Theory If

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theory If
imports FOL
begin

(*  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))