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| -rw-r--r-- | coq/KnasterTarski.v | 29 |
1 files changed, 29 insertions, 0 deletions
diff --git a/coq/KnasterTarski.v b/coq/KnasterTarski.v new file mode 100644 index 00000000..c377c1a3 --- /dev/null +++ b/coq/KnasterTarski.v @@ -0,0 +1,29 @@ +(* 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. + +Theorem Tarski_lemma : Eq M (f M). +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. |