Theory Propositional

theory Propositional
imports LK
(*  Title:      Sequents/LK/Propositional.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)


header {* Classical sequent calculus: examples with propositional connectives *}

theory Propositional
imports LK
begin

text "absorptive laws of & and | "

lemma "|- P & P <-> P"
by fast_prop

lemma "|- P | P <-> P"
by fast_prop


text "commutative laws of & and | "

lemma "|- P & Q <-> Q & P"
by fast_prop

lemma "|- P | Q <-> Q | P"
by fast_prop


text "associative laws of & and | "

lemma "|- (P & Q) & R <-> P & (Q & R)"
by fast_prop

lemma "|- (P | Q) | R <-> P | (Q | R)"
by fast_prop


text "distributive laws of & and | "

lemma "|- (P & Q) | R <-> (P | R) & (Q | R)"
by fast_prop

lemma "|- (P | Q) & R <-> (P & R) | (Q & R)"
by fast_prop


text "Laws involving implication"

lemma "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"
by fast_prop

lemma "|- (P & Q --> R) <-> (P--> (Q-->R))"
by fast_prop

lemma "|- (P --> Q & R) <-> (P-->Q) & (P-->R)"
by fast_prop


text "Classical theorems"

lemma "|- P|Q --> P| ~P&Q"
by fast_prop

lemma "|- (P-->Q)&(~P-->R) --> (P&Q | R)"
by fast_prop

lemma "|- P&Q | ~P&R <-> (P-->Q)&(~P-->R)"
by fast_prop

lemma "|- (P-->Q) | (P-->R) <-> (P --> Q | R)"
by fast_prop


(*If and only if*)

lemma "|- (P<->Q) <-> (Q<->P)"
by fast_prop

lemma "|- ~ (P <-> ~P)"
by fast_prop


(*Sample problems from
F. J. Pelletier,
Seventy-Five Problems for Testing Automatic Theorem Provers,
J. Automated Reasoning 2 (1986), 191-216.
Errata, JAR 4 (1988), 236-236.
*)


(*1*)
lemma "|- (P-->Q) <-> (~Q --> ~P)"
by fast_prop

(*2*)
lemma "|- ~ ~ P <-> P"
by fast_prop

(*3*)
lemma "|- ~(P-->Q) --> (Q-->P)"
by fast_prop

(*4*)
lemma "|- (~P-->Q) <-> (~Q --> P)"
by fast_prop

(*5*)
lemma "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))"
by fast_prop

(*6*)
lemma "|- P | ~ P"
by fast_prop

(*7*)
lemma "|- P | ~ ~ ~ P"
by fast_prop

(*8. Peirce's law*)
lemma "|- ((P-->Q) --> P) --> P"
by fast_prop

(*9*)
lemma "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
by fast_prop

(*10*)
lemma "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q"
by fast_prop

(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
lemma "|- P<->P"
by fast_prop

(*12. "Dijkstra's law"*)
lemma "|- ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"
by fast_prop

(*13. Distributive law*)
lemma "|- P | (Q & R) <-> (P | Q) & (P | R)"
by fast_prop

(*14*)
lemma "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))"
by fast_prop

(*15*)
lemma "|- (P --> Q) <-> (~P | Q)"
by fast_prop

(*16*)
lemma "|- (P-->Q) | (Q-->P)"
by fast_prop

(*17*)
lemma "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
by fast_prop

end