real : THEORY
BEGIN
%(sturm) and (mono-poly) incorporate some nice decision procedures 
IMPORTING Sturm@strategies
%IMPORTING Tarski@strategies

% Subtypes D1 and D2 which is the set {1,2,3,4,5}
D1: TYPE = {i: nat | i = 1 OR i= 2 OR i=3 OR i=4 OR i=5}
D2: TYPE = {i: nat | 1 <= i AND i <= 5}

conja1: CONJECTURE (FORALL (x:D1) : x*x >= x)

% The abobe proves wih (grind) or the proof below (saved with with C-c 2p):
%|- conja1 : PROOF
%|- (then (skeep) (typepred "x") (assert))
%|- QED

conja2: CONJECTURE (FORALL (x:D2) : x*x >= x)

%|- conja2 : PROOF
%|- (then (skeep) (lemma "both_sides_div_pos_neg_ge1") (grind))
%|- QED
% Lemma is: FORALL (n0z: nonzero_real, x, y: real):
%         IF n0z > 0 THEN x/n0z >= y/n0z ELSE y/n0z >= x/n0z ENDIF
%          IFF x >= y

% We could also prove the above automatically in one step using (sturm). 

% The following should not prove because e.g. x = 0.5 is a counter-example
conjb: CONJECTURE (FORALL (x:real) : x^2 >= x)

conjc: CONJECTURE (FORALL (x: real): x >= 1.0 IMPLIES x * x >= x)

% (sturm) proves the above automatically.
% Alternatively, manip introduces additional rules such as (div-by):
%|- conj4 : PROOF
%|- (then (skeep) (div-by 1 "x"))
%|- QED

  example_14 : LEMMA 
    FORALL (x:real) : x ## [| 0, oo |] IMPLIES -x^3 <= 0
%|- example_14 : PROOF
%|- (sturm)
%|- QED

  mono_example_4: LEMMA
    FORALL (x,y:real): x /= y IMPLIES x^3 /= y^3
%|- mono_example_4 : PROOF
%|- (mono-poly)
%|- QED


END real