diff options
Diffstat (limited to 'theories/Numbers/Natural/BigN/Nbasic.v')
| -rw-r--r-- | theories/Numbers/Natural/BigN/Nbasic.v | 64 |
1 files changed, 32 insertions, 32 deletions
diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v index ae2cfd30..d42db97d 100644 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ b/theories/Numbers/Natural/BigN/Nbasic.v @@ -8,7 +8,7 @@ (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) -(*i $Id: Nbasic.v 10964 2008-05-22 11:08:13Z letouzey $ i*) +(*i $Id$ i*) Require Import ZArith. Require Import BigNumPrelude. @@ -21,7 +21,7 @@ Require Import DoubleCyclic. (* To compute the necessary height *) Fixpoint plength (p: positive) : positive := - match p with + match p with xH => xH | xO p1 => Psucc (plength p1) | xI p1 => Psucc (plength p1) @@ -34,10 +34,10 @@ rewrite Zpower_exp; auto with zarith. rewrite Zpos_succ_morphism; unfold Zsucc; auto with zarith. intros p; elim p; simpl plength; auto. intros p1 Hp1; rewrite F; repeat rewrite Zpos_xI. -assert (tmp: (forall p, 2 * p = p + p)%Z); +assert (tmp: (forall p, 2 * p = p + p)%Z); try repeat rewrite tmp; auto with zarith. intros p1 Hp1; rewrite F; rewrite (Zpos_xO p1). -assert (tmp: (forall p, 2 * p = p + p)%Z); +assert (tmp: (forall p, 2 * p = p + p)%Z); try repeat rewrite tmp; auto with zarith. rewrite Zpower_1_r; auto with zarith. Qed. @@ -73,7 +73,7 @@ case (Z_mod_lt (Zpos p) (Zpos q) H1); auto with zarith. intros q1 H2. replace (Zpos p - Zpos q * Zpos q1) with (Zpos p mod Zpos q). 2: pattern (Zpos p) at 2; rewrite H2; auto with zarith. -generalize H2 (Z_mod_lt (Zpos p) (Zpos q) H1); clear H2; +generalize H2 (Z_mod_lt (Zpos p) (Zpos q) H1); clear H2; case Zmod. intros HH _; rewrite HH; auto with zarith. intros r1 HH (_,HH1); rewrite HH; rewrite Zpos_succ_morphism. @@ -121,9 +121,9 @@ Definition zn2z_word_comm : forall w n, zn2z (word w n) = word (zn2z w) n. Defined. Fixpoint extend (n:nat) {struct n} : forall w:Type, zn2z w -> word w (S n) := - match n return forall w:Type, zn2z w -> word w (S n) with + match n return forall w:Type, zn2z w -> word w (S n) with | O => fun w x => x - | S m => + | S m => let aux := extend m in fun w x => WW W0 (aux w x) end. @@ -169,7 +169,7 @@ Fixpoint diff_l (m n : nat) {struct m} : fst (diff m n) + n = max m n := | S n1 => let v := fst (diff m1 n1) + n1 in let v1 := fst (diff m1 n1) + S n1 in - eq_ind v (fun n => v1 = S n) + eq_ind v (fun n => v1 = S n) (eq_ind v1 (fun n => v1 = n) (refl_equal v1) (S v) (plusnS _ _)) _ (diff_l _ _) end @@ -182,7 +182,7 @@ Fixpoint diff_r (m n: nat) {struct m}: snd (diff m n) + m = max m n := | 0 => refl_equal _ | S _ => plusn0 _ end - | S m => + | S m => match n return (snd (diff (S m) n) + S m = max (S m) n) with | 0 => refl_equal (snd (diff (S m) 0) + S m) | S n1 => @@ -253,9 +253,9 @@ Section ReduceRec. | WW xh xl => match xh with | W0 => @reduce_n m xl - | _ => @c (S m) x + | _ => @c (S m) x end - end + end end. End ReduceRec. @@ -276,14 +276,14 @@ Section CompareRec. Variable compare_m : wm -> w -> comparison. Fixpoint compare0_mn (n:nat) : word wm n -> comparison := - match n return word wm n -> comparison with - | O => compare0_m + match n return word wm n -> comparison with + | O => compare0_m | S m => fun x => match x with | W0 => Eq - | WW xh xl => + | WW xh xl => match compare0_mn m xh with - | Eq => compare0_mn m xl + | Eq => compare0_mn m xl | r => Lt end end @@ -296,7 +296,7 @@ Section CompareRec. Variable spec_compare0_m: forall x, match compare0_m x with Eq => w_to_Z w_0 = wm_to_Z x - | Lt => w_to_Z w_0 < wm_to_Z x + | Lt => w_to_Z w_0 < wm_to_Z x | Gt => w_to_Z w_0 > wm_to_Z x end. Variable wm_to_Z_pos: forall x, 0 <= wm_to_Z x < base wm_base. @@ -341,14 +341,14 @@ Section CompareRec. Qed. Fixpoint compare_mn_1 (n:nat) : word wm n -> w -> comparison := - match n return word wm n -> w -> comparison with - | O => compare_m - | S m => fun x y => + match n return word wm n -> w -> comparison with + | O => compare_m + | S m => fun x y => match x with | W0 => compare w_0 y - | WW xh xl => + | WW xh xl => match compare0_mn m xh with - | Eq => compare_mn_1 m xl y + | Eq => compare_mn_1 m xl y | r => Gt end end @@ -366,7 +366,7 @@ Section CompareRec. | Lt => wm_to_Z x < w_to_Z y | Gt => wm_to_Z x > w_to_Z y end. - Variable wm_base_lt: forall x, + Variable wm_base_lt: forall x, 0 <= w_to_Z x < base (wm_base). Let double_wB_lt: forall n x, @@ -385,7 +385,7 @@ Section CompareRec. unfold Zpower_pos; simpl; ring. Qed. - + Lemma spec_compare_mn_1: forall n x y, match compare_mn_1 n x y with Eq => double_to_Z n x = w_to_Z y @@ -434,7 +434,7 @@ Section AddS. | C1 z => match incr hy with C0 z1 => C0 (WW z1 z) | C1 z1 => C1 (WW z1 z) - end + end end end. @@ -458,12 +458,12 @@ End AddS. Fixpoint length_pos x := match x with xH => O | xO x1 => S (length_pos x1) | xI x1 => S (length_pos x1) end. - + Theorem length_pos_lt: forall x y, (length_pos x < length_pos y)%nat -> Zpos x < Zpos y. Proof. intros x; elim x; clear x; [intros x1 Hrec | intros x1 Hrec | idtac]; - intros y; case y; clear y; intros y1 H || intros H; simpl length_pos; + intros y; case y; clear y; intros y1 H || intros H; simpl length_pos; try (rewrite (Zpos_xI x1) || rewrite (Zpos_xO x1)); try (rewrite (Zpos_xI y1) || rewrite (Zpos_xO y1)); try (inversion H; fail); @@ -492,20 +492,20 @@ End AddS. Qed. Theorem make_zop: forall w (x: znz_op w), - znz_to_Z (mk_zn2z_op x) = - fun z => match z with + znz_to_Z (mk_zn2z_op x) = + fun z => match z with W0 => 0 - | WW xh xl => znz_to_Z x xh * base (znz_digits x) + | WW xh xl => znz_to_Z x xh * base (znz_digits x) + znz_to_Z x xl end. intros ww x; auto. Qed. Theorem make_kzop: forall w (x: znz_op w), - znz_to_Z (mk_zn2z_op_karatsuba x) = - fun z => match z with + znz_to_Z (mk_zn2z_op_karatsuba x) = + fun z => match z with W0 => 0 - | WW xh xl => znz_to_Z x xh * base (znz_digits x) + | WW xh xl => znz_to_Z x xh * base (znz_digits x) + znz_to_Z x xl end. intros ww x; auto. |