doc-src/ZF/If.thy
 author boehmes Tue, 07 Dec 2010 15:44:38 +0100 changeset 41064 0c447a17770a parent 35416 d8d7d1b785af child 42637 381fdcab0f36 permissions -rw-r--r--
updated SMT certificates
```
(*  Title:      FOL/ex/If.ML
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

First-Order Logic: the 'if' example.
*)

theory If imports 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]} *}
--{* @{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]} *}
--{* @{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]} *}
--{* @{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]} *}