diff options
| author | glondu | 2009-09-17 15:58:14 +0000 |
|---|---|---|
| committer | glondu | 2009-09-17 15:58:14 +0000 |
| commit | 61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch) | |
| tree | 961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/Numbers | |
| parent | 6d3fbdf36c6a47b49c2a4b16f498972c93c07574 (diff) | |
Delete trailing whitespaces in all *.{v,ml*} files
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12337 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Numbers')
28 files changed, 1310 insertions, 1310 deletions
diff --git a/theories/Numbers/BigNumPrelude.v b/theories/Numbers/BigNumPrelude.v index a08c6e62f9..3a64a8dc11 100644 --- a/theories/Numbers/BigNumPrelude.v +++ b/theories/Numbers/BigNumPrelude.v @@ -30,8 +30,8 @@ Declare ML Module "numbers_syntax_plugin". *) -Open Local Scope Z_scope. - +Open Local Scope Z_scope. + (* For compatibility of scripts, weaker version of some lemmas of Zdiv *) Lemma Zlt0_not_eq : forall n, 0<n -> n<>0. @@ -45,14 +45,14 @@ Definition Z_div_plus_l a b c H := Zdiv.Z_div_plus_full_l a b c (Zlt0_not_eq _ H (* Automation *) -Hint Extern 2 (Zle _ _) => +Hint Extern 2 (Zle _ _) => (match goal with |- Zpos _ <= Zpos _ => exact (refl_equal _) | H: _ <= ?p |- _ <= ?p => apply Zle_trans with (2 := H) | H: _ < ?p |- _ <= ?p => apply Zlt_le_weak; apply Zle_lt_trans with (2 := H) end). -Hint Extern 2 (Zlt _ _) => +Hint Extern 2 (Zlt _ _) => (match goal with |- Zpos _ < Zpos _ => exact (refl_equal _) | H: _ <= ?p |- _ <= ?p => apply Zlt_le_trans with (2 := H) @@ -62,13 +62,13 @@ Hint Extern 2 (Zlt _ _) => Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. -(************************************** +(************************************** Properties of order and product **************************************) - Theorem beta_lex: forall a b c d beta, - a * beta + b <= c * beta + d -> - 0 <= b < beta -> 0 <= d < beta -> + Theorem beta_lex: forall a b c d beta, + a * beta + b <= c * beta + d -> + 0 <= b < beta -> 0 <= d < beta -> a <= c. Proof. intros a b c d beta H1 (H3, H4) (H5, H6). @@ -80,15 +80,15 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. Theorem beta_lex_inv: forall a b c d beta, a < c -> 0 <= b < beta -> - 0 <= d < beta -> - a * beta + b < c * beta + d. + 0 <= d < beta -> + a * beta + b < c * beta + d. Proof. intros a b c d beta H1 (H3, H4) (H5, H6). case (Zle_or_lt (c * beta + d) (a * beta + b)); auto with zarith. intros H7; contradict H1;apply Zle_not_lt;apply beta_lex with (1 := H7);auto. Qed. - Lemma beta_mult : forall h l beta, + Lemma beta_mult : forall h l beta, 0 <= h < beta -> 0 <= l < beta -> 0 <= h*beta+l < beta^2. Proof. intros h l beta H1 H2;split. auto with zarith. @@ -96,7 +96,7 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. apply beta_lex_inv;auto with zarith. Qed. - Lemma Zmult_lt_b : + Lemma Zmult_lt_b : forall b x y, 0 <= x < b -> 0 <= y < b -> 0 <= x * y <= b^2 - 2*b + 1. Proof. intros b x y (Hx1,Hx2) (Hy1,Hy2);split;auto with zarith. @@ -106,17 +106,17 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. Qed. Lemma sum_mul_carry : forall xh xl yh yl wc cc beta, - 1 < beta -> + 1 < beta -> 0 <= wc < beta -> 0 <= xh < beta -> 0 <= xl < beta -> 0 <= yh < beta -> 0 <= yl < beta -> 0 <= cc < beta^2 -> - wc*beta^2 + cc = xh*yl + xl*yh -> + wc*beta^2 + cc = xh*yl + xl*yh -> 0 <= wc <= 1. Proof. - intros xh xl yh yl wc cc beta U H1 H2 H3 H4 H5 H6 H7. + intros xh xl yh yl wc cc beta U H1 H2 H3 H4 H5 H6 H7. assert (H8 := Zmult_lt_b beta xh yl H2 H5). assert (H9 := Zmult_lt_b beta xl yh H3 H4). split;auto with zarith. @@ -134,7 +134,7 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. apply Zle_lt_trans with ((beta-1)*(beta-1)+(beta-1)); auto with zarith. apply Zplus_le_compat; auto with zarith. apply Zmult_le_compat; auto with zarith. - repeat (rewrite Zmult_minus_distr_l || rewrite Zmult_minus_distr_r); + repeat (rewrite Zmult_minus_distr_l || rewrite Zmult_minus_distr_r); rewrite Zpower_2; auto with zarith. Qed. @@ -149,7 +149,7 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. apply Zle_lt_trans with ((beta-1)*(beta-1)+(2*beta-2));auto with zarith. apply Zplus_le_compat; auto with zarith. apply Zmult_le_compat; auto with zarith. - repeat (rewrite Zmult_minus_distr_l || rewrite Zmult_minus_distr_r); + repeat (rewrite Zmult_minus_distr_l || rewrite Zmult_minus_distr_r); rewrite Zpower_2; auto with zarith. Qed. @@ -201,9 +201,9 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. apply Zplus_le_lt_compat; auto with zarith. replace b with ((b - a) + a); try ring. rewrite Zpower_exp; auto with zarith. - pattern (2 ^a) at 4; rewrite <- (Zmult_1_l (2 ^a)); + pattern (2 ^a) at 4; rewrite <- (Zmult_1_l (2 ^a)); try rewrite <- Zmult_minus_distr_r. - rewrite (Zmult_comm (2 ^(b - a))); rewrite Zmult_mod_distr_l; + rewrite (Zmult_comm (2 ^(b - a))); rewrite Zmult_mod_distr_l; auto with zarith. rewrite (Zmult_comm (2 ^a)); apply Zmult_le_compat_r; auto with zarith. match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; @@ -224,22 +224,22 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. split; auto with zarith. assert (0 <= 2 ^a * r); auto with zarith. apply Zplus_le_0_compat; auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; + match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. pattern (2 ^ b) at 2;replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); try ring. apply Zplus_le_lt_compat; auto with zarith. replace b with ((b - a) + a); try ring. rewrite Zpower_exp; auto with zarith. - pattern (2 ^a) at 4; rewrite <- (Zmult_1_l (2 ^a)); + pattern (2 ^a) at 4; rewrite <- (Zmult_1_l (2 ^a)); try rewrite <- Zmult_minus_distr_r. repeat rewrite (fun x => Zmult_comm x (2 ^ a)); rewrite Zmult_mod_distr_l; auto with zarith. apply Zmult_le_compat_l; auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; + match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. Qed. - Theorem Zdiv_shift_r: + Theorem Zdiv_shift_r: forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a -> (r * 2 ^a + t) / (2 ^ b) = (r * 2 ^a) / (2 ^ b). Proof. @@ -253,7 +253,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. rewrite <- Zmod_shift_r; auto with zarith. rewrite (Zmult_comm (2 ^ b)); rewrite Z_div_plus_full_l; auto with zarith. rewrite (fun x y => @Zdiv_small (x mod y)); auto with zarith. - match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; + match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. Qed. @@ -264,8 +264,8 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. a * 2^p = a / 2^(n - p) * 2^n + (a*2^p) mod 2^n. Proof. intros n p a H1 H2. - pattern (a*2^p) at 1;replace (a*2^p) with - (a*2^p/2^n * 2^n + a*2^p mod 2^n). + pattern (a*2^p) at 1;replace (a*2^p) with + (a*2^p/2^n * 2^n + a*2^p mod 2^n). 2:symmetry;rewrite (Zmult_comm (a*2^p/2^n));apply Z_div_mod_eq. replace (a * 2 ^ p / 2 ^ n) with (a / 2 ^ (n - p));trivial. replace (2^n) with (2^(n-p)*2^p). @@ -279,8 +279,8 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Qed. - Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n -> - ((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) = + Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n -> + ((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) = a mod 2 ^ p. Proof. intros. @@ -312,16 +312,16 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Lemma div_le_0 : forall p x, 0 <= x -> 0 <= x / 2 ^ p. Proof. intros p x Hle;destruct (Z_le_gt_dec 0 p). - apply Zdiv_le_lower_bound;auto with zarith. + apply Zdiv_le_lower_bound;auto with zarith. replace (2^p) with 0. destruct x;compute;intro;discriminate. destruct p;trivial;discriminate z. Qed. - + Lemma div_lt : forall p x y, 0 <= x < y -> x / 2^p < y. Proof. intros p x y H;destruct (Z_le_gt_dec 0 p). - apply Zdiv_lt_upper_bound;auto with zarith. + apply Zdiv_lt_upper_bound;auto with zarith. apply Zlt_le_trans with y;auto with zarith. rewrite <- (Zmult_1_r y);apply Zmult_le_compat;auto with zarith. assert (0 < 2^p);auto with zarith. @@ -357,7 +357,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. compute; auto. Qed. - Lemma Zdiv_gcd_zero : forall a b, b / Zgcd a b = 0 -> b <> 0 -> + Lemma Zdiv_gcd_zero : forall a b, b / Zgcd a b = 0 -> b <> 0 -> Zgcd a b = 0. Proof. intros. @@ -369,7 +369,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. intros; subst k; simpl in *; subst b; elim H0; auto. Qed. - Lemma Zgcd_mult_rel_prime : forall a b c, + Lemma Zgcd_mult_rel_prime : forall a b c, Zgcd a c = 1 -> Zgcd b c = 1 -> Zgcd (a*b) c = 1. Proof. intros. @@ -378,7 +378,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Qed. Lemma Zcompare_gt : forall (A:Type)(a a':A)(p q:Z), - match (p?=q)%Z with Gt => a | _ => a' end = + match (p?=q)%Z with Gt => a | _ => a' end = if Z_le_gt_dec p q then a' else a. Proof. intros. diff --git a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v index b7a427532d..32d1503314 100644 --- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v +++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v @@ -12,8 +12,8 @@ (** * Signature and specification of a bounded integer structure *) -(** This file specifies how to represent [Z/nZ] when [n=2^d], - [d] being the number of digits of these bounded integers. *) +(** This file specifies how to represent [Z/nZ] when [n=2^d], + [d] being the number of digits of these bounded integers. *) Set Implicit Arguments. @@ -33,7 +33,7 @@ Section Z_nZ_Op. Record znz_op := mk_znz_op { (* Conversion functions with Z *) - znz_digits : positive; + znz_digits : positive; znz_zdigits: znz; znz_to_Z : znz -> Z; znz_of_pos : positive -> N * znz; (* Euclidean division by [2^digits] *) @@ -78,12 +78,12 @@ Section Z_nZ_Op. znz_div : znz -> znz -> znz * znz; znz_mod_gt : znz -> znz -> znz; (* specialized version of [znz_mod] *) - znz_mod : znz -> znz -> znz; + znz_mod : znz -> znz -> znz; znz_gcd_gt : znz -> znz -> znz; (* specialized version of [znz_gcd] *) - znz_gcd : znz -> znz -> znz; + znz_gcd : znz -> znz -> znz; (* [znz_add_mul_div p i j] is a combination of the [(digits-p)] - low bits of [i] above the [p] high bits of [j]: + low bits of [i] above the [p] high bits of [j]: [znz_add_mul_div p i j = i*2^p+j/2^(digits-p)] *) znz_add_mul_div : znz -> znz -> znz -> znz; (* [znz_pos_mod p i] is [i mod 2^p] *) @@ -135,7 +135,7 @@ Section Z_nZ_Spec. Let w_mul_c := w_op.(znz_mul_c). Let w_mul := w_op.(znz_mul). Let w_square_c := w_op.(znz_square_c). - + Let w_div21 := w_op.(znz_div21). Let w_div_gt := w_op.(znz_div_gt). Let w_div := w_op.(znz_div). @@ -229,25 +229,25 @@ Section Z_nZ_Spec. spec_div : forall a b, 0 < [|b|] -> let (q,r) := w_div a b in [|a|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]; - + 0 <= [|r|] < [|b|]; + spec_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> [|w_mod_gt a b|] = [|a|] mod [|b|]; spec_mod : forall a b, 0 < [|b|] -> [|w_mod a b|] = [|a|] mod [|b|]; - + spec_gcd_gt : forall a b, [|a|] > [|b|] -> Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]; spec_gcd : forall a b, Zis_gcd [|a|] [|b|] [|w_gcd a b|]; - + (* shift operations *) spec_head00: forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits; spec_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB; + wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB; spec_tail00: forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits; - spec_tail0 : forall x, 0 < [|x|] -> - exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|w_tail0 x|]) ; + spec_tail0 : forall x, 0 < [|x|] -> + exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|w_tail0 x|]) ; spec_add_mul_div : forall x y p, [|p|] <= Zpos w_digits -> [| w_add_mul_div p x y |] = @@ -272,23 +272,23 @@ End Z_nZ_Spec. (** Generic construction of double words *) Section WW. - + Variable w : Type. Variable w_op : znz_op w. Variable op_spec : znz_spec w_op. - + Let wB := base w_op.(znz_digits). Let w_to_Z := w_op.(znz_to_Z). Let w_eq0 := w_op.(znz_eq0). Let w_0 := w_op.(znz_0). - Definition znz_W0 h := + Definition znz_W0 h := if w_eq0 h then W0 else WW h w_0. - Definition znz_0W l := + Definition znz_0W l := if w_eq0 l then W0 else WW w_0 l. - Definition znz_WW h l := + Definition znz_WW h l := if w_eq0 h then znz_0W l else WW h l. Lemma spec_W0 : forall h, @@ -300,7 +300,7 @@ Section WW. unfold w_0; rewrite op_spec.(spec_0); auto with zarith. Qed. - Lemma spec_0W : forall l, + Lemma spec_0W : forall l, zn2z_to_Z wB w_to_Z (znz_0W l) = w_to_Z l. Proof. unfold zn2z_to_Z, znz_0W, w_to_Z; simpl; intros. @@ -309,7 +309,7 @@ Section WW. unfold w_0; rewrite op_spec.(spec_0); auto with zarith. Qed. - Lemma spec_WW : forall h l, + Lemma spec_WW : forall h l, zn2z_to_Z wB w_to_Z (znz_WW h l) = (w_to_Z h)*wB + w_to_Z l. Proof. unfold znz_WW, w_to_Z; simpl; intros. @@ -324,7 +324,7 @@ End WW. (** Injecting [Z] numbers into a cyclic structure *) Section znz_of_pos. - + Variable w : Type. Variable w_op : znz_op w. Variable op_spec : znz_spec w_op. @@ -349,7 +349,7 @@ Section znz_of_pos. apply Zle_trans with X; auto with zarith end. match goal with |- ?X <= _ => - pattern X at 1; rewrite <- (Zmult_1_l); + pattern X at 1; rewrite <- (Zmult_1_l); apply Zmult_le_compat_r; auto with zarith end. case p1; simpl; intros; red; simpl; intros; discriminate. diff --git a/theories/Numbers/Cyclic/Abstract/NZCyclic.v b/theories/Numbers/Cyclic/Abstract/NZCyclic.v index 125fd3f127..5891593903 100644 --- a/theories/Numbers/Cyclic/Abstract/NZCyclic.v +++ b/theories/Numbers/Cyclic/Abstract/NZCyclic.v @@ -17,9 +17,9 @@ Require Import CyclicAxioms. (** * From [CyclicType] to [NZAxiomsSig] *) -(** A [Z/nZ] representation given by a module type [CyclicType] - implements [NZAxiomsSig], e.g. the common properties between - N and Z with no ordering. Notice that the [n] in [Z/nZ] is +(** A [Z/nZ] representation given by a module type [CyclicType] + implements [NZAxiomsSig], e.g. the common properties between + N and Z with no ordering. Notice that the [n] in [Z/nZ] is a power of 2. *) @@ -98,7 +98,7 @@ Notation "x * y" := (NZmul x y) : IntScope. Theorem gt_wB_1 : 1 < wB. Proof. -unfold base. +unfold base. apply Zpower_gt_1; unfold Zlt; auto with zarith. Qed. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v index d60af33ecb..b4f6a81604 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v @@ -36,10 +36,10 @@ Section DoubleAdd. Definition ww_succ_c x := match x with | W0 => C0 ww_1 - | WW xh xl => + | WW xh xl => match w_succ_c xl with | C0 l => C0 (WW xh l) - | C1 l => + | C1 l => match w_succ_c xh with | C0 h => C0 (WW h w_0) | C1 h => C1 W0 @@ -47,13 +47,13 @@ Section DoubleAdd. end end. - Definition ww_succ x := + Definition ww_succ x := match x with | W0 => ww_1 | WW xh xl => match w_succ_c xl with | C0 l => WW xh l - | C1 l => w_W0 (w_succ xh) + | C1 l => w_W0 (w_succ xh) end end. @@ -63,12 +63,12 @@ Section DoubleAdd. | _, W0 => C0 x | WW xh xl, WW yh yl => match w_add_c xl yl with - | C0 l => + | C0 l => match w_add_c xh yh with | C0 h => C0 (WW h l) | C1 h => C1 (w_WW h l) - end - | C1 l => + end + | C1 l => match w_add_carry_c xh yh with | C0 h => C0 (WW h l) | C1 h => C1 (w_WW h l) @@ -85,12 +85,12 @@ Section DoubleAdd. | _, W0 => f0 x | WW xh xl, WW yh yl => match w_add_c xl yl with - | C0 l => + | C0 l => match w_add_c xh yh with | C0 h => f0 (WW h l) | C1 h => f1 (w_WW h l) - end - | C1 l => + end + | C1 l => match w_add_carry_c xh yh with | C0 h => f0 (WW h l) | C1 h => f1 (w_WW h l) @@ -118,12 +118,12 @@ Section DoubleAdd. | WW xh xl, W0 => ww_succ_c (WW xh xl) | WW xh xl, WW yh yl => match w_add_carry_c xl yl with - | C0 l => + | C0 l => match w_add_c xh yh with | C0 h => C0 (WW h l) | C1 h => C1 (WW h l) end - | C1 l => + | C1 l => match w_add_carry_c xh yh with | C0 h => C0 (WW h l) | C1 h => C1 (w_WW h l) @@ -131,7 +131,7 @@ Section DoubleAdd. end end. - Definition ww_add_carry x y := + Definition ww_add_carry x y := match x, y with | W0, W0 => ww_1 | W0, WW yh yl => ww_succ (WW yh yl) @@ -146,7 +146,7 @@ Section DoubleAdd. (*Section DoubleProof.*) Variable w_digits : positive. Variable w_to_Z : w -> Z. - + Notation wB := (base w_digits). Notation wwB := (base (ww_digits w_digits)). @@ -157,11 +157,11 @@ Section DoubleAdd. (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[+[ c ]]" := + (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). Variable spec_w_0 : [|w_0|] = 0. @@ -172,7 +172,7 @@ Section DoubleAdd. Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. Variable spec_w_succ_c : forall x, [+|w_succ_c x|] = [|x|] + 1. Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_w_add_carry_c : + Variable spec_w_add_carry_c : forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1. Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB. Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. @@ -187,11 +187,11 @@ Section DoubleAdd. rewrite <- Zplus_assoc;rewrite <- H;rewrite Zmult_1_l. assert ([|l|] = 0). generalize (spec_to_Z xl)(spec_to_Z l);omega. rewrite H0;generalize (spec_w_succ_c xh);destruct (w_succ_c xh) as [h|h]; - intro H1;unfold interp_carry in H1. + intro H1;unfold interp_carry in H1. simpl;rewrite H1;rewrite spec_w_0;ring. unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB. assert ([|xh|] = wB - 1). generalize (spec_to_Z xh)(spec_to_Z h);omega. - rewrite H2;ring. + rewrite H2;ring. Qed. Lemma spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. @@ -222,12 +222,12 @@ Section DoubleAdd. Proof. destruct x as [ |xh xl];simpl;trivial. apply spec_f0;trivial. - destruct y as [ |yh yl];simpl. + destruct y as [ |yh yl];simpl. apply spec_f0;simpl;rewrite Zplus_0_r;trivial. generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; intros H;unfold interp_carry in H. generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *. + intros H1;unfold interp_carry in *. apply spec_f0. simpl;rewrite H;rewrite H1;ring. apply spec_f1. simpl;rewrite spec_w_WW;rewrite H. rewrite Zplus_assoc;rewrite wwB_wBwB. rewrite Zpower_2; rewrite <- Zmult_plus_distr_l. @@ -236,12 +236,12 @@ Section DoubleAdd. as [h|h]; intros H1;unfold interp_carry in *. apply spec_f0;simpl;rewrite H1. rewrite Zmult_plus_distr_l. rewrite <- Zplus_assoc;rewrite H;ring. - apply spec_f1. simpl;rewrite spec_w_WW;rewrite wwB_wBwB. - rewrite Zplus_assoc; rewrite Zpower_2; rewrite <- Zmult_plus_distr_l. + apply spec_f1. simpl;rewrite spec_w_WW;rewrite wwB_wBwB. + rewrite Zplus_assoc; rewrite Zpower_2; rewrite <- Zmult_plus_distr_l. rewrite Zmult_1_l in H1;rewrite H1. rewrite Zmult_plus_distr_l. rewrite <- Zplus_assoc;rewrite H;ring. Qed. - + End Cont. Lemma spec_ww_add_carry_c : @@ -251,16 +251,16 @@ Section DoubleAdd. exact (spec_ww_succ_c y). destruct y as [ |yh yl];simpl. rewrite Zplus_0_r;exact (spec_ww_succ_c (WW xh xl)). - replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) + replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. - generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) + generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H. generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. unfold interp_carry;repeat rewrite Zmult_1_l;simpl;rewrite wwB_wBwB;ring. rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h];intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. + generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) + as [h|h];intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. unfold interp_carry;rewrite spec_w_WW; repeat rewrite Zmult_1_l;simpl;rewrite wwB_wBwB;ring. Qed. @@ -287,9 +287,9 @@ Section DoubleAdd. rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. destruct y as [ |yh yl]. change [[W0]] with 0;rewrite Zplus_0_r. - rewrite Zmod_small;trivial. + rewrite Zmod_small;trivial. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)). - simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) + simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring. generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; unfold interp_carry;intros H;simpl;rewrite <- H. @@ -305,14 +305,14 @@ Section DoubleAdd. exact (spec_ww_succ y). destruct y as [ |yh yl]. change [[W0]] with 0;rewrite Zplus_0_r. exact (spec_ww_succ (WW xh xl)). - simpl;replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) + simpl;replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. - generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) + generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) as [l|l];unfold interp_carry;intros H;rewrite <- H;simpl ww_to_Z. rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. - Qed. + Qed. (* End DoubleProof. *) End DoubleAdd. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v index 37b9f47b49..82480fa2ef 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v @@ -29,8 +29,8 @@ Section DoubleBase. Variable w_zdigits: w. Variable w_add: w -> w -> zn2z w. Variable w_to_Z : w -> Z. - Variable w_compare : w -> w -> comparison. - + Variable w_compare : w -> w -> comparison. + Definition ww_digits := xO w_digits. Definition ww_zdigits := w_add w_zdigits w_zdigits. @@ -46,7 +46,7 @@ Section DoubleBase. | W0, W0 => W0 | _, _ => WW xh xl end. - + Definition ww_W0 h : zn2z (zn2z w) := match h with | W0 => W0 @@ -58,10 +58,10 @@ Section DoubleBase. | W0 => W0 | _ => WW W0 l end. - - Definition double_WW (n:nat) := - match n return word w n -> word w n -> word w (S n) with - | O => w_WW + + Definition double_WW (n:nat) := + match n return word w n -> word w n -> word w (S n) with + | O => w_WW | S n => fun (h l : zn2z (word w n)) => match h, l with @@ -70,8 +70,8 @@ Section DoubleBase. end end. - Fixpoint double_digits (n:nat) : positive := - match n with + Fixpoint double_digits (n:nat) : positive := + match n with | O => w_digits | S n => xO (double_digits n) end. @@ -80,7 +80,7 @@ Section DoubleBase. Fixpoint double_to_Z (n:nat) : word w n -> Z := match n return word w n -> Z with - | O => w_to_Z + | O => w_to_Z | S n => zn2z_to_Z (double_wB n) (double_to_Z n) end. @@ -98,21 +98,21 @@ Section DoubleBase. end. Definition double_0 n : word w n := - match n return word w n with + match n return word w n with | O => w_0 | S _ => W0 end. - + Definition double_split (n:nat) (x:zn2z (word w n)) := - match x with - | W0 => - match n return word w n * word w n with + match x with + | W0 => + match n return word w n * word w n with | O => (w_0,w_0) | S _ => (W0, W0) end | WW h l => (h,l) end. - + Definition ww_compare x y := match x, y with | W0, W0 => Eq @@ -148,15 +148,15 @@ Section DoubleBase. end end. - + Section DoubleProof. Notation wB := (base w_digits). Notation wwB := (base ww_digits). Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z x) (at level 0, x at level 99). - Notation "[+[ c ]]" := + Notation "[+[ c ]]" := (interp_carry 1 wwB ww_to_Z c) (at level 0, x at level 99). - Notation "[-[ c ]]" := + Notation "[-[ c ]]" := (interp_carry (-1) wwB ww_to_Z c) (at level 0, x at level 99). Notation "[! n | x !]" := (double_to_Z n x) (at level 0, x at level 99). @@ -188,7 +188,7 @@ Section DoubleBase. Proof. simpl;rewrite spec_w_Bm1;rewrite wwB_wBwB;ring. Qed. Lemma lt_0_wB : 0 < wB. - Proof. + Proof. unfold base;apply Zpower_gt_0. unfold Zlt;reflexivity. unfold Zle;intros H;discriminate H. Qed. @@ -197,25 +197,25 @@ Section DoubleBase. Proof. rewrite wwB_wBwB; rewrite Zpower_2; apply Zmult_lt_0_compat;apply lt_0_wB. Qed. Lemma wB_pos: 1 < wB. - Proof. + Proof. unfold base;apply Zlt_le_trans with (2^1). unfold Zlt;reflexivity. apply Zpower_le_monotone. unfold Zlt;reflexivity. split;unfold Zle;intros H. discriminate H. clear spec_w_0W w_0W spec_w_Bm1 spec_to_Z spec_w_WW w_WW. destruct w_digits; discriminate H. Qed. - - Lemma wwB_pos: 1 < wwB. + + Lemma wwB_pos: 1 < wwB. Proof. assert (H:= wB_pos);rewrite wwB_wBwB;rewrite <-(Zmult_1_r 1). rewrite Zpower_2. apply Zmult_lt_compat2;(split;[unfold Zlt;reflexivity|trivial]). - apply Zlt_le_weak;trivial. + apply Zlt_le_weak;trivial. Qed. Theorem wB_div_2: 2 * (wB / 2) = wB. Proof. - clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W + clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W spec_to_Z;unfold base. assert (2 ^ Zpos w_digits = 2 * (2 ^ (Zpos w_digits - 1))). pattern 2 at 2; rewrite <- Zpower_1_r. @@ -228,7 +228,7 @@ Section DoubleBase. Theorem wwB_div_2 : wwB / 2 = wB / 2 * wB. Proof. - clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W + clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W spec_to_Z. rewrite wwB_wBwB; rewrite Zpower_2. pattern wB at 1; rewrite <- wB_div_2; auto. @@ -236,11 +236,11 @@ Section DoubleBase. repeat (rewrite (Zmult_comm 2); rewrite Z_div_mult); auto with zarith. Qed. - Lemma mod_wwB : forall z x, + Lemma mod_wwB : forall z x, (z*wB + [|x|]) mod wwB = (z mod wB)*wB + [|x|]. Proof. intros z x. - rewrite Zplus_mod. + rewrite Zplus_mod. pattern wwB at 1;rewrite wwB_wBwB; rewrite Zpower_2. rewrite Zmult_mod_distr_r;try apply lt_0_wB. rewrite (Zmod_small [|x|]). @@ -260,8 +260,8 @@ Section DoubleBase. destruct (spec_to_Z x);trivial. Qed. - Lemma wB_div_plus : forall x y p, - 0 <= p -> + Lemma wB_div_plus : forall x y p, + 0 <= p -> ([|x|]*wB + [|y|]) / 2^(Zpos w_digits + p) = [|x|] / 2^p. Proof. clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. @@ -277,7 +277,7 @@ Section DoubleBase. assert (0 < Zpos w_digits). compute;reflexivity. unfold ww_digits;rewrite Zpos_xO;auto with zarith. Qed. - + Lemma w_to_Z_wwB : forall x, x < wB -> x < wwB. Proof. intros x H;apply Zlt_trans with wB;trivial;apply lt_wB_wwB. @@ -298,7 +298,7 @@ Section DoubleBase. Proof. intros n;unfold double_wB;simpl. unfold base;rewrite (Zpos_xO (double_digits n)). - replace (2 * Zpos (double_digits n)) with + replace (2 * Zpos (double_digits n)) with (Zpos (double_digits n) + Zpos (double_digits n)). symmetry; apply Zpower_exp;intro;discriminate. ring. @@ -327,7 +327,7 @@ Section DoubleBase. unfold base; auto with zarith. Qed. - Lemma spec_double_to_Z : + Lemma spec_double_to_Z : forall n (x:word w n), 0 <= [!n | x!] < double_wB n. Proof. clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. @@ -347,7 +347,7 @@ Section DoubleBase. Qed. Lemma spec_get_low: - forall n x, + forall n x, [!n | x!] < wB -> [|get_low n x|] = [!n | x!]. Proof. clear spec_w_1 spec_w_Bm1. @@ -380,19 +380,19 @@ Section DoubleBase. Qed. Lemma spec_extend_aux : forall n x, [!S n|extend_aux n x!] = [[x]]. - Proof. induction n;simpl;trivial. Qed. + Proof. induction n;simpl;trivial. Qed. Lemma spec_extend : forall n x, [!S n|extend n x!] = [|x|]. - Proof. + Proof. intros n x;assert (H:= spec_w_0W x);unfold extend. - destruct (w_0W x);simpl;trivial. + destruct (w_0W x);simpl;trivial. rewrite <- H;exact (spec_extend_aux n (WW w0 w1)). Qed. Lemma spec_double_0 : forall n, [!n|double_0 n!] = 0. Proof. destruct n;trivial. Qed. - Lemma spec_double_split : forall n x, + Lemma spec_double_split : forall n x, let (h,l) := double_split n x in [!S n|x!] = [!n|h!] * double_wB n + [!n|l!]. Proof. @@ -401,9 +401,9 @@ Section DoubleBase. rewrite spec_w_0;trivial. Qed. - Lemma wB_lex_inv: forall a b c d, - a < c -> - a * wB + [|b|] < c * wB + [|d|]. + Lemma wB_lex_inv: forall a b c d, + a < c -> + a * wB + [|b|] < c * wB + [|d|]. Proof. intros a b c d H1; apply beta_lex_inv with (1 := H1); auto. Qed. @@ -420,7 +420,7 @@ Section DoubleBase. intros H;rewrite spec_w_0 in H. rewrite <- H;simpl;rewrite <- spec_w_0;apply spec_w_compare. change 0 with (0*wB+0);pattern 0 at 2;rewrite <- spec_w_0. - apply wB_lex_inv;trivial. + apply wB_lex_inv;trivial. absurd (0 <= [|yh|]). apply Zgt_not_le;trivial. destruct (spec_to_Z yh);trivial. generalize (spec_w_compare xh w_0);destruct (w_compare xh w_0); @@ -429,8 +429,8 @@ Section DoubleBase. absurd (0 <= [|xh|]). apply Zgt_not_le;apply Zlt_gt;trivial. destruct (spec_to_Z xh);trivial. apply Zlt_gt;change 0 with (0*wB+0);pattern 0 at 2;rewrite <- spec_w_0. - apply wB_lex_inv;apply Zgt_lt;trivial. - + apply wB_lex_inv;apply Zgt_lt;trivial. + generalize (spec_w_compare xh yh);destruct (w_compare xh yh);intros H. rewrite H;generalize (spec_w_compare xl yl);destruct (w_compare xl yl); intros H1;[rewrite H1|apply Zplus_lt_compat_l|apply Zplus_gt_compat_l]; @@ -439,7 +439,7 @@ Section DoubleBase. apply Zlt_gt;apply wB_lex_inv;apply Zgt_lt;trivial. Qed. - + End DoubleProof. End DoubleBase. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v index b590e9b3ce..db3b622b00 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v @@ -22,7 +22,7 @@ Require Import DoubleMul. Require Import DoubleSqrt. Require Import DoubleLift. Require Import DoubleDivn1. -Require Import DoubleDiv. +Require Import DoubleDiv. Require Import CyclicAxioms. Open Local Scope Z_scope. @@ -80,7 +80,7 @@ Section Z_2nZ. Let w_gcd_gt := w_op.(znz_gcd_gt). Let w_gcd := w_op.(znz_gcd). - Let w_add_mul_div := w_op.(znz_add_mul_div). + Let w_add_mul_div := w_op.(znz_add_mul_div). Let w_pos_mod := w_op.(znz_pos_mod). @@ -93,7 +93,7 @@ Section Z_2nZ. Let wB := base w_digits. Let w_Bm2 := w_pred w_Bm1. - + Let ww_1 := ww_1 w_0 w_1. Let ww_Bm1 := ww_Bm1 w_Bm1. @@ -112,16 +112,16 @@ Section Z_2nZ. Let ww_of_pos p := match w_of_pos p with | (N0, l) => (N0, WW w_0 l) - | (Npos ph,l) => + | (Npos ph,l) => let (n,h) := w_of_pos ph in (n, w_WW h l) end. Let head0 := - Eval lazy beta delta [ww_head0] in + Eval lazy beta delta [ww_head0] in ww_head0 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits. Let tail0 := - Eval lazy beta delta [ww_tail0] in + Eval lazy beta delta [ww_tail0] in ww_tail0 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits. Let ww_WW := Eval lazy beta delta [ww_WW] in (@ww_WW w). @@ -132,7 +132,7 @@ Section Z_2nZ. Let compare := Eval lazy beta delta[ww_compare] in ww_compare w_0 w_compare. - Let eq0 (x:zn2z w) := + Let eq0 (x:zn2z w) := match x with | W0 => true | _ => false @@ -147,7 +147,7 @@ Section Z_2nZ. Let opp_carry := Eval lazy beta delta [ww_opp_carry] in ww_opp_carry w_WW ww_Bm1 w_opp_carry. - + (* ** Additions ** *) Let succ_c := @@ -157,16 +157,16 @@ Section Z_2nZ. Eval lazy beta delta [ww_add_c] in ww_add_c w_WW w_add_c w_add_carry_c. Let add_carry_c := - Eval lazy beta iota delta [ww_add_carry_c ww_succ_c] in + Eval lazy beta iota delta [ww_add_carry_c ww_succ_c] in ww_add_carry_c w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c. - Let succ := + Let succ := Eval lazy beta delta [ww_succ] in ww_succ w_W0 ww_1 w_succ_c w_succ. Let add := Eval lazy beta delta [ww_add] in ww_add w_add_c w_add w_add_carry. - Let add_carry := + Let add_carry := Eval lazy beta iota delta [ww_add_carry ww_succ] in ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ w_add w_add_carry. @@ -174,9 +174,9 @@ Section Z_2nZ. Let pred_c := Eval lazy beta delta [ww_pred_c] in ww_pred_c w_Bm1 w_WW ww_Bm1 w_pred_c. - + Let sub_c := - Eval lazy beta iota delta [ww_sub_c ww_opp_c] in + Eval lazy beta iota delta [ww_sub_c ww_opp_c] in ww_sub_c w_0 w_WW w_opp_c w_opp_carry w_sub_c w_sub_carry_c. Let sub_carry_c := @@ -186,8 +186,8 @@ Section Z_2nZ. Let pred := Eval lazy beta delta [ww_pred] in ww_pred w_Bm1 w_WW ww_Bm1 w_pred_c w_pred. - Let sub := - Eval lazy beta iota delta [ww_sub ww_opp] in + Let sub := + Eval lazy beta iota delta [ww_sub ww_opp] in ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry. Let sub_carry := @@ -204,7 +204,7 @@ Section Z_2nZ. Let karatsuba_c := Eval lazy beta iota delta [ww_karatsuba_c double_mul_c kara_prod] in - ww_karatsuba_c w_0 w_1 w_WW w_W0 w_compare w_add w_sub w_mul_c + ww_karatsuba_c w_0 w_1 w_WW w_W0 w_compare w_add w_sub w_mul_c add_c add add_carry sub_c sub. Let mul := @@ -219,7 +219,7 @@ Section Z_2nZ. Let div32 := Eval lazy beta iota delta [w_div32] in - w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c + w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c. Let div21 := @@ -234,40 +234,40 @@ Section Z_2nZ. Let div_gt := Eval lazy beta delta [ww_div_gt] in - ww_div_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp + ww_div_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits. Let div := Eval lazy beta delta [ww_div] in ww_div ww_1 compare div_gt. - + Let mod_gt := Eval lazy beta delta [ww_mod_gt] in ww_mod_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_mod_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div w_zdigits. - Let mod_ := + Let mod_ := Eval lazy beta delta [ww_mod] in ww_mod compare mod_gt. - Let pos_mod := - Eval lazy beta delta [ww_pos_mod] in + Let pos_mod := + Eval lazy beta delta [ww_pos_mod] in ww_pos_mod w_0 w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits. - Let is_even := + Let is_even := Eval lazy beta delta [ww_is_even] in ww_is_even w_is_even. - Let sqrt2 := + Let sqrt2 := Eval lazy beta delta [ww_sqrt2] in ww_sqrt2 w_is_even w_compare w_0 w_1 w_Bm1 w_0W w_sub w_square_c w_div21 w_add_mul_div w_zdigits w_add_c w_sqrt2 w_pred pred_c pred add_c add sub_c add_mul_div. - Let sqrt := + Let sqrt := Eval lazy beta delta [ww_sqrt] in ww_sqrt w_is_even w_0 w_sub w_add_mul_div w_zdigits _ww_zdigits w_sqrt2 pred add_mul_div head0 compare low. - Let gcd_gt_fix := + Let gcd_gt_fix := Eval cbv beta delta [ww_gcd_gt_aux ww_gcd_gt_body] in ww_gcd_gt_aux w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt @@ -278,7 +278,7 @@ Section Z_2nZ. Eval lazy beta delta [gcd_cont] in gcd_cont ww_1 w_1 w_compare. Let gcd_gt := - Eval lazy beta delta [ww_gcd_gt] in + Eval lazy beta delta [ww_gcd_gt] in ww_gcd_gt w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont. Let gcd := @@ -286,18 +286,18 @@ Section Z_2nZ. ww_gcd compare w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont. (* ** Record of operators on 2 words *) - - Definition mk_zn2z_op := + + Definition mk_zn2z_op := mk_znz_op _ww_digits _ww_zdigits to_Z ww_of_pos head0 tail0 W0 ww_1 ww_Bm1 compare eq0 opp_c opp opp_carry - succ_c add_c add_carry_c - succ add add_carry - pred_c sub_c sub_carry_c + succ_c add_c add_carry_c + succ add add_carry + pred_c sub_c sub_carry_c pred sub sub_carry - mul_c mul square_c + mul_c mul square_c div21 div_gt div mod_gt mod_ gcd_gt gcd @@ -307,17 +307,17 @@ Section Z_2nZ. sqrt2 sqrt. - Definition mk_zn2z_op_karatsuba := + Definition mk_zn2z_op_karatsuba := mk_znz_op _ww_digits _ww_zdigits to_Z ww_of_pos head0 tail0 W0 ww_1 ww_Bm1 compare eq0 opp_c opp opp_carry - succ_c add_c add_carry_c - succ add add_carry - pred_c sub_c sub_carry_c + succ_c add_c add_carry_c + succ add add_carry + pred_c sub_c sub_carry_c pred sub sub_carry - karatsuba_c mul square_c + karatsuba_c mul square_c div21 div_gt div mod_gt mod_ gcd_gt gcd @@ -330,7 +330,7 @@ Section Z_2nZ. (* Proof *) Variable op_spec : znz_spec w_op. - Hint Resolve + Hint Resolve (spec_to_Z op_spec) (spec_of_pos op_spec) (spec_0 op_spec) @@ -358,13 +358,13 @@ Section Z_2nZ. (spec_square_c op_spec) (spec_div21 op_spec) (spec_div_gt op_spec) - (spec_div op_spec) + (spec_div op_spec) (spec_mod_gt op_spec) - (spec_mod op_spec) + (spec_mod op_spec) (spec_gcd_gt op_spec) - (spec_gcd op_spec) - (spec_head0 op_spec) - (spec_tail0 op_spec) + (spec_gcd op_spec) + (spec_head0 op_spec) + (spec_tail0 op_spec) (spec_add_mul_div op_spec) (spec_pos_mod) (spec_is_even) @@ -417,20 +417,20 @@ Section Z_2nZ. Let spec_ww_Bm1 : [|ww_Bm1|] = wwB - 1. Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);auto. Qed. - Let spec_ww_compare : + Let spec_ww_compare : forall x y, match compare x y with | Eq => [|x|] = [|y|] | Lt => [|x|] < [|y|] | Gt => [|x|] > [|y|] end. - Proof. - refine (spec_ww_compare w_0 w_digits w_to_Z w_compare _ _ _);auto. - exact (spec_compare op_spec). + Proof. + refine (spec_ww_compare w_0 w_digits w_to_Z w_compare _ _ _);auto. + exact (spec_compare op_spec). Qed. Let spec_ww_eq0 : forall x, eq0 x = true -> [|x|] = 0. - Proof. destruct x;simpl;intros;trivial;discriminate. Qed. + Proof. destruct x;simpl;intros;trivial;discriminate. Qed. Let spec_ww_opp_c : forall x, [-|opp_c x|] = -[|x|]. Proof. @@ -440,7 +440,7 @@ Section Z_2nZ. Let spec_ww_opp : forall x, [|opp x|] = (-[|x|]) mod wwB. Proof. - refine(spec_ww_opp w_0 w_0 W0 w_opp_c w_opp_carry w_opp + refine(spec_ww_opp w_0 w_0 W0 w_opp_c w_opp_carry w_opp w_digits w_to_Z _ _ _ _ _); auto. Qed. @@ -480,25 +480,25 @@ Section Z_2nZ. Let spec_ww_add_carry : forall x y, [|add_carry x y|]=([|x|]+[|y|]+1)mod wwB. Proof. - refine (spec_ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ + refine (spec_ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ w_add w_add_carry w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto. Qed. Let spec_ww_pred_c : forall x, [-|pred_c x|] = [|x|] - 1. Proof. - refine (spec_ww_pred_c w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_digits w_to_Z + refine (spec_ww_pred_c w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_digits w_to_Z _ _ _ _ _);wwauto. Qed. Let spec_ww_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|]. Proof. - refine (spec_ww_sub_c w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c + refine (spec_ww_sub_c w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _);wwauto. Qed. Let spec_ww_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|]-[|y|]-1. Proof. - refine (spec_ww_sub_carry_c w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c + refine (spec_ww_sub_carry_c w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_c w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto. Qed. @@ -533,17 +533,17 @@ Section Z_2nZ. _ _ _ _ _ _ _ _ _ _ _ _); wwauto. unfold w_digits; apply spec_more_than_1_digit; auto. exact (spec_compare op_spec). - Qed. + Qed. Let spec_ww_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wwB. Proof. refine (spec_ww_mul w_W0 w_add w_mul_c w_mul add w_digits w_to_Z _ _ _ _ _); - wwauto. + wwauto. Qed. Let spec_ww_square_c : forall x, [[square_c x]] = [|x|] * [|x|]. Proof. - refine (spec_ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add + refine (spec_ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add add_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _);wwauto. Qed. @@ -574,7 +574,7 @@ Section Z_2nZ. 0 <= [|r|] < [|b|]. Proof. refine (spec_ww_div21 w_0 w_0W div32 ww_1 compare sub w_digits w_to_Z - _ _ _ _ _ _ _);wwauto. + _ _ _ _ _ _ _);wwauto. Qed. Let spec_add2: forall x y, @@ -602,7 +602,7 @@ Section Z_2nZ. unfold wB, base; auto with zarith. Qed. - Let spec_ww_digits: + Let spec_ww_digits: [|_ww_zdigits|] = Zpos (xO w_digits). Proof. unfold w_to_Z, _ww_zdigits. @@ -615,7 +615,7 @@ Section Z_2nZ. Let spec_ww_head00 : forall x, [|x|] = 0 -> [|head0 x|] = Zpos _ww_digits. Proof. - refine (spec_ww_head00 w_0 w_0W + refine (spec_ww_head00 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ (refl_equal _ww_digits) _ _ _ _); auto. exact (spec_compare op_spec). @@ -626,8 +626,8 @@ Section Z_2nZ. Let spec_ww_head0 : forall x, 0 < [|x|] -> wwB/ 2 <= 2 ^ [|head0 x|] * [|x|] < wwB. Proof. - refine (spec_ww_head0 w_0 w_0W w_compare w_head0 - w_add2 w_zdigits _ww_zdigits + refine (spec_ww_head0 w_0 w_0W w_compare w_head0 + w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ _ _ _ _);wwauto. exact (spec_compare op_spec). exact (spec_zdigits op_spec). @@ -635,7 +635,7 @@ Section Z_2nZ. Let spec_ww_tail00 : forall x, [|x|] = 0 -> [|tail0 x|] = Zpos _ww_digits. Proof. - refine (spec_ww_tail00 w_0 w_0W + refine (spec_ww_tail00 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ (refl_equal _ww_digits) _ _ _ _); wwauto. exact (spec_compare op_spec). @@ -647,7 +647,7 @@ Section Z_2nZ. Let spec_ww_tail0 : forall x, 0 < [|x|] -> exists y, 0 <= y /\ [|x|] = (2 * y + 1) * 2 ^ [|tail0 x|]. Proof. - refine (spec_ww_tail0 (w_digits := w_digits) w_0 w_0W w_compare w_tail0 + refine (spec_ww_tail0 (w_digits := w_digits) w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ _ _ _ _);wwauto. exact (spec_compare op_spec). exact (spec_zdigits op_spec). @@ -659,19 +659,19 @@ Section Z_2nZ. ([|x|] * (2 ^ [|p|]) + [|y|] / (2 ^ ((Zpos _ww_digits) - [|p|]))) mod wwB. Proof. - refine (@spec_ww_add_mul_div w w_0 w_WW w_W0 w_0W compare w_add_mul_div + refine (@spec_ww_add_mul_div w w_0 w_WW w_W0 w_0W compare w_add_mul_div sub w_digits w_zdigits low w_to_Z _ _ _ _ _ _ _ _ _ _ _);wwauto. exact (spec_zdigits op_spec). Qed. - Let spec_ww_div_gt : forall a b, + Let spec_ww_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> let (q,r) := div_gt a b in [|a|] = [|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|]. Proof. -refine -(@spec_ww_div_gt w w_digits w_0 w_WW w_0W w_compare w_eq0 +refine +(@spec_ww_div_gt w w_digits w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @@ -707,14 +707,14 @@ refine refine (spec_ww_div w_digits ww_1 compare div_gt w_to_Z _ _ _ _);auto. Qed. - Let spec_ww_mod_gt : forall a b, + Let spec_ww_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> [|mod_gt a b|] = [|a|] mod [|b|]. Proof. - refine (@spec_ww_mod_gt w w_digits w_0 w_WW w_0W w_compare w_eq0 + refine (@spec_ww_mod_gt w w_digits w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt w_mod_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div - w_zdigits w_to_Z + w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div + w_zdigits w_to_Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. exact (spec_compare op_spec). exact (spec_div_gt op_spec). @@ -731,12 +731,12 @@ refine Let spec_ww_gcd_gt : forall a b, [|a|] > [|b|] -> Zis_gcd [|a|] [|b|] [|gcd_gt a b|]. Proof. - refine (@spec_ww_gcd_gt w w_digits W0 w_to_Z _ + refine (@spec_ww_gcd_gt w w_digits W0 w_to_Z _ w_0 w_0 w_eq0 w_gcd_gt _ww_digits _ gcd_gt_fix _ _ _ _ gcd_cont _);auto. refine (@spec_ww_gcd_gt_aux w w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0 - w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z + w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. exact (spec_compare op_spec). exact (spec_div21 op_spec). @@ -753,7 +753,7 @@ refine _ww_digits _ gcd_gt_fix _ _ _ _ gcd_cont _);auto. refine (@spec_ww_gcd_gt_aux w w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0 - w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z + w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. exact (spec_compare op_spec). exact (spec_div21 op_spec). @@ -798,7 +798,7 @@ refine Let spec_ww_sqrt : forall x, [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2. Proof. - refine (@spec_ww_sqrt w w_is_even w_0 w_1 w_Bm1 + refine (@spec_ww_sqrt w w_is_even w_0 w_1 w_Bm1 w_sub w_add_mul_div w_digits w_zdigits _ww_zdigits w_sqrt2 pred add_mul_div head0 compare _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto. @@ -814,7 +814,7 @@ refine apply mk_znz_spec;auto. exact spec_ww_add_mul_div. - refine (@spec_ww_pos_mod w w_0 w_digits w_zdigits w_WW + refine (@spec_ww_pos_mod w w_0 w_digits w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z _ _ _ _ _ _ _ _ _ _ _ _);wwauto. exact (spec_pos_mod op_spec). @@ -828,7 +828,7 @@ refine Proof. apply mk_znz_spec;auto. exact spec_ww_add_mul_div. - refine (@spec_ww_pos_mod w w_0 w_digits w_zdigits w_WW + refine (@spec_ww_pos_mod w w_0 w_digits w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z _ _ _ _ _ _ _ _ _ _ _ _);wwauto. exact (spec_pos_mod op_spec). @@ -838,10 +838,10 @@ refine rewrite <- Zpos_xO; exact spec_ww_digits. Qed. -End Z_2nZ. - +End Z_2nZ. + Section MulAdd. - + Variable w: Type. Variable op: znz_op w. Variable sop: znz_spec op. @@ -870,7 +870,7 @@ Section MulAdd. End MulAdd. -(** Modular versions of DoubleCyclic *) +(** Modular versions of DoubleCyclic *) Module DoubleCyclic (C:CyclicType) <: CyclicType. Definition w := zn2z C.w. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v index d3dfd2505c..03c6114422 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v @@ -41,13 +41,13 @@ Section POS_MOD. Variable ww_zdigits : zn2z w. - Definition ww_pos_mod p x := + Definition ww_pos_mod p x := let zdigits := w_0W w_zdigits in match x with | W0 => W0 | WW xh xl => match ww_compare p zdigits with - | Eq => w_WW w_0 xl + | Eq => w_WW w_0 xl | Lt => w_WW w_0 (w_pos_mod (low p) xl) | Gt => match ww_compare p ww_zdigits with @@ -87,7 +87,7 @@ Section POS_MOD. | Lt => [[x]] < [[y]] | Gt => [[x]] > [[y]] end. - Variable spec_ww_sub: forall x y, + Variable spec_ww_sub: forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. @@ -106,7 +106,7 @@ Section POS_MOD. unfold ww_pos_mod; case w1. simpl; rewrite Zmod_small; split; auto with zarith. intros xh xl; generalize (spec_ww_compare p (w_0W w_zdigits)); - case ww_compare; + case ww_compare; rewrite spec_w_0W; rewrite spec_zdigits; fold wB; intros H1. rewrite H1; simpl ww_to_Z. @@ -135,13 +135,13 @@ Section POS_MOD. autorewrite with w_rewrite rm10. rewrite Zmod_mod; auto with zarith. generalize (spec_ww_compare p ww_zdigits); - case ww_compare; rewrite spec_ww_zdigits; + case ww_compare; rewrite spec_ww_zdigits; rewrite spec_zdigits; intros H2. replace (2^[[p]]) with wwB. rewrite Zmod_small; auto with zarith. unfold base; rewrite H2. rewrite spec_ww_digits; auto. - assert (HH0: [|low (ww_sub p (w_0W w_zdigits))|] = + assert (HH0: [|low (ww_sub p (w_0W w_zdigits))|] = [[p]] - Zpos w_digits). rewrite spec_low. rewrite spec_ww_sub. @@ -152,11 +152,11 @@ generalize (spec_ww_compare p ww_zdigits); apply Zlt_le_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_le_lin; auto with zarith. exists wB; unfold base; rewrite <- Zpower_exp; auto with zarith. - rewrite spec_ww_digits; + rewrite spec_ww_digits; apply f_equal with (f := Zpower 2); rewrite Zpos_xO; auto with zarith. simpl ww_to_Z; autorewrite with w_rewrite. rewrite spec_pos_mod; rewrite HH0. - pattern [|xh|] at 2; + pattern [|xh|] at 2; rewrite Z_div_mod_eq with (b := 2 ^ ([[p]] - Zpos w_digits)); auto with zarith. rewrite (fun x => (Zmult_comm (2 ^ x))); rewrite Zmult_plus_distr_l. @@ -196,7 +196,7 @@ generalize (spec_ww_compare p ww_zdigits); split; auto with zarith. rewrite Zpos_xO; auto with zarith. Qed. - + End POS_MOD. Section DoubleDiv32. @@ -222,24 +222,24 @@ Section DoubleDiv32. match w_compare a1 b1 with | Lt => let (q,r) := w_div21 a1 a2 b1 in - match ww_sub_c (w_WW r a3) (w_mul_c q b2) with + match ww_sub_c (w_WW r a3) (w_mul_c q b2) with | C0 r1 => (q,r1) | C1 r1 => let q := w_pred q in - ww_add_c_cont w_WW w_add_c w_add_carry_c + ww_add_c_cont w_WW w_add_c w_add_carry_c (fun r2=>(w_pred q, ww_add w_add_c w_add w_add_carry r2 (WW b1 b2))) (fun r2 => (q,r2)) r1 (WW b1 b2) end | Eq => - ww_add_c_cont w_WW w_add_c w_add_carry_c + ww_add_c_cont w_WW w_add_c w_add_carry_c (fun r => (w_Bm2, ww_add w_add_c w_add w_add_carry r (WW b1 b2))) (fun r => (w_Bm1,r)) (WW (w_sub a2 b2) a3) (WW b1 b2) | Gt => (w_0, W0) (* cas absurde *) end. - (* Proof *) + (* Proof *) Variable w_digits : positive. Variable w_to_Z : w -> Z. @@ -253,8 +253,8 @@ Section DoubleDiv32. (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). @@ -273,7 +273,7 @@ Section DoubleDiv32. | Gt => [|x|] > [|y|] end. Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_w_add_carry_c : + Variable spec_w_add_carry_c : forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1. Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. @@ -315,8 +315,8 @@ Section DoubleDiv32. wB/2 <= [|b1|] -> [[WW a1 a2]] < [[WW b1 b2]] -> let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ + [|a1|] * wwB + [|a2|] * wB + [|a3|] = + [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ 0 <= [[r]] < [|b1|] * wB + [|b2|]. Proof. intros a1 a2 a3 b1 b2 Hle Hlt. @@ -327,17 +327,17 @@ Section DoubleDiv32. match w_compare a1 b1 with | Lt => let (q,r) := w_div21 a1 a2 b1 in - match ww_sub_c (w_WW r a3) (w_mul_c q b2) with + match ww_sub_c (w_WW r a3) (w_mul_c q b2) with | C0 r1 => (q,r1) | C1 r1 => let q := w_pred q in - ww_add_c_cont w_WW w_add_c w_add_carry_c + ww_add_c_cont w_WW w_add_c w_add_carry_c (fun r2=>(w_pred q, ww_add w_add_c w_add w_add_carry r2 (WW b1 b2))) (fun r2 => (q,r2)) r1 (WW b1 b2) end | Eq => - ww_add_c_cont w_WW w_add_c w_add_carry_c + ww_add_c_cont w_WW w_add_c w_add_carry_c (fun r => (w_Bm2, ww_add w_add_c w_add w_add_carry r (WW b1 b2))) (fun r => (w_Bm1,r)) (WW (w_sub a2 b2) a3) (WW b1 b2) @@ -360,7 +360,7 @@ Section DoubleDiv32. [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ 0 <= [[r]] < [|b1|] * wB + [|b2|]);eauto. rewrite H0;intros r. - repeat + repeat (rewrite spec_ww_add;eauto || rewrite spec_w_Bm1 || rewrite spec_w_Bm2); simpl ww_to_Z;try rewrite Zmult_1_l;intros H1. assert (0<= ([[r]] + ([|b1|] * wB + [|b2|])) - wwB < [|b1|] * wB + [|b2|]). @@ -385,7 +385,7 @@ Section DoubleDiv32. 1 ([[r]] + ([|b1|] * wB + [|b2|]) - wwB));zarith;try (ring;fail). split. rewrite H1;rewrite Hcmp;ring. trivial. Spec_ww_to_Z (WW b1 b2). simpl in HH4;zarith. - rewrite H0;intros r;repeat + rewrite H0;intros r;repeat (rewrite spec_w_Bm1 || rewrite spec_w_Bm2); simpl ww_to_Z;try rewrite Zmult_1_l;intros H1. assert ([[r]]=([|a2|]-[|b2|])*wB+[|a3|]+([|b1|]*wB+[|b2|])). zarith. @@ -409,7 +409,7 @@ Section DoubleDiv32. as [r1|r1];repeat (rewrite spec_w_WW || rewrite spec_mul_c); unfold interp_carry;intros H1. rewrite H1. - split. ring. split. + split. ring. split. rewrite <- H1;destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z r1);trivial. apply Zle_lt_trans with ([|r|] * wB + [|a3|]). assert ( 0 <= [|q|] * [|b2|]);zarith. @@ -418,7 +418,7 @@ Section DoubleDiv32. rewrite <- H1;ring. Spec_ww_to_Z r1; assert (0 <= [|r|]*wB). zarith. assert (0 < [|q|] * [|b2|]). zarith. - assert (0 < [|q|]). + assert (0 < [|q|]). apply Zmult_lt_0_reg_r_2 with [|b2|];zarith. eapply spec_ww_add_c_cont with (P := fun (x y:zn2z w) (res:w*zn2z w) => @@ -440,18 +440,18 @@ Section DoubleDiv32. wwB * 1 + ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2 * ([|b1|] * wB + [|b2|]))). rewrite H7;rewrite H2;ring. - assert - ([|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) + assert + ([|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) < [|b1|]*wB + [|b2|]). Spec_ww_to_Z r2;omega. Spec_ww_to_Z (WW b1 b2). simpl in HH5. - assert - (0 <= [|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) + assert + (0 <= [|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) < wwB). split;try omega. replace (2*([|b1|]*wB+[|b2|])) with ((2*[|b1|])*wB+2*[|b2|]). 2:ring. assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). rewrite wwB_wBwB; rewrite Zpower_2; zarith. omega. - rewrite <- (Zmod_unique + rewrite <- (Zmod_unique ([[r2]] + ([|b1|] * wB + [|b2|])) wwB 1 @@ -486,7 +486,7 @@ Section DoubleDiv21. Definition ww_div21 a1 a2 b := match a1 with - | W0 => + | W0 => match ww_compare a2 b with | Gt => (ww_1, ww_sub a2 b) | Eq => (ww_1, W0) @@ -529,8 +529,8 @@ Section DoubleDiv21. Notation wwB := (base (ww_digits w_digits)). Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). Variable spec_w_0 : [|w_0|] = 0. @@ -540,8 +540,8 @@ Section DoubleDiv21. wB/2 <= [|b1|] -> [[WW a1 a2]] < [[WW b1 b2]] -> let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ + [|a1|] * wwB + [|a2|] * wB + [|a3|] = + [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ 0 <= [[r]] < [|b1|] * wB + [|b2|]. Variable spec_ww_1 : [[ww_1]] = 1. Variable spec_ww_compare : forall x y, @@ -591,10 +591,10 @@ Section DoubleDiv21. intros Hlt H; match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); intros q1 r H0 - end; (assert (Eq1: wB / 2 <= [|b1|]);[ + end; (assert (Eq1: wB / 2 <= [|b1|]);[ apply (@beta_lex (wB / 2) 0 [|b1|] [|b2|] wB); auto with zarith; autorewrite with rm10;repeat rewrite (Zmult_comm wB); - rewrite <- wwB_div_2; trivial + rewrite <- wwB_div_2; trivial | generalize (H0 Eq1 Hlt);clear H0;destruct r as [ |r1 r2];simpl; try rewrite spec_w_0; try rewrite spec_w_0W;repeat rewrite Zplus_0_r; intros (H1,H2) ]). @@ -611,10 +611,10 @@ Section DoubleDiv21. rewrite <- wwB_wBwB;rewrite H1. rewrite spec_w_0 in H4;rewrite Zplus_0_r in H4. repeat rewrite Zmult_plus_distr_l. rewrite <- (Zmult_assoc [|r1|]). - rewrite <- Zpower_2; rewrite <- wwB_wBwB;rewrite H4;simpl;ring. + rewrite <- Zpower_2; rewrite <- wwB_wBwB;rewrite H4;simpl;ring. split;[rewrite wwB_wBwB | split;zarith]. - replace (([|a1h|] * wB + [|a1l|]) * wB^2 + ([|a3|] * wB + [|a4|])) - with (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB+ [|a4|]). + replace (([|a1h|] * wB + [|a1l|]) * wB^2 + ([|a3|] * wB + [|a4|])) + with (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB+ [|a4|]). rewrite H1;ring. rewrite wwB_wBwB;ring. change [|a4|] with (0*wB+[|a4|]);apply beta_lex_inv;zarith. assert (1 <= wB/2);zarith. @@ -624,7 +624,7 @@ Section DoubleDiv21. intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end. split;trivial. replace (([|a1h|] * wB + [|a1l|]) * wwB + ([|a3|] * wB + [|a4|])) with - (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB + [|a4|]); + (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB + [|a4|]); [rewrite H1 | rewrite wwB_wBwB;ring]. replace (([|q1|]*([|b1|]*wB+[|b2|])+([|r1|]*wB+[|r2|]))*wB+[|a4|]) with (([|q1|]*([|b1|]*wB+[|b2|]))*wB+([|r1|]*wwB+[|r2|]*wB+[|a4|])); @@ -666,22 +666,22 @@ Section DoubleDivGt. Eval lazy beta iota delta [ww_sub ww_opp] in let p := w_head0 bh in match w_compare p w_0 with - | Gt => + | Gt => let b1 := w_add_mul_div p bh bl in let b2 := w_add_mul_div p bl w_0 in let a1 := w_add_mul_div p w_0 ah in let a2 := w_add_mul_div p ah al in let a3 := w_add_mul_div p al w_0 in let (q,r) := w_div32 a1 a2 a3 b1 b2 in - (WW w_0 q, ww_add_mul_div + (WW w_0 q, ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r) | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)) end. - Definition ww_div_gt a b := - Eval lazy beta iota delta [ww_div_gt_aux double_divn1 + Definition ww_div_gt a b := + Eval lazy beta iota delta [ww_div_gt_aux double_divn1 double_divn1_p double_divn1_p_aux double_divn1_0 double_divn1_0_aux double_split double_0 double_WW] in match a, b with @@ -691,11 +691,11 @@ Section DoubleDivGt. if w_eq0 ah then let (q,r) := w_div_gt al bl in (WW w_0 q, w_0W r) - else + else match w_compare w_0 bh with - | Eq => + | Eq => let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 + double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 a bl in (q, w_0W r) | Lt => ww_div_gt_aux ah al bh bl @@ -707,7 +707,7 @@ Section DoubleDivGt. Eval lazy beta iota delta [ww_sub ww_opp] in let p := w_head0 bh in match w_compare p w_0 with - | Gt => + | Gt => let b1 := w_add_mul_div p bh bl in let b2 := w_add_mul_div p bl w_0 in let a1 := w_add_mul_div p w_0 ah in @@ -716,13 +716,13 @@ Section DoubleDivGt. let (q,r) := w_div32 a1 a2 a3 b1 b2 in ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r - | _ => + | _ => ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry (WW ah al) (WW bh bl) end. - Definition ww_mod_gt a b := - Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 + Definition ww_mod_gt a b := + Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux double_split double_0 double_WW snd] in match a, b with @@ -730,10 +730,10 @@ Section DoubleDivGt. | _, W0 => W0 | WW ah al, WW bh bl => if w_eq0 ah then w_0W (w_mod_gt al bl) - else + else match w_compare w_0 bh with - | Eq => - w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 + | Eq => + w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 w_compare w_sub 1 a bl) | Lt => ww_mod_gt_aux ah al bh bl | Gt => W0 (* cas absurde *) @@ -741,14 +741,14 @@ Section DoubleDivGt. end. Definition ww_gcd_gt_body (cont: w->w->w->w->zn2z w) (ah al bh bl: w) := - Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 + Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux double_split double_0 double_WW snd] in match w_compare w_0 bh with | Eq => match w_compare w_0 bl with | Eq => WW ah al (* normalement n'arrive pas si forme normale *) - | Lt => + | Lt => let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW ah al) bl in WW w_0 (w_gcd_gt bl m) @@ -757,14 +757,14 @@ Section DoubleDivGt. | Lt => let m := ww_mod_gt_aux ah al bh bl in match m with - | W0 => WW bh bl + | W0 => WW bh bl | WW mh ml => match w_compare w_0 mh with | Eq => match w_compare w_0 ml with | Eq => WW bh bl - | _ => - let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 + | _ => + let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW bh bl) ml in WW w_0 (w_gcd_gt ml r) end @@ -779,18 +779,18 @@ Section DoubleDivGt. end | Gt => W0 (* absurde *) end. - - Fixpoint ww_gcd_gt_aux - (p:positive) (cont: w -> w -> w -> w -> zn2z w) (ah al bh bl : w) + + Fixpoint ww_gcd_gt_aux + (p:positive) (cont: w -> w -> w -> w -> zn2z w) (ah al bh bl : w) {struct p} : zn2z w := - ww_gcd_gt_body + ww_gcd_gt_body (fun mh ml rh rl => match p with | xH => cont mh ml rh rl | xO p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl | xI p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl end) ah al bh bl. - + (* Proof *) Variable w_to_Z : w -> Z. @@ -816,7 +816,7 @@ Section DoubleDivGt. | Gt => [|x|] > [|y|] end. Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. - + Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB. Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1. @@ -854,8 +854,8 @@ Section DoubleDivGt. wB/2 <= [|b1|] -> [[WW a1 a2]] < [[WW b1 b2]] -> let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ + [|a1|] * wwB + [|a2|] * wB + [|a3|] = + [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ 0 <= [[r]] < [|b1|] * wB + [|b2|]. Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits. @@ -899,14 +899,14 @@ Section DoubleDivGt. change (let (q, r) := let p := w_head0 bh in match w_compare p w_0 with - | Gt => + | Gt => let b1 := w_add_mul_div p bh bl in let b2 := w_add_mul_div p bl w_0 in let a1 := w_add_mul_div p w_0 ah in let a2 := w_add_mul_div p ah al in let a3 := w_add_mul_div p al w_0 in let (q,r) := w_div32 a1 a2 a3 b1 b2 in - (WW w_0 q, ww_add_mul_div + (WW w_0 q, ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r) | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c @@ -945,11 +945,11 @@ Section DoubleDivGt. (spec_add_mul_div bl w_0 Hb); rewrite spec_w_0; repeat rewrite Zmult_0_l;repeat rewrite Zplus_0_l; rewrite Zdiv_0_l;repeat rewrite Zplus_0_r. - Spec_w_to_Z ah;Spec_w_to_Z bh. + Spec_w_to_Z ah;Spec_w_to_Z bh. unfold base;repeat rewrite Zmod_shift_r;zarith. assert (H3:=to_Z_div_minus_p ah HHHH);assert(H4:=to_Z_div_minus_p al HHHH); assert (H5:=to_Z_div_minus_p bl HHHH). - rewrite Zmult_comm in Hh. + rewrite Zmult_comm in Hh. assert (2^[|w_head0 bh|] < wB). unfold base;apply Zpower_lt_monotone;zarith. unfold base in H0;rewrite Zmod_small;zarith. fold wB; rewrite (Zmod_small ([|bh|] * 2 ^ [|w_head0 bh|]));zarith. @@ -964,15 +964,15 @@ Section DoubleDivGt. (w_add_mul_div (w_head0 bh) al w_0) (w_add_mul_div (w_head0 bh) bh bl) (w_add_mul_div (w_head0 bh) bl w_0)) as (q,r). - rewrite V1;rewrite V2. rewrite Zmult_plus_distr_l. - rewrite <- (Zplus_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)). + rewrite V1;rewrite V2. rewrite Zmult_plus_distr_l. + rewrite <- (Zplus_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)). unfold base;rewrite <- shift_unshift_mod;zarith. fold wB. replace ([|bh|] * 2 ^ [|w_head0 bh|] * wB + [|bl|] * 2 ^ [|w_head0 bh|]) with ([[WW bh bl]] * 2^[|w_head0 bh|]). 2:simpl;ring. fold wwB. rewrite wwB_wBwB. rewrite Zpower_2. rewrite U1;rewrite U2;rewrite U3. - rewrite Zmult_assoc. rewrite Zmult_plus_distr_l. + rewrite Zmult_assoc. rewrite Zmult_plus_distr_l. rewrite (Zplus_assoc ([|ah|] / 2^(Zpos(w_digits) - [|w_head0 bh|])*wB * wB)). - rewrite <- Zmult_plus_distr_l. rewrite <- Zplus_assoc. + rewrite <- Zmult_plus_distr_l. rewrite <- Zplus_assoc. unfold base;repeat rewrite <- shift_unshift_mod;zarith. fold wB. replace ([|ah|] * 2 ^ [|w_head0 bh|] * wB + [|al|] * 2 ^ [|w_head0 bh|]) with ([[WW ah al]] * 2^[|w_head0 bh|]). 2:simpl;ring. @@ -1027,7 +1027,7 @@ Section DoubleDivGt. [[a]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]. Proof. - intros a b Hgt Hpos;unfold ww_div_gt. + intros a b Hgt Hpos;unfold ww_div_gt. change (let (q,r) := match a, b with | W0, _ => (W0,W0) | _, W0 => (W0,W0) @@ -1035,23 +1035,23 @@ Section DoubleDivGt. if w_eq0 ah then let (q,r) := w_div_gt al bl in (WW w_0 q, w_0W r) - else + else match w_compare w_0 bh with - | Eq => + | Eq => let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 + double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 a bl in (q, w_0W r) | Lt => ww_div_gt_aux ah al bh bl | Gt => (W0,W0) (* cas absurde *) end - end in [[a]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]). + end in [[a]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]). destruct a as [ |ah al]. simpl in Hgt;omega. destruct b as [ |bh bl]. simpl in Hpos;omega. Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl. assert (H:=@spec_eq0 ah);destruct (w_eq0 ah). simpl ww_to_Z;rewrite H;trivial. simpl in Hgt;rewrite H in Hgt;trivial. - assert ([|bh|] <= 0). + assert ([|bh|] <= 0). apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;rewrite H1;simpl in Hgt. simpl. simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos. @@ -1066,7 +1066,7 @@ Section DoubleDivGt. w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hpos). unfold double_to_Z,double_wB,double_digits in H2. - destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 + destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW ah al) bl). rewrite spec_w_0W;unfold ww_to_Z;trivial. @@ -1104,26 +1104,26 @@ Section DoubleDivGt. rewrite Zmult_comm in H;destruct H. symmetry;apply Zmod_unique with [|q|];trivial. Qed. - + Lemma spec_ww_mod_gt_eq : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> [[ww_mod_gt a b]] = [[snd (ww_div_gt a b)]]. Proof. intros a b Hgt Hpos. - change (ww_mod_gt a b) with + change (ww_mod_gt a b) with (match a, b with | W0, _ => W0 | _, W0 => W0 | WW ah al, WW bh bl => if w_eq0 ah then w_0W (w_mod_gt al bl) - else + else match w_compare w_0 bh with - | Eq => - w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 + | Eq => + w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 w_compare w_sub 1 a bl) | Lt => ww_mod_gt_aux ah al bh bl | Gt => W0 (* cas absurde *) end end). - change (ww_div_gt a b) with + change (ww_div_gt a b) with (match a, b with | W0, _ => (W0,W0) | _, W0 => (W0,W0) @@ -1131,11 +1131,11 @@ Section DoubleDivGt. if w_eq0 ah then let (q,r) := w_div_gt al bl in (WW w_0 q, w_0W r) - else + else match w_compare w_0 bh with - | Eq => + | Eq => let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 + double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 a bl in (q, w_0W r) | Lt => ww_div_gt_aux ah al bh bl @@ -1147,7 +1147,7 @@ Section DoubleDivGt. Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl. assert (H:=@spec_eq0 ah);destruct (w_eq0 ah). simpl in Hgt;rewrite H in Hgt;trivial. - assert ([|bh|] <= 0). + assert ([|bh|] <= 0). apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos. @@ -1155,7 +1155,7 @@ Section DoubleDivGt. destruct (w_div_gt al bl);simpl;rewrite spec_w_0W;trivial. clear H. assert (H2 := spec_compare w_0 bh);destruct (w_compare w_0 bh). - rewrite (@spec_double_modn1_aux w w_zdigits w_0 w_WW w_head0 w_add_mul_div + rewrite (@spec_double_modn1_aux w w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_w_0 spec_compare 1 (WW ah al) bl). destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW ah al) bl);simpl;trivial. @@ -1174,7 +1174,7 @@ Section DoubleDivGt. rewrite Zmult_comm;trivial. Qed. - Lemma Zis_gcd_mod : forall a b d, + Lemma Zis_gcd_mod : forall a b d, 0 < b -> Zis_gcd b (a mod b) d -> Zis_gcd a b d. Proof. intros a b d H H1; apply Zis_gcd_for_euclid with (a/b). @@ -1182,12 +1182,12 @@ Section DoubleDivGt. ring_simplify (b * (a / b) + a mod b - a / b * b);trivial. zarith. Qed. - Lemma spec_ww_gcd_gt_aux_body : + Lemma spec_ww_gcd_gt_aux_body : forall ah al bh bl n cont, - [[WW bh bl]] <= 2^n -> + [[WW bh bl]] <= 2^n -> [[WW ah al]] > [[WW bh bl]] -> - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^(n-1) -> + (forall xh xl yh yl, + [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^(n-1) -> Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> Zis_gcd [[WW ah al]] [[WW bh bl]] [[ww_gcd_gt_body cont ah al bh bl]]. Proof. @@ -1196,7 +1196,7 @@ Section DoubleDivGt. | Eq => match w_compare w_0 bl with | Eq => WW ah al (* normalement n'arrive pas si forme normale *) - | Lt => + | Lt => let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW ah al) bl in WW w_0 (w_gcd_gt bl m) @@ -1205,14 +1205,14 @@ Section DoubleDivGt. | Lt => let m := ww_mod_gt_aux ah al bh bl in match m with - | W0 => WW bh bl + | W0 => WW bh bl | WW mh ml => match w_compare w_0 mh with | Eq => match w_compare w_0 ml with | Eq => WW bh bl - | _ => - let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 + | _ => + let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW bh bl) ml in WW w_0 (w_gcd_gt ml r) end @@ -1227,10 +1227,10 @@ Section DoubleDivGt. end | Gt => W0 (* absurde *) end). - assert (Hbh := spec_compare w_0 bh);destruct (w_compare w_0 bh). + assert (Hbh := spec_compare w_0 bh);destruct (w_compare w_0 bh). simpl ww_to_Z in *. rewrite spec_w_0 in Hbh;rewrite <- Hbh; rewrite Zmult_0_l;rewrite Zplus_0_l. - assert (Hbl := spec_compare w_0 bl); destruct (w_compare w_0 bl). + assert (Hbl := spec_compare w_0 bl); destruct (w_compare w_0 bl). rewrite spec_w_0 in Hbl;rewrite <- Hbl;apply Zis_gcd_0. simpl;rewrite spec_w_0;rewrite Zmult_0_l;rewrite Zplus_0_l. rewrite spec_w_0 in Hbl. @@ -1239,54 +1239,54 @@ Section DoubleDivGt. rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hbl). - apply spec_gcd_gt. - rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + apply spec_gcd_gt. + rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. + apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. rewrite spec_w_0 in Hbl;Spec_w_to_Z bl;elimtype False;omega. rewrite spec_w_0 in Hbh;assert (H:= spec_ww_mod_gt_aux _ _ _ Hgt Hbh). - assert (H2 : 0 < [[WW bh bl]]). + assert (H2 : 0 < [[WW bh bl]]). simpl;Spec_w_to_Z bl. apply Zlt_le_trans with ([|bh|]*wB);zarith. apply Zmult_lt_0_compat;zarith. apply Zis_gcd_mod;trivial. rewrite <- H. simpl in *;destruct (ww_mod_gt_aux ah al bh bl) as [ |mh ml]. - simpl;apply Zis_gcd_0;zarith. - assert (Hmh := spec_compare w_0 mh);destruct (w_compare w_0 mh). + simpl;apply Zis_gcd_0;zarith. + assert (Hmh := spec_compare w_0 mh);destruct (w_compare w_0 mh). simpl;rewrite spec_w_0 in Hmh; rewrite <- Hmh;simpl. - assert (Hml := spec_compare w_0 ml);destruct (w_compare w_0 ml). + assert (Hml := spec_compare w_0 ml);destruct (w_compare w_0 ml). rewrite <- Hml;rewrite spec_w_0;simpl;apply Zis_gcd_0. - simpl;rewrite spec_w_0;simpl. + simpl;rewrite spec_w_0;simpl. rewrite spec_w_0 in Hml. apply Zis_gcd_mod;zarith. change ([|bh|] * wB + [|bl|]) with (double_to_Z w_digits w_to_Z 1 (WW bh bl)). rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div spec_div21 spec_compare spec_sub 1 (WW bh bl) ml Hml). - apply spec_gcd_gt. - rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + apply spec_gcd_gt. + rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. + apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. rewrite spec_w_0 in Hml;Spec_w_to_Z ml;elimtype False;omega. rewrite spec_w_0 in Hmh. assert ([[WW bh bl]] > [[WW mh ml]]). - rewrite H;simpl; apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + rewrite H;simpl; apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. assert (H1:= spec_ww_mod_gt_aux _ _ _ H0 Hmh). - assert (H3 : 0 < [[WW mh ml]]). + assert (H3 : 0 < [[WW mh ml]]). simpl;Spec_w_to_Z ml. apply Zlt_le_trans with ([|mh|]*wB);zarith. apply Zmult_lt_0_compat;zarith. apply Zis_gcd_mod;zarith. simpl in *;rewrite <- H1. destruct (ww_mod_gt_aux bh bl mh ml) as [ |rh rl]. simpl; apply Zis_gcd_0. simpl;apply Hcont. simpl in H1;rewrite H1. - apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. - apply Zle_trans with (2^n/2). - apply Zdiv_le_lower_bound;zarith. + apply Zle_trans with (2^n/2). + apply Zdiv_le_lower_bound;zarith. apply Zle_trans with ([|bh|] * wB + [|bl|]);zarith. assert (H3' := Z_div_mod_eq [[WW bh bl]] [[WW mh ml]] (Zlt_gt _ _ H3)). assert (H4' : 0 <= [[WW bh bl]]/[[WW mh ml]]). apply Zge_le;apply Z_div_ge0;zarith. simpl in *;rewrite H1. pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3'. destruct (Zle_lt_or_eq _ _ H4'). - assert (H6' : [[WW bh bl]] mod [[WW mh ml]] = + assert (H6' : [[WW bh bl]] mod [[WW mh ml]] = [[WW bh bl]] - [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). simpl;pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3';ring. simpl in H6'. assert ([[WW mh ml]] <= [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). @@ -1304,10 +1304,10 @@ Section DoubleDivGt. rewrite spec_w_0 in Hbh;Spec_w_to_Z bh;elimtype False;zarith. Qed. - Lemma spec_ww_gcd_gt_aux : + Lemma spec_ww_gcd_gt_aux : forall p cont n, - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> + (forall xh xl yh yl, + [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^n -> Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] -> @@ -1334,7 +1334,7 @@ Section DoubleDivGt. apply Zle_trans with (2 ^ (Zpos p + n -1));zarith. apply Zpower_le_monotone2;zarith. apply Zle_trans with (2 ^ (2*Zpos p + n -1));zarith. - apply Zpower_le_monotone2;zarith. + apply Zpower_le_monotone2;zarith. apply spec_ww_gcd_gt_aux_body with (n := n+1);trivial. rewrite Zplus_comm;trivial. ring_simplify (n + 1 - 1);trivial. @@ -1352,16 +1352,16 @@ Section DoubleDiv. Variable ww_div_gt : zn2z w -> zn2z w -> zn2z w * zn2z w. Variable ww_mod_gt : zn2z w -> zn2z w -> zn2z w. - Definition ww_div a b := - match ww_compare a b with - | Gt => ww_div_gt a b + Definition ww_div a b := + match ww_compare a b with + | Gt => ww_div_gt a b | Eq => (ww_1, W0) | Lt => (W0, a) end. - Definition ww_mod a b := - match ww_compare a b with - | Gt => ww_mod_gt a b + Definition ww_mod a b := + match ww_compare a b with + | Gt => ww_mod_gt a b | Eq => W0 | Lt => a end. @@ -1401,7 +1401,7 @@ Section DoubleDiv. Proof. intros a b Hpos;unfold ww_div. assert (H:=spec_ww_compare a b);destruct (ww_compare a b). - simpl;rewrite spec_ww_1;split;zarith. + simpl;rewrite spec_ww_1;split;zarith. simpl;split;[ring|Spec_ww_to_Z a;zarith]. apply spec_ww_div_gt;trivial. Qed. @@ -1409,7 +1409,7 @@ Section DoubleDiv. Lemma spec_ww_mod : forall a b, 0 < [[b]] -> [[ww_mod a b]] = [[a]] mod [[b]]. Proof. - intros a b Hpos;unfold ww_mod. + intros a b Hpos;unfold ww_mod. assert (H := spec_ww_compare a b);destruct (ww_compare a b). simpl;apply Zmod_unique with 1;try rewrite H;zarith. Spec_ww_to_Z a;symmetry;apply Zmod_small;zarith. @@ -1424,8 +1424,8 @@ Section DoubleDiv. Variable w_gcd_gt : w -> w -> w. Variable _ww_digits : positive. Variable spec_ww_digits_ : _ww_digits = xO w_digits. - Variable ww_gcd_gt_fix : - positive -> (w -> w -> w -> w -> zn2z w) -> + Variable ww_gcd_gt_fix : + positive -> (w -> w -> w -> w -> zn2z w) -> w -> w -> w -> w -> zn2z w. Variable spec_w_0 : [|w_0|] = 0. @@ -1440,10 +1440,10 @@ Section DoubleDiv. Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. Variable spec_gcd_gt : forall a b, [|a|] > [|b|] -> Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]. - Variable spec_gcd_gt_fix : + Variable spec_gcd_gt_fix : forall p cont n, - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> + (forall xh xl yh yl, + [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^n -> Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] -> @@ -1451,20 +1451,20 @@ Section DoubleDiv. Zis_gcd [[WW ah al]] [[WW bh bl]] [[ww_gcd_gt_fix p cont ah al bh bl]]. - Definition gcd_cont (xh xl yh yl:w) := + Definition gcd_cont (xh xl yh yl:w) := match w_compare w_1 yl with - | Eq => ww_1 + | Eq => ww_1 | _ => WW xh xl end. - Lemma spec_gcd_cont : forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> + Lemma spec_gcd_cont : forall xh xl yh yl, + [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 1 -> Zis_gcd [[WW xh xl]] [[WW yh yl]] [[gcd_cont xh xl yh yl]]. Proof. intros xh xl yh yl Hgt' Hle. simpl in Hle. assert ([|yh|] = 0). - change 1 with (0*wB+1) in Hle. + change 1 with (0*wB+1) in Hle. assert (0 <= 1 < wB). split;zarith. apply wB_pos. assert (H1:= beta_lex _ _ _ _ _ Hle (spec_to_Z yl) H). Spec_w_to_Z yh;zarith. @@ -1478,15 +1478,15 @@ Section DoubleDiv. rewrite H0;simpl;apply Zis_gcd_0;trivial. Qed. - + Variable cont : w -> w -> w -> w -> zn2z w. - Variable spec_cont : forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> + Variable spec_cont : forall xh xl yh yl, + [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 1 -> Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]. - - Definition ww_gcd_gt a b := - match a, b with + + Definition ww_gcd_gt a b := + match a, b with | W0, _ => b | _, W0 => a | WW ah al, WW bh bl => @@ -1509,8 +1509,8 @@ Section DoubleDiv. destruct a as [ |ah al]. simpl;apply Zis_gcd_sym;apply Zis_gcd_0. destruct b as [ |bh bl]. simpl;apply Zis_gcd_0. simpl in Hgt. generalize (@spec_eq0 ah);destruct (w_eq0 ah);intros. - simpl;rewrite H in Hgt;trivial;rewrite H;trivial;rewrite spec_w_0;simpl. - assert ([|bh|] <= 0). + simpl;rewrite H in Hgt;trivial;rewrite H;trivial;rewrite spec_w_0;simpl. + assert ([|bh|] <= 0). apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. Spec_w_to_Z bh;assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. rewrite H1;simpl;auto. clear H. @@ -1522,7 +1522,7 @@ Section DoubleDiv. Lemma spec_ww_gcd : forall a b, Zis_gcd [[a]] [[b]] [[ww_gcd a b]]. Proof. intros a b. - change (ww_gcd a b) with + change (ww_gcd a b) with (match ww_compare a b with | Gt => ww_gcd_gt a b | Eq => a diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v index 1f1d609f1d..fd6718e4e9 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v @@ -31,19 +31,19 @@ Section GENDIVN1. Variable w_div21 : w -> w -> w -> w * w. Variable w_compare : w -> w -> comparison. Variable w_sub : w -> w -> w. - - + + (* ** For proofs ** *) Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). + + Notation wB := (base w_digits). Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) + Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) (at level 0, x at level 99). Notation "[[ x ]]" := (zn2z_to_Z wB w_to_Z x) (at level 0, x at level 99). - + Variable spec_to_Z : forall x, 0 <= [| x |] < wB. Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits. Variable spec_0 : [|w_0|] = 0. @@ -68,10 +68,10 @@ Section GENDIVN1. | Lt => [|x|] < [|y|] | Gt => [|x|] > [|y|] end. - Variable spec_sub: forall x y, + Variable spec_sub: forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - + Section DIVAUX. Variable b2p : w. @@ -85,10 +85,10 @@ Section GENDIVN1. Fixpoint double_divn1_0 (n:nat) : w -> word w n -> word w n * w := match n return w -> word w n -> word w n * w with - | O => fun r x => w_div21 r x b2p - | S n => double_divn1_0_aux n (double_divn1_0 n) + | O => fun r x => w_div21 r x b2p + | S n => double_divn1_0_aux n (double_divn1_0 n) end. - + Lemma spec_split : forall (n : nat) (x : zn2z (word w n)), let (h, l) := double_split w_0 n x in [!S n | x!] = [!n | h!] * double_wB w_digits n + [!n | l!]. @@ -132,11 +132,11 @@ Section GENDIVN1. induction n;simpl;intros;trivial. unfold double_modn1_0_aux, double_divn1_0_aux. destruct (double_split w_0 n x) as (hh,hl). - rewrite (IHn r hh). + rewrite (IHn r hh). destruct (double_divn1_0 n r hh) as (qh,rh);simpl. rewrite IHn. destruct (double_divn1_0 n rh hl);trivial. Qed. - + Variable p : w. Variable p_bounded : [|p|] <= Zpos w_digits. @@ -148,18 +148,18 @@ Section GENDIVN1. intros;apply spec_add_mul_div;auto. Qed. - Definition double_divn1_p_aux n - (divn1 : w -> word w n -> word w n -> word w n * w) r h l := + Definition double_divn1_p_aux n + (divn1 : w -> word w n -> word w n -> word w n * w) r h l := let (hh,hl) := double_split w_0 n h in - let (lh,ll) := double_split w_0 n l in + let (lh,ll) := double_split w_0 n l in let (qh,rh) := divn1 r hh hl in let (ql,rl) := divn1 rh hl lh in (double_WW w_WW n qh ql, rl). Fixpoint double_divn1_p (n:nat) : w -> word w n -> word w n -> word w n * w := match n return w -> word w n -> word w n -> word w n * w with - | O => fun r h l => w_div21 r (w_add_mul_div p h l) b2p - | S n => double_divn1_p_aux n (double_divn1_p n) + | O => fun r h l => w_div21 r (w_add_mul_div p h l) b2p + | S n => double_divn1_p_aux n (double_divn1_p n) end. Lemma p_lt_double_digits : forall n, [|p|] <= Zpos (double_digits w_digits n). @@ -175,8 +175,8 @@ Section GENDIVN1. Lemma spec_double_divn1_p : forall n r h l, [|r|] < [|b2p|] -> let (q,r') := double_divn1_p n r h l in - [|r|] * double_wB w_digits n + - ([!n|h!]*2^[|p|] + + [|r|] * double_wB w_digits n + + ([!n|h!]*2^[|p|] + [!n|l!] / (2^(Zpos(double_digits w_digits n) - [|p|]))) mod double_wB w_digits n = [!n|q!] * [|b2p|] + [|r'|] /\ 0 <= [|r'|] < [|b2p|]. @@ -198,26 +198,26 @@ Section GENDIVN1. ([!n|lh!] * double_wB w_digits n + [!n|ll!]) / 2^(Zpos (double_digits w_digits (S n)) - [|p|])) mod (double_wB w_digits n * double_wB w_digits n)) with - (([|r|] * double_wB w_digits n + ([!n|hh!] * 2^[|p|] + + (([|r|] * double_wB w_digits n + ([!n|hh!] * 2^[|p|] + [!n|hl!] / 2^(Zpos (double_digits w_digits n) - [|p|])) mod double_wB w_digits n) * double_wB w_digits n + - ([!n|hl!] * 2^[|p|] + - [!n|lh!] / 2^(Zpos (double_digits w_digits n) - [|p|])) mod + ([!n|hl!] * 2^[|p|] + + [!n|lh!] / 2^(Zpos (double_digits w_digits n) - [|p|])) mod double_wB w_digits n). generalize (IHn r hh hl H);destruct (double_divn1_p n r hh hl) as (qh,rh); intros (H3,H4);rewrite H3. - assert ([|rh|] < [|b2p|]). omega. + assert ([|rh|] < [|b2p|]). omega. replace (([!n|qh!] * [|b2p|] + [|rh|]) * double_wB w_digits n + ([!n|hl!] * 2 ^ [|p|] + [!n|lh!] / 2 ^ (Zpos (double_digits w_digits n) - [|p|])) mod - double_wB w_digits n) with + double_wB w_digits n) with ([!n|qh!] * [|b2p|] *double_wB w_digits n + ([|rh|]*double_wB w_digits n + ([!n|hl!] * 2 ^ [|p|] + [!n|lh!] / 2 ^ (Zpos (double_digits w_digits n) - [|p|])) mod double_wB w_digits n)). 2:ring. generalize (IHn rh hl lh H0);destruct (double_divn1_p n rh hl lh) as (ql,rl); intros (H5,H6);rewrite H5. - split;[rewrite spec_double_WW;trivial;ring|trivial]. + split;[rewrite spec_double_WW;trivial;ring|trivial]. assert (Uhh := spec_double_to_Z w_digits w_to_Z spec_to_Z n hh); unfold double_wB,base in Uhh. assert (Uhl := spec_double_to_Z w_digits w_to_Z spec_to_Z n hl); @@ -228,37 +228,37 @@ Section GENDIVN1. unfold double_wB,base in Ull. unfold double_wB,base. assert (UU:=p_lt_double_digits n). - rewrite Zdiv_shift_r;auto with zarith. - 2:change (Zpos (double_digits w_digits (S n))) + rewrite Zdiv_shift_r;auto with zarith. + 2:change (Zpos (double_digits w_digits (S n))) with (2*Zpos (double_digits w_digits n));auto with zarith. replace (2 ^ (Zpos (double_digits w_digits (S n)) - [|p|])) with (2^(Zpos (double_digits w_digits n) - [|p|])*2^Zpos (double_digits w_digits n)). rewrite Zdiv_mult_cancel_r;auto with zarith. - rewrite Zmult_plus_distr_l with (p:= 2^[|p|]). + rewrite Zmult_plus_distr_l with (p:= 2^[|p|]). pattern ([!n|hl!] * 2^[|p|]) at 2; rewrite (shift_unshift_mod (Zpos(double_digits w_digits n))([|p|])([!n|hl!])); auto with zarith. - rewrite Zplus_assoc. - replace + rewrite Zplus_assoc. + replace ([!n|hh!] * 2^Zpos (double_digits w_digits n)* 2^[|p|] + ([!n|hl!] / 2^(Zpos (double_digits w_digits n)-[|p|])* 2^Zpos(double_digits w_digits n))) - with - (([!n|hh!] *2^[|p|] + double_to_Z w_digits w_to_Z n hl / + with + (([!n|hh!] *2^[|p|] + double_to_Z w_digits w_to_Z n hl / 2^(Zpos (double_digits w_digits n)-[|p|])) * 2^Zpos(double_digits w_digits n));try (ring;fail). rewrite <- Zplus_assoc. rewrite <- (Zmod_shift_r ([|p|]));auto with zarith. - replace + replace (2 ^ Zpos (double_digits w_digits n) * 2 ^ Zpos (double_digits w_digits n)) with (2 ^ (Zpos (double_digits w_digits n) + Zpos (double_digits w_digits n))). rewrite (Zmod_shift_r (Zpos (double_digits w_digits n)));auto with zarith. replace (2 ^ (Zpos (double_digits w_digits n) + Zpos (double_digits w_digits n))) - with (2^Zpos(double_digits w_digits n) *2^Zpos(double_digits w_digits n)). + with (2^Zpos(double_digits w_digits n) *2^Zpos(double_digits w_digits n)). rewrite (Zmult_comm (([!n|hh!] * 2 ^ [|p|] + [!n|hl!] / 2 ^ (Zpos (double_digits w_digits n) - [|p|])))). rewrite Zmult_mod_distr_l;auto with zarith. - ring. + ring. rewrite Zpower_exp;auto with zarith. assert (0 < Zpos (double_digits w_digits n)). unfold Zlt;reflexivity. auto with zarith. @@ -267,24 +267,24 @@ Section GENDIVN1. split;auto with zarith. apply Zdiv_lt_upper_bound;auto with zarith. rewrite <- Zpower_exp;auto with zarith. - replace ([|p|] + (Zpos (double_digits w_digits n) - [|p|])) with + replace ([|p|] + (Zpos (double_digits w_digits n) - [|p|])) with (Zpos(double_digits w_digits n));auto with zarith. rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (double_digits w_digits (S n)) - [|p|]) with - (Zpos (double_digits w_digits n) - [|p|] + + replace (Zpos (double_digits w_digits (S n)) - [|p|]) with + (Zpos (double_digits w_digits n) - [|p|] + Zpos (double_digits w_digits n));trivial. - change (Zpos (double_digits w_digits (S n))) with + change (Zpos (double_digits w_digits (S n))) with (2*Zpos (double_digits w_digits n)). ring. Qed. Definition double_modn1_p_aux n (modn1 : w -> word w n -> word w n -> w) r h l:= let (hh,hl) := double_split w_0 n h in - let (lh,ll) := double_split w_0 n l in + let (lh,ll) := double_split w_0 n l in modn1 (modn1 r hh hl) hl lh. Fixpoint double_modn1_p (n:nat) : w -> word w n -> word w n -> w := match n return w -> word w n -> word w n -> w with - | O => fun r h l => snd (w_div21 r (w_add_mul_div p h l) b2p) + | O => fun r h l => snd (w_div21 r (w_add_mul_div p h l) b2p) | S n => double_modn1_p_aux n (double_modn1_p n) end. @@ -302,8 +302,8 @@ Section GENDIVN1. Fixpoint high (n:nat) : word w n -> w := match n return word w n -> w with - | O => fun a => a - | S n => + | O => fun a => a + | S n => fun (a:zn2z (word w n)) => match a with | W0 => w_0 @@ -314,20 +314,20 @@ Section GENDIVN1. Lemma spec_double_digits:forall n, Zpos w_digits <= Zpos (double_digits w_digits n). Proof. induction n;simpl;auto with zarith. - change (Zpos (xO (double_digits w_digits n))) with + change (Zpos (xO (double_digits w_digits n))) with (2*Zpos (double_digits w_digits n)). assert (0 < Zpos w_digits);auto with zarith. exact (refl_equal Lt). Qed. - Lemma spec_high : forall n (x:word w n), + Lemma spec_high : forall n (x:word w n), [|high n x|] = [!n|x!] / 2^(Zpos (double_digits w_digits n) - Zpos w_digits). Proof. induction n;intros. unfold high,double_digits,double_to_Z. replace (Zpos w_digits - Zpos w_digits) with 0;try ring. simpl. rewrite <- (Zdiv_unique [|x|] 1 [|x|] 0);auto with zarith. - assert (U2 := spec_double_digits n). + assert (U2 := spec_double_digits n). assert (U3 : 0 < Zpos w_digits). exact (refl_equal Lt). destruct x;unfold high;fold high. unfold double_to_Z,zn2z_to_Z;rewrite spec_0. @@ -337,31 +337,31 @@ Section GENDIVN1. simpl [!S n|WW w0 w1!]. unfold double_wB,base;rewrite Zdiv_shift_r;auto with zarith. replace (2 ^ (Zpos (double_digits w_digits (S n)) - Zpos w_digits)) with - (2^(Zpos (double_digits w_digits n) - Zpos w_digits) * + (2^(Zpos (double_digits w_digits n) - Zpos w_digits) * 2^Zpos (double_digits w_digits n)). rewrite Zdiv_mult_cancel_r;auto with zarith. rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (double_digits w_digits n) - Zpos w_digits + + replace (Zpos (double_digits w_digits n) - Zpos w_digits + Zpos (double_digits w_digits n)) with (Zpos (double_digits w_digits (S n)) - Zpos w_digits);trivial. - change (Zpos (double_digits w_digits (S n))) with + change (Zpos (double_digits w_digits (S n))) with (2*Zpos (double_digits w_digits n));ring. - change (Zpos (double_digits w_digits (S n))) with + change (Zpos (double_digits w_digits (S n))) with (2*Zpos (double_digits w_digits n)); auto with zarith. Qed. - - Definition double_divn1 (n:nat) (a:word w n) (b:w) := + + Definition double_divn1 (n:nat) (a:word w n) (b:w) := let p := w_head0 b in match w_compare p w_0 with | Gt => let b2p := w_add_mul_div p b w_0 in let ha := high n a in let k := w_sub w_zdigits p in - let lsr_n := w_add_mul_div k w_0 in + let lsr_n := w_add_mul_div k w_0 in let r0 := w_add_mul_div p w_0 ha in let (q,r) := double_divn1_p b2p p n r0 a (double_0 w_0 n) in (q, lsr_n r) - | _ => double_divn1_0 b n w_0 a + | _ => double_divn1_0 b n w_0 a end. Lemma spec_double_divn1 : forall n a b, @@ -392,21 +392,21 @@ Section GENDIVN1. apply Zmult_le_compat;auto with zarith. assert (wB <= 2^[|w_head0 b|]). unfold base;apply Zpower_le_monotone;auto with zarith. omega. - assert ([|w_add_mul_div (w_head0 b) b w_0|] = + assert ([|w_add_mul_div (w_head0 b) b w_0|] = 2 ^ [|w_head0 b|] * [|b|]). rewrite (spec_add_mul_div b w_0); auto with zarith. rewrite spec_0;rewrite Zdiv_0_l; try omega. rewrite Zplus_0_r; rewrite Zmult_comm. rewrite Zmod_small; auto with zarith. assert (H5 := spec_to_Z (high n a)). - assert + assert ([|w_add_mul_div (w_head0 b) w_0 (high n a)|] <[|w_add_mul_div (w_head0 b) b w_0|]). rewrite H4. rewrite spec_add_mul_div;auto with zarith. rewrite spec_0;rewrite Zmult_0_l;rewrite Zplus_0_l. assert (([|high n a|]/2^(Zpos w_digits - [|w_head0 b|])) < wB). - apply Zdiv_lt_upper_bound;auto with zarith. + apply Zdiv_lt_upper_bound;auto with zarith. apply Zlt_le_trans with wB;auto with zarith. pattern wB at 1;replace wB with (wB*1);try ring. apply Zmult_le_compat;auto with zarith. @@ -420,8 +420,8 @@ Section GENDIVN1. apply Zmult_le_compat;auto with zarith. pattern 2 at 1;rewrite <- Zpower_1_r. apply Zpower_le_monotone;split;auto with zarith. - rewrite <- H4 in H0. - assert (Hb3: [|w_head0 b|] <= Zpos w_digits); auto with zarith. + rewrite <- H4 in H0. + assert (Hb3: [|w_head0 b|] <= Zpos w_digits); auto with zarith. assert (H7:= spec_double_divn1_p H0 Hb3 n a (double_0 w_0 n) H6). destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n (w_add_mul_div (w_head0 b) w_0 (high n a)) a @@ -436,7 +436,7 @@ Section GENDIVN1. rewrite Zmod_small;auto with zarith. rewrite spec_high. rewrite Zdiv_Zdiv;auto with zarith. rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (double_digits w_digits n) - Zpos w_digits + + replace (Zpos (double_digits w_digits n) - Zpos w_digits + (Zpos w_digits - [|w_head0 b|])) with (Zpos (double_digits w_digits n) - [|w_head0 b|]);trivial;ring. assert (H8 := Zpower_gt_0 2 (Zpos w_digits - [|w_head0 b|]));auto with zarith. @@ -448,11 +448,11 @@ Section GENDIVN1. rewrite H8 in H7;unfold double_wB,base in H7. rewrite <- shift_unshift_mod in H7;auto with zarith. rewrite H4 in H7. - assert ([|w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r|] + assert ([|w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r|] = [|r|]/2^[|w_head0 b|]). rewrite spec_add_mul_div. rewrite spec_0;rewrite Zmult_0_l;rewrite Zplus_0_l. - replace (Zpos w_digits - [|w_sub w_zdigits (w_head0 b)|]) + replace (Zpos w_digits - [|w_sub w_zdigits (w_head0 b)|]) with ([|w_head0 b|]). rewrite Zmod_small;auto with zarith. assert (H9 := spec_to_Z r). @@ -474,11 +474,11 @@ Section GENDIVN1. split. rewrite <- (Z_div_mult [!n|a!] (2^[|w_head0 b|]));auto with zarith. rewrite H71;rewrite H9. - replace ([!n|q!] * (2 ^ [|w_head0 b|] * [|b|])) + replace ([!n|q!] * (2 ^ [|w_head0 b|] * [|b|])) with ([!n|q!] *[|b|] * 2^[|w_head0 b|]); try (ring;fail). rewrite Z_div_plus_l;auto with zarith. - assert (H10 := spec_to_Z + assert (H10 := spec_to_Z (w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r));split; auto with zarith. rewrite H9. @@ -487,19 +487,19 @@ Section GENDIVN1. exact (spec_double_to_Z w_digits w_to_Z spec_to_Z n a). Qed. - - Definition double_modn1 (n:nat) (a:word w n) (b:w) := + + Definition double_modn1 (n:nat) (a:word w n) (b:w) := let p := w_head0 b in match w_compare p w_0 with | Gt => let b2p := w_add_mul_div p b w_0 in let ha := high n a in let k := w_sub w_zdigits p in - let lsr_n := w_add_mul_div k w_0 in + let lsr_n := w_add_mul_div k w_0 in let r0 := w_add_mul_div p w_0 ha in let r := double_modn1_p b2p p n r0 a (double_0 w_0 n) in lsr_n r - | _ => double_modn1_0 b n w_0 a + | _ => double_modn1_0 b n w_0 a end. Lemma spec_double_modn1_aux : forall n a b, @@ -525,4 +525,4 @@ Section GENDIVN1. destruct H1 as (h1,h2);rewrite h1;ring. Qed. -End GENDIVN1. +End GENDIVN1. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v index d9c2340936..28dff1a29a 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v @@ -61,13 +61,13 @@ Section DoubleLift. (* 0 < p < ww_digits *) - Definition ww_add_mul_div p x y := + Definition ww_add_mul_div p x y := let zdigits := w_0W w_zdigits in match x, y with | W0, W0 => W0 | W0, WW yh yl => match ww_compare p zdigits with - | Eq => w_0W yh + | Eq => w_0W yh | Lt => w_0W (w_add_mul_div (low p) w_0 yh) | Gt => let n := low (ww_sub p zdigits) in @@ -75,15 +75,15 @@ Section DoubleLift. end | WW xh xl, W0 => match ww_compare p zdigits with - | Eq => w_W0 xl + | Eq => w_W0 xl | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl w_0) | Gt => let n := low (ww_sub p zdigits) in - w_W0 (w_add_mul_div n xl w_0) + w_W0 (w_add_mul_div n xl w_0) end | WW xh xl, WW yh yl => match ww_compare p zdigits with - | Eq => w_WW xl yh + | Eq => w_WW xl yh | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl yh) | Gt => let n := low (ww_sub p zdigits) in @@ -93,7 +93,7 @@ Section DoubleLift. Section DoubleProof. Variable w_to_Z : w -> Z. - + Notation wB := (base w_digits). Notation wwB := (base (ww_digits w_digits)). Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). @@ -122,21 +122,21 @@ Section DoubleLift. Variable spec_w_head0 : forall x, 0 < [|x|] -> wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB. Variable spec_w_tail00 : forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits. - Variable spec_w_tail0 : forall x, 0 < [|x|] -> + Variable spec_w_tail0 : forall x, 0 < [|x|] -> exists y, 0 <= y /\ [|x|] = (2* y + 1) * (2 ^ [|w_tail0 x|]). Variable spec_w_add_mul_div : forall x y p, [|p|] <= Zpos w_digits -> [| w_add_mul_div p x y |] = ([|x|] * (2 ^ [|p|]) + [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_w_add: forall x y, + Variable spec_w_add: forall x y, [[w_add x y]] = [|x|] + [|y|]. - Variable spec_ww_sub: forall x y, + Variable spec_ww_sub: forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. Variable spec_low: forall x, [| low x|] = [[x]] mod wB. - + Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos ww_Digits. Hint Resolve div_le_0 div_lt w_to_Z_wwB: lift. @@ -168,7 +168,7 @@ Section DoubleLift. rewrite spec_w_0; auto with zarith. rewrite spec_w_0; auto with zarith. Qed. - + Lemma spec_ww_head0 : forall x, 0 < [[x]] -> wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. Proof. @@ -179,7 +179,7 @@ Section DoubleLift. assert (H0 := spec_compare w_0 xh);rewrite spec_w_0 in H0. destruct (w_compare w_0 xh). rewrite <- H0. simpl Zplus. rewrite <- H0 in H;simpl in H. - case (spec_to_Z w_zdigits); + case (spec_to_Z w_zdigits); case (spec_to_Z (w_head0 xl)); intros HH1 HH2 HH3 HH4. rewrite spec_w_add. rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. @@ -209,7 +209,7 @@ Section DoubleLift. rewrite <- Zmult_assoc; apply Zmult_lt_compat_l; zarith. rewrite <- (Zplus_0_r (2^(Zpos w_digits - p)*wB));apply beta_lex_inv;zarith. apply Zmult_lt_reg_r with (2 ^ p); zarith. - rewrite <- Zpower_exp;zarith. + rewrite <- Zpower_exp;zarith. rewrite Zmult_comm;ring_simplify (Zpos w_digits - p + p);fold wB;zarith. assert (H1 := spec_to_Z xh);zarith. Qed. @@ -293,8 +293,8 @@ Section DoubleLift. Qed. Hint Rewrite Zdiv_0_l Zmult_0_l Zplus_0_l Zmult_0_r Zplus_0_r - spec_w_W0 spec_w_0W spec_w_WW spec_w_0 - (wB_div w_digits w_to_Z spec_to_Z) + spec_w_W0 spec_w_0W spec_w_WW spec_w_0 + (wB_div w_digits w_to_Z spec_to_Z) (wB_div_plus w_digits w_to_Z spec_to_Z) : w_rewrite. Ltac w_rewrite := autorewrite with w_rewrite;trivial. @@ -303,12 +303,12 @@ Section DoubleLift. [[p]] <= Zpos (xO w_digits) -> [[match ww_compare p zdigits with | Eq => w_WW xl yh - | Lt => w_WW (w_add_mul_div (low p) xh xl) + | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl yh) | Gt => let n := low (ww_sub p zdigits) in w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl) - end]] = + end]] = ([[WW xh xl]] * (2^[[p]]) + [[WW yh yl]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. Proof. @@ -317,7 +317,7 @@ Section DoubleLift. case (spec_to_w_Z p); intros Hv1 Hv2. replace (Zpos (xO w_digits)) with (Zpos w_digits + Zpos w_digits). 2 : rewrite Zpos_xO;ring. - replace (Zpos w_digits + Zpos w_digits - [[p]]) with + replace (Zpos w_digits + Zpos w_digits - [[p]]) with (Zpos w_digits + (Zpos w_digits - [[p]])). 2:ring. intros Hp; assert (Hxh := spec_to_Z xh);assert (Hxl:=spec_to_Z xl); assert (Hx := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)); @@ -330,7 +330,7 @@ Section DoubleLift. fold wB. rewrite Zmult_plus_distr_l;rewrite <- Zmult_assoc;rewrite <- Zplus_assoc. rewrite <- Zpower_2. - rewrite <- wwB_wBwB;apply Zmod_unique with [|xh|]. + rewrite <- wwB_wBwB;apply Zmod_unique with [|xh|]. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xl yh)). ring. simpl ww_to_Z; w_rewrite;zarith. assert (HH0: [|low p|] = [[p]]). @@ -353,7 +353,7 @@ Section DoubleLift. rewrite Zmult_plus_distr_l. pattern ([|xl|] * 2 ^ [[p]]) at 2; rewrite shift_unshift_mod with (n:= Zpos w_digits);fold wB;zarith. - replace ([|xh|] * wB * 2^[[p]]) with ([|xh|] * 2^[[p]] * wB). 2:ring. + replace ([|xh|] * wB * 2^[[p]]) with ([|xh|] * 2^[[p]] * wB). 2:ring. rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. rewrite <- Zplus_assoc. unfold base at 5;rewrite <- Zmod_shift_r;zarith. unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); @@ -387,8 +387,8 @@ Section DoubleLift. lazy zeta; simpl ww_to_Z; w_rewrite;zarith. repeat rewrite spec_w_add_mul_div;zarith. rewrite HH0. - pattern wB at 5;replace wB with - (2^(([[p]] - Zpos w_digits) + pattern wB at 5;replace wB with + (2^(([[p]] - Zpos w_digits) + (Zpos w_digits - ([[p]] - Zpos w_digits)))). rewrite Zpower_exp;zarith. rewrite Zmult_assoc. rewrite Z_div_plus_l;zarith. @@ -401,28 +401,28 @@ Section DoubleLift. repeat rewrite <- Zplus_assoc. unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); fold wB;fold wwB;zarith. - unfold base;rewrite Zmod_shift_r with (a:= Zpos w_digits) + unfold base;rewrite Zmod_shift_r with (a:= Zpos w_digits) (b:= Zpos w_digits);fold wB;fold wwB;zarith. rewrite wwB_wBwB; rewrite Zpower_2; rewrite Zmult_mod_distr_r;zarith. rewrite Zmult_plus_distr_l. - replace ([|xh|] * wB * 2 ^ u) with + replace ([|xh|] * wB * 2 ^ u) with ([|xh|]*2^u*wB). 2:ring. - repeat rewrite <- Zplus_assoc. + repeat rewrite <- Zplus_assoc. rewrite (Zplus_comm ([|xh|] * 2 ^ u * wB)). rewrite Z_mod_plus;zarith. rewrite Z_mod_mult;zarith. unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. - unfold u; split;zarith. + unfold u; split;zarith. split;zarith. unfold u; apply Zdiv_lt_upper_bound;zarith. rewrite <- Zpower_exp;zarith. - fold u. - ring_simplify (u + (Zpos w_digits - u)); fold + fold u. + ring_simplify (u + (Zpos w_digits - u)); fold wB;zarith. unfold ww_digits;rewrite Zpos_xO;zarith. unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. unfold u; split;zarith. unfold u; split;zarith. apply Zdiv_lt_upper_bound;zarith. rewrite <- Zpower_exp;zarith. - fold u. + fold u. ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith. unfold u;zarith. unfold u;zarith. @@ -446,7 +446,7 @@ Section DoubleLift. clear H1;w_rewrite);simpl ww_add_mul_div. replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. intros Heq;rewrite <- Heq;clear Heq; auto. - generalize (spec_ww_compare p (w_0W w_zdigits)); + generalize (spec_ww_compare p (w_0W w_zdigits)); case ww_compare; intros H1; w_rewrite. rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. generalize H1; w_rewrite; rewrite spec_zdigits; clear H1; intros H1. @@ -459,7 +459,7 @@ Section DoubleLift. rewrite HH0; auto with zarith. replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. intros Heq;rewrite <- Heq;clear Heq. - generalize (spec_ww_compare p (w_0W w_zdigits)); + generalize (spec_ww_compare p (w_0W w_zdigits)); case ww_compare; intros H1; w_rewrite. rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. rewrite Zpos_xO in H;zarith. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v index cc32214017..b215f6a868 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v @@ -45,7 +45,7 @@ Section DoubleMul. (* (xh*B+xl) (yh*B + yl) xh*yh = hh = |hhh|hhl|B2 xh*yl +xl*yh = cc = |cch|ccl|B - xl*yl = ll = |llh|lll + xl*yl = ll = |llh|lll *) Definition double_mul_c (cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w) x y := @@ -56,7 +56,7 @@ Section DoubleMul. let hh := w_mul_c xh yh in let ll := w_mul_c xl yl in let (wc,cc) := cross xh xl yh yl hh ll in - match cc with + match cc with | W0 => WW (ww_add hh (w_W0 wc)) ll | WW cch ccl => match ww_add_c (w_W0 ccl) ll with @@ -67,8 +67,8 @@ Section DoubleMul. end. Definition ww_mul_c := - double_mul_c - (fun xh xl yh yl hh ll=> + double_mul_c + (fun xh xl yh yl hh ll=> match ww_add_c (w_mul_c xh yl) (w_mul_c xl yh) with | C0 cc => (w_0, cc) | C1 cc => (w_1, cc) @@ -77,11 +77,11 @@ Section DoubleMul. Definition w_2 := w_add w_1 w_1. Definition kara_prod xh xl yh yl hh ll := - match ww_add_c hh ll with + match ww_add_c hh ll with C0 m => match w_compare xl xh with Eq => (w_0, m) - | Lt => + | Lt => match w_compare yl yh with Eq => (w_0, m) | Lt => (w_0, ww_sub m (w_mul_c (w_sub xh xl) (w_sub yh yl))) @@ -89,7 +89,7 @@ Section DoubleMul. C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1) end end - | Gt => + | Gt => match w_compare yl yh with Eq => (w_0, m) | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with @@ -101,17 +101,17 @@ Section DoubleMul. | C1 m => match w_compare xl xh with Eq => (w_1, m) - | Lt => + | Lt => match w_compare yl yh with Eq => (w_1, m) | Lt => match ww_sub_c m (w_mul_c (w_sub xh xl) (w_sub yh yl)) with C0 m1 => (w_1, m1) | C1 m1 => (w_0, m1) - end + end | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1) end end - | Gt => + | Gt => match w_compare yl yh with Eq => (w_1, m) | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with @@ -129,8 +129,8 @@ Section DoubleMul. Definition ww_mul x y := match x, y with | W0, _ => W0 - | _, W0 => W0 - | WW xh xl, WW yh yl => + | _, W0 => W0 + | WW xh xl, WW yh yl => let ccl := w_add (w_mul xh yl) (w_mul xl yh) in ww_add (w_W0 ccl) (w_mul_c xl yl) end. @@ -161,9 +161,9 @@ Section DoubleMul. Variable w_mul_add : w -> w -> w -> w * w. Fixpoint double_mul_add_n1 (n:nat) : word w n -> w -> w -> w * word w n := - match n return word w n -> w -> w -> w * word w n with - | O => w_mul_add - | S n1 => + match n return word w n -> w -> w -> w * word w n with + | O => w_mul_add + | S n1 => let mul_add := double_mul_add_n1 n1 in fun x y r => match x with @@ -183,11 +183,11 @@ Section DoubleMul. Variable wn_0W : wn -> zn2z wn. Variable wn_WW : wn -> wn -> zn2z wn. Variable w_mul_add_n1 : wn -> w -> w -> w*wn. - Fixpoint double_mul_add_mn1 (m:nat) : + Fixpoint double_mul_add_mn1 (m:nat) : word wn m -> w -> w -> w*word wn m := - match m return word wn m -> w -> w -> w*word wn m with - | O => w_mul_add_n1 - | S m1 => + match m return word wn m -> w -> w -> w*word wn m with + | O => w_mul_add_n1 + | S m1 => let mul_add := double_mul_add_mn1 m1 in fun x y r => match x with @@ -207,11 +207,11 @@ Section DoubleMul. | WW h l => match w_add_c l r with | C0 lr => (h,lr) - | C1 lr => (w_succ h, lr) + | C1 lr => (w_succ h, lr) end end. - + (*Section DoubleProof. *) Variable w_digits : positive. Variable w_to_Z : w -> Z. @@ -225,11 +225,11 @@ Section DoubleMul. (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[+[ c ]]" := + (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). Notation "[|| x ||]" := @@ -269,8 +269,8 @@ Section DoubleMul. forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB. Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - - + + Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB. Proof. intros x;apply spec_ww_to_Z;auto. Qed. @@ -281,21 +281,21 @@ Section DoubleMul. Ltac zarith := auto with zarith mult. Lemma wBwB_lex: forall a b c d, - a * wB^2 + [[b]] <= c * wB^2 + [[d]] -> + a * wB^2 + [[b]] <= c * wB^2 + [[d]] -> a <= c. - Proof. + Proof. intros a b c d H; apply beta_lex with [[b]] [[d]] (wB^2);zarith. Qed. - Lemma wBwB_lex_inv: forall a b c d, - a < c -> - a * wB^2 + [[b]] < c * wB^2 + [[d]]. + Lemma wBwB_lex_inv: forall a b c d, + a < c -> + a * wB^2 + [[b]] < c * wB^2 + [[d]]. Proof. intros a b c d H; apply beta_lex_inv; zarith. Qed. Lemma sum_mul_carry : forall xh xl yh yl wc cc, - [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] -> + [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] -> 0 <= [|wc|] <= 1. Proof. intros. @@ -303,14 +303,14 @@ Section DoubleMul. apply wB_pos. Qed. - Theorem mult_add_ineq: forall xH yH crossH, + Theorem mult_add_ineq: forall xH yH crossH, 0 <= [|xH|] * [|yH|] + [|crossH|] < wwB. Proof. intros;rewrite wwB_wBwB;apply mult_add_ineq;zarith. Qed. - + Hint Resolve mult_add_ineq : mult. - + Lemma spec_mul_aux : forall xh xl yh yl wc (cc:zn2z w) hh ll, [[hh]] = [|xh|] * [|yh|] -> [[ll]] = [|xl|] * [|yl|] -> @@ -325,9 +325,9 @@ Section DoubleMul. end||] = ([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|]). Proof. intros;assert (U1 := wB_pos w_digits). - replace (([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|])) with + replace (([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|])) with ([|xh|]*[|yh|]*wB^2 + ([|xh|]*[|yl|] + [|xl|]*[|yh|])*wB + [|xl|]*[|yl|]). - 2:ring. rewrite <- H1;rewrite <- H;rewrite <- H0. + 2:ring. rewrite <- H1;rewrite <- H;rewrite <- H0. assert (H2 := sum_mul_carry _ _ _ _ _ _ H1). destruct cc as [ | cch ccl]; simpl zn2z_to_Z; simpl ww_to_Z. rewrite spec_ww_add;rewrite spec_w_W0;rewrite Zmod_small; @@ -346,7 +346,7 @@ Section DoubleMul. rewrite <- Zmult_plus_distr_l. assert (((2 * wB - 4) + 2)*wB <= ([|wc|] * wB + [|cch|])*wB). apply Zmult_le_compat;zarith. - rewrite Zmult_plus_distr_l in H3. + rewrite Zmult_plus_distr_l in H3. intros. assert (U2 := spec_to_Z ccl);omega. generalize (spec_ww_add_c (w_W0 ccl) ll);destruct (ww_add_c (w_W0 ccl) ll) as [l|l];unfold interp_carry;rewrite spec_w_W0;try rewrite Zmult_1_l; @@ -363,8 +363,8 @@ Section DoubleMul. (forall xh xl yh yl hh ll, [[hh]] = [|xh|]*[|yh|] -> [[ll]] = [|xl|]*[|yl|] -> - let (wc,cc) := cross xh xl yh yl hh ll in - [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]) -> + let (wc,cc) := cross xh xl yh yl hh ll in + [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]) -> forall x y, [||double_mul_c cross x y||] = [[x]] * [[y]]. Proof. intros cross Hcross x y;destruct x as [ |xh xl];simpl;trivial. @@ -376,7 +376,7 @@ Section DoubleMul. rewrite <- wwB_wBwB;trivial. Qed. - Lemma spec_ww_mul_c : forall x y, [||ww_mul_c x y||] = [[x]] * [[y]]. + Lemma spec_ww_mul_c : forall x y, [||ww_mul_c x y||] = [[x]] * [[y]]. Proof. intros x y;unfold ww_mul_c;apply spec_double_mul_c. intros xh xl yh yl hh ll H1 H2. @@ -402,9 +402,9 @@ Section DoubleMul. let (wc,cc) := kara_prod xh xl yh yl hh ll in [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]. Proof. - intros xh xl yh yl hh ll H H0; rewrite <- kara_prod_aux; + intros xh xl yh yl hh ll H H0; rewrite <- kara_prod_aux; rewrite <- H; rewrite <- H0; unfold kara_prod. - assert (Hxh := (spec_to_Z xh)); assert (Hxl := (spec_to_Z xl)); + assert (Hxh := (spec_to_Z xh)); assert (Hxl := (spec_to_Z xl)); assert (Hyh := (spec_to_Z yh)); assert (Hyl := (spec_to_Z yl)). generalize (spec_ww_add_c hh ll); case (ww_add_c hh ll); intros z Hz; rewrite <- Hz; unfold interp_carry; assert (Hz1 := (spec_ww_to_Z z)). @@ -412,7 +412,7 @@ Section DoubleMul. try rewrite Hxlh; try rewrite spec_w_0; try (ring; fail). generalize (spec_w_compare yl yh); case (w_compare yl yh); intros Hylh. rewrite Hylh; rewrite spec_w_0; try (ring; fail). - rewrite spec_w_0; try (ring; fail). + rewrite spec_w_0; try (ring; fail). repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). split; auto with zarith. @@ -508,8 +508,8 @@ Section DoubleMul. repeat rewrite Zmod_small; auto with zarith; try (ring; fail). Qed. - Lemma sub_carry : forall xh xl yh yl z, - 0 <= z -> + Lemma sub_carry : forall xh xl yh yl z, + 0 <= z -> [|xh|]*[|yl|] + [|xl|]*[|yh|] = wwB + z -> z < wwB. Proof. @@ -519,7 +519,7 @@ Section DoubleMul. generalize (Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)). rewrite <- wwB_wBwB;intros H1 H2. assert (H3 := wB_pos w_digits). - assert (2*wB <= wwB). + assert (2*wB <= wwB). rewrite wwB_wBwB; rewrite Zpower_2; apply Zmult_le_compat;zarith. omega. Qed. @@ -528,7 +528,7 @@ Section DoubleMul. let H:= fresh "H" in assert (H:= spec_ww_to_Z x). - Ltac Zmult_lt_b x y := + Ltac Zmult_lt_b x y := let H := fresh "H" in assert (H := Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)). @@ -582,7 +582,7 @@ Section DoubleMul. Variable w_mul_add : w -> w -> w -> w * w. Variable spec_w_mul_add : forall x y r, let (h,l):= w_mul_add x y r in - [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. + [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. Lemma spec_double_mul_add_n1 : forall n x y r, let (h,l) := double_mul_add_n1 w_mul_add n x y r in @@ -590,7 +590,7 @@ Section DoubleMul. Proof. induction n;intros x y r;trivial. exact (spec_w_mul_add x y r). - unfold double_mul_add_n1;destruct x as[ |xh xl]; + unfold double_mul_add_n1;destruct x as[ |xh xl]; fold(double_mul_add_n1 w_mul_add). rewrite spec_w_0;rewrite spec_extend;simpl;trivial. assert(H:=IHn xl y r);destruct (double_mul_add_n1 w_mul_add n xl y r)as(rl,l). @@ -599,13 +599,13 @@ Section DoubleMul. rewrite Zmult_plus_distr_l;rewrite <- Zplus_assoc;rewrite <- H. rewrite Zmult_assoc;rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. rewrite U;ring. - Qed. - + Qed. + End DoubleMulAddn1Proof. - Lemma spec_w_mul_add : forall x y r, + Lemma spec_w_mul_add : forall x y r, let (h,l):= w_mul_add x y r in - [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. + [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. Proof. intros x y r;unfold w_mul_add;assert (H:=spec_w_mul_c x y); destruct (w_mul_c x y) as [ |h l];simpl;rewrite <- H. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v index c72abed619..ac2232cc0d 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v @@ -52,7 +52,7 @@ Section DoubleSqrt. Let wwBm1 := ww_Bm1 w_Bm1. - Definition ww_is_even x := + Definition ww_is_even x := match x with | W0 => true | WW xh xl => w_is_even xl @@ -62,7 +62,7 @@ Section DoubleSqrt. match w_compare x z with | Eq => match w_compare y z with - Eq => (C1 w_1, w_0) + Eq => (C1 w_1, w_0) | Gt => (C1 w_1, w_sub y z) | Lt => (C1 w_0, y) end @@ -120,7 +120,7 @@ Section DoubleSqrt. let ( q, r) := w_sqrt2 x1 x2 in let (q1, r1) := w_div2s r y1 q in match q1 with - C0 q1 => + C0 q1 => let q2 := w_square_c q1 in let a := WW q q1 in match r1 with @@ -132,9 +132,9 @@ Section DoubleSqrt. | C0 r2 => match ww_sub_c (WW r2 y2) q2 with C0 r3 => (a, C0 r3) - | C1 r3 => + | C1 r3 => let a2 := ww_add_mul_div (w_0W w_1) a W0 in - match ww_pred_c a2 with + match ww_pred_c a2 with C0 a3 => (ww_pred a, ww_add_c a3 r3) | C1 a3 => @@ -166,20 +166,20 @@ Section DoubleSqrt. | Gt => match ww_add_mul_div p x W0 with W0 => W0 - | WW x1 x2 => + | WW x1 x2 => let (r, _) := w_sqrt2 x1 x2 in - WW w_0 (w_add_mul_div - (w_sub w_zdigits + WW w_0 (w_add_mul_div + (w_sub w_zdigits (low (ww_add_mul_div (ww_pred ww_zdigits) W0 p))) w_0 r) end - | _ => + | _ => match x with W0 => W0 | WW x1 x2 => WW w_0 (fst (w_sqrt2 x1 x2)) end end. - + Variable w_to_Z : w -> Z. @@ -192,11 +192,11 @@ Section DoubleSqrt. (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[+[ c ]]" := + (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). Notation "[|| x ||]" := @@ -274,7 +274,7 @@ Section DoubleSqrt. Lemma spec_ww_is_even : forall x, if ww_is_even x then [[x]] mod 2 = 0 else [[x]] mod 2 = 1. -clear spec_more_than_1_digit. +clear spec_more_than_1_digit. intros x; case x; simpl ww_is_even. simpl. rewrite Zmod_small; auto with zarith. @@ -377,8 +377,8 @@ intros x; case x; simpl ww_is_even. end. rewrite Zpower_1_r; rewrite Zmod_small; auto with zarith. destruct (spec_to_Z w1) as [H1 H2];auto with zarith. - split; auto with zarith. - apply Zdiv_lt_upper_bound; auto with zarith. + split; auto with zarith. + apply Zdiv_lt_upper_bound; auto with zarith. rewrite Hp; ring. Qed. @@ -400,7 +400,7 @@ intros x; case x; simpl ww_is_even. rewrite Zmax_right; auto with zarith. rewrite Zpower_1_r; rewrite Zmod_small; auto with zarith. destruct (spec_to_Z w1) as [H1 H2];auto with zarith. - split; auto with zarith. + split; auto with zarith. unfold base. match goal with |- _ < _ ^ ?X => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; @@ -432,7 +432,7 @@ intros x; case x; simpl ww_is_even. intros w1. rewrite spec_ww_add_mul_div; auto with zarith. autorewrite with w_rewrite rm10. - rewrite spec_w_0W; rewrite spec_w_1. + rewrite spec_w_0W; rewrite spec_w_1. rewrite Zpower_1_r; auto with zarith. rewrite Zmult_comm; auto. rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. @@ -456,7 +456,7 @@ intros x; case x; simpl ww_is_even. match goal with |- 0 <= ?X - 1 => assert (0 < X); auto with zarith end. - apply Zpower_gt_0; auto with zarith. + apply Zpower_gt_0; auto with zarith. match goal with |- 0 <= ?X - 1 => assert (0 < X); auto with zarith; red; reflexivity end. @@ -540,7 +540,7 @@ intros x; case x; simpl ww_is_even. rewrite add_mult_div_2_plus_1; unfold base. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; + rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; try rewrite Zpower_1_r; auto with zarith end. rewrite Zpos_minus; auto with zarith. @@ -557,7 +557,7 @@ intros x; case x; simpl ww_is_even. unfold base. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; + rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; try rewrite Zpower_1_r; auto with zarith end. rewrite Zpos_minus; auto with zarith. @@ -590,7 +590,7 @@ intros x; case x; simpl ww_is_even. rewrite H1; unfold base. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; + rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; try rewrite Zpower_1_r; auto with zarith end. rewrite Zpos_minus; auto with zarith. @@ -609,7 +609,7 @@ intros x; case x; simpl ww_is_even. rewrite H1; unfold base. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; + rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; try rewrite Zpower_1_r; auto with zarith end. rewrite Zpos_minus; auto with zarith. @@ -680,7 +680,7 @@ intros x; case x; simpl ww_is_even. rewrite Zsquare_mult; replace (p * p) with ((- p) * (- p)); try ring. apply Zmult_le_0_compat; auto with zarith. Qed. - + Lemma spec_split: forall x, [|fst (split x)|] * wB + [|snd (split x)|] = [[x]]. intros x; case x; simpl; autorewrite with w_rewrite; @@ -749,7 +749,7 @@ intros x; case x; simpl ww_is_even. match goal with |- ?X <= ?Y => replace Y with (2 * (wB/ 2 - 1)); auto with zarith end. - pattern wB at 2; rewrite <- wB_div_2; auto with zarith. + pattern wB at 2; rewrite <- wB_div_2; auto with zarith. match type of H1 with ?X = _ => assert (U5: X < wB / 4 * wB) end. @@ -762,9 +762,9 @@ intros x; case x; simpl ww_is_even. destruct (spec_to_Z w3);auto with zarith. generalize (@spec_w_div2s c w0 w4 U1 H2). case (w_div2s c w0 w4). - intros c0; case c0; intros w5; + intros c0; case c0; intros w5; repeat (rewrite C0_id || rewrite C1_plus_wB). - intros c1; case c1; intros w6; + intros c1; case c1; intros w6; repeat (rewrite C0_id || rewrite C1_plus_wB). intros (H3, H4). match goal with |- context [ww_sub_c ?y ?z] => @@ -1036,7 +1036,7 @@ intros x; case x; simpl ww_is_even. end. apply Zle_not_lt; rewrite <- H3; auto with zarith. rewrite Zmult_plus_distr_l. - apply Zlt_le_trans with ((2 * [|w4|]) * wB + 0); + apply Zlt_le_trans with ((2 * [|w4|]) * wB + 0); auto with zarith. apply beta_lex_inv; auto with zarith. destruct (spec_to_Z w0);auto with zarith. @@ -1117,9 +1117,9 @@ intros x; case x; simpl ww_is_even. auto with zarith. simpl ww_to_Z. assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z x);auto with zarith. - Qed. - - Lemma wwB_4_2: 2 * (wwB / 4) = wwB/ 2. + Qed. + + Lemma wwB_4_2: 2 * (wwB / 4) = wwB/ 2. pattern wwB at 1; rewrite wwB_wBwB; rewrite Zpower_2. rewrite <- wB_div_2. match goal with |- context[(2 * ?X) * (2 * ?Z)] => @@ -1132,7 +1132,7 @@ intros x; case x; simpl ww_is_even. Lemma spec_ww_head1 - : forall x : zn2z w, + : forall x : zn2z w, (ww_is_even (ww_head1 x) = true) /\ (0 < [[x]] -> wwB / 4 <= 2 ^ [[ww_head1 x]] * [[x]] < wwB). assert (U := wB_pos w_digits). @@ -1165,7 +1165,7 @@ intros x; case x; simpl ww_is_even. rewrite Hp. rewrite Zminus_mod; auto with zarith. rewrite H2; repeat rewrite Zmod_small; auto with zarith. - intros H3; rewrite Hp. + intros H3; rewrite Hp. case (spec_ww_head0 x); auto; intros Hv3 Hv4. assert (Hu: forall u, 0 < u -> 2 * 2 ^ (u - 1) = 2 ^u). intros u Hu. @@ -1187,7 +1187,7 @@ intros x; case x; simpl ww_is_even. apply sym_equal; apply Zdiv_unique with 0; auto with zarith. rewrite Zmult_assoc; rewrite wB_div_4; auto with zarith. - rewrite wwB_wBwB; ring. + rewrite wwB_wBwB; ring. Qed. Lemma spec_ww_sqrt : forall x, @@ -1196,14 +1196,14 @@ intros x; case x; simpl ww_is_even. intro x; unfold ww_sqrt. generalize (spec_ww_is_zero x); case (ww_is_zero x). simpl ww_to_Z; simpl Zpower; unfold Zpower_pos; simpl; - auto with zarith. + auto with zarith. intros H1. generalize (spec_ww_compare (ww_head1 x) W0); case ww_compare; simpl ww_to_Z; autorewrite with rm10. generalize H1; case x. intros HH; contradict HH; simpl ww_to_Z; auto with zarith. intros w0 w1; simpl ww_to_Z; autorewrite with w_rewrite rm10. - intros H2; case (spec_ww_head1 (WW w0 w1)); intros H3 H4 H5. + intros H2; case (spec_ww_head1 (WW w0 w1)); intros H3 H4 H5. generalize (H4 H2); clear H4; rewrite H5; clear H5; autorewrite with rm10. intros (H4, H5). assert (V: wB/4 <= [|w0|]). @@ -1239,7 +1239,7 @@ intros x; case x; simpl ww_is_even. apply Zle_not_lt; unfold base. apply Zle_trans with (2 ^ [[ww_head1 x]]). apply Zpower_le_monotone; auto with zarith. - pattern (2 ^ [[ww_head1 x]]) at 1; + pattern (2 ^ [[ww_head1 x]]) at 1; rewrite <- (Zmult_1_r (2 ^ [[ww_head1 x]])). apply Zmult_le_compat_l; auto with zarith. generalize (spec_ww_add_mul_div x W0 (ww_head1 x) Hv2); @@ -1281,13 +1281,13 @@ intros x; case x; simpl ww_is_even. rewrite Zmod_small; auto with zarith. split; auto with zarith. apply Zlt_le_trans with (Zpos (xO w_digits)); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. + unfold base; apply Zpower2_le_lin; auto with zarith. assert (Hv4: [[ww_head1 x]]/2 < wB). apply Zle_lt_trans with (Zpos w_digits). apply Zmult_le_reg_r with 2; auto with zarith. repeat rewrite (fun x => Zmult_comm x 2). rewrite <- Hv0; rewrite <- Zpos_xO; auto. - unfold base; apply Zpower2_lt_lin; auto with zarith. + unfold base; apply Zpower2_lt_lin; auto with zarith. assert (Hv5: [[(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))]] = [[ww_head1 x]]/2). rewrite spec_ww_add_mul_div. @@ -1328,14 +1328,14 @@ intros x; case x; simpl ww_is_even. rewrite tmp; clear tmp. apply Zpower_le_monotone3; auto with zarith. split; auto with zarith. - pattern [|w2|] at 2; + pattern [|w2|] at 2; rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]] / 2))); auto with zarith. match goal with |- ?X <= ?X + ?Y => assert (0 <= Y); auto with zarith end. case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]] / 2))); auto with zarith. - case c; unfold interp_carry; autorewrite with rm10; + case c; unfold interp_carry; autorewrite with rm10; intros w3; assert (V3 := spec_to_Z w3);auto with zarith. apply Zmult_lt_reg_r with (2 ^ [[ww_head1 x]]); auto with zarith. rewrite H4. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v index 638bf69160..d3a08c6e00 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v @@ -39,7 +39,7 @@ Section DoubleSub. Definition ww_opp_c x := match x with | W0 => C0 W0 - | WW xh xl => + | WW xh xl => match w_opp_c xl with | C0 _ => match w_opp_c xh with @@ -53,7 +53,7 @@ Section DoubleSub. Definition ww_opp x := match x with | W0 => W0 - | WW xh xl => + | WW xh xl => match w_opp_c xl with | C0 _ => WW (w_opp xh) w_0 | C1 l => WW (w_opp_carry xh) l @@ -72,14 +72,14 @@ Section DoubleSub. | WW xh xl => match w_pred_c xl with | C0 l => C0 (w_WW xh l) - | C1 _ => - match w_pred_c xh with + | C1 _ => + match w_pred_c xh with | C0 h => C0 (WW h w_Bm1) | C1 _ => C1 ww_Bm1 end end end. - + Definition ww_pred x := match x with | W0 => ww_Bm1 @@ -89,19 +89,19 @@ Section DoubleSub. | C1 l => WW (w_pred xh) w_Bm1 end end. - + Definition ww_sub_c x y := match y, x with | W0, _ => C0 x | WW yh yl, W0 => ww_opp_c (WW yh yl) | WW yh yl, WW xh xl => match w_sub_c xl yl with - | C0 l => + | C0 l => match w_sub_c xh yh with | C0 h => C0 (w_WW h l) | C1 h => C1 (WW h l) end - | C1 l => + | C1 l => match w_sub_carry_c xh yh with | C0 h => C0 (WW h l) | C1 h => C1 (WW h l) @@ -109,7 +109,7 @@ Section DoubleSub. end end. - Definition ww_sub x y := + Definition ww_sub x y := match y, x with | W0, _ => x | WW yh yl, W0 => ww_opp (WW yh yl) @@ -127,7 +127,7 @@ Section DoubleSub. | WW yh yl, W0 => C1 (ww_opp_carry (WW yh yl)) | WW yh yl, WW xh xl => match w_sub_carry_c xl yl with - | C0 l => + | C0 l => match w_sub_c xh yh with | C0 h => C0 (w_WW h l) | C1 h => C1 (WW h l) @@ -155,7 +155,7 @@ Section DoubleSub. (*Section DoubleProof.*) Variable w_digits : positive. Variable w_to_Z : w -> Z. - + Notation wB := (base w_digits). Notation wwB := (base (ww_digits w_digits)). @@ -166,13 +166,13 @@ Section DoubleSub. (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[+[ c ]]" := + (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - + Variable spec_w_0 : [|w_0|] = 0. Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. Variable spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1. @@ -187,7 +187,7 @@ Section DoubleSub. Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|]. Variable spec_sub_carry_c : forall x y, [-|w_sub_carry_c x y|] = [|x|] - [|y|] - 1. - + Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB. Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. Variable spec_sub_carry : @@ -197,12 +197,12 @@ Section DoubleSub. Lemma spec_ww_opp_c : forall x, [-[ww_opp_c x]] = -[[x]]. Proof. destruct x as [ |xh xl];simpl. reflexivity. - rewrite Zopp_plus_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) + rewrite Zopp_plus_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H; - rewrite Zopp_mult_distr_l. + rewrite Zopp_mult_distr_l. assert ([|l|] = 0). assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh) + rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh) as [h|h];intros H1;unfold interp_carry in *;rewrite <- H1. assert ([|h|] = 0). assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. @@ -216,7 +216,7 @@ Section DoubleSub. Proof. destruct x as [ |xh xl];simpl. reflexivity. rewrite Zopp_plus_distr;rewrite Zopp_mult_distr_l. - generalize (spec_opp_c xl);destruct (w_opp_c xl) + generalize (spec_opp_c xl);destruct (w_opp_c xl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. rewrite spec_w_0;rewrite Zplus_0_r;rewrite wwB_wBwB. assert ([|l|] = 0). @@ -247,7 +247,7 @@ Section DoubleSub. assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. rewrite H0;change ([|xh|] + -1) with ([|xh|] - 1). generalize (spec_pred_c xh);destruct (w_pred_c xh) as [h|h]; - intros H1;unfold interp_carry in H1;rewrite <- H1. + intros H1;unfold interp_carry in H1;rewrite <- H1. simpl;rewrite spec_w_Bm1;ring. assert ([|h|] = wB - 1). assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. @@ -258,14 +258,14 @@ Section DoubleSub. Proof. destruct y as [ |yh yl];simpl. ring. destruct x as [ |xh xl];simpl. exact (spec_ww_opp_c (WW yh yl)). - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|])). 2:ring. generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl) as [l|l];intros H; unfold interp_carry in H;rewrite <- H. generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1;simpl ww_to_Z; @@ -275,37 +275,37 @@ Section DoubleSub. Lemma spec_ww_sub_carry_c : forall x y, [-[ww_sub_carry_c x y]] = [[x]] - [[y]] - 1. Proof. - destruct y as [ |yh yl];simpl. + destruct y as [ |yh yl];simpl. unfold Zminus;simpl;rewrite Zplus_0_r;exact (spec_ww_pred_c x). destruct x as [ |xh xl]. unfold interp_carry;rewrite spec_w_WW;simpl ww_to_Z;rewrite wwB_wBwB; repeat rewrite spec_opp_carry;ring. simpl ww_to_Z. - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|]-1)). 2:ring. - generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl) + generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H. generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1;try rewrite spec_w_WW; simpl ww_to_Z; try rewrite wwB_wBwB;ring. - Qed. - + Qed. + Lemma spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. Proof. - destruct x as [ |xh xl];simpl. + destruct x as [ |xh xl];simpl. apply Zmod_unique with (-1). apply spec_ww_to_Z;trivial. rewrite spec_ww_Bm1;ring. replace ([|xh|]*wB + [|xl|] - 1) with ([|xh|]*wB + ([|xl|] - 1)). 2:ring. generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l];intro H; unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. - rewrite Zmod_small. apply spec_w_WW. + rewrite Zmod_small. apply spec_w_WW. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh l)). - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. change ([|xh|] + -1) with ([|xh|] - 1). assert ([|l|] = wB - 1). assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega. @@ -318,7 +318,7 @@ Section DoubleSub. destruct y as [ |yh yl];simpl. ring_simplify ([[x]] - 0);rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. destruct x as [ |xh xl];simpl. exact (spec_ww_opp (WW yh yl)). - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|])). 2:ring. generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl)as[l|l];intros H; unfold interp_carry in H;rewrite <- H. @@ -338,7 +338,7 @@ Section DoubleSub. apply spec_ww_to_Z;trivial. fold (ww_opp_carry (WW yh yl)). rewrite (spec_ww_opp_carry (WW yh yl));simpl ww_to_Z;ring. - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|] - 1)). 2:ring. generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl)as[l|l]; intros H;unfold interp_carry in H;rewrite <- H;rewrite spec_w_WW. @@ -354,4 +354,4 @@ End DoubleSub. - + diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v index 73fd266e42..3bd4b8127a 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v @@ -37,10 +37,10 @@ Section Zn2Z. Variable znz : Type. - (** From a type [znz] representing a cyclic structure Z/nZ, + (** From a type [znz] representing a cyclic structure Z/nZ, we produce a representation of Z/2nZ by pairs of elements of [znz] - (plus a special case for zero). High half of the new number comes - first. + (plus a special case for zero). High half of the new number comes + first. *) Inductive zn2z := @@ -57,10 +57,10 @@ End Zn2Z. Implicit Arguments W0 [znz]. -(** From a cyclic representation [w], we iterate the [zn2z] construct - [n] times, gaining the type of binary trees of depth at most [n], - whose leafs are either W0 (if depth < n) or elements of w - (if depth = n). +(** From a cyclic representation [w], we iterate the [zn2z] construct + [n] times, gaining the type of binary trees of depth at most [n], + whose leafs are either W0 (if depth < n) or elements of w + (if depth = n). *) Fixpoint word (w:Type) (n:nat) : Type := diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v index 3835c6cde7..6e71bad82c 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v @@ -34,9 +34,9 @@ Section Basics. Lemma iszero_eq0 : forall x, iszero x = true -> x=0. Proof. destruct x; simpl; intros. - repeat - match goal with H:(if ?d then _ else _) = true |- _ => - destruct d; try discriminate + repeat + match goal with H:(if ?d then _ else _) = true |- _ => + destruct d; try discriminate end. reflexivity. Qed. @@ -46,26 +46,26 @@ Section Basics. intros x H Eq; rewrite Eq in H; simpl in *; discriminate. Qed. - Lemma sneakl_shiftr : forall x, + Lemma sneakl_shiftr : forall x, x = sneakl (firstr x) (shiftr x). Proof. destruct x; simpl; auto. Qed. - Lemma sneakr_shiftl : forall x, + Lemma sneakr_shiftl : forall x, x = sneakr (firstl x) (shiftl x). Proof. destruct x; simpl; auto. Qed. - Lemma twice_zero : forall x, + Lemma twice_zero : forall x, twice x = 0 <-> twice_plus_one x = 1. Proof. - destruct x; simpl in *; split; + destruct x; simpl in *; split; intro H; injection H; intros; subst; auto. Qed. - Lemma twice_or_twice_plus_one : forall x, + Lemma twice_or_twice_plus_one : forall x, x = twice (shiftr x) \/ x = twice_plus_one (shiftr x). Proof. intros; case_eq (firstr x); intros. @@ -79,13 +79,13 @@ Section Basics. Definition nshiftr n x := iter_nat n _ shiftr x. - Lemma nshiftr_S : + Lemma nshiftr_S : forall n x, nshiftr (S n) x = shiftr (nshiftr n x). Proof. reflexivity. Qed. - Lemma nshiftr_S_tail : + Lemma nshiftr_S_tail : forall n x, nshiftr (S n) x = nshiftr n (shiftr x). Proof. induction n; simpl; auto. @@ -103,7 +103,7 @@ Section Basics. destruct x; simpl; auto. Qed. - Lemma nshiftr_above_size : forall k x, size<=k -> + Lemma nshiftr_above_size : forall k x, size<=k -> nshiftr k x = 0. Proof. intros. @@ -117,13 +117,13 @@ Section Basics. Definition nshiftl n x := iter_nat n _ shiftl x. - Lemma nshiftl_S : + Lemma nshiftl_S : forall n x, nshiftl (S n) x = shiftl (nshiftl n x). Proof. reflexivity. Qed. - Lemma nshiftl_S_tail : + Lemma nshiftl_S_tail : forall n x, nshiftl (S n) x = nshiftl n (shiftl x). Proof. induction n; simpl; auto. @@ -141,7 +141,7 @@ Section Basics. destruct x; simpl; auto. Qed. - Lemma nshiftl_above_size : forall k x, size<=k -> + Lemma nshiftl_above_size : forall k x, size<=k -> nshiftl k x = 0. Proof. intros. @@ -151,27 +151,27 @@ Section Basics. simpl; rewrite nshiftl_S, IHn; auto. Qed. - Lemma firstr_firstl : + Lemma firstr_firstl : forall x, firstr x = firstl (nshiftl (pred size) x). Proof. destruct x; simpl; auto. Qed. - Lemma firstl_firstr : + Lemma firstl_firstr : forall x, firstl x = firstr (nshiftr (pred size) x). Proof. destruct x; simpl; auto. Qed. - + (** More advanced results about [nshiftr] *) - Lemma nshiftr_predsize_0_firstl : forall x, + Lemma nshiftr_predsize_0_firstl : forall x, nshiftr (pred size) x = 0 -> firstl x = D0. Proof. destruct x; compute; intros H; injection H; intros; subst; auto. Qed. - Lemma nshiftr_0_propagates : forall n p x, n <= p -> + Lemma nshiftr_0_propagates : forall n p x, n <= p -> nshiftr n x = 0 -> nshiftr p x = 0. Proof. intros. @@ -181,7 +181,7 @@ Section Basics. simpl; rewrite nshiftr_S; rewrite IHn0; auto. Qed. - Lemma nshiftr_0_firstl : forall n x, n < size -> + Lemma nshiftr_0_firstl : forall n x, n < size -> nshiftr n x = 0 -> firstl x = D0. Proof. intros. @@ -194,8 +194,8 @@ Section Basics. (** Not used for the moment. Are they really useful ? *) Lemma int31_ind_sneakl : forall P : int31->Prop, - P 0 -> - (forall x d, P x -> P (sneakl d x)) -> + P 0 -> + (forall x d, P x -> P (sneakl d x)) -> forall x, P x. Proof. intros. @@ -210,10 +210,10 @@ Section Basics. change x with (nshiftr (size-size) x); auto. Qed. - Lemma int31_ind_twice : forall P : int31->Prop, - P 0 -> - (forall x, P x -> P (twice x)) -> - (forall x, P x -> P (twice_plus_one x)) -> + Lemma int31_ind_twice : forall P : int31->Prop, + P 0 -> + (forall x, P x -> P (twice x)) -> + (forall x, P x -> P (twice_plus_one x)) -> forall x, P x. Proof. induction x using int31_ind_sneakl; auto. @@ -224,21 +224,21 @@ Section Basics. (** * Some generic results about [recr] *) Section Recr. - + (** [recr] satisfies the fixpoint equation used for its definition. *) Variable (A:Type)(case0:A)(caserec:digits->int31->A->A). - - Lemma recr_aux_eqn : forall n x, iszero x = false -> - recr_aux (S n) A case0 caserec x = + + Lemma recr_aux_eqn : forall n x, iszero x = false -> + recr_aux (S n) A case0 caserec x = caserec (firstr x) (shiftr x) (recr_aux n A case0 caserec (shiftr x)). Proof. intros; simpl; rewrite H; auto. Qed. - Lemma recr_aux_converges : + Lemma recr_aux_converges : forall n p x, n <= size -> n <= p -> - recr_aux n A case0 caserec (nshiftr (size - n) x) = + recr_aux n A case0 caserec (nshiftr (size - n) x) = recr_aux p A case0 caserec (nshiftr (size - n) x). Proof. induction n. @@ -255,8 +255,8 @@ Section Basics. apply IHn; auto with arith. Qed. - Lemma recr_eqn : forall x, iszero x = false -> - recr A case0 caserec x = + Lemma recr_eqn : forall x, iszero x = false -> + recr A case0 caserec x = caserec (firstr x) (shiftr x) (recr A case0 caserec (shiftr x)). Proof. intros. @@ -265,11 +265,11 @@ Section Basics. rewrite (recr_aux_converges size (S size)); auto with arith. rewrite recr_aux_eqn; auto. Qed. - - (** [recr] is usually equivalent to a variant [recrbis] + + (** [recr] is usually equivalent to a variant [recrbis] written without [iszero] check. *) - Fixpoint recrbis_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A) + Fixpoint recrbis_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A) (i:int31) : A := match n with | O => case0 @@ -277,7 +277,7 @@ Section Basics. let si := shiftr i in caserec (firstr i) si (recrbis_aux next A case0 caserec si) end. - + Definition recrbis := recrbis_aux size. Hypothesis case0_caserec : caserec D0 0 case0 = case0. @@ -291,8 +291,8 @@ Section Basics. replace (recrbis_aux n A case0 caserec 0) with case0; auto. clear H IHn; induction n; simpl; congruence. Qed. - - Lemma recrbis_equiv : forall x, + + Lemma recrbis_equiv : forall x, recrbis A case0 caserec x = recr A case0 caserec x. Proof. intros; apply recrbis_aux_equiv; auto. @@ -348,7 +348,7 @@ Section Basics. rewrite incr_eqn1; destruct x; simpl; auto. Qed. - Lemma incr_twice_plus_one_firstl : + Lemma incr_twice_plus_one_firstl : forall x, firstl x = D0 -> incr (twice_plus_one x) = twice (incr x). Proof. intros. @@ -356,9 +356,9 @@ Section Basics. f_equal; f_equal. destruct x; simpl in *; rewrite H; auto. Qed. - - (** The previous result is actually true even without the - constraint on [firstl], but this is harder to prove + + (** The previous result is actually true even without the + constraint on [firstl], but this is harder to prove (see later). *) End Incr. @@ -369,9 +369,9 @@ Section Basics. (** Variant of [phi] via [recrbis] *) - Let Phi := fun b (_:int31) => + Let Phi := fun b (_:int31) => match b with D0 => Zdouble | D1 => Zdouble_plus_one end. - + Definition phibis_aux n x := recrbis_aux n _ Z0 Phi x. Lemma phibis_aux_equiv : forall x, phibis_aux size x = phi x. @@ -382,7 +382,7 @@ Section Basics. (** Recursive equations satisfied by [phi] *) - Lemma phi_eqn1 : forall x, firstr x = D0 -> + Lemma phi_eqn1 : forall x, firstr x = D0 -> phi x = Zdouble (phi (shiftr x)). Proof. intros. @@ -392,7 +392,7 @@ Section Basics. rewrite H; auto. Qed. - Lemma phi_eqn2 : forall x, firstr x = D1 -> + Lemma phi_eqn2 : forall x, firstr x = D1 -> phi x = Zdouble_plus_one (phi (shiftr x)). Proof. intros. @@ -402,7 +402,7 @@ Section Basics. rewrite H; auto. Qed. - Lemma phi_twice_firstl : forall x, firstl x = D0 -> + Lemma phi_twice_firstl : forall x, firstl x = D0 -> phi (twice x) = Zdouble (phi x). Proof. intros. @@ -411,7 +411,7 @@ Section Basics. destruct x; simpl in *; rewrite H; auto. Qed. - Lemma phi_twice_plus_one_firstl : forall x, firstl x = D0 -> + Lemma phi_twice_plus_one_firstl : forall x, firstl x = D0 -> phi (twice_plus_one x) = Zdouble_plus_one (phi x). Proof. intros. @@ -427,23 +427,23 @@ Section Basics. Lemma phibis_aux_pos : forall n x, (0 <= phibis_aux n x)%Z. Proof. induction n. - simpl; unfold phibis_aux; simpl; auto with zarith. + simpl; unfold phibis_aux; simpl; auto with zarith. intros. - unfold phibis_aux, recrbis_aux; fold recrbis_aux; + unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux n (shiftr x)). destruct (firstr x). specialize IHn with (shiftr x); rewrite Zdouble_mult; omega. specialize IHn with (shiftr x); rewrite Zdouble_plus_one_mult; omega. Qed. - Lemma phibis_aux_bounded : - forall n x, n <= size -> + Lemma phibis_aux_bounded : + forall n x, n <= size -> (phibis_aux n (nshiftr (size-n) x) < 2 ^ (Z_of_nat n))%Z. Proof. induction n. simpl; unfold phibis_aux; simpl; auto with zarith. intros. - unfold phibis_aux, recrbis_aux; fold recrbis_aux; + unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux n (shiftr (nshiftr (size - S n) x))). assert (shiftr (nshiftr (size - S n) x) = nshiftr (size-n) x). replace (size - n)%nat with (S (size - (S n))) by omega. @@ -468,8 +468,8 @@ Section Basics. apply phibis_aux_bounded; auto. Qed. - Lemma phibis_aux_lowerbound : - forall n x, firstr (nshiftr n x) = D1 -> + Lemma phibis_aux_lowerbound : + forall n x, firstr (nshiftr n x) = D1 -> (2 ^ Z_of_nat n <= phibis_aux (S n) x)%Z. Proof. induction n. @@ -480,7 +480,7 @@ Section Basics. intros. remember (S n) as m. - unfold phibis_aux, recrbis_aux; fold recrbis_aux; + unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux m (shiftr x)). subst m. rewrite inj_S, Zpower_Zsucc; auto with zarith. @@ -488,13 +488,13 @@ Section Basics. apply IHn. rewrite <- nshiftr_S_tail; auto. destruct (firstr x). - change (Zdouble (phibis_aux (S n) (shiftr x))) with + change (Zdouble (phibis_aux (S n) (shiftr x))) with (2*(phibis_aux (S n) (shiftr x)))%Z. omega. rewrite Zdouble_plus_one_mult; omega. Qed. - Lemma phi_lowerbound : + Lemma phi_lowerbound : forall x, firstl x = D1 -> (2^(Z_of_nat (pred size)) <= phi x)%Z. Proof. intros. @@ -508,9 +508,9 @@ Section Basics. Section EqShiftL. - (** After killing [n] bits at the left, are the numbers equal ?*) + (** After killing [n] bits at the left, are the numbers equal ?*) - Definition EqShiftL n x y := + Definition EqShiftL n x y := nshiftl n x = nshiftl n y. Lemma EqShiftL_zero : forall x y, EqShiftL O x y <-> x = y. @@ -523,7 +523,7 @@ Section Basics. red; intros; rewrite 2 nshiftl_above_size; auto. Qed. - Lemma EqShiftL_le : forall k k' x y, k <= k' -> + Lemma EqShiftL_le : forall k k' x y, k <= k' -> EqShiftL k x y -> EqShiftL k' x y. Proof. unfold EqShiftL; intros. @@ -534,18 +534,18 @@ Section Basics. rewrite 2 nshiftl_S; f_equal; auto. Qed. - Lemma EqShiftL_firstr : forall k x y, k < size -> + Lemma EqShiftL_firstr : forall k x y, k < size -> EqShiftL k x y -> firstr x = firstr y. Proof. intros. rewrite 2 firstr_firstl. f_equal. - apply EqShiftL_le with k; auto. + apply EqShiftL_le with k; auto. unfold size. auto with arith. Qed. - Lemma EqShiftL_twice : forall k x y, + Lemma EqShiftL_twice : forall k x y, EqShiftL k (twice x) (twice y) <-> EqShiftL (S k) x y. Proof. intros; unfold EqShiftL. @@ -553,7 +553,7 @@ Section Basics. Qed. (** * From int31 to list of digits. *) - + (** Lower (=rightmost) bits comes first. *) Definition i2l := recrbis _ nil (fun d _ rec => d::rec). @@ -561,10 +561,10 @@ Section Basics. Lemma i2l_length : forall x, length (i2l x) = size. Proof. intros; reflexivity. - Qed. + Qed. - Fixpoint lshiftl l x := - match l with + Fixpoint lshiftl l x := + match l with | nil => x | d::l => sneakl d (lshiftl l x) end. @@ -576,19 +576,19 @@ Section Basics. destruct x; compute; auto. Qed. - Lemma i2l_sneakr : forall x d, + Lemma i2l_sneakr : forall x d, i2l (sneakr d x) = tail (i2l x) ++ d::nil. Proof. destruct x; compute; auto. Qed. - Lemma i2l_sneakl : forall x d, + Lemma i2l_sneakl : forall x d, i2l (sneakl d x) = d :: removelast (i2l x). Proof. destruct x; compute; auto. Qed. - Lemma i2l_l2i : forall l, length l = size -> + Lemma i2l_l2i : forall l, length l = size -> i2l (l2i l) = l. Proof. repeat (destruct l as [ |? l]; [intros; discriminate | ]). @@ -596,9 +596,9 @@ Section Basics. intros _; compute; auto. Qed. - Fixpoint cstlist (A:Type)(a:A) n := - match n with - | O => nil + Fixpoint cstlist (A:Type)(a:A) n := + match n with + | O => nil | S n => a::cstlist _ a n end. @@ -612,7 +612,7 @@ Section Basics. induction (i2l x); simpl; f_equal; auto. rewrite H0; clear H0. reflexivity. - + intros. rewrite nshiftl_S. unfold shiftl; rewrite i2l_sneakl. @@ -657,10 +657,10 @@ Section Basics. f_equal; auto. Qed. - (** This equivalence allows to prove easily the following delicate + (** This equivalence allows to prove easily the following delicate result *) - Lemma EqShiftL_twice_plus_one : forall k x y, + Lemma EqShiftL_twice_plus_one : forall k x y, EqShiftL k (twice_plus_one x) (twice_plus_one y) <-> EqShiftL (S k) x y. Proof. intros. @@ -683,7 +683,7 @@ Section Basics. subst lx n; rewrite i2l_length; omega. Qed. - Lemma EqShiftL_shiftr : forall k x y, EqShiftL k x y -> + Lemma EqShiftL_shiftr : forall k x y, EqShiftL k x y -> EqShiftL (S k) (shiftr x) (shiftr y). Proof. intros. @@ -704,41 +704,41 @@ Section Basics. omega. Qed. - Lemma EqShiftL_incrbis : forall n k x y, n<=size -> + Lemma EqShiftL_incrbis : forall n k x y, n<=size -> (n+k=S size)%nat -> - EqShiftL k x y -> + EqShiftL k x y -> EqShiftL k (incrbis_aux n x) (incrbis_aux n y). Proof. induction n; simpl; intros. red; auto. - destruct (eq_nat_dec k size). + destruct (eq_nat_dec k size). subst k; apply EqShiftL_size; auto. - unfold incrbis_aux; simpl; + unfold incrbis_aux; simpl; fold (incrbis_aux n (shiftr x)); fold (incrbis_aux n (shiftr y)). rewrite (EqShiftL_firstr k x y); auto; try omega. case_eq (firstr y); intros. rewrite EqShiftL_twice_plus_one. apply EqShiftL_shiftr; auto. - + rewrite EqShiftL_twice. apply IHn; try omega. apply EqShiftL_shiftr; auto. Qed. - Lemma EqShiftL_incr : forall x y, + Lemma EqShiftL_incr : forall x y, EqShiftL 1 x y -> EqShiftL 1 (incr x) (incr y). Proof. intros. rewrite <- 2 incrbis_aux_equiv. apply EqShiftL_incrbis; auto. Qed. - + End EqShiftL. (** * More equations about [incr] *) - Lemma incr_twice_plus_one : + Lemma incr_twice_plus_one : forall x, incr (twice_plus_one x) = twice (incr x). Proof. intros. @@ -757,7 +757,7 @@ Section Basics. destruct (incr (shiftr x)); simpl; discriminate. Qed. - Lemma incr_inv : forall x y, + Lemma incr_inv : forall x y, incr x = twice_plus_one y -> x = twice y. Proof. intros. @@ -777,7 +777,7 @@ Section Basics. (** First, recursive equations *) - Lemma phi_inv_double_plus_one : forall z, + Lemma phi_inv_double_plus_one : forall z, phi_inv (Zdouble_plus_one z) = twice_plus_one (phi_inv z). Proof. destruct z; simpl; auto. @@ -789,14 +789,14 @@ Section Basics. auto. Qed. - Lemma phi_inv_double : forall z, + Lemma phi_inv_double : forall z, phi_inv (Zdouble z) = twice (phi_inv z). Proof. destruct z; simpl; auto. rewrite incr_twice_plus_one; auto. Qed. - Lemma phi_inv_incr : forall z, + Lemma phi_inv_incr : forall z, phi_inv (Zsucc z) = incr (phi_inv z). Proof. destruct z. @@ -816,19 +816,19 @@ Section Basics. rewrite incr_twice_plus_one; auto. Qed. - (** [phi_inv o inv], the always-exact and easy-to-prove trip : + (** [phi_inv o inv], the always-exact and easy-to-prove trip : from int31 to Z and then back to int31. *) - Lemma phi_inv_phi_aux : - forall n x, n <= size -> - phi_inv (phibis_aux n (nshiftr (size-n) x)) = + Lemma phi_inv_phi_aux : + forall n x, n <= size -> + phi_inv (phibis_aux n (nshiftr (size-n) x)) = nshiftr (size-n) x. Proof. induction n. intros; simpl. rewrite nshiftr_size; auto. intros. - unfold phibis_aux, recrbis_aux; fold recrbis_aux; + unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux n (shiftr (nshiftr (size-S n) x))). assert (shiftr (nshiftr (size - S n) x) = nshiftr (size-n) x). replace (size - n)%nat with (S (size - (S n))); auto; omega. @@ -863,10 +863,10 @@ Section Basics. (** * [positive_to_int31] *) - (** A variant of [p2i] with [twice] and [twice_plus_one] instead of + (** A variant of [p2i] with [twice] and [twice_plus_one] instead of [2*i] and [2*i+1] *) - Fixpoint p2ibis n p : (N*int31)%type := + Fixpoint p2ibis n p : (N*int31)%type := match n with | O => (Npos p, On) | S n => match p with @@ -876,7 +876,7 @@ Section Basics. end end. - Lemma p2ibis_bounded : forall n p, + Lemma p2ibis_bounded : forall n p, nshiftr n (snd (p2ibis n p)) = 0. Proof. induction n. @@ -906,20 +906,20 @@ Section Basics. replace (shiftr In) with 0; auto. apply nshiftr_n_0. Qed. - + Lemma p2ibis_spec : forall n p, n<=size -> - Zpos p = ((Z_of_N (fst (p2ibis n p)))*2^(Z_of_nat n) + + Zpos p = ((Z_of_N (fst (p2ibis n p)))*2^(Z_of_nat n) + phi (snd (p2ibis n p)))%Z. Proof. induction n; intros. simpl; rewrite Pmult_1_r; auto. - replace (2^(Z_of_nat (S n)))%Z with (2*2^(Z_of_nat n))%Z by - (rewrite <- Zpower_Zsucc, <- Zpos_P_of_succ_nat; + replace (2^(Z_of_nat (S n)))%Z with (2*2^(Z_of_nat n))%Z by + (rewrite <- Zpower_Zsucc, <- Zpos_P_of_succ_nat; auto with zarith). rewrite (Zmult_comm 2). assert (n<=size) by omega. - destruct p; simpl; [ | | auto]; - specialize (IHn p H0); + destruct p; simpl; [ | | auto]; + specialize (IHn p H0); generalize (p2ibis_bounded n p); destruct (p2ibis n p) as (r,i); simpl in *; intros. @@ -937,25 +937,25 @@ Section Basics. (** We now prove that this [p2ibis] is related to [phi_inv_positive] *) - Lemma phi_inv_positive_p2ibis : forall n p, (n<=size)%nat -> + Lemma phi_inv_positive_p2ibis : forall n p, (n<=size)%nat -> EqShiftL (size-n) (phi_inv_positive p) (snd (p2ibis n p)). Proof. induction n. intros. apply EqShiftL_size; auto. intros. - simpl p2ibis; destruct p; [ | | red; auto]; - specialize IHn with p; - destruct (p2ibis n p); simpl snd in *; simpl phi_inv_positive; - rewrite ?EqShiftL_twice_plus_one, ?EqShiftL_twice; - replace (S (size - S n))%nat with (size - n)%nat by omega; + simpl p2ibis; destruct p; [ | | red; auto]; + specialize IHn with p; + destruct (p2ibis n p); simpl snd in *; simpl phi_inv_positive; + rewrite ?EqShiftL_twice_plus_one, ?EqShiftL_twice; + replace (S (size - S n))%nat with (size - n)%nat by omega; apply IHn; omega. Qed. (** This gives the expected result about [phi o phi_inv], at least for the positive case. *) - Lemma phi_phi_inv_positive : forall p, + Lemma phi_phi_inv_positive : forall p, phi (phi_inv_positive p) = (Zpos p) mod (2^(Z_of_nat size)). Proof. intros. @@ -975,12 +975,12 @@ Section Basics. Lemma double_twice_firstl : forall x, firstl x = D0 -> Twon*x = twice x. Proof. - intros. + intros. unfold mul31. rewrite <- Zdouble_mult, <- phi_twice_firstl, phi_inv_phi; auto. Qed. - Lemma double_twice_plus_one_firstl : forall x, firstl x = D0 -> + Lemma double_twice_plus_one_firstl : forall x, firstl x = D0 -> Twon*x+In = twice_plus_one x. Proof. intros. @@ -989,14 +989,14 @@ Section Basics. rewrite phi_twice_firstl, <- Zdouble_plus_one_mult, <- phi_twice_plus_one_firstl, phi_inv_phi; auto. Qed. - - Lemma p2i_p2ibis : forall n p, (n<=size)%nat -> + + Lemma p2i_p2ibis : forall n p, (n<=size)%nat -> p2i n p = p2ibis n p. Proof. induction n; simpl; auto; intros. - destruct p; auto; specialize IHn with p; - generalize (p2ibis_bounded n p); - rewrite IHn; try omega; destruct (p2ibis n p); simpl; intros; + destruct p; auto; specialize IHn with p; + generalize (p2ibis_bounded n p); + rewrite IHn; try omega; destruct (p2ibis n p); simpl; intros; f_equal; auto. apply double_twice_plus_one_firstl. apply (nshiftr_0_firstl n); auto; omega. @@ -1004,7 +1004,7 @@ Section Basics. apply (nshiftr_0_firstl n); auto; omega. Qed. - Lemma positive_to_int31_phi_inv_positive : forall p, + Lemma positive_to_int31_phi_inv_positive : forall p, snd (positive_to_int31 p) = phi_inv_positive p. Proof. intros; unfold positive_to_int31. @@ -1014,8 +1014,8 @@ Section Basics. apply (phi_inv_positive_p2ibis size); auto. Qed. - Lemma positive_to_int31_spec : forall p, - Zpos p = ((Z_of_N (fst (positive_to_int31 p)))*2^(Z_of_nat size) + + Lemma positive_to_int31_spec : forall p, + Zpos p = ((Z_of_N (fst (positive_to_int31 p)))*2^(Z_of_nat size) + phi (snd (positive_to_int31 p)))%Z. Proof. unfold positive_to_int31. @@ -1023,11 +1023,11 @@ Section Basics. apply p2ibis_spec; auto. Qed. - (** Thanks to the result about [phi o phi_inv_positive], we can - now establish easily the most general results about + (** Thanks to the result about [phi o phi_inv_positive], we can + now establish easily the most general results about [phi o twice] and so one. *) - - Lemma phi_twice : forall x, + + Lemma phi_twice : forall x, phi (twice x) = (Zdouble (phi x)) mod 2^(Z_of_nat size). Proof. intros. @@ -1041,7 +1041,7 @@ Section Basics. compute in H; elim H; auto. Qed. - Lemma phi_twice_plus_one : forall x, + Lemma phi_twice_plus_one : forall x, phi (twice_plus_one x) = (Zdouble_plus_one (phi x)) mod 2^(Z_of_nat size). Proof. intros. @@ -1055,14 +1055,14 @@ Section Basics. compute in H; elim H; auto. Qed. - Lemma phi_incr : forall x, + Lemma phi_incr : forall x, phi (incr x) = (Zsucc (phi x)) mod 2^(Z_of_nat size). Proof. intros. pattern x at 1; rewrite <- (phi_inv_phi x). rewrite <- phi_inv_incr. assert (0 <= Zsucc (phi x))%Z. - change (Zsucc (phi x)) with ((phi x)+1)%Z; + change (Zsucc (phi x)) with ((phi x)+1)%Z; generalize (phi_bounded x); omega. destruct (Zsucc (phi x)). simpl; auto. @@ -1070,10 +1070,10 @@ Section Basics. compute in H; elim H; auto. Qed. - (** With the previous results, we can deal with [phi o phi_inv] even + (** With the previous results, we can deal with [phi o phi_inv] even in the negative case *) - Lemma phi_phi_inv_negative : + Lemma phi_phi_inv_negative : forall p, phi (incr (complement_negative p)) = (Zneg p) mod 2^(Z_of_nat size). Proof. induction p. @@ -1091,11 +1091,11 @@ Section Basics. rewrite incr_twice_plus_one, phi_twice. remember (phi (incr (complement_negative p))) as q. rewrite Zdouble_mult, IHp, Zmult_mod_idemp_r; auto with zarith. - + simpl; auto. Qed. - Lemma phi_phi_inv : + Lemma phi_phi_inv : forall z, phi (phi_inv z) = z mod 2 ^ (Z_of_nat size). Proof. destruct z. @@ -1120,7 +1120,7 @@ Let w_pos_mod p i := end. (** Parity test *) -Let w_iseven i := +Let w_iseven i := let (_,r) := i/2 in match r ?= 0 with Eq => true | _ => false end. @@ -1181,14 +1181,14 @@ Definition int31_op := (mk_znz_op End Int31_Op. Section Int31_Spec. - + Open Local Scope Z_scope. Notation "[| x |]" := (phi x) (at level 0, x at level 99). Notation Local wB := (2 ^ (Z_of_nat size)). - - Lemma wB_pos : wB > 0. + + Lemma wB_pos : wB > 0. Proof. auto with zarith. Qed. @@ -1216,12 +1216,12 @@ Section Int31_Spec. Proof. reflexivity. Qed. - + Lemma spec_1 : [| 1 |] = 1. Proof. reflexivity. Qed. - + Lemma spec_Bm1 : [| Tn |] = wB - 1. Proof. reflexivity. @@ -1252,16 +1252,16 @@ Section Int31_Spec. destruct (Z_lt_le_dec (X+Y) wB). contradict H1; auto using Zmod_small with zarith. rewrite <- (Z_mod_plus_full (X+Y) (-1) wB). - rewrite Zmod_small; romega. + rewrite Zmod_small; romega. generalize (Zcompare_Eq_eq ((X+Y) mod wB) (X+Y)); intros Heq. - destruct Zcompare; intros; + destruct Zcompare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. Lemma spec_succ_c : forall x, [+|add31c x 1|] = [|x|] + 1. Proof. - intros; apply spec_add_c. + intros; apply spec_add_c. Qed. Lemma spec_add_carry_c : forall x y, [+|add31carryc x y|] = [|x|] + [|y|] + 1. @@ -1279,7 +1279,7 @@ Section Int31_Spec. rewrite Zmod_small; romega. generalize (Zcompare_Eq_eq ((X+Y+1) mod wB) (X+Y+1)); intros Heq. - destruct Zcompare; intros; + destruct Zcompare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. @@ -1304,7 +1304,7 @@ Section Int31_Spec. (** Substraction *) Lemma spec_sub_c : forall x y, [-|sub31c x y|] = [|x|] - [|y|]. - Proof. + Proof. unfold sub31c, sub31, interp_carry; intros. rewrite phi_phi_inv. generalize (phi_bounded x)(phi_bounded y); intros. @@ -1337,7 +1337,7 @@ Section Int31_Spec. contradict H1; apply Zmod_small; romega. generalize (Zcompare_Eq_eq ((X-Y-1) mod wB) (X-Y-1)); intros Heq. - destruct Zcompare; intros; + destruct Zcompare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. @@ -1355,7 +1355,7 @@ Section Int31_Spec. Qed. Lemma spec_opp_c : forall x, [-|sub31c 0 x|] = -[|x|]. - Proof. + Proof. intros; apply spec_sub_c. Qed. @@ -1402,7 +1402,7 @@ Section Int31_Spec. change (wB*wB) with (wB^2); ring. unfold phi_inv2. - destruct x; unfold zn2z_to_Z; rewrite ?phi_phi_inv; + destruct x; unfold zn2z_to_Z; rewrite ?phi_phi_inv; change base with wB; auto. Qed. @@ -1426,7 +1426,7 @@ Section Int31_Spec. intros; apply spec_mul_c. Qed. - (** Division *) + (** Division *) Lemma spec_div21 : forall a1 a2 b, wB/2 <= [|b|] -> @@ -1537,7 +1537,7 @@ Section Int31_Spec. intros (H,_); compute in H; elim H; auto. Qed. - Lemma iter_int31_iter_nat : forall A f i a, + Lemma iter_int31_iter_nat : forall A f i a, iter_int31 i A f a = iter_nat (Zabs_nat [|i|]) A f a. Proof. intros. @@ -1548,17 +1548,17 @@ Section Int31_Spec. revert i a; induction size. simpl; auto. simpl; intros. - case_eq (firstr i); intros H; rewrite 2 IHn; + case_eq (firstr i); intros H; rewrite 2 IHn; unfold phibis_aux; simpl; rewrite H; fold (phibis_aux n (shiftr i)); - generalize (phibis_aux_pos n (shiftr i)); intros; - set (z := phibis_aux n (shiftr i)) in *; clearbody z; + generalize (phibis_aux_pos n (shiftr i)); intros; + set (z := phibis_aux n (shiftr i)) in *; clearbody z; rewrite <- iter_nat_plus. f_equal. rewrite Zdouble_mult, Zmult_comm, <- Zplus_diag_eq_mult_2. symmetry; apply Zabs_nat_Zplus; auto with zarith. - change (iter_nat (S (Zabs_nat z + Zabs_nat z)) A f a = + change (iter_nat (S (Zabs_nat z + Zabs_nat z)) A f a = iter_nat (Zabs_nat (Zdouble_plus_one z)) A f a); f_equal. rewrite Zdouble_plus_one_mult, Zmult_comm, <- Zplus_diag_eq_mult_2. rewrite Zabs_nat_Zplus; auto with zarith. @@ -1566,13 +1566,13 @@ Section Int31_Spec. change (Zabs_nat 1) with 1%nat; omega. Qed. - Fixpoint addmuldiv31_alt n i j := - match n with - | O => i + Fixpoint addmuldiv31_alt n i j := + match n with + | O => i | S n => addmuldiv31_alt n (sneakl (firstl j) i) (shiftl j) end. - Lemma addmuldiv31_equiv : forall p x y, + Lemma addmuldiv31_equiv : forall p x y, addmuldiv31 p x y = addmuldiv31_alt (Zabs_nat [|p|]) x y. Proof. intros. @@ -1588,7 +1588,7 @@ Section Int31_Spec. Qed. Lemma spec_add_mul_div : forall x y p, [|p|] <= Zpos 31 -> - [| addmuldiv31 p x y |] = + [| addmuldiv31 p x y |] = ([|x|] * (2 ^ [|p|]) + [|y|] / (2 ^ ((Zpos 31) - [|p|]))) mod wB. Proof. intros. @@ -1626,7 +1626,7 @@ Section Int31_Spec. replace (31-Z_of_nat n) with (Zsucc(31-Zsucc(Z_of_nat n))) by ring. rewrite Zpower_Zsucc, <- Zdiv_Zdiv; auto with zarith. rewrite Zmult_comm, Z_div_mult; auto with zarith. - + rewrite phi_twice_plus_one, Zdouble_plus_one_mult. rewrite phi_twice; auto. change (Zdouble [|y|]) with (2*[|y|]). @@ -1644,7 +1644,7 @@ Section Int31_Spec. unfold wB'. rewrite <- Zpower_Zsucc, <- inj_S by (auto with zarith). f_equal. rewrite H1. - replace wB with (2^(Z_of_nat n)*2^(31-Z_of_nat n)) by + replace wB with (2^(Z_of_nat n)*2^(31-Z_of_nat n)) by (rewrite <- Zpower_exp; auto with zarith; f_equal; unfold size; ring). unfold Zminus; rewrite Zopp_mult_distr_l. rewrite Z_div_plus; auto with zarith. @@ -1669,8 +1669,8 @@ Section Int31_Spec. apply Zlt_le_trans with wB; auto with zarith. apply Zpower_le_monotone; auto with zarith. intros. - case_eq ([|p|] ?= 31); intros; - [ apply H; rewrite (Zcompare_Eq_eq _ _ H0); auto with zarith | | + case_eq ([|p|] ?= 31); intros; + [ apply H; rewrite (Zcompare_Eq_eq _ _ H0); auto with zarith | | apply H; change ([|p|]>31)%Z in H0; auto with zarith ]. change ([|p|]<31) in H0. rewrite spec_add_mul_div by auto with zarith. @@ -1701,16 +1701,16 @@ Section Int31_Spec. simpl; auto. Qed. - Fixpoint head031_alt n x := - match n with + Fixpoint head031_alt n x := + match n with | O => 0%nat - | S n => match firstl x with + | S n => match firstl x with | D0 => S (head031_alt n (shiftl x)) | D1 => 0%nat end end. - Lemma head031_equiv : + Lemma head031_equiv : forall x, [|head031 x|] = Z_of_nat (head031_alt size x). Proof. intros. @@ -1720,10 +1720,10 @@ Section Int31_Spec. unfold head031, recl. change On with (phi_inv (Z_of_nat (31-size))). - replace (head031_alt size x) with + replace (head031_alt size x) with (head031_alt size x + (31 - size))%nat by (apply plus_0_r; auto). assert (size <= 31)%nat by auto with arith. - + revert x H; induction size; intros. simpl; auto. unfold recl_aux; fold recl_aux. @@ -1748,7 +1748,7 @@ Section Int31_Spec. change [|In|] with 1. replace (31-n)%nat with (S (31 - S n))%nat by omega. rewrite inj_S; ring. - + clear - H H2. rewrite (sneakr_shiftl x) in H. rewrite H2 in H. @@ -1793,7 +1793,7 @@ Section Int31_Spec. rewrite (sneakr_shiftl x), H1, H; auto. rewrite <- nshiftl_S_tail; auto. - + change (2^(Z_of_nat 0)) with 1; rewrite Zmult_1_l. generalize (phi_bounded x); unfold size; split; auto with zarith. change (2^(Z_of_nat 31)/2) with (2^(Z_of_nat (pred size))). @@ -1809,16 +1809,16 @@ Section Int31_Spec. simpl; auto. Qed. - Fixpoint tail031_alt n x := - match n with + Fixpoint tail031_alt n x := + match n with | O => 0%nat - | S n => match firstr x with + | S n => match firstr x with | D0 => S (tail031_alt n (shiftr x)) | D1 => 0%nat end end. - Lemma tail031_equiv : + Lemma tail031_equiv : forall x, [|tail031 x|] = Z_of_nat (tail031_alt size x). Proof. intros. @@ -1828,10 +1828,10 @@ Section Int31_Spec. unfold tail031, recr. change On with (phi_inv (Z_of_nat (31-size))). - replace (tail031_alt size x) with + replace (tail031_alt size x) with (tail031_alt size x + (31 - size))%nat by (apply plus_0_r; auto). assert (size <= 31)%nat by auto with arith. - + revert x H; induction size; intros. simpl; auto. unfold recr_aux; fold recr_aux. @@ -1856,7 +1856,7 @@ Section Int31_Spec. change [|In|] with 1. replace (31-n)%nat with (S (31 - S n))%nat by omega. rewrite inj_S; ring. - + clear - H H2. rewrite (sneakl_shiftr x) in H. rewrite H2 in H. @@ -1864,7 +1864,7 @@ Section Int31_Spec. rewrite (iszero_eq0 _ H0) in H; discriminate. Qed. - Lemma spec_tail0 : forall x, 0 < [|x|] -> + Lemma spec_tail0 : forall x, 0 < [|x|] -> exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|tail031 x|]). Proof. intros. @@ -1882,23 +1882,23 @@ Section Int31_Spec. case_eq (firstr x); intros. rewrite (inj_S (tail031_alt n (shiftr x))), Zpower_Zsucc; auto with zarith. destruct (IHn (shiftr x)) as (y & Hy1 & Hy2). - + rewrite phi_nz; rewrite phi_nz in H; contradict H. rewrite (sneakl_shiftr x), H1, H; auto. rewrite <- nshiftr_S_tail; auto. - + exists y; split; auto. rewrite phi_eqn1; auto. rewrite Zdouble_mult, Hy2; ring. - + exists [|shiftr x|]. split. generalize (phi_bounded (shiftr x)); auto with zarith. rewrite phi_eqn2; auto. rewrite Zdouble_plus_one_mult; simpl; ring. Qed. - + (* Sqrt *) (* Direct transcription of an old proof @@ -1910,23 +1910,23 @@ Section Int31_Spec. intros H1; rewrite Zmod_eq_full; auto with zarith. Qed. - Lemma sqrt_main_trick j k: 0 <= j -> 0 <= k -> + Lemma sqrt_main_trick j k: 0 <= j -> 0 <= k -> (j * k) + j <= ((j + k)/2 + 1) ^ 2. Proof. - intros j k Hj; generalize Hj k; pattern j; apply natlike_ind; + intros j k Hj; generalize Hj k; pattern j; apply natlike_ind; auto; clear k j Hj. intros _ k Hk; repeat rewrite Zplus_0_l. apply Zmult_le_0_compat; generalize (Z_div_pos k 2); auto with zarith. intros j Hj Hrec _ k Hk; pattern k; apply natlike_ind; auto; clear k Hk. rewrite Zmult_0_r, Zplus_0_r, Zplus_0_l. - generalize (sqr_pos (Zsucc j / 2)) (quotient_by_2 (Zsucc j)); + generalize (sqr_pos (Zsucc j / 2)) (quotient_by_2 (Zsucc j)); unfold Zsucc. rewrite Zpower_2, Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. auto with zarith. intros k Hk _. replace ((Zsucc j + Zsucc k) / 2) with ((j + k)/2 + 1). generalize (Hrec Hj k Hk) (quotient_by_2 (j + k)). - unfold Zsucc; repeat rewrite Zpower_2; + unfold Zsucc; repeat rewrite Zpower_2; repeat rewrite Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. repeat rewrite Zmult_1_l; repeat rewrite Zmult_1_r. auto with zarith. @@ -1991,7 +1991,7 @@ Section Int31_Spec. Qed. Lemma sqrt31_step_def rec i j: - sqrt31_step rec i j = + sqrt31_step rec i j = match (fst (i/j) ?= j)%int31 with Lt => rec i (fst ((j + fst(i/j))/2))%int31 | _ => j @@ -2008,8 +2008,8 @@ Section Int31_Spec. rewrite H1; ring. Qed. - Lemma sqrt31_step_correct rec i j: - 0 < [|i|] -> 0 < [|j|] -> [|i|] < ([|j|] + 1) ^ 2 -> + Lemma sqrt31_step_correct rec i j: + 0 < [|i|] -> 0 < [|j|] -> [|i|] < ([|j|] + 1) ^ 2 -> 2 * [|j|] < wB -> (forall j1 : int31, 0 < [|j1|] < [|j|] -> [|i|] < ([|j1|] + 1) ^ 2 -> @@ -2018,14 +2018,14 @@ Section Int31_Spec. Proof. assert (Hp2: 0 < [|2|]) by exact (refl_equal Lt). intros rec i j Hi Hj Hij H31 Hrec; rewrite sqrt31_step_def. - generalize (spec_compare (fst (i/j)%int31) j); case compare31; + generalize (spec_compare (fst (i/j)%int31) j); case compare31; rewrite div31_phi; auto; intros Hc; try (split; auto; apply sqrt_test_true; auto with zarith; fail). apply Hrec; repeat rewrite div31_phi; auto with zarith. replace [|(j + fst (i / j)%int31)|] with ([|j|] + [|i|] / [|j|]). split. case (Zle_lt_or_eq 1 [|j|]); auto with zarith; intros Hj1. - replace ([|j|] + [|i|]/[|j|]) with + replace ([|j|] + [|i|]/[|j|]) with (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])); try ring. rewrite Z_div_plus_full_l; auto with zarith. assert (0 <= [|i|]/ [|j|]) by (apply Z_div_pos; auto with zarith). @@ -2048,7 +2048,7 @@ Section Int31_Spec. Lemma iter31_sqrt_correct n rec i j: 0 < [|i|] -> 0 < [|j|] -> [|i|] < ([|j|] + 1) ^ 2 -> 2 * [|j|] < 2 ^ (Z_of_nat size) -> - (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> + (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> [|i|] < ([|j1|] + 1) ^ 2 -> 2 * [|j1|] < 2 ^ (Z_of_nat size) -> [|rec i j1|] ^ 2 <= [|i|] < ([|rec i j1|] + 1) ^ 2) -> [|iter31_sqrt n rec i j|] ^ 2 <= [|i|] < ([|iter31_sqrt n rec i j|] + 1) ^ 2. @@ -2098,7 +2098,7 @@ Section Int31_Spec. Qed. Lemma sqrt312_step_def rec ih il j: - sqrt312_step rec ih il j = + sqrt312_step rec ih il j = match (ih ?= j)%int31 with Eq => j | Gt => j @@ -2116,7 +2116,7 @@ Section Int31_Spec. simpl; case compare31; auto. Qed. - Lemma sqrt312_lower_bound ih il j: + Lemma sqrt312_lower_bound ih il j: phi2 ih il < ([|j|] + 1) ^ 2 -> [|ih|] <= [|j|]. Proof. intros ih il j H1. @@ -2140,11 +2140,11 @@ Section Int31_Spec. simpl fst; apply trans_equal with (1 := Hq); ring. Qed. - Lemma sqrt312_step_correct rec ih il j: - 2 ^ 29 <= [|ih|] -> 0 < [|j|] -> phi2 ih il < ([|j|] + 1) ^ 2 -> + Lemma sqrt312_step_correct rec ih il j: + 2 ^ 29 <= [|ih|] -> 0 < [|j|] -> phi2 ih il < ([|j|] + 1) ^ 2 -> (forall j1, 0 < [|j1|] < [|j|] -> phi2 ih il < ([|j1|] + 1) ^ 2 -> [|rec ih il j1|] ^ 2 <= phi2 ih il < ([|rec ih il j1|] + 1) ^ 2) -> - [|sqrt312_step rec ih il j|] ^ 2 <= phi2 ih il + [|sqrt312_step rec ih il j|] ^ 2 <= phi2 ih il < ([|sqrt312_step rec ih il j|] + 1) ^ 2. Proof. assert (Hp2: (0 < [|2|])%Z) by exact (refl_equal Lt). @@ -2174,7 +2174,7 @@ Section Int31_Spec. case (Zle_lt_or_eq 1 ([|j|])); auto with zarith; intros Hf2. 2: contradict Hc; apply Zle_not_lt; rewrite <- Hf2, Zdiv_1_r; auto with zarith. assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2). - replace ([|j|] + phi2 ih il/ [|j|])%Z with + replace ([|j|] + phi2 ih il/ [|j|])%Z with (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring. rewrite Z_div_plus_full_l; auto with zarith. assert (0 <= ([|j|] - 2 + phi2 ih il / [|j|]) / 2) ; auto with zarith. @@ -2213,7 +2213,7 @@ Section Int31_Spec. rewrite div31_phi; change (phi 2) with 2%Z; auto. change (2 ^Z_of_nat size) with (base/2 + phi v30). assert (phi r / 2 < base/2); auto with zarith. - apply Zmult_gt_0_lt_reg_r with 2; auto with zarith. + apply Zmult_gt_0_lt_reg_r with 2; auto with zarith. change (base/2 * 2) with base. apply Zle_lt_trans with (phi r). rewrite Zmult_comm; apply Z_mult_div_ge; auto with zarith. @@ -2234,12 +2234,12 @@ Section Int31_Spec. apply Zge_le; apply Z_div_ge; auto with zarith. Qed. - Lemma iter312_sqrt_correct n rec ih il j: - 2^29 <= [|ih|] -> 0 < [|j|] -> phi2 ih il < ([|j|] + 1) ^ 2 -> - (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> - phi2 ih il < ([|j1|] + 1) ^ 2 -> + Lemma iter312_sqrt_correct n rec ih il j: + 2^29 <= [|ih|] -> 0 < [|j|] -> phi2 ih il < ([|j|] + 1) ^ 2 -> + (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> + phi2 ih il < ([|j1|] + 1) ^ 2 -> [|rec ih il j1|] ^ 2 <= phi2 ih il < ([|rec ih il j1|] + 1) ^ 2) -> - [|iter312_sqrt n rec ih il j|] ^ 2 <= phi2 ih il + [|iter312_sqrt n rec ih il j|] ^ 2 <= phi2 ih il < ([|iter312_sqrt n rec ih il j|] + 1) ^ 2. Proof. intros n; elim n; unfold iter312_sqrt; fold iter312_sqrt; clear n. @@ -2265,7 +2265,7 @@ Section Int31_Spec. Proof. intros ih il Hih; unfold sqrt312. change [||WW ih il||] with (phi2 ih il). - assert (Hbin: forall s, s * s + 2* s + 1 = (s + 1) ^ 2) by + assert (Hbin: forall s, s * s + 2* s + 1 = (s + 1) ^ 2) by (intros s; ring). assert (Hb: 0 <= base) by (red; intros HH; discriminate). assert (Hi2: phi2 ih il < (phi Tn + 1) ^ 2). @@ -2428,9 +2428,9 @@ Section Int31_Spec. apply Zcompare_Eq_eq. now destruct ([|x|] ?= 0). Qed. - + (* Even *) - + Let w_is_even := int31_op.(znz_is_even). Lemma spec_is_even : forall x, @@ -2460,13 +2460,13 @@ Section Int31_Spec. exact spec_more_than_1_digit. exact spec_0. - exact spec_1. + exact spec_1. exact spec_Bm1. exact spec_compare. exact spec_eq0. - exact spec_opp_c. + exact spec_opp_c. exact spec_opp. exact spec_opp_carry. @@ -2500,7 +2500,7 @@ Section Int31_Spec. exact spec_head00. exact spec_head0. - exact spec_tail00. + exact spec_tail00. exact spec_tail0. exact spec_add_mul_div. diff --git a/theories/Numbers/Cyclic/Int31/Int31.v b/theories/Numbers/Cyclic/Int31/Int31.v index 12c0cc2642..1168e7fd6d 100644 --- a/theories/Numbers/Cyclic/Int31/Int31.v +++ b/theories/Numbers/Cyclic/Int31/Int31.v @@ -17,7 +17,7 @@ Require Export DoubleType. Unset Boxed Definitions. -(** * 31-bit integers *) +(** * 31-bit integers *) (** This file contains basic definitions of a 31-bit integer arithmetic. In fact it is more general than that. The only reason @@ -36,8 +36,8 @@ Definition size := 31%nat. Inductive digits : Type := D0 | D1. (** The type of 31-bit integers *) - -(** The type [int31] has a unique constructor [I31] that expects + +(** The type [int31] has a unique constructor [I31] that expects 31 arguments of type [digits]. *) Inductive int31 : Type := I31 : nfun digits size int31. @@ -69,26 +69,26 @@ Definition Twon : int31 := Eval compute in (napply_cst _ _ D0 (size-2) I31) D1 D (** * Bits manipulation *) -(** [sneakr b x] shifts [x] to the right by one bit. +(** [sneakr b x] shifts [x] to the right by one bit. Rightmost digit is lost while leftmost digit becomes [b]. - Pseudo-code is + Pseudo-code is [ match x with (I31 d0 ... dN) => I31 b d0 ... d(N-1) end ] *) Definition sneakr : digits -> int31 -> int31 := Eval compute in fun b => int31_rect _ (napply_except_last _ _ (size-1) (I31 b)). -(** [sneakl b x] shifts [x] to the left by one bit. +(** [sneakl b x] shifts [x] to the left by one bit. Leftmost digit is lost while rightmost digit becomes [b]. - Pseudo-code is + Pseudo-code is [ match x with (I31 d0 ... dN) => I31 d1 ... dN b end ] *) -Definition sneakl : digits -> int31 -> int31 := Eval compute in +Definition sneakl : digits -> int31 -> int31 := Eval compute in fun b => int31_rect _ (fun _ => napply_then_last _ _ b (size-1) I31). -(** [shiftl], [shiftr], [twice] and [twice_plus_one] are direct +(** [shiftl], [shiftr], [twice] and [twice_plus_one] are direct consequences of [sneakl] and [sneakr]. *) Definition shiftl := sneakl D0. @@ -96,31 +96,31 @@ Definition shiftr := sneakr D0. Definition twice := sneakl D0. Definition twice_plus_one := sneakl D1. -(** [firstl x] returns the leftmost digit of number [x]. +(** [firstl x] returns the leftmost digit of number [x]. Pseudo-code is [ match x with (I31 d0 ... dN) => d0 end ] *) -Definition firstl : int31 -> digits := Eval compute in +Definition firstl : int31 -> digits := Eval compute in int31_rect _ (fun d => napply_discard _ _ d (size-1)). -(** [firstr x] returns the rightmost digit of number [x]. +(** [firstr x] returns the rightmost digit of number [x]. Pseudo-code is [ match x with (I31 d0 ... dN) => dN end ] *) -Definition firstr : int31 -> digits := Eval compute in +Definition firstr : int31 -> digits := Eval compute in int31_rect _ (napply_discard _ _ (fun d=>d) (size-1)). -(** [iszero x] is true iff [x = I31 D0 ... D0]. Pseudo-code is +(** [iszero x] is true iff [x = I31 D0 ... D0]. Pseudo-code is [ match x with (I31 D0 ... D0) => true | _ => false end ] *) -Definition iszero : int31 -> bool := Eval compute in - let f d b := match d with D0 => b | D1 => false end +Definition iszero : int31 -> bool := Eval compute in + let f d b := match d with D0 => b | D1 => false end in int31_rect _ (nfold_bis _ _ f true size). -(* NB: DO NOT transform the above match in a nicer (if then else). +(* NB: DO NOT transform the above match in a nicer (if then else). It seems to work, but later "unfold iszero" takes forever. *) -(** [base] is [2^31], obtained via iterations of [Zdouble]. - It can also be seen as the smallest b > 0 s.t. phi_inv b = 0 +(** [base] is [2^31], obtained via iterations of [Zdouble]. + It can also be seen as the smallest b > 0 s.t. phi_inv b = 0 (see below) *) Definition base := Eval compute in @@ -140,7 +140,7 @@ Fixpoint recl_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A) caserec (firstl i) si (recl_aux next A case0 caserec si) end. -Fixpoint recr_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A) +Fixpoint recr_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A) (i:int31) : A := match n with | O => case0 @@ -159,22 +159,22 @@ Definition recr := recr_aux size. (** From int31 to Z, we simply iterates [Zdouble] or [Zdouble_plus_one]. *) -Definition phi : int31 -> Z := +Definition phi : int31 -> Z := recr Z (0%Z) (fun b _ => match b with D0 => Zdouble | D1 => Zdouble_plus_one end). -(** From positive to int31. An abstract definition could be : - [ phi_inv (2n) = 2*(phi_inv n) /\ +(** From positive to int31. An abstract definition could be : + [ phi_inv (2n) = 2*(phi_inv n) /\ phi_inv 2n+1 = 2*(phi_inv n) + 1 ] *) -Fixpoint phi_inv_positive p := +Fixpoint phi_inv_positive p := match p with | xI q => twice_plus_one (phi_inv_positive q) | xO q => twice (phi_inv_positive q) | xH => In end. -(** The negative part : 2-complement *) +(** The negative part : 2-complement *) Fixpoint complement_negative p := match p with @@ -186,9 +186,9 @@ Fixpoint complement_negative p := (** A simple incrementation function *) Definition incr : int31 -> int31 := - recr int31 In - (fun b si rec => match b with - | D0 => sneakl D1 si + recr int31 In + (fun b si rec => match b with + | D0 => sneakl D1 si | D1 => sneakl D0 rec end). (** We can now define the conversion from Z to int31. *) @@ -196,11 +196,11 @@ Definition incr : int31 -> int31 := Definition phi_inv : Z -> int31 := fun n => match n with | Z0 => On - | Zpos p => phi_inv_positive p + | Zpos p => phi_inv_positive p | Zneg p => incr (complement_negative p) end. -(** [phi_inv2] is similar to [phi_inv] but returns a double word +(** [phi_inv2] is similar to [phi_inv] but returns a double word [zn2z int31] *) Definition phi_inv2 n := @@ -211,7 +211,7 @@ Definition phi_inv2 n := (** [phi2] is similar to [phi] but takes a double word (two args) *) -Definition phi2 nh nl := +Definition phi2 nh nl := ((phi nh)*base+(phi nl))%Z. (** * Addition *) @@ -227,11 +227,11 @@ Notation "n + m" := (add31 n m) : int31_scope. (* mode, (phi n)+(phi m) is computed twice*) (* it may be considered to optimize it *) -Definition add31c (n m : int31) := +Definition add31c (n m : int31) := let npm := n+m in - match (phi npm ?= (phi n)+(phi m))%Z with - | Eq => C0 npm - | _ => C1 npm + match (phi npm ?= (phi n)+(phi m))%Z with + | Eq => C0 npm + | _ => C1 npm end. Notation "n '+c' m" := (add31c n m) (at level 50, no associativity) : int31_scope. @@ -254,7 +254,7 @@ Notation "n - m" := (sub31 n m) : int31_scope. (** Subtraction with carry (thus exact) *) -Definition sub31c (n m : int31) := +Definition sub31c (n m : int31) := let nmm := n-m in match (phi nmm ?= (phi n)-(phi m))%Z with | Eq => C0 nmm @@ -290,13 +290,13 @@ Notation "n '*c' m" := (mul31c n m) (at level 40, no associativity) : int31_scop (** Division of a double size word modulo [2^31] *) -Definition div3121 (nh nl m : int31) := +Definition div3121 (nh nl m : int31) := let (q,r) := Zdiv_eucl (phi2 nh nl) (phi m) in (phi_inv q, phi_inv r). (** Division modulo [2^31] *) -Definition div31 (n m : int31) := +Definition div31 (n m : int31) := let (q,r) := Zdiv_eucl (phi n) (phi m) in (phi_inv q, phi_inv r). Notation "n / m" := (div31 n m) : int31_scope. @@ -308,12 +308,12 @@ Definition compare31 (n m : int31) := ((phi n)?=(phi m))%Z. Notation "n ?= m" := (compare31 n m) (at level 70, no associativity) : int31_scope. -(** Computing the [i]-th iterate of a function: +(** Computing the [i]-th iterate of a function: [iter_int31 i A f = f^i] *) Definition iter_int31 i A f := - recr (A->A) (fun x => x) - (fun b si rec => match b with + recr (A->A) (fun x => x) + (fun b si rec => match b with | D0 => fun x => rec (rec x) | D1 => fun x => f (rec (rec x)) end) @@ -322,9 +322,9 @@ Definition iter_int31 i A f := (** Combining the [(31-p)] low bits of [i] above the [p] high bits of [j]: [addmuldiv31 p i j = i*2^p+j/2^(31-p)] (modulo [2^31]) *) -Definition addmuldiv31 p i j := - let (res, _ ) := - iter_int31 p (int31*int31) +Definition addmuldiv31 p i j := + let (res, _ ) := + iter_int31 p (int31*int31) (fun ij => let (i,j) := ij in (sneakl (firstl j) i, shiftl j)) (i,j) in @@ -346,7 +346,7 @@ Register addmuldiv31 as int31 addmuldiv in "coq_int31" by True. Definition gcd31 (i j:int31) := (fix euler (guard:nat) (i j:int31) {struct guard} := - match guard with + match guard with | O => In | S p => match j ?= On with | Eq => i @@ -370,17 +370,17 @@ Eval lazy delta [Twon] in | _ => j end. -Fixpoint iter31_sqrt (n: nat) (rec: int31 -> int31 -> int31) +Fixpoint iter31_sqrt (n: nat) (rec: int31 -> int31 -> int31) (i j: int31) {struct n} : int31 := - sqrt31_step + sqrt31_step (match n with O => rec | S n => (iter31_sqrt n (iter31_sqrt n rec)) end) i j. -Definition sqrt31 i := +Definition sqrt31 i := Eval lazy delta [On In Twon] in - match compare31 In i with + match compare31 In i with Gt => On | Eq => In | Lt => iter31_sqrt 31 (fun i j => j) i (fst (i/Twon)) @@ -388,7 +388,7 @@ Eval lazy delta [On In Twon] in Definition v30 := Eval compute in (addmuldiv31 (phi_inv (Z_of_nat size - 1)) In On). -Definition sqrt312_step (rec: int31 -> int31 -> int31 -> int31) +Definition sqrt312_step (rec: int31 -> int31 -> int31 -> int31) (ih il j: int31) := Eval lazy delta [Twon v30] in match ih ?= j with Eq => j | Gt => j | _ => @@ -401,28 +401,28 @@ Eval lazy delta [Twon v30] in | _ => j end end. -Fixpoint iter312_sqrt (n: nat) - (rec: int31 -> int31 -> int31 -> int31) +Fixpoint iter312_sqrt (n: nat) + (rec: int31 -> int31 -> int31 -> int31) (ih il j: int31) {struct n} : int31 := - sqrt312_step + sqrt312_step (match n with O => rec | S n => (iter312_sqrt n (iter312_sqrt n rec)) end) ih il j. -Definition sqrt312 ih il := +Definition sqrt312 ih il := Eval lazy delta [On In] in let s := iter312_sqrt 31 (fun ih il j => j) ih il Tn in match s *c s with W0 => (On, C0 On) (* impossible *) | WW ih1 il1 => match il -c il1 with - C0 il2 => + C0 il2 => match ih ?= ih1 with Gt => (s, C1 il2) | _ => (s, C0 il2) end - | C1 il2 => + | C1 il2 => match (ih - In) ?= ih1 with (* we could parametrize ih - 1 *) Gt => (s, C1 il2) | _ => (s, C0 il2) @@ -431,7 +431,7 @@ Eval lazy delta [On In] in end. -Fixpoint p2i n p : (N*int31)%type := +Fixpoint p2i n p : (N*int31)%type := match n with | O => (Npos p, On) | S n => match p with @@ -444,26 +444,26 @@ Fixpoint p2i n p : (N*int31)%type := Definition positive_to_int31 (p:positive) := p2i size p. (** Constant 31 converted into type int31. - It is used as default answer for numbers of zeros + It is used as default answer for numbers of zeros in [head0] and [tail0] *) Definition T31 : int31 := Eval compute in phi_inv (Z_of_nat size). Definition head031 (i:int31) := - recl _ (fun _ => T31) - (fun b si rec n => match b with + recl _ (fun _ => T31) + (fun b si rec n => match b with | D0 => rec (add31 n In) | D1 => n end) i On. Definition tail031 (i:int31) := - recr _ (fun _ => T31) - (fun b si rec n => match b with + recr _ (fun _ => T31) + (fun b si rec n => match b with | D0 => rec (add31 n In) | D1 => n end) i On. Register head031 as int31 head0 in "coq_int31" by True. -Register tail031 as int31 tail0 in "coq_int31" by True. +Register tail031 as int31 tail0 in "coq_int31" by True. diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v index 7373acc9ad..1b1283400b 100644 --- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v +++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v @@ -8,11 +8,11 @@ (* $Id$ *) -(** * Type [Z] viewed modulo a particular constant corresponds to [Z/nZ] +(** * Type [Z] viewed modulo a particular constant corresponds to [Z/nZ] as defined abstractly in CyclicAxioms. *) -(** Even if the construction provided here is not reused for building - the efficient arbitrary precision numbers, it provides a simple +(** Even if the construction provided here is not reused for building + the efficient arbitrary precision numbers, it provides a simple implementation of CyclicAxioms, hence ensuring its coherence. *) Set Implicit Arguments. @@ -56,9 +56,9 @@ Section ZModulo. destruct 1; auto. Qed. Let digits_gt_1 := spec_more_than_1_digit. - + Lemma wB_pos : wB > 0. - Proof. + Proof. unfold wB, base; auto with zarith. Qed. Hint Resolve wB_pos. @@ -79,7 +79,7 @@ Section ZModulo. auto. Qed. - Definition znz_of_pos x := + Definition znz_of_pos x := let (q,r) := Zdiv_eucl_POS x wB in (N_of_Z q, r). Lemma spec_of_pos : forall p, @@ -90,10 +90,10 @@ Section ZModulo. destruct (Zdiv_eucl_POS p wB); simpl; destruct 1. unfold znz_to_Z; rewrite Zmod_small; auto. assert (0 <= z). - replace z with (Zpos p / wB) by + replace z with (Zpos p / wB) by (symmetry; apply Zdiv_unique with z0; auto). apply Z_div_pos; auto with zarith. - replace (Z_of_N (N_of_Z z)) with z by + replace (Z_of_N (N_of_Z z)) with z by (destruct z; simpl; auto; elim H1; auto). rewrite Zmult_comm; auto. Qed. @@ -110,7 +110,7 @@ Section ZModulo. Definition znz_0 := 0. Definition znz_1 := 1. Definition znz_Bm1 := wB - 1. - + Lemma spec_0 : [|znz_0|] = 0. Proof. unfold znz_to_Z, znz_0. @@ -121,7 +121,7 @@ Section ZModulo. Proof. unfold znz_to_Z, znz_1. apply Zmod_small; split; auto with zarith. - unfold wB, base. + unfold wB, base. apply Zlt_trans with (Zpos digits); auto. apply Zpower2_lt_lin; auto with zarith. Qed. @@ -138,7 +138,7 @@ Section ZModulo. Definition znz_compare x y := Zcompare [|x|] [|y|]. - Lemma spec_compare : forall x y, + Lemma spec_compare : forall x y, match znz_compare x y with | Eq => [|x|] = [|y|] | Lt => [|x|] < [|y|] @@ -150,19 +150,19 @@ Section ZModulo. intros; apply Zcompare_Eq_eq; auto. Qed. - Definition znz_eq0 x := + Definition znz_eq0 x := match [|x|] with Z0 => true | _ => false end. - + Lemma spec_eq0 : forall x, znz_eq0 x = true -> [|x|] = 0. Proof. unfold znz_eq0; intros; now destruct [|x|]. Qed. - Definition znz_opp_c x := + Definition znz_opp_c x := if znz_eq0 x then C0 0 else C1 (- x). Definition znz_opp x := - x. Definition znz_opp_carry x := - x - 1. - + Lemma spec_opp_c : forall x, [-|znz_opp_c x|] = -[|x|]. Proof. intros; unfold znz_opp_c, znz_to_Z; auto. @@ -180,7 +180,7 @@ Section ZModulo. change ((- x) mod wB = (0 - (x mod wB)) mod wB). rewrite Zminus_mod_idemp_r; simpl; auto. Qed. - + Lemma spec_opp_carry : forall x, [|znz_opp_carry x|] = wB - [|x|] - 1. Proof. intros; unfold znz_opp_carry, znz_to_Z; auto. @@ -194,15 +194,15 @@ Section ZModulo. generalize (Z_mod_lt x wB wB_pos); omega. Qed. - Definition znz_succ_c x := - let y := Zsucc x in + Definition znz_succ_c x := + let y := Zsucc x in if znz_eq0 y then C1 0 else C0 y. - Definition znz_add_c x y := - let z := [|x|] + [|y|] in + Definition znz_add_c x y := + let z := [|x|] + [|y|] in if Z_lt_le_dec z wB then C0 z else C1 (z-wB). - Definition znz_add_carry_c x y := + Definition znz_add_carry_c x y := let z := [|x|]+[|y|]+1 in if Z_lt_le_dec z wB then C0 z else C1 (z-wB). @@ -210,7 +210,7 @@ Section ZModulo. Definition znz_add := Zplus. Definition znz_add_carry x y := x + y + 1. - Lemma Zmod_equal : + Lemma Zmod_equal : forall x y z, z>0 -> (x-y) mod z = 0 -> x mod z = y mod z. Proof. intros. @@ -225,12 +225,12 @@ Section ZModulo. Proof. intros; unfold znz_succ_c, znz_to_Z, Zsucc. case_eq (znz_eq0 (x+1)); intros; unfold interp_carry. - + rewrite Zmult_1_l. replace (wB + 0 mod wB) with wB by auto with zarith. symmetry; rewrite Zeq_plus_swap. assert ((x+1) mod wB = 0) by (apply spec_eq0; auto). - replace (wB-1) with ((wB-1) mod wB) by + replace (wB-1) with ((wB-1) mod wB) by (apply Zmod_small; generalize wB_pos; omega). rewrite <- Zminus_mod_idemp_l; rewrite Z_mod_same; simpl; auto. apply Zmod_equal; auto. @@ -289,15 +289,15 @@ Section ZModulo. rewrite Zplus_mod_idemp_l; auto. Qed. - Definition znz_pred_c x := + Definition znz_pred_c x := if znz_eq0 x then C1 (wB-1) else C0 (x-1). - Definition znz_sub_c x y := - let z := [|x|]-[|y|] in + Definition znz_sub_c x y := + let z := [|x|]-[|y|] in if Z_lt_le_dec z 0 then C1 (wB+z) else C0 z. - Definition znz_sub_carry_c x y := - let z := [|x|]-[|y|]-1 in + Definition znz_sub_carry_c x y := + let z := [|x|]-[|y|]-1 in if Z_lt_le_dec z 0 then C1 (wB+z) else C0 z. Definition znz_pred := Zpred. @@ -323,7 +323,7 @@ Section ZModulo. Proof. intros; unfold znz_sub_c, znz_to_Z, interp_carry. destruct Z_lt_le_dec. - replace ((wB + (x mod wB - y mod wB)) mod wB) with + replace ((wB + (x mod wB - y mod wB)) mod wB) with (wB + (x mod wB - y mod wB)). omega. symmetry; apply Zmod_small. @@ -337,7 +337,7 @@ Section ZModulo. Proof. intros; unfold znz_sub_carry_c, znz_to_Z, interp_carry. destruct Z_lt_le_dec. - replace ((wB + (x mod wB - y mod wB - 1)) mod wB) with + replace ((wB + (x mod wB - y mod wB - 1)) mod wB) with (wB + (x mod wB - y mod wB -1)). omega. symmetry; apply Zmod_small. @@ -358,7 +358,7 @@ Section ZModulo. intros; unfold znz_sub, znz_to_Z; apply Zminus_mod. Qed. - Lemma spec_sub_carry : + Lemma spec_sub_carry : forall x y, [|znz_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB. Proof. intros; unfold znz_sub_carry, znz_to_Z. @@ -367,15 +367,15 @@ Section ZModulo. rewrite Zminus_mod_idemp_l. auto. Qed. - - Definition znz_mul_c x y := + + Definition znz_mul_c x y := let (h,l) := Zdiv_eucl ([|x|]*[|y|]) wB in if znz_eq0 h then if znz_eq0 l then W0 else WW h l else WW h l. Definition znz_mul := Zmult. Definition znz_square_c x := znz_mul_c x x. - + Lemma spec_mul_c : forall x y, [|| znz_mul_c x y ||] = [|x|] * [|y|]. Proof. intros; unfold znz_mul_c, zn2z_to_Z. @@ -426,7 +426,7 @@ Section ZModulo. destruct Zdiv_eucl as (q,r); destruct 1; intros. injection H1; clear H1; intros. assert ([|r|]=r). - apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; + apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; auto with zarith. assert ([|q|]=q). apply Zmod_small. @@ -453,7 +453,7 @@ Section ZModulo. Definition znz_mod x y := [|x|] mod [|y|]. Definition znz_mod_gt x y := [|x|] mod [|y|]. - + Lemma spec_mod : forall a b, 0 < [|b|] -> [|znz_mod a b|] = [|a|] mod [|b|]. Proof. @@ -469,7 +469,7 @@ Section ZModulo. Proof. intros; apply spec_mod; auto. Qed. - + Definition znz_gcd x y := Zgcd [|x|] [|y|]. Definition znz_gcd_gt x y := Zgcd [|x|] [|y|]. @@ -516,7 +516,7 @@ Section ZModulo. intros. apply spec_gcd; auto. Qed. - Definition znz_div21 a1 a2 b := + Definition znz_div21 a1 a2 b := Zdiv_eucl ([|a1|]*wB+[|a2|]) [|b|]. Lemma spec_div21 : forall a1 a2 b, @@ -537,7 +537,7 @@ Section ZModulo. destruct Zdiv_eucl as (q,r); destruct 1; intros. injection H4; clear H4; intros. assert ([|r|]=r). - apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; + apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; auto with zarith. assert ([|q|]=q). apply Zmod_small. @@ -576,7 +576,7 @@ Section ZModulo. apply Zmod_le; auto with zarith. Qed. - Definition znz_is_even x := + Definition znz_is_even x := if Z_eq_dec ([|x|] mod 2) 0 then true else false. Lemma spec_is_even : forall x, @@ -586,7 +586,7 @@ Section ZModulo. generalize (Z_mod_lt [|x|] 2); omega. Qed. - Definition znz_sqrt x := Zsqrt_plain [|x|]. + Definition znz_sqrt x := Zsqrt_plain [|x|]. Lemma spec_sqrt : forall x, [|znz_sqrt x|] ^ 2 <= [|x|] < ([|znz_sqrt x|] + 1) ^ 2. Proof. @@ -609,12 +609,12 @@ Section ZModulo. generalize wB_pos; auto with zarith. Qed. - Definition znz_sqrt2 x y := - let z := [|x|]*wB+[|y|] in - match z with + Definition znz_sqrt2 x y := + let z := [|x|]*wB+[|y|] in + match z with | Z0 => (0, C0 0) - | Zpos p => - let (s,r,_,_) := sqrtrempos p in + | Zpos p => + let (s,r,_,_) := sqrtrempos p in (s, if Z_lt_le_dec r wB then C0 r else C1 (r-wB)) | Zneg _ => (0, C0 0) end. @@ -651,7 +651,7 @@ Section ZModulo. rewrite Zpower_2; auto with zarith. replace [|r-wB|] with (r-wB) by (symmetry; apply Zmod_small; auto with zarith). rewrite Zpower_2; omega. - + assert (0<=Zneg p). rewrite Heqz; generalize wB_pos; auto with zarith. compute in H0; elim H0; auto. @@ -665,8 +665,8 @@ Section ZModulo. apply two_power_pos_correct. Qed. - Definition znz_head0 x := match [|x|] with - | Z0 => znz_zdigits + Definition znz_head0 x := match [|x|] with + | Z0 => znz_zdigits | Zpos p => znz_zdigits - log_inf p - 1 | _ => 0 end. @@ -695,7 +695,7 @@ Section ZModulo. change (Zpos x~0) with (2*(Zpos x)) in H. replace p with (Zsucc (p-1)) in H; auto with zarith. rewrite Zpower_Zsucc in H; auto with zarith. - + simpl; intros; destruct p; compute; auto with zarith. Qed. @@ -730,8 +730,8 @@ Section ZModulo. by ring. unfold wB, base, znz_zdigits; auto with zarith. apply Zmult_le_compat; auto with zarith. - - apply Zlt_le_trans + + apply Zlt_le_trans with (2^(znz_zdigits - log_inf p - 1)*(2^(Zsucc (log_inf p)))). apply Zmult_lt_compat_l; auto with zarith. rewrite <- Zpower_exp; auto with zarith. @@ -740,17 +740,17 @@ Section ZModulo. unfold wB, base, znz_zdigits; auto with zarith. Qed. - Fixpoint Ptail p := match p with + Fixpoint Ptail p := match p with | xO p => (Ptail p)+1 | _ => 0 - end. + end. Lemma Ptail_pos : forall p, 0 <= Ptail p. Proof. induction p; simpl; auto with zarith. Qed. Hint Resolve Ptail_pos. - + Lemma Ptail_bounded : forall p d, Zpos p < 2^(Zpos d) -> Ptail p < Zpos d. Proof. induction p; try (compute; auto; fail). @@ -775,7 +775,7 @@ Section ZModulo. Qed. Definition znz_tail0 x := - match [|x|] with + match [|x|] with | Z0 => znz_zdigits | Zpos p => Ptail p | Zneg _ => 0 @@ -788,7 +788,7 @@ Section ZModulo. apply spec_zdigits. Qed. - Lemma spec_tail0 : forall x, 0 < [|x|] -> + Lemma spec_tail0 : forall x, 0 < [|x|] -> exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|znz_tail0 x|]). Proof. intros; unfold znz_tail0. @@ -818,7 +818,7 @@ Section ZModulo. (** Let's now group everything in two records *) - Definition zmod_op := mk_znz_op + Definition zmod_op := mk_znz_op (znz_digits : positive) (znz_zdigits: znz) (znz_to_Z : znz -> Z) @@ -859,11 +859,11 @@ Section ZModulo. (znz_div_gt : znz -> znz -> znz * znz) (znz_div : znz -> znz -> znz * znz) - (znz_mod_gt : znz -> znz -> znz) - (znz_mod : znz -> znz -> znz) + (znz_mod_gt : znz -> znz -> znz) + (znz_mod : znz -> znz -> znz) (znz_gcd_gt : znz -> znz -> znz) - (znz_gcd : znz -> znz -> znz) + (znz_gcd : znz -> znz -> znz) (znz_add_mul_div : znz -> znz -> znz -> znz) (znz_pos_mod : znz -> znz -> znz) @@ -878,54 +878,54 @@ Section ZModulo. spec_more_than_1_digit spec_0 - spec_1 - spec_Bm1 - - spec_compare - spec_eq0 - - spec_opp_c - spec_opp - spec_opp_carry - - spec_succ_c - spec_add_c - spec_add_carry_c - spec_succ - spec_add - spec_add_carry - - spec_pred_c - spec_sub_c - spec_sub_carry_c - spec_pred - spec_sub - spec_sub_carry - - spec_mul_c - spec_mul - spec_square_c - - spec_div21 - spec_div_gt - spec_div - - spec_mod_gt - spec_mod - - spec_gcd_gt - spec_gcd - - spec_head00 - spec_head0 - spec_tail00 - spec_tail0 - - spec_add_mul_div - spec_pos_mod - - spec_is_even - spec_sqrt2 + spec_1 + spec_Bm1 + + spec_compare + spec_eq0 + + spec_opp_c + spec_opp + spec_opp_carry + + spec_succ_c + spec_add_c + spec_add_carry_c + spec_succ + spec_add + spec_add_carry + + spec_pred_c + spec_sub_c + spec_sub_carry_c + spec_pred + spec_sub + spec_sub_carry + + spec_mul_c + spec_mul + spec_square_c + + spec_div21 + spec_div_gt + spec_div + + spec_mod_gt + spec_mod + + spec_gcd_gt + spec_gcd + + spec_head00 + spec_head0 + spec_tail00 + spec_tail0 + + spec_add_mul_div + spec_pos_mod + + spec_is_even + spec_sqrt2 spec_sqrt. End ZModulo. @@ -934,7 +934,7 @@ End ZModulo. Module Type PositiveNotOne. Parameter p : positive. - Axiom not_one : p<> 1%positive. + Axiom not_one : p<> 1%positive. End PositiveNotOne. Module ZModuloCyclicType (P:PositiveNotOne) <: CyclicType. diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v index cbf6f701f2..dc22256348 100644 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ b/theories/Numbers/Integer/BigZ/ZMake.v @@ -17,31 +17,31 @@ Require Import ZSig. Open Scope Z_scope. -(** * ZMake - - A generic transformation from a structure of natural numbers +(** * ZMake + + A generic transformation from a structure of natural numbers [NSig.NType] to a structure of integers [ZSig.ZType]. *) Module Make (N:NType) <: ZType. - - Inductive t_ := + + Inductive t_ := | Pos : N.t -> t_ | Neg : N.t -> t_. - + Definition t := t_. Definition zero := Pos N.zero. Definition one := Pos N.one. Definition minus_one := Neg N.one. - Definition of_Z x := + Definition of_Z x := match x with | Zpos x => Pos (N.of_N (Npos x)) | Z0 => zero | Zneg x => Neg (N.of_N (Npos x)) end. - + Definition to_Z x := match x with | Pos nx => N.to_Z nx @@ -99,13 +99,13 @@ Module Make (N:NType) <: ZType. unfold compare, to_Z; intros x y; case x; case y; clear x y; intros x y; auto; generalize (N.spec_pos x) (N.spec_pos y). generalize (N.spec_compare y x); case N.compare; auto with zarith. - generalize (N.spec_compare y N.zero); case N.compare; + generalize (N.spec_compare y N.zero); case N.compare; try rewrite N.spec_0; auto with zarith. generalize (N.spec_compare x N.zero); case N.compare; rewrite N.spec_0; auto with zarith. generalize (N.spec_compare x N.zero); case N.compare; rewrite N.spec_0; auto with zarith. - generalize (N.spec_compare N.zero y); case N.compare; + generalize (N.spec_compare N.zero y); case N.compare; try rewrite N.spec_0; auto with zarith. generalize (N.spec_compare N.zero x); case N.compare; rewrite N.spec_0; auto with zarith. @@ -114,7 +114,7 @@ Module Make (N:NType) <: ZType. generalize (N.spec_compare x y); case N.compare; auto with zarith. Qed. - Definition eq_bool x y := + Definition eq_bool x y := match compare x y with | Eq => true | _ => false @@ -128,9 +128,9 @@ Module Make (N:NType) <: ZType. Definition cmp_sign x y := match x, y with - | Pos nx, Neg ny => - if N.eq_bool ny N.zero then Eq else Gt - | Neg nx, Pos ny => + | Pos nx, Neg ny => + if N.eq_bool ny N.zero then Eq else Gt + | Neg nx, Pos ny => if N.eq_bool nx N.zero then Eq else Lt | _, _ => Eq end. @@ -150,7 +150,7 @@ Module Make (N:NType) <: ZType. rewrite N.spec_0; unfold to_Z. generalize (N.spec_pos x) (N.spec_pos y); auto with zarith. Qed. - + Definition to_N x := match x with | Pos nx => nx @@ -164,9 +164,9 @@ Module Make (N:NType) <: ZType. simpl; rewrite Zabs_eq; auto. simpl; rewrite Zabs_non_eq; simpl; auto with zarith. Qed. - - Definition opp x := - match x with + + Definition opp x := + match x with | Pos nx => Neg nx | Neg nx => Pos nx end. @@ -174,7 +174,7 @@ Module Make (N:NType) <: ZType. Theorem spec_opp: forall x, to_Z (opp x) = - to_Z x. intros x; case x; simpl; auto with zarith. Qed. - + Definition succ x := match x with | Pos n => Pos (N.succ n) @@ -188,7 +188,7 @@ Module Make (N:NType) <: ZType. Theorem spec_succ: forall n, to_Z (succ n) = to_Z n + 1. intros x; case x; clear x; intros x. exact (N.spec_succ x). - simpl; generalize (N.spec_compare N.zero x); case N.compare; + simpl; generalize (N.spec_compare N.zero x); case N.compare; rewrite N.spec_0; simpl. intros HH; rewrite <- HH; rewrite N.spec_1; ring. intros HH; rewrite N.spec_pred; auto with zarith. @@ -212,7 +212,7 @@ Module Make (N:NType) <: ZType. end | Neg nx, Neg ny => Neg (N.add nx ny) end. - + Theorem spec_add: forall x y, to_Z (add x y) = to_Z x + to_Z y. unfold add, to_Z; intros [x | x] [y | y]. exact (N.spec_add x y). @@ -239,7 +239,7 @@ Module Make (N:NType) <: ZType. Theorem spec_pred: forall x, to_Z (pred x) = to_Z x - 1. unfold pred, to_Z, minus_one; intros [x | x]. - generalize (N.spec_compare N.zero x); case N.compare; + generalize (N.spec_compare N.zero x); case N.compare; rewrite N.spec_0; try rewrite N.spec_1; auto with zarith. intros H; exact (N.spec_pred _ H). generalize (N.spec_pos x); auto with zarith. @@ -248,7 +248,7 @@ Module Make (N:NType) <: ZType. Definition sub x y := match x, y with - | Pos nx, Pos ny => + | Pos nx, Pos ny => match N.compare nx ny with | Gt => Pos (N.sub nx ny) | Eq => zero @@ -256,7 +256,7 @@ Module Make (N:NType) <: ZType. end | Pos nx, Neg ny => Pos (N.add nx ny) | Neg nx, Pos ny => Neg (N.add nx ny) - | Neg nx, Neg ny => + | Neg nx, Neg ny => match N.compare nx ny with | Gt => Neg (N.sub nx ny) | Eq => zero @@ -278,7 +278,7 @@ Module Make (N:NType) <: ZType. intros; rewrite N.spec_sub; try ring; auto with zarith. Qed. - Definition mul x y := + Definition mul x y := match x, y with | Pos nx, Pos ny => Pos (N.mul nx ny) | Pos nx, Neg ny => Neg (N.mul nx ny) @@ -291,7 +291,7 @@ Module Make (N:NType) <: ZType. unfold mul, to_Z; intros [x | x] [y | y]; rewrite N.spec_mul; ring. Qed. - Definition square x := + Definition square x := match x with | Pos nx => Pos (N.square nx) | Neg nx => Pos (N.square nx) @@ -304,7 +304,7 @@ Module Make (N:NType) <: ZType. Definition power_pos x p := match x with | Pos nx => Pos (N.power_pos nx p) - | Neg nx => + | Neg nx => match p with | xH => x | xO _ => Pos (N.power_pos nx p) @@ -315,7 +315,7 @@ Module Make (N:NType) <: ZType. Theorem spec_power_pos: forall x n, to_Z (power_pos x n) = to_Z x ^ Zpos n. assert (F0: forall x, (-x)^2 = x^2). intros x; rewrite Zpower_2; ring. - unfold power_pos, to_Z; intros [x | x] [p | p |]; + unfold power_pos, to_Z; intros [x | x] [p | p |]; try rewrite N.spec_power_pos; try ring. assert (F: 0 <= 2 * Zpos p). assert (0 <= Zpos p); auto with zarith. @@ -336,7 +336,7 @@ Module Make (N:NType) <: ZType. end. - Theorem spec_sqrt: forall x, 0 <= to_Z x -> + Theorem spec_sqrt: forall x, 0 <= to_Z x -> to_Z (sqrt x) ^ 2 <= to_Z x < (to_Z (sqrt x) + 1) ^ 2. unfold to_Z, sqrt; intros [x | x] H. exact (N.spec_sqrt x). @@ -381,7 +381,7 @@ Module Make (N:NType) <: ZType. generalize (N.spec_pos y); auto with zarith. generalize (N.spec_div_eucl x y HH); case N.div_eucl; auto. intros q r; generalize (N.spec_pos x) HH; unfold Zdiv_eucl; - case_eq (N.to_Z x); case_eq (N.to_Z y); + case_eq (N.to_Z x); case_eq (N.to_Z y); try (intros; apply False_ind; auto with zarith; fail). intros p He1 He2 _ _ H1; injection H1; intros H2 H3. generalize (N.spec_compare N.zero r); case N.compare; @@ -407,13 +407,13 @@ Module Make (N:NType) <: ZType. assert (N.to_Z r = (Zpos p1 mod (Zpos p))). unfold Zmod, Zdiv_eucl; rewrite <- H3; auto. case (Z_mod_lt (Zpos p1) (Zpos p)); auto with zarith. - rewrite N.spec_0; intros H2; generalize (N.spec_pos r); + rewrite N.spec_0; intros H2; generalize (N.spec_pos r); intros; apply False_ind; auto with zarith. assert (HH: 0 < N.to_Z y). generalize (N.spec_pos y); auto with zarith. generalize (N.spec_div_eucl x y HH); case N.div_eucl; auto. intros q r; generalize (N.spec_pos x) HH; unfold Zdiv_eucl; - case_eq (N.to_Z x); case_eq (N.to_Z y); + case_eq (N.to_Z x); case_eq (N.to_Z y); try (intros; apply False_ind; auto with zarith; fail). intros p He1 He2 _ _ H1; injection H1; intros H2 H3. generalize (N.spec_compare N.zero r); case N.compare; @@ -443,7 +443,7 @@ Module Make (N:NType) <: ZType. generalize (N.spec_pos y); auto with zarith. generalize (N.spec_div_eucl x y H1); case N.div_eucl; auto. intros q r; generalize (N.spec_pos x) H1; unfold Zdiv_eucl; - case_eq (N.to_Z x); case_eq (N.to_Z y); + case_eq (N.to_Z x); case_eq (N.to_Z y); try (intros; apply False_ind; auto with zarith; fail). change (-0) with 0; lazy iota beta; auto. intros p _ _ _ _ H2; injection H2. @@ -478,7 +478,7 @@ Module Make (N:NType) <: ZType. | Pos nx, Pos ny => Pos (N.gcd nx ny) | Pos nx, Neg ny => Pos (N.gcd nx ny) | Neg nx, Pos ny => Pos (N.gcd nx ny) - | Neg nx, Neg ny => Pos (N.gcd nx ny) + | Neg nx, Neg ny => Pos (N.gcd nx ny) end. Theorem spec_gcd: forall a b, to_Z (gcd a b) = Zgcd (to_Z a) (to_Z b). diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v index 4e45939831..00e292db0f 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSig.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSig.v @@ -58,7 +58,7 @@ Module Type ZType. Parameter spec_eq_bool: forall x y, if eq_bool x y then [x] = [y] else [x] <> [y]. - + Parameter succ : t -> t. Parameter spec_succ: forall n, [succ n] = [n] + 1. @@ -93,21 +93,21 @@ Module Type ZType. Parameter sqrt : t -> t. - Parameter spec_sqrt: forall x, 0 <= [x] -> + Parameter spec_sqrt: forall x, 0 <= [x] -> [sqrt x] ^ 2 <= [x] < ([sqrt x] + 1) ^ 2. Parameter div_eucl : t -> t -> t * t. Parameter spec_div_eucl: forall x y, [y] <> 0 -> let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y]. - + Parameter div : t -> t -> t. Parameter spec_div: forall x y, [y] <> 0 -> [div x y] = [x] / [y]. Parameter modulo : t -> t -> t. - Parameter spec_modulo: forall x y, [y] <> 0 -> + Parameter spec_modulo: forall x y, [y] <> 0 -> [modulo x y] = [x] mod [y]. Parameter gcd : t -> t -> t. diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v index 4d1054553f..030c589ff9 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v @@ -27,7 +27,7 @@ Infix "-" := Z.sub : IntScope. Infix "*" := Z.mul : IntScope. Notation "- x" := (Z.opp x) : IntScope. -Hint Rewrite +Hint Rewrite Z.spec_0 Z.spec_1 Z.spec_add Z.spec_sub Z.spec_pred Z.spec_succ Z.spec_mul Z.spec_opp Z.spec_of_Z : Zspec. @@ -91,7 +91,7 @@ Section Induction. Variable A : Z.t -> Prop. Hypothesis A_wd : predicate_wd Z.eq A. Hypothesis A0 : A 0. -Hypothesis AS : forall n, A n <-> A (Z.succ n). +Hypothesis AS : forall n, A n <-> A (Z.succ n). Add Morphism A with signature Z.eq ==> iff as A_morph. Proof. apply A_wd. Qed. @@ -214,7 +214,7 @@ Proof. Qed. Add Morphism Z.compare with signature Z.eq ==> Z.eq ==> (@eq comparison) as compare_wd. -Proof. +Proof. intros x x' Hx y y' Hy. rewrite 2 spec_compare_alt; unfold Z.eq in *; rewrite Hx, Hy; intuition. Qed. diff --git a/theories/Numbers/NaryFunctions.v b/theories/Numbers/NaryFunctions.v index feb7a49166..a8adf49af6 100644 --- a/theories/Numbers/NaryFunctions.v +++ b/theories/Numbers/NaryFunctions.v @@ -16,19 +16,19 @@ Require Import List. (** * Generic dependently-typed operators about [n]-ary functions *) -(** The type of [n]-ary function: [nfun A n B] is +(** The type of [n]-ary function: [nfun A n B] is [A -> ... -> A -> B] with [n] occurences of [A] in this type. *) -Fixpoint nfun A n B := +Fixpoint nfun A n B := match n with - | O => B + | O => B | S n => A -> (nfun A n B) - end. + end. Notation " A ^^ n --> B " := (nfun A n B) (at level 50, n at next level) : type_scope. -(** [napply_cst _ _ a n f] iterates [n] times the application of a +(** [napply_cst _ _ a n f] iterates [n] times the application of a particular constant [a] to the [n]-ary function [f]. *) Fixpoint napply_cst (A B:Type)(a:A) n : (A^^n-->B) -> B := @@ -40,47 +40,47 @@ Fixpoint napply_cst (A B:Type)(a:A) n : (A^^n-->B) -> B := (** A generic transformation from an n-ary function to another one.*) -Fixpoint nfun_to_nfun (A B C:Type)(f:B -> C) n : +Fixpoint nfun_to_nfun (A B C:Type)(f:B -> C) n : (A^^n-->B) -> (A^^n-->C) := - match n return (A^^n-->B) -> (A^^n-->C) with + match n return (A^^n-->B) -> (A^^n-->C) with | O => f | S n => fun g a => nfun_to_nfun _ _ _ f n (g a) end. -(** [napply_except_last _ _ n f] expects [n] arguments of type [A], - applies [n-1] of them to [f] and discard the last one. *) +(** [napply_except_last _ _ n f] expects [n] arguments of type [A], + applies [n-1] of them to [f] and discard the last one. *) -Definition napply_except_last (A B:Type) := +Definition napply_except_last (A B:Type) := nfun_to_nfun A B (A->B) (fun b a => b). -(** [napply_then_last _ _ a n f] expects [n] arguments of type [A], - applies them to [f] and then apply [a] to the result. *) +(** [napply_then_last _ _ a n f] expects [n] arguments of type [A], + applies them to [f] and then apply [a] to the result. *) -Definition napply_then_last (A B:Type)(a:A) := +Definition napply_then_last (A B:Type)(a:A) := nfun_to_nfun A (A->B) B (fun fab => fab a). -(** [napply_discard _ b n] expects [n] arguments, discards then, +(** [napply_discard _ b n] expects [n] arguments, discards then, and returns [b]. *) Fixpoint napply_discard (A B:Type)(b:B) n : A^^n-->B := - match n return A^^n-->B with + match n return A^^n-->B with | O => b | S n => fun _ => napply_discard _ _ b n end. (** A fold function *) -Fixpoint nfold A B (f:A->B->B)(b:B) n : (A^^n-->B) := - match n return (A^^n-->B) with +Fixpoint nfold A B (f:A->B->B)(b:B) n : (A^^n-->B) := + match n return (A^^n-->B) with | O => b | S n => fun a => (nfold _ _ f (f a b) n) end. -(** [n]-ary products : [nprod A n] is [A*...*A*unit], +(** [n]-ary products : [nprod A n] is [A*...*A*unit], with [n] occurrences of [A] in this type. *) -Fixpoint nprod A n : Type := match n with +Fixpoint nprod A n : Type := match n with | O => unit | S n => (A * nprod A n)%type end. @@ -89,54 +89,54 @@ Notation "A ^ n" := (nprod A n) : type_scope. (** [n]-ary curryfication / uncurryfication *) -Fixpoint ncurry (A B:Type) n : (A^n -> B) -> (A^^n-->B) := - match n return (A^n -> B) -> (A^^n-->B) with +Fixpoint ncurry (A B:Type) n : (A^n -> B) -> (A^^n-->B) := + match n return (A^n -> B) -> (A^^n-->B) with | O => fun x => x tt | S n => fun f a => ncurry _ _ n (fun p => f (a,p)) end. -Fixpoint nuncurry (A B:Type) n : (A^^n-->B) -> (A^n -> B) := +Fixpoint nuncurry (A B:Type) n : (A^^n-->B) -> (A^n -> B) := match n return (A^^n-->B) -> (A^n -> B) with | O => fun x _ => x | S n => fun f p => let (x,p) := p in nuncurry _ _ n (f x) p end. -(** Earlier functions can also be defined via [ncurry/nuncurry]. +(** Earlier functions can also be defined via [ncurry/nuncurry]. For instance : *) Definition nfun_to_nfun_bis A B C (f:B->C) n : - (A^^n-->B) -> (A^^n-->C) := + (A^^n-->B) -> (A^^n-->C) := fun anb => ncurry _ _ n (fun an => f ((nuncurry _ _ n anb) an)). -(** We can also us it to obtain another [fold] function, +(** We can also us it to obtain another [fold] function, equivalent to the previous one, but with a nicer expansion (see for instance Int31.iszero). *) -Fixpoint nfold_bis A B (f:A->B->B)(b:B) n : (A^^n-->B) := - match n return (A^^n-->B) with +Fixpoint nfold_bis A B (f:A->B->B)(b:B) n : (A^^n-->B) := + match n return (A^^n-->B) with | O => b - | S n => fun a => + | S n => fun a => nfun_to_nfun_bis _ _ _ (f a) n (nfold_bis _ _ f b n) end. (** From [nprod] to [list] *) -Fixpoint nprod_to_list (A:Type) n : A^n -> list A := - match n with +Fixpoint nprod_to_list (A:Type) n : A^n -> list A := + match n with | O => fun _ => nil | S n => fun p => let (x,p) := p in x::(nprod_to_list _ n p) end. (** From [list] to [nprod] *) -Fixpoint nprod_of_list (A:Type)(l:list A) : A^(length l) := - match l return A^(length l) with +Fixpoint nprod_of_list (A:Type)(l:list A) : A^(length l) := + match l return A^(length l) with | nil => tt | x::l => (x, nprod_of_list _ l) end. (** This gives an additional way to write the fold *) -Definition nfold_list (A B:Type)(f:A->B->B)(b:B) n : (A^^n-->B) := +Definition nfold_list (A B:Type)(f:A->B->B)(b:B) n : (A^^n-->B) := ncurry _ _ n (fun p => fold_right f b (nprod_to_list _ _ p)). diff --git a/theories/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v index 1ef7809866..a9c023856f 100644 --- a/theories/Numbers/NatInt/NZAxioms.v +++ b/theories/Numbers/NatInt/NZAxioms.v @@ -23,7 +23,7 @@ Parameter Inline NZadd : NZ -> NZ -> NZ. Parameter Inline NZsub : NZ -> NZ -> NZ. Parameter Inline NZmul : NZ -> NZ -> NZ. -(* Unary subtraction (opp) is not defined on natural numbers, so we have +(* Unary subtraction (opp) is not defined on natural numbers, so we have it for integers only *) Axiom NZeq_equiv : equiv NZ NZeq. diff --git a/theories/Numbers/Natural/Abstract/NOrder.v b/theories/Numbers/Natural/Abstract/NOrder.v index 5212e63814..f02baca2cf 100644 --- a/theories/Numbers/Natural/Abstract/NOrder.v +++ b/theories/Numbers/Natural/Abstract/NOrder.v @@ -309,7 +309,7 @@ Proof NZgt_wf. Theorem lt_wf_0 : well_founded lt. Proof. -setoid_replace lt with (fun n m : N => 0 <= n /\ n < m) +setoid_replace lt with (fun n m : N => 0 <= n /\ n < m) using relation (@relations_eq N N). apply lt_wf. intros x y; split. diff --git a/theories/Numbers/Natural/BigN/NMake_gen.ml b/theories/Numbers/Natural/BigN/NMake_gen.ml index 7424d877bb..c22680be3a 100644 --- a/theories/Numbers/Natural/BigN/NMake_gen.ml +++ b/theories/Numbers/Natural/BigN/NMake_gen.ml @@ -15,7 +15,7 @@ (*s The two parameters that control the generation: *) -let size = 6 (* how many times should we repeat the Z/nZ --> Z/2nZ +let size = 6 (* how many times should we repeat the Z/nZ --> Z/2nZ process before relying on a generic construct *) let gen_proof = true (* should we generate proofs ? *) @@ -27,18 +27,18 @@ let c = "N" let pz n = if n == 0 then "w_0" else "W0" let rec gen2 n = if n == 0 then "1" else if n == 1 then "2" else "2 * " ^ (gen2 (n - 1)) -let rec genxO n s = +let rec genxO n s = if n == 0 then s else " (xO" ^ (genxO (n - 1) s) ^ ")" -(* NB: in ocaml >= 3.10, we could use Printf.ifprintf for printing to - /dev/null, but for being compatible with earlier ocaml and not - relying on system-dependent stuff like open_out "/dev/null", +(* NB: in ocaml >= 3.10, we could use Printf.ifprintf for printing to + /dev/null, but for being compatible with earlier ocaml and not + relying on system-dependent stuff like open_out "/dev/null", let's use instead a magical hack *) (* Standard printer, with a final newline *) let pr s = Printf.printf (s^^"\n") (* Printing to /dev/null *) -let pn = (fun s -> Obj.magic (fun _ _ _ _ _ _ _ _ _ _ _ _ _ _ -> ()) +let pn = (fun s -> Obj.magic (fun _ _ _ _ _ _ _ _ _ _ _ _ _ _ -> ()) : ('a, out_channel, unit) format -> 'a) (* Proof printer : prints iff gen_proof is true *) let pp = if gen_proof then pr else pn @@ -51,7 +51,7 @@ let pp0 = if gen_proof then pr0 else pn (*s The actual printing *) -let _ = +let _ = pr "(************************************************************************)"; pr "(* v * The Coq Proof Assistant / The Coq Development Team *)"; @@ -67,7 +67,7 @@ let _ = pr ""; pr "(** From a cyclic Z/nZ representation to arbitrary precision natural numbers.*)"; pr ""; - pr "(** Remark: File automatically generated by NMake_gen.ml, DO NOT EDIT ! *)"; + pr "(** Remark: File automatically generated by NMake_gen.ml, DO NOT EDIT ! *)"; pr ""; pr "Require Import BigNumPrelude."; pr "Require Import ZArith."; @@ -132,7 +132,7 @@ let _ = pr ""; pr " Inductive %s_ :=" t; - for i = 0 to size do + for i = 0 to size do pr " | %s%i : w%i -> %s_" c i i t done; pr " | %sn : forall n, word w%i (S n) -> %s_." c size t; @@ -167,7 +167,7 @@ let _ = pr " Definition to_N x := Zabs_N (to_Z x)."; pr ""; - + pr " Definition eq x y := (to_Z x = to_Z y)."; pr ""; @@ -191,7 +191,7 @@ let _ = for i = 0 to size do pp " Let nmake_op%i := nmake_op _ w%i_op." i i; pp " Let eval%in n := znz_to_Z (nmake_op%i n)." i i; - if i == 0 then + if i == 0 then pr " Let extend%i := DoubleBase.extend (WW w_0)." i else pr " Let extend%i := DoubleBase.extend (WW (W0: w%i))." i i; @@ -280,7 +280,7 @@ let _ = pp " Let w0_spec: znz_spec w0_op := W0.w_spec."; for i = 1 to 3 do - pp " Let w%i_spec: znz_spec w%i_op := mk_znz2_spec w%i_spec." i i (i-1) + pp " Let w%i_spec: znz_spec w%i_op := mk_znz2_spec w%i_spec." i i (i-1) done; for i = 4 to size + 3 do pp " Let w%i_spec : znz_spec w%i_op := mk_znz2_karatsuba_spec w%i_spec." i i (i-1) @@ -309,14 +309,14 @@ let _ = for i = 0 to size do - pp " Theorem digits_w%i: znz_digits w%i_op = znz_digits (nmake_op _ w0_op %i)." i i i; + pp " Theorem digits_w%i: znz_digits w%i_op = znz_digits (nmake_op _ w0_op %i)." i i i; if i == 0 then pp " auto." else pp " rewrite digits_nmake; rewrite <- digits_w%i; auto." (i - 1); pp " Qed."; pp ""; - pp " Let spec_double_eval%in: forall n, eval%in n = DoubleBase.double_to_Z (znz_digits w%i_op) (znz_to_Z w%i_op) n." i i i i; + pp " Let spec_double_eval%in: forall n, eval%in n = DoubleBase.double_to_Z (znz_digits w%i_op) (znz_to_Z w%i_op) n." i i i i; pp " Proof."; pp " intros n; exact (nmake_double n w%i w%i_op)." i i; pp " Qed."; @@ -325,7 +325,7 @@ let _ = for i = 0 to size do for j = 0 to (size - i) do - pp " Theorem digits_w%in%i: znz_digits w%i_op = znz_digits (nmake_op _ w%i_op %i)." i j (i + j) i j; + pp " Theorem digits_w%in%i: znz_digits w%i_op = znz_digits (nmake_op _ w%i_op %i)." i j (i + j) i j; pp " Proof."; if j == 0 then if i == 0 then @@ -346,7 +346,7 @@ let _ = end; pp " Qed."; pp ""; - pp " Let spec_eval%in%i: forall x, [%s%i x] = eval%in %i x." i j c (i + j) i j; + pp " Let spec_eval%in%i: forall x, [%s%i x] = eval%in %i x." i j c (i + j) i j; pp " Proof."; if j == 0 then pp " intros x; rewrite spec_double_eval%in; unfold DoubleBase.double_to_Z, to_Z; auto." i @@ -363,7 +363,7 @@ let _ = pp " Qed."; if i + j <> size then begin - pp " Let spec_extend%in%i: forall x, [%s%i x] = [%s%i (extend%i %i x)]." i (i + j + 1) c i c (i + j + 1) i j; + pp " Let spec_extend%in%i: forall x, [%s%i x] = [%s%i (extend%i %i x)]." i (i + j + 1) c i c (i + j + 1) i j; if j == 0 then begin pp " intros x; change (extend%i 0 x) with (WW (znz_0 w%i_op) x)." i (i + j); @@ -393,7 +393,7 @@ let _ = pp " Qed."; pp ""; - pp " Let spec_eval%in%i: forall x, [%sn 0 x] = eval%in %i x." i (size - i + 1) c i (size - i + 1); + pp " Let spec_eval%in%i: forall x, [%sn 0 x] = eval%in %i x." i (size - i + 1) c i (size - i + 1); pp " Proof."; pp " intros x; case x."; pp " auto."; @@ -405,7 +405,7 @@ let _ = pp " Qed."; pp ""; - pp " Let spec_eval%in%i: forall x, [%sn 1 x] = eval%in %i x." i (size - i + 2) c i (size - i + 2); + pp " Let spec_eval%in%i: forall x, [%sn 1 x] = eval%in %i x." i (size - i + 2) c i (size - i + 2); pp " intros x; case x."; pp " auto."; pp " intros xh xl; unfold to_Z; rewrite znz_to_Z_%i." (size + 2); @@ -430,7 +430,7 @@ let _ = pp " Qed."; pp ""; - pp " Let spec_eval%in: forall n x, [%sn n x] = eval%in (S n) x." size c size; + pp " Let spec_eval%in: forall n x, [%sn n x] = eval%in (S n) x." size c size; pp " intros n; elim n; clear n."; pp " exact spec_eval%in1." size; pp " intros n Hrec x; case x; clear x."; @@ -446,7 +446,7 @@ let _ = pp " Qed."; pp ""; - pp " Let spec_extend%in: forall n x, [%s%i x] = [%sn n (extend%i n x)]." size c size c size ; + pp " Let spec_extend%in: forall n x, [%s%i x] = [%sn n (extend%i n x)]." size c size c size ; pp " intros n; elim n; clear n."; pp " intros x; change (extend%i 0 x) with (WW (znz_0 w%i_op) x)." size size; pp " unfold to_Z."; @@ -578,14 +578,14 @@ let _ = pr " | %s%i wx, %s%i wy => f%i (extend%i %i wx) wy" c i c j j i (j - i - 1); done; if i == size then - pr " | %s%i wx, %sn m wy => fnn m (extend%i m wx) wy" c size c size - else + pr " | %s%i wx, %sn m wy => fnn m (extend%i m wx) wy" c size c size + else pr " | %s%i wx, %sn m wy => fnn m (extend%i m (extend%i %i wx)) wy" c i c size i (size - i - 1); done; for i = 0 to size do if i == size then - pr " | %sn n wx, %s%i wy => fnn n wx (extend%i n wy)" c c size size - else + pr " | %sn n wx, %s%i wy => fnn n wx (extend%i n wy)" c c size size + else pr " | %sn n wx, %s%i wy => fnn n wx (extend%i n (extend%i %i wy))" c c i size i (size - i - 1); done; pr " | %sn n wx, Nn m wy =>" c; @@ -611,14 +611,14 @@ let _ = done; if i == size then pp " intros m y; rewrite (spec_extend%in m); apply Pfnn." size - else + else pp " intros m y; rewrite spec_extend%in%i; rewrite (spec_extend%in m); apply Pfnn." i size size; done; pp " intros n x y; case y; clear y."; for i = 0 to size do if i == size then pp " intros y; rewrite (spec_extend%in n); apply Pfnn." size - else + else pp " intros y; rewrite spec_extend%in%i; rewrite (spec_extend%in n); apply Pfnn." i size size; done; pp " intros m y; rewrite <- (spec_cast_l n m x); "; @@ -644,7 +644,7 @@ let _ = pr " match y with"; for j = 0 to i - 1 do pr " | %s%i wy =>" c j; - if j == 0 then + if j == 0 then pr " if w0_eq0 wy then ft0 x else"; pr " f%i wx (extend%i %i wy)" i j (i - j -1); done; @@ -653,8 +653,8 @@ let _ = pr " | %s%i wy => f%i (extend%i %i wx) wy" c j j i (j - i - 1); done; if i == size then - pr " | %sn m wy => fnn m (extend%i m wx) wy" c size - else + pr " | %sn m wy => fnn m (extend%i m wx) wy" c size + else pr " | %sn m wy => fnn m (extend%i m (extend%i %i wx)) wy" c size i (size - i - 1); pr" end"; done; @@ -665,8 +665,8 @@ let _ = if i == 0 then pr " if w0_eq0 wy then ft0 x else"; if i == size then - pr " fnn n wx (extend%i n wy)" size - else + pr " fnn n wx (extend%i n wy)" size + else pr " fnn n wx (extend%i n (extend%i %i wy))" size i (size - i - 1); done; pr " | %sn m wy =>" c; @@ -707,7 +707,7 @@ let _ = done; if i == size then pp " intros m y; rewrite (spec_extend%in m); apply Pfnn." size - else + else pp " intros m y; rewrite spec_extend%in%i; rewrite (spec_extend%in m); apply Pfnn." i size size; done; pp " intros n x y; case y; clear y."; @@ -721,7 +721,7 @@ let _ = end; if i == size then pp " rewrite (spec_extend%in n); apply Pfnn." size - else + else pp " rewrite spec_extend%in%i; rewrite (spec_extend%in n); apply Pfnn." i size size; done; pp " intros m y; rewrite <- (spec_cast_l n m x); "; @@ -748,14 +748,14 @@ let _ = pr " | %s%i wx, %s%i wy => f%in %i wx wy" c i c j i (j - i - 1); done; if i == size then - pr " | %s%i wx, %sn m wy => f%in m wx wy" c size c size - else + pr " | %s%i wx, %sn m wy => f%in m wx wy" c size c size + else pr " | %s%i wx, %sn m wy => f%in m (extend%i %i wx) wy" c i c size i (size - i - 1); done; for i = 0 to size do if i == size then - pr " | %sn n wx, %s%i wy => fn%i n wx wy" c c size size - else + pr " | %sn n wx, %s%i wy => fn%i n wx wy" c c size size + else pr " | %sn n wx, %s%i wy => fn%i n wx (extend%i %i wy)" c c i size i (size - i - 1); done; pr " | %sn n wx, %sn m wy => fnm n m wx wy" c c; @@ -779,14 +779,14 @@ let _ = done; if i == size then pp " intros m y; rewrite spec_eval%in; apply Pf%in." size size - else + else pp " intros m y; rewrite spec_extend%in%i; rewrite spec_eval%in; apply Pf%in." i size size size; done; pp " intros n x y; case y; clear y."; for i = 0 to size do if i == size then pp " intros y; rewrite spec_eval%in; apply Pfn%i." size size - else + else pp " intros y; rewrite spec_extend%in%i; rewrite spec_eval%in; apply Pfn%i." i size size size; done; pp " intros m y; apply Pfnm."; @@ -820,8 +820,8 @@ let _ = pr " | %s%i wy => f%in %i wx wy" c j i (j - i - 1); done; if i == size then - pr " | %sn m wy => f%in m wx wy" c size - else + pr " | %sn m wy => f%in m wx wy" c size + else pr " | %sn m wy => f%in m (extend%i %i wx) wy" c size i (size - i - 1); pr " end"; done; @@ -832,8 +832,8 @@ let _ = if i == 0 then pr " if w0_eq0 wy then ft0 x else"; if i == size then - pr " fn%i n wx wy" size - else + pr " fn%i n wx wy" size + else pr " fn%i n wx (extend%i %i wy)" size i (size - i - 1); done; pr " | %sn m wy => fnm n m wx wy" c; @@ -869,7 +869,7 @@ let _ = done; if i == size then pp " intros m y; rewrite spec_eval%in; apply Pf%in." size size - else + else pp " intros m y; rewrite spec_extend%in%i; rewrite spec_eval%in; apply Pf%in." i size size size; done; pp " intros n x y; case y; clear y."; @@ -883,7 +883,7 @@ let _ = end; if i == size then pp " rewrite spec_eval%in; apply Pfn%i." size size - else + else pp " rewrite spec_extend%in%i; rewrite spec_eval%in; apply Pfn%i." i size size size; done; pp " intros m y; apply Pfnm."; @@ -902,20 +902,20 @@ let _ = pr " (***************************************************************)"; pr ""; - pr " Definition reduce_0 (x:w) := %s0 x." c; + pr " Definition reduce_0 (x:w) := %s0 x." c; pr " Definition reduce_1 :="; pr " Eval lazy beta iota delta[reduce_n1] in"; pr " reduce_n1 _ _ zero w0_eq0 %s0 %s1." c c; for i = 2 to size do pr " Definition reduce_%i :=" i; pr " Eval lazy beta iota delta[reduce_n1] in"; - pr " reduce_n1 _ _ zero w%i_eq0 reduce_%i %s%i." + pr " reduce_n1 _ _ zero w%i_eq0 reduce_%i %s%i." (i-1) (i-1) c i done; pr " Definition reduce_%i :=" (size+1); pr " Eval lazy beta iota delta[reduce_n1] in"; - pr " reduce_n1 _ _ zero w%i_eq0 reduce_%i (%sn 0)." - size size c; + pr " reduce_n1 _ _ zero w%i_eq0 reduce_%i (%sn 0)." + size size c; pr " Definition reduce_n n := "; pr " Eval lazy beta iota delta[reduce_n] in"; @@ -940,7 +940,7 @@ let _ = pp " intros x1 y1."; pp " generalize (spec_w%i_eq0 x1); " (i - 1); pp " case w%i_eq0; intros H1; auto." (i - 1); - if i <> 1 then + if i <> 1 then pp " rewrite spec_reduce_%i." (i - 1); pp " unfold to_Z; rewrite znz_to_Z_%i." i; pp " unfold to_Z in H1; rewrite H1; auto."; @@ -983,19 +983,19 @@ let _ = for i = 0 to size-1 do pr " | %s%i wx =>" c i; pr " match w%i_succ_c wx with" i; - pr " | C0 r => %s%i r" c i; + pr " | C0 r => %s%i r" c i; pr " | C1 r => %s%i (WW one%i r)" c (i+1) i; pr " end"; done; pr " | %s%i wx =>" c size; pr " match w%i_succ_c wx with" size; - pr " | C0 r => %s%i r" c size; + pr " | C0 r => %s%i r" c size; pr " | C1 r => %sn 0 (WW one%i r)" c size ; pr " end"; pr " | %sn n wx =>" c; pr " let op := make_op n in"; pr " match op.(znz_succ_c) wx with"; - pr " | C0 r => %sn n r" c; + pr " | C0 r => %sn n r" c; pr " | C1 r => %sn (S n) (WW op.(znz_1) r)" c; pr " end"; pr " end."; @@ -1033,7 +1033,7 @@ let _ = pr ""; for i = 0 to size do - pr " Definition w%i_add_c := znz_add_c w%i_op." i i; + pr " Definition w%i_add_c := znz_add_c w%i_op." i i; pr " Definition w%i_add x y :=" i; pr " match w%i_add_c x y with" i; pr " | C0 r => %s%i r" c i; @@ -1057,7 +1057,7 @@ let _ = pp " Proof."; pp " intros n m; unfold to_Z, w%i_add, w%i_add_c." i i; pp " generalize (spec_add_c w%i_spec n m); case znz_add_c; auto." i; - pp " intros ww H; rewrite <- H."; + pp " intros ww H; rewrite <- H."; pp " rewrite znz_to_Z_%i; unfold interp_carry;" (i + 1); pp " apply f_equal2 with (f := Zplus); auto;"; pp " apply f_equal2 with (f := Zmult); auto;"; @@ -1070,7 +1070,7 @@ let _ = pp " Proof."; pp " intros k n m; unfold to_Z, addn."; pp " generalize (spec_add_c (wn_spec k) n m); case znz_add_c; auto."; - pp " intros ww H; rewrite <- H."; + pp " intros ww H; rewrite <- H."; pp " rewrite (znz_to_Z_n k); unfold interp_carry;"; pp " apply f_equal2 with (f := Zplus); auto;"; pp " apply f_equal2 with (f := Zmult); auto;"; @@ -1116,14 +1116,14 @@ let _ = for i = 0 to size do pr " | %s%i wx =>" c i; pr " match w%i_pred_c wx with" i; - pr " | C0 r => reduce_%i r" i; + pr " | C0 r => reduce_%i r" i; pr " | C1 r => zero"; pr " end"; done; pr " | %sn n wx =>" c; pr " let op := make_op n in"; pr " match op.(znz_pred_c) wx with"; - pr " | C0 r => reduce_n n r"; + pr " | C0 r => reduce_n n r"; pr " | C1 r => zero"; pr " end"; pr " end."; @@ -1153,7 +1153,7 @@ let _ = pp " unfold to_Z in H1; auto with zarith."; pp " Qed."; pp " "; - + pp " Let spec_pred0: forall x, [x] = 0 -> [pred x] = 0."; pp " Proof."; pp " intros x; case x; unfold pred."; @@ -1187,7 +1187,7 @@ let _ = done; pr ""; - for i = 0 to size do + for i = 0 to size do pr " Definition w%i_sub x y :=" i; pr " match w%i_sub_c x y with" i; pr " | C0 r => reduce_%i r" i; @@ -1209,7 +1209,7 @@ let _ = pp " Proof."; pp " intros n m; unfold w%i_sub, w%i_sub_c." i i; pp " generalize (spec_sub_c w%i_spec n m); case znz_sub_c; " i; - if i == 0 then + if i == 0 then pp " intros x; auto." else pp " intros x; try rewrite spec_reduce_%i; auto." i; @@ -1219,7 +1219,7 @@ let _ = pp " Qed."; pp ""; done; - + pp " Let spec_wn_sub: forall n x y, [%sn n y] <= [%sn n x] -> [subn n x y] = [%sn n x] - [%sn n y]." c c c c; pp " Proof."; pp " intros k n m; unfold subn."; @@ -1299,7 +1299,7 @@ let _ = pr " Definition comparen_%i :=" i; pr " compare_mn_1 w%i w%i %s compare_%i (compare_%i %s) compare_%i." i i (pz i) i i (pz i) i done; - pr ""; + pr ""; pr " Definition comparenm n m wx wy :="; pr " let mn := Max.max n m in"; @@ -1337,7 +1337,7 @@ let _ = pp " unfold compare_%i, to_Z; exact (spec_compare w%i_spec)." i i; pp " Qed."; pp ""; - + pp " Let spec_comparen_%i:" i; pp " forall (n : nat) (x : word w%i n) (y : w%i)," i i; pp " match comparen_%i n x y with" i; @@ -1387,12 +1387,12 @@ let _ = pp " (fun n => comparen_%i (S n)) _ _ _" i; done; pp " comparenm _)."; - + for i = 0 to size - 1 do pp " exact spec_compare_%i." i; pp " intros n x y H;apply spec_opp_compare; apply spec_comparen_%i." i; pp " intros n x y H; exact (spec_comparen_%i (S n) x y)." i; - done; + done; pp " exact spec_compare_%i." size; pp " intros n x y;apply spec_opp_compare; apply spec_comparen_%i." size; pp " intros n; exact (spec_comparen_%i (S n))." size; @@ -1461,7 +1461,7 @@ let _ = pr " match n return word w%i (S n) -> t_ with" i; for j = 0 to size - i do if (i + j) == size then - begin + begin pr " | %i%s => fun x => %sn 0 x" j "%nat" c; pr " | %i%s => fun x => %sn 1 x" (j + 1) "%nat" c end @@ -1471,7 +1471,7 @@ let _ = pr " | _ => fun _ => N0 w_0"; pr " end."; pr ""; - done; + done; for i = 0 to size - 1 do @@ -1486,7 +1486,7 @@ let _ = pp " repeat rewrite inj_S; unfold Zsucc; auto with zarith."; pp " Qed."; pp ""; - done; + done; for i = 0 to size do @@ -1497,8 +1497,8 @@ let _ = pr " if w%i_eq0 w then %sn n r" i c; pr " else %sn (S n) (WW (extend%i n w) r)." c i; end - else - begin + else + begin pr " if w%i_eq0 w then to_Z%i n r" i i; pr " else to_Z%i (S n) (WW (extend%i n w) r)." i i; end; @@ -1556,7 +1556,7 @@ let _ = pp " Qed."; pp ""; done; - + pp " Lemma nmake_op_WW: forall ww ww1 n x y,"; pp " znz_to_Z (nmake_op ww ww1 (S n)) (WW x y) ="; pp " znz_to_Z (nmake_op ww ww1 n) x * base (znz_digits (nmake_op ww ww1 n)) +"; @@ -1564,7 +1564,7 @@ let _ = pp " auto."; pp " Qed."; pp ""; - + for i = 0 to size do pp " Lemma extend%in_spec: forall n x1," i; pp " znz_to_Z (nmake_op _ w%i_op (S n)) (extend%i n x1) = " i i; @@ -1573,12 +1573,12 @@ let _ = pp " intros n1 x2; rewrite nmake_double."; pp " unfold extend%i." i; pp " rewrite DoubleBase.spec_extend; auto."; - if i == 0 then + if i == 0 then pp " intros l; simpl; unfold w_0; rewrite (spec_0 w0_spec); ring."; pp " Qed."; pp ""; done; - + pp " Lemma spec_muln:"; pp " forall n (x: word _ (S n)) y,"; pp " [%sn (S n) (znz_mul_c (make_op n) x y)] = [%sn n x] * [%sn n y]." c c c; @@ -1614,7 +1614,7 @@ let _ = pp " generalize (spec_w%i_eq0 x1); case w%i_eq0; intros HH." i i; pp " unfold to_Z in HH; rewrite HH."; if i == size then - begin + begin pp " rewrite spec_eval%in; unfold eval%in, nmake_op%i; auto." i i i; pp " rewrite spec_eval%in; unfold eval%in, nmake_op%i." i i i end @@ -1708,7 +1708,7 @@ let _ = pr " (* Power *)"; pr " (* *)"; pr " (***************************************************************)"; - pr ""; + pr ""; pr " Fixpoint power_pos (x:%s) (p:positive) {struct p} : %s :=" t t; pr " match p with"; @@ -1719,7 +1719,7 @@ let _ = pr ""; pr " Theorem spec_power_pos: forall x n, [power_pos x n] = [x] ^ Zpos n."; - pa " Admitted."; + pa " Admitted."; pp " Proof."; pp " intros x n; generalize x; elim n; clear n x; simpl power_pos."; pp " intros; rewrite spec_mul; rewrite spec_square; rewrite H."; @@ -1775,7 +1775,7 @@ let _ = pr " (* Division *)"; pr " (* *)"; pr " (***************************************************************)"; - pr ""; + pr ""; for i = 0 to size do pr " Definition w%i_div_gt := w%i_op.(znz_div_gt)." i i @@ -1844,7 +1844,7 @@ let _ = pr " Definition div_gt := Eval lazy beta delta [iter] in"; pr " (iter _ "; - for i = 0 to size do + for i = 0 to size do pr " div_gt%i" i; pr " (fun n x y => div_gt%i x (DoubleBase.get_low %s (S n) y))" i (pz i); pr " w%i_divn1" i; @@ -1862,10 +1862,10 @@ let _ = pp " forall x y, [x] > [y] -> 0 < [y] ->"; pp " let (q,r) := div_gt x y in"; pp " [x] = [q] * [y] + [r] /\\ 0 <= [r] < [y])."; - pp " refine (spec_iter (t_*t_) (fun x y res => x > y -> 0 < y ->"; + pp " refine (spec_iter (t_*t_) (fun x y res => x > y -> 0 < y ->"; pp " let (q,r) := res in"; pp " x = [q] * y + [r] /\\ 0 <= [r] < y)"; - for i = 0 to size do + for i = 0 to size do pp " div_gt%i" i; pp " (fun n x y => div_gt%i x (DoubleBase.get_low %s (S n) y))" i (pz i); pp " w%i_divn1 _ _ _" i; @@ -1883,7 +1883,7 @@ let _ = pp " (DoubleBase.get_low %s (S n) y))." (pz i); pp0 " "; for j = 0 to i do - pp0 "unfold w%i; " (i-j); + pp0 "unfold w%i; " (i-j); done; pp "case znz_div_gt."; pp " intros xx yy H4; repeat rewrite spec_reduce_%i." i; @@ -1897,7 +1897,7 @@ let _ = pp " (spec_divn1 w%i w%i_op w%i_spec (S n) x y H3)." i i i; pp0 " unfold w%i_divn1; " i; for j = 0 to i do - pp0 "unfold w%i; " (i-j); + pp0 "unfold w%i; " (i-j); done; pp "case double_divn1."; pp " intros xx yy H4."; @@ -1990,7 +1990,7 @@ let _ = pr " (* Modulo *)"; pr " (* *)"; pr " (***************************************************************)"; - pr ""; + pr ""; for i = 0 to size do pr " Definition w%i_mod_gt := w%i_op.(znz_mod_gt)." i i @@ -2063,7 +2063,7 @@ let _ = pp " rewrite <- (spec_get_end%i (S n) y x) in H3; auto with zarith." i; if i == size then pp " intros n x y H2 H3; rewrite spec_reduce_%i." i - else + else pp " intros n x y H1 H2 H3; rewrite spec_reduce_%i." i; pp " unfold w%i_modn1, to_Z; rewrite spec_double_eval%in." i i; pp " apply (spec_modn1 _ _ w%i_spec); auto." i; @@ -2110,7 +2110,7 @@ let _ = pr " (* Gcd *)"; pr " (* *)"; pr " (***************************************************************)"; - pr ""; + pr ""; pr " Definition digits x :="; pr " match x with"; @@ -2423,7 +2423,7 @@ let _ = pr " (* Shift *)"; pr " (* *)"; pr " (***************************************************************)"; - pr ""; + pr ""; (* Head0 *) pr " Definition head0 w := match w with"; @@ -2513,7 +2513,7 @@ let _ = pr " Definition %sdigits x :=" c; pr " match x with"; pr " | %s0 _ => %s0 w0_op.(znz_zdigits)" c c; - for i = 1 to size do + for i = 1 to size do pr " | %s%i _ => reduce_%i w%i_op.(znz_zdigits)" c i i i; done; pr " | %sn n _ => reduce_n n (make_op n).(znz_zdigits)" c; @@ -2644,7 +2644,7 @@ let _ = pp " apply F4 with (3:=(wn_spec m))(4:=wn_spec m)(5:=w%i_spec); auto with zarith." size; pp " try (apply sym_equal; exact (spec_extend%in m x))." size; end - else + else begin pp " intros m y; unfold shiftrn, Ndigits."; pp " repeat rewrite spec_reduce_n; unfold to_Z; intros H1."; @@ -2857,7 +2857,7 @@ let _ = pp " apply F4 with (3:=(wn_spec m))(4:=wn_spec m)(5:=w%i_spec); auto with zarith." size; pp " try (apply sym_equal; exact (spec_extend%in m x))." size; end - else + else begin pp " intros m y; unfold shiftln, head0."; pp " repeat rewrite spec_reduce_n; unfold to_Z; intros H1."; @@ -3030,7 +3030,7 @@ let _ = pr " (forall x, 2 ^ (Zpos p + 1) <= [head0 x]->"; pr " [cont n x] = [x] * 2 ^ [n]) ->"; pr " [safe_shiftl_aux_body cont n x] = [x] * 2 ^ [n]."; - pa " Admitted."; + pa " Admitted."; pp " Proof."; pp " intros n p x cont H1 H2; unfold safe_shiftl_aux_body."; pp " generalize (spec_compare n (head0 x)); case compare; intros H."; diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v index c3fdd1bf4a..d42db97d57 100644 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ b/theories/Numbers/Natural/BigN/Nbasic.v @@ -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. diff --git a/theories/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v index e53e627eca..5295aaec2a 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSig.v +++ b/theories/Numbers/Natural/SpecViaZ/NSig.v @@ -58,7 +58,7 @@ Module Type NType. Parameter spec_eq_bool: forall x y, if eq_bool x y then [x] = [y] else [x] <> [y]. - + Parameter succ : t -> t. Parameter spec_succ: forall n, [succ n] = [n] + 1. @@ -98,7 +98,7 @@ Module Type NType. Parameter spec_div_eucl: forall x y, 0 < [y] -> let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y]. - + Parameter div : t -> t -> t. Parameter spec_div: forall x y, 0 < [y] -> [div x y] = [x] / [y]. diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 773807120b..578cb62561 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v @@ -97,7 +97,7 @@ Section Induction. Variable A : N.t -> Prop. Hypothesis A_wd : predicate_wd N.eq A. Hypothesis A0 : A 0. -Hypothesis AS : forall n, A n <-> A (N.succ n). +Hypothesis AS : forall n, A n <-> A (N.succ n). Add Morphism A with signature N.eq ==> iff as A_morph. Proof. apply A_wd. Qed. @@ -221,7 +221,7 @@ Proof. Qed. Add Morphism N.compare with signature N.eq ==> N.eq ==> (@eq comparison) as compare_wd. -Proof. +Proof. intros x x' Hx y y' Hy. rewrite 2 spec_compare_alt. unfold N.eq in *. rewrite Hx, Hy; intuition. Qed. diff --git a/theories/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v index 67411eac81..0973b7d8d9 100644 --- a/theories/Numbers/Rational/BigQ/QMake.v +++ b/theories/Numbers/Rational/BigQ/QMake.v @@ -28,27 +28,27 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. number y interpreted as x/y. The pairs (x,0) and (0,y) are all interpreted as 0. *) - Inductive t_ := + Inductive t_ := | Qz : Z.t -> t_ | Qq : Z.t -> N.t -> t_. Definition t := t_. - (** Specification with respect to [QArith] *) + (** Specification with respect to [QArith] *) Open Local Scope Q_scope. Definition of_Z x: t := Qz (Z.of_Z x). - Definition of_Q (q:Q) : t := - let (x,y) := q in - match y with + Definition of_Q (q:Q) : t := + let (x,y) := q in + match y with | 1%positive => Qz (Z.of_Z x) | _ => Qq (Z.of_Z x) (N.of_N (Npos y)) end. - Definition to_Q (q: t) := - match q with + Definition to_Q (q: t) := + match q with | Qz x => Z.to_Z x # 1 | Qq x y => if N.eq_bool y N.zero then 0 else Z.to_Z x # Z2P (N.to_Z y) @@ -59,11 +59,11 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Theorem strong_spec_of_Q: forall q: Q, [of_Q q] = q. Proof. intros(x,y); destruct y; simpl; rewrite Z.spec_of_Z; auto. - generalize (N.spec_eq_bool (N.of_N (Npos y~1)) N.zero); + generalize (N.spec_eq_bool (N.of_N (Npos y~1)) N.zero); case N.eq_bool; auto; rewrite N.spec_0. rewrite N.spec_of_N; discriminate. rewrite N.spec_of_N; auto. - generalize (N.spec_eq_bool (N.of_N (Npos y~0)) N.zero); + generalize (N.spec_eq_bool (N.of_N (Npos y~0)) N.zero); case N.eq_bool; auto; rewrite N.spec_0. rewrite N.spec_of_N; discriminate. rewrite N.spec_of_N; auto. @@ -98,77 +98,77 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition compare (x y: t) := match x, y with | Qz zx, Qz zy => Z.compare zx zy - | Qz zx, Qq ny dy => + | Qz zx, Qq ny dy => if N.eq_bool dy N.zero then Z.compare zx Z.zero else Z.compare (Z.mul zx (Z_of_N dy)) ny - | Qq nx dx, Qz zy => - if N.eq_bool dx N.zero then Z.compare Z.zero zy + | Qq nx dx, Qz zy => + if N.eq_bool dx N.zero then Z.compare Z.zero zy else Z.compare nx (Z.mul zy (Z_of_N dx)) | Qq nx dx, Qq ny dy => match N.eq_bool dx N.zero, N.eq_bool dy N.zero with | true, true => Eq | true, false => Z.compare Z.zero ny | false, true => Z.compare nx Z.zero - | false, false => Z.compare (Z.mul nx (Z_of_N dy)) + | false, false => Z.compare (Z.mul nx (Z_of_N dy)) (Z.mul ny (Z_of_N dx)) end end. - Lemma Zcompare_spec_alt : + Lemma Zcompare_spec_alt : forall z z', Z.compare z z' = (Z.to_Z z ?= Z.to_Z z')%Z. Proof. intros; generalize (Z.spec_compare z z'); destruct Z.compare; auto. intro H; rewrite H; symmetry; apply Zcompare_refl. Qed. - - Lemma Ncompare_spec_alt : + + Lemma Ncompare_spec_alt : forall n n', N.compare n n' = (N.to_Z n ?= N.to_Z n')%Z. Proof. intros; generalize (N.spec_compare n n'); destruct N.compare; auto. intro H; rewrite H; symmetry; apply Zcompare_refl. Qed. - Lemma N_to_Z2P : forall n, N.to_Z n <> 0%Z -> + Lemma N_to_Z2P : forall n, N.to_Z n <> 0%Z -> Zpos (Z2P (N.to_Z n)) = N.to_Z n. Proof. intros; apply Z2P_correct. generalize (N.spec_pos n); romega. Qed. - Hint Rewrite - Zplus_0_r Zplus_0_l Zmult_0_r Zmult_0_l Zmult_1_r Zmult_1_l + Hint Rewrite + Zplus_0_r Zplus_0_l Zmult_0_r Zmult_0_l Zmult_1_r Zmult_1_l Z.spec_0 N.spec_0 Z.spec_1 N.spec_1 Z.spec_m1 Z.spec_opp Zcompare_spec_alt Ncompare_spec_alt - Z.spec_add N.spec_add Z.spec_mul N.spec_mul + Z.spec_add N.spec_add Z.spec_mul N.spec_mul Z.spec_gcd N.spec_gcd Zgcd_Zabs Zgcd_1 spec_Z_of_N spec_Zabs_N : nz. Ltac nzsimpl := autorewrite with nz in *. Ltac destr_neq_bool := repeat - (match goal with |- context [N.eq_bool ?x ?y] => + (match goal with |- context [N.eq_bool ?x ?y] => generalize (N.spec_eq_bool x y); case N.eq_bool end). - + Ltac destr_zeq_bool := repeat - (match goal with |- context [Z.eq_bool ?x ?y] => + (match goal with |- context [Z.eq_bool ?x ?y] => generalize (Z.spec_eq_bool x y); case Z.eq_bool end). Ltac simpl_ndiv := rewrite N.spec_div by (nzsimpl; romega). - Tactic Notation "simpl_ndiv" "in" "*" := + Tactic Notation "simpl_ndiv" "in" "*" := rewrite N.spec_div in * by (nzsimpl; romega). Ltac simpl_zdiv := rewrite Z.spec_div by (nzsimpl; romega). - Tactic Notation "simpl_zdiv" "in" "*" := + Tactic Notation "simpl_zdiv" "in" "*" := rewrite Z.spec_div in * by (nzsimpl; romega). - Ltac qsimpl := try red; unfold to_Q; simpl; intros; + Ltac qsimpl := try red; unfold to_Q; simpl; intros; destr_neq_bool; destr_zeq_bool; simpl; nzsimpl; auto; intros. Theorem spec_compare: forall q1 q2, (compare q1 q2) = ([q1] ?= [q2]). Proof. - intros [z1 | x1 y1] [z2 | x2 y2]; + intros [z1 | x1 y1] [z2 | x2 y2]; unfold Qcompare, compare; qsimpl; rewrite ! N_to_Z2P; auto. Qed. @@ -177,7 +177,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition min n m := match compare n m with Gt => m | _ => n end. Definition max n m := match compare n m with Lt => m | _ => n end. - Definition eq_bool n m := + Definition eq_bool n m := match compare n m with Eq => true | _ => false end. Theorem spec_eq_bool: forall x y, @@ -196,9 +196,9 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (** Normalisation function *) Definition norm n d : t := - let gcd := N.gcd (Zabs_N n) d in + let gcd := N.gcd (Zabs_N n) d in match N.compare N.one gcd with - | Lt => + | Lt => let n := Z.div n (Z_of_N gcd) in let d := N.div d gcd in match N.compare d N.one with @@ -249,7 +249,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Theorem strong_spec_norm : forall p q, [norm p q] = Qred [Qq p q]. Proof. intros. - replace (Qred [Qq p q]) with (Qred [norm p q]) by + replace (Qred [Qq p q]) with (Qred [norm p q]) by (apply Qred_complete; apply spec_norm). symmetry; apply Qred_identity. unfold norm. @@ -282,10 +282,10 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. simpl; auto with zarith. Qed. - (** Reduction function : producing irreducible fractions *) + (** Reduction function : producing irreducible fractions *) - Definition red (x : t) : t := - match x with + Definition red (x : t) : t := + match x with | Qz z => x | Qq n d => norm n d end. @@ -307,18 +307,18 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. symmetry; apply Qred_identity; simpl; auto with zarith. unfold red; apply strong_spec_norm. Qed. - + Definition add (x y: t): t := match x with | Qz zx => match y with | Qz zy => Qz (Z.add zx zy) - | Qq ny dy => - if N.eq_bool dy N.zero then x + | Qq ny dy => + if N.eq_bool dy N.zero then x else Qq (Z.add (Z.mul zx (Z_of_N dy)) ny) dy end | Qq nx dx => - if N.eq_bool dx N.zero then y + if N.eq_bool dx N.zero then y else match y with | Qz zy => Qq (Z.add nx (Z.mul zy (Z_of_N dx))) dx | Qq ny dy => @@ -352,12 +352,12 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. | Qz zx => match y with | Qz zy => Qz (Z.add zx zy) - | Qq ny dy => - if N.eq_bool dy N.zero then x + | Qq ny dy => + if N.eq_bool dy N.zero then x else norm (Z.add (Z.mul zx (Z_of_N dy)) ny) dy end | Qq nx dx => - if N.eq_bool dx N.zero then y + if N.eq_bool dx N.zero then y else match y with | Qz zy => norm (Z.add nx (Z.mul zy (Z_of_N dx))) dx | Qq ny dy => @@ -372,16 +372,16 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Theorem spec_add_norm : forall x y, [add_norm x y] == [x] + [y]. Proof. intros x y; rewrite <- spec_add. - destruct x; destruct y; unfold add_norm, add; + destruct x; destruct y; unfold add_norm, add; destr_neq_bool; auto using Qeq_refl, spec_norm. Qed. - Theorem strong_spec_add_norm : forall x y : t, + Theorem strong_spec_add_norm : forall x y : t, Reduced x -> Reduced y -> Reduced (add_norm x y). Proof. unfold Reduced; intros. rewrite strong_spec_red. - rewrite <- (Qred_complete [add x y]); + rewrite <- (Qred_complete [add x y]); [ | rewrite spec_add, spec_add_norm; apply Qeq_refl ]. rewrite <- strong_spec_red. destruct x as [zx|nx dx]; destruct y as [zy|ny dy]. @@ -404,7 +404,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Proof. intros [z | x y]; simpl. rewrite Z.spec_opp; auto. - match goal with |- context[N.eq_bool ?X ?Y] => + match goal with |- context[N.eq_bool ?X ?Y] => generalize (N.spec_eq_bool X Y); case N.eq_bool end; auto; rewrite N.spec_0. rewrite Z.spec_opp; auto. @@ -438,7 +438,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite spec_opp; ring. Qed. - Theorem strong_spec_sub_norm : forall x y, + Theorem strong_spec_sub_norm : forall x y, Reduced x -> Reduced y -> Reduced (sub_norm x y). Proof. intros. @@ -470,7 +470,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. generalize (N.spec_pos dy); omega. Qed. - Lemma norm_denum : forall n d, + Lemma norm_denum : forall n d, [if N.eq_bool d N.one then Qz n else Qq n d] == [Qq n d]. Proof. intros; simpl; qsimpl. @@ -478,15 +478,15 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite N_to_Z2P, H0; auto with zarith. Qed. - Definition irred n d := + Definition irred n d := let gcd := N.gcd (Zabs_N n) d in - match N.compare gcd N.one with + match N.compare gcd N.one with | Gt => (Z.div n (Z_of_N gcd), N.div d gcd) | _ => (n, d) end. - Lemma spec_irred : forall n d, exists g, - let (n',d') := irred n d in + Lemma spec_irred : forall n d, exists g, + let (n',d') := irred n d in (Z.to_Z n' * g = Z.to_Z n)%Z /\ (N.to_Z d' * g = N.to_Z d)%Z. Proof. intros. @@ -511,7 +511,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite Zmult_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. Qed. - Lemma spec_irred_zero : forall n d, + Lemma spec_irred_zero : forall n d, (N.to_Z d = 0)%Z <-> (N.to_Z (snd (irred n d)) = 0)%Z. Proof. intros. @@ -535,8 +535,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. compute in H1; elim H1; auto. Qed. - Lemma strong_spec_irred : forall n d, - (N.to_Z d <> 0%Z) -> + Lemma strong_spec_irred : forall n d, + (N.to_Z d <> 0%Z) -> let (n',d') := irred n d in Zgcd (Z.to_Z n') (N.to_Z d') = 1%Z. Proof. unfold irred; intros. @@ -554,31 +554,31 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply Zgcd_is_gcd; auto. Qed. - Definition mul_norm_Qz_Qq z n d := - if Z.eq_bool z Z.zero then zero + Definition mul_norm_Qz_Qq z n d := + if Z.eq_bool z Z.zero then zero else let gcd := N.gcd (Zabs_N z) d in match N.compare gcd N.one with - | Gt => + | Gt => let z := Z.div z (Z_of_N gcd) in let d := N.div d gcd in if N.eq_bool d N.one then Qz (Z.mul z n) else Qq (Z.mul z n) d | _ => Qq (Z.mul z n) d end. - Definition mul_norm (x y: t): t := + Definition mul_norm (x y: t): t := match x, y with | Qz zx, Qz zy => Qz (Z.mul zx zy) | Qz zx, Qq ny dy => mul_norm_Qz_Qq zx ny dy | Qq nx dx, Qz zy => mul_norm_Qz_Qq zy nx dx - | Qq nx dx, Qq ny dy => - let (nx, dy) := irred nx dy in - let (ny, dx) := irred ny dx in + | Qq nx dx, Qq ny dy => + let (nx, dy) := irred nx dy in + let (ny, dx) := irred ny dx in let d := N.mul dx dy in if N.eq_bool d N.one then Qz (Z.mul ny nx) else Qq (Z.mul ny nx) d end. - Lemma spec_mul_norm_Qz_Qq : forall z n d, + Lemma spec_mul_norm_Qz_Qq : forall z n d, [mul_norm_Qz_Qq z n d] == [Qq (Z.mul z n) d]. Proof. intros z n d; unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. @@ -599,14 +599,14 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite <- Zgcd_div_swap0; auto with zarith; ring. Qed. - Lemma strong_spec_mul_norm_Qz_Qq : forall z n d, + Lemma strong_spec_mul_norm_Qz_Qq : forall z n d, Reduced (Qq n d) -> Reduced (mul_norm_Qz_Qq z n d). Proof. unfold Reduced; intros z n d. rewrite 2 strong_spec_red, 2 Qred_iff. simpl; nzsimpl. destr_neq_bool; intros Hd H; simpl in *; nzsimpl. - + unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. destr_zeq_bool; intros Hz; simpl; nzsimpl; simpl; auto. destruct Z_le_gt_dec. @@ -670,7 +670,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destruct (spec_irred ny dx) as (g' & Hg'). assert (Hz := spec_irred_zero nx dy). assert (Hz':= spec_irred_zero ny dx). - destruct irred as (n1,d1); destruct irred as (n2,d2). + destruct irred as (n1,d1); destruct irred as (n2,d2). simpl snd in *; destruct Hg as [Hg1 Hg2]; destruct Hg' as [Hg1' Hg2']. rewrite norm_denum. qsimpl. @@ -686,10 +686,10 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite 2 Z2P_correct. rewrite <- Hg1, <- Hg2, <- Hg1', <- Hg2'; ring. - assert (0 <= N.to_Z d2 * N.to_Z d1)%Z + assert (0 <= N.to_Z d2 * N.to_Z d1)%Z by (apply Zmult_le_0_compat; apply N.spec_pos). romega. - assert (0 <= N.to_Z dx * N.to_Z dy)%Z + assert (0 <= N.to_Z dx * N.to_Z dy)%Z by (apply Zmult_le_0_compat; apply N.spec_pos). romega. Qed. @@ -712,7 +712,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. assert (Hz':= spec_irred_zero ny dx). assert (Hgc := strong_spec_irred nx dy). assert (Hgc' := strong_spec_irred ny dx). - destruct irred as (n1,d1); destruct irred as (n2,d2). + destruct irred as (n1,d1); destruct irred as (n2,d2). simpl snd in *; destruct Hg as [Hg1 Hg2]; destruct Hg' as [Hg1' Hg2']. destr_neq_bool; simpl; nzsimpl; intros; auto. destr_neq_bool; simpl; nzsimpl; intros; auto. @@ -729,7 +729,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply Zgcd_mult_rel_prime; rewrite Zgcd_comm; apply Zgcd_mult_rel_prime; rewrite Zgcd_comm; auto. - + rewrite Zgcd_1_rel_prime in *. apply bezout_rel_prime. destruct (rel_prime_bezout _ _ H4) as [u v Huv]. @@ -747,15 +747,15 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. romega. Qed. - Definition inv (x: t): t := + Definition inv (x: t): t := match x with - | Qz z => - match Z.compare Z.zero z with + | Qz z => + match Z.compare Z.zero z with | Eq => zero | Lt => Qq Z.one (Zabs_N z) | Gt => Qq Z.minus_one (Zabs_N z) end - | Qq n d => + | Qq n d => match Z.compare Z.zero n with | Eq => zero | Lt => Qq (Z_of_N d) (Zabs_N n) @@ -827,25 +827,25 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite T, Zpos_mult_morphism, N_to_Z2P; auto; ring. Qed. - Definition inv_norm (x: t): t := + Definition inv_norm (x: t): t := match x with - | Qz z => - match Z.compare Z.zero z with + | Qz z => + match Z.compare Z.zero z with | Eq => zero | Lt => Qq Z.one (Zabs_N z) | Gt => Qq Z.minus_one (Zabs_N z) end - | Qq n d => - if N.eq_bool d N.zero then zero else - match Z.compare Z.zero n with + | Qq n d => + if N.eq_bool d N.zero then zero else + match Z.compare Z.zero n with | Eq => zero - | Lt => - match Z.compare n Z.one with + | Lt => + match Z.compare n Z.one with | Gt => Qq (Z_of_N d) (Zabs_N n) | _ => Qz (Z_of_N d) end - | Gt => - match Z.compare n Z.minus_one with + | Gt => + match Z.compare n Z.minus_one with | Lt => Qq (Z.opp (Z_of_N d)) (Zabs_N n) | _ => Qz (Z.opp (Z_of_N d)) end @@ -882,7 +882,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Theorem strong_spec_inv_norm : forall x, Reduced x -> Reduced (inv_norm x). Proof. - unfold Reduced. + unfold Reduced. intros. destruct x as [ z | n d ]. (* Qz *) @@ -952,8 +952,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply Qeq_refl. apply spec_inv_norm; auto. Qed. - - Theorem strong_spec_div_norm : forall x y, + + Theorem strong_spec_div_norm : forall x y, Reduced x -> Reduced y -> Reduced (div_norm x y). Proof. intros; unfold div_norm. @@ -980,7 +980,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite H in H0; simpl in H0; elim H0; auto. assert (0 < N.to_Z d)%Z by (generalize (N.spec_pos d); romega). clear H H0. - rewrite Z.spec_square, N.spec_square. + rewrite Z.spec_square, N.spec_square. red; simpl. rewrite Zpos_mult_morphism; rewrite !Z2P_correct; auto. apply Zmult_lt_0_compat; auto. @@ -991,7 +991,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. | Qz zx => Qz (Z.power_pos zx p) | Qq nx dx => Qq (Z.power_pos nx p) (N.power_pos dx p) end. - + Theorem spec_power_pos : forall x p, [power_pos x p] == [x] ^ Zpos p. Proof. intros [ z | n d ] p; unfold power_pos. @@ -1019,7 +1019,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite N.spec_power_pos. auto. Qed. - Theorem strong_spec_power_pos : forall x p, + Theorem strong_spec_power_pos : forall x p, Reduced x -> Reduced (power_pos x p). Proof. destruct x as [z | n d]; simpl; intros. @@ -1040,8 +1040,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply rel_prime_Zpower; auto with zarith. Qed. - Definition power (x : t) (z : Z) : t := - match z with + Definition power (x : t) (z : Z) : t := + match z with | Z0 => one | Zpos p => power_pos x p | Zneg p => inv (power_pos x p) @@ -1056,8 +1056,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite spec_inv, spec_power_pos; apply Qeq_refl. Qed. - Definition power_norm (x : t) (z : Z) : t := - match z with + Definition power_norm (x : t) (z : Z) : t := + match z with | Z0 => one | Zpos p => power_pos x p | Zneg p => inv_norm (power_pos x p) @@ -1072,7 +1072,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite spec_inv_norm, spec_power_pos; apply Qeq_refl. Qed. - Theorem strong_spec_power_norm : forall x z, + Theorem strong_spec_power_norm : forall x z, Reduced x -> Reduced (power_norm x z). Proof. destruct z; simpl. @@ -1085,7 +1085,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (** Interaction with [Qcanon.Qc] *) - + Open Scope Qc_scope. Definition of_Qc q := of_Q (this q). @@ -1166,7 +1166,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply Qplus_comp; apply Qeq_sym; apply Qred_correct. Qed. - Theorem spec_add_normc_bis : forall x y : Qc, + Theorem spec_add_normc_bis : forall x y : Qc, [add_norm (of_Qc x) (of_Qc y)] = x+y. Proof. intros. @@ -1189,7 +1189,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite spec_oppc; ring. Qed. - Theorem spec_sub_normc_bis : forall x y : Qc, + Theorem spec_sub_normc_bis : forall x y : Qc, [sub_norm (of_Qc x) (of_Qc y)] = x-y. Proof. intros. @@ -1228,7 +1228,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply Qmult_comp; apply Qeq_sym; apply Qred_correct. Qed. - Theorem spec_mul_normc_bis : forall x y : Qc, + Theorem spec_mul_normc_bis : forall x y : Qc, [mul_norm (of_Qc x) (of_Qc y)] = x*y. Proof. intros. @@ -1266,7 +1266,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply Qinv_comp; apply Qeq_sym; apply Qred_correct. Qed. - Theorem spec_inv_normc_bis : forall x : Qc, + Theorem spec_inv_normc_bis : forall x : Qc, [inv_norm (of_Qc x)] = /x. Proof. intros. @@ -1280,7 +1280,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Proof. intros x y; unfold div; rewrite spec_mulc; auto. unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. + apply spec_invc; auto. Qed. Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. @@ -1290,7 +1290,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. apply spec_inv_normc; auto. Qed. - Theorem spec_div_normc_bis : forall x y : Qc, + Theorem spec_div_normc_bis : forall x y : Qc, [div_norm (of_Qc x) (of_Qc y)] = x/y. Proof. intros. diff --git a/theories/Numbers/Rational/SpecViaQ/QSig.v b/theories/Numbers/Rational/SpecViaQ/QSig.v index 7c88d25aae..8be66493e5 100644 --- a/theories/Numbers/Rational/SpecViaQ/QSig.v +++ b/theories/Numbers/Rational/SpecViaQ/QSig.v @@ -48,12 +48,12 @@ Module Type QType. Definition max n m := match compare n m with Lt => m | _ => n end. Parameter eq_bool : t -> t -> bool. - - Parameter spec_eq_bool : forall x y, + + Parameter spec_eq_bool : forall x y, if eq_bool x y then [x]==[y] else ~([x]==[y]). Parameter red : t -> t. - + Parameter spec_red : forall x, [red x] == [x]. Parameter strong_spec_red : forall x, [red x] = Qred [x]. |