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