src/FOL/ex/int.ML
author lcp
Tue, 18 Jan 1994 16:58:41 +0100
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(*  Title: 	FOL/ex/int
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    ID:         $Id$
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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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Intuitionistic First-Order Logic
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Single-step commands:
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by (Int.step_tac 1);
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by (biresolve_tac safe_brls 1);
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by (biresolve_tac haz_brls 1);
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by (assume_tac 1);
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by (Int.safe_tac 1);
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by (Int.mp_tac 1);
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by (Int.fast_tac 1);
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*)
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writeln"File FOL/ex/int.";
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(*Note: for PROPOSITIONAL formulae...
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  ~A is classically provable iff it is intuitionistically provable.  
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  Therefore A is classically provable iff ~~A is intuitionistically provable.
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Let Q be the conjuction of the propositions A|~A, one for each atom A in
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P.  If P is provable classically, then clearly P&Q is provable
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intuitionistically, so ~~(P&Q) is also provable intuitionistically.
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The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
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since ~~Q is intuitionistically provable.  Finally, if P is a negation then
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~~P is intuitionstically equivalent to P.  [Andy Pitts]
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*)
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goal IFOL.thy "~~(P&Q) <-> ~~P & ~~Q";
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by (Int.fast_tac 1);
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result();
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goal IFOL.thy "~~~P <-> ~P";
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by (Int.fast_tac 1);
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result();
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goal IFOL.thy "~~((P --> Q | R)  -->  (P-->Q) | (P-->R))";
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by (Int.fast_tac 1);
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result();
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goal IFOL.thy "(P<->Q) <-> (Q<->P)";
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by (Int.fast_tac 1);
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result();
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writeln"Lemmas for the propositional double-negation translation";
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goal IFOL.thy "P --> ~~P";
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by (Int.fast_tac 1);
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result();
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goal IFOL.thy "~~(~~P --> P)";
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by (Int.fast_tac 1);
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result();
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goal IFOL.thy "~~P & ~~(P --> Q) --> ~~Q";
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by (Int.fast_tac 1);
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result();
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writeln"The following are classically but not constructively valid.";
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(*The attempt to prove them terminates quickly!*)
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goal IFOL.thy "((P-->Q) --> P)  -->  P";
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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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(*Check that subgoals remain: proof failed.*)
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getgoal 1;  
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goal IFOL.thy "(P&Q-->R)  -->  (P-->R) | (Q-->R)";
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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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getgoal 1;  
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writeln"Intuitionistic FOL: propositional problems based on Pelletier.";
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writeln"Problem ~~1";
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goal IFOL.thy "~~((P-->Q)  <->  (~Q --> ~P))";
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by (Int.fast_tac 1);
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result();
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(*5 secs*)
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writeln"Problem ~~2";
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goal IFOL.thy "~~(~~P  <->  P)";
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by (Int.fast_tac 1);
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result();
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(*1 secs*)
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writeln"Problem 3";
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goal IFOL.thy "~(P-->Q) --> (Q-->P)";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~4";
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goal IFOL.thy "~~((~P-->Q)  <->  (~Q --> P))";
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by (Int.fast_tac 1);
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result();
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(*9 secs*)
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writeln"Problem ~~5";
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goal IFOL.thy "~~((P|Q-->P|R) --> P|(Q-->R))";
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by (Int.fast_tac 1);
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result();
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(*10 secs*)
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writeln"Problem ~~6";
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goal IFOL.thy "~~(P | ~P)";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~7";
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goal IFOL.thy "~~(P | ~~~P)";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~8.  Peirce's law";
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goal IFOL.thy "~~(((P-->Q) --> P)  -->  P)";
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by (Int.fast_tac 1);
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result();
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writeln"Problem 9";
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goal IFOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
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by (Int.fast_tac 1);
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result();
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(*9 secs*)
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writeln"Problem 10";
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goal IFOL.thy "(Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)";
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by (Int.fast_tac 1);
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result();
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writeln"11.  Proved in each direction (incorrectly, says Pelletier!!) ";
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goal IFOL.thy "P<->P";
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by (Int.fast_tac 1);
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writeln"Problem ~~12.  Dijkstra's law  ";
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goal IFOL.thy "~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))";
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by (Int.fast_tac 1);
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result();
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goal IFOL.thy "((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))";
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by (Int.fast_tac 1);
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result();
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writeln"Problem 13.  Distributive law";
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goal IFOL.thy "P | (Q & R)  <-> (P | Q) & (P | R)";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~14";
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goal IFOL.thy "~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~15";
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goal IFOL.thy "~~((P --> Q) <-> (~P | Q))";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~16";
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goal IFOL.thy "~~((P-->Q) | (Q-->P))";
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by (Int.fast_tac 1);
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result();
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writeln"Problem ~~17";
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goal IFOL.thy
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  "~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))";
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by (Int.fast_tac 1);    
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result();
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(*Dijkstra's "Golden Rule"*)
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goal IFOL.thy "(P&Q) <-> P <-> Q <-> (P|Q)";
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by (Int.fast_tac 1);
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result();
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writeln"****Examples with quantifiers****";
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writeln"The converse is classical in the following implications...";
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goal IFOL.thy "(EX x.P(x)-->Q)  -->  (ALL x.P(x)) --> Q";
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by (Int.fast_tac 1); 
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result();  
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goal IFOL.thy "((ALL x.P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)";
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by (Int.fast_tac 1); 
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result();  
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goal IFOL.thy "((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))";
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   197
by (Int.fast_tac 1); 
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parents:
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   198
result();  
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parents:
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   199
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   200
goal IFOL.thy "(ALL x.P(x)) | Q  -->  (ALL x. P(x) | Q)";
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parents:
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   201
by (Int.fast_tac 1); 
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parents:
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   202
result();  
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parents:
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   203
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goal IFOL.thy "(EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))";
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parents:
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   205
by (Int.fast_tac 1);
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parents:
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   206
result();  
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clasohm
parents:
diff changeset
   207
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clasohm
parents:
diff changeset
   208
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clasohm
parents:
diff changeset
   209
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clasohm
parents:
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   210
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writeln"The following are not constructively valid!";
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parents:
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(*The attempt to prove them terminates quickly!*)
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parents:
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   213
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   214
goal IFOL.thy "((ALL x.P(x))-->Q) --> (EX x.P(x)-->Q)";
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parents:
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   215
by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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clasohm
parents:
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   216
getgoal 1; 
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clasohm
parents:
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   217
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parents:
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   218
goal IFOL.thy "(P --> (EX x.Q(x))) --> (EX x. P-->Q(x))";
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clasohm
parents:
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   219
by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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clasohm
parents:
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   220
getgoal 1; 
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parents:
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   221
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parents:
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   222
goal IFOL.thy "(ALL x. P(x) | Q) --> ((ALL x.P(x)) | Q)";
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clasohm
parents:
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   223
by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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clasohm
parents:
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   224
getgoal 1; 
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clasohm
parents:
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   225
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parents:
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   226
goal IFOL.thy "(ALL x. ~~P(x)) --> ~~(ALL x. P(x))";
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clasohm
parents:
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   227
by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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clasohm
parents:
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   228
getgoal 1; 
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clasohm
parents:
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   229
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parents:
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   230
(*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
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parents:
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   231
goal IFOL.thy "EX x. Q(x) --> (ALL x. Q(x))";
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clasohm
parents:
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   232
by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
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clasohm
parents:
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   233
getgoal 1; 
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clasohm
parents:
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   234
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clasohm
parents:
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   235
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parents:
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   236
writeln"Hard examples with quantifiers";
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clasohm
parents:
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   237
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parents:
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   238
(*The ones that have not been proved are not known to be valid!
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parents:
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   239
  Some will require quantifier duplication -- not currently available*)
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clasohm
parents:
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   240
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parents:
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   241
writeln"Problem ~~18";
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parents:
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   242
goal IFOL.thy "~~(EX y. ALL x. P(y)-->P(x))";
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clasohm
parents:
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   243
(*NOT PROVED*)
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parents:
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   244
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parents:
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   245
writeln"Problem ~~19";
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parents:
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   246
goal IFOL.thy "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))";
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clasohm
parents:
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   247
(*NOT PROVED*)
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clasohm
parents:
diff changeset
   248
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parents:
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   249
writeln"Problem 20";
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parents:
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   250
goal IFOL.thy "(ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))     \
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parents:
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   251
\   --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))";
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parents:
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   252
by (Int.fast_tac 1); 
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parents:
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   253
result();
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clasohm
parents:
diff changeset
   254
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parents:
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   255
writeln"Problem 21";
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parents:
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   256
goal IFOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))";
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parents:
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   257
(*NOT PROVED*)
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clasohm
parents:
diff changeset
   258
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parents:
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   259
writeln"Problem 22";
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parents:
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   260
goal IFOL.thy "(ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))";
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parents:
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   261
by (Int.fast_tac 1); 
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clasohm
parents:
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   262
result();
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clasohm
parents:
diff changeset
   263
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parents:
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   264
writeln"Problem ~~23";
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parents:
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   265
goal IFOL.thy "~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))";
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parents:
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   266
by (Int.best_tac 1);  
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parents:
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   267
result();
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clasohm
parents:
diff changeset
   268
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parents:
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   269
writeln"Problem 24";
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parents:
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   270
goal IFOL.thy "~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &  \
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parents:
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   271
\    ~(EX x.P(x)) --> (EX x.Q(x)) & (ALL x. Q(x)|R(x) --> S(x))  \
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parents:
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   272
\   --> (EX x. P(x)&R(x))";
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parents:
diff changeset
   273
by (Int.fast_tac 1); 
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parents:
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   274
result();
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clasohm
parents:
diff changeset
   275
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parents:
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   276
writeln"Problem 25";
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parents:
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   277
goal IFOL.thy "(EX x. P(x)) &  \
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clasohm
parents:
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   278
\       (ALL x. L(x) --> ~ (M(x) & R(x))) &  \
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clasohm
parents:
diff changeset
   279
\       (ALL x. P(x) --> (M(x) & L(x))) &   \
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clasohm
parents:
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   280
\       ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))  \
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clasohm
parents:
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   281
\   --> (EX x. Q(x)&P(x))";
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clasohm
parents:
diff changeset
   282
by (Int.best_tac 1); 
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parents:
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   283
result();
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clasohm
parents:
diff changeset
   284
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clasohm
parents:
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   285
writeln"Problem ~~26";
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clasohm
parents:
diff changeset
   286
goal IFOL.thy "(~~(EX x. p(x)) <-> ~~(EX x. q(x))) &	\
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clasohm
parents:
diff changeset
   287
\     (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y)))	\
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clasohm
parents:
diff changeset
   288
\ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))";
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clasohm
parents:
diff changeset
   289
(*NOT PROVED*)
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clasohm
parents:
diff changeset
   290
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parents:
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   291
writeln"Problem 27";
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clasohm
parents:
diff changeset
   292
goal IFOL.thy "(EX x. P(x) & ~Q(x)) &   \
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clasohm
parents:
diff changeset
   293
\             (ALL x. P(x) --> R(x)) &   \
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clasohm
parents:
diff changeset
   294
\             (ALL x. M(x) & L(x) --> P(x)) &   \
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clasohm
parents:
diff changeset
   295
\             ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x)))  \
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clasohm
parents:
diff changeset
   296
\         --> (ALL x. M(x) --> ~L(x))";
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clasohm
parents:
diff changeset
   297
by (Int.fast_tac 1);   (*44 secs*)
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clasohm
parents:
diff changeset
   298
result();
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clasohm
parents:
diff changeset
   299
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clasohm
parents:
diff changeset
   300
writeln"Problem ~~28.  AMENDED";
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clasohm
parents:
diff changeset
   301
goal IFOL.thy "(ALL x. P(x) --> (ALL x. Q(x))) &   \
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clasohm
parents:
diff changeset
   302
\       (~~(ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) &  \
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clasohm
parents:
diff changeset
   303
\       (~~(EX x.S(x)) --> (ALL x. L(x) --> M(x)))  \
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clasohm
parents:
diff changeset
   304
\   --> (ALL x. P(x) & L(x) --> M(x))";
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clasohm
parents:
diff changeset
   305
by (Int.fast_tac 1);  (*101 secs*)
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clasohm
parents:
diff changeset
   306
result();
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clasohm
parents:
diff changeset
   307
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clasohm
parents:
diff changeset
   308
writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
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clasohm
parents:
diff changeset
   309
goal IFOL.thy "(EX x. P(x)) & (EX y. Q(y))  \
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clasohm
parents:
diff changeset
   310
\   --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->     \
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clasohm
parents:
diff changeset
   311
\        (ALL x y. P(x) & Q(y) --> R(x) & S(y)))";
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clasohm
parents:
diff changeset
   312
by (Int.fast_tac 1); 
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clasohm
parents:
diff changeset
   313
result();
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clasohm
parents:
diff changeset
   314
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clasohm
parents:
diff changeset
   315
writeln"Problem ~~30";
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clasohm
parents:
diff changeset
   316
goal IFOL.thy "(ALL x. (P(x) | Q(x)) --> ~ R(x)) & \
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clasohm
parents:
diff changeset
   317
\       (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
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clasohm
parents:
diff changeset
   318
\   --> (ALL x. ~~S(x))";
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clasohm
parents:
diff changeset
   319
by (Int.fast_tac 1);  
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clasohm
parents:
diff changeset
   320
result();
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clasohm
parents:
diff changeset
   321
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clasohm
parents:
diff changeset
   322
writeln"Problem 31";
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clasohm
parents:
diff changeset
   323
goal IFOL.thy "~(EX x.P(x) & (Q(x) | R(x))) & \
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clasohm
parents:
diff changeset
   324
\       (EX x. L(x) & P(x)) & \
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clasohm
parents:
diff changeset
   325
\       (ALL x. ~ R(x) --> M(x))  \
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clasohm
parents:
diff changeset
   326
\   --> (EX x. L(x) & M(x))";
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clasohm
parents:
diff changeset
   327
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   328
result();
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clasohm
parents:
diff changeset
   329
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clasohm
parents:
diff changeset
   330
writeln"Problem 32";
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clasohm
parents:
diff changeset
   331
goal IFOL.thy "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \
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clasohm
parents:
diff changeset
   332
\       (ALL x. S(x) & R(x) --> L(x)) & \
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clasohm
parents:
diff changeset
   333
\       (ALL x. M(x) --> R(x))  \
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clasohm
parents:
diff changeset
   334
\   --> (ALL x. P(x) & M(x) --> L(x))";
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clasohm
parents:
diff changeset
   335
by (Int.best_tac 1);
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clasohm
parents:
diff changeset
   336
result();
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clasohm
parents:
diff changeset
   337
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clasohm
parents:
diff changeset
   338
writeln"Problem ~~33";
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clasohm
parents:
diff changeset
   339
goal IFOL.thy "(ALL x. ~~(P(a) & (P(x)-->P(b))-->P(c)))  <->    \
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clasohm
parents:
diff changeset
   340
\    (ALL x. ~~((~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c))))";
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clasohm
parents:
diff changeset
   341
by (Int.best_tac 1);
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clasohm
parents:
diff changeset
   342
result();
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clasohm
parents:
diff changeset
   343
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clasohm
parents:
diff changeset
   344
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clasohm
parents:
diff changeset
   345
writeln"Problem 36";
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clasohm
parents:
diff changeset
   346
goal IFOL.thy 
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clasohm
parents:
diff changeset
   347
     "(ALL x. EX y. J(x,y)) & \
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clasohm
parents:
diff changeset
   348
\     (ALL x. EX y. G(x,y)) & \
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clasohm
parents:
diff changeset
   349
\     (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z)))   \
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clasohm
parents:
diff changeset
   350
\ --> (ALL x. EX y. H(x,y))";
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clasohm
parents:
diff changeset
   351
by (Int.fast_tac 1);  (*35 secs*)
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clasohm
parents:
diff changeset
   352
result();
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clasohm
parents:
diff changeset
   353
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clasohm
parents:
diff changeset
   354
writeln"Problem 37";
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clasohm
parents:
diff changeset
   355
goal IFOL.thy
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clasohm
parents:
diff changeset
   356
       "(ALL z. EX w. ALL x. EX y. \
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clasohm
parents:
diff changeset
   357
\          ~~(P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u.Q(u,w)))) & \
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clasohm
parents:
diff changeset
   358
\       (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \
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clasohm
parents:
diff changeset
   359
\       (~~(EX x y. Q(x,y)) --> (ALL x. R(x,x)))  \
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clasohm
parents:
diff changeset
   360
\   --> ~~(ALL x. EX y. R(x,y))";
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clasohm
parents:
diff changeset
   361
(*NOT PROVED*)
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clasohm
parents:
diff changeset
   362
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clasohm
parents:
diff changeset
   363
writeln"Problem 39";
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clasohm
parents:
diff changeset
   364
goal IFOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
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clasohm
parents:
diff changeset
   365
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   366
result();
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clasohm
parents:
diff changeset
   367
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clasohm
parents:
diff changeset
   368
writeln"Problem 40.  AMENDED";
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clasohm
parents:
diff changeset
   369
goal IFOL.thy "(EX y. ALL x. F(x,y) <-> F(x,x)) -->  \
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clasohm
parents:
diff changeset
   370
\             ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
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clasohm
parents:
diff changeset
   371
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   372
result();
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clasohm
parents:
diff changeset
   373
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clasohm
parents:
diff changeset
   374
writeln"Problem 44";
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clasohm
parents:
diff changeset
   375
goal IFOL.thy "(ALL x. f(x) -->					\
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clasohm
parents:
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   376
\             (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &   	\
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clasohm
parents:
diff changeset
   377
\             (EX x. j(x) & (ALL y. g(y) --> h(x,y)))			\
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clasohm
parents:
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   378
\             --> (EX x. j(x) & ~f(x))";
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clasohm
parents:
diff changeset
   379
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   380
result();
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clasohm
parents:
diff changeset
   381
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clasohm
parents:
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   382
writeln"Problem 48";
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clasohm
parents:
diff changeset
   383
goal IFOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
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clasohm
parents:
diff changeset
   384
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   385
result();
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clasohm
parents:
diff changeset
   386
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clasohm
parents:
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   387
writeln"Problem 51";
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clasohm
parents:
diff changeset
   388
goal IFOL.thy
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clasohm
parents:
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   389
    "(EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->  \
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clasohm
parents:
diff changeset
   390
\    (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)";
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clasohm
parents:
diff changeset
   391
by (Int.best_tac 1);  (*60 seconds*)
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clasohm
parents:
diff changeset
   392
result();
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clasohm
parents:
diff changeset
   393
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clasohm
parents:
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   394
writeln"Problem 52";
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clasohm
parents:
diff changeset
   395
(*Almost the same as 51. *)
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clasohm
parents:
diff changeset
   396
goal IFOL.thy
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clasohm
parents:
diff changeset
   397
    "(EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->  \
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clasohm
parents:
diff changeset
   398
\    (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)";
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clasohm
parents:
diff changeset
   399
by (Int.best_tac 1);  (*60 seconds*)
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clasohm
parents:
diff changeset
   400
result();
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clasohm
parents:
diff changeset
   401
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clasohm
parents:
diff changeset
   402
writeln"Problem 56";
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clasohm
parents:
diff changeset
   403
goal IFOL.thy
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clasohm
parents:
diff changeset
   404
    "(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
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clasohm
parents:
diff changeset
   405
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   406
result();
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clasohm
parents:
diff changeset
   407
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clasohm
parents:
diff changeset
   408
writeln"Problem 57";
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clasohm
parents:
diff changeset
   409
goal IFOL.thy
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clasohm
parents:
diff changeset
   410
    "P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
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clasohm
parents:
diff changeset
   411
\    (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))";
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clasohm
parents:
diff changeset
   412
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   413
result();
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clasohm
parents:
diff changeset
   414
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clasohm
parents:
diff changeset
   415
writeln"Problem 60";
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clasohm
parents:
diff changeset
   416
goal IFOL.thy
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clasohm
parents:
diff changeset
   417
    "ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
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clasohm
parents:
diff changeset
   418
by (Int.fast_tac 1);
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clasohm
parents:
diff changeset
   419
result();
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clasohm
parents:
diff changeset
   420
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clasohm
parents:
diff changeset
   421
writeln"Reached end of file.";