lean2/tests/lean/run/elab_bug1.lean

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--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Jeremy Avigad
----------------------------------------------------------------------------------------------------
import logic
import function
using function
namespace congruence
-- TODO: delete this
axiom sorry {P : Prop} : P
-- TODO: move this somewhere else
abbreviation reflexive {T : Type} (R : T → T → Type) : Prop := ∀x, R x x
-- Congruence classes for unary and binary functions
-- -------------------------------------------------
inductive congruence {T1 : Type} {T2 : Type} (R1 : T1 → T1 → Prop) (R2 : T2 → T2 → Prop)
(f : T1 → T2) : Prop :=
| mk : (∀x y : T1, R1 x y → R2 (f x) (f y)) → congruence R1 R2 f
-- to trigger class inference
theorem congr_app {T1 : Type} {T2 : Type} (R1 : T1 → T1 → Prop) (R2 : T2 → T2 → Prop)
(f : T1 → T2) {C : congruence R1 R2 f} {x y : T1} : R1 x y → R2 (f x) (f y) :=
congruence_rec id C x y
-- General tools to build instances
-- --------------------------------
theorem congr_trivial [instance] {T : Type} (R : T → T → Prop) : congruence R R id :=
mk (take x y H, H)
theorem congr_const {T2 : Type} (R2 : T2 → T2 → Prop) (H : reflexive R2) :
∀(T1 : Type) (R1 : T1 → T1 → Prop) (c : T2), congruence R1 R2 (const T1 c) :=
take T1 R1 c, mk (take x y H1, H c)
-- congruences for logic
theorem congr_const_iff [instance] (T1 : Type) (R1 : T1 → T1 → Prop) (c : Prop) :
congruence R1 iff (const T1 c) := congr_const iff iff_refl T1 R1 c
theorem congr_or [instance] (T : Type) (R : T → T → Prop) (f1 f2 : T → Prop)
(H1 : congruence R iff f1) (H2 : congruence R iff f2) :
congruence R iff (λx, f1 x f2 x) := sorry
theorem congr_implies [instance] (T : Type) (R : T → T → Prop) (f1 f2 : T → Prop)
(H1 : congruence R iff f1) (H2 : congruence R iff f2) :
congruence R iff (λx, f1 x → f2 x) := sorry
theorem congr_iff [instance] (T : Type) (R : T → T → Prop) (f1 f2 : T → Prop)
(H1 : congruence R iff f1) (H2 : congruence R iff f2) :
congruence R iff (λx, f1 x ↔ f2 x) := sorry
theorem congr_not [instance] (T : Type) (R : T → T → Prop) (f : T → Prop)
(H : congruence R iff f) :
congruence R iff (λx, ¬ f x) := sorry
theorem subst_iff {T : Type} {R : T → T → Prop} {P : T → Prop} {C : congruence R iff P}
{a b : T} (H : R a b) (H1 : P a) : P b :=
-- iff_mp_left (congruence_rec id C a b H) H1
iff_mp_left (@congr_app _ _ R iff P C a b H) H1
theorem test2 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a c → ¬(d → a)) : b c → ¬(d → b) :=
subst_iff H1 H2