1 include "logic/equality.ma".
3 (* Inclusion of: GRP167-5.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP167-5 : TPTP v3.7.0. Bugfixed v1.2.1. *)
9 (* Domain : Group Theory (Lattice Ordered) *)
11 (* Problem : Product of positive and negative parts *)
13 (* Version : [Fuc94] (equality) axioms : Augmented. *)
15 (* English : Each element in a lattice ordered group can be stated as a *)
17 (* product of it's positive and it's negative part. *)
19 (* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
21 (* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
23 (* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
29 (* Status : Unsatisfiable *)
31 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.20 v2.0.0 *)
33 (* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 1 RR) *)
35 (* Number of atoms : 21 ( 21 equality) *)
37 (* Maximal clause size : 1 ( 1 average) *)
39 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
41 (* Number of functors : 8 ( 2 constant; 0-2 arity) *)
43 (* Number of variables : 43 ( 2 singleton) *)
45 (* Maximal term depth : 3 ( 2 average) *)
47 (* Comments : ORDERING LPO inverse > greatest_lower_bound > *)
49 (* least_upper_bound > product > negative_part > positive_part > *)
53 (* : This is a standardized version of the problem that appears in *)
57 (* : [Dah95] suggested the addition of p10 as a useful lemma. *)
59 (* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
61 (* : v1.2.1 - Clause p10 fixed. *)
63 (* -------------------------------------------------------------------------- *)
65 (* ----Include equality group theory axioms *)
67 (* Inclusion of: Axioms/GRP004-0.ax *)
69 (* -------------------------------------------------------------------------- *)
71 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
73 (* Domain : Group Theory *)
75 (* Axioms : Group theory (equality) axioms *)
77 (* Version : [MOW76] (equality) axioms : *)
79 (* Reduced > Complete. *)
83 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
85 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
93 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
95 (* Number of atoms : 3 ( 3 equality) *)
97 (* Maximal clause size : 1 ( 1 average) *)
99 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
101 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
103 (* Number of variables : 5 ( 0 singleton) *)
105 (* Maximal term depth : 3 ( 2 average) *)
107 (* Comments : [MOW76] also contains redundant right_identity and *)
109 (* right_inverse axioms. *)
111 (* : These axioms are also used in [Wos88] p.186, also with *)
113 (* right_identity and right_inverse. *)
115 (* -------------------------------------------------------------------------- *)
117 (* ----For any x and y in the group x*y is also in the group. No clause *)
119 (* ----is needed here since this is an instance of reflexivity *)
121 (* ----There exists an identity element *)
123 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
125 (* ----= identity. *)
127 (* ----The operation '*' is associative *)
129 (* -------------------------------------------------------------------------- *)
131 (* ----Include Lattice ordered group (equality) axioms *)
133 (* Inclusion of: Axioms/GRP004-2.ax *)
135 (* -------------------------------------------------------------------------- *)
137 (* File : GRP004-2 : TPTP v3.7.0. Bugfixed v1.2.0. *)
139 (* Domain : Group Theory (Lattice Ordered) *)
141 (* Axioms : Lattice ordered group (equality) axioms *)
143 (* Version : [Fuc94] (equality) axioms. *)
147 (* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
149 (* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
151 (* Source : [Sch95] *)
157 (* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
159 (* Number of atoms : 12 ( 12 equality) *)
161 (* Maximal clause size : 1 ( 1 average) *)
163 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
165 (* Number of functors : 3 ( 0 constant; 2-2 arity) *)
167 (* Number of variables : 28 ( 2 singleton) *)
169 (* Maximal term depth : 3 ( 2 average) *)
171 (* Comments : Requires GRP004-0.ax *)
173 (* -------------------------------------------------------------------------- *)
175 (* ----Specification of the least upper bound and greatest lower bound *)
177 (* ----Monotony of multiply *)
179 (* -------------------------------------------------------------------------- *)
181 (* -------------------------------------------------------------------------- *)
183 (* ----Extra lemma *)
185 (∀Univ:Type.∀A:Univ.∀B:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
187 ∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
189 ∀inverse:∀_:Univ.Univ.
190 ∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
191 ∀multiply:∀_:Univ.∀_:Univ.Univ.
192 ∀negative_part:∀_:Univ.Univ.
193 ∀positive_part:∀_:Univ.Univ.
194 ∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (least_upper_bound Y Z)) (least_upper_bound (greatest_lower_bound X Y) (greatest_lower_bound X Z)).
195 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (least_upper_bound X Y) (least_upper_bound X Z)).
196 ∀H2:∀X:Univ.eq Univ (negative_part X) (greatest_lower_bound X identity).
197 ∀H3:∀X:Univ.eq Univ (positive_part X) (least_upper_bound X identity).
198 ∀H4:∀A:Univ.∀B:Univ.eq Univ (inverse (least_upper_bound A B)) (greatest_lower_bound (inverse A) (inverse B)).
199 ∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
200 ∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
201 ∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
202 ∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
203 ∀H9:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
204 ∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
205 ∀H11:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
206 ∀H12:∀X:Univ.eq Univ (least_upper_bound X X) X.
207 ∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
208 ∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
209 ∀H15:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
210 ∀H16:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
211 ∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
212 ∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
213 ∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a (multiply (positive_part a) (negative_part a)))
222 #greatest_lower_bound ##.
225 #least_upper_bound ##.
249 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19 ##;
250 ntry (nassumption) ##;
253 (* -------------------------------------------------------------------------- *)