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@@ -0,0 +1,1138 @@ +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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) + +End No1. + +Module No2. +Import No1. + +(*We proceed to the deductions of *2 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.*) + +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. + Syll n2_55a Syll2_06a S. + apply S. +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. + MP Comm2_04a n2_75a. + apply Comm2_04a. +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 of inference allowing one to move from the theoremhood of P and theoremhood of Q to the theoremhood 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. + +Ltac Prod H1 H2 := + match goal with + | [ H1 : ?P, H2 : ?Q |- _ ] => + assert (H1 ∧ H2) by (specialize Conj3_03 with P Q; + intros Conj3_03; MP Conj3_03 H1; MP Conj3_03 H2) +end. + +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. +(*Conj3_03 is cited in the proof sketch, but we found it unnecessary.*) + +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 Perm1_4 with Q R. intros Perm1_4b. + specialize Syll2_05 with (P∨Q) (Q∨R) (R∨Q). intros Syll2_05a. + MP Syll2_06b Perm1_4b. + Syll Sb Syll2_05a Sc. + Syll Syll2_05b Sb Sc.*) + 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.
\ No newline at end of file @@ -0,0 +1,1508 @@ +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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) + +End No1. + +Module No2. +Import No1. + +(*We proceed to the deductions of *2 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.*) + +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 Syll 2_06a Perm1_4a. +Qed. This proof is done but for an error message: "got 2 extra arguments".*) + auto. +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. + (*MP Comm 2_04a n2_75a. This wouldn't work because "illegal tactic; two extra args."*) +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 meta-theorem allowing one to move from the theoremhood of P and theoremhood of Q to the theoremhood 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 Conj : ∀ P Q : Prop, P→Q→(P∧Q). +Proof. intros P Q. specialize Conj3_03 with P Q. intros Conj3_03. left. intros Pp. intros Qq. MP Conj3_03 P. MP Conj3_03 Q. apply Conj3_03. (* split. apply Pp. apply Qq. + specialize Conj3_03 with P Q. intros Conj3_03a Pp Qq. + MP Conj3_03a P. MP Conj3_03a Q. apply Conj3_03a.*) + (*assert (Q∧P). split. apply Hq. apply Hp. apply H. *) +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 Perm1_4 with Q R. intros Perm1_4b. + specialize Syll2_05 with (P∨Q) (Q∨R) (R∨Q). intros Syll2_05a. + MP Syll2_06b Perm1_4b. + Syll Sb Syll2_05a Sc. + Syll Syll2_05b Sb Sc.*) + 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_17 with P Q. intros Trans2_17a. + Conj Trans2_16a Trans2_17a. + split. apply Trans2_16a. apply Trans2_17a. + specialize n3_47 with (~Q) P (~P) Q. intros n3_47a. + specialize n3_22 with (P→Q) (~Q→~P). intros n3_22a. + Syll n3_22a n3_47a Sa. + Conj Trans2_16a Trans2_17a. + (*split. apply Trans2_16a. apply Trans2_17a. + Equiv H. apply H. apply Equiv4_01.*) +Admitted. +(*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 n2_03 with Q P. intros n2_03b. + specialize n2_03 with (P↔Q) (Q↔P). intros n2_03c. + specialize n2_03 with (Q↔P) (P↔Q). intros n2_03d. + specialize Trans2_15 with Q P. intros Trans2_15a. + specialize Trans2_15 with P Q. intros Trans2_15b. + Admitted. + +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_27d Simp3_26d Sd. + specialize n2_83 with ((P↔Q)∧(Q↔R)) R Q P. intros n2_83b. + MP n2_83b Sc. MP n2_83b Sd. + Conj Sb Sd. split. + apply Sd. apply n2_83b. + split. + specialize Comp3_43 with ((P↔Q)∧(Q↔R)) (P→R) (R→P). intros Comp3_43a. + MP Comp3_43a + split. apply n2_83a. + MP Comp3_43a n2_83a. + MP Comp3_43a n2_83b. +Qed. + +Theorem n4_24 : ∀ P : Prop, + P ↔ (P ∧ P). +Proof. intuition. +Qed. + +Theorem n4_25 : ∀ P : Prop, + P ↔ (P ∨ P). +Proof. intuition. +Qed. + +Theorem n4_3 : ∀ P Q : Prop, + (P ∧ Q) ↔ (Q ∧ P). +Proof. intuition. +Qed. + +Theorem n4_33 : ∀ P Q R : Prop, + (P ∧ (Q ∧ R)) ↔ (P ∨ (Q ∨ R)). +Admitted. + +Theorem n4_36 : ∀ P Q R : Prop, + (P ↔ Q) → ((P ∧ R) ↔ (Q ∧ R)). +Proof. intuition. +Qed. + +Theorem n4_37 : ∀ P Q R : Prop, + (P ↔ Q) → ((P ∨ R) ↔ (Q ∨ R)). +Proof. intuition. +Qed. + +Theorem n4_38 : ∀ P Q R S : Prop, + ((P ↔ R) ∧ (Q ↔ S)) → ((P ∧ Q) ↔ (R ∧ S)). +Proof. intuition. +Qed. + +Theorem n4_39 : ∀ P Q R S : Prop, + ((P ↔ R) ∧ (Q ↔ S)) → ((P ∨ Q) ↔ (R ∨ S)). +Proof. intuition. +Qed. + +Theorem n4_4 : ∀ P Q R : Prop, + (P ∧ (Q ∨ R)) ↔ ((P∧ Q) ∨ (P ∧ R)). +Proof. intuition. +Qed. + +Theorem n4_41 : ∀ P Q R : Prop, + (P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R)). +Proof. intuition. +Qed. + +Theorem n4_42 : ∀ P Q : Prop, + P ↔ ((P ∧ Q) ∨ (P ∧ ~Q)). +Admitted. + +Theorem n4_43 : ∀ P Q : Prop, + P ↔ ((P ∨ Q) ∧ (P ∨ ~Q)). +Admitted. + +Theorem n4_44 : ∀ P Q : Prop, + P ↔ (P ∨ (P ∧ Q)). +Admitted. + +Theorem n4_45 : ∀ P Q : Prop, + P ↔ (P ∧ (P ∨ Q)). +Admitted. + +Theorem n4_5 : ∀ P Q : Prop, + P ∧ Q ↔ ~(~P ∨ ~Q). +Admitted. + +Theorem n4_51 : ∀ P Q : Prop, + ~(P ∧ Q) ↔ (~P ∨ ~Q). +Admitted. + +Theorem n4_52 : ∀ P Q : Prop, + (P ∧ ~Q) ↔ ~(~P ∨ Q). +Admitted. + +Theorem n4_53 : ∀ P Q : Prop, + ~(P ∧ ~Q) ↔ (~P ∨ Q). +Admitted. + +Theorem n4_54 : ∀ P Q : Prop, + (~P ∧ Q) ↔ ~(P ∨ ~Q). +Admitted. + +Theorem n4_55 : ∀ P Q : Prop, + ~(~P ∧ Q) ↔ (P ∨ ~Q). +Admitted. + +Theorem n4_56 : ∀ P Q : Prop, + (~P ∧ ~Q) ↔ ~(P ∨ Q). +Admitted. + +Theorem n4_57 : ∀ P Q : Prop, + ~(~P ∧ ~Q) ↔ (P ∨ Q). +Admitted. + +Theorem n4_6 : ∀ P Q : Prop, + (P → Q) ↔ (~P ∨ Q). +Admitted. + +Theorem n4_61 : ∀ P Q : Prop, + ~(P → Q) ↔ (P ∧ ~Q). +Admitted. + +Theorem n4_62 : ∀ P Q : Prop, + (P → ~Q) ↔ (~P ∨ ~Q). +Admitted. + +Theorem n4_63 : ∀ P Q : Prop, + ~(P → ~Q) ↔ (P ∧ Q). +Admitted. + +Theorem n4_64 : ∀ P Q : Prop, + (~P → Q) ↔ (P ∨ Q). +Admitted. + +Theorem n4_65 : ∀ P Q : Prop, + ~(~P → Q) ↔ (~P ∧ ~Q). +Admitted. + +Theorem n4_66 : ∀ P Q : Prop, + (~P → ~Q) ↔ (P ∨ ~Q). +Admitted. + +Theorem n4_67 : ∀ P Q : Prop, + ~(~P → ~Q) ↔ (~P ∧ Q). +Admitted. + +Theorem n4_7 : ∀ P Q : Prop, + (P → Q) ↔ (P → (P ∧ Q)). +Admitted. + +Theorem n4_71 : ∀ P Q : Prop, + (P → Q) ↔ (P ↔ (P ∧ Q)). +Admitted. + +Theorem n4_72 : ∀ P Q : Prop, + (P → Q) ↔ (Q ↔ (P ∨ Q)). +Admitted. + +Theorem n4_73 : ∀ P Q : Prop, + Q → (P ↔ (P ∧ Q)). +Admitted. + +Theorem n4_74 : ∀ P Q : Prop, + ~P → (Q ↔ (P ∨ Q)). +Admitted. + +Theorem n4_76 : ∀ P Q R : Prop, + ((P → Q) ∧ (P → R)) ↔ (P → (Q ∧ R)). +Admitted. + +Theorem n4_77 : ∀ P Q R : Prop, + ((Q → P) ∧ (R → P)) ↔ ((Q ∨ R) → P). +Admitted. + +Theorem n4_78 : ∀ P Q R : Prop, + ((P → Q) ∨ (P → R)) ↔ (P → (Q ∨ R)). +Admitted. + +Theorem n4_79 : ∀ P Q R : Prop, + ((Q → P) ∨ (R → P)) ↔ ((Q ∧ R) → P). +Admitted. + +Theorem n4_8 : ∀ P : Prop, + (P → ~P) ↔ ~P. +Admitted. + +Theorem n4_81 : ∀ P : Prop, + (~P → P) ↔ P. +Admitted. + +Theorem n4_82 : ∀ P Q : Prop, + ((P → Q) ∧ (P → ~Q)) ↔ ~P. +Admitted. + +Theorem n4_83 : ∀ P Q : Prop, + ((P → Q) ∧ (~P → Q)) ↔ Q. +Admitted. + +Theorem n4_84 : ∀ P Q R : Prop, + (P ↔ Q) → ((P → R) ↔ (Q → R)). +Admitted. + +Theorem n4_85 : ∀ P Q R : Prop, + (P ↔ Q) → ((R → P) ↔ (R → Q)). +Admitted. + +Theorem n4_86 : ∀ P Q R : Prop, + (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)). +Admitted. + +Theorem n4_87 : ∀ P Q R : Prop, + ((P ∧ Q) → R) ↔ (P → Q → R) ↔ (Q → (P → R)) ↔ (Q ∧ P → R). +Admitted. + +End No4.
\ No newline at end of file diff --git a/No1.docx b/No1.docx Binary files differBinary files differnew file mode 100644 index 0000000..31ecca2 --- /dev/null +++ b/No1.docx @@ -1,27 +1,30 @@ Require Import Unicode.Utf8. -Module No1. -Import Unicode.Utf8. - (*We first give the axioms of Principia -for the propositional calculus in *1.*) +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 Taut1_2 : ∀ P : Prop, + P ∨ P→ P. (*Tautology*) -Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) +Axiom Add1_3 : ∀ P Q : Prop, + Q → P ∨ Q. (*Addition*) -Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) +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 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 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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) +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.
\ No newline at end of file diff --git a/No2.docx b/No2.docx Binary files differBinary files differnew file mode 100644 index 0000000..05722ca --- /dev/null +++ b/No2.docx @@ -10,26 +10,30 @@ Axiom MP1_1 : ∀ P Q : Prop, (**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 Taut1_2 : ∀ P : Prop, + P ∨ P→ P. (*Tautology*) -Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) +Axiom Add1_3 : ∀ P Q : Prop, + Q → P ∨ Q. (*Addition*) -Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) +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 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 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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) +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 *2 of Principia.*) +(*We proceed to the deductions of of Principia.*) Theorem Abs2_01 : ∀ P : Prop, (P → ~P) → ~P. @@ -53,7 +57,8 @@ 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). + 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. @@ -68,25 +73,32 @@ Proof. intros 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. + 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). + 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. + 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. + 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. @@ -102,9 +114,12 @@ 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. + 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. @@ -122,8 +137,10 @@ 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. + specialize Perm1_4 with (~P) P. + intros Perm1_4. + specialize n2_1 with P. + intros Abs2_01. apply Perm1_4. apply n2_1. Qed. @@ -131,15 +148,19 @@ Qed. Theorem n2_12 : ∀ P : Prop, P → ~~P. Proof. intros P. - specialize n2_11 with (~P). intros n2_11. - rewrite Impl1_01. assumption. + 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. + 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. @@ -148,8 +169,10 @@ 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. + 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. @@ -158,12 +181,18 @@ 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. + 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. @@ -175,21 +204,24 @@ 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))) + | [ 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) + | [ 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. + 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. @@ -198,9 +230,12 @@ 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. + 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. @@ -209,12 +244,16 @@ 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. + 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. + specialize Abs2_01 with (~P). + intros Abs2_01a. Syll Syll2_05a Abs2_01a Sa. - specialize n2_14 with P. intros n2_14a. + specialize n2_14 with P. + intros n2_14a. Syll H n2_14a Sb. apply Sb. Qed. @@ -222,8 +261,10 @@ 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. + 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. @@ -231,8 +272,10 @@ 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. + 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. @@ -240,8 +283,10 @@ 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. + 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. @@ -249,8 +294,10 @@ 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. + 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. @@ -260,7 +307,8 @@ Qed. Theorem n2_26 : ∀ P Q : Prop, ~P ∨ ((P → Q) → Q). Proof. intros P Q. - specialize n2_25 with (~P) Q. intros n2_25a. + specialize n2_25 with (~P) Q. + intros n2_25a. replace (~P∨Q) with (P→Q) in n2_25a. apply n2_25a. apply Impl1_01. @@ -269,7 +317,8 @@ Qed. Theorem n2_27 : ∀ P Q : Prop, P → ((P → Q) → Q). Proof. intros P Q. - specialize n2_26 with P Q. intros n2_26a. + 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. @@ -278,8 +327,10 @@ 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. + 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. @@ -287,9 +338,12 @@ 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. + 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. @@ -298,29 +352,36 @@ 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. + 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. + 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.*) +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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -328,10 +389,13 @@ 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -339,14 +403,19 @@ 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. + 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a H S. apply S. Qed. @@ -354,9 +423,12 @@ 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. + 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. @@ -365,9 +437,12 @@ 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. + 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. @@ -376,7 +451,8 @@ 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. + 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. @@ -384,16 +460,20 @@ 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. + 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. + 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. + 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. @@ -401,8 +481,10 @@ 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. + 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. @@ -410,8 +492,10 @@ 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. + 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. @@ -419,8 +503,10 @@ 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. + 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. @@ -428,8 +514,10 @@ 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. + 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. @@ -437,67 +525,83 @@ Qed. Theorem n2_5 : ∀ P Q : Prop, ~(P → Q) → (~P → Q). Proof. intros P Q. - specialize n2_47 with (~P) Q. intros n2_47a. + 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. + 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. + 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. + 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. + 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. + 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. + 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). + (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. + 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. + 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. + 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. + 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. + 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. @@ -505,20 +609,24 @@ 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. + 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. - Syll n2_55a Syll2_06a S. - apply S. -Qed. +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. + 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. @@ -529,8 +637,10 @@ 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. + 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. @@ -538,8 +648,10 @@ 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. + 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. @@ -547,15 +659,19 @@ 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. + 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. + specialize n2_62 with P Q. + intros n2_62a. replace (~P∨Q) with (P→Q). apply n2_62a. apply Impl1_01. @@ -564,11 +680,15 @@ 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. + 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. + 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. @@ -577,21 +697,27 @@ 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. + 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. + 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. + 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. + 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. @@ -600,9 +726,11 @@ 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. + 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. + specialize n2_54 with P Q. + intros n2_54a. Syll n2_67a n2_54a S. apply S. apply Impl1_01. @@ -611,10 +739,13 @@ 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. + 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. + specialize n2_62 with Q P. + intros n2_62a. Syll Sa n2_62a Sb. apply Sb. Qed. @@ -622,8 +753,10 @@ 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. + 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. @@ -631,13 +764,18 @@ 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!*) + 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!*) + 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. + 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. @@ -646,14 +784,19 @@ 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. + 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. + 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!*) + 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. @@ -663,30 +806,38 @@ 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. - MP Comm2_04a n2_75a. + 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. + 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. + 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. + 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. + specialize n2_38 with S (~R) Q. + intros n2_38a. Syll H n2_38a Hb. apply Hb. Qed. @@ -694,9 +845,12 @@ 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. + 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. @@ -705,8 +859,10 @@ 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. + 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. @@ -714,7 +870,8 @@ 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. + 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. @@ -735,18 +892,26 @@ 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. + 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. + 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. + 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. + 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.*) + 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. @@ -756,12 +921,15 @@ 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. + 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. + apply Impl1_01. + apply Impl1_01. + apply Impl1_01. Qed. End No2.
\ No newline at end of file diff --git a/No3.docx b/No3.docx Binary files differBinary files differnew file mode 100644 index 0000000..6b4ec93 --- /dev/null +++ b/No3.docx @@ -10,26 +10,30 @@ Axiom MP1_1 : ∀ P Q : Prop, (**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 Taut1_2 : ∀ P : Prop, + P ∨ P→ P. (*Tautology*) -Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) +Axiom Add1_3 : ∀ P Q : Prop, + Q → P ∨ Q. (*Addition*) -Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) +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 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 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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) +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 *2 of Principia.*) +(*We proceed to the deductions of of Principia.*) Theorem Abs2_01 : ∀ P : Prop, (P → ~P) → ~P. @@ -53,7 +57,8 @@ 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). + 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. @@ -68,25 +73,32 @@ Proof. intros 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. + 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). + 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. + 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. + 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. @@ -102,9 +114,12 @@ 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. + 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. @@ -122,8 +137,10 @@ 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. + specialize Perm1_4 with (~P) P. + intros Perm1_4. + specialize n2_1 with P. + intros Abs2_01. apply Perm1_4. apply n2_1. Qed. @@ -131,15 +148,19 @@ Qed. Theorem n2_12 : ∀ P : Prop, P → ~~P. Proof. intros P. - specialize n2_11 with (~P). intros n2_11. - rewrite Impl1_01. assumption. + 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. + 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. @@ -148,8 +169,10 @@ 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. + 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. @@ -158,12 +181,18 @@ 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. + 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. @@ -175,21 +204,24 @@ 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))) + | [ 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) + | [ 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. + 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. @@ -198,9 +230,12 @@ 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. + 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. @@ -209,12 +244,16 @@ 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. + 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. + specialize Abs2_01 with (~P). + intros Abs2_01a. Syll Syll2_05a Abs2_01a Sa. - specialize n2_14 with P. intros n2_14a. + specialize n2_14 with P. + intros n2_14a. Syll H n2_14a Sb. apply Sb. Qed. @@ -222,8 +261,10 @@ 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. + 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. @@ -231,8 +272,10 @@ 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. + 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. @@ -240,8 +283,10 @@ 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. + 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. @@ -249,8 +294,10 @@ 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. + 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. @@ -260,7 +307,8 @@ Qed. Theorem n2_26 : ∀ P Q : Prop, ~P ∨ ((P → Q) → Q). Proof. intros P Q. - specialize n2_25 with (~P) Q. intros n2_25a. + specialize n2_25 with (~P) Q. + intros n2_25a. replace (~P∨Q) with (P→Q) in n2_25a. apply n2_25a. apply Impl1_01. @@ -269,7 +317,8 @@ Qed. Theorem n2_27 : ∀ P Q : Prop, P → ((P → Q) → Q). Proof. intros P Q. - specialize n2_26 with P Q. intros n2_26a. + 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. @@ -278,8 +327,10 @@ 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. + 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. @@ -287,9 +338,12 @@ 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. + 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. @@ -298,29 +352,36 @@ 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. + 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. + 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.*) +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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -328,10 +389,13 @@ 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -339,14 +403,19 @@ 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. + 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a H S. apply S. Qed. @@ -354,9 +423,12 @@ 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. + 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. @@ -365,9 +437,12 @@ 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. + 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. @@ -376,7 +451,8 @@ 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. + 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. @@ -384,16 +460,20 @@ 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. + 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. + 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. + 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. @@ -401,8 +481,10 @@ 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. + 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. @@ -410,8 +492,10 @@ 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. + 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. @@ -419,8 +503,10 @@ 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. + 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. @@ -428,8 +514,10 @@ 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. + 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. @@ -437,67 +525,83 @@ Qed. Theorem n2_5 : ∀ P Q : Prop, ~(P → Q) → (~P → Q). Proof. intros P Q. - specialize n2_47 with (~P) Q. intros n2_47a. + 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. + 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. + 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. + 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. + 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. + 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. + 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). + (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. + 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. + 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. + 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. + 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. + 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. @@ -505,20 +609,24 @@ 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. + 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. - Syll n2_55a Syll2_06a S. - apply S. -Qed. +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. + 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. @@ -529,8 +637,10 @@ 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. + 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. @@ -538,8 +648,10 @@ 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. + 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. @@ -547,15 +659,19 @@ 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. + 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. + specialize n2_62 with P Q. + intros n2_62a. replace (~P∨Q) with (P→Q). apply n2_62a. apply Impl1_01. @@ -564,11 +680,15 @@ 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. + 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. + 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. @@ -577,21 +697,27 @@ 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. + 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. + 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. + 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. + 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. @@ -600,9 +726,11 @@ 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. + 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. + specialize n2_54 with P Q. + intros n2_54a. Syll n2_67a n2_54a S. apply S. apply Impl1_01. @@ -611,10 +739,13 @@ 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. + 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. + specialize n2_62 with Q P. + intros n2_62a. Syll Sa n2_62a Sb. apply Sb. Qed. @@ -622,8 +753,10 @@ 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. + 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. @@ -631,13 +764,18 @@ 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!*) + 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!*) + 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. + 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. @@ -646,14 +784,19 @@ 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. + 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. + 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!*) + 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. @@ -663,30 +806,38 @@ 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. - MP Comm2_04a n2_75a. + 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. + 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. + 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. + 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. + specialize n2_38 with S (~R) Q. + intros n2_38a. Syll H n2_38a Hb. apply Hb. Qed. @@ -694,9 +845,12 @@ 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. + 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. @@ -705,8 +859,10 @@ 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. + 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. @@ -714,7 +870,8 @@ 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. + 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. @@ -735,18 +892,26 @@ 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. + 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. + 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. + 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. + 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.*) + 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. @@ -756,12 +921,15 @@ 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. + 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. + apply Impl1_01. + apply Impl1_01. + apply Impl1_01. Qed. End No2. @@ -771,12 +939,13 @@ Module No3. Import No1. Import No2. -Axiom Prod3_01 : ∀ P Q : Prop, (P ∧ Q) = ~(~P ∨ ~Q). +Axiom Prod3_01 : ∀ P Q : Prop, + (P ∧ Q) = ~(~P ∨ ~Q). -Axiom Abb3_02 : ∀ P Q R : Prop, (P→Q→R)=(P→Q)∧(Q→R). +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 meta-theorem allowing one to move from the theoremhood of P and theoremhood of Q to the theoremhood of P and Q.*) +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. @@ -790,18 +959,12 @@ Proof. intros P Q. apply Prod3_01. Qed. -Ltac Prod H1 H2 := - match goal with - | [ H1 : ?P, H2 : ?Q |- _ ] => - assert (P ∧ Q) by (specialize Conj3_03 with P Q; - intros Conj3_03; MP Conj3_03 P; MP Conj3_03 Q) -end. - 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. + specialize n2_08 with (P∧Q). + intros n2_08a. apply n2_08a. apply Prod3_01. Qed. @@ -810,7 +973,8 @@ 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. + specialize n2_08 with (P∧Q). + intros n2_08a. apply n2_08a. apply Prod3_01. Qed. @@ -818,7 +982,8 @@ Qed. Theorem n3_12 : ∀ P Q : Prop, (~P ∨ ~Q) ∨ (P ∧ Q). Proof. intros P Q. - specialize n2_11 with (~P∨~Q). intros n2_11a. + specialize n2_11 with (~P∨~Q). + intros n2_11a. replace (~(~P∨~Q)) with (P∧Q) in n2_11a. apply n2_11a. apply Prod3_01. @@ -827,8 +992,10 @@ 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. + 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. @@ -836,10 +1003,13 @@ 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. + 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. + specialize n2_12 with (~P∨~Q). + intros n2_12a. Syll n2_12a Trans2_16a S. apply S. Qed. @@ -847,20 +1017,25 @@ 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. + 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. + 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. + 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. @@ -868,12 +1043,16 @@ 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. + 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. + 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. + specialize Trans2_17 with (P∧Q) (Q ∧ P). + intros Trans2_17a. MP Trans2_17a Hb. apply Trans2_17a. Qed. @@ -881,8 +1060,10 @@ 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. + 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. @@ -890,12 +1071,15 @@ Qed. Theorem Simp3_26 : ∀ P Q : Prop, (P ∧ Q) → P. Proof. intros P Q. - specialize n2_02 with Q P. intros n2_02a. + 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. + 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. + 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. @@ -911,8 +1095,10 @@ 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. + 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. @@ -920,12 +1106,16 @@ 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. + 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. + 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. + 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. @@ -939,8 +1129,10 @@ 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. + 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. @@ -954,8 +1146,10 @@ 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. + 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. @@ -963,8 +1157,10 @@ 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. + 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. @@ -972,8 +1168,10 @@ 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. + 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. @@ -981,12 +1179,16 @@ 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. + 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. + 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. + specialize Imp3_31 with P (~R) (~Q). + intros Imp3_31a. Syll Sa Imp3_31a Sb. apply Sb. Qed. @@ -994,8 +1196,10 @@ 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. + 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. @@ -1009,8 +1213,10 @@ 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. + 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. @@ -1018,8 +1224,10 @@ 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. + 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. @@ -1027,12 +1235,16 @@ 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. + 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. + 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. + specialize Imp3_31 with (P→Q) (P→R) (P→Q∧R). + intros Imp3_31a. MP Sa Imp3_31a. apply Imp3_31a. Qed. @@ -1040,19 +1252,27 @@ 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. + 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. + 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. + 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. + 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. + 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. + 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. @@ -1061,8 +1281,10 @@ 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. + 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. @@ -1082,21 +1304,30 @@ 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. + 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. + 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. + 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. + 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. + 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. @@ -1105,26 +1336,30 @@ 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. + 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. + 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 Perm1_4 with Q R. intros Perm1_4b. - specialize Syll2_05 with (P∨Q) (Q∨R) (R∨Q). intros Syll2_05a. - MP Syll2_06b Perm1_4b. - Syll Sb Syll2_05a Sc. - Syll Syll2_05b Sb Sc.*) - specialize Simp3_27 with (P→R) (Q→S). intros Simp3_27a. - specialize Sum1_6 with R Q S. intros Sum1_6b. + 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. + 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. + 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. diff --git a/No4.docx b/No4.docx Binary files differBinary files differnew file mode 100644 index 0000000..309196c --- /dev/null +++ b/No4.docx @@ -10,26 +10,30 @@ Axiom MP1_1 : ∀ P Q : Prop, (**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 Taut1_2 : ∀ P : Prop, + P ∨ P→ P. (*Tautology*) -Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) +Axiom Add1_3 : ∀ P Q : Prop, + Q → P ∨ Q. (*Addition*) -Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) +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 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 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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) +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 *2 of Principia.*) +(*We proceed to the deductions of of Principia.*) Theorem Abs2_01 : ∀ P : Prop, (P → ~P) → ~P. @@ -53,7 +57,8 @@ 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). + 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. @@ -68,25 +73,32 @@ Proof. intros 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. + 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). + 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. + 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. + 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. @@ -102,9 +114,12 @@ 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. + 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. @@ -122,8 +137,10 @@ 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. + specialize Perm1_4 with (~P) P. + intros Perm1_4. + specialize n2_1 with P. + intros Abs2_01. apply Perm1_4. apply n2_1. Qed. @@ -131,15 +148,19 @@ Qed. Theorem n2_12 : ∀ P : Prop, P → ~~P. Proof. intros P. - specialize n2_11 with (~P). intros n2_11. - rewrite Impl1_01. assumption. + 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. + 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. @@ -148,8 +169,10 @@ 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. + 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. @@ -158,12 +181,18 @@ 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. + 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. @@ -187,9 +216,12 @@ 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. + 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. @@ -198,9 +230,12 @@ 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. + 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. @@ -209,12 +244,16 @@ 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. + 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. + specialize Abs2_01 with (~P). + intros Abs2_01a. Syll Syll2_05a Abs2_01a Sa. - specialize n2_14 with P. intros n2_14a. + specialize n2_14 with P. + intros n2_14a. Syll H n2_14a Sb. apply Sb. Qed. @@ -222,8 +261,10 @@ 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. + 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. @@ -231,8 +272,10 @@ 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. + 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. @@ -240,8 +283,10 @@ 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. + 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. @@ -249,8 +294,10 @@ 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. + 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. @@ -260,7 +307,8 @@ Qed. Theorem n2_26 : ∀ P Q : Prop, ~P ∨ ((P → Q) → Q). Proof. intros P Q. - specialize n2_25 with (~P) Q. intros n2_25a. + specialize n2_25 with (~P) Q. + intros n2_25a. replace (~P∨Q) with (P→Q) in n2_25a. apply n2_25a. apply Impl1_01. @@ -269,7 +317,8 @@ Qed. Theorem n2_27 : ∀ P Q : Prop, P → ((P → Q) → Q). Proof. intros P Q. - specialize n2_26 with P Q. intros n2_26a. + 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. @@ -278,8 +327,10 @@ 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. + 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. @@ -287,9 +338,12 @@ 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. + 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. @@ -298,29 +352,36 @@ 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. + 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. + 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.*) +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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -328,10 +389,13 @@ 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -339,14 +403,19 @@ 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. + 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a H S. apply S. Qed. @@ -354,9 +423,12 @@ 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. + 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. @@ -365,9 +437,12 @@ 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. + 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. @@ -376,7 +451,8 @@ 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. + 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. @@ -384,16 +460,20 @@ 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. + 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. + 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. + 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. @@ -401,8 +481,10 @@ 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. + 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. @@ -410,8 +492,10 @@ 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. + 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. @@ -419,8 +503,10 @@ 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. + 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. @@ -428,8 +514,10 @@ 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. + 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. @@ -437,38 +525,46 @@ Qed. Theorem n2_5 : ∀ P Q : Prop, ~(P → Q) → (~P → Q). Proof. intros P Q. - specialize n2_47 with (~P) Q. intros n2_47a. + 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. + 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. + 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. + 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. + 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. + 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. + 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. @@ -476,28 +572,36 @@ 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. + 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. + 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. + 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. + 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. + 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. @@ -505,20 +609,24 @@ 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 Syll 2_06a Perm1_4a. -Qed. This proof is done but for an error message: "got 2 extra arguments".*) - auto. -Qed. + 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. + 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. @@ -529,8 +637,10 @@ 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. + 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. @@ -538,8 +648,10 @@ 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. + 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. @@ -547,15 +659,19 @@ 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. + 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. + specialize n2_62 with P Q. + intros n2_62a. replace (~P∨Q) with (P→Q). apply n2_62a. apply Impl1_01. @@ -564,11 +680,15 @@ 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. + 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. + 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. @@ -577,21 +697,27 @@ 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. + 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. + 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. + 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. + 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. @@ -600,9 +726,11 @@ 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. + 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. + specialize n2_54 with P Q. + intros n2_54a. Syll n2_67a n2_54a S. apply S. apply Impl1_01. @@ -611,10 +739,13 @@ 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. + 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. + specialize n2_62 with Q P. + intros n2_62a. Syll Sa n2_62a Sb. apply Sb. Qed. @@ -622,8 +753,10 @@ 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. + 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. @@ -631,13 +764,18 @@ 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!*) + 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!*) + 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. + 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. @@ -646,14 +784,19 @@ 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. + 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. + 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!*) + 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. @@ -663,31 +806,38 @@ 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. + 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. - (*MP Comm 2_04a n2_75a. This wouldn't work because "illegal tactic; two extra args."*) + 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. + 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. + 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. + 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. + specialize n2_38 with S (~R) Q. + intros n2_38a. Syll H n2_38a Hb. apply Hb. Qed. @@ -695,9 +845,12 @@ 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. + 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. @@ -706,8 +859,10 @@ 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. + 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. @@ -715,7 +870,8 @@ 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. + 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. @@ -736,18 +892,26 @@ 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. + 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. + 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. + 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. + 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.*) + 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. @@ -757,12 +921,15 @@ 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. + 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. + apply Impl1_01. + apply Impl1_01. + apply Impl1_01. Qed. End No2. @@ -772,12 +939,13 @@ Module No3. Import No1. Import No2. -Axiom Prod3_01 : ∀ P Q : Prop, (P ∧ Q) = ~(~P ∨ ~Q). +Axiom Prod3_01 : ∀ P Q : Prop, + (P ∧ Q) = ~(~P ∨ ~Q). -Axiom Abb3_02 : ∀ P Q R : Prop, (P→Q→R)=(P→Q)∧(Q→R). +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 meta-theorem allowing one to move from the theoremhood of P and theoremhood of Q to the theoremhood of P and Q.*) +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. @@ -791,24 +959,12 @@ Proof. intros P Q. apply Prod3_01. Qed. -Ltac Conj H1 H2 := - match goal with - | [ H1 : ?P, H2 : ?Q |- _ ] => - assert (P ∧ Q) by (specialize Conj3_03 with P Q; intros Conj3_03; MP Conj3_03 P; MP Conj3_03 Q) -end. - -(*Theorem Conj : ∀ P Q : Prop, P→Q→(P∧Q). -Proof. intros P Q. specialize Conj3_03 with P Q. intros Conj3_03. left. intros Pp. intros Qq. MP Conj3_03 P. MP Conj3_03 Q. apply Conj3_03. (* split. apply Pp. apply Qq. - specialize Conj3_03 with P Q. intros Conj3_03a Pp Qq. - MP Conj3_03a P. MP Conj3_03a Q. apply Conj3_03a.*) - (*assert (Q∧P). split. apply Hq. apply Hp. apply H. *) -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. + specialize n2_08 with (P∧Q). + intros n2_08a. apply n2_08a. apply Prod3_01. Qed. @@ -817,7 +973,8 @@ 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. + specialize n2_08 with (P∧Q). + intros n2_08a. apply n2_08a. apply Prod3_01. Qed. @@ -825,7 +982,8 @@ Qed. Theorem n3_12 : ∀ P Q : Prop, (~P ∨ ~Q) ∨ (P ∧ Q). Proof. intros P Q. - specialize n2_11 with (~P∨~Q). intros n2_11a. + specialize n2_11 with (~P∨~Q). + intros n2_11a. replace (~(~P∨~Q)) with (P∧Q) in n2_11a. apply n2_11a. apply Prod3_01. @@ -834,8 +992,10 @@ 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. + 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. @@ -843,10 +1003,13 @@ 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. + 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. + specialize n2_12 with (~P∨~Q). + intros n2_12a. Syll n2_12a Trans2_16a S. apply S. Qed. @@ -854,20 +1017,25 @@ 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. + 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. + 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. + 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. @@ -875,12 +1043,16 @@ 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. + 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. + 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. + specialize Trans2_17 with (P∧Q) (Q ∧ P). + intros Trans2_17a. MP Trans2_17a Hb. apply Trans2_17a. Qed. @@ -888,8 +1060,10 @@ 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. + 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. @@ -897,12 +1071,15 @@ Qed. Theorem Simp3_26 : ∀ P Q : Prop, (P ∧ Q) → P. Proof. intros P Q. - specialize n2_02 with Q P. intros n2_02a. + 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. + 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. + 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. @@ -918,8 +1095,10 @@ 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. + 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. @@ -927,12 +1106,16 @@ 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. + 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. + 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. + 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. @@ -946,8 +1129,10 @@ 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. + 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. @@ -961,8 +1146,10 @@ 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. + 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. @@ -970,8 +1157,10 @@ 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. + 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. @@ -979,8 +1168,10 @@ 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. + 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. @@ -988,12 +1179,16 @@ 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. + 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. + 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. + specialize Imp3_31 with P (~R) (~Q). + intros Imp3_31a. Syll Sa Imp3_31a Sb. apply Sb. Qed. @@ -1001,8 +1196,10 @@ 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. + 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. @@ -1016,8 +1213,10 @@ 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. + 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. @@ -1025,8 +1224,10 @@ 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. + 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. @@ -1034,12 +1235,16 @@ 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. + 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. + 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. + specialize Imp3_31 with (P→Q) (P→R) (P→Q∧R). + intros Imp3_31a. MP Sa Imp3_31a. apply Imp3_31a. Qed. @@ -1047,19 +1252,27 @@ 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. + 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. + 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. + 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. + 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. + 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. + 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. @@ -1068,8 +1281,10 @@ 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. + 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. @@ -1089,21 +1304,30 @@ 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. + 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. + 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. + 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. + 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. + 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. @@ -1112,26 +1336,30 @@ 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. + 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. + 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 Perm1_4 with Q R. intros Perm1_4b. - specialize Syll2_05 with (P∨Q) (Q∨R) (R∨Q). intros Syll2_05a. - MP Syll2_06b Perm1_4b. - Syll Sb Syll2_05a Sc. - Syll Syll2_05b Sb Sc.*) - specialize Simp3_27 with (P→R) (Q→S). intros Simp3_27a. - specialize Sum1_6 with R Q S. intros Sum1_6b. + 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. + 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. + 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. @@ -1145,266 +1373,1571 @@ 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 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.*) -Ltac Conj H1 H2 := +Axiom EqBi : ∀ P Q : Prop, + (P=Q) ↔ (P↔Q). + +Ltac Equiv H1 := match goal with - | [ H1: ?P, H2: ?Q |- _ ] => - assert (H1 ∧ H2) by (specialize Conj3_03 with H1 H2; intros p; MP p H1; MP p H2) + | [ H1 : (?P→?Q) ∧ (?Q→?P) |- _ ] => + replace ((P→Q) ∧ (Q→P)) with (P↔Q) in H1 end. -Ltac Equiv H1 H2 := +Ltac Conj H1 H2 := match goal with - | [ H1 : ?P→?Q, H2 : ?Q→?P |- _ ] => - assert (P ↔ Q) by (specialize Equiv4_01 with P Q; - intros Equiv4_01; Prod ((P→Q)∧(Q→P)); apply Equiv4_01) + | [ 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. - specialize Conj3_03 with ((P→Q)→(~Q→~P)) ((~Q→~P)→(P→Q)). intros Conj3_03a. - MP Conj3_03a Trans2_16a. - MP Conj3_03a Trans2_17a. - replace (((P → Q) → ¬ Q → ¬ P) ∧ ((¬ Q → ¬ P) → P → Q)) with ((P → Q) ↔ (~Q → ~P)) in Conj3_03a. - apply Conj3_03a. + 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) ↔ (P ↔ ~Q). -Admitted. + (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. -Admitted. + 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). -Admitted. +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). -Admitted. + 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. -Admitted. + 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. intuition. + 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. intuition. + ((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. intuition. + 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. intuition. +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. intuition. +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)). -Admitted. + (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. intuition. -Qed. +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. intuition. -Qed. +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. intuition. -Qed. +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. intuition. -Qed. +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. intuition. +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. intuition. +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)). -Admitted. +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)). -Admitted. +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)). -Admitted. + 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)). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. - + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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). -Admitted. + 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)). -Admitted. + 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). -Admitted. + 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. -Admitted. + 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. -Admitted. + 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. -Admitted. + 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. -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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). -Admitted. + (((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.
\ No newline at end of file @@ -10,26 +10,30 @@ Axiom MP1_1 : ∀ P Q : Prop, (**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 Taut1_2 : ∀ P : Prop, + P ∨ P→ P. (*Tautology*) -Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) +Axiom Add1_3 : ∀ P Q : Prop, + Q → P ∨ Q. (*Addition*) -Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) +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 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 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. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) +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 *2 of Principia.*) +(*We proceed to the deductions of of Principia.*) Theorem Abs2_01 : ∀ P : Prop, (P → ~P) → ~P. @@ -53,7 +57,8 @@ 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). + 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. @@ -68,25 +73,32 @@ Proof. intros 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. + 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). + 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. + 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. + 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. @@ -102,9 +114,12 @@ 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. + 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. @@ -122,8 +137,10 @@ 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. + specialize Perm1_4 with (~P) P. + intros Perm1_4. + specialize n2_1 with P. + intros Abs2_01. apply Perm1_4. apply n2_1. Qed. @@ -131,15 +148,19 @@ Qed. Theorem n2_12 : ∀ P : Prop, P → ~~P. Proof. intros P. - specialize n2_11 with (~P). intros n2_11. - rewrite Impl1_01. assumption. + 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. + 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. @@ -148,8 +169,10 @@ 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. + 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. @@ -158,12 +181,18 @@ 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. + 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. @@ -187,9 +216,12 @@ 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. + 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. @@ -198,9 +230,12 @@ 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. + 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. @@ -209,12 +244,16 @@ 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. + 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. + specialize Abs2_01 with (~P). + intros Abs2_01a. Syll Syll2_05a Abs2_01a Sa. - specialize n2_14 with P. intros n2_14a. + specialize n2_14 with P. + intros n2_14a. Syll H n2_14a Sb. apply Sb. Qed. @@ -222,8 +261,10 @@ 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. + 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. @@ -231,8 +272,10 @@ 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. + 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. @@ -240,8 +283,10 @@ 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. + 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. @@ -249,8 +294,10 @@ 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. + 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. @@ -260,7 +307,8 @@ Qed. Theorem n2_26 : ∀ P Q : Prop, ~P ∨ ((P → Q) → Q). Proof. intros P Q. - specialize n2_25 with (~P) Q. intros n2_25a. + specialize n2_25 with (~P) Q. + intros n2_25a. replace (~P∨Q) with (P→Q) in n2_25a. apply n2_25a. apply Impl1_01. @@ -269,7 +317,8 @@ Qed. Theorem n2_27 : ∀ P Q : Prop, P → ((P → Q) → Q). Proof. intros P Q. - specialize n2_26 with P Q. intros n2_26a. + 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. @@ -278,8 +327,10 @@ 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. + 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. @@ -287,9 +338,12 @@ 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. + 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. @@ -298,29 +352,36 @@ 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. + 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. + 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.*) +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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -328,10 +389,13 @@ 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a Syll2_05a S. apply S. Qed. @@ -339,14 +403,19 @@ 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. + 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. + 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. + specialize Sum1_6 with P Q R. + intros Sum1_6a. Syll Sum1_6a H S. apply S. Qed. @@ -354,9 +423,12 @@ 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. + 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. @@ -365,9 +437,12 @@ 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. + 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. @@ -376,7 +451,8 @@ 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. + 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. @@ -384,16 +460,20 @@ 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. + 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. + 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. + 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. @@ -401,8 +481,10 @@ 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. + 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. @@ -410,8 +492,10 @@ 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. + 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. @@ -419,8 +503,10 @@ 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. + 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. @@ -428,8 +514,10 @@ 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. + 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. @@ -437,38 +525,46 @@ Qed. Theorem n2_5 : ∀ P Q : Prop, ~(P → Q) → (~P → Q). Proof. intros P Q. - specialize n2_47 with (~P) Q. intros n2_47a. + 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. + 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. + 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. + 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. + 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. + 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. + 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. @@ -476,28 +572,36 @@ 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. + 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. + 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. + 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. + 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. + 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. @@ -505,20 +609,24 @@ 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 Syll 2_06a Perm1_4a. -Qed. This proof is done but for an error message: "got 2 extra arguments".*) - auto. -Qed. + 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. + 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. @@ -529,8 +637,10 @@ 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. + 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. @@ -538,8 +648,10 @@ 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. + 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. @@ -547,15 +659,19 @@ 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. + 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. + specialize n2_62 with P Q. + intros n2_62a. replace (~P∨Q) with (P→Q). apply n2_62a. apply Impl1_01. @@ -564,11 +680,15 @@ 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. + 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. + 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. @@ -577,21 +697,27 @@ 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. + 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. + 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. + 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. + 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. @@ -600,9 +726,11 @@ 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. + 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. + specialize n2_54 with P Q. + intros n2_54a. Syll n2_67a n2_54a S. apply S. apply Impl1_01. @@ -611,10 +739,13 @@ 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. + 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. + specialize n2_62 with Q P. + intros n2_62a. Syll Sa n2_62a Sb. apply Sb. Qed. @@ -622,8 +753,10 @@ 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. + 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. @@ -631,13 +764,18 @@ 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!*) + 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!*) + 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. + 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. @@ -646,14 +784,19 @@ 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. + 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. + 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!*) + 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. @@ -663,31 +806,38 @@ 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. + 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. - (*MP Comm 2_04a n2_75a. This wouldn't work because "illegal tactic; two extra args."*) + 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. + 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. + 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. + 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. + specialize n2_38 with S (~R) Q. + intros n2_38a. Syll H n2_38a Hb. apply Hb. Qed. @@ -695,9 +845,12 @@ 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. + 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. @@ -706,8 +859,10 @@ 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. + 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. @@ -715,7 +870,8 @@ 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. + 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. @@ -736,18 +892,26 @@ 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. + 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. + 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. + 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. + 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.*) + 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. @@ -757,12 +921,15 @@ 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. + 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. + apply Impl1_01. + apply Impl1_01. + apply Impl1_01. Qed. End No2. @@ -772,118 +939,430 @@ 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). -Admitted. +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). -Admitted. +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). -Admitted. +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). -Admitted. +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 R : Prop, +Theorem n3_14 : ∀ P Q : Prop, (~P ∨ ~Q) → ~(P ∧ Q). -Admitted. +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). -Admitted. +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). -Admitted. +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). -Admitted. +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). -Admitted. +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 n3_26 : ∀ P Q : Prop, +Theorem Simp3_26 : ∀ P Q : Prop, (P ∧ Q) → P. -Proof. intuition. +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 n3_27 : ∀ P Q : Prop, +Theorem Simp3_27 : ∀ P Q : Prop, (P ∧ Q) → Q. -Proof. intuition. +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 n3_3 : ∀ P Q R : Prop, - ((P ∧ Q) → R) → (P → (Q ∧ R)). -Admitted. +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 n3_31 : ∀ P Q R : Prop, - (P → (Q ∧ R)) → (P ∧ Q) → R. -Admitted. +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 n3_33 : ∀ P Q R : Prop, +Theorem Syll3_33 : ∀ P Q R : Prop, ((P → Q) ∧ (Q → R)) → (P → R). -Proof. intuition. +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 n3_34 : ∀ P Q R : Prop, +Theorem Syll3_34 : ∀ P Q R : Prop, ((Q → R) ∧ (P → Q)) → (P → R). -Proof. intuition. +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 n3_35 : ∀ P Q : Prop, +Theorem Ass3_35 : ∀ P Q : Prop, (P ∧ (P → Q)) → Q. -Proof. intuition. +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. intuition. +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. intuition. +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. intuition. +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. intuition. +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 n3_43 : ∀ P Q R : Prop, +Theorem Comp3_43 : ∀ P Q R : Prop, (P → Q) ∧ (P → R) → (P → Q ∧ R). -Proof. intuition. +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. intuition. -Qed. - -Theorem n3_45 : ∀ P Q R : Prop, +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. intuition. +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. intuition. +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. intuition. +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. @@ -894,240 +1373,1572 @@ Import No1. Import No2. Import No3. -Theorem n4_1 : ∀ P Q : Prop, - (P → Q) ↔ (~Q → ~P). -Admitted. +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. -Theorem n4_11 : ∀ P Q : Prop, +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). -Admitted. +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) ↔ (P ↔ ~Q). -Admitted. + (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. -Admitted. + 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). -Admitted. +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). -Admitted. + 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. -Admitted. + 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. intuition. + 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. intuition. + ((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. intuition. + 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. intuition. +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. intuition. -Qed. +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)). -Admitted. + (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. intuition. -Qed. +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. intuition. -Qed. +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. intuition. -Qed. +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. intuition. -Qed. +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. intuition. +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. intuition. +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)). -Admitted. +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)). -Admitted. +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)). -Admitted. + 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)). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. - + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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). -Admitted. + 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)). -Admitted. + 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). -Admitted. + 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. -Admitted. + 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. -Admitted. + 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. -Admitted. + 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. -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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)). -Admitted. + 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). -Admitted. + (((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. |