0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.02/0.24 % Computer : n002.star.cs.uiowa.edu 0.02/0.24 % Model : x86_64 x86_64 0.02/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.24 % Memory : 32218.625MB 0.02/0.24 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.24 % CPULimit : 300 0.02/0.24 % DateTime : Sat Jul 14 06:04:40 CDT 2018 0.02/0.24 % CPUTime : 15.11/15.38 % SZS status Theorem 15.11/15.38 15.11/15.38 % SZS output start Proof 15.11/15.38 Take the following subset of the input axioms: 15.11/15.38 fof(fact_Lin__irrefl, axiom, 15.11/15.38 ![V_L_2, V_ba_2, V_aa_2]: 15.11/15.38 ((~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.38 tc_Arrow__Order__Mirabelle_Oalt)), 15.11/15.38 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.38 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.38 V_ba_2), 15.11/15.38 V_aa_2)), 15.11/15.38 V_L_2)) 15.11/15.38 <= hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.38 tc_Arrow__Order__Mirabelle_Oalt)), 15.11/15.38 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.38 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.38 V_aa_2), 15.11/15.38 V_ba_2)), 15.11/15.38 V_L_2))) 15.11/15.38 <= hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.38 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.38 tc_HOL_Obool)), 15.11/15.38 V_L_2), 15.11/15.38 c_Arrow__Order__Mirabelle_OLin)))). 15.11/15.38 fof(fact_Nil2__notin__lex, axiom, 15.11/15.38 ![T_a, V_r_2, V_xs_2]: 15.11/15.38 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 15.11/15.38 tc_List_Olist(T_a))), 15.11/15.38 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 15.11/15.38 tc_List_Olist(T_a)), 15.11/15.38 V_xs_2), 15.11/15.38 c_List_Olist_ONil(T_a))), 15.11/15.38 c_List_Olex(T_a, V_r_2)))). 15.11/15.38 fof(fact_Nil__notin__lex, axiom, 15.11/15.38 ![T_a, V_r_2, V_ys_2]: 15.11/15.39 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a))), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a)), 15.11/15.39 c_List_Olist_ONil(T_a)), 15.11/15.39 V_ys_2)), 15.11/15.39 c_List_Olex(T_a, V_r_2)))). 15.11/15.39 fof(fact_dropWhile__eq__Cons__conv, axiom, 15.11/15.39 ![V_y_2, T_a, V_ys_2, V_xs_2, V_Pa_2]: 15.11/15.39 (hAPP(hAPP(c_List_Olist_OCons(T_a), V_y_2), 15.11/15.39 V_ys_2)=c_List_OdropWhile(T_a, V_Pa_2, V_xs_2) 15.11/15.39 <=> (~hBOOL(hAPP(V_Pa_2, V_y_2)) 15.11/15.39 & V_xs_2=hAPP(hAPP(c_List_Oappend(T_a), 15.11/15.39 c_List_OtakeWhile(T_a, V_Pa_2, V_xs_2)), 15.11/15.39 hAPP(hAPP(c_List_Olist_OCons(T_a), V_y_2), V_ys_2))))). 15.11/15.39 fof(fact_ext, axiom, 15.11/15.39 ![V_g_2, V_f_2]: 15.11/15.39 (V_f_2=V_g_2 <= ![B_x]: hAPP(V_g_2, B_x)=hAPP(V_f_2, B_x))). 15.11/15.39 fof(fact_impossible__Cons, axiom, 15.11/15.39 ![T_a, V_x, V_xs, V_ys]: 15.11/15.39 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, 15.11/15.39 c_Nat_Osize__class_Osize(tc_List_Olist(T_a), V_xs), 15.11/15.39 c_Nat_Osize__class_Osize(tc_List_Olist(T_a), V_ys)) 15.11/15.39 => hAPP(hAPP(c_List_Olist_OCons(T_a), V_x), V_ys)!=V_xs)). 15.11/15.39 fof(fact_in__measures_I1_J, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 ~hBOOL(hAPP(hAPP(c_member(tc_prod(T_a, T_a)), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(T_a, T_a), V_x_2), V_y_2)), 15.11/15.39 c_List_Omeasures(T_a, 15.11/15.39 c_List_Olist_ONil(tc_fun(T_a, tc_Nat_Onat)))))). 15.11/15.39 fof(fact_in__mkbot, axiom, 15.11/15.39 ![V_y_2, V_x_2, V_z_2, V_L_2]: 15.11/15.39 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt)), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.39 V_x_2), 15.11/15.39 V_y_2)), 15.11/15.39 c_Arrow__Order__Mirabelle_Omkbot(V_L_2, V_z_2))) 15.11/15.39 <=> (V_z_2!=V_y_2 15.11/15.39 & ((V_x_2=V_z_2 => V_x_2!=V_y_2) 15.11/15.39 & (V_x_2!=V_z_2 15.11/15.39 => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt)), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.39 V_x_2), 15.11/15.39 V_y_2)), 15.11/15.39 V_L_2))))))). 15.11/15.39 fof(fact_in__mktop, axiom, 15.11/15.39 ![V_y_2, V_x_2, V_z_2, V_L_2]: 15.11/15.39 (((V_y_2!=V_x_2 <= V_z_2=V_y_2) 15.11/15.39 & ((hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt)), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.39 V_x_2), 15.11/15.39 V_y_2)), 15.11/15.39 V_L_2)) 15.11/15.39 <= V_z_2!=V_y_2) 15.11/15.39 & V_x_2!=V_z_2)) 15.11/15.39 <=> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt)), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 15.11/15.39 tc_Arrow__Order__Mirabelle_Oalt), 15.11/15.39 V_x_2), 15.11/15.39 V_y_2)), 15.11/15.39 c_Arrow__Order__Mirabelle_Omktop(V_L_2, V_z_2))))). 15.11/15.39 fof(fact_irrefl__def, axiom, 15.11/15.39 ![T_a, V_r_2]: 15.11/15.39 (c_Relation_Oirrefl(T_a, V_r_2) 15.11/15.39 <=> ![B_x]: 15.11/15.39 ~hBOOL(hAPP(hAPP(c_member(tc_prod(T_a, T_a)), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(T_a, T_a), B_x), B_x)), 15.11/15.39 V_r_2)))). 15.11/15.39 fof(fact_leD, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 (class_Orderings_Olinorder(T_a) 15.11/15.39 => (~c_Orderings_Oord__class_Oless(T_a, V_x, V_y) 15.11/15.39 <= c_Orderings_Oord__class_Oless__eq(T_a, V_y, V_x)))). 15.11/15.39 fof(fact_less__fun__def, axiom, 15.11/15.39 ![T_a, V_g_2, V_f_2, T_b]: 15.11/15.39 (class_Orderings_Oord(T_b) 15.11/15.39 => ((~c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, T_b), V_g_2, 15.11/15.39 V_f_2) 15.11/15.39 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, T_b), V_f_2, 15.11/15.39 V_g_2)) 15.11/15.39 <=> c_Orderings_Oord__class_Oless(tc_fun(T_a, T_b), V_f_2, 15.11/15.39 V_g_2)))). 15.11/15.39 fof(fact_less__imp__neq, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 (class_Orderings_Oorder(T_a) 15.11/15.39 => (V_x!=V_y <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 15.11/15.39 fof(fact_less__irrefl__nat, axiom, 15.11/15.39 ![V_n]: ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, V_n)). 15.11/15.39 fof(fact_less__le__not__le, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 (class_Orderings_Opreorder(T_a) 15.11/15.39 => (c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 15.11/15.39 <=> (c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 15.11/15.39 & ~c_Orderings_Oord__class_Oless__eq(T_a, V_y_2, V_x_2))))). 15.11/15.39 fof(fact_less__not__refl2, axiom, 15.11/15.39 ![V_m, V_n]: 15.11/15.39 (c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, V_m) 15.11/15.39 => V_n!=V_m)). 15.11/15.39 fof(fact_less__not__refl3, axiom, 15.11/15.39 ![V_t, V_s]: 15.11/15.39 (V_s!=V_t 15.11/15.39 <= c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_s, V_t))). 15.11/15.39 fof(fact_lexord__Nil__right, axiom, 15.11/15.39 ![V_x_2, T_a, V_r_2]: 15.11/15.39 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a))), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a)), 15.11/15.39 V_x_2), 15.11/15.39 c_List_Olist_ONil(T_a))), 15.11/15.39 c_List_Olexord(T_a, V_r_2)))). 15.11/15.39 fof(fact_linorder__antisym__conv2, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 (((~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 15.11/15.39 <=> V_x_2=V_y_2) 15.11/15.39 <= c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2)) 15.11/15.39 <= class_Orderings_Olinorder(T_a))). 15.11/15.39 fof(fact_linorder__neq__iff, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 (class_Orderings_Olinorder(T_a) 15.11/15.39 => ((c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 15.11/15.39 | c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2)) 15.11/15.39 <=> V_y_2!=V_x_2))). 15.11/15.39 fof(fact_linorder__not__le, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 15.11/15.39 <=> c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2)) 15.11/15.39 <= class_Orderings_Olinorder(T_a))). 15.11/15.39 fof(fact_linorder__not__less, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 15.11/15.39 <=> c_Orderings_Oord__class_Oless__eq(T_a, V_y_2, V_x_2)) 15.11/15.39 <= class_Orderings_Olinorder(T_a))). 15.11/15.39 fof(fact_list_Osimps_I2_J, axiom, 15.11/15.39 ![T_a, V_list_H, V_a_H]: 15.11/15.39 c_List_Olist_ONil(T_a)!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_a_H), 15.11/15.39 V_list_H)). 15.11/15.39 fof(fact_list_Osimps_I3_J, axiom, 15.11/15.39 ![T_a, V_list_H, V_a_H]: 15.11/15.39 c_List_Olist_ONil(T_a)!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_a_H), 15.11/15.39 V_list_H)). 15.11/15.39 fof(fact_nat__less__cases, axiom, 15.11/15.39 ![V_Pa_2, V_m_2, V_n_2]: 15.11/15.39 (((V_m_2=V_n_2 => hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2))) 15.11/15.39 => ((c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n_2, V_m_2) 15.11/15.39 => hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2))) 15.11/15.39 => hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2)))) 15.11/15.39 <= (c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2) 15.11/15.39 => hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2))))). 15.11/15.39 fof(fact_nat__neq__iff, axiom, 15.11/15.39 ![V_m_2, V_n_2]: 15.11/15.39 ((c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2) 15.11/15.39 | c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n_2, V_m_2)) 15.11/15.39 <=> V_n_2!=V_m_2)). 15.11/15.39 fof(fact_not__Cons__self, axiom, 15.11/15.39 ![T_a, V_x, V_xs]: 15.11/15.39 V_xs!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_x), V_xs)). 15.11/15.39 fof(fact_not__Cons__self2, axiom, 15.11/15.39 ![T_a, V_x, V_xs]: 15.11/15.39 hAPP(hAPP(c_List_Olist_OCons(T_a), V_x), V_xs)!=V_xs). 15.11/15.39 fof(fact_not__Nil__listrel1, axiom, 15.11/15.39 ![T_a, V_r_2, V_xs_2]: 15.11/15.39 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a))), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a)), 15.11/15.39 c_List_Olist_ONil(T_a)), 15.11/15.39 V_xs_2)), 15.11/15.39 c_List_Olistrel1(T_a, V_r_2)))). 15.11/15.39 fof(fact_not__less__iff__gr__or__eq, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 15.11/15.39 <=> (c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2) 15.11/15.39 | V_x_2=V_y_2)) 15.11/15.39 <= class_Orderings_Olinorder(T_a))). 15.11/15.39 fof(fact_not__listrel1__Nil, axiom, 15.11/15.39 ![T_a, V_r_2, V_xs_2]: 15.11/15.39 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a))), 15.11/15.39 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 15.11/15.39 tc_List_Olist(T_a)), 15.11/15.39 V_xs_2), 15.11/15.39 c_List_Olist_ONil(T_a))), 15.11/15.39 c_List_Olistrel1(T_a, V_r_2)))). 15.11/15.39 fof(fact_order__less__asym, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless(T_a, V_y, V_x) 15.11/15.39 <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 15.11/15.39 <= class_Orderings_Opreorder(T_a))). 15.11/15.39 fof(fact_order__less__asym_H, axiom, 15.11/15.39 ![T_a, V_a, V_b]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless(T_a, V_b, V_a) 15.11/15.39 <= c_Orderings_Oord__class_Oless(T_a, V_a, V_b)) 15.11/15.39 <= class_Orderings_Opreorder(T_a))). 15.11/15.39 fof(fact_order__less__imp__not__eq, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 (class_Orderings_Oorder(T_a) 15.11/15.39 => (V_y!=V_x <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 15.11/15.39 fof(fact_order__less__imp__not__eq2, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 (class_Orderings_Oorder(T_a) 15.11/15.39 => (V_x!=V_y <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 15.11/15.39 fof(fact_order__less__imp__not__less, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 (class_Orderings_Opreorder(T_a) 15.11/15.39 => (~c_Orderings_Oord__class_Oless(T_a, V_y, V_x) 15.11/15.39 <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 15.11/15.39 fof(fact_order__less__irrefl, axiom, 15.11/15.39 ![T_a, V_x]: 15.11/15.39 (~c_Orderings_Oord__class_Oless(T_a, V_x, V_x) 15.11/15.39 <= class_Orderings_Opreorder(T_a))). 15.11/15.39 fof(fact_order__less__le, axiom, 15.11/15.39 ![V_y_2, V_x_2, T_a]: 15.11/15.39 (((c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 15.11/15.39 & V_x_2!=V_y_2) 15.11/15.39 <=> c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2)) 15.11/15.39 <= class_Orderings_Oorder(T_a))). 15.11/15.39 fof(fact_order__less__not__sym, axiom, 15.11/15.39 ![T_a, V_y, V_x]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless(T_a, V_y, V_x) 15.11/15.39 <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 15.11/15.39 <= class_Orderings_Opreorder(T_a))). 15.11/15.39 fof(fact_psubset__eq, axiom, 15.11/15.39 ![T_a, V_A_2, V_B_2]: 15.11/15.39 (c_Orderings_Oord__class_Oless(tc_fun(T_a, tc_HOL_Obool), V_A_2, 15.11/15.39 V_B_2) 15.11/15.39 <=> (V_B_2!=V_A_2 15.11/15.39 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, tc_HOL_Obool), 15.11/15.39 V_A_2, V_B_2)))). 15.11/15.39 fof(fact_snoc__eq__iff__butlast, axiom, 15.11/15.39 ![V_x_2, T_a, V_ys_2, V_xs_2]: 15.11/15.39 ((V_xs_2=c_List_Obutlast(T_a, V_ys_2) 15.11/15.39 & (V_x_2=c_List_Olast(T_a, V_ys_2) 15.11/15.39 & V_ys_2!=c_List_Olist_ONil(T_a))) 15.11/15.39 <=> hAPP(hAPP(c_List_Oappend(T_a), V_xs_2), 15.11/15.39 hAPP(hAPP(c_List_Olist_OCons(T_a), V_x_2), 15.11/15.39 c_List_Olist_ONil(T_a)))=V_ys_2)). 15.11/15.39 fof(fact_xt1_I9_J, axiom, 15.11/15.39 ![T_a, V_a, V_b]: 15.11/15.39 ((~c_Orderings_Oord__class_Oless(T_a, V_a, V_b) 15.11/15.39 <= c_Orderings_Oord__class_Oless(T_a, V_b, V_a)) 15.11/15.39 <= class_Orderings_Oorder(T_a))). 15.11/15.39 fof(help_c__COMBI__1, axiom, 15.11/15.39 ![T_a, V_P]: V_P=hAPP(c_COMBI(T_a), V_P)). 15.11/15.39 fof(help_c__COMBS__1, axiom, 15.11/15.39 ![T_a, T_b, V_Pa_2, V_R_2, T_c, V_Q_2]: 15.11/15.39 hAPP(hAPP(hAPP(c_COMBS(T_b, T_c, T_a), V_Pa_2), V_Q_2), 15.11/15.39 V_R_2)=hAPP(hAPP(V_Pa_2, V_R_2), hAPP(V_Q_2, V_R_2))). 15.11/15.39 fof(help_c__fNot__1, axiom, 15.11/15.39 ![V_Pa_2]: (~hBOOL(V_Pa_2) | ~hBOOL(hAPP(c_fNot, V_Pa_2)))). 15.11/15.39 15.11/15.39 Now clausify the problem and encode Horn clauses using encoding 3 of 15.11/15.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 15.11/15.39 We repeatedly replace C & s=t => u=v by the two clauses: 15.11/15.39 $$fresh(y, y, x1...xn) = u 15.11/15.39 C => $$fresh(s, t, x1...xn) = v 15.11/15.39 where $$fresh is a fresh function symbol and x1..xn are the free 15.11/15.39 variables of u and v. 15.11/15.39 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 15.11/15.39 input problem has no model of domain size 1). 15.11/15.39 15.11/15.39 The encoding turns the above axioms into the following unit equations and goals: 15.11/15.39 15.11/15.39 Axiom 841 (help_c__COMBS__1): hAPP(hAPP(hAPP(c_COMBS(X, Y, Z), W), V), U) = hAPP(hAPP(W, U), hAPP(V, U)). 15.11/15.39 Axiom 983 (fact_ext): $$fresh7(hAPP(X, sK27_fact_ext_B_x(X, Y)), hAPP(Y, sK27_fact_ext_B_x(X, Y)), X, Y) = Y. 15.11/15.39 Axiom 993 (help_c__COMBI__1): X = hAPP(c_COMBI(Y), X). 15.11/15.39 15.11/15.39 Lemma 994: $$fresh7(sK27_fact_ext_B_x(c_COMBI(Y), X), hAPP(X, sK27_fact_ext_B_x(c_COMBI(Y), X)), c_COMBI(Y), X) = X. 15.11/15.39 Proof: 15.11/15.39 $$fresh7(sK27_fact_ext_B_x(c_COMBI(Y), X), hAPP(X, sK27_fact_ext_B_x(c_COMBI(Y), X)), c_COMBI(Y), X) 15.11/15.39 = { by axiom 993 (help_c__COMBI__1) } 15.11/15.39 $$fresh7(hAPP(c_COMBI(Y), sK27_fact_ext_B_x(c_COMBI(Y), X)), hAPP(X, sK27_fact_ext_B_x(c_COMBI(Y), X)), c_COMBI(Y), X) 15.11/15.39 = { by axiom 983 (fact_ext) } 15.11/15.39 X 15.11/15.39 15.11/15.39 Goal 1 (fact_not__Cons__self): X = hAPP(hAPP(c_List_Olist_OCons(Y), Z), X). 15.11/15.39 The goal is true when: 15.11/15.39 X = hAPP(hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?))) 15.11/15.39 Y = ? 15.11/15.39 Z = hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)) 15.11/15.39 where "?" stands for an arbitrary term of your choice. 15.11/15.39 15.11/15.39 Proof: 15.11/15.39 hAPP(hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?))) 15.11/15.39 = { by axiom 841 (help_c__COMBS__1) } 15.11/15.39 hAPP(hAPP(hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?)), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?))), hAPP(c_COMBI(?), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)))) 15.11/15.39 = { by axiom 993 (help_c__COMBI__1) } 15.11/15.39 hAPP(hAPP(hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?)), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?))), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?))) 15.11/15.39 = { by axiom 841 (help_c__COMBS__1) } 15.11/15.39 hAPP(hAPP(c_List_Olist_OCons(?), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?))), hAPP(hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)), hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)))) 15.11/15.39 % SZS output end Proof 15.11/15.39 15.11/15.39 RESULT: Theorem (the conjecture is true). 15.23/15.42 EOF