%--------------------------------------------------------------------------
% File     : CAT004-4 : TPTP v2.1.0. Released v1.0.0.
% Domain   : Category Theory
% Problem  : X and Y epimorphisms, XY well-defined => XY epimorphism
% Version  : [Sco79] axioms : Reduced > Complete.
% English  : If x and y are epimorphisms and xy is well-defined, then 
%            xy is an epimorphism.

% Refs     : [Sco79] Scott (1979), Identity and Existence in Intuitionist L
% Source   : [TPTP]
% Names    : 

% Status   : unsatisfiable
% Rating   : 0.33 v2.1.0, 0.40 v2.0.0
% Syntax   : Number of clauses    :   17 (   0 non-Horn;   7 unit;  14 RR)
%            Number of literals   :   31 (  15 equality)
%            Maximal clause size  :    3 (   1 average)
%            Number of predicates :    3 (   0 propositional; 1-2 arity)
%            Number of functors   :    7 (   4 constant; 0-2 arity)
%            Number of variables  :   25 (   2 singleton)
%            Maximal term depth   :    3 (   1 average)

% Comments : The dependent axioms have been removed.
%          : tptp2X -f tptp -t rm_equality:rstfp CAT004-4.p 
%--------------------------------------------------------------------------
input_clause(equivalence_implies_existence1,axiom,
    [-- equivalent(X, Y),
     ++ there_exists(X)]).

input_clause(equivalence_implies_existence2,axiom,
    [-- equivalent(X, Y),
     ++ equal(X, Y)]).

input_clause(existence_and_equality_implies_equivalence1,axiom,
    [-- there_exists(X),
     -- equal(X, Y),
     ++ equivalent(X, Y)]).

input_clause(domain_has_elements,axiom,
    [-- there_exists(domain(X)),
     ++ there_exists(X)]).

input_clause(codomain_has_elements,axiom,
    [-- there_exists(codomain(X)),
     ++ there_exists(X)]).

input_clause(composition_implies_domain,axiom,
    [-- there_exists(compose(X, Y)),
     ++ there_exists(domain(X))]).

input_clause(domain_codomain_composition1,axiom,
    [-- there_exists(compose(X, Y)),
     ++ equal(domain(X), codomain(Y))]).

input_clause(domain_codomain_composition2,axiom,
    [-- there_exists(domain(X)),
     -- equal(domain(X), codomain(Y)),
     ++ there_exists(compose(X, Y))]).

input_clause(associativity_of_compose,axiom,
    [++ equal(compose(X, compose(Y, Z)), compose(compose(X, Y), Z))]).

input_clause(compose_domain,axiom,
    [++ equal(compose(X, domain(X)), X)]).

input_clause(compose_codomain,axiom,
    [++ equal(compose(codomain(X), X), X)]).

input_clause(assume_ab_exists,hypothesis,
    [++ there_exists(compose(a, b))]).

input_clause(cancellation_for_product1,hypothesis,
    [-- equal(compose(X, a), Y),
     -- equal(compose(Z, a), Y),
     ++ equal(X, Z)]).

input_clause(cancellation_for_product2,hypothesis,
    [-- equal(compose(X, b), Y),
     -- equal(compose(Z, b), Y),
     ++ equal(X, Z)]).

input_clause(assume_h_exists,hypothesis,
    [++ there_exists(h)]).

input_clause(h_ab_equals_g_ab,hypothesis,
    [++ equal(compose(h, compose(a, b)), compose(g, compose(a, b)))]).

input_clause(prove_h_equals_g,conjecture,
    [-- equal(h, g)]).
%--------------------------------------------------------------------------