author | nipkow |
Tue, 17 Oct 2000 08:00:46 +0200 | |
changeset 10228 | e653cb933293 |
parent 3836 | f1a1817659e6 |
child 15531 | 08c8dad8e399 |
permissions | -rw-r--r-- |
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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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writeln"File FOLP/ex/int.ML"; |
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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 IFOLP.thy "?p : ~~(P&Q) <-> ~~P & ~~Q"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOLP.thy "?p : ~~~P <-> ~P"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?p : P --> ~~P"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOLP.thy "?p : ~~(~~P --> P)"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy |
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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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?p : (ALL x. P(x) | Q) --> ((ALL 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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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 IFOLP.thy "?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); |
0 | 248 |
result(); |
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writeln"Problem 21"; |
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goal IFOLP.thy |
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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 IFOLP.thy "?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(); |
259 |
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writeln"Problem ~~23"; |
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goal IFOLP.thy "?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 IFOLP.thy "?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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by (IntPr.fast_tac 1); |
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by (IntPr.fast_tac 1); |
0 | 274 |
result(); |
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writeln"Problem 25"; |
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goal IFOLP.thy "?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))) & \ |
|
280 |
\ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ |
|
281 |
\ --> (EX x. Q(x)&P(x))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
282 |
by (IntPr.best_tac 1); |
0 | 283 |
result(); |
284 |
||
285 |
writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; |
|
286 |
goal IFOLP.thy "?p : (EX x. P(x)) & (EX y. Q(y)) \ |
|
287 |
\ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ |
|
288 |
\ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
289 |
by (IntPr.fast_tac 1); |
0 | 290 |
result(); |
291 |
||
292 |
writeln"Problem ~~30"; |
|
293 |
goal IFOLP.thy "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) & \ |
|
294 |
\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ |
|
295 |
\ --> (ALL x. ~~S(x))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
296 |
by (IntPr.fast_tac 1); |
0 | 297 |
result(); |
298 |
||
299 |
writeln"Problem 31"; |
|
3836 | 300 |
goal IFOLP.thy "?p : ~(EX x. P(x) & (Q(x) | R(x))) & \ |
0 | 301 |
\ (EX x. L(x) & P(x)) & \ |
302 |
\ (ALL x. ~ R(x) --> M(x)) \ |
|
303 |
\ --> (EX x. L(x) & M(x))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
304 |
by (IntPr.fast_tac 1); |
0 | 305 |
result(); |
306 |
||
307 |
writeln"Problem 32"; |
|
308 |
goal IFOLP.thy "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ |
|
309 |
\ (ALL x. S(x) & R(x) --> L(x)) & \ |
|
310 |
\ (ALL x. M(x) --> R(x)) \ |
|
311 |
\ --> (ALL x. P(x) & M(x) --> L(x))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
312 |
by (IntPr.best_tac 1); (*SLOW*) |
0 | 313 |
result(); |
314 |
||
315 |
writeln"Problem 39"; |
|
316 |
goal IFOLP.thy "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
317 |
by (IntPr.fast_tac 1); |
0 | 318 |
result(); |
319 |
||
320 |
writeln"Problem 40. AMENDED"; |
|
321 |
goal IFOLP.thy "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) --> \ |
|
322 |
\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; |
|
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
323 |
by (IntPr.fast_tac 1); |
0 | 324 |
result(); |
325 |
||
326 |
writeln"Problem 44"; |
|
1459 | 327 |
goal IFOLP.thy "?p : (ALL x. f(x) --> \ |
328 |
\ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ |
|
329 |
\ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ |
|
0 | 330 |
\ --> (EX x. j(x) & ~f(x))"; |
2603
4988dda71c0b
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
331 |
by (IntPr.fast_tac 1); |
0 | 332 |
result(); |
333 |
||
334 |
writeln"Problem 48"; |
|
335 |
goal IFOLP.thy "?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
|
336 |
by (IntPr.fast_tac 1); |
0 | 337 |
result(); |
338 |
||
339 |
writeln"Problem 51"; |
|
340 |
goal IFOLP.thy |
|
341 |
"?p : (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
342 |
\ (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
|
343 |
by (IntPr.best_tac 1); (*60 seconds*) |
0 | 344 |
result(); |
345 |
||
346 |
writeln"Problem 56"; |
|
347 |
goal IFOLP.thy |
|
348 |
"?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
|
349 |
by (IntPr.fast_tac 1); |
0 | 350 |
result(); |
351 |
||
352 |
writeln"Problem 57"; |
|
353 |
goal IFOLP.thy |
|
354 |
"?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
|
355 |
\ (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
|
356 |
by (IntPr.fast_tac 1); |
0 | 357 |
result(); |
358 |
||
359 |
writeln"Problem 60"; |
|
360 |
goal IFOLP.thy |
|
361 |
"?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
|
362 |
by (IntPr.fast_tac 1); |
0 | 363 |
result(); |
364 |
||
365 |
writeln"Reached end of file."; |