0.12/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.12/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.14/0.35 % Computer : n017.cluster.edu 0.14/0.35 % Model : x86_64 x86_64 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.14/0.35 % Memory : 8042.1875MB 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.14/0.35 % CPULimit : 960 0.14/0.35 % WCLimit : 120 0.14/0.35 % DateTime : Thu Jul 2 07:49:51 EDT 2020 0.14/0.35 % CPUTime : 29.85/4.29 % SZS status Theorem 29.85/4.29 29.85/4.29 % SZS output start Proof 29.85/4.29 Take the following subset of the input axioms: 29.85/4.30 fof(conj_0, conjecture, hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_F, v_Q____)))). 29.85/4.30 fof(fact__096Q_A_058_AProf_096, axiom, hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi, tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), tc_HOL_Obool))), v_Q____), c_Arrow__Order__Mirabelle_OProf))). 29.85/4.30 fof(fact__096_B_Bi_O_Ab_A_060_092_060_094bsub_062Q_Ai_092_060_094esub_062_Ab_H_096, axiom, ![V_i_2]: hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_Q____, V_i_2)))). 29.85/4.30 fof(fact_u, axiom, c_Arrow__Order__Mirabelle_Ounanimity(v_F)). 29.85/4.30 fof(fact_unanimity__def, axiom, ![V_Fa_2]: (![B_x]: (hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi, tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), tc_HOL_Obool))), B_x), c_Arrow__Order__Mirabelle_OProf)) => ![B_a, B_b]: (![B_i]: hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), B_a), B_b)), hAPP(B_x, B_i))) => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), B_a), B_b)), hAPP(V_Fa_2, B_x))))) <=> c_Arrow__Order__Mirabelle_Ounanimity(V_Fa_2))). 29.85/4.30 29.85/4.30 Now clausify the problem and encode Horn clauses using encoding 3 of 29.85/4.30 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 29.85/4.30 We repeatedly replace C & s=t => u=v by the two clauses: 29.85/4.30 fresh(y, y, x1...xn) = u 29.85/4.30 C => fresh(s, t, x1...xn) = v 29.85/4.30 where fresh is a fresh function symbol and x1..xn are the free 29.85/4.30 variables of u and v. 29.85/4.30 A predicate p(X) is encoded as p(X)=true (this is sound, because the 29.85/4.30 input problem has no model of domain size 1). 29.85/4.30 29.85/4.30 The encoding turns the above axioms into the following unit equations and goals: 29.85/4.30 29.85/4.30 Axiom 1 (fact_IIA__def_5): fresh582(X, X, Y, Z, W, V, U) = hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), V), U)), hAPP(Y, Z))). 29.85/4.30 Axiom 2 (fact_extensional__funcset__mem): fresh357(X, X, Y, Z, W, V, U, T) = hBOOL(hAPP(hAPP(c_member(U), hAPP(V, Y)), Z)). 29.85/4.30 Axiom 3 (fact_in__mono): fresh344(X, X, Y, Z, W, V) = hBOOL(hAPP(hAPP(c_member(V), Y), Z)). 29.85/4.30 Axiom 4 (fact_unanimity__def_2): fresh612(X, X, Y, Z, W, V) = hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), W), V)), hAPP(Y, Z))). 29.85/4.30 Axiom 5 (fact_unanimity__def_2): fresh112(X, X, Y, Z, W, V) = true2. 29.85/4.30 Axiom 6 (fact_unanimity__def_2): fresh611(X, X, Y, Z, W, V) = fresh612(hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi, tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), tc_HOL_Obool))), Z), c_Arrow__Order__Mirabelle_OProf)), true2, Y, Z, W, V). 29.85/4.30 Axiom 7 (fact__096Q_A_058_AProf_096): hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi, tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), tc_HOL_Obool))), v_Q____), c_Arrow__Order__Mirabelle_OProf)) = true2. 29.85/4.30 Axiom 8 (fact_unanimity__def_2): fresh611(c_Arrow__Order__Mirabelle_Ounanimity(X), true2, X, Y, Z, W) = fresh112(hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), Z), W)), hAPP(Y, sK67_fact_unanimity__def_B_i(Y, Z, W)))), true2, X, Y, Z, W). 29.85/4.30 Axiom 9 (fact_u): c_Arrow__Order__Mirabelle_Ounanimity(v_F) = true2. 29.85/4.30 Axiom 10 (fact__096_B_Bi_O_Ab_A_060_092_060_094bsub_062Q_Ai_092_060_094esub_062_Ab_H_096): hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_Q____, X))) = true2. 29.85/4.30 29.85/4.30 Lemma 11: fresh357(?, ?, V, hAPP(Y, Z), ?, hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), W), tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), ?) = fresh612(?, ?, Y, Z, W, V). 29.85/4.30 Proof: 29.85/4.30 fresh357(?, ?, V, hAPP(Y, Z), ?, hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), W), tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), ?) 29.85/4.30 = { by axiom 2 (fact_extensional__funcset__mem) } 29.85/4.30 hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), W), V)), hAPP(Y, Z))) 29.85/4.30 = { by axiom 4 (fact_unanimity__def_2) } 29.85/4.30 fresh612(?, ?, Y, Z, W, V) 29.85/4.30 29.85/4.30 Lemma 12: fresh582(?, ?, X, Y, ?, Z, W) = fresh612(?, ?, X, Y, Z, W). 29.85/4.30 Proof: 29.85/4.30 fresh582(?, ?, X, Y, ?, Z, W) 29.85/4.30 = { by axiom 1 (fact_IIA__def_5) } 29.85/4.30 hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), Z), W)), hAPP(X, Y))) 29.85/4.30 = { by axiom 2 (fact_extensional__funcset__mem) } 29.85/4.30 fresh357(?, ?, W, hAPP(X, Y), ?, hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), Z), tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), ?) 29.85/4.30 = { by lemma 11 } 29.85/4.30 fresh612(?, ?, X, Y, Z, W) 29.85/4.30 29.85/4.30 Goal 1 (conj_0): hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_F, v_Q____))) = true2. 29.85/4.30 Proof: 29.85/4.30 hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_F, v_Q____))) 29.85/4.30 = { by axiom 3 (fact_in__mono) } 29.85/4.30 fresh344(?, ?, hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____), hAPP(v_F, v_Q____), ?, tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)) 29.85/4.30 = { by axiom 3 (fact_in__mono) } 29.85/4.30 hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_F, v_Q____))) 29.85/4.30 = { by axiom 4 (fact_unanimity__def_2) } 29.85/4.30 fresh612(true2, true2, v_F, v_Q____, v_b____, v_b_H____) 29.85/4.30 = { by axiom 7 (fact__096Q_A_058_AProf_096) } 29.85/4.30 fresh612(hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi, tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), tc_HOL_Obool))), v_Q____), c_Arrow__Order__Mirabelle_OProf)), true2, v_F, v_Q____, v_b____, v_b_H____) 29.85/4.30 = { by axiom 6 (fact_unanimity__def_2) } 29.85/4.30 fresh611(true2, true2, v_F, v_Q____, v_b____, v_b_H____) 29.85/4.30 = { by axiom 9 (fact_u) } 29.85/4.30 fresh611(c_Arrow__Order__Mirabelle_Ounanimity(v_F), true2, v_F, v_Q____, v_b____, v_b_H____) 29.85/4.30 = { by axiom 8 (fact_unanimity__def_2) } 29.85/4.30 fresh112(hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt)), hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, tc_Arrow__Order__Mirabelle_Oalt), v_b____), v_b_H____)), hAPP(v_Q____, sK67_fact_unanimity__def_B_i(v_Q____, v_b____, v_b_H____)))), true2, v_F, v_Q____, v_b____, v_b_H____) 29.85/4.30 = { by axiom 10 (fact__096_B_Bi_O_Ab_A_060_092_060_094bsub_062Q_Ai_092_060_094esub_062_Ab_H_096) } 29.85/4.30 fresh112(true2, true2, v_F, v_Q____, v_b____, v_b_H____) 29.85/4.30 = { by axiom 5 (fact_unanimity__def_2) } 29.85/4.30 true2 29.85/4.30 % SZS output end Proof 29.85/4.30 29.85/4.30 RESULT: Theorem (the conjecture is true). 29.85/4.32 EOF