TPTP Problem File: RNG006-2.p
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%--------------------------------------------------------------------------
% File : RNG006-2 : TPTP v8.2.0. Released v1.0.0.
% Domain : Ring Theory
% Problem : X*(Y+ -Z) = (X*Y) + -(X*Z)
% Version : [Wos65] axioms : Reduced & Augmented > Incomplete.
% English :
% Refs : [Wos65] Wos (1965), Unpublished Note
% : [WM76] Wilson & Minker (1976), Resolution, Refinements, and S
% Source : [SPRFN]
% Names : Problem 25 [Wos65]
% : wos25 [WM76]
% Status : Unsatisfiable
% Rating : 0.09 v8.2.0, 0.14 v8.1.0, 0.00 v7.4.0, 0.17 v7.3.0, 0.00 v6.1.0, 0.14 v6.0.0, 0.22 v5.5.0, 0.06 v5.4.0, 0.11 v5.3.0, 0.25 v5.2.0, 0.08 v5.1.0, 0.06 v5.0.0, 0.07 v4.0.1, 0.14 v3.4.0, 0.20 v3.3.0, 0.00 v2.7.0, 0.12 v2.6.0, 0.14 v2.4.0, 0.14 v2.3.0, 0.29 v2.2.1, 0.33 v2.2.0, 0.22 v2.1.0, 0.14 v2.0.0
% Syntax : Number of clauses : 36 ( 13 unt; 0 nHn; 26 RR)
% Number of literals : 87 ( 0 equ; 52 neg)
% Maximal clause size : 5 ( 2 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 3 ( 3 usr; 0 prp; 2-3 aty)
% Number of functors : 11 ( 11 usr; 8 con; 0-2 aty)
% Number of variables : 114 ( 2 sgn)
% SPC : CNF_UNS_RFO_NEQ_HRN
% Comments : These are the same axioms as in [MOW76].
%--------------------------------------------------------------------------
%----Include ring theory axioms
%include('Axioms/RNG001-0.ax').
%--------------------------------------------------------------------------
%----Equality axioms for additive operator
cnf(additive_inverse_substitution,axiom,
( ~ equalish(X,Y)
| equalish(additive_inverse(X),additive_inverse(Y)) ) ).
cnf(add_substitution1,axiom,
( ~ equalish(X,Y)
| equalish(add(X,W),add(Y,W)) ) ).
%----This axiom omited in this version
%input_clause(add_substitution2,axiom,
% [--equalish(X,Y),
% ++equalish(add(W,X),add(W,Y))]).
cnf(sum_substitution1,axiom,
( ~ equalish(X,Y)
| ~ sum(X,W,Z)
| sum(Y,W,Z) ) ).
cnf(sum_substitution2,axiom,
( ~ equalish(X,Y)
| ~ sum(W,X,Z)
| sum(W,Y,Z) ) ).
cnf(sum_substitution3,axiom,
( ~ equalish(X,Y)
| ~ sum(W,Z,X)
| sum(W,Z,Y) ) ).
%----Equality axioms for multiplicative operator
cnf(multiply_substitution1,axiom,
( ~ equalish(X,Y)
| equalish(multiply(X,W),multiply(Y,W)) ) ).
%----This axiom omited in this version
%input_clause(multiply_substitution2,axiom,
% [--equalish(X,Y),
% ++equalish(multiply(W,X),multiply(W,Y))]).
cnf(product_substitution1,axiom,
( ~ equalish(X,Y)
| ~ product(X,W,Z)
| product(Y,W,Z) ) ).
cnf(product_substitution2,axiom,
( ~ equalish(X,Y)
| ~ product(W,X,Z)
| product(W,Y,Z) ) ).
cnf(product_substitution3,axiom,
( ~ equalish(X,Y)
| ~ product(W,Z,X)
| product(W,Z,Y) ) ).
cnf(additive_identity1,axiom,
sum(additive_identity,X,X) ).
cnf(additive_identity2,axiom,
sum(X,additive_identity,X) ).
cnf(closure_of_multiplication,axiom,
product(X,Y,multiply(X,Y)) ).
cnf(closure_of_addition,axiom,
sum(X,Y,add(X,Y)) ).
cnf(left_inverse,axiom,
sum(additive_inverse(X),X,additive_identity) ).
cnf(right_inverse,axiom,
sum(X,additive_inverse(X),additive_identity) ).
cnf(associativity_of_addition1,axiom,
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) ) ).
cnf(associativity_of_addition2,axiom,
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W)
| sum(U,Z,W) ) ).
cnf(commutativity_of_addition,axiom,
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ).
cnf(associativity_of_multiplication1,axiom,
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) ) ).
cnf(associativity_of_multiplication2,axiom,
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) ) ).
cnf(distributivity1,axiom,
( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4)
| sum(V1,V2,V4) ) ).
cnf(distributivity2,axiom,
( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ sum(V1,V2,V4)
| product(X,V3,V4) ) ).
cnf(distributivity3,axiom,
( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| sum(V1,V2,V4) ) ).
cnf(distributivity4,axiom,
( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ sum(V1,V2,V4)
| product(V3,X,V4) ) ).
%-----Equality axioms for operators
cnf(addition_is_well_defined,axiom,
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| equalish(U,V) ) ).
cnf(multiplication_is_well_defined,axiom,
( ~ product(X,Y,U)
| ~ product(X,Y,V)
| equalish(U,V) ) ).
cnf(multiplicative_identity1,axiom,
product(additive_identity,X,additive_identity) ).
cnf(multiplicative_identity2,axiom,
product(X,additive_identity,additive_identity) ).
cnf(product_lemma1,axiom,
( ~ product(A,B,C)
| product(A,additive_inverse(B),additive_inverse(C)) ) ).
cnf(product_lemma2,axiom,
( ~ product(A,B,C)
| product(additive_inverse(A),B,additive_inverse(C)) ) ).
cnf(product_lemma3,axiom,
( ~ product(A,B,C)
| product(additive_inverse(A),additive_inverse(B),C) ) ).
cnf(b_plus_inverse_c,hypothesis,
sum(b,additive_inverse(c),bS_Ic) ).
cnf(a_times_b,hypothesis,
product(a,b,aPb) ).
cnf(a_times_c,hypothesis,
product(a,c,aPc) ).
cnf(aPb_plus_IaPc,hypothesis,
sum(aPb,additive_inverse(aPc),aPb_S_IaPc) ).
cnf(prove_a_times_bS_Ic_is_aPb_S__IaPc,negated_conjecture,
~ product(a,bS_Ic,aPb_S_IaPc) ).
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