0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n011.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 13:10:12 EDT 2019 0.13/0.34 % CPUTime : 1.15/1.40 ============================== Prover9 =============================== 1.15/1.40 Prover9 (32) version 2009-11A, November 2009. 1.15/1.40 Process 32130 was started by sandbox2 on n011.cluster.edu, 1.15/1.40 Thu Aug 29 13:10:13 2019 1.15/1.40 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 180 -f /tmp/Prover9_31977_n011.cluster.edu". 1.15/1.40 ============================== end of head =========================== 1.15/1.40 1.15/1.40 ============================== INPUT ================================= 1.15/1.40 1.15/1.40 % Reading from file /tmp/Prover9_31977_n011.cluster.edu 1.15/1.40 1.15/1.40 set(prolog_style_variables). 1.15/1.40 set(auto2). 1.15/1.40 % set(auto2) -> set(auto). 1.15/1.40 % set(auto) -> set(auto_inference). 1.15/1.40 % set(auto) -> set(auto_setup). 1.15/1.40 % set(auto_setup) -> set(predicate_elim). 1.15/1.40 % set(auto_setup) -> assign(eq_defs, unfold). 1.15/1.40 % set(auto) -> set(auto_limits). 1.15/1.40 % set(auto_limits) -> assign(max_weight, "100.000"). 1.15/1.40 % set(auto_limits) -> assign(sos_limit, 20000). 1.15/1.40 % set(auto) -> set(auto_denials). 1.15/1.40 % set(auto) -> set(auto_process). 1.15/1.40 % set(auto2) -> assign(new_constants, 1). 1.15/1.40 % set(auto2) -> assign(fold_denial_max, 3). 1.15/1.40 % set(auto2) -> assign(max_weight, "200.000"). 1.15/1.40 % set(auto2) -> assign(max_hours, 1). 1.15/1.40 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.15/1.40 % set(auto2) -> assign(max_seconds, 0). 1.15/1.40 % set(auto2) -> assign(max_minutes, 5). 1.15/1.40 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.15/1.40 % set(auto2) -> set(sort_initial_sos). 1.15/1.40 % set(auto2) -> assign(sos_limit, -1). 1.15/1.40 % set(auto2) -> assign(lrs_ticks, 3000). 1.15/1.40 % set(auto2) -> assign(max_megs, 400). 1.15/1.40 % set(auto2) -> assign(stats, some). 1.15/1.40 % set(auto2) -> clear(echo_input). 1.15/1.40 % set(auto2) -> set(quiet). 1.15/1.40 % set(auto2) -> clear(print_initial_clauses). 1.15/1.40 % set(auto2) -> clear(print_given). 1.15/1.40 assign(lrs_ticks,-1). 1.15/1.40 assign(sos_limit,10000). 1.15/1.40 assign(order,kbo). 1.15/1.40 set(lex_order_vars). 1.15/1.40 clear(print_given). 1.15/1.40 1.15/1.40 % formulas(sos). % not echoed (533 formulas) 1.15/1.40 1.15/1.40 ============================== end of input ========================== 1.15/1.40 1.15/1.40 % From the command line: assign(max_seconds, 180). 1.15/1.40 1.15/1.40 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.15/1.40 1.15/1.40 % Formulas that are not ordinary clauses: 1.15/1.40 1 (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_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))) = 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))))) # label(fact_setsum__head__upt__Suc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 2 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) | c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 3 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n)) # label(fact_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 4 (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,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)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 5 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 6 (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]. 1.15/1.40 7 (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]. 1.15/1.40 8 (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]. 1.15/1.40 9 (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_OatMost(T_a,V_less__eq_2,V_k_2)) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)))) # label(fact_ord_OatMost__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 10 (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]. 1.15/1.40 11 (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]. 1.15/1.40 12 (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]. 1.15/1.40 13 (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]. 1.15/1.40 14 (all V_A_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_A_2) -> c_Set_Oinsert(T_a,V_a_2,V_A_2) = V_A_2)) # label(fact_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 15 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (V_a_2 = V_c_2 & V_d_2 = V_b_2 | V_b_2 = V_c_2 & V_a_2 = V_d_2 <-> c_Set_Oinsert(T_a,V_c_2,c_Set_Oinsert(T_a,V_d_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 16 (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]. 1.15/1.40 17 (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]. 1.15/1.40 18 (all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (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)) <-> -c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)))) # label(fact_atLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 19 (all T_b all T_a (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))) <-> 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))))) # label(fact_finite__Plus__UNIV__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 20 (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]. 1.15/1.40 21 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x))))) # label(fact_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 22 (all V_x all T_a (class_Orderings_Obot(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Orderings_Obot__class_Obot(T_a),V_x))) # label(fact_bot__least) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 23 (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]. 1.15/1.40 24 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 25 (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]. 1.15/1.40 26 (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]. 1.15/1.40 27 (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]. 1.15/1.40 28 (all V_n all V_m (V_m = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n)) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 29 (all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(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))))) # label(fact_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 30 (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]. 1.15/1.40 31 (all V_b_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 32 (all V_n_2 all V_m_2 (V_n_2 = 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,V_m_2,V_n_2))) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 33 (all V_A_2 all V_f_2 all T_b all T_a (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_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 34 (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]. 1.15/1.40 35 (all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> -(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 36 (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]. 1.15/1.40 37 (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]. 1.15/1.40 38 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b = V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 39 (all V_n all V_m (V_n = V_m -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 40 (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]. 1.15/1.40 41 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_A_2)) # label(fact_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 42 (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]. 1.15/1.40 43 (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]. 1.15/1.40 44 (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]. 1.15/1.40 45 (all V_nat_H c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 46 (all V_m all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m) -> V_n != V_m)) # label(fact_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 47 (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_a,V_A_2) = c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,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]. 1.15/1.40 48 (all V_x_2 all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> (c_member(T_a,V_x_2,V_A_2) <-> c_member(T_a,V_x_2,V_B_2)))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 49 (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]. 1.15/1.40 50 (all V_n V_n != hAPP(c_Nat_OSuc,V_n)) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 1.15/1.40 51 (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]. 1.15/1.40 52 (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]. 1.15/1.40 53 (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]. 1.15/1.41 54 (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]. 1.15/1.41 55 (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]. 1.15/1.41 56 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_prod(T_2,T_1)))) # label(arity_prod__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 57 (all V_n_2 all V_m_2 all V_k_2 (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) <-> V_m_2 = V_n_2)) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 58 (all T_2 all T_1 (class_Orderings_Oord(T_1) -> class_Orderings_Oord(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 59 (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]. 1.15/1.41 60 (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]. 1.15/1.41 61 (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]. 1.15/1.41 62 (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_x_2 = V_y_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]. 1.15/1.41 63 (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]. 1.15/1.41 64 (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]. 1.15/1.41 65 (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]. 1.15/1.41 66 (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(T_a,V_y,V_x) | V_y = V_x)) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 67 (all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_y = V_x)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 68 (all T_2 all T_1 (class_Orderings_Obot(T_1) -> class_Orderings_Obot(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Obot) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 69 (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]. 1.15/1.41 70 (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]. 1.15/1.41 71 (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]. 1.15/1.41 72 (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]. 1.15/1.41 73 (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]. 1.15/1.41 74 (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]. 1.15/1.41 75 (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]. 1.15/1.41 76 (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_y_2 = V_x_2))) # label(fact_injD) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 77 (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) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> (exists B_x (hAPP(V_f_2,B_x) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_xa != B_x -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & c_member(T_a,B_x,V_A_2)))))) # label(fact_setsum__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 78 (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_a_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_d_2)))) # label(fact_atLeastLessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 79 (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]. 1.15/1.41 80 (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]. 1.15/1.41 81 (all V_n_2 all V_m_2 ((exists B_k V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k))) <-> 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]. 1.15/1.41 82 (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,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),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_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 83 (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]. 1.15/1.41 84 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 85 (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]. 1.15/1.41 86 (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]. 1.15/1.41 87 (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]. 1.15/1.41 88 (all V_x_2 all T_a c_member(T_a,V_x_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_UNIV__I) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 89 (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_b,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_c,V_a) -> V_b = V_c))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 90 (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_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_setsum_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 91 (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]. 1.15/1.41 92 (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,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_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 93 (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__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 94 (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,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 95 (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]. 1.15/1.41 96 (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]. 1.15/1.41 97 (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]. 1.15/1.41 98 (all V_C_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)))) # label(fact_subset__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 99 (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]. 1.15/1.41 100 (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_b != V_a -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 101 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_b = V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 102 (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]. 1.15/1.41 103 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_SetInterval_Oord__class_OgreaterThan(T_a,V_x_2) = c_SetInterval_Oord__class_OgreaterThan(T_a,V_y_2) <-> V_y_2 = V_x_2))) # label(fact_greaterThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 104 (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) -> V_x_2 = hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2))))) # label(fact_the__inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 105 (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) <-> c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2)))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 106 (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]. 1.15/1.41 107 (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]. 1.15/1.41 108 (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]. 1.15/1.41 109 (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]. 1.15/1.41 110 (all V_n_2 all 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)) <-> 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))) # label(fact_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 111 (all V_x_2 all T_a hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_x_2))) # label(fact_top1I) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 112 (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]. 1.15/1.41 113 (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_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != V_A_2 -> 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]. 1.15/1.41 114 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 115 (all V_A_2 all T_a all V_y_2 all V_x_2 (V_x_2 = V_y_2 -> (c_member(T_a,V_x_2,V_A_2) <-> c_member(T_a,V_y_2,V_A_2)))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 116 (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]. 1.15/1.41 117 (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__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 118 (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) & c_Finite__Set_Ofinite(T_b,V_B_2))) # label(fact_finite__Plus__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 119 (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]. 1.15/1.41 120 (all V_y_2 all V_x_2 (-hBOOL(hAPP(c_fequal(V_x_2),V_y_2)) | V_x_2 = V_y_2)) # label(help_c__fequal__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 121 (all V_n_2 all 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)) <-> V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 122 (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]. 1.15/1.41 123 (all V_A_2 all T_a ((exists B_x c_member(T_a,B_x,V_A_2)) <-> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 124 (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]. 1.15/1.41 125 (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]. 1.15/1.41 126 (all V_y all V_x (hAPP(c_Nat_OSuc,V_x) = hAPP(c_Nat_OSuc,V_y) -> V_y = V_x)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 127 (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]. 1.15/1.41 128 (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) -> (V_B_2 = c_Set_Oimage(T_a,T_b,V_f_2,V_A_2) -> ((all B_a (c_member(T_a,B_a,V_A_2) -> hAPP(V_g_2,B_a) = hAPP(V_h_2,hAPP(V_f_2,B_a)))) -> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c,V_h_2,V_B_2) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_c,V_g_2,V_A_2)))))) # label(fact_setsum__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 129 (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]. 1.15/1.41 130 (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_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) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 131 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_y = V_x -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 132 (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]. 1.15/1.41 133 (all V_y_2 all V_x_2 (hBOOL(hAPP(c_fequal(V_x_2),V_y_2)) | V_x_2 != V_y_2)) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 134 (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_m_2,V_k_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_k_2))) # label(fact_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 135 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n))) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 136 (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,V_a_2,c_Groups_Ozero__class_Ozero(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)))))) # label(fact_Ints__odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 137 (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]. 1.15/1.41 138 (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]. 1.15/1.41 139 (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]. 1.15/1.41 140 (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]. 1.15/1.41 141 (all T_2 all T_1 (class_Orderings_Otop(T_1) -> class_Orderings_Otop(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Otop) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 142 (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]. 1.15/1.41 143 (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]. 1.15/1.41 144 (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]. 1.15/1.41 145 (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]. 1.15/1.41 146 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> V_x != V_y))) # label(fact_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 147 (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]. 1.15/1.41 148 (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(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2))))) # label(fact_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 149 (all V_y_2 all V_x_2 all T_a (V_y_2 = V_x_2 <-> c_member(T_a,V_x_2,c_fequal(V_y_2)))) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 1.15/1.41 150 (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]. 1.19/1.41 151 (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]. 1.19/1.41 152 (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) & c_Set_Oimage(T_b,T_a,V_f_2,B_AA) = V_B_2)) <-> 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]. 1.19/1.41 153 (all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_equalityD1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.41 154 (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_Ozero__class_Ozero(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)))))) # label(fact_setsum__cl__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.41 155 (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_c),V_b) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.41 156 (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_n_2 = V_m_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]. 1.19/1.41 157 (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]. 1.19/1.41 158 (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]. 1.19/1.41 159 (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]. 1.19/1.41 160 (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))) -> ((exists B_x ((all B_y (V_b_2 = hAPP(V_f_2,B_y) -> B_x = B_y)) & hAPP(V_f_2,B_x) = V_b_2)) <-> 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))))))) # label(fact_range__ex1__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 161 (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]. 1.19/1.42 162 (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]. 1.19/1.42 163 (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]. 1.19/1.42 164 (all V_x all T_a (class_Orderings_Otop(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)))) # label(fact_top__greatest) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 165 (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]. 1.19/1.42 166 (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]. 1.19/1.42 167 (all V_x_2 all T_b all T_a (class_Orderings_Obot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_x_2))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 168 (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]. 1.19/1.42 169 (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]. 1.19/1.42 170 (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]. 1.19/1.42 171 (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)) <-> (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)) | V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 172 (all V_f_2 all T_b all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> (c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) <-> c_Finite__Set_Ocard(T_a,V_A_2) = c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))))) # label(fact_inj__on__iff__eq__card) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 173 (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) & V_A_2 != V_B_2)) # label(fact_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 174 (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]. 1.19/1.42 175 (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]. 1.19/1.42 176 (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]. 1.19/1.42 177 (all V_u_2 all V_l_2 all V_less_2 all V_i_2 all T_a (c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThanLessThan(T_a,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_2,V_l_2),V_i_2)))) # label(fact_ord_OgreaterThanLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 178 (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,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_image__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 179 (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_c_2) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2) <-> V_c_2 = V_b_2))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 180 (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]. 1.19/1.42 181 (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]. 1.19/1.42 182 (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]. 1.19/1.42 183 (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]. 1.19/1.42 184 (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]. 1.19/1.42 185 (all V_c_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2)))) # label(fact_subsetD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 186 (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]. 1.19/1.42 187 (all V_b_2 all V_a_2 all T_a (c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) -> V_a_2 = V_b_2)) # label(fact_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 188 (all T_a c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_UNIV__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 189 (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]. 1.19/1.42 190 (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]. 1.19/1.42 191 (all V_x_2 all V_y_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2))))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 192 (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]. 1.19/1.42 193 (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]. 1.19/1.42 194 (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) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_B_2 & V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_Plus__eq__empty__conv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 195 (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]. 1.19/1.42 196 (all V_B_2 all V_a_2 all V_f_2 all T_a all T_b c_Set_Oimage(T_b,T_a,V_f_2,c_Set_Oinsert(T_b,V_a_2,V_B_2)) = 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))) # label(fact_image__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 197 (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) <-> V_y_2 = V_x_2 | c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2)))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 198 (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]. 1.19/1.42 199 (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]. 1.19/1.42 200 (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]. 1.19/1.42 201 (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]. 1.19/1.42 202 (all V_x_2 all V_Q_2 all V_P_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2) -> (hBOOL(hAPP(V_P_2,V_x_2)) -> hBOOL(hAPP(V_Q_2,V_x_2))))) # label(fact_predicate1D) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 203 (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]. 1.19/1.42 204 (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_OatLeastLessThan(T_a,V_a_2,V_b_2) <-> -c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)))) # label(fact_atLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 205 (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_b,V_B_2) & c_Finite__Set_Ofinite(T_a,V_A_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]. 1.19/1.42 206 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 207 (all V_A_2 all V_f_2 all T_b all V_z_2 all T_a (c_member(T_a,V_z_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)) <-> (exists B_x (hAPP(V_f_2,B_x) = V_z_2 & c_member(T_b,B_x,V_A_2))))) # label(fact_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 208 (all V_n all V_m (V_m = V_n | 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]. 1.19/1.42 209 (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]. 1.19/1.42 210 (all V_x_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) | c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = V_A_2)) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 211 (all V_k_2 all V_less__eq_2 all V_i_2 all T_a (hBOOL(hAPP(hAPP(V_less__eq_2,V_k_2),V_i_2)) <-> c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeast(T_a,V_less__eq_2,V_k_2)))) # label(fact_ord_OatLeast__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 212 (all V_A_2 all V_a_2 all T_a (c_Finite__Set_Ofinite(T_a,c_Set_Oinsert(T_a,V_a_2,V_A_2)) <-> c_Finite__Set_Ofinite(T_a,V_A_2))) # label(fact_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 213 (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_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))) = 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)))))) # label(fact_setsum__add__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 214 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a))) # label(fact_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 215 (all V_k all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_k)) # label(fact_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 216 (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_a,V_A_2) = c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))) # label(fact_card__image) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 217 (all V_x_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) -> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2))))) # label(fact_le__funE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 218 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2)) # label(fact_subset__refl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 219 (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]. 1.19/1.42 220 (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]. 1.19/1.42 221 (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_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)) = 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)))))) # label(fact_setsum__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 222 (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]. 1.19/1.42 223 (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_y_2) = hAPP(V_f_2,V_x_2) -> (c_member(T_a,V_x_2,V_A_2) -> (c_member(T_a,V_y_2,V_A_2) -> V_y_2 = V_x_2))))) # label(fact_inj__onD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 224 (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]. 1.19/1.42 225 (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]. 1.19/1.42 226 (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_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 227 (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]. 1.19/1.42 228 (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]. 1.19/1.42 229 (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]. 1.19/1.42 230 (all V_t all V_s (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_s,V_t) -> V_t != V_s)) # label(fact_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 231 (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) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> V_x_2 = V_y_2)))) # label(fact_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 232 (all V_k_2 all V_f_2 (hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = 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,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,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)))) # label(fact_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 233 (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 hAPP(c_Nat_OSuc,B_m) = V_n_2))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 234 (all V_Q_2 all T_a all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_x_2)) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2) -> hBOOL(hAPP(V_Q_2,V_x_2))))) # label(fact_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 235 (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]. 1.19/1.42 236 (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]. 1.19/1.42 237 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,hAPP(c_Nat_OSuc,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 238 (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]. 1.19/1.42 239 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> ((all B_x c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x))) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 240 (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]. 1.19/1.42 241 (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]. 1.19/1.42 242 (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_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__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 243 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(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))))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 244 (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]. 1.19/1.42 245 (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__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 246 (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]. 1.19/1.42 247 (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]. 1.19/1.42 248 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x))) # label(fact_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 249 (all V_A_2 all V_a_2 all T_a c_Set_Oinsert(T_a,V_a_2,V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) # label(fact_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 250 (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]. 1.19/1.42 251 (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(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2))))) # label(fact_folding__image__simple__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 252 (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]. 1.19/1.42 253 (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]. 1.19/1.42 254 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 255 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z))))) # label(fact_order__trans) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 256 (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_d_2 = V_b_2))))) # label(fact_atLeastLessThan__inj_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 257 (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]. 1.19/1.42 258 (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]. 1.19/1.42 259 (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_g_2,V_f_2) & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_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]. 1.19/1.42 260 (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]. 1.19/1.42 261 (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]. 1.19/1.42 262 (all V_nat c_Nat_Osize__class_Osize(tc_Nat_Onat,hAPP(c_Nat_OSuc,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)))) # label(fact_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 263 (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]. 1.19/1.42 264 (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]. 1.19/1.42 265 (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]. 1.19/1.42 266 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 267 (all V_n_2 all V_m_2 (V_m_2 = V_n_2 | c_Orderings_Oord__class_Oless(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) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 268 (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_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_x_2,V_y_2) <-> 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]. 1.19/1.42 269 (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]. 1.19/1.42 270 (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]. 1.19/1.42 271 (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]. 1.19/1.42 272 (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]. 1.19/1.42 273 (all V_a_2 all T_a all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2 -> -c_member(T_a,V_a_2,V_A_2))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 274 (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]. 1.19/1.42 275 (all V_P_2 all 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))) <-> (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)))))) # label(fact_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 276 (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]. 1.19/1.42 277 (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]. 1.19/1.42 278 (all V_n_2 all V_m_2 ((exists B_k c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k) = V_n_2) <-> 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]. 1.19/1.42 279 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 280 (all V_x_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_x_2,V_A_2) -> c_member(T_a,V_x_2,V_B_2)))) # label(fact_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 281 (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]. 1.19/1.42 282 (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_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)) <-> 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)))) # label(fact_ord_OatLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 283 (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(V_Fa_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_Fa_2,V_B_2)),hAPP(V_Fa_2,V_A_2)))))) # label(fact_folding__image__simple__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 284 (all T_2 all T_1 (class_Orderings_Oorder(T_1) -> class_Orderings_Oorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 285 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a)) # label(fact_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 286 (all V_a_2 all V_b_2 all T_a (c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) <-> V_a_2 = V_b_2)) # label(fact_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 287 (all V_a_2 all V_D_2 all V_C_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_C_2),c_Set_Oinsert(T_a,V_a_2,V_D_2)))) # label(fact_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.19/1.42 288 (all V_f_2 all V_A_2 all T_a (c_Finite__Set_Ofinite(T_a,V_A_2) -> ((exists B_x ((all B_xa (c_member(T_a,B_xa,V_A_2) -> (B_xa != B_x -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(V_f_2,B_x) & c_member(T_a,B_x,V_A_2))) <-> c_Groups_Oone__class_Oone(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__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 289 (all T_b all T_a 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_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]. 1.19/1.43 290 (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_Orderings_Oord(T_a) & class_Groups_Ocomm__monoid__add(T_b) -> (V_c_2 = V_a_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_f_2,c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2)) = 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))))))) # label(fact_setsum__ivl__cong) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 291 (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]. 1.19/1.43 292 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_fun(T_2,T_1)))) # label(arity_fun__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 293 (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]. 1.19/1.43 294 (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]. 1.19/1.43 295 (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(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__eq(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_a_2,V_b_2)))) # label(fact_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 296 (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]. 1.19/1.43 297 (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]. 1.19/1.43 298 (all V_a_2 all T_a -c_member(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_emptyE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 299 (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]. 1.19/1.43 300 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (V_c = V_b -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 301 (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_b = V_c -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 302 (all V_c_2 all V_a_2 all V_b_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (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) <-> V_b_2 = V_c_2))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 303 (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]. 1.19/1.43 304 (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]. 1.19/1.43 305 (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]. 1.19/1.43 306 (all V_x_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) -> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2))))) # label(fact_le__funD) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 307 (all V_c_2 all T_a all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> ((c_member(T_a,V_c_2,V_A_2) -> -c_member(T_a,V_c_2,V_B_2)) -> -(-c_member(T_a,V_c_2,V_A_2) -> c_member(T_a,V_c_2,V_B_2))))) # label(fact_equalityCE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 308 (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]. 1.19/1.43 309 (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]. 1.19/1.43 310 (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]. 1.19/1.43 311 (all T_a all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2))) # label(fact_equalityD2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 312 (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,V_m_2,V_n_2) & V_m_2 != V_n_2)) # label(fact_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 313 (all V_A_2 all V_y_2 all V_x_2 all T_a c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_y_2,V_A_2)) = c_Set_Oinsert(T_a,V_y_2,c_Set_Oinsert(T_a,V_x_2,V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 314 (all V_B_2 all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_2,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> c_member(T_a,V_x_2,V_B_2)))) # label(fact_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 315 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 316 (all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) <-> V_A_2 = V_B_2 | c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) # label(fact_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 317 (all V_A_2 all T_a (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) | c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2)) # label(fact_card__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 318 (all V_A_H_2 all T_b all T_a all V_A_2 (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) -> ((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 c_Set_Oimage(T_b,T_a,B_g,V_A_H_2) = V_A_2)))) # label(fact_inj__on__iff__surj) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 319 (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]. 1.19/1.43 320 (all V_n c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 321 (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]. 1.19/1.43 322 (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]. 1.19/1.43 323 (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]. 1.19/1.43 324 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_a_2 <-> 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]. 1.19/1.43 325 (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]. 1.19/1.43 326 (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]. 1.19/1.43 327 (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]. 1.19/1.43 328 (all V_a_2 all V_b_2 all T_a (c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) -> V_a_2 = V_b_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 329 (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]. 1.19/1.43 330 (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]. 1.19/1.43 331 (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__eq_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_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]. 1.19/1.43 332 (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_b_2 = V_d_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]. 1.19/1.43 333 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 334 (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]. 1.19/1.43 335 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (V_x_2 != V_y_2 & c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 336 (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_Ozero__class_Ozero(T_a)) & (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_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_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]. 1.19/1.43 337 (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]. 1.19/1.43 338 (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]. 1.19/1.43 339 (all V_z all V_y all V_x c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,V_z))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 340 (all V_c_2 all T_a -c_member(T_a,V_c_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 341 (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]. 1.19/1.43 342 (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_Oplus__class_Oplus(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(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))))) # label(fact_card__Plus) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 343 (all V_k_2 all V_f_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = 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,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]. 1.19/1.43 344 (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_A_2 = V_B_2))) # label(fact_inj__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 345 (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_OlessThan(T_a,V_less_2,V_k_2)) <-> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_k_2)))) # label(fact_ord_OlessThan__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 346 (all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 1.19/1.43 347 (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]. 1.19/1.43 348 (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]. 1.19/1.43 349 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 350 (all V_n_2 all 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)) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Groups_Ozero__class_Ozero(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]. 1.19/1.43 351 (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]. 1.19/1.43 352 (all V_x_2 all T_a c_Set_Othe__elem(T_a,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_x_2) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 353 (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) -> (c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != V_N_2 -> hAPP(V_h_2,hAPP(V_Fa_2,V_N_2)) = hAPP(V_Fa_2,c_Set_Oimage(T_a,T_a,V_h_2,V_N_2))))))) # label(fact_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 354 (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]. 1.19/1.43 355 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 356 (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__eq(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 357 (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_c_2 = V_a_2))))) # label(fact_atLeastLessThan__inj_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 358 (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]. 1.19/1.43 359 (all V_k_2 all V_A_2 all T_a ((exists B_b exists B_B (V_A_2 = c_Set_Oinsert(T_a,B_b,B_B) & -c_member(T_a,B_b,B_B) & c_Finite__Set_Ocard(T_a,B_B) = V_k_2 & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> B_B = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) <-> c_Finite__Set_Ocard(T_a,V_A_2) = hAPP(c_Nat_OSuc,V_k_2))) # label(fact_card__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 360 (all V_m_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,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_atLeastLessThan0) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 361 (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]. 1.19/1.43 362 (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_y_2 = V_x_2))) # label(fact_inj__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 363 (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]. 1.19/1.43 364 (all V_x_2 all T_a (class_Groups_Oone(T_a) -> (V_x_2 = c_Groups_Oone__class_Oone(T_a) <-> c_Groups_Oone__class_Oone(T_a) = V_x_2))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 365 (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_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))) = 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))))) # label(fact_setsum__head__Suc) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 366 (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]. 1.19/1.43 367 (all V_A_2 all T_a (-c_Finite__Set_Ofinite(T_a,V_A_2) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Finite__Set_Ocard(T_a,V_A_2))) # label(fact_card__infinite) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 368 (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_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))) & (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,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_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]. 1.19/1.43 369 (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]. 1.19/1.43 370 (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 ((all B_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_i,B_k) -> -hBOOL(hAPP(V_P_2,B_i)))) & hBOOL(hAPP(V_P_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B_k,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) & 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]. 1.19/1.43 371 (all V_n V_n = c_Nat_Osize__class_Osize(tc_Nat_Onat,V_n)) # label(fact_nat__size) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 372 (all T_a c_Finite__Set_Ocard(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 373 (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]. 1.19/1.43 374 (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]. 1.19/1.43 375 (all T_2 all T_1 (class_Orderings_Opreorder(T_1) -> class_Orderings_Opreorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 376 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 377 (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) -> (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) -> 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]. 1.19/1.43 378 (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]. 1.19/1.43 379 (all V_A_2 all V_f_2 all T_b all T_a ((all B_x (c_member(T_a,B_x,V_A_2) -> (all B_xa (c_member(T_a,B_xa,V_A_2) -> (hAPP(V_f_2,B_xa) = hAPP(V_f_2,B_x) -> B_x = B_xa))))) <-> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2))) # label(fact_inj__on__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 380 (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]. 1.19/1.43 381 (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]. 1.19/1.43 382 (all V_B_2 all V_A_2 all V_x_2 all T_a (-c_member(T_a,V_x_2,V_A_2) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_x_2,V_B_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)))) # label(fact_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 383 (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]. 1.19/1.43 384 (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]. 1.19/1.43 385 (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]. 1.19/1.43 386 (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) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> V_x_2 = V_y_2)))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 387 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (V_y != V_x -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 388 (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]. 1.19/1.43 389 (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_B_2 = V_A_2 <-> c_Set_Oinsert(T_a,V_x_2,V_A_2) = c_Set_Oinsert(T_a,V_x_2,V_B_2))))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 390 (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]. 1.19/1.43 391 (all V_a_2 all V_B_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oinsert(T_a,V_a_2,V_B_2))) # label(fact_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 392 (all T_b all T_a c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_a,T_b),tc_HOL_Obool)) = 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)))) # label(fact_UNIV__Plus__UNIV) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 393 (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]. 1.19/1.43 394 (all V_nat_H_2 all V_nat_2 (V_nat_H_2 = V_nat_2 <-> hAPP(c_Nat_OSuc,V_nat_H_2) = hAPP(c_Nat_OSuc,V_nat_2))) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 1.19/1.43 395 (all V_m V_m = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 396 (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]. 1.19/1.44 397 (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]. 1.19/1.44 398 (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]. 1.19/1.44 399 (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]. 1.19/1.44 400 (all V_x_2 all T_a (class_Groups_Ozero(T_a) -> (V_x_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = V_x_2))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 401 (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]. 1.19/1.44 402 (all V_b_2 all V_B_2 all V_a_2 all T_a (c_member(T_a,V_a_2,V_B_2) -> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)))) # label(fact_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 403 (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]. 1.19/1.44 404 (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]. 1.19/1.44 405 (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_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)) = 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)))))) # label(fact_setsum_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 406 (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(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_Oinsert) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 407 (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_Ozero__class_Ozero(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)))) # label(fact_Ints__odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 408 (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]. 1.19/1.44 409 (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_y_2) = hAPP(V_f_2,V_x_2) <-> V_x_2 = V_y_2))))) # label(fact_inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 410 (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]. 1.19/1.44 411 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 412 (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]. 1.19/1.44 413 (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]. 1.19/1.44 414 (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]. 1.19/1.44 415 (all V_A_2 all V_b_2 all V_a_2 all T_a (c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)) -> (V_a_2 != V_b_2 -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 416 (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(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,c_Set_Oinsert(T_b,V_x_2,V_A_2))))) # label(fact_folding__image__simple__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 417 (all V_c_2 all V_b_2 all V_a_2 all T_a (class_Orderings_Oorder(T_a) -> (V_a_2 = V_b_2 & V_c_2 = V_b_2 <-> c_Set_Oinsert(T_a,V_c_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__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 418 (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]. 1.19/1.44 419 (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]. 1.19/1.44 420 (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]. 1.19/1.44 421 (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]. 1.19/1.44 422 (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]. 1.19/1.44 423 (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]. 1.19/1.44 424 (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]. 1.19/1.44 425 (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) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2))))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 426 (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]. 1.19/1.44 427 (all V_B_2 all V_a_2 all T_a c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_a_2,V_B_2))) # label(fact_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 428 (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]. 1.19/1.44 429 (all T_a all V_A_2 all T_b 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_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]. 1.19/1.44 430 (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__eq(T_a,V_a_2,V_b_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_d_2) & (c_Orderings_Oord__class_Oless(T_a,V_c_2,V_a_2) | c_Orderings_Oord__class_Oless(T_a,V_b_2,V_d_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]. 1.19/1.44 431 (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]. 1.19/1.44 432 (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]. 1.19/1.44 433 (all V_x_2 all V_B_2 all V_A_2 all T_a (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) -> (c_member(T_a,V_x_2,V_A_2) -> c_member(T_a,V_x_2,V_B_2)))) # label(fact_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 434 (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]. 1.19/1.44 435 (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]. 1.19/1.44 436 (all V_j_2 all V_i_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)) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2))) # label(fact_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 437 (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]. 1.19/1.44 438 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 439 (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]. 1.19/1.44 440 (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_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 441 (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]. 1.19/1.44 442 (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)) <-> 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_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 443 (all V_A_2 all V_x_2 all T_a c_Set_Oinsert(T_a,V_x_2,V_A_2) = c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_x_2,V_A_2))) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 444 (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_a_2,V_b_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__empty__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 445 (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]. 1.19/1.44 446 (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]. 1.19/1.44 447 (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]. 1.19/1.44 448 (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]. 1.19/1.44 449 (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]. 1.19/1.44 450 (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]. 1.19/1.44 451 (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_a_2 = V_b_2 <-> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2)))) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 452 (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]. 1.19/1.44 453 (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]. 1.19/1.44 454 (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]. 1.19/1.44 455 (all V_A_2 all T_a ((all B_x -c_member(T_a,B_x,V_A_2)) <-> c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2)) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 456 (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]. 1.19/1.44 457 (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]. 1.19/1.44 458 (all V_x_2 all T_a c_member(T_a,V_x_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_iso__tuple__UNIV__I) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 459 (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]. 1.19/1.44 460 (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_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)))) # label(fact_ord_OatLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 461 (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]. 1.19/1.44 462 (all V_n_2 all V_m_2 all V_k_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)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 463 (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_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)) <-> (exists B_f (c_Fun_Oinj__on(T_a,T_b,B_f,V_A_2) & 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))))))) # label(fact_inj__on__iff__card__le) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 464 (all V_A_2 all T_a c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_subset__UNIV) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 465 (all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 466 (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]. 1.19/1.44 467 (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]. 1.19/1.44 468 (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]. 1.19/1.44 469 (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]. 1.19/1.44 470 (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]. 1.19/1.44 471 (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]. 1.19/1.44 472 (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,V_a_2,V_b_2) <-> 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))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 473 (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]. 1.19/1.44 474 (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_a_2,V_b_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__iff2) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 475 (all V_A_2 all V_x_2 all T_a (c_member(T_a,V_x_2,V_A_2) <-> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 476 (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]. 1.19/1.44 477 (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,V_c,c_Groups_Oplus__class_Oplus(T_a,V_a,V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 478 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> (hAPP(c_Nat_OSuc,V_m) != V_n -> 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]. 1.19/1.44 479 (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_Ocard(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) != hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> 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]. 1.19/1.44 480 (all V_f_2 all V_Fa_2 all T_a (c_Finite__Set_Ofinite(T_a,V_Fa_2) -> (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) <-> (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)))))) # label(fact_setsum__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 481 (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]. 1.19/1.44 482 (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]. 1.19/1.44 483 (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]. 1.19/1.44 484 (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]. 1.19/1.44 485 (all V_A_2 all T_a all V_b_2 all V_a_2 (V_b_2 = V_a_2 -> (c_member(T_a,V_b_2,V_A_2) -> c_member(T_a,V_a_2,V_A_2)))) # label(fact_eq__mem__trans) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 486 (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]. 1.19/1.44 487 (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]. 1.19/1.44 488 (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]. 1.19/1.44 489 (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]. 1.19/1.44 490 (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_Oplus__class_Oplus(T_a,V_a_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.19/1.44 491 (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]. 1.22/1.46 1.22/1.46 ============================== end of process non-clausal formulas === 1.22/1.46 1.22/1.46 ============================== PROCESS INITIAL CLAUSES =============== 1.22/1.46 1.22/1.46 ============================== PREDICATE ELIMINATION ================= 1.22/1.46 492 -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(7)]. 1.22/1.46 493 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 1.22/1.46 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(492,a,493,a)]. 1.22/1.46 494 -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(44)]. 1.22/1.46 495 -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(70)]. 1.22/1.46 496 -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(106)]. 1.22/1.46 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(496,a,493,a)]. 1.22/1.46 497 -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(198)]. 1.22/1.46 498 -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(268)]. 1.22/1.46 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(498,a,493,a)]. 1.22/1.46 499 -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(268)]. 1.22/1.46 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(499,a,493,a)]. 1.22/1.49 500 -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(268)]. 1.22/1.49 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(500,a,493,a)]. 1.22/1.49 501 -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(270)]. 1.22/1.49 502 -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(277)]. 1.22/1.49 503 -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(299)]. 1.22/1.49 504 -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(361)]. 1.22/1.49 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(504,a,493,a)]. 1.22/1.49 505 -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(373)]. 1.22/1.49 506 -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(380)]. 1.22/1.49 507 -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(381)]. 1.22/1.49 508 class_Groups_Ocomm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 1.22/1.49 509 -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(1)]. 1.22/1.49 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(508,a,509,a)]. 1.22/1.49 510 -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(74)]. 1.22/1.49 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(510,a,508,a)]. 1.22/1.49 511 -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(79)]. 1.22/1.49 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(511,a,508,a)]. 1.22/1.49 512 -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(90)]. 1.22/1.49 513 -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(102)]. 1.22/1.49 514 -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,f6(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(128)]. 1.22/1.49 Derived: -c_Fun_Oinj__on(A,B,C,D) | c_Set_Oimage(A,B,C,D) != E | c_member(A,f6(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(514,a,508,a)]. 1.22/1.49 515 -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,f6(V6,V7,F,E,D,C,B,A))) != hAPP(V7,f6(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(128)]. 1.22/1.49 Derived: -c_Fun_Oinj__on(A,B,C,D) | c_Set_Oimage(A,B,C,D) != E | hAPP(F,hAPP(C,f6(F,V6,E,D,C,B,A,tc_Nat_Onat))) != hAPP(V6,f6(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(515,a,508,a)]. 1.22/1.49 516 -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(147)]. 1.22/1.49 517 -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(154)]. 1.22/1.49 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(517,a,508,a)]. 1.22/1.49 518 -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(154)]. 1.22/1.50 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(518,a,508,a)]. 1.22/1.50 519 -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(213)]. 1.22/1.50 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(519,a,508,a)]. 1.22/1.50 520 -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(221)]. 1.22/1.50 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(520,a,508,a)]. 1.22/1.50 521 -class_Orderings_Oord(A) | -class_Groups_Ocomm__monoid__add(B) | C != D | E != F | c_Orderings_Oord__class_Oless__eq(A,D,f25(V6,V7,E,F,D,C,A,B)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,B,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,D,E)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,B,V7,c_SetInterval_Oord__class_OatLeastLessThan(A,C,F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(290)]. 1.22/1.50 Derived: -class_Orderings_Oord(A) | B != C | D != E | c_Orderings_Oord__class_Oless__eq(A,C,f25(F,V6,D,E,C,B,A,tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,F,c_SetInterval_Oord__class_OatLeastLessThan(A,C,D)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,B,E)). [resolve(521,b,508,a)]. 1.22/1.50 522 -class_Orderings_Oord(A) | -class_Groups_Ocomm__monoid__add(B) | C != D | E != F | c_Orderings_Oord__class_Oless(A,f25(V6,V7,E,F,D,C,A,B),E) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,B,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,D,E)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,B,V7,c_SetInterval_Oord__class_OatLeastLessThan(A,C,F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(290)]. 1.22/1.50 Derived: -class_Orderings_Oord(A) | B != C | D != E | c_Orderings_Oord__class_Oless(A,f25(F,V6,D,E,C,B,A,tc_Nat_Onat),D) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,F,c_SetInterval_Oord__class_OatLeastLessThan(A,C,D)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,B,E)). [resolve(522,b,508,a)]. 1.22/1.50 523 -class_Orderings_Oord(A) | -class_Groups_Ocomm__monoid__add(B) | C != D | E != F | hAPP(V6,f25(V6,V7,E,F,D,C,A,B)) != hAPP(V7,f25(V6,V7,E,F,D,C,A,B)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,B,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,D,E)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,B,V7,c_SetInterval_Oord__class_OatLeastLessThan(A,C,F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(290)]. 1.22/1.50 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f25(F,V6,D,E,C,B,A,tc_Nat_Onat)) != hAPP(V6,f25(F,V6,D,E,C,B,A,tc_Nat_Onat)) | c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,F,c_SetInterval_Oord__class_OatLeastLessThan(A,C,D)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat,V6,c_SetInterval_Oord__class_OatLeastLessThan(A,B,E)). [resolve(523,b,508,a)]. 1.22/1.50 524 -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(336)]. 1.22/1.50 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(524,a,508,a)]. 1.22/1.50 525 -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(336)]. 1.22/1.50 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(525,a,508,a)]. 1.22/1.50 526 -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(343)]. 1.22/1.50 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(526,a,508,a)]. 1.22/1.50 527 -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(365)]. 1.22/1.51 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(527,a,508,a)]. 1.22/1.51 528 -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,f37(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(377)]. 1.22/1.51 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,f37(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(528,b,508,a)]. 1.22/1.51 529 -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,f37(E,D,C,B,A)),hAPP(E,f37(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(377)]. 1.22/1.51 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,f37(D,C,B,A,tc_Nat_Onat)),hAPP(D,f37(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(529,b,508,a)]. 1.22/1.51 530 -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(405)]. 1.22/1.51 531 -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(438)]. 1.22/1.51 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(531,a,508,a)]. 1.22/1.51 532 -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(469)]. 1.22/1.51 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(532,a,508,a)]. 1.31/1.53 533 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 1.31/1.53 534 -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(2)]. 1.31/1.53 535 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | B = C # label(fact_linorder__less__linear) # label(axiom). [clausify(66)]. 1.31/1.53 536 -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(75)]. 1.31/1.53 537 -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,B) | c_Orderings_Oord__class_Oless__eq(A,D,B) # label(fact_atLeastLessThan__subset__iff) # label(axiom). [clausify(78)]. 1.31/1.53 538 -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,B) | c_Orderings_Oord__class_Oless__eq(A,C,E) # label(fact_atLeastLessThan__subset__iff) # label(axiom). [clausify(78)]. 1.31/1.53 539 -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(84)]. 1.31/1.53 540 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OgreaterThan(A,B) != c_SetInterval_Oord__class_OgreaterThan(A,C) | B = C # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(103)]. 1.31/1.53 541 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OgreaterThan(A,B) = c_SetInterval_Oord__class_OgreaterThan(A,C) | B != C # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(103)]. 1.31/1.53 542 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(105)]. 1.31/1.53 543 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(105)]. 1.31/1.53 544 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__neqE) # label(axiom). [clausify(124)]. 1.31/1.53 545 -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(157)]. 1.31/1.53 546 -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(157)]. 1.31/1.53 547 -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(162)]. 1.31/1.53 548 -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(162)]. 1.31/1.53 549 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,C,B) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neq__iff) # label(axiom). [clausify(182)]. 1.31/1.53 550 -class_Orderings_Olinorder(A) | B != C | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__neq__iff) # label(axiom). [clausify(182)]. 1.31/1.53 551 -class_Orderings_Olinorder(A) | B != C | -c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neq__iff) # label(axiom). [clausify(182)]. 1.31/1.53 552 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C = B | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(197)]. 1.31/1.53 553 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | C != B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(197)]. 1.31/1.53 554 -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(197)]. 1.31/1.53 555 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | B = C # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(231)]. 1.31/1.53 556 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) | B != C # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(231)]. 1.31/1.53 557 -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(250)]. 1.31/1.53 558 -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,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | E = C # label(fact_atLeastLessThan__inj_I2_J) # label(axiom). [clausify(256)]. 1.31/1.53 559 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | B != D | E != C | c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) = c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(332)]. 1.31/1.53 560 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | B = D | c_SetInterval_Oord__class_OatLeastLessThan(A,B,C) != c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(332)]. 1.31/1.53 561 -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,B,C) != c_SetInterval_Oord__class_OatLeastLessThan(A,D,E) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(332)]. 1.31/1.53 562 -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,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | B = D # label(fact_atLeastLessThan__inj_I1_J) # label(axiom). [clausify(357)]. 1.31/1.53 563 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,B,C) | C = B # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(386)]. 1.31/1.53 564 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,B,C) | C != B # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(386)]. 1.31/1.53 565 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | B = C | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__cases) # label(axiom). [clausify(387)]. 1.31/1.53 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(533,a,534,a)]. 1.31/1.53 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A) | A = B. [resolve(533,a,535,a)]. 1.31/1.53 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(533,a,536,a)]. 1.31/1.53 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,A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,A). [resolve(533,a,537,a)]. 1.31/1.56 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,A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,D). [resolve(533,a,538,a)]. 1.31/1.56 Derived: c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,A) != c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,B) | A = B. [resolve(533,a,540,a)]. 1.31/1.56 Derived: c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,A) = c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,B) | A != B. [resolve(533,a,541,a)]. 1.31/1.56 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(533,a,542,a)]. 1.31/1.56 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(533,a,545,a)]. 1.31/1.56 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(533,a,546,a)]. 1.31/1.56 Derived: A != B | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(533,a,550,a)]. 1.31/1.56 Derived: A != B | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B). [resolve(533,a,551,a)]. 1.31/1.56 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(533,a,554,a)]. 1.31/1.56 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,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | D = B. [resolve(533,a,558,a)]. 1.31/1.56 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | A != C | D != B | c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D). [resolve(533,a,559,a)]. 1.31/1.56 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | A = C | c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,B) != c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C,D). [resolve(533,a,560,a)]. 1.31/1.56 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | B = A. [resolve(533,a,563,a)]. 1.31/1.56 566 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C != B | c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(425)]. 1.31/1.56 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | B != A | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B). [resolve(566,a,533,a)]. 1.31/1.56 567 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C = B | -c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(425)]. 1.31/1.56 568 -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(461)]. 1.31/1.56 569 -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(470)]. 1.31/1.56 570 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]. 1.31/1.56 571 -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__cancel__right) # label(axiom). [clausify(4)]. 1.31/1.56 572 -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__cancel__right) # label(axiom). [clausify(4)]. 1.31/1.56 573 -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(37)]. 1.31/1.56 574 -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(137)]. 1.31/1.56 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(570,a,571,a)]. 1.31/1.56 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,C,B)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,C). [resolve(570,a,574,a)]. 1.31/1.56 575 -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(419)]. 1.31/1.56 576 -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(420)]. 1.31/1.56 577 -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(439)]. 1.31/1.56 578 -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(439)]. 1.31/1.56 579 -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(453)]. 1.31/1.56 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(579,a,570,a)]. 1.31/1.56 580 -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(453)]. 1.31/1.56 581 -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,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__less__cancel__left) # label(axiom). [clausify(472)]. 1.31/1.56 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,C,A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B)). [resolve(581,a,570,a)]. 1.31/1.56 582 -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,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__less__cancel__left) # label(axiom). [clausify(472)]. 1.31/1.56 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,C,A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B)). [resolve(582,a,570,a)]. 1.35/1.61 583 class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 1.35/1.61 584 -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(11)]. 1.35/1.61 585 -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(64)]. 1.35/1.61 586 -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(203)]. 1.35/1.61 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(583,a,585,a)]. 1.35/1.61 587 class_Rings_Ocomm__semiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.35/1.61 588 -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(24)]. 1.35/1.61 589 -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(72)]. 1.35/1.61 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(589,a,587,a)]. 1.35/1.61 590 -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(82)]. 1.35/1.61 591 -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(94)]. 1.35/1.61 592 -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(155)]. 1.35/1.61 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(592,a,587,a)]. 1.35/1.61 593 -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(315)]. 1.35/1.61 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) = A. [resolve(593,a,587,a)]. 1.35/1.61 594 -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(399)]. 1.49/1.73 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(594,a,587,a)]. 1.49/1.73 595 -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(477)]. 1.49/1.73 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(595,a,587,a)]. 1.49/1.73 596 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.49/1.73 597 -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(25)]. 1.49/1.73 598 -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(176)]. 1.49/1.73 599 -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(189)]. 1.49/1.73 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(596,a,597,a)]. 1.49/1.73 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(596,a,598,a)]. 1.49/1.73 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(596,a,599,a)]. 1.49/1.73 600 -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(319)]. 1.49/1.73 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(600,a,596,a)]. 1.49/1.73 601 -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(401)]. 1.49/1.73 602 -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(412)]. 1.49/1.73 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(602,a,596,a)]. 1.49/1.73 603 -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(445)]. 1.49/1.73 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(603,a,596,a)]. 1.49/1.73 604 -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(413)]. 1.49/1.73 605 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 1.49/1.73 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C) | B = C. [resolve(604,a,605,a)]. 1.49/1.73 606 -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(214)]. 1.75/1.99 607 class_Rings_Ozero__neq__one(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ozero__neq__one) # label(axiom). [assumption]. 1.75/1.99 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(606,a,607,a)]. 1.75/1.99 608 -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(486)]. 1.75/1.99 609 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 1.75/1.99 610 -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(89)]. 1.75/1.99 611 -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(179)]. 1.75/1.99 612 -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(179)]. 1.75/1.99 613 -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__cancel) # label(axiom). [clausify(302)]. 1.75/1.99 614 -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__cancel) # label(axiom). [clausify(302)]. 1.75/1.99 615 -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(354)]. 1.75/1.99 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B) | A = C. [resolve(609,a,610,a)]. 1.75/1.99 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C) | B != C. [resolve(609,a,612,a)]. 1.75/1.99 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B) | A != C. [resolve(609,a,614,a)]. 1.75/1.99 616 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 1.75/1.99 617 -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(92)]. 1.75/1.99 618 class_Groups_Omonoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__add) # label(axiom). [assumption]. 1.75/1.99 619 -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(285)]. 1.75/1.99 620 -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(376)]. 1.75/1.99 621 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]. 1.75/1.99 622 -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(325)]. 1.75/1.99 623 -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(325)]. 1.75/1.99 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(621,a,622,a)]. 1.75/1.99 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(621,a,623,a)]. 1.75/1.99 1.75/1.99 ============================== end predicate elimination ============= 1.75/1.99 1.75/1.99 Auto_denials: (non-Horn, no changes). 1.75/1.99 1.75/1.99 Term ordering decisions: 1.75/1.99 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. f3=1. f4=1. f5=1. f7=1. f8=1. f12=1. f13=1. f18=1. f19=1. f20=1. f21=1. f22=1. f26=1. f34=1. f35=1. f36=1. f41=1. f42=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. f14=1. f16=1. f27=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. f1=1. f24=1. f32=1. f33=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. f2=1. f11=1. f17=1. f23=1. f28=1. f29=1. f38=1. f39=1. f40=1. f43=1. c_SetInterval_Oord_OatLeastLessThan=1. c_SetInterval_Oord_OgreaterThanAtMost=1. f9=1. f10=1. f15=1. f37=1. f6=1. f25=1. 60.99/61.26 60.99/61.26 ============================== end of process initial clauses ======== 60.99/61.26 60.99/61.26 ============================== CLAUSES FOR SEARCH ==================== 60.99/61.26 60.99/61.26 ============================== end of clauses for search ============= 60.99/61.26 60.99/61.26 ============================== SEARCH ================================ 60.99/61.26 60.99/61.26 % Starting search at 0.64 seconds. 60.99/61.26 60.99/61.26 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 35 (0.00 of 0.84 sec). 60.99/61.26 60.99/61.26 Low Water (keep): wt=37.000, iters=3493 60.99/61.26 60.99/61.26 Low Water (keep): wt=35.000, iters=3412 60.99/61.26 60.99/61.26 Low Water (keep): wt=32.000, iters=3467 60.99/61.26 60.99/61.26 Low Water (keep): wt=30.000, iters=3391 60.99/61.26 60.99/61.26 Low Water (keep): wt=27.000, iters=3408 60.99/61.26 60.99/61.26 Low Water (keep): wt=26.000, iters=3537 60.99/61.26 60.99/61.26 Low Water (keep): wt=24.000, iters=3598 60.99/61.26 60.99/61.26 Low Water (keep): wt=22.000, iters=3358 60.99/61.26 60.99/61.26 Low Water (keep): wt=21.000, iters=3349 60.99/61.26 60.99/61.26 Low Water (keep): wt=20.000, iters=3363 60.99/61.26 60.99/61.26 Low Water (keep): wt=19.000, iters=3359 60.99/61.26 60.99/61.26 Low Water (keep): wt=18.000, iters=3494 60.99/61.26 60.99/61.26 Low Water (keep): wt=17.000, iters=3382 60.99/61.26 60.99/61.26 Low Water (keep): wt=16.000, iters=3377 60.99/61.26 60.99/61.26 Low Water (keep): wt=15.000, iters=3548 60.99/61.26 60.99/61.26 Low Water (keep): wt=14.000, iters=3451 60.99/61.26 60.99/61.26 Low Water (keep): wt=12.000, iters=3348 60.99/61.26 60.99/61.26 Low Water (keep): wt=11.000, iters=3353 60.99/61.26 60.99/61.26 Low Water (keep): wt=10.000, iters=3340 60.99/61.26 60.99/61.26 Low Water (keep): wt=9.000, iters=3368 60.99/61.26 60.99/61.26 Low Water (keep): wt=8.000, iters=3339 60.99/61.26 60.99/61.26 Low Water (displace): id=2805, wt=78.000 60.99/61.26 60.99/61.26 Low Water (displace): id=2806, wt=76.000 60.99/61.26 60.99/61.26 Low Water (displace): id=4518, wt=69.000 60.99/61.26 60.99/61.26 Low Water (displace): id=3362, wt=58.000 60.99/61.26 60.99/61.26 Low Water (displace): id=3075, wt=57.000 60.99/61.26 60.99/61.26 Low Water (displace): id=12413, wt=44.000 60.99/61.26 60.99/61.26 Low Water (displace): id=12171, wt=43.000 60.99/61.26 60.99/61.26 Low Water (displace): id=12411, wt=42.000 60.99/61.26 60.99/61.26 Low Water (displace): id=12169, wt=41.000 60.99/61.26 60.99/61.26 Low Water (displace): id=12065, wt=40.000 60.99/61.26 60.99/61.26 Low Water (displace): id=11987, wt=39.000 60.99/61.26 60.99/61.26 Low Water (displace): id=11919, wt=38.000 60.99/61.26 60.99/61.26 Low Water (displace): id=11413, wt=37.000 60.99/61.26 60.99/61.26 Low Water (displace): id=13393, wt=36.000 60.99/61.26 60.99/61.26 Low Water (displace): id=12501, wt=35.000 60.99/61.26 60.99/61.26 Low Water (displace): id=14103, wt=34.000 60.99/61.26 60.99/61.26 Low Water (displace): id=14355, wt=33.000 60.99/61.26 60.99/61.26 Low Water (displace): id=14126, wt=32.000 60.99/61.26 60.99/61.26 Low Water (displace): id=14903, wt=31.000 60.99/61.26 60.99/61.26 Low Water (displace): id=15311, wt=30.000 60.99/61.26 60.99/61.26 Low Water (displace): id=14894, wt=29.000 60.99/61.26 60.99/61.26 Low Water (displace): id=15929, wt=28.000 60.99/61.26 60.99/61.26 Low Water (displace): id=16183, wt=27.000 60.99/61.26 60.99/61.26 Low Water (displace): id=17287, wt=26.000 60.99/61.26 60.99/61.26 ============================== PROOF ================================= 60.99/61.26 % SZS status Theorem 60.99/61.26 % SZS output start Refutation 60.99/61.26 60.99/61.26 % Proof 1 at 57.81 (+ 2.11) seconds. 60.99/61.26 % Length of proof is 110. 60.99/61.26 % Level of proof is 12. 60.99/61.26 % Maximum clause weight is 25.000. 60.99/61.26 % Given clauses 19369. 60.99/61.26 60.99/61.26 10 (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]. 60.99/61.26 14 (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]. 60.99/61.26 36 (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]. 60.99/61.26 41 (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]. 60.99/61.26 53 (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]. 60.99/61.26 85 (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]. 60.99/61.26 87 (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]. 60.99/61.26 105 (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) <-> c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2)))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 111 (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]. 60.99/61.26 123 (all V_A_2 all T_a ((exists B_x c_member(T_a,B_x,V_A_2)) <-> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 129 (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]. 60.99/61.26 145 (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]. 60.99/61.26 167 (all V_x_2 all T_b all T_a (class_Orderings_Obot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_x_2))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 219 (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]. 60.99/61.26 227 (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]. 60.99/61.26 237 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,hAPP(c_Nat_OSuc,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 238 (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]. 60.99/61.26 298 (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]. 60.99/61.26 305 (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]. 60.99/61.26 317 (all V_A_2 all T_a (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) | c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = V_A_2)) # label(fact_card__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 366 (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]. 60.99/61.26 372 (all T_a c_Finite__Set_Ocard(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 374 (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]. 60.99/61.26 395 (all V_m V_m = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 398 (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]. 60.99/61.26 427 (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]. 60.99/61.26 444 (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_a_2,V_b_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__empty__iff) # label(axiom) # label(non_clause). [assumption]. 60.99/61.26 475 (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]. 60.99/61.26 533 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 60.99/61.26 542 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(105)]. 60.99/61.26 632 -c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),hAPP(c_Nat_OSuc,B)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) # label(fact_Suc__less__SucD) # label(axiom). [clausify(10)]. 60.99/61.26 635 -c_member(A,B,C) | c_Set_Oinsert(A,B,C) = C # label(fact_insert__absorb) # label(axiom). [clausify(14)]. 60.99/61.26 666 c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A)) # label(fact_not__less__eq) # label(axiom). [clausify(36)]. 60.99/61.26 667 -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A)) # label(fact_not__less__eq) # label(axiom). [clausify(36)]. 60.99/61.26 670 c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)),B) # label(fact_empty__subsetI) # label(axiom). [clausify(41)]. 60.99/61.26 684 -c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),A,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,c_Finite__Set_Ocard(tc_Nat_Onat,A)))) | c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,c_Finite__Set_Ocard(tc_Nat_Onat,A))) = A # label(fact_subset__card__intvl__is__intvl) # label(axiom). [clausify(53)]. 60.99/61.26 722 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]. 60.99/61.26 724 -class_Finite__Set_Ofinite(A) | c_Finite__Set_Ofinite(A,B) # label(fact_finite__code) # label(axiom). [clausify(85)]. 60.99/61.26 727 c_Finite__Set_Ofinite(tc_Nat_Onat,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) # label(fact_finite__atLeastAtMost) # label(axiom). [clausify(87)]. 60.99/61.26 742 hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),B)) # label(fact_top1I) # label(axiom). [clausify(111)]. 60.99/61.26 766 -c_member(A,B,C) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != C # label(fact_ex__in__conv) # label(axiom). [clausify(123)]. 60.99/61.26 767 c_member(A,f5(B,A),B) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B # label(fact_ex__in__conv) # label(axiom). [clausify(123)]. 60.99/61.26 772 -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(129)]. 60.99/61.26 792 -class_Orderings_Otop(A) | c_Orderings_Otop__class_Otop(A) = hAPP(c_Orderings_Otop__class_Otop(tc_fun(B,A)),C) # label(fact_top__apply) # label(axiom). [clausify(145)]. 60.99/61.26 793 -class_Orderings_Otop(A) | hAPP(c_Orderings_Otop__class_Otop(tc_fun(B,A)),C) = c_Orderings_Otop__class_Otop(A). [copy(792),flip(b)]. 60.99/61.26 815 -class_Orderings_Obot(A) | c_Orderings_Obot__class_Obot(A) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(B,A)),C) # label(fact_bot__apply) # label(axiom). [clausify(167)]. 60.99/61.26 816 -class_Orderings_Obot(A) | hAPP(c_Orderings_Obot__class_Obot(tc_fun(B,A)),C) = c_Orderings_Obot__class_Obot(A). [copy(815),flip(b)]. 60.99/61.26 874 c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,A) # label(fact_nat__add__commute) # label(axiom). [clausify(219)]. 60.99/61.26 879 -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_less__nat__zero__code) # label(axiom). [clausify(227)]. 60.99/61.26 887 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,B)) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom). [clausify(237)]. 60.99/61.26 888 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,B)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B). [copy(887),flip(a)]. 60.99/61.26 889 A != B | hBOOL(hAPP(c_Set_Oinsert(C,A,D),B)) # label(fact_insert__code) # label(axiom). [clausify(238)]. 60.99/61.26 891 A = B | hBOOL(hAPP(C,B)) | -hBOOL(hAPP(c_Set_Oinsert(D,A,C),B)) # label(fact_insert__code) # label(axiom). [clausify(238)]. 60.99/61.26 937 class_Orderings_Otop(tc_HOL_Obool) # label(arity_HOL__Obool__Orderings_Otop) # label(axiom). [assumption]. 60.99/61.26 969 -c_member(A,B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))) # label(fact_emptyE) # label(axiom). [clausify(298)]. 60.99/61.26 975 -c_member(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,f27(B)) | -c_Finite__Set_Ofinite(tc_Nat_Onat,B) # label(fact_finite__nat__set__iff__bounded) # label(axiom). [clausify(305)]. 60.99/61.26 982 c_Finite__Set_Ocard(A,B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != B # label(fact_card__eq__0__iff) # label(axiom). [clausify(317)]. 60.99/61.26 1030 -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(366)]. 60.99/61.26 1040 class_Finite__Set_Ofinite(tc_Product__Type_Ounit) # label(arity_Product____Type__Ounit__Finite__Set_Ofinite) # label(axiom). [assumption]. 60.99/61.26 1042 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(372)]. 60.99/61.26 1043 -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),C) # label(fact_less__trans__Suc) # label(axiom). [clausify(374)]. 60.99/61.26 1064 class_Orderings_Oorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Oorder) # label(axiom). [assumption]. 60.99/61.26 1066 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(395)]. 60.99/61.26 1068 c_Set_Oinsert(tc_Nat_Onat,A,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,A)) # label(fact_atLeastLessThan__singleton) # label(axiom). [clausify(398)]. 60.99/61.26 1069 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A) = c_Set_Oinsert(tc_Nat_Onat,A,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))). [copy(1068),rewrite([888(10)]),flip(a)]. 60.99/61.26 1075 class_Orderings_Obot(tc_HOL_Obool) # label(arity_HOL__Obool__Orderings_Obot) # label(axiom). [assumption]. 60.99/61.26 1087 c_member(A,B,c_Set_Oinsert(A,B,C)) # label(fact_insertI1) # label(axiom). [clausify(427)]. 60.99/61.26 1109 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,C) != c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) # label(fact_atLeastatMost__empty__iff) # label(axiom). [clausify(444)]. 60.99/61.26 1133 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]. 60.99/61.26 1135 -c_member(A,B,C) | hBOOL(hAPP(C,B)) # label(fact_mem__def) # label(axiom). [clausify(475)]. 60.99/61.26 1136 c_member(A,B,C) | -hBOOL(hAPP(C,B)) # label(fact_mem__def) # label(axiom). [clausify(475)]. 60.99/61.26 1202 c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(533,a,542,a)]. 60.99/61.26 1311 c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,A). [resolve(684,a,670,a),rewrite([1042(8),1066(5)]),flip(a)]. 60.99/61.26 1328 -c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [para(722(a,1),632(a,3)),unit_del(b,879)]. 60.99/61.26 1329 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,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)). [para(722(a,1),666(b,3))]. 60.99/61.26 1330 -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,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)). [para(722(a,1),667(b,3))]. 60.99/61.26 1384 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Set_Oinsert(A,f5(B,A),B) = B. [resolve(767,a,635,a)]. 60.99/61.26 1651 hBOOL(hAPP(c_Set_Oinsert(A,B,C),B)). [xx_res(889,a)]. 60.99/61.26 1754 hAPP(c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),B) = c_Orderings_Otop__class_Otop(tc_HOL_Obool). [resolve(937,a,793,a)]. 60.99/61.26 1756 hBOOL(c_Orderings_Otop__class_Otop(tc_HOL_Obool)). [back_rewrite(742),rewrite([1754(4)])]. 60.99/61.26 2055 c_Finite__Set_Ofinite(tc_Product__Type_Ounit,A). [resolve(1040,a,724,a)]. 60.99/61.26 2178 c_Finite__Set_Ocard(A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1066,a,982,b(flip)),rewrite([874(7)])]. 60.99/61.26 2262 hAPP(c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)),B) = c_Orderings_Obot__class_Obot(tc_HOL_Obool). [resolve(1075,a,816,a)]. 60.99/61.26 2304 c_member(tc_Nat_Onat,A,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A)). [para(1069(a,2),1087(a,3))]. 60.99/61.26 3274 c_member(tc_Product__Type_Ounit,A,B) | c_Finite__Set_Ocard(tc_Product__Type_Ounit,c_Set_Oinsert(tc_Product__Type_Ounit,A,B)) = hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(tc_Product__Type_Ounit,B)). [resolve(2055,a,1030,a)]. 60.99/61.26 3284 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(2055,a,772,a),rewrite([1133(6)]),flip(a)]. 60.99/61.26 3390 -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,c_Groups_Oone__class_Oone(tc_Nat_Onat),A). [resolve(1330,b,1202,b)]. 60.99/61.26 5402 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,A) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B,B). [para(1311(a,1),1311(a,1))]. 60.99/61.26 5403 c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A,A) = c_0. [new_symbol(5402)]. 60.99/61.26 5464 c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_0. [back_rewrite(1311),rewrite([5403(6)])]. 60.99/61.26 6985 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,A). [resolve(1329,b,1043,b),unit_del(c,1328)]. 60.99/61.26 10289 c_member(A,B,c_Set_Oinsert(C,B,D)). [resolve(1651,a,1136,b)]. 60.99/61.26 10307 hBOOL(hAPP(c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A),A)). [resolve(2304,a,1135,a)]. 60.99/61.26 15683 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != c_Set_Oinsert(B,C,D). [resolve(10289,a,766,a)]. 60.99/61.26 15685 c_member(A,B,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,B)). [resolve(10307,a,1136,b)]. 60.99/61.26 15688 c_Set_Oinsert(A,B,C) != c_0. [para(5464(a,1),15683(a,1)),flip(a)]. 60.99/61.26 15691 c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = c_0. [resolve(15688,a,1384,b)]. 60.99/61.26 15730 hAPP(c_0,A) = c_Orderings_Obot__class_Obot(tc_HOL_Obool). [back_rewrite(2262),rewrite([15691(3)])]. 60.99/61.26 15732 c_Finite__Set_Ocard(A,c_0) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [back_rewrite(2178),rewrite([15691(6),874(5),1066(5)])]. 60.99/61.26 15739 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,C) != c_0. [back_rewrite(1109),rewrite([15691(6)])]. 60.99/61.26 15741 -c_member(A,B,c_0). [back_rewrite(969),rewrite([15691(3)])]. 60.99/61.26 15748 -hBOOL(c_Orderings_Obot__class_Obot(tc_HOL_Obool)). [para(15730(a,1),1136(b,1)),unit_del(a,15741)]. 60.99/61.26 16235 c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,f27(c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A))). [resolve(15685,a,975,a),unit_del(b,727)]. 60.99/61.26 26883 c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),f27(c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A))). [resolve(6985,b,16235,a)]. 60.99/61.26 27360 c_Finite__Set_Ocard(tc_Product__Type_Ounit,c_Set_Oinsert(tc_Product__Type_Ounit,A,c_0)) = c_Groups_Oone__class_Oone(tc_Nat_Onat). [resolve(3274,a,15741,a),rewrite([15732(9),722(9)])]. 60.99/61.26 31500 c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),f27(c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A))). [resolve(26883,a,3390,a)]. 60.99/61.26 31510 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(27360,a,3284,a)]. 60.99/61.26 31559 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),f27(c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,A))) != c_0. [resolve(31500,a,15739,b),unit_del(a,1064)]. 60.99/61.26 31576 A = B. [para(31510(a,2),891(c,1,1)),rewrite([15730(3),1754(9)]),unit_del(b,15748),unit_del(c,1756)]. 60.99/61.26 31577 $F. [resolve(31576,a,31559,a)]. 60.99/61.26 60.99/61.26 % SZS output end Refutation 60.99/61.26 ============================== end of proof ========================== 60.99/61.26 60.99/61.26 ============================== STATISTICS ============================ 60.99/61.26 60.99/61.26 Given=19369. Generated=4125201. Kept=30858. proofs=1. 60.99/61.26 Usable=18910. Sos=7656. Demods=182. Limbo=10, Disabled=5113. Hints=0. 60.99/61.26 Megabytes=35.58. 60.99/61.26 User_CPU=57.81, System_CPU=2.11, Wall_clock=60. 60.99/61.26 60.99/61.26 ============================== end of statistics ===================== 60.99/61.26 60.99/61.26 ============================== end of search ========================= 60.99/61.26 60.99/61.26 THEOREM PROVED 60.99/61.26 % SZS status Theorem 60.99/61.26 60.99/61.26 Exiting with 1 proof. 60.99/61.26 60.99/61.26 Process 32130 exit (max_proofs) Thu Aug 29 13:11:13 2019 60.99/61.26 Prover9 interrupted 60.99/61.26 EOF