--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP170-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP170-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p03b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c > d *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p03b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound c d) c.
+∀H1:eq Univ (greatest_lower_bound a b) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀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).
+∀H11:∀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).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply a c) (multiply b d)) (multiply a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
projects / helm.git / blobdiff