---------------------------------------------------------------------------------------------------- --- 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.connectives.basic logic.connectives.eq struc.relation using relation -- Congruences for logic -- --------------------- theorem congr_not : congr.class 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.class2 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.class2 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.class2 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.class2 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 -- Generalized substitution -- ------------------------ namespace gensubst -- TODO: note that the target has to be "iff". Otherwise, there is not enough -- information to infer an mp-like relation. theorem subst {T : Type} {R : T → T → Prop} {P : T → Prop} {C : congr.class R iff P} {a b : T} (H : R a b) (H1 : P a) : P b := iff_elim_left (congr.app C H) H1 infixr `▸`:75 := subst end -- gensubst -- = is an equivalence relation -- ---------------------------- theorem is_reflexive_eq [instance] (T : Type) : relation.is_reflexive.class (@eq T) := relation.is_reflexive.mk (@refl T) theorem is_symmetric_eq [instance] (T : Type) : relation.is_symmetric.class (@eq T) := relation.is_symmetric.mk (@symm T) theorem is_transitive_eq [instance] (T : Type) : relation.is_transitive.class (@eq T) := relation.is_transitive.mk (@trans T) -- iff is an equivalence relation -- ------------------------------ theorem is_reflexive_iff [instance] : relation.is_reflexive.class iff := relation.is_reflexive.mk (@iff_refl) theorem is_symmetric_iff [instance] : relation.is_symmetric.class iff := relation.is_symmetric.mk (@iff_symm) theorem is_transitive_iff [instance] : relation.is_transitive.class iff := relation.is_transitive.mk (@iff_trans) -- Mp-like for iff -- --------------- theorem mp_like_iff [instance] (a b : Prop) (H : a ↔ b) : relation.mp_like.class H := relation.mp_like.mk (iff_elim_left H) -- Tests -- ----- namespace logic_instances_tests section using relation.operations theorem test1 (a b : Prop) (H : a ↔ b) (H1 : a) : b := mp H H1 end section using gensubst theorem test2 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) := subst H1 H2 theorem test3 (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) := H1 ▸ H2 end theorem test4 (a b c d e : Prop) (H1 : a ↔ b) : (a ∨ c → ¬(d → a)) ↔ (b ∨ c → ¬(d → b)) := congr.infer iff iff (λa, (a ∨ c → ¬(d → a))) H1 section using relation.symbols theorem test5 (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d := H1 ⬝ H2⁻¹ ⬝ H3 theorem test6 (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d := H1 ⬝ (H2⁻¹ ⬝ H3) end end -- Boolean calculations -- -------------------- -- TODO: move these to new file -- TODO: declare trans theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b := calc (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _ ... ↔ a ∨ (c ∨ b) : congr.infer iff iff _ (or_comm b c) ... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _) -- TODO: add or_left_comm, and_right_comm, and_left_comm