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Require Import Arith Unicode.Utf8. | |
Definition f (n : nat) := | |
n - 1. | |
Inductive reaches_1 : nat → Prop := | |
| term_done : reaches_1 1 | |
| term_more (n : nat) : reaches_1 (f n) → reaches_1 n. | |
Check reaches_1_ind. | |
Theorem a01 : ∀ n, n <> 0 -> reaches_1 n. | |
Proof. | |
induction n; [now auto | intros _]. | |
destruct n; [now apply term_done | apply term_more]. | |
unfold f. simpl. | |
apply IHn. | |
now apply Nat.neq_succ_0. | |
Qed. | |
Theorem not_reaches_1_0_aux : forall n, n=0 -> ~ reaches_1 n. | |
Proof. | |
intros n n0 rn. | |
induction rn; [now auto |]. | |
apply IHrn. unfold f. now rewrite n0. | |
Qed. | |
Theorem not_reaches_1_0 : ~ reaches_1 0. | |
Proof. | |
now apply (not_reaches_1_0_aux 0). | |
Qed. |
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