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(* A sample tarski theorem proof, for f: A -> A.
Syntax is for coq v8. *)
Parameter A : Set.
Variable R : A -> A -> Prop.
Variable Eq : A -> A -> Prop.
Axiom Assym : forall x y : A, R x y -> R y x -> Eq x y.
Axiom Trans : forall x y z : A, R x y -> R y z -> R x z.
Variable f : A -> A.
Axiom Incr : forall x y : A, R x y -> R (f x) (f y).
Variable M : A.
Hypothesis Up : forall x : A, R x (f x) -> R x M.
Hypothesis Least : forall x : A, (forall y : A, R y (f y) -> R y x) -> R M x.
Hint Resolve Up Assym Incr Least Incr Up Trans : db.
Theorem Tarski_lemma : Eq M (f M).
(* We can prove the theorem in one line: *)
(* eauto 15 with db. *)
(* But we rather use basic tactics in this sample file: *)
cut (R M (f M)).
intro.
apply Assym; trivial.
apply Up.
apply Incr; trivial.
apply Least.
intros.
apply Trans with (f y); trivial.
apply Incr.
apply Up; trivial.
Qed.
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