Theory Prolog

(*  Title:      FOL/ex/Prolog.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge
*)

section ‹First-Order Logic: PROLOG examples›

theory Prolog
imports FOL
begin

typedecl 'a list
instance list :: (‹term›) ‹term› ..

axiomatization
  Nil     :: ‹'a list› and
  Cons    :: ‹['a, 'a list]=> 'a list›    (infixr ‹:› 60) and
  app     :: ‹['a list, 'a list, 'a list] => o› and
  rev     :: ‹['a list, 'a list] => o›
where
  appNil:  ‹app(Nil,ys,ys)› and
  appCons: ‹app(xs,ys,zs) ==> app(x:xs, ys, x:zs)› and
  revNil:  ‹rev(Nil,Nil)› and
  revCons: ‹[| rev(xs,ys);  app(ys, x:Nil, zs) |] ==> rev(x:xs, zs)›

schematic_goal ‹app(a:b:c:Nil, d:e:Nil, ?x)›
apply (rule appNil appCons)
apply (rule appNil appCons)
apply (rule appNil appCons)
apply (rule appNil appCons)
done

schematic_goal ‹app(?x, c:d:Nil, a:b:c:d:Nil)›
apply (rule appNil appCons)+
done

schematic_goal ‹app(?x, ?y, a:b:c:d:Nil)›
apply (rule appNil appCons)+
back
back
back
back
done

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

lemmas rules = appNil appCons revNil revCons

schematic_goal ‹rev(a:b:c:d:Nil, ?x)›
apply (rule rules)+
done

schematic_goal ‹rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:Nil, ?w)›
apply (rule rules)+
done

schematic_goal ‹rev(?x, a:b:c:Nil)›
apply (rule rules)+  ― ‹does not solve it directly!›
back
back
done

(*backtracking version*)
ML ‹
fun prolog_tac ctxt =
  DEPTH_FIRST Thm.no_prems (resolve_tac ctxt @{thms rules} 1)
›

schematic_goal ‹rev(?x, a:b:c:Nil)›
apply (tactic ‹prolog_tac \<^context>›)
done

schematic_goal ‹rev(a:?x:c:?y:Nil, d:?z:b:?u)›
apply (tactic ‹prolog_tac \<^context>›)
done

(*rev([a..p], ?w) requires 153 inferences *)
schematic_goal ‹rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil, ?w)›
apply (tactic ‹
  DEPTH_SOLVE (resolve_tac \<^context> ([@{thm refl}, @{thm conjI}] @ @{thms rules}) 1)›)
done

(*?x has 16, ?y has 32;  rev(?y,?w) requires 561 (rather large) inferences
  total inferences = 2 + 1 + 17 + 561 = 581*)
schematic_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)›
apply (tactic ‹
  DEPTH_SOLVE (resolve_tac \<^context> ([@{thm refl}, @{thm conjI}] @ @{thms rules}) 1)›)
done

end