author | paulson |
Fri, 28 Feb 1997 15:44:32 +0100 | |
changeset 2687 | b7c86d3c9d0a |
parent 2601 | b301958c465d |
child 3835 | 9a5a4e123859 |
permissions | -rw-r--r-- |
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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 (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 FOL/ex/int."; |
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(*Metatheorem (for PROPOSITIONAL formulae...): |
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P is classically provable iff ~~P is intuitionistically provable. |
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Therefore ~P is classically provable iff it is intuitionistically provable. |
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Proof: Let Q be the conjuction of the propositions A|~A, one for each atom A |
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in P. Now ~~Q is intuitionistically provable because ~~(A|~A) is and because |
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~~ distributes over &. If P is provable classically, then clearly Q-->P is |
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provable intuitionistically, so ~~(Q-->P) is also provable intuitionistically. |
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The latter is intuitionistically equivalent to ~~Q-->~~P, hence to ~~P, since |
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~~Q is intuitionistically provable. Finally, if P is a negation then ~~P is |
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intuitionstically equivalent to P. [Andy Pitts] *) |
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goal IFOL.thy "~~(P&Q) <-> ~~P & ~~Q"; |
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by (IntPr.fast_tac 1); |
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result(); |
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(* ~~ does NOT distribute over | *) |
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goal IFOL.thy "~~(P-->Q) <-> (~~P --> ~~Q)"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOL.thy "~~~P <-> ~P"; |
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by (IntPr.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 (IntPr.fast_tac 1); |
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result(); |
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goal IFOL.thy "(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 IFOL.thy "P --> ~~P"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOL.thy "~~(~~P --> P)"; |
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by (IntPr.fast_tac 1); |
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result(); |
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goal IFOL.thy "~~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 IFOL.thy "((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 IFOL.thy "(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"de Bruijn formulae"; |
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(*de Bruijn formula with three predicates*) |
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goal IFOL.thy "((P<->Q) --> P&Q&R) & \ |
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\ ((Q<->R) --> P&Q&R) & \ |
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\ ((R<->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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(*de Bruijn formula with five predicates*) |
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goal IFOL.thy "((P<->Q) --> P&Q&R&S&T) & \ |
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\ ((Q<->R) --> P&Q&R&S&T) & \ |
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\ ((R<->S) --> P&Q&R&S&T) & \ |
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\ ((S<->T) --> P&Q&R&S&T) & \ |
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\ ((T<->P) --> P&Q&R&S&T) --> P&Q&R&S&T"; |
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by (IntPr.fast_tac 1); |
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result(); |
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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 (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 IFOL.thy "~~(~~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 IFOL.thy "~(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 IFOL.thy "~~((~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 IFOL.thy "~~((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 IFOL.thy "~~(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 IFOL.thy "~~(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 IFOL.thy "~~(((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 IFOL.thy "((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 IFOL.thy "(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 IFOL.thy "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 IFOL.thy "~~(((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 IFOL.thy "((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 IFOL.thy "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 IFOL.thy "~~((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 IFOL.thy "~~((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 IFOL.thy "~~((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 IFOL.thy |
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"~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"; |
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by (IntPr.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 (IntPr.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 (IntPr.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 (IntPr.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 (IntPr.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 (IntPr.fast_tac 1); |
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result(); |
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goal IFOL.thy "(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 IFOL.thy "((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; |
243 |
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goal IFOL.thy "(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; |
247 |
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goal IFOL.thy "(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"; |
0 | 250 |
getgoal 1; |
251 |
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goal IFOL.thy "(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; |
255 |
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(*Classically but not intuitionistically valid. Proved by a bug in 1986!*) |
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goal IFOL.thy "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; |
260 |
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261 |
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writeln"Hard examples with quantifiers"; |
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263 |
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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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266 |
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writeln"Problem ~~18"; |
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goal IFOL.thy "~~(EX y. ALL x. P(y)-->P(x))"; |
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(*NOT PROVED*) |
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270 |
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writeln"Problem ~~19"; |
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272 |
goal IFOL.thy "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))"; |
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273 |
(*NOT PROVED*) |
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274 |
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writeln"Problem 20"; |
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goal IFOL.thy "(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(); |
280 |
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writeln"Problem 21"; |
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goal IFOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))"; |
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(*NOT PROVED; needs quantifier duplication*) |
0 | 284 |
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writeln"Problem 22"; |
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goal IFOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; |
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by (IntPr.fast_tac 1); |
0 | 288 |
result(); |
289 |
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writeln"Problem ~~23"; |
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291 |
goal IFOL.thy "~~ ((ALL x. P | Q(x)) <-> (P | (ALL x. Q(x))))"; |
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292 |
by (IntPr.best_tac 1); |
0 | 293 |
result(); |
294 |
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writeln"Problem 24"; |
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goal IFOL.thy "~(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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298 |
\ --> ~~(EX x. P(x)&R(x))"; |
f3e04805895a
Correction to Problem 24 (with unsatisfactory proof)
paulson
parents:
1459
diff
changeset
|
299 |
(*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*) |
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
300 |
by IntPr.safe_tac; |
2573
f3e04805895a
Correction to Problem 24 (with unsatisfactory proof)
paulson
parents:
1459
diff
changeset
|
301 |
by (etac impE 1); |
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
302 |
by (IntPr.fast_tac 1); |
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
303 |
by (IntPr.fast_tac 1); |
0 | 304 |
result(); |
305 |
||
306 |
writeln"Problem 25"; |
|
307 |
goal IFOL.thy "(EX x. P(x)) & \ |
|
308 |
\ (ALL x. L(x) --> ~ (M(x) & R(x))) & \ |
|
309 |
\ (ALL x. P(x) --> (M(x) & L(x))) & \ |
|
310 |
\ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ |
|
311 |
\ --> (EX x. Q(x)&P(x))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
312 |
by (IntPr.best_tac 1); |
0 | 313 |
result(); |
314 |
||
315 |
writeln"Problem ~~26"; |
|
1459 | 316 |
goal IFOL.thy "(~~(EX x. p(x)) <-> ~~(EX x. q(x))) & \ |
317 |
\ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \ |
|
0 | 318 |
\ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"; |
319 |
(*NOT PROVED*) |
|
320 |
||
321 |
writeln"Problem 27"; |
|
322 |
goal IFOL.thy "(EX x. P(x) & ~Q(x)) & \ |
|
323 |
\ (ALL x. P(x) --> R(x)) & \ |
|
324 |
\ (ALL x. M(x) & L(x) --> P(x)) & \ |
|
325 |
\ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \ |
|
326 |
\ --> (ALL x. M(x) --> ~L(x))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
327 |
by (IntPr.fast_tac 1); (*21 secs*) |
0 | 328 |
result(); |
329 |
||
330 |
writeln"Problem ~~28. AMENDED"; |
|
331 |
goal IFOL.thy "(ALL x. P(x) --> (ALL x. Q(x))) & \ |
|
332 |
\ (~~(ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \ |
|
333 |
\ (~~(EX x.S(x)) --> (ALL x. L(x) --> M(x))) \ |
|
334 |
\ --> (ALL x. P(x) & L(x) --> M(x))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
335 |
by (IntPr.fast_tac 1); (*48 secs*) |
0 | 336 |
result(); |
337 |
||
338 |
writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; |
|
339 |
goal IFOL.thy "(EX x. P(x)) & (EX y. Q(y)) \ |
|
340 |
\ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ |
|
341 |
\ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
342 |
by (IntPr.fast_tac 1); |
0 | 343 |
result(); |
344 |
||
345 |
writeln"Problem ~~30"; |
|
346 |
goal IFOL.thy "(ALL x. (P(x) | Q(x)) --> ~ R(x)) & \ |
|
347 |
\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ |
|
348 |
\ --> (ALL x. ~~S(x))"; |
|
2601
b301958c465d
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 31"; |
|
353 |
goal IFOL.thy "~(EX x.P(x) & (Q(x) | R(x))) & \ |
|
354 |
\ (EX x. L(x) & P(x)) & \ |
|
355 |
\ (ALL x. ~ R(x) --> M(x)) \ |
|
356 |
\ --> (EX x. L(x) & M(x))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
357 |
by (IntPr.fast_tac 1); |
0 | 358 |
result(); |
359 |
||
360 |
writeln"Problem 32"; |
|
361 |
goal IFOL.thy "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ |
|
362 |
\ (ALL x. S(x) & R(x) --> L(x)) & \ |
|
363 |
\ (ALL x. M(x) --> R(x)) \ |
|
364 |
\ --> (ALL x. P(x) & M(x) --> L(x))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
365 |
by (IntPr.best_tac 1); |
0 | 366 |
result(); |
367 |
||
368 |
writeln"Problem ~~33"; |
|
369 |
goal IFOL.thy "(ALL x. ~~(P(a) & (P(x)-->P(b))-->P(c))) <-> \ |
|
370 |
\ (ALL x. ~~((~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c))))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
371 |
by (IntPr.best_tac 1); |
0 | 372 |
result(); |
373 |
||
374 |
||
375 |
writeln"Problem 36"; |
|
376 |
goal IFOL.thy |
|
377 |
"(ALL x. EX y. J(x,y)) & \ |
|
378 |
\ (ALL x. EX y. G(x,y)) & \ |
|
379 |
\ (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z))) \ |
|
380 |
\ --> (ALL x. EX y. H(x,y))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
381 |
by (IntPr.fast_tac 1); (*35 secs*) |
0 | 382 |
result(); |
383 |
||
384 |
writeln"Problem 37"; |
|
385 |
goal IFOL.thy |
|
386 |
"(ALL z. EX w. ALL x. EX y. \ |
|
387 |
\ ~~(P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u.Q(u,w)))) & \ |
|
388 |
\ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \ |
|
389 |
\ (~~(EX x y. Q(x,y)) --> (ALL x. R(x,x))) \ |
|
390 |
\ --> ~~(ALL x. EX y. R(x,y))"; |
|
391 |
(*NOT PROVED*) |
|
392 |
||
393 |
writeln"Problem 39"; |
|
394 |
goal IFOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
395 |
by (IntPr.fast_tac 1); |
0 | 396 |
result(); |
397 |
||
398 |
writeln"Problem 40. AMENDED"; |
|
399 |
goal IFOL.thy "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \ |
|
400 |
\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
401 |
by (IntPr.fast_tac 1); |
0 | 402 |
result(); |
403 |
||
404 |
writeln"Problem 44"; |
|
1459 | 405 |
goal IFOL.thy "(ALL x. f(x) --> \ |
406 |
\ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ |
|
407 |
\ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ |
|
0 | 408 |
\ --> (EX x. j(x) & ~f(x))"; |
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
409 |
by (IntPr.fast_tac 1); |
0 | 410 |
result(); |
411 |
||
412 |
writeln"Problem 48"; |
|
413 |
goal IFOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
414 |
by (IntPr.fast_tac 1); |
0 | 415 |
result(); |
416 |
||
417 |
writeln"Problem 51"; |
|
418 |
goal IFOL.thy |
|
419 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
420 |
\ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
421 |
by (IntPr.best_tac 1); (*34 seconds*) |
0 | 422 |
result(); |
423 |
||
424 |
writeln"Problem 52"; |
|
425 |
(*Almost the same as 51. *) |
|
426 |
goal IFOL.thy |
|
427 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
428 |
\ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
429 |
by (IntPr.best_tac 1); (*34 seconds*) |
0 | 430 |
result(); |
431 |
||
432 |
writeln"Problem 56"; |
|
433 |
goal IFOL.thy |
|
434 |
"(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
435 |
by (IntPr.fast_tac 1); |
0 | 436 |
result(); |
437 |
||
438 |
writeln"Problem 57"; |
|
439 |
goal IFOL.thy |
|
440 |
"P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
|
441 |
\ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
442 |
by (IntPr.fast_tac 1); |
0 | 443 |
result(); |
444 |
||
445 |
writeln"Problem 60"; |
|
446 |
goal IFOL.thy |
|
447 |
"ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; |
|
2601
b301958c465d
Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
paulson
parents:
2573
diff
changeset
|
448 |
by (IntPr.fast_tac 1); |
0 | 449 |
result(); |
450 |
||
451 |
writeln"Reached end of file."; |