## TPTP Axioms File: ANA002-0.ax

```%--------------------------------------------------------------------------
% File     : ANA002-0 : TPTP v7.5.0. 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 (   2 non-Horn;  11 unit;  13 RR)
%            Number of atoms      :   45 (   6 equality)
%            Maximal clause size  :    3 (   2 average)
%            Number of predicates :    2 (   0 propositional; 2-2 arity)
%            Number of functors   :    5 (   1 constant; 0-2 arity)
%            Number of variables  :   59 (   4 singleton)
%            Maximal term depth   :    3 (   2 average)
% 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,

%----Clause 8.1
cnf(absolute_sum_less_or_equal_sum_of_absolutes2,axiom,

%----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,

%----Clause 11.1
cnf(half_plus_half_less_or_equal_whole,axiom,

%----Clause 11.2
cnf(whole_less_or_equal_half_plus_half,axiom,

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

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

%----Clause 13
( ~ less_or_equal(X,Y)

%----Clause 13.1
( ~ less_or_equal(X,Y)
| ~ less_or_equal(Z,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

%----Clause 16_1

%----Clause 17

%----Clause 17_1