0.00/0.03 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.03/0.23 % Computer : n181.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.23 % CPULimit : 300 0.03/0.23 % DateTime : Sat Jul 14 05:25:09 CDT 2018 0.03/0.23 % CPUTime : 0.72/0.98 ============================== Prover9 =============================== 0.72/0.98 Prover9 (32) version 2009-11A, November 2009. 0.72/0.98 Process 26259 was started by sandbox2 on n181.star.cs.uiowa.edu, 0.72/0.98 Sat Jul 14 05:25:10 2018 0.72/0.98 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_26210_n181.star.cs.uiowa.edu". 0.72/0.98 ============================== end of head =========================== 0.72/0.98 0.72/0.98 ============================== INPUT ================================= 0.72/0.98 0.72/0.98 % Reading from file /tmp/Prover9_26210_n181.star.cs.uiowa.edu 0.72/0.98 0.72/0.98 set(prolog_style_variables). 0.72/0.98 set(auto2). 0.72/0.98 % set(auto2) -> set(auto). 0.72/0.98 % set(auto) -> set(auto_inference). 0.72/0.98 % set(auto) -> set(auto_setup). 0.72/0.98 % set(auto_setup) -> set(predicate_elim). 0.72/0.98 % set(auto_setup) -> assign(eq_defs, unfold). 0.72/0.98 % set(auto) -> set(auto_limits). 0.72/0.98 % set(auto_limits) -> assign(max_weight, "100.000"). 0.72/0.98 % set(auto_limits) -> assign(sos_limit, 20000). 0.72/0.98 % set(auto) -> set(auto_denials). 0.72/0.98 % set(auto) -> set(auto_process). 0.72/0.98 % set(auto2) -> assign(new_constants, 1). 0.72/0.98 % set(auto2) -> assign(fold_denial_max, 3). 0.72/0.98 % set(auto2) -> assign(max_weight, "200.000"). 0.72/0.98 % set(auto2) -> assign(max_hours, 1). 0.72/0.98 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.72/0.98 % set(auto2) -> assign(max_seconds, 0). 0.72/0.98 % set(auto2) -> assign(max_minutes, 5). 0.72/0.98 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.72/0.98 % set(auto2) -> set(sort_initial_sos). 0.72/0.98 % set(auto2) -> assign(sos_limit, -1). 0.72/0.98 % set(auto2) -> assign(lrs_ticks, 3000). 0.72/0.98 % set(auto2) -> assign(max_megs, 400). 0.72/0.98 % set(auto2) -> assign(stats, some). 0.72/0.98 % set(auto2) -> clear(echo_input). 0.72/0.98 % set(auto2) -> set(quiet). 0.72/0.98 % set(auto2) -> clear(print_initial_clauses). 0.72/0.98 % set(auto2) -> clear(print_given). 0.72/0.98 assign(lrs_ticks,-1). 0.72/0.98 assign(sos_limit,10000). 0.72/0.98 assign(order,kbo). 0.72/0.98 set(lex_order_vars). 0.72/0.98 clear(print_given). 0.72/0.98 0.72/0.98 % formulas(sos). % not echoed (533 formulas) 0.72/0.98 0.72/0.98 ============================== end of input ========================== 0.72/0.98 0.72/0.98 % From the command line: assign(max_seconds, 300). 0.72/0.98 0.72/0.98 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.72/0.98 0.72/0.98 % Formulas that are not ordinary clauses: 0.72/0.98 1 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 2 (all V_N_2 c_Fun_Oinj__on(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,V_N_2)) # label(fact_inj__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 3 (all V_f_2 all V_m_2 all V_n_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)),hAPP(V_f_2,hAPP(c_Nat_OSuc,V_n_2)))) & (c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2) -> c_Groups_Ozero__class_Ozero(T_a) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))))) # label(fact_setsum__cl__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 4 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 5 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 6 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2)))) # label(fact_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 7 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))))) # label(fact_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 8 (all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,V_A_2)) -> c_Finite__Set_Ofinite(T_a,V_A_2))) # label(fact_card__ge__0__finite) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 9 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 10 (all V_n V_n = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n)) # label(fact_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 11 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__increasing) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 12 (all V_k all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k))) # label(fact_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 13 (all V_f_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oimage(T_a,T_a,V_f_2,V_A_2)) -> c_Fun_Oinj__on(T_a,T_a,V_f_2,V_A_2)))) # label(fact_finite__surj__inj) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 14 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) -> (V_x_2 = V_y_2 <-> -c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2))))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 15 (all V_x_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (-c_member(T_a,V_x_2,V_A_2) -> hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(T_a,V_A_2)) = c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2))))) # label(fact_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 16 (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_x_2,V_A_2) <-> c_member(T_a,V_y_2,V_A_2)))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 17 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) -> (-c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> hAPP(c_Nat_OSuc,V_n) = V_m))) # label(fact_le__SucE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 18 (all V_l all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l))))) # label(fact_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 19 (all V_A_2 all T_a (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> (exists B_x c_member(T_a,B_x,V_A_2)))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 20 (all V_A_2 all V_f_2 all T_b all T_a (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)),c_Finite__Set_Ocard(T_b,V_A_2)) -> -c_Fun_Oinj__on(T_b,T_a,V_f_2,V_A_2))) # label(fact_pigeonhole) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 21 (all V_a_2 all T_a (class_Int_Oring__char__0(T_a) -> (c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a)) -> (c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_Ints__double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 22 (all V_N_2 ((exists B_m all B_x (c_member(tc_Nat_Onat,B_x,V_N_2) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_x,B_m))) <-> c_Finite__Set_Ofinite(tc_Nat_Onat,V_N_2))) # label(fact_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 23 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2))))) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 24 (all V_b_2 all V_B_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_B_2) -> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_insertI2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 25 (all V_x_2 all T_a (class_Groups_Oone(T_a) -> (V_x_2 = c_Groups_Oone__class_Oone(T_a) <-> V_x_2 = c_Groups_Oone__class_Oone(T_a)))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 26 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 27 (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]. 0.72/0.98 28 (all V_m all V_i c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_m)))) # label(fact_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 29 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2) -> V_b_2 = V_d_2))))) # label(fact_atLeastLessThan__inj_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 30 (all V_c all V_a all V_b all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_c,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_b,V_a) -> V_c = V_b))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 31 (all T_a (class_Rings_Olinordered__semidom(T_a) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 32 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)))) # label(fact_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 33 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 34 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_psubset__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 35 (all V_A_2 all T_a (class_Finite__Set_Ofinite(T_a) -> c_Finite__Set_Ofinite(T_a,V_A_2))) # label(fact_finite__code) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 36 (all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n))) # label(fact_gr0I) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 37 (all V_x_2 all V_y_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) -> (V_y_2 = V_x_2 <-> -c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2))))) # label(fact_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 38 (all V_A_2 all V_B_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2))))) # label(fact_inj__on__strict__subset) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 39 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 40 (all V_A_2 all V_B_2 all T_a (c_Finite__Set_Ofinite(T_a,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,V_B_2)) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))))) # label(fact_card__psubset) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 41 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) # label(fact_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 42 (all V_n_2 all V_m_2 (V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 <-> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_add__is__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 43 (all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) & V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,V_A_2)))) # label(fact_card__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 44 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (V_x != V_y -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 45 (all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) = c_Groups_Oplus__class_Oplus(T_a,V_c,V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 46 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 47 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2) -> (V_c_2 = V_a_2 & V_d_2 = V_b_2 <-> c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2)))))) # label(fact_atLeastLessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 48 (all V_a_2 all V_b_2 all T_a (c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> V_a_2 = V_b_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 49 (all V_A_2 all V_B_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)))) # label(fact_subset__inj__on) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 50 (all V_N_2 ((exists B_m all B_x (c_member(tc_Nat_Onat,B_x,V_N_2) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_x,B_m))) <-> c_Finite__Set_Ofinite(tc_Nat_Onat,V_N_2))) # label(fact_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 51 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)))) # label(fact_le__SucI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 52 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> c_Orderings_Oord__class_Oless(T_a,V_b,V_a))))) # label(fact_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 53 (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]. 0.72/0.98 54 (all V_B_2 all V_A_2 all T_b all T_a (c_Finite__Set_Ofinite(T_b,V_B_2) & c_Finite__Set_Ofinite(T_a,V_A_2) <-> c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)))) # label(fact_finite__Plus__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 55 (all V_A_2 all T_a -c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_not__psubset__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 56 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z))))) # label(fact_order__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 57 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2) -> V_a_2 = V_c_2))))) # label(fact_atLeastLessThan__inj_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 58 (all V_n all V_k all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n))) # label(fact_add__leD2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 59 (all V_y all V_z all V_x (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_z) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)))) # label(fact_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 60 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_y != V_x))) # label(fact_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 61 (all V_A_2 all V_b_2 all V_a_2 all T_a (c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)) -> (V_a_2 != V_b_2 -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 62 (all T_b all T_a c_Sum__Type_OPlus(T_a,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_a,T_b),tc_HOL_Obool))) # label(fact_UNIV__Plus__UNIV) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 63 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) -> V_b = V_c))) # label(fact_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 64 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 65 (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]. 0.72/0.98 66 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b != V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 67 (all V_y all V_x (V_y != V_x -> (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_y,V_x)))) # label(fact_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 68 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 69 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_psubset__subset__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 70 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_x_2 = V_y_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 71 (all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))))) # label(fact_less__add__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 72 (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_x_2,V_y_2) & V_x_2 != V_y_2 <-> c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 73 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 74 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (V_x_2 != V_y_2 -> (c_member(T_a,V_x_2,V_A_2) -> (c_member(T_a,V_y_2,V_A_2) -> hAPP(V_f_2,V_y_2) != hAPP(V_f_2,V_x_2)))))) # label(fact_inj__on__contraD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 75 (all V_x_2 all V_A_2 all V_Fa_2 all V_g_2 all V_z_2 all V_f_2 all T_b all T_a (c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_b,V_A_2) -> hAPP(V_Fa_2,c_Set_Oinsert(T_b,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_Fa_2,V_A_2))))) # label(fact_folding__image__simple__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 76 (all V_m_2 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) # label(fact_atLeastLessThan0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 77 (all V_k_2 all V_less_2 all V_i_2 all T_a (c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThan(T_a,V_less_2,V_k_2)) <-> hBOOL(hAPP(hAPP(V_less_2,V_k_2),V_i_2)))) # label(fact_ord_OgreaterThan__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 78 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x))))) # label(fact_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 79 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2) -> V_x_2 = V_y_2))) # label(fact_injD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 80 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 81 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_add__0__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 82 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> -c_Orderings_Oord__class_Oless(T_a,V_x,V_x))) # label(fact_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 83 (all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n)) # label(fact_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 84 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 85 (all V_i all V_j -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_i),V_i)) # label(fact_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 86 (all V_n_2 all V_m_2 ((c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) -> c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = c_Set_Oinsert(tc_Nat_Onat,V_n_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2))) & (-c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_atLeastLessThanSuc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 87 (all V_B_2 all V_A_2 all T_b all T_a (c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) -> c_Finite__Set_Ofinite(T_b,V_B_2))) # label(fact_finite__PlusD_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 88 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) | c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 89 (all V_B_2 all V_A_2 all V_Fa_2 all V_g_2 all V_z_2 all V_f_2 all T_b all T_a (c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_b,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),V_B_2,V_A_2) -> hAPP(hAPP(V_f_2,hAPP(V_Fa_2,V_B_2)),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2))))) # label(fact_folding__image__simple__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 90 (all V_z all V_y all V_x c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,V_z)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 91 (all V_n_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) <-> (exists B_m V_n_2 = hAPP(c_Nat_OSuc,B_m)))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 92 (all T_a c_Finite__Set_Ofinite(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_finite_OemptyI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 93 (all V_m_2 all V_n_2 (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,V_m_2)) <-> V_m_2 = V_n_2))) # label(fact_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 94 (all V_n_2 all V_m_2 (hAPP(c_Nat_OSuc,V_n_2) = V_m_2 | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))) # label(fact_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 95 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 96 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_subset__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 97 (all V_j_2 all V_i_2 c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2))) # label(fact_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 98 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OgreaterThan(T_a,V_x_2),c_SetInterval_Oord__class_OgreaterThan(T_a,V_y_2))))) # label(fact_greaterThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 99 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))) # label(fact_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 100 (all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),V_a_2),V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_Ints__odd__less__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 101 (all V_k_2 all V_less__eq_2 all V_i_2 all T_a (c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeast(T_a,V_less__eq_2,V_k_2)) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_k_2),V_i_2)))) # label(fact_ord_OatLeast__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 102 (all V_h_2 all T_b all V_Fa_2 all T_a (c_Finite__Set_Ofinite(T_a,V_Fa_2) -> c_Finite__Set_Ofinite(T_b,c_Set_Oimage(T_a,T_b,V_h_2,V_Fa_2)))) # label(fact_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 103 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 104 (all V_n all V_m hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n))) # label(fact_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 105 (all V_B_2 all V_A_2 all V_x_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_x_2,V_A_2),V_B_2) <-> c_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_insert__subset) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 106 (all V_f_2 all T_b all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_a,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_setsum__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 107 (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]. 0.72/0.98 108 (all V_A_2 all V_y_2 all V_x_2 all T_a c_Set_Oinsert(T_a,V_y_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_y_2,V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 109 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 110 (all V_k_2 all V_f_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)))) # label(fact_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 111 (all V_nat c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_Osize__class_Osize(tc_Nat_Onat,V_nat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Nat_Osize__class_Osize(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_nat))) # label(fact_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 112 (all V_k_2 all V_A_2 (c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),V_A_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,c_Finite__Set_Ocard(tc_Nat_Onat,V_A_2)))) -> c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,c_Finite__Set_Ocard(tc_Nat_Onat,V_A_2))) = V_A_2)) # label(fact_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 113 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (V_y_2 = V_x_2 <-> c_SetInterval_Oord__class_OgreaterThan(T_a,V_x_2) = c_SetInterval_Oord__class_OgreaterThan(T_a,V_y_2)))) # label(fact_greaterThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 114 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> -c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_leD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 115 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) -> (c_member(T_a,V_x_2,V_A_2) -> (c_member(T_a,V_y_2,V_A_2) -> V_x_2 = V_y_2))))) # label(fact_inj__onD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 116 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> -(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 117 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)))) # label(fact_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 118 (all V_g_2 all V_f_2 all V_A_2 all T_a all T_b (class_Groups_Oordered__cancel__ab__semigroup__add(T_b) & class_Groups_Ocomm__monoid__add(T_b) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> ((all B_x (c_member(T_a,B_x,V_A_2) -> c_Orderings_Oord__class_Oless(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x)))) -> c_Orderings_Oord__class_Oless(T_b,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,V_A_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2))))))) # label(fact_setsum__strict__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 119 (all V_g_2 all V_x_2 all V_A_2 all T_a all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (-c_member(T_a,V_x_2,V_A_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Groups_Oplus__class_Oplus(T_b,hAPP(V_g_2,V_x_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2)))))) # label(fact_setsum_Oinsert) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 120 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 121 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 122 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),hAPP(c_Nat_OSuc,V_n)) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 123 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 124 (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]. 0.72/0.98 125 (all V_m all V_i c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_i)))) # label(fact_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 126 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_add__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 127 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 128 (all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_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) -> (hAPP(V_f_2,B_x) = hAPP(V_f_2,B_xa) -> B_xa = B_x))))))) # label(fact_inj__on__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 129 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__ab__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) -> V_b = V_c))) # label(fact_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 130 (all V_k_2 all V_f_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))))) # label(fact_setsum__shift__lb__Suc0__0__upt) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 131 (all V_m all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n))) # label(fact_le__add2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 132 (all V_B_2 all V_a_2 all V_f_2 all T_a all T_b c_Set_Oinsert(T_a,hAPP(V_f_2,V_a_2),c_Set_Oimage(T_b,T_a,V_f_2,V_B_2)) = c_Set_Oimage(T_b,T_a,V_f_2,c_Set_Oinsert(T_b,V_a_2,V_B_2))) # label(fact_image__insert) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 133 (all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (V_b_2 = V_c_2 & V_a_2 = V_b_2 <-> c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Set_Oinsert(T_a,V_c_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_atLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 134 (all V_f_2 all V_n_2 all V_m_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(V_f_2,V_m_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2)))))) # label(fact_setsum__head__upt__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 135 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2)) -> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2) | c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2)))) # label(fact_atLeastLessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 136 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2)))) # label(fact_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 137 (all V_a all V_b all V_c all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 138 (all V_y_2 all V_x_2 all T_a (V_x_2 = V_y_2 <-> c_member(T_a,V_x_2,c_fequal(V_y_2)))) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 139 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> ((all B_x c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x))) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 140 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 141 (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_A_2) <-> c_member(T_a,V_x_2,V_B_2)))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 142 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_b all T_a all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_Set_Oimage(T_a,T_b,V_f_2,V_A_2) = V_B_2 -> ((all B_a (c_member(T_a,B_a,V_A_2) -> hAPP(V_h_2,hAPP(V_f_2,B_a)) = hAPP(V_g_2,B_a))) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_c,V_g_2,V_A_2) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c,V_h_2,V_B_2)))))) # label(fact_setsum__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 143 (all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_n)) # label(fact_le__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 144 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_2) & class_Finite__Set_Ofinite(T_1) -> class_Finite__Set_Ofinite(tc_fun(T_2,T_1)))) # label(arity_fun__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 145 (all V_nat_H_1 c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H_1)) # label(fact_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 146 (all T_a (class_Rings_Oring__1(T_a) -> c_member(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Oring__1__class_OInts(T_a)))) # label(fact_Ints__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 147 (all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Finite__Set_Ocard(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Finite__Set_Ocard(T_a,V_A_2) -> c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = V_A_2))) # label(fact_card__eq__UNIV__imp__eq__UNIV) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 148 (all T_a all V_A_2 all T_b all V_x_2 all V_f_2 all V_b_2 (hAPP(V_f_2,V_x_2) = V_b_2 -> (c_member(T_b,V_x_2,V_A_2) -> c_member(T_a,V_b_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))))) # label(fact_image__eqI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 149 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_a (hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) <-> c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThanAtMost(T_a,V_less__eq_2,V_less_2,V_l_2,V_u_2)))) # label(fact_ord_OgreaterThanAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 150 (all V_j all V_i -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_i)) # label(fact_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 151 (all V_n -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n),V_n)) # label(fact_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 152 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) | V_n_2 = V_m_2)) # label(fact_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 153 (all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 154 (all V_P_2 all V_n_2 ((exists B_m (hBOOL(hAPP(V_P_2,B_m)) & c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m,V_n_2))) <-> (exists B_x (hBOOL(hAPP(V_P_2,B_x)) & c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))))) # label(fact_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 155 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2) <-> -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 156 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_c,c_Groups_Oplus__class_Oplus(T_a,V_a,V_d)) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 157 (all V_x_2 all V_y_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_x_2 = V_y_2)))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 158 (all V_g_2 all T_b all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_a,V_g_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_setsum_Oempty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 159 (all V_g_2 all V_f_2 ((all B_x hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)) -> V_f_2 = V_g_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 160 (all V_z all V_y all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)))) # label(fact_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 161 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_d))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 162 (all V_n hAPP(c_Nat_OSuc,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 163 (all T_b all T_a (c_Finite__Set_Ofinite(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) & c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) <-> c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_a,T_b),tc_HOL_Obool))))) # label(fact_finite__Plus__UNIV__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 164 (all V_x_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (-c_member(T_a,V_x_2,V_A_2) -> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,V_A_2)) = c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2))))) # label(fact_card_Oinsert) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 165 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)))) # label(fact_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 166 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_leI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 167 (all V_n c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,hAPP(c_Nat_OSuc,V_n))) # label(fact_lessI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 168 (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]. 0.72/0.98 169 (all V_A_2 all T_a (class_Finite__Set_Ofinite(T_a) -> c_Finite__Set_Ofinite(T_a,V_A_2))) # label(fact_finite) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 170 (all V_x_2 all V_A_2 all V_Fa_2 all V_f_2 all T_a (c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> hAPP(V_Fa_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)))))) # label(fact_folding__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 171 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_add__0__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 172 (all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 173 (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]. 0.72/0.98 174 (all V_k_2 all V_less__eq_2 all V_i_2 all T_a (hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)) <-> c_member(T_a,V_i_2,c_SetInterval_Oord_OatMost(T_a,V_less__eq_2,V_k_2)))) # label(fact_ord_OatMost__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 175 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2) | -c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_c_2,V_d_2))))) # label(fact_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 176 (all V_y_2 all V_x_2 all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & -c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 177 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),V_B_2) -> (c_Finite__Set_Ofinite(T_b,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2)))))) # label(fact_card__inj__on__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 178 (all V_k_2 all V_A_2 all T_a (hAPP(c_Nat_OSuc,V_k_2) = c_Finite__Set_Ocard(T_a,V_A_2) <-> (exists B_b exists B_B (c_Set_Oinsert(T_a,B_b,B_B) = V_A_2 & -c_member(T_a,B_b,B_B) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> B_B = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) & c_Finite__Set_Ocard(T_a,B_B) = V_k_2)))) # label(fact_card__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 179 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_y != V_x))) # label(fact_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 180 (all V_b_2 all V_a_2 all T_a (class_Rings_Oring__1(T_a) -> (c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a)) -> (c_member(T_a,V_b_2,c_Int_Oring__1__class_OInts(T_a)) -> c_member(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2),c_Int_Oring__1__class_OInts(T_a)))))) # label(fact_Ints__add) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 181 (all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (((exists B_x c_member(T_a,B_x,V_A_2)) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Finite__Set_Ocard(T_a,V_A_2)))) # label(fact_card_Oneutral) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 182 (all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> c_Set_Oinsert(T_a,V_a_2,V_A_2) = V_A_2)) # label(fact_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 183 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))))) # label(fact_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 184 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 185 (all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 186 (all V_B_2 all V_A_2 all V_Fa_2 all V_f_2 all T_a (c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) -> hAPP(hAPP(V_f_2,hAPP(V_Fa_2,V_B_2)),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2)))))) # label(fact_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 187 (all V_n_2 all V_P_2 (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> (hBOOL(hAPP(V_P_2,V_n_2)) -> (exists B_k (hBOOL(hAPP(V_P_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B_k,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) & (all B_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_i,B_k) -> -hBOOL(hAPP(V_P_2,B_i)))) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_k,V_n_2)))))) # label(fact_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 188 (all T_a (class_Rings_Oring__1(T_a) -> c_member(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Oring__1__class_OInts(T_a)))) # label(fact_Ints__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 189 (all V_f_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)),c_Finite__Set_Ocard(T_a,V_A_2)))) # label(fact_card__image__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 190 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) -> V_m = V_n))) # label(fact_le__antisym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 191 (all V_x_2 all V_Fa_2 all V_g_2 all V_z_2 all V_f_2 all T_b all T_a (c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_Fa_2) -> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2)) # label(fact_folding__image__simple__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 192 (all V_l all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_l) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l))))) # label(fact_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 193 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 194 (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_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_insertCI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 195 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 196 (all V_n all V_m hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 197 (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_Set_Oimage(T_a,T_b,V_f_2,V_A_2) = c_Set_Oinsert(T_b,hAPP(V_f_2,V_x_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))) # label(fact_insert__image) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 198 (all T_a all V_B_2 all V_A_2 (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> V_A_2 = V_B_2)) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 199 (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_x = V_y)))) # label(fact_order__antisym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 200 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)))) # label(fact_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.98 201 (all V_n all V_k all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n) -> -(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n)))) # label(fact_add__leE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 202 (all V_c_2 all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> ((c_member(T_a,V_c_2,V_A_2) -> -c_member(T_a,V_c_2,V_B_2)) -> -(-c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2))))) # label(fact_equalityCE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 203 (all V_m all V_n (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,hAPP(c_Nat_OSuc,V_m)) -> V_m = V_n))) # label(fact_less__antisym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 204 (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]. 0.72/0.99 205 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_y = V_x | c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 206 (all V_A_2 all V_a_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_member(T_a,V_a_2,V_A_2) <-> c_member(T_b,hAPP(V_f_2,V_a_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))))) # label(fact_inj__image__mem__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 207 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 208 (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]. 0.72/0.99 209 (all V_c_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_psubsetD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 210 (all V_A_H_2 all T_b all T_a all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != V_A_2 -> ((exists B_f (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,B_f,V_A_2),V_A_H_2) & c_Fun_Oinj__on(T_a,T_b,B_f,V_A_2))) <-> (exists B_g V_A_2 = c_Set_Oimage(T_b,T_a,B_g,V_A_H_2))))) # label(fact_inj__on__iff__surj) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 211 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_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_inj__image__subset__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 212 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 213 (all V_y_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_member(T_b,V_y_2,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) -> V_y_2 = hAPP(V_f_2,hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2))))) # label(fact_f__the__inv__into__f) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 214 (all V_A_2 all V_f_2 all T_a all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2)) # label(fact_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 215 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 216 (all V_f_2 all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Set_Oimage(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_finite__UNIV__surj__inj) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 217 (all V_n_2 all V_m_2 (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_add__is__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 218 (all T_b all T_a all V_x_2 all V_f_2 all V_b_2 (V_b_2 = hAPP(V_f_2,V_x_2) -> c_member(T_a,V_b_2,c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_range__eqI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 219 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_A_2)) # label(fact_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 220 (all V_m_2 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_m_2)) = c_Set_Oinsert(tc_Nat_Onat,V_m_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_atLeastLessThan__singleton) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 221 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 222 (all V_n all V_m (V_n = V_m | c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 223 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 224 (all V_n hAPP(c_Nat_OSuc,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),V_n)) # label(fact_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 225 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)))) # label(fact_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 226 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n))) # label(fact_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 227 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (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_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2)))) # label(fact_atLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 228 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (V_x != V_y -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 229 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) | V_y_2 = V_x_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)))) # label(fact_order__le__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 230 (all V_x_2 all V_A_2 all V_y_2 all T_a (V_x_2 = V_y_2 | hBOOL(hAPP(V_A_2,V_x_2)) <-> hBOOL(hAPP(c_Set_Oinsert(T_a,V_y_2,V_A_2),V_x_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 231 (all V_B_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Finite__Set_Ofinite(T_b,V_B_2) -> ((exists B_f (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,B_f,V_A_2),V_B_2) & c_Fun_Oinj__on(T_a,T_b,B_f,V_A_2))) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2)))))) # label(fact_inj__on__iff__card__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 232 (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_set__mp) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 233 (all V_n_2 all V_A_2 all V_f_2 all T_a (hAPP(c_Nat_OSuc,V_n_2) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_A_2) -> (exists B_x (c_member(T_a,B_x,V_A_2) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(V_f_2,B_x)))))) # label(fact_setsum__SucD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 234 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 235 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 236 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> V_n_2 != V_m_2 & c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 237 (all T_a (class_Rings_Olinordered__semidom(T_a) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))))) # label(fact_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 238 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_not__less0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 239 (all T_a (class_Rings_Olinordered__semidom(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 240 (all V_B_2 all V_x_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_member(T_b,V_x_2,c_Set_Oimage(T_a,T_b,V_f_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,hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_x_2),V_B_2))))) # label(fact_the__inv__into__into) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 241 (all V_n_2 all V_m_2 all V_k_2 (V_m_2 = V_n_2 <-> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2))) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 242 (all V_f_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_A_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat) <-> (exists B_x (c_member(T_a,B_x,V_A_2) & c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(V_f_2,B_x) & (all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_xa != B_x -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,B_xa))))))))) # label(fact_setsum__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 243 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 244 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,V_m_2)) <-> V_m_2 = V_n_2))) # label(fact_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 245 (all V_m all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m) -> V_m != V_n)) # label(fact_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 246 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) & -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_g_2,V_f_2) <-> c_Orderings_Oord__class_Oless(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 247 (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]. 0.72/0.99 248 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 249 (all V_P_2 all V_n_2 all V_m_2 ((c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> ((V_m_2 = V_n_2 -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> ((c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)))))) # label(fact_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 250 (all V_a_2 all V_b_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_b_2 = c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2) <-> c_Groups_Ozero__class_Ozero(T_a) = V_a_2))) # label(fact_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 251 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) <-> -c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)))) # label(fact_atLeastatMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 252 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n)) # label(fact_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 253 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> (V_m != V_n -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)))) # label(fact_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 254 (all V_A_2 all T_a (-c_Finite__Set_Ofinite(T_a,V_A_2) | V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_card__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 255 (all V_A_2 all V_B_2 all T_a (c_Finite__Set_Ofinite(T_a,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Finite__Set_Ofinite(T_a,V_A_2)))) # label(fact_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 256 (all V_n_2 all V_k_2 all V_m_2 (V_n_2 = V_m_2 <-> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_k_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_k_2))) # label(fact_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 257 (all T_1 (class_Finite__Set_Ofinite(T_1) -> class_Finite__Set_Ofinite(tc_Option_Ooption(T_1)))) # label(arity_Option__Ooption__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 258 (all V_u_2 all V_l_2 all V_less__eq_2 all V_i_2 all T_a (c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeastAtMost(T_a,V_less__eq_2,V_l_2,V_u_2)) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)))) # label(fact_ord_OatLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 259 (all V_A_2 all V_f_2 all T_b all T_a (c_Finite__Set_Ofinite(T_a,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)) -> (c_Fun_Oinj__on(T_b,T_a,V_f_2,V_A_2) -> c_Finite__Set_Ofinite(T_b,V_A_2)))) # label(fact_finite__imageD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 260 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 261 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_member(T_a,V_x_2,V_A_2) -> (c_member(T_a,V_y_2,V_A_2) -> (hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) <-> V_y_2 = V_x_2))))) # label(fact_inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 262 (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]. 0.72/0.99 263 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)))) # label(fact_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 264 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 265 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Set_Oimage(T_a,T_b,V_f_2,V_B_2) = c_Set_Oimage(T_a,T_b,V_f_2,V_A_2) <-> V_B_2 = V_A_2))) # label(fact_inj__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 266 (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]. 0.72/0.99 267 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (V_b != V_a -> c_Orderings_Oord__class_Oless(T_a,V_b,V_a))))) # label(fact_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 268 (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(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 269 (all V_A_2 all V_a_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) <-> c_Finite__Set_Ofinite(T_a,c_Set_Oinsert(T_a,V_a_2,V_A_2)))) # label(fact_finite__insert) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 270 (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_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_subset__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 271 (all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_a_2))) # label(fact_atLeastAtMost__singleton) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 272 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_c_2,V_d_2)) <-> ((c_Orderings_Oord__class_Oless(T_a,V_b_2,V_d_2) | c_Orderings_Oord__class_Oless(T_a,V_c_2,V_a_2)) & c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2) | -c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)) & c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2)))) # label(fact_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 273 (all V_x all T_a (class_Orderings_Obot(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Orderings_Obot__class_Obot(T_a),V_x))) # label(fact_bot__least) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 274 (all V_t all V_s (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_s,V_t) -> V_s != V_t)) # label(fact_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 275 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2)))) # label(fact_atLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 276 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 277 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)))) # label(fact_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 278 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 279 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 280 (all V_y all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_y))) # label(fact_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 281 (all V_a_2 all V_b_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2) -> c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_atLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 282 (all V_y_2 all V_x_2 (V_y_2 = V_x_2 | -hBOOL(hAPP(c_fequal(V_x_2),V_y_2)))) # label(help_c__fequal__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 283 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a))) # label(fact_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 284 (all V_b_2 all V_c_2 all V_a_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))))) # label(fact_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 285 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)))) # label(fact_less__SucI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 286 (all V_x_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> V_x_2 = hAPP(c_Fun_Othe__inv__into(T_a,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_f_2),hAPP(V_f_2,V_x_2)))) # label(fact_the__inv__f__f) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 287 (all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2) <-> V_b_2 = V_c_2))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 288 (all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 289 (all V_f_2 all V_B_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),V_B_2,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) -> c_Finite__Set_Ofinite(T_b,V_B_2)))) # label(fact_finite__surj) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 290 (all V_n c_Nat_Osize__class_Osize(tc_Nat_Onat,V_n) = V_n) # label(fact_nat__size) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 291 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 | (exists B_j (c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_j,V_n_2) & hAPP(c_Nat_OSuc,B_j) = V_m_2)))) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 292 (all V_A_2 all V_f_2 all T_b all V_B_2 all T_a ((exists B_AA (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),B_AA,V_A_2) & V_B_2 = c_Set_Oimage(T_b,T_a,V_f_2,B_AA))) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)))) # label(fact_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 293 (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]. 0.72/0.99 294 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n)) # label(fact_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 295 (all V_x_2 all V_A_2 all V_Fa_2 all V_f_2 all T_a (c_Finite__Set_Ofolding__one(T_a,V_f_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (-c_member(T_a,V_x_2,V_A_2) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,c_Set_Oinsert(T_a,V_x_2,V_A_2))))))) # label(fact_folding__one_Oinsert) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 296 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 297 (all V_f_2 all V_p_2 all V_n_2 all V_m_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_p_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_p_2)) = c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2,V_p_2))))))) # label(fact_setsum__add__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 298 (all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2))) # label(fact_card__image) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 299 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) -> (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> V_m = V_n))) # label(fact_less__SucE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 300 (all V_nat_H hAPP(c_Nat_OSuc,V_nat_H) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 301 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a))) # label(fact_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 302 (all V_b all V_a all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> -c_Orderings_Oord__class_Oless(T_a,V_b,V_a)))) # label(fact_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 303 (all V_x_2 all V_A_2 all V_Fa_2 all V_f_2 all T_a (c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_member(T_a,V_x_2,V_A_2) -> hAPP(V_Fa_2,V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)))))) # label(fact_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 304 (all V_x_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_member(T_a,V_x_2,V_A_2) -> hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2))) # label(fact_the__inv__into__f__f) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 305 (all V_n all V_m (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = V_m -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n)) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 306 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 307 (all V_B_2 all V_A_2 all T_b all T_a (c_Orderings_Obot__class_Obot(tc_fun(tc_sum(T_a,T_b),tc_HOL_Obool)) = c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) & c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_B_2)) # label(fact_Plus__eq__empty__conv) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 308 (all V_n all V_m (V_m = V_n -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 309 (all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> c_Fun_Oinj__on(T_b,T_a,c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))) # label(fact_inj__on__the__inv__into) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 310 (all V_B_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Finite__Set_Ofinite(T_b,V_B_2) -> c_Finite__Set_Ocard(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2))))) # label(fact_card__Plus) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 311 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 312 (all V_p_2 all V_f_2 all V_n_2 all V_m_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,c_Groups_Oone__class_Oone(tc_Nat_Onat))) -> c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_p_2)))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_p_2)))))) # label(fact_setsum__ub__add__nat) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 313 (all V_x_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_member(T_a,V_x_2,V_A_2) -> c_Finite__Set_Ocard(T_a,V_A_2) = c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2))) & (-c_member(T_a,V_x_2,V_A_2) -> c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(T_a,V_A_2))))) # label(fact_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 314 (all V_P_2 all V_n_2 ((all B_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_m,V_n_2) -> hBOOL(hAPP(V_P_2,B_m)))) <-> (all B_x (c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 315 (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]. 0.72/0.99 316 (all V_f_2 all V_A_2 all T_a all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (-c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Groups_Ozero__class_Ozero(T_b) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,V_A_2)))) # label(fact_setsum__infinite) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 317 (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]. 0.72/0.99 318 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_not__leE) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 319 (all V_P_2 all V_n_2 ((exists B_x (hBOOL(hAPP(V_P_2,B_x)) & c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))) <-> (exists B_m (hBOOL(hAPP(V_P_2,B_m)) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_m,V_n_2))))) # label(fact_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 320 (all V_y_2 all V_x_2 (V_y_2 != V_x_2 | hBOOL(hAPP(c_fequal(V_x_2),V_y_2)))) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 321 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)))) # label(fact_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 322 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 323 (all V_y all V_x (hAPP(c_Nat_OSuc,V_y) = hAPP(c_Nat_OSuc,V_x) -> V_y = V_x)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 324 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))))) # label(fact_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 325 (all V_u_2 all V_l_2 all V_less_2 all V_i_2 all T_a (hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)) <-> c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThanLessThan(T_a,V_less_2,V_l_2,V_u_2)))) # label(fact_ord_OgreaterThanLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 326 (all V_b_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 327 (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]. 0.72/0.99 328 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 329 (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]. 0.72/0.99 330 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)))) # label(fact_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 331 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 332 (all V_B_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Finite__Set_Ofinite(T_b,V_B_2) -> c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2))))) # label(fact_finite__Plus) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 333 (all V_n all V_k all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_add__leD1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 334 (all V_f_2 all V_Fa_2 all T_a (c_Finite__Set_Ofinite(T_a,V_Fa_2) -> ((all B_x (c_member(T_a,B_x,V_Fa_2) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) <-> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_Fa_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_setsum__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 335 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 336 (all V_b_2 all V_a_2 all T_a (c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) -> V_b_2 = V_a_2)) # label(fact_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 337 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_j,V_k) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i),V_k)))) # label(fact_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 338 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_subset__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 339 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2)))) # label(fact_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 340 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_a (c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeastLessThan(T_a,V_less__eq_2,V_less_2,V_l_2,V_u_2)) <-> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)))) # label(fact_ord_OatLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 341 (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]. 0.72/0.99 342 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)))) # label(fact_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 343 (all V_k_2 all V_i_2 all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_k_2,V_i_2) <-> c_member(T_a,V_i_2,c_SetInterval_Oord__class_OgreaterThan(T_a,V_k_2))))) # label(fact_greaterThan__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 344 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b = V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 345 (all V_f_2 all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Fun_Oinj__on(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> c_Set_Oimage(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_finite__UNIV__inj__surj) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 346 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_psubset__imp__subset) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 347 (all V_c_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_subsetD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 348 (all B_h (c_Fun_Oinj__on(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,B_h,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) -> (c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))) = c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,B_h,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) -> v_thesis____))) # label(conj_0) # label(hypothesis) # label(non_clause). [assumption]. 0.72/0.99 349 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 350 (all T_2 all T_1 (class_Orderings_Oorder(T_1) -> class_Orderings_Oorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 351 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> -c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 352 (all V_nat c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_Onat_Onat__size(V_nat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Nat_Onat_Onat__size(hAPP(c_Nat_OSuc,V_nat))) # label(fact_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 353 (all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n)) # label(fact_le0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 354 (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]. 0.72/0.99 355 (all V_z all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)))) # label(fact_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 356 (all T_2 all T_1 (class_Orderings_Obot(T_1) -> class_Orderings_Obot(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Obot) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 357 (all V_x_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) | V_A_2 = c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 358 (all V_A_2 all V_B_2 all T_a (c_Finite__Set_Ofinite(T_a,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_Nat_Onat,c_Finite__Set_Ocard(T_a,V_B_2),c_Finite__Set_Ocard(T_a,V_A_2)) -> V_B_2 = V_A_2)))) # label(fact_card__seteq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 359 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Set_Oinsert(T_a,V_c_2,c_Set_Oinsert(T_a,V_d_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) <-> V_d_2 = V_a_2 & V_c_2 = V_b_2 | V_c_2 = V_a_2 & V_d_2 = V_b_2)) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 360 (all V_c_2 all T_a -c_member(T_a,V_c_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_empty__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 361 (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]. 0.72/0.99 362 (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]. 0.72/0.99 363 (all V_a_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Finite__Set_Ofinite(T_a,c_Set_Oinsert(T_a,V_a_2,V_A_2)))) # label(fact_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 364 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> (V_b = V_c -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 365 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 366 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (hAPP(V_f_2,V_x_2) = V_y_2 -> (c_member(T_a,V_x_2,V_A_2) -> hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2) = V_x_2)))) # label(fact_the__inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 367 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2))))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 368 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (V_c = V_b -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 369 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c))) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 370 (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]. 0.72/0.99 371 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_2) & class_Finite__Set_Ofinite(T_1) -> class_Finite__Set_Ofinite(tc_prod(T_2,T_1)))) # label(arity_prod__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 372 (all V_N_2 all V_h_2 all V_Fa_2 all V_f_2 all T_a (c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(V_f_2,B_x),B_y)) = hAPP(hAPP(V_f_2,hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (c_Finite__Set_Ofinite(T_a,V_N_2) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> hAPP(V_Fa_2,c_Set_Oimage(T_a,T_a,V_h_2,V_N_2)) = hAPP(V_h_2,hAPP(V_Fa_2,V_N_2))))))) # label(fact_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 373 (all T_2 all T_1 (class_Orderings_Oord(T_1) -> class_Orderings_Oord(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 374 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) | V_x_2 = V_y_2))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 375 (all V_A_2 all V_a_2 all T_a 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_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 376 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 377 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_x != V_y))) # label(fact_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 378 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 379 (all V_B_2 all V_A_2 all T_b all T_a (c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) -> c_Finite__Set_Ofinite(T_a,V_A_2))) # label(fact_finite__PlusD_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 380 (all V_x_2 all V_A_2 all V_Fa_2 all V_g_2 all V_z_2 all V_f_2 all T_b all T_a (c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_Fa_2) -> (c_Finite__Set_Ofinite(T_b,V_A_2) -> (c_member(T_b,V_x_2,V_A_2) -> hAPP(V_Fa_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_Fa_2,V_A_2)))))) # label(fact_folding__image__simple__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 381 (all V_f_2 all T_b all T_a c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_inj__on__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 382 (all V_u_2 all V_l_2 c_Finite__Set_Ofinite(tc_Nat_Onat,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2))) # label(fact_finite__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 383 (all V_f_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2) -> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)))) # label(fact_eq__card__imp__inj__on) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 384 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_atLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 385 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (V_y_2 != V_x_2 <-> c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) | c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2)))) # label(fact_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 386 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_k))) # label(fact_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 387 (all V_a_2 all V_b_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) -> c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_atLeastatMost__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 388 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) -> c_Set_Oinsert(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))) # label(fact_atLeastAtMostSuc__conv) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 389 (all T_b all T_a (c_Finite__Set_Ofinite(tc_fun(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool))) -> (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != c_Finite__Set_Ocard(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_finite__fun__UNIVD1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 390 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) <-> -c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 391 (all V_n_2 all V_m_2 (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,V_m_2)))) # label(fact_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 392 (all V_y all V_z all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_z) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)))) # label(fact_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 393 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 394 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (V_b = V_c -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 395 (all V_n_2 all V_m_2 ((exists B_k V_n_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 396 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),hAPP(c_Nat_OSuc,V_n)))) # label(fact_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 397 (all V_a_2 all V_D_2 all V_C_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_C_2),c_Set_Oinsert(T_a,V_a_2,V_D_2)))) # label(fact_insert__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 398 (all V_f_2 all T_a all T_b c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_image__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 399 (all V_n all V_m all V_l all V_k (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l) -> (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_l) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)))) # label(fact_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 400 (all V_n c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n))) # label(fact_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 401 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 402 (all V_P_2 all V_n_2 ((all B_x (c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)) -> hBOOL(hAPP(V_P_2,B_x)))) <-> (all B_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m,V_n_2) -> hBOOL(hAPP(V_P_2,B_m)))))) # label(fact_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 403 (all V_nat_H_2 all V_nat_2 (hAPP(c_Nat_OSuc,V_nat_H_2) = hAPP(c_Nat_OSuc,V_nat_2) <-> V_nat_H_2 = V_nat_2)) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 404 (all V_x_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2))))) # label(fact_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 405 (all T_b all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Finite__Set_Ofinite(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> c_Finite__Set_Ofinite(tc_prod(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)))))) # label(fact_finite__Prod__UNIV) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 406 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n))) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 407 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j,V_k) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_k)))) # label(fact_le__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 408 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 409 (all V_a_2 all V_B_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oinsert(T_a,V_a_2,V_B_2))) # label(fact_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 410 (all V_u_2 all V_l_2 c_Finite__Set_Ofinite(tc_Nat_Onat,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,V_u_2))) # label(fact_finite__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 0.72/0.99 411 (all V_f_2 all V_a_2 all V_Fa_2 all T_a all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (c_Finite__Set_Ofinite(T_a,V_Fa_2) -> (-c_member(T_a,V_a_2,V_Fa_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,c_Set_Oinsert(T_a,V_a_2,V_Fa_2)) = c_Groups_Oplus__class_Oplus(T_b,hAPP(V_f_2,V_a_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,V_Fa_2)))))) # label(fact_setsum__insert) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 412 (all V_f_2 all V_m_2 all V_n_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)),hAPP(V_f_2,V_n_2))) & (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) -> c_Groups_Ozero__class_Ozero(T_a) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))))) # label(fact_setsum__op__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 413 (all V_f_2 all V_n_2 all V_m_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(V_f_2,V_m_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2)))))) # label(fact_setsum__head__Suc) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 414 (all T_b all V_f_2 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) -> (V_b_2 = hAPP(V_f_2,V_x_2) -> c_member(T_b,V_b_2,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))))) # label(fact_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 415 (all V_A_2 all T_a (-c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_card__infinite) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 416 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 417 (all V_A_2 all V_f_2 all T_b all V_z_2 all T_a ((exists B_x (c_member(T_b,B_x,V_A_2) & V_z_2 = hAPP(V_f_2,B_x))) <-> c_member(T_a,V_z_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)))) # label(fact_image__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 418 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) | 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_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 419 (all T_a c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Finite__Set_Ocard(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 420 (all V_f_2 all T_b 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_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)))) # label(fact_image__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 421 (all V_g_2 all V_A_2 all T_a all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (-c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Groups_Ozero__class_Ozero(T_b) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2)))) # label(fact_setsum_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 422 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_sum(T_2,T_1)))) # label(arity_sum__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 423 (all V_A_2 all V_a_2 all T_a c_Set_Oinsert(T_a,V_a_2,V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 424 (all T_b all V_x_2 all V_f_2 all T_a c_member(T_a,hAPP(V_f_2,V_x_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_rangeI) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 425 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (V_a != V_b -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 426 (all T_a (class_Finite__Set_Ofinite(T_a) -> c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_finite__UNIV) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 427 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))))) # label(fact_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 428 (all V_B_2 all T_b all V_A_2 all T_a ((-(c_Finite__Set_Ofinite(T_a,V_A_2) & c_Finite__Set_Ofinite(T_b,V_B_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Finite__Set_Ocard(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2))) & (c_Finite__Set_Ofinite(T_a,V_A_2) & c_Finite__Set_Ofinite(T_b,V_B_2) -> c_Finite__Set_Ocard(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2))))) # label(fact_card__Plus__conv__if) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 429 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> (V_n != hAPP(c_Nat_OSuc,V_m) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)))) # label(fact_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 430 (all T_b all T_a (c_Finite__Set_Ofinite(tc_fun(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool))) -> c_Finite__Set_Ofinite(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_finite__fun__UNIVD2) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 431 (all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),V_B_2) -> (c_Fun_Oinj__on(T_b,T_a,V_g_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_b,T_a,V_g_2,V_B_2),V_A_2) -> (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Finite__Set_Ofinite(T_b,V_B_2) -> c_Finite__Set_Ocard(T_b,V_B_2) = c_Finite__Set_Ocard(T_a,V_A_2)))))))) # label(fact_card__bij__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 432 (all V_b_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_member(T_b,V_b_2,c_Set_Oimage(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) <-> (exists B_x ((all B_y (V_b_2 = hAPP(V_f_2,B_y) -> B_y = B_x)) & hAPP(V_f_2,B_x) = V_b_2))))) # label(fact_range__ex1__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 433 (all T_2 all T_1 (class_Orderings_Opreorder(T_1) -> class_Orderings_Opreorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 434 (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]. 0.72/1.00 435 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 436 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (V_x != V_y -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 437 (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]. 0.72/1.00 438 (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_member(T_b,hAPP(V_f_2,V_x_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))) # label(fact_imageI) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 439 (all T_a c_Finite__Set_Ofinite(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_finite_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 440 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_b))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 441 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 442 (all V_A_2 all V_B_2 all T_a (c_Finite__Set_Ofinite(T_a,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,V_B_2))))) # label(fact_psubset__card__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 443 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 444 (all V_x_2 all V_Fa_2 all V_f_2 all T_a (c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2) -> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2)) # label(fact_folding__one__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 445 (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_A_2 = V_B_2))) # label(fact_equalityI) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 446 (all V_A_2 all V_B_2 all T_a (c_Finite__Set_Ofinite(T_a,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_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,V_B_2))))) # label(fact_card__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 447 (all V_j_2 all V_i_2 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2)) = c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2,V_j_2))) # label(fact_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 448 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x))) # label(fact_order__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 449 (all V_a_2 all T_a all V_A_2 (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> -c_member(T_a,V_a_2,V_A_2))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 450 (all V_k_2 all V_less_2 all V_i_2 all T_a (hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_k_2)) <-> c_member(T_a,V_i_2,c_SetInterval_Oord_OlessThan(T_a,V_less_2,V_k_2)))) # label(fact_ord_OlessThan__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 451 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),hAPP(c_Nat_OSuc,V_m_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_m_2))) # label(fact_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 452 (all V_a_2 all T_a (class_Int_Oring__char__0(T_a) -> (c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a)) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),V_a_2),V_a_2) != c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_Ints__odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 453 (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]. 0.72/1.00 454 (all V_A_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) -> c_Set_Oimage(T_b,T_a,c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = V_A_2)) # label(fact_the__inv__into__onto) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 455 (all V_B_2 all V_a_2 all T_a c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_a_2,V_B_2))) # label(fact_insertI1) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 456 (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_Finite__Set_Ofinite(T_a,V_B_2) -> c_Finite__Set_Ofinite(T_a,V_A_2)))) # label(fact_finite__subset) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 457 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_subset__psubset__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 458 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 459 (all V_c_2 all V_a_2 all V_b_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (V_b_2 = V_c_2 <-> c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2) = c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2)))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 460 (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]. 0.72/1.00 461 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) <-> V_n_2 != V_m_2)) # label(fact_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 462 (all V_n_2 all V_m_2 ((exists B_k hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k)) = V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 463 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_a (c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2) <-> V_x_2 = V_y_2))) # label(fact_inj__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 464 (all V_n_2 all V_m_2 (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n_2 & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 465 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 466 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> V_m_2 = V_n_2 | c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 467 (all V_f_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_a,T_a,V_f_2,V_A_2),V_A_2) -> (c_Fun_Oinj__on(T_a,T_a,V_f_2,V_A_2) -> V_A_2 = c_Set_Oimage(T_a,T_a,V_f_2,V_A_2))))) # label(fact_endo__inj__surj) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 468 (all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_2,V_A_2) <-> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 469 (all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) <-> c_Finite__Set_Ofinite(tc_Option_Ooption(T_a),c_Orderings_Otop__class_Otop(tc_fun(tc_Option_Ooption(T_a),tc_HOL_Obool))))) # label(fact_finite__option__UNIV) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 470 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2)))) # label(fact_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 471 (all V_x_2 all T_b all T_a (class_Orderings_Otop(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_b,T_a)),V_x_2))) # label(fact_top__apply) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 472 (all V_g_2 all V_f_2 all V_d_2 all V_b_2 all V_c_2 all V_a_2 all T_a all T_b (class_Groups_Ocomm__monoid__add(T_b) & class_Orderings_Oord(T_a) -> (V_a_2 = V_c_2 -> (V_b_2 = V_d_2 -> ((all B_x (c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,B_x) -> (c_Orderings_Oord__class_Oless(T_a,B_x,V_d_2) -> hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)))) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2))))))) # label(fact_setsum__ivl__cong) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 473 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),hAPP(c_Nat_OSuc,V_n_2)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 474 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 475 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,hAPP(c_Nat_OSuc,V_u_2))) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 476 (all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_finite__UNIV__card__ge__0) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 477 (all V_f_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> ((exists B_x (c_member(T_a,B_x,V_A_2) & (all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_x != B_xa -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & hAPP(V_f_2,B_x) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) <-> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_A_2)))) # label(fact_setsum__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 478 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_y) | V_y = V_x | c_Orderings_Oord__class_Oless(T_a,V_y,V_x))) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 479 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) <-> -c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)))) # label(fact_atLeastatMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 480 (all V_y_2 all V_x_2 all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y_2) -> (c_Groups_Oplus__class_Oplus(T_a,V_x_2,V_y_2) = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = V_y_2 & c_Groups_Ozero__class_Ozero(T_a) = V_x_2))))) # label(fact_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 481 (all V_A_2 all V_f_2 all T_b all T_a (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 482 (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(T_a,V_y,V_x))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 483 (all V_x_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_member(T_a,V_x_2,V_A_2) -> c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,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))))))))) # label(fact_card_Oremove) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 484 (all V_a_2 all T_a -c_member(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_emptyE) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 485 (all V_f_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2) <-> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)))) # label(fact_inj__on__iff__eq__card) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 486 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 487 (all V_B_2 all V_A_2 all T_a (V_A_2 != V_B_2 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 488 (all V_b_2 all V_c_2 all V_a_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 489 (all V_m all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m))) # label(fact_le__add1) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 490 (all V_x_2 all T_a (class_Groups_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_x_2 <-> c_Groups_Ozero__class_Ozero(T_a) = V_x_2))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 491 (all V_A_2 all V_b_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) | V_b_2 = V_a_2 <-> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)))) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/1.00 0.72/1.00 ============================== end of process non-clausal formulas === 0.72/1.00 0.72/1.00 ============================== PROCESS INITIAL CLAUSES =============== 0.72/1.00 0.72/1.00 ============================== PREDICATE ELIMINATION ================= 0.72/1.00 492 class_Groups_Ocomm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 0.72/1.00 493 -class_Groups_Ocomm__monoid__add(A) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),C) | c_Groups_Oplus__class_Oplus(A,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,B)),hAPP(D,hAPP(c_Nat_OSuc,B))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__cl__ivl__Suc) # label(axiom). [clausify(3)]. 0.72/1.00 494 -class_Groups_Ocomm__monoid__add(A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),C) | c_Groups_Ozero__class_Ozero(A) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__cl__ivl__Suc) # label(axiom). [clausify(3)]. 0.72/1.00 495 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setsum__empty) # label(axiom). [clausify(106)]. 0.72/1.00 496 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -c_Finite__Set_Ofinite(B,C) | c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)) = C | c_member(B,f7(D,E,C,B,A),C) | c_Orderings_Oord__class_Oless(A,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,E,C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,D,C)) # label(fact_setsum__strict__mono) # label(axiom). [clausify(118)]. 0.72/1.00 497 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -c_Finite__Set_Ofinite(B,C) | c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)) = C | -c_Orderings_Oord__class_Oless(A,hAPP(D,f7(E,D,C,B,A)),hAPP(E,f7(E,D,C,B,A))) | c_Orderings_Oord__class_Oless(A,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,D,C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,E,C)) # label(fact_setsum__strict__mono) # label(axiom). [clausify(118)]. 0.72/1.00 498 -class_Groups_Ocomm__monoid__add(A) | -c_Finite__Set_Ofinite(B,C) | c_member(B,D,C) | c_Groups_Oplus__class_Oplus(A,hAPP(E,D),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,E,C)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,E,c_Set_Oinsert(B,D,C)) # label(fact_setsum_Oinsert) # label(axiom). [clausify(119)]. 0.72/1.00 499 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ozero__class_Ozero(A),B) = B # label(fact_add__0) # label(axiom). [clausify(126)]. 0.72/1.00 500 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,B,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,B,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) # label(fact_setsum__shift__lb__Suc0__0__upt) # label(axiom). [clausify(130)]. 0.72/1.00 501 -class_Groups_Ocomm__monoid__add(A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Groups_Oplus__class_Oplus(A,hAPP(D,B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),C))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,C)) # label(fact_setsum__head__upt__Suc) # label(axiom). [clausify(134)]. 0.72/1.00 502 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | c_Set_Oimage(B,C,D,E) != F | c_member(B,f11(V6,V7,F,E,D,C,B,A),E) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A,V6,F) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V7,E) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(142)]. 0.72/1.00 503 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | c_Set_Oimage(B,C,D,E) != F | hAPP(V6,hAPP(D,f11(V6,V7,F,E,D,C,B,A))) != hAPP(V7,f11(V6,V7,F,E,D,C,B,A)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A,V6,F) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V7,E) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(142)]. 0.72/1.00 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A)),hAPP(C,hAPP(c_Nat_OSuc,A))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(492,a,493,a)]. 0.72/1.00 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(492,a,494,a)]. 0.72/1.00 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(492,a,495,a)]. 0.72/1.00 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) | -c_Finite__Set_Ofinite(A,B) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_member(A,f7(C,D,B,A,tc_Nat_Onat),B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,D,B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,C,B)). [resolve(492,a,496,b)]. 0.72/1.00 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) | -c_Finite__Set_Ofinite(A,B) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(C,f7(D,C,B,A,tc_Nat_Onat)),hAPP(D,f7(D,C,B,A,tc_Nat_Onat))) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,C,B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,D,B)). [resolve(492,a,497,b)]. 0.72/1.00 Derived: -c_Finite__Set_Ofinite(A,B) | c_member(A,C,B) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(D,C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,D,B)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,D,c_Set_Oinsert(A,C,B)). [resolve(492,a,498,a)]. 0.72/1.00 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) = A. [resolve(492,a,499,a)]. 0.72/1.00 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,A,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,A,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)). [resolve(492,a,500,a)]. 0.72/1.00 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(C,A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B)). [resolve(492,a,501,a)]. 0.72/1.00 Derived: -c_Fun_Oinj__on(A,B,C,D) | c_Set_Oimage(A,B,C,D) != E | c_member(A,f11(F,V6,E,D,C,B,A,tc_Nat_Onat),D) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat,F,E) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,D). [resolve(492,a,502,a)]. 0.72/1.00 Derived: -c_Fun_Oinj__on(A,B,C,D) | c_Set_Oimage(A,B,C,D) != E | hAPP(F,hAPP(C,f11(F,V6,E,D,C,B,A,tc_Nat_Onat))) != hAPP(V6,f11(F,V6,E,D,C,B,A,tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat,F,E) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,D). [resolve(492,a,503,a)]. 0.72/1.01 504 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setsum_Oempty) # label(axiom). [clausify(158)]. 0.72/1.01 505 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add_Ocomm__neutral) # label(axiom). [clausify(221)]. 0.72/1.01 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(505,a,492,a)]. 0.72/1.01 506 -class_Groups_Ocomm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | c_Groups_Oplus__class_Oplus(A,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,E,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,C)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,E,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,E,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,D)) # label(fact_setsum__add__nat__ivl) # label(axiom). [clausify(297)]. 0.72/1.01 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,C))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,C)). [resolve(506,a,492,a)]. 0.72/1.01 507 -class_Groups_Ocomm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,c_Groups_Oone__class_Oone(tc_Nat_Onat))) | c_Groups_Oplus__class_Oplus(A,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,E)))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,E))) # label(fact_setsum__ub__add__nat) # label(axiom). [clausify(312)]. 0.72/1.01 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,c_Groups_Oone__class_Oone(tc_Nat_Onat))) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D))). [resolve(507,a,492,a)]. 0.72/1.01 508 -class_Groups_Ocomm__monoid__add(A) | c_Finite__Set_Ofinite(B,C) | c_Groups_Ozero__class_Ozero(A) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,D,C) # label(fact_setsum__infinite) # label(axiom). [clausify(316)]. 0.72/1.01 Derived: c_Finite__Set_Ofinite(A,B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,C,B). [resolve(508,a,492,a)]. 0.72/1.02 509 -class_Groups_Ocomm__monoid__add(A) | -c_Finite__Set_Ofinite(B,C) | c_member(B,D,C) | c_Groups_Oplus__class_Oplus(A,hAPP(E,D),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,E,C)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,E,c_Set_Oinsert(B,D,C)) # label(fact_setsum__insert) # label(axiom). [clausify(411)]. 0.72/1.02 510 -class_Groups_Ocomm__monoid__add(A) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Groups_Oplus__class_Oplus(A,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,B)),hAPP(D,B)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__op__ivl__Suc) # label(axiom). [clausify(412)]. 0.72/1.02 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,A)),hAPP(C,A)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(510,a,492,a)]. 0.72/1.02 511 -class_Groups_Ocomm__monoid__add(A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Groups_Ozero__class_Ozero(A) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__op__ivl__Suc) # label(axiom). [clausify(412)]. 0.72/1.02 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(511,a,492,a)]. 0.72/1.02 512 -class_Groups_Ocomm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Groups_Oplus__class_Oplus(A,hAPP(D,B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),C))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A,D,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) # label(fact_setsum__head__Suc) # label(axiom). [clausify(413)]. 0.72/1.02 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(C,A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,C,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)). [resolve(512,a,492,a)]. 0.72/1.02 513 -class_Groups_Ocomm__monoid__add(A) | c_Finite__Set_Ofinite(B,C) | c_Groups_Ozero__class_Ozero(A) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,D,C) # label(fact_setsum_Oinfinite) # label(axiom). [clausify(421)]. 0.72/1.02 514 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | c_Orderings_Oord__class_Oless__eq(B,C,f42(V6,V7,E,F,C,D,B,A)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V6,c_SetInterval_Oord__class_OatLeastLessThan(B,C,E)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V7,c_SetInterval_Oord__class_OatLeastLessThan(B,D,F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(472)]. 0.72/1.02 Derived: -class_Orderings_Oord(A) | B != C | D != E | c_Orderings_Oord__class_Oless__eq(A,B,f42(F,V6,D,E,B,C,A,tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,F,c_SetInterval_Oord__class_OatLeastLessThan(A,B,D)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,C,E)). [resolve(514,a,492,a)]. 0.72/1.02 515 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | c_Orderings_Oord__class_Oless(B,f42(V6,V7,E,F,C,D,B,A),E) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V6,c_SetInterval_Oord__class_OatLeastLessThan(B,C,E)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V7,c_SetInterval_Oord__class_OatLeastLessThan(B,D,F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(472)]. 0.72/1.07 Derived: -class_Orderings_Oord(A) | B != C | D != E | c_Orderings_Oord__class_Oless(A,f42(F,V6,D,E,B,C,A,tc_Nat_Onat),D) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,F,c_SetInterval_Oord__class_OatLeastLessThan(A,B,D)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,C,E)). [resolve(515,a,492,a)]. 0.72/1.07 516 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hAPP(V6,f42(V6,V7,E,F,C,D,B,A)) != hAPP(V7,f42(V6,V7,E,F,C,D,B,A)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V6,c_SetInterval_Oord__class_OatLeastLessThan(B,C,E)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A,V7,c_SetInterval_Oord__class_OatLeastLessThan(B,D,F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(472)]. 0.72/1.07 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f42(F,V6,D,E,B,C,A,tc_Nat_Onat)) != hAPP(V6,f42(F,V6,D,E,B,C,A,tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,F,c_SetInterval_Oord__class_OatLeastLessThan(A,B,D)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,C,E)). [resolve(516,a,492,a)]. 0.72/1.07 517 -class_Rings_Olinordered__semidom(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oone__class_Oone(A)) # label(fact_zero__less__one) # label(axiom). [clausify(31)]. 0.72/1.07 518 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 0.72/1.07 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(517,a,518,a)]. 0.72/1.07 519 -class_Rings_Olinordered__semidom(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oone__class_Oone(A))) # label(fact_less__add__one) # label(axiom). [clausify(71)]. 0.72/1.07 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(519,a,518,a)]. 0.72/1.07 520 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),c_Groups_Ozero__class_Ozero(A)) # label(fact_not__one__less__zero) # label(axiom). [clausify(223)]. 0.72/1.07 521 -class_Rings_Olinordered__semidom(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,c_Groups_Oone__class_Oone(A),c_Groups_Oone__class_Oone(A))) # label(fact_zero__less__two) # label(axiom). [clausify(237)]. 0.72/1.07 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(521,a,518,a)]. 0.72/1.07 522 -class_Rings_Olinordered__semidom(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oone__class_Oone(A)) # label(fact_zero__le__one) # label(axiom). [clausify(239)]. 0.72/1.07 523 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A),c_Groups_Ozero__class_Ozero(A)) # label(fact_not__one__le__zero) # label(axiom). [clausify(349)]. 0.72/1.07 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(523,a,518,a)]. 0.72/1.07 524 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,D) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_pos__add__strict) # label(axiom). [clausify(443)]. 0.72/1.07 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(524,a,518,a)]. 0.86/1.09 525 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 0.86/1.09 526 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_add__increasing) # label(axiom). [clausify(11)]. 0.86/1.09 527 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__neg__neg) # label(axiom). [clausify(80)]. 0.86/1.09 528 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,D) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_add__strict__increasing2) # label(axiom). [clausify(103)]. 0.86/1.09 529 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Oplus__class_Oplus(A,D,B)) # label(fact_add__increasing2) # label(axiom). [clausify(137)]. 0.86/1.09 530 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__neg__nonpos) # label(axiom). [clausify(207)]. 0.86/1.09 531 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__pos__pos) # label(axiom). [clausify(215)]. 0.86/1.09 532 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_add__strict__increasing) # label(axiom). [clausify(243)]. 0.86/1.09 533 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__nonneg__pos) # label(axiom). [clausify(278)]. 0.86/1.09 534 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__nonpos__neg) # label(axiom). [clausify(328)]. 0.86/1.09 535 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__nonpos__nonpos) # label(axiom). [clausify(378)]. 0.86/1.09 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(525,a,526,a)]. 0.86/1.09 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(525,a,527,a)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(525,a,528,a)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,A)). [resolve(525,a,529,a)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(525,a,530,a)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(525,a,532,a)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(525,a,534,a)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(525,a,535,a)]. 0.86/1.10 536 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__nonneg__nonneg) # label(axiom). [clausify(416)]. 0.86/1.10 537 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__pos__nonneg) # label(axiom). [clausify(441)]. 0.86/1.10 538 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Groups_Ozero__class_Ozero(A) != c_Groups_Oplus__class_Oplus(A,B,C) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(480)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(538,a,525,a)]. 0.86/1.10 539 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Groups_Ozero__class_Ozero(A) != c_Groups_Oplus__class_Oplus(A,B,C) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(480)]. 0.86/1.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(539,a,525,a)]. 0.86/1.11 540 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Groups_Ozero__class_Ozero(A) = c_Groups_Oplus__class_Oplus(A,B,C) | c_Groups_Ozero__class_Ozero(A) != C | c_Groups_Ozero__class_Ozero(A) != B # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(480)]. 0.86/1.11 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A. [resolve(540,a,525,a)]. 0.86/1.11 541 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 0.86/1.11 542 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | B != C | -c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(14)]. 0.86/1.11 543 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | B = C | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(14)]. 0.86/1.11 544 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) != c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) | -c_Orderings_Oord__class_Oless(A,D,E) | -c_Orderings_Oord__class_Oless(A,B,C) | C = E # label(fact_atLeastLessThan__inj_I2_J) # label(axiom). [clausify(29)]. 0.86/1.11 545 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C != B | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(37)]. 0.86/1.11 546 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C = B | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(37)]. 0.86/1.11 547 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__less) # label(axiom). [clausify(39)]. 0.86/1.11 548 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__less) # label(axiom). [clausify(39)]. 0.86/1.11 549 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C = B | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__cases) # label(axiom). [clausify(44)]. 0.86/1.11 550 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | D != B | E != C | c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) = c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(47)]. 0.86/1.11 551 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | D = B | c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) != c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(47)]. 0.86/1.11 552 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | E = C | c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) != c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(47)]. 0.86/1.11 553 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) != c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) | -c_Orderings_Oord__class_Oless(A,D,E) | -c_Orderings_Oord__class_Oless(A,B,C) | B = D # label(fact_atLeastLessThan__inj_I1_J) # label(axiom). [clausify(57)]. 0.86/1.11 554 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | B = C # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(70)]. 0.86/1.11 555 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,B,C) | B != C # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(70)]. 0.86/1.11 556 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__linear) # label(axiom). [clausify(88)]. 0.86/1.11 557 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool),c_SetInterval_Oord__class_OgreaterThan(A,C),c_SetInterval_Oord__class_OgreaterThan(A,B)) # label(fact_greaterThan__subset__iff) # label(axiom). [clausify(98)]. 0.86/1.11 558 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool),c_SetInterval_Oord__class_OgreaterThan(A,C),c_SetInterval_Oord__class_OgreaterThan(A,B)) # label(fact_greaterThan__subset__iff) # label(axiom). [clausify(98)]. 0.86/1.11 Derived: c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B) != c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | B = D. [resolve(541,a,544,a)]. 0.86/1.11 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(541,a,547,a)]. 0.86/1.11 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(541,a,548,a)]. 0.86/1.11 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | C != A | D != B | c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B). [resolve(541,a,550,a)]. 0.86/1.11 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | C = A | c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D) != c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B). [resolve(541,a,551,a)]. 0.86/1.11 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,B),c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,A)). [resolve(541,a,557,a)]. 0.86/1.11 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,B),c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,A)). [resolve(541,a,558,a)]. 0.86/1.11 559 -class_Orderings_Olinorder(A) | B != C | c_SetInterval_Oord__class_OgreaterThan(A,B) = c_SetInterval_Oord__class_OgreaterThan(A,C) # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(113)]. 0.86/1.11 Derived: A != B | c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,A) = c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,B). [resolve(559,a,541,a)]. 0.86/1.11 560 -class_Orderings_Olinorder(A) | B = C | c_SetInterval_Oord__class_OgreaterThan(A,B) != c_SetInterval_Oord__class_OgreaterThan(A,C) # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(113)]. 0.86/1.11 Derived: A = B | c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,A) != c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,B). [resolve(560,a,541,a)]. 0.86/1.11 561 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_leD) # label(axiom). [clausify(114)]. 0.86/1.11 562 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastLessThan(A,B,C),c_SetInterval_Oord__class_OatLeastLessThan(A,D,E)) | c_Orderings_Oord__class_Oless__eq(A,C,E) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_atLeastLessThan__subset__iff) # label(axiom). [clausify(135)]. 0.86/1.11 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,D) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(562,a,541,a)]. 0.86/1.11 563 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastLessThan(A,B,C),c_SetInterval_Oord__class_OatLeastLessThan(A,D,E)) | c_Orderings_Oord__class_Oless__eq(A,D,B) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_atLeastLessThan__subset__iff) # label(axiom). [clausify(135)]. 0.94/1.18 Derived: -c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(563,a,541,a)]. 0.94/1.18 564 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_leI) # label(axiom). [clausify(166)]. 0.94/1.18 565 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__leE) # label(axiom). [clausify(318)]. 0.94/1.18 566 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | B = C # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(374)]. 0.94/1.18 567 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(374)]. 0.94/1.18 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(567,a,541,a)]. 0.94/1.18 568 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | B != C # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(374)]. 0.94/1.18 569 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__neq__iff) # label(axiom). [clausify(385)]. 0.94/1.18 570 -class_Orderings_Olinorder(A) | B != C | -c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neq__iff) # label(axiom). [clausify(385)]. 0.94/1.18 571 -class_Orderings_Olinorder(A) | B != C | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__neq__iff) # label(axiom). [clausify(385)]. 0.94/1.18 572 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(390)]. 0.94/1.18 573 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(390)]. 0.94/1.18 574 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,C,B) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neqE) # label(axiom). [clausify(436)]. 0.94/1.18 575 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__le__cases) # label(axiom). [clausify(458)]. 0.94/1.18 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(575,a,541,a)]. 0.94/1.18 576 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C = B | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__less__linear) # label(axiom). [clausify(478)]. 0.94/1.18 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | B = A | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(576,a,541,a)]. 0.94/1.18 577 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__le__less__linear) # label(axiom). [clausify(482)]. 0.94/1.18 578 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 0.94/1.18 579 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,D,C) | B = D # label(fact_add__right__imp__eq) # label(axiom). [clausify(30)]. 0.94/1.18 580 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,B,D) | C = D # label(fact_add__left__imp__eq) # label(axiom). [clausify(63)]. 0.94/1.18 581 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,B,D) | C = D # label(fact_add__left__cancel) # label(axiom). [clausify(287)]. 1.01/1.24 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C) | B = C. [resolve(581,a,578,a)]. 1.01/1.24 582 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) = c_Groups_Oplus__class_Oplus(A,B,D) | C != D # label(fact_add__left__cancel) # label(axiom). [clausify(287)]. 1.01/1.24 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C) | B != C. [resolve(582,a,578,a)]. 1.01/1.24 583 -class_Groups_Ocancel__semigroup__add(A) | B != C | c_Groups_Oplus__class_Oplus(A,B,D) = c_Groups_Oplus__class_Oplus(A,C,D) # label(fact_add__right__cancel) # label(axiom). [clausify(459)]. 1.01/1.24 Derived: A != B | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C). [resolve(583,a,578,a)]. 1.01/1.24 584 -class_Groups_Ocancel__semigroup__add(A) | B = C | c_Groups_Oplus__class_Oplus(A,B,D) != c_Groups_Oplus__class_Oplus(A,C,D) # label(fact_add__right__cancel) # label(axiom). [clausify(459)]. 1.01/1.24 Derived: A = B | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C). [resolve(584,a,578,a)]. 1.01/1.24 585 -class_Groups_Oab__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(369)]. 1.01/1.24 586 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 1.01/1.24 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)). [resolve(585,a,586,a)]. 1.01/1.24 587 -class_Groups_Oordered__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__left__mono) # label(axiom). [clausify(324)]. 1.01/1.24 588 class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 1.01/1.24 589 -class_Groups_Oordered__ab__semigroup__add(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_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_add__mono) # label(axiom). [clausify(408)]. 1.01/1.24 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)). [resolve(589,a,588,a)]. 1.01/1.24 590 -class_Groups_Oordered__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__right__mono) # label(axiom). [clausify(427)]. 1.01/1.24 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)). [resolve(590,a,588,a)]. 1.01/1.24 591 class_Rings_Ocomm__semiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.01/1.24 592 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,C) = c_Groups_Oplus__class_Oplus(A,C,B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(45)]. 1.01/1.24 593 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(84)]. 1.01/1.24 594 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,E)) = c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(123)]. 1.09/1.33 595 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) = c_Groups_Oplus__class_Oplus(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(156)]. 1.09/1.33 596 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(161)]. 1.09/1.33 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,A). [resolve(591,a,592,a)]. 1.09/1.33 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,D)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)). [resolve(591,a,594,a)]. 1.09/1.33 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(591,a,595,a)]. 1.09/1.33 597 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(212)]. 1.09/1.33 598 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ozero__class_Ozero(A),B) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(264)]. 1.09/1.33 599 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,D),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(440)]. 1.09/1.33 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),B). [resolve(599,a,591,a)]. 1.09/1.33 600 class_Groups_Omonoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__add) # label(axiom). [assumption]. 1.09/1.33 601 -class_Groups_Omonoid__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ozero__class_Ozero(A),B) = B # label(fact_add__0__left) # label(axiom). [clausify(81)]. 1.09/1.33 602 -class_Groups_Omonoid__add(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add__0__right) # label(axiom). [clausify(171)]. 1.09/1.33 603 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 1.09/1.33 604 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless(A,C,D) # label(fact_add__less__imp__less__left) # label(axiom). [clausify(109)]. 1.09/1.33 605 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_add__le__imp__le__left) # label(axiom). [clausify(225)]. 1.09/1.33 606 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,C)) | c_Orderings_Oord__class_Oless(A,B,D) # label(fact_add__less__imp__less__right) # label(axiom). [clausify(276)]. 1.09/1.33 607 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(284)]. 1.15/1.44 608 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(284)]. 1.15/1.44 609 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless(A,C,D) # label(fact_add__less__cancel__left) # label(axiom). [clausify(306)]. 1.15/1.44 610 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | -c_Orderings_Oord__class_Oless(A,C,D) # label(fact_add__less__cancel__left) # label(axiom). [clausify(306)]. 1.15/1.44 611 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_add__le__cancel__left) # label(axiom). [clausify(330)]. 1.15/1.44 612 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | -c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_add__le__cancel__left) # label(axiom). [clausify(330)]. 1.15/1.44 613 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,C)) | c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_add__le__imp__le__right) # label(axiom). [clausify(342)]. 1.15/1.44 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(603,a,605,a)]. 1.15/1.44 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,C). [resolve(603,a,606,a)]. 1.15/1.44 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)). [resolve(603,a,608,a)]. 1.15/1.44 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(603,a,612,a)]. 1.15/1.44 614 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__less__cancel__right) # label(axiom). [clausify(488)]. 1.15/1.44 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)). [resolve(614,a,603,a)]. 1.15/1.44 615 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__less__cancel__right) # label(axiom). [clausify(488)]. 1.15/1.44 616 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 1.15/1.44 617 -class_Groups_Ocancel__ab__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,B,D) | C = D # label(fact_add__imp__eq) # label(axiom). [clausify(129)]. 1.15/1.44 618 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Nat_Onat) # label(arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 1.15/1.44 619 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Oplus__class_Oplus(A,B,C) != B | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__0__iff) # label(axiom). [clausify(250)]. 2.96/3.24 620 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Oplus__class_Oplus(A,B,C) = B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_add__0__iff) # label(axiom). [clausify(250)]. 2.96/3.24 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(618,a,619,a)]. 2.96/3.24 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(618,a,620,a)]. 2.96/3.24 621 class_Rings_Ozero__neq__one(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ozero__neq__one) # label(axiom). [assumption]. 2.96/3.24 622 -class_Rings_Ozero__neq__one(A) | c_Groups_Oone__class_Oone(A) != c_Groups_Ozero__class_Ozero(A) # label(fact_zero__neq__one) # label(axiom). [clausify(283)]. 2.96/3.24 623 -class_Rings_Ozero__neq__one(A) | c_Groups_Oone__class_Oone(A) != c_Groups_Ozero__class_Ozero(A) # label(fact_one__neq__zero) # label(axiom). [clausify(301)]. 2.96/3.24 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(621,a,622,a)]. 2.96/3.24 2.96/3.24 ============================== end predicate elimination ============= 2.96/3.24 2.96/3.24 Auto_denials: (non-Horn, no changes). 2.96/3.24 2.96/3.24 Term ordering decisions: 2.96/3.24 Function symbol KB weights: tc_Nat_Onat=1. tc_HOL_Obool=1. c_Nat_OSuc=1. tc_Arrow__Order__Mirabelle_Oindi=1. tc_Product__Type_Ounit=1. c_fTrue=1. c_fFalse=1. hAPP=1. tc_fun=1. c_Finite__Set_Ocard=1. tc_sum=1. c_SetInterval_Oord__class_OgreaterThan=1. c_Nat_Osize__class_Osize=1. tc_prod=1. c_Set_Othe__elem=1. f1=1. f2=1. f4=1. f12=1. f13=1. f14=1. f17=1. f18=1. f25=1. f26=1. f28=1. f29=1. f30=1. f31=1. f35=1. f36=1. f37=1. f41=1. c_Groups_Ozero__class_Ozero=1. c_Orderings_Obot__class_Obot=1. c_Orderings_Otop__class_Otop=1. c_Groups_Oone__class_Oone=1. tc_Option_Ooption=1. c_fequal=1. c_Nat_Onat_Onat__size=1. c_HOL_Obool_Obool__size=1. f3=1. f5=1. f6=1. c_Groups_Oplus__class_Oplus=1. c_Set_Oinsert=1. c_SetInterval_Oord__class_OatLeastAtMost=1. c_SetInterval_Oord__class_OatLeastLessThan=1. c_SetInterval_Oord_OatLeast=1. c_SetInterval_Oord_OatMost=1. c_SetInterval_Oord_OgreaterThan=1. c_SetInterval_Oord_OlessThan=1. c_Groups_Ominus__class_Ominus=1. f15=1. f16=1. f23=1. f32=1. f44=1. c_Set_Oimage=1. c_Big__Operators_Ocomm__monoid__add__class_Osetsum=1. c_Sum__Type_OPlus=1. c_Fun_Othe__inv__into=1. c_SetInterval_Oord_OatLeastAtMost=1. c_SetInterval_Oord_OgreaterThanLessThan=1. f8=1. f9=1. f10=1. f19=1. f20=1. f21=1. f22=1. f24=1. f39=1. f43=1. c_SetInterval_Oord_OatLeastLessThan=1. c_SetInterval_Oord_OgreaterThanAtMost=1. f7=1. f27=1. f38=1. f40=1. f11=1. f42=1. 2.96/3.24 2.96/3.24 ============================== end of process initial clauses ======== 2.96/3.24 2.96/3.24 ============================== CLAUSES FOR SEARCH ==================== 2.96/3.24 2.96/3.24 ============================== end of clauses for search ============= 2.96/3.24 2.96/3.24 ============================== SEARCH ================================ 2.96/3.24 2.96/3.24 % Starting search at 0.62 seconds. 2.96/3.24 2.96/3.24 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 31 (0.00 of 0.88 sec). 2.96/3.24 2.96/3.24 Low Water (keep): wt=47.000, iters=3639 2.96/3.24 2.96/3.24 Low Water (keep): wt=39.000, iters=3625 2.96/3.24 2.96/3.24 Low Water (keep): wt=30.000, iters=3451 2.96/3.24 2.96/3.24 Low Water (keep): wt=27.000, iters=3432 2.96/3.24 2.96/3.24 Low Water (keep): wt=26.000, iters=3379 2.96/3.24 2.96/3.24 Low Water (keep): wt=25.000, iters=3495 2.96/3.24 2.96/3.24 Low Water (keep): wt=23.000, iters=3357 2.96/3.24 2.96/3.24 Low Water (keep): wt=22.000, iters=3429 2.96/3.24 2.96/3.24 Low Water (keep): wt=21.000, iters=3342 2.96/3.24 2.96/3.24 Low Water (keep): wt=20.000, iters=3343 2.96/3.24 2.96/3.24 Low Water (keep): wt=19.000, iters=3348 2.96/3.24 2.96/3.24 Low Water (keep): wt=18.000, iters=3336 2.96/3.24 2.96/3.24 Low Water (keep): wt=17.000, iters=3386 2.96/3.24 2.96/3.24 Low Water (keep): wt=16.000, iters=3421 2.96/3.24 2.96/3.24 Low Water (keep): wt=15.000, iters=3384 2.96/3.24 2.96/3.24 Low Water (keep): wt=14.000, iters=3454 2.96/3.24 2.96/3.24 Low Water (keep): wt=12.000, iters=3344 2.96/3.24 2.96/3.24 Low Water (keep): wt=10.000, iters=3342 2.96/3.24 2.96/3.24 Low Water (keep): wt=8.000, iters=3346 2.96/3.24 2.96/3.24 Low Water (displace): id=2640, wt=78.000 2.96/3.24 2.96/3.24 Low Water (displace): id=2558, wt=76.000 2.96/3.24 2.96/3.24 Low Water (displace): id=3357, wt=58.000 2.96/3.24 2.96/3.24 Low Water (displace): id=2472, wt=57.000 2.96/3.24 2.96/3.24 Low Water (displace): id=11401, wt=39.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12402, wt=38.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12346, wt=37.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12405, wt=36.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12349, wt=35.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12400, wt=34.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12344, wt=33.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12542, wt=32.000 65.25/65.51 65.25/65.51 Low Water (displace): id=12650, wt=31.000 65.25/65.51 65.25/65.51 Low Water (displace): id=13071, wt=30.000 65.25/65.51 65.25/65.51 Low Water (displace): id=13764, wt=29.000 65.25/65.51 65.25/65.51 Low Water (displace): id=13964, wt=28.000 65.25/65.51 65.25/65.51 Low Water (displace): id=14485, wt=27.000 65.25/65.51 65.25/65.51 Low Water (displace): id=16470, wt=26.000 65.25/65.51 65.25/65.51 ============================== PROOF ================================= 65.25/65.51 % SZS status Theorem 65.25/65.51 % SZS output start Refutation 65.25/65.51 65.25/65.51 % Proof 1 at 62.75 (+ 1.82) seconds. 65.25/65.51 % Length of proof is 88. 65.25/65.51 % Level of proof is 12. 65.25/65.51 % Maximum clause weight is 22.000. 65.25/65.51 % Given clauses 19359. 65.25/65.51 65.25/65.51 15 (all V_x_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (-c_member(T_a,V_x_2,V_A_2) -> hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(T_a,V_A_2)) = c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2))))) # label(fact_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 19 (all V_A_2 all T_a (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) <-> (exists B_x c_member(T_a,B_x,V_A_2)))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 35 (all V_A_2 all T_a (class_Finite__Set_Ofinite(T_a) -> c_Finite__Set_Ofinite(T_a,V_A_2))) # label(fact_finite__code) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 61 (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]. 65.25/65.51 92 (all T_a c_Finite__Set_Ofinite(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_finite_OemptyI) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 143 (all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_n)) # label(fact_le__refl) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 147 (all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) -> (c_Finite__Set_Ocard(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Finite__Set_Ocard(T_a,V_A_2) -> c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = V_A_2))) # label(fact_card__eq__UNIV__imp__eq__UNIV) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 176 (all V_y_2 all V_x_2 all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & -c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 182 (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]. 65.25/65.51 185 (all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 220 (all V_m_2 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_m_2)) = c_Set_Oinsert(tc_Nat_Onat,V_m_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_atLeastLessThan__singleton) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 230 (all V_x_2 all V_A_2 all V_y_2 all T_a (V_x_2 = V_y_2 | hBOOL(hAPP(V_A_2,V_x_2)) <-> hBOOL(hAPP(c_Set_Oinsert(T_a,V_y_2,V_A_2),V_x_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 247 (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]. 65.25/65.51 272 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_c_2,V_d_2)) <-> ((c_Orderings_Oord__class_Oless(T_a,V_b_2,V_d_2) | c_Orderings_Oord__class_Oless(T_a,V_c_2,V_a_2)) & c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2) | -c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)) & c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2)))) # label(fact_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 275 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2)))) # label(fact_atLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 281 (all V_a_2 all V_b_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2) -> c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_atLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 350 (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]. 65.25/65.51 360 (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]. 65.25/65.51 362 (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]. 65.25/65.51 387 (all V_a_2 all V_b_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) -> c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) # label(fact_atLeastatMost__empty) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 400 (all V_n c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n))) # label(fact_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 419 (all T_a c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Finite__Set_Ocard(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 448 (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]. 65.25/65.51 468 (all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_2,V_A_2) <-> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 475 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,hAPP(c_Nat_OSuc,V_u_2))) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 65.25/65.51 634 -c_Finite__Set_Ofinite(A,B) | c_member(A,C,B) | c_Finite__Set_Ocard(A,c_Set_Oinsert(A,C,B)) = hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(A,B)) # label(fact_card__insert__disjoint) # label(axiom). [clausify(15)]. 65.25/65.51 640 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != B | -c_member(A,C,B) # label(fact_ex__in__conv) # label(axiom). [clausify(19)]. 65.25/65.51 642 class_Finite__Set_Ofinite(tc_Product__Type_Ounit) # label(arity_Product____Type__Ounit__Finite__Set_Ofinite) # label(axiom). [assumption]. 65.25/65.51 655 -class_Finite__Set_Ofinite(A) | c_Finite__Set_Ofinite(A,B) # label(fact_finite__code) # label(axiom). [clausify(35)]. 65.25/65.51 690 -c_member(A,B,c_Set_Oinsert(A,C,D)) | C = B | c_member(A,B,D) # label(fact_insertE) # label(axiom). [clausify(61)]. 65.25/65.51 716 c_Finite__Set_Ofinite(A,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))) # label(fact_finite_OemptyI) # label(axiom). [clausify(92)]. 65.25/65.51 746 class_Orderings_Oorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Oorder) # label(axiom). [assumption]. 65.25/65.51 769 c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,A) # label(fact_le__refl) # label(axiom). [clausify(143)]. 65.25/65.51 772 -c_Finite__Set_Ofinite(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool))) | c_Finite__Set_Ocard(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool))) != c_Finite__Set_Ocard(A,B) | c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)) = B # label(fact_card__eq__UNIV__imp__eq__UNIV) # label(axiom). [clausify(147)]. 65.25/65.51 811 -class_Orderings_Opreorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_less__le__not__le) # label(axiom). [clausify(176)]. 65.25/65.51 823 -c_member(A,B,C) | c_Set_Oinsert(A,B,C) = C # label(fact_insert__absorb) # label(axiom). [clausify(182)]. 65.25/65.51 826 c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A # label(fact_Nat_Oadd__0__right) # label(axiom). [clausify(185)]. 65.25/65.51 870 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,A)) = c_Set_Oinsert(tc_Nat_Onat,A,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) # label(fact_atLeastLessThan__singleton) # label(axiom). [clausify(220)]. 65.25/65.51 877 A != B | hBOOL(hAPP(c_Set_Oinsert(C,B,D),A)) # label(fact_insert__code) # label(axiom). [clausify(230)]. 65.25/65.51 911 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_member(A,f25(B,A),B) # label(fact_all__not__in__conv) # label(axiom). [clausify(247)]. 65.25/65.51 922 class_Orderings_Oorder(tc_HOL_Obool) # label(arity_HOL__Obool__Orderings_Oorder) # label(axiom). [assumption]. 65.25/65.51 946 -class_Orderings_Oorder(A) | c_Orderings_Oord__class_Oless(tc_fun(A,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(A,B,C),c_SetInterval_Oord__class_OatLeastAtMost(A,D,E)) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) # label(fact_atLeastatMost__psubset__iff) # label(axiom). [clausify(272)]. 65.25/65.51 950 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) != c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) # label(fact_atLeastLessThan__empty__iff) # label(axiom). [clausify(275)]. 65.25/65.51 952 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_SetInterval_Oord__class_OatLeastLessThan(A,C,B) = c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) # label(fact_atLeastLessThan__empty) # label(axiom). [clausify(281)]. 65.25/65.51 994 class_Orderings_Opreorder(tc_HOL_Obool) # label(arity_HOL__Obool__Orderings_Opreorder) # label(axiom). [assumption]. 65.25/65.51 1018 -class_Orderings_Oorder(A) | class_Orderings_Oorder(tc_fun(B,A)) # label(arity_fun__Orderings_Oorder) # label(axiom). [clausify(350)]. 65.25/65.51 1033 -c_member(A,B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))) # label(fact_empty__iff) # label(axiom). [clausify(360)]. 65.25/65.51 1034 hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(tc_Nat_Onat) # label(fact_One__nat__def) # label(axiom). [assumption]. 65.25/65.51 1035 hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),B)) # label(fact_top1I) # label(axiom). [clausify(362)]. 65.25/65.51 1043 c_Finite__Set_Ocard(tc_Product__Type_Ounit,c_Orderings_Otop__class_Otop(tc_fun(tc_Product__Type_Ounit,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(tc_Nat_Onat) # label(fact_card__UNIV__unit) # label(axiom). [assumption]. 65.25/65.51 1051 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(A,C,B) # label(fact_atLeastatMost__empty) # label(axiom). [clausify(387)]. 65.25/65.51 1068 c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)) # label(fact_zero__less__Suc) # label(axiom). [clausify(400)]. 65.25/65.51 1086 c_Finite__Set_Ocard(A,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) # label(fact_card_Oempty) # label(axiom). [clausify(419)]. 65.25/65.51 1105 -class_Orderings_Opreorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,B) # label(fact_order__refl) # label(axiom). [clausify(448)]. 65.25/65.51 1131 c_member(A,B,C) | -hBOOL(hAPP(C,B)) # label(fact_mem__def) # label(axiom). [clausify(468)]. 65.25/65.51 1140 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,B)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom). [clausify(475)]. 65.25/65.51 1158 class_Orderings_Opreorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Opreorder) # label(axiom). [assumption]. 65.25/65.51 1275 c_Set_Oinsert(tc_Nat_Onat,A,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A). [back_rewrite(870),rewrite([1140(4)]),flip(a)]. 65.25/65.51 1298 c_Finite__Set_Ofinite(tc_Product__Type_Ounit,A). [resolve(655,a,642,a)]. 65.25/65.51 1359 c_Finite__Set_Ocard(A,c_Set_Oinsert(A,B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) = c_Groups_Oone__class_Oone(tc_Nat_Onat). [resolve(716,a,634,a),rewrite([1086(14),1034(13)]),unit_del(a,1033)]. 65.25/65.51 1620 hBOOL(hAPP(c_Set_Oinsert(A,B,C),B)). [resolve(877,a,826,a),rewrite([826(5)])]. 65.25/65.51 1651 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Set_Oinsert(A,f25(B,A),B) = B. [resolve(911,b,823,a)]. 65.25/65.51 1768 c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,C)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B). [resolve(946,d,769,a),unit_del(a,746)]. 65.25/65.51 1953 class_Orderings_Oorder(tc_fun(A,tc_HOL_Obool)). [resolve(1018,a,922,a)]. 65.25/65.51 2172 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)). [resolve(1068,a,1051,b),flip(b),unit_del(a,746)]. 65.25/65.51 2183 -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1068,a,811,b),unit_del(a,1158)]. 65.25/65.51 2391 c_Orderings_Oord__class_Oless__eq(tc_HOL_Obool,A,A). [resolve(1105,a,994,a)]. 65.25/65.51 2447 c_member(A,B,c_Orderings_Otop__class_Otop(tc_fun(C,tc_HOL_Obool))). [resolve(1131,b,1035,a)]. 65.25/65.51 3242 c_Finite__Set_Ocard(tc_Product__Type_Ounit,A) != c_Groups_Oone__class_Oone(tc_Nat_Onat) | c_Orderings_Otop__class_Otop(tc_fun(tc_Product__Type_Ounit,tc_HOL_Obool)) = A. [resolve(1298,a,772,a),rewrite([1043(6)]),flip(a)]. 65.25/65.51 3383 c_SetInterval_Oord__class_OatLeastLessThan(tc_HOL_Obool,A,A) = c_Orderings_Obot__class_Obot(tc_fun(tc_HOL_Obool,tc_HOL_Obool)). [resolve(2391,a,952,b),unit_del(a,922)]. 65.25/65.51 10556 c_member(A,B,c_Set_Oinsert(C,B,D)). [resolve(1620,a,1131,b)]. 65.25/65.51 15329 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != c_Set_Oinsert(B,C,D). [resolve(10556,a,640,b)]. 65.25/65.51 23493 c_SetInterval_Oord__class_OatLeastLessThan(tc_HOL_Obool,A,A) != c_Set_Oinsert(B,C,D). [para(3383(a,2),15329(a,1))]. 65.25/65.51 23990 c_SetInterval_Oord__class_OatLeastLessThan(tc_HOL_Obool,A,A) = c_SetInterval_Oord__class_OatLeastLessThan(tc_HOL_Obool,B,B). [para(3383(a,2),3383(a,2))]. 65.25/65.51 23991 c_SetInterval_Oord__class_OatLeastLessThan(tc_HOL_Obool,A,A) = c_0. [new_symbol(23990)]. 65.25/65.51 24446 c_Set_Oinsert(A,B,C) != c_0. [back_rewrite(23493),rewrite([23991(2)]),flip(a)]. 65.25/65.51 25037 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = c_0. [resolve(24446,a,1651,b)]. 65.25/65.51 26027 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_0. [back_rewrite(2172),rewrite([25037(10)])]. 65.25/65.51 26045 c_Finite__Set_Ocard(A,c_Set_Oinsert(A,B,c_0)) = c_Groups_Oone__class_Oone(tc_Nat_Onat). [back_rewrite(1359),rewrite([25037(3)])]. 65.25/65.51 26048 c_Set_Oinsert(tc_Nat_Onat,A,c_0) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A). [back_rewrite(1275),rewrite([25037(5)])]. 65.25/65.51 26051 -c_member(A,B,c_0). [back_rewrite(1033),rewrite([25037(3)])]. 65.25/65.51 26052 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) != c_0. [back_rewrite(950),rewrite([25037(6)])]. 65.25/65.51 33032 c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_0,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A)). [para(26027(a,1),1768(a,2)),unit_del(b,2183)]. 65.25/65.51 33048 c_Orderings_Otop__class_Otop(tc_fun(tc_Product__Type_Ounit,tc_HOL_Obool)) = c_Set_Oinsert(tc_Product__Type_Ounit,A,c_0). [resolve(26045,a,3242,a)]. 65.25/65.51 34195 c_SetInterval_Oord__class_OatLeastLessThan(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_0,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A)) != c_0. [resolve(33032,a,26052,b),unit_del(a,1953)]. 65.25/65.51 34202 c_SetInterval_Oord__class_OatLeastLessThan(tc_fun(tc_Nat_Onat,tc_HOL_Obool),c_0,c_Set_Oinsert(tc_Nat_Onat,A,c_0)) != c_0. [para(26048(a,2),34195(a,1,3))]. 65.25/65.51 34219 A = B. [para(33048(a,2),690(a,3)),unit_del(a,2447),unit_del(c,26051)]. 65.25/65.51 34220 $F. [resolve(34219,a,34202,a)]. 65.25/65.51 65.25/65.51 % SZS output end Refutation 65.25/65.51 ============================== end of proof ========================== 65.25/65.51 65.25/65.51 ============================== STATISTICS ============================ 65.25/65.51 65.25/65.51 Given=19359. Generated=3582921. Kept=33491. proofs=1. 65.25/65.51 Usable=16870. Sos=8496. Demods=168. Limbo=10, Disabled=8948. Hints=0. 65.25/65.51 Megabytes=37.50. 65.25/65.51 User_CPU=62.75, System_CPU=1.82, Wall_clock=65. 65.25/65.51 65.25/65.51 ============================== end of statistics ===================== 65.25/65.51 65.25/65.51 ============================== end of search ========================= 65.25/65.51 65.25/65.51 THEOREM PROVED 65.25/65.51 % SZS status Theorem 65.25/65.51 65.25/65.51 Exiting with 1 proof. 65.25/65.51 65.25/65.51 Process 26259 exit (max_proofs) Sat Jul 14 05:26:15 2018 65.25/65.51 Prover9 interrupted 65.25/65.52 EOF