chore: reordering in Array.Basic

This is to simplify the manual's cross-referencing.

src/Init/ByCases.lean

@@ -40,21 +40,23 @@ theorem apply_ite (f : α → β) (P : Prop) [Decidable P] (x y : α) :
 /-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/
 @[simp] theorem dite_eq_ite [Decidable P] : (dite P (fun _ => a) fun _ => b) = ite P a b := rfl
 
--- We don't mark this as `simp` as it is already handled by `ite_eq_right_iff`.
+@[deprecated "Use `ite_eq_right_iff`" (since := "2024-09-18")]
 theorem ite_some_none_eq_none [Decidable P] :
     (if P then some x else none) = none ↔ ¬ P := by
   simp only [ite_eq_right_iff, reduceCtorEq]
   rfl
 
-@[simp] theorem ite_some_none_eq_some [Decidable P] :
+@[deprecated "Use `Option.ite_none_right_eq_some`" (since := "2024-09-18")]
+theorem ite_some_none_eq_some [Decidable P] :
     (if P then some x else none) = some y ↔ P ∧ x = y := by
   split <;> simp_all
 
--- This is not marked as `simp` as it is already handled by `dite_eq_right_iff`.
+@[deprecated "Use `dite_eq_right_iff" (since := "2024-09-18")]
 theorem dite_some_none_eq_none [Decidable P] {x : P → α} :
     (if h : P then some (x h) else none) = none ↔ ¬P := by
   simp
 
-@[simp] theorem dite_some_none_eq_some [Decidable P] {x : P → α} {y : α} :
+@[deprecated "Use `Option.dite_none_right_eq_some`" (since := "2024-09-18")]
+theorem dite_some_none_eq_some [Decidable P] {x : P → α} {y : α} :
     (if h : P then some (x h) else none) = some y ↔ ∃ h : P, x h = y := by
   by_cases h : P <;> simp [h]

src/Init/Control/Basic.lean

@@ -28,7 +28,7 @@ Important instances include
 * `Option`, where `failure := none` and `<|>` returns the left-most `some`.
 * Parser combinators typically provide an `Applicative` instance for error-handling and
   backtracking.
-  
+
 Error recovery and state can interact subtly. For example, the implementation of `Alternative` for `OptionT (StateT σ Id)` keeps modifications made to the state while recovering from failure, while `StateT σ (OptionT Id)` discards them.
 -/
 -- NB: List instance is in mathlib. Once upstreamed, add

src/Init/Core.lean

@@ -823,6 +823,8 @@ theorem iff_iff_implies_and_implies {a b : Prop} : (a ↔ b) ↔ (a → b) ∧ (
 protected theorem Iff.rfl {a : Prop} : a ↔ a :=
   Iff.refl a
 
+macro_rules | `(tactic| rfl) => `(tactic| exact Iff.rfl)
+
 theorem Iff.of_eq (h : a = b) : a ↔ b := h ▸ Iff.rfl
 
 theorem Iff.trans (h₁ : a ↔ b) (h₂ : b ↔ c) : a ↔ c :=
@@ -1191,6 +1193,21 @@ end
 
 /-! # Product -/
 
+instance [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (α × β) :=
+  Nonempty.elim h1 fun x =>
+    Nonempty.elim h2 fun y =>
+      ⟨(x, y)⟩
+
+instance [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (MProd α β) :=
+  Nonempty.elim h1 fun x =>
+    Nonempty.elim h2 fun y =>
+      ⟨⟨x, y⟩⟩
+
+instance [h1 : Nonempty α] [h2 : Nonempty β] : Nonempty (PProd α β) :=
+  Nonempty.elim h1 fun x =>
+    Nonempty.elim h2 fun y =>
+      ⟨⟨x, y⟩⟩
+
 instance [Inhabited α] [Inhabited β] : Inhabited (α × β) where
   default := (default, default)
 

src/Init/Data/Array.lean

@@ -15,3 +15,4 @@ import Init.Data.Array.BasicAux
 import Init.Data.Array.Lemmas
 import Init.Data.Array.TakeDrop
 import Init.Data.Array.Bootstrap
+import Init.Data.Array.GetLit

src/Init/Data/Array/Basic.lean

@@ -13,43 +13,76 @@ import Init.Data.ToString.Basic
 import Init.GetElem
 universe u v w
 
-namespace Array
+/-! ### Array literal syntax -/
+
+syntax "#[" withoutPosition(sepBy(term, ", ")) "]" : term
+
+macro_rules
+  | `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
+
 variable {α : Type u}
 
+namespace Array
+
+/-! ### Preliminary theorems -/
+
+@[simp] theorem size_set (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
+  List.length_set ..
+
+@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
+  List.length_concat ..
+
+theorem ext (a b : Array α)
+    (h₁ : a.size = b.size)
+    (h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
+    : a = b := by
+  let rec extAux (a b : List α)
+      (h₁ : a.length = b.length)
+      (h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
+      : a = b := by
+    induction a generalizing b with
+    | nil =>
+      cases b with
+      | nil       => rfl
+      | cons b bs => rw [List.length_cons] at h₁; injection h₁
+    | cons a as ih =>
+      cases b with
+      | nil => rw [List.length_cons] at h₁; injection h₁
+      | cons b bs =>
+        have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
+        have hz₂ : 0 < (b::bs).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
+        have headEq : a = b := h₂ 0 hz₁ hz₂
+        have h₁' : as.length = bs.length := by rw [List.length_cons, List.length_cons] at h₁; injection h₁
+        have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get ⟨i, hi₁⟩ = bs.get ⟨i, hi₂⟩ := by
+          intro i hi₁ hi₂
+          have hi₁' : i+1 < (a::as).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
+          have hi₂' : i+1 < (b::bs).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
+          have : (a::as).get ⟨i+1, hi₁'⟩ = (b::bs).get ⟨i+1, hi₂'⟩ := h₂ (i+1) hi₁' hi₂'
+          apply this
+        have tailEq : as = bs := ih bs h₁' h₂'
+        rw [headEq, tailEq]
+  cases a; cases b
+  apply congrArg
+  apply extAux
+  assumption
+  assumption
+
+theorem ext' {as bs : Array α} (h : as.toList = bs.toList) : as = bs := by
+  cases as; cases bs; simp at h; rw [h]
+
+@[simp] theorem toArrayAux_eq (as : List α) (acc : Array α) : (as.toArrayAux acc).toList = acc.toList ++ as := by
+  induction as generalizing acc <;> simp [*, List.toArrayAux, Array.push, List.append_assoc, List.concat_eq_append]
+
+@[simp] theorem toList_toArray (as : List α) : as.toArray.toList = as := by
+  simp [List.toArray, Array.mkEmpty]
+
+@[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [size]
+
+@[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @toList_toArray
+
 @[deprecated Array.toList (since := "2024-09-10")] abbrev Array.data := @Array.toList
 
-@[extern "lean_mk_array"]
-def mkArray {α : Type u} (n : Nat) (v : α) : Array α where
-  toList := List.replicate n v
-
-/--
-`ofFn f` with `f : Fin n → α` returns the list whose ith element is `f i`.
-```
-ofFn f = #[f 0, f 1, ... , f(n - 1)]
-``` -/
-def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
-  /-- Auxiliary for `ofFn`. `ofFn.go f i acc = acc ++ #[f i, ..., f(n - 1)]` -/
-  go (i : Nat) (acc : Array α) : Array α :=
-    if h : i < n then go (i+1) (acc.push (f ⟨i, h⟩)) else acc
-termination_by n - i
-decreasing_by simp_wf; decreasing_trivial_pre_omega
-
-/-- The array `#[0, 1, ..., n - 1]`. -/
-def range (n : Nat) : Array Nat :=
-  n.fold (flip Array.push) (mkEmpty n)
-
-@[simp] theorem size_mkArray (n : Nat) (v : α) : (mkArray n v).size = n :=
-  List.length_replicate ..
-
-instance : EmptyCollection (Array α) := ⟨Array.empty⟩
-instance : Inhabited (Array α) where
-  default := Array.empty
-
-@[simp] def isEmpty (a : Array α) : Bool :=
-  a.size = 0
-
-def singleton (v : α) : Array α :=
-  mkArray 1 v
+/-! ### Externs -/
 
 /-- 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
@@ -65,29 +98,6 @@ def usize (a : @& Array α) : USize := a.size.toUSize
 def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
   a[i.toNat]
 
-instance : GetElem (Array α) USize α fun xs i => i.toNat < xs.size where
-  getElem xs i h := xs.uget i h
-
-def back [Inhabited α] (a : Array α) : α :=
-  a.get! (a.size - 1)
-
-def get? (a : Array α) (i : Nat) : Option α :=
-  if h : i < a.size then some a[i] else none
-
-def back? (a : Array α) : Option α :=
-  a.get? (a.size - 1)
-
--- auxiliary declaration used in the equation compiler when pattern matching array literals.
-abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
-  have := h₁.symm ▸ h₂
-  a[i]
-
-@[simp] theorem size_set (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
-  List.length_set ..
-
-@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
-  List.length_concat ..
-
 /-- Low-level version of `fset` which is as fast as a C array fset.
    `Fin` values are represented as tag pointers in the Lean runtime. Thus,
    `fset` may be slightly slower than `uset`. -/
@@ -95,6 +105,19 @@ abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size =
 def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
   a.set ⟨i.toNat, h⟩ v
 
+@[extern "lean_array_pop"]
+def pop (a : Array α) : Array α where
+  toList := a.toList.dropLast
+
+@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 := by
+  match a with
+  | ⟨[]⟩ => rfl
+  | ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
+
+@[extern "lean_mk_array"]
+def mkArray {α : Type u} (n : Nat) (v : α) : Array α where
+  toList := List.replicate n v
+
 /--
 Swaps two entries in an array.
 
@@ -108,6 +131,10 @@ def swap (a : Array α) (i j : @& Fin a.size) : Array α :=
   let a'  := a.set i v₂
   a'.set (size_set a i v₂ ▸ j) v₁
 
+@[simp] theorem size_swap (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size := by
+  show ((a.set i (a.get j)).set (size_set a i _ ▸ j) (a.get i)).size = a.size
+  rw [size_set, size_set]
+
 /--
 Swaps two entries in an array, or returns the array unchanged if either index is out of bounds.
 
@@ -121,6 +148,66 @@ def swap! (a : Array α) (i j : @& Nat) : Array α :=
   else a
   else a
 
+/-! ### GetElem instance for `USize`, backed by `uget` -/
+
+instance : GetElem (Array α) USize α fun xs i => i.toNat < xs.size where
+  getElem xs i h := xs.uget i h
+
+/-! ### Definitions -/
+
+instance : EmptyCollection (Array α) := ⟨Array.empty⟩
+instance : Inhabited (Array α) where
+  default := Array.empty
+
+@[simp] def isEmpty (a : Array α) : Bool :=
+  a.size = 0
+
+-- TODO(Leo): cleanup
+@[specialize]
+def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
+  if h : i < a.size then
+     have : i < b.size := hsz ▸ h
+     p a[i] b[i] && isEqvAux a b hsz p (i+1)
+  else
+    true
+decreasing_by simp_wf; decreasing_trivial_pre_omega
+
+@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
+  if h : a.size = b.size then
+    isEqvAux a b h p 0
+  else
+    false
+
+instance [BEq α] : BEq (Array α) :=
+  ⟨fun a b => isEqv a b BEq.beq⟩
+
+/--
+`ofFn f` with `f : Fin n → α` returns the list whose ith element is `f i`.
+```
+ofFn f = #[f 0, f 1, ... , f(n - 1)]
+``` -/
+def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
+  /-- Auxiliary for `ofFn`. `ofFn.go f i acc = acc ++ #[f i, ..., f(n - 1)]` -/
+  go (i : Nat) (acc : Array α) : Array α :=
+    if h : i < n then go (i+1) (acc.push (f ⟨i, h⟩)) else acc
+decreasing_by simp_wf; decreasing_trivial_pre_omega
+
+/-- The array `#[0, 1, ..., n - 1]`. -/
+def range (n : Nat) : Array Nat :=
+  n.fold (flip Array.push) (mkEmpty n)
+
+def singleton (v : α) : Array α :=
+  mkArray 1 v
+
+def back [Inhabited α] (a : Array α) : α :=
+  a.get! (a.size - 1)
+
+def get? (a : Array α) (i : Nat) : Option α :=
+  if h : i < a.size then some a[i] else none
+
+def back? (a : Array α) : Option α :=
+  a.get? (a.size - 1)
+
 @[inline] def swapAt (a : Array α) (i : Fin a.size) (v : α) : α × Array α :=
   let e := a.get i
   let a := a.set i v
@@ -134,10 +221,6 @@ def swapAt! (a : Array α) (i : Nat) (v : α) : α × Array α :=
     have : Inhabited α := ⟨v⟩
     panic! ("index " ++ toString i ++ " out of bounds")
 
-@[extern "lean_array_pop"]
-def pop (a : Array α) : Array α where
-  toList := a.toList.dropLast
-
 def shrink (a : Array α) (n : Nat) : Array α :=
   let rec loop
     | 0,   a => a
@@ -311,7 +394,6 @@ def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α
       map (i+1) (r.push (← f as[i]))
     else
       pure r
-  termination_by as.size - i
   decreasing_by simp_wf; decreasing_trivial_pre_omega
   map 0 (mkEmpty as.size)
 
@@ -384,7 +466,6 @@ def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as :
           loop (j+1)
       else
         pure false
-      termination_by stop - j
       decreasing_by simp_wf; decreasing_trivial_pre_omega
     loop start
   if h : stop ≤ as.size then
@@ -470,12 +551,22 @@ def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
     if h : j < as.size then
       if p as[j] then some j else loop (j + 1)
     else none
-    termination_by as.size - j
     decreasing_by simp_wf; decreasing_trivial_pre_omega
   loop 0
 
 def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
-a.findIdx? fun a => a == v
+  a.findIdx? fun a => a == v
+
+def indexOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
+  if h : i < a.size then
+    let idx : Fin a.size := ⟨i, h⟩;
+    if a.get idx == v then some idx
+    else indexOfAux a v (i+1)
+  else none
+decreasing_by simp_wf; decreasing_trivial_pre_omega
+
+def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
+  indexOfAux a v 0
 
 @[inline]
 def any (as : Array α) (p : α → Bool) (start := 0) (stop := as.size) : Bool :=
@@ -491,13 +582,6 @@ def contains [BEq α] (as : Array α) (a : α) : Bool :=
 def elem [BEq α] (a : α) (as : Array α) : Bool :=
   as.contains a
 
-@[inline] def getEvenElems (as : Array α) : Array α :=
-  (·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
-    if even then
-      (false, r.push a)
-    else
-      (true, r)
-
 /-- Convert a `Array α` into an `List α`. This is O(n) in the size of the array.  -/
 -- This function is exported to C, where it is called by `Array.toList`
 -- (the projection) to implement this functionality.
@@ -510,17 +594,6 @@ def toListImpl (as : Array α) : List α :=
 def toListAppend (as : Array α) (l : List α) : List α :=
   as.foldr List.cons l
 
-instance {α : Type u} [Repr α] : Repr (Array α) where
-  reprPrec a _ :=
-    let _ : Std.ToFormat α := ⟨repr⟩
-    if a.size == 0 then
-      "#[]"
-    else
-      Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
-
-instance [ToString α] : ToString (Array α) where
-  toString a := "#" ++ toString a.toList
-
 protected def append (as : Array α) (bs : Array α) : Array α :=
   bs.foldl (init := as) fun r v => r.push v
 
@@ -546,44 +619,13 @@ def concatMap (f : α → Array β) (as : Array α) : Array β :=
 def flatten (as : Array (Array α)) : Array α :=
   as.foldl (init := empty) fun r a => r ++ a
 
-end Array
-
-export Array (mkArray)
-
-syntax "#[" withoutPosition(sepBy(term, ", ")) "]" : term
-
-macro_rules
-  | `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
-
-namespace Array
-
--- TODO(Leo): cleanup
-@[specialize]
-def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
-  if h : i < a.size then
-     have : i < b.size := hsz ▸ h
-     p a[i] b[i] && isEqvAux a b hsz p (i+1)
-  else
-    true
-termination_by a.size - i
-decreasing_by simp_wf; decreasing_trivial_pre_omega
-
-@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
-  if h : a.size = b.size then
-    isEqvAux a b h p 0
-  else
-    false
-
-instance [BEq α] : BEq (Array α) :=
-  ⟨fun a b => isEqv a b BEq.beq⟩
-
 @[inline]
 def filter (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Array α :=
   as.foldl (init := #[]) (start := start) (stop := stop) fun r a =>
     if p a then r.push a else r
 
 @[inline]
-def filterM [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
+def filterM {α : Type} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
   as.foldlM (init := #[]) (start := start) (stop := stop) fun r a => do
     if (← p a) then return r.push a else return r
 
@@ -618,92 +660,23 @@ def partition (p : α → Bool) (as : Array α) : Array α × Array α := Id.run
       cs := cs.push a
   return (bs, cs)
 
-theorem ext (a b : Array α)
-    (h₁ : a.size = b.size)
-    (h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
-    : a = b := by
-  let rec extAux (a b : List α)
-      (h₁ : a.length = b.length)
-      (h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
-      : a = b := by
-    induction a generalizing b with
-    | nil =>
-      cases b with
-      | nil       => rfl
-      | cons b bs => rw [List.length_cons] at h₁; injection h₁
-    | cons a as ih =>
-      cases b with
-      | nil => rw [List.length_cons] at h₁; injection h₁
-      | cons b bs =>
-        have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
-        have hz₂ : 0 < (b::bs).length := by rw [List.length_cons]; apply Nat.zero_lt_succ
-        have headEq : a = b := h₂ 0 hz₁ hz₂
-        have h₁' : as.length = bs.length := by rw [List.length_cons, List.length_cons] at h₁; injection h₁
-        have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get ⟨i, hi₁⟩ = bs.get ⟨i, hi₂⟩ := by
-          intro i hi₁ hi₂
-          have hi₁' : i+1 < (a::as).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
-          have hi₂' : i+1 < (b::bs).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
-          have : (a::as).get ⟨i+1, hi₁'⟩ = (b::bs).get ⟨i+1, hi₂'⟩ := h₂ (i+1) hi₁' hi₂'
-          apply this
-        have tailEq : as = bs := ih bs h₁' h₂'
-        rw [headEq, tailEq]
-  cases a; cases b
-  apply congrArg
-  apply extAux
-  assumption
-  assumption
-
-theorem extLit {n : Nat}
-    (a b : Array α)
-    (hsz₁ : a.size = n) (hsz₂ : b.size = n)
-    (h : (i : Nat) → (hi : i < n) → a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
-  Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ _ => h i (hsz₁ ▸ hi₁)
-
-end Array
-
--- CLEANUP the following code
-namespace Array
-
-def indexOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
-  if h : i < a.size then
-    let idx : Fin a.size := ⟨i, h⟩;
-    if a.get idx == v then some idx
-    else indexOfAux a v (i+1)
-  else none
-termination_by a.size - i
-decreasing_by simp_wf; decreasing_trivial_pre_omega
-
-def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
-  indexOfAux a v 0
-
-@[simp] theorem size_swap (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size := by
-  show ((a.set i (a.get j)).set (size_set a i _ ▸ j) (a.get i)).size = a.size
-  rw [size_set, size_set]
-
-@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 := by
-  match a with
-  | ⟨[]⟩ => rfl
-  | ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]
-
-theorem reverse.termination {i j : Nat} (h : i < j) : j - 1 - (i + 1) < j - i := by
-  rw [Nat.sub_sub, Nat.add_comm]
-  exact Nat.lt_of_le_of_lt (Nat.pred_le _) (Nat.sub_succ_lt_self _ _ h)
-
 def reverse (as : Array α) : Array α :=
   if h : as.size ≤ 1 then
     as
   else
     loop as 0 ⟨as.size - 1, Nat.pred_lt (mt (fun h : as.size = 0 => h ▸ by decide) h)⟩
 where
+  termination {i j : Nat} (h : i < j) : j - 1 - (i + 1) < j - i := by
+    rw [Nat.sub_sub, Nat.add_comm]
+    exact Nat.lt_of_le_of_lt (Nat.pred_le _) (Nat.sub_succ_lt_self _ _ h)
   loop (as : Array α) (i : Nat) (j : Fin as.size) :=
     if h : i < j then
-      have := reverse.termination h
+      have := termination h
       let as := as.swap ⟨i, Nat.lt_trans h j.2⟩ j
       have : j-1 < as.size := by rw [size_swap]; exact Nat.lt_of_le_of_lt (Nat.pred_le _) j.2
       loop as (i+1) ⟨j-1, this⟩
     else
       as
-termination_by j - i
 
 def popWhile (p : α → Bool) (as : Array α) : Array α :=
   if h : as.size > 0 then
@@ -713,7 +686,6 @@ def popWhile (p : α → Bool) (as : Array α) : Array α :=
       as
   else
     as
-termination_by as.size
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def takeWhile (p : α → Bool) (as : Array α) : Array α :=
@@ -726,7 +698,6 @@ def takeWhile (p : α → Bool) (as : Array α) : Array α :=
         r
     else
       r
-    termination_by as.size - i
     decreasing_by simp_wf; decreasing_trivial_pre_omega
   go 0 #[]
 
@@ -744,6 +715,7 @@ def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
 termination_by a.size - i.val
 decreasing_by simp_wf; exact Nat.sub_succ_lt_self _ _ i.isLt
 
+-- This is required in `Lean.Data.PersistentHashMap`.
 theorem size_feraseIdx (a : Array α) (i : Fin a.size) : (a.feraseIdx i).size = a.size - 1 := by
   induction a, i using Array.feraseIdx.induct with
   | @case1 a i h a' _ ih =>
@@ -774,7 +746,6 @@ def erase [BEq α] (as : Array α) (a : α) : Array α :=
       loop as ⟨j', by rw [size_swap]; exact j'.2⟩
     else
       as
-    termination_by j.1
     decreasing_by simp_wf; decreasing_trivial_pre_omega
   let j := as.size
   let as := as.push a
@@ -786,41 +757,6 @@ def insertAt! (as : Array α) (i : Nat) (a : α) : Array α :=
     insertAt as ⟨i, Nat.lt_succ_of_le h⟩ a
   else panic! "invalid index"
 
-def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
-  | 0,     _,  acc => acc
-  | (i+1), hi, acc => toListLitAux a n hsz i (Nat.le_of_succ_le hi) (a.getLit i hsz (Nat.lt_of_lt_of_eq (Nat.lt_of_lt_of_le (Nat.lt_succ_self i) hi) hsz) :: acc)
-
-def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
-  List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.le_refl _) []
-
-theorem ext' {as bs : Array α} (h : as.toList = bs.toList) : as = bs := by
-  cases as; cases bs; simp at h; rw [h]
-
-@[simp] theorem toArrayAux_eq (as : List α) (acc : Array α) : (as.toArrayAux acc).toList = acc.toList ++ as := by
-  induction as generalizing acc <;> simp [*, List.toArrayAux, Array.push, List.append_assoc, List.concat_eq_append]
-
-@[simp] theorem toList_toArray (as : List α) : as.toArray.toList = as := by
-  simp [List.toArray, Array.mkEmpty]
-
-@[deprecated toList_toArray (since := "2024-09-09")] abbrev data_toArray := @toList_toArray
-
-@[simp] theorem size_toArray (as : List α) : as.toArray.size = as.length := by simp [size]
-
-theorem toArrayLit_eq (as : Array α) (n : Nat) (hsz : as.size = n) : as = toArrayLit as n hsz := by
-  apply ext'
-  simp [toArrayLit, toList_toArray]
-  have hle : n ≤ as.size := hsz ▸ Nat.le_refl _
-  have hge : as.size ≤ n := hsz ▸ Nat.le_refl _
-  have := go n hle
-  rw [List.drop_eq_nil_of_le hge] at this
-  rw [this]
-where
-  getLit_eq (as : Array α) (i : Nat) (h₁ : as.size = n) (h₂ : i < n) : as.getLit i h₁ h₂ = getElem as.toList i ((id (α := as.toList.length = n) h₁) ▸ h₂) :=
-    rfl
-
-  go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.toList.drop i) = as.toList := by
-    induction i <;> simp [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, *]
-
 def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) (i : Nat) : Bool :=
   if h : i < as.size then
     let a := as[i]
@@ -832,7 +768,6 @@ def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) (i : N
       false
   else
     true
-termination_by as.size - i
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 /-- Return true iff `as` is a prefix of `bs`.
@@ -843,23 +778,6 @@ def isPrefixOf [BEq α] (as bs : Array α) : Bool :=
   else
     false
 
-private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat), i < as.size → Bool
-  | 0,   _ => true
-  | i+1, h =>
-    have : i < as.size := Nat.lt_trans (Nat.lt_succ_self _) h;
-    a != as[i] && allDiffAuxAux as a i this
-
-private def allDiffAux [BEq α] (as : Array α) (i : Nat) : Bool :=
-  if h : i < as.size then
-    allDiffAuxAux as as[i] i h && allDiffAux as (i+1)
-  else
-    true
-termination_by as.size - i
-decreasing_by simp_wf; decreasing_trivial_pre_omega
-
-def allDiff [BEq α] (as : Array α) : Bool :=
-  allDiffAux as 0
-
 @[specialize] def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) : Array γ :=
   if h : i < as.size then
     let a := as[i]
@@ -870,7 +788,6 @@ def allDiff [BEq α] (as : Array α) : Bool :=
       cs
   else
     cs
-termination_by as.size - i
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 @[inline] def zipWith (as : Array α) (bs : Array β) (f : α → β → γ) : Array γ :=
@@ -886,4 +803,47 @@ def split (as : Array α) (p : α → Bool) : Array α × Array α :=
   as.foldl (init := (#[], #[])) fun (as, bs) a =>
     if p a then (as.push a, bs) else (as, bs.push a)
 
+/-! ### Auxiliary functions used in metaprogramming.
+
+We do not intend to provide verification theorems for these functions.
+-/
+
+private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat), i < as.size → Bool
+  | 0,   _ => true
+  | i+1, h =>
+    have : i < as.size := Nat.lt_trans (Nat.lt_succ_self _) h;
+    a != as[i] && allDiffAuxAux as a i this
+
+private def allDiffAux [BEq α] (as : Array α) (i : Nat) : Bool :=
+  if h : i < as.size then
+    allDiffAuxAux as as[i] i h && allDiffAux as (i+1)
+  else
+    true
+decreasing_by simp_wf; decreasing_trivial_pre_omega
+
+def allDiff [BEq α] (as : Array α) : Bool :=
+  allDiffAux as 0
+
+@[inline] def getEvenElems (as : Array α) : Array α :=
+  (·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
+    if even then
+      (false, r.push a)
+    else
+      (true, r)
+
+/-! ### Repr and ToString -/
+
+instance {α : Type u} [Repr α] : Repr (Array α) where
+  reprPrec a _ :=
+    let _ : Std.ToFormat α := ⟨repr⟩
+    if a.size == 0 then
+      "#[]"
+    else
+      Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
+
+instance [ToString α] : ToString (Array α) where
+  toString a := "#" ++ toString a.toList
+
 end Array
+
+export Array (mkArray)

src/Init/Data/Array/GetLit.lean

@@ -0,0 +1,46 @@
+/-
+Copyright (c) 2018 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+
+prelude
+import Init.Data.Array.Basic
+
+namespace Array
+
+/-! ### getLit -/
+
+-- auxiliary declaration used in the equation compiler when pattern matching array literals.
+abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
+  have := h₁.symm ▸ h₂
+  a[i]
+
+theorem extLit {n : Nat}
+    (a b : Array α)
+    (hsz₁ : a.size = n) (hsz₂ : b.size = n)
+    (h : (i : Nat) → (hi : i < n) → a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
+  Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ _ => h i (hsz₁ ▸ hi₁)
+
+def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
+  | 0,     _,  acc => acc
+  | (i+1), hi, acc => toListLitAux a n hsz i (Nat.le_of_succ_le hi) (a.getLit i hsz (Nat.lt_of_lt_of_eq (Nat.lt_of_lt_of_le (Nat.lt_succ_self i) hi) hsz) :: acc)
+
+def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
+  List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.le_refl _) []
+
+theorem toArrayLit_eq (as : Array α) (n : Nat) (hsz : as.size = n) : as = toArrayLit as n hsz := by
+  apply ext'
+  simp [toArrayLit, toList_toArray]
+  have hle : n ≤ as.size := hsz ▸ Nat.le_refl _
+  have hge : as.size ≤ n := hsz ▸ Nat.le_refl _
+  have := go n hle
+  rw [List.drop_eq_nil_of_le hge] at this
+  rw [this]
+where
+  getLit_eq (as : Array α) (i : Nat) (h₁ : as.size = n) (h₂ : i < n) : as.getLit i h₁ h₂ = getElem as.toList i ((id (α := as.toList.length = n) h₁) ▸ h₂) :=
+    rfl
+  go (i : Nat) (hi : i ≤ as.size) : toListLitAux as n hsz i hi (as.toList.drop i) = as.toList := by
+    induction i <;> simp [getLit_eq, List.get_drop_eq_drop, toListLitAux, List.drop, *]
+
+end Array

src/Init/Data/Array/Lemmas.lean

@@ -19,7 +19,7 @@ This file contains some theorems about `Array` and `List` needed for `Init.Data.
 
 namespace Array
 
-attribute [simp] data_toArray uset
+attribute [simp] uset
 
 @[simp] theorem singleton_def (v : α) : singleton v = #[v] := rfl
 
@@ -271,6 +271,9 @@ termination_by n - i
 
 /-- # mkArray -/
 
+@[simp] theorem size_mkArray (n : Nat) (v : α) : (mkArray n v).size = n :=
+  List.length_replicate ..
+
 @[simp] theorem toList_mkArray (n : Nat) (v : α) : (mkArray n v).toList = List.replicate n v := rfl
 
 @[deprecated toList_mkArray (since := "2024-09-09")]
@@ -495,7 +498,6 @@ abbrev size_eq_length_data := @size_eq_length_toList
   let rec go (as : Array α) (i j) : (reverse.loop as i j).size = as.size := by
     rw [reverse.loop]
     if h : i < j then
-      have := reverse.termination h
       simp [(go · (i+1) ⟨j-1, ·⟩), h]
     else simp [h]
     termination_by j - i
@@ -527,9 +529,8 @@ set_option linter.deprecated false in
       (H : ∀ k, as.toList.get? k = if i ≤ k ∧ k ≤ j then a.toList.get? k else a.toList.reverse.get? k)
       (k) : (reverse.loop as i ⟨j, hj⟩).toList.get? k = a.toList.reverse.get? k := by
     rw [reverse.loop]; dsimp; split <;> rename_i h₁
-    · have p := reverse.termination h₁
-      match j with | j+1 => ?_
-      simp only [Nat.add_sub_cancel] at p ⊢
+    · match j with | j+1 => ?_
+      simp only [Nat.add_sub_cancel]
       rw [(go · (i+1) j)]
       · rwa [Nat.add_right_comm i]
       · simp [size_swap, h₂]
@@ -828,7 +829,7 @@ theorem get_append_right {as bs : Array α} {h : i < (as ++ bs).size} (hle : as.
     (as ++ bs)[i] = bs[i - as.size] := by
   simp only [getElem_eq_toList_getElem]
   have h' : i < (as.toList ++ bs.toList).length := by rwa [← toList_length, append_toList] at h
-  conv => rhs; rw [← List.getElem_append_right (h' := h') (h := Nat.not_lt_of_ge hle)]
+  conv => rhs; rw [← List.getElem_append_right (h₁ := hle) (h₂ := h')]
   apply List.get_of_eq; rw [append_toList]
 
 @[simp] theorem append_nil (as : Array α) : as ++ #[] = as := by
@@ -1113,5 +1114,4 @@ theorem swap_comm (a : Array α) {i j : Fin a.size} : a.swap i j = a.swap j i :=
     · split <;> simp_all
     · split <;> simp_all
 
-
 end Array

src/Init/Data/BitVec/Basic.lean

@@ -64,7 +64,7 @@ protected def ofNatLt {n : Nat} (i : Nat) (p : i < 2^n) : BitVec n where
 /-- The `BitVec` with value `i mod 2^n`. -/
 @[match_pattern]
 protected def ofNat (n : Nat) (i : Nat) : BitVec n where
-  toFin := Fin.ofNat' i (Nat.two_pow_pos n)
+  toFin := Fin.ofNat' (2^n) i
 
 instance instOfNat : OfNat (BitVec n) i where ofNat := .ofNat n i
 instance natCastInst : NatCast (BitVec w) := ⟨BitVec.ofNat w⟩
@@ -173,6 +173,9 @@ instance : GetElem (BitVec w) Nat Bool fun _ i => i < w where
 theorem getElem_eq_testBit_toNat (x : BitVec w) (i : Nat) (h : i < w) :
   x[i] = x.toNat.testBit i := rfl
 
+theorem getLsbD_eq_getElem {x : BitVec w} {i : Nat} (h : i < w) :
+    x.getLsbD i = x[i] := rfl
+
 end getElem
 
 section Int
@@ -450,13 +453,15 @@ SMT-Lib name: `extract`.
 def extractLsb (hi lo : Nat) (x : BitVec n) : BitVec (hi - lo + 1) := extractLsb' lo _ x
 
 /--
-A version of `zeroExtend` that requires a proof, but is a noop.
+A version of `setWidth` that requires a proof, but is a noop.
 -/
-def zeroExtend' {n w : Nat} (le : n ≤ w) (x : BitVec n) : BitVec w :=
+def setWidth' {n w : Nat} (le : n ≤ w) (x : BitVec n) : BitVec w :=
   x.toNat#'(by
     apply Nat.lt_of_lt_of_le x.isLt
     exact Nat.pow_le_pow_of_le_right (by trivial) le)
 
+@[deprecated setWidth' (since := "2024-09-18"), inherit_doc setWidth'] abbrev zeroExtend' := @setWidth'
+
 /--
 `shiftLeftZeroExtend x n` returns `zeroExtend (w+n) x <<< n` without
 needing to compute `x % 2^(2+n)`.
@@ -469,22 +474,35 @@ def shiftLeftZeroExtend (msbs : BitVec w) (m : Nat) : BitVec (w + m) :=
   (msbs.toNat <<< m)#'(shiftLeftLt msbs.isLt m)
 
 /--
-Zero extend vector `x` of length `w` by adding zeros in the high bits until it has length `v`.
-If `v < w` then it truncates the high bits instead.
+Transform `x` of length `w` into a bitvector of length `v`, by either:
+- zero extending, that is, adding zeros in the high bits until it has length `v`, if `v > w`, or
+- truncating the high bits, if `v < w`.
 
 SMT-Lib name: `zero_extend`.
 -/
-def zeroExtend (v : Nat) (x : BitVec w) : BitVec v :=
+def setWidth (v : Nat) (x : BitVec w) : BitVec v :=
   if h : w ≤ v then
-    zeroExtend' h x
+    setWidth' h x
   else
     .ofNat v x.toNat
 
 /--
-Truncate the high bits of bitvector `x` of length `w`, resulting in a vector of length `v`.
-If `v > w` then it zero-extends the vector instead.
+Transform `x` of length `w` into a bitvector of length `v`, by either:
+- zero extending, that is, adding zeros in the high bits until it has length `v`, if `v > w`, or
+- truncating the high bits, if `v < w`.
+
+SMT-Lib name: `zero_extend`.
 -/
-abbrev truncate := @zeroExtend
+abbrev zeroExtend := @setWidth
+
+/--
+Transform `x` of length `w` into a bitvector of length `v`, by either:
+- zero extending, that is, adding zeros in the high bits until it has length `v`, if `v > w`, or
+- truncating the high bits, if `v < w`.
+
+SMT-Lib name: `zero_extend`.
+-/
+abbrev truncate := @setWidth
 
 /--
 Sign extend a vector of length `w`, extending with `i` additional copies of the most significant
@@ -635,7 +653,7 @@ input is on the left, so `0xAB#8 ++ 0xCD#8 = 0xABCD#16`.
 SMT-Lib name: `concat`.
 -/
 def append (msbs : BitVec n) (lsbs : BitVec m) : BitVec (n+m) :=
-  shiftLeftZeroExtend msbs m ||| zeroExtend' (Nat.le_add_left m n) lsbs
+  shiftLeftZeroExtend msbs m ||| setWidth' (Nat.le_add_left m n) lsbs
 
 instance : HAppend (BitVec w) (BitVec v) (BitVec (w + v)) := ⟨.append⟩
 

src/Init/Data/BitVec/Bitblast.lean

@@ -132,18 +132,18 @@ theorem toNat_add_of_and_eq_zero {x y : BitVec w} (h : x &&& y = 0#w) :
   simp [not_eq_true, carry_of_and_eq_zero h]
 
 /-- Carry function for bitwise addition. -/
-def adcb (x y c : Bool) : Bool × Bool := (atLeastTwo x y c, Bool.xor x (Bool.xor y c))
+def adcb (x y c : Bool) : Bool × Bool := (atLeastTwo x y c, x ^^ (y ^^ c))
 
 /-- Bitwise addition implemented via a ripple carry adder. -/
 def adc (x y : BitVec w) : Bool → Bool × BitVec w :=
   iunfoldr fun (i : Fin w) c => adcb (x.getLsbD i) (y.getLsbD i) c
 
 theorem getLsbD_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool) :
-    getLsbD (x + y + zeroExtend w (ofBool c)) i =
-      Bool.xor (getLsbD x i) (Bool.xor (getLsbD y i) (carry i x y c)) := by
+    getLsbD (x + y + setWidth w (ofBool c)) i =
+      (getLsbD x i ^^ (getLsbD y i ^^ carry i x y c)) := by
   let ⟨x, x_lt⟩ := x
   let ⟨y, y_lt⟩ := y
-  simp only [getLsbD, toNat_add, toNat_zeroExtend, i_lt, toNat_ofFin, toNat_ofBool,
+  simp only [getLsbD, toNat_add, toNat_setWidth, i_lt, toNat_ofFin, toNat_ofBool,
     Nat.mod_add_mod, Nat.add_mod_mod]
   apply Eq.trans
   rw [← Nat.div_add_mod x (2^i), ← Nat.div_add_mod y (2^i)]
@@ -161,15 +161,15 @@ theorem getLsbD_add_add_bool {i : Nat} (i_lt : i < w) (x y : BitVec w) (c : Bool
 
 theorem getLsbD_add {i : Nat} (i_lt : i < w) (x y : BitVec w) :
     getLsbD (x + y) i =
-      Bool.xor (getLsbD x i) (Bool.xor (getLsbD y i) (carry i x y false)) := by
+      (getLsbD x i ^^ (getLsbD y i ^^ carry i x y false)) := by
   simpa using getLsbD_add_add_bool i_lt x y false
 
 theorem adc_spec (x y : BitVec w) (c : Bool) :
-    adc x y c = (carry w x y c, x + y + zeroExtend w (ofBool c)) := by
+    adc x y c = (carry w x y c, x + y + setWidth w (ofBool c)) := by
   simp only [adc]
   apply iunfoldr_replace
           (fun i => carry i x y c)
-          (x + y + zeroExtend w (ofBool c))
+          (x + y + setWidth w (ofBool c))
           c
   case init =>
     simp [carry, Nat.mod_one]
@@ -306,12 +306,12 @@ theorem mulRec_succ_eq (x y : BitVec w) (s : Nat) :
 Recurrence lemma: truncating to `i+1` bits and then zero extending to `w`
 equals truncating upto `i` bits `[0..i-1]`, and then adding the `i`th bit of `x`.
 -/
-theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow (x : BitVec w) (i : Nat) :
-    zeroExtend w (x.truncate (i + 1)) =
-      zeroExtend w (x.truncate i) + (x &&& twoPow w i) := by
+theorem setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (x : BitVec w) (i : Nat) :
+    setWidth w (x.setWidth (i + 1)) =
+      setWidth w (x.setWidth i) + (x &&& twoPow w i) := by
   rw [add_eq_or_of_and_eq_zero]
   · ext k
-    simp only [getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
+    simp only [getLsbD_setWidth, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
     by_cases hik : i = k
     · subst hik
       simp
@@ -322,27 +322,32 @@ theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow (x : BitVec w
       · have hik'' : ¬ (k < i) := by omega
         simp [hik', hik'']
   · ext k
-    simp only [and_twoPow, getLsbD_and, getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and,
+    simp only [and_twoPow, getLsbD_and, getLsbD_setWidth, Fin.is_lt, decide_True, Bool.true_and,
       getLsbD_zero, and_eq_false_imp, and_eq_true, decide_eq_true_eq, and_imp]
     by_cases hi : x.getLsbD i <;> simp [hi] <;> omega
 
+@[deprecated setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow (since := "2024-09-18"),
+  inherit_doc setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow]
+abbrev zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow :=
+  @setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow
+
 /--
 Recurrence lemma: multiplying `x` with the first `s` bits of `y` is the
 same as truncating `y` to `s` bits, then zero extending to the original length,
 and performing the multplication. -/
-theorem mulRec_eq_mul_signExtend_truncate (x y : BitVec w) (s : Nat) :
-    mulRec x y s = x * ((y.truncate (s + 1)).zeroExtend w) := by
+theorem mulRec_eq_mul_signExtend_setWidth (x y : BitVec w) (s : Nat) :
+    mulRec x y s = x * ((y.setWidth (s + 1)).setWidth w) := by
   induction s
   case zero =>
     simp only [mulRec_zero_eq, ofNat_eq_ofNat, Nat.reduceAdd]
     by_cases y.getLsbD 0
     case pos hy =>
-      simp only [hy, ↓reduceIte, truncate, zeroExtend_one_eq_ofBool_getLsb_zero,
+      simp only [hy, ↓reduceIte, setWidth_one_eq_ofBool_getLsb_zero,
         ofBool_true, ofNat_eq_ofNat]
-      rw [zeroExtend_ofNat_one_eq_ofNat_one_of_lt (by omega)]
+      rw [setWidth_ofNat_one_eq_ofNat_one_of_lt (by omega)]
       simp
     case neg hy =>
-      simp [hy, zeroExtend_one_eq_ofBool_getLsb_zero]
+      simp [hy, setWidth_one_eq_ofBool_getLsb_zero]
   case succ s' hs =>
     rw [mulRec_succ_eq, hs]
     have heq :
@@ -350,13 +355,16 @@ theorem mulRec_eq_mul_signExtend_truncate (x y : BitVec w) (s : Nat) :
         (x * (y &&& (BitVec.twoPow w (s' + 1)))) := by
       simp only [ofNat_eq_ofNat, and_twoPow]
       by_cases hy : y.getLsbD (s' + 1) <;> simp [hy]
-    rw [heq, ← BitVec.mul_add, ← zeroExtend_truncate_succ_eq_zeroExtend_truncate_add_twoPow]
+    rw [heq, ← BitVec.mul_add, ← setWidth_setWidth_succ_eq_setWidth_setWidth_add_twoPow]
+
+@[deprecated mulRec_eq_mul_signExtend_setWidth (since := "2024-09-18"),
+  inherit_doc mulRec_eq_mul_signExtend_setWidth]
+abbrev mulRec_eq_mul_signExtend_truncate := @mulRec_eq_mul_signExtend_setWidth
 
 theorem getLsbD_mul (x y : BitVec w) (i : Nat) :
     (x * y).getLsbD i = (mulRec x y w).getLsbD i := by
-  simp only [mulRec_eq_mul_signExtend_truncate]
-  rw [truncate, ← truncate_eq_zeroExtend, ← truncate_eq_zeroExtend,
-    truncate_truncate_of_le]
+  simp only [mulRec_eq_mul_signExtend_setWidth]
+  rw [setWidth_setWidth_of_le]
   · simp
   · omega
 
@@ -402,22 +410,22 @@ theorem shiftLeft_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
 `shiftLeftRec x y n` shifts `x` to the left by the first `n` bits of `y`.
 -/
 theorem shiftLeftRec_eq {x : BitVec w₁} {y : BitVec w₂} {n : Nat} :
-    shiftLeftRec x y n = x <<< (y.truncate (n + 1)).zeroExtend w₂ := by
+    shiftLeftRec x y n = x <<< (y.setWidth (n + 1)).setWidth w₂ := by
   induction n generalizing x y
   case zero =>
     ext i
-    simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, truncate_one,
-      and_one_eq_zeroExtend_ofBool_getLsbD]
+    simp only [shiftLeftRec_zero, twoPow_zero, Nat.reduceAdd, setWidth_one,
+      and_one_eq_setWidth_ofBool_getLsbD]
   case succ n ih =>
     simp only [shiftLeftRec_succ, and_twoPow]
     rw [ih]
     by_cases h : y.getLsbD (n + 1)
     · simp only [h, ↓reduceIte]
-      rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true h,
+      rw [setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true h,
         shiftLeft_or_of_and_eq_zero]
       simp [and_twoPow]
     · simp only [h, false_eq_true, ↓reduceIte, shiftLeft_zero']
-      rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false (i := n + 1)]
+      rw [setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false (i := n + 1)]
       simp [h]
 
 /--
@@ -466,18 +474,18 @@ theorem sshiftRight'_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
     toNat_add_of_and_eq_zero h, sshiftRight_add]
 
 theorem sshiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
-    sshiftRightRec x y n = x.sshiftRight' ((y.truncate (n + 1)).zeroExtend w₂) := by
+    sshiftRightRec x y n = x.sshiftRight' ((y.setWidth (n + 1)).setWidth w₂) := by
   induction n generalizing x y
   case zero =>
     ext i
-    simp [twoPow_zero, Nat.reduceAdd, and_one_eq_zeroExtend_ofBool_getLsbD, truncate_one]
+    simp [twoPow_zero, Nat.reduceAdd, and_one_eq_setWidth_ofBool_getLsbD, setWidth_one]
   case succ n ih =>
     simp only [sshiftRightRec_succ_eq, and_twoPow, ih]
     by_cases h : y.getLsbD (n + 1)
-    · rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true h,
+    · rw [setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true h,
         sshiftRight'_or_of_and_eq_zero (by simp [and_twoPow]), h]
       simp
-    · rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false (i := n + 1)
+    · rw [setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false (i := n + 1)
         (by simp [h])]
       simp [h]
 
@@ -529,20 +537,20 @@ theorem ushiftRight'_or_of_and_eq_zero {x : BitVec w₁} {y z : BitVec w₂}
   simp [← add_eq_or_of_and_eq_zero _ _ h, toNat_add_of_and_eq_zero h, shiftRight_add]
 
 theorem ushiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
-    ushiftRightRec x y n = x >>> (y.truncate (n + 1)).zeroExtend w₂ := by
+    ushiftRightRec x y n = x >>> (y.setWidth (n + 1)).setWidth w₂ := by
   induction n generalizing x y
   case zero =>
     ext i
     simp only [ushiftRightRec_zero, twoPow_zero, Nat.reduceAdd,
-      and_one_eq_zeroExtend_ofBool_getLsbD, truncate_one]
+      and_one_eq_setWidth_ofBool_getLsbD, setWidth_one]
   case succ n ih =>
     simp only [ushiftRightRec_succ, and_twoPow]
     rw [ih]
     by_cases h : y.getLsbD (n + 1) <;> simp only [h, ↓reduceIte]
-    · rw [zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true h,
+    · rw [setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true h,
         ushiftRight'_or_of_and_eq_zero]
       simp [and_twoPow]
-    · simp [zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false, h]
+    · simp [setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false, h]
 
 /--
 Show that `x >>> y` can be written in terms of `ushiftRightRec`.

src/Init/Data/BitVec/Folds.lean

@@ -48,7 +48,7 @@ private theorem iunfoldr.eq_test
     simp only [init, eq_nil]
   case step =>
     intro i
-    simp_all [truncate_succ]
+    simp_all [setWidth_succ]
 
 theorem iunfoldr_getLsbD' {f : Fin w → α → α × Bool} (state : Nat → α)
     (ind : ∀(i : Fin w), (f i (state i.val)).fst = state (i.val+1)) :

src/Init/Data/BitVec/Lemmas.lean

@@ -31,13 +31,13 @@ namespace BitVec
   simp only [Bool.and_eq_false_imp, decide_eq_true_eq]
   omega
 
-theorem lt_of_getLsbD (x : BitVec w) (i : Nat) : getLsbD x i = true → i < w := by
+theorem lt_of_getLsbD {x : BitVec w} {i : Nat} : getLsbD x i = true → i < w := by
   if h : i < w then
     simp [h]
   else
     simp [Nat.ge_of_not_lt h]
 
-theorem lt_of_getMsbD (x : BitVec w) (i : Nat) : getMsbD x i = true → i < w := by
+theorem lt_of_getMsbD {x : BitVec w} {i : Nat} : getMsbD x i = true → i < w := by
   if h : i < w then
     simp [h]
   else
@@ -234,6 +234,15 @@ theorem ofBool_eq_iff_eq : ∀ {b b' : Bool}, BitVec.ofBool b = BitVec.ofBool b'
 
 @[simp] theorem not_ofBool : ~~~ (ofBool b) = ofBool (!b) := by cases b <;> rfl
 
+@[simp] theorem ofBool_and_ofBool : ofBool b &&& ofBool b' = ofBool (b && b') := by
+  cases b <;> cases b' <;> rfl
+
+@[simp] theorem ofBool_or_ofBool : ofBool b ||| ofBool b' = ofBool (b || b') := by
+  cases b <;> cases b' <;> rfl
+
+@[simp] theorem ofBool_xor_ofBool : ofBool b ^^^ ofBool b' = ofBool (b ^^ b') := by
+  cases b <;> cases b' <;> rfl
+
 @[simp, bv_toNat] theorem toNat_ofFin (x : Fin (2^n)) : (BitVec.ofFin x).toNat = x.val := rfl
 
 @[simp] theorem toNat_ofNatLt (x : Nat) (p : x < 2^w) : (x#'p).toNat = x := rfl
@@ -245,7 +254,7 @@ theorem ofBool_eq_iff_eq : ∀ {b b' : Bool}, BitVec.ofBool b = BitVec.ofBool b'
 @[simp, bv_toNat] theorem toNat_ofNat (x w : Nat) : (BitVec.ofNat w x).toNat = x % 2^w := by
   simp [BitVec.toNat, BitVec.ofNat, Fin.ofNat']
 
-@[simp] theorem toFin_ofNat (x : Nat) : toFin (BitVec.ofNat w x) = Fin.ofNat' x (Nat.two_pow_pos w) := rfl
+@[simp] theorem toFin_ofNat (x : Nat) : toFin (BitVec.ofNat w x) = Fin.ofNat' (2^w) x := rfl
 
 -- Remark: we don't use `[simp]` here because simproc` subsumes it for literals.
 -- If `x` and `n` are not literals, applying this theorem eagerly may not be a good idea.
@@ -273,8 +282,31 @@ theorem getLsbD_ofNat (n : Nat) (x : Nat) (i : Nat) :
 private theorem lt_two_pow_of_le {x m n : Nat} (lt : x < 2 ^ m) (le : m ≤ n) : x < 2 ^ n :=
   Nat.lt_of_lt_of_le lt (Nat.pow_le_pow_of_le_right (by trivial : 0 < 2) le)
 
+@[simp] theorem getElem_zero_ofNat_zero (i : Nat) (h : i < w) : (BitVec.ofNat w 0)[i] = false := by
+  simp [getElem_eq_testBit_toNat]
+
+@[simp] theorem getElem_zero_ofNat_one (h : 0 < w) : (BitVec.ofNat w 1)[0] = true := by
+  simp [getElem_eq_testBit_toNat, h]
+
+theorem getElem?_zero_ofNat_zero : (BitVec.ofNat (w+1) 0)[0]? = some false := by
+  simp
+
+theorem getElem?_zero_ofNat_one : (BitVec.ofNat (w+1) 1)[0]? = some true := by
+  simp
+
+-- This does not need to be a `@[simp]` theorem as it is already handled by `getElem?_eq_getElem`.
+theorem getElem?_zero_ofBool (b : Bool) : (ofBool b)[0]? = some b := by
+  simp [ofBool, cond_eq_if]
+  split <;> simp_all
+
+@[simp] theorem getElem_zero_ofBool (b : Bool) : (ofBool b)[0] = b := by
+  rw [getElem_eq_iff, getElem?_zero_ofBool]
+
+theorem getElem?_succ_ofBool (b : Bool) (i : Nat) : (ofBool b)[i + 1]? = none := by
+  simp
+
 @[simp]
-theorem getLsbD_ofBool (b : Bool) (i : Nat) : (BitVec.ofBool b).getLsbD i = ((i = 0) && b) := by
+theorem getLsbD_ofBool (b : Bool) (i : Nat) : (ofBool b).getLsbD i = ((i = 0) && b) := by
   rcases b with rfl | rfl
   · simp [ofBool]
   · simp only [ofBool, ofNat_eq_ofNat, cond_true, getLsbD_ofNat, Bool.and_true]
@@ -330,6 +362,10 @@ theorem toNat_ge_of_msb_true {x : BitVec n} (p : BitVec.msb x = true) : x.toNat
 
 @[simp] theorem getMsbD_cast (h : w = v) (x : BitVec w) : (cast h x).getMsbD i = x.getMsbD i := by
   subst h; simp
+
+@[simp] theorem getElem_cast (h : w = v) (x : BitVec w) (p : i < v) : (cast h x)[i] = x[i] := by
+  subst h; simp
+
 @[simp] theorem msb_cast (h : w = v) (x : BitVec w) : (cast h x).msb = x.msb := by
   simp [BitVec.msb]
 
@@ -412,46 +448,46 @@ theorem toInt_pos_iff {w : Nat} {x : BitVec w} :
     0 ≤ BitVec.toInt x ↔ 2 * x.toNat < 2 ^ w := by
   simp [toInt_eq_toNat_cond]; omega
 
-/-! ### zeroExtend and truncate -/
+/-! ### setWidth, zeroExtend and truncate -/
 
-theorem truncate_eq_zeroExtend {v : Nat} {x : BitVec w} :
-  truncate v x = zeroExtend v x := rfl
+@[simp]
+theorem truncate_eq_setWidth {v : Nat} {x : BitVec w} :
+  truncate v x = setWidth v x := rfl
 
-@[simp, bv_toNat] theorem toNat_zeroExtend' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
-    (zeroExtend' p x).toNat = x.toNat := by
-  simp [zeroExtend']
+@[simp]
+theorem zeroExtend_eq_setWidth {v : Nat} {x : BitVec w} :
+  zeroExtend v x = setWidth v x := rfl
 
-@[bv_toNat] theorem toNat_zeroExtend (i : Nat) (x : BitVec n) :
-    BitVec.toNat (zeroExtend i x) = x.toNat % 2^i := by
+@[simp, bv_toNat] theorem toNat_setWidth' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
+    (setWidth' p x).toNat = x.toNat := by
+  simp [setWidth']
+
+@[simp, bv_toNat] theorem toNat_setWidth (i : Nat) (x : BitVec n) :
+    BitVec.toNat (setWidth i x) = x.toNat % 2^i := by
   let ⟨x, lt_n⟩ := x
-  simp only [zeroExtend]
+  simp only [setWidth]
   if n_le_i : n ≤ i then
     have x_lt_two_i : x < 2 ^ i := lt_two_pow_of_le lt_n n_le_i
     simp [n_le_i, Nat.mod_eq_of_lt, x_lt_two_i]
   else
     simp [n_le_i, toNat_ofNat]
 
-theorem zeroExtend'_eq {x : BitVec w} (h : w ≤ v) : x.zeroExtend' h = x.zeroExtend v := by
+theorem setWidth'_eq {x : BitVec w} (h : w ≤ v) : x.setWidth' h = x.setWidth v := by
   apply eq_of_toNat_eq
-  rw [toNat_zeroExtend, toNat_zeroExtend']
+  rw [toNat_setWidth, toNat_setWidth']
   rw [Nat.mod_eq_of_lt]
   exact Nat.lt_of_lt_of_le x.isLt (Nat.pow_le_pow_right (Nat.zero_lt_two) h)
 
-@[simp, bv_toNat] theorem toNat_truncate (x : BitVec n) : (truncate i x).toNat = x.toNat % 2^i :=
-  toNat_zeroExtend i x
-
-@[simp] theorem zeroExtend_eq (x : BitVec n) : zeroExtend n x = x := by
+@[simp] theorem setWidth_eq (x : BitVec n) : setWidth n x = x := by
   apply eq_of_toNat_eq
   let ⟨x, lt_n⟩ := x
-  simp [truncate, zeroExtend]
+  simp [setWidth]
 
-@[simp] theorem zeroExtend_zero (m n : Nat) : zeroExtend m 0#n = 0#m := by
+@[simp] theorem setWidth_zero (m n : Nat) : setWidth m 0#n = 0#m := by
   apply eq_of_toNat_eq
-  simp [toNat_zeroExtend]
+  simp [toNat_setWidth]
 
-theorem truncate_eq (x : BitVec n) : truncate n x = x := zeroExtend_eq x
-
-@[simp] theorem ofNat_toNat (m : Nat) (x : BitVec n) : BitVec.ofNat m x.toNat = truncate m x := by
+@[simp] theorem ofNat_toNat (m : Nat) (x : BitVec n) : BitVec.ofNat m x.toNat = setWidth m x := by
   apply eq_of_toNat_eq
   simp
 
@@ -470,33 +506,33 @@ theorem nat_eq_toNat {x : BitVec w} {y : Nat}
   rw [@eq_comm _ _ x.toNat]
   apply toNat_eq_nat
 
-theorem getElem_zeroExtend' (x : BitVec w) (i : Nat) (h : w ≤ v) (hi : i < v) :
-    (zeroExtend' h x)[i] = x.getLsbD i := by
-  rw [getElem_eq_testBit_toNat, toNat_zeroExtend', getLsbD]
+theorem getElem_setWidth' (x : BitVec w) (i : Nat) (h : w ≤ v) (hi : i < v) :
+    (setWidth' h x)[i] = x.getLsbD i := by
+  rw [getElem_eq_testBit_toNat, toNat_setWidth', getLsbD]
 
-theorem getElem_zeroExtend (m : Nat) (x : BitVec n) (i : Nat) (h : i < m) :
-    (zeroExtend m x)[i] = x.getLsbD i := by
-  rw [zeroExtend]
+theorem getElem_setWidth (m : Nat) (x : BitVec n) (i : Nat) (h : i < m) :
+    (setWidth m x)[i] = x.getLsbD i := by
+  rw [setWidth]
   split
-  · rw [getElem_zeroExtend']
+  · rw [getElem_setWidth']
   · simp [getElem_eq_testBit_toNat, getLsbD]
     omega
 
-theorem getElem?_zeroExtend' (x : BitVec w) (i : Nat) (h : w ≤ v) :
-    (zeroExtend' h x)[i]? = if i < v then some (x.getLsbD i) else none := by
-  simp [getElem?_eq, getElem_zeroExtend']
+theorem getElem?_setWidth' (x : BitVec w) (i : Nat) (h : w ≤ v) :
+    (setWidth' h x)[i]? = if i < v then some (x.getLsbD i) else none := by
+  simp [getElem?_eq, getElem_setWidth']
 
-theorem getElem?_zeroExtend (m : Nat) (x : BitVec n) (i : Nat) :
-    (x.zeroExtend m)[i]? = if i < m then some (x.getLsbD i) else none := by
-  simp [getElem?_eq, getElem_zeroExtend]
+theorem getElem?_setWidth (m : Nat) (x : BitVec n) (i : Nat) :
+    (x.setWidth m)[i]? = if i < m then some (x.getLsbD i) else none := by
+  simp [getElem?_eq, getElem_setWidth]
 
-@[simp] theorem getLsbD_zeroExtend' (ge : m ≥ n) (x : BitVec n) (i : Nat) :
-    getLsbD (zeroExtend' ge x) i = getLsbD x i := by
-  simp [getLsbD, toNat_zeroExtend']
+@[simp] theorem getLsbD_setWidth' (ge : m ≥ n) (x : BitVec n) (i : Nat) :
+    getLsbD (setWidth' ge x) i = getLsbD x i := by
+  simp [getLsbD, toNat_setWidth']
 
-@[simp] theorem getMsbD_zeroExtend' (ge : m ≥ n) (x : BitVec n) (i : Nat) :
-    getMsbD (zeroExtend' ge x) i = (decide (i ≥ m - n) && getMsbD x (i - (m - n))) := by
-  simp only [getMsbD, getLsbD_zeroExtend', gt_iff_lt]
+@[simp] theorem getMsbD_setWidth' (ge : m ≥ n) (x : BitVec n) (i : Nat) :
+    getMsbD (setWidth' ge x) i = (decide (i ≥ m - n) && getMsbD x (i - (m - n))) := by
+  simp only [getMsbD, getLsbD_setWidth', gt_iff_lt]
   by_cases h₁ : decide (i < m) <;> by_cases h₂ : decide (i ≥ m - n) <;> by_cases h₃ : decide (i - (m - n) < n) <;>
     by_cases h₄ : n - 1 - (i - (m - n)) = m - 1 - i
   all_goals
@@ -507,15 +543,15 @@ theorem getElem?_zeroExtend (m : Nat) (x : BitVec n) (i : Nat) :
     (try apply (getLsbD_ge _ _ _).symm) <;>
     omega
 
-@[simp] theorem getLsbD_zeroExtend (m : Nat) (x : BitVec n) (i : Nat) :
-    getLsbD (zeroExtend m x) i = (decide (i < m) && getLsbD x i) := by
-  simp [getLsbD, toNat_zeroExtend, Nat.testBit_mod_two_pow]
+@[simp] theorem getLsbD_setWidth (m : Nat) (x : BitVec n) (i : Nat) :
+    getLsbD (setWidth m x) i = (decide (i < m) && getLsbD x i) := by
+  simp [getLsbD, toNat_setWidth, Nat.testBit_mod_two_pow]
 
-@[simp] theorem getMsbD_zeroExtend_add {x : BitVec w} (h : k ≤ i) :
-    (x.zeroExtend (w + k)).getMsbD i = x.getMsbD (i - k) := by
+@[simp] theorem getMsbD_setWidth_add {x : BitVec w} (h : k ≤ i) :
+    (x.setWidth (w + k)).getMsbD i = x.getMsbD (i - k) := by
   by_cases h : w = 0
   · subst h; simp [of_length_zero]
-  simp only [getMsbD, getLsbD_zeroExtend]
+  simp only [getMsbD, getLsbD_setWidth]
   by_cases h₁ : i < w + k <;> by_cases h₂ : i - k < w <;> by_cases h₃ : w + k - 1 - i < w + k
     <;> simp [h₁, h₂, h₃]
   · congr 1
@@ -523,75 +559,63 @@ theorem getElem?_zeroExtend (m : Nat) (x : BitVec n) (i : Nat) :
   all_goals (first | apply getLsbD_ge | apply Eq.symm; apply getLsbD_ge)
     <;> omega
 
-theorem getLsbD_truncate (m : Nat) (x : BitVec n) (i : Nat) :
-    getLsbD (truncate m x) i = (decide (i < m) && getLsbD x i) :=
-  getLsbD_zeroExtend m x i
-
-theorem msb_truncate (x : BitVec w) : (x.truncate (k + 1)).msb = x.getLsbD k := by
-  simp [BitVec.msb, getMsbD]
-
-@[simp] theorem cast_zeroExtend (h : v = v') (x : BitVec w) :
-    cast h (zeroExtend v x) = zeroExtend v' x := by
+@[simp] theorem cast_setWidth (h : v = v') (x : BitVec w) :
+    cast h (setWidth v x) = setWidth v' x := by
   subst h
   ext
   simp
 
-@[simp] theorem cast_truncate (h : v = v') (x : BitVec w) :
-    cast h (truncate v x) = truncate v' x := by
-  subst h
-  ext
-  simp
-
-@[simp] theorem zeroExtend_zeroExtend_of_le (x : BitVec w) (h : k ≤ l) :
-    (x.zeroExtend l).zeroExtend k = x.zeroExtend k := by
+@[simp] theorem setWidth_setWidth_of_le (x : BitVec w) (h : k ≤ l) :
+    (x.setWidth l).setWidth k = x.setWidth k := by
   ext i
-  simp only [getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and]
-  have p := lt_of_getLsbD x i
+  simp only [getLsbD_setWidth, Fin.is_lt, decide_True, Bool.true_and]
+  have p := lt_of_getLsbD (x := x) (i := i)
   revert p
   cases getLsbD x i <;> simp; omega
 
-@[simp] theorem truncate_truncate_of_le (x : BitVec w) (h : k ≤ l) :
-    (x.truncate l).truncate k = x.truncate k :=
-  zeroExtend_zeroExtend_of_le x h
-
-/-- Truncating by the bitwidth has no effect. -/
--- This doesn't need to be a `@[simp]` lemma, as `zeroExtend_eq` applies.
-theorem truncate_eq_self {x : BitVec w} : x.truncate w = x := zeroExtend_eq _
-
-@[simp] theorem truncate_cast {h : w = v} : (cast h x).truncate k = x.truncate k := by
+@[simp] theorem setWidth_cast {h : w = v} : (cast h x).setWidth k = x.setWidth k := by
   apply eq_of_getLsbD_eq
   simp
 
-theorem msb_zeroExtend (x : BitVec w) : (x.zeroExtend v).msb = (decide (0 < v) && x.getLsbD (v - 1)) := by
+theorem msb_setWidth (x : BitVec w) : (x.setWidth v).msb = (decide (0 < v) && x.getLsbD (v - 1)) := by
   rw [msb_eq_getLsbD_last]
-  simp only [getLsbD_zeroExtend]
+  simp only [getLsbD_setWidth]
   cases getLsbD x (v - 1) <;> simp; omega
 
-theorem msb_zeroExtend' (x : BitVec w) (h : w ≤ v) : (x.zeroExtend' h).msb = (decide (0 < v) && x.getLsbD (v - 1)) := by
-  rw [zeroExtend'_eq, msb_zeroExtend]
+theorem msb_setWidth' (x : BitVec w) (h : w ≤ v) : (x.setWidth' h).msb = (decide (0 < v) && x.getLsbD (v - 1)) := by
+  rw [setWidth'_eq, msb_setWidth]
+
+theorem msb_setWidth'' (x : BitVec w) : (x.setWidth (k + 1)).msb = x.getLsbD k := by
+  simp [BitVec.msb, getMsbD]
 
 /-- zero extending a bitvector to width 1 equals the boolean of the lsb. -/
-theorem zeroExtend_one_eq_ofBool_getLsb_zero (x : BitVec w) :
-    x.zeroExtend 1 = BitVec.ofBool (x.getLsbD 0) := by
+theorem setWidth_one_eq_ofBool_getLsb_zero (x : BitVec w) :
+    x.setWidth 1 = BitVec.ofBool (x.getLsbD 0) := by
   ext i
-  simp [getLsbD_zeroExtend, Fin.fin_one_eq_zero i]
+  simp [getLsbD_setWidth, Fin.fin_one_eq_zero i]
 
 /-- Zero extending `1#v` to `1#w` equals `1#w` when `v > 0`. -/
-theorem zeroExtend_ofNat_one_eq_ofNat_one_of_lt {v w : Nat} (hv : 0 < v) :
-    (BitVec.ofNat v 1).zeroExtend w = BitVec.ofNat w 1 := by
+theorem setWidth_ofNat_one_eq_ofNat_one_of_lt {v w : Nat} (hv : 0 < v) :
+    (BitVec.ofNat v 1).setWidth w = BitVec.ofNat w 1 := by
   ext ⟨i, hilt⟩
-  simp only [getLsbD_zeroExtend, hilt, decide_True, getLsbD_ofNat, Bool.true_and,
+  simp only [getLsbD_setWidth, hilt, decide_True, getLsbD_ofNat, Bool.true_and,
     Bool.and_iff_right_iff_imp, decide_eq_true_eq]
   intros hi₁
   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.getLsbD 0) := by
+theorem setWidth_one {x : BitVec w} :
+    x.setWidth 1 = ofBool (x.getLsbD 0) := by
   ext i
   simp [show i = 0 by omega]
 
+@[simp] theorem setWidth_ofNat_of_le (h : v ≤ w) (x : Nat) : setWidth v (BitVec.ofNat w x) = BitVec.ofNat v x := by
+  apply BitVec.eq_of_toNat_eq
+  simp only [toNat_setWidth, toNat_ofNat]
+  rw [Nat.mod_mod_of_dvd]
+  exact Nat.pow_dvd_pow_iff_le_right'.mpr h
+
 /-! ## extractLsb -/
 
 @[simp]
@@ -633,6 +657,9 @@ theorem extractLsb'_eq_extractLsb {w : Nat} (x : BitVec w) (start len : Nat) (h
 @[simp] theorem getLsbD_allOnes : (allOnes v).getLsbD i = decide (i < v) := by
   simp [allOnes]
 
+@[simp] theorem getElem_allOnes (i : Nat) (h : i < v) : (allOnes v)[i] = true := by
+  simp [getElem_eq_testBit_toNat, h]
+
 @[simp] theorem ofFin_add_rev (x : Fin (2^n)) : ofFin (x + x.rev) = allOnes n := by
   ext
   simp only [Fin.rev, getLsbD_ofFin, getLsbD_allOnes, Fin.is_lt, decide_True]
@@ -660,11 +687,14 @@ theorem extractLsb'_eq_extractLsb {w : Nat} (x : BitVec w) (start len : Nat) (h
   simp only [getMsbD]
   by_cases h : i < w <;> simp [h]
 
+@[simp] theorem getElem_or {x y : BitVec w} {i : Nat} (h : i < w) : (x ||| y)[i] = (x[i] || y[i]) := by
+  simp [getElem_eq_testBit_toNat]
+
 @[simp] theorem msb_or {x y : BitVec w} : (x ||| y).msb = (x.msb || y.msb) := by
   simp [BitVec.msb]
 
-@[simp] theorem truncate_or {x y : BitVec w} :
-    (x ||| y).truncate k = x.truncate k ||| y.truncate k := by
+@[simp] theorem setWidth_or {x y : BitVec w} :
+    (x ||| y).setWidth k = x.setWidth k ||| y.setWidth k := by
   ext
   simp
 
@@ -698,11 +728,14 @@ instance : Std.Commutative (fun (x y : BitVec w) => x ||| y) := ⟨BitVec.or_com
   simp only [getMsbD]
   by_cases h : i < w <;> simp [h]
 
+@[simp] theorem getElem_and {x y : BitVec w} {i : Nat} (h : i < w) : (x &&& y)[i] = (x[i] && y[i]) := by
+  simp [getElem_eq_testBit_toNat]
+
 @[simp] theorem msb_and {x y : BitVec w} : (x &&& y).msb = (x.msb && y.msb) := by
   simp [BitVec.msb]
 
-@[simp] theorem truncate_and {x y : BitVec w} :
-    (x &&& y).truncate k = x.truncate k &&& y.truncate k := by
+@[simp] theorem setWidth_and {x y : BitVec w} :
+    (x &&& y).setWidth k = x.setWidth k &&& y.setWidth k := by
   ext
   simp
 
@@ -729,21 +762,24 @@ instance : Std.Commutative (fun (x y : BitVec w) => x &&& y) := ⟨BitVec.and_co
   exact (Nat.mod_eq_of_lt <| Nat.xor_lt_two_pow x.isLt y.isLt).symm
 
 @[simp] theorem getLsbD_xor {x y : BitVec v} :
-    (x ^^^ y).getLsbD i = (xor (x.getLsbD i) (y.getLsbD i)) := by
+    (x ^^^ y).getLsbD i = ((x.getLsbD i) ^^ (y.getLsbD i)) := by
   rw [← testBit_toNat, getLsbD, getLsbD]
   simp
 
 @[simp] theorem getMsbD_xor {x y : BitVec w} :
-    (x ^^^ y).getMsbD i = (xor (x.getMsbD i) (y.getMsbD i)) := by
+    (x ^^^ y).getMsbD i = (x.getMsbD i ^^ y.getMsbD i) := by
   simp only [getMsbD]
   by_cases h : i < w <;> simp [h]
 
+@[simp] theorem getElem_xor {x y : BitVec w} {i : Nat} (h : i < w) : (x ^^^ y)[i] = (x[i] ^^ y[i]) := by
+  simp [getElem_eq_testBit_toNat]
+
 @[simp] theorem msb_xor {x y : BitVec w} :
-    (x ^^^ y).msb = (xor x.msb y.msb) := by
+    (x ^^^ y).msb = (x.msb ^^ y.msb) := by
   simp [BitVec.msb]
 
-@[simp] theorem truncate_xor {x y : BitVec w} :
-    (x ^^^ y).truncate k = x.truncate k ^^^ y.truncate k := by
+@[simp] theorem setWidth_xor {x y : BitVec w} :
+    (x ^^^ y).setWidth k = x.setWidth k ^^^ y.setWidth k := by
   ext
   simp
 
@@ -791,8 +827,14 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
 @[simp] theorem getLsbD_not {x : BitVec v} : (~~~x).getLsbD i = (decide (i < v) && ! x.getLsbD i) := by
   by_cases h' : i < v <;> simp_all [not_def]
 
-@[simp] theorem truncate_not {x : BitVec w} (h : k ≤ w) :
-    (~~~x).truncate k = ~~~(x.truncate k) := by
+@[simp] theorem getElem_not {x : BitVec w} {i : Nat} (h : i < w) : (~~~x)[i] = !x[i] := by
+  simp only [getElem_eq_testBit_toNat, toNat_not]
+  rw [← Nat.sub_add_eq, Nat.add_comm 1]
+  rw [Nat.testBit_two_pow_sub_succ x.isLt]
+  simp [h]
+
+@[simp] theorem setWidth_not {x : BitVec w} (h : k ≤ w) :
+    (~~~x).setWidth k = ~~~(x.setWidth k) := by
   ext
   simp [h]
   omega
@@ -826,7 +868,7 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
   BitVec.toNat_ofNat _ _
 
 @[simp] theorem toFin_shiftLeft {n : Nat} (x : BitVec w) :
-    BitVec.toFin (x <<< n) = Fin.ofNat' (x.toNat <<< n) (Nat.two_pow_pos w) := rfl
+    BitVec.toFin (x <<< n) = Fin.ofNat' (2^w) (x.toNat <<< n) := rfl
 
 @[simp]
 theorem shiftLeft_zero_eq (x : BitVec w) : x <<< 0 = x := by
@@ -880,9 +922,9 @@ theorem shiftLeft_or_distrib (x y : BitVec w) (n : Nat) :
     <;> omega
 
 theorem shiftLeftZeroExtend_eq {x : BitVec w} :
-    shiftLeftZeroExtend x n = zeroExtend (w+n) x <<< n := by
+    shiftLeftZeroExtend x n = setWidth (w+n) x <<< n := by
   apply eq_of_toNat_eq
-  rw [shiftLeftZeroExtend, zeroExtend]
+  rw [shiftLeftZeroExtend, setWidth]
   split
   · simp
     rw [Nat.mod_eq_of_lt]
@@ -893,7 +935,7 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
 @[simp] theorem getLsbD_shiftLeftZeroExtend (x : BitVec m) (n : Nat) :
     getLsbD (shiftLeftZeroExtend x n) i = ((! decide (i < n)) && getLsbD x (i - n)) := by
   rw [shiftLeftZeroExtend_eq]
-  simp only [getLsbD_shiftLeft, getLsbD_zeroExtend]
+  simp only [getLsbD_shiftLeft, getLsbD_setWidth]
   cases h₁ : decide (i < n) <;> cases h₂ : decide (i - n < m + n) <;> cases h₃ : decide (i < m + n)
     <;> simp_all
     <;> (rw [getLsbD_ge]; omega)
@@ -947,6 +989,10 @@ theorem getLsbD_shiftLeft' {x : BitVec w₁} {y : BitVec w₂} {i : Nat} :
     getLsbD (x >>> i) j = getLsbD x (i+j) := by
   unfold getLsbD ; simp
 
+@[simp] theorem getElem_ushiftRight (x : BitVec w) (i n : Nat) (h : i < w) :
+    (x >>> n)[i] = x.getLsbD (n + i) := by
+  simp [getElem_eq_testBit_toNat, toNat_ushiftRight, Nat.testBit_shiftRight, getLsbD]
+
 theorem ushiftRight_xor_distrib (x y : BitVec w) (n : Nat) :
     (x ^^^ y) >>> n = (x >>> n) ^^^ (y >>> n) := by
   ext
@@ -1027,7 +1073,7 @@ theorem sshiftRight_eq_of_msb_true {x : BitVec w} {s : Nat} (h : x.msb = true) :
     · simp only [hi, decide_False, Bool.not_false, Bool.true_and, Bool.iff_and_self,
         decide_eq_true_eq]
       intros hlsb
-      apply BitVec.lt_of_getLsbD _ _ hlsb
+      apply BitVec.lt_of_getLsbD hlsb
   · by_cases hi : i ≥ w
     · simp [hi]
     · simp only [sshiftRight_eq_of_msb_true hmsb, getLsbD_not, getLsbD_ushiftRight, Bool.not_and,
@@ -1132,15 +1178,15 @@ private theorem Int.negSucc_emod (m : Nat) (n : Int) :
     -(m + 1) % n = Int.subNatNat (Int.natAbs n) ((m % Int.natAbs n) + 1) := rfl
 
 /-- The sign extension is the same as zero extending when `msb = false`. -/
-theorem signExtend_eq_not_zeroExtend_not_of_msb_false {x : BitVec w} {v : Nat} (hmsb : x.msb = false) :
-    x.signExtend v = x.zeroExtend v := by
+theorem signExtend_eq_not_setWidth_not_of_msb_false {x : BitVec w} {v : Nat} (hmsb : x.msb = false) :
+    x.signExtend v = x.setWidth v := by
   ext i
   by_cases hv : i < v
-  · simp only [signExtend, getLsbD, getLsbD_zeroExtend, hv, decide_True, Bool.true_and, toNat_ofInt,
+  · simp only [signExtend, getLsbD, getLsbD_setWidth, hv, decide_True, Bool.true_and, toNat_ofInt,
       BitVec.toInt_eq_msb_cond, hmsb, ↓reduceIte, reduceCtorEq]
     rw [Int.ofNat_mod_ofNat, Int.toNat_ofNat, Nat.testBit_mod_two_pow]
     simp [BitVec.testBit_toNat]
-  · simp only [getLsbD_zeroExtend, hv, decide_False, Bool.false_and]
+  · simp only [getLsbD_setWidth, hv, decide_False, Bool.false_and]
     apply getLsbD_ge
     omega
 
@@ -1148,11 +1194,11 @@ theorem signExtend_eq_not_zeroExtend_not_of_msb_false {x : BitVec w} {v : Nat} (
 The sign extension is a bitwise not, followed by a zero extend, followed by another bitwise not
 when `msb = true`. The double bitwise not ensures that the high bits are '1',
 and the lower bits are preserved. -/
-theorem signExtend_eq_not_zeroExtend_not_of_msb_true {x : BitVec w} {v : Nat} (hmsb : x.msb = true) :
-    x.signExtend v = ~~~((~~~x).zeroExtend v) := by
+theorem signExtend_eq_not_setWidth_not_of_msb_true {x : BitVec w} {v : Nat} (hmsb : x.msb = true) :
+    x.signExtend v = ~~~((~~~x).setWidth v) := by
   apply BitVec.eq_of_toNat_eq
   simp only [signExtend, BitVec.toInt_eq_msb_cond, toNat_ofInt, toNat_not,
-    toNat_truncate, hmsb, ↓reduceIte]
+    toNat_setWidth, hmsb, ↓reduceIte]
   norm_cast
   rw [Int.ofNat_sub_ofNat_of_lt, Int.negSucc_emod]
   simp only [Int.natAbs_ofNat, Nat.succ_eq_add_one]
@@ -1168,27 +1214,27 @@ theorem signExtend_eq_not_zeroExtend_not_of_msb_true {x : BitVec w} {v : Nat} (h
 @[simp] theorem getLsbD_signExtend (x  : BitVec w) {v i : Nat} :
     (x.signExtend v).getLsbD i = (decide (i < v) && if i < w then x.getLsbD i else x.msb) := by
   rcases hmsb : x.msb with rfl | rfl
-  · rw [signExtend_eq_not_zeroExtend_not_of_msb_false hmsb]
+  · rw [signExtend_eq_not_setWidth_not_of_msb_false hmsb]
     by_cases (i < v) <;> by_cases (i < w) <;> simp_all <;> omega
-  · rw [signExtend_eq_not_zeroExtend_not_of_msb_true hmsb]
+  · rw [signExtend_eq_not_setWidth_not_of_msb_true hmsb]
     by_cases (i < v) <;> by_cases (i < w) <;> simp_all <;> omega
 
 /-- Sign extending to a width smaller than the starting width is a truncation. -/
-theorem signExtend_eq_truncate_of_lt (x : BitVec w) {v : Nat} (hv : v ≤ w):
-  x.signExtend v = x.truncate v := by
+theorem signExtend_eq_setWidth_of_lt (x : BitVec w) {v : Nat} (hv : v ≤ w):
+  x.signExtend v = x.setWidth v := by
   ext i
-  simp only [getLsbD_signExtend, Fin.is_lt, decide_True, Bool.true_and, getLsbD_zeroExtend,
+  simp only [getLsbD_signExtend, Fin.is_lt, decide_True, Bool.true_and, getLsbD_setWidth,
     ite_eq_left_iff, Nat.not_lt]
   omega
 
 /-- Sign extending to the same bitwidth is a no op. -/
 theorem signExtend_eq (x : BitVec w) : x.signExtend w = x := by
-  rw [signExtend_eq_truncate_of_lt _ (Nat.le_refl _), truncate_eq]
+  rw [signExtend_eq_setWidth_of_lt _ (Nat.le_refl _), setWidth_eq]
 
 /-! ### append -/
 
 theorem append_def (x : BitVec v) (y : BitVec w) :
-    x ++ y = (shiftLeftZeroExtend x w ||| zeroExtend' (Nat.le_add_left w v) y) := rfl
+    x ++ y = (shiftLeftZeroExtend x w ||| setWidth' (Nat.le_add_left w v) y) := rfl
 
 @[simp] theorem toNat_append (x : BitVec m) (y : BitVec n) :
     (x ++ y).toNat = x.toNat <<< n ||| y.toNat :=
@@ -1196,7 +1242,7 @@ theorem append_def (x : BitVec v) (y : BitVec w) :
 
 @[simp] theorem getLsbD_append {x : BitVec n} {y : BitVec m} :
     getLsbD (x ++ y) i = bif i < m then getLsbD y i else getLsbD x (i - m) := by
-  simp only [append_def, getLsbD_or, getLsbD_shiftLeftZeroExtend, getLsbD_zeroExtend']
+  simp only [append_def, getLsbD_or, getLsbD_shiftLeftZeroExtend, getLsbD_setWidth']
   by_cases h : i < m
   · simp [h]
   · simp [h]; simp_all
@@ -1211,7 +1257,7 @@ theorem append_def (x : BitVec v) (y : BitVec w) :
 theorem msb_append {x : BitVec w} {y : BitVec v} :
     (x ++ y).msb = bif (w == 0) then (y.msb) else (x.msb) := by
   rw [← append_eq, append]
-  simp [msb_zeroExtend']
+  simp [msb_setWidth']
   by_cases h : w = 0
   · subst h
     simp [BitVec.msb, getMsbD]
@@ -1228,7 +1274,7 @@ theorem msb_append {x : BitVec w} {y : BitVec v} :
 @[simp] theorem zero_width_append (x : BitVec 0) (y : BitVec v) : x ++ y = cast (by omega) y := by
   ext
   rw [getLsbD_append]
-  simpa using lt_of_getLsbD _ _
+  simpa using lt_of_getLsbD
 
 @[simp] theorem cast_append_right (h : w + v = w + v') (x : BitVec w) (y : BitVec v) :
     cast h (x ++ y) = x ++ cast (by omega) y := by
@@ -1246,11 +1292,11 @@ theorem msb_append {x : BitVec w} {y : BitVec v} :
   ext
   simp
 
-theorem truncate_append {x : BitVec w} {y : BitVec v} :
-    (x ++ y).truncate k = if h : k ≤ v then y.truncate k else (x.truncate (k - v) ++ y).cast (by omega) := by
+theorem setWidth_append {x : BitVec w} {y : BitVec v} :
+    (x ++ y).setWidth k = if h : k ≤ v then y.setWidth k else (x.setWidth (k - v) ++ y).cast (by omega) := by
   apply eq_of_getLsbD_eq
   intro i
-  simp only [getLsbD_zeroExtend, Fin.is_lt, decide_True, getLsbD_append, Bool.true_and]
+  simp only [getLsbD_setWidth, Fin.is_lt, decide_True, getLsbD_append, Bool.true_and]
   split
   · have t : i < v := by omega
     simp [t]
@@ -1259,8 +1305,20 @@ theorem truncate_append {x : BitVec w} {y : BitVec v} :
     · have t' : i - v < k - v := by omega
       simp [t, t']
 
-@[simp] theorem truncate_cons {x : BitVec w} : (cons a x).truncate w = x := by
-  simp [cons, truncate_append]
+@[simp] theorem setWidth_append_of_eq {x : BitVec v} {y : BitVec w} (h : w' = w) : setWidth (v' + w') (x ++ y) = setWidth v' x ++ setWidth w' y := by
+  subst h
+  ext i
+  simp only [getLsbD_setWidth, Fin.is_lt, decide_True, getLsbD_append, cond_eq_if,
+    decide_eq_true_eq, Bool.true_and, setWidth_eq]
+  split
+  · simp_all
+  · simp_all only [Bool.iff_and_self, decide_eq_true_eq]
+    intro h
+    have := BitVec.lt_of_getLsbD h
+    omega
+
+@[simp] theorem setWidth_cons {x : BitVec w} : (cons a x).setWidth w = x := by
+  simp [cons, setWidth_append]
 
 @[simp] theorem not_append {x : BitVec w} {y : BitVec v} : ~~~ (x ++ y) = (~~~ x) ++ (~~~ y) := by
   ext i
@@ -1347,18 +1405,18 @@ theorem toNat_cons' {x : BitVec w} :
 @[simp] theorem getMsbD_cons_succ : (cons a x).getMsbD (i + 1) = x.getMsbD i := by
   simp [cons, Nat.le_add_left 1 i]
 
-theorem truncate_succ (x : BitVec w) :
-    truncate (i+1) x = cons (getLsbD x i) (truncate i x) := by
+theorem setWidth_succ (x : BitVec w) :
+    setWidth (i+1) x = cons (getLsbD x i) (setWidth i x) := by
   apply eq_of_getLsbD_eq
   intro j
-  simp only [getLsbD_truncate, getLsbD_cons, j.isLt, decide_True, Bool.true_and]
+  simp only [getLsbD_setWidth, getLsbD_cons, j.isLt, decide_True, Bool.true_and]
   if j_eq : j.val = i then
     simp [j_eq]
   else
     have j_lt : j.val < i := Nat.lt_of_le_of_ne (Nat.le_of_succ_le_succ j.isLt) j_eq
     simp [j_eq, j_lt]
 
-theorem eq_msb_cons_truncate (x : BitVec (w+1)) : x = (cons x.msb (x.truncate w)) := by
+theorem eq_msb_cons_setWidth (x : BitVec (w+1)) : x = (cons x.msb (x.setWidth w)) := by
   ext i
   simp
   split <;> rename_i h
@@ -1372,15 +1430,18 @@ theorem eq_msb_cons_truncate (x : BitVec (w+1)) : x = (cons x.msb (x.truncate w)
 
 @[simp] theorem cons_or_cons (x y : BitVec w) (a b : Bool) :
     (cons a x) ||| (cons b y) = cons (a || b) (x ||| y) := by
-  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+  ext i
+  simp [cons]
 
 @[simp] theorem cons_and_cons (x y : BitVec w) (a b : Bool) :
     (cons a x) &&& (cons b y) = cons (a && b) (x &&& y) := by
-  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+  ext i
+  simp [cons]
 
 @[simp] theorem cons_xor_cons (x y : BitVec w) (a b : Bool) :
-    (cons a x) ^^^ (cons b y) = cons (xor a b) (x ^^^ y) := by
-  ext i; cases i using Fin.succRecOn <;> simp <;> split <;> rfl
+    (cons a x) ^^^ (cons b y) = cons (a ^^ b) (x ^^^ y) := by
+  ext i
+  simp [cons]
 
 /-! ### concat -/
 
@@ -1418,7 +1479,7 @@ theorem getLsbD_concat (x : BitVec w) (b : Bool) (i : Nat) :
   ext i; cases i using Fin.succRecOn <;> simp
 
 @[simp] theorem concat_xor_concat (x y : BitVec w) (a b : Bool) :
-    (concat x a) ^^^ (concat y b) = concat (x ^^^ y) (xor a b) := by
+    (concat x a) ^^^ (concat y b) = concat (x ^^^ y) (a ^^ b) := by
   ext i; cases i using Fin.succRecOn <;> simp
 
 /-! ### add -/
@@ -1436,7 +1497,8 @@ Definition of bitvector addition as a nat.
   x + .ofFin y = .ofFin (x.toFin + y) := rfl
 
 theorem ofNat_add {n} (x y : Nat) : BitVec.ofNat n (x + y) = BitVec.ofNat n x + BitVec.ofNat n y := by
-  apply eq_of_toNat_eq ; simp [BitVec.ofNat]
+  apply eq_of_toNat_eq
+  simp [BitVec.ofNat, Fin.ofNat'_add]
 
 theorem ofNat_add_ofNat {n} (x y : Nat) : BitVec.ofNat n x + BitVec.ofNat n y = BitVec.ofNat n (x + y) :=
   (ofNat_add x y).symm
@@ -1456,8 +1518,8 @@ instance : Std.LawfulIdentity (α := BitVec n) (· + ·) 0#n where
   left_id := BitVec.zero_add
   right_id := BitVec.add_zero
 
-theorem truncate_add (x y : BitVec w) (h : i ≤ w) :
-    (x + y).truncate i = x.truncate i + y.truncate i := by
+theorem setWidth_add (x y : BitVec w) (h : i ≤ w) :
+    (x + y).setWidth i = x.setWidth i + y.setWidth i := by
   have dvd : 2^i ∣ 2^w := Nat.pow_dvd_pow _ h
   simp [bv_toNat, h, Nat.mod_mod_of_dvd _ dvd]
 
@@ -1490,10 +1552,12 @@ theorem sub_def {n} (x y : BitVec n) : x - y = .ofNat n ((2^n - y.toNat) + x.toN
   rfl
 @[simp] theorem sub_ofFin (x : BitVec n) (y : Fin (2^n)) : x - .ofFin y = .ofFin (x.toFin - y) :=
   rfl
+
 -- Remark: we don't use `[simp]` here because simproc` subsumes it for literals.
 -- If `x` and `n` are not literals, applying this theorem eagerly may not be a good idea.
 theorem ofNat_sub_ofNat {n} (x y : Nat) : BitVec.ofNat n x - BitVec.ofNat n y = .ofNat n ((2^n - y % 2^n) + x) := by
-  apply eq_of_toNat_eq ; simp [BitVec.ofNat]
+  apply eq_of_toNat_eq
+  simp [BitVec.ofNat, Fin.ofNat'_sub]
 
 @[simp] protected theorem sub_zero (x : BitVec n) : x - 0#n = x := by apply eq_of_toNat_eq ; simp
 
@@ -1508,7 +1572,7 @@ theorem ofNat_sub_ofNat {n} (x y : Nat) : BitVec.ofNat n x - BitVec.ofNat n y =
   simp [Neg.neg, BitVec.neg]
 
 @[simp] theorem toFin_neg (x : BitVec n) :
-    (-x).toFin = Fin.ofNat' (2^n - x.toNat) (Nat.two_pow_pos _) :=
+    (-x).toFin = Fin.ofNat' (2^n) (2^n - x.toNat) :=
   rfl
 
 theorem sub_toAdd {n} (x y : BitVec n) : x - y = x + - y := by
@@ -1628,7 +1692,7 @@ theorem ofInt_mul {n} (x y : Int) : BitVec.ofInt n (x * y) =
   x < BitVec.ofFin y ↔ x.toFin < y := Iff.rfl
 @[simp] theorem ofFin_lt {x : Fin (2^n)} {y : BitVec n} :
   BitVec.ofFin x < y ↔ x < y.toFin := Iff.rfl
-@[simp] theorem ofNat_lt_ofNat {n} (x y : Nat) : BitVec.ofNat n x < BitVec.ofNat n y ↔ x % 2^n < y % 2^n := by
+@[simp] theorem ofNat_lt_ofNat {n} {x y : Nat} : BitVec.ofNat n x < BitVec.ofNat n y ↔ x % 2^n < y % 2^n := by
   simp [lt_def]
 
 @[simp] protected theorem not_le {x y : BitVec n} : ¬ x ≤ y ↔ y < x := by
@@ -1913,18 +1977,18 @@ theorem twoPow_zero {w : Nat} : twoPow w 0 = 1#w := by
 theorem getLsbD_one {w i : Nat} : (1#w).getLsbD i = (decide (0 < w) && decide (0 = i)) := by
   rw [← twoPow_zero, getLsbD_twoPow]
 
-/- ### zeroExtend, truncate, and bitwise operations -/
+/- ### setWidth, setWidth, and bitwise operations -/
 
 /--
 When the `(i+1)`th bit of `x` is false,
 keeping the lower `(i + 1)` bits of `x` equals keeping the lower `i` bits.
 -/
-theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false
+theorem setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false
   {x : BitVec w} {i : Nat} (hx : x.getLsbD i = false) :
-    zeroExtend w (x.truncate (i + 1)) =
-      zeroExtend w (x.truncate i) := by
+    setWidth w (x.setWidth (i + 1)) =
+      setWidth w (x.setWidth i) := by
   ext k
-  simp only [getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
+  simp only [getLsbD_setWidth, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
   by_cases hik : i = k
   · subst hik
     simp [hx]
@@ -1935,22 +1999,22 @@ When the `(i+1)`th bit of `x` is true,
 keeping the lower `(i + 1)` bits of `x` equalsk eeping the lower `i` bits
 and then performing bitwise-or with `twoPow i = (1 << i)`,
 -/
-theorem zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true
+theorem setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true
     {x : BitVec w} {i : Nat} (hx : x.getLsbD i = true) :
-    zeroExtend w (x.truncate (i + 1)) =
-      zeroExtend w (x.truncate i) ||| (twoPow w i) := by
+    setWidth w (x.setWidth (i + 1)) =
+      setWidth w (x.setWidth i) ||| (twoPow w i) := by
   ext k
-  simp only [getLsbD_zeroExtend, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
+  simp only [getLsbD_setWidth, Fin.is_lt, decide_True, Bool.true_and, getLsbD_or, getLsbD_and]
   by_cases hik : i = k
   · subst hik
     simp [hx]
   · by_cases hik' : k < i + 1 <;> simp [hik, hik'] <;> omega
 
 /-- Bitwise and of `(x : BitVec w)` with `1#w` equals zero extending `x.lsb` to `w`. -/
-theorem and_one_eq_zeroExtend_ofBool_getLsbD {x : BitVec w} :
-    (x &&& 1#w) = zeroExtend w (ofBool (x.getLsbD 0)) := by
+theorem and_one_eq_setWidth_ofBool_getLsbD {x : BitVec w} :
+    (x &&& 1#w) = setWidth w (ofBool (x.getLsbD 0)) := by
   ext i
-  simp only [getLsbD_and, getLsbD_one, getLsbD_zeroExtend, Fin.is_lt, decide_True, getLsbD_ofBool,
+  simp only [getLsbD_and, getLsbD_one, getLsbD_setWidth, Fin.is_lt, decide_True, getLsbD_ofBool,
     Bool.true_and]
   by_cases h : (0 = (i : Nat)) <;> simp [h] <;> omega
 
@@ -2062,4 +2126,143 @@ theorem toNat_sub_of_le {x y : BitVec n} (h : y ≤ x) :
   · have : 2 ^ n - y.toNat + x.toNat = 2 ^ n + (x.toNat - y.toNat) := by omega
     rw [this, Nat.add_mod_left, Nat.mod_eq_of_lt (by omega)]
 
+/-! ### Deprecations -/
+
+set_option linter.missingDocs false
+
+@[deprecated truncate_eq_setWidth (since := "2024-09-18")]
+abbrev truncate_eq_zeroExtend := @truncate_eq_setWidth
+
+@[deprecated toNat_setWidth' (since := "2024-09-18")]
+abbrev toNat_zeroExtend' := @toNat_setWidth'
+
+@[deprecated toNat_setWidth (since := "2024-09-18")]
+abbrev toNat_zeroExtend := @toNat_setWidth
+
+@[deprecated toNat_setWidth (since := "2024-09-18")]
+abbrev toNat_truncate := @toNat_setWidth
+
+@[deprecated setWidth_eq (since := "2024-09-18")]
+abbrev zeroExtend_eq := @setWidth_eq
+
+@[deprecated setWidth_eq (since := "2024-09-18")]
+abbrev truncate_eq := @setWidth_eq
+
+@[deprecated setWidth_zero (since := "2024-09-18")]
+abbrev zeroExtend_zero := @setWidth_zero
+
+@[deprecated getElem_setWidth (since := "2024-09-18")]
+abbrev getElem_zeroExtend := @getElem_setWidth
+
+@[deprecated getElem_setWidth' (since := "2024-09-18")]
+abbrev getElem_zeroExtend' := @getElem_setWidth'
+
+@[deprecated getElem?_setWidth (since := "2024-09-18")]
+abbrev getElem?_zeroExtend := @getElem?_setWidth
+
+@[deprecated getElem?_setWidth' (since := "2024-09-18")]
+abbrev getElem?_zeroExtend' := @getElem?_setWidth'
+
+@[deprecated getLsbD_setWidth (since := "2024-09-18")]
+abbrev getLsbD_zeroExtend := @getLsbD_setWidth
+
+@[deprecated getLsbD_setWidth' (since := "2024-09-18")]
+abbrev getLsbD_zeroExtend' := @getLsbD_setWidth'
+
+@[deprecated getMsbD_setWidth_add (since := "2024-09-18")]
+abbrev getMsbD_zeroExtend_add := @getMsbD_setWidth_add
+
+@[deprecated getMsbD_setWidth' (since := "2024-09-18")]
+abbrev getMsbD_zeroExtend' := @getMsbD_setWidth'
+
+@[deprecated getElem_setWidth (since := "2024-09-18")]
+abbrev getElem_truncate := @getElem_setWidth
+
+@[deprecated getElem?_setWidth (since := "2024-09-18")]
+abbrev getElem?_truncate := @getElem?_setWidth
+
+@[deprecated getLsbD_setWidth (since := "2024-09-18")]
+abbrev getLsbD_truncate := @getLsbD_setWidth
+
+@[deprecated msb_setWidth (since := "2024-09-18")]
+abbrev msb_truncate := @msb_setWidth
+
+@[deprecated cast_setWidth (since := "2024-09-18")]
+abbrev cast_zeroExtend := @cast_setWidth
+
+@[deprecated cast_setWidth (since := "2024-09-18")]
+abbrev cast_truncate := @cast_setWidth
+
+@[deprecated setWidth_setWidth_of_le (since := "2024-09-18")]
+abbrev zeroExtend_zeroExtend_of_le := @setWidth_setWidth_of_le
+
+@[deprecated setWidth_eq (since := "2024-09-18")]
+abbrev truncate_eq_self := @setWidth_eq
+
+@[deprecated setWidth_cast (since := "2024-09-18")]
+abbrev truncate_cast := @setWidth_cast
+
+@[deprecated msb_setWidth (since := "2024-09-18")]
+abbrev mbs_zeroExtend := @msb_setWidth
+
+@[deprecated msb_setWidth' (since := "2024-09-18")]
+abbrev mbs_zeroExtend' := @msb_setWidth'
+
+@[deprecated setWidth_one_eq_ofBool_getLsb_zero (since := "2024-09-18")]
+abbrev zeroExtend_one_eq_ofBool_getLsb_zero := @setWidth_one_eq_ofBool_getLsb_zero
+
+@[deprecated setWidth_ofNat_one_eq_ofNat_one_of_lt (since := "2024-09-18")]
+abbrev zeroExtend_ofNat_one_eq_ofNat_one_of_lt := @setWidth_ofNat_one_eq_ofNat_one_of_lt
+
+@[deprecated setWidth_one (since := "2024-09-18")]
+abbrev truncate_one := @setWidth_one
+
+@[deprecated setWidth_ofNat_of_le (since := "2024-09-18")]
+abbrev truncate_ofNat_of_le := @setWidth_ofNat_of_le
+
+@[deprecated setWidth_or (since := "2024-09-18")]
+abbrev truncate_or := @setWidth_or
+
+@[deprecated setWidth_and (since := "2024-09-18")]
+abbrev truncate_and := @setWidth_and
+
+@[deprecated setWidth_xor (since := "2024-09-18")]
+abbrev truncate_xor := @setWidth_xor
+
+@[deprecated setWidth_not (since := "2024-09-18")]
+abbrev truncate_not := @setWidth_not
+
+@[deprecated signExtend_eq_not_setWidth_not_of_msb_false  (since := "2024-09-18")]
+abbrev signExtend_eq_not_zeroExtend_not_of_msb_false  := @signExtend_eq_not_setWidth_not_of_msb_false
+
+@[deprecated signExtend_eq_not_setWidth_not_of_msb_true (since := "2024-09-18")]
+abbrev signExtend_eq_not_zeroExtend_not_of_msb_true := @signExtend_eq_not_setWidth_not_of_msb_true
+
+@[deprecated signExtend_eq_setWidth_of_lt (since := "2024-09-18")]
+abbrev signExtend_eq_truncate_of_lt := @signExtend_eq_setWidth_of_lt
+
+@[deprecated truncate_append (since := "2024-09-18")]
+abbrev truncate_append := @setWidth_append
+
+@[deprecated truncate_append_of_eq (since := "2024-09-18")]
+abbrev truncate_append_of_eq := @setWidth_append_of_eq
+
+@[deprecated truncate_cons (since := "2024-09-18")]
+abbrev truncate_cons := @setWidth_cons
+
+@[deprecated truncate_succ (since := "2024-09-18")]
+abbrev truncate_succ := @setWidth_succ
+
+@[deprecated truncate_add (since := "2024-09-18")]
+abbrev truncate_add := @setWidth_add
+
+@[deprecated setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false (since := "2024-09-18")]
+abbrev zeroExtend_truncate_succ_eq_zeroExtend_truncate_of_getLsbD_false := @setWidth_setWidth_succ_eq_setWidth_setWidth_of_getLsbD_false
+
+@[deprecated setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true (since := "2024-09-18")]
+abbrev zeroExtend_truncate_succ_eq_zeroExtend_truncate_or_twoPow_of_getLsbD_true := @setWidth_setWidth_succ_eq_setWidth_setWidth_or_twoPow_of_getLsbD_true
+
+@[deprecated and_one_eq_setWidth_ofBool_getLsbD (since := "2024-09-18")]
+abbrev and_one_eq_zeroExtend_ofBool_getLsbD := @and_one_eq_setWidth_ofBool_getLsbD
+
 end BitVec

src/Init/Data/Bool.lean

@@ -6,16 +6,13 @@ Authors: F. G. Dorais
 prelude
 import Init.NotationExtra
 
-/-- Boolean exclusive or -/
-abbrev xor : Bool → Bool → Bool := bne
 
 namespace Bool
 
-/- Namespaced versions that can be used instead of prefixing `_root_` -/
-@[inherit_doc not] protected abbrev not := not
-@[inherit_doc or]  protected abbrev or  := or
-@[inherit_doc and] protected abbrev and := and
-@[inherit_doc xor] protected abbrev xor := xor
+/-- Boolean exclusive or -/
+abbrev xor : Bool → Bool → Bool := bne
+
+@[inherit_doc] infixl:33 " ^^ " => xor
 
 instance (p : Bool → Prop) [inst : DecidablePred p] : Decidable (∀ x, p x) :=
   match inst true, inst false with
@@ -150,8 +147,8 @@ theorem and_or_distrib_right : ∀ (x y z : Bool), ((x || y) && z) = (x && z ||
 theorem or_and_distrib_left  : ∀ (x y z : Bool), (x || y && z) = ((x || y) && (x || z)) := by decide
 theorem or_and_distrib_right : ∀ (x y z : Bool), (x && y || z) = ((x || z) && (y || z)) := by decide
 
-theorem and_xor_distrib_left  : ∀ (x y z : Bool), (x && xor y z) = xor (x && y) (x && z) := by decide
-theorem and_xor_distrib_right : ∀ (x y z : Bool), (xor x y && z) = xor (x && z) (y && z) := by decide
+theorem and_xor_distrib_left  : ∀ (x y z : Bool), (x && (y ^^ z)) = ((x && y) ^^ (x && z)) := by decide
+theorem and_xor_distrib_right : ∀ (x y z : Bool), ((x ^^ y) && z) = ((x && z) ^^ (y && z)) := by decide
 
 /-- De Morgan's law for boolean and -/
 @[simp] theorem not_and : ∀ (x y : Bool), (!(x && y)) = (!x || !y) := by decide
@@ -257,15 +254,6 @@ theorem beq_eq_decide_eq [BEq α] [LawfulBEq α] [DecidableEq α] (a b : α) :
 theorem eq_not : ∀ {a b : Bool}, (a = (!b)) ↔ (a ≠ b) := by decide
 theorem not_eq : ∀ {a b : Bool}, ((!a) = b) ↔ (a ≠ b) := by decide
 
-@[simp] theorem not_eq_not : ∀ {a b : Bool}, ¬a = !b ↔ a = b := by decide
-@[simp] theorem not_not_eq : ∀ {a b : Bool}, ¬(!a) = b ↔ a = b := by decide
-
-/--
-We move `!` from the left hand side of an equality to the right hand side.
-This helps confluence, and also helps combining pairs of `!`s.
--/
-@[simp] theorem not_eq_eq_eq_not : ∀ {a b : Bool}, ((!a) = b) ↔ (a = !b) := by decide
-
 @[simp] theorem coe_iff_coe : ∀{a b : Bool}, (a ↔ b) ↔ a = b := by decide
 
 @[simp] theorem coe_true_iff_false  : ∀{a b : Bool}, (a ↔ b = false) ↔ a = (!b) := by decide
@@ -279,37 +267,37 @@ theorem beq_comm {α} [BEq α] [LawfulBEq α] {a b : α} : (a == b) = (b == a) :
 
 /-! ### xor -/
 
-theorem false_xor : ∀ (x : Bool), xor false x = x := false_bne
+theorem false_xor : ∀ (x : Bool), (false ^^ x) = x := false_bne
 
-theorem xor_false : ∀ (x : Bool), xor x false = x := bne_false
+theorem xor_false : ∀ (x : Bool), (x ^^ false) = x := bne_false
 
-theorem true_xor : ∀ (x : Bool), xor true x = !x := true_bne
+theorem true_xor : ∀ (x : Bool), (true ^^ x) = !x := true_bne
 
-theorem xor_true : ∀ (x : Bool), xor x true = !x := bne_true
+theorem xor_true : ∀ (x : Bool), (x ^^ true) = !x := bne_true
 
-theorem not_xor_self : ∀ (x : Bool), xor (!x) x = true := not_bne_self
+theorem not_xor_self : ∀ (x : Bool), (!x ^^ x) = true := not_bne_self
 
-theorem xor_not_self : ∀ (x : Bool), xor x (!x) = true := bne_not_self
+theorem xor_not_self : ∀ (x : Bool), (x ^^ !x) = true := bne_not_self
 
-theorem not_xor : ∀ (x y : Bool), xor (!x) y = !(xor x y) := by decide
+theorem not_xor : ∀ (x y : Bool), (!x ^^ y) = !(x ^^ y) := by decide
 
-theorem xor_not : ∀ (x y : Bool), xor x (!y) = !(xor x y) := by decide
+theorem xor_not : ∀ (x y : Bool), (x ^^ !y) = !(x ^^ y) := by decide
 
-theorem not_xor_not : ∀ (x y : Bool), xor (!x) (!y) = (xor x y) := not_bne_not
+theorem not_xor_not : ∀ (x y : Bool), (!x ^^ !y) = (x ^^ y) := not_bne_not
 
-theorem xor_self : ∀ (x : Bool), xor x x = false := by decide
+theorem xor_self : ∀ (x : Bool), (x ^^ x) = false := by decide
 
-theorem xor_comm : ∀ (x y : Bool), xor x y = xor y x := by decide
+theorem xor_comm : ∀ (x y : Bool), (x ^^ y) = (y ^^ x) := by decide
 
-theorem xor_left_comm : ∀ (x y z : Bool), xor x (xor y z) = xor y (xor x z) := by decide
+theorem xor_left_comm : ∀ (x y z : Bool), (x ^^ (y ^^ z)) = (y ^^ (x ^^ z)) := by decide
 
-theorem xor_right_comm : ∀ (x y z : Bool), xor (xor x y) z = xor (xor x z) y := by decide
+theorem xor_right_comm : ∀ (x y z : Bool), ((x ^^ y) ^^ z) = ((x ^^ z) ^^ y) := by decide
 
-theorem xor_assoc : ∀ (x y z : Bool), xor (xor x y) z = xor x (xor y z) := bne_assoc
+theorem xor_assoc : ∀ (x y z : Bool), ((x ^^ y) ^^ z) = (x ^^ (y ^^ z)) := bne_assoc
 
-theorem xor_left_inj : ∀ {x y z : Bool}, xor x y = xor x z ↔ y = z := bne_left_inj
+theorem xor_left_inj : ∀ {x y z : Bool}, (x ^^ y) = (x ^^ z) ↔ y = z := bne_left_inj
 
-theorem xor_right_inj : ∀ {x y z : Bool}, xor x z = xor y z ↔ x = y := bne_right_inj
+theorem xor_right_inj : ∀ {x y z : Bool}, (x ^^ z) = (y ^^ z) ↔ x = y := bne_right_inj
 
 /-! ### le/lt -/
 
@@ -597,7 +585,7 @@ theorem decide_beq_decide (p q : Prop) [dpq : Decidable (p ↔ q)] [dp : Decidab
 
 end Bool
 
-export Bool (cond_eq_if)
+export Bool (cond_eq_if xor and or not)
 
 /-! ### decide -/
 

src/Init/Data/Fin/Basic.lean

@@ -31,7 +31,7 @@ This differs from addition, which wraps around:
 (2 : Fin 3) + 1 = (0 : Fin 3)
 ```
 -/
-def succ : Fin n → Fin n.succ
+def succ : Fin n → Fin (n + 1)
   | ⟨i, h⟩ => ⟨i+1, Nat.succ_lt_succ h⟩
 
 variable {n : Nat}
@@ -39,16 +39,20 @@ variable {n : Nat}
 /--
 Returns `a` modulo `n + 1` as a `Fin n.succ`.
 -/
-protected def ofNat {n : Nat} (a : Nat) : Fin n.succ :=
+protected def ofNat {n : Nat} (a : Nat) : Fin (n + 1) :=
   ⟨a % (n+1), Nat.mod_lt _ (Nat.zero_lt_succ _)⟩
 
 /--
 Returns `a` modulo `n` as a `Fin n`.
 
-The assumption `n > 0` ensures that `Fin n` is nonempty.
+The assumption `NeZero n` ensures that `Fin n` is nonempty.
 -/
-protected def ofNat' {n : Nat} (a : Nat) (h : n > 0) : Fin n :=
-  ⟨a % n, Nat.mod_lt _ h⟩
+protected def ofNat' (n : Nat) [NeZero n] (a : Nat) : Fin n :=
+  ⟨a % n, Nat.mod_lt _ (pos_of_neZero n)⟩
+
+-- We intend to deprecate `Fin.ofNat` in favor of `Fin.ofNat'` (and later rename).
+-- This is waiting on https://github.com/leanprover/lean4/pull/5323
+-- attribute [deprecated Fin.ofNat' (since := "2024-09-16")] Fin.ofNat
 
 private theorem mlt {b : Nat} : {a : Nat} → a < n → b % n < n
   | 0,   h => Nat.mod_lt _ h
@@ -141,10 +145,10 @@ instance : ShiftLeft (Fin n) where
 instance : ShiftRight (Fin n) where
   shiftRight := Fin.shiftRight
 
-instance instOfNat {n : Nat} [NeZero n] {i : Nat} : OfNat (Fin (no_index n)) i where
-  ofNat := Fin.ofNat' i (pos_of_neZero _)
+instance instOfNat {n : Nat} [NeZero n] {i : Nat} : OfNat (Fin n) i where
+  ofNat := Fin.ofNat' n i
 
-instance : Inhabited (Fin (no_index (n+1))) where
+instance instInhabited {n : Nat} [NeZero n] : Inhabited (Fin n) where
   default := 0
 
 @[simp] theorem zero_eta : (⟨0, Nat.zero_lt_succ _⟩ : Fin (n + 1)) = 0 := rfl

src/Init/Data/Fin/Lemmas.lean

@@ -51,10 +51,15 @@ theorem eq_mk_iff_val_eq {a : Fin n} {k : Nat} {hk : k < n} :
 
 theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
 
-@[simp] theorem val_ofNat' (a : Nat) (is_pos : n > 0) :
-  (Fin.ofNat' a is_pos).val = a % n := rfl
+@[simp] theorem val_ofNat' (n : Nat) [NeZero n] (a : Nat) :
+  (Fin.ofNat' n a).val = a % n := rfl
 
-@[simp] theorem ofNat'_val_eq_self (x : Fin n) (h) : (Fin.ofNat' x h) = x := by
+@[simp] theorem ofNat'_self {n : Nat} [NeZero n] : Fin.ofNat' n n = 0 := by
+  ext
+  simp
+  congr
+
+@[simp] theorem ofNat'_val_eq_self [NeZero n] (x : Fin n) : (Fin.ofNat' n x) = x := by
   ext
   rw [val_ofNat', Nat.mod_eq_of_lt]
   exact x.2
@@ -68,6 +73,9 @@ theorem mk_val (i : Fin n) : (⟨i, i.isLt⟩ : Fin n) = i := Fin.eta ..
 @[simp] theorem modn_val (a : Fin n) (b : Nat) : (a.modn b).val = a.val % b :=
   rfl
 
+@[simp] theorem val_eq_zero (a : Fin 1) : a.val = 0 :=
+  Nat.eq_zero_of_le_zero <| Nat.le_of_lt_succ a.isLt
+
 theorem ite_val {n : Nat} {c : Prop} [Decidable c] {x : c → Fin n} (y : ¬c → Fin n) :
     (if h : c then x h else y h).val = if h : c then (x h).val else (y h).val := by
   by_cases c <;> simp [*]
@@ -120,7 +128,7 @@ theorem mk_le_of_le_val {b : Fin n} {a : Nat} (h : a ≤ b) :
 
 @[simp] theorem mk_lt_mk {x y : Nat} {hx hy} : (⟨x, hx⟩ : Fin n) < ⟨y, hy⟩ ↔ x < y := .rfl
 
-@[simp] theorem val_zero (n : Nat) : (0 : Fin (n + 1)).1 = 0 := rfl
+@[simp] theorem val_zero (n : Nat) [NeZero n] : ((0 : Fin n) : Nat) = 0 := rfl
 
 @[simp] theorem mk_zero : (⟨0, Nat.succ_pos n⟩ : Fin (n + 1)) = 0 := rfl
 
@@ -167,8 +175,24 @@ theorem rev_eq {n a : Nat} (i : Fin (n + 1)) (h : n = a + i) :
 @[simp] theorem rev_lt_rev {i j : Fin n} : rev i < rev j ↔ j < i := by
   rw [← Fin.not_le, ← Fin.not_le, rev_le_rev]
 
+/-! ### last -/
+
 @[simp] theorem val_last (n : Nat) : last n = n := rfl
 
+@[simp] theorem last_zero : (Fin.last 0 : Fin 1) = 0 := by
+  ext
+  simp
+
+@[simp] theorem zero_eq_last_iff {n : Nat} : (0 : Fin (n + 1)) = last n ↔ n = 0 := by
+  constructor
+  · intro h
+    simp_all [Fin.ext_iff]
+  · rintro rfl
+    simp
+
+@[simp] theorem last_eq_zero_iff {n : Nat} : Fin.last n = 0 ↔ n = 0 := by
+  simp [eq_comm (a := Fin.last n)]
+
 theorem le_last (i : Fin (n + 1)) : i ≤ last n := Nat.le_of_lt_succ i.is_lt
 
 theorem last_pos : (0 : Fin (n + 2)) < last (n + 1) := Nat.succ_pos _
@@ -202,10 +226,28 @@ instance subsingleton_one : Subsingleton (Fin 1) := subsingleton_iff_le_one.2 (b
 
 theorem fin_one_eq_zero (a : Fin 1) : a = 0 := Subsingleton.elim a 0
 
+@[simp] theorem zero_eq_one_iff {n : Nat} [NeZero n] : (0 : Fin n) = 1 ↔ n = 1 := by
+  constructor
+  · intro h
+    simp [Fin.ext_iff] at h
+    change 0 % n = 1 % n at h
+    rw [eq_comm] at h
+    simpa using h
+  · rintro rfl
+    simp
+
+@[simp] theorem one_eq_zero_iff {n : Nat} [NeZero n] : (1 : Fin n) = 0 ↔ n = 1 := by
+  rw [eq_comm]
+  simp
+
 theorem add_def (a b : Fin n) : a + b = Fin.mk ((a + b) % n) (Nat.mod_lt _ a.size_pos) := rfl
 
 theorem val_add (a b : Fin n) : (a + b).val = (a.val + b.val) % n := rfl
 
+@[simp] protected theorem zero_add {n : Nat} [NeZero n] (i : Fin n) : (0 : Fin n) + i = i := by
+  ext
+  simp [Fin.add_def, Nat.mod_eq_of_lt i.2]
+
 theorem val_add_one_of_lt {n : Nat} {i : Fin n.succ} (h : i < last _) : (i + 1).1 = i + 1 := by
   match n with
   | 0 => cases h
@@ -329,6 +371,10 @@ theorem succ_succ_ne_one (a : Fin n) : Fin.succ (Fin.succ a) ≠ 1 :=
 
 @[simp] theorem cast_mk (h : n = m) (i : Nat) (hn : i < n) : cast h ⟨i, hn⟩ = ⟨i, h ▸ hn⟩ := rfl
 
+@[simp] theorem cast_refl (n : Nat) (h : n = n) : cast h = id := by
+  ext
+  simp
+
 @[simp] theorem cast_trans {k : Nat} (h : n = m) (h' : m = k) {i : Fin n} :
     cast h' (cast h i) = cast (Eq.trans h h') i := rfl
 
@@ -437,6 +483,10 @@ theorem succ_castSucc {n : Nat} (i : Fin n) : i.castSucc.succ = castSucc i.succ
 
 @[simp] theorem coe_addNat (m : Nat) (i : Fin n) : (addNat i m : Nat) = i + m := rfl
 
+@[simp] theorem addNat_zero (n : Nat) (i : Fin n) : addNat i 0 = i := by
+  ext
+  simp
+
 @[simp] theorem addNat_one {i : Fin n} : addNat i 1 = i.succ := rfl
 
 theorem le_coe_addNat (m : Nat) (i : Fin n) : m ≤ addNat i m :=
@@ -466,7 +516,7 @@ theorem cast_addNat_left {n n' m : Nat} (i : Fin n') (h : n' + m = n + m) :
 
 theorem le_coe_natAdd (m : Nat) (i : Fin n) : m ≤ natAdd m i := Nat.le_add_right ..
 
-theorem natAdd_zero {n : Nat} : natAdd 0 = cast (Nat.zero_add n).symm := by ext; simp
+@[simp] theorem natAdd_zero {n : Nat} : natAdd 0 = cast (Nat.zero_add n).symm := by ext; simp
 
 /-- For rewriting in the reverse direction, see `Fin.cast_natAdd_right`. -/
 theorem natAdd_cast {n n' : Nat} (m : Nat) (i : Fin n') (h : n' = n) :
@@ -504,9 +554,19 @@ theorem cast_addNat {n : Nat} (m : Nat) (i : Fin n) :
 
 @[simp] theorem natAdd_last {m n : Nat} : natAdd n (last m) = last (n + m) := rfl
 
+@[simp] theorem addNat_last (n : Nat) :
+    addNat (last n) m = cast (by omega) (last (n + m)) := by
+  ext
+  simp
+
 theorem natAdd_castSucc {m n : Nat} {i : Fin m} : natAdd n (castSucc i) = castSucc (natAdd n i) :=
   rfl
 
+@[simp] theorem natAdd_eq_addNat (n : Nat) (i : Fin n) : Fin.natAdd n i = i.addNat n := by
+  ext
+  simp
+  omega
+
 theorem rev_castAdd (k : Fin n) (m : Nat) : rev (castAdd m k) = addNat (rev k) m := Fin.ext <| by
   rw [val_rev, coe_castAdd, coe_addNat, val_rev, Nat.sub_add_comm (Nat.succ_le_of_lt k.is_lt)]
 
@@ -572,6 +632,15 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
 @[simp] theorem subNat_mk {i : Nat} (h₁ : i < n + m) (h₂ : m ≤ i) :
     subNat m ⟨i, h₁⟩ h₂ = ⟨i - m, Nat.sub_lt_right_of_lt_add h₂ h₁⟩ := rfl
 
+@[simp] theorem subNat_zero (i : Fin n) (h : 0 ≤ (i : Nat)): Fin.subNat 0 i h = i := by
+  ext
+  simp
+
+@[simp] theorem subNat_one_succ (i : Fin (n + 1)) (h : 1 ≤ ↑i) : (subNat 1 i h).succ = i := by
+  ext
+  simp
+  omega
+
 @[simp] theorem pred_castSucc_succ (i : Fin n) :
     pred (castSucc i.succ) (Fin.ne_of_gt (castSucc_pos i.succ_pos)) = castSucc i := rfl
 
@@ -582,7 +651,7 @@ theorem pred_add_one (i : Fin (n + 2)) (h : (i : Nat) < n + 1) :
     subNat m (addNat i m) h = i := Fin.ext <| Nat.add_sub_cancel i m
 
 @[simp] theorem natAdd_subNat_cast {i : Fin (n + m)} (h : n ≤ i) :
-    natAdd n (subNat n (cast (Nat.add_comm ..) i) h) = i := by simp [← cast_addNat]; rfl
+    natAdd n (subNat n (cast (Nat.add_comm ..) i) h) = i := by simp [← cast_addNat]
 
 /-! ### recursion and induction principles -/
 
@@ -750,13 +819,13 @@ theorem addCases_right {m n : Nat} {motive : Fin (m + n) → Sort _} {left right
 
 /-! ### add -/
 
-@[simp] theorem ofNat'_add (x : Nat) (lt : 0 < n) (y : Fin n) :
-    Fin.ofNat' x lt + y = Fin.ofNat' (x + y.val) lt := by
+theorem ofNat'_add [NeZero n] (x : Nat) (y : Fin n) :
+    Fin.ofNat' n x + y = Fin.ofNat' n (x + y.val) := by
   apply Fin.eq_of_val_eq
   simp [Fin.ofNat', Fin.add_def]
 
-@[simp] theorem add_ofNat' (x : Fin n) (y : Nat) (lt : 0 < n) :
-    x + Fin.ofNat' y lt = Fin.ofNat' (x.val + y) lt := by
+theorem add_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
+    x + Fin.ofNat' n y = Fin.ofNat' n (x.val + y) := by
   apply Fin.eq_of_val_eq
   simp [Fin.ofNat', Fin.add_def]
 
@@ -765,16 +834,21 @@ theorem addCases_right {m n : Nat} {motive : Fin (m + n) → Sort _} {left right
 protected theorem coe_sub (a b : Fin n) : ((a - b : Fin n) : Nat) = ((n - b) + a) % n := by
   cases a; cases b; rfl
 
-@[simp] theorem ofNat'_sub (x : Nat) (lt : 0 < n) (y : Fin n) :
-    Fin.ofNat' x lt - y = Fin.ofNat' ((n - y.val) + x) lt := by
+theorem ofNat'_sub [NeZero n] (x : Nat) (y : Fin n) :
+    Fin.ofNat' n x - y = Fin.ofNat' n ((n - y.val) + x) := by
   apply Fin.eq_of_val_eq
   simp [Fin.ofNat', Fin.sub_def]
 
-@[simp] theorem sub_ofNat' (x : Fin n) (y : Nat) (lt : 0 < n) :
-    x - Fin.ofNat' y lt = Fin.ofNat' ((n - y % n) + x.val) lt := by
+theorem sub_ofNat' [NeZero n] (x : Fin n) (y : Nat) :
+    x - Fin.ofNat' n y = Fin.ofNat' n ((n - y % n) + x.val) := by
   apply Fin.eq_of_val_eq
   simp [Fin.ofNat', Fin.sub_def]
 
+@[simp] protected theorem sub_self [NeZero n] {x : Fin n} : x - x = 0 := by
+  ext
+  rw [Fin.sub_def]
+  simp
+
 private theorem _root_.Nat.mod_eq_sub_of_lt_two_mul {x n} (h₁ : n ≤ x) (h₂ : x < 2 * n) :
     x % n = x - n := by
   rw [Nat.mod_eq, if_pos (by omega), Nat.mod_eq_of_lt (by omega)]

src/Init/Data/List/Attach.lean

@@ -48,6 +48,8 @@ Unsafe implementation of `attachWith`, taking advantage of the fact that the rep
 
 @[simp] theorem attach_nil : ([] : List α).attach = [] := rfl
 
+@[simp] theorem attachWith_nil : ([] : List α).attachWith P H = [] := 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
@@ -81,7 +83,12 @@ theorem attach_congr {l₁ l₂ : List α} (h : l₁ = l₂) :
   subst h
   simp
 
-@[simp] theorem attach_cons (x : α) (xs : List α) :
+theorem attachWith_congr {l₁ l₂ : List α} (w : l₁ = l₂) {P : α → Prop} {H : ∀ x ∈ l₁, P x} :
+    l₁.attachWith P H = l₂.attachWith P fun x h => H _ (w ▸ h) := by
+  subst w
+  simp
+
+@[simp] theorem attach_cons {x : α} {xs : List α} :
     (x :: xs).attach =
       ⟨x, mem_cons_self x xs⟩ :: xs.attach.map fun ⟨y, h⟩ => ⟨y, mem_cons_of_mem x h⟩ := by
   simp only [attach, attachWith, pmap, map_pmap, cons.injEq, true_and]
@@ -89,6 +96,12 @@ theorem attach_congr {l₁ l₂ : List α} (h : l₁ = l₂) :
   intros a _ m' _
   rfl
 
+@[simp]
+theorem attachWith_cons {x : α} {xs : List α} {p : α → Prop} (h : ∀ a ∈ x :: xs, p a) :
+    (x :: xs).attachWith p h = ⟨x, h x (mem_cons_self x xs)⟩ ::
+      xs.attachWith p (fun a ha ↦ h a (mem_cons_of_mem x ha)) :=
+  rfl
+
 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_left l fun _ _ _ _ => rfl
@@ -104,14 +117,18 @@ theorem attach_map_val (l : List α) (f : α → β) : (l.attach.map fun i => f
 theorem attach_map_subtype_val (l : List α) : l.attach.map Subtype.val = l :=
   (attach_map_coe _ _).trans (List.map_id _)
 
-theorem countP_attach (l : List α) (p : α → Bool) :
-    l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
-  simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attach_map_subtype_val]
+theorem attachWith_map_coe {p : α → Prop} (f : α → β) (l : List α) (H : ∀ a ∈ l, p a) :
+    ((l.attachWith p H).map fun (i : { i // p i}) => f i) = l.map f := by
+  rw [attachWith, map_pmap]; exact pmap_eq_map _ _ _ _
+
+theorem attachWith_map_val {p : α → Prop} (f : α → β) (l : List α) (H : ∀ a ∈ l, p a) :
+    ((l.attachWith p H).map fun i => f i.val) = l.map f :=
+  attachWith_map_coe _ _ _
 
 @[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 _ _
+theorem attachWith_map_subtype_val {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) :
+    (l.attachWith p H).map Subtype.val = l :=
+  (attachWith_map_coe _ _ _).trans (List.map_id _)
 
 @[simp]
 theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
@@ -137,7 +154,11 @@ theorem length_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H} : length (pm
   · simp only [*, pmap, length]
 
 @[simp]
-theorem length_attach (L : List α) : L.attach.length = L.length :=
+theorem length_attach {L : List α} : L.attach.length = L.length :=
+  length_pmap
+
+@[simp]
+theorem length_attachWith {p : α → Prop} {l H} : length (l.attachWith p H) = length l :=
   length_pmap
 
 @[simp]
@@ -155,6 +176,15 @@ theorem attach_eq_nil_iff {l : List α} : l.attach = [] ↔ l = [] :=
 theorem attach_ne_nil_iff {l : List α} : l.attach ≠ [] ↔ l ≠ [] :=
   pmap_ne_nil_iff _ _
 
+@[simp]
+theorem attachWith_eq_nil_iff {l : List α} {P : α → Prop} {H : ∀ a ∈ l, P a} :
+    l.attachWith P H = [] ↔ l = [] :=
+  pmap_eq_nil_iff
+
+theorem attachWith_ne_nil_iff {l : List α} {P : α → Prop} {H : ∀ a ∈ l, P a} :
+    l.attachWith P H ≠ [] ↔ l ≠ [] :=
+  pmap_ne_nil_iff _ _
+
 @[deprecated pmap_eq_nil_iff (since := "2024-09-06")] abbrev pmap_eq_nil := @pmap_eq_nil_iff
 @[deprecated pmap_ne_nil_iff (since := "2024-09-06")] abbrev pmap_ne_nil := @pmap_ne_nil_iff
 @[deprecated attach_eq_nil_iff (since := "2024-09-06")] abbrev attach_eq_nil := @attach_eq_nil_iff
@@ -187,7 +217,7 @@ theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h
     (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
+        (h _ (getElem_mem (@length_pmap _ _ p f l h ▸ hn))) := by
   induction l generalizing n with
   | nil =>
     simp only [length, pmap] at hn
@@ -205,34 +235,26 @@ theorem get_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h :
   simp only [get_eq_getElem]
   simp [getElem_pmap]
 
+@[simp]
+theorem getElem?_attachWith {xs : List α} {i : Nat} {P : α → Prop} {H : ∀ a ∈ xs, P a} :
+    (xs.attachWith P H)[i]? = xs[i]?.pmap Subtype.mk (fun _ a => H _ (getElem?_mem a)) :=
+  getElem?_pmap ..
+
 @[simp]
 theorem getElem?_attach {xs : List α} {i : Nat} :
-    xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => getElem?_mem a) := by
-  induction xs generalizing i with
-  | nil => simp
-  | cons x xs ih =>
-    rcases i with ⟨i⟩
-    · simp only [attach_cons, Option.pmap]
-      split <;> simp_all
-    · simp only [attach_cons, getElem?_cons_succ, getElem?_map, ih]
-      simp only [Option.pmap]
-      split <;> split <;> simp_all
+    xs.attach[i]? = xs[i]?.pmap Subtype.mk (fun _ a => getElem?_mem a) :=
+  getElem?_attachWith
+
+@[simp]
+theorem getElem_attachWith {xs : List α} {P : α → Prop} {H : ∀ a ∈ xs, P a}
+    {i : Nat} (h : i < (xs.attachWith P H).length) :
+    (xs.attachWith P H)[i] = ⟨xs[i]'(by simpa using h), H _ (getElem_mem (by simpa using h))⟩ :=
+  getElem_pmap ..
 
 @[simp]
 theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
-    xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem xs i (by simpa using h)⟩ := by
-  apply Option.some.inj
-  rw [← getElem?_eq_getElem]
-  rw [getElem?_attach]
-  simp only [Option.pmap]
-  split <;> rename_i h' _
-  · simp at h
-    simp at h'
-    exfalso
-    exact Nat.lt_irrefl _ (Nat.lt_of_le_of_lt h' h)
-  · simp only [Option.some.injEq, Subtype.mk.injEq]
-    apply Option.some.inj
-    rw [← getElem?_eq_getElem, h']
+    xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
+  getElem_attachWith h
 
 @[simp] theorem head?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
     (H : ∀ (a : α), a ∈ xs → P a) :
@@ -250,20 +272,112 @@ theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
   | nil => simp at h
   | cons x xs ih => simp [head_pmap, ih]
 
+@[simp] theorem head?_attachWith {P : α → Prop} {xs : List α}
+    (H : ∀ (a : α), a ∈ xs → P a) :
+    (xs.attachWith P H).head? = xs.head?.pbind (fun a h => some ⟨a, H _ (mem_of_mem_head? h)⟩) := by
+  cases xs <;> simp_all
+
+@[simp] theorem head_attachWith {P : α → Prop} {xs : List α}
+    {H : ∀ (a : α), a ∈ xs → P a} (h : xs.attachWith P H ≠ []) :
+    (xs.attachWith P H).head h = ⟨xs.head (by simpa using h), H _ (head_mem _)⟩ := by
+  cases xs with
+  | nil => simp at h
+  | cons x xs => simp [head_attachWith, h]
+
 @[simp] theorem head?_attach (xs : List α) :
     xs.attach.head? = xs.head?.pbind (fun a h => some ⟨a, mem_of_mem_head? h⟩) := by
   cases xs <;> simp_all
 
-theorem head_attach {xs : List α} (h) :
+@[simp] theorem head_attach {xs : List α} (h) :
     xs.attach.head h = ⟨xs.head (by simpa using h), head_mem (by simpa using h)⟩ := by
   cases xs with
   | nil => simp at h
   | cons x xs => simp [head_attach, h]
 
+@[simp] theorem tail_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
+    (H : ∀ (a : α), a ∈ xs → P a) :
+    (xs.pmap f H).tail = xs.tail.pmap f (fun a h => H a (mem_of_mem_tail h)) := by
+  cases xs <;> simp
+
+@[simp] theorem tail_attachWith {P : α → Prop} {xs : List α}
+    {H : ∀ (a : α), a ∈ xs → P a} :
+    (xs.attachWith P H).tail = xs.tail.attachWith P (fun a h => H a (mem_of_mem_tail h)) := by
+  cases xs <;> simp
+
+@[simp] theorem tail_attach (xs : List α) :
+    xs.attach.tail = xs.tail.attach.map (fun ⟨x, h⟩ => ⟨x, mem_of_mem_tail h⟩) := by
+  cases xs <;> simp
+
+theorem foldl_pmap (l : List α) {P : α → Prop} (f : (a : α) → P a → β)
+  (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
+    (l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
+  rw [pmap_eq_map_attach, foldl_map]
+
+theorem foldr_pmap (l : List α) {P : α → Prop} (f : (a : α) → P a → β)
+  (H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) :
+    (l.pmap f H).foldr g x = l.attach.foldr (fun a acc => g (f a.1 (H _ a.2)) acc) x := by
+  rw [pmap_eq_map_attach, foldr_map]
+
+/--
+If we fold over `l.attach` with a function that ignores the membership predicate,
+we get the same results as folding over `l` directly.
+
+This is useful when we need to use `attach` to show termination.
+
+Unfortunately this can't be applied by `simp` because of the higher order unification problem,
+and even when rewriting we need to specify the function explicitly.
+-/
+theorem foldl_attach (l : List α) (f : β → α → β) (b : β) :
+    l.attach.foldl (fun acc t => f acc t.1) b = l.foldl f b := by
+  induction l generalizing b with
+  | nil => simp
+  | cons a l ih => rw [foldl_cons, attach_cons, foldl_cons, foldl_map, ih]
+
+/--
+If we fold over `l.attach` with a function that ignores the membership predicate,
+we get the same results as folding over `l` directly.
+
+This is useful when we need to use `attach` to show termination.
+
+Unfortunately this can't be applied by `simp` because of the higher order unification problem,
+and even when rewriting we need to specify the function explicitly.
+-/
+theorem foldr_attach (l : List α) (f : α → β → β) (b : β) :
+    l.attach.foldr (fun t acc => f t.1 acc) b = l.foldr f b := by
+  induction l generalizing b with
+  | nil => simp
+  | cons a l ih => rw [foldr_cons, attach_cons, foldr_cons, foldr_map, ih]
+
 theorem attach_map {l : List α} (f : α → β) :
     (l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) := by
   induction l <;> simp [*]
 
+theorem attachWith_map {l : List α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ l.map f → P b} :
+    (l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun a h => H _ (mem_map_of_mem f h))).map
+      fun ⟨x, h⟩ => ⟨f x, h⟩ := by
+  induction l <;> simp [*]
+
+theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
+    (f : { x // P x } → β) :
+    (l.attachWith P H).map f =
+      l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [attachWith_cons, map_cons, ih, pmap, cons.injEq, true_and]
+    apply pmap_congr_left
+    simp
+
+/-- See also `pmap_eq_map_attach` for writing `pmap` in terms of `map` and `attach`. -/
+theorem map_attach {l : List α} (f : { x // x ∈ l } → β) :
+    l.attach.map f = l.pmap (fun a h => f ⟨a, h⟩) (fun _ => id) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [attach_cons, map_cons, map_map, Function.comp_apply, pmap, cons.injEq, true_and, ih]
+    apply pmap_congr_left
+    simp
+
 theorem attach_filterMap {l : List α} {f : α → Option β} :
     (l.filterMap f).attach = l.attach.filterMap
       fun ⟨x, h⟩ => (f x).pbind (fun b m => some ⟨b, mem_filterMap.mpr ⟨x, h, m⟩⟩) := by
@@ -304,6 +418,9 @@ theorem attach_filter {l : List α} (p : α → Bool) :
   ext1
   split <;> simp
 
+-- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
+-- Also missing are `filterMap_attach`, `filter_attach`, `filterMap_attachWith` and `filter_attachWith`.
+
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
     pmap f (pmap g l H₁) H₂ =
       pmap (α := { x // x ∈ l }) (fun a h => f (g a h) (H₂ (g a h) (mem_pmap_of_mem a.2))) l.attach
@@ -334,6 +451,12 @@ theorem pmap_append' {p : α → Prop} (f : ∀ a : α, p a → β) (l₁ l₂ :
   congr 1 <;>
   exact pmap_congr_left _ fun _ _ _ _ => rfl
 
+@[simp] theorem attachWith_append {P : α → Prop} {xs ys : List α}
+    {H : ∀ (a : α), a ∈ xs ++ ys → P a} :
+    (xs ++ ys).attachWith P H = xs.attachWith P (fun a h => H a (mem_append_of_mem_left ys h)) ++
+      ys.attachWith P (fun a h => H a (mem_append_of_mem_right xs h)) := by
+  simp only [attachWith, attach_append, map_pmap, pmap_append]
+
 @[simp] theorem pmap_reverse {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
     (H : ∀ (a : α), a ∈ xs.reverse → P a) :
     xs.reverse.pmap f H = (xs.pmap f (fun a h => H a (by simpa using h))).reverse := by
@@ -344,6 +467,17 @@ theorem reverse_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List
     (xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h)) := by
   rw [pmap_reverse]
 
+@[simp] theorem attachWith_reverse {P : α → Prop} {xs : List α}
+    {H : ∀ (a : α), a ∈ xs.reverse → P a} :
+    xs.reverse.attachWith P H =
+      (xs.attachWith P (fun a h => H a (by simpa using h))).reverse :=
+  pmap_reverse ..
+
+theorem reverse_attachWith {P : α → Prop} {xs : List α}
+    {H : ∀ (a : α), a ∈ xs → P a} :
+    (xs.attachWith P H).reverse = (xs.reverse.attachWith P (fun a h => H a (by simpa using h))) :=
+  reverse_pmap ..
+
 @[simp] theorem attach_reverse (xs : List α) :
     xs.reverse.attach = xs.attach.reverse.map fun ⟨x, h⟩ => ⟨x, by simpa using h⟩ := by
   simp only [attach, attachWith, reverse_pmap, map_pmap]
@@ -358,17 +492,6 @@ theorem reverse_attach (xs : List α) :
   intros
   rfl
 
-@[simp]
-theorem getLast?_attach {xs : List α} :
-    xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast?_eq_some h⟩ := by
-  rw [getLast?_eq_head?_reverse, reverse_attach, head?_map, head?_attach]
-  simp
-
-@[simp]
-theorem getLast_attach {xs : List α} (h : xs.attach ≠ []) :
-    xs.attach.getLast h = ⟨xs.getLast (by simpa using h), getLast_mem (by simpa using h)⟩ := by
-  simp only [getLast_eq_head_reverse, reverse_attach, head_map, head_attach]
-
 @[simp] theorem getLast?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).getLast? = xs.attach.getLast?.map fun ⟨a, m⟩ => f a (H a m) := by
@@ -383,4 +506,46 @@ theorem getLast_attach {xs : List α} (h : xs.attach ≠ []) :
   simp only [getLast_eq_head_reverse]
   simp only [reverse_pmap, head_pmap, head_reverse]
 
+@[simp] theorem getLast?_attachWith {P : α → Prop} {xs : List α}
+    {H : ∀ (a : α), a ∈ xs → P a} :
+    (xs.attachWith P H).getLast? = xs.getLast?.pbind (fun a h => some ⟨a, H _ (mem_of_getLast?_eq_some h)⟩) := by
+  rw [getLast?_eq_head?_reverse, reverse_attachWith, head?_attachWith]
+  simp
+
+@[simp] theorem getLast_attachWith {P : α → Prop} {xs : List α}
+    {H : ∀ (a : α), a ∈ xs → P a} (h : xs.attachWith P H ≠ []) :
+    (xs.attachWith P H).getLast h = ⟨xs.getLast (by simpa using h), H _ (getLast_mem _)⟩ := by
+  simp only [getLast_eq_head_reverse, reverse_attachWith, head_attachWith, head_map]
+
+@[simp]
+theorem getLast?_attach {xs : List α} :
+    xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast?_eq_some h⟩ := by
+  rw [getLast?_eq_head?_reverse, reverse_attach, head?_map, head?_attach]
+  simp
+
+@[simp]
+theorem getLast_attach {xs : List α} (h : xs.attach ≠ []) :
+    xs.attach.getLast h = ⟨xs.getLast (by simpa using h), getLast_mem (by simpa using h)⟩ := by
+  simp only [getLast_eq_head_reverse, reverse_attach, head_map, head_attach]
+
+@[simp]
+theorem countP_attach (l : List α) (p : α → Bool) :
+    l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
+  simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attach_map_subtype_val]
+
+@[simp]
+theorem countP_attachWith {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) (q : α → Bool) :
+    (l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
+  simp only [← Function.comp_apply (g := Subtype.val), ← countP_map, attachWith_map_subtype_val]
+
+@[simp]
+theorem count_attach [DecidableEq α] (l : List α) (a : {x // x ∈ l}) :
+    l.attach.count a = l.count ↑a :=
+  Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attach _ _
+
+@[simp]
+theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : List α) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
+    (l.attachWith p H).count a = l.count ↑a :=
+  Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attachWith _ _ _
+
 end List

src/Init/Data/List/Basic.lean

@@ -1588,6 +1588,14 @@ such that adjacent elements are related by `R`.
   | []    => []
   | a::as => loop as a [] []
 where
+  /--
+  The arguments of `groupBy.loop l ag g gs` represent the following:
+
+  - `l : List α` are the elements which we still need to group.
+  - `ag : α` is the previous element for which a comparison was performed.
+  - `g : List α` is the group currently being assembled, in **reverse order**.
+  - `gs : List (List α)` is all of the groups that have been completed, in **reverse order**.
+  -/
   @[specialize] loop : List α → α → List α → List (List α) → List (List α)
   | a::as, ag, g, gs => match R ag a with
     | true  => loop as a (ag::g) gs

src/Init/Data/List/BasicAux.lean

@@ -155,7 +155,7 @@ def mapMono (as : List α) (f : α → α) : List α :=
 
 /-! ## Additional lemmas required for bootstrapping `Array`. -/
 
-theorem getElem_append_left (as bs : List α) (h : i < as.length) {h'} : (as ++ bs)[i] = as[i] := by
+theorem getElem_append_left {as bs : List α} (h : i < as.length) {h'} : (as ++ bs)[i] = as[i] := by
   induction as generalizing i with
   | nil => trivial
   | cons a as ih =>
@@ -163,12 +163,14 @@ theorem getElem_append_left (as bs : List α) (h : i < as.length) {h'} : (as ++
     | zero => rfl
     | succ i => apply ih
 
-theorem getElem_append_right (as bs : List α) (h : ¬ i < as.length) {h' h''} : (as ++ bs)[i]'h' = bs[i - as.length]'h'' := by
+theorem getElem_append_right {as bs : List α} {i : Nat} (h₁ : as.length ≤ i) {h₂} :
+    (as ++ bs)[i]'h₂ =
+      bs[i - as.length]'(by rw [length_append] at h₂; exact Nat.sub_lt_left_of_lt_add h₁ h₂) := by
   induction as generalizing i with
   | nil => trivial
   | cons a as ih =>
-    cases i with simp [get, Nat.succ_sub_succ] <;> simp [Nat.succ_sub_succ] at h
-    | succ i => apply ih; simp [h]
+    cases i with simp [get, Nat.succ_sub_succ] <;> simp [Nat.succ_sub_succ] at h₁
+    | succ i => apply ih; simp [h₁]
 
 theorem get_last {as : List α} {i : Fin (length (as ++ [a]))} (h : ¬ i.1 < as.length) : (as ++ [a] : List _).get i = a := by
   cases i; rename_i i h'

src/Init/Data/List/Count.lean

@@ -115,6 +115,13 @@ theorem IsPrefix.countP_le (s : l₁ <+: l₂) : countP p l₁ ≤ countP p l₂
 theorem IsSuffix.countP_le (s : l₁ <:+ l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
 theorem IsInfix.countP_le (s : l₁ <:+: l₂) : countP p l₁ ≤ countP p l₂ := s.sublist.countP_le _
 
+-- See `Init.Data.List.Nat.Count` for `Sublist.le_countP : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁`.
+
+theorem countP_tail_le (l) : countP p l.tail ≤ countP p l :=
+  (tail_sublist l).countP_le _
+
+-- See `Init.Data.List.Nat.Count` for `le_countP_tail : countP p l - 1 ≤ countP p l.tail`.
+
 theorem countP_filter (l : List α) :
     countP p (filter q l) = countP (fun a => p a && q a) l := by
   simp only [countP_eq_length_filter, filter_filter]
@@ -207,6 +214,13 @@ theorem IsPrefix.count_le (h : l₁ <+: l₂) (a : α) : count a l₁ ≤ count
 theorem IsSuffix.count_le (h : l₁ <:+ l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
 theorem IsInfix.count_le (h : l₁ <:+: l₂) (a : α) : count a l₁ ≤ count a l₂ := h.sublist.count_le _
 
+-- See `Init.Data.List.Nat.Count` for `Sublist.le_count : count a l₂ - (l₂.length - l₁.length) ≤ countP a l₁`.
+
+theorem count_tail_le (a : α) (l) : count a l.tail ≤ count a l :=
+  (tail_sublist l).count_le _
+
+-- See `Init.Data.List.Nat.Count` for `le_count_tail : count a l - 1 ≤ count a l.tail`.
+
 theorem count_le_count_cons (a b : α) (l : List α) : count a l ≤ count a (b :: l) :=
   (sublist_cons_self _ _).count_le _
 

src/Init/Data/List/Erase.lean

@@ -109,6 +109,10 @@ protected theorem Sublist.eraseP : l₁ <+ l₂ → l₁.eraseP p <+ l₂.eraseP
 theorem length_eraseP_le (l : List α) : (l.eraseP p).length ≤ l.length :=
   l.eraseP_sublist.length_le
 
+theorem le_length_eraseP (l : List α) : l.length - 1 ≤ (l.eraseP p).length := by
+  rw [length_eraseP]
+  split <;> simp
+
 theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (eraseP_subset _ ·)
 
 @[simp] theorem mem_eraseP_of_neg {l : List α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
@@ -332,6 +336,10 @@ theorem IsPrefix.erase (a : α) {l₁ l₂ : List α} (h : l₁ <+: l₂) : l₁
 theorem length_erase_le (a : α) (l : List α) : (l.erase a).length ≤ l.length :=
   (erase_sublist a l).length_le
 
+theorem le_length_erase [LawfulBEq α] (a : α) (l : List α) : l.length - 1 ≤ (l.erase a).length := by
+  rw [length_erase]
+  split <;> simp
+
 theorem mem_of_mem_erase {a b : α} {l : List α} (h : a ∈ l.erase b) : a ∈ l := erase_subset _ _ h
 
 @[simp] theorem mem_erase_of_ne [LawfulBEq α] {a b : α} {l : List α} (ab : a ≠ b) :
@@ -452,13 +460,22 @@ 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)]
+theorem length_eraseIdx (l : List α) (i : Nat) :
+    (l.eraseIdx i).length = if i < l.length then l.length - 1 else l.length := by
+  induction l generalizing i with
+  | nil => simp
+  | cons x l ih =>
+    cases i with
+    | zero => simp
+    | succ i =>
+      simp only [eraseIdx, length_cons, ih, add_one_lt_add_one_iff, Nat.add_one_sub_one]
+      split
+      · cases l <;> simp_all
+      · rfl
+
+theorem length_eraseIdx_of_lt {l : List α} {i} (h : i < length l) :
+    (l.eraseIdx i).length = length l - 1 := by
+  simp [length_eraseIdx, h]
 
 @[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
 
@@ -468,6 +485,8 @@ theorem eraseIdx_eq_take_drop_succ :
   | a::l, 0 => by simp
   | a::l, i + 1 => by simp [eraseIdx_eq_take_drop_succ l i]
 
+-- See `Init.Data.List.Nat.Erase` for `getElem?_eraseIdx` and `getElem_eraseIdx`.
+
 @[simp] theorem eraseIdx_eq_nil {l : List α} {i : Nat} : eraseIdx l i = [] ↔ l = [] ∨ (length l = 1 ∧ i = 0) := by
   match l, i with
   | [], _
@@ -499,6 +518,13 @@ theorem eraseIdx_eq_self : ∀ {l : List α} {k : Nat}, eraseIdx l k = l ↔ len
 theorem eraseIdx_of_length_le {l : List α} {k : Nat} (h : length l ≤ k) : eraseIdx l k = l := by
   rw [eraseIdx_eq_self.2 h]
 
+theorem length_eraseIdx_le (l : List α) (i : Nat) : length (l.eraseIdx i) ≤ length l :=
+  (eraseIdx_sublist l i).length_le
+
+theorem le_length_eraseIdx (l : List α) (i : Nat) : length l - 1 ≤ length (l.eraseIdx i) := by
+  rw [length_eraseIdx]
+  split <;> simp
+
 theorem eraseIdx_append_of_lt_length {l : List α} {k : Nat} (hk : k < length l) (l' : List α) :
     eraseIdx (l ++ l') k = eraseIdx l k ++ l' := by
   induction l generalizing k with
@@ -520,7 +546,7 @@ theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤
 theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
     (replicate n a).eraseIdx k = if k < n then replicate (n - 1) a else replicate n a := by
   split <;> rename_i h
-  · rw [eq_replicate_iff, length_eraseIdx (by simpa using h)]
+  · rw [eq_replicate_iff, length_eraseIdx_of_lt (by simpa using h)]
     simp only [length_replicate, true_and]
     intro b m
     replace m := mem_of_mem_eraseIdx m

src/Init/Data/List/Find.lean

@@ -224,7 +224,7 @@ theorem find?_eq_some : xs.find? p = some b ↔ p b ∧ ∃ as bs, xs = as ++ b
           simp only [cons_append] at h₁
           obtain ⟨rfl, -⟩ := h₁
           simp_all
-    · simp only [ih, Bool.not_eq_true', exists_and_right, and_congr_right_iff]
+    · simp only [ih, Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, and_congr_right_iff]
       intro pb
       constructor
       · rintro ⟨as, ⟨⟨bs, rfl⟩, h₁⟩⟩
@@ -620,6 +620,18 @@ theorem IsPrefix.findIdx_eq_of_findIdx_lt_length {l₁ l₂ : List α} {p : α
   · rfl
   · simp_all
 
+theorem findIdx_le_findIdx {l : List α} {p q : α → Bool} (h : ∀ x ∈ l, p x → q x) : l.findIdx q ≤ l.findIdx p := by
+  induction l with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx_cons, cond_eq_if]
+    split
+    · simp
+    · split
+      · simp_all
+      · simp only [Nat.add_le_add_iff_right]
+        exact ih fun _ m w => h _ (mem_cons_of_mem x m) w
+
 /-! ### findIdx? -/
 
 @[simp] theorem findIdx?_nil : ([] : List α).findIdx? p i = none := rfl
@@ -803,7 +815,7 @@ theorem findIdx?_join {l : List (List α)} {p : α → Bool} :
     simp only [replicate, findIdx?_cons, Nat.zero_add, findIdx?_succ, zero_lt_succ, true_and]
     split <;> simp_all
 
-theorem findIdx?_eq_enum_findSome? {xs : List α} {p : α → Bool} :
+theorem findIdx?_eq_findSome?_enum {xs : List α} {p : α → Bool} :
     xs.findIdx? p = xs.enum.findSome? fun ⟨i, a⟩ => if p a then some i else none := by
   induction xs with
   | nil => simp
@@ -814,6 +826,30 @@ theorem findIdx?_eq_enum_findSome? {xs : List α} {p : α → Bool} :
     · simp_all only [enumFrom_cons, ite_false, Option.isNone_none, findSome?_cons_of_isNone, reduceCtorEq]
       simp [Function.comp_def, ← map_fst_add_enum_eq_enumFrom, findSome?_map]
 
+theorem findIdx?_eq_fst_find?_enum {xs : List α} {p : α → Bool} :
+    xs.findIdx? p = (xs.enum.find? fun ⟨_, x⟩ => p x).map (·.1) := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [findIdx?_cons, Nat.zero_add, findIdx?_start_succ, enum_cons]
+    split
+    · simp_all
+    · simp only [Option.map_map, enumFrom_eq_map_enum, Bool.false_eq_true, not_false_eq_true,
+        find?_cons_of_neg, find?_map, *]
+      congr
+
+-- See also `findIdx_le_findIdx`.
+theorem findIdx?_eq_none_of_findIdx?_eq_none {xs : List α} {p q : α → Bool} (w : ∀ x ∈ xs, p x → q x) :
+    xs.findIdx? q = none → xs.findIdx? p = none := by
+  simp only [findIdx?_eq_none_iff]
+  intro h x m
+  cases z : p x
+  · rfl
+  · exfalso
+    specialize w x m z
+    specialize h x m
+    simp_all
+
 theorem Sublist.findIdx?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
     (l₁.findIdx? p).isSome → (l₂.findIdx? p).isSome := by
   simp only [List.findIdx?_isSome, any_eq_true]
@@ -878,7 +914,7 @@ theorem lookup_eq_some_iff {l : List (α × β)} {k : α} {b : β} :
   simp only [lookup_eq_findSome?, findSome?_eq_some_iff]
   constructor
   · rintro ⟨l₁, a, l₂, rfl, h₁, h₂⟩
-    simp only [beq_iff_eq, ite_some_none_eq_some] at h₁
+    simp only [beq_iff_eq, Option.ite_none_right_eq_some, Option.some.injEq] at h₁
     obtain ⟨rfl, rfl⟩ := h₁
     simp at h₂
     exact ⟨l₁, l₂, rfl, by simpa using h₂⟩

src/Init/Data/List/Lemmas.lean

@@ -109,6 +109,9 @@ theorem cons_eq_cons {a b : α} {l l' : List α} : a :: l = b :: l' ↔ a = b
 theorem exists_cons_of_ne_nil : ∀ {l : List α}, l ≠ [] → ∃ b L, l = b :: L
   | c :: l', _ => ⟨c, l', rfl⟩
 
+theorem singleton_inj {α : Type _} {a b : α} : [a] = [b] ↔ a = b := by
+  simp
+
 /-! ### length -/
 
 theorem eq_nil_of_length_eq_zero (_ : length l = 0) : l = [] := match l with | [] => rfl
@@ -263,9 +266,15 @@ theorem get!_len_le [Inhabited α] : ∀ {l : List α} {n}, length l ≤ n → l
 theorem getElem?_eq_some_iff {l : List α} : l[n]? = some a ↔ ∃ h : n < l.length, l[n] = a := by
   simp only [← get?_eq_getElem?, get?_eq_some, get_eq_getElem]
 
+theorem some_eq_getElem?_iff {l : List α} : some a = l[n]? ↔ ∃ h : n < l.length, l[n] = a := by
+  rw [eq_comm, getElem?_eq_some_iff]
+
 @[simp] theorem getElem?_eq_none_iff : l[n]? = none ↔ length l ≤ n := by
   simp only [← get?_eq_getElem?, get?_eq_none]
 
+@[simp] theorem none_eq_getElem?_iff {l : List α} {n : Nat} : none = l[n]? ↔ length l ≤ n := by
+  simp [eq_comm (a := none)]
+
 theorem getElem?_eq_none (h : length l ≤ n) : l[n]? = none := getElem?_eq_none_iff.mpr h
 
 theorem getElem?_eq (l : List α) (i : Nat) :
@@ -480,9 +489,9 @@ theorem getElem?_of_mem {a} {l : List α} (h : a ∈ l) : ∃ n : Nat, l[n]? = s
 theorem get?_of_mem {a} {l : List α} (h : a ∈ l) : ∃ n, l.get? n = some a :=
   let ⟨⟨n, _⟩, e⟩ := get_of_mem h; ⟨n, e ▸ get?_eq_get _⟩
 
-theorem getElem_mem : ∀ (l : List α) n (h : n < l.length), l[n]'h ∈ l
+theorem getElem_mem : ∀ {l : List α} {n} (h : n < l.length), l[n]'h ∈ l
   | _ :: _, 0, _ => .head ..
-  | _ :: l, _+1, _ => .tail _ (getElem_mem l ..)
+  | _ :: l, _+1, _ => .tail _ (getElem_mem (l := l) ..)
 
 theorem get_mem : ∀ (l : List α) n h, get l ⟨n, h⟩ ∈ l
   | _ :: _, 0, _ => .head ..
@@ -524,7 +533,7 @@ theorem forall_getElem {l : List α} {p : α → Prop} :
       · simpa
       · apply w
         simp only [getElem_cons_succ]
-        exact getElem_mem l n (lt_of_succ_lt_succ h)
+        exact getElem_mem (lt_of_succ_lt_succ h)
 
 @[simp] theorem decide_mem_cons [BEq α] [LawfulBEq α] {l : List α} :
     decide (y ∈ a :: l) = (y == a || decide (y ∈ l)) := by
@@ -729,6 +738,45 @@ theorem mem_or_eq_of_mem_set : ∀ {l : List α} {n : Nat} {a b : α}, a ∈ l.s
 
 -- See also `set_eq_take_append_cons_drop` in `Init.Data.List.TakeDrop`.
 
+/-! ### BEq -/
+
+@[simp] theorem reflBEq_iff [BEq α] : ReflBEq (List α) ↔ ReflBEq α := by
+  constructor
+  · intro h
+    constructor
+    intro a
+    suffices ([a] == [a]) = true by
+      simpa only [List.instBEq, List.beq, Bool.and_true]
+    simp
+  · intro h
+    constructor
+    intro a
+    induction a with
+    | nil => simp only [List.instBEq, List.beq]
+    | cons a as ih =>
+      simp [List.instBEq, List.beq]
+      exact ih
+
+@[simp] theorem lawfulBEq_iff [BEq α] : LawfulBEq (List α) ↔ LawfulBEq α := by
+  constructor
+  · intro h
+    constructor
+    · intro a b h
+      apply singleton_inj.1
+      apply eq_of_beq
+      simp only [List.instBEq, List.beq]
+      simpa
+    · intro a
+      suffices ([a] == [a]) = true by
+        simpa only [List.instBEq, List.beq, Bool.and_true]
+      simp
+  · intro h
+    constructor
+    · intro a b h
+      simpa using h
+    · intro a
+      simp
+
 /-! ### Lexicographic ordering -/
 
 protected theorem lt_irrefl [LT α] (lt_irrefl : ∀ x : α, ¬x < x) (l : List α) : ¬l < l := by
@@ -929,6 +977,10 @@ theorem getLast_mem : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ l
   | [_], _ => .head ..
   | _::a::l, _ => .tail _ <| getLast_mem (cons_ne_nil a l)
 
+theorem getLast_mem_getLast? : ∀ {l : List α} (h : l ≠ []), getLast l h ∈ getLast? l
+  | [], h => by contradiction
+  | a :: l, _ => rfl
+
 theorem getLastD_mem_cons : ∀ (l : List α) (a : α), getLastD l a ∈ a::l
   | [], _ => .head ..
   | _::_, _ => .tail _ <| getLast_mem _
@@ -999,6 +1051,11 @@ theorem head?_eq_getElem? : ∀ l : List α, head? l = l[0]?
   | [] => rfl
   | a :: l => by simp
 
+theorem head_eq_getElem (l : List α) (h : l ≠ []) : head l h = l[0]'(length_pos.mpr h) := by
+  cases l with
+  | nil => simp at h
+  | cons _ _ => simp
+
 theorem head_eq_iff_head?_eq_some {xs : List α} (h) : xs.head h = a ↔ xs.head? = some a := by
   cases xs with
   | nil => simp at h
@@ -1026,6 +1083,10 @@ theorem mem_of_mem_head? : ∀ {l : List α} {a : α}, a ∈ l.head? → a ∈ l
     cases h
     exact mem_cons_self a l
 
+theorem head_mem_head? : ∀ {l : List α} (h : l ≠ []), head l h ∈ head? l
+  | [], h => by contradiction
+  | a :: l, _ => rfl
+
 theorem head?_concat {a : α} : (l ++ [a]).head? = l.head?.getD a := by
   cases l <;> simp
 
@@ -1055,6 +1116,55 @@ theorem tail_eq_tail? (l) : @tail α l = (tail? l).getD [] := by simp [tail_eq_t
 theorem mem_of_mem_tail {a : α} {l : List α} (h : a ∈ tail l) : a ∈ l := by
   induction l <;> simp_all
 
+theorem ne_nil_of_tail_ne_nil {l : List α} : l.tail ≠ [] → l ≠ [] := by
+  cases l <;> simp
+
+@[simp] theorem getElem_tail (l : List α) (i : Nat) (h : i < l.tail.length) :
+    (tail l)[i] = l[i + 1]'(add_lt_of_lt_sub (by simpa using h)) := by
+  cases l with
+  | nil => simp at h
+  | cons _ l => simp
+
+@[simp] theorem getElem?_tail (l : List α) (i : Nat) :
+    (tail l)[i]? = l[i + 1]? := by
+  cases l <;> simp
+
+@[simp] theorem set_tail (l : List α) (i : Nat) (a : α) :
+    l.tail.set i a = (l.set (i + 1) a).tail := by
+  cases l <;> simp
+
+theorem one_lt_length_of_tail_ne_nil {l : List α} (h : l.tail ≠ []) : 1 < l.length := by
+  cases l with
+  | nil => simp at h
+  | cons _ l =>
+    simp only [tail_cons, ne_eq] at h
+    exact Nat.lt_add_of_pos_left (length_pos.mpr h)
+
+@[simp] theorem head_tail (l : List α) (h : l.tail ≠ []) :
+    (tail l).head h = l[1]'(one_lt_length_of_tail_ne_nil h) := by
+  cases l with
+  | nil => simp at h
+  | cons _ l => simp [head_eq_getElem]
+
+@[simp] theorem head?_tail (l : List α) : (tail l).head? = l[1]? := by
+  simp [head?_eq_getElem?]
+
+@[simp] theorem getLast_tail (l : List α) (h : l.tail ≠ []) :
+    (tail l).getLast h = l.getLast (ne_nil_of_tail_ne_nil h) := by
+  simp only [getLast_eq_getElem, length_tail, getElem_tail]
+  congr
+  match l with
+  | _ :: _ :: l => simp
+
+theorem getLast?_tail (l : List α) : (tail l).getLast? = if l.length = 1 then none else l.getLast? := by
+  match l with
+  | [] => simp
+  | [a] => simp
+  | _ :: _ :: l =>
+    simp only [tail_cons, length_cons, getLast?_cons_cons]
+    rw [if_neg]
+    rintro ⟨⟩
+
 /-! ## Basic operations -/
 
 /-! ### map -/
@@ -1507,10 +1617,11 @@ theorem filterMap_eq_cons_iff {l} {b} {bs} :
 
 /-! ### append -/
 
-theorem getElem_append : ∀ {l₁ l₂ : List α} (n : Nat) (h : n < l₁.length),
-    (l₁ ++ l₂)[n]'(length_append .. ▸ Nat.lt_add_right _ h) = l₁[n]
-| a :: l, _, 0, h => rfl
-| a :: l, _, n+1, h => by simp only [get, cons_append]; apply getElem_append
+theorem getElem_append {l₁ l₂ : List α} (n : Nat) (h) :
+    (l₁ ++ l₂)[n] = if h' : n < l₁.length then l₁[n] else l₂[n - l₁.length]'(by simp at h h'; exact Nat.sub_lt_left_of_lt_add h' h) := by
+  split <;> rename_i h'
+  · rw [getElem_append_left h']
+  · rw [getElem_append_right (by simpa using h')]
 
 theorem getElem?_append_left {l₁ l₂ : List α} {n : Nat} (hn : n < l₁.length) :
     (l₁ ++ l₂)[n]? = l₁[n]? := by
@@ -1536,12 +1647,13 @@ theorem get?_append_right {l₁ l₂ : List α} {n : Nat} (h : l₁.length ≤ n
     (l₁ ++ l₂).get? n = l₂.get? (n - l₁.length) := by
   simp [getElem?_append_right, h]
 
-theorem getElem_append_right' {l₁ l₂ : List α} {n : Nat} (h₁ : l₁.length ≤ n) (h₂) :
-    (l₁ ++ l₂)[n]'h₂ =
-      l₂[n - l₁.length]'(by rw [length_append] at h₂; exact Nat.sub_lt_left_of_lt_add h₁ h₂) :=
-  Option.some.inj <| by rw [← getElem?_eq_getElem, ← getElem?_eq_getElem, getElem?_append_right h₁]
+/-- Variant of `getElem_append_left` useful for rewriting from the small list to the big list. -/
+theorem getElem_append_left' (l₂ : List α) {l₁ : List α} {n : Nat} (hn : n < l₁.length) :
+    l₁[n] = (l₁ ++ l₂)[n]'(by simpa using Nat.lt_add_right l₂.length hn) := by
+  rw [getElem_append_left] <;> simp
 
-theorem getElem_append_right'' (l₁ : List α) {l₂ : List α} {n : Nat} (hn : n < l₂.length) :
+/-- Variant of `getElem_append_right` useful for rewriting from the small list to the big list. -/
+theorem getElem_append_right' (l₁ : List α) {l₂ : List α} {n : Nat} (hn : n < l₂.length) :
     l₂[n] = (l₁ ++ l₂)[n + l₁.length]'(by simpa [Nat.add_comm] using Nat.add_lt_add_left hn _) := by
   rw [getElem_append_right] <;> simp [*, le_add_left]
 
@@ -1552,7 +1664,7 @@ theorem get_append_right_aux {l₁ l₂ : List α} {n : Nat}
   exact Nat.sub_lt_left_of_lt_add h₁ h₂
 
 set_option linter.deprecated false in
-@[deprecated getElem_append_right' (since := "2024-06-12")]
+@[deprecated getElem_append_right (since := "2024-06-12")]
 theorem get_append_right' {l₁ l₂ : List α} {n : Nat} (h₁ : l₁.length ≤ n) (h₂) :
     (l₁ ++ l₂).get ⟨n, h₂⟩ = l₂.get ⟨n - l₁.length, get_append_right_aux h₁ h₂⟩ :=
   Option.some.inj <| by rw [← get?_eq_get, ← get?_eq_get, get?_append_right h₁]
@@ -1644,7 +1756,7 @@ theorem get_append_left (as bs : List α) (h : i < as.length) {h'} :
   simp [getElem_append_left, h, h']
 
 @[deprecated getElem_append_right (since := "2024-06-12")]
-theorem get_append_right (as bs : List α) (h : ¬ i < as.length) {h' h''} :
+theorem get_append_right (as bs : List α) (h : as.length ≤ i) {h' h''} :
     (as ++ bs).get ⟨i, h'⟩ = bs.get ⟨i - as.length, h''⟩ := by
   simp [getElem_append_right, h, h', h'']
 
@@ -1840,7 +1952,7 @@ theorem map_eq_append_iff {f : α → β} :
     map f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
   rw [← filterMap_eq_map, filterMap_eq_append_iff]
 
-theorem append_eq_map_iff (f : α → β) :
+theorem append_eq_map_iff {f : α → β} :
     L₁ ++ L₂ = map f l ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = L₁ ∧ map f l₂ = L₂ := by
   rw [eq_comm, map_eq_append_iff]
 
@@ -2323,6 +2435,47 @@ theorem bind_replicate {β} (f : α → List β) : (replicate n a).bind f = (rep
 @[simp] theorem isEmpty_replicate : (replicate n a).isEmpty = decide (n = 0) := by
   cases n <;> simp [replicate_succ]
 
+/-- Every list is either empty, a non-empty `replicate`, or begins with a non-empty `replicate`
+followed by a different element. -/
+theorem eq_replicate_or_eq_replicate_append_cons {α : Type _} (l : List α) :
+    (l = []) ∨ (∃ n a, l = replicate n a ∧ 0 < n) ∨
+      (∃ n a b l', l = replicate n a ++ b :: l' ∧ 0 < n ∧ a ≠ b) := by
+  induction l with
+  | nil => simp
+  | cons x l ih =>
+    right
+    rcases ih with rfl | ⟨n, a, rfl, h⟩ | ⟨n, a, b, l', rfl, h⟩
+    · left
+      exact ⟨1, x, rfl, by decide⟩
+    · by_cases h' : x = a
+      · subst h'
+        left
+        exact ⟨n + 1, x, rfl, by simp⟩
+      · right
+        refine ⟨1, x, a, replicate (n - 1) a, ?_, by decide, h'⟩
+        match n with | n + 1 => simp [replicate_succ]
+    · right
+      by_cases h' : x = a
+      · subst h'
+        refine ⟨n + 1, x, b, l', by simp [replicate_succ], by simp, h.2⟩
+      · refine ⟨1, x, a, replicate (n - 1) a ++ b :: l', ?_, by decide, h'⟩
+        match n with | n + 1 => simp [replicate_succ]
+
+/-- An induction principle for lists based on contiguous runs of identical elements. -/
+-- A `Sort _` valued version would require a different design. (And associated `@[simp]` lemmas.)
+theorem replicateRecOn {α : Type _} {p : List α → Prop} (m : List α)
+    (h0 : p []) (hr : ∀ a n, 0 < n → p (replicate n a))
+    (hi : ∀ a b n l, a ≠ b → 0 < n → p (b :: l) → p (replicate n a ++ b :: l)) : p m := by
+  rcases eq_replicate_or_eq_replicate_append_cons m with
+    rfl | ⟨n, a, rfl, hn⟩ | ⟨n, a, b, l', w, hn, h⟩
+  · exact h0
+  · exact hr _ _ hn
+  · have : (b :: l').length < m.length := by
+      simpa [w] using Nat.lt_add_of_pos_left hn
+    subst w
+    exact hi _ _ _ _ h hn (replicateRecOn (b :: l') h0 hr hi)
+termination_by m.length
+
 /-! ### reverse -/
 
 @[simp] theorem length_reverse (as : List α) : (as.reverse).length = as.length := by
@@ -2754,6 +2907,12 @@ theorem dropLast_append_cons : dropLast (l₁ ++ b :: l₂) = l₁ ++ dropLast (
     dropLast (a :: replicate n a) = replicate n a := by
   rw [← replicate_succ, dropLast_replicate, Nat.add_sub_cancel]
 
+@[simp] theorem tail_reverse (l : List α) : l.reverse.tail = l.dropLast.reverse := by
+  apply ext_getElem
+  · simp
+  · intro i h₁ h₂
+    simp [Nat.add_comm i, Nat.sub_add_eq]
+
 /-!
 ### splitAt
 

src/Init/Data/List/Nat.lean

@@ -10,3 +10,5 @@ import Init.Data.List.Nat.Range
 import Init.Data.List.Nat.Sublist
 import Init.Data.List.Nat.TakeDrop
 import Init.Data.List.Nat.Count
+import Init.Data.List.Nat.Erase
+import Init.Data.List.Nat.Find

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

@@ -5,6 +5,7 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Count
+import Init.Data.List.Find
 import Init.Data.List.MinMax
 import Init.Data.Nat.Lemmas
 
@@ -18,6 +19,26 @@ open Nat
 
 namespace List
 
+/-! ### dropLast -/
+
+theorem tail_dropLast (l : List α) : tail (dropLast l) = dropLast (tail l) := by
+  ext1
+  simp only [getElem?_tail, getElem?_dropLast, length_tail]
+  split <;> split
+  · rfl
+  · omega
+  · omega
+  · rfl
+
+@[simp] theorem dropLast_reverse (l : List α) : l.reverse.dropLast = l.tail.reverse := by
+  apply ext_getElem
+  · simp
+  · intro i h₁ h₂
+    simp only [getElem_dropLast, getElem_reverse, length_tail, getElem_tail]
+    congr
+    simp only [length_dropLast, length_reverse, length_tail] at h₁ h₂
+    omega
+
 /-! ### filter -/
 
 theorem length_filter_lt_length_iff_exists {l} :
@@ -37,7 +58,8 @@ theorem getElem_eq_getElem_reverse {l : List α} {i} (h : i < l.length) :
 
 /-- The length of the List returned by `List.leftpad n a l` is equal
   to the larger of `n` and `l.length` -/
-@[simp]
+-- We don't mark this as a `@[simp]` lemma since we allow `simp` to unfold `leftpad`,
+-- so the left hand side simplifies directly to `n - l.length + l.length`.
 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]
@@ -97,6 +119,53 @@ theorem minimum?_cons' {a : Nat} {l : List Nat} :
         specialize le b h
         split <;> omega
 
+theorem foldl_min
+    {α : Type _} [Min α] [Std.IdempotentOp (min : α → α → α)] [Std.Associative (min : α → α → α)]
+    {l : List α} {a : α} :
+    l.foldl (init := a) min = min a (l.minimum?.getD a) := by
+  cases l with
+  | nil => simp [Std.IdempotentOp.idempotent]
+  | cons b l =>
+    simp only [minimum?]
+    induction l generalizing a b with
+    | nil => simp
+    | cons c l ih => simp [ih, Std.Associative.assoc]
+
+theorem foldl_min_right {α β : Type _}
+    [Min β] [Std.IdempotentOp (min : β → β → β)] [Std.Associative (min : β → β → β)]
+    {l : List α} {b : β} {f : α → β} :
+    (l.foldl (init := b) fun acc a => min acc (f a)) = min b ((l.map f).minimum?.getD b) := by
+  rw [← foldl_map, foldl_min]
+
+theorem foldl_min_le {l : List Nat} {a : Nat} : l.foldl (init := a) min ≤ a := by
+  induction l generalizing a with
+  | nil => simp
+  | cons c l ih =>
+    simp only [foldl_cons]
+    exact Nat.le_trans ih (Nat.min_le_left _ _)
+
+theorem foldl_min_min_of_le {l : List Nat} {a b : Nat} (h : a ≤ b) :
+    l.foldl (init := a) min ≤ b :=
+  Nat.le_trans (foldl_min_le) h
+
+theorem minimum?_getD_le_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
+    l.minimum?.getD k ≤ a := by
+  cases l with
+  | nil => simp at h
+  | cons b l =>
+    simp [minimum?_cons]
+    simp at h
+    rcases h with (rfl | h)
+    · exact foldl_min_le
+    · induction l generalizing b with
+      | nil => simp_all
+      | cons c l ih =>
+        simp only [foldl_cons]
+        simp at h
+        rcases h with (rfl | h)
+        · exact foldl_min_min_of_le (Nat.min_le_right _ _)
+        · exact ih _ h
+
 /-! ### maximum? -/
 
 -- A specialization of `maximum?_eq_some_iff` to Nat.
@@ -130,4 +199,51 @@ theorem maximum?_cons' {a : Nat} {l : List Nat} :
         specialize le b h
         split <;> omega
 
+theorem foldl_max
+    {α : Type _} [Max α] [Std.IdempotentOp (max : α → α → α)] [Std.Associative (max : α → α → α)]
+    {l : List α} {a : α} :
+    l.foldl (init := a) max = max a (l.maximum?.getD a) := by
+  cases l with
+  | nil => simp [Std.IdempotentOp.idempotent]
+  | cons b l =>
+    simp only [maximum?]
+    induction l generalizing a b with
+    | nil => simp
+    | cons c l ih => simp [ih, Std.Associative.assoc]
+
+theorem foldl_max_right {α β : Type _}
+    [Max β] [Std.IdempotentOp (max : β → β → β)] [Std.Associative (max : β → β → β)]
+    {l : List α} {b : β} {f : α → β} :
+    (l.foldl (init := b) fun acc a => max acc (f a)) = max b ((l.map f).maximum?.getD b) := by
+  rw [← foldl_map, foldl_max]
+
+theorem le_foldl_max {l : List Nat} {a : Nat} : a ≤ l.foldl (init := a) max := by
+  induction l generalizing a with
+  | nil => simp
+  | cons c l ih =>
+    simp only [foldl_cons]
+    exact Nat.le_trans (Nat.le_max_left _ _) ih
+
+theorem le_foldl_max_of_le {l : List Nat} {a b : Nat} (h : a ≤ b) :
+    a ≤ l.foldl (init := b) max :=
+  Nat.le_trans h (le_foldl_max)
+
+theorem le_maximum?_getD_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
+    a ≤ l.maximum?.getD k := by
+  cases l with
+  | nil => simp at h
+  | cons b l =>
+    simp [maximum?_cons]
+    simp at h
+    rcases h with (rfl | h)
+    · exact le_foldl_max
+    · induction l generalizing b with
+      | nil => simp_all
+      | cons c l ih =>
+        simp only [foldl_cons]
+        simp at h
+        rcases h with (rfl | h)
+        · exact le_foldl_max_of_le (Nat.le_max_right b a)
+        · exact ih _ h
+
 end List

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

@@ -28,4 +28,59 @@ theorem count_set [BEq α] (a b : α) (l : List α) (i : Nat) (h : i < l.length)
     (l.set i a).count b = l.count b - (if l[i] == b then 1 else 0) + (if a == b then 1 else 0) := by
   simp [count_eq_countP, countP_set, h]
 
+/--
+The number of elements satisfying a predicate in a sublist is at least the number of elements satisfying the predicate in the list,
+minus the difference in the lengths.
+-/
+theorem Sublist.le_countP (s : l₁ <+ l₂) (p) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ := by
+  match s with
+  | .slnil => simp
+  | .cons a s =>
+    rename_i l
+    simp only [countP_cons, length_cons]
+    have := s.le_countP p
+    have := s.length_le
+    split <;> omega
+  | .cons₂ a s =>
+    rename_i l₁ l₂
+    simp only [countP_cons, length_cons]
+    have := s.le_countP p
+    have := s.length_le
+    split <;> omega
+
+theorem IsPrefix.le_countP (s : l₁ <+: l₂) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ :=
+  s.sublist.le_countP _
+
+theorem IsSuffix.le_countP (s : l₁ <:+ l₂) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ :=
+  s.sublist.le_countP _
+
+theorem IsInfix.le_countP (s : l₁ <:+: l₂) : countP p l₂ - (l₂.length - l₁.length) ≤ countP p l₁ :=
+  s.sublist.le_countP _
+
+/--
+The number of elements satisfying a predicate in the tail of a list is
+at least one less than the number of elements satisfying the predicate in the list.
+-/
+theorem le_countP_tail (l) : countP p l - 1 ≤ countP p l.tail := by
+  have := (tail_sublist l).le_countP p
+  simp only [length_tail] at this
+  omega
+
+variable [BEq α]
+
+theorem Sublist.le_count (s : l₁ <+ l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
+  s.le_countP _
+
+theorem IsPrefix.le_count (s : l₁ <+: l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
+  s.sublist.le_count _
+
+theorem IsSuffix.le_count (s : l₁ <:+ l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
+  s.sublist.le_count _
+
+theorem IsInfix.le_count (s : l₁ <:+: l₂) (a : α) : count a l₂ - (l₂.length - l₁.length) ≤ count a l₁ :=
+  s.sublist.le_count _
+
+theorem le_count_tail (a : α) (l) : count a l - 1 ≤ count a l.tail :=
+  le_countP_tail _
+
 end List

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

@@ -0,0 +1,66 @@
+/-
+Copyright (c) 2024 Kim Morrison. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Nat.TakeDrop
+import Init.Data.List.Erase
+
+namespace List
+
+theorem getElem?_eraseIdx (l : List α) (i : Nat) (j : Nat) :
+    (l.eraseIdx i)[j]? = if h : j < i then l[j]? else l[j + 1]? := by
+  rw [eraseIdx_eq_take_drop_succ, getElem?_append]
+  split <;> rename_i h
+  · rw [getElem?_take]
+    split
+    · rfl
+    · simp_all
+      omega
+  · rw [getElem?_drop]
+    split <;> rename_i h'
+    · simp only [length_take, Nat.min_def, Nat.not_lt] at h
+      split at h
+      · omega
+      · simp_all [getElem?_eq_none]
+        omega
+    · simp only [length_take]
+      simp only [length_take, Nat.min_def, Nat.not_lt] at h
+      split at h
+      · congr 1
+        omega
+      · rw [getElem?_eq_none, getElem?_eq_none] <;> omega
+
+theorem getElem?_eraseIdx_of_lt (l : List α) (i : Nat) (j : Nat) (h : j < i) :
+    (l.eraseIdx i)[j]? = l[j]? := by
+  rw [getElem?_eraseIdx]
+  simp [h]
+
+theorem getElem?_eraseIdx_of_ge (l : List α) (i : Nat) (j : Nat) (h : i ≤ j) :
+    (l.eraseIdx i)[j]? = l[j + 1]? := by
+  rw [getElem?_eraseIdx]
+  simp only [dite_eq_ite, ite_eq_right_iff]
+  intro h'
+  omega
+
+theorem getElem_eraseIdx (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) :
+    (l.eraseIdx i)[j] = if h' : j < i then
+        l[j]'(by have := length_eraseIdx_le l i; omega)
+      else
+        l[j + 1]'(by rw [length_eraseIdx] at h; split at h <;> omega) := by
+  apply Option.some.inj
+  rw [← getElem?_eq_getElem, getElem?_eraseIdx]
+  split <;> simp
+
+theorem getElem_eraseIdx_of_lt (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) (h' : j < i) :
+    (l.eraseIdx i)[j] = l[j]'(by have := length_eraseIdx_le l i; omega) := by
+  rw [getElem_eraseIdx]
+  simp only [dite_eq_left_iff, Nat.not_lt]
+  intro h'
+  omega
+
+theorem getElem_eraseIdx_of_ge (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) (h' : i ≤ j) :
+    (l.eraseIdx i)[j] = l[j + 1]'(by rw [length_eraseIdx] at h; split at h <;> omega) := by
+  rw [getElem_eraseIdx, dif_neg]
+  omega

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

@@ -0,0 +1,32 @@
+/-
+Copyright (c) 2024 Kim Morrison. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+prelude
+import Init.Data.List.Nat.Range
+import Init.Data.List.Find
+
+namespace List
+
+theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : List α} {p q : α → Bool} (w : ∀ x ∈ xs, p x → q x) {i : Nat}
+    (h : xs.findIdx? p = some i) : ∃ j, j ≤ i ∧ xs.findIdx? q = some j := by
+  simp only [findIdx?_eq_findSome?_enum] at h
+  rw [findSome?_eq_some_iff] at h
+  simp only [Option.ite_none_right_eq_some, Option.some.injEq, ite_eq_right_iff, reduceCtorEq,
+    imp_false, Bool.not_eq_true, Prod.forall, exists_and_right, Prod.exists] at h
+  obtain ⟨h, h₁, b, ⟨es, h₂⟩, ⟨hb, rfl⟩, h₃⟩ := h
+  rw [enum_eq_enumFrom, enumFrom_eq_append_iff] at h₂
+  obtain ⟨l₁', l₂', rfl, rfl, h₂⟩ := h₂
+  rw [eq_comm, enumFrom_eq_cons_iff] at h₂
+  obtain ⟨a, as, rfl, h₂, rfl⟩ := h₂
+  simp only [Nat.zero_add, Prod.mk.injEq] at h₂
+  obtain ⟨rfl, rfl⟩ := h₂
+  simp only [findIdx?_append]
+  match h : findIdx? q l₁' with
+  | some j =>
+    refine ⟨j, ?_, by simp⟩
+    rw [findIdx?_eq_some_iff_findIdx_eq] at h
+    omega
+  | none =>
+    refine ⟨l₁'.length, by simp, by simp_all⟩

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

@@ -109,7 +109,8 @@ theorem range'_eq_append_iff : range' s n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs =
 @[simp] theorem find?_range'_eq_some {s n : Nat} {i : Nat} {p : Nat → Bool} :
     (range' s n).find? p = some i ↔ p i ∧ i ∈ range' s n ∧ ∀ j, s ≤ j → j < i → !p j := by
   rw [find?_eq_some]
-  simp only [Bool.not_eq_true', exists_and_right, mem_range'_1, and_congr_right_iff]
+  simp only [Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, mem_range'_1,
+    and_congr_right_iff]
   simp only [range'_eq_append_iff, eq_comm (a := i :: _), range'_eq_cons_iff]
   intro h
   constructor
@@ -176,7 +177,7 @@ theorem pairwise_le_range (n : Nat) : Pairwise (· ≤ ·) (range n) :=
 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]
+  · 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']
@@ -258,6 +259,9 @@ theorem nodup_iota (n : Nat) : Nodup (iota n) :=
   | zero => simp at h
   | succ n => simp
 
+@[simp] theorem tail_iota (n : Nat) : (iota n).tail = iota (n - 1) := by
+  cases n <;> simp
+
 @[simp] theorem reverse_iota : reverse (iota n) = range' 1 n := by
   induction n with
   | zero => simp
@@ -272,15 +276,15 @@ theorem nodup_iota (n : Nat) : Nodup (iota n) :=
   rw [getLast_eq_head_reverse]
   simp
 
-theorem find?_iota_eq_none {n : Nat} (p : Nat → Bool) :
+theorem find?_iota_eq_none {n : Nat} {p : Nat → Bool} :
     (iota n).find? p = none ↔ ∀ i, 0 < i → i ≤ n → !p i := by
   simp
 
 @[simp] theorem find?_iota_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
     (iota n).find? p = some i ↔ p i ∧ i ∈ iota n ∧ ∀ j, i < j → j ≤ n → !p j := by
   rw [find?_eq_some]
-  simp only [iota_eq_reverse_range', reverse_eq_append_iff, reverse_cons, append_assoc,
-    singleton_append, Bool.not_eq_true', exists_and_right, mem_reverse, mem_range'_1,
+  simp only [iota_eq_reverse_range', reverse_eq_append_iff, reverse_cons, append_assoc, cons_append,
+    nil_append, Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, mem_reverse, mem_range'_1,
     and_congr_right_iff]
   intro h
   constructor
@@ -354,17 +358,6 @@ 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
 
@@ -387,10 +380,6 @@ theorem mem_enumFrom {x : α} {i j : Nat} {xs : List α} (h : (i, x) ∈ xs.enum
       x = xs[i - j]'(by have := le_fst_of_mem_enumFrom h; have := fst_lt_add_of_mem_enumFrom h; omega) :=
   ⟨le_fst_of_mem_enumFrom h, fst_lt_add_of_mem_enumFrom h, snd_eq_of_mem_enumFrom h⟩
 
-theorem enumFrom_cons' (n : Nat) (x : α) (xs : List α) :
-    enumFrom n (x :: xs) = (n, x) :: (enumFrom n xs).map (Prod.map (· + 1) id) := by
-  rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]
-
 theorem enumFrom_map (n : Nat) (l : List α) (f : α → β) :
     enumFrom n (l.map f) = (enumFrom n l).map (Prod.map id f) := by
   induction l with
@@ -407,22 +396,39 @@ theorem enumFrom_append (xs ys : List α) (n : Nat) :
     rw [cons_append, enumFrom_cons, IH, ← cons_append, ← enumFrom_cons, length, Nat.add_right_comm,
       Nat.add_assoc]
 
-theorem enumFrom_eq_zip_range' (l : List α) {n : Nat} : l.enumFrom n = (range' n l.length).zip l :=
-  zip_of_prod (enumFrom_map_fst _ _) (enumFrom_map_snd _ _)
+theorem enumFrom_eq_cons_iff {l : List α} {n : Nat} :
+    l.enumFrom n = x :: l' ↔ ∃ a as, l = a :: as ∧ x = (n, a) ∧ l' = enumFrom (n + 1) as := by
+  rw [enumFrom_eq_zip_range', zip_eq_cons_iff]
+  constructor
+  · rintro ⟨l₁, l₂, h, rfl, rfl⟩
+    rw [range'_eq_cons_iff] at h
+    obtain ⟨rfl, -, rfl⟩ := h
+    exact ⟨x.2, l₂, by simp [enumFrom_eq_zip_range']⟩
+  · rintro ⟨a, as, rfl, rfl, rfl⟩
+    refine ⟨range' (n+1) as.length, as, ?_⟩
+    simp [enumFrom_eq_zip_range', range'_succ]
 
-@[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']
+theorem enumFrom_eq_append_iff {l : List α} {n : Nat} :
+    l.enumFrom n = l₁ ++ l₂ ↔
+      ∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.enumFrom n ∧ l₂ = l₂'.enumFrom (n + l₁'.length) := by
+  rw [enumFrom_eq_zip_range', zip_eq_append_iff]
+  constructor
+  · rintro ⟨w, x, y, z, h, h', rfl, rfl, rfl⟩
+    rw [range'_eq_append_iff] at h'
+    obtain ⟨k, -, rfl, rfl⟩ := h'
+    simp only [length_range'] at h
+    obtain rfl := h
+    refine ⟨y, z, rfl, ?_⟩
+    simp only [enumFrom_eq_zip_range', length_append, true_and]
+    congr
+    omega
+  · rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
+    simp only [enumFrom_eq_zip_range']
+    refine ⟨range' n l₁'.length, range' (n + l₁'.length) l₂'.length, l₁', l₂', ?_⟩
+    simp [Nat.add_comm]
 
 /-! ### 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
 
@@ -448,6 +454,9 @@ theorem getElem_enum (l : List α) (i : Nat) (h : i < l.enum.length) :
     l.enum.getLast? = l.getLast?.map fun a => (l.length - 1, a) := by
   simp [getLast?_eq_getElem?]
 
+@[simp] theorem tail_enum (l : List α) : (enum l).tail = enumFrom 1 l.tail := by
+  simp [enum]
+
 theorem mk_mem_enum_iff_getElem? {i : Nat} {x : α} {l : List α} : (i, x) ∈ enum l ↔ l[i]? = x := by
   simp [enum, mk_mem_enumFrom_iff_le_and_getElem?_sub]
 

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

@@ -36,23 +36,23 @@ theorem length_take_of_le (h : n ≤ length l) : length (take n l) = n := by sim
 
 /-- 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) :
+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 ..
+  getElem_of_eq (take_append_drop j L).symm _ ▸ getElem_append_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 small list to the big list. -/
-theorem getElem_take' (L : List α) {j i : Nat} {h : i < (L.take j).length} :
+theorem getElem_take (L : List α) {j i : Nat} {h : i < (L.take j).length} :
     (L.take j)[i] =
     L[i]'(Nat.lt_of_lt_of_le h (length_take_le' _ _)) := by
-  rw [length_take, Nat.lt_min] at h; rw [getElem_take L _ h.1]
+  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")]
+@[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]
+  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. -/
@@ -60,7 +60,7 @@ length `> i`. Version designed to rewrite from the small list to the big list. -
 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']
+  simp [getElem_take]
 
 theorem getElem?_take_eq_none {l : List α} {n m : Nat} (h : n ≤ m) :
     (l.take n)[m]? = none :=
@@ -110,7 +110,7 @@ theorem getLast?_take {l : List α} : (l.take n).getLast? = if n = 0 then none e
 
 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']
+  rw [getLast_eq_getElem, getElem_take]
   simp [length_take, Nat.min_def]
   simp at h
   split
@@ -196,7 +196,7 @@ theorem dropLast_take {n : Nat} {l : List α} (h : n < l.length) :
 theorem take_prefix_take_left (l : List α) {m n : Nat} (h : m ≤ n) : take m l <+: take n l := by
   rw [isPrefix_iff]
   intro i w
-  rw [getElem?_take_of_lt, getElem_take', getElem?_eq_getElem]
+  rw [getElem?_take_of_lt, getElem_take, getElem?_eq_getElem]
   simp only [length_take] at w
   exact Nat.lt_of_lt_of_le (Nat.lt_of_lt_of_le w (Nat.min_le_left _ _)) h
 
@@ -219,8 +219,9 @@ dropping the first `i` elements. Version designed to rewrite from the big list t
 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]
+  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]
+  · simp [Nat.min_eq_left this, 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. -/
@@ -267,9 +268,9 @@ theorem mem_take_iff_getElem {l : List α} {a : α} :
   constructor
   · rintro ⟨i, hm, rfl⟩
     simp at hm
-    refine ⟨i, by omega, by rw [getElem_take']⟩
+    refine ⟨i, by omega, by rw [getElem_take]⟩
   · rintro ⟨i, hm, rfl⟩
-    refine ⟨i, by simpa, by rw [getElem_take']⟩
+    refine ⟨i, by simpa, by rw [getElem_take]⟩
 
 theorem mem_drop_iff_getElem {l : List α} {a : α} :
     a ∈ l.drop n ↔ ∃ (i : Nat) (hm : i + n < l.length), l[n + i] = a := by
@@ -478,7 +479,7 @@ theorem false_of_mem_take_findIdx {xs : List α} {p : α → Bool} (h : x ∈ xs
     · simp only [take_succ_cons, findIdx?_cons]
       split
       · simp
-      · simp [ih, Option.guard_comp]
+      · simp [ih, Option.guard_comp, Option.bind_map]
 
 @[simp] theorem min_findIdx_findIdx {xs : List α} {p q : α → Bool} :
     min (xs.findIdx p) (xs.findIdx q) = xs.findIdx (fun a => p a || q a) := by

src/Init/Data/List/Range.lean

@@ -5,6 +5,7 @@ Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, M
 -/
 prelude
 import Init.Data.List.Pairwise
+import Init.Data.List.Zip
 
 /-!
 # Lemmas about `List.range` and `List.enum`
@@ -35,11 +36,16 @@ theorem range'_succ (s n step) : range' s (n + 1) step = s :: range' (s + step)
 theorem range'_ne_nil (s : Nat) {n : Nat} : range' s n ≠ [] ↔ n ≠ 0 := by
   cases n <;> simp
 
-@[simp] theorem range'_zero : range' s 0 = [] := by
+@[simp] theorem range'_zero : range' s 0 step = [] := by
   simp
 
 @[simp] theorem range'_one {s step : Nat} : range' s 1 step = [s] := rfl
 
+@[simp] theorem tail_range' (n : Nat) : (range' s n step).tail = range' (s + step) (n - 1) step := by
+  cases n with
+  | zero => simp
+  | succ n => simp [range'_succ]
+
 @[simp] theorem range'_inj : range' s n = range' s' n' ↔ n = n' ∧ (n = 0 ∨ s = s') := by
   constructor
   · intro h
@@ -153,6 +159,9 @@ theorem range'_eq_map_range (s n : Nat) : range' s n = map (s + ·) (range n) :=
 theorem range_ne_nil {n : Nat} : range n ≠ [] ↔ n ≠ 0 := by
   cases n <;> simp
 
+@[simp] theorem tail_range (n : Nat) : (range n).tail = range' 1 (n - 1) := by
+  rw [range_eq_range', tail_range']
+
 @[simp]
 theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by
   simp only [range_eq_range', range'_sublist_right]
@@ -219,6 +228,12 @@ theorem getElem_enumFrom (l : List α) (n) (i : Nat) (h : i < (l.enumFrom n).len
   simp only [getElem?_enumFrom, getElem?_eq_getElem h]
   simp
 
+@[simp]
+theorem tail_enumFrom (l : List α) (n : Nat) : (enumFrom n l).tail = enumFrom (n + 1) l.tail := by
+  induction l generalizing n with
+  | nil => simp
+  | cons _ l ih => simp [ih, enumFrom_cons]
+
 theorem map_fst_add_enumFrom_eq_enumFrom (l : List α) (n k : Nat) :
     map (Prod.map (· + n) id) (enumFrom k l) = enumFrom (n + k) l :=
   ext_getElem? fun i ↦ by simp [(· ∘ ·), Nat.add_comm, Nat.add_left_comm]; rfl
@@ -227,4 +242,47 @@ theorem map_fst_add_enum_eq_enumFrom (l : List α) (n : Nat) :
     map (Prod.map (· + n) id) (enum l) = enumFrom n l :=
   map_fst_add_enumFrom_eq_enumFrom l _ _
 
+theorem enumFrom_cons' (n : Nat) (x : α) (xs : List α) :
+    enumFrom n (x :: xs) = (n, x) :: (enumFrom n xs).map (Prod.map (· + 1) id) := by
+  rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]
+
+@[simp]
+theorem enumFrom_map_fst (n) :
+    ∀ (l : List α), map Prod.fst (enumFrom n l) = range' n l.length
+  | [] => rfl
+  | _ :: _ => congrArg (cons _) (enumFrom_map_fst _ _)
+
+@[simp]
+theorem enumFrom_map_snd : ∀ (n) (l : List α), map Prod.snd (enumFrom n l) = l
+  | _, [] => rfl
+  | _, _ :: _ => congrArg (cons _) (enumFrom_map_snd _ _)
+
+theorem enumFrom_eq_zip_range' (l : List α) {n : Nat} : l.enumFrom n = (range' n l.length).zip l :=
+  zip_of_prod (enumFrom_map_fst _ _) (enumFrom_map_snd _ _)
+
+@[simp]
+theorem unzip_enumFrom_eq_prod (l : List α) {n : Nat} :
+    (l.enumFrom n).unzip = (range' n l.length, l) := by
+  simp only [enumFrom_eq_zip_range', unzip_zip, length_range']
+
+/-! ### enum -/
+
+theorem enum_cons : (a::as).enum = (0, a) :: as.enumFrom 1 := rfl
+
+theorem enum_cons' (x : α) (xs : List α) :
+    enum (x :: xs) = (0, x) :: (enum xs).map (Prod.map (· + 1) id) :=
+  enumFrom_cons' _ _ _
+
+theorem enum_eq_enumFrom {l : List α} : l.enum = l.enumFrom 0 := rfl
+
+theorem enumFrom_eq_map_enum (l : List α) (n : Nat) :
+    enumFrom n l = (enum l).map (Prod.map (· + n) id) := by
+  induction l generalizing n with
+  | nil => simp
+  | cons x xs ih =>
+    simp only [enumFrom_cons, ih, enum_cons, map_cons, Prod.map_apply, Nat.zero_add, id_eq, map_map,
+      cons.injEq, map_inj_left, Function.comp_apply, Prod.forall, Prod.mk.injEq, and_true, true_and]
+    intro a b _
+    exact (succ_add a n).symm
+
 end List

src/Init/Data/List/Sort/Lemmas.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Lean FRO. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Morrison
+Authors: Kim Morrison, Eric Wieser, François G. Dorais
 -/
 prelude
 import Init.Data.List.Perm
@@ -114,31 +114,40 @@ theorem enumLE_trans (trans : ∀ a b c, le a b → le b c → le a c)
   · simp_all
   · simp_all
 
-theorem enumLE_total (total : ∀ a b, !le a b → le b a)
-    (a b : Nat × α) : !enumLE le a b → enumLE le b a := by
+theorem enumLE_total (total : ∀ a b, le a b || le b a)
+    (a b : Nat × α) : enumLE le a b || enumLE le b a := by
   simp only [enumLE]
   split <;> split
-  · simpa using Nat.le_of_lt
+  · simpa using Nat.le_total a.fst b.fst
   · simp
   · simp
-  · simp_all [total a.2 b.2]
+  · have := total a.2 b.2
+    simp_all
 
 /-! ### merge -/
 
-theorem merge_stable : ∀ (xs ys) (_ : ∀ x y, x ∈ xs → y ∈ ys → x.1 ≤ y.1),
-    (merge xs ys (enumLE le)).map (·.2) = merge (xs.map (·.2)) (ys.map (·.2)) le
-  | [], ys, _ => by simp [merge]
-  | xs, [], _ => by simp [merge]
-  | (i, x) :: xs, (j, y) :: ys, h => by
-    simp only [merge, enumLE, map_cons]
-    split <;> rename_i w
-    · rw [if_pos (by simp [h _ _ (mem_cons_self ..) (mem_cons_self ..)])]
-      simp only [map_cons, cons.injEq, true_and]
-      rw [merge_stable, map_cons]
-      exact fun x' y' mx my => h x' y' (mem_cons_of_mem (i, x) mx) my
-    · simp only [↓reduceIte, map_cons, cons.injEq, true_and, reduceCtorEq]
-      rw [merge_stable, map_cons]
-      exact fun x' y' mx my => h x' y' mx (mem_cons_of_mem (j, y) my)
+theorem cons_merge_cons (s : α → α → Bool) (a b l r) :
+    merge (a::l) (b::r) s = if s a b then a :: merge l (b::r) s else b :: merge (a::l) r s := by
+  simp only [merge]
+
+@[simp] theorem cons_merge_cons_pos (s : α → α → Bool) (l r) (h : s a b) :
+    merge (a::l) (b::r) s = a :: merge l (b::r) s := by
+  rw [cons_merge_cons, if_pos h]
+
+@[simp] theorem cons_merge_cons_neg (s : α → α → Bool) (l r) (h : ¬ s a b) :
+    merge (a::l) (b::r) s = b :: merge (a::l) r s := by
+  rw [cons_merge_cons, if_neg h]
+
+@[simp] theorem length_merge (s : α → α → Bool) (l r) :
+    (merge l r s).length = l.length + r.length := by
+  match l, r with
+  | [], r => simp
+  | l, [] => simp
+  | a::l, b::r =>
+    rw [cons_merge_cons]
+    split
+    · simp_arith [length_merge s l (b::r)]
+    · simp_arith [length_merge s (a::l) r]
 
 /--
 The elements of `merge le xs ys` are exactly the elements of `xs` and `ys`.
@@ -158,16 +167,37 @@ theorem mem_merge {a : α} {xs ys : List α} : a ∈ merge xs ys le ↔ a ∈ xs
         apply or_congr_left
         simp only [or_comm (a := a = y), or_assoc]
 
+theorem mem_merge_left (s : α → α → Bool) (h : x ∈ l) : x ∈ merge l r s :=
+  mem_merge.2 <| .inl h
+
+theorem mem_merge_right (s : α → α → Bool) (h : x ∈ r) : x ∈ merge l r s :=
+  mem_merge.2 <| .inr h
+
+theorem merge_stable : ∀ (xs ys) (_ : ∀ x y, x ∈ xs → y ∈ ys → x.1 ≤ y.1),
+    (merge xs ys (enumLE le)).map (·.2) = merge (xs.map (·.2)) (ys.map (·.2)) le
+  | [], ys, _ => by simp [merge]
+  | xs, [], _ => by simp [merge]
+  | (i, x) :: xs, (j, y) :: ys, h => by
+    simp only [merge, enumLE, map_cons]
+    split <;> rename_i w
+    · rw [if_pos (by simp [h _ _ (mem_cons_self ..) (mem_cons_self ..)])]
+      simp only [map_cons, cons.injEq, true_and]
+      rw [merge_stable, map_cons]
+      exact fun x' y' mx my => h x' y' (mem_cons_of_mem (i, x) mx) my
+    · simp only [↓reduceIte, map_cons, cons.injEq, true_and, reduceCtorEq]
+      rw [merge_stable, map_cons]
+      exact fun x' y' mx my => h x' y' mx (mem_cons_of_mem (j, y) my)
+
 -- We enable this instance locally so we can write `Pairwise le` instead of `Pairwise (le · ·)` everywhere.
 attribute [local instance] boolRelToRel
 
 /--
-If the ordering relation `le` is transitive and total (i.e. `le a b ∨ le b a` for all `a, b`)
+If the ordering relation `le` is transitive and total (i.e. `le a b || le b a` for all `a, b`)
 then the `merge` of two sorted lists is sorted.
 -/
 theorem sorted_merge
     (trans : ∀ (a b c : α), le a b → le b c → le a c)
-    (total : ∀ (a b : α), !le a b → le b a)
+    (total : ∀ (a b : α), le a b || le b a)
     (l₁ l₂ : List α) (h₁ : l₁.Pairwise le) (h₂ : l₂.Pairwise le) : (merge l₁ l₂ le).Pairwise le := by
   induction l₁ generalizing l₂ with
   | nil => simpa only [merge]
@@ -188,9 +218,10 @@ theorem sorted_merge
       · apply Pairwise.cons
         · intro z m
           rw [mem_merge, mem_cons] at m
+          simp only [Bool.not_eq_true] at h
           rcases m with (⟨rfl|m⟩|m)
-          · exact total _ _ (by simpa using h)
-          · exact trans _ _ _ (total _ _ (by simpa using h)) (rel_of_pairwise_cons h₁ m)
+          · simpa [h] using total y z
+          · exact trans _ _ _ (by simpa [h] using total x y) (rel_of_pairwise_cons h₁ m)
           · exact rel_of_pairwise_cons h₂ m
         · exact ih₂ h₂.tail
 
@@ -234,7 +265,7 @@ theorem mergeSort_perm : ∀ (l : List α) (le), mergeSort l le ~ l
         (Perm.of_eq (splitInTwo_fst_append_splitInTwo_snd _)))
 termination_by l => l.length
 
-@[simp] theorem mergeSort_length (l : List α) : (mergeSort l le).length = l.length :=
+@[simp] theorem length_mergeSort (l : List α) : (mergeSort l le).length = l.length :=
   (mergeSort_perm l le).length_eq
 
 @[simp] theorem mem_mergeSort {a : α} {l : List α} : a ∈ mergeSort l le ↔ a ∈ l :=
@@ -243,13 +274,13 @@ termination_by l => l.length
 /--
 The result of `mergeSort` is sorted,
 as long as the comparison function is transitive (`le a b → le b c → le a c`)
-and total in the sense that `le a b ∨ le b a`.
+and total in the sense that `le a b || le b a`.
 
 The comparison function need not be irreflexive, i.e. `le a b` and `le b a` is allowed even when `a ≠ b`.
 -/
 theorem sorted_mergeSort
     (trans : ∀ (a b c : α), le a b → le b c → le a c)
-    (total : ∀ (a b : α), !le a b → le b a) :
+    (total : ∀ (a b : α), le a b || le b a) :
     (l : List α) → (mergeSort l le).Pairwise le
   | [] => by simp [mergeSort]
   | [a] => by simp [mergeSort]
@@ -317,7 +348,7 @@ termination_by _ l => l.length
 
 theorem mergeSort_cons {le : α → α → Bool}
     (trans : ∀ (a b c : α), le a b → le b c → le a c)
-    (total : ∀ (a b : α), !le a b → le b a)
+    (total : ∀ (a b : α), le a b || le b a)
     (a : α) (l : List α) :
     ∃ l₁ l₂, mergeSort (a :: l) le = l₁ ++ a :: l₂ ∧ mergeSort l le = l₁ ++ l₂ ∧
       ∀ b, b ∈ l₁ → !le a b := by
@@ -376,7 +407,7 @@ then `c` is still a sublist of `mergeSort le l`.
 -/
 theorem sublist_mergeSort
     (trans : ∀ (a b c : α), le a b → le b c → le a c)
-    (total : ∀ (a b : α), !le a b → le b a) :
+    (total : ∀ (a b : α), le a b || le b a) :
     ∀ {c : List α} (_ : c.Pairwise le) (_ : c <+ l),
     c <+ mergeSort l le
   | _, _, .slnil => nil_sublist _
@@ -407,8 +438,45 @@ then `[a, b]` is still a sublist of `mergeSort le l`.
 -/
 theorem pair_sublist_mergeSort
     (trans : ∀ (a b c : α), le a b → le b c → le a c)
-    (total : ∀ (a b : α), !le a b → le b a)
+    (total : ∀ (a b : α), le a b || le b a)
     (hab : le a b) (h : [a, b] <+ l) : [a, b] <+ mergeSort l le :=
   sublist_mergeSort trans total (pairwise_pair.mpr hab) h
 
 @[deprecated (since := "2024-09-02")] abbrev mergeSort_stable_pair := @pair_sublist_mergeSort
+
+theorem map_merge {f : α → β} {r : α → α → Bool} {s : β → β → Bool} {l l' : List α}
+    (hl : ∀ a ∈ l, ∀ b ∈ l', r a b = s (f a) (f b)) :
+    (l.merge l' r).map f = (l.map f).merge (l'.map f) s := by
+  match l, l' with
+  | [], x' => simp
+  | x, [] => simp
+  | x :: xs, x' :: xs' =>
+    simp only [List.forall_mem_cons] at hl
+    simp only [forall_and] at hl
+    simp only [List.map, List.cons_merge_cons]
+    rw [← hl.1.1]
+    split
+    · rw [List.map, map_merge, List.map]
+      simp only [List.forall_mem_cons, forall_and]
+      exact ⟨hl.2.1, hl.2.2⟩
+    · rw [List.map, map_merge, List.map]
+      simp only [List.forall_mem_cons]
+      exact ⟨hl.1.2, hl.2.2⟩
+
+theorem map_mergeSort {r : α → α → Bool} {s : β → β → Bool} {f : α → β} {l : List α}
+    (hl : ∀ a ∈ l, ∀ b ∈ l, r a b = s (f a) (f b)) :
+    (l.mergeSort r).map f = (l.map f).mergeSort s :=
+  match l with
+  | [] => by simp
+  | [x] => by simp
+  | a :: b :: l => by
+    simp only [mergeSort, splitInTwo_fst, splitInTwo_snd, map_cons]
+    rw [map_merge (fun a am b bm => hl a (mem_of_mem_take (by simpa using am))
+      b (mem_of_mem_drop (by simpa using bm)))]
+    rw [map_mergeSort (s := s) (fun a am b bm => hl a (mem_of_mem_take (by simpa using am))
+      b (mem_of_mem_take (by simpa using bm)))]
+    rw [map_mergeSort (s := s) (fun a am b bm => hl a (mem_of_mem_drop (by simpa using am))
+      b (mem_of_mem_drop (by simpa using bm)))]
+    rw [map_take, map_drop]
+    simp
+  termination_by length l

src/Init/Data/List/Sublist.lean

@@ -667,7 +667,7 @@ theorem IsSuffix.length_le (h : l₁ <:+ l₂) : l₁.length ≤ l₂.length :=
 theorem IsPrefix.getElem {x y : List α} (h : x <+: y) {n} (hn : n < x.length) :
     x[n] = y[n]'(Nat.le_trans hn h.length_le) := by
   obtain ⟨_, rfl⟩ := h
-  exact (List.getElem_append n hn).symm
+  exact (List.getElem_append_left hn).symm
 
 -- See `Init.Data.List.Nat.Sublist` for `IsSuffix.getElem`.
 

src/Init/Data/List/Zip.lean

@@ -16,83 +16,6 @@ open Nat
 
 /-! ## Zippers -/
 
-/-! ### zip -/
-
-theorem zip_map (f : α → γ) (g : β → δ) :
-    ∀ (l₁ : List α) (l₂ : List β), zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
-  | [], l₂ => rfl
-  | l₁, [] => by simp only [map, zip_nil_right]
-  | a :: l₁, b :: l₂ => by
-    simp only [map, zip_cons_cons, zip_map, Prod.map]; constructor
-
-theorem zip_map_left (f : α → γ) (l₁ : List α) (l₂ : List β) :
-    zip (l₁.map f) l₂ = (zip l₁ l₂).map (Prod.map f id) := by rw [← zip_map, map_id]
-
-theorem zip_map_right (f : β → γ) (l₁ : List α) (l₂ : List β) :
-    zip l₁ (l₂.map f) = (zip l₁ l₂).map (Prod.map id f) := by rw [← zip_map, map_id]
-
-theorem zip_append :
-    ∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
-      zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
-  | [], r₁, l₂, r₂, h => by simp only [eq_nil_of_length_eq_zero h.symm]; rfl
-  | l₁, r₁, [], r₂, h => by simp only [eq_nil_of_length_eq_zero h]; rfl
-  | a :: l₁, r₁, b :: l₂, r₂, h => by
-    simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]
-
-theorem zip_map' (f : α → β) (g : α → γ) :
-    ∀ l : List α, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
-  | [] => rfl
-  | a :: l => by simp only [map, zip_cons_cons, zip_map']
-
-theorem of_mem_zip {a b} : ∀ {l₁ : List α} {l₂ : List β}, (a, b) ∈ zip l₁ l₂ → a ∈ l₁ ∧ b ∈ l₂
-  | _ :: l₁, _ :: l₂, h => by
-    cases h
-    case head => simp
-    case tail h =>
-    · have := of_mem_zip h
-      exact ⟨Mem.tail _ this.1, Mem.tail _ this.2⟩
-
-@[deprecated of_mem_zip (since := "2024-07-28")] abbrev mem_zip := @of_mem_zip
-
-theorem map_fst_zip :
-    ∀ (l₁ : List α) (l₂ : List β), l₁.length ≤ l₂.length → map Prod.fst (zip l₁ l₂) = l₁
-  | [], bs, _ => rfl
-  | _ :: as, _ :: bs, h => by
-    simp [Nat.succ_le_succ_iff] at h
-    show _ :: map Prod.fst (zip as bs) = _ :: as
-    rw [map_fst_zip as bs h]
-  | a :: as, [], h => by simp at h
-
-theorem map_snd_zip :
-    ∀ (l₁ : List α) (l₂ : List β), l₂.length ≤ l₁.length → map Prod.snd (zip l₁ l₂) = l₂
-  | _, [], _ => by
-    rw [zip_nil_right]
-    rfl
-  | [], b :: bs, h => by simp at h
-  | a :: as, b :: bs, h => by
-    simp [Nat.succ_le_succ_iff] at h
-    show _ :: map Prod.snd (zip as bs) = _ :: bs
-    rw [map_snd_zip as bs h]
-
-theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
-    (l.map fun x => (x, f x)) = l.zip (l.map f) := by
-  rw [← zip_map']
-  congr
-  simp
-
-theorem map_prod_right_eq_zip {l : List α} (f : α → β) :
-    (l.map fun x => (f x, x)) = (l.map f).zip l := by
-  rw [← zip_map']
-  congr
-  simp
-
-/-- See also `List.zip_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
-@[simp] theorem zip_replicate' {a : α} {b : β} {n : Nat} :
-    zip (replicate n a) (replicate n b) = replicate n (a, b) := by
-  induction n with
-  | zero => rfl
-  | succ n ih => simp [replicate_succ, ih]
-
 /-! ### zipWith -/
 
 theorem zipWith_comm (f : α → β → γ) :
@@ -229,6 +152,7 @@ theorem drop_zipWith : (zipWith f l l').drop n = zipWith f (l.drop n) (l'.drop n
 
 @[deprecated drop_zipWith (since := "2024-07-26")] abbrev zipWith_distrib_drop := @drop_zipWith
 
+@[simp]
 theorem tail_zipWith : (zipWith f l l').tail = zipWith f l.tail l'.tail := by
   rw [← drop_one]; simp [drop_zipWith]
 
@@ -248,6 +172,65 @@ theorem zipWith_append (f : α → β → γ) (l la : List α) (l' lb : List β)
       simp only [length_cons, Nat.succ.injEq] at h
       simp [ih _ h]
 
+theorem zipWith_eq_cons_iff {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
+    zipWith f l₁ l₂ = g :: l ↔
+      ∃ a l₁' b l₂', l₁ = a :: l₁' ∧ l₂ = b :: l₂' ∧ g = f a b ∧ l = zipWith f l₁' l₂' := by
+  match l₁, l₂ with
+  | [], [] => simp
+  | [], b :: l₂ => simp
+  | a :: l₁, [] => simp
+  | a' :: l₁, b' :: l₂ =>
+    simp only [zip_cons_cons, cons.injEq, Prod.mk.injEq]
+    constructor
+    · rintro ⟨⟨rfl, rfl⟩, rfl⟩
+      refine ⟨a', l₁, b', l₂, by simp⟩
+    · rintro ⟨a, l₁, b, l₂, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl, rfl⟩
+      simp
+
+theorem zipWith_eq_append_iff {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
+    zipWith f l₁ l₂ = l₁' ++ l₂' ↔
+      ∃ w x y z, w.length = y.length ∧ l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zipWith f w y ∧ l₂' = zipWith f x z := by
+  induction l₁ generalizing l₂ l₁' with
+  | nil =>
+    simp
+    constructor
+    · rintro ⟨rfl, rfl⟩
+      exact ⟨[], [], [], by simp⟩
+    · rintro ⟨_, _, _, -, ⟨rfl, rfl⟩, _, rfl, rfl, rfl⟩
+      simp
+  | cons x₁ l₁ ih₁ =>
+    cases l₂ with
+    | nil =>
+      constructor
+      · simp only [zipWith_nil_right, nil_eq, append_eq_nil, exists_and_left, and_imp]
+        rintro rfl  rfl
+        exact ⟨[], x₁ :: l₁, [], by simp⟩
+      · rintro ⟨w, x, y, z, h₁, _, h₃, rfl, rfl⟩
+        simp only [nil_eq, append_eq_nil] at h₃
+        obtain ⟨rfl, rfl⟩ := h₃
+        simp
+    | cons x₂ l₂ =>
+      simp only [zipWith_cons_cons]
+      rw [cons_eq_append_iff]
+      constructor
+      · rintro (⟨rfl, rfl⟩ | ⟨l₁'', rfl, h⟩)
+        · exact ⟨[], x₁ :: l₁, [], x₂ :: l₂, by simp⟩
+        · rw [ih₁] at h
+          obtain ⟨w, x, y, z, h, rfl, rfl, h', rfl⟩ := h
+          refine ⟨x₁ :: w, x, x₂ :: y, z, by simp [h, h']⟩
+      · rintro ⟨w, x, y, z, h₁, h₂, h₃, rfl, rfl⟩
+        rw [cons_eq_append_iff] at h₂
+        rw [cons_eq_append_iff] at h₃
+        obtain (⟨rfl, rfl⟩ | ⟨w', rfl, rfl⟩) := h₂
+        · simp only [zipWith_nil_left, true_and, nil_eq, reduceCtorEq, false_and, exists_const,
+          or_false]
+          obtain (⟨rfl, rfl⟩ | ⟨y', rfl, rfl⟩) := h₃
+          · simp
+          · simp_all
+        · obtain (⟨rfl, rfl⟩ | ⟨y', rfl, rfl⟩) := h₃
+          · simp_all
+          · simp_all [zipWith_append, Nat.succ_inj']
+
 /-- See also `List.zipWith_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
 @[simp] theorem zipWith_replicate' {a : α} {b : β} {n : Nat} :
     zipWith f (replicate n a) (replicate n b) = replicate n (f a b) := by
@@ -255,6 +238,113 @@ theorem zipWith_append (f : α → β → γ) (l la : List α) (l' lb : List β)
   | zero => rfl
   | succ n ih => simp [replicate_succ, ih]
 
+/-! ### zip -/
+
+theorem zip_eq_zipWith : ∀ (l₁ : List α) (l₂ : List β), zip l₁ l₂ = zipWith Prod.mk l₁ l₂
+  | [], _ => rfl
+  | _, [] => rfl
+  | a :: l₁, b :: l₂ => by simp [zip_cons_cons, zip_eq_zipWith l₁ l₂]
+
+theorem zip_map (f : α → γ) (g : β → δ) :
+    ∀ (l₁ : List α) (l₂ : List β), zip (l₁.map f) (l₂.map g) = (zip l₁ l₂).map (Prod.map f g)
+  | [], 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]
+
+@[simp] theorem tail_zip (l₁ : List α) (l₂ : List β) :
+    (zip l₁ l₂).tail = zip l₁.tail l₂.tail := by
+  cases l₁ <;> cases l₂ <;> simp
+
+theorem zip_append :
+    ∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
+      zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
+  | [], r₁, l₂, r₂, h => by simp only [eq_nil_of_length_eq_zero h.symm]; rfl
+  | l₁, r₁, [], r₂, h => by simp only [eq_nil_of_length_eq_zero h]; rfl
+  | a :: l₁, r₁, b :: l₂, r₂, h => by
+    simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)]
+
+theorem zip_map' (f : α → β) (g : α → γ) :
+    ∀ l : List α, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
+  | [] => rfl
+  | a :: l => by simp only [map, zip_cons_cons, zip_map']
+
+theorem of_mem_zip {a b} : ∀ {l₁ : List α} {l₂ : List β}, (a, b) ∈ zip l₁ l₂ → a ∈ l₁ ∧ b ∈ l₂
+  | _ :: l₁, _ :: l₂, h => by
+    cases h
+    case head => simp
+    case tail h =>
+    · have := of_mem_zip h
+      exact ⟨Mem.tail _ this.1, Mem.tail _ this.2⟩
+
+@[deprecated of_mem_zip (since := "2024-07-28")] abbrev mem_zip := @of_mem_zip
+
+theorem map_fst_zip :
+    ∀ (l₁ : List α) (l₂ : List β), l₁.length ≤ l₂.length → map Prod.fst (zip l₁ l₂) = l₁
+  | [], bs, _ => rfl
+  | _ :: as, _ :: bs, h => by
+    simp [Nat.succ_le_succ_iff] at h
+    show _ :: map Prod.fst (zip as bs) = _ :: as
+    rw [map_fst_zip as bs h]
+  | a :: as, [], h => by simp at h
+
+theorem map_snd_zip :
+    ∀ (l₁ : List α) (l₂ : List β), l₂.length ≤ l₁.length → map Prod.snd (zip l₁ l₂) = l₂
+  | _, [], _ => by
+    rw [zip_nil_right]
+    rfl
+  | [], b :: bs, h => by simp at h
+  | a :: as, b :: bs, h => by
+    simp [Nat.succ_le_succ_iff] at h
+    show _ :: map Prod.snd (zip as bs) = _ :: bs
+    rw [map_snd_zip as bs h]
+
+theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
+    (l.map fun x => (x, f x)) = l.zip (l.map f) := by
+  rw [← zip_map']
+  congr
+  simp
+
+theorem map_prod_right_eq_zip {l : List α} (f : α → β) :
+    (l.map fun x => (f x, x)) = (l.map f).zip l := by
+  rw [← zip_map']
+  congr
+  simp
+
+@[simp] theorem zip_eq_nil_iff {l₁ : List α} {l₂ : List β} :
+    zip l₁ l₂ = [] ↔ l₁ = [] ∨ l₂ = [] := by
+  simp [zip_eq_zipWith]
+
+theorem zip_eq_cons_iff {l₁ : List α} {l₂ : List β} :
+    zip l₁ l₂ = (a, b) :: l ↔
+      ∃ l₁' l₂', l₁ = a :: l₁' ∧ l₂ = b :: l₂' ∧ l = zip l₁' l₂' := by
+  simp only [zip_eq_zipWith, zipWith_eq_cons_iff]
+  constructor
+  · rintro ⟨a, l₁, b, l₂, rfl, rfl, h, rfl, rfl⟩
+    simp only [Prod.mk.injEq] at h
+    obtain ⟨rfl, rfl⟩ := h
+    simp
+  · rintro ⟨l₁', l₂', rfl, rfl, rfl⟩
+    refine ⟨a, l₁', b, l₂', by simp⟩
+
+theorem zip_eq_append_iff {l₁ : List α} {l₂ : List β} :
+    zip l₁ l₂ = l₁' ++ l₂' ↔
+      ∃ w x y z, w.length = y.length ∧ l₁ = w ++ x ∧ l₂ = y ++ z ∧ l₁' = zip w y ∧ l₂' = zip x z := by
+  simp [zip_eq_zipWith, zipWith_eq_append_iff]
+
+/-- See also `List.zip_replicate` in `Init.Data.List.TakeDrop` for a generalization with different lengths. -/
+@[simp] theorem zip_replicate' {a : α} {b : β} {n : Nat} :
+    zip (replicate n a) (replicate n b) = replicate n (a, b) := by
+  induction n with
+  | zero => rfl
+  | succ n ih => simp [replicate_succ, ih]
+
 /-! ### zipWithAll -/
 
 theorem getElem?_zipWithAll {f : Option α → Option β → γ} {i : Nat} :
@@ -284,12 +374,16 @@ theorem head?_zipWithAll {f : Option α → Option β → γ} :
       | none, none => .none | a?, b? => some (f a? b?) := by
   simp [head?_eq_getElem?, getElem?_zipWithAll]
 
-theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
+@[simp] theorem head_zipWithAll {f : Option α → Option β → γ} (h) :
     (zipWithAll f as bs).head h = f as.head? bs.head? := by
   apply Option.some.inj
   rw [← head?_eq_head, head?_zipWithAll]
   split <;> simp_all
 
+@[simp] theorem tail_zipWithAll {f : Option α → Option β → γ} :
+    (zipWithAll f as bs).tail = zipWithAll f as.tail bs.tail := by
+  cases as <;> cases bs <;> simp
+
 theorem zipWithAll_map {μ} (f : Option γ → Option δ → μ) (g : α → γ) (h : β → δ) (l₁ : List α) (l₂ : List β) :
     zipWithAll f (l₁.map g) (l₂.map h) = zipWithAll (fun a b => f (g <$> a) (h <$> b)) l₁ l₂ := by
   induction l₁ generalizing l₂ <;> cases l₂ <;> simp_all
@@ -358,6 +452,12 @@ theorem zip_of_prod {l : List α} {l' : List β} {lp : List (α × β)} (hl : lp
     (hr : lp.map Prod.snd = l') : lp = l.zip l' := by
   rw [← hl, ← hr, ← zip_unzip lp, ← unzip_fst, ← unzip_snd, zip_unzip, zip_unzip]
 
+theorem tail_zip_fst {l : List (α × β)} : l.unzip.1.tail = l.tail.unzip.1 := by
+  simp
+
+theorem tail_zip_snd {l : List (α × β)} : l.unzip.2.tail = l.tail.unzip.2 := by
+  simp
+
 @[simp] theorem unzip_replicate {n : Nat} {a : α} {b : β} :
     unzip (replicate n (a, b)) = (replicate n a, replicate n b) := by
   ext1 <;> simp

src/Init/Data/Nat/Basic.lean

@@ -514,6 +514,10 @@ protected theorem add_lt_add_left {n m : Nat} (h : n < m) (k : Nat) : k + n < k
 protected theorem add_lt_add_right {n m : Nat} (h : n < m) (k : Nat) : n + k < m + k :=
   Nat.add_comm k m ▸ Nat.add_comm k n ▸ Nat.add_lt_add_left h k
 
+protected theorem lt_add_of_pos_left (h : 0 < k) : n < k + n := by
+  rw [Nat.add_comm]
+  exact Nat.add_lt_add_left h n
+
 protected theorem lt_add_of_pos_right (h : 0 < k) : n < n + k :=
   Nat.add_lt_add_left h n
 
@@ -718,7 +722,7 @@ protected theorem zero_ne_one : 0 ≠ (1 : Nat) :=
 
 theorem succ_ne_zero (n : Nat) : succ n ≠ 0 := by simp
 
-instance {n : Nat} : NeZero (succ n) := ⟨succ_ne_zero n⟩
+instance instNeZeroSucc {n : Nat} : NeZero (n + 1) := ⟨succ_ne_zero n⟩
 
 /-! # mul + order -/
 

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

@@ -226,18 +226,18 @@ private theorem succ_mod_two : succ x % 2 = 1 - x % 2 := by
     simp [Nat.mod_eq (x+2) 2, p, hyp]
     cases Nat.mod_two_eq_zero_or_one x with | _ p => simp [p]
 
-private theorem testBit_succ_zero : testBit (x + 1) 0 = not (testBit x 0) := by
+private theorem testBit_succ_zero : testBit (x + 1) 0 = !(testBit x 0) := by
   simp [testBit_to_div_mod, succ_mod_two]
   cases Nat.mod_two_eq_zero_or_one x with | _ p =>
     simp [p]
 
-theorem testBit_two_pow_add_eq (x i : Nat) : testBit (2^i + x) i = not (testBit x i) := by
+theorem testBit_two_pow_add_eq (x i : Nat) : testBit (2^i + x) i = !(testBit x i) := by
   simp [testBit_to_div_mod, add_div_left, Nat.two_pow_pos, succ_mod_two]
   cases mod_two_eq_zero_or_one (x / 2 ^ i) with
   | _ p => simp [p]
 
 theorem testBit_mul_two_pow_add_eq (a b i : Nat) :
-    testBit (2^i*a + b) i = Bool.xor (a%2 = 1) (testBit b i) := by
+    testBit (2^i*a + b) i = (a%2 = 1 ^^ testBit b i) := by
   match a with
   | 0 => simp
   | a+1 =>
@@ -570,7 +570,7 @@ theorem or_div_two : (a ||| b) / 2 = a / 2 ||| b / 2 := by
 /-! ### xor -/
 
 @[simp] theorem testBit_xor (x y i : Nat) :
-    (x ^^^ y).testBit i = Bool.xor (x.testBit i) (y.testBit i) := by
+    (x ^^^ y).testBit i = ((x.testBit i) ^^ (y.testBit i)) := by
   simp [HXor.hXor, Xor.xor, xor, testBit_bitwise ]
 
 @[simp] theorem zero_xor (x : Nat) : 0 ^^^ x = x := by

src/Init/Data/Nat/Div.lean

@@ -134,6 +134,19 @@ theorem mod_eq_of_lt {a b : Nat} (h : a < b) : a % b = a :=
     if_neg h'
   (mod_eq a b).symm ▸ this
 
+@[simp] theorem one_mod_eq_zero_iff {n : Nat} : 1 % n = 0 ↔ n = 1 := by
+  match n with
+  | 0 => simp
+  | 1 => simp
+  | n + 2 =>
+    rw [mod_eq_of_lt (by exact Nat.lt_of_sub_eq_succ rfl)]
+    simp only [add_one_ne_zero, false_iff, ne_eq]
+    exact ne_of_beq_eq_false rfl
+
+@[simp] theorem Nat.zero_eq_one_mod_iff {n : Nat} : 0 = 1 % n ↔ n = 1 := by
+  rw [eq_comm]
+  simp
+
 theorem mod_eq_sub_mod {a b : Nat} (h : a ≥ b) : a % b = (a - b) % b :=
   match eq_zero_or_pos b with
   | Or.inl h₁ => h₁.symm ▸ (Nat.sub_zero a).symm ▸ rfl
@@ -157,6 +170,13 @@ theorem mod_lt (x : Nat) {y : Nat} : y > 0 → x % y < y := by
     rw [mod_eq_sub_mod h₁]
     exact h₂ h₃
 
+@[simp] protected theorem sub_mod_add_mod_cancel (a b : Nat) [NeZero a] : a - b % a + b % a = a := by
+  rw [Nat.sub_add_cancel]
+  cases a with
+  | zero => simp_all
+  | succ a =>
+    exact Nat.le_of_lt (mod_lt b (zero_lt_succ a))
+
 theorem mod_le (x y : Nat) : x % y ≤ x := by
   match Nat.lt_or_ge x y with
   | Or.inl h₁ => rw [mod_eq_of_lt h₁]; apply Nat.le_refl
@@ -197,7 +217,6 @@ decreasing_by apply div_rec_lemma; assumption
 theorem div_eq_sub_div (h₁ : 0 < b) (h₂ : b ≤ a) : a / b = (a - b) / b + 1 := by
  rw [div_eq a, if_pos]; constructor <;> assumption
 
-
 theorem mod_add_div (m k : Nat) : m % k + k * (m / k) = m := by
   induction m, k using mod.inductionOn with rw [div_eq, mod_eq]
   | base x y h => simp [h]

src/Init/Data/Nat/Lemmas.lean

@@ -84,9 +84,6 @@ protected theorem add_lt_add_of_lt_of_le {a b c d : Nat} (hlt : a < b) (hle : c
     a + c < b + d :=
   Nat.lt_of_le_of_lt (Nat.add_le_add_left hle _) (Nat.add_lt_add_right hlt _)
 
-protected theorem lt_add_of_pos_left : 0 < k → n < k + n := by
-  rw [Nat.add_comm]; exact Nat.lt_add_of_pos_right
-
 protected theorem pos_of_lt_add_right (h : n < n + k) : 0 < k :=
   Nat.lt_of_add_lt_add_left h
 
@@ -233,6 +230,17 @@ instance : Std.Associative (α := Nat) min := ⟨Nat.min_assoc⟩
 @[simp] protected theorem min_self_assoc' {m n : Nat} : min n (min m n) = min n m := by
   rw [Nat.min_comm m n, ← Nat.min_assoc, Nat.min_self]
 
+@[simp] theorem min_add_left {a b : Nat} : min a (b + a) = a := by
+  rw [Nat.min_def]
+  simp
+@[simp] theorem min_add_right {a b : Nat} : min a (a + b) = a := by
+  rw [Nat.min_def]
+  simp
+@[simp] theorem add_left_min {a b : Nat} : min (b + a) a = a := by
+  rw [Nat.min_comm, min_add_left]
+@[simp] theorem add_right_min {a b : Nat} : min (a + b) a = a := by
+  rw [Nat.min_comm, min_add_right]
+
 protected theorem sub_sub_eq_min : ∀ (a b : Nat), a - (a - b) = min a b
   | 0, _ => by rw [Nat.zero_sub, Nat.zero_min]
   | _, 0 => by rw [Nat.sub_zero, Nat.sub_self, Nat.min_zero]
@@ -287,6 +295,17 @@ protected theorem max_assoc : ∀ (a b c : Nat), max (max a b) c = max a (max b
   | _+1, _+1, _+1 => by simp only [Nat.succ_max_succ]; exact congrArg succ <| Nat.max_assoc ..
 instance : Std.Associative (α := Nat) max := ⟨Nat.max_assoc⟩
 
+@[simp] theorem max_add_left {a b : Nat} : max a (b + a) = b + a := by
+  rw [Nat.max_def]
+  simp
+@[simp] theorem max_add_right {a b : Nat} : max a (a + b) = a + b := by
+  rw [Nat.max_def]
+  simp
+@[simp] theorem add_left_max {a b : Nat} : max (b + a) a = b + a := by
+  rw [Nat.max_comm, max_add_left]
+@[simp] theorem add_right_max {a b : Nat} : max (a + b) a = a + b := by
+  rw [Nat.max_comm, max_add_right]
+
 protected theorem sub_add_eq_max (a b : Nat) : a - b + b = max a b := by
   match Nat.le_total a b with
   | .inl hl => rw [Nat.max_eq_right hl, Nat.sub_eq_zero_iff_le.mpr hl, Nat.zero_add]

src/Init/Data/Option/Lemmas.lean

@@ -6,6 +6,7 @@ Authors: Mario Carneiro
 prelude
 import Init.Data.Option.BasicAux
 import Init.Data.Option.Instances
+import Init.Data.BEq
 import Init.Classical
 import Init.Ext
 
@@ -13,7 +14,7 @@ namespace Option
 
 theorem mem_iff {a : α} {b : Option α} : a ∈ b ↔ b = some a := .rfl
 
-@[simp] theorem mem_some {a b : α} : a ∈ some b ↔ b = a := by simp [mem_iff]
+theorem mem_some {a b : α} : a ∈ some b ↔ b = a := by simp
 
 theorem mem_some_self (a : α) : a ∈ some a := mem_some.2 rfl
 
@@ -230,7 +231,7 @@ theorem isSome_filter_of_isSome (p : α → Bool) (o : Option α) (h : (o.filter
     o.isSome := by
   cases o <;> simp at h ⊢
 
-@[simp] theorem filter_eq_none (p : α → Bool) :
+@[simp] theorem filter_eq_none {p : α → Bool} :
     Option.filter p o = none ↔ o = none ∨ ∀ a, a ∈ o → ¬ p a := by
   cases o <;> simp [filter_some]
 
@@ -247,7 +248,7 @@ theorem isSome_filter_of_isSome (p : α → Bool) (o : Option α) (h : (o.filter
 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
 
-@[simp] theorem bind_map {f : α → β} {g : β → Option γ} {x : Option α} :
+theorem bind_map {f : α → β} {g : β → Option γ} {x : Option α} :
     (x.map f).bind g = x.bind (g ∘ f) := by cases x <;> simp
 
 @[simp] theorem map_bind {f : α → Option β} {g : β → γ} {x : Option α} :
@@ -411,27 +412,42 @@ variable [BEq α]
 @[simp] theorem some_beq_none (a : α) : ((some a : Option α) == none) = false := rfl
 @[simp] theorem some_beq_some {a b : α} : (some a == some b) = (a == b) := rfl
 
+@[simp] theorem reflBEq_iff : ReflBEq (Option α) ↔ ReflBEq α := by
+  constructor
+  · intro h
+    constructor
+    intro a
+    suffices (some a == some a) = true by
+      simpa only [some_beq_some]
+    simp
+  · intro h
+    constructor
+    · rintro (_ | a) <;> simp
+
+@[simp] theorem lawfulBEq_iff : LawfulBEq (Option α) ↔ LawfulBEq α := by
+  constructor
+  · intro h
+    constructor
+    · intro a b h
+      apply Option.some.inj
+      apply eq_of_beq
+      simpa
+    · intro a
+      suffices (some a == some a) = true by
+        simpa only [some_beq_some]
+      simp
+  · intro h
+    constructor
+    · intro a b h
+      simpa using h
+    · intro a
+      simp
+
 end beq
 
 /-! ### ite -/
 section ite
 
-@[simp] theorem mem_dite_none_left {x : α} [Decidable p] {l : ¬ p → Option α} :
-    (x ∈ if h : p then none else l h) ↔ ∃ h : ¬ p, x ∈ l h := by
-  split <;> simp_all
-
-@[simp] theorem mem_dite_none_right {x : α} [Decidable p] {l : p → Option α} :
-    (x ∈ if h : p then l h else none) ↔ ∃ h : p, x ∈ l h := by
-  split <;> simp_all
-
-@[simp] theorem mem_ite_none_left {x : α} [Decidable p] {l : Option α} :
-    (x ∈ if p then none else l) ↔ ¬ p ∧ x ∈ l := by
-  split <;> simp_all
-
-@[simp] theorem mem_ite_none_right {x : α} [Decidable p] {l : Option α} :
-    (x ∈ if p then l else none) ↔ p ∧ x ∈ l := by
-  split <;> simp_all
-
 @[simp] theorem dite_none_left_eq_some {p : Prop} [Decidable p] {b : ¬p → Option β} :
     (if h : p then none else b h) = some a ↔ ∃ h, b h = some a := by
   split <;> simp_all
@@ -464,6 +480,22 @@ section ite
     some a = (if p then b else none) ↔ p ∧ some a = b := by
   split <;> simp_all
 
+theorem mem_dite_none_left {x : α} [Decidable p] {l : ¬ p → Option α} :
+    (x ∈ if h : p then none else l h) ↔ ∃ h : ¬ p, x ∈ l h := by
+  simp
+
+theorem mem_dite_none_right {x : α} [Decidable p] {l : p → Option α} :
+    (x ∈ if h : p then l h else none) ↔ ∃ h : p, x ∈ l h := by
+  simp
+
+theorem mem_ite_none_left {x : α} [Decidable p] {l : Option α} :
+    (x ∈ if p then none else l) ↔ ¬ p ∧ x ∈ l := by
+  simp
+
+theorem mem_ite_none_right {x : α} [Decidable p] {l : Option α} :
+    (x ∈ if p then l else none) ↔ p ∧ x ∈ l := by
+  simp
+
 @[simp] theorem isSome_dite {p : Prop} [Decidable p] {b : p → β} :
     (if h : p then some (b h) else none).isSome = true ↔ p := by
   split <;> simpa

src/Init/Data/UInt/Basic.lean

@@ -290,11 +290,17 @@ instance (a b : UInt64) : Decidable (a ≤ b) := UInt64.decLe a b
 instance : Max UInt64 := maxOfLe
 instance : Min UInt64 := minOfLe
 
+-- This instance would interfere with the global instance `NeZero (n + 1)`,
+-- so we only enable it locally.
+@[local instance]
+private def instNeZeroUSizeSize : NeZero USize.size := ⟨add_one_ne_zero _⟩
+
+@[deprecated (since := "2024-09-16")]
 theorem usize_size_gt_zero : USize.size > 0 :=
   Nat.zero_lt_succ ..
 
 @[extern "lean_usize_of_nat"]
-def USize.ofNat (n : @& Nat) : USize := ⟨Fin.ofNat' n usize_size_gt_zero⟩
+def USize.ofNat (n : @& Nat) : USize := ⟨Fin.ofNat'  _ n⟩
 abbrev Nat.toUSize := USize.ofNat
 @[extern "lean_usize_to_nat"]
 def USize.toNat (n : USize) : Nat := n.val.val

src/Init/Meta.lean

@@ -7,7 +7,7 @@ Additional goodies for writing macros
 -/
 prelude
 import Init.MetaTypes
-import Init.Data.Array.Basic
+import Init.Data.Array.GetLit
 import Init.Data.Option.BasicAux
 
 namespace Lean

src/Init/Prelude.lean

@@ -1014,7 +1014,7 @@ with `Or : Prop → Prop → Prop`, which is the propositional connective).
 It is `@[macro_inline]` because it has C-like short-circuiting behavior:
 if `x` is true then `y` is not evaluated.
 -/
-@[macro_inline] def or (x y : Bool) : Bool :=
+@[macro_inline] def Bool.or (x y : Bool) : Bool :=
   match x with
   | true  => true
   | false => y
@@ -1025,7 +1025,7 @@ with `And : Prop → Prop → Prop`, which is the propositional connective).
 It is `@[macro_inline]` because it has C-like short-circuiting behavior:
 if `x` is false then `y` is not evaluated.
 -/
-@[macro_inline] def and (x y : Bool) : Bool :=
+@[macro_inline] def Bool.and (x y : Bool) : Bool :=
   match x with
   | false => false
   | true  => y
@@ -1034,10 +1034,12 @@ if `x` is false then `y` is not evaluated.
 `not x`, or `!x`, is the boolean "not" operation (not to be confused
 with `Not : Prop → Prop`, which is the propositional connective).
 -/
-@[inline] def not : Bool → Bool
+@[inline] def Bool.not : Bool → Bool
   | true  => false
   | false => true
 
+export Bool (or and not)
+
 /--
 The type of natural numbers, starting at zero. It is defined as an
 inductive type freely generated by "zero is a natural number" and

src/Init/SimpLemmas.lean

@@ -231,8 +231,21 @@ instance : Std.Associative (· || ·) := ⟨Bool.or_assoc⟩
 @[simp] theorem Bool.not_false : (!false) = true := by decide
 @[simp] theorem beq_true  (b : Bool) : (b == true)  =  b := by cases b <;> rfl
 @[simp] theorem beq_false (b : Bool) : (b == false) = !b := by cases b <;> rfl
-@[simp] theorem Bool.not_eq_true'  (b : Bool) : ((!b) = true) = (b = false) := by cases b <;> simp
-@[simp] theorem Bool.not_eq_false' (b : Bool) : ((!b) = false) = (b = true) := by cases b <;> simp
+
+
+/--
+We move `!` from the left hand side of an equality to the right hand side.
+This helps confluence, and also helps combining pairs of `!`s.
+-/
+@[simp] theorem Bool.not_eq_eq_eq_not {a b : Bool} : ((!a) = b) ↔ (a = !b) := by
+  cases a <;> cases b <;> simp
+
+@[simp] theorem Bool.not_eq_not {a b : Bool} : ¬a = !b ↔ a = b := by
+  cases a <;> cases b <;> simp
+theorem Bool.not_not_eq {a b : Bool} : ¬(!a) = b ↔ a = b := by simp
+
+theorem Bool.not_eq_true'  (b : Bool) : ((!b) = true) = (b = false) := by simp
+theorem Bool.not_eq_false' (b : Bool) : ((!b) = false) = (b = true) := by simp
 
 @[simp] theorem Bool.not_eq_true (b : Bool) : (¬(b = true)) = (b = false) := by cases b <;> decide
 @[simp] theorem Bool.not_eq_false (b : Bool) : (¬(b = false)) = (b = true) := by cases b <;> decide

src/Init/Tactics.lean

@@ -773,8 +773,9 @@ macro_rules
 macro "refine_lift' " e:term : tactic => `(tactic| focus (refine' no_implicit_lambda% $e; rotate_right))
 /-- Similar to `have`, but using `refine'` -/
 macro "have' " d:haveDecl : tactic => `(tactic| refine_lift' have $d:haveDecl; ?_)
-/-- Similar to `have`, but using `refine'` -/
+set_option linter.missingDocs false in -- OK, because `tactic_alt` causes inheritance of docs
 macro (priority := high) "have'" x:ident " := " p:term : tactic => `(tactic| have' $x:ident : _ := $p)
+attribute [tactic_alt tacticHave'_] «tacticHave'_:=_»
 /-- Similar to `let`, but using `refine'` -/
 macro "let' " d:letDecl : tactic => `(tactic| refine_lift' let $d:letDecl; ?_)
 

src/Lean/Compiler/IR/Borrow.lean

@@ -68,7 +68,7 @@ namespace InitParamMap
 def initBorrow (ps : Array Param) : Array Param :=
   ps.map fun p => { p with borrow := p.ty.isObj }
 
-/-- We do perform borrow inference for constants marked as `export`.
+/-- We do not perform borrow inference for constants marked as `export`.
    Reason: we current write wrappers in C++ for using exported functions.
    These wrappers use smart pointers such as `object_ref`.
    When writing a new wrapper we need to know whether an argument is a borrow

src/Lean/Compiler/IR/RC.lean

@@ -91,7 +91,7 @@ private def isBorrowParamAux (x : VarId) (ys : Array Arg) (consumeParamPred : Na
     | Arg.var y      => x == y && !consumeParamPred i
 
 private def isBorrowParam (x : VarId) (ys : Array Arg) (ps : Array Param) : Bool :=
-  isBorrowParamAux x ys fun i => not ps[i]!.borrow
+  isBorrowParamAux x ys fun i => ! ps[i]!.borrow
 
 /--
 Return `n`, the number of times `x` is consumed.
@@ -124,7 +124,7 @@ private def addIncBeforeAux (ctx : Context) (xs : Array Arg) (consumeParamPred :
         addInc ctx x b numIncs
 
 private def addIncBefore (ctx : Context) (xs : Array Arg) (ps : Array Param) (b : FnBody) (liveVarsAfter : LiveVarSet) : FnBody :=
-  addIncBeforeAux ctx xs (fun i => not ps[i]!.borrow) b liveVarsAfter
+  addIncBeforeAux ctx xs (fun i => ! ps[i]!.borrow) b liveVarsAfter
 
 /-- See `addIncBeforeAux`/`addIncBefore` for the procedure that inserts `inc` operations before an application.  -/
 private def addDecAfterFullApp (ctx : Context) (xs : Array Arg) (ps : Array Param) (b : FnBody) (bLiveVars : LiveVarSet) : FnBody :=

src/Lean/Data/PersistentArray.lean

@@ -317,8 +317,8 @@ variable {m : Type → Type w} [Monad m]
   anyMAux p t.root <||> t.tail.anyM p
 
 @[inline] def allM (a : PersistentArray α) (p : α → m Bool) : m Bool := do
-  let b ← anyM a (fun v => do let b ← p v; pure (not b))
-  pure (not b)
+  let b ← anyM a (fun v => do let b ← p v; pure (!b))
+  pure (!b)
 
 end
 

src/Lean/Elab/InfoTree/Main.lean

@@ -183,7 +183,7 @@ def UserWidgetInfo.format (info : UserWidgetInfo) : Format :=
   f!"UserWidget {info.id}\n{Std.ToFormat.format <| info.props.run' {}}"
 
 def FVarAliasInfo.format (info : FVarAliasInfo) : Format :=
-  f!"FVarAlias {info.userName.eraseMacroScopes}"
+  f!"FVarAlias {info.userName.eraseMacroScopes}: {info.id.name} -> {info.baseId.name}"
 
 def FieldRedeclInfo.format (ctx : ContextInfo) (info : FieldRedeclInfo) : Format :=
   f!"FieldRedecl @ {formatStxRange ctx info.stx}"

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

@@ -599,7 +599,7 @@ partial def solve {m} {α} [Monad m] (measures : Array α)
             all_le := false
             break
       -- No progress here? Try the next measure
-      if not (has_lt && all_le) then continue
+      if !(has_lt && all_le) then continue
       -- We found a suitable measure, remove it from the list (mild optimization)
       let measures' := measures.eraseIdx measureIdx
       return ← go measures' todo (acc.push measure)

src/Lean/Elab/Tactic/BVDecide/Frontend/BVCheck.lean

@@ -41,9 +41,9 @@ def lratChecker (cfg : TacticContext) (bvExpr : BVLogicalExpr) : MetaM Expr := d
 
 @[inherit_doc Lean.Parser.Tactic.bvCheck]
 def bvCheck (g : MVarId) (cfg : TacticContext) : MetaM Unit := do
-  let unsatProver : UnsatProver := fun bvExpr _ => do
+  let unsatProver : UnsatProver := fun reflectionResult _ => do
     withTraceNode `sat (fun _ => return "Preparing LRAT reflection term") do
-      let proof ← lratChecker cfg bvExpr
+      let proof ← lratChecker cfg reflectionResult.bvExpr
       return ⟨proof, ""⟩
   let _ ← closeWithBVReflection g unsatProver
   return ()

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

@@ -74,19 +74,111 @@ def reconstructCounterExample (var2Cnf : Std.HashMap BVBit Nat) (assignment : Ar
     finalMap := finalMap.push (atomExpr, ⟨BitVec.ofNat currentBit value⟩)
   return finalMap
 
+structure ReflectionResult where
+  bvExpr : BVLogicalExpr
+  proveFalse : Expr → M Expr
+  unusedHypotheses : Std.HashSet FVarId
+
 structure UnsatProver.Result where
   proof : Expr
   lratCert : LratCert
 
-abbrev UnsatProver := BVLogicalExpr → Std.HashMap Nat Expr → MetaM UnsatProver.Result
+abbrev UnsatProver := ReflectionResult → Std.HashMap Nat Expr → MetaM UnsatProver.Result
 
-def lratBitblaster (cfg : TacticContext) (bv : BVLogicalExpr)
+/--
+Contains values that will be used to diagnose spurious counter examples.
+-/
+structure DiagnosisInput where
+  unusedHypotheses : Std.HashSet FVarId
+  atomsAssignment : Std.HashMap Nat Expr
+
+/--
+The result of a spurious counter example diagnosis.
+-/
+structure Diagnosis where
+  uninterpretedSymbols : Std.HashSet Expr := {}
+  unusedRelevantHypotheses : Std.HashSet FVarId := {}
+
+abbrev DiagnosisM : Type → Type := ReaderT DiagnosisInput <| StateRefT Diagnosis MetaM
+
+namespace DiagnosisM
+
+def run (x : DiagnosisM Unit) (unusedHypotheses : Std.HashSet FVarId)
+    (atomsAssignment : Std.HashMap Nat Expr) : MetaM Diagnosis := do
+  let (_, issues) ← ReaderT.run x { unusedHypotheses, atomsAssignment } |>.run {}
+  return issues
+
+def unusedHyps : DiagnosisM (Std.HashSet FVarId) := do
+  return (← read).unusedHypotheses
+
+def atomsAssignment : DiagnosisM (Std.HashMap Nat Expr) := do
+  return (← read).atomsAssignment
+
+def addUninterpretedSymbol (e : Expr) : DiagnosisM Unit :=
+  modify fun s => { s with uninterpretedSymbols := s.uninterpretedSymbols.insert e }
+
+def addUnusedRelevantHypothesis (fvar : FVarId) : DiagnosisM Unit :=
+  modify fun s => { s with unusedRelevantHypotheses := s.unusedRelevantHypotheses.insert fvar }
+
+def checkRelevantHypsUsed (fvar : FVarId) : DiagnosisM Unit := do
+  for hyp in ← unusedHyps do
+    if (← hyp.getType).containsFVar fvar then
+      addUnusedRelevantHypothesis hyp
+
+/--
+Diagnose spurious counter examples, currently this checks:
+- Whether uninterpreted symbols were used
+- Whether all hypotheses which contain any variable that was bitblasted were included
+-/
+def diagnose : DiagnosisM Unit := do
+  for (_, expr) in ← atomsAssignment do
+    match_expr expr with
+    | BitVec.ofBool x =>
+      match x with
+      | .fvar fvarId => checkRelevantHypsUsed fvarId
+      | _ => addUninterpretedSymbol expr
+    | _ =>
+      match expr with
+      | .fvar fvarId => checkRelevantHypsUsed fvarId
+      | _ => addUninterpretedSymbol expr
+
+end DiagnosisM
+
+def uninterpretedExplainer (d : Diagnosis) : Option MessageData := do
+  guard !d.uninterpretedSymbols.isEmpty
+  let symList := d.uninterpretedSymbols.toList
+  return m!"It abstracted the following unsupported expressions as opaque variables: {symList}"
+
+def unusedRelevantHypothesesExplainer (d : Diagnosis) : Option MessageData := do
+  guard !d.unusedRelevantHypotheses.isEmpty
+  let hypList := d.unusedRelevantHypotheses.toList.map mkFVar
+  return m!"The following potentially relevant hypotheses could not be used: {hypList}"
+
+def explainers : List (Diagnosis → Option MessageData) :=
+  [uninterpretedExplainer, unusedRelevantHypothesesExplainer]
+
+def explainCounterExampleQuality (unusedHypotheses : Std.HashSet FVarId)
+    (atomsAssignment : Std.HashMap Nat Expr) : MetaM MessageData := do
+  let diagnosis ← DiagnosisM.run DiagnosisM.diagnose unusedHypotheses atomsAssignment
+  let folder acc explainer := if let some m := explainer diagnosis then acc.push m else acc
+  let explanations := explainers.foldl (init := #[]) folder
+
+  if explanations.isEmpty then
+    return m!"The prover found a counterexample, consider the following assignment:\n"
+  else
+    let mut err := m!"The prover found a potentially spurious counterexample:\n"
+    err := err ++ explanations.foldl (init := m!"") (fun acc exp => acc ++ m!"- " ++ exp ++ m!"\n")
+    err := err ++ m!"Consider the following assignment:\n"
+    return err
+
+def lratBitblaster (cfg : TacticContext) (reflectionResult : ReflectionResult)
     (atomsAssignment : Std.HashMap Nat Expr) :
     MetaM UnsatProver.Result := do
+  let bvExpr := reflectionResult.bvExpr
   let entry ←
     withTraceNode `bv (fun _ => return "Bitblasting BVLogicalExpr to AIG") do
       -- lazyPure to prevent compiler lifting
-      IO.lazyPure (fun _ => bv.bitblast)
+      IO.lazyPure (fun _ => bvExpr.bitblast)
   let aigSize := entry.aig.decls.size
   trace[Meta.Tactic.bv] s!"AIG has {aigSize} nodes."
 
@@ -108,18 +200,25 @@ def lratBitblaster (cfg : TacticContext) (bv : BVLogicalExpr)
 
   match res with
   | .ok cert =>
-    let proof ← cert.toReflectionProof cfg bv ``verifyBVExpr ``unsat_of_verifyBVExpr_eq_true
+    let proof ← cert.toReflectionProof cfg bvExpr ``verifyBVExpr ``unsat_of_verifyBVExpr_eq_true
     return ⟨proof, cert⟩
   | .error assignment =>
     let reconstructed := reconstructCounterExample map assignment aigSize atomsAssignment
-    let mut error := m!"The prover found a potential counterexample, consider the following assignment:\n"
+    let mut error ← explainCounterExampleQuality reflectionResult.unusedHypotheses atomsAssignment
     for (var, value) in reconstructed do
       error := error ++ m!"{var} = {value.bv}\n"
     throwError error
 
-def reflectBV (g : MVarId) : M (BVLogicalExpr × (Expr → M Expr)) := g.withContext do
-  let hyps ← getLocalHyps
-  let sats ← hyps.filterMapM SatAtBVLogical.of
+
+def reflectBV (g : MVarId) : M ReflectionResult := g.withContext do
+  let hyps ← getPropHyps
+  let mut sats := #[]
+  let mut unusedHypotheses := {}
+  for hyp in hyps do
+    if let some reflected ← SatAtBVLogical.of (mkFVar hyp) then
+      sats := sats.push reflected
+    else
+      unusedHypotheses := unusedHypotheses.insert hyp
   if sats.size = 0 then
     let mut error := "None of the hypotheses are in the supported BitVec fragment.\n"
     error := error ++ "There are two potential fixes for this:\n"
@@ -128,27 +227,32 @@ def reflectBV (g : MVarId) : M (BVLogicalExpr × (Expr → M Expr)) := g.withCon
     error := error ++ "   Consider expressing it in terms of different operations that are better supported."
     throwError error
   let sat := sats.foldl (init := SatAtBVLogical.trivial) SatAtBVLogical.and
-  return (sat.bvExpr, sat.proveFalse)
+  return {
+    bvExpr := sat.bvExpr,
+    proveFalse := sat.proveFalse,
+    unusedHypotheses := unusedHypotheses
+  }
+
 
 def closeWithBVReflection (g : MVarId) (unsatProver : UnsatProver) :
     MetaM LratCert := M.run do
   g.withContext do
-    let (bvLogicalExpr, f) ←
+    let reflectionResult ←
       withTraceNode `bv (fun _ => return "Reflecting goal into BVLogicalExpr") do
         reflectBV g
-    trace[Meta.Tactic.bv] "Reflected bv logical expression: {bvLogicalExpr}"
+    trace[Meta.Tactic.bv] "Reflected bv logical expression: {reflectionResult.bvExpr}"
 
     let atomsPairs := (← getThe State).atoms.toList.map (fun (expr, _, ident) => (ident, expr))
     let atomsAssignment := Std.HashMap.ofList atomsPairs
-    let ⟨bvExprUnsat, cert⟩ ← unsatProver bvLogicalExpr atomsAssignment
-    let proveFalse ← f bvExprUnsat
+    let ⟨bvExprUnsat, cert⟩ ← unsatProver reflectionResult atomsAssignment
+    let proveFalse ← reflectionResult.proveFalse bvExprUnsat
     g.assign proveFalse
     return cert
 
 def bvUnsat (g : MVarId) (cfg : TacticContext) : MetaM LratCert := M.run do
-  let unsatProver : UnsatProver := fun bvExpr atomsAssignment => do
+  let unsatProver : UnsatProver := fun reflectionResult atomsAssignment => do
     withTraceNode `bv (fun _ => return "Preparing LRAT reflection term") do
-      lratBitblaster cfg bvExpr atomsAssignment
+      lratBitblaster cfg reflectionResult atomsAssignment
   closeWithBVReflection g unsatProver
 
 structure Result where

src/Lean/Elab/Tactic/BVDecide/Frontend/BVDecide/ReifiedBVLogical.lean

@@ -56,7 +56,7 @@ partial def of (t : Expr) : M (Option ReifiedBVLogical) := do
     let expr := mkApp2 (mkConst ``BoolExpr.const) (mkConst ``BVPred) (toExpr Bool.false)
     let proof := return mkRefl (mkConst ``Bool.false)
     return some ⟨boolExpr, proof, expr⟩
-  | not subExpr =>
+  | Bool.not subExpr =>
     let some sub ← of subExpr | return none
     let boolExpr := .not sub.bvExpr
     let expr := mkApp2 (mkConst ``BoolExpr.not) (mkConst ``BVPred) sub.expr
@@ -65,9 +65,9 @@ partial def of (t : Expr) : M (Option ReifiedBVLogical) := do
       let subProof ← sub.evalsAtAtoms
       return mkApp3 (mkConst ``Std.Tactic.BVDecide.Reflect.Bool.not_congr) subExpr subEvalExpr subProof
     return some ⟨boolExpr, proof, expr⟩
-  | or lhsExpr rhsExpr => gateReflection lhsExpr rhsExpr .or ``Std.Tactic.BVDecide.Reflect.Bool.or_congr
-  | and lhsExpr rhsExpr => gateReflection lhsExpr rhsExpr .and ``Std.Tactic.BVDecide.Reflect.Bool.and_congr
-  | xor lhsExpr rhsExpr => gateReflection lhsExpr rhsExpr .xor ``Std.Tactic.BVDecide.Reflect.Bool.xor_congr
+  | Bool.or lhsExpr rhsExpr => gateReflection lhsExpr rhsExpr .or ``Std.Tactic.BVDecide.Reflect.Bool.or_congr
+  | Bool.and lhsExpr rhsExpr => gateReflection lhsExpr rhsExpr .and ``Std.Tactic.BVDecide.Reflect.Bool.and_congr
+  | Bool.xor lhsExpr rhsExpr => gateReflection lhsExpr rhsExpr .xor ``Std.Tactic.BVDecide.Reflect.Bool.xor_congr
   | BEq.beq α _ lhsExpr rhsExpr =>
     match_expr α with
     | Bool => gateReflection lhsExpr rhsExpr .beq ``Std.Tactic.BVDecide.Reflect.Bool.beq_congr

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

@@ -422,10 +422,11 @@ partial def addFact (p : MetaProblem) (h : Expr) : OmegaM (MetaProblem × Nat) :
     trace[omega] "adding fact: {t}"
     match t with
     | .forallE _ x y _ =>
-      if (← isProp x) && (← isProp y) then
+      if ← pure t.isArrow <&&> isProp x <&&> isProp y then
         p.addFact (mkApp4 (.const ``Decidable.not_or_of_imp []) x y
           (.app (.const ``Classical.propDecidable []) x) h)
       else
+        trace[omega] "rejecting forall: it's not an arrow, or not propositional"
         return (p, 0)
     | .app _ _ =>
       match_expr t with

src/Lean/Elab/Term.lean

@@ -1439,7 +1439,7 @@ def resolveLocalName (n : Name) : TermElabM (Option (Expr × List String)) := do
     let localDecl? := lctx.decls.findSomeRev? fun localDecl? => do
       let localDecl ← localDecl?
       if localDecl.isAuxDecl then
-        guard (not skipAuxDecl)
+        guard (!skipAuxDecl)
         if let some fullDeclName := auxDeclToFullName.find? localDecl.fvarId then
           matchAuxRecDecl? localDecl fullDeclName givenNameView
         else
@@ -1497,7 +1497,7 @@ def resolveLocalName (n : Name) : TermElabM (Option (Expr × List String)) := do
       foo := 10
     ```
     -/
-    match findLocalDecl? givenNameView (skipAuxDecl := globalDeclFound && not projs.isEmpty) with
+    match findLocalDecl? givenNameView (skipAuxDecl := globalDeclFound && !projs.isEmpty) with
     | some decl => return some (decl.toExpr, projs)
     | none => match n with
       | .str pre s => loop pre (s::projs) globalDeclFoundNext

src/Lean/Linter/UnusedVariables.lean

@@ -447,7 +447,10 @@ def unusedVariables : Linter where
     let fvarAliases : Std.HashMap FVarId FVarId := s.fvarAliases.fold (init := {}) fun m id baseId =>
       m.insert id (followAliases s.fvarAliases baseId)
 
+    let getCanonVar (id : FVarId) : FVarId := fvarAliases.getD id id
+
     -- Collect all non-alias fvars corresponding to `fvarUses` by resolving aliases in the list.
+    -- Unlike `s.fvarUses`, `fvarUsesRef` is guaranteed to contain no aliases.
     let fvarUsesRef ← IO.mkRef <| fvarAliases.fold (init := s.fvarUses) fun fvarUses id baseId =>
       if fvarUses.contains id then fvarUses.insert baseId else fvarUses
 
@@ -461,7 +464,7 @@ def unusedVariables : Linter where
       let fvarUses ← fvarUsesRef.get
       -- If any of the `fvar`s corresponding to this declaration is (an alias of) a variable in
       -- `fvarUses`, then it is used
-      if aliases.any fun id => fvarUses.contains (fvarAliases.getD id id) then continue
+      if aliases.any fun id => fvarUses.contains (getCanonVar id) then continue
       -- If this is a global declaration then it is (potentially) used after the command
       if s.constDecls.contains range then continue
 
@@ -493,10 +496,12 @@ def unusedVariables : Linter where
       if !initializedMVars then
         -- collect additional `fvarUses` from tactic assignments
         visitAssignments (← IO.mkRef {}) fvarUsesRef s.assignments
+        -- Resolve potential aliases again to preserve `fvarUsesRef` invariant
+        fvarUsesRef.modify fun fvarUses => fvarUses.fold (·.insert <| getCanonVar ·) {}
         initializedMVars := true
         let fvarUses ← fvarUsesRef.get
         -- Redo the initial check because `fvarUses` could be bigger now
-        if aliases.any fun id => fvarUses.contains (fvarAliases.getD id id) then continue
+        if aliases.any fun id => fvarUses.contains (getCanonVar id) then continue
 
       -- If we made it this far then the variable is unused and not ignored
       unused := unused.push (declStx, userName)

src/Lean/Meta/Check.lean

@@ -82,7 +82,7 @@ where
         return (a, b)
       else if a.getAppNumArgs != b.getAppNumArgs then
         return (a, b)
-      else if not (← isDefEq a.getAppFn b.getAppFn) then
+      else if !(← isDefEq a.getAppFn b.getAppFn) then
         return (a, b)
       else
         let fType ← inferType a.getAppFn

src/Lean/Meta/DiscrTree.lean

@@ -211,7 +211,7 @@ private def ignoreArg (a : Expr) (i : Nat) (infos : Array ParamInfo) : MetaM Boo
     if info.isInstImplicit then
       return true
     else if info.isImplicit || info.isStrictImplicit then
-      return not (← isType a)
+      return !(← isType a)
     else
       isProof a
   else

src/Lean/Meta/ExprDefEq.lean

@@ -418,7 +418,7 @@ This method assumes that for any `xs[i]` and `xs[j]` where `i < j`, we have that
 where the index is the position in the local context.
 -/
 private partial def mkLambdaFVarsWithLetDeps (xs : Array Expr) (v : Expr) : MetaM (Option Expr) := do
-  if not (← hasLetDeclsInBetween) then
+  if !(← hasLetDeclsInBetween) then
     mkLambdaFVars xs v (etaReduce := true)
   else
     let ys ← addLetDeps

src/Lean/Meta/LazyDiscrTree.lean

@@ -77,7 +77,7 @@ private def ignoreArg (a : Expr) (i : Nat) (infos : Array ParamInfo) : MetaM Boo
     if info.isInstImplicit then
       return true
     else if info.isImplicit || info.isStrictImplicit then
-      return not (← isType a)
+      return !(← isType a)
     else
       isProof a
   else

src/Lean/Meta/Match/MatcherApp/Transform.lean

@@ -199,9 +199,8 @@ Performs a possibly type-changing transformation to a `MatcherApp`.
 If `useSplitter` is true, the matcher is replaced with the splitter.
 NB: Not all operations on `MatcherApp` can handle one `matcherName` is a splitter.
 
-The array `addEqualities`, if provided, indicates for which of the discriminants an equality
-connecting the discriminant to the parameters of the alternative (like in `match h : x with …`)
-should be added (if it is isn't already there).
+If `addEqualities` is true, then equalities connecting the discriminant to the parameters of the
+alternative (like in `match h : x with …`) are be added, if not already there.
 
 This function works even if the the type of alternatives do *not* fit the inferred type. This
 allows you to post-process the `MatcherApp` with `MatcherApp.inferMatchType`, which will
@@ -212,20 +211,13 @@ def transform
     [AddMessageContext n] [MonadOptions n]
     (matcherApp : MatcherApp)
     (useSplitter := false)
-    (addEqualities : Array Bool := mkArray matcherApp.discrs.size false)
+    (addEqualities : Bool := false)
     (onParams : Expr → n Expr := pure)
     (onMotive : Array Expr → Expr → n Expr := fun _ e => pure e)
     (onAlt : Expr → Expr → n Expr := fun _ e => pure e)
     (onRemaining : Array Expr → n (Array Expr) := pure) :
     n MatcherApp := do
 
-  if addEqualities.size != matcherApp.discrs.size then
-    throwError "MatcherApp.transform: addEqualities has wrong size"
-
-  -- Do not add equalities when the matcher already does so
-  let addEqualities := Array.zipWith addEqualities matcherApp.discrInfos fun b di =>
-    if di.hName?.isSome then false else b
-
   -- We also handle CasesOn applications here, and need to treat them specially in a
   -- few places.
   -- TODO: Expand MatcherApp with the necessary fields to make this more uniform
@@ -241,17 +233,26 @@ def transform
   let params' ← matcherApp.params.mapM onParams
   let discrs' ← matcherApp.discrs.mapM onParams
 
-
-  let (motive', uElim) ← lambdaTelescope matcherApp.motive fun motiveArgs motiveBody => do
+  let (motive', uElim, addHEqualities) ← lambdaTelescope matcherApp.motive fun motiveArgs motiveBody => do
     unless motiveArgs.size == matcherApp.discrs.size do
       throwError "unexpected matcher application, motive must be lambda expression with #{matcherApp.discrs.size} arguments"
     let mut motiveBody' ← onMotive motiveArgs motiveBody
 
-    -- Prepend (x = e) → to the motive when an equality is requested
-    for arg in motiveArgs, discr in discrs', b in addEqualities do if b then
-      motiveBody' ← liftMetaM <| mkArrow (← mkEq discr arg) motiveBody'
+    -- Prepend `(x = e) →` or `(HEq x e) → ` to the motive when an equality is requested
+    -- and not already present, and remember whether we added an Eq or a HEq
+    let mut addHEqualities : Array (Option Bool) := #[]
+    for arg in motiveArgs, discr in discrs', di in matcherApp.discrInfos do
+      if addEqualities && di.hName?.isNone then
+        if ← isProof arg then
+          addHEqualities := addHEqualities.push none
+        else
+          let heq ← mkEqHEq discr arg
+          motiveBody' ← liftMetaM <| mkArrow heq motiveBody'
+          addHEqualities := addHEqualities.push heq.isHEq
+      else
+        addHEqualities := addHEqualities.push none
 
-    return (← mkLambdaFVars motiveArgs motiveBody', ← getLevel motiveBody')
+    return (← mkLambdaFVars motiveArgs motiveBody', ← getLevel motiveBody', addHEqualities)
 
   let matcherLevels ← match matcherApp.uElimPos? with
     | none     => pure matcherApp.matcherLevels
@@ -261,15 +262,14 @@ def transform
   -- (and count them along the way)
   let mut remaining' := #[]
   let mut extraEqualities : Nat := 0
-  for discr in discrs'.reverse, b in addEqualities.reverse do if b then
-    remaining' := remaining'.push (← mkEqRefl discr)
-    extraEqualities := extraEqualities + 1
+  for discr in discrs'.reverse, b in addHEqualities.reverse do
+    match b with
+    | none => pure ()
+    | some is_heq =>
+        remaining' := remaining'.push (← (if is_heq then mkHEqRefl else mkEqRefl) discr)
+        extraEqualities := extraEqualities + 1
 
   if useSplitter && !isCasesOn then
-    -- We replace the matcher with the splitter
-    let matchEqns ← Match.getEquationsFor matcherApp.matcherName
-    let splitter := matchEqns.splitterName
-
     let aux1 := mkAppN (mkConst matcherApp.matcherName matcherLevels.toList) params'
     let aux1 := mkApp aux1 motive'
     let aux1 := mkAppN aux1 discrs'
@@ -278,6 +278,10 @@ def transform
       check aux1
     let origAltTypes ← inferArgumentTypesN matcherApp.alts.size aux1
 
+    -- We replace the matcher with the splitter
+    let matchEqns ← Match.getEquationsFor matcherApp.matcherName
+    let splitter := matchEqns.splitterName
+
     let aux2 := mkAppN (mkConst splitter matcherLevels.toList) params'
     let aux2 := mkApp aux2 motive'
     let aux2 := mkAppN aux2 discrs'

src/Lean/Meta/Tactic/FunInd.lean

@@ -8,7 +8,6 @@ prelude
 import Lean.Meta.Basic
 import Lean.Meta.Match.MatcherApp.Transform
 import Lean.Meta.Check
-import Lean.Meta.Tactic.Cleanup
 import Lean.Meta.Tactic.Subst
 import Lean.Meta.Injective -- for elimOptParam
 import Lean.Meta.ArgsPacker
@@ -402,19 +401,51 @@ def assertIHs (vals : Array Expr) (mvarid : MVarId) : MetaM MVarId := do
     mvarid ← mvarid.assert (.mkSimple s!"ih{i+1}") (← inferType v) v
   return mvarid
 
-
 /--
-Substitutes equations, but makes sure to only substitute variables introduced after the motives
-(given by the index) as the motive could depend on anything before, and `substVar` would happily
-drop equations about these fixed parameters.
+Goal cleanup:
+Substitutes equations (with `substVar`) to remove superfluous varialbes, and clears unused
+let bindings.
+
+Substitutes from the outside in so that the inner-bound variable name wins, but does a first pass
+looking only at variables with names with macro scope, so that preferably they disappear.
+
+Careful to only touch the context after the motives (given by the index) as the motive could depend
+on anything before, and `substVar` would happily drop equations about these fixed parameters.
 -/
-def substVarAfter (mvarId : MVarId) (index : Nat) : MetaM MVarId := do
-  mvarId.withContext do
-    let mut mvarId := mvarId
-    for localDecl in (← getLCtx) do
-      if localDecl.index > index then
-        mvarId ← trySubstVar mvarId localDecl.fvarId
-    return mvarId
+partial def cleanupAfter (mvarId : MVarId) (index : Nat) : MetaM MVarId := do
+  let mvarId ← go mvarId index true
+  let mvarId ← go mvarId index false
+  return mvarId
+where
+  go (mvarId : MVarId) (index : Nat) (firstPass : Bool) : MetaM MVarId := do
+    if let some mvarId ← cleanupAfter? mvarId index firstPass then
+      go mvarId index firstPass
+    else
+      return mvarId
+
+  allHeqToEq (mvarId : MVarId) (index : Nat) : MetaM MVarId :=
+    mvarId.withContext do
+      let mut mvarId := mvarId
+      for localDecl in (← getLCtx) do
+        if localDecl.index > index then
+          let (_, mvarId') ← heqToEq mvarId localDecl.fvarId
+          mvarId := mvarId'
+      return mvarId
+
+  cleanupAfter? (mvarId : MVarId) (index : Nat) (firstPass : Bool)  : MetaM (Option MVarId) := do
+    mvarId.withContext do
+      for localDecl in (← getLCtx) do
+        if localDecl.index > index && (!firstPass || localDecl.userName.hasMacroScopes) then
+          if localDecl.isLet then
+            if let some mvarId' ← observing? <| mvarId.clear localDecl.fvarId then
+              return some mvarId'
+          if let some mvarId' ← substVar? mvarId localDecl.fvarId then
+            -- After substituting, some HEq might turn into Eqs, and we want to be able to substitute
+            -- them as well
+            let mvarId' ← allHeqToEq mvarId' index
+            return some mvarId'
+      return none
+
 
 /--
 Second helper monad collecting the cases as mvars
@@ -429,7 +460,7 @@ def M2.branch {α} (act : M2 α) : M2 α :=
 
 
 /-- Base case of `buildInductionBody`: Construct a case for the final induction hypthesis.  -/
-def buildInductionCase (oldIH newIH : FVarId) (isRecCall : Expr → Option Expr) (toClear toPreserve : Array FVarId)
+def buildInductionCase (oldIH newIH : FVarId) (isRecCall : Expr → Option Expr) (toClear : Array FVarId)
     (goal : Expr)  (e : Expr) : M2 Expr := do
   let _e' ← foldAndCollect oldIH newIH isRecCall e
   let IHs : Array Expr ← M.ask
@@ -441,8 +472,6 @@ def buildInductionCase (oldIH newIH : FVarId) (isRecCall : Expr → Option Expr)
   trace[Meta.FunInd] "Goal before cleanup:{mvarId}"
   for fvarId in toClear do
     mvarId ← mvarId.clear fvarId
-  mvarId ← mvarId.cleanup (toPreserve := toPreserve)
-  trace[Meta.FunInd] "Goal after cleanup (toClear := {toClear.map mkFVar}) (toPreserve := {toPreserve.map mkFVar}):{mvarId}"
   modify (·.push mvarId)
   let mvar ← instantiateMVars mvar
   pure mvar
@@ -457,7 +486,7 @@ Like `mkLambdaFVars (usedOnly := true)`, but
 The result `r` can be applied with `r.beta (maskArray mask args)`.
 
 We use this when generating the functional induction principle to refine the goal through a `match`,
-here `xs` are the discriminans of the `match`.
+here `xs` are the discriminants of the `match`.
 We do not expect non-trivial discriminants to appear in the goal (and if they do, the user will
 get a helpful equality into the context).
 -/
@@ -487,7 +516,7 @@ Builds an expression of type `goal` by replicating the expression `e` into its t
 where it calls `buildInductionCase`. Collects the cases of the final induction hypothesis
 as `MVars` as it goes.
 -/
-partial def buildInductionBody (toClear toPreserve : Array FVarId) (goal : Expr)
+partial def buildInductionBody (toClear : Array FVarId) (goal : Expr)
     (oldIH newIH : FVarId) (isRecCall : Expr → Option Expr) (e : Expr) : M2 Expr := do
 
   -- if-then-else cause case split:
@@ -496,10 +525,10 @@ partial def buildInductionBody (toClear toPreserve : Array FVarId) (goal : Expr)
     let c' ← foldAndCollect oldIH newIH isRecCall c
     let h' ← foldAndCollect oldIH newIH isRecCall h
     let t' ← withLocalDecl `h .default c' fun h => M2.branch do
-      let t' ← buildInductionBody toClear (toPreserve.push h.fvarId!) goal oldIH newIH isRecCall t
+      let t' ← buildInductionBody toClear goal oldIH newIH isRecCall t
       mkLambdaFVars #[h] t'
     let f' ← withLocalDecl `h .default (mkNot c') fun h => M2.branch do
-      let f' ← buildInductionBody toClear (toPreserve.push h.fvarId!) goal oldIH newIH isRecCall f
+      let f' ← buildInductionBody toClear goal oldIH newIH isRecCall f
       mkLambdaFVars #[h] f'
     let u ← getLevel goal
     return mkApp5 (mkConst ``dite [u]) goal c' h' t' f'
@@ -508,11 +537,11 @@ partial def buildInductionBody (toClear toPreserve : Array FVarId) (goal : Expr)
     let h' ← foldAndCollect oldIH newIH isRecCall h
     let t' ← withLocalDecl `h .default c' fun h => M2.branch do
       let t ← instantiateLambda t #[h]
-      let t' ← buildInductionBody toClear (toPreserve.push h.fvarId!) goal oldIH newIH isRecCall t
+      let t' ← buildInductionBody toClear goal oldIH newIH isRecCall t
       mkLambdaFVars #[h] t'
     let f' ← withLocalDecl `h .default (mkNot c') fun h => M2.branch do
       let f ← instantiateLambda f #[h]
-      let f' ← buildInductionBody toClear (toPreserve.push h.fvarId!) goal oldIH newIH isRecCall f
+      let f' ← buildInductionBody toClear goal oldIH newIH isRecCall f
       mkLambdaFVars #[h] f'
     let u ← getLevel goal
     return mkApp5 (mkConst ``dite [u]) goal c' h' t' f'
@@ -523,8 +552,8 @@ partial def buildInductionBody (toClear toPreserve : Array FVarId) (goal : Expr)
   match_expr goal with
   | And goal₁ goal₂ => match_expr e with
     | PProd.mk _α _β e₁ e₂ =>
-      let e₁' ← buildInductionBody toClear toPreserve goal₁ oldIH newIH isRecCall e₁
-      let e₂' ← buildInductionBody toClear toPreserve goal₂ oldIH newIH isRecCall e₂
+      let e₁' ← buildInductionBody toClear goal₁ oldIH newIH isRecCall e₁
+      let e₂' ← buildInductionBody toClear goal₂ oldIH newIH isRecCall e₂
       return mkApp4 (.const ``And.intro []) goal₁ goal₂ e₁' e₂'
     | _ =>
       throwError "Goal is PProd, but expression is:{indentExpr e}"
@@ -543,14 +572,14 @@ partial def buildInductionBody (toClear toPreserve : Array FVarId) (goal : Expr)
     -- so we need to replace that IH
     if matcherApp.remaining.size == 1 && matcherApp.remaining[0]!.isFVarOf oldIH then
       let matcherApp' ← matcherApp.transform (useSplitter := true)
-        (addEqualities := mask.map not)
+        (addEqualities := true)
         (onParams := (foldAndCollect oldIH newIH isRecCall ·))
         (onMotive := fun xs _body => pure (absMotiveBody.beta (maskArray mask xs)))
         (onAlt := fun expAltType alt => M2.branch do
           removeLamda alt fun oldIH' alt => do
             forallBoundedTelescope expAltType (some 1) fun newIH' goal' => do
               let #[newIH'] := newIH' | unreachable!
-              let alt' ← buildInductionBody (toClear.push newIH'.fvarId!) toPreserve goal' oldIH' newIH'.fvarId! isRecCall alt
+              let alt' ← buildInductionBody (toClear.push newIH'.fvarId!) goal' oldIH' newIH'.fvarId! isRecCall alt
               mkLambdaFVars #[newIH'] alt')
         (onRemaining := fun _ => pure #[.fvar newIH])
       return matcherApp'.toExpr
@@ -562,32 +591,34 @@ partial def buildInductionBody (toClear toPreserve : Array FVarId) (goal : Expr)
       let (mask, absMotiveBody) ← mkLambdaFVarsMasked matcherApp.discrs goal
 
       let matcherApp' ← matcherApp.transform (useSplitter := true)
-        (addEqualities := mask.map not)
+        (addEqualities := true)
         (onParams := (foldAndCollect oldIH newIH isRecCall ·))
         (onMotive := fun xs _body => pure (absMotiveBody.beta (maskArray mask xs)))
         (onAlt := fun expAltType alt => M2.branch do
-          buildInductionBody toClear toPreserve expAltType oldIH newIH isRecCall alt)
+          buildInductionBody toClear expAltType oldIH newIH isRecCall alt)
       return matcherApp'.toExpr
 
   if let .letE n t v b _ := e then
     let t' ← foldAndCollect oldIH newIH isRecCall t
     let v' ← foldAndCollect oldIH newIH isRecCall v
     return ← withLetDecl n t' v' fun x => M2.branch do
-      let b' ← buildInductionBody toClear toPreserve goal oldIH newIH isRecCall (b.instantiate1 x)
+      let b' ← buildInductionBody toClear goal oldIH newIH isRecCall (b.instantiate1 x)
       mkLetFVars #[x] b'
 
   if let some (n, t, v, b) := e.letFun? then
     let t' ← foldAndCollect oldIH newIH isRecCall t
     let v' ← foldAndCollect oldIH newIH isRecCall v
     return ← withLocalDecl n .default t' fun x => M2.branch do
-      let b' ← buildInductionBody toClear toPreserve goal oldIH newIH isRecCall (b.instantiate1 x)
+      let b' ← buildInductionBody toClear goal oldIH newIH isRecCall (b.instantiate1 x)
       mkLetFun x v' b'
 
-  liftM <| buildInductionCase oldIH newIH isRecCall toClear toPreserve goal e
+  liftM <| buildInductionCase oldIH newIH isRecCall toClear goal e
 
 /--
-Given an expression `e` with metavariables
-* collects all these meta-variables,
+Given an expression `e` with metavariables `mvars`
+* performs more cleanup:
+  * removes unused let-expressions after index `index`
+  * tries to substitute variables after index `index`
 * lifts them to the current context by reverting all local declarations after index `index`
 * introducing a local variable for each of the meta variable
 * assigning that local variable to the mvar
@@ -605,7 +636,7 @@ do not handle delayed assignemnts correctly.
 def abstractIndependentMVars (mvars : Array MVarId) (index : Nat) (e : Expr) : MetaM Expr := do
   trace[Meta.FunInd] "abstractIndependentMVars, to revert after {index}, original mvars: {mvars}"
   let mvars ← mvars.mapM fun mvar => do
-    let mvar ← substVarAfter mvar index
+    let mvar ← cleanupAfter mvar index
     mvar.withContext do
       let fvarIds := (← getLCtx).foldl (init := #[]) (start := index+1) fun fvarIds decl => fvarIds.push decl.fvarId
       let (_, mvar) ← mvar.revert fvarIds
@@ -662,7 +693,7 @@ def deriveUnaryInduction (name : Name) : MetaM Name := do
             let body ← instantiateLambda body targets
             removeLamda body fun oldIH body => do
               let body ← instantiateLambda body extraParams
-              let body' ← buildInductionBody #[genIH.fvarId!] #[] goal oldIH genIH.fvarId! isRecCall body
+              let body' ← buildInductionBody #[genIH.fvarId!] goal oldIH genIH.fvarId! isRecCall body
               if body'.containsFVar oldIH then
                 throwError m!"Did not fully eliminate {mkFVar oldIH} from induction principle body:{indentExpr body}"
               mkLambdaFVars (targets.push genIH) (← mkLambdaFVars extraParams body')
@@ -972,7 +1003,7 @@ def deriveInductionStructural (names : Array Name) (numFixed : Nat) : MetaM Unit
                 removeLamda body fun oldIH body => do
                   trace[Meta.FunInd] "replacing {Expr.fvar oldIH} with {genIH}"
                   let body ← instantiateLambda body extraParams
-                  let body' ← buildInductionBody #[genIH.fvarId!] #[] goal oldIH genIH.fvarId! isRecCall body
+                  let body' ← buildInductionBody #[genIH.fvarId!] goal oldIH genIH.fvarId! isRecCall body
                   if body'.containsFVar oldIH then
                     throwError m!"Did not fully eliminate {mkFVar oldIH} from induction principle body:{indentExpr body}"
                   mkLambdaFVars (targets.push genIH) (← mkLambdaFVars extraParams body')

src/Lean/Meta/Tactic/Simp/BuiltinSimprocs/BitVec.lean

@@ -51,7 +51,7 @@ Helper function for reducing homogenous binary bitvector operators.
   else
     return .continue
 
-/-- Simplification procedure for `zeroExtend` and `signExtend` on `BitVec`s. -/
+/-- Simplification procedure for `setWidth`, `zeroExtend` and `signExtend` on `BitVec`s. -/
 @[inline] def reduceExtend (declName : Name)
     (op : {n : Nat} → (m : Nat) → BitVec n → BitVec m) (e : Expr) : SimpM DStep := do
   unless e.isAppOfArity declName 3 do return .continue
@@ -253,13 +253,13 @@ builtin_dsimproc [simp, seval] reduceSLT (BitVec.slt _ _) :=
 builtin_dsimproc [simp, seval] reduceSLE (BitVec.sle _ _) :=
   reduceBoolPred ``BitVec.sle 3 BitVec.sle
 
-/-- Simplification procedure for `zeroExtend'` on `BitVec`s. -/
-builtin_dsimproc [simp, seval] reduceZeroExtend' (zeroExtend' _ _) := fun e => do
-  let_expr zeroExtend' _ w _ v ← e | return .continue
+/-- Simplification procedure for `setWidth'` on `BitVec`s. -/
+builtin_dsimproc [simp, seval] reduceSetWidth' (setWidth' _ _) := fun e => do
+  let_expr setWidth' _ w _ v ← e | return .continue
   let some v ← fromExpr? v | return .continue
   let some w ← Nat.fromExpr? w | return .continue
   if h : v.n ≤ w then
-    return .done <| toExpr (v.value.zeroExtend' h)
+    return .done <| toExpr (v.value.setWidth' h)
   else
     return .continue
 
@@ -285,6 +285,9 @@ builtin_dsimproc [simp, seval] reduceReplicate (replicate _ _) := fun e => do
   let some i ← Nat.fromExpr? i | return .continue
   return .done <| toExpr (v.value.replicate i)
 
+/-- Simplification procedure for `setWidth` on `BitVec`s. -/
+builtin_dsimproc [simp, seval] reduceSetWidth (setWidth _ _) := reduceExtend ``setWidth setWidth
+
 /-- Simplification procedure for `zeroExtend` on `BitVec`s. -/
 builtin_dsimproc [simp, seval] reduceZeroExtend (zeroExtend _ _) := reduceExtend ``zeroExtend zeroExtend
 

src/Lean/Meta/Tactic/Simp/BuiltinSimprocs/Fin.lean

@@ -77,8 +77,9 @@ builtin_dsimproc [simp, seval] reduceFinMk (Fin.mk _ _)  := fun e => do
   let_expr Fin.mk n v _ ← e | return .continue
   let some n ← evalNat n |>.run | return .continue
   let some v ← getNatValue? v | return .continue
-  if h : n > 0 then
-    return .done <| toExpr (Fin.ofNat' v h)
+  if h : n ≠ 0 then
+    have : NeZero n := ⟨h⟩
+    return .done <| toExpr (Fin.ofNat' n v)
   else
     return .continue
 

src/Lean/Parser/Term.lean

@@ -451,6 +451,15 @@ def withAnonymousAntiquot := leading_parser
 @[builtin_term_parser] def «trailing_parser» := leading_parser:leadPrec
   "trailing_parser" >> optExprPrecedence >> optExprPrecedence >> ppSpace >> termParser
 
+/-- 
+Indicates that an argument to a function marked `@[extern]` is borrowed.
+
+Being borrowed only affects the ABI and runtime behavior of the function when compiled or interpreted. From the perspective of Lean's type system, this annotation has no effect. It similarly has no effect on functions not marked `@[extern]`.
+
+When a function argument is borrowed, the function does not consume the value. This means that the function will not decrement the value's reference count or deallocate it, and the caller is responsible for doing so.
+
+Please see https://lean-lang.org/lean4/doc/dev/ffi.html#borrowing for a complete description.
+-/
 @[builtin_term_parser] def borrowed   := leading_parser
   "@& " >> termParser leadPrec
 /-- A literal of type `Name`. -/
@@ -754,7 +763,7 @@ We use them to implement `macro_rules` and `elab_rules`
 def namedArgument  := leading_parser (withAnonymousAntiquot := false)
   atomic ("(" >> ident >> " := ") >> withoutPosition termParser >> ")"
 def ellipsis       := leading_parser (withAnonymousAntiquot := false)
-  ".." >> notFollowedBy "." "`.` immediately after `..`"
+  ".." >> notFollowedBy (checkNoWsBefore >> ".") "`.` immediately after `..`"
 def argument       :=
   checkWsBefore "expected space" >>
   checkColGt "expected to be indented" >>

src/Lean/PrettyPrinter/Delaborator/TopDownAnalyze.lean

@@ -137,7 +137,7 @@ def isNonConstFun (motive : Expr) : MetaM Bool := do
   | _ => return motive.hasLooseBVars
 
 def isSimpleHOFun (motive : Expr) : MetaM Bool :=
-  return not (← returnsPi motive) && not (← isNonConstFun motive)
+  return !(← returnsPi motive) && !(← isNonConstFun motive)
 
 def isType2Type (motive : Expr) : MetaM Bool := do
   match ← inferType motive with
@@ -278,7 +278,7 @@ where
   inspectAux (fType mType : Expr) (i : Nat) (args mvars : Array Expr) := do
     let fType ← whnf fType
     let mType ← whnf mType
-    if not (i < args.size) then return ()
+    if !(i < args.size) then return ()
     match fType, mType with
     | Expr.forallE _ fd fb _, Expr.forallE _ _  mb _ => do
       -- TODO: do I need to check (← okBottomUp? args[i] mvars[i] fuel).isSafe here?
@@ -324,7 +324,7 @@ def withKnowing (knowsType knowsLevel : Bool) (x : AnalyzeM α) : AnalyzeM α :=
 builtin_initialize analyzeFailureId : InternalExceptionId ← registerInternalExceptionId `analyzeFailure
 
 def checkKnowsType : AnalyzeM Unit := do
-  if not (← read).knowsType then
+  if !(← read).knowsType then
     throw $ Exception.internal analyzeFailureId
 
 def annotateBoolAt (n : Name) (pos : Pos) : AnalyzeM Unit := do
@@ -425,7 +425,7 @@ mutual
         funBinders   := mkArray args.size false
       }
 
-      if not rest.isEmpty then
+      if !rest.isEmpty then
         -- Note: this shouldn't happen for type-correct terms
         if !args.isEmpty then
           analyzeAppStaged (mkAppN f args) rest
@@ -501,11 +501,11 @@ mutual
     collectHigherOrders := do
       let { args, mvars, bInfos, ..} ← read
       for i in [:args.size] do
-        if not (bInfos[i]! == BinderInfo.implicit || bInfos[i]! == BinderInfo.strictImplicit) then continue
-        if not (← isHigherOrder (← inferType args[i]!)) then continue
+        if !(bInfos[i]! == BinderInfo.implicit || bInfos[i]! == BinderInfo.strictImplicit) then continue
+        if !(← isHigherOrder (← inferType args[i]!)) then continue
         if getPPAnalyzeTrustId (← getOptions) && isIdLike args[i]! then continue
 
-        if getPPAnalyzeTrustKnownFOType2TypeHOFuns (← getOptions) && not (← valUnknown mvars[i]!)
+        if getPPAnalyzeTrustKnownFOType2TypeHOFuns (← getOptions) && !(← valUnknown mvars[i]!)
           && (← isType2Type (args[i]!)) && (← isFOLike (args[i]!)) then continue
 
         tryUnify args[i]! mvars[i]!
@@ -602,7 +602,7 @@ mutual
               | _                 => annotateNamedArg (← mvarName mvars[i]!)
             else annotateBool `pp.analysis.skip; provided := false
             modify fun s => { s with provideds := s.provideds.set! i provided }
-          if (← get).provideds[i]! then withKnowing (not (← typeUnknown mvars[i]!)) true analyze
+          if (← get).provideds[i]! then withKnowing (!(← typeUnknown mvars[i]!)) true analyze
           tryUnify mvars[i]! args[i]!
 
     maybeSetExplicit := do

src/Lean/Util/ShareCommon.lean

@@ -5,19 +5,18 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.ShareCommon
-import Lean.Data.HashSet
-import Lean.Data.HashMap
+import Std.Data.HashSet
+import Std.Data.HashMap
 import Lean.Data.PersistentHashMap
 import Lean.Data.PersistentHashSet
 
 open ShareCommon
 namespace Lean.ShareCommon
 
-set_option linter.deprecated false in
 def objectFactory :=
   StateFactory.mk {
-    Map := HashMap, mkMap := (mkHashMap ·), mapFind? := (·.find?), mapInsert := (·.insert)
-    Set := HashSet, mkSet := (mkHashSet ·), setFind? := (·.find?), setInsert := (·.insert)
+    Map := Std.HashMap, mkMap := (Std.HashMap.empty ·), mapFind? := (·.get?), mapInsert := (·.insert)
+    Set := Std.HashSet, mkSet := (Std.HashSet.empty ·), setFind? := (·.get?), setInsert := (·.insert)
   }
 
 def persistentObjectFactory :=

src/Std/Data/DHashMap/Internal/Model.lean

@@ -98,14 +98,14 @@ theorem exists_bucket_of_uset [BEq α] [Hashable α]
     refine ⟨?_, ?_⟩
     · apply List.containsKey_bind_eq_false
       intro j hj
-      rw [← List.getElem_append (l₂ := self[i] :: l₂), getElem_congr_coll h₁.symm]
+      rw [List.getElem_append_left' (l₂ := self[i] :: l₂), getElem_congr_coll h₁.symm]
       apply (h.hashes_to j _).containsKey_eq_false h₀ k
       omega
     · apply List.containsKey_bind_eq_false
       intro j hj
       rw [← List.getElem_cons_succ self[i] _ _
         (by simp only [Array.ugetElem_eq_getElem, List.length_cons]; omega)]
-      rw [List.getElem_append_right'' l₁, getElem_congr_coll h₁.symm]
+      rw [List.getElem_append_right' l₁, getElem_congr_coll h₁.symm]
       apply (h.hashes_to (j + 1 + l₁.length) _).containsKey_eq_false h₀ k
       omega
 

src/Std/Sat/AIG/CNF.lean

@@ -67,7 +67,7 @@ theorem atomToCNF_eval :
 theorem gateToCNF_eval :
     (gateToCNF output lhs rhs linv rinv).eval assign
       =
-    (assign output == ((xor (assign lhs) linv) && (xor (assign rhs) rinv))) := by
+    (assign output == (((assign lhs) ^^ linv) && ((assign rhs) ^^ rinv))) := by
   simp only [CNF.eval, gateToCNF, CNF.Clause.eval, List.all_cons, List.any_cons, beq_false,
     List.any_nil, Bool.or_false, beq_true, List.all_nil, Bool.and_true]
   cases assign output

src/Std/Sat/AIG/CachedGatesLemmas.lean

@@ -32,7 +32,7 @@ private theorem or_as_aig : ∀ (a b : Bool), (!(!a && !b)) = (a || b) := by
 /--
 Encoding of XOR as boolean expression in AIG form.
 -/
-private theorem xor_as_aig : ∀ (a b : Bool), (!(a && b) && !(!a && !b)) = (xor a b) := by
+private theorem xor_as_aig : ∀ (a b : Bool), (!(a && b) && !(!a && !b)) = (a ^^ b) := by
   decide
 
 /--

src/Std/Sat/AIG/Lemmas.lean

@@ -92,11 +92,7 @@ instance : LawfulOperator α GateInput mkGate where
 theorem denote_mkGate {aig : AIG α} {input : GateInput aig} :
     ⟦aig.mkGate input, assign⟧
       =
-    (
-      (xor ⟦aig, input.lhs.ref, assign⟧ input.lhs.inv)
-        &&
-      (xor ⟦aig, input.rhs.ref, assign⟧ input.rhs.inv)
-    ) := by
+    ((⟦aig, input.lhs.ref, assign⟧ ^^ input.lhs.inv) && (⟦aig, input.rhs.ref, assign⟧ ^^ input.rhs.inv)) := by
   conv =>
     lhs
     unfold denote denote.go
@@ -224,9 +220,9 @@ theorem denote_idx_gate {aig : AIG α} {hstart} (h : aig.decls[start] = .gate lh
     ⟦aig, ⟨start, hstart⟩, assign⟧
       =
     (
-      (xor ⟦aig, ⟨lhs, by have := aig.invariant hstart h; omega⟩, assign⟧ linv)
+      (⟦aig, ⟨lhs, by have := aig.invariant hstart h; omega⟩, assign⟧ ^^ linv)
         &&
-      (xor ⟦aig, ⟨rhs, by have := aig.invariant hstart h; omega⟩, assign⟧ rinv)
+      (⟦aig, ⟨rhs, by have := aig.invariant hstart h; omega⟩, assign⟧ ^^ rinv)
     ) := by
   unfold denote
   conv =>

src/Std/Sat/CNF/Literal.lean

@@ -21,7 +21,7 @@ namespace Literal
 /--
 Flip the polarity of `l`.
 -/
-def negate (l : Literal α) : Literal α := (l.1, not l.2)
+def negate (l : Literal α) : Literal α := (l.1, !l.2)
 
 end Literal
 

src/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Expr.lean

@@ -67,7 +67,7 @@ theorem go_denote_eq (aig : AIG BVBit) (expr : BVExpr w) (assign : Assignment) :
     simp [go, hidx, denote_blastVar]
   | zeroExtend v inner ih =>
     simp only [go, denote_blastZeroExtend, ih, dite_eq_ite, Bool.if_false_right,
-      eval_zeroExtend, BitVec.getLsbD_zeroExtend, hidx, decide_True, Bool.true_and,
+      eval_zeroExtend, BitVec.getLsbD_setWidth, hidx, decide_True, Bool.true_and,
       Bool.and_iff_right_iff_imp, decide_eq_true_eq]
     apply BitVec.lt_of_getLsbD
   | append lhs rhs lih rih =>
@@ -93,9 +93,9 @@ theorem go_denote_eq (aig : AIG BVBit) (expr : BVExpr w) (assign : Assignment) :
       rw [blastSignExtend_empty_eq_zeroExtend] at hgo
       · rw [← hgo]
         simp only [eval_signExtend]
-        rw [BitVec.signExtend_eq_not_zeroExtend_not_of_msb_false]
+        rw [BitVec.signExtend_eq_not_setWidth_not_of_msb_false]
         · simp only [denote_blastZeroExtend, ih, dite_eq_ite, Bool.if_false_right,
-            BitVec.getLsbD_zeroExtend, hidx, decide_True, Bool.true_and, Bool.and_iff_right_iff_imp,
+            BitVec.getLsbD_setWidth, hidx, decide_True, Bool.true_and, Bool.and_iff_right_iff_imp,
             decide_eq_true_eq]
           apply BitVec.lt_of_getLsbD
         · subst heq

src/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Add.lean

@@ -27,7 +27,7 @@ namespace blastAdd
 theorem denote_mkFullAdderOut (assign : α → Bool) (aig : AIG α) (input : FullAdderInput aig) :
     ⟦mkFullAdderOut aig input, assign⟧
       =
-    xor (xor ⟦aig, input.lhs, assign⟧ ⟦aig, input.rhs, assign⟧) ⟦aig, input.cin, assign⟧
+    ((⟦aig, input.lhs, assign⟧ ^^ ⟦aig, input.rhs, assign⟧) ^^ ⟦aig, input.cin, assign⟧)
     := by
   simp only [mkFullAdderOut, Ref.cast_eq, denote_mkXorCached, denote_projected_entry, Bool.bne_assoc,
     Bool.bne_left_inj]
@@ -37,15 +37,8 @@ theorem denote_mkFullAdderOut (assign : α → Bool) (aig : AIG α) (input : Ful
 theorem denote_mkFullAdderCarry (assign : α → Bool) (aig : AIG α) (input : FullAdderInput aig) :
     ⟦mkFullAdderCarry aig input, assign⟧
       =
-    or
-      (and
-        (xor
-          ⟦aig, input.lhs, assign⟧
-          ⟦aig, input.rhs, assign⟧)
-        ⟦aig, input.cin, assign⟧)
-      (and
-        ⟦aig, input.lhs, assign⟧
-        ⟦aig, input.rhs, assign⟧)
+      ((⟦aig, input.lhs, assign⟧ ^^ ⟦aig, input.rhs, assign⟧) && ⟦aig, input.cin, assign⟧ ||
+       ⟦aig, input.lhs, assign⟧ && ⟦aig, input.rhs, assign⟧)
     := by
   simp only [mkFullAdderCarry, Ref.cast_eq, Int.reduceNeg, denote_mkOrCached,
     LawfulOperator.denote_input_entry, denote_mkAndCached, denote_projected_entry',
@@ -137,7 +130,7 @@ theorem go_get (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (cin : Ref aig)
 theorem atLeastTwo_eq_halfAdder (lhsBit rhsBit carry : Bool) :
     Bool.atLeastTwo lhsBit rhsBit carry
       =
-    (((xor lhsBit rhsBit) && carry) || (lhsBit && rhsBit)) := by
+    (((lhsBit ^^ rhsBit) && carry) || (lhsBit && rhsBit)) := by
   revert lhsBit rhsBit carry
   decide
 

src/Std/Tactic/BVDecide/Bitblast/BoolExpr/Basic.lean

@@ -33,7 +33,7 @@ def toString : Gate → String
 def eval : Gate → Bool → Bool → Bool
   | and => (· && ·)
   | or => (· || ·)
-  | xor => _root_.xor
+  | xor => (· ^^ ·)
   | beq => (· == ·)
   | imp => (!· || ·)
 

src/Std/Tactic/BVDecide/LRAT/Checker.lean

@@ -72,7 +72,7 @@ theorem check_sound (lratProof : Array IntAction) (cnf : CNF Nat) :
           simp [← h2, WellFormedAction]
         . simp only [Option.some.injEq] at h2
           simp [← h2, WellFormedAction]
-        . simp only [ite_some_none_eq_some] at h2
+        . simp only [Option.ite_none_right_eq_some, Option.some.injEq] at h2
           rcases h2 with ⟨hleft, hright⟩
           simp [WellFormedAction, hleft, ← hright, Clause.limplies_iff_mem]
       )

src/Std/Tactic/BVDecide/LRAT/Internal/CNF.lean

@@ -21,12 +21,11 @@ theorem sat_negate_iff_not_sat {p : α → Bool} {l : Literal α} : p ⊨ Litera
   simp only [Literal.negate, sat_iff]
   constructor
   · intro h pl
-    rw [sat_iff, h, not.eq_def] at pl
-    split at pl <;> simp_all
+    rw [sat_iff, h] at pl
+    simp at pl
   · intro h
     rw [sat_iff] at h
-    rw [not.eq_def]
-    split <;> simp_all
+    cases h : p l.fst <;> simp_all
 
 theorem unsat_of_limplies_complement [Entails α t] (x : t) (l : Literal α) :
     Limplies α x l → Limplies α x (Literal.negate l) → Unsatisfiable α x := by

src/Std/Tactic/BVDecide/LRAT/Internal/Clause.lean

@@ -168,7 +168,7 @@ Attempts to add the literal `(idx, b)` to clause `c`. Returns none if doing so w
 tautology.
 -/
 def insert (c : DefaultClause n) (l : Literal (PosFin n)) : Option (DefaultClause n) :=
-  if heq1 : c.clause.contains (l.1, not l.2) then
+  if heq1 : c.clause.contains (l.1, !l.2) then
     none -- Adding l would make c a tautology
   else if heq2 : c.clause.contains l then
     some c
@@ -353,7 +353,7 @@ def reduce_fold_fn (assignments : Array Assignment) (acc : ReduceResult (PosFin
         else
           .reducedToEmpty
       | .neg =>
-        if not l.2 then
+        if !l.2 then
           .reducedToUnit l
         else
           .reducedToEmpty
@@ -367,7 +367,7 @@ def reduce_fold_fn (assignments : Array Assignment) (acc : ReduceResult (PosFin
         else
           .reducedToUnit l'
       | .neg =>
-        if not l.2 then
+        if !l.2 then
           .reducedToNonunit -- Assignment fails to refute both l and l'
         else
           .reducedToUnit l'

src/Std/Tactic/BVDecide/LRAT/Internal/Formula/Implementation.lean

@@ -219,7 +219,7 @@ def performRupAdd {n : Nat} (f : DefaultFormula n) (c : DefaultClause n) (rupHin
     let (f, derivedLits, derivedEmpty, encounteredError) := performRupCheck f rupHints
     if encounteredError then
       (f, false)
-    else if not derivedEmpty then
+    else if !derivedEmpty then
       (f, false)
     else -- derivedEmpty is true and encounteredError is false
       let ⟨clauses, rupUnits, ratUnits, assignments⟩ := f
@@ -284,7 +284,7 @@ def performRatCheck {n : Nat} (f : DefaultFormula n) (negPivot : Literal (PosFin
           -- assignments should now be the same as it was before the performRupCheck call
           let f := clearRatUnits ⟨clauses, rupUnits, ratUnits, assignments⟩
           -- f should now be the same as it was before insertRatUnits
-          if encounteredError || not derivedEmpty then (f, false)
+          if encounteredError || !derivedEmpty then (f, false)
           else (f, true)
     | none => (⟨clauses, rupUnits, ratUnits, assignments⟩, false)
 
@@ -309,7 +309,7 @@ def performRatAdd {n : Nat} (f : DefaultFormula n) (c : DefaultClause n)
           if allChecksPassed then performRatCheck f (Literal.negate pivot) ratHint
           else (f, false)
         let (f, allChecksPassed) := ratHints.foldl fold_fn (f, true)
-        if not allChecksPassed then (f, false)
+        if !allChecksPassed then (f, false)
         else
           match f with
           | ⟨clauses, rupUnits, ratUnits, assignments⟩ =>

src/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean

@@ -507,8 +507,8 @@ theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFor
                 Fin.eta, Array.get_eq_getElem, Array.getElem_eq_toList_getElem, decidableGetElem?] at heq
               rw [hidx, hl] at heq
               simp only [unit, Option.some.injEq, DefaultClause.mk.injEq, List.cons.injEq, and_true] at heq
-              simp only [← heq, not] at l_ne_b
-              split at l_ne_b <;> simp at l_ne_b
+              simp only [← heq] at l_ne_b
+              simp at l_ne_b
             · next id_ne_idx => simp [id_ne_idx]
           · exact hf
         · exact Or.inr hf

src/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean

@@ -57,8 +57,8 @@ theorem entails_of_irrelevant_assignment {n : Nat} {p : (PosFin n) → Bool} {c
     · simp [Clause.toList, delete_iff, negl_ne_v, v_in_c_del_l]
     · split
       · next heq =>
-        simp only [heq, Literal.negate, not, ne_eq, Prod.mk.injEq, true_and] at negl_ne_v
-        split at negl_ne_v <;> simp_all
+        simp only [heq, Literal.negate, ne_eq, Prod.mk.injEq, true_and] at negl_ne_v
+        simp_all
       · next hne =>
         exact pv
   · exists v
@@ -67,8 +67,8 @@ theorem entails_of_irrelevant_assignment {n : Nat} {p : (PosFin n) → Bool} {c
     · simp [Clause.toList, delete_iff, negl_ne_v, v_in_c_del_l]
     · split
       · next heq =>
-        simp only [heq, Literal.negate, not, ne_eq, Prod.mk.injEq, true_and] at negl_ne_v
-        split at negl_ne_v <;> simp_all
+        simp only [heq, Literal.negate, ne_eq, Prod.mk.injEq, true_and] at negl_ne_v
+        simp_all
       · next hne =>
         exact pv
 

src/Std/Tactic/BVDecide/Normalize/BitVec.lean

@@ -56,8 +56,8 @@ attribute [bv_normalize] BitVec.zero_add
 attribute [bv_normalize] BitVec.neg_zero
 attribute [bv_normalize] BitVec.sub_self
 attribute [bv_normalize] BitVec.sub_zero
-attribute [bv_normalize] BitVec.zeroExtend_eq
-attribute [bv_normalize] BitVec.zeroExtend_zero
+attribute [bv_normalize] BitVec.setWidth_eq
+attribute [bv_normalize] BitVec.setWidth_zero
 attribute [bv_normalize] BitVec.getLsbD_zero
 attribute [bv_normalize] BitVec.getLsbD_zero_length
 attribute [bv_normalize] BitVec.getLsbD_concat_zero
@@ -206,5 +206,17 @@ theorem BitVec.max_ult' (a : BitVec w) : (BitVec.ult (-1#w) a) = false := by
 
 attribute [bv_normalize] BitVec.replicate_zero_eq
 
+@[bv_normalize]
+theorem BitVec.ofBool_getLsbD (a : BitVec w) (i : Nat) :
+    BitVec.ofBool (a.getLsbD i) = a.extractLsb' i 1 := by
+  ext j
+  simp
+
+@[bv_normalize]
+theorem BitVec.ofBool_getElem (a : BitVec w) (i : Nat) (h : i < w) :
+    BitVec.ofBool a[i] = a.extractLsb' i 1 := by
+  rw [← BitVec.getLsbD_eq_getElem]
+  apply ofBool_getLsbD
+
 end Normalize
 end Std.Tactic.BVDecide

src/Std/Tactic/BVDecide/Normalize/Bool.lean

@@ -46,7 +46,7 @@ attribute [bv_normalize] Bool.and_self_left
 attribute [bv_normalize] Bool.and_self_right
 
 @[bv_normalize]
-theorem Bool.not_xor : ∀ (a b : Bool), !(xor a b) = (a == b) := by decide
+theorem Bool.not_xor : ∀ (a b : Bool), !(a ^^ b) = (a == b) := by decide
 
 end Normalize
 end Std.Tactic.BVDecide

src/Std/Tactic/BVDecide/Reflect.lean

@@ -126,7 +126,7 @@ theorem or_congr (lhs rhs lhs' rhs' : Bool) (h1 : lhs' = lhs) (h2 : rhs' = rhs)
   simp[*]
 
 theorem xor_congr (lhs rhs lhs' rhs' : Bool) (h1 : lhs' = lhs) (h2 : rhs' = rhs) :
-    (xor lhs' rhs') = (xor lhs rhs) := by
+    (lhs' ^^ rhs') = (lhs ^^ rhs) := by
   simp[*]
 
 theorem beq_congr (lhs rhs lhs' rhs' : Bool) (h1 : lhs' = lhs) (h2 : rhs' = rhs) :