TSTP Solution File: NUM925+2 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : NUM925+2 : TPTP v8.1.0. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Mon Jul 18 13:33:19 EDT 2022
% Result : Theorem 1.76s 1.99s
% Output : Refutation 1.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : NUM925+2 : TPTP v8.1.0. Released v5.3.0.
% 0.06/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.13/0.31 % Computer : n016.cluster.edu
% 0.13/0.31 % Model : x86_64 x86_64
% 0.13/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.31 % Memory : 8042.1875MB
% 0.13/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.31 % CPULimit : 300
% 0.13/0.31 % WCLimit : 600
% 0.13/0.31 % DateTime : Wed Jul 6 19:39:05 EDT 2022
% 0.13/0.31 % CPUTime :
% 1.07/1.36 ============================== Prover9 ===============================
% 1.07/1.36 Prover9 (32) version 2009-11A, November 2009.
% 1.07/1.36 Process 632 was started by sandbox on n016.cluster.edu,
% 1.07/1.36 Wed Jul 6 19:39:06 2022
% 1.07/1.36 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_458_n016.cluster.edu".
% 1.07/1.36 ============================== end of head ===========================
% 1.07/1.36
% 1.07/1.36 ============================== INPUT =================================
% 1.07/1.36
% 1.07/1.36 % Reading from file /tmp/Prover9_458_n016.cluster.edu
% 1.07/1.36
% 1.07/1.36 set(prolog_style_variables).
% 1.07/1.36 set(auto2).
% 1.07/1.36 % set(auto2) -> set(auto).
% 1.07/1.36 % set(auto) -> set(auto_inference).
% 1.07/1.36 % set(auto) -> set(auto_setup).
% 1.07/1.36 % set(auto_setup) -> set(predicate_elim).
% 1.07/1.36 % set(auto_setup) -> assign(eq_defs, unfold).
% 1.07/1.36 % set(auto) -> set(auto_limits).
% 1.07/1.36 % set(auto_limits) -> assign(max_weight, "100.000").
% 1.07/1.36 % set(auto_limits) -> assign(sos_limit, 20000).
% 1.07/1.36 % set(auto) -> set(auto_denials).
% 1.07/1.36 % set(auto) -> set(auto_process).
% 1.07/1.36 % set(auto2) -> assign(new_constants, 1).
% 1.07/1.36 % set(auto2) -> assign(fold_denial_max, 3).
% 1.07/1.36 % set(auto2) -> assign(max_weight, "200.000").
% 1.07/1.36 % set(auto2) -> assign(max_hours, 1).
% 1.07/1.36 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 1.07/1.36 % set(auto2) -> assign(max_seconds, 0).
% 1.07/1.36 % set(auto2) -> assign(max_minutes, 5).
% 1.07/1.36 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 1.07/1.36 % set(auto2) -> set(sort_initial_sos).
% 1.07/1.36 % set(auto2) -> assign(sos_limit, -1).
% 1.07/1.36 % set(auto2) -> assign(lrs_ticks, 3000).
% 1.07/1.36 % set(auto2) -> assign(max_megs, 400).
% 1.07/1.36 % set(auto2) -> assign(stats, some).
% 1.07/1.36 % set(auto2) -> clear(echo_input).
% 1.07/1.36 % set(auto2) -> set(quiet).
% 1.07/1.36 % set(auto2) -> clear(print_initial_clauses).
% 1.07/1.36 % set(auto2) -> clear(print_given).
% 1.07/1.36 assign(lrs_ticks,-1).
% 1.07/1.36 assign(sos_limit,10000).
% 1.07/1.36 assign(order,kbo).
% 1.07/1.36 set(lex_order_vars).
% 1.07/1.36 clear(print_given).
% 1.07/1.36
% 1.07/1.36 % formulas(sos). % not echoed (708 formulas)
% 1.07/1.36
% 1.07/1.36 ============================== end of input ==========================
% 1.07/1.36
% 1.07/1.36 % From the command line: assign(max_seconds, 300).
% 1.07/1.36
% 1.07/1.36 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 1.07/1.36
% 1.07/1.36 % Formulas that are not ordinary clauses:
% 1.07/1.36 1 (all B_1_1 all B_2_1 is_bool(hAPP_int_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 2 (all B_1_1 all B_2_1 is_bool(hAPP_nat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 3 (all B_1_1 all B_2_1 is_bool(hAPP_real_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 4 (all Xa all Ya (hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int <-> Xa = zero_zero_int & Ya = zero_zero_int)) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 5 (all Xa all Ya (hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real <-> Xa = zero_zero_real & Ya = zero_zero_real)) # label(fact_3_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 6 (all A_24 (hAPP_nat_int(power_power_int(A_24),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int <-> A_24 = zero_zero_int)) # label(fact_10_zero__eq__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 7 (all A_24 (hAPP_nat_real(power_power_real(A_24),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real <-> A_24 = zero_zero_real)) # label(fact_11_zero__eq__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 8 (all W_8 hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_8)) = number_number_of_int(hAPP_int_int(plus_plus_int(bit1(pls)),W_8))) # label(fact_12_add__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 9 (all W_8 hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_8)) = number267125858f_real(hAPP_int_int(plus_plus_int(bit1(pls)),W_8))) # label(fact_13_add__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 10 (all V_11 hAPP_int_int(plus_plus_int(number_number_of_int(V_11)),one_one_int) = number_number_of_int(hAPP_int_int(plus_plus_int(V_11),bit1(pls)))) # label(fact_14_add__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 11 (all V_11 hAPP_real_real(plus_plus_real(number267125858f_real(V_11)),one_one_real) = number267125858f_real(hAPP_int_int(plus_plus_int(V_11),bit1(pls)))) # label(fact_15_add__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 12 (all X hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit0(bit1(pls)))))) # label(fact_21_quartic__square__square) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 13 (all W_7 ((number_number_of_nat(W_7) = zero_zero_nat -> hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_7)) = one_one_int) & (number_number_of_nat(W_7) != zero_zero_nat -> hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_7)) = zero_zero_int))) # label(fact_22_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 14 (all W_7 ((number_number_of_nat(W_7) = zero_zero_nat -> hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_7)) = one_one_nat) & (number_number_of_nat(W_7) != zero_zero_nat -> hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_7)) = zero_zero_nat))) # label(fact_23_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 15 (all W_7 ((number_number_of_nat(W_7) = zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_7)) = one_one_real) & (number_number_of_nat(W_7) != zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_7)) = zero_zero_real))) # label(fact_24_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 16 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y)) | X = Y | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)))) # label(fact_30_zless__linear) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 17 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K_2)),number_number_of_int(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)))) # label(fact_31_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 18 (all V_6 all W hAPP_int_int(plus_plus_int(number_number_of_int(V_6)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_6),W))) # label(fact_32_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 19 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_33_less__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 20 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_34_less__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 21 (all M_1 all N_1 hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_1)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_1),N_1))) # label(fact_36_zpower__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 22 (all M_1 all N_1 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_1),N_1)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_1)),N_1)) # label(fact_37_int__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 23 (all M_1 all N_1 all Z_2 hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N_1)),Z_2)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M_1),N_1))),Z_2)) # label(fact_38_zadd__int__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 24 (all M_1 all N_1 hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M_1),N_1))) # label(fact_39_zadd__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 25 (all Na (hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int <-> Na = zero_zero_nat)) # label(fact_43_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 26 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_46_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 27 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K_2)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)))) # label(fact_47_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 28 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_49_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 29 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K_2)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)))) # label(fact_50_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 30 (all K all I_2 all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(I_2),K)),hAPP_int_int(plus_plus_int(J_2),K))))) # label(fact_51_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 31 (all V_10 all V_6 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),number_number_of_nat(V_10)) = number_number_of_nat(V_10)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_10),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),number_number_of_nat(V_10)) = number_number_of_nat(V_6)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_10),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),number_number_of_nat(V_10)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V_6),V_10)))))) # label(fact_52_add__nat__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 32 (all Xa all Ya (number_number_of_int(Xa) = number_number_of_int(Ya) <-> Xa = Ya)) # label(fact_56_eq__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 33 (all Xa all Ya (number267125858f_real(Xa) = number267125858f_real(Ya) <-> Xa = Ya)) # label(fact_57_eq__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 34 (all Wa all Xa (number_number_of_nat(Wa) = Xa <-> Xa = number_number_of_nat(Wa))) # label(fact_58_number__of__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 35 (all Wa all Xa (number_number_of_int(Wa) = Xa <-> Xa = number_number_of_int(Wa))) # label(fact_59_number__of__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 36 (all Wa all Xa (number267125858f_real(Wa) = Xa <-> Xa = number267125858f_real(Wa))) # label(fact_60_number__of__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 37 (all K_2 all L_1 (bit1(K_2) = bit1(L_1) <-> K_2 = L_1)) # label(fact_61_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 38 (all K_2 all L_1 (bit0(K_2) = bit0(L_1) <-> K_2 = L_1)) # label(fact_62_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 39 (all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_24),A_24)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),zero_zero_int)))) # label(fact_63_even__less__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 40 (all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_24),A_24)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),zero_zero_real)))) # label(fact_64_even__less__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 41 (all Z1 all Z2 all Z3 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(Z1),Z2)),Z3) = hAPP_int_int(plus_plus_int(Z1),hAPP_int_int(plus_plus_int(Z2),Z3))) # label(fact_65_zadd__assoc) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 42 (all X all Y all Z_2 hAPP_int_int(plus_plus_int(X),hAPP_int_int(plus_plus_int(Y),Z_2)) = hAPP_int_int(plus_plus_int(Y),hAPP_int_int(plus_plus_int(X),Z_2))) # label(fact_66_zadd__left__commute) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 43 (all Z_2 all W hAPP_int_int(plus_plus_int(Z_2),W) = hAPP_int_int(plus_plus_int(W),Z_2)) # label(fact_67_zadd__commute) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 44 (all Ma all Na (hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na) <-> Ma = Na)) # label(fact_68_int__int__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 45 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)))) # label(fact_69_less__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 46 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)))) # label(fact_70_less__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 47 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)))) # label(fact_71_less__special_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 48 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)))) # label(fact_72_less__special_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 49 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K_2)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),pls)))) # label(fact_73_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 50 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_74_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 51 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K_2)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)))) # label(fact_75_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 52 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K_2)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),pls)))) # label(fact_76_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 53 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit0(K_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_2)))) # label(fact_77_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 54 (all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(Wa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)))) # label(fact_78_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.36 55 (all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(Wa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)))) # label(fact_80_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 56 (all Wa all Z_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_int_int(plus_plus_int(Z_3),one_one_int))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_3)) | Wa = Z_3)) # label(fact_82_zless__add1__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 57 (all K -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K)),zero_zero_int))) # label(fact_83_int__less__0__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 58 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))))) # label(fact_84_less__special_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 59 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),one_one_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))))) # label(fact_85_less__special_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 60 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)))) # label(fact_86_less__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 61 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)))) # label(fact_87_less__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 62 (all Z_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_3)),Z_3)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),zero_zero_int)))) # label(fact_88_odd__less__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 63 (all A_24 (hAPP_int_int(plus_plus_int(A_24),A_24) = zero_zero_int <-> A_24 = zero_zero_int)) # label(fact_89_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 64 (all A_24 (hAPP_real_real(plus_plus_real(A_24),A_24) = zero_zero_real <-> A_24 = zero_zero_real)) # label(fact_90_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 65 (all K bit1(K) != pls) # label(fact_91_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 66 (all L pls != bit1(L)) # label(fact_92_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 67 (all K all L bit1(K) != bit0(L)) # label(fact_93_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 68 (all K all L bit0(K) != bit1(L)) # label(fact_94_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 69 (all K_2 (bit0(K_2) = pls <-> K_2 = pls)) # label(fact_95_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 70 (all L_1 (pls = bit0(L_1) <-> pls = L_1)) # label(fact_96_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 71 (all K hAPP_int_int(plus_plus_int(K),pls) = K) # label(fact_100_add__Pls__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 72 (all K hAPP_int_int(plus_plus_int(pls),K) = K) # label(fact_101_add__Pls) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 73 (all K all L hAPP_int_int(plus_plus_int(bit0(K)),bit0(L)) = bit0(hAPP_int_int(plus_plus_int(K),L))) # label(fact_102_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 74 (all K bit0(K) = hAPP_int_int(plus_plus_int(K),K)) # label(fact_103_Bit0__def) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 75 (all Z_2 hAPP_int_int(plus_plus_int(Z_2),zero_zero_int) = Z_2) # label(fact_104_zadd__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 76 (all Z_2 hAPP_int_int(plus_plus_int(zero_zero_int),Z_2) = Z_2) # label(fact_105_zadd__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 77 (all A_68 hAPP_int_int(plus_plus_int(number_number_of_int(pls)),A_68) = A_68) # label(fact_113_add__numeral__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 78 (all A_68 hAPP_real_real(plus_plus_real(number267125858f_real(pls)),A_68) = A_68) # label(fact_114_add__numeral__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 79 (all A_67 hAPP_int_int(plus_plus_int(A_67),number_number_of_int(pls)) = A_67) # label(fact_115_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 80 (all A_67 hAPP_real_real(plus_plus_real(A_67),number267125858f_real(pls)) = A_67) # label(fact_116_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 81 (all A_24 all Wa (hAPP_nat_int(power_power_int(A_24),number_number_of_nat(Wa)) = zero_zero_int <-> A_24 = zero_zero_int & number_number_of_nat(Wa) != zero_zero_nat)) # label(fact_117_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 82 (all A_24 all Wa (hAPP_nat_nat(power_power_nat(A_24),number_number_of_nat(Wa)) = zero_zero_nat <-> A_24 = zero_zero_nat & number_number_of_nat(Wa) != zero_zero_nat)) # label(fact_118_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 83 (all A_24 all Wa (hAPP_nat_real(power_power_real(A_24),number_number_of_nat(Wa)) = zero_zero_real <-> A_24 = zero_zero_real & number_number_of_nat(Wa) != zero_zero_nat)) # label(fact_119_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 84 (all V_9 all W_6 all Z_5 hAPP_int_int(plus_plus_int(number_number_of_int(V_9)),hAPP_int_int(plus_plus_int(number_number_of_int(W_6)),Z_5)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_9),W_6))),Z_5)) # label(fact_120_add__number__of__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 85 (all V_9 all W_6 all Z_5 hAPP_real_real(plus_plus_real(number267125858f_real(V_9)),hAPP_real_real(plus_plus_real(number267125858f_real(W_6)),Z_5)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_9),W_6))),Z_5)) # label(fact_121_add__number__of__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 86 (all V_8 all W_5 hAPP_int_int(plus_plus_int(number_number_of_int(V_8)),number_number_of_int(W_5)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_8),W_5))) # label(fact_122_add__number__of__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 87 (all V_8 all W_5 hAPP_real_real(plus_plus_real(number267125858f_real(V_8)),number267125858f_real(W_5)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_8),W_5))) # label(fact_123_add__number__of__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 88 (all V_7 all W_4 number_number_of_int(hAPP_int_int(plus_plus_int(V_7),W_4)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_7)),number_number_of_int(W_4))) # label(fact_124_number__of__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 89 (all V_7 all W_4 number267125858f_real(hAPP_int_int(plus_plus_int(V_7),W_4)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_7)),number267125858f_real(W_4))) # label(fact_125_number__of__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 90 (all K all L hAPP_int_int(plus_plus_int(bit1(K)),bit0(L)) = bit1(hAPP_int_int(plus_plus_int(K),L))) # label(fact_126_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 91 (all K all L hAPP_int_int(plus_plus_int(bit0(K)),bit1(L)) = bit1(hAPP_int_int(plus_plus_int(K),L))) # label(fact_127_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 92 (all K bit1(K) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),K)),K)) # label(fact_128_Bit1__def) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 93 (all Z_2 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_2)),Z_2) != zero_zero_int) # label(fact_129_odd__nonzero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 94 (all N_29 number_number_of_nat(hAPP_nat_int(semiri1621563631at_int,N_29)) = hAPP_nat_nat(semiri984289939at_nat,N_29)) # label(fact_130_number__of__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 95 (all N_29 number_number_of_int(hAPP_nat_int(semiri1621563631at_int,N_29)) = hAPP_nat_int(semiri1621563631at_int,N_29)) # label(fact_131_number__of__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 96 (all N_29 number267125858f_real(hAPP_nat_int(semiri1621563631at_int,N_29)) = hAPP_nat_real(semiri132038758t_real,N_29)) # label(fact_132_number__of__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 97 (all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_24),number_number_of_nat(bit0(bit1(pls)))))) <-> A_24 != zero_zero_int)) # label(fact_133_zero__less__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 98 (all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_24),number_number_of_nat(bit0(bit1(pls)))))) <-> A_24 != zero_zero_real)) # label(fact_134_zero__less__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 99 (all A_66 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_66),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int))) # label(fact_135_power2__less__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 100 (all A_66 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_66),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real))) # label(fact_136_power2__less__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 101 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls))))))) <-> Xa != zero_zero_int | Ya != zero_zero_int)) # label(fact_137_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 102 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))))) <-> Xa != zero_zero_real | Ya != zero_zero_real)) # label(fact_138_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 103 (all X_16 all Y_13 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_16),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_13),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int))) # label(fact_139_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 104 (all X_16 all Y_13 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_16),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_13),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real))) # label(fact_140_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 105 (all W_3 number_number_of_int(bit0(W_3)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(zero_zero_int),number_number_of_int(W_3))),number_number_of_int(W_3))) # label(fact_141_number__of__Bit0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 106 (all W_3 number267125858f_real(bit0(W_3)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(zero_zero_real),number267125858f_real(W_3))),number267125858f_real(W_3))) # label(fact_142_number__of__Bit0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 107 (all W_2 number_number_of_int(bit1(W_2)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_2))),number_number_of_int(W_2))) # label(fact_143_number__of__Bit1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 108 (all W_2 number267125858f_real(bit1(W_2)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_2))),number267125858f_real(W_2))) # label(fact_144_number__of__Bit1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 109 (all N_28 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_28)),hAPP_nat_int(power_power_int(number_number_of_int(bit0(bit1(pls)))),N_28)))) # label(fact_149_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 110 (all N_28 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_28)),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_28)))) # label(fact_150_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 111 (all Xa all Ya (hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real <-> Xa = zero_zero_real & Ya = zero_zero_real)) # label(fact_153_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 112 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_154_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 113 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_155_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 114 (all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_156_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 115 (all N_27 all A_65 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_65)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_27)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_65),N_27)))))) # label(fact_157_one__less__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 116 (all N_27 all A_65 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_65)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_27)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_65),N_27)))))) # label(fact_158_one__less__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 117 (all N_27 all A_65 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_65)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_27)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_65),N_27)))))) # label(fact_159_one__less__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 118 (all N_26 ((N_26 = zero_zero_nat -> hAPP_nat_int(power_power_int(zero_zero_int),N_26) = one_one_int) & (N_26 != zero_zero_nat -> hAPP_nat_int(power_power_int(zero_zero_int),N_26) = zero_zero_int))) # label(fact_160_power__0__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 119 (all N_26 ((N_26 = zero_zero_nat -> hAPP_nat_nat(power_power_nat(zero_zero_nat),N_26) = one_one_nat) & (N_26 != zero_zero_nat -> hAPP_nat_nat(power_power_nat(zero_zero_nat),N_26) = zero_zero_nat))) # label(fact_161_power__0__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 120 (all N_26 ((N_26 = zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),N_26) = one_one_real) & (N_26 != zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),N_26) = zero_zero_real))) # label(fact_162_power__0__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 121 (all A_64 all N_25 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_25),N_24)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_64)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_64),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_64),N_24)),hAPP_nat_int(power_power_int(A_64),N_25))))))) # label(fact_163_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 122 (all A_64 all N_25 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_25),N_24)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_64)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_64),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_64),N_24)),hAPP_nat_nat(power_power_nat(A_64),N_25))))))) # label(fact_164_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 123 (all A_64 all N_25 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_25),N_24)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_64)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_64),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_64),N_24)),hAPP_nat_real(power_power_real(A_64),N_25))))))) # label(fact_165_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 124 (all P_1 all K_2 all I (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),I)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(K_2),one_one_int))) -> ((all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),I_1)) -> (hBOOL(hAPP_int_bool(P_1,I_1)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_1),one_one_int)))))) -> hBOOL(hAPP_int_bool(P_1,I)))))) # label(fact_169_int__gr__induct) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 125 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_171_less__zeroE) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 126 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1))) # label(fact_173_less__not__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 127 (all I_2 all J_2 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),J_2)),I_2))) # label(fact_174_not__add__less1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 128 (all J_2 all I_2 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J_2),I_2)),I_2))) # label(fact_175_not__add__less2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 129 (all K number_number_of_int(K) = K) # label(fact_176_number__of__is__id) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 130 (all Ma all Na (Ma != Na <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)))) # label(fact_177_nat__neq__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 131 (all M_1 all N_1 hAPP_nat_nat(plus_plus_nat(M_1),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),M_1)) # label(fact_178_nat__add__commute) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 132 (all X all Y all Z_2 hAPP_nat_nat(plus_plus_nat(X),hAPP_nat_nat(plus_plus_nat(Y),Z_2)) = hAPP_nat_nat(plus_plus_nat(Y),hAPP_nat_nat(plus_plus_nat(X),Z_2))) # label(fact_179_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 133 (all M_1 all N_1 all K hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M_1),N_1)),K) = hAPP_nat_nat(plus_plus_nat(M_1),hAPP_nat_nat(plus_plus_nat(N_1),K))) # label(fact_180_nat__add__assoc) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 134 (all K_2 all Ma all Na (hAPP_nat_nat(plus_plus_nat(K_2),Ma) = hAPP_nat_nat(plus_plus_nat(K_2),Na) <-> Ma = Na)) # label(fact_181_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 135 (all Ma all K_2 all Na (hAPP_nat_nat(plus_plus_nat(Ma),K_2) = hAPP_nat_nat(plus_plus_nat(Na),K_2) <-> Ma = Na)) # label(fact_182_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 136 (all K_2 all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K_2),Ma)),hAPP_nat_nat(plus_plus_nat(K_2),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_183_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 137 (all X all Y (X != Y -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X))))) # label(fact_184_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 138 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1))) # label(fact_185_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 139 (all N_1 all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1)) -> M_1 != N_1)) # label(fact_186_less__not__refl2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 140 (all S all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T)) -> S != T)) # label(fact_187_less__not__refl3) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 141 (all M_1 all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(plus_plus_nat(J_2),M_1))))) # label(fact_188_trans__less__add1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 142 (all M_1 all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(plus_plus_nat(M_1),J_2))))) # label(fact_189_trans__less__add2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 143 (all K all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),K))))) # label(fact_190_add__less__mono1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 144 (all K all L all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),L)))))) # label(fact_191_add__less__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 145 (all M_1 all N_1 all K all L (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L)) -> (hAPP_nat_nat(plus_plus_nat(M_1),L) = hAPP_nat_nat(plus_plus_nat(K),N_1) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))))) # label(fact_192_less__add__eq__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 146 (all I_2 all J_2 all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),J_2)),K)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K)))) # label(fact_193_add__lessD1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 147 (all P_1 all Ma all Na ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> ((Ma = Na -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)))))) # label(fact_194_nat__less__cases) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 148 (all N_1 (N_1 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_195_gr0I) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 149 (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> N_1 != zero_zero_nat)) # label(fact_196_gr__implies__not0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 150 (all M_1 all N_1 all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(I_2),M_1)),hAPP_nat_nat(power_power_nat(I_2),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))))) # label(fact_197_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 151 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_198_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 152 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | Na = zero_zero_nat)) # label(fact_199_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 153 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(Ma),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_200_add__gr__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 154 (all Na (Na != zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_201_neq0__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 155 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_202_not__less0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 156 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na))) <-> Na = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)))) # label(fact_203_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 157 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_204_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 158 (all Xa all Wa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),number_number_of_nat(Wa)))) <-> number_number_of_nat(Wa) = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)))) # label(fact_205_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 159 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_206_zless__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 160 (all Va all V (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Va)),number_number_of_nat(V))) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V)))) # label(fact_207_less__nat__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 161 (all X_15 all Y_12 (X_15 != Y_12 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_15),Y_12)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_12),X_15))))) # label(fact_208_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 162 (all X_15 all Y_12 (X_15 != Y_12 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_15),Y_12)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_15))))) # label(fact_209_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 163 (all M_1 all N_1 (hAPP_nat_nat(plus_plus_nat(M_1),N_1) = M_1 -> N_1 = zero_zero_nat)) # label(fact_210_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 164 (all Ma all Na (hAPP_nat_nat(plus_plus_nat(Ma),Na) = zero_zero_nat <-> Ma = zero_zero_nat & Na = zero_zero_nat)) # label(fact_211_add__is__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 165 (all M_1 hAPP_nat_nat(plus_plus_nat(M_1),zero_zero_nat) = M_1) # label(fact_212_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 166 (all N_1 hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1) # label(fact_213_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 167 (all A_63 hAPP_nat_real(power_power_real(A_63),one_one_nat) = A_63) # label(fact_214_power__one__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 168 (all A_63 hAPP_nat_nat(power_power_nat(A_63),one_one_nat) = A_63) # label(fact_215_power__one__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 169 (all A_63 hAPP_nat_int(power_power_int(A_63),one_one_nat) = A_63) # label(fact_216_power__one__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 170 (all Ma all Na (hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,Na) <-> Ma = Na)) # label(fact_217_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 171 (all Ma all Na (hAPP_nat_nat(semiri984289939at_nat,Ma) = hAPP_nat_nat(semiri984289939at_nat,Na) <-> Ma = Na)) # label(fact_218_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 172 (all Ma all Na (hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na) <-> Ma = Na)) # label(fact_219_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 173 (all Xa all Ya (hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,Ya) <-> Xa = Ya)) # label(fact_220_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 174 (all Xa all Ya all P_1 ((hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,Xa),Ya))) & (-hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(if_nat(P_1,Xa),Ya))))) # label(fact_221_int__if__cong) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 175 (all Va (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(Va))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Va)))) # label(fact_222_less__0__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 176 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_223_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 177 (all N_23 all A_62 (A_62 != zero_zero_real -> hAPP_nat_real(power_power_real(A_62),N_23) != zero_zero_real)) # label(fact_230_field__power__not__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 178 (all N_23 all A_62 (A_62 != zero_zero_int -> hAPP_nat_int(power_power_int(A_62),N_23) != zero_zero_int)) # label(fact_231_field__power__not__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 179 (all N_22 hAPP_nat_real(power_power_real(one_one_real),N_22) = one_one_real) # label(fact_232_power__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 180 (all N_22 hAPP_nat_nat(power_power_nat(one_one_nat),N_22) = one_one_nat) # label(fact_233_power__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 181 (all N_22 hAPP_nat_int(power_power_int(one_one_int),N_22) = one_one_int) # label(fact_234_power__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 182 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_235_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 183 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_236_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 184 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_237_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 185 (all M_9 all N_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_21)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_21))))) # label(fact_238_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 186 (all M_9 all N_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_21)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_21))))) # label(fact_239_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 187 (all M_9 all N_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_21)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_21))))) # label(fact_240_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 188 (all M_8 all N_20 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_8)),hAPP_nat_real(semiri132038758t_real,N_20))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_8),N_20)))) # label(fact_241_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 189 (all M_8 all N_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_8)),hAPP_nat_nat(semiri984289939at_nat,N_20))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_8),N_20)))) # label(fact_242_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 190 (all M_8 all N_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_8)),hAPP_nat_int(semiri1621563631at_int,N_20))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_8),N_20)))) # label(fact_243_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 191 (all M_7 all N_19 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(plus_plus_nat(M_7),N_19)) = hAPP_real_real(plus_plus_real(hAPP_nat_real(semiri132038758t_real,M_7)),hAPP_nat_real(semiri132038758t_real,N_19))) # label(fact_244_of__nat__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 192 (all M_7 all N_19 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(plus_plus_nat(M_7),N_19)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(semiri984289939at_nat,M_7)),hAPP_nat_nat(semiri984289939at_nat,N_19))) # label(fact_245_of__nat__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 193 (all M_7 all N_19 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(M_7),N_19)) = hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_7)),hAPP_nat_int(semiri1621563631at_int,N_19))) # label(fact_246_of__nat__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 194 (all M_6 all N_18 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(power_power_nat(M_6),N_18)) = hAPP_nat_real(power_power_real(hAPP_nat_real(semiri132038758t_real,M_6)),N_18)) # label(fact_250_of__nat__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 195 (all M_6 all N_18 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(power_power_nat(M_6),N_18)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(semiri984289939at_nat,M_6)),N_18)) # label(fact_251_of__nat__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 196 (all M_6 all N_18 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_6),N_18)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_6)),N_18)) # label(fact_252_of__nat__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.37 197 (all X all Y hAPP_int_int(plus_plus_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(plus_plus_nat(X),Y))) # label(fact_254_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 198 (all X all N_1 hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,X)),N_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(X),N_1))) # label(fact_255_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 199 (all B_41 all C_31 all A_61 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_61)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),C_31)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),hAPP_real_real(plus_plus_real(A_61),C_31)))))) # label(fact_256_pos__add__strict) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 200 (all B_41 all C_31 all A_61 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_61)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),C_31)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),hAPP_nat_nat(plus_plus_nat(A_61),C_31)))))) # label(fact_257_pos__add__strict) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 201 (all B_41 all C_31 all A_61 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_61)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),C_31)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),hAPP_int_int(plus_plus_int(A_61),C_31)))))) # label(fact_258_pos__add__strict) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 202 (all N_17 all A_60 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_60)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_60),N_17))))) # label(fact_265_zero__less__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 203 (all N_17 all A_60 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_60)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_60),N_17))))) # label(fact_266_zero__less__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 204 (all N_17 all A_60 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_60)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_60),N_17))))) # label(fact_267_zero__less__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 205 (all A_59 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_59),hAPP_real_real(plus_plus_real(A_59),one_one_real)))) # label(fact_268_less__add__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 206 (all A_59 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_59),hAPP_nat_nat(plus_plus_nat(A_59),one_one_nat)))) # label(fact_269_less__add__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 207 (all A_59 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_59),hAPP_int_int(plus_plus_int(A_59),one_one_int)))) # label(fact_270_less__add__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 208 (all Ma all Na all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_24)) -> (hAPP_nat_real(power_power_real(A_24),Ma) = hAPP_nat_real(power_power_real(A_24),Na) <-> Ma = Na))) # label(fact_271_power__inject__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 209 (all Ma all Na all A_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_24)) -> (hAPP_nat_nat(power_power_nat(A_24),Ma) = hAPP_nat_nat(power_power_nat(A_24),Na) <-> Ma = Na))) # label(fact_272_power__inject__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 210 (all Ma all Na all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_24)) -> (hAPP_nat_int(power_power_int(A_24),Ma) = hAPP_nat_int(power_power_int(A_24),Na) <-> Ma = Na))) # label(fact_273_power__inject__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 211 (all Xa all Ya all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B_21),Xa)),hAPP_nat_real(power_power_real(B_21),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))))) # label(fact_274_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 212 (all Xa all Ya all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B_21),Xa)),hAPP_nat_nat(power_power_nat(B_21),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))))) # label(fact_275_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 213 (all Xa all Ya all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B_21),Xa)),hAPP_nat_int(power_power_int(B_21),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))))) # label(fact_276_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 214 (all M_5 all N_16 all A_58 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_58)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_58),M_5)),hAPP_nat_real(power_power_real(A_58),N_16))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_16))))) # label(fact_277_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 215 (all M_5 all N_16 all A_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_58)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_58),M_5)),hAPP_nat_nat(power_power_nat(A_58),N_16))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_16))))) # label(fact_278_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 216 (all M_5 all N_16 all A_58 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_58)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_58),M_5)),hAPP_nat_int(power_power_int(A_58),N_16))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_16))))) # label(fact_279_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 217 (all A_57 all N_15 all N_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),N_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_57)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_57),N_15)),hAPP_nat_real(power_power_real(A_57),N_14)))))) # label(fact_280_power__strict__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 218 (all A_57 all N_15 all N_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),N_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_57)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_57),N_15)),hAPP_nat_nat(power_power_nat(A_57),N_14)))))) # label(fact_281_power__strict__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 219 (all A_57 all N_15 all N_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),N_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_57)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_57),N_15)),hAPP_nat_int(power_power_int(A_57),N_14)))))) # label(fact_282_power__strict__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 220 (all A_24 all Na (hAPP_nat_real(power_power_real(A_24),Na) = zero_zero_real <-> A_24 = zero_zero_real & Na != zero_zero_nat)) # label(fact_283_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 221 (all A_24 all Na (hAPP_nat_nat(power_power_nat(A_24),Na) = zero_zero_nat <-> A_24 = zero_zero_nat & Na != zero_zero_nat)) # label(fact_284_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 222 (all A_24 all Na (hAPP_nat_int(power_power_int(A_24),Na) = zero_zero_int <-> A_24 = zero_zero_int & Na != zero_zero_nat)) # label(fact_285_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 223 (all M_4 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_4)),zero_zero_real))) # label(fact_286_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 224 (all M_4 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_4)),zero_zero_nat))) # label(fact_287_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 225 (all M_4 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_4)),zero_zero_int))) # label(fact_288_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 226 (all A_56 hAPP_nat_real(power_power_real(A_56),zero_zero_nat) = one_one_real) # label(fact_289_power__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 227 (all A_56 hAPP_nat_nat(power_power_nat(A_56),zero_zero_nat) = one_one_nat) # label(fact_290_power__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 228 (all A_56 hAPP_nat_int(power_power_int(A_56),zero_zero_nat) = one_one_int) # label(fact_291_power__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 229 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K)) -> (exists N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) & K = hAPP_nat_int(semiri1621563631at_int,N))))) # label(fact_297_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 230 (all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2)) -> (exists K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) & hAPP_nat_nat(plus_plus_nat(I_2),K_1) = J_2)))) # label(fact_298_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 231 (all Xa all Na (hAPP_nat_nat(power_power_nat(Xa),Na) = one_one_nat <-> Xa = one_one_nat | Na = zero_zero_nat)) # label(fact_299_exp__eq__1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 232 (all X_14 hAPP_nat_real(power_power_real(X_14),zero_zero_nat) = one_one_real) # label(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 233 (all X_14 hAPP_nat_nat(power_power_nat(X_14),zero_zero_nat) = one_one_nat) # label(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 234 (all X_14 hAPP_nat_int(power_power_int(X_14),zero_zero_nat) = one_one_int) # label(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 235 (all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_24),A_24))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_24)))) # label(fact_303_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 236 (all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_24),A_24))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_24)))) # label(fact_304_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 237 (all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_24),A_24)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),zero_zero_real)))) # label(fact_305_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 238 (all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_24),A_24)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),zero_zero_int)))) # label(fact_306_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 239 (all Xa (zero_zero_real = Xa <-> Xa = zero_zero_real)) # label(fact_307_zero__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 240 (all Xa (zero_zero_nat = Xa <-> Xa = zero_zero_nat)) # label(fact_308_zero__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 241 (all Xa (zero_zero_int = Xa <-> Xa = zero_zero_int)) # label(fact_309_zero__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 242 (all B_40 all A_55 all C_30 (hAPP_real_real(plus_plus_real(B_40),A_55) = hAPP_real_real(plus_plus_real(C_30),A_55) -> B_40 = C_30)) # label(fact_310_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 243 (all B_40 all A_55 all C_30 (hAPP_nat_nat(plus_plus_nat(B_40),A_55) = hAPP_nat_nat(plus_plus_nat(C_30),A_55) -> B_40 = C_30)) # label(fact_311_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 244 (all B_40 all A_55 all C_30 (hAPP_int_int(plus_plus_int(B_40),A_55) = hAPP_int_int(plus_plus_int(C_30),A_55) -> B_40 = C_30)) # label(fact_312_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 245 (all A_54 all B_39 all C_29 (hAPP_real_real(plus_plus_real(A_54),B_39) = hAPP_real_real(plus_plus_real(A_54),C_29) -> B_39 = C_29)) # label(fact_313_add__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 246 (all A_54 all B_39 all C_29 (hAPP_nat_nat(plus_plus_nat(A_54),B_39) = hAPP_nat_nat(plus_plus_nat(A_54),C_29) -> B_39 = C_29)) # label(fact_314_add__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 247 (all A_54 all B_39 all C_29 (hAPP_int_int(plus_plus_int(A_54),B_39) = hAPP_int_int(plus_plus_int(A_54),C_29) -> B_39 = C_29)) # label(fact_315_add__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 248 (all A_53 all B_38 all C_28 (hAPP_real_real(plus_plus_real(A_53),B_38) = hAPP_real_real(plus_plus_real(A_53),C_28) -> B_38 = C_28)) # label(fact_316_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 249 (all A_53 all B_38 all C_28 (hAPP_nat_nat(plus_plus_nat(A_53),B_38) = hAPP_nat_nat(plus_plus_nat(A_53),C_28) -> B_38 = C_28)) # label(fact_317_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 250 (all A_53 all B_38 all C_28 (hAPP_int_int(plus_plus_int(A_53),B_38) = hAPP_int_int(plus_plus_int(A_53),C_28) -> B_38 = C_28)) # label(fact_318_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 251 (all A_52 all B_37 all C_27 all D_6 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_52),B_37)),hAPP_real_real(plus_plus_real(C_27),D_6)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_52),C_27)),hAPP_real_real(plus_plus_real(B_37),D_6))) # label(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 252 (all A_52 all B_37 all C_27 all D_6 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_52),B_37)),hAPP_nat_nat(plus_plus_nat(C_27),D_6)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_52),C_27)),hAPP_nat_nat(plus_plus_nat(B_37),D_6))) # label(fact_320_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 253 (all A_52 all B_37 all C_27 all D_6 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_52),B_37)),hAPP_int_int(plus_plus_int(C_27),D_6)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_52),C_27)),hAPP_int_int(plus_plus_int(B_37),D_6))) # label(fact_321_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 254 (all B_21 all A_24 all C_15 (hAPP_real_real(plus_plus_real(B_21),A_24) = hAPP_real_real(plus_plus_real(C_15),A_24) <-> B_21 = C_15)) # label(fact_322_add__right__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 255 (all B_21 all A_24 all C_15 (hAPP_nat_nat(plus_plus_nat(B_21),A_24) = hAPP_nat_nat(plus_plus_nat(C_15),A_24) <-> B_21 = C_15)) # label(fact_323_add__right__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 256 (all B_21 all A_24 all C_15 (hAPP_int_int(plus_plus_int(B_21),A_24) = hAPP_int_int(plus_plus_int(C_15),A_24) <-> B_21 = C_15)) # label(fact_324_add__right__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 257 (all A_24 all B_21 all C_15 (hAPP_real_real(plus_plus_real(A_24),B_21) = hAPP_real_real(plus_plus_real(A_24),C_15) <-> B_21 = C_15)) # label(fact_325_add__left__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 258 (all A_24 all B_21 all C_15 (hAPP_nat_nat(plus_plus_nat(A_24),B_21) = hAPP_nat_nat(plus_plus_nat(A_24),C_15) <-> B_21 = C_15)) # label(fact_326_add__left__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 259 (all A_24 all B_21 all C_15 (hAPP_int_int(plus_plus_int(A_24),B_21) = hAPP_int_int(plus_plus_int(A_24),C_15) <-> B_21 = C_15)) # label(fact_327_add__left__cancel) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 260 (all A_51 all B_36 all C_26 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_51),B_36)),C_26) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_51),C_26)),B_36)) # label(fact_328_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 261 (all A_51 all B_36 all C_26 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_51),B_36)),C_26) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_51),C_26)),B_36)) # label(fact_329_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 262 (all A_51 all B_36 all C_26 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_51),B_36)),C_26) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_51),C_26)),B_36)) # label(fact_330_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 263 (all A_50 all B_35 all C_25 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_50),B_35)),C_25) = hAPP_real_real(plus_plus_real(A_50),hAPP_real_real(plus_plus_real(B_35),C_25))) # label(fact_331_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 264 (all A_50 all B_35 all C_25 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_50),B_35)),C_25) = hAPP_nat_nat(plus_plus_nat(A_50),hAPP_nat_nat(plus_plus_nat(B_35),C_25))) # label(fact_332_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 265 (all A_50 all B_35 all C_25 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_50),B_35)),C_25) = hAPP_int_int(plus_plus_int(A_50),hAPP_int_int(plus_plus_int(B_35),C_25))) # label(fact_333_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 266 (all A_49 all B_34 all C_24 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_49),B_34)),C_24) = hAPP_real_real(plus_plus_real(A_49),hAPP_real_real(plus_plus_real(B_34),C_24))) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 267 (all A_49 all B_34 all C_24 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_49),B_34)),C_24) = hAPP_nat_nat(plus_plus_nat(A_49),hAPP_nat_nat(plus_plus_nat(B_34),C_24))) # label(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 268 (all A_49 all B_34 all C_24 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_49),B_34)),C_24) = hAPP_int_int(plus_plus_int(A_49),hAPP_int_int(plus_plus_int(B_34),C_24))) # label(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 269 (all A_48 all C_23 all D_5 hAPP_real_real(plus_plus_real(A_48),hAPP_real_real(plus_plus_real(C_23),D_5)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_48),C_23)),D_5)) # label(fact_337_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 270 (all A_48 all C_23 all D_5 hAPP_nat_nat(plus_plus_nat(A_48),hAPP_nat_nat(plus_plus_nat(C_23),D_5)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_48),C_23)),D_5)) # label(fact_338_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 271 (all A_48 all C_23 all D_5 hAPP_int_int(plus_plus_int(A_48),hAPP_int_int(plus_plus_int(C_23),D_5)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_48),C_23)),D_5)) # label(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 272 (all A_47 all C_22 all D_4 hAPP_real_real(plus_plus_real(A_47),hAPP_real_real(plus_plus_real(C_22),D_4)) = hAPP_real_real(plus_plus_real(C_22),hAPP_real_real(plus_plus_real(A_47),D_4))) # label(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 273 (all A_47 all C_22 all D_4 hAPP_nat_nat(plus_plus_nat(A_47),hAPP_nat_nat(plus_plus_nat(C_22),D_4)) = hAPP_nat_nat(plus_plus_nat(C_22),hAPP_nat_nat(plus_plus_nat(A_47),D_4))) # label(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 274 (all A_47 all C_22 all D_4 hAPP_int_int(plus_plus_int(A_47),hAPP_int_int(plus_plus_int(C_22),D_4)) = hAPP_int_int(plus_plus_int(C_22),hAPP_int_int(plus_plus_int(A_47),D_4))) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 275 (all A_46 all C_21 hAPP_real_real(plus_plus_real(A_46),C_21) = hAPP_real_real(plus_plus_real(C_21),A_46)) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 276 (all A_46 all C_21 hAPP_nat_nat(plus_plus_nat(A_46),C_21) = hAPP_nat_nat(plus_plus_nat(C_21),A_46)) # label(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 277 (all A_46 all C_21 hAPP_int_int(plus_plus_int(A_46),C_21) = hAPP_int_int(plus_plus_int(C_21),A_46)) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 278 (all Xa (one_one_real = Xa <-> Xa = one_one_real)) # label(fact_346_one__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 279 (all Xa (one_one_nat = Xa <-> Xa = one_one_nat)) # label(fact_347_one__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 280 (all Xa (one_one_int = Xa <-> Xa = one_one_int)) # label(fact_348_one__reorient) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 281 (all B_21 all A_24 (B_21 = hAPP_real_real(plus_plus_real(B_21),A_24) <-> A_24 = zero_zero_real)) # label(fact_349_add__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 282 (all B_21 all A_24 (B_21 = hAPP_nat_nat(plus_plus_nat(B_21),A_24) <-> A_24 = zero_zero_nat)) # label(fact_350_add__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 283 (all B_21 all A_24 (B_21 = hAPP_int_int(plus_plus_int(B_21),A_24) <-> A_24 = zero_zero_int)) # label(fact_351_add__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 284 (all A_45 hAPP_real_real(plus_plus_real(A_45),zero_zero_real) = A_45) # label(fact_352_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 285 (all A_45 hAPP_nat_nat(plus_plus_nat(A_45),zero_zero_nat) = A_45) # label(fact_353_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 286 (all A_45 hAPP_int_int(plus_plus_int(A_45),zero_zero_int) = A_45) # label(fact_354_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 287 (all A_44 hAPP_real_real(plus_plus_real(A_44),zero_zero_real) = A_44) # label(fact_355_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 288 (all A_44 hAPP_nat_nat(plus_plus_nat(A_44),zero_zero_nat) = A_44) # label(fact_356_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 289 (all A_44 hAPP_int_int(plus_plus_int(A_44),zero_zero_int) = A_44) # label(fact_357_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 290 (all A_43 hAPP_real_real(plus_plus_real(A_43),zero_zero_real) = A_43) # label(fact_358_add__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 291 (all A_43 hAPP_nat_nat(plus_plus_nat(A_43),zero_zero_nat) = A_43) # label(fact_359_add__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 292 (all A_43 hAPP_int_int(plus_plus_int(A_43),zero_zero_int) = A_43) # label(fact_360_add__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 293 (all A_24 (zero_zero_real = hAPP_real_real(plus_plus_real(A_24),A_24) <-> A_24 = zero_zero_real)) # label(fact_361_double__zero__sym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 294 (all A_24 (zero_zero_int = hAPP_int_int(plus_plus_int(A_24),A_24) <-> A_24 = zero_zero_int)) # label(fact_362_double__zero__sym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 295 (all A_42 hAPP_real_real(plus_plus_real(zero_zero_real),A_42) = A_42) # label(fact_363_add__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 296 (all A_42 hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_42) = A_42) # label(fact_364_add__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 297 (all A_42 hAPP_int_int(plus_plus_int(zero_zero_int),A_42) = A_42) # label(fact_365_add__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 298 (all A_41 hAPP_real_real(plus_plus_real(zero_zero_real),A_41) = A_41) # label(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 299 (all A_41 hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_41) = A_41) # label(fact_367_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 300 (all A_41 hAPP_int_int(plus_plus_int(zero_zero_int),A_41) = A_41) # label(fact_368_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 301 (all A_40 hAPP_real_real(plus_plus_real(zero_zero_real),A_40) = A_40) # label(fact_369_add__0__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 302 (all A_40 hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_40) = A_40) # label(fact_370_add__0__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 303 (all A_40 hAPP_int_int(plus_plus_int(zero_zero_int),A_40) = A_40) # label(fact_371_add__0__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 304 (all C_20 all A_39 all B_33 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_20),A_39)),hAPP_real_real(plus_plus_real(C_20),B_33))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_39),B_33)))) # label(fact_372_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 305 (all C_20 all A_39 all B_33 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_20),A_39)),hAPP_nat_nat(plus_plus_nat(C_20),B_33))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_39),B_33)))) # label(fact_373_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 306 (all C_20 all A_39 all B_33 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_20),A_39)),hAPP_int_int(plus_plus_int(C_20),B_33))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_39),B_33)))) # label(fact_374_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 307 (all A_38 all C_19 all B_32 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_38),C_19)),hAPP_real_real(plus_plus_real(B_32),C_19))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_38),B_32)))) # label(fact_375_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 308 (all A_38 all C_19 all B_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_38),C_19)),hAPP_nat_nat(plus_plus_nat(B_32),C_19))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_38),B_32)))) # label(fact_376_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 309 (all A_38 all C_19 all B_32 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_38),C_19)),hAPP_int_int(plus_plus_int(B_32),C_19))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_38),B_32)))) # label(fact_377_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 310 (all C_18 all D_3 all A_37 all B_31 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_37),B_31)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_18),D_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_37),C_18)),hAPP_real_real(plus_plus_real(B_31),D_3)))))) # label(fact_378_add__strict__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 311 (all C_18 all D_3 all A_37 all B_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_37),B_31)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_18),D_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_37),C_18)),hAPP_nat_nat(plus_plus_nat(B_31),D_3)))))) # label(fact_379_add__strict__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 312 (all C_18 all D_3 all A_37 all B_31 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_37),B_31)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_18),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_37),C_18)),hAPP_int_int(plus_plus_int(B_31),D_3)))))) # label(fact_380_add__strict__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 313 (all C_17 all A_36 all B_30 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_36),B_30)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_17),A_36)),hAPP_real_real(plus_plus_real(C_17),B_30))))) # label(fact_381_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 314 (all C_17 all A_36 all B_30 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),B_30)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_17),A_36)),hAPP_nat_nat(plus_plus_nat(C_17),B_30))))) # label(fact_382_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 315 (all C_17 all A_36 all B_30 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),B_30)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_17),A_36)),hAPP_int_int(plus_plus_int(C_17),B_30))))) # label(fact_383_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 316 (all C_16 all A_35 all B_29 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_35),B_29)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_35),C_16)),hAPP_real_real(plus_plus_real(B_29),C_16))))) # label(fact_384_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 317 (all C_16 all A_35 all B_29 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_35),B_29)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_35),C_16)),hAPP_nat_nat(plus_plus_nat(B_29),C_16))))) # label(fact_385_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 318 (all C_16 all A_35 all B_29 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_35),B_29)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_35),C_16)),hAPP_int_int(plus_plus_int(B_29),C_16))))) # label(fact_386_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 319 (all C_15 all A_24 all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(C_15),A_24)),hAPP_real_real(plus_plus_real(C_15),B_21))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),B_21)))) # label(fact_387_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 320 (all C_15 all A_24 all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(C_15),A_24)),hAPP_nat_nat(plus_plus_nat(C_15),B_21))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_24),B_21)))) # label(fact_388_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 321 (all C_15 all A_24 all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(C_15),A_24)),hAPP_int_int(plus_plus_int(C_15),B_21))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),B_21)))) # label(fact_389_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 322 (all A_24 all C_15 all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_24),C_15)),hAPP_real_real(plus_plus_real(B_21),C_15))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_24),B_21)))) # label(fact_390_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 323 (all A_24 all C_15 all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_24),C_15)),hAPP_nat_nat(plus_plus_nat(B_21),C_15))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_24),B_21)))) # label(fact_391_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 324 (all A_24 all C_15 all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_24),C_15)),hAPP_int_int(plus_plus_int(B_21),C_15))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_24),B_21)))) # label(fact_392_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 325 (all X_13 hAPP_nat_real(power_power_real(X_13),one_one_nat) = X_13) # label(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 326 (all X_13 hAPP_nat_nat(power_power_nat(X_13),one_one_nat) = X_13) # label(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 327 (all X_13 hAPP_nat_int(power_power_int(X_13),one_one_nat) = X_13) # label(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 328 (all Ma all Na (hAPP_nat_nat(power_power_nat(Ma),Na) = zero_zero_nat <-> Na != zero_zero_nat & Ma = zero_zero_nat)) # label(fact_396_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 329 (all B_28 all A_34 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_34),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_28),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_34),B_28)),zero_zero_real))))) # label(fact_397_add__neg__neg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 330 (all B_28 all A_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_34),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_28),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_34),B_28)),zero_zero_nat))))) # label(fact_398_add__neg__neg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 331 (all B_28 all A_34 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_34),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_28),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_34),B_28)),zero_zero_int))))) # label(fact_399_add__neg__neg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 332 (all B_27 all A_33 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_33)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_27)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_33),B_27)))))) # label(fact_400_add__pos__pos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 333 (all B_27 all A_33 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_33)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_27)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_33),B_27)))))) # label(fact_401_add__pos__pos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 334 (all B_27 all A_33 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_33)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_27)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_33),B_27)))))) # label(fact_402_add__pos__pos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 335 (all A all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) -> (exists X_12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_12)) & hAPP_nat_real(power_power_real(X_12),N_1) = A & (all Y_11 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_11)) & hAPP_nat_real(power_power_real(Y_11),N_1) = A -> Y_11 = X_12))))))) # label(fact_403_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 336 (all A all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) -> (exists R (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R)) & hAPP_nat_real(power_power_real(R),N_1) = A))))) # label(fact_404_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 337 (all V_6 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),one_one_nat) = one_one_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_6)),one_one_nat) = number_number_of_nat(succ(V_6))))) # label(fact_406_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 338 (all V_6 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V_6)) = one_one_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V_6)) = number_number_of_nat(succ(V_6))))) # label(fact_407_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 339 (all A_24 all C_15 all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_24),C_15)),hAPP_real_real(plus_plus_real(B_21),C_15))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_21)))) # label(fact_408_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 340 (all A_24 all C_15 all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_24),C_15)),hAPP_nat_nat(plus_plus_nat(B_21),C_15))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_21)))) # label(fact_409_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 341 (all A_24 all C_15 all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_24),C_15)),hAPP_int_int(plus_plus_int(B_21),C_15))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_21)))) # label(fact_410_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 342 (all C_15 all A_24 all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_15),A_24)),hAPP_real_real(plus_plus_real(C_15),B_21))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_21)))) # label(fact_411_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 343 (all C_15 all A_24 all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_15),A_24)),hAPP_nat_nat(plus_plus_nat(C_15),B_21))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_21)))) # label(fact_412_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 344 (all C_15 all A_24 all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_15),A_24)),hAPP_int_int(plus_plus_int(C_15),B_21))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_21)))) # label(fact_413_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 345 (all C_14 all A_32 all B_26 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_32),B_26)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_32),C_14)),hAPP_real_real(plus_plus_real(B_26),C_14))))) # label(fact_414_add__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 346 (all C_14 all A_32 all B_26 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_32),B_26)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_32),C_14)),hAPP_nat_nat(plus_plus_nat(B_26),C_14))))) # label(fact_415_add__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 347 (all C_14 all A_32 all B_26 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_32),B_26)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_32),C_14)),hAPP_int_int(plus_plus_int(B_26),C_14))))) # label(fact_416_add__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 348 (all C_13 all A_31 all B_25 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_31),B_25)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_13),A_31)),hAPP_real_real(plus_plus_real(C_13),B_25))))) # label(fact_417_add__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 349 (all C_13 all A_31 all B_25 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_31),B_25)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_13),A_31)),hAPP_nat_nat(plus_plus_nat(C_13),B_25))))) # label(fact_418_add__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 350 (all C_13 all A_31 all B_25 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_31),B_25)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_13),A_31)),hAPP_int_int(plus_plus_int(C_13),B_25))))) # label(fact_419_add__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 351 (all C_12 all D_2 all A_30 all B_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_30),B_24)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_12),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_30),C_12)),hAPP_real_real(plus_plus_real(B_24),D_2)))))) # label(fact_420_add__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 352 (all C_12 all D_2 all A_30 all B_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_24)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),D_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_30),C_12)),hAPP_nat_nat(plus_plus_nat(B_24),D_2)))))) # label(fact_421_add__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 353 (all C_12 all D_2 all A_30 all B_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_30),B_24)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),D_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_30),C_12)),hAPP_int_int(plus_plus_int(B_24),D_2)))))) # label(fact_422_add__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.38 354 (all A_29 all C_11 all B_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_29),C_11)),hAPP_real_real(plus_plus_real(B_23),C_11))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_29),B_23)))) # label(fact_423_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 355 (all A_29 all C_11 all B_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_29),C_11)),hAPP_nat_nat(plus_plus_nat(B_23),C_11))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_29),B_23)))) # label(fact_424_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 356 (all A_29 all C_11 all B_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_29),C_11)),hAPP_int_int(plus_plus_int(B_23),C_11))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_29),B_23)))) # label(fact_425_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 357 (all C_10 all A_28 all B_22 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(C_10),A_28)),hAPP_real_real(plus_plus_real(C_10),B_22))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_28),B_22)))) # label(fact_426_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 358 (all C_10 all A_28 all B_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(C_10),A_28)),hAPP_nat_nat(plus_plus_nat(C_10),B_22))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_28),B_22)))) # label(fact_427_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 359 (all C_10 all A_28 all B_22 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(C_10),A_28)),hAPP_int_int(plus_plus_int(C_10),B_22))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_28),B_22)))) # label(fact_428_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 360 (all A_27 all N_13 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_13),N_12)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_27)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_27),N_13)),hAPP_nat_real(power_power_real(A_27),N_12)))))) # label(fact_429_power__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 361 (all A_27 all N_13 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_13),N_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_27)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_27),N_13)),hAPP_nat_nat(power_power_nat(A_27),N_12)))))) # label(fact_430_power__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 362 (all A_27 all N_13 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_13),N_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_27)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_27),N_13)),hAPP_nat_int(power_power_int(A_27),N_12)))))) # label(fact_431_power__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 363 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_432_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 364 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K_2)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)))) # label(fact_433_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 365 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_435_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 366 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K_2)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)))) # label(fact_436_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 367 (all W hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W))) # label(fact_438_zle__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 368 (all Z_2 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_2)))) # label(fact_439_zle__linear) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 369 (all K all I_2 all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K))))) # label(fact_440_zle__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 370 (all Z_2 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_2)) -> Z_2 = W))) # label(fact_441_zle__antisym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 371 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))) # label(fact_442_le__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 372 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))) # label(fact_443_le__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 373 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_444_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 374 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_445_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 375 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_446_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 376 (all Z_3 all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),Wa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_3),Wa)) & Z_3 != Wa)) # label(fact_447_zless__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 377 (all K all I_2 all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(K),I_2)),hAPP_int_int(plus_plus_int(K),J_2))))) # label(fact_448_zadd__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 378 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K_2)),number_number_of_int(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)))) # label(fact_449_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 379 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)))) # label(fact_450_le__special_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 380 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)))) # label(fact_451_le__special_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 381 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)))) # label(fact_452_le__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 382 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)))) # label(fact_453_le__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 383 (all A_26 all N_11 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_26)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_26),N_10)),hAPP_nat_real(power_power_real(A_26),N_11))))))) # label(fact_454_power__decreasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 384 (all A_26 all N_11 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_26)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_26),N_10)),hAPP_nat_nat(power_power_nat(A_26),N_11))))))) # label(fact_455_power__decreasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 385 (all A_26 all N_11 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_26)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_26),N_10)),hAPP_nat_int(power_power_int(A_26),N_11))))))) # label(fact_456_power__decreasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 386 (all Xa all Ya all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B_21),Xa)),hAPP_nat_real(power_power_real(B_21),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_457_power__increasing__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 387 (all Xa all Ya all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B_21),Xa)),hAPP_nat_nat(power_power_nat(B_21),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_458_power__increasing__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 388 (all Xa all Ya all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B_21),Xa)),hAPP_nat_int(power_power_int(B_21),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_459_power__increasing__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 389 (all M_3 all N_9 all A_25 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_25)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_25),M_3)),hAPP_nat_real(power_power_real(A_25),N_9))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_9))))) # label(fact_460_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 390 (all M_3 all N_9 all A_25 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_25)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_25),M_3)),hAPP_nat_nat(power_power_nat(A_25),N_9))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_9))))) # label(fact_461_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 391 (all M_3 all N_9 all A_25 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_25)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_25),M_3)),hAPP_nat_int(power_power_int(A_25),N_9))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_9))))) # label(fact_462_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 392 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)))) # label(fact_463_le__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 393 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)))) # label(fact_464_le__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 394 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),one_one_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))))) # label(fact_465_le__special_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 395 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))))) # label(fact_466_le__special_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 396 (all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_24),A_24))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_24)))) # label(fact_467_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 397 (all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_24),A_24))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_24)))) # label(fact_468_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 398 (all A_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_24),A_24)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),zero_zero_real)))) # label(fact_469_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 399 (all A_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_24),A_24)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),zero_zero_int)))) # label(fact_470_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 400 (all B_20 all A_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_23)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_20)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_23),B_20)))))) # label(fact_471_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 401 (all B_20 all A_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_20)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_23),B_20)))))) # label(fact_472_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 402 (all B_20 all A_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_23)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_20)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_23),B_20)))))) # label(fact_473_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 403 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) -> (hAPP_real_real(plus_plus_real(Xa),Ya) = zero_zero_real <-> Xa = zero_zero_real & Ya = zero_zero_real)))) # label(fact_474_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 404 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Ya)) -> (hAPP_nat_nat(plus_plus_nat(Xa),Ya) = zero_zero_nat <-> Xa = zero_zero_nat & Ya = zero_zero_nat)))) # label(fact_475_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 405 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (hAPP_int_int(plus_plus_int(Xa),Ya) = zero_zero_int <-> Xa = zero_zero_int & Ya = zero_zero_int)))) # label(fact_476_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 406 (all B_19 all C_9 all A_22 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_22)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),C_9)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),hAPP_real_real(plus_plus_real(A_22),C_9)))))) # label(fact_477_add__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 407 (all B_19 all C_9 all A_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),C_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),hAPP_nat_nat(plus_plus_nat(A_22),C_9)))))) # label(fact_478_add__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 408 (all B_19 all C_9 all A_22 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_22)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),C_9)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),hAPP_int_int(plus_plus_int(A_22),C_9)))))) # label(fact_479_add__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 409 (all B_18 all A_21 all C_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_8)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),A_21)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),hAPP_real_real(plus_plus_real(A_21),C_8)))))) # label(fact_480_add__increasing2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 410 (all B_18 all A_21 all C_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),A_21)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),hAPP_nat_nat(plus_plus_nat(A_21),C_8)))))) # label(fact_481_add__increasing2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 411 (all B_18 all A_21 all C_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_8)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),A_21)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),hAPP_int_int(plus_plus_int(A_21),C_8)))))) # label(fact_482_add__increasing2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 412 (all B_17 all A_20 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_20),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_17),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(A_20),B_17)),zero_zero_real))))) # label(fact_483_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 413 (all B_17 all A_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_17),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(A_20),B_17)),zero_zero_nat))))) # label(fact_484_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 414 (all B_17 all A_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_17),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A_20),B_17)),zero_zero_int))))) # label(fact_485_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 415 (all C_7 all D_1 all A_19 all B_16 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_19),B_16)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_7),D_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_19),C_7)),hAPP_real_real(plus_plus_real(B_16),D_1)))))) # label(fact_486_add__less__le__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 416 (all C_7 all D_1 all A_19 all B_16 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_19),B_16)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_7),D_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_19),C_7)),hAPP_nat_nat(plus_plus_nat(B_16),D_1)))))) # label(fact_487_add__less__le__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 417 (all C_7 all D_1 all A_19 all B_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),B_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_7),D_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_19),C_7)),hAPP_int_int(plus_plus_int(B_16),D_1)))))) # label(fact_488_add__less__le__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 418 (all C_6 all D all A_18 all B_15 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_18),B_15)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_6),D)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_18),C_6)),hAPP_real_real(plus_plus_real(B_15),D)))))) # label(fact_489_add__le__less__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 419 (all C_6 all D all A_18 all B_15 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),B_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_6),D)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_18),C_6)),hAPP_nat_nat(plus_plus_nat(B_15),D)))))) # label(fact_490_add__le__less__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 420 (all C_6 all D all A_18 all B_15 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),B_15)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_6),D)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_18),C_6)),hAPP_int_int(plus_plus_int(B_15),D)))))) # label(fact_491_add__le__less__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 421 (all Va all Wa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Va)),number267125858f_real(Wa))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Wa)),number267125858f_real(Va))))) # label(fact_498_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 422 (all Va all Wa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(Wa))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Wa)),number_number_of_nat(Va))))) # label(fact_499_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 423 (all Va all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Va)),number_number_of_int(Wa))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Wa)),number_number_of_int(Va))))) # label(fact_500_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 424 (all N_8 all A_17 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_17)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_17),N_8))))) # label(fact_501_zero__le__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 425 (all N_8 all A_17 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_17),N_8))))) # label(fact_502_zero__le__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 426 (all N_8 all A_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_17)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_17),N_8))))) # label(fact_503_zero__le__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 427 (all N_7 all A_16 all B_14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_16),B_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_16),N_7)),hAPP_nat_real(power_power_real(B_14),N_7)))))) # label(fact_504_power__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 428 (all N_7 all A_16 all B_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_16),B_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_16),N_7)),hAPP_nat_nat(power_power_nat(B_14),N_7)))))) # label(fact_505_power__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 429 (all N_7 all A_16 all B_14 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_16),B_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_16),N_7)),hAPP_nat_int(power_power_int(B_14),N_7)))))) # label(fact_506_power__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 430 (all N_6 all A_15 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_15)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(A_15),N_6))))) # label(fact_507_one__le__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 431 (all N_6 all A_15 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_15),N_6))))) # label(fact_508_one__le__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 432 (all N_6 all A_15 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(power_power_int(A_15),N_6))))) # label(fact_509_one__le__power) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 433 (all M_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_2)))) # label(fact_510_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 434 (all M_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_2)))) # label(fact_511_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 435 (all M_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_2)))) # label(fact_512_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 436 (all N_5 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_5)))) # label(fact_513_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 437 (all N_5 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_5)))) # label(fact_514_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 438 (all N_5 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_5)))) # label(fact_515_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 439 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit1(K_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_2)))) # label(fact_516_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 440 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K_2)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)))) # label(fact_517_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 441 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_518_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 442 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit0(K_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_2)))) # label(fact_519_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 443 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K_2)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),pls)))) # label(fact_520_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 444 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(X),Y)))))) # label(fact_522_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 445 (all Z_4 all Z_2 all W_1 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_1),Z_4)),hAPP_int_int(plus_plus_int(W),Z_2)))))) # label(fact_523_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 446 (all Z_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_2)))) # label(fact_524_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 447 (all P_1 ((exists X_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_12)) & hBOOL(hAPP_int_bool(P_1,X_12)))) <-> (exists X_12 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_12)))))) # label(fact_525_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 448 (all P_1 ((all X_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_12)) -> hBOOL(hAPP_int_bool(P_1,X_12)))) <-> (all X_12 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_12)))))) # label(fact_526_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 449 (all N_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1)))) # label(fact_527_zero__zle__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 450 (all N_1 all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(X),N_1))))) # label(fact_528_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 451 (all Wa all Z_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_3)) <-> (exists N Z_3 = hAPP_int_int(plus_plus_int(Wa),hAPP_nat_int(semiri1621563631at_int,N))))) # label(fact_529_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 452 (all V_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_5)) -> hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(V_5)) = number267125858f_real(succ(V_5)))) # label(fact_530_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 453 (all V_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_5)) -> hAPP_nat_nat(plus_plus_nat(one_one_nat),number_number_of_nat(V_5)) = number_number_of_nat(succ(V_5)))) # label(fact_531_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 454 (all V_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_5)) -> hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(V_5)) = number_number_of_int(succ(V_5)))) # label(fact_532_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 455 (all V_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_4)) -> hAPP_real_real(plus_plus_real(number267125858f_real(V_4)),one_one_real) = number267125858f_real(succ(V_4)))) # label(fact_533_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 456 (all V_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_4)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_4)),one_one_nat) = number_number_of_nat(succ(V_4)))) # label(fact_534_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 457 (all V_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_4)) -> hAPP_int_int(plus_plus_int(number_number_of_int(V_4)),one_one_int) = number_number_of_int(succ(V_4)))) # label(fact_535_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 458 (all B_13 all A_14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_13)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_14),B_13)))))) # label(fact_536_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 459 (all B_13 all A_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_13)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_14),B_13)))))) # label(fact_537_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 460 (all B_13 all A_14 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_13)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_14),B_13)))))) # label(fact_538_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 461 (all B_12 all A_13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_13)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_12)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(A_13),B_12)))))) # label(fact_539_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 462 (all B_12 all A_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_13)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_12)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(A_13),B_12)))))) # label(fact_540_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 463 (all B_12 all A_13 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_13)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_12)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(A_13),B_12)))))) # label(fact_541_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 464 (all B_11 all C_5 all A_12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_12)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_11),C_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_11),hAPP_real_real(plus_plus_real(A_12),C_5)))))) # label(fact_542_add__strict__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 465 (all B_11 all C_5 all A_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_11),C_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_11),hAPP_nat_nat(plus_plus_nat(A_12),C_5)))))) # label(fact_543_add__strict__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 466 (all B_11 all C_5 all A_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_11),C_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_11),hAPP_int_int(plus_plus_int(A_12),C_5)))))) # label(fact_544_add__strict__increasing) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 467 (all B_10 all C_4 all A_11 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_11)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_10),C_4)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_10),hAPP_real_real(plus_plus_real(A_11),C_4)))))) # label(fact_545_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 468 (all B_10 all C_4 all A_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_11)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_10),C_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_10),hAPP_nat_nat(plus_plus_nat(A_11),C_4)))))) # label(fact_546_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 469 (all B_10 all C_4 all A_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_11)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),C_4)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),hAPP_int_int(plus_plus_int(A_11),C_4)))))) # label(fact_547_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 470 (all B_9 all A_10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_10),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_9),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_10),B_9)),zero_zero_real))))) # label(fact_548_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 471 (all B_9 all A_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_10),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_9),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_10),B_9)),zero_zero_nat))))) # label(fact_549_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 472 (all B_9 all A_10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_10),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_9),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_10),B_9)),zero_zero_int))))) # label(fact_550_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 473 (all B_8 all A_9 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_8),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_9),B_8)),zero_zero_real))))) # label(fact_551_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 474 (all B_8 all A_9 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_8),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(A_9),B_8)),zero_zero_nat))))) # label(fact_552_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 475 (all B_8 all A_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_8),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_9),B_8)),zero_zero_int))))) # label(fact_553_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 476 (all A_8 all N_4 all B_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_8),N_4)),hAPP_nat_real(power_power_real(B_7),N_4))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_8),B_7))))) # label(fact_554_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 477 (all A_8 all N_4 all B_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_8),N_4)),hAPP_nat_nat(power_power_nat(B_7),N_4))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_8),B_7))))) # label(fact_555_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 478 (all A_8 all N_4 all B_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_8),N_4)),hAPP_nat_int(power_power_int(B_7),N_4))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_8),B_7))))) # label(fact_556_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 479 (all Va (zero_zero_nat = number_number_of_nat(Va) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)))) # label(fact_557_eq__0__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 480 (all Va (number_number_of_nat(Va) = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)))) # label(fact_558_eq__number__of__0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 481 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_2)))) # label(fact_559_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 482 (all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K_2)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),pls)))) # label(fact_560_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 483 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K_2)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),L_1)))) # label(fact_561_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 484 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_562_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 485 (all K_2 all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K_2)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),L_1)))) # label(fact_563_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.39 486 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_564_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 487 (all Z_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_3)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_3)))) # label(fact_565_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 488 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int)) <-> Na = zero_zero_nat)) # label(fact_566_int__le__0__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 489 (all K succ(bit0(K)) = bit1(K)) # label(fact_568_succ__Bit0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 490 (all K succ(bit1(K)) = bit0(succ(K))) # label(fact_569_succ__Bit1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 491 (all Wa all Z_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_int_int(plus_plus_int(Z_3),one_one_int))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_3)))) # label(fact_570_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 492 (all Wa all Z_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(Wa),one_one_int)),Z_3)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_3)))) # label(fact_571_add1__zle__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 493 (all W all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W),one_one_int)),Z_2)))) # label(fact_572_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 494 (all K succ(K) = hAPP_int_int(plus_plus_int(K),one_one_int)) # label(fact_573_succ__def) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 495 (all A_7 all N_3 all B_6 (hAPP_nat_real(power_power_real(A_7),N_3) = hAPP_nat_real(power_power_real(B_6),N_3) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3)) -> A_7 = B_6))))) # label(fact_574_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 496 (all A_7 all N_3 all B_6 (hAPP_nat_nat(power_power_nat(A_7),N_3) = hAPP_nat_nat(power_power_nat(B_6),N_3) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_7)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3)) -> A_7 = B_6))))) # label(fact_575_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 497 (all A_7 all N_3 all B_6 (hAPP_nat_int(power_power_int(A_7),N_3) = hAPP_nat_int(power_power_int(B_6),N_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3)) -> A_7 = B_6))))) # label(fact_576_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 498 (all V_3 all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_3)) -> hAPP_real_real(plus_plus_real(number267125858f_real(V_2)),number267125858f_real(V_3)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_2),V_3))))) # label(fact_577_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 499 (all V_3 all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_3)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_2)),number_number_of_nat(V_3)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V_2),V_3))))) # label(fact_578_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 500 (all V_3 all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_3)) -> hAPP_int_int(plus_plus_int(number_number_of_int(V_2)),number_number_of_int(V_3)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_2),V_3))))) # label(fact_579_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 501 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),Z_2))))) # label(fact_581_le__imp__0__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 502 (all K_3 number267125858f_real(succ(K_3)) = hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(K_3))) # label(fact_582_number__of__succ) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 503 (all K_3 number_number_of_int(succ(K_3)) = hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(K_3))) # label(fact_583_number__of__succ) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 504 (all K all L hAPP_int_int(plus_plus_int(bit1(K)),bit1(L)) = bit0(hAPP_int_int(plus_plus_int(K),succ(L)))) # label(fact_584_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 505 (all N_2 all A_6 all B_5 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_6),B_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_6),N_2)),hAPP_nat_real(power_power_real(B_5),N_2))))))) # label(fact_585_power__strict__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 506 (all N_2 all A_6 all B_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),B_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_6),N_2)),hAPP_nat_nat(power_power_nat(B_5),N_2))))))) # label(fact_586_power__strict__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 507 (all N_2 all A_6 all B_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_6),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_6),N_2)),hAPP_nat_int(power_power_int(B_5),N_2))))))) # label(fact_587_power__strict__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 508 (all X hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_589_power2__ge__self) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 509 (all A_5 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_5),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_590_zero__le__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 510 (all A_5 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_5),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_591_zero__le__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 511 (all X_11 all Y_10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_10),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_10)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_11),Y_10))))) # label(fact_592_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 512 (all X_11 all Y_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_10),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_10)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_10))))) # label(fact_593_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 513 (all X_11 all Y_10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_10),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),Y_10))))) # label(fact_594_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 514 (all X_10 all Y_9 (hAPP_nat_real(power_power_real(X_10),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_real(power_power_real(Y_9),number_number_of_nat(bit0(bit1(pls)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_9)) -> X_10 = Y_9)))) # label(fact_595_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 515 (all X_10 all Y_9 (hAPP_nat_nat(power_power_nat(X_10),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(power_power_nat(Y_9),number_number_of_nat(bit0(bit1(pls)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_9)) -> X_10 = Y_9)))) # label(fact_596_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 516 (all X_10 all Y_9 (hAPP_nat_int(power_power_int(X_10),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(Y_9),number_number_of_nat(bit0(bit1(pls)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_9)) -> X_10 = Y_9)))) # label(fact_597_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 517 (all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1))) -> hAPP_nat_real(semiri132038758t_real,number_number_of_nat(V_1)) = number267125858f_real(V_1)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1))) -> hAPP_nat_real(semiri132038758t_real,number_number_of_nat(V_1)) = zero_zero_real))) # label(fact_598_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 518 (all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1))) -> hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V_1)) = number_number_of_int(V_1)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(V_1))) -> hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V_1)) = zero_zero_int))) # label(fact_599_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 519 (all X_9 all Y_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X_9),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_8),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_9),Y_8))))) # label(fact_600_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 520 (all X_9 all Y_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(X_9),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_8),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_8))))) # label(fact_601_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 521 (all X_9 all Y_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(X_9),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_8),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_8))))) # label(fact_602_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 522 (all X_8 all Y_7 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_8),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_7),number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_603_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 523 (all X_8 all Y_7 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_8),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_7),number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_604_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 524 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) <-> Xa = zero_zero_real & Ya = zero_zero_real)) # label(fact_605_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 525 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) <-> Xa = zero_zero_int & Ya = zero_zero_int)) # label(fact_606_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 526 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),number_number_of_int(bit0(bit1(pls))))) -> X = zero_zero_int | X = one_one_int))) # label(fact_607_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 527 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N_1)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N_1)))))) # label(fact_608_number__of1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 528 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (exists N K = hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_610_zero__le__imp__eq__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 529 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> -(all N K != hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_611_nonneg__int__cases) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 530 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1))) # label(fact_612_le0) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 531 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1))) # label(fact_613_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 532 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat)) <-> Na = zero_zero_nat)) # label(fact_614_le__0__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 533 (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) | M_1 = N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)))) # label(fact_615_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 534 (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) -> (M_1 != N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))))) # label(fact_616_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 535 (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)))) # label(fact_617_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 536 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | Ma = Na)) # label(fact_618_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 537 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) & Ma != Na)) # label(fact_619_nat__less__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 538 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1))) # label(fact_620_le__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 539 (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)))) # label(fact_621_nat__le__linear) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 540 (all M_1 all N_1 (M_1 = N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)))) # label(fact_622_eq__imp__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 541 (all K all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K))))) # label(fact_623_le__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 542 (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) -> M_1 = N_1))) # label(fact_624_le__antisym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 543 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) | Xa = Ya)) # label(fact_625_less__eq__real__def) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 544 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) & Xa != Ya)) # label(fact_626_real__less__def) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 545 (all N_1 all M_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(M_1),N_1)))) # label(fact_627_le__add2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 546 (all N_1 all M_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(N_1),M_1)))) # label(fact_628_le__add1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 547 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> (exists K_1 Na = hAPP_nat_nat(plus_plus_nat(Ma),K_1)))) # label(fact_629_le__iff__add) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 548 (all K_2 all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K_2),Ma)),hAPP_nat_nat(plus_plus_nat(K_2),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_630_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 549 (all M_1 all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(plus_plus_nat(J_2),M_1))))) # label(fact_631_trans__le__add1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 550 (all M_1 all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(plus_plus_nat(M_1),J_2))))) # label(fact_632_trans__le__add2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 551 (all K all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),K))))) # label(fact_633_add__le__mono1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 552 (all K all L all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),hAPP_nat_nat(plus_plus_nat(J_2),L)))))) # label(fact_634_add__le__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 553 (all M_1 all K all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M_1),K)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)))) # label(fact_635_add__leD2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 554 (all M_1 all K all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M_1),K)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)))) # label(fact_636_add__leD1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 555 (all M_1 all K all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M_1),K)),N_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1))))) # label(fact_637_add__leE) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 556 (all Z_2 all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Z_2),X)),hAPP_real_real(plus_plus_real(Z_2),Y))))) # label(fact_638_real__add__left__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 557 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)))) # label(fact_639_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 558 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_640_zle__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 559 (all Va all V (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(V))) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls))))) # label(fact_641_le__nat__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 560 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_1)))) # label(fact_642_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 561 (all P_2 all P_1 all Xa ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(P_1) <-> hBOOL(P_2))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) & hBOOL(P_1) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) & hBOOL(P_2)))) # label(fact_643_conj__le__cong) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 562 (all P_2 all P_1 all Xa ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(P_1) <-> hBOOL(P_2))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(P_1)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(P_2))))) # label(fact_644_imp__le__cong) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 563 (all Na all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,Na)) -> (exists K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),Na)) & (all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1)) -> -hBOOL(hAPP_nat_bool(P_1,I_1)))) & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(plus_plus_nat(K_1),one_one_nat)))))))) # label(fact_645_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 564 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> -(all M Z_2 != hAPP_nat_int(semiri1621563631at_int,M)))) # label(fact_646_nonneg__eq__int) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 565 (all Ma all K_2 all F ((all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M)),hAPP_nat_nat(F,N))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(F,Ma)),K_2)),hAPP_nat_nat(F,hAPP_nat_nat(plus_plus_nat(Ma),K_2)))))) # label(fact_647_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 566 (all W hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W))) # label(fact_648_real__le__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 567 (all Z_2 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_2),W)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_2)))) # label(fact_649_real__le__linear) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 568 (all K all I_2 all J_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),J_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),K))))) # label(fact_650_real__le__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 569 (all Z_2 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_2),W)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_2)) -> Z_2 = W))) # label(fact_651_real__le__antisym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 570 (all P_1 all K_2 all I (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),I)) -> (hBOOL(hAPP_int_bool(P_1,K_2)) -> ((all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_2),I_1)) -> (hBOOL(hAPP_int_bool(P_1,I_1)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(plus_plus_int(I_1),one_one_int)))))) -> hBOOL(hAPP_int_bool(P_1,I)))))) # label(fact_652_int__ge__induct) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 571 (all Na all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,Na)) -> (exists K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Na)) & (all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1)) -> -hBOOL(hAPP_nat_bool(P_1,I_1)))) & hBOOL(hAPP_nat_bool(P_1,K_1))))))) # label(fact_653_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 572 (all I all J all F ((all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_1))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I)),hAPP_nat_nat(F,J)))))) # label(fact_654_less__mono__imp__le__mono) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 573 (all X_7 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_7),X_7))) # label(fact_655_order__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 574 (all X_7 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),X_7))) # label(fact_656_order__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 575 (all X_7 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),X_7))) # label(fact_657_order__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 576 (all X_6 all Y_6 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_6),Y_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_6),X_6)))) # label(fact_658_linorder__le__cases) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 577 (all X_6 all Y_6 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_6),Y_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_6),X_6)))) # label(fact_659_linorder__le__cases) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 578 (all X_6 all Y_6 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_6),Y_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),X_6)))) # label(fact_660_linorder__le__cases) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 579 (all Z_1 all Y_5 all X_5 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_5),X_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),Y_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),X_5))))) # label(fact_661_xt1_I6_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 580 (all Z_1 all Y_5 all X_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_1),Y_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_1),X_5))))) # label(fact_662_xt1_I6_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 581 (all Z_1 all Y_5 all X_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),Y_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),X_5))))) # label(fact_663_xt1_I6_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 582 (all Y_4 all X_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_4),X_4)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_4),Y_4)) -> X_4 = Y_4))) # label(fact_664_xt1_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 583 (all Y_4 all X_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_4),X_4)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Y_4)) -> X_4 = Y_4))) # label(fact_665_xt1_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 584 (all Y_4 all X_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_4),X_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),Y_4)) -> X_4 = Y_4))) # label(fact_666_xt1_I5_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 585 (all Z all X_3 all Y_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_3),Y_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),Z)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_3),Z))))) # label(fact_667_order__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 586 (all Z all X_3 all Y_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),Y_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),Z)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),Z))))) # label(fact_668_order__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 587 (all Z all X_3 all Y_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),Y_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),Z)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_3),Z))))) # label(fact_669_order__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 588 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_2)) -> X_2 = Y_2))) # label(fact_670_order__antisym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 589 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_2)) -> X_2 = Y_2))) # label(fact_671_order__antisym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 590 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),X_2)) -> X_2 = Y_2))) # label(fact_672_order__antisym) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 591 (all C_3 all B_4 all A_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),A_4)) -> (B_4 = C_3 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_3),A_4))))) # label(fact_673_xt1_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 592 (all C_3 all B_4 all A_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),A_4)) -> (B_4 = C_3 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_3),A_4))))) # label(fact_674_xt1_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 593 (all C_3 all B_4 all A_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),A_4)) -> (B_4 = C_3 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_3),A_4))))) # label(fact_675_xt1_I4_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 594 (all C_2 all A_3 all B_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_3)) -> (B_3 = C_2 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),C_2))))) # label(fact_676_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 595 (all C_2 all A_3 all B_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),B_3)) -> (B_3 = C_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),C_2))))) # label(fact_677_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 596 (all C_2 all A_3 all B_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),B_3)) -> (B_3 = C_2 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),C_2))))) # label(fact_678_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 597 (all C_1 all A_2 all B_2 (A_2 = B_2 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),B_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),A_2))))) # label(fact_679_xt1_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 598 (all C_1 all A_2 all B_2 (A_2 = B_2 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_2))))) # label(fact_680_xt1_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 599 (all C_1 all A_2 all B_2 (A_2 = B_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),A_2))))) # label(fact_681_xt1_I3_J) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 600 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),C)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),C))))) # label(fact_682_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 601 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_1),C)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),C))))) # label(fact_683_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 602 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),C)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),C))))) # label(fact_684_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 603 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> Xa = Ya))) # label(fact_685_order__antisym__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 604 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> Xa = Ya))) # label(fact_686_order__antisym__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 605 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> Xa = Ya))) # label(fact_687_order__antisym__conv) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 606 (all X_1 all Y_1 (X_1 = Y_1 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)))) # label(fact_688_order__eq__refl) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 607 (all D2 all D1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2)) -> (exists E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2))))))) # label(fact_689_real__lbound__gt__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 608 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),sqrt(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y),number_number_of_nat(bit0(bit1(pls))))))))) # label(fact_690_real__sqrt__sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 609 (all A all B hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(A),B)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_691_zadd__power2) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 610 (all K all L times_times_int(bit1(K),L) = hAPP_int_int(plus_plus_int(bit0(times_times_int(K,L))),L)) # label(fact_692_mult__Bit1) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 611 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),sqrt(X))))) # label(fact_693_real__sqrt__gt__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 612 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> sqrt(X) != zero_zero_real)) # label(fact_694_real__sqrt__not__eq__zero) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 613 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,sqrt(Xa)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)))) # label(fact_695_real__sqrt__lt__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 614 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),sqrt(Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)))) # label(fact_696_real__sqrt__gt__0__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 615 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,sqrt(Xa)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)))) # label(fact_697_real__sqrt__lt__1__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 616 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),sqrt(Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Ya)))) # label(fact_698_real__sqrt__gt__1__iff) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 617 (all X all Y hAPP_nat_nat(if_nat(fTrue,X),Y) = X) # label(help_If_1_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption].
% 1.07/1.40 618 (all X all Y hAPP_nat_nat(if_nat(fFalse,X),Y) = Y) # label(help_If_2_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 619 (all P (is_bool(P) -> P = fTrue | P = fFalse)) # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 620 -(hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) != zero_zero_int) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption].
% 1.76/1.99
% 1.76/1.99 ============================== end of process non-clausal formulas ===
% 1.76/1.99
% 1.76/1.99 ============================== PROCESS INITIAL CLAUSES ===============
% 1.76/1.99
% 1.76/1.99 ============================== PREDICATE ELIMINATION =================
% 1.76/1.99 621 -is_bool(A) | A = fTrue | A = fFalse # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom). [clausify(619)].
% 1.76/1.99 622 is_bool(fFalse) # label(gsy_c_fFalse) # label(axiom). [assumption].
% 1.76/1.99 623 is_bool(fTrue) # label(gsy_c_fTrue) # label(axiom). [assumption].
% 1.76/1.99 624 is_bool(hAPP_int_bool(A,B)) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) # label(axiom). [clausify(1)].
% 1.76/1.99 625 is_bool(hAPP_nat_bool(A,B)) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom). [clausify(2)].
% 1.76/1.99 626 is_bool(hAPP_real_bool(A,B)) # label(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool) # label(axiom). [clausify(3)].
% 1.76/1.99 Derived: fFalse = fTrue | fFalse = fFalse. [resolve(621,a,622,a)].
% 1.76/1.99 Derived: fTrue = fTrue | fTrue = fFalse. [resolve(621,a,623,a)].
% 1.76/1.99 Derived: hAPP_int_bool(A,B) = fTrue | hAPP_int_bool(A,B) = fFalse. [resolve(621,a,624,a)].
% 1.76/1.99 Derived: hAPP_nat_bool(A,B) = fTrue | hAPP_nat_bool(A,B) = fFalse. [resolve(621,a,625,a)].
% 1.76/1.99 Derived: hAPP_real_bool(A,B) = fTrue | hAPP_real_bool(A,B) = fFalse. [resolve(621,a,626,a)].
% 1.76/1.99
% 1.76/1.99 ============================== end predicate elimination =============
% 1.76/1.99
% 1.76/1.99 Auto_denials: (non-Horn, no changes).
% 1.76/1.99
% 1.76/1.99 Term ordering decisions:
% 1.76/1.99 Function symbol KB weights: ord_less_eq_int=1. ord_less_int=1. pls=1. zero_zero_int=1. zero_zero_nat=1. ord_less_nat=1. zero_zero_real=1. ord_less_eq_nat=1. ord_less_real=1. ord_less_eq_real=1. semiri1621563631at_int=1. one_one_int=1. one_one_nat=1. one_one_real=1. semiri132038758t_real=1. semiri984289939at_nat=1. fFalse=1. fTrue=1. n=1. t=1. m1=1. tn=1. hAPP_int_bool=1. hAPP_i1948725293t_bool=1. hAPP_nat_bool=1. hAPP_n1699378549t_bool=1. hAPP_nat_nat=1. hAPP_real_bool=1. hAPP_r1134773055l_bool=1. hAPP_int_int=1. hAPP_nat_int=1. hAPP_real_real=1. hAPP_nat_real=1. if_nat=1. times_times_int=1. f3=1. f4=1. f5=1. f10=1. f13=1. f14=1. f19=1. f22=1. plus_plus_int=1. bit1=1. plus_plus_nat=1. number_number_of_nat=1. bit0=1. plus_plus_real=1. number_number_of_int=1. power_power_int=1. power_power_real=1. power_power_nat=1. number267125858f_real=1. succ=1. sqrt=1. f2=1. f6=1. f7=1. f8=1. f9=1. f11=1. f12=1. f15=1. f1=1. f16=1. f17=1. f18=1. f20=1. f21=1.
% 1.76/1.99
% 1.76/1.99 ============================== end of process initial clauses ========
% 1.76/1.99
% 1.76/1.99 ============================== CLAUSES FOR SEARCH ====================
% 1.76/1.99
% 1.76/1.99 ============================== end of clauses for search =============
% 1.76/1.99
% 1.76/1.99 ============================== SEARCH ================================
% 1.76/1.99
% 1.76/1.99 % Starting search at 0.24 seconds.
% 1.76/1.99
% 1.76/1.99 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 45 (0.00 of 0.69 sec).
% 1.76/1.99
% 1.76/1.99 ============================== PROOF =================================
% 1.76/1.99 % SZS status Theorem
% 1.76/1.99 % SZS output start Refutation
% 1.76/1.99
% 1.76/1.99 % Proof 1 at 0.70 (+ 0.01) seconds.
% 1.76/1.99 % Length of proof is 25.
% 1.76/1.99 % Level of proof is 4.
% 1.76/1.99 % Maximum clause weight is 17.000.
% 1.76/1.99 % Given clauses 512.
% 1.76/1.99
% 1.76/1.99 74 (all K bit0(K) = hAPP_int_int(plus_plus_int(K),K)) # label(fact_103_Bit0__def) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 129 (all K number_number_of_int(K) = K) # label(fact_176_number__of__is__id) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 178 (all N_23 all A_62 (A_62 != zero_zero_int -> hAPP_nat_int(power_power_int(A_62),N_23) != zero_zero_int)) # label(fact_231_field__power__not__zero) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 376 (all Z_3 all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),Wa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_3),Wa)) & Z_3 != Wa)) # label(fact_447_zless__le) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 494 (all K succ(K) = hAPP_int_int(plus_plus_int(K),one_one_int)) # label(fact_573_succ__def) # label(axiom) # label(non_clause). [assumption].
% 1.76/1.99 620 -(hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) != zero_zero_int) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption].
% 1.76/1.99 627 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) # label(fact_0_n1pos) # label(axiom). [assumption].
% 1.76/1.99 672 one_one_int = number_number_of_int(bit1(pls)) # label(fact_25_semiring__norm_I110_J) # label(axiom). [assumption].
% 1.76/1.99 673 number_number_of_int(bit1(pls)) = one_one_int. [copy(672),flip(a)].
% 1.76/1.99 788 pls = zero_zero_int # label(fact_98_Pls__def) # label(axiom). [assumption].
% 1.76/1.99 789 zero_zero_int = pls. [copy(788),flip(a)].
% 1.76/1.99 796 bit0(A) = hAPP_int_int(plus_plus_int(A),A) # label(fact_103_Bit0__def) # label(axiom). [clausify(74)].
% 1.76/1.99 957 number_number_of_int(A) = A # label(fact_176_number__of__is__id) # label(axiom). [clausify(129)].
% 1.76/1.99 1047 A = zero_zero_int | hAPP_nat_int(power_power_int(A),B) != zero_zero_int # label(fact_231_field__power__not__zero) # label(axiom). [clausify(178)].
% 1.76/1.99 1048 pls = A | hAPP_nat_int(power_power_int(A),B) != pls. [copy(1047),rewrite([789(1),789(5)]),flip(a)].
% 1.76/1.99 1355 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) | A != B # label(fact_447_zless__le) # label(axiom). [clausify(376)].
% 1.76/1.99 1608 succ(A) = hAPP_int_int(plus_plus_int(A),one_one_int) # label(fact_573_succ__def) # label(axiom). [clausify(494)].
% 1.76/1.99 1609 hAPP_int_int(plus_plus_int(A),one_one_int) = succ(A). [copy(1608),flip(a)].
% 1.76/1.99 1789 hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int # label(conj_0) # label(negated_conjecture). [clausify(620)].
% 1.76/1.99 1790 hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls)))) = pls. [copy(1789),rewrite([796(10),789(16)])].
% 1.76/1.99 1814 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))). [back_rewrite(627),rewrite([789(2)])].
% 1.76/1.99 1839 bit1(pls) = one_one_int. [back_rewrite(673),rewrite([957(3)])].
% 1.76/1.99 1901 hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(succ(one_one_int))) = pls. [back_rewrite(1790),rewrite([1839(9),1839(11),1609(11)])].
% 1.76/1.99 4648 hAPP_int_int(plus_plus_int(one_one_int),hAPP_nat_int(semiri1621563631at_int,n)) != pls. [resolve(1814,a,1355,a),flip(a)].
% 1.76/1.99 5184 $F. [resolve(1901,a,1048,b),flip(a),unit_del(a,4648)].
% 1.76/1.99
% 1.76/1.99 % SZS output end Refutation
% 1.76/1.99 ============================== end of proof ==========================
% 1.76/1.99
% 1.76/1.99 ============================== STATISTICS ============================
% 1.76/1.99
% 1.76/1.99 Given=512. Generated=9937. Kept=4032. proofs=1.
% 1.76/1.99 Usable=508. Sos=3175. Demods=191. Limbo=0, Disabled=1300. Hints=0.
% 1.76/1.99 Megabytes=11.94.
% 1.76/1.99 User_CPU=0.70, System_CPU=0.01, Wall_clock=1.
% 1.76/1.99
% 1.76/1.99 ============================== end of statistics =====================
% 1.76/1.99
% 1.76/1.99 ============================== end of search =========================
% 1.76/1.99
% 1.76/1.99 THEOREM PROVED
% 1.76/1.99 % SZS status Theorem
% 1.76/1.99
% 1.76/1.99 Exiting with 1 proof.
% 1.76/1.99
% 1.76/1.99 Process 632 exit (max_proofs) Wed Jul 6 19:39:07 2022
% 1.76/1.99 Prover9 interrupted
%------------------------------------------------------------------------------