diff options
Diffstat (limited to 'theories/Numbers/Cyclic/Int31/Cyclic31.v')
| -rw-r--r-- | theories/Numbers/Cyclic/Int31/Cyclic31.v | 464 |
1 files changed, 232 insertions, 232 deletions
diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v index 6da1c6ec..8addf5b9 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Cyclic31.v 11907 2009-02-10 23:54:28Z letouzey $ i*) +(*i $Id$ i*) (** * Int31 numbers defines indeed a cyclic structure : Z/(2^31)Z *) @@ -24,8 +24,8 @@ Require Import BigNumPrelude. Require Import CyclicAxioms. Require Import ROmega. -Open Scope nat_scope. -Open Scope int31_scope. +Local Open Scope nat_scope. +Local Open Scope int31_scope. Section Basics. @@ -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. @@ -1140,7 +1140,7 @@ Definition int31_op := (mk_znz_op w_iszero (* Basic arithmetic operations *) (fun i => 0 -c i) - (fun i => 0 - i) + opp31 (fun i => 0-i-1) (fun i => i +c 1) add31c @@ -1181,14 +1181,14 @@ Definition int31_op := (mk_znz_op End Int31_Op. Section Int31_Spec. - - Open Local Scope Z_scope. + + Local Open 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. + Local Notation wB := (2 ^ (Z_of_nat size)). + + 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 @@ -1906,27 +1906,27 @@ Section Int31_Spec. Lemma quotient_by_2 a: a - 1 <= (a/2) + (a/2). Proof. - intros a; case (Z_mod_lt a 2); auto with zarith. + case (Z_mod_lt a 2); auto with zarith. 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 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. @@ -1936,7 +1936,7 @@ Section Int31_Spec. Lemma sqrt_main i j: 0 <= i -> 0 < j -> i < ((j + (i/j))/2 + 1) ^ 2. Proof. - intros i j Hi Hj. + intros Hi Hj. assert (Hij: 0 <= i/j) by (apply Z_div_pos; auto with zarith). apply Zlt_le_trans with (2 := sqrt_main_trick _ _ (Zlt_le_weak _ _ Hj) Hij). pattern i at 1; rewrite (Z_div_mod_eq i j); case (Z_mod_lt i j); auto with zarith. @@ -1944,7 +1944,7 @@ Section Int31_Spec. Lemma sqrt_init i: 1 < i -> i < (i/2 + 1) ^ 2. Proof. - intros i Hi. + intros Hi. assert (H1: 0 <= i - 2) by auto with zarith. assert (H2: 1 <= (i / 2) ^ 2); auto with zarith. replace i with (1* 2 + (i - 2)); auto with zarith. @@ -1962,14 +1962,14 @@ Section Int31_Spec. Lemma sqrt_test_true i j: 0 <= i -> 0 < j -> i/j >= j -> j ^ 2 <= i. Proof. - intros i j Hi Hj Hd; rewrite Zpower_2. + intros Hi Hj Hd; rewrite Zpower_2. apply Zle_trans with (j * (i/j)); auto with zarith. apply Z_mult_div_ge; auto with zarith. Qed. Lemma sqrt_test_false i j: 0 <= i -> 0 < j -> i/j < j -> (j + (i/j))/2 < j. Proof. - intros i j Hi Hj H; case (Zle_or_lt j ((j + (i/j))/2)); auto. + intros Hi Hj H; case (Zle_or_lt j ((j + (i/j))/2)); auto. intros H1; contradict H; apply Zle_not_lt. assert (2 * j <= j + (i/j)); auto with zarith. apply Zle_trans with (2 * ((j + (i/j))/2)); auto with zarith. @@ -1984,32 +1984,32 @@ Section Int31_Spec. Lemma Zcompare_spec i j: ZcompareSpec i j (i ?= j). Proof. - intros i j; case_eq (Zcompare i j); intros H. + case_eq (Zcompare i j); intros H. apply ZcompareSpecEq; apply Zcompare_Eq_eq; auto. apply ZcompareSpecLt; auto. apply ZcompareSpecGt; apply Zgt_lt; auto. 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 end. Proof. - intros rec i j; unfold sqrt31_step; case div31; intros. + unfold sqrt31_step; case div31; intros. simpl; case compare31; auto. Qed. Lemma div31_phi i j: 0 < [|j|] -> [|fst (i/j)%int31|] = [|i|]/[|j|]. - intros i j Hj; generalize (spec_div i j Hj). + intros Hj; generalize (spec_div i j Hj). case div31; intros q r; simpl fst. intros (H1,H2); apply Zdiv_unique with [|r|]; auto with zarith. 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 -> @@ -2017,15 +2017,15 @@ Section Int31_Spec. [|sqrt31_step rec i j|] ^ 2 <= [|i|] < ([|sqrt31_step rec i j|] + 1) ^ 2. 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; + intros Hi Hj Hij H31 Hrec; rewrite sqrt31_step_def. + 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). @@ -2048,12 +2048,12 @@ 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. Proof. - intros n; elim n; unfold iter31_sqrt; fold iter31_sqrt; clear n. + revert rec i j; elim n; unfold iter31_sqrt; fold iter31_sqrt; clear n. intros rec i j Hi Hj Hij H31 Hrec; apply sqrt31_step_correct; auto with zarith. intros; apply Hrec; auto with zarith. rewrite Zpower_0_r; auto with zarith. @@ -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 @@ -2112,14 +2112,14 @@ Section Int31_Spec. end end. Proof. - intros rec ih il j; unfold sqrt312_step; case div3121; intros. + unfold sqrt312_step; case div3121; intros. 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. + intros H1. case (phi_bounded j); intros Hbj _. case (phi_bounded il); intros Hbil _. case (phi_bounded ih); intros Hbih Hbih1. @@ -2133,22 +2133,22 @@ Section Int31_Spec. Lemma div312_phi ih il j: (2^30 <= [|j|] -> [|ih|] < [|j|] -> [|fst (div3121 ih il j)|] = phi2 ih il/[|j|])%Z. Proof. - intros ih il j Hj Hj1. + intros Hj Hj1. generalize (spec_div21 ih il j Hj Hj1). case div3121; intros q r (Hq, Hr). apply Zdiv_unique with (phi r); auto with zarith. 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). - intros rec ih il j Hih Hj Hij Hrec; rewrite sqrt312_step_def. + intros Hih Hj Hij Hrec; rewrite sqrt312_step_def. assert (H1: ([|ih|] <= [|j|])%Z) by (apply sqrt312_lower_bound with il; auto). case (phi_bounded ih); intros Hih1 _. case (phi_bounded il); intros Hil1 _. @@ -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. @@ -2234,15 +2234,15 @@ 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. + revert rec ih il j; elim n; unfold iter312_sqrt; fold iter312_sqrt; clear n. intros rec ih il j Hi Hj Hij Hrec; apply sqrt312_step_correct; auto with zarith. intros; apply Hrec; auto with zarith. rewrite Zpower_0_r; auto with zarith. @@ -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. |