src/FOLP/ex/int.ML
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(*  Title:      FOLP/ex/int.ML
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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 (IntPr.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 (IntPr.safe_tac 1);
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by (IntPr.mp_tac 1);
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by (IntPr.fast_tac 1);
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*)
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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 (theory "IFOLP") "?p : ~~(P&Q) <-> ~~P & ~~Q";
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by (IntPr.fast_tac 1);
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result();
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goal (theory "IFOLP") "?p : ~~~P <-> ~P";
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by (IntPr.fast_tac 1);
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result();
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goal (theory "IFOLP") "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))";
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by (IntPr.fast_tac 1);
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result();
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goal (theory "IFOLP") "?p : (P<->Q) <-> (Q<->P)";
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by (IntPr.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 (theory "IFOLP") "?p : P --> ~~P";
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by (IntPr.fast_tac 1);
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result();
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goal (theory "IFOLP") "?p : ~~(~~P --> P)";
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by (IntPr.fast_tac 1);
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result();
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goal (theory "IFOLP") "?p : ~~P & ~~(P --> Q) --> ~~Q";
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by (IntPr.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 (theory "IFOLP") "?p : ((P-->Q) --> P)  -->  P";
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by (IntPr.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 (theory "IFOLP") "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)";
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by (IntPr.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 (theory "IFOLP") "?p : ~~((P-->Q)  <->  (~Q --> ~P))";
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by (IntPr.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 (theory "IFOLP") "?p : ~~(~~P  <->  P)";
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by (IntPr.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 (theory "IFOLP") "?p : ~(P-->Q) --> (Q-->P)";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~4";
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goal (theory "IFOLP") "?p : ~~((~P-->Q)  <->  (~Q --> P))";
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by (IntPr.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 (theory "IFOLP") "?p : ~~((P|Q-->P|R) --> P|(Q-->R))";
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by (IntPr.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 (theory "IFOLP") "?p : ~~(P | ~P)";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~7";
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goal (theory "IFOLP") "?p : ~~(P | ~~~P)";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~8.  Peirce's law";
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goal (theory "IFOLP") "?p : ~~(((P-->Q) --> P)  -->  P)";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem 9";
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goal (theory "IFOLP") "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
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by (IntPr.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 (theory "IFOLP") "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)";
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by (IntPr.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 (theory "IFOLP") "?p : P<->P";
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by (IntPr.fast_tac 1);
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writeln"Problem ~~12.  Dijkstra's law  ";
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goal (theory "IFOLP") "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))";
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by (IntPr.fast_tac 1);
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result();
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goal (theory "IFOLP") "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem 13.  Distributive law";
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goal (theory "IFOLP") "?p : P | (Q & R)  <-> (P | Q) & (P | R)";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~14";
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goal (theory "IFOLP") "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~15";
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goal (theory "IFOLP") "?p : ~~((P --> Q) <-> (~P | Q))";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~16";
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goal (theory "IFOLP") "?p : ~~((P-->Q) | (Q-->P))";
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by (IntPr.fast_tac 1);
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result();
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writeln"Problem ~~17";
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goal (theory "IFOLP")
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  "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))";
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by (IntPr.fast_tac 1);
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(*over 5 minutes?? -- printing the proof term takes 40 secs!!*)
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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 (theory "IFOLP") "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q";
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by (IntPr.fast_tac 1); 
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result();  
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goal (theory "IFOLP") "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)";
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by (IntPr.fast_tac 1); 
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result();  
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goal (theory "IFOLP") "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))";
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by (IntPr.fast_tac 1); 
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result();  
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goal (theory "IFOLP") "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)";
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by (IntPr.fast_tac 1); 
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result();  
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goal (theory "IFOLP") "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))";
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by (IntPr.fast_tac 1);
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result();  
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writeln"The following are not constructively valid!";
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(*The attempt to prove them terminates quickly!*)
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goal (theory "IFOLP") "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)";
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by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";  
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getgoal 1; 
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goal (theory "IFOLP") "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))";
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by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";  
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getgoal 1; 
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goal (theory "IFOLP") "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)";
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   216
by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";  
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getgoal 1; 
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goal (theory "IFOLP") "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))";
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by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";  
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getgoal 1; 
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(*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
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goal (theory "IFOLP") "?p : EX x. Q(x) --> (ALL x. Q(x))";
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by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";  
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getgoal 1; 
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writeln"Hard examples with quantifiers";
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(*The ones that have not been proved are not known to be valid!
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  Some will require quantifier duplication -- not currently available*)
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writeln"Problem ~~18";
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goal (theory "IFOLP") "?p : ~~(EX y. ALL x. P(y)-->P(x))";
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(*NOT PROVED*)
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writeln"Problem ~~19";
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goal (theory "IFOLP") "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))";
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(*NOT PROVED*)
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writeln"Problem 20";
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goal (theory "IFOLP") "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))     \
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\   --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))";
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by (IntPr.fast_tac 1); 
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result();
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writeln"Problem 21";
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goal (theory "IFOLP")
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    "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))";
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(*NOT PROVED*)
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writeln"Problem 22";
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goal (theory "IFOLP") "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))";
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by (IntPr.fast_tac 1); 
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result();
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writeln"Problem ~~23";
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goal (theory "IFOLP") "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))";
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by (IntPr.best_tac 1);  
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result();
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writeln"Problem 24";
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goal (theory "IFOLP") "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &  \
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\    (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))  \
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\   --> ~~(EX x. P(x)&R(x))";
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(*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
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by IntPr.safe_tac; 
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by (etac impE 1);
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   270
by (IntPr.fast_tac 1); 
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by (IntPr.fast_tac 1); 
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result();
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writeln"Problem 25";
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goal (theory "IFOLP") "?p : (EX x. P(x)) &  \
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\       (ALL x. L(x) --> ~ (M(x) & R(x))) &  \
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\       (ALL x. P(x) --> (M(x) & L(x))) &   \
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\       ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))  \
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\   --> (EX x. Q(x)&P(x))";
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by (IntPr.best_tac 1);
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result();
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writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
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goal (theory "IFOLP") "?p : (EX x. P(x)) & (EX y. Q(y))  \
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\   --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->     \
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\        (ALL x y. P(x) & Q(y) --> R(x) & S(y)))";
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   287
by (IntPr.fast_tac 1); 
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result();
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writeln"Problem ~~30";
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   291
goal (theory "IFOLP") "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) & \
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\       (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
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\   --> (ALL x. ~~S(x))";
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   294
by (IntPr.fast_tac 1);  
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result();
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writeln"Problem 31";
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goal (theory "IFOLP") "?p : ~(EX x. P(x) & (Q(x) | R(x))) & \
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   299
\       (EX x. L(x) & P(x)) & \
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\       (ALL x. ~ R(x) --> M(x))  \
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\   --> (EX x. L(x) & M(x))";
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   302
by (IntPr.fast_tac 1);
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result();
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writeln"Problem 32";
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   306
goal (theory "IFOLP") "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \
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   307
\       (ALL x. S(x) & R(x) --> L(x)) & \
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   308
\       (ALL x. M(x) --> R(x))  \
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   309
\   --> (ALL x. P(x) & M(x) --> L(x))";
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   310
by (IntPr.best_tac 1);  (*SLOW*)
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result();
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writeln"Problem 39";
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goal (theory "IFOLP") "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
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diff changeset
   315
by (IntPr.fast_tac 1);
0
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result();
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writeln"Problem 40.  AMENDED";
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   319
goal (theory "IFOLP") "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->  \
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\             ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
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diff changeset
   321
by (IntPr.fast_tac 1);
0
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result();
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writeln"Problem 44";
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   325
goal (theory "IFOLP") "?p : (ALL x. f(x) -->                                   \
1459
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   326
\             (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &       \
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   327
\             (EX x. j(x) & (ALL y. g(y) --> h(x,y)))                   \
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\             --> (EX x. j(x) & ~f(x))";
2603
4988dda71c0b Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents: 2573
diff changeset
   329
by (IntPr.fast_tac 1);
0
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clasohm
parents:
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   330
result();
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clasohm
parents:
diff changeset
   331
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parents:
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   332
writeln"Problem 48";
17480
fd19f77dcf60 converted to Isar theory format;
wenzelm
parents: 15661
diff changeset
   333
goal (theory "IFOLP") "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c";
2603
4988dda71c0b Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents: 2573
diff changeset
   334
by (IntPr.fast_tac 1);
0
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clasohm
parents:
diff changeset
   335
result();
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clasohm
parents:
diff changeset
   336
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clasohm
parents:
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   337
writeln"Problem 51";
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wenzelm
parents: 15661
diff changeset
   338
goal (theory "IFOLP")
0
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clasohm
parents:
diff changeset
   339
    "?p : (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->  \
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clasohm
parents:
diff changeset
   340
\    (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)";
2603
4988dda71c0b Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents: 2573
diff changeset
   341
by (IntPr.best_tac 1);  (*60 seconds*)
0
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clasohm
parents:
diff changeset
   342
result();
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clasohm
parents:
diff changeset
   343
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parents:
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   344
writeln"Problem 56";
17480
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wenzelm
parents: 15661
diff changeset
   345
goal (theory "IFOLP")
0
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clasohm
parents:
diff changeset
   346
    "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
2603
4988dda71c0b Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents: 2573
diff changeset
   347
by (IntPr.fast_tac 1);
0
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clasohm
parents:
diff changeset
   348
result();
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clasohm
parents:
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   349
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clasohm
parents:
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   350
writeln"Problem 57";
17480
fd19f77dcf60 converted to Isar theory format;
wenzelm
parents: 15661
diff changeset
   351
goal (theory "IFOLP")
0
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clasohm
parents:
diff changeset
   352
    "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   353
\    (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))";
2603
4988dda71c0b Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents: 2573
diff changeset
   354
by (IntPr.fast_tac 1);
0
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clasohm
parents:
diff changeset
   355
result();
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clasohm
parents:
diff changeset
   356
a5a9c433f639 Initial revision
clasohm
parents:
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   357
writeln"Problem 60";
17480
fd19f77dcf60 converted to Isar theory format;
wenzelm
parents: 15661
diff changeset
   358
goal (theory "IFOLP")
0
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clasohm
parents:
diff changeset
   359
    "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
2603
4988dda71c0b Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents: 2573
diff changeset
   360
by (IntPr.fast_tac 1);
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   361
result();