assign(max_seconds, 30). %------------------------------------------------------------------------------ % File : SET668+3 : TPTP v3.0.1. Released v2.2.0. % Domain : Set Theory (Relations) % Problem : Id on X subset of R => X is domain R & X subset of range R % Version : [Wor89] axioms : Reduced > Incomplete. % English : If the identity relation on X is a subset of a relation R from % X to Y then X is the domain of R and X is a subset of the range % of R. % Refs : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int % : [Wor89] Woronowicz (1989), Relations Defined on Sets % Source : [ILF] % Names : RELSET_1 (31) [Wor89] % Status : Theorem % Rating : 0.22 v2.7.0, 0.17 v2.6.0, 0.14 v2.5.0, 0.12 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1 % Syntax : Number of formulae : 33 ( 2 unit) % Number of atoms : 130 ( 7 equality) % Maximal formula depth : 11 ( 6 average) % Number of connectives : 101 ( 4 ~ ; 0 |; 13 &) % ( 10 <=>; 74 =>; 0 <=) % ( 0 <~>; 0 ~|; 0 ~&) % Number of predicates : 6 ( 0 propositional; 1-2 arity) % Number of functors : 13 ( 2 constant; 0-3 arity) % Number of variables : 74 ( 0 singleton; 68 !; 6 ?) % Maximal term depth : 3 ( 1 average) % Comments : % : tptp2X -f mace4 SET668+3.p %------------------------------------------------------------------------------ %----NOTE WELL: conjecture has been negated set(prolog_style_variables). formulas(assumptions). % p1, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( ( subset(B,C) & subset(C,B) ) -> B = C ) ) ) ) ). % p2, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,set_type) -> ( all E ( ilf_type(E,relation_type(B,C)) -> ( subset(identity_relation_of(D),E) -> ( subset(D,domain(B,C,E)) & subset(D,range(B,C,E)) ) ) ) ) ) ) ) ) ) ). % p3, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,set_type) -> ( member(ordered_pair(C,D),identity_relation_of(B)) <-> ( member(C,B) & C = D ) ) ) ) ) ) ) ). % p4, axiom. ( all B ( ilf_type(B,set_type) -> ilf_type(identity_relation_of(B),binary_relation_type) ) ). % p5, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( ( all D ( ilf_type(D,subset_type(cross_product(B,C))) -> ilf_type(D,relation_type(B,C)) ) ) & ( all E ( ilf_type(E,relation_type(B,C)) -> ilf_type(E,subset_type(cross_product(B,C))) ) ) ) ) ) ) ). % p6, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( exists D ilf_type(D,relation_type(C,B)) ) ) ) ) ). % p7, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( subset(B,C) <-> ( all D ( ilf_type(D,set_type) -> ( member(D,B) -> member(D,C) ) ) ) ) ) ) ) ). % p8, axiom. ( all B ( ilf_type(B,binary_relation_type) -> ( all C ( ilf_type(C,binary_relation_type) -> ( subset(B,C) <-> ( all D ( ilf_type(D,set_type) -> ( all E ( ilf_type(E,set_type) -> ( member(ordered_pair(D,E),B) -> member(ordered_pair(D,E),C) ) ) ) ) ) ) ) ) ) ). % p9, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( B = C <-> ( subset(B,C) & subset(C,B) ) ) ) ) ) ). % p10, axiom. ( all B ( ilf_type(B,binary_relation_type) -> ilf_type(domain_of(B),set_type) ) ). % p11, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ilf_type(cross_product(B,C),set_type) ) ) ) ). % p12, axiom. ( all B ( ilf_type(B,binary_relation_type) -> ilf_type(range_of(B),set_type) ) ). % p13, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ilf_type(ordered_pair(B,C),set_type) ) ) ) ). % p14, axiom. ( all B ( ilf_type(B,set_type) -> ( ilf_type(B,binary_relation_type) <-> ( relation_like(B) & ilf_type(B,set_type) ) ) ) ). % p15, axiom. ( exists B ilf_type(B,binary_relation_type) ). % p16, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( ilf_type(C,subset_type(B)) <-> ilf_type(C,member_type(power_set(B))) ) ) ) ) ). % p17, axiom. ( all B ( ilf_type(B,set_type) -> ( exists C ilf_type(C,subset_type(B)) ) ) ). % p18, axiom. ( all B ( ilf_type(B,set_type) -> subset(B,B) ) ). % p19, axiom. ( all B ( ilf_type(B,binary_relation_type) -> subset(B,B) ) ). % p20, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( member(B,power_set(C)) <-> ( all D ( ilf_type(D,set_type) -> ( member(D,B) -> member(D,C) ) ) ) ) ) ) ) ). % p21, axiom. ( all B ( ilf_type(B,set_type) -> ( -(empty(power_set(B))) & ilf_type(power_set(B),set_type) ) ) ). % p22, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ( -(empty(C)) & ilf_type(C,set_type) ) -> ( ilf_type(B,member_type(C)) <-> member(B,C) ) ) ) ) ). % p23, axiom. ( all B ( ( -(empty(B)) & ilf_type(B,set_type) ) -> ( exists C ilf_type(C,member_type(B)) ) ) ). % p24, axiom. ( all B ( ilf_type(B,set_type) -> ( relation_like(B) <-> ( all C ( ilf_type(C,set_type) -> ( member(C,B) -> ( exists D ( ilf_type(D,set_type) & ( exists E ( ilf_type(E,set_type) & C = ordered_pair(D,E) ) ) ) ) ) ) ) ) ) ). % p25, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,subset_type(cross_product(B,C))) -> relation_like(D) ) ) ) ) ) ). % p26, axiom. ( all B ( ilf_type(B,set_type) -> ( empty(B) <-> ( all C ( ilf_type(C,set_type) -> -(member(C,B)) ) ) ) ) ). % p27, axiom. ( all B ( ( empty(B) & ilf_type(B,set_type) ) -> relation_like(B) ) ). % p28, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,relation_type(B,C)) -> domain(B,C,D) = domain_of(D) ) ) ) ) ) ). % p29, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,relation_type(B,C)) -> ilf_type(domain(B,C,D),subset_type(B)) ) ) ) ) ) ). % p30, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,relation_type(B,C)) -> range(B,C,D) = range_of(D) ) ) ) ) ) ). % p31, axiom. ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,relation_type(B,C)) -> ilf_type(range(B,C,D),subset_type(C)) ) ) ) ) ) ). % p32, axiom. ( all B ilf_type(B,set_type) ). % prove_relset_1_31, negated_conjecture. -(( ( all B ( ilf_type(B,set_type) -> ( all C ( ilf_type(C,set_type) -> ( all D ( ilf_type(D,relation_type(C,B)) -> ( subset(identity_relation_of(C),D) -> ( C = domain(C,B,D) & subset(C,range(C,B,D)) ) ) ) ) ) ) ) ) )). end_of_list. %------------------------------------------------------------------------------