%--------------------------------------------------------------------------
% File     : RNG001-0 : TPTP v9.0.0. Released v1.0.0.
% Domain   : Ring Theory
% Axioms   : Ring theory axioms
% Version  : [MOW76] axioms.
% English  :

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
%          : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr
% Source   : [MOW76]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses     :   17 (   6 unt;   0 nHn;  11 RR)
%            Number of literals    :   50 (   2 equ;  33 neg)
%            Maximal clause size   :    5 (   2 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 2-3 aty)
%            Number of functors    :    4 (   4 usr;   1 con; 0-2 aty)
%            Number of variables   :   71 (   0 sgn)
% SPC      : 

% Comments : These axioms are used in [Wos88] p.201.
%--------------------------------------------------------------------------
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)
    | U = V ) ).

cnf(multiplication_is_well_defined,axiom,
    ( ~ product(X,Y,U)
    | ~ product(X,Y,V)
    | U = V ) ).

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