SET007 Axioms: SET007+429.ax
%------------------------------------------------------------------------------
% File : SET007+429 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to Circuits, II
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : circuit2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 39 ( 0 unt; 0 def)
% Number of atoms : 557 ( 55 equ)
% Maximal formula atoms : 27 ( 14 avg)
% Number of connectives : 598 ( 80 ~; 6 |; 313 &)
% ( 6 <=>; 193 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 14 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-4 aty)
% Number of functors : 54 ( 54 usr; 4 con; 0-5 aty)
% Number of variables : 193 ( 192 !; 1 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m3_pboole(C,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,k11_msafree(A,B)))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_funcop_1(D) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F)
& v1_finseq_1(F)
& v6_trees_3(F) )
=> ! [G] :
( m1_subset_1(G,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),E))
=> ~ ( r2_hidden(E,k4_msafree2(A))
& G = k4_trees_4(k4_tarski(k5_msafree2(A,E),u1_struct_0(A)),F)
& r1_msualg_3(A,k11_msafree(A,B),k11_msafree(A,B),C)
& D = k5_relat_1(k1_msualg_1(A,k5_msafree2(A,E)),C)
& ! [H] :
( ( v1_relat_1(H)
& v1_funct_1(H)
& v1_finseq_1(H)
& v6_trees_3(H) )
=> ~ ( H = k15_pralg_1(D,F)
& k8_funct_2(k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),E),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),E),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,k11_msafree(A,B)),C,E),G) = k4_trees_4(k4_tarski(k5_msafree2(A,E),u1_struct_0(A)),H) ) ) ) ) ) ) ) ) ) ) ).
fof(d1_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),k13_msafree(A,u4_msualg_1(A,B)),u4_msualg_1(A,k6_msafree2(A,B)))
=> ( D = k2_circuit2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(E,k2_msafree2(A))
=> k1_msualg_3(u1_struct_0(A),k13_msafree(A,u4_msualg_1(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),D,E) = k2_pre_circ(k12_msafree(A,u4_msualg_1(A,B),E),k1_trees_4(k4_tarski(k1_funct_1(C,E),E))) )
& ( r2_hidden(E,k1_msafree2(A))
=> k1_msualg_3(u1_struct_0(A),k13_msafree(A,u4_msualg_1(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),D,E) = k2_pre_circ(k12_msafree(A,u4_msualg_1(A,B),E),k1_trees_4(k4_tarski(k5_msafree2(A,E),u1_struct_0(A)))) )
& ( r2_hidden(E,k6_subset_1(u1_struct_0(A),k4_msafree2(A),k1_msafree2(A)))
=> k1_msualg_3(u1_struct_0(A),k13_msafree(A,u4_msualg_1(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),D,E) = k6_relat_1(k12_msafree(A,u4_msualg_1(A,B),E)) ) ) ) ) ) ) ) ) ).
fof(d2_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)))
=> ( D = k3_circuit2(A,B,C)
<=> ( r1_msualg_3(A,k6_msafree2(A,B),k6_msafree2(A,B),D)
& r2_pboole(u1_struct_0(A),k2_circuit2(A,B,C),D) ) ) ) ) ) ) ).
fof(t2_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ! [F] :
( ( r2_hidden(D,k6_subset_1(u1_struct_0(A),k4_msafree2(A),k1_msafree2(A)))
& E = k1_trees_4(k4_tarski(F,D)) )
=> k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),D),E) = E ) ) ) ) ) ) ).
fof(t3_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,B),D))
=> ( r2_hidden(D,k2_msafree2(A))
=> k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),D),k1_trees_4(k4_tarski(E,D))) = k1_trees_4(k4_tarski(k1_funct_1(C,D),D)) ) ) ) ) ) ) ).
fof(t4_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F)
& v1_finseq_1(F)
& v6_trees_3(F) )
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G)
& v1_finseq_1(G)
& v6_trees_3(G) )
=> ( ( r2_hidden(D,k4_msafree2(A))
& E = k4_trees_4(k4_tarski(k5_msafree2(A,D),u1_struct_0(A)),F)
& k4_finseq_1(F) = k4_finseq_1(G)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k4_finseq_1(F))
=> k1_funct_1(G,H) = k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),k4_finseq_4(k5_numbers,u1_struct_0(A),k1_msualg_1(A,k5_msafree2(A,D)),H)),k1_funct_1(F,H)) ) ) )
=> k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),D),E) = k4_trees_4(k4_tarski(k5_msafree2(A,D),u1_struct_0(A)),G) ) ) ) ) ) ) ) ) ).
fof(t5_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ( r2_hidden(D,k1_msafree2(A))
=> k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),D),E) = k1_trees_4(k4_tarski(k5_msafree2(A,D),u1_struct_0(A))) ) ) ) ) ) ) ).
fof(t6_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ! [F] :
( m1_subset_1(F,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G)
& v3_trees_2(G) )
=> ! [H] :
( ( v1_relat_1(H)
& v1_funct_1(H)
& v3_trees_2(H) )
=> ( ( G = E
& H = F
& F = k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),D),E) )
=> k1_relat_1(G) = k1_relat_1(H) ) ) ) ) ) ) ) ) ) ).
fof(t7_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msafree2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ! [F] :
( m1_subset_1(F,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D))
=> ( F = k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),D),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,C),D),E)
=> k4_card_1(E) = k4_card_1(F) ) ) ) ) ) ) ) ).
fof(d3_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C))
=> ( E = k4_circuit2(A,B,C,D)
<=> ? [F] :
( m1_subset_1(F,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C))
& k4_card_1(F) = k4_circuit1(A,B,C)
& E = k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,D),C),F) ) ) ) ) ) ) ) ).
fof(t8_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_msafree2(D,A,B)
=> k4_circuit2(A,B,C,D) = k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B)),k3_circuit2(A,B,D),C),k4_circuit2(A,B,C,D)) ) ) ) ) ).
fof(t9_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E)
& v6_trees_3(E) )
=> ( ( r2_hidden(C,k4_msafree2(A))
& k4_finseq_1(E) = k4_finseq_1(k1_msualg_1(A,k5_msafree2(A,C)))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k4_finseq_1(E))
=> k1_funct_1(E,F) = k4_circuit2(A,B,k4_finseq_4(k5_numbers,u1_struct_0(A),k1_msualg_1(A,k5_msafree2(A,C)),F),D) ) ) )
=> k4_circuit2(A,B,C,D) = k4_trees_4(k4_tarski(k5_msafree2(A,C),u1_struct_0(A)),E) ) ) ) ) ) ) ).
fof(d4_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_msafree2(D,A,B)
=> k5_circuit2(A,B,C,D) = k8_funct_2(k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C),k1_funct_1(u4_msualg_1(A,B),C),k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B),k7_msafree2(A,B),C),k4_circuit2(A,B,C,D)) ) ) ) ) ).
fof(t10_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ( r2_hidden(C,k2_msafree2(A))
=> k5_circuit2(A,B,C,D) = k1_funct_1(D,C) ) ) ) ) ) ).
fof(t11_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ( r2_hidden(C,k1_msafree2(A))
=> k5_circuit2(A,B,C,D) = k1_funct_1(k1_circuit1(A,B),C) ) ) ) ) ) ).
fof(d5_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
=> ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ( D = k6_circuit2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(E,k2_msafree2(A))
=> k1_funct_1(D,E) = k1_funct_1(C,E) )
& ( r2_hidden(E,k4_msafree2(A))
=> k1_funct_1(D,E) = k8_funct_2(k3_msualg_1(A,k5_msafree2(A,E),B),k4_msualg_1(A,k5_msafree2(A,E),B),k5_msualg_1(A,k5_msafree2(A,E),B),k3_circuit1(A,B,C,k5_msafree2(A,E))) ) ) ) ) ) ) ) ) ).
fof(t12_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ( r1_tarski(D,C)
=> r1_tarski(D,k6_circuit2(A,B,C)) ) ) ) ) ) ).
fof(d6_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
=> ( v1_circuit2(C,A,B)
<=> C = k6_circuit2(A,B,C) ) ) ) ) ).
fof(d7_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
=> ! [D] :
( m1_msafree2(D,A,B)
=> k7_circuit2(A,B,C,D) = k6_circuit2(A,B,k1_circuit2(A,B,C,D)) ) ) ) ) ).
fof(d8_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> k8_circuit2(A,B,C,D) = k1_funct_4(k1_circuit2(A,B,D,k2_circuit1(A,B,C,np__0)),k1_circuit1(A,B)) ) ) ) ) ).
fof(d9_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k4_card_3(u4_msualg_1(A,B)))
& m2_relset_1(E,k5_numbers,k4_card_3(u4_msualg_1(A,B))) )
=> ( E = k9_circuit2(A,B,C,D)
<=> ( k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),E,np__0) = k8_circuit2(A,B,C,D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),E,k1_nat_1(F,np__1)) = k7_circuit2(A,B,k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),E,F),k2_circuit1(A,B,C,k1_nat_1(F,np__1))) ) ) ) ) ) ) ) ) ).
fof(t13_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r1_xreal_0(k6_circuit1(A,B,F),E)
=> k1_funct_1(C,F) = k5_circuit2(A,B,F,D) ) )
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r1_xreal_0(k6_circuit1(A,B,F),k1_nat_1(E,np__1))
=> k1_funct_1(k6_circuit2(A,B,C),F) = k5_circuit2(A,B,F,D) ) ) ) ) ) ) ) ) ).
fof(t14_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ( v5_seqm_3(k10_funct_6(C))
=> ( v1_xboole_0(k2_msafree2(A))
| ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ! [E] :
( m1_msafree2(E,A,B)
=> ( E = k1_funct_1(k10_funct_6(C),np__0)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> r1_tarski(E,k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,D),F)) ) ) ) ) ) ) ) ) ) ).
fof(t15_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( v5_seqm_3(k10_funct_6(C))
& v1_circuit2(k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,D),E),A,B) )
=> ( v1_xboole_0(k2_msafree2(A))
| ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(E,F)
=> k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,D),E) = k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,D),F) ) ) ) ) ) ) ) ) ) ).
fof(t16_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ( v5_seqm_3(k10_funct_6(C))
=> ( v1_xboole_0(k2_msafree2(A))
| ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ! [E] :
( m1_msafree2(E,A,B)
=> ( E = k1_funct_1(k10_funct_6(C),np__0)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( r1_xreal_0(k6_circuit1(A,B,G),F)
=> k1_funct_1(k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,D),F),G) = k5_circuit2(A,B,G,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ! [D] :
( m1_msafree2(D,A,B)
=> ( ( v5_seqm_3(k10_funct_6(C))
& D = k1_funct_1(k10_funct_6(C),np__0) )
=> ( v1_xboole_0(k2_msafree2(A))
| ! [E] :
( m1_subset_1(E,k4_card_3(u4_msualg_1(A,B)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( G = k7_circuit1(A,B)
=> k1_funct_1(k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,E),G),F) = k5_circuit2(A,B,F,D) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ( v5_seqm_3(k10_funct_6(C))
=> ( v1_xboole_0(k2_msafree2(A))
| ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( E = k7_circuit1(A,B)
=> v1_circuit2(k8_funct_2(k5_numbers,k4_card_3(u4_msualg_1(A,B)),k9_circuit2(A,B,C,D),E),A,B) ) ) ) ) ) ) ) ) ).
fof(t19_circuit2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
=> ( v5_seqm_3(k10_funct_6(C))
=> ( v1_xboole_0(k2_msafree2(A))
| ! [D] :
( m1_subset_1(D,k4_card_3(u4_msualg_1(A,B)))
=> ! [E] :
( m1_subset_1(E,k4_card_3(u4_msualg_1(A,B)))
=> k1_funct_1(k9_circuit2(A,B,C,D),k7_circuit1(A,B)) = k1_funct_1(k9_circuit2(A,B,C,E),k7_circuit1(A,B)) ) ) ) ) ) ) ) ).
fof(dt_k1_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
& m1_msafree2(D,A,B) )
=> m1_subset_1(k1_circuit2(A,B,C,D),k4_card_3(u4_msualg_1(A,B))) ) ).
fof(idempotence_k1_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
& m1_msafree2(D,A,B) )
=> k1_circuit2(A,B,C,C) = C ) ).
fof(redefinition_k1_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
& m1_msafree2(D,A,B) )
=> k1_circuit2(A,B,C,D) = k1_funct_4(C,D) ) ).
fof(dt_k2_circuit2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_msafree2(C,A,B) )
=> m3_pboole(k2_circuit2(A,B,C),u1_struct_0(A),k13_msafree(A,u4_msualg_1(A,B)),u4_msualg_1(A,k6_msafree2(A,B))) ) ).
fof(dt_k3_circuit2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_msafree2(C,A,B) )
=> m3_pboole(k3_circuit2(A,B,C),u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,k6_msafree2(A,B))) ) ).
fof(dt_k4_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_msafree2(D,A,B) )
=> m1_subset_1(k4_circuit2(A,B,C,D),k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C)) ) ).
fof(dt_k5_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_msafree2(D,A,B) )
=> m1_subset_1(k5_circuit2(A,B,C,D),k1_funct_1(u4_msualg_1(A,B),C)) ) ).
fof(dt_k6_circuit2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,k4_card_3(u4_msualg_1(A,B))) )
=> m1_subset_1(k6_circuit2(A,B,C),k4_card_3(u4_msualg_1(A,B))) ) ).
fof(dt_k7_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,k4_card_3(u4_msualg_1(A,B)))
& m1_msafree2(D,A,B) )
=> m1_subset_1(k7_circuit2(A,B,C,D),k4_card_3(u4_msualg_1(A,B))) ) ).
fof(dt_k8_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
& m1_subset_1(D,k4_card_3(u4_msualg_1(A,B))) )
=> m1_subset_1(k8_circuit2(A,B,C,D),k4_card_3(u4_msualg_1(A,B))) ) ).
fof(dt_k9_circuit2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v2_msafree2(A)
& v5_msafree2(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v4_msafree2(B,A)
& l3_msualg_1(B,A)
& m3_pboole(C,k2_msafree2(A),k2_pre_circ(k2_msafree2(A),k5_numbers),k3_pre_circ(u1_struct_0(A),u4_msualg_1(A,B),k2_msafree2(A)))
& m1_subset_1(D,k4_card_3(u4_msualg_1(A,B))) )
=> ( v1_funct_1(k9_circuit2(A,B,C,D))
& v1_funct_2(k9_circuit2(A,B,C,D),k5_numbers,k4_card_3(u4_msualg_1(A,B)))
& m2_relset_1(k9_circuit2(A,B,C,D),k5_numbers,k4_card_3(u4_msualg_1(A,B))) ) ) ).
%------------------------------------------------------------------------------