semireducible

src/Init/Data/Array/Basic.lean

@@ -162,19 +162,16 @@ instance : Inhabited (Array α) where
 @[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
+def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) :
+    ∀ (i : Nat) (_ : i ≤ a.size), Bool
+  | 0, _ => true
+  | i+1, h =>
+    p a[i] (b[i]'(hsz ▸ h)) && isEqvAux a b hsz p i (Nat.le_trans (Nat.le_add_right i 1) h)
 
 @[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
   if h : a.size = b.size then
-    isEqvAux a b h p 0
+    isEqvAux a b h p a.size (Nat.le_refl a.size)
   else
     false
 
@@ -188,9 +185,10 @@ ofFn f = #[f 0, f 1, ... , f(n - 1)]
 ``` -/
 def ofFn {n} (f : Fin n → α) : Array α := go 0 (mkEmpty n) where
   /-- Auxiliary for `ofFn`. `ofFn.go f i acc = acc ++ #[f i, ..., f(n - 1)]` -/
+  @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
   go (i : Nat) (acc : Array α) : Array α :=
     if h : i < n then go (i+1) (acc.push (f ⟨i, h⟩)) else acc
-decreasing_by simp_wf; decreasing_trivial_pre_omega
+  decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 /-- The array `#[0, 1, ..., n - 1]`. -/
 def range (n : Nat) : Array Nat :=
@@ -389,11 +387,12 @@ unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad
 def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
   -- Note: we cannot use `foldlM` here for the reference implementation because this calls
   -- `bind` and `pure` too many times. (We are not assuming `m` is a `LawfulMonad`)
-  let rec map (i : Nat) (r : Array β) : m (Array β) := do
-    if hlt : i < as.size then
-      map (i+1) (r.push (← f as[i]))
-    else
-      pure r
+  let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
+    map (i : Nat) (r : Array β) : m (Array β) := do
+      if hlt : i < as.size then
+        map (i+1) (r.push (← f as[i]))
+      else
+        pure r
   decreasing_by simp_wf; decreasing_trivial_pre_omega
   map 0 (mkEmpty as.size)
 
@@ -457,7 +456,8 @@ unsafe def anyMUnsafe {α : Type u} {m : Type → Type w} [Monad m] (p : α →
 @[implemented_by anyMUnsafe]
 def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
   let any (stop : Nat) (h : stop ≤ as.size) :=
-    let rec loop (j : Nat) : m Bool := do
+    let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
+    loop (j : Nat) : m Bool := do
       if hlt : j < stop then
         have : j < as.size := Nat.lt_of_lt_of_le hlt h
         if (← p as[j]) then
@@ -547,7 +547,8 @@ def findRev? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
 
 @[inline]
 def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
-  let rec loop (j : Nat) :=
+  let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
+  loop (j : Nat) :=
     if h : j < as.size then
       if p as[j] then some j else loop (j + 1)
     else none
@@ -557,6 +558,7 @@ def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
 def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
   a.findIdx? fun a => a == v
 
+@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
 def indexOfAux [BEq α] (a : Array α) (v : α) (i : Nat) : Option (Fin a.size) :=
   if h : i < a.size then
     let idx : Fin a.size := ⟨i, h⟩;
@@ -678,6 +680,7 @@ where
     else
       as
 
+@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
 def popWhile (p : α → Bool) (as : Array α) : Array α :=
   if h : as.size > 0 then
     if p (as.get ⟨as.size - 1, Nat.sub_lt h (by decide)⟩) then
@@ -689,7 +692,8 @@ def popWhile (p : α → Bool) (as : Array α) : Array α :=
 decreasing_by simp_wf; decreasing_trivial_pre_omega
 
 def takeWhile (p : α → Bool) (as : Array α) : Array α :=
-  let rec go (i : Nat) (r : Array α) : Array α :=
+  let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
+  go (i : Nat) (r : Array α) : Array α :=
     if h : i < as.size then
       let a := as.get ⟨i, h⟩
       if p a then
@@ -705,6 +709,7 @@ def takeWhile (p : α → Bool) (as : Array α) : Array α :=
 
   This function takes worst case O(n) time because
   it has to backshift all elements at positions greater than `i`.-/
+@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
 def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
   if h : i.val + 1 < a.size then
     let a' := a.swap ⟨i.val + 1, h⟩ i
@@ -739,7 +744,8 @@ def erase [BEq α] (as : Array α) (a : α) : Array α :=
 
 /-- Insert element `a` at position `i`. -/
 @[inline] def insertAt (as : Array α) (i : Fin (as.size + 1)) (a : α) : Array α :=
-  let rec loop (as : Array α) (j : Fin as.size) :=
+  let rec @[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
+  loop (as : Array α) (j : Fin as.size) :=
     if i.1 < j then
       let j' := ⟨j-1, Nat.lt_of_le_of_lt (Nat.pred_le _) j.2⟩
       let as := as.swap j' j
@@ -757,6 +763,7 @@ def insertAt! (as : Array α) (i : Nat) (a : α) : Array α :=
     insertAt as ⟨i, Nat.lt_succ_of_le h⟩ a
   else panic! "invalid index"
 
+@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
 def 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]
@@ -778,7 +785,8 @@ def isPrefixOf [BEq α] (as bs : Array α) : Bool :=
   else
     false
 
-@[specialize] def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) : Array γ :=
+@[semireducible, specialize] -- This is otherwise irreducible because it uses well-founded recursion.
+def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) : Array γ :=
   if h : i < as.size then
     let a := as[i]
     if h : i < bs.size then
@@ -814,6 +822,7 @@ private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat),
     have : i < as.size := Nat.lt_trans (Nat.lt_succ_self _) h;
     a != as[i] && allDiffAuxAux as a i this
 
+@[semireducible] -- This is otherwise irreducible because it uses well-founded recursion.
 private def allDiffAux [BEq α] (as : Array α) (i : Nat) : Bool :=
   if h : i < as.size then
     allDiffAuxAux as as[i] i h && allDiffAux as (i+1)

src/Init/Data/Array/DecidableEq.lean

@@ -9,37 +9,37 @@ import Init.ByCases
 
 namespace Array
 
-theorem eq_of_isEqvAux [DecidableEq α] (a b : Array α) (hsz : a.size = b.size) (i : Nat) (hi : i ≤ a.size) (heqv : Array.isEqvAux a b hsz (fun x y => x = y) i) (j : Nat) (low : i ≤ j) (high : j < a.size) : a[j] = b[j]'(hsz ▸ high) := by
-  by_cases h : i < a.size
-  · unfold Array.isEqvAux at heqv
-    simp [h] at heqv
-    have hind := eq_of_isEqvAux a b hsz (i+1) (Nat.succ_le_of_lt h) heqv.2
-    by_cases heq : i = j
-    · subst heq; exact heqv.1
-    · exact hind j (Nat.succ_le_of_lt (Nat.lt_of_le_of_ne low heq)) high
-  · have heq : i = a.size := Nat.le_antisymm hi (Nat.ge_of_not_lt h)
-    subst heq
-    exact absurd (Nat.lt_of_lt_of_le high low) (Nat.lt_irrefl j)
-termination_by a.size - i
-decreasing_by decreasing_trivial_pre_omega
-
+theorem eq_of_isEqvAux
+    [DecidableEq α] (a b : Array α) (hsz : a.size = b.size) (i : Nat) (hi : i ≤ a.size)
+    (heqv : Array.isEqvAux a b hsz (fun x y => x = y) i hi)
+    (j : Nat) (hj : j < i) : a[j]'(Nat.lt_of_lt_of_le hj hi) = b[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi)) := by
+  induction i with
+  | zero => contradiction
+  | succ i ih =>
+    simp only [Array.isEqvAux, Bool.and_eq_true, decide_eq_true_eq] at heqv
+    by_cases hj' : j < i
+    next =>
+      exact ih _ heqv.right hj'
+    next =>
+      replace hj' : j = i := Nat.eq_of_le_of_lt_succ (Nat.not_lt.mp hj') hj
+      subst hj'
+      exact heqv.left
 
 theorem eq_of_isEqv [DecidableEq α] (a b : Array α) : Array.isEqv a b (fun x y => x = y) → a = b := by
   simp [Array.isEqv]
   split
   next hsz =>
    intro h
-   have aux := eq_of_isEqvAux a b hsz 0 (Nat.zero_le ..) h
-   exact ext a b hsz fun i h _ => aux i (Nat.zero_le ..) _
+   have aux := eq_of_isEqvAux a b hsz a.size (Nat.le_refl ..) h
+   exact ext a b hsz fun i h _ => aux i h
   next => intro; contradiction
 
-theorem isEqvAux_self [DecidableEq α] (a : Array α) (i : Nat) : Array.isEqvAux a a rfl (fun x y => x = y) i = true := by
-  unfold Array.isEqvAux
-  split
-  next h => simp [h, isEqvAux_self a (i+1)]
-  next h => simp [h]
-termination_by a.size - i
-decreasing_by decreasing_trivial_pre_omega
+theorem isEqvAux_self [DecidableEq α] (a : Array α) (i : Nat) (h : i ≤ a.size) :
+    Array.isEqvAux a a rfl (fun x y => x = y) i h = true := by
+  induction i with
+  | zero => simp [Array.isEqvAux]
+  | succ i ih =>
+    simp_all only [isEqvAux, decide_True, Bool.and_self]
 
 theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (fun x y => x = y) = true := by
   simp [isEqv, isEqvAux_self]

src/Lean/Elab/PatternVar.lean

@@ -156,9 +156,11 @@ private def processVar (idStx : Syntax) : M Syntax := do
   modify fun s => { s with vars := s.vars.push idStx, found := s.found.insert id }
   return idStx
 
-private def samePatternsVariables (startingAt : Nat) (s₁ s₂ : State) : Bool :=
-  if h : s₁.vars.size = s₂.vars.size then
-    Array.isEqvAux s₁.vars s₂.vars h (.==.) startingAt
+private def samePatternsVariables (startingAt : Nat) (s₁ s₂ : State) : Bool := Id.run do
+  if h₁ : s₁.vars.size = s₂.vars.size then
+    for h₂ : i in [startingAt:s₁.vars.size] do
+      if s₁.vars[i] != s₂.vars[i]'(by obtain ⟨_, y⟩ := h₂; simp_all) then return false
+    true
   else
     false
 

tests/lean/run/array_isEqvAux.lean

@@ -0,0 +1,45 @@
+
+/-!
+Because `Array.isEqvAux` was defined by well-founded recursion, this used to fail with
+```
+tactic 'decide' failed for proposition
+  #[0, 1] = #[0, 1]
+since its 'Decidable' instance
+  #[0, 1].instDecidableEq #[0, 1]
+did not reduce to 'isTrue' or 'isFalse'.
+
+After unfolding the instances 'instDecidableEqNat', 'Array.instDecidableEq' and 'Nat.decEq', reduction got stuck at the 'Decidable' instance
+  match h : #[0, 1].isEqv #[0, 1] fun a b => decide (a = b) with
+  | true => isTrue ⋯
+  | false => isFalse ⋯
+```
+-/
+
+example : #[0, 1] = #[0, 1] := by decide
+
+/-!
+There are other `Array` functions that use well-founded recursion,
+which we've marked as `@[semireducible]`. We test that `decide` can unfold them here.
+-/
+
+example : Array.ofFn (id : Fin 2 → Fin 2) = #[0, 1] := by decide
+
+example : #[0, 1].map (· + 1) = #[1, 2] := by decide
+
+example : #[0, 1].any (· % 2 = 0) := by decide
+
+example : #[0, 1].findIdx? (· % 2 = 0) = some 0 := by decide
+
+example : #[0, 1, 2].popWhile (· % 2 = 0) = #[0, 1] := by decide
+
+example : #[0, 1, 2].takeWhile (· % 2 = 0) = #[0] := by decide
+
+example : #[0, 1, 2].feraseIdx ⟨1, by decide⟩ = #[0, 2] := by decide
+
+example : #[0, 1, 2].insertAt ⟨1, by decide⟩ 3 = #[0, 3, 1, 2] := by decide
+
+example : #[0, 1, 2].isPrefixOf #[0, 1, 2, 3] = true := by decide
+
+example : #[0, 1, 2].zipWith #[3, 4, 5] (· + ·) = #[3, 5, 7] := by decide
+
+example : #[0, 1, 2].allDiff = true := by decide

tests/lean/run/overAndPartialAppsAtWF.lean

@@ -1,7 +1,17 @@
-theorem eq_of_isEqvAux [DecidableEq α] (a b : Array α) (hsz : a.size = b.size) (i : Nat) (hi : i ≤ a.size) (heqv : Array.isEqvAux a b hsz (fun x y => x = y) i) : ∀ (j : Nat) (hl : i ≤ j) (hj : j < a.size), a.get ⟨j, hj⟩ = b.get ⟨j, hsz ▸ hj⟩ := by
+@[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
+
+theorem eq_of_isEqvAux [DecidableEq α] (a b : Array α) (hsz : a.size = b.size) (i : Nat) (hi : i ≤ a.size) (heqv : isEqvAux a b hsz (fun x y => x = y) i) : ∀ (j : Nat) (hl : i ≤ j) (hj : j < a.size), a.get ⟨j, hj⟩ = b.get ⟨j, hsz ▸ hj⟩ := by
   intro j low high
   by_cases h : i < a.size
-  · unfold Array.isEqvAux at heqv
+  · unfold isEqvAux at heqv
     simp [h] at heqv
     have hind := eq_of_isEqvAux a b hsz (i+1) (Nat.succ_le_of_lt h) heqv.2
     by_cases heq : i = j