1 set "baseuri" "cic:/matita/TPTP/HEN011-3".
2 include "logic/equality.ma".
4 (* Inclusion of: HEN011-3.p *)
6 (* -------------------------------------------------------------------------- *)
8 (* File : HEN011-3 : TPTP v3.2.0. Released v1.0.0. *)
10 (* Domain : Henkin Models *)
12 (* Problem : This operation is commutative *)
14 (* Version : [MOW76] axioms. *)
16 (* English : Define & on the set of Z', where Z' = identity/Z, *)
18 (* by X' & Y'=X'/(identity/Y'). The operation is commutative. *)
20 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
24 (* Names : HP11 [ANL] *)
26 (* Status : Unsatisfiable *)
28 (* Rating : 0.43 v3.2.0, 0.14 v3.1.0, 0.33 v2.7.0, 0.17 v2.6.0, 0.43 v2.5.0, 0.40 v2.4.0, 0.50 v2.3.0, 0.33 v2.2.1, 0.67 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
30 (* Syntax : Number of clauses : 13 ( 0 non-Horn; 10 unit; 9 RR) *)
32 (* Number of atoms : 17 ( 9 equality) *)
34 (* Maximal clause size : 3 ( 1 average) *)
36 (* Number of predicates : 2 ( 0 propositional; 2-2 arity) *)
38 (* Number of functors : 9 ( 8 constant; 0-2 arity) *)
40 (* Number of variables : 13 ( 3 singleton) *)
42 (* Maximal term depth : 4 ( 2 average) *)
46 (* -------------------------------------------------------------------------- *)
48 (* ----Include Henkin model axioms for equality formulation *)
50 (* Inclusion of: Axioms/HEN002-0.ax *)
52 (* -------------------------------------------------------------------------- *)
54 (* File : HEN002-0 : TPTP v3.2.0. Released v1.0.0. *)
56 (* Domain : Henkin Models *)
58 (* Axioms : Henkin model axioms *)
60 (* Version : [MOW76] axioms. *)
64 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
72 (* Syntax : Number of clauses : 7 ( 0 non-Horn; 4 unit; 3 RR) *)
74 (* Number of literals : 11 ( 3 equality) *)
76 (* Maximal clause size : 3 ( 2 average) *)
78 (* Number of predicates : 2 ( 0 propositional; 2-2 arity) *)
80 (* Number of functors : 3 ( 2 constant; 0-2 arity) *)
82 (* Number of variables : 13 ( 3 singleton) *)
84 (* Maximal term depth : 3 ( 1 average) *)
88 (* -------------------------------------------------------------------------- *)
90 (* ----A0: Definition of less_equal *)
92 (* ----A1: x/y <= x *)
94 (* ----A2: (x/z) / (y/z) <= (x/y) / z *)
98 (* ----A4: x <= y and y <= x implies that x = y *)
100 (* ----A5: x <= identity (Thus an implicative model with unit ) *)
102 (* ----Implicit in equality formulation: '/' is well defined *)
104 (* -------------------------------------------------------------------------- *)
106 (* -------------------------------------------------------------------------- *)
107 theorem prove_commutativity:
108 ∀Univ:Set.∀X:Univ.∀Y:Univ.∀Z:Univ.∀a:Univ.∀b:Univ.∀c:Univ.∀d:Univ.∀divide:∀_:Univ.∀_:Univ.Univ.∀e:Univ.∀g:Univ.∀identity:Univ.∀less_equal:∀_:Univ.∀_:Univ.Prop.∀zero:Univ.∀H0:eq Univ (divide identity d) g.∀H1:eq Univ (divide identity c) e.∀H2:eq Univ (divide identity b) d.∀H3:eq Univ (divide identity a) c.∀H4:eq Univ (divide (divide identity a) (divide identity (divide identity b))) (divide (divide identity b) (divide identity (divide identity a))).∀H5:∀X:Univ.less_equal X identity.∀H6:∀X:Univ.∀Y:Univ.∀_:less_equal Y X.∀_:less_equal X Y.eq Univ X Y.∀H7:∀X:Univ.less_equal zero X.∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.less_equal (divide (divide X Z) (divide Y Z)) (divide (divide X Y) Z).∀H9:∀X:Univ.∀Y:Univ.less_equal (divide X Y) X.∀H10:∀X:Univ.∀Y:Univ.∀_:eq Univ (divide X Y) zero.less_equal X Y.∀H11:∀X:Univ.∀Y:Univ.∀_:less_equal X Y.eq Univ (divide X Y) zero.eq Univ (divide c g) (divide d e)
111 autobatch depth=5 width=5 size=20 timeout=10;
116 (* -------------------------------------------------------------------------- *)