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% Bernoulli's inequality.   1 + nx <= (1+x)^n 
%  here n is a natural number and x is real.
% so the typings are  as follows:
% type(R, ^(R,N)).
%  type(R, +(R,R))  (We never need to add exponents)
% type(R, *(R,R))   (We never need to multiply exponents)
% type(o,N)
% type(1,R)
% type(0,R)
% type(R,i(N))    (i injects N into R)
% type(N, s(N))

set(simplify_rule).
set(binary_res).
set(hyper_res).
set(lambda).
set(induction).
set(para_into).
set(para_all).
assign(pick_given_ratio,4).
%set(very_verbose).

op(450, xfy, +).  
op(440, fx, -).    %unary minus should be parenthesized a + (-b) + c
op(400, xfx, /).
op(350, fy, ~).   
op(400, xfy, *).  
op(300, xfx, ^).

assign(max_weight,36).
assign(max_distinct_vars,0).
weight_list(pick_and_purge).
weight(g($(0)),1).   %every g-term counts like a constant
weight(i($(1)),0).    % don't count 'i' as a symbol, it's just an injection of N into R
weight(junk,50).
end_of_list.

list(usable).
x = x.
x <=x.
x + 1 != 0.
-(x<x).
-(0<x) |  x^o = 1.
x + 1 != y + 1 | x = y.
-ap(y,o) | ap(y,g(y)) | ap(y,z).              %induction 
-ap(y,o) | -ap(y,s(g(y))) | ap(y,z).         
x <= y | y < x.
- (x<y)  | - (z <= x) | z < y.           % transitivity
x < y | x = y | y < x.
-(x<= y) | -(y <x) .
0 < a+1.
-(x <= y) | - (0 < z) | z*x <= z*y.
-(x <= y) | - (0 < z) | x*z <= y*z.
-(x <= y) | - (0 < z) | x*z <= z*y.
-(x <= y) | - (0 < z) | z*x <= y*z.
0 <= x^2.
- (x<y) | x + (- (y)) < 0. 
(x <= y) | y < x.
(x < y) | y <= x.
x^o = 1 | x = 0.
0 <= i(x).       % i embeds N into R
end_of_list.

list(demodulators).
x/y = junk.           % Division is irrelevant in this problem.
i(o) = 0.
i(s(x)) = i(x)+1.
x^s(y)= x*x^y.
x*0 = 0.
x+0=x.
0+x = x.
0*x = 0.
 % The following demodulators are used only on terms created by Simplify,  and converts them to correctly typed terms.
 % The second two are never actually used but are theoretically necessary to keep + out of the exponent.
x^ (y+1) = x*x^y.  
x^ (y+z) = (x^y) * (x^z).  
x^1 = x.  
end_of_list.

list(sos).
 -( 1 + i(n)*a <= (1+a)^n ).  
 end_of_list.