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@@ -1,2943 +0,0 @@ -Require Import Unicode.Utf8. - -Module No1. -Import Unicode.Utf8. - (*We first give the axioms of Principia -for the propositional calculus in *1.*) - -Axiom MP1_1 : ∀ P Q : Prop, - (P → Q) → P → Q. (*Modus ponens*) - - (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*) - -Axiom Taut1_2 : ∀ P : Prop, - P ∨ P→ P. (*Tautology*) - -Axiom Add1_3 : ∀ P Q : Prop, - Q → P ∨ Q. (*Addition*) - -Axiom Perm1_4 : ∀ P Q : Prop, - P ∨ Q → Q ∨ P. (*Permutation*) - -Axiom Assoc1_5 : ∀ P Q R : Prop, - P ∨ (Q ∨ R) → Q ∨ (P ∨ R). - -Axiom Sum1_6: ∀ P Q R : Prop, - (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*) - -Axiom Impl1_01 : ∀ P Q : Prop, - (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*) - -End No1. - -Module No2. -Import No1. - -(*We proceed to the deductions of of Principia.*) - -Theorem Abs2_01 : ∀ P : Prop, - (P → ~P) → ~P. -Proof. intros P. - specialize Taut1_2 with (~P). - replace (~P ∨ ~P) with (P → ~P). - apply MP1_1. - apply Impl1_01. -Qed. - -Theorem n2_02 : ∀ P Q : Prop, - Q → (P → Q). -Proof. intros P Q. - specialize Add1_3 with (~P) Q. - replace (~P ∨ Q) with (P → Q). - apply (MP1_1 Q (P → Q)). - apply Impl1_01. -Qed. - -Theorem n2_03 : ∀ P Q : Prop, - (P → ~Q) → (Q → ~P). -Proof. intros P Q. - specialize Perm1_4 with (~P) (~Q). - replace (~P ∨ ~Q) with (P → ~Q). - replace (~Q ∨ ~P) with (Q → ~P). - apply (MP1_1 (P → ~Q) (Q → ~P)). - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Comm2_04 : ∀ P Q R : Prop, - (P → (Q → R)) → (Q → (P → R)). -Proof. intros P Q R. - specialize Assoc1_5 with (~P) (~Q) R. - replace (~Q ∨ R) with (Q → R). - replace (~P ∨ (Q → R)) with (P → (Q → R)). - replace (~P ∨ R) with (P → R). - replace (~Q ∨ (P → R)) with (Q → (P → R)). - apply (MP1_1 (P → Q → R) (Q → P → R)). - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Syll2_05 : ∀ P Q R : Prop, - (Q → R) → ((P → Q) → (P → R)). -Proof. intros P Q R. - specialize Sum1_6 with (~P) Q R. - replace (~P ∨ Q) with (P → Q). - replace (~P ∨ R) with (P → R). - apply (MP1_1 (Q → R) ((P → Q) → (P → R))). - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Syll2_06 : ∀ P Q R : Prop, - (P → Q) → ((Q → R) → (P → R)). -Proof. intros P Q R. - specialize Comm2_04 with (Q → R) (P → Q) (P → R). - intros Comm2_04. - specialize Syll2_05 with P Q R. - intros Syll2_05. - specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). - intros MP1_1. - apply MP1_1. - apply Comm2_04. - apply Syll2_05. -Qed. - -Theorem n2_07 : ∀ P : Prop, - P → (P ∨ P). -Proof. intros P. - specialize Add1_3 with P P. - apply MP1_1. -Qed. - -Theorem n2_08 : ∀ P : Prop, - P → P. -Proof. intros P. - specialize Syll2_05 with P (P ∨ P) P. - intros Syll2_05. - specialize Taut1_2 with P. - intros Taut1_2. - specialize MP1_1 with ((P ∨ P) → P) (P → P). - intros MP1_1. - apply Syll2_05. - apply Taut1_2. - apply n2_07. -Qed. - -Theorem n2_1 : ∀ P : Prop, - (~P) ∨ P. -Proof. intros P. - specialize n2_08 with P. - replace (~P ∨ P) with (P → P). - apply MP1_1. - apply Impl1_01. -Qed. - -Theorem n2_11 : ∀ P : Prop, - P ∨ ~P. -Proof. intros P. - specialize Perm1_4 with (~P) P. - intros Perm1_4. - specialize n2_1 with P. - intros Abs2_01. - apply Perm1_4. - apply n2_1. -Qed. - -Theorem n2_12 : ∀ P : Prop, - P → ~~P. -Proof. intros P. - specialize n2_11 with (~P). - intros n2_11. - rewrite Impl1_01. - assumption. -Qed. - -Theorem n2_13 : ∀ P : Prop, - P ∨ ~~~P. -Proof. intros P. - specialize Sum1_6 with P (~P) (~~~P). - intros Sum1_6. - specialize n2_12 with (~P). - intros n2_12. - apply Sum1_6. - apply n2_12. - apply n2_11. -Qed. - -Theorem n2_14 : ∀ P : Prop, - ~~P → P. -Proof. intros P. - specialize Perm1_4 with P (~~~P). - intros Perm1_4. - specialize n2_13 with P. - intros n2_13. - rewrite Impl1_01. - apply Perm1_4. - apply n2_13. -Qed. - -Theorem Trans2_15 : ∀ P Q : Prop, - (~P → Q) → (~Q → P). -Proof. intros P Q. - specialize Syll2_05 with (~P) Q (~~Q). - intros Syll2_05a. - specialize n2_12 with Q. - intros n2_12. - specialize n2_03 with (~P) (~Q). - intros n2_03. - specialize Syll2_05 with (~Q) (~~P) P. - intros Syll2_05b. - specialize Syll2_05 with (~P → Q) (~P → ~~Q) (~Q → ~~P). - intros Syll2_05c. - specialize Syll2_05 with (~P → Q) (~Q → ~~P) (~Q → P). - intros Syll2_05d. - apply Syll2_05d. - apply Syll2_05b. - apply n2_14. - apply Syll2_05c. - apply n2_03. - apply Syll2_05a. - apply n2_12. -Qed. - -Ltac Syll H1 H2 S := - let S := fresh S in match goal with - | [ H1 : ?P → ?Q, H2 : ?Q → ?R |- _ ] => - assert (S : P → R) by (intros p; apply (H2 (H1 p))) -end. - -Ltac MP H1 H2 := - match goal with - | [ H1 : ?P → ?Q, H2 : ?P |- _ ] => specialize (H1 H2) -end. - -Theorem Trans2_16 : ∀ P Q : Prop, - (P → Q) → (~Q → ~P). -Proof. intros P Q. - specialize n2_12 with Q. - intros n2_12a. - specialize Syll2_05 with P Q (~~Q). - intros Syll2_05a. - specialize n2_03 with P (~Q). - intros n2_03a. - MP n2_12a Syll2_05a. - Syll Syll2_05a n2_03a S. - apply S. -Qed. - -Theorem Trans2_17 : ∀ P Q : Prop, - (~Q → ~P) → (P → Q). -Proof. intros P Q. - specialize n2_03 with (~Q) P. - intros n2_03a. - specialize n2_14 with Q. - intros n2_14a. - specialize Syll2_05 with P (~~Q) Q. - intros Syll2_05a. - MP n2_14a Syll2_05a. - Syll n2_03a Syll2_05a S. - apply S. -Qed. - -Theorem n2_18 : ∀ P : Prop, - (~P → P) → P. -Proof. intros P. - specialize n2_12 with P. - intro n2_12a. - specialize Syll2_05 with (~P) P (~~P). - intro Syll2_05a. - MP Syll2_05a n2_12. - specialize Abs2_01 with (~P). - intros Abs2_01a. - Syll Syll2_05a Abs2_01a Sa. - specialize n2_14 with P. - intros n2_14a. - Syll H n2_14a Sb. - apply Sb. -Qed. - -Theorem n2_2 : ∀ P Q : Prop, - P → (P ∨ Q). -Proof. intros P Q. - specialize Add1_3 with Q P. - intros Add1_3a. - specialize Perm1_4 with Q P. - intros Perm1_4a. - Syll Add1_3a Perm1_4a S. - apply S. -Qed. - -Theorem n2_21 : ∀ P Q : Prop, - ~P → (P → Q). -Proof. intros P Q. - specialize n2_2 with (~P) Q. - intros n2_2a. - specialize Impl1_01 with P Q. - intros Impl1_01a. - replace (~P∨Q) with (P→Q) in n2_2a. - apply n2_2a. -Qed. - -Theorem n2_24 : ∀ P Q : Prop, - P → (~P → Q). -Proof. intros P Q. - specialize n2_21 with P Q. - intros n2_21a. - specialize Comm2_04 with (~P) P Q. - intros Comm2_04a. - apply Comm2_04a. - apply n2_21a. -Qed. - -Theorem n2_25 : ∀ P Q : Prop, - P ∨ ((P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_1 with (P ∨ Q). - intros n2_1a. - specialize Assoc1_5 with (~(P∨Q)) P Q. - intros Assoc1_5a. - MP Assoc1_5a n2_1a. - replace (~(P∨Q)∨Q) with (P∨Q→Q) in Assoc1_5a. - apply Assoc1_5a. - apply Impl1_01. -Qed. - -Theorem n2_26 : ∀ P Q : Prop, - ~P ∨ ((P → Q) → Q). -Proof. intros P Q. - specialize n2_25 with (~P) Q. - intros n2_25a. - replace (~P∨Q) with (P→Q) in n2_25a. - apply n2_25a. - apply Impl1_01. -Qed. - -Theorem n2_27 : ∀ P Q : Prop, - P → ((P → Q) → Q). -Proof. intros P Q. - specialize n2_26 with P Q. - intros n2_26a. - replace (~P∨((P→Q)→Q)) with (P→(P→Q)→Q) in n2_26a. - apply n2_26a. - apply Impl1_01. -Qed. - -Theorem n2_3 : ∀ P Q R : Prop, - (P ∨ (Q ∨ R)) → (P ∨ (R ∨ Q)). -Proof. intros P Q R. - specialize Perm1_4 with Q R. - intros Perm1_4a. - specialize Sum1_6 with P (Q∨R) (R∨Q). - intros Sum1_6a. - MP Sum1_6a Perm1_4a. - apply Sum1_6a. -Qed. - -Theorem n2_31 : ∀ P Q R : Prop, - (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R). -Proof. intros P Q R. - specialize n2_3 with P Q R. - intros n2_3a. - specialize Assoc1_5 with P R Q. - intros Assoc1_5a. - specialize Perm1_4 with R (P∨Q). - intros Perm1_4a. - Syll Assoc1_5a Perm1_4a Sa. - Syll n2_3a Sa Sb. - apply Sb. -Qed. - -Theorem n2_32 : ∀ P Q R : Prop, - ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R)). -Proof. intros P Q R. - specialize Perm1_4 with (P∨Q) R. - intros Perm1_4a. - specialize Assoc1_5 with R P Q. - intros Assoc1_5a. - specialize n2_3 with P R Q. - intros n2_3a. - specialize Syll2_06 with ((P∨Q)∨R) (R∨P∨Q) (P∨R∨Q). - intros Syll2_06a. - MP Syll2_06a Perm1_4a. - MP Syll2_06a Assoc1_5a. - specialize Syll2_06 with ((P∨Q)∨R) (P∨R∨Q) (P∨Q∨R). - intros Syll2_06b. - MP Syll2_06b Syll2_06a. - MP Syll2_06b n2_3a. - apply Syll2_06b. -Qed. - -Axiom n2_33 : ∀ P Q R : Prop, - (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*) - -Theorem n2_36 : ∀ P Q R : Prop, - (Q → R) → ((P ∨ Q) → (R ∨ P)). -Proof. intros P Q R. - specialize Perm1_4 with P R. - intros Perm1_4a. - specialize Syll2_05 with (P∨Q) (P∨R) (R∨P). - intros Syll2_05a. - MP Syll2_05a Perm1_4a. - specialize Sum1_6 with P Q R. - intros Sum1_6a. - Syll Sum1_6a Syll2_05a S. - apply S. -Qed. - -Theorem n2_37 : ∀ P Q R : Prop, - (Q → R) → ((Q ∨ P) → (P ∨ R)). -Proof. intros P Q R. - specialize Perm1_4 with Q P. - intros Perm1_4a. - specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). - intros Syll2_06a. - MP Syll2_05a Perm1_4a. - specialize Sum1_6 with P Q R. - intros Sum1_6a. - Syll Sum1_6a Syll2_05a S. - apply S. -Qed. - -Theorem n2_38 : ∀ P Q R : Prop, - (Q → R) → ((Q ∨ P) → (R ∨ P)). -Proof. intros P Q R. - specialize Perm1_4 with P R. - intros Perm1_4a. - specialize Syll2_05 with (Q∨P) (P∨R) (R∨P). - intros Syll2_05a. - MP Syll2_05a Perm1_4a. - specialize Perm1_4 with Q P. - intros Perm1_4b. - specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). - intros Syll2_06a. - MP Syll2_06a Perm1_4b. - Syll Syll2_06a Syll2_05a H. - specialize Sum1_6 with P Q R. - intros Sum1_6a. - Syll Sum1_6a H S. - apply S. -Qed. - -Theorem n2_4 : ∀ P Q : Prop, - (P ∨ (P ∨ Q)) → (P ∨ Q). -Proof. intros P Q. - specialize n2_31 with P P Q. - intros n2_31a. - specialize Taut1_2 with P. - intros Taut1_2a. - specialize n2_38 with Q (P∨P) P. - intros n2_38a. - MP n2_38a Taut1_2a. - Syll n2_31a n2_38a S. - apply S. -Qed. - -Theorem n2_41 : ∀ P Q : Prop, - (Q ∨ (P ∨ Q)) → (P ∨ Q). -Proof. intros P Q. - specialize Assoc1_5 with Q P Q. - intros Assoc1_5a. - specialize Taut1_2 with Q. - intros Taut1_2a. - specialize Sum1_6 with P (Q∨Q) Q. - intros Sum1_6a. - MP Sum1_6a Taut1_2a. - Syll Assoc1_5a Sum1_6a S. - apply S. -Qed. - -Theorem n2_42 : ∀ P Q : Prop, - (~P ∨ (P → Q)) → (P → Q). -Proof. intros P Q. - specialize n2_4 with (~P) Q. - intros n2_4a. - replace (~P∨Q) with (P→Q) in n2_4a. - apply n2_4a. apply Impl1_01. -Qed. - -Theorem n2_43 : ∀ P Q : Prop, - (P → (P → Q)) → (P → Q). -Proof. intros P Q. - specialize n2_42 with P Q. - intros n2_42a. - replace (~P ∨ (P→Q)) with (P→(P→Q)) in n2_42a. - apply n2_42a. - apply Impl1_01. -Qed. - -Theorem n2_45 : ∀ P Q : Prop, - ~(P ∨ Q) → ~P. -Proof. intros P Q. - specialize n2_2 with P Q. - intros n2_2a. - specialize Trans2_16 with P (P∨Q). - intros Trans2_16a. - MP n2_2 Trans2_16a. - apply Trans2_16a. -Qed. - -Theorem n2_46 : ∀ P Q : Prop, - ~(P ∨ Q) → ~Q. -Proof. intros P Q. - specialize Add1_3 with P Q. - intros Add1_3a. - specialize Trans2_16 with Q (P∨Q). - intros Trans2_16a. - MP Add1_3a Trans2_16a. - apply Trans2_16a. -Qed. - -Theorem n2_47 : ∀ P Q : Prop, - ~(P ∨ Q) → (~P ∨ Q). -Proof. intros P Q. - specialize n2_45 with P Q. - intros n2_45a. - specialize n2_2 with (~P) Q. - intros n2_2a. - Syll n2_45a n2_2a S. - apply S. -Qed. - -Theorem n2_48 : ∀ P Q : Prop, - ~(P ∨ Q) → (P ∨ ~Q). -Proof. intros P Q. - specialize n2_46 with P Q. - intros n2_46a. - specialize Add1_3 with P (~Q). - intros Add1_3a. - Syll n2_46a Add1_3a S. - apply S. -Qed. - -Theorem n2_49 : ∀ P Q : Prop, - ~(P ∨ Q) → (~P ∨ ~Q). -Proof. intros P Q. - specialize n2_45 with P Q. - intros n2_45a. - specialize n2_2 with (~P) (~Q). - intros n2_2a. - Syll n2_45a n2_2a S. - apply S. -Qed. - -Theorem n2_5 : ∀ P Q : Prop, - ~(P → Q) → (~P → Q). -Proof. intros P Q. - specialize n2_47 with (~P) Q. - intros n2_47a. - replace (~P∨Q) with (P→Q) in n2_47a. - replace (~~P∨Q) with (~P→Q) in n2_47a. - apply n2_47a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_51 : ∀ P Q : Prop, - ~(P → Q) → (P → ~Q). -Proof. intros P Q. - specialize n2_48 with (~P) Q. - intros n2_48a. - replace (~P∨Q) with (P→Q) in n2_48a. - replace (~P∨~Q) with (P→~Q) in n2_48a. - apply n2_48a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_52 : ∀ P Q : Prop, - ~(P → Q) → (~P → ~Q). -Proof. intros P Q. - specialize n2_49 with (~P) Q. - intros n2_49a. - replace (~P∨Q) with (P→Q) in n2_49a. - replace (~~P∨~Q) with (~P→~Q) in n2_49a. - apply n2_49a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_521 : ∀ P Q : Prop, - ~(P→Q)→(Q→P). -Proof. intros P Q. - specialize n2_52 with P Q. - intros n2_52a. - specialize Trans2_17 with Q P. - intros Trans2_17a. - Syll n2_52a Trans2_17a S. - apply S. -Qed. - -Theorem n2_53 : ∀ P Q : Prop, - (P ∨ Q) → (~P → Q). -Proof. intros P Q. - specialize n2_12 with P. - intros n2_12a. - specialize n2_38 with Q P (~~P). - intros n2_38a. - MP n2_38a n2_12a. - replace (~~P∨Q) with (~P→Q) in n2_38a. - apply n2_38a. - apply Impl1_01. -Qed. - -Theorem n2_54 : ∀ P Q : Prop, - (~P → Q) → (P ∨ Q). -Proof. intros P Q. - specialize n2_14 with P. - intros n2_14a. - specialize n2_38 with Q (~~P) P. - intros n2_38a. - MP n2_38a n2_12a. - replace (~~P∨Q) with (~P→Q) in n2_38a. - apply n2_38a. - apply Impl1_01. -Qed. - -Theorem n2_55 : ∀ P Q : Prop, - ~P → ((P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_53 with P Q. - intros n2_53a. - specialize Comm2_04 with (P∨Q) (~P) Q. - intros Comm2_04a. - MP n2_53a Comm2_04a. - apply Comm2_04a. -Qed. - -Theorem n2_56 : ∀ P Q : Prop, - ~Q → ((P ∨ Q) → P). -Proof. intros P Q. - specialize n2_55 with Q P. - intros n2_55a. - specialize Perm1_4 with P Q. - intros Perm1_4a. - specialize Syll2_06 with (P∨Q) (Q∨P) P. - intros Syll2_06a. - MP Syll2_06a Perm1_4a. -Qed. - -Theorem n2_6 : ∀ P Q : Prop, - (~P→Q) → ((P → Q) → Q). -Proof. intros P Q. - specialize n2_38 with Q (~P) Q. - intros n2_38a. - specialize Taut1_2 with Q. - intros Taut1_2a. - specialize Syll2_05 with (~P∨Q) (Q∨Q) Q. - intros Syll2_05a. - MP Syll2_05a Taut1_2a. - Syll n2_38a Syll2_05a S. - replace (~P∨Q) with (P→Q) in S. - apply S. - apply Impl1_01. -Qed. - -Theorem n2_61 : ∀ P Q : Prop, - (P → Q) → ((~P → Q) → Q). -Proof. intros P Q. - specialize n2_6 with P Q. - intros n2_6a. - specialize Comm2_04 with (~P→Q) (P→Q) Q. - intros Comm2_04a. - MP Comm2_04a n2_6a. - apply Comm2_04a. -Qed. - -Theorem n2_62 : ∀ P Q : Prop, - (P ∨ Q) → ((P → Q) → Q). -Proof. intros P Q. - specialize n2_53 with P Q. - intros n2_53a. - specialize n2_6 with P Q. - intros n2_6a. - Syll n2_53a n2_6a S. - apply S. -Qed. - -Theorem n2_621 : ∀ P Q : Prop, - (P → Q) → ((P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_62 with P Q. - intros n2_62a. - specialize Comm2_04 with (P ∨ Q) (P→Q) Q. - intros Comm2_04a. - MP Comm2_04a n2_62a. - apply Comm2_04a. -Qed. - -Theorem n2_63 : ∀ P Q : Prop, - (P ∨ Q) → ((~P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_62 with P Q. - intros n2_62a. - replace (~P∨Q) with (P→Q). - apply n2_62a. - apply Impl1_01. -Qed. - -Theorem n2_64 : ∀ P Q : Prop, - (P ∨ Q) → ((P ∨ ~Q) → P). -Proof. intros P Q. - specialize n2_63 with Q P. - intros n2_63a. - specialize Perm1_4 with P Q. - intros Perm1_4a. - Syll n2_63a Perm1_4a Ha. - specialize Syll2_06 with (P∨~Q) (~Q∨P) P. - intros Syll2_06a. - specialize Perm1_4 with P (~Q). - intros Perm1_4b. - MP Syll2_05a Perm1_4b. - Syll Syll2_05a Ha S. - apply S. -Qed. - -Theorem n2_65 : ∀ P Q : Prop, - (P → Q) → ((P → ~Q) → ~P). -Proof. intros P Q. - specialize n2_64 with (~P) Q. - intros n2_64a. - replace (~P∨Q) with (P→Q) in n2_64a. - replace (~P∨~Q) with (P→~Q) in n2_64a. - apply n2_64a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_67 : ∀ P Q : Prop, - ((P ∨ Q) → Q) → (P → Q). -Proof. intros P Q. - specialize n2_54 with P Q. - intros n2_54a. - specialize Syll2_06 with (~P→Q) (P∨Q) Q. - intros Syll2_06a. - MP Syll2_06a n2_54a. - specialize n2_24 with P Q. - intros n2_24. - specialize Syll2_06 with P (~P→Q) Q. - intros Syll2_06b. - MP Syll2_06b n2_24a. - Syll Syll2_06b Syll2_06a S. - apply S. -Qed. - -Theorem n2_68 : ∀ P Q : Prop, - ((P → Q) → Q) → (P ∨ Q). -Proof. intros P Q. - specialize n2_67 with (~P) Q. - intros n2_67a. - replace (~P∨Q) with (P→Q) in n2_67a. - specialize n2_54 with P Q. - intros n2_54a. - Syll n2_67a n2_54a S. - apply S. - apply Impl1_01. -Qed. - -Theorem n2_69 : ∀ P Q : Prop, - ((P → Q) → Q) → ((Q → P) → P). -Proof. intros P Q. - specialize n2_68 with P Q. - intros n2_68a. - specialize Perm1_4 with P Q. - intros Perm1_4a. - Syll n2_68a Perm1_4a Sa. - specialize n2_62 with Q P. - intros n2_62a. - Syll Sa n2_62a Sb. - apply Sb. -Qed. - -Theorem n2_73 : ∀ P Q R : Prop, - (P → Q) → (((P ∨ Q) ∨ R) → (Q ∨ R)). -Proof. intros P Q R. - specialize n2_621 with P Q. - intros n2_621a. - specialize n2_38 with R (P∨Q) Q. - intros n2_38a. - Syll n2_621a n2_38a S. - apply S. -Qed. - -Theorem n2_74 : ∀ P Q R : Prop, - (Q → P) → ((P ∨ Q) ∨ R) → (P ∨ R). -Proof. intros P Q R. - specialize n2_73 with Q P R. - intros n2_73a. - specialize Assoc1_5 with P Q R. - intros Assoc1_5a. - specialize n2_31 with Q P R. - intros n2_31a. (*not cited explicitly!*) - Syll Assoc1_5a n2_31a Sa. - specialize n2_32 with P Q R. - intros n2_32a. (*not cited explicitly!*) - Syll n2_32a Sa Sb. - specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). - intros Syll2_06a. - MP Syll2_06a Sb. - Syll n2_73a Syll2_05a H. - apply H. -Qed. - -Theorem n2_75 : ∀ P Q R : Prop, - (P ∨ Q) → ((P ∨ (Q → R)) → (P ∨ R)). -Proof. intros P Q R. - specialize n2_74 with P (~Q) R. - intros n2_74a. - specialize n2_53 with Q P. - intros n2_53a. - Syll n2_53a n2_74a Sa. - specialize n2_31 with P (~Q) R. - intros n2_31a. - specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R) (P∨R). - intros Syll2_06a. - MP Syll2_06a n2_31a. - Syll Sa Syll2_06a Sb. - specialize Perm1_4 with P Q. - intros Perm1_4a. (*not cited!*) - Syll Perm1_4a Sb Sc. - replace (~Q∨R) with (Q→R) in Sc. - apply Sc. - apply Impl1_01. -Qed. - -Theorem n2_76 : ∀ P Q R : Prop, - (P ∨ (Q → R)) → ((P ∨ Q) → (P ∨ R)). -Proof. intros P Q R. - specialize n2_75 with P Q R. - intros n2_75a. - specialize Comm2_04 with (P∨Q) (P∨(Q→R)) (P∨R). - intros Comm2_04a. - apply Comm2_04a. - apply n2_75a. -Qed. - -Theorem n2_77 : ∀ P Q R : Prop, - (P → (Q → R)) → ((P → Q) → (P → R)). -Proof. intros P Q R. - specialize n2_76 with (~P) Q R. - intros n2_76a. - replace (~P∨(Q→R)) with (P→Q→R) in n2_76a. - replace (~P∨Q) with (P→Q) in n2_76a. - replace (~P∨R) with (P→R) in n2_76a. - apply n2_76a. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_8 : ∀ Q R S : Prop, - (Q ∨ R) → ((~R ∨ S) → (Q ∨ S)). -Proof. intros Q R S. - specialize n2_53 with R Q. - intros n2_53a. - specialize Perm1_4 with Q R. - intros Perm1_4a. - Syll Perm1_4a n2_53a Ha. - specialize n2_38 with S (~R) Q. - intros n2_38a. - Syll H n2_38a Hb. - apply Hb. -Qed. - -Theorem n2_81 : ∀ P Q R S : Prop, - (Q → (R → S)) → ((P ∨ Q) → ((P ∨ R) → (P ∨ S))). -Proof. intros P Q R S. - specialize Sum1_6 with P Q (R→S). - intros Sum1_6a. - specialize n2_76 with P R S. - intros n2_76a. - specialize Syll2_05 with (P∨Q) (P∨(R→S)) ((P∨R)→(P∨S)). - intros Syll2_05a. - MP Syll2_05a n2_76a. - Syll Sum1_6a Syll2_05a H. - apply H. -Qed. - -Theorem n2_82 : ∀ P Q R S : Prop, - (P ∨ Q ∨ R)→((P ∨ ~R ∨ S)→(P ∨ Q ∨ S)). -Proof. intros P Q R S. - specialize n2_8 with Q R S. - intros n2_8a. - specialize n2_81 with P (Q∨R) (~R∨S) (Q∨S). - intros n2_81a. - MP n2_81a n2_8a. - apply n2_81a. -Qed. - -Theorem n2_83 : ∀ P Q R S : Prop, - (P→(Q→R))→((P→(R→S))→(P→(Q→S))). -Proof. intros P Q R S. - specialize n2_82 with (~P) (~Q) R S. - intros n2_82a. - replace (~Q∨R) with (Q→R) in n2_82a. - replace (~P∨(Q→R)) with (P→Q→R) in n2_82a. - replace (~R∨S) with (R→S) in n2_82a. - replace (~P∨(R→S)) with (P→R→S) in n2_82a. - replace (~Q∨S) with (Q→S) in n2_82a. - replace (~Q∨S) with (Q→S) in n2_82a. - replace (~P∨(Q→S)) with (P→Q→S) in n2_82a. - apply n2_82a. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_85 : ∀ P Q R : Prop, - ((P ∨ Q) → (P ∨ R)) → (P ∨ (Q → R)). -Proof. intros P Q R. - specialize Add1_3 with P Q. - intros Add1_3a. - specialize Syll2_06 with Q (P∨Q) R. - intros Syll2_06a. - MP Syll2_06a Add1_3a. - specialize n2_55 with P R. - intros n2_55a. - specialize Syll2_05 with (P∨Q) (P∨R) R. - intros Syll2_05a. - Syll n2_55a Syll2_05a Ha. - specialize n2_83 with (~P) ((P∨Q)→(P∨R)) ((P∨Q)→R) (Q→R). - intros n2_83a. - MP n2_83a Ha. - specialize Comm2_04 with (~P) (P∨Q→P∨R) (Q→R). - intros Comm2_04a. - Syll Ha Comm2_04a Hb. - specialize n2_54 with P (Q→R). - intros n2_54a. - specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). - intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*) - MP Syll2_06a n2_02a. - MP Hb n2_02a. - Syll Hb n2_54a Hc. - apply Hc. -Qed. - -Theorem n2_86 : ∀ P Q R : Prop, - ((P → Q) → (P → R)) → (P → (Q → R)). -Proof. intros P Q R. - specialize n2_85 with (~P) Q R. - intros n2_85a. - replace (~P∨Q) with (P→Q) in n2_85a. - replace (~P∨R) with (P→R) in n2_85a. - replace (~P∨(Q→R)) with (P→Q→R) in n2_85a. - apply n2_85a. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -End No2. - -Module No3. - -Import No1. -Import No2. - -Axiom Prod3_01 : ∀ P Q : Prop, - (P ∧ Q) = ~(~P ∨ ~Q). - -Axiom Abb3_02 : ∀ P Q R : Prop, - (P→Q→R)=(P→Q)∧(Q→R). - -Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧Q). (*3.03 is a derived rule permitting an inference from the theoremhood of P and that of Q to that of P and Q.*) -Proof. intros P Q. - specialize n2_11 with (~P∨~Q). intros n2_11a. - specialize n2_32 with (~P) (~Q) (~(~P ∨ ~Q)). intros n2_32a. - MP n2_32a n2_11a. - replace (~(~P∨~Q)) with (P∧Q) in n2_32a. - replace (~Q ∨ (P∧Q)) with (Q→(P∧Q)) in n2_32a. - replace (~P ∨ (Q → (P∧Q))) with (P→Q→(P∧Q)) in n2_32a. - apply n2_32a. - apply Impl1_01. - apply Impl1_01. - apply Prod3_01. -Qed. - -Theorem n3_1 : ∀ P Q : Prop, - (P ∧ Q) → ~(~P ∨ ~Q). -Proof. intros P Q. - replace (~(~P∨~Q)) with (P∧Q). - specialize n2_08 with (P∧Q). - intros n2_08a. - apply n2_08a. - apply Prod3_01. -Qed. - -Theorem n3_11 : ∀ P Q : Prop, - ~(~P ∨ ~Q) → (P ∧ Q). -Proof. intros P Q. - replace (~(~P∨~Q)) with (P∧Q). - specialize n2_08 with (P∧Q). - intros n2_08a. - apply n2_08a. - apply Prod3_01. -Qed. - -Theorem n3_12 : ∀ P Q : Prop, - (~P ∨ ~Q) ∨ (P ∧ Q). -Proof. intros P Q. - specialize n2_11 with (~P∨~Q). - intros n2_11a. - replace (~(~P∨~Q)) with (P∧Q) in n2_11a. - apply n2_11a. - apply Prod3_01. -Qed. - -Theorem n3_13 : ∀ P Q : Prop, - ~(P ∧ Q) → (~P ∨ ~Q). -Proof. intros P Q. - specialize n3_11 with P Q. - intros n3_11a. - specialize Trans2_15 with (~P∨~Q) (P∧Q). - intros Trans2_15a. - MP Trans2_16a n3_11a. - apply Trans2_15a. -Qed. - -Theorem n3_14 : ∀ P Q : Prop, - (~P ∨ ~Q) → ~(P ∧ Q). -Proof. intros P Q. - specialize n3_1 with P Q. - intros n3_1a. - specialize Trans2_16 with (P∧Q) (~(~P∨~Q)). - intros Trans2_16a. - MP Trans2_16a n3_1a. - specialize n2_12 with (~P∨~Q). - intros n2_12a. - Syll n2_12a Trans2_16a S. - apply S. -Qed. - -Theorem n3_2 : ∀ P Q : Prop, - P → Q → (P ∧ Q). -Proof. intros P Q. - specialize n3_12 with P Q. - intros n3_12a. - specialize n2_32 with (~P) (~Q) (P∧Q). - intros n2_32a. - MP n3_32a n3_12a. - replace (~Q ∨ P ∧ Q) with (Q→P∧Q) in n2_32a. - replace (~P ∨ (Q → P ∧ Q)) with (P→Q→P∧Q) in n2_32a. - apply n2_32a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n3_21 : ∀ P Q : Prop, - Q → P → (P ∧ Q). -Proof. intros P Q. - specialize n3_2 with P Q. - intros n3_2a. - specialize Comm2_04 with P Q (P∧Q). - intros Comm2_04a. - MP Comm2_04a n3_2a. - apply Comm2_04a. -Qed. - -Theorem n3_22 : ∀ P Q : Prop, - (P ∧ Q) → (Q ∧ P). -Proof. intros P Q. - specialize n3_13 with Q P. - intros n3_13a. - specialize Perm1_4 with (~Q) (~P). - intros Perm1_4a. - Syll n3_13a Perm1_4a Ha. - specialize n3_14 with P Q. - intros n3_14a. - Syll Ha n3_14a Hb. - specialize Trans2_17 with (P∧Q) (Q ∧ P). - intros Trans2_17a. - MP Trans2_17a Hb. - apply Trans2_17a. -Qed. - -Theorem n3_24 : ∀ P : Prop, - ~(P ∧ ~P). -Proof. intros P. - specialize n2_11 with (~P). - intros n2_11a. - specialize n3_14 with P (~P). - intros n3_14a. - MP n3_14a n2_11a. - apply n3_14a. -Qed. - -Theorem Simp3_26 : ∀ P Q : Prop, - (P ∧ Q) → P. -Proof. intros P Q. - specialize n2_02 with Q P. - intros n2_02a. - replace (P→(Q→P)) with (~P∨(Q→P)) in n2_02a. - replace (Q→P) with (~Q∨P) in n2_02a. - specialize n2_31 with (~P) (~Q) P. - intros n2_31a. - MP n2_31a n2_02a. - specialize n2_53 with (~P∨~Q) P. - intros n2_53a. - MP n2_53a n2_02a. - replace (~(~P∨~Q)) with (P∧Q) in n2_53a. - apply n2_53a. - apply Prod3_01. - replace (~Q∨P) with (Q→P). - reflexivity. - apply Impl1_01. - replace (~P∨(Q→P)) with (P→Q→P). - reflexivity. - apply Impl1_01. -Qed. - -Theorem Simp3_27 : ∀ P Q : Prop, - (P ∧ Q) → Q. -Proof. intros P Q. - specialize n3_22 with P Q. - intros n3_22a. - specialize Simp3_26 with Q P. - intros Simp3_26a. - Syll n3_22a Simp3_26a S. - apply S. -Qed. - -Theorem Exp3_3 : ∀ P Q R : Prop, - ((P ∧ Q) → R) → (P → (Q → R)). -Proof. intros P Q R. - specialize Trans2_15 with (~P∨~Q) R. - intros Trans2_15a. - replace (~R→(~P∨~Q)) with (~R→(P→~Q)) in Trans2_15a. - specialize Comm2_04 with (~R) P (~Q). - intros Comm2_04a. - Syll Trans2_15a Comm2_04a Sa. - specialize Trans2_17 with Q R. - intros Trans2_17a. - specialize Syll2_05 with P (~R→~Q) (Q→R). - intros Syll2_05a. - MP Syll2_05a Trans2_17a. - Syll Sa Syll2_05a Sb. - replace (~(~P∨~Q)) with (P∧Q) in Sb. - apply Sb. - apply Prod3_01. - replace (~P∨~Q) with (P→~Q). - reflexivity. - apply Impl1_01. -Qed. - -Theorem Imp3_31 : ∀ P Q R : Prop, - (P → (Q → R)) → (P ∧ Q) → R. -Proof. intros P Q R. - specialize n2_31 with (~P) (~Q) R. - intros n2_31a. - specialize n2_53 with (~P∨~Q) R. - intros n2_53a. - Syll n2_31a n2_53a S. - replace (~Q∨R) with (Q→R) in S. - replace (~P∨(Q→R)) with (P→Q→R) in S. - replace (~(~P∨~Q)) with (P∧Q) in S. - apply S. - apply Prod3_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Syll3_33 : ∀ P Q R : Prop, - ((P → Q) ∧ (Q → R)) → (P → R). -Proof. intros P Q R. - specialize Syll2_06 with P Q R. - intros Syll2_06a. - specialize Imp3_31 with (P→Q) (Q→R) (P→R). - intros Imp3_31a. - MP Imp3_31a Syll2_06a. - apply Imp3_31a. -Qed. - -Theorem Syll3_34 : ∀ P Q R : Prop, - ((Q → R) ∧ (P → Q)) → (P → R). -Proof. intros P Q R. - specialize Syll2_05 with P Q R. - intros Syll2_05a. - specialize Imp3_31 with (Q→R) (P→Q) (P→R). - intros Imp3_31a. - MP Imp3_31a Syll2_05a. - apply Imp3_31a. -Qed. - -Theorem Ass3_35 : ∀ P Q : Prop, - (P ∧ (P → Q)) → Q. -Proof. intros P Q. - specialize n2_27 with P Q. - intros n2_27a. - specialize Imp3_31 with P (P→Q) Q. - intros Imp3_31a. - MP Imp3_31a n2_27a. - apply Imp3_31a. -Qed. - -Theorem n3_37 : ∀ P Q R : Prop, - (P ∧ Q → R) → (P ∧ ~R → ~Q). -Proof. intros P Q R. - specialize Trans2_16 with Q R. - intros Trans2_16a. - specialize Syll2_05 with P (Q→R) (~R→~Q). - intros Syll2_05a. - MP Syll2_05a Trans2_16a. - specialize Exp3_3 with P Q R. - intros Exp3_3a. - Syll Exp3_3a Syll2_05a Sa. - specialize Imp3_31 with P (~R) (~Q). - intros Imp3_31a. - Syll Sa Imp3_31a Sb. - apply Sb. -Qed. - -Theorem n3_4 : ∀ P Q : Prop, - (P ∧ Q) → P → Q. -Proof. intros P Q. - specialize n2_51 with P Q. - intros n2_51a. - specialize Trans2_15 with (P→Q) (P→~Q). - intros Trans2_15a. - MP Trans2_15a n2_51a. - replace (P→~Q) with (~P∨~Q) in Trans2_15a. - replace (~(~P∨~Q)) with (P∧Q) in Trans2_15a. - apply Trans2_15a. - apply Prod3_01. - replace (~P∨~Q) with (P→~Q). - reflexivity. - apply Impl1_01. -Qed. - -Theorem n3_41 : ∀ P Q R : Prop, - (P → R) → (P ∧ Q → R). -Proof. intros P Q R. - specialize Simp3_26 with P Q. - intros Simp3_26a. - specialize Syll2_06 with (P∧Q) P R. - intros Syll2_06a. - MP Simp3_26a Syll2_06a. - apply Syll2_06a. -Qed. - -Theorem n3_42 : ∀ P Q R : Prop, - (Q → R) → (P ∧ Q → R). -Proof. intros P Q R. - specialize Simp3_27 with P Q. - intros Simp3_27a. - specialize Syll2_06 with (P∧Q) Q R. - intros Syll2_06a. - MP Syll2_05a Simp3_27a. - apply Syll2_06a. -Qed. - -Theorem Comp3_43 : ∀ P Q R : Prop, - (P → Q) ∧ (P → R) → (P → Q ∧ R). -Proof. intros P Q R. - specialize n3_2 with Q R. - intros n3_2a. - specialize Syll2_05 with P Q (R→Q∧R). - intros Syll2_05a. - MP Syll2_05a n3_2a. - specialize n2_77 with P R (Q∧R). - intros n2_77a. - Syll Syll2_05a n2_77a Sa. - specialize Imp3_31 with (P→Q) (P→R) (P→Q∧R). - intros Imp3_31a. - MP Sa Imp3_31a. - apply Imp3_31a. -Qed. - -Theorem n3_44 : ∀ P Q R : Prop, - (Q → P) ∧ (R → P) → (Q ∨ R → P). -Proof. intros P Q R. - specialize Syll3_33 with (~Q) R P. - intros Syll3_33a. - specialize n2_6 with Q P. - intros n2_6a. - Syll Syll3_33a n2_6a Sa. - specialize Exp3_3 with (~Q→R) (R→P) ((Q→P)→P). - intros Exp3_3a. - MP Exp3_3a Sa. - specialize Comm2_04 with (R→P) (Q→P) P. - intros Comm2_04a. - Syll Exp3_3a Comm2_04a Sb. - specialize Imp3_31 with (Q→P) (R→P) P. - intros Imp3_31a. - Syll Sb Imp3_31a Sc. - specialize Comm2_04 with (~Q→R) ((Q→P)∧(R→P)) P. - intros Comm2_04b. - MP Comm2_04b Sc. - specialize n2_53 with Q R. - intros n2_53a. - specialize Syll2_06 with (Q∨R) (~Q→R) P. - intros Syll2_06a. - MP Syll2_06a n2_53a. - Syll Comm2_04b Syll2_06a Sd. - apply Sd. -Qed. - -Theorem Fact3_45 : ∀ P Q R : Prop, - (P → Q) → (P ∧ R) → (Q ∧ R). -Proof. intros P Q R. - specialize Syll2_06 with P Q (~R). - intros Syll2_06a. - specialize Trans2_16 with (Q→~R) (P→~R). - intros Trans2_16a. - Syll Syll2_06a Trans2_16a S. - replace (P→~R) with (~P∨~R) in S. - replace (Q→~R) with (~Q∨~R) in S. - replace (~(~P∨~R)) with (P∧R) in S. - replace (~(~Q∨~R)) with (Q∧R) in S. - apply S. - apply Prod3_01. - apply Prod3_01. - replace (~Q∨~R) with (Q→~R). - reflexivity. - apply Impl1_01. - replace (~P∨~R) with (P→~R). - reflexivity. - apply Impl1_01. -Qed. - -Theorem n3_47 : ∀ P Q R S : Prop, - ((P → R) ∧ (Q → S)) → (P ∧ Q) → R ∧ S. -Proof. intros P Q R S. - specialize Simp3_26 with (P→R) (Q→S). - intros Simp3_26a. - specialize Fact3_45 with P R Q. - intros Fact3_45a. - Syll Simp3_26a Fact3_45a Sa. - specialize n3_22 with R Q. - intros n3_22a. - specialize Syll2_05 with (P∧Q) (R∧Q) (Q∧R). - intros Syll2_05a. - MP Syll2_05a n3_22a. - Syll Sa Syll2_05a Sb. - specialize Simp3_27 with (P→R) (Q→S). - intros Simp3_27a. - specialize Fact3_45 with Q S R. - intros Fact3_45b. - Syll Simp3_27a Fact3_45b Sc. - specialize n3_22 with S R. - intros n3_22b. - specialize Syll2_05 with (Q∧R) (S∧R) (R∧S). - intros Syll2_05b. - MP Syll2_05b n3_22b. - Syll Sc Syll2_05b Sd. - specialize n2_83 with ((P→R)∧(Q→S)) (P∧Q) (Q∧R) (R∧S). - intros n2_83a. - MP n2_83a Sb. - MP n2_83 Sd. - apply n2_83a. -Qed. - -Theorem n3_48 : ∀ P Q R S : Prop, - ((P → R) ∧ (Q → S)) → (P ∨ Q) → R ∨ S. -Proof. intros P Q R S. - specialize Simp3_26 with (P→R) (Q→S). - intros Simp3_26a. - specialize Sum1_6 with Q P R. - intros Sum1_6a. - Syll Simp3_26a Sum1_6a Sa. - specialize Perm1_4 with P Q. - intros Perm1_4a. - specialize Syll2_06 with (P∨Q) (Q∨P) (Q∨R). - intros Syll2_06a. - MP Syll2_06a Perm1_4a. - Syll Sa Syll2_06a Sb. - specialize Simp3_27 with (P→R) (Q→S). - intros Simp3_27a. - specialize Sum1_6 with R Q S. - intros Sum1_6b. - Syll Simp3_27a Sum1_6b Sc. - specialize Perm1_4 with Q R. - intros Perm1_4b. - specialize Syll2_06 with (Q∨R) (R∨Q) (R∨S). - intros Syll2_06b. - MP Syll2_06b Perm1_4b. - Syll Sc Syll2_06a Sd. - specialize n2_83 with ((P→R)∧(Q→S)) (P∨Q) (Q∨R) (R∨S). - intros n2_83a. - MP n2_83a Sb. - MP n2_83a Sd. - apply n2_83a. -Qed. - -End No3. - -Module No4. - -Import No1. -Import No2. -Import No3. - -Axiom Equiv4_01 : ∀ P Q : Prop, - (P↔Q)=((P→Q) ∧ (Q→P)). (*n4_02 defines P iff Q iff R as P iff Q AND Q iff R.*) - -Axiom EqBi : ∀ P Q : Prop, - (P=Q) ↔ (P↔Q). - -Ltac Equiv H1 := - match goal with - | [ H1 : (?P→?Q) ∧ (?Q→?P) |- _ ] => - replace ((P→Q) ∧ (Q→P)) with (P↔Q) in H1 -end. - -Ltac Conj H1 H2 := - match goal with - | [ H1 : ?P, H2 : ?Q |- _ ] => - assert (P ∧ Q) -end. - -Theorem Trans4_1 : ∀ P Q : Prop, - (P → Q) ↔ (~Q → ~P). -Proof. intros P Q. - specialize Trans2_16 with P Q. - intros Trans2_16a. - specialize Trans2_17 with P Q. - intros Trans2_17a. - Conj Trans2_16a Trans2_17a. - split. - apply Trans2_16a. - apply Trans2_17a. - Equiv H. - apply H. - apply Equiv4_01. -Qed. - -Theorem Trans4_11 : ∀ P Q : Prop, - (P ↔ Q) ↔ (~P ↔ ~Q). -Proof. intros P Q. - specialize Trans2_16 with P Q. - intros Trans2_16a. - specialize Trans2_16 with Q P. - intros Trans2_16b. - Conj Trans2_16a Trans2_16b. - split. - apply Trans2_16a. - apply Trans2_16b. - specialize n3_47 with (P→Q) (Q→P) (~Q→~P) (~P→~Q). - intros n3_47a. - MP n3_47 H. - specialize n3_22 with (¬ Q → ¬ P) (¬ P → ¬ Q). - intros n3_22a. - Syll n3_47a n3_22a Sa. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sa. - replace ((¬ P → ¬ Q) ∧ (¬ Q → ¬ P)) with (~P↔~Q) in Sa. - clear Trans2_16a. clear H. clear Trans2_16b. clear n3_22a. clear n3_47a. - specialize Trans2_17 with Q P. - intros Trans2_17a. - specialize Trans2_17 with P Q. - intros Trans2_17b. - Conj Trans2_17a Trans2_17b. - split. - apply Trans2_17a. - apply Trans2_17b. - specialize n3_47 with (~P→~Q) (~Q→~P) (Q→P) (P→Q). - intros n3_47a. - MP n3_47a H. - specialize n3_22 with (Q→P) (P→Q). - intros n3_22a. - Syll n3_47a n3_22a Sb. - clear Trans2_17a. clear Trans2_17b. clear H. clear n3_47a. clear n3_22a. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sb. - replace ((¬ P → ¬ Q) ∧ (¬ Q → ¬ P)) with (~P↔~Q) in Sb. - Conj Sa Sb. - split. - apply Sa. - apply Sb. - Equiv H. - apply H. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. -Qed. - -Theorem n4_12 : ∀ P Q : Prop, - (P ↔ ~Q) ↔ (Q ↔ ~P). - Proof. intros P Q. - specialize n2_03 with P Q. - intros n2_03a. - specialize Trans2_15 with Q P. - intros Trans2_15a. - Conj n2_03a Trans2_15a. - split. - apply n2_03a. - apply Trans2_15a. - specialize n3_47 with (P→~Q) (~Q→P) (Q→~P) (~P→Q). - intros n3_47a. - MP n3_47a H. - specialize n2_03 with Q P. - intros n2_03b. - specialize Trans2_15 with P Q. - intros Trans2_15b. - Conj n2_03b Trans2_15b. - split. - apply n2_03b. - apply Trans2_15b. - specialize n3_47 with (Q→~P) (~P→Q) (P→~Q) (~Q→P). - intros n3_47b. - MP n3_47b H0. - clear n2_03a. clear Trans2_15a. clear H. clear n2_03b. clear Trans2_15b. clear H0. - replace ((P → ¬ Q) ∧ (~Q → P)) with (P↔~Q) in n3_47a. - replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47a. - replace ((P → ¬ Q) ∧ (~Q → P)) with (P↔~Q) in n3_47b. - replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47b. - Conj n3_47a n3_47b. - split. - apply n3_47a. - apply n3_47b. - Equiv H. - apply H. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - Qed. - -Theorem n4_13 : ∀ P : Prop, - P ↔ ~~P. - Proof. intros P. - specialize n2_12 with P. - intros n2_12a. - specialize n2_14 with P. - intros n2_14a. - Conj n2_12a n2_14a. - split. - apply n2_12a. - apply n2_14a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_14 : ∀ P Q R : Prop, - ((P ∧ Q) → R) ↔ ((P ∧ ~R) → ~Q). -Proof. intros P Q R. -specialize n3_37 with P Q R. -intros n3_37a. -specialize n3_37 with P (~R) (~Q). -intros n3_37b. -Conj n3_37a n3_37b. -split. apply n3_37a. -apply n3_37b. -specialize n4_13 with Q. -intros n4_13a. -specialize n4_13 with R. -intros n4_13b. -replace (~~Q) with Q in H. -replace (~~R) with R in H. -Equiv H. -apply H. -apply Equiv4_01. -apply EqBi. -apply n4_13b. -apply EqBi. -apply n4_13a. -Qed. - -Theorem n4_15 : ∀ P Q R : Prop, - ((P ∧ Q) → ~R) ↔ ((Q ∧ R) → ~P). - Proof. intros P Q R. - specialize n4_14 with Q P (~R). - intros n4_14a. - specialize n3_22 with Q P. - intros n3_22a. - specialize Syll2_06 with (Q∧P) (P∧Q) (~R). - intros Syll2_06a. - MP Syll2_06a n3_22a. - specialize n4_13 with R. - intros n4_13a. - replace (~~R) with R in n4_14a. - rewrite Equiv4_01 in n4_14a. - specialize Simp3_26 with ((Q ∧ P → ¬ R) → Q ∧ R → ¬ P) ((Q ∧ R → ¬ P) → Q ∧ P → ¬ R). - intros Simp3_26a. - MP Simp3_26a n4_14a. - Syll Syll2_06a Simp3_26a Sa. - specialize Simp3_27 with ((Q ∧ P → ¬ R) → Q ∧ R → ¬ P) ((Q ∧ R → ¬ P) → Q ∧ P → ¬ R). - intros Simp3_27a. - MP Simp3_27a n4_14a. - specialize n3_22 with P Q. - intros n3_22b. - specialize Syll2_06 with (P∧Q) (Q∧P) (~R). - intros Syll2_06b. - MP Syll2_06b n3_22b. - Syll Syll2_06b Simp3_27a Sb. - split. - apply Sa. - apply Sb. - apply EqBi. - apply n4_13a. - Qed. - -Theorem n4_2 : ∀ P : Prop, - P ↔ P. - Proof. intros P. - specialize n3_2 with (P→P) (P→P). - intros n3_2a. - specialize n2_08 with P. - intros n2_08a. - MP n3_2a n2_08a. - MP n3_2a n2_08a. - Equiv n3_2a. - apply n3_2a. - apply Equiv4_01. - Qed. - -Theorem n4_21 : ∀ P Q : Prop, - (P ↔ Q) ↔ (Q ↔ P). - Proof. intros P Q. - specialize n3_22 with (P→Q) (Q→P). - intros n3_22a. - specialize Equiv4_01 with P Q. - intros Equiv4_01a. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22a. - specialize Equiv4_01 with Q P. - intros Equiv4_01b. - replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22a. - specialize n3_22 with (Q→P) (P→Q). - intros n3_22b. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22b. - replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22b. - Conj n3_22a n3_22b. - split. - apply Equiv4_01b. - apply n3_22b. - split. - apply n3_22a. - apply n3_22b. -Qed. - -Theorem n4_22 : ∀ P Q R : Prop, - ((P ↔ Q) ∧ (Q ↔ R)) → (P ↔ R). -Proof. intros P Q R. - specialize Simp3_26 with (P↔Q) (Q↔R). - intros Simp3_26a. - specialize Simp3_26 with (P→Q) (Q→P). - intros Simp3_26b. - replace ((P→Q) ∧ (Q→P)) with (P↔Q) in Simp3_26b. - Syll Simp3_26a Simp3_26b Sa. - specialize Simp3_27 with (P↔Q) (Q↔R). - intros Simp3_27a. - specialize Simp3_26 with (Q→R) (R→Q). - intros Simp3_26c. - replace ((Q→R) ∧ (R→Q)) with (Q↔R) in Simp3_26c. - Syll Simp3_27a Simp3_26c Sb. - specialize n2_83 with ((P↔Q)∧(Q↔R)) P Q R. - intros n2_83a. - MP n2_83a Sa. - MP n2_83a Sb. - specialize Simp3_27 with (P↔Q) (Q↔R). - intros Simp3_27b. - specialize Simp3_27 with (Q→R) (R→Q). - intros Simp3_27c. - replace ((Q→R) ∧ (R→Q)) with (Q↔R) in Simp3_27c. - Syll Simp3_27b Simp3_27c Sc. - specialize Simp3_26 with (P↔Q) (Q↔R). - intros Simp3_26d. - specialize Simp3_27 with (P→Q) (Q→P). - intros Simp3_27d. - replace ((P→Q) ∧ (Q→P)) with (P↔Q) in Simp3_27d. - Syll Simp3_26d Simp3_27d Sd. - specialize n2_83 with ((P↔Q)∧(Q↔R)) R Q P. - intros n2_83b. - MP n2_83b Sc. MP n2_83b Sd. - clear Sd. clear Sb. clear Sc. clear Sa. clear Simp3_26a. clear Simp3_26b. clear Simp3_26c. clear Simp3_26d. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d. - Conj n2_83a n2_83b. - split. - apply n2_83a. - apply n2_83b. - specialize Comp3_43 with ((P↔Q)∧(Q↔R)) (P→R) (R→P). - intros Comp3_43a. - MP Comp3_43a H. - replace ((P→R) ∧ (R→P)) with (P↔R) in Comp3_43a. - apply Comp3_43a. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. -Qed. - -Theorem n4_24 : ∀ P : Prop, - P ↔ (P ∧ P). - Proof. intros P. - specialize n3_2 with P P. - intros n3_2a. - specialize n2_43 with P (P ∧ P). - intros n2_43a. - MP n3_2a n2_43a. - specialize Simp3_26 with P P. - intros Simp3_26a. - Conj n2_43a Simp3_26a. - split. - apply n2_43a. - apply Simp3_26a. - Equiv H. - apply H. - apply Equiv4_01. -Qed. - -Theorem n4_25 : ∀ P : Prop, - P ↔ (P ∨ P). -Proof. intros P. - specialize Add1_3 with P P. - intros Add1_3a. - specialize Taut1_2 with P. - intros Taut1_2a. - Conj Add1_3a Taut1_2a. - split. - apply Add1_3a. - apply Taut1_2a. - Equiv H. apply H. - apply Equiv4_01. -Qed. - -Theorem n4_3 : ∀ P Q : Prop, - (P ∧ Q) ↔ (Q ∧ P). -Proof. intros P Q. - specialize n3_22 with P Q. - intros n3_22a. - specialize n3_22 with Q P. - intros n3_22b. - Conj n3_22a n3_22b. - split. - apply n3_22a. - apply n3_22b. - Equiv H. apply H. - apply Equiv4_01. -Qed. - -Theorem n4_31 : ∀ P Q : Prop, - (P ∨ Q) ↔ (Q ∨ P). - Proof. intros P Q. - specialize Perm1_4 with P Q. - intros Perm1_4a. - specialize Perm1_4 with Q P. - intros Perm1_4b. - Conj Perm1_4a Perm1_4b. - split. - apply Perm1_4a. - apply Perm1_4b. - Equiv H. apply H. - apply Equiv4_01. -Qed. - - Theorem n4_32 : ∀ P Q R : Prop, - ((P ∧ Q) ∧ R) ↔ (P ∧ (Q ∧ R)). - Proof. intros P Q R. - specialize n4_15 with P Q R. - intros n4_15a. - specialize Trans4_1 with P (~(Q ∧ R)). - intros Trans4_1a. - replace (~~(Q ∧ R)) with (Q ∧ R) in Trans4_1a. - replace (Q ∧ R→~P) with (P→~(Q ∧ R)) in n4_15a. - specialize Trans4_11 with (P ∧ Q → ¬ R) (P → ¬ (Q ∧ R)). - intros Trans4_11a. - replace ((P ∧ Q → ¬ R) ↔ (P → ¬ (Q ∧ R))) with (¬ (P ∧ Q → ¬ R) ↔ ¬ (P → ¬ (Q ∧ R))) in n4_15a. - replace (P ∧ Q → ¬ R) with (~(P ∧ Q ) ∨ ¬ R) in n4_15a. - replace (P → ¬ (Q ∧ R)) with (~P ∨ ~(Q ∧ R)) in n4_15a. - replace (¬ (¬ (P ∧ Q) ∨ ¬ R)) with ((P ∧ Q) ∧ R) in n4_15a. - replace (¬ (¬ P ∨ ¬ (Q ∧ R))) with (P ∧ (Q ∧ R )) in n4_15a. - apply n4_15a. - apply Prod3_01. - apply Prod3_01. - rewrite Impl1_01. - reflexivity. - rewrite Impl1_01. - reflexivity. - replace (¬ (P ∧ Q → ¬ R) ↔ ¬ (P → ¬ (Q ∧ R))) with ((P ∧ Q → ¬ R) ↔ (P → ¬ (Q ∧ R))). - reflexivity. - apply EqBi. - apply Trans4_11a. - apply EqBi. - apply Trans4_1a. - apply EqBi. - apply n4_13. - Qed. (*Note that the actual proof uses n4_12, but that transposition involves transforming a biconditional into a conditional. This way of doing it - using Trans4_1 to transpose a conditional and then applying n4_13 to double negate - is easier without a derived rule for replacing a biconditional with one of its equivalent implications.*) - -Theorem n4_33 : ∀ P Q R : Prop, - (P ∨ (Q ∨ R)) ↔ ((P ∨ Q) ∨ R). - Proof. intros P Q R. - specialize n2_31 with P Q R. - intros n2_31a. - specialize n2_32 with P Q R. - intros n2_32a. - split. apply n2_31a. - apply n2_32a. - Qed. - - Axiom n4_34 : ∀ P Q R : Prop, - P ∧ Q ∧ R = ((P ∧ Q) ∧ R). (*This axiom ensures left association of brackets. Coq's default is right association. But Principia proves associativity of logical product as n4_32. So in effect, this axiom gives us a derived rule that allows us to shift between Coq's and Principia's default rules for brackets of logical products.*) - -Theorem n4_36 : ∀ P Q R : Prop, - (P ↔ Q) → ((P ∧ R) ↔ (Q ∧ R)). -Proof. intros P Q R. - specialize Fact3_45 with P Q R. - intros Fact3_45a. - specialize Fact3_45 with Q P R. - intros Fact3_45b. - Conj Fact3_45a Fact3_45b. - split. - apply Fact3_45a. - apply Fact3_45b. - specialize n3_47 with (P→Q) (Q→P) (P ∧ R → Q ∧ R) (Q ∧ R → P ∧ R). - intros n3_47a. - MP n3_47 H. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a. - replace ((P ∧ R → Q ∧ R) ∧ (Q ∧ R → P ∧ R)) with (P ∧ R ↔ Q ∧ R) in n3_47a. - apply n3_47a. - apply Equiv4_01. - apply Equiv4_01. - Qed. - -Theorem n4_37 : ∀ P Q R : Prop, - (P ↔ Q) → ((P ∨ R) ↔ (Q ∨ R)). -Proof. intros P Q R. - specialize Sum1_6 with R P Q. - intros Sum1_6a. - specialize Sum1_6 with R Q P. - intros Sum1_6b. - Conj Sum1_6a Sum1_6b. - split. - apply Sum1_6a. - apply Sum1_6b. - specialize n3_47 with (P → Q) (Q → P) (R ∨ P → R ∨ Q) (R ∨ Q → R ∨ P). - intros n3_47a. - MP n3_47 H. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a. - replace ((R ∨ P → R ∨ Q) ∧ (R ∨ Q → R ∨ P)) with (R ∨ P ↔ R ∨ Q) in n3_47a. - replace (R ∨ P) with (P ∨ R) in n3_47a. - replace (R ∨ Q) with (Q ∨ R) in n3_47a. - apply n3_47a. - apply EqBi. - apply n4_31. - apply EqBi. - apply n4_31. - apply Equiv4_01. - apply Equiv4_01. - Qed. - -Theorem n4_38 : ∀ P Q R S : Prop, - ((P ↔ R) ∧ (Q ↔ S)) → ((P ∧ Q) ↔ (R ∧ S)). -Proof. intros P Q R S. - specialize n3_47 with P Q R S. - intros n3_47a. - specialize n3_47 with R S P Q. - intros n3_47b. - Conj n3_47a n3_47b. - split. - apply n3_47a. - apply n3_47b. - specialize n3_47 with ((P→R) ∧ (Q→S)) ((R→P) ∧ (S→Q)) (P ∧ Q → R ∧ S) (R ∧ S → P ∧ Q). - intros n3_47c. - MP n3_47c H. - specialize n4_32 with (P→R) (Q→S) ((R→P) ∧ (S → Q)). - intros n4_32a. - replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) in n3_47c. - specialize n4_32 with (Q→S) (R→P) (S → Q). - intros n4_32b. - replace ((Q → S) ∧ (R → P) ∧ (S → Q)) with (((Q → S) ∧ (R → P)) ∧ (S → Q)) in n3_47c. - specialize n3_22 with (Q→S) (R→P). - intros n3_22a. - specialize n3_22 with (R→P) (Q→S). - intros n3_22b. - Conj n3_22a n3_22b. - split. - apply n3_22a. - apply n3_22b. - Equiv H0. - replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47c. - specialize n4_32 with (R → P) (Q → S) (S → Q). - intros n4_32c. - replace (((R → P) ∧ (Q → S)) ∧ (S → Q)) with ((R → P) ∧ (Q → S) ∧ (S → Q)) in n3_47c. - specialize n4_32 with (P→R) (R → P) ((Q → S)∧(S → Q)). - intros n4_32d. - replace ((P → R) ∧ (R → P) ∧ (Q → S) ∧ (S → Q)) with (((P → R) ∧ (R → P)) ∧ (Q → S) ∧ (S → Q)) in n3_47c. - replace ((P→R) ∧ (R → P)) with (P↔R) in n3_47c. - replace ((Q → S) ∧ (S → Q)) with (Q↔S) in n3_47c. - replace ((P ∧ Q → R ∧ S) ∧ (R ∧ S → P ∧ Q)) with ((P ∧ Q) ↔ (R ∧ S)) in n3_47c. - apply n3_47c. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - apply EqBi. - apply n4_32d. - replace ((R → P) ∧ (Q → S) ∧ (S → Q)) with (((R → P) ∧ (Q → S)) ∧ (S → Q)). - reflexivity. - apply EqBi. - apply n4_32c. - replace ((R → P) ∧ (Q → S)) with ((Q → S) ∧ (R → P)). - reflexivity. - apply EqBi. - apply H0. - apply Equiv4_01. - apply EqBi. - apply n4_32b. - replace ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) with (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)). - reflexivity. - apply EqBi. - apply n4_32a. - Qed. - -Theorem n4_39 : ∀ P Q R S : Prop, - ((P ↔ R) ∧ (Q ↔ S)) → ((P ∨ Q) ↔ (R ∨ S)). -Proof. intros P Q R S. - specialize n3_48 with P Q R S. - intros n3_48a. - specialize n3_48 with R S P Q. - intros n3_48b. - Conj n3_48a n3_48b. - split. - apply n3_48a. - apply n3_48b. - specialize n3_47 with ((P → R) ∧ (Q → S)) ((R → P) ∧ (S → Q)) (P ∨ Q → R ∨ S) (R ∨ S → P ∨ Q). - intros n3_47a. - MP n3_47a H. - replace ((P ∨ Q → R ∨ S) ∧ (R ∨ S → P ∨ Q)) with ((P ∨ Q) ↔ (R ∨ S)) in n3_47a. - specialize n4_32 with ((P → R) ∧ (Q → S)) (R → P) (S → Q). - intros n4_32a. - replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((((P → R) ∧ (Q → S)) ∧ (R → P)) ∧ (S → Q)) in n3_47a. - specialize n4_32 with (P → R) (Q → S) (R → P). - intros n4_32b. - replace (((P → R) ∧ (Q → S)) ∧ (R → P)) with ((P → R) ∧ (Q → S) ∧ (R → P)) in n3_47a. - specialize n3_22 with (Q → S) (R → P). - intros n3_22a. - specialize n3_22 with (R → P) (Q → S). - intros n3_22b. - Conj n3_22a n3_22b. - split. - apply n3_22a. - apply n3_22b. - Equiv H0. - replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47a. - specialize n4_32 with (P → R) (R → P) (Q → S). - intros n4_32c. - replace ((P → R) ∧ (R → P) ∧ (Q → S)) with (((P → R) ∧ (R → P)) ∧ (Q → S)) in n3_47a. - replace ((P → R) ∧ (R → P)) with (P↔R) in n3_47a. - specialize n4_32 with (P↔R) (Q→S) (S→Q). - intros n4_32d. - replace (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)) with ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) in n3_47a. - replace ((Q → S) ∧ (S → Q)) with (Q ↔ S) in n3_47a. - apply n3_47a. - apply Equiv4_01. - replace ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) with (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)). - reflexivity. - apply EqBi. - apply n4_32d. - apply Equiv4_01. - apply EqBi. - apply n4_32c. - replace ((R → P) ∧ (Q → S)) with ((Q → S) ∧ (R → P)). - reflexivity. - apply EqBi. - apply H0. - apply Equiv4_01. - replace ((P → R) ∧ (Q → S) ∧ (R → P)) with (((P → R) ∧ (Q → S)) ∧ (R → P)). - reflexivity. - apply EqBi. - apply n4_32b. - apply EqBi. - apply n4_32a. - apply Equiv4_01. - Qed. - -Theorem n4_4 : ∀ P Q R : Prop, - (P ∧ (Q ∨ R)) ↔ ((P∧ Q) ∨ (P ∧ R)). -Proof. intros P Q R. - specialize n3_2 with P Q. - intros n3_2a. - specialize n3_2 with P R. - intros n3_2b. - Conj n3_2a n3_2b. - split. - apply n3_2a. - apply n3_2b. - specialize Comp3_43 with P (Q→P∧Q) (R→P∧R). - intros Comp3_43a. - MP Comp3_43a H. - specialize n3_48 with Q R (P∧Q) (P∧R). - intros n3_48a. - Syll Comp3_43a n3_48a Sa. - specialize Imp3_31 with P (Q∨R) ((P∧ Q) ∨ (P ∧ R)). - intros Imp3_31a. - MP Imp3_31a Sa. - specialize Simp3_26 with P Q. - intros Simp3_26a. - specialize Simp3_26 with P R. - intros Simp3_26b. - Conj Simp3_26a Simp3_26b. - split. - apply Simp3_26a. - apply Simp3_26b. - specialize n3_44 with P (P∧Q) (P∧R). - intros n3_44a. - MP n3_44a H0. - specialize Simp3_27 with P Q. - intros Simp3_27a. - specialize Simp3_27 with P R. - intros Simp3_27b. - Conj Simp3_27a Simp3_27b. - split. - apply Simp3_27a. - apply Simp3_27b. - specialize n3_48 with (P∧Q) (P∧R) Q R. - intros n3_48b. - MP n3_48b H1. - clear H1. clear Simp3_27a. clear Simp3_27b. - Conj n3_44a n3_48b. - split. - apply n3_44a. - apply n3_48b. - specialize Comp3_43 with (P ∧ Q ∨ P ∧ R) P (Q∨R). - intros Comp3_43b. - MP Comp3_43b H1. - clear H1. clear H0. clear n3_44a. clear n3_48b. clear Simp3_26a. clear Simp3_26b. - Conj Imp3_31a Comp3_43b. - split. -apply Imp3_31a. -apply Comp3_43b. -Equiv H0. -apply H0. -apply Equiv4_01. -Qed. - -Theorem n4_41 : ∀ P Q R : Prop, - (P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R)). -Proof. intros P Q R. - specialize Simp3_26 with Q R. - intros Simp3_26a. - specialize Sum1_6 with P (Q ∧ R) Q. - intros Sum1_6a. - MP Simp3_26a Sum1_6a. - specialize Simp3_27 with Q R. - intros Simp3_27a. - specialize Sum1_6 with P (Q ∧ R) R. - intros Sum1_6b. - MP Simp3_27a Sum1_6b. - clear Simp3_26a. clear Simp3_27a. - Conj Sum1_6a Sum1_6b. - split. - apply Sum1_6a. - apply Sum1_6b. - specialize Comp3_43 with (P ∨ Q ∧ R) (P ∨ Q) (P ∨ R). - intros Comp3_43a. - MP Comp3_43a H. - specialize n2_53 with P Q. - intros n2_53a. - specialize n2_53 with P R. - intros n2_53b. - Conj n2_53a n2_53b. - split. - apply n2_53a. - apply n2_53b. - specialize n3_47 with (P ∨ Q) (P ∨ R) (¬ P → Q) (¬ P → R). - intros n3_47a. - MP n3_47a H0. - specialize Comp3_43 with (~P) Q R. - intros Comp3_43b. - Syll n3_47a Comp3_43b Sa. - specialize n2_54 with P (Q∧R). - intros n2_54a. - Syll Sa n2_54a Sb. - split. - apply Comp3_43a. - apply Sb. -Qed. - -Theorem n4_42 : ∀ P Q : Prop, - P ↔ ((P ∧ Q) ∨ (P ∧ ~Q)). -Proof. intros P Q. - specialize n3_21 with P (Q ∨ ~Q). - intros n3_21a. - specialize n2_11 with Q. - intros n2_11a. - MP n3_21a n2_11a. - specialize Simp3_26 with P (Q ∨ ~Q). - intros Simp3_26a. clear n2_11a. - Conj n3_21a Simp3_26a. - split. - apply n3_21a. - apply Simp3_26a. - Equiv H. - specialize n4_4 with P Q (~Q). - intros n4_4a. - replace (P ∧ (Q ∨ ¬ Q)) with P in n4_4a. - apply n4_4a. - apply EqBi. - apply H. - apply Equiv4_01. -Qed. - -Theorem n4_43 : ∀ P Q : Prop, - P ↔ ((P ∨ Q) ∧ (P ∨ ~Q)). -Proof. intros P Q. - specialize n2_2 with P Q. - intros n2_2a. - specialize n2_2 with P (~Q). - intros n2_2b. - Conj n2_2a n2_2b. - split. - apply n2_2a. - apply n2_2b. - specialize Comp3_43 with P (P∨Q) (P∨~Q). - intros Comp3_43a. - MP Comp3_43a H. - specialize n2_53 with P Q. - intros n2_53a. - specialize n2_53 with P (~Q). - intros n2_53b. - Conj n2_53a n2_53b. - split. - apply n2_53a. - apply n2_53b. - specialize n3_47 with (P∨Q) (P∨~Q) (~P→Q) (~P→~Q). - intros n3_47a. - MP n3_47a H0. - specialize n2_65 with (~P) Q. - intros n2_65a. - replace (~~P) with P in n2_65a. - specialize Imp3_31 with (¬ P → Q) (¬ P → ¬ Q) (P). - intros Imp3_31a. - MP Imp3_31a n2_65a. - Syll n3_47a Imp3_31a Sa. - clear n2_2a. clear n2_2b. clear H. clear n2_53a. clear n2_53b. clear H0. clear n2_65a. clear n3_47a. clear Imp3_31a. - Conj Comp3_43a Sa. - split. - apply Comp3_43a. - apply Sa. - Equiv H. - apply H. - apply Equiv4_01. - apply EqBi. - apply n4_13. -Qed. - -Theorem n4_44 : ∀ P Q : Prop, - P ↔ (P ∨ (P ∧ Q)). - Proof. intros P Q. - specialize n2_2 with P (P∧Q). - intros n2_2a. - specialize n2_08 with P. - intros n2_08a. - specialize Simp3_26 with P Q. - intros Simp3_26a. - Conj n2_08a Simp3_26a. - split. - apply n2_08a. - apply Simp3_26a. - specialize n3_44 with P P (P ∧ Q). - intros n3_44a. - MP n3_44a H. - clear H. clear n2_08a. clear Simp3_26a. - Conj n2_2a n3_44a. - split. - apply n2_2a. - apply n3_44a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_45 : ∀ P Q : Prop, - P ↔ (P ∧ (P ∨ Q)). - Proof. intros P Q. - specialize n2_2 with (P ∧ P) (P ∧ Q). - intros n2_2a. - replace (P ∧ P ∨ P ∧ Q) with (P ∧ (P ∨ Q)) in n2_2a. - replace (P ∧ P) with P in n2_2a. - specialize Simp3_26 with P (P ∨ Q). - intros Simp3_26a. - split. - apply n2_2a. - apply Simp3_26a. - apply EqBi. - apply n4_24. - apply EqBi. - apply n4_4. -Qed. - -Theorem n4_5 : ∀ P Q : Prop, - P ∧ Q ↔ ~(~P ∨ ~Q). - Proof. intros P Q. - specialize n4_2 with (P ∧ Q). - intros n4_2a. - rewrite Prod3_01. - replace (~(~P ∨ ~Q)) with (P ∧ Q). - apply n4_2a. - apply Prod3_01. - Qed. - -Theorem n4_51 : ∀ P Q : Prop, - ~(P ∧ Q) ↔ (~P ∨ ~Q). - Proof. intros P Q. - specialize n4_5 with P Q. - intros n4_5a. - specialize n4_12 with (P ∧ Q) (¬ P ∨ ¬ Q). - intros n4_12a. - replace ((P ∧ Q ↔ ¬ (¬ P ∨ ¬ Q)) ↔ (¬ P ∨ ¬ Q ↔ ¬ (P ∧ Q))) with ((P ∧ Q ↔ ¬ (¬ P ∨ ¬ Q)) = (¬ P ∨ ¬ Q ↔ ¬ (P ∧ Q))) in n4_12a. - replace (P ∧ Q ↔ ¬ (¬ P ∨ ¬ Q)) with (¬ P ∨ ¬ Q ↔ ¬ (P ∧ Q)) in n4_5a. - replace (¬ P ∨ ¬ Q ↔ ¬ (P ∧ Q)) with (~(P ∧ Q) ↔ (~P ∨ ~Q)) in n4_5a. - apply n4_5a. - specialize n4_21 with (¬ (P ∧ Q)) (¬ P ∨ ¬ Q). - intros n4_21a. - apply EqBi. - apply n4_21. - apply EqBi. - apply EqBi. - Qed. - -Theorem n4_52 : ∀ P Q : Prop, - (P ∧ ~Q) ↔ ~(~P ∨ Q). - Proof. intros P Q. - specialize n4_5 with P (~Q). - intros n4_5a. - replace (~~Q) with Q in n4_5a. - apply n4_5a. - specialize n4_13 with Q. - intros n4_13a. - apply EqBi. - apply n4_13a. - Qed. - -Theorem n4_53 : ∀ P Q : Prop, - ~(P ∧ ~Q) ↔ (~P ∨ Q). - Proof. intros P Q. - specialize n4_52 with P Q. - intros n4_52a. - specialize n4_12 with ( P ∧ ¬ Q) ((¬ P ∨ Q)). - intros n4_12a. - replace ((P ∧ ¬ Q ↔ ¬ (¬ P ∨ Q)) ↔ (¬ P ∨ Q ↔ ¬ (P ∧ ¬ Q))) with ((P ∧ ¬ Q ↔ ¬ (¬ P ∨ Q)) = (¬ P ∨ Q ↔ ¬ (P ∧ ¬ Q))) in n4_12a. - replace (P ∧ ¬ Q ↔ ¬ (¬ P ∨ Q)) with (¬ P ∨ Q ↔ ¬ (P ∧ ¬ Q)) in n4_52a. - replace (¬ P ∨ Q ↔ ¬ (P ∧ ¬ Q)) with (~(P ∧ ~Q) ↔ (~P ∨ Q)) in n4_52a. - apply n4_52a. - specialize n4_21 with (¬ (P ∧ ¬ Q)) (¬ P ∨ Q). - intros n4_21a. - apply EqBi. - apply n4_21a. - apply EqBi. - apply EqBi. - Qed. - -Theorem n4_54 : ∀ P Q : Prop, - (~P ∧ Q) ↔ ~(P ∨ ~Q). - Proof. intros P Q. - specialize n4_5 with (~P) Q. - intros n4_5a. - specialize n4_13 with P. - intros n4_13a. - replace (~~P) with P in n4_5a. - apply n4_5a. - apply EqBi. - apply n4_13a. - Qed. - -Theorem n4_55 : ∀ P Q : Prop, - ~(~P ∧ Q) ↔ (P ∨ ~Q). - Proof. intros P Q. - specialize n4_54 with P Q. - intros n4_54a. - specialize n4_12 with (~P ∧ Q) (P ∨ ~Q). - intros n4_12a. - replace (¬ P ∧ Q ↔ ¬ (P ∨ ¬ Q)) with (P ∨ ¬ Q ↔ ¬ (¬ P ∧ Q)) in n4_54a. - replace (P ∨ ¬ Q ↔ ¬ (¬ P ∧ Q)) with (~(~P ∧ Q) ↔ (P ∨ ~Q)) in n4_54a. - apply n4_54a. - specialize n4_21 with (~(~P ∧ Q)) (P ∨ ~Q). - intros n4_21a. - apply EqBi. - apply n4_21a. - replace ((¬ P ∧ Q ↔ ¬ (P ∨ ¬ Q)) ↔ (P ∨ ¬ Q ↔ ¬ (¬ P ∧ Q))) with ((¬ P ∧ Q ↔ ¬ (P ∨ ¬ Q)) = (P ∨ ¬ Q ↔ ¬ (¬ P ∧ Q))) in n4_12a. - rewrite n4_12a. - reflexivity. - apply EqBi. - apply EqBi. - Qed. - -Theorem n4_56 : ∀ P Q : Prop, - (~P ∧ ~Q) ↔ ~(P ∨ Q). - Proof. intros P Q. - specialize n4_54 with P (~Q). - intros n4_54a. - replace (~~Q) with Q in n4_54a. - apply n4_54a. - apply EqBi. - apply n4_13. - Qed. - -Theorem n4_57 : ∀ P Q : Prop, - ~(~P ∧ ~Q) ↔ (P ∨ Q). - Proof. intros P Q. - specialize n4_56 with P Q. - intros n4_56a. - specialize n4_12 with (¬ P ∧ ¬ Q) (P ∨ Q). - intros n4_12a. - replace (¬ P ∧ ¬ Q ↔ ¬ (P ∨ Q)) with (P ∨ Q ↔ ¬ (¬ P ∧ ¬ Q)) in n4_56a. - replace (P ∨ Q ↔ ¬ (¬ P ∧ ¬ Q)) with (¬ (¬ P ∧ ¬ Q) ↔ P ∨ Q) in n4_56a. - apply n4_56a. - specialize n4_21 with (¬ (¬ P ∧ ¬ Q)) (P ∨ Q). - intros n4_21a. - apply EqBi. - apply n4_21a. - replace ((¬ P ∧ ¬ Q ↔ ¬ (P ∨ Q)) ↔ (P ∨ Q ↔ ¬ (¬ P ∧ ¬ Q))) with ((P ∨ Q ↔ ¬ (¬ P ∧ ¬ Q)) ↔ (¬ P ∧ ¬ Q ↔ ¬ (P ∨ Q))) in n4_12a. - apply EqBi. - apply n4_12a. - apply EqBi. - specialize n4_21 with (P ∨ Q ↔ ¬ (¬ P ∧ ¬ Q)) (¬ P ∧ ¬ Q ↔ ¬ (P ∨ Q)). - intros n4_21b. - apply n4_21b. - Qed. - -Theorem n4_6 : ∀ P Q : Prop, - (P → Q) ↔ (~P ∨ Q). - Proof. intros P Q. - specialize n4_2 with (~P∨ Q). - intros n4_2a. - rewrite Impl1_01. - apply n4_2a. - Qed. - -Theorem n4_61 : ∀ P Q : Prop, - ~(P → Q) ↔ (P ∧ ~Q). - Proof. intros P Q. - specialize n4_6 with P Q. - intros n4_6a. - specialize Trans4_11 with (P→Q) (~P∨Q). - intros Trans4_11a. - specialize n4_52 with P Q. - intros n4_52a. - replace ((P → Q) ↔ ¬ P ∨ Q) with (¬ (P → Q) ↔ ¬ (¬ P ∨ Q)) in n4_6a. - replace (¬ (¬ P ∨ Q)) with (P ∧ ¬ Q) in n4_6a. - apply n4_6a. - apply EqBi. - apply n4_52a. - replace (((P → Q) ↔ ¬ P ∨ Q) ↔ (¬ (P → Q) ↔ ¬ (¬ P ∨ Q))) with ((¬ (P → Q) ↔ ¬ (¬ P ∨ Q)) ↔ ((P → Q) ↔ ¬ P ∨ Q)) in Trans4_11a. - apply EqBi. - apply Trans4_11a. - apply EqBi. - apply n4_21. - Qed. - -Theorem n4_62 : ∀ P Q : Prop, - (P → ~Q) ↔ (~P ∨ ~Q). - Proof. intros P Q. - specialize n4_6 with P (~Q). - intros n4_6a. - apply n4_6a. - Qed. - -Theorem n4_63 : ∀ P Q : Prop, - ~(P → ~Q) ↔ (P ∧ Q). - Proof. intros P Q. - specialize n4_62 with P Q. - intros n4_62a. - specialize Trans4_11 with (P → ¬ Q) (¬ P ∨ ¬ Q). - intros Trans4_11a. - specialize n4_5 with P Q. - intros n4_5a. - replace (¬ (¬ P ∨ ¬ Q)) with (P ∧ Q) in Trans4_11a. - replace ((P → ¬ Q) ↔ ¬ P ∨ ¬ Q) with ((¬ (P → ¬ Q) ↔ P ∧ Q)) in n4_62a. - apply n4_62a. - replace (((P → ¬ Q) ↔ ¬ P ∨ ¬ Q) ↔ (¬ (P → ¬ Q) ↔ P ∧ Q)) with ((¬ (P → ¬ Q) ↔ P ∧ Q) ↔ ((P → ¬ Q) ↔ ¬ P ∨ ¬ Q)) in Trans4_11a. - apply EqBi. - apply Trans4_11a. - specialize n4_21 with (¬ (P → ¬ Q) ↔ P ∧ Q) ((P → ¬ Q) ↔ ¬ P ∨ ¬ Q). - intros n4_21a. - apply EqBi. - apply n4_21a. - apply EqBi. - apply n4_5a. - Qed. - -Theorem n4_64 : ∀ P Q : Prop, - (~P → Q) ↔ (P ∨ Q). - Proof. intros P Q. - specialize n2_54 with P Q. - intros n2_54a. - specialize n2_53 with P Q. - intros n2_53a. - Conj n2_54a n2_53a. - split. - apply n2_54a. - apply n2_53a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_65 : ∀ P Q : Prop, - ~(~P → Q) ↔ (~P ∧ ~Q). - Proof. intros P Q. - specialize n4_64 with P Q. - intros n4_64a. - specialize Trans4_11 with(¬ P → Q) (P ∨ Q). - intros Trans4_11a. - specialize n4_56 with P Q. - intros n4_56a. - replace (((¬ P → Q) ↔ P ∨ Q) ↔ (¬ (¬ P → Q) ↔ ¬ (P ∨ Q))) with ((¬ (¬ P → Q) ↔ ¬ (P ∨ Q)) ↔ ((¬ P → Q) ↔ P ∨ Q)) in Trans4_11a. - replace ((¬ P → Q) ↔ P ∨ Q) with (¬ (¬ P → Q) ↔ ¬ (P ∨ Q)) in n4_64a. - replace (¬ (P ∨ Q)) with (¬ P ∧ ¬ Q) in n4_64a. - apply n4_64a. - apply EqBi. - apply n4_56a. - apply EqBi. - apply Trans4_11a. - apply EqBi. - apply n4_21. - Qed. - -Theorem n4_66 : ∀ P Q : Prop, - (~P → ~Q) ↔ (P ∨ ~Q). - Proof. intros P Q. - specialize n4_64 with P (~Q). - intros n4_64a. - apply n4_64a. - Qed. - -Theorem n4_67 : ∀ P Q : Prop, - ~(~P → ~Q) ↔ (~P ∧ Q). - Proof. intros P Q. - specialize n4_66 with P Q. - intros n4_66a. - specialize Trans4_11 with (¬ P → ¬ Q) (P ∨ ¬ Q). - intros Trans4_11a. - replace ((¬ P → ¬ Q) ↔ P ∨ ¬ Q) with (¬ (¬ P → ¬ Q) ↔ ¬ (P ∨ ¬ Q)) in n4_66a. - specialize n4_54 with P Q. - intros n4_54a. - replace (¬ (P ∨ ¬ Q)) with (¬ P ∧ Q) in n4_66a. - apply n4_66a. - apply EqBi. - apply n4_54a. - replace (((¬ P → ¬ Q) ↔ P ∨ ¬ Q) ↔ (¬ (¬ P → ¬ Q) ↔ ¬ (P ∨ ¬ Q))) with ((¬ (¬ P → ¬ Q) ↔ ¬ (P ∨ ¬ Q)) ↔ ((¬ P → ¬ Q) ↔ P ∨ ¬ Q)) in Trans4_11a. - apply EqBi. - apply Trans4_11a. - apply EqBi. - apply n4_21. - Qed. - -Theorem n4_7 : ∀ P Q : Prop, - (P → Q) ↔ (P → (P ∧ Q)). - Proof. intros P Q. - specialize Comp3_43 with P P Q. - intros Comp3_43a. - specialize Exp3_3 with (P → P) (P → Q) (P → P ∧ Q). - intros Exp3_3a. - MP Exp3_3a Comp3_43a. - specialize n2_08 with P. - intros n2_08a. - MP Exp3_3a n2_08a. - specialize Simp3_27 with P Q. - intros Simp3_27a. - specialize Syll2_05 with P (P ∧ Q) Q. - intros Syll2_05a. - MP Syll2_05a Simp3_26a. - clear n2_08a. clear Comp3_43a. clear Simp3_27a. - Conj Syll2_05a Exp3_3a. - split. - apply Exp3_3a. - apply Syll2_05a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_71 : ∀ P Q : Prop, - (P → Q) ↔ (P ↔ (P ∧ Q)). - Proof. intros P Q. - specialize n4_7 with P Q. - intros n4_7a. - specialize n3_21 with (P→(P∧Q)) ((P∧Q)→P). - intros n3_21a. - replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in n3_21a. - specialize Simp3_26 with P Q. - intros Simp3_26a. - MP n3_21a Simp3_26a. - specialize Simp3_26 with (P→(P∧Q)) ((P∧Q)→P). - intros Simp3_26b. - replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in Simp3_26b. clear Simp3_26a. - Conj n3_21a Simp3_26b. - split. - apply n3_21a. - apply Simp3_26b. - Equiv H. - clear n3_21a. clear Simp3_26b. - Conj n4_7a H. - split. - apply n4_7a. - apply H. - specialize n4_22 with (P → Q) (P → P ∧ Q) (P ↔ P ∧ Q). - intros n4_22a. - MP n4_22a H0. - apply n4_22a. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - Qed. - -Theorem n4_72 : ∀ P Q : Prop, - (P → Q) ↔ (Q ↔ (P ∨ Q)). - Proof. intros P Q. - specialize Trans4_1 with P Q. - intros Trans4_1a. - specialize n4_71 with (~Q) (~P). - intros n4_71a. - Conj Trans4_1a n4_71a. - split. - apply Trans4_1a. - apply n4_71a. - specialize n4_22 with (P→Q) (~Q→~P) (~Q↔~Q ∧ ~ P). - intros n4_22a. - MP n4_22a H. - specialize n4_21 with (~Q) (~Q ∧ ~P). - intros n4_21a. - Conj n4_22a n4_21a. - split. - apply n4_22a. - apply n4_21a. - specialize n4_22 with (P→Q) (¬ Q ↔ ¬ Q ∧ ¬ P) (¬ Q ∧ ¬ P ↔ ¬ Q). - intros n4_22b. - MP n4_22b H0. - specialize n4_12 with (~Q ∧ ~ P) (Q). - intros n4_12a. - Conj n4_22b n4_12a. - split. - apply n4_22b. - apply n4_12a. - specialize n4_22 with (P → Q) ((~Q ∧ ~ P) ↔ ~Q) (Q ↔ ~(¬ Q ∧ ¬ P)). - intros n4_22c. - MP n4_22b H0. - specialize n4_57 with Q P. - intros n4_57a. - replace (~(~Q ∧ ~P)) with (Q ∨ P) in n4_22c. - specialize n4_31 with P Q. - intros n4_31a. - replace (Q ∨ P) with (P ∨ Q) in n4_22c. - apply n4_22c. - apply EqBi. - apply n4_31a. - apply EqBi. - replace (¬ (¬ Q ∧ ¬ P) ↔ Q ∨ P) with (Q ∨ P ↔¬ (¬ Q ∧ ¬ P)) in n4_57a. - apply n4_57a. - apply EqBi. - apply n4_21. - Qed. - -Theorem n4_73 : ∀ P Q : Prop, - Q → (P ↔ (P ∧ Q)). - Proof. intros P Q. - specialize n2_02 with P Q. - intros n2_02a. - specialize n4_71 with P Q. - intros n4_71a. - replace ((P → Q) ↔ (P ↔ P ∧ Q)) with (((P → Q) → (P ↔ P ∧ Q)) ∧ ((P ↔ P ∧ Q)→(P→Q))) in n4_71a. - specialize Simp3_26 with ((P → Q) → P ↔ P ∧ Q) (P ↔ P ∧ Q → P → Q). - intros Simp3_26a. - MP Simp3_26a n4_71a. - Syll n2_02a Simp3_26a Sa. - apply Sa. - apply Equiv4_01. - Qed. - -Theorem n4_74 : ∀ P Q : Prop, - ~P → (Q ↔ (P ∨ Q)). - Proof. intros P Q. - specialize n2_21 with P Q. - intros n2_21a. - specialize n4_72 with P Q. - intros n4_72a. - replace (P → Q) with (Q ↔ P ∨ Q) in n2_21a. - apply n2_21a. - apply EqBi. - replace ((P → Q) ↔ (Q ↔ P ∨ Q)) with ((Q ↔ P ∨ Q) ↔ (P → Q)) in n4_72a. - apply n4_72a. - apply EqBi. - apply n4_21. - Qed. - -Theorem n4_76 : ∀ P Q R : Prop, - ((P → Q) ∧ (P → R)) ↔ (P → (Q ∧ R)). - Proof. intros P Q R. - specialize n4_41 with (~P) Q R. - intros n4_41a. - replace (~P ∨ Q) with (P→Q) in n4_41a. - replace (~P ∨ R) with (P→R) in n4_41a. - replace (¬ P ∨ Q ∧ R) with (P → Q ∧ R) in n4_41a. - replace ((P → Q ∧ R) ↔ (P → Q) ∧ (P → R)) with ((P → Q) ∧ (P → R) ↔ (P → Q ∧ R)) in n4_41a. - apply n4_41a. - apply EqBi. - apply n4_21. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - Qed. - -Theorem n4_77 : ∀ P Q R : Prop, - ((Q → P) ∧ (R → P)) ↔ ((Q ∨ R) → P). - Proof. intros P Q R. - specialize n3_44 with P Q R. - intros n3_44a. - split. - apply n3_44a. - split. - specialize n2_2 with Q R. - intros n2_2a. - Syll n2_2a H Sa. - apply Sa. - specialize Add1_3 with Q R. - intros Add1_3a. - Syll Add1_3a H Sb. - apply Sb. - Qed. (*Note that we used the split tactic on a conditional, effectively introducing an assumption for conditional proof. It remains to prove that (AvB)→C and A→(AvB) together imply A→C, and similarly that (AvB)→C and B→(AvB) together imply B→C. This can be proved by Syll, but we need a rule of replacement in the context of ((AvB)→C)→(A→C)/\(B→C).*) - -Theorem n4_78 : ∀ P Q R : Prop, - ((P → Q) ∨ (P → R)) ↔ (P → (Q ∨ R)). - Proof. intros P Q R. - specialize n4_2 with ((P→Q) ∨ (P → R)). - intros n4_2a. - replace (((P → Q) ∨ (P → R))↔((P → Q) ∨ (P → R))) with (((P → Q) ∨ (P → R))↔((¬ P ∨ Q) ∨ ¬ P ∨ R)) in n4_2a. - specialize n4_33 with (~P) Q (~P ∨ R). - intros n4_33a. - replace ((¬ P ∨ Q) ∨ ¬ P ∨ R) with (¬ P ∨ Q ∨ ¬ P ∨ R) in n4_2a. - specialize n4_31 with (~P) Q. - intros n4_31a. - specialize n4_37 with (~P∨Q) (Q ∨ ~P) R. - intros n4_37a. - MP n4_37a n4_31a. - replace (Q ∨ ¬ P ∨ R) with ((Q ∨ ¬ P) ∨ R) in n4_2a. - replace ((Q ∨ ¬ P) ∨ R) with ((¬ P ∨ Q) ∨ R) in n4_2a. - specialize n4_33 with (~P) (~P∨Q) R. - intros n4_33b. - replace (¬ P ∨ (¬ P ∨ Q) ∨ R) with ((¬ P ∨ (¬ P ∨ Q)) ∨ R) in n4_2a. - specialize n4_25 with (~P). - intros n4_25a. - specialize n4_37 with (~P) (~P ∨ ~P) (Q ∨ R). - intros n4_37b. - MP n4_37b n4_25a. - replace (¬ P ∨ ¬ P ∨ Q) with ((¬ P ∨ ¬ P) ∨ Q) in n4_2a. - replace (((¬ P ∨ ¬ P) ∨ Q) ∨ R) with ((¬ P ∨ ¬ P) ∨ Q ∨ R) in n4_2a. - replace ((¬ P ∨ ¬ P) ∨ Q ∨ R) with ((¬ P) ∨ (Q ∨ R)) in n4_2a. - replace (¬ P ∨ Q ∨ R) with (P → (Q ∨ R)) in n4_2a. - apply n4_2a. - apply Impl1_01. - apply EqBi. - apply n4_37b. - apply n2_33. - replace ((¬ P ∨ ¬ P) ∨ Q) with (¬ P ∨ ¬ P ∨ Q). - reflexivity. - apply n2_33. - replace ((¬ P ∨ ¬ P ∨ Q) ∨ R) with (¬ P ∨ (¬ P ∨ Q) ∨ R). - reflexivity. - apply EqBi. - apply n4_33b. - apply EqBi. - apply n4_37a. - replace ((Q ∨ ¬ P) ∨ R) with (Q ∨ ¬ P ∨ R). - reflexivity. - apply n2_33. - apply EqBi. - apply n4_33a. - replace (¬ P ∨ Q) with (P→Q). - replace (¬ P ∨ R) with (P→R). - reflexivity. - apply Impl1_01. - apply Impl1_01. - Qed. - -Theorem n4_79 : ∀ P Q R : Prop, - ((Q → P) ∨ (R → P)) ↔ ((Q ∧ R) → P). - Proof. intros P Q R. - specialize Trans4_1 with Q P. - intros Trans4_1a. - specialize Trans4_1 with R P. - intros Trans4_1b. - Conj Trans4_1a Trans4_1b. - split. - apply Trans4_1a. - apply Trans4_1b. - specialize n4_39 with (Q→P) (R→P) (~P→~Q) (~P→~R). - intros n4_39a. - MP n4_39a H. - specialize n4_78 with (~P) (~Q) (~R). - intros n4_78a. - replace ((¬ P → ¬ Q) ∨ (¬ P → ¬ R)) with (¬ P → ¬ Q ∨ ¬ R) in n4_39a. - specialize Trans2_15 with P (~Q ∨ ~R). - intros Trans2_15a. - replace (¬ P → ¬ Q ∨ ¬ R) with (¬ (¬ Q ∨ ¬ R) → P) in n4_39a. - replace (~(~Q ∨ ~R)) with (Q ∧ R) in n4_39a. - apply n4_39a. - apply Prod3_01. - replace (¬ (¬ Q ∨ ¬ R) → P) with (¬ P → ¬ Q ∨ ¬ R). - reflexivity. - apply EqBi. - split. - apply Trans2_15a. - apply Trans2_15. - replace (¬ P → ¬ Q ∨ ¬ R) with ((¬ P → ¬ Q) ∨ (¬ P → ¬ R)). - reflexivity. - apply EqBi. - apply n4_78a. - Qed. - -Theorem n4_8 : ∀ P : Prop, - (P → ~P) ↔ ~P. - Proof. intros P. - specialize Abs2_01 with P. - intros Abs2_01a. - specialize n2_02 with P (~P). - intros n2_02a. - Conj Abs2_01a n2_02a. - split. - apply Abs2_01a. - apply n2_02a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_81 : ∀ P : Prop, - (~P → P) ↔ P. - Proof. intros P. - specialize n2_18 with P. - intros n2_18a. - specialize n2_02 with (~P) P. - intros n2_02a. - Conj n2_18a n2_02a. - split. - apply n2_18a. - apply n2_02a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_82 : ∀ P Q : Prop, - ((P → Q) ∧ (P → ~Q)) ↔ ~P. - Proof. intros P Q. - specialize n2_65 with P Q. - intros n2_65a. - specialize Imp3_31 with (P→Q) (P→~Q) (~P). - intros Imp3_31a. - MP Imp3_31a n2_65a. - specialize n2_21 with P Q. - intros n2_21a. - specialize n2_21 with P (~Q). - intros n2_21b. - Conj n2_21a n2_21b. - split. - apply n2_21a. - apply n2_21b. - specialize Comp3_43 with (~P) (P→Q) (P→~Q). - intros Comp3_43a. - MP Comp3_43a H. - clear n2_65a. clear n2_21a. clear n2_21b. - clear H. - Conj Imp3_31a Comp3_43a. - split. - apply Imp3_31a. - apply Comp3_43a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_83 : ∀ P Q : Prop, - ((P → Q) ∧ (~P → Q)) ↔ Q. - Proof. intros P Q. - specialize n2_61 with P Q. - intros n2_61a. - specialize Imp3_31 with (P→Q) (~P→Q) (Q). - intros Imp3_31a. - MP Imp3_31a n2_61a. - specialize n2_02 with P Q. - intros n2_02a. - specialize n2_02 with (~P) Q. - intros n2_02b. - Conj n2_02a n2_02b. - split. - apply n2_02a. - apply n2_02b. - specialize Comp3_43 with Q (P→Q) (~P→Q). - intros Comp3_43a. - MP Comp3_43a H. - clear n2_61a. clear n2_02a. clear n2_02b. - clear H. - Conj Imp3_31a Comp3_43a. - split. - apply Imp3_31a. - apply Comp3_43a. - Equiv H. - apply H. - apply Equiv4_01. - Qed. - -Theorem n4_84 : ∀ P Q R : Prop, - (P ↔ Q) → ((P → R) ↔ (Q → R)). - Proof. intros P Q R. - specialize Syll2_06 with P Q R. - intros Syll2_06a. - specialize Syll2_06 with Q P R. - intros Syll2_06b. - Conj Syll2_06a Syll2_06b. - split. - apply Syll2_06a. - apply Syll2_06b. - specialize n3_47 with (P→Q) (Q→P) ((Q→R)→P→R) ((P→R)→Q→R). - intros n3_47a. - MP n3_47a H. - replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a. - replace (((Q → R) → P → R) ∧ ((P → R) → Q → R)) with ((Q → R) ↔ (P → R)) in n3_47a. - replace ((Q → R) ↔ (P → R)) with ((P→ R) ↔ (Q → R)) in n3_47a. - apply n3_47a. - apply EqBi. - apply n4_21. - apply Equiv4_01. - apply Equiv4_01. - Qed. - -Theorem n4_85 : ∀ P Q R : Prop, - (P ↔ Q) → ((R → P) ↔ (R → Q)). - Proof. intros P Q R. - specialize Syll2_05 with R P Q. - intros Syll2_05a. - specialize Syll2_05 with R Q P. - intros Syll2_05b. - Conj Syll2_05a Syll2_05b. - split. - apply Syll2_05a. - apply Syll2_05b. - specialize n3_47 with (P→Q) (Q→P) ((R→P)→R→Q) ((R→Q)→R→P). - intros n3_47a. - MP n3_47a H. - replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a. - replace (((R → P) → R → Q) ∧ ((R → Q) → R → P)) with ((R → P) ↔ (R → Q)) in n3_47a. - apply n3_47a. - apply Equiv4_01. - apply Equiv4_01. -Qed. - -Theorem n4_86 : ∀ P Q R : Prop, - (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)). - Proof. intros P Q R. - split. - split. - replace (P↔Q) with (Q↔P) in H. - Conj H H0. - split. - apply H. - apply H0. - specialize n4_22 with Q P R. - intros n4_22a. - MP n4_22a H1. - replace (Q ↔ R) with ((Q→R) ∧ (R→Q)) in n4_22a. - specialize Simp3_26 with (Q→R) (R→Q). - intros Simp3_26a. - MP Simp3_26a n4_22a. - apply Simp3_26a. - apply Equiv4_01. - apply EqBi. - apply n4_21. - replace (P↔R) with (R↔P) in H0. - Conj H0 H. - split. - apply H. - apply H0. - replace ((P ↔ Q) ∧ (R ↔ P)) with ((R ↔ P) ∧ (P ↔ Q)) in H1. - specialize n4_22 with R P Q. - intros n4_22a. - MP n4_22a H1. - replace (R ↔ Q) with ((R→Q) ∧ (Q→R)) in n4_22a. - specialize Simp3_26 with (R→Q) (Q→R). - intros Simp3_26a. - MP Simp3_26a n4_22a. - apply Simp3_26a. - apply Equiv4_01. - apply EqBi. - apply n4_3. - apply EqBi. - apply n4_21. - split. - Conj H H0. - split. - apply H. - apply H0. - specialize n4_22 with P Q R. - intros n4_22a. - MP n4_22a H1. - replace (P↔R) with ((P→R)∧(R→P)) in n4_22a. - specialize Simp3_26 with (P→R) (R→P). - intros Simp3_26a. - MP Simp3_26a n4_22a. - apply Simp3_26a. - apply Equiv4_01. - Conj H H0. - split. - apply H. - apply H0. - specialize n4_22 with P Q R. - intros n4_22a. - MP n4_22a H1. - replace (P↔R) with ((P→R)∧(R→P)) in n4_22a. - specialize Simp3_27 with (P→R) (R→P). - intros Simp3_27a. - MP Simp3_27a n4_22a. - apply Simp3_27a. - apply Equiv4_01. - Qed. - -Theorem n4_87 : ∀ P Q R : Prop, - (((P ∧ Q) → R) ↔ (P → Q → R)) ↔ ((Q → (P → R)) ↔ (Q ∧ P → R)). - Proof. intros P Q R. - specialize Exp3_3 with P Q R. - intros Exp3_3a. - specialize Imp3_31 with P Q R. - intros Imp3_31a. - Conj Exp3_3a Imp3_31a. - split. - apply Exp3_3a. - apply Imp3_31a. - Equiv H. - specialize Exp3_3 with Q P R. - intros Exp3_3b. - specialize Imp3_31 with Q P R. - intros Imp3_31b. - Conj Exp3_3b Imp3_31b. - split. - apply Exp3_3b. - apply Imp3_31b. - Equiv H0. - specialize Comm2_04 with P Q R. - intros Comm2_04a. - specialize Comm2_04 with Q P R. - intros Comm2_04b. - Conj Comm2_04a Comm2_04b. - split. - apply Comm2_04a. - apply Comm2_04b. - Equiv H1. - clear Exp3_3a. clear Imp3_31a. clear Exp3_3b. clear Imp3_31b. clear Comm2_04a. clear Comm2_04b. - replace (P ∧ Q → R) with (P → Q → R). - replace (Q ∧ P → R) with (Q → P → R). - replace (Q → P → R) with (P → Q → R). - specialize n4_2 with ((P → Q → R) ↔ (P → Q → R)). - intros n4_2a. - apply n4_2a. - apply EqBi. - apply H1. - replace (Q → P → R) with (Q ∧ P → R). - reflexivity. - apply EqBi. - apply H0. - replace (P → Q → R) with (P ∧ Q → R). - reflexivity. - apply EqBi. - apply H. - apply Equiv4_01. - apply Equiv4_01. - apply Equiv4_01. - Qed. - -End No4.
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