0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.13	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.34	% Computer   : n002.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 1200
0.12/0.34	% WCLimit    : 120
0.12/0.34	% DateTime   : Tue Jul 13 15:41:18 EDT 2021
0.12/0.34	% CPUTime    : 
32.43/4.47	% SZS status Theorem
32.43/4.47	
32.43/4.48	% SZS output start Proof
32.43/4.48	Take the following subset of the input axioms:
32.43/4.48	  fof(thm_2Ebool_2EEQ__CLAUSES, axiom, ![V0t_2E0]: ((p(s(tyop_2Emin_2Ebool, V0t_2E0)) <=> s(tyop_2Emin_2Ebool, V0t_2E0)=s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0)) & ((~p(s(tyop_2Emin_2Ebool, V0t_2E0)) <=> s(tyop_2Emin_2Ebool, c_2Ebool_2EF_2E0)=s(tyop_2Emin_2Ebool, V0t_2E0)) & ((s(tyop_2Emin_2Ebool, c_2Ebool_2EF_2E0)=s(tyop_2Emin_2Ebool, V0t_2E0) <=> ~p(s(tyop_2Emin_2Ebool, V0t_2E0))) & (p(s(tyop_2Emin_2Ebool, V0t_2E0)) <=> s(tyop_2Emin_2Ebool, V0t_2E0)=s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0)))))).
32.43/4.48	  fof(thm_2Ebool_2Ebool__case__thm, axiom, ![A_27a]: (![V2t1_2E0, V3t2_2E0]: s(A_27a, V3t2_2E0)=s(A_27a, c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Ebool_2EF_2E0), s(A_27a, V2t1_2E0), s(A_27a, V3t2_2E0))) & ![V0t1_2E0, V1t2_2E0]: s(A_27a, V0t1_2E0)=s(A_27a, c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0), s(A_27a, V0t1_2E0), s(A_27a, V1t2_2E0))))).
32.43/4.48	  fof(thm_2Ewords_2EWORD__LITERAL__ADD, axiom, ![A_27a, A_27b]: (![V0m_2E0, V1n_2E0]: s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__add_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, V0m_2E0))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, V1n_2E0)))))))=s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum, V0m_2E0), s(tyop_2Enum_2Enum, V1n_2E0))))))) & ![V2m_2E0, V3n_2E0]: s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, V3n_2E0), s(tyop_2Enum_2Enum, V2m_2E0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, V2m_2E0), s(tyop_2Enum_2Enum, V3n_2E0))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, V3n_2E0), s(tyop_2Enum_2Enum, V2m_2E0)))))))))=s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2Eword__add_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, V2m_2E0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27b), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, V3n_2E0))))))))).
32.43/4.48	  fof(thm_2Ewords_2En2w__sub, conjecture, ![A_27a, V0a_2E0, V1b_2E0]: (s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__sub_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, V0a_2E0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, V1b_2E0)))))=s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, V0a_2E0), s(tyop_2Enum_2Enum, V1b_2E0))))) <= p(s(tyop_2Emin_2Ebool, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, V1b_2E0), s(tyop_2Enum_2Enum, V0a_2E0)))))).
32.43/4.48	  fof(thm_2Ewords_2Eword__sub__def, axiom, ![A_27a, V0v_2E0, V1w_2E0]: s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__add_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), V0v_2E0), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), V1w_2E0)))))=s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), c_2Ewords_2Eword__sub_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), V0v_2E0), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, A_27a), V1w_2E0)))).
32.43/4.48	
32.43/4.48	Now clausify the problem and encode Horn clauses using encoding 3 of
32.43/4.48	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
32.43/4.48	We repeatedly replace C & s=t => u=v by the two clauses:
32.43/4.48	  fresh(y, y, x1...xn) = u
32.43/4.48	  C => fresh(s, t, x1...xn) = v
32.43/4.48	where fresh is a fresh function symbol and x1..xn are the free
32.43/4.48	variables of u and v.
32.43/4.48	A predicate p(X) is encoded as p(X)=true (this is sound, because the
32.43/4.48	input problem has no model of domain size 1).
32.43/4.48	
32.43/4.48	The encoding turns the above axioms into the following unit equations and goals:
32.43/4.48	
32.43/4.48	Axiom 1 (thm_2Ebool_2EEQ__CLAUSES_6): fresh7(X, X, Y) = s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0).
32.43/4.48	Axiom 2 (thm_2Ebool_2EEQ__CLAUSES_6): fresh7(p(s(tyop_2Emin_2Ebool, X)), true2, X) = s(tyop_2Emin_2Ebool, X).
32.43/4.48	Axiom 3 (thm_2Ewords_2En2w__sub): p(s(tyop_2Emin_2Ebool, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0)))) = true2.
32.43/4.48	Axiom 4 (thm_2Ebool_2Ebool__case__thm_1): s(X, Y) = s(X, c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0), s(X, Y), s(X, Z))).
32.43/4.48	Axiom 5 (thm_2Ewords_2Eword__sub__def): s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2Eword__add_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), Y), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), Z))))) = s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2Eword__sub_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), Y), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), Z))).
32.43/4.48	Axiom 6 (thm_2Ewords_2EWORD__LITERAL__ADD_1): s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, Y), s(tyop_2Enum_2Enum, Z))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, Z), s(tyop_2Enum_2Enum, Y))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, Y), s(tyop_2Enum_2Enum, Z))))))))) = s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2Eword__add_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, Z))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, X), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, Y))))))).
32.43/4.48	
32.43/4.48	Goal 1 (thm_2Ewords_2En2w__sub_1): s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__sub_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, v0a_2e0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, v1b_2e0))))) = s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v0a_2e0), s(tyop_2Enum_2Enum, v1b_2e0))))).
32.43/4.48	Proof:
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__sub_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, v0a_2e0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, v1b_2e0)))))
32.43/4.48	= { by axiom 5 (thm_2Ewords_2Eword__sub__def) R->L }
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__add_2E2(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, v0a_2e0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, v1b_2e0)))))))
32.43/4.48	= { by axiom 6 (thm_2Ewords_2EWORD__LITERAL__ADD_1) R->L }
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v0a_2e0), s(tyop_2Enum_2Enum, v1b_2e0))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0)))))))))
32.43/4.48	= { by axiom 2 (thm_2Ebool_2EEQ__CLAUSES_6) R->L }
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ebool_2ECOND_2E3(fresh7(p(s(tyop_2Emin_2Ebool, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0)))), true2, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v0a_2e0), s(tyop_2Enum_2Enum, v1b_2e0))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0)))))))))
32.43/4.48	= { by axiom 3 (thm_2Ewords_2En2w__sub) }
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ebool_2ECOND_2E3(fresh7(true2, true2, c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v0a_2e0), s(tyop_2Enum_2Enum, v1b_2e0))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0)))))))))
32.43/4.48	= { by axiom 1 (thm_2Ebool_2EEQ__CLAUSES_6) }
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v0a_2e0), s(tyop_2Enum_2Enum, v1b_2e0))))), s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2Eword__2comp_2E1(s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v1b_2e0), s(tyop_2Enum_2Enum, v0a_2e0)))))))))
32.43/4.48	= { by axiom 4 (thm_2Ebool_2Ebool__case__thm_1) R->L }
32.43/4.48	  s(tyop_2Efcp_2Ecart(tyop_2Emin_2Ebool, a_27a), c_2Ewords_2En2w_2E1(s(tyop_2Enum_2Enum, c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum, v0a_2e0), s(tyop_2Enum_2Enum, v1b_2e0)))))
32.43/4.48	% SZS output end Proof
32.43/4.48	
32.43/4.48	RESULT: Theorem (the conjecture is true).
32.43/4.49	EOF