0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.02/0.23 % Computer : n061.star.cs.uiowa.edu 0.02/0.23 % Model : x86_64 x86_64 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.23 % Memory : 32218.625MB 0.02/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.23 % CPULimit : 300 0.02/0.23 % DateTime : Sat Jul 14 05:46:10 CDT 2018 0.02/0.23 % CPUTime : 0.79/0.99 ============================== Prover9 =============================== 0.79/0.99 Prover9 (32) version 2009-11A, November 2009. 0.79/0.99 Process 18833 was started by sandbox2 on n061.star.cs.uiowa.edu, 0.79/0.99 Sat Jul 14 05:46:11 2018 0.79/0.99 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_18784_n061.star.cs.uiowa.edu". 0.79/0.99 ============================== end of head =========================== 0.79/0.99 0.79/0.99 ============================== INPUT ================================= 0.79/0.99 0.79/0.99 % Reading from file /tmp/Prover9_18784_n061.star.cs.uiowa.edu 0.79/0.99 0.79/0.99 set(prolog_style_variables). 0.79/0.99 set(auto2). 0.79/0.99 % set(auto2) -> set(auto). 0.79/0.99 % set(auto) -> set(auto_inference). 0.79/0.99 % set(auto) -> set(auto_setup). 0.79/0.99 % set(auto_setup) -> set(predicate_elim). 0.79/0.99 % set(auto_setup) -> assign(eq_defs, unfold). 0.79/0.99 % set(auto) -> set(auto_limits). 0.79/0.99 % set(auto_limits) -> assign(max_weight, "100.000"). 0.79/0.99 % set(auto_limits) -> assign(sos_limit, 20000). 0.79/0.99 % set(auto) -> set(auto_denials). 0.79/0.99 % set(auto) -> set(auto_process). 0.79/0.99 % set(auto2) -> assign(new_constants, 1). 0.79/0.99 % set(auto2) -> assign(fold_denial_max, 3). 0.79/0.99 % set(auto2) -> assign(max_weight, "200.000"). 0.79/0.99 % set(auto2) -> assign(max_hours, 1). 0.79/0.99 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.79/0.99 % set(auto2) -> assign(max_seconds, 0). 0.79/0.99 % set(auto2) -> assign(max_minutes, 5). 0.79/0.99 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.79/0.99 % set(auto2) -> set(sort_initial_sos). 0.79/0.99 % set(auto2) -> assign(sos_limit, -1). 0.79/0.99 % set(auto2) -> assign(lrs_ticks, 3000). 0.79/0.99 % set(auto2) -> assign(max_megs, 400). 0.79/0.99 % set(auto2) -> assign(stats, some). 0.79/0.99 % set(auto2) -> clear(echo_input). 0.79/0.99 % set(auto2) -> set(quiet). 0.79/0.99 % set(auto2) -> clear(print_initial_clauses). 0.79/0.99 % set(auto2) -> clear(print_given). 0.79/0.99 assign(lrs_ticks,-1). 0.79/0.99 assign(sos_limit,10000). 0.79/0.99 assign(order,kbo). 0.79/0.99 set(lex_order_vars). 0.79/0.99 clear(print_given). 0.79/0.99 0.79/0.99 % formulas(sos). % not echoed (530 formulas) 0.79/0.99 0.79/0.99 ============================== end of input ========================== 0.79/0.99 0.79/0.99 % From the command line: assign(max_seconds, 300). 0.79/0.99 0.79/0.99 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.79/0.99 0.79/0.99 % Formulas that are not ordinary clauses: 0.79/0.99 1 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 2 (all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,V_r_2) & c_Relation_Otrans(T_a,V_r_2) <-> c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2))) # label(fact_preorder__on__def) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 3 (all V_r_2 all V_b_2 all T_b all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2) -> c_member(T_b,V_b_2,c_Relation_OImage(T_a,T_b,V_r_2,V_A_2))))) # label(fact_rev__ImageI) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 4 (all V_xa_2 all V_La_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_xa_2),c_Arrow__Order__Mirabelle_OLin))) # label(fact_mktop__Lin) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 5 (all V_r_2 all T_a all T_b c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) = c_Relation_ORange(T_b,T_a,V_r_2)) # label(fact_Range__def) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 6 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2))) # label(fact_equalityD2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 7 (all V_b_2 all V_a_2 all T_b all T_a all V_f_2 (hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))) -> hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2)))) # label(fact_curryI) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 8 (all V_a_2 all V_H_2 all T_b all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> hAPP(hAPP(V_H_2,c_Recdef_Ocut(T_a,T_b,c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2),V_r_2,V_a_2)),V_a_2) = hAPP(c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2),V_a_2))) # label(fact_wfrec) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 9 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Olinorder(T_a) -> (V_xa_2 != V_ya_2 <-> c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) | c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2)))) # label(fact_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 10 (all T_a all V_R_2 all V_ya_2 all V_y_H_2 all T_b all V_xa_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_xa_2)) -> (c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_y_H_2,V_ya_2),hAPP(V_R_2,V_xa_2)) -> c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_y_H_2),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_ya_2)),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2))))) # label(fact_same__fstI) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 11 (all V_b_2 all V_B_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_B_2) -> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_insertI2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 12 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_y = V_x -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 13 (all V_a2_2 all V_a1_2 all V_F_2 all V_R_2 all T_b all T_a (c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_a1_2,hAPP(hAPP(V_F_2,V_a2_2),V_a1_2)) <-> (exists B_g (hAPP(hAPP(V_F_2,B_g),V_a1_2) = hAPP(hAPP(V_F_2,V_a2_2),V_a1_2) & (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(B_g,B_z)))))))) # label(fact_wfrec__rel_Oequations) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 14 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) | V_xa_2 = V_ya_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)))) # label(fact_order__le__less) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 15 (all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId(T_a)) <-> V_a_2 = V_b_2)) # label(fact_pair__in__Id__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 16 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_x != V_y))) # label(fact_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 17 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)) -> -(c_member(T_a,V_a_2,V_A_2) -> -c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2))))) # label(fact_SigmaE2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 18 (all V_A_2 all V_xa_2 all T_a c_Set_Oinsert(T_a,V_xa_2,c_Set_Oinsert(T_a,V_xa_2,V_A_2)) = c_Set_Oinsert(T_a,V_xa_2,V_A_2)) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 19 (all V_r_2 all V_xa_2 all T_a ((all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_xa_2),V_r_2) -> c_member(T_a,B_y,c_Wellfounded_Oacc(T_a,V_r_2)))) -> c_member(T_a,V_xa_2,c_Wellfounded_Oacc(T_a,V_r_2)))) # label(fact_acc_OaccI) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 20 (all V_B_2 all V_xa_2 all V_A_2 all T_a ((-c_member(T_a,V_xa_2,V_B_2) -> (-c_member(T_a,V_xa_2,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) & (c_member(T_a,V_xa_2,V_A_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2))) & (c_member(T_a,V_xa_2,V_B_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) <-> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2)))) # label(fact_psubset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 21 (all V_a_2 all V_D_2 all V_C_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_C_2),c_Set_Oinsert(T_a,V_a_2,V_D_2)))) # label(fact_insert__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 22 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_z)),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z))))) # label(fact_distrib__inf__le) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 23 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)) -> c_member(T_a,V_a_2,V_A_2))) # label(fact_SigmaD1) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 24 (all V_r_2 all T_a ((all B_x -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))) -> c_Wellfounded_Oacyclic(T_a,V_r_2))) # label(fact_acyclicI) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 25 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) & -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_g_2,V_f_2) <-> c_Orderings_Oord__class_Oless(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 26 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_x != V_y))) # label(fact_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 27 (all V_a_2 all T_a -c_member(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_emptyE) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 28 (all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_x = V_y)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 29 (all V_S_2 all V_R_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_S_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) -> c_Transitive__Closure_Ortrancl(T_a,V_S_2) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)))) # label(fact_rtrancl__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 30 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_psubset__subset__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 31 (all V_B_2 all V_A_2 all V_xa_2 all T_a (-c_member(T_a,V_xa_2,V_A_2) -> (-c_member(T_a,V_xa_2,V_B_2) -> (c_Set_Oinsert(T_a,V_xa_2,V_B_2) = c_Set_Oinsert(T_a,V_xa_2,V_A_2) <-> V_B_2 = V_A_2)))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 32 (all V_A_2 all V_S_2 all V_R_2 all T_a all T_b c_Relation_OImage(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_R_2,V_S_2),V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,V_A_2),c_Relation_OImage(T_b,T_a,V_S_2,V_A_2))) # label(fact_Un__Image) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 33 (all V_s_2 all V_r_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2) -> (c_Relation_Oantisym(T_a,V_s_2) -> c_Relation_Oantisym(T_a,V_r_2)))) # label(fact_antisym__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 34 (all V_R_2 all T_a all T_b c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_OImage(T_b,T_a,V_R_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Image__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 35 (all V_f_2 all V_R_2 all T_b all T_a c_Relation_Oconverse(T_a,T_a,hAPP(c_Relation_Oinv__image(T_b,T_a,V_R_2),V_f_2)) = hAPP(c_Relation_Oinv__image(T_b,T_a,c_Relation_Oconverse(T_b,T_b,V_R_2)),V_f_2)) # label(fact_converse__inv__image) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 36 (all V_rb_2 all T_b all V_ra_2 all T_a (c_Wellfounded_Owf(T_a,V_ra_2) -> (c_Wellfounded_Owf(T_b,V_rb_2) -> c_Wellfounded_Owf(tc_prod(T_a,T_b),c_Wellfounded_Olex__prod(T_a,T_b,V_ra_2,V_rb_2))))) # label(fact_wf__lex__prod) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 37 (all V_P_2 all V_r_2 all V_by_2 all V_bx_2 all V_ay_2 all V_ax_2 all T_b all T_a (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) -> (hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2)) -> ((all B_a all B_b (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,B_a,B_b)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) -> (all B_aa all B_ba (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba)),V_r_2) -> (hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)) -> hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba))))))) -> hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2)))))) # label(fact_rtrancl__induct2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 38 (all V_B_2 all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_Un__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 39 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2) -> c_Order__Relation_Ostrict__linear__order__on(T_a,V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))))) # label(fact_strict__linear__order__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 40 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),V_z))) # label(fact_inf__sup__aci_I6_J) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 41 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_x = V_y | c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 42 (all V_A_2 all V_a_2 all T_a c_Set_Oinsert(T_a,V_a_2,V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 43 (all V_a_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2))) # label(fact_wf__irrefl) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 44 (exists B_L c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),B_L,c_Arrow__Order__Mirabelle_OLin)) # label(fact_linear__alt) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 45 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 46 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_sup__ge1) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 47 (all V_r_2 all V_a_2 all T_a c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) # label(fact_rtrancl_Ortrancl__refl) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 48 (all V_A_2 all T_a c_Relation_Otrans(T_a,c_Relation_OId__on(T_a,V_A_2))) # label(fact_trans__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 49 (all V_B_2 all V_a_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 50 (all V_r_2 all T_a c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) # label(fact_rtrancl__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 51 (all V_r_2 all V_yx_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),V_yx_2,c_Relation_Oconverse(T_b,T_a,V_r_2)) -> -(all B_x all B_y (c_Product__Type_OPair(T_a,T_b,B_y,B_x) = V_yx_2 -> -c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,B_y),V_r_2))))) # label(fact_converseE) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 52 (all V_A_2 all T_a ((all B_x -c_member(T_a,B_x,V_A_2)) <-> c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2)) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 53 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_x))) # label(fact_inf__sup__ord_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 54 (all T_2 all T_1 (class_Lattices_Olattice(T_1) -> class_Lattices_Osemilattice__sup(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Osemilattice__sup) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 55 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) -> -(c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_DiffE) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 56 (all V_A_2 all V_xa_2 all T_a (-c_member(T_a,V_xa_2,V_A_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_A_2)) # label(fact_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 57 (all V_A_2 all V_B_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Un__upper2) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 58 (all V_xa_2 all V_a_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_xa_2),V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),V_r_2)))) # label(fact_wf__not__sym) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 59 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Owell__order__on(T_a,V_A_2,V_r_2) <-> c_Wellfounded_Owf(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) & c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2))) # label(fact_well__order__on__def) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 60 (all T_a c_Order__Relation_Opreorder__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_preorder__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/0.99 61 (all T_a c_Relation_Otrans(T_a,c_Relation_OId(T_a))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 62 (all V_A_2 all T_a V_A_2 = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 63 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))))) # label(fact_rtrancl_Ortrancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 64 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_equalityD1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 65 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Opreorder__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2)) <-> c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2))) # label(fact_preorder__on__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 66 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (V_x != V_y -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 67 (all V_xa_2 all V_a_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_xa_2),V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),V_r_2)))) # label(fact_wf__asym) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 68 (all V_A_2 all V_b_2 all V_a_2 all T_a (c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)) -> (V_a_2 != V_b_2 -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 69 (all T_a c_Relation_OId(T_a) = c_Transitive__Closure_Ortrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_rtrancl__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 70 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) <-> c_member(T_a,V_c_2,V_A_2) & -c_member(T_a,V_c_2,V_B_2))) # label(fact_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 71 (all T_2 all T_1 (class_Lattices_Olattice(T_1) -> class_Lattices_Olattice(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Olattice) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 72 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_trancl__rtrancl__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 73 (all T_a all V_A_2 all V_xa_2 all V_B_2 ((-hBOOL(hAPP(V_B_2,V_xa_2)) -> hBOOL(hAPP(V_A_2,V_xa_2))) -> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)))) # label(fact_sup1CI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 74 (all V_B_2 all V_A_2 all V_R_2 all T_b all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,V_A_2),c_Relation_OImage(T_b,T_a,V_R_2,V_B_2)))) # label(fact_Image__Int__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 75 (all V_A_2 all V_c_2 all T_a (c_member(tc_prod(T_a,T_a),V_c_2,c_Relation_OId__on(T_a,V_A_2)) -> -(all B_x (c_member(T_a,B_x,V_A_2) -> V_c_2 != c_Product__Type_OPair(T_a,T_a,B_x,B_x))))) # label(fact_Id__onE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 76 (all V_B_2 all V_A_2 all V_xa_2 all T_a (-c_member(T_a,V_xa_2,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2))))) # label(fact_subset__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 77 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_b all T_a (V_b_H_2 = V_b_2 & V_a_H_2 = V_a_2 <-> c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2) = c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))) # label(fact_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 78 (all V_z_2 all V_r_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_rtrancl__trancl__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 79 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> V_y = V_x)))) # label(fact_order__antisym) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 80 (all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)))) # label(fact_trancl_Or__into__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 81 (all V_B_2 all T_a all V_xa_2 all V_A_2 (hBOOL(hAPP(V_A_2,V_xa_2)) -> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)))) # label(fact_sup1I1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 82 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_trancl__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 83 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_x))) # label(fact_inf__le1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 84 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (V_b_2 = V_c_2 & V_d_2 = V_a_2 | V_d_2 = V_b_2 & V_a_2 = V_c_2 <-> c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Set_Oinsert(T_a,V_c_2,c_Set_Oinsert(T_a,V_d_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 85 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x))) # label(fact_order__refl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 86 (all V_a_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2))) # label(fact_wf__not__refl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 87 (all V_z_2 all V_ya_2 all V_xa_2 all V_r_2 all T_a (c_Relation_Osingle__valued(T_a,T_a,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_z_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))))) # label(fact_single__valued__confluent) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 88 (all V_B_2 all V_A_2 all V_xa_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2) <-> c_member(T_a,V_xa_2,V_B_2) & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_insert__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 89 (all V_Q_2 all V_P_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_P_2,V_Q_2))) -> ((all B_x (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_P_2)) -> (all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_Q_2) -> B_y = B_x)))) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_P_2))))) # label(fact_rtrancl__Un__separator__converseE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 90 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) -> (-c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) <-> V_ya_2 = V_xa_2)))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 91 (all V_z_2 all V_ya_2 all V_xa_2 all V_r_2 all T_b all T_a (c_Relation_Osingle__valued(T_a,T_b,V_r_2) -> (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_ya_2),V_r_2) -> (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_z_2),V_r_2) -> V_z_2 = V_ya_2)))) # label(fact_single__valuedD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 92 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) -> -c_member(T_a,V_c_2,V_B_2))) # label(fact_DiffD2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 93 (all V_s_2 all V_r_2 all T_b all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_r_2),c_Relation_ODomain(T_a,T_b,V_s_2)))) # label(fact_Domain__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 94 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_subset__psubset__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 95 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_inf__sup__aci_I8_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 96 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)),c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z))))) # label(fact_distrib__sup__le) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 97 (all T_b all T_a all B_M all B_R (c_Wellfounded_Owf(T_a,B_R) -> (all B_x hAPP(c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_x) = hAPP(hAPP(B_M,c_Recdef_Ocut(T_a,T_b,c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_R,B_x)),B_x)))) # label(fact_tfl__wfrec) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 98 (all V_R2_2 all T_b all V_R1_2 all T_a (c_Relation_Otrans(T_a,V_R1_2) -> (c_Relation_Otrans(T_b,V_R2_2) -> c_Relation_Otrans(tc_prod(T_a,T_b),c_Wellfounded_Olex__prod(T_a,T_b,V_R1_2,V_R2_2))))) # label(fact_trans__lex__prod) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 99 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 100 (all V_r_2 all V_a_2 all T_a ((all B_x (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,V_a_2),V_r_2) -> c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2)))) <-> c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2)))) # label(fact_acc_Osimps) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 101 (all V_b all V_x all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_x))))) # label(fact_le__supI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 102 (all V_r_2 all T_a (c_Relation_Oantisym(T_a,V_r_2) <-> c_Relation_Oantisym(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))) # label(fact_antisym__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 103 (all V_r_2 all T_a all T_b V_r_2 = c_Relation_Oconverse(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2))) # label(fact_converse__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 104 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,V_A_2) -> (-c_member(T_a,V_c_2,V_B_2) -> c_member(T_a,V_c_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))))) # label(fact_DiffI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 105 (all V_xa_2 all V_r_2 all T_a (c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Oconverse(T_a,T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_xa_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)))) # label(fact_irrefl__tranclI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 106 (all V_B_2 all V_A_2 all V_xa_2 all T_a (c_member(T_a,V_xa_2,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_member(T_a,V_xa_2,V_B_2)))) # label(fact_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 107 (all V_A_2 all V_r_2 all T_b all V_b_2 all T_a (c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,V_A_2)) -> -(all B_x (c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2) -> -c_member(T_b,B_x,V_A_2))))) # label(fact_ImageE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 108 (all T_a c_Relation_Orefl__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_refl__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 109 (all V_b_2 all V_a_2 all T_a (c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) -> V_a_2 = V_b_2)) # label(fact_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 110 (all V_a_2 all V_b_2 all V_r_2 all V_A_2 all T_a (c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) -> (c_member(T_a,V_b_2,V_A_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2))))) # label(fact_subset__equiv__class) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 111 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_leI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 112 (all T_a c_Relation_Oinv__image(tc_Nat_Onat,T_a,c_Wellfounded_Oless__than) = c_Wellfounded_Omeasure(T_a)) # label(fact_measure__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 113 (all T_2 all T_1 (class_Orderings_Obot(T_1) -> class_Orderings_Obot(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Obot) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 114 (all V_A_2 all V_xa_2 all T_a (c_member(T_a,V_xa_2,V_A_2) <-> hBOOL(hAPP(V_A_2,V_xa_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 115 (all V_a all V_b all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b))))) # label(fact_less__supI2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 116 (all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = V_x))) # label(fact_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 117 (all V_c_2 all V_b_2 all V_a_2 all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),V_r_2))))) # label(fact_transD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 118 (all V_r_2 all T_a c_Relation_Oirrefl(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a)))) # label(fact_irrefl__diff__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 119 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (V_a != V_b -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 120 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> V_B_2 = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Un__absorb1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 121 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)))) # label(fact_Diff__subset__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 122 (all V_f_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),hAPP(c_Wellfounded_Omeasure(T_a),V_f_2)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2)))) # label(fact_in__measure) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 123 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2) -> (c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2)) -> (c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2)))))) # label(fact_subset__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 124 (all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = V_A_2) # label(fact_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 125 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 126 (all V_b_2 all V_a_2 all V_f_2 all T_a all T_c all T_b c_Product__Type_Ocurry(T_b,T_c,T_a,V_f_2,V_a_2,V_b_2) = hAPP(V_f_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2))) # label(fact_curry__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 127 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) <-> c_Relation_Otrans(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))) # label(fact_trans__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 128 (all V_B_2 all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_A_2)) # label(fact_Diff__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 129 (all V_A_2 all V_a_2 all T_a c_Set_Oinsert(T_a,V_a_2,V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 130 (all T_2 all T_1 (class_Groups_Ominus(T_1) -> class_Groups_Ominus(tc_fun(T_2,T_1)))) # label(arity_fun__Groups_Ominus) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 131 (all T_2 all T_1 (class_Orderings_Oorder(T_1) -> class_Orderings_Oorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 132 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2) <-> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 133 (all V_r_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))) # label(fact_rtrancl__converseD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 134 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 135 (all V_B_2 all V_A_2 all T_a all T_b c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ORange(T_b,T_a,V_A_2),c_Relation_ORange(T_b,T_a,V_B_2)) = c_Relation_ORange(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Range__Un__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 136 (all V_B_2 all V_A_2 all T_b all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_A_2),c_Relation_ODomain(T_a,T_b,V_B_2)) = c_Relation_ODomain(T_a,T_b,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Domain__Un__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 137 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2) & c_Relation_Oantisym(T_a,V_r_2) <-> c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2))) # label(fact_partial__order__on__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 138 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_psubset__imp__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 139 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 140 (all T_2 all T_1 (class_Lattices_Obounded__lattice(T_1) -> class_Lattices_Obounded__lattice__bot(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 141 (all V_S_2 all V_R_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_S_2)),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)))) # label(fact_rtrancl__Un__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 142 (all V_A_2 all V_B_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) -> V_A_2 = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Un__absorb2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 143 (all V_b_2 all V_B_2 all V_a_2 all T_a ((-c_member(T_a,V_a_2,V_B_2) -> V_b_2 = V_a_2) -> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_insertCI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 144 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b = V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 145 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) <-> V_ya_2 = V_xa_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 146 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)))) # label(fact_sup__left__commute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 147 (all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_Wellfounded_Oacyclic(T_a,V_r_2))) # label(fact_wf__acyclic) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 148 (all V_R_2 all T_a (c_Wellfounded_Owf(T_a,V_R_2) -> c_Wellfounded_OwfP(T_a,c_FunDef_Oin__rel(T_a,T_a,V_R_2)))) # label(fact_wf__in__rel) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 149 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_y))) # label(fact_inf__le2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 150 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2) -> c_member(T_b,V_b_2,c_Relation_ORange(T_a,T_b,V_r_2)))) # label(fact_RangeI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 151 (all V_R2_2 all V_R1_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R1_2,V_R2_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Oacc(T_a,V_R2_2),c_Wellfounded_Oacc(T_a,V_R1_2)))) # label(fact_acc__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 152 (all V_a_2 all V_r_2 all V_b_2 all T_a (c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))))) # label(fact_acc__downward) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 153 (all V_r_2 all T_a c_Relation_ODomain(T_a,T_a,V_r_2) = c_Relation_ODomain(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) # label(fact_trancl__domain) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 154 (all V_A_2 all T_a V_A_2 = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 155 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2))))) # label(fact_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 156 (all V_B_2 all V_xa_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2) -> (c_member(T_a,V_xa_2,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2))))) # label(fact_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 157 (all V_xa_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = V_A_2 | c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2)) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 158 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 159 (all V_r_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_r_2) = c_Transitive__Closure_Otrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a)))) # label(fact_trancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 160 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))))) # label(fact_rtrancl__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 161 (all V_ya_2 all V_xa_2 all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) <-> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_xa_2,V_ya_2) = V_xa_2))) # label(fact_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 162 (all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (V_a_2 != V_b_2 -> -(all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_y),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_b_2),V_r_2)))))) # label(fact_rtranclE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 163 (all V_r_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Otrancl(T_a,V_r_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2)) # label(fact_reflcl__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 164 (all V_c all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_c) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)))) # label(fact_sup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 165 (all V_c_2 all V_R_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_R_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_R_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_R_2))))) # label(fact_r__r__into__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 166 (all V_s_2 all V_r_2 all T_b all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2) -> (c_Relation_Osingle__valued(T_a,T_b,V_s_2) -> c_Relation_Osingle__valued(T_a,T_b,V_r_2)))) # label(fact_single__valued__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 167 (all V_r_2 all T_a (c_Relation_Oantisym(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) <-> c_Relation_Oantisym(T_a,V_r_2))) # label(fact_antisym__reflcl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 168 (all V_b_2 all V_a_2 all V_c_2 all T_a all T_c all T_b c_Product__Type_Ointernal__split(T_b,T_c,T_a,V_c_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_c_2,V_a_2),V_b_2)) # label(fact_internal__split__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 169 (all V_z_2 all V_La_2 all V_ya_2 all V_xa_2 ((V_z_2 != V_ya_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2)) & (V_ya_2 = V_z_2 -> V_xa_2 != V_ya_2) & V_xa_2 != V_z_2 <-> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_z_2)))) # label(fact_in__mktop) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 170 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_x))) # label(fact_sup__commute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 171 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> -c_Orderings_Oord__class_Oless(T_a,V_x,V_x))) # label(fact_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 172 (all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_Wellfounded_Owf(T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)))) # label(fact_wf__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 173 (all V_ya_2 all V_xa_2 all V_r_2 all T_a ((all B_x -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_r_2) -> V_ya_2 != V_xa_2))) # label(fact_irrefl__trancl__rD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 174 (all V_P_2 all V_B_2 all V_A_2 all T_a ((exists B_x (hBOOL(hAPP(V_P_2,B_x)) & c_member(T_a,B_x,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) <-> (exists B_x (c_member(T_a,B_x,V_B_2) & hBOOL(hAPP(V_P_2,B_x)))) | (exists B_x (hBOOL(hAPP(V_P_2,B_x)) & c_member(T_a,B_x,V_A_2))))) # label(fact_bex__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 175 (all V_r_2 all V_xa_2 all V_ya_2 all T_a (-c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) & c_Wellfounded_Oacyclic(T_a,V_r_2) <-> c_Wellfounded_Oacyclic(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),V_r_2)))) # label(fact_acyclic__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 176 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2)) = V_A_2))) # label(fact_double__diff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 177 (all V_A_2 all T_a c_Relation_ODomain(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2) # label(fact_Domain__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 178 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Relation_Oconverse(T_b,T_a,V_r_2)) -> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),V_r_2))) # label(fact_converseD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 179 (all T_b all V_a_2 all V_xa_2 all T_a all B_f all B_R (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),B_R) -> hAPP(c_Recdef_Ocut(T_a,T_b,B_f,B_R,V_a_2),V_xa_2) = hAPP(B_f,V_xa_2))) # label(fact_tfl__cut__apply) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 180 (all T_2 all T_1 (class_Lattices_Obounded__lattice(T_1) -> class_Lattices_Obounded__lattice(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Obounded__lattice) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 181 (all V_r_2 all T_a (c_Wellfounded_Oacyclic(T_a,V_r_2) -> c_Relation_Oantisym(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)))) # label(fact_acyclic__impl__antisym__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 182 (all V_a_2 all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,V_r_2) -> (c_member(T_a,V_a_2,V_A_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2)))) # label(fact_refl__onD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 183 (all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Otrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)) # label(fact_trancl__rtrancl__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 184 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Ostrict__linear__order__on(T_a,V_A_2,V_r_2) <-> c_Relation_Oirrefl(T_a,V_r_2) & c_Relation_Ototal__on(T_a,V_A_2,V_r_2) & c_Relation_Otrans(T_a,V_r_2))) # label(fact_strict__linear__order__on__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 185 (all V_A_2 all T_a V_A_2 = c_Relation_OImage(T_a,T_a,c_Relation_OId(T_a),V_A_2)) # label(fact_Image__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 186 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 187 (all V_x all T_a (class_Orderings_Obot(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Orderings_Obot__class_Obot(T_a),V_x))) # label(fact_bot__least) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 188 (all V_r_2 all T_b all V_a_2 all T_a (c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) <-> (exists B_y c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,B_y),V_r_2)))) # label(fact_Domain__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 189 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> V_A_2 = V_B_2 | c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 190 (all V_z_2 all V_La_2 all V_ya_2 all V_xa_2 (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_z_2)) <-> V_ya_2 != V_z_2 & (V_xa_2 = V_z_2 -> V_xa_2 != V_ya_2) & (V_xa_2 != V_z_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2)))) # label(fact_in__mkbot) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 191 (all V_A_2 all T_a all V_b_2 all V_a_2 (V_b_2 = V_a_2 -> (c_member(T_a,V_a_2,V_A_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId__on(T_a,V_A_2))))) # label(fact_Id__on__eqI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 192 (all V_xa_2 all V_A_2 all V_ya_2 all T_a (hBOOL(hAPP(c_Set_Oinsert(T_a,V_ya_2,V_A_2),V_xa_2)) <-> V_xa_2 = V_ya_2 | hBOOL(hAPP(V_A_2,V_xa_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 193 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) <-> c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2)))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 194 (all V_s_2 all V_r_2 all T_a c_Relation_OField(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_OField(T_a,V_r_2),c_Relation_OField(T_a,V_s_2))) # label(fact_Field__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 195 (all V_r_2 all T_a ((all B_x all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2) -> (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_z),V_r_2))))) -> c_Relation_Otrans(T_a,V_r_2))) # label(fact_transI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 196 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Un__assoc) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 197 (all V_xa_2 all T_a c_Set_Othe__elem(T_a,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_xa_2) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 198 (all T_a c_Wellfounded_Owf(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_wf__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 199 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) <-> c_member(T_a,V_c_2,V_A_2) | c_member(T_a,V_c_2,V_B_2))) # label(fact_Un__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 200 (all V_A_2 all V_B_2 all V_c_2 all T_a ((-c_member(T_a,V_c_2,V_B_2) -> c_member(T_a,V_c_2,V_A_2)) -> c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_UnCI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 201 (all V_a_2 all V_b_2 all T_a (c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> V_a_2 = V_b_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 202 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) -> c_member(T_a,V_c_2,V_A_2))) # label(fact_DiffD1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 203 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_y))) # label(fact_inf__sup__ord_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 204 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) <-> (all B_x all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2) -> (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_z),V_r_2))))))) # label(fact_trans__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 205 (all V_xa_2 all V_F_2 all V_R_2 all T_a all T_b hAPP(hAPP(V_F_2,c_Recdef_Ocut(T_b,T_a,c_Nitpick_Owf__wfrec(T_b,T_a,V_R_2,V_F_2),V_R_2,V_xa_2)),V_xa_2) = c_Nitpick_Owf__wfrec_H(T_b,T_a,V_R_2,V_F_2,V_xa_2)) # label(fact_Nitpick_Owf__wfrec_H__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 206 (all V_A_2 all T_a (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_subset__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 207 (all V_r_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_a,V_r_2),c_Relation_ORange(T_a,T_a,V_r_2)) = c_Relation_OField(T_a,V_r_2)) # label(fact_Field__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 208 (all V_ya_2 all V_R_2 all V_xa_2 all T_a (-c_member(T_a,V_xa_2,c_Relation_ODomain(T_a,T_a,V_R_2)) -> (V_ya_2 = V_xa_2 <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))))) # label(fact_Not__Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 209 (all V_B_2 all V_A_2 all T_b all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Relation_ORange(T_b,T_a,V_A_2),c_Relation_ORange(T_b,T_a,V_B_2)),c_Relation_ORange(T_b,T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_Range__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 210 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,V_A_2) -> (c_member(T_a,V_c_2,V_B_2) -> c_member(T_a,V_c_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))))) # label(fact_IntI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 211 (all V_ya_2 all V_xa_2 all V_La_2 all V_b_2 all V_a_2 (V_b_2 != V_a_2 -> (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> ((V_b_2 != V_xa_2 -> (V_b_2 = V_ya_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_a_2),V_La_2) | V_a_2 = V_xa_2) & (V_ya_2 != V_b_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2))) & (V_xa_2 = V_b_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_a_2,V_ya_2),V_La_2)) & V_ya_2 != V_xa_2 <-> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Oabove(V_La_2,V_a_2,V_b_2)))))) # label(fact_in__above) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 212 (all V_B_2 all V_xa_2 all V_A_2 all T_a ((-c_member(T_a,V_xa_2,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) & (c_member(T_a,V_xa_2,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2)))) # label(fact_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 213 (all T_a c_Relation_Oantisym(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_antisym__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 214 (all V_ya_2 all V_z_2 all V_f_2 all T_b all V_A_2 all V_xa_2 all T_a (c_member(T_a,V_xa_2,V_A_2) -> (c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_ya_2) -> c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,V_A_2,hAPP(hAPP(V_f_2,V_xa_2),V_ya_2))))) # label(fact_fold__graph_H_Ointros_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 215 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Olinear__order__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2)) <-> c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2))) # label(fact_linear__order__on__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 216 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2) -> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),c_Relation_Oconverse(T_a,T_b,V_r_2)))) # label(fact_converseI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 217 (all V_S_2 all V_R_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),V_xa_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) | c_member(tc_prod(T_a,T_a),V_xa_2,c_Transitive__Closure_Ortrancl(T_a,V_S_2)) -> c_member(tc_prod(T_a,T_a),V_xa_2,c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2))))) # label(fact_in__rtrancl__UnI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 218 (all V_r_2 all T_a (c_Wellfounded_Oacyclic(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) <-> c_Wellfounded_Oacyclic(T_a,V_r_2))) # label(fact_acyclic__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 219 (all V_a2_2 all V_a1_2 all T_b all T_a c_Nitpick_Ofold__graph_H(T_a,T_b,V_a1_2,V_a2_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_a2_2)) # label(fact_fold__graph_H_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 220 (all T_a c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) = c_Relation_OId__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_Id__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 221 (all V_r_2 all T_a c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2)) # label(fact_rtrancl__idemp) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 222 (all V_ya_2 all V_xa_2 all V_La_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> (V_ya_2 != V_xa_2 -> (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_ya_2,V_xa_2),V_La_2) <-> -c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2))))) # label(fact_notin__Lin__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 223 (all V_g_2 all V_F_2 all T_b all V_R_2 all V_xa_2 all T_a ((all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_xa_2),V_R_2) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(V_g_2,B_z)))) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_xa_2,hAPP(hAPP(V_F_2,V_g_2),V_xa_2)))) # label(fact_wfrec__rel_Ointros) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 224 (all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_sup__ge2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 225 (all V_c all V_a all V_b all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_c)))) # label(fact_sup_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 226 (all V_f_2 all T_b all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_Wellfounded_Owf(T_b,hAPP(c_Relation_Oinv__image(T_a,T_b,V_r_2),V_f_2)))) # label(fact_wf__inv__image) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 227 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Un__Diff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 228 (all V_b_2 all V_r_2 all V_a_2 all T_a (c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2))))) # label(fact_acc__downwards) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 229 (all V_f_2 all V_r_2 all T_b all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2)),V_r_2) <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),hAPP(c_Relation_Oinv__image(T_b,T_a,V_r_2),V_f_2)))) # label(fact_in__inv__image) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 230 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 231 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_B_2)) # label(fact_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 232 (all V_r_2 all T_b all T_a ((all B_x all B_y (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2) -> (all B_z (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2) -> B_z = B_y)))) -> c_Relation_Osingle__valued(T_a,T_b,V_r_2))) # label(fact_single__valuedI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 233 (all T_a c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId(T_a))) # label(fact_single__valued__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 234 (all V_ya_2 all V_xa_2 all V_La_2 all V_b_2 all V_a_2 (V_b_2 != V_a_2 -> (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> (V_xa_2 != V_ya_2 & (V_ya_2 != V_a_2 -> (V_xa_2 != V_a_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2)) & (V_xa_2 = V_a_2 -> V_b_2 = V_ya_2 | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_b_2,V_ya_2),V_La_2))) & (V_a_2 = V_ya_2 -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_b_2),V_La_2)) <-> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Obelow(V_La_2,V_a_2,V_b_2)))))) # label(fact_in__below) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 235 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 236 (all V_a5_2 all V_a1_2 all V_a2_2 all V_a4_2 all V_a3_2 all T_b all T_a (c_Nitpick_Ofold__graph_H(T_a,T_b,V_a3_2,V_a4_2,V_a2_2,hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2)) <-> (exists B_x exists B_y (c_Nitpick_Ofold__graph_H(T_a,T_b,V_a3_2,V_a4_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_a2_2,c_Set_Oinsert(T_a,B_x,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),B_y) & c_member(T_a,B_x,V_a2_2) & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y))) | V_a4_2 = hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) & c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_a2_2)) # label(fact_fold__graph_H_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 237 (all V_B_2 all V_A_2 all V_c_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),V_c_2,c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)) -> -(all B_x (c_member(T_a,B_x,V_A_2) -> (all B_y (c_member(T_b,B_y,hAPP(V_B_2,B_x)) -> V_c_2 != c_Product__Type_OPair(T_a,T_b,B_x,B_y))))))) # label(fact_SigmaE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 238 (all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2))) # label(fact_Un__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 239 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_y = V_x | c_Orderings_Oord__class_Oless(T_a,V_y,V_x) | c_Orderings_Oord__class_Oless(T_a,V_x,V_y))) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 240 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)))) # label(fact_inf__sup__aci_I7_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 241 (all V_r_2 all V_b_2 all V_a_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OField(T_a,V_r_2)) = c_Relation_OField(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2))) # label(fact_Field__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 242 (all V_r_2 all V_p_2 all T_a (c_member(tc_prod(T_a,T_a),V_p_2,V_r_2) -> c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_r_2)))) # label(fact_r__into__trancl_H) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 243 (all V_S_2 all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)) = c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_S_2)))) # label(fact_rtrancl__Un__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 244 (all V_B_2 all V_b_2 all T_b all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> (c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) -> c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))))) # label(fact_SigmaI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 245 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_m_2,V_n_2),c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat)))) # label(fact_less__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 246 (all V_r_2 all T_a (c_Relation_Oantisym(T_a,V_r_2) <-> (all B_x all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2) -> B_y = B_x))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 247 (all T_2 all T_1 (class_Lattices_Olattice(T_1) -> class_Lattices_Osemilattice__inf(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Osemilattice__inf) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 248 (all V_xa_2 all V_B_2 all V_A_2 all T_b all T_a (class_Groups_Ominus(T_a) -> c_Groups_Ominus__class_Ominus(T_a,hAPP(V_A_2,V_xa_2),hAPP(V_B_2,V_xa_2)) = hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_xa_2))) # label(fact_minus__apply) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 249 (all V_A_2 all V_B_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2),V_A_2)) # label(fact_Un__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 250 (all T_b all T_a c_Relation_ODomain(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Domain__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.00 251 (all V_z_2 all V_ya_2 all V_xa_2 all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_z_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_z_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_xa_2,V_ya_2),V_z_2)))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 252 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2) <-> c_Order__Relation_Opartial__order__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2)))) # label(fact_partial__order__on__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 253 (all V_R_2 all T_a all V_ya_2 all V_xa_2 all V_f_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)))) # label(fact_mlex__less) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 254 (all V_r_2 all T_a ((all B_Q ((exists B_x c_member(T_a,B_x,B_Q)) -> (exists B_x ((all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2) -> -c_member(T_a,B_y,B_Q))) & c_member(T_a,B_x,B_Q))))) <-> c_Wellfounded_Owf(T_a,V_r_2))) # label(fact_wf__eq__minimal) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 255 (all B_x -c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_x),v_L)) # label(conj_1_AndLHS) # label(hypothesis) # label(non_clause). [assumption]. 0.79/1.01 256 (all B_x all B_y (B_x != B_y -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_x),v_L) | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L))) # label(conj_1_AndRHS_AndLHS) # label(hypothesis) # label(non_clause). [assumption]. 0.79/1.01 257 (all B_x all B_y (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L) -> (all B_z (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_z),v_L) -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_z),v_L))))) # label(conj_1_AndRHS_AndRHS) # label(hypothesis) # label(non_clause). [assumption]. 0.79/1.01 258 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a c_Set_Oinsert(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) = c_Relation_ODomain(T_a,T_b,c_Set_Oinsert(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2))) # label(fact_Domain__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 259 (all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_x) = V_x)) # label(fact_sup__idem) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 260 (all V_P_2 all V_r_2 all V_by_2 all V_bx_2 all V_ay_2 all V_ax_2 all T_b all T_a (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) -> (hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2)) -> ((all B_a all B_b all B_aa all B_ba (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba)),V_r_2) -> (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) -> (hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba)) -> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)))))) -> hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2)))))) # label(fact_converse__rtrancl__induct2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 261 (all V_ya_2 all V_xa_2 all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_r_2) -> c_member(T_a,V_ya_2,V_A_2)))) # label(fact_refl__onD2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 262 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_subset__refl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 263 (all V_R_2 all T_a all V_ya_2 all V_xa_2 all V_f_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_R_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2))))) # label(fact_mlex__leq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 264 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) | c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 265 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Relation_Oantisym(T_a,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2) -> V_a_2 = V_b_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 266 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_m_2,V_n_2),c_Transitive__Closure_Ortrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat)))) # label(fact_pred__nat__trancl__eq__le) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 267 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> -c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 268 (all V_A_2 all T_a all V_xa_2 all V_B_2 (hBOOL(hAPP(V_B_2,V_xa_2)) -> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)))) # label(fact_sup1I2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 269 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> ((all B_x c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x))) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 270 (all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_OField(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_Field__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 271 (all V_P_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (hBOOL(hAPP(V_P_2,V_a_2)) -> ((all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_y),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2) -> (hBOOL(hAPP(V_P_2,B_y)) -> hBOOL(hAPP(V_P_2,B_z))))))) -> hBOOL(hAPP(V_P_2,V_b_2)))))) # label(fact_rtrancl__induct) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 272 (all T_a all V_R_2 all T_b all V_P_2 ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> c_Wellfounded_Owf(T_b,hAPP(V_R_2,B_x)))) -> c_Wellfounded_Owf(tc_prod(T_a,T_b),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2)))) # label(fact_wf__same__fst) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 273 (all V_b_2 all V_a_2 all V_La_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_a_2,V_b_2),V_La_2) -> -c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_b_2,V_a_2),V_La_2)))) # label(fact_Lin__irrefl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 274 (all V_f_2 all T_b all V_r_2 all V_a_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),V_r_2) -> hAPP(c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2),V_xa_2) = hAPP(V_f_2,V_xa_2))) # label(fact_cut__apply) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 275 (all V_f_2 all T_a c_Wellfounded_Owf(T_a,hAPP(c_Wellfounded_Omeasure(T_a),V_f_2))) # label(fact_wf__measure) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 276 (all V_A_2 all T_a V_A_2 = c_Relation_ORange(T_a,T_a,c_Relation_OId__on(T_a,V_A_2))) # label(fact_Range__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 277 (all V_r_2 all T_a c_Relation_OField(T_a,V_r_2) = c_Relation_OField(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) # label(fact_Field__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 278 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> V_x = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Orderings_Obot__class_Obot(T_a)))) # label(fact_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 279 (all V_La_2 all V_ya_2 all V_xa_2 (V_ya_2 != V_xa_2 -> (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Arrow__Order__Mirabelle_Obelow(V_La_2,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_OLin)))) # label(fact_below__Lin) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 280 (all V_r_2 all T_a c_Relation_Otrans(T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) # label(fact_trans__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 281 (all V_r_2 all V_a_2 all V_b_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2)) -> c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2))))) # label(fact_acc__downwards__aux) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 282 (all V_B_2 all T_a all V_A_2 (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) -> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_less__by__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 283 (all T_a c_Transitive__Closure_Otrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) # label(fact_trancl__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 284 (all V_r_2 all T_b all T_a (c_Relation_Osingle__valued(T_a,T_b,V_r_2) <-> (all B_x all B_y (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2) -> (all B_z (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2) -> B_z = B_y)))))) # label(fact_single__valued__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 285 (all V_D_2 all V_B_2 all V_C_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_D_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2))))) # label(fact_Un__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 286 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_inf__sup__ord_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 287 (all V_s_2 all V_r_2 all V_p_2 all T_a (c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2) -> c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_s_2))))) # label(fact_trancl__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 288 (all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)) # label(fact_rtrancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 289 (all V_f_2 all V_R_2 all T_a (c_Wellfounded_Owf(T_a,V_R_2) -> c_Wellfounded_Owf(T_a,c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)))) # label(fact_wf__mlex) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 290 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_rtrancl__into__trancl2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 291 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_UnI1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 292 (all V_B_2 all V_A_2 all V_R_2 all T_a all T_b c_Relation_OImage(T_b,T_a,V_R_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,V_A_2),c_Relation_OImage(T_b,T_a,V_R_2,V_B_2))) # label(fact_Image__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 293 (all V_r_2 all V_A_2 all T_a (c_Relation_Ototal__on(T_a,V_A_2,V_r_2) <-> c_Relation_Ototal__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2)))) # label(fact_total__on__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 294 (all V_A_2 all V_a_2 all T_a c_Set_Oinsert(T_a,V_a_2,V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_A_2)) # label(fact_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 295 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) <-> c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2) | V_xa_2 = V_ya_2))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 296 (all V_A_2 all V_ya_2 all V_xa_2 all T_a (c_member(T_a,V_xa_2,V_A_2) & V_ya_2 = V_xa_2 <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Relation_OId__on(T_a,V_A_2)))) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 297 (all V_ya_2 all V_xa_2 all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> (c_Orderings_Obot__class_Obot(T_a) = V_xa_2 & c_Orderings_Obot__class_Obot(T_a) = V_ya_2 <-> c_Orderings_Obot__class_Obot(T_a) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_xa_2,V_ya_2)))) # label(fact_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 298 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = V_y))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 299 (all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) <-> (all B_x c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2))))) # label(fact_wf__acc__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 300 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 301 (all V_b_2 all V_a_2 all V_f_2 all T_b all T_a (hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2)) -> hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))))) # label(fact_curryE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 302 (all V_xa_2 all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> (c_member(T_a,V_xa_2,V_A_2) <-> c_member(T_a,V_xa_2,V_B_2)))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 303 (all V_A_2 all T_a all V_ya_2 all V_xa_2 (V_xa_2 = V_ya_2 -> (c_member(T_a,V_xa_2,V_A_2) <-> c_member(T_a,V_ya_2,V_A_2)))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 304 (all V_c_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_subsetD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 305 (all V_A_2 all T_a c_Relation_Oantisym(T_a,c_Relation_OId__on(T_a,V_A_2))) # label(fact_antisym__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 306 (all V_f_2 all T_b all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> c_Relation_Otrans(T_b,hAPP(c_Relation_Oinv__image(T_a,T_b,V_r_2),V_f_2)))) # label(fact_trans__inv__image) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 307 (all V_P_2 all V_B_2 all V_A_2 all T_a ((all B_x (c_member(T_a,B_x,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) -> hBOOL(hAPP(V_P_2,B_x)))) <-> (all B_x (c_member(T_a,B_x,V_B_2) -> hBOOL(hAPP(V_P_2,B_x)))) & (all B_x (c_member(T_a,B_x,V_A_2) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_ball__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 308 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),V_z))) # label(fact_sup__assoc) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 309 (all V_A_2 all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_A_2)) # label(fact_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 310 (all V_R_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)) # label(fact_rtrancl__reflcl__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 311 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)) -> c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)))) # label(fact_SigmaD2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 312 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_A_2)) # label(fact_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 313 (all V_r_2 all T_a all T_b c_Relation_ORange(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = c_Relation_ODomain(T_a,T_b,V_r_2)) # label(fact_Range__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 314 (all V_P_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (hBOOL(hAPP(V_P_2,V_b_2)) -> ((all B_y all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (hBOOL(hAPP(V_P_2,B_z)) -> hBOOL(hAPP(V_P_2,B_y)))))) -> hBOOL(hAPP(V_P_2,V_a_2)))))) # label(fact_converse__rtrancl__induct) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 315 (all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) # label(fact_Un__left__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 316 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) | c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 317 (all V_r_2 all T_a (c_Relation_Oirrefl(T_a,V_r_2) <-> (all B_x -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),V_r_2)))) # label(fact_irrefl__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 318 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_trancl__into__trancl2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 319 (all V_xa_2 all V_B_2 all V_A_2 all T_a (hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)) -> -(hBOOL(hAPP(V_A_2,V_xa_2)) -> -hBOOL(hAPP(V_B_2,V_xa_2))))) # label(fact_inf1E) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 320 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (V_a != V_b -> c_Orderings_Oord__class_Oless(T_a,V_b,V_a))))) # label(fact_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 321 (all V_x all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_x) -> -(c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x) -> -c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x))))) # label(fact_le__supE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 322 (all V_r_2 all T_a ((all B_x c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2))) -> c_Wellfounded_Owf(T_a,V_r_2))) # label(fact_acc__wfI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 323 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) -> (V_b_2 = V_a_2 <-> V_d_2 = V_c_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 324 (all V_r_2 all V_xa_2 all V_ya_2 all T_a (-c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) & c_Wellfounded_Owf(T_a,V_r_2) <-> c_Wellfounded_Owf(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),V_r_2)))) # label(fact_wf__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 325 (all V_r_2 all V_z_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> (-c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),V_r_2) -> -(all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_y),V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))))) # label(fact_converse__tranclE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 326 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_psubset__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 327 (all V_g_2 all V_f_2 ((all B_x hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)) -> V_f_2 = V_g_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 328 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_subset__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 329 (all V_A_2 all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 330 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) -> -(c_member(T_a,V_c_2,V_A_2) -> -c_member(T_a,V_c_2,V_B_2)))) # label(fact_IntE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 331 (all V_s_2 all V_r_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)))) # label(fact_rtrancl__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 332 (all V_z all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z),V_x))))) # label(fact_sup__least) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 333 (all V_r_2 all T_a c_Transitive__Closure_Ortrancl(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_r_2)) # label(fact_rtrancl__r__diff__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 334 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (V_b = V_c -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 335 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) & -c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2)))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 336 (all T_a c_Order__Relation_Olinear__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_lnear__order__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 337 (all V_A_2 all V_A_H_2 all V_r_2 all V_r_H_2 all T_b all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_H_2,V_r_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_H_2,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Relation_OImage(T_a,T_b,V_r_H_2,V_A_H_2),c_Relation_OImage(T_a,T_b,V_r_2,V_A_2))))) # label(fact_Image__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 338 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_not__leE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 339 (all V_g_2 all V_xa_2 all V_r_2 all V_f_2 all T_b all T_a (c_Recdef_Ocut(T_a,T_b,V_g_2,V_r_2,V_xa_2) = c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_xa_2) <-> (all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_xa_2),V_r_2) -> hAPP(V_f_2,B_y) = hAPP(V_g_2,B_y))))) # label(fact_cuts__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 340 (all V_xa_2 all V_B_2 all V_A_2 all T_a (hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)) -> (-hBOOL(hAPP(V_A_2,V_xa_2)) -> hBOOL(hAPP(V_B_2,V_xa_2))))) # label(fact_sup1E) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 341 (all V_B_2 all V_a_2 all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2)) # label(fact_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 342 (all V_xa_2 all T_b all T_a (class_Orderings_Obot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_xa_2))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 343 (all T_b all T_a all V_P_2 ((all B_a all B_b hBOOL(hAPP(V_P_2,c_Product__Type_OPair(T_a,T_b,B_a,B_b)))) <-> (all B_x1 hBOOL(hAPP(V_P_2,B_x1))))) # label(fact_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 344 (all V_x all V_y all T_a (class_Lattices_Olattice(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_inf__sup__ord_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 345 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_x) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y))) # label(fact_inf__sup__aci_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 346 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Relation_Orefl__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2) -> (c_Relation_Oantisym(T_a,V_r_2) -> (c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2)) -> (c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2)) -> (c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) <-> V_b_2 = V_a_2)))))) # label(fact_Refl__antisym__eq__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 347 (all V_r_2 all T_a c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) # label(fact_trancl__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 348 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> -c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_leD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 349 (all V_a_2 all T_a all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2 -> -c_member(T_a,V_a_2,V_A_2))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 350 (all V_B_2 all V_D_2 all V_C_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_D_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2))))) # label(fact_Diff__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 351 (all V_B_2 all V_A_2 all T_a (V_A_2 != V_B_2 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 352 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2))) # label(fact_Un__left__commute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 353 (all V_C_2 all V_B_2 all V_a_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_B_2),V_C_2) = c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 354 (all V_r_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_trancl__converseD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 355 (all V_r_2 all V_b_2 all V_a_2 all T_a all T_b c_Relation_ORange(T_b,T_a,c_Set_Oinsert(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_a_2,V_b_2),V_r_2)) = c_Set_Oinsert(T_a,V_b_2,c_Relation_ORange(T_b,T_a,V_r_2))) # label(fact_Range__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 356 (all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))) # label(fact_trancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 357 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(T_a,V_a_2,V_A_2) & c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) <-> c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)))) # label(fact_mem__Sigma__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 358 (all V_A_2 all T_a -c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_not__psubset__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 359 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> (c_Relation_Oantisym(T_a,V_r_2) -> c_Relation_Otrans(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a)))))) # label(fact_trans__diff__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 360 (all V_r_2 all T_a ((all B_x all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2) -> B_x = B_y))) -> c_Relation_Oantisym(T_a,V_r_2))) # label(fact_antisymI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 361 (all V_r_2 all T_a all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ORange(T_b,T_a,V_r_2) <-> c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)) = V_r_2)) # label(fact_Range__empty__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 362 (all V_B_2 all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Un__upper1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 363 (all V_r_2 all T_b all T_a c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) = c_Relation_ORange(T_b,T_a,V_r_2)) # label(fact_Domain__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 364 (all V_R_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) -> (exists B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_ya_2),V_R_2) & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_z),c_Transitive__Closure_Ortrancl(T_a,V_R_2)))))) # label(fact_tranclD2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 365 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Relation_Oconverse(T_b,T_a,V_r_2)) <-> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),V_r_2))) # label(fact_converse__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 366 (all V_B_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = V_B_2) # label(fact_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 367 (all V_xa_2 all V_ya_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) -> (V_ya_2 = V_xa_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2))))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 368 (all V_r_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))))) # label(fact_trancl__converseI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 369 (all V_A_2 all V_B_2 all V_xa_2 all T_a ((c_member(T_a,V_xa_2,V_B_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2)) & (-c_member(T_a,V_xa_2,V_B_2) -> c_Set_Oinsert(T_a,V_xa_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2)))) # label(fact_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 370 (all V_r_2 all V_xa_2 all V_ya_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))))) # label(fact_rtrancl__converseI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 371 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_Transitive__Closure_Otrancl__into__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 372 (all V_b_2 all V_a_2 all V_f1_2 all T_a all T_c all T_b c_Product__Type_Oprod_Oprod__rec(T_b,T_c,T_a,V_f1_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_prod_Orecs) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 373 (all V_r_2 all V_zb_2 all V_za_2 all V_xb_2 all V_xa_2 all T_b all T_a (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2),c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) -> (c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2) != c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2) -> -(all B_a all B_b (c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2),c_Product__Type_OPair(T_a,T_b,B_a,B_b)),V_r_2) -> -c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))))))) # label(fact_converse__rtranclE2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 374 (all V_r_2 all T_b all V_a_2 all T_a (c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) -> -(all B_y -c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,B_y),V_r_2)))) # label(fact_DomainE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 375 (all V_s_2 all V_r_2 all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_b all T_a (V_a_H_2 = V_a_2 & c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_b_2,V_b_H_2),V_s_2) | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_H_2),V_r_2) <-> c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2)),c_Wellfounded_Olex__prod(T_a,T_b,V_r_2,V_s_2)))) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 376 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b != V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 377 (all V_ya_2 all V_xa_2 all T_a (V_xa_2 = V_ya_2 <-> c_member(T_a,V_xa_2,c_fequal(V_ya_2)))) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 378 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Oorder(T_a) -> (V_ya_2 = V_xa_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 379 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Orderings_Obot__class_Obot(T_a),V_x) = V_x)) # label(fact_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 380 (all T_a c_Relation_OId(T_a) = c_Relation_Oconverse(T_a,T_a,c_Relation_OId(T_a))) # label(fact_converse__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 381 (all V_B_2 all V_A_2 all T_b all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_A_2),c_Relation_ODomain(T_a,T_b,V_B_2)),c_Relation_ODomain(T_a,T_b,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_Domain__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 382 (all V_xa_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_member(T_a,V_xa_2,c_Wellfounded_Oacc(T_a,V_r_2)))) # label(fact_acc__wfD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 383 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_sup__left__idem) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 384 (all V_A_2 all T_a all V_b_2 all V_a_2 (V_a_2 = V_b_2 -> (c_member(T_a,V_b_2,V_A_2) -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_eq__mem__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 385 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z))))) # label(fact_order__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 386 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2) <-> c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2) & c_Relation_Ototal__on(T_a,V_A_2,V_r_2))) # label(fact_linear__order__on__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 387 (all V_r_2 all V_s_2 all T_a (c_Wellfounded_Oacyclic(T_a,V_s_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2) -> c_Wellfounded_Oacyclic(T_a,V_r_2)))) # label(fact_acyclic__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 388 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) = V_B_2)) # label(fact_Diff__partition) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 389 (all V_B_2 all V_a_2 all T_a c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_a_2,V_B_2))) # label(fact_insertI1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 390 (all V_La_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) <-> c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Relation_Oconverse(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_La_2),c_Arrow__Order__Mirabelle_OLin))) # label(fact_converse__in__Lin) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 391 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2) -> c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)))) # label(fact_DomainI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 392 (all V_r_2 all T_a (c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) -> c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_wf__converse__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 393 (all T_2 all T_1 (class_Orderings_Opreorder(T_1) -> class_Orderings_Opreorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 394 (all V_Q_2 all V_P_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_P_2,V_Q_2))) -> ((all B_x (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_x),c_Transitive__Closure_Ortrancl(T_a,V_P_2)) -> (all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_Q_2) -> B_y = B_x)))) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_P_2))))) # label(fact_rtrancl__Un__separatorE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 395 (all V_B_2 all V_C_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2)))) # label(fact_Un__least) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 396 (all V_b_2 all V_a_2 all V_f_2 all T_b all T_a (hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2)) -> hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))))) # label(fact_curryD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 397 (all V_a_2 all T_a c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),c_Relation_OId(T_a))) # label(fact_IdI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 398 (all V_y_2 all V_x_2 (V_x_2 = V_y_2 | -hBOOL(hAPP(c_fequal(V_x_2),V_y_2)))) # label(help_c__fequal__1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 399 (all V_X_2 all V_r_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,V_X_2),V_X_2) -> c_Relation_OImage(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2),V_X_2) = V_X_2)) # label(fact_Image__closed__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 400 (all V_d all V_b all V_c all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_d) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),c_Lattices_Osemilattice__sup__class_Osup(T_a,V_c,V_d)))))) # label(fact_sup__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 401 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 402 (all V_p_2 all T_a (c_member(tc_prod(T_a,T_a),V_p_2,c_Relation_OId(T_a)) -> -(all B_x V_p_2 != c_Product__Type_OPair(T_a,T_a,B_x,B_x)))) # label(fact_IdE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 403 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> -(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 404 (all V_xa_2 all V_La_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) -> c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_xa_2),c_Arrow__Order__Mirabelle_OLin))) # label(fact_mkbot__Lin) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 405 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) -> V_B_2 = V_A_2))) # label(fact_equalityI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 406 (all V_A_2 all V_B_2 all V_c_2 all T_a (c_member(T_a,V_c_2,V_B_2) -> c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_UnI2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 407 (all V_c_2 all T_a -c_member(T_a,V_c_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_empty__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 408 (all V_c_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_psubsetD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 409 (all T_a all V_B_2 all V_xa_2 all V_A_2 (hBOOL(hAPP(V_A_2,V_xa_2)) -> (hBOOL(hAPP(V_B_2,V_xa_2)) -> hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2))))) # label(fact_inf1I) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 410 (all V_A_2 all V_r_2 all T_b all V_b_2 all T_a ((exists B_x (c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2) & c_member(T_b,B_x,V_A_2))) <-> c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,V_A_2)))) # label(fact_Image__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 411 (all T_a c_Order__Relation_Opartial__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_partial__order__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 412 (all T_a all T_b c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ORange(T_b,T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)))) # label(fact_Range__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 413 (all V_ya_2 all V_xa_2 all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) <-> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_xa_2,V_ya_2) = V_ya_2))) # label(fact_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 414 (all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) # label(fact_Un__commute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 415 (all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2)) <-> c_Relation_Orefl__on(T_a,V_A_2,V_r_2))) # label(fact_refl__on__converse) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 416 (all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> V_a = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_a))) # label(fact_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 417 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> c_Orderings_Oord__class_Oless(T_a,V_b,V_a))))) # label(fact_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 418 (all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != V_A_2 <-> (exists B_x c_member(T_a,B_x,V_A_2)))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 419 (all V_xa_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_xa_2,V_A_2) -> c_member(T_a,V_xa_2,V_B_2)))) # label(fact_set__mp) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 420 (all V_A_2 all V_b_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) | V_b_2 = V_a_2 <-> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)))) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 421 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_a) -> c_Orderings_Oord__class_Oless(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b))))) # label(fact_less__supI1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 422 (all V_s_2 all V_r_2 all T_a all T_b c_Relation_Oconverse(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_r_2,V_s_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),c_Relation_Oconverse(T_b,T_a,V_r_2),c_Relation_Oconverse(T_b,T_a,V_s_2))) # label(fact_converse__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 423 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_x != V_y))) # label(fact_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 424 (all V_xa_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) -> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2))))) # label(fact_le__funD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 425 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_rtrancl__into__trancl1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 426 (all V_a_2 all V_H_2 all V_r_2 all T_b all T_a all V_f_2 (V_f_2 = c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2) -> (c_Wellfounded_Owf(T_a,V_r_2) -> hAPP(hAPP(V_H_2,c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2)),V_a_2) = hAPP(V_f_2,V_a_2)))) # label(fact_def__wfrec) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 427 (all V_B_2 all V_A_2 all V_c_2 all T_a (c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) -> (-c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_UnE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 428 (all V_A_2 all T_a c_Relation_Orefl__on(T_a,V_A_2,c_Relation_OId__on(T_a,V_A_2))) # label(fact_refl__on__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 429 (all V_r_2 all T_a c_Relation_Ototal__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2)) # label(fact_total__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 430 (all V_R_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) -> (exists B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_z),V_R_2))))) # label(fact_tranclD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 431 (all V_a_2 all V_r_2 all T_b all V_b_2 all T_a (c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,c_Set_Oinsert(T_b,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) <-> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_a_2,V_b_2),V_r_2))) # label(fact_Image__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 432 (all V_xa_2 all V_Q_2 all V_P_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2) -> (hBOOL(hAPP(V_P_2,V_xa_2)) -> hBOOL(hAPP(V_Q_2,V_xa_2))))) # label(fact_predicate1D) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 433 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> V_r_2 = c_Transitive__Closure_Otrancl(T_a,V_r_2))) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 434 (all V_B_2 all V_a_2 all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 435 (all V_B_2 all T_b all T_a c_Product__Type_OSigma(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool))) # label(fact_Sigma__empty1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 436 (all V_r_2 all T_b all V_a_2 all T_a ((exists B_y c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_y,V_a_2),V_r_2)) <-> c_member(T_a,V_a_2,c_Relation_ORange(T_b,T_a,V_r_2)))) # label(fact_Range__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 437 (all V_Q_2 all T_a all V_xa_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_xa_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2) -> hBOOL(hAPP(V_Q_2,V_xa_2))))) # label(fact_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 438 (all V_a_2 all V_b_2 all T_a (V_b_2 = V_a_2 <-> c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 439 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2))))) # label(fact_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 440 (all T_2 all T_1 (class_Orderings_Oord(T_1) -> class_Orderings_Oord(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 441 (all V_y_2 all V_x_2 (V_x_2 != V_y_2 | hBOOL(hAPP(c_fequal(V_x_2),V_y_2)))) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 442 (all V_r_2 all T_b all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ODomain(T_a,T_b,V_r_2) <-> c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) = V_r_2)) # label(fact_Domain__empty__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 443 (all V_p_2 all V_r_2 all T_a (c_Wellfounded_OwfP(T_a,V_r_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),V_p_2,V_r_2) -> c_Wellfounded_OwfP(T_a,V_p_2)))) # label(fact_wfP__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 444 (all T_a c_Relation_Oantisym(T_a,c_Relation_OId(T_a))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 445 (all V_A_2 all V_R_2 all T_a (c_Wellfounded_Owf(T_a,V_R_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Relation_OImage(T_a,T_a,V_R_2,V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2))) # label(fact_wfE__pf) # label(axiom) # label(non_clause). [assumption]. 0.79/1.01 446 (all V_b all V_a all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> -c_Orderings_Oord__class_Oless(T_a,V_b,V_a)))) # label(fact_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 447 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (V_c = V_b -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 448 (all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> V_A_2 = c_Set_Oinsert(T_a,V_a_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))))) # label(fact_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 449 (all V_a2_2 all V_a1_2 all V_F_2 all V_R_2 all T_b all T_a ((exists B_g (V_a2_2 = hAPP(hAPP(V_F_2,B_g),V_a1_2) & (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(B_g,B_z)))))) <-> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_a1_2,V_a2_2))) # label(fact_wfrec__rel_Osimps) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 450 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b))))) # label(fact_le__supI1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 451 (all V_xa_2 all V_ya_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2) -> (V_xa_2 = V_ya_2 <-> -c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2))))) # label(fact_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 452 (all V_A_2 all T_a c_Relation_Oconverse(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = c_Relation_OId__on(T_a,V_A_2)) # label(fact_converse__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 453 (all V_r_2 all T_a c_Relation_Otrans(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2))) # label(fact_trans__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 454 (all V_A_2 all V_ya_2 all V_xa_2 all T_a c_Set_Oinsert(T_a,V_ya_2,c_Set_Oinsert(T_a,V_xa_2,V_A_2)) = c_Set_Oinsert(T_a,V_xa_2,c_Set_Oinsert(T_a,V_ya_2,V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 455 (all V_r_2 all V_p_2 all T_a (c_member(tc_prod(T_a,T_a),V_p_2,V_r_2) -> c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Ortrancl(T_a,V_r_2)))) # label(fact_r__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 456 (all V_r_2 all V_A_2 all T_a (c_Relation_Ototal__on(T_a,V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) <-> c_Relation_Ototal__on(T_a,V_A_2,V_r_2))) # label(fact_total__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 457 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x))))) # label(fact_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 458 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 459 (all V_ya_2 all V_xa_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_xa_2,V_ya_2) <-> c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_xa_2,V_ya_2),c_Wellfounded_Oless__than))) # label(fact_less__than__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 460 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (V_b = V_c -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 461 (all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> c_Set_Oinsert(T_a,V_a_2,V_A_2) = V_A_2)) # label(fact_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 462 (all V_s_2 all V_B_2 all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,V_r_2) -> (c_Relation_Orefl__on(T_a,V_B_2,V_s_2) -> c_Relation_Orefl__on(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2))))) # label(fact_refl__on__Un) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 463 (all V_R_2 all V_xa_2 all T_a (-c_member(T_a,V_xa_2,c_Wellfounded_Oacc(T_a,V_R_2)) -> -(all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_xa_2),V_R_2) -> c_member(T_a,B_z,c_Wellfounded_Oacc(T_a,V_R_2)))))) # label(fact_not__acc__down) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 464 (all V_r_2 all V_z_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (V_xa_2 != V_z_2 -> -(all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_y),V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))))))) # label(fact_converse__rtranclE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 465 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (V_y != V_x -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 466 (all V_y all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_y))) # label(fact_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 467 (all T_a c_Order__Relation_Owell__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_well__order__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 468 (all V_c_2 all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> ((c_member(T_a,V_c_2,V_A_2) -> -c_member(T_a,V_c_2,V_B_2)) -> -(-c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2))))) # label(fact_equalityCE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 469 (all V_z_2 all V_f_2 all T_b all T_a c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_z_2)) # label(fact_fold__graph_H_Ointros_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 470 (all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b))) # label(fact_sup_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 471 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))))) # label(fact_converse__rtrancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 472 (all V_A_2 all V_a_2 all T_a c_Set_Oinsert(T_a,V_a_2,V_A_2) = c_Set_Oinsert(T_a,V_a_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 473 (all V_r_2 all T_a (c_Nitpick_Orefl_H(T_a,V_r_2) <-> (all B_x c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),V_r_2)))) # label(fact_Nitpick_Orefl_H__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 474 (all V_R_2 all T_a c_Transitive__Closure_Otrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)) # label(fact_rtrancl__trancl__absorb) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 475 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 476 (all V_p_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_p_2,V_r_2) -> c_Wellfounded_Owf(T_a,V_p_2)))) # label(fact_wf__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 477 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2)))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 478 (all V_r_2 all V_A_2 all T_a (c_Relation_Ototal__on(T_a,V_A_2,V_r_2) <-> (all B_x (c_member(T_a,B_x,V_A_2) -> (all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_xa != B_x -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_xa),V_r_2) | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2)))))))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 479 (all V_B_2 all V_A_2 all V_r_2 all T_a (c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Relation_OField(T_a,V_r_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Relation_OField(T_a,V_r_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,V_A_2),c_Relation_OImage(T_a,T_a,V_r_2,V_B_2)) <-> (all B_x (c_member(T_a,B_x,V_A_2) -> (exists B_xa (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2) & c_member(T_a,B_xa,V_B_2)))))))))) # label(fact_subset__Image__Image__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 480 (all V_B_2 all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 481 (all V_s_2 all V_r_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Transitive__Closure_Ortrancl(T_a,V_s_2)) -> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)))) # label(fact_rtrancl__subset__rtrancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 482 (all V_r_2 all T_a (c_Wellfounded_Oacyclic(T_a,V_r_2) <-> (all B_x -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_acyclic__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 483 (all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 484 (all V_r_2 all T_a c_Relation_ORange(T_a,T_a,V_r_2) = c_Relation_ORange(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) # label(fact_trancl__range) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 485 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Order__Relation_Opartial__order__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2) -> (c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2)) -> (c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2)) -> (V_b_2 = V_a_2 <-> c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))))))) # label(fact_Partial__order__eq__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 486 (all V_a_2 all V_B_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oinsert(T_a,V_a_2,V_B_2))) # label(fact_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 487 (all V_A_2 all T_a c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId__on(T_a,V_A_2))) # label(fact_single__valued__Id__on) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 488 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> c_Relation_Otrans(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))))) # label(fact_trans__reflclI) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 489 (all V_b_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 490 (all V_ya_2 all V_xa_2 all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_r_2) -> c_member(T_a,V_xa_2,V_A_2)))) # label(fact_refl__onD1) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 491 (all V_xa_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) -> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2))))) # label(fact_le__funE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 492 (all V_R_2 all V_ya_2 all V_xa_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) <-> V_xa_2 = V_ya_2 | V_xa_2 != V_ya_2 & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)))) # label(fact_rtrancl__eq__or__trancl) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 493 (all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_a))) # label(fact_sup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 494 (all V_ya_2 all V_xa_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2) & V_ya_2 != V_xa_2))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 495 (all V_ya_2 all V_xa_2 all V_R_2 all T_b all T_a (hBOOL(hAPP(hAPP(c_FunDef_Oin__rel(T_a,T_b,V_R_2),V_xa_2),V_ya_2)) <-> c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_ya_2),V_R_2))) # label(fact_in__rel__def) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 496 (all V_r_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> (-c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> -(all B_c (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_c),c_Transitive__Closure_Otrancl(T_a,V_r_2)) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_c,V_b_2),V_r_2)))))) # label(fact_tranclE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 497 (all V_xa_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_xa_2,V_A_2) -> c_member(T_a,V_xa_2,V_B_2)))) # label(fact_in__mono) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 498 (all V_a all V_b all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b))))) # label(fact_le__supI2) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 499 (all V_xa_2 all V_g_2 all V_f_2 all T_b all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) = hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a),V_f_2,V_g_2),V_xa_2))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 500 (all V_R_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) & V_a_2 != V_b_2 | V_a_2 = V_b_2)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 501 (all V_r_2 all T_b all V_b_2 all T_a (c_member(T_a,V_b_2,c_Relation_ORange(T_b,T_a,V_r_2)) -> -(all B_x -c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2)))) # label(fact_RangeE) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 502 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> (V_b = V_c -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 503 (all V_b_H all V_a_H all V_b all V_a all T_b all T_a (c_Product__Type_OPair(T_a,T_b,V_a,V_b) = c_Product__Type_OPair(T_a,T_b,V_a_H,V_b_H) -> -(V_a = V_a_H -> V_b_H != V_b))) # label(fact_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 0.79/1.02 504 (all V_A_2 all V_B_2 all V_xa_2 all T_a (c_member(T_a,V_xa_2,V_B_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 2.03/2.21 505 (all V_s_2 all V_r_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OField(T_a,V_r_2),c_Relation_OField(T_a,V_s_2)))) # label(fact_mono__Field) # label(axiom) # label(non_clause). [assumption]. 2.03/2.21 506 -((all B_x all B_y ((v_y = B_x & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_x,B_y),v_L) & v_y != B_y -> (all B_z ((B_y = v_x -> v_y != B_z) & (v_y = B_z -> -c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,v_x),v_L)) & (v_y != B_z & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_z),v_L) -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_x,B_z),v_L))))) & (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_x),v_L) & B_x != v_y & v_y = B_y -> (all B_z (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_x,B_z),v_L) & v_y != B_z -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_z),v_L)))) & (v_y != B_x & v_y != B_y & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L) -> (all B_z ((v_x = B_y & v_y = B_z -> B_x = v_x | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_x),v_L)) & (c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,v_x),v_L) & B_z = v_y -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_x),v_L) | v_x = B_x) & (v_y != B_z & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_z),v_L) -> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_z),v_L))))))) & (all B_x all B_y (B_y != B_x -> v_x = B_x & v_y = B_y | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_x),v_L) & v_y != B_x & v_y = B_y | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_x,B_y),v_L) & v_y != B_y & B_x = v_y | B_x = v_y & v_x = B_y | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,v_x),v_L) & v_y != B_y & B_x = v_y | v_y = B_y & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_x,B_x),v_L) & B_x != v_y | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_x),v_L) & B_y != v_y & B_x != v_y | v_y != B_y & v_y != B_x & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L)))) # label(conj_2) # label(negated_conjecture) # label(non_clause). [assumption]. 2.03/2.27 2.03/2.27 ============================== end of process non-clausal formulas === 2.03/2.27 2.03/2.27 ============================== PROCESS INITIAL CLAUSES =============== 2.03/2.27 2.03/2.27 ============================== PREDICATE ELIMINATION ================= 2.03/2.27 507 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 2.03/2.27 508 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,C,B) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neq__iff) # label(axiom). [clausify(9)]. 2.03/2.27 509 -class_Orderings_Olinorder(A) | B != C | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__neq__iff) # label(axiom). [clausify(9)]. 2.03/2.27 510 -class_Orderings_Olinorder(A) | B != C | -c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neq__iff) # label(axiom). [clausify(9)]. 2.03/2.27 Derived: A = B | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B). [resolve(507,a,508,a)]. 2.03/2.27 Derived: A != B | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(507,a,509,a)]. 2.03/2.27 Derived: A != B | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B). [resolve(507,a,510,a)]. 2.03/2.27 511 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__neqE) # label(axiom). [clausify(66)]. 2.03/2.27 512 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,B,C) | C = B # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(90)]. 2.03/2.27 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | B = A. [resolve(512,a,507,a)]. 2.03/2.27 513 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,B,C) | C != B # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(90)]. 2.03/2.27 514 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_leI) # label(axiom). [clausify(111)]. 2.03/2.27 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(514,a,507,a)]. 2.03/2.27 515 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | C = B # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(145)]. 2.03/2.27 516 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,B,C) | C != B # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(145)]. 2.03/2.27 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | B != A. [resolve(516,a,507,a)]. 2.03/2.27 517 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(193)]. 2.03/2.27 518 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(193)]. 2.03/2.27 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(518,a,507,a)]. 2.03/2.27 519 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,C,B) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__less__linear) # label(axiom). [clausify(239)]. 2.03/2.27 520 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__le__less__linear) # label(axiom). [clausify(264)]. 4.24/4.49 521 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | C = B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(295)]. 4.24/4.49 522 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(295)]. 4.24/4.49 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(522,a,507,a)]. 4.24/4.49 523 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | C != B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(295)]. 4.24/4.49 524 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__le__cases) # label(axiom). [clausify(300)]. 4.24/4.49 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(524,a,507,a)]. 4.24/4.49 525 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__linear) # label(axiom). [clausify(316)]. 4.24/4.49 526 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__leE) # label(axiom). [clausify(338)]. 4.24/4.49 527 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_leD) # label(axiom). [clausify(348)]. 4.24/4.49 528 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | B != C | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(451)]. 4.24/4.49 529 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | B = C | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(451)]. 4.24/4.49 530 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | B = C | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__cases) # label(axiom). [clausify(465)]. 4.24/4.49 531 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__less) # label(axiom). [clausify(477)]. 4.24/4.49 532 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__less) # label(axiom). [clausify(477)]. 4.24/4.49 533 -c_Order__Relation_Ostrict__linear__order__on(A,B,C) | c_Relation_Oirrefl(A,C) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(184)]. 4.24/4.49 534 -c_Order__Relation_Olinear__order__on(A,B,C) | c_Order__Relation_Ostrict__linear__order__on(A,B,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) # label(fact_strict__linear__order__on__diff__Id) # label(axiom). [clausify(39)]. 4.24/4.49 Derived: c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),B,c_Relation_OId(A))) | -c_Order__Relation_Olinear__order__on(A,C,B). [resolve(533,a,534,b)]. 4.24/4.49 535 -c_Order__Relation_Ostrict__linear__order__on(A,B,C) | c_Relation_Ototal__on(A,B,C) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(184)]. 4.24/4.49 Derived: c_Relation_Ototal__on(A,B,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) | -c_Order__Relation_Olinear__order__on(A,B,C). [resolve(535,a,534,b)]. 4.24/4.49 536 -c_Order__Relation_Ostrict__linear__order__on(A,B,C) | c_Relation_Otrans(A,C) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(184)]. 4.24/4.49 Derived: c_Relation_Otrans(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),B,c_Relation_OId(A))) | -c_Order__Relation_Olinear__order__on(A,C,B). [resolve(536,a,534,b)]. 4.24/4.49 537 c_Order__Relation_Ostrict__linear__order__on(A,B,C) | -c_Relation_Oirrefl(A,C) | -c_Relation_Ototal__on(A,B,C) | -c_Relation_Otrans(A,C) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(184)]. 4.24/4.49 538 -class_Lattices_Osemilattice__inf(A) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__inf__class_Oinf(A,B,C),B) # label(fact_inf__le1) # label(axiom). [clausify(83)]. 5.43/5.62 539 class_Lattices_Osemilattice__inf(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Osemilattice__inf) # label(axiom). [assumption]. 5.43/5.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),A). [resolve(538,a,539,a)]. 5.43/5.62 540 class_Lattices_Osemilattice__inf(tc_Nat_Onat) # label(arity_Nat__Onat__Lattices_Osemilattice__inf) # label(axiom). [assumption]. 5.43/5.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Lattices_Osemilattice__inf__class_Oinf(tc_Nat_Onat,A,B),A). [resolve(540,a,538,a)]. 5.43/5.62 541 -class_Lattices_Osemilattice__inf(A) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__inf__class_Oinf(A,B,C),C) # label(fact_inf__le2) # label(axiom). [clausify(149)]. 5.43/5.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),B). [resolve(541,a,539,a)]. 5.43/5.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Lattices_Osemilattice__inf__class_Oinf(tc_Nat_Onat,A,B),B). [resolve(541,a,540,a)]. 5.43/5.62 542 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) = B # label(fact_le__iff__inf) # label(axiom). [clausify(161)]. 5.43/5.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) = A. [resolve(542,a,539,a)]. 5.43/5.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Lattices_Osemilattice__inf__class_Oinf(tc_Nat_Onat,A,B) = A. [resolve(542,a,540,a)]. 5.43/5.62 543 -class_Lattices_Osemilattice__inf(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) != B # label(fact_le__iff__inf) # label(axiom). [clausify(161)]. 5.43/5.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) != A. [resolve(543,a,539,a)]. 5.43/5.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Lattices_Osemilattice__inf__class_Oinf(tc_Nat_Onat,A,B) != A. [resolve(543,a,540,a)]. 5.43/5.62 544 -class_Lattices_Olattice(A) | class_Lattices_Osemilattice__inf(tc_fun(B,A)) # label(arity_fun__Lattices_Osemilattice__inf) # label(axiom). [clausify(247)]. 5.43/5.62 Derived: -class_Lattices_Olattice(A) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D),C). [resolve(544,b,538,a)]. 5.43/5.62 Derived: -class_Lattices_Olattice(A) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D),D). [resolve(544,b,541,a)]. 5.43/5.62 Derived: -class_Lattices_Olattice(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) = C. [resolve(544,b,542,a)]. 5.43/5.62 Derived: -class_Lattices_Olattice(A) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) != C. [resolve(544,b,543,a)]. 5.43/5.62 545 -class_Lattices_Olattice(A) | class_Lattices_Osemilattice__sup(tc_fun(B,A)) # label(arity_fun__Lattices_Osemilattice__sup) # label(axiom). [clausify(54)]. 5.43/5.62 546 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,B,C)) # label(fact_sup__ge1) # label(axiom). [clausify(46)]. 5.43/5.62 Derived: -class_Lattices_Olattice(A) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,D)). [resolve(545,b,546,a)]. 5.43/5.62 547 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,C) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,D),C) # label(fact_le__supI) # label(axiom). [clausify(101)]. 5.43/5.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),E,D) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,E),D) | -class_Lattices_Olattice(B). [resolve(547,a,545,b)]. 5.43/5.63 548 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,D,C)) # label(fact_less__supI2) # label(axiom). [clausify(115)]. 5.43/5.63 Derived: -c_Orderings_Oord__class_Oless(tc_fun(A,B),C,D) | c_Orderings_Oord__class_Oless(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),E,D)) | -class_Lattices_Olattice(B). [resolve(548,a,545,b)]. 5.43/5.63 549 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__sup__class_Osup(A,C,B) = C # label(fact_sup__absorb1) # label(axiom). [clausify(116)]. 5.43/5.63 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,C) = D | -class_Lattices_Olattice(B). [resolve(549,a,545,b)]. 5.43/5.63 550 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,D)) = c_Lattices_Osemilattice__sup__class_Osup(A,C,c_Lattices_Osemilattice__sup__class_Osup(A,B,D)) # label(fact_sup__left__commute) # label(axiom). [clausify(146)]. 5.43/5.63 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,E)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,E)) | -class_Lattices_Olattice(B). [resolve(550,a,545,b)]. 5.43/5.63 551 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,C),D) = c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,D)) # label(fact_sup_Oassoc) # label(axiom). [clausify(164)]. 5.43/5.63 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D),E) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,E)) | -class_Lattices_Olattice(B). [resolve(551,a,545,b)]. 5.43/5.63 552 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) = c_Lattices_Osemilattice__sup__class_Osup(A,C,B) # label(fact_sup__commute) # label(axiom). [clausify(170)]. 5.43/5.63 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,C) | -class_Lattices_Olattice(B). [resolve(552,a,545,b)]. 5.43/5.63 553 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,B)) # label(fact_sup__ge2) # label(axiom). [clausify(224)]. 5.43/5.63 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,C)) | -class_Lattices_Olattice(B). [resolve(553,a,545,b)]. 5.43/5.63 554 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,D)) = c_Lattices_Osemilattice__sup__class_Osup(A,C,c_Lattices_Osemilattice__sup__class_Osup(A,B,D)) # label(fact_sup_Oleft__commute) # label(axiom). [clausify(225)]. 5.43/5.63 555 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,C) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,D,B),C) # label(fact_le__sup__iff) # label(axiom). [clausify(251)]. 5.43/5.63 556 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,D,B),C) # label(fact_le__sup__iff) # label(axiom). [clausify(251)]. 5.43/5.63 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),E,C),D) | -class_Lattices_Olattice(B). [resolve(556,a,545,b)]. 5.43/5.63 557 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,D),C) # label(fact_le__sup__iff) # label(axiom). [clausify(251)]. 5.43/5.63 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,E),D) | -class_Lattices_Olattice(B). [resolve(557,a,545,b)]. 5.43/5.65 558 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,B) = B # label(fact_sup__idem) # label(axiom). [clausify(259)]. 5.43/5.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,C) = C | -class_Lattices_Olattice(B). [resolve(558,a,545,b)]. 5.43/5.65 559 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) = C # label(fact_sup__absorb2) # label(axiom). [clausify(298)]. 5.43/5.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) = D | -class_Lattices_Olattice(B). [resolve(559,a,545,b)]. 5.43/5.65 560 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,C),D) = c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,D)) # label(fact_sup__assoc) # label(axiom). [clausify(308)]. 5.43/5.65 561 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,C),D) | c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_le__supE) # label(axiom). [clausify(321)]. 5.43/5.65 562 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,C),D) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_le__supE) # label(axiom). [clausify(321)]. 5.43/5.65 563 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,C) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,D),C) # label(fact_sup__least) # label(axiom). [clausify(332)]. 5.43/5.65 564 class_Lattices_Osemilattice__sup(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Osemilattice__sup) # label(axiom). [assumption]. 5.43/5.65 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B)). [resolve(564,a,546,a)]. 5.43/5.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,C,B) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C),B). [resolve(564,a,547,a)]. 5.43/5.65 Derived: -c_Orderings_Oord__class_Oless(tc_HOL_Obool,A,B) | c_Orderings_Oord__class_Oless(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,C,B)). [resolve(564,a,548,a)]. 5.43/5.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A) = B. [resolve(564,a,549,a)]. 5.43/5.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,C)) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C)). [resolve(564,a,550,a)]. 5.43/5.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B),C) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,C)). [resolve(564,a,551,a)]. 5.43/5.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A). [resolve(564,a,552,a)]. 5.43/5.65 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A)). [resolve(564,a,553,a)]. 5.43/5.65 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,C,A),B). [resolve(564,a,556,a)]. 5.43/5.65 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C),B). [resolve(564,a,557,a)]. 5.43/5.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,A) = A. [resolve(564,a,558,a)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = B. [resolve(564,a,559,a)]. 5.43/5.67 565 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,B,C)) = c_Lattices_Osemilattice__sup__class_Osup(A,B,C) # label(fact_sup__left__idem) # label(axiom). [clausify(383)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) | -class_Lattices_Olattice(B). [resolve(565,a,545,b)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B)) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B). [resolve(565,a,564,a)]. 5.43/5.67 566 class_Lattices_Osemilattice__sup(tc_Nat_Onat) # label(arity_Nat__Onat__Lattices_Osemilattice__sup) # label(axiom). [assumption]. 5.43/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B)). [resolve(566,a,546,a)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,C),B). [resolve(566,a,547,a)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,C,B)). [resolve(566,a,548,a)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,A) = B. [resolve(566,a,549,a)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,C)) = c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,C)). [resolve(566,a,550,a)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B),C) = c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,C)). [resolve(566,a,551,a)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B) = c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,A). [resolve(566,a,552,a)]. 5.43/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,A)). [resolve(566,a,553,a)]. 5.43/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,C,A),B). [resolve(566,a,556,a)]. 5.43/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,C),B). [resolve(566,a,557,a)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,A) = A. [resolve(566,a,558,a)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B) = B. [resolve(566,a,559,a)]. 5.43/5.67 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B)) = c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B). [resolve(566,a,565,a)]. 5.43/5.67 567 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,D),c_Lattices_Osemilattice__sup__class_Osup(A,C,E)) # label(fact_sup__mono) # label(axiom). [clausify(400)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),E,F) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,E),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,F)) | -class_Lattices_Olattice(B). [resolve(567,a,545,b)]. 5.43/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,C,D) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C),c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,D)). [resolve(567,a,564,a)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,C),c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,D)). [resolve(567,a,566,a)]. 5.43/5.69 568 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) = C # label(fact_le__iff__sup) # label(axiom). [clausify(413)]. 5.43/5.69 569 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) != C # label(fact_le__iff__sup) # label(axiom). [clausify(413)]. 5.43/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) != D | -class_Lattices_Olattice(B). [resolve(569,a,545,b)]. 5.43/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) != B. [resolve(569,a,564,a)]. 5.43/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,A,B) != B. [resolve(569,a,566,a)]. 5.43/5.69 570 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,B) = B # label(fact_sup_Oidem) # label(axiom). [clausify(416)]. 5.43/5.69 571 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,D)) # label(fact_less__supI1) # label(axiom). [clausify(421)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless(tc_fun(A,B),C,D) | c_Orderings_Oord__class_Oless(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,E)) | -class_Lattices_Olattice(B). [resolve(571,a,545,b)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless(tc_HOL_Obool,A,B) | c_Orderings_Oord__class_Oless(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,C)). [resolve(571,a,564,a)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,C)). [resolve(571,a,566,a)]. 5.43/5.69 572 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,C,D)) # label(fact_le__supI1) # label(axiom). [clausify(450)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,E)) | -class_Lattices_Olattice(B). [resolve(572,a,545,b)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,C)). [resolve(572,a,564,a)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,B,C)). [resolve(572,a,566,a)]. 5.43/5.69 573 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,B,C)) = c_Lattices_Osemilattice__sup__class_Osup(A,B,C) # label(fact_sup_Oleft__idem) # label(axiom). [clausify(470)]. 5.43/5.69 574 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) = c_Lattices_Osemilattice__sup__class_Osup(A,C,B) # label(fact_sup_Ocommute) # label(axiom). [clausify(493)]. 5.43/5.69 575 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__sup__class_Osup(A,D,C)) # label(fact_le__supI2) # label(axiom). [clausify(498)]. 5.43/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),E,D)) | -class_Lattices_Olattice(B). [resolve(575,a,545,b)]. 9.93/10.14 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,C,B)). [resolve(575,a,564,a)]. 9.93/10.14 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat,C,B)). [resolve(575,a,566,a)]. 9.93/10.14 576 c_Order__Relation_Owell__order__on(A,B,C) | -c_Wellfounded_Owf(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) | -c_Order__Relation_Olinear__order__on(A,B,C) # label(fact_well__order__on__def) # label(axiom). [clausify(59)]. 9.93/10.14 577 -c_Order__Relation_Owell__order__on(A,B,C) | c_Wellfounded_Owf(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) # label(fact_well__order__on__def) # label(axiom). [clausify(59)]. 9.93/10.14 578 -c_Order__Relation_Owell__order__on(A,B,C) | c_Order__Relation_Olinear__order__on(A,B,C) # label(fact_well__order__on__def) # label(axiom). [clausify(59)]. 9.93/10.14 579 c_Order__Relation_Owell__order__on(A,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(A,A),tc_HOL_Obool))) # label(fact_well__order__on__empty) # label(axiom). [clausify(467)]. 9.93/10.14 Derived: c_Wellfounded_Owf(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(A,A),tc_HOL_Obool)),c_Relation_OId(A))). [resolve(579,a,577,a)]. 9.93/10.14 Derived: c_Order__Relation_Olinear__order__on(A,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(A,A),tc_HOL_Obool))). [resolve(579,a,578,a)]. 9.93/10.14 580 -c_Relation_Oirrefl(A,B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),B) # label(fact_irrefl__def) # label(axiom). [clausify(317)]. 9.93/10.14 581 c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),B,c_Relation_OId(A))) # label(fact_irrefl__diff__Id) # label(axiom). [clausify(118)]. 9.93/10.14 Derived: -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,B,B),c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))). [resolve(580,a,581,a)]. 9.93/10.14 582 c_Relation_Oirrefl(A,B) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f60(B,A),f60(B,A)),B) # label(fact_irrefl__def) # label(axiom). [clausify(317)]. 9.93/10.14 Derived: c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f60(B,A),f60(B,A)),B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),B). [resolve(582,a,580,a)]. 9.93/10.14 583 c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),B,c_Relation_OId(A))) | -c_Order__Relation_Olinear__order__on(A,C,B). [resolve(533,a,534,b)]. 9.93/10.14 584 -class_Lattices_Obounded__lattice__bot(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Orderings_Obot__class_Obot(A)) = B # label(fact_sup__bot__right) # label(axiom). [clausify(278)]. 9.93/10.14 585 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__bot(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom). [clausify(140)]. 9.93/10.14 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,c_Orderings_Obot__class_Obot(tc_fun(A,B))) = C | -class_Lattices_Obounded__lattice(B). [resolve(584,a,585,b)]. 9.93/10.14 586 class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 9.93/10.14 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = A. [resolve(586,a,584,a)]. 9.93/10.14 587 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != B | c_Orderings_Obot__class_Obot(A) != C | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) = c_Orderings_Obot__class_Obot(A) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(297)]. 9.93/10.14 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != D | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) = c_Orderings_Obot__class_Obot(tc_fun(A,B)) | -class_Lattices_Obounded__lattice(B). [resolve(587,a,585,b)]. 15.53/15.77 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != B | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = c_Orderings_Obot__class_Obot(tc_HOL_Obool). [resolve(587,a,586,a)]. 15.53/15.77 588 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = B | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) != c_Orderings_Obot__class_Obot(A) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(297)]. 15.53/15.77 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) != c_Orderings_Obot__class_Obot(tc_fun(A,B)) | -class_Lattices_Obounded__lattice(B). [resolve(588,a,585,b)]. 15.53/15.77 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) != c_Orderings_Obot__class_Obot(tc_HOL_Obool). [resolve(588,a,586,a)]. 15.53/15.77 589 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = B | c_Lattices_Osemilattice__sup__class_Osup(A,C,B) != c_Orderings_Obot__class_Obot(A) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(297)]. 15.53/15.77 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,C) != c_Orderings_Obot__class_Obot(tc_fun(A,B)) | -class_Lattices_Obounded__lattice(B). [resolve(589,a,585,b)]. 15.53/15.77 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A) != c_Orderings_Obot__class_Obot(tc_HOL_Obool). [resolve(589,a,586,a)]. 15.53/15.77 590 -class_Lattices_Obounded__lattice__bot(A) | c_Lattices_Osemilattice__sup__class_Osup(A,c_Orderings_Obot__class_Obot(A),B) = B # label(fact_sup__bot__left) # label(axiom). [clausify(379)]. 15.53/15.77 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),c_Orderings_Obot__class_Obot(tc_fun(A,B)),C) = C | -class_Lattices_Obounded__lattice(B). [resolve(590,a,585,b)]. 15.53/15.77 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Orderings_Obot__class_Obot(tc_HOL_Obool),A) = A. [resolve(590,a,586,a)]. 15.53/15.77 591 c_Nitpick_Orefl_H(A,B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f85(B,A),f85(B,A)),B) # label(fact_Nitpick_Orefl_H__def) # label(axiom). [clausify(473)]. 15.53/15.77 592 -c_Nitpick_Orefl_H(A,B) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),B) # label(fact_Nitpick_Orefl_H__def) # label(axiom). [clausify(473)]. 15.53/15.77 Derived: -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f85(B,A),f85(B,A)),B) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),B). [resolve(591,a,592,a)]. 15.53/15.77 15.53/15.77 ============================== end predicate elimination ============= 15.53/15.77 15.53/15.77 Auto_denials: (non-Horn, no changes). 15.53/15.77 15.53/15.77 Term ordering decisions: 15.53/15.77 Function symbol KB weights: tc_Arrow__Order__Mirabelle_Oalt=1. v_L=1. v_y=1. v_x=1. tc_HOL_Obool=1. tc_Nat_Onat=1. c_Arrow__Order__Mirabelle_OLin=1. c_Wellfounded_Oless__than=1. c_Wellfounded_Opred__nat=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. tc_prod=1. tc_fun=1. hAPP=1. c_Transitive__Closure_Ortrancl=1. c_Transitive__Closure_Otrancl=1. c_Relation_OField=1. c_Wellfounded_Oacc=1. c_Relation_OId__on=1. c_Arrow__Order__Mirabelle_Omkbot=1. c_Arrow__Order__Mirabelle_Omktop=1. c_Set_Othe__elem=1. f4=1. f11=1. f23=1. f24=1. f25=1. f26=1. f27=1. f28=1. f37=1. f38=1. f39=1. f40=1. f54=1. f60=1. f61=1. f63=1. f68=1. f69=1. f76=1. f78=1. f85=1. f90=1. c_Orderings_Obot__class_Obot=1. c_Relation_OId=1. c_Wellfounded_Omeasure=1. c_fequal=1. c_Lattices_Osemilattice__sup__class_Osup=1. c_Set_Oinsert=1. c_Groups_Ominus__class_Ominus=1. c_Relation_Oconverse=1. c_Lattices_Osemilattice__inf__class_Oinf=1. c_Relation_ODomain=1. c_Relation_ORange=1. c_Arrow__Order__Mirabelle_Obelow=1. c_Arrow__Order__Mirabelle_Oabove=1. c_Relation_Oinv__image=1. c_FunDef_Oin__rel=1. c_Wellfounded_Omlex__prod=1. f3=1. f12=1. f15=1. f30=1. f31=1. f32=1. f41=1. f42=1. f51=1. f52=1. f53=1. f65=1. f66=1. f67=1. f83=1. f86=1. f87=1. c_Product__Type_OPair=1. c_Relation_OImage=1. c_Product__Type_OSigma=1. c_Wellfounded_Olex__prod=1. c_Recdef_Owfrec=1Cputime limit exceeded (core dumped) 300.00/300.22 EOF