(* Title: FOL/ex/If.ML ID: $Id: If.thy,v 1.5 2005/06/17 14:15:09 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1991 University of Cambridge *) header {* First-Order Logic: the 'if' example *} theory If imports FOL begin constdefs if :: "[o,o,o]=>o" "if(P,Q,R) == P&Q | ~P&R" lemma ifI: "[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)" apply (simp add: if_def, blast) done lemma ifE: "[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S" apply (simp add: if_def, blast) done lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))" apply (rule iffI) apply (erule ifE) apply (erule ifE) apply (rule ifI) apply (rule ifI) oops text{*Trying again from the beginning in order to use @{text blast}*} declare ifI [intro!] declare ifE [elim!] lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))" by blast lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))" by blast text{*Trying again from the beginning in order to prove from the definitions*} lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))" by (simp add: if_def, blast) text{*An invalid formula. High-level rules permit a simpler diagnosis*} lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))" apply auto -- {*The next step will fail unless subgoals remain*} apply (tactic all_tac) oops text{*Trying again from the beginning in order to prove from the definitions*} lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))" apply (simp add: if_def, auto) -- {*The next step will fail unless subgoals remain*} apply (tactic all_tac) oops end
lemma ifI:
[| P ==> Q; ¬ P ==> R |] ==> if(P, Q, R)
lemma ifE:
[| if(P, Q, R); [| P; Q |] ==> S; [| ¬ P; R |] ==> S |] ==> S
lemma if_commute:
if(P, if(Q, A, B), if(Q, C, D)) <-> if(Q, if(P, A, C), if(P, B, D))
lemma
if(if(P, Q, R), A, B) <-> if(P, if(Q, A, B), if(R, A, B))
lemma
if(if(P, Q, R), A, B) <-> if(P, if(Q, A, B), if(R, A, B))