1 set "baseuri" "cic:/matita/TPTP/CAT018-4".
2 include "logic/equality.ma".
4 (* Inclusion of: CAT018-4.p *)
6 (* -------------------------------------------------------------------------- *)
8 (* File : CAT018-4 : TPTP v3.2.0. Released v1.0.0. *)
10 (* Domain : Category Theory *)
12 (* Problem : If xy and yz exist, then so does x(yz) *)
14 (* Version : [Sco79] axioms : Reduced > Complete. *)
18 (* Refs : [Sco79] Scott (1979), Identity and Existence in Intuitionist L *)
24 (* Status : Unsatisfiable *)
26 (* Rating : 0.29 v3.1.0, 0.33 v2.7.0, 0.17 v2.6.0, 0.43 v2.5.0, 0.20 v2.4.0, 0.17 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.80 v2.0.0 *)
28 (* Syntax : Number of clauses : 14 ( 0 non-Horn; 6 unit; 11 RR) *)
30 (* Number of atoms : 24 ( 7 equality) *)
32 (* Maximal clause size : 3 ( 2 average) *)
34 (* Number of predicates : 3 ( 0 propositional; 1-2 arity) *)
36 (* Number of functors : 6 ( 3 constant; 0-2 arity) *)
38 (* Number of variables : 19 ( 2 singleton) *)
40 (* Maximal term depth : 3 ( 2 average) *)
42 (* Comments : The dependent axioms have been removed. *)
44 (* -------------------------------------------------------------------------- *)
46 (* ----Include Scott's axioms for category theory *)
48 (* Inclusion of: Axioms/CAT004-0.ax *)
50 (* -------------------------------------------------------------------------- *)
52 (* File : CAT004-0 : TPTP v3.2.0. Released v1.0.0. *)
54 (* Domain : Category Theory *)
56 (* Axioms : Category theory axioms *)
58 (* Version : [Sco79] axioms : Reduced > Complete. *)
62 (* Refs : [Sco79] Scott (1979), Identity and Existence in Intuitionist L *)
70 (* Syntax : Number of clauses : 11 ( 0 non-Horn; 3 unit; 8 RR) *)
72 (* Number of literals : 21 ( 7 equality) *)
74 (* Maximal clause size : 3 ( 2 average) *)
76 (* Number of predicates : 3 ( 0 propositional; 1-2 arity) *)
78 (* Number of functors : 3 ( 0 constant; 1-2 arity) *)
80 (* Number of variables : 19 ( 2 singleton) *)
82 (* Maximal term depth : 3 ( 2 average) *)
84 (* Comments : The dependent axioms have been removed. *)
86 (* -------------------------------------------------------------------------- *)
88 (* ----Non-standard in that E(x) stands for "x exists", i.e. a system -for *)
90 (* ----intuitionistic logic. Viewed classically, this can be -taken *)
92 (* ----as a partitioning of models M into elements of E and -other elements *)
94 (* ----of M; then Scott's quantifiers are relativized -to range only over *)
96 (* ----E, whereas free variables range over all of -M. *)
98 (* ----Quaife has this written: (this looks really weird, but results -from *)
100 (* ----having = here stand for equivalence, and it is a strange -fact from *)
102 (* ----Scott's conception that all non-existent things are -equivalent. all *)
104 (* ----x all y ((x = y) | E(x) | E(y))). *)
106 (* -----Restricted equality axioms *)
108 (* -----Category theory axioms *)
110 (* -------------------------------------------------------------------------- *)
112 (* -------------------------------------------------------------------------- *)
113 theorem prove_a_bc_exists:
114 ∀Univ:Set.∀X:Univ.∀Y:Univ.∀Z:Univ.∀a:Univ.∀b:Univ.∀c:Univ.∀codomain:∀_:Univ.Univ.∀compose:∀_:Univ.∀_:Univ.Univ.∀domain:∀_:Univ.Univ.∀equivalent:∀_:Univ.∀_:Univ.Prop.∀there_exists:∀_:Univ.Prop.∀H0:there_exists (compose b c).∀H1:there_exists (compose a b).∀H2:∀X:Univ.eq Univ (compose (codomain X) X) X.∀H3:∀X:Univ.eq Univ (compose X (domain X)) X.∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (compose X (compose Y Z)) (compose (compose X Y) Z).∀H5:∀X:Univ.∀Y:Univ.∀_:eq Univ (domain X) (codomain Y).∀_:there_exists (domain X).there_exists (compose X Y).∀H6:∀X:Univ.∀Y:Univ.∀_:there_exists (compose X Y).eq Univ (domain X) (codomain Y).∀H7:∀X:Univ.∀Y:Univ.∀_:there_exists (compose X Y).there_exists (domain X).∀H8:∀X:Univ.∀_:there_exists (codomain X).there_exists X.∀H9:∀X:Univ.∀_:there_exists (domain X).there_exists X.∀H10:∀X:Univ.∀Y:Univ.∀_:eq Univ X Y.∀_:there_exists X.equivalent X Y.∀H11:∀X:Univ.∀Y:Univ.∀_:equivalent X Y.eq Univ X Y.∀H12:∀X:Univ.∀Y:Univ.∀_:equivalent X Y.there_exists X.there_exists (compose a (compose b c))
117 autobatch depth=5 width=5 size=20 timeout=10;
122 (* -------------------------------------------------------------------------- *)