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(**** FOL examples ****)
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Pretty.setmargin 72; (*existing macros just allow this margin*)
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print_depth 0;
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(*** Intuitionistic examples ***)
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(*Quantifier example from Logic&Computation*)
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goal Int_Rule.thy "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))";
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by (resolve_tac [impI] 1);
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by (resolve_tac [allI] 1);
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by (resolve_tac [exI] 1);
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by (eresolve_tac [exE] 1);
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choplev 2;
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by (eresolve_tac [exE] 1);
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by (resolve_tac [exI] 1);
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by (eresolve_tac [allE] 1);
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by (assume_tac 1);
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(*Example of Dyckhoff's method*)
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goalw Int_Rule.thy [not_def] "~ ~ ((P-->Q) | (Q-->P))";
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by (resolve_tac [impI] 1);
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by (eresolve_tac [disj_impE] 1);
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by (eresolve_tac [imp_impE] 1);
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by (eresolve_tac [imp_impE] 1);
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by (REPEAT (eresolve_tac [FalseE] 2));
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by (assume_tac 1);
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- goal Int_Rule.thy "(EX ay. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))";
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Level 0
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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- by (resolve_tac [impI] 1);
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Level 1
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. EX y. ALL x. Q(x,y) ==> ALL x. EX y. Q(x,y)
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- by (resolve_tac [allI] 1);
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Level 2
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x. EX y. ALL x. Q(x,y) ==> EX y. Q(x,y)
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- by (resolve_tac [exI] 1);
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Level 3
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x. EX y. ALL x. Q(x,y) ==> Q(x,?y2(x))
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- by (eresolve_tac [exE] 1);
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Level 4
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x y. ALL x. Q(x,y) ==> Q(x,?y2(x))
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- choplev 2;
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Level 2
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x. EX y. ALL x. Q(x,y) ==> EX y. Q(x,y)
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- by (eresolve_tac [exE] 1);
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Level 3
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x y. ALL x. Q(x,y) ==> EX y. Q(x,y)
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- by (resolve_tac [exI] 1);
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Level 4
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x y. ALL x. Q(x,y) ==> Q(x,?y3(x,y))
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- by (eresolve_tac [allE] 1);
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Level 5
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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1. !!x y. Q(?x4(x,y),y) ==> Q(x,?y3(x,y))
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- by (assume_tac 1);
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Level 6
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(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))
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No subgoals!
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> goalw Int_Rule.thy [not_def] "~ ~ ((P-->Q) | (Q-->P))";
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Level 0
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~ ~ ((P --> Q) | (Q --> P))
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1. ((P --> Q) | (Q --> P) --> False) --> False
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> by (resolve_tac [impI] 1);
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Level 1
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~ ~ ((P --> Q) | (Q --> P))
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1. (P --> Q) | (Q --> P) --> False ==> False
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> by (eresolve_tac [disj_impE] 1);
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Level 2
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~ ~ ((P --> Q) | (Q --> P))
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1. [| (P --> Q) --> False; (Q --> P) --> False |] ==> False
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> by (eresolve_tac [imp_impE] 1);
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Level 3
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~ ~ ((P --> Q) | (Q --> P))
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1. [| (Q --> P) --> False; P; Q --> False |] ==> Q
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2. [| (Q --> P) --> False; False |] ==> False
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> by (eresolve_tac [imp_impE] 1);
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Level 4
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~ ~ ((P --> Q) | (Q --> P))
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1. [| P; Q --> False; Q; P --> False |] ==> P
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2. [| P; Q --> False; False |] ==> Q
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3. [| (Q --> P) --> False; False |] ==> False
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> by (REPEAT (eresolve_tac [FalseE] 2));
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Level 5
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~ ~ ((P --> Q) | (Q --> P))
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1. [| P; Q --> False; Q; P --> False |] ==> P
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> by (assume_tac 1);
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Level 6
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~ ~ ((P --> Q) | (Q --> P))
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No subgoals!
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(*** Classical examples ***)
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goal cla_thy "EX y. ALL x. P(y)-->P(x)";
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by (resolve_tac [exCI] 1);
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by (resolve_tac [allI] 1);
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by (resolve_tac [impI] 1);
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by (eresolve_tac [allE] 1);
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prth (allI RSN (2,swap));
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by (eresolve_tac [it] 1);
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by (resolve_tac [impI] 1);
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by (eresolve_tac [notE] 1);
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by (assume_tac 1);
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goal cla_thy "EX y. ALL x. P(y)-->P(x)";
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by (best_tac FOL_dup_cs 1);
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- goal cla_thy "EX y. ALL x. P(y)-->P(x)";
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Level 0
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EX y. ALL x. P(y) --> P(x)
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1. EX y. ALL x. P(y) --> P(x)
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- by (resolve_tac [exCI] 1);
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Level 1
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EX y. ALL x. P(y) --> P(x)
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1. ALL y. ~(ALL x. P(y) --> P(x)) ==> ALL x. P(?a) --> P(x)
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- by (resolve_tac [allI] 1);
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Level 2
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EX y. ALL x. P(y) --> P(x)
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1. !!x. ALL y. ~(ALL x. P(y) --> P(x)) ==> P(?a) --> P(x)
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- by (resolve_tac [impI] 1);
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Level 3
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EX y. ALL x. P(y) --> P(x)
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1. !!x. [| ALL y. ~(ALL x. P(y) --> P(x)); P(?a) |] ==> P(x)
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- by (eresolve_tac [allE] 1);
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Level 4
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EX y. ALL x. P(y) --> P(x)
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1. !!x. [| P(?a); ~(ALL xa. P(?y3(x)) --> P(xa)) |] ==> P(x)
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- prth (allI RSN (2,swap));
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[| ~(ALL x. ?P1(x)); !!x. ~?Q ==> ?P1(x) |] ==> ?Q
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- by (eresolve_tac [it] 1);
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Level 5
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EX y. ALL x. P(y) --> P(x)
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1. !!x xa. [| P(?a); ~P(x) |] ==> P(?y3(x)) --> P(xa)
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- by (resolve_tac [impI] 1);
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Level 6
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EX y. ALL x. P(y) --> P(x)
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1. !!x xa. [| P(?a); ~P(x); P(?y3(x)) |] ==> P(xa)
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- by (eresolve_tac [notE] 1);
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Level 7
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EX y. ALL x. P(y) --> P(x)
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1. !!x xa. [| P(?a); P(?y3(x)) |] ==> P(x)
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- by (assume_tac 1);
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Level 8
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EX y. ALL x. P(y) --> P(x)
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No subgoals!
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- goal cla_thy "EX y. ALL x. P(y)-->P(x)";
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Level 0
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EX y. ALL x. P(y) --> P(x)
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1. EX y. ALL x. P(y) --> P(x)
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- by (best_tac FOL_dup_cs 1);
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Level 1
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EX y. ALL x. P(y) --> P(x)
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No subgoals!
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(**** finally, the example FOL/ex/if.ML ****)
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> val prems = goalw if_thy [if_def]
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# "[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)";
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Level 0
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if(P,Q,R)
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1. P & Q | ~P & R
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> by (Classical.fast_tac (FOL_cs addIs prems) 1);
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Level 1
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if(P,Q,R)
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No subgoals!
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> val ifI = result();
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> val major::prems = goalw if_thy [if_def]
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# "[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S";
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Level 0
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S
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1. S
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> by (cut_facts_tac [major] 1);
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Level 1
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S
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1. P & Q | ~P & R ==> S
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> by (Classical.fast_tac (FOL_cs addIs prems) 1);
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Level 2
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S
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No subgoals!
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> val ifE = result();
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> goal if_thy "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))";
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Level 0
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. 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 (resolve_tac [iffI] 1);
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Level 1
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. if(P,if(Q,A,B),if(Q,C,D)) ==> if(Q,if(P,A,C),if(P,B,D))
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2. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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> by (eresolve_tac [ifE] 1);
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Level 2
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. [| P; if(Q,A,B) |] ==> if(Q,if(P,A,C),if(P,B,D))
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2. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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3. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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> by (eresolve_tac [ifE] 1);
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Level 3
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. [| P; Q; A |] ==> if(Q,if(P,A,C),if(P,B,D))
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2. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D))
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3. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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4. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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> by (resolve_tac [ifI] 1);
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Level 4
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. [| P; Q; A; Q |] ==> if(P,A,C)
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2. [| P; Q; A; ~Q |] ==> if(P,B,D)
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3. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D))
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4. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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5. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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> by (resolve_tac [ifI] 1);
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Level 5
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. [| P; Q; A; Q; P |] ==> A
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2. [| P; Q; A; Q; ~P |] ==> C
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3. [| P; Q; A; ~Q |] ==> if(P,B,D)
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4. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D))
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5. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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6. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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> choplev 0;
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Level 0
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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1. if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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> val if_cs = FOL_cs addSIs [ifI] addSEs[ifE];
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> by (Classical.fast_tac if_cs 1);
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Level 1
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if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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No subgoals!
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> val if_commute = result();
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> goal if_thy "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))";
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Level 0
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if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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> by (Classical.fast_tac if_cs 1);
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Level 1
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if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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No subgoals!
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> val nested_ifs = result();
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> choplev 0;
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Level 0
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if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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> by (rewrite_goals_tac [if_def]);
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Level 1
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if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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273 |
1. (P & Q | ~P & R) & A | ~(P & Q | ~P & R) & B <->
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274 |
P & (Q & A | ~Q & B) | ~P & (R & A | ~R & B)
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275 |
> by (Classical.fast_tac FOL_cs 1);
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276 |
Level 2
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277 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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278 |
No subgoals!
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279 |
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280 |
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281 |
> goal if_thy "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))";
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282 |
Level 0
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283 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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284 |
1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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285 |
> by (REPEAT (Classical.step_tac if_cs 1));
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286 |
Level 1
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287 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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288 |
1. [| A; ~P; R; ~P; R |] ==> B
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289 |
2. [| B; ~P; ~R; ~P; ~R |] ==> A
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290 |
3. [| ~P; R; B; ~P; R |] ==> A
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291 |
4. [| ~P; ~R; A; ~B; ~P |] ==> R
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292 |
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293 |
> choplev 0;
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294 |
Level 0
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295 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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296 |
1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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297 |
> by (rewrite_goals_tac [if_def]);
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298 |
Level 1
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299 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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300 |
1. (P & Q | ~P & R) & A | ~(P & Q | ~P & R) & B <->
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301 |
P & (Q & A | ~Q & B) | ~P & (R & B | ~R & A)
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302 |
> by (Classical.fast_tac FOL_cs 1);
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303 |
by: tactic failed
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304 |
Exception- ERROR raised
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305 |
Exception failure raised
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306 |
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307 |
> by (REPEAT (Classical.step_tac FOL_cs 1));
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308 |
Level 2
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309 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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310 |
1. [| A; ~P; R; ~P; R; ~False |] ==> B
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311 |
2. [| A; ~P; R; R; ~False; ~B; ~B |] ==> Q
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312 |
3. [| B; ~P; ~R; ~P; ~A |] ==> R
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313 |
4. [| B; ~P; ~R; ~Q; ~A |] ==> R
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314 |
5. [| B; ~R; ~P; ~A; ~R; Q; ~False |] ==> A
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315 |
6. [| ~P; R; B; ~P; R; ~False |] ==> A
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316 |
7. [| ~P; ~R; A; ~B; ~R |] ==> P
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317 |
8. [| ~P; ~R; A; ~B; ~R |] ==> Q
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