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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
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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

section

parameters {T1 : Type} {T2 : Type} (R1 : T1 → T1 → Prop) (R2 : T2 → T2 → Prop) (f : T1 → T2)

definition congruence : Prop := ∀x y : T1, R1 x y → R2 (f x) (f y)

theorem congr_app {H1 : congruence} {x y : T1} (H2 : R1 x y) : R2 (f x) (f y) := H1 x y H2

end

theorem congr_trivial [instance] {T : Type} (R : T → T → Prop) : congruence R R id := 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 x y H1, H c

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_and [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_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 test1 (a b c d e : Prop) (H1 : a ↔ b) : (a ∨ c → ¬(d → a)) ↔ (b ∨ c → ¬(d → b)) :=
  congr_app iff iff _ H1

theorem subst_iff {T : Type} {R : T → T → Prop} {P : T → Prop} {Hcongr : congruence R iff P}
    {a b : T} (H : R a b) (H1 : P a) : P b :=
iff_mp_left (@congr_app _ _ R iff P Hcongr _ _ 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