2 include "logic/equality.ma".
3 (* Inclusion of: ROB030-1.p *)
4 (* ------------------------------------------------------------------------------ *)
5 (* File : ROB030-1 : TPTP v3.1.1. Released v3.1.0. *)
6 (* Domain : Robbins Algebra *)
7 (* Problem : Exists absorbed element => Exists absorbed within negation element *)
8 (* Version : [Win90] (equality) axioms. *)
9 (* Theorem formulation : Denies Huntington's axiom. *)
10 (* English : If there are elements c and d such that c+d=d, then the *)
11 (* algebra is Boolean. *)
12 (* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
13 (* : [Loe04] Loechner (2004), Email to Geoff Sutcliffe *)
14 (* Source : [Loe04] *)
15 (* Names : (1) [Loe04] *)
16 (* Status : Unsatisfiable *)
17 (* Rating : 0.00 v3.1.0 *)
18 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
19 (* Number of atoms : 5 ( 5 equality) *)
20 (* Maximal clause size : 1 ( 1 average) *)
21 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
22 (* Number of functors : 4 ( 2 constant; 0-2 arity) *)
23 (* Number of variables : 9 ( 1 singleton) *)
24 (* Maximal term depth : 6 ( 2 average) *)
26 (* ------------------------------------------------------------------------------ *)
27 (* ----Include axioms for Robbins algebra *)
28 (* Inclusion of: Axioms/ROB001-0.ax *)
29 (* -------------------------------------------------------------------------- *)
30 (* File : ROB001-0 : TPTP v3.1.1. Released v1.0.0. *)
31 (* Domain : Robbins algebra *)
32 (* Axioms : Robbins algebra axioms *)
33 (* Version : [Win90] (equality) axioms. *)
35 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
36 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
37 (* Source : [OTTER] *)
38 (* Names : Lemma 2.2 [Win90] *)
40 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
41 (* Number of literals : 3 ( 3 equality) *)
42 (* Maximal clause size : 1 ( 1 average) *)
43 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
44 (* Number of functors : 2 ( 0 constant; 1-2 arity) *)
45 (* Number of variables : 7 ( 0 singleton) *)
46 (* Maximal term depth : 6 ( 3 average) *)
48 (* -------------------------------------------------------------------------- *)
49 (* -------------------------------------------------------------------------- *)
50 (* ------------------------------------------------------------------------------ *)
51 theorem prove_absorption_within_negation:
53 \forall add:\forall _:Univ.\forall _:Univ.Univ.
56 \forall negate:\forall _:Univ.Univ.
57 \forall H0:eq Univ (add c d) d.
58 \forall H1:\forall X:Univ.\forall Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
59 \forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
60 \forall H3:\forall X:Univ.\forall Y:Univ.eq Univ (add X Y) (add Y X).\exist A:Univ.\exist B:Univ.eq Univ (negate (add A B)) (negate B)
67 autobatch paramodulation timeout=100;
75 (* ------------------------------------------------------------------------------ *)