lean2/library/standard/congr.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
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import logic
import function

using function

namespace congr

-- 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
-- -------------------------------------------------

-- TODO: call this 'class', so outside it is congruence.class
inductive struc {T1 : Type} (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
    (f : T1 → T2) : Prop :=
| mk : (∀x y : T1, R1 x y → R2 (f x) (f y)) → struc R1 R2 f

abbreviation app {T1 : Type} {R1 : T1 → T1 → Prop} {T2 : Type} {R2 : T2 → T2 → Prop}
    {f : T1 → T2} (C : struc R1 R2 f) ⦃x y : T1⦄ : R1 x y → R2 (f x) (f y) :=
struc_rec id C x y

-- to trigger class inference
theorem infer {T1 : Type} (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
    (f : T1 → T2) {C : struc R1 R2 f} ⦃x y : T1⦄ : R1 x y → R2 (f x) (f y) :=
struc_rec id C x y

-- for binary functions
inductive struc2 {T1 : Type}  (R1 : T1 → T1 → Prop) {T2 : Type} (R2 : T2 → T2 → Prop)
    {T3 : Type} (R3 : T3 → T3 → Prop) (f : T1 → T2 → T3) : Prop :=
| mk2 : (∀(x1 y1 : T1) (x2 y2 : T2), R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2)) →
    struc2 R1 R2 R3 f

abbreviation app2 {T1 : Type} {R1 : T1 → T1 → Prop} {T2 : Type} {R2 : T2 → T2 → Prop}
    {T3 : Type} {R3 : T3 → T3 → Prop}
    {f : T1 → T2 → T3} (C : struc2 R1 R2 R3 f) ⦃x1 y1 : T1⦄ ⦃x2 y2 : T2⦄
  : R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2) :=
struc2_rec id C x1 y1 x2 y2


-- General tools to build instances
-- --------------------------------

theorem compose
    {T2 : Type} {R2 : T2 → T2 → Prop}
    {T3 : Type} {R3 : T3 → T3 → Prop}
    {g : T2 → T3} (C2 : congr.struc R2 R3 g)
    {{T1 : Type}} {R1 : T1 → T1 → Prop}
    {f : T1 → T2} (C1 : congr.struc R1 R2 f) :
  congr.struc R1 R3 (λx, g (f x)) := mk (take x1 x2 H, app C2 (app C1 H))

theorem compose21
    {T2 : Type} {R2 : T2 → T2 → Prop}
    {T3 : Type} {R3 : T3 → T3 → Prop}
    {T4 : Type} {R4 : T4 → T4 → Prop}
    {g : T2 → T3 → T4} (C3 : congr.struc2 R2 R3 R4 g)
    ⦃T1 : Type⦄ {R1 : T1 → T1 → Prop}
    {f1 : T1 → T2} (C1 : congr.struc R1 R2 f1)
    {f2 : T1 → T3} (C2 : congr.struc R1 R3 f2) :
  congr.struc R1 R4 (λx, g (f1 x) (f2 x)) := mk (take x1 x2 H, app2 C3 (app C1 H) (app C2 H))

theorem trivial [instance] {T : Type} (R : T → T → Prop) : struc R R id :=
mk (take x y H, H)

theorem const {T2 : Type} (R2 : T2 → T2 → Prop) (H : reflexive R2) :
  ∀(T1 : Type) (R1 : T1 → T1 → Prop) (c : T2), struc R1 R2 (function.const T1 c) :=
take T1 R1 c, mk (take x y H1, H c)

-- instances for logic
-- -------------------

-- TODO: swap order for and_elim?

abbreviation imp (a b : Prop) : Prop := a → b

theorem and_imp_and {a b c d : Prop} (H1 : a ∧ b) (H2 : a → c) (H3 : b → d) : c ∧ d :=
and_elim (assume Ha : a, assume Hb : b, and_intro (H2 Ha) (H3 Hb)) H1

theorem imp_and_left {a b c : Prop} (H1 : a ∧ c) (H : a → b) : b ∧ c :=
and_elim (assume Ha : a, assume Hc : c, and_intro (H Ha) Hc) H1

theorem imp_and_right {a b c : Prop} (H1 : c ∧ a) (H : a → b) : c ∧ b :=
and_elim (assume Hc : c, assume Ha : a, and_intro Hc (H Ha)) H1

theorem congr_not : congr.struc iff iff not :=
congr.mk
  (take a b,
    assume H : a ↔ b, iff_intro
      (assume H1 : ¬a, assume H2 : b, H1 (iff_elim_right H H2))
      (assume H1 : ¬b, assume H2 : a, H1 (iff_elim_left H H2)))

theorem congr_and : congr.struc2 iff iff iff and :=
congr.mk2
  (take a1 b1 a2 b2,
    assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
    iff_intro
      (assume H3 : a1 ∧ a2, and_imp_and H3 (iff_elim_left H1) (iff_elim_left H2))
      (assume H3 : b1 ∧ b2, and_imp_and H3 (iff_elim_right H1) (iff_elim_right H2)))

theorem congr_or : congr.struc2 iff iff iff or :=
congr.mk2
  (take a1 b1 a2 b2,
    assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
    iff_intro
      (assume H3 : a1 ∨ a2, or_imp_or H3 (iff_elim_left H1) (iff_elim_left H2))
      (assume H3 : b1 ∨ b2, or_imp_or H3 (iff_elim_right H1) (iff_elim_right H2)))

theorem congr_imp : congr.struc2 iff iff iff imp :=
congr.mk2
  (take a1 b1 a2 b2,
    assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
    iff_intro
      (assume H3 : a1 → a2, assume Hb1 : b1, iff_elim_left H2 (H3 ((iff_elim_right H1) Hb1)))
      (assume H3 : b1 → b2, assume Ha1 : a1, iff_elim_right H2 (H3 ((iff_elim_left H1) Ha1))))

theorem congr_iff : congr.struc2 iff iff iff iff :=
congr.mk2
  (take a1 b1 a2 b2,
    assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
    iff_intro
      (assume H3 : a1 ↔ a2, iff_trans (iff_symm H1) (iff_trans H3 H2))
      (assume H3 : b1 ↔ b2, iff_trans H1 (iff_trans H3 (iff_symm H2))))

theorem congr_const_iff [instance] := congr.const iff iff_refl
theorem congr_not_compose [instance] := congr.compose congr_not
theorem congr_and_compose [instance] := congr.compose21 congr_and
theorem congr_or_compose [instance] := congr.compose21 congr_or
theorem congr_implies_compose [instance] := congr.compose21 congr_imp
theorem congr_iff_compose [instance] := congr.compose21 congr_iff

theorem subst_iff {T : Type} {R : T → T → Prop} {P : T → Prop} {C : struc R iff P}
  {a b : T} (H : R a b) (H1 : P a) : P b := iff_mp_left (app C H) H1

theorem test1 (a b c d e : Prop) (H1 : a ↔ b) : (a ∨ c → ¬(d → a)) ↔ (b ∨ c → ¬(d → b)) :=
congr.infer iff iff _ 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