% Mizar problem: t35_hilbert2,hilbert2,1759,52 fof(t35_hilbert2,conjecture,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ! [B] : ( m1_trees_1(B,k1_relat_1(k3_hilbert2(A))) => k7_trees_2(k1_hilbert1,k3_hilbert2(A),B) = k3_hilbert2(k3_trees_2(k1_hilbert1,k3_hilbert2(A),B)) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc10_trees_3,cc1_finseq_1,cc1_fraenkel,cc1_funct_7,cc1_hilbert1,cc1_nat_1,cc1_relset_1,cc2_funct_7,cc2_nat_1,cc3_hilbert1,cc3_nat_1,cc4_trees_3,cc5_trees_3,cc7_trees_3,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,d10_hilbert2,d11_trees_2,d1_hilbert2,d1_tarski,d6_hilbert2,d7_hilbert1,d7_hilbert2,d8_hilbert1,d8_hilbert2,d9_hilbert1,d9_trees_1,dt_k10_trees_4,dt_k12_finseq_1,dt_k15_trees_3,dt_k1_funct_1,dt_k1_hilbert1,dt_k1_hilbert2,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_trees_4,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_hilbert1,dt_k2_hilbert2,dt_k2_trees_1,dt_k2_trees_4,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_hilbert1,dt_k3_hilbert2,dt_k3_trees_2,dt_k4_hilbert1,dt_k4_trees_1,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_trees_2,dt_k5_trees_3,dt_k6_finseq_1,dt_k6_trees_4,dt_k7_finseq_1,dt_k7_trees_2,dt_k8_finseq_1,dt_k9_finseq_1,dt_m1_finseq_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m1_trees_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,dt_m3_trees_2,dt_m4_trees_3,dt_m5_trees_3,existence_m1_finseq_1,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m1_trees_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,existence_m3_trees_2,existence_m4_trees_3,existence_m5_trees_3,fc10_funct_7,fc10_trees_3,fc12_finseq_1,fc13_finseq_1,fc13_trees_3,fc14_finseq_1,fc16_trees_3,fc17_finseq_1,fc1_hilbert1,fc1_hilbert2,fc1_nat_1,fc1_ordinal2,fc1_subset_1,fc25_trees_3,fc26_trees_3,fc2_finseq_1,fc2_hilbert1,fc2_subset_1,fc2_trees_2,fc3_finseq_1,fc3_funct_7,fc3_nat_1,fc4_finseq_1,fc4_funct_7,fc4_hilbert2,fc4_nat_1,fc4_prvect_1,fc4_subset_1,fc5_trees_2,fc6_trees_3,fc7_trees_3,fc8_funct_7,fc8_trees_3,fc9_trees_3,rc1_finseq_1,rc1_fraenkel,rc1_hilbert1,rc1_nat_1,rc1_subset_1,rc2_finseq_1,rc2_nat_1,rc2_subset_1,rc2_trees_3,rc3_finseq_1,rc3_nat_1,rc4_finseq_1,rc4_trees_3,rc5_trees_2,rc5_trees_3,rc6_finseq_1,rc6_trees_2,rc6_trees_3,rc7_finseq_1,rc7_trees_2,rc7_trees_3,rc8_finseq_1,rc8_trees_2,rc8_trees_3,rc9_trees_2,redefinition_k10_trees_4,redefinition_k12_finseq_1,redefinition_k1_nat_1,redefinition_k2_trees_4,redefinition_k3_trees_2,redefinition_k5_numbers,redefinition_k7_trees_2,redefinition_k8_finseq_1,redefinition_k9_finseq_1,redefinition_m1_trees_1,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_m5_trees_3,reflexivity_r1_tarski,s1_hilbert2__e4_46__hilbert2,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,t14_trees_4,t1_arithm,t1_numerals,t1_subset,t1_trees_9,t2_subset,t30_hilbert2,t31_hilbert2,t32_hilbert2,t33_hilbert2,t34_hilbert2,t3_subset,t3_trees_4,t3_trees_9,t4_subset,t56_trees_1,t5_hilbert2,t5_subset,t6_boole,t7_boole,t7_hilbert2,t8_boole,t8_hilbert2,t9_hilbert2]), [file(hilbert2,t35_hilbert2)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_trees_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v3_trees_3(A) ) => ! [B] : ( m1_finseq_1(B,A) => v6_trees_3(B) ) ) ), file(trees_3,cc10_trees_3), []). fof(cc1_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_fraenkel,axiom,( ! [A] : ( v1_fraenkel(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ) ), file(fraenkel,cc1_fraenkel), []). fof(cc1_funct_7,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_funct_7(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_funcop_1(A) ) ) ), file(funct_7,cc1_funct_7), []). fof(cc1_hilbert1,axiom,( ! [A] : ( v5_hilbert1(A) => ( ~ v1_xboole_0(A) & v1_hilbert1(A) & v2_hilbert1(A) & v3_hilbert1(A) & v4_hilbert1(A) ) ) ), file(hilbert1,cc1_hilbert1), []). fof(cc1_nat_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) ) ) ), file(nat_1,cc1_nat_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_funct_7,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_funcop_1(A) & v1_funct_7(A) ) ) ), file(funct_7,cc2_funct_7), []). fof(cc2_nat_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) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc3_hilbert1,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ) ), file(hilbert1,cc3_hilbert1), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc4_trees_3,axiom,( ! [A] : ( v3_trees_3(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v3_trees_3(B) ) ) ), file(trees_3,cc4_trees_3), []). fof(cc5_trees_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m4_trees_3(B,A) => v3_trees_3(B) ) ) ), file(trees_3,cc5_trees_3), []). fof(cc7_trees_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_trees_3(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v4_trees_3(A) ) ) ), file(trees_3,cc7_trees_3), []). 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_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(d10_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => k3_hilbert2(A) = k1_funct_1(k2_hilbert2,A) ) ), file(hilbert2,d10_hilbert2), []). fof(d11_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( C = k5_trees_2(A,B) <=> ( k1_relat_1(C) = k4_trees_1(k1_relat_1(A),B) & ! [D] : ( m2_finseq_1(D,k5_numbers) => ( r2_hidden(D,k4_trees_1(k1_relat_1(A),B)) => k1_funct_1(C,D) = k1_funct_1(A,k8_finseq_1(k5_numbers,B,D)) ) ) ) ) ) ) ) ), file(trees_2,d11_trees_2), []). fof(d1_hilbert2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_hilbert2(A) = k12_finseq_1(k5_numbers,k1_nat_1(3,A)) ) ), file(hilbert2,d1_hilbert2), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d6_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ( v1_hilbert2(A) <=> ? [B] : ( m1_subset_1(B,k1_hilbert1) & ? [C] : ( m1_subset_1(C,k1_hilbert1) & A = k4_hilbert1(B,C) ) ) ) ) ), file(hilbert2,d6_hilbert2), []). fof(d7_hilbert1,axiom,( k2_hilbert1 = k12_finseq_1(k5_numbers,0) ), file(hilbert1,d7_hilbert1), []). fof(d7_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ( v2_hilbert2(A) <=> ? [B] : ( m1_subset_1(B,k1_hilbert1) & ? [C] : ( m1_subset_1(C,k1_hilbert1) & A = k3_hilbert1(B,C) ) ) ) ) ), file(hilbert2,d7_hilbert2), []). fof(d8_hilbert1,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ! [B] : ( m1_subset_1(B,k1_hilbert1) => k3_hilbert1(A,B) = k7_finseq_1(k7_finseq_1(k12_finseq_1(k5_numbers,1),A),B) ) ) ), file(hilbert1,d8_hilbert1), []). fof(d8_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ( v3_hilbert2(A) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k1_hilbert2(B) ) ) ) ), file(hilbert2,d8_hilbert2), []). fof(d9_hilbert1,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ! [B] : ( m1_subset_1(B,k1_hilbert1) => k4_hilbert1(A,B) = k7_finseq_1(k7_finseq_1(k12_finseq_1(k5_numbers,2),A),B) ) ) ), file(hilbert1,d9_hilbert1), []). fof(d9_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ( r2_hidden(B,A) => ! [C] : ( ( ~ v1_xboole_0(C) & v1_trees_1(C) ) => ( C = k4_trees_1(A,B) <=> ! [D] : ( m2_finseq_1(D,k5_numbers) => ( r2_hidden(D,C) <=> r2_hidden(k8_finseq_1(k5_numbers,B,D),A) ) ) ) ) ) ) ) ), file(trees_1,d9_trees_1), []). fof(dt_k10_trees_4,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & v1_funct_1(C) & v3_trees_2(C) & m3_trees_2(C,A) & v1_funct_1(D) & v3_trees_2(D) & m3_trees_2(D,A) ) => ( v1_funct_1(k10_trees_4(A,B,C,D)) & v3_trees_2(k10_trees_4(A,B,C,D)) & m3_trees_2(k10_trees_4(A,B,C,D),A) ) ) ), file(trees_4,k10_trees_4), []). 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_k15_trees_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & ~ v1_xboole_0(B) & v1_trees_1(B) ) => ( ~ v1_xboole_0(k15_trees_3(A,B)) & v1_trees_1(k15_trees_3(A,B)) ) ) ), file(trees_3,k15_trees_3), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_hilbert1,axiom,( $true ), file(hilbert1,k1_hilbert1), []). fof(dt_k1_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k1_hilbert2(A),k1_hilbert1) ) ), file(hilbert2,k1_hilbert2), []). 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_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_trees_4,axiom,( ! [A] : ( v1_relat_1(k1_trees_4(A)) & v1_funct_1(k1_trees_4(A)) & v3_trees_2(k1_trees_4(A)) ) ), file(trees_4,k1_trees_4), []). 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_k2_hilbert1,axiom,( m1_subset_1(k2_hilbert1,k1_hilbert1) ), file(hilbert1,k2_hilbert1), []). fof(dt_k2_hilbert2,axiom,( m1_pboole(k2_hilbert2,k1_hilbert1) ), file(hilbert2,k2_hilbert2), []). fof(dt_k2_trees_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k2_trees_1(A)) & v1_finset_1(k2_trees_1(A)) & v1_trees_1(k2_trees_1(A)) ) ) ), file(trees_1,k2_trees_1), []). fof(dt_k2_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m5_trees_3(k2_trees_4(A,B),A,k5_trees_3(A)) ) ), file(trees_4,k2_trees_4), []). 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_hilbert1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_hilbert1) & m1_subset_1(B,k1_hilbert1) ) => m1_subset_1(k3_hilbert1(A,B),k1_hilbert1) ) ), file(hilbert1,k3_hilbert1), []). fof(dt_k3_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ( v1_funct_1(k3_hilbert2(A)) & v3_trees_2(k3_hilbert2(A)) & m3_trees_2(k3_hilbert2(A),k1_hilbert1) ) ) ), file(hilbert2,k3_hilbert2), []). fof(dt_k3_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => m1_subset_1(k3_trees_2(A,B,C),A) ) ), file(trees_2,k3_trees_2), []). fof(dt_k4_hilbert1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_hilbert1) & m1_subset_1(B,k1_hilbert1) ) => m1_subset_1(k4_hilbert1(A,B),k1_hilbert1) ) ), file(hilbert1,k4_hilbert1), []). fof(dt_k4_trees_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & m1_finseq_1(B,k5_numbers) ) => ( ~ v1_xboole_0(k4_trees_1(A,B)) & v1_trees_1(k4_trees_1(A,B)) ) ) ), file(trees_1,k4_trees_1), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_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_trees_2,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) & m1_finseq_1(B,k5_numbers) ) => ( v1_relat_1(k5_trees_2(A,B)) & v1_funct_1(k5_trees_2(A,B)) & v3_trees_2(k5_trees_2(A,B)) ) ) ), file(trees_2,k5_trees_2), []). fof(dt_k5_trees_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => m4_trees_3(k5_trees_3(A),A) ) ), file(trees_3,k5_trees_3), []). fof(dt_k6_finseq_1,axiom,( ! [A] : ( v1_xboole_0(k6_finseq_1(A)) & m2_finseq_1(k6_finseq_1(A),A) ) ), file(finseq_1,k6_finseq_1), []). fof(dt_k6_trees_4,axiom,( ! [A,B,C] : ( ( v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) & v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( v1_relat_1(k6_trees_4(A,B,C)) & v1_funct_1(k6_trees_4(A,B,C)) & v3_trees_2(k6_trees_4(A,B,C)) ) ) ), file(trees_4,k6_trees_4), []). 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_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => ( v1_funct_1(k7_trees_2(A,B,C)) & v3_trees_2(k7_trees_2(A,B,C)) & m3_trees_2(k7_trees_2(A,B,C),A) ) ) ), file(trees_2,k7_trees_2), []). 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_k9_finseq_1,axiom,( ! [A] : ( v1_relat_1(k9_finseq_1(A)) & v1_funct_1(k9_finseq_1(A)) ) ), file(finseq_1,k9_finseq_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_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ), file(pboole,m1_pboole), []). 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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) => m2_finseq_1(B,k5_numbers) ) ) ), file(trees_1,m1_trees_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_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_m3_trees_2,axiom,( ! [A,B] : ( m3_trees_2(B,A) => v1_relat_1(B) ) ), file(trees_2,m3_trees_2), []). fof(dt_m4_trees_3,axiom,( $true ), file(trees_3,m4_trees_3), []). fof(dt_m5_trees_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m4_trees_3(B,A) ) => ! [C] : ( m5_trees_3(C,A,B) => ( v1_funct_1(C) & v3_trees_2(C) & m3_trees_2(C,A) ) ) ) ), file(trees_3,m5_trees_3), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_pboole,axiom,( ! [A] : ? [B] : m1_pboole(B,A) ), file(pboole,m1_pboole), []). 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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : m1_trees_1(B,A) ) ), file(trees_1,m1_trees_1), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). 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(existence_m3_trees_2,axiom,( ! [A] : ? [B] : m3_trees_2(B,A) ), file(trees_2,m3_trees_2), []). fof(existence_m4_trees_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : m4_trees_3(B,A) ) ), file(trees_3,m4_trees_3), []). fof(existence_m5_trees_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m4_trees_3(B,A) ) => ? [C] : m5_trees_3(C,A,B) ) ), file(trees_3,m5_trees_3), []). fof(fc10_funct_7,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) & v1_funcop_1(k5_finseq_1(A)) & v1_funct_7(k5_finseq_1(A)) ) ) ), file(funct_7,fc10_funct_7), []). fof(fc10_trees_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & ~ v1_xboole_0(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) & v4_trees_3(k5_finseq_1(A)) ) ) ), file(trees_3,fc10_trees_3), []). fof(fc12_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc12_finseq_1), []). fof(fc13_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(A,B)) & v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,fc13_finseq_1), []). fof(fc13_trees_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) ) => ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & ~ v1_xboole_0(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) & v4_trees_3(k5_finseq_1(A)) & v5_trees_3(k5_finseq_1(A)) ) ) ), file(trees_3,fc13_trees_3), []). fof(fc14_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(B,A)) & v1_relat_1(k7_finseq_1(B,A)) & v1_funct_1(k7_finseq_1(B,A)) & v1_finset_1(k7_finseq_1(B,A)) & v1_finseq_1(k7_finseq_1(B,A)) ) ) ), file(finseq_1,fc14_finseq_1), []). fof(fc16_trees_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & ~ v1_xboole_0(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) & v6_trees_3(k5_finseq_1(A)) ) ) ), file(trees_3,fc16_trees_3), []). fof(fc17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_1), []). fof(fc1_hilbert1,axiom, ( ~ v1_xboole_0(k1_hilbert1) & v1_hilbert1(k1_hilbert1) & v2_hilbert1(k1_hilbert1) & v3_hilbert1(k1_hilbert1) & v4_hilbert1(k1_hilbert1) & v5_hilbert1(k1_hilbert1) ), file(hilbert1,fc1_hilbert1), []). fof(fc1_hilbert2,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) & v4_trees_3(k1_xboole_0) ), file(hilbert2,fc1_hilbert2), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). 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(fc25_trees_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) & v1_trees_1(B) ) => ( ~ v1_xboole_0(k15_trees_3(A,B)) & v1_finset_1(k15_trees_3(A,B)) & v1_trees_1(k15_trees_3(A,B)) ) ) ), file(trees_3,fc25_trees_3), []). fof(fc26_trees_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_trees_3(A)) ) => ( ~ v1_xboole_0(k1_relat_1(B)) & v1_finset_1(k1_relat_1(B)) & v1_trees_1(k1_relat_1(B)) ) ) ), file(trees_3,fc26_trees_3), []). fof(fc2_finseq_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) ), file(finseq_1,fc2_finseq_1), []). fof(fc2_hilbert1,axiom, ( ~ v1_xboole_0(k1_hilbert1) & v1_fraenkel(k1_hilbert1) & v1_hilbert1(k1_hilbert1) & v2_hilbert1(k1_hilbert1) & v3_hilbert1(k1_hilbert1) & v4_hilbert1(k1_hilbert1) & v5_hilbert1(k1_hilbert1) ), file(hilbert1,fc2_hilbert1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ( ~ v1_xboole_0(k1_relat_1(A)) & v1_trees_1(k1_relat_1(A)) ) ) ), file(trees_2,fc2_trees_2), []). fof(fc3_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) ) ), file(finseq_1,fc3_finseq_1), []). fof(fc3_funct_7,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) ), file(funct_7,fc3_funct_7), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc4_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc4_finseq_1), []). fof(fc4_funct_7,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) & v1_funcop_1(k5_finseq_1(A)) ) ) ), file(funct_7,fc4_funct_7), []). fof(fc4_hilbert2,axiom,( ! [A,B,C] : ( ( v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) & v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( v1_relat_1(k6_trees_4(A,B,C)) & v1_funct_1(k6_trees_4(A,B,C)) & v3_trees_2(k6_trees_4(A,B,C)) & ~ v1_trees_9(k6_trees_4(A,B,C)) ) ) ), file(hilbert2,fc4_hilbert2), []). fof(fc4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc4_prvect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) ) => ~ v1_xboole_0(k1_relat_1(A)) ) ), file(prvect_1,fc4_prvect_1), []). 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_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(trees_2,fc5_trees_2), []). fof(fc6_trees_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ( ~ v1_xboole_0(k5_trees_3(A)) & v3_trees_3(k5_trees_3(A)) ) ) ), file(trees_3,fc6_trees_3), []). fof(fc7_trees_3,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v4_trees_3(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) & v4_trees_3(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) & v4_trees_3(k7_finseq_1(A,B)) ) ) ), file(trees_3,fc7_trees_3), []). fof(fc8_funct_7,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_funcop_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) & v1_funcop_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) & v1_funcop_1(k7_finseq_1(A,B)) ) ) ), file(funct_7,fc8_funct_7), []). fof(fc8_trees_3,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v5_trees_3(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) & v5_trees_3(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) & v4_trees_3(k7_finseq_1(A,B)) & v5_trees_3(k7_finseq_1(A,B)) ) ) ), file(trees_3,fc8_trees_3), []). fof(fc9_trees_3,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v6_trees_3(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) & v6_trees_3(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) & v6_trees_3(k7_finseq_1(A,B)) ) ) ), file(trees_3,fc9_trees_3), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_fraenkel,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_fraenkel(A) ) ), file(fraenkel,rc1_fraenkel), []). fof(rc1_hilbert1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_hilbert1(A) & v2_hilbert1(A) & v3_hilbert1(A) & v4_hilbert1(A) & v5_hilbert1(A) ) ), file(hilbert1,rc1_hilbert1), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_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(rc2_finseq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_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_trees_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v3_trees_3(A) ) ), file(trees_3,rc2_trees_3), []). fof(rc3_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc4_finseq_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), []). fof(rc4_trees_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m4_trees_3(B,A) & ~ v1_xboole_0(B) & v1_finset_1(B) & v3_trees_3(B) ) ) ), file(trees_3,rc4_trees_3), []). fof(rc5_trees_2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) ), file(trees_2,rc5_trees_2), []). fof(rc5_trees_3,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v1_finset_1(A) & v1_finseq_1(A) & v4_trees_3(A) & v5_trees_3(A) ) ), file(trees_3,rc5_trees_3), []). fof(rc6_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(rc6_trees_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) ) ) ), file(trees_2,rc6_trees_2), []). fof(rc6_trees_3,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v1_finset_1(A) & v1_finseq_1(A) & v6_trees_3(A) ) ), file(trees_3,rc6_trees_3), []). fof(rc7_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(rc7_trees_2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v3_trees_2(A) ) ), file(trees_2,rc7_trees_2), []). fof(rc7_trees_3,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v4_trees_3(A) & v5_trees_3(A) ) ), file(trees_3,rc7_trees_3), []). fof(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(rc8_trees_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v3_trees_2(B) ) ) ), file(trees_2,rc8_trees_2), []). fof(rc8_trees_3,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v6_trees_3(A) ) ), file(trees_3,rc8_trees_3), []). fof(rc9_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & ~ v1_xboole_0(C) ) ) ), file(trees_2,rc9_trees_2), []). fof(redefinition_k10_trees_4,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & v1_funct_1(C) & v3_trees_2(C) & m3_trees_2(C,A) & v1_funct_1(D) & v3_trees_2(D) & m3_trees_2(D,A) ) => k10_trees_4(A,B,C,D) = k6_trees_4(B,C,D) ) ), file(trees_4,k10_trees_4), []). 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_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_k2_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k2_trees_4(A,B) = k1_trees_4(B) ) ), file(trees_4,k2_trees_4), []). fof(redefinition_k3_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => k3_trees_2(A,B,C) = k1_funct_1(B,C) ) ), file(trees_2,k3_trees_2), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k7_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => k7_trees_2(A,B,C) = k5_trees_2(B,C) ) ), file(trees_2,k7_trees_2), []). 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_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) <=> m1_subset_1(B,A) ) ) ), file(trees_1,m1_trees_1), []). 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_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(redefinition_m5_trees_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m4_trees_3(B,A) ) => ! [C] : ( m5_trees_3(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(trees_3,m5_trees_3), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(s1_hilbert2__e4_46__hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ( ( ? [B] : ( m1_subset_1(B,k1_hilbert1) & B = k1_funct_1(k3_hilbert2(A),k6_finseq_1(k5_numbers)) & k5_trees_2(k3_hilbert2(A),k6_finseq_1(k5_numbers)) = k3_hilbert2(B) ) & ! [C] : ( m1_trees_1(C,k1_relat_1(k3_hilbert2(A))) => ( ? [D] : ( m1_subset_1(D,k1_hilbert1) & D = k1_funct_1(k3_hilbert2(A),C) & k5_trees_2(k3_hilbert2(A),C) = k3_hilbert2(D) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,E)),k1_relat_1(k3_hilbert2(A))) => ? [F] : ( m1_subset_1(F,k1_hilbert1) & F = k1_funct_1(k3_hilbert2(A),k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,E))) & k5_trees_2(k3_hilbert2(A),k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,E))) = k3_hilbert2(F) ) ) ) ) ) ) => ! [C] : ( m1_trees_1(C,k1_relat_1(k3_hilbert2(A))) => ? [G] : ( m1_subset_1(G,k1_hilbert1) & G = k1_funct_1(k3_hilbert2(A),C) & k5_trees_2(k3_hilbert2(A),C) = k3_hilbert2(G) ) ) ) ) ), file(hilbert2,s1_hilbert2__e4_46__hilbert2), []). 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(t14_trees_4,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => k1_relat_1(k6_trees_4(A,B,C)) = k15_trees_3(k1_relat_1(B),k1_relat_1(C)) ) ) ), file(trees_4,t14_trees_4), []). 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_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_trees_9,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => k5_trees_2(A,k6_finseq_1(k5_numbers)) = A ) ), file(trees_9,t1_trees_9), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t30_hilbert2,axiom,( k3_hilbert2(k2_hilbert1) = k2_trees_4(k1_hilbert1,k2_hilbert1) ), file(hilbert2,t30_hilbert2), []). fof(t31_hilbert2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k3_hilbert2(k1_hilbert2(A)) = k2_trees_4(k1_hilbert1,k1_hilbert2(A)) ) ), file(hilbert2,t31_hilbert2), []). fof(t32_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ! [B] : ( m1_subset_1(B,k1_hilbert1) => k3_hilbert2(k4_hilbert1(A,B)) = k10_trees_4(k1_hilbert1,k4_hilbert1(A,B),k3_hilbert2(A),k3_hilbert2(B)) ) ) ), file(hilbert2,t32_hilbert2), []). fof(t33_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ! [B] : ( m1_subset_1(B,k1_hilbert1) => k3_hilbert2(k3_hilbert1(A,B)) = k10_trees_4(k1_hilbert1,k3_hilbert1(A,B),k3_hilbert2(A),k3_hilbert2(B)) ) ) ), file(hilbert2,t33_hilbert2), []). fof(t34_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => k1_funct_1(k3_hilbert2(A),k1_xboole_0) = A ) ), file(hilbert2,t34_hilbert2), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_trees_4,axiom,( ! [A] : ( k1_relat_1(k1_trees_4(A)) = k2_trees_1(0) & k1_funct_1(k1_trees_4(A),k1_xboole_0) = A ) ), file(trees_4,t3_trees_4), []). fof(t3_trees_9,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ! [C] : ( m2_finseq_1(C,k5_numbers) => ( r2_hidden(k8_finseq_1(k5_numbers,B,C),k1_relat_1(A)) => k5_trees_2(A,k8_finseq_1(k5_numbers,B,C)) = k5_trees_2(k5_trees_2(A,B),C) ) ) ) ) ), file(trees_9,t3_trees_9), []). 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(t56_trees_1,axiom,( k2_trees_1(0) = k1_tarski(k1_xboole_0) ), file(trees_1,t56_trees_1), []). fof(t5_hilbert2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) ) => ! [C] : ( ( ~ v1_xboole_0(C) & v1_trees_1(C) ) => ( r2_hidden(k9_finseq_1(A),k15_trees_3(B,C)) <=> ( A = 0 | A = 1 ) ) ) ) ), file(hilbert2,t5_hilbert2), []). 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(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_hilbert2,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( k1_funct_1(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,0)) = k1_funct_1(B,k1_xboole_0) & k1_funct_1(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,1)) = k1_funct_1(C,k1_xboole_0) ) ) ) ), file(hilbert2,t7_hilbert2), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_hilbert2,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v3_trees_2(C) ) => ( k5_trees_2(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,0)) = B & k5_trees_2(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,1)) = C ) ) ) ), file(hilbert2,t8_hilbert2), []). fof(t9_hilbert2,axiom,( ! [A] : ( m1_subset_1(A,k1_hilbert1) => ~ ( ~ v1_hilbert2(A) & ~ v2_hilbert2(A) & ~ v3_hilbert2(A) & A != k2_hilbert1 ) ) ), file(hilbert2,t9_hilbert2), []).