diff options
| author | Landon D. C. Elkind | 2021-02-03 16:10:08 -0700 |
|---|---|---|
| committer | Landon D. C. Elkind | 2021-02-03 16:10:08 -0700 |
| commit | a32839d6c14a7d9e689e52dfe321091c8fafb517 (patch) | |
| tree | cb1e8c3a358f78d26491023761fe5bdd3e58a44b /PL.v | |
| parent | cbf59b2290f89979e500faacaccaa4d171469c37 (diff) | |
Deleted superseded files, fixed typos
Diffstat (limited to 'PL.v')
| -rw-r--r-- | PL.v | 1503 |
1 files changed, 996 insertions, 507 deletions
@@ -5,13 +5,22 @@ Import Unicode.Utf8. (*We first give the axioms of Principia for the propositional calculus in *1.*) +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.*) + 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*) + (*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*) + P ∨ P → P. (*Tautology*) Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) @@ -20,13 +29,12 @@ Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) Axiom Assoc1_5 : ∀ P Q R : Prop, - P ∨ (Q ∨ R) → Q ∨ (P ∨ R). + P ∨ (Q ∨ R) → Q ∨ (P ∨ R). (*Association*) 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.*) + (Q → R) → (P ∨ Q → P ∨ R). (*Summation*) + +(*These are all the propositional axioms of Principia.*) End No1. @@ -44,7 +52,7 @@ Proof. intros P. apply Impl1_01. Qed. -Theorem n2_02 : ∀ P Q : Prop, +Theorem Simp2_02 : ∀ P Q : Prop, Q → (P → Q). Proof. intros P Q. specialize Add1_3 with (~P) Q. @@ -97,7 +105,8 @@ Proof. intros P Q 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))). + 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. @@ -140,7 +149,7 @@ Proof. intros P. specialize Perm1_4 with (~P) P. intros Perm1_4. specialize n2_1 with P. - intros Abs2_01. + intros n2_1. apply Perm1_4. apply n2_1. Qed. @@ -151,7 +160,7 @@ Proof. intros P. specialize n2_11 with (~P). intros n2_11. rewrite Impl1_01. - assumption. + apply n2_11. Qed. Theorem n2_13 : ∀ P : Prop, @@ -163,6 +172,8 @@ Proof. intros P. intros n2_12. apply Sum1_6. apply n2_12. + specialize n2_11 with P. + intros n2_11. apply n2_11. Qed. @@ -195,6 +206,8 @@ Proof. intros P Q. intros Syll2_05d. apply Syll2_05d. apply Syll2_05b. + specialize n2_14 with P. + intros n2_14. apply n2_14. apply Syll2_05c. apply n2_03. @@ -369,8 +382,10 @@ Proof. intros P Q R. 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.*) +Axiom Abb2_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.*) Theorem n2_36 : ∀ P Q R : Prop, (Q → R) → ((P ∨ Q) → (R ∨ P)). @@ -393,10 +408,10 @@ Proof. intros P Q R. intros Perm1_4a. specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). intros Syll2_06a. - MP Syll2_05a Perm1_4a. + MP Syll2_06a Perm1_4a. specialize Sum1_6 with P Q R. intros Sum1_6a. - Syll Sum1_6a Syll2_05a S. + Syll Sum1_6a Syll2_06a S. apply S. Qed. @@ -691,8 +706,8 @@ Proof. intros P Q. intros Syll2_06a. specialize Perm1_4 with P (~Q). intros Perm1_4b. - MP Syll2_05a Perm1_4b. - Syll Syll2_05a Ha S. + MP Syll2_06a Perm1_4b. + Syll Syll2_06a Ha S. apply S. Qed. @@ -771,10 +786,10 @@ Proof. intros P Q R. specialize Assoc1_5 with P Q R. intros Assoc1_5a. specialize n2_31 with Q P R. - intros n2_31a. (*not cited explicitly!*) + intros n2_31a. (*not cited*) Syll Assoc1_5a n2_31a Sa. specialize n2_32 with P Q R. - intros n2_32a. (*not cited explicitly!*) + intros n2_32a. (*not cited*) Syll n2_32a Sa Sb. specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). intros Syll2_06a. @@ -793,12 +808,12 @@ Proof. intros P Q R. 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). + 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!*) + intros Perm1_4a. (*not cited*) Syll Perm1_4a Sb Sc. replace (~Q∨R) with (Q→R) in Sc. apply Sc. @@ -912,10 +927,11 @@ Proof. intros P Q R. 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. + specialize Simp2_02 with (~P) ((P∨Q→R)→(Q→R)). + intros Simp2_02a. (*Not cited*) + (*Greg's suggestion per the BRS list on June 25, 2017.*) + MP Syll2_06a Simp2_02a. + MP Hb Simp2_02a. Syll Hb n2_54a Hc. apply Hc. Qed. @@ -945,9 +961,9 @@ Axiom Prod3_01 : ∀ P Q : Prop, (P ∧ Q) = ~(~P ∨ ~Q). Axiom Abb3_02 : ∀ P Q R : Prop, - (P→Q→R)=(P→Q)∧(Q→R). + (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.*) +Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧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. @@ -960,6 +976,14 @@ Proof. intros P Q. apply Impl1_01. apply Prod3_01. Qed. +(*3.03 is permits the inference from the theoremhood + of P and that of Q to the theoremhood of P and Q. So:*) + +Ltac Conj H1 H2 := + match goal with + | [ H1 : ?P, H2 : ?Q |- _ ] => + assert (P ∧ Q) +end. Theorem n3_1 : ∀ P Q : Prop, (P ∧ Q) → ~(~P ∨ ~Q). @@ -998,7 +1022,7 @@ Proof. intros P Q. intros n3_11a. specialize Trans2_15 with (~P∨~Q) (P∧Q). intros Trans2_15a. - MP Trans2_16a n3_11a. + MP Trans2_15a n3_11a. apply Trans2_15a. Qed. @@ -1073,16 +1097,16 @@ 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 Simp2_02 with Q P. + intros Simp2_02a. + replace (P→(Q→P)) with (~P∨(Q→P)) in Simp2_02a. + replace (Q→P) with (~Q∨P) in Simp2_02a. specialize n2_31 with (~P) (~Q) P. intros n2_31a. - MP n2_31a n2_02a. + MP n2_31a Simp2_02a. specialize n2_53 with (~P∨~Q) P. intros n2_53a. - MP n2_53a n2_02a. + MP n2_53a Simp2_02a. replace (~(~P∨~Q)) with (P∧Q) in n2_53a. apply n2_53a. apply Prod3_01. @@ -1110,9 +1134,9 @@ Theorem Exp3_3 : ∀ P Q R : Prop, 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. + replace (P → ~Q) with (~P ∨ ~Q) in Comm2_04a. Syll Trans2_15a Comm2_04a Sa. specialize Trans2_17 with Q R. intros Trans2_17a. @@ -1120,13 +1144,15 @@ Proof. intros P Q R. intros Syll2_05a. MP Syll2_05a Trans2_17a. Syll Sa Syll2_05a Sb. - replace (~(~P∨~Q)) with (P∧Q) in 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. +(*The proof sketch cites n2_08, but + we did not seem to need it.*) Theorem Imp3_31 : ∀ P Q R : Prop, (P → (Q → R)) → (P ∧ Q) → R. @@ -1144,6 +1170,8 @@ Proof. intros P Q R. apply Impl1_01. apply Impl1_01. Qed. +(*The proof sketch cites n2_08, but + we did not seem to need it.*) Theorem Syll3_33 : ∀ P Q R : Prop, ((P → Q) ∧ (Q → R)) → (P → R). @@ -1230,7 +1258,7 @@ Proof. intros P Q R. intros Simp3_27a. specialize Syll2_06 with (P∧Q) Q R. intros Syll2_06a. - MP Syll2_05a Simp3_27a. + MP Syll2_06a Simp3_27a. apply Syll2_06a. Qed. @@ -1376,10 +1404,13 @@ 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.*) + (P ↔ Q) = ((P → Q) ∧ (Q → P)). + +Axiom Abb4_02 : ∀ P Q R : Prop, + (P ↔ Q ↔ R) = ((P ↔ Q) ∧ (Q ↔ R)). Axiom EqBi : ∀ P Q : Prop, - (P=Q) ↔ (P↔Q). + (P = Q) ↔ (P ↔ Q). Ltac Equiv H1 := match goal with @@ -1387,12 +1418,6 @@ Ltac Equiv H1 := 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. @@ -1428,7 +1453,8 @@ Proof. intros P Q. 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. + 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. @@ -1443,7 +1469,8 @@ Proof. intros P Q. 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. + 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. @@ -1484,7 +1511,8 @@ Theorem n4_12 : ∀ P Q : Prop, 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. + 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. @@ -1557,11 +1585,13 @@ Theorem n4_15 : ∀ P Q R : Prop, 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). + 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). + 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. @@ -1570,9 +1600,16 @@ Theorem n4_15 : ∀ P Q R : Prop, intros Syll2_06b. MP Syll2_06b n3_22b. Syll Syll2_06b Simp3_27a Sb. + clear n4_14a. clear n3_22a. clear Syll2_06a. + clear n4_13a. clear Simp3_26a. clear n3_22b. + clear Simp3_27a. clear Syll2_06b. + Conj Sa Sb. split. apply Sa. apply Sb. + Equiv H. + apply H. + apply Equiv4_01. apply EqBi. apply n4_13a. Qed. @@ -1596,11 +1633,7 @@ Theorem n4_21 : ∀ P Q : Prop, 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. @@ -1608,11 +1641,15 @@ Theorem n4_21 : ∀ P Q : Prop, 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. + Equiv H. + apply H. + apply Equiv4_01. + apply Equiv4_01. + apply Equiv4_01. + apply Equiv4_01. + apply Equiv4_01. Qed. Theorem n4_22 : ∀ P Q R : Prop, @@ -1649,7 +1686,10 @@ Proof. intros P Q R. 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. + 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. @@ -1741,11 +1781,16 @@ Qed. 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. + 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. @@ -1753,15 +1798,26 @@ Qed. reflexivity. rewrite Impl1_01. reflexivity. - replace (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) with ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))). + 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.*) + specialize n4_13 with (Q ∧ R). + intros n4_13a. + apply n4_13a. + 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). @@ -1770,12 +1826,23 @@ Theorem n4_33 : ∀ P Q R : Prop, intros n2_31a. specialize n2_32 with P Q R. intros n2_32a. - split. apply n2_31a. + Conj n2_31a n2_32a. + split. + apply n2_31a. apply n2_32a. + Equiv H. + apply H. + apply Equiv4_01. 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.*) +Axiom Abb4_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)). @@ -1788,11 +1855,13 @@ Proof. intros P Q R. split. apply Fact3_45a. apply Fact3_45b. - specialize n3_47 with (P→Q) (Q→P) (P ∧ R → Q ∧ R) (Q ∧ R → P ∧ R). + 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. + 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. @@ -1809,18 +1878,24 @@ Proof. intros P Q R. split. apply Sum1_6a. apply Sum1_6b. - specialize n3_47 with (P → Q) (Q → P) (R ∨ P → R ∨ Q) (R ∨ Q → R ∨ P). + 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 → 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. + specialize n4_31 with Q R. + intros n4_31a. + apply n4_31a. apply EqBi. - apply n4_31. + specialize n4_31 with P R. + intros n4_31b. + apply n4_31b. apply Equiv4_01. apply Equiv4_01. Qed. @@ -1836,15 +1911,18 @@ Proof. intros P Q R S. 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). + 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. + 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. + 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). @@ -1854,23 +1932,28 @@ Proof. intros P Q R S. apply n3_22a. apply n3_22b. Equiv H0. - replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47c. + 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. + 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) ∧ (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. + 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)). + replace ((R → P) ∧ (Q → S) ∧ (S → Q)) with + (((R → P) ∧ (Q → S)) ∧ (S → Q)). reflexivity. apply EqBi. apply n4_32c. @@ -1881,7 +1964,8 @@ Proof. intros P Q R S. 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)). + replace ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) with + (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)). reflexivity. apply EqBi. apply n4_32a. @@ -1898,16 +1982,20 @@ Proof. intros P Q R S. 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). + 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. + 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. + 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. + 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). @@ -1917,18 +2005,22 @@ Proof. intros P Q R S. apply n3_22a. apply n3_22b. Equiv H0. - replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47a. + 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) ∧ (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 (((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)). + replace ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) with + (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)). reflexivity. apply EqBi. apply n4_32d. @@ -1940,7 +2032,8 @@ Proof. intros P Q R S. apply EqBi. apply H0. apply Equiv4_01. - replace ((P → R) ∧ (Q → S) ∧ (R → P)) with (((P → R) ∧ (Q → S)) ∧ (R → P)). + replace ((P → R) ∧ (Q → S) ∧ (R → P)) with + (((P → R) ∧ (Q → S)) ∧ (R → P)). reflexivity. apply EqBi. apply n4_32b. @@ -1999,14 +2092,15 @@ Proof. intros P Q R. 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. + 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. + apply Imp3_31a. + apply Comp3_43b. + Equiv H0. + apply H0. + apply Equiv4_01. Qed. Theorem n4_41 : ∀ P Q R : Prop, @@ -2023,7 +2117,7 @@ Proof. intros P Q R. intros Sum1_6b. MP Simp3_27a Sum1_6b. clear Simp3_26a. clear Simp3_27a. - Conj Sum1_6a Sum1_6b. + Conj Sum1_6a Sum1_6a. split. apply Sum1_6a. apply Sum1_6b. @@ -2047,9 +2141,16 @@ Proof. intros P Q R. specialize n2_54 with P (Q∧R). intros n2_54a. Syll Sa n2_54a Sb. + clear Sum1_6a. clear Sum1_6b. clear H. clear n2_53a. + clear n2_53b. clear H0. clear n3_47a. clear Sa. + clear Comp3_43b. clear n2_54a. + Conj Comp3_43a Sb. split. apply Comp3_43a. apply Sb. + Equiv H. + apply H. + apply Equiv4_01. Qed. Theorem n4_42 : ∀ P Q : Prop, @@ -2108,7 +2209,8 @@ Proof. intros P Q. 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. + 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. @@ -2117,7 +2219,9 @@ Proof. intros P Q. apply H. apply Equiv4_01. apply EqBi. - apply n4_13. + specialize n4_13 with P. + intros n4_13a. + apply n4_13a. Qed. Theorem n4_44 : ∀ P Q : Prop, @@ -2155,13 +2259,21 @@ Theorem n4_45 : ∀ P Q : Prop, replace (P ∧ P) with P in n2_2a. specialize Simp3_26 with P (P ∨ Q). intros Simp3_26a. + Conj n2_2a Simp3_26a. split. apply n2_2a. apply Simp3_26a. + Equiv H. + apply H. + apply Equiv4_01. + specialize n4_24 with P. + intros n4_24a. apply EqBi. - apply n4_24. + apply n4_24a. + specialize n4_4 with P P Q. + intros n4_4a. apply EqBi. - apply n4_4. + apply n4_4a. Qed. Theorem n4_5 : ∀ P Q : Prop, @@ -2182,14 +2294,19 @@ Theorem n4_51 : ∀ P Q : Prop, 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. + 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. + specialize n4_21 with (~(P∧Q)) (~P ∨ ~Q). + intros n4_21b. + apply n4_21b. apply EqBi. apply EqBi. Qed. @@ -2214,9 +2331,12 @@ Theorem n4_53 : ∀ P Q : Prop, 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. + 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. @@ -2246,14 +2366,17 @@ Theorem n4_55 : ∀ P Q : Prop, 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. + 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. + 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. @@ -2268,7 +2391,9 @@ Theorem n4_56 : ∀ P Q : Prop, replace (~~Q) with Q in n4_54a. apply n4_54a. apply EqBi. - apply n4_13. + specialize n4_13 with Q. + intros n4_13a. + apply n4_13a. Qed. Theorem n4_57 : ∀ P Q : Prop, @@ -2278,18 +2403,22 @@ Theorem n4_57 : ∀ P Q : Prop, 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. + 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. + 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)). + specialize n4_21 with + (P ∨ Q ↔ ~(~P ∧ ~Q)) (~P ∧ ~Q ↔ ~(P ∨ Q)). intros n4_21b. apply n4_21b. Qed. @@ -2312,16 +2441,21 @@ Theorem n4_61 : ∀ P Q : Prop, 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) ↔ ~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. + 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. + specialize n4_21 with (~(P→Q)↔~(~P∨Q)) + ((P→Q)↔(~P∨Q)). + intros n4_21a. + apply n4_21a. Qed. Theorem n4_62 : ∀ P Q : Prop, @@ -2342,12 +2476,15 @@ Theorem n4_63 : ∀ P Q : Prop, 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. + 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. + 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). + specialize n4_21 with + (~(P → ~Q) ↔ P ∧ Q) ((P → ~Q) ↔ ~P ∨ ~Q). intros n4_21a. apply EqBi. apply n4_21a. @@ -2380,8 +2517,10 @@ Theorem n4_65 : ∀ P Q : Prop, 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)↔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. @@ -2389,7 +2528,10 @@ Theorem n4_65 : ∀ P Q : Prop, apply EqBi. apply Trans4_11a. apply EqBi. - apply n4_21. + specialize n4_21 with (~(~P → Q)↔~(P ∨ Q)) + ((~P → Q)↔(P ∨ Q)). + intros n4_21a. + apply n4_21a. Qed. Theorem n4_66 : ∀ P Q : Prop, @@ -2407,18 +2549,23 @@ Theorem n4_67 : ∀ P Q : Prop, 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. + 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. + 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. + specialize n4_21 with (~(~P → ~Q)↔~(P ∨ ~Q)) + ((~P → ~Q)↔(P ∨ ~Q)). + intros n4_21a. + apply n4_21a. Qed. Theorem n4_7 : ∀ P Q : Prop, @@ -2426,7 +2573,8 @@ Theorem n4_7 : ∀ P Q : Prop, 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). + specialize Exp3_3 with + (P → P) (P → Q) (P → P ∧ Q). intros Exp3_3a. MP Exp3_3a Comp3_43a. specialize n2_08 with P. @@ -2436,7 +2584,7 @@ Theorem n4_7 : ∀ P Q : Prop, intros Simp3_27a. specialize Syll2_05 with P (P ∧ Q) Q. intros Syll2_05a. - MP Syll2_05a Simp3_26a. + MP Syll2_05a Simp3_27a. clear n2_08a. clear Comp3_43a. clear Simp3_27a. Conj Syll2_05a Exp3_3a. split. @@ -2454,13 +2602,15 @@ Theorem n4_71 : ∀ P Q : Prop, 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. + 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. + 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. @@ -2491,7 +2641,8 @@ Theorem n4_72 : ∀ P Q : Prop, split. apply Trans4_1a. apply n4_71a. - specialize n4_22 with (P→Q) (~Q→~P) (~Q↔~Q ∧ ~ P). + 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). @@ -2500,16 +2651,18 @@ Theorem n4_72 : ∀ P Q : Prop, split. apply n4_22a. apply n4_21a. - specialize n4_22 with (P→Q) (~Q ↔ ~Q ∧ ~P) (~Q ∧ ~P ↔ ~Q). + 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). + 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)). + 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. @@ -2522,24 +2675,29 @@ Theorem n4_72 : ∀ P Q : Prop, apply EqBi. apply n4_31a. apply EqBi. - replace (~(~Q ∧ ~P) ↔ Q ∨ P) with (Q ∨ P ↔~(~Q ∧ ~P)) in n4_57a. + replace (~(~Q ∧ ~P) ↔ Q ∨ P) with + (Q ∨ P ↔~(~Q ∧ ~P)) in n4_57a. apply n4_57a. apply EqBi. - apply n4_21. + specialize n4_21 with (Q ∨ P) (~(~Q ∧ ~P)). + intros n4_21b. + apply n4_21b. Qed. Theorem n4_73 : ∀ P Q : Prop, Q → (P ↔ (P ∧ Q)). Proof. intros P Q. - specialize n2_02 with P Q. - intros n2_02a. + specialize Simp2_02 with P Q. + intros Simp2_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). + 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. + Syll Simp2_02a Simp3_26a Sa. apply Sa. apply Equiv4_01. Qed. @@ -2554,10 +2712,13 @@ Theorem n4_74 : ∀ P Q : Prop, 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. + replace ((P → Q) ↔ (Q ↔ P ∨ Q)) with + ((Q ↔ P ∨ Q) ↔ (P → Q)) in n4_72a. apply n4_72a. apply EqBi. - apply n4_21. + specialize n4_21 with (Q↔(P ∨ Q)) (P → Q). + intros n4_21a. + apply n4_21a. Qed. Theorem n4_76 : ∀ P Q R : Prop, @@ -2568,10 +2729,13 @@ Theorem n4_76 : ∀ P Q R : Prop, 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. + 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. + specialize n4_21 with ((P → Q) ∧ (P → R)) (P → Q ∧ R). + intros n4_21a. + apply n4_21a. apply Impl1_01. apply Impl1_01. apply Impl1_01. @@ -2582,56 +2746,110 @@ Theorem n4_77 : ∀ P Q R : Prop, Proof. intros P Q R. specialize n3_44 with P Q R. intros n3_44a. + specialize n2_08 with (Q ∨ R → P). + intros n2_08a. (*Not cited*) + replace ((Q ∨ R → P) → (Q ∨ R → P)) with + ((Q ∨ R → P) → (~(Q ∨ R) ∨ P)) in n2_08a. + replace (~(Q ∨ R)) with (~Q ∧ ~R) in n2_08a. + replace (~Q ∧ ~R ∨ P) with + ((~Q ∨ P) ∧ (~R ∨ P)) in n2_08a. + replace (~Q ∨ P) with (Q → P) in n2_08a. + replace (~R ∨ P) with (R → P) in n2_08a. + Conj n3_44a n2_08a. 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).*) + apply n2_08a. + Equiv H. + apply H. + apply Equiv4_01. + apply Impl1_01. + apply Impl1_01. + specialize n4_41 with P (~Q) (~R). + intros n4_41a. (*Not cited*) + replace (P ∨ ~Q) with + (~Q ∨ P) in n4_41a. + replace (P ∨ ~R) with + (~R ∨ P) in n4_41a. + replace (P ∨ ~Q ∧ ~R) with (~Q ∧ ~R ∨ P) in n4_41a. + replace (~Q ∧ ~R ∨ P ↔ (~Q ∨ P) ∧ (~R ∨ P)) with + ((~Q ∨ P) ∧ (~R ∨ P) ↔ ~Q ∧ ~R ∨ P) in n4_41a. + apply EqBi. + apply n4_41a. + apply EqBi. + specialize n4_21 + with ((~Q ∨ P) ∧ (~R ∨ P)) (~Q ∧ ~R ∨ P). + intros n4_21a. (*Not cited*) + apply n4_21a. + specialize n4_31 with (~Q ∧ ~R) P. + intros n4_31a. (*Not cited*) + apply EqBi. + apply n4_31a. + specialize n4_31 with (~R) P. + intros n4_31b. (*Not cited*) + apply EqBi. + apply n4_31b. + specialize n4_31 with (~Q) P. + intros n4_31c. (*Not cited*) + apply EqBi. + apply n4_31c. (*Not cited*) + apply EqBi. + specialize n4_56 with Q R. + intros n4_56a. (*Not cited*) + apply n4_56a. + replace (~(Q ∨ R) ∨ P) with (Q∨R→P). + reflexivity. + apply Impl1_01. (*Not cited*) + Qed. + (*Proof sketch cites Add1_3 + n2_2.*) Theorem n4_78 : ∀ P Q R : Prop, ((P → Q) ∨ (P → R)) ↔ (P → (Q ∨ R)). Proof. intros P Q R. specialize n4_2 with ((P→Q) ∨ (P → R)). intros n4_2a. - replace (((P → Q) ∨ (P → R))↔((P → Q) ∨ (P → R))) with (((P → Q) ∨ (P → R))↔((~P ∨ Q) ∨ ~P ∨ R)) in n4_2a. + 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. + 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. + 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. + 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). + 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. + 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. + apply Abb2_33. replace ((~P ∨ ~P) ∨ Q) with (~P ∨ ~P ∨ Q). reflexivity. - apply n2_33. - replace ((~P ∨ ~P ∨ Q) ∨ R) with (~P ∨ (~P ∨ Q) ∨ R). + apply Abb2_33. + replace ((~P ∨ ~P ∨ Q) ∨ R) with + (~P ∨ (~P ∨ Q) ∨ R). reflexivity. apply EqBi. apply n4_33b. @@ -2639,7 +2857,7 @@ Theorem n4_78 : ∀ P Q R : Prop, apply n4_37a. replace ((Q ∨ ~P) ∨ R) with (Q ∨ ~P ∨ R). reflexivity. - apply n2_33. + apply Abb2_33. apply EqBi. apply n4_33a. replace (~P ∨ Q) with (P→Q). @@ -2660,41 +2878,52 @@ Theorem n4_79 : ∀ P Q R : Prop, split. apply Trans4_1a. apply Trans4_1b. - specialize n4_39 with (Q→P) (R→P) (~P→~Q) (~P→~R). + 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. + replace ((~P → ~Q) ∨ (~P → ~R)) with + (~P → ~Q ∨ ~R) in n4_39a. + specialize Trans4_1 with (~P) (~Q ∨ ~R). + intros Trans4_1c. + replace (~P → ~Q ∨ ~R) with + (~(~Q ∨ ~R) → ~~P) in n4_39a. + replace (~(~Q ∨ ~R)) with + (Q ∧ R) in n4_39a. + replace (~~P) with P in n4_39a. apply n4_39a. + specialize n4_13 with P. + intros n4_13a. + apply EqBi. + apply n4_13a. apply Prod3_01. - replace (~(~Q ∨ ~R) → P) with (~P → ~Q ∨ ~R). + 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)). + apply Trans4_1c. + replace (~P → ~Q ∨ ~R) with + ((~P → ~Q) ∨ (~P → ~R)). reflexivity. apply EqBi. apply n4_78a. Qed. + (*The proof sketch cites Trans2_15, but we did + not need Trans2_15 as a lemma here.*) Theorem n4_8 : ∀ P : Prop, (P → ~P) ↔ ~P. Proof. intros P. specialize Abs2_01 with P. intros Abs2_01a. - specialize n2_02 with P (~P). - intros n2_02a. - Conj Abs2_01a n2_02a. + specialize Simp2_02 with P (~P). + intros Simp2_02a. + Conj Abs2_01a Simp2_02a. split. apply Abs2_01a. - apply n2_02a. + apply Simp2_02a. Equiv H. apply H. apply Equiv4_01. @@ -2705,12 +2934,12 @@ Theorem n4_81 : ∀ P : Prop, 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. + specialize Simp2_02 with (~P) P. + intros Simp2_02a. + Conj n2_18a Simp2_02a. split. apply n2_18a. - apply n2_02a. + apply Simp2_02a. Equiv H. apply H. apply Equiv4_01. @@ -2754,18 +2983,18 @@ Theorem n4_83 : ∀ P Q : Prop, 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. + specialize Simp2_02 with P Q. + intros Simp2_02a. + specialize Simp2_02 with (~P) Q. + intros Simp2_02b. + Conj Simp2_02a Simp2_02b. split. - apply n2_02a. - apply n2_02b. + apply Simp2_02a. + apply Simp2_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 n2_61a. clear Simp2_02a. clear Simp2_02b. clear H. Conj Imp3_31a Comp3_43a. split. @@ -2787,15 +3016,21 @@ Theorem n4_84 : ∀ P Q R : Prop, split. apply Syll2_06a. apply Syll2_06b. - specialize n3_47 with (P→Q) (Q→P) ((Q→R)→P→R) ((P→R)→Q→R). + 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. + 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. + specialize n4_21 with (P→R) (Q→R). + intros n4_21a. + apply n4_21a. apply Equiv4_01. apply Equiv4_01. Qed. @@ -2811,87 +3046,55 @@ Theorem n4_85 : ∀ P Q R : Prop, split. apply Syll2_05a. apply Syll2_05b. - specialize n3_47 with (P→Q) (Q→P) ((R→P)→R→Q) ((R→Q)→R→P). + 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. + 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)). + (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)). Proof. intros P Q R. - split. - split. - replace (P↔Q) with (Q↔P) in H. - Conj H H0. - split. - apply H. - apply H0. specialize n4_22 with Q P R. intros n4_22a. - MP n4_22a H1. - replace (Q ↔ R) with ((Q→R) ∧ (R→Q)) in n4_22a. - specialize Simp3_26 with (Q→R) (R→Q). - intros Simp3_26a. - MP Simp3_26a n4_22a. - apply Simp3_26a. - apply Equiv4_01. - apply EqBi. - apply n4_21. - replace (P↔R) with (R↔P) in H0. - Conj H0 H. + specialize Exp3_3 with (Q↔P) (P↔R) (Q↔R). + intros Exp3_3a. (*Not cited*) + MP Exp3_3a n4_22a. + specialize n4_22 with P Q R. + intros n4_22b. + specialize Exp3_3 with (P↔Q) (Q↔R) (P↔R). + intros Exp3_3b. + MP Exp3_3b n4_22b. + clear n4_22a. + clear n4_22b. + replace (Q↔P) with (P↔Q) in Exp3_3a. + Conj Exp3_3a Exp3_3b. 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 Exp3_3a. + apply Exp3_3b. + specialize Comp3_43 with (P↔Q) + ((P↔R)→(Q↔R)) ((Q↔R)→(P↔R)). + intros Comp3_43a. (*Not cited*) + MP Comp3_43a H. + replace (((P↔R)→(Q↔R))∧((Q↔R)→(P↔R))) + with ((P↔R)↔(Q↔R)) in Comp3_43a. + apply Comp3_43a. 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. + specialize n4_21 with P Q. + intros n4_21a. + apply n4_21a. Qed. Theorem n4_87 : ∀ P Q R : Prop, - (((P ∧ Q) → R) ↔ (P → Q → R)) ↔ ((Q → (P → R)) ↔ (Q ∧ P → R)). + (((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. @@ -2920,11 +3123,14 @@ Theorem n4_87 : ∀ P Q R : Prop, 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. + 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)). + specialize n4_2 with + ((P → Q → R) ↔ (P → Q → R)). intros n4_2a. apply n4_2a. apply EqBi. @@ -2967,71 +3173,85 @@ Theorem n5_1 : ∀ P Q : Prop, apply n3_4a. apply Sa. specialize n4_76 with (P∧Q) (P→Q) (Q→P). - intros n4_76a. - replace ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)) with (P ∧ Q → (P → Q) ∧ (Q → P)) in H. + intros n4_76a. (*Not cited*) + replace ((P∧Q→P→Q)∧(P∧Q→Q→P)) with + (P ∧ Q → (P → Q) ∧ (Q → P)) in H. replace ((P→Q)∧(Q→P)) with (P↔Q) in H. apply H. apply Equiv4_01. - replace (P ∧ Q → (P → Q) ∧ (Q → P)) with ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)). + replace (P ∧ Q → (P → Q) ∧ (Q → P)) with + ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)). reflexivity. apply EqBi. apply n4_76a. - Qed. (*Note that n4_76 is not cited, but it is used to move from ((a→b) ∧ (a→c)) to (a→ (b ∧ c)).*) + Qed. Theorem n5_11 : ∀ P Q : Prop, (P → Q) ∨ (~P → Q). Proof. intros P Q. specialize n2_5 with P Q. intros n2_5a. - specialize n2_54 with ((P → Q)) (~ P → Q). + specialize n2_54 with ((P → Q)) (~P → Q). intros n2_54a. MP n2_54a n2_5a. apply n2_54a. Qed. + (*The proof sketch cites n2_51, + but this may be a misprint.*) Theorem n5_12 : ∀ P Q : Prop, (P → Q) ∨ (P → ~Q). Proof. intros P Q. specialize n2_51 with P Q. intros n2_51a. - specialize n2_54 with ((P → Q)) (P → ~ Q). + specialize n2_54 with ((P → Q)) (P → ~Q). intros n2_54a. MP n2_54a n2_5a. apply n2_54a. Qed. + (*The proof sketch cites n2_52, + but this may be a misprint.*) Theorem n5_13 : ∀ P Q : Prop, (P → Q) ∨ (Q → P). Proof. intros P Q. specialize n2_521 with P Q. intros n2_521a. - replace (~ (P → Q) → Q → P) with (~~ (P → Q) ∨ (Q → P)) in n2_521a. + replace (~(P → Q) → Q → P) with + (~~(P → Q) ∨ (Q → P)) in n2_521a. replace (~~(P→Q)) with (P→Q) in n2_521a. apply n2_521a. apply EqBi. - apply n4_13. - replace (~~ (P → Q) ∨ (Q → P)) with (~ (P → Q) → Q → P). + specialize n4_13 with (P→Q). + intros n4_13a. (*Not cited*) + apply n4_13a. + replace (~~(P → Q) ∨ (Q → P)) with + (~(P → Q) → Q → P). reflexivity. apply Impl1_01. - Qed. (*n4_13 is not cited, but is needed for double negation elimination.*) + Qed. Theorem n5_14 : ∀ P Q R : Prop, (P → Q) ∨ (Q → R). Proof. intros P Q R. - specialize n2_02 with P Q. - intros n2_02a. + specialize Simp2_02 with P Q. + intros Simp2_02a. specialize Trans2_16 with Q (P→Q). intros Trans2_16a. - MP Trans2_16a n2_02a. + MP Trans2_16a Simp2_02a. specialize n2_21 with Q R. intros n2_21a. Syll Trans2_16a n2_21a Sa. - replace (~(P→Q)→(Q→R)) with (~~(P→Q)∨(Q→R)) in Sa. + replace (~(P→Q)→(Q→R)) with + (~~(P→Q)∨(Q→R)) in Sa. replace (~~(P→Q)) with (P→Q) in Sa. apply Sa. apply EqBi. - apply n4_13. - replace (~~(P→Q)∨(Q→R)) with (~(P→Q)→(Q→R)). + specialize n4_13 with (P→Q). + intros n4_13a. + apply n4_13a. + replace (~~(P→Q)∨(Q→R)) with + (~(P→Q)→(Q→R)). reflexivity. apply Impl1_01. Qed. @@ -3041,8 +3261,10 @@ Theorem n5_15 : ∀ P Q : Prop, Proof. intros P Q. specialize n4_61 with P Q. intros n4_61a. - replace (~ (P → Q) ↔ P ∧ ~ Q) with ((~ (P → Q) → P ∧ ~ Q) ∧ ((P ∧ ~ Q) → ~ (P → Q))) in n4_61a. - specialize Simp3_26 with (~ (P → Q) → P ∧ ~ Q) ((P ∧ ~ Q) → ~ (P → Q)). + replace (~(P → Q) ↔ P ∧ ~Q) with + ((~(P→Q)→P∧~Q)∧((P∧~Q)→~(P→Q))) in n4_61a. + specialize Simp3_26 with + (~(P → Q) → P ∧ ~Q) ((P ∧ ~Q) → ~(P → Q)). intros Simp3_26a. MP Simp3_26a n4_61a. specialize n5_1 with P (~Q). @@ -3053,8 +3275,10 @@ Theorem n5_15 : ∀ P Q : Prop, MP n2_54a Sa. specialize n4_61 with Q P. intros n4_61b. - replace ((~(Q → P)) ↔ (Q ∧ ~P)) with (((~(Q → P)) → (Q ∧ ~P)) ∧ ((Q ∧ ~P) → (~(Q → P)))) in n4_61b. - specialize Simp3_26 with (~(Q → P)→ (Q ∧ ~P)) ((Q ∧ ~P)→ (~(Q → P))). + replace ((~(Q → P)) ↔ (Q ∧ ~P)) with + (((~(Q→P))→(Q∧~P))∧((Q∧~P)→(~(Q→P)))) in n4_61b. + specialize Simp3_26 with + (~(Q → P)→ (Q ∧ ~P)) ((Q ∧ ~P)→ (~(Q → P))). intros Simp3_26b. MP Simp3_26b n4_61b. specialize n5_1 with Q (~P). @@ -3066,10 +3290,14 @@ Theorem n5_15 : ∀ P Q : Prop, specialize n2_54 with (Q→P) (P↔~Q). intros n2_54b. MP n2_54b Sb. - clear n4_61a. clear Simp3_26a. clear n5_1a. clear n2_54a. clear n4_61b. clear Simp3_26b. clear n5_1b. clear n4_12a. clear n2_54b. - replace (~(P → Q) → P ↔ ~Q) with (~~ (P → Q) ∨ (P ↔ ~Q)) in Sa. + clear n4_61a. clear Simp3_26a. clear n5_1a. + clear n2_54a. clear n4_61b. clear Simp3_26b. + clear n5_1b. clear n4_12a. clear n2_54b. + replace (~(P → Q) → P ↔ ~Q) with + (~~(P → Q) ∨ (P ↔ ~Q)) in Sa. replace (~~(P→Q)) with (P→Q) in Sa. - replace (~(Q → P) → (P ↔ ~Q)) with (~~(Q → P) ∨ (P ↔ ~Q)) in Sb. + replace (~(Q → P) → (P ↔ ~Q)) with + (~~(Q → P) ∨ (P ↔ ~Q)) in Sb. replace (~~(Q→P)) with (Q→P) in Sb. Conj Sa Sb. split. @@ -3077,11 +3305,15 @@ Theorem n5_15 : ∀ P Q : Prop, apply Sb. specialize n4_41 with (P↔~Q) (P→Q) (Q→P). intros n4_41a. - replace ((P → Q) ∨ (P ↔ ~Q)) with ((P ↔ ~Q) ∨ (P → Q)) in H. - replace ((Q → P) ∨ (P ↔ ~Q)) with ((P ↔ ~Q) ∨ (Q → P)) in H. - replace (((P ↔ ~Q) ∨ (P → Q)) ∧ ((P ↔ ~Q) ∨ (Q → P))) with ((P ↔ ~Q) ∨ (P → Q) ∧ (Q → P)) in H. + replace ((P → Q) ∨ (P ↔ ~Q)) with + ((P ↔ ~Q) ∨ (P → Q)) in H. + replace ((Q → P) ∨ (P ↔ ~Q)) with + ((P ↔ ~Q) ∨ (Q → P)) in H. + replace (((P↔~Q)∨(P→Q))∧((P↔~Q)∨(Q→P))) with + ((P ↔ ~Q) ∨ (P → Q) ∧ (Q → P)) in H. replace ((P→Q) ∧ (Q → P)) with (P↔Q) in H. - replace ((P ↔ ~Q) ∨ (P ↔ Q)) with ((P ↔ Q) ∨ (P ↔ ~Q)) in H. + replace ((P ↔ ~Q) ∨ (P ↔ Q)) with + ((P ↔ Q) ∨ (P ↔ ~Q)) in H. apply H. apply EqBi. apply n4_31. @@ -3093,13 +3325,19 @@ Theorem n5_15 : ∀ P Q : Prop, apply EqBi. apply n4_31. apply EqBi. - apply n4_13. - replace (~~(Q → P) ∨ (P ↔ ~Q)) with (~(Q → P) → (P ↔ ~Q)). + specialize n4_13 with (Q→P). + intros n4_13a. + apply n4_13a. + replace (~~(Q → P) ∨ (P ↔ ~Q)) with + (~(Q → P) → (P ↔ ~Q)). reflexivity. apply Impl1_01. apply EqBi. - apply n4_13. - replace (~~ (P → Q) ∨ (P ↔ ~Q)) with (~(P → Q) → P ↔ ~Q). + specialize n4_13 with (P→Q). + intros n4_13b. + apply n4_13b. + replace (~~(P → Q) ∨ (P ↔ ~Q)) with + (~(P → Q) → P ↔ ~Q). reflexivity. apply Impl1_01. apply EqBi. @@ -3115,80 +3353,110 @@ Theorem n5_16 : ∀ P Q : Prop, intros Simp3_26a. specialize n2_08 with ((P ↔ Q) ∧ (P → ~Q)). intros n2_08a. - replace (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)) with ((P → Q) ∧ ((P → ~Q) ∧ (Q → P))) in Simp3_26a. - replace ((P → ~Q) ∧ (Q → P)) with ((Q → P) ∧ (P → ~Q)) in Simp3_26a. - replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_26a. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Simp3_26a. + replace (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)) with + ((P→Q)∧((P→~Q)∧(Q→P))) in Simp3_26a. + replace ((P → ~Q) ∧ (Q → P)) with + ((Q → P) ∧ (P → ~Q)) in Simp3_26a. + replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with + (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_26a. + replace ((P → Q) ∧ (Q → P)) with + (P↔Q) in Simp3_26a. Syll n2_08a Simp3_26a Sa. specialize n4_82 with P Q. intros n4_82a. replace ((P → Q) ∧ (P → ~Q)) with (~P) in Sa. - specialize Simp3_27 with (P→Q) ((Q→P)∧ (P → ~Q)). + specialize Simp3_27 with + (P→Q) ((Q→P)∧ (P → ~Q)). intros Simp3_27a. - replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_27a. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Simp3_27a. + replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with + (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_27a. + replace ((P → Q) ∧ (Q → P)) with + (P↔Q) in Simp3_27a. specialize Syll3_33 with Q P (~Q). intros Syll3_33a. Syll Simp3_27a Syll2_06a Sb. specialize Abs2_01 with Q. intros Abs2_01a. Syll Sb Abs2_01a Sc. - clear Sb. clear Simp3_26a. clear n2_08a. clear n4_82a. clear Simp3_27a. clear Syll3_33a. clear Abs2_01a. + clear Sb. clear Simp3_26a. clear n2_08a. + clear n4_82a. clear Simp3_27a. clear Syll3_33a. + clear Abs2_01a. Conj Sa Sc. split. apply Sa. apply Sc. - specialize Comp3_43 with ((P ↔ Q) ∧ (P → ~Q)) (~P) (~Q). + specialize Comp3_43 with + ((P ↔ Q) ∧ (P → ~Q)) (~P) (~Q). intros Comp3_43a. MP Comp3_43a H. specialize n4_65 with Q P. intros n4_65a. replace (~Q ∧ ~P) with (~P ∧ ~Q) in n4_65a. - replace (~P ∧ ~Q) with (~(~Q→P)) in Comp3_43a. - specialize Exp3_3 with (P↔Q) (P→~Q) (~(~Q→P)). + replace (~P ∧ ~Q) with + (~(~Q→P)) in Comp3_43a. + specialize Exp3_3 with + (P↔Q) (P→~Q) (~(~Q→P)). intros Exp3_3a. MP Exp3_3a Comp3_43a. - replace ((P→~Q)→~(~Q→P)) with (~(P→~Q)∨~(~Q→P)) in Exp3_3a. + replace ((P→~Q)→~(~Q→P)) with + (~(P→~Q)∨~(~Q→P)) in Exp3_3a. specialize n4_51 with (P→~Q) (~Q→P). intros n4_51a. - replace (~(P → ~Q) ∨ ~(~Q → P)) with (~((P → ~Q) ∧ (~Q → P))) in Exp3_3a. - replace ((P→~Q) ∧ (~ Q → P)) with (P↔~Q) in Exp3_3a. - replace ((P↔Q)→~(P↔~Q)) with (~(P↔Q)∨~(P↔~Q)) in Exp3_3a. + replace (~(P → ~Q) ∨ ~(~Q → P)) with + (~((P → ~Q) ∧ (~Q → P))) in Exp3_3a. + replace ((P→~Q) ∧ (~Q → P)) with + (P↔~Q) in Exp3_3a. + replace ((P↔Q)→~(P↔~Q)) with + (~(P↔Q)∨~(P↔~Q)) in Exp3_3a. specialize n4_51 with (P↔Q) (P↔~Q). intros n4_51b. - replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with (~((P ↔ Q) ∧ (P ↔ ~Q))) in Exp3_3a. + replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with + (~((P ↔ Q) ∧ (P ↔ ~Q))) in Exp3_3a. apply Exp3_3a. apply EqBi. apply n4_51b. - replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with (P ↔ Q → ~(P ↔ ~Q)). + replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with + (P ↔ Q → ~(P ↔ ~Q)). reflexivity. apply Impl1_01. apply Equiv4_01. apply EqBi. apply n4_51a. - replace (~(P → ~Q) ∨ ~(~Q → P)) with ((P → ~Q) → ~(~Q → P)). + replace (~(P → ~Q) ∨ ~(~Q → P)) with + ((P → ~Q) → ~(~Q → P)). reflexivity. apply Impl1_01. apply EqBi. apply n4_65a. apply EqBi. - apply n4_3. + specialize n4_3 with (~P) (~Q). + intros n4_3a. + apply n4_3a. apply Equiv4_01. apply EqBi. - apply n4_32. + specialize n4_32 with (P→Q) (Q→P) (P→~Q). + intros n4_32a. + apply n4_32a. replace (~P) with ((P → Q) ∧ (P → ~Q)). reflexivity. apply EqBi. apply n4_82a. apply Equiv4_01. apply EqBi. - apply n4_32. + specialize n4_32 with (P→Q) (Q→P) (P→~Q). + intros n4_32b. + apply n4_32b. apply EqBi. - apply n4_3. - replace ((P → Q) ∧ (P → ~Q) ∧ (Q → P)) with (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)). + specialize n4_3 with (Q→P) (P→~Q). + intros n4_3b. + apply n4_3b. + replace ((P → Q) ∧ (P → ~Q) ∧ (Q → P)) with + (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)). reflexivity. apply EqBi. - apply n4_32. + specialize n4_32 with (P→Q) (P→~Q) (Q→P). + intros n4_32a. + apply n4_32a. Qed. Theorem n5_17 : ∀ P Q : Prop, @@ -3198,24 +3466,30 @@ Theorem n5_17 : ∀ P Q : Prop, intros n4_64a. specialize n4_21 with (Q∨P) (~Q→P). intros n4_21a. - replace ((~Q→P)↔(Q∨P)) with ((Q∨P)↔(~Q→P)) in n4_64a. + replace ((~Q→P)↔(Q∨P)) with + ((Q∨P)↔(~Q→P)) in n4_64a. replace (Q∨P) with (P∨Q) in n4_64a. specialize n4_63 with P Q. intros n4_63a. - replace (~(P → ~Q) ↔ P ∧ Q) with (P ∧ Q ↔ ~(P → ~Q)) in n4_63a. + replace (~(P → ~Q) ↔ P ∧ Q) with + (P ∧ Q ↔ ~(P → ~Q)) in n4_63a. specialize Trans4_11 with (P∧Q) (~(P→~Q)). intros Trans4_11a. - replace (~~(P→~Q)) with (P→~Q) in Trans4_11a. - replace (P ∧ Q ↔ ~(P → ~Q)) with (~(P ∧ Q) ↔ (P → ~Q)) in n4_63a. + replace (~~(P→~Q)) with + (P→~Q) in Trans4_11a. + replace (P ∧ Q ↔ ~(P → ~Q)) with + (~(P ∧ Q) ↔ (P → ~Q)) in n4_63a. clear Trans4_11a. clear n4_21a. Conj n4_64a n4_63a. split. apply n4_64a. apply n4_63a. - specialize n4_38 with (P ∨ Q) (~(P ∧ Q)) (~Q → P) (P → ~Q). + specialize n4_38 with + (P ∨ Q) (~(P ∧ Q)) (~Q → P) (P → ~Q). intros n4_38a. MP n4_38a H. - replace ((~Q→P) ∧ (P → ~Q)) with (~Q↔P) in n4_38a. + replace ((~Q→P) ∧ (P → ~Q)) with + (~Q↔P) in n4_38a. specialize n4_21 with P (~Q). intros n4_21b. replace (~Q↔P) with (P↔~Q) in n4_38a. @@ -3223,16 +3497,23 @@ Theorem n5_17 : ∀ P Q : Prop, apply EqBi. apply n4_21b. apply Equiv4_01. - replace (~(P ∧ Q) ↔ (P → ~Q)) with (P ∧ Q ↔ ~(P → ~Q)). + replace (~(P ∧ Q) ↔ (P → ~Q)) with + (P ∧ Q ↔ ~(P → ~Q)). reflexivity. apply EqBi. apply Trans4_11a. apply EqBi. - apply n4_13. + specialize n4_13 with (P→~Q). + intros n4_13a. + apply n4_13a. apply EqBi. - apply n4_21. + specialize n4_21 with (P ∧ Q) (~(P→~Q)). + intros n4_21b. + apply n4_21b. apply EqBi. - apply n4_31. + specialize n4_31 with P Q. + intros n4_31a. + apply n4_31a. apply EqBi. apply n4_21a. Qed. @@ -3250,7 +3531,8 @@ Theorem n5_18 : ∀ P Q : Prop, apply n5_16a. specialize n5_17 with (P↔Q) (P↔~Q). intros n5_17a. - replace ((P ↔ Q) ↔ ~(P ↔ ~Q)) with (((P ↔ Q) ∨ (P ↔ ~Q)) ∧ ~((P ↔ Q) ∧ (P ↔ ~Q))). + replace ((P ↔ Q) ↔ ~(P ↔ ~Q)) with + (((P↔Q)∨(P↔~Q))∧~((P↔Q)∧(P↔~Q))). apply H. apply EqBi. apply n5_17a. @@ -3293,13 +3575,16 @@ Theorem n5_22 : ∀ P Q : Prop, split. apply n4_61a. apply n4_61b. - specialize n4_39 with (~(P → Q)) (~(Q → P)) (P ∧ ~Q) (Q ∧ ~P). + specialize n4_39 with + (~(P → Q)) (~(Q → P)) (P ∧ ~Q) (Q ∧ ~P). intros n4_39a. MP n4_39a H. specialize n4_51 with (P→Q) (Q→P). intros n4_51a. - replace (~(P → Q) ∨ ~(Q → P)) with (~((P → Q) ∧ (Q → P))) in n4_39a. - replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n4_39a. + replace (~(P → Q) ∨ ~(Q → P)) with + (~((P → Q) ∧ (Q → P))) in n4_39a. + replace ((P → Q) ∧ (Q → P)) with + (P↔Q) in n4_39a. apply n4_39a. apply Equiv4_01. apply EqBi. @@ -3315,19 +3600,24 @@ Theorem n5_23 : ∀ P Q : Prop, intros n5_22a. specialize n4_13 with Q. intros n4_13a. - replace (~(P↔~Q)) with ((P ∧ ~~Q) ∨ (~Q ∧ ~P)) in n5_18a. + replace (~(P↔~Q)) with + ((P ∧ ~~Q) ∨ (~Q ∧ ~P)) in n5_18a. replace (~~Q) with Q in n5_18a. replace (~Q ∧ ~P) with (~P ∧ ~Q) in n5_18a. apply n5_18a. apply EqBi. - apply n4_3. (*with (~P) (~Q)*) + specialize n4_3 with (~P) (~Q). + intros n4_3a. + apply n4_3a. (*with (~P) (~Q)*) apply EqBi. apply n4_13a. replace (P∧~~Q∨~Q∧~P) with (~(P↔~Q)). reflexivity. apply EqBi. apply n5_22a. - Qed. (*The proof sketch in Principia offers n4_36, but we found it far simpler to simply use the commutativity of conjunction (n4_3).*) + Qed. + (*The proof sketch in Principia offers n4_36, + but we found it far simpler to to use n4_3.*) Theorem n5_24 : ∀ P Q : Prop, ~((P ∧ Q) ∨ (~P ∧ ~Q)) ↔ ((P ∧ ~Q) ∨ (Q ∧ ~P)). @@ -3336,20 +3626,24 @@ Theorem n5_24 : ∀ P Q : Prop, intros n5_22a. specialize n5_23 with P Q. intros n5_23a. - replace ((P↔Q)↔((P∧ Q) ∨(~P ∧ ~Q))) with ((~(P↔Q)↔~((P∧ Q) ∨(~P ∧ ~Q)))) in n5_23a. - replace (~(P↔Q)) with (~((P ∧ Q) ∨ (~P ∧ ~Q))) in n5_22a. + replace ((P↔Q)↔((P∧ Q) ∨(~P ∧ ~Q))) with + ((~(P↔Q)↔~((P∧ Q)∨(~P ∧ ~Q)))) in n5_23a. + replace (~(P↔Q)) with + (~((P ∧ Q) ∨ (~P ∧ ~Q))) in n5_22a. apply n5_22a. - replace (~((P ∧ Q) ∨ (~P ∧ ~Q))) with (~(P ↔ Q)). + replace (~((P ∧ Q)∨(~P ∧ ~Q))) with (~(P↔Q)). reflexivity. apply EqBi. apply n5_23a. - replace (~(P ↔ Q) ↔ ~(P ∧ Q ∨ ~P ∧ ~Q)) with ((P ↔ Q) ↔ P ∧ Q ∨ ~P ∧ ~Q). + replace (~(P ↔ Q) ↔ ~(P ∧ Q ∨ ~P ∧ ~Q)) with + ((P ↔ Q) ↔ P ∧ Q ∨ ~P ∧ ~Q). reflexivity. - specialize Trans4_11 with (P↔Q) (P ∧ Q ∨ ~P ∧ ~Q). + specialize Trans4_11 with + (P↔Q) (P ∧ Q ∨ ~P ∧ ~Q). intros Trans4_11a. apply EqBi. - apply Trans4_11a. - Qed. (*Note that Trans4_11 is not cited explicitly.*) + apply Trans4_11a. (*Not cited*) + Qed. Theorem n5_25 : ∀ P Q : Prop, (P ∨ Q) ↔ ((P → Q) → Q). @@ -3372,8 +3666,9 @@ Theorem n5_3 : ∀ P Q R : Prop, Proof. intros P Q R. specialize Comp3_43 with (P ∧ Q) P R. intros Comp3_43a. - specialize Exp3_3 with (P ∧ Q → P) (P ∧ Q →R) (P ∧ Q → P ∧ R). - intros Exp3_3a. + specialize Exp3_3 with + (P ∧ Q → P) (P ∧ Q →R) (P ∧ Q → P ∧ R). + intros Exp3_3a. (*Not cited*) MP Exp3_3a Comp3_43a. specialize Simp3_26 with P Q. intros Simp3_26a. @@ -3383,7 +3678,8 @@ Theorem n5_3 : ∀ P Q R : Prop, specialize Simp3_27 with P R. intros Simp3_27a. MP Syll2_05a Simp3_27a. - clear Comp3_43a. clear Simp3_27a. clear Simp3_26a. + clear Comp3_43a. clear Simp3_27a. + clear Simp3_26a. Conj Exp3_3a Syll2_05a. split. apply Exp3_3a. @@ -3391,27 +3687,28 @@ Theorem n5_3 : ∀ P Q R : Prop, Equiv H. apply H. apply Equiv4_01. - Qed. (*Note that Exp is not cited in the proof sketch, but seems necessary.*) + Qed. Theorem n5_31 : ∀ P Q R : Prop, (R ∧ (P → Q)) → (P → (Q ∧ R)). Proof. intros P Q R. specialize Comp3_43 with P Q R. intros Comp3_43a. - specialize n2_02 with P R. - intros n2_02a. - replace ((P→Q) ∧ (P→R)) with ((P→R) ∧ (P→Q)) in Comp3_43a. - specialize Exp3_3 with (P→R) (P→Q) (P→(Q ∧ R)). - intros Exp3_3a. - MP Exp3_3a Comp3_43a. - Syll n2_02a Exp3_3a Sa. + specialize Simp2_02 with P R. + intros Simp2_02a. + specialize Exp3_3 with + (P→R) (P→Q) (P→(Q ∧ R)). + intros Exp3_3a. (*Not cited*) + specialize n3_22 with (P → R) (P → Q). (*Not cited*) + intros n3_22a. + Syll n3_22a Comp3_43a Sa. + MP Exp3_3a Sa. + Syll Simp2_02a Exp3_3a Sb. specialize Imp3_31 with R (P→Q) (P→(Q ∧ R)). - intros Imp3_31a. - MP Imp3_31a Sa. + intros Imp3_31a. (*Not cited*) + MP Imp3_31a Sb. apply Imp3_31a. - apply EqBi. - apply n4_3. (*with (P→R)∧(P→Q)).*) - Qed. (*Note that Exp, Imp, and n4_3 are not cited in the proof sketch.*) + Qed. Theorem n5_32 : ∀ P Q R : Prop, (P → (Q ↔ R)) ↔ ((P ∧ Q) ↔ (P ∧ R)). @@ -3444,13 +3741,18 @@ Theorem n5_32 : ∀ P Q R : Prop, replace (P∧Q→R) with (P∧Q→P∧R) in n4_76a. replace (P→R→Q) with (P∧R→Q) in n4_76a. replace (P∧R→Q) with (P∧R→P∧Q) in n4_76a. - replace ((P∧Q→P∧R)∧(P∧R→P∧Q)) with ((P∧Q)↔(P∧R)) in n4_76a. - replace ((P∧Q ↔ P∧R)↔(P→(Q→R)∧(R→Q))) with ((P→(Q→R)∧(R→Q))↔(P∧Q ↔ P∧R)) in n4_76a. + replace ((P∧Q→P∧R)∧(P∧R→P∧Q)) with + ((P∧Q)↔(P∧R)) in n4_76a. + replace ((P∧Q↔P∧R)↔(P→(Q→R)∧(R→Q))) with + ((P→(Q→R)∧(R→Q))↔(P∧Q ↔ P∧R)) in n4_76a. replace ((Q→R)∧(R→Q)) with (Q↔R) in n4_76a. apply n4_76a. apply Equiv4_01. apply EqBi. - apply n4_3. (*to commute the biconditional to get the theorem.*) + specialize n4_21 with + (P→((Q→R)∧(R→Q))) ((P∧Q)↔(P∧R)). + intros n4_21a. + apply n4_21a. (*to commute the biconditional*) apply Equiv4_01. replace (P ∧ R → P ∧ Q) with (P ∧ R → Q). reflexivity. @@ -3473,9 +3775,17 @@ Theorem n5_33 : ∀ P Q R : Prop, Proof. intros P Q R. specialize n5_32 with P (Q→R) ((P∧Q)→R). intros n5_32a. - replace ((P→(Q→R)↔(P∧Q→R))↔(P∧(Q→R)↔P∧(P∧Q→R))) with (((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R)))∧((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R))))) in n5_32a. - specialize Simp3_26 with ((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R))) ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R)))). (*Not cited.*) - intros Simp3_26a. + replace + ((P→(Q→R)↔(P∧Q→R))↔(P∧(Q→R)↔P∧(P∧Q→R))) + with + (((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R))) + ∧ + ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R))))) + in n5_32a. + specialize Simp3_26 with + ((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R))) + ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R)))). + intros Simp3_26a. (*Not cited*) MP Simp3_26a n5_32a. specialize n4_73 with Q P. intros n4_73a. @@ -3486,7 +3796,9 @@ Theorem n5_33 : ∀ P Q R : Prop, MP Simp3_26a Sa. apply Simp3_26a. apply EqBi. - apply n4_3. (*Not cited.*) + specialize n4_3 with P Q. + intros n4_3a. + apply n4_3a. (*Not cited*) apply Equiv4_01. Qed. @@ -3511,31 +3823,39 @@ Theorem n5_36 : ∀ P Q : Prop, intros n5_32a. specialize n2_08 with (P↔Q). intros n2_08a. - replace (P↔Q→P↔Q) with ((P↔Q)∧P↔(P↔Q)∧Q) in n2_08a. + replace (P↔Q→P↔Q) with + ((P↔Q)∧P↔(P↔Q)∧Q) in n2_08a. replace ((P↔Q)∧P) with (P∧(P↔Q)) in n2_08a. replace ((P↔Q)∧Q) with (Q∧(P↔Q)) in n2_08a. apply n2_08a. apply EqBi. - apply n4_3. + specialize n4_3 with Q (P↔Q). + intros n4_3a. + apply n4_3a. apply EqBi. - apply n4_3. - replace ((P ↔ Q) ∧ P ↔ (P ↔ Q) ∧ Q) with (P ↔ Q → P ↔ Q). + specialize n4_3 with P (P↔Q). + intros n4_3b. + apply n4_3b. + replace ((P ↔ Q) ∧ P ↔ (P ↔ Q) ∧ Q) with + (P ↔ Q → P ↔ Q). reflexivity. apply EqBi. apply n5_32a. - Qed. (*The proof sketch cites Ass3_35 and n4_38. Since I couldn't decipher how that proof would go, I used a different one invoking other theorems.*) + Qed. + (*The proof sketch cites Ass3_35 and n4_38, but + the sketch was indecipherable.*) Theorem n5_4 : ∀ P Q : Prop, (P → (P → Q)) ↔ (P → Q). Proof. intros P Q. specialize n2_43 with P Q. intros n2_43a. - specialize n2_02 with (P) (P→Q). - intros n2_02a. - Conj n2_43a n2_02a. + specialize Simp2_02 with (P) (P→Q). + intros Simp2_02a. + Conj n2_43a Simp2_02a. split. apply n2_43a. - apply n2_02a. + apply Simp2_02a. Equiv H. apply H. apply Equiv4_01. @@ -3567,7 +3887,8 @@ Theorem n5_42 : ∀ P Q R : Prop, replace ((P∧Q)→R) with (P→Q→R) in n5_3a. specialize n4_87 with P Q (P∧R). intros n4_87b. - replace ((P∧Q)→(P∧R)) with (P→Q→(P∧R)) in n5_3a. + replace ((P∧Q)→(P∧R)) with + (P→Q→(P∧R)) in n5_3a. apply n5_3a. specialize Imp3_31 with P Q (P∧R). intros Imp3_31b. @@ -3593,49 +3914,66 @@ Theorem n5_42 : ∀ P Q R : Prop, apply EqBi. apply H. apply Equiv4_01. - Qed. (*The law n4_87 is really unwieldy to use in Coq. It is actually easier to introduce the subformula of the importation-exportation law required and apply that biconditional. It may be worthwhile in later parts of PM to prove a derived rule that allows us to manipulate a biconditional's subformulas that are biconditionals.*) + Qed. Theorem n5_44 : ∀ P Q R : Prop, (P→Q) → ((P → R) ↔ (P → (Q ∧ R))). Proof. intros P Q R. specialize n4_76 with P Q R. intros n4_76a. - replace ((P→Q)∧(P→R)↔(P→Q∧R)) with (((P→Q)∧(P→R)→(P→Q∧R))∧((P→Q∧R)→(P→Q)∧(P→R))) in n4_76a. - specialize Simp3_26 with ((P→Q)∧(P→R)→(P→Q∧R)) ((P→Q∧R)→(P→Q)∧(P→R)). - intros Simp3_26a. (*Not cited.*) + replace ((P→Q)∧(P→R)↔(P→Q∧R)) with + (((P→Q)∧(P→R)→(P→Q∧R)) + ∧ + ((P→Q∧R)→(P→Q)∧(P→R))) in n4_76a. + specialize Simp3_26 with + ((P→Q)∧(P→R)→(P→Q∧R)) + ((P→Q∧R)→(P→Q)∧(P→R)). + intros Simp3_26a. (*Not cited*) MP Simp3_26a n4_76a. specialize Exp3_3 with (P→Q) (P→R) (P→Q∧R). - intros Exp3_3a. (*Not cited.*) + intros Exp3_3a. (*Not cited*) MP Exp3_3a Simp3_26a. - specialize Simp3_27 with ((P→Q)∧(P→R)→(P→Q∧R)) ((P→Q∧R)→(P→Q)∧(P→R)). - intros Simp3_27a. (*Not cited.*) + specialize Simp3_27 with + ((P→Q)∧(P→R)→(P→Q∧R)) + ((P→Q∧R)→(P→Q)∧(P→R)). + intros Simp3_27a. (*Not cited*) MP Simp3_27a n4_76a. specialize Simp3_26 with (P→R) (P→Q). intros Simp3_26b. - replace ((P→Q)∧(P→R)) with ((P→R)∧(P→Q)) in Simp3_27a. + replace ((P→Q)∧(P→R)) with + ((P→R)∧(P→Q)) in Simp3_27a. Syll Simp3_27a Simp3_26b Sa. - specialize n2_02 with (P→Q) ((P→Q∧R)→P→R). - intros n2_02a. (*Not cited.*) - MP n2_02a Sa. - clear Sa. clear Simp3_26b. clear Simp3_26a. clear n4_76a. clear Simp3_27a. - Conj Exp3_3a n2_02a. + specialize Simp2_02 with (P→Q) ((P→Q∧R)→P→R). + intros Simp2_02a. (*Not cited*) + MP Simp2_02a Sa. + clear Sa. clear Simp3_26b. clear Simp3_26a. + clear n4_76a. clear Simp3_27a. + Conj Exp3_3a Simp2_02a. split. apply Exp3_3a. - apply n2_02a. - specialize n4_76 with (P→Q) ((P→R)→(P→(Q∧R))) ((P→(Q∧R))→(P→R)). - intros n4_76b. - replace (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)) with ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) in H. - replace (((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with ((P→R)↔(P→Q∧R)) in H. + apply Simp2_02a. + specialize n4_76 with (P→Q) + ((P→R)→(P→(Q∧R))) ((P→(Q∧R))→(P→R)). + intros n4_76b. (*Second use not indicated*) + replace + (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)) + with + ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) in H. + replace (((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with + ((P→R)↔(P→Q∧R)) in H. apply H. apply Equiv4_01. - replace ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)). + replace ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with + (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)). reflexivity. apply EqBi. apply n4_76b. apply EqBi. - apply n4_3. (*Not cited.*) + specialize n4_3 with (P→R) (P→Q). + intros n4_3a. + apply n4_3a. (*Not cited*) apply Equiv4_01. - Qed. (*This proof does not use either n5_3 or n5_32. It instead uses four propositions not cited in the proof sketch, plus a second use of n4_76.*) + Qed. Theorem n5_5 : ∀ P Q : Prop, P → ((P → Q) ↔ Q). @@ -3645,15 +3983,15 @@ Theorem n5_5 : ∀ P Q : Prop, specialize Exp3_3 with P (P→Q) Q. intros Exp3_3a. MP Exp3_3a Ass3_35a. - specialize n2_02 with P Q. - intros n2_02a. + specialize Simp2_02 with P Q. + intros Simp2_02a. specialize Exp3_3 with P Q (P→Q). intros Exp3_3b. - specialize n3_42 with P Q (P→Q). (*Not mentioned explicitly.*) + specialize n3_42 with P Q (P→Q). (*Not cited*) intros n3_42a. - MP n3_42a n2_02a. + MP n3_42a Simp2_02a. MP Exp3_3b n3_42a. - clear n3_42a. clear n2_02a. clear Ass3_35a. + clear n3_42a. clear Simp2_02a. clear Ass3_35a. Conj Exp3_3a Exp3_3b. split. apply Exp3_3a. @@ -3662,11 +4000,14 @@ Theorem n5_5 : ∀ P Q : Prop, intros n3_47a. MP n3_47a H. replace (P∧P) with P in n3_47a. - replace (((P→Q)→Q)∧(Q→(P→Q))) with ((P→Q)↔Q) in n3_47a. + replace (((P→Q)→Q)∧(Q→(P→Q))) with + ((P→Q)↔Q) in n3_47a. apply n3_47a. apply Equiv4_01. apply EqBi. - apply n4_24. (*with P.*) + specialize n4_24 with P. + intros n4_24a. (*Not cited*) + apply n4_24a. Qed. Theorem n5_501 : ∀ P Q : Prop, @@ -3680,33 +4021,41 @@ Theorem n5_501 : ∀ P Q : Prop, specialize Ass3_35 with P Q. intros Ass3_35a. specialize Simp3_26 with (P∧(P→Q)) (Q→P). - intros Simp3_26a. (*Not cited.*) + intros Simp3_26a. (*Not cited*) Syll Simp3_26a Ass3_35a Sa. - replace ((P∧(P→Q))∧(Q→P)) with (P∧((P→Q)∧(Q→P))) in Sa. + replace ((P∧(P→Q))∧(Q→P)) with + (P∧((P→Q)∧(Q→P))) in Sa. replace ((P→Q)∧(Q→P)) with (P↔Q) in Sa. specialize Exp3_3 with P (P↔Q) Q. intros Exp3_3b. MP Exp3_3b Sa. - clear n5_1a. clear Ass3_35a. clear Simp3_26a. clear Sa. + clear n5_1a. clear Ass3_35a. + clear Simp3_26a. clear Sa. Conj Exp3_3a Exp3_3b. split. apply Exp3_3a. apply Exp3_3b. specialize n4_76 with P (Q→(P↔Q)) ((P↔Q)→Q). - intros n4_76a. (*Not cited.*) - replace ((P→Q→P↔Q)∧(P→P↔Q→Q)) with ((P→(Q→P↔Q)∧(P↔Q→Q))) in H. - replace ((Q→(P↔Q))∧((P↔Q)→Q)) with (Q↔(P↔Q)) in H. + intros n4_76a. (*Not cited*) + replace ((P→Q→P↔Q)∧(P→P↔Q→Q)) with + ((P→(Q→P↔Q)∧(P↔Q→Q))) in H. + replace ((Q→(P↔Q))∧((P↔Q)→Q)) with + (Q↔(P↔Q)) in H. apply H. apply Equiv4_01. - replace (P→(Q→P↔Q)∧(P↔Q→Q)) with ((P→Q→P↔Q)∧(P→P↔Q→Q)). + replace (P→(Q→P↔Q)∧(P↔Q→Q)) with + ((P→Q→P↔Q)∧(P→P↔Q→Q)). reflexivity. apply EqBi. apply n4_76a. apply Equiv4_01. - replace (P∧(P→Q)∧(Q→P)) with ((P∧(P→Q))∧(Q→P)). + replace (P∧(P→Q)∧(Q→P)) with + ((P∧(P→Q))∧(Q→P)). reflexivity. apply EqBi. - apply n4_32. (*Not cited.*) + specialize n4_32 with P (P→Q) (Q→P). + intros n4_32a. (*Not cited*) + apply n4_32a. Qed. Theorem n5_53 : ∀ P Q R S : Prop, @@ -3716,11 +4065,18 @@ Theorem n5_53 : ∀ P Q R S : Prop, intros n4_77a. specialize n4_77 with S P Q. intros n4_77b. - replace (P ∨ Q → S) with ((P→S)∧(Q→S)) in n4_77a. - replace ((((P→S)∧(Q→S))∧(R→S))↔(((P∨Q)∨R)→S)) with ((((P∨Q)∨R)→S)↔(((P→S)∧(Q→S))∧(R→S))) in n4_77a. + replace (P ∨ Q → S) with + ((P→S)∧(Q→S)) in n4_77a. + replace ((((P→S)∧(Q→S))∧(R→S))↔(((P∨Q)∨R)→S)) + with + ((((P∨Q)∨R)→S)↔(((P→S)∧(Q→S))∧(R→S))) + in n4_77a. apply n4_77a. apply EqBi. - apply n4_3. (*Not cited explicitly.*) + specialize n4_21 with ((P ∨ Q) ∨ R → S) + (((P → S) ∧ (Q → S)) ∧ (R → S)). + intros n4_21a. + apply n4_21a. (*Not cited*) apply EqBi. apply n4_77b. Qed. @@ -3738,13 +4094,16 @@ Theorem n5_54 : ∀ P Q : Prop, specialize Trans4_11 with Q (Q∨(P∧Q)). intros Trans4_11a. replace (Q∧P) with (P∧Q) in n4_44a. - replace (Q↔Q∨P∧Q) with (~Q↔~(Q∨P∧Q)) in n4_44a. + replace (Q↔Q∨P∧Q) with + (~Q↔~(Q∨P∧Q)) in n4_44a. replace (~Q) with (~(Q∨P∧Q)) in Trans2_16a. - replace (~(Q∨P∧Q)) with (~Q∧~(P∧Q)) in Trans2_16a. + replace (~(Q∨P∧Q)) with + (~Q∧~(P∧Q)) in Trans2_16a. specialize n5_1 with (~Q) (~(P∧Q)). intros n5_1a. Syll Trans2_16a n5_1a Sa. - replace (~(P↔P∧Q)→(~Q↔~(P∧Q))) with (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) in Sa. + replace (~(P↔P∧Q)→(~Q↔~(P∧Q))) with + (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) in Sa. replace (~~(P↔P∧Q)) with (P↔P∧Q) in Sa. specialize Trans4_11 with Q (P∧Q). intros Trans4_11b. @@ -3753,18 +4112,27 @@ Theorem n5_54 : ∀ P Q : Prop, replace (P↔(P∧Q)) with ((P∧Q)↔P) in Sa. apply Sa. apply EqBi. - apply n4_21. (*Not cited.*) + specialize n4_21 with (P∧Q) P. + intros n4_21a. (*Not cited*) + apply n4_21a. apply EqBi. - apply n4_21. + specialize n4_21 with (P∧Q) Q. + intros n4_21b. (*Not cited*) + apply n4_21b. apply EqBi. apply Trans4_11b. apply EqBi. - apply n4_13. (*Not cited.*) - replace (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) with (~(P↔P∧Q)→~Q↔~(P∧Q)). + specialize n4_13 with (P ↔ (P∧Q)). + intros n4_13a. (*Not cited*) + apply n4_13a. + replace (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) with + (~(P↔P∧Q)→~Q↔~(P∧Q)). reflexivity. - apply Impl1_01. (*Not cited.*) + apply Impl1_01. (*Not cited*) apply EqBi. - apply n4_56. (*Not cited.*) + specialize n4_56 with Q (P∧Q). + intros n4_56a. (*Not cited*) + apply n4_56a. replace (~(Q∨P∧Q)) with (~Q). reflexivity. apply EqBi. @@ -3774,7 +4142,9 @@ Theorem n5_54 : ∀ P Q : Prop, apply EqBi. apply Trans4_11a. apply EqBi. - apply n4_3. (*Not cited.*) + specialize n4_3 with P Q. + intros n4_3a. (*Not cited*) + apply n4_3a. Qed. Theorem n5_55 : ∀ P Q : Prop, @@ -3791,40 +4161,61 @@ Theorem n5_55 : ∀ P Q : Prop, specialize n4_74 with P Q. intros n4_74a. specialize Trans2_15 with P (Q↔P∨Q). - intros Trans2_15a. (*Not cited.*) + intros Trans2_15a. (*Not cited*) MP Trans2_15a n4_74a. Syll Trans2_15a Sa Sb. - replace (~(Q↔(P∨Q))→(P↔(P∨Q))) with (~~(Q↔(P∨Q))∨(P↔(P∨Q))) in Sb. + replace (~(Q↔(P∨Q))→(P↔(P∨Q))) with + (~~(Q↔(P∨Q))∨(P↔(P∨Q))) in Sb. replace (~~(Q↔(P∨Q))) with (Q↔(P∨Q)) in Sb. replace (Q↔(P∨Q)) with ((P∨Q)↔Q) in Sb. replace (P↔(P∨Q)) with ((P∨Q)↔P) in Sb. - replace ((P∨Q↔Q)∨(P∨Q↔P)) with ((P∨Q↔P)∨(P∨Q↔Q)) in Sb. + replace ((P∨Q↔Q)∨(P∨Q↔P)) with + ((P∨Q↔P)∨(P∨Q↔Q)) in Sb. apply Sb. apply EqBi. - apply n4_31. (*Not cited.*) + specialize n4_31 with (P ∨ Q ↔ P) (P ∨ Q ↔ Q). + intros n4_31a. (*Not cited*) + apply n4_31a. apply EqBi. - apply n4_21. (*Not cited.*) + specialize n4_21 with (P ∨ Q) P. + intros n4_21a. (*Not cited*) + apply n4_21a. apply EqBi. - apply n4_21. + specialize n4_21 with (P ∨ Q) Q. + intros n4_21b. (*Not cited*) + apply n4_21b. apply EqBi. - apply n4_13. (*Not cited.*) - replace (~~(Q↔P∨Q)∨(P↔P∨Q)) with (~(Q↔P∨Q)→P↔P∨Q). + specialize n4_13 with (Q ↔ P ∨ Q). + intros n4_13a. (*Not cited*) + apply n4_13a. + replace (~~(Q↔P∨Q)∨(P↔P∨Q)) with + (~(Q↔P∨Q)→P↔P∨Q). reflexivity. apply Impl1_01. apply EqBi. - apply n4_31. + specialize n4_31 with P Q. + intros n4_31b. + apply n4_31b. apply EqBi. - apply n4_25. (*Not cited.*) + specialize n4_25 with P. + intros n4_25a. (*Not cited*) + apply n4_25a. replace ((P∨P)∧(Q∨P)) with ((P∧Q)∨P). reflexivity. replace ((P∧Q)∨P) with (P∨(P∧Q)). replace (Q∨P) with (P∨Q). apply EqBi. - apply n4_41. (*Not cited.*) + specialize n4_41 with P P Q. + intros n4_41a. (*Not cited*) + apply n4_41a. apply EqBi. - apply n4_31. + specialize n4_31 with P Q. + intros n4_31c. + apply n4_31c. apply EqBi. - apply n4_31. + specialize n4_31 with P (P ∧ Q). + intros n4_31d. (*Not cited*) + apply n4_31d. Qed. Theorem n5_6 : ∀ P Q R : Prop, @@ -3836,8 +4227,15 @@ Theorem n5_6 : ∀ P Q R : Prop, intros n4_64a. specialize n4_85 with P Q R. intros n4_85a. - replace (((P ∧ ~Q → R) ↔ (P → ~Q → R)) ↔ ((~Q → P → R) ↔ (~Q ∧ P → R))) with ((((P ∧ ~Q → R) ↔ (P → ~Q → R)) → ((~Q → P → R) ↔ (~Q ∧ P → R)))∧((((~Q → P → R) ↔ (~Q ∧ P → R)))→(((P ∧ ~Q → R) ↔ (P → ~Q → R))))) in n4_87a. - specialize Simp3_27 with (((P ∧ ~Q → R) ↔ (P → ~Q → R) → (~Q → P → R) ↔ (~Q ∧ P → R))) (((~Q → P → R) ↔ (~Q ∧ P → R) → (P ∧ ~Q → R) ↔ (P → ~Q → R))). + replace (((P∧~Q→R)↔(P→~Q→R))↔((~Q→P→R)↔(~Q∧P→R))) + with + ((((P∧~Q→R)↔(P→~Q→R))→((~Q→P→R)↔(~Q∧P→R))) + ∧ + ((((~Q→P→R)↔(~Q∧P→R)))→(((P∧~Q→R)↔(P→~Q→R))))) + in n4_87a. + specialize Simp3_27 with + (((P∧~Q→R)↔(P→~Q→R)→(~Q→P→R)↔(~Q∧P→R))) + (((~Q→P→R)↔(~Q∧P→R)→(P∧~Q→R)↔(P→~Q→R))). intros Simp3_27a. MP Simp3_27a n4_87a. specialize Imp3_31 with (~Q) P R. @@ -3858,7 +4256,7 @@ Theorem n5_6 : ∀ P Q R : Prop, apply n4_64a. apply Equiv4_01. apply Equiv4_01. - Qed. (*A fair amount of manipulation was needed here to pull the relevant biconditional out of the biconditional of biconditionals.*) + Qed. Theorem n5_61 : ∀ P Q : Prop, ((P ∨ Q) ∧ ~Q) ↔ (P ∧ ~Q). @@ -3867,21 +4265,32 @@ Theorem n5_61 : ∀ P Q : Prop, intros n4_74a. specialize n5_32 with (~Q) P (Q∨P). intros n5_32a. - replace (~Q → P ↔ Q ∨ P) with (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) in n4_74a. + replace (~Q → P ↔ Q ∨ P) with + (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) in n4_74a. replace (~Q∧P) with (P∧~Q) in n4_74a. replace (~Q∧(Q∨P)) with ((Q∨P)∧~Q) in n4_74a. replace (Q∨P) with (P∨Q) in n4_74a. - replace (P ∧ ~Q ↔ (P ∨ Q) ∧ ~Q) with ((P ∨ Q) ∧ ~Q ↔ P ∧ ~Q) in n4_74a. + replace (P ∧ ~Q ↔ (P ∨ Q) ∧ ~Q) with + ((P ∨ Q) ∧ ~Q ↔ P ∧ ~Q) in n4_74a. apply n4_74a. apply EqBi. - apply n4_3. (*Not cited exlicitly.*) + specialize n4_21 with ((P∨Q)∧~Q) (P∧~Q). + intros n4_21a. (*Not cited*) + apply n4_21a. apply EqBi. - apply n4_31. (*Not cited explicitly.*) + specialize n4_31 with P Q. + intros n4_31a. (*Not cited*) + apply n4_31a. apply EqBi. - apply n4_3. (*Not cited explicitly.*) + specialize n4_3 with (Q∨P) (~Q). + intros n4_3a. (*Not cited*) + apply n4_3a. apply EqBi. - apply n4_3. (*Not cited explicitly.*) - replace (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) with (~Q → P ↔ Q ∨ P). + specialize n4_3 with P (~Q). + intros n4_3b. (*Not cited*) + apply n4_3b. + replace (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) with + (~Q → P ↔ Q ∨ P). reflexivity. apply EqBi. apply n5_32a. @@ -3897,24 +4306,37 @@ Theorem n5_62 : ∀ P Q : Prop, replace (~Q∨(Q∧P)) with ((Q∧P)∨~Q) in n4_7a. replace (~Q∨P) with (P∨~Q) in n4_7a. replace (Q∧P) with (P∧Q) in n4_7a. - replace (P ∨ ~Q ↔ P ∧ Q ∨ ~Q) with (P ∧ Q ∨ ~Q ↔ P ∨ ~Q) in n4_7a. + replace (P ∨ ~Q ↔ P ∧ Q ∨ ~Q) with + (P ∧ Q ∨ ~Q ↔ P ∨ ~Q) in n4_7a. apply n4_7a. apply EqBi. - apply n4_21. (*Not cited explicitly.*) + specialize n4_21 with (P ∧ Q ∨ ~Q) (P ∨ ~Q). + intros n4_21a. (*Not cited*) + apply n4_21a. apply EqBi. - apply n4_3. (*Not cited explicitly.*) + specialize n4_3 with P Q. + intros n4_3a. (*Not cited*) + apply n4_3a. apply EqBi. - apply n4_31. (*Not cited explicitly.*) + specialize n4_31 with P (~Q). + intros n4_31a. (*Not cited*) + apply n4_31a. apply EqBi. - apply n4_31. (*Not cited explicitly.*) + specialize n4_31 with (Q ∧ P) (~Q). + intros n4_31b. (*Not cited*) + apply n4_31b. replace (~Q ∨ Q ∧ P) with (Q → Q ∧ P). reflexivity. apply EqBi. - apply n4_6. (*Not cited explicitly.*) + specialize n4_6 with Q (Q∧P). + intros n4_6a. (*Not cited*) + apply n4_6a. replace (~Q ∨ P) with (Q → P). reflexivity. apply EqBi. - apply n4_6. (*Not cited explicitly.*) + specialize n4_6 with Q P. + intros n4_6b. (*Not cited*) + apply n4_6b. Qed. Theorem n5_63 : ∀ P Q : Prop, @@ -3925,60 +4347,92 @@ Theorem n5_63 : ∀ P Q : Prop, replace (~~P) with P in n5_62a. replace (Q ∨ P) with (P ∨ Q) in n5_62a. replace ((Q∧~P)∨P) with (P∨(Q∧~P)) in n5_62a. - replace (P ∨ Q ∧ ~ P ↔ P ∨ Q) with (P ∨ Q ↔ P ∨ Q ∧ ~ P) in n5_62a. + replace (P ∨ Q ∧ ~P ↔ P ∨ Q) with + (P ∨ Q ↔ P ∨ Q ∧ ~P) in n5_62a. replace (Q∧~P) with (~P∧Q) in n5_62a. apply n5_62a. apply EqBi. - apply n4_3. (*Not cited explicitly.*) + specialize n4_3 with (~P) Q. + intros n4_3a. + apply n4_3a. (*Not cited*) apply EqBi. - apply n4_21. (*Not cited explicitly.*) + specialize n4_21 with (P∨Q) (P∨(Q∧~P)). + intros n4_21a. (*Not cited*) + apply n4_21a. apply EqBi. - apply n4_31. (*Not cited explicitly.*) + specialize n4_31 with P (Q∧~P). + intros n4_31a. (*Not cited*) + apply n4_31a. apply EqBi. - apply n4_31. (*Not cited explicitly.*) + specialize n4_31 with P Q. + intros n4_31b. (*Not cited*) + apply n4_31b. apply EqBi. - apply n4_13. (*Not cited explicitly.*) + specialize n4_13 with P. + intros n4_13a. (*Not cited*) + apply n4_13a. Qed. Theorem n5_7 : ∀ P Q R : Prop, ((P ∨ R) ↔ (Q ∨ R)) ↔ (R ∨ (P ↔ Q)). Proof. intros P Q R. specialize n5_32 with (~R) (~P) (~Q). - intros n5_32a. (*Not cited.*) + intros n5_32a. (*Not cited*) replace (~R∧~P) with (~(R∨P)) in n5_32a. replace (~R∧~Q) with (~(R∨Q)) in n5_32a. - replace ((~(R∨P))↔(~(R∨Q))) with ((R∨P)↔(R∨Q)) in n5_32a. + replace ((~(R∨P))↔(~(R∨Q))) with + ((R∨P)↔(R∨Q)) in n5_32a. replace ((~P)↔(~Q)) with (P↔Q) in n5_32a. - replace (~R→(P↔Q)) with (~~R∨(P↔Q)) in n5_32a. + replace (~R→(P↔Q)) with + (~~R∨(P↔Q)) in n5_32a. replace (~~R) with R in n5_32a. replace (R∨P) with (P∨R) in n5_32a. replace (R∨Q) with (Q∨R) in n5_32a. - replace ((R∨(P↔Q))↔(P∨R↔Q∨R)) with ((P∨R↔Q∨R)↔(R∨(P↔Q))) in n5_32a. - apply n5_32a. (*Not cited.*) + replace ((R∨(P↔Q))↔(P∨R↔Q∨R)) with + ((P∨R↔Q∨R)↔(R∨(P↔Q))) in n5_32a. + apply n5_32a. (*Not cited*) apply EqBi. - apply n4_21. (*Not cited.*) + specialize n4_21 with ((P∨R)↔(Q∨R)) (R∨(P↔Q)). + intros n4_21a. + apply n4_21a. (*Not cited*) apply EqBi. - apply n4_31. + specialize n4_31 with Q R. + intros n4_31a. (*Not cited*) + apply n4_31a. apply EqBi. - apply n4_31. + specialize n4_31 with P R. + intros n4_31b. (*Not cited*) + apply n4_31b. apply EqBi. - apply n4_13. (*Not cited.*) + specialize n4_13 with R. + intros n4_13a. (*Not cited*) + apply n4_13a. replace (~~R∨(P↔Q)) with (~R→P↔Q). reflexivity. - apply Impl1_01. (*Not cited.*) + apply Impl1_01. (*Not cited*) apply EqBi. - apply Trans4_11. (*Not cited.*) + specialize Trans4_11 with P Q. + intros Trans4_11a. (*Not cited*) + apply Trans4_11a. apply EqBi. - apply Trans4_11. + specialize Trans4_11 with (R ∨ P) (R ∨ Q). + intros Trans4_11a. (*Not cited*) + apply Trans4_11a. replace (~(R∨Q)) with (~R∧~Q). reflexivity. apply EqBi. - apply n4_56. (*Not cited.*) + specialize n4_56 with R Q. + intros n4_56a. (*Not cited*) + apply n4_56a. replace (~(R∨P)) with (~R∧~P). reflexivity. apply EqBi. - apply n4_56. - Qed. (*The proof sketch was indecipherable, but an easy proof was available through n5_32.*) + specialize n4_56 with R P. + intros n4_56b. (*Not cited*) + apply n4_56b. + Qed. + (*The proof sketch was indecipherable, but an + easy proof was available through n5_32.*) Theorem n5_71 : ∀ P Q R : Prop, (Q → ~R) → (((P ∨ Q) ∧ R) ↔ (P ∧ R)). @@ -3990,8 +4444,12 @@ Theorem n5_71 : ∀ P Q R : Prop, specialize n4_51 with Q R. intros n4_51a. replace (~Q∨~R) with (~(Q∧R)) in n4_62a. - replace ((Q→~R)↔~(Q∧R)) with (((Q→~R)→~(Q∧R))∧(~(Q∧R)→(Q→~R))) in n4_62a. - specialize Simp3_26 with ((Q→~R)→~(Q∧R)) (~(Q∧R)→(Q→~R)). + replace ((Q→~R)↔~(Q∧R)) with + (((Q→~R)→~(Q∧R)) + ∧ + (~(Q∧R)→(Q→~R))) in n4_62a. + specialize Simp3_26 with + ((Q→~R)→~(Q∧R)) (~(Q∧R)→(Q→~R)). intros Simp3_26a. MP Simp3_26a n4_62a. specialize n4_74 with (Q∧R) (P∧R). @@ -3999,23 +4457,35 @@ Theorem n5_71 : ∀ P Q R : Prop, Syll Simp3_26a n4_74a Sa. replace (R∧P) with (P∧R) in n4_4a. replace (R∧Q) with (Q∧R) in n4_4a. - replace ((P∧R)∨(Q∧R)) with ((Q∧R)∨(P∧R)) in n4_4a. + replace ((P∧R)∨(Q∧R)) with + ((Q∧R)∨(P∧R)) in n4_4a. replace ((Q∧R)∨(P∧R)) with (R∧(P∨Q)) in Sa. replace (R∧(P∨Q)) with ((P∨Q)∧R) in Sa. - replace ((P∧R)↔((P∨Q)∧R)) with (((P∨Q)∧R)↔(P∧R)) in Sa. + replace ((P∧R)↔((P∨Q)∧R)) with + (((P∨Q)∧R)↔(P∧R)) in Sa. apply Sa. apply EqBi. - apply n4_21. (*Not cited.*) + specialize n4_21 with ((P∨Q)∧R) (P∧R). + intros n4_21a. (*Not cited*) + apply n4_21a. apply EqBi. - apply n4_3. (*Not cited.*) + specialize n4_3 with (P∨Q) R. + intros n4_3a. + apply n4_3a. (*Not cited*) apply EqBi. - apply n4_4a. (*Not cited.*) + apply n4_4a. (*Not cited*) apply EqBi. - apply n4_31. (*Not cited.*) + specialize n4_31 with (Q∧R) (P∧R). + intros n4_31a. (*Not cited*) + apply n4_31a. apply EqBi. - apply n4_3. (*Not cited.*) + specialize n4_3 with Q R. + intros n4_3a. (*Not cited*) + apply n4_3a. apply EqBi. - apply n4_3. (*Not cited.*) + specialize n4_3 with P R. + intros n4_3b. (*Not cited*) + apply n4_3b. apply Equiv4_01. apply EqBi. apply n4_51a. @@ -4032,18 +4502,25 @@ Theorem n5_74 : ∀ P Q R : Prop, split. apply n5_41a. apply n5_41b. - specialize n4_38 with ((P→Q)→(P→R)) ((P→R)→(P→Q)) (P→Q→R) (P→R→Q). + specialize n4_38 with + ((P→Q)→(P→R)) ((P→R)→(P→Q)) + (P→Q→R) (P→R→Q). intros n4_38a. MP n4_38a H. - replace (((P→Q)→(P→R))∧((P→R)→(P→Q))) with ((P→Q)↔(P→R)) in n4_38a. + replace (((P→Q)→(P→R))∧((P→R)→(P→Q))) with + ((P→Q)↔(P→R)) in n4_38a. specialize n4_76 with P (Q→R) (R→Q). intros n4_76a. replace ((Q→R)∧(R→Q)) with (Q↔R) in n4_76a. - replace ((P→Q→R)∧(P→R→Q)) with (P→(Q↔R)) in n4_38a. - replace (((P→Q)↔(P→R))↔(P→Q↔R)) with ((P→(Q↔R))↔((P→Q)↔(P→R))) in n4_38a. + replace ((P→Q→R)∧(P→R→Q)) with + (P→(Q↔R)) in n4_38a. + replace (((P→Q)↔(P→R))↔(P→Q↔R)) with + ((P→(Q↔R))↔((P→Q)↔(P→R))) in n4_38a. apply n4_38a. apply EqBi. - apply n4_21. (*Not cited.*) + specialize n4_21 with (P→Q↔R) ((P→Q)↔(P→R)). + intros n4_21a. (*Not cited*) + apply n4_21a. replace (P→Q↔R) with ((P→Q→R)∧(P→R→Q)). reflexivity. apply EqBi. @@ -4057,20 +4534,26 @@ Theorem n5_75 : ∀ P Q R : Prop, Proof. intros P Q R. specialize n5_6 with P Q R. intros n5_6a. - replace ((P∧~Q→R)↔(P→Q∨R)) with (((P∧~Q→R)→(P→Q∨R))∧((P→Q∨R)→(P∧~Q→R))) in n5_6a. - specialize Simp3_27 with ((P∧~Q→R)→(P→Q∨R)) ((P→Q∨R)→(P∧~Q→R)). + replace ((P∧~Q→R)↔(P→Q∨R)) with + (((P∧~Q→R)→(P→Q∨R)) + ∧ + ((P→Q∨R)→(P∧~Q→R))) in n5_6a. + specialize Simp3_27 with + ((P∧~Q→R)→(P→Q∨R)) ((P→Q∨R)→(P∧~Q→R)). intros Simp3_27a. MP Simp3_27a n5_6a. specialize Simp3_26 with (P→(Q∨R)) ((Q∨R)→P). intros Simp3_26a. - replace ((P→(Q∨R))∧((Q∨R)→P)) with (P↔(Q∨R)) in Simp3_26a. + replace ((P→(Q∨R))∧((Q∨R)→P)) with + (P↔(Q∨R)) in Simp3_26a. Syll Simp3_26a Simp3_27a Sa. specialize Simp3_27 with (R→~Q) (P↔(Q∨R)). intros Simp3_27b. Syll Simp3_27b Sa Sb. specialize Simp3_27 with (P→(Q∨R)) ((Q∨R)→P). intros Simp3_27c. - replace ((P→(Q∨R))∧((Q∨R)→P)) with (P↔(Q∨R)) in Simp3_27c. + replace ((P→(Q∨R))∧((Q∨R)→P)) with + (P↔(Q∨R)) in Simp3_27c. Syll Simp3_27b Simp3_27c Sc. specialize n4_77 with P Q R. intros n4_77a. @@ -4084,21 +4567,27 @@ Theorem n5_75 : ∀ P Q R : Prop, split. apply Sd. apply Simp3_26b. - specialize Comp3_43 with ((R→~Q)∧(P↔(Q∨R))) (R→P) (R→~Q). + specialize Comp3_43 with + ((R→~Q)∧(P↔(Q∨R))) (R→P) (R→~Q). intros Comp3_43a. MP Comp3_43a H. specialize Comp3_43 with R P (~Q). intros Comp3_43b. Syll Comp3_43a Comp3_43b Se. - clear n5_6a. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d. clear Simp3_26a. clear Simp3_26b. clear Comp3_43a. clear Comp3_43b. clear Sa. clear Sc. clear Sd. clear H. clear n4_77a. + clear n5_6a. clear Simp3_27a. clear Simp3_27b. + clear Simp3_27c. clear Simp3_27d. clear Simp3_26a. + clear Simp3_26b. clear Comp3_43a. clear Comp3_43b. + clear Sa. clear Sc. clear Sd. clear H. clear n4_77a. Conj Sb Se. split. apply Sb. apply Se. - specialize Comp3_43 with ((R→~Q)∧(P↔Q∨R)) (P∧~Q→R) (R→P∧~Q). + specialize Comp3_43 with + ((R→~Q)∧(P↔Q∨R)) (P∧~Q→R) (R→P∧~Q). intros Comp3_43c. MP Comp3_43c H. - replace ((P∧~Q→R)∧(R→P∧~Q)) with (P∧~Q↔R) in Comp3_43c. + replace ((P∧~Q→R)∧(R→P∧~Q)) with + (P∧~Q↔R) in Comp3_43c. apply Comp3_43c. apply Equiv4_01. apply EqBi. |