%------------------------------------------------------------------------------
% File     : ANA034-2 : TPTP v8.2.0. Released v3.2.0.
% Domain   : Analysis
% Problem  : Problem about Big-O notation
% Version  : [Pau06] axioms : Reduced > Especial.
% English  :

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

% Status   : Unsatisfiable
% Rating   : 0.00 v7.3.0, 0.11 v7.2.0, 0.00 v6.0.0, 0.33 v5.5.0, 0.38 v5.4.0, 0.33 v5.3.0, 0.42 v5.2.0, 0.25 v5.1.0, 0.14 v4.1.0, 0.11 v4.0.1, 0.17 v4.0.0, 0.33 v3.5.0, 0.17 v3.3.0, 0.29 v3.2.0
% Syntax   : Number of clauses     :   15 (   6 unt;   0 nHn;  13 RR)
%            Number of literals    :   31 (   2 equ;  17 neg)
%            Maximal clause size   :    6 (   2 avg)
%            Maximal term depth    :    5 (   2 avg)
%            Number of predicates  :    8 (   7 usr;   0 prp; 1-3 aty)
%            Number of functors    :   11 (  11 usr;   5 con; 0-3 aty)
%            Number of variables   :   20 (   1 sgn)
% SPC      : CNF_UNS_RFO_SEQ_HRN

% Comments : The problems in the [Pau06] collection each have very many axioms,
%            of which only a small selection are required for the refutation.
%            The mission is to find those few axioms, after which a refutation
%            can be quite easily found. This version has only the necessary
%            axioms.
%------------------------------------------------------------------------------
cnf(cls_Ring__and__Field_Oabs__mult_0,axiom,
    ( ~ class_Ring__and__Field_Oordered__idom(T_a)
    | c_HOL_Oabs(c_times(V_a,V_b,T_a),T_a) = c_times(c_HOL_Oabs(V_a,T_a),c_HOL_Oabs(V_b,T_a),T_a) ) ).

cnf(cls_Ring__and__Field_Omult__mono_0,axiom,
    ( ~ class_Ring__and__Field_Opordered__semiring(T_a)
    | ~ c_lessequals(V_c,V_d,T_a)
    | ~ c_lessequals(V_a,V_b,T_a)
    | ~ c_lessequals(c_0,V_c,T_a)
    | ~ c_lessequals(c_0,V_b,T_a)
    | c_lessequals(c_times(V_a,V_c,T_a),c_times(V_b,V_d,T_a),T_a) ) ).

cnf(cls_Ring__and__Field_Omult__nonneg__nonneg_0,axiom,
    ( ~ class_Ring__and__Field_Opordered__cancel__semiring(T_a)
    | ~ c_lessequals(c_0,V_b,T_a)
    | ~ c_lessequals(c_0,V_a,T_a)
    | c_lessequals(c_0,c_times(V_a,V_b,T_a),T_a) ) ).

cnf(cls_Orderings_Oorder__less__imp__le_0,axiom,
    ( ~ class_Orderings_Oorder(T_a)
    | ~ c_less(V_x,V_y,T_a)
    | c_lessequals(V_x,V_y,T_a) ) ).

cnf(cls_OrderedGroup_Oabs__ge__zero_0,axiom,
    ( ~ class_OrderedGroup_Olordered__ab__group__abs(T_a)
    | c_lessequals(c_0,c_HOL_Oabs(V_a,T_a),T_a) ) ).

cnf(cls_conjecture_0,negated_conjecture,
    c_less(c_0,v_c,t_b) ).

cnf(cls_conjecture_2,negated_conjecture,
    c_lessequals(c_HOL_Oabs(v_a(v_x),t_b),c_times(v_c,c_HOL_Oabs(v_f(v_x),t_b),t_b),t_b) ).

cnf(cls_conjecture_3,negated_conjecture,
    c_lessequals(c_HOL_Oabs(v_b(v_x),t_b),c_times(v_ca,c_HOL_Oabs(v_g(v_x),t_b),t_b),t_b) ).

cnf(cls_conjecture_4,negated_conjecture,
    c_times(c_times(v_c,v_ca,t_b),c_HOL_Oabs(c_times(v_f(v_x),v_g(v_x),t_b),t_b),t_b) = c_times(c_times(v_c,c_HOL_Oabs(v_f(v_x),t_b),t_b),c_times(v_ca,c_HOL_Oabs(v_g(v_x),t_b),t_b),t_b) ).

cnf(cls_conjecture_5,negated_conjecture,
    ~ c_lessequals(c_HOL_Oabs(c_times(v_a(v_x),v_b(v_x),t_b),t_b),c_times(c_times(v_c,v_ca,t_b),c_HOL_Oabs(c_times(v_f(v_x),v_g(v_x),t_b),t_b),t_b),t_b) ).

cnf(tfree_tcs,negated_conjecture,
    class_Ring__and__Field_Oordered__idom(t_b) ).

cnf(clsrel_OrderedGroup_Olordered__ab__group__abs_17,axiom,
    ( ~ class_OrderedGroup_Olordered__ab__group__abs(T)
    | class_Orderings_Oorder(T) ) ).

cnf(clsrel_Ring__and__Field_Oordered__idom_40,axiom,
    ( ~ class_Ring__and__Field_Oordered__idom(T)
    | class_Ring__and__Field_Opordered__cancel__semiring(T) ) ).

cnf(clsrel_Ring__and__Field_Oordered__idom_42,axiom,
    ( ~ class_Ring__and__Field_Oordered__idom(T)
    | class_Ring__and__Field_Opordered__semiring(T) ) ).

cnf(clsrel_Ring__and__Field_Oordered__idom_50,axiom,
    ( ~ class_Ring__and__Field_Oordered__idom(T)
    | class_OrderedGroup_Olordered__ab__group__abs(T) ) ).

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