TPTP Problem File: SWV582-1.p
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%------------------------------------------------------------------------------
% File : SWV582-1 : TPTP v8.2.0. Released v4.1.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform 074_16
% Version : Especial.
% English : Formalization of a functional implementation of the FFT algorithm
% over the complex numbers, and its inverse. Both are shown
% equivalent to the usual definitions of these operations through
% Vandermonde matrices. They are also shown to be inverse to each
% other, more precisely, that composition of the inverse and the
% transformation yield the identity up to a scalar.
% Refs : [Nip10] Nipkow (2010), Email to Geoff Sutcliffe
% : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% Source : [Nip10]
% Names : FFT-074_16 [Nip10]
% Status : Unsatisfiable
% Rating : 0.80 v8.2.0, 0.86 v8.1.0, 0.84 v7.4.0, 0.88 v7.3.0, 0.83 v7.0.0, 0.87 v6.3.0, 0.82 v6.2.0, 0.80 v6.1.0, 0.93 v6.0.0, 0.80 v5.5.0, 0.95 v5.3.0, 0.94 v5.0.0, 0.93 v4.1.0
% Syntax : Number of clauses : 1088 ( 132 unt; 157 nHn; 581 RR)
% Number of literals : 3157 ( 701 equ;1902 neg)
% Maximal clause size : 6 ( 2 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 72 ( 71 usr; 0 prp; 1-9 aty)
% Number of functors : 44 ( 44 usr; 14 con; 0-6 aty)
% Number of variables : 5287 (1755 sgn)
% SPC : CNF_UNS_RFO_SEQ_NHN
% Comments :
%------------------------------------------------------------------------------
cnf(cls_minus__mult__right_0,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I20_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I21_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_le__eq__neg_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b))) ) ).
cnf(cls_le__eq__neg_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I18_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I19_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| V_a = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_mult__left__idem_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__idem__mult(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)) = hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y) ) ).
cnf(cls_neg__le__0__iff__le_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_neg__le__0__iff__le_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I25_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_a)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_fcomp__apply_0,axiom,
hAPP(c_Fun_Ofcomp(V_f,V_g,T_b,T_c,T_a),V_x) = hAPP(V_g,hAPP(V_f,V_x)) ).
cnf(cls_mult__le__0__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__le__0__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__le__0__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__le__0__iff_3,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__neg__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__pos__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_mult__pos__neg2_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_pordered__ring_Osplit__mult__pos__le_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_zero),hAPP(hAPP(V_times,V_a),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_zero),V_b))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_zero),V_a))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Osplit__mult__pos__le_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_zero),hAPP(hAPP(V_times,V_a),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_b),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),V_zero))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Omult__left__mono__neg_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_times,V_c),V_a)),hAPP(hAPP(V_times,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_c),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_b),V_a))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Omult__nonpos__nonpos_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_zero),hAPP(hAPP(V_times,V_a),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_b),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),V_zero))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Omult__right__mono__neg_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_c),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_b),V_a))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_setsum__op__ivl__Suc_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,hAPP(c_Suc,V_n),tc_nat),tc_nat,T_a) = c_HOL_Ozero__class_Ozero(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m)) ) ).
cnf(cls_pos__add__strict_0,axiom,
( ~ class_Ring__and__Field_Oordered__semidom(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_mult__mono1_0,axiom,
( ~ class_Ring__and__Field_Omult__mono1(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_mult__left__mono_0,axiom,
( ~ class_Ring__and__Field_Omult__mono(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_mult__right__mono_0,axiom,
( ~ class_Ring__and__Field_Omult__mono(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_mult__left__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a)) ) ).
cnf(cls_mult__right__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a)) ) ).
cnf(cls_mult_Oadditive__right_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| c_RealVector_Oadditive(c_HOL_Otimes__class_Otimes(V_a,T_a),T_a,T_a) ) ).
cnf(cls_sup__eq__if_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_a,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a) = V_a
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_class__ringb_Oadd__scale__eq__noteq_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(c_HOL_Otimes__class_Otimes(V_r,T_a),V_c)) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(c_HOL_Otimes__class_Otimes(V_r,T_a),V_d))
| V_c = V_d
| V_r = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sum__squares__eq__zero__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y)) != c_HOL_Ozero__class_Ozero(T_a)
| V_x = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sum__squares__eq__zero__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y)) != c_HOL_Ozero__class_Ozero(T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I16_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I17_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_n__not__Suc__n_0,axiom,
V_n != hAPP(c_Suc,V_n) ).
cnf(cls_Suc__n__not__n_0,axiom,
hAPP(c_Suc,V_n) != V_n ).
cnf(cls_divide__less__eq_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a)) ) ).
cnf(cls_divide__less__eq_6,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_less__divide__eq_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a))) ) ).
cnf(cls_less__divide__eq_6,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_mult__imp__div__pos__less_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_z,T_a),V_y)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y)) ) ).
cnf(cls_mult__imp__less__div__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_z,T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_z,T_a),V_y),T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y)) ) ).
cnf(cls_ring_Oring__simps_I33_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b),T_b),V_c) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I31_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c)) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I18_J_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I17_J_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_le__supE_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a),T_a),V_x)) ) ).
cnf(cls_le__supE_1,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a),T_a),V_x)) ) ).
cnf(cls_le__supI1_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_a)) ) ).
cnf(cls_le__supI2_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_b)) ) ).
cnf(cls_le__sup__iff_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),T_a),V_z)) ) ).
cnf(cls_le__sup__iff_1,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_y,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),T_a),V_z)) ) ).
cnf(cls_field_Ofield__eq__simps_I35_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c)) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I37_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b),T_b),V_c) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I21_J_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I22_J_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_zero__less__divide__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_zero__less__divide__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_zero__less__divide__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_zero__less__divide__iff_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_divide__nonneg__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_divide__nonpos__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_finite__atMost_0,axiom,
c_Finite__Set_Ofinite(c_SetInterval_Oord__class_OatMost(V_k,tc_nat),tc_nat) ).
cnf(cls_mult__idem_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__idem__mult(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x) = V_x ) ).
cnf(cls_mult__strict__mono_H_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_pordered__ring_Oring__simps_I27_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I27_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I28_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I28_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_comp__cong_0,axiom,
( hAPP(V_f,hAPP(V_g,V_x)) != hAPP(V_f_H,hAPP(V_g_H,V_x_H))
| hAPP(c_Fun_Ocomp(V_f,V_g,T_b,T_a,T_c),V_x) = hAPP(c_Fun_Ocomp(V_f_H,V_g_H,T_d,T_a,T_e),V_x_H) ) ).
cnf(cls_o__eq__dest_0,axiom,
( c_Fun_Ocomp(V_a,V_b,T_c,T_b,T_a) != c_Fun_Ocomp(V_c,V_d,T_d,T_b,T_a)
| hAPP(V_a,hAPP(V_b,V_v)) = hAPP(V_c,hAPP(V_d,V_v)) ) ).
cnf(cls_minus__mult__left_0,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) = hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b) ) ).
cnf(cls_mult__frac__frac_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_z,V_w,T_a)) = c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_w),T_a) ) ).
cnf(cls_bounded__bilinear_Oprod__diff__prod_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_c),hAPP(hAPP(V_prod,V_x),V_y)),hAPP(hAPP(V_prod,V_a),V_b)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_c),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_c),hAPP(hAPP(V_prod,hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_a)),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_y),V_b))),hAPP(hAPP(V_prod,hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_a)),V_b))),hAPP(hAPP(V_prod,V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_y),V_b)))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_lessI_0,axiom,
hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),hAPP(c_Suc,V_n))) ).
cnf(cls_less__Suc__eq_2,axiom,
hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),hAPP(c_Suc,V_x))) ).
cnf(cls_less__trans__Suc_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_Suc,V_i),tc_nat),V_k))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_j,tc_nat),V_k))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_i,tc_nat),V_j)) ) ).
cnf(cls_option_Osimps_I4_J_0,axiom,
c_Option_Ooption_Ooption__case(V_f1,V_f2,c_Option_Ooption_ONone(T_b),T_a,T_b) = V_f1 ).
cnf(cls_lessThan__Suc__atMost_0,axiom,
c_SetInterval_Oord__class_OlessThan(hAPP(c_Suc,V_k),tc_nat) = c_SetInterval_Oord__class_OatMost(V_k,tc_nat) ).
cnf(cls_scaleR__right_Odiff_0,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_x)),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_y)) ) ).
cnf(cls_scaleR_Odiff__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_b_H)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_b)),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_b_H)) ) ).
cnf(cls_linorder__neqE__ordered__idom_0,axiom,
( ~ class_Ring__and__Field_Oordered__idom(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| V_x = V_y ) ).
cnf(cls_not__less__iff__gr__or__eq_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| V_x = V_y
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_linorder__antisym__conv3_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| V_x = V_y
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x)) ) ).
cnf(cls_linorder__less__linear_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x))
| V_x = V_y
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_order__antisym__conv_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| V_x = V_y
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_le__antisym_0,axiom,
( V_m = V_n
| ~ hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_m))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_nat__neq__iff_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| V_m = V_n ) ).
cnf(cls_nat__less__cases_0,axiom,
( hBOOL(hAPP(hAPP(V_P,V_n),V_m))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m))
| V_m = V_n
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_linorder__neqE__nat_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,tc_nat),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),V_y))
| V_x = V_y ) ).
cnf(cls_order__eq__iff_2,axiom,
( ~ class_Orderings_Oorder(T_a)
| V_x = V_y
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_linorder__neqE_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| V_x = V_y ) ).
cnf(cls_order__antisym_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| V_x = V_y
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_less__eq__Suc__le__raw_0,axiom,
c_HOL_Oord__class_Oless(v_n,tc_nat) = c_lessequals(hAPP(c_Suc,v_n),tc_nat) ).
cnf(cls_compl__eq__compl__iff_0,axiom,
( ~ class_Lattices_Oboolean__algebra(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x) != hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_y)
| V_x = V_y ) ).
cnf(cls_neg__equal__iff__equal_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a) != hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)
| V_a = V_b ) ).
cnf(cls_eq__divide__imp_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| V_c = c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),V_c,T_a) ) ).
cnf(cls_divide__eq__imp_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| V_c = c_HOL_Ozero__class_Ozero(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),V_c,T_a) = V_a ) ).
cnf(cls_eq__divide__eq_3,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| V_a = c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),V_c,T_a)
| V_c = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide__eq__eq_3,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),V_c,T_a) = V_a
| V_c = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_field_Ononzero__mult__divide__mult__cancel__right2_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_c),V_b)) = hAPP(hAPP(V_divide,V_a),V_b)
| V_c = V_zero
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__mult__divide__mult__cancel__right_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c)) = hAPP(hAPP(V_divide,V_a),V_b)
| V_c = V_zero
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__mult__divide__mult__cancel__left2_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_c),V_a)),hAPP(hAPP(V_times,V_b),V_c)) = hAPP(hAPP(V_divide,V_a),V_b)
| V_c = V_zero
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__mult__divide__mult__cancel__left_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_c),V_a)),hAPP(hAPP(V_times,V_c),V_b)) = hAPP(hAPP(V_divide,V_a),V_b)
| V_c = V_zero
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__divide__mult__cancel__right_0,axiom,
( hAPP(hAPP(V_divide,V_b),hAPP(hAPP(V_times,V_a),V_b)) = hAPP(hAPP(V_divide,V_one),V_a)
| V_b = V_zero
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__divide__mult__cancel__left_0,axiom,
( hAPP(hAPP(V_divide,V_a),hAPP(hAPP(V_times,V_a),V_b)) = hAPP(hAPP(V_divide,V_one),V_b)
| V_b = V_zero
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Otimes__divide__eq__right_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_divide,V_b),V_c)) = hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Otimes__divide__eq__left_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_divide,V_b),V_c)),V_a) = hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_b),V_a)),V_c)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odiff__divide__distrib_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_divide,V_a),V_c)),hAPP(hAPP(V_divide,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Oadd__divide__distrib_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,hAPP(hAPP(V_divide,V_a),V_c)),hAPP(hAPP(V_divide,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ominus__divide__left_0,axiom,
( hAPP(V_uminus,hAPP(hAPP(V_divide,V_a),V_b)) = hAPP(hAPP(V_divide,hAPP(V_uminus,V_a)),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Oinverse__eq__divide_0,axiom,
( hAPP(V_inverse,V_a) = hAPP(hAPP(V_divide,V_one),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__minus__left_0,axiom,
( hAPP(hAPP(V_divide,hAPP(V_uminus,V_a)),V_b) = hAPP(V_uminus,hAPP(hAPP(V_divide,V_a),V_b))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__zero__left_0,axiom,
( hAPP(hAPP(V_divide,V_zero),V_a) = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__inverse_0,axiom,
( hAPP(hAPP(V_divide,V_a),V_b) = hAPP(hAPP(V_times,V_a),hAPP(V_inverse,V_b))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odiff__frac__eq_0,axiom,
( hAPP(hAPP(V_minus,hAPP(hAPP(V_divide,V_x),V_y)),hAPP(hAPP(V_divide,V_w),V_z)) = hAPP(hAPP(V_divide,hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_x),V_z)),hAPP(hAPP(V_times,V_w),V_y))),hAPP(hAPP(V_times,V_y),V_z))
| V_z = V_zero
| V_y = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Oinverse__add_0,axiom,
( hAPP(hAPP(V_plus,hAPP(V_inverse,V_a)),hAPP(V_inverse,V_b)) = hAPP(hAPP(V_times,hAPP(hAPP(V_times,hAPP(hAPP(V_plus,V_a),V_b)),hAPP(V_inverse,V_a))),hAPP(V_inverse,V_b))
| V_b = V_zero
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Oadd__frac__eq_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_divide,V_x),V_y)),hAPP(hAPP(V_divide,V_w),V_z)) = hAPP(hAPP(V_divide,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_x),V_z)),hAPP(hAPP(V_times,V_w),V_y))),hAPP(hAPP(V_times,V_y),V_z))
| V_z = V_zero
| V_y = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__1_0,axiom,
( hAPP(hAPP(V_divide,V_a),V_one) = V_a
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I18_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_times,V_a),V_b)),V_c) = hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I17_J_0,axiom,
( hAPP(hAPP(V_times,V_a),V_b) = hAPP(hAPP(V_times,V_b),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I16_J_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_times,V_b),V_c)) = hAPP(hAPP(V_times,V_b),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I15_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I14_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),V_b) = hAPP(hAPP(V_plus,V_b),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I13_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,V_b),hAPP(hAPP(V_plus,V_a),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I12_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(V_uminus,V_b)) = hAPP(hAPP(V_minus,V_a),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I11_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(V_uminus,V_a)),V_b) = hAPP(hAPP(V_minus,V_b),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I10_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I9_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I8_J_0,axiom,
( ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_a = hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_b) ) ).
cnf(cls_field_Ofield__eq__simps_I8_J_1,axiom,
( ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_field_Ofield__eq__simps_I7_J_0,axiom,
( ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_field_Ofield__eq__simps_I7_J_1,axiom,
( ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_a = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_b) ) ).
cnf(cls_field_Ofield__eq__simps_I6_J_0,axiom,
( hAPP(hAPP(V_minus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I5_J_0,axiom,
( hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I4_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I3_J_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_b)),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I2_J_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_a),V_b)),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I1_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I26_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I26_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I25_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I25_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_class__semiring_Osemiring__rules_I15_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ly),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),V_ry)) = hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ly),T_a),V_ry)) ) ).
cnf(cls_class__semiring_Osemiring__rules_I14_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ly),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),V_ry)) = hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_ly,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),V_ry))) ) ).
cnf(cls_class__semiring_Osemiring__rules_I13_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ly),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),V_ry)) = hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_rx),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_ly,T_a),V_ry)) ) ).
cnf(cls_neg__equal__0__iff__equal_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sup__0__imp__0_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_a,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sum__squares__le__zero__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_setsum__def_1,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,V_A,T_b,T_a) = c_HOL_Ozero__class_Ozero(T_a)
| c_Finite__Set_Ofinite(V_A,T_b) ) ).
cnf(cls_setsum__infinite_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_b)
| c_Finite__Set_Osetsum(V_f,V_A,T_a,T_b) = c_HOL_Ozero__class_Ozero(T_b)
| c_Finite__Set_Ofinite(V_A,T_a) ) ).
cnf(cls_add__nonpos__neg_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_add__neg__nonpos_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_neg__less__iff__less_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_neg__less__iff__less_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_le__divide__eq_8,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_divide__le__eq_8,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_pordered__ab__group__add_Oless__iff__diff__less__0_1,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),V_b))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_minus,V_a),V_b)),V_zero))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oless__iff__diff__less__0_0,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_minus,V_a),V_b)),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),V_b))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__less__0__iff__less_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_a)),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less,V_zero),V_a))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__less__0__iff__less_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_zero),V_a))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_a)),V_zero))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__0__less__iff__less_1,axiom,
( hBOOL(hAPP(hAPP(V_less,V_zero),hAPP(V_uminus,V_a)))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),V_zero))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__0__less__iff__less_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less,V_zero),hAPP(V_uminus,V_a)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__less__iff__less_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_b)),hAPP(V_uminus,V_a)))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),V_b))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__less__iff__less_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),V_b))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_b)),hAPP(V_uminus,V_a)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__iff__diff__le__0_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),V_b))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_minus,V_a),V_b)),V_zero))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__iff__diff__le__0_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_minus,V_a),V_b)),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),V_b))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__le__0__iff__le_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_a)),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_zero),V_a))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__le__0__iff__le_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_zero),V_a))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_a)),V_zero))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__0__le__iff__le_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_zero),hAPP(V_uminus,V_a)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),V_zero))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__0__le__iff__le_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),V_zero))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_zero),hAPP(V_uminus,V_a)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_less__SucI_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),hAPP(c_Suc,V_n)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_Suc__lessD_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_Suc,V_m),tc_nat),V_n)) ) ).
cnf(cls_less__eqI_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) != hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x_H),V_y_H)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x_H,T_a),V_y_H)) ) ).
cnf(cls_less__eqI_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) != hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x_H),V_y_H)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x_H,T_a),V_y_H))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_le__eqI_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) != hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x_H),V_y_H)
| hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_y_H,T_a),V_x_H)) ) ).
cnf(cls_le__eqI_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) != hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x_H),V_y_H)
| hBOOL(hAPP(c_lessequals(V_y_H,T_a),V_x_H))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_order__less__le_2,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| V_x = V_y
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_order__le__less_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| V_x = V_y
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_le__neq__implies__less_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| V_m = V_n
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_le__eq__less__or__eq_0,axiom,
( V_m = V_n
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_nat__less__le_2,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| V_m = V_n
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_order__le__neq__trans_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| V_a = V_b
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_order__neq__le__trans_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| V_a = V_b ) ).
cnf(cls_linorder__antisym__conv1_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| V_x = V_y
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_linorder__antisym__conv2_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| V_x = V_y
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_xt1_I12_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a))
| V_a = V_b ) ).
cnf(cls_xt1_I11_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| V_a = V_b
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a)) ) ).
cnf(cls_divide__right__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_c,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_divide__right__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_c,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_split__mult__pos__le_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_split__mult__pos__le_1,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__nonpos__nonpos_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_zero__le__square_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_a))) ) ).
cnf(cls_zero__le__mult__iff_4,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_zero__le__mult__iff_5,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__nonneg__nonneg_0,axiom,
( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_ab__semigroup__mult__class_Omult__ac_I1_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),V_c) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)) ) ).
cnf(cls_class__semiring_Osemiring__rules_I16_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ly),T_a),V_rx) = hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_rx),T_a),V_ly) ) ).
cnf(cls_class__semiring_Osemiring__rules_I17_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ly),T_a),V_rx) = hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_ly,T_a),V_rx)) ) ).
cnf(cls_class__semiring_Osemiring__rules_I18_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),V_ry)) = hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_rx),T_a),V_ry) ) ).
cnf(cls_class__semiring_Osemiring__rules_I19_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),V_ry)) = hAPP(c_HOL_Otimes__class_Otimes(V_rx,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_lx,T_a),V_ry)) ) ).
cnf(cls_class__semiring_Omul__a_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_z)) = hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y),T_a),V_z) ) ).
cnf(cls_frac__less_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_x,V_z,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_y,V_w,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_w,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_w))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_zero__less__double__add__iff__zero__less__single__add_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a))) ) ).
cnf(cls_zero__less__double__add__iff__zero__less__single__add_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_less__supI1_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_a)) ) ).
cnf(cls_less__supI2_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_b)) ) ).
cnf(cls_minus__divide__right_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)) = c_HOL_Oinverse__class_Odivide(V_a,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a) ) ).
cnf(cls_minus__divide__left_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)) = c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),V_b,T_a) ) ).
cnf(cls_zero__le__mult__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b))) ) ).
cnf(cls_zero__le__mult__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b))) ) ).
cnf(cls_zero__le__mult__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b))) ) ).
cnf(cls_zero__le__mult__iff_3,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b))) ) ).
cnf(cls_neg__equal__zero_0,axiom,
( ~ class_OrderedGroup_Oordered__ab__group__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a) != V_a
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sum__squares__gt__zero__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))))) ) ).
cnf(cls_pordered__ring_Oring__simps_I21_J_0,axiom,
( ~ class_Ring__and__Field_Oring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I22_J_0,axiom,
( ~ class_Ring__and__Field_Oring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_map__comp__def_0,axiom,
c_Map_Omap__comp(V_f,V_g,v_k,T_c,T_b,t_a) = c_Option_Ooption_Ooption__case(c_Option_Ooption_ONone(T_b),V_f,hAPP(V_g,v_k),tc_Option_Ooption(T_b),T_c) ).
cnf(cls_map__comp__None__iff_2,axiom,
( hAPP(V_m2,V_k) != c_Option_Ooption_ONone(T_b)
| c_Map_Omap__comp(V_m1,V_m2,V_k,T_b,T_a,T_c) = c_Option_Ooption_ONone(T_a) ) ).
cnf(cls_map__comp__simps_I1_J_0,axiom,
( hAPP(V_m2,V_k) != c_Option_Ooption_ONone(T_a)
| c_Map_Omap__comp(V_m1,V_m2,V_k,T_a,T_c,T_b) = c_Option_Ooption_ONone(T_c) ) ).
cnf(cls_diff__frac__eq_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)),c_HOL_Oinverse__class_Odivide(V_w,V_z,T_a)) = c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z)),hAPP(c_HOL_Otimes__class_Otimes(V_w,T_a),V_y)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_z),T_a)
| V_z = c_HOL_Ozero__class_Ozero(T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult_Ominus__left_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) ) ).
cnf(cls_mult__left_Ominus_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x),T_a),V_y) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)) ) ).
cnf(cls_mult_Ominus__right_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) ) ).
cnf(cls_mult__right_Ominus_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),V_x)) ) ).
cnf(cls_square__eq__iff_1,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_b) ) ).
cnf(cls_minus__mult__minus_0,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b) ) ).
cnf(cls_mult__mono_0,axiom,
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_mult__mono_H_0,axiom,
( ~ class_Ring__and__Field_Opordered__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_divide__less__eq_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a)) ) ).
cnf(cls_divide__less__eq_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_divide__less__eq_8,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_divide__less__eq_9,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_less__divide__eq_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a))) ) ).
cnf(cls_less__divide__eq_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_less__divide__eq_8,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_less__divide__eq_9,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_pordered__ring_Oring__simps_I33_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I34_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_ring_Oring__simps_I30_J_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_b)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I29_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_a)),V_b) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_a)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_not__less__less__Suc__eq_1,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),hAPP(c_Suc,V_x))) ) ).
cnf(cls_not__less__eq_1,axiom,
( ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),hAPP(c_Suc,V_m))) ) ).
cnf(cls_not__less__eq_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),hAPP(c_Suc,V_m)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_le__SucE_0,axiom,
( V_m = hAPP(c_Suc,V_n)
| hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),hAPP(c_Suc,V_n))) ) ).
cnf(cls_finite__greaterThanLessThan_0,axiom,
c_Finite__Set_Ofinite(c_SetInterval_Oord__class_OgreaterThanLessThan(V_l,V_u,tc_nat),tc_nat) ).
cnf(cls_mult__less__cancel__left__disj_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__less__cancel__left__disj_3,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__less__cancel__right__disj_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_mult__less__cancel__right__disj_3,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_mod__minus__eq_0,axiom,
( ~ class_Divides_Oring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),V_b,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_Divides_Odiv__class_Omod(V_a,V_b,T_a)),V_b,T_a) ) ).
cnf(cls_divide__neg__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__pos__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_zero__less__divide__iff_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_zero__less__divide__iff_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_additive_Odiff_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(V_f,hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(V_f,V_x)),hAPP(V_f,V_y))
| ~ c_RealVector_Oadditive(V_f,T_a,T_b) ) ).
cnf(cls_group__add__class_Odiff__0_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_HOL_Ozero__class_Ozero(T_a)),V_a) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a) ) ).
cnf(cls_scaleR_Oadditive__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| c_RealVector_Oadditive(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),T_a,T_a) ) ).
cnf(cls_add__frac__num_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)),V_z) = c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(c_HOL_Otimes__class_Otimes(V_z,T_a),V_y)),V_y,T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_add__num__frac_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_z),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)) = c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(c_HOL_Otimes__class_Otimes(V_z,T_a),V_y)),V_y,T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_vector__space_Oscale__eq__0__iff_1,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| hAPP(hAPP(V_scale,c_HOL_Ozero__class_Ozero(T_a)),V_x) = c_HOL_Ozero__class_Ozero(T_b) ) ).
cnf(cls_vector__space_Oscale__eq__0__iff_2,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| hAPP(hAPP(V_scale,V_a),c_HOL_Ozero__class_Ozero(T_b)) = c_HOL_Ozero__class_Ozero(T_b) ) ).
cnf(cls_vector__space_Oscale__zero__left_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,c_HOL_Ozero__class_Ozero(T_a)),V_x) = c_HOL_Ozero__class_Ozero(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_vector__space_Oscale__minus__left_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),V_x) = hAPP(c_HOL_Ouminus__class_Ouminus(T_b),hAPP(hAPP(V_scale,V_a),V_x))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_vector__space_Oscale__zero__right_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),c_HOL_Ozero__class_Ozero(T_b)) = c_HOL_Ozero__class_Ozero(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_vector__space_Oscale__cancel__left_1,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| hAPP(hAPP(V_scale,c_HOL_Ozero__class_Ozero(T_a)),V_x) = hAPP(hAPP(V_scale,c_HOL_Ozero__class_Ozero(T_a)),V_y) ) ).
cnf(cls_vector__space_Oscale__minus__right_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_x)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_b),hAPP(hAPP(V_scale,V_a),V_x))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_vector__space_Oscale__cancel__right_1,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| hAPP(hAPP(V_scale,V_a),c_HOL_Ozero__class_Ozero(T_b)) = hAPP(hAPP(V_scale,V_b),c_HOL_Ozero__class_Ozero(T_b)) ) ).
cnf(cls_diff__def_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_y)) ) ).
cnf(cls_class__ring_Osub__add_0,axiom,
( ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_y)) ) ).
cnf(cls_pordered__ab__group__add_Ominus__less__iff_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_a)),V_b))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_b)),V_a))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ominus__less__iff_0,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_b)),V_a))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(V_uminus,V_a)),V_b))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oless__minus__iff_1,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),hAPP(V_uminus,V_b)))
| ~ hBOOL(hAPP(hAPP(V_less,V_b),hAPP(V_uminus,V_a)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oless__minus__iff_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_b),hAPP(V_uminus,V_a)))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),hAPP(V_uminus,V_b)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oneg__le__iff__le_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),V_b))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_b)),hAPP(V_uminus,V_a)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__imp__neg__le_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_b)),hAPP(V_uminus,V_a)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),V_b))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ominus__le__iff_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_a)),V_b))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_b)),V_a))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ominus__le__iff_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_b)),V_a))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(V_uminus,V_a)),V_b))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oless__diff__eq_1,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Oless__diff__eq_0,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__minus__iff_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(V_uminus,V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_b),hAPP(V_uminus,V_a)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__minus__iff_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_b),hAPP(V_uminus,V_a)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(V_uminus,V_b)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Odiff__less__eq_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Odiff__less__eq_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__diff__eq_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ole__diff__eq_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Odiff__le__eq_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Odiff__le__eq_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_not__sum__squares__lt__zero_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y)),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I19_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I18_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| V_a = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_zero__less__mult__pos2_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_a))) ) ).
cnf(cls_zero__less__mult__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b))) ) ).
cnf(cls_mult__pos__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_mult__neg__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_double__add__less__zero__iff__single__less__zero_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_double__add__less__zero__iff__single__less__zero_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_even__less__0__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__idom(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_even__less__0__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__idom(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__eq__0__iff_0,axiom,
( ~ class_Ring__and__Field_Oring__no__zero__divisors(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| V_b = c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_no__zero__divisors_0,axiom,
( ~ class_Ring__and__Field_Ono__zero__divisors(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| V_b = c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_no__zero__divirors__neq0_0,axiom,
( ~ class_Ring__and__Field_Ono__zero__divisors(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a)
| V_b = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_ring_Oring__simps_I3_J_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_b)),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I4_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I5_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_times,V_a),V_b)),V_c) = hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I6_J_0,axiom,
( hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I7_J_0,axiom,
( hAPP(hAPP(V_minus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I8_J_1,axiom,
( ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| V_a = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_b) ) ).
cnf(cls_ring_Oring__simps_I8_J_0,axiom,
( ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_ring_Oring__simps_I9_J_1,axiom,
( ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_ring_Oring__simps_I9_J_0,axiom,
( ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| V_a = hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_b) ) ).
cnf(cls_ring_Oring__simps_I10_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I11_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I12_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(V_uminus,V_a)),V_b) = hAPP(hAPP(V_minus,V_b),V_a)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I13_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(V_uminus,V_b)) = hAPP(hAPP(V_minus,V_a),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I14_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,V_b),hAPP(hAPP(V_plus,V_a),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I15_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),V_b) = hAPP(hAPP(V_plus,V_b),V_a)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I16_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oright__diff__distrib_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_a),V_b)),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Ominus__mult__commute_0,axiom,
( hAPP(hAPP(V_times,hAPP(V_uminus,V_a)),V_b) = hAPP(hAPP(V_times,V_a),hAPP(V_uminus,V_b))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oleft__diff__distrib_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Ominus__mult__right_0,axiom,
( hAPP(V_uminus,hAPP(hAPP(V_times,V_a),V_b)) = hAPP(hAPP(V_times,V_a),hAPP(V_uminus,V_b))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Ominus__mult__minus_0,axiom,
( hAPP(hAPP(V_times,hAPP(V_uminus,V_a)),hAPP(V_uminus,V_b)) = hAPP(hAPP(V_times,V_a),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Ominus__mult__left_0,axiom,
( hAPP(V_uminus,hAPP(hAPP(V_times,V_a),V_b)) = hAPP(hAPP(V_times,hAPP(V_uminus,V_a)),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oeq__add__iff2_1,axiom,
( ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_b),V_a)),V_e)),V_d)) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d) ) ).
cnf(cls_ring_Oeq__add__iff1_1,axiom,
( ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_e)),V_c)) ) ).
cnf(cls_diff__divide__distrib_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),V_c,T_a) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_HOL_Oinverse__class_Odivide(V_a,V_c,T_a)),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)) ) ).
cnf(cls_divide_Odiff_0,axiom,
( ~ class_RealVector_Oreal__normed__field(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y),V_ya,T_a) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_HOL_Oinverse__class_Odivide(V_x,V_ya,T_a)),c_HOL_Oinverse__class_Odivide(V_y,V_ya,T_a)) ) ).
cnf(cls_add__nonneg__eq__0__iff_1,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y) != c_HOL_Ozero__class_Ozero(T_a)
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x))
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_add__nonneg__eq__0__iff_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y) != c_HOL_Ozero__class_Ozero(T_a)
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x))
| V_x = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_eq__iff__diff__eq__0_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_x) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_right__minus__eq_1,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_x) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_diff__0__right_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),c_HOL_Ozero__class_Ozero(T_a)) = V_a ) ).
cnf(cls_diff__self_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_class__ringb_Osubr0__iff_1,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_x) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_square__eq__iff_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_a) != hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_b)
| V_a = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)
| V_a = V_b ) ).
cnf(cls_mult__less__imp__less__right_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_mult__less__imp__less__left_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__right__less__imp__less_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_mult__left__less__imp__less_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__imp__le__div__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(V_z,T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_z,T_a),V_y),T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y)) ) ).
cnf(cls_mult__imp__div__pos__le_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_z,T_a),V_y)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y)) ) ).
cnf(cls_le__divide__eq_6,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_le__divide__eq_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a))) ) ).
cnf(cls_divide__le__eq_6,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_divide__le__eq_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a)) ) ).
cnf(cls_sum__squares__ge__zero_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y)))) ) ).
cnf(cls_le__add__iff2_1,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a),T_a),V_e)),V_d))) ) ).
cnf(cls_le__add__iff2_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(V_c,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a),T_a),V_e)),V_d)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d))) ) ).
cnf(cls_le__add__iff1_1,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),V_e)),V_c),T_a),V_d)) ) ).
cnf(cls_le__add__iff1_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),V_e)),V_c),T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d))) ) ).
cnf(cls_divide__le__0__iff_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__le__0__iff_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_ring_Oring__simps_I25_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_ring_Oring__simps_I26_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| V_a = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_Suc__leI_0,axiom,
( hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_m),tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_Suc__le__eq_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_m),tc_nat),V_n)) ) ).
cnf(cls_less__eq__Suc__le_0,axiom,
( hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),V_m))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m)) ) ).
cnf(cls_less__eq__Suc__le_1,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),V_m)) ) ).
cnf(cls_add__increasing2_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_add__increasing_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_ring_Oring__simps_I32_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_a)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_scaleR__minus__right_0,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_x)) ) ).
cnf(cls_scaleR_Ominus__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_b)) ) ).
cnf(cls_eq__add__iff2_1,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a),T_a),V_e)),V_d)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d) ) ).
cnf(cls_neg__le__iff__le_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_le__imp__neg__le_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_mult__strict__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_zmod__simps_I4_J_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(c_Divides_Odiv__class_Omod(V_a,V_c,T_a),T_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),V_c,T_a) ) ).
cnf(cls_bounded__bilinear_Oadditive__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| c_RealVector_Oadditive(hAPP(V_prod,V_a),T_b,T_c)
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_diff__minus_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) ) ).
cnf(cls_ab__diff__minus_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) ) ).
cnf(cls_nonzero__minus__divide__divide_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a) = c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)
| V_b = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide__strict__right__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_c,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_divide__strict__right__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_c,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a)) ) ).
cnf(cls_field_Ofield__eq__simps_I33_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_a)),V_b) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I34_J_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_b)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_mod__diff__eq_0,axiom,
( ~ class_Divides_Oring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(V_a,V_c,T_a)),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) ) ).
cnf(cls_divide__nonneg__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_divide__nonpos__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__self__if_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(c_HOL_Ozero__class_Ozero(T_a),c_HOL_Ozero__class_Ozero(T_a),T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_finite__lessThan_0,axiom,
c_Finite__Set_Ofinite(c_SetInterval_Oord__class_OlessThan(V_k,tc_nat),tc_nat) ).
cnf(cls_pordered__ring_Oring__simps_I1_J_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I1_J_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I2_J_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I2_J_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I3_J_1,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I3_J_0,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_minus,V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I4_J_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I4_J_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_a),hAPP(hAPP(V_plus,V_c),V_b)))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_minus,V_a),V_b)),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Ole__add__iff1_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_e)),V_c)),V_d))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Ole__add__iff1_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_e)),V_c)),V_d))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Ole__add__iff2_0,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,V_c),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_b),V_a)),V_e)),V_d)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Ole__add__iff2_1,axiom,
( hBOOL(hAPP(hAPP(V_less__eq,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ hBOOL(hAPP(hAPP(V_less__eq,V_c),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_b),V_a)),V_e)),V_d)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oless__add__iff1_0,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_e)),V_c)),V_d))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oless__add__iff1_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_e)),V_c)),V_d))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oless__add__iff2_0,axiom,
( hBOOL(hAPP(hAPP(V_less,V_c),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_b),V_a)),V_e)),V_d)))
| ~ hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oless__add__iff2_1,axiom,
( hBOOL(hAPP(hAPP(V_less,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c)),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)))
| ~ hBOOL(hAPP(hAPP(V_less,V_c),hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_b),V_a)),V_e)),V_d)))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_not__square__less__zero_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__le__0__iff_4,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_mult__le__0__iff_5,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_split__mult__neg__le_0,axiom,
( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_split__mult__neg__le_1,axiom,
( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__nonneg__nonpos_0,axiom,
( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_mult__nonpos__nonneg_0,axiom,
( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__nonneg__nonpos2_0,axiom,
( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_add__nonneg__pos_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_add__pos__nonneg_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_pordered__ring_Oring__simps_I31_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| V_a = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_pordered__ring_Oring__simps_I32_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_inf__sup__ord_I3_J_0,axiom,
( ~ class_Lattices_Olattice(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a))) ) ).
cnf(cls_inf__sup__ord_I4_J_0,axiom,
( ~ class_Lattices_Olattice(T_a)
| hBOOL(hAPP(c_lessequals(V_y,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a))) ) ).
cnf(cls_le__supI_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a),T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_x)) ) ).
cnf(cls_sup__ge1_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a))) ) ).
cnf(cls_sup__ge2_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(V_y,T_a),c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a))) ) ).
cnf(cls_sup__least_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_y,V_z,T_a),T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_z,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_le__sup__iff_2,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| hBOOL(hAPP(c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_z)) ) ).
cnf(cls_divide__le__eq_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_divide__le__eq_7,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_le__divide__eq_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_le__divide__eq_7,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_mult__right_Osetsum_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),c_Finite__Set_Osetsum(V_g,V_A,T_b,T_a)) = c_Finite__Set_Osetsum(c_COMBB(c_HOL_Otimes__class_Otimes(V_x,T_a),V_g,T_a,T_a,T_b),V_A,T_b,T_a) ) ).
cnf(cls_setsum__right__distrib_0,axiom,
( ~ class_Ring__and__Field_Osemiring__0(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_r,T_a),c_Finite__Set_Osetsum(V_f,V_A,T_b,T_a)) = c_Finite__Set_Osetsum(c_COMBB(c_HOL_Otimes__class_Otimes(V_r,T_a),V_f,T_a,T_a,T_b),V_A,T_b,T_a) ) ).
cnf(cls_neg__0__le__iff__le_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_neg__0__le__iff__le_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult_OscaleR__left_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_a),T_a),V_b) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) ) ).
cnf(cls_mult__left_OscaleR_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_x),T_a),V_y) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)) ) ).
cnf(cls_mult_OscaleR__right_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_b)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) ) ).
cnf(cls_mult__right_OscaleR_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_x)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),V_x)) ) ).
cnf(cls_mult__scaleR__left_0,axiom,
( ~ class_RealVector_Oreal__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_x),T_a),V_y) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)) ) ).
cnf(cls_mult__scaleR__right_0,axiom,
( ~ class_RealVector_Oreal__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_y)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)) ) ).
cnf(cls_mod__diff__left__eq_0,axiom,
( ~ class_Divides_Oring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(V_a,V_c,T_a)),V_b),V_c,T_a) ) ).
cnf(cls_mod__diff__right__eq_0,axiom,
( ~ class_Divides_Oring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) ) ).
cnf(cls_add__nonneg__nonneg_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_less__Suc__eq__le_1,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),hAPP(c_Suc,V_n)))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_less__Suc__eq__le_0,axiom,
( hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),hAPP(c_Suc,V_n))) ) ).
cnf(cls_le__less__Suc__eq_1,axiom,
( ~ hBOOL(hAPP(c_lessequals(V_x,tc_nat),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),hAPP(c_Suc,V_x))) ) ).
cnf(cls_divide_OscaleR_0,axiom,
( ~ class_RealVector_Oreal__normed__field(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_x),V_y,T_a) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)) ) ).
cnf(cls_sup__eq__if_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_a,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_zero__le__double__add__iff__zero__le__single__add_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a))) ) ).
cnf(cls_zero__le__double__add__iff__zero__le__single__add_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_DERIV__mult__lemma_0,axiom,
( ~ class_RealVector_Oreal__field(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_d)),V_h,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_d),V_h,T_a))),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_c),V_h,T_a),T_a),V_d)) ) ).
cnf(cls_mod__mod__trivial_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(c_Divides_Odiv__class_Omod(V_a,V_b,T_a),V_b,T_a) = c_Divides_Odiv__class_Omod(V_a,V_b,T_a) ) ).
cnf(cls_add__strict__increasing_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_pordered__ring_Oring__simps_I38_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_a)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_setsum__subtractf_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| c_Finite__Set_Osetsum(c_COMBS(c_COMBB(c_HOL_Ominus__class_Ominus(T_a),V_f,T_a,tc_fun(T_a,T_a),T_b),V_g,T_b,T_a,T_a),V_A,T_b,T_a) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_Finite__Set_Osetsum(V_f,V_A,T_b,T_a)),c_Finite__Set_Osetsum(V_g,V_A,T_b,T_a)) ) ).
cnf(cls_le__minus__self__iff_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_le__minus__self__iff_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_minus__le__self__iff_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_a)) ) ).
cnf(cls_minus__le__self__iff_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_less__eq__neg__nonpos_0,axiom,
( ~ class_OrderedGroup_Oordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_less__eq__neg__nonpos_1,axiom,
( ~ class_OrderedGroup_Oordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_neg__less__eq__nonneg_0,axiom,
( ~ class_OrderedGroup_Oordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_a)) ) ).
cnf(cls_neg__less__eq__nonneg_1,axiom,
( ~ class_OrderedGroup_Oordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_eq__add__iff1_1,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),V_e)),V_c)) ) ).
cnf(cls_pordered__ring_Oring__simps_I31_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ring_Oring__simps_I32_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| V_a = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_bounded__bilinear_Odiff__left_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_a_H)),V_b) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_c),hAPP(hAPP(V_prod,V_a),V_b)),hAPP(hAPP(V_prod,V_a_H),V_b))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_bounded__bilinear_Odiff__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_b_H)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_c),hAPP(hAPP(V_prod,V_a),V_b)),hAPP(hAPP(V_prod,V_a),V_b_H))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_eq__add__iff1_0,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),V_e)),V_c) = V_d ) ).
cnf(cls_eq__add__iff2_0,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d)
| V_c = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a),T_a),V_e)),V_d) ) ).
cnf(cls_pordered__ring_Oring__simps_I29_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I30_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_scaleR_OscaleR__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_b)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_b)) ) ).
cnf(cls_scaleR__left__commute_0,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_b,T_a),V_x)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_b,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_x)) ) ).
cnf(cls_scaleR__right_OscaleR_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_ra,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_x)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_ra,T_a),V_x)) ) ).
cnf(cls_mod__by__0_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(V_a,c_HOL_Ozero__class_Ozero(T_a),T_a) = V_a ) ).
cnf(cls_mod__self_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(V_a,V_a,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide__le__0__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__le__0__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__left__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_c,V_a,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_c,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_divide__left__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_c,V_a,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_c,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a)) ) ).
cnf(cls_ring_Oring__simps_I23_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I24_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_frac__less2_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_x,V_z,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_y,V_w,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_w,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_w))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_nonzero__minus__divide__right_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)) = c_HOL_Oinverse__class_Odivide(V_a,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a)
| V_b = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_neg__less__0__iff__less_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_neg__less__0__iff__less_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mod__0_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(c_HOL_Ozero__class_Ozero(T_a),V_a,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I26_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b),T_b),V_c) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I24_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c)) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I28_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I27_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_add__nonneg__eq__0__iff_2,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Ozero__class_Ozero(T_a)),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_class__semiring_Omul__c_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y) = hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_x) ) ).
cnf(cls_class__semiring_Osemiring__rules_I7_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_a) ) ).
cnf(cls_sup__idem_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,V_x,T_a) = V_x ) ).
cnf(cls_xt1_I10_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_z,T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_z,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x)) ) ).
cnf(cls_order__less__trans_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_neg__0__equal__iff__equal_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| c_HOL_Ozero__class_Ozero(T_a) != hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)
| c_HOL_Ozero__class_Ozero(T_a) = V_a ) ).
cnf(cls_COMBB__def_0,axiom,
hAPP(c_COMBB(V_P,V_Q,T_b,T_a,T_c),V_R) = hAPP(V_P,hAPP(V_Q,V_R)) ).
cnf(cls_COMBS__def_0,axiom,
hAPP(c_COMBS(V_P,V_Q,T_b,T_c,T_a),V_R) = hAPP(hAPP(V_P,V_R),hAPP(V_Q,V_R)) ).
cnf(cls_less__le__not__le_1,axiom,
( ~ class_Orderings_Opreorder(T_a)
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_not__leE_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_linorder__antisym__conv2_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_x)) ) ).
cnf(cls_linorder__antisym__conv1_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_x))
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_x)) ) ).
cnf(cls_linorder__not__less_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_linorder__not__less_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_linorder__not__le_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x))
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_linorder__not__le_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x)) ) ).
cnf(cls_option_Osplit_0,axiom,
( ~ hBOOL(hAPP(V_P,c_Option_Ooption_Ooption__case(V_f1,V_f2,c_Option_Ooption_ONone(t_b),t_a,t_b)))
| hBOOL(hAPP(V_P,V_f1)) ) ).
cnf(cls_zero__le__divide__iff_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_zero__le__divide__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_zero__le__divide__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_zero__le__divide__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a))) ) ).
cnf(cls_atLeastSucLessThan__greaterThanLessThan_0,axiom,
c_SetInterval_Oord__class_OatLeastLessThan(hAPP(c_Suc,V_l),V_u,tc_nat) = c_SetInterval_Oord__class_OgreaterThanLessThan(V_l,V_u,tc_nat) ).
cnf(cls_add__strict__increasing2_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_field_Ofield__eq__simps_I30_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_a = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_field_Ofield__eq__simps_I29_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_eq__eqI_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_x) != hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x_H),V_y_H)
| V_x_H = V_y_H ) ).
cnf(cls_eq__eqI_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_xa),V_y) != hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_x)
| V_xa = V_y ) ).
cnf(cls_field_Ofield__eq__simps_I36_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_mult__right__le__imp__le_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_mult__left__le__imp__le_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__le__cancel__left__pos_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_mult__le__cancel__left__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_mult__le__cancel__left__neg_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__le__cancel__left__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_vector__space_Oscale__right__imp__eq_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),V_x) != hAPP(hAPP(V_scale,V_b),V_x)
| V_x = c_HOL_Ozero__class_Ozero(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| V_a = V_b ) ).
cnf(cls_vector__space_Oscale__cancel__right_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),V_x) != hAPP(hAPP(V_scale,V_b),V_x)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| V_x = c_HOL_Ozero__class_Ozero(T_b)
| V_a = V_b ) ).
cnf(cls_vector__space_Oscale__left__imp__eq_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),V_x) != hAPP(hAPP(V_scale,V_a),V_y)
| V_a = c_HOL_Ozero__class_Ozero(T_a)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| V_x = V_y ) ).
cnf(cls_vector__space_Oscale__cancel__left_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),V_x) != hAPP(hAPP(V_scale,V_a),V_y)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| V_a = c_HOL_Ozero__class_Ozero(T_a)
| V_x = V_y ) ).
cnf(cls_vector__space_Oscale__eq__0__iff_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),V_x) != c_HOL_Ozero__class_Ozero(T_b)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b)
| V_x = c_HOL_Ozero__class_Ozero(T_b)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sup__left__commute_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,c_Lattices_Oupper__semilattice__class_Osup(V_y,V_z,T_a),T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_y,c_Lattices_Oupper__semilattice__class_Osup(V_x,V_z,T_a),T_a) ) ).
cnf(cls_sup__assoc_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),V_z,T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_x,c_Lattices_Oupper__semilattice__class_Osup(V_y,V_z,T_a),T_a) ) ).
cnf(cls_inf__sup__aci_I7_J_0,axiom,
( ~ class_Lattices_Olattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,c_Lattices_Oupper__semilattice__class_Osup(V_y,V_z,T_a),T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_y,c_Lattices_Oupper__semilattice__class_Osup(V_x,V_z,T_a),T_a) ) ).
cnf(cls_inf__sup__aci_I6_J_0,axiom,
( ~ class_Lattices_Olattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),V_z,T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_x,c_Lattices_Oupper__semilattice__class_Osup(V_y,V_z,T_a),T_a) ) ).
cnf(cls_minus__divide__divide_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a) = c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a) ) ).
cnf(cls_divide_Ominus_0,axiom,
( ~ class_RealVector_Oreal__normed__field(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x),V_y,T_a) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)) ) ).
cnf(cls_mult_Oprod__diff__prod_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_a),T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_y),V_b))),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_a),T_a),V_b))),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_y),V_b))) ) ).
cnf(cls_scaleR_Ozero__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_scaleR__eq__0__iff_2,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_lessThan__eq__iff_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| c_SetInterval_Oord__class_OlessThan(V_x,T_a) != c_SetInterval_Oord__class_OlessThan(V_y,T_a)
| V_x = V_y ) ).
cnf(cls_eq__neg__iff__add__eq__0_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)),V_b) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_ab__left__minus_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),V_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_right__minus_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_left__minus_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),V_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I32_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I31_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_add__neg__neg_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_eq__iff__diff__eq__0_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = V_b ) ).
cnf(cls_right__minus__eq_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = V_b ) ).
cnf(cls_class__ringb_Osubr0__iff_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y) != c_HOL_Ozero__class_Ozero(T_a)
| V_x = V_y ) ).
cnf(cls_sum__squares__le__zero__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a)
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y)),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_sum__squares__le__zero__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| V_x = c_HOL_Ozero__class_Ozero(T_a)
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y)),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__right_Odiff_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),V_y)) ) ).
cnf(cls_mult_Odiff__right_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_b_H)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b_H)) ) ).
cnf(cls_mult_Odiff__left_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_a_H),T_a),V_b) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a_H,T_a),V_b)) ) ).
cnf(cls_mult__left_Odiff_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_x),V_y),T_a),V_ya) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_ya)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_ya)) ) ).
cnf(cls_divide__zero__left_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| c_HOL_Oinverse__class_Odivide(c_HOL_Ozero__class_Ozero(T_a),V_a,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide__eq__eq_1,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(V_b,c_HOL_Ozero__class_Ozero(T_a),T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide__zero_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(V_a,c_HOL_Ozero__class_Ozero(T_a),T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide_Ozero_0,axiom,
( ~ class_RealVector_Oreal__normed__field(T_a)
| c_HOL_Oinverse__class_Odivide(c_HOL_Ozero__class_Ozero(T_a),V_y,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_scaleR__right_Osetsum_0,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),c_Finite__Set_Osetsum(V_g,V_A,T_b,T_a)) = c_Finite__Set_Osetsum(c_COMBB(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_g,T_a,T_a,T_b),V_A,T_b,T_a) ) ).
cnf(cls_Suc__lessI_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_Suc,V_m),tc_nat),V_n))
| hAPP(c_Suc,V_m) = V_n
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_not__less__eq__eq_0,axiom,
( hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),V_m))
| hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_not__less__eq__eq_1,axiom,
( ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),V_m)) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I15_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I14_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),V_b) = hAPP(hAPP(V_plus,V_b),V_a)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I13_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,V_b),hAPP(hAPP(V_plus,V_a),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I12_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(V_uminus,V_b)) = hAPP(hAPP(V_minus,V_a),V_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I11_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(V_uminus,V_a)),V_b) = hAPP(hAPP(V_minus,V_b),V_a)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I10_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_c)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I9_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I8_J_0,axiom,
( ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| V_a = hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_b) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I8_J_1,axiom,
( ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I7_J_0,axiom,
( ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I7_J_1,axiom,
( ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a)
| V_a = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_b) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I6_J_0,axiom,
( hAPP(hAPP(V_minus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I5_J_0,axiom,
( hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_divide__le__0__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__le__0__iff_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_pordered__ring_Oring__simps_I5_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I6_J_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_a),V_b)),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I7_J_0,axiom,
( hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I8_J_0,axiom,
( hAPP(hAPP(V_minus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I9_J_1,axiom,
( ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| V_a = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_b) ) ).
cnf(cls_pordered__ring_Oring__simps_I9_J_0,axiom,
( ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ring_Oring__simps_I10_J_1,axiom,
( ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_c),V_b)),V_b) = V_c ) ).
cnf(cls_pordered__ring_Oring__simps_I10_J_0,axiom,
( ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a)
| V_a = hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_b) ) ).
cnf(cls_pordered__ring_Oring__simps_I11_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_minus,V_a),V_b)),V_c) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I12_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_minus,V_b),V_c)) = hAPP(hAPP(V_minus,hAPP(hAPP(V_plus,V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I13_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(V_uminus,V_a)),V_b) = hAPP(hAPP(V_minus,V_b),V_a)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I14_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(V_uminus,V_b)) = hAPP(hAPP(V_minus,V_a),V_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I15_J_0,axiom,
( hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_b)),hAPP(hAPP(V_times,V_a),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I16_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_c)),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I17_J_0,axiom,
( hAPP(hAPP(V_times,hAPP(hAPP(V_times,V_a),V_b)),V_c) = hAPP(hAPP(V_times,V_a),hAPP(hAPP(V_times,V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I18_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c)) = hAPP(hAPP(V_plus,V_b),hAPP(hAPP(V_plus,V_a),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I19_J_0,axiom,
( hAPP(hAPP(V_plus,V_a),V_b) = hAPP(hAPP(V_plus,V_b),V_a)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I20_J_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_bounded__bilinear_Ozero__left_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| ~ class_RealVector_Oreal__normed__vector(T_b)
| hAPP(hAPP(V_prod,c_HOL_Ozero__class_Ozero(T_a)),V_b) = c_HOL_Ozero__class_Ozero(T_c)
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_bounded__bilinear_Ominus__left_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),V_b) = hAPP(c_HOL_Ouminus__class_Ouminus(T_c),hAPP(hAPP(V_prod,V_a),V_b))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_bounded__bilinear_Ozero__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| ~ class_RealVector_Oreal__normed__vector(T_b)
| hAPP(hAPP(V_prod,V_a),c_HOL_Ozero__class_Ozero(T_b)) = c_HOL_Ozero__class_Ozero(T_c)
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_bounded__bilinear_Ominus__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_b)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_c),hAPP(hAPP(V_prod,V_a),V_b))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_mod__minus__cong_0,axiom,
( ~ class_Divides_Oring__div(T_a)
| c_Divides_Odiv__class_Omod(V_a,V_b,T_a) != c_Divides_Odiv__class_Omod(V_a_H,V_b,T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),V_b,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a_H),V_b,T_a) ) ).
cnf(cls_mod__mult__eq_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(c_Divides_Odiv__class_Omod(V_a,V_c,T_a),T_a),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) ) ).
cnf(cls_divide__less__eq_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a)) ) ).
cnf(cls_divide__less__eq_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_divide__less__eq_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_divide__less__eq_7,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_less__divide__eq_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a))) ) ).
cnf(cls_less__divide__eq_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_less__divide__eq_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_less__divide__eq_7,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_field_Ofield__eq__simps_I23_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I23_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I24_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I24_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_o__eq__dest__lhs_0,axiom,
hAPP(V_a,hAPP(V_b,V_v)) = hAPP(c_Fun_Ocomp(V_a,V_b,T_c,T_b,T_a),V_v) ).
cnf(cls_o__apply_0,axiom,
hAPP(c_Fun_Ocomp(V_f,V_g,T_b,T_a,T_c),V_x) = hAPP(V_f,hAPP(V_g,V_x)) ).
cnf(cls_nat_Oinject_0,axiom,
( hAPP(c_Suc,V_nat) != hAPP(c_Suc,V_nat_H)
| V_nat = V_nat_H ) ).
cnf(cls_Suc__inject_0,axiom,
( hAPP(c_Suc,V_x) != hAPP(c_Suc,V_y)
| V_x = V_y ) ).
cnf(cls_less__add__iff1_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),V_e)),V_c),T_a),V_d))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d))) ) ).
cnf(cls_less__add__iff1_1,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),V_e)),V_c),T_a),V_d)) ) ).
cnf(cls_less__add__iff2_0,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a),T_a),V_e)),V_d)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d))) ) ).
cnf(cls_less__add__iff2_1,axiom,
( ~ class_Ring__and__Field_Opordered__ring(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_d)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a),T_a),V_e)),V_d))) ) ).
cnf(cls_eq__divide__eq_4,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Ozero__class_Ozero(T_a) = c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c),V_c,T_a) ) ).
cnf(cls_vector__space_Oscale__left__commute_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),hAPP(hAPP(V_scale,V_b),V_x)) = hAPP(hAPP(V_scale,V_b),hAPP(hAPP(V_scale,V_a),V_x))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_minus__unique_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a) = V_b ) ).
cnf(cls_eq__neg__iff__add__eq__0_1,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b) ) ).
cnf(cls_additive_Osetsum_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(V_f,c_Finite__Set_Osetsum(V_g,V_A,T_c,T_a)) = c_Finite__Set_Osetsum(c_COMBB(V_f,V_g,T_a,T_b,T_c),V_A,T_c,T_b)
| ~ c_RealVector_Oadditive(V_f,T_a,T_b) ) ).
cnf(cls_not__less__less__Suc__eq_0,axiom,
( V_n = V_m
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),hAPP(c_Suc,V_m)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m)) ) ).
cnf(cls_less__antisym_0,axiom,
( V_m = V_n
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),hAPP(c_Suc,V_m)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m)) ) ).
cnf(cls_less__SucE_0,axiom,
( V_m = V_n
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),hAPP(c_Suc,V_n))) ) ).
cnf(cls_mod__mult__cong_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(V_b,V_c,T_a) != c_Divides_Odiv__class_Omod(V_b_H,V_c,T_a)
| c_Divides_Odiv__class_Omod(V_a,V_c,T_a) != c_Divides_Odiv__class_Omod(V_a_H,V_c,T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a_H,T_a),V_b_H),V_c,T_a) ) ).
cnf(cls_add__frac__eq_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a)),c_HOL_Oinverse__class_Odivide(V_w,V_z,T_a)) = c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z)),hAPP(c_HOL_Otimes__class_Otimes(V_w,T_a),V_y)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_z),T_a)
| V_z = c_HOL_Ozero__class_Ozero(T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_divide__eq__eq_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| V_b = hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_c)
| V_c = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_eq__divide__eq_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_c) = V_b
| V_c = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_class__semiring_Osemiring__rules_I9_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_class__semiring_Osemiring__rules_I10_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__left_Ozero_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult_Ozero__left_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult_Ozero__right_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__right_Ozero_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_class__semiring_Omul__0_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__zero__left_0,axiom,
( ~ class_Ring__and__Field_Omult__zero(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__zero__right_0,axiom,
( ~ class_Ring__and__Field_Omult__zero(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__eq__0__iff_1,axiom,
( ~ class_Ring__and__Field_Oring__no__zero__divisors(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__eq__0__iff_2,axiom,
( ~ class_Ring__and__Field_Oring__no__zero__divisors(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_linorder__neq__iff_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_x)) ) ).
cnf(cls_order__less__le_1,axiom,
( ~ class_Orderings_Oorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_x)) ) ).
cnf(cls_less__not__refl_0,axiom,
~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_n)) ).
cnf(cls_nat__less__le_1,axiom,
~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),V_x)) ).
cnf(cls_order__less__irrefl_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_x)) ) ).
cnf(cls_less__le__not__le_2,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I22_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_a)),V_b) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_a)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I23_J_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_b)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_add__pos__pos_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_semigroup__add_Oadd__assoc_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_plus,V_a),V_b)),V_c) = hAPP(hAPP(V_plus,V_a),hAPP(hAPP(V_plus,V_b),V_c))
| ~ c_OrderedGroup_Osemigroup__add(V_plus,T_a) ) ).
cnf(cls_mod__diff__cong_0,axiom,
( ~ class_Divides_Oring__div(T_a)
| c_Divides_Odiv__class_Omod(V_b,V_c,T_a) != c_Divides_Odiv__class_Omod(V_b_H,V_c,T_a)
| c_Divides_Odiv__class_Omod(V_a,V_c,T_a) != c_Divides_Odiv__class_Omod(V_a_H,V_c,T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a_H),V_b_H),V_c,T_a) ) ).
cnf(cls_le__iff__diff__le__0_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_le__iff__diff__le__0_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_ring_Oring__simps_I20_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I20_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I19_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I19_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_lessequals(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_order__le__less_1,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_nat__less__le_0,axiom,
( hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_order__less__imp__le_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_termination__basic__simps_I5_J_0,axiom,
( hBOOL(hAPP(c_lessequals(V_x,tc_nat),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),V_y)) ) ).
cnf(cls_le__iff__sup_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) = V_y
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_le__iff__sup_1,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) != V_y
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_sup__absorb1_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) = V_x
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_zero__le__divide__iff_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_zero__le__divide__iff_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_frac__le_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_x,V_z,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_y,V_w,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_w,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_w))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_less__iff__diff__less__0_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_less__iff__diff__less__0_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_bounded__bilinear_OscaleR__left_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_a),V_a)),V_b) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_c),hAPP(hAPP(V_prod,V_a),V_b))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_bounded__bilinear_OscaleR__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,V_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_b),V_b)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_r,T_c),hAPP(hAPP(V_prod,V_a),V_b))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_mod__mult__right__eq_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) ) ).
cnf(cls_mod__mult__left__eq_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(c_Divides_Odiv__class_Omod(V_a,V_c,T_a),T_a),V_b),V_c,T_a) ) ).
cnf(cls_ring_Oring__simps_I25_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| V_a = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_ring_Oring__simps_I26_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_sum__squares__gt__zero__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y))))
| V_y = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_sum__squares__gt__zero__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_y))))
| V_x = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I20_J_0,axiom,
( ~ class_Ring__and__Field_Oring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I19_J_0,axiom,
( ~ class_Ring__and__Field_Oring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_sup__commute_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_y,V_x,T_a) ) ).
cnf(cls_inf__sup__aci_I5_J_0,axiom,
( ~ class_Lattices_Olattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_y,V_x,T_a) ) ).
cnf(cls_ring_Oeq__add__iff2_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c) != hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| V_c = hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_b),V_a)),V_e)),V_d) ) ).
cnf(cls_ring_Oeq__add__iff1_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_a),V_e)),V_c) != hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_b),V_e)),V_d)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a)
| hAPP(hAPP(V_plus,hAPP(hAPP(V_times,hAPP(hAPP(V_minus,V_a),V_b)),V_e)),V_c) = V_d ) ).
cnf(cls_divide__less__0__iff_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__less__0__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__less__0__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__less__0__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_fcomp__def_0,axiom,
hAPP(c_Fun_Ofcomp(V_f,V_g,t_a,T_c,T_b),v_x) = hAPP(V_g,hAPP(V_f,v_x)) ).
cnf(cls_field_Ofield__eq__simps_I30_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)),V_b) = V_c ) ).
cnf(cls_field_Ofield__eq__simps_I29_J_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_a = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_b) ) ).
cnf(cls_mult__divide__mult__cancel__right_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c),T_a) = c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)
| V_c = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__divide__mult__cancel__left_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b),T_a) = c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a)
| V_c = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_le__Suc__eq_2,axiom,
hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),hAPP(c_Suc,V_n))) ).
cnf(cls_field_Ofrac__eq__eq_1,axiom,
( hAPP(hAPP(V_times,V_x),V_z) != hAPP(hAPP(V_times,V_w),V_y)
| V_z = V_zero
| V_y = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_divide,V_x),V_y) = hAPP(hAPP(V_divide,V_w),V_z) ) ).
cnf(cls_field_Ofrac__eq__eq_0,axiom,
( hAPP(hAPP(V_divide,V_x),V_y) != hAPP(hAPP(V_divide,V_w),V_z)
| V_z = V_zero
| V_y = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_times,V_x),V_z) = hAPP(hAPP(V_times,V_w),V_y) ) ).
cnf(cls_sup__left__idem_0,axiom,
( ~ class_Lattices_Oupper__semilattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) ) ).
cnf(cls_inf__sup__aci_I8_J_0,axiom,
( ~ class_Lattices_Olattice(T_a)
| c_Lattices_Oupper__semilattice__class_Osup(V_x,c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a),T_a) = c_Lattices_Oupper__semilattice__class_Osup(V_x,V_y,T_a) ) ).
cnf(cls_finite_0,axiom,
( ~ class_Finite__Set_Ofinite_Ofinite(T_a)
| c_Finite__Set_Ofinite(V_A,T_a) ) ).
cnf(cls_mult__frac__num_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),V_z) = c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z),V_y,T_a) ) ).
cnf(cls_mult__strict__left__mono__comm_0,axiom,
( ~ class_Ring__and__Field_Oordered__comm__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_mult__less__cancel__right__disj_5,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__less__cancel__right__disj_4,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_mult__strict__right__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_mult__strict__right__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a)) ) ).
cnf(cls_mult__less__cancel__left__disj_5,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__less__cancel__left__disj_4,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_mult__strict__left__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_mult__strict__left__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a)) ) ).
cnf(cls_mult__less__cancel__left__pos_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_mult__less__cancel__left__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_mult__less__cancel__left__neg_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_mult__less__cancel__left__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_diff__minus__eq__add_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b) ) ).
cnf(cls_mod__mult__self2_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)),V_b,T_a) = c_Divides_Odiv__class_Omod(V_a,V_b,T_a) ) ).
cnf(cls_mod__mult__self1_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b)),V_b,T_a) = c_Divides_Odiv__class_Omod(V_a,V_b,T_a) ) ).
cnf(cls_setsum__negf_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| c_Finite__Set_Osetsum(c_COMBB(c_HOL_Ouminus__class_Ouminus(T_a),V_f,T_a,T_a,T_b),V_A,T_b,T_a) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_Finite__Set_Osetsum(V_f,V_A,T_b,T_a)) ) ).
cnf(cls_le__divide__eq_9,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b)) ) ).
cnf(cls_le__divide__eq_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_le__divide__eq_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a))) ) ).
cnf(cls_divide__le__eq_9,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c))) ) ).
cnf(cls_divide__le__eq_3,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_divide__le__eq_2,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a)) ) ).
cnf(cls_minus__equation__iff_1,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = V_b ) ).
cnf(cls_equation__minus__iff_1,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| V_a = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)) ) ).
cnf(cls_equation__minus__iff_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| V_b = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) ) ).
cnf(cls_double__compl_0,axiom,
( ~ class_Lattices_Oboolean__algebra(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x)) = V_x ) ).
cnf(cls_minus__minus_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)) = V_a ) ).
cnf(cls_ring_Oring__simps_I27_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c)),V_b)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I28_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I36_J_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_b)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I35_J_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Ouminus__class_Ouminus(T_b),V_a)),V_b) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_b),V_a)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_minus__diff__eq_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_b),V_a) ) ).
cnf(cls_add__less__le__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__cancel__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_add__le__less__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__cancel__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_Suc__mono_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_Suc,V_m),tc_nat),hAPP(c_Suc,V_n)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_Suc__less__eq_0,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_Suc,V_m),tc_nat),hAPP(c_Suc,V_n))) ) ).
cnf(cls_less__Suc__eq__0__disj_3,axiom,
( hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_Suc,V_x),tc_nat),hAPP(c_Suc,V_n)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,tc_nat),V_n)) ) ).
cnf(cls_minus__less__iff_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),V_a)) ) ).
cnf(cls_minus__less__iff_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b)) ) ).
cnf(cls_less__minus__iff_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_less__minus__iff_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b))) ) ).
cnf(cls_minus__le__iff_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),V_a)) ) ).
cnf(cls_minus__le__iff_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b)) ) ).
cnf(cls_le__minus__iff_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_le__minus__iff_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b))) ) ).
cnf(cls_minus__add__distrib_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) ) ).
cnf(cls_minus__add_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)) ) ).
cnf(cls_le__add__right__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c))) ) ).
cnf(cls_add__strict__right__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__cancel__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_add__less__cancel__right_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_add__less__cancel__right_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c))) ) ).
cnf(cls_add__strict__left__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__cancel__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_add__less__cancel__left_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_add__less__cancel__left_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_b))) ) ).
cnf(cls_minus__add__cancel_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)) = V_b ) ).
cnf(cls_add__cancel__end_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_z)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_z)) = V_y ) ).
cnf(cls_add__strict__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__cancel__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_ring_Oring__simps_I35_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_a)
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_class__ringb_Oadd__mul__solve_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_w,T_a),V_y)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z)) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_w,T_a),V_z)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y))
| V_y = V_z
| V_w = V_x ) ).
cnf(cls_class__ringb_Onoteq__reduce_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_d)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))
| V_c = V_d
| V_a = V_b ) ).
cnf(cls_field_Ofield__eq__simps_I39_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_a)
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_mod__add__cong_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(V_b,V_c,T_a) != c_Divides_Odiv__class_Omod(V_b_H,V_c,T_a)
| c_Divides_Odiv__class_Omod(V_a,V_c,T_a) != c_Divides_Odiv__class_Omod(V_a_H,V_c,T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a_H),V_b_H),V_c,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I42_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I40_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_add__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I28_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_a)
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_class__semiring_Osemiring__rules_I1_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_m)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_m)) = hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),V_m) ) ).
cnf(cls_comm__semiring__class_Odistrib_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)) ) ).
cnf(cls_class__semiring_Omul__d_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z)) ) ).
cnf(cls_mult_Oadd__left_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a_H),T_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a_H,T_a),V_b)) ) ).
cnf(cls_mult__left_Oadd_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y),T_a),V_ya) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_ya)),hAPP(c_HOL_Otimes__class_Otimes(V_y,T_a),V_ya)) ) ).
cnf(cls_divide__strict__left__mono_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_c,V_a,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_c,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a)) ) ).
cnf(cls_divide__strict__left__mono__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_c,V_a,T_a),T_a),c_HOL_Oinverse__class_Odivide(V_c,V_b,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_field_Ofield__eq__simps_I25_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I25_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I26_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I26_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_neg__0__equal__iff__equal_1,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| c_HOL_Ozero__class_Ozero(T_a) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Ozero__class_Ozero(T_a)) ) ).
cnf(cls_mult__le__less__imp__less_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_mult__less__le__imp__less_0,axiom,
( ~ class_Ring__and__Field_Oordered__semiring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_d)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_lessequals(V_c,T_a),V_d))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_o__assoc_0,axiom,
c_Fun_Ocomp(V_f,c_Fun_Ocomp(V_g,V_h,T_d,T_c,T_a),T_c,T_b,T_a) = c_Fun_Ocomp(c_Fun_Ocomp(V_f,V_g,T_c,T_b,T_d),V_h,T_d,T_b,T_a) ).
cnf(cls_minus__mult__commute_0,axiom,
( ~ class_Ring__and__Field_Oring(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a),T_a),V_b) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) ) ).
cnf(cls_vector__space_Oscale__scale_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),hAPP(hAPP(V_scale,V_b),V_x)) = hAPP(hAPP(V_scale,hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)),V_x)
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_le__SucI_0,axiom,
( hBOOL(hAPP(c_lessequals(V_m,tc_nat),hAPP(c_Suc,V_n)))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_Suc__leD_0,axiom,
( hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_m),tc_nat),V_n)) ) ).
cnf(cls_minus__zero_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_neg__equal__zero_1,axiom,
( ~ class_OrderedGroup_Oordered__ab__group__add(T_a)
| hAPP(c_HOL_Ouminus__class_Ouminus(T_a),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I23_J_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I24_J_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I37_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c)) = hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_pordered__ring_Oring__simps_I39_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__mult(T_b)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),V_b),T_b),V_c) = hAPP(c_HOL_Otimes__class_Otimes(V_a,T_b),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_b),V_c))
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_Suc__n__not__le__n_0,axiom,
~ hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),V_n)) ).
cnf(cls_o__def_0,axiom,
hAPP(c_Fun_Ocomp(V_f,V_g,T_c,T_b,t_a),v_x) = hAPP(V_f,hAPP(V_g,v_x)) ).
cnf(cls_vector__space_Oscale__left__diff__distrib_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b)),V_x) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(V_scale,V_a),V_x)),hAPP(hAPP(V_scale,V_b),V_x))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_vector__space_Oscale__right__diff__distrib_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_x),V_y)) = hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),hAPP(hAPP(V_scale,V_a),V_x)),hAPP(hAPP(V_scale,V_a),V_y))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_double__add__le__zero__iff__single__add__le__zero_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_double__add__le__zero__iff__single__add__le__zero_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_Lim_Ominus__diff__minus_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b)) ) ).
cnf(cls_divide__le__eq_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a)) ) ).
cnf(cls_divide__le__eq_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a),T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),V_b))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_le__divide__eq_1,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a))) ) ).
cnf(cls_le__divide__eq_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c)) ) ).
cnf(cls_divide__neg__pos_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_divide__pos__neg_0,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_x)) ) ).
cnf(cls_divide__less__0__iff_4,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_a)) ) ).
cnf(cls_divide__less__0__iff_5,axiom,
( ~ class_Ring__and__Field_Oordered__field(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Oinverse__class_Odivide(V_a,V_b,T_a),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_order__eq__refl_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_x)) ) ).
cnf(cls_order__eq__iff_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_x)) ) ).
cnf(cls_eq__imp__le_0,axiom,
hBOOL(hAPP(c_lessequals(V_x,tc_nat),V_x)) ).
cnf(cls_le__trans_0,axiom,
( hBOOL(hAPP(c_lessequals(V_i,tc_nat),V_k))
| ~ hBOOL(hAPP(c_lessequals(V_j,tc_nat),V_k))
| ~ hBOOL(hAPP(c_lessequals(V_i,tc_nat),V_j)) ) ).
cnf(cls_le__refl_0,axiom,
hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_n)) ).
cnf(cls_order__le__less__trans_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_order__less__le__trans_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_z))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_order__trans_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_z))
| ~ hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_add__nonpos__nonpos_0,axiom,
( ~ class_OrderedGroup_Opordered__comm__monoid__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_xt1_I8_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_z,T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_z,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_xt1_I7_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_z,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_z,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x)) ) ).
cnf(cls_xt1_I6_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| hBOOL(hAPP(c_lessequals(V_z,T_a),V_x))
| ~ hBOOL(hAPP(c_lessequals(V_z,T_a),V_y))
| ~ hBOOL(hAPP(c_lessequals(V_y,T_a),V_x)) ) ).
cnf(cls_field_Odivide__self_0,axiom,
( hAPP(hAPP(V_divide,V_a),V_a) = V_one
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__eq__imp_0,axiom,
( V_c = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_c)),V_c) = V_a ) ).
cnf(cls_field_Oeq__divide__imp_0,axiom,
( V_c = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_a = hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_c)),V_c) ) ).
cnf(cls_field_Ofield__inverse_0,axiom,
( hAPP(hAPP(V_times,hAPP(V_inverse,V_a)),V_a) = V_one
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Oright__inverse__eq_0,axiom,
( V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,hAPP(hAPP(V_divide,V_a),V_b),V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_a = V_b ) ).
cnf(cls_field_Oright__inverse__eq_1,axiom,
( V_x = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_divide,V_x),V_x) = V_one ) ).
cnf(cls_field_Oadd__divide__eq__iff_0,axiom,
( hAPP(hAPP(V_plus,V_x),hAPP(hAPP(V_divide,V_y),V_z)) = hAPP(hAPP(V_divide,hAPP(hAPP(V_plus,hAPP(hAPP(V_times,V_z),V_x)),V_y)),V_z)
| V_z = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__add__eq__iff_0,axiom,
( hAPP(hAPP(V_plus,hAPP(hAPP(V_divide,V_x),V_z)),V_y) = hAPP(hAPP(V_divide,hAPP(hAPP(V_plus,V_x),hAPP(hAPP(V_times,V_z),V_y))),V_z)
| V_z = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odiff__divide__eq__iff_0,axiom,
( hAPP(hAPP(V_minus,V_x),hAPP(hAPP(V_divide,V_y),V_z)) = hAPP(hAPP(V_divide,hAPP(hAPP(V_minus,hAPP(hAPP(V_times,V_z),V_x)),V_y)),V_z)
| V_z = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Odivide__diff__eq__iff_0,axiom,
( hAPP(hAPP(V_minus,hAPP(hAPP(V_divide,V_x),V_z)),V_y) = hAPP(hAPP(V_divide,hAPP(hAPP(V_minus,V_x),hAPP(hAPP(V_times,V_z),V_y))),V_z)
| V_z = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__divide__eq__eq_0,axiom,
( V_c = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| V_b = hAPP(hAPP(V_times,hAPP(hAPP(V_divide,V_b),V_c)),V_c) ) ).
cnf(cls_field_Ononzero__eq__divide__eq_0,axiom,
( V_c = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a)
| hAPP(hAPP(V_times,hAPP(hAPP(V_divide,V_b),V_c)),V_c) = V_b ) ).
cnf(cls_field_Ononzero__inverse__eq__divide_0,axiom,
( hAPP(V_inverse,V_a) = hAPP(hAPP(V_divide,V_one),V_a)
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__minus__divide__right_0,axiom,
( hAPP(V_uminus,hAPP(hAPP(V_divide,V_a),V_b)) = hAPP(hAPP(V_divide,V_a),hAPP(V_uminus,V_b))
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__minus__divide__divide_0,axiom,
( hAPP(hAPP(V_divide,hAPP(V_uminus,V_a)),hAPP(V_uminus,V_b)) = hAPP(hAPP(V_divide,V_a),V_b)
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__mult__divide__cancel__left_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_b)),V_a) = V_b
| V_a = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ononzero__mult__divide__cancel__right_0,axiom,
( hAPP(hAPP(V_divide,hAPP(hAPP(V_times,V_a),V_b)),V_b) = V_a
| V_b = V_zero
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_atMost__eq__iff_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| c_SetInterval_Oord__class_OatMost(V_x,T_a) != c_SetInterval_Oord__class_OatMost(V_y,T_a)
| V_x = V_y ) ).
cnf(cls_mod__mult__self1__is__0_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_a),V_b,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mod__mult__self2__is__0_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b),V_b,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_additive_Ominus_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(V_f,hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_x)) = hAPP(c_HOL_Ouminus__class_Ouminus(T_b),hAPP(V_f,V_x))
| ~ c_RealVector_Oadditive(V_f,T_a,T_b) ) ).
cnf(cls_additive_Ozero_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(V_f,c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_b)
| ~ c_RealVector_Oadditive(V_f,T_a,T_b) ) ).
cnf(cls_le__less__Suc__eq_0,axiom,
( V_n = V_m
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),hAPP(c_Suc,V_m)))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_scaleR__cancel__right_1,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)) = hAPP(c_RealVector_OscaleR__class_OscaleR(V_b,T_a),c_HOL_Ozero__class_Ozero(T_a)) ) ).
cnf(cls_scaleR__cancel__right_0,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_x) != hAPP(c_RealVector_OscaleR__class_OscaleR(V_b,T_a),V_x)
| V_x = c_HOL_Ozero__class_Ozero(T_a)
| V_a = V_b ) ).
cnf(cls_sum__squares__eq__zero__iff_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))),hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),c_HOL_Ozero__class_Ozero(T_a))) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mult__less__cancel__left__disj_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__less__cancel__left__disj_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b))) ) ).
cnf(cls_mult__less__cancel__right__disj_1,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_mult__less__cancel__right__disj_2,axiom,
( ~ class_Ring__and__Field_Oordered__ring__strict(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_c,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c))) ) ).
cnf(cls_eq__divide__eq_2,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Ozero__class_Ozero(T_a) = c_HOL_Oinverse__class_Odivide(V_b,c_HOL_Ozero__class_Ozero(T_a),T_a) ) ).
cnf(cls_neg__0__less__iff__less_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_neg__0__less__iff__less_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_HOL_Ozero__class_Ozero(T_a),T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_frac__eq__eq_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a) != c_HOL_Oinverse__class_Odivide(V_w,V_z,T_a)
| V_z = c_HOL_Ozero__class_Ozero(T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z) = hAPP(c_HOL_Otimes__class_Otimes(V_w,T_a),V_y) ) ).
cnf(cls_frac__eq__eq_1,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z) != hAPP(c_HOL_Otimes__class_Otimes(V_w,T_a),V_y)
| V_z = c_HOL_Ozero__class_Ozero(T_a)
| V_y = c_HOL_Ozero__class_Ozero(T_a)
| c_HOL_Oinverse__class_Odivide(V_x,V_y,T_a) = c_HOL_Oinverse__class_Odivide(V_w,V_z,T_a) ) ).
cnf(cls_mod__mult__mult1_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_a),hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),V_b),T_a) = hAPP(c_HOL_Otimes__class_Otimes(V_c,T_a),c_Divides_Odiv__class_Omod(V_a,V_b,T_a)) ) ).
cnf(cls_mod__mult__mult2_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c),T_a) = hAPP(c_HOL_Otimes__class_Otimes(c_Divides_Odiv__class_Omod(V_a,V_b,T_a),T_a),V_c) ) ).
cnf(cls_ring_Oring__simps_I22_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I22_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I21_J_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I21_J_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__group__add(T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_b),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_b),V_c),V_b)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b),T_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_less__minus__self__iff_0,axiom,
( ~ class_Ring__and__Field_Oordered__idom(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a))) ) ).
cnf(cls_less__minus__self__iff_1,axiom,
( ~ class_Ring__and__Field_Oordered__idom(T_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),c_HOL_Ozero__class_Ozero(T_a))) ) ).
cnf(cls_Suc__le__mono_1,axiom,
( hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),hAPP(c_Suc,V_m)))
| ~ hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_m)) ) ).
cnf(cls_Suc__le__mono_0,axiom,
( hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_m))
| ~ hBOOL(hAPP(c_lessequals(hAPP(c_Suc,V_n),tc_nat),hAPP(c_Suc,V_m))) ) ).
cnf(cls_divide__eq__eq_4,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| ~ class_Ring__and__Field_Odivision__by__zero(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(c_HOL_Otimes__class_Otimes(c_HOL_Ozero__class_Ozero(T_a),T_a),V_c),V_c,T_a) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_fcomp__assoc_0,axiom,
c_Fun_Ofcomp(c_Fun_Ofcomp(V_f,V_g,T_a,T_d,T_c),V_h,T_a,T_c,T_b) = c_Fun_Ofcomp(V_f,c_Fun_Ofcomp(V_g,V_h,T_d,T_c,T_b),T_a,T_d,T_b) ).
cnf(cls_order__less__asym_H_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b)) ) ).
cnf(cls_order__less__asym_0,axiom,
( ~ class_Orderings_Opreorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y)) ) ).
cnf(cls_linorder__linear_0,axiom,
( ~ class_Orderings_Olinorder(T_a)
| hBOOL(hAPP(c_lessequals(V_y,T_a),V_x))
| hBOOL(hAPP(c_lessequals(V_x,T_a),V_y)) ) ).
cnf(cls_nat__le__linear_0,axiom,
( hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_m))
| hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_not__less__iff__gr__or__eq_1,axiom,
( ~ class_Orderings_Olinorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_x,T_a),V_y))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_y,T_a),V_x)) ) ).
cnf(cls_xt1_I9_J_0,axiom,
( ~ class_Orderings_Oorder(T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_b,T_a),V_a)) ) ).
cnf(cls_mult_Oadd__right_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_b_H)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b)),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_b_H)) ) ).
cnf(cls_mult__right_Oadd_0,axiom,
( ~ class_RealVector_Oreal__normed__algebra(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),V_x)),hAPP(c_HOL_Otimes__class_Otimes(V_xa,T_a),V_y)) ) ).
cnf(cls_class__semiring_Osemiring__rules_I8_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_c)) ) ).
cnf(cls_zmod__simps_I1_J_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(V_a,V_c,T_a)),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) ) ).
cnf(cls_zmod__simps_I2_J_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) ) ).
cnf(cls_add__minus__cancel_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_a)),V_b)) = V_b ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I29_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_pordered__ab__group__add_Ogroup__simps_I27_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c))
| ~ c_OrderedGroup_Opordered__ab__group__add(V_minus,V_uminus,V_zero,V_plus,V_less__eq,V_less,T_a) ) ).
cnf(cls_semigroup__add__class_Oaxioms_0,axiom,
( ~ class_OrderedGroup_Osemigroup__add(T_a)
| c_OrderedGroup_Osemigroup__add(c_HOL_Oplus__class_Oplus(T_a),T_a) ) ).
cnf(cls_mod__add__self2_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_b,T_a) = c_Divides_Odiv__class_Omod(V_a,V_b,T_a) ) ).
cnf(cls_mod__add__self1_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_a),V_b,T_a) = c_Divides_Odiv__class_Omod(V_a,V_b,T_a) ) ).
cnf(cls_add__le__cancel__right_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_add__le__cancel__right_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c))) ) ).
cnf(cls_add__le__cancel__left_1,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_add__le__cancel__left_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(T_a)
| hBOOL(hAPP(c_lessequals(V_a,T_a),V_b))
| ~ hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_b))) ) ).
cnf(cls_add__right__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_add__left__mono_0,axiom,
( ~ class_OrderedGroup_Opordered__ab__semigroup__add(T_a)
| hBOOL(hAPP(c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a),T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_b)))
| ~ hBOOL(hAPP(c_lessequals(V_a,T_a),V_b)) ) ).
cnf(cls_pordered__ring_Oring__simps_I41_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_a)
| ~ c_Ring__and__Field_Opordered__ring(V_plus,V_zero,V_less__eq,V_less,V_times,V_minus,V_uminus,T_a) ) ).
cnf(cls_vector__space_Oscale__right__distrib_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_x),V_y)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(V_scale,V_a),V_x)),hAPP(hAPP(V_scale,V_a),V_y))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_vector__space_Oscale__left__distrib_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_b)
| ~ class_Ring__and__Field_Ofield(T_a)
| hAPP(hAPP(V_scale,hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),V_x) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(V_scale,V_a),V_x)),hAPP(hAPP(V_scale,V_b),V_x))
| ~ c_RealVector_Ovector__space(V_scale,T_a,T_b) ) ).
cnf(cls_double__eq__0__iff_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a) != c_HOL_Ozero__class_Ozero(T_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_class__ringb_Oadd__r0__iff_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| V_x != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_a)
| V_a = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_ring_Oring__simps_I34_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_ring_Oring__simps_I36_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_Ring__and__Field_Oring(V_minus,V_uminus,V_zero,V_plus,V_times,T_a) ) ).
cnf(cls_additive_Oadd_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| ~ class_OrderedGroup_Oab__group__add(T_b)
| hAPP(V_f,hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(V_f,V_x)),hAPP(V_f,V_y))
| ~ c_RealVector_Oadditive(V_f,T_a,T_b) ) ).
cnf(cls_class__ringb_Oadd__mul__solve_1,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_z)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_y)) ) ).
cnf(cls_class__ringb_Onoteq__reduce_1,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_c)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_d)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_d)),hAPP(c_HOL_Otimes__class_Otimes(V_x,T_a),V_c)) ) ).
cnf(cls_diff__add__cancel_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b)),V_b) = V_a ) ).
cnf(cls_double__eq__0__iff_1,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Ozero__class_Ozero(T_a)),c_HOL_Ozero__class_Ozero(T_a)) = c_HOL_Ozero__class_Ozero(T_a) ) ).
cnf(cls_mod__add__eq_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(V_a,V_c,T_a)),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) ) ).
cnf(cls_add__sup__distrib__right_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add__join(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a)),V_c) = c_Lattices_Oupper__semilattice__class_Osup(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c),T_a) ) ).
cnf(cls_add__sup__distrib__left_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add__join(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_Lattices_Oupper__semilattice__class_Osup(V_b,V_c,T_a)) = c_Lattices_Oupper__semilattice__class_Osup(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a) ) ).
cnf(cls_lordered__ab__group__add__class_Oadd__sup__inf__distribs_I4_J_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_Lattices_Oupper__semilattice__class_Osup(V_b,V_c,T_a)) = c_Lattices_Oupper__semilattice__class_Osup(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),T_a) ) ).
cnf(cls_lordered__ab__group__add__class_Oadd__sup__inf__distribs_I3_J_0,axiom,
( ~ class_OrderedGroup_Olordered__ab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Lattices_Oupper__semilattice__class_Osup(V_a,V_b,T_a)),V_c) = c_Lattices_Oupper__semilattice__class_Osup(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c),T_a) ) ).
cnf(cls_mod__add__right__eq_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(V_b,V_c,T_a)),V_c,T_a) ) ).
cnf(cls_mod__add__left__eq_0,axiom,
( ~ class_Divides_Osemiring__div(T_a)
| c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) = c_Divides_Odiv__class_Omod(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(V_a,V_c,T_a)),V_b),V_c,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I40_J_0,axiom,
( ~ class_OrderedGroup_Osemigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_field_Ofield__eq__simps_I38_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_b)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_a),V_c))
| ~ c_Ring__and__Field_Ofield(V_inverse,V_divide,V_one,V_times,V_minus,V_uminus,V_zero,V_plus,T_a) ) ).
cnf(cls_cancel__ab__semigroup__add__class_Oaxioms_0,axiom,
( ~ class_OrderedGroup_Ocancel__ab__semigroup__add(T_a)
| c_OrderedGroup_Ocancel__ab__semigroup__add__axioms(c_HOL_Oplus__class_Oplus(T_a),T_a) ) ).
cnf(cls_combine__common__factor_0,axiom,
( ~ class_Ring__and__Field_Osemiring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_a,T_a),V_e)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(V_b,T_a),V_e)),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_HOL_Otimes__class_Otimes(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),T_a),V_e)),V_c) ) ).
cnf(cls_Deriv_Oadd__diff__add_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_a),V_b)),hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),V_c),V_d)) ) ).
cnf(cls_scaleR__right_Oadd_0,axiom,
( ~ class_RealVector_Oreal__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_x)),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_y)) ) ).
cnf(cls_scaleR_Oadd__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_b_H)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_b)),hAPP(c_RealVector_OscaleR__class_OscaleR(V_a,T_a),V_b_H)) ) ).
cnf(cls_add__cancel__end_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_z)) != V_y
| V_x = hAPP(c_HOL_Ouminus__class_Ouminus(T_a),V_z) ) ).
cnf(cls_monoid__add__class_Oadd__0__right_0,axiom,
( ~ class_OrderedGroup_Omonoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_HOL_Ozero__class_Ozero(T_a)) = V_a ) ).
cnf(cls_add__0__left_0,axiom,
( ~ class_OrderedGroup_Omonoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Ozero__class_Ozero(T_a)),V_a) = V_a ) ).
cnf(cls_comm__monoid__add_Omult__1__right_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_HOL_Ozero__class_Ozero(T_a)) = V_a ) ).
cnf(cls_class__ringb_Oadd__r0__iff_1,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| V_x = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),c_HOL_Ozero__class_Ozero(T_a)) ) ).
cnf(cls_class__semiring_Oadd__0_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Ozero__class_Ozero(T_a)),V_x) = V_x ) ).
cnf(cls_comm__monoid__add_Omult__1_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Ozero__class_Ozero(T_a)),V_a) = V_a ) ).
cnf(cls_class__semiring_Osemiring__rules_I6_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),c_HOL_Ozero__class_Ozero(T_a)) = V_a ) ).
cnf(cls_class__semiring_Osemiring__rules_I5_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Ozero__class_Ozero(T_a)),V_a) = V_a ) ).
cnf(cls_add__diff__cancel_0,axiom,
( ~ class_OrderedGroup_Ogroup__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),V_b) = V_a ) ).
cnf(cls_bounded__bilinear_Oadd__right_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_b),V_b),V_b_H)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_c),hAPP(hAPP(V_prod,V_a),V_b)),hAPP(hAPP(V_prod,V_a),V_b_H))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_bounded__bilinear_Oadd__left_0,axiom,
( ~ class_RealVector_Oreal__normed__vector(T_b)
| ~ class_RealVector_Oreal__normed__vector(T_c)
| ~ class_RealVector_Oreal__normed__vector(T_a)
| hAPP(hAPP(V_prod,hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_a_H)),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_c),hAPP(hAPP(V_prod,V_a),V_b)),hAPP(hAPP(V_prod,V_a_H),V_b))
| ~ c_RealVector_Obounded__bilinear(V_prod,T_a,T_b,T_c) ) ).
cnf(cls_add__divide__distrib_0,axiom,
( ~ class_Ring__and__Field_Ofield(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b),V_c,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Oinverse__class_Odivide(V_a,V_c,T_a)),c_HOL_Oinverse__class_Odivide(V_b,V_c,T_a)) ) ).
cnf(cls_divide_Oadd_0,axiom,
( ~ class_RealVector_Oreal__normed__field(T_a)
| c_HOL_Oinverse__class_Odivide(hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y),V_ya,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_HOL_Oinverse__class_Odivide(V_x,V_ya,T_a)),c_HOL_Oinverse__class_Odivide(V_y,V_ya,T_a)) ) ).
cnf(cls_setsum__addf_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(T_a),V_f,T_a,tc_fun(T_a,T_a),T_b),V_g,T_b,T_a,T_a),V_A,T_b,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Finite__Set_Osetsum(V_f,V_A,T_b,T_a)),c_Finite__Set_Osetsum(V_g,V_A,T_b,T_a)) ) ).
cnf(cls_setsum__cong2_1,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_b)
| hAPP(V_f,c_ATP__Linkup_Osko__Finite__Set__Xsetsum__cong2__1__1(V_A,V_f,V_g,T_a,T_b)) != hAPP(V_g,c_ATP__Linkup_Osko__Finite__Set__Xsetsum__cong2__1__1(V_A,V_f,V_g,T_a,T_b))
| c_Finite__Set_Osetsum(V_f,V_A,T_a,T_b) = c_Finite__Set_Osetsum(V_g,V_A,T_a,T_b) ) ).
cnf(cls_setsum__cong_1,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_b)
| hAPP(V_f,c_ATP__Linkup_Osko__Finite__Set__Xsetsum__cong__1__1(V_x,V_f,V_g,T_a,T_b)) != hAPP(V_g,c_ATP__Linkup_Osko__Finite__Set__Xsetsum__cong__1__1(V_x,V_f,V_g,T_a,T_b))
| c_Finite__Set_Osetsum(V_f,V_x,T_a,T_b) = c_Finite__Set_Osetsum(V_g,V_x,T_a,T_b) ) ).
cnf(cls_setsum__lessThan__Suc_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OlessThan(hAPP(c_Suc,V_n),tc_nat),tc_nat,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OlessThan(V_n,tc_nat),tc_nat,T_a)),hAPP(V_f,V_n)) ) ).
cnf(cls_setsum__atMost__Suc_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatMost(hAPP(c_Suc,V_n),tc_nat),tc_nat,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatMost(V_n,tc_nat),tc_nat,T_a)),hAPP(V_f,hAPP(c_Suc,V_n))) ) ).
cnf(cls_setsum__add__split__nat__ivl__singleton_1,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(t_a),hAPP(V_f,V_m)),c_Finite__Set_Osetsum(V_g,c_SetInterval_Oord__class_OgreaterThanLessThan(V_m,V_n,tc_nat),tc_nat,t_a)) = c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,t_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(v_sko__FFT__Mirabelle__Xsetsum__add__split__nat__ivl__singleton__1(V_f,V_g,V_m,V_n),tc_nat),V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_setsum__add__split__nat__ivl__singleton_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(t_a),hAPP(V_f,V_m)),c_Finite__Set_Osetsum(V_g,c_SetInterval_Oord__class_OgreaterThanLessThan(V_m,V_n,tc_nat),tc_nat,t_a)) = c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,t_a)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),v_sko__FFT__Mirabelle__Xsetsum__add__split__nat__ivl__singleton__1(V_f,V_g,V_m,V_n)))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_setsum__add__split__nat__ivl__singleton_2,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(V_g,v_sko__FFT__Mirabelle__Xsetsum__add__split__nat__ivl__singleton__1(V_f,V_g,V_m,V_n)) != hAPP(V_f,v_sko__FFT__Mirabelle__Xsetsum__add__split__nat__ivl__singleton__1(V_f,V_g,V_m,V_n))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n))
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(t_a),hAPP(V_f,V_m)),c_Finite__Set_Osetsum(V_g,c_SetInterval_Oord__class_OgreaterThanLessThan(V_m,V_n,tc_nat),tc_nat,t_a)) = c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,t_a) ) ).
cnf(cls_setsum__ivl__cong_1,axiom,
( ~ class_HOL_Oord(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_b)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_xa,V_x,T_a),T_a,T_b) = c_Finite__Set_Osetsum(V_g,c_SetInterval_Oord__class_OatLeastLessThan(V_xa,V_x,T_a),T_a,T_b)
| hBOOL(hAPP(c_HOL_Oord__class_Oless(c_List_Osko__SetInterval__Xsetsum__ivl__cong__1__1(V_xa,V_x,V_f,V_g,T_a,T_b),T_a),V_x)) ) ).
cnf(cls_setsum__ivl__cong_0,axiom,
( ~ class_HOL_Oord(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_b)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_xa,V_x,T_a),T_a,T_b) = c_Finite__Set_Osetsum(V_g,c_SetInterval_Oord__class_OatLeastLessThan(V_xa,V_x,T_a),T_a,T_b)
| hBOOL(hAPP(c_lessequals(V_xa,T_a),c_List_Osko__SetInterval__Xsetsum__ivl__cong__1__1(V_xa,V_x,V_f,V_g,T_a,T_b))) ) ).
cnf(cls_setsum__shift__bounds__Suc__ivl_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(hAPP(c_Suc,V_m),hAPP(c_Suc,V_n),tc_nat),tc_nat,T_a) = c_Finite__Set_Osetsum(c_COMBB(V_f,c_Suc,tc_nat,T_a,tc_nat),c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,T_a) ) ).
cnf(cls_setsum__diff__nat__ivl_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Ominus__class_Ominus(T_a),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_p,tc_nat),tc_nat,T_a)),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,T_a)) = c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_n,V_p,tc_nat),tc_nat,T_a)
| ~ hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_p))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_finite__atLeastLessThan_0,axiom,
c_Finite__Set_Ofinite(c_SetInterval_Oord__class_OatLeastLessThan(V_l,V_u,tc_nat),tc_nat) ).
cnf(cls_fcomp__def__raw_0,axiom,
hAPP(c_Fun_Ofcomp(v_f,v_g,t_a,t_b,t_c),v_x) = hAPP(v_g,hAPP(v_f,v_x)) ).
cnf(cls_comp__def__raw_0,axiom,
hAPP(c_Fun_Ocomp(v_f,v_g,t_a,t_b,t_c),v_x) = hAPP(v_f,hAPP(v_g,v_x)) ).
cnf(cls_map__comp__def__raw_0,axiom,
c_Map_Omap__comp(v_f,v_g,v_k,t_a,t_b,t_c) = c_Option_Ooption_Ooption__case(c_Option_Ooption_ONone(t_b),v_f,hAPP(v_g,v_k),tc_Option_Ooption(t_b),t_a) ).
cnf(cls_setsum__head__upt__Suc_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(V_f,V_m)),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(hAPP(c_Suc,V_m),V_n,tc_nat),tc_nat,T_a))
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_setsum__op__ivl__Suc_1,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,hAPP(c_Suc,V_n),tc_nat),tc_nat,T_a) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,T_a)),hAPP(V_f,V_n))
| hBOOL(hAPP(c_HOL_Oord__class_Oless(V_n,tc_nat),V_m)) ) ).
cnf(cls_setsum__add__nat__ivl__singleton_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(V_f,V_m)),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OgreaterThanLessThan(V_m,V_n,tc_nat),tc_nat,T_a)) = c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,T_a)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_m,tc_nat),V_n)) ) ).
cnf(cls_setsum__ivl__cong_2,axiom,
( ~ class_HOL_Oord(T_a)
| ~ class_OrderedGroup_Ocomm__monoid__add(T_b)
| hAPP(V_f,c_List_Osko__SetInterval__Xsetsum__ivl__cong__1__1(V_xa,V_x,V_f,V_g,T_a,T_b)) != hAPP(V_g,c_List_Osko__SetInterval__Xsetsum__ivl__cong__1__1(V_xa,V_x,V_f,V_g,T_a,T_b))
| c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_xa,V_x,T_a),T_a,T_b) = c_Finite__Set_Osetsum(V_g,c_SetInterval_Oord__class_OatLeastLessThan(V_xa,V_x,T_a),T_a,T_b) ) ).
cnf(cls_le_I2_J_0,axiom,
hBOOL(hAPP(c_lessequals(v_k,tc_nat),v_n)) ).
cnf(cls_g_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(v_g,V_i) = hAPP(v_f,V_i)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_i,tc_nat),v_k))
| ~ hBOOL(hAPP(c_lessequals(v_m,tc_nat),V_i)) ) ).
cnf(cls_le_I1_J_0,axiom,
hBOOL(hAPP(c_lessequals(v_m,tc_nat),v_k)) ).
cnf(cls_setsum__add__nat__ivl_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_n,tc_nat),tc_nat,T_a)),c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_n,V_p,tc_nat),tc_nat,T_a)) = c_Finite__Set_Osetsum(V_f,c_SetInterval_Oord__class_OatLeastLessThan(V_m,V_p,tc_nat),tc_nat,T_a)
| ~ hBOOL(hAPP(c_lessequals(V_n,tc_nat),V_p))
| ~ hBOOL(hAPP(c_lessequals(V_m,tc_nat),V_n)) ) ).
cnf(cls_h_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(v_h,V_i) = hAPP(v_f,V_i)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_i,tc_nat),v_n))
| ~ hBOOL(hAPP(c_lessequals(v_k,tc_nat),V_i)) ) ).
cnf(cls_comm__monoid__add_Omult__commute_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_a) ) ).
cnf(cls_class__semiring_Oadd__c_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_x) ) ).
cnf(cls_class__semiring_Osemiring__rules_I24_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a) ) ).
cnf(cls_add__cancel__21_0,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_z)) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_u)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_z) = V_u ) ).
cnf(cls_add__right__cancel_0,axiom,
( ~ class_OrderedGroup_Ocancel__semigroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_a) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_a)
| V_b = V_c ) ).
cnf(cls_add__left__cancel_0,axiom,
( ~ class_OrderedGroup_Ocancel__semigroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)
| V_b = V_c ) ).
cnf(cls_add__imp__eq_0,axiom,
( ~ class_OrderedGroup_Ocancel__ab__semigroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)
| V_b = V_c ) ).
cnf(cls_class__ringb_Oadd__cancel_0,axiom,
( ~ class_Ring__and__Field_Oidom(T_a)
| ~ class_Int_Onumber__ring(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_z)
| V_y = V_z ) ).
cnf(cls_class__semiring_Osemiring__rules_I20_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_d)) ) ).
cnf(cls_comm__monoid__add_Omult__left__commute_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)) ) ).
cnf(cls_comm__monoid__add_Omult__assoc_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)) ) ).
cnf(cls_class__semiring_Oadd__a_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_y)),V_z) ) ).
cnf(cls_add__cancel__21_1,axiom,
( ~ class_OrderedGroup_Oab__group__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_y),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_x),V_z)) ) ).
cnf(cls_ab__semigroup__add__class_Oadd__ac_I1_J_0,axiom,
( ~ class_OrderedGroup_Oab__semigroup__add(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)) ) ).
cnf(cls_class__semiring_Osemiring__rules_I25_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)),V_d) ) ).
cnf(cls_class__semiring_Osemiring__rules_I23_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_c)),V_b) ) ).
cnf(cls_class__semiring_Osemiring__rules_I22_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_d)) ) ).
cnf(cls_class__semiring_Osemiring__rules_I21_J_0,axiom,
( ~ class_Ring__and__Field_Ocomm__semiring__1(T_a)
| hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_HOL_Oplus__class_Oplus(T_a),V_b),V_c)) ) ).
cnf(cls_CHAINED_0,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(v_h,V_i) = hAPP(v_f,V_i)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_i,tc_nat),v_n))
| ~ hBOOL(hAPP(c_lessequals(v_k,tc_nat),V_i)) ) ).
cnf(cls_CHAINED_0_01,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(t_a)
| hAPP(v_g,V_i) = hAPP(v_f,V_i)
| ~ hBOOL(hAPP(c_HOL_Oord__class_Oless(V_i,tc_nat),v_k))
| ~ hBOOL(hAPP(c_lessequals(v_m,tc_nat),V_i)) ) ).
cnf(cls_CHAINED_0_02,axiom,
hBOOL(hAPP(c_lessequals(v_k,tc_nat),v_n)) ).
cnf(cls_CHAINED_0_03,axiom,
hBOOL(hAPP(c_lessequals(v_m,tc_nat),v_k)) ).
cnf(tfree_tcs,negated_conjecture,
class_OrderedGroup_Ocomm__monoid__add(t_a) ).
cnf(cls_conjecture_0,negated_conjecture,
hAPP(hAPP(c_HOL_Oplus__class_Oplus(t_a),c_Finite__Set_Osetsum(v_g,c_SetInterval_Oord__class_OatLeastLessThan(v_m,v_k,tc_nat),tc_nat,t_a)),c_Finite__Set_Osetsum(v_h,c_SetInterval_Oord__class_OatLeastLessThan(v_k,v_n,tc_nat),tc_nat,t_a)) != c_Finite__Set_Osetsum(v_f,c_SetInterval_Oord__class_OatLeastLessThan(v_m,v_n,tc_nat),tc_nat,t_a) ).
cnf(clsrel_OrderedGroup_Ocomm__monoid__add_OrderedGroup_Oab__semigroup__add,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T)
| class_OrderedGroup_Oab__semigroup__add(T) ) ).
cnf(clsrel_OrderedGroup_Ocomm__monoid__add_OrderedGroup_Osemigroup__add,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T)
| class_OrderedGroup_Osemigroup__add(T) ) ).
cnf(clsrel_OrderedGroup_Ocomm__monoid__add_OrderedGroup_Omonoid__add,axiom,
( ~ class_OrderedGroup_Ocomm__monoid__add(T)
| class_OrderedGroup_Omonoid__add(T) ) ).
cnf(clsarity_fun__Lattices_Oupper__semilattice,axiom,
( class_Lattices_Oupper__semilattice(tc_fun(T_2,T_1))
| ~ class_Lattices_Olattice(T_1) ) ).
cnf(clsarity_fun__Lattices_Oboolean__algebra,axiom,
( class_Lattices_Oboolean__algebra(tc_fun(T_2,T_1))
| ~ class_Lattices_Oboolean__algebra(T_1) ) ).
cnf(clsarity_fun__Finite__Set_Ofinite_Ofinite,axiom,
( class_Finite__Set_Ofinite_Ofinite(tc_fun(T_2,T_1))
| ~ class_Finite__Set_Ofinite_Ofinite(T_1)
| ~ class_Finite__Set_Ofinite_Ofinite(T_2) ) ).
cnf(clsarity_fun__Orderings_Opreorder,axiom,
( class_Orderings_Opreorder(tc_fun(T_2,T_1))
| ~ class_Orderings_Opreorder(T_1) ) ).
cnf(clsarity_fun__Lattices_Olattice,axiom,
( class_Lattices_Olattice(tc_fun(T_2,T_1))
| ~ class_Lattices_Olattice(T_1) ) ).
cnf(clsarity_fun__Orderings_Oorder,axiom,
( class_Orderings_Oorder(tc_fun(T_2,T_1))
| ~ class_Orderings_Oorder(T_1) ) ).
cnf(clsarity_fun__HOL_Oord,axiom,
( class_HOL_Oord(tc_fun(T_2,T_1))
| ~ class_HOL_Oord(T_1) ) ).
cnf(clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add,axiom,
class_OrderedGroup_Opordered__cancel__ab__semigroup__add(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add__imp__le,axiom,
class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Oordered__comm__semiring__strict,axiom,
class_Ring__and__Field_Oordered__comm__semiring__strict(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Opordered__cancel__semiring,axiom,
class_Ring__and__Field_Opordered__cancel__semiring(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Oordered__semiring__strict,axiom,
class_Ring__and__Field_Oordered__semiring__strict(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add,axiom,
class_OrderedGroup_Opordered__ab__semigroup__add(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Opordered__comm__monoid__add,axiom,
class_OrderedGroup_Opordered__comm__monoid__add(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add,axiom,
class_OrderedGroup_Ocancel__ab__semigroup__add(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Ocancel__semigroup__add,axiom,
class_OrderedGroup_Ocancel__semigroup__add(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Opordered__semiring,axiom,
class_Ring__and__Field_Opordered__semiring(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Oordered__semiring,axiom,
class_Ring__and__Field_Oordered__semiring(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Ono__zero__divisors,axiom,
class_Ring__and__Field_Ono__zero__divisors(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Oordered__semidom,axiom,
class_Ring__and__Field_Oordered__semidom(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Ocomm__semiring__1,axiom,
class_Ring__and__Field_Ocomm__semiring__1(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Oab__semigroup__mult,axiom,
class_OrderedGroup_Oab__semigroup__mult(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Oab__semigroup__add,axiom,
class_OrderedGroup_Oab__semigroup__add(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Ocomm__semiring,axiom,
class_Ring__and__Field_Ocomm__semiring(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Ocomm__monoid__add,axiom,
class_OrderedGroup_Ocomm__monoid__add(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Osemigroup__mult,axiom,
class_OrderedGroup_Osemigroup__mult(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Osemigroup__add,axiom,
class_OrderedGroup_Osemigroup__add(tc_nat) ).
cnf(clsarity_nat__Lattices_Oupper__semilattice,axiom,
class_Lattices_Oupper__semilattice(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Osemiring__0,axiom,
class_Ring__and__Field_Osemiring__0(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Omult__mono1,axiom,
class_Ring__and__Field_Omult__mono1(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Omult__zero,axiom,
class_Ring__and__Field_Omult__zero(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Omult__mono,axiom,
class_Ring__and__Field_Omult__mono(tc_nat) ).
cnf(clsarity_nat__Ring__and__Field_Osemiring,axiom,
class_Ring__and__Field_Osemiring(tc_nat) ).
cnf(clsarity_nat__OrderedGroup_Omonoid__add,axiom,
class_OrderedGroup_Omonoid__add(tc_nat) ).
cnf(clsarity_nat__Divides_Osemiring__div,axiom,
class_Divides_Osemiring__div(tc_nat) ).
cnf(clsarity_nat__Orderings_Opreorder,axiom,
class_Orderings_Opreorder(tc_nat) ).
cnf(clsarity_nat__Orderings_Olinorder,axiom,
class_Orderings_Olinorder(tc_nat) ).
cnf(clsarity_nat__Lattices_Olattice,axiom,
class_Lattices_Olattice(tc_nat) ).
cnf(clsarity_nat__Orderings_Oorder,axiom,
class_Orderings_Oorder(tc_nat) ).
cnf(clsarity_nat__HOL_Oord,axiom,
class_HOL_Oord(tc_nat) ).
cnf(clsarity_bool__Lattices_Oupper__semilattice,axiom,
class_Lattices_Oupper__semilattice(tc_bool) ).
cnf(clsarity_bool__Lattices_Oboolean__algebra,axiom,
class_Lattices_Oboolean__algebra(tc_bool) ).
cnf(clsarity_bool__Finite__Set_Ofinite_Ofinite,axiom,
class_Finite__Set_Ofinite_Ofinite(tc_bool) ).
cnf(clsarity_bool__Orderings_Opreorder,axiom,
class_Orderings_Opreorder(tc_bool) ).
cnf(clsarity_bool__Lattices_Olattice,axiom,
class_Lattices_Olattice(tc_bool) ).
cnf(clsarity_bool__Orderings_Oorder,axiom,
class_Orderings_Oorder(tc_bool) ).
cnf(clsarity_bool__HOL_Oord,axiom,
class_HOL_Oord(tc_bool) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Opordered__cancel__ab__semigroup__add,axiom,
class_OrderedGroup_Opordered__cancel__ab__semigroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Opordered__ab__semigroup__add__imp__le,axiom,
class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__comm__semiring__strict,axiom,
class_Ring__and__Field_Oordered__comm__semiring__strict(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Opordered__cancel__semiring,axiom,
class_Ring__and__Field_Opordered__cancel__semiring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__join,axiom,
class_OrderedGroup_Olordered__ab__group__add__join(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__semiring__strict,axiom,
class_Ring__and__Field_Oordered__semiring__strict(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Opordered__ab__semigroup__add,axiom,
class_OrderedGroup_Opordered__ab__semigroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Opordered__comm__monoid__add,axiom,
class_OrderedGroup_Opordered__comm__monoid__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oring__no__zero__divisors,axiom,
class_Ring__and__Field_Oring__no__zero__divisors(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Ocancel__ab__semigroup__add,axiom,
class_OrderedGroup_Ocancel__ab__semigroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__ring__strict,axiom,
class_Ring__and__Field_Oordered__ring__strict(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Opordered__ab__group__add,axiom,
class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add,axiom,
class_OrderedGroup_Olordered__ab__group__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Oordered__ab__group__add,axiom,
class_OrderedGroup_Oordered__ab__group__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Ocancel__semigroup__add,axiom,
class_OrderedGroup_Ocancel__semigroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Opordered__semiring,axiom,
class_Ring__and__Field_Opordered__semiring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__semiring,axiom,
class_Ring__and__Field_Oordered__semiring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Ono__zero__divisors,axiom,
class_Ring__and__Field_Ono__zero__divisors(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Odivision__by__zero,axiom,
class_Ring__and__Field_Odivision__by__zero(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__semidom,axiom,
class_Ring__and__Field_Oordered__semidom(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Ocomm__semiring__1,axiom,
class_Ring__and__Field_Ocomm__semiring__1(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__RealVector_Oreal__normed__algebra,axiom,
class_RealVector_Oreal__normed__algebra(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Oab__semigroup__mult,axiom,
class_OrderedGroup_Oab__semigroup__mult(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__RealVector_Oreal__normed__vector,axiom,
class_RealVector_Oreal__normed__vector(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Oab__semigroup__add,axiom,
class_OrderedGroup_Oab__semigroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Opordered__ring,axiom,
class_Ring__and__Field_Opordered__ring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__field,axiom,
class_Ring__and__Field_Oordered__field(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Ocomm__semiring,axiom,
class_Ring__and__Field_Ocomm__semiring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__RealVector_Oreal__normed__field,axiom,
class_RealVector_Oreal__normed__field(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Ocomm__monoid__add,axiom,
class_OrderedGroup_Ocomm__monoid__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oordered__idom,axiom,
class_Ring__and__Field_Oordered__idom(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Osemigroup__mult,axiom,
class_OrderedGroup_Osemigroup__mult(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Osemigroup__add,axiom,
class_OrderedGroup_Osemigroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Lattices_Oupper__semilattice,axiom,
class_Lattices_Oupper__semilattice(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Osemiring__0,axiom,
class_Ring__and__Field_Osemiring__0(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Omult__mono1,axiom,
class_Ring__and__Field_Omult__mono1(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Oab__group__add,axiom,
class_OrderedGroup_Oab__group__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Omult__zero,axiom,
class_Ring__and__Field_Omult__zero(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Omult__mono,axiom,
class_Ring__and__Field_Omult__mono(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Osemiring,axiom,
class_Ring__and__Field_Osemiring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__RealVector_Oreal__algebra,axiom,
class_RealVector_Oreal__algebra(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Omonoid__add,axiom,
class_OrderedGroup_Omonoid__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__RealVector_Oreal__vector,axiom,
class_RealVector_Oreal__vector(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__OrderedGroup_Ogroup__add,axiom,
class_OrderedGroup_Ogroup__add(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__RealVector_Oreal__field,axiom,
class_RealVector_Oreal__field(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Ofield,axiom,
class_Ring__and__Field_Ofield(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oring,axiom,
class_Ring__and__Field_Oring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Ring__and__Field_Oidom,axiom,
class_Ring__and__Field_Oidom(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Orderings_Opreorder,axiom,
class_Orderings_Opreorder(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Orderings_Olinorder,axiom,
class_Orderings_Olinorder(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Lattices_Olattice,axiom,
class_Lattices_Olattice(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Orderings_Oorder,axiom,
class_Orderings_Oorder(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__Int_Onumber__ring,axiom,
class_Int_Onumber__ring(tc_RealDef_Oreal) ).
cnf(clsarity_RealDef__Oreal__HOL_Oord,axiom,
class_HOL_Oord(tc_RealDef_Oreal) ).
cnf(clsarity_Option__Ooption__Finite__Set_Ofinite_Ofinite,axiom,
( class_Finite__Set_Ofinite_Ofinite(tc_Option_Ooption(T_1))
| ~ class_Finite__Set_Ofinite_Ofinite(T_1) ) ).
cnf(cls_ATP__Linkup_Oequal__imp__fequal_0,axiom,
c_fequal(V_x,V_x,T_a) ).
cnf(cls_ATP__Linkup_Ofequal__imp__equal_0,axiom,
( V_X = V_Y
| ~ c_fequal(V_X,V_Y,T_a) ) ).
%------------------------------------------------------------------------------