This commit is contained in:
Thomas Baruchel 2022-12-10 16:46:54 +01:00
parent 1ea4f6d502
commit e5e7f7f6a0

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@ -1117,6 +1117,47 @@ Proof.
+ simpl in H. symmetry in H. apply Bool.diff_true_false in H. contradiction H. + simpl in H. symmetry in H. apply Bool.diff_true_false in H. contradiction H.
Qed. Qed.
Lemma lt_split_bits : forall n m k j,
0 < k -> j < 2^m -> k*2^m < 2^n -> k*2^m+j < 2^n.
Proof.
intros n m k j. intros H I J.
assert (K: 2^m <= k*2^m). rewrite <- Nat.mul_1_l at 1.
apply Nat.mul_le_mono_r. rewrite Nat.le_succ_l. assumption.
assert (L:2^m < 2^n). generalize J. generalize K. apply Nat.le_lt_trans.
assert (k < 2^(n-m)). rewrite Nat.mul_lt_mono_pos_r with (p := 2^m).
rewrite <- Nat.pow_add_r. rewrite Nat.sub_add. assumption.
apply Nat.pow_lt_mono_r_iff in L. apply Nat.lt_le_incl. assumption.
apply Nat.lt_1_2. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
replace (2^(n-m)) with (S (2^(n-m)-1)) in H0. rewrite Nat.lt_succ_r in H0.
apply Nat.mul_le_mono_r with (p := 2^m) in H0.
rewrite Nat.mul_sub_distr_r in H0. rewrite Nat.mul_1_l in H0.
rewrite <- Nat.pow_add_r in H0. rewrite Nat.sub_add in H0.
rewrite Nat.add_le_mono_r with (p := j) in H0.
assert (2^n - 2^m + j < 2^n).
rewrite Nat.add_lt_mono_l with (p := 2^n) in I.
rewrite Nat.add_lt_mono_r with (p := 2^m).
rewrite <- Nat.add_assoc. rewrite <- Nat.add_sub_swap.
rewrite Nat.add_assoc. rewrite Nat.add_sub. assumption.
apply Nat.lt_le_incl. assumption.
generalize H1. generalize H0. apply Nat.le_lt_trans.
apply Nat.lt_le_incl. rewrite <- Nat.pow_lt_mono_r_iff in L. assumption.
apply Nat.lt_1_2. rewrite <- Nat.add_1_r. apply Nat.sub_add.
rewrite Nat.le_succ_l. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
Qed.
(* TODO: supprimer lt_even_even après avoir réussi ce truc ??? *)
Lemma lt_split_bits : forall n m k j, Lemma lt_split_bits : forall n m k j,
0 < k -> j < m -> k * 2^m < 2^n -> k * 2^m +2^j < 2^n. 0 < k -> j < m -> k * 2^m < 2^n -> k * 2^m +2^j < 2^n.
Proof. Proof.
@ -1173,6 +1214,8 @@ Proof.
Qed. Qed.
(* TODO: truc suivant inutile ? *) (* TODO: truc suivant inutile ? *)
(* TODO: réécrire en plus puissant avec i et j < 2^m et on ajoute
désormais direcctement i et j au lieu d'ajouter 2^i et 2^j *)
Lemma tm_step_repeating_patterns : Lemma tm_step_repeating_patterns :
forall (n m i j : nat), forall (n m i j : nat),
i < m -> j < m i < m -> j < m