0.03/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n006.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 960 0.12/0.33 % WCLimit : 120 0.12/0.33 % DateTime : Tue Aug 9 05:41:43 EDT 2022 0.12/0.34 % CPUTime : 1.12/1.43 ============================== Prover9 =============================== 1.12/1.43 Prover9 (32) version 2009-11A, November 2009. 1.12/1.43 Process 21682 was started by sandbox on n006.cluster.edu, 1.12/1.43 Tue Aug 9 05:41:45 2022 1.12/1.43 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 960 -f /tmp/Prover9_21434_n006.cluster.edu". 1.12/1.43 ============================== end of head =========================== 1.12/1.43 1.12/1.43 ============================== INPUT ================================= 1.12/1.43 1.12/1.43 % Reading from file /tmp/Prover9_21434_n006.cluster.edu 1.12/1.43 1.12/1.43 set(prolog_style_variables). 1.12/1.43 set(auto2). 1.12/1.43 % set(auto2) -> set(auto). 1.12/1.43 % set(auto) -> set(auto_inference). 1.12/1.43 % set(auto) -> set(auto_setup). 1.12/1.43 % set(auto_setup) -> set(predicate_elim). 1.12/1.43 % set(auto_setup) -> assign(eq_defs, unfold). 1.12/1.43 % set(auto) -> set(auto_limits). 1.12/1.43 % set(auto_limits) -> assign(max_weight, "100.000"). 1.12/1.43 % set(auto_limits) -> assign(sos_limit, 20000). 1.12/1.43 % set(auto) -> set(auto_denials). 1.12/1.43 % set(auto) -> set(auto_process). 1.12/1.43 % set(auto2) -> assign(new_constants, 1). 1.12/1.43 % set(auto2) -> assign(fold_denial_max, 3). 1.12/1.43 % set(auto2) -> assign(max_weight, "200.000"). 1.12/1.43 % set(auto2) -> assign(max_hours, 1). 1.12/1.43 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.12/1.43 % set(auto2) -> assign(max_seconds, 0). 1.12/1.43 % set(auto2) -> assign(max_minutes, 5). 1.12/1.43 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.12/1.43 % set(auto2) -> set(sort_initial_sos). 1.12/1.43 % set(auto2) -> assign(sos_limit, -1). 1.12/1.43 % set(auto2) -> assign(lrs_ticks, 3000). 1.12/1.43 % set(auto2) -> assign(max_megs, 400). 1.12/1.43 % set(auto2) -> assign(stats, some). 1.12/1.43 % set(auto2) -> clear(echo_input). 1.12/1.43 % set(auto2) -> set(quiet). 1.12/1.43 % set(auto2) -> clear(print_initial_clauses). 1.12/1.43 % set(auto2) -> clear(print_given). 1.12/1.43 assign(lrs_ticks,-1). 1.12/1.43 assign(sos_limit,10000). 1.12/1.43 assign(order,kbo). 1.12/1.43 set(lex_order_vars). 1.12/1.43 clear(print_given). 1.12/1.43 1.12/1.43 % formulas(sos). % not echoed (527 formulas) 1.12/1.43 1.12/1.43 ============================== end of input ========================== 1.12/1.43 1.12/1.43 % From the command line: assign(max_seconds, 960). 1.12/1.43 1.12/1.43 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.12/1.43 1.12/1.43 % Formulas that are not ordinary clauses: 1.12/1.43 1 (all V_x all V_z all V_y all T_a (class_Lattices_Odistrib__lattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z),V_x) = c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_x),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_z,V_x)))) # label(fact_inf__sup__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 2 (all V_A_2 all T_a c_Relation_ORange(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2) # label(fact_Range__Id__on) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 3 (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_Otrans(T_a,V_r_2) & c_Relation_Orefl__on(T_a,V_A_2,V_r_2))) # label(fact_preorder__on__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 4 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_b all T_a (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) <-> V_a_H_2 = V_a_2 & V_b_2 = V_b_H_2)) # label(fact_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 5 (all V_x all V_y all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y) = V_y))) # label(fact_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 6 (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]. 1.12/1.43 7 (all V_B_2 all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_B_2)) # label(fact_Un__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 8 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> V_x = c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Orderings_Otop__class_Otop(T_a),V_x))) # label(fact_inf__top__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 9 (all V_A_2 all V_B_2 all V_x_2 all T_a (c_member(T_a,V_x_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_x_2,V_A_2),V_B_2))) # label(fact_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 10 (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]. 1.12/1.43 11 (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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> V_A_2 = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 12 (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_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,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_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)))))))))) # label(fact_subset__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 13 (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,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]. 1.12/1.43 14 (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]. 1.12/1.43 15 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) = hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2))) # label(fact_inf__apply) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 16 (all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_2,V_A_2) <-> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 17 (all V_R_2 all V_x_2 all T_a (-c_member(T_a,V_x_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_x_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]. 1.12/1.43 18 (all V_y_2 all V_x_2 all V_L_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_2,c_Arrow__Order__Mirabelle_OLin) -> (V_y_2 != V_x_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_y_2,V_x_2),V_L_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_x_2,V_y_2),V_L_2))))) # label(fact_notin__Lin__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 19 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)))))) # label(fact_inf__greatest) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 20 (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]. 1.12/1.43 21 (all V_y_2 all V_x_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_member(T_a,V_x_2,V_A_2) -> (c_member(T_a,V_y_2,V_A_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),V_r_2) <-> c_Equiv__Relations_Oquotient(T_a,c_Set_Oinsert(T_a,V_y_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_r_2) = c_Equiv__Relations_Oquotient(T_a,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_r_2)))))) # label(fact_eq__equiv__class__iff2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 22 (all V_x_2 all V_Fa_2 all V_R_2 all T_a all T_b hAPP(hAPP(V_Fa_2,c_Recdef_Ocut(T_b,T_a,c_Nitpick_Owf__wfrec(T_b,T_a,V_R_2,V_Fa_2),V_R_2,V_x_2)),V_x_2) = c_Nitpick_Owf__wfrec_H(T_b,T_a,V_R_2,V_Fa_2,V_x_2)) # label(fact_Nitpick_Owf__wfrec_H__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 23 (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]. 1.12/1.43 24 (all V_B_2 all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2)) # label(fact_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 25 (all V_x_2 all T_a c_Set_Othe__elem(T_a,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_x_2) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 26 (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]. 1.12/1.43 27 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_x = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y)))) # label(fact_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 28 (all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_R_2) = 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))) # label(fact_rtrancl__reflcl__absorb) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 29 (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_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_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__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 30 (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)) -> (V_a_2 = V_b_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_Refl__antisym__eq__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 31 (all V_B_2 all V_A_2 all T_a 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_A_2,V_B_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Un__Diff__Int) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 32 (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]. 1.12/1.43 33 (all V_x_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_x_2),hAPP(V_B_2,V_x_2)) = hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_x_2))) # label(fact_minus__apply) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 34 (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]. 1.12/1.43 35 (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]. 1.12/1.43 36 (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_y = V_x)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 37 (all V_a_2 all T_a all V_A_2 (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> -c_member(T_a,V_a_2,V_A_2))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 38 (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]. 1.12/1.43 39 (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]. 1.12/1.43 40 (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]. 1.12/1.43 41 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(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_B_2,V_C_2)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2)) = 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),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2))) # label(fact_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 42 (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]. 1.12/1.43 43 (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 & 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)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 44 (all V_x_2 all T_b all T_a (class_Orderings_Otop(T_a) -> hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_b,T_a)),V_x_2) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_top__apply) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 45 (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]. 1.12/1.43 46 (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]. 1.12/1.43 47 (all V_r_2 all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)) = c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_quotient__is__empty2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 48 (all T_2 all T_1 (class_Orderings_Otop(T_1) -> class_Orderings_Otop(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Otop) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 49 (all V_a_2 all V_r_2 all T_b all V_b_2 all T_a (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) <-> 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))))))) # label(fact_Image__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 50 (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,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)))) # label(fact_Diff__partition) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 51 (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]. 1.12/1.43 52 (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]. 1.12/1.43 53 (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]. 1.12/1.43 54 (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]. 1.12/1.43 55 (all V_b_2 all V_a_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_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_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)))))))) # label(fact_equiv__class__subset) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 56 (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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_A_2)) # label(fact_Int__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 57 (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]. 1.12/1.43 58 (all V_A_2 all V_B_2 all T_a 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) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) # label(fact_Un__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 59 (all V_A_2 all V_x_2 all T_a (-c_member(T_a,V_x_2,V_A_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_x_2,V_A_2),c_Set_Oinsert(T_a,V_x_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]. 1.12/1.43 60 (all V_f_2 all T_a c_Wellfounded_Owf(T_a,c_Wellfounded_Omeasure(T_a,V_f_2))) # label(fact_wf__measure) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 61 (all V_x_2 all T_a c_member(T_a,V_x_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_iso__tuple__UNIV__I) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 62 (all V_y_2 all V_x_2 all T_a (V_x_2 = V_y_2 <-> c_member(T_a,V_x_2,c_fequal(V_y_2)))) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 63 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 64 (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]. 1.12/1.43 65 (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]. 1.12/1.43 66 (all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> V_x = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_x))) # label(fact_inf__idem) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 67 (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_B_2))) # label(fact_IntD2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 68 (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)) <-> (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)))))) # label(fact_acc_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 69 (all T_a all V_B_2 all V_x_2 all V_A_2 (hBOOL(hAPP(V_A_2,V_x_2)) -> (hBOOL(hAPP(V_B_2,V_x_2)) -> hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2))))) # label(fact_inf1I) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 70 (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))) # label(fact_IntD1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 71 (all V_y_2 all V_x_2 (hBOOL(hAPP(c_fequal(V_x_2),V_y_2)) | V_y_2 != V_x_2)) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 72 (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]. 1.12/1.43 73 (all V_r_2 all T_a c_Relation_Orefl__on(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) # label(fact_refl__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 74 (all V_R_2 all V_y_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_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_y_2),V_R_2) & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,B_z),c_Transitive__Closure_Ortrancl(T_a,V_R_2)))))) # label(fact_tranclD2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 75 (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]. 1.12/1.43 76 (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]. 1.12/1.43 77 (all V_b all V_x all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),V_x)))) # label(fact_le__infI1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 78 (all V_r_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_r_2) = 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))) # label(fact_reflcl__trancl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 79 (all T_a c_Relation_ORange(T_a,T_a,c_Relation_OId(T_a)) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Range__Id) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 80 (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_Relation_ORange(T_b,T_a,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2)),c_Lattices_Osemilattice__inf__class_Oinf(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)))) # label(fact_Range__Int__subset) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 81 (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]. 1.12/1.43 82 (all V_A_2 all V_C_2 all V_B_2 all T_a 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_B_2,V_A_2),c_Lattices_Osemilattice__inf__class_Oinf(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),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2),V_A_2)) # label(fact_Int__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 83 (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__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) = 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)))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 84 (all V_x_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_x_2),V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_a_2),V_r_2)))) # label(fact_wf__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 85 (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]. 1.12/1.43 86 (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) <-> (all B_x c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x)))))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 87 (all V_Y_2 all V_X_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_member(tc_fun(T_a,tc_HOL_Obool),V_X_2,c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)) -> (c_member(tc_fun(T_a,tc_HOL_Obool),V_Y_2,c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_X_2,V_Y_2) | V_Y_2 = V_X_2)))) # label(fact_quotient__disj) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 88 (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]. 1.12/1.43 89 (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]. 1.12/1.43 90 (all V_b_2 all V_a_2 all V_x_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_member(T_a,V_x_2,c_Lattices_Osemilattice__inf__class_Oinf(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_a_2,V_b_2),V_r_2)))) # label(fact_equiv__class__nondisjoint) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 91 (all V_z_2 all V_L_2 all V_y_2 all V_x_2 ((V_z_2 = V_x_2 -> V_y_2 != V_x_2) & (V_z_2 != V_x_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_x_2,V_y_2),V_L_2)) & V_y_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_x_2,V_y_2),c_Arrow__Order__Mirabelle_Omkbot(V_L_2,V_z_2)))) # label(fact_in__mkbot) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 92 (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]. 1.12/1.43 93 (all V_y_2 all V_x_2 all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> (c_Orderings_Obot__class_Obot(T_a) = V_x_2 & V_y_2 = c_Orderings_Obot__class_Obot(T_a) <-> c_Orderings_Obot__class_Obot(T_a) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x_2,V_y_2)))) # label(fact_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 94 (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]. 1.12/1.43 95 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)))) # label(fact_inf__assoc) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 96 (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]. 1.12/1.43 97 (all T_a c_Relation_Otrans(T_a,c_Relation_OId(T_a))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 98 (all V_y_2 all V_z_2 all V_f_2 all T_b all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_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_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_y_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_x_2),V_y_2))))) # label(fact_fold__graph_H_Ointros_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 99 (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]. 1.12/1.43 100 (all V_A_2 all V_y_2 all V_x_2 all T_a (c_member(T_a,V_x_2,V_A_2) & V_y_2 = V_x_2 <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Relation_OId__on(T_a,V_A_2)))) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 101 (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]. 1.12/1.43 102 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(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) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Int__assoc) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 103 (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_xa,B_x),V_r_2) | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_xa),V_r_2)))))))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 104 (all V_y_2 all V_x_2 all V_L_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_L_2,c_Arrow__Order__Mirabelle_OLin) -> (V_x_2 != V_y_2 & (V_b_2 != V_x_2 -> (V_b_2 != V_y_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_x_2,V_y_2),V_L_2)) & (V_y_2 = V_b_2 -> V_a_2 = V_x_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_x_2,V_a_2),V_L_2))) & (V_x_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_y_2),V_L_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_x_2,V_y_2),c_Arrow__Order__Mirabelle_Oabove(V_L_2,V_a_2,V_b_2)))))) # label(fact_in__above) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 105 (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]. 1.12/1.43 106 (all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = V_A_2) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 107 (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_za_2,V_zb_2) != c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_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]. 1.12/1.43 108 (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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_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))) # label(fact_Int__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 109 (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_Order__Relation_Olinear__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))))) # label(fact_well__order__on__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 110 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,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__inf__class_Oinf(T_a,V_y,V_z)))) # label(fact_inf__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 111 (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]. 1.12/1.43 112 (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]. 1.12/1.43 113 (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]. 1.12/1.43 114 (all V_A_2 all V_b_2 all V_a_2 all V_r_2 all T_a (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_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2) -> (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_eq__equiv__class) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 115 (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__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]. 1.12/1.43 116 (all T_a c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) = c_Transitive__Closure_Otrancl(T_a,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]. 1.12/1.43 117 (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]. 1.12/1.43 118 (all T_b all V_a_2 all V_x_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_x_2,V_a_2),B_R) -> hAPP(B_f,V_x_2) = hAPP(c_Recdef_Ocut(T_a,T_b,B_f,B_R,V_a_2),V_x_2))) # label(fact_tfl__cut__apply) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 119 (all V_r_2 all V_x_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_x_2),V_r_2) -> c_member(T_a,B_y,c_Wellfounded_Oacc(T_a,V_r_2)))) -> c_member(T_a,V_x_2,c_Wellfounded_Oacc(T_a,V_r_2)))) # label(fact_acc_OaccI) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 120 (all V_x_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_x_2)) -> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_inf1D1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 121 (all V_x_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_x_2)) -> hBOOL(hAPP(V_B_2,V_x_2)))) # label(fact_inf1D2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 122 (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,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)))) # label(fact_sup__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 123 (all V_B_2 all V_A_2 all T_a ((all B_x (c_member(T_a,B_x,V_A_2) -> (all B_xa (c_member(T_a,B_xa,V_B_2) -> B_x != B_xa)))) <-> c_Orderings_Obot__class_Obot(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))) # label(fact_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 124 (all T_a c_Relation_OField(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(T_a,tc_HOL_Obool))) # label(fact_Field__empty) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 125 (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]. 1.12/1.43 126 (all V_x_2 all V_A_2 all V_y_2 all T_a (hBOOL(hAPP(c_Set_Oinsert(T_a,V_y_2,V_A_2),V_x_2)) <-> hBOOL(hAPP(V_A_2,V_x_2)) | V_x_2 = V_y_2)) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 127 (all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = V_A_2) # label(fact_Int__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 128 (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_Groups_Ominus__class_Ominus(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__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 129 (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]. 1.12/1.43 130 (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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_A_2)) # label(fact_Int__lower1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 131 (all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_R_2) = c_Transitive__Closure_Otrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2))) # label(fact_rtrancl__trancl__absorb) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 132 (all V_f_2 all V_r_2 all T_b all V_y_2 all V_x_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_x_2),hAPP(V_f_2,V_y_2)),V_r_2) <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),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]. 1.12/1.43 133 (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]. 1.12/1.43 134 (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_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_total__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 135 (all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_R_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,c_Relation_OId(T_a)))) # label(fact_rtrancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 136 (all V_x_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_x_2),V_r_2) -> -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_a_2),V_r_2)))) # label(fact_wf__asym) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 137 (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]. 1.12/1.43 138 (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_Relation_ODomain(T_a,T_b,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,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_ODomain(T_a,T_b,V_A_2),c_Relation_ODomain(T_a,T_b,V_B_2)))) # label(fact_Domain__Int__subset) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 139 (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]. 1.12/1.43 140 (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]. 1.12/1.43 141 (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_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 142 (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]. 1.12/1.43 143 (all V_L_2 all V_y_2 all V_x_2 (V_x_2 != V_y_2 -> (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_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_L_2,V_x_2,V_y_2),c_Arrow__Order__Mirabelle_OLin)))) # label(fact_below__Lin) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 144 (all V_R_2 all V_y_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_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,V_x_2,B_z),V_R_2) & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)))))) # label(fact_tranclD) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 145 (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_Int__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 146 (all V_B_2 all T_a all V_x_2 all V_A_2 (hBOOL(hAPP(V_A_2,V_x_2)) -> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)))) # label(fact_sup1I1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 147 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> c_Transitive__Closure_Otrancl(T_a,V_r_2) = V_r_2)) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 148 (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]. 1.12/1.43 149 (all V_B_2 all V_A_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_Un__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 150 (all V_x_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_member(T_a,V_x_2,c_Wellfounded_Oacc(T_a,V_r_2)))) # label(fact_acc__wfD) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 151 (all V_r_2 all T_a c_Relation_ORange(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ORange(T_a,T_a,V_r_2)) # label(fact_trancl__range) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 152 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_sup__top__right) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 153 (all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2 <-> 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]. 1.12/1.43 154 (all V_x_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_x_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)) | c_Set_Oinsert(T_a,V_x_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]. 1.12/1.43 155 (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]. 1.12/1.43 156 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> V_x = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)))) # label(fact_inf__top__right) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 157 (all V_B_2 all V_A_2 all T_a 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_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 158 (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]. 1.12/1.43 159 (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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_B_2)) # label(fact_Int__lower2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 160 (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]. 1.12/1.43 161 (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]. 1.12/1.43 162 (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]. 1.12/1.43 163 (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]. 1.12/1.43 164 (all V_B_2 all T_a V_B_2 = 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)) # label(fact_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 165 (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]. 1.12/1.43 166 (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]. 1.12/1.43 167 (all V_R_2 all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ORange(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2))) # label(fact_Range__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 168 (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]. 1.12/1.43 169 (all T_a all V_R_2 all V_y_2 all V_y_H_2 all T_b all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_x_2)) -> (c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_y_H_2,V_y_2),hAPP(V_R_2,V_x_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_x_2,V_y_H_2),c_Product__Type_OPair(T_a,T_b,V_x_2,V_y_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]. 1.12/1.43 170 (all V_A_2 all V_x_2 all T_a c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Set_Oinsert(T_a,V_x_2,V_A_2)) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 171 (all V_B_2 all V_C_2 all V_a_2 all T_a ((c_member(T_a,V_a_2,V_C_2) -> c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_B_2),V_C_2)) & (-c_member(T_a,V_a_2,V_C_2) -> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)))) # label(fact_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 172 (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]. 1.12/1.43 173 (all V_r_H_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_Wellfounded_Owf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_r_H_2)))) # label(fact_wf__Int1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 174 (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]. 1.12/1.43 175 (all V_A_2 all V_a_2 all T_a 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))))) = c_Set_Oinsert(T_a,V_a_2,V_A_2)) # label(fact_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 176 (all V_y_2 all V_x_2 all V_R_2 all T_b all T_a (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_x_2,V_y_2),V_R_2) <-> c_FunDef_Oin__rel(T_a,T_b,V_R_2,V_x_2,V_y_2))) # label(fact_in__rel__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 177 (all V_a2_2 all V_a1_2 all V_Fa_2 all V_R_2 all T_b all T_a (c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,V_a1_2,hAPP(hAPP(V_Fa_2,V_a2_2),V_a1_2)) <-> (exists B_g ((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_Fa_2,B_z,hAPP(B_g,B_z)))) & hAPP(hAPP(V_Fa_2,B_g),V_a1_2) = hAPP(hAPP(V_Fa_2,V_a2_2),V_a1_2))))) # label(fact_wfrec__rel_Oequations) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 178 (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]. 1.12/1.43 179 (all V_P_2 all V_B_2 all V_A_2 all T_a ((exists 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)))) <-> (exists B_x (hBOOL(hAPP(V_P_2,B_x)) & c_member(T_a,B_x,V_B_2))) | (exists B_x (c_member(T_a,B_x,V_A_2) & hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_bex__Un) # label(axiom) # label(non_clause). [assumption]. 1.12/1.43 180 (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]. 1.12/1.43 181 (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]. 1.12/1.43 182 (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]. 1.12/1.43 183 (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]. 1.12/1.44 184 (all V_y_2 all V_x_2 all V_L_2 all V_b_2 all V_a_2 (V_a_2 != V_b_2 -> (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_2,c_Arrow__Order__Mirabelle_OLin) -> ((V_a_2 != V_y_2 -> (V_x_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_b_2,V_y_2),V_L_2) | V_y_2 = V_b_2) & (V_x_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_x_2,V_y_2),V_L_2))) & (V_y_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_x_2,V_b_2),V_L_2)) & V_x_2 != V_y_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_x_2,V_y_2),c_Arrow__Order__Mirabelle_Obelow(V_L_2,V_a_2,V_b_2)))))) # label(fact_in__below) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 185 (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]. 1.12/1.44 186 (all V_B_2 all V_A_2 all V_x_2 all T_a (-c_member(T_a,V_x_2,V_A_2) -> (-c_member(T_a,V_x_2,V_B_2) -> (c_Set_Oinsert(T_a,V_x_2,V_A_2) = c_Set_Oinsert(T_a,V_x_2,V_B_2) <-> V_B_2 = V_A_2)))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 187 (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]. 1.12/1.44 188 (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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2))))) # label(fact_Int__mono) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 189 (all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) # label(fact_Int__commute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 190 (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]. 1.12/1.44 191 (all V_x_2 all V_L_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_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_L_2,V_x_2),c_Arrow__Order__Mirabelle_OLin))) # label(fact_mkbot__Lin) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 192 (all V_s_2 all V_r_2 all T_a 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)) = 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))) # label(fact_Field__Un) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 193 (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]. 1.12/1.44 194 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y))) # label(fact_inf__sup__aci_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 195 (all V_a2_2 all V_a1_2 all V_Fa_2 all V_R_2 all T_b all T_a (c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,V_a1_2,V_a2_2) <-> (exists B_g ((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_Fa_2,B_z,hAPP(B_g,B_z)))) & hAPP(hAPP(V_Fa_2,B_g),V_a1_2) = V_a2_2)))) # label(fact_wfrec__rel_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 196 (all V_B_2 all V_A_2 all V_C_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_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_Lattices_Osemilattice__inf__class_Oinf(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_C_2,V_B_2))) # label(fact_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 197 (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]. 1.12/1.44 198 (all V_r_2 all T_a c_Relation_Orefl__on(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),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_refl__reflcl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 199 (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]. 1.12/1.44 200 (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]. 1.12/1.44 201 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Orderings_Obot__class_Obot(T_a)) = V_x)) # label(fact_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 202 (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]. 1.12/1.44 203 (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]. 1.12/1.44 204 (all V_A_2 all T_a all V_y_2 all V_x_2 (V_x_2 = V_y_2 -> (c_member(T_a,V_x_2,V_A_2) <-> c_member(T_a,V_y_2,V_A_2)))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 205 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(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_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Diff__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 206 (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]. 1.12/1.44 207 (all V_R_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_R_2) = c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Otrancl(T_a,V_R_2))) # label(fact_trancl__rtrancl__absorb) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 208 (all V_r_2 all V_x_2 all V_y_2 all T_a (c_Wellfounded_Oacyclic(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_x_2),V_r_2)) <-> c_Wellfounded_Oacyclic(T_a,V_r_2) & -c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))) # label(fact_acyclic__insert) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 209 (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]. 1.12/1.44 210 (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]. 1.12/1.44 211 (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]. 1.12/1.44 212 (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),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__inf__class_Oinf(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_A_2,V_C_2))) # label(fact_Diff__Un) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 213 (all V_a_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_member(T_a,V_a_2,V_A_2) -> c_member(T_a,V_a_2,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_equiv__class__self) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 214 (all V_y_2 all V_x_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_x_2,V_y_2),V_r_2) -> V_x_2 != V_y_2))) # label(fact_irrefl__trancl__rD) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 215 (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]. 1.12/1.44 216 (all V_z all V_y all V_x all T_a (class_Lattices_Odistrib__lattice(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_sup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 217 (all V_y_2 all V_x_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_member(T_a,V_x_2,V_A_2) -> (c_member(T_a,V_y_2,V_A_2) -> (c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_y_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_x_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_x_2,V_y_2),V_r_2)))))) # label(fact_eq__equiv__class__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 218 (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) = V_B_2 <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 219 (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]. 1.12/1.44 220 (all V_b_2 all V_a_2 all T_a (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_a_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]. 1.12/1.44 221 (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]. 1.12/1.44 222 (all V_x_2 all V_y_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_y_2 = V_x_2)))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 223 (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]. 1.12/1.44 224 (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__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]. 1.12/1.44 225 (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_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]. 1.12/1.44 226 (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]. 1.12/1.44 227 (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]. 1.12/1.44 228 (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]. 1.12/1.44 229 (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]. 1.12/1.44 230 (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]. 1.12/1.44 231 (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]. 1.12/1.44 232 (all V_A_2 all V_r_2 all T_b all V_b_2 all T_a ((exists B_x (c_member(T_b,B_x,V_A_2) & 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_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]. 1.12/1.44 233 (all V_g_2 all V_Fa_2 all T_b all V_R_2 all V_x_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_x_2),V_R_2) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,B_z,hAPP(V_g_2,B_z)))) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,V_x_2,hAPP(hAPP(V_Fa_2,V_g_2),V_x_2)))) # label(fact_wfrec__rel_Ointros) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 234 (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]. 1.12/1.44 235 (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,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]. 1.12/1.44 236 (all V_x_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_x_2,V_A_2) -> c_member(T_a,V_x_2,V_B_2)))) # label(fact_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 237 (all V_R_2 all T_a c_Relation_ODomain(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 238 (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]. 1.12/1.44 239 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Orderings_Otop__class_Otop(T_a),V_x) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_sup__top__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 240 (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]. 1.12/1.44 241 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> V_x = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_inf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 242 (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]. 1.12/1.44 243 (all V_y_2 all V_x_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_x_2,V_y_2),V_r_2) -> c_member(T_a,V_x_2,V_A_2)))) # label(fact_refl__onD1) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 244 (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]. 1.12/1.44 245 (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]. 1.12/1.44 246 (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]. 1.12/1.44 247 (all V_y_2 all V_x_2 all T_a (class_Lattices_Osemilattice__sup(T_a) -> (V_y_2 = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)))) # label(fact_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 248 (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]. 1.12/1.44 249 (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]. 1.12/1.44 250 (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]. 1.12/1.44 251 (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]. 1.12/1.44 252 (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]. 1.12/1.44 253 (all V_z_2 all V_y_2 all V_x_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_x_2,V_y_2),V_r_2) -> (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_x_2,V_z_2),V_r_2) -> V_y_2 = V_z_2)))) # label(fact_single__valuedD) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 254 (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]. 1.12/1.44 255 (all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 256 (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]. 1.12/1.44 257 (all V_B_2 all V_C_2 all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Lattices_Osemilattice__inf__class_Oinf(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),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),V_B_2)) # label(fact_Diff__Int2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 258 (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) -> V_y = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 259 (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]. 1.12/1.44 260 (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]. 1.12/1.44 261 (all V_g_2 all V_f_2 ((all B_x hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)) -> V_g_2 = V_f_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 262 (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]. 1.12/1.44 263 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)))) # label(fact_inf__sup__aci_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 264 (all V_a all V_x all V_b all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),V_x)))) # label(fact_le__infI2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 265 (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]. 1.12/1.44 266 (all T_a all V_B_2 all 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) <-> V_A_2 = V_B_2)) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 267 (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]. 1.12/1.44 268 (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]. 1.12/1.44 269 (all V_L_2 all V_y_2 all V_x_2 (V_x_2 != V_y_2 -> (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_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_Oabove(V_L_2,V_x_2,V_y_2),c_Arrow__Order__Mirabelle_OLin)))) # label(fact_above__Lin) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 270 (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]. 1.12/1.44 271 (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]. 1.12/1.44 272 (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]. 1.12/1.44 273 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_subset__UNIV) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 274 (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]. 1.12/1.44 275 (all B_x (c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,B_x)) & c_Relation_Ototal__on(tc_Arrow__Order__Mirabelle_Oalt,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(v_P,B_x)) & c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,B_x)))) -> c_Relation_Ototal__on(tc_Arrow__Order__Mirabelle_Oalt,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),v_F(v_P)) & c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) & c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) # label(conj_2) # label(hypothesis) # label(non_clause). [assumption]. 1.12/1.44 276 (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]. 1.12/1.44 277 (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]. 1.12/1.44 278 (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]. 1.12/1.44 279 (all V_A_2 all V_S_2 all V_R_2 all T_a all T_b 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)) = 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)) # label(fact_Un__Image) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 280 (all V_r_2 all T_a c_Transitive__Closure_Ortrancl(T_a,V_r_2) = c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2))) # label(fact_rtrancl__idemp) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 281 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_x) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y))) # label(fact_inf__commute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 282 (all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_a) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b))) # label(fact_sup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 283 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_x) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y))) # label(fact_inf__sup__aci_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 284 (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 != V_b_H))) # label(fact_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 285 (all V_r_2 all T_a c_Equiv__Relations_Oquotient(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool))) # label(fact_quotient__empty) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 286 (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]. 1.12/1.44 287 (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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__inf__class_Oinf(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_A_2,V_C_2))) # label(fact_Un__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 288 (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]. 1.12/1.44 289 (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]. 1.12/1.44 290 (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_c = V_b -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 291 (all T_a all V_A_2 all V_x_2 all V_B_2 ((-hBOOL(hAPP(V_B_2,V_x_2)) -> hBOOL(hAPP(V_A_2,V_x_2))) -> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)))) # label(fact_sup1CI) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 292 (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]. 1.12/1.44 293 (all V_z_2 all V_r_2 all V_y_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_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_y_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_x_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))))) # label(fact_rtrancl__trancl__trancl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 294 (all T_a c_Relation_Orefl__on(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),c_Relation_OId(T_a))) # label(fact_refl__Id) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 295 (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]. 1.12/1.44 296 (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]. 1.12/1.44 297 (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]. 1.12/1.44 298 (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]. 1.12/1.44 299 (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]. 1.12/1.44 300 (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 (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)) <-> 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) | 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))) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 301 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,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__inf__class_Oinf(T_a,V_y,V_z)))) # label(fact_inf__sup__aci_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 302 (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]. 1.12/1.44 303 (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]. 1.12/1.44 304 (all V_B_2 all V_C_2 all V_a_2 all T_a (-c_member(T_a,V_a_2,V_C_2) -> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 305 (all V_A_2 all V_b_2 all V_a_2 all T_a (V_a_2 = V_b_2 | c_member(T_a,V_a_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]. 1.12/1.44 306 (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]. 1.12/1.44 307 (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_b_2 = V_a_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 308 (all T_a c_Relation_Oantisym(T_a,c_Relation_OId(T_a))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 309 (all V_C_2 all V_B_2 all V_a_2 all T_a 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)) = 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)) # label(fact_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 310 (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]. 1.12/1.44 311 (all V_g_2 all V_x_2 all V_r_2 all V_f_2 all T_b 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_x_2),V_r_2) -> hAPP(V_f_2,B_y) = hAPP(V_g_2,B_y))) <-> c_Recdef_Ocut(T_a,T_b,V_g_2,V_r_2,V_x_2) = c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_x_2))) # label(fact_cuts__eq) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 312 (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]. 1.12/1.44 313 (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]. 1.12/1.44 314 (all V_y_2 all V_R_2 all V_x_2 all T_a (-c_member(T_a,V_x_2,c_Relation_ODomain(T_a,T_a,V_R_2)) -> (V_x_2 = V_y_2 <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))))) # label(fact_Not__Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 315 (all T_a all T_b c_Relation_ORange(T_b,T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Range__empty) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 316 (all V_A_2 all V_C_2 all V_B_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),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_C_2,V_A_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_B_2,V_C_2),V_A_2)) # label(fact_Un__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 317 (all V_B_2 all V_x_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_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2) -> (c_member(T_a,V_x_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_x_2,V_B_2))))) # label(fact_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 318 (all V_c all V_a all V_b all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,V_c)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_c)))) # label(fact_inf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 319 (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_x = B_y))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 320 (all V_d all V_b all V_c all V_a all T_a (class_Lattices_Osemilattice__inf(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__inf__class_Oinf(T_a,V_a,V_b),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_c,V_d)))))) # label(fact_inf__mono) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 321 (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__inf__class_Oinf(T_a,V_x,V_y)) = V_x)) # label(fact_sup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 322 (all V_A_2 all V_B_2 all V_x_2 all T_a ((-c_member(T_a,V_x_2,V_B_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_x_2,V_A_2),V_B_2) = c_Set_Oinsert(T_a,V_x_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) & (c_member(T_a,V_x_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_x_2,V_A_2),V_B_2)))) # label(fact_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 323 (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]. 1.12/1.44 324 (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_A_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)))))) # label(fact_ball__Un) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 325 (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]. 1.12/1.44 326 (all V_z_2 all V_y_2 all V_x_2 all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_z_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y_2,V_z_2))))) # label(fact_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 327 (all V_x_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_x_2)) -> -(hBOOL(hAPP(V_A_2,V_x_2)) -> -hBOOL(hAPP(V_B_2,V_x_2))))) # label(fact_inf1E) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 328 (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]. 1.12/1.44 329 (all V_b_2 all V_a_2 all 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),c_Relation_OId(T_a)))) # label(fact_pair__in__Id__conv) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 330 (all V_z_2 all V_y_2 all V_x_2 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_x_2,V_y_2),V_z_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_z_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_z_2)))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 331 (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,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_c))) # label(fact_sup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 332 (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]. 1.12/1.44 333 (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]. 1.12/1.44 334 (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_c_2 = V_d_2 <-> V_b_2 = V_a_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 335 (all V_z_2 all V_L_2 all V_y_2 all V_x_2 (V_x_2 != V_z_2 & (V_y_2 = V_z_2 -> V_y_2 != V_x_2) & (V_y_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_x_2,V_y_2),V_L_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_x_2,V_y_2),c_Arrow__Order__Mirabelle_Omktop(V_L_2,V_z_2)))) # label(fact_in__mktop) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 336 (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]. 1.12/1.44 337 (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]. 1.12/1.44 338 (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]. 1.12/1.44 339 (all V_A_2 all T_a all V_x_2 all V_B_2 (hBOOL(hAPP(V_B_2,V_x_2)) -> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)))) # label(fact_sup1I2) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 340 (all V_r_2 all V_z_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_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_x_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_x_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]. 1.12/1.44 341 (all V_B_2 all V_A_2 all V_a_2 all T_a ((c_member(T_a,V_a_2,V_A_2) -> c_Lattices_Osemilattice__inf__class_Oinf(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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) & (-c_member(T_a,V_a_2,V_A_2) -> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 342 (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]. 1.12/1.44 343 (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),V_r_2)) <-> c_Relation_Oirrefl(T_a,V_r_2))) # label(fact_irrefl__def) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 344 (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]. 1.12/1.44 345 (all V_S_2 all V_R_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),V_x_2,c_Transitive__Closure_Ortrancl(T_a,V_S_2)) | c_member(tc_prod(T_a,T_a),V_x_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) -> c_member(tc_prod(T_a,T_a),V_x_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]. 1.12/1.44 346 (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]. 1.12/1.44 347 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b)) -> -(c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a) -> -c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b))))) # label(fact_le__infE) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 348 (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]. 1.12/1.44 349 (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]. 1.12/1.44 350 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_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__inf__class_Oinf(T_a,V_a,V_b)))))) # label(fact_le__infI) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 351 (all V_B_2 all V_x_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_x_2,V_B_2)) <-> (-c_member(T_a,V_x_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_x_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_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2)))) # label(fact_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.12/1.44 352 (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]. 1.12/1.44 353 (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]. 1.17/1.44 354 (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]. 1.17/1.44 355 (all V_y all V_x 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__eq(T_a,V_x,V_y))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 1.17/1.44 356 (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]. 1.17/1.44 357 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (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)))) <-> V_d_2 = V_a_2 & V_c_2 = V_b_2 | V_b_2 = V_d_2 & V_c_2 = V_a_2)) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.44 358 (all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.17/1.44 359 (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]. 1.17/1.44 360 (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]. 1.17/1.44 361 (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]. 1.17/1.44 362 (all V_y_2 all V_x_2 all T_a (class_Lattices_Osemilattice__inf(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_x_2 = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x_2,V_y_2)))) # label(fact_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 1.17/1.44 363 (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]. 1.17/1.44 364 (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]. 1.17/1.44 365 (all V_x_2 all T_a hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_x_2))) # label(fact_top1I) # label(axiom) # label(non_clause). [assumption]. 1.17/1.44 366 (all V_B_2 all V_C_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_C_2) -> c_Lattices_Osemilattice__inf__class_Oinf(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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)))) # label(fact_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 367 (all V_r_2 all T_b all T_a (V_r_2 = 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)) = c_Relation_ODomain(T_a,T_b,V_r_2))) # label(fact_Domain__empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 368 (all V_x_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_x_2),hAPP(V_g_2,V_x_2))))) # label(fact_le__funD) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 369 (all V_x_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_x_2,V_A_2) -> c_member(T_a,V_x_2,V_B_2)))) # label(fact_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 370 (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]. 1.17/1.45 371 (all V_r_2 all V_x_2 all V_y_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_x_2,V_y_2),c_Transitive__Closure_Ortrancl(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_y_2,V_x_2),V_r_2)))) # label(fact_wf__insert) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 372 (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]. 1.17/1.45 373 (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]. 1.17/1.45 374 (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]. 1.17/1.45 375 (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]. 1.17/1.45 376 (all V_Q_2 all T_a all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_x_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_x_2))))) # label(fact_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 377 (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 ((exists B_x exists B_y (hAPP(hAPP(V_a3_2,B_x),B_y) = hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) & 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))) | 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 <-> 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)))) # label(fact_fold__graph_H_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 378 (all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = V_A_2) # label(fact_Int__absorb) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 379 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(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_C_2)) = c_Groups_Ominus__class_Ominus(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_Int__Diff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 380 (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_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]. 1.17/1.45 381 (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]. 1.17/1.45 382 (all V_y_2 all V_x_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_x_2,V_y_2),V_r_2) -> c_member(T_a,V_y_2,V_A_2)))) # label(fact_refl__onD2) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 383 (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]. 1.17/1.45 384 (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]. 1.17/1.45 385 (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]. 1.17/1.45 386 (all T_2 all T_1 (class_Lattices_Odistrib__lattice(T_1) -> class_Lattices_Odistrib__lattice(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Odistrib__lattice) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 387 (all V_x_2 all V_L_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_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_L_2,V_x_2),c_Arrow__Order__Mirabelle_OLin))) # label(fact_mktop__Lin) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 388 (all V_r_2 all T_b all V_a_2 all T_a ((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)) <-> c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)))) # label(fact_Domain__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 389 (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]. 1.17/1.45 390 (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_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]. 1.17/1.45 391 (all V_x all V_z all V_y all T_a (class_Lattices_Odistrib__lattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_x),c_Lattices_Osemilattice__sup__class_Osup(T_a,V_z,V_x)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z),V_x))) # label(fact_sup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 392 (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]. 1.17/1.45 393 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y)))) # label(fact_inf__left__idem) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 394 (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]. 1.17/1.45 395 (all B_x (c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,B_x)) & c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,B_x)) & c_Relation_Ototal__on(tc_Arrow__Order__Mirabelle_Oalt,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(v_P,B_x)))) -> (all B_a all B_b (B_a != B_b -> (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_a,B_b),hAPP(v_P,v_i)) -> 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_a,B_b),v_F(v_P))))) # label(conj_3) # label(hypothesis) # label(non_clause). [assumption]. 1.17/1.45 396 (all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,V_a) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b))) # label(fact_inf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 397 (all T_a c_Relation_ODomain(T_a,T_a,c_Relation_OId(T_a)) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Domain__Id) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 398 (all V_r_H_2 all V_r_2 all T_a (c_Wellfounded_Owf(T_a,V_r_2) -> c_Wellfounded_Owf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_H_2,V_r_2)))) # label(fact_wf__Int2) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 399 (all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) # label(fact_Int__left__absorb) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 400 (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]. 1.17/1.45 401 (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]. 1.17/1.45 402 (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]. 1.17/1.45 403 (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]. 1.17/1.45 404 (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]. 1.17/1.45 405 (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]. 1.17/1.45 406 (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]. 1.17/1.45 407 (all V_z_2 all V_y_2 all V_x_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_x_2,V_y_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_x_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_y_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_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))))) # label(fact_single__valued__confluent) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 408 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_x = V_y -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 409 (all V_A_2 all T_a V_A_2 = c_Relation_ODomain(T_a,T_a,c_Relation_OId__on(T_a,V_A_2))) # label(fact_Domain__Id__on) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 410 (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]. 1.17/1.45 411 (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]. 1.17/1.45 412 (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]. 1.17/1.45 413 (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_x = B_y)))) -> 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]. 1.17/1.45 414 (all V_A_2 all V_y_2 all V_x_2 all T_a c_Set_Oinsert(T_a,V_y_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_y_2,V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 415 (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_B_2 = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Int__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 416 (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]. 1.17/1.45 417 (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))) # label(fact_equalityD2) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 418 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2))) # label(fact_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 419 (all B_x (c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,B_x)) & c_Relation_Ototal__on(tc_Arrow__Order__Mirabelle_Oalt,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(v_P,B_x)) & c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,B_x)))) # label(conj_0) # label(hypothesis) # label(non_clause). [assumption]. 1.17/1.45 420 (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_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]. 1.17/1.45 421 (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]. 1.17/1.45 422 (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]. 1.17/1.45 423 (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]. 1.17/1.45 424 (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]. 1.17/1.45 425 (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_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]. 1.17/1.45 426 (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),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__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 427 (all V_y_2 all V_x_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_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),V_r_2) <-> c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_x_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_y_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) & c_member(T_a,V_y_2,V_A_2) & c_member(T_a,V_x_2,V_A_2)))) # label(fact_equiv__class__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 428 (all V_b_2 all V_a_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_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_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))))))) # label(fact_equiv__class__eq) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 429 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Orderings_Obot__class_Obot(T_a)) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 430 (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]. 1.17/1.45 431 (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]. 1.17/1.45 432 (all V_x_2 all T_b all T_a (class_Orderings_Obot(T_a) -> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_x_2) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 433 (all V_B_2 all V_A_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_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))))) # label(fact_Int__greatest) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 434 (all V_R_2 all V_y_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) <-> V_y_2 = V_x_2 | V_x_2 != V_y_2 & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)))) # label(fact_rtrancl__eq__or__trancl) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 435 (all V_x all T_a (class_Orderings_Otop(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)))) # label(fact_top__greatest) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 436 (all V_f_2 all T_b all V_r_2 all V_a_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_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_x_2) = hAPP(V_f_2,V_x_2))) # label(fact_cut__apply) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 437 (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]. 1.17/1.45 438 (all V_B_2 all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_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_x_2,V_B_2)))) # label(fact_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 439 (all V_b_2 all V_a_2 all V_c_2 all T_a all T_c all T_b hAPP(hAPP(V_c_2,V_a_2),V_b_2) = 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))) # label(fact_internal__split__conv) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 440 (all V_x_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_x_2),hAPP(V_g_2,V_x_2))))) # label(fact_le__funE) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 441 (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]. 1.17/1.45 442 (all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_UNIV__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 443 (all V_x_2 all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> (c_member(T_a,V_x_2,V_B_2) <-> c_member(T_a,V_x_2,V_A_2)))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 444 (all V_B_2 all V_A_2 all V_x_2 all T_a (-c_member(T_a,V_x_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_x_2,V_B_2))))) # label(fact_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 445 (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]. 1.17/1.45 446 (all V_B_2 all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> c_Lattices_Osemilattice__inf__class_Oinf(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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 447 (all V_r_2 all T_a all T_b (c_Relation_ORange(T_b,T_a,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> 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]. 1.17/1.45 448 (all T_a c_Transitive__Closure_Ortrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Relation_OId(T_a)) # label(fact_rtrancl__empty) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 449 (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]. 1.17/1.45 450 (all V_y_2 all V_x_2 all T_a (class_Lattices_Obounded__lattice__top(T_a) -> (V_y_2 = c_Orderings_Otop__class_Otop(T_a) & c_Orderings_Otop__class_Otop(T_a) = V_x_2 <-> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x_2,V_y_2) = c_Orderings_Otop__class_Otop(T_a)))) # label(fact_inf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 451 (all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_a) = V_a)) # label(fact_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 452 (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]. 1.17/1.45 453 (all V_A_2 all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_Un__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 454 (all V_x_2 all T_a c_member(T_a,V_x_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_UNIV__I) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 455 (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_Relation_Oantisym(T_a,V_r_2) & c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2))) # label(fact_partial__order__on__def) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 456 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_a (class_Lattices_Olattice(T_a) -> hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2) = c_Lattices_Osemilattice__sup__class_Osup(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 457 (all V_c all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,V_c)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),V_c))) # label(fact_inf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 458 (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]. 1.17/1.45 459 (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]. 1.17/1.45 460 (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) -> V_A_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))))) # label(fact_double__diff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 461 (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]. 1.17/1.45 462 (all V_z all V_y all V_x all T_a (class_Lattices_Odistrib__lattice(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_inf__sup__distrib1) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 463 (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_x,V_y)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y))) # label(fact_sup__left__idem) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 464 (all T_2 all T_1 (class_Lattices_Obounded__lattice(T_1) -> class_Lattices_Obounded__lattice__top(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 465 (all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Diff__UNIV) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 466 (all V_B_2 all V_A_2 all V_a_2 all T_a (-c_member(T_a,V_a_2,V_A_2) -> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 467 (all V_r_2 all T_a c_Relation_OField(T_a,V_r_2) = 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))) # label(fact_Field__def) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 468 (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) <-> 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)) = V_A_2)) # label(fact_Un__empty) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 469 (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_b_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)) -> -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]. 1.17/1.45 470 (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]. 1.17/1.45 471 (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]. 1.17/1.45 472 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Orderings_Obot__class_Obot(T_a),V_x) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 473 (all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b)))) # label(fact_inf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 474 (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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2))))) # label(fact_refl__on__Int) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 475 (all V_B_2 all V_A_2 all V_x_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_x_2,V_A_2),V_B_2) <-> c_member(T_a,V_x_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]. 1.17/1.45 476 (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]. 1.17/1.45 477 (all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2 <-> (all B_x -c_member(T_a,B_x,V_A_2)))) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 478 (all V_x_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_x_2)) -> (-hBOOL(hAPP(V_A_2,V_x_2)) -> hBOOL(hAPP(V_B_2,V_x_2))))) # label(fact_sup1E) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 479 (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__inf__class_Oinf(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_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 480 (all V_r_2 all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2 <-> c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_quotient__is__empty) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 481 (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]. 1.17/1.45 482 (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]. 1.17/1.45 483 (all V_s_2 all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> (c_Relation_Otrans(T_a,V_s_2) -> c_Relation_Otrans(T_a,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2))))) # label(fact_trans__Int) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 484 (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]. 1.17/1.45 485 (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]. 1.17/1.45 486 (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_y = B_z)))))) # label(fact_single__valued__def) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 487 (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]. 1.17/1.45 488 (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]. 1.17/1.45 489 (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_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 490 (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_b_2 != V_a_2 -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 491 (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,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),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__assoc) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 492 (all V_r_2 all V_z_2 all V_x_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) -> (V_z_2 != V_x_2 -> -(all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_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]. 1.17/1.45 493 (all V_x_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_x_2)) -> hBOOL(hAPP(V_Q_2,V_x_2))))) # label(fact_predicate1D) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 494 (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]. 1.17/1.45 495 (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]. 1.17/1.45 496 (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]. 1.17/1.45 497 (all V_c_2 all T_a all V_B_2 all V_A_2 (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)) -> -(-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]. 1.17/1.45 498 (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]. 1.17/1.45 499 (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]. 1.17/1.45 500 (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_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_A_2,V_C_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Diff__Int) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 501 (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]. 1.17/1.45 502 (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_b_2 = V_a_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 503 (all V_b_2 all V_a_2 all V_L_2 (c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_L_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_L_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_L_2)))) # label(fact_Lin__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 504 (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]. 1.17/1.45 505 (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]. 1.17/1.45 506 (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]. 1.17/1.45 507 (all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> V_a = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_a))) # label(fact_inf_Oidem) # label(axiom) # label(non_clause). [assumption]. 1.17/1.45 508 (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]. 1.17/1.45 509 (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_y = B_z)))) -> c_Relation_Osingle__valued(T_a,T_b,V_r_2))) # label(fact_single__valuedI) # label(axiom) # label(non_clause). [assumption]. 1.17/1.46 510 (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]. 1.17/1.46 511 (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]. 1.17/1.46 512 (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]. 1.17/1.46 1.17/1.46 ============================== end of process non-clausal formulas === 1.17/1.46 1.17/1.46 ============================== PROCESS INITIAL CLAUSES =============== 1.17/1.46 1.17/1.46 ============================== PREDICATE ELIMINATION ================= 1.17/1.46 513 -class_Lattices_Olattice(A) | class_Lattices_Osemilattice__inf(tc_fun(B,A)) # label(arity_fun__Lattices_Osemilattice__inf) # label(axiom). [clausify(238)]. 1.17/1.46 514 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Lattices_Osemilattice__inf__class_Oinf(A,C,B) = B # label(fact_inf__absorb2) # label(axiom). [clausify(5)]. 1.17/1.46 515 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,D) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) # label(fact_inf__greatest) # label(axiom). [clausify(19)]. 1.17/1.46 516 -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_inf__absorb1) # label(axiom). [clausify(27)]. 1.17/1.46 517 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,B) = B # label(fact_inf__idem) # label(axiom). [clausify(66)]. 1.17/1.46 518 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__inf__class_Oinf(A,B,D),C) # label(fact_le__infI1) # label(axiom). [clausify(77)]. 1.17/1.46 519 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,c_Lattices_Osemilattice__inf__class_Oinf(A,B,C),D) = c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) # label(fact_inf__assoc) # label(axiom). [clausify(95)]. 1.17/1.46 520 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) = c_Lattices_Osemilattice__inf__class_Oinf(A,C,c_Lattices_Osemilattice__inf__class_Oinf(A,B,D)) # label(fact_inf__left__commute) # label(axiom). [clausify(110)]. 1.17/1.46 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),D,C) = C. [resolve(513,b,514,a)]. 1.17/1.46 Derived: -class_Lattices_Olattice(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,E) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,E)). [resolve(513,b,515,a)]. 1.17/1.46 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(513,b,516,a)]. 1.17/1.46 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,C) = C. [resolve(513,b,517,a)]. 1.17/1.47 Derived: -class_Lattices_Olattice(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,E),D). [resolve(513,b,518,a)]. 1.17/1.47 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D),E) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,E)). [resolve(513,b,519,a)]. 1.17/1.47 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,E)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,E)). [resolve(513,b,520,a)]. 1.17/1.47 521 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__inf__class_Oinf(A,D,B),C) # label(fact_le__infI2) # label(axiom). [clausify(264)]. 1.17/1.47 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__inf__class_Oinf(tc_fun(A,B),E,C),D) | -class_Lattices_Olattice(B). [resolve(521,a,513,b)]. 1.17/1.47 522 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) = c_Lattices_Osemilattice__inf__class_Oinf(A,C,B) # label(fact_inf__commute) # label(axiom). [clausify(281)]. 1.17/1.47 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,D) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),D,C) | -class_Lattices_Olattice(B). [resolve(522,a,513,b)]. 1.17/1.47 523 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) = c_Lattices_Osemilattice__inf__class_Oinf(A,C,c_Lattices_Osemilattice__inf__class_Oinf(A,B,D)) # label(fact_inf_Oleft__commute) # label(axiom). [clausify(318)]. 1.17/1.47 524 -class_Lattices_Osemilattice__inf(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__inf__class_Oinf(A,B,D),c_Lattices_Osemilattice__inf__class_Oinf(A,C,E)) # label(fact_inf__mono) # label(axiom). [clausify(320)]. 1.17/1.47 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__inf__class_Oinf(tc_fun(A,B),C,E),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),D,F)) | -class_Lattices_Olattice(B). [resolve(524,a,513,b)]. 1.17/1.47 525 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,D) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) # label(fact_le__inf__iff) # label(axiom). [clausify(326)]. 1.17/1.47 526 -class_Lattices_Osemilattice__inf(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) # label(fact_le__inf__iff) # label(axiom). [clausify(326)]. 1.17/1.47 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__inf__class_Oinf(tc_fun(A,B),D,E)) | -class_Lattices_Olattice(B). [resolve(526,a,513,b)]. 1.17/1.47 527 -class_Lattices_Osemilattice__inf(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,D,C)) # label(fact_le__inf__iff) # label(axiom). [clausify(326)]. 1.17/1.47 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__inf__class_Oinf(tc_fun(A,B),E,D)) | -class_Lattices_Olattice(B). [resolve(527,a,513,b)]. 1.17/1.47 528 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) | c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_le__infE) # label(axiom). [clausify(347)]. 1.17/1.47 529 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) | c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_le__infE) # label(axiom). [clausify(347)]. 1.17/1.47 530 -class_Lattices_Osemilattice__inf(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,D) | c_Orderings_Oord__class_Oless__eq(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) # label(fact_le__infI) # label(axiom). [clausify(350)]. 1.17/1.47 531 -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(362)]. 1.17/1.47 532 -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(362)]. 1.17/1.47 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,D) != C | -class_Lattices_Olattice(B). [resolve(532,a,513,b)]. 1.17/1.47 533 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,B,C)) = c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) # label(fact_inf__left__idem) # label(axiom). [clausify(393)]. 1.17/1.47 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,D)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,D) | -class_Lattices_Olattice(B). [resolve(533,a,513,b)]. 1.17/1.47 534 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) = c_Lattices_Osemilattice__inf__class_Oinf(A,C,B) # label(fact_inf_Ocommute) # label(axiom). [clausify(396)]. 1.17/1.47 535 -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(402)]. 1.17/1.47 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,D),C) | -class_Lattices_Olattice(B). [resolve(535,a,513,b)]. 1.17/1.47 536 class_Lattices_Osemilattice__inf(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Osemilattice__inf) # label(axiom). [assumption]. 1.17/1.47 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,A) = A. [resolve(536,a,514,a)]. 1.17/1.47 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,C) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,C)). [resolve(536,a,515,a)]. 1.17/1.47 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(536,a,516,a)]. 1.17/1.47 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,A) = A. [resolve(536,a,517,a)]. 1.17/1.47 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,C),B). [resolve(536,a,518,a)]. 1.17/1.47 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),C) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,C)). [resolve(536,a,519,a)]. 1.17/1.47 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,C)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,C)). [resolve(536,a,520,a)]. 1.17/1.47 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,C,A),B). [resolve(536,a,521,a)]. 1.17/1.47 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,A). [resolve(536,a,522,a)]. 1.17/1.47 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__inf__class_Oinf(tc_HOL_Obool,A,C),c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,D)). [resolve(536,a,524,a)]. 1.17/1.50 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__inf__class_Oinf(tc_HOL_Obool,B,C)). [resolve(536,a,526,a)]. 1.17/1.50 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__inf__class_Oinf(tc_HOL_Obool,C,B)). [resolve(536,a,527,a)]. 1.17/1.50 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(536,a,532,a)]. 1.17/1.50 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B). [resolve(536,a,533,a)]. 1.17/1.50 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),A). [resolve(536,a,535,a)]. 1.17/1.50 537 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,c_Lattices_Osemilattice__inf__class_Oinf(A,B,C),D) = c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,C,D)) # label(fact_inf_Oassoc) # label(axiom). [clausify(457)]. 1.17/1.50 538 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Lattices_Osemilattice__inf__class_Oinf(A,B,C)) = c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) # label(fact_inf_Oleft__idem) # label(axiom). [clausify(473)]. 1.17/1.50 539 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,B) = B # label(fact_inf_Oidem) # label(axiom). [clausify(507)]. 1.17/1.50 540 -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(508)]. 1.17/1.50 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,D),D) | -class_Lattices_Olattice(B). [resolve(540,a,513,b)]. 1.17/1.50 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),B). [resolve(540,a,536,a)]. 1.17/1.50 541 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__top(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom). [clausify(464)]. 1.17/1.50 542 -class_Lattices_Obounded__lattice__top(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,c_Orderings_Otop__class_Otop(A),B) = B # label(fact_inf__top__left) # label(axiom). [clausify(8)]. 1.17/1.50 543 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Orderings_Otop__class_Otop(A)) # label(fact_sup__top__right) # label(axiom). [clausify(152)]. 1.17/1.50 544 -class_Lattices_Obounded__lattice__top(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Orderings_Otop__class_Otop(A)) = B # label(fact_inf__top__right) # label(axiom). [clausify(156)]. 1.17/1.50 545 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = c_Lattices_Osemilattice__sup__class_Osup(A,c_Orderings_Otop__class_Otop(A),B) # label(fact_sup__top__left) # label(axiom). [clausify(239)]. 1.17/1.50 546 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) != B | c_Orderings_Otop__class_Otop(A) != C | c_Orderings_Otop__class_Otop(A) = c_Lattices_Osemilattice__inf__class_Oinf(A,C,B) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(450)]. 1.17/1.50 547 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Orderings_Otop__class_Otop(A) != c_Lattices_Osemilattice__inf__class_Oinf(A,C,B) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(450)]. 1.17/1.50 548 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Orderings_Otop__class_Otop(A) != c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(450)]. 1.17/1.50 Derived: -class_Lattices_Obounded__lattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),c_Orderings_Otop__class_Otop(tc_fun(B,A)),C) = C. [resolve(541,b,542,a)]. 1.17/1.54 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,c_Orderings_Otop__class_Otop(tc_fun(B,A))). [resolve(541,b,543,a)]. 1.17/1.54 Derived: -class_Lattices_Obounded__lattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,c_Orderings_Otop__class_Otop(tc_fun(B,A))) = C. [resolve(541,b,544,a)]. 1.17/1.54 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),c_Orderings_Otop__class_Otop(tc_fun(B,A)),C). [resolve(541,b,545,a)]. 1.17/1.54 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != C | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != D | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,C). [resolve(541,b,546,a)]. 1.17/1.54 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = C | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,C). [resolve(541,b,547,a)]. 1.17/1.54 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = C | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D). [resolve(541,b,548,a)]. 1.17/1.54 549 class_Lattices_Obounded__lattice__top(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__top) # label(axiom). [assumption]. 1.17/1.54 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,c_Orderings_Otop__class_Otop(tc_HOL_Obool),A) = A. [resolve(549,a,542,a)]. 1.17/1.54 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Orderings_Otop__class_Otop(tc_HOL_Obool)). [resolve(549,a,543,a)]. 1.17/1.54 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Orderings_Otop__class_Otop(tc_HOL_Obool)) = A. [resolve(549,a,544,a)]. 1.17/1.54 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Orderings_Otop__class_Otop(tc_HOL_Obool),A). [resolve(549,a,545,a)]. 1.17/1.54 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) != A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != B | c_Orderings_Otop__class_Otop(tc_HOL_Obool) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,A). [resolve(549,a,546,a)]. 1.17/1.54 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,A). [resolve(549,a,547,a)]. 1.17/1.54 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B). [resolve(549,a,548,a)]. 1.17/1.54 550 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != B | c_Orderings_Obot__class_Obot(A) != C | c_Orderings_Obot__class_Obot(A) = c_Lattices_Osemilattice__sup__class_Osup(A,B,C) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(93)]. 1.17/1.54 551 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__bot(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom). [clausify(26)]. 1.17/1.54 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != D | c_Orderings_Obot__class_Obot(tc_fun(A,B)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) | -class_Lattices_Obounded__lattice(B). [resolve(550,a,551,b)]. 1.17/1.54 552 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = B | c_Orderings_Obot__class_Obot(A) != c_Lattices_Osemilattice__sup__class_Osup(A,B,C) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(93)]. 1.17/1.54 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) | -class_Lattices_Obounded__lattice(B). [resolve(552,a,551,b)]. 1.17/1.54 553 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = B | c_Orderings_Obot__class_Obot(A) != c_Lattices_Osemilattice__sup__class_Osup(A,C,B) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(93)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),D,C) | -class_Lattices_Obounded__lattice(B). [resolve(553,a,551,b)]. 1.17/1.56 554 -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(201)]. 1.17/1.56 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(554,a,551,b)]. 1.17/1.56 555 class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != B | c_Orderings_Obot__class_Obot(tc_HOL_Obool) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B). [resolve(555,a,550,a)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B). [resolve(555,a,552,a)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A). [resolve(555,a,553,a)]. 1.17/1.56 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = A. [resolve(555,a,554,a)]. 1.17/1.56 556 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Orderings_Obot__class_Obot(A)) # label(fact_inf__bot__right) # label(axiom). [clausify(429)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,c_Orderings_Obot__class_Obot(tc_fun(A,B))) | -class_Lattices_Obounded__lattice(B). [resolve(556,a,551,b)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Orderings_Obot__class_Obot(tc_HOL_Obool)). [resolve(556,a,555,a)]. 1.17/1.56 557 -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(430)]. 1.17/1.56 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(557,a,551,b)]. 1.17/1.56 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Orderings_Obot__class_Obot(tc_HOL_Obool),A) = A. [resolve(557,a,555,a)]. 1.17/1.56 558 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = c_Lattices_Osemilattice__inf__class_Oinf(A,c_Orderings_Obot__class_Obot(A),B) # label(fact_inf__bot__left) # label(axiom). [clausify(472)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),c_Orderings_Obot__class_Obot(tc_fun(A,B)),C) | -class_Lattices_Obounded__lattice(B). [resolve(558,a,551,b)]. 1.17/1.56 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,c_Orderings_Obot__class_Obot(tc_HOL_Obool),A). [resolve(558,a,555,a)]. 1.17/1.56 559 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(65)]. 1.17/1.56 560 -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(65)]. 1.17/1.56 561 -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(65)]. 1.17/1.56 562 -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(65)]. 1.17/1.59 563 -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(259)]. 1.17/1.59 Derived: -c_Order__Relation_Olinear__order__on(A,B,C) | 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))). [resolve(563,b,561,a)]. 1.17/1.59 Derived: -c_Order__Relation_Olinear__order__on(A,B,C) | c_Relation_Otrans(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))). [resolve(563,b,562,a)]. 1.17/1.59 564 -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(99)]. 1.17/1.59 565 -class_Lattices_Olattice(A) | class_Lattices_Osemilattice__sup(tc_fun(B,A)) # label(arity_fun__Lattices_Osemilattice__sup) # label(axiom). [clausify(72)]. 1.17/1.59 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(564,a,565,b)]. 1.17/1.59 566 -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(122)]. 1.17/1.59 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(566,a,565,b)]. 1.17/1.59 567 -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(162)]. 1.17/1.59 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(567,a,565,b)]. 1.17/1.59 568 class_Lattices_Osemilattice__sup(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Osemilattice__sup) # label(axiom). [assumption]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A). [resolve(568,a,564,a)]. 1.17/1.59 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(568,a,566,a)]. 1.17/1.59 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(568,a,567,a)]. 1.17/1.59 569 -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(202)]. 1.17/1.59 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D)) | -class_Lattices_Olattice(B). [resolve(569,a,565,b)]. 1.17/1.59 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B)). [resolve(569,a,568,a)]. 1.17/1.59 570 -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(211)]. 1.17/1.59 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(570,a,565,b)]. 1.17/1.59 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(570,a,568,a)]. 1.17/1.59 571 -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(242)]. 1.17/1.59 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(571,a,565,b)]. 1.17/1.59 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(571,a,568,a)]. 1.17/1.59 572 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) != C | c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_le__iff__sup) # label(axiom). [clausify(247)]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) != D | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -class_Lattices_Olattice(B). [resolve(572,a,565,b)]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) != B | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B). [resolve(572,a,568,a)]. 1.17/1.59 573 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,C) = C | -c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_le__iff__sup) # label(axiom). [clausify(247)]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D) = D | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -class_Lattices_Olattice(B). [resolve(573,a,565,b)]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = B | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B). [resolve(573,a,568,a)]. 1.17/1.59 574 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,B) = B # label(fact_sup__idem) # label(axiom). [clausify(254)]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,C) = C | -class_Lattices_Olattice(B). [resolve(574,a,565,b)]. 1.17/1.59 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,A) = A. [resolve(574,a,568,a)]. 1.17/1.59 575 -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(258)]. 1.17/1.59 576 -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(260)]. 1.17/1.59 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(576,a,565,b)]. 1.17/1.59 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(576,a,568,a)]. 1.17/1.59 577 -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(282)]. 1.17/1.59 578 -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(302)]. 1.17/1.59 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(578,a,565,b)]. 1.17/1.59 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(578,a,568,a)]. 1.17/1.60 579 -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(310)]. 1.17/1.60 580 -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__sup__iff) # label(axiom). [clausify(330)]. 1.17/1.60 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D),E) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,E) | -class_Lattices_Olattice(B). [resolve(580,a,565,b)]. 1.17/1.60 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B),C) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,C). [resolve(580,a,568,a)]. 1.17/1.60 581 -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__sup__iff) # label(axiom). [clausify(330)]. 1.17/1.60 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D),E) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),D,E) | -class_Lattices_Olattice(B). [resolve(581,a,565,b)]. 1.17/1.60 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B),C) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,B,C). [resolve(581,a,568,a)]. 1.17/1.60 582 -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) | -c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_le__sup__iff) # label(axiom). [clausify(330)]. 1.17/1.60 Derived: c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,D),E) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,E) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),D,E) | -class_Lattices_Olattice(B). [resolve(582,a,565,b)]. 1.17/1.60 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B),C) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,C) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,B,C). [resolve(582,a,568,a)]. 1.17/1.60 583 -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(331)]. 1.17/1.60 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(583,a,565,b)]. 1.17/1.60 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(583,a,568,a)]. 1.17/1.60 584 -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(375)]. 1.17/1.60 585 -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(411)]. 1.17/1.60 586 -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(411)]. 1.39/1.67 587 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,B) = B # label(fact_sup_Oidem) # label(axiom). [clausify(451)]. 1.39/1.67 588 -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(463)]. 1.39/1.67 589 -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(491)]. 1.39/1.67 590 -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(510)]. 1.39/1.67 591 -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(512)]. 1.39/1.67 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(591,a,565,b)]. 1.39/1.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A)). [resolve(591,a,568,a)]. 1.39/1.67 592 c_Order__Relation_Owell__order__on(A,B,C) | -c_Order__Relation_Olinear__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(109)]. 1.39/1.67 593 -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(109)]. 1.39/1.67 594 -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(109)]. 1.39/1.67 595 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(288)]. 1.39/1.67 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(595,a,593,a)]. 1.39/1.67 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(595,a,594,a)]. 1.39/1.67 596 -c_Order__Relation_Olinear__order__on(A,B,C) | c_Order__Relation_Opartial__order__on(A,B,C) # label(fact_linear__order__on__def) # label(axiom). [clausify(270)]. 1.39/1.67 597 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))) # label(fact_lnear__order__on__empty) # label(axiom). [clausify(161)]. 1.39/1.67 Derived: c_Order__Relation_Opartial__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(596,a,597,a)]. 1.39/1.67 598 -c_Order__Relation_Olinear__order__on(A,B,C) | c_Relation_Ototal__on(A,B,C) # label(fact_linear__order__on__def) # label(axiom). [clausify(270)]. 1.39/1.67 599 c_Order__Relation_Olinear__order__on(A,B,C) | -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C) # label(fact_linear__order__on__def) # label(axiom). [clausify(270)]. 1.39/1.67 600 -c_Order__Relation_Olinear__order__on(A,B,C) | 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))). [resolve(563,b,561,a)]. 1.39/1.67 Derived: c_Relation_Ototal__on(A,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)),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(600,a,597,a)]. 1.60/1.90 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_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C). [resolve(600,a,599,a)]. 1.60/1.90 601 -c_Order__Relation_Olinear__order__on(A,B,C) | c_Relation_Otrans(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))). [resolve(563,b,562,a)]. 1.60/1.90 Derived: c_Relation_Otrans(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(601,a,597,a)]. 1.60/1.90 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_Opartial__order__on(A,C,B) | -c_Relation_Ototal__on(A,C,B). [resolve(601,a,599,a)]. 1.60/1.90 602 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(595,a,593,a)]. 1.60/1.90 Derived: c_Relation_Ototal__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(602,a,598,a)]. 1.60/1.90 603 c_member(tc_prod(A,B),c_Product__Type_OPair(A,B,C,D),E) | -c_FunDef_Oin__rel(A,B,E,C,D) # label(fact_in__rel__def) # label(axiom). [clausify(176)]. 1.60/1.90 604 -c_member(tc_prod(A,B),c_Product__Type_OPair(A,B,C,D),E) | c_FunDef_Oin__rel(A,B,E,C,D) # label(fact_in__rel__def) # label(axiom). [clausify(176)]. 1.60/1.90 605 c_Nitpick_Orefl_H(A,B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f46(B,A),f46(B,A)),B) # label(fact_Nitpick_Orefl_H__def) # label(axiom). [clausify(276)]. 1.60/1.90 606 -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(276)]. 1.60/1.90 Derived: -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f46(B,A),f46(B,A)),B) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),B). [resolve(605,a,606,a)]. 1.60/1.90 1.60/1.90 ============================== end predicate elimination ============= 1.60/1.90 1.60/1.90 Auto_denials: (non-Horn, no changes). 1.60/1.90 1.60/1.90 Term ordering decisions: 1.60/1.90 Function symbol KB weights: tc_HOL_Obool=1. tc_Arrow__Order__Mirabelle_Oalt=1. c_Arrow__Order__Mirabelle_OLin=1. v_P=1. v_a=1. v_b=1. v_i=1. c1=1. c2=1. c3=1. tc_fun=1. tc_prod=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. c_Wellfounded_Omeasure=1. f2=1. f3=1. f4=1. f5=1. f6=1. f12=1. f13=1. f14=1. f17=1. f41=1. f44=1. f46=1. f47=1. f48=1. f49=1. f51=1. f52=1. f57=1. f71=1. c_Orderings_Obot__class_Obot=1. c_Orderings_Otop__class_Otop=1. c_Relation_OId=1. v_F=1. c_fequal=1. c_Lattices_Osemilattice__sup__class_Osup=1. c_Lattices_Osemilattice__inf__class_Oinf=1. c_Set_Oinsert=1. c_Groups_Ominus__class_Ominus=1. c_Relation_ODomain=1. c_Relation_ORange=1. c_Arrow__Order__Mirabelle_Oabove=1. c_Arrow__Order__Mirabelle_Obelow=1. c_Equiv__Relations_Oquotient=1. f1=1. f7=1. f10=1. f11=1. f18=1. f19=1. f20=1. f21=1. f67=1. f68=1. f69=1. f76=1. f77=1. f78=1. f81=1. f82=1. f83=1. c_Product__Type_OPair=1. c_Relation_OImage=1. c_Wellfounded_Olex__prod=1. c_Recdef_Owfrec=1. c_Relation_Oinv__image=1. c_Recdef_Osame__fst=1. c_Nitpick_Owf__wfrec=1. f8=1. f9=1. f22=1. f29=1. f30=1. f31=1. f34=1. f35=1. f36=1. f38=1. f53=1. f54=1. f55=1. f56=1. f58=1. f61=1. f62=1. f70=1. f79=1. f80=1. c_Recdef_Ocut=1. c_Nitpick_Owf__wfrec_H=1. c_Product__Type_Ointernal__split=1. c_Product__Type_Oprod_Oprod__rec=1. f37=1. f39=1. f42=1. f43=1. f45=1. f63=1. f64=1. f65=1. f66=1. f84=1. f85=1. f27=1. f32=1. f40=1. f50=1. f15=1. f16=1. f28=1. f33=1. f59=1. f60=1. f23=1. f24=1. f25=1. f26=1. f72=1. f73=1. f74=1. f75=1. 1.60/1.90 1.60/1.90 ============================== end of process initial clauses ======== 1.60/1.90 1.60/1.90 ============================== CLAUSES FOR SEARCH ==================== 1.60/1.90 1.60/1.90 ============================== end of clauses forAlarm clock 118.58/120.04 Prover9 interrupted 118.58/120.04 EOF