TPTP Problem File: ANA002-2.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : ANA002-2 : TPTP v8.2.0. Bugfixed v1.0.1.
% Domain : Analysis
% Problem : Intermediate value theorem
% Version : [WB87] axioms : Reduced & Augmented > Complete.
% Theorem formulation : Ground form of goal.
% English : If a function f is continuous in a real closed interval
% [a,b], where f(a)<=0 and 0<=f(b), then there exists X such
% that f(X) = 0.
% Refs : [WB87] Wang & Bledsoe (1987), Hierarchical Deduction
% Source : [TPTP]
% Names :
% Status : Unsatisfiable
% Rating : 0.33 v8.2.0, 0.29 v8.1.0, 0.14 v7.5.0, 0.17 v7.1.0, 0.33 v7.0.0, 0.38 v6.3.0, 0.14 v6.2.0, 0.22 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.70 v5.2.0, 0.60 v5.1.0, 0.55 v5.0.0, 0.50 v4.1.0, 0.38 v4.0.1, 0.40 v4.0.0, 0.43 v3.4.0, 0.25 v3.3.0, 0.33 v2.7.0, 0.25 v2.6.0, 0.33 v2.5.0, 0.40 v2.3.0, 0.33 v2.2.1, 0.25 v2.1.0, 0.75 v2.0.0
% Syntax : Number of clauses : 18 ( 5 unt; 5 nHn; 14 RR)
% Number of literals : 42 ( 0 equ; 20 neg)
% Maximal clause size : 5 ( 2 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 2 ( 2 usr; 0 prp; 2-3 aty)
% Number of functors : 9 ( 9 usr; 4 con; 0-2 aty)
% Number of variables : 23 ( 0 sgn)
% SPC : CNF_UNS_RFO_NEQ_NHN
% Comments : The l comes from an instantiation of the least-up-bound axiom,
% which says that there exists a point l such that f(l) = 0.
% : In this version the clause in_interval is used.
% Bugfixes : v1.0.1 - Clause crossover3_and_g_function2, literal 2, fixed.
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%----Definition of in an interval
cnf(in_interval,axiom,
( in_interval(Lower,X,Upper)
| ~ less_than_or_equal(Lower,X)
| ~ less_than_or_equal(X,Upper) ) ).
%----Inequality axioms
cnf(reflexivity_of_less,axiom,
less_than_or_equal(X,X) ).
cnf(totality_of_less,axiom,
( less_than_or_equal(X,Y)
| less_than_or_equal(Y,X) ) ).
cnf(transitivity_of_less,axiom,
( less_than_or_equal(X,Z)
| ~ less_than_or_equal(X,Y)
| ~ less_than_or_equal(Y,Z) ) ).
%----Interpolation axioms
cnf(interpolation1,axiom,
( ~ less_than_or_equal(X,q(Y,X))
| less_than_or_equal(X,Y) ) ).
cnf(interpolation2,axiom,
( ~ less_than_or_equal(q(X,Y),X)
| less_than_or_equal(Y,X) ) ).
%----Continuity axioms
cnf(continuity1,axiom,
( less_than_or_equal(f(X),n0)
| ~ less_than_or_equal(X,h(X))
| ~ in_interval(lower_bound,X,upper_bound) ) ).
cnf(continuity2,axiom,
( less_than_or_equal(f(X),n0)
| ~ less_than_or_equal(Y,X)
| ~ less_than_or_equal(f(Y),n0)
| less_than_or_equal(Y,h(X))
| ~ in_interval(lower_bound,X,upper_bound) ) ).
cnf(continuity3,axiom,
( less_than_or_equal(n0,f(X))
| ~ less_than_or_equal(k(X),X)
| ~ in_interval(lower_bound,X,upper_bound) ) ).
cnf(continuity4,axiom,
( less_than_or_equal(n0,f(X))
| ~ less_than_or_equal(X,Y)
| ~ less_than_or_equal(n0,f(Y))
| less_than_or_equal(k(X),Y)
| ~ in_interval(lower_bound,X,upper_bound) ) ).
%----Least upper bound axioms
cnf(crossover1,axiom,
( less_than_or_equal(X,l)
| ~ less_than_or_equal(X,upper_bound)
| ~ less_than_or_equal(f(X),n0) ) ).
cnf(crossover2_and_g_function1,axiom,
( less_than_or_equal(g(X),upper_bound)
| less_than_or_equal(l,X) ) ).
cnf(crossover3_and_g_function2,axiom,
( less_than_or_equal(f(g(X)),n0)
| less_than_or_equal(l,X) ) ).
cnf(crossover4_and_g_function3,axiom,
( ~ less_than_or_equal(g(X),X)
| less_than_or_equal(l,X) ) ).
%----Enpoints of the interval
cnf(the_interval,hypothesis,
less_than_or_equal(lower_bound,upper_bound) ).
cnf(lower_mapping,hypothesis,
less_than_or_equal(f(lower_bound),n0) ).
cnf(upper_mapping,hypothesis,
less_than_or_equal(n0,f(upper_bound)) ).
cnf(prove_there_is_x_which_crosses,negated_conjecture,
~ in_interval(f(l),n0,f(l)) ).
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