# Theory Test

theory Test
imports HOHH
```(*  Title:    HOL/Prolog/Test.thy
Author:   David von Oheimb (based on a lecture on Lambda Prolog by Nadathur)
*)

section ‹Basic examples›

theory Test
imports HOHH
begin

typedecl nat

typedecl 'a list

consts
Nil   :: "'a list"                                  ("[]")
Cons  :: "'a => 'a list => 'a list"                 (infixr "#"  65)

syntax
(* list Enumeration *)
"_list"     :: "args => 'a list"                          ("[(_)]")

translations
"[x, xs]"     == "x#[xs]"
"[x]"         == "x#[]"

typedecl person

axiomatization
append  :: "['a list, 'a list, 'a list]            => bool" and
reverse :: "['a list, 'a list]                     => bool" and

mappred :: "[('a => 'b => bool), 'a list, 'b list] => bool" and
mapfun  :: "[('a => 'b), 'a list, 'b list]         => bool" and

bob     :: person and
sue     :: person and
ned     :: person and

nat23   :: nat          ("23") and
nat24   :: nat          ("24") and
nat25   :: nat          ("25") and

age     :: "[person, nat]                          => bool" and

eq      :: "['a, 'a]                               => bool" and

empty   :: "['b]                                   => bool" and
add     :: "['a, 'b, 'b]                           => bool" and
remove  :: "['a, 'b, 'b]                           => bool" and
bag_appl:: "['a, 'a, 'a, 'a]                       => bool"
where
append:  "⋀x xs ys zs. append  []    xs  xs    ..
append (x#xs) ys (x#zs) :- append xs ys zs" and
reverse: "⋀L1 L2. reverse L1 L2 :- (∀rev_aux.
(∀L.          rev_aux  []    L  L )..
(∀X L1 L2 L3. rev_aux (X#L1) L2 L3 :- rev_aux L1 L2 (X#L3))
=> rev_aux L1 L2 [])" and

mappred: "⋀x xs y ys P. mappred P  []     []    ..
mappred P (x#xs) (y#ys) :- P x y  &  mappred P xs ys" and
mapfun:  "⋀x xs ys f. mapfun f  []     []      ..
mapfun f (x#xs) (f x#ys) :- mapfun f xs ys" and

age:     "age bob 24 ..
age sue 23 ..
age ned 23" and

eq:      "⋀x. eq x x" and

(* actual definitions of empty and add is hidden -> yields abstract data type *)

bag_appl: "⋀A B X Y. bag_appl A B X Y:- (∃S1 S2 S3 S4 S5.
empty    S1    &
remove X S3 S4 &
remove Y S4 S5 &
empty    S5)"

lemmas prog_Test = append reverse mappred mapfun age eq bag_appl

schematic_goal "append ?x ?y [a,b,c,d]"
apply (prolog prog_Test)
back
back
back
back
done

schematic_goal "append [a,b] y ?L"
apply (prolog prog_Test)
done

schematic_goal "∀y. append [a,b] y (?L y)"
apply (prolog prog_Test)
done

schematic_goal "reverse [] ?L"
apply (prolog prog_Test)
done

schematic_goal "reverse [23] ?L"
apply (prolog prog_Test)
done

schematic_goal "reverse [23,24,?x] ?L"
apply (prolog prog_Test)
done

schematic_goal "reverse ?L [23,24,?x]"
apply (prolog prog_Test)
done

schematic_goal "mappred age ?x [23,24]"
apply (prolog prog_Test)
back
done

schematic_goal "mappred (λx y. ∃z. age z y) ?x [23,24]"
apply (prolog prog_Test)
done

schematic_goal "mappred ?P [bob,sue] [24,23]"
apply (prolog prog_Test)
done

schematic_goal "mapfun f [bob,bob,sue] [?x,?y,?z]"
apply (prolog prog_Test)
done

schematic_goal "mapfun (%x. h x 25) [bob,sue] ?L"
apply (prolog prog_Test)
done

schematic_goal "mapfun ?F [24,25] [h bob 24,h bob 25]"
apply (prolog prog_Test)
done

schematic_goal "mapfun ?F [24] [h 24 24]"
apply (prolog prog_Test)
back
back
back
done

lemma "True"
apply (prolog prog_Test)
done

schematic_goal "age ?x 24 & age ?y 23"
apply (prolog prog_Test)
back
done

schematic_goal "age ?x 24 | age ?x 23"
apply (prolog prog_Test)
back
back
done

lemma "∃x y. age x y"
apply (prolog prog_Test)
done

lemma "∀x y. append [] x x"
apply (prolog prog_Test)
done

schematic_goal "age sue 24 .. age bob 23 => age ?x ?y"
apply (prolog prog_Test)
back
back
back
back
done

(*set trace_DEPTH_FIRST;*)
lemma "age bob 25 :- age bob 24 => age bob 25"
apply (prolog prog_Test)
done
(*reset trace_DEPTH_FIRST;*)

schematic_goal "(∀x. age x 25 :- age x 23) => age ?x 25 & age ?y 25"
apply (prolog prog_Test)
back
back
back
done

lemma "∀x. ∃y. eq x y"
apply (prolog prog_Test)
done

schematic_goal "∃P. P ∧ eq P ?x"
apply (prolog prog_Test)
(*
back
back
back
back
back
back
back
back
*)
done

lemma "∃P. eq P (True & True) ∧ P"
apply (prolog prog_Test)
done

lemma "∃P. eq P (∨) ∧ P k True"
apply (prolog prog_Test)
done

lemma "∃P. eq P (Q => True) ∧ P"
apply (prolog prog_Test)
done

(* P flexible: *)
lemma "(∀P k l. P k l :- eq P Q) => Q a b"
apply (prolog prog_Test)
done
(*
loops:
lemma "(!P k l. P k l :- eq P (%x y. x | y)) => a | b"
*)

(* implication and disjunction in atom: *)
lemma "∃Q. (∀p q. R(q :- p) => R(Q p q)) ∧ Q (t | s) (s | t)"
by fast

(* disjunction in atom: *)
lemma "(∀P. g P :- (P => b ∨ a)) => g(a ∨ b)"
apply (tactic "step_tac (put_claset HOL_cs @{context}) 1")
apply (tactic "step_tac (put_claset HOL_cs @{context}) 1")
apply (tactic "step_tac (put_claset HOL_cs @{context}) 1")
prefer 2
apply fast
apply fast
done

(*
hangs:
lemma "(!P. g P :- (P => b | a)) => g(a | b)"
by fast
*)

schematic_goal "∀Emp Stk.(
empty    (Emp::'b) ..
(∀(X::nat) S. add    X (S::'b)         (Stk X S)) ..
(∀(X::nat) S. remove X ((Stk X S)::'b) S))
=> bag_appl 23 24 ?X ?Y"
oops

schematic_goal "∀Qu. (
(∀L.            empty    (Qu L L)) ..
(∀(X::nat) L K. add    X (Qu L (X#K)) (Qu L K)) ..
(∀(X::nat) L K. remove X (Qu (X#L) K) (Qu L K)))
=> bag_appl 23 24 ?X ?Y"
oops

lemma "D ∧ (∀y. E) :- (∀x. True ∧ True) :- True => D"
apply (prolog prog_Test)
done

schematic_goal "P x .. P y => P ?X"
apply (prolog prog_Test)
back
done

end
```