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DecidableEq.lean
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DecidableEq.lean
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/-
Copyright (c) 2022 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
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
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
case inr => intro; contradiction
case inl 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 ..) _
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
case inl h => simp [h, isEqvAux_self a (i+1)]
case inr h => simp [h]
termination_by a.size - i
theorem isEqv_self [DecidableEq α] (a : Array α) : Array.isEqv a a (fun x y => x = y) = true := by
simp [isEqv, isEqvAux_self]
instance [DecidableEq α] : DecidableEq (Array α) :=
fun a b =>
match h:isEqv a b (fun a b => a = b) with
| true => isTrue (eq_of_isEqv a b h)
| false => isFalse fun h' => by subst h'; rw [isEqv_self] at h; contradiction
end Array