Modified datatype com.
Added (part of) relative completeness proof for Hoare logic.
(* Title: FOLP/ex/prop.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
First-Order Logic: propositional examples (intuitionistic and classical)
Needs declarations of the theory "thy" and the tactic "tac"
*)
writeln"File FOLP/ex/prop.ML";
writeln"commutative laws of & and | ";
goal thy "?p : P & Q --> Q & P";
by tac;
result();
goal thy "?p : P | Q --> Q | P";
by tac;
result();
writeln"associative laws of & and | ";
goal thy "?p : (P & Q) & R --> P & (Q & R)";
by tac;
result();
goal thy "?p : (P | Q) | R --> P | (Q | R)";
by tac;
result();
writeln"distributive laws of & and | ";
goal thy "?p : (P & Q) | R --> (P | R) & (Q | R)";
by tac;
result();
goal thy "?p : (P | R) & (Q | R) --> (P & Q) | R";
by tac;
result();
goal thy "?p : (P | Q) & R --> (P & R) | (Q & R)";
by tac;
result();
goal thy "?p : (P & R) | (Q & R) --> (P | Q) & R";
by tac;
result();
writeln"Laws involving implication";
goal thy "?p : (P-->R) & (Q-->R) <-> (P|Q --> R)";
by tac;
result();
goal thy "?p : (P & Q --> R) <-> (P--> (Q-->R))";
by tac;
result();
goal thy "?p : ((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R";
by tac;
result();
goal thy "?p : ~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)";
by tac;
result();
goal thy "?p : (P --> Q & R) <-> (P-->Q) & (P-->R)";
by tac;
result();
writeln"Propositions-as-types";
(*The combinator K*)
goal thy "?p : P --> (Q --> P)";
by tac;
result();
(*The combinator S*)
goal thy "?p : (P-->Q-->R) --> (P-->Q) --> (P-->R)";
by tac;
result();
(*Converse is classical*)
goal thy "?p : (P-->Q) | (P-->R) --> (P --> Q | R)";
by tac;
result();
goal thy "?p : (P-->Q) --> (~Q --> ~P)";
by tac;
result();
writeln"Schwichtenberg's examples (via T. Nipkow)";
(* stab-imp *)
goal thy "?p : (((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q";
by tac;
result();
(* stab-to-peirce *)
goal thy "?p : (((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q) \
\ --> ((P --> Q) --> P) --> P";
by tac;
result();
(* peirce-imp1 *)
goal thy "?p : (((Q --> R) --> Q) --> Q) \
\ --> (((P --> Q) --> R) --> P --> Q) --> P --> Q";
by tac;
result();
(* peirce-imp2 *)
goal thy "?p : (((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P";
by tac;
result();
(* mints *)
goal thy "?p : ((((P --> Q) --> P) --> P) --> Q) --> Q";
by tac;
result();
(* mints-solovev *)
goal thy "?p : (P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R";
by tac;
result();
(* tatsuta *)
goal thy "?p : (((P7 --> P1) --> P10) --> P4 --> P5) \
\ --> (((P8 --> P2) --> P9) --> P3 --> P10) \
\ --> (P1 --> P8) --> P6 --> P7 \
\ --> (((P3 --> P2) --> P9) --> P4) \
\ --> (P1 --> P3) --> (((P6 --> P1) --> P2) --> P9) --> P5";
by tac;
result();
(* tatsuta1 *)
goal thy "?p : (((P8 --> P2) --> P9) --> P3 --> P10) \
\ --> (((P3 --> P2) --> P9) --> P4) \
\ --> (((P6 --> P1) --> P2) --> P9) \
\ --> (((P7 --> P1) --> P10) --> P4 --> P5) \
\ --> (P1 --> P3) --> (P1 --> P8) --> P6 --> P7 --> P5";
by tac;
result();
writeln"Reached end of file.";