#--------------------------------------------------------------------------
# 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 setheo:sign -t rm_equality:rstfp CAT004-4.p 
#--------------------------------------------------------------------------
# equivalence_implies_existence1, axiom.
there_exists(X) <- 
    equivalent(X, Y).

# equivalence_implies_existence2, axiom.
equal(X, Y) <- 
    equivalent(X, Y).

# existence_and_equality_implies_equivalence1, axiom.
equivalent(X, Y) <- 
    there_exists(X),
    equal(X, Y).

# domain_has_elements, axiom.
there_exists(X) <- 
    there_exists(domain(X)).

# codomain_has_elements, axiom.
there_exists(X) <- 
    there_exists(codomain(X)).

# composition_implies_domain, axiom.
there_exists(domain(X)) <- 
    there_exists(compose(X, Y)).

# domain_codomain_composition1, axiom.
equal(domain(X), codomain(Y)) <- 
    there_exists(compose(X, Y)).

# domain_codomain_composition2, axiom.
there_exists(compose(X, Y)) <- 
    there_exists(domain(X)),
    equal(domain(X), codomain(Y)).

# associativity_of_compose, axiom.
equal(compose(X, compose(Y, Z)), compose(compose(X, Y), Z)) <- .

# compose_domain, axiom.
equal(compose(X, domain(X)), X) <- .

# compose_codomain, axiom.
equal(compose(codomain(X), X), X) <- .

# assume_ab_exists, hypothesis.
there_exists(compose(a, b)) <- .

# cancellation_for_product1, hypothesis.
equal(X, Z) <- 
    equal(compose(X, a), Y),
    equal(compose(Z, a), Y).

# cancellation_for_product2, hypothesis.
equal(X, Z) <- 
    equal(compose(X, b), Y),
    equal(compose(Z, b), Y).

# assume_h_exists, hypothesis.
there_exists(h) <- .

# h_ab_equals_g_ab, hypothesis.
equal(compose(h, compose(a, b)), compose(g, compose(a, b))) <- .

# prove_h_equals_g, conjecture.
 <- equal(h, g).

#--------------------------------------------------------------------------