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@@ -1,937 +0,0 @@ -Require Import Unicode.Utf8. - -Module No1. -Import Unicode.Utf8. - (*We first give the axioms of Principia -for the propositional calculus in *1.*) - -Axiom MP1_1 : ∀ P Q : Prop, - (P → Q) → P → Q. (*Modus ponens*) - - (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*) - -Axiom Taut1_2 : ∀ P : Prop, - P ∨ P→ P. (*Tautology*) - -Axiom Add1_3 : ∀ P Q : Prop, - Q → P ∨ Q. (*Addition*) - -Axiom Perm1_4 : ∀ P Q : Prop, - P ∨ Q → Q ∨ P. (*Permutation*) - -Axiom Assoc1_5 : ∀ P Q R : Prop, - P ∨ (Q ∨ R) → Q ∨ (P ∨ R). - -Axiom Sum1_6: ∀ P Q R : Prop, - (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*) - -Axiom Impl1_01 : ∀ P Q : Prop, - (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*) - -End No1. - -Module No2. -Import No1. - -(*We proceed to the deductions of of Principia.*) - -Theorem Abs2_01 : ∀ P : Prop, - (P → ~P) → ~P. -Proof. intros P. - specialize Taut1_2 with (~P). - replace (~P ∨ ~P) with (P → ~P). - apply MP1_1. - apply Impl1_01. -Qed. - -Theorem n2_02 : ∀ P Q : Prop, - Q → (P → Q). -Proof. intros P Q. - specialize Add1_3 with (~P) Q. - replace (~P ∨ Q) with (P → Q). - apply (MP1_1 Q (P → Q)). - apply Impl1_01. -Qed. - -Theorem n2_03 : ∀ P Q : Prop, - (P → ~Q) → (Q → ~P). -Proof. intros P Q. - specialize Perm1_4 with (~P) (~Q). - replace (~P ∨ ~Q) with (P → ~Q). - replace (~Q ∨ ~P) with (Q → ~P). - apply (MP1_1 (P → ~Q) (Q → ~P)). - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Comm2_04 : ∀ P Q R : Prop, - (P → (Q → R)) → (Q → (P → R)). -Proof. intros P Q R. - specialize Assoc1_5 with (~P) (~Q) R. - replace (~Q ∨ R) with (Q → R). - replace (~P ∨ (Q → R)) with (P → (Q → R)). - replace (~P ∨ R) with (P → R). - replace (~Q ∨ (P → R)) with (Q → (P → R)). - apply (MP1_1 (P → Q → R) (Q → P → R)). - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Syll2_05 : ∀ P Q R : Prop, - (Q → R) → ((P → Q) → (P → R)). -Proof. intros P Q R. - specialize Sum1_6 with (~P) Q R. - replace (~P ∨ Q) with (P → Q). - replace (~P ∨ R) with (P → R). - apply (MP1_1 (Q → R) ((P → Q) → (P → R))). - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem Syll2_06 : ∀ P Q R : Prop, - (P → Q) → ((Q → R) → (P → R)). -Proof. intros P Q R. - specialize Comm2_04 with (Q → R) (P → Q) (P → R). - intros Comm2_04. - specialize Syll2_05 with P Q R. - intros Syll2_05. - specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). - intros MP1_1. - apply MP1_1. - apply Comm2_04. - apply Syll2_05. -Qed. - -Theorem n2_07 : ∀ P : Prop, - P → (P ∨ P). -Proof. intros P. - specialize Add1_3 with P P. - apply MP1_1. -Qed. - -Theorem n2_08 : ∀ P : Prop, - P → P. -Proof. intros P. - specialize Syll2_05 with P (P ∨ P) P. - intros Syll2_05. - specialize Taut1_2 with P. - intros Taut1_2. - specialize MP1_1 with ((P ∨ P) → P) (P → P). - intros MP1_1. - apply Syll2_05. - apply Taut1_2. - apply n2_07. -Qed. - -Theorem n2_1 : ∀ P : Prop, - (~P) ∨ P. -Proof. intros P. - specialize n2_08 with P. - replace (~P ∨ P) with (P → P). - apply MP1_1. - apply Impl1_01. -Qed. - -Theorem n2_11 : ∀ P : Prop, - P ∨ ~P. -Proof. intros P. - specialize Perm1_4 with (~P) P. - intros Perm1_4. - specialize n2_1 with P. - intros Abs2_01. - apply Perm1_4. - apply n2_1. -Qed. - -Theorem n2_12 : ∀ P : Prop, - P → ~~P. -Proof. intros P. - specialize n2_11 with (~P). - intros n2_11. - rewrite Impl1_01. - assumption. -Qed. - -Theorem n2_13 : ∀ P : Prop, - P ∨ ~~~P. -Proof. intros P. - specialize Sum1_6 with P (~P) (~~~P). - intros Sum1_6. - specialize n2_12 with (~P). - intros n2_12. - apply Sum1_6. - apply n2_12. - apply n2_11. -Qed. - -Theorem n2_14 : ∀ P : Prop, - ~~P → P. -Proof. intros P. - specialize Perm1_4 with P (~~~P). - intros Perm1_4. - specialize n2_13 with P. - intros n2_13. - rewrite Impl1_01. - apply Perm1_4. - apply n2_13. -Qed. - -Theorem Trans2_15 : ∀ P Q : Prop, - (~P → Q) → (~Q → P). -Proof. intros P Q. - specialize Syll2_05 with (~P) Q (~~Q). - intros Syll2_05a. - specialize n2_12 with Q. - intros n2_12. - specialize n2_03 with (~P) (~Q). - intros n2_03. - specialize Syll2_05 with (~Q) (~~P) P. - intros Syll2_05b. - specialize Syll2_05 with (~P → Q) (~P → ~~Q) (~Q → ~~P). - intros Syll2_05c. - specialize Syll2_05 with (~P → Q) (~Q → ~~P) (~Q → P). - intros Syll2_05d. - apply Syll2_05d. - apply Syll2_05b. - apply n2_14. - apply Syll2_05c. - apply n2_03. - apply Syll2_05a. - apply n2_12. -Qed. - -Ltac Syll H1 H2 S := - let S := fresh S in match goal with - | [ H1 : ?P → ?Q, H2 : ?Q → ?R |- _ ] => - assert (S : P → R) by (intros p; apply (H2 (H1 p))) -end. - -Ltac MP H1 H2 := - match goal with - | [ H1 : ?P → ?Q, H2 : ?P |- _ ] => specialize (H1 H2) -end. - -Theorem Trans2_16 : ∀ P Q : Prop, - (P → Q) → (~Q → ~P). -Proof. intros P Q. - specialize n2_12 with Q. - intros n2_12a. - specialize Syll2_05 with P Q (~~Q). - intros Syll2_05a. - specialize n2_03 with P (~Q). - intros n2_03a. - MP n2_12a Syll2_05a. - Syll Syll2_05a n2_03a S. - apply S. -Qed. - -Theorem Trans2_17 : ∀ P Q : Prop, - (~Q → ~P) → (P → Q). -Proof. intros P Q. - specialize n2_03 with (~Q) P. - intros n2_03a. - specialize n2_14 with Q. - intros n2_14a. - specialize Syll2_05 with P (~~Q) Q. - intros Syll2_05a. - MP n2_14a Syll2_05a. - Syll n2_03a Syll2_05a S. - apply S. -Qed. - -Theorem n2_18 : ∀ P : Prop, - (~P → P) → P. -Proof. intros P. - specialize n2_12 with P. - intro n2_12a. - specialize Syll2_05 with (~P) P (~~P). - intro Syll2_05a. - MP Syll2_05a n2_12. - specialize Abs2_01 with (~P). - intros Abs2_01a. - Syll Syll2_05a Abs2_01a Sa. - specialize n2_14 with P. - intros n2_14a. - Syll H n2_14a Sb. - apply Sb. -Qed. - -Theorem n2_2 : ∀ P Q : Prop, - P → (P ∨ Q). -Proof. intros P Q. - specialize Add1_3 with Q P. - intros Add1_3a. - specialize Perm1_4 with Q P. - intros Perm1_4a. - Syll Add1_3a Perm1_4a S. - apply S. -Qed. - -Theorem n2_21 : ∀ P Q : Prop, - ~P → (P → Q). -Proof. intros P Q. - specialize n2_2 with (~P) Q. - intros n2_2a. - specialize Impl1_01 with P Q. - intros Impl1_01a. - replace (~P∨Q) with (P→Q) in n2_2a. - apply n2_2a. -Qed. - -Theorem n2_24 : ∀ P Q : Prop, - P → (~P → Q). -Proof. intros P Q. - specialize n2_21 with P Q. - intros n2_21a. - specialize Comm2_04 with (~P) P Q. - intros Comm2_04a. - apply Comm2_04a. - apply n2_21a. -Qed. - -Theorem n2_25 : ∀ P Q : Prop, - P ∨ ((P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_1 with (P ∨ Q). - intros n2_1a. - specialize Assoc1_5 with (~(P∨Q)) P Q. - intros Assoc1_5a. - MP Assoc1_5a n2_1a. - replace (~(P∨Q)∨Q) with (P∨Q→Q) in Assoc1_5a. - apply Assoc1_5a. - apply Impl1_01. -Qed. - -Theorem n2_26 : ∀ P Q : Prop, - ~P ∨ ((P → Q) → Q). -Proof. intros P Q. - specialize n2_25 with (~P) Q. - intros n2_25a. - replace (~P∨Q) with (P→Q) in n2_25a. - apply n2_25a. - apply Impl1_01. -Qed. - -Theorem n2_27 : ∀ P Q : Prop, - P → ((P → Q) → Q). -Proof. intros P Q. - specialize n2_26 with P Q. - intros n2_26a. - replace (~P∨((P→Q)→Q)) with (P→(P→Q)→Q) in n2_26a. - apply n2_26a. - apply Impl1_01. -Qed. - -Theorem n2_3 : ∀ P Q R : Prop, - (P ∨ (Q ∨ R)) → (P ∨ (R ∨ Q)). -Proof. intros P Q R. - specialize Perm1_4 with Q R. - intros Perm1_4a. - specialize Sum1_6 with P (Q∨R) (R∨Q). - intros Sum1_6a. - MP Sum1_6a Perm1_4a. - apply Sum1_6a. -Qed. - -Theorem n2_31 : ∀ P Q R : Prop, - (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R). -Proof. intros P Q R. - specialize n2_3 with P Q R. - intros n2_3a. - specialize Assoc1_5 with P R Q. - intros Assoc1_5a. - specialize Perm1_4 with R (P∨Q). - intros Perm1_4a. - Syll Assoc1_5a Perm1_4a Sa. - Syll n2_3a Sa Sb. - apply Sb. -Qed. - -Theorem n2_32 : ∀ P Q R : Prop, - ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R)). -Proof. intros P Q R. - specialize Perm1_4 with (P∨Q) R. - intros Perm1_4a. - specialize Assoc1_5 with R P Q. - intros Assoc1_5a. - specialize n2_3 with P R Q. - intros n2_3a. - specialize Syll2_06 with ((P∨Q)∨R) (R∨P∨Q) (P∨R∨Q). - intros Syll2_06a. - MP Syll2_06a Perm1_4a. - MP Syll2_06a Assoc1_5a. - specialize Syll2_06 with ((P∨Q)∨R) (P∨R∨Q) (P∨Q∨R). - intros Syll2_06b. - MP Syll2_06b Syll2_06a. - MP Syll2_06b n2_3a. - apply Syll2_06b. -Qed. - -Axiom n2_33 : ∀ P Q R : Prop, - (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*) - -Theorem n2_36 : ∀ P Q R : Prop, - (Q → R) → ((P ∨ Q) → (R ∨ P)). -Proof. intros P Q R. - specialize Perm1_4 with P R. - intros Perm1_4a. - specialize Syll2_05 with (P∨Q) (P∨R) (R∨P). - intros Syll2_05a. - MP Syll2_05a Perm1_4a. - specialize Sum1_6 with P Q R. - intros Sum1_6a. - Syll Sum1_6a Syll2_05a S. - apply S. -Qed. - -Theorem n2_37 : ∀ P Q R : Prop, - (Q → R) → ((Q ∨ P) → (P ∨ R)). -Proof. intros P Q R. - specialize Perm1_4 with Q P. - intros Perm1_4a. - specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). - intros Syll2_06a. - MP Syll2_05a Perm1_4a. - specialize Sum1_6 with P Q R. - intros Sum1_6a. - Syll Sum1_6a Syll2_05a S. - apply S. -Qed. - -Theorem n2_38 : ∀ P Q R : Prop, - (Q → R) → ((Q ∨ P) → (R ∨ P)). -Proof. intros P Q R. - specialize Perm1_4 with P R. - intros Perm1_4a. - specialize Syll2_05 with (Q∨P) (P∨R) (R∨P). - intros Syll2_05a. - MP Syll2_05a Perm1_4a. - specialize Perm1_4 with Q P. - intros Perm1_4b. - specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). - intros Syll2_06a. - MP Syll2_06a Perm1_4b. - Syll Syll2_06a Syll2_05a H. - specialize Sum1_6 with P Q R. - intros Sum1_6a. - Syll Sum1_6a H S. - apply S. -Qed. - -Theorem n2_4 : ∀ P Q : Prop, - (P ∨ (P ∨ Q)) → (P ∨ Q). -Proof. intros P Q. - specialize n2_31 with P P Q. - intros n2_31a. - specialize Taut1_2 with P. - intros Taut1_2a. - specialize n2_38 with Q (P∨P) P. - intros n2_38a. - MP n2_38a Taut1_2a. - Syll n2_31a n2_38a S. - apply S. -Qed. - -Theorem n2_41 : ∀ P Q : Prop, - (Q ∨ (P ∨ Q)) → (P ∨ Q). -Proof. intros P Q. - specialize Assoc1_5 with Q P Q. - intros Assoc1_5a. - specialize Taut1_2 with Q. - intros Taut1_2a. - specialize Sum1_6 with P (Q∨Q) Q. - intros Sum1_6a. - MP Sum1_6a Taut1_2a. - Syll Assoc1_5a Sum1_6a S. - apply S. -Qed. - -Theorem n2_42 : ∀ P Q : Prop, - (~P ∨ (P → Q)) → (P → Q). -Proof. intros P Q. - specialize n2_4 with (~P) Q. - intros n2_4a. - replace (~P∨Q) with (P→Q) in n2_4a. - apply n2_4a. apply Impl1_01. -Qed. - -Theorem n2_43 : ∀ P Q : Prop, - (P → (P → Q)) → (P → Q). -Proof. intros P Q. - specialize n2_42 with P Q. - intros n2_42a. - replace (~P ∨ (P→Q)) with (P→(P→Q)) in n2_42a. - apply n2_42a. - apply Impl1_01. -Qed. - -Theorem n2_45 : ∀ P Q : Prop, - ~(P ∨ Q) → ~P. -Proof. intros P Q. - specialize n2_2 with P Q. - intros n2_2a. - specialize Trans2_16 with P (P∨Q). - intros Trans2_16a. - MP n2_2 Trans2_16a. - apply Trans2_16a. -Qed. - -Theorem n2_46 : ∀ P Q : Prop, - ~(P ∨ Q) → ~Q. -Proof. intros P Q. - specialize Add1_3 with P Q. - intros Add1_3a. - specialize Trans2_16 with Q (P∨Q). - intros Trans2_16a. - MP Add1_3a Trans2_16a. - apply Trans2_16a. -Qed. - -Theorem n2_47 : ∀ P Q : Prop, - ~(P ∨ Q) → (~P ∨ Q). -Proof. intros P Q. - specialize n2_45 with P Q. - intros n2_45a. - specialize n2_2 with (~P) Q. - intros n2_2a. - Syll n2_45a n2_2a S. - apply S. -Qed. - -Theorem n2_48 : ∀ P Q : Prop, - ~(P ∨ Q) → (P ∨ ~Q). -Proof. intros P Q. - specialize n2_46 with P Q. - intros n2_46a. - specialize Add1_3 with P (~Q). - intros Add1_3a. - Syll n2_46a Add1_3a S. - apply S. -Qed. - -Theorem n2_49 : ∀ P Q : Prop, - ~(P ∨ Q) → (~P ∨ ~Q). -Proof. intros P Q. - specialize n2_45 with P Q. - intros n2_45a. - specialize n2_2 with (~P) (~Q). - intros n2_2a. - Syll n2_45a n2_2a S. - apply S. -Qed. - -Theorem n2_5 : ∀ P Q : Prop, - ~(P → Q) → (~P → Q). -Proof. intros P Q. - specialize n2_47 with (~P) Q. - intros n2_47a. - replace (~P∨Q) with (P→Q) in n2_47a. - replace (~~P∨Q) with (~P→Q) in n2_47a. - apply n2_47a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_51 : ∀ P Q : Prop, - ~(P → Q) → (P → ~Q). -Proof. intros P Q. - specialize n2_48 with (~P) Q. - intros n2_48a. - replace (~P∨Q) with (P→Q) in n2_48a. - replace (~P∨~Q) with (P→~Q) in n2_48a. - apply n2_48a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_52 : ∀ P Q : Prop, - ~(P → Q) → (~P → ~Q). -Proof. intros P Q. - specialize n2_49 with (~P) Q. - intros n2_49a. - replace (~P∨Q) with (P→Q) in n2_49a. - replace (~~P∨~Q) with (~P→~Q) in n2_49a. - apply n2_49a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_521 : ∀ P Q : Prop, - ~(P→Q)→(Q→P). -Proof. intros P Q. - specialize n2_52 with P Q. - intros n2_52a. - specialize Trans2_17 with Q P. - intros Trans2_17a. - Syll n2_52a Trans2_17a S. - apply S. -Qed. - -Theorem n2_53 : ∀ P Q : Prop, - (P ∨ Q) → (~P → Q). -Proof. intros P Q. - specialize n2_12 with P. - intros n2_12a. - specialize n2_38 with Q P (~~P). - intros n2_38a. - MP n2_38a n2_12a. - replace (~~P∨Q) with (~P→Q) in n2_38a. - apply n2_38a. - apply Impl1_01. -Qed. - -Theorem n2_54 : ∀ P Q : Prop, - (~P → Q) → (P ∨ Q). -Proof. intros P Q. - specialize n2_14 with P. - intros n2_14a. - specialize n2_38 with Q (~~P) P. - intros n2_38a. - MP n2_38a n2_12a. - replace (~~P∨Q) with (~P→Q) in n2_38a. - apply n2_38a. - apply Impl1_01. -Qed. - -Theorem n2_55 : ∀ P Q : Prop, - ~P → ((P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_53 with P Q. - intros n2_53a. - specialize Comm2_04 with (P∨Q) (~P) Q. - intros Comm2_04a. - MP n2_53a Comm2_04a. - apply Comm2_04a. -Qed. - -Theorem n2_56 : ∀ P Q : Prop, - ~Q → ((P ∨ Q) → P). -Proof. intros P Q. - specialize n2_55 with Q P. - intros n2_55a. - specialize Perm1_4 with P Q. - intros Perm1_4a. - specialize Syll2_06 with (P∨Q) (Q∨P) P. - intros Syll2_06a. - MP Syll2_06a Perm1_4a. - Syll n2_55a Syll2_06a Sa. - apply Sa. - Qed. - -Theorem n2_6 : ∀ P Q : Prop, - (~P→Q) → ((P → Q) → Q). -Proof. intros P Q. - specialize n2_38 with Q (~P) Q. - intros n2_38a. - specialize Taut1_2 with Q. - intros Taut1_2a. - specialize Syll2_05 with (~P∨Q) (Q∨Q) Q. - intros Syll2_05a. - MP Syll2_05a Taut1_2a. - Syll n2_38a Syll2_05a S. - replace (~P∨Q) with (P→Q) in S. - apply S. - apply Impl1_01. -Qed. - -Theorem n2_61 : ∀ P Q : Prop, - (P → Q) → ((~P → Q) → Q). -Proof. intros P Q. - specialize n2_6 with P Q. - intros n2_6a. - specialize Comm2_04 with (~P→Q) (P→Q) Q. - intros Comm2_04a. - MP Comm2_04a n2_6a. - apply Comm2_04a. -Qed. - -Theorem n2_62 : ∀ P Q : Prop, - (P ∨ Q) → ((P → Q) → Q). -Proof. intros P Q. - specialize n2_53 with P Q. - intros n2_53a. - specialize n2_6 with P Q. - intros n2_6a. - Syll n2_53a n2_6a S. - apply S. -Qed. - -Theorem n2_621 : ∀ P Q : Prop, - (P → Q) → ((P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_62 with P Q. - intros n2_62a. - specialize Comm2_04 with (P ∨ Q) (P→Q) Q. - intros Comm2_04a. - MP Comm2_04a n2_62a. - apply Comm2_04a. -Qed. - -Theorem n2_63 : ∀ P Q : Prop, - (P ∨ Q) → ((~P ∨ Q) → Q). -Proof. intros P Q. - specialize n2_62 with P Q. - intros n2_62a. - replace (~P∨Q) with (P→Q). - apply n2_62a. - apply Impl1_01. -Qed. - -Theorem n2_64 : ∀ P Q : Prop, - (P ∨ Q) → ((P ∨ ~Q) → P). -Proof. intros P Q. - specialize n2_63 with Q P. - intros n2_63a. - specialize Perm1_4 with P Q. - intros Perm1_4a. - Syll n2_63a Perm1_4a Ha. - specialize Syll2_06 with (P∨~Q) (~Q∨P) P. - intros Syll2_06a. - specialize Perm1_4 with P (~Q). - intros Perm1_4b. - MP Syll2_05a Perm1_4b. - Syll Syll2_05a Ha S. - apply S. -Qed. - -Theorem n2_65 : ∀ P Q : Prop, - (P → Q) → ((P → ~Q) → ~P). -Proof. intros P Q. - specialize n2_64 with (~P) Q. - intros n2_64a. - replace (~P∨Q) with (P→Q) in n2_64a. - replace (~P∨~Q) with (P→~Q) in n2_64a. - apply n2_64a. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_67 : ∀ P Q : Prop, - ((P ∨ Q) → Q) → (P → Q). -Proof. intros P Q. - specialize n2_54 with P Q. - intros n2_54a. - specialize Syll2_06 with (~P→Q) (P∨Q) Q. - intros Syll2_06a. - MP Syll2_06a n2_54a. - specialize n2_24 with P Q. - intros n2_24. - specialize Syll2_06 with P (~P→Q) Q. - intros Syll2_06b. - MP Syll2_06b n2_24a. - Syll Syll2_06b Syll2_06a S. - apply S. -Qed. - -Theorem n2_68 : ∀ P Q : Prop, - ((P → Q) → Q) → (P ∨ Q). -Proof. intros P Q. - specialize n2_67 with (~P) Q. - intros n2_67a. - replace (~P∨Q) with (P→Q) in n2_67a. - specialize n2_54 with P Q. - intros n2_54a. - Syll n2_67a n2_54a S. - apply S. - apply Impl1_01. -Qed. - -Theorem n2_69 : ∀ P Q : Prop, - ((P → Q) → Q) → ((Q → P) → P). -Proof. intros P Q. - specialize n2_68 with P Q. - intros n2_68a. - specialize Perm1_4 with P Q. - intros Perm1_4a. - Syll n2_68a Perm1_4a Sa. - specialize n2_62 with Q P. - intros n2_62a. - Syll Sa n2_62a Sb. - apply Sb. -Qed. - -Theorem n2_73 : ∀ P Q R : Prop, - (P → Q) → (((P ∨ Q) ∨ R) → (Q ∨ R)). -Proof. intros P Q R. - specialize n2_621 with P Q. - intros n2_621a. - specialize n2_38 with R (P∨Q) Q. - intros n2_38a. - Syll n2_621a n2_38a S. - apply S. -Qed. - -Theorem n2_74 : ∀ P Q R : Prop, - (Q → P) → ((P ∨ Q) ∨ R) → (P ∨ R). -Proof. intros P Q R. - specialize n2_73 with Q P R. - intros n2_73a. - specialize Assoc1_5 with P Q R. - intros Assoc1_5a. - specialize n2_31 with Q P R. - intros n2_31a. (*not cited explicitly!*) - Syll Assoc1_5a n2_31a Sa. - specialize n2_32 with P Q R. - intros n2_32a. (*not cited explicitly!*) - Syll n2_32a Sa Sb. - specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). - intros Syll2_06a. - MP Syll2_06a Sb. - Syll n2_73a Syll2_05a H. - apply H. -Qed. - -Theorem n2_75 : ∀ P Q R : Prop, - (P ∨ Q) → ((P ∨ (Q → R)) → (P ∨ R)). -Proof. intros P Q R. - specialize n2_74 with P (~Q) R. - intros n2_74a. - specialize n2_53 with Q P. - intros n2_53a. - Syll n2_53a n2_74a Sa. - specialize n2_31 with P (~Q) R. - intros n2_31a. - specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R) (P∨R). - intros Syll2_06a. - MP Syll2_06a n2_31a. - Syll Sa Syll2_06a Sb. - specialize Perm1_4 with P Q. - intros Perm1_4a. (*not cited!*) - Syll Perm1_4a Sb Sc. - replace (~Q∨R) with (Q→R) in Sc. - apply Sc. - apply Impl1_01. -Qed. - -Theorem n2_76 : ∀ P Q R : Prop, - (P ∨ (Q → R)) → ((P ∨ Q) → (P ∨ R)). -Proof. intros P Q R. - specialize n2_75 with P Q R. - intros n2_75a. - specialize Comm2_04 with (P∨Q) (P∨(Q→R)) (P∨R). - intros Comm2_04a. - apply Comm2_04a. - apply n2_75a. -Qed. - -Theorem n2_77 : ∀ P Q R : Prop, - (P → (Q → R)) → ((P → Q) → (P → R)). -Proof. intros P Q R. - specialize n2_76 with (~P) Q R. - intros n2_76a. - replace (~P∨(Q→R)) with (P→Q→R) in n2_76a. - replace (~P∨Q) with (P→Q) in n2_76a. - replace (~P∨R) with (P→R) in n2_76a. - apply n2_76a. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_8 : ∀ Q R S : Prop, - (Q ∨ R) → ((~R ∨ S) → (Q ∨ S)). -Proof. intros Q R S. - specialize n2_53 with R Q. - intros n2_53a. - specialize Perm1_4 with Q R. - intros Perm1_4a. - Syll Perm1_4a n2_53a Ha. - specialize n2_38 with S (~R) Q. - intros n2_38a. - Syll H n2_38a Hb. - apply Hb. -Qed. - -Theorem n2_81 : ∀ P Q R S : Prop, - (Q → (R → S)) → ((P ∨ Q) → ((P ∨ R) → (P ∨ S))). -Proof. intros P Q R S. - specialize Sum1_6 with P Q (R→S). - intros Sum1_6a. - specialize n2_76 with P R S. - intros n2_76a. - specialize Syll2_05 with (P∨Q) (P∨(R→S)) ((P∨R)→(P∨S)). - intros Syll2_05a. - MP Syll2_05a n2_76a. - Syll Sum1_6a Syll2_05a H. - apply H. -Qed. - -Theorem n2_82 : ∀ P Q R S : Prop, - (P ∨ Q ∨ R)→((P ∨ ~R ∨ S)→(P ∨ Q ∨ S)). -Proof. intros P Q R S. - specialize n2_8 with Q R S. - intros n2_8a. - specialize n2_81 with P (Q∨R) (~R∨S) (Q∨S). - intros n2_81a. - MP n2_81a n2_8a. - apply n2_81a. -Qed. - -Theorem n2_83 : ∀ P Q R S : Prop, - (P→(Q→R))→((P→(R→S))→(P→(Q→S))). -Proof. intros P Q R S. - specialize n2_82 with (~P) (~Q) R S. - intros n2_82a. - replace (~Q∨R) with (Q→R) in n2_82a. - replace (~P∨(Q→R)) with (P→Q→R) in n2_82a. - replace (~R∨S) with (R→S) in n2_82a. - replace (~P∨(R→S)) with (P→R→S) in n2_82a. - replace (~Q∨S) with (Q→S) in n2_82a. - replace (~Q∨S) with (Q→S) in n2_82a. - replace (~P∨(Q→S)) with (P→Q→S) in n2_82a. - apply n2_82a. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -Theorem n2_85 : ∀ P Q R : Prop, - ((P ∨ Q) → (P ∨ R)) → (P ∨ (Q → R)). -Proof. intros P Q R. - specialize Add1_3 with P Q. - intros Add1_3a. - specialize Syll2_06 with Q (P∨Q) R. - intros Syll2_06a. - MP Syll2_06a Add1_3a. - specialize n2_55 with P R. - intros n2_55a. - specialize Syll2_05 with (P∨Q) (P∨R) R. - intros Syll2_05a. - Syll n2_55a Syll2_05a Ha. - specialize n2_83 with (~P) ((P∨Q)→(P∨R)) ((P∨Q)→R) (Q→R). - intros n2_83a. - MP n2_83a Ha. - specialize Comm2_04 with (~P) (P∨Q→P∨R) (Q→R). - intros Comm2_04a. - Syll Ha Comm2_04a Hb. - specialize n2_54 with P (Q→R). - intros n2_54a. - specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). - intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*) - MP Syll2_06a n2_02a. - MP Hb n2_02a. - Syll Hb n2_54a Hc. - apply Hc. -Qed. - -Theorem n2_86 : ∀ P Q R : Prop, - ((P → Q) → (P → R)) → (P → (Q → R)). -Proof. intros P Q R. - specialize n2_85 with (~P) Q R. - intros n2_85a. - replace (~P∨Q) with (P→Q) in n2_85a. - replace (~P∨R) with (P→R) in n2_85a. - replace (~P∨(Q→R)) with (P→Q→R) in n2_85a. - apply n2_85a. - apply Impl1_01. - apply Impl1_01. - apply Impl1_01. -Qed. - -End No2.
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