%------------------------------------------------------------------------------
% File     : ANA003-0 : TPTP v7.2.0. Released v3.2.0.
% Domain   : Analysis
% Axioms   : A theory of Big-O notation
% Version  : [Pau06] axioms.
% English  :

% Refs     : [Pau06] Paulson (2006), Email to G. Sutcliffe
% Source   : [Pau06]
% Names    : BigO_simp.ax [Pau06]

% Status   : Satisfiable
% Syntax   : Number of clauses     :   53 (   0 non-Horn;   0 unit;  15 RR)
%            Number of atoms       :  122 (   0 equality)
%            Maximal clause size   :    4 (   2 average)
%            Number of predicates  :   32 (   0 propositional; 1-3 arity)
%            Number of functors    :    6 (   0 constant; 1-3 arity)
%            Number of variables   :  123 (  30 singleton)
%            Maximal term depth    :    3 (   2 average)
% SPC      : 

% Comments : Requires MSC001-0.ax, MSC001-1.ax
%------------------------------------------------------------------------------
cnf(cls_SetsAndFunctions_Oset__plus__intro2_0,axiom,
    ( ~ class_HOL_Oplus(T_a)
    | ~ c_in(V_b,V_C,T_a)
    | c_in(c_plus(V_a,V_b,T_a),c_SetsAndFunctions_Oelt__set__plus(V_a,V_C,T_a),T_a) )).

cnf(cls_SetsAndFunctions_Oset__plus__intro_0,axiom,
    ( ~ class_HOL_Oplus(T_a)
    | ~ c_in(V_b,V_D,T_a)
    | ~ c_in(V_a,V_C,T_a)
    | c_in(c_plus(V_a,V_b,T_a),c_plus(V_C,V_D,tc_set(T_a)),T_a) )).

cnf(cls_SetsAndFunctions_Oset__plus__mono2_0,axiom,
    ( ~ class_HOL_Oplus(T_a)
    | ~ c_lessequals(V_E,V_F,tc_set(T_a))
    | ~ c_lessequals(V_C,V_D,tc_set(T_a))
    | c_lessequals(c_plus(V_C,V_E,tc_set(T_a)),c_plus(V_D,V_F,tc_set(T_a)),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__plus__mono3_0,axiom,
    ( ~ class_HOL_Oplus(T_a)
    | ~ c_in(V_a,V_C,T_a)
    | c_lessequals(c_SetsAndFunctions_Oelt__set__plus(V_a,V_D,T_a),c_plus(V_C,V_D,tc_set(T_a)),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__plus__mono4_0,axiom,
    ( ~ class_OrderedGroup_Ocomm__monoid__add(T_a)
    | ~ c_in(V_a,V_C,T_a)
    | c_lessequals(c_SetsAndFunctions_Oelt__set__plus(V_a,V_D,T_a),c_plus(V_D,V_C,tc_set(T_a)),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__plus__mono_0,axiom,
    ( ~ class_HOL_Oplus(T_a)
    | ~ c_lessequals(V_C,V_D,tc_set(T_a))
    | c_lessequals(c_SetsAndFunctions_Oelt__set__plus(V_a,V_C,T_a),c_SetsAndFunctions_Oelt__set__plus(V_a,V_D,T_a),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__times__intro2_0,axiom,
    ( ~ class_HOL_Otimes(T_a)
    | ~ c_in(V_b,V_C,T_a)
    | c_in(c_times(V_a,V_b,T_a),c_SetsAndFunctions_Oelt__set__times(V_a,V_C,T_a),T_a) )).

cnf(cls_SetsAndFunctions_Oset__times__intro_0,axiom,
    ( ~ class_HOL_Otimes(T_a)
    | ~ c_in(V_b,V_D,T_a)
    | ~ c_in(V_a,V_C,T_a)
    | c_in(c_times(V_a,V_b,T_a),c_times(V_C,V_D,tc_set(T_a)),T_a) )).

cnf(cls_SetsAndFunctions_Oset__times__mono2_0,axiom,
    ( ~ class_HOL_Otimes(T_a)
    | ~ c_lessequals(V_E,V_F,tc_set(T_a))
    | ~ c_lessequals(V_C,V_D,tc_set(T_a))
    | c_lessequals(c_times(V_C,V_E,tc_set(T_a)),c_times(V_D,V_F,tc_set(T_a)),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__times__mono3_0,axiom,
    ( ~ class_HOL_Otimes(T_a)
    | ~ c_in(V_a,V_C,T_a)
    | c_lessequals(c_SetsAndFunctions_Oelt__set__times(V_a,V_D,T_a),c_times(V_C,V_D,tc_set(T_a)),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__times__mono4_0,axiom,
    ( ~ class_OrderedGroup_Ocomm__monoid__mult(T_a)
    | ~ c_in(V_a,V_C,T_a)
    | c_lessequals(c_SetsAndFunctions_Oelt__set__times(V_a,V_D,T_a),c_times(V_D,V_C,tc_set(T_a)),tc_set(T_a)) )).

cnf(cls_SetsAndFunctions_Oset__times__mono_0,axiom,
    ( ~ class_HOL_Otimes(T_a)
    | ~ c_lessequals(V_C,V_D,tc_set(T_a))
    | c_lessequals(c_SetsAndFunctions_Oelt__set__times(V_a,V_C,T_a),c_SetsAndFunctions_Oelt__set__times(V_a,V_D,T_a),tc_set(T_a)) )).

cnf(clsarity_fun_10,axiom,
    ( class_OrderedGroup_Ocancel__semigroup__add(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Oab__group__add(T_1) )).

cnf(clsarity_fun_11,axiom,
    ( class_OrderedGroup_Ocancel__ab__semigroup__add(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Oab__group__add(T_1) )).

cnf(clsarity_fun_12,axiom,
    ( class_OrderedGroup_Oab__group__add(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Oab__group__add(T_1) )).

cnf(clsarity_fun_13,axiom,
    ( class_OrderedGroup_Osemigroup__mult(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Osemigroup__mult(T_1) )).

cnf(clsarity_fun_14,axiom,
    ( class_OrderedGroup_Oab__semigroup__mult(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__mult(T_1) )).

cnf(clsarity_fun_15,axiom,
    ( class_OrderedGroup_Omonoid__mult(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__mult(T_1) )).

cnf(clsarity_fun_16,axiom,
    ( class_OrderedGroup_Ocomm__monoid__mult(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__mult(T_1) )).

cnf(clsarity_fun_17,axiom,
    ( class_Ring__and__Field_Osemiring(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_18,axiom,
    ( class_Ring__and__Field_Ocomm__semiring(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_19,axiom,
    ( class_Ring__and__Field_Osemiring__0(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_2,axiom,
    ( class_HOL_Oplus(tc_fun(T_2,T_1))
    | ~ class_HOL_Oplus(T_1) )).

cnf(clsarity_fun_20,axiom,
    ( class_Ring__and__Field_Ocomm__semiring__0(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_21,axiom,
    ( class_Ring__and__Field_Osemiring__0__cancel(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_22,axiom,
    ( class_Ring__and__Field_Oring(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_23,axiom,
    ( class_Ring__and__Field_Ocomm__semiring__0__cancel(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_24,axiom,
    ( class_Ring__and__Field_Ocomm__ring(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_25,axiom,
    ( class_Ring__and__Field_Oaxclass__0__neq__1(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_26,axiom,
    ( class_Ring__and__Field_Osemiring__1(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_27,axiom,
    ( class_Ring__and__Field_Ocomm__semiring__1(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_28,axiom,
    ( class_Ring__and__Field_Osemiring__1__cancel(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_29,axiom,
    ( class_Ring__and__Field_Oring__1(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_3,axiom,
    ( class_HOL_Otimes(tc_fun(T_2,T_1))
    | ~ class_HOL_Otimes(T_1) )).

cnf(clsarity_fun_30,axiom,
    ( class_Ring__and__Field_Ocomm__semiring__1__cancel(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_31,axiom,
    ( class_Ring__and__Field_Ocomm__ring__1(tc_fun(T_2,T_1))
    | ~ class_Ring__and__Field_Ocomm__ring__1(T_1) )).

cnf(clsarity_fun_4,axiom,
    ( class_HOL_Ominus(tc_fun(T_2,T_1))
    | ~ class_HOL_Ominus(T_1) )).

cnf(clsarity_fun_5,axiom,
    ( class_HOL_Ozero(tc_fun(T_2,T_1))
    | ~ class_HOL_Ozero(T_1) )).

cnf(clsarity_fun_6,axiom,
    ( class_HOL_Oone(tc_fun(T_2,T_1))
    | ~ class_HOL_Oone(T_1) )).

cnf(clsarity_fun_7,axiom,
    ( class_OrderedGroup_Osemigroup__add(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Osemigroup__add(T_1) )).

cnf(clsarity_fun_8,axiom,
    ( class_OrderedGroup_Oab__semigroup__add(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__add(T_1) )).

cnf(clsarity_fun_9,axiom,
    ( class_OrderedGroup_Ocomm__monoid__add(tc_fun(T_2,T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__add(T_1) )).

cnf(clsarity_set_10,axiom,
    ( class_OrderedGroup_Osemigroup__mult(tc_set(T_1))
    | ~ class_OrderedGroup_Osemigroup__mult(T_1) )).

cnf(clsarity_set_11,axiom,
    ( class_OrderedGroup_Oab__semigroup__add(tc_set(T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__add(T_1) )).

cnf(clsarity_set_12,axiom,
    ( class_OrderedGroup_Ocomm__monoid__add(tc_set(T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__add(T_1) )).

cnf(clsarity_set_13,axiom,
    ( class_OrderedGroup_Oab__semigroup__mult(tc_set(T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__mult(T_1) )).

cnf(clsarity_set_14,axiom,
    ( class_OrderedGroup_Omonoid__mult(tc_set(T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__mult(T_1) )).

cnf(clsarity_set_15,axiom,
    ( class_OrderedGroup_Ocomm__monoid__mult(tc_set(T_1))
    | ~ class_OrderedGroup_Ocomm__monoid__mult(T_1) )).

cnf(clsarity_set_5,axiom,
    ( class_HOL_Oplus(tc_set(T_1))
    | ~ class_HOL_Oplus(T_1) )).

cnf(clsarity_set_6,axiom,
    ( class_HOL_Otimes(tc_set(T_1))
    | ~ class_HOL_Otimes(T_1) )).

cnf(clsarity_set_7,axiom,
    ( class_HOL_Ozero(tc_set(T_1))
    | ~ class_HOL_Ozero(T_1) )).

cnf(clsarity_set_8,axiom,
    ( class_HOL_Oone(tc_set(T_1))
    | ~ class_HOL_Oone(T_1) )).

cnf(clsarity_set_9,axiom,
    ( class_OrderedGroup_Osemigroup__add(tc_set(T_1))
    | ~ class_OrderedGroup_Osemigroup__add(T_1) )).

%------------------------------------------------------------------------------