TPTP Axioms File: NUM003-0.ax
%--------------------------------------------------------------------------
% File : NUM003-0 : TPTP v9.0.0. Bugfixed v1.2.1.
% Domain : Number Theory
% Axioms : Number theory axioms, based on Godel set theory
% Version : [BL+86] axioms.
% English :
% Refs : [BL+86] Boyer et al. (1986), Set Theory in First-Order Logic:
% : [McC92] McCune (1992), Email to G. Sutcliffe
% Source : [McC92]
% Names :
% Status : Satisfiable
% Syntax : Number of clauses : 54 ( 0 unt; 32 nHn; 54 RR)
% Number of literals : 215 ( 16 equ; 116 neg)
% Maximal clause size : 7 ( 3 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-3 aty)
% Number of functors : 29 ( 29 usr; 7 con; 0-3 aty)
% Number of variables : 90 ( 0 sgn)
% SPC :
% Comments : Requires SET003-0.ax ALG001-0.ax
% Bugfixes : v1.2.1 - Clauses finite3 and finite5 fixed.
%--------------------------------------------------------------------------
%----Definition of natural_numbers (natural numbers)
cnf(natural_numbers1,axiom,
( ~ member(Z,natural_numbers)
| ~ little_set(Xs)
| ~ member(empty_set,Xs)
| member(f43(Z,Xs),Xs)
| member(Z,Xs) ) ).
cnf(natural_numbers2,axiom,
( ~ member(Z,natural_numbers)
| ~ little_set(Xs)
| ~ member(empty_set,Xs)
| ~ member(successor(f43(Z,Xs)),Xs)
| member(Z,Xs) ) ).
cnf(natural_numbers3,axiom,
( member(Z,natural_numbers)
| ~ little_set(Z)
| little_set(f44(Z)) ) ).
cnf(natural_numbers4,axiom,
( member(Z,natural_numbers)
| ~ little_set(Z)
| member(empty_set,f44(Z)) ) ).
cnf(natural_numbers5,axiom,
( member(Z,natural_numbers)
| ~ little_set(Z)
| ~ member(Xk,f44(Z))
| member(successor(Xk),f44(Z)) ) ).
cnf(natural_numbers6,axiom,
( member(Z,natural_numbers)
| ~ member(Z,f44(Z)) ) ).
%----Definition of plus
cnf(plus1,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| member(f45(Z,Xs),natural_numbers)
| member(f46(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(plus2,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| member(f45(Z,Xs),natural_numbers)
| member(f47(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(plus3,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| member(f45(Z,Xs),natural_numbers)
| member(f48(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(plus4,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| member(f45(Z,Xs),natural_numbers)
| member(ordered_pair(ordered_pair(f46(Z,Xs),f47(Z,Xs)),f48(Z,Xs)),Xs)
| member(Z,Xs) ) ).
cnf(plus5,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| member(f45(Z,Xs),natural_numbers)
| ~ member(ordered_pair(ordered_pair(successor(f46(Z,Xs)),f47(Z,Xs)),successor(f48(Z,Xs))),Xs)
| member(Z,Xs) ) ).
cnf(plus6,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f45(Z,Xs)),f45(Z,Xs)),Xs)
| member(f46(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(plus7,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f45(Z,Xs)),f45(Z,Xs)),Xs)
| member(f47(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(plus8,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f45(Z,Xs)),f45(Z,Xs)),Xs)
| member(f48(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(plus9,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f45(Z,Xs)),f45(Z,Xs)),Xs)
| member(ordered_pair(ordered_pair(f46(Z,Xs),f47(Z,Xs)),f48(Z,Xs)),Xs)
| member(Z,Xs) ) ).
cnf(plus10,axiom,
( ~ member(Z,plus)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f45(Z,Xs)),f45(Z,Xs)),Xs)
| ~ member(ordered_pair(ordered_pair(successor(f46(Z,Xs)),f47(Z,Xs)),successor(f48(Z,Xs))),Xs)
| member(Z,Xs) ) ).
cnf(plus11,axiom,
( member(Z,plus)
| ~ little_set(Z)
| little_set(f49(Z)) ) ).
cnf(plus12,axiom,
( member(Z,plus)
| ~ little_set(Z)
| ~ member(Xi,natural_numbers)
| member(ordered_pair(ordered_pair(empty_set,Xi),Xi),f49(Z)) ) ).
cnf(plus13,axiom,
( member(Z,plus)
| ~ little_set(Z)
| ~ member(Uu1,natural_numbers)
| ~ member(Xj,natural_numbers)
| ~ member(Xk,natural_numbers)
| ~ member(ordered_pair(ordered_pair(Uu1,Xj),Xk),f49(Z))
| member(ordered_pair(ordered_pair(successor(Uu1),Xj),successor(Xk)),f49(Z)) ) ).
cnf(plus14,axiom,
( member(Z,plus)
| ~ member(Z,f49(Z)) ) ).
%----Definition of times
cnf(times1,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| member(f50(Z,Xs),natural_numbers)
| member(f51(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(times2,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| member(f50(Z,Xs),natural_numbers)
| member(f52(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(times3,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| member(f50(Z,Xs),natural_numbers)
| member(f53(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(times4,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| member(f50(Z,Xs),natural_numbers)
| member(ordered_pair(ordered_pair(f51(Z,Xs),f52(Z,Xs)),f53(Z,Xs)),Xs)
| member(Z,Xs) ) ).
cnf(times5,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| member(f50(Z,Xs),natural_numbers)
| ~ member(ordered_pair(ordered_pair(successor(f51(Z,Xs)),f52(Z,Xs)),apply_to_two_arguments(plus,f53(Z,Xs),f52(Z,Xs))),Xs)
| member(Z,Xs) ) ).
cnf(times6,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f50(Z,Xs)),empty_set),Xs)
| member(f51(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(times7,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f50(Z,Xs)),empty_set),Xs)
| member(f52(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(times8,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f50(Z,Xs)),empty_set),Xs)
| member(f53(Z,Xs),natural_numbers)
| member(Z,Xs) ) ).
cnf(times9,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f50(Z,Xs)),empty_set),Xs)
| member(ordered_pair(ordered_pair(f51(Z,Xs),f52(Z,Xs)),f53(Z,Xs)),Xs)
| member(Z,Xs) ) ).
cnf(times10,axiom,
( ~ member(Z,times)
| ~ little_set(Xs)
| ~ member(ordered_pair(ordered_pair(empty_set,f50(Z,Xs)),empty_set),Xs)
| ~ member(ordered_pair(ordered_pair(successor(f51(Z,Xs)),f52(Z,Xs)),apply_to_two_arguments(plus,f53(Z,Xs),f52(Z,Xs))),Xs)
| member(Z,Xs) ) ).
cnf(times11,axiom,
( member(Z,times)
| ~ little_set(Z)
| little_set(f54(Z)) ) ).
cnf(times12,axiom,
( member(Z,times)
| ~ little_set(Z)
| ~ member(Xi,natural_numbers)
| member(ordered_pair(ordered_pair(empty_set,Xi),empty_set),f54(Z)) ) ).
cnf(times13,axiom,
( member(Z,times)
| ~ little_set(Z)
| ~ member(Uu2,natural_numbers)
| ~ member(Xj,natural_numbers)
| ~ member(Xk,natural_numbers)
| ~ member(ordered_pair(ordered_pair(Uu2,Xj),Xk),f54(Z))
| member(ordered_pair(ordered_pair(successor(Uu2),Xj),apply_to_two_arguments(plus,Xk,Xj)),f54(Z)) ) ).
cnf(times14,axiom,
( member(Z,times)
| ~ member(Z,f54(Z)) ) ).
%----Definition of prime_numbers
cnf(prime_numbers1,axiom,
( ~ member(Z,prime_numbers)
| member(Z,natural_numbers) ) ).
cnf(prime_numbers2,axiom,
( ~ member(Z,prime_numbers)
| Z != empty_set ) ).
cnf(prime_numbers3,axiom,
( ~ member(Z,prime_numbers)
| Z != successor(empty_set) ) ).
cnf(prime_numbers4,axiom,
( ~ member(Z,prime_numbers)
| ~ member(U,natural_numbers)
| ~ member(V,natural_numbers)
| apply_to_two_arguments(times,U,V) != Z
| member(U,non_ordered_pair(successor(empty_set),Z)) ) ).
cnf(prime_numbers5,axiom,
( member(Z,prime_numbers)
| ~ member(Z,natural_numbers)
| Z = empty_set
| Z = successor(empty_set)
| member(f55(Z),natural_numbers) ) ).
cnf(prime_numbers6,axiom,
( member(Z,prime_numbers)
| ~ member(Z,natural_numbers)
| Z = empty_set
| Z = successor(empty_set)
| member(f56(Z),natural_numbers) ) ).
cnf(prime_numbers7,axiom,
( member(Z,prime_numbers)
| ~ member(Z,natural_numbers)
| Z = empty_set
| Z = successor(empty_set)
| apply_to_two_arguments(times,f55(Z),f56(Z)) = Z ) ).
cnf(prime_numbers8,axiom,
( member(Z,prime_numbers)
| ~ member(Z,natural_numbers)
| Z = empty_set
| Z = successor(empty_set)
| ~ member(f55(Z),non_ordered_pair(successor(empty_set),Z)) ) ).
%----Definition of finite
cnf(finite1,axiom,
( ~ finite(X)
| member(f57(X),natural_numbers) ) ).
cnf(finite2,axiom,
( ~ finite(X)
| maps(f58(X),f57(X),X) ) ).
cnf(finite3,axiom,
( ~ finite(X)
| range_of(f58(X)) = X ) ).
cnf(finite4,axiom,
( ~ finite(X)
| one_to_one_function(f58(X)) ) ).
cnf(finite5,axiom,
( finite(X)
| ~ member(Xn,natural_numbers)
| ~ maps(Xf,Xn,X)
| range_of(Xf) != X
| ~ one_to_one_function(Xf) ) ).
%----Definition of twin prime_numbers
cnf(twin_primes1,axiom,
( ~ member(Z,twin_prime_numbers)
| member(Z,prime_numbers) ) ).
cnf(twin_primes2,axiom,
( ~ member(Z,twin_prime_numbers)
| member(successor(successor(Z)),prime_numbers) ) ).
cnf(twin_primes3,axiom,
( member(Z,twin_prime_numbers)
| ~ member(Z,prime_numbers)
| ~ member(successor(successor(Z)),prime_numbers) ) ).
%----Definition of even_numbers (even natural numbers)
cnf(even_numbers1,axiom,
( ~ member(Z,even_numbers)
| member(Z,natural_numbers) ) ).
cnf(even_numbers2,axiom,
( ~ member(Z,even_numbers)
| member(f59(Z),natural_numbers) ) ).
cnf(even_numbers3,axiom,
( ~ member(Z,even_numbers)
| apply_to_two_arguments(plus,f59(Z),f59(Z)) = Z ) ).
cnf(even_numbers4,axiom,
( member(Z,even_numbers)
| ~ member(Z,natural_numbers)
| ~ member(X,natural_numbers)
| apply_to_two_arguments(plus,X,X) != Z ) ).
%--------------------------------------------------------------------------