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Thierry Coquand - The computational content of classical logic (coq)
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From Coq Require Import Unicode.Utf8. | |
From Coq Require Import Arith. | |
Section CoquandCompContext. | |
Hypothesis f: nat -> nat. | |
Context (x0: nat). | |
Hypothesis H1: forall x, exists y, x < f y. | |
Hypothesis H2: f 0 <= x0. | |
Definition P := exists n, f n <= x0 /\ x0 < f (S n). | |
Definition F: Type -> Type := fun _ => P. | |
Inductive M(F: Type -> Type): Type -> Type := | |
| ret: forall t, t -> M F t | |
| bind: forall a b, M F a -> (a -> M F b) -> M F b | |
| K: forall t, F t -> M F t. | |
Definition Δ := M F. | |
Lemma lem1: forall n X, f n <= x0 -> x0 < f (S n) -> Δ X. | |
Proof. | |
intros. | |
apply K. | |
unfold F, P. | |
exists n. split; auto. | |
Defined. | |
Lemma lem: forall n, Δ (f n <= x0). | |
Proof. | |
induction n. | |
- apply ret. apply H2. | |
- eapply bind. eapply IHn. | |
intro H. | |
destruct (le_gt_dec (f (S n)) x0). | |
+ apply ret. apply l. | |
+ eapply lem1. apply H. | |
apply g. | |
Defined. | |
Lemma lem2': forall X, Δ X -> P + X. | |
Proof. | |
intros. | |
induction X0. | |
- unfold F in *. left. apply f0. | |
- right. apply t0. | |
- destruct IHX0. | |
+ left. apply p. | |
+ apply X. apply a0. | |
Defined. | |
Lemma lem2: forall X, ~ X -> Δ X -> P. | |
Proof. | |
intros. | |
apply lem2' in X0. | |
destruct X0. | |
+ apply p. | |
+ contradiction. | |
Defined. | |
Lemma final: P. | |
Proof. | |
destruct (H1 x0). | |
rename x into y0. | |
eapply (@lem2 (f y0 <= x0)). | |
- intros Hcontra. | |
apply (le_not_lt (f y0) x0). | |
apply Hcontra. | |
apply H. | |
- apply lem. | |
Defined. | |
End CoquandCompContext. |
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