc, Nat.sub_eq_zero_of_le (Nat.le_add_right ..)]; rfl
 
-/- ## add/sub injectivity -/
-
-@[simp]
-protected theorem add_right_inj (i j k : Int) : (i + k = j + k) ↔ i = j := by
-  apply Iff.intro
-  · intro p
-    rw [←Int.add_sub_cancel i k, ←Int.add_sub_cancel j k, p]
-  · exact congrArg (· + k)
-
-@[simp]
-protected theorem add_left_inj (i j k : Int) : (k + i = k + j) ↔ i = j := by
-  simp [Int.add_comm k]
-
-@[simp]
-protected theorem sub_left_inj (i j k : Int) : (k - i = k - j) ↔ i = j := by
-  simp [Int.sub_eq_add_neg, Int.neg_inj]
-
-@[simp]
-protected theorem sub_right_inj (i j k : Int) : (i - k = j - k) ↔ i = j := by
-  simp [Int.sub_eq_add_neg]
-
 /- ## Ring properties -/
 
 @[simp] theorem ofNat_mul_negSucc (m n : Nat) : (m : Int) * -[n+1] = -↑(m * succ n) := rfl
@@ -390,7 +370,7 @@ 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 [sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h), ← neg_ofNat_succ, Nat.mul_sub_left_distrib,
+      rw [succ_pred_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
@@ -405,7 +385,7 @@ 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 [sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h), Nat.mul_sub_left_distrib]
+    simp [succ_pred_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)
@@ -421,12 +401,12 @@ 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 [← Nat.left_distrib, Nat.add_left_comm, Nat.add_assoc]
+  | (m:Nat), -[n+1],  -[k+1]  => by simp; rw [← Nat.left_distrib, succ_add]; rfl
   | -[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 [← Nat.left_distrib, Nat.add_left_comm, Nat.add_assoc]
+  | -[m+1],  -[n+1],  -[k+1]  => by simp; rw [← Nat.left_distrib, succ_add]; rfl
 
 protected theorem add_mul (a b c : Int) : (a + b) * c = a * c + b * c := by
   simp [Int.mul_comm, Int.mul_add]
@@ -498,6 +478,10 @@ 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]
 
+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]
+
 /-! NatCast lemmas -/
 
 /-!

src/Init/Data/Int/Order.lean

@@ -6,7 +6,6 @@ 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.
@@ -193,11 +192,6 @@ protected theorem min_le_right (a b : Int) : min a b ≤ b := by rw [Int.min_def
 
 protected theorem min_le_left (a b : Int) : min a b ≤ a := Int.min_comm .. ▸ Int.min_le_right ..
 
-protected theorem min_eq_left {a b : Int} (h : a ≤ b) : min a b = a := by simp [Int.min_def, h]
-
-protected theorem min_eq_right {a b : Int} (h : b ≤ a) : min a b = b := by
-  rw [Int.min_comm a b]; exact Int.min_eq_left h
-
 protected theorem le_min {a b c : Int} : a ≤ min b c ↔ a ≤ b ∧ a ≤ c :=
   ⟨fun h => ⟨Int.le_trans h (Int.min_le_left ..), Int.le_trans h (Int.min_le_right ..)⟩,
    fun ⟨h₁, h₂⟩ => by rw [Int.min_def]; split <;> assumption⟩
@@ -216,12 +210,6 @@ protected theorem max_le {a b c : Int} : max a b ≤ c ↔ a ≤ c ∧ b ≤ c :
   ⟨fun h => ⟨Int.le_trans (Int.le_max_left ..) h, Int.le_trans (Int.le_max_right ..) h⟩,
    fun ⟨h₁, h₂⟩ => by rw [Int.max_def]; split <;> assumption⟩
 
-protected theorem max_eq_right {a b : Int} (h : a ≤ b) : max a b = b := by
-  simp [Int.max_def, h, Int.not_lt.2 h]
-
-protected theorem max_eq_left {a b : Int} (h : b ≤ a) : max a b = a := by
-  rw [← Int.max_comm b a]; exact Int.max_eq_right h
-
 theorem eq_natAbs_of_zero_le {a : Int} (h : 0 ≤ a) : a = natAbs a := by
   let ⟨n, e⟩ := eq_ofNat_of_zero_le h
   rw [e]; rfl
@@ -448,575 +436,3 @@ theorem natAbs_of_nonneg {a : Int} (H : 0 ≤ a) : (natAbs a : Int) = a :=
 
 theorem ofNat_natAbs_of_nonpos {a : Int} (H : a ≤ 0) : (natAbs a : Int) = -a := by
   rw [← natAbs_neg, natAbs_of_nonneg (Int.neg_nonneg_of_nonpos H)]
-
-/-! ### toNat -/
-
-theorem toNat_eq_max : ∀ a : Int, (toNat a : Int) = max a 0
-  | (n : Nat) => (Int.max_eq_left (ofNat_zero_le n)).symm
-  | -[n+1] => (Int.max_eq_right (Int.le_of_lt (negSucc_lt_zero n))).symm
-
-@[simp] theorem toNat_zero : (0 : Int).toNat = 0 := rfl
-
-@[simp] theorem toNat_one : (1 : Int).toNat = 1 := rfl
-
-@[simp] theorem toNat_of_nonneg {a : Int} (h : 0 ≤ a) : (toNat a : Int) = a := by
-  rw [toNat_eq_max, Int.max_eq_left h]
-
-@[simp] theorem toNat_ofNat (n : Nat) : toNat ↑n = n := rfl
-
-@[simp] theorem toNat_ofNat_add_one {n : Nat} : ((n : Int) + 1).toNat = n + 1 := rfl
-
-theorem self_le_toNat (a : Int) : a ≤ toNat a := by rw [toNat_eq_max]; apply Int.le_max_left
-
-@[simp] theorem le_toNat {n : Nat} {z : Int} (h : 0 ≤ z) : n ≤ z.toNat ↔ (n : Int) ≤ z := by
-  rw [← Int.ofNat_le, Int.toNat_of_nonneg h]
-
-@[simp] theorem toNat_lt {n : Nat} {z : Int} (h : 0 ≤ z) : z.toNat < n ↔ z < (n : Int) := by
-  rw [← Int.not_le, ← Nat.not_le, Int.le_toNat h]
-
-theorem toNat_add {a b : Int} (ha : 0 ≤ a) (hb : 0 ≤ b) : (a + b).toNat = a.toNat + b.toNat :=
-  match a, b, eq_ofNat_of_zero_le ha, eq_ofNat_of_zero_le hb with
-  | _, _, ⟨_, rfl⟩, ⟨_, rfl⟩ => rfl
-
-theorem toNat_add_nat {a : Int} (ha : 0 ≤ a) (n : Nat) : (a + n).toNat = a.toNat + n :=
-  match a, eq_ofNat_of_zero_le ha with | _, ⟨_, rfl⟩ => rfl
-
-@[simp] theorem pred_toNat : ∀ i : Int, (i - 1).toNat = i.toNat - 1
-  | 0 => rfl
-  | (n+1:Nat) => by simp [ofNat_add]
-  | -[n+1] => rfl
-
-@[simp] theorem toNat_sub_toNat_neg : ∀ n : Int, ↑n.toNat - ↑(-n).toNat = n
-  | 0 => rfl
-  | (_+1:Nat) => Int.sub_zero _
-  | -[_+1] => Int.zero_sub _
-
-@[simp] theorem toNat_add_toNat_neg_eq_natAbs : ∀ n : Int, n.toNat + (-n).toNat = n.natAbs
-  | 0 => rfl
-  | (_+1:Nat) => Nat.add_zero _
-  | -[_+1] => Nat.zero_add _
-
-@[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

@@ -1,44 +0,0 @@
-/-
-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

@@ -727,9 +727,9 @@ inductive lt [LT α] : List α → List α → Prop where
 instance [LT α] : LT (List α) := ⟨List.lt⟩
 
 instance hasDecidableLt [LT α] [h : DecidableRel (α:=α) (·<·)] : (l₁ l₂ : List α) → Decidable (l₁ < l₂)
-  | [],    []    => isFalse nofun
+  | [],    []    => isFalse (fun h => nomatch h)
   | [],    _::_  => isTrue (List.lt.nil _ _)
-  | _::_, []     => isFalse nofun
+  | _::_, []     => isFalse (fun h => nomatch h)
   | a::as, b::bs =>
     match h a b with
     | isTrue h₁  => isTrue (List.lt.head _ _ h₁)

src/Init/Data/List/BasicAux.lean

@@ -5,6 +5,9 @@ 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

@@ -6,8 +6,9 @@ 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.Basic
+import Init.Control.Lawful
 import Init.Hints
 
 namespace List
@@ -68,7 +69,7 @@ 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 [-not_or]
+  cases l <;> simp
 
 /-! ### append -/
 
@@ -450,9 +451,9 @@ 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_all [or_and_left]
-    · simp_all [or_and_right]
+    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
 
 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

@@ -16,6 +16,3 @@ import Init.Data.Nat.Power2
 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

@@ -10,29 +10,6 @@ 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`
@@ -148,12 +125,9 @@ theorem add_succ (n m : Nat) : n + succ m = succ (n + m) :=
 theorem add_one (n : Nat) : n + 1 = succ n :=
   rfl
 
-@[simp] theorem succ_eq_add_one (n : Nat) : succ n = n + 1 :=
+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
@@ -215,7 +189,7 @@ protected theorem mul_comm : ∀ (n m : Nat), n * m = m * n
   Nat.mul_comm n 1 ▸ Nat.mul_one n
 
 protected theorem left_distrib (n m k : Nat) : n * (m + k) = n * m + n * k := by
-  induction n with
+  induction n generalizing m k with
   | zero      => repeat rw [Nat.zero_mul]
   | succ n ih => simp [succ_mul, ih]; rw [Nat.add_assoc, Nat.add_assoc (n*m)]; apply congrArg; apply Nat.add_left_comm
 
@@ -235,9 +209,6 @@ 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
@@ -253,7 +224,7 @@ 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
 
-@[simp] theorem pred_le : ∀ (n : Nat), pred n ≤ n
+theorem pred_le : ∀ (n : Nat), pred n ≤ n
   | zero   => Nat.le.refl
   | succ _ => le_succ _
 
@@ -286,7 +257,7 @@ 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 only [Nat.add_succ, Nat.sub_succ, ih]
+  | succ c ih => simp [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
@@ -327,8 +298,7 @@ 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)
-
-@[simp] theorem lt_succ_self (n : Nat) : n < succ n := lt.base n
+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
@@ -367,12 +337,6 @@ 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⟩
@@ -462,9 +426,6 @@ 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
 
@@ -490,137 +451,6 @@ 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 :=
@@ -632,7 +462,7 @@ protected theorem one_ne_zero : 1 ≠ (0 : Nat) :=
 protected theorem zero_ne_one : 0 ≠ (1 : Nat) :=
   fun h => Nat.noConfusion h
 
-@[simp] theorem succ_ne_zero (n : Nat) : succ n ≠ 0 :=
+theorem succ_ne_zero (n : Nat) : succ n ≠ 0 :=
   fun h => Nat.noConfusion h
 
 /-! # mul + order -/
@@ -673,10 +503,10 @@ theorem eq_of_mul_eq_mul_right {n m k : Nat} (hm : 0 < m) (h : n * m = k * m) :
 
 /-! # power -/
 
-protected theorem pow_succ (n m : Nat) : n^(succ m) = n^m * n :=
+theorem pow_succ (n m : Nat) : n^(succ m) = n^m * n :=
   rfl
 
-protected theorem pow_zero (n : Nat) : n^0 = 1 := rfl
+theorem pow_zero (n : Nat) : n^0 = 1 := rfl
 
 theorem pow_le_pow_of_le_left {n m : Nat} (h : n ≤ m) : ∀ (i : Nat), n^i ≤ m^i
   | 0      => Nat.le_refl _
@@ -743,11 +573,6 @@ 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
@@ -770,7 +595,7 @@ 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_sub_self_left]; decide
+    | Or.inl h => injection h with h; subst h; rw [←Nat.add_one, 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 _ _)
@@ -784,7 +609,7 @@ 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 only [Nat.sub_zero, ne_eq, not_false_eq_true]
+  | 0, succ b, _ => by simp
   | 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)
 
@@ -802,7 +627,7 @@ 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 [← Nat.add_assoc, ih]
+  | succ k ih => simp [add_succ, add_succ, succ_sub_succ, 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]
@@ -915,7 +740,7 @@ 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.add_succ, Nat.sub_succ, ih]
+  | succ k ih => rw [Nat.add_succ, Nat.sub_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

@@ -51,26 +51,6 @@ 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

@@ -6,7 +6,6 @@ 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
@@ -24,13 +23,26 @@ 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
-  omega
+  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
 
 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
-  omega
+  match n with
+  | 0 => contradiction
+  | n + 1 => simp [Nat.mul_succ, Nat.mul_add_mod, mod_eq_of_lt]
 
 /-! ### Preliminaries -/
 
@@ -87,11 +99,6 @@ 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 =>
@@ -232,7 +239,7 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
     rw [Nat.sub_eq_zero_iff_le] at i_sub_j_eq
     exact Nat.not_le_of_gt j_lt_i i_sub_j_eq
   | d+1 =>
-    simp [Nat.pow_succ, Nat.mul_comm _ 2,  Nat.mul_add_mod]
+    simp [pow_succ, Nat.mul_comm _ 2,  Nat.mul_add_mod]
 
 @[simp] theorem testBit_mod_two_pow (x j i : Nat) :
     testBit (x % 2^j) i = (decide (i < j) && testBit x i) := by
@@ -262,28 +269,31 @@ 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 =>
-      simp [not_decide_mod_two_eq_one]
+      -- 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
       omega
   | succ i ih =>
     simp only [testBit_succ]
     match n with
     | 0 =>
-      simp [decide_eq_false]
+      simp only [pow_zero, succ_sub_succ_eq_sub, Nat.zero_sub, Nat.zero_div, zero_testBit]
+      rw [decide_eq_false] <;> simp
     | n+1 =>
       rw [Nat.two_pow_succ_sub_succ_div_two, ih]
       · simp [Nat.succ_lt_succ_iff]
-      · omega
+      · rw [Nat.pow_succ'] at h₂
+        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]
@@ -334,7 +344,7 @@ 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])
+    (fun p => by simp [p, Nat.zero_lt_succ])
 
 private theorem eq_0_of_lt (x : Nat) : x < 2^ 0 ↔ x = 0 := eq_0_of_lt_one x
 
@@ -342,7 +352,7 @@ private theorem eq_0_of_lt (x : Nat) : x < 2^ 0 ↔ x = 0 := eq_0_of_lt_one x
 private theorem zero_lt_pow (n : Nat) : 0 < 2^n := by
   induction n
   case zero => simp [eq_0_of_lt]
-  case succ n hyp => simpa [Nat.pow_succ]
+  case succ n hyp => simpa [pow_succ]
 
 private theorem div_two_le_of_lt_two {m n : Nat} (p : m < 2 ^ succ n) : m / 2 < 2^n := by
   simp [div_lt_iff_lt_mul Nat.zero_lt_two]
@@ -367,7 +377,7 @@ theorem bitwise_lt_two_pow (left : x < 2^n) (right : y < 2^n) : (Nat.bitwise f x
       simp only [x_zero, y_zero, if_neg]
       have hyp1 := hyp (div_two_le_of_lt_two left) (div_two_le_of_lt_two right)
       by_cases p : f (decide (x % 2 = 1)) (decide (y % 2 = 1)) = true <;>
-        simp [p, Nat.pow_succ, mul_succ, Nat.add_assoc]
+        simp [p, pow_succ, mul_succ, Nat.add_assoc]
       case pos =>
         apply lt_of_succ_le
         simp only [← Nat.succ_add]
@@ -437,8 +447,12 @@ theorem testBit_mul_pow_two_add (a : Nat) {b i : Nat} (b_lt : b < 2^i) (j : Nat)
   cases Nat.lt_or_ge j i with
   | inl j_lt =>
     simp only [j_lt]
-    have i_def : i = j + succ (pred (i-j)) := by
-      rw [succ_pred_eq_of_pos] <;> omega
+    have i_ge := Nat.le_of_lt j_lt
+    have i_sub_j_nez : i-j ≠ 0 := Nat.sub_ne_zero_of_lt j_lt
+    have i_def : i = j + succ (pred (i-j)) :=
+          calc i = j + (i-j) := (Nat.add_sub_cancel' i_ge).symm
+               _ = j + succ (pred (i-j)) := by
+                 rw [← congrArg (j+·) (Nat.succ_pred i_sub_j_nez)]
     rw [i_def]
     simp only [testBit_to_div_mod, Nat.pow_add, Nat.mul_assoc]
     simp only [Nat.mul_add_div (Nat.two_pow_pos _), Nat.mul_add_mod]

src/Init/Data/Nat/Compare.lean

@@ -1,57 +0,0 @@
-/-
-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

@@ -10,13 +10,6 @@ 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
 
@@ -35,7 +28,7 @@ theorem div_eq (x y : Nat) : x / y = if 0 < y ∧ y ≤ x then (x - y) / y + 1 e
   rw [Nat.div]
   rfl
 
-def div.inductionOn.{u}
+theorem 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)
@@ -102,7 +95,7 @@ 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]
 
-def mod.inductionOn.{u}
+theorem 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)
@@ -205,33 +198,13 @@ 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 only [add_one, succ_mul, false_iff, Nat.not_le]
-    refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_left ..)
+    simp [not_succ_le_zero x, succ_mul, Nat.add_comm]
+    refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_right ..)
     exact Nat.not_le.1 fun h' => h ⟨k0, h'⟩
   | ind y k h IH =>
-    rw [Nat.add_le_add_iff_right, IH k0, succ_mul,
+    rw [← add_one, 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 _)
@@ -293,7 +266,7 @@ 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 [Nat.pred_succ, mul_succ, Nat.sub_sub]
+      simp [add_one, 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
@@ -334,50 +307,4 @@ 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

@@ -5,10 +5,16 @@ 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⟩
@@ -91,42 +97,4 @@ 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

@@ -1,12 +1,10 @@
 /-
 Copyright (c) 2021 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
+Authors: Leonardo de Moura
 -/
 prelude
 import Init.Data.Nat.Dvd
-import Init.NotationExtra
-import Init.RCases
 
 namespace Nat
 
@@ -16,8 +14,8 @@ 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
@@ -71,166 +69,4 @@ 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

@@ -1,66 +0,0 @@
-/-
-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
 
@@ -19,6 +19,131 @@ 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
@@ -66,6 +191,15 @@ 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
 
@@ -175,6 +309,8 @@ 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
 
@@ -334,32 +470,6 @@ 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 ..)
@@ -408,6 +518,10 @@ 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 ..⟩
@@ -505,6 +619,68 @@ 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
@@ -516,6 +692,12 @@ 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]
@@ -556,6 +738,12 @@ 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 [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
@@ -695,17 +883,37 @@ theorem lt_log2_self : n < 2 ^ (n.log2 + 1) :=
 
 /-! ### dvd -/
 
-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]
+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 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 {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_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 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
 
 theorem pow_dvd_pow_iff_pow_le_pow {k l : Nat} :
     ∀ {x : Nat}, 0 < x → (x ^ k ∣ x ^ l ↔ x ^ k ≤ x ^ l)
@@ -729,6 +937,18 @@ 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 =>
@@ -763,6 +983,10 @@ 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
@@ -778,9 +1002,19 @@ 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 [← Nat.add_assoc, shiftLeft_shiftLeft _ _ k, shiftLeft_succ]
+  | 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]
 
 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

@@ -4,7 +4,10 @@ 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
@@ -580,7 +583,7 @@ 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
+  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) = 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

@@ -4,6 +4,7 @@ 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/Nat/Mod.lean

@@ -1,76 +0,0 @@
-/-
-Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
--/
-prelude
-import Init.Omega
-
-/-!
-# Further results about `mod`.
-
-This file proves some results about `mod` that are useful for bitblasting,
-in particular
-`Nat.mod_mul : x % (a * b) = x % a + a * (x / a % b)`
-and its corollary
-`Nat.mod_pow_succ : x % b ^ (k + 1) = x % b ^ k + b ^ k * ((x / b ^ k) % b)`.
-
-It contains the necesssary preliminary results relating order and `*` and `/`,
-which should probably be moved to their own file.
--/
-
-namespace Nat
-
-@[simp] protected theorem mul_lt_mul_left (a0 : 0 < a) : a * b < a * c ↔ b < c := by
-  induction a with
-  | zero => simp_all
-  | succ a ih =>
-    cases a
-    · simp
-    · simp_all [succ_eq_add_one, Nat.right_distrib]
-      omega
-
-@[simp] protected theorem mul_lt_mul_right (a0 : 0 < a) : b * a < c * a ↔ b < c := by
-  rw [Nat.mul_comm b a, Nat.mul_comm c a, Nat.mul_lt_mul_left a0]
-
-protected theorem lt_of_mul_lt_mul_left {a b c : Nat} (h : a * b < a * c) : b < c := by
-  cases a <;> simp_all
-
-protected theorem lt_of_mul_lt_mul_right {a b c : Nat} (h : b * a < c * a) : b < c := by
-  rw [Nat.mul_comm b a, Nat.mul_comm c a] at h
-  exact Nat.lt_of_mul_lt_mul_left h
-
-protected theorem div_lt_of_lt_mul {m n k : Nat} (h : m < n * k) : m / n < k :=
-  Nat.lt_of_mul_lt_mul_left <|
-    calc
-      n * (m / n) ≤ m % n + n * (m / n) := Nat.le_add_left _ _
-      _ = m := mod_add_div _ _
-      _ < n * k := h
-
-theorem mod_mul_right_div_self (m n k : Nat) : m % (n * k) / n = m / n % k := by
-  rcases Nat.eq_zero_or_pos n with (rfl | hn); simp [mod_zero]
-  rcases Nat.eq_zero_or_pos k with (rfl | hk); simp [mod_zero]
-  conv => rhs; rw [← mod_add_div m (n * k)]
-  rw [Nat.mul_assoc, add_mul_div_left _ _ hn, add_mul_mod_self_left,
-    mod_eq_of_lt (Nat.div_lt_of_lt_mul (mod_lt _ (Nat.mul_pos hn hk)))]
-
-theorem mod_mul_left_div_self (m n k : Nat) : m % (k * n) / n = m / n % k := by
-  rw [Nat.mul_comm k n, mod_mul_right_div_self]
-
-@[simp 1100]
-theorem mod_mul_right_mod (a b c : Nat) : a % (b * c) % b = a % b :=
-  Nat.mod_mod_of_dvd a (Nat.dvd_mul_right b c)
-
-@[simp 1100]
-theorem mod_mul_left_mod (a b c : Nat) : a % (b * c) % c = a % c :=
-  Nat.mod_mod_of_dvd a (Nat.mul_comm _ _ ▸ Nat.dvd_mul_left c b)
-
-theorem mod_mul {a b x : Nat} : x % (a * b) = x % a + a * (x / a % b) := by
-  rw [Nat.add_comm, ← Nat.div_add_mod (x % (a*b)) a, Nat.mod_mul_right_mod,
-    Nat.mod_mul_right_div_self]
-
-theorem mod_pow_succ {x b k : Nat} :
-    x % b ^ (k + 1) = x % b ^ k + b ^ k * ((x / b ^ k) % b) := by
-  rw [Nat.pow_succ, Nat.mod_mul]
-
-end Nat

src/Init/Data/Ord.lean

@@ -5,6 +5,7 @@ Authors: Dany Fabian, Sebastian Ullrich
 -/
 
 prelude
+import Init.Data.Int
 import Init.Data.String
 
 inductive Ordering where

src/Init/Data/Random.lean

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

src/Init/NotationExtra.lean

@@ -9,7 +9,6 @@ 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 =>
@@ -124,7 +123,7 @@ calc abc
   _ = xyz := pwxyz
 ```
 
-`calc` works as a term, as a tactic or as a `conv` tactic.
+`calc` has term mode and tactic mode variants. This is the term mode variant.
 
 See [Theorem Proving in Lean 4][tpil4] for more information.
 
@@ -132,12 +131,44 @@ See [Theorem Proving in Lean 4][tpil4] for more information.
 -/
 syntax (name := calc) "calc" calcSteps : term
 
-@[inherit_doc «calc»]
-syntax (name := calcTactic) "calc" calcSteps : tactic
+/-- 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
+```
 
-@[inherit_doc «calc»]
-macro tk:"calc" steps:calcSteps : conv =>
-  `(conv| tactic => calc%$tk $steps)
+`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
+-/
+syntax (name := calcTactic) "calc" calcSteps : tactic
 
 @[app_unexpander Unit.unit] def unexpandUnit : Lean.PrettyPrinter.Unexpander
   | `($(_)) => `(())

src/Init/Omega/Int.lean

@@ -4,8 +4,9 @@ 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`.
@@ -48,7 +49,7 @@ 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 only [Nat.shiftRight_eq_div_pow, Int.ofNat_ediv]
+  simp [Nat.shiftRight_eq_div_pow]
 
 -- 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

@@ -5,8 +5,6 @@ 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

@@ -1485,7 +1485,6 @@ 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)
@@ -1636,8 +1635,8 @@ instance : LT Nat where
   lt := Nat.lt
 
 theorem Nat.not_succ_le_zero : ∀ (n : Nat), LE.le (succ n) 0 → False
-  | 0      => nofun
-  | succ _ => nofun
+  | 0,      h => nomatch h
+  | succ _, h => nomatch h
 
 theorem Nat.not_lt_zero (n : Nat) : Not (LT.lt n 0) :=
   not_succ_le_zero n
@@ -2036,7 +2035,7 @@ instance : Inhabited UInt64 where
   default := UInt64.ofNatCore 0 (by decide)
 
 /--
-The size of type `USize`, that is, `2^System.Platform.numBits`, which may
+The size of type `UInt16`, 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
@@ -2054,7 +2053,7 @@ instance : OfNat (Fin (n+1)) i where
   ofNat := Fin.ofNat i
 ```
 -/
-abbrev USize.size : Nat := hAdd (hSub (hPow 2 System.Platform.numBits) 1) 1
+abbrev USize.size : Nat := Nat.succ (Nat.sub (hPow 2 System.Platform.numBits) 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

src/Init/PropLemmas.lean

@@ -11,21 +11,6 @@ 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))
@@ -119,62 +104,10 @@ 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)
-
-/-
-`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_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
@@ -335,14 +268,7 @@ end quantifiers
 
 /-! ## decidable -/
 
-@[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.not_not [Decidable p] : ¬¬p ↔ p := ⟨of_not_not, not_not_intro⟩
 
 theorem Decidable.by_contra [Decidable p] : (¬p → False) → p := of_not_not
 
@@ -384,7 +310,7 @@ 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⟩
 
-@[simp] theorem Decidable.not_imp_self [Decidable a] : (¬a → a) ↔ a := by
+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) :=
@@ -463,12 +389,8 @@ 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 :=
-  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 _)
+  ⟨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⟩
 
 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]
@@ -513,53 +435,3 @@ 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

@@ -15,15 +15,12 @@ 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 :=
@@ -127,7 +124,6 @@ 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 -/
 
@@ -177,11 +173,6 @@ 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
@@ -208,9 +199,9 @@ 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) :
@@ -221,14 +212,11 @@ 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 [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 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 heq_eq_eq (a b : α) : HEq a b = (a = b) := propext <| Iff.intro eq_of_heq heq_of_eq
+@[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 cond_true (a b : α) : cond true a b = a := rfl
 @[simp] theorem cond_false (a b : α) : cond false a b = b := rfl
@@ -240,29 +228,11 @@ 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

@@ -31,43 +31,22 @@ 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.
 -/
@@ -107,60 +86,33 @@ 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) := $body) => do
-     return mkNullNode <|
-       #[(← `($[$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) := $body))]
-       ++ (← mkAttributeCmds kind pre? ids? n)
+     let mut cmds := #[(← `($[$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
 
 macro_rules
   | `($[$doc?:docComment]? $kind:attrKind builtin_simproc $[$pre?]? $n:ident ($pattern:term) := $body) => do
@@ -174,16 +126,4 @@ macro_rules
       attribute [$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

@@ -4,6 +4,7 @@ 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

@@ -673,13 +673,12 @@ It makes sure the "continuation" `?_` is the main goal after refining.
 macro "refine_lift " e:term : tactic => `(tactic| focus (refine no_implicit_lambda% $e; rotate_right))
 
 /--
-The `have` tactic is for adding hypotheses to the local context of the main goal.
-* `have h : t := e` adds the hypothesis `h : t` if `e` is a term of type `t`.
-* `have h := e` uses the type of `e` for `t`.
-* `have : t := e` and `have := e` use `this` for the name of the hypothesis.
-* `have pat := e` for a pattern `pat` is equivalent to `match e with | pat => _`,
-  where `_` stands for the tactics that follow this one.
-  It is convenient for types that have only one applicable constructor.
+`have h : t := e` adds the hypothesis `h : t` to the current goal if `e` a term
+of type `t`.
+* If `t` is omitted, it will be inferred.
+* If `h` is omitted, the name `this` is used.
+* The variant `have pattern := e` is equivalent to `match e with | pattern => _`,
+  and it is convenient for types that have only one applicable constructor.
   For example, given `h : p ∧ q ∧ r`, `have ⟨h₁, h₂, h₃⟩ := h` produces the
   hypotheses `h₁ : p`, `h₂ : q`, and `h₃ : r`.
 -/
@@ -694,15 +693,12 @@ If `h :` is omitted, the name `this` is used.
  -/
 macro "suffices " d:sufficesDecl : tactic => `(tactic| refine_lift suffices $d; ?_)
 /--
-The `let` tactic is for adding definitions to the local context of the main goal.
-* `let x : t := e` adds the definition `x : t := e` if `e` is a term of type `t`.
-* `let x := e` uses the type of `e` for `t`.
-* `let : t := e` and `let := e` use `this` for the name of the hypothesis.
-* `let pat := e` for a pattern `pat` is equivalent to `match e with | pat => _`,
-  where `_` stands for the tactics that follow this one.
-  It is convenient for types that let only one applicable constructor.
-  For example, given `p : α × β × γ`, `let ⟨x, y, z⟩ := p` produces the
-  local variables `x : α`, `y : β`, and `z : γ`.
+`let h : t := e` adds the hypothesis `h : t := e` to the current goal if `e` a term of type `t`.
+If `t` is omitted, it will be inferred.
+The variant `let pattern := e` is equivalent to `match e with | pattern => _`,
+and it is convenient for types that have only applicable constructor.
+Example: given `h : p ∧ q ∧ r`, `let ⟨h₁, h₂, h₃⟩ := h` produces the hypotheses
+`h₁ : p`, `h₂ : q`, and `h₃ : r`.
 -/
 macro "let " d:letDecl : tactic => `(tactic| refine_lift let $d:letDecl; ?_)
 /--
@@ -1310,26 +1306,6 @@ used when closing the goal.
 -/
 syntax (name := apply?) "apply?" (" using " (colGt term),+)? : tactic
 
-/--
-`show_term tac` runs `tac`, then prints the generated term in the form
-"exact X Y Z" or "refine X ?_ Z" if there are remaining subgoals.
-
-(For some tactics, the printed term will not be human readable.)
--/
-syntax (name := showTerm) "show_term " tacticSeq : tactic
-
-/--
-`show_term e` elaborates `e`, then prints the generated term.
--/
-macro (name := showTermElab) tk:"show_term " t:term : term =>
-  `(term| no_implicit_lambda% (show_term_elab%$tk $t))
-
-/--
-The command `by?` will print a suggestion for replacing the proof block with a proof term
-using `show_term`.
--/
-macro (name := by?) tk:"by?" t:tacticSeq : term => `(show_term%$tk by%$tk $t)
-
 end Tactic
 
 namespace Attr

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)
 
-noncomputable def recursion {C : α → Sort v} (a : α) (h : ∀ x, (∀ y, r y x → C y) → C x) : C a := by
+theorem 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 noncomputable def strongInductionOn
+protected theorem 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 noncomputable def caseStrongInductionOn
+protected theorem caseStrongInductionOn
     {motive : Nat → Sort u}
     (a : Nat)
     (zero : motive 0)

src/Lean/CoreM.lean

@@ -289,9 +289,6 @@ 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

@@ -116,22 +116,6 @@ 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 ⟩ =>
@@ -141,10 +125,9 @@ 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)
-  | insertIfNewWff : ∀ m a b, WellFormed m → WellFormed (insertIfNew 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)
+  | eraseWff  : ∀ m a,   WellFormed m → WellFormed (erase m a)
 
 end HashMapImp
 
@@ -173,22 +156,13 @@ 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 flag indicating whether an existing entry has been replaced with `a -> b`. -/
+/-- Similar to `insert`, but also returns a Boolean flad 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/PersistentHashMap.lean

@@ -84,14 +84,14 @@ partial def insertAtCollisionNodeAux [BEq α] : CollisionNode α β → Nat →
       else insertAtCollisionNodeAux n (i+1) k v
     else
       ⟨Node.collision (keys.push k) (vals.push v) (size_push heq k v), IsCollisionNode.mk _ _ _⟩
-  | ⟨Node.entries _, h⟩, _, _, _ => nomatch h
+  | ⟨Node.entries _, h⟩, _, _, _ => False.elim (nomatch h)
 
 def insertAtCollisionNode [BEq α] : CollisionNode α β → α → β → CollisionNode α β :=
   fun n k v => insertAtCollisionNodeAux n 0 k v
 
 def getCollisionNodeSize : CollisionNode α β → Nat
   | ⟨Node.collision keys _ _, _⟩ => keys.size
-  | ⟨Node.entries _, h⟩          => nomatch h
+  | ⟨Node.entries _, h⟩          => False.elim (nomatch h)
 
 def mkCollisionNode (k₁ : α) (v₁ : β) (k₂ : α) (v₂ : β) : Node α β :=
   let ks : Array α := Array.mkEmpty maxCollisions
@@ -105,7 +105,7 @@ partial def insertAux [BEq α] [Hashable α] : Node α β → USize → USize
     let newNode := insertAtCollisionNode ⟨Node.collision keys vals heq, IsCollisionNode.mk _ _ _⟩ k v
     if depth >= maxDepth || getCollisionNodeSize newNode < maxCollisions then newNode.val
     else match newNode with
-      | ⟨Node.entries _, h⟩ => nomatch h
+      | ⟨Node.entries _, h⟩ => False.elim (nomatch h)
       | ⟨Node.collision keys vals heq, _⟩ =>
         let rec traverse (i : Nat) (entries : Node α β) : Node α β :=
           if h : i < keys.size then

src/Lean/Elab.lean

@@ -49,4 +49,3 @@ import Lean.Elab.InheritDoc
 import Lean.Elab.ParseImportsFast
 import Lean.Elab.GuardMsgs
 import Lean.Elab.CheckTactic
-import Lean.Elab.MatchExpr

src/Lean/Elab/CheckTactic.lean

@@ -7,7 +7,6 @@ prelude
 import Lean.Elab.Tactic.ElabTerm
 import Lean.Elab.Command
 import Lean.Elab.Tactic.Meta
-import Lean.Meta.CheckTactic
 
 /-!
 Commands to validate tactic results.
@@ -19,6 +18,15 @@ 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
@@ -26,10 +34,11 @@ def elabCheckTactic : CommandElab := fun stx => do
     runTermElabM $ fun _vars => do
       let u ← Lean.Elab.Term.elabTerm t none
       let type ← inferType u
-      let checkGoalType ← mkCheckGoalType u type
+      let lvl ← mkFreshLevelMVar
+      let checkGoalType : Expr := mkApp2 (mkConst ``CheckGoalType [lvl]) type u
       let mvar ← mkFreshExprMVar (.some checkGoalType)
-      let expTerm ← Lean.Elab.Term.elabTerm result (.some type)
       let (goals, _) ← Lean.Elab.runTactic mvar.mvarId! tac.raw
+      let expTerm ← Lean.Elab.Term.elabTerm result (.some type)
       match goals with
       | [] =>
         throwErrorAt stx
@@ -42,6 +51,7 @@ 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
@@ -50,7 +60,8 @@ def elabCheckTacticFailure : CommandElab := fun stx => do
     runTermElabM $ fun _vars => do
       let val ← Lean.Elab.Term.elabTerm t none
       let type ← inferType val
-      let checkGoalType ← mkCheckGoalType val type
+      let lvl ← mkFreshLevelMVar
+      let checkGoalType : Expr := mkApp2 (mkConst ``CheckGoalType [lvl]) type val
       let mvar ← mkFreshExprMVar (.some checkGoalType)
       let act := Lean.Elab.runTactic mvar.mvarId! tactic.raw
       match ← try (Term.withoutErrToSorry (some <$> act)) catch _ => pure none with
@@ -62,12 +73,12 @@ def elabCheckTacticFailure : CommandElab := fun stx => do
                 pure m!"{indentExpr val}"
           let msg ←
             match gls with
-              | [] => pure m!"{tactic} expected to fail on {t}, but closed goal."
+              | [] => pure m!"{tactic} expected to fail on {val}, but closed goal."
               | [g] =>
-                pure <| m!"{tactic} expected to fail on {t}, but returned: {←ppGoal g}"
+                pure <| m!"{tactic} expected to fail on {val}, but returned: {←ppGoal g}"
               | gls =>
                 let app m g := do pure <| m ++ (←ppGoal g)
-                let init := m!"{tactic} expected to fail on {t}, but returned goals:"
+                let init := m!"{tactic} expected to fail on {val}, but returned goals:"
                 gls.foldlM (init := init) app
           throwErrorAt stx msg
 

src/Lean/Elab/Do.lean

@@ -131,31 +131,12 @@ abbrev Var := Syntax  -- TODO: should be `Ident`
 
 /-- A `doMatch` alternative. `vars` is the array of variables declared by `patterns`. -/
 structure Alt (σ : Type) where
-  ref      : Syntax
-  vars     : Array Var
+  ref : Syntax
+  vars : Array Var
   patterns : Syntax
-  rhs      : σ
+  rhs : σ
   deriving Inhabited
 
-/-- A `doMatchExpr` alternative. -/
-structure AltExpr (σ : Type) where
-  ref     : Syntax
-  var?    : Option Var
-  funName : Syntax
-  pvars   : Array Syntax
-  rhs     : σ
-  deriving Inhabited
-
-def AltExpr.vars (alt : AltExpr σ) : Array Var := Id.run do
-  let mut vars := #[]
-  if let some var := alt.var? then
-    vars := vars.push var
-  for pvar in alt.pvars do
-    match pvar with
-    | `(_) => pure ()
-    | _ => vars := vars.push pvar
-  return vars
-
 /--
   Auxiliary datastructure for representing a `do` code block, and compiling "reassignments" (e.g., `x := x + 1`).
   We convert `Code` into a `Syntax` term representing the:
@@ -217,7 +198,6 @@ inductive Code where
   /-- Recall that an if-then-else may declare a variable using `optIdent` for the branches `thenBranch` and `elseBranch`. We store the variable name at `var?`. -/
   | ite          (ref : Syntax) (h? : Option Var) (optIdent : Syntax) (cond : Syntax) (thenBranch : Code) (elseBranch : Code)
   | match        (ref : Syntax) (gen : Syntax) (discrs : Syntax) (optMotive : Syntax) (alts : Array (Alt Code))
-  | matchExpr    (ref : Syntax) (meta : Bool) (discr : Syntax) (alts : Array (AltExpr Code)) (elseBranch : Code)
   | jmp          (ref : Syntax) (jpName : Name) (args : Array Syntax)
   deriving Inhabited
 
@@ -232,7 +212,6 @@ def Code.getRef? : Code → Option Syntax
   | .return ref _        => ref
   | .ite ref ..          => ref
   | .match ref ..        => ref
-  | .matchExpr ref ..    => ref
   | .jmp ref ..          => ref
 
 abbrev VarSet := RBMap Name Syntax Name.cmp
@@ -264,28 +243,19 @@ partial def CodeBlocl.toMessageData (codeBlock : CodeBlock) : MessageData :=
     | .match _ _ ds _ alts   =>
       m!"match {ds} with"
       ++ alts.foldl (init := m!"") fun acc alt => acc ++ m!"\n| {alt.patterns} => {loop alt.rhs}"
-    | .matchExpr _ meta d alts elseCode =>
-      let r := m!"match_expr {if meta then "" else "(meta := false)"} {d} with"
-      let r := r ++ alts.foldl (init := m!"") fun acc alt =>
-            let acc := acc ++ m!"\n| {if let some var := alt.var? then m!"{var}@" else ""}"
-            let acc := acc ++ m!"{alt.funName}"
-            let acc := acc ++ alt.pvars.foldl (init := m!"") fun acc pvar => acc ++ m!" {pvar}"
-            acc ++ m!" => {loop alt.rhs}"
-      r ++ m!"| _ => {loop elseCode}"
   loop codeBlock.code
 
 /-- Return true if the give code contains an exit point that satisfies `p` -/
 partial def hasExitPointPred (c : Code) (p : Code → Bool) : Bool :=
   let rec loop : Code → Bool
-    | .decl _ _ k             => loop k
-    | .reassign _ _ k         => loop k
-    | .joinpoint _ _ b k      => loop b || loop k
-    | .seq _ k                => loop k
-    | .ite _ _ _ _ t e        => loop t || loop e
-    | .match _ _ _ _ alts     => alts.any (loop ·.rhs)
-    | .matchExpr _ _ _ alts e => alts.any (loop ·.rhs) || loop e
-    | .jmp ..                 => false
-    | c                       => p c
+    | .decl _ _ k         => loop k
+    | .reassign _ _ k     => loop k
+    | .joinpoint _ _ b k  => loop b || loop k
+    | .seq _ k            => loop k
+    | .ite _ _ _ _ t e    => loop t || loop e
+    | .match _ _ _ _ alts => alts.any (loop ·.rhs)
+    | .jmp ..             => false
+    | c                   => p c
   loop c
 
 def hasExitPoint (c : Code) : Bool :=
@@ -330,18 +300,13 @@ partial def convertTerminalActionIntoJmp (code : Code) (jp : Name) (xs : Array V
     | .joinpoint n ps b k    => return .joinpoint n ps (← loop b) (← loop k)
     | .seq e k               => return .seq e (← loop k)
     | .ite ref x? h c t e    => return .ite ref x? h c (← loop t) (← loop e)
+    | .match ref g ds t alts => return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← loop alt.rhs) })
     | .action e              => mkAuxDeclFor e fun y =>
       let ref := e
       -- We jump to `jp` with xs **and** y
       let jmpArgs := xs.push y
       return Code.jmp ref jp jmpArgs
-    | .match ref g ds t alts =>
-      return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← loop alt.rhs) })
-    | .matchExpr ref meta d alts e => do
-      let alts ← alts.mapM fun alt => do pure { alt with rhs := (← loop alt.rhs) }
-      let e ← loop e
-      return .matchExpr ref meta d alts e
-    | c => return c
+    | c                            => return c
   loop code
 
 structure JPDecl where
@@ -407,13 +372,14 @@ def mkJmp (ref : Syntax) (rs : VarSet) (val : Syntax) (mkJPBody : Syntax → Mac
   return Code.jmp ref jp args
 
 /-- `pullExitPointsAux rs c` auxiliary method for `pullExitPoints`, `rs` is the set of update variable in the current path.  -/
-partial def pullExitPointsAux (rs : VarSet) (c : Code) : StateRefT (Array JPDecl) TermElabM Code := do
+partial def pullExitPointsAux (rs : VarSet) (c : Code) : StateRefT (Array JPDecl) TermElabM Code :=
   match c with
   | .decl xs stx k         => return .decl xs stx (← pullExitPointsAux (eraseVars rs xs) k)
   | .reassign xs stx k     => return .reassign xs stx (← pullExitPointsAux (insertVars rs xs) k)
   | .joinpoint j ps b k    => return .joinpoint j ps (← pullExitPointsAux rs b) (← pullExitPointsAux rs k)
   | .seq e k               => return .seq e (← pullExitPointsAux rs k)
   | .ite ref x? o c t e    => return .ite ref x? o c (← pullExitPointsAux (eraseOptVar rs x?) t) (← pullExitPointsAux (eraseOptVar rs x?) e)
+  | .match ref g ds t alts => return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← pullExitPointsAux (eraseVars rs alt.vars) alt.rhs) })
   | .jmp ..                => return  c
   | .break ref             => mkSimpleJmp ref rs (.break ref)
   | .continue ref          => mkSimpleJmp ref rs (.continue ref)
@@ -423,13 +389,6 @@ partial def pullExitPointsAux (rs : VarSet) (c : Code) : StateRefT (Array JPDecl
     mkAuxDeclFor e fun y =>
       let ref := e
       mkJmp ref rs y (fun yFresh => return .action (← ``(Pure.pure $yFresh)))
-  | .match ref g ds t alts =>
-    let alts ← alts.mapM fun alt => do pure { alt with rhs := (← pullExitPointsAux (eraseVars rs alt.vars) alt.rhs) }
-    return .match ref g ds t alts
-  | .matchExpr ref meta d alts e =>
-    let alts ← alts.mapM fun alt => do pure { alt with rhs := (← pullExitPointsAux (eraseVars rs alt.vars) alt.rhs) }
-    let e ← pullExitPointsAux rs e
-    return .matchExpr ref meta d alts e
 
 /--
 Auxiliary operation for adding new variables to the collection of updated variables in a CodeBlock.
@@ -498,14 +457,6 @@ partial def extendUpdatedVarsAux (c : Code) (ws : VarSet) : TermElabM Code :=
         pullExitPoints c
       else
         return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← update alt.rhs) })
-    | .matchExpr ref meta d alts e =>
-      if alts.any fun alt => alt.vars.any fun x => ws.contains x.getId then
-        -- If a pattern variable is shadowing a variable in ws, we `pullExitPoints`
-        pullExitPoints c
-      else
-        let alts ← alts.mapM fun alt => do pure { alt with rhs := (← update alt.rhs) }
-        let e ← update e
-        return .matchExpr ref meta d alts e
     | .ite ref none o c t e => return .ite ref none o c (← update t) (← update e)
     | .ite ref (some h) o cond t e =>
       if ws.contains h.getId then
@@ -619,16 +570,6 @@ def mkMatch (ref : Syntax) (genParam : Syntax) (discrs : Syntax) (optMotive : Sy
     return { ref := alt.ref, vars := alt.vars, patterns := alt.patterns, rhs := rhs.code : Alt Code }
   return { code := .match ref genParam discrs optMotive alts, uvars := ws }
 
-def mkMatchExpr (ref : Syntax) (meta : Bool) (discr : Syntax) (alts : Array (AltExpr CodeBlock)) (elseBranch : CodeBlock) : TermElabM CodeBlock := do
-  -- nary version of homogenize
-  let ws := alts.foldl (union · ·.rhs.uvars) {}
-  let ws := union ws elseBranch.uvars
-  let alts ← alts.mapM fun alt => do
-    let rhs ← extendUpdatedVars alt.rhs ws
-    return { alt with rhs := rhs.code : AltExpr Code }
-  let elseBranch ← extendUpdatedVars elseBranch ws
-  return { code := .matchExpr ref meta discr alts elseBranch.code, uvars := ws }
-
 /-- Return a code block that executes `terminal` and then `k` with the value produced by `terminal`.
    This method assumes `terminal` is a terminal -/
 def concat (terminal : CodeBlock) (kRef : Syntax) (y? : Option Var) (k : CodeBlock) : TermElabM CodeBlock := do
@@ -765,19 +706,6 @@ private def expandDoIf? (stx : Syntax) : MacroM (Option Syntax) := match stx wit
     return some e
   | _ => pure none
 
-/--
-  If the given syntax is a `doLetExpr` or `doLetMetaExpr`, return an equivalent `doIf` that has an `else` but no `else if`s or `if let`s.  -/
-private def expandDoLetExpr? (stx : Syntax) (doElems : List Syntax) : MacroM (Option Syntax) := match stx with
-  | `(doElem| let_expr $pat:matchExprPat := $discr:term | $elseBranch:doSeq) =>
-    return some (← `(doElem| match_expr (meta := false) $discr:term with
-                             | $pat:matchExprPat => $(mkDoSeq doElems.toArray)
-                             | _ => $elseBranch))
-  | `(doElem| let_expr $pat:matchExprPat ← $discr:term | $elseBranch:doSeq) =>
-    return some (← `(doElem| match_expr $discr:term with
-                             | $pat:matchExprPat => $(mkDoSeq doElems.toArray)
-                             | _ => $elseBranch))
-  | _ => return none
-
 structure DoIfView where
   ref        : Syntax
   optIdent   : Syntax
@@ -1149,26 +1077,10 @@ where
       let mut termAlts := #[]
       for alt in alts do
         let rhs ← toTerm alt.rhs
-        let termAlt := mkNode ``Parser.Term.matchAlt #[mkAtomFrom alt.ref "|", mkNullNode #[alt.patterns], mkAtomFrom alt.ref "=>", rhs]
+        let termAlt := mkNode `Lean.Parser.Term.matchAlt #[mkAtomFrom alt.ref "|", mkNullNode #[alt.patterns], mkAtomFrom alt.ref "=>", rhs]
         termAlts := termAlts.push termAlt
-      let termMatchAlts := mkNode ``Parser.Term.matchAlts #[mkNullNode termAlts]
-      return mkNode ``Parser.Term.«match» #[mkAtomFrom ref "match", genParam, optMotive, discrs, mkAtomFrom ref "with", termMatchAlts]
-    | .matchExpr ref meta d alts elseBranch => withFreshMacroScope do
-      let d' ← `(discr)
-      let mut termAlts := #[]
-      for alt in alts do
-        let rhs ← `(($(← toTerm alt.rhs) : $((← read).m) _))
-        let optVar := if let some var := alt.var? then mkNullNode #[var, mkAtomFrom var "@"] else mkNullNode #[]
-        let pat := mkNode ``Parser.Term.matchExprPat #[optVar, alt.funName, mkNullNode alt.pvars]
-        let termAlt := mkNode ``Parser.Term.matchExprAlt #[mkAtomFrom alt.ref "|", pat, mkAtomFrom alt.ref "=>", rhs]
-        termAlts := termAlts.push termAlt
-      let elseBranch := mkNode ``Parser.Term.matchExprElseAlt #[mkAtomFrom ref "|", mkHole ref, mkAtomFrom ref "=>", (← toTerm elseBranch)]
-      let termMatchExprAlts := mkNode ``Parser.Term.matchExprAlts #[mkNullNode termAlts, elseBranch]
-      let body := mkNode ``Parser.Term.matchExpr #[mkAtomFrom ref "match_expr", d', mkAtomFrom ref "with", termMatchExprAlts]
-      if meta then
-        `(Bind.bind (instantiateMVarsIfMVarApp $d) fun discr => $body)
-      else
-        `(let discr := $d; $body)
+      let termMatchAlts := mkNode `Lean.Parser.Term.matchAlts #[mkNullNode termAlts]
+      return mkNode `Lean.Parser.Term.«match» #[mkAtomFrom ref "match", genParam, optMotive, discrs, mkAtomFrom ref "with", termMatchAlts]
 
 def run (code : Code) (m : Syntax) (returnType : Syntax) (uvars : Array Var := #[]) (kind := Kind.regular) : MacroM Syntax :=
   toTerm code { m, returnType, kind, uvars }
@@ -1621,24 +1533,6 @@ mutual
     let matchCode ← mkMatch ref genParam discrs optMotive alts
     concatWith matchCode doElems
 
-  /-- Generate `CodeBlock` for `doMatchExpr; doElems` -/
-  partial def doMatchExprToCode (doMatchExpr : Syntax) (doElems: List Syntax) : M CodeBlock := do
-    let ref       := doMatchExpr
-    let meta      := doMatchExpr[1].isNone
-    let discr     := doMatchExpr[2]
-    let alts      := doMatchExpr[4][0].getArgs -- Array of `doMatchExprAlt`
-    let alts ← alts.mapM fun alt => do
-      let pat      := alt[1]
-      let var?     := if pat[0].isNone then none else some pat[0][0]
-      let funName  := pat[1]
-      let pvars    := pat[2].getArgs
-      let rhs      := alt[3]
-      let rhs ← doSeqToCode (getDoSeqElems rhs)
-      pure { ref, var?, funName, pvars, rhs }
-    let elseBranch ← doSeqToCode (getDoSeqElems doMatchExpr[4][1][3])
-    let matchCode ← mkMatchExpr ref meta discr alts elseBranch
-    concatWith matchCode doElems
-
   /--
     Generate `CodeBlock` for `doTry; doElems`
     ```
@@ -1708,9 +1602,6 @@ mutual
       | none =>
       match (← liftMacroM <| expandDoIf? doElem) with
       | some doElem => doSeqToCode (doElem::doElems)
-      | none =>
-      match (← liftMacroM <| expandDoLetExpr? doElem doElems) with
-      | some doElem => doSeqToCode [doElem]
       | none =>
         let (liftedDoElems, doElem) ← expandLiftMethod doElem
         if !liftedDoElems.isEmpty then
@@ -1749,8 +1640,6 @@ mutual
             doForToCode doElem doElems
           else if k == ``Parser.Term.doMatch then
             doMatchToCode doElem doElems
-          else if k == ``Parser.Term.doMatchExpr then
-            doMatchExprToCode doElem doElems
           else if k == ``Parser.Term.doTry then
             doTryToCode doElem doElems
           else if k == ``Parser.Term.doBreak then

src/Lean/Elab/Extra.lean

@@ -488,10 +488,8 @@ 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 lhsStx := stx[2]
-    let rhsStx := stx[3]
-    let lhs ← withRef lhsStx <| toTree lhsStx
-    let rhs ← withRef rhsStx <| toTree rhsStx
+    let lhs ← withRef stx[2] <| toTree stx[2]
+    let rhs ← withRef stx[3] <| toTree stx[3]
     let tree := .binop stx .regular f lhs rhs
     let r ← analyze tree none
     trace[Elab.binrel] "hasUncomparable: {r.hasUncomparable}, maxType: {r.max?}"
@@ -499,10 +497,10 @@ def elabBinRelCore (noProp : Bool) (stx : Syntax) (expectedType? : Option Expr)
       -- Use default elaboration strategy + `toBoolIfNecessary`
       let lhs ← toExprCore lhs
       let rhs ← toExprCore rhs
-      let lhs ← withRef lhsStx <| toBoolIfNecessary lhs
-      let rhs ← withRef rhsStx <| toBoolIfNecessary rhs
+      let lhs ← toBoolIfNecessary lhs
+      let rhs ← toBoolIfNecessary rhs
       let lhsType ← inferType lhs
-      let rhs ← withRef rhsStx <| ensureHasType lhsType rhs
+      let rhs ← 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/GuardMsgs.lean

@@ -73,28 +73,9 @@ 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 |> removeTrailingWhitespaceMarker
+    let expected : String := (← dc?.mapM (getDocStringText ·)).getD "" |>.trim
     let specFn ← parseGuardMsgsSpec spec?
     let initMsgs ← modifyGet fun st => (st.messages, { st with messages := {} })
     elabCommandTopLevel cmd
@@ -107,7 +88,8 @@ def equalUpToNewlines (exp res : String) : Bool :=
       | .drop        => pure ()
       | .passthrough => toPassthrough := toPassthrough.add msg
     let res := "---\n".intercalate (← toCheck.toList.mapM (messageToStringWithoutPos ·)) |>.trim
-    if equalUpToNewlines expected res then
+    -- We do some whitespace normalization here to allow users to break long lines.
+    if expected.replace "\n" " " == res.replace "\n" " " then
       -- Passed. Only put toPassthrough messages back on the message log
       modify fun st => { st with messages := initMsgs ++ toPassthrough }
     else
@@ -137,7 +119,6 @@ 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/MatchExpr.lean

@@ -1,217 +0,0 @@
-/-
-Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Leonardo de Moura
--/
-prelude
-import Lean.Elab.Term
-
-namespace Lean.Elab.Term
-namespace MatchExpr
-/--
-`match_expr` alternative. Recall that it has the following structure.
-```
-| (ident "@")? ident bindeIdent* => rhs
-```
-
-Example:
-```
-| c@Eq _ a b => f c a b
-```
--/
-structure Alt where
-  /--
-  `some c` if there is a variable binding to the function symbol being matched.
-  `c` is the variable name.
-  -/
-  var?    : Option Ident
-  /-- Function being matched. -/
-  funName : Ident
-  /-- Pattern variables. The list uses `none` for representing `_`, and `some a` for pattern variable `a`. -/
-  pvars   : List (Option Ident)
-  /-- right-hand-side for the alternative. -/
-  rhs     : Syntax
-  /-- Store the auxliary continuation function for each right-hand-side. -/
-  k       : Ident := ⟨.missing⟩
-  /-- Actual value to be passed as an argument. -/
-  actuals : List Term := []
-
-/--
-`match_expr` else-alternative. Recall that it has the following structure.
-```
-| _ => rhs
-```
--/
-structure ElseAlt where
-  rhs : Syntax
-
-open Parser Term
-
-/--
-Converts syntax representing a `match_expr` else-alternative into an `ElseAlt`.
--/
-def toElseAlt? (stx : Syntax) : Option ElseAlt :=
-  if !stx.isOfKind ``matchExprElseAlt then none else
-  some { rhs := stx[3] }
-
-/--
-Converts syntax representing a `match_expr` alternative into an `Alt`.
--/
-def toAlt? (stx : Syntax) : Option Alt :=
-  if !stx.isOfKind ``matchExprAlt then none else
-  match stx[1] with
-  | `(matchExprPat| $[$var? @]? $funName:ident $pvars*) =>
-    let pvars := pvars.toList.reverse.map fun arg =>
-      match arg.raw with
-      | `(_) => none
-      | _ => some ⟨arg⟩
-    let rhs := stx[3]
-    some { var?, funName, pvars, rhs }
-  | _ => none
-
-/--
-Returns the function names of alternatives that do not have any pattern variable left.
--/
-def getFunNamesToMatch (alts : List Alt) : List Ident := Id.run do
-  let mut funNames := #[]
-  for alt in alts do
-    if alt.pvars.isEmpty then
-      if Option.isNone <| funNames.find? fun funName => funName.getId == alt.funName.getId then
-        funNames := funNames.push alt.funName
-  return funNames.toList
-
-/--
-Returns `true` if there is at least one alternative whose next pattern variable is not a `_`.
--/
-def shouldSaveActual (alts : List Alt) : Bool :=
-  alts.any fun alt => alt.pvars matches some _ :: _
-
-/--
-Returns the first alternative whose function name is `funName` **and**
-does not have pattern variables left to match.
--/
-def getAltFor? (alts : List Alt) (funName : Ident) : Option Alt :=
-  alts.find? fun alt => alt.funName.getId == funName.getId && alt.pvars.isEmpty
-
-/--
-Removes alternatives that do not have any pattern variable left to be matched.
-For the ones that still have pattern variables, remove the first one, and
-save `actual` if the removed pattern variable is not a `_`.
--/
-def next (alts : List Alt) (actual : Term) : List Alt :=
-  alts.filterMap fun alt =>
-    if let some _ :: pvars := alt.pvars then
-      some { alt with pvars, actuals := actual :: alt.actuals }
-    else if let none :: pvars := alt.pvars then
-      some { alt with pvars }
-    else
-      none
-
-/--
-Creates a fresh identifier for representing the continuation function used to
-execute the RHS of the given alternative, and stores it in the field `k`.
--/
-def initK (alt : Alt) : MacroM Alt := withFreshMacroScope do
-  -- Remark: the compiler frontend implemented in C++ currently detects jointpoints created by
-  -- the "do" notation by testing the name. See hack at method `visit_let` at `lcnf.cpp`
-  -- We will remove this hack when we re-implement the compiler frontend in Lean.
-  let k : Ident ← `(__do_jp)
-  return { alt with k }
-
-/--
-Generates parameters for the continuation function used to execute
-the RHS of the given alternative.
--/
-def getParams (alt : Alt) : MacroM (Array (TSyntax ``bracketedBinder)) := do
-  let mut params := #[]
-  if let some var := alt.var? then
-    params := params.push (← `(bracketedBinderF| ($var : Expr)))
-  params := params ++ (← alt.pvars.toArray.reverse.filterMapM fun
-    | none => return none
-    | some arg => return some (← `(bracketedBinderF| ($arg : Expr))))
-  if params.isEmpty then
-    return #[(← `(bracketedBinderF| (_ : Unit)))]
-  return params
-
-/--
-Generates the actual arguments for invoking the auxiliary continuation function
-associated with the given alternative. The arguments are the actuals stored in `alt`.
-`discr` is also an argument if `alt.var?` is not none.
--/
-def getActuals (discr : Term) (alt : Alt) : MacroM (Array Term) := do
-  let mut actuals := #[]
-  if alt.var?.isSome then
-    actuals := actuals.push discr
-  actuals := actuals ++ alt.actuals.toArray
-  if actuals.isEmpty then
-    return #[← `(())]
-  return actuals
-
-def toDoubleQuotedName (ident : Ident) : Term :=
-  ⟨mkNode ``Parser.Term.doubleQuotedName #[mkAtom "`", mkAtom "`", ident]⟩
-
-/--
-Generates an `if-then-else` tree for implementing a `match_expr` with discriminant `discr`,
-alternatives `alts`, and else-alternative `elseAlt`.
--/
-partial def generate (discr : Term) (alts : List Alt) (elseAlt : ElseAlt) : MacroM Syntax := do
-  let alts ← alts.mapM initK
-  let discr' ← `(__discr)
-  -- Remark: the compiler frontend implemented in C++ currently detects jointpoints created by
-  -- the "do" notation by testing the name. See hack at method `visit_let` at `lcnf.cpp`
-  -- We will remove this hack when we re-implement the compiler frontend in Lean.
-  let kElse ← `(__do_jp)
-  let rec loop (discr : Term) (alts : List Alt) : MacroM Term := withFreshMacroScope do
-    let funNamesToMatch := getFunNamesToMatch alts
-    let saveActual := shouldSaveActual alts
-    let actual ← if saveActual then `(a) else `(_)
-    let altsNext := next alts actual
-    let body ← if altsNext.isEmpty then
-      `($kElse ())
-    else
-      let discr' ← `(__discr)
-      let body ← loop discr' altsNext
-      if saveActual then
-        `(if h : ($discr).isApp then let a := Expr.appArg $discr h; let __discr := Expr.appFnCleanup $discr h; $body else $kElse ())
-      else
-        `(if h : ($discr).isApp then let __discr := Expr.appFnCleanup $discr h; $body else $kElse ())
-    let mut result := body
-    for funName in funNamesToMatch do
-      if let some alt := getAltFor? alts funName then
-        let actuals ← getActuals discr alt
-        result ← `(if ($discr).isConstOf $(toDoubleQuotedName funName) then $alt.k $actuals* else $result)
-    return result
-  let body ← loop discr' alts
-  let mut result ← `(let_delayed __do_jp (_ : Unit) := $(⟨elseAlt.rhs⟩):term; let __discr := Expr.cleanupAnnotations $discr:term; $body:term)
-  for alt in alts do
-    let params ← getParams alt
-    result ← `(let_delayed $alt.k:ident $params:bracketedBinder* := $(⟨alt.rhs⟩):term; $result:term)
-  return result
-
-def main (discr : Term) (alts : Array Syntax) (elseAlt : Syntax) : MacroM Syntax := do
-  let alts ← alts.toList.mapM fun alt =>
-    if let some alt := toAlt? alt then
-      pure alt
-    else
-      Macro.throwErrorAt alt "unexpected `match_expr` alternative"
-  let some elseAlt := toElseAlt? elseAlt
-    | Macro.throwErrorAt elseAlt "unexpected `match_expr` else-alternative"
-  generate discr alts elseAlt
-
-end MatchExpr
-
-@[builtin_macro Lean.Parser.Term.matchExpr] def expandMatchExpr : Macro := fun stx =>
-  match stx with
-  | `(match_expr $discr:term with $alts) =>
-    MatchExpr.main discr alts.raw[0].getArgs alts.raw[1]
-  | _ => Macro.throwUnsupported
-
-@[builtin_macro Lean.Parser.Term.letExpr] def expandLetExpr : Macro := fun stx =>
-  match stx with
-  | `(let_expr $pat:matchExprPat := $discr:term | $elseBranch:term; $body:term) =>
-    `(match_expr $discr with
-      | $pat:matchExprPat => $body
-      | _ => $elseBranch)
-  | _ => Macro.throwUnsupported
-
-end Lean.Elab.Term

src/Lean/Elab/MutualDef.lean

@@ -68,6 +68,8 @@ 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
@@ -644,9 +646,6 @@ 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
@@ -660,14 +659,10 @@ 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 {

src/Lean/Elab/PreDefinition/Eqns.lean

@@ -380,7 +380,7 @@ 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 ++ `def
+      let name := baseName ++ `_unfold
       addDecl <| Declaration.thmDecl {
         name, type, value
         levelParams := info.levelParams

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

@@ -68,7 +68,7 @@ def mkEqns (info : EqnInfo) : MetaM (Array Name) :=
   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

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

@@ -117,7 +117,7 @@ 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

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

@@ -302,11 +302,7 @@ def GuessLexRel.toNatRel : GuessLexRel → Expr
   | le => mkAppN (mkConst ``LE.le [levelZero]) #[mkConst ``Nat, mkConst ``instLENat]
   | no_idea => unreachable!
 
-/--
-Given an expression `e`, produce `sizeOf e` with a suitable instance.
-NB: We must use the instance of the type of the function parameter!
-The concrete argument at hand may have a different (still def-eq) typ.
--/
+/-- Given an expression `e`, produce `sizeOf e` with a suitable instance. -/
 def mkSizeOf (e : Expr) : MetaM Expr := do
   let ty ← inferType e
   let lvl ← getLevel ty
@@ -319,8 +315,8 @@ def mkSizeOf (e : Expr) : MetaM Expr := do
 For a given recursive call, and a choice of parameter and argument index,
 try to prove equality, < or ≤.
 -/
-def evalRecCall (decrTactic? : Option DecreasingBy) (rcc : RecCallWithContext)
-  (paramIdx argIdx : Nat) : MetaM GuessLexRel := do
+def evalRecCall (decrTactic? : Option DecreasingBy) (rcc : RecCallWithContext) (paramIdx argIdx : Nat) :
+    MetaM GuessLexRel := do
   rcc.ctxt.run do
     let param := rcc.params[paramIdx]!
     let arg := rcc.args[argIdx]!
@@ -411,7 +407,7 @@ def inspectCall (rc : RecCallCache) : MutualMeasure → MetaM GuessLexRel
       return .eq
 
 /--
-Given a predefinition with value `fun (x₁ ... xₙ) (y₁ : α₁)... (yₘ : αₘ) => ...`,
+Given a predefinition with value `fun (x_₁ ... xₙ) (y_₁ : α₁)... (yₘ : αₘ) => ...`,
 where `n = fixedPrefixSize`, return an array `A` s.t. `i ∈ A` iff `sizeOf yᵢ` reduces to a literal.
 This is the case for types such as `Prop`, `Type u`, etc.
 These arguments should not be considered when guessing a well-founded relation.
@@ -429,47 +425,6 @@ def getForbiddenByTrivialSizeOf (fixedPrefixSize : Nat) (preDef : PreDefinition)
         result := result.push i
     return result
 
-/--
-Given a predefinition with value `fun (x₁ ... xₙ) (y₁ : α₁)... (yₘ : αₘ) => ...`,
-where `n = fixedPrefixSize`, return an array `A` s.t. `i ∈ A` iff the
-`WellFoundedRelation` of `aᵢ` goes via `SizeOf`, and `aᵢ` does not depend on `y₁`….
-
-These are the parameters for which we omit an explicit call to `sizeOf` in the termination argument.
-
-We only use this in the non-mutual case; in the mutual case we would have to additional check
-if the parameters that line up in the actual `TerminationWF` have the same type.
--/
-def getSizeOfParams (fixedPrefixSize : Nat) (preDef : PreDefinition) : MetaM (Array Nat) :=
-  lambdaTelescope preDef.value fun xs _ => do
-    let xs  : Array Expr := xs[fixedPrefixSize:]
-    let mut result := #[]
-    for x in xs, i in [:xs.size] do
-      try
-        let t ← inferType x
-        if t.hasAnyFVar (fun fvar => xs.contains (.fvar fvar)) then continue
-        let u ← getLevel t
-        let wfi ← synthInstance (.app (.const ``WellFoundedRelation [u]) t)
-        let soi ← synthInstance (.app (.const ``SizeOf [u]) t)
-        if ← isDefEq wfi (mkApp2 (.const ``sizeOfWFRel [u]) t soi) then
-          result := result.push i
-      catch _ =>
-        pure ()
-    return result
-
-/--
-Given a predefinition with value `fun (x₁ ... xₙ) (y₁ : α₁)... (yₘ : αₘ) => ...`,
-where `n = fixedPrefixSize`, return an array `A` s.t. `i ∈ A` iff `aᵢ` is `Nat`.
-These are parameters where we can definitely omit the call to `sizeOf`.
--/
-def getNatParams (fixedPrefixSize : Nat) (preDef : PreDefinition) : MetaM (Array Nat) :=
-  lambdaTelescope preDef.value fun xs _ => do
-    let xs  : Array Expr := xs[fixedPrefixSize:]
-    let mut result := #[]
-    for x in xs, i in [:xs.size] do
-      let t ← inferType x
-      if ← withReducible (isDefEq t (.const `Nat [])) then
-        result := result.push i
-    return result
 
 /--
 Generate all combination of arguments, skipping those that are forbidden.
@@ -584,26 +539,23 @@ combination of these measures. The parameters are
 * `measures`: The measures to be used.
 -/
 def buildTermWF (originalVarNamess : Array (Array Name)) (varNamess : Array (Array Name))
-  (needsNoSizeOf : Array (Array Nat)) (measures : Array MutualMeasure) : MetaM TerminationWF := do
+  (measures : Array MutualMeasure) : MetaM TerminationWF := do
   varNamess.mapIdxM fun funIdx varNames => do
     let idents := varNames.map mkIdent
     let measureStxs ← measures.mapM fun
       | .args varIdxs => do
           let varIdx := varIdxs[funIdx]!
           let v := idents[varIdx]!
-          if needsNoSizeOf[funIdx]!.contains varIdx then
-            `($v)
-          else
-            -- Print `sizeOf` as such, unless it is shadowed.
-            -- Shadowing by a `def` in the current namespace is handled by `unresolveNameGlobal`.
-            -- But it could also be shadowed by an earlier parameter (including the fixed prefix),
-            -- so look for unqualified (single tick) occurrences in `originalVarNames`
-            let sizeOfIdent :=
-              if originalVarNamess[funIdx]!.any (· = `sizeOf) then
-                mkIdent ``sizeOf -- fully qualified
-              else
-                mkIdent (← unresolveNameGlobal ``sizeOf)
-            `($sizeOfIdent $v)
+          -- Print `sizeOf` as such, unless it is shadowed.
+          -- Shadowing by a `def` in the current namespace is handled by `unresolveNameGlobal`.
+          -- But it could also be shadowed by an earlier parameter (including the fixed prefix),
+          -- so look for unqualified (single tick) occurrences in `originalVarNames`
+          let sizeOfIdent :=
+            if originalVarNamess[funIdx]!.any (· = `sizeOf) then
+              mkIdent ``sizeOf -- fully qualified
+            else
+              mkIdent (← unresolveNameGlobal ``sizeOf)
+          `($sizeOfIdent $v)
       | .func funIdx' => if funIdx' == funIdx then `(1) else `(0)
     let body ← mkTupleSyntax measureStxs
     return { ref := .missing, vars := idents, body, synthetic := true }
@@ -716,20 +668,11 @@ def explainFailure (declNames : Array Name) (varNamess : Array (Array Name))
     r := r ++ (← explainMutualFailure declNames varNamess rcs)
   return r
 
-/--
-Shows the termination measure used to the user, and implements `termination_by?`
--/
-def reportWF (preDefs : Array PreDefinition) (wf : TerminationWF) : MetaM Unit := do
-  let extraParamss := preDefs.map (·.termination.extraParams)
-  let wf' := trimTermWF extraParamss wf
-  for preDef in preDefs, term in wf' do
-    if showInferredTerminationBy.get (← getOptions) then
-      logInfoAt preDef.ref m!"Inferred termination argument:\n{← term.unexpand}"
-    if let some ref := preDef.termination.terminationBy?? then
-      Tactic.TryThis.addSuggestion ref (← term.unexpand)
+end Lean.Elab.WF.GuessLex
 
-end GuessLex
-open GuessLex
+namespace Lean.Elab.WF
+
+open Lean.Elab.WF.GuessLex
 
 /--
 Main entry point of this module:
@@ -740,17 +683,14 @@ terminates. See the module doc string for a high-level overview.
 def guessLex (preDefs : Array PreDefinition) (unaryPreDef : PreDefinition)
     (fixedPrefixSize : Nat) :
     MetaM TerminationWF := do
+  let extraParamss := preDefs.map (·.termination.extraParams)
   let originalVarNamess ← preDefs.mapM originalVarNames
   let varNamess ← originalVarNamess.mapM (naryVarNames fixedPrefixSize ·)
   let arities := varNamess.map (·.size)
   trace[Elab.definition.wf] "varNames is: {varNamess}"
 
-  let forbiddenArgs ← preDefs.mapM (getForbiddenByTrivialSizeOf fixedPrefixSize)
-  let needsNoSizeOf ←
-    if preDefs.size = 1 then
-      preDefs.mapM (getSizeOfParams fixedPrefixSize)
-    else
-      preDefs.mapM (getNatParams fixedPrefixSize)
+  let forbiddenArgs ← preDefs.mapM fun preDef =>
+    getForbiddenByTrivialSizeOf fixedPrefixSize preDef
 
   -- The list of measures, including the measures that order functions.
   -- The function ordering measures come last
@@ -758,9 +698,7 @@ def guessLex (preDefs : Array PreDefinition) (unaryPreDef : PreDefinition)
 
   -- If there is only one plausible measure, use that
   if let #[solution] := measures then
-    let wf ← buildTermWF originalVarNamess varNamess needsNoSizeOf #[solution]
-    reportWF preDefs wf
-    return wf
+    return ← buildTermWF originalVarNamess varNamess #[solution]
 
   -- Collect all recursive calls and extract their context
   let recCalls ← collectRecCalls unaryPreDef fixedPrefixSize arities
@@ -770,8 +708,15 @@ def guessLex (preDefs : Array PreDefinition) (unaryPreDef : PreDefinition)
 
   match ← liftMetaM <| solve measures callMatrix with
   | .some solution => do
-    let wf ← buildTermWF originalVarNamess varNamess needsNoSizeOf solution
-    reportWF preDefs wf
+    let wf ← buildTermWF originalVarNamess varNamess solution
+
+    let wf' := trimTermWF extraParamss wf
+    for preDef in preDefs, term in wf' do
+      if showInferredTerminationBy.get (← getOptions) then
+        logInfoAt preDef.ref m!"Inferred termination argument:\n{← term.unexpand}"
+      if let some ref := preDef.termination.terminationBy?? then
+        Tactic.TryThis.addSuggestion ref (← term.unexpand)
+
     return wf
   | .none =>
     let explanation ← explainFailure (preDefs.map (·.declName)) varNamess rcs

src/Lean/Elab/Print.lean

@@ -5,7 +5,6 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Lean.Util.FoldConsts
-import Lean.Meta.Eqns
 import Lean.Elab.Command
 
 namespace Lean.Elab.Command
@@ -129,18 +128,4 @@ private def printAxiomsOf (constName : Name) : CommandElabM Unit := do
     cs.forM printAxiomsOf
   | _ => throwUnsupportedSyntax
 
-private def printEqnsOf (constName : Name) : CommandElabM Unit := do
-  let some eqns ← liftTermElabM <| Meta.getEqnsFor? constName (nonRec := true) |
-    logInfo m!"'{constName}' does not have equations"
-  let mut m := m!"equations:"
-  for eq in eqns do
-    let cinfo ← getConstInfo eq
-    m := m ++ Format.line ++ (← mkHeader "theorem" eq cinfo.levelParams cinfo.type .safe)
-  logInfo m
-
-@[builtin_command_elab «printEqns»] def elabPrintEqns : CommandElab := fun stx => do
-  let id := stx[2]
-  let cs ← resolveGlobalConstWithInfos id
-  cs.forM printEqnsOf
-
 end Lean.Elab.Command

src/Lean/Elab/StructInst.lean

@@ -802,8 +802,10 @@ partial def mkDefaultValueAux? (struct : Struct) : Expr → TermElabM (Option Ex
         let arg ← mkFreshExprMVar d
         mkDefaultValueAux? struct (b.instantiate1 arg)
   | e =>
-    let_expr id _ a := e | return some e
-    return some a
+    if e.isAppOfArity ``id 2 then
+      return some e.appArg!
+    else
+      return some e
 
 def mkDefaultValue? (struct : Struct) (cinfo : ConstantInfo) : TermElabM (Option Expr) :=
   withRef struct.ref do

src/Lean/Elab/Syntax.lean

@@ -382,6 +382,7 @@ def addMacroScopeIfLocal [MonadQuotation m] [Monad m] (name : Name) (attrKind :
   let name ← match name? with
     | some name => pure name.getId
     | none => addMacroScopeIfLocal (← liftMacroM <| mkNameFromParserSyntax cat syntaxParser) attrKind
+  trace[Meta.debug] "name: {name}"
   let prio ← liftMacroM <| evalOptPrio prio?
   let idRef := (name?.map (·.raw)).getD tk
   let stxNodeKind := (← getCurrNamespace) ++ name

src/Lean/Elab/Tactic.lean

@@ -37,4 +37,3 @@ import Lean.Elab.Tactic.NormCast
 import Lean.Elab.Tactic.Symm
 import Lean.Elab.Tactic.SolveByElim
 import Lean.Elab.Tactic.LibrarySearch
-import Lean.Elab.Tactic.ShowTerm

src/Lean/Elab/Tactic/BuiltinTactic.lean

@@ -352,21 +352,12 @@ def renameInaccessibles (mvarId : MVarId) (hs : TSyntaxArray ``binderIdent) : Ta
     let mut info  := #[]
     let mut found : NameSet := {}
     let n := lctx.numIndices
-    -- hypotheses are inaccessible if their scopes are different from the caller's (we assume that
-    -- the scopes are the same for all the hypotheses in `hs`, which is reasonable to expect in
-    -- practice and otherwise the expected semantics of `rename_i` really are not clear)
-    let some callerScopes := hs.findSome? (fun
-        | `(binderIdent| $h:ident) => some <| extractMacroScopes h.getId
-        | _ => none)
-      | return mvarId
     for i in [:n] do
       let j := n - i - 1
       match lctx.getAt? j with
       | none => pure ()
       | some localDecl =>
-        let inaccessible := !(extractMacroScopes localDecl.userName |>.equalScope callerScopes)
-        let shadowed := found.contains localDecl.userName
-        if inaccessible || shadowed then
+        if localDecl.userName.hasMacroScopes || found.contains localDecl.userName then
           if let `(binderIdent| $h:ident) := hs.back then
             let newName := h.getId
             lctx := lctx.setUserName localDecl.fvarId newName

src/Lean/Elab/Tactic/Induction.lean

@@ -534,9 +534,9 @@ private def elabTermForElim (stx : Syntax) : TermElabM Expr := do
       return e
 
 -- `optElimId` is of the form `("using" term)?`
-private def getElimNameInfo (optElimId : Syntax) (targets : Array Expr) (induction : Bool) : TacticM ElimInfo := do
+private def getElimNameInfo (optElimId : Syntax) (targets : Array Expr) (induction : Bool): TacticM ElimInfo := do
   if optElimId.isNone then
-    if let some elimName ← getCustomEliminator? targets induction then
+    if let some elimName ← getCustomEliminator? targets then
       return ← getElimInfo elimName
     unless targets.size == 1 do
       throwError "eliminator must be provided when multiple targets are used (use 'using <eliminator-name>'), and no default eliminator has been registered using attribute `[eliminator]`"

src/Lean/Elab/Tactic/Omega/Frontend.lean

@@ -358,7 +358,6 @@ def addIntInequality (p : MetaProblem) (h y : Expr) : OmegaM MetaProblem := do
 /-- Given a fact `h` with type `¬ P`, return a more useful fact obtained by pushing the negation. -/
 def pushNot (h P : Expr) : MetaM (Option Expr) := do
   let P ← whnfR P
-  trace[omega] "pushing negation: {P}"
   match P with
   | .forallE _ t b _ =>
     if (← isProp t) && (← isProp b) then
@@ -367,42 +366,43 @@ def pushNot (h P : Expr) : MetaM (Option Expr) := do
     else
       return none
   | .app _ _ =>
-    match_expr P with
-    | LT.lt α _ x y => match_expr α with
-      | Nat => return some (mkApp3 (.const ``Nat.le_of_not_lt []) x y h)
-      | Int => return some (mkApp3 (.const ``Int.le_of_not_lt []) x y h)
-      | Fin n => return some (mkApp4 (.const ``Fin.le_of_not_lt []) n x y h)
-      | _ => return none
-    | LE.le α _ x y => match_expr α with
-      | Nat => return some (mkApp3 (.const ``Nat.lt_of_not_le []) x y h)
-      | Int => return some (mkApp3 (.const ``Int.lt_of_not_le []) x y h)
-      | Fin n => return some (mkApp4 (.const ``Fin.lt_of_not_le []) n x y h)
-      | _ => return none
-    | Eq α x y => match_expr α with
-      | Nat => return some (mkApp3 (.const ``Nat.lt_or_gt_of_ne []) x y h)
-      | Int => return some (mkApp3 (.const ``Int.lt_or_gt_of_ne []) x y h)
-      | Fin n => return some (mkApp4 (.const ``Fin.lt_or_gt_of_ne []) n x y h)
-      | _ => return none
-    | Dvd.dvd α _ k x => match_expr α with
-      | Nat => return some (mkApp3 (.const ``Nat.emod_pos_of_not_dvd []) k x h)
-      | Int =>
-        -- This introduces a disjunction that could be avoided by checking `k ≠ 0`.
-        return some (mkApp3 (.const ``Int.emod_pos_of_not_dvd []) k x h)
-      | _ => return none
-    | Prod.Lex _ _ _ _ _ _ => return some (← mkAppM ``Prod.of_not_lex #[h])
-    | Not P =>
-      return some (mkApp3 (.const ``Decidable.of_not_not []) P
-        (.app (.const ``Classical.propDecidable []) P) h)
-    | And P Q =>
-      return some (mkApp5 (.const ``Decidable.or_not_not_of_not_and []) P Q
-        (.app (.const ``Classical.propDecidable []) P)
-        (.app (.const ``Classical.propDecidable []) Q) h)
-    | Or P Q =>
-      return some (mkApp3 (.const ``and_not_not_of_not_or []) P Q h)
-    | Iff P Q =>
-      return some (mkApp5 (.const ``Decidable.and_not_or_not_and_of_not_iff []) P Q
-        (.app (.const ``Classical.propDecidable []) P)
-        (.app (.const ``Classical.propDecidable []) Q) h)
+    match P.getAppFnArgs with
+    | (``LT.lt, #[.const ``Int [], _, x, y]) =>
+      return some (mkApp3 (.const ``Int.le_of_not_lt []) x y h)
+    | (``LE.le, #[.const ``Int [], _, x, y]) =>
+      return some (mkApp3 (.const ``Int.lt_of_not_le []) x y h)
+    | (``LT.lt, #[.const ``Nat [], _, x, y]) =>
+      return some (mkApp3 (.const ``Nat.le_of_not_lt []) x y h)
+    | (``LE.le, #[.const ``Nat [], _, x, y]) =>
+      return some (mkApp3 (.const ``Nat.lt_of_not_le []) x y h)
+    | (``LT.lt, #[.app (.const ``Fin []) n, _, x, y]) =>
+      return some (mkApp4 (.const ``Fin.le_of_not_lt []) n x y h)
+    | (``LE.le, #[.app (.const ``Fin []) n, _, x, y]) =>
+      return some (mkApp4 (.const ``Fin.lt_of_not_le []) n x y h)
+    | (``Eq, #[.const ``Nat [], x, y]) =>
+      return some (mkApp3 (.const ``Nat.lt_or_gt_of_ne []) x y h)
+    | (``Eq, #[.const ``Int [], x, y]) =>
+      return some (mkApp3 (.const ``Int.lt_or_gt_of_ne []) x y h)
+    | (``Prod.Lex, _) => return some (← mkAppM ``Prod.of_not_lex #[h])
+    | (``Eq, #[.app (.const ``Fin []) n, x, y]) =>
+      return some (mkApp4 (.const ``Fin.lt_or_gt_of_ne []) n x y h)
+    | (``Dvd.dvd, #[.const ``Nat [], _, k, x]) =>
+      return some (mkApp3 (.const ``Nat.emod_pos_of_not_dvd []) k x h)
+    | (``Dvd.dvd, #[.const ``Int [], _, k, x]) =>
+      -- This introduces a disjunction that could be avoided by checking `k ≠ 0`.
+      return some (mkApp3 (.const ``Int.emod_pos_of_not_dvd []) k x h)
+    | (``Or, #[P₁, P₂]) => return some (mkApp3 (.const ``and_not_not_of_not_or []) P₁ P₂ h)
+    | (``And, #[P₁, P₂]) =>
+      return some (mkApp5 (.const ``Decidable.or_not_not_of_not_and []) P₁ P₂
+        (.app (.const ``Classical.propDecidable []) P₁)
+        (.app (.const ``Classical.propDecidable []) P₂) h)
+    | (``Not, #[P']) =>
+      return some (mkApp3 (.const ``Decidable.of_not_not []) P'
+        (.app (.const ``Classical.propDecidable []) P') h)
+    | (``Iff, #[P₁, P₂]) =>
+      return some (mkApp5 (.const ``Decidable.and_not_or_not_and_of_not_iff []) P₁ P₂
+        (.app (.const ``Classical.propDecidable []) P₁)
+        (.app (.const ``Classical.propDecidable []) P₂) h)
     | _ => return none
   | _ => return none
 

src/Lean/Elab/Tactic/Omega/MinNatAbs.lean

@@ -5,10 +5,8 @@ Authors: Scott Morrison
 -/
 prelude
 import Init.BinderPredicates
-import Init.Data.Int.Order
-import Init.Data.List.Lemmas
-import Init.Data.Nat.MinMax
-import Init.Data.Option.Lemmas
+import Init.Data.List
+import Init.Data.Option
 
 /-!
 # `List.nonzeroMinimum`, `List.minNatAbs`, `List.maxNatAbs`

src/Lean/Elab/Tactic/Omega/OmegaM.lean

@@ -7,9 +7,8 @@ prelude
 import Init.Omega.LinearCombo
 import Init.Omega.Int
 import Init.Omega.Logic
-import Init.Data.BitVec.Basic
+import Init.Data.BitVec
 import Lean.Meta.AppBuilder
-import Lean.Meta.Canonicalizer
 
 /-!
 # The `OmegaM` state monad.
@@ -55,7 +54,7 @@ structure State where
   atoms : HashMap Expr Nat := {}
 
 /-- An intermediate layer in the `OmegaM` monad. -/
-abbrev OmegaM' := StateRefT State (ReaderT Context CanonM)
+abbrev OmegaM' := StateRefT State (ReaderT Context MetaM)
 
 /--
 Cache of expressions that have been visited, and their reflection as a linear combination.
@@ -71,7 +70,7 @@ abbrev OmegaM := StateRefT Cache OmegaM'
 
 /-- Run a computation in the `OmegaM` monad, starting with no recorded atoms. -/
 def OmegaM.run (m : OmegaM α) (cfg : OmegaConfig) : MetaM α :=
-  m.run' HashMap.empty |>.run' {} { cfg } |>.run
+  m.run' HashMap.empty |>.run' {} { cfg }
 
 /-- Retrieve the user-specified configuration options. -/
 def cfg : OmegaM OmegaConfig := do pure (← read).cfg
@@ -177,7 +176,7 @@ def analyzeAtom (e : Expr) : OmegaM (HashSet Expr) := do
       | _, (``Fin.val, #[n, i]) =>
         r := r.insert (mkApp2 (.const ``Fin.isLt []) n i)
       | _, (``BitVec.toNat, #[n, x]) =>
-        r := r.insert (mkApp2 (.const ``BitVec.isLt []) n x)
+        r := r.insert (mkApp2 (.const ``BitVec.toNat_lt []) n x)
       | _, _ => pure ()
     return r
   | (``HDiv.hDiv, #[_, _, _, _, x, k]) => match natCast? k with
@@ -245,7 +244,6 @@ Return its index, and, if it is new, a collection of interesting facts about the
 -/
 def lookup (e : Expr) : OmegaM (Nat × Option (HashSet Expr)) := do
   let c ← getThe State
-  let e ← canon e
   match c.atoms.find? e with
   | some i => return (i, none)
   | none =>

src/Lean/Elab/Tactic/ShowTerm.lean

@@ -1,28 +0,0 @@
-/-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Mario Carneiro
--/
-prelude
-import Lean.Elab.ElabRules
-import Lean.Meta.Tactic.TryThis
-
-namespace Std.Tactic
-open Lean Elab Term Tactic Meta.Tactic.TryThis Parser.Tactic
-
-@[builtin_tactic showTerm] def evalShowTerm : Tactic := fun stx =>
-  match stx with
-  | `(tactic| show_term%$tk $t) => withMainContext do
-    let g ← getMainGoal
-    evalTactic t
-    addExactSuggestion tk (← instantiateMVars (mkMVar g)).headBeta (origSpan? := ← getRef)
-  | _ => throwUnsupportedSyntax
-
-/-- Implementation of `show_term` term elaborator. -/
-@[builtin_term_elab showTermElabImpl] def elabShowTerm : TermElab
-  | `(show_term_elab%$tk $t), ty => do
-    let e ← Term.elabTermEnsuringType t ty
-    Term.synthesizeSyntheticMVarsNoPostponing
-    addTermSuggestion tk (← instantiateMVars e).headBeta (origSpan? := ← getRef)
-    pure e
-  | _, _ => throwUnsupportedSyntax

src/Lean/Elab/Tactic/Simp.lean

@@ -434,7 +434,7 @@ where
   if tactic.simp.trace.get (← getOptions) then
     traceSimpCall stx usedSimps
 
-def dsimpLocation (ctx : Simp.Context) (simprocs : Simp.SimprocsArray) (loc : Location) : TacticM Unit := do
+def dsimpLocation (ctx : Simp.Context) (loc : Location) : TacticM Unit := do
   match loc with
   | Location.targets hyps simplifyTarget =>
     withMainContext do
@@ -446,7 +446,7 @@ def dsimpLocation (ctx : Simp.Context) (simprocs : Simp.SimprocsArray) (loc : Lo
 where
   go (fvarIdsToSimp : Array FVarId) (simplifyTarget : Bool) : TacticM Unit := do
     let mvarId ← getMainGoal
-    let (result?, usedSimps) ← dsimpGoal mvarId ctx simprocs (simplifyTarget := simplifyTarget) (fvarIdsToSimp := fvarIdsToSimp)
+    let (result?, usedSimps) ← dsimpGoal mvarId ctx (simplifyTarget := simplifyTarget) (fvarIdsToSimp := fvarIdsToSimp)
     match result? with
     | none => replaceMainGoal []
     | some mvarId => replaceMainGoal [mvarId]
@@ -454,8 +454,8 @@ where
       mvarId.withContext <| traceSimpCall (← getRef) usedSimps
 
 @[builtin_tactic Lean.Parser.Tactic.dsimp] def evalDSimp : Tactic := fun stx => do
-  let { ctx, simprocs, .. } ← withMainContext <| mkSimpContext stx (eraseLocal := false) (kind := .dsimp)
-  dsimpLocation ctx simprocs (expandOptLocation stx[5])
+  let { ctx, .. } ← withMainContext <| mkSimpContext stx (eraseLocal := false) (kind := .dsimp)
+  dsimpLocation ctx (expandOptLocation stx[5])
 
 end Lean.Elab.Tactic
 

src/Lean/Elab/Tactic/SimpTrace.lean

@@ -56,8 +56,7 @@ def mkSimpCallStx (stx : Syntax) (usedSimps : UsedSimps) : MetaM (TSyntax `tacti
   | _ => throwUnsupportedSyntax
 
 /-- Implementation of `dsimp?`. -/
-def dsimpLocation' (ctx : Simp.Context) (simprocs : SimprocsArray) (loc : Location) :
-    TacticM Simp.UsedSimps := do
+def dsimpLocation' (ctx : Simp.Context) (loc : Location) : TacticM Simp.UsedSimps := do
   match loc with
   | Location.targets hyps simplifyTarget =>
     withMainContext do
@@ -70,8 +69,8 @@ where
   /-- Implementation of `dsimp?`. -/
   go (fvarIdsToSimp : Array FVarId) (simplifyTarget : Bool) : TacticM Simp.UsedSimps := do
     let mvarId ← getMainGoal
-    let (result?, usedSimps) ← dsimpGoal mvarId ctx simprocs (simplifyTarget := simplifyTarget)
-      (fvarIdsToSimp := fvarIdsToSimp)
+    let (result?, usedSimps) ←
+      dsimpGoal mvarId ctx (simplifyTarget := simplifyTarget) (fvarIdsToSimp := fvarIdsToSimp)
     match result? with
     | none => replaceMainGoal []
     | some mvarId => replaceMainGoal [mvarId]
@@ -84,9 +83,8 @@ where
       `(tactic| dsimp!%$tk $(config)? $[only%$o]? $[[$args,*]]? $(loc)?)
     else
       `(tactic| dsimp%$tk $(config)? $[only%$o]? $[[$args,*]]? $(loc)?)
-    let { ctx, simprocs, .. } ←
-      withMainContext <| mkSimpContext stx (eraseLocal := false) (kind := .dsimp)
-    let usedSimps ← dsimpLocation' ctx simprocs <| (loc.map expandLocation).getD (.targets #[] true)
+    let { ctx, .. } ← withMainContext <| mkSimpContext stx (eraseLocal := false) (kind := .dsimp)
+    let usedSimps ← dsimpLocation' ctx <| (loc.map expandLocation).getD (.targets #[] true)
     let stx ← mkSimpCallStx stx usedSimps
     addSuggestion tk stx (origSpan? := ← getRef)
   | _ => throwUnsupportedSyntax

src/Lean/Elab/Tactic/Simproc.lean

@@ -26,11 +26,10 @@ def elabSimprocKeys (stx : Syntax) : MetaM (Array Meta.SimpTheoremKey) := do
   let pattern ← elabSimprocPattern stx
   DiscrTree.mkPath pattern simpDtConfig
 
-def checkSimprocType (declName : Name) : CoreM Bool := do
+def checkSimprocType (declName : Name) : CoreM Unit := do
   let decl ← getConstInfo declName
   match decl.type with
-  | .const ``Simproc _ => pure false
-  | .const ``DSimproc _ => pure true
+  | .const ``Simproc _ => pure ()
   | _ => throwError "unexpected type at '{declName}', 'Simproc' expected"
 
 namespace Command
@@ -39,7 +38,7 @@ namespace Command
   let `(simproc_pattern% $pattern => $declName) := stx | throwUnsupportedSyntax
   let declName ← resolveGlobalConstNoOverload declName
   liftTermElabM do
-    discard <| checkSimprocType declName
+    checkSimprocType declName
     let keys ← elabSimprocKeys pattern
     registerSimproc declName keys
 
@@ -47,10 +46,9 @@ namespace Command
   let `(builtin_simproc_pattern% $pattern => $declName) := stx | throwUnsupportedSyntax
   let declName ← resolveGlobalConstNoOverload declName
   liftTermElabM do
-    let dsimp ← checkSimprocType declName
+    checkSimprocType declName
     let keys ← elabSimprocKeys pattern
-    let registerProcName := if dsimp then ``registerBuiltinDSimproc else ``registerBuiltinSimproc
-    let val := mkAppN (mkConst registerProcName) #[toExpr declName, toExpr keys, mkConst declName]
+    let val := mkAppN (mkConst ``registerBuiltinSimproc) #[toExpr declName, toExpr keys, mkConst declName]
     let initDeclName ← mkFreshUserName (declName ++ `declare)
     declareBuiltin initDeclName val
 

src/Lean/Elab/Term.lean

@@ -793,10 +793,10 @@ def mkCoe (expectedType : Expr) (e : Expr) (f? : Option Expr := none) (errorMsgH
     | _            => throwTypeMismatchError errorMsgHeader? expectedType (← inferType e) e f?
 
 /--
-If `expectedType?` is `some t`, then ensures `t` and `eType` are definitionally equal by inserting a coercion if necessary.
+  If `expectedType?` is `some t`, then ensure `t` and `eType` are definitionally equal.
+  If they are not, then try coercions.
 
-Argument `f?` is used only for generating error messages when inserting coercions fails.
--/
+  Argument `f?` is used only for generating error messages. -/
 def ensureHasType (expectedType? : Option Expr) (e : Expr)
     (errorMsgHeader? : Option String := none) (f? : Option Expr := none) : TermElabM Expr := do
   let some expectedType := expectedType? | return e
@@ -1432,22 +1432,9 @@ def addDotCompletionInfo (stx : Syntax) (e : Expr) (expectedType? : Option Expr)
 def elabTerm (stx : Syntax) (expectedType? : Option Expr) (catchExPostpone := true) (implicitLambda := true) : TermElabM Expr :=
   withRef stx <| elabTermAux expectedType? catchExPostpone implicitLambda stx
 
-/--
-Similar to `Lean.Elab.Term.elabTerm`, but ensures that the type of the elaborated term is `expectedType?`
-by inserting coercions if necessary.
-
-If `errToSorry` is true, then if coercion insertion fails, this function returns `sorry` and logs the error.
-Otherwise, it throws the error.
--/
 def elabTermEnsuringType (stx : Syntax) (expectedType? : Option Expr) (catchExPostpone := true) (implicitLambda := true) (errorMsgHeader? : Option String := none) : TermElabM Expr := do
   let e ← elabTerm stx expectedType? catchExPostpone implicitLambda
-  try
-    withRef stx <| ensureHasType expectedType? e errorMsgHeader?
-  catch ex =>
-    if (← read).errToSorry && ex matches .error .. then
-      exceptionToSorry ex expectedType?
-    else
-      throw ex
+  withRef stx <| ensureHasType expectedType? e errorMsgHeader?
 
 /-- Execute `x` and return `some` if no new errors were recorded or exceptions were thrown. Otherwise, return `none`. -/
 def commitIfNoErrors? (x : TermElabM α) : TermElabM (Option α) := do

src/Lean/Elab/Util.lean

@@ -24,13 +24,6 @@ def MacroScopesView.format (view : MacroScopesView) (mainModule : Name) : Format
     else
       view.scopes.foldl Name.mkNum (view.name ++ view.imported ++ view.mainModule)
 
-/--
-Two names are from the same lexical scope if their scoping information modulo `MacroScopesView.name`
-is equal.
--/
-def MacroScopesView.equalScope (a b : MacroScopesView) : Bool :=
-  a.scopes == b.scopes && a.mainModule == b.mainModule && a.imported == b.imported
-
 namespace Elab
 
 def expandOptNamedPrio (stx : Syntax) : MacroM Nat :=

src/Lean/Environment.lean

@@ -230,7 +230,6 @@ inductive KernelException where
   | exprTypeMismatch (env : Environment) (lctx : LocalContext) (expr : Expr) (expectedType : Expr)
   | appTypeMismatch  (env : Environment) (lctx : LocalContext) (app : Expr) (funType : Expr) (argType : Expr)
   | invalidProj      (env : Environment) (lctx : LocalContext) (proj : Expr)
-  | thmTypeIsNotProp (env : Environment) (name : Name) (type : Expr)
   | other            (msg : String)
   | deterministicTimeout
   | excessiveMemory
@@ -770,33 +769,6 @@ partial def importModulesCore (imports : Array Import) : ImportStateM Unit := do
       moduleNames := s.moduleNames.push i.module
     }
 
-/--
-Return `true` if `cinfo₁` and `cinfo₂` are theorems with the same name, universe parameters,
-and types. We allow different modules to prove the same theorem.
-
-Motivation: We want to generate equational theorems on demand and potentially
-in different files, and we want them to have non-private names.
-We may add support for other kinds of definitions in the future. For now, theorems are
-sufficient for our purposes.
-
-We may have to revise this design decision and eagerly generate equational theorems when
-we implement the module system.
-
-Remark: we do not check whether the theorem `value` field match. This feature is useful and
-ensures the proofs for equational theorems do not need to be identical. This decision
-relies on the fact that theorem types are propositions, we have proof irrelevance,
-and theorems are (mostly) opaque in Lean. For `Acc.rec`, we may unfold theorems
-during type-checking, but we are assuming this is not an issue in practice,
-and we are planning to address this issue in the future.
--/
-private def equivInfo (cinfo₁ cinfo₂ : ConstantInfo) : Bool := Id.run do
-  let .thmInfo tval₁ := cinfo₁ | false
-  let .thmInfo tval₂ := cinfo₂ | false
-  return tval₁.name == tval₂.name
-    && tval₁.type == tval₂.type
-    && tval₁.levelParams == tval₂.levelParams
-    && tval₁.all == tval₂.all
-
 /--
   Construct environment from `importModulesCore` results.
 
@@ -815,13 +787,11 @@ def finalizeImport (s : ImportState) (imports : Array Import) (opts : Options) (
   for h:modIdx in [0:s.moduleData.size] do
     let mod := s.moduleData[modIdx]'h.upper
     for cname in mod.constNames, cinfo in mod.constants do
-      match constantMap.insertIfNew cname cinfo with
-      | (constantMap', cinfoPrev?) =>
+      match constantMap.insert' cname cinfo with
+      | (constantMap', replaced) =>
         constantMap := constantMap'
-        if let some cinfoPrev := cinfoPrev? then
-          -- Recall that the map has not been modified when `cinfoPrev? = some _`.
-          unless equivInfo cinfoPrev cinfo do
-            throwAlreadyImported s const2ModIdx modIdx cname
+        if replaced then
+          throwAlreadyImported s const2ModIdx modIdx cname
       const2ModIdx := const2ModIdx.insert cname modIdx
     for cname in mod.extraConstNames do
       const2ModIdx := const2ModIdx.insert cname modIdx

src/Lean/Exception.lean

@@ -69,7 +69,7 @@ protected def throwError [Monad m] [MonadError m] (msg : MessageData) : m α :=
   let (ref, msg) ← AddErrorMessageContext.add ref msg
   throw <| Exception.error ref msg
 
-/-- Throw an unknown constant error message. -/
+/-- Thrown an unknown constant error message. -/
 def throwUnknownConstant [Monad m] [MonadError m] (constName : Name) : m α :=
   Lean.throwError m!"unknown constant '{mkConst constName}'"
 

src/Lean/Expr.lean

@@ -801,7 +801,7 @@ def isType0 : Expr → Bool
 
 /-- Return `true` if the given expression is `.sort .zero` -/
 def isProp : Expr → Bool
-  | sort .zero => true
+  | sort (.zero ..) => true
   | _ => false
 
 /-- Return `true` if the given expression is a bound variable. -/
@@ -904,14 +904,6 @@ def appArg!' : Expr → Expr
   | app _ a   => a
   | _         => panic! "application expected"
 
-def appArg (e : Expr) (h : e.isApp) : Expr :=
-  match e, h with
-  | .app _ a, _ => a
-
-def appFn (e : Expr) (h : e.isApp) : Expr :=
-  match e, h with
-  | .app f _, _ => f
-
 def sortLevel! : Expr → Level
   | sort u => u
   | _      => panic! "sort expected"
@@ -924,7 +916,7 @@ def isRawNatLit : Expr → Bool
   | lit (Literal.natVal _) => true
   | _                      => false
 
-def rawNatLit? : Expr → Option Nat
+def natLit? : Expr → Option Nat
   | lit (Literal.natVal v) => v
   | _                      => none
 
@@ -1075,6 +1067,33 @@ def isAppOfArity' : Expr → Name → Nat → Bool
   | app f _,    n, a+1 => isAppOfArity' f n a
   | _,          _,  _   => false
 
+/--
+Checks if an expression is a "natural number numeral in normal form",
+i.e. of type `Nat`, and explicitly of the form `OfNat.ofNat n`
+where `n` matches `.lit (.natVal n)` for some literal natural number `n`.
+and if so returns `n`.
+-/
+-- Note that `Expr.lit (.natVal n)` is not considered in normal form!
+def nat? (e : Expr) : Option Nat := do
+  guard <| e.isAppOfArity ``OfNat.ofNat 3
+  let lit (.natVal n) := e.appFn!.appArg! | none
+  n
+
+/--
+Checks if an expression is an "integer numeral in normal form",
+i.e. of type `Nat` or `Int`, and either a natural number numeral in normal form (as specified by `nat?`),
+or the negation of a positive natural number numberal in normal form,
+and if so returns the integer.
+-/
+def int? (e : Expr) : Option Int :=
+  if e.isAppOfArity ``Neg.neg 3 then
+    match e.appArg!.nat? with
+    | none => none
+    | some 0 => none
+    | some n => some (-n)
+  else
+    e.nat?
+
 private def getAppNumArgsAux : Expr → Nat → Nat
   | app f _, n => getAppNumArgsAux f (n+1)
   | _,       n => n
@@ -1597,45 +1616,12 @@ partial def cleanupAnnotations (e : Expr) : Expr :=
   let e' := e.consumeMData.consumeTypeAnnotations
   if e' == e then e else cleanupAnnotations e'
 
-/--
-Similar to `appFn`, but also applies `cleanupAnnotations` to resulting function.
-This function is used compile the `match_expr` term.
--/
-def appFnCleanup (e : Expr) (h : e.isApp) : Expr :=
-  match e, h with
-  | .app f _, _ => f.cleanupAnnotations
-
 def isFalse (e : Expr) : Bool :=
   e.cleanupAnnotations.isConstOf ``False
 
 def isTrue (e : Expr) : Bool :=
   e.cleanupAnnotations.isConstOf ``True
 
-/--
-Checks if an expression is a "natural number numeral in normal form",
-i.e. of type `Nat`, and explicitly of the form `OfNat.ofNat n`
-where `n` matches `.lit (.natVal n)` for some literal natural number `n`.
-and if so returns `n`.
--/
--- Note that `Expr.lit (.natVal n)` is not considered in normal form!
-def nat? (e : Expr) : Option Nat := do
-  let_expr OfNat.ofNat _ n _ := e | failure
-  let lit (.natVal n) := n | failure
-  n
-
-/--
-Checks if an expression is an "integer numeral in normal form",
-i.e. of type `Nat` or `Int`, and either a natural number numeral in normal form (as specified by `nat?`),
-or the negation of a positive natural number numberal in normal form,
-and if so returns the integer.
--/
-def int? (e : Expr) : Option Int :=
-  let_expr Neg.neg _ _ a := e | e.nat?
-  match a.nat? with
-  | none => none
-  | some 0 => none
-  | some n => some (-n)
-
 /-- Return true iff `e` contains a free variable which satisfies `p`. -/
 @[inline] def hasAnyFVar (e : Expr) (p : FVarId → Bool) : Bool :=
   let rec @[specialize] visit (e : Expr) := if !e.hasFVar then false else
@@ -2003,38 +1989,4 @@ def mkEM (p : Expr) : Expr := mkApp (mkConst ``Classical.em) p
 /-- Return `p ↔ q` -/
 def mkIff (p q : Expr) : Expr := mkApp2 (mkConst ``Iff) p q
 
-private def natAddFn : Expr :=
-  let nat := mkConst ``Nat
-  mkApp4 (mkConst ``HAdd.hAdd [0, 0, 0]) nat nat nat (mkApp2 (mkConst ``instHAdd [0]) nat (mkConst ``instAddNat))
-
-private def natMulFn : Expr :=
-  let nat := mkConst ``Nat
-  mkApp4 (mkConst ``HMul.hMul [0, 0, 0]) nat nat nat (mkApp2 (mkConst ``instHMul [0]) nat (mkConst ``instMulNat))
-
-/-- Given `a : Nat`, returns `Nat.succ a` -/
-def mkNatSucc (a : Expr) : Expr :=
-  mkApp (mkConst ``Nat.succ) a
-
-/-- Given `a b : Nat`, returns `a + b` -/
-def mkNatAdd (a b : Expr) : Expr :=
-  mkApp2 natAddFn a b
-
-/-- Given `a b : Nat`, returns `a * b` -/
-def mkNatMul (a b : Expr) : Expr :=
-  mkApp2 natMulFn a b
-
-private def natLEPred : Expr :=
-  mkApp2 (mkConst ``LE.le [0]) (mkConst ``Nat) (mkConst ``instLENat)
-
-/-- Given `a b : Nat`, return `a ≤ b` -/
-def mkNatLE (a b : Expr) : Expr :=
-  mkApp2 natLEPred a b
-
-private def natEqPred : Expr :=
-  mkApp (mkConst ``Eq [1]) (mkConst ``Nat)
-
-/-- Given `a b : Nat`, return `a = b` -/
-def mkNatEq (a b : Expr) : Expr :=
-  mkApp2 natEqPred a b
-
 end Lean

src/Lean/Message.lean

@@ -366,7 +366,6 @@ def toMessageData (e : KernelException) (opts : Options) : MessageData :=
   | appTypeMismatch  env lctx e fnType argType =>
     mkCtx env lctx opts m!"application type mismatch{indentExpr e}\nargument has type{indentExpr argType}\nbut function has type{indentExpr fnType}"
   | invalidProj env lctx e              => mkCtx env lctx opts m!"(kernel) invalid projection{indentExpr e}"
-  | thmTypeIsNotProp env constName type => mkCtx env {} opts m!"(kernel) type of theorem '{constName}' is not a proposition{indentExpr type}"
   | other msg                           => m!"(kernel) {msg}"
   | deterministicTimeout                => "(kernel) deterministic timeout"
   | excessiveMemory                     => "(kernel) excessive memory consumption detected"

src/Lean/Meta.lean

@@ -47,5 +47,3 @@ import Lean.Meta.CoeAttr
 import Lean.Meta.Iterator
 import Lean.Meta.LazyDiscrTree
 import Lean.Meta.LitValues
-import Lean.Meta.CheckTactic
-import Lean.Meta.Canonicalizer

src/Lean/Meta/Basic.lean

@@ -254,7 +254,7 @@ structure PostponedEntry where
   ref  : Syntax
   lhs  : Level
   rhs  : Level
-  /-- Context for the surrounding `isDefEq` call when the entry was created. -/
+  /-- Context for the surrounding `isDefEq` call when entry was created. -/
   ctx? : Option DefEqContext
   deriving Inhabited
 
@@ -264,7 +264,7 @@ structure PostponedEntry where
 structure State where
   mctx             : MetavarContext := {}
   cache            : Cache := {}
-  /-- When `trackZetaDelta == true`, then any let-decl free variable that is zetaDelta-expanded by `MetaM` is stored in `zetaDeltaFVarIds`. -/
+  /-- When `trackZetaDelta == true`, then any let-decl free variable that is zetaDelta expansion performed by `MetaM` is stored in `zetaDeltaFVarIds`. -/
   zetaDeltaFVarIds : FVarIdSet := {}
   /-- Array of postponed universe level constraints -/
   postponed        : PersistentArray PostponedEntry := {}
@@ -1347,16 +1347,6 @@ private def withNewMCtxDepthImp (allowLevelAssignments : Bool) (x : MetaM α) :
   finally
     modify fun s => { s with mctx := saved.mctx, postponed := saved.postponed }
 
-/--
-Removes `fvarId` from the local context, and replaces occurrences of it with `e`.
-It is the responsibility of the caller to ensure that `e` is well-typed in the context
-of any occurrence of `fvarId`.
--/
-def withReplaceFVarId {α} (fvarId : FVarId) (e : Expr) : MetaM α → MetaM α :=
-  withReader fun ctx => { ctx with
-    lctx := ctx.lctx.replaceFVarId fvarId e
-    localInstances := ctx.localInstances.erase fvarId }
-
 /--
   `withNewMCtxDepth k` executes `k` with a higher metavariable context depth,
   where metavariables created outside the `withNewMCtxDepth` (with a lower depth) cannot be assigned.
@@ -1747,15 +1737,6 @@ def isDefEqNoConstantApprox (t s : Expr) : MetaM Bool :=
 def etaExpand (e : Expr) : MetaM Expr :=
   withDefault do forallTelescopeReducing (← inferType e) fun xs _ => mkLambdaFVars xs (mkAppN e xs)
 
-/--
-If `e` is of the form `?m ...` instantiate metavars
--/
-def instantiateMVarsIfMVarApp (e : Expr) : MetaM Expr := do
-  if e.getAppFn.isMVar then
-    instantiateMVars e
-  else
-    return e
-
 end Meta
 
 builtin_initialize

src/Lean/Meta/Canonicalizer.lean

@@ -1,146 +0,0 @@
-/-
-Copyright (c) 2024 Amazon.com, Inc. or its affiliates. All Rights Reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Leonardo de Moura
--/
-prelude
-import Lean.Util.ShareCommon
-import Lean.Data.HashMap
-import Lean.Meta.Basic
-import Lean.Meta.FunInfo
-
-namespace Lean.Meta
-namespace Canonicalizer
-
-/-!
-Applications have implicit arguments. Thus, two terms that may look identical when pretty-printed can be structurally different.
-For example, `@id (Id Nat) x` and `@id Nat x` are structurally different but are both pretty-printed as `id x`.
-Moreover, these two terms are definitionally equal since `Id Nat` reduces to `Nat`. This may create situations
-that are counterintuitive to our users. Furthermore, several tactics (e.g., `omega`) need to collect unique atoms in a goal.
-One simple approach is to maintain a list of atoms found so far, and whenever a new atom is discovered, perform a
-linear scan to test whether it is definitionally equal to a previously found one. However, this method is too costly,
-even if the definitional equality test were inexpensive.
-
-This module aims to efficiently identify terms that are structurally different, definitionally equal, and structurally equal
-when we disregard implicit arguments like `@id (Id Nat) x` and `@id Nat x`. The procedure is straightforward. For each atom,
-we create a new abstracted atom by erasing all implicit information. We refer to this abstracted atom as a 'key.' For the two
-terms mentioned, the key would be `@id _ x`, where `_` denotes a placeholder for a dummy term. To preserve any
-pre-existing directed acyclic graph (DAG) structure and prevent exponential blowups while constructing the key, we employ
-unsafe techniques, such as pointer equality. Additionally, we maintain a mapping from keys to lists of terms, where each
-list contains terms sharing the same key but not definitionally equal. We posit that these lists will be small in practice.
--/
-
-/--
-Auxiliary structure for creating a pointer-equality mapping from `Expr` to `Key`.
-We use this mapping to ensure we preserve the dag-structure of input expressions.
--/
-structure ExprVisited where
-  e : Expr
-  deriving Inhabited
-
-unsafe instance : BEq ExprVisited where
-  beq a b := ptrAddrUnsafe a == ptrAddrUnsafe b
-
-unsafe instance : Hashable ExprVisited where
-  hash a := USize.toUInt64 (ptrAddrUnsafe a)
-
-abbrev Key := ExprVisited
-
-/--
-State for the `CanonM` monad.
--/
-structure State where
-  /-- "Set" of all keys created so far. This is a hash-consing helper structure available in Lean. -/
-  keys       : ShareCommon.State.{0} Lean.ShareCommon.objectFactory := ShareCommon.State.mk Lean.ShareCommon.objectFactory
-  /-- Mapping from `Expr` to `Key`. See comment at `ExprVisited`. -/
-  -- We use `HashMapImp` to ensure we don't have to tag `State` as `unsafe`.
-  cache      : HashMapImp ExprVisited Key := mkHashMapImp
-  /--
-  Given a key `k` and `keyToExprs.find? k = some es`, we have that all `es` share key `k`, and
-  are not definitionally equal modulo the transparency setting used. -/
-  keyToExprs : HashMapImp Key (List Expr) := mkHashMapImp
-
-instance : Inhabited State where
-  default := {}
-
-abbrev CanonM := ReaderT TransparencyMode $ StateRefT State MetaM
-
-/--
-The definitionally equality tests are performed using the given transparency mode.
-We claim `TransparencyMode.instances` is a good setting for most applications.
--/
-def CanonM.run (x : CanonM α) (transparency := TransparencyMode.instances) : MetaM α :=
-  StateRefT'.run' (x transparency) {}
-
-private def shareCommon (a : α) : CanonM α :=
-  modifyGet fun { keys, cache, keyToExprs } =>
-    let (a, keys) := ShareCommon.State.shareCommon keys a
-    (a, { keys, cache, keyToExprs })
-
-private partial def mkKey (e : Expr) : CanonM Key := do
-  if let some key := unsafe (← get).cache.find? { e } then
-    return key
-  else
-    let key ← match e with
-      | .sort .. | .fvar .. | .bvar .. | .const .. | .lit .. =>
-        pure { e := (← shareCommon e) }
-      | .mvar .. =>
-        -- We instantiate assigned metavariables because the
-        -- pretty-printer also instantiates them.
-        let eNew ← instantiateMVars e
-        if eNew == e then pure { e := (← shareCommon e) }
-        else mkKey eNew
-      | .mdata _ a => mkKey a
-      | .app .. =>
-        let f := (← mkKey e.getAppFn).e
-        if f.isMVar then
-          let eNew ← instantiateMVars e
-          unless eNew == e do
-            return (← mkKey eNew)
-        let info ← getFunInfo f
-        let args ← e.getAppArgs.mapIdxM fun i arg => do
-          if h : i < info.paramInfo.size then
-            let info := info.paramInfo[i]
-            if info.isExplicit then
-              pure (← mkKey arg).e
-            else
-              pure (mkSort 0) -- some dummy value for erasing implicit
-          else
-            pure (← mkKey arg).e
-        pure { e := (← shareCommon (mkAppN f args)) }
-      | .lam n t b i =>
-        pure { e := (← shareCommon (.lam n (← mkKey t).e (← mkKey b).e i)) }
-      | .forallE n t b i =>
-        pure { e := (← shareCommon (.forallE n (← mkKey t).e (← mkKey b).e i)) }
-      | .letE n t v b d =>
-        pure { e := (← shareCommon (.letE n (← mkKey t).e (← mkKey v).e (← mkKey b).e d)) }
-      | .proj t i s =>
-        pure { e := (← shareCommon (.proj t i (← mkKey s).e)) }
-    unsafe modify fun { keys, cache, keyToExprs} => { keys, keyToExprs, cache := cache.insert { e } key |>.1 }
-    return key
-
-/--
-"Canonicalize" the given expression.
--/
-def canon (e : Expr) : CanonM Expr := do
-  let k ← mkKey e
-  -- Find all expressions canonicalized before that have the same key.
-  if let some es' := unsafe (← get).keyToExprs.find? k then
-    withTransparency (← read) do
-      for e' in es' do
-        -- Found an expression `e'` that is definitionally equal to `e` and share the same key.
-        if (← isDefEq e e') then
-          return e'
-      -- `e` is not definitionally equal to any expression in `es'`. We claim this should be rare.
-      unsafe modify fun { keys, cache, keyToExprs } => { keys, cache, keyToExprs := keyToExprs.insert k (e :: es') |>.1 }
-      return e
-  else
-    -- `e` is the first expression we found with key `k`.
-    unsafe modify fun { keys, cache, keyToExprs } => { keys, cache, keyToExprs := keyToExprs.insert k [e] |>.1 }
-    return e
-
-end Canonicalizer
-
-export Canonicalizer (CanonM canon)
-
-end Lean.Meta

src/Lean/Meta/CheckTactic.lean

@@ -1,24 +0,0 @@
-/-
-Copyright (c) 2024 Lean FRO. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joe Hendrix
--/
-prelude
-import Lean.Meta.Basic
-
-namespace Lean.Meta.CheckTactic
-
-def mkCheckGoalType (val type : Expr) : MetaM Expr := do
-  let lvl ← mkFreshLevelMVar
-  pure <| mkApp2 (mkConst ``CheckGoalType [lvl]) type val
-
-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)
-
-end Lean.Meta.CheckTactic

src/Lean/Meta/Coe.lean

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
+import Lean.Meta.WHNF
 import Lean.Meta.Transform
 import Lean.Meta.SynthInstance
 import Lean.Meta.AppBuilder

src/Lean/Meta/CtorRecognizer.lean

@@ -5,7 +5,6 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Lean.Meta.LitValues
-import Lean.Meta.Offset
 
 namespace Lean.Meta
 
@@ -65,17 +64,9 @@ def constructorApp? (e : Expr) : MetaM (Option (ConstructorVal × Array Expr)) :
 
 /--
 Similar to `constructorApp?`, but on failure it puts `e` in WHNF and tries again.
-It also `isOffset?`
 -/
 def constructorApp'? (e : Expr) : MetaM (Option (ConstructorVal × Array Expr)) := do
-  if let some (e, k) ← isOffset? e then
-    if k = 0 then
-      return none
-    else
-      let .ctorInfo val ← getConstInfo ``Nat.succ | return none
-      if k = 1 then return some (val, #[e])
-      else return some (val, #[mkNatAdd e (toExpr (k-1))])
-  else if let some r ← constructorApp? e then
+  if let some r ← constructorApp? e then
     return some r
   else
     constructorApp? (← whnf e)