0.00/0.03 % 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 0.03/0.23 % Computer : n150.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.23 % CPULimit : 300 0.03/0.23 % DateTime : Sat Jul 14 05:53:54 CDT 2018 0.03/0.23 % CPUTime : 16.12/16.35 % SZS status Theorem 16.12/16.35 16.12/16.35 % SZS output start Proof 16.12/16.35 Take the following subset of the input axioms: 16.12/16.35 fof(fact_Lin__irrefl, axiom, 16.12/16.35 ![V_L_2, V_ba_2, V_aa_2]: 16.12/16.35 (hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 tc_HOL_Obool)), 16.12/16.35 V_L_2), 16.12/16.35 c_Arrow__Order__Mirabelle_OLin)) 16.12/16.35 => (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 V_aa_2), 16.12/16.35 V_ba_2)), 16.12/16.35 V_L_2)) 16.12/16.35 => ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 V_ba_2), 16.12/16.35 V_aa_2)), 16.12/16.35 V_L_2))))). 16.12/16.35 fof(fact_Nil2__notin__lex, axiom, 16.12/16.35 ![T_a, V_r_2, V_xs_2]: 16.12/16.35 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 16.12/16.35 tc_List_Olist(T_a))), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 16.12/16.35 tc_List_Olist(T_a)), 16.12/16.35 V_xs_2), 16.12/16.35 c_List_Olist_ONil(T_a))), 16.12/16.35 c_List_Olex(T_a, V_r_2)))). 16.12/16.35 fof(fact_Nil__notin__lex, axiom, 16.12/16.35 ![T_a, V_r_2, V_ys_2]: 16.12/16.35 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 16.12/16.35 tc_List_Olist(T_a))), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 16.12/16.35 tc_List_Olist(T_a)), 16.12/16.35 c_List_Olist_ONil(T_a)), 16.12/16.35 V_ys_2)), 16.12/16.35 c_List_Olex(T_a, V_r_2)))). 16.12/16.35 fof(fact_dropWhile__eq__Cons__conv, axiom, 16.12/16.35 ![T_a, V_xs_2, V_y_2, V_ys_2, V_Pa_2]: 16.12/16.35 ((~hBOOL(hAPP(V_Pa_2, V_y_2)) 16.12/16.35 & V_xs_2=hAPP(hAPP(c_List_Oappend(T_a), 16.12/16.35 c_List_OtakeWhile(T_a, V_Pa_2, V_xs_2)), 16.12/16.35 hAPP(hAPP(c_List_Olist_OCons(T_a), V_y_2), V_ys_2))) 16.12/16.35 <=> hAPP(hAPP(c_List_Olist_OCons(T_a), V_y_2), 16.12/16.35 V_ys_2)=c_List_OdropWhile(T_a, V_Pa_2, V_xs_2))). 16.12/16.35 fof(fact_ext, axiom, 16.12/16.35 ![V_g_2, V_f_2]: 16.12/16.35 (V_f_2=V_g_2 <= ![B_x]: hAPP(V_g_2, B_x)=hAPP(V_f_2, B_x))). 16.12/16.35 fof(fact_impossible__Cons, axiom, 16.12/16.35 ![V_x, V_ys, V_xs, T_a]: 16.12/16.35 (V_xs!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_x), V_ys) 16.12/16.35 <= c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, 16.12/16.35 c_Nat_Osize__class_Osize(tc_List_Olist(T_a), V_xs), 16.12/16.35 c_Nat_Osize__class_Osize(tc_List_Olist(T_a), V_ys)))). 16.12/16.35 fof(fact_in__measures_I1_J, axiom, 16.12/16.35 ![T_a, V_y_2, V_x_2]: 16.12/16.35 ~hBOOL(hAPP(hAPP(c_member(tc_prod(T_a, T_a)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(T_a, T_a), V_x_2), V_y_2)), 16.12/16.35 c_List_Omeasures(T_a, 16.12/16.35 c_List_Olist_ONil(tc_fun(T_a, tc_Nat_Onat)))))). 16.12/16.35 fof(fact_in__mkbot, axiom, 16.12/16.35 ![V_z_2, V_y_2, V_L_2, V_x_2]: 16.12/16.35 ((V_z_2!=V_y_2 16.12/16.35 & ((V_y_2!=V_x_2 <= V_x_2=V_z_2) 16.12/16.35 & (V_x_2!=V_z_2 16.12/16.35 => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 V_x_2), 16.12/16.35 V_y_2)), 16.12/16.35 V_L_2))))) 16.12/16.35 <=> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 V_x_2), 16.12/16.35 V_y_2)), 16.12/16.35 c_Arrow__Order__Mirabelle_Omkbot(V_L_2, V_z_2))))). 16.12/16.35 fof(fact_in__mktop, axiom, 16.12/16.35 ![V_z_2, V_y_2, V_L_2, V_x_2]: 16.12/16.35 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 V_x_2), 16.12/16.35 V_y_2)), 16.12/16.35 c_Arrow__Order__Mirabelle_Omktop(V_L_2, V_z_2))) 16.12/16.35 <=> ((V_y_2!=V_x_2 <= V_z_2=V_y_2) 16.12/16.35 & ((hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt, 16.12/16.35 tc_Arrow__Order__Mirabelle_Oalt), 16.12/16.35 V_x_2), 16.12/16.35 V_y_2)), 16.12/16.35 V_L_2)) 16.12/16.35 <= V_z_2!=V_y_2) 16.12/16.35 & V_x_2!=V_z_2)))). 16.12/16.35 fof(fact_irrefl__def, axiom, 16.12/16.35 ![T_a, V_r_2]: 16.12/16.35 (c_Relation_Oirrefl(T_a, V_r_2) 16.12/16.35 <=> ![B_x]: 16.12/16.35 ~hBOOL(hAPP(hAPP(c_member(tc_prod(T_a, T_a)), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(T_a, T_a), B_x), B_x)), 16.12/16.35 V_r_2)))). 16.12/16.35 fof(fact_leD, axiom, 16.12/16.35 ![V_x, T_a, V_y]: 16.12/16.35 (class_Orderings_Olinorder(T_a) 16.12/16.35 => (c_Orderings_Oord__class_Oless__eq(T_a, V_y, V_x) 16.12/16.35 => ~c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 16.12/16.35 fof(fact_less__fun__def, axiom, 16.12/16.35 ![T_a, V_g_2, V_f_2, T_b]: 16.12/16.35 (class_Orderings_Oord(T_b) 16.12/16.35 => ((~c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, T_b), V_g_2, 16.12/16.35 V_f_2) 16.12/16.35 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, T_b), V_f_2, 16.12/16.35 V_g_2)) 16.12/16.35 <=> c_Orderings_Oord__class_Oless(tc_fun(T_a, T_b), V_f_2, 16.12/16.35 V_g_2)))). 16.12/16.35 fof(fact_less__imp__neq, axiom, 16.12/16.35 ![V_x, T_a, V_y]: 16.12/16.35 ((V_y!=V_x <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 16.12/16.35 <= class_Orderings_Oorder(T_a))). 16.12/16.35 fof(fact_less__le__not__le, axiom, 16.12/16.35 ![T_a, V_y_2, V_x_2]: 16.12/16.35 (class_Orderings_Opreorder(T_a) 16.12/16.35 => ((c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 16.12/16.35 & ~c_Orderings_Oord__class_Oless__eq(T_a, V_y_2, V_x_2)) 16.12/16.35 <=> c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2)))). 16.12/16.35 fof(fact_less__not__refl2, axiom, 16.12/16.35 ![V_m, V_n]: 16.12/16.35 (V_n!=V_m 16.12/16.35 <= c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, V_m))). 16.12/16.35 fof(fact_less__not__refl3, axiom, 16.12/16.35 ![V_t, V_s]: 16.12/16.35 (V_s!=V_t 16.12/16.35 <= c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_s, V_t))). 16.12/16.35 fof(fact_lexord__Nil__right, axiom, 16.12/16.35 ![T_a, V_r_2, V_x_2]: 16.12/16.35 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 16.12/16.35 tc_List_Olist(T_a))), 16.12/16.35 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 16.12/16.35 tc_List_Olist(T_a)), 16.12/16.35 V_x_2), 16.12/16.35 c_List_Olist_ONil(T_a))), 16.12/16.35 c_List_Olexord(T_a, V_r_2)))). 16.12/16.35 fof(fact_linorder__antisym__conv2, axiom, 16.12/16.35 ![T_a, V_y_2, V_x_2]: 16.12/16.35 (class_Orderings_Olinorder(T_a) 16.12/16.35 => (c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 16.12/16.35 => (~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 16.12/16.35 <=> V_y_2=V_x_2)))). 16.12/16.35 fof(fact_linorder__neq__iff, axiom, 16.12/16.35 ![T_a, V_y_2, V_x_2]: 16.12/16.35 (class_Orderings_Olinorder(T_a) 16.12/16.35 => (V_y_2!=V_x_2 16.12/16.35 <=> (c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2) 16.12/16.35 | c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2))))). 16.12/16.35 fof(fact_linorder__not__le, axiom, 16.12/16.35 ![T_a, V_y_2, V_x_2]: 16.12/16.35 ((~c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 16.12/16.35 <=> c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2)) 16.12/16.35 <= class_Orderings_Olinorder(T_a))). 16.12/16.35 fof(fact_linorder__not__less, axiom, 16.12/16.35 ![T_a, V_y_2, V_x_2]: 16.12/16.35 ((c_Orderings_Oord__class_Oless__eq(T_a, V_y_2, V_x_2) 16.12/16.35 <=> ~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2)) 16.12/16.35 <= class_Orderings_Olinorder(T_a))). 16.12/16.35 fof(fact_list_Osimps_I2_J, axiom, 16.12/16.35 ![T_a, V_list_H, V_a_H]: 16.12/16.35 c_List_Olist_ONil(T_a)!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_a_H), 16.12/16.35 V_list_H)). 16.12/16.35 fof(fact_list_Osimps_I3_J, axiom, 16.12/16.35 ![T_a, V_list_H, V_a_H]: 16.12/16.35 c_List_Olist_ONil(T_a)!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_a_H), 16.12/16.35 V_list_H)). 16.12/16.35 fof(fact_nat__less__cases, axiom, 16.12/16.35 ![V_Pa_2, V_n_2, V_m_2]: 16.12/16.35 ((c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2) 16.12/16.35 => hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2))) 16.12/16.35 => ((V_m_2=V_n_2 => hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2))) 16.12/16.35 => (hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2)) 16.12/16.35 <= (hBOOL(hAPP(hAPP(V_Pa_2, V_n_2), V_m_2)) 16.12/16.35 <= c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n_2, V_m_2)))))). 16.12/16.35 fof(fact_not__Cons__self, axiom, 16.12/16.35 ![V_x, V_xs, T_a]: 16.12/16.36 hAPP(hAPP(c_List_Olist_OCons(T_a), V_x), V_xs)!=V_xs). 16.12/16.36 fof(fact_not__Cons__self2, axiom, 16.12/16.36 ![V_x, V_xs, T_a]: 16.12/16.36 V_xs!=hAPP(hAPP(c_List_Olist_OCons(T_a), V_x), V_xs)). 16.12/16.36 fof(fact_not__Nil__listrel1, axiom, 16.12/16.36 ![T_a, V_r_2, V_xs_2]: 16.12/16.36 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 16.12/16.36 tc_List_Olist(T_a))), 16.12/16.36 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 16.12/16.36 tc_List_Olist(T_a)), 16.12/16.36 c_List_Olist_ONil(T_a)), 16.12/16.36 V_xs_2)), 16.12/16.36 c_List_Olistrel1(T_a, V_r_2)))). 16.12/16.36 fof(fact_not__less__iff__gr__or__eq, axiom, 16.12/16.36 ![T_a, V_y_2, V_x_2]: 16.12/16.36 ((~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 16.12/16.36 <=> (V_y_2=V_x_2 16.12/16.36 | c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2))) 16.12/16.36 <= class_Orderings_Olinorder(T_a))). 16.12/16.36 fof(fact_not__listrel1__Nil, axiom, 16.12/16.36 ![T_a, V_r_2, V_xs_2]: 16.12/16.36 ~hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_a), 16.12/16.36 tc_List_Olist(T_a))), 16.12/16.36 hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_a), 16.12/16.36 tc_List_Olist(T_a)), 16.12/16.36 V_xs_2), 16.12/16.36 c_List_Olist_ONil(T_a))), 16.12/16.36 c_List_Olistrel1(T_a, V_r_2)))). 16.12/16.36 fof(fact_order__less__asym, axiom, 16.12/16.36 ![V_x, T_a, V_y]: 16.12/16.36 ((~c_Orderings_Oord__class_Oless(T_a, V_y, V_x) 16.12/16.36 <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 16.12/16.36 <= class_Orderings_Opreorder(T_a))). 16.12/16.36 fof(fact_order__less__asym_H, axiom, 16.12/16.36 ![T_a, V_a, V_b]: 16.12/16.36 ((~c_Orderings_Oord__class_Oless(T_a, V_b, V_a) 16.12/16.36 <= c_Orderings_Oord__class_Oless(T_a, V_a, V_b)) 16.12/16.36 <= class_Orderings_Opreorder(T_a))). 16.12/16.36 fof(fact_order__less__imp__not__eq, axiom, 16.12/16.36 ![V_x, T_a, V_y]: 16.12/16.36 ((c_Orderings_Oord__class_Oless(T_a, V_x, V_y) => V_y!=V_x) 16.12/16.36 <= class_Orderings_Oorder(T_a))). 16.12/16.36 fof(fact_order__less__imp__not__eq2, axiom, 16.12/16.36 ![V_x, T_a, V_y]: 16.12/16.36 ((c_Orderings_Oord__class_Oless(T_a, V_x, V_y) => V_x!=V_y) 16.12/16.36 <= class_Orderings_Oorder(T_a))). 16.12/16.36 fof(fact_order__less__imp__not__less, axiom, 16.12/16.36 ![V_x, T_a, V_y]: 16.12/16.36 (class_Orderings_Opreorder(T_a) 16.12/16.36 => (c_Orderings_Oord__class_Oless(T_a, V_x, V_y) 16.12/16.36 => ~c_Orderings_Oord__class_Oless(T_a, V_y, V_x)))). 16.12/16.36 fof(fact_order__less__irrefl, axiom, 16.12/16.36 ![V_x, T_a]: 16.12/16.36 (~c_Orderings_Oord__class_Oless(T_a, V_x, V_x) 16.12/16.36 <= class_Orderings_Opreorder(T_a))). 16.12/16.36 fof(fact_order__less__le, axiom, 16.12/16.36 ![T_a, V_y_2, V_x_2]: 16.12/16.36 ((c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 16.12/16.36 <=> (c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 16.12/16.36 & V_x_2!=V_y_2)) 16.12/16.36 <= class_Orderings_Oorder(T_a))). 16.12/16.36 fof(fact_order__less__not__sym, axiom, 16.12/16.36 ![V_x, T_a, V_y]: 16.12/16.36 ((c_Orderings_Oord__class_Oless(T_a, V_x, V_y) 16.12/16.36 => ~c_Orderings_Oord__class_Oless(T_a, V_y, V_x)) 16.12/16.36 <= class_Orderings_Opreorder(T_a))). 16.12/16.36 fof(fact_psubset__eq, axiom, 16.12/16.36 ![T_a, V_A_2, V_B_2]: 16.12/16.36 (c_Orderings_Oord__class_Oless(tc_fun(T_a, tc_HOL_Obool), V_A_2, 16.12/16.36 V_B_2) 16.12/16.36 <=> (V_B_2!=V_A_2 16.12/16.36 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, tc_HOL_Obool), 16.12/16.36 V_A_2, V_B_2)))). 16.12/16.36 fof(fact_snoc__eq__iff__butlast, axiom, 16.12/16.36 ![T_a, V_xs_2, V_x_2, V_ys_2]: 16.12/16.36 ((c_List_Olist_ONil(T_a)!=V_ys_2 16.12/16.36 & (V_xs_2=c_List_Obutlast(T_a, V_ys_2) 16.12/16.36 & c_List_Olast(T_a, V_ys_2)=V_x_2)) 16.12/16.36 <=> hAPP(hAPP(c_List_Oappend(T_a), V_xs_2), 16.12/16.36 hAPP(hAPP(c_List_Olist_OCons(T_a), V_x_2), 16.12/16.36 c_List_Olist_ONil(T_a)))=V_ys_2)). 16.12/16.36 fof(fact_xt1_I9_J, axiom, 16.12/16.36 ![T_a, V_a, V_b]: 16.12/16.36 (class_Orderings_Oorder(T_a) 16.12/16.36 => (~c_Orderings_Oord__class_Oless(T_a, V_a, V_b) 16.12/16.36 <= c_Orderings_Oord__class_Oless(T_a, V_b, V_a)))). 16.12/16.36 fof(help_c__COMBI__1, axiom, 16.12/16.36 ![T_a, V_P]: hAPP(c_COMBI(T_a), V_P)=V_P). 16.12/16.36 fof(help_c__COMBS__1, axiom, 16.12/16.36 ![T_a, T_b, V_Pa_2, V_Q_2, V_R_2, T_c]: 16.12/16.36 hAPP(hAPP(V_Pa_2, V_R_2), 16.12/16.36 hAPP(V_Q_2, V_R_2))=hAPP(hAPP(hAPP(c_COMBS(T_b, T_c, T_a), V_Pa_2), 16.12/16.36 V_Q_2), 16.12/16.36 V_R_2)). 16.12/16.36 fof(help_c__fNot__1, axiom, 16.12/16.36 ![V_Pa_2]: (~hBOOL(hAPP(c_fNot, V_Pa_2)) | ~hBOOL(V_Pa_2))). 16.12/16.36 16.12/16.36 Now clausify the problem and encode Horn clauses using encoding 3 of 16.12/16.36 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 16.12/16.36 We repeatedly replace C & s=t => u=v by the two clauses: 16.12/16.36 $$fresh(y, y, x1...xn) = u 16.12/16.36 C => $$fresh(s, t, x1...xn) = v 16.12/16.36 where $$fresh is a fresh function symbol and x1..xn are the free 16.12/16.36 variables of u and v. 16.12/16.36 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 16.12/16.36 input problem has no model of domain size 1). 16.12/16.36 16.12/16.36 The encoding turns the above axioms into the following unit equations and goals: 16.12/16.36 16.12/16.36 Axiom 1011 (help_c__COMBI__1): hAPP(c_COMBI(X), Y) = Y. 16.12/16.36 Axiom 1080 (help_c__COMBS__1): hAPP(hAPP(X, Y), hAPP(Z, Y)) = hAPP(hAPP(hAPP(c_COMBS(W, V, U), X), Z), Y). 16.12/16.36 Axiom 1081 (fact_ext): $$fresh7(hAPP(X, sK5_fact_ext_B_x(X, Y)), hAPP(Y, sK5_fact_ext_B_x(X, Y)), X, Y) = Y. 16.12/16.36 16.12/16.36 Lemma 1082: $$fresh7(sK5_fact_ext_B_x(c_COMBI(Y), X), hAPP(X, sK5_fact_ext_B_x(c_COMBI(Y), X)), c_COMBI(Y), X) = X. 16.12/16.36 Proof: 16.12/16.36 $$fresh7(sK5_fact_ext_B_x(c_COMBI(Y), X), hAPP(X, sK5_fact_ext_B_x(c_COMBI(Y), X)), c_COMBI(Y), X) 16.12/16.36 = { by axiom 1011 (help_c__COMBI__1) } 16.12/16.36 $$fresh7(hAPP(c_COMBI(Y), sK5_fact_ext_B_x(c_COMBI(Y), X)), hAPP(X, sK5_fact_ext_B_x(c_COMBI(Y), X)), c_COMBI(Y), X) 16.12/16.36 = { by axiom 1081 (fact_ext) } 16.12/16.36 X 16.12/16.36 16.12/16.36 Goal 1 (fact_not__Cons__self): hAPP(hAPP(c_List_Olist_OCons(X), Y), Z) = Z. 16.12/16.36 The goal is true when: 16.12/16.36 X = ? 16.12/16.36 Y = hAPP(hAPP(c_COMBS(?, ?, ?), hAPP(c_COMBS(?, ?, ?), c_List_Olist_OCons(?))), c_COMBI(?)) 16.12/16.36 Z = 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(?))) 16.12/16.36 where "?" stands for an arbitrary term of your choice. 16.12/16.36 16.12/16.36 Proof: 16.12/16.36 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(?)))) 16.12/16.36 = { by axiom 1080 (help_c__COMBS__1) } 16.12/16.36 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(?))) 16.12/16.36 = { by axiom 1011 (help_c__COMBI__1) } 16.12/16.36 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(?)))) 16.12/16.36 = { by axiom 1080 (help_c__COMBS__1) } 16.12/16.36 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(?))) 16.12/16.36 % SZS output end Proof 16.12/16.36 16.12/16.36 RESULT: Theorem (the conjecture is true). 16.12/16.38 EOF