% Mizar ND problem: t1_alg_1,alg_1,37,51 fof(dh_c1_1__alg_1,definition, ( ( ~ v1_xboole_0(c1_1__alg_1) => ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,c1_1__alg_1) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,c1_1__alg_1,A) & m2_relset_1(C,c1_1__alg_1,A) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,A,B,C)) = k4_finseq_1(B) & k3_finseq_1(k5_finseqop(c1_1__alg_1,A,B,C)) = k3_finseq_1(B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(k5_finseqop(c1_1__alg_1,A,B,C))) => k1_funct_1(k5_finseqop(c1_1__alg_1,A,B,C),D) = k1_funct_1(C,k1_funct_1(B,D)) ) ) ) ) ) ) ) => ! [E] : ( ~ v1_xboole_0(E) => ! [F] : ( ~ v1_xboole_0(F) => ! [G] : ( m2_finseq_1(G,E) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,E,F) & m2_relset_1(H,E,F) ) => ( k4_finseq_1(k5_finseqop(E,F,G,H)) = k4_finseq_1(G) & k3_finseq_1(k5_finseqop(E,F,G,H)) = k3_finseq_1(G) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( r2_hidden(I,k4_finseq_1(k5_finseqop(E,F,G,H))) => k1_funct_1(k5_finseqop(E,F,G,H),I) = k1_funct_1(H,k1_funct_1(G,I)) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_1__alg_1),file(alg_1,c1_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c1_1__alg_1)]). fof(dh_c2_1__alg_1,definition, ( ( ~ v1_xboole_0(c2_1__alg_1) => ! [A] : ( m2_finseq_1(A,c1_1__alg_1) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(B,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B)) = k4_finseq_1(A) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B)) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B),C) = k1_funct_1(B,k1_funct_1(A,C)) ) ) ) ) ) ) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( m2_finseq_1(E,c1_1__alg_1) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,c1_1__alg_1,D) & m2_relset_1(F,c1_1__alg_1,D) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,D,E,F)) = k4_finseq_1(E) & k3_finseq_1(k5_finseqop(c1_1__alg_1,D,E,F)) = k3_finseq_1(E) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r2_hidden(G,k4_finseq_1(k5_finseqop(c1_1__alg_1,D,E,F))) => k1_funct_1(k5_finseqop(c1_1__alg_1,D,E,F),G) = k1_funct_1(F,k1_funct_1(E,G)) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_1__alg_1),file(alg_1,c2_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c2_1__alg_1)]). fof(dh_c3_1__alg_1,definition, ( ( m2_finseq_1(c3_1__alg_1,c1_1__alg_1) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(A,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A)) = k3_finseq_1(c3_1__alg_1) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(B,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A),B) = k1_funct_1(A,k1_funct_1(c3_1__alg_1,B)) ) ) ) ) ) => ! [C] : ( m2_finseq_1(C,c1_1__alg_1) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(D,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,C,D)) = k4_finseq_1(C) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,C,D)) = k3_finseq_1(C) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(E,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,C,D))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,C,D),E) = k1_funct_1(D,k1_funct_1(C,E)) ) ) ) ) ) ), introduced(definition,[new_symbol(c3_1__alg_1),file(alg_1,c3_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c3_1__alg_1)]). fof(dh_c4_1__alg_1,definition, ( ( ( v1_funct_1(c4_1__alg_1) & v1_funct_2(c4_1__alg_1,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(c4_1__alg_1,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k3_finseq_1(c3_1__alg_1) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),A) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,A)) ) ) ) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(B,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,B)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,B)) = k3_finseq_1(c3_1__alg_1) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,B))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,B),C) = k1_funct_1(B,k1_funct_1(c3_1__alg_1,C)) ) ) ) ) ), introduced(definition,[new_symbol(c4_1__alg_1),file(alg_1,c4_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c4_1__alg_1)]). fof(cc1_arytm_3,theorem,( ! [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), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). fof(cc2_arytm_3,theorem,( ! [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), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_funct_2,theorem,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_partfun1(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) ) ) ) ), file(funct_2,cc1_funct_2), [interesting(0.9),axiom,file(funct_2,cc1_funct_2)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc1_relset_1,theorem,( ! [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), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_arytm_3,theorem,( ! [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), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). fof(fc6_membered,theorem, ( v1_xboole_0(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) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,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), [interesting(0.9),axiom,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), [interesting(0.9),axiom,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), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_finseq_1,definition,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). 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), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,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), [interesting(0.9),axiom,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), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc5_funct_2,theorem,( ! [A,B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc5_funct_2), [interesting(0.9),axiom,file(funct_2,cc5_funct_2)]). fof(cc6_funct_2,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & ~ v1_xboole_0(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc6_funct_2), [interesting(0.9),axiom,file(funct_2,cc6_funct_2)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(rc1_funct_2,theorem,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,A,B) ) ), file(funct_2,rc1_funct_2), [interesting(0.9),axiom,file(funct_2,rc1_funct_2)]). fof(rc2_partfun1,theorem,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) ) ), file(partfun1,rc2_partfun1), [interesting(0.9),axiom,file(partfun1,rc2_partfun1)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(redefinition_k4_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]). fof(redefinition_k4_relset_1,definition,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(redefinition_k5_finseqop,definition,( ! [A,B,C,D] : ( ( m1_finseq_1(C,A) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) ) => k5_finseqop(A,B,C,D) = k5_relat_1(C,D) ) ), file(finseqop,k5_finseqop), [interesting(0.9),axiom,file(finseqop,k5_finseqop)]). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), [interesting(0.9),axiom,file(relat_1,k1_relat_1)]). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), [interesting(0.9),axiom,file(relat_1,k2_relat_1)]). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(dt_k5_finseqop,axiom,( ! [A,B,C,D] : ( ( m1_finseq_1(C,A) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) ) => m2_finseq_1(k5_finseqop(A,B,C,D),B) ) ), file(finseqop,k5_finseqop), [interesting(0.9),axiom,file(finseqop,k5_finseqop)]). fof(dt_k5_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_1), [interesting(0.9),axiom,file(relat_1,k5_relat_1)]). fof(dt_c1_1__alg_1,assumption,( ~ v1_xboole_0(c1_1__alg_1) ), introduced(assumption,[file(alg_1,c1_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c1_1__alg_1)]). fof(dt_c2_1__alg_1,assumption,( ~ v1_xboole_0(c2_1__alg_1) ), introduced(assumption,[file(alg_1,c2_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c2_1__alg_1)]). fof(dt_c3_1__alg_1,assumption,( m2_finseq_1(c3_1__alg_1,c1_1__alg_1) ), introduced(assumption,[file(alg_1,c3_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c3_1__alg_1)]). fof(dt_c4_1__alg_1,assumption, ( v1_funct_1(c4_1__alg_1) & v1_funct_2(c4_1__alg_1,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(c4_1__alg_1,c1_1__alg_1,c2_1__alg_1) ), introduced(assumption,[file(alg_1,c4_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c4_1__alg_1)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(d4_finseq_1,definition,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( m1_finseq_1(B,A) <=> r1_tarski(k2_relat_1(B),A) ) ) ), file(finseq_1,d4_finseq_1), [interesting(0.9),axiom,file(finseq_1,d4_finseq_1)]). fof(e1_1__alg_1,plain,( r1_tarski(k2_relat_1(c3_1__alg_1),c1_1__alg_1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_1__alg_1,dt_c3_1__alg_1])],[cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,existence_m1_relset_1,dt_k1_numbers,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,cc1_relset_1,cc3_arytm_3,cc6_membered,fc2_membered,fc5_membered,rc2_partfun1,antisymmetry_r2_hidden,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k5_numbers,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc9_membered,fc6_membered,rc1_membered,t1_subset,t4_subset,t5_subset,t8_boole,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_zfmisc_1,dt_m1_subset_1,dt_m2_finseq_1,cc15_membered,t2_subset,t6_boole,t7_boole,reflexivity_r1_tarski,existence_m1_finseq_1,dt_k2_relat_1,dt_m1_finseq_1,dt_c1_1__alg_1,dt_c3_1__alg_1,t3_subset,d4_finseq_1]), [interesting(0.8),file(alg_1,e1_1__alg_1),[file(alg_1,e1_1__alg_1)]]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [interesting(0.9),axiom,file(funct_2,d1_funct_2)]). fof(t12_relset_1,theorem,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) ) ), file(relset_1,t12_relset_1), [interesting(0.9),axiom,file(relset_1,t12_relset_1)]). fof(e2_1__alg_1,plain, ( k4_relset_1(c1_1__alg_1,c2_1__alg_1,c4_1__alg_1) = c1_1__alg_1 & r1_tarski(k2_relat_1(c4_1__alg_1),c2_1__alg_1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c4_1__alg_1])],[antisymmetry_r2_hidden,cc1_funct_2,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,rc1_funct_2,rc1_membered,rc2_partfun1,t2_subset,t7_boole,t8_boole,reflexivity_r1_tarski,existence_m2_relset_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k2_relat_1,dt_k4_relset_1,dt_m2_relset_1,dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c4_1__alg_1,fc6_membered,t3_subset,t6_boole,d1_funct_2,t12_relset_1]), [interesting(0.8),file(alg_1,e2_1__alg_1),[file(alg_1,e2_1__alg_1)]]). fof(t46_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(k2_relat_1(A),k1_relat_1(B)) => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ), file(relat_1,t46_relat_1), [interesting(0.9),axiom,file(relat_1,t46_relat_1)]). fof(e4_1__alg_1,plain,( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k4_finseq_1(c3_1__alg_1) ), inference(mizar_by,[status(thm),assumptions([dt_c3_1__alg_1,dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c4_1__alg_1])],[cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_numbers,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,fc2_membered,fc5_membered,fc6_membered,rc1_membered,t1_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,cc15_membered,cc5_funct_2,cc6_funct_2,cc9_membered,rc1_funct_2,rc2_partfun1,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,redefinition_k4_finseq_1,redefinition_k4_relset_1,redefinition_k5_finseqop,dt_k1_relat_1,dt_k2_relat_1,dt_k4_finseq_1,dt_k4_relset_1,dt_k5_finseqop,dt_k5_relat_1,dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,t3_subset,e1_1__alg_1,e2_1__alg_1,t46_relat_1]), [interesting(0.8),file(alg_1,e4_1__alg_1),[file(alg_1,e4_1__alg_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), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_m2_subset_1,definition,( ! [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), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), [interesting(0.9),axiom,file(card_1,k1_card_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), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k3_finseq_1,definition,( ! [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), [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]). 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), [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]). fof(t45_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ), file(relat_1,t45_relat_1), [interesting(0.9),axiom,file(relat_1,t45_relat_1)]). fof(e3_1__alg_1,plain, ( r1_tarski(k2_relat_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)),k2_relat_1(c4_1__alg_1)) & k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k4_finseq_1(c3_1__alg_1) ), inference(mizar_by,[status(thm),assumptions([dt_c2_1__alg_1,dt_c4_1__alg_1,dt_c1_1__alg_1,dt_c3_1__alg_1])],[cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_numbers,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,fc2_membered,fc5_membered,fc6_membered,rc1_membered,t1_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,cc15_membered,cc5_funct_2,cc6_funct_2,cc9_membered,rc1_funct_2,rc2_partfun1,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,redefinition_k4_finseq_1,redefinition_k4_relset_1,redefinition_k5_finseqop,dt_k1_relat_1,dt_k2_relat_1,dt_k4_finseq_1,dt_k4_relset_1,dt_k5_finseqop,dt_k5_relat_1,dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,t3_subset,e2_1__alg_1,e1_1__alg_1,t45_relat_1,t46_relat_1]), [interesting(0.8),file(alg_1,e3_1__alg_1),[file(alg_1,e3_1__alg_1)]]). fof(t31_finseq_3,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( k3_finseq_1(A) = k3_finseq_1(B) <=> k4_finseq_1(A) = k4_finseq_1(B) ) ) ) ), file(finseq_3,t31_finseq_3), [interesting(0.9),axiom,file(finseq_3,t31_finseq_3)]). fof(e5_1__alg_1,plain,( k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k3_finseq_1(c3_1__alg_1) ), inference(mizar_by,[status(thm),assumptions([dt_c2_1__alg_1,dt_c4_1__alg_1,dt_c1_1__alg_1,dt_c3_1__alg_1])],[cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,fc5_membered,fc6_membered,rc1_membered,t1_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_relat_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,cc15_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc2_membered,rc1_funct_2,rc2_partfun1,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,redefinition_k3_finseq_1,redefinition_k4_finseq_1,redefinition_k5_finseqop,dt_k2_relat_1,dt_k3_finseq_1,dt_k4_finseq_1,dt_k5_finseqop,dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,t3_subset,e3_1__alg_1,t31_finseq_3]), [interesting(0.8),file(alg_1,e5_1__alg_1),[file(alg_1,e5_1__alg_1)]]). fof(dh_c5_1__alg_1,definition, ( ( m2_subset_1(c5_1__alg_1,k1_numbers,k5_numbers) => ( r2_hidden(c5_1__alg_1,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),c5_1__alg_1) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,c5_1__alg_1)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),A) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,A)) ) ) ), introduced(definition,[new_symbol(c5_1__alg_1),file(alg_1,c5_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c5_1__alg_1)]). fof(e6_1__alg_1,assumption,( r2_hidden(c5_1__alg_1,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) ), introduced(assumption,[file(alg_1,e6_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,e6_1__alg_1)]). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(dt_c5_1__alg_1,assumption,( m2_subset_1(c5_1__alg_1,k1_numbers,k5_numbers) ), introduced(assumption,[file(alg_1,c5_1__alg_1)]), [interesting(0.8),axiom,file(alg_1,c5_1__alg_1)]). fof(t22_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B))) => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ), file(funct_1,t22_funct_1), [interesting(0.9),axiom,file(funct_1,t22_funct_1)]). fof(e7_1__alg_1,plain,( k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),c5_1__alg_1) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,c5_1__alg_1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,dt_c5_1__alg_1,e6_1__alg_1])],[cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,fc5_membered,fc6_membered,rc1_membered,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,cc15_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc2_membered,rc1_funct_2,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k4_finseq_1,redefinition_k5_finseqop,dt_k1_funct_1,dt_k1_relat_1,dt_k4_finseq_1,dt_k5_finseqop,dt_k5_relat_1,dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,dt_c5_1__alg_1,t1_subset,t7_boole,e6_1__alg_1,t22_funct_1]), [interesting(0.8),file(alg_1,e7_1__alg_1),[file(alg_1,e7_1__alg_1)]]). fof(i8_1__alg_1,theorem,( $true ), introduced(tautology,[file(alg_1,i8_1__alg_1)]), [interesting(0.8),trivial,file(alg_1,i8_1__alg_1)]). fof(i7_1__alg_1,plain,( k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),c5_1__alg_1) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,c5_1__alg_1)) ), inference(conclusion,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,dt_c5_1__alg_1,e6_1__alg_1])],[e7_1__alg_1,i8_1__alg_1]), [interesting(0.8),file(alg_1,i7_1__alg_1),[file(alg_1,i7_1__alg_1)]]). fof(i6_1__alg_1,plain, ( r2_hidden(c5_1__alg_1,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),c5_1__alg_1) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,c5_1__alg_1)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1,dt_c5_1__alg_1]),discharge_asm(discharge,[e6_1__alg_1])],[e6_1__alg_1,i7_1__alg_1]), [interesting(0.8),file(alg_1,i6_1__alg_1),[file(alg_1,i6_1__alg_1)]]). fof(i6_1_tmp__alg_1,plain, ( m2_subset_1(c5_1__alg_1,k1_numbers,k5_numbers) => ( r2_hidden(c5_1__alg_1,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),c5_1__alg_1) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,c5_1__alg_1)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1]),discharge_asm(discharge,[dt_c5_1__alg_1])],[dt_c5_1__alg_1,i6_1__alg_1]), [interesting(0.8),i5_1__alg_1]). fof(i5_1__alg_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),A) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,A)) ) ) ), inference(let,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1])],[i6_1_tmp__alg_1,dh_c5_1__alg_1]), [interesting(0.8),file(alg_1,i5_1__alg_1),[file(alg_1,i5_1__alg_1)]]). fof(i4_1__alg_1,plain, ( k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k3_finseq_1(c3_1__alg_1) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),A) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,A)) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1])],[e5_1__alg_1,i5_1__alg_1]), [interesting(0.8),file(alg_1,i4_1__alg_1),[file(alg_1,i4_1__alg_1)]]). fof(i3_1__alg_1,plain, ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k3_finseq_1(c3_1__alg_1) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),A) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,A)) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1,dt_c4_1__alg_1])],[e4_1__alg_1,i4_1__alg_1]), [interesting(0.8),file(alg_1,i3_1__alg_1),[file(alg_1,i3_1__alg_1)]]). fof(i3_1_tmp__alg_1,plain, ( ( v1_funct_1(c4_1__alg_1) & v1_funct_2(c4_1__alg_1,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(c4_1__alg_1,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1)) = k3_finseq_1(c3_1__alg_1) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,c4_1__alg_1),A) = k1_funct_1(c4_1__alg_1,k1_funct_1(c3_1__alg_1,A)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1]),discharge_asm(discharge,[dt_c4_1__alg_1])],[dt_c4_1__alg_1,i3_1__alg_1]), [interesting(0.8),i2_1__alg_1]). fof(i2_1__alg_1,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(A,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A)) = k3_finseq_1(c3_1__alg_1) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(B,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A),B) = k1_funct_1(A,k1_funct_1(c3_1__alg_1,B)) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1,dt_c3_1__alg_1])],[i3_1_tmp__alg_1,dh_c4_1__alg_1]), [interesting(0.8),file(alg_1,i2_1__alg_1),[file(alg_1,i2_1__alg_1)]]). fof(i2_1_tmp__alg_1,plain, ( m2_finseq_1(c3_1__alg_1,c1_1__alg_1) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(A,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A)) = k4_finseq_1(c3_1__alg_1) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A)) = k3_finseq_1(c3_1__alg_1) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(B,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,c3_1__alg_1,A),B) = k1_funct_1(A,k1_funct_1(c3_1__alg_1,B)) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1]),discharge_asm(discharge,[dt_c3_1__alg_1])],[dt_c3_1__alg_1,i2_1__alg_1]), [interesting(0.8),i1_1__alg_1]). fof(i1_1__alg_1,plain,( ! [A] : ( m2_finseq_1(A,c1_1__alg_1) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(B,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B)) = k4_finseq_1(A) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B)) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B),C) = k1_funct_1(B,k1_funct_1(A,C)) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_1__alg_1,dt_c2_1__alg_1])],[i2_1_tmp__alg_1,dh_c3_1__alg_1]), [interesting(0.8),file(alg_1,i1_1__alg_1),[file(alg_1,i1_1__alg_1)]]). fof(i1_1_tmp__alg_1,plain, ( ( ~ v1_xboole_0(c1_1__alg_1) & ~ v1_xboole_0(c2_1__alg_1) ) => ! [A] : ( m2_finseq_1(A,c1_1__alg_1) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,c1_1__alg_1,c2_1__alg_1) & m2_relset_1(B,c1_1__alg_1,c2_1__alg_1) ) => ( k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B)) = k4_finseq_1(A) & k3_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B)) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B))) => k1_funct_1(k5_finseqop(c1_1__alg_1,c2_1__alg_1,A,B),C) = k1_funct_1(B,k1_funct_1(A,C)) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_1__alg_1,dt_c2_1__alg_1])],[dt_c1_1__alg_1,dt_c2_1__alg_1,i1_1__alg_1]), [interesting(1),t1_alg_1]). fof(t1_alg_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_finseq_1(C,A) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( k4_finseq_1(k5_finseqop(A,B,C,D)) = k4_finseq_1(C) & k3_finseq_1(k5_finseqop(A,B,C,D)) = k3_finseq_1(C) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(E,k4_finseq_1(k5_finseqop(A,B,C,D))) => k1_funct_1(k5_finseqop(A,B,C,D),E) = k1_funct_1(D,k1_funct_1(C,E)) ) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_1_tmp__alg_1,dh_c1_1__alg_1,dh_c2_1__alg_1]), [interesting(1),file(alg_1,t1_alg_1),[file(alg_1,t1_alg_1)]]).