(* Title: Doc/ZF/If.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1991 University of Cambridge First-Order Logic: the 'if' example. *) theory If imports "~~/src/FOL/FOL" begin definition "if" :: "[o,o,o]=>o" where "if(P,Q,R) == P&Q | ~P&R" lemma ifI: "[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)" --{* @{subgoals[display,indent=0,margin=65]} *} apply (simp add: if_def) --{* @{subgoals[display,indent=0,margin=65]} *} apply blast done lemma ifE: "[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S" --{* @{subgoals[display,indent=0,margin=65]} *} apply (simp add: if_def) --{* @{subgoals[display,indent=0,margin=65]} *} apply 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))" --{* @{subgoals[display,indent=0,margin=65]} *} apply (rule iffI) --{* @{subgoals[display,indent=0,margin=65]} *} apply (erule ifE) --{* @{subgoals[display,indent=0,margin=65]} *} apply (erule ifE) --{* @{subgoals[display,indent=0,margin=65]} *} apply (rule ifI) --{* @{subgoals[display,indent=0,margin=65]} *} apply (rule ifI) --{* @{subgoals[display,indent=0,margin=65]} *} 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))" --{* @{subgoals[display,indent=0,margin=65]} *} 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))" --{* @{subgoals[display,indent=0,margin=65]} *} apply (simp add: if_def) --{* @{subgoals[display,indent=0,margin=65]} *} apply blast done 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))" --{* @{subgoals[display,indent=0,margin=65]} *} apply auto --{* @{subgoals[display,indent=0,margin=65]} *} (*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))" --{* @{subgoals[display,indent=0,margin=65]} *} apply (simp add: if_def) --{* @{subgoals[display,indent=0,margin=65]} *} apply (auto) --{* @{subgoals[display,indent=0,margin=65]} *} (*The next step will fail unless subgoals remain*) apply (tactic all_tac) oops end