% Mizar problem: t42_turing_1,turing_1,2332,46 fof(t42_turing_1,conjecture,( ! [A] : ( m1_subset_1(A,k3_zfmisc_1(u2_turing_1(k20_turing_1),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1)))) => ! [B] : ( m2_fraenkel(B,k4_numbers,u1_turing_1(k20_turing_1),k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_finseq_1(D,k5_numbers) => ( ( r1_xreal_0(3,k3_finseq_1(D)) & A = k4_domain_1(k5_numbers,k5_numbers,k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1)),0,C,B) & r2_turing_1(k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,C),D),k20_turing_1,B) ) => ( v2_turing_1(A,k20_turing_1) & k6_mcart_1(u2_turing_1(k20_turing_1),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1)),k11_turing_1(k20_turing_1,A)) = k1_nat_1(k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),k4_finseq_4(k5_numbers,k5_numbers,D,2)),4) & r2_turing_1(k2_finseq_4(k5_numbers,k1_nat_1(k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),k4_finseq_4(k5_numbers,k5_numbers,D,2)),4),k4_finseq_4(k5_numbers,k5_numbers,D,3)),k20_turing_1,k7_mcart_1(u2_turing_1(k20_turing_1),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1)),k11_turing_1(k20_turing_1,A))) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_turing_1,antisymmetry_r2_hidden,cc1_arytm_3,cc1_finset_1,cc1_funct_1,cc1_int_1,cc1_relset_1,cc2_arytm_3,cc2_finset_1,cc2_funct_1,cc2_int_1,cc3_arytm_3,cc3_int_1,cc4_int_1,commutativity_k1_nat_1,commutativity_k2_tarski,commutativity_k2_xcmplx_0,commutativity_k7_domain_1,connectedness_r1_xreal_0,d10_turing_1,d13_turing_1,d1_turing_1,d21_turing_1,d22_turing_1,d2_int_1,d3_turing_1,d4_turing_1,d5_tarski,d5_turing_1,d6_turing_1,d8_turing_1,d9_turing_1,dt_g1_turing_1,dt_k10_finseq_1,dt_k10_turing_1,dt_k11_turing_1,dt_k12_finseq_1,dt_k12_mcart_1,dt_k12_turing_1,dt_k13_mcart_1,dt_k19_turing_1,dt_k1_card_1,dt_k1_domain_1,dt_k1_enumset1,dt_k1_fraenkel,dt_k1_funct_1,dt_k1_funct_2,dt_k1_funct_4,dt_k1_nat_1,dt_k1_numbers,dt_k1_real_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_turing_1,dt_k2_finseq_4,dt_k2_tarski,dt_k2_turing_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_cqc_lang,dt_k3_finseq_1,dt_k3_mcart_1,dt_k3_turing_1,dt_k3_zfmisc_1,dt_k4_domain_1,dt_k4_finseq_4,dt_k4_numbers,dt_k4_tarski,dt_k4_xcmplx_0,dt_k5_finseq_1,dt_k5_mcart_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_turing_1,dt_k6_mcart_1,dt_k6_turing_1,dt_k7_domain_1,dt_k7_finseq_1,dt_k7_mcart_1,dt_k7_turing_1,dt_k8_domain_1,dt_k8_finseq_1,dt_k8_funct_2,dt_k8_turing_1,dt_k9_turing_1,dt_l1_turing_1,dt_m1_finseq_1,dt_m1_fraenkel,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_fraenkel,dt_m2_relset_1,dt_m2_subset_1,dt_u1_turing_1,dt_u2_turing_1,dt_u3_turing_1,dt_u4_turing_1,dt_u5_turing_1,existence_l1_turing_1,existence_m1_finseq_1,existence_m1_fraenkel,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_fraenkel,existence_m2_relset_1,existence_m2_subset_1,fc14_finset_1,fc15_finset_1,fc1_cqc_lang,fc1_finset_1,fc1_int_1,fc1_numbers,fc1_ordinal2,fc1_subset_1,fc1_turing_1,fc1_xboole_0,fc2_comput_1,fc2_finset_1,fc2_mod_2,fc2_subset_1,fc3_finset_1,fc3_int_1,fc3_subset_1,fc4_numbers,fc4_subset_1,fc5_comput_1,fc5_int_1,fc5_subset_1,fc6_int_1,fraenkel_a_1_0_turing_1,free_g1_turing_1,idempotence_k1_funct_4,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,l74_turing_1,l75_turing_1,l76_turing_1,l77_turing_1,rc10_comput_1,rc1_arytm_3,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_subset_1,rc1_turing_1,rc1_xboole_0,rc2_finset_1,rc2_funct_1,rc2_int_1,rc2_subset_1,rc2_xboole_0,rc3_finset_1,rc3_funct_1,rc4_comput_1,rc4_finset_1,rc4_funct_1,rc9_comput_1,redefinition_k12_finseq_1,redefinition_k12_mcart_1,redefinition_k13_mcart_1,redefinition_k1_domain_1,redefinition_k1_fraenkel,redefinition_k1_nat_1,redefinition_k1_real_1,redefinition_k2_finseq_4,redefinition_k2_turing_1,redefinition_k3_finseq_1,redefinition_k4_domain_1,redefinition_k5_numbers,redefinition_k7_domain_1,redefinition_k8_domain_1,redefinition_k8_finseq_1,redefinition_k8_funct_2,redefinition_m2_finseq_1,redefinition_m2_fraenkel,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,spc4_boole,spc4_numerals,spc6_boole,spc6_numerals,t12_turing_1,t13_turing_1,t19_turing_1,t1_arithm,t1_numerals,t1_real,t1_subset,t26_turing_1,t27_turing_1,t28_turing_1,t29_nat_1,t2_real,t2_subset,t2_tarski,t2_xreal_1,t31_xreal_1,t37_nat_1,t3_real,t3_subset,t41_turing_1,t4_real,t4_subset,t5_real,t5_subset,t68_mcart_1,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,t9_xreal_1]), [file(turing_1,t42_turing_1)]). fof(abstractness_v1_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( v1_turing_1(A) => A = g1_turing_1(u1_turing_1(A),u2_turing_1(A),u3_turing_1(A),u4_turing_1(A),u5_turing_1(A)) ) ) ), file(turing_1,v1_turing_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_arytm_3,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_int_1,axiom,( ! [A] : ( m1_subset_1(A,k4_numbers) => v1_int_1(A) ) ), file(int_1,cc1_int_1), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc2_arytm_3,axiom,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), []). fof(cc2_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc3_arytm_3,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(commutativity_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(commutativity_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k7_domain_1(A,C,B) ) ), file(domain_1,k7_domain_1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d10_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => ( v2_turing_1(B,A) => ! [C] : ( m1_subset_1(C,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => ( C = k11_turing_1(A,B) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & C = k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),k10_turing_1(A,B),D) & k5_mcart_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),k10_turing_1(A,B),D)) = u5_turing_1(A) ) ) ) ) ) ) ), file(turing_1,d10_turing_1), []). fof(d13_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m2_fraenkel(B,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ! [C] : ( v1_int_1(C) => ! [D] : ( v1_int_1(D) => ( r1_turing_1(A,B,C,D) <=> ( k1_funct_1(B,C) = 0 & k1_funct_1(B,D) = 0 & ! [E] : ( v1_int_1(E) => ~ ( ~ r1_xreal_0(E,C) & ~ r1_xreal_0(D,E) & k1_funct_1(B,E) != 1 ) ) ) ) ) ) ) ) ), file(turing_1,d13_turing_1), []). fof(d1_turing_1,axiom,( ! [A] : ( v4_ordinal2(A) => k3_turing_1(A) = a_1_0_turing_1(A) ) ), file(turing_1,d1_turing_1), []). fof(d21_turing_1,axiom,( k19_turing_1 = k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k1_funct_4(k12_turing_1(k12_mcart_1(k5_numbers,k5_numbers,k3_turing_1(3),k7_domain_1(k5_numbers,0,1)),k8_domain_1(k1_numbers,k1_real_1(1),0,1),0),k2_turing_1(k2_zfmisc_1(k5_numbers,k5_numbers),k3_zfmisc_1(k5_numbers,k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,0,0),k4_domain_1(k5_numbers,k5_numbers,k5_numbers,1,0,1))),k2_turing_1(k2_zfmisc_1(k5_numbers,k5_numbers),k3_zfmisc_1(k5_numbers,k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,1,1),k4_domain_1(k5_numbers,k5_numbers,k5_numbers,1,0,1))),k2_turing_1(k2_zfmisc_1(k5_numbers,k5_numbers),k3_zfmisc_1(k5_numbers,k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,1,0),k4_domain_1(k5_numbers,k5_numbers,k5_numbers,2,0,1))),k2_turing_1(k2_zfmisc_1(k5_numbers,k5_numbers),k3_zfmisc_1(k5_numbers,k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,2,1),k4_domain_1(k5_numbers,k5_numbers,k5_numbers,2,0,1))),k2_turing_1(k2_zfmisc_1(k5_numbers,k5_numbers),k3_zfmisc_1(k5_numbers,k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,2,0),k4_domain_1(k5_numbers,k5_numbers,k5_numbers,3,0,0))) ), file(turing_1,d21_turing_1), []). fof(d22_turing_1,axiom,( ! [A] : ( ( v1_turing_1(A) & l1_turing_1(A) ) => ( A = k20_turing_1 <=> ( u1_turing_1(A) = k7_domain_1(k5_numbers,0,1) & u2_turing_1(A) = k3_turing_1(3) & u3_turing_1(A) = k19_turing_1 & u4_turing_1(A) = 0 & u5_turing_1(A) = 3 ) ) ) ), file(turing_1,d22_turing_1), []). fof(d2_int_1,axiom,( ! [A] : ( v1_int_1(A) <=> r2_hidden(A,k4_numbers) ) ), file(int_1,d2_int_1), []). fof(d3_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m2_fraenkel(B,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ! [C] : ( v1_int_1(C) => ! [D] : ( m1_subset_1(D,u1_turing_1(A)) => k5_turing_1(A,B,C,D) = k1_funct_4(B,k3_cqc_lang(C,D)) ) ) ) ) ), file(turing_1,d3_turing_1), []). fof(d4_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) => k6_turing_1(A,B) = k7_mcart_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1),B) ) ) ), file(turing_1,d4_turing_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d5_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => k7_turing_1(A,B) = k6_mcart_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),B) ) ) ), file(turing_1,d5_turing_1), []). fof(d6_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => k8_turing_1(A,B) = k1_funct_1(u3_turing_1(A),k4_tarski(k5_mcart_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),B),k1_funct_1(k7_mcart_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),B),k7_turing_1(A,B)))) ) ) ), file(turing_1,d6_turing_1), []). fof(d8_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) & m2_relset_1(C,k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) => ( C = k10_turing_1(A,B) <=> ( k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),C,0) = B & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),C,k1_nat_1(D,1)) = k9_turing_1(A,k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),C,D)) ) ) ) ) ) ) ), file(turing_1,d8_turing_1), []). fof(d9_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => ( v2_turing_1(B,A) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & k5_mcart_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),k10_turing_1(A,B),C)) = u5_turing_1(A) ) ) ) ) ), file(turing_1,d9_turing_1), []). fof(dt_g1_turing_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_relset_1(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_subset_1(D,B) & m1_subset_1(E,B) ) => ( v1_turing_1(g1_turing_1(A,B,C,D,E)) & l1_turing_1(g1_turing_1(A,B,C,D,E)) ) ) ), file(turing_1,g1_turing_1), []). fof(dt_k10_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), []). fof(dt_k10_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) => ( v1_funct_1(k10_turing_1(A,B)) & v1_funct_2(k10_turing_1(A,B),k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) & m2_relset_1(k10_turing_1(A,B),k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) ) ), file(turing_1,k10_turing_1), []). fof(dt_k11_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) => m1_subset_1(k11_turing_1(A,B),k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) ), file(turing_1,k11_turing_1), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => m1_subset_1(k12_mcart_1(A,B,C,D),k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(mcart_1,k12_mcart_1), []). fof(dt_k12_turing_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( v1_funct_1(k12_turing_1(A,B,C)) & v1_funct_2(k12_turing_1(A,B,C),A,k2_zfmisc_1(A,B)) & m2_relset_1(k12_turing_1(A,B,C),A,k2_zfmisc_1(A,B)) ) ) ), file(turing_1,k12_turing_1), []). fof(dt_k13_mcart_1,axiom,( ! [A,B,C,D,E,F] : ( ( m1_subset_1(D,k1_zfmisc_1(A)) & m1_subset_1(E,k1_zfmisc_1(B)) & m1_subset_1(F,k1_zfmisc_1(C)) ) => m1_subset_1(k13_mcart_1(A,B,C,D,E,F),k1_zfmisc_1(k3_zfmisc_1(A,B,C))) ) ), file(mcart_1,k13_mcart_1), []). fof(dt_k19_turing_1,axiom, ( v1_funct_1(k19_turing_1) & v1_funct_2(k19_turing_1,k12_mcart_1(k5_numbers,k5_numbers,k3_turing_1(3),k7_domain_1(k5_numbers,0,1)),k13_mcart_1(k5_numbers,k5_numbers,k1_numbers,k3_turing_1(3),k7_domain_1(k5_numbers,0,1),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m2_relset_1(k19_turing_1,k12_mcart_1(k5_numbers,k5_numbers,k3_turing_1(3),k7_domain_1(k5_numbers,0,1)),k13_mcart_1(k5_numbers,k5_numbers,k1_numbers,k3_turing_1(3),k7_domain_1(k5_numbers,0,1),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ), file(turing_1,k19_turing_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_enumset1,axiom,( $true ), file(enumset1,k1_enumset1), []). fof(dt_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => m1_fraenkel(k1_fraenkel(A,B),A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_funct_2,axiom,( $true ), file(funct_2,k1_funct_2), []). fof(dt_k1_funct_4,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k1_funct_4(A,B)) & v1_funct_1(k1_funct_4(A,B)) ) ) ), file(funct_4,k1_funct_4), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k20_turing_1,axiom, ( v1_turing_1(k20_turing_1) & l1_turing_1(k20_turing_1) ), file(turing_1,k20_turing_1), []). fof(dt_k2_finseq_4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m2_finseq_1(k2_finseq_4(A,B,C),A) ) ), file(finseq_4,k2_finseq_4), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_turing_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => ( v1_funct_1(k2_turing_1(A,B,C,D)) & m2_relset_1(k2_turing_1(A,B,C,D),A,B) ) ) ), file(turing_1,k2_turing_1), []). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_cqc_lang,axiom,( $true ), file(cqc_lang,k3_cqc_lang), []). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_mcart_1,axiom,( $true ), file(mcart_1,k3_mcart_1), []). fof(dt_k3_turing_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k3_turing_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(turing_1,k3_turing_1), []). fof(dt_k3_zfmisc_1,axiom,( $true ), file(zfmisc_1,k3_zfmisc_1), []). fof(dt_k4_domain_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & m1_subset_1(D,A) & m1_subset_1(E,B) & m1_subset_1(F,C) ) => m1_subset_1(k4_domain_1(A,B,C,D,E,F),k3_zfmisc_1(A,B,C)) ) ), file(domain_1,k4_domain_1), []). fof(dt_k4_finseq_4,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k4_finseq_4(A,B,C,D),B) ) ), file(finseq_4,k4_finseq_4), []). fof(dt_k4_numbers,axiom,( $true ), file(numbers,k4_numbers), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => m1_subset_1(k5_mcart_1(A,B,C,D),A) ) ), file(mcart_1,k5_mcart_1), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_turing_1,axiom,( ! [A,B,C,D] : ( ( l1_turing_1(A) & m1_subset_1(B,k1_fraenkel(k4_numbers,u1_turing_1(A))) & v1_int_1(C) & m1_subset_1(D,u1_turing_1(A)) ) => m2_fraenkel(k5_turing_1(A,B,C,D),k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) ) ), file(turing_1,k5_turing_1), []). fof(dt_k6_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => m1_subset_1(k6_mcart_1(A,B,C,D),B) ) ), file(mcart_1,k6_mcart_1), []). fof(dt_k6_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) => v1_int_1(k6_turing_1(A,B)) ) ), file(turing_1,k6_turing_1), []). fof(dt_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k7_domain_1(A,B,C),k1_zfmisc_1(A)) ) ), file(domain_1,k7_domain_1), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k7_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => m1_subset_1(k7_mcart_1(A,B,C,D),C) ) ), file(mcart_1,k7_mcart_1), []). fof(dt_k7_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) => v1_int_1(k7_turing_1(A,B)) ) ), file(turing_1,k7_turing_1), []). fof(dt_k8_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) ) => m1_subset_1(k8_domain_1(A,B,C,D),k1_zfmisc_1(A)) ) ), file(domain_1,k8_domain_1), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) => m1_subset_1(k8_turing_1(A,B),k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ), file(turing_1,k8_turing_1), []). fof(dt_k9_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) => m1_subset_1(k9_turing_1(A,B),k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) ) ), file(turing_1,k9_turing_1), []). fof(dt_l1_turing_1,axiom,( $true ), file(turing_1,l1_turing_1), []). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_fraenkel,axiom,( ! [A,B,C] : ( m1_fraenkel(C,A,B) => ( ~ v1_xboole_0(C) & v1_fraenkel(C) ) ) ), file(fraenkel,m1_fraenkel), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ! [D] : ( m2_fraenkel(D,A,B,C) => ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) ) ) ), file(fraenkel,m2_fraenkel), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), []). fof(dt_u1_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( ~ v1_xboole_0(u1_turing_1(A)) & v1_finset_1(u1_turing_1(A)) ) ) ), file(turing_1,u1_turing_1), []). fof(dt_u2_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( ~ v1_xboole_0(u2_turing_1(A)) & v1_finset_1(u2_turing_1(A)) ) ) ), file(turing_1,u2_turing_1), []). fof(dt_u3_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( v1_funct_1(u3_turing_1(A)) & v1_funct_2(u3_turing_1(A),k2_zfmisc_1(u2_turing_1(A),u1_turing_1(A)),k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m2_relset_1(u3_turing_1(A),k2_zfmisc_1(u2_turing_1(A),u1_turing_1(A)),k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ) ), file(turing_1,u3_turing_1), []). fof(dt_u4_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => m1_subset_1(u4_turing_1(A),u2_turing_1(A)) ) ), file(turing_1,u4_turing_1), []). fof(dt_u5_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => m1_subset_1(u5_turing_1(A),u2_turing_1(A)) ) ), file(turing_1,u5_turing_1), []). fof(existence_l1_turing_1,axiom,( ? [A] : l1_turing_1(A) ), file(turing_1,l1_turing_1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_fraenkel,axiom,( ! [A,B] : ? [C] : m1_fraenkel(C,A,B) ), file(fraenkel,m1_fraenkel), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ? [D] : m2_fraenkel(D,A,B,C) ) ), file(fraenkel,m2_fraenkel), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), []). fof(fc14_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). fof(fc15_finset_1,axiom,( ! [A,B,C] : ( ( v1_finset_1(A) & v1_finset_1(B) & v1_finset_1(C) ) => v1_finset_1(k3_zfmisc_1(A,B,C)) ) ), file(finset_1,fc15_finset_1), []). fof(fc1_cqc_lang,axiom,( ! [A,B] : ( v1_relat_1(k3_cqc_lang(A,B)) & v1_funct_1(k3_cqc_lang(A,B)) ) ), file(cqc_lang,fc1_cqc_lang), []). fof(fc1_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_numbers,axiom,( ~ v1_xboole_0(k1_numbers) ), file(numbers,fc1_numbers), []). fof(fc1_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_turing_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ v1_xboole_0(k3_turing_1(A)) & v1_finset_1(k3_turing_1(A)) ) ) ), file(turing_1,fc1_turing_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc2_comput_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) & v1_funct_7(k1_xboole_0) & v4_comput_1(k1_xboole_0) ), file(comput_1,fc2_comput_1), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(fc2_mod_2,axiom,( ! [A,B,C] : ~ v1_xboole_0(k1_enumset1(A,B,C)) ), file(mod_2,fc2_mod_2), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc3_finset_1,axiom,( ! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) ), file(finset_1,fc3_finset_1), []). fof(fc3_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc4_numbers,axiom,( ~ v1_xboole_0(k4_numbers) ), file(numbers,fc4_numbers), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_comput_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) & v1_funct_7(k1_xboole_0) & v4_comput_1(k1_xboole_0) & v5_comput_1(k1_xboole_0) ), file(comput_1,fc5_comput_1), []). fof(fc5_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), []). fof(fc5_subset_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) ) => ~ v1_xboole_0(k3_zfmisc_1(A,B,C)) ) ), file(subset_1,fc5_subset_1), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fraenkel_a_1_0_turing_1,axiom,( ! [A,B] : ( v4_ordinal2(B) => ( r2_hidden(A,a_1_0_turing_1(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r1_xreal_0(C,B) ) ) ) ), file(turing_1,a_1_0_turing_1), []). fof(free_g1_turing_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_relset_1(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_subset_1(D,B) & m1_subset_1(E,B) ) => ! [F,G,H,I,J] : ( g1_turing_1(A,B,C,D,E) = g1_turing_1(F,G,H,I,J) => ( A = F & B = G & C = H & D = I & E = J ) ) ) ), file(turing_1,g1_turing_1), []). fof(idempotence_k1_funct_4,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => k1_funct_4(A,A) = A ) ), file(funct_4,k1_funct_4), []). fof(involutiveness_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), []). fof(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(l74_turing_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(A,3) => m1_subset_1(A,u2_turing_1(k20_turing_1)) ) ) ), file(turing_1,l74_turing_1), []). fof(l75_turing_1,axiom, ( r2_hidden(0,u1_turing_1(k20_turing_1)) & r2_hidden(1,u1_turing_1(k20_turing_1)) ), file(turing_1,l75_turing_1), []). fof(l76_turing_1,axiom,( ! [A] : ( m1_subset_1(A,k3_zfmisc_1(u2_turing_1(k20_turing_1),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1)))) => ! [B,C,D] : ( A = k3_mcart_1(B,C,D) => ( B = 3 | k9_turing_1(k20_turing_1,A) = k3_mcart_1(k5_mcart_1(u2_turing_1(k20_turing_1),u1_turing_1(k20_turing_1),k8_domain_1(k1_numbers,k1_real_1(1),0,1),k8_turing_1(k20_turing_1,A)),k2_xcmplx_0(k7_turing_1(k20_turing_1,A),k6_turing_1(k20_turing_1,k8_turing_1(k20_turing_1,A))),k5_turing_1(k20_turing_1,k7_mcart_1(u2_turing_1(k20_turing_1),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(k20_turing_1)),A),k7_turing_1(k20_turing_1,A),k6_mcart_1(u2_turing_1(k20_turing_1),u1_turing_1(k20_turing_1),k8_domain_1(k1_numbers,k1_real_1(1),0,1),k8_turing_1(k20_turing_1,A)))) ) ) ) ), file(turing_1,l76_turing_1), []). fof(l77_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => ! [C] : ( m1_subset_1(C,u2_turing_1(A)) => ! [D] : ( m1_subset_1(D,k4_numbers) => ! [E] : ( m2_fraenkel(E,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ~ ( G = D & m1_subset_1(0,u1_turing_1(A)) & B = k4_domain_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),C,D,E) & k1_funct_1(u3_turing_1(A),k1_domain_1(u2_turing_1(A),k5_numbers,C,1)) = k4_domain_1(u2_turing_1(A),k5_numbers,k5_numbers,C,0,1) & C != u5_turing_1(A) & ! [H] : ( v1_int_1(H) => ( r1_xreal_0(G,H) => ( r1_xreal_0(k1_nat_1(G,F),H) | k1_funct_1(E,H) = 1 ) ) ) & ! [H] : ( m2_fraenkel(H,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ~ ( k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),k10_turing_1(A,B),F) = k4_domain_1(u2_turing_1(A),k5_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)),C,k1_nat_1(G,F),H) & ! [I] : ( v1_int_1(I) => ( r1_xreal_0(G,I) => ( r1_xreal_0(k1_nat_1(G,F),I) | k1_funct_1(H,I) = 0 ) ) ) & ! [I] : ( v1_int_1(I) => ( ~ ( r1_xreal_0(G,I) & ~ r1_xreal_0(k1_nat_1(G,F),I) ) => k1_funct_1(H,I) = k1_funct_1(E,I) ) ) ) ) ) ) ) ) ) ) ) ) ), file(turing_1,l77_turing_1), []). fof(rc10_comput_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) & v5_comput_1(B) ) ), file(comput_1,rc10_comput_1), []). fof(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_turing_1,axiom,( ? [A] : ( l1_turing_1(A) & v1_turing_1(A) ) ), file(turing_1,rc1_turing_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc4_comput_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v4_comput_1(A) ) ), file(comput_1,rc4_comput_1), []). fof(rc4_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). fof(rc4_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). fof(rc9_comput_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v5_comput_1(A) ) ), file(comput_1,rc9_comput_1), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => k12_mcart_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ), file(mcart_1,k12_mcart_1), []). fof(redefinition_k13_mcart_1,axiom,( ! [A,B,C,D,E,F] : ( ( m1_subset_1(D,k1_zfmisc_1(A)) & m1_subset_1(E,k1_zfmisc_1(B)) & m1_subset_1(F,k1_zfmisc_1(C)) ) => k13_mcart_1(A,B,C,D,E,F) = k3_zfmisc_1(D,E,F) ) ), file(mcart_1,k13_mcart_1), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). fof(redefinition_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => k1_fraenkel(A,B) = k1_funct_2(A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), []). fof(redefinition_k2_finseq_4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k2_finseq_4(A,B,C) = k10_finseq_1(B,C) ) ), file(finseq_4,k2_finseq_4), []). fof(redefinition_k2_turing_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k2_turing_1(A,B,C,D) = k3_cqc_lang(C,D) ) ), file(turing_1,k2_turing_1), []). fof(redefinition_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k4_domain_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & m1_subset_1(D,A) & m1_subset_1(E,B) & m1_subset_1(F,C) ) => k4_domain_1(A,B,C,D,E,F) = k3_mcart_1(D,E,F) ) ), file(domain_1,k4_domain_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k2_tarski(B,C) ) ), file(domain_1,k7_domain_1), []). fof(redefinition_k8_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) ) => k8_domain_1(A,B,C,D) = k1_enumset1(B,C,D) ) ), file(domain_1,k8_domain_1), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ! [D] : ( m2_fraenkel(D,A,B,C) <=> m1_subset_1(D,C) ) ) ), file(fraenkel,m2_fraenkel), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(spc3_boole,axiom,( ~ v1_xboole_0(3) ), file(boole,spc3_boole), []). fof(spc3_numerals,axiom, ( v2_xreal_0(3) & m1_subset_1(3,k5_numbers) ), file(numerals,spc3_numerals), []). fof(spc4_boole,axiom,( ~ v1_xboole_0(4) ), file(boole,spc4_boole), []). fof(spc4_numerals,axiom, ( v2_xreal_0(4) & m1_subset_1(4,k5_numbers) ), file(numerals,spc4_numerals), []). fof(spc6_boole,axiom,( ~ v1_xboole_0(6) ), file(boole,spc6_boole), []). fof(spc6_numerals,axiom, ( v2_xreal_0(6) & m1_subset_1(6,k5_numbers) ), file(numerals,spc6_numerals), []). fof(t12_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A))),k10_turing_1(A,B),1) = k9_turing_1(A,B) ) ) ), file(turing_1,t12_turing_1), []). fof(t13_turing_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( l1_turing_1(C) => ! [D] : ( m1_subset_1(D,k3_zfmisc_1(u2_turing_1(C),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(C)))) => k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(C),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(C))),k10_turing_1(C,D),k1_nat_1(A,B)) = k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(C),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(C))),k10_turing_1(C,k8_funct_2(k5_numbers,k3_zfmisc_1(u2_turing_1(C),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(C))),k10_turing_1(C,D),A)),B) ) ) ) ) ), file(turing_1,t13_turing_1), []). fof(t19_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m2_fraenkel(B,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_turing_1(A,B,C,k1_nat_1(k1_nat_1(C,D),2)) => r2_turing_1(k2_finseq_4(k5_numbers,C,D),A,B) ) ) ) ) ) ), file(turing_1,t19_turing_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t26_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m2_fraenkel(B,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_finseq_1(D,k5_numbers) => ( ( r1_xreal_0(3,k3_finseq_1(D)) & r2_turing_1(k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,C),D),A,B) ) => ( r1_turing_1(A,B,C,k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),2)) & r1_turing_1(A,B,k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),2),k1_nat_1(k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),k4_finseq_4(k5_numbers,k5_numbers,D,2)),4)) & r1_turing_1(A,B,k1_nat_1(k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),k4_finseq_4(k5_numbers,k5_numbers,D,2)),4),k1_nat_1(k1_nat_1(k1_nat_1(k1_nat_1(C,k4_finseq_4(k5_numbers,k5_numbers,D,1)),k4_finseq_4(k5_numbers,k5_numbers,D,2)),k4_finseq_4(k5_numbers,k5_numbers,D,3)),6)) ) ) ) ) ) ) ), file(turing_1,t26_turing_1), []). fof(t27_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m1_subset_1(B,k3_zfmisc_1(u2_turing_1(A),k4_numbers,k1_fraenkel(k4_numbers,u1_turing_1(A)))) => ! [C,D,E] : ( B = k3_mcart_1(C,D,E) => k7_turing_1(A,B) = D ) ) ) ), file(turing_1,t27_turing_1), []). fof(t28_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( m2_fraenkel(B,k4_numbers,u1_turing_1(A),k1_fraenkel(k4_numbers,u1_turing_1(A))) => ! [C] : ( v1_int_1(C) => ! [D] : ( m1_subset_1(D,u1_turing_1(A)) => ( k1_funct_1(B,C) = D => k5_turing_1(A,B,C,D) = B ) ) ) ) ) ), file(turing_1,t28_turing_1), []). fof(t29_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t2_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t2_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), []). fof(t31_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), []). fof(t37_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(A,B) => r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ), file(nat_1,t37_nat_1), []). fof(t3_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t41_turing_1,axiom, ( k1_funct_1(k19_turing_1,k1_domain_1(k5_numbers,k5_numbers,0,0)) = k4_domain_1(k5_numbers,k5_numbers,k5_numbers,1,0,1) & k1_funct_1(k19_turing_1,k1_domain_1(k5_numbers,k5_numbers,1,1)) = k4_domain_1(k5_numbers,k5_numbers,k5_numbers,1,0,1) & k1_funct_1(k19_turing_1,k1_domain_1(k5_numbers,k5_numbers,1,0)) = k4_domain_1(k5_numbers,k5_numbers,k5_numbers,2,0,1) & k1_funct_1(k19_turing_1,k1_domain_1(k5_numbers,k5_numbers,2,1)) = k4_domain_1(k5_numbers,k5_numbers,k5_numbers,2,0,1) & k1_funct_1(k19_turing_1,k1_domain_1(k5_numbers,k5_numbers,2,0)) = k4_domain_1(k5_numbers,k5_numbers,k5_numbers,3,0,0) ), file(turing_1,t41_turing_1), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t68_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => ~ ( A != k1_xboole_0 & B != k1_xboole_0 & C != k1_xboole_0 & ? [E,F,G] : ( D = k3_mcart_1(E,F,G) & ~ ( k5_mcart_1(A,B,C,D) = E & k6_mcart_1(A,B,C,D) = F & k7_mcart_1(A,B,C,D) = G ) ) ) ) ), file(mcart_1,t68_mcart_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t9_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), []).