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(* Title: FOL/ex/If.ML
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1991 University of Cambridge
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First-Order Logic: the 'if' example.
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*)
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paulson@14205
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haftmann@16417
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theory If imports FOL begin
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haftmann@35413
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definition "if" :: "[o,o,o]=>o" where
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wenzelm@19792
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"if(P,Q,R) == P&Q | ~P&R"
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paulson@14205
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lemma ifI:
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"[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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paulson@14205
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apply (simp add: if_def)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply blast
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done
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lemma ifE:
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"[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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paulson@14205
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apply (simp add: if_def)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply blast
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done
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lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (rule iffI)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (erule ifE)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (erule ifE)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (rule ifI)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (rule ifI)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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oops
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text{*Trying again from the beginning in order to use @{text blast}*}
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declare ifI [intro!]
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declare ifE [elim!]
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lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))"
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by blast
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lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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by blast
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text{*Trying again from the beginning in order to prove from the definitions*}
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lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (simp add: if_def)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply blast
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done
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text{*An invalid formula. High-level rules permit a simpler diagnosis*}
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lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply auto
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--{* @{subgoals[display,indent=0,margin=65]} *}
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(*The next step will fail unless subgoals remain*)
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apply (tactic all_tac)
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oops
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text{*Trying again from the beginning in order to prove from the definitions*}
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lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))"
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (simp add: if_def)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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apply (auto)
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--{* @{subgoals[display,indent=0,margin=65]} *}
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(*The next step will fail unless subgoals remain*)
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apply (tactic all_tac)
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oops
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end
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