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163 lines
2.4 KiB
Coq
163 lines
2.4 KiB
Coq
(**
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算術演算関連の補題
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*)
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Require Import Omega NArith Euclid.
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(** ** 算術演算 *)
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Lemma mult_S_lt_reg_l :
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forall n m p, 0 < n -> n * m < n * p -> m < p.
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Proof.
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intros.
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destruct n.
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inversion H.
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elim (le_or_lt m p).
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intro.
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inversion H1.
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rewrite H2 in H0.
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elim (lt_irrefl _ H0).
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omega.
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intro.
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apply (mult_S_lt_compat_l n _ _) in H1.
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omega.
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Qed.
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Lemma plus_elim: forall p a b,
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a + p < b -> a < b.
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Proof.
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intros.
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omega.
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Qed.
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(** ** pow *)
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Fixpoint pow (n : nat) :=
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match n with
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| 0 =>
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1
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| S n' =>
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2 * pow n'
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end.
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Lemma pow_lt_O : forall n,
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0 < pow n.
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Proof.
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induction n; simpl; omega.
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Qed.
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Hint Resolve pow_lt_O.
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Lemma pow_add: forall n m,
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pow n * pow m = pow (n + m).
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Proof.
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induction n; intros.
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simpl in *.
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omega.
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simpl.
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repeat rewrite plus_0_r.
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rewrite <- IHn, mult_plus_distr_r.
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reflexivity.
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Qed.
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(** ** divmod *)
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Definition divmod (n m : nat) (P : 0 < m) :=
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eucl_dev m P n.
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Lemma divmod_lt_q : forall (n m q r s t: nat),
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n < pow (s + t) ->
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n = q * pow s + r ->
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q < pow t.
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Proof.
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intros.
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rewrite H0 in H.
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apply plus_elim in H.
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rewrite <- pow_add, mult_comm in H.
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apply mult_S_lt_reg_l in H.
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assumption.
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apply pow_lt_O.
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Qed.
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Lemma divmod_not_O: forall n m q r,
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0 < n ->
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0 < m ->
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n = q * m + r ->
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0 < q \/ 0 < r.
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Proof.
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intros.
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rewrite H1 in H.
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destruct q.
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simpl in H.
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right.
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assumption.
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left.
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omega.
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Qed.
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Lemma divmod_O: forall n q r,
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0 = q * n + r ->
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n <> 0 ->
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q = 0 /\ r = 0.
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Proof.
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intros.
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destruct q; destruct n; destruct r; try omega.
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simpl in H.
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discriminate.
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simpl in H.
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discriminate.
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Qed.
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Lemma divmod_O_pow: forall n q r,
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0 = q * pow n + r ->
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q = 0 /\ r = 0.
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Proof.
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intros.
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apply (divmod_O (pow n) _ _); auto.
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intro.
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generalize (pow_lt_O n); intros.
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omega.
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Qed.
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Lemma plus_O : forall n,
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n + 0 = n.
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Proof.
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Admitted.
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Lemma plus_double : forall n,
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n < n + n.
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Admitted.
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Lemma pow_lt : forall n m,
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n < m ->
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pow n < pow m.
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Proof.
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induction n; induction m; simpl; intros.
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inversion H.
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destruct m; auto.
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transitivity (pow (S m));
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[ apply IHm | idtac];
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omega.
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inversion H.
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simpl in IHm.
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inversion H; simpl.
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repeat (rewrite plus_O in *).
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apply (Plus.plus_lt_compat_r O _).
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transitivity (pow n); auto.
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apply plus_double.
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assert (S n < m); auto.
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apply IHm in H2.
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transitivity (pow m); auto.
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rewrite plus_O.
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apply plus_double.
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Qed.
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Hint Resolve pow_lt pow_lt_O: pow.
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