Library Bignums.BigN.Nbasic
Require Import ZArith Ndigits.
Require Import BigNumPrelude.
Require Import Max.
Require Import DoubleType.
Require Import DoubleBase.
Require Import CyclicAxioms.
Require Import DoubleCyclic.
Arguments mk_zn2z_ops [t] ops.
Arguments mk_zn2z_ops_karatsuba [t] ops.
Arguments mk_zn2z_specs [t ops] specs.
Arguments mk_zn2z_specs_karatsuba [t ops] specs.
Arguments ZnZ.digits [t] Ops.
Arguments ZnZ.zdigits [t] Ops.
Lemma Pshiftl_nat_Zpower : forall n p,
Zpos (Pos.shiftl_nat p n) = Zpos p * 2 ^ Z.of_nat n.
Proof.
intros.
rewrite Z.mul_comm.
induction n. simpl; auto.
transitivity (2 * (2 ^ Z.of_nat n * Zpos p)).
rewrite <- IHn. auto.
rewrite Z.mul_assoc.
rewrite Nat2Z.inj_succ.
rewrite <- Z.pow_succ_r; auto with zarith.
Qed.
Fixpoint plength (p: positive) : positive :=
match p with
xH => xH
| xO p1 => Pos.succ (plength p1)
| xI p1 => Pos.succ (plength p1)
end.
Theorem plength_correct: forall p, (Zpos p < 2 ^ Zpos (plength p))%Z.
assert (F: (forall p, 2 ^ (Zpos (Pos.succ p)) = 2 * 2 ^ Zpos p)%Z).
intros p; replace (Zpos (Pos.succ p)) with (1 + Zpos p)%Z.
rewrite Zpower_exp; auto with zarith.
rewrite Pos2Z.inj_succ; unfold Z.succ; auto with zarith.
intros p; elim p; simpl plength; auto.
intros p1 Hp1; rewrite F; repeat rewrite Pos2Z.inj_xI.
assert (tmp: (forall p, 2 * p = p + p)%Z);
try repeat rewrite tmp; auto with zarith.
intros p1 Hp1; rewrite F; rewrite (Pos2Z.inj_xO p1).
assert (tmp: (forall p, 2 * p = p + p)%Z);
try repeat rewrite tmp; auto with zarith.
rewrite Z.pow_1_r; auto with zarith.
Qed.
Theorem plength_pred_correct: forall p, (Zpos p <= 2 ^ Zpos (plength (Pos.pred p)))%Z.
intros p; case (Pos.succ_pred_or p); intros H1.
subst; simpl plength.
rewrite Z.pow_1_r; auto with zarith.
pattern p at 1; rewrite <- H1.
rewrite Pos2Z.inj_succ; unfold Z.succ; auto with zarith.
generalize (plength_correct (Pos.pred p)); auto with zarith.
Qed.
Definition Pdiv p q :=
match Z.div (Zpos p) (Zpos q) with
Zpos q1 => match (Zpos p) - (Zpos q) * (Zpos q1) with
Z0 => q1
| _ => (Pos.succ q1)
end
| _ => xH
end.
Theorem Pdiv_le: forall p q,
Zpos p <= Zpos q * Zpos (Pdiv p q).
intros p q.
unfold Pdiv.
assert (H1: Zpos q > 0); auto with zarith.
assert (H1b: Zpos p >= 0); auto with zarith.
generalize (Z_div_ge0 (Zpos p) (Zpos q) H1 H1b).
generalize (Z_div_mod_eq (Zpos p) (Zpos q) H1); case Z.div.
intros HH _; rewrite HH; rewrite Z.mul_0_r; rewrite Z.mul_1_r; simpl.
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;
case Z.modulo.
intros HH _; rewrite HH; auto with zarith.
intros r1 HH (_,HH1); rewrite HH; rewrite Pos2Z.inj_succ.
unfold Z.succ; rewrite Z.mul_add_distr_l; auto with zarith.
intros r1 _ (HH,_); case HH; auto.
intros q1 HH; rewrite HH.
unfold Z.ge; simpl Z.compare; intros HH1; case HH1; auto.
Qed.
Definition is_one p := match p with xH => true | _ => false end.
Theorem is_one_one: forall p, is_one p = true -> p = xH.
intros p; case p; auto; intros p1 H1; discriminate H1.
Qed.
Definition get_height digits p :=
let r := Pdiv p digits in
if is_one r then xH else Pos.succ (plength (Pos.pred r)).
Theorem get_height_correct:
forall digits N,
Zpos N <= Zpos digits * (2 ^ (Zpos (get_height digits N) -1)).
intros digits N.
unfold get_height.
assert (H1 := Pdiv_le N digits).
case_eq (is_one (Pdiv N digits)); intros H2.
rewrite (is_one_one _ H2) in H1.
rewrite Z.mul_1_r in H1.
change (2^(1-1))%Z with 1; rewrite Z.mul_1_r; auto.
clear H2.
apply Z.le_trans with (1 := H1).
apply Z.mul_le_mono_nonneg_l; auto with zarith.
rewrite Pos2Z.inj_succ; unfold Z.succ.
rewrite Z.add_comm; rewrite Z.add_simpl_l.
apply plength_pred_correct.
Qed.
Definition zn2z_word_comm : forall w n, zn2z (word w n) = word (zn2z w) n.
fix zn2z_word_comm 2.
intros w n; case n.
reflexivity.
intros n0;simpl.
case (zn2z_word_comm w n0).
reflexivity.
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
| O => fun w x => x
| S m =>
let aux := extend m in
fun w x => WW W0 (aux w x)
end.
Section ExtendMax.
Open Scope nat_scope.
Fixpoint plusnS (n m: nat) {struct n} : (n + S m = S (n + m))%nat :=
match n return (n + S m = S (n + m))%nat with
| 0 => eq_refl (S m)
| S n1 =>
let v := S (S n1 + m) in
eq_ind_r (fun n => S n = v) (eq_refl v) (plusnS n1 m)
end.
Fixpoint plusn0 n : n + 0 = n :=
match n return (n + 0 = n) with
| 0 => eq_refl 0
| S n1 =>
let v := S n1 in
eq_ind_r (fun n : nat => S n = v) (eq_refl v) (plusn0 n1)
end.
Fixpoint diff (m n: nat) {struct m}: nat * nat :=
match m, n with
O, n => (O, n)
| m, O => (m, O)
| S m1, S n1 => diff m1 n1
end.
Fixpoint diff_l (m n : nat) {struct m} : fst (diff m n) + n = max m n :=
match m return fst (diff m n) + n = max m n with
| 0 =>
match n return (n = max 0 n) with
| 0 => eq_refl _
| S n0 => eq_refl _
end
| S m1 =>
match n return (fst (diff (S m1) n) + n = max (S m1) n)
with
| 0 => plusn0 _
| 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 v1 (fun n => v1 = n) (eq_refl v1) (S v) (plusnS _ _))
_ (diff_l _ _)
end
end.
Fixpoint diff_r (m n: nat) {struct m}: snd (diff m n) + m = max m n :=
match m return (snd (diff m n) + m = max m n) with
| 0 =>
match n return (snd (diff 0 n) + 0 = max 0 n) with
| 0 => eq_refl _
| S _ => plusn0 _
end
| S m =>
match n return (snd (diff (S m) n) + S m = max (S m) n) with
| 0 => eq_refl (snd (diff (S m) 0) + S m)
| S n1 =>
let v := S (max m n1) in
eq_ind_r (fun n => n = v)
(eq_ind_r (fun n => S n = v)
(eq_refl v) (diff_r _ _)) (plusnS _ _)
end
end.
Variable w: Type.
Definition castm (m n: nat) (H: m = n) (x: word w (S m)):
(word w (S n)) :=
match H in (_ = y) return (word w (S y)) with
| eq_refl => x
end.
Variable m: nat.
Variable v: (word w (S m)).
Fixpoint extend_tr (n : nat) {struct n}: (word w (S (n + m))) :=
match n return (word w (S (n + m))) with
| O => v
| S n1 => WW W0 (extend_tr n1)
end.
End ExtendMax.
Arguments extend_tr [w m] v n.
Arguments castm [w m n] H x.
Section Reduce.
Variable w : Type.
Variable nT : Type.
Variable N0 : nT.
Variable eq0 : w -> bool.
Variable reduce_n : w -> nT.
Variable zn2z_to_Nt : zn2z w -> nT.
Definition reduce_n1 (x:zn2z w) :=
match x with
| W0 => N0
| WW xh xl =>
if eq0 xh then reduce_n xl
else zn2z_to_Nt x
end.
End Reduce.
Section ReduceRec.
Variable w : Type.
Variable nT : Type.
Variable N0 : nT.
Variable reduce_1n : zn2z w -> nT.
Variable c : forall n, word w (S n) -> nT.
Fixpoint reduce_n (n:nat) : word w (S n) -> nT :=
match n return word w (S n) -> nT with
| O => reduce_1n
| S m => fun x =>
match x with
| W0 => N0
| WW xh xl =>
match xh with
| W0 => @reduce_n m xl
| _ => @c (S m) x
end
end
end.
End ReduceRec.
Section CompareRec.
Variable wm w : Type.
Variable w_0 : w.
Variable compare : w -> w -> comparison.
Variable compare0_m : wm -> comparison.
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
| S m => fun x =>
match x with
| W0 => Eq
| WW xh xl =>
match compare0_mn m xh with
| Eq => compare0_mn m xl
| r => Lt
end
end
end.
Variable wm_base: positive.
Variable wm_to_Z: wm -> Z.
Variable w_to_Z: w -> Z.
Variable w_to_Z_0: w_to_Z w_0 = 0.
Variable spec_compare0_m: forall x,
compare0_m x = (w_to_Z w_0 ?= wm_to_Z x).
Variable wm_to_Z_pos: forall x, 0 <= wm_to_Z x < base wm_base.
Let double_to_Z := double_to_Z wm_base wm_to_Z.
Let double_wB := double_wB wm_base.
Lemma base_xO: forall n, base (xO n) = (base n)^2.
Proof.
intros n1; unfold base.
rewrite (Pos2Z.inj_xO n1); rewrite Z.mul_comm; rewrite Z.pow_mul_r; auto with zarith.
Qed.
Let double_to_Z_pos: forall n x, 0 <= double_to_Z n x < double_wB n :=
(spec_double_to_Z wm_base wm_to_Z wm_to_Z_pos).
Declare Equivalent Keys compare0_mn compare0_m.
Lemma spec_compare0_mn: forall n x,
compare0_mn n x = (0 ?= double_to_Z n x).
Proof.
intros n; elim n; clear n; auto.
intros x; rewrite spec_compare0_m; rewrite w_to_Z_0; auto.
intros n Hrec x; case x; unfold compare0_mn; fold compare0_mn; auto.
fold word in *.
intros xh xl.
rewrite 2 Hrec.
simpl double_to_Z.
set (wB := DoubleBase.double_wB wm_base n).
case Z.compare_spec; intros Cmp.
rewrite <- Cmp. reflexivity.
symmetry. apply Z.gt_lt, Z.lt_gt. assert (0 < wB).
unfold wB, DoubleBase.double_wB, base; auto with zarith.
change 0 with (0 + 0); apply Z.add_lt_le_mono; auto with zarith.
apply Z.mul_pos_pos; auto with zarith.
case (double_to_Z_pos n xl); auto with zarith.
case (double_to_Z_pos n xh); intros; exfalso; omega.
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 x with
| W0 => compare w_0 y
| WW xh xl =>
match compare0_mn m xh with
| Eq => compare_mn_1 m xl y
| r => Gt
end
end
end.
Variable spec_compare: forall x y,
compare x y = Z.compare (w_to_Z x) (w_to_Z y).
Variable spec_compare_m: forall x y,
compare_m x y = Z.compare (wm_to_Z x) (w_to_Z y).
Variable wm_base_lt: forall x,
0 <= w_to_Z x < base (wm_base).
Let double_wB_lt: forall n x,
0 <= w_to_Z x < (double_wB n).
Proof.
intros n x; elim n; simpl; auto; clear n.
intros n (H0, H); split; auto.
apply Z.lt_le_trans with (1:= H).
unfold double_wB, DoubleBase.double_wB; simpl.
rewrite base_xO.
set (u := base (Pos.shiftl_nat wm_base n)).
assert (0 < u).
unfold u, base; auto with zarith.
replace (u^2) with (u * u); simpl; auto with zarith.
apply Z.le_trans with (1 * u); auto with zarith.
unfold Z.pow_pos; simpl; ring.
Qed.
Lemma spec_compare_mn_1: forall n x y,
compare_mn_1 n x y = Z.compare (double_to_Z n x) (w_to_Z y).
Proof.
intros n; elim n; simpl; auto; clear n.
intros n Hrec x; case x; clear x; auto.
intros y; rewrite spec_compare; rewrite w_to_Z_0. reflexivity.
intros xh xl y; simpl;
rewrite spec_compare0_mn, Hrec. case Z.compare_spec.
intros H1b.
rewrite <- H1b; rewrite Z.mul_0_l; rewrite Z.add_0_l; auto.
symmetry. apply Z.lt_gt.
case (double_wB_lt n y); intros _ H0.
apply Z.lt_le_trans with (1:= H0).
fold double_wB.
case (double_to_Z_pos n xl); intros H1 H2.
apply Z.le_trans with (double_to_Z n xh * double_wB n); auto with zarith.
apply Z.le_trans with (1 * double_wB n); auto with zarith.
case (double_to_Z_pos n xh); intros; exfalso; omega.
Qed.
End CompareRec.
Section AddS.
Variable w wm : Type.
Variable incr : wm -> carry wm.
Variable addr : w -> wm -> carry wm.
Variable injr : w -> zn2z wm.
Variable w_0 u: w.
Fixpoint injs (n:nat): word w (S n) :=
match n return (word w (S n)) with
O => WW w_0 u
| S n1 => (WW W0 (injs n1))
end.
Definition adds x y :=
match y with
W0 => C0 (injr x)
| WW hy ly => match addr x ly with
C0 z => C0 (WW hy z)
| C1 z => match incr hy with
C0 z1 => C0 (WW z1 z)
| C1 z1 => C1 (WW z1 z)
end
end
end.
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;
try (rewrite (Pos2Z.inj_xI x1) || rewrite (Pos2Z.inj_xO x1));
try (rewrite (Pos2Z.inj_xI y1) || rewrite (Pos2Z.inj_xO y1));
try (inversion H; fail);
try (assert (Zpos x1 < Zpos y1); [apply Hrec; apply lt_S_n | idtac]; auto with zarith);
assert (0 < Zpos y1); auto with zarith; red; auto.
Qed.
Theorem cancel_app: forall A B (f g: A -> B) x, f = g -> f x = g x.
Proof.
intros A B f g x H; rewrite H; auto.
Qed.
Section SimplOp.
Variable w: Type.
Theorem digits_zop: forall t (ops : ZnZ.Ops t),
ZnZ.digits (mk_zn2z_ops ops) = xO (ZnZ.digits ops).
Proof.
intros ww x; auto.
Qed.
Theorem digits_kzop: forall t (ops : ZnZ.Ops t),
ZnZ.digits (mk_zn2z_ops_karatsuba ops) = xO (ZnZ.digits ops).
Proof.
intros ww x; auto.
Qed.
Theorem make_zop: forall t (ops : ZnZ.Ops t),
@ZnZ.to_Z _ (mk_zn2z_ops ops) =
fun z => match z with
| W0 => 0
| WW xh xl => ZnZ.to_Z xh * base (ZnZ.digits ops)
+ ZnZ.to_Z xl
end.
Proof.
intros ww x; auto.
Qed.
Theorem make_kzop: forall t (ops: ZnZ.Ops t),
@ZnZ.to_Z _ (mk_zn2z_ops_karatsuba ops) =
fun z => match z with
| W0 => 0
| WW xh xl => ZnZ.to_Z xh * base (ZnZ.digits ops)
+ ZnZ.to_Z xl
end.
Proof.
intros ww x; auto.
Qed.
End SimplOp.
Abstract vision of a datatype of arbitrary-large numbers.
Concrete operations can be derived from these generic
fonctions, in particular from iter_t and same_level.
The domains: a sequence of Z/nZ structures
Parameter dom_t : nat -> Type.
Declare Instance dom_op n : ZnZ.Ops (dom_t n).
Declare Instance dom_spec n : ZnZ.Specs (dom_op n).
Axiom digits_dom_op : forall n,
ZnZ.digits (dom_op n) = Pos.shiftl_nat (ZnZ.digits (dom_op 0)) n.
The type t of arbitrary-large numbers, with abstract constructor mk_t
and destructor destr_t and iterator iter_t
Parameter t : Type.
Parameter mk_t : forall (n:nat), dom_t n -> t.
Inductive View_t : t -> Prop :=
Mk_t : forall n (x : dom_t n), View_t (mk_t n x).
Axiom destr_t : forall x, View_t x.
Parameter iter_t : forall {A:Type}(f : forall n, dom_t n -> A), t -> A.
Axiom iter_mk_t : forall A (f:forall n, dom_t n -> A),
forall n x, iter_t f (mk_t n x) = f n x.
Conversion to ZArith
Parameter to_Z : t -> Z.
Local Notation "[ x ]" := (to_Z x).
Axiom spec_mk_t : forall n x, [mk_t n x] = ZnZ.to_Z x.
reduce is like mk_t, but try to minimise the level of the number
Parameter reduce : forall (n:nat), dom_t n -> t.
Axiom spec_reduce : forall n x, [reduce n x] = ZnZ.to_Z x.
Number of level in the tree representation of a number.
NB: This function isn't a morphism for setoid eq.
same_level and its rich specification, indexed by level
Parameter same_level : forall {A:Type}
(f : forall n, dom_t n -> dom_t n -> A), t -> t -> A.
Axiom spec_same_level_dep :
forall res
(P : nat -> Z -> Z -> res -> Prop)
(Pantimon : forall n m z z' r, (n <= m)%nat -> P m z z' r -> P n z z' r)
(f : forall n, dom_t n -> dom_t n -> res)
(Pf: forall n x y, P n (ZnZ.to_Z x) (ZnZ.to_Z y) (f n x y)),
forall x y, P (level x) [x] [y] (same_level f x y).
mk_t_S : building a number of the next level