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(* Title: HOL/Prolog/Test.thy
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ID: $Id$
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Author: David von Oheimb (based on a lecture on Lambda Prolog by Nadathur)
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*)
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header {* Basic examples *}
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theory Test
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imports HOHH
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begin
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typedecl nat
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typedecl 'a list
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consts
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Nil :: "'a list" ("[]")
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Cons :: "'a => 'a list => 'a list" (infixr "#" 65)
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syntax
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(* list Enumeration *)
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"@list" :: "args => 'a list" ("[(_)]")
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translations
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"[x, xs]" == "x#[xs]"
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"[x]" == "x#[]"
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typedecl person
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consts
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append :: "['a list, 'a list, 'a list] => bool"
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reverse :: "['a list, 'a list] => bool"
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mappred :: "[('a => 'b => bool), 'a list, 'b list] => bool"
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mapfun :: "[('a => 'b), 'a list, 'b list] => bool"
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bob :: person
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sue :: person
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ned :: person
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"23" :: nat ("23")
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"24" :: nat ("24")
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"25" :: nat ("25")
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age :: "[person, nat] => bool"
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eq :: "['a, 'a] => bool"
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empty :: "['b] => bool"
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add :: "['a, 'b, 'b] => bool"
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remove :: "['a, 'b, 'b] => bool"
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bag_appl:: "['a, 'a, 'a, 'a] => bool"
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axioms
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append: "append [] xs xs ..
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append (x#xs) ys (x#zs) :- append xs ys zs"
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reverse: "reverse L1 L2 :- (!rev_aux.
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(!L. rev_aux [] L L )..
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(!X L1 L2 L3. rev_aux (X#L1) L2 L3 :- rev_aux L1 L2 (X#L3))
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=> rev_aux L1 L2 [])"
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mappred: "mappred P [] [] ..
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mappred P (x#xs) (y#ys) :- P x y & mappred P xs ys"
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mapfun: "mapfun f [] [] ..
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mapfun f (x#xs) (f x#ys) :- mapfun f xs ys"
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age: "age bob 24 ..
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age sue 23 ..
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age ned 23"
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eq: "eq x x"
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(* actual definitions of empty and add is hidden -> yields abstract data type *)
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bag_appl: "bag_appl A B X Y:- (? S1 S2 S3 S4 S5.
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empty S1 &
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add A S1 S2 &
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add B S2 S3 &
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remove X S3 S4 &
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remove Y S4 S5 &
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empty S5)"
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lemmas prog_Test = append reverse mappred mapfun age eq bag_appl
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lemma "append ?x ?y [a,b,c,d]"
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apply (prolog prog_Test)
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back
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back
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back
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back
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done
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lemma "append [a,b] y ?L"
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apply (prolog prog_Test)
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done
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lemma "!y. append [a,b] y (?L y)"
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apply (prolog prog_Test)
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done
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lemma "reverse [] ?L"
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apply (prolog prog_Test)
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done
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lemma "reverse [23] ?L"
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apply (prolog prog_Test)
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done
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lemma "reverse [23,24,?x] ?L"
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apply (prolog prog_Test)
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done
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lemma "reverse ?L [23,24,?x]"
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apply (prolog prog_Test)
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done
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lemma "mappred age ?x [23,24]"
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apply (prolog prog_Test)
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back
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done
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lemma "mappred (%x y. ? z. age z y) ?x [23,24]"
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apply (prolog prog_Test)
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done
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lemma "mappred ?P [bob,sue] [24,23]"
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apply (prolog prog_Test)
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done
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lemma "mapfun f [bob,bob,sue] [?x,?y,?z]"
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apply (prolog prog_Test)
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done
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lemma "mapfun (%x. h x 25) [bob,sue] ?L"
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apply (prolog prog_Test)
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done
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lemma "mapfun ?F [24,25] [h bob 24,h bob 25]"
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apply (prolog prog_Test)
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done
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lemma "mapfun ?F [24] [h 24 24]"
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apply (prolog prog_Test)
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back
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back
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back
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done
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lemma "True"
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apply (prolog prog_Test)
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done
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lemma "age ?x 24 & age ?y 23"
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apply (prolog prog_Test)
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back
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done
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lemma "age ?x 24 | age ?x 23"
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apply (prolog prog_Test)
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back
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back
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done
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lemma "? x y. age x y"
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apply (prolog prog_Test)
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done
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lemma "!x y. append [] x x"
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apply (prolog prog_Test)
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done
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lemma "age sue 24 .. age bob 23 => age ?x ?y"
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apply (prolog prog_Test)
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back
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back
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back
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back
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done
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(*set trace_DEPTH_FIRST;*)
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lemma "age bob 25 :- age bob 24 => age bob 25"
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apply (prolog prog_Test)
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done
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(*reset trace_DEPTH_FIRST;*)
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lemma "(!x. age x 25 :- age x 23) => age ?x 25 & age ?y 25"
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apply (prolog prog_Test)
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back
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back
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back
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done
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lemma "!x. ? y. eq x y"
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apply (prolog prog_Test)
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done
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lemma "? P. P & eq P ?x"
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apply (prolog prog_Test)
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(*
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back
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back
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back
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back
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back
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back
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back
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back
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*)
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done
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lemma "? P. eq P (True & True) & P"
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apply (prolog prog_Test)
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done
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lemma "? P. eq P op | & P k True"
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apply (prolog prog_Test)
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done
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lemma "? P. eq P (Q => True) & P"
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apply (prolog prog_Test)
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done
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(* P flexible: *)
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lemma "(!P k l. P k l :- eq P Q) => Q a b"
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apply (prolog prog_Test)
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done
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(*
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loops:
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lemma "(!P k l. P k l :- eq P (%x y. x | y)) => a | b"
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*)
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(* implication and disjunction in atom: *)
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lemma "? Q. (!p q. R(q :- p) => R(Q p q)) & Q (t | s) (s | t)"
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by fast
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(* disjunction in atom: *)
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lemma "(!P. g P :- (P => b | a)) => g(a | b)"
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apply (tactic "step_tac HOL_cs 1")
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apply (tactic "step_tac HOL_cs 1")
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apply (tactic "step_tac HOL_cs 1")
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prefer 2
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apply fast
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apply fast
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done
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(*
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hangs:
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lemma "(!P. g P :- (P => b | a)) => g(a | b)"
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by fast
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*)
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lemma "!Emp Stk.(
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empty (Emp::'b) ..
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(!(X::nat) S. add X (S::'b) (Stk X S)) ..
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(!(X::nat) S. remove X ((Stk X S)::'b) S))
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=> bag_appl 23 24 ?X ?Y"
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oops
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lemma "!Qu. (
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(!L. empty (Qu L L)) ..
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(!(X::nat) L K. add X (Qu L (X#K)) (Qu L K)) ..
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(!(X::nat) L K. remove X (Qu (X#L) K) (Qu L K)))
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=> bag_appl 23 24 ?X ?Y"
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oops
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lemma "D & (!y. E) :- (!x. True & True) :- True => D"
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apply (prolog prog_Test)
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done
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lemma "P x .. P y => P ?X"
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apply (prolog prog_Test)
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back
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done
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(*
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back
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-> problem with DEPTH_SOLVE:
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Exception- THM ("dest_state", 1, ["P x & P y --> P y"]) raised
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Exception raised at run-time
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*)
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end
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