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% An element z is called nilpotent if z^n = 0 for some positive integer n. 
%  Show that 0 is the only nilpotent in a domain.
% This file shows that Otter-lambda can use lambda-unification to find the required instance of induction.

op(450, xfx, +).  % sum
op(350, fy, ~).   % minus
op(400, xfx, *).  % product

set(lambda).
set(induction).
set(hyper_res).
set(binary_res).
set(knuth_bendix).

assign(max_weight,32).
assign(pick_given_ratio,4).

%Typings:  R +(R,R)
%          R *(R,R)
%          R^  inv(R)  where R^ means R union "undefined"
%          R pow(N,R)
%          Prop ap(PF,N).
%          PF lambda(N,Prop).
%  i.e.  all propositional functions take integer variables


list(usable).
x=x.
% ring axioms
 0 + x = x.
 x + 0 = x.
 ~x + x = 0.
 x + ~x = 0.
 (x + y) + z = x + (y + z).
 x + y = y + x.
(x * y) * z = x * (y * z).     % product is associative
 x * (y + z) = x * y + x * z.   % right distributivity
 (y + z) * x = y * x + z * x.   % left distributivity
 x * 1 = x.
 one * x = x.
 x = 0 | x * inv(x) = 1.
 x = 0 |  inv(x) * x = one.
1 != 0.
 x * y != 0  | x = 0 | y = 0.    %  no zero divisors  ("integral domain")
pow(s(y),z) = z * pow(y,z).
pow(o,z) = 1.

%axioms for natural numbers.  Use lower case o for 0 to preserve typing.  s is successor, p is predecessor.
x = o | x = s(p(x)).     % every nonzero number is the successor of its predecessor.

ap(y,o) | -ap(y,g(z,y)) | -ap(y,z).         % These two are the clausal form of mathematical induction
ap(y,o) | ap(y,s(g(z,y))) | -ap(y,z).
b != 0.
end_of_list.
 
list(sos).
pow(n,b) = 0.       % The negated goal
end_of_list.