#--------------------------------------------------------------------------
# File     : CAT005-3 : TPTP v2.1.0. Released v1.0.0.
# Domain   : Category Theory
# Problem  : Domain is the unique right identity
# Version  : [Sco79] axioms : Reduced > Complete.
# English  : domain(x) is the unique identity i such that xi is defined.

# Refs     : [Sco79] Scott (1979), Identity and Existence in Intuitionist L
# Source   : [ANL]
# Names    : p5.ver3.in [ANL]

# Status   : unsatisfiable
# Rating   : 0.73 v2.1.0, 0.56 v2.0.0
# Syntax   : Number of clauses    :   21 (   2 non-Horn;   5 unit;  16 RR)
#            Number of literals   :   43 (  18 equality)
#            Maximal clause size  :    4 (   2 average)
#            Number of predicates :    3 (   0 propositional; 1-2 arity)
#            Number of functors   :    6 (   2 constant; 0-2 arity)
#            Number of variables  :   33 (   4 singleton)
#            Maximal term depth   :    3 (   1 average)

# Comments : Axioms simplified by Art Quaife.
#          : tptp2X -f setheo:sign -t rm_equality:rstfp CAT005-3.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) <- .

# equivalence_implies_existence3, axiom.
there_exists(Y) <- 
    equivalent(X, Y).

# existence_and_equality_implies_equivalence2, axiom.
equivalent(X, Y) <- 
    there_exists(X),
    there_exists(Y),
    equal(X, Y).

# composition_implies_codomain, axiom.
there_exists(codomain(X)) <- 
    there_exists(compose(X, Y)).

# indiscernibles1, axiom.
there_exists(f1(X, Y));
equal(X, Y) <- .

# indiscernibles2, axiom.
equal(X, f1(X, Y));
equal(Y, f1(X, Y));
equal(X, Y) <- .

# indiscernibles3, axiom.
equal(X, Y) <- 
    equal(X, f1(X, Y)),
    equal(Y, f1(X, Y)).

# ad_exists, hypothesis.
there_exists(compose(a, d)) <- .

# xd_equals_x, hypothesis.
equal(compose(X, d), X) <- 
    there_exists(compose(X, d)).

# dy_equals_y, hypothesis.
equal(compose(d, Y), Y) <- 
    there_exists(compose(d, Y)).

# prove_domain_of_a_is_d, conjecture.
 <- equal(domain(a), d).

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