%--------------------------------------------------------------------------
% File     : ANA002-0 : TPTP v8.2.0. Released v1.0.0.
% Domain   : Analysis (Limits)
% Axioms   : Analysis (limits) axioms for continuous functions
% Version  : [Ble90] axioms.
% English  :

% Refs     : [Ble90] Bledsoe (1990), Challenge Problems in Elementary Calcu
%          : [Ble92] Bledsoe (1992), Email to G. Sutcliffe
% Source   : [Ble92]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses     :   26 (  11 unt;   2 nHn;  13 RR)
%            Number of literals    :   45 (   6 equ;  17 neg)
%            Maximal clause size   :    3 (   1 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    5 (   5 usr;   1 con; 0-2 aty)
%            Number of variables   :   59 (   4 sgn)
% SPC      : 

% Comments : Based on the theorem in calculus that the sum of two continuous
%            functions is continuous.
%          : Used some ideas from [SETHEO] to format this.
%--------------------------------------------------------------------------
%----|X + Y| <= |X| + |Y|.
%----Clause 8
cnf(absolute_sum_less_or_equal_sum_of_absolutes1,axiom,
    less_or_equal(absolute(add(X,Y)),add(absolute(X),absolute(Y))) ).

%----Clause 8.1
cnf(absolute_sum_less_or_equal_sum_of_absolutes2,axiom,
    ( ~ less_or_equal(add(absolute(X),absolute(Y)),Z)
    | less_or_equal(absolute(add(X,Y)),Z) ) ).

%----Properties of minimum.
%----Clause 9
cnf(minimum1,axiom,
    ( ~ less_or_equal(X,Y)
    | minimum(X,Y) = X ) ).

%----Clause 9.1
cnf(minimum2,axiom,
    less_or_equal(minimum(X,Y),X) ).

%----Clause 9.11
cnf(minimum3,axiom,
    ( ~ less_or_equal(Z,minimum(X,Y))
    | less_or_equal(Z,X) ) ).

%----Clause 9.2
cnf(minimum4,axiom,
    ( ~ less_or_equal(X,Y)
    | less_or_equal(X,minimum(X,Y)) ) ).

%----Clause 10
cnf(minimum5,axiom,
    ( ~ less_or_equal(Y,X)
    | minimum(X,Y) = Y ) ).

%----Clause 10.1
cnf(minimum6,axiom,
    less_or_equal(minimum(X,Y),Y) ).

%----Clause 10.11
cnf(minimum7,axiom,
    ( ~ less_or_equal(Z,minimum(X,Y))
    | less_or_equal(Z,Y) ) ).

%----Clause 10.2
cnf(minimum8,axiom,
    ( ~ less_or_equal(Y,X)
    | less_or_equal(Y,minimum(X,Y)) ) ).

%----Clause 10.3
cnf(minimum9,axiom,
    ( less_or_equal(X,n0)
    | less_or_equal(Y,n0)
    | ~ less_or_equal(minimum(X,Y),n0) ) ).

%----Properties of half.
%----Clause 11
cnf(half_plus_half_is_whole,axiom,
    add(half(X),half(X)) = X ).

%----Clause 11.1
cnf(half_plus_half_less_or_equal_whole,axiom,
    less_or_equal(add(half(X),half(X)),X) ).

%----Clause 11.2
cnf(whole_less_or_equal_half_plus_half,axiom,
    less_or_equal(X,add(half(X),half(X))) ).

%----Clause 11.3
cnf(less_or_equal_sum_of_halves,axiom,
    ( ~ less_or_equal(X,half(Z))
    | ~ less_or_equal(Y,half(Z))
    | less_or_equal(add(X,Y),Z) ) ).

%----Clause 12
cnf(zero_and_half,axiom,
    ( less_or_equal(X,n0)
    | ~ less_or_equal(half(X),n0) ) ).

%----Properties of add.
%----Clause 13
cnf(add_to_both_sides_of_less_equal1,axiom,
    ( ~ less_or_equal(X,Y)
    | less_or_equal(add(X,Z),add(Y,Z)) ) ).

%----Clause 13.1
cnf(add_to_both_sides_of_less_equal2,axiom,
    ( ~ less_or_equal(X,Y)
    | ~ less_or_equal(Z,W)
    | less_or_equal(add(X,Z),add(Y,W)) ) ).

%----Clause 14
cnf(commutativity_of_less_or_equal,axiom,
    ( less_or_equal(X,Y)
    | less_or_equal(Y,X) ) ).

%----Clause 15
cnf(transitivity_of_less_or_equal,axiom,
    ( ~ less_or_equal(X,Y)
    | ~ less_or_equal(Y,Z)
    | less_or_equal(X,Z) ) ).

%----Clause 15.1 omitted - it's the same as Clause 15

%----Clause 16
cnf(commutativity_of_add,axiom,
    add(X,Y) = add(Y,X) ).

%----Clause 16_1
cnf(commutativity_of_add_for_less_or_equal,axiom,
    less_or_equal(add(X,Y),add(Y,X)) ).

%----Clause 17
cnf(associativity_of_add,axiom,
    add(add(X,Y),Z) = add(X,add(Y,Z)) ).

%----Clause 17_1
cnf(associativity_of_add_for_less_or_equal1,axiom,
    less_or_equal(add(add(X,Y),Z),add(X,add(Y,Z))) ).

%----Clause 17_2
cnf(associativity_of_add_for_less_or_equal2,axiom,
    less_or_equal(add(X,add(Y,Z)),add(add(X,Y),Z)) ).

%----Clause 18
cnf(equal_implies_less_or_equal,axiom,
    ( X != Y
    | less_or_equal(X,Y) ) ).

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