0.11/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n012.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 1200 0.13/0.34 % DateTime : Tue Jul 13 16:44:46 EDT 2021 0.19/0.34 % CPUTime : 1.19/1.51 ============================== Prover9 =============================== 1.19/1.51 Prover9 (32) version 2009-11A, November 2009. 1.19/1.51 Process 11852 was started by sandbox on n012.cluster.edu, 1.19/1.51 Tue Jul 13 16:44:47 2021 1.19/1.51 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1200 -f /tmp/Prover9_11447_n012.cluster.edu". 1.19/1.51 ============================== end of head =========================== 1.19/1.51 1.19/1.51 ============================== INPUT ================================= 1.19/1.51 1.19/1.51 % Reading from file /tmp/Prover9_11447_n012.cluster.edu 1.19/1.51 1.19/1.51 set(prolog_style_variables). 1.19/1.51 set(auto2). 1.19/1.51 % set(auto2) -> set(auto). 1.19/1.51 % set(auto) -> set(auto_inference). 1.19/1.51 % set(auto) -> set(auto_setup). 1.19/1.51 % set(auto_setup) -> set(predicate_elim). 1.19/1.51 % set(auto_setup) -> assign(eq_defs, unfold). 1.19/1.51 % set(auto) -> set(auto_limits). 1.19/1.51 % set(auto_limits) -> assign(max_weight, "100.000"). 1.19/1.51 % set(auto_limits) -> assign(sos_limit, 20000). 1.19/1.51 % set(auto) -> set(auto_denials). 1.19/1.51 % set(auto) -> set(auto_process). 1.19/1.51 % set(auto2) -> assign(new_constants, 1). 1.19/1.51 % set(auto2) -> assign(fold_denial_max, 3). 1.19/1.51 % set(auto2) -> assign(max_weight, "200.000"). 1.19/1.51 % set(auto2) -> assign(max_hours, 1). 1.19/1.51 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.19/1.51 % set(auto2) -> assign(max_seconds, 0). 1.19/1.51 % set(auto2) -> assign(max_minutes, 5). 1.19/1.51 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.19/1.51 % set(auto2) -> set(sort_initial_sos). 1.19/1.51 % set(auto2) -> assign(sos_limit, -1). 1.19/1.51 % set(auto2) -> assign(lrs_ticks, 3000). 1.19/1.51 % set(auto2) -> assign(max_megs, 400). 1.19/1.51 % set(auto2) -> assign(stats, some). 1.19/1.51 % set(auto2) -> clear(echo_input). 1.19/1.51 % set(auto2) -> set(quiet). 1.19/1.51 % set(auto2) -> clear(print_initial_clauses). 1.19/1.51 % set(auto2) -> clear(print_given). 1.19/1.51 assign(lrs_ticks,-1). 1.19/1.51 assign(sos_limit,10000). 1.19/1.51 assign(order,kbo). 1.19/1.51 set(lex_order_vars). 1.19/1.51 clear(print_given). 1.19/1.51 1.19/1.51 % formulas(sos). % not echoed (527 formulas) 1.19/1.51 1.19/1.51 ============================== end of input ========================== 1.19/1.51 1.19/1.51 % From the command line: assign(max_seconds, 1200). 1.19/1.51 1.19/1.51 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.19/1.51 1.19/1.51 % Formulas that are not ordinary clauses: 1.19/1.51 1 (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]. 1.19/1.51 2 (all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ODomain(T_a,T_a,c_Relation_OId(T_a))) # label(fact_Domain__Id) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 3 (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]. 1.19/1.51 4 (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]. 1.19/1.51 5 (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]. 1.19/1.51 6 (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]. 1.19/1.51 7 (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]. 1.19/1.51 8 (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]. 1.19/1.51 9 (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]. 1.19/1.51 10 (all V_y_2 all V_x_2 all T_a (class_Lattices_Osemilattice__inf(T_a) -> (V_x_2 = c_Lattices_Osemilattice__inf__class_Oinf(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__inf) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 11 (all V_A_2 all T_a all V_b_2 all V_a_2 (V_b_2 = V_a_2 -> (c_member(T_a,V_a_2,V_A_2) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId__on(T_a,V_A_2))))) # label(fact_Id__on__eqI) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 12 (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]. 1.19/1.51 13 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Orderings_Obot__class_Obot(T_a),V_x))) # label(fact_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 14 (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]. 1.19/1.51 15 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> -(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 16 (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_y,V_x))) # label(fact_inf__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 17 (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]. 1.19/1.51 18 (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]. 1.19/1.51 19 (all V_B_2 all V_A_2 all T_a all T_b 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)) = 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))) # label(fact_Range__Un__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 20 (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]. 1.19/1.51 21 (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]. 1.19/1.51 22 (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]. 1.19/1.51 23 (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]. 1.19/1.51 24 (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]. 1.19/1.51 25 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_a 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)) = c_Set_Oinsert(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2))) # label(fact_Domain__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 26 (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]. 1.19/1.51 27 (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)) -> (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)))) <-> V_b_2 = V_a_2))))) # label(fact_Partial__order__eq__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 28 (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]. 1.19/1.51 29 (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]. 1.19/1.51 30 (all V_z all V_y all V_x 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_x,V_y),c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)))) # label(fact_sup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 31 (all V_r_2 all V_A_2 all T_a (c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2) & c_Relation_Oantisym(T_a,V_r_2) <-> c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2))) # label(fact_partial__order__on__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 32 (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]. 1.19/1.51 33 (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]. 1.19/1.51 34 (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]. 1.19/1.51 35 (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_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_A_2)) # label(fact_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 36 (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]. 1.19/1.51 37 (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]. 1.19/1.51 38 (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_z_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_Ortrancl(T_a,V_r_2)))))) # label(fact_single__valued__confluent) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 39 (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]. 1.19/1.51 40 (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]. 1.19/1.51 41 (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]. 1.19/1.51 42 (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]. 1.19/1.51 43 (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,V_B_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_member(T_a,V_a_2,V_A_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_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))))) # label(fact_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 44 (all V_c all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_c) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)))) # label(fact_sup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 45 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_b = V_a -> (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]. 1.19/1.51 46 (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]. 1.19/1.51 47 (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]. 1.19/1.51 48 (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]. 1.19/1.51 49 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Orderings_Otop__class_Otop(T_a),V_x) = V_x)) # label(fact_inf__top__left) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 50 (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]. 1.19/1.51 51 (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]. 1.19/1.51 52 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) | c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 53 (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]. 1.19/1.51 54 (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]. 1.19/1.51 55 (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]. 1.19/1.51 56 (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]. 1.19/1.51 57 (all V_a_2 all T_a all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2 -> -c_member(T_a,V_a_2,V_A_2))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 58 (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_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)))) # label(fact_inf__sup__aci_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 59 (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]. 1.19/1.51 60 (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]. 1.19/1.51 61 (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]. 1.19/1.51 62 (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]. 1.19/1.51 63 (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]. 1.19/1.51 64 (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]. 1.19/1.51 65 (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) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2))))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 66 (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]. 1.19/1.51 67 (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]. 1.19/1.51 68 (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]. 1.19/1.51 69 (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) -> c_Product__Type_OPair(T_a,T_a,B_x,B_x) != V_c_2)))) # label(fact_Id__onE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 70 (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_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y) = V_x))) # label(fact_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 71 (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]. 1.19/1.51 72 (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]. 1.19/1.51 73 (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]. 1.19/1.51 74 (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]. 1.19/1.51 75 (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]. 1.19/1.51 76 (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]. 1.19/1.51 77 (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_x_2,V_A_2) & c_member(T_a,V_y_2,V_A_2)))) # label(fact_equiv__class__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 78 (all V_y_2 all V_x_2 all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> (V_x_2 = c_Orderings_Obot__class_Obot(T_a) & 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]. 1.19/1.51 79 (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]. 1.19/1.51 80 (all V_R_2 all T_a c_Transitive__Closure_Otrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)) # label(fact_rtrancl__trancl__absorb) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 81 (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]. 1.19/1.51 82 (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_x_2 != V_b_2 -> (V_b_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_x_2,V_a_2),V_L_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_b_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_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]. 1.19/1.51 83 (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]. 1.19/1.51 84 (all V_z all V_y all V_x all T_a (class_Lattices_Odistrib__lattice(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(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__inf__class_Oinf(T_a,V_x,V_y),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_z)))) # label(fact_inf__sup__distrib1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 85 (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]. 1.19/1.51 86 (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]. 1.19/1.51 87 (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]. 1.19/1.51 88 (all V_R_2 all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ODomain(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2))) # label(fact_Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.51 89 (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]. 1.19/1.51 90 (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]. 1.19/1.51 91 (all V_B_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = V_B_2) # label(fact_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 92 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2))) # label(fact_Un__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 93 (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]. 1.19/1.52 94 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = V_y))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 95 (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]. 1.19/1.52 96 (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]. 1.19/1.52 97 (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]. 1.19/1.52 98 (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]. 1.19/1.52 99 (all V_x all V_z all V_y 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_y,V_z),V_x) = 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)))) # label(fact_sup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 100 (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]. 1.19/1.52 101 (all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) # label(fact_UNIV__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 102 (all V_R_2 all V_b_2 all V_a_2 all T_a (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) & V_b_2 != V_a_2 | V_b_2 = V_a_2)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 103 (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]. 1.19/1.52 104 (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)) <-> (exists B_x (c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2) & c_member(T_b,B_x,V_A_2))))) # label(fact_Image__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 105 (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,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y_2,V_z_2)) <-> 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)))) # label(fact_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 106 (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]. 1.19/1.52 107 (all T_b all T_a all B_M all B_R (c_Wellfounded_Owf(T_a,B_R) -> (all 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) = hAPP(c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_x)))) # label(fact_tfl__wfrec) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 108 (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]. 1.19/1.52 109 (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]. 1.19/1.52 110 (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]. 1.19/1.52 111 (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]. 1.19/1.52 112 (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]. 1.19/1.52 113 (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]. 1.19/1.52 114 (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_Ototal__on(T_a,V_A_2,V_r_2) <-> c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2))) # label(fact_linear__order__on__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 115 (all V_r_2 all T_a c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2)) # label(fact_rtrancl__idemp) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 116 (all V_x_2 all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_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]. 1.19/1.52 117 (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]. 1.19/1.52 118 (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_b,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_c)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)))) # label(fact_sup_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 119 (all V_A_2 all T_a V_A_2 = 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)))) # label(fact_Int__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 120 (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_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]. 1.19/1.52 121 (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]. 1.19/1.52 122 (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]. 1.19/1.52 123 (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]. 1.19/1.52 124 (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]. 1.19/1.52 125 (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]. 1.19/1.52 126 (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__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]. 1.19/1.52 127 (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]. 1.19/1.52 128 (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]. 1.19/1.52 129 (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]. 1.19/1.52 130 (all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_a) = V_a)) # label(fact_inf_Oidem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 131 (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__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 132 (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]. 1.19/1.52 133 (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]. 1.19/1.52 134 (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]. 1.19/1.52 135 (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_x,V_y)) = 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]. 1.19/1.52 136 (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)))) -> 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)) = V_A_2)) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 137 (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]. 1.19/1.52 138 (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]. 1.19/1.52 139 (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]. 1.19/1.52 140 (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_R_2) = c_Transitive__Closure_Ortrancl(T_a,V_S_2)))) # label(fact_rtrancl__subset) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 141 (all V_y_2 all V_x_2 all T_a (class_Lattices_Osemilattice__sup(T_a) -> (c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x_2,V_y_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]. 1.19/1.52 142 (all V_R_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2)) # label(fact_rtrancl__reflcl__absorb) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 143 (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__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_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Int__Diff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 144 (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))) # label(fact_equalityD1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 145 (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]. 1.19/1.52 146 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> V_r_2 = c_Transitive__Closure_Otrancl(T_a,V_r_2))) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 147 (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]. 1.19/1.52 148 (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]. 1.19/1.52 149 (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]. 1.19/1.52 150 (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]. 1.19/1.52 151 (all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_x) = V_x)) # label(fact_inf__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 152 (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]. 1.19/1.52 153 (all V_y_2 all V_x_2 all T_a (c_member(T_a,V_x_2,c_fequal(V_y_2)) <-> V_y_2 = V_x_2)) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 154 (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]. 1.19/1.52 155 (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]. 1.19/1.52 156 (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]. 1.19/1.52 157 (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]. 1.19/1.52 158 (all V_r_2 all T_a 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))) = c_Transitive__Closure_Ortrancl(T_a,V_r_2)) # label(fact_trancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 159 (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]. 1.19/1.52 160 (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]. 1.19/1.52 161 (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]. 1.19/1.52 162 (all V_a2_2 all V_a1_2 all V_Fa_2 all V_R_2 all T_b all T_a ((exists B_g (hAPP(hAPP(V_Fa_2,B_g),V_a1_2) = V_a2_2 & (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,B_z,hAPP(B_g,B_z)))))) <-> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,V_a1_2,V_a2_2))) # label(fact_wfrec__rel_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 163 (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]. 1.19/1.52 164 (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_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 165 (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]. 1.19/1.52 166 (all V_r_2 all T_b all T_a ((all B_x all B_y (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2) -> (all B_z (c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2) -> B_z = B_y)))) -> c_Relation_Osingle__valued(T_a,T_b,V_r_2))) # label(fact_single__valuedI) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 167 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 168 (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]. 1.19/1.52 169 (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]. 1.19/1.52 170 (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]. 1.19/1.52 171 (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]. 1.19/1.52 172 (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]. 1.19/1.52 173 (all V_A_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_Relation_OId__on(T_a,V_A_2)) <-> c_member(T_a,V_x_2,V_A_2) & V_y_2 = V_x_2)) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 174 (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]. 1.19/1.52 175 (all V_r_2 all T_b all T_a (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) = V_r_2 <-> 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]. 1.19/1.52 176 (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_A_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 177 (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]. 1.19/1.52 178 (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]. 1.19/1.52 179 (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_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_A_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2))) # label(fact_Un__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 180 (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]. 1.19/1.52 181 (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]. 1.19/1.52 182 (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]. 1.19/1.52 183 (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) -> (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)) <-> (V_a_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_b_2),V_L_2)) & (V_y_2 != V_a_2 -> (V_a_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_b_2,V_y_2),V_L_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_y_2),V_L_2))) & V_y_2 != V_x_2)))) # label(fact_in__below) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 184 (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]. 1.19/1.52 185 (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]. 1.19/1.52 186 (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]. 1.19/1.52 187 (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]. 1.19/1.52 188 (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]. 1.19/1.52 189 (all V_c_2 all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> ((c_member(T_a,V_c_2,V_A_2) -> -c_member(T_a,V_c_2,V_B_2)) -> -(-c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2))))) # label(fact_equalityCE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 190 (all V_A_2 all T_a all V_y_2 all V_x_2 (V_y_2 = V_x_2 -> (c_member(T_a,V_y_2,V_A_2) <-> c_member(T_a,V_x_2,V_A_2)))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 191 (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]. 1.19/1.52 192 (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]. 1.19/1.52 193 (all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)))) # label(fact_sup_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.52 194 (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]. 1.19/1.52 195 (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]. 1.19/1.52 196 (all V_r_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,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2)) # label(fact_quotient__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 197 (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)) <-> c_member(T_a,V_a_2,V_A_2) | V_a_2 = V_b_2)) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 198 (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]. 1.19/1.53 199 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2))))) # label(fact_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 200 (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__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]. 1.19/1.53 201 (all V_r_2 all V_A_2 all T_a (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> 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))) # label(fact_quotient__is__empty2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 202 (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]. 1.19/1.53 203 (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),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_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__Int2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 204 (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]. 1.19/1.53 205 (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]. 1.19/1.53 206 (all V_R_2 all T_a c_Relation_ORange(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_Range__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 207 (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]. 1.19/1.53 208 (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]. 1.19/1.53 209 (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),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) = 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))) # label(fact_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 210 (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]. 1.19/1.53 211 (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]. 1.19/1.53 212 (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]. 1.19/1.53 213 (all V_L_2 all V_y_2 all V_x_2 (V_y_2 != V_x_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]. 1.19/1.53 214 (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]. 1.19/1.53 215 (all V_L_2 all V_y_2 all V_x_2 (V_y_2 != V_x_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]. 1.19/1.53 216 (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]. 1.19/1.53 217 (all V_y_2 all V_x_2 all T_a (class_Lattices_Obounded__lattice__top(T_a) -> (V_x_2 = c_Orderings_Otop__class_Otop(T_a) & V_y_2 = c_Orderings_Otop__class_Otop(T_a) <-> c_Orderings_Otop__class_Otop(T_a) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x_2,V_y_2)))) # label(fact_inf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 218 (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]. 1.19/1.53 219 (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_c = V_b -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 220 (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]. 1.19/1.53 221 (all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_OField(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_Field__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 222 (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]. 1.19/1.53 223 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Relation_Orefl__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2) -> (c_Relation_Oantisym(T_a,V_r_2) -> (c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2)) -> (c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2)) -> (c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_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)))) <-> V_a_2 = V_b_2)))))) # label(fact_Refl__antisym__eq__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 224 (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_R_2)) | 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,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]. 1.19/1.53 225 (all V_a2_2 all V_a1_2 all V_Fa_2 all V_R_2 all T_b all T_a ((exists B_g (hAPP(hAPP(V_Fa_2,V_a2_2),V_a1_2) = hAPP(hAPP(V_Fa_2,B_g),V_a1_2) & (all B_z (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2) -> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_Fa_2,B_z,hAPP(B_g,B_z)))))) <-> 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)))) # label(fact_wfrec__rel_Oequations) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 226 (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]. 1.19/1.53 227 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Orderings_Obot__class_Obot(T_a)))) # label(fact_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 228 (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]. 1.19/1.53 229 (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 c_Product__Type_OPair(T_a,T_a,B_x,B_x) != V_p_2))) # label(fact_IdE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 230 (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]. 1.19/1.53 231 (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]. 1.19/1.53 232 (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]. 1.19/1.53 233 (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]. 1.19/1.53 234 (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]. 1.19/1.53 235 (all V_A_2 all V_x_2 all T_a 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_x_2,V_A_2))) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 236 (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]. 1.19/1.53 237 (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]. 1.19/1.53 238 (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]. 1.19/1.53 239 (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]. 1.19/1.53 240 (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]. 1.19/1.53 241 (all V_r_2 all V_A_2 all T_a (c_Relation_Otrans(T_a,V_r_2) & c_Relation_Ototal__on(T_a,V_A_2,V_r_2) & c_Relation_Oirrefl(T_a,V_r_2) <-> c_Order__Relation_Ostrict__linear__order__on(T_a,V_A_2,V_r_2))) # label(fact_strict__linear__order__on__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 242 (all V_A_2 all V_y_2 all V_x_2 all T_a c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_y_2,V_A_2)) = c_Set_Oinsert(T_a,V_y_2,c_Set_Oinsert(T_a,V_x_2,V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 243 (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]. 1.19/1.53 244 (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]. 1.19/1.53 245 (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]. 1.19/1.53 246 (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]. 1.19/1.53 247 (all V_r_2 all V_A_2 all T_a (c_Relation_Ototal__on(T_a,V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) <-> c_Relation_Ototal__on(T_a,V_A_2,V_r_2))) # label(fact_total__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 248 (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]. 1.19/1.53 249 (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]. 1.19/1.53 250 (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_1) # label(hypothesis) # label(non_clause). [assumption]. 1.19/1.53 251 (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]. 1.19/1.53 252 (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_B_2,V_A_2)) # label(fact_Un__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 253 (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]. 1.19/1.53 254 (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]. 1.19/1.53 255 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b = V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 256 (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]. 1.19/1.53 257 (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]. 1.19/1.53 258 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> c_Orderings_Otop__class_Otop(T_a) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)))) # label(fact_sup__top__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 259 (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]. 1.19/1.53 260 (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_a_2,V_b_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) -> (V_d_2 = V_c_2 <-> V_a_2 = V_b_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 261 (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]. 1.19/1.53 262 (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]. 1.19/1.53 263 (all V_y_2 all V_x_2 (V_x_2 != V_y_2 | hBOOL(hAPP(c_fequal(V_x_2),V_y_2)))) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 264 (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]. 1.19/1.53 265 (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_x_2) & (V_x_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)) & V_z_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_Omkbot(V_L_2,V_z_2)))) # label(fact_in__mkbot) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 266 (all V_B_2 all V_A_2 all V_C_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_C_2,V_A_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_B_2)) = 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))) # label(fact_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 267 (all V_y_2 all V_x_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_x_2 = V_y_2))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 268 (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]. 1.19/1.53 269 (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]. 1.19/1.53 270 (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]. 1.19/1.53 271 (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]. 1.19/1.53 272 (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,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__left__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 273 (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]. 1.19/1.53 274 (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]. 1.19/1.53 275 (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]. 1.19/1.53 276 (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]. 1.19/1.53 277 (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]. 1.19/1.53 278 (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_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_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_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_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 279 (all V_g_2 all V_f_2 ((all B_x hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)) -> V_f_2 = V_g_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 280 (all V_P_2 all V_B_2 all V_A_2 all T_a ((exists B_x (c_member(T_a,B_x,V_B_2) & hBOOL(hAPP(V_P_2,B_x)))) | (exists B_x (c_member(T_a,B_x,V_A_2) & hBOOL(hAPP(V_P_2,B_x)))) <-> (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)))))) # label(fact_bex__Un) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 281 (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]. 1.19/1.53 282 (all B_P ((all B_x (c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(B_P,B_x)) & c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(B_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(B_P,B_x)))) -> (all B_a all B_b (B_b != B_a -> (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(B_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(B_P))))))) # label(conj_0) # label(hypothesis) # label(non_clause). [assumption]. 1.19/1.53 283 (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) -> V_x = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 284 (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]. 1.19/1.53 285 (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]. 1.19/1.53 286 (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]. 1.19/1.53 287 (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_a,V_b)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b))) # label(fact_inf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 288 (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]. 1.19/1.53 289 (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_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_Transitive__Closure_Otrancl(T_a,V_R_2)))) # label(fact_rtrancl__eq__or__trancl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 290 (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]. 1.19/1.53 291 (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]. 1.19/1.53 292 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) = V_B_2)) # label(fact_Diff__partition) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 293 (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]. 1.19/1.53 294 (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]. 1.19/1.53 295 (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]. 1.19/1.53 296 (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]. 1.19/1.53 297 (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]. 1.19/1.53 298 (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),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)))) = 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))) # label(fact_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 299 (all V_r_2 all V_A_2 all T_a (c_Relation_Ototal__on(T_a,V_A_2,V_r_2) <-> (all B_x (c_member(T_a,B_x,V_A_2) -> (all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_xa != B_x -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_xa),V_r_2) | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2)))))))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 300 (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]. 1.19/1.53 301 (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]. 1.19/1.53 302 (all V_r_2 all T_a (c_Relation_Oantisym(T_a,V_r_2) <-> (all B_x all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2) -> B_y = B_x))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 303 (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]. 1.19/1.53 304 (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]. 1.19/1.53 305 (all V_x_2 all V_Fa_2 all V_R_2 all T_a all T_b c_Nitpick_Owf__wfrec_H(T_b,T_a,V_R_2,V_Fa_2,V_x_2) = 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)) # label(fact_Nitpick_Owf__wfrec_H__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 306 (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]. 1.19/1.53 307 (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]. 1.19/1.53 308 (all V_C_2 all V_B_2 all V_A_2 all T_a c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),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__assoc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.53 309 (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]. 1.19/1.53 310 (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]. 1.19/1.54 311 (all V_b_2 all V_a_2 all V_c_2 all T_a all T_c all T_b c_Product__Type_Ointernal__split(T_b,T_c,T_a,V_c_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_c_2,V_a_2),V_b_2)) # label(fact_internal__split__conv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 312 (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]. 1.19/1.54 313 (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]. 1.19/1.54 314 (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]. 1.19/1.54 315 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (V_a_2 = V_c_2 & V_b_2 = V_d_2 | V_b_2 = V_c_2 & V_d_2 = V_a_2 <-> 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)))) = 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)))))) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 316 (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]. 1.19/1.54 317 (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]. 1.19/1.54 318 (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))) & (-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]. 1.19/1.54 319 (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]. 1.19/1.54 320 (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_y = B_x))) -> c_Relation_Oantisym(T_a,V_r_2))) # label(fact_antisymI) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 321 (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]. 1.19/1.54 322 (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]. 1.19/1.54 323 (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]. 1.19/1.54 324 (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]. 1.19/1.54 325 (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]. 1.19/1.54 326 (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]. 1.19/1.54 327 (all T_b all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ODomain(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)))) # label(fact_Domain__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 328 (all V_A_2 all T_a c_Relation_ODomain(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2) # label(fact_Domain__Id__on) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 329 (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),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,V_B_2)))) # label(fact_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 330 (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),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__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Diff__Un) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 331 (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]. 1.19/1.54 332 (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]. 1.19/1.54 333 (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]. 1.19/1.54 334 (all V_A_2 all V_x_2 all T_a (hBOOL(hAPP(V_A_2,V_x_2)) <-> c_member(T_a,V_x_2,V_A_2))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 335 (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]. 1.19/1.54 336 (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]. 1.19/1.54 337 (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]. 1.19/1.54 338 (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]. 1.19/1.54 339 (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]. 1.19/1.54 340 (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]. 1.19/1.54 341 (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_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_z)))) # label(fact_inf__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 342 (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]. 1.19/1.54 343 (all V_B_2 all V_A_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_R_2,V_B_2)) = 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))) # label(fact_Image__Un) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 344 (all V_z_2 all V_L_2 all V_y_2 all V_x_2 (V_x_2 != V_z_2 & (V_z_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_z_2 -> 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_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]. 1.19/1.54 345 (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]. 1.19/1.54 346 (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]. 1.19/1.54 347 (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]. 1.19/1.54 348 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2)) = V_A_2))) # label(fact_double__diff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 349 (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]. 1.19/1.54 350 (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]. 1.19/1.54 351 (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]. 1.19/1.54 352 (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]. 1.19/1.54 353 (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]. 1.19/1.54 354 (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]. 1.19/1.54 355 (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]. 1.19/1.54 356 (all V_A_2 all T_a 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) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 357 (all V_a_2 all V_b_2 all T_a (V_a_2 = V_b_2 <-> c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 358 (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]. 1.19/1.54 359 (all V_B_2 all V_a_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_A_2,V_B_2)) = 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))) # label(fact_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 360 (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]. 1.19/1.54 361 (all V_A_2 all T_a V_A_2 = c_Relation_ORange(T_a,T_a,c_Relation_OId__on(T_a,V_A_2))) # label(fact_Range__Id__on) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 362 (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,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_A_2)) # label(fact_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 363 (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_x_2 != V_z_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]. 1.19/1.54 364 (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_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_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]. 1.19/1.54 365 (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]. 1.19/1.54 366 (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]. 1.19/1.54 367 (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__sup__class_Osup(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) = hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 368 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)))) # label(fact_sup__left__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 369 (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]. 1.19/1.54 370 (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]. 1.19/1.54 371 (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]. 1.19/1.54 372 (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]. 1.19/1.54 373 (all V_r_2 all V_A_2 all T_a (c_Relation_Orefl__on(T_a,V_A_2,V_r_2) & c_Relation_Otrans(T_a,V_r_2) <-> c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2))) # label(fact_preorder__on__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 374 (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]. 1.19/1.54 375 (all V_A_2 all T_a V_A_2 = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 376 (all V_A_2 all V_b_2 all V_a_2 all T_a (c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)) -> (V_a_2 != V_b_2 -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 377 (all V_r_2 all T_a all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Relation_ORange(T_b,T_a,V_r_2) <-> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)))) # label(fact_Range__empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 378 (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]. 1.19/1.54 379 (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]. 1.19/1.54 380 (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]. 1.19/1.54 381 (all V_a_2 all V_H_2 all V_r_2 all T_b all T_a all V_f_2 (c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2) = V_f_2 -> (c_Wellfounded_Owf(T_a,V_r_2) -> hAPP(V_f_2,V_a_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)))) # label(fact_def__wfrec) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 382 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> ((all B_x c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x))) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 383 (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]. 1.19/1.54 384 (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]. 1.19/1.54 385 (all V_B_2 all V_A_2 all T_a (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> V_B_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)) = V_A_2)) # label(fact_Un__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 386 (all V_x all V_z all V_y 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_y,V_x),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_z,V_x)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z),V_x))) # label(fact_inf__sup__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 387 (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]. 1.19/1.54 388 (all V_B_2 all V_A_2 all T_a c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 389 (all V_r_2 all T_a ((all B_x c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2))) <-> c_Wellfounded_Owf(T_a,V_r_2))) # label(fact_wf__acc__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 390 (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]. 1.19/1.54 391 (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]. 1.19/1.54 392 (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]. 1.19/1.54 393 (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]. 1.19/1.54 394 (all V_A_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_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_A_2) = c_Set_Oinsert(T_a,V_a_2,V_A_2)) # label(fact_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 395 (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__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)) = 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))) # label(fact_Un__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 396 (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]. 1.19/1.54 397 (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]. 1.19/1.54 398 (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]. 1.19/1.54 399 (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]. 1.19/1.54 400 (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]. 1.19/1.54 401 (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]. 1.19/1.54 402 (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,c_Lattices_Osemilattice__sup__class_Osup(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_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2))) # label(fact_Diff__subset__conv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 403 (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]. 1.19/1.54 404 (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]. 1.19/1.54 405 (all V_g_2 all V_x_2 all V_r_2 all V_f_2 all T_b all T_a (c_Recdef_Ocut(T_a,T_b,V_g_2,V_r_2,V_x_2) = c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_x_2) <-> (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))))) # label(fact_cuts__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 406 (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]. 1.19/1.54 407 (all V_x_2 all T_a V_x_2 = 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))))) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 408 (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]. 1.19/1.54 409 (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),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),c_Set_Oinsert(T_a,V_x_2,V_A_2),V_B_2))) # label(fact_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 410 (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]. 1.19/1.54 411 (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]. 1.19/1.54 412 (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]. 1.19/1.54 413 (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_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_Transitive__Closure_Ortrancl(T_a,V_R_2))))) # label(fact_Not__Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 414 (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]. 1.19/1.54 415 (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)) <-> V_x_2 = V_y_2 | hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 416 (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 (V_a2_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = V_a4_2 | (exists B_x exists B_y (c_Nitpick_Ofold__graph_H(T_a,T_b,V_a3_2,V_a4_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_a2_2,c_Set_Oinsert(T_a,B_x,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),B_y) & c_member(T_a,B_x,V_a2_2) & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y))) <-> 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]. 1.19/1.54 417 (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]. 1.19/1.54 418 (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]. 1.19/1.54 419 (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_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 420 (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,V_x_2,V_z_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_z_2) <-> 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)))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 421 (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]. 1.19/1.54 422 (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]. 1.19/1.54 423 (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]. 1.19/1.54 424 (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]. 1.19/1.54 425 (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]. 1.19/1.54 426 (all 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)),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)),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_3) # label(hypothesis) # label(non_clause). [assumption]. 1.19/1.54 427 (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]. 1.19/1.54 428 (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),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)) = 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))) # label(fact_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 429 (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]. 1.19/1.54 430 (all V_x_2 all V_B_2 all V_A_2 all T_b all T_a (class_Groups_Ominus(T_a) -> hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_x_2) = c_Groups_Ominus__class_Ominus(T_a,hAPP(V_A_2,V_x_2),hAPP(V_B_2,V_x_2)))) # label(fact_minus__apply) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 431 (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]. 1.19/1.54 432 (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(V_f_2,V_x_2) = hAPP(c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2),V_x_2))) # label(fact_cut__apply) # label(axiom) # label(non_clause). [assumption]. 1.19/1.54 433 (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]. 1.19/1.54 434 (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]. 1.19/1.55 435 (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]. 1.19/1.55 436 (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]. 1.19/1.55 437 (all V_a_2 all V_b_2 all T_a (c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> V_a_2 = V_b_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 438 (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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = V_A_2) # label(fact_Un__Diff__Int) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 439 (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_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))) # label(fact_quotient__is__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 440 (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]. 1.19/1.55 441 (all V_A_2 all T_a c_Relation_OImage(T_a,T_a,c_Relation_OId(T_a),V_A_2) = V_A_2) # label(fact_Image__Id) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 442 (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) | 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) & V_a_H_2 = V_a_2)) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 443 (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]. 1.19/1.55 444 (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_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_z))) # label(fact_inf__assoc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 445 (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]. 1.19/1.55 446 (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),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__assoc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 447 (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]. 1.19/1.55 448 (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]. 1.19/1.55 449 (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) -> (V_A_2 = V_B_2 <-> c_Set_Oinsert(T_a,V_x_2,V_B_2) = c_Set_Oinsert(T_a,V_x_2,V_A_2))))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 450 (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_y_2 != V_x_2))) # label(fact_irrefl__trancl__rD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 451 (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]. 1.19/1.55 452 (all T_a c_Relation_Otrans(T_a,c_Relation_OId(T_a))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 453 (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)) -> ((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))))) <-> 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))))))) # label(fact_subset__Image__Image__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 454 (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]. 1.19/1.55 455 (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]. 1.19/1.55 456 (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]. 1.19/1.55 457 (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]. 1.19/1.55 458 (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]. 1.19/1.55 459 (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]. 1.19/1.55 460 (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]. 1.19/1.55 461 (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]. 1.19/1.55 462 (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]. 1.19/1.55 463 (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]. 1.19/1.55 464 (all V_A_2 all T_a c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 465 (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]. 1.19/1.55 466 (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]. 1.19/1.55 467 (all V_B_2 all T_a all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) = V_A_2 -> 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]. 1.19/1.55 468 (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,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),V_c) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,V_c)))) # label(fact_inf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 469 (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]. 1.19/1.55 470 (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]. 1.19/1.55 471 (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]. 1.19/1.55 472 (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_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__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))) # label(fact_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 473 (all V_a_2 all V_r_2 all T_b all V_b_2 all T_a (c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,c_Set_Oinsert(T_b,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) <-> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_a_2,V_b_2),V_r_2))) # label(fact_Image__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 474 (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]. 1.19/1.55 475 (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_2,V_b_2) = c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2) <-> V_a_2 = V_a_H_2 & V_b_2 = V_b_H_2)) # label(fact_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 476 (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_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))))))))) # label(fact_eq__equiv__class__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 477 (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]. 1.19/1.55 478 (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]. 1.19/1.55 479 (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]. 1.19/1.55 480 (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]. 1.19/1.55 481 (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,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_inf__sup__aci_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 482 (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]. 1.19/1.55 483 (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]. 1.19/1.55 484 (all T_a c_Relation_OId(T_a) = c_Transitive__Closure_Ortrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) # label(fact_rtrancl__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 485 (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]. 1.19/1.55 486 (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]. 1.19/1.55 487 (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__inf__class_Oinf(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)))) # label(fact_inf__apply) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 488 (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]. 1.19/1.55 489 (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]. 1.19/1.55 490 (all V_P_2 all V_B_2 all V_A_2 all T_a ((all B_x (c_member(T_a,B_x,V_B_2) -> hBOOL(hAPP(V_P_2,B_x)))) & (all B_x (c_member(T_a,B_x,V_A_2) -> hBOOL(hAPP(V_P_2,B_x)))) <-> (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)))))) # label(fact_ball__Un) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 491 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Relation_Oantisym(T_a,V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2) -> V_a_2 = V_b_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 492 (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]. 1.19/1.55 493 (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]. 1.19/1.55 494 (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_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_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)))) # label(fact_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 495 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_B_2)) # label(fact_Un__absorb1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 496 (all T_b all T_a all V_P_2 ((all B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> (all B_a all B_b hBOOL(hAPP(V_P_2,c_Product__Type_OPair(T_a,T_b,B_a,B_b)))))) # label(fact_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 497 (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]. 1.19/1.55 498 (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]. 1.19/1.55 499 (all V_b_2 all V_a_2 all V_r_2 all T_a (c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2) -> (c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2)) -> (c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) <-> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2)))))) # label(fact_subset__Image1__Image1__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 500 (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]. 1.19/1.55 501 (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]. 1.19/1.55 502 (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_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__separator__converseE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 503 (all T_a c_Relation_Oantisym(T_a,c_Relation_OId(T_a))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 504 (all V_b_2 all V_a_2 all V_f1_2 all T_a all T_c all T_b hAPP(hAPP(V_f1_2,V_a_2),V_b_2) = 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))) # label(fact_prod_Orecs) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 505 (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]. 1.19/1.55 506 (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]. 1.19/1.55 507 (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]. 1.19/1.55 508 (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]. 1.19/1.55 509 (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]. 1.19/1.55 510 (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]. 1.19/1.55 511 (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,c_Set_Oinsert(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_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.55 512 (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]. 1.19/1.55 1.19/1.55 ============================== end of process non-clausal formulas === 1.19/1.55 1.19/1.55 ============================== PROCESS INITIAL CLAUSES =============== 1.19/1.55 1.19/1.55 ============================== PREDICATE ELIMINATION ================= 1.19/1.55 513 class_Lattices_Osemilattice__inf(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Osemilattice__inf) # label(axiom). [assumption]. 1.19/1.55 514 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) != B | c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_le__iff__inf) # label(axiom). [clausify(10)]. 1.19/1.55 515 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) = B | -c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_le__iff__inf) # label(axiom). [clausify(10)]. 1.19/1.55 516 -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(16)]. 1.19/1.55 517 -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(70)]. 1.19/1.55 518 -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(75)]. 1.19/1.55 519 -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(76)]. 1.19/1.55 520 -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(85)]. 1.19/1.55 521 -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(96)]. 1.19/1.55 522 -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__inf__iff) # label(axiom). [clausify(105)]. 1.19/1.55 523 -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__inf__iff) # label(axiom). [clausify(105)]. 1.19/1.55 524 -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) | -c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_le__inf__iff) # label(axiom). [clausify(105)]. 1.19/1.56 525 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,B) = B # label(fact_inf_Oidem) # label(axiom). [clausify(130)]. 1.19/1.56 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) != A | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B). [resolve(513,a,514,a)]. 1.19/1.56 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) = A | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B). [resolve(513,a,515,a)]. 1.19/1.56 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) = c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,A). [resolve(513,a,516,a)]. 1.19/1.56 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),A). [resolve(513,a,518,a)]. 1.19/1.56 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(513,a,519,a)]. 1.19/1.56 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(513,a,520,a)]. 1.19/1.56 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B),B). [resolve(513,a,521,a)]. 1.19/1.56 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,C)) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B). [resolve(513,a,522,a)]. 1.19/1.56 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,C)) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,C). [resolve(513,a,523,a)]. 1.19/1.56 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,A) = A. [resolve(513,a,525,a)]. 1.19/1.56 526 -class_Lattices_Osemilattice__inf(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,B) = B # label(fact_inf__idem) # label(axiom). [clausify(151)]. 1.19/1.56 527 -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(159)]. 1.19/1.56 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(527,a,513,a)]. 1.19/1.56 528 -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(178)]. 1.19/1.56 529 -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(264)]. 1.19/1.56 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(529,a,513,a)]. 1.19/1.56 530 -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(272)]. 1.19/1.56 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(530,a,513,a)]. 1.19/1.56 531 -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(287)]. 1.19/1.57 532 -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(319)]. 1.19/1.57 533 -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(341)]. 1.19/1.57 534 -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(372)]. 1.19/1.57 535 -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(372)]. 1.19/1.57 536 -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(410)]. 1.19/1.57 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,513,a)]. 1.19/1.57 537 -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__assoc) # label(axiom). [clausify(444)]. 1.19/1.57 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(537,a,513,a)]. 1.19/1.57 538 -class_Lattices_Olattice(A) | class_Lattices_Osemilattice__inf(tc_fun(B,A)) # label(arity_fun__Lattices_Osemilattice__inf) # label(axiom). [clausify(465)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) != C | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D). [resolve(538,b,514,a)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) = C | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D). [resolve(538,b,515,a)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,C). [resolve(538,b,516,a)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D),C). [resolve(538,b,518,a)]. 1.19/1.57 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),E,F) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,E),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,F)). [resolve(538,b,519,a)]. 1.19/1.57 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(538,b,520,a)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D),D). [resolve(538,b,521,a)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,E)) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D). [resolve(538,b,522,a)]. 1.19/1.57 Derived: -class_Lattices_Olattice(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,E)) | c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,E). [resolve(538,b,523,a)]. 1.19/1.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,C) = C. [resolve(538,b,525,a)]. 1.19/1.59 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(538,b,527,a)]. 1.19/1.59 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(538,b,529,a)]. 1.19/1.59 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),C,D)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D). [resolve(538,b,530,a)]. 1.19/1.59 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(538,b,536,a)]. 1.19/1.59 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(538,b,537,a)]. 1.19/1.59 539 -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(468)]. 1.19/1.59 540 -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(498)]. 1.19/1.59 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(540,a,513,a)]. 1.19/1.59 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),E,C),D) | -class_Lattices_Olattice(B). [resolve(540,a,538,b)]. 1.19/1.59 541 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__bot(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom). [clausify(122)]. 1.19/1.59 542 -class_Lattices_Obounded__lattice__bot(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,c_Orderings_Obot__class_Obot(A),B) = c_Orderings_Obot__class_Obot(A) # label(fact_inf__bot__left) # label(axiom). [clausify(13)]. 1.19/1.59 543 -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(78)]. 1.19/1.59 544 -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(78)]. 1.19/1.59 545 -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(78)]. 1.19/1.59 Derived: -class_Lattices_Obounded__lattice(A) | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),c_Orderings_Obot__class_Obot(tc_fun(B,A)),C) = c_Orderings_Obot__class_Obot(tc_fun(B,A)). [resolve(541,b,542,a)]. 1.19/1.59 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Obot__class_Obot(tc_fun(B,A)) != C | c_Orderings_Obot__class_Obot(tc_fun(B,A)) != D | c_Orderings_Obot__class_Obot(tc_fun(B,A)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,D). [resolve(541,b,543,a)]. 1.19/1.59 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Obot__class_Obot(tc_fun(B,A)) = C | c_Orderings_Obot__class_Obot(tc_fun(B,A)) != c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,D). [resolve(541,b,544,a)]. 1.19/1.64 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Obot__class_Obot(tc_fun(B,A)) = C | c_Orderings_Obot__class_Obot(tc_fun(B,A)) != c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),D,C). [resolve(541,b,545,a)]. 1.19/1.64 546 -class_Lattices_Obounded__lattice__bot(A) | c_Lattices_Osemilattice__inf__class_Oinf(A,B,c_Orderings_Obot__class_Obot(A)) = c_Orderings_Obot__class_Obot(A) # label(fact_inf__bot__right) # label(axiom). [clausify(227)]. 1.19/1.64 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B),C,c_Orderings_Obot__class_Obot(tc_fun(A,B))) = c_Orderings_Obot__class_Obot(tc_fun(A,B)) | -class_Lattices_Obounded__lattice(B). [resolve(546,a,541,b)]. 1.19/1.64 547 -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(230)]. 1.19/1.64 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(547,a,541,b)]. 1.19/1.64 548 class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 1.19/1.64 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,c_Orderings_Obot__class_Obot(tc_HOL_Obool),A) = c_Orderings_Obot__class_Obot(tc_HOL_Obool). [resolve(548,a,542,a)]. 1.19/1.64 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(548,a,543,a)]. 1.19/1.64 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(548,a,544,a)]. 1.19/1.64 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(548,a,545,a)]. 1.19/1.64 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = c_Orderings_Obot__class_Obot(tc_HOL_Obool). [resolve(548,a,546,a)]. 1.19/1.64 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Orderings_Obot__class_Obot(tc_HOL_Obool),A) = A. [resolve(548,a,547,a)]. 1.19/1.64 549 -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(337)]. 1.19/1.64 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(549,a,541,b)]. 1.19/1.64 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = A. [resolve(549,a,548,a)]. 1.19/1.64 550 -class_Lattices_Olattice(A) | class_Lattices_Osemilattice__sup(tc_fun(B,A)) # label(arity_fun__Lattices_Osemilattice__sup) # label(axiom). [clausify(103)]. 1.19/1.64 551 -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(28)]. 1.19/1.64 552 -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(44)]. 1.19/1.64 553 -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(90)]. 1.19/1.64 554 -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(94)]. 1.19/1.65 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),D,E)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),D,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,E)). [resolve(550,b,551,a)]. 1.19/1.65 Derived: -class_Lattices_Olattice(A) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,D),E) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),D,E)). [resolve(550,b,552,a)]. 1.19/1.65 Derived: -class_Lattices_Olattice(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(B,A),C,D) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,D) = D. [resolve(550,b,554,a)]. 1.19/1.65 555 -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(109)]. 1.19/1.65 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(555,a,550,b)]. 1.19/1.65 556 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,B) = B # label(fact_sup__idem) # label(axiom). [clausify(112)]. 1.19/1.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,C) = C | -class_Lattices_Olattice(B). [resolve(556,a,550,b)]. 1.19/1.65 557 -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(118)]. 1.19/1.65 558 -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(141)]. 1.19/1.65 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(558,a,550,b)]. 1.19/1.65 559 -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(141)]. 1.19/1.65 560 -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(191)]. 1.19/1.65 561 -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(193)]. 1.19/1.65 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(561,a,550,b)]. 1.19/1.65 562 -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(212)]. 1.19/1.65 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(562,a,550,b)]. 1.19/1.65 563 -class_Lattices_Osemilattice__sup(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,C) | c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,D),C) # label(fact_sup__least) # label(axiom). [clausify(244)]. 1.19/1.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),C,D) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),E,D) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,B),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B),C,E),D) | -class_Lattices_Olattice(B). [resolve(563,a,550,b)]. 1.19/1.65 564 -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(246)]. 1.19/1.65 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(564,a,550,b)]. 1.19/1.65 565 -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(277)]. 1.19/1.65 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(565,a,550,b)]. 1.19/1.65 566 -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(283)]. 1.19/1.65 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(566,a,550,b)]. 1.19/1.65 567 class_Lattices_Osemilattice__sup(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Osemilattice__sup) # label(axiom). [assumption]. 1.19/1.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,C)) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C)). [resolve(567,a,551,a)]. 1.19/1.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B),C) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,C)). [resolve(567,a,552,a)]. 1.19/1.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = B. [resolve(567,a,554,a)]. 1.19/1.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B) = c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A). [resolve(567,a,555,a)]. 1.19/1.65 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,A) = A. [resolve(567,a,556,a)]. 1.19/1.65 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(567,a,558,a)]. 1.19/1.65 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(567,a,561,a)]. 1.19/1.65 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(567,a,562,a)]. 1.19/1.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,C,B) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C),B). [resolve(567,a,563,a)]. 1.19/1.65 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(567,a,564,a)]. 1.19/1.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,C,D) | c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,C),c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,D)). [resolve(567,a,565,a)]. 1.19/1.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,B) | c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A) = B. [resolve(567,a,566,a)]. 1.19/1.65 568 -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(368)]. 1.19/1.68 569 -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,B) = B # label(fact_sup_Oidem) # label(axiom). [clausify(374)]. 1.19/1.68 570 -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(378)]. 1.19/1.68 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(570,a,550,b)]. 1.19/1.68 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,B)). [resolve(570,a,567,a)]. 1.19/1.68 571 -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(408)]. 1.19/1.68 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(571,a,550,b)]. 1.19/1.68 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(571,a,567,a)]. 1.19/1.68 572 -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(408)]. 1.19/1.68 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(572,a,550,b)]. 1.19/1.68 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(572,a,567,a)]. 1.19/1.68 573 -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__sup__iff) # label(axiom). [clausify(420)]. 1.19/1.68 574 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,B,D),C) # label(fact_le__sup__iff) # label(axiom). [clausify(420)]. 1.19/1.68 575 -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Lattices_Osemilattice__sup__class_Osup(A,D,B),C) # label(fact_le__sup__iff) # label(axiom). [clausify(420)]. 1.19/1.68 576 -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(431)]. 1.19/1.68 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(576,a,550,b)]. 1.19/1.68 Derived: c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,B,A)). [resolve(576,a,567,a)]. 1.19/1.68 577 -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(454)]. 1.19/1.68 578 class_Lattices_Obounded__lattice__top(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__top) # label(axiom). [assumption]. 1.19/1.68 579 -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(49)]. 1.19/1.68 580 -class_Lattices_Obounded__lattice__top(A) | c_Lattices_Osemilattice__sup__class_Osup(A,c_Orderings_Otop__class_Otop(A),B) = c_Orderings_Otop__class_Otop(A) # label(fact_sup__top__left) # label(axiom). [clausify(64)]. 1.19/1.69 581 -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(106)]. 1.19/1.69 582 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) != B | c_Orderings_Otop__class_Otop(A) != C | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) = c_Orderings_Otop__class_Otop(A) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(217)]. 1.19/1.69 583 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Lattices_Osemilattice__inf__class_Oinf(A,B,C) != c_Orderings_Otop__class_Otop(A) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(217)]. 1.19/1.69 584 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Lattices_Osemilattice__inf__class_Oinf(A,C,B) != c_Orderings_Otop__class_Otop(A) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(217)]. 1.19/1.69 585 -class_Lattices_Obounded__lattice__top(A) | c_Lattices_Osemilattice__sup__class_Osup(A,B,c_Orderings_Otop__class_Otop(A)) = c_Orderings_Otop__class_Otop(A) # label(fact_sup__top__right) # label(axiom). [clausify(258)]. 1.19/1.69 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,c_Orderings_Otop__class_Otop(tc_HOL_Obool),A) = A. [resolve(578,a,579,a)]. 1.19/1.69 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,c_Orderings_Otop__class_Otop(tc_HOL_Obool),A) = c_Orderings_Otop__class_Otop(tc_HOL_Obool). [resolve(578,a,580,a)]. 1.19/1.69 Derived: c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,c_Orderings_Otop__class_Otop(tc_HOL_Obool)) = A. [resolve(578,a,581,a)]. 1.19/1.69 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) != A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != B | c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) = c_Orderings_Otop__class_Otop(tc_HOL_Obool). [resolve(578,a,582,a)]. 1.19/1.69 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,A,B) != c_Orderings_Otop__class_Otop(tc_HOL_Obool). [resolve(578,a,583,a)]. 1.19/1.69 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool,B,A) != c_Orderings_Otop__class_Otop(tc_HOL_Obool). [resolve(578,a,584,a)]. 1.19/1.69 Derived: c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool,A,c_Orderings_Otop__class_Otop(tc_HOL_Obool)) = c_Orderings_Otop__class_Otop(tc_HOL_Obool). [resolve(578,a,585,a)]. 1.19/1.69 586 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__top(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom). [clausify(433)]. 1.19/1.69 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(586,b,579,a)]. 1.19/1.69 Derived: -class_Lattices_Obounded__lattice(A) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),c_Orderings_Otop__class_Otop(tc_fun(B,A)),C) = c_Orderings_Otop__class_Otop(tc_fun(B,A)). [resolve(586,b,580,a)]. 1.19/1.69 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(586,b,581,a)]. 1.19/1.69 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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) = c_Orderings_Otop__class_Otop(tc_fun(B,A)). [resolve(586,b,582,a)]. 1.19/1.69 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = C | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),C,D) != c_Orderings_Otop__class_Otop(tc_fun(B,A)). [resolve(586,b,583,a)]. 1.19/1.69 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = C | c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A),D,C) != c_Orderings_Otop__class_Otop(tc_fun(B,A)). [resolve(586,b,584,a)]. 1.19/1.69 Derived: -class_Lattices_Obounded__lattice(A) | c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A),C,c_Orderings_Otop__class_Otop(tc_fun(B,A))) = c_Orderings_Otop__class_Otop(tc_fun(B,A)). [resolve(586,b,585,a)]. 1.19/1.81 587 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(113)]. 1.19/1.81 588 -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(113)]. 1.19/1.81 589 c_Order__Relation_Opartial__order__on(A,B,C) | -c_Order__Relation_Olinear__order__on(A,B,C) # label(fact_linear__order__on__def) # label(axiom). [clausify(114)]. 1.19/1.81 590 -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C) | c_Order__Relation_Olinear__order__on(A,B,C) # label(fact_linear__order__on__def) # label(axiom). [clausify(114)]. 1.19/1.81 591 c_Relation_Ototal__on(A,B,C) | -c_Order__Relation_Olinear__order__on(A,B,C) # label(fact_linear__order__on__def) # label(axiom). [clausify(114)]. 1.19/1.81 592 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(188)]. 1.19/1.81 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(592,a,589,b)]. 1.19/1.81 593 -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(234)]. 1.19/1.81 Derived: 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))) | -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C). [resolve(593,a,590,c)]. 1.19/1.81 Derived: c_Order__Relation_Ostrict__linear__order__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(593,a,592,a)]. 1.19/1.81 594 -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(248)]. 1.19/1.81 Derived: -c_Order__Relation_Owell__order__on(A,B,C) | c_Order__Relation_Opartial__order__on(A,B,C). [resolve(594,b,589,b)]. 1.19/1.81 Derived: -c_Order__Relation_Owell__order__on(A,B,C) | c_Relation_Ototal__on(A,B,C). [resolve(594,b,591,b)]. 1.19/1.81 Derived: -c_Order__Relation_Owell__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))). [resolve(594,b,593,a)]. 1.19/1.81 595 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(248)]. 1.19/1.81 Derived: c_Order__Relation_Owell__order__on(A,B,C) | -c_Wellfounded_Owf(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) | -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C). [resolve(595,b,590,c)]. 1.19/1.81 596 c_Relation_Otrans(A,B) | -c_Order__Relation_Ostrict__linear__order__on(A,C,B) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(241)]. 1.19/1.81 597 -c_Relation_Otrans(A,B) | -c_Relation_Ototal__on(A,C,B) | -c_Relation_Oirrefl(A,B) | c_Order__Relation_Ostrict__linear__order__on(A,C,B) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(241)]. 1.19/1.81 598 c_Relation_Ototal__on(A,B,C) | -c_Order__Relation_Ostrict__linear__order__on(A,B,C) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(241)]. 1.19/1.81 599 c_Relation_Oirrefl(A,B) | -c_Order__Relation_Ostrict__linear__order__on(A,C,B) # label(fact_strict__linear__order__on__def) # label(axiom). [clausify(241)]. 1.19/1.81 600 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))) | -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C). [resolve(593,a,590,c)]. 1.19/1.83 Derived: -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__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(600,a,596,b)]. 1.19/1.83 Derived: -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__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(600,a,598,b)]. 1.19/1.83 Derived: -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C) | c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))). [resolve(600,a,599,b)]. 1.19/1.83 601 c_Order__Relation_Ostrict__linear__order__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(593,a,592,a)]. 1.19/1.83 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,596,b)]. 1.19/1.83 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(601,a,598,b)]. 1.19/1.83 Derived: c_Relation_Oirrefl(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,599,b)]. 1.19/1.83 602 -c_Order__Relation_Owell__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))). [resolve(594,b,593,a)]. 1.19/1.83 Derived: -c_Order__Relation_Owell__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(602,b,596,b)]. 1.19/1.83 Derived: -c_Order__Relation_Owell__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(602,b,598,b)]. 1.19/1.83 Derived: -c_Order__Relation_Owell__order__on(A,B,C) | c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))). [resolve(602,b,599,b)]. 1.19/1.83 603 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(427)]. 1.19/1.83 604 -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(248)]. 1.19/1.83 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(603,a,604,a)]. 1.19/1.83 605 -c_Order__Relation_Owell__order__on(A,B,C) | c_Order__Relation_Opartial__order__on(A,B,C). [resolve(594,b,589,b)]. 1.19/1.83 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(605,a,603,a)]. 1.19/1.83 606 -c_Order__Relation_Owell__order__on(A,B,C) | c_Relation_Ototal__on(A,B,C). [resolve(594,b,591,b)]. 1.19/1.83 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(606,a,603,a)]. 1.19/1.83 607 c_Order__Relation_Owell__order__on(A,B,C) | -c_Wellfounded_Owf(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) | -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C). [resolve(595,b,590,c)]. 1.66/1.94 608 -c_Order__Relation_Owell__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(602,b,596,b)]. 1.66/1.94 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(608,a,603,a)]. 1.66/1.94 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_Wellfounded_Owf(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(608,a,607,a)]. 1.66/1.94 609 -c_Order__Relation_Owell__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(602,b,598,b)]. 1.66/1.94 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(609,a,603,a)]. 1.66/1.94 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_Wellfounded_Owf(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))) | -c_Order__Relation_Opartial__order__on(A,B,C) | -c_Relation_Ototal__on(A,B,C). [resolve(609,a,607,a)]. 1.66/1.94 610 -c_Order__Relation_Owell__order__on(A,B,C) | c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),C,c_Relation_OId(A))). [resolve(602,b,599,b)]. 1.66/1.94 Derived: c_Relation_Oirrefl(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(610,a,603,a)]. 1.66/1.94 Derived: c_Relation_Oirrefl(A,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool),B,c_Relation_OId(A))) | -c_Wellfounded_Owf(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(610,a,607,a)]. 1.66/1.94 611 c_Nitpick_Orefl_H(A,B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f57(B,A),f57(B,A)),B) # label(fact_Nitpick_Orefl_H__def) # label(axiom). [clausify(387)]. 1.66/1.94 612 -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(387)]. 1.66/1.94 Derived: -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f57(B,A),f57(B,A)),B) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),B). [resolve(611,a,612,a)]. 1.66/1.94 1.66/1.94 ============================== end predicate elimination ============= 1.66/1.94 1.66/1.94 Auto_denials: (non-Horn, no changes). 1.66/1.94 1.66/1.94 Term ordering decisions: 1.66/1.94 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. 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. f1=1. f26=1. f32=1. f33=1. f37=1. f45=1. f46=1. f47=1. f48=1. f49=1. f51=1. f52=1. f53=1. f57=1. f58=1. f60=1. f61=1. f62=1. f73=1. c_Orderings_Obot__class_Obot=1. c_Orderings_Otop__class_Otop=1. c_Relation_OId=1. v_F=1. c_fequal=1. f41=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. f2=1. f3=1. f4=1. f12=1. f18=1. f19=1. f22=1. f23=1. f24=1. f42=1. f43=1. f44=1. f72=1. f77=1. f78=1. f79=1. f80=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. f11=1. f13=1. f14=1. f16=1. f25=1. f27=1. f38=1. f39=1. f40=1. f50=1. f54=1. f55=1. f56=1. f59=1. f68=1. f69=1. f71=1. f74=1. f75=1. f76=1. c_Recdef_Ocut=1. c_Nitpick_Owf__wfrec_H=1. c_Product__Type_Ointernal__split=1. c_Product__Type_Oprod_Oprod__rec=1. f15=1. f28=1. f29=1. f34=1. f35=1. f36=1. f64=1. f65=1. f70=1. f81=1. f82=1. f17=1. f21=1. f31=1. f63=1. f5=1. f6=1. f20=1. f30=1. f66=1. f67=1. f7=1. f8=1. f9=1. f10=1. f83=1. f84=1. f85=1. f86=1. 22.01/22.30 22.01/22.30 ============================== end of process initial clauses ======== 22.01/22.30 22.01/22.30 ============================== CLAUSES FOR SEARCH ==================== 22.01/22.30 22.01/22.30 ============================== end of clauses for search ============= 22.01/22.30 22.01/22.30 ============================== SEARCH ================================ 22.01/22.30 22.01/22.30 % Starting search at 0.50 seconds. 22.01/22.30 22.01/22.30 Low Water (keep): wt=67.000, iters=3389 22.01/22.30 22.01/22.30 Low Water (keep): wt=66.000, iters=3431 22.01/22.30 22.01/22.30 Low Water (keep): wt=64.000, iters=3367 22.01/22.30 22.01/22.30 Low Water (keep): wt=58.000, iters=3346 22.01/22.30 22.01/22.30 Low Water (keep): wt=56.000, iters=3414 22.01/22.30 22.01/22.30 Low Water (keep): wt=54.000, iters=3337 22.01/22.30 22.01/22.30 Low Water (keep): wt=52.000, iters=3339 22.01/22.30 22.01/22.30 Low Water (keep): wt=51.000, iters=3390 22.01/22.30 22.01/22.30 Low Water (keep): wt=50.000, iters=3346 22.01/22.30 22.01/22.30 Low Water (keep): wt=49.000, iters=3360 22.01/22.30 22.01/22.30 Low Water (keep): wt=48.000, iters=3428 22.01/22.30 22.01/22.30 Low Water (keep): wt=47.000, iters=3380 22.01/22.30 22.01/22.30 Low Water (keep): wt=46.000, iters=3353 22.01/22.30 22.01/22.30 Low Water (keep): wt=45.000, iters=3397 22.01/22.30 22.01/22.30 Low Water (keep): wt=44.000, iters=3343 22.01/22.30 22.01/22.30 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 32 (0.00 of 1.47 sec). 22.01/22.30 22.01/22.30 Low Water (keep): wt=43.000, iters=3347 22.01/22.30 22.01/22.30 Low Water (keep): wt=42.000, iters=3364 22.01/22.30 22.01/22.30 Low Water (keep): wt=40.000, iters=3335 22.01/22.30 22.01/22.30 Low Water (keep): wt=39.000, iters=3418 22.01/22.30 22.01/22.30 Low Water (keep): wt=37.000, iters=3352 22.01/22.30 22.01/22.30 Low Water (keep): wt=36.000, iters=3454 22.01/22.30 22.01/22.30 Low Water (keep): wt=35.000, iters=3425 22.01/22.30 22.01/22.30 Low Water (keep): wt=34.000, iters=3456 22.01/22.30 22.01/22.30 Low Water (keep): wt=33.000, iters=3359 22.01/22.30 22.01/22.30 Low Water (keep): wt=32.000, iters=3349 22.01/22.30 22.01/22.30 Low Water (keep): wt=31.000, iters=3359 22.01/22.30 22.01/22.30 Low Water (keep): wt=30.000, iters=3364 22.01/22.30 22.01/22.30 Low Water (keep): wt=29.000, iters=3343 22.01/22.30 22.01/22.30 Low Water (keep): wt=28.000, iters=3357 22.01/22.30 22.01/22.30 Low Water (keep): wt=27.000, iters=3355 22.01/22.30 22.01/22.30 Low Water (keep): wt=26.000, iters=3412 22.01/22.30 22.01/22.30 Low Water (keep): wt=25.000, iters=3355 22.01/22.30 22.01/22.30 Low Water (keep): wt=24.000, iters=3360 22.01/22.30 22.01/22.30 Low Water (keep): wt=23.000, iters=3343 22.01/22.30 22.01/22.30 Low Water (keep): wt=22.000, iters=3349 22.01/22.30 22.01/22.30 Low Water (keep): wt=21.000, iters=3335 22.01/22.30 22.01/22.30 Low Water (keep): wt=20.000, iters=3368 22.01/22.30 22.01/22.30 Low Water (keep): wt=19.000, iters=3339 22.01/22.30 22.01/22.30 Low Water (keep): wt=18.000, iters=3340 22.01/22.30 22.01/22.30 Low Water (displace): id=4746, wt=182.000 22.01/22.30 22.01/22.30 Low Water (displace): id=4721, wt=173.000 22.01/22.30 22.01/22.30 Low Water (displace): id=4284, wt=170.000 22.01/22.30 22.01/22.30 Low Water (displace): id=4732, wt=162.000 22.01/22.30 22.01/22.30 Low Water (displace): id=13371, wt=15.000 22.01/22.30 22.01/22.30 Low Water (displace): id=13536, wt=14.000 22.01/22.30 22.01/22.30 Low Water (displace): id=14128, wt=13.000 22.01/22.30 22.01/22.30 Low Water (displace): id=14155, wt=12.000 22.01/22.30 22.01/22.30 Low Water (keep): wt=17.000, iters=3345 22.01/22.30 22.01/22.30 Low Water (keep): wt=16.000, iters=3341 22.01/22.30 22.01/22.30 Low Water (keep): wt=15.000, iters=3333 22.01/22.30 22.01/22.30 ============================== PROOF ================================= 22.01/22.30 % SZS status Theorem 22.01/22.30 % SZS output start Refutation 22.01/22.30 22.01/22.30 % Proof 1 at 20.46 (+ 0.38) seconds. 22.01/22.30 % Length of proof is 54. 22.01/22.30 % Level of proof is 9. 22.01/22.30 % Maximum clause weight is 48.000. 22.01/22.30 % Given clauses 9716. 22.01/22.30 22.01/22.30 12 (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]. 22.01/22.30 55 (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]. 22.01/22.30 108 (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]. 22.01/22.30 146 (all V_r_2 all T_a (c_Relation_Otrans(T_a,V_r_2) -> V_r_2 = c_Transitive__Closure_Otrancl(T_a,V_r_2))) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 22.01/22.30 147 (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]. 22.01/22.30 150 (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]. 22.01/22.30 250 (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_1) # label(hypothesis) # label(non_clause). [assumption]. 22.01/22.30 282 (all B_P ((all B_x (c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(B_P,B_x)) & c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(B_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(B_P,B_x)))) -> (all B_a all B_b (B_b != B_a -> (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(B_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(B_P))))))) # label(conj_0) # label(hypothesis) # label(non_clause). [assumption]. 22.01/22.30 299 (all V_r_2 all V_A_2 all T_a (c_Relation_Ototal__on(T_a,V_A_2,V_r_2) <-> (all B_x (c_member(T_a,B_x,V_A_2) -> (all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_xa != B_x -> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_xa),V_r_2) | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2)))))))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 22.01/22.30 302 (all V_r_2 all T_a (c_Relation_Oantisym(T_a,V_r_2) <-> (all B_x all B_y (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2) -> (c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2) -> B_y = B_x))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 22.01/22.30 304 (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]. 22.01/22.30 324 (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]. 22.01/22.30 426 (all 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)),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)),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_3) # label(hypothesis) # label(non_clause). [assumption]. 22.01/22.30 450 (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_y_2 != V_x_2))) # label(fact_irrefl__trancl__rD) # label(axiom) # label(non_clause). [assumption]. 22.01/22.30 623 -c_Wellfounded_Oacyclic(A,B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,C),c_Transitive__Closure_Otrancl(A,B)) # label(fact_acyclic__def) # label(axiom). [clausify(12)]. 22.01/22.30 624 c_Wellfounded_Oacyclic(A,B) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f1(B,A),f1(B,A)),c_Transitive__Closure_Otrancl(A,B)) # label(fact_acyclic__def) # label(axiom). [clausify(12)]. 22.01/22.31 675 c_member(A,B,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool))) # label(fact_UNIV__I) # label(axiom). [clausify(55)]. 22.01/22.31 742 c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool),B,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool),B,C)) # label(fact_Un__upper1) # label(axiom). [clausify(108)]. 22.01/22.31 782 -c_Relation_Otrans(A,B) | c_Transitive__Closure_Otrancl(A,B) = B # label(fact_trancl__id) # label(axiom). [clausify(146)]. 22.01/22.31 783 -c_Wellfounded_Oacyclic(A,B) | c_Relation_Oantisym(A,c_Transitive__Closure_Ortrancl(A,B)) # label(fact_acyclic__impl__antisym__rtrancl) # label(axiom). [clausify(147)]. 22.01/22.31 784 c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(A,A),tc_HOL_Obool),c_Transitive__Closure_Otrancl(A,B),c_Relation_OId(A)) = c_Transitive__Closure_Ortrancl(A,B) # label(fact_reflcl__trancl) # label(axiom). [clausify(150)]. 22.01/22.31 902 c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,A)) # label(conj_1) # label(hypothesis). [clausify(250)]. 22.01/22.31 903 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,A)) # label(conj_1) # label(hypothesis). [clausify(250)]. 22.01/22.31 904 c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,A)) # label(conj_1) # label(hypothesis). [clausify(250)]. 22.01/22.31 952 -c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(A,f41(A))) | -c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(A,f41(A))) | -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(A,f41(A))) | B = C | -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,C,B),hAPP(A,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,C,B),v_F(A)) # label(conj_0) # label(hypothesis). [clausify(282)]. 22.01/22.31 969 -c_Relation_Ototal__on(A,B,C) | -c_member(A,D,B) | -c_member(A,E,B) | E = D | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,D,E),C) | c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,E,D),C) # label(fact_total__on__def) # label(axiom). [clausify(299)]. 22.01/22.31 979 -c_Relation_Oantisym(A,B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,D),B) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,D,C),B) | D = C # label(fact_antisym__def) # label(axiom). [clausify(302)]. 22.01/22.31 984 -c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(A,A),tc_HOL_Obool),B,C) | -c_Relation_Oantisym(A,C) | c_Relation_Oantisym(A,B) # label(fact_antisym__subset) # label(axiom). [clausify(304)]. 22.01/22.31 1004 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,v_b),v_F(v_P)) # label(conj_2) # label(hypothesis). [assumption]. 22.01/22.31 1012 -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,B,B),C) | -c_Relation_Oirrefl(A,C) # label(fact_irrefl__def) # label(axiom). [clausify(324)]. 22.01/22.31 1150 -c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,c2)) | -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,c2)) | -c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,c2)) | c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) # label(conj_3) # label(hypothesis). [clausify(426)]. 22.01/22.31 1151 c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)). [copy(1150),unit_del(a,904),unit_del(b,903),unit_del(c,902)]. 22.01/22.31 1152 -c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,c2)) | -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,c2)) | -c_Relation_Oirrefl(tc_Arrow__Order__Mirabelle_Oalt,hAPP(v_P,c2)) | c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) # label(conj_3) # label(hypothesis). [clausify(426)]. 22.01/22.31 1153 c_Relation_Otrans(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)). [copy(1152),unit_del(a,904),unit_del(b,903),unit_del(c,902)]. 22.01/22.31 1171 c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,f68(B,C,D,A),f68(B,C,D,A)),c_Transitive__Closure_Otrancl(A,D)) | -c_member(tc_prod(A,A),c_Product__Type_OPair(A,A,C,B),D) | B != C # label(fact_irrefl__trancl__rD) # label(axiom). [clausify(450)]. 22.01/22.31 1248 -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,v_b),hAPP(v_P,v_i)) # label(conj_4) # label(negated_conjecture). [assumption]. 22.01/22.31 1884 c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(A,A),tc_HOL_Obool),c_Transitive__Closure_Otrancl(A,B),c_Transitive__Closure_Ortrancl(A,B)). [para(784(a,1),742(a,3))]. 22.01/22.31 3503 A = 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),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),v_F(v_P)). [resolve(952,c,903,a),unit_del(a,904),unit_del(b,902)]. 22.01/22.31 3709 A = 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),hAPP(v_P,C)) | 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,A,B),hAPP(v_P,C)). [resolve(969,a,903,a),unit_del(a,675),unit_del(b,675)]. 22.01/22.31 4748 -c_Relation_Oantisym(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) | -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,v_a),v_F(v_P)) | v_b = v_a. [resolve(1004,a,979,c),flip(c)]. 22.01/22.31 5208 -c_Relation_Oirrefl(A,c_Transitive__Closure_Otrancl(A,B)) | c_Wellfounded_Oacyclic(A,B). [resolve(1012,a,624,b)]. 22.01/22.31 6238 c_Transitive__Closure_Otrancl(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) = v_F(v_P). [resolve(1153,a,782,a)]. 22.01/22.31 6387 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,f68(v_b,v_a,v_F(v_P),tc_Arrow__Order__Mirabelle_Oalt),f68(v_b,v_a,v_F(v_P),tc_Arrow__Order__Mirabelle_Oalt)),v_F(v_P)) | v_b != v_a. [resolve(1171,b,1004,a),rewrite([6238(22)])]. 22.01/22.31 7745 -c_Wellfounded_Oacyclic(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) | -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,A,A),v_F(v_P)). [para(6238(a,1),623(b,3))]. 22.01/22.31 8003 c_Wellfounded_Oacyclic(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)). [para(6238(a,1),5208(a,2)),unit_del(a,1151)]. 22.01/22.31 8006 -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,A,A),v_F(v_P)). [back_unit_del(7745),unit_del(a,8003)]. 22.01/22.31 8007 v_b != v_a. [back_unit_del(6387),unit_del(a,8006)]. 22.01/22.31 8020 -c_Relation_Oantisym(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)) | -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,v_a),v_F(v_P)). [back_unit_del(4748),unit_del(c,8007)]. 22.01/22.31 8022 c_Relation_Oantisym(tc_Arrow__Order__Mirabelle_Oalt,c_Transitive__Closure_Ortrancl(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P))). [resolve(8003,a,783,a)]. 22.01/22.31 14311 -c_Relation_Oantisym(A,c_Transitive__Closure_Ortrancl(A,B)) | c_Relation_Oantisym(A,c_Transitive__Closure_Otrancl(A,B)). [resolve(1884,a,984,a)]. 22.01/22.31 14331 c_Relation_Oantisym(tc_Arrow__Order__Mirabelle_Oalt,v_F(v_P)). [resolve(14311,a,8022,a),rewrite([6238(5)])]. 22.01/22.31 14333 -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,v_a),v_F(v_P)). [back_unit_del(8020),unit_del(a,14331)]. 22.01/22.31 35943 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,v_a),hAPP(v_P,v_i)). [resolve(3709,b,1248,a),unit_del(a,8007)]. 22.01/22.31 35958 $F. [resolve(35943,a,3503,b),flip(a),unit_del(a,8007),unit_del(b,14333)]. 22.01/22.31 22.01/22.31 % SZS output end Refutation 22.01/22.31 ============================== end of proof ========================== 22.01/22.31 22.01/22.31 ============================== STATISTICS ============================ 22.01/22.31 22.01/22.31 Given=9716. Generated=638329. Kept=35260. proofs=1. 22.01/22.31 Usable=8733. Sos=9990. Demods=908. Limbo=14, Disabled=17418. Hints=0. 22.01/22.31 Megabytes=58.93. 22.01/22.31 User_CPU=20.46, System_CPU=0.38, Wall_clock=21. 22.01/22.31 22.01/22.31 ============================== end of statistics ===================== 22.01/22.31 22.01/22.31 ============================== end of search ========================= 22.01/22.31 22.01/22.31 THEOREM PROVED 22.01/22.31 % SZS status Theorem 22.01/22.31 22.01/22.31 Exiting with 1 proof. 22.01/22.31 22.01/22.31 Process 11852 exit (max_proofs) Tue Jul 13 16:45:08 2021 22.01/22.31 Prover9 interrupted 22.01/22.31 EOF