# File Prolog.ML

    (*  Title:      FOL/ex/prolog.ML
        ID:         $Id: Prolog.ML,v 1.4 2005/09/03 15:15:52 wenzelm Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1992  University of Cambridge
    *)

    Goal "app(a:b:c:Nil, d:e:Nil, ?x)";
    by (resolve_tac [appNil,appCons] 1);
    by (resolve_tac [appNil,appCons] 1);
    by (resolve_tac [appNil,appCons] 1);
    by (resolve_tac [appNil,appCons] 1);
    prth (result());

    Goal "app(?x, c:d:Nil, a:b:c:d:Nil)";
    by (REPEAT (resolve_tac [appNil,appCons] 1));
    result();

    Goal "app(?x, ?y, a:b:c:d:Nil)";
    by (REPEAT (resolve_tac [appNil,appCons] 1));
    back();
    back();
    back();
    back();
    result();

    (*app([x1,...,xn], y, ?z) requires (n+1) inferences*)
    (*rev([x1,...,xn], ?y) requires (n+1)(n+2)/2 inferences*)

    Goal "rev(a:b:c:d:Nil, ?x)";
    val rules = [appNil,appCons,revNil,revCons];
    by (REPEAT (resolve_tac rules 1));
    result();

    Goal "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:Nil, ?w)";
    by (REPEAT (resolve_tac rules 1));
    result();

    Goal "rev(?x, a:b:c:Nil)";
    by (REPEAT (resolve_tac rules 1)); (*does not solve it directly!*)
    back();
    back();

    (*backtracking version*)
    val prolog_tac = DEPTH_FIRST (has_fewer_prems 1) (resolve_tac rules 1);

    choplev 0;
    by prolog_tac;
    result();

    Goal "rev(a:?x:c:?y:Nil, d:?z:b:?u)";
    by prolog_tac;
    result();

    (*rev([a..p], ?w) requires 153 inferences *)
    Goal "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil, ?w)";
    by (DEPTH_SOLVE (resolve_tac ([refl,conjI]@rules) 1));
    (*Poly/ML: 4 secs >> 38 lips*)
    result();

    (*?x has 16, ?y has 32;  rev(?y,?w) requires 561 (rather large) inferences;
      total inferences = 2 + 1 + 17 + 561 = 581*)
    Goal
        "a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil = ?x & app(?x,?x,?y) & rev(?y,?w)";
    by (DEPTH_SOLVE (resolve_tac ([refl,conjI]@rules) 1));
    (*Poly/ML: 29 secs >> 20 lips*)
    result();