(* Title: FOL/ex/Propositional_Int.thy ID: $Id: Propositional_Int.thy,v 1.1 2007/07/22 20:01:30 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1991 University of Cambridge *) header {* First-Order Logic: propositional examples (intuitionistic version) *} theory Propositional_Int imports IFOL begin text {* commutative laws of @{text "&"} and @{text "|"} *} lemma "P & Q --> Q & P" by (tactic "IntPr.fast_tac 1") lemma "P | Q --> Q | P" by (tactic "IntPr.fast_tac 1") text {* associative laws of @{text "&"} and @{text "|"} *} lemma "(P & Q) & R --> P & (Q & R)" by (tactic "IntPr.fast_tac 1") lemma "(P | Q) | R --> P | (Q | R)" by (tactic "IntPr.fast_tac 1") text {* distributive laws of @{text "&"} and @{text "|"} *} lemma "(P & Q) | R --> (P | R) & (Q | R)" by (tactic "IntPr.fast_tac 1") lemma "(P | R) & (Q | R) --> (P & Q) | R" by (tactic "IntPr.fast_tac 1") lemma "(P | Q) & R --> (P & R) | (Q & R)" by (tactic "IntPr.fast_tac 1") lemma "(P & R) | (Q & R) --> (P | Q) & R" by (tactic "IntPr.fast_tac 1") text {* Laws involving implication *} lemma "(P-->R) & (Q-->R) <-> (P|Q --> R)" by (tactic "IntPr.fast_tac 1") lemma "(P & Q --> R) <-> (P--> (Q-->R))" by (tactic "IntPr.fast_tac 1") lemma "((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R" by (tactic "IntPr.fast_tac 1") lemma "~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)" by (tactic "IntPr.fast_tac 1") lemma "(P --> Q & R) <-> (P-->Q) & (P-->R)" by (tactic "IntPr.fast_tac 1") text {* Propositions-as-types *} -- {* The combinator K *} lemma "P --> (Q --> P)" by (tactic "IntPr.fast_tac 1") -- {* The combinator S *} lemma "(P-->Q-->R) --> (P-->Q) --> (P-->R)" by (tactic "IntPr.fast_tac 1") -- {* Converse is classical *} lemma "(P-->Q) | (P-->R) --> (P --> Q | R)" by (tactic "IntPr.fast_tac 1") lemma "(P-->Q) --> (~Q --> ~P)" by (tactic "IntPr.fast_tac 1") text {* Schwichtenberg's examples (via T. Nipkow) *} lemma stab_imp: "(((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q" by (tactic "IntPr.fast_tac 1") lemma stab_to_peirce: "(((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q) --> ((P --> Q) --> P) --> P" by (tactic "IntPr.fast_tac 1") lemma peirce_imp1: "(((Q --> R) --> Q) --> Q) --> (((P --> Q) --> R) --> P --> Q) --> P --> Q" by (tactic "IntPr.fast_tac 1") lemma peirce_imp2: "(((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P" by (tactic "IntPr.fast_tac 1") lemma mints: "((((P --> Q) --> P) --> P) --> Q) --> Q" by (tactic "IntPr.fast_tac 1") lemma mints_solovev: "(P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R" by (tactic "IntPr.fast_tac 1") lemma tatsuta: "(((P7 --> P1) --> P10) --> P4 --> P5) --> (((P8 --> P2) --> P9) --> P3 --> P10) --> (P1 --> P8) --> P6 --> P7 --> (((P3 --> P2) --> P9) --> P4) --> (P1 --> P3) --> (((P6 --> P1) --> P2) --> P9) --> P5" by (tactic "IntPr.fast_tac 1") lemma tatsuta1: "(((P8 --> P2) --> P9) --> P3 --> P10) --> (((P3 --> P2) --> P9) --> P4) --> (((P6 --> P1) --> P2) --> P9) --> (((P7 --> P1) --> P10) --> P4 --> P5) --> (P1 --> P3) --> (P1 --> P8) --> P6 --> P7 --> P5" by (tactic "IntPr.fast_tac 1") end
lemma
P ∧ Q --> Q ∧ P
lemma
P ∨ Q --> Q ∨ P
lemma
(P ∧ Q) ∧ R --> P ∧ Q ∧ R
lemma
(P ∨ Q) ∨ R --> P ∨ Q ∨ R
lemma
P ∧ Q ∨ R --> (P ∨ R) ∧ (Q ∨ R)
lemma
(P ∨ R) ∧ (Q ∨ R) --> P ∧ Q ∨ R
lemma
(P ∨ Q) ∧ R --> P ∧ R ∨ Q ∧ R
lemma
P ∧ R ∨ Q ∧ R --> (P ∨ Q) ∧ R
lemma
(P --> R) ∧ (Q --> R) <-> P ∨ Q --> R
lemma
(P ∧ Q --> R) <-> P --> Q --> R
lemma
((P --> R) --> R) --> ((Q --> R) --> R) --> (P ∧ Q --> R) --> R
lemma
¬ (P --> R) --> ¬ (Q --> R) --> ¬ (P ∧ Q --> R)
lemma
(P --> Q ∧ R) <-> (P --> Q) ∧ (P --> R)
lemma
P --> Q --> P
lemma
(P --> Q --> R) --> (P --> Q) --> P --> R
lemma
(P --> Q) ∨ (P --> R) --> P --> Q ∨ R
lemma
(P --> Q) --> ¬ Q --> ¬ P
lemma stab_imp:
(((Q --> R) --> R) --> Q) --> (((P --> Q) --> R) --> R) --> P --> Q
lemma stab_to_peirce:
(((P --> R) --> R) --> P) -->
(((Q --> R) --> R) --> Q) --> ((P --> Q) --> P) --> P
lemma peirce_imp1:
(((Q --> R) --> Q) --> Q) --> (((P --> Q) --> R) --> P --> Q) --> P --> Q
lemma peirce_imp2:
(((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P
lemma mints:
((((P --> Q) --> P) --> P) --> Q) --> Q
lemma mints_solovev:
(P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R
lemma tatsuta:
(((P7.0 --> P1.0) --> P10.0) --> P4.0 --> P5.0) -->
(((P8.0 --> P2.0) --> P9.0) --> P3.0 --> P10.0) -->
(P1.0 --> P8.0) -->
P6.0 -->
P7.0 -->
(((P3.0 --> P2.0) --> P9.0) --> P4.0) -->
(P1.0 --> P3.0) --> (((P6.0 --> P1.0) --> P2.0) --> P9.0) --> P5.0
lemma tatsuta1:
(((P8.0 --> P2.0) --> P9.0) --> P3.0 --> P10.0) -->
(((P3.0 --> P2.0) --> P9.0) --> P4.0) -->
(((P6.0 --> P1.0) --> P2.0) --> P9.0) -->
(((P7.0 --> P1.0) --> P10.0) --> P4.0 --> P5.0) -->
(P1.0 --> P3.0) --> (P1.0 --> P8.0) --> P6.0 --> P7.0 --> P5.0