Merge remote-tracking branch 'origin/master' into array_swap_docstring

It addresses a performance issue at
https://github.com/leanprover/LNSym/blob/proof_size_expt/Proofs/SHA512/Experiments/Sym20.lean

.github/workflows/ci.yml

@@ -298,8 +298,8 @@ jobs:
         uses: msys2/setup-msys2@v2
         with:
           msystem: clang64
-          # `:p` means prefix with appropriate msystem prefix
-          pacboy: "make python cmake:p clang:p ccache:p gmp:p git zip unzip diffutils binutils tree zstd:p tar"
+          # `:` means do not prefix with msystem
+          pacboy: "make: python: cmake clang ccache gmp git: zip: unzip: diffutils: binutils: tree: zstd tar:"
         if: runner.os == 'Windows'
       - name: Install Brew Packages
         run: |
@@ -426,7 +426,7 @@ jobs:
         if: matrix.test-speedcenter
       - name: Check Stage 3
         run: |
-          make -C build -j$NPROC stage3
+          make -C build -j$NPROC check-stage3
         if: matrix.test-speedcenter
       - name: Test Speedcenter Benchmarks
         run: |
@@ -455,12 +455,24 @@ jobs:
     # mark as merely cancelled not failed if builds are cancelled
     if: ${{ !cancelled() }}
     steps:
+    - if: ${{ contains(needs.*.result, 'failure') && github.repository == 'leanprover/lean4' && github.ref_name == 'master' }}
+      uses: zulip/github-actions-zulip/send-message@v1
+      with:
+        api-key: ${{ secrets.ZULIP_BOT_KEY }}
+        email: "github-actions-bot@lean-fro.zulipchat.com"
+        organization-url: "https://lean-fro.zulipchat.com"
+        to: "infrastructure"
+        topic: "Github actions"
+        type: "stream"
+        content: |
+          A build of `${{ github.ref_name }}`, triggered by event `${{ github.event_name }}`, [failed](https://github.com/${{ github.repository }}/actions/runs/${{ github.run_id }}).
     - if: contains(needs.*.result, 'failure')
       uses: actions/github-script@v7
       with:
         script: |
             core.setFailed('Some jobs failed')
 
+
   # This job creates releases from tags
   # (whether they are "unofficial" releases for experiments, or official releases when the tag is "v" followed by a semver string.)
   # We do not attempt to automatically construct a changelog here:
@@ -533,3 +545,8 @@ jobs:
           gh workflow -R leanprover/release-index run update-index.yml
         env:
           GITHUB_TOKEN: ${{ secrets.RELEASE_INDEX_TOKEN }}
+      - name: Update toolchain on mathlib4's nightly-testing branch
+        run: |
+          gh workflow -R leanprover-community/mathlib4 run nightly_bump_toolchain.yml
+        env:
+          GITHUB_TOKEN: ${{ secrets.MATHLIB4_BOT }}

.github/workflows/jira.yml

@@ -0,0 +1,34 @@
+name: Jira sync
+
+on:
+  issues:
+    types: [closed]
+
+jobs:
+  jira-sync:
+    runs-on: ubuntu-latest
+
+    steps:
+      - name: Move Jira issue to Done
+        env:
+          JIRA_API_TOKEN: ${{ secrets.JIRA_API_TOKEN }}
+          JIRA_USERNAME: ${{ secrets.JIRA_USERNAME }}
+          JIRA_BASE_URL: ${{ secrets.JIRA_BASE_URL }}
+        run: |
+          issue_number=${{ github.event.issue.number }}
+
+          jira_issue_key=$(curl -s -u "${JIRA_USERNAME}:${JIRA_API_TOKEN}" \
+            -X GET -H "Content-Type: application/json" \
+            "${JIRA_BASE_URL}/rest/api/2/search?jql=summary~\"${issue_number}\"" | \
+            jq -r '.issues[0].key')
+
+          if [ -z "$jira_issue_key" ]; then
+            exit
+          fi
+
+          curl -s -u "${JIRA_USERNAME}:${JIRA_API_TOKEN}" \
+            -X POST -H "Content-Type: application/json" \
+            --data "{\"transition\": {\"id\": \"41\"}}" \
+            "${JIRA_BASE_URL}/rest/api/2/issue/${jira_issue_key}/transitions"
+
+          echo "Moved Jira issue ${jira_issue_key} to Done"

CONTRIBUTING.md

@@ -63,6 +63,20 @@ Because the change will be squashed, there is no need to polish the commit messa
 Reviews and Feedback:
 ----
 
+The lean4 repo is managed by the Lean FRO's *triage team* that aims to provide initial feedback on new bug reports, PRs, and RFCs weekly.
+This feedback generally consists of prioritizing the ticket using one of the following categories:
+* label `P-high`: We will work on this issue
+* label `P-medium`: We may work on this issue if we find the time
+* label `P-low`: We are not planning to work on this issue
+* *closed*: This issue is already fixed, it is not an issue, or is not sufficiently compatible with our roadmap for the project and we will not work on it nor accept external contributions on it
+
+For *bug reports*, the listed priority reflects our commitment to fixing the issue.
+It is generally indicative but not necessarily identical to the priority an external contribution addressing this bug would receive.
+For *PRs* and *RFCs*, the priority reflects our commitment to reviewing them and getting them to an acceptable state.
+Accepted RFCs are marked with the label `RFC accepted` and afterwards assigned a new "implementation" priority as with bug reports.
+
+General guidelines for interacting with reviews and feedback:
+
 **Be Patient**: Given the limited number of full-time maintainers and the volume of PRs, reviews may take some time.
 
 **Engage Constructively**: Always approach feedback positively and constructively. Remember, reviews are about ensuring the best quality for the project, not personal criticism.

doc/examples/interp.lean

@@ -149,4 +149,4 @@ def fact : Expr ctx (Ty.fn Ty.int Ty.int) :=
            (op (·*·) (delay fun _ => app fact (op (·-·) (var stop) (val 1))) (var stop)))
   decreasing_by sorry
 
-#eval fact.interp Env.nil 10
+#eval! fact.interp Env.nil 10

doc/int.md

@@ -13,7 +13,7 @@ Recall that nonnegative numerals are considered to be a `Nat` if there are no ty
 
 The operator `/` for `Int` implements integer division.
 ```lean
-#eval -10 / 4 -- -2
+#eval -10 / 4 -- -3
 ```
 
 Similar to `Nat`, the internal representation of `Int` is optimized. Small integers are

doc/quickstart.md

@@ -7,12 +7,17 @@ See [Setup](./setup.md) for supported platforms and other ways to set up Lean 4.
 
 1. Launch VS Code and install the `lean4` extension by clicking on the "Extensions" sidebar entry and searching for "lean4".
 
-    ![installing the vscode-lean4 extension](images/code-ext.png)
+   ![installing the vscode-lean4 extension](images/code-ext.png)
 
-1. Open the Lean 4 setup guide by creating a new text file using "File > New Text File" (`Ctrl+N`), clicking on the ∀-symbol in the top right and selecting "Documentation… > Setup: Show Setup Guide".
+1. Open the Lean 4 setup guide by creating a new text file using "File > New Text File" (`Ctrl+N` / `Cmd+N`), clicking on the ∀-symbol in the top right and selecting "Documentation… > Docs: Show Setup Guide".
 
-    ![show setup guide](images/show-setup-guide.png)
+   ![show setup guide](images/show-setup-guide.png)
 
-1. Follow the Lean 4 setup guide. It will walk you through learning resources for Lean 4, teach you how to set up Lean's dependencies on your platform, install Lean 4 for you at the click of a button and help you set up your first project.
+1. Follow the Lean 4 setup guide. It will:
 
-    ![setup guide](images/setup_guide.png)
+   - walk you through learning resources for Lean,
+   - teach you how to set up Lean's dependencies on your platform,
+   - install Lean 4 for you at the click of a button,
+   - help you set up your first project.
+
+   ![setup guide](images/setup_guide.png)

releases_drafts/mutualStructural.md

@@ -0,0 +1,65 @@
+* Structural recursion can now be explicitly requested using
+  ```
+  termination_by structural x
+  ```
+  in analogy to the existing `termination_by x` syntax that causes well-founded recursion to be used.
+  (#4542)
+
+* The `termination_by?` syntax no longer forces the use of well-founded recursion, and when structural
+  recursion is inferred, will print the result using the `termination_by` syntax.
+
+* Mutual structural recursion is supported now. This supports both mutual recursion over a non-mutual
+  data type, as well as recursion over mutual or nested data types:
+
+  ```lean
+  mutual
+  def Even : Nat → Prop
+    | 0 => True
+    | n+1 => Odd n
+
+  def Odd : Nat → Prop
+    | 0 => False
+    | n+1 => Even n
+  end
+
+  mutual
+  inductive A
+  | other : B → A
+  | empty
+  inductive B
+  | other : A → B
+  | empty
+  end
+
+  mutual
+  def A.size : A → Nat
+  | .other b => b.size + 1
+  | .empty => 0
+
+  def B.size : B → Nat
+  | .other a => a.size + 1
+  | .empty => 0
+  end
+
+  inductive Tree where | node : List Tree → Tree
+
+  mutual
+  def Tree.size : Tree → Nat
+  | node ts => Tree.list_size ts
+
+  def Tree.list_size : List Tree → Nat
+  | [] => 0
+  | t::ts => Tree.size t + Tree.list_size ts
+  end
+  ```
+
+  Functional induction principles are generated for these functions as well (`A.size.induct`, `A.size.mutual_induct`).
+
+  Nested structural recursion is still not supported.
+
+  PRs #4639, #4715, #4642, #4656, #4684, #4715, #4728, #4575, #4731, #4658, #4734, #4738, #4718,
+  #4733, #4787, #4788, #4789, #4807, #4772
+
+* A bugfix in the structural recursion code may in some cases break existing code, when a parameter
+  of the type of the recursive argument is bound behind indices of that type. This can usually be
+  fixed by reordering the parameters of the function (PR #4672)

src/CMakeLists.txt

@@ -1,5 +1,6 @@
 cmake_minimum_required(VERSION 3.10)
 cmake_policy(SET CMP0054 NEW)
+cmake_policy(SET CMP0110 NEW)
 if(NOT (${CMAKE_GENERATOR} MATCHES "Unix Makefiles"))
   message(FATAL_ERROR "The only supported CMake generator at the moment is 'Unix Makefiles'")
 endif()

src/Init/ByCases.lean

@@ -67,12 +67,8 @@ theorem ite_some_none_eq_none [Decidable P] :
 -- This is not marked as `simp` as it is already handled by `dite_eq_right_iff`.
 theorem dite_some_none_eq_none [Decidable P] {x : P → α} :
     (if h : P then some (x h) else none) = none ↔ ¬P := by
-  simp only [dite_eq_right_iff]
-  rfl
+  simp
 
 @[simp] theorem dite_some_none_eq_some [Decidable P] {x : P → α} {y : α} :
     (if h : P then some (x h) else none) = some y ↔ ∃ h : P, x h = y := by
-  by_cases h : P <;> simp only [h, dite_cond_eq_true, dite_cond_eq_false, Option.some.injEq,
-    false_iff, not_exists]
-  case pos => exact ⟨fun h_eq ↦ Exists.intro h h_eq, fun h_exists => h_exists.2⟩
-  case neg => exact fun h_false _ ↦ h_false
+  by_cases h : P <;> simp [h]

src/Init/Core.lean

@@ -474,6 +474,8 @@ class LawfulSingleton (α : Type u) (β : Type v) [EmptyCollection β] [Insert
   insert_emptyc_eq (x : α) : (insert x ∅ : β) = singleton x
 export LawfulSingleton (insert_emptyc_eq)
 
+attribute [simp] insert_emptyc_eq
+
 /-- Type class used to implement the notation `{ a ∈ c | p a }` -/
 class Sep (α : outParam <| Type u) (γ : Type v) where
   /-- Computes `{ a ∈ c | p a }`. -/
@@ -701,7 +703,7 @@ theorem Ne.elim (h : a ≠ b) : a = b → False := h
 
 theorem Ne.irrefl (h : a ≠ a) : False := h rfl
 
-theorem Ne.symm (h : a ≠ b) : b ≠ a := fun h₁ => h (h₁.symm)
+@[symm] theorem Ne.symm (h : a ≠ b) : b ≠ a := fun h₁ => h (h₁.symm)
 
 theorem ne_comm {α} {a b : α} : a ≠ b ↔ b ≠ a := ⟨Ne.symm, Ne.symm⟩
 
@@ -754,7 +756,7 @@ noncomputable def HEq.elim {α : Sort u} {a : α} {p : α → Sort v} {b : α} (
 theorem HEq.subst {p : (T : Sort u) → T → Prop} (h₁ : HEq a b) (h₂ : p α a) : p β b :=
   HEq.ndrecOn h₁ h₂
 
-theorem HEq.symm (h : HEq a b) : HEq b a :=
+@[symm] theorem HEq.symm (h : HEq a b) : HEq b a :=
   h.rec (HEq.refl a)
 
 theorem heq_of_eq (h : a = a') : HEq a a' :=
@@ -810,15 +812,15 @@ instance : Trans Iff Iff Iff where
 theorem Eq.comm {a b : α} : a = b ↔ b = a := Iff.intro Eq.symm Eq.symm
 theorem eq_comm {a b : α} : a = b ↔ b = a := Eq.comm
 
-theorem Iff.symm (h : a ↔ b) : b ↔ a := Iff.intro h.mpr h.mp
+@[symm] theorem Iff.symm (h : a ↔ b) : b ↔ a := Iff.intro h.mpr h.mp
 theorem Iff.comm: (a ↔ b) ↔ (b ↔ a) := Iff.intro Iff.symm Iff.symm
 theorem iff_comm : (a ↔ b) ↔ (b ↔ a) := Iff.comm
 
-theorem And.symm : a ∧ b → b ∧ a := fun ⟨ha, hb⟩ => ⟨hb, ha⟩
+@[symm] theorem And.symm : a ∧ b → b ∧ a := fun ⟨ha, hb⟩ => ⟨hb, ha⟩
 theorem And.comm : a ∧ b ↔ b ∧ a := Iff.intro And.symm And.symm
 theorem and_comm : a ∧ b ↔ b ∧ a := And.comm
 
-theorem Or.symm : a ∨ b → b ∨ a := .rec .inr .inl
+@[symm] theorem Or.symm : a ∨ b → b ∨ a := .rec .inr .inl
 theorem Or.comm : a ∨ b ↔ b ∨ a := Iff.intro Or.symm Or.symm
 theorem or_comm : a ∨ b ↔ b ∨ a := Or.comm
 
@@ -1105,6 +1107,7 @@ inductive Relation.TransGen {α : Sort u} (r : α → α → Prop) : α → α
 /-! # Subtype -/
 
 namespace Subtype
+
 theorem existsOfSubtype {α : Type u} {p : α → Prop} : { x // p x } → Exists (fun x => p x)
   | ⟨a, h⟩ => ⟨a, h⟩
 
@@ -1201,9 +1204,13 @@ def Prod.map {α₁ : Type u₁} {α₂ : Type u₂} {β₁ : Type v₁} {β₂
 
 /-! # Dependent products -/
 
-theorem ex_of_PSigma {α : Type u} {p : α → Prop} : (PSigma (fun x => p x)) → Exists (fun x => p x)
+theorem PSigma.exists {α : Sort u} {p : α → Prop} : (PSigma (fun x => p x)) → Exists (fun x => p x)
   | ⟨x, hx⟩ => ⟨x, hx⟩
 
+@[deprecated PSigma.exists (since := "2024-07-27")]
+theorem ex_of_PSigma {α : Type u} {p : α → Prop} : (PSigma (fun x => p x)) → Exists (fun x => p x) :=
+  PSigma.exists
+
 protected theorem PSigma.eta {α : Sort u} {β : α → Sort v} {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂}
     (h₁ : a₁ = a₂) (h₂ : Eq.ndrec b₁ h₁ = b₂) : PSigma.mk a₁ b₁ = PSigma.mk a₂ b₂ := by
   subst h₁
@@ -1545,7 +1552,7 @@ protected abbrev rec
     (q : Quot r) : motive q :=
   Eq.ndrecOn (Quot.liftIndepPr1 f h q) ((lift (Quot.indep f) (Quot.indepCoherent f h) q).2)
 
-@[inherit_doc Quot.rec] protected abbrev recOn
+@[inherit_doc Quot.rec, elab_as_elim] protected abbrev recOn
     (q : Quot r)
     (f : (a : α) → motive (Quot.mk r a))
     (h : (a b : α) → (p : r a b) → Eq.ndrec (f a) (sound p) = f b)
@@ -1556,7 +1563,7 @@ protected abbrev rec
 Dependent induction principle for a quotient, when the target type is a `Subsingleton`.
 In this case the quotient's side condition is trivial so any function can be lifted.
 -/
-protected abbrev recOnSubsingleton
+@[elab_as_elim] protected abbrev recOnSubsingleton
     [h : (a : α) → Subsingleton (motive (Quot.mk r a))]
     (q : Quot r)
     (f : (a : α) → motive (Quot.mk r a))

src/Init/Data.lean

@@ -36,3 +36,4 @@ import Init.Data.Channel
 import Init.Data.Cast
 import Init.Data.Sum
 import Init.Data.BEq
+import Init.Data.Subtype

src/Init/Data/Array/Basic.lean

@@ -50,6 +50,13 @@ instance : Inhabited (Array α) where
 def singleton (v : α) : Array α :=
   mkArray 1 v
 
+/-- Low-level version of `size` that directly queries the C array object cached size.
+    While this is not provable, `usize` always returns the exact size of the array since
+    the implementation only supports arrays of size less than `USize.size`.
+-/
+@[extern "lean_array_size", simp]
+def usize (a : @& Array α) : USize := a.size.toUSize
+
 /-- Low-level version of `fget` which is as fast as a C array read.
    `Fin` values are represented as tag pointers in the Lean runtime. Thus,
    `fget` may be slightly slower than `uget`. -/
@@ -101,7 +108,7 @@ def swap (a : Array α) (i j : @& Fin a.size) : Array α :=
   a'.set (size_set a i v₂ ▸ j) v₁
 
 /--
-Swaps two entries in an array, or panics if either index is out of bounds.
+Swaps two entries in an array, or returns the array unchanged if either index is out of bounds.
 
 This will perform the update destructively provided that `a` has a reference
 count of 1 when called.
@@ -174,7 +181,7 @@ def modifyOp (self : Array α) (idx : Nat) (f : α → α) : Array α :=
 
   This kind of low level trick can be removed with a little bit of compiler support. For example, if the compiler simplifies `as.size < usizeSz` to true. -/
 @[inline] unsafe def forInUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
-  let sz := USize.ofNat as.size
+  let sz := as.usize
   let rec @[specialize] loop (i : USize) (b : β) : m β := do
     if i < sz then
       let a := as.uget i lcProof
@@ -280,7 +287,7 @@ def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
 /-- See comment at `forInUnsafe` -/
 @[inline]
 unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
-  let sz := USize.ofNat as.size
+  let sz := as.usize
   let rec @[specialize] map (i : USize) (r : Array NonScalar) : m (Array PNonScalar.{v}) := do
     if i < sz then
      let v    := r.uget i lcProof

src/Init/Data/Array/Lemmas.lean

@@ -6,7 +6,7 @@ Authors: Mario Carneiro
 prelude
 import Init.Data.Nat.MinMax
 import Init.Data.Nat.Lemmas
-import Init.Data.List.Lemmas
+import Init.Data.List.Monadic
 import Init.Data.Fin.Basic
 import Init.Data.Array.Mem
 import Init.TacticsExtra

src/Init/Data/Array/TakeDrop.lean

@@ -5,7 +5,7 @@ Authors: Markus Himmel
 -/
 prelude
 import Init.Data.Array.Lemmas
-import Init.Data.List.TakeDrop
+import Init.Data.List.Nat.TakeDrop
 
 namespace Array
 

src/Init/Data/BitVec/Bitblast.lean

@@ -98,6 +98,37 @@ theorem carry_succ (i : Nat) (x y : BitVec w) (c : Bool) :
     exact mod_two_pow_add_mod_two_pow_add_bool_lt_two_pow_succ ..
   cases x.toNat.testBit i <;> cases y.toNat.testBit i <;> (simp; omega)
 
+/--
+If `x &&& y = 0`, then the carry bit `(x + y + 0)` is always `false` for any index `i`.
+Intuitively, this is because a carry is only produced when at least two of `x`, `y`, and the
+previous carry are true. However, since `x &&& y = 0`, at most one of `x, y` can be true,
+and thus we never have a previous carry, which means that the sum cannot produce a carry.
+-/
+theorem carry_of_and_eq_zero {x y : BitVec w} (h : x &&& y = 0#w) : carry i x y false = false := by
+  induction i with
+  | zero => simp
+  | succ i ih =>
+    replace h := congrArg (·.getLsb i) h
+    simp_all [carry_succ]
+
+/-- The final carry bit when computing `x + y + c` is `true` iff `x.toNat + y.toNat + c.toNat ≥ 2^w`. -/
+theorem carry_width {x y : BitVec w} :
+    carry w x y c = decide (x.toNat + y.toNat + c.toNat ≥ 2^w) := by
+  simp [carry]
+
+/--
+If `x &&& y = 0`, then addition does not overflow, and thus `(x + y).toNat = x.toNat + y.toNat`.
+-/
+theorem toNat_add_of_and_eq_zero {x y : BitVec w} (h : x &&& y = 0#w) :
+    (x + y).toNat = x.toNat + y.toNat := by
+  rw [toNat_add]
+  apply Nat.mod_eq_of_lt
+  suffices ¬ decide (x.toNat + y.toNat + false.toNat ≥ 2^w) by
+    simp only [decide_eq_true_eq] at this
+    omega
+  rw [← carry_width]
+  simp [not_eq_true, carry_of_and_eq_zero h]
+
 /-- Carry function for bitwise addition. -/
 def adcb (x y c : Bool) : Bool × Bool := (atLeastTwo x y c, Bool.xor x (Bool.xor y c))
 
@@ -290,7 +321,7 @@ theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow (x : BitVec w
         simp [hik', hik'']
   · ext k
     simp
-    omega
+    by_cases hi : x.getLsb i <;> simp [hi] <;> omega
 
 /--
 Recurrence lemma: multiplying `l` with the first `s` bits of `r` is the
@@ -314,7 +345,7 @@ theorem mulRec_eq_mul_signExtend_truncate (l r : BitVec w) (s : Nat) :
     have heq :
       (if r.getLsb (s' + 1) = true then l <<< (s' + 1) else 0) =
         (l * (r &&& (BitVec.twoPow w (s' + 1)))) := by
-      simp only [ofNat_eq_ofNat, and_twoPow_eq]
+      simp only [ofNat_eq_ofNat, and_twoPow]
       by_cases hr : r.getLsb (s' + 1) <;> simp [hr]
     rw [heq, ← BitVec.mul_add, ← zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow]
 
@@ -326,4 +357,78 @@ theorem getLsb_mul (x y : BitVec w) (i : Nat) :
   · simp
   · omega
 
+/-! ## shiftLeft recurrence for bitblasting -/
+
+/--
+`shiftLeftRec x y n` shifts `x` to the left by the first `n` bits of `y`.
+
+The theorem `shiftLeft_eq_shiftLeftRec` proves the equivalence of `(x <<< y)` and `shiftLeftRec`.
+
+Together with equations `shiftLeftRec_zero`, `shiftLeftRec_succ`,
+this allows us to unfold `shiftLeft` into a circuit for bitblasting.
+ -/
+def shiftLeftRec (x : BitVec w₁) (y : BitVec w₂) (n : Nat) : BitVec w₁ :=
+  let shiftAmt := (y &&& (twoPow w₂ n))
+  match n with
+  | 0 => x <<< shiftAmt
+  | n + 1 => (shiftLeftRec x y n) <<< shiftAmt
+
+@[simp]
+theorem shiftLeftRec_zero {x : BitVec w₁} {y : BitVec w₂} :
+    shiftLeftRec x y 0 = x <<< (y &&& twoPow w₂ 0)  := by
+  simp [shiftLeftRec]
+
+@[simp]
+theorem shiftLeftRec_succ {x : BitVec w₁} {y : BitVec w₂} :
+    shiftLeftRec x y (n + 1) = (shiftLeftRec x y n) <<< (y &&& twoPow w₂ (n + 1)) := by
+  simp [shiftLeftRec]
+
+/--
+If `y &&& z = 0`, `x <<< (y ||| z) = x <<< y <<< z`.
+This follows as `y &&& z = 0` implies `y ||| z = y + z`,
+and thus `x <<< (y ||| z) = x <<< (y + z) = x <<< y <<< z`.
+-/
+theorem shiftLeft_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
+    (h : y &&& z = 0#w₂) :
+    x <<< (y ||| z) = x <<< y <<< z := by
+  rw [← add_eq_or_of_and_eq_zero _ _ h,
+    shiftLeft_eq', toNat_add_of_and_eq_zero h]
+  simp [shiftLeft_add]
+
+/--
+`shiftLeftRec x y n` shifts `x` to the left by the first `n` bits of `y`.
+-/
+theorem shiftLeftRec_eq {x : BitVec w₁} {y : BitVec w₂} {n : Nat} :
+    shiftLeftRec x y n = x <<< (y.truncate (n + 1)).zeroExtend w₂ := by
+  induction n generalizing x y
+  case zero =>
+    ext i
+    simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, truncate_one]
+    suffices (y &&& 1#w₂) = zeroExtend w₂ (ofBool (y.getLsb 0)) by simp [this]
+    ext i
+    by_cases h : (↑i : Nat) = 0
+    · simp [h, Bool.and_comm]
+    · simp [h]; omega
+  case succ n ih =>
+    simp only [shiftLeftRec_succ, and_twoPow]
+    rw [ih]
+    by_cases h : y.getLsb (n + 1)
+    · simp only [h, ↓reduceIte]
+      rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsb_true h,
+        shiftLeft_or_of_and_eq_zero]
+      simp
+    · simp only [h, false_eq_true, ↓reduceIte, shiftLeft_zero']
+      rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsb_false (i := n + 1)]
+      simp [h]
+
+/--
+Show that `x <<< y` can be written in terms of `shiftLeftRec`.
+This can be unfolded in terms of `shiftLeftRec_zero`, `shiftLeftRec_succ` for bitblasting.
+-/
+theorem shiftLeft_eq_shiftLeftRec (x : BitVec w₁) (y : BitVec w₂) :
+    x <<< y = shiftLeftRec x y (w₂ - 1) := by
+  rcases w₂ with rfl | w₂
+  · simp [of_length_zero]
+  · simp [shiftLeftRec_eq]
+
 end BitVec

src/Init/Data/BitVec/Lemmas.lean

@@ -436,6 +436,12 @@ theorem zeroExtend_ofNat_one_eq_ofNat_one_of_lt {v w : Nat} (hv : 0 < v) :
   have hv := Nat.testBit_one_eq_true_iff_self_eq_zero.mp hi₁
   omega
 
+/-- Truncating to width 1 produces a bitvector equal to the least significant bit. -/
+theorem truncate_one {x : BitVec w} :
+    x.truncate 1 = ofBool (x.getLsb 0) := by
+  ext i
+  simp [show i = 0 by omega]
+
 /-! ## extractLsb -/
 
 @[simp]
@@ -531,6 +537,11 @@ theorem and_assoc (x y z : BitVec w) :
   ext i
   simp [Bool.and_assoc]
 
+theorem and_comm (x y : BitVec w) :
+    x &&& y = y &&& x := by
+  ext i
+  simp [Bool.and_comm]
+
 /-! ### xor -/
 
 @[simp] theorem toNat_xor (x y : BitVec v) :
@@ -626,6 +637,10 @@ theorem shiftLeft_zero_eq (x : BitVec w) : x <<< 0 = x := by
   apply eq_of_toNat_eq
   simp
 
+@[simp]
+theorem zero_shiftLeft (n : Nat) : 0#w <<< n = 0#w := by
+  simp [bv_toNat]
+
 @[simp] theorem getLsb_shiftLeft (x : BitVec m) (n) :
     getLsb (x <<< n) i = (decide (i < m) && !decide (i < n) && getLsb x (i - n)) := by
   rw [← testBit_toNat, getLsb]
@@ -691,6 +706,22 @@ theorem shiftLeft_shiftLeft {w : Nat} (x : BitVec w) (n m : Nat) :
     (x <<< n) <<< m = x <<< (n + m) := by
   rw [shiftLeft_add]
 
+/-! ### shiftLeft reductions from BitVec to Nat -/
+
+@[simp]
+theorem shiftLeft_eq' {x : BitVec w₁} {y : BitVec w₂} : x <<< y = x <<< y.toNat := by rfl
+
+@[simp]
+theorem shiftLeft_zero' {x : BitVec w₁} : x <<< 0#w₂ = x := by simp
+
+theorem shiftLeft_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {z : BitVec w₃} :
+    x <<< y <<< z = x <<< (y.toNat + z.toNat) := by
+  simp [shiftLeft_add]
+
+theorem getLsb_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {i : Nat} :
+    (x <<< y).getLsb i = (decide (i < w₁) && !decide (i < y.toNat) && x.getLsb (i - y.toNat)) := by
+  simp [shiftLeft_eq', getLsb_shiftLeft]
+
 /-! ### ushiftRight -/
 
 @[simp, bv_toNat] theorem toNat_ushiftRight (x : BitVec n) (i : Nat) :
@@ -1452,12 +1483,18 @@ theorem getLsb_twoPow (i j : Nat) : (twoPow w i).getLsb j = ((i < w) && (i = j))
           simp at hi
         simp_all
 
-theorem and_twoPow_eq (x : BitVec w) (i : Nat) :
+@[simp]
+theorem and_twoPow (x : BitVec w) (i : Nat) :
     x &&& (twoPow w i) = if x.getLsb i then twoPow w i else 0#w := by
   ext j
   simp only [getLsb_and, getLsb_twoPow]
   by_cases hj : i = j <;> by_cases hx : x.getLsb i <;> simp_all
 
+@[simp]
+theorem twoPow_and (x : BitVec w) (i : Nat) :
+    (twoPow w i) &&& x = if x.getLsb i then twoPow w i else 0#w := by
+  rw [BitVec.and_comm, and_twoPow]
+
 @[simp]
 theorem mul_twoPow_eq_shiftLeft (x : BitVec w) (i : Nat) :
     x * (twoPow w i) = x <<< i := by
@@ -1471,6 +1508,14 @@ theorem mul_twoPow_eq_shiftLeft (x : BitVec w) (i : Nat) :
       apply Nat.pow_dvd_pow 2 (by omega)
     simp [Nat.mul_mod, hpow]
 
+theorem twoPow_zero {w : Nat} : twoPow w 0 = 1#w := by
+  apply eq_of_toNat_eq
+  simp
+
+@[simp]
+theorem getLsb_one {w i : Nat} : (1#w).getLsb i = (decide (0 < w) && decide (0 = i)) := by
+  rw [← twoPow_zero, getLsb_twoPow]
+
 /- ### zeroExtend, truncate, and bitwise operations -/
 
 /--

src/Init/Data/ByteArray/Basic.lean

@@ -37,6 +37,10 @@ def push : ByteArray → UInt8 → ByteArray
 def size : (@& ByteArray) → Nat
   | ⟨bs⟩ => bs.size
 
+@[extern "lean_sarray_size", simp]
+def usize (a : @& ByteArray) : USize :=
+  a.size.toUSize
+
 @[extern "lean_byte_array_uget"]
 def uget : (a : @& ByteArray) → (i : USize) → i.toNat < a.size → UInt8
   | ⟨bs⟩, i, h => bs[i]
@@ -119,7 +123,7 @@ def toList (bs : ByteArray) : List UInt8 :=
   TODO: avoid code duplication in the future after we improve the compiler.
 -/
 @[inline] unsafe def forInUnsafe {β : Type v} {m : Type v → Type w} [Monad m] (as : ByteArray) (b : β) (f : UInt8 → β → m (ForInStep β)) : m β :=
-  let sz := USize.ofNat as.size
+  let sz := as.usize
   let rec @[specialize] loop (i : USize) (b : β) : m β := do
     if i < sz then
       let a := as.uget i lcProof

src/Init/Data/FloatArray/Basic.lean

@@ -37,6 +37,10 @@ def push : FloatArray → Float → FloatArray
 def size : (@& FloatArray) → Nat
   | ⟨ds⟩ => ds.size
 
+@[extern "lean_sarray_size", simp]
+def usize (a : @& FloatArray) : USize :=
+  a.size.toUSize
+
 @[extern "lean_float_array_uget"]
 def uget : (a : @& FloatArray) → (i : USize) → i.toNat < a.size → Float
   | ⟨ds⟩, i, h => ds[i]
@@ -90,7 +94,7 @@ partial def toList (ds : FloatArray) : List Float :=
 -/
 -- TODO: avoid code duplication in the future after we improve the compiler.
 @[inline] unsafe def forInUnsafe {β : Type v} {m : Type v → Type w} [Monad m] (as : FloatArray) (b : β) (f : Float → β → m (ForInStep β)) : m β :=
-  let sz := USize.ofNat as.size
+  let sz := as.usize
   let rec @[specialize] loop (i : USize) (b : β) : m β := do
     if i < sz then
       let a := as.uget i lcProof

src/Init/Data/List.lean

@@ -4,11 +4,20 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
+import Init.Data.List.Attach
 import Init.Data.List.Basic
 import Init.Data.List.BasicAux
 import Init.Data.List.Control
-import Init.Data.List.Lemmas
-import Init.Data.List.Attach
+import Init.Data.List.Count
+import Init.Data.List.Erase
+import Init.Data.List.Find
 import Init.Data.List.Impl
-import Init.Data.List.TakeDrop
+import Init.Data.List.Lemmas
+import Init.Data.List.MinMax
+import Init.Data.List.Monadic
+import Init.Data.List.Nat
 import Init.Data.List.Notation
+import Init.Data.List.Pairwise
+import Init.Data.List.Sublist
+import Init.Data.List.TakeDrop
+import Init.Data.List.Zip

src/Init/Data/List/Attach.lean

@@ -4,7 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
 prelude
-import Init.Data.List.Lemmas
+import Init.Data.List.Count
+import Init.Data.Subtype
 
 namespace List
 
@@ -44,3 +45,155 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
   | nil, hL' => rfl
   | cons _ L', hL' => congrArg _ <| go L' fun _ hx => hL' (.tail _ hx)
   exact go L h'
+
+@[simp] theorem attach_nil : ([] : List α).attach = [] := rfl
+
+@[simp]
+theorem pmap_eq_map (p : α → Prop) (f : α → β) (l : List α) (H) :
+    @pmap _ _ p (fun a _ => f a) l H = map f l := by
+  induction l
+  · rfl
+  · simp only [*, pmap, map]
+
+theorem pmap_congr {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (l : List α) {H₁ H₂}
+    (h : ∀ a ∈ l, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f l H₁ = pmap g l H₂ := by
+  induction l with
+  | nil => rfl
+  | cons x l ih => rw [pmap, pmap, h _ (mem_cons_self _ _), ih fun a ha => h a (mem_cons_of_mem _ ha)]
+
+theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l H) :
+    map g (pmap f l H) = pmap (fun a h => g (f a h)) l H := by
+  induction l
+  · rfl
+  · simp only [*, pmap, map]
+
+theorem pmap_map {p : β → Prop} (g : ∀ b, p b → γ) (f : α → β) (l H) :
+    pmap g (map f l) H = pmap (fun a h => g (f a) h) l fun a h => H _ (mem_map_of_mem _ h) := by
+  induction l
+  · rfl
+  · simp only [*, pmap, map]
+
+theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
+    pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
+  rw [attach, attachWith, map_pmap]; exact pmap_congr l fun _ _ _ _ => rfl
+
+theorem attach_map_coe (l : List α) (f : α → β) :
+    (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
+  rw [attach, attachWith, map_pmap]; exact pmap_eq_map _ _ _ _
+
+theorem attach_map_val (l : List α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
+  attach_map_coe _ _
+
+@[simp]
+theorem attach_map_subtype_val (l : List α) : l.attach.map Subtype.val = l :=
+  (attach_map_coe _ _).trans l.map_id
+
+theorem countP_attach (l : List α) (p : α → Bool) : l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
+  simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attach_map_subtype_val]
+
+@[simp]
+theorem count_attach [DecidableEq α] (l : List α) (a : {x // x ∈ l}) : l.attach.count a = l.count ↑a :=
+  Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attach _ _
+
+@[simp]
+theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
+  | ⟨a, h⟩ => by
+    have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
+    rcases this with ⟨⟨_, _⟩, m, rfl⟩
+    exact m
+
+@[simp]
+theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
+    b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
+  simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, Subtype.exists, eq_comm]
+
+@[simp]
+theorem length_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : length (pmap f l H) = length l := by
+  induction l
+  · rfl
+  · simp only [*, pmap, length]
+
+@[simp]
+theorem length_attach (L : List α) : L.attach.length = L.length :=
+  length_pmap
+
+@[simp]
+theorem pmap_eq_nil {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = [] ↔ l = [] := by
+  rw [← length_eq_zero, length_pmap, length_eq_zero]
+
+@[simp]
+theorem attach_eq_nil (l : List α) : l.attach = [] ↔ l = [] :=
+  pmap_eq_nil
+
+theorem getLast_pmap (p : α → Prop) (f : ∀ a, p a → β) (l : List α)
+    (hl₁ : ∀ a ∈ l, p a) (hl₂ : l ≠ []) :
+    (l.pmap f hl₁).getLast (mt List.pmap_eq_nil.1 hl₂) =
+      f (l.getLast hl₂) (hl₁ _ (List.getLast_mem hl₂)) := by
+  induction l with
+  | nil => apply (hl₂ rfl).elim
+  | cons l_hd l_tl l_ih =>
+    by_cases hl_tl : l_tl = []
+    · simp [hl_tl]
+    · simp only [pmap]
+      rw [getLast_cons, l_ih _ hl_tl]
+      simp only [getLast_cons hl_tl]
+
+theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
+    (pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (getElem?_mem H) := by
+  induction l generalizing n with
+  | nil => simp
+  | cons hd tl hl =>
+    rcases n with ⟨n⟩
+    · simp only [Option.pmap]
+      split <;> simp_all
+    · simp only [hl, pmap, Option.pmap, getElem?_cons_succ]
+      split <;> rename_i h₁ _ <;> split <;> rename_i h₂ _
+      · simp_all
+      · simp at h₂
+        simp_all
+      · simp_all
+      · simp_all
+
+theorem get?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : Nat) :
+    get? (pmap f l h) n = Option.pmap f (get? l n) fun x H => h x (get?_mem H) := by
+  simp only [get?_eq_getElem?]
+  simp [getElem?_pmap, h]
+
+theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) {n : Nat}
+    (hn : n < (pmap f l h).length) :
+    (pmap f l h)[n] =
+      f (l[n]'(@length_pmap _ _ p f l h ▸ hn))
+        (h _ (getElem_mem l n (@length_pmap _ _ p f l h ▸ hn))) := by
+  induction l generalizing n with
+  | nil =>
+    simp only [length, pmap] at hn
+    exact absurd hn (Nat.not_lt_of_le n.zero_le)
+  | cons hd tl hl =>
+    cases n
+    · simp
+    · simp [hl]
+
+theorem get_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) {n : Nat}
+    (hn : n < (pmap f l h).length) :
+    get (pmap f l h) ⟨n, hn⟩ =
+      f (get l ⟨n, @length_pmap _ _ p f l h ▸ hn⟩)
+        (h _ (get_mem l n (@length_pmap _ _ p f l h ▸ hn))) := by
+  simp only [get_eq_getElem]
+  simp [getElem_pmap]
+
+theorem pmap_append {p : ι → Prop} (f : ∀ a : ι, p a → α) (l₁ l₂ : List ι)
+    (h : ∀ a ∈ l₁ ++ l₂, p a) :
+    (l₁ ++ l₂).pmap f h =
+      (l₁.pmap f fun a ha => h a (mem_append_left l₂ ha)) ++
+        l₂.pmap f fun a ha => h a (mem_append_right l₁ ha) := by
+  induction l₁ with
+  | nil => rfl
+  | cons _ _ ih =>
+    dsimp only [pmap, cons_append]
+    rw [ih]
+
+theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ l₂ : List α)
+    (h₁ : ∀ a ∈ l₁, p a) (h₂ : ∀ a ∈ l₂, p a) :
+    ((l₁ ++ l₂).pmap f fun a ha => (List.mem_append.1 ha).elim (h₁ a) (h₂ a)) =
+      l₁.pmap f h₁ ++ l₂.pmap f h₂ :=
+  pmap_append f l₁ l₂ _

src/Init/Data/List/Basic.lean

@@ -27,24 +27,32 @@ Recall that `length`, `get`, `set`, `foldl`, and `concat` have already been defi
 The operations are organized as follow:
 * Equality: `beq`, `isEqv`.
 * Lexicographic ordering: `lt`, `le`, and instances.
+* Head and tail operators: `head`, `head?`, `headD?`, `tail`, `tail?`, `tailD`.
 * Basic operations:
-  `map`, `filter`, `filterMap`, `foldr`, `append`, `join`, `pure`, `bind`, `replicate`, and `reverse`.
+  `map`, `filter`, `filterMap`, `foldr`, `append`, `join`, `pure`, `bind`, `replicate`, and
+  `reverse`.
+* Additional functions defined in terms of these: `leftpad`, `rightPad`, and `reduceOption`.
 * List membership: `isEmpty`, `elem`, `contains`, `mem` (and the `∈` notation),
   and decidability for predicates quantifying over membership in a `List`.
 * Sublists: `take`, `drop`, `takeWhile`, `dropWhile`, `partition`, `dropLast`,
-  `isPrefixOf`, `isPrefixOf?`, `isSuffixOf`, `isSuffixOf?`, `Subset`, `Sublist`, `rotateLeft` and `rotateRight`.
-* Manipulating elements: `replace`, `insert`, `erase`, `eraseP`, `eraseIdx`, `find?`, `findSome?`, and `lookup`.
+  `isPrefixOf`, `isPrefixOf?`, `isSuffixOf`, `isSuffixOf?`, `Subset`, `Sublist`,
+  `rotateLeft` and `rotateRight`.
+* Manipulating elements: `replace`, `insert`, `erase`, `eraseP`, `eraseIdx`.
+* Finding elements: `find?`, `findSome?`, `findIdx`, `indexOf`, `findIdx?`, `indexOf?`,
+ `countP`, `count`, and `lookup`.
 * Logic: `any`, `all`, `or`, and `and`.
 * Zippers: `zipWith`, `zip`, `zipWithAll`, and `unzip`.
 * Ranges and enumeration: `range`, `iota`, `enumFrom`, and `enum`.
 * Minima and maxima: `minimum?` and `maximum?`.
-* Other functions: `intersperse`, `intercalate`, `eraseDups`, `eraseReps`, `span`, `groupBy`, `removeAll`
+* Other functions: `intersperse`, `intercalate`, `eraseDups`, `eraseReps`, `span`, `groupBy`,
+  `removeAll`
   (currently these functions are mostly only used in meta code,
   and do not have API suitable for verification).
 
-Further operations are defined in `Init.Data.List.BasicAux` (because they use `Array` in their implementations), namely:
+Further operations are defined in `Init.Data.List.BasicAux`
+(because they use `Array` in their implementations), namely:
 * Variant getters: `get!`, `get?`, `getD`, `getLast`, `getLast!`, `getLast?`, and `getLastD`.
-* Head and tail: `head`, `head!`, `head?`, `headD`, `tail!`, `tail?`, and `tailD`.
+* Head and tail: `head!`, `tail!`.
 * Other operations on sublists: `partitionMap`, `rotateLeft`, and `rotateRight`.
 -/
 
@@ -315,6 +323,16 @@ def headD : (as : List α) → (fallback : α) → α
 @[simp 1100] theorem headD_nil : @headD α [] d = d := rfl
 @[simp 1100] theorem headD_cons : @headD α (a::l) d = a := rfl
 
+/-! ### tail -/
+
+/-- Get the tail of a nonempty list, or return `[]` for `[]`. -/
+def tail : List α → List α
+  | []    => []
+  | _::as => as
+
+@[simp] theorem tail_nil : @tail α [] = [] := rfl
+@[simp] theorem tail_cons : @tail α (a::as) = as := rfl
+
 /-! ### tail? -/
 
 /--
@@ -577,6 +595,28 @@ theorem replicate_succ (a : α) (n) : replicate (n+1) a = a :: replicate n a :=
   | zero => simp
   | succ n ih => simp only [ih, replicate_succ, length_cons, Nat.succ_eq_add_one]
 
+/-! ## Additional functions -/
+
+/-! ### leftpad and rightpad -/
+
+/--
+Pads `l : List α` on the left with repeated occurrences of `a : α` until it is of length `n`.
+If `l` is initially larger than `n`, just return `l`.
+-/
+def leftpad (n : Nat) (a : α) (l : List α) : List α := replicate (n - length l) a ++ l
+
+/--
+Pads `l : List α` on the right with repeated occurrences of `a : α` until it is of length `n`.
+If `l` is initially larger than `n`, just return `l`.
+-/
+def rightpad (n : Nat) (a : α) (l : List α) : List α := l ++ replicate (n - length l) a
+
+/-! ### reduceOption -/
+
+/-- Drop `none`s from a list, and replace each remaining `some a` with `a`. -/
+@[inline] def reduceOption {α} : List (Option α) → List α :=
+  List.filterMap id
+
 /-! ## List membership
 
 * `L.contains a : Bool` determines, using a `[BEq α]` instance, whether `L` contains an element `· == a`.
@@ -719,7 +759,7 @@ def take : Nat → List α → List α
 
 @[simp] theorem take_nil : ([] : List α).take i = [] := by cases i <;> rfl
 @[simp] theorem take_zero (l : List α) : l.take 0 = [] := rfl
-@[simp] theorem take_cons_succ : (a::as).take (i+1) = a :: as.take i := rfl
+@[simp] theorem take_succ_cons : (a::as).take (i+1) = a :: as.take i := rfl
 
 /-! ### drop -/
 
@@ -826,46 +866,6 @@ def dropLast {α} : List α → List α
     have ih := length_dropLast_cons b bs
     simp [dropLast, ih]
 
-/-! ### isPrefixOf -/
-
-/--  `isPrefixOf l₁ l₂` returns `true` Iff `l₁` is a prefix of `l₂`.
-That is, there exists a `t` such that `l₂ == l₁ ++ t`. -/
-def isPrefixOf [BEq α] : List α → List α → Bool
-  | [],    _     => true
-  | _,     []    => false
-  | a::as, b::bs => a == b && isPrefixOf as bs
-
-@[simp] theorem isPrefixOf_nil_left [BEq α] : isPrefixOf ([] : List α) l = true := by
-  simp [isPrefixOf]
-@[simp] theorem isPrefixOf_cons_nil [BEq α] : isPrefixOf (a::as) ([] : List α) = false := rfl
-theorem isPrefixOf_cons₂ [BEq α] {a : α} :
-    isPrefixOf (a::as) (b::bs) = (a == b && isPrefixOf as bs) := rfl
-
-/-! ### isPrefixOf? -/
-
-/-- `isPrefixOf? l₁ l₂` returns `some t` when `l₂ == l₁ ++ t`. -/
-def isPrefixOf? [BEq α] : List α → List α → Option (List α)
-  | [], l₂ => some l₂
-  | _, [] => none
-  | (x₁ :: l₁), (x₂ :: l₂) =>
-    if x₁ == x₂ then isPrefixOf? l₁ l₂ else none
-
-/-! ### isSuffixOf -/
-
-/--  `isSuffixOf l₁ l₂` returns `true` Iff `l₁` is a suffix of `l₂`.
-That is, there exists a `t` such that `l₂ == t ++ l₁`. -/
-def isSuffixOf [BEq α] (l₁ l₂ : List α) : Bool :=
-  isPrefixOf l₁.reverse l₂.reverse
-
-@[simp] theorem isSuffixOf_nil_left [BEq α] : isSuffixOf ([] : List α) l = true := by
-  simp [isSuffixOf]
-
-/-! ### isSuffixOf? -/
-
-/-- `isSuffixOf? l₁ l₂` returns `some t` when `l₂ == t ++ l₁`.-/
-def isSuffixOf? [BEq α] (l₁ l₂ : List α) : Option (List α) :=
-  Option.map List.reverse <| isPrefixOf? l₁.reverse l₂.reverse
-
 /-! ### Subset -/
 
 /--
@@ -900,6 +900,68 @@ def isSublist [BEq α] : List α → List α → Bool
     then tl₁.isSublist tl₂
     else l₁.isSublist tl₂
 
+/-! ### IsPrefix / isPrefixOf / isPrefixOf? -/
+
+/--
+`IsPrefix l₁ l₂`, or `l₁ <+: l₂`, means that `l₁` is a prefix of `l₂`,
+that is, `l₂` has the form `l₁ ++ t` for some `t`.
+-/
+def IsPrefix (l₁ : List α) (l₂ : List α) : Prop := Exists fun t => l₁ ++ t = l₂
+
+@[inherit_doc] infixl:50 " <+: " => IsPrefix
+
+/--  `isPrefixOf l₁ l₂` returns `true` Iff `l₁` is a prefix of `l₂`.
+That is, there exists a `t` such that `l₂ == l₁ ++ t`. -/
+def isPrefixOf [BEq α] : List α → List α → Bool
+  | [],    _     => true
+  | _,     []    => false
+  | a::as, b::bs => a == b && isPrefixOf as bs
+
+@[simp] theorem isPrefixOf_nil_left [BEq α] : isPrefixOf ([] : List α) l = true := by
+  simp [isPrefixOf]
+@[simp] theorem isPrefixOf_cons_nil [BEq α] : isPrefixOf (a::as) ([] : List α) = false := rfl
+theorem isPrefixOf_cons₂ [BEq α] {a : α} :
+    isPrefixOf (a::as) (b::bs) = (a == b && isPrefixOf as bs) := rfl
+
+/-- `isPrefixOf? l₁ l₂` returns `some t` when `l₂ == l₁ ++ t`. -/
+def isPrefixOf? [BEq α] : List α → List α → Option (List α)
+  | [], l₂ => some l₂
+  | _, [] => none
+  | (x₁ :: l₁), (x₂ :: l₂) =>
+    if x₁ == x₂ then isPrefixOf? l₁ l₂ else none
+
+/-! ### IsSuffix / isSuffixOf / isSuffixOf? -/
+
+/--  `isSuffixOf l₁ l₂` returns `true` Iff `l₁` is a suffix of `l₂`.
+That is, there exists a `t` such that `l₂ == t ++ l₁`. -/
+def isSuffixOf [BEq α] (l₁ l₂ : List α) : Bool :=
+  isPrefixOf l₁.reverse l₂.reverse
+
+@[simp] theorem isSuffixOf_nil_left [BEq α] : isSuffixOf ([] : List α) l = true := by
+  simp [isSuffixOf]
+
+/-- `isSuffixOf? l₁ l₂` returns `some t` when `l₂ == t ++ l₁`.-/
+def isSuffixOf? [BEq α] (l₁ l₂ : List α) : Option (List α) :=
+  Option.map List.reverse <| isPrefixOf? l₁.reverse l₂.reverse
+
+/--
+`IsSuffix l₁ l₂`, or `l₁ <:+ l₂`, means that `l₁` is a suffix of `l₂`,
+that is, `l₂` has the form `t ++ l₁` for some `t`.
+-/
+def IsSuffix (l₁ : List α) (l₂ : List α) : Prop := Exists fun t => t ++ l₁ = l₂
+
+@[inherit_doc] infixl:50 " <:+ " => IsSuffix
+
+/-! ### IsInfix -/
+
+/--
+`IsInfix l₁ l₂`, or `l₁ <:+: l₂`, means that `l₁` is a contiguous
+substring of `l₂`, that is, `l₂` has the form `s ++ l₁ ++ t` for some `s, t`.
+-/
+def IsInfix (l₁ : List α) (l₂ : List α) : Prop := Exists fun s => Exists fun t => s ++ l₁ ++ t = l₂
+
+@[inherit_doc] infixl:50 " <:+: " => IsInfix
+
 /-! ### rotateLeft -/
 
 /--
@@ -1058,6 +1120,8 @@ def eraseIdx : List α → Nat → List α
 @[simp] theorem eraseIdx_cons_zero : (a::as).eraseIdx 0 = as := rfl
 @[simp] theorem eraseIdx_cons_succ : (a::as).eraseIdx (i+1) = a :: as.eraseIdx i := rfl
 
+/-! Finding elements -/
+
 /-! ### find? -/
 
 /--
@@ -1095,6 +1159,50 @@ theorem findSome?_cons {f : α → Option β} :
     (a::as).findSome? f = match f a with | some b => some b | none => as.findSome? f :=
   rfl
 
+/-! ### findIdx -/
+
+/-- Returns the index of the first element satisfying `p`, or the length of the list otherwise. -/
+@[inline] def findIdx (p : α → Bool) (l : List α) : Nat := go l 0 where
+  /-- Auxiliary for `findIdx`: `findIdx.go p l n = findIdx p l + n` -/
+  @[specialize] go : List α → Nat → Nat
+  | [], n => n
+  | a :: l, n => bif p a then n else go l (n + 1)
+
+@[simp] theorem findIdx_nil {α : Type _} (p : α → Bool) : [].findIdx p = 0 := rfl
+
+/-! ### indexOf -/
+
+/-- Returns the index of the first element equal to `a`, or the length of the list otherwise. -/
+def indexOf [BEq α] (a : α) : List α → Nat := findIdx (· == a)
+
+@[simp] theorem indexOf_nil [BEq α] : ([] : List α).indexOf x = 0 := rfl
+
+/-! ### findIdx? -/
+
+/-- Return the index of the first occurrence of an element satisfying `p`. -/
+def findIdx? (p : α → Bool) : List α → (start : Nat := 0) → Option Nat
+| [], _ => none
+| a :: l, i => if p a then some i else findIdx? p l (i + 1)
+
+/-! ### indexOf? -/
+
+/-- Return the index of the first occurrence of `a` in the list. -/
+@[inline] def indexOf? [BEq α] (a : α) : List α → Option Nat := findIdx? (· == a)
+
+/-! ### countP -/
+
+/-- `countP p l` is the number of elements of `l` that satisfy `p`. -/
+@[inline] def countP (p : α → Bool) (l : List α) : Nat := go l 0 where
+  /-- Auxiliary for `countP`: `countP.go p l acc = countP p l + acc`. -/
+  @[specialize] go : List α → Nat → Nat
+  | [], acc => acc
+  | x :: xs, acc => bif p x then go xs (acc + 1) else go xs acc
+
+/-! ### count -/
+
+/-- `count a l` is the number of occurrences of `a` in `l`. -/
+@[inline] def count [BEq α] (a : α) : List α → Nat := countP (· == a)
+
 /-! ### lookup -/
 
 /--
@@ -1236,6 +1344,14 @@ def unzip : List (α × β) → List α × List β
 
 /-! ## Ranges and enumeration -/
 
+/-- Sum of a list of natural numbers. -/
+-- This is not in the `List` namespace as later `List.sum` will be defined polymorphically.
+protected def _root_.Nat.sum (l : List Nat) : Nat := l.foldr (·+·) 0
+
+@[simp] theorem _root_.Nat.sum_nil : Nat.sum ([] : List Nat) = 0 := rfl
+@[simp] theorem _root_.Nat.sum_cons (a : Nat) (l : List Nat) :
+    Nat.sum (a::l) = a + Nat.sum l := rfl
+
 /-! ### range -/
 
 /--
@@ -1251,6 +1367,14 @@ where
 
 @[simp] theorem range_zero : range 0 = [] := rfl
 
+/-! ### range' -/
+
+/-- `range' start len step` is the list of numbers `[start, start+step, ..., start+(len-1)*step]`.
+  It is intended mainly for proving properties of `range` and `iota`. -/
+def range' : (start len : Nat) → (step : Nat := 1) → List Nat
+  | _, 0, _ => []
+  | s, n+1, step => s :: range' (s+step) n step
+
 /-! ### iota -/
 
 /--

src/Init/Data/List/Control.lean

@@ -127,12 +127,12 @@ results `y` for which `f x` returns `some y`.
 @[inline]
 def filterMapM {m : Type u → Type v} [Monad m] {α β : Type u} (f : α → m (Option β)) (as : List α) : m (List β) :=
   let rec @[specialize] loop
-    | [],     bs => pure bs
+    | [],     bs => pure bs.reverse
     | a :: as, bs => do
       match (← f a) with
       | none   => loop as bs
       | some b => loop as (b::bs)
-  loop as.reverse []
+  loop as []
 
 /--
 Folds a monadic function over a list from left to right:
@@ -227,6 +227,8 @@ def findSomeM? {m : Type u → Type v} [Monad m] {α : Type w} {β : Type u} (f
 instance : ForIn m (List α) α where
   forIn := List.forIn
 
+@[simp] theorem forIn_eq_forIn [Monad m] : @List.forIn α β m _ = forIn := rfl
+
 @[simp] theorem forIn_nil [Monad m] (f : α → β → m (ForInStep β)) (b : β) : forIn [] b f = pure b :=
   rfl
 

src/Init/Data/List/Count.lean

@@ -0,0 +1,242 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Sublist
+
+/-!
+# Lemmas about `List.countP` and `List.count`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ### countP -/
+section countP
+
+variable (p q : α → Bool)
+
+@[simp] theorem countP_nil : countP p [] = 0 := rfl
+
+protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
+  induction l generalizing n with
+  | nil => rfl
+  | cons head tail ih =>
+    unfold countP.go
+    rw [ih (n := n + 1), ih (n := n), ih (n := 1)]
+    if h : p head then simp [h, Nat.add_assoc] else simp [h]
+
+@[simp] theorem countP_cons_of_pos (l) (pa : p a) : countP p (a :: l) = countP p l + 1 := by
+  have : countP.go p (a :: l) 0 = countP.go p l 1 := show cond .. = _ by rw [pa]; rfl
+  unfold countP
+  rw [this, Nat.add_comm, List.countP_go_eq_add]
+
+@[simp] theorem countP_cons_of_neg (l) (pa : ¬p a) : countP p (a :: l) = countP p l := by
+  simp [countP, countP.go, pa]
+
+theorem countP_cons (a : α) (l) : countP p (a :: l) = countP p l + if p a then 1 else 0 := by
+  by_cases h : p a <;> simp [h]
+
+theorem length_eq_countP_add_countP (l) : length l = countP p l + countP (fun a => ¬p a) l := by
+  induction l with
+  | nil => rfl
+  | cons x h ih =>
+    if h : p x then
+      rw [countP_cons_of_pos _ _ h, countP_cons_of_neg _ _ _, length, ih]
+      · rw [Nat.add_assoc, Nat.add_comm _ 1, Nat.add_assoc]
+      · simp only [h, not_true_eq_false, decide_False, not_false_eq_true]
+    else
+      rw [countP_cons_of_pos (fun a => ¬p a) _ _, countP_cons_of_neg _ _ h, length, ih]
+      · rfl
+      · simp only [h, not_false_eq_true, decide_True]
+
+theorem countP_eq_length_filter (l) : countP p l = length (filter p l) := by
+  induction l with
+  | nil => rfl
+  | cons x l ih =>
+    if h : p x
+    then rw [countP_cons_of_pos p l h, ih, filter_cons_of_pos h, length]
+    else rw [countP_cons_of_neg p l h, ih, filter_cons_of_neg h]
+
+theorem countP_le_length : countP p l ≤ l.length := by
+  simp only [countP_eq_length_filter]
+  apply length_filter_le
+
+@[simp] theorem countP_append (l₁ l₂) : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂ := by
+  simp only [countP_eq_length_filter, filter_append, length_append]
+
+theorem countP_pos : 0 < countP p l ↔ ∃ a ∈ l, p a := by
+  simp only [countP_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]
+
+theorem countP_eq_zero : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by
+  simp only [countP_eq_length_filter, length_eq_zero, filter_eq_nil]
+
+theorem countP_eq_length : countP p l = l.length ↔ ∀ a ∈ l, p a := by
+  rw [countP_eq_length_filter, filter_length_eq_length]
+
+theorem Sublist.countP_le (s : l₁ <+ l₂) : countP p l₁ ≤ countP p l₂ := by
+  simp only [countP_eq_length_filter]
+  apply s.filter _ |>.length_le
+
+theorem countP_filter (l : List α) :
+    countP p (filter q l) = countP (fun a => p a ∧ q a) l := by
+  simp only [countP_eq_length_filter, filter_filter]
+
+@[simp] theorem countP_true {l : List α} : (l.countP fun _ => true) = l.length := by
+  rw [countP_eq_length]
+  simp
+
+@[simp] theorem countP_false {l : List α} : (l.countP fun _ => false) = 0 := by
+  rw [countP_eq_zero]
+  simp
+
+@[simp] theorem countP_map (p : β → Bool) (f : α → β) :
+    ∀ l, countP p (map f l) = countP (p ∘ f) l
+  | [] => rfl
+  | a :: l => by rw [map_cons, countP_cons, countP_cons, countP_map p f l]; rfl
+
+variable {p q}
+
+theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by
+  induction l with
+  | nil => apply Nat.le_refl
+  | cons a l ihl =>
+    rw [forall_mem_cons] at h
+    have ⟨ha, hl⟩ := h
+    simp [countP_cons]
+    cases h : p a
+    · simp only [Bool.false_eq_true, ↓reduceIte, Nat.add_zero]
+      apply Nat.le_trans ?_ (Nat.le_add_right _ _)
+      apply ihl hl
+    · simp only [↓reduceIte, ha h, succ_le_succ_iff]
+      apply ihl hl
+
+theorem countP_congr (h : ∀ x ∈ l, p x ↔ q x) : countP p l = countP q l :=
+  Nat.le_antisymm
+    (countP_mono_left fun x hx => (h x hx).1)
+    (countP_mono_left fun x hx => (h x hx).2)
+
+end countP
+
+/-! ### count -/
+section count
+
+variable [BEq α]
+
+@[simp] theorem count_nil (a : α) : count a [] = 0 := rfl
+
+theorem count_cons (a b : α) (l : List α) :
+    count a (b :: l) = count a l + if b == a then 1 else 0 := by
+  simp [count, countP_cons]
+
+theorem count_tail : ∀ (l : List α) (a : α) (h : l ≠ []),
+      l.tail.count a = l.count a - if l.head h == a then 1 else 0
+  | head :: tail, a, _ => by simp [count_cons]
+
+theorem count_le_length (a : α) (l : List α) : count a l ≤ l.length := countP_le_length _
+
+theorem Sublist.count_le (h : l₁ <+ l₂) (a : α) : count a l₁ ≤ count a l₂ := h.countP_le _
+
+theorem count_le_count_cons (a b : α) (l : List α) : count a l ≤ count a (b :: l) :=
+  (sublist_cons_self _ _).count_le _
+
+theorem count_singleton (a b : α) : count a [b] = if b == a then 1 else 0 := by
+  simp [count_cons]
+
+@[simp] theorem count_append (a : α) : ∀ l₁ l₂, count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
+  countP_append _
+
+variable [LawfulBEq α]
+
+@[simp] theorem count_cons_self (a : α) (l : List α) : count a (a :: l) = count a l + 1 := by
+  simp [count_cons]
+
+@[simp] theorem count_cons_of_ne (h : a ≠ b) (l : List α) : count a (b :: l) = count a l := by
+  simp only [count_cons, cond_eq_if, beq_iff_eq]
+  split <;> simp_all
+
+theorem count_singleton_self (a : α) : count a [a] = 1 := by simp
+
+theorem count_concat_self (a : α) (l : List α) :
+    count a (concat l a) = (count a l) + 1 := by simp
+
+@[simp]
+theorem count_pos_iff_mem {a : α} {l : List α} : 0 < count a l ↔ a ∈ l := by
+  simp only [count, countP_pos, beq_iff_eq, exists_eq_right]
+
+theorem count_eq_zero_of_not_mem {a : α} {l : List α} (h : a ∉ l) : count a l = 0 :=
+  Decidable.byContradiction fun h' => h <| count_pos_iff_mem.1 (Nat.pos_of_ne_zero h')
+
+theorem not_mem_of_count_eq_zero {a : α} {l : List α} (h : count a l = 0) : a ∉ l :=
+  fun h' => Nat.ne_of_lt (count_pos_iff_mem.2 h') h.symm
+
+theorem count_eq_zero {l : List α} : count a l = 0 ↔ a ∉ l :=
+  ⟨not_mem_of_count_eq_zero, count_eq_zero_of_not_mem⟩
+
+theorem count_eq_length {l : List α} : count a l = l.length ↔ ∀ b ∈ l, a = b := by
+  rw [count, countP_eq_length]
+  refine ⟨fun h b hb => Eq.symm ?_, fun h b hb => ?_⟩
+  · simpa using h b hb
+  · rw [h b hb, beq_self_eq_true]
+
+@[simp] theorem count_replicate_self (a : α) (n : Nat) : count a (replicate n a) = n :=
+  (count_eq_length.2 <| fun _ h => (eq_of_mem_replicate h).symm).trans (length_replicate ..)
+
+theorem count_replicate (a b : α) (n : Nat) : count a (replicate n b) = if b == a then n else 0 := by
+  split <;> (rename_i h; simp only [beq_iff_eq] at h)
+  · exact ‹b = a› ▸ count_replicate_self ..
+  · exact count_eq_zero.2 <| mt eq_of_mem_replicate (Ne.symm h)
+
+theorem filter_beq (l : List α) (a : α) : l.filter (· == a) = replicate (count a l) a := by
+  simp only [count, countP_eq_length_filter, eq_replicate, mem_filter, beq_iff_eq]
+  exact ⟨trivial, fun _ h => h.2⟩
+
+theorem filter_eq {α} [DecidableEq α] (l : List α) (a : α) : l.filter (· = a) = replicate (count a l) a :=
+  filter_beq l a
+
+theorem le_count_iff_replicate_sublist {l : List α} : n ≤ count a l ↔ replicate n a <+ l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · exact ((replicate_sublist_replicate a).2 h).trans <| filter_beq l a ▸ filter_sublist _
+  · simpa only [count_replicate_self] using h.count_le a
+
+theorem replicate_count_eq_of_count_eq_length {l : List α} (h : count a l = length l) :
+    replicate (count a l) a = l :=
+  (le_count_iff_replicate_sublist.mp (Nat.le_refl _)).eq_of_length <|
+    (length_replicate (count a l) a).trans h
+
+@[simp] theorem count_filter {l : List α} (h : p a) : count a (filter p l) = count a l := by
+  rw [count, countP_filter]; congr; funext b
+  simp; rintro rfl; exact h
+
+theorem count_le_count_map [DecidableEq β] (l : List α) (f : α → β) (x : α) :
+    count x l ≤ count (f x) (map f l) := by
+  rw [count, count, countP_map]
+  apply countP_mono_left; simp (config := { contextual := true })
+
+theorem count_erase (a b : α) :
+    ∀ l : List α, count a (l.erase b) = count a l - if b == a then 1 else 0
+  | [] => by simp
+  | c :: l => by
+    rw [erase_cons]
+    if hc : c = b then
+      have hc_beq := (beq_iff_eq _ _).mpr hc
+      rw [if_pos hc_beq, hc, count_cons, Nat.add_sub_cancel]
+    else
+      have hc_beq := beq_false_of_ne hc
+      simp only [hc_beq, if_false, count_cons, count_cons, count_erase a b l]
+      if ha : b = a then
+        rw [ha, eq_comm] at hc
+        rw [if_pos ((beq_iff_eq _ _).2 ha), if_neg (by simpa using Ne.symm hc), Nat.add_zero, Nat.add_zero]
+      else
+        rw [if_neg (by simpa using ha), Nat.sub_zero, Nat.sub_zero]
+
+@[simp] theorem count_erase_self (a : α) (l : List α) :
+    count a (List.erase l a) = count a l - 1 := by rw [count_erase, if_pos (by simp)]
+
+@[simp] theorem count_erase_of_ne (ab : a ≠ b) (l : List α) : count a (l.erase b) = count a l := by
+  rw [count_erase, if_neg (by simpa using ab.symm), Nat.sub_zero]
+
+end count

src/Init/Data/List/Erase.lean

@@ -0,0 +1,445 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
+  Yury Kudryashov
+-/
+prelude
+import Init.Data.List.Pairwise
+
+/-!
+# Lemmas about `List.eraseP` and `List.erase`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ### eraseP -/
+
+@[simp] theorem eraseP_nil : [].eraseP p = [] := rfl
+
+theorem eraseP_cons (a : α) (l : List α) :
+    (a :: l).eraseP p = bif p a then l else a :: l.eraseP p := rfl
+
+@[simp] theorem eraseP_cons_of_pos {l : List α} {p} (h : p a) : (a :: l).eraseP p = l := by
+  simp [eraseP_cons, h]
+
+@[simp] theorem eraseP_cons_of_neg {l : List α} {p} (h : ¬p a) :
+    (a :: l).eraseP p = a :: l.eraseP p := by simp [eraseP_cons, h]
+
+theorem eraseP_of_forall_not {l : List α} (h : ∀ a, a ∈ l → ¬p a) : l.eraseP p = l := by
+  induction l with
+  | nil => rfl
+  | cons _ _ ih => simp [h _ (.head ..), ih (forall_mem_cons.1 h).2]
+
+theorem exists_of_eraseP : ∀ {l : List α} {a} (al : a ∈ l) (pa : p a),
+    ∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.eraseP p = l₁ ++ l₂
+  | b :: l, a, al, pa =>
+    if pb : p b then
+      ⟨b, [], l, forall_mem_nil _, pb, by simp [pb]⟩
+    else
+      match al with
+      | .head .. => nomatch pb pa
+      | .tail _ al =>
+        let ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩ := exists_of_eraseP al pa
+        ⟨c, b::l₁, l₂, (forall_mem_cons ..).2 ⟨pb, h₁⟩,
+          h₂, by rw [h₃, cons_append], by simp [pb, h₄]⟩
+
+theorem exists_or_eq_self_of_eraseP (p) (l : List α) :
+    l.eraseP p = l ∨
+    ∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.eraseP p = l₁ ++ l₂ :=
+  if h : ∃ a ∈ l, p a then
+    let ⟨_, ha, pa⟩ := h
+    .inr (exists_of_eraseP ha pa)
+  else
+    .inl (eraseP_of_forall_not (h ⟨·, ·, ·⟩))
+
+@[simp] theorem length_eraseP_of_mem (al : a ∈ l) (pa : p a) :
+    length (l.eraseP p) = length l - 1 := by
+  let ⟨_, l₁, l₂, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
+  rw [e₂]; simp [length_append, e₁]; rfl
+
+theorem length_eraseP {l : List α} : (l.eraseP p).length = if l.any p then l.length - 1 else l.length := by
+  split <;> rename_i h
+  · simp only [any_eq_true] at h
+    obtain ⟨x, m, h⟩ := h
+    simp [length_eraseP_of_mem m h]
+  · simp only [any_eq_true] at h
+    rw [eraseP_of_forall_not]
+    simp_all
+
+theorem eraseP_sublist (l : List α) : l.eraseP p <+ l := by
+  match exists_or_eq_self_of_eraseP p l with
+  | .inl h => rw [h]; apply Sublist.refl
+  | .inr ⟨c, l₁, l₂, _, _, h₃, h₄⟩ => rw [h₄, h₃]; simp
+
+theorem eraseP_subset (l : List α) : l.eraseP p ⊆ l := (eraseP_sublist l).subset
+
+protected theorem Sublist.eraseP : l₁ <+ l₂ → l₁.eraseP p <+ l₂.eraseP p
+  | .slnil => Sublist.refl _
+  | .cons a s => by
+    by_cases h : p a
+    · simpa [h] using s.eraseP.trans (eraseP_sublist _)
+    · simpa [h] using s.eraseP.cons _
+  | .cons₂ a s => by
+    by_cases h : p a
+    · simpa [h] using s
+    · simpa [h] using s.eraseP
+
+theorem length_eraseP_le (l : List α) : (l.eraseP p).length ≤ l.length :=
+  l.eraseP_sublist.length_le
+
+theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (eraseP_subset _ ·)
+
+@[simp] theorem mem_eraseP_of_neg {l : List α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
+  refine ⟨mem_of_mem_eraseP, fun al => ?_⟩
+  match exists_or_eq_self_of_eraseP p l with
+  | .inl h => rw [h]; assumption
+  | .inr ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩ =>
+    rw [h₄]; rw [h₃] at al
+    have : a ≠ c := fun h => (h ▸ pa).elim h₂
+    simp [this] at al; simp [al]
+
+@[simp] theorem eraseP_eq_self_iff {p} {l : List α} : l.eraseP p = l ↔ ∀ a ∈ l, ¬ p a := by
+  rw [← Sublist.length_eq (eraseP_sublist l), length_eraseP]
+  split <;> rename_i h
+  · simp only [any_eq_true, length_eq_zero] at h
+    constructor
+    · intro; simp_all [Nat.sub_one_eq_self]
+    · intro; obtain ⟨x, m, h⟩ := h; simp_all
+  · simp_all
+
+theorem eraseP_map (f : β → α) : ∀ (l : List β), (map f l).eraseP p = map f (l.eraseP (p ∘ f))
+  | [] => rfl
+  | b::l => by by_cases h : p (f b) <;> simp [h, eraseP_map f l, eraseP_cons_of_pos]
+
+theorem eraseP_filterMap (f : α → Option β) : ∀ (l : List α),
+    (filterMap f l).eraseP p = filterMap f (l.eraseP (fun x => match f x with | some y => p y | none => false))
+  | [] => rfl
+  | a::l => by
+    rw [filterMap_cons, eraseP_cons]
+    split <;> rename_i h
+    · simp [h, eraseP_filterMap]
+    · rename_i b
+      rw [h, eraseP_cons]
+      by_cases w : p b
+      · simp [w]
+      · simp only [w, cond_false]
+        rw [filterMap_cons_some h, eraseP_filterMap]
+
+theorem eraseP_filter (f : α → Bool) (l : List α) :
+    (filter f l).eraseP p = filter f (l.eraseP (fun x => p x && f x)) := by
+  rw [← filterMap_eq_filter, eraseP_filterMap]
+  congr
+  ext x
+  simp only [Option.guard]
+  split <;> split at * <;> simp_all
+
+theorem eraseP_append_left {a : α} (pa : p a) :
+    ∀ {l₁ : List α} l₂, a ∈ l₁ → (l₁++l₂).eraseP p = l₁.eraseP p ++ l₂
+  | x :: xs, l₂, h => by
+    by_cases h' : p x <;> simp [h']
+    rw [eraseP_append_left pa l₂ ((mem_cons.1 h).resolve_left (mt _ h'))]
+    intro | rfl => exact pa
+
+theorem eraseP_append_right :
+    ∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁++l₂) = l₁ ++ l₂.eraseP p
+  | [],      l₂, _ => rfl
+  | x :: xs, l₂, h => by
+    simp [(forall_mem_cons.1 h).1, eraseP_append_right _ (forall_mem_cons.1 h).2]
+
+theorem eraseP_append (l₁ l₂ : List α) :
+    (l₁ ++ l₂).eraseP p = if l₁.any p then l₁.eraseP p ++ l₂ else l₁ ++ l₂.eraseP p := by
+  split <;> rename_i h
+  · simp only [any_eq_true] at h
+    obtain ⟨x, m, h⟩ := h
+    rw [eraseP_append_left h _ m]
+  · simp only [any_eq_true] at h
+    rw [eraseP_append_right _]
+    simp_all
+
+theorem eraseP_eq_iff {p} {l : List α} :
+    l.eraseP p = l' ↔
+      ((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
+        ∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l' = l₁ ++ l₂ := by
+  cases exists_or_eq_self_of_eraseP p l with
+  | inl h =>
+    constructor
+    · intro h'
+      left
+      exact ⟨eraseP_eq_self_iff.1 h, by simp_all⟩
+    · rintro (⟨-, rfl⟩ | ⟨a, l₁, l₂, h₁, h₂, rfl, rfl⟩)
+      · assumption
+      · rw [eraseP_append_right _ h₁, eraseP_cons_of_pos h₂]
+  | inr h =>
+    obtain ⟨a, l₁, l₂, h₁, h₂, w₁, w₂⟩ := h
+    rw [w₂]
+    subst w₁
+    constructor
+    · rintro rfl
+      right
+      refine ⟨a, l₁, l₂, ?_⟩
+      simp_all
+    · rintro (h | h)
+      · simp_all
+      · obtain ⟨a', l₁', l₂', h₁', h₂', h, rfl⟩ := h
+        have p : l₁ = l₁' := by
+          have q : l₁ = takeWhile (fun x => !p x) (l₁ ++ a :: l₂) := by
+            rw [takeWhile_append_of_pos (by simp_all),
+              takeWhile_cons_of_neg (by simp [h₂]), append_nil]
+          have q' : l₁' = takeWhile (fun x => !p x) (l₁' ++ a' :: l₂') := by
+            rw [takeWhile_append_of_pos (by simpa using h₁'),
+              takeWhile_cons_of_neg (by simp [h₂']), append_nil]
+          simp [h] at q
+          rw [q', q]
+        subst p
+        simp_all
+
+@[simp] theorem eraseP_replicate_of_pos {n : Nat} {a : α} (h : p a) :
+    (replicate n a).eraseP p = replicate (n - 1) a := by
+  cases n <;> simp [replicate_succ, h]
+
+@[simp] theorem eraseP_replicate_of_neg {n : Nat} {a : α} (h : ¬p a) :
+    (replicate n a).eraseP p = replicate n a := by
+  rw [eraseP_of_forall_not (by simp_all)]
+
+theorem Nodup.eraseP (p) : Nodup l → Nodup (l.eraseP p) :=
+  Nodup.sublist <| eraseP_sublist _
+
+theorem eraseP_comm {l : List α} (h : ∀ a ∈ l, ¬ p a ∨ ¬ q a) :
+    (l.eraseP p).eraseP q = (l.eraseP q).eraseP p := by
+  induction l with
+  | nil => rfl
+  | cons a l ih =>
+    simp only [eraseP_cons]
+    by_cases h₁ : p a
+    · by_cases h₂ : q a
+      · simp_all
+      · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
+    · by_cases h₂ : q a
+      · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
+      · simp [h₁, h₂, ih (fun b m => h b (mem_cons_of_mem _ m))]
+
+/-! ### erase -/
+section erase
+variable [BEq α]
+
+@[simp] theorem erase_cons_head [LawfulBEq α] (a : α) (l : List α) : (a :: l).erase a = l := by
+  simp [erase_cons]
+
+@[simp] theorem erase_cons_tail {a b : α} {l : List α} (h : ¬(b == a)) :
+    (b :: l).erase a = b :: l.erase a := by simp only [erase_cons, if_neg h]
+
+theorem erase_of_not_mem [LawfulBEq α] {a : α} : ∀ {l : List α}, a ∉ l → l.erase a = l
+  | [], _ => rfl
+  | b :: l, h => by
+    rw [mem_cons, not_or] at h
+    simp only [erase_cons, if_neg, erase_of_not_mem h.2, beq_iff_eq, Ne.symm h.1, not_false_eq_true]
+
+theorem erase_eq_eraseP' (a : α) (l : List α) : l.erase a = l.eraseP (· == a) := by
+  induction l
+  · simp
+  · next b t ih =>
+    rw [erase_cons, eraseP_cons, ih]
+    if h : b == a then simp [h] else simp [h]
+
+theorem erase_eq_eraseP [LawfulBEq α] (a : α) : ∀ l : List α,  l.erase a = l.eraseP (a == ·)
+  | [] => rfl
+  | b :: l => by
+    if h : a = b then simp [h] else simp [h, Ne.symm h, erase_eq_eraseP a l]
+
+theorem exists_erase_eq [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
+    ∃ l₁ l₂, a ∉ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l.erase a = l₁ ++ l₂ := by
+  let ⟨_, l₁, l₂, h₁, e, h₂, h₃⟩ := exists_of_eraseP h (beq_self_eq_true _)
+  rw [erase_eq_eraseP]; exact ⟨l₁, l₂, fun h => h₁ _ h (beq_self_eq_true _), eq_of_beq e ▸ h₂, h₃⟩
+
+@[simp] theorem length_erase_of_mem [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
+    length (l.erase a) = length l - 1 := by
+  rw [erase_eq_eraseP]; exact length_eraseP_of_mem h (beq_self_eq_true a)
+
+theorem length_erase [LawfulBEq α] (a : α) (l : List α) :
+    length (l.erase a) = if a ∈ l then length l - 1 else length l := by
+  rw [erase_eq_eraseP, length_eraseP]
+  split <;> split <;> simp_all
+
+theorem erase_sublist (a : α) (l : List α) : l.erase a <+ l :=
+  erase_eq_eraseP' a l ▸ eraseP_sublist ..
+
+theorem erase_subset (a : α) (l : List α) : l.erase a ⊆ l := (erase_sublist a l).subset
+
+theorem Sublist.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+ l₂) : l₁.erase a <+ l₂.erase a := by
+  simp only [erase_eq_eraseP']; exact h.eraseP
+
+theorem length_erase_le (a : α) (l : List α) : (l.erase a).length ≤ l.length :=
+  (erase_sublist a l).length_le
+
+theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈ l := erase_subset _ _ h
+
+@[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {l : List α} (ab : a ≠ b) :
+    a ∈ l.erase b ↔ a ∈ l :=
+  erase_eq_eraseP b l ▸ mem_eraseP_of_neg (mt eq_of_beq ab.symm)
+
+@[simp] theorem erase_eq_self_iff [LawfulBEq α] {l : List α} : l.erase a = l ↔ a ∉ l := by
+  rw [erase_eq_eraseP', eraseP_eq_self_iff]
+  simp
+
+theorem erase_filter [LawfulBEq α] (f : α → Bool) (l : List α) :
+    (filter f l).erase a = filter f (l.erase a) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    by_cases h : a = x
+    · rw [erase_cons]
+      simp only [h, beq_self_eq_true, ↓reduceIte]
+      rw [filter_cons]
+      split
+      · rw [erase_cons_head]
+      · rw [erase_of_not_mem]
+        simp_all [mem_filter]
+    · rw [erase_cons_tail (by simpa using Ne.symm h), filter_cons, filter_cons]
+      split
+      · rw [erase_cons_tail (by simpa using Ne.symm h), ih]
+      · rw [ih]
+
+theorem erase_append_left [LawfulBEq α] {l₁ : List α} (l₂) (h : a ∈ l₁) :
+    (l₁ ++ l₂).erase a = l₁.erase a ++ l₂ := by
+  simp [erase_eq_eraseP]; exact eraseP_append_left (beq_self_eq_true a) l₂ h
+
+theorem erase_append_right [LawfulBEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : a ∉ l₁) :
+    (l₁ ++ l₂).erase a = (l₁ ++ l₂.erase a) := by
+  rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right]
+  intros b h' h''; rw [eq_of_beq h''] at h; exact h h'
+
+theorem erase_append [LawfulBEq α] {a : α} {l₁ l₂ : List α} :
+    (l₁ ++ l₂).erase a = if a ∈ l₁ then l₁.erase a ++ l₂ else l₁ ++ l₂.erase a := by
+  simp [erase_eq_eraseP, eraseP_append]
+
+theorem erase_comm [LawfulBEq α] (a b : α) (l : List α) :
+    (l.erase a).erase b = (l.erase b).erase a := by
+  if ab : a == b then rw [eq_of_beq ab] else ?_
+  if ha : a ∈ l then ?_ else
+    simp only [erase_of_not_mem ha, erase_of_not_mem (mt mem_of_mem_erase ha)]
+  if hb : b ∈ l then ?_ else
+    simp only [erase_of_not_mem hb, erase_of_not_mem (mt mem_of_mem_erase hb)]
+  match l, l.erase a, exists_erase_eq ha with
+  | _, _, ⟨l₁, l₂, ha', rfl, rfl⟩ =>
+    if h₁ : b ∈ l₁ then
+      rw [erase_append_left _ h₁, erase_append_left _ h₁,
+          erase_append_right _ (mt mem_of_mem_erase ha'), erase_cons_head]
+    else
+      rw [erase_append_right _ h₁, erase_append_right _ h₁, erase_append_right _ ha',
+          erase_cons_tail ab, erase_cons_head]
+
+theorem erase_eq_iff [LawfulBEq α] {a : α} {l : List α} :
+    l.erase a = l' ↔
+      (a ∉ l ∧ l = l') ∨
+        ∃ l₁ l₂, a ∉ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l' = l₁ ++ l₂ := by
+  rw [erase_eq_eraseP', eraseP_eq_iff]
+  simp only [beq_iff_eq, forall_mem_ne', exists_and_left]
+  constructor
+  · rintro (⟨h, rfl⟩ | ⟨a', l', h, rfl, x, rfl, rfl⟩)
+    · left; simp_all
+    · right; refine ⟨l', h, x, by simp⟩
+  · rintro (⟨h, rfl⟩ | ⟨l₁, h, x, rfl, rfl⟩)
+    · left; simp_all
+    · right; refine ⟨a, l₁, h, by simp⟩
+
+@[simp] theorem erase_replicate_self [LawfulBEq α] {a : α} :
+    (replicate n a).erase a = replicate (n - 1) a := by
+  cases n <;> simp [replicate_succ]
+
+@[simp] theorem erase_replicate_ne [LawfulBEq α] {a b : α} (h : !b == a) :
+    (replicate n a).erase b = replicate n a := by
+  rw [erase_of_not_mem]
+  simp_all
+
+theorem Nodup.erase_eq_filter [BEq α] [LawfulBEq α] {l} (d : Nodup l) (a : α) : l.erase a = l.filter (· != a) := by
+  induction d with
+  | nil => rfl
+  | cons m _n ih =>
+    rename_i b l
+    by_cases h : b = a
+    · subst h
+      rw [erase_cons_head, filter_cons_of_neg (by simp)]
+      apply Eq.symm
+      rw [filter_eq_self]
+      simpa [@eq_comm α] using m
+    · simp [beq_false_of_ne h, ih, h]
+
+theorem Nodup.mem_erase_iff [BEq α] [LawfulBEq α] {a : α} (d : Nodup l) : a ∈ l.erase b ↔ a ≠ b ∧ a ∈ l := by
+  rw [Nodup.erase_eq_filter d, mem_filter, and_comm, bne_iff_ne]
+
+theorem Nodup.not_mem_erase [BEq α] [LawfulBEq α] {a : α} (h : Nodup l) : a ∉ l.erase a := fun H => by
+  simpa using ((Nodup.mem_erase_iff h).mp H).left
+
+theorem Nodup.erase [BEq α] [LawfulBEq α] (a : α) : Nodup l → Nodup (l.erase a) :=
+  Nodup.sublist <| erase_sublist _ _
+
+end erase
+
+/-! ### eraseIdx -/
+
+theorem length_eraseIdx : ∀ {l i}, i < length l → length (@eraseIdx α l i) = length l - 1
+  | [], _, _ => rfl
+  | _::_, 0, _ => by simp [eraseIdx]
+  | x::xs, i+1, h => by
+    have : i < length xs := Nat.lt_of_succ_lt_succ h
+    simp [eraseIdx, ← Nat.add_one]
+    rw [length_eraseIdx this, Nat.sub_add_cancel (Nat.lt_of_le_of_lt (Nat.zero_le _) this)]
+
+@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
+
+theorem eraseIdx_eq_take_drop_succ :
+    ∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
+  | nil, _ => by simp
+  | a::l, 0 => by simp
+  | a::l, i + 1 => by simp [eraseIdx_eq_take_drop_succ l i]
+
+theorem eraseIdx_sublist : ∀ (l : List α) (k : Nat), eraseIdx l k <+ l
+  | [], _ => by simp
+  | a::l, 0 => by simp
+  | a::l, k + 1 => by simp [eraseIdx_sublist l k]
+
+theorem eraseIdx_subset (l : List α) (k : Nat) : eraseIdx l k ⊆ l := (eraseIdx_sublist l k).subset
+
+@[simp]
+theorem eraseIdx_eq_self : ∀ {l : List α} {k : Nat}, eraseIdx l k = l ↔ length l ≤ k
+  | [], _ => by simp
+  | a::l, 0 => by simp [(cons_ne_self _ _).symm]
+  | a::l, k + 1 => by simp [eraseIdx_eq_self]
+
+theorem eraseIdx_of_length_le {l : List α} {k : Nat} (h : length l ≤ k) : eraseIdx l k = l := by
+  rw [eraseIdx_eq_self.2 h]
+
+theorem eraseIdx_append_of_lt_length {l : List α} {k : Nat} (hk : k < length l) (l' : List α) :
+    eraseIdx (l ++ l') k = eraseIdx l k ++ l' := by
+  induction l generalizing k with
+  | nil => simp_all
+  | cons x l ih =>
+    cases k with
+    | zero => rfl
+    | succ k => simp_all [eraseIdx_cons_succ, Nat.succ_lt_succ_iff]
+
+theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤ k) (l' : List α) :
+    eraseIdx (l ++ l') k = l ++ eraseIdx l' (k - length l) := by
+  induction l generalizing k with
+  | nil => simp_all
+  | cons x l ih =>
+    cases k with
+    | zero => simp_all
+    | succ k => simp_all [eraseIdx_cons_succ, Nat.succ_sub_succ]
+
+protected theorem IsPrefix.eraseIdx {l l' : List α} (h : l <+: l') (k : Nat) :
+    eraseIdx l k <+: eraseIdx l' k := by
+  rcases h with ⟨t, rfl⟩
+  if hkl : k < length l then
+    simp [eraseIdx_append_of_lt_length hkl]
+  else
+    rw [Nat.not_lt] at hkl
+    simp [eraseIdx_append_of_length_le hkl, eraseIdx_of_length_le hkl]
+
+-- See also `mem_eraseIdx_iff_getElem` and `mem_eraseIdx_iff_getElem?` in
+-- `Init/Data/List/Nat/Basic.lean`.
+
+end List

src/Init/Data/List/Find.lean

@@ -0,0 +1,229 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Lemmas
+
+/-!
+# Lemmas about `List.find?`, `List.findSome?`, `List.findIdx`, `List.findIdx?`, and `List.indexOf`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ### find? -/
+
+@[simp] theorem find?_cons_of_pos (l) (h : p a) : find? p (a :: l) = some a := by
+  simp [find?, h]
+
+@[simp] theorem find?_cons_of_neg (l) (h : ¬p a) : find? p (a :: l) = find? p l := by
+  simp [find?, h]
+
+@[simp] theorem find?_eq_none : find? p l = none ↔ ∀ x ∈ l, ¬ p x := by
+  induction l <;> simp [find?_cons]; split <;> simp [*]
+
+theorem find?_some : ∀ {l}, find? p l = some a → p a
+  | b :: l, H => by
+    by_cases h : p b <;> simp [find?, h] at H
+    · exact H ▸ h
+    · exact find?_some H
+
+@[simp] theorem mem_of_find?_eq_some : ∀ {l}, find? p l = some a → a ∈ l
+  | b :: l, H => by
+    by_cases h : p b <;> simp [find?, h] at H
+    · exact H ▸ .head _
+    · exact .tail _ (mem_of_find?_eq_some H)
+
+@[simp] theorem find?_map (f : β → α) (l : List β) : find? p (l.map f) = (l.find? (p ∘ f)).map f := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [map_cons, find?]
+    by_cases h : p (f x) <;> simp [h, ih]
+
+theorem find?_replicate : find? p (replicate n a) = if n = 0 then none else if p a then some a else none := by
+  cases n
+  · simp
+  · by_cases p a <;> simp_all [replicate_succ]
+
+@[simp] theorem find?_replicate_of_length_pos (h : 0 < n) : find? p (replicate n a) = if p a then some a else none := by
+  simp [find?_replicate, Nat.ne_of_gt h]
+
+@[simp] theorem find?_replicate_of_pos (h : p a) : find? p (replicate n a) = if n = 0 then none else some a := by
+  simp [find?_replicate, h]
+
+@[simp] theorem find?_replicate_of_neg (h : ¬ p a) : find? p (replicate n a) = none := by
+  simp [find?_replicate, h]
+
+theorem find?_isSome_of_sublist {l₁ l₂ : List α} (h : l₁ <+ l₂) : (l₁.find? p).isSome → (l₂.find? p).isSome := by
+  induction h with
+  | slnil => simp
+  | cons a h ih
+  | cons₂ a h ih =>
+    simp only [find?]
+    split <;> simp_all
+
+/-! ### findSome? -/
+
+@[simp] theorem findSome?_cons_of_isSome (l) (h : (f a).isSome) : findSome? f (a :: l) = f a := by
+  simp only [findSome?]
+  split <;> simp_all
+
+@[simp] theorem findSome?_cons_of_isNone (l) (h : (f a).isNone) : findSome? f (a :: l) = findSome? f l := by
+  simp only [findSome?]
+  split <;> simp_all
+
+theorem exists_of_findSome?_eq_some {l : List α} {f : α → Option β} (w : l.findSome? f = some b) :
+    ∃ a, a ∈ l ∧ f a = b := by
+  induction l with
+  | nil => simp_all
+  | cons h l ih =>
+    simp_all only [findSome?_cons, mem_cons, exists_eq_or_imp]
+    split at w <;> simp_all
+
+@[simp] theorem findSome?_map (f : β → γ) (l : List β) : findSome? p (l.map f) = l.findSome? (p ∘ f) := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [map_cons, findSome?]
+    split <;> simp_all
+
+theorem findSome?_replicate : findSome? f (replicate n a) = if n = 0 then none else f a := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [replicate_succ, findSome?_cons]
+    split <;> simp_all
+
+@[simp] theorem findSome?_replicate_of_pos (h : 0 < n) : findSome? f (replicate n a) = f a := by
+  simp [findSome?_replicate, Nat.ne_of_gt h]
+
+-- Argument is unused, but used to decide whether `simp` should unfold.
+@[simp] theorem find?_replicate_of_isSome (_ : (f a).isSome) : findSome? f (replicate n a) = if n = 0 then none else f a := by
+  simp [findSome?_replicate]
+
+@[simp] theorem find?_replicate_of_isNone (h : (f a).isNone) : findSome? f (replicate n a) = none := by
+  rw [Option.isNone_iff_eq_none] at h
+  simp [findSome?_replicate, h]
+
+theorem findSome?_isSome_of_sublist {l₁ l₂ : List α} (h : l₁ <+ l₂) :
+    (l₁.findSome? f).isSome → (l₂.findSome? f).isSome := by
+  induction h with
+  | slnil => simp
+  | cons a h ih
+  | cons₂ a h ih =>
+    simp only [findSome?]
+    split <;> simp_all
+
+/-! ### findIdx -/
+
+theorem findIdx_cons (p : α → Bool) (b : α) (l : List α) :
+    (b :: l).findIdx p = bif p b then 0 else (l.findIdx p) + 1 := by
+  cases H : p b with
+  | true => simp [H, findIdx, findIdx.go]
+  | false => simp [H, findIdx, findIdx.go, findIdx_go_succ]
+where
+  findIdx_go_succ (p : α → Bool) (l : List α) (n : Nat) :
+      List.findIdx.go p l (n + 1) = (findIdx.go p l n) + 1 := by
+    cases l with
+    | nil => unfold findIdx.go; exact Nat.succ_eq_add_one n
+    | cons head tail =>
+      unfold findIdx.go
+      cases p head <;> simp only [cond_false, cond_true]
+      exact findIdx_go_succ p tail (n + 1)
+
+theorem findIdx_of_get?_eq_some {xs : List α} (w : xs.get? (xs.findIdx p) = some y) : p y := by
+  induction xs with
+  | nil => simp_all
+  | cons x xs ih => by_cases h : p x <;> simp_all [findIdx_cons]
+
+theorem findIdx_get {xs : List α} {w : xs.findIdx p < xs.length} :
+    p (xs.get ⟨xs.findIdx p, w⟩) :=
+  xs.findIdx_of_get?_eq_some (get?_eq_get w)
+
+theorem findIdx_lt_length_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
+    xs.findIdx p < xs.length := by
+  induction xs with
+  | nil => simp_all
+  | cons x xs ih =>
+    by_cases p x
+    · simp_all only [forall_exists_index, and_imp, mem_cons, exists_eq_or_imp, true_or,
+        findIdx_cons, cond_true, length_cons]
+      apply Nat.succ_pos
+    · simp_all [findIdx_cons]
+      refine Nat.succ_lt_succ ?_
+      obtain ⟨x', m', h'⟩ := h
+      exact ih x' m' h'
+
+theorem findIdx_get?_eq_get_of_exists {xs : List α} (h : ∃ x ∈ xs, p x) :
+    xs.get? (xs.findIdx p) = some (xs.get ⟨xs.findIdx p, xs.findIdx_lt_length_of_exists h⟩) :=
+  get?_eq_get (findIdx_lt_length_of_exists h)
+
+  /-! ### findIdx? -/
+
+@[simp] theorem findIdx?_nil : ([] : List α).findIdx? p i = none := rfl
+
+@[simp] theorem findIdx?_cons :
+    (x :: xs).findIdx? p i = if p x then some i else findIdx? p xs (i + 1) := rfl
+
+@[simp] theorem findIdx?_succ :
+    (xs : List α).findIdx? p (i+1) = (xs.findIdx? p i).map fun i => i + 1 := by
+  induction xs generalizing i with simp
+  | cons _ _ _ => split <;> simp_all
+
+theorem findIdx?_eq_some_iff (xs : List α) (p : α → Bool) :
+    xs.findIdx? p = some i ↔ (xs.take (i + 1)).map p = replicate i false ++ [true] := by
+  induction xs generalizing i with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx?_cons, Nat.zero_add, findIdx?_succ, take_succ_cons, map_cons]
+    split <;> cases i <;> simp_all [replicate_succ, succ_inj']
+
+theorem findIdx?_of_eq_some {xs : List α} {p : α → Bool} (w : xs.findIdx? p = some i) :
+    match xs.get? i with | some a => p a | none => false := by
+  induction xs generalizing i with
+  | nil => simp_all
+  | cons x xs ih =>
+    simp_all only [findIdx?_cons, Nat.zero_add, findIdx?_succ]
+    split at w <;> cases i <;> simp_all [succ_inj']
+
+theorem findIdx?_of_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p = none) :
+    ∀ i, match xs.get? i with | some a => ¬ p a | none => true := by
+  intro i
+  induction xs generalizing i with
+  | nil => simp_all
+  | cons x xs ih =>
+    simp_all only [Bool.not_eq_true, findIdx?_cons, Nat.zero_add, findIdx?_succ]
+    cases i with
+    | zero =>
+      split at w <;> simp_all
+    | succ i =>
+      simp only [get?_cons_succ]
+      apply ih
+      split at w <;> simp_all
+
+@[simp] theorem findIdx?_append :
+    (xs ++ ys : List α).findIdx? p =
+      (xs.findIdx? p <|> (ys.findIdx? p).map fun i => i + xs.length) := by
+  induction xs with simp
+  | cons _ _ _ => split <;> simp_all [Option.map_orElse, Option.map_map]; rfl
+
+@[simp] theorem findIdx?_replicate :
+    (replicate n a).findIdx? p = if 0 < n ∧ p a then some 0 else none := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [replicate, findIdx?_cons, Nat.zero_add, findIdx?_succ, Nat.zero_lt_succ, true_and]
+    split <;> simp_all
+
+/-! ### indexOf -/
+
+theorem indexOf_cons [BEq α] :
+    (x :: xs : List α).indexOf y = bif x == y then 0 else xs.indexOf y + 1 := by
+  dsimp [indexOf]
+  simp [findIdx_cons]
+
+end List

src/Init/Data/List/Impl.lean

@@ -193,6 +193,17 @@ theorem replicateTR_loop_eq : ∀ n, replicateTR.loop a n acc = replicate n a ++
   apply funext; intro α; apply funext; intro n; apply funext; intro a
   exact (replicateTR_loop_replicate_eq _ 0 n).symm
 
+/-! ## Additional functions -/
+
+/-! ### leftpad -/
+
+/-- Optimized version of `leftpad`. -/
+@[inline] def leftpadTR (n : Nat) (a : α) (l : List α) : List α :=
+  replicateTR.loop a (n - length l) l
+
+@[csimp] theorem leftpad_eq_leftpadTR : @leftpad = @leftpadTR := by
+  funext α n a l; simp [leftpad, leftpadTR, replicateTR_loop_eq]
+
 /-! ## Sublists -/
 
 /-! ### take -/
@@ -366,6 +377,26 @@ def unzipTR (l : List (α × β)) : List α × List β :=
 
 /-! ## Ranges and enumeration -/
 
+/-! ### range' -/
+
+/-- Optimized version of `range'`. -/
+@[inline] def range'TR (s n : Nat) (step : Nat := 1) : List Nat := go n (s + step * n) [] where
+  /-- Auxiliary for `range'TR`: `range'TR.go n e = [e-n, ..., e-1] ++ acc`. -/
+  go : Nat → Nat → List Nat → List Nat
+  | 0, _, acc => acc
+  | n+1, e, acc => go n (e-step) ((e-step) :: acc)
+
+@[csimp] theorem range'_eq_range'TR : @range' = @range'TR := by
+  funext s n step
+  let rec go (s) : ∀ n m,
+    range'TR.go step n (s + step * n) (range' (s + step * n) m step) = range' s (n + m) step
+  | 0, m => by simp [range'TR.go]
+  | n+1, m => by
+    simp [range'TR.go]
+    rw [Nat.mul_succ, ← Nat.add_assoc, Nat.add_sub_cancel, Nat.add_right_comm n]
+    exact go s n (m + 1)
+  exact (go s n 0).symm
+
 /-! ### iota -/
 
 /-- Tail-recursive version of `List.iota`. -/

src/Init/Data/List/MinMax.lean

@@ -0,0 +1,153 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Lemmas
+
+/-!
+# Lemmas about `List.minimum?` and `List.maximum?.
+-/
+
+namespace List
+
+open Nat
+
+/-! ## Minima and maxima -/
+
+/-! ### minimum? -/
+
+@[simp] theorem minimum?_nil [Min α] : ([] : List α).minimum? = none := rfl
+
+-- We don't put `@[simp]` on `minimum?_cons`,
+-- because the definition in terms of `foldl` is not useful for proofs.
+theorem minimum?_cons [Min α] {xs : List α} : (x :: xs).minimum? = foldl min x xs := rfl
+
+@[simp] theorem minimum?_eq_none_iff {xs : List α} [Min α] : xs.minimum? = none ↔ xs = [] := by
+  cases xs <;> simp [minimum?]
+
+theorem minimum?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
+    {xs : List α} → xs.minimum? = some a → a ∈ xs := by
+  intro xs
+  match xs with
+  | nil => simp
+  | x :: xs =>
+    simp only [minimum?_cons, Option.some.injEq, List.mem_cons]
+    intro eq
+    induction xs generalizing x with
+    | nil =>
+      simp at eq
+      simp [eq]
+    | cons y xs ind =>
+      simp at eq
+      have p := ind _ eq
+      cases p with
+      | inl p =>
+        cases min_eq_or x y with | _ q => simp [p, q]
+      | inr p => simp [p, mem_cons]
+
+-- See also `Init.Data.List.Nat.Basic` for specialisations of the next two results to `Nat`.
+
+theorem le_minimum?_iff [Min α] [LE α]
+    (le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) :
+    {xs : List α} → xs.minimum? = some a → ∀ x, x ≤ a ↔ ∀ b, b ∈ xs → x ≤ b
+  | nil => by simp
+  | cons x xs => by
+    rw [minimum?]
+    intro eq y
+    simp only [Option.some.injEq] at eq
+    induction xs generalizing x with
+    | nil =>
+      simp at eq
+      simp [eq]
+    | cons z xs ih =>
+      simp at eq
+      simp [ih _ eq, le_min_iff, and_assoc]
+
+-- This could be refactored by designing appropriate typeclasses to replace `le_refl`, `min_eq_or`,
+-- and `le_min_iff`.
+theorem minimum?_eq_some_iff [Min α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
+    (le_refl : ∀ a : α, a ≤ a)
+    (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b)
+    (le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) {xs : List α} :
+    xs.minimum? = some a ↔ a ∈ xs ∧ ∀ b, b ∈ xs → a ≤ b := by
+  refine ⟨fun h => ⟨minimum?_mem min_eq_or h, (le_minimum?_iff le_min_iff h _).1 (le_refl _)⟩, ?_⟩
+  intro ⟨h₁, h₂⟩
+  cases xs with
+  | nil => simp at h₁
+  | cons x xs =>
+    exact congrArg some <| anti.1
+      ((le_minimum?_iff le_min_iff (xs := x::xs) rfl _).1 (le_refl _) _ h₁)
+      (h₂ _ (minimum?_mem min_eq_or (xs := x::xs) rfl))
+
+theorem minimum?_replicate [Min α] {n : Nat} {a : α} (w : min a a = a) :
+    (replicate n a).minimum? = if n = 0 then none else some a := by
+  induction n with
+  | zero => rfl
+  | succ n ih => cases n <;> simp_all [replicate_succ, minimum?_cons]
+
+@[simp] theorem minimum?_replicate_of_pos [Min α] {n : Nat} {a : α} (w : min a a = a) (h : 0 < n) :
+    (replicate n a).minimum? = some a := by
+  simp [minimum?_replicate, Nat.ne_of_gt h, w]
+
+/-! ### maximum? -/
+
+@[simp] theorem maximum?_nil [Max α] : ([] : List α).maximum? = none := rfl
+
+-- We don't put `@[simp]` on `maximum?_cons`,
+-- because the definition in terms of `foldl` is not useful for proofs.
+theorem maximum?_cons [Max α] {xs : List α} : (x :: xs).maximum? = foldl max x xs := rfl
+
+@[simp] theorem maximum?_eq_none_iff {xs : List α} [Max α] : xs.maximum? = none ↔ xs = [] := by
+  cases xs <;> simp [maximum?]
+
+theorem maximum?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
+    {xs : List α} → xs.maximum? = some a → a ∈ xs
+  | nil => by simp
+  | cons x xs => by
+    rw [maximum?]; rintro ⟨⟩
+    induction xs generalizing x with simp at *
+    | cons y xs ih =>
+      rcases ih (max x y) with h | h <;> simp [h]
+      simp [← or_assoc, min_eq_or x y]
+
+-- See also `Init.Data.List.Nat.Basic` for specialisations of the next two results to `Nat`.
+
+theorem maximum?_le_iff [Max α] [LE α]
+    (max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) :
+    {xs : List α} → xs.maximum? = some a → ∀ x, a ≤ x ↔ ∀ b ∈ xs, b ≤ x
+  | nil => by simp
+  | cons x xs => by
+    rw [maximum?]; rintro ⟨⟩ y
+    induction xs generalizing x with
+    | nil => simp
+    | cons y xs ih => simp [ih, max_le_iff, and_assoc]
+
+-- This could be refactored by designing appropriate typeclasses to replace `le_refl`, `max_eq_or`,
+-- and `le_min_iff`.
+theorem maximum?_eq_some_iff [Max α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
+    (le_refl : ∀ a : α, a ≤ a)
+    (max_eq_or : ∀ a b : α, max a b = a ∨ max a b = b)
+    (max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) {xs : List α} :
+    xs.maximum? = some a ↔ a ∈ xs ∧ ∀ b ∈ xs, b ≤ a := by
+  refine ⟨fun h => ⟨maximum?_mem max_eq_or h, (maximum?_le_iff max_le_iff h _).1 (le_refl _)⟩, ?_⟩
+  intro ⟨h₁, h₂⟩
+  cases xs with
+  | nil => simp at h₁
+  | cons x xs =>
+    exact congrArg some <| anti.1
+      (h₂ _ (maximum?_mem max_eq_or (xs := x::xs) rfl))
+      ((maximum?_le_iff max_le_iff (xs := x::xs) rfl _).1 (le_refl _) _ h₁)
+
+theorem maximum?_replicate [Max α] {n : Nat} {a : α} (w : max a a = a) :
+    (replicate n a).maximum? = if n = 0 then none else some a := by
+  induction n with
+  | zero => rfl
+  | succ n ih => cases n <;> simp_all [replicate_succ, maximum?_cons]
+
+@[simp] theorem maximum?_replicate_of_pos [Max α] {n : Nat} {a : α} (w : max a a = a) (h : 0 < n) :
+    (replicate n a).maximum? = some a := by
+  simp [maximum?_replicate, Nat.ne_of_gt h, w]
+
+end List

src/Init/Data/List/Monadic.lean

@@ -0,0 +1,69 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.TakeDrop
+
+/-!
+# Lemmas about `List.mapM` and `List.forM`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ## Monadic operations -/
+
+-- We may want to replace these `simp` attributes with explicit equational lemmas,
+-- as we already have for all the non-monadic functions.
+attribute [simp] mapA forA filterAuxM firstM anyM allM findM? findSomeM?
+
+-- Previously `mapM.loop`, `filterMapM.loop`, `forIn.loop`, `forIn'.loop`
+-- had attribute `@[simp]`.
+-- We don't currently provide simp lemmas,
+-- as this is an internal implementation and they don't seem to be needed.
+
+/-! ### mapM -/
+
+/-- Alternate (non-tail-recursive) form of mapM for proofs. -/
+def mapM' [Monad m] (f : α → m β) : List α → m (List β)
+  | [] => pure []
+  | a :: l => return (← f a) :: (← l.mapM' f)
+
+@[simp] theorem mapM'_nil [Monad m] {f : α → m β} : mapM' f [] = pure [] := rfl
+@[simp] theorem mapM'_cons [Monad m] {f : α → m β} :
+    mapM' f (a :: l) = return ((← f a) :: (← l.mapM' f)) :=
+  rfl
+
+theorem mapM'_eq_mapM [Monad m] [LawfulMonad m] (f : α → m β) (l : List α) :
+    mapM' f l = mapM f l := by simp [go, mapM] where
+  go : ∀ l acc, mapM.loop f l acc = return acc.reverse ++ (← mapM' f l)
+    | [], acc => by simp [mapM.loop, mapM']
+    | a::l, acc => by simp [go l, mapM.loop, mapM']
+
+@[simp] theorem mapM_nil [Monad m] (f : α → m β) : [].mapM f = pure [] := rfl
+
+@[simp] theorem mapM_cons [Monad m] [LawfulMonad m] (f : α → m β) :
+    (a :: l).mapM f = (return (← f a) :: (← l.mapM f)) := by simp [← mapM'_eq_mapM, mapM']
+
+@[simp] theorem mapM_append [Monad m] [LawfulMonad m] (f : α → m β) {l₁ l₂ : List α} :
+    (l₁ ++ l₂).mapM f = (return (← l₁.mapM f) ++ (← l₂.mapM f)) := by induction l₁ <;> simp [*]
+
+/-! ### forM -/
+
+-- We use `List.forM` as the simp normal form, rather that `ForM.forM`.
+-- As such we need to replace `List.forM_nil` and `List.forM_cons`:
+
+@[simp] theorem forM_nil' [Monad m] : ([] : List α).forM f = (pure .unit : m PUnit) := rfl
+
+@[simp] theorem forM_cons' [Monad m] :
+    (a::as).forM f = (f a >>= fun _ => as.forM f : m PUnit) :=
+  List.forM_cons _ _ _
+
+@[simp] theorem forM_append [Monad m] [LawfulMonad m] (l₁ l₂ : List α) (f : α → m PUnit) :
+    (l₁ ++ l₂).forM f = (do l₁.forM f; l₂.forM f) := by
+  induction l₁ <;> simp [*]
+
+end List

src/Init/Data/List/Nat.lean

@@ -0,0 +1,10 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Nat.Basic
+import Init.Data.List.Nat.Pairwise
+import Init.Data.List.Nat.Range
+import Init.Data.List.Nat.TakeDrop

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

@@ -0,0 +1,125 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Count
+import Init.Data.List.MinMax
+import Init.Data.Nat.Lemmas
+
+/-!
+# Miscellaneous `List` lemmas, that require more `Nat` lemmas than are available in `Init.Data.List.Lemmas`.
+
+In particular, `omega` is available here.
+-/
+
+open Nat
+
+namespace List
+
+/-! ### filter -/
+
+theorem length_filter_lt_length_iff_exists (l) :
+    length (filter p l) < length l ↔ ∃ x ∈ l, ¬p x := by
+  simpa [length_eq_countP_add_countP p l, countP_eq_length_filter] using
+  countP_pos (fun x => ¬p x) (l := l)
+
+/-! ### leftpad -/
+
+/-- The length of the List returned by `List.leftpad n a l` is equal
+  to the larger of `n` and `l.length` -/
+@[simp]
+theorem leftpad_length (n : Nat) (a : α) (l : List α) :
+    (leftpad n a l).length = max n l.length := by
+  simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max]
+
+/-! ### eraseIdx -/
+
+theorem mem_eraseIdx_iff_getElem {x : α} :
+    ∀ {l} {k}, x ∈ eraseIdx l k ↔ ∃ i h, i ≠ k ∧ l[i]'h = x
+  | [], _ => by
+    simp only [eraseIdx, not_mem_nil, false_iff]
+    rintro ⟨i, h, -⟩
+    exact Nat.not_lt_zero _ h
+  | a::l, 0 => by simp [mem_iff_getElem, Nat.succ_lt_succ_iff]
+  | a::l, k+1 => by
+    rw [← Nat.or_exists_add_one]
+    simp [mem_eraseIdx_iff_getElem, @eq_comm _ a, succ_inj', Nat.succ_lt_succ_iff]
+
+theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l[i]? = some x := by
+  simp only [mem_eraseIdx_iff_getElem, getElem_eq_iff, exists_and_left]
+  refine exists_congr fun i => and_congr_right' ?_
+  constructor
+  · rintro ⟨_, h⟩; exact h
+  · rintro h;
+    obtain ⟨h', -⟩ := getElem?_eq_some.1 h
+    exact ⟨h', h⟩
+
+/-! ### minimum? -/
+
+-- A specialization of `minimum?_eq_some_iff` to Nat.
+theorem minimum?_eq_some_iff' {xs : List Nat} :
+    xs.minimum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
+  minimum?_eq_some_iff
+    (le_refl := Nat.le_refl)
+    (min_eq_or := fun _ _ => by omega)
+    (le_min_iff := fun _ _ _ => by omega)
+
+-- This could be generalized,
+-- but will first require further work on order typeclasses in the core repository.
+theorem minimum?_cons' {a : Nat} {l : List Nat} :
+    (a :: l).minimum? = some (match l.minimum? with
+    | none => a
+    | some m => min a m) := by
+  rw [minimum?_eq_some_iff']
+  split <;> rename_i h m
+  · simp_all
+  · rw [minimum?_eq_some_iff'] at m
+    obtain ⟨m, le⟩ := m
+    rw [Nat.min_def]
+    constructor
+    · split
+      · exact mem_cons_self a l
+      · exact mem_cons_of_mem a m
+    · intro b m
+      cases List.mem_cons.1 m with
+      | inl => split <;> omega
+      | inr h =>
+        specialize le b h
+        split <;> omega
+
+/-! ### maximum? -/
+
+-- A specialization of `maximum?_eq_some_iff` to Nat.
+theorem maximum?_eq_some_iff' {xs : List Nat} :
+    xs.maximum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, b ≤ a) :=
+  maximum?_eq_some_iff
+    (le_refl := Nat.le_refl)
+    (max_eq_or := fun _ _ => by omega)
+    (max_le_iff := fun _ _ _ => by omega)
+
+-- This could be generalized,
+-- but will first require further work on order typeclasses in the core repository.
+theorem maximum?_cons' {a : Nat} {l : List Nat} :
+    (a :: l).maximum? = some (match l.maximum? with
+    | none => a
+    | some m => max a m) := by
+  rw [maximum?_eq_some_iff']
+  split <;> rename_i h m
+  · simp_all
+  · rw [maximum?_eq_some_iff'] at m
+    obtain ⟨m, le⟩ := m
+    rw [Nat.max_def]
+    constructor
+    · split
+      · exact mem_cons_of_mem a m
+      · exact mem_cons_self a l
+    · intro b m
+      cases List.mem_cons.1 m with
+      | inl => split <;> omega
+      | inr h =>
+        specialize le b h
+        split <;> omega
+
+end List

src/Init/Data/List/Nat/Pairwise.lean

@@ -0,0 +1,73 @@
+/-
+Copyright (c) 2018 Mario Carneiro. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro, James Gallicchio
+-/
+prelude
+import Init.Data.Fin.Lemmas
+import Init.Data.List.Nat.TakeDrop
+import Init.Data.List.Pairwise
+
+/-!
+# Lemmas about `List.Pairwise`
+-/
+
+namespace List
+
+/-- Given a list `is` of monotonically increasing indices into `l`, getting each index
+  produces a sublist of `l`.  -/
+theorem map_getElem_sublist {l : List α} {is : List (Fin l.length)} (h : is.Pairwise (· < ·)) :
+    is.map (l[·]) <+ l := by
+  suffices ∀ n l', l' = l.drop n → (∀ i ∈ is, n ≤ i) → map (l[·]) is <+ l'
+    from this 0 l (by simp) (by simp)
+  rintro n l' rfl his
+  induction is generalizing n with
+  | nil => simp
+  | cons hd tl IH =>
+    simp only [Fin.getElem_fin, map_cons]
+    have := IH h.of_cons (hd+1) (pairwise_cons.mp h).1
+    specialize his hd (.head _)
+    have := (drop_eq_getElem_cons ..).symm ▸ this.cons₂ (get l hd)
+    have := Sublist.append (nil_sublist (take hd l |>.drop n)) this
+    rwa [nil_append, ← (drop_append_of_le_length ?_), take_append_drop] at this
+    simp [Nat.min_eq_left (Nat.le_of_lt hd.isLt), his]
+
+@[deprecated map_getElem_sublist (since := "2024-07-30")]
+theorem map_get_sublist {l : List α} {is : List (Fin l.length)} (h : is.Pairwise (·.val < ·.val)) :
+    is.map (get l) <+ l := by
+  simpa using map_getElem_sublist h
+
+/-- Given a sublist `l' <+ l`, there exists an increasing list of indices `is` such that
+  `l' = is.map fun i => l[i]`. -/
+theorem sublist_eq_map_getElem {l l' : List α} (h : l' <+ l) : ∃ is : List (Fin l.length),
+    l' = is.map (l[·]) ∧ is.Pairwise (· < ·) := by
+  induction h with
+  | slnil => exact ⟨[], by simp⟩
+  | cons _ _ IH =>
+    let ⟨is, IH⟩ := IH
+    refine ⟨is.map (·.succ), ?_⟩
+    simpa [Function.comp_def, pairwise_map]
+  | cons₂ _ _ IH =>
+    rcases IH with ⟨is,IH⟩
+    refine ⟨⟨0, by simp [Nat.zero_lt_succ]⟩ :: is.map (·.succ), ?_⟩
+    simp [Function.comp_def, pairwise_map, IH, ← get_eq_getElem]
+
+@[deprecated sublist_eq_map_getElem (since := "2024-07-30")]
+theorem sublist_eq_map_get (h : l' <+ l) : ∃ is : List (Fin l.length),
+    l' = map (get l) is ∧ is.Pairwise (· < ·) := by
+  simpa using sublist_eq_map_getElem h
+
+theorem pairwise_iff_getElem : Pairwise R l ↔
+    ∀ (i j : Nat) (_hi : i < l.length) (_hj : j < l.length) (_hij : i < j), R l[i] l[j] := by
+  rw [pairwise_iff_forall_sublist]
+  constructor <;> intro h
+  · intros i j hi hj h'
+    apply h
+    simpa [h'] using map_getElem_sublist (is := [⟨i, hi⟩, ⟨j, hj⟩])
+  · intros a b h'
+    have ⟨is, h', hij⟩ := sublist_eq_map_getElem h'
+    rcases is with ⟨⟩ | ⟨a', ⟨⟩ | ⟨b', ⟨⟩⟩⟩ <;> simp at h'
+    rcases h' with ⟨rfl, rfl⟩
+    apply h; simpa using hij
+
+end List

src/Init/Data/List/Nat/Range.lean

@@ -0,0 +1,387 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Nat.TakeDrop
+import Init.Data.List.Pairwise
+
+/-!
+# Lemmas about `List.range` and `List.enum`
+-/
+
+namespace List
+
+open Nat
+
+/-! ## Ranges and enumeration -/
+
+/-! ### range' -/
+
+theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step) n step := by
+  simp [range', Nat.add_succ, Nat.mul_succ]
+
+@[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
+
+@[simp] theorem length_range' (s step) : ∀ n : Nat, length (range' s n step) = n
+  | 0 => rfl
+  | _ + 1 => congrArg succ (length_range' _ _ _)
+
+@[simp] theorem range'_eq_nil : range' s n step = [] ↔ n = 0 := by
+  rw [← length_eq_zero, length_range']
+
+theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step * i
+  | 0 => by simp [range', Nat.not_lt_zero]
+  | n + 1 => by
+    have h (i) : i ≤ n ↔ i = 0 ∨ ∃ j, i = succ j ∧ j < n := by
+      cases i <;> simp [Nat.succ_le, Nat.succ_inj']
+    simp [range', mem_range', Nat.lt_succ, h]; simp only [← exists_and_right, and_assoc]
+    rw [exists_comm]; simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
+
+@[simp] theorem mem_range'_1 : m ∈ range' s n ↔ s ≤ m ∧ m < s + n := by
+  simp [mem_range']; exact ⟨
+    fun ⟨i, h, e⟩ => e ▸ ⟨Nat.le_add_right .., Nat.add_lt_add_left h _⟩,
+    fun ⟨h₁, h₂⟩ => ⟨m - s, Nat.sub_lt_left_of_lt_add h₁ h₂, (Nat.add_sub_cancel' h₁).symm⟩⟩
+
+theorem pairwise_lt_range' s n (step := 1) (pos : 0 < step := by simp) :
+    Pairwise (· < ·) (range' s n step) :=
+  match s, n, step, pos with
+  | _, 0, _, _ => Pairwise.nil
+  | s, n + 1, step, pos => by
+    simp only [range'_succ, pairwise_cons]
+    constructor
+    · intros n m
+      rw [mem_range'] at m
+      omega
+    · exact pairwise_lt_range' (s + step) n step pos
+
+theorem pairwise_le_range' s n (step := 1) :
+    Pairwise (· ≤ ·) (range' s n step) :=
+  match s, n, step with
+  | _, 0, _ => Pairwise.nil
+  | s, n + 1, step => by
+    simp only [range'_succ, pairwise_cons]
+    constructor
+    · intros n m
+      rw [mem_range'] at m
+      omega
+    · exact pairwise_le_range' (s + step) n step
+
+theorem nodup_range' (s n : Nat) (step := 1) (h : 0 < step := by simp) : Nodup (range' s n step) :=
+  (pairwise_lt_range' s n step h).imp Nat.ne_of_lt
+
+@[simp]
+theorem map_add_range' (a) : ∀ s n step, map (a + ·) (range' s n step) = range' (a + s) n step
+  | _, 0, _ => rfl
+  | s, n + 1, step => by simp [range', map_add_range' _ (s + step) n step, Nat.add_assoc]
+
+theorem map_sub_range' (a s n : Nat) (h : a ≤ s) :
+    map (· - a) (range' s n step) = range' (s - a) n step := by
+  conv => lhs; rw [← Nat.add_sub_cancel' h]
+  rw [← map_add_range', map_map, (?_ : _∘_ = _), map_id]
+  funext x; apply Nat.add_sub_cancel_left
+
+theorem range'_append : ∀ s m n step : Nat,
+    range' s m step ++ range' (s + step * m) n step = range' s (n + m) step
+  | s, 0, n, step => rfl
+  | s, m + 1, n, step => by
+    simpa [range', Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
+      using range'_append (s + step) m n step
+
+@[simp] theorem range'_append_1 (s m n : Nat) :
+    range' s m ++ range' (s + m) n = range' s (n + m) := by simpa using range'_append s m n 1
+
+theorem range'_sublist_right {s m n : Nat} : range' s m step <+ range' s n step ↔ m ≤ n :=
+  ⟨fun h => by simpa only [length_range'] using h.length_le,
+   fun h => by rw [← Nat.sub_add_cancel h, ← range'_append]; apply sublist_append_left⟩
+
+theorem range'_subset_right {s m n : Nat} (step0 : 0 < step) :
+    range' s m step ⊆ range' s n step ↔ m ≤ n := by
+  refine ⟨fun h => Nat.le_of_not_lt fun hn => ?_, fun h => (range'_sublist_right.2 h).subset⟩
+  have ⟨i, h', e⟩ := mem_range'.1 <| h <| mem_range'.2 ⟨_, hn, rfl⟩
+  exact Nat.ne_of_gt h' (Nat.eq_of_mul_eq_mul_left step0 (Nat.add_left_cancel e))
+
+theorem range'_subset_right_1 {s m n : Nat} : range' s m ⊆ range' s n ↔ m ≤ n :=
+  range'_subset_right (by decide)
+
+theorem getElem?_range' (s step) :
+    ∀ {m n : Nat}, m < n → (range' s n step)[m]? = some (s + step * m)
+  | 0, n + 1, _ => by simp [range'_succ]
+  | m + 1, n + 1, h => by
+    simp only [range'_succ, getElem?_cons_succ]
+    exact (getElem?_range' (s + step) step (Nat.lt_of_add_lt_add_right h)).trans <| by
+      simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]
+
+@[simp] theorem getElem_range' {n m step} (i) (H : i < (range' n m step).length) :
+    (range' n m step)[i] = n + step * i :=
+  (getElem?_eq_some.1 <| getElem?_range' n step (by simpa using H)).2
+
+theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ [s + step * n] := by
+  rw [Nat.add_comm n 1]; exact (range'_append s n 1 step).symm
+
+theorem range'_1_concat (s n : Nat) : range' s (n + 1) = range' s n ++ [s + n] := by
+  simp [range'_concat]
+
+/-! ### range -/
+
+theorem range_loop_range' : ∀ s n : Nat, range.loop s (range' s n) = range' 0 (n + s)
+  | 0, n => rfl
+  | s + 1, n => by rw [← Nat.add_assoc, Nat.add_right_comm n s 1]; exact range_loop_range' s (n + 1)
+
+theorem range_eq_range' (n : Nat) : range n = range' 0 n :=
+  (range_loop_range' n 0).trans <| by rw [Nat.zero_add]
+
+theorem range_succ_eq_map (n : Nat) : range (n + 1) = 0 :: map succ (range n) := by
+  rw [range_eq_range', range_eq_range', range', Nat.add_comm, ← map_add_range']
+  congr; exact funext (Nat.add_comm 1)
+
+theorem reverse_range' : ∀ s n : Nat, reverse (range' s n) = map (s + n - 1 - ·) (range n)
+  | s, 0 => rfl
+  | s, n + 1 => by
+    rw [range'_1_concat, reverse_append, range_succ_eq_map,
+      show s + (n + 1) - 1 = s + n from rfl, map, map_map]
+    simp [reverse_range', Nat.sub_right_comm, Nat.sub_sub]
+
+theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) := by
+  rw [range_eq_range', map_add_range']; rfl
+
+@[simp] theorem length_range (n : Nat) : length (range n) = n := by
+  simp only [range_eq_range', length_range']
+
+@[simp] theorem range_eq_nil {n : Nat} : range n = [] ↔ n = 0 := by
+  rw [← length_eq_zero, length_range]
+
+@[simp]
+theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by
+  simp only [range_eq_range', range'_sublist_right]
+
+@[simp]
+theorem range_subset {m n : Nat} : range m ⊆ range n ↔ m ≤ n := by
+  simp only [range_eq_range', range'_subset_right, lt_succ_self]
+
+@[simp]
+theorem mem_range {m n : Nat} : m ∈ range n ↔ m < n := by
+  simp only [range_eq_range', mem_range'_1, Nat.zero_le, true_and, Nat.zero_add]
+
+theorem not_mem_range_self {n : Nat} : n ∉ range n := by simp
+
+theorem self_mem_range_succ (n : Nat) : n ∈ range (n + 1) := by simp
+
+theorem pairwise_lt_range (n : Nat) : Pairwise (· < ·) (range n) := by
+  simp (config := {decide := true}) only [range_eq_range', pairwise_lt_range']
+
+theorem pairwise_le_range (n : Nat) : Pairwise (· ≤ ·) (range n) :=
+  Pairwise.imp Nat.le_of_lt (pairwise_lt_range _)
+
+theorem getElem?_range {m n : Nat} (h : m < n) : (range n)[m]? = some m := by
+  simp [range_eq_range', getElem?_range' _ _ h]
+
+@[simp] theorem getElem_range {n : Nat} (m) (h : m < (range n).length) : (range n)[m] = m := by
+  simp [range_eq_range']
+
+theorem range_succ (n : Nat) : range (succ n) = range n ++ [n] := by
+  simp only [range_eq_range', range'_1_concat, Nat.zero_add]
+
+theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + ·) := by
+  rw [← range'_eq_map_range]
+  simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm
+
+theorem take_range (m n : Nat) : take m (range n) = range (min m n) := by
+  apply List.ext_getElem
+  · simp
+  · simp (config := { contextual := true }) [← getElem_take, Nat.lt_min]
+
+theorem nodup_range (n : Nat) : Nodup (range n) := by
+  simp (config := {decide := true}) only [range_eq_range', nodup_range']
+
+/-! ### iota -/
+
+theorem iota_eq_reverse_range' : ∀ n : Nat, iota n = reverse (range' 1 n)
+  | 0 => rfl
+  | n + 1 => by simp [iota, range'_concat, iota_eq_reverse_range' n, reverse_append, Nat.add_comm]
+
+@[simp] theorem length_iota (n : Nat) : length (iota n) = n := by simp [iota_eq_reverse_range']
+
+@[simp]
+theorem mem_iota {m n : Nat} : m ∈ iota n ↔ 1 ≤ m ∧ m ≤ n := by
+  simp [iota_eq_reverse_range', Nat.add_comm, Nat.lt_succ]
+
+theorem pairwise_gt_iota (n : Nat) : Pairwise (· > ·) (iota n) := by
+  simpa only [iota_eq_reverse_range', pairwise_reverse] using pairwise_lt_range' 1 n
+
+theorem nodup_iota (n : Nat) : Nodup (iota n) :=
+  (pairwise_gt_iota n).imp Nat.ne_of_gt
+
+/-! ### enumFrom -/
+
+@[simp]
+theorem enumFrom_singleton (x : α) (n : Nat) : enumFrom n [x] = [(n, x)] :=
+  rfl
+
+@[simp]
+theorem enumFrom_eq_nil {n : Nat} {l : List α} : List.enumFrom n l = [] ↔ l = [] := by
+  cases l <;> simp
+
+@[simp] theorem enumFrom_length : ∀ {n} {l : List α}, (enumFrom n l).length = l.length
+  | _, [] => rfl
+  | _, _ :: _ => congrArg Nat.succ enumFrom_length
+
+@[simp]
+theorem getElem?_enumFrom :
+    ∀ n (l : List α) m, (enumFrom n l)[m]? = l[m]?.map fun a => (n + m, a)
+  | n, [], m => rfl
+  | n, a :: l, 0 => by simp
+  | n, a :: l, m + 1 => by
+    simp only [enumFrom_cons, getElem?_cons_succ]
+    exact (getElem?_enumFrom (n + 1) l m).trans <| by rw [Nat.add_right_comm]; rfl
+
+@[simp]
+theorem getElem_enumFrom (l : List α) (n) (i : Nat) (h : i < (l.enumFrom n).length) :
+    (l.enumFrom n)[i] = (n + i, l[i]'(by simpa [enumFrom_length] using h)) := by
+  simp only [enumFrom_length] at h
+  rw [getElem_eq_getElem?]
+  simp only [getElem?_enumFrom, getElem?_eq_getElem h]
+  simp
+
+theorem mk_add_mem_enumFrom_iff_getElem? {n i : Nat} {x : α} {l : List α} :
+    (n + i, x) ∈ enumFrom n l ↔ l[i]? = some x := by
+  simp [mem_iff_get?]
+
+theorem mk_mem_enumFrom_iff_le_and_getElem?_sub {n i : Nat} {x : α} {l : List α} :
+    (i, x) ∈ enumFrom n l ↔ n ≤ i ∧ l[i - n]? = x := by
+  if h : n ≤ i then
+    rcases Nat.exists_eq_add_of_le h with ⟨i, rfl⟩
+    simp [mk_add_mem_enumFrom_iff_getElem?, Nat.add_sub_cancel_left]
+  else
+    have : ∀ k, n + k ≠ i := by rintro k rfl; simp at h
+    simp [h, mem_iff_get?, this]
+
+theorem le_fst_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) :
+    n ≤ x.1 :=
+  (mk_mem_enumFrom_iff_le_and_getElem?_sub.1 h).1
+
+theorem fst_lt_add_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) :
+    x.1 < n + length l := by
+  rcases mem_iff_get.1 h with ⟨i, rfl⟩
+  simpa using i.isLt
+
+theorem map_enumFrom (f : α → β) (n : Nat) (l : List α) :
+    map (Prod.map id f) (enumFrom n l) = enumFrom n (map f l) := by
+  induction l generalizing n <;> simp_all
+
+@[simp]
+theorem enumFrom_map_fst (n) :
+    ∀ (l : List α), map Prod.fst (enumFrom n l) = range' n l.length
+  | [] => rfl
+  | _ :: _ => congrArg (cons _) (enumFrom_map_fst _ _)
+
+@[simp]
+theorem enumFrom_map_snd : ∀ (n) (l : List α), map Prod.snd (enumFrom n l) = l
+  | _, [] => rfl
+  | _, _ :: _ => congrArg (cons _) (enumFrom_map_snd _ _)
+
+theorem snd_mem_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) : x.2 ∈ l :=
+  enumFrom_map_snd n l ▸ mem_map_of_mem _ h
+
+theorem mem_enumFrom {x : α} {i j : Nat} (xs : List α) (h : (i, x) ∈ xs.enumFrom j) :
+    j ≤ i ∧ i < j + xs.length ∧ x ∈ xs :=
+  ⟨le_fst_of_mem_enumFrom h, fst_lt_add_of_mem_enumFrom h, snd_mem_of_mem_enumFrom h⟩
+
+theorem map_fst_add_enumFrom_eq_enumFrom (l : List α) (n k : Nat) :
+    map (Prod.map (· + n) id) (enumFrom k l) = enumFrom (n + k) l :=
+  ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
+
+theorem map_fst_add_enum_eq_enumFrom (l : List α) (n : Nat) :
+    map (Prod.map (· + n) id) (enum l) = enumFrom n l :=
+  map_fst_add_enumFrom_eq_enumFrom l _ _
+
+theorem enumFrom_cons' (n : Nat) (x : α) (xs : List α) :
+    enumFrom n (x :: xs) = (n, x) :: (enumFrom n xs).map (Prod.map (· + 1) id) := by
+  rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]
+
+theorem enumFrom_map (n : Nat) (l : List α) (f : α → β) :
+    enumFrom n (l.map f) = (enumFrom n l).map (Prod.map id f) := by
+  induction l with
+  | nil => rfl
+  | cons hd tl IH =>
+    rw [map_cons, enumFrom_cons', enumFrom_cons', map_cons, map_map, IH, map_map]
+    rfl
+
+theorem enumFrom_append (xs ys : List α) (n : Nat) :
+    enumFrom n (xs ++ ys) = enumFrom n xs ++ enumFrom (n + xs.length) ys := by
+  induction xs generalizing ys n with
+  | nil => simp
+  | cons x xs IH =>
+    rw [cons_append, enumFrom_cons, IH, ← cons_append, ← enumFrom_cons, length, Nat.add_right_comm,
+      Nat.add_assoc]
+
+theorem enumFrom_eq_zip_range' (l : List α) {n : Nat} : l.enumFrom n = (range' n l.length).zip l :=
+  zip_of_prod (enumFrom_map_fst _ _) (enumFrom_map_snd _ _)
+
+@[simp]
+theorem unzip_enumFrom_eq_prod (l : List α) {n : Nat} :
+    (l.enumFrom n).unzip = (range' n l.length, l) := by
+  simp only [enumFrom_eq_zip_range', unzip_zip, length_range']
+
+/-! ### enum -/
+
+theorem enum_cons : (a::as).enum = (0, a) :: as.enumFrom 1 := rfl
+
+theorem enum_cons' (x : α) (xs : List α) :
+    enum (x :: xs) = (0, x) :: (enum xs).map (Prod.map (· + 1) id) :=
+  enumFrom_cons' _ _ _
+
+@[simp]
+theorem enum_eq_nil {l : List α} : List.enum l = [] ↔ l = [] := enumFrom_eq_nil
+
+@[simp] theorem enum_singleton (x : α) : enum [x] = [(0, x)] := rfl
+
+@[simp] theorem enum_length : (enum l).length = l.length :=
+  enumFrom_length
+
+@[simp]
+theorem getElem?_enum (l : List α) (n : Nat) : (enum l)[n]? = l[n]?.map fun a => (n, a) := by
+  rw [enum, getElem?_enumFrom, Nat.zero_add]
+
+@[simp]
+theorem getElem_enum (l : List α) (i : Nat) (h : i < l.enum.length) :
+    l.enum[i] = (i, l[i]'(by simpa [enum_length] using h)) := by
+  simp [enum]
+
+theorem mk_mem_enum_iff_getElem? {i : Nat} {x : α} {l : List α} : (i, x) ∈ enum l ↔ l[i]? = x := by
+  simp [enum, mk_mem_enumFrom_iff_le_and_getElem?_sub]
+
+theorem mem_enum_iff_getElem? {x : Nat × α} {l : List α} : x ∈ enum l ↔ l[x.1]? = some x.2 :=
+  mk_mem_enum_iff_getElem?
+
+theorem fst_lt_of_mem_enum {x : Nat × α} {l : List α} (h : x ∈ enum l) : x.1 < length l := by
+  simpa using fst_lt_add_of_mem_enumFrom h
+
+theorem snd_mem_of_mem_enum {x : Nat × α} {l : List α} (h : x ∈ enum l) : x.2 ∈ l :=
+  snd_mem_of_mem_enumFrom h
+
+theorem map_enum (f : α → β) (l : List α) : map (Prod.map id f) (enum l) = enum (map f l) :=
+  map_enumFrom f 0 l
+
+@[simp] theorem enum_map_fst (l : List α) : map Prod.fst (enum l) = range l.length := by
+  simp only [enum, enumFrom_map_fst, range_eq_range']
+
+@[simp]
+theorem enum_map_snd (l : List α) : map Prod.snd (enum l) = l :=
+  enumFrom_map_snd _ _
+
+theorem enum_map (l : List α) (f : α → β) : (l.map f).enum = l.enum.map (Prod.map id f) :=
+  enumFrom_map _ _ _
+
+theorem enum_append (xs ys : List α) : enum (xs ++ ys) = enum xs ++ enumFrom xs.length ys := by
+  simp [enum, enumFrom_append]
+
+theorem enum_eq_zip_range (l : List α) : l.enum = (range l.length).zip l :=
+  zip_of_prod (enum_map_fst _) (enum_map_snd _)
+
+@[simp]
+theorem unzip_enum_eq_prod (l : List α) : l.enum.unzip = (range l.length, l) := by
+  simp only [enum_eq_zip_range, unzip_zip, length_range]
+
+end List

src/Init/Data/List/Nat/TakeDrop.lean

@@ -0,0 +1,503 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Zip
+import Init.Data.Nat.Lemmas
+
+/-!
+# Further lemmas about `List.take`, `List.drop`, `List.zip` and `List.zipWith`.
+
+These are in a separate file from most of the list lemmas
+as they required importing more lemmas about natural numbers, and use `omega`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ### take -/
+
+@[simp] theorem length_take : ∀ (i : Nat) (l : List α), length (take i l) = min i (length l)
+  | 0, l => by simp [Nat.zero_min]
+  | succ n, [] => by simp [Nat.min_zero]
+  | succ n, _ :: l => by simp [Nat.succ_min_succ, length_take]
+
+theorem length_take_le (n) (l : List α) : length (take n l) ≤ n := by simp [Nat.min_le_left]
+
+theorem length_take_le' (n) (l : List α) : length (take n l) ≤ l.length :=
+  by simp [Nat.min_le_right]
+
+theorem length_take_of_le (h : n ≤ length l) : length (take n l) = n := by simp [Nat.min_eq_left h]
+
+/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
+length `> i`. Version designed to rewrite from the big list to the small list. -/
+theorem getElem_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
+    L[i] = (L.take j)[i]'(length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩) :=
+  getElem_of_eq (take_append_drop j L).symm _ ▸ getElem_append ..
+
+/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
+length `> i`. Version designed to rewrite from the small list to the big list. -/
+theorem getElem_take' (L : List α) {j i : Nat} {h : i < (L.take j).length} :
+    (L.take j)[i] =
+    L[i]'(Nat.lt_of_lt_of_le h (length_take_le' _ _)) := by
+  rw [length_take, Nat.lt_min] at h; rw [getElem_take L _ h.1]
+
+/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
+length `> i`. Version designed to rewrite from the big list to the small list. -/
+@[deprecated getElem_take (since := "2024-06-12")]
+theorem get_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
+    get L ⟨i, hi⟩ = get (L.take j) ⟨i, length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩⟩ := by
+  simp [getElem_take _ hi hj]
+
+/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
+length `> i`. Version designed to rewrite from the small list to the big list. -/
+@[deprecated getElem_take (since := "2024-06-12")]
+theorem get_take' (L : List α) {j i} :
+    get (L.take j) i =
+    get L ⟨i.1, Nat.lt_of_lt_of_le i.2 (length_take_le' _ _)⟩ := by
+  simp [getElem_take']
+
+theorem getElem?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
+    (l.take n)[m]? = none :=
+  getElem?_eq_none <| Nat.le_trans (length_take_le _ _) h
+
+@[deprecated getElem?_take_eq_none (since := "2024-06-12")]
+theorem get?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
+    (l.take n).get? m = none := by
+  simp [getElem?_take_eq_none h]
+
+theorem getElem?_take_eq_if {l : List α} {n m : Nat} :
+    (l.take n)[m]? = if m < n then l[m]? else none := by
+  split
+  · next h => exact getElem?_take h
+  · next h => exact getElem?_take_eq_none (Nat.le_of_not_lt h)
+
+@[deprecated getElem?_take_eq_if (since := "2024-06-12")]
+theorem get?_take_eq_if {l : List α} {n m : Nat} :
+    (l.take n).get? m = if m < n then l.get? m else none := by
+  simp [getElem?_take_eq_if]
+
+theorem head?_take {l : List α} {n : Nat} :
+    (l.take n).head? = if n = 0 then none else l.head? := by
+  simp [head?_eq_getElem?, getElem?_take_eq_if]
+  split
+  · rw [if_neg (by omega)]
+  · rw [if_pos (by omega)]
+
+theorem head_take {l : List α} {n : Nat} (h : l.take n ≠ []) :
+    (l.take n).head h = l.head (by simp_all) := by
+  apply Option.some_inj.1
+  rw [← head?_eq_head, ← head?_eq_head, head?_take, if_neg]
+  simp_all
+
+theorem getLast?_take {l : List α} : (l.take n).getLast? = if n = 0 then none else l[n - 1]?.or l.getLast? := by
+  rw [getLast?_eq_getElem?, getElem?_take_eq_if, length_take]
+  split
+  · rw [if_neg (by omega)]
+    rw [Nat.min_def]
+    split
+    · rw [getElem?_eq_getElem (by omega)]
+      simp
+    · rw [← getLast?_eq_getElem?, getElem?_eq_none (by omega)]
+      simp
+  · rw [if_pos]
+    omega
+
+theorem getLast_take {l : List α} (h : l.take n ≠ []) :
+    (l.take n).getLast h = l[n - 1]?.getD (l.getLast (by simp_all)) := by
+  rw [getLast_eq_getElem, getElem_take']
+  simp [length_take, Nat.min_def]
+  simp at h
+  split
+  · rw [getElem?_eq_getElem (by omega)]
+    simp
+  · rw [getElem?_eq_none (by omega), getLast_eq_getElem]
+    simp
+
+theorem take_take : ∀ (n m) (l : List α), take n (take m l) = take (min n m) l
+  | n, 0, l => by rw [Nat.min_zero, take_zero, take_nil]
+  | 0, m, l => by rw [Nat.zero_min, take_zero, take_zero]
+  | succ n, succ m, nil => by simp only [take_nil]
+  | succ n, succ m, a :: l => by
+    simp only [take, succ_min_succ, take_take n m l]
+
+theorem take_set_of_lt (a : α) {n m : Nat} (l : List α) (h : m < n) :
+    (l.set n a).take m = l.take m :=
+  List.ext_getElem? fun i => by
+    rw [getElem?_take_eq_if, getElem?_take_eq_if]
+    split
+    · next h' => rw [getElem?_set_ne (by omega)]
+    · rfl
+
+@[simp] theorem take_replicate (a : α) : ∀ n m : Nat, take n (replicate m a) = replicate (min n m) a
+  | n, 0 => by simp [Nat.min_zero]
+  | 0, m => by simp [Nat.zero_min]
+  | succ n, succ m => by simp [replicate_succ, succ_min_succ, take_replicate]
+
+@[simp] theorem drop_replicate (a : α) : ∀ n m : Nat, drop n (replicate m a) = replicate (m - n) a
+  | n, 0 => by simp
+  | 0, m => by simp
+  | succ n, succ m => by simp [replicate_succ, succ_sub_succ, drop_replicate]
+
+/-- Taking the first `n` elements in `l₁ ++ l₂` is the same as appending the first `n` elements
+of `l₁` to the first `n - l₁.length` elements of `l₂`. -/
+theorem take_append_eq_append_take {l₁ l₂ : List α} {n : Nat} :
+    take n (l₁ ++ l₂) = take n l₁ ++ take (n - l₁.length) l₂ := by
+  induction l₁ generalizing n
+  · simp
+  · cases n
+    · simp [*]
+    · simp only [cons_append, take_succ_cons, length_cons, succ_eq_add_one, cons.injEq,
+        append_cancel_left_eq, true_and, *]
+      congr 1
+      omega
+
+theorem take_append_of_le_length {l₁ l₂ : List α} {n : Nat} (h : n ≤ l₁.length) :
+    (l₁ ++ l₂).take n = l₁.take n := by
+  simp [take_append_eq_append_take, Nat.sub_eq_zero_of_le h]
+
+/-- Taking the first `l₁.length + i` elements in `l₁ ++ l₂` is the same as appending the first
+`i` elements of `l₂` to `l₁`. -/
+theorem take_append {l₁ l₂ : List α} (i : Nat) :
+    take (l₁.length + i) (l₁ ++ l₂) = l₁ ++ take i l₂ := by
+  rw [take_append_eq_append_take, take_of_length_le (Nat.le_add_right _ _), Nat.add_sub_cancel_left]
+
+@[simp]
+theorem take_eq_take :
+    ∀ {l : List α} {m n : Nat}, l.take m = l.take n ↔ min m l.length = min n l.length
+  | [], m, n => by simp [Nat.min_zero]
+  | _ :: xs, 0, 0 => by simp
+  | x :: xs, m + 1, 0 => by simp [Nat.zero_min, succ_min_succ]
+  | x :: xs, 0, n + 1 => by simp [Nat.zero_min, succ_min_succ]
+  | x :: xs, m + 1, n + 1 => by simp [succ_min_succ, take_eq_take]
+
+theorem take_add (l : List α) (m n : Nat) : l.take (m + n) = l.take m ++ (l.drop m).take n := by
+  suffices take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l) by
+    rw [take_append_drop] at this
+    assumption
+  rw [take_append_eq_append_take, take_of_length_le, append_right_inj]
+  · simp only [take_eq_take, length_take, length_drop]
+    omega
+  apply Nat.le_trans (m := m)
+  · apply length_take_le
+  · apply Nat.le_add_right
+
+theorem dropLast_take {n : Nat} {l : List α} (h : n < l.length) :
+    (l.take n).dropLast = l.take (n - 1) := by
+  simp only [dropLast_eq_take, length_take, Nat.le_of_lt h, Nat.min_eq_left, take_take, sub_le]
+
+theorem map_eq_append_split {f : α → β} {l : List α} {s₁ s₂ : List β}
+    (h : map f l = s₁ ++ s₂) : ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = s₁ ∧ map f l₂ = s₂ := by
+  have := h
+  rw [← take_append_drop (length s₁) l] at this ⊢
+  rw [map_append] at this
+  refine ⟨_, _, rfl, append_inj this ?_⟩
+  rw [length_map, length_take, Nat.min_eq_left]
+  rw [← length_map l f, h, length_append]
+  apply Nat.le_add_right
+
+/-! ### drop -/
+
+theorem lt_length_drop (L : List α) {i j : Nat} (h : i + j < L.length) : j < (L.drop i).length := by
+  have A : i < L.length := Nat.lt_of_le_of_lt (Nat.le.intro rfl) h
+  rw [(take_append_drop i L).symm] at h
+  simpa only [Nat.le_of_lt A, Nat.min_eq_left, Nat.add_lt_add_iff_left, length_take,
+    length_append] using h
+
+/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
+dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
+theorem getElem_drop (L : List α) {i j : Nat} (h : i + j < L.length) :
+    L[i + j] = (L.drop i)[j]'(lt_length_drop L h) := by
+  have : i ≤ L.length := Nat.le_trans (Nat.le_add_right _ _) (Nat.le_of_lt h)
+  rw [getElem_of_eq (take_append_drop i L).symm h, getElem_append_right'] <;>
+    simp [Nat.min_eq_left this, Nat.add_sub_cancel_left, Nat.le_add_right]
+
+/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
+dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
+@[deprecated getElem_drop (since := "2024-06-12")]
+theorem get_drop (L : List α) {i j : Nat} (h : i + j < L.length) :
+    get L ⟨i + j, h⟩ = get (L.drop i) ⟨j, lt_length_drop L h⟩ := by
+  simp [getElem_drop]
+
+/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
+dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
+theorem getElem_drop' (L : List α) {i : Nat} {j : Nat} {h : j < (L.drop i).length} :
+    (L.drop i)[j] = L[i + j]'(by
+      rw [Nat.add_comm]
+      exact Nat.add_lt_of_lt_sub (length_drop i L ▸ h)) := by
+  rw [getElem_drop]
+
+/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
+dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
+@[deprecated getElem_drop' (since := "2024-06-12")]
+theorem get_drop' (L : List α) {i j} :
+    get (L.drop i) j = get L ⟨i + j, by
+      rw [Nat.add_comm]
+      exact Nat.add_lt_of_lt_sub (length_drop i L ▸ j.2)⟩ := by
+  simp [getElem_drop']
+
+@[simp]
+theorem getElem?_drop (L : List α) (i j : Nat) : (L.drop i)[j]? = L[i + j]? := by
+  ext
+  simp only [getElem?_eq_some, getElem_drop', Option.mem_def]
+  constructor <;> intro ⟨h, ha⟩
+  · exact ⟨_, ha⟩
+  · refine ⟨?_, ha⟩
+    rw [length_drop]
+    rw [Nat.add_comm] at h
+    apply Nat.lt_sub_of_add_lt h
+
+@[deprecated getElem?_drop (since := "2024-06-12")]
+theorem get?_drop (L : List α) (i j : Nat) : get? (L.drop i) j = get? L (i + j) := by
+  simp
+
+theorem head?_drop (l : List α) (n : Nat) :
+    (l.drop n).head? = l[n]? := by
+  rw [head?_eq_getElem?, getElem?_drop, Nat.add_zero]
+
+theorem head_drop {l : List α} {n : Nat} (h : l.drop n ≠ []) :
+    (l.drop n).head h = l[n]'(by simp_all) := by
+  have w : n < l.length := length_lt_of_drop_ne_nil h
+  simpa [head?_eq_head, getElem?_eq_getElem, h, w] using head?_drop l n
+
+theorem getLast?_drop {l : List α} : (l.drop n).getLast? = if l.length ≤ n then none else l.getLast? := by
+  rw [getLast?_eq_getElem?, getElem?_drop]
+  rw [length_drop]
+  split
+  · rw [getElem?_eq_none (by omega)]
+  · rw [getLast?_eq_getElem?]
+    congr
+    omega
+
+theorem getLast_drop {l : List α} (h : l.drop n ≠ []) :
+    (l.drop n).getLast h = l.getLast (ne_nil_of_length_pos (by simp at h; omega)) := by
+  simp only [ne_eq, drop_eq_nil_iff_le] at h
+  apply Option.some_inj.1
+  simp only [← getLast?_eq_getLast, getLast?_drop, ite_eq_right_iff]
+  omega
+
+theorem drop_length_cons {l : List α} (h : l ≠ []) (a : α) :
+    (a :: l).drop l.length = [l.getLast h] := by
+  induction l generalizing a with
+  | nil =>
+    cases h rfl
+  | cons y l ih =>
+    simp only [drop, length]
+    by_cases h₁ : l = []
+    · simp [h₁]
+    rw [getLast_cons h₁]
+    exact ih h₁ y
+
+/-- Dropping the elements up to `n` in `l₁ ++ l₂` is the same as dropping the elements up to `n`
+in `l₁`, dropping the elements up to `n - l₁.length` in `l₂`, and appending them. -/
+theorem drop_append_eq_append_drop {l₁ l₂ : List α} {n : Nat} :
+    drop n (l₁ ++ l₂) = drop n l₁ ++ drop (n - l₁.length) l₂ := by
+  induction l₁ generalizing n
+  · simp
+  · cases n
+    · simp [*]
+    · simp only [cons_append, drop_succ_cons, length_cons, succ_eq_add_one, append_cancel_left_eq, *]
+      congr 1
+      omega
+
+theorem drop_append_of_le_length {l₁ l₂ : List α} {n : Nat} (h : n ≤ l₁.length) :
+    (l₁ ++ l₂).drop n = l₁.drop n ++ l₂ := by
+  simp [drop_append_eq_append_drop, Nat.sub_eq_zero_of_le h]
+
+/-- Dropping the elements up to `l₁.length + i` in `l₁ + l₂` is the same as dropping the elements
+up to `i` in `l₂`. -/
+@[simp]
+theorem drop_append {l₁ l₂ : List α} (i : Nat) : drop (l₁.length + i) (l₁ ++ l₂) = drop i l₂ := by
+  rw [drop_append_eq_append_drop, drop_eq_nil_of_le] <;>
+    simp [Nat.add_sub_cancel_left, Nat.le_add_right]
+
+theorem set_eq_take_append_cons_drop {l : List α} {n : Nat} {a : α} :
+    l.set n a = if n < l.length then l.take n ++ a :: l.drop (n + 1) else l := by
+  split <;> rename_i h
+  · ext1 m
+    by_cases h' : m < n
+    · rw [getElem?_append (by simp [length_take]; omega), getElem?_set_ne (by omega),
+        getElem?_take h']
+    · by_cases h'' : m = n
+      · subst h''
+        rw [getElem?_set_eq ‹_›, getElem?_append_right, length_take,
+          Nat.min_eq_left (by omega), Nat.sub_self, getElem?_cons_zero]
+        rw [length_take]
+        exact Nat.min_le_left m l.length
+      · have h''' : n < m := by omega
+        rw [getElem?_set_ne (by omega), getElem?_append_right, length_take,
+          Nat.min_eq_left (by omega)]
+        · obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h'''
+          have p : n + k + 1 - n = k + 1 := by omega
+          rw [p]
+          rw [getElem?_cons_succ, getElem?_drop]
+          congr 1
+          omega
+        · rw [length_take]
+          exact Nat.le_trans (Nat.min_le_left _ _) (by omega)
+  · rw [set_eq_of_length_le]
+    omega
+
+theorem exists_of_set {n : Nat} {a' : α} {l : List α} (h : n < l.length) :
+    ∃ l₁ l₂, l = l₁ ++ l[n] :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂ := by
+  refine ⟨l.take n, l.drop (n + 1), ⟨by simp, ⟨length_take_of_le (Nat.le_of_lt h), ?_⟩⟩⟩
+  simp [set_eq_take_append_cons_drop, h]
+
+theorem drop_set_of_lt (a : α) {n m : Nat} (l : List α)
+    (hnm : n < m) : drop m (l.set n a) = l.drop m :=
+  ext_getElem? fun k => by simpa only [getElem?_drop] using getElem?_set_ne (by omega)
+
+theorem drop_take : ∀ (m n : Nat) (l : List α), drop n (take m l) = take (m - n) (drop n l)
+  | 0, _, _ => by simp
+  | _, 0, _ => by simp
+  | _, _, [] => by simp
+  | m+1, n+1, h :: t => by
+    simp [take_succ_cons, drop_succ_cons, drop_take m n t]
+    congr 1
+    omega
+
+theorem take_reverse {α} {xs : List α} {n : Nat} (h : n ≤ xs.length) :
+    xs.reverse.take n = (xs.drop (xs.length - n)).reverse := by
+  induction xs generalizing n <;>
+    simp only [reverse_cons, drop, reverse_nil, Nat.zero_sub, length, take_nil]
+  next xs_hd xs_tl xs_ih =>
+  cases Nat.lt_or_eq_of_le h with
+  | inl h' =>
+    have h' := Nat.le_of_succ_le_succ h'
+    rw [take_append_of_le_length, xs_ih h']
+    rw [show xs_tl.length + 1 - n = succ (xs_tl.length - n) from _, drop]
+    · rwa [succ_eq_add_one, Nat.sub_add_comm]
+    · rwa [length_reverse]
+  | inr h' =>
+    subst h'
+    rw [length, Nat.sub_self, drop]
+    suffices xs_tl.length + 1 = (xs_tl.reverse ++ [xs_hd]).length by
+      rw [this, take_length, reverse_cons]
+    rw [length_append, length_reverse]
+    rfl
+
+@[deprecated (since := "2024-06-15")] abbrev reverse_take := @take_reverse
+
+theorem drop_reverse {α} {xs : List α} {n : Nat} (h : n ≤ xs.length) :
+    xs.reverse.drop n = (xs.take (xs.length - n)).reverse := by
+  conv =>
+    rhs
+    rw [← reverse_reverse xs]
+  rw [← reverse_reverse xs] at h
+  generalize xs.reverse = xs' at h ⊢
+  rw [take_reverse]
+  · simp only [length_reverse, reverse_reverse] at *
+    congr
+    omega
+  · simp only [length_reverse, sub_le]
+
+/-! ### rotateLeft -/
+
+@[simp] theorem rotateLeft_replicate (n) (a : α) : rotateLeft (replicate m a) n = replicate m a := by
+  cases n with
+  | zero => simp
+  | succ n =>
+    suffices 1 < m → m - (n + 1) % m + min ((n + 1) % m) m = m by
+      simpa [rotateLeft]
+    intro h
+    rw [Nat.min_eq_left (Nat.le_of_lt (Nat.mod_lt _ (by omega)))]
+    have : (n + 1) % m < m := Nat.mod_lt _ (by omega)
+    omega
+
+/-! ### rotateRight -/
+
+@[simp] theorem rotateRight_replicate (n) (a : α) : rotateRight (replicate m a) n = replicate m a := by
+  cases n with
+  | zero => simp
+  | succ n =>
+    suffices 1 < m → m - (m - (n + 1) % m) + min (m - (n + 1) % m) m = m by
+      simpa [rotateRight]
+    intro h
+    have : (n + 1) % m < m := Nat.mod_lt _ (by omega)
+    rw [Nat.min_eq_left (by omega)]
+    omega
+
+/-! ### zipWith -/
+
+@[simp] theorem length_zipWith (f : α → β → γ) (l₁ l₂) :
+    length (zipWith f l₁ l₂) = min (length l₁) (length l₂) := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;>
+    simp_all [succ_min_succ, Nat.zero_min, Nat.min_zero]
+
+theorem lt_length_left_of_zipWith {f : α → β → γ} {i : Nat} {l : List α} {l' : List β}
+    (h : i < (zipWith f l l').length) : i < l.length := by rw [length_zipWith] at h; omega
+
+theorem lt_length_right_of_zipWith {f : α → β → γ} {i : Nat} {l : List α} {l' : List β}
+    (h : i < (zipWith f l l').length) : i < l'.length := by rw [length_zipWith] at h; omega
+
+@[simp]
+theorem getElem_zipWith {f : α → β → γ} {l : List α} {l' : List β}
+    {i : Nat} {h : i < (zipWith f l l').length} :
+    (zipWith f l l')[i] =
+      f (l[i]'(lt_length_left_of_zipWith h))
+        (l'[i]'(lt_length_right_of_zipWith h)) := by
+  rw [← Option.some_inj, ← getElem?_eq_getElem, getElem?_zipWith_eq_some]
+  exact
+    ⟨l[i]'(lt_length_left_of_zipWith h), l'[i]'(lt_length_right_of_zipWith h),
+      by rw [getElem?_eq_getElem], by rw [getElem?_eq_getElem]; exact ⟨rfl, rfl⟩⟩
+
+theorem zipWith_eq_zipWith_take_min : ∀ (l₁ : List α) (l₂ : List β),
+    zipWith f l₁ l₂ = zipWith f (l₁.take (min l₁.length l₂.length)) (l₂.take (min l₁.length l₂.length))
+  | [], _ => by simp
+  | _, [] => by simp
+  | a :: l₁, b :: l₂ => by simp [succ_min_succ, zipWith_eq_zipWith_take_min l₁ l₂]
+
+theorem reverse_zipWith (h : l.length = l'.length) :
+    (zipWith f l l').reverse = zipWith f l.reverse l'.reverse := by
+  induction l generalizing l' with
+  | nil => simp
+  | cons hd tl hl =>
+    cases l' with
+    | nil => simp
+    | cons hd' tl' =>
+      simp only [Nat.add_right_cancel_iff, length] at h
+      have : tl.reverse.length = tl'.reverse.length := by simp [h]
+      simp [hl h, zipWith_append _ _ _ _ _ this]
+
+@[deprecated reverse_zipWith (since := "2024-07-28")] abbrev zipWith_distrib_reverse := @reverse_zipWith
+
+@[simp] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
+    zipWith f (replicate m a) (replicate n b) = replicate (min m n) (f a b) := by
+  rw [zipWith_eq_zipWith_take_min]
+  simp
+
+/-! ### zip -/
+
+@[simp] theorem length_zip (l₁ : List α) (l₂ : List β) :
+    length (zip l₁ l₂) = min (length l₁) (length l₂) := by
+  simp [zip]
+
+theorem lt_length_left_of_zip {i : Nat} {l : List α} {l' : List β} (h : i < (zip l l').length) :
+    i < l.length :=
+  lt_length_left_of_zipWith h
+
+theorem lt_length_right_of_zip {i : Nat} {l : List α} {l' : List β} (h : i < (zip l l').length) :
+    i < l'.length :=
+  lt_length_right_of_zipWith h
+
+@[simp]
+theorem getElem_zip {l : List α} {l' : List β} {i : Nat} {h : i < (zip l l').length} :
+    (zip l l')[i] =
+      (l[i]'(lt_length_left_of_zip h), l'[i]'(lt_length_right_of_zip h)) :=
+  getElem_zipWith (h := h)
+
+theorem zip_eq_zip_take_min : ∀ (l₁ : List α) (l₂ : List β),
+    zip l₁ l₂ = zip (l₁.take (min l₁.length l₂.length)) (l₂.take (min l₁.length l₂.length))
+  | [], _ => by simp
+  | _, [] => by simp
+  | a :: l₁, b :: l₂ => by simp [succ_min_succ, zip_eq_zip_take_min l₁ l₂]
+
+@[simp] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
+    zip (replicate m a) (replicate n b) = replicate (min m n) (a, b) := by
+  rw [zip_eq_zip_take_min]
+  simp
+
+end List

src/Init/Data/List/Pairwise.lean

@@ -0,0 +1,269 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.Sublist
+
+/-!
+# Lemmas about `List.Pairwise` and `List.Nodup`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ## Pairwise and Nodup -/
+
+/-! ### Pairwise -/
+
+theorem Pairwise.sublist : l₁ <+ l₂ → l₂.Pairwise R → l₁.Pairwise R
+  | .slnil, h => h
+  | .cons _ s, .cons _ h₂ => h₂.sublist s
+  | .cons₂ _ s, .cons h₁ h₂ => (h₂.sublist s).cons fun _ h => h₁ _ (s.subset h)
+
+theorem Pairwise.imp {α R S} (H : ∀ {a b}, R a b → S a b) :
+    ∀ {l : List α}, l.Pairwise R → l.Pairwise S
+  | _, .nil => .nil
+  | _, .cons h₁ h₂ => .cons (H ∘ h₁ ·) (h₂.imp H)
+
+theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
+  (pairwise_cons.1 p).1 _
+
+theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
+  (pairwise_cons.1 p).2
+
+theorem Pairwise.tail : ∀ {l : List α} (_p : Pairwise R l), Pairwise R l.tail
+  | [], h => h
+  | _ :: _, h => h.of_cons
+
+theorem Pairwise.imp_of_mem {S : α → α → Prop}
+    (H : ∀ {a b}, a ∈ l → b ∈ l → R a b → S a b) (p : Pairwise R l) : Pairwise S l := by
+  induction p with
+  | nil => constructor
+  | @cons a l r _ ih =>
+    constructor
+    · exact fun x h => H (mem_cons_self ..) (mem_cons_of_mem _ h) <| r x h
+    · exact ih fun m m' => H (mem_cons_of_mem _ m) (mem_cons_of_mem _ m')
+
+theorem Pairwise.and (hR : Pairwise R l) (hS : Pairwise S l) :
+    l.Pairwise fun a b => R a b ∧ S a b := by
+  induction hR with
+  | nil => simp only [Pairwise.nil]
+  | cons R1 _ IH =>
+    simp only [Pairwise.nil, pairwise_cons] at hS ⊢
+    exact ⟨fun b bl => ⟨R1 b bl, hS.1 b bl⟩, IH hS.2⟩
+
+theorem pairwise_and_iff : l.Pairwise (fun a b => R a b ∧ S a b) ↔ Pairwise R l ∧ Pairwise S l :=
+  ⟨fun h => ⟨h.imp fun h => h.1, h.imp fun h => h.2⟩, fun ⟨hR, hS⟩ => hR.and hS⟩
+
+theorem Pairwise.imp₂ (H : ∀ a b, R a b → S a b → T a b)
+    (hR : Pairwise R l) (hS : l.Pairwise S) : l.Pairwise T :=
+  (hR.and hS).imp fun ⟨h₁, h₂⟩ => H _ _ h₁ h₂
+
+theorem Pairwise.iff_of_mem {S : α → α → Prop} {l : List α}
+    (H : ∀ {a b}, a ∈ l → b ∈ l → (R a b ↔ S a b)) : Pairwise R l ↔ Pairwise S l :=
+  ⟨Pairwise.imp_of_mem fun m m' => (H m m').1, Pairwise.imp_of_mem fun m m' => (H m m').2⟩
+
+theorem Pairwise.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {l : List α} :
+    Pairwise R l ↔ Pairwise S l :=
+  Pairwise.iff_of_mem fun _ _ => H ..
+
+theorem pairwise_of_forall {l : List α} (H : ∀ x y, R x y) : Pairwise R l := by
+  induction l <;> simp [*]
+
+theorem Pairwise.and_mem {l : List α} :
+    Pairwise R l ↔ Pairwise (fun x y => x ∈ l ∧ y ∈ l ∧ R x y) l :=
+  Pairwise.iff_of_mem <| by simp (config := { contextual := true })
+
+theorem Pairwise.imp_mem {l : List α} :
+    Pairwise R l ↔ Pairwise (fun x y => x ∈ l → y ∈ l → R x y) l :=
+  Pairwise.iff_of_mem <| by simp (config := { contextual := true })
+
+theorem Pairwise.forall_of_forall_of_flip (h₁ : ∀ x ∈ l, R x x) (h₂ : Pairwise R l)
+    (h₃ : l.Pairwise (flip R)) : ∀ ⦃x⦄, x ∈ l → ∀ ⦃y⦄, y ∈ l → R x y := by
+  induction l with
+  | nil => exact forall_mem_nil _
+  | cons a l ih =>
+    rw [pairwise_cons] at h₂ h₃
+    simp only [mem_cons]
+    rintro x (rfl | hx) y (rfl | hy)
+    · exact h₁ _ (l.mem_cons_self _)
+    · exact h₂.1 _ hy
+    · exact h₃.1 _ hx
+    · exact ih (fun x hx => h₁ _ <| mem_cons_of_mem _ hx) h₂.2 h₃.2 hx hy
+
+theorem pairwise_singleton (R) (a : α) : Pairwise R [a] := by simp
+
+theorem pairwise_pair {a b : α} : Pairwise R [a, b] ↔ R a b := by simp
+
+theorem pairwise_map {l : List α} :
+    (l.map f).Pairwise R ↔ l.Pairwise fun a b => R (f a) (f b) := by
+  induction l
+  · simp
+  · simp only [map, pairwise_cons, forall_mem_map, *]
+
+theorem Pairwise.of_map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, S (f a) (f b) → R a b)
+    (p : Pairwise S (map f l)) : Pairwise R l :=
+  (pairwise_map.1 p).imp (H _ _)
+
+theorem Pairwise.map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
+    (p : Pairwise R l) : Pairwise S (map f l) :=
+  pairwise_map.2 <| p.imp (H _ _)
+
+theorem pairwise_filterMap (f : β → Option α) {l : List β} :
+    Pairwise R (filterMap f l) ↔ Pairwise (fun a a' : β => ∀ b ∈ f a, ∀ b' ∈ f a', R b b') l := by
+  let _S (a a' : β) := ∀ b ∈ f a, ∀ b' ∈ f a', R b b'
+  simp only [Option.mem_def]
+  induction l with
+  | nil => simp only [filterMap, Pairwise.nil]
+  | cons a l IH => ?_
+  match e : f a with
+  | none =>
+    rw [filterMap_cons_none e, pairwise_cons]
+    simp only [e, false_implies, implies_true, true_and, IH]
+  | some b =>
+    rw [filterMap_cons_some e]
+    simpa [IH, e] using fun _ =>
+      ⟨fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab⟩
+
+theorem Pairwise.filterMap {S : β → β → Prop} (f : α → Option β)
+    (H : ∀ a a' : α, R a a' → ∀ b ∈ f a, ∀ b' ∈ f a', S b b') {l : List α} (p : Pairwise R l) :
+    Pairwise S (filterMap f l) :=
+  (pairwise_filterMap _).2 <| p.imp (H _ _)
+
+@[deprecated Pairwise.filterMap (since := "2024-07-29")] abbrev Pairwise.filter_map := @Pairwise.filterMap
+
+theorem pairwise_filter (p : α → Prop) [DecidablePred p] {l : List α} :
+    Pairwise R (filter p l) ↔ Pairwise (fun x y => p x → p y → R x y) l := by
+  rw [← filterMap_eq_filter, pairwise_filterMap]
+  simp
+
+theorem Pairwise.filter (p : α → Bool) : Pairwise R l → Pairwise R (filter p l) :=
+  Pairwise.sublist (filter_sublist _)
+
+theorem pairwise_append {l₁ l₂ : List α} :
+    (l₁ ++ l₂).Pairwise R ↔ l₁.Pairwise R ∧ l₂.Pairwise R ∧ ∀ a ∈ l₁, ∀ b ∈ l₂, R a b := by
+  induction l₁ <;> simp [*, or_imp, forall_and, and_assoc, and_left_comm]
+
+theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {l₁ l₂ : List α} :
+    Pairwise R (l₁ ++ l₂) ↔ Pairwise R (l₂ ++ l₁) := by
+  have (l₁ l₂ : List α) (H : ∀ x : α, x ∈ l₁ → ∀ y : α, y ∈ l₂ → R x y)
+    (x : α) (xm : x ∈ l₂) (y : α) (ym : y ∈ l₁) : R x y := s (H y ym x xm)
+  simp only [pairwise_append, and_left_comm]; rw [Iff.intro (this l₁ l₂) (this l₂ l₁)]
+
+theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {a : α} {l₁ l₂ : List α} :
+    Pairwise R (l₁ ++ a :: l₂) ↔ Pairwise R (a :: (l₁ ++ l₂)) := by
+  show Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
+  rw [← append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l₁), pairwise_append_comm s]
+  simp only [mem_append, or_comm]
+
+theorem pairwise_join {L : List (List α)} :
+    Pairwise R (join L) ↔
+      (∀ l ∈ L, Pairwise R l) ∧ Pairwise (fun l₁ l₂ => ∀ x ∈ l₁, ∀ y ∈ l₂, R x y) L := by
+  induction L with
+  | nil => simp
+  | cons l L IH =>
+    simp only [join, pairwise_append, IH, mem_join, exists_imp, and_imp, forall_mem_cons,
+      pairwise_cons, and_assoc, and_congr_right_iff]
+    rw [and_comm, and_congr_left_iff]
+    intros; exact ⟨fun h a b c d e => h c d e a b, fun h c d e a b => h a b c d e⟩
+
+theorem pairwise_bind {R : β → β → Prop} {l : List α} {f : α → List β} :
+    List.Pairwise R (l.bind f) ↔
+      (∀ a ∈ l, Pairwise R (f a)) ∧ Pairwise (fun a₁ a₂ => ∀ x ∈ f a₁, ∀ y ∈ f a₂, R x y) l := by
+  simp [List.bind, pairwise_join, pairwise_map]
+
+theorem pairwise_reverse {l : List α} :
+    l.reverse.Pairwise R ↔ l.Pairwise (fun a b => R b a) := by
+  induction l <;> simp [*, pairwise_append, and_comm]
+
+@[simp] theorem pairwise_replicate {n : Nat} {a : α} :
+    (replicate n a).Pairwise R ↔ n ≤ 1 ∨ R a a := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [replicate_succ, pairwise_cons, mem_replicate, ne_eq, and_imp,
+      forall_eq_apply_imp_iff, ih]
+    constructor
+    · rintro ⟨h, h' | h'⟩
+      · by_cases w : n = 0
+        · left
+          subst w
+          simp
+        · right
+          exact h w
+      · right
+        exact h'
+    · rintro (h | h)
+      · obtain rfl := eq_zero_of_le_zero (le_of_lt_succ h)
+        simp
+      · exact ⟨fun _ => h, Or.inr h⟩
+
+theorem Pairwise.drop {l : List α} {n : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.drop n) :=
+  h.sublist (drop_sublist _ _)
+
+theorem Pairwise.take {l : List α} {n : Nat} (h : List.Pairwise R l) : List.Pairwise R (l.take n) :=
+  h.sublist (take_sublist _ _)
+
+theorem pairwise_iff_forall_sublist : l.Pairwise R ↔ (∀ {a b}, [a,b] <+ l → R a b) := by
+  induction l with
+  | nil => simp
+  | cons hd tl IH =>
+    rw [List.pairwise_cons]
+    constructor <;> intro h
+    · intro
+      | a, b, .cons _ hab => exact IH.mp h.2 hab
+      | _, b, .cons₂ _ hab => refine h.1 _ (hab.subset ?_); simp
+    · constructor
+      · intro x hx
+        apply h
+        rw [List.cons_sublist_cons, List.singleton_sublist]
+        exact hx
+      · apply IH.mpr
+        intro a b hab
+        apply h; exact hab.cons _
+
+/-! ### Nodup -/
+
+@[simp]
+theorem nodup_nil : @Nodup α [] :=
+  Pairwise.nil
+
+@[simp]
+theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by
+  simp only [Nodup, pairwise_cons, forall_mem_ne]
+
+theorem Nodup.sublist : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
+  Pairwise.sublist
+
+theorem Sublist.nodup : l₁ <+ l₂ → Nodup l₂ → Nodup l₁ :=
+  Nodup.sublist
+
+theorem getElem?_inj {xs : List α}
+    (h₀ : i < xs.length) (h₁ : Nodup xs) (h₂ : xs[i]? = xs[j]?) : i = j := by
+  induction xs generalizing i j with
+  | nil => cases h₀
+  | cons x xs ih =>
+    match i, j with
+    | 0, 0 => rfl
+    | i+1, j+1 =>
+      cases h₁ with
+      | cons ha h₁ =>
+        simp only [getElem?_cons_succ] at h₂
+        exact congrArg (· + 1) (ih (Nat.lt_of_succ_lt_succ h₀) h₁ h₂)
+    | i+1, 0 => ?_
+    | 0, j+1 => ?_
+    all_goals
+      simp only [get?_eq_getElem?, getElem?_cons_zero, getElem?_cons_succ] at h₂
+      cases h₁; rename_i h' h
+      have := h x ?_ rfl; cases this
+      rw [mem_iff_get?]
+      simp only [get?_eq_getElem?]
+    exact ⟨_, h₂⟩; exact ⟨_ , h₂.symm⟩
+
+@[simp] theorem nodup_replicate {n : Nat} {a : α} :
+    (replicate n a).Nodup ↔ n ≤ 1 := by simp [Nodup]
+
+end List

src/Init/Data/List/Sublist.lean

@@ -0,0 +1,754 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.TakeDrop
+
+/-!
+# Lemmas about `List.Subset`, `List.Sublist`, `List.IsPrefix`, `List.IsSuffix`, and `List.IsInfix`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ### isPrefixOf -/
+section isPrefixOf
+variable [BEq α]
+
+@[simp] theorem isPrefixOf_cons₂_self [LawfulBEq α] {a : α} :
+    isPrefixOf (a::as) (a::bs) = isPrefixOf as bs := by simp [isPrefixOf_cons₂]
+
+@[simp] theorem isPrefixOf_length_pos_nil {L : List α} (h : 0 < L.length) : isPrefixOf L [] = false := by
+  cases L <;> simp_all [isPrefixOf]
+
+@[simp] theorem isPrefixOf_replicate {a : α} :
+    isPrefixOf l (replicate n a) = (decide (l.length ≤ n) && l.all (· == a)) := by
+  induction l generalizing n with
+  | nil => simp
+  | cons h t ih =>
+    cases n
+    · simp
+    · simp [replicate_succ, isPrefixOf_cons₂, ih, Nat.succ_le_succ_iff, Bool.and_left_comm]
+
+end isPrefixOf
+
+/-! ### isSuffixOf -/
+section isSuffixOf
+variable [BEq α]
+
+@[simp] theorem isSuffixOf_cons_nil : isSuffixOf (a::as) ([] : List α) = false := by
+  simp [isSuffixOf]
+
+@[simp] theorem isSuffixOf_replicate {a : α} :
+    isSuffixOf l (replicate n a) = (decide (l.length ≤ n) && l.all (· == a)) := by
+  simp [isSuffixOf, all_eq]
+
+end isSuffixOf
+
+/-! ### Subset -/
+
+/-! ### List subset -/
+
+theorem subset_def {l₁ l₂ : List α} : l₁ ⊆ l₂ ↔ ∀ {a : α}, a ∈ l₁ → a ∈ l₂ := .rfl
+
+@[simp] theorem nil_subset (l : List α) : [] ⊆ l := nofun
+
+@[simp] theorem Subset.refl (l : List α) : l ⊆ l := fun _ i => i
+
+theorem Subset.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=
+  fun _ i => h₂ (h₁ i)
+
+instance : Trans (Membership.mem : α → List α → Prop) Subset Membership.mem :=
+  ⟨fun h₁ h₂ => h₂ h₁⟩
+
+instance : Trans (Subset : List α → List α → Prop) Subset Subset :=
+  ⟨Subset.trans⟩
+
+@[simp] theorem subset_cons_self (a : α) (l : List α) : l ⊆ a :: l := fun _ => Mem.tail _
+
+theorem subset_of_cons_subset {a : α} {l₁ l₂ : List α} : a :: l₁ ⊆ l₂ → l₁ ⊆ l₂ :=
+  fun s _ i => s (mem_cons_of_mem _ i)
+
+theorem subset_cons_of_subset (a : α) {l₁ l₂ : List α} : l₁ ⊆ l₂ → l₁ ⊆ a :: l₂ :=
+  fun s _ i => .tail _ (s i)
+
+theorem cons_subset_cons {l₁ l₂ : List α} (a : α) (s : l₁ ⊆ l₂) : a :: l₁ ⊆ a :: l₂ :=
+  fun _ => by simp only [mem_cons]; exact Or.imp_right (@s _)
+
+@[simp] theorem cons_subset : a :: l ⊆ m ↔ a ∈ m ∧ l ⊆ m := by
+  simp only [subset_def, mem_cons, or_imp, forall_and, forall_eq]
+
+@[simp] theorem subset_nil {l : List α} : l ⊆ [] ↔ l = [] :=
+  ⟨fun h => match l with | [] => rfl | _::_ => (nomatch h (.head ..)), fun | rfl => Subset.refl _⟩
+
+theorem map_subset {l₁ l₂ : List α} {f : α → β} (h : l₁ ⊆ l₂) : map f l₁ ⊆ map f l₂ :=
+  fun x => by simp only [mem_map]; exact .imp fun a => .imp_left (@h _)
+
+theorem filter_subset {l₁ l₂ : List α} (p : α → Bool) (H : l₁ ⊆ l₂) : filter p l₁ ⊆ filter p l₂ :=
+  fun x => by simp_all [mem_filter, subset_def.1 H]
+
+theorem filterMap_subset {l₁ l₂ : List α} (f : α → Option β) (H : l₁ ⊆ l₂) :
+    filterMap f l₁ ⊆ filterMap f l₂ := by
+  intro x
+  simp only [mem_filterMap]
+  rintro ⟨a, h, w⟩
+  exact ⟨a, H h, w⟩
+
+@[simp] theorem subset_append_left (l₁ l₂ : List α) : l₁ ⊆ l₁ ++ l₂ := fun _ => mem_append_left _
+
+@[simp] theorem subset_append_right (l₁ l₂ : List α) : l₂ ⊆ l₁ ++ l₂ := fun _ => mem_append_right _
+
+theorem subset_append_of_subset_left (l₂ : List α) : l ⊆ l₁ → l ⊆ l₁ ++ l₂ :=
+fun s => Subset.trans s <| subset_append_left _ _
+
+theorem subset_append_of_subset_right (l₁ : List α) : l ⊆ l₂ → l ⊆ l₁ ++ l₂ :=
+fun s => Subset.trans s <| subset_append_right _ _
+
+@[simp] theorem append_subset {l₁ l₂ l : List α} :
+    l₁ ++ l₂ ⊆ l ↔ l₁ ⊆ l ∧ l₂ ⊆ l := by simp [subset_def, or_imp, forall_and]
+
+theorem replicate_subset {n : Nat} {a : α} {l : List α} : replicate n a ⊆ l ↔ n = 0 ∨ a ∈ l := by
+  induction n with
+  | zero => simp
+  | succ n ih => simp (config := {contextual := true}) [replicate_succ, ih, cons_subset]
+
+theorem subset_replicate {n : Nat} {a : α} {l : List α} (h : n ≠ 0) : l ⊆ replicate n a ↔ ∀ x ∈ l, x = a := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [cons_subset, mem_replicate, ne_eq, ih, mem_cons, forall_eq_or_imp,
+      and_congr_left_iff, and_iff_right_iff_imp]
+    solve_by_elim
+
+@[simp] theorem reverse_subset {l₁ l₂ : List α} : reverse l₁ ⊆ l₂ ↔ l₁ ⊆ l₂ := by
+  simp [subset_def]
+
+@[simp] theorem subset_reverse {l₁ l₂ : List α} : l₁ ⊆ reverse l₂ ↔ l₁ ⊆ l₂ := by
+  simp [subset_def]
+
+/-! ### Sublist and isSublist -/
+
+@[simp] theorem nil_sublist : ∀ l : List α, [] <+ l
+  | [] => .slnil
+  | a :: l => (nil_sublist l).cons a
+
+@[simp] theorem Sublist.refl : ∀ l : List α, l <+ l
+  | [] => .slnil
+  | a :: l => (Sublist.refl l).cons₂ a
+
+theorem Sublist.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ <+ l₂) (h₂ : l₂ <+ l₃) : l₁ <+ l₃ := by
+  induction h₂ generalizing l₁ with
+  | slnil => exact h₁
+  | cons _ _ IH => exact (IH h₁).cons _
+  | @cons₂ l₂ _ a _ IH =>
+    generalize e : a :: l₂ = l₂'
+    match e ▸ h₁ with
+    | .slnil => apply nil_sublist
+    | .cons a' h₁' => cases e; apply (IH h₁').cons
+    | .cons₂ a' h₁' => cases e; apply (IH h₁').cons₂
+
+instance : Trans (@Sublist α) Sublist Sublist := ⟨Sublist.trans⟩
+
+@[simp] theorem sublist_cons_self (a : α) (l : List α) : l <+ a :: l := (Sublist.refl l).cons _
+
+theorem sublist_of_cons_sublist : a :: l₁ <+ l₂ → l₁ <+ l₂ :=
+  (sublist_cons_self a l₁).trans
+
+@[simp]
+theorem cons_sublist_cons : a :: l₁ <+ a :: l₂ ↔ l₁ <+ l₂ :=
+  ⟨fun | .cons _ s => sublist_of_cons_sublist s | .cons₂ _ s => s, .cons₂ _⟩
+
+theorem sublist_or_mem_of_sublist (h : l <+ l₁ ++ a :: l₂) : l <+ l₁ ++ l₂ ∨ a ∈ l := by
+  induction l₁ generalizing l with
+  | nil => match h with
+    | .cons _ h => exact .inl h
+    | .cons₂ _ h => exact .inr (.head ..)
+  | cons b l₁ IH =>
+    match h with
+    | .cons _ h => exact (IH h).imp_left (Sublist.cons _)
+    | .cons₂ _ h => exact (IH h).imp (Sublist.cons₂ _) (.tail _)
+
+theorem Sublist.subset : l₁ <+ l₂ → l₁ ⊆ l₂
+  | .slnil, _, h => h
+  | .cons _ s, _, h => .tail _ (s.subset h)
+  | .cons₂ .., _, .head .. => .head ..
+  | .cons₂ _ s, _, .tail _ h => .tail _ (s.subset h)
+
+instance : Trans (@Sublist α) Subset Subset :=
+  ⟨fun h₁ h₂ => trans h₁.subset h₂⟩
+
+instance : Trans Subset (@Sublist α) Subset :=
+  ⟨fun h₁ h₂ => trans h₁ h₂.subset⟩
+
+instance : Trans (Membership.mem : α → List α → Prop) Sublist Membership.mem :=
+  ⟨fun h₁ h₂ => h₂.subset h₁⟩
+
+theorem mem_of_cons_sublist {a : α} {l₁ l₂ : List α} (s : a :: l₁ <+ l₂) : a ∈ l₂ :=
+  (cons_subset.1 s.subset).1
+
+@[simp] theorem sublist_nil {l : List α} : l <+ [] ↔ l = [] :=
+  ⟨fun s => subset_nil.1 s.subset, fun H => H ▸ Sublist.refl _⟩
+
+theorem Sublist.length_le : l₁ <+ l₂ → length l₁ ≤ length l₂
+  | .slnil => Nat.le_refl 0
+  | .cons _l s => le_succ_of_le (length_le s)
+  | .cons₂ _ s => succ_le_succ (length_le s)
+
+theorem Sublist.eq_of_length : l₁ <+ l₂ → length l₁ = length l₂ → l₁ = l₂
+  | .slnil, _ => rfl
+  | .cons a s, h => nomatch Nat.not_lt.2 s.length_le (h ▸ lt_succ_self _)
+  | .cons₂ a s, h => by rw [s.eq_of_length (succ.inj h)]
+
+theorem Sublist.eq_of_length_le (s : l₁ <+ l₂) (h : length l₂ ≤ length l₁) : l₁ = l₂ :=
+  s.eq_of_length <| Nat.le_antisymm s.length_le h
+
+theorem Sublist.length_eq (s : l₁ <+ l₂) : length l₁ = length l₂ ↔ l₁ = l₂ :=
+  ⟨s.eq_of_length, congrArg _⟩
+
+protected theorem Sublist.map (f : α → β) {l₁ l₂} (s : l₁ <+ l₂) : map f l₁ <+ map f l₂ := by
+  induction s with
+  | slnil => simp
+  | cons a s ih =>
+    simpa using cons (f a) ih
+  | cons₂ a s ih =>
+    simpa using cons₂ (f a) ih
+
+protected theorem Sublist.filterMap (f : α → Option β) (s : l₁ <+ l₂) :
+    filterMap f l₁ <+ filterMap f l₂ := by
+  induction s <;> simp [filterMap_cons] <;> split <;> simp [*, cons, cons₂]
+
+protected theorem Sublist.filter (p : α → Bool) {l₁ l₂} (s : l₁ <+ l₂) : filter p l₁ <+ filter p l₂ := by
+  rw [← filterMap_eq_filter]; apply s.filterMap
+
+theorem sublist_filterMap_iff {l₁ : List β} {f : α → Option β} :
+    l₁ <+ l₂.filterMap f ↔ ∃ l', l' <+ l₂ ∧ l₁ = l'.filterMap f := by
+  induction l₂ generalizing l₁ with
+  | nil => simp
+  | cons a l₂ ih =>
+    simp only [filterMap_cons]
+    split
+    · simp only [ih]
+      constructor
+      · rintro ⟨l', h, rfl⟩
+        exact ⟨l', Sublist.cons a h, rfl⟩
+      · rintro ⟨l', h, rfl⟩
+        cases h with
+        | cons _ h =>
+          exact ⟨l', h, rfl⟩
+        | cons₂ _ h =>
+          rename_i l'
+          exact ⟨l', h, by simp_all⟩
+    · constructor
+      · intro w
+        cases w with
+        | cons _ h =>
+          obtain ⟨l', s, rfl⟩ := ih.1 h
+          exact ⟨l', Sublist.cons a s, rfl⟩
+        | cons₂ _ h =>
+          rename_i l'
+          obtain ⟨l', s, rfl⟩ := ih.1 h
+          refine ⟨a :: l', Sublist.cons₂ a s, ?_⟩
+          rwa [filterMap_cons_some]
+      · rintro ⟨l', h, rfl⟩
+        replace h := h.filterMap f
+        rwa [filterMap_cons_some] at h
+        assumption
+
+theorem sublist_map_iff {l₁ : List β} {f : α → β} :
+    l₁ <+ l₂.map f ↔ ∃ l', l' <+ l₂ ∧ l₁ = l'.map f := by
+  simp only [← filterMap_eq_map, sublist_filterMap_iff]
+
+theorem sublist_filter_iff {l₁ : List α} {p : α → Bool} :
+    l₁ <+ l₂.filter p ↔ ∃ l', l' <+ l₂ ∧ l₁ = l'.filter p := by
+  simp only [← filterMap_eq_filter, sublist_filterMap_iff]
+
+@[simp] theorem sublist_append_left : ∀ l₁ l₂ : List α, l₁ <+ l₁ ++ l₂
+  | [], _ => nil_sublist _
+  | _ :: l₁, l₂ => (sublist_append_left l₁ l₂).cons₂ _
+
+@[simp] theorem sublist_append_right : ∀ l₁ l₂ : List α, l₂ <+ l₁ ++ l₂
+  | [], _ => Sublist.refl _
+  | _ :: l₁, l₂ => (sublist_append_right l₁ l₂).cons _
+
+@[simp] theorem singleton_sublist {a : α} {l} : [a] <+ l ↔ a ∈ l := by
+  refine ⟨fun h => h.subset (mem_singleton_self _), fun h => ?_⟩
+  obtain ⟨_, _, rfl⟩ := append_of_mem h
+  exact ((nil_sublist _).cons₂ _).trans (sublist_append_right ..)
+
+theorem sublist_append_of_sublist_left (s : l <+ l₁) : l <+ l₁ ++ l₂ :=
+  s.trans <| sublist_append_left ..
+
+theorem sublist_append_of_sublist_right (s : l <+ l₂) : l <+ l₁ ++ l₂ :=
+  s.trans <| sublist_append_right ..
+
+@[simp] theorem append_sublist_append_left : ∀ l, l ++ l₁ <+ l ++ l₂ ↔ l₁ <+ l₂
+  | [] => Iff.rfl
+  | _ :: l => cons_sublist_cons.trans (append_sublist_append_left l)
+
+theorem Sublist.append_left : l₁ <+ l₂ → ∀ l, l ++ l₁ <+ l ++ l₂ :=
+  fun h l => (append_sublist_append_left l).mpr h
+
+theorem Sublist.append_right : l₁ <+ l₂ → ∀ l, l₁ ++ l <+ l₂ ++ l
+  | .slnil, _ => Sublist.refl _
+  | .cons _ h, _ => (h.append_right _).cons _
+  | .cons₂ _ h, _ => (h.append_right _).cons₂ _
+
+theorem Sublist.append (hl : l₁ <+ l₂) (hr : r₁ <+ r₂) : l₁ ++ r₁ <+ l₂ ++ r₂ :=
+  (hl.append_right _).trans ((append_sublist_append_left _).2 hr)
+
+theorem sublist_cons_iff {a : α} {l l'} :
+    l <+ a :: l' ↔ l <+ l' ∨ ∃ r, l = a :: r ∧ r <+ l' := by
+  constructor
+  · intro h
+    cases h with
+    | cons _ h => exact Or.inl h
+    | cons₂ _ h => exact Or.inr ⟨_, rfl, h⟩
+  · rintro (h | ⟨r, rfl, h⟩)
+    · exact h.cons _
+    · exact h.cons₂ _
+
+theorem cons_sublist_iff {a : α} {l l'} :
+    a :: l <+ l' ↔ ∃ r₁ r₂, l' = r₁ ++ r₂ ∧ a ∈ r₁ ∧ l <+ r₂ := by
+  induction l' with
+  | nil => simp
+  | cons a' l' ih =>
+    constructor
+    · intro w
+      cases w with
+      | cons _ w =>
+        obtain ⟨r₁, r₂, rfl, h₁, h₂⟩ := ih.1 w
+        exact ⟨a' :: r₁, r₂, by simp, mem_cons_of_mem a' h₁, h₂⟩
+      | cons₂ _ w =>
+        exact ⟨[a], l', by simp, mem_singleton_self _, w⟩
+    · rintro ⟨r₁, r₂, w, h₁, h₂⟩
+      rw [w, ← singleton_append]
+      exact Sublist.append (by simpa) h₂
+
+theorem sublist_append_iff {l : List α} :
+    l <+ r₁ ++ r₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁ <+ r₁ ∧ l₂ <+ r₂ := by
+  induction r₁ generalizing l with
+  | nil =>
+    constructor
+    · intro w
+      refine ⟨[], l, by simp_all⟩
+    · rintro ⟨l₁, l₂, rfl, w₁, w₂⟩
+      simp_all
+  | cons r r₁ ih =>
+    constructor
+    · intro w
+      simp only [cons_append] at w
+      cases w with
+      | cons _ w =>
+        obtain ⟨l₁, l₂, rfl, w₁, w₂⟩ := ih.1 w
+        exact ⟨l₁, l₂, rfl, Sublist.cons r w₁, w₂⟩
+      | cons₂ _ w =>
+        rename_i l
+        obtain ⟨l₁, l₂, rfl, w₁, w₂⟩ := ih.1 w
+        refine ⟨r :: l₁, l₂, by simp, cons_sublist_cons.mpr w₁, w₂⟩
+    · rintro ⟨l₁, l₂, rfl, w₁, w₂⟩
+      cases w₁ with
+      | cons _ w₁ =>
+        exact Sublist.cons _ (Sublist.append w₁ w₂)
+      | cons₂ _ w₁ =>
+        rename_i l
+        exact Sublist.cons₂ _ (Sublist.append w₁ w₂)
+
+theorem append_sublist_iff {l₁ l₂ : List α} :
+    l₁ ++ l₂ <+ r ↔ ∃ r₁ r₂, r = r₁ ++ r₂ ∧ l₁ <+ r₁ ∧ l₂ <+ r₂ := by
+  induction l₁ generalizing r with
+  | nil =>
+    constructor
+    · intro w
+      refine ⟨[], r, by simp_all⟩
+    · rintro ⟨r₁, r₂, rfl, -, w₂⟩
+      simp only [nil_append]
+      exact sublist_append_of_sublist_right w₂
+  | cons a l₁ ih =>
+    constructor
+    · rw [cons_append, cons_sublist_iff]
+      rintro ⟨r₁, r₂, rfl, h₁, h₂⟩
+      obtain ⟨s₁, s₂, rfl, t₁, t₂⟩ := ih.1 h₂
+      refine ⟨r₁ ++ s₁, s₂, by simp, ?_, t₂⟩
+      rw [← singleton_append]
+      exact Sublist.append (by simpa) t₁
+    · rintro ⟨r₁, r₂, rfl, h₁, h₂⟩
+      exact Sublist.append h₁ h₂
+
+theorem Sublist.reverse : l₁ <+ l₂ → l₁.reverse <+ l₂.reverse
+  | .slnil => Sublist.refl _
+  | .cons _ h => by rw [reverse_cons]; exact sublist_append_of_sublist_left h.reverse
+  | .cons₂ _ h => by rw [reverse_cons, reverse_cons]; exact h.reverse.append_right _
+
+@[simp] theorem reverse_sublist : l₁.reverse <+ l₂.reverse ↔ l₁ <+ l₂ :=
+  ⟨fun h => l₁.reverse_reverse ▸ l₂.reverse_reverse ▸ h.reverse, Sublist.reverse⟩
+
+theorem sublist_reverse_iff : l₁ <+ l₂.reverse ↔ l₁.reverse <+ l₂ :=
+  by rw [← reverse_sublist, reverse_reverse]
+
+@[simp] theorem append_sublist_append_right (l) : l₁ ++ l <+ l₂ ++ l ↔ l₁ <+ l₂ :=
+  ⟨fun h => by
+    have := h.reverse
+    simp only [reverse_append, append_sublist_append_left, reverse_sublist] at this
+    exact this,
+   fun h => h.append_right l⟩
+
+@[simp] theorem replicate_sublist_replicate {m n} (a : α) :
+    replicate m a <+ replicate n a ↔ m ≤ n := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · have := h.length_le; simp only [length_replicate] at this ⊢; exact this
+  · induction h with
+    | refl => apply Sublist.refl
+    | step => simp [*, replicate, Sublist.cons]
+
+theorem sublist_replicate_iff : l <+ replicate m a ↔ ∃ n, n ≤ m ∧ l = replicate n a := by
+  induction l generalizing m with
+  | nil =>
+    simp only [nil_sublist, true_iff]
+    exact ⟨0, zero_le m, by simp⟩
+  | cons b l ih =>
+    constructor
+    · intro w
+      cases m with
+      | zero => simp at w
+      | succ m =>
+        simp [replicate_succ] at w
+        cases w with
+        | cons _ w =>
+          obtain ⟨n, le, rfl⟩ := ih.1 (sublist_of_cons_sublist w)
+          obtain rfl := (mem_replicate.1 (mem_of_cons_sublist w)).2
+          exact ⟨n+1, Nat.add_le_add_right le 1, rfl⟩
+        | cons₂ _ w =>
+          obtain ⟨n, le, rfl⟩ := ih.1 w
+          refine ⟨n+1, Nat.add_le_add_right le 1, by simp [replicate_succ]⟩
+    · rintro ⟨n, le, w⟩
+      rw [w]
+      exact (replicate_sublist_replicate a).2 le
+
+theorem sublist_join_of_mem {L : List (List α)} {l} (h : l ∈ L) : l <+ L.join := by
+  induction L with
+  | nil => cases h
+  | cons l' L ih =>
+    rcases mem_cons.1 h with (rfl | h)
+    · simp [h]
+    · simp [ih h, join_cons, sublist_append_of_sublist_right]
+
+theorem sublist_join_iff {L : List (List α)} {l} :
+    l <+ L.join ↔
+      ∃ L' : List (List α), l = L'.join ∧ ∀ i (_ : i < L'.length), L'[i] <+ L[i]?.getD [] := by
+  induction L generalizing l with
+  | nil =>
+    constructor
+    · intro w
+      simp only [join_nil, sublist_nil] at w
+      subst w
+      exact ⟨[], by simp, fun i x => by cases x⟩
+    · rintro ⟨L', rfl, h⟩
+      simp only [join_nil, sublist_nil, join_eq_nil_iff]
+      simp only [getElem?_nil, Option.getD_none, sublist_nil] at h
+      exact (forall_getElem L' (· = [])).1 h
+  | cons l' L ih =>
+    simp only [join_cons, sublist_append_iff, ih]
+    constructor
+    · rintro ⟨l₁, l₂, rfl, s, L', rfl, h⟩
+      refine ⟨l₁ :: L', by simp, ?_⟩
+      intro i lt
+      cases i <;> simp_all
+    · rintro ⟨L', rfl, h⟩
+      cases L' with
+      | nil =>
+        exact ⟨[], [], by simp, by simp, [], by simp, fun i x => by cases x⟩
+      | cons l₁ L' =>
+        exact ⟨l₁, L'.join, by simp, by simpa using h 0 (by simp), L', rfl,
+          fun i lt => by simpa using h (i+1) (Nat.add_lt_add_right lt 1)⟩
+
+theorem join_sublist_iff {L : List (List α)} {l} :
+    L.join <+ l ↔
+      ∃ L' : List (List α), l = L'.join ∧ ∀ i (_ : i < L.length), L[i] <+ L'[i]?.getD [] := by
+  induction L generalizing l with
+  | nil =>
+    constructor
+    · intro _
+      exact ⟨[l], by simp, fun i x => by cases x⟩
+    · rintro ⟨L', rfl, _⟩
+      simp only [join_nil, nil_sublist]
+  | cons l' L ih =>
+    simp only [join_cons, append_sublist_iff, ih]
+    constructor
+    · rintro ⟨l₁, l₂, rfl, s, L', rfl, h⟩
+      refine ⟨l₁ :: L', by simp, ?_⟩
+      intro i lt
+      cases i <;> simp_all
+    · rintro ⟨L', rfl, h⟩
+      cases L' with
+      | nil =>
+        exact ⟨[], [], by simp, by simpa using h 0 (by simp), [], by simp,
+          fun i x => by simpa using h (i+1) (Nat.add_lt_add_right x 1)⟩
+      | cons l₁ L' =>
+        exact ⟨l₁, L'.join, by simp, by simpa using h 0 (by simp), L', rfl,
+          fun i lt => by simpa using h (i+1) (Nat.add_lt_add_right lt 1)⟩
+
+@[simp] theorem isSublist_iff_sublist [BEq α] [LawfulBEq α] {l₁ l₂ : List α} :
+    l₁.isSublist l₂ ↔ l₁ <+ l₂ := by
+  cases l₁ <;> cases l₂ <;> simp [isSublist]
+  case cons.cons hd₁ tl₁ hd₂ tl₂ =>
+    if h_eq : hd₁ = hd₂ then
+      simp [h_eq, cons_sublist_cons, isSublist_iff_sublist]
+    else
+      simp only [beq_iff_eq, h_eq]
+      constructor
+      · intro h_sub
+        apply Sublist.cons
+        exact isSublist_iff_sublist.mp h_sub
+      · intro h_sub
+        cases h_sub
+        case cons h_sub =>
+          exact isSublist_iff_sublist.mpr h_sub
+        case cons₂ =>
+          contradiction
+
+instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <+ l₂) :=
+  decidable_of_iff (l₁.isSublist l₂) isSublist_iff_sublist
+
+/-! ### IsPrefix / IsSuffix / IsInfix -/
+
+@[simp] theorem prefix_append (l₁ l₂ : List α) : l₁ <+: l₁ ++ l₂ := ⟨l₂, rfl⟩
+
+@[simp] theorem suffix_append (l₁ l₂ : List α) : l₂ <:+ l₁ ++ l₂ := ⟨l₁, rfl⟩
+
+theorem infix_append (l₁ l₂ l₃ : List α) : l₂ <:+: l₁ ++ l₂ ++ l₃ := ⟨l₁, l₃, rfl⟩
+
+@[simp] theorem infix_append' (l₁ l₂ l₃ : List α) : l₂ <:+: l₁ ++ (l₂ ++ l₃) := by
+  rw [← List.append_assoc]; apply infix_append
+
+theorem IsPrefix.isInfix : l₁ <+: l₂ → l₁ <:+: l₂ := fun ⟨t, h⟩ => ⟨[], t, h⟩
+
+theorem IsSuffix.isInfix : l₁ <:+ l₂ → l₁ <:+: l₂ := fun ⟨t, h⟩ => ⟨t, [], by rw [h, append_nil]⟩
+
+@[simp] theorem nil_prefix (l : List α) : [] <+: l := ⟨l, rfl⟩
+
+@[simp] theorem nil_suffix (l : List α) : [] <:+ l := ⟨l, append_nil _⟩
+
+@[simp] theorem nil_infix (l : List α) : [] <:+: l := (nil_prefix _).isInfix
+
+@[simp] theorem prefix_refl (l : List α) : l <+: l := ⟨[], append_nil _⟩
+
+@[simp] theorem suffix_refl (l : List α) : l <:+ l := ⟨[], rfl⟩
+
+@[simp] theorem infix_refl (l : List α) : l <:+: l := (prefix_refl l).isInfix
+
+@[simp] theorem suffix_cons (a : α) : ∀ l, l <:+ a :: l := suffix_append [a]
+
+theorem infix_cons : l₁ <:+: l₂ → l₁ <:+: a :: l₂ := fun ⟨L₁, L₂, h⟩ => ⟨a :: L₁, L₂, h ▸ rfl⟩
+
+theorem infix_concat : l₁ <:+: l₂ → l₁ <:+: concat l₂ a := fun ⟨L₁, L₂, h⟩ =>
+  ⟨L₁, concat L₂ a, by simp [← h, concat_eq_append, append_assoc]⟩
+
+theorem IsPrefix.trans : ∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₂ → l₂ <+: l₃ → l₁ <+: l₃
+  | _, _, _, ⟨r₁, rfl⟩, ⟨r₂, rfl⟩ => ⟨r₁ ++ r₂, (append_assoc _ _ _).symm⟩
+
+theorem IsSuffix.trans : ∀ {l₁ l₂ l₃ : List α}, l₁ <:+ l₂ → l₂ <:+ l₃ → l₁ <:+ l₃
+  | _, _, _, ⟨l₁, rfl⟩, ⟨l₂, rfl⟩ => ⟨l₂ ++ l₁, append_assoc _ _ _⟩
+
+theorem IsInfix.trans : ∀ {l₁ l₂ l₃ : List α}, l₁ <:+: l₂ → l₂ <:+: l₃ → l₁ <:+: l₃
+  | l, _, _, ⟨l₁, r₁, rfl⟩, ⟨l₂, r₂, rfl⟩ => ⟨l₂ ++ l₁, r₁ ++ r₂, by simp only [append_assoc]⟩
+
+protected theorem IsInfix.sublist : l₁ <:+: l₂ → l₁ <+ l₂
+  | ⟨_, _, h⟩ => h ▸ (sublist_append_right ..).trans (sublist_append_left ..)
+
+protected theorem IsInfix.subset (hl : l₁ <:+: l₂) : l₁ ⊆ l₂ :=
+  hl.sublist.subset
+
+protected theorem IsPrefix.sublist (h : l₁ <+: l₂) : l₁ <+ l₂ :=
+  h.isInfix.sublist
+
+protected theorem IsPrefix.subset (hl : l₁ <+: l₂) : l₁ ⊆ l₂ :=
+  hl.sublist.subset
+
+protected theorem IsSuffix.sublist (h : l₁ <:+ l₂) : l₁ <+ l₂ :=
+  h.isInfix.sublist
+
+protected theorem IsSuffix.subset (hl : l₁ <:+ l₂) : l₁ ⊆ l₂ :=
+  hl.sublist.subset
+
+@[simp] theorem reverse_suffix : reverse l₁ <:+ reverse l₂ ↔ l₁ <+: l₂ :=
+  ⟨fun ⟨r, e⟩ => ⟨reverse r, by rw [← reverse_reverse l₁, ← reverse_append, e, reverse_reverse]⟩,
+   fun ⟨r, e⟩ => ⟨reverse r, by rw [← reverse_append, e]⟩⟩
+
+@[simp] theorem reverse_prefix : reverse l₁ <+: reverse l₂ ↔ l₁ <:+ l₂ := by
+  rw [← reverse_suffix]; simp only [reverse_reverse]
+
+@[simp] theorem reverse_infix : reverse l₁ <:+: reverse l₂ ↔ l₁ <:+: l₂ := by
+  refine ⟨fun ⟨s, t, e⟩ => ⟨reverse t, reverse s, ?_⟩, fun ⟨s, t, e⟩ => ⟨reverse t, reverse s, ?_⟩⟩
+  · rw [← reverse_reverse l₁, append_assoc, ← reverse_append, ← reverse_append, e,
+      reverse_reverse]
+  · rw [append_assoc, ← reverse_append, ← reverse_append, e]
+
+theorem IsInfix.length_le (h : l₁ <:+: l₂) : l₁.length ≤ l₂.length :=
+  h.sublist.length_le
+
+theorem IsPrefix.length_le (h : l₁ <+: l₂) : l₁.length ≤ l₂.length :=
+  h.sublist.length_le
+
+theorem IsSuffix.length_le (h : l₁ <:+ l₂) : l₁.length ≤ l₂.length :=
+  h.sublist.length_le
+
+@[simp] theorem infix_nil : l <:+: [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ infix_refl _)⟩
+
+@[simp] theorem prefix_nil : l <+: [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ prefix_refl _)⟩
+
+@[simp] theorem suffix_nil : l <:+ [] ↔ l = [] := ⟨(sublist_nil.1 ·.sublist), (· ▸ suffix_refl _)⟩
+
+theorem infix_iff_prefix_suffix (l₁ l₂ : List α) : l₁ <:+: l₂ ↔ ∃ t, l₁ <+: t ∧ t <:+ l₂ :=
+  ⟨fun ⟨_, t, e⟩ => ⟨l₁ ++ t, ⟨_, rfl⟩, e ▸ append_assoc .. ▸ ⟨_, rfl⟩⟩,
+    fun ⟨_, ⟨t, rfl⟩, s, e⟩ => ⟨s, t, append_assoc .. ▸ e⟩⟩
+
+theorem IsInfix.eq_of_length (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
+  h.sublist.eq_of_length
+
+theorem IsPrefix.eq_of_length (h : l₁ <+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
+  h.sublist.eq_of_length
+
+theorem IsSuffix.eq_of_length (h : l₁ <:+ l₂) : l₁.length = l₂.length → l₁ = l₂ :=
+  h.sublist.eq_of_length
+
+theorem prefix_of_prefix_length_le :
+    ∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
+  | [], l₂, _, _, _, _ => nil_prefix _
+  | a :: l₁, b :: l₂, _, ⟨r₁, rfl⟩, ⟨r₂, e⟩, ll => by
+    injection e with _ e'; subst b
+    rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩ (le_of_succ_le_succ ll) with ⟨r₃, rfl⟩
+    exact ⟨r₃, rfl⟩
+
+theorem prefix_or_prefix_of_prefix (h₁ : l₁ <+: l₃) (h₂ : l₂ <+: l₃) : l₁ <+: l₂ ∨ l₂ <+: l₁ :=
+  (Nat.le_total (length l₁) (length l₂)).imp (prefix_of_prefix_length_le h₁ h₂)
+    (prefix_of_prefix_length_le h₂ h₁)
+
+theorem suffix_of_suffix_length_le
+    (h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) (ll : length l₁ ≤ length l₂) : l₁ <:+ l₂ :=
+  reverse_prefix.1 <|
+    prefix_of_prefix_length_le (reverse_prefix.2 h₁) (reverse_prefix.2 h₂) (by simp [ll])
+
+theorem suffix_or_suffix_of_suffix (h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) : l₁ <:+ l₂ ∨ l₂ <:+ l₁ :=
+  (prefix_or_prefix_of_prefix (reverse_prefix.2 h₁) (reverse_prefix.2 h₂)).imp reverse_prefix.1
+    reverse_prefix.1
+
+theorem prefix_cons_iff : l₁ <+: a :: l₂ ↔ l₁ = [] ∨ ∃ t, l₁ = a :: t ∧ t <+: l₂ := by
+  cases l₁ with
+  | nil => simp
+  | cons a' l₁ =>
+    constructor
+    · rintro ⟨t, h⟩
+      simp at h
+      obtain ⟨rfl, rfl⟩ := h
+      exact Or.inr ⟨l₁, rfl, prefix_append l₁ t⟩
+    · rintro (h | ⟨t, w, ⟨s, h'⟩⟩)
+      · simp [h]
+      · simp only [w]
+        refine ⟨s, by simp [h']⟩
+
+@[simp] theorem cons_prefix_cons : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
+  simp only [prefix_cons_iff, cons.injEq, false_or]
+  constructor
+  · rintro ⟨t, ⟨rfl, rfl⟩, h⟩
+    exact ⟨rfl, h⟩
+  · rintro ⟨rfl, h⟩
+    exact ⟨l₁, ⟨rfl, rfl⟩, h⟩
+
+theorem suffix_cons_iff : l₁ <:+ a :: l₂ ↔ l₁ = a :: l₂ ∨ l₁ <:+ l₂ := by
+  constructor
+  · rintro ⟨⟨hd, tl⟩, hl₃⟩
+    · exact Or.inl hl₃
+    · simp only [cons_append] at hl₃
+      injection hl₃ with _ hl₄
+      exact Or.inr ⟨_, hl₄⟩
+  · rintro (rfl | hl₁)
+    · exact (a :: l₂).suffix_refl
+    · exact hl₁.trans (l₂.suffix_cons _)
+
+theorem infix_cons_iff : l₁ <:+: a :: l₂ ↔ l₁ <+: a :: l₂ ∨ l₁ <:+: l₂ := by
+  constructor
+  · rintro ⟨⟨hd, tl⟩, t, hl₃⟩
+    · exact Or.inl ⟨t, hl₃⟩
+    · simp only [cons_append] at hl₃
+      injection hl₃ with _ hl₄
+      exact Or.inr ⟨_, t, hl₄⟩
+  · rintro (h | hl₁)
+    · exact h.isInfix
+    · exact infix_cons hl₁
+
+theorem infix_of_mem_join : ∀ {L : List (List α)}, l ∈ L → l <:+: join L
+  | l' :: _, h =>
+    match h with
+    | List.Mem.head .. => infix_append [] _ _
+    | List.Mem.tail _ hlMemL =>
+      IsInfix.trans (infix_of_mem_join hlMemL) <| (suffix_append _ _).isInfix
+
+theorem prefix_append_right_inj (l) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ :=
+  exists_congr fun r => by rw [append_assoc, append_right_inj]
+
+@[simp]
+theorem prefix_cons_inj (a) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ :=
+  prefix_append_right_inj [a]
+
+theorem take_prefix (n) (l : List α) : take n l <+: l :=
+  ⟨_, take_append_drop _ _⟩
+
+theorem drop_suffix (n) (l : List α) : drop n l <:+ l :=
+  ⟨_, take_append_drop _ _⟩
+
+theorem take_sublist (n) (l : List α) : take n l <+ l :=
+  (take_prefix n l).sublist
+
+theorem drop_sublist (n) (l : List α) : drop n l <+ l :=
+  (drop_suffix n l).sublist
+
+theorem take_subset (n) (l : List α) : take n l ⊆ l :=
+  (take_sublist n l).subset
+
+theorem drop_subset (n) (l : List α) : drop n l ⊆ l :=
+  (drop_sublist n l).subset
+
+theorem mem_of_mem_take {l : List α} (h : a ∈ l.take n) : a ∈ l :=
+  take_subset n l h
+
+theorem mem_of_mem_drop {n} {l : List α} (h : a ∈ l.drop n) : a ∈ l :=
+  drop_subset _ _ h
+
+theorem IsPrefix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
+    l₁.filter p <+: l₂.filter p := by
+  obtain ⟨xs, rfl⟩ := h
+  rw [filter_append]; apply prefix_append
+
+theorem IsSuffix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+ l₂) :
+    l₁.filter p <:+ l₂.filter p := by
+  obtain ⟨xs, rfl⟩ := h
+  rw [filter_append]; apply suffix_append
+
+theorem IsInfix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <:+: l₂) :
+    l₁.filter p <:+: l₂.filter p := by
+  obtain ⟨xs, ys, rfl⟩ := h
+  rw [filter_append, filter_append]; apply infix_append _
+
+@[simp] theorem isPrefixOf_iff_prefix [BEq α] [LawfulBEq α] {l₁ l₂ : List α} :
+    l₁.isPrefixOf l₂ ↔ l₁ <+: l₂ := by
+  induction l₁ generalizing l₂ with
+  | nil => simp
+  | cons a l₁ ih =>
+    cases l₂ with
+    | nil => simp
+    | cons a' l₂ => simp [isPrefixOf, ih]
+
+instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <+: l₂) :=
+  decidable_of_iff (l₁.isPrefixOf l₂) isPrefixOf_iff_prefix
+
+@[simp] theorem isSuffixOf_iff_suffix [BEq α] [LawfulBEq α] {l₁ l₂ : List α} :
+    l₁.isSuffixOf l₂ ↔ l₁ <:+ l₂ := by
+  simp [isSuffixOf]
+
+instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <:+ l₂) :=
+  decidable_of_iff (l₁.isSuffixOf l₂) isSuffixOf_iff_suffix
+
+end List

src/Init/Data/List/TakeDrop.lean

@@ -5,500 +5,443 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Lemmas
-import Init.Data.Nat.Lemmas
 
 /-!
-# Further lemmas about `List.take`, `List.drop`, `List.zip` and `List.zipWith`.
-
-These are in a separate file from most of the list lemmas
-as they required importing more lemmas about natural numbers, and use `omega`.
+# Lemmas about `List.zip`, `List.zipWith`, `List.zipWithAll`, and `List.unzip`.
 -/
 
 namespace List
 
 open Nat
 
-/-! ### take -/
+/-! ### take and drop
 
-@[simp] theorem length_take : ∀ (i : Nat) (l : List α), length (take i l) = min i (length l)
-  | 0, l => by simp [Nat.zero_min]
-  | succ n, [] => by simp [Nat.min_zero]
-  | succ n, _ :: l => by simp [Nat.succ_min_succ, length_take]
-
-theorem length_take_le (n) (l : List α) : length (take n l) ≤ n := by simp [Nat.min_le_left]
-
-theorem length_take_le' (n) (l : List α) : length (take n l) ≤ l.length :=
-  by simp [Nat.min_le_right]
-
-theorem length_take_of_le (h : n ≤ length l) : length (take n l) = n := by simp [Nat.min_eq_left h]
-
-theorem take_take : ∀ (n m) (l : List α), take n (take m l) = take (min n m) l
-  | n, 0, l => by rw [Nat.min_zero, take_zero, take_nil]
-  | 0, m, l => by rw [Nat.zero_min, take_zero, take_zero]
-  | succ n, succ m, nil => by simp only [take_nil]
-  | succ n, succ m, a :: l => by
-    simp only [take, succ_min_succ, take_take n m l]
-
-@[simp] theorem take_replicate (a : α) : ∀ n m : Nat, take n (replicate m a) = replicate (min n m) a
-  | n, 0 => by simp [Nat.min_zero]
-  | 0, m => by simp [Nat.zero_min]
-  | succ n, succ m => by simp [replicate_succ, succ_min_succ, take_replicate]
-
-@[simp] theorem drop_replicate (a : α) : ∀ n m : Nat, drop n (replicate m a) = replicate (m - n) a
-  | n, 0 => by simp
-  | 0, m => by simp
-  | succ n, succ m => by simp [replicate_succ, succ_sub_succ, drop_replicate]
-
-/-- Taking the first `n` elements in `l₁ ++ l₂` is the same as appending the first `n` elements
-of `l₁` to the first `n - l₁.length` elements of `l₂`. -/
-theorem take_append_eq_append_take {l₁ l₂ : List α} {n : Nat} :
-    take n (l₁ ++ l₂) = take n l₁ ++ take (n - l₁.length) l₂ := by
-  induction l₁ generalizing n
-  · simp
-  · cases n
-    · simp [*]
-    · simp only [cons_append, take_cons_succ, length_cons, succ_eq_add_one, cons.injEq,
-        append_cancel_left_eq, true_and, *]
-      congr 1
-      omega
-
-theorem take_append_of_le_length {l₁ l₂ : List α} {n : Nat} (h : n ≤ l₁.length) :
-    (l₁ ++ l₂).take n = l₁.take n := by
-  simp [take_append_eq_append_take, Nat.sub_eq_zero_of_le h]
-
-/-- Taking the first `l₁.length + i` elements in `l₁ ++ l₂` is the same as appending the first
-`i` elements of `l₂` to `l₁`. -/
-theorem take_append {l₁ l₂ : List α} (i : Nat) :
-    take (l₁.length + i) (l₁ ++ l₂) = l₁ ++ take i l₂ := by
-  rw [take_append_eq_append_take, take_all_of_le (Nat.le_add_right _ _), Nat.add_sub_cancel_left]
-
-/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
-length `> i`. Version designed to rewrite from the big list to the small list. -/
-theorem getElem_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
-    L[i] = (L.take j)[i]'(length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩) :=
-  getElem_of_eq (take_append_drop j L).symm _ ▸ getElem_append ..
-
-/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
-length `> i`. Version designed to rewrite from the small list to the big list. -/
-theorem getElem_take' (L : List α) {j i : Nat} {h : i < (L.take j).length} :
-    (L.take j)[i] =
-    L[i]'(Nat.lt_of_lt_of_le h (length_take_le' _ _)) := by
-  rw [length_take, Nat.lt_min] at h; rw [getElem_take L _ h.1]
-
-/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
-length `> i`. Version designed to rewrite from the big list to the small list. -/
-@[deprecated getElem_take (since := "2024-06-12")]
-theorem get_take (L : List α) {i j : Nat} (hi : i < L.length) (hj : i < j) :
-    get L ⟨i, hi⟩ = get (L.take j) ⟨i, length_take .. ▸ Nat.lt_min.mpr ⟨hj, hi⟩⟩ := by
-  simp [getElem_take _ hi hj]
-
-/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
-length `> i`. Version designed to rewrite from the small list to the big list. -/
-@[deprecated getElem_take (since := "2024-06-12")]
-theorem get_take' (L : List α) {j i} :
-    get (L.take j) i =
-    get L ⟨i.1, Nat.lt_of_lt_of_le i.2 (length_take_le' _ _)⟩ := by
-  simp [getElem_take']
-
-theorem getElem?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
-    (l.take n)[m]? = none :=
-  getElem?_eq_none <| Nat.le_trans (length_take_le _ _) h
-
-@[deprecated getElem?_take_eq_none (since := "2024-06-12")]
-theorem get?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
-    (l.take n).get? m = none := by
-  simp [getElem?_take_eq_none h]
-
-theorem getElem?_take_eq_if {l : List α} {n m : Nat} :
-    (l.take n)[m]? = if m < n then l[m]? else none := by
-  split
-  · next h => exact getElem?_take h
-  · next h => exact getElem?_take_eq_none (Nat.le_of_not_lt h)
-
-@[deprecated getElem?_take_eq_if (since := "2024-06-12")]
-theorem get?_take_eq_if {l : List α} {n m : Nat} :
-    (l.take n).get? m = if m < n then l.get? m else none := by
-  simp [getElem?_take_eq_if]
-
-theorem head?_take {l : List α} {n : Nat} :
-    (l.take n).head? = if n = 0 then none else l.head? := by
-  simp [head?_eq_getElem?, getElem?_take_eq_if]
-  split
-  · rw [if_neg (by omega)]
-  · rw [if_pos (by omega)]
-
-theorem head_take {l : List α} {n : Nat} (h : l.take n ≠ []) :
-    (l.take n).head h = l.head (by simp_all) := by
-  apply Option.some_inj.1
-  rw [← head?_eq_head, ← head?_eq_head, head?_take, if_neg]
-  simp_all
-
-theorem getLast?_take {l : List α} : (l.take n).getLast? = if n = 0 then none else l[n - 1]?.or l.getLast? := by
-  rw [getLast?_eq_getElem?, getElem?_take_eq_if, length_take]
-  split
-  · rw [if_neg (by omega)]
-    rw [Nat.min_def]
-    split
-    · rw [getElem?_eq_getElem (by omega)]
-      simp
-    · rw [← getLast?_eq_getElem?, getElem?_eq_none (by omega)]
-      simp
-  · rw [if_pos]
-    omega
-
-theorem getLast_take {l : List α} (h : l.take n ≠ []) :
-    (l.take n).getLast h = l[n - 1]?.getD (l.getLast (by simp_all)) := by
-  rw [getLast_eq_getElem, getElem_take']
-  simp [length_take, Nat.min_def]
-  simp at h
-  split
-  · rw [getElem?_eq_getElem (by omega)]
-    simp
-  · rw [getElem?_eq_none (by omega), getLast_eq_getElem]
-    simp
+Further results on `List.take` and `List.drop`, which rely on stronger automation in `Nat`,
+are given in `Init.Data.List.TakeDrop`.
+-/
 
 @[simp]
-theorem take_eq_take :
-    ∀ {l : List α} {m n : Nat}, l.take m = l.take n ↔ min m l.length = min n l.length
-  | [], m, n => by simp [Nat.min_zero]
-  | _ :: xs, 0, 0 => by simp
-  | x :: xs, m + 1, 0 => by simp [Nat.zero_min, succ_min_succ]
-  | x :: xs, 0, n + 1 => by simp [Nat.zero_min, succ_min_succ]
-  | x :: xs, m + 1, n + 1 => by simp [succ_min_succ, take_eq_take]; omega
+theorem drop_one : ∀ l : List α, drop 1 l = tail l
+  | [] | _ :: _ => rfl
 
-theorem take_add (l : List α) (m n : Nat) : l.take (m + n) = l.take m ++ (l.drop m).take n := by
-  suffices take (m + n) (take m l ++ drop m l) = take m l ++ take n (drop m l) by
-    rw [take_append_drop] at this
-    assumption
-  rw [take_append_eq_append_take, take_all_of_le, append_right_inj]
-  · simp only [take_eq_take, length_take, length_drop]
-    omega
-  apply Nat.le_trans (m := m)
-  · apply length_take_le
-  · apply Nat.le_add_right
+@[simp] theorem take_append_drop : ∀ (n : Nat) (l : List α), take n l ++ drop n l = l
+  | 0, _ => rfl
+  | _+1, [] => rfl
+  | n+1, x :: xs => congrArg (cons x) <| take_append_drop n xs
 
-theorem dropLast_take {n : Nat} {l : List α} (h : n < l.length) :
-    (l.take n).dropLast = l.take n.pred := by
-  simp only [dropLast_eq_take, length_take, Nat.le_of_lt h, take_take, pred_le, Nat.min_eq_left]
+@[simp] theorem length_drop : ∀ (i : Nat) (l : List α), length (drop i l) = length l - i
+  | 0, _ => rfl
+  | succ i, [] => Eq.symm (Nat.zero_sub (succ i))
+  | succ i, x :: l => calc
+    length (drop (succ i) (x :: l)) = length l - i := length_drop i l
+    _ = succ (length l) - succ i := (Nat.succ_sub_succ_eq_sub (length l) i).symm
 
-theorem map_eq_append_split {f : α → β} {l : List α} {s₁ s₂ : List β}
-    (h : map f l = s₁ ++ s₂) : ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = s₁ ∧ map f l₂ = s₂ := by
-  have := h
-  rw [← take_append_drop (length s₁) l] at this ⊢
-  rw [map_append] at this
-  refine ⟨_, _, rfl, append_inj this ?_⟩
-  rw [length_map, length_take, Nat.min_eq_left]
-  rw [← length_map l f, h, length_append]
-  apply Nat.le_add_right
+theorem drop_of_length_le {l : List α} (h : l.length ≤ i) : drop i l = [] :=
+  length_eq_zero.1 (length_drop .. ▸ Nat.sub_eq_zero_of_le h)
 
-/-! ### drop -/
+theorem length_lt_of_drop_ne_nil {l : List α} {n} (h : drop n l ≠ []) : n < l.length :=
+  gt_of_not_le (mt drop_of_length_le h)
 
-theorem drop_length_cons {l : List α} (h : l ≠ []) (a : α) :
-    (a :: l).drop l.length = [l.getLast h] := by
-  induction l generalizing a with
-  | nil =>
-    cases h rfl
-  | cons y l ih =>
-    simp only [drop, length]
-    by_cases h₁ : l = []
-    · simp [h₁]
-    rw [getLast_cons' _ h₁]
-    exact ih h₁ y
+theorem take_of_length_le {l : List α} (h : l.length ≤ i) : take i l = l := by
+  have := take_append_drop i l
+  rw [drop_of_length_le h, append_nil] at this; exact this
 
-/-- Dropping the elements up to `n` in `l₁ ++ l₂` is the same as dropping the elements up to `n`
-in `l₁`, dropping the elements up to `n - l₁.length` in `l₂`, and appending them. -/
-theorem drop_append_eq_append_drop {l₁ l₂ : List α} {n : Nat} :
-    drop n (l₁ ++ l₂) = drop n l₁ ++ drop (n - l₁.length) l₂ := by
-  induction l₁ generalizing n
-  · simp
-  · cases n
-    · simp [*]
-    · simp only [cons_append, drop_succ_cons, length_cons, succ_eq_add_one, append_cancel_left_eq, *]
-      congr 1
-      omega
+theorem lt_length_of_take_ne_self {l : List α} {n} (h : l.take n ≠ l) : n < l.length :=
+  gt_of_not_le (mt take_of_length_le h)
 
-theorem drop_append_of_le_length {l₁ l₂ : List α} {n : Nat} (h : n ≤ l₁.length) :
-    (l₁ ++ l₂).drop n = l₁.drop n ++ l₂ := by
-  simp [drop_append_eq_append_drop, Nat.sub_eq_zero_of_le h]
+@[deprecated drop_of_length_le (since := "2024-07-07")] abbrev drop_length_le := @drop_of_length_le
+@[deprecated take_of_length_le (since := "2024-07-07")] abbrev take_length_le := @take_of_length_le
 
+@[simp] theorem drop_length (l : List α) : drop l.length l = [] := drop_of_length_le (Nat.le_refl _)
 
-/-- Dropping the elements up to `l₁.length + i` in `l₁ + l₂` is the same as dropping the elements
-up to `i` in `l₂`. -/
-@[simp]
-theorem drop_append {l₁ l₂ : List α} (i : Nat) : drop (l₁.length + i) (l₁ ++ l₂) = drop i l₂ := by
-  rw [drop_append_eq_append_drop, drop_eq_nil_of_le] <;>
-    simp [Nat.add_sub_cancel_left, Nat.le_add_right]
-
-theorem lt_length_drop (L : List α) {i j : Nat} (h : i + j < L.length) : j < (L.drop i).length := by
-  have A : i < L.length := Nat.lt_of_le_of_lt (Nat.le.intro rfl) h
-  rw [(take_append_drop i L).symm] at h
-  simpa only [Nat.le_of_lt A, Nat.min_eq_left, Nat.add_lt_add_iff_left, length_take,
-    length_append] using h
-
-/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
-dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
-theorem getElem_drop (L : List α) {i j : Nat} (h : i + j < L.length) :
-    L[i + j] = (L.drop i)[j]'(lt_length_drop L h) := by
-  have : i ≤ L.length := Nat.le_trans (Nat.le_add_right _ _) (Nat.le_of_lt h)
-  rw [getElem_of_eq (take_append_drop i L).symm h, getElem_append_right'] <;>
-    simp [Nat.min_eq_left this, Nat.add_sub_cancel_left, Nat.le_add_right]
-
-/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
-dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
-@[deprecated getElem_drop (since := "2024-06-12")]
-theorem get_drop (L : List α) {i j : Nat} (h : i + j < L.length) :
-    get L ⟨i + j, h⟩ = get (L.drop i) ⟨j, lt_length_drop L h⟩ := by
-  simp [getElem_drop]
-
-/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
-dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
-theorem getElem_drop' (L : List α) {i : Nat} {j : Nat} {h : j < (L.drop i).length} :
-    (L.drop i)[j] = L[i + j]'(by
-      rw [Nat.add_comm]
-      exact Nat.add_lt_of_lt_sub (length_drop i L ▸ h)) := by
-  rw [getElem_drop]
-
-/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
-dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
-@[deprecated getElem_drop' (since := "2024-06-12")]
-theorem get_drop' (L : List α) {i j} :
-    get (L.drop i) j = get L ⟨i + j, by
-      rw [Nat.add_comm]
-      exact Nat.add_lt_of_lt_sub (length_drop i L ▸ j.2)⟩ := by
-  simp [getElem_drop']
+@[simp] theorem take_length (l : List α) : take l.length l = l := take_of_length_le (Nat.le_refl _)
 
 @[simp]
-theorem getElem?_drop (L : List α) (i j : Nat) : (L.drop i)[j]? = L[i + j]? := by
-  ext
-  simp only [getElem?_eq_some, getElem_drop', Option.mem_def]
-  constructor <;> intro ⟨h, ha⟩
-  · exact ⟨_, ha⟩
-  · refine ⟨?_, ha⟩
-    rw [length_drop]
-    rw [Nat.add_comm] at h
-    apply Nat.lt_sub_of_add_lt h
+theorem getElem_cons_drop : ∀ (l : List α) (i : Nat) (h : i < l.length),
+    l[i] :: drop (i + 1) l = drop i l
+  | _::_, 0, _ => rfl
+  | _::_, i+1, _ => getElem_cons_drop _ i _
 
-@[deprecated getElem?_drop (since := "2024-06-12")]
-theorem get?_drop (L : List α) (i j : Nat) : get? (L.drop i) j = get? L (i + j) := by
+@[deprecated getElem_cons_drop (since := "2024-06-12")]
+theorem get_cons_drop (l : List α) (i) : get l i :: drop (i + 1) l = drop i l := by
   simp
 
-theorem head?_drop (l : List α) (n : Nat) :
-    (l.drop n).head? = l[n]? := by
-  rw [head?_eq_getElem?, getElem?_drop, Nat.add_zero]
+theorem drop_eq_getElem_cons {n} {l : List α} (h) : drop n l = l[n] :: drop (n + 1) l :=
+  (getElem_cons_drop _ n h).symm
 
-theorem head_drop {l : List α} {n : Nat} (h : l.drop n ≠ []) :
-    (l.drop n).head h = l[n]'(by simp_all) := by
-  have w : n < l.length := length_lt_of_drop_ne_nil h
-  simpa [head?_eq_head, getElem?_eq_getElem, h, w] using head?_drop l n
+@[deprecated drop_eq_getElem_cons (since := "2024-06-12")]
+theorem drop_eq_get_cons {n} {l : List α} (h) : drop n l = get l ⟨n, h⟩ :: drop (n + 1) l := by
+  simp [drop_eq_getElem_cons]
 
-theorem getLast?_drop {l : List α} : (l.drop n).getLast? = if l.length ≤ n then none else l.getLast? := by
-  rw [getLast?_eq_getElem?, getElem?_drop]
-  rw [length_drop]
-  split
-  · rw [getElem?_eq_none (by omega)]
-  · rw [getLast?_eq_getElem?]
-    congr
-    omega
+@[simp]
+theorem getElem?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n)[m]? = l[m]? := by
+  induction n generalizing l m with
+  | zero =>
+    exact absurd h (Nat.not_lt_of_le m.zero_le)
+  | succ _ hn =>
+    cases l with
+    | nil => simp only [take_nil]
+    | cons hd tl =>
+      cases m
+      · simp
+      · simpa using hn (Nat.lt_of_succ_lt_succ h)
 
-theorem getLast_drop {l : List α} (h : l.drop n ≠ []) :
-    (l.drop n).getLast h = l.getLast (ne_nil_of_length_pos (by simp at h; omega)) := by
-  simp only [ne_eq, drop_eq_nil_iff_le] at h
-  apply Option.some_inj.1
-  simp only [← getLast?_eq_getLast, getLast?_drop, ite_eq_right_iff]
-  omega
+@[deprecated getElem?_take (since := "2024-06-12")]
+theorem get?_take {l : List α} {n m : Nat} (h : m < n) : (l.take n).get? m = l.get? m := by
+  simp [getElem?_take, h]
 
-theorem set_eq_take_append_cons_drop {l : List α} {n : Nat} {a : α} :
-    l.set n a = if n < l.length then l.take n ++ a :: l.drop (n + 1) else l := by
-  split <;> rename_i h
-  · ext1 m
-    by_cases h' : m < n
-    · rw [getElem?_append (by simp [length_take]; omega), getElem?_set_ne (by omega),
-        getElem?_take h']
-    · by_cases h'' : m = n
-      · subst h''
-        rw [getElem?_set_eq (by simp; omega), getElem?_append_right, length_take,
-          Nat.min_eq_left (by omega), Nat.sub_self, getElem?_cons_zero]
-        rw [length_take]
-        exact Nat.min_le_left m l.length
-      · have h''' : n < m := by omega
-        rw [getElem?_set_ne (by omega), getElem?_append_right, length_take,
-          Nat.min_eq_left (by omega)]
-        · obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h'''
-          have p : n + k + 1 - n = k + 1 := by omega
-          rw [p]
-          rw [getElem?_cons_succ, getElem?_drop]
-          congr 1
-          omega
-        · rw [length_take]
-          exact Nat.le_trans (Nat.min_le_left _ _) (by omega)
-  · rw [set_eq_of_length_le]
-    omega
+@[simp]
+theorem getElem?_take_of_succ {l : List α} {n : Nat} : (l.take (n + 1))[n]? = l[n]? :=
+  getElem?_take (Nat.lt_succ_self n)
 
-theorem exists_of_set {n : Nat} {a' : α} {l : List α} (h : n < l.length) :
-    ∃ l₁ l₂, l = l₁ ++ l[n] :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂ := by
-  refine ⟨l.take n, l.drop (n + 1), ⟨by simp, ⟨length_take_of_le (Nat.le_of_lt h), ?_⟩⟩⟩
-  simp [set_eq_take_append_cons_drop, h]
+theorem tail_drop (l : List α) (n : Nat) : (l.drop n).tail = l.drop (n + 1) := by
+  induction l generalizing n with
+  | nil => simp
+  | cons hd tl hl =>
+    cases n
+    · simp
+    · simp [hl]
 
-theorem drop_set_of_lt (a : α) {n m : Nat} (l : List α)
-    (hnm : n < m) : drop m (l.set n a) = l.drop m :=
-  ext_getElem? fun k => by simpa only [getElem?_drop] using getElem?_set_ne (by omega)
+@[simp]
+theorem drop_eq_nil_iff_le {l : List α} {k : Nat} : l.drop k = [] ↔ l.length ≤ k := by
+  refine' ⟨fun h => _, drop_eq_nil_of_le⟩
+  induction k generalizing l with
+  | zero =>
+    simp only [drop] at h
+    simp [h]
+  | succ k hk =>
+    cases l
+    · simp
+    · simp only [drop] at h
+      simpa [Nat.succ_le_succ_iff] using hk h
 
-theorem drop_take : ∀ (m n : Nat) (l : List α), drop n (take m l) = take (m - n) (drop n l)
+@[simp]
+theorem take_eq_nil_iff {l : List α} {k : Nat} : l.take k = [] ↔ k = 0 ∨ l = [] := by
+  cases l <;> cases k <;> simp [Nat.succ_ne_zero]
+
+theorem drop_eq_nil_of_eq_nil : ∀ {as : List α} {i}, as = [] → as.drop i = []
+  | _, _, rfl => drop_nil
+
+theorem ne_nil_of_drop_ne_nil {as : List α} {i : Nat} (h: as.drop i ≠ []) : as ≠ [] :=
+  mt drop_eq_nil_of_eq_nil h
+
+theorem take_eq_nil_of_eq_nil : ∀ {as : List α} {i}, as = [] → as.take i = []
+  | _, _, rfl => take_nil
+
+theorem ne_nil_of_take_ne_nil {as : List α} {i : Nat} (h : as.take i ≠ []) : as ≠ [] :=
+  mt take_eq_nil_of_eq_nil h
+
+theorem set_take {l : List α} {n m : Nat} {a : α} :
+    (l.set m a).take n = (l.take n).set m a := by
+  induction n generalizing l m with
+  | zero => simp
+  | succ _ hn =>
+    cases l with
+    | nil => simp
+    | cons hd tl => cases m <;> simp_all
+
+theorem drop_set {l : List α} {n m : Nat} {a : α} :
+    (l.set m a).drop n = if m < n then l.drop n else (l.drop n).set (m - n) a := by
+  induction n generalizing l m with
+  | zero => simp
+  | succ _ hn =>
+    cases l with
+    | nil => simp
+    | cons hd tl =>
+      cases m
+      · simp_all
+      · simp only [hn, set_cons_succ, drop_succ_cons, succ_lt_succ_iff]
+        congr 2
+        exact (Nat.add_sub_add_right ..).symm
+
+theorem set_drop {l : List α} {n m : Nat} {a : α} :
+    (l.drop n).set m a = (l.set (n + m) a).drop n := by
+  rw [drop_set, if_neg, add_sub_self_left n m]
+  exact (Nat.not_lt).2 (le_add_right n m)
+
+theorem take_concat_get (l : List α) (i : Nat) (h : i < l.length) :
+    (l.take i).concat l[i] = l.take (i+1) :=
+  Eq.symm <| (append_left_inj _).1 <| (take_append_drop (i+1) l).trans <| by
+    rw [concat_eq_append, append_assoc, singleton_append, get_drop_eq_drop, take_append_drop]
+
+@[deprecated take_succ_cons (since := "2024-07-25")]
+theorem take_cons_succ : (a::as).take (i+1) = a :: as.take i := rfl
+
+@[deprecated take_of_length_le (since := "2024-07-25")]
+theorem take_all_of_le {n} {l : List α} (h : length l ≤ n) : take n l = l :=
+  take_of_length_le h
+
+theorem drop_left : ∀ l₁ l₂ : List α, drop (length l₁) (l₁ ++ l₂) = l₂
+  | [], _ => rfl
+  | _ :: l₁, l₂ => drop_left l₁ l₂
+
+@[simp]
+theorem drop_left' {l₁ l₂ : List α} {n} (h : length l₁ = n) : drop n (l₁ ++ l₂) = l₂ := by
+  rw [← h]; apply drop_left
+
+theorem take_left : ∀ l₁ l₂ : List α, take (length l₁) (l₁ ++ l₂) = l₁
+  | [], _ => rfl
+  | a :: l₁, l₂ => congrArg (cons a) (take_left l₁ l₂)
+
+@[simp]
+theorem take_left' {l₁ l₂ : List α} {n} (h : length l₁ = n) : take n (l₁ ++ l₂) = l₁ := by
+  rw [← h]; apply take_left
+
+theorem take_succ {l : List α} {n : Nat} : l.take (n + 1) = l.take n ++ l[n]?.toList := by
+  induction l generalizing n with
+  | nil =>
+    simp only [take_nil, Option.toList, getElem?_nil, append_nil]
+  | cons hd tl hl =>
+    cases n
+    · simp only [take, Option.toList, getElem?_cons_zero, nil_append]
+    · simp only [take, hl, getElem?_cons_succ, cons_append]
+
+@[deprecated (since := "2024-07-25")]
+theorem drop_sizeOf_le [SizeOf α] (l : List α) (n : Nat) : sizeOf (l.drop n) ≤ sizeOf l := by
+  induction l generalizing n with
+  | nil => rw [drop_nil]; apply Nat.le_refl
+  | cons _ _ lih =>
+    induction n with
+    | zero => apply Nat.le_refl
+    | succ n =>
+      exact Trans.trans (lih _) (Nat.le_add_left _ _)
+
+theorem dropLast_eq_take (l : List α) : l.dropLast = l.take (l.length - 1) := by
+  cases l with
+  | nil => simp [dropLast]
+  | cons x l =>
+    induction l generalizing x <;> simp_all [dropLast]
+
+@[simp] theorem map_take (f : α → β) :
+    ∀ (L : List α) (i : Nat), (L.take i).map f = (L.map f).take i
+  | [], i => by simp
+  | _, 0 => by simp
+  | h :: t, n + 1 => by dsimp; rw [map_take f t n]
+
+@[simp] theorem map_drop (f : α → β) :
+    ∀ (L : List α) (i : Nat), (L.drop i).map f = (L.map f).drop i
+  | [], i => by simp
+  | L, 0 => by simp
+  | h :: t, n + 1 => by
+    dsimp
+    rw [map_drop f t]
+
+@[simp] theorem drop_drop (n : Nat) : ∀ (m) (l : List α), drop n (drop m l) = drop (n + m) l
+  | m, [] => by simp
+  | 0, l => by simp
+  | m + 1, a :: l =>
+    calc
+      drop n (drop (m + 1) (a :: l)) = drop n (drop m l) := rfl
+      _ = drop (n + m) l := drop_drop n m l
+      _ = drop (n + (m + 1)) (a :: l) := rfl
+
+theorem take_drop : ∀ (m n : Nat) (l : List α), take n (drop m l) = drop m (take (m + n) l)
   | 0, _, _ => by simp
-  | _, 0, _ => by simp
   | _, _, [] => by simp
-  | m+1, n+1, h :: t => by
-    simp [take_succ_cons, drop_succ_cons, drop_take m n t]
-    congr 1
-    omega
+  | _+1, _, _ :: _ => by simpa [Nat.succ_add, take_succ_cons, drop_succ_cons] using take_drop ..
 
-theorem take_reverse {α} {xs : List α} (n : Nat) (h : n ≤ xs.length) :
-    xs.reverse.take n = (xs.drop (xs.length - n)).reverse := by
-  induction xs generalizing n <;>
-    simp only [reverse_cons, drop, reverse_nil, Nat.zero_sub, length, take_nil]
-  next xs_hd xs_tl xs_ih =>
-  cases Nat.lt_or_eq_of_le h with
-  | inl h' =>
-    have h' := Nat.le_of_succ_le_succ h'
-    rw [take_append_of_le_length, xs_ih _ h']
-    rw [show xs_tl.length + 1 - n = succ (xs_tl.length - n) from _, drop]
-    · rwa [succ_eq_add_one, Nat.sub_add_comm]
-    · rwa [length_reverse]
-  | inr h' =>
-    subst h'
-    rw [length, Nat.sub_self, drop]
-    suffices xs_tl.length + 1 = (xs_tl.reverse ++ [xs_hd]).length by
-      rw [this, take_length, reverse_cons]
-    rw [length_append, length_reverse]
-    rfl
+@[deprecated drop_drop (since := "2024-06-15")]
+theorem drop_add (m n) (l : List α) : drop (m + n) l = drop m (drop n l) := by
+  simp [drop_drop]
 
-@[deprecated (since := "2024-06-15")] abbrev reverse_take := @take_reverse
+/-! ### takeWhile and dropWhile -/
+
+theorem takeWhile_cons (p : α → Bool) (a : α) (l : List α) :
+    (a :: l).takeWhile p = if p a then a :: l.takeWhile p else [] := by
+  simp only [takeWhile]
+  by_cases h: p a <;> simp [h]
+
+@[simp] theorem takeWhile_cons_of_pos {p : α → Bool} {a : α} {l : List α} (h : p a) :
+    (a :: l).takeWhile p = a :: l.takeWhile p := by
+  simp [takeWhile_cons, h]
+
+@[simp] theorem takeWhile_cons_of_neg {p : α → Bool} {a : α} {l : List α} (h : ¬ p a) :
+    (a :: l).takeWhile p = [] := by
+  simp [takeWhile_cons, h]
+
+theorem dropWhile_cons :
+    (x :: xs : List α).dropWhile p = if p x then xs.dropWhile p else x :: xs := by
+  split <;> simp_all [dropWhile]
+
+@[simp] theorem dropWhile_cons_of_pos {a : α} {l : List α} (h : p a) :
+    (a :: l).dropWhile p = l.dropWhile p := by
+  simp [dropWhile_cons, h]
+
+@[simp] theorem dropWhile_cons_of_neg {a : α} {l : List α} (h : ¬ p a) :
+    (a :: l).dropWhile p = a :: l := by
+  simp [dropWhile_cons, h]
+
+theorem head?_takeWhile (p : α → Bool) (l : List α) : (l.takeWhile p).head? = l.head?.filter p := by
+  cases l with
+  | nil => rfl
+  | cons x xs =>
+    simp only [takeWhile_cons, head?_cons, Option.filter_some]
+    split <;> simp
+
+theorem head_takeWhile (p : α → Bool) (l : List α) (w) :
+    (l.takeWhile p).head w = l.head (by rintro rfl; simp_all) := by
+  cases l with
+  | nil => rfl
+  | cons x xs =>
+    simp only [takeWhile_cons, head_cons]
+    simp only [takeWhile_cons] at w
+    split <;> simp_all
+
+theorem head?_dropWhile_not (p : α → Bool) (l : List α) :
+    match (l.dropWhile p).head? with | some x => p x = false | none => True := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [dropWhile_cons]
+    split <;> rename_i h <;> split at h <;> simp_all
+
+theorem head_dropWhile_not (p : α → Bool) (l : List α) (w) :
+    p ((l.dropWhile p).head w) = false := by
+  simpa [head?_eq_head, w] using head?_dropWhile_not p l
+
+theorem takeWhile_map (f : α → β) (p : β → Bool) (l : List α) :
+    (l.map f).takeWhile p = (l.takeWhile (p ∘ f)).map f := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [map_cons, takeWhile_cons]
+    split <;> simp_all
+
+theorem dropWhile_map (f : α → β) (p : β → Bool) (l : List α) :
+    (l.map f).dropWhile p = (l.dropWhile (p ∘ f)).map f := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [map_cons, dropWhile_cons]
+    split <;> simp_all
+
+theorem takeWhile_filterMap (f : α → Option β) (p : β → Bool) (l : List α) :
+    (l.filterMap f).takeWhile p = (l.takeWhile fun a => (f a).all p).filterMap f := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [filterMap_cons]
+    split <;> rename_i h
+    · simp only [takeWhile_cons, h]
+      split <;> simp_all
+    · simp [takeWhile_cons, h, ih]
+      split <;> simp_all [filterMap_cons]
+
+theorem dropWhile_filterMap (f : α → Option β) (p : β → Bool) (l : List α) :
+    (l.filterMap f).dropWhile p = (l.dropWhile fun a => (f a).all p).filterMap f := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [filterMap_cons]
+    split <;> rename_i h
+    · simp only [dropWhile_cons, h]
+      split <;> simp_all
+    · simp [dropWhile_cons, h, ih]
+      split <;> simp_all [filterMap_cons]
+
+theorem takeWhile_filter (p q : α → Bool) (l : List α) :
+    (l.filter p).takeWhile q = (l.takeWhile fun a => !p a || q a).filter p := by
+  simp [← filterMap_eq_filter, takeWhile_filterMap]
+
+theorem dropWhile_filter (p q : α → Bool) (l : List α) :
+    (l.filter p).dropWhile q = (l.dropWhile fun a => !p a || q a).filter p := by
+  simp [← filterMap_eq_filter, dropWhile_filterMap]
+
+@[simp] theorem takeWhile_append_dropWhile (p : α → Bool) :
+    ∀ (l : List α), takeWhile p l ++ dropWhile p l = l
+  | [] => rfl
+  | x :: xs => by simp [takeWhile, dropWhile]; cases p x <;> simp [takeWhile_append_dropWhile p xs]
+
+theorem takeWhile_append {xs ys : List α} :
+    (xs ++ ys).takeWhile p =
+      if (xs.takeWhile p).length = xs.length then xs ++ ys.takeWhile p else xs.takeWhile p := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [cons_append, takeWhile_cons]
+    split
+    · simp_all only [length_cons, add_one_inj]
+      split <;> rfl
+    · simp_all
+
+@[simp] theorem takeWhile_append_of_pos {p : α → Bool} {l₁ l₂ : List α} (h : ∀ a ∈ l₁, p a) :
+    (l₁ ++ l₂).takeWhile p = l₁ ++ l₂.takeWhile p := by
+  induction l₁ with
+  | nil => simp
+  | cons x xs ih => simp_all [takeWhile_cons]
+
+theorem dropWhile_append {xs ys : List α} :
+    (xs ++ ys).dropWhile p =
+      if (xs.dropWhile p).isEmpty then ys.dropWhile p else xs.dropWhile p ++ ys := by
+  induction xs with
+  | nil => simp
+  | cons h t ih =>
+    simp only [cons_append, dropWhile_cons]
+    split <;> simp_all
+
+@[simp] theorem dropWhile_append_of_pos {p : α → Bool} {l₁ l₂ : List α} (h : ∀ a ∈ l₁, p a) :
+    (l₁ ++ l₂).dropWhile p = l₂.dropWhile p := by
+  induction l₁ with
+  | nil => simp
+  | cons x xs ih => simp_all [dropWhile_cons]
+
+@[simp] theorem takeWhile_replicate_eq_filter (p : α → Bool) :
+    (replicate n a).takeWhile p = (replicate n a).filter p := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [replicate_succ, takeWhile_cons]
+    split <;> simp_all
+
+theorem takeWhile_replicate (p : α → Bool) :
+    (replicate n a).takeWhile p = if p a then replicate n a else [] := by
+  rw [takeWhile_replicate_eq_filter, filter_replicate]
+
+@[simp] theorem dropWhile_replicate_eq_filter_not (p : α → Bool) :
+    (replicate n a).dropWhile p = (replicate n a).filter (fun a => !p a) := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [replicate_succ, dropWhile_cons]
+    split <;> simp_all
+
+theorem dropWhile_replicate (p : α → Bool) :
+    (replicate n a).dropWhile p = if p a then [] else replicate n a := by
+  simp only [dropWhile_replicate_eq_filter_not, filter_replicate]
+  split <;> simp_all
+
+theorem take_takeWhile {l : List α} (p : α → Bool) n :
+    (l.takeWhile p).take n = (l.takeWhile p).take n := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    by_cases h : p x <;> simp [takeWhile_cons, h, ih]
+
+@[simp] theorem all_takeWhile {l : List α} : (l.takeWhile p).all p = true := by
+  induction l with
+  | nil => rfl
+  | cons h t ih => by_cases p h <;> simp_all
+
+@[simp] theorem any_dropWhile {l : List α} :
+    (l.dropWhile p).any (fun x => !p x) = !l.all p := by
+  induction l with
+  | nil => rfl
+  | cons h t ih => by_cases p h <;> simp_all
 
 /-! ### rotateLeft -/
 
-@[simp] theorem rotateLeft_replicate (n) (a : α) : rotateLeft (replicate m a) n = replicate m a := by
-  cases n with
-  | zero => simp
-  | succ n =>
-    suffices 1 < m → m - (n + 1) % m + min ((n + 1) % m) m = m by
-      simpa [rotateLeft]
-    intro h
-    rw [Nat.min_eq_left (Nat.le_of_lt (Nat.mod_lt _ (by omega)))]
-    have : (n + 1) % m < m := Nat.mod_lt _ (by omega)
-    omega
+@[simp] theorem rotateLeft_zero (l : List α) : rotateLeft l 0 = l := by
+  simp [rotateLeft]
+
+-- TODO Batteries defines its own `getElem?_rotate`, which we need to adapt.
+-- TODO Prove `map_rotateLeft`, using `ext` and `getElem?_rotateLeft`.
 
 /-! ### rotateRight -/
 
-@[simp] theorem rotateRight_replicate (n) (a : α) : rotateRight (replicate m a) n = replicate m a := by
-  cases n with
-  | zero => simp
-  | succ n =>
-    suffices 1 < m → m - (m - (n + 1) % m) + min (m - (n + 1) % m) m = m by
-      simpa [rotateRight]
-    intro h
-    have : (n + 1) % m < m := Nat.mod_lt _ (by omega)
-    rw [Nat.min_eq_left (by omega)]
-    omega
+@[simp] theorem rotateRight_zero (l : List α) : rotateRight l 0 = l := by
+  simp [rotateRight]
 
-/-! ### zipWith -/
-
-@[simp] theorem length_zipWith (f : α → β → γ) (l₁ l₂) :
-    length (zipWith f l₁ l₂) = min (length l₁) (length l₂) := by
-  induction l₁ generalizing l₂ <;> cases l₂ <;>
-    simp_all [succ_min_succ, Nat.zero_min, Nat.min_zero]
-
-theorem zipWith_eq_zipWith_take_min : ∀ (l₁ : List α) (l₂ : List β),
-    zipWith f l₁ l₂ = zipWith f (l₁.take (min l₁.length l₂.length)) (l₂.take (min l₁.length l₂.length))
-  | [], _ => by simp
-  | _, [] => by simp
-  | a :: l₁, b :: l₂ => by simp [succ_min_succ, zipWith_eq_zipWith_take_min l₁ l₂]
-
-@[simp] theorem zipWith_replicate {a : α} {b : β} {m n : Nat} :
-    zipWith f (replicate m a) (replicate n b) = replicate (min m n) (f a b) := by
-  rw [zipWith_eq_zipWith_take_min]
-  simp
-
-/-! ### zip -/
-
-@[simp] theorem length_zip (l₁ : List α) (l₂ : List β) :
-    length (zip l₁ l₂) = min (length l₁) (length l₂) := by
-  simp [zip]
-
-theorem zip_eq_zip_take_min : ∀ (l₁ : List α) (l₂ : List β),
-    zip l₁ l₂ = zip (l₁.take (min l₁.length l₂.length)) (l₂.take (min l₁.length l₂.length))
-  | [], _ => by simp
-  | _, [] => by simp
-  | a :: l₁, b :: l₂ => by simp [succ_min_succ, zip_eq_zip_take_min l₁ l₂]
-
-@[simp] theorem zip_replicate {a : α} {b : β} {m n : Nat} :
-    zip (replicate m a) (replicate n b) = replicate (min m n) (a, b) := by
-  rw [zip_eq_zip_take_min]
-  simp
-
-/-! ### minimum? -/
-
--- A specialization of `minimum?_eq_some_iff` to Nat.
-theorem minimum?_eq_some_iff' {xs : List Nat} :
-    xs.minimum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
-  minimum?_eq_some_iff
-    (le_refl := Nat.le_refl)
-    (min_eq_or := fun _ _ => by omega)
-    (le_min_iff := fun _ _ _ => by omega)
-
--- This could be generalized,
--- but will first require further work on order typeclasses in the core repository.
-theorem minimum?_cons' {a : Nat} {l : List Nat} :
-    (a :: l).minimum? = some (match l.minimum? with
-    | none => a
-    | some m => min a m) := by
-  rw [minimum?_eq_some_iff']
-  split <;> rename_i h m
-  · simp_all
-  · rw [minimum?_eq_some_iff'] at m
-    obtain ⟨m, le⟩ := m
-    rw [Nat.min_def]
-    constructor
-    · split
-      · exact mem_cons_self a l
-      · exact mem_cons_of_mem a m
-    · intro b m
-      cases List.mem_cons.1 m with
-      | inl => split <;> omega
-      | inr h =>
-        specialize le b h
-        split <;> omega
-
-/-! ### maximum? -/
-
--- A specialization of `maximum?_eq_some_iff` to Nat.
-theorem maximum?_eq_some_iff' {xs : List Nat} :
-    xs.maximum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, b ≤ a) :=
-  maximum?_eq_some_iff
-    (le_refl := Nat.le_refl)
-    (max_eq_or := fun _ _ => by omega)
-    (max_le_iff := fun _ _ _ => by omega)
-
--- This could be generalized,
--- but will first require further work on order typeclasses in the core repository.
-theorem maximum?_cons' {a : Nat} {l : List Nat} :
-    (a :: l).maximum? = some (match l.maximum? with
-    | none => a
-    | some m => max a m) := by
-  rw [maximum?_eq_some_iff']
-  split <;> rename_i h m
-  · simp_all
-  · rw [maximum?_eq_some_iff'] at m
-    obtain ⟨m, le⟩ := m
-    rw [Nat.max_def]
-    constructor
-    · split
-      · exact mem_cons_of_mem a m
-      · exact mem_cons_self a l
-    · intro b m
-      cases List.mem_cons.1 m with
-      | inl => split <;> omega
-      | inr h =>
-        specialize le b h
-        split <;> omega
+-- TODO Batteries defines its own `getElem?_rotate`, which we need to adapt.
+-- TODO Prove `map_rotateRight`, using `ext` and `getElem?_rotateRight`.
 
 end List

src/Init/Data/List/Zip.lean

@@ -0,0 +1,363 @@
+/-
+Copyright (c) 2014 Parikshit Khanna. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
+-/
+prelude
+import Init.Data.List.TakeDrop
+
+/-!
+# Lemmas about `List.zip`, `List.zipWith`, `List.zipWithAll`, and `List.unzip`.
+-/
+
+namespace List
+
+open Nat
+
+/-! ## Zippers -/
+
+/-! ### zip -/
+
+theorem zip_map (f : α → γ) (g : β → δ) :
+    ∀ (l₁ : List α) (l₂ : List β), zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
+  | [], l₂ => rfl
+  | l₁, [] => by simp only [map, zip_nil_right]
+  | a :: l₁, b :: l₂ => by
+    simp only [map, zip_cons_cons, zip_map, Prod.map]; constructor
+
+theorem zip_map_left (f : α → γ) (l₁ : List α) (l₂ : List β) :
+    zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]
+
+theorem zip_map_right (f : β → γ) (l₁ : List α) (l₂ : List β) :
+    zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]
+
+theorem zip_append :
+    ∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
+      zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
+  | [], r₁, l₂, r₂, h => by simp only [eq_nil_of_length_eq_zero h.symm]; rfl
+  | l₁, r₁, [], r₂, h => by simp only [eq_nil_of_length_eq_zero h]; rfl
+  | a :: l₁, r₁, b :: l₂, r₂, h => by
+    simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]
+
+theorem zip_map' (f : α → β) (g : α → γ) :
+    ∀ l : List α, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
+  | [] => rfl
+  | a :: l => by simp only [map, zip_cons_cons, zip_map']
+
+theorem of_mem_zip {a b} : ∀ {l₁ : List α} {l₂ : List β}, (a, b) ∈ zip l₁ l₂ → a ∈ l₁ ∧ b ∈ l₂
+  | _ :: l₁, _ :: l₂, h => by
+    cases h
+    case head => simp
+    case tail h =>
+    · have := of_mem_zip h
+      exact ⟨Mem.tail _ this.1, Mem.tail _ this.2⟩
+
+@[deprecated of_mem_zip (since := "2024-07-28")] abbrev mem_zip := @of_mem_zip
+
+theorem map_fst_zip :
+    ∀ (l₁ : List α) (l₂ : List β), l₁.length ≤ l₂.length → map Prod.fst (zip l₁ l₂) = l₁
+  | [], bs, _ => rfl
+  | _ :: as, _ :: bs, h => by
+    simp [Nat.succ_le_succ_iff] at h
+    show _ :: map Prod.fst (zip as bs) = _ :: as
+    rw [map_fst_zip as bs h]
+  | a :: as, [], h => by simp at h
+
+theorem map_snd_zip :
+    ∀ (l₁ : List α) (l₂ : List β), l₂.length ≤ l₁.length → map Prod.snd (zip l₁ l₂) = l₂
+  | _, [], _ => by
+    rw [zip_nil_right]
+    rfl
+  | [], b :: bs, h => by simp at h
+  | a :: as, b :: bs, h => by
+    simp [Nat.succ_le_succ_iff] at h
+    show _ :: map Prod.snd (zip as bs) = _ :: bs
+    rw [map_snd_zip as bs h]
+
+theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
+    (l.map fun x => (x, f x)) = l.zip (l.map f) := by
+  rw [← zip_map']
+  congr
+  exact map_id _
+
+theorem map_prod_right_eq_zip {l : List α} (f : α → β) :
+    (l.map fun x => (f x, x)) = (l.map f).zip l := by
+  rw [← zip_map']
+  congr
+  exact map_id _
+
+/-- See also `List.zip_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
+@[simp] theorem zip_replicate' {a : α} {b : β} {n : Nat} :
+    zip (replicate n a) (replicate n b) = replicate n (a, b) := by
+  induction n with
+  | zero => rfl
+  | succ n ih => simp [replicate_succ, ih]
+
+/-! ### zipWith -/
+
+theorem zipWith_comm (f : α → β → γ) :
+    ∀ (la : List α) (lb : List β), zipWith f la lb = zipWith (fun b a => f a b) lb la
+  | [], _ => List.zipWith_nil_right.symm
+  | _ :: _, [] => rfl
+  | _ :: as, _ :: bs => congrArg _ (zipWith_comm f as bs)
+
+theorem zipWith_comm_of_comm (f : α → α → β) (comm : ∀ x y : α, f x y = f y x) (l l' : List α) :
+    zipWith f l l' = zipWith f l' l := by
+  rw [zipWith_comm]
+  simp only [comm]
+
+@[simp]
+theorem zipWith_same (f : α → α → δ) : ∀ l : List α, zipWith f l l = l.map fun a => f a a
+  | [] => rfl
+  | _ :: xs => congrArg _ (zipWith_same f xs)
+
+/--
+See also `getElem?_zipWith'` for a variant
+using `Option.map` and `Option.bind` rather than a `match`.
+-/
+theorem getElem?_zipWith {f : α → β → γ} {i : Nat} :
+    (List.zipWith f as bs)[i]? = match as[i]?, bs[i]? with
+      | some a, some b => some (f a b) | _, _ => none := by
+  induction as generalizing bs i with
+  | nil => cases bs with
+    | nil => simp
+    | cons b bs => simp
+  | cons a as aih => cases bs with
+    | nil => simp
+    | cons b bs => cases i <;> simp_all
+
+/-- Variant of `getElem?_zipWith` using `Option.map` and `Option.bind` rather than a `match`. -/
+theorem getElem?_zipWith' {f : α → β → γ} {i : Nat} :
+    (zipWith f l₁ l₂)[i]? = (l₁[i]?.map f).bind fun g => l₂[i]?.map g := by
+  induction l₁ generalizing l₂ i with
+  | nil => rw [zipWith] <;> simp
+  | cons head tail =>
+    cases l₂
+    · simp
+    · cases i <;> simp_all
+
+theorem getElem?_zipWith_eq_some (f : α → β → γ) (l₁ : List α) (l₂ : List β) (z : γ) (i : Nat) :
+    (zipWith f l₁ l₂)[i]? = some z ↔
+      ∃ x y, l₁[i]? = some x ∧ l₂[i]? = some y ∧ f x y = z := by
+  induction l₁ generalizing l₂ i
+  · simp
+  · cases l₂ <;> cases i <;> simp_all
+
+theorem getElem?_zip_eq_some (l₁ : List α) (l₂ : List β) (z : α × β) (i : Nat) :
+    (zip l₁ l₂)[i]? = some z ↔ l₁[i]? = some z.1 ∧ l₂[i]? = some z.2 := by
+  cases z
+  rw [zip, getElem?_zipWith_eq_some]; constructor
+  · rintro ⟨x, y, h₀, h₁, h₂⟩
+    simpa [h₀, h₁] using h₂
+  · rintro ⟨h₀, h₁⟩
+    exact ⟨_, _, h₀, h₁, rfl⟩
+
+@[deprecated getElem?_zipWith (since := "2024-06-12")]
+theorem get?_zipWith {f : α → β → γ} :
+    (List.zipWith f as bs).get? i = match as.get? i, bs.get? i with
+      | some a, some b => some (f a b) | _, _ => none := by
+  simp [getElem?_zipWith]
+
+set_option linter.deprecated false in
+@[deprecated getElem?_zipWith (since := "2024-06-07")] abbrev zipWith_get? := @get?_zipWith
+
+theorem head?_zipWith {f : α → β → γ} :
+    (List.zipWith f as bs).head? = match as.head?, bs.head? with
+      | some a, some b => some (f a b) | _, _ => none := by
+  simp [head?_eq_getElem?, getElem?_zipWith]
+
+theorem head_zipWith {f : α → β → γ} (h):
+    (List.zipWith f as bs).head h = f (as.head (by rintro rfl; simp_all)) (bs.head (by rintro rfl; simp_all)) := by
+  apply Option.some.inj
+  rw [← head?_eq_head, head?_zipWith, head?_eq_head, head?_eq_head]
+
+@[simp]
+theorem zipWith_map {μ} (f : γ → δ → μ) (g : α → γ) (h : β → δ) (l₁ : List α) (l₂ : List β) :
+    zipWith f (l₁.map g) (l₂.map h) = zipWith (fun a b => f (g a) (h b)) l₁ l₂ := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem zipWith_map_left (l₁ : List α) (l₂ : List β) (f : α → α') (g : α' → β → γ) :
+    zipWith g (l₁.map f) l₂ = zipWith (fun a b => g (f a) b) l₁ l₂ := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem zipWith_map_right (l₁ : List α) (l₂ : List β) (f : β → β') (g : α → β' → γ) :
+    zipWith g l₁ (l₂.map f) = zipWith (fun a b => g a (f b)) l₁ l₂ := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem zipWith_foldr_eq_zip_foldr {f : α → β → γ} (i : δ):
+    (zipWith f l₁ l₂).foldr g i = (zip l₁ l₂).foldr (fun p r => g (f p.1 p.2) r) i := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem zipWith_foldl_eq_zip_foldl {f : α → β → γ} (i : δ):
+    (zipWith f l₁ l₂).foldl g i = (zip l₁ l₂).foldl (fun r p => g r (f p.1 p.2)) i := by
+  induction l₁ generalizing i l₂ <;> cases l₂ <;> simp_all
+
+@[simp]
+theorem zipWith_eq_nil_iff {f : α → β → γ} {l l'} : zipWith f l l' = [] ↔ l = [] ∨ l' = [] := by
+  cases l <;> cases l' <;> simp
+
+theorem map_zipWith {δ : Type _} (f : α → β) (g : γ → δ → α) (l : List γ) (l' : List δ) :
+    map f (zipWith g l l') = zipWith (fun x y => f (g x y)) l l' := by
+  induction l generalizing l' with
+  | nil => simp
+  | cons hd tl hl =>
+    · cases l'
+      · simp
+      · simp [hl]
+
+theorem take_zipWith : (zipWith f l l').take n = zipWith f (l.take n) (l'.take n) := by
+  induction l generalizing l' n with
+  | nil => simp
+  | cons hd tl hl =>
+    cases l'
+    · simp
+    · cases n
+      · simp
+      · simp [hl]
+
+@[deprecated take_zipWith (since := "2024-07-26")] abbrev zipWith_distrib_take := @take_zipWith
+
+theorem drop_zipWith : (zipWith f l l').drop n = zipWith f (l.drop n) (l'.drop n) := by
+  induction l generalizing l' n with
+  | nil => simp
+  | cons hd tl hl =>
+    · cases l'
+      · simp
+      · cases n
+        · simp
+        · simp [hl]
+
+@[deprecated drop_zipWith (since := "2024-07-26")] abbrev zipWith_distrib_drop := @drop_zipWith
+
+theorem tail_zipWith : (zipWith f l l').tail = zipWith f l.tail l'.tail := by
+  rw [← drop_one]; simp [drop_zipWith]
+
+@[deprecated tail_zipWith (since := "2024-07-28")] abbrev zipWith_distrib_tail := @tail_zipWith
+
+theorem zipWith_append (f : α → β → γ) (l la : List α) (l' lb : List β)
+    (h : l.length = l'.length) :
+    zipWith f (l ++ la) (l' ++ lb) = zipWith f l l' ++ zipWith f la lb := by
+  induction l generalizing l' with
+  | nil =>
+    have : l' = [] := eq_nil_of_length_eq_zero (by simpa using h.symm)
+    simp [this]
+  | cons hl tl ih =>
+    cases l' with
+    | nil => simp at h
+    | cons head tail =>
+      simp only [length_cons, Nat.succ.injEq] at h
+      simp [ih _ h]
+
+/-- See also `List.zipWith_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
+@[simp] theorem zipWith_replicate' {a : α} {b : β} {n : Nat} :
+    zipWith f (replicate n a) (replicate n b) = replicate n (f a b) := by
+  induction n with
+  | zero => rfl
+  | succ n ih => simp [replicate_succ, ih]
+
+/-! ### zipWithAll -/
+
+theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
+    (zipWithAll f as bs)[i]? = match as[i]?, bs[i]? with
+      | none, none => .none | a?, b? => some (f a? b?) := by
+  induction as generalizing bs i with
+  | nil => induction bs generalizing i with
+    | nil => simp
+    | cons b bs bih => cases i <;> simp_all
+  | cons a as aih => cases bs with
+    | nil =>
+      specialize @aih []
+      cases i <;> simp_all
+    | cons b bs => cases i <;> simp_all
+
+@[deprecated getElem?_zipWithAll (since := "2024-06-12")]
+theorem get?_zipWithAll {f : Option α → Option β → γ} :
+    (zipWithAll f as bs).get? i = match as.get? i, bs.get? i with
+      | none, none => .none | a?, b? => some (f a? b?) := by
+  simp [getElem?_zipWithAll]
+
+set_option linter.deprecated false in
+@[deprecated getElem?_zipWithAll (since := "2024-06-07")] abbrev zipWithAll_get? := @get?_zipWithAll
+
+theorem head?_zipWithAll {f : Option α → Option β → γ} :
+    (zipWithAll f as bs).head? = match as.head?, bs.head? with
+      | none, none => .none | a?, b? => some (f a? b?) := by
+  simp [head?_eq_getElem?, getElem?_zipWithAll]
+
+theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
+    (zipWithAll f as bs).head h = f as.head? bs.head? := by
+  apply Option.some.inj
+  rw [← head?_eq_head, head?_zipWithAll]
+  split <;> simp_all
+
+theorem zipWithAll_map {μ} (f : Option γ → Option δ → μ) (g : α → γ) (h : β → δ) (l₁ : List α) (l₂ : List β) :
+    zipWithAll f (l₁.map g) (l₂.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) l₁ l₂ := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem zipWithAll_map_left (l₁ : List α) (l₂ : List β) (f : α → α') (g : Option α' → Option β → γ) :
+    zipWithAll g (l₁.map f) l₂ = zipWithAll (fun a b => g (f <$> a) b) l₁ l₂ := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem zipWithAll_map_right (l₁ : List α) (l₂ : List β) (f : β → β') (g : Option α → Option β' → γ) :
+    zipWithAll g l₁ (l₂.map f) = zipWithAll (fun a b => g a (f <$> b)) l₁ l₂ := by
+  induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
+
+theorem map_zipWithAll {δ : Type _} (f : α → β) (g : Option γ → Option δ → α) (l : List γ) (l' : List δ) :
+    map f (zipWithAll g l l') = zipWithAll (fun x y => f (g x y)) l l' := by
+  induction l generalizing l' with
+  | nil => simp
+  | cons hd tl hl =>
+    cases l' <;> simp_all
+
+@[simp] theorem zipWithAll_replicate {a : α} {b : β} {n : Nat} :
+    zipWithAll f (replicate n a) (replicate n b) = replicate n (f a b) := by
+  induction n with
+  | zero => rfl
+  | succ n ih => simp [replicate_succ, ih]
+
+/-! ### unzip -/
+
+@[simp] theorem unzip_fst : (unzip l).fst = l.map Prod.fst := by
+  induction l <;> simp_all
+
+@[simp] theorem unzip_snd : (unzip l).snd = l.map Prod.snd := by
+  induction l <;> simp_all
+
+@[deprecated unzip_fst (since := "2024-07-28")] abbrev unzip_left := @unzip_fst
+@[deprecated unzip_snd (since := "2024-07-28")] abbrev unzip_right := @unzip_snd
+
+theorem unzip_eq_map : ∀ l : List (α × β), unzip l = (l.map Prod.fst, l.map Prod.snd)
+  | [] => rfl
+  | (a, b) :: l => by simp only [unzip_cons, map_cons, unzip_eq_map l]
+
+theorem zip_unzip : ∀ l : List (α × β), zip (unzip l).1 (unzip l).2 = l
+  | [] => rfl
+  | (a, b) :: l => by simp only [unzip_cons, zip_cons_cons, zip_unzip l]
+
+theorem unzip_zip_left :
+    ∀ {l₁ : List α} {l₂ : List β}, length l₁ ≤ length l₂ → (unzip (zip l₁ l₂)).1 = l₁
+  | [], l₂, _ => rfl
+  | l₁, [], h => by rw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_le_zero h)]; rfl
+  | a :: l₁, b :: l₂, h => by
+    simp only [zip_cons_cons, unzip_cons, unzip_zip_left (le_of_succ_le_succ h)]
+
+theorem unzip_zip_right :
+    ∀ {l₁ : List α} {l₂ : List β}, length l₂ ≤ length l₁ → (unzip (zip l₁ l₂)).2 = l₂
+  | [], l₂, _ => by simp_all
+  | l₁, [], _ => by simp
+  | a :: l₁, b :: l₂, h => by
+    simp only [zip_cons_cons, unzip_cons, unzip_zip_right (le_of_succ_le_succ h)]
+
+theorem unzip_zip {l₁ : List α} {l₂ : List β} (h : length l₁ = length l₂) :
+    unzip (zip l₁ l₂) = (l₁, l₂) := by
+  ext
+  · rw [unzip_zip_left (Nat.le_of_eq h)]
+  · rw [unzip_zip_right (Nat.le_of_eq h.symm)]
+
+theorem zip_of_prod {l : List α} {l' : List β} {lp : List (α × β)} (hl : lp.map Prod.fst = l)
+    (hr : lp.map Prod.snd = l') : lp = l.zip l' := by
+  rw [← hl, ← hr, ← zip_unzip lp, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]
+
+@[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
+    unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
+  ext1 <;> simp

src/Init/Data/Nat/Basic.lean

@@ -102,6 +102,13 @@ def blt (a b : Nat) : Bool :=
 attribute [simp] Nat.zero_le
 attribute [simp] Nat.not_lt_zero
 
+theorem and_forall_add_one {p : Nat → Prop} : p 0 ∧ (∀ n, p (n + 1)) ↔ ∀ n, p n :=
+  ⟨fun h n => Nat.casesOn n h.1 h.2, fun h => ⟨h _, fun _ => h _⟩⟩
+
+theorem or_exists_add_one : p 0 ∨ (Exists fun n => p (n + 1)) ↔ Exists p :=
+  ⟨fun h => h.elim (fun h0 => ⟨0, h0⟩) fun ⟨n, hn⟩ => ⟨n + 1, hn⟩,
+    fun ⟨n, h⟩ => match n with | 0 => Or.inl h | n+1 => Or.inr ⟨n, h⟩⟩
+
 /-! # Helper "packing" theorems -/
 
 @[simp] theorem zero_eq : Nat.zero = 0 := rfl
@@ -388,11 +395,11 @@ theorem le_or_eq_of_le_succ {m n : Nat} (h : m ≤ succ n) : m ≤ n ∨ m = suc
 theorem le_or_eq_of_le_add_one {m n : Nat} (h : m ≤ n + 1) : m ≤ n ∨ m = n + 1 :=
   le_or_eq_of_le_succ h
 
-theorem le_add_right : ∀ (n k : Nat), n ≤ n + k
+@[simp] theorem le_add_right : ∀ (n k : Nat), n ≤ n + k
   | n, 0   => Nat.le_refl n
   | n, k+1 => le_succ_of_le (le_add_right n k)
 
-theorem le_add_left (n m : Nat): n ≤ m + n :=
+@[simp] theorem le_add_left (n m : Nat): n ≤ m + n :=
   Nat.add_comm n m ▸ le_add_right n m
 
 theorem le_of_add_right_le {n m k : Nat} (h : n + k ≤ m) : n ≤ m :=
@@ -528,7 +535,7 @@ protected theorem le_of_add_le_add_right {a b c : Nat} : a + b ≤ c + b → a
   rw [Nat.add_comm _ b, Nat.add_comm _ b]
   apply Nat.le_of_add_le_add_left
 
-protected theorem add_le_add_iff_right {n : Nat} : m + n ≤ k + n ↔ m ≤ k :=
+@[simp] 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 -/

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

@@ -265,8 +265,8 @@ theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
       have x_eq : x = y + 2^j := Nat.eq_add_of_sub_eq x_ge_j y_eq
       simp only [Nat.two_pow_pos, x_eq, Nat.le_add_left, true_and, ite_true]
       have y_lt_x : y < x := by
-            simp [x_eq]
-            exact Nat.lt_add_of_pos_right (Nat.two_pow_pos j)
+        simp only [x_eq, Nat.lt_add_right_iff_pos]
+        exact Nat.two_pow_pos j
       simp only [hyp y y_lt_x]
       if i_lt_j : i < j then
         rw [Nat.add_comm _ (2^_), testBit_two_pow_add_gt i_lt_j]

src/Init/Data/Nat/Gcd.lean

@@ -46,6 +46,9 @@ theorem gcd_succ (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x) := by
 theorem gcd_add_one (x y : Nat) : gcd (x + 1) y = gcd (y % (x + 1)) (x + 1) := by
   rw [gcd]; rfl
 
+theorem gcd_def (x y : Nat) : gcd x y = if x = 0 then y else gcd (y % x) x := by
+  cases x <;> simp [Nat.gcd_add_one]
+
 @[simp] theorem gcd_one_left (n : Nat) : gcd 1 n = 1 := by
   rw [gcd_succ, mod_one]
   rfl

src/Init/Data/Nat/Lemmas.lean

@@ -19,6 +19,14 @@ and later these lemmas should be organised into other files more systematically.
 -/
 
 namespace Nat
+
+@[deprecated and_forall_add_one (since := "2024-07-30")] abbrev and_forall_succ := @and_forall_add_one
+@[deprecated or_exists_add_one (since := "2024-07-30")] abbrev or_exists_succ := @or_exists_add_one
+
+@[simp] theorem exists_ne_zero {P : Nat → Prop} : (∃ n, ¬ n = 0 ∧ P n) ↔ ∃ n, P (n + 1) :=
+  ⟨fun ⟨n, h, w⟩ => by cases n with | zero => simp at h | succ n => exact ⟨n, w⟩,
+    fun ⟨n, w⟩ => ⟨n + 1, by simp, w⟩⟩
+
 /-! ## add -/
 
 protected theorem add_add_add_comm (a b c d : Nat) : (a + b) + (c + d) = (a + c) + (b + d) := by
@@ -36,13 +44,13 @@ protected theorem eq_zero_of_add_eq_zero_right (h : n + m = 0) : n = 0 :=
 protected theorem add_eq_zero_iff : n + m = 0 ↔ n = 0 ∧ m = 0 :=
   ⟨Nat.eq_zero_of_add_eq_zero, fun ⟨h₁, h₂⟩ => h₂.symm ▸ h₁⟩
 
-protected theorem add_left_cancel_iff {n : Nat} : n + m = n + k ↔ m = k :=
+@[simp] protected theorem add_left_cancel_iff {n : Nat} : n + m = n + k ↔ m = k :=
   ⟨Nat.add_left_cancel, fun | rfl => rfl⟩
 
-protected theorem add_right_cancel_iff {n : Nat} : m + n = k + n ↔ m = k :=
+@[simp] protected theorem add_right_cancel_iff {n : Nat} : m + n = k + n ↔ m = k :=
   ⟨Nat.add_right_cancel, fun | rfl => rfl⟩
 
-protected theorem add_le_add_iff_left {n : Nat} : n + m ≤ n + k ↔ m ≤ k :=
+@[simp] protected theorem add_le_add_iff_left {n : Nat} : n + m ≤ n + k ↔ m ≤ k :=
   ⟨Nat.le_of_add_le_add_left, fun h => Nat.add_le_add_left h _⟩
 
 protected theorem lt_of_add_lt_add_right : ∀ {n : Nat}, k + n < m + n → k < m
@@ -52,10 +60,10 @@ protected theorem lt_of_add_lt_add_right : ∀ {n : Nat}, k + n < m + n → k <
 protected theorem lt_of_add_lt_add_left {n : Nat} : n + k < n + m → k < m := by
   rw [Nat.add_comm n, Nat.add_comm n]; exact Nat.lt_of_add_lt_add_right
 
-protected theorem add_lt_add_iff_left {k n m : Nat} : k + n < k + m ↔ n < m :=
+@[simp] protected theorem add_lt_add_iff_left {k n m : Nat} : k + n < k + m ↔ n < m :=
   ⟨Nat.lt_of_add_lt_add_left, fun h => Nat.add_lt_add_left h _⟩
 
-protected theorem add_lt_add_iff_right {k n m : Nat} : n + k < m + k ↔ n < m :=
+@[simp] protected theorem add_lt_add_iff_right {k n m : Nat} : n + k < m + k ↔ n < m :=
   ⟨Nat.lt_of_add_lt_add_right, fun h => Nat.add_lt_add_right h _⟩
 
 protected theorem add_lt_add_of_le_of_lt {a b c d : Nat} (hle : a ≤ b) (hlt : c < d) :
@@ -75,10 +83,10 @@ protected theorem pos_of_lt_add_right (h : n < n + k) : 0 < k :=
 protected theorem pos_of_lt_add_left : n < k + n → 0 < k := by
   rw [Nat.add_comm]; exact Nat.pos_of_lt_add_right
 
-protected theorem lt_add_right_iff_pos : n < n + k ↔ 0 < k :=
+@[simp] protected theorem lt_add_right_iff_pos : n < n + k ↔ 0 < k :=
   ⟨Nat.pos_of_lt_add_right, Nat.lt_add_of_pos_right⟩
 
-protected theorem lt_add_left_iff_pos : n < k + n ↔ 0 < k :=
+@[simp] protected theorem lt_add_left_iff_pos : n < k + n ↔ 0 < k :=
   ⟨Nat.pos_of_lt_add_left, Nat.lt_add_of_pos_left⟩
 
 protected theorem add_pos_left (h : 0 < m) (n) : 0 < m + n :=

src/Init/Data/Nat/Linear.lean

@@ -173,13 +173,13 @@ instance : LawfulBEq PolyCnstr where
   eq_of_beq {a b} h := by
     cases a; rename_i eq₁ lhs₁ rhs₁
     cases b; rename_i eq₂ lhs₂ rhs₂
-    have h : eq₁ == eq₂ && lhs₁ == lhs₂ && rhs₁ == rhs₂ := h
+    have h : eq₁ == eq₂ && (lhs₁ == lhs₂ && rhs₁ == rhs₂) := h
     simp at h
-    have ⟨⟨h₁, h₂⟩, h₃⟩ := h
+    have ⟨h₁, h₂, h₃⟩ := h
     rw [h₁, h₂, h₃]
   rfl {a} := by
     cases a; rename_i eq lhs rhs
-    show (eq == eq && lhs == lhs && rhs == rhs) = true
+    show (eq == eq && (lhs == lhs && rhs == rhs)) = true
     simp [LawfulBEq.rfl]
 
 def PolyCnstr.mul (k : Nat) (c : PolyCnstr) : PolyCnstr :=

src/Init/Data/Option/Basic.lean

@@ -212,6 +212,9 @@ instance (α) [BEq α] [LawfulBEq α] : LawfulBEq (Option α) where
 @[simp] theorem all_none : Option.all p none = true := rfl
 @[simp] theorem all_some : Option.all p (some x) = p x := rfl
 
+@[simp] theorem any_none : Option.any p none = false := rfl
+@[simp] theorem any_some : Option.any p (some x) = p x := rfl
+
 /-- The minimum of two optional values. -/
 protected def min [Min α] : Option α → Option α → Option α
   | some x, some y => some (Min.min x y)

src/Init/Data/Option/Lemmas.lean

@@ -193,6 +193,16 @@ theorem mem_map_of_mem (g : α → β) (h : a ∈ x) : g a ∈ Option.map g x :=
 @[simp] theorem filter_none (p : α → Bool) : none.filter p = none := rfl
 theorem filter_some : Option.filter p (some a) = if p a then some a else none := rfl
 
+@[simp] theorem all_guard (p : α → Prop) [DecidablePred p] (a : α) :
+    Option.all q (guard p a) = (!p a || q a) := by
+  simp only [guard]
+  split <;> simp_all
+
+@[simp] theorem any_guard (p : α → Prop) [DecidablePred p] (a : α) :
+    Option.any q (guard p a) = (p a && q a) := by
+  simp only [guard]
+  split <;> simp_all
+
 theorem bind_map_comm {α β} {x : Option (Option α)} {f : α → β} :
     x.bind (Option.map f) = (x.map (Option.map f)).bind id := by cases x <;> simp
 

src/Init/Data/Subtype.lean

@@ -0,0 +1,27 @@
+/-
+Copyright (c) 2017 Johannes Hölzl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johannes Hölzl
+-/
+prelude
+import Init.Ext
+
+namespace Subtype
+
+universe u
+variable {α : Sort u} {p q : α → Prop}
+
+@[ext]
+protected theorem ext : ∀ {a1 a2 : { x // p x }}, (a1 : α) = (a2 : α) → a1 = a2
+  | ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
+
+@[simp]
+protected theorem «forall» {q : { a // p a } → Prop} : (∀ x, q x) ↔ ∀ a b, q ⟨a, b⟩ :=
+  ⟨fun h a b ↦ h ⟨a, b⟩, fun h ⟨a, b⟩ ↦ h a b⟩
+
+@[simp]
+protected theorem «exists» {q : { a // p a } → Prop} :
+    (Exists fun x => q x) ↔ Exists fun a => Exists fun b => q ⟨a, b⟩ :=
+  ⟨fun ⟨⟨a, b⟩, h⟩ ↦ ⟨a, b, h⟩, fun ⟨a, b, h⟩ ↦ ⟨⟨a, b⟩, h⟩⟩
+
+end Subtype

src/Init/Ext.lean

@@ -24,15 +24,16 @@ syntax extFlat := atomic("(" &"flat" " := " &"false" ")")
 /--
 Registers an extensionality theorem.
 
-* When `@[ext]` is applied to a structure, it generates `.ext` and `.ext_iff` theorems and registers
-  them for the `ext` tactic.
-
-* When `@[ext]` is applied to a theorem, the theorem is registered for the `ext` tactic, and it generates an `ext_iff` theorem.
+* When `@[ext]` is applied to a theorem, the theorem is registered for the `ext` tactic, and it generates an "`ext_iff`" theorem.
   The name of the theorem is from adding the suffix `_iff` to the theorem name.
 
-* An optional natural number argument, e.g. `@[ext 9000]`, specifies a priority for the lemma. Higher-priority lemmas are chosen first, and the default is `1000`.
+* When `@[ext]` is applied to a structure, it generates an `.ext` theorem and applies the `@[ext]` attribute to it.
+  The result is an `.ext` and an `.ext_iff` theorem with the `.ext` theorem registered for the `ext` tactic.
 
-* The flag `@[ext (iff := false)]` prevents it from generating an `ext_iff` theorem.
+* An optional natural number argument, e.g. `@[ext 9000]`, specifies a priority for the `ext` lemma.
+  Higher-priority lemmas are chosen first, and the default is `1000`.
+
+* The flag `@[ext (iff := false)]` disables generating an `ext_iff` theorem.
 
 * The flag `@[ext (flat := false)]` causes generated structure extensionality theorems to show inherited fields based on their representation,
   rather than flattening the parents' fields into the lemma's equality hypotheses.

src/Init/Meta.lean

@@ -399,9 +399,16 @@ def setTailInfo (stx : Syntax) (info : SourceInfo) : Syntax :=
   | some stx => stx
   | none     => stx
 
+/--
+Replaces the trailing whitespace in `stx`, if any, with an empty substring.
+
+The trailing substring's `startPos` and `str` are preserved in order to ensure that the result could
+have been produced by the parser, in case any syntax consumers rely on such an assumption.
+-/
 def unsetTrailing (stx : Syntax) : Syntax :=
   match stx.getTailInfo with
-  | SourceInfo.original lead pos _ endPos => stx.setTailInfo (SourceInfo.original lead pos "".toSubstring endPos)
+  | SourceInfo.original lead pos trail endPos =>
+    stx.setTailInfo (SourceInfo.original lead pos { trail with stopPos := trail.startPos } endPos)
   | _                                     => stx
 
 @[specialize] private partial def updateFirst {α} [Inhabited α] (a : Array α) (f : α → Option α) (i : Nat) : Option (Array α) :=

src/Init/MetaTypes.lean

@@ -219,13 +219,13 @@ structure Config where
   -/
   index             : Bool := true
   /--
-  When `true` (default: `false`), `simp` will **not** create a proof for a rewriting rule associated
+  When `true` (default: `true`), `simp` will **not** create a proof for a rewriting rule associated
   with an `rfl`-theorem.
   Rewriting rules are provided by users by annotating theorems with the attribute `@[simp]`.
   If the proof of the theorem is just `rfl` (reflexivity), and `implicitDefEqProofs := true`, `simp`
   will **not** create a proof term which is an application of the annotated theorem.
   -/
-  implicitDefEqProofs : Bool := false
+  implicitDefEqProofs : Bool := true
   deriving Inhabited, BEq
 
 -- Configuration object for `simp_all`

src/Init/Omega/IntList.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 -/
 prelude
-import Init.Data.List.Lemmas
+import Init.Data.List.Zip
 import Init.Data.Int.DivModLemmas
 import Init.Data.Nat.Gcd
 

src/Init/Prelude.lean

@@ -320,7 +320,7 @@ Because this is in the `Eq` namespace, if you have a variable `h : a = b`,
 
 For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
 -/
-theorem Eq.symm {α : Sort u} {a b : α} (h : Eq a b) : Eq b a :=
+@[symm] theorem Eq.symm {α : Sort u} {a b : α} (h : Eq a b) : Eq b a :=
   h ▸ rfl
 
 /--
@@ -2214,12 +2214,17 @@ def Char.utf8Size (c : Char) : Nat :=
 or `none`. In functional programming languages, this type is used to represent
 the possibility of failure, or sometimes nullability.
 
-For example, the function `HashMap.find? : HashMap α β → α → Option β` looks up
+For example, the function `HashMap.get? : HashMap α β → α → Option β` looks up
 a specified key `a : α` inside the map. Because we do not know in advance
 whether the key is actually in the map, the return type is `Option β`, where
 `none` means the value was not in the map, and `some b` means that the value
 was found and `b` is the value retrieved.
 
+The `xs[i]` syntax, which is used to index into collections, has a variant
+`xs[i]?` that returns an optional value depending on whether the given index
+is valid. For example, if `m : HashMap α β` and `a : α`, then `m[a]?` is
+equivalent to `HashMap.get? m a`.
+
 To extract a value from an `Option α`, we use pattern matching:
 ```
 def map (f : α → β) (x : Option α) : Option β :=

src/Init/PropLemmas.lean

@@ -202,6 +202,17 @@ theorem exists_imp : ((∃ x, p x) → b) ↔ ∀ x, p x → b := forall_exists_
 @[simp] theorem exists_const (α) [i : Nonempty α] : (∃ _ : α, b) ↔ b :=
   ⟨fun ⟨_, h⟩ => h, i.elim Exists.intro⟩
 
+@[congr]
+theorem exists_prop_congr {p p' : Prop} {q q' : p → Prop} (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') :
+    Exists q ↔ ∃ h : p', q' (hp.2 h) :=
+  ⟨fun ⟨_, _⟩ ↦ ⟨hp.1 ‹_›, (hq _).1 ‹_›⟩, fun ⟨_, _⟩ ↦ ⟨_, (hq _).2 ‹_›⟩⟩
+
+theorem exists_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (Exists fun h' : p => q h') ↔ q h :=
+  @exists_const (q h) p ⟨h⟩
+
+@[simp] theorem exists_true_left (p : True → Prop) : Exists p ↔ p True.intro :=
+  exists_prop_of_true _
+
 section forall_congr
 
 theorem forall_congr' (h : ∀ a, p a ↔ q a) : (∀ a, p a) ↔ ∀ a, q a :=
@@ -309,6 +320,11 @@ theorem not_forall_of_exists_not {p : α → Prop} : (∃ x, ¬p x) → ¬∀ x,
 @[simp] theorem exists_eq_right_right' : (∃ (a : α), p a ∧ q a ∧ a' = a) ↔ p a' ∧ q a' := by
   simp [@eq_comm _ a']
 
+@[simp] theorem exists_or_eq_left (y : α) (p : α → Prop) : ∃ x : α, x = y ∨ p x := ⟨y, .inl rfl⟩
+@[simp] theorem exists_or_eq_right (y : α) (p : α → Prop) : ∃ x : α, p x ∨ x = y := ⟨y, .inr rfl⟩
+@[simp] theorem exists_or_eq_left' (y : α) (p : α → Prop) : ∃ x : α, y = x ∨ p x := ⟨y, .inl rfl⟩
+@[simp] theorem exists_or_eq_right' (y : α) (p : α → Prop) : ∃ x : α, p x ∨ y = x := ⟨y, .inr rfl⟩
+
 @[simp] theorem exists_prop : (∃ _h : a, b) ↔ a ∧ b :=
   ⟨fun ⟨hp, hq⟩ => ⟨hp, hq⟩, fun ⟨hp, hq⟩ => ⟨hp, hq⟩⟩
 

src/Init/Util.lean

@@ -45,6 +45,13 @@ def dbgSleep {α : Type u} (ms : UInt32) (f : Unit → α) : α := f ()
 @[extern "lean_ptr_addr"]
 unsafe opaque ptrAddrUnsafe {α : Type u} (a : @& α) : USize
 
+/--
+Returns `true` if `a` is an exclusive object.
+We say an object is exclusive if it is single-threaded and its reference counter is 1.
+-/
+@[extern "lean_is_exclusive_obj"]
+unsafe opaque isExclusiveUnsafe {α : Type u} (a : @& α) : Bool
+
 set_option linter.unusedVariables.funArgs false in
 @[inline] unsafe def withPtrAddrUnsafe {α : Type u} {β : Type v} (a : α) (k : USize → β) (h : ∀ u₁ u₂, k u₁ = k u₂) : β :=
   k (ptrAddrUnsafe a)

src/Lean/AuxRecursor.lean

@@ -37,7 +37,7 @@ def isAuxRecursor (env : Environment) (declName : Name) : Bool :=
 
 def isAuxRecursorWithSuffix (env : Environment) (declName : Name) (suffix : String) : Bool :=
   match declName with
-  | .str _ s => s == suffix && isAuxRecursor env declName
+  | .str _ s => (s == suffix || s.startsWith s!"{suffix}_") && isAuxRecursor env declName
   | _ => false
 
 def isCasesOnRecursor (env : Environment) (declName : Name) : Bool :=

src/Lean/Elab/App.lean

@@ -5,6 +5,7 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Lean.Util.FindMVar
+import Lean.Util.CollectFVars
 import Lean.Parser.Term
 import Lean.Meta.KAbstract
 import Lean.Meta.Tactic.ElimInfo
@@ -711,6 +712,12 @@ structure Context where
   ```
   theorem Eq.subst' {α} {motive : α → Prop} {a b : α} (h : a = b) : motive a → motive b
   ```
+  For another example, the term `isEmptyElim (α := α)` is an underapplied eliminator, and it needs
+  argument `α` to be elaborated eagerly to create a type-correct motive.
+  ```
+  def isEmptyElim [IsEmpty α] {p : α → Sort _} (a : α) : p a := ...
+  example {α : Type _} [IsEmpty α] : id (α → False) := isEmptyElim (α := α)
+  ```
   -/
   extraArgsPos : Array Nat
 
@@ -724,8 +731,8 @@ structure State where
   namedArgs    : List NamedArg
   /-- User-provided arguments that still have to be processed. -/
   args         : List Arg
-  /-- Discriminants processed so far. -/
-  discrs       : Array Expr := #[]
+  /-- Discriminants (targets) processed so far. -/
+  discrs       : Array (Option Expr)
   /-- Instance implicit arguments collected so far. -/
   instMVars    : Array MVarId := #[]
   /-- Position of the next argument to be processed. We use it to decide whether the argument is the motive or a discriminant. -/
@@ -736,7 +743,7 @@ structure State where
 abbrev M := ReaderT Context $ StateRefT State TermElabM
 
 /-- Infer the `motive` using the expected type by `kabstract`ing the discriminants. -/
-def mkMotive (discrs : Array Expr) (expectedType : Expr): MetaM Expr := do
+def mkMotive (discrs : Array Expr) (expectedType : Expr) : MetaM Expr := do
   discrs.foldrM (init := expectedType) fun discr motive => do
     let discr ← instantiateMVars discr
     let motiveBody ← kabstract motive discr
@@ -758,7 +765,7 @@ def revertArgs (args : List Arg) (f : Expr) (expectedType : Expr) : TermElabM (E
     return (mkApp f val, mkForall (← mkFreshBinderName) BinderInfo.default valType expectedTypeBody)
 
 /--
-Construct the resulting application after all discriminants have bee elaborated, and we have
+Construct the resulting application after all discriminants have been elaborated, and we have
 consumed as many given arguments as possible.
 -/
 def finalize : M Expr := do
@@ -766,29 +773,50 @@ def finalize : M Expr := do
     throwError "failed to elaborate eliminator, unused named arguments: {(← get).namedArgs.map (·.name)}"
   let some motive := (← get).motive?
     | throwError "failed to elaborate eliminator, insufficient number of arguments"
+  trace[Elab.app.elab_as_elim] "motive: {motive}"
   forallTelescope (← get).fType fun xs _ => do
+    trace[Elab.app.elab_as_elim] "xs: {xs}"
     let mut expectedType := (← read).expectedType
+    trace[Elab.app.elab_as_elim] "expectedType:{indentD expectedType}"
+    let throwInsufficient := do
+      throwError "failed to elaborate eliminator, insufficient number of arguments, expected type:{indentExpr expectedType}"
     let mut f := (← get).f
     if xs.size > 0 then
+      -- under-application, specialize the expected type using `xs`
       assert! (← get).args.isEmpty
-      try
-        expectedType ← instantiateForall expectedType xs
-      catch _ =>
-        throwError "failed to elaborate eliminator, insufficient number of arguments, expected type:{indentExpr expectedType}"
+      for x in xs do
+        let .forallE _ t b _ ← whnf expectedType | throwInsufficient
+        unless ← fullApproxDefEq <| isDefEq t (← inferType x) do
+          -- We can't assume that these binding domains were supposed to line up, so report insufficient arguments
+          throwInsufficient
+        expectedType := b.instantiate1 x
+      trace[Elab.app.elab_as_elim] "xs after specialization of expected type: {xs}"
     else
-      -- over-application, simulate `revert`
+      -- over-application, simulate `revert` while generalizing the values of these arguments in the expected type
       (f, expectedType) ← revertArgs (← get).args f expectedType
+      unless ← isTypeCorrect expectedType do
+        throwError "failed to elaborate eliminator, after generalizing over-applied arguments, expected type is type incorrect:{indentExpr expectedType}"
+    trace[Elab.app.elab_as_elim] "expectedType after processing:{indentD expectedType}"
     let result := mkAppN f xs
+    trace[Elab.app.elab_as_elim] "result:{indentD result}"
     let mut discrs := (← get).discrs
     let idx := (← get).idx
-    if (← get).discrs.size < (← read).elimInfo.targetsPos.size then
+    if discrs.any Option.isNone then
       for i in [idx:idx + xs.size], x in xs do
-        if (← read).elimInfo.targetsPos.contains i then
-          discrs := discrs.push x
-    let motiveVal ← mkMotive discrs expectedType
+        if let some tidx := (← read).elimInfo.targetsPos.indexOf? i then
+          discrs := discrs.set! tidx x
+    if let some idx := discrs.findIdx? Option.isNone then
+      -- This should not happen.
+      trace[Elab.app.elab_as_elim] "Internal error, missing target with index {idx}"
+      throwError "failed to elaborate eliminator, insufficient number of arguments"
+    trace[Elab.app.elab_as_elim] "discrs: {discrs.map Option.get!}"
+    let motiveVal ← mkMotive (discrs.map Option.get!) expectedType
+    unless (← isTypeCorrect motiveVal) do
+      throwError "failed to elaborate eliminator, motive is not type correct:{indentD motiveVal}"
     unless (← isDefEq motive motiveVal) do
       throwError "failed to elaborate eliminator, invalid motive{indentExpr motiveVal}"
     synthesizeAppInstMVars (← get).instMVars result
+    trace[Elab.app.elab_as_elim] "completed motive:{indentD motive}"
     let result ← mkLambdaFVars xs (← instantiateMVars result)
     return result
 
@@ -816,9 +844,9 @@ def getNextArg? (binderName : Name) (binderInfo : BinderInfo) : M (LOption Arg)
 def setMotive (motive : Expr) : M Unit :=
   modify fun s => { s with motive? := motive }
 
-/-- Push the given expression into the `discrs` field in the state. -/
-def addDiscr (discr : Expr) : M Unit :=
-  modify fun s => { s with discrs := s.discrs.push discr }
+/-- Push the given expression into the `discrs` field in the state, where `i` is which target it is for. -/
+def addDiscr (i : Nat) (discr : Expr) : M Unit :=
+  modify fun s => { s with discrs := s.discrs.set! i discr }
 
 /-- Elaborate the given argument with the given expected type. -/
 private def elabArg (arg : Arg) (argExpectedType : Expr) : M Expr := do
@@ -850,14 +878,23 @@ partial def main : M Expr := do
     main
   let idx := (← get).idx
   if (← read).elimInfo.motivePos == idx then
-    let motive ← mkImplicitArg binderType binderInfo
+    let motive ←
+      match (← getNextArg? binderName binderInfo) with
+      | .some arg =>
+        /- Due to `Lean.Elab.Term.elabAppArgs.elabAsElim?`, this must be a positional argument that is the syntax `_`. -/
+        elabArg arg binderType
+      | .none | .undef =>
+        /- Note: undef occurs when the motive is explicit but missing.
+           In this case, we treat it as if it were an implicit argument
+           to support writing `h.rec` when `h : False`, rather than requiring `h.rec _`. -/
+        mkImplicitArg binderType binderInfo
     setMotive motive
     addArgAndContinue motive
-  else if (← read).elimInfo.targetsPos.contains idx then
+  else if let some tidx := (← read).elimInfo.targetsPos.indexOf? idx then
     match (← getNextArg? binderName binderInfo) with
-    | .some arg => let discr ← elabArg arg binderType; addDiscr discr; addArgAndContinue discr
+    | .some arg => let discr ← elabArg arg binderType; addDiscr tidx discr; addArgAndContinue discr
     | .undef => finalize
-    | .none => let discr ← mkImplicitArg binderType binderInfo; addDiscr discr; addArgAndContinue discr
+    | .none => let discr ← mkImplicitArg binderType binderInfo; addDiscr tidx discr; addArgAndContinue discr
   else match (← getNextArg? binderName binderInfo) with
     | .some (.stx stx) =>
       if (← read).extraArgsPos.contains idx then
@@ -919,10 +956,12 @@ def elabAppArgs (f : Expr) (namedArgs : Array NamedArg) (args : Array Arg)
     let expectedType ← instantiateMVars expectedType
     if expectedType.getAppFn.isMVar then throwError "failed to elaborate eliminator, expected type is not available"
     let extraArgsPos ← getElabAsElimExtraArgsPos elimInfo
+    trace[Elab.app.elab_as_elim] "extraArgsPos: {extraArgsPos}"
     ElabElim.main.run { elimInfo, expectedType, extraArgsPos } |>.run' {
       f, fType
       args := args.toList
       namedArgs := namedArgs.toList
+      discrs := mkArray elimInfo.targetsPos.size none
     }
   else
     ElabAppArgs.main.run { explicit, ellipsis, resultIsOutParamSupport } |>.run' {
@@ -940,31 +979,60 @@ where
     unless (← shouldElabAsElim declName) do return none
     let elimInfo ← getElimInfo declName
     forallTelescopeReducing (← inferType f) fun xs _ => do
-      if h : elimInfo.motivePos < xs.size then
-        let x := xs[elimInfo.motivePos]
+      /- Process arguments similar to `Lean.Elab.Term.ElabElim.main` to see if the motive has been
+         provided, in which case we use the standard app elaborator.
+         If the motive is explicit (like for `False.rec`), then a positional `_` counts as "not provided". -/
+      let mut args := args.toList
+      let mut namedArgs := namedArgs.toList
+      for x in xs[0:elimInfo.motivePos] do
         let localDecl ← x.fvarId!.getDecl
-        if findBinderName? namedArgs.toList localDecl.userName matches some _ then
+        match findBinderName? namedArgs localDecl.userName with
+        | some _ => namedArgs := eraseNamedArg namedArgs localDecl.userName
+        | none   => if localDecl.binderInfo.isExplicit then args := args.tailD []
+      -- Invariant: `elimInfo.motivePos < xs.size` due to construction of `elimInfo`.
+      let some x := xs[elimInfo.motivePos]? | unreachable!
+      let localDecl ← x.fvarId!.getDecl
+      if findBinderName? namedArgs localDecl.userName matches some _ then
+        -- motive has been explicitly provided, so we should use standard app elaborator
+        return none
+      else
+        match localDecl.binderInfo.isExplicit, args with
+        | true, .expr _ :: _  =>
           -- motive has been explicitly provided, so we should use standard app elaborator
           return none
-        return some elimInfo
-      else
-        return none
+        | true, .stx arg :: _ =>
+          if arg.isOfKind ``Lean.Parser.Term.hole then
+            return some elimInfo
+          else
+            -- positional motive is not `_`, so we should use standard app elaborator
+            return none
+        | _, _ => return some elimInfo
 
   /--
   Collect extra argument positions that must be elaborated eagerly when using `elab_as_elim`.
-  The idea is that the contribute to motive inference. See comment at `ElamElim.Context.extraArgsPos`.
+  The idea is that they contribute to motive inference. See comment at `ElamElim.Context.extraArgsPos`.
   -/
   getElabAsElimExtraArgsPos (elimInfo : ElimInfo) : MetaM (Array Nat) := do
     forallTelescope elimInfo.elimType fun xs type => do
-      let resultArgs := type.getAppArgs
+      let targets := type.getAppArgs
+      /- Compute transitive closure of fvars appearing in the motive and the targets. -/
+      let initMotiveFVars : CollectFVars.State := targets.foldl (init := {}) collectFVars
+      let motiveFVars ← xs.size.foldRevM (init := initMotiveFVars) fun i s => do
+        let x := xs[i]!
+        if elimInfo.motivePos == i || elimInfo.targetsPos.contains i || s.fvarSet.contains x.fvarId! then
+          return collectFVars s (← inferType x)
+        else
+          return s
+      /- Collect the extra argument positions -/
       let mut extraArgsPos := #[]
       for i in [:xs.size] do
         let x := xs[i]!
-        unless elimInfo.targetsPos.contains i do
-          let xType ← inferType x
+        unless elimInfo.motivePos == i || elimInfo.targetsPos.contains i do
+          let xType ← x.fvarId!.getType
           /- We only consider "first-order" types because we can reliably "extract" information from them. -/
-          if isFirstOrder xType
-             && Option.isSome (xType.find? fun e => e.isFVar && resultArgs.contains e) then
+          if motiveFVars.fvarSet.contains x.fvarId!
+             || (isFirstOrder xType
+                 && Option.isSome (xType.find? fun e => e.isFVar && motiveFVars.fvarSet.contains e.fvarId!)) then
             extraArgsPos := extraArgsPos.push i
       return extraArgsPos
 
@@ -1314,9 +1382,17 @@ private partial def resolveDotName (id : Syntax) (expectedType? : Option Expr) :
   tryPostponeIfNoneOrMVar expectedType?
   let some expectedType := expectedType?
     | throwError "invalid dotted identifier notation, expected type must be known"
-  forallTelescopeReducing expectedType fun _ resultType => do
+  withForallBody expectedType fun resultType => do
     go resultType expectedType #[]
 where
+  /-- A weak version of forallTelescopeReducing that only uses whnfCore, to avoid unfolding definitions except by `unfoldDefinition?` below. -/
+  withForallBody {α} (type : Expr) (k : Expr → TermElabM α) : TermElabM α :=
+    forallTelescope type fun _ body => do
+      let body ← whnfCore body
+      if body.isForall then
+        withForallBody body k
+      else
+        k body
   go (resultType : Expr) (expectedType : Expr) (previousExceptions : Array Exception) : TermElabM Name := do
     let resultType ← instantiateMVars resultType
     let resultTypeFn := resultType.cleanupAnnotations.getAppFn
@@ -1332,7 +1408,8 @@ where
       | ex@(.error ..) =>
         match (← unfoldDefinition? resultType) with
         | some resultType =>
-          go (← whnfCore resultType) expectedType (previousExceptions.push ex)
+          withForallBody resultType fun resultType => do
+            go resultType expectedType (previousExceptions.push ex)
         | none =>
           previousExceptions.forM fun ex => logException ex
           throw ex
@@ -1525,5 +1602,6 @@ builtin_initialize
   registerTraceClass `Elab.app.args (inherited := true)
   registerTraceClass `Elab.app.propagateExpectedType (inherited := true)
   registerTraceClass `Elab.app.finalize (inherited := true)
+  registerTraceClass `Elab.app.elab_as_elim (inherited := true)
 
 end Lean.Elab.Term

src/Lean/Elab/BuiltinCommand.lean

@@ -5,6 +5,7 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Lean.Util.CollectLevelParams
+import Lean.Util.CollectAxioms
 import Lean.Meta.Reduce
 import Lean.Elab.DeclarationRange
 import Lean.Elab.Eval
@@ -340,8 +341,7 @@ private def mkRunEval (e : Expr) : MetaM Expr := do
   let instVal ← mkEvalInstCore ``Lean.Eval e
   instantiateMVars (mkAppN (mkConst ``Lean.runEval [u]) #[α, instVal, mkSimpleThunk e])
 
-unsafe def elabEvalUnsafe : CommandElab
-  | `(#eval%$tk $term) => do
+unsafe def elabEvalCoreUnsafe (bang : Bool) (tk term : Syntax): CommandElabM Unit := do
     let declName := `_eval
     let addAndCompile (value : Expr) : TermElabM Unit := do
       let value ← Term.levelMVarToParam (← instantiateMVars value)
@@ -358,6 +358,13 @@ unsafe def elabEvalUnsafe : CommandElab
       }
       Term.ensureNoUnassignedMVars decl
       addAndCompile decl
+    -- Check for sorry axioms
+    let checkSorry (declName : Name) : MetaM Unit := do
+      unless bang do
+        let axioms ← collectAxioms declName
+        if axioms.contains ``sorryAx then
+          throwError ("cannot evaluate expression that depends on the `sorry` axiom.\nUse `#eval!` to " ++
+            "evaluate nevertheless (which may cause lean to crash).")
     -- Elaborate `term`
     let elabEvalTerm : TermElabM Expr := do
       let e ← Term.elabTerm term none
@@ -386,6 +393,7 @@ unsafe def elabEvalUnsafe : CommandElab
           else
             let e ← mkRunMetaEval e
             addAndCompile e
+            checkSorry declName
             let act ← evalConst (Environment → Options → IO (String × Except IO.Error Environment)) declName
             pure <| Sum.inr act
       match act with
@@ -402,6 +410,7 @@ unsafe def elabEvalUnsafe : CommandElab
       -- modify e to `runEval e`
       let e ← mkRunEval (← elabEvalTerm)
       addAndCompile e
+      checkSorry declName
       let act ← evalConst (IO (String × Except IO.Error Unit)) declName
       let (out, res) ← liftM (m := IO) act
       logInfoAt tk out
@@ -412,10 +421,19 @@ unsafe def elabEvalUnsafe : CommandElab
       elabMetaEval
     else
       elabEval
+
+@[implemented_by elabEvalCoreUnsafe]
+opaque elabEvalCore (bang : Bool) (tk term : Syntax): CommandElabM Unit
+
+@[builtin_command_elab «eval»]
+def elabEval : CommandElab
+  | `(#eval%$tk $term) => elabEvalCore false tk term
   | _ => throwUnsupportedSyntax
 
-@[builtin_command_elab «eval», implemented_by elabEvalUnsafe]
-opaque elabEval : CommandElab
+@[builtin_command_elab evalBang]
+def elabEvalBang : CommandElab
+  | `(Parser.Command.evalBang|#eval!%$tk $term) => elabEvalCore true tk term
+  | _ => throwUnsupportedSyntax
 
 private def checkImportsForRunCmds : CommandElabM Unit := do
   unless (← getEnv).contains ``CommandElabM do

src/Lean/Elab/BuiltinTerm.lean

@@ -362,4 +362,7 @@ private opaque evalFilePath (stx : Syntax) : TermElabM System.FilePath
     mkStrLit <$> IO.FS.readFile path
   | _, _ => throwUnsupportedSyntax
 
+@[builtin_term_elab Lean.Parser.Term.namedPattern] def elabNamedPatternErr : TermElab := fun stx _ =>
+  throwError "`<identifier>@<term>` is a named pattern and can only be used in pattern matching contexts{indentD stx}"
+
 end Lean.Elab.Term

src/Lean/Elab/Command.lean

@@ -12,17 +12,62 @@ import Lean.Language.Basic
 
 namespace Lean.Elab.Command
 
+/--
+A `Scope` records the part of the `CommandElabM` state that respects scoping,
+such as the data for `universe`, `open`, and `variable` declarations, the current namespace,
+and currently enabled options.
+The `CommandElabM` state contains a stack of scopes, and only the top `Scope`
+on the stack is read from or modified. There is always at least one `Scope` on the stack,
+even outside any `section` or `namespace`, and each new pushed `Scope`
+starts as a modified copy of the previous top scope.
+-/
 structure Scope where
+  /--
+  The component of the `namespace` or `section` that this scope is associated to.
+  For example, `section a.b.c` and `namespace a.b.c` each create three scopes with headers
+  named `a`, `b`, and `c`.
+  This is used for checking the `end` command. The "base scope" has `""` as its header.
+  -/
   header        : String
+  /--
+  The current state of all set options at this point in the scope. Note that this is the
+  full current set of options and does *not* simply contain the options set
+  while this scope has been active.
+  -/
   opts          : Options := {}
+  /-- The current namespace. The top-level namespace is represented by `Name.anonymous`. -/
   currNamespace : Name := Name.anonymous
+  /-- All currently `open`ed namespaces and names. -/
   openDecls     : List OpenDecl := []
+  /-- The current list of names for universe level variables to use for new declarations. This is managed by the `universe` command. -/
   levelNames    : List Name := []
-  /-- section variables -/
+  /--
+  The current list of binders to use for new declarations.
+  This is managed by the `variable` command.
+  Each binder is represented in `Syntax` form, and it is re-elaborated
+  within each command that uses this information.
+
+  This is also used by commands, such as `#check`, to create an initial local context,
+  even if they do not work with binders per se.
+  -/
   varDecls      : Array (TSyntax ``Parser.Term.bracketedBinder) := #[]
-  /-- Globally unique internal identifiers for the `varDecls` -/
+  /--
+  Globally unique internal identifiers for the `varDecls`.
+  There is one identifier per variable introduced by the binders
+  (recall that a binder such as `(a b c : Ty)` can produce more than one variable),
+  and each identifier is the user-provided variable name with a macro scope.
+  This is used by `TermElabM` in `Lean.Elab.Term.Context` to help with processing macros
+  that capture these variables.
+  -/
   varUIds       : Array Name := #[]
-  /-- noncomputable sections automatically add the `noncomputable` modifier to any declaration we cannot generate code for. -/
+  /--
+  If true (default: false), all declarations that fail to compile
+  automatically receive the `noncomputable` modifier.
+  A scope with this flag set is created by `noncomputable section`.
+
+  Recall that a new scope inherits all values from its parent scope,
+  so all sections and namespaces nested within a `noncomputable` section also have this flag set.
+  -/
   isNoncomputable : Bool := false
   deriving Inhabited
 
@@ -230,6 +275,7 @@ private def ioErrorToMessage (ctx : Context) (ref : Syntax) (err : IO.Error) : M
 instance : MonadLiftT IO CommandElabM where
   monadLift := liftIO
 
+/-- Return the current scope. -/
 def getScope : CommandElabM Scope := do pure (← get).scopes.head!
 
 instance : MonadResolveName CommandElabM where
@@ -479,7 +525,7 @@ def elabCommandTopLevel (stx : Syntax) : CommandElabM Unit := withRef stx do pro
       -- should be true iff the command supports incrementality
       if (← IO.hasFinished snap.new.result) then
         trace[Elab.snapshotTree]
-          Language.ToSnapshotTree.toSnapshotTree snap.new.result.get |>.format
+          (←Language.ToSnapshotTree.toSnapshotTree snap.new.result.get |>.format)
     modify fun st => { st with
       messages := initMsgs ++ msgs
       infoState := { st.infoState with trees := initInfoTrees ++ st.infoState.trees }
@@ -612,6 +658,11 @@ Interrupt and abort exceptions are caught but not logged.
 private def liftAttrM {α} (x : AttrM α) : CommandElabM α := do
   liftCoreM x
 
+/--
+Return the stack of all currently active scopes:
+the base scope always comes last; new scopes are prepended in the front.
+In particular, the current scope is always the first element.
+-/
 def getScopes : CommandElabM (List Scope) := do
   pure (← get).scopes
 

src/Lean/Elab/Deriving/BEq.lean

@@ -43,12 +43,10 @@ where
         let mut ctorArgs1 := #[]
         let mut ctorArgs2 := #[]
         let mut rhs ← `(true)
-        -- add `_` for inductive parameters, they are inaccessible
-        for _ in [:indVal.numParams] do
-          ctorArgs1 := ctorArgs1.push (← `(_))
-          ctorArgs2 := ctorArgs2.push (← `(_))
+        let mut rhs_empty := true
         for i in [:ctorInfo.numFields] do
-          let x := xs[indVal.numParams + i]!
+          let pos := indVal.numParams + ctorInfo.numFields - i - 1
+          let x := xs[pos]!
           if type.containsFVar x.fvarId! then
             -- If resulting type depends on this field, we don't need to compare
             ctorArgs1 := ctorArgs1.push (← `(_))
@@ -62,11 +60,32 @@ where
             if (← isProp xType) then
               continue
             if xType.isAppOf indVal.name then
-              rhs ← `($rhs && $(mkIdent auxFunName):ident $a:ident $b:ident)
+              if rhs_empty then
+                rhs ← `($(mkIdent auxFunName):ident $a:ident $b:ident)
+                rhs_empty := false
+              else
+                rhs ← `($(mkIdent auxFunName):ident $a:ident $b:ident && $rhs)
+            /- If `x` appears in the type of another field, use `eq_of_beq` to
+               unify the types of the subsequent variables -/
+            else if ← xs[pos+1:].anyM
+                (fun fvar => (Expr.containsFVar · x.fvarId!) <$> (inferType fvar)) then
+              rhs ← `(if h : $a:ident == $b:ident then by
+                        cases (eq_of_beq h)
+                        exact $rhs
+                      else false)
+              rhs_empty := false
             else
-              rhs ← `($rhs && $a:ident == $b:ident)
-        patterns := patterns.push (← `(@$(mkIdent ctorName):ident $ctorArgs1:term*))
-        patterns := patterns.push (← `(@$(mkIdent ctorName):ident $ctorArgs2:term*))
+              if rhs_empty then
+                rhs ← `($a:ident == $b:ident)
+                rhs_empty := false
+              else
+                rhs ← `($a:ident == $b:ident && $rhs)
+          -- add `_` for inductive parameters, they are inaccessible
+        for _ in [:indVal.numParams] do
+          ctorArgs1 := ctorArgs1.push (← `(_))
+          ctorArgs2 := ctorArgs2.push (← `(_))
+        patterns := patterns.push (← `(@$(mkIdent ctorName):ident $ctorArgs1.reverse:term*))
+        patterns := patterns.push (← `(@$(mkIdent ctorName):ident $ctorArgs2.reverse:term*))
         `(matchAltExpr| | $[$patterns:term],* => $rhs:term)
       alts := alts.push alt
     alts := alts.push (← mkElseAlt)

src/Lean/Elab/Deriving/DecEq.lean

@@ -91,8 +91,14 @@ def mkAuxFunction (ctx : Context) (auxFunName : Name) (indVal : InductiveVal): T
   let header  ← mkDecEqHeader indVal
   let body    ← mkMatch ctx header indVal
   let binders := header.binders
-  let type    ← `(Decidable ($(mkIdent header.targetNames[0]!) = $(mkIdent header.targetNames[1]!)))
-  `(private def $(mkIdent auxFunName):ident $binders:bracketedBinder* : $type:term := $body:term)
+  let target₁ := mkIdent header.targetNames[0]!
+  let target₂ := mkIdent header.targetNames[1]!
+  let termSuffix ← if indVal.isRec
+    then `(Parser.Termination.suffix|termination_by structural $target₁)
+    else `(Parser.Termination.suffix|)
+  let type    ← `(Decidable ($target₁ = $target₂))
+  `(private def $(mkIdent auxFunName):ident $binders:bracketedBinder* : $type:term := $body:term
+    $termSuffix:suffix)
 
 def mkAuxFunctions (ctx : Context) : TermElabM (TSyntax `command) := do
   let mut res : Array (TSyntax `command) := #[]

src/Lean/Elab/PreDefinition/Eqns.lean

@@ -333,7 +333,7 @@ def tryContradiction (mvarId : MVarId) : MetaM Bool := do
 partial def mkUnfoldProof (declName : Name) (mvarId : MVarId) : MetaM Unit := do
   let some eqs ← getEqnsFor? declName | throwError "failed to generate equations for '{declName}'"
   let tryEqns (mvarId : MVarId) : MetaM Bool :=
-    eqs.anyM fun eq => commitWhen do
+    eqs.anyM fun eq => commitWhen do checkpointDefEq (mayPostpone := false) do
       try
         let subgoals ← mvarId.apply (← mkConstWithFreshMVarLevels eq)
         subgoals.allM fun subgoal => do

src/Lean/Elab/PreDefinition/Main.lean

@@ -114,7 +114,7 @@ def checkTerminationByHints (preDefs : Array PreDefinition) : CoreM Unit := do
       if !structural && !preDefsWithout.isEmpty then
         let m := MessageData.andList (preDefsWithout.toList.map (m!"{·.declName}"))
         let doOrDoes := if preDefsWithout.size = 1 then "does" else "do"
-        logErrorAt termBy.ref (m!"Incomplete set of `termination_by` annotations:\n"++
+        logErrorAt termBy.ref (m!"incomplete set of `termination_by` annotations:\n"++
           m!"This function is mutually with {m}, which {doOrDoes} not have " ++
           m!"a `termination_by` clause.\n" ++
           m!"The present clause is ignored.")

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

@@ -5,11 +5,11 @@ Authors: Leonardo de Moura, Joachim Breitner
 -/
 prelude
 import Lean.Util.HasConstCache
+import Lean.Meta.PProdN
 import Lean.Meta.Match.MatcherApp.Transform
 import Lean.Elab.RecAppSyntax
 import Lean.Elab.PreDefinition.Basic
 import Lean.Elab.PreDefinition.Structural.Basic
-import Lean.Elab.PreDefinition.Structural.FunPacker
 import Lean.Elab.PreDefinition.Structural.RecArgInfo
 
 namespace Lean.Elab.Structural
@@ -21,11 +21,11 @@ private def throwToBelowFailed : MetaM α :=
 partial def searchPProd (e : Expr) (F : Expr) (k : Expr → Expr → MetaM α) : MetaM α := do
   match (← whnf e) with
   | .app (.app (.const `PProd _) d1) d2 =>
-        (do searchPProd d1 (← mkAppM ``PProd.fst #[F]) k)
-    <|> (do searchPProd d2 (← mkAppM `PProd.snd #[F]) k)
+        (do searchPProd d1 (.proj ``PProd 0 F) k)
+    <|> (do searchPProd d2 (.proj ``PProd 1 F) k)
   | .app (.app (.const `And _) d1) d2 =>
-        (do searchPProd d1 (← mkAppM `And.left #[F]) k)
-    <|> (do searchPProd d2 (← mkAppM `And.right #[F]) k)
+        (do searchPProd d1 (.proj `And 0 F) k)
+    <|> (do searchPProd d2 (.proj `And 1 F) k)
   | .const `PUnit _
   | .const `True _ => throwToBelowFailed
   | _ => k e F
@@ -85,7 +85,7 @@ private def withBelowDict [Inhabited α] (below : Expr) (numIndParams : Nat)
         return ((← mkFreshUserName `C), fun _ => pure t)
     withLocalDeclsD CDecls fun Cs => do
       -- We have to pack these canary motives like we packed the real motives
-      let packedCs ← positions.mapMwith packMotives motiveTypes Cs
+      let packedCs ← positions.mapMwith PProdN.packLambdas motiveTypes Cs
       let belowDict := mkAppN pre packedCs
       let belowDict := mkAppN belowDict finalArgs
       trace[Elab.definition.structural] "initial belowDict for {Cs}:{indentExpr belowDict}"
@@ -245,23 +245,12 @@ def mkBRecOnConst (recArgInfos : Array RecArgInfo) (positions : Positions)
       decLevel brecOnUniv
     else
       pure brecOnUniv
-  let brecOnCons := fun idx  =>
-    let brecOn :=
-      if let .some n := indGroup.all[idx]? then
-        if useBInductionOn then .const (mkBInductionOnName n) indGroup.levels
-        else                    .const (mkBRecOnName n) (brecOnUniv :: indGroup.levels)
-      else
-        let n := indGroup.all[0]!
-        let j := idx - indGroup.all.size + 1
-        if useBInductionOn then .const (mkBInductionOnName n |>.appendIndexAfter j) indGroup.levels
-        else                    .const (mkBRecOnName n |>.appendIndexAfter j) (brecOnUniv :: indGroup.levels)
-    mkAppN brecOn indGroup.params
-
+  let brecOnCons := fun idx => indGroup.brecOn useBInductionOn brecOnUniv idx
   -- Pick one as a prototype
   let brecOnAux := brecOnCons 0
   -- Infer the type of the packed motive arguments
   let packedMotiveTypes ← inferArgumentTypesN indGroup.numMotives brecOnAux
-  let packedMotives ← positions.mapMwith packMotives packedMotiveTypes motives
+  let packedMotives ← positions.mapMwith PProdN.packLambdas packedMotiveTypes motives
 
   return fun n => mkAppN (brecOnCons n) packedMotives
 
@@ -300,12 +289,11 @@ def mkBrecOnApp (positions : Positions) (fnIdx : Nat) (brecOnConst : Nat → Exp
     let brecOn := brecOnConst recArgInfo.indIdx
     let brecOn := mkAppN brecOn indexMajorArgs
     let packedFTypes ← inferArgumentTypesN positions.size brecOn
-    let packedFArgs ← positions.mapMwith packFArgs packedFTypes FArgs
+    let packedFArgs ← positions.mapMwith PProdN.mkLambdas packedFTypes FArgs
     let brecOn := mkAppN brecOn packedFArgs
     let some poss := positions.find? (·.contains fnIdx)
       | throwError "mkBrecOnApp: Could not find {fnIdx} in {positions}"
-    let brecOn ← if poss.size = 1 then pure brecOn else
-      mkPProdProjN (poss.getIdx? fnIdx).get! brecOn
+    let brecOn ← PProdN.proj poss.size (poss.getIdx? fnIdx).get! brecOn
     mkLambdaFVars ys (mkAppN brecOn otherArgs)
 
 end Lean.Elab.Structural

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

@@ -21,6 +21,7 @@ namespace Structural
 structure EqnInfo extends EqnInfoCore where
   recArgPos : Nat
   declNames : Array Name
+  numFixed  : Nat
   deriving Inhabited
 
 private partial def mkProof (declName : Name) (type : Expr) : MetaM Expr := do
@@ -81,9 +82,11 @@ def mkEqns (info : EqnInfo) : MetaM (Array Name) :=
 
 builtin_initialize eqnInfoExt : MapDeclarationExtension EqnInfo ← mkMapDeclarationExtension
 
-def registerEqnsInfo (preDef : PreDefinition) (declNames : Array Name) (recArgPos : Nat) : CoreM Unit := do
+def registerEqnsInfo (preDef : PreDefinition) (declNames : Array Name) (recArgPos : Nat)
+    (numFixed : Nat) : CoreM Unit := do
   ensureEqnReservedNamesAvailable preDef.declName
-  modifyEnv fun env => eqnInfoExt.insert env preDef.declName { preDef with recArgPos, declNames }
+  modifyEnv fun env => eqnInfoExt.insert env preDef.declName
+    { preDef with recArgPos, declNames, numFixed }
 
 def getEqnsFor? (declName : Name) : MetaM (Option (Array Name)) := do
   if let some info := eqnInfoExt.find? (← getEnv) declName then

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

@@ -68,9 +68,7 @@ def getRecArgInfo (fnName : Name) (numFixed : Nat) (xs : Array Expr) (i : Nat) :
       throwError "it is a let-binding"
     let xType ← whnfD localDecl.type
     matchConstInduct xType.getAppFn (fun _ => throwError "its type is not an inductive") fun indInfo us => do
-    if !(← hasConst (mkBRecOnName indInfo.name)) then
-      throwError "its type {indInfo.name} does not have a recursor"
-    else if indInfo.isReflexive && !(← hasConst (mkBInductionOnName indInfo.name)) && !(← isInductivePredicate indInfo.name) then
+    if indInfo.isReflexive && !(← hasConst (mkBInductionOnName indInfo.name)) && !(← isInductivePredicate indInfo.name) then
       throwError "its type {indInfo.name} is a reflexive inductive, but {mkBInductionOnName indInfo.name} does not exist and it is not an inductive predicate"
     else
       let indArgs    : Array Expr := xType.getAppArgs
@@ -263,6 +261,11 @@ def tryAllArgs (fnNames : Array Name) (xs : Array Expr) (values : Array Expr)
     if let some combs := allCombinations recArgInfoss' then
       for comb in combs do
         try
+          -- Check that the group actually has a brecOn (we used to check this in getRecArgInfo,
+          -- but in the first phase we do not want to rule-out non-recursive types like `Array`, which
+          -- are ok in a nested group. This logic can maybe simplified)
+          unless (← hasConst (group.brecOnName false 0)) do
+            throwError "the type {group} does not have a `.brecOn` recursor"
           -- TODO: Here we used to save and restore the state. But should the `try`-`catch`
           -- not suffice?
           let r ← k comb

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

@@ -1,126 +0,0 @@
-/-
-Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joachim Breitner
--/
-prelude
-import Lean.Meta.InferType
-
-/-!
-This module contains the logic that packs the motives and FArgs of multiple functions into one,
-to allow structural mutual recursion where the number of functions is not exactly the same
-as the number of inductive data types in the mutual inductive group.
-
-The private helper functions related to `PProd` here should at some point be moved to their own
-module, so that they can be used elsewhere (e.g. `FunInd`), and possibly unified with the similar
-constructions for well-founded recursion (see `ArgsPacker` module).
--/
-
-namespace Lean.Elab.Structural
-open Meta
-
-private def mkPUnit : Level → Expr
-  | .zero => .const ``True []
-  | lvl   => .const ``PUnit [lvl]
-
-private def mkPProd (e1 e2 : Expr) : MetaM Expr := do
-  let lvl1 ← getLevel e1
-  let lvl2 ← getLevel e2
-  if lvl1 matches .zero && lvl2 matches .zero then
-    return mkApp2 (.const `And []) e1 e2
-  else
-    return mkApp2 (.const ``PProd [lvl1, lvl2]) e1 e2
-
-private def mkNProd (lvl : Level) (es : Array Expr) : MetaM Expr :=
-  es.foldrM (init := mkPUnit lvl) mkPProd
-
-private def mkPUnitMk : Level → Expr
-  | .zero => .const ``True.intro []
-  | lvl   => .const ``PUnit.unit [lvl]
-
-private def mkPProdMk (e1 e2 : Expr) : MetaM Expr := do
-  let t1 ← inferType e1
-  let t2 ← inferType e2
-  let lvl1 ← getLevel t1
-  let lvl2 ← getLevel t2
-  if lvl1 matches .zero && lvl2 matches .zero then
-    return mkApp4 (.const ``And.intro []) t1 t2 e1 e2
-  else
-    return mkApp4 (.const ``PProd.mk [lvl1, lvl2]) t1 t2 e1 e2
-
-private def mkNProdMk (lvl : Level) (es : Array Expr) : MetaM Expr :=
-  es.foldrM (init := mkPUnitMk lvl) mkPProdMk
-
-/-- `PProd.fst` or `And.left` (as projections) -/
-private def mkPProdFst (e : Expr) : MetaM Expr := do
-  let t ← whnf (← inferType e)
-  match_expr t with
-  | PProd _ _ => return .proj ``PProd 0 e
-  | And _ _ =>   return .proj ``And 0 e
-  | _ => throwError "Cannot project .1 out of{indentExpr e}\nof type{indentExpr t}"
-
-/-- `PProd.snd` or `And.right` (as projections) -/
-private def mkPProdSnd (e : Expr) : MetaM Expr := do
-  let t ← whnf (← inferType e)
-  match_expr t with
-  | PProd _ _ => return .proj ``PProd 1 e
-  | And _ _ =>   return .proj ``And 1 e
-  | _ => throwError "Cannot project .2 out of{indentExpr e}\nof type{indentExpr t}"
-
-/-- Given a proof of `P₁ ∧ … ∧ Pᵢ ∧ … ∧ Pₙ ∧ True`, return the proof of `Pᵢ` -/
-def mkPProdProjN (i : Nat) (e : Expr) : MetaM Expr := do
-  let mut value := e
-  for _ in [:i] do
-      value ← mkPProdSnd value
-  value ← mkPProdFst value
-  return value
-
-/--
-Combines motives from different functions that recurse on the same parameter type into a single
-function returning a `PProd` type.
-
-For example
-```
-packMotives (Nat → Sort u) #[(fun (n : Nat) => Nat), (fun (n : Nat) => Fin n -> Fin n )]
-```
-will return
-```
-fun (n : Nat) (PProd Nat (Fin n → Fin n))
-```
-
-It is the identity if `motives.size = 1`.
-
-It returns a dummy motive `(xs : ) → PUnit` or `(xs : … ) → True` if no motive is given.
-(this is the reason we need the expected type in the `motiveType` parameter).
-
--/
-def packMotives (motiveType : Expr) (motives : Array Expr) : MetaM Expr := do
-  if motives.size = 1 then
-    return motives[0]!
-  trace[Elab.definition.structural] "packing Motives\nexpected: {motiveType}\nmotives: {motives}"
-  forallTelescope motiveType fun xs sort => do
-    unless sort.isSort do
-      throwError "packMotives: Unexpected motiveType {motiveType}"
-    -- NB: Use beta, not instantiateLambda; when constructing the belowDict below
-    -- we pass `C`, a plain FVar, here
-    let motives := motives.map (·.beta xs)
-    let packedMotives ← mkNProd sort.sortLevel! motives
-    mkLambdaFVars xs packedMotives
-
-/--
-Combines the F-args from different functions that recurse on the same parameter type into a single
-function returning a `PProd` value. See `packMotives`
-
-It is the identity if `motives.size = 1`.
--/
-def packFArgs (FArgType : Expr) (FArgs : Array Expr) : MetaM Expr := do
-  if FArgs.size = 1 then
-    return FArgs[0]!
-  forallTelescope FArgType fun xs body => do
-    let lvl ← getLevel body
-    let FArgs := FArgs.map (·.beta xs)
-    let packedFArgs ← mkNProdMk lvl FArgs
-    mkLambdaFVars xs packedFArgs
-
-
-end Lean.Elab.Structural

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

@@ -36,6 +36,16 @@ def IndGroupInfo.ofInductiveVal (indInfo : InductiveVal) : IndGroupInfo where
 def IndGroupInfo.numMotives (group : IndGroupInfo) : Nat :=
   group.all.size + group.numNested
 
+/-- Instantiates the right `.brecOn` or `.bInductionOn` for the given type former index,
+including universe parameters and fixed prefix.  -/
+partial def IndGroupInfo.brecOnName (info : IndGroupInfo) (ind : Bool) (idx : Nat) : Name :=
+  if let .some n := info.all[idx]? then
+      if ind then mkBInductionOnName n
+      else        mkBRecOnName n
+    else
+      let j := idx - info.all.size + 1
+      info.brecOnName ind 0 |>.appendIndexAfter j
+
 /--
 An instance of an mutually inductive group of inductives, identified by the `all` array
 and the level and expressions parameters.
@@ -62,6 +72,13 @@ def IndGroupInst.isDefEq (igi1 igi2 : IndGroupInst) : MetaM Bool := do
   unless (← (igi1.params.zip igi2.params).allM (fun (e₁, e₂) => Meta.isDefEqGuarded e₁ e₂)) do return false
   return true
 
+/-- Instantiates the right `.brecOn` or `.bInductionOn` for the given type former index,
+including universe parameters and fixed prefix.  -/
+def IndGroupInst.brecOn (group : IndGroupInst) (ind : Bool) (lvl : Level) (idx : Nat) : Expr :=
+  let n := group.brecOnName ind idx
+  let us := if ind then group.levels else lvl :: group.levels
+  mkAppN (.const n us) group.params
+
 /--
 Figures out the nested type formers of an inductive group, with parameters instantiated
 and indices still forall-abstracted.

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

@@ -12,8 +12,35 @@ import Lean.Elab.PreDefinition.Structural.RecArgInfo
 namespace Lean.Elab.Structural
 open Meta
 
-private partial def replaceIndPredRecApps (recArgInfo : RecArgInfo) (motive : Expr) (e : Expr) : M Expr := do
-  let maxDepth := IndPredBelow.maxBackwardChainingDepth.get (← getOptions)
+private def replaceIndPredRecApp (numFixed : Nat) (funType : Expr) (e : Expr) : M Expr := do
+  withoutProofIrrelevance do
+  withTraceNode `Elab.definition.structural (fun _ => pure m!"eliminating recursive call {e}") do
+    -- We want to replace `e` with an expression of the same type
+    let main ← mkFreshExprSyntheticOpaqueMVar (← inferType e)
+    let args : Array Expr := e.getAppArgs[numFixed:]
+    let lctx ← getLCtx
+    let r ← lctx.anyM fun localDecl => do
+      if localDecl.isAuxDecl then return false
+      let (mvars, _, t) ← forallMetaTelescope localDecl.type -- NB: do not reduce, we want to see the `funType`
+      unless t.getAppFn == funType do return false
+      withTraceNodeBefore `Elab.definition.structural (do pure m!"trying {mkFVar localDecl.fvarId} : {localDecl.type}") do
+        if args.size < t.getAppNumArgs then
+          trace[Elab.definition.structural] "too few arguments. Underapplied recursive call?"
+          return false
+        if (← (t.getAppArgs.zip args).allM (fun (t,s) => isDefEq t s)) then
+          main.mvarId!.assign (mkAppN (mkAppN localDecl.toExpr mvars) args[t.getAppNumArgs:])
+          return ← mvars.allM fun v => do
+            unless (← v.mvarId!.isAssigned) do
+              trace[Elab.definition.structural] "Cannot use {mkFVar localDecl.fvarId}: parameter {v} remains unassigned"
+              return false
+            return true
+        trace[Elab.definition.structural] "Arguments do not match"
+        return false
+    unless r do
+      throwError "Could not eliminate recursive call {e}"
+    instantiateMVars main
+
+private partial def replaceIndPredRecApps (recArgInfo : RecArgInfo) (funType : Expr) (motive : Expr) (e : Expr) : M Expr := do
   let rec loop (e : Expr) : M Expr := do
     match e with
     | Expr.lam n d b c =>
@@ -35,12 +62,7 @@ private partial def replaceIndPredRecApps (recArgInfo : RecArgInfo) (motive : Ex
       let processApp (e : Expr) : M Expr := do
         e.withApp fun f args => do
           if f.isConstOf recArgInfo.fnName then
-            let ty ← inferType e
-            let main ← mkFreshExprSyntheticOpaqueMVar ty
-            if (← IndPredBelow.backwardsChaining main.mvarId! maxDepth) then
-              pure main
-            else
-              throwError "could not solve using backwards chaining {MessageData.ofGoal main.mvarId!}"
+            replaceIndPredRecApp recArgInfo.numFixed funType e
           else
             return mkAppN (← loop f) (← args.mapM loop)
       match (← matchMatcherApp? e) with
@@ -79,33 +101,36 @@ def mkIndPredBRecOn (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do
     let type  := (← inferType value).headBeta
     let (indexMajorArgs, otherArgs) := recArgInfo.pickIndicesMajor ys
     trace[Elab.definition.structural] "numFixed: {recArgInfo.numFixed}, indexMajorArgs: {indexMajorArgs}, otherArgs: {otherArgs}"
-    let motive ← mkForallFVars otherArgs type
-    let motive ← mkLambdaFVars indexMajorArgs motive
-    trace[Elab.definition.structural] "brecOn motive: {motive}"
-    let brecOn := Lean.mkConst (mkBRecOnName recArgInfo.indName!) recArgInfo.indGroupInst.levels
-    let brecOn := mkAppN brecOn recArgInfo.indGroupInst.params
-    let brecOn := mkApp brecOn motive
-    let brecOn := mkAppN brecOn indexMajorArgs
-    check brecOn
-    let brecOnType ← inferType brecOn
-    trace[Elab.definition.structural] "brecOn     {brecOn}"
-    trace[Elab.definition.structural] "brecOnType {brecOnType}"
-    -- we need to close the telescope here, because the local context is used:
-    -- The root cause was, that this copied code puts an ih : FType into the
-    -- local context and later, when we use the local context to build the recursive
-    -- call, it uses this ih. But that ih doesn't exist in the actual brecOn call.
-    -- That's why it must go.
-    let FType ← forallBoundedTelescope brecOnType (some 1) fun F _ => do
-      let F := F[0]!
-      let FType ← inferType F
-      trace[Elab.definition.structural] "FType: {FType}"
-      instantiateForall FType indexMajorArgs
-    forallBoundedTelescope FType (some 1) fun below _ => do
-      let below := below[0]!
-      let valueNew     ← replaceIndPredRecApps recArgInfo motive value
-      let Farg         ← mkLambdaFVars (indexMajorArgs ++ #[below] ++ otherArgs) valueNew
-      let brecOn       := mkApp brecOn Farg
-      let brecOn       := mkAppN brecOn otherArgs
-      mkLambdaFVars ys brecOn
+    let funType ← mkLambdaFVars ys type
+    withLetDecl `funType (← inferType funType) funType fun funType => do
+      let motive ← mkForallFVars otherArgs (mkAppN funType ys)
+      let motive ← mkLambdaFVars indexMajorArgs motive
+      trace[Elab.definition.structural] "brecOn motive: {motive}"
+      let brecOn := Lean.mkConst (mkBRecOnName recArgInfo.indName!) recArgInfo.indGroupInst.levels
+      let brecOn := mkAppN brecOn recArgInfo.indGroupInst.params
+      let brecOn := mkApp brecOn motive
+      let brecOn := mkAppN brecOn indexMajorArgs
+      check brecOn
+      let brecOnType ← inferType brecOn
+      trace[Elab.definition.structural] "brecOn     {brecOn}"
+      trace[Elab.definition.structural] "brecOnType {brecOnType}"
+      -- we need to close the telescope here, because the local context is used:
+      -- The root cause was, that this copied code puts an ih : FType into the
+      -- local context and later, when we use the local context to build the recursive
+      -- call, it uses this ih. But that ih doesn't exist in the actual brecOn call.
+      -- That's why it must go.
+      let FType ← forallBoundedTelescope brecOnType (some 1) fun F _ => do
+        let F := F[0]!
+        let FType ← inferType F
+        trace[Elab.definition.structural] "FType: {FType}"
+        instantiateForall FType indexMajorArgs
+      forallBoundedTelescope FType (some 1) fun below _ => do
+        let below := below[0]!
+        let valueNew     ← replaceIndPredRecApps recArgInfo funType motive value
+        let Farg         ← mkLambdaFVars (indexMajorArgs ++ #[below] ++ otherArgs) valueNew
+        let brecOn       := mkApp brecOn Farg
+        let brecOn       := mkAppN brecOn otherArgs
+        let brecOn       ← mkLetFVars #[funType] brecOn
+        mkLambdaFVars ys brecOn
 
 end Lean.Elab.Structural

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

@@ -128,7 +128,7 @@ private def elimMutualRecursion (preDefs : Array PreDefinition) (xs : Array Expr
   return (Array.zip preDefs valuesNew).map fun ⟨preDef, valueNew⟩ => { preDef with value := valueNew }
 
 private def inferRecArgPos (preDefs : Array PreDefinition) (termArg?s : Array (Option TerminationArgument)) :
-    M (Array Nat × Array PreDefinition) := do
+    M (Array Nat × (Array PreDefinition) × Nat) := do
   withoutModifyingEnv do
     preDefs.forM (addAsAxiom ·)
     let fnNames := preDefs.map (·.declName)
@@ -154,7 +154,7 @@ private def inferRecArgPos (preDefs : Array PreDefinition) (termArg?s : Array (O
         withErasedFVars (xs.extract numFixed xs.size |>.map (·.fvarId!)) do
           let xs := xs[:numFixed]
           let preDefs' ← elimMutualRecursion preDefs xs recArgInfos
-          return (recArgPoss, preDefs')
+          return (recArgPoss, preDefs', numFixed)
 
 def reportTermArg (preDef : PreDefinition) (recArgPos : Nat) : MetaM Unit := do
   if let some ref := preDef.termination.terminationBy?? then
@@ -167,7 +167,7 @@ def reportTermArg (preDef : PreDefinition) (recArgPos : Nat) : MetaM Unit := do
 
 def structuralRecursion (preDefs : Array PreDefinition) (termArg?s : Array (Option TerminationArgument)) : TermElabM Unit := do
   let names := preDefs.map (·.declName)
-  let ((recArgPoss, preDefsNonRec), state) ← run <| inferRecArgPos preDefs termArg?s
+  let ((recArgPoss, preDefsNonRec, numFixed), state) ← run <| inferRecArgPos preDefs termArg?s
   for recArgPos in recArgPoss, preDef in preDefs do
     reportTermArg preDef recArgPos
   state.addMatchers.forM liftM
@@ -190,7 +190,7 @@ def structuralRecursion (preDefs : Array PreDefinition) (termArg?s : Array (Opti
         for theorems and definitions that are propositions.
         See issue #2327
         -/
-        registerEqnsInfo preDef (preDefs.map (·.declName)) recArgPos
+        registerEqnsInfo preDef (preDefs.map (·.declName)) recArgPos numFixed
     addSmartUnfoldingDef preDef recArgPos
     markAsRecursive preDef.declName
   applyAttributesOf preDefsNonRec AttributeApplicationTime.afterCompilation

src/Lean/Elab/PreDefinition/TerminationHint.lean

@@ -57,18 +57,18 @@ structure TerminationHints where
 
 def TerminationHints.none : TerminationHints := ⟨.missing, .none, .none, .none, 0⟩
 
-/-- Logs warnings when the `TerminationHints` are present.  -/
+/-- Logs warnings when the `TerminationHints` are unexpectedly present.  -/
 def TerminationHints.ensureNone (hints : TerminationHints) (reason : String) : CoreM Unit := do
   match hints.terminationBy??, hints.terminationBy?, hints.decreasingBy? with
   | .none, .none, .none => pure ()
   | .none, .none, .some dec_by =>
-    logErrorAt dec_by.ref m!"unused `decreasing_by`, function is {reason}"
+    logWarningAt dec_by.ref m!"unused `decreasing_by`, function is {reason}"
   | .some term_by?, .none, .none =>
-    logErrorAt term_by? m!"unused `termination_by?`, function is {reason}"
+    logWarningAt term_by? m!"unused `termination_by?`, function is {reason}"
   | .none, .some term_by, .none =>
-    logErrorAt term_by.ref m!"unused `termination_by`, function is {reason}"
+    logWarningAt term_by.ref m!"unused `termination_by`, function is {reason}"
   | _, _, _ =>
-    logErrorAt hints.ref m!"unused termination hints, function is {reason}"
+    logWarningAt hints.ref m!"unused termination hints, function is {reason}"
 
 /-- True if any form of termination hint is present. -/
 def TerminationHints.isNotNone (hints : TerminationHints) : Bool :=

src/Lean/Elab/Print.lean

@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
-import Lean.Util.FoldConsts
 import Lean.Meta.Eqns
+import Lean.Util.CollectAxioms
 import Lean.Elab.Command
 
 namespace Lean.Elab.Command
@@ -120,40 +120,12 @@ private def printId (id : Syntax) : CommandElabM Unit := do
   | `(#print%$tk $s:str)    => logInfoAt tk s.getString
   | _                       => throwError "invalid #print command"
 
-namespace CollectAxioms
-
-structure State where
-  visited : NameSet    := {}
-  axioms  : Array Name := #[]
-
-abbrev M := ReaderT Environment $ StateM State
-
-partial def collect (c : Name) : M Unit := do
-  let collectExpr (e : Expr) : M Unit := e.getUsedConstants.forM collect
-  let s ← get
-  unless s.visited.contains c do
-    modify fun s => { s with visited := s.visited.insert c }
-    let env ← read
-    match env.find? c with
-    | some (ConstantInfo.axiomInfo _)  => modify fun s => { s with axioms := s.axioms.push c }
-    | some (ConstantInfo.defnInfo v)   => collectExpr v.type *> collectExpr v.value
-    | some (ConstantInfo.thmInfo v)    => collectExpr v.type *> collectExpr v.value
-    | some (ConstantInfo.opaqueInfo v) => collectExpr v.type *> collectExpr v.value
-    | some (ConstantInfo.quotInfo _)   => pure ()
-    | some (ConstantInfo.ctorInfo v)   => collectExpr v.type
-    | some (ConstantInfo.recInfo v)    => collectExpr v.type
-    | some (ConstantInfo.inductInfo v) => collectExpr v.type *> v.ctors.forM collect
-    | none                             => pure ()
-
-end CollectAxioms
-
 private def printAxiomsOf (constName : Name) : CommandElabM Unit := do
-  let env ← getEnv
-  let (_, s) := ((CollectAxioms.collect constName).run env).run {}
-  if s.axioms.isEmpty then
+  let axioms ← collectAxioms constName
+  if axioms.isEmpty then
     logInfo m!"'{constName}' does not depend on any axioms"
   else
-    logInfo m!"'{constName}' depends on axioms: {s.axioms.qsort Name.lt |>.toList}"
+    logInfo m!"'{constName}' depends on axioms: {axioms.qsort Name.lt |>.toList}"
 
 @[builtin_command_elab «printAxioms»] def elabPrintAxioms : CommandElab
   | `(#print%$tk axioms $id) => withRef tk do

src/Lean/Elab/Tactic/Basic.lean

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Sebastian Ullrich
 -/
 prelude
+import Lean.Meta.Tactic.Util
 import Lean.Elab.Term
 
 namespace Lean.Elab
@@ -155,7 +156,9 @@ partial def evalTactic (stx : Syntax) : TacticM Unit := do
         -- Macro writers create a sequence of tactics `t₁ ... tₙ` using `mkNullNode #[t₁, ..., tₙ]`
         -- We could support incrementality here by allocating `n` new snapshot bundles but the
         -- practical value is not clear
-        Term.withoutTacticIncrementality true do
+        -- NOTE: `withTacticInfoContext` is used to preserve the invariant of `elabTactic` producing
+        -- exactly one info tree, which is necessary for using `getInfoTreeWithContext`.
+        Term.withoutTacticIncrementality true <| withTacticInfoContext stx do
           stx.getArgs.forM evalTactic
       else withTraceNode `Elab.step (fun _ => return stx) (tag := stx.getKind.toString) do
         let evalFns := tacticElabAttribute.getEntries (← getEnv) stx.getKind
@@ -222,14 +225,18 @@ where
                     snap.new.resolve <| .mk {
                       stx := stx'
                       diagnostics := .empty
-                      finished := .pure { state? := (← Tactic.saveState) }
-                    } #[{ range? := stx'.getRange?, task := promise.result }]
+                      finished := .pure {
+                        diagnostics := .empty
+                        state? := (← Tactic.saveState)
+                      }
+                      next := #[{ range? := stx'.getRange?, task := promise.result }]
+                    }
                     -- Update `tacSnap?` to old unfolding
                     withTheReader Term.Context ({ · with tacSnap? := some {
                       new := promise
                       old? := do
                         let old ← old?
-                        return ⟨old.data.stx, (← old.next.get? 0)⟩
+                        return ⟨old.data.stx, (← old.data.next.get? 0)⟩
                     } }) do
                       evalTactic stx'
                   return
@@ -398,12 +405,19 @@ def ensureHasNoMVars (e : Expr) : TacticM Unit := do
   if e.hasExprMVar then
     throwError "tactic failed, resulting expression contains metavariables{indentExpr e}"
 
-/-- Close main goal using the given expression. If `checkUnassigned == true`, then `val` must not contain unassigned metavariables. -/
-def closeMainGoal (val : Expr) (checkUnassigned := true): TacticM Unit := do
+/--
+Closes main goal using the given expression.
+If `checkUnassigned == true`, then `val` must not contain unassigned metavariables.
+Returns `true` if `val` was successfully used to close the goal.
+-/
+def closeMainGoal (tacName : Name) (val : Expr) (checkUnassigned := true): TacticM Unit := do
   if checkUnassigned then
     ensureHasNoMVars val
-  (← getMainGoal).assign val
-  replaceMainGoal []
+  let mvarId ← getMainGoal
+  if (← mvarId.checkedAssign val) then
+    replaceMainGoal []
+  else
+    throwTacticEx tacName mvarId m!"attempting to close the goal using{indentExpr val}\nthis is often due occurs-check failure"
 
 @[inline] def liftMetaMAtMain (x : MVarId → MetaM α) : TacticM α := do
   withMainContext do x (← getMainGoal)

src/Lean/Elab/Tactic/BuiltinTactic.lean

@@ -60,7 +60,7 @@ where
     if let some snap := (← readThe Term.Context).tacSnap? then
       if let some old := snap.old? then
         let oldParsed := old.val.get
-        oldInner? := oldParsed.next.get? 0 |>.map (⟨oldParsed.data.stx, ·⟩)
+        oldInner? := oldParsed.data.inner? |>.map (⟨oldParsed.data.stx, ·⟩)
     -- compare `stx[0]` for `finished`/`next` reuse, focus on remainder of script
     Term.withNarrowedTacticReuse (stx := stx) (fun stx => (stx[0], mkNullNode stx.getArgs[1:])) fun stxs => do
       let some snap := (← readThe Term.Context).tacSnap?
@@ -73,29 +73,47 @@ where
         if let some state := oldParsed.data.finished.get.state? then
           reusableResult? := some ((), state)
           -- only allow `next` reuse in this case
-          oldNext? := oldParsed.next.get? 1 |>.map (⟨old.stx, ·⟩)
+          oldNext? := oldParsed.data.next.get? 0 |>.map (⟨old.stx, ·⟩)
 
+      -- For `tac`'s snapshot task range, disregard synthetic info as otherwise
+      -- `SnapshotTree.findInfoTreeAtPos` might choose the wrong snapshot: for example, when
+      -- hovering over a `show` tactic, we should choose the info tree in `finished` over that in
+      -- `inner`, which points to execution of the synthesized `refine` step and does not contain
+      -- the full info. In most other places, siblings in the snapshot tree have disjoint ranges and
+      -- so this issue does not occur.
+      let mut range? := tac.getRange? (canonicalOnly := true)
+      -- Include trailing whitespace in the range so that `goalsAs?` does not have to wait for more
+      -- snapshots than necessary.
+      if let some range := range? then
+        range? := some { range with stop := ⟨range.stop.byteIdx + tac.getTrailingSize⟩ }
       withAlwaysResolvedPromise fun next => do
         withAlwaysResolvedPromise fun finished => do
           withAlwaysResolvedPromise fun inner => do
             snap.new.resolve <| .mk {
+              diagnostics := .empty
               stx := tac
-              diagnostics := (← Language.Snapshot.Diagnostics.ofMessageLog
-                (← Core.getAndEmptyMessageLog))
-              finished := finished.result
-            } #[
-              {
-                range? := tac.getRange?
-                task := inner.result },
-              {
-                range? := stxs |>.getRange?
-                task := next.result }]
-            let (_, state) ← withRestoreOrSaveFull reusableResult?
-                -- set up nested reuse; `evalTactic` will check for `isIncrementalElab`
-                (tacSnap? := some { old? := oldInner?, new := inner }) do
-              Term.withReuseContext tac do
-                evalTactic tac
-            finished.resolve { state? := state }
+              inner? := some { range?, task := inner.result }
+              finished := { range?, task := finished.result }
+              next := #[{ range? := stxs.getRange?, task := next.result }]
+            }
+            -- Run `tac` in a fresh info tree state and store resulting state in snapshot for
+            -- incremental reporting, then add back saved trees. Here we rely on `evalTactic`
+            -- producing at most one info tree as otherwise `getInfoTreeWithContext?` would panic.
+            let trees ← getResetInfoTrees
+            try
+              let (_, state) ← withRestoreOrSaveFull reusableResult?
+                  -- set up nested reuse; `evalTactic` will check for `isIncrementalElab`
+                  (tacSnap? := some { old? := oldInner?, new := inner }) do
+                Term.withReuseContext tac do
+                  evalTactic tac
+              finished.resolve {
+                diagnostics := (← Language.Snapshot.Diagnostics.ofMessageLog
+                  (← Core.getAndEmptyMessageLog))
+                infoTree? := (← Term.getInfoTreeWithContext?)
+                state? := state
+              }
+            finally
+              modifyInfoState fun s => { s with trees := trees ++ s.trees }
 
         withTheReader Term.Context ({ · with tacSnap? := some {
           new := next

src/Lean/Elab/Tactic/ElabTerm.lean

@@ -56,9 +56,9 @@ def elabTermEnsuringType (stx : Syntax) (expectedType? : Option Expr) (mayPostpo
     return e
 
 /-- Try to close main goal using `x target`, where `target` is the type of the main goal.  -/
-def closeMainGoalUsing (x : Expr → TacticM Expr) (checkUnassigned := true) : TacticM Unit :=
+def closeMainGoalUsing (tacName : Name) (x : Expr → TacticM Expr) (checkUnassigned := true) : TacticM Unit :=
   withMainContext do
-    closeMainGoal (checkUnassigned := checkUnassigned) (← x (← getMainTarget))
+    closeMainGoal (tacName := tacName) (checkUnassigned := checkUnassigned) (← x (← getMainTarget))
 
 def logUnassignedAndAbort (mvarIds : Array MVarId) : TacticM Unit := do
    if (← Term.logUnassignedUsingErrorInfos mvarIds) then
@@ -68,13 +68,14 @@ def filterOldMVars (mvarIds : Array MVarId) (mvarCounterSaved : Nat) : MetaM (Ar
   let mctx ← getMCtx
   return mvarIds.filter fun mvarId => (mctx.getDecl mvarId |>.index) >= mvarCounterSaved
 
-@[builtin_tactic «exact»] def evalExact : Tactic := fun stx =>
+@[builtin_tactic «exact»] def evalExact : Tactic := fun stx => do
   match stx with
-  | `(tactic| exact $e) => closeMainGoalUsing (checkUnassigned := false) fun type => do
-    let mvarCounterSaved := (← getMCtx).mvarCounter
-    let r ← elabTermEnsuringType e type
-    logUnassignedAndAbort (← filterOldMVars (← getMVars r) mvarCounterSaved)
-    return r
+  | `(tactic| exact $e) =>
+    closeMainGoalUsing `exact (checkUnassigned := false) fun type => do
+      let mvarCounterSaved := (← getMCtx).mvarCounter
+      let r ← elabTermEnsuringType e type
+      logUnassignedAndAbort (← filterOldMVars (← getMVars r) mvarCounterSaved)
+      return r
   | _ => throwUnsupportedSyntax
 
 def sortMVarIdArrayByIndex [MonadMCtx m] [Monad m] (mvarIds : Array MVarId) : m (Array MVarId) := do
@@ -393,7 +394,7 @@ private partial def blameDecideReductionFailure (inst : Expr) : MetaM Expr := do
   return inst
 
 @[builtin_tactic Lean.Parser.Tactic.decide] def evalDecide : Tactic := fun _ =>
-  closeMainGoalUsing fun expectedType => do
+  closeMainGoalUsing `decide fun expectedType => do
     let expectedType ← preprocessPropToDecide expectedType
     let d ← mkDecide expectedType
     let d ← instantiateMVars d
@@ -472,7 +473,7 @@ private def mkNativeAuxDecl (baseName : Name) (type value : Expr) : TermElabM Na
   pure auxName
 
 @[builtin_tactic Lean.Parser.Tactic.nativeDecide] def evalNativeDecide : Tactic := fun _ =>
-  closeMainGoalUsing fun expectedType => do
+  closeMainGoalUsing `nativeDecide fun expectedType => do
     let expectedType ← preprocessPropToDecide expectedType
     let d ← mkDecide expectedType
     let auxDeclName ← mkNativeAuxDecl `_nativeDecide (Lean.mkConst `Bool) d

src/Lean/Elab/Tactic/Ext.lean

@@ -74,16 +74,16 @@ def mkExtIffType (extThmName : Name) : MetaM Expr := withLCtx {} {} do
     let some (_, x, y) := ty.eq? | failNotEq
     let some xIdx := args.findIdx? (· == x) | failNotEq
     let some yIdx := args.findIdx? (· == y) | failNotEq
-    unless xIdx == yIdx + 1 || xIdx + 1 == yIdx do
+    unless xIdx + 1 == yIdx do
       throwError "expecting {x} and {y} to be consecutive arguments"
-    let startIdx := max xIdx yIdx + 1
+    let startIdx := yIdx + 1
     let toRevert := args[startIdx:].toArray
     let fvars ← toRevert.foldlM (init := {}) (fun st e => return collectFVars st (← inferType e))
     for fvar in toRevert do
       unless ← Meta.isProof fvar do
-        throwError "argument {fvar} is not a proof, which is not supported"
+        throwError "argument {fvar} is not a proof, which is not supported for arguments after {x} and {y}"
       if fvars.fvarSet.contains fvar.fvarId! then
-        throwError "argument {fvar} is depended upon, which is not supported"
+        throwError "argument {fvar} is depended upon, which is not supported for arguments after {x} and {y}"
     let conj := mkAndN (← toRevert.mapM (inferType ·)).toList
     -- Make everything implicit except for inst implicits
     let mut newBis := #[]
@@ -104,27 +104,31 @@ def realizeExtTheorem (structName : Name) (flat : Bool) : Elab.Command.CommandEl
     throwError "'{structName}' is not a structure"
   let extName := structName.mkStr "ext"
   unless (← getEnv).contains extName do
-    Elab.Command.liftTermElabM <| withoutErrToSorry <| withDeclName extName do
-      let type ← mkExtType structName flat
-      let pf ← withSynthesize do
-        let indVal ← getConstInfoInduct structName
-        let params := Array.mkArray indVal.numParams (← `(_))
-        Elab.Term.elabTermEnsuringType (expectedType? := type) (implicitLambda := false)
-          -- introduce the params, do cases on 'x' and 'y', and then substitute each equation
-          (← `(by intro $params* {..} {..}; intros; subst_eqs; rfl))
-      let pf ← instantiateMVars pf
-      if pf.hasMVar then throwError "(internal error) synthesized ext proof contains metavariables{indentD pf}"
-      let info ← getConstInfo structName
-      addDecl <| Declaration.thmDecl {
-        name := extName
-        type
-        value := pf
-        levelParams := info.levelParams
-      }
-      modifyEnv fun env => addProtected env extName
-      Lean.addDeclarationRanges extName {
-        range := ← getDeclarationRange (← getRef)
-        selectionRange := ← getDeclarationRange (← getRef) }
+    try
+      Elab.Command.liftTermElabM <| withoutErrToSorry <| withDeclName extName do
+        let type ← mkExtType structName flat
+        let pf ← withSynthesize do
+          let indVal ← getConstInfoInduct structName
+          let params := Array.mkArray indVal.numParams (← `(_))
+          Elab.Term.elabTermEnsuringType (expectedType? := type) (implicitLambda := false)
+            -- introduce the params, do cases on 'x' and 'y', and then substitute each equation
+            (← `(by intro $params* {..} {..}; intros; subst_eqs; rfl))
+        let pf ← instantiateMVars pf
+        if pf.hasMVar then throwError "(internal error) synthesized ext proof contains metavariables{indentD pf}"
+        let info ← getConstInfo structName
+        addDecl <| Declaration.thmDecl {
+          name := extName
+          type
+          value := pf
+          levelParams := info.levelParams
+        }
+        modifyEnv fun env => addProtected env extName
+        Lean.addDeclarationRanges extName {
+          range := ← getDeclarationRange (← getRef)
+          selectionRange := ← getDeclarationRange (← getRef) }
+    catch e =>
+      throwError m!"\
+        Failed to generate an 'ext' theorem for '{MessageData.ofConstName structName}': {e.toMessageData}"
   return extName
 
 /--
@@ -138,29 +142,35 @@ def realizeExtIffTheorem (extName : Name) : Elab.Command.CommandElabM Name := do
     | .str n s => .str n (s ++ "_iff")
     | _ => .str extName "ext_iff"
   unless (← getEnv).contains extIffName do
-    let info ← getConstInfo extName
-    Elab.Command.liftTermElabM <| withoutErrToSorry <| withDeclName extIffName do
-      let type ← mkExtIffType extName
-      let pf ← withSynthesize do
-        Elab.Term.elabTermEnsuringType (expectedType? := type) <| ← `(by
-          intros
-          refine ⟨?_, ?_⟩
-          · intro h; cases h; and_intros <;> (intros; first | rfl | simp | fail "Failed to prove converse of ext theorem")
-          · intro; (repeat cases ‹_ ∧ _›); apply $(mkCIdent extName) <;> assumption)
-      let pf ← instantiateMVars pf
-      if pf.hasMVar then throwError "(internal error) synthesized ext_iff proof contains metavariables{indentD pf}"
-      addDecl <| Declaration.thmDecl {
-        name := extIffName
-        type
-        value := pf
-        levelParams := info.levelParams
-      }
-      -- Only declarations in a namespace can be protected:
-      unless extIffName.isAtomic do
-        modifyEnv fun env => addProtected env extIffName
-      Lean.addDeclarationRanges extIffName {
-        range := ← getDeclarationRange (← getRef)
-        selectionRange := ← getDeclarationRange (← getRef) }
+    try
+      let info ← getConstInfo extName
+      Elab.Command.liftTermElabM <| withoutErrToSorry <| withDeclName extIffName do
+        let type ← mkExtIffType extName
+        let pf ← withSynthesize do
+          Elab.Term.elabTermEnsuringType (expectedType? := type) <| ← `(by
+            intros
+            refine ⟨?_, ?_⟩
+            · intro h; cases h; and_intros <;> (intros; first | rfl | simp | fail "Failed to prove converse of ext theorem")
+            · intro; (repeat cases ‹_ ∧ _›); apply $(mkCIdent extName) <;> assumption)
+        let pf ← instantiateMVars pf
+        if pf.hasMVar then throwError "(internal error) synthesized ext_iff proof contains metavariables{indentD pf}"
+        addDecl <| Declaration.thmDecl {
+          name := extIffName
+          type
+          value := pf
+          levelParams := info.levelParams
+        }
+        -- Only declarations in a namespace can be protected:
+        unless extIffName.isAtomic do
+          modifyEnv fun env => addProtected env extIffName
+        Lean.addDeclarationRanges extIffName {
+          range := ← getDeclarationRange (← getRef)
+          selectionRange := ← getDeclarationRange (← getRef) }
+    catch e =>
+      throwError m!"\
+        Failed to generate an 'ext_iff' theorem from '{MessageData.ofConstName extName}': {e.toMessageData}\n\
+        \n\
+        Try '@[ext (iff := false)]' to prevent generating an 'ext_iff' theorem."
   return extIffName
 
 

src/Lean/Elab/Tactic/Induction.lean

@@ -263,9 +263,10 @@ where
               -- save all relevant syntax here for comparison with next document version
               stx := mkNullNode altStxs
               diagnostics := .empty
-              finished := finished.result
-            } (altStxs.zipWith altPromises fun stx prom =>
-                { range? := stx.getRange?, task := prom.result })
+              finished := { range? := none, task := finished.result }
+              next := altStxs.zipWith altPromises fun stx prom =>
+                { range? := stx.getRange?, task := prom.result }
+            }
             goWithIncremental <| altPromises.mapIdx fun i prom => {
               old? := do
                 let old ← tacSnap.old?
@@ -274,10 +275,10 @@ where
                 let old := old.val.get
                 -- use old version of `mkNullNode altsSyntax` as guard, will be compared with new
                 -- version and picked apart in `applyAltStx`
-                return ⟨old.data.stx, (← old.next[i]?)⟩
+                return ⟨old.data.stx, (← old.data.next[i]?)⟩
               new := prom
             }
-            finished.resolve { state? := (← saveState) }
+            finished.resolve { diagnostics := .empty, state? := (← saveState) }
         return
 
     goWithIncremental #[]

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

@@ -6,7 +6,7 @@ Authors: Scott Morrison
 prelude
 import Init.BinderPredicates
 import Init.Data.Int.Order
-import Init.Data.List.Lemmas
+import Init.Data.List.MinMax
 import Init.Data.Nat.MinMax
 import Init.Data.Option.Lemmas
 

src/Lean/Elab/Term.lean

@@ -190,33 +190,38 @@ structure SavedState where
   term   : Term.SavedState
   tactic : State
 
-/-- State after finishing execution of a tactic. -/
-structure TacticFinished where
-  /-- Reusable state, if no fatal exception occurred. -/
+/-- Snapshot after finishing execution of a tactic. -/
+structure TacticFinishedSnapshot extends Language.Snapshot where
+  /-- State saved for reuse, if no fatal exception occurred. -/
   state? : Option SavedState
 deriving Inhabited
+instance : ToSnapshotTree TacticFinishedSnapshot where
+  toSnapshotTree s := ⟨s.toSnapshot, #[]⟩
 
 /-- Snapshot just before execution of a tactic. -/
-structure TacticParsedSnapshotData extends Language.Snapshot where
+structure TacticParsedSnapshotData (TacticParsedSnapshot : Type) extends Language.Snapshot where
   /-- Syntax tree of the tactic, stored and compared for incremental reuse. -/
   stx      : Syntax
+  /-- Task for nested incrementality, if enabled for tactic. -/
+  inner?   : Option (SnapshotTask TacticParsedSnapshot) := none
   /-- Task for state after tactic execution. -/
-  finished : Task TacticFinished
+  finished : SnapshotTask TacticFinishedSnapshot
+  /-- Tasks for subsequent, potentially parallel, tactic steps. -/
+  next     : Array (SnapshotTask TacticParsedSnapshot) := #[]
 deriving Inhabited
 
 /-- State after execution of a single synchronous tactic step. -/
 inductive TacticParsedSnapshot where
-  | mk (data : TacticParsedSnapshotData) (next : Array (SnapshotTask TacticParsedSnapshot))
+  | mk (data : TacticParsedSnapshotData TacticParsedSnapshot)
 deriving Inhabited
-abbrev TacticParsedSnapshot.data : TacticParsedSnapshot → TacticParsedSnapshotData
-  | .mk data _ => data
-/-- Potential, potentially parallel, follow-up tactic executions. -/
--- In the first, non-parallel version, each task will depend on its predecessor
-abbrev TacticParsedSnapshot.next : TacticParsedSnapshot → Array (SnapshotTask TacticParsedSnapshot)
-  | .mk _ next => next
+abbrev TacticParsedSnapshot.data : TacticParsedSnapshot → TacticParsedSnapshotData TacticParsedSnapshot
+  | .mk data => data
 partial instance : ToSnapshotTree TacticParsedSnapshot where
   toSnapshotTree := go where
-    go := fun ⟨s, next⟩ => ⟨s.toSnapshot, next.map (·.map (sync := true) go)⟩
+    go := fun ⟨s⟩ => ⟨s.toSnapshot,
+      s.inner?.toArray.map (·.map (sync := true) go) ++
+      #[s.finished.map (sync := true) toSnapshotTree] ++
+      s.next.map (·.map (sync := true) go)⟩
 
 end Snapshot
 end Tactic
@@ -630,6 +635,32 @@ private def withoutModifyingStateWithInfoAndMessagesImpl (x : TermElabM α) : Te
     let saved := { saved with meta.core.infoState := (← getInfoState), meta.core.messages := (← getThe Core.State).messages }
     restoreState saved
 
+/--
+Wraps the trees returned from `getInfoTrees`, if any, in an `InfoTree.context` node based on the
+current monadic context and state. This is mainly used to report info trees early via
+`Snapshot.infoTree?`. The trees are not removed from the `getInfoTrees` state as the final info tree
+of the elaborated command should be complete and not depend on whether parts have been reported
+early.
+
+As `InfoTree.context` can have only one child, this function panics if `trees` contains more than 1
+tree. Also, `PartialContextInfo.parentDeclCtx` is not currently generated as that information is not
+available in the monadic context and only needed for the final info tree.
+-/
+def getInfoTreeWithContext? : TermElabM (Option InfoTree) := do
+  let st ← getInfoState
+  if st.trees.size > 1 then
+    return panic! "getInfoTreeWithContext: overfull tree"
+  let some t := st.trees[0]? |
+    return none
+  let t := t.substitute st.assignment
+  let ctx ← readThe Core.Context
+  let s ← getThe Core.State
+  let ctx := PartialContextInfo.commandCtx {
+    env := s.env, fileMap := ctx.fileMap, mctx := {}, currNamespace := ctx.currNamespace,
+    openDecls := ctx.openDecls, options := ctx.options, ngen := s.ngen
+  }
+  return InfoTree.context ctx t
+
 /-- For testing `TermElabM` methods. The #eval command will sign the error. -/
 def throwErrorIfErrors : TermElabM Unit := do
   if (← MonadLog.hasErrors) then

src/Lean/Language/Basic.lean

@@ -12,6 +12,7 @@ prelude
 import Init.System.Promise
 import Lean.Message
 import Lean.Parser.Types
+import Lean.Elab.InfoTree
 
 set_option linter.missingDocs true
 
@@ -46,6 +47,8 @@ def Snapshot.Diagnostics.empty : Snapshot.Diagnostics where
   The base class of all snapshots: all the generic information the language server needs about a
   snapshot. -/
 structure Snapshot where
+  /-- Debug description shown by `trace.Elab.snapshotTree`, defaults to the caller's decl name. -/
+  desc : String := by exact decl_name%.toString
   /--
     The messages produced by this step. The union of message logs of all finished snapshots is
     reported to the user. -/
@@ -71,7 +74,7 @@ structure SnapshotTask (α : Type) where
   range? : Option String.Range
   /-- Underlying task producing the snapshot. -/
   task : Task α
-deriving Nonempty
+deriving Nonempty, Inhabited
 
 /-- Creates a snapshot task from a reporting range and a `BaseIO` action. -/
 def SnapshotTask.ofIO (range? : Option String.Range) (act : BaseIO α) : BaseIO (SnapshotTask α) := do
@@ -203,15 +206,19 @@ abbrev SnapshotTree.children : SnapshotTree → Array (SnapshotTask SnapshotTree
   | mk _ children => children
 
 /-- Produces debug tree format of given snapshot tree, synchronously waiting on all children. -/
-partial def SnapshotTree.format : SnapshotTree → Format := go none
-where go range? s :=
-  let range := match range? with
-    | some range => f!"{range.start}..{range.stop} "
-    | none => ""
-  let element := f!"{s.element.diagnostics.msgLog.unreported.size} diagnostics"
-  let children := Std.Format.prefixJoin .line <|
-    s.children.toList.map fun c => go c.range? c.get
-  .nestD f!"• {range}{element}{children}"
+partial def SnapshotTree.format [Monad m] [MonadFileMap m] [MonadLiftT IO m] :
+    SnapshotTree → m Format :=
+  go none
+where go range? s := do
+  let file ← getFileMap
+  let mut desc := f!"• {s.element.desc}"
+  if let some range := range? then
+    desc := desc ++ f!"{file.toPosition range.start}-{file.toPosition range.stop} "
+  desc := desc ++ .prefixJoin "\n• " (← s.element.diagnostics.msgLog.toList.mapM (·.toString))
+  if let some t := s.element.infoTree? then
+    desc := desc ++ f!"\n{← t.format}"
+  desc := desc ++ .prefixJoin "\n" (← s.children.toList.mapM fun c => go c.range? c.get)
+  return .nestD desc
 
 /--
   Helper class for projecting a heterogeneous hierarchy of snapshot classes to a homogeneous

src/Lean/Language/Lean.lean

@@ -362,16 +362,16 @@ where
   parseHeader (old? : Option HeaderParsedSnapshot) : LeanProcessingM HeaderParsedSnapshot := do
     let ctx ← read
     let ictx := ctx.toInputContext
-    let unchanged old newParserState :=
+    let unchanged old newStx newParserState :=
       -- when header syntax is unchanged, reuse import processing task as is and continue with
       -- parsing the first command, synchronously if possible
-      -- NOTE: even if the syntax tree is functionally unchanged, the new parser state may still
-      -- have changed because of trailing whitespace and comments etc., so it is passed separately
-      -- from `old`
+      -- NOTE: even if the syntax tree is functionally unchanged, its concrete structure and the new
+      -- parser state may still have changed because of trailing whitespace and comments etc., so
+      -- they are passed separately from `old`
       if let some oldSuccess := old.result? then
         return {
           ictx
-          stx := old.stx
+          stx := newStx
           diagnostics := old.diagnostics
           cancelTk? := ctx.newCancelTk
           result? := some { oldSuccess with
@@ -394,7 +394,7 @@ where
           if let some nextCom ← processed.firstCmdSnap.get? then
             if (← isBeforeEditPos nextCom.data.parserState.pos) then
               -- ...go immediately to next snapshot
-              return (← unchanged old oldSuccess.parserState)
+              return (← unchanged old old.stx oldSuccess.parserState)
 
     withHeaderExceptions ({ · with
         ictx, stx := .missing, result? := none, cancelTk? := none }) do
@@ -408,16 +408,19 @@ where
           cancelTk? := none
         }
 
-      -- semi-fast path: go to next snapshot if syntax tree is unchanged AND we're still in front
-      -- of the edit location
-      -- TODO: dropping the second condition would require adjusting positions in the state
-      -- NOTE: as `parserState.pos` includes trailing whitespace, this forces reprocessing even if
-      -- only that whitespace changes, which is wasteful but still necessary because it may
-      -- influence the range of error messages such as from a trailing `exact`
+      let trimmedStx := stx.unsetTrailing
+      -- semi-fast path: go to next snapshot if syntax tree is unchanged
+      -- NOTE: We compare modulo `unsetTrailing` in order to ensure that changes in trailing
+      -- whitespace do not invalidate the header. This is safe because we only pass the trimmed
+      -- syntax tree to `processHeader` below, so there cannot be any references to the trailing
+      -- whitespace in its result. We still store the untrimmed syntax tree in the snapshot in order
+      -- to uphold the invariant that concatenating all top-level snapshots' syntax trees results in
+      -- the original file.
       if let some old := old? then
-        if (← isBeforeEditPos parserState.pos) && old.stx == stx then
-          -- Here we must make sure to pass the *new* parser state; see NOTE in `unchanged`
-          return (← unchanged old parserState)
+        if trimmedStx.eqWithInfo old.stx.unsetTrailing then
+          -- Here we must make sure to pass the *new* syntax and parser state; see NOTE in
+          -- `unchanged`
+          return (← unchanged old stx parserState)
         -- on first change, make sure to cancel old invocation
         if let some tk := ctx.oldCancelTk? then
           tk.set
@@ -426,7 +429,7 @@ where
         diagnostics := (← Snapshot.Diagnostics.ofMessageLog msgLog)
         result? := some {
           parserState
-          processedSnap := (← processHeader stx parserState)
+          processedSnap := (← processHeader trimmedStx parserState)
         }
         cancelTk? := ctx.newCancelTk
       }
@@ -523,7 +526,10 @@ where
 
       -- semi-fast path
       if let some old := old? then
-        if (← isBeforeEditPos parserState.pos ctx) && old.data.stx == stx then
+        -- NOTE: as `parserState.pos` includes trailing whitespace, this forces reprocessing even if
+        -- only that whitespace changes, which is wasteful but still necessary because it may
+        -- influence the range of error messages such as from a trailing `exact`
+        if stx.eqWithInfo old.data.stx then
           -- Here we must make sure to pass the *new* parser state; see NOTE in `unchanged`
           return (← unchanged old parserState)
         -- on first change, make sure to cancel old invocation

src/Lean/Meta/AppBuilder.lean

@@ -665,27 +665,6 @@ def mkIffOfEq (h : Expr) : MetaM Expr := do
   else
     mkAppM ``Iff.of_eq #[h]
 
-/--
-Given proofs of `P₁`, …, `Pₙ`, returns a proof of `P₁ ∧ … ∧ Pₙ`.
-If `n = 0` returns a proof of `True`.
-If `n = 1` returns the proof of `P₁`.
--/
-def mkAndIntroN : Array Expr → MetaM Expr
-| #[] => return mkConst ``True.intro []
-| #[e] => return e
-| es => es.foldrM (start := es.size - 1) (fun a b => mkAppM ``And.intro #[a,b]) es.back
-
-
-/-- Given a proof of `P₁ ∧ … ∧ Pᵢ ∧ … ∧ Pₙ`, return the proof of `Pᵢ` -/
-def mkProjAndN (n i : Nat) (e : Expr) : Expr := Id.run do
-  let mut value := e
-  for _ in [:i] do
-      value := mkProj ``And 1 value
-  if i + 1 < n then
-      value := mkProj ``And 0 value
-  return value
-
-
 builtin_initialize do
   registerTraceClass `Meta.appBuilder
   registerTraceClass `Meta.appBuilder.result (inherited := true)

src/Lean/Meta/ArgsPacker.lean

@@ -6,6 +6,7 @@ Authors: Joachim Breitner
 
 prelude
 import Lean.Meta.AppBuilder
+import Lean.Meta.PProdN
 import Lean.Meta.ArgsPacker.Basic
 
 /-!
@@ -518,7 +519,7 @@ def curry (argsPacker : ArgsPacker) (e : Expr) : MetaM Expr := do
   let mut es := #[]
   for i in [:argsPacker.numFuncs] do
     es := es.push (← argsPacker.curryProj e i)
-  mkAndIntroN es
+  PProdN.mk 0 es
 
 /--
 Given type `(a ⊗' b ⊕' c ⊗' d) → e`, brings `a → b → e` and `c → d → e`
@@ -533,7 +534,7 @@ where
   | [], acc => k acc
   | t::ts, acc => do
     let name := if argsPacker.numFuncs = 1 then name else .mkSimple s!"{name}{acc.size+1}"
-    withLocalDecl name .default t fun x => do
+    withLocalDeclD name t fun x => do
       go ts (acc.push x)
 
 /--

src/Lean/Meta/Basic.lean

@@ -1865,9 +1865,13 @@ abbrev isDefEqGuarded (t s : Expr) : MetaM Bool :=
 def isDefEqNoConstantApprox (t s : Expr) : MetaM Bool :=
   approxDefEq <| isDefEq t s
 
-/-- Shorthand for `isDefEq (mkMVar mvarId) val` -/
-def _root_.Lean.MVarId.checkedAssign (mvarId : MVarId) (val : Expr) : MetaM Bool :=
-  isDefEq (mkMVar mvarId) val
+/--
+Returns `true` if `mvarId := val` was successfully assigned.
+This method uses the same assignment validation performed by `isDefEq`, but it does not check whether the types match.
+-/
+-- Remark: this method is implemented at `ExprDefEq`
+@[extern "lean_checked_assign"]
+opaque _root_.Lean.MVarId.checkedAssign (mvarId : MVarId) (val : Expr) : MetaM Bool
 
 /--
   Eta expand the given expression.

src/Lean/Meta/Constructions/BRecOn.lean

@@ -8,67 +8,18 @@ import Lean.Meta.InferType
 import Lean.AuxRecursor
 import Lean.AddDecl
 import Lean.Meta.CompletionName
+import Lean.Meta.PProdN
 
 namespace Lean
 open Meta
 
-section PProd
-
-/--!
-Helpers to construct types and values of `PProd` and project out of them, set up to use `And`
-instead of `PProd` if the universes allow. Maybe be extracted into a Utils module when useful
-elsewhere.
--/
-
-private def mkPUnit : Level → Expr
-  | .zero => .const ``True []
-  | lvl   => .const ``PUnit [lvl]
-
-private def mkPProd (e1 e2 : Expr) : MetaM Expr := do
-  let lvl1 ← getLevel e1
-  let lvl2 ← getLevel e2
-  if lvl1 matches .zero && lvl2 matches .zero then
-    return mkApp2 (.const `And []) e1 e2
-  else
-    return mkApp2 (.const ``PProd [lvl1, lvl2]) e1 e2
-
-private def mkNProd (lvl : Level) (es : Array Expr) : MetaM Expr :=
-  es.foldrM (init := mkPUnit lvl) mkPProd
-
-private def mkPUnitMk : Level → Expr
-  | .zero => .const ``True.intro []
-  | lvl   => .const ``PUnit.unit [lvl]
-
-private def mkPProdMk (e1 e2 : Expr) : MetaM Expr := do
-  let t1 ← inferType e1
-  let t2 ← inferType e2
-  let lvl1 ← getLevel t1
-  let lvl2 ← getLevel t2
-  if lvl1 matches .zero && lvl2 matches .zero then
-    return mkApp4 (.const ``And.intro []) t1 t2 e1 e2
-  else
-    return mkApp4 (.const ``PProd.mk [lvl1, lvl2]) t1 t2 e1 e2
-
-private def mkNProdMk (lvl : Level) (es : Array Expr) : MetaM Expr :=
-  es.foldrM (init := mkPUnitMk lvl) mkPProdMk
-
-/-- `PProd.fst` or `And.left` (as projections) -/
-private def mkPProdFst (e : Expr) : MetaM Expr := do
-  let t ← whnf (← inferType e)
-  match_expr t with
-  | PProd _ _ => return .proj ``PProd 0 e
-  | And _ _ =>   return .proj ``And 0 e
-  | _ => throwError "Cannot project .1 out of{indentExpr e}\nof type{indentExpr t}"
-
-/-- `PProd.snd` or `And.right` (as projections) -/
-private def mkPProdSnd (e : Expr) : MetaM Expr := do
-  let t ← whnf (← inferType e)
-  match_expr t with
-  | PProd _ _ => return .proj ``PProd 1 e
-  | And _ _ =>   return .proj ``And 1 e
-  | _ => throwError "Cannot project .2 out of{indentExpr e}\nof type{indentExpr t}"
-
-end PProd
+/-- Transforms `e : xᵢ → (t₁ ×' t₂)` into `(xᵢ → t₁) ×' (xᵢ → t₂) -/
+private def etaPProd (xs : Array Expr) (e : Expr) : MetaM Expr := do
+  if xs.isEmpty then return e
+  let r := mkAppN e xs
+  let r₁ ← mkLambdaFVars xs (← mkPProdFst r)
+  let r₂ ← mkLambdaFVars xs (← mkPProdSnd r)
+  mkPProdMk r₁ r₂
 
 /--
 If `minorType` is the type of a minor premies of a recursor, such as
@@ -91,7 +42,7 @@ private def buildBelowMinorPremise (rlvl : Level) (motives : Array Expr) (minorT
 where
   ibelow := rlvl matches .zero
   go (prods : Array Expr) : List Expr → MetaM Expr
-  | [] => mkNProd rlvl prods
+  | [] => PProdN.pack rlvl prods
   | arg::args => do
     let argType ← inferType arg
     forallTelescope argType fun arg_args arg_type => do
@@ -243,7 +194,7 @@ private def buildBRecOnMinorPremise (rlvl : Level) (motives : Array Expr)
   forallTelescope minorType fun minor_args minor_type => do
     let rec go (prods : Array Expr) : List Expr → MetaM Expr
       | [] => minor_type.withApp fun minor_type_fn minor_type_args => do
-          let b ← mkNProdMk rlvl prods
+          let b ← PProdN.mk rlvl prods
           let .some ⟨idx, _⟩ := motives.indexOf? minor_type_fn
             | throwError m!"Did not find {minor_type} in {motives}"
           mkPProdMk (mkAppN fs[idx]! (minor_type_args.push b)) b
@@ -256,14 +207,8 @@ private def buildBRecOnMinorPremise (rlvl : Level) (motives : Array Expr)
               let type' ← mkForallFVars arg_args
                 (← mkPProd arg_type (mkAppN belows[idx]! arg_type_args) )
               withLocalDeclD name type' fun arg' => do
-                if arg_args.isEmpty then
-                  mkLambdaFVars #[arg'] (← go (prods.push arg') args)
-                else
-                  let r := mkAppN arg' arg_args
-                  let r₁ ← mkLambdaFVars arg_args (← mkPProdFst r)
-                  let r₂ ← mkLambdaFVars arg_args (← mkPProdSnd r)
-                  let r ← mkPProdMk r₁ r₂
-                  mkLambdaFVars #[arg'] (← go (prods.push r) args)
+                let r ← etaPProd arg_args arg'
+                mkLambdaFVars #[arg'] (← go (prods.push r) args)
             else
               mkLambdaFVars #[arg] (← go prods args)
     go #[] minor_args.toList

src/Lean/Meta/ExprDefEq.lean

@@ -1046,6 +1046,15 @@ def checkAssignment (mvarId : MVarId) (fvars : Array Expr) (v : Expr) : MetaM (O
       return none
     return some v
 
+-- Implementation for `_root_.Lean.MVarId.checkedAssign`
+@[export lean_checked_assign]
+def checkedAssignImpl (mvarId : MVarId) (val : Expr) : MetaM Bool := do
+  if let some val ← checkAssignment mvarId #[] val then
+    mvarId.assign val
+    return true
+  else
+    return false
+
 private def processAssignmentFOApproxAux (mvar : Expr) (args : Array Expr) (v : Expr) : MetaM Bool :=
   match v with
   | .mdata _ e   => processAssignmentFOApproxAux mvar args e

src/Lean/Meta/IndPredBelow.lean

@@ -6,6 +6,7 @@ Authors: Dany Fabian
 prelude
 import Lean.Meta.Constructions.CasesOn
 import Lean.Meta.Match.Match
+import Lean.Meta.Tactic.SolveByElim
 
 namespace Lean.Meta.IndPredBelow
 open Match
@@ -230,22 +231,28 @@ def mkBelowDecl (ctx : Context) : MetaM Declaration := do
     ctx.typeInfos[0]!.isUnsafe
 
 partial def backwardsChaining (m : MVarId) (depth : Nat) : MetaM Bool := do
-  if depth = 0 then return false
-  else
-    m.withContext do
-    let lctx ← getLCtx
+  m.withContext do
     let mTy ← m.getType
-    lctx.anyM fun localDecl =>
-      if localDecl.isAuxDecl then
-        return false
-      else
-        commitWhen do
-        let (mvars, _, t) ← forallMetaTelescope localDecl.type
-        if ←isDefEq mTy t then
-          m.assign (mkAppN localDecl.toExpr mvars)
-          mvars.allM fun v =>
-            v.mvarId!.isAssigned <||> backwardsChaining v.mvarId! (depth - 1)
-        else return false
+    if depth = 0 then
+      trace[Meta.IndPredBelow.search] "searching for {mTy}: ran out of max depth"
+      return false
+    else
+      let lctx ← getLCtx
+      let r ← lctx.anyM fun localDecl =>
+        if localDecl.isAuxDecl then
+          return false
+        else
+          commitWhen do
+          let (mvars, _, t) ← forallMetaTelescope localDecl.type
+          if (← isDefEq mTy t) then
+            trace[Meta.IndPredBelow.search] "searching for {mTy}: trying {mkFVar localDecl.fvarId} : {localDecl.type}"
+            m.assign (mkAppN localDecl.toExpr mvars)
+            mvars.allM fun v =>
+              v.mvarId!.isAssigned <||> backwardsChaining v.mvarId! (depth - 1)
+          else return false
+      unless r do
+        trace[Meta.IndPredBelow.search] "searching for {mTy} failed"
+      return r
 
 partial def proveBrecOn (ctx : Context) (indVal : InductiveVal) (type : Expr) : MetaM Expr := do
   let main ← mkFreshExprSyntheticOpaqueMVar type
@@ -563,7 +570,7 @@ def findBelowIdx (xs : Array Expr) (motive : Expr) : MetaM $ Option (Expr × Nat
       let below ← mkFreshExprSyntheticOpaqueMVar belowTy
       try
         trace[Meta.IndPredBelow.match] "{←Meta.ppGoal below.mvarId!}"
-        if (← backwardsChaining below.mvarId! 10) then
+        if (← below.mvarId!.applyRules { backtracking := false, maxDepth := 1 } []).isEmpty then
           trace[Meta.IndPredBelow.match] "Found below term in the local context: {below}"
           if (← xs.anyM (isDefEq below)) then pure none else pure (below, idx.val)
         else
@@ -596,5 +603,6 @@ def mkBelow (declName : Name) : MetaM Unit := do
 builtin_initialize
   registerTraceClass `Meta.IndPredBelow
   registerTraceClass `Meta.IndPredBelow.match
+  registerTraceClass `Meta.IndPredBelow.search
 
 end Lean.Meta.IndPredBelow

src/Lean/Meta/PProdN.lean

@@ -0,0 +1,145 @@
+/-
+Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+
+prelude
+import Lean.Meta.InferType
+
+/-!
+This module provides functios to pack and unpack values using nested `PProd` or `And`,
+as used in the `.below` construction, in the `.brecOn` construction for mutual recursion and
+and the `mutual_induct` construction.
+
+It uses `And` (equivalent to `PProd.{0}` when possible).
+
+The nesting is `t₁ ×' (t₂ ×' t₃)`, not `t₁ ×' (t₂ ×' (t₃ ×' PUnit))`. This is more readable,
+slightly shorter, and means that the packing is the identity if `n=1`, which we rely on in some
+places. It comes at the expense that hat projection needs to know `n`.
+
+Packing an empty list uses `True` or `PUnit` depending on the given `lvl`.
+
+Also see `Lean.Meta.ArgsPacker` for a similar module for `PSigma` and `PSum`, used by well-founded recursion.
+-/
+
+
+namespace Lean.Meta
+
+/-- Given types `t₁` and `t₂`, produces `t₁ ×' t₂` (or `t₁ ∧ t₂` if the universes allow) -/
+def mkPProd (e1 e2 : Expr) : MetaM Expr := do
+  let lvl1 ← getLevel e1
+  let lvl2 ← getLevel e2
+  if lvl1 matches .zero && lvl2 matches .zero then
+    return mkApp2 (.const `And []) e1 e2
+  else
+    return mkApp2 (.const ``PProd [lvl1, lvl2]) e1 e2
+
+/-- Given values of typs `t₁` and `t₂`, produces value of type `t₁ ×' t₂` (or `t₁ ∧ t₂` if the universes allow) -/
+def mkPProdMk (e1 e2 : Expr) : MetaM Expr := do
+  let t1 ← inferType e1
+  let t2 ← inferType e2
+  let lvl1 ← getLevel t1
+  let lvl2 ← getLevel t2
+  if lvl1 matches .zero && lvl2 matches .zero then
+    return mkApp4 (.const ``And.intro []) t1 t2 e1 e2
+  else
+    return mkApp4 (.const ``PProd.mk [lvl1, lvl2]) t1 t2 e1 e2
+
+/-- `PProd.fst` or `And.left` (using `.proj`) -/
+def mkPProdFst (e : Expr) : MetaM Expr := do
+  let t ← whnf (← inferType e)
+  match_expr t with
+  | PProd _ _ => return .proj ``PProd 0 e
+  | And _ _ => return .proj ``And 0 e
+  | _ => panic! "mkPProdFst: cannot handle{indentExpr e}\nof type{indentExpr t}"
+
+/-- `PProd.snd` or `And.right` (using `.proj`) -/
+def mkPProdSnd (e : Expr) : MetaM Expr := do
+  let t ← whnf (← inferType e)
+  match_expr t with
+  | PProd _ _ => return .proj ``PProd 1 e
+  | And _ _ => return .proj ``And 1 e
+  | _ => panic! "mkPProdSnd: cannot handle{indentExpr e}\nof type{indentExpr t}"
+
+
+
+namespace PProdN
+
+/-- Given types `tᵢ`, produces `t₁ ×' t₂ ×' t₃` -/
+def pack (lvl : Level) (xs : Array Expr) : MetaM Expr := do
+  if xs.size = 0 then
+    if lvl matches .zero then return .const ``True []
+                         else return .const ``PUnit [lvl]
+  let xBack := xs.back
+  xs.pop.foldrM mkPProd xBack
+
+/-- Given values `xᵢ` of type `tᵢ`, produces value of type `t₁ ×' t₂ ×' t₃` -/
+def mk (lvl : Level) (xs : Array Expr) : MetaM Expr := do
+  if xs.size = 0 then
+    if lvl matches .zero then return .const ``True.intro []
+                         else return .const ``PUnit.unit [lvl]
+  let xBack := xs.back
+  xs.pop.foldrM mkPProdMk xBack
+
+/-- Given a value of type `t₁ ×' … ×' tᵢ ×' … ×' tₙ`, return a value of type `tᵢ` -/
+def proj (n i : Nat) (e : Expr) : MetaM Expr := do
+  let mut value := e
+  for _ in [:i] do
+      value ← mkPProdSnd value
+  if i+1 < n then
+    mkPProdFst value
+  else
+    pure value
+
+
+
+/--
+Packs multiple type-forming lambda expressions taking the same parameters using `PProd`.
+
+The parameter `type` is the common type of the these expressions
+
+For example
+```
+packLambdas (Nat → Sort u) #[(fun (n : Nat) => Nat), (fun (n : Nat) => Fin n -> Fin n )]
+```
+will return
+```
+fun (n : Nat) => (Nat ×' (Fin n → Fin n))
+```
+
+It is the identity if `es.size = 1`.
+
+It returns a dummy motive `(xs : ) → PUnit` or `(xs : … ) → True` if no expressions are given.
+(this is the reason we need the expected type in the `type` parameter).
+
+-/
+def packLambdas (type : Expr) (es : Array Expr) : MetaM Expr := do
+  if es.size = 1 then
+    return es[0]!
+  forallTelescope type fun xs sort => do
+    assert! sort.isSort
+    -- NB: Use beta, not instantiateLambda; when constructing the belowDict below
+    -- we pass `C`, a plain FVar, here
+    let es' := es.map (·.beta xs)
+    let packed ← PProdN.pack sort.sortLevel! es'
+    mkLambdaFVars xs packed
+
+/--
+The value analogue to `PProdN.packLambdas`.
+
+It is the identity if `es.size = 1`.
+-/
+def mkLambdas (type : Expr) (es : Array Expr) : MetaM Expr := do
+  if es.size = 1 then
+    return es[0]!
+  forallTelescope type fun xs body => do
+    let lvl ← getLevel body
+    let es' := es.map (·.beta xs)
+    let packed ← PProdN.mk lvl es'
+    mkLambdaFVars xs packed
+
+
+end PProdN
+
+end Lean.Meta

src/Lean/Parser/Command.lean

@@ -269,7 +269,7 @@ corresponding `end <id>` or the end of the file.
   "namespace " >> checkColGt >> ident
 /--
 `end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed
-with `end <id>`.
+with `end <id>`. The `end` command is optional at the end of a file.
 -/
 @[builtin_command_parser] def «end»          := leading_parser
   "end" >> optional (ppSpace >> checkColGt >> ident)
@@ -437,6 +437,8 @@ structure Pair (α : Type u) (β : Type v) : Type (max u v) where
   "#check_failure " >> termParser -- Like `#check`, but succeeds only if term does not type check
 @[builtin_command_parser] def eval           := leading_parser
   "#eval " >> termParser
+@[builtin_command_parser] def evalBang       := leading_parser
+  "#eval! " >> termParser
 @[builtin_command_parser] def synth          := leading_parser
   "#synth " >> termParser
 @[builtin_command_parser] def exit           := leading_parser

src/Lean/PrettyPrinter/Delaborator/Builtins.lean

@@ -339,19 +339,22 @@ inductive AppImplicitArg
   | skip
   /-- A regular argument. -/
   | regular (s : Term)
+  /-- A regular argument that, if it comes as the last argument, may be omitted. -/
+  | optional (name : Name) (s : Term)
   /-- It's a named argument. Named arguments inhibit applying unexpanders. -/
   | named (s : TSyntax ``Parser.Term.namedArgument)
   deriving Inhabited
 
 /-- Whether unexpanding is allowed with this argument. -/
 def AppImplicitArg.canUnexpand : AppImplicitArg → Bool
-  | .regular .. | .skip => true
+  | .regular .. | .optional .. | .skip => true
   | .named .. => false
 
 /-- If the argument has associated syntax, returns it. -/
 def AppImplicitArg.syntax? : AppImplicitArg → Option Syntax
   | .skip => none
   | .regular s => s
+  | .optional _ s => s
   | .named s => s
 
 /--
@@ -371,13 +374,13 @@ def delabAppImplicitCore (unexpand : Bool) (numArgs : Nat) (delabHead : Delab) (
       appFieldNotationCandidate?
     else
       pure none
-  let (fnStx, args) ←
+  let (fnStx, args') ←
     withBoundedAppFnArgs numArgs
       (do return ((← delabHead), Array.mkEmpty numArgs))
-      (fun (fnStx, args) => do
-        let idx := args.size
-        let arg ← mkArg (numArgs - idx - 1) paramKinds[idx]!
-        return (fnStx, args.push arg))
+      (fun (fnStx, args) => return (fnStx, args.push (← mkArg paramKinds[args.size]!)))
+
+  -- Strip off optional arguments. We save the original `args'` for structure instance notation
+  let args := args'.popWhile (· matches .optional ..)
 
   -- App unexpanders
   if ← pure unexpand <&&> getPPOption getPPNotation then
@@ -385,11 +388,10 @@ def delabAppImplicitCore (unexpand : Bool) (numArgs : Nat) (delabHead : Delab) (
     if let some stx ← (some <$> tryAppUnexpanders fnStx args) <|> pure none then
       return stx
 
-  let stx := Syntax.mkApp fnStx (args.filterMap (·.syntax?))
-
   -- Structure instance notation
-  if ← pure (unexpand && args.all (·.canUnexpand)) <&&> getPPOption getPPStructureInstances then
+  if ← pure (unexpand && args'.all (·.canUnexpand)) <&&> getPPOption getPPStructureInstances then
     -- Try using the structure instance unexpander.
+    let stx := Syntax.mkApp fnStx (args'.filterMap (·.syntax?))
     if let some stx ← (some <$> unexpandStructureInstance stx) <|> pure none then
       return stx
 
@@ -416,7 +418,7 @@ def delabAppImplicitCore (unexpand : Bool) (numArgs : Nat) (delabHead : Delab) (
         return Syntax.mkApp head (args'.filterMap (·.syntax?))
 
   -- Normal application
-  return stx
+  return Syntax.mkApp fnStx (args.filterMap (·.syntax?))
 where
   mkNamedArg (name : Name) : DelabM AppImplicitArg :=
     return .named <| ← `(Parser.Term.namedArgument| ($(mkIdent name) := $(← delab)))
@@ -424,15 +426,16 @@ where
   Delaborates the current argument.
   The argument `remainingArgs` is the number of arguments in the application after this one.
   -/
-  mkArg (remainingArgs : Nat) (param : ParamKind) : DelabM AppImplicitArg := do
+  mkArg (param : ParamKind) : DelabM AppImplicitArg := do
     let arg ← getExpr
     if ← getPPOption getPPAnalysisSkip then return .skip
     else if ← getPPOption getPPAnalysisHole then return .regular (← `(_))
     else if ← getPPOption getPPAnalysisNamedArg then
       mkNamedArg param.name
-    else if param.defVal.isSome && remainingArgs == 0 && param.defVal.get! == arg then
-      -- Assumption: `useAppExplicit` has already detected whether it is ok to omit this argument
-      return .skip
+    else if param.defVal.isSome && param.defVal.get! == arg then
+      -- Assumption: `useAppExplicit` has already detected whether it is ok to omit this argument, if it is the last one.
+      -- We will later remove all optional arguments from the end.
+      return .optional param.name (← delab)
     else if param.bInfo.isExplicit then
       return .regular (← delab)
     else if ← pure (param.name == `motive) <&&> shouldShowMotive arg (← getOptions) then

src/Lean/PrettyPrinter/Parenthesizer.lean

@@ -350,15 +350,17 @@ def term.parenthesizer : CategoryParenthesizer | prec => do
   maybeParenthesize `term true wrapParens prec $
     parenthesizeCategoryCore `term prec
 where
-  /-- Wraps the term `stx` in parentheses and then copies its `SourceInfo` to the result.
-  The purpose of this is to copy synthetic delaborator positions from the `stx` node to the parentheses node,
-  which causes the info view to view both of these nodes as referring to the same expression.
-  If we did not copy info, the info view would consider the parentheses to belong to the outer term.
+  /-- Wraps the term `stx` in parentheses and then moves its `SourceInfo` to the result.
+  The purpose of this is to move synthetic delaborator positions from the `stx` node to the parentheses node,
+  which causes the info view to view the node with parentheses as referring to the parenthesized expression.
+  If we did not move info, the info view would consider the parentheses to belong to the outer term.
   Note: we do not do `withRef stx` because that causes the "(" and ")" tokens to have source info as well,
   causing the info view to highlight each parenthesis as an independent expression. -/
   wrapParens (stx : Syntax) : Syntax := Unhygienic.run do
+    let stxInfo := SourceInfo.fromRef stx
+    let stx := stx.setInfo .none
     let pstx ← `(($(⟨stx⟩)))
-    return pstx.raw.setInfo (SourceInfo.fromRef stx)
+    return pstx.raw.setInfo stxInfo
 
 @[builtin_category_parenthesizer tactic]
 def tactic.parenthesizer : CategoryParenthesizer | prec => do

src/Lean/Server/FileWorker/RequestHandling.lean

@@ -234,31 +234,27 @@ def getInteractiveGoals (p : Lsp.PlainGoalParams) : RequestM (RequestTask (Optio
   let doc ← readDoc
   let text := doc.meta.text
   let hoverPos := text.lspPosToUtf8Pos p.position
-  -- NOTE: use `>=` since the cursor can be *after* the input
-  withWaitFindSnap doc (fun s => s.endPos >= hoverPos)
-    (notFoundX := return none) fun snap => do
-      if let rs@(_ :: _) := snap.infoTree.goalsAt? doc.meta.text hoverPos then
-        let goals : List Widget.InteractiveGoals ← rs.mapM fun { ctxInfo := ci, tacticInfo := ti, useAfter := useAfter, .. } => do
-          let ciAfter := { ci with mctx := ti.mctxAfter }
-          let ci := if useAfter then ciAfter else { ci with mctx := ti.mctxBefore }
-          -- compute the interactive goals
-          let goals ← ci.runMetaM {} (do
-            let goals := List.toArray <| if useAfter then ti.goalsAfter else ti.goalsBefore
-            let goals ← goals.mapM Widget.goalToInteractive
-            return {goals}
-          )
-          -- compute the goal diff
-          let goals ← ciAfter.runMetaM {} (do
-              try
-                Widget.diffInteractiveGoals useAfter ti goals
-              catch _ =>
-                -- fail silently, since this is just a bonus feature
-                return goals
-          )
-          return goals
-        return some <| goals.foldl (· ++ ·) ∅
-      else
-        return none
+  mapTask (findInfoTreeAtPos doc hoverPos) <| Option.bindM fun infoTree => do
+    let rs@(_ :: _) := infoTree.goalsAt? doc.meta.text hoverPos
+      | return none
+    let goals : List Widget.InteractiveGoals ← rs.mapM fun { ctxInfo := ci, tacticInfo := ti, useAfter := useAfter, .. } => do
+      let ciAfter := { ci with mctx := ti.mctxAfter }
+      let ci := if useAfter then ciAfter else { ci with mctx := ti.mctxBefore }
+      -- compute the interactive goals
+      let goals ← ci.runMetaM {} (do
+        let goals := List.toArray <| if useAfter then ti.goalsAfter else ti.goalsBefore
+        let goals ← goals.mapM Widget.goalToInteractive
+        return {goals}
+      )
+      -- compute the goal diff
+      ciAfter.runMetaM {} (do
+          try
+            Widget.diffInteractiveGoals useAfter ti goals
+          catch _ =>
+            -- fail silently, since this is just a bonus feature
+            return goals
+      )
+    return some <| goals.foldl (· ++ ·) ∅
 
 open Elab in
 def handlePlainGoal (p : PlainGoalParams)
@@ -280,19 +276,17 @@ def getInteractiveTermGoal (p : Lsp.PlainTermGoalParams)
   let doc ← readDoc
   let text := doc.meta.text
   let hoverPos := text.lspPosToUtf8Pos p.position
-  withWaitFindSnap doc (fun s => s.endPos > hoverPos)
-    (notFoundX := pure none) fun snap => do
-      if let some {ctx := ci, info := i@(Elab.Info.ofTermInfo ti), ..} := snap.infoTree.termGoalAt? hoverPos then
-        let ty ← ci.runMetaM i.lctx do
-          instantiateMVars <| ti.expectedType?.getD (← Meta.inferType ti.expr)
-        -- for binders, hide the last hypothesis (the binder itself)
-        let lctx' := if ti.isBinder then i.lctx.pop else i.lctx
-        let goal ← ci.runMetaM lctx' do
-          Widget.goalToInteractive (← Meta.mkFreshExprMVar ty).mvarId!
-        let range := if let some r := i.range? then r.toLspRange text else ⟨p.position, p.position⟩
-        return some { goal with range, term := ⟨ti⟩ }
-      else
-        return none
+  mapTask (findInfoTreeAtPos doc hoverPos) <| Option.bindM fun infoTree => do
+    let some {ctx := ci, info := i@(Elab.Info.ofTermInfo ti), ..} := infoTree.termGoalAt? hoverPos
+      | return none
+    let ty ← ci.runMetaM i.lctx do
+      instantiateMVars <| ti.expectedType?.getD (← Meta.inferType ti.expr)
+    -- for binders, hide the last hypothesis (the binder itself)
+    let lctx' := if ti.isBinder then i.lctx.pop else i.lctx
+    let goal ← ci.runMetaM lctx' do
+      Widget.goalToInteractive (← Meta.mkFreshExprMVar ty).mvarId!
+    let range := if let some r := i.range? then r.toLspRange text else ⟨p.position, p.position⟩
+    return some { goal with range, term := ⟨ti⟩ }
 
 def handlePlainTermGoal (p : PlainTermGoalParams)
     : RequestM (RequestTask (Option PlainTermGoal)) := do

src/Lean/Server/GoTo.lean

@@ -37,6 +37,8 @@ def moduleFromDocumentUri (srcSearchPath : SearchPath) (uri : DocumentUri)
 open Elab in
 def locationLinksFromDecl (srcSearchPath : SearchPath) (uri : DocumentUri) (n : Name)
     (originRange? : Option Range) : MetaM (Array LocationLink) := do
+  -- Potentially this name is a builtin that has not been imported yet:
+  unless (← getEnv).contains n do return #[]
   let mod? ← findModuleOf? n
   let modUri? ← match mod? with
     | some modName => documentUriFromModule srcSearchPath modName

src/Lean/Server/Requests.lean

@@ -16,6 +16,32 @@ import Lean.Server.FileWorker.Utils
 
 import Lean.Server.Rpc.Basic
 
+namespace Lean.Language
+
+/--
+Finds the first (in pre-order) snapshot task in `tree` whose `range?` contains `pos` and which
+contains an info tree, and then returns that info tree, waiting for any snapshot tasks on the way.
+Subtrees that do not contain the position are skipped without forcing their tasks.
+-/
+partial def SnapshotTree.findInfoTreeAtPos (tree : SnapshotTree) (pos : String.Pos) :
+    Task (Option Elab.InfoTree) :=
+  goSeq tree.children.toList
+where
+  goSeq
+    | [] => .pure none
+    | t::ts =>
+      if t.range?.any (·.contains pos) then
+        t.task.bind (sync := true) fun tree => Id.run do
+          if let some infoTree := tree.element.infoTree? then
+            return .pure infoTree
+          tree.findInfoTreeAtPos pos |>.bind (sync := true) fun
+            | some infoTree => .pure (some infoTree)
+            | none => goSeq ts
+      else
+        goSeq ts
+
+end Lean.Language
+
 namespace Lean.Server
 
 structure RequestError where
@@ -144,6 +170,45 @@ def withWaitFindSnapAtPos
     (notFoundX := throw ⟨.invalidParams, s!"no snapshot found at {lspPos}"⟩)
     (x := f)
 
+open Language.Lean in
+/-- Finds the first `CommandParsedSnapshot` fulfilling `p`, asynchronously. -/
+partial def findCmdParsedSnap (doc : EditableDocument) (p : CommandParsedSnapshot → Bool) :
+    Task (Option CommandParsedSnapshot) := Id.run do
+  let some headerParsed := doc.initSnap.result?
+    | .pure none
+  headerParsed.processedSnap.task.bind (sync := true) fun headerProcessed => Id.run do
+    let some headerSuccess := headerProcessed.result?
+      | return .pure none
+    headerSuccess.firstCmdSnap.task.bind (sync := true) go
+where
+  go cmdParsed :=
+    if p cmdParsed then
+      .pure (some cmdParsed)
+    else
+      match cmdParsed.nextCmdSnap? with
+      | some next => next.task.bind (sync := true) go
+      | none => .pure none
+
+open Language in
+/--
+Finds the info tree of the first snapshot task containing `pos`, asynchronously. The info tree may
+be from a nested snapshot, such as a single tactic.
+
+See `SnapshotTree.findInfoTreeAtPos` for details on how the search is done.
+-/
+partial def findInfoTreeAtPos (doc : EditableDocument) (pos : String.Pos) :
+    Task (Option Elab.InfoTree) :=
+  -- NOTE: use `>=` since the cursor can be *after* the input (and there is no interesting info on
+  -- the first character of the subsequent command if any)
+  findCmdParsedSnap doc (·.data.parserState.pos ≥ pos) |>.bind (sync := true) fun
+    | some cmdParsed => toSnapshotTree cmdParsed |>.findInfoTreeAtPos pos |>.bind (sync := true) fun
+      | some infoTree => .pure <| some infoTree
+      | none          => cmdParsed.data.finishedSnap.task.map (sync := true) fun s =>
+        -- the parser returns exactly one command per snapshot, and the elaborator creates exactly one node per command
+        assert! s.cmdState.infoState.trees.size == 1
+        some s.cmdState.infoState.trees[0]!
+    | none => .pure none
+
 open Elab.Command in
 def runCommandElabM (snap : Snapshot) (c : RequestT CommandElabM α) : RequestM α := do
   let rc ← readThe RequestContext

src/Lean/Util/CollectAxioms.lean

@@ -0,0 +1,45 @@
+/-
+Copyright (c) 2020 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Lean.MonadEnv
+import Lean.Util.FoldConsts
+
+namespace Lean
+
+namespace CollectAxioms
+
+
+structure State where
+  visited : NameSet    := {}
+  axioms  : Array Name := #[]
+
+abbrev M := ReaderT Environment $ StateM State
+
+partial def collect (c : Name) : M Unit := do
+  let collectExpr (e : Expr) : M Unit := e.getUsedConstants.forM collect
+  let s ← get
+  unless s.visited.contains c do
+    modify fun s => { s with visited := s.visited.insert c }
+    let env ← read
+    match env.find? c with
+    | some (ConstantInfo.axiomInfo _)  => modify fun s => { s with axioms := s.axioms.push c }
+    | some (ConstantInfo.defnInfo v)   => collectExpr v.type *> collectExpr v.value
+    | some (ConstantInfo.thmInfo v)    => collectExpr v.type *> collectExpr v.value
+    | some (ConstantInfo.opaqueInfo v) => collectExpr v.type *> collectExpr v.value
+    | some (ConstantInfo.quotInfo _)   => pure ()
+    | some (ConstantInfo.ctorInfo v)   => collectExpr v.type
+    | some (ConstantInfo.recInfo v)    => collectExpr v.type
+    | some (ConstantInfo.inductInfo v) => collectExpr v.type *> v.ctors.forM collect
+    | none                             => pure ()
+
+end CollectAxioms
+
+def collectAxioms [Monad m] [MonadEnv m] (constName : Name) : m (Array Name) := do
+  let env ← getEnv
+  let (_, s) := ((CollectAxioms.collect constName).run env).run {}
+  pure s.axioms
+
+end Lean

src/Lean/Util/FindExpr.lean

@@ -5,52 +5,14 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Lean.Expr
-import Lean.Util.PtrSet
 
 namespace Lean
 namespace Expr
-namespace FindImpl
 
-unsafe abbrev FindM := StateT (PtrSet Expr) Id
+@[extern "lean_find_expr"]
+opaque findImpl? (p : @& (Expr → Bool)) (e : @& Expr) : Option Expr
 
-@[inline] unsafe def checkVisited (e : Expr) : OptionT FindM Unit := do
-  if (← get).contains e then
-    failure
-  modify fun s => s.insert e
-
-unsafe def findM? (p : Expr → Bool) (e : Expr) : OptionT FindM Expr :=
-  let rec visit (e : Expr) := do
-    checkVisited e
-    if p e then
-      pure e
-    else match e with
-      | .forallE _ d b _ => visit d <|> visit b
-      | .lam _ d b _     => visit d <|> visit b
-      | .mdata _ b       => visit b
-      | .letE _ t v b _  => visit t <|> visit v <|> visit b
-      | .app f a         => visit f <|> visit a
-      | .proj _ _ b      => visit b
-      | _                => failure
-  visit e
-
-unsafe def findUnsafe? (p : Expr → Bool) (e : Expr) : Option Expr :=
-  Id.run <| findM? p e |>.run' mkPtrSet
-
-end FindImpl
-
-@[implemented_by FindImpl.findUnsafe?]
-def find? (p : Expr → Bool) (e : Expr) : Option Expr :=
-  /- This is a reference implementation for the unsafe one above -/
-  if p e then
-    some e
-  else match e with
-    | .forallE _ d b _ => find? p d <|> find? p b
-    | .lam _ d b _     => find? p d <|> find? p b
-    | .mdata _ b       => find? p b
-    | .letE _ t v b _  => find? p t <|> find? p v <|> find? p b
-    | .app f a         => find? p f <|> find? p a
-    | .proj _ _ b      => find? p b
-    | _                => none
+@[inline] def find? (p : Expr → Bool) (e : Expr) : Option Expr := findImpl? p e
 
 /-- Return true if `e` occurs in `t` -/
 def occurs (e : Expr) (t : Expr) : Bool :=
@@ -64,41 +26,13 @@ inductive FindStep where
   /-- Search subterms -/ | visit
   /-- Do not search subterms -/ | done
 
-namespace FindExtImpl
-
-unsafe def findM? (p : Expr → FindStep) (e : Expr) : OptionT FindImpl.FindM Expr :=
-  visit e
-where
-  visitApp (e : Expr) :=
-    match e with
-    | .app f a .. => visitApp f <|> visit a
-    | e => visit e
-
-  visit (e : Expr) := do
-    FindImpl.checkVisited e
-    match p e with
-      | .done  => failure
-      | .found => pure e
-      | .visit =>
-        match e with
-        | .forallE _ d b _ => visit d <|> visit b
-        | .lam _ d b _     => visit d <|> visit b
-        | .mdata _ b       => visit b
-        | .letE _ t v b _  => visit t <|> visit v <|> visit b
-        | .app ..          => visitApp e
-        | .proj _ _ b      => visit b
-        | _                => failure
-
-unsafe def findUnsafe? (p : Expr → FindStep) (e : Expr) : Option Expr :=
-  Id.run <| findM? p e |>.run' mkPtrSet
-
-end FindExtImpl
+@[extern "lean_find_ext_expr"]
+opaque findExtImpl? (p : @& (Expr → FindStep)) (e : @& Expr) : Option Expr
 
 /--
   Similar to `find?`, but `p` can return `FindStep.done` to interrupt the search on subterms.
   Remark: Differently from `find?`, we do not invoke `p` for partial applications of an application. -/
-@[implemented_by FindExtImpl.findUnsafe?]
-opaque findExt? (p : Expr → FindStep) (e : Expr) : Option Expr
+@[inline] def findExt? (p : Expr → FindStep) (e : Expr) : Option Expr := findExtImpl? p e
 
 end Expr
 end Lean

src/Lean/Util/FoldConsts.lean

@@ -11,52 +11,35 @@ namespace Lean
 namespace Expr
 namespace FoldConstsImpl
 
-abbrev cacheSize : USize := 8192 - 1
+unsafe structure State where
+ visited       : PtrSet Expr := mkPtrSet
+ visitedConsts : NameHashSet := {}
 
-structure State where
-  visitedTerms  : Array Expr  -- Remark: cache based on pointer address. Our "unsafe" implementation relies on the fact that `()` is not a valid Expr
-  visitedConsts : NameHashSet -- cache based on structural equality
+unsafe abbrev FoldM := StateM State
 
-abbrev FoldM := StateM State
-
-unsafe def visited (e : Expr) (size : USize) : FoldM Bool := do
-  let s ← get
-  let h := ptrAddrUnsafe e
-  let i := h % size
-  let k := s.visitedTerms.uget i lcProof
-  if ptrAddrUnsafe k == h then pure true
-  else do
-    modify fun s => { s with visitedTerms := s.visitedTerms.uset i e lcProof }
-    pure false
-
-unsafe def fold {α : Type} (f : Name → α → α) (size : USize) (e : Expr) (acc : α) : FoldM α :=
+unsafe def fold {α : Type} (f : Name → α → α) (e : Expr) (acc : α) : FoldM α :=
   let rec visit (e : Expr) (acc : α) : FoldM α := do
-    if (← visited e size) then
-      pure acc
-    else
-      match e with
-      | Expr.forallE _ d b _   => visit b (← visit d acc)
-      | Expr.lam _ d b _       => visit b (← visit d acc)
-      | Expr.mdata _ b         => visit b acc
-      | Expr.letE _ t v b _    => visit b (← visit v (← visit t acc))
-      | Expr.app f a           => visit a (← visit f acc)
-      | Expr.proj _ _ b        => visit b acc
-      | Expr.const c _         =>
-        let s ← get
-        if s.visitedConsts.contains c then
-          pure acc
-        else do
-          modify fun s => { s with visitedConsts := s.visitedConsts.insert c };
-          pure $ f c acc
-      | _                      => pure acc
+    if (← get).visited.contains e then
+      return acc
+    modify fun s => { s with visited := s.visited.insert e }
+    match e with
+    | .forallE _ d b _   => visit b (← visit d acc)
+    | .lam _ d b _       => visit b (← visit d acc)
+    | .mdata _ b         => visit b acc
+    | .letE _ t v b _    => visit b (← visit v (← visit t acc))
+    | .app f a           => visit a (← visit f acc)
+    | .proj _ _ b        => visit b acc
+    | .const c _         =>
+      if (← get).visitedConsts.contains c then
+        return acc
+      else
+        modify fun s => { s with visitedConsts := s.visitedConsts.insert c };
+        return f c acc
+    | _ => return acc
   visit e acc
 
-unsafe def initCache : State :=
-  { visitedTerms  := mkArray cacheSize.toNat (cast lcProof ()),
-    visitedConsts := {} }
-
 @[inline] unsafe def foldUnsafe {α : Type} (e : Expr) (init : α) (f : Name → α → α) : α :=
-  (fold f cacheSize e init).run' initCache
+  (fold f e init).run' {}
 
 end FoldConstsImpl