; Deps: sat.plf smt.plf th_base.plf

(check
(% s sort
(% a (term s)
(% b (term s)
(% c (term s)
(% f (term (arrow s s))

; -------------------- declaration of input formula -----------------------------------

(% A1 (th_holds (= s a b))
(% A2 (th_holds (= s b c))
(% A3 (th_holds (not (= s a c)))

; ------------------- specify that the following is a proof of the empty clause -----------------

(: (holds cln)

; ---------- use atoms (a1, a2, a3) to map theory literals to boolean literals (v1, v2, v3) ------

(decl_atom (= s a b) (\ v1 (\ a1
(decl_atom (= s b c) (\ v2 (\ a2
(decl_atom (= s a c) (\ v3 (\ a3

; --------------------------- proof of theory lemma ---------------------------------------------

(satlem _ _ (ast _ _ _ a1 (\ l1 (ast _ _ _ a2 (\ l2 (asf _ _ _ a3 (\ l3  (clausify_false (contra _ (trans _ _ _ _ l1 l2) l3)))))))) (\ CT1

; -------------------- clausification of input formulas -----------------------------------------

(satlem _ _
(asf _ _ _ a1 (\ l1
(clausify_false
  (contra _ A1 l1)
))) (\ C1
; C1 is the clause ( v1 )

(satlem _ _
(asf _ _ _ a2 (\ l2
(clausify_false
   (contra _ A2 l2)
))) (\ C2
; C2 is the clause ( v2 )

(satlem _ _
(ast _ _ _ a3 (\ l3
(clausify_false
  (contra _ l3 A3)
))) (\ C3
; C3 is the clause ( ~v3 )


; -------------------- resolution proof ------------------------------------------------------------

(satlem_simplify _ _ _

(R _ _
(R _ _ C2
(R _ _ C1 CT1 v1) v2) C3 v3)

(\ x x))

)))))))))))))))))))))))))))