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theory Propositional_Int(* Title: FOLP/ex/Propositional_Int.thy ID: $Id: Propositional_Int.thy,v 1.1 2008/03/26 21:38:55 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1991 University of Cambridge *) header {* First-Order Logic: propositional examples *} theory Propositional_Int imports IFOLP begin text "commutative laws of & and | " lemma "?p : P & Q --> Q & P" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : P | Q --> Q | P" by (tactic {* IntPr.fast_tac 1 *}) text "associative laws of & and | " lemma "?p : (P & Q) & R --> P & (Q & R)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P | Q) | R --> P | (Q | R)" by (tactic {* IntPr.fast_tac 1 *}) text "distributive laws of & and | " lemma "?p : (P & Q) | R --> (P | R) & (Q | R)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P | R) & (Q | R) --> (P & Q) | R" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P | Q) & R --> (P & R) | (Q & R)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P & R) | (Q & R) --> (P | Q) & R" by (tactic {* IntPr.fast_tac 1 *}) text "Laws involving implication" lemma "?p : (P-->R) & (Q-->R) <-> (P|Q --> R)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P & Q --> R) <-> (P--> (Q-->R))" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : ((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : ~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P --> Q & R) <-> (P-->Q) & (P-->R)" by (tactic {* IntPr.fast_tac 1 *}) text "Propositions-as-types" (*The combinator K*) lemma "?p : P --> (Q --> P)" by (tactic {* IntPr.fast_tac 1 *}) (*The combinator S*) lemma "?p : (P-->Q-->R) --> (P-->Q) --> (P-->R)" by (tactic {* IntPr.fast_tac 1 *}) (*Converse is classical*) lemma "?p : (P-->Q) | (P-->R) --> (P --> Q | R)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (P-->Q) --> (~Q --> ~P)" by (tactic {* IntPr.fast_tac 1 *}) text "Schwichtenberg's examples (via T. Nipkow)" lemma stab_imp: "?p : (((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q" by (tactic {* IntPr.fast_tac 1 *}) lemma stab_to_peirce: "?p : (((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q) --> ((P --> Q) --> P) --> P" by (tactic {* IntPr.fast_tac 1 *}) lemma peirce_imp1: "?p : (((Q --> R) --> Q) --> Q) --> (((P --> Q) --> R) --> P --> Q) --> P --> Q" by (tactic {* IntPr.fast_tac 1 *}) lemma peirce_imp2: "?p : (((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P" by (tactic {* IntPr.fast_tac 1 *}) lemma mints: "?p : ((((P --> Q) --> P) --> P) --> Q) --> Q" by (tactic {* IntPr.fast_tac 1 *}) lemma mints_solovev: "?p : (P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R" by (tactic {* IntPr.fast_tac 1 *}) lemma tatsuta: "?p : (((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: "?p : (((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