# Theory If

Up to index of Isabelle/FOL/ex

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))