feat(library/standard/congruence.lean): add class to infer that a function is a congruence with respect to two relations

library/standard/congruence.lean

@@ -0,0 +1,67 @@
+----------------------------------------------------------------------------------------------------
+--- 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
+
+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