diff options
Diffstat (limited to 'theories/Numbers/Rational/BigQ')
| -rw-r--r-- | theories/Numbers/Rational/BigQ/BigQ.v | 188 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/Q0Make.v | 1412 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/QMake.v | 1345 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/QMake_base.v | 34 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/QbiMake.v | 1066 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/QifMake.v | 979 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/QpMake.v | 901 | ||||
| -rw-r--r-- | theories/Numbers/Rational/BigQ/QvMake.v | 1151 |
8 files changed, 1518 insertions, 5558 deletions
diff --git a/theories/Numbers/Rational/BigQ/BigQ.v b/theories/Numbers/Rational/BigQ/BigQ.v index 39e120f7..21f2513f 100644 --- a/theories/Numbers/Rational/BigQ/BigQ.v +++ b/theories/Numbers/Rational/BigQ/BigQ.v @@ -8,19 +8,35 @@ (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) -(*i $Id: BigQ.v 11028 2008-06-01 17:34:19Z letouzey $ i*) +(*i $Id: BigQ.v 11208 2008-07-04 16:57:46Z letouzey $ i*) -Require Export QMake_base. -Require Import QpMake. -Require Import QvMake. -Require Import Q0Make. -Require Import QifMake. -Require Import QbiMake. +Require Import Field Qfield BigN BigZ QSig QMake. -(* We choose for Q the implemention with - multiple representation of 0: 0, 1/0, 2/0 etc *) +(** We choose for BigQ an implemention with + multiple representation of 0: 0, 1/0, 2/0 etc. + See [QMake.v] *) -Module BigQ <: QSig.QType := Q0. +(** First, we provide translations functions between [BigN] and [BigZ] *) + +Module BigN_BigZ <: NType_ZType BigN.BigN BigZ. + Definition Z_of_N := BigZ.Pos. + Lemma spec_Z_of_N : forall n, BigZ.to_Z (Z_of_N n) = BigN.to_Z n. + Proof. + reflexivity. + Qed. + Definition Zabs_N := BigZ.to_N. + Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Zabs (BigZ.to_Z z). + Proof. + unfold Zabs_N; intros. + rewrite BigZ.spec_to_Z, Zmult_comm; apply Zsgn_Zabs. + Qed. +End BigN_BigZ. + +(** This allows to build [BigQ] out of [BigN] and [BigQ] via [QMake] *) + +Module BigQ <: QSig.QType := QMake.Make BigN BigZ BigN_BigZ. + +(** Notations about [BigQ] *) Notation bigQ := BigQ.t. @@ -28,8 +44,150 @@ Delimit Scope bigQ_scope with bigQ. Bind Scope bigQ_scope with bigQ. Bind Scope bigQ_scope with BigQ.t. -Notation " i + j " := (BigQ.add i j) : bigQ_scope. -Notation " i - j " := (BigQ.sub i j) : bigQ_scope. -Notation " i * j " := (BigQ.mul i j) : bigQ_scope. -Notation " i / j " := (BigQ.div i j) : bigQ_scope. -Notation " i ?= j " := (BigQ.compare i j) : bigQ_scope. +Infix "+" := BigQ.add : bigQ_scope. +Infix "-" := BigQ.sub : bigQ_scope. +Notation "- x" := (BigQ.opp x) : bigQ_scope. +Infix "*" := BigQ.mul : bigQ_scope. +Infix "/" := BigQ.div : bigQ_scope. +Infix "^" := BigQ.power : bigQ_scope. +Infix "?=" := BigQ.compare : bigQ_scope. +Infix "==" := BigQ.eq : bigQ_scope. +Infix "<" := BigQ.lt : bigQ_scope. +Infix "<=" := BigQ.le : bigQ_scope. +Notation "x > y" := (BigQ.lt y x)(only parsing) : bigQ_scope. +Notation "x >= y" := (BigQ.le y x)(only parsing) : bigQ_scope. +Notation "[ q ]" := (BigQ.to_Q q) : bigQ_scope. + +Open Scope bigQ_scope. + +(** [BigQ] is a setoid *) + +Add Relation BigQ.t BigQ.eq + reflexivity proved by (fun x => Qeq_refl [x]) + symmetry proved by (fun x y => Qeq_sym [x] [y]) + transitivity proved by (fun x y z => Qeq_trans [x] [y] [z]) +as BigQeq_rel. + +Add Morphism BigQ.add with signature BigQ.eq ==> BigQ.eq ==> BigQ.eq as BigQadd_wd. +Proof. + unfold BigQ.eq; intros; rewrite !BigQ.spec_add; rewrite H, H0; apply Qeq_refl. +Qed. + +Add Morphism BigQ.opp with signature BigQ.eq ==> BigQ.eq as BigQopp_wd. +Proof. + unfold BigQ.eq; intros; rewrite !BigQ.spec_opp; rewrite H; apply Qeq_refl. +Qed. + +Add Morphism BigQ.sub with signature BigQ.eq ==> BigQ.eq ==> BigQ.eq as BigQsub_wd. +Proof. + unfold BigQ.eq; intros; rewrite !BigQ.spec_sub; rewrite H, H0; apply Qeq_refl. +Qed. + +Add Morphism BigQ.mul with signature BigQ.eq ==> BigQ.eq ==> BigQ.eq as BigQmul_wd. +Proof. + unfold BigQ.eq; intros; rewrite !BigQ.spec_mul; rewrite H, H0; apply Qeq_refl. +Qed. + +Add Morphism BigQ.inv with signature BigQ.eq ==> BigQ.eq as BigQinv_wd. +Proof. + unfold BigQ.eq; intros; rewrite !BigQ.spec_inv; rewrite H; apply Qeq_refl. +Qed. + +Add Morphism BigQ.div with signature BigQ.eq ==> BigQ.eq ==> BigQ.eq as BigQdiv_wd. +Proof. + unfold BigQ.eq; intros; rewrite !BigQ.spec_div; rewrite H, H0; apply Qeq_refl. +Qed. + +(* TODO : fix this. For the moment it's useless (horribly slow) +Hint Rewrite + BigQ.spec_0 BigQ.spec_1 BigQ.spec_m1 BigQ.spec_compare + BigQ.spec_red BigQ.spec_add BigQ.spec_sub BigQ.spec_opp + BigQ.spec_mul BigQ.spec_inv BigQ.spec_div BigQ.spec_power_pos + BigQ.spec_square : bigq. *) + + +(** [BigQ] is a field *) + +Lemma BigQfieldth : + field_theory BigQ.zero BigQ.one BigQ.add BigQ.mul BigQ.sub BigQ.opp BigQ.div BigQ.inv BigQ.eq. +Proof. +constructor. +constructor; intros; red. +rewrite BigQ.spec_add, BigQ.spec_0; ring. +rewrite ! BigQ.spec_add; ring. +rewrite ! BigQ.spec_add; ring. +rewrite BigQ.spec_mul, BigQ.spec_1; ring. +rewrite ! BigQ.spec_mul; ring. +rewrite ! BigQ.spec_mul; ring. +rewrite BigQ.spec_add, ! BigQ.spec_mul, BigQ.spec_add; ring. +unfold BigQ.sub; apply Qeq_refl. +rewrite BigQ.spec_add, BigQ.spec_0, BigQ.spec_opp; ring. +compute; discriminate. +intros; red. +unfold BigQ.div; apply Qeq_refl. +intros; red. +rewrite BigQ.spec_mul, BigQ.spec_inv, BigQ.spec_1; field. +rewrite <- BigQ.spec_0; auto. +Qed. + +Lemma BigQpowerth : + power_theory BigQ.one BigQ.mul BigQ.eq Z_of_N BigQ.power. +Proof. +constructor. +intros; red. +rewrite BigQ.spec_power. +replace ([r] ^ Z_of_N n)%Q with (pow_N 1 Qmult [r] n)%Q. +destruct n. +simpl; compute; auto. +induction p; simpl; auto; try rewrite !BigQ.spec_mul, !IHp; apply Qeq_refl. +destruct n; reflexivity. +Qed. + +Lemma BigQ_eq_bool_correct : + forall x y, BigQ.eq_bool x y = true -> x==y. +Proof. +intros; generalize (BigQ.spec_eq_bool x y); rewrite H; auto. +Qed. + +Lemma BigQ_eq_bool_complete : + forall x y, x==y -> BigQ.eq_bool x y = true. +Proof. +intros; generalize (BigQ.spec_eq_bool x y). +destruct BigQ.eq_bool; auto. +Qed. + +(* TODO : improve later the detection of constants ... *) + +Ltac BigQcst t := + match t with + | BigQ.zero => BigQ.zero + | BigQ.one => BigQ.one + | BigQ.minus_one => BigQ.minus_one + | _ => NotConstant + end. + +Add Field BigQfield : BigQfieldth + (decidable BigQ_eq_bool_correct, + completeness BigQ_eq_bool_complete, + constants [BigQcst], + power_tac BigQpowerth [Qpow_tac]). + +Section Examples. + +Let ex1 : forall x y z, (x+y)*z == (x*z)+(y*z). + intros. + ring. +Qed. + +Let ex8 : forall x, x ^ 1 == x. + intro. + ring. +Qed. + +Let ex10 : forall x y, ~(y==BigQ.zero) -> (x/y)*y == x. +intros. +field. +auto. +Qed. + +End Examples.
\ No newline at end of file diff --git a/theories/Numbers/Rational/BigQ/Q0Make.v b/theories/Numbers/Rational/BigQ/Q0Make.v deleted file mode 100644 index 93f52c03..00000000 --- a/theories/Numbers/Rational/BigQ/Q0Make.v +++ /dev/null @@ -1,1412 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(*i $Id: Q0Make.v 11028 2008-06-01 17:34:19Z letouzey $ i*) - -Require Import Bool. -Require Import ZArith. -Require Import Znumtheory. -Require Import BigNumPrelude. -Require Import Arith. -Require Export BigN. -Require Export BigZ. -Require Import QArith. -Require Import Qcanon. -Require Import Qpower. -Require Import QSig. -Require Import QMake_base. - -Module Q0 <: QType. - - Import BinInt Zorder. - - (** The notation of a rational number is either an integer x, - interpreted as itself or a pair (x,y) of an integer x and a natural - number y interpreted as x/y. The pairs (x,0) and (0,y) are all - interpreted as 0. *) - - Definition t := q_type. - - (** Specification with respect to [QArith] *) - - Open Local Scope Q_scope. - - Definition of_Z x: t := Qz (BigZ.of_Z x). - - Definition of_Q q: t := - match q with x # y => - Qq (BigZ.of_Z x) (BigN.of_N (Npos y)) - end. - - Definition to_Q (q: t) := - match q with - Qz x => BigZ.to_Z x # 1 - |Qq x y => if BigN.eq_bool y BigN.zero then 0 - else BigZ.to_Z x # Z2P (BigN.to_Z y) - end. - - Notation "[ x ]" := (to_Q x). - - Theorem strong_spec_of_Q: forall q: Q, [of_Q q] = q. - Proof. - intros (x,y); simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigN.spec_of_pos; intros HH; discriminate HH. - rewrite BigZ.spec_of_Z; simpl. - rewrite (BigN.spec_of_pos); auto. - Qed. - - Theorem spec_of_Q: forall q: Q, [of_Q q] == q. - Proof. - intros; rewrite strong_spec_of_Q; red; auto. - Qed. - - Definition eq x y := [x] == [y]. - - Definition zero: t := Qz BigZ.zero. - Definition one: t := Qz BigZ.one. - Definition minus_one: t := Qz BigZ.minus_one. - - Lemma spec_0: [zero] == 0. - Proof. - reflexivity. - Qed. - - Lemma spec_1: [one] == 1. - Proof. - reflexivity. - Qed. - - Lemma spec_m1: [minus_one] == -(1). - Proof. - reflexivity. - Qed. - - Definition opp (x: t): t := - match x with - | Qz zx => Qz (BigZ.opp zx) - | Qq nx dx => Qq (BigZ.opp nx) dx - end. - - Theorem strong_spec_opp: forall q, [opp q] = -[q]. - Proof. - intros [z | x y]; simpl. - rewrite BigZ.spec_opp; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigZ.spec_opp; auto. - Qed. - - Theorem spec_opp : forall q, [opp q] == -[q]. - Proof. - intros; rewrite strong_spec_opp; red; auto. - Qed. - - Definition compare (x y: t) := - match x, y with - | Qz zx, Qz zy => BigZ.compare zx zy - | Qz zx, Qq ny dy => - if BigN.eq_bool dy BigN.zero then BigZ.compare zx BigZ.zero - else BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny - | Qq nx dx, Qz zy => - if BigN.eq_bool dx BigN.zero then BigZ.compare BigZ.zero zy - else BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx)) - | Qq nx dx, Qq ny dy => - match BigN.eq_bool dx BigN.zero, BigN.eq_bool dy BigN.zero with - | true, true => Eq - | true, false => BigZ.compare BigZ.zero ny - | false, true => BigZ.compare nx BigZ.zero - | false, false => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) - end - end. - - Theorem spec_compare: forall q1 q2, (compare q1 q2) = ([q1] ?= [q2]). - Proof. - intros [z1 | x1 y1] [z2 | x2 y2]; - unfold Qcompare, compare, to_Q, Qnum, Qden. - repeat rewrite Zmult_1_r. - generalize (BigZ.spec_compare z1 z2); case BigZ.compare; intros H; auto. - rewrite H; rewrite Zcompare_refl; auto. - rewrite Zmult_1_r. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - rewrite Zmult_1_r; generalize (BigZ.spec_compare z1 BigZ.zero); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH1; rewrite HH1; rewrite Zcompare_refl; auto. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (z1 * BigZ.Pos y2) x2)%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare BigZ.zero z2); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH1; rewrite <- HH1; rewrite Zcompare_refl; auto. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y1); auto with zarith. - rewrite Zmult_1_r. - generalize (BigZ.spec_compare x1 (z2 * BigZ.Pos y1))%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - rewrite Zcompare_refl; auto. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare BigZ.zero x2); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH2; rewrite <- HH2; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare x1 BigZ.zero)%bigZ; case BigZ.compare; - auto; rewrite BigZ.spec_0. - intros HH2; rewrite <- HH2; rewrite Zcompare_refl; auto. - repeat rewrite Z2P_correct. - 2: generalize (BigN.spec_pos y1); auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (x1 * BigZ.Pos y2) - (x2 * BigZ.Pos y1))%bigZ; case BigZ.compare; - repeat rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - Qed. - - Definition lt n m := compare n m = Lt. - Definition le n m := compare n m <> Gt. - 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. - -(* Je pense que cette fonction normalise bien ... *) - Definition norm n d: t := - let gcd := BigN.gcd (BigZ.to_N n) d in - match BigN.compare BigN.one gcd with - | Lt => - let n := BigZ.div n (BigZ.Pos gcd) in - let d := BigN.div d gcd in - match BigN.compare d BigN.one with - | Gt => Qq n d - | Eq => Qz n - | Lt => zero - end - | Eq => Qq n d - | Gt => zero (* gcd = 0 => both numbers are 0 *) - end. - - Theorem spec_norm: forall n q, [norm n q] == [Qq n q]. - Proof. - intros p q; unfold norm. - assert (Hp := BigN.spec_pos (BigZ.to_N p)). - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; auto; rewrite BigN.spec_1; rewrite BigN.spec_gcd; intros H1. - apply Qeq_refl. - generalize (BigN.spec_pos (q / BigN.gcd (BigZ.to_N p) q)%bigN). - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; auto; rewrite BigN.spec_1; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; intros H2 HH. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H3; simpl; - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - auto with zarith. - generalize H2; rewrite H3; - rewrite Zdiv_0_l; auto with zarith. - generalize H1 H2 H3 (BigN.spec_pos q); clear H1 H2 H3. - rewrite spec_to_N. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 H4; rewrite Z2P_correct; auto with zarith. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H3; simpl. - case H3. - generalize H1 H2 H3 HH; clear H1 H2 H3 HH. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 HH. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto with zarith. - case (Zle_lt_or_eq _ _ HH); auto with zarith. - intros HH1; rewrite <- HH1; ring. - generalize (Zgcd_is_gcd a b); intros HH1; inversion HH1; auto. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; intros H3. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H4. - case H3; rewrite H4; rewrite Zdiv_0_l; auto with zarith. - simpl. - assert (FF := BigN.spec_pos q). - rewrite Z2P_correct; auto with zarith. - rewrite <- BigN.spec_gcd; rewrite <- BigN.spec_div; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; auto with zarith. - simpl; rewrite BigZ.spec_div; simpl. - rewrite BigN.spec_gcd; auto with zarith. - generalize H1 H2 H3 H4 HH FF; clear H1 H2 H3 H4 HH FF. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 H4 HH FF. - rewrite spec_to_N; fold a. - rewrite Zgcd_div_swap; auto with zarith. - rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_gcd; auto with zarith. - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (BigN.gcd (BigZ.to_N p) q))); - rewrite BigN.spec_gcd; auto with zarith. - intros; apply False_ind; auto with zarith. - intros HH2; assert (FF1 := Zgcd_inv_0_l _ _ (sym_equal HH2)). - assert (FF2 := Zgcd_inv_0_l _ _ (sym_equal HH2)). - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2; simpl. - rewrite spec_to_N. - rewrite FF2; ring. - Qed. - - - Definition add (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (BigZ.add zx zy) - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - end - | Qq nx dx => - if BigN.eq_bool dx BigN.zero then y - else match y with - | Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - Qq n d - end - end. - - Theorem spec_add : forall x y, [add x y] == [x] + [y]. - Proof. - intros [x | nx dx] [y | ny dy]; unfold Qplus; simpl. - rewrite BigZ.spec_add; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - assert (F1:= BigN.spec_pos dy). - rewrite Zmult_1_r; red; simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH; simpl; try ring. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH1; simpl; try ring. - case HH; auto. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl; auto. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH; simpl; try ring. - rewrite Zmult_1_r; apply Qeq_refl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH1; simpl; try ring. - case HH; auto. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl; auto. - rewrite Zmult_1_r; rewrite Pmult_1_r. - apply Qeq_refl. - assert (F1:= BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - apply Qeq_refl. - case HH2; auto. - simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - case HH2; auto. - case HH1; auto. - rewrite Zmult_1_r; apply Qeq_refl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - simpl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - case HH; auto. - rewrite Zmult_1_r; rewrite Zplus_0_r; rewrite Pmult_1_r. - apply Qeq_refl. - simpl. - generalize (BigN.spec_eq_bool (dx * dy)%bigN BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_mul; - rewrite BigN.spec_0; intros HH2. - (case (Zmult_integral _ _ HH2); intros HH3); - [case HH| case HH1]; auto. - rewrite BigZ.spec_add; repeat rewrite BigZ.spec_mul; simpl. - assert (Fx: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (Fy: (0 < BigN.to_Z dy)%Z). - generalize (BigN.spec_pos dy); auto with zarith. - red; simpl; rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto with zarith. - apply Zmult_lt_0_compat; auto. - Qed. - - Definition add_norm (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (BigZ.add zx zy) - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - end - | Qq nx dx => - if BigN.eq_bool dx BigN.zero then y - else match y with - | Qz zy => norm (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - norm n d - end - end. - - Theorem spec_add_norm : forall x y, [add_norm x y] == [x] + [y]. - Proof. - intros x y; rewrite <- spec_add; auto. - case x; case y; clear x y; unfold add_norm, add. - intros; apply Qeq_refl. - intros p1 n p2. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - simpl. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - apply Qeq_refl. - apply Qeq_refl. - intros p1 p2 n. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - apply Qeq_refl. - intros p1 q1 p2 q2. - generalize (BigN.spec_eq_bool q2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - apply Qeq_refl. - generalize (BigN.spec_eq_bool q1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - apply Qeq_refl. - Qed. - - Definition sub x y := add x (opp y). - - Theorem spec_sub : forall x y, [sub x y] == [x] - [y]. - Proof. - intros x y; unfold sub; rewrite spec_add; auto. - rewrite spec_opp; ring. - Qed. - - Definition sub_norm x y := add_norm x (opp y). - - Theorem spec_sub_norm : forall x y, [sub_norm x y] == [x] - [y]. - Proof. - intros x y; unfold sub_norm; rewrite spec_add_norm; auto. - rewrite spec_opp; ring. - Qed. - - Definition mul (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => Qq (BigZ.mul nx ny) (BigN.mul dx dy) - end. - - Theorem spec_mul : forall x y, [mul x y] == [x] * [y]. - Proof. - intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl. - rewrite BigZ.spec_mul; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH1. - red; simpl; ring. - rewrite BigZ.spec_mul; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH1. - red; simpl; ring. - rewrite BigZ.spec_mul; rewrite Pmult_1_r. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_mul; - intros HH1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH2. - red; simpl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH3. - red; simpl; ring. - case (Zmult_integral _ _ HH1); intros HH. - case HH2; auto. - case HH3; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH2. - case HH1; rewrite HH2; ring. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH3. - case HH1; rewrite HH3; ring. - rewrite BigZ.spec_mul. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - apply Qeq_refl. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dy); auto with zarith. - Qed. - -Definition mul_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => - if BigZ.eq_bool zx BigZ.zero then zero - else - let gcd := BigN.gcd (BigZ.to_N zx) dy in - match BigN.compare gcd BigN.one with - Gt => - let zx := BigZ.div zx (BigZ.Pos gcd) in - let d := BigN.div dy gcd in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul zx ny) - else Qq (BigZ.mul zx ny) d - | _ => Qq (BigZ.mul zx ny) dy - end - | Qq nx dx, Qz zy => - if BigZ.eq_bool zy BigZ.zero then zero - else - let gcd := BigN.gcd (BigZ.to_N zy) dx in - match BigN.compare gcd BigN.one with - Gt => - let zy := BigZ.div zy (BigZ.Pos gcd) in - let d := BigN.div dx gcd in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul zy nx) - else Qq (BigZ.mul zy nx) d - | _ => Qq (BigZ.mul zy nx) dx - end - | Qq nx dx, Qq ny dy => - let (nx, dy) := - let gcd := BigN.gcd (BigZ.to_N nx) dy in - match BigN.compare gcd BigN.one with - Gt => (BigZ.div nx (BigZ.Pos gcd), BigN.div dy gcd) - | _ => (nx, dy) - end in - let (ny, dx) := - let gcd := BigN.gcd (BigZ.to_N ny) dx in - match BigN.compare gcd BigN.one with - Gt => (BigZ.div ny (BigZ.Pos gcd), BigN.div dx gcd) - | _ => (ny, dx) - end in - let d := (BigN.mul dx dy) in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul ny nx) - else Qq (BigZ.mul ny nx) d - end. - - Theorem spec_mul_norm : forall x y, [mul_norm x y] == [x] * [y]. - Proof. - intros x y; rewrite <- spec_mul; auto. - unfold mul_norm, mul; case x; case y; clear x y. - intros; apply Qeq_refl. - intros p1 n p2. - set (a := BigN.to_Z (BigZ.to_N p2)). - set (b := BigN.to_Z n). - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; unfold zero, to_Q; repeat rewrite BigZ.spec_0; intros H. - case BigN.eq_bool; try apply Qeq_refl. - rewrite BigZ.spec_mul; rewrite H. - red; simpl; ring. - assert (F: (0 < a)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos (BigZ.to_N p2))); auto. - intros H1; case H; rewrite spec_to_N; rewrite <- H1; ring. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; rewrite BigN.spec_gcd; - fold a b; intros H1. - apply Qeq_refl. - apply Qeq_refl. - assert (F0 : (0 < (Zgcd a b))%Z). - apply Zlt_trans with 1%Z. - red; auto. - apply Zgt_lt; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; - fold a b; intros H2. - assert (F1: b = Zgcd a b). - pattern b at 1; rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); - auto with zarith. - rewrite H2; ring. - assert (FF := Zgcd_is_gcd a b); inversion FF; auto. - assert (F2: (0 < b)%Z). - rewrite F1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; fold b; intros H3. - rewrite H3 in F2; discriminate F2. - rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - rewrite BigZ.spec_mul. - red; simpl; rewrite Z2P_correct; auto. - rewrite Zmult_1_r; rewrite spec_to_N; fold a b. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p1)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; fold a b; auto; intros H3. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H4. - apply Qeq_refl. - case H4; fold b. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite H3; ring. - assert (FF := Zgcd_is_gcd a b); inversion FF; auto. - simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; fold b; intros H4. - case H3; rewrite H4; rewrite Zdiv_0_l; auto. - rewrite BigZ.spec_mul; rewrite BigZ.spec_div; simpl; - rewrite BigN.spec_gcd; fold a b; auto with zarith. - assert (F1: (0 < b)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos n)); fold b; auto with zarith. - red; simpl. - rewrite BigZ.spec_mul. - repeat rewrite Z2P_correct; auto. - rewrite spec_to_N; fold a. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p1)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - ring. - apply Zgcd_div_pos; auto. - intros p1 p2 n. - set (a := BigN.to_Z (BigZ.to_N p1)). - set (b := BigN.to_Z n). - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; unfold zero, to_Q; repeat rewrite BigZ.spec_0; intros H. - case BigN.eq_bool; try apply Qeq_refl. - rewrite BigZ.spec_mul; rewrite H. - red; simpl; ring. - assert (F: (0 < a)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos (BigZ.to_N p1))); auto. - intros H1; case H; rewrite spec_to_N; rewrite <- H1; ring. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; rewrite BigN.spec_gcd; - fold a b; intros H1. - repeat rewrite BigZ.spec_mul; rewrite Zmult_comm. - apply Qeq_refl. - repeat rewrite BigZ.spec_mul; rewrite Zmult_comm. - apply Qeq_refl. - assert (F0 : (0 < (Zgcd a b))%Z). - apply Zlt_trans with 1%Z. - red; auto. - apply Zgt_lt; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; - fold a b; intros H2. - assert (F1: b = Zgcd a b). - pattern b at 1; rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); - auto with zarith. - rewrite H2; ring. - assert (FF := Zgcd_is_gcd a b); inversion FF; auto. - assert (F2: (0 < b)%Z). - rewrite F1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; fold b; intros H3. - rewrite H3 in F2; discriminate F2. - rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - rewrite BigZ.spec_mul. - red; simpl; rewrite Z2P_correct; auto. - rewrite Zmult_1_r; rewrite spec_to_N; fold a b. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p2)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; fold a b; auto; intros H3. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H4. - apply Qeq_refl. - case H4; fold b. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite H3; ring. - assert (FF := Zgcd_is_gcd a b); inversion FF; auto. - simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; fold b; intros H4. - case H3; rewrite H4; rewrite Zdiv_0_l; auto. - rewrite BigZ.spec_mul; rewrite BigZ.spec_div; simpl; - rewrite BigN.spec_gcd; fold a b; auto with zarith. - assert (F1: (0 < b)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos n)); fold b; auto with zarith. - red; simpl. - rewrite BigZ.spec_mul. - repeat rewrite Z2P_correct; auto. - rewrite spec_to_N; fold a. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p2)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - ring. - apply Zgcd_div_pos; auto. - set (f := fun p t => - match (BigN.gcd (BigZ.to_N p) t ?= BigN.one)%bigN with - | Eq => (p, t) - | Lt => (p, t) - | Gt => - ((p / BigZ.Pos (BigN.gcd (BigZ.to_N p) t))%bigZ, - (t / BigN.gcd (BigZ.to_N p) t)%bigN) - end). - assert (F: forall p t, - let (n, d) := f p t in [Qq p t] == [Qq n d]). - intros p t1; unfold f. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; rewrite BigN.spec_gcd; intros H1. - apply Qeq_refl. - apply Qeq_refl. - set (a := BigN.to_Z (BigZ.to_N p)). - set (b := BigN.to_Z t1). - fold a b in H1. - assert (F0 : (0 < (Zgcd a b))%Z). - apply Zlt_trans with 1%Z. - red; auto. - apply Zgt_lt; auto. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; fold b; intros HH1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; fold b; intros HH2. - simpl; ring. - case HH2. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; fold a b; auto. - rewrite HH1; rewrite Zdiv_0_l; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; - rewrite BigN.spec_div; rewrite BigN.spec_gcd; fold a b; auto; - intros HH2. - case HH1. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite HH2; ring. - assert (FF := Zgcd_is_gcd a b); inversion FF; auto. - simpl. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; fold a b; auto with zarith. - assert (F1: (0 < b)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos t1)); fold b; auto with zarith. - intros HH; case HH1; auto. - repeat rewrite Z2P_correct; auto. - rewrite spec_to_N; fold a. - rewrite Zgcd_div_swap; auto. - apply Zgcd_div_pos; auto. - intros HH; rewrite HH in F0; discriminate F0. - intros p1 n1 p2 n2. - change ([let (nx , dy) := f p2 n1 in - let (ny, dx) := f p1 n2 in - if BigN.eq_bool (dx * dy)%bigN BigN.one - then Qz (ny * nx) - else Qq (ny * nx) (dx * dy)] == [Qq (p2 * p1) (n2 * n1)]). - generalize (F p2 n1) (F p1 n2). - case f; case f. - intros u1 u2 v1 v2 Hu1 Hv1. - apply Qeq_trans with [mul (Qq p2 n1) (Qq p1 n2)]. - rewrite spec_mul; rewrite Hu1; rewrite Hv1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; rewrite BigN.spec_mul; intros HH1. - assert (F1: BigN.to_Z u2 = 1%Z). - case (Zmult_1_inversion_l _ _ HH1); auto. - generalize (BigN.spec_pos u2); auto with zarith. - assert (F2: BigN.to_Z v2 = 1%Z). - rewrite Zmult_comm in HH1. - case (Zmult_1_inversion_l _ _ HH1); auto. - generalize (BigN.spec_pos v2); auto with zarith. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - rewrite H1 in F2; discriminate F2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2. - rewrite H2 in F1; discriminate F1. - simpl; rewrite BigZ.spec_mul. - rewrite F1; rewrite F2; simpl; ring. - rewrite Qmult_comm; rewrite <- spec_mul. - apply Qeq_refl. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_mul; - rewrite Zmult_comm; intros H1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_mul; intros H2; auto. - case H2; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_mul; intros H2; auto. - case H1; auto. - Qed. - - -Definition inv (x: t): t := - match x with - | Qz (BigZ.Pos n) => Qq BigZ.one n - | Qz (BigZ.Neg n) => Qq BigZ.minus_one n - | Qq (BigZ.Pos n) d => Qq (BigZ.Pos d) n - | Qq (BigZ.Neg n) d => Qq (BigZ.Neg d) n - end. - - Theorem spec_inv : forall x, [inv x] == /[x]. - Proof. - intros [ [x | x] | [nx | nx] dx]; unfold inv, Qinv; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - rewrite H1; apply Qeq_refl. - generalize H1 (BigN.spec_pos x); case (BigN.to_Z x); auto. - intros HH; case HH; auto. - intros; red; simpl; auto. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - rewrite H1; apply Qeq_refl. - generalize H1 (BigN.spec_pos x); case (BigN.to_Z x); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl; auto. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - apply Qeq_refl. - rewrite H1; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - rewrite H2; red; simpl; auto. - generalize H1 (BigN.spec_pos nx); case (BigN.to_Z nx); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - apply Qeq_refl. - rewrite H1; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - rewrite H2; red; simpl; auto. - generalize H1 (BigN.spec_pos nx); case (BigN.to_Z nx); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p _ HH; case HH; auto. - Qed. - -Definition inv_norm (x: t): t := - match x with - | Qz (BigZ.Pos n) => - match BigN.compare n BigN.one with - Gt => Qq BigZ.one n - | _ => x - end - | Qz (BigZ.Neg n) => - match BigN.compare n BigN.one with - Gt => Qq BigZ.minus_one n - | _ => x - end - | Qq (BigZ.Pos n) d => - match BigN.compare n BigN.one with - Gt => Qq (BigZ.Pos d) n - | Eq => Qz (BigZ.Pos d) - | Lt => Qz (BigZ.zero) - end - | Qq (BigZ.Neg n) d => - match BigN.compare n BigN.one with - Gt => Qq (BigZ.Neg d) n - | Eq => Qz (BigZ.Neg d) - | Lt => Qz (BigZ.zero) - end - end. - - Theorem spec_inv_norm : forall x, [inv_norm x] == /[x]. - Proof. - intros [ [x | x] | [nx | nx] dx]; unfold inv_norm, Qinv. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H. - simpl; rewrite H; apply Qeq_refl. - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); simpl. - generalize H; case BigN.to_Z. - intros _ HH; discriminate HH. - intros p; case p; auto. - intros p1 HH; discriminate HH. - intros p1 HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; discriminate HH. - intros HH; rewrite <- HH. - apply Qeq_refl. - generalize H; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - rewrite H1; intros HH; discriminate. - generalize H; case BigN.to_Z. - intros HH; discriminate HH. - intros; red; simpl; auto. - intros p HH; discriminate HH. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H. - simpl; rewrite H; apply Qeq_refl. - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); simpl. - generalize H; case BigN.to_Z. - intros _ HH; discriminate HH. - intros p; case p; auto. - intros p1 HH; discriminate HH. - intros p1 HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; discriminate HH. - intros HH; rewrite <- HH. - apply Qeq_refl. - generalize H; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - rewrite H1; intros HH; discriminate. - generalize H; case BigN.to_Z. - intros HH; discriminate HH. - intros; red; simpl; auto. - intros p HH; discriminate HH. - simpl Qnum. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; simpl. - case BigN.compare; red; simpl; auto. - rewrite H1; auto. - case BigN.eq_bool; auto. - simpl; rewrite H1; auto. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H2. - rewrite H2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - red; simpl. - rewrite Zmult_1_r; rewrite Pmult_1_r; rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - generalize H2 (BigN.spec_pos nx); case (BigN.to_Z nx). - intros; apply Qeq_refl. - intros p; case p; clear p. - intros p HH; discriminate HH. - intros p HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - simpl; generalize H2; case (BigN.to_Z nx). - intros HH; discriminate HH. - intros p Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H4. - rewrite H4 in H2; discriminate H2. - red; simpl. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - intros p HH; discriminate HH. - simpl Qnum. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; simpl. - case BigN.compare; red; simpl; auto. - rewrite H1; auto. - case BigN.eq_bool; auto. - simpl; rewrite H1; auto. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H2. - rewrite H2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zmult_1_r; rewrite Pmult_1_r; rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - generalize H2 (BigN.spec_pos nx); case (BigN.to_Z nx). - intros; apply Qeq_refl. - intros p; case p; clear p. - intros p HH; discriminate HH. - intros p HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - simpl; generalize H2; case (BigN.to_Z nx). - intros HH; discriminate HH. - intros p Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H4. - rewrite H4 in H2; discriminate H2. - red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p HH; discriminate HH. - Qed. - - Definition div x y := mul x (inv y). - - Theorem spec_div x y: [div x y] == [x] / [y]. - Proof. - intros x y; unfold div; rewrite spec_mul; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Definition div_norm x y := mul_norm x (inv y). - - Theorem spec_div_norm x y: [div_norm x y] == [x] / [y]. - Proof. - intros x y; unfold div_norm; rewrite spec_mul_norm; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Definition square (x: t): t := - match x with - | Qz zx => Qz (BigZ.square zx) - | Qq nx dx => Qq (BigZ.square nx) (BigN.square dx) - end. - - Theorem spec_square : forall x, [square x] == [x] ^ 2. - Proof. - intros [ x | nx dx]; unfold square. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - simpl Qpower. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H. - red; simpl. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_square; - intros H1. - case H1; rewrite H; auto. - red; simpl. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_square; - intros H1. - case H; case (Zmult_integral _ _ H1); auto. - simpl. - rewrite BigZ.spec_square. - rewrite Zpos_mult_morphism. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dx); auto with zarith. - Qed. - - Definition power_pos (x: t) p: t := - match x with - | Qz zx => Qz (BigZ.power_pos zx p) - | Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p) - end. - - Theorem spec_power_pos : forall x p, [power_pos x p] == [x] ^ Zpos p. - Proof. - intros [x | nx dx] p; unfold power_pos. - unfold power_pos; red; simpl. - generalize (Qpower_decomp p (BigZ.to_Z x) 1). - unfold Qeq; simpl. - rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Zmult_1_r. - intros H; rewrite H. - rewrite BigZ.spec_power_pos; simpl; ring. - simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_power_pos; intros H1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2. - elim p; simpl. - intros; red; simpl; auto. - intros p1 Hp1; rewrite <- Hp1; red; simpl; auto. - apply Qeq_refl. - case H2; generalize H1. - elim p; simpl. - intros p1 Hrec. - change (xI p1) with (1 + (xO p1))%positive. - rewrite Zpower_pos_is_exp; rewrite Zpower_pos_1_r. - intros HH; case (Zmult_integral _ _ HH); auto. - rewrite <- Pplus_diag. - rewrite Zpower_pos_is_exp. - intros HH1; case (Zmult_integral _ _ HH1); auto. - intros p1 Hrec. - rewrite <- Pplus_diag. - rewrite Zpower_pos_is_exp. - intros HH1; case (Zmult_integral _ _ HH1); auto. - rewrite Zpower_pos_1_r; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2. - case H1; rewrite H2; auto. - simpl; rewrite Zpower_pos_0_l; auto. - assert (F1: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (F2: (0 < BigN.to_Z dx ^ ' p)%Z). - unfold Zpower; apply Zpower_pos_pos; auto. - unfold power_pos; red; simpl. - generalize (Qpower_decomp p (BigZ.to_Z nx) - (Z2P (BigN.to_Z dx))). - unfold Qeq; simpl. - repeat rewrite Z2P_correct; auto. - unfold Qeq; simpl; intros HH. - rewrite HH. - rewrite BigZ.spec_power_pos; simpl; ring. - Qed. - - (** Interaction with [Qcanon.Qc] *) - - Open Scope Qc_scope. - - Definition of_Qc q := of_Q (this q). - - Definition to_Qc q := !!(to_Q q). - - Notation "[[ x ]]" := (to_Qc x). - - Theorem spec_of_Qc: forall q, [[of_Qc q]] = q. - Proof. - intros (x, Hx); unfold of_Qc, to_Qc; simpl. - apply Qc_decomp; simpl. - intros. - rewrite <- H0 at 2; apply Qred_complete. - apply spec_of_Q. - Qed. - - Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. - Proof. - intros q; unfold Qcopp, to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - rewrite spec_opp. - rewrite <- Qred_opp. - rewrite Qred_correct; red; auto. - Qed. - - Theorem spec_comparec: forall q1 q2, - compare q1 q2 = ([[q1]] ?= [[q2]]). - Proof. - unfold Qccompare, to_Qc. - intros q1 q2; rewrite spec_compare; simpl; auto. - apply Qcompare_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_addc x y: - [[add x y]] = [[x]] + [[y]]. - Proof. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_add_normc x y: - [[add_norm x y]] = [[x]] + [[y]]. - Proof. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add_norm; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_subc x y: [[sub x y]] = [[x]] - [[y]]. - Proof. - intros x y; unfold sub; rewrite spec_addc; auto. - rewrite spec_oppc; ring. - Qed. - - Theorem spec_sub_normc x y: - [[sub_norm x y]] = [[x]] - [[y]]. - intros x y; unfold sub_norm; rewrite spec_add_normc; auto. - rewrite spec_oppc; ring. - Qed. - - Theorem spec_mulc x y: - [[mul x y]] = [[x]] * [[y]]. - Proof. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_mul_normc x y: - [[mul_norm x y]] = [[x]] * [[y]]. - Proof. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul_norm; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_invc x: - [[inv x]] = /[[x]]. - Proof. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_inv_normc x: - [[inv_norm x]] = /[[x]]. - Proof. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv_norm; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_divc x y: [[div x y]] = [[x]] / [[y]]. - Proof. - intros x y; unfold div; rewrite spec_mulc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. - Proof. - intros x y; unfold div_norm; rewrite spec_mul_normc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - Theorem spec_squarec x: [[square x]] = [[x]]^2. - Proof. - intros x; unfold to_Qc. - apply trans_equal with (!! ([x]^2)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_square; auto. - simpl Qcpower. - replace (!! [x] * 1) with (!![x]); try ring. - simpl. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Theorem spec_power_posc x p: - [[power_pos x p]] = [[x]] ^ nat_of_P p. - Proof. - intros x p; unfold to_Qc. - apply trans_equal with (!! ([x]^Zpos p)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_power_pos; auto. - pattern p; apply Pind; clear p. - simpl; ring. - intros p Hrec. - rewrite nat_of_P_succ_morphism; simpl Qcpower. - rewrite <- Hrec. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; - unfold this. - apply Qred_complete. - assert (F: [x] ^ ' Psucc p == [x] * [x] ^ ' p). - simpl; case x; simpl; clear x Hrec. - intros x; simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - intros nx dx. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - unfold Qpower_positive. - assert (tmp: forall p, pow_pos Qmult 0%Q p = 0%Q). - intros p1; elim p1; simpl; auto; clear p1. - intros p1 Hp1; rewrite Hp1; auto. - intros p1 Hp1; rewrite Hp1; auto. - repeat rewrite tmp; intros; red; simpl; auto. - intros H1. - assert (F1: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - repeat rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto. - 2: apply Zpower_pos_pos; auto. - 2: apply Zpower_pos_pos; auto. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - rewrite F. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - -End Q0. diff --git a/theories/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v new file mode 100644 index 00000000..494420bd --- /dev/null +++ b/theories/Numbers/Rational/BigQ/QMake.v @@ -0,0 +1,1345 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) +(************************************************************************) + +(*i $Id: QMake.v 11208 2008-07-04 16:57:46Z letouzey $ i*) + +Require Import BigNumPrelude ROmega. +Require Import QArith Qcanon Qpower. +Require Import NSig ZSig QSig. + +Module Type NType_ZType (N:NType)(Z:ZType). + Parameter Z_of_N : N.t -> Z.t. + Parameter spec_Z_of_N : forall n, Z.to_Z (Z_of_N n) = N.to_Z n. + Parameter Zabs_N : Z.t -> N.t. + Parameter spec_Zabs_N : forall z, N.to_Z (Zabs_N z) = Zabs (Z.to_Z z). +End NType_ZType. + +Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. + + (** The notation of a rational number is either an integer x, + interpreted as itself or a pair (x,y) of an integer x and a natural + number y interpreted as x/y. The pairs (x,0) and (0,y) are all + interpreted as 0. *) + + Inductive t_ := + | Qz : Z.t -> t_ + | Qq : Z.t -> N.t -> t_. + + Definition t := t_. + + (** 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 + | 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 + | 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) + end. + + Notation "[ x ]" := (to_Q x). + + 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); + case N.eq_bool; auto; rewrite N.spec_0. + rewrite N.spec_of_N; intros; discriminate. + rewrite N.spec_of_N; auto. + 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; intros; discriminate. + rewrite N.spec_of_N; auto. + Qed. + + Theorem spec_of_Q: forall q: Q, [of_Q q] == q. + Proof. + intros; rewrite strong_spec_of_Q; red; auto. + Qed. + + Definition eq x y := [x] == [y]. + + Definition zero: t := Qz Z.zero. + Definition one: t := Qz Z.one. + Definition minus_one: t := Qz Z.minus_one. + + Lemma spec_0: [zero] == 0. + Proof. + simpl; rewrite Z.spec_0; reflexivity. + Qed. + + Lemma spec_1: [one] == 1. + Proof. + simpl; rewrite Z.spec_1; reflexivity. + Qed. + + Lemma spec_m1: [minus_one] == -(1). + Proof. + simpl; rewrite Z.spec_m1; reflexivity. + Qed. + + Definition compare (x y: t) := + match x, y with + | Qz zx, Qz zy => Z.compare zx zy + | 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 + 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)) + (Z.mul ny (Z_of_N dx)) + end + end. + + 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 : + 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 -> + 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 + 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_gcd N.spec_gcd Zgcd_Zabs + 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] => + 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] => + 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" "*" := + rewrite N.spec_div in * by (nzsimpl; romega). + + Ltac simpl_zdiv := rewrite Z.spec_div by (nzsimpl; romega). + Tactic Notation "simpl_zdiv" "in" "*" := + rewrite Z.spec_div in * by (nzsimpl; romega). + + 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]; + unfold Qcompare, compare; qsimpl; rewrite ! N_to_Z2P; auto. + Qed. + + Definition lt n m := compare n m = Lt. + Definition le n m := compare n m <> Gt. + 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 := + match compare n m with Eq => true | _ => false end. + + Theorem spec_eq_bool: forall x y, + if eq_bool x y then [x] == [y] else ~([x] == [y]). + Proof. + intros. + unfold eq_bool. + rewrite spec_compare. + generalize (Qeq_alt [x] [y]). + destruct Qcompare. + intros H; rewrite H; auto. + intros H H'; rewrite H in H'; discriminate. + intros H H'; rewrite H in H'; discriminate. + Qed. + + (** Normalisation function *) + + Definition norm n d : t := + let gcd := N.gcd (Zabs_N n) d in + match N.compare N.one gcd with + | 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 + | Gt => Qq n d + | Eq => Qz n + | Lt => zero + end + | Eq => Qq n d + | Gt => zero (* gcd = 0 => both numbers are 0 *) + end. + + Theorem spec_norm: forall n q, [norm n q] == [Qq n q]. + Proof. + intros p q; unfold norm. + assert (Hp := N.spec_pos (Zabs_N p)). + assert (Hq := N.spec_pos q). + nzsimpl. + destr_zcompare. + qsimpl. + + simpl_ndiv. + destr_zcompare. + qsimpl. + rewrite H1 in *; rewrite Zdiv_0_l in H0; discriminate. + rewrite N_to_Z2P; auto. + simpl_zdiv; nzsimpl. + rewrite Zgcd_div_swap0, H0; romega. + + qsimpl. + assert (0 < N.to_Z q / Zgcd (Z.to_Z p) (N.to_Z q))%Z. + apply Zgcd_div_pos; romega. + romega. + + qsimpl. + simpl_ndiv in *; nzsimpl; romega. + simpl_ndiv in *. + rewrite H1, Zdiv_0_l in H2; elim H2; auto. + rewrite 2 N_to_Z2P; auto. + simpl_ndiv; simpl_zdiv; nzsimpl. + apply Zgcd_div_swap0; romega. + + qsimpl. + assert (H' : Zgcd (Z.to_Z p) (N.to_Z q) = 0%Z). + generalize (Zgcd_is_pos (Z.to_Z p) (N.to_Z q)); romega. + symmetry; apply (Zgcd_inv_0_l _ _ H'); auto. + Qed. + + 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 + (apply Qred_complete; apply spec_norm). + symmetry; apply Qred_identity. + unfold norm. + assert (Hp := N.spec_pos (Zabs_N p)). + assert (Hq := N.spec_pos q). + nzsimpl. + destr_zcompare. + (* Eq *) + simpl. + destr_neq_bool; nzsimpl; simpl; auto. + intros. + rewrite N_to_Z2P; auto. + (* Lt *) + simpl_ndiv. + destr_zcompare. + qsimpl; auto. + qsimpl. + qsimpl. + simpl_zdiv; nzsimpl. + rewrite N_to_Z2P; auto. + clear H1. + simpl_ndiv; nzsimpl. + rewrite Zgcd_1_rel_prime. + destruct (Z_lt_le_dec 0 (N.to_Z q)). + apply Zis_gcd_rel_prime; auto with zarith. + apply Zgcd_is_gcd. + replace (N.to_Z q) with 0%Z in * by romega. + rewrite Zdiv_0_l in H0; discriminate. + (* Gt *) + simpl; auto. + Qed. + + (** Reduction function : producing irreducible fractions *) + + Definition red (x : t) : t := + match x with + | Qz z => x + | Qq n d => norm n d + end. + + Definition Reduced x := [red x] = [x]. + + Theorem spec_red : forall x, [red x] == [x]. + Proof. + intros [ z | n d ]. + auto with qarith. + unfold red. + apply spec_norm. + Qed. + + Theorem strong_spec_red : forall x, [red x] = Qred [x]. + Proof. + intros [ z | n d ]. + unfold red. + symmetry; apply Qred_identity; simpl; auto. + 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 + 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 + else match y with + | Qz zy => Qq (Z.add nx (Z.mul zy (Z_of_N dx))) dx + | Qq ny dy => + if N.eq_bool dy N.zero then x + else + let n := Z.add (Z.mul nx (Z_of_N dy)) (Z.mul ny (Z_of_N dx)) in + let d := N.mul dx dy in + Qq n d + end + end. + + Theorem spec_add : forall x y, [add x y] == [x] + [y]. + Proof. + intros [x | nx dx] [y | ny dy]; unfold Qplus; qsimpl. + intuition. + rewrite N_to_Z2P; auto. + intuition. + rewrite Pmult_1_r, N_to_Z2P; auto. + intuition. + rewrite Pmult_1_r, N_to_Z2P; auto. + destruct (Zmult_integral _ _ H); intuition. + rewrite Zpos_mult_morphism, 2 N_to_Z2P; auto. + rewrite (Z2P_correct (N.to_Z dx * N.to_Z dy)); auto. + apply Zmult_lt_0_compat. + generalize (N.spec_pos dx); romega. + generalize (N.spec_pos dy); romega. + Qed. + + Definition add_norm (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 + 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 + else match y with + | Qz zy => norm (Z.add nx (Z.mul zy (Z_of_N dx))) dx + | Qq ny dy => + if N.eq_bool dy N.zero then x + else + let n := Z.add (Z.mul nx (Z_of_N dy)) (Z.mul ny (Z_of_N dx)) in + let d := N.mul dx dy in + norm n d + end + end. + + 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; + destr_neq_bool; auto using Qeq_refl, spec_norm. + Qed. + + 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 spec_add, spec_add_norm; apply Qeq_refl ]. + rewrite <- strong_spec_red. + destruct x as [zx|nx dx]; destruct y as [zy|ny dy]. + simpl in *; auto. + simpl; intros. + destr_neq_bool; nzsimpl; simpl; auto. + simpl; intros. + destr_neq_bool; nzsimpl; simpl; auto. + simpl; intros. + destr_neq_bool; nzsimpl; simpl; auto. + Qed. + + Definition opp (x: t): t := + match x with + | Qz zx => Qz (Z.opp zx) + | Qq nx dx => Qq (Z.opp nx) dx + end. + + Theorem strong_spec_opp: forall q, [opp q] = -[q]. + Proof. + intros [z | x y]; simpl. + rewrite Z.spec_opp; auto. + 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. + Qed. + + Theorem spec_opp : forall q, [opp q] == -[q]. + Proof. + intros; rewrite strong_spec_opp; red; auto. + Qed. + + Theorem strong_spec_opp_norm : forall q, Reduced q -> Reduced (opp q). + Proof. + unfold Reduced; intros. + rewrite strong_spec_opp, <- H, !strong_spec_red, <- Qred_opp. + apply Qred_complete; apply spec_opp. + Qed. + + Definition sub x y := add x (opp y). + + Theorem spec_sub : forall x y, [sub x y] == [x] - [y]. + Proof. + intros x y; unfold sub; rewrite spec_add; auto. + rewrite spec_opp; ring. + Qed. + + Definition sub_norm x y := add_norm x (opp y). + + Theorem spec_sub_norm : forall x y, [sub_norm x y] == [x] - [y]. + Proof. + intros x y; unfold sub_norm; rewrite spec_add_norm; auto. + rewrite spec_opp; ring. + Qed. + + Theorem strong_spec_sub_norm : forall x y, + Reduced x -> Reduced y -> Reduced (sub_norm x y). + Proof. + intros. + unfold sub_norm. + apply strong_spec_add_norm; auto. + apply strong_spec_opp_norm; auto. + Qed. + + Definition mul (x y: t): t := + match x, y with + | Qz zx, Qz zy => Qz (Z.mul zx zy) + | Qz zx, Qq ny dy => Qq (Z.mul zx ny) dy + | Qq nx dx, Qz zy => Qq (Z.mul nx zy) dx + | Qq nx dx, Qq ny dy => Qq (Z.mul nx ny) (N.mul dx dy) + end. + + Theorem spec_mul : forall x y, [mul x y] == [x] * [y]. + Proof. + intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl; qsimpl. + rewrite Pmult_1_r, N_to_Z2P; auto. + destruct (Zmult_integral _ _ H1); intuition. + rewrite H0 in H1; elim H1; auto. + rewrite H0 in H1; elim H1; auto. + rewrite H in H1; nzsimpl; elim H1; auto. + rewrite Zpos_mult_morphism, 2 N_to_Z2P; auto. + rewrite (Z2P_correct (N.to_Z dx * N.to_Z dy)); auto. + apply Zmult_lt_0_compat. + generalize (N.spec_pos dx); omega. + generalize (N.spec_pos dy); omega. + Qed. + + 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. + rewrite H0 in H; discriminate. + rewrite N_to_Z2P, H0; auto with zarith. + Qed. + + Definition irred n d := + let gcd := N.gcd (Zabs_N n) d in + 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 + (Z.to_Z n' * g = Z.to_Z n)%Z /\ (N.to_Z d' * g = N.to_Z d)%Z. + Proof. + intros. + unfold irred; nzsimpl; simpl. + destr_zcompare. + exists 1%Z; nzsimpl; auto. + exists 0%Z; nzsimpl. + assert (Zgcd (Z.to_Z n) (N.to_Z d) = 0%Z). + generalize (Zgcd_is_pos (Z.to_Z n) (N.to_Z d)); romega. + clear H. + split. + symmetry; apply (Zgcd_inv_0_l _ _ H0). + symmetry; apply (Zgcd_inv_0_r _ _ H0). + exists (Zgcd (Z.to_Z n) (N.to_Z d)). + simpl. + split. + simpl_zdiv; nzsimpl. + destruct (Zgcd_is_gcd (Z.to_Z n) (N.to_Z d)). + rewrite Zmult_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. + simpl_ndiv; nzsimpl. + destruct (Zgcd_is_gcd (Z.to_Z n) (N.to_Z d)). + rewrite Zmult_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. + Qed. + + Lemma spec_irred_zero : forall n d, + (N.to_Z d = 0)%Z <-> (N.to_Z (snd (irred n d)) = 0)%Z. + Proof. + intros. + unfold irred. + split. + nzsimpl; intros. + destr_zcompare; auto. + simpl. + simpl_ndiv; nzsimpl. + rewrite H, Zdiv_0_l; auto. + nzsimpl; destr_zcompare; simpl; auto. + simpl_ndiv; nzsimpl. + intros. + generalize (N.spec_pos d); intros. + destruct (N.to_Z d); auto. + assert (0 < 0)%Z. + rewrite <- H0 at 2. + apply Zgcd_div_pos; auto with zarith. + compute; auto. + discriminate. + compute in H1; elim H1; auto. + Qed. + + 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. + nzsimpl. + destr_zcompare; simpl; auto. + elim H. + apply (Zgcd_inv_0_r (Z.to_Z n)). + generalize (Zgcd_is_pos (Z.to_Z n) (N.to_Z d)); romega. + + simpl_ndiv; simpl_zdiv; nzsimpl. + rewrite Zgcd_1_rel_prime. + apply Zis_gcd_rel_prime. + generalize (N.spec_pos d); romega. + generalize (Zgcd_is_pos (Z.to_Z n) (N.to_Z d)); romega. + apply Zgcd_is_gcd; auto. + Qed. + + 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 => + 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 := + 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 + 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, + [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. + destr_zeq_bool; intros Hz; nzsimpl. + qsimpl; rewrite Hz; auto. + assert (Hd := N.spec_pos d). + destruct Z_le_gt_dec. + qsimpl. + rewrite norm_denum. + qsimpl. + simpl_ndiv in *; nzsimpl. + rewrite (Zdiv_gcd_zero _ _ H0 H) in z0; discriminate. + simpl_ndiv in *; nzsimpl. + rewrite H, Zdiv_0_l in H0; elim H0; auto. + rewrite 2 N_to_Z2P; auto. + simpl_ndiv; simpl_zdiv; nzsimpl. + rewrite (Zmult_comm (Z.to_Z z)), <- 2 Zmult_assoc. + rewrite <- Zgcd_div_swap0; auto with zarith; ring. + Qed. + + 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. + simpl; nzsimpl. + destr_neq_bool; simpl; nzsimpl; auto. + intros H'; elim H'; auto. + destr_neq_bool; simpl; nzsimpl. + simpl_ndiv; nzsimpl; rewrite Hd, Zdiv_0_l; intros; discriminate. + intros _. + destr_neq_bool; simpl; nzsimpl; auto. + simpl_ndiv; nzsimpl; rewrite Hd, Zdiv_0_l; intro H'; elim H'; auto. + + rewrite N_to_Z2P in H; auto. + unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. + destr_zeq_bool; intros Hz; simpl; nzsimpl; simpl; auto. + destruct Z_le_gt_dec as [H'|H']. + simpl; nzsimpl. + destr_neq_bool; simpl; nzsimpl; auto. + intros. + rewrite N_to_Z2P; auto. + apply Zgcd_mult_rel_prime; auto. + generalize (Zgcd_inv_0_l (Z.to_Z z) (N.to_Z d)) + (Zgcd_is_pos (Z.to_Z z) (N.to_Z d)); romega. + destr_neq_bool; simpl; nzsimpl; auto. + simpl_ndiv; simpl_zdiv; nzsimpl. + intros. + destr_neq_bool; simpl; nzsimpl; auto. + simpl_ndiv in *; nzsimpl. + intros. + rewrite Z2P_correct. + apply Zgcd_mult_rel_prime. + rewrite Zgcd_1_rel_prime. + apply Zis_gcd_rel_prime. + generalize (N.spec_pos d); romega. + generalize (Zgcd_is_pos (Z.to_Z z) (N.to_Z d)); romega. + apply Zgcd_is_gcd. + destruct (Zgcd_is_gcd (Z.to_Z z) (N.to_Z d)) as [ (z0,Hz0) (d0,Hd0) Hzd]. + replace (N.to_Z d / Zgcd (Z.to_Z z) (N.to_Z d))%Z with d0. + rewrite Zgcd_1_rel_prime in *. + apply bezout_rel_prime. + destruct (rel_prime_bezout _ _ H) as [u v Huv]. + apply Bezout_intro with u (v*(Zgcd (Z.to_Z z) (N.to_Z d)))%Z. + rewrite <- Huv; rewrite Hd0 at 2; ring. + rewrite Hd0 at 1. + symmetry; apply Z_div_mult_full; auto with zarith. + apply Zgcd_div_pos. + generalize (N.spec_pos d); romega. + generalize (Zgcd_is_pos (Z.to_Z z) (N.to_Z d)); romega. + Qed. + + Theorem spec_mul_norm : forall x y, [mul_norm x y] == [x] * [y]. + Proof. + intros x y; rewrite <- spec_mul; auto. + unfold mul_norm, mul; destruct x; destruct y. + apply Qeq_refl. + apply spec_mul_norm_Qz_Qq. + rewrite spec_mul_norm_Qz_Qq; qsimpl; ring. + + rename t0 into nx, t3 into dy, t2 into ny, t1 into dx. + destruct (spec_irred nx dy) as (g & Hg). + 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). + simpl snd in *; destruct Hg as [Hg1 Hg2]; destruct Hg' as [Hg1' Hg2']. + rewrite norm_denum. + qsimpl. + + elim H; destruct (Zmult_integral _ _ H0) as [Eq|Eq]. + rewrite <- Hz' in Eq; rewrite Eq; simpl; auto. + rewrite <- Hz in Eq; rewrite Eq; nzsimpl; auto. + + elim H0; destruct (Zmult_integral _ _ H) as [Eq|Eq]. + rewrite Hz' in Eq; rewrite Eq; simpl; auto. + rewrite Hz in Eq; rewrite Eq; nzsimpl; auto. + + rewrite 2 Z2P_correct. + rewrite <- Hg1, <- Hg2, <- Hg1', <- Hg2'; ring. + + 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 + by (apply Zmult_le_0_compat; apply N.spec_pos). + romega. + Qed. + + Theorem strong_spec_mul_norm : forall x y, + Reduced x -> Reduced y -> Reduced (mul_norm x y). + Proof. + unfold Reduced; intros. + rewrite strong_spec_red, Qred_iff. + destruct x as [zx|nx dx]; destruct y as [zy|ny dy]. + simpl in *; auto. + simpl. + rewrite <- Qred_iff, <- strong_spec_red, strong_spec_mul_norm_Qz_Qq; auto. + simpl. + rewrite <- Qred_iff, <- strong_spec_red, strong_spec_mul_norm_Qz_Qq; auto. + simpl. + destruct (spec_irred nx dy) as [g Hg]. + destruct (spec_irred ny dx) as [g' Hg']. + assert (Hz := spec_irred_zero nx dy). + 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). + simpl snd in *; destruct Hg as [Hg1 Hg2]; destruct Hg' as [Hg1' Hg2']. + destr_neq_bool; simpl; nzsimpl; intros. + apply Zis_gcd_gcd; auto with zarith; apply Zis_gcd_1. + destr_neq_bool; simpl; nzsimpl; intros. + auto. + + revert H H0. + rewrite 2 strong_spec_red, 2 Qred_iff; simpl. + destr_neq_bool; simpl; nzsimpl; intros. + rewrite Hz in H; rewrite H in H2; nzsimpl; elim H2; auto. + rewrite Hz' in H0; rewrite H0 in H2; nzsimpl; elim H2; auto. + rewrite Hz in H; rewrite H in H2; nzsimpl; elim H2; auto. + + rewrite N_to_Z2P in *; auto. + rewrite Z2P_correct. + + apply Zgcd_mult_rel_prime; rewrite Zgcd_sym; + apply Zgcd_mult_rel_prime; rewrite Zgcd_sym; auto. + + rewrite Zgcd_1_rel_prime in *. + apply bezout_rel_prime. + destruct (rel_prime_bezout _ _ H4) as [u v Huv]. + apply Bezout_intro with (u*g')%Z (v*g)%Z. + rewrite <- Huv, <- Hg1', <- Hg2. ring. + + rewrite Zgcd_1_rel_prime in *. + apply bezout_rel_prime. + destruct (rel_prime_bezout _ _ H3) as [u v Huv]. + apply Bezout_intro with (u*g)%Z (v*g')%Z. + rewrite <- Huv, <- Hg2', <- Hg1. ring. + + assert (0 <= N.to_Z d2 * N.to_Z d1)%Z. + apply Zmult_le_0_compat; apply N.spec_pos. + romega. + Qed. + + Definition inv (x: t): t := + match x 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 => + match Z.compare Z.zero n with + | Eq => zero + | Lt => Qq (Z_of_N d) (Zabs_N n) + | Gt => Qq (Z.opp (Z_of_N d)) (Zabs_N n) + end + end. + + Theorem spec_inv : forall x, [inv x] == /[x]. + Proof. + destruct x as [ z | n d ]. + (* Qz z *) + simpl. + rewrite Zcompare_spec_alt; destr_zcompare. + (* 0 = z *) + rewrite <- H. + simpl; nzsimpl; compute; auto. + (* 0 < z *) + simpl. + destr_neq_bool; nzsimpl; [ intros; rewrite Zabs_eq in *; romega | intros _ ]. + set (z':=Z.to_Z z) in *; clearbody z'. + red; simpl. + rewrite Zabs_eq by romega. + rewrite Z2P_correct by auto. + unfold Qinv; simpl; destruct z'; simpl; auto; discriminate. + (* 0 > z *) + simpl. + destr_neq_bool; nzsimpl; [ intros; rewrite Zabs_non_eq in *; romega | intros _ ]. + set (z':=Z.to_Z z) in *; clearbody z'. + red; simpl. + rewrite Zabs_non_eq by romega. + rewrite Z2P_correct by romega. + unfold Qinv; simpl; destruct z'; simpl; auto; discriminate. + (* Qq n d *) + simpl. + rewrite Zcompare_spec_alt; destr_zcompare. + (* 0 = n *) + rewrite <- H. + simpl; nzsimpl. + destr_neq_bool; intros; compute; auto. + (* 0 < n *) + simpl. + destr_neq_bool; nzsimpl; intros. + intros; rewrite Zabs_eq in *; romega. + intros; rewrite Zabs_eq in *; romega. + clear H1. + rewrite H0. + compute; auto. + clear H1. + set (n':=Z.to_Z n) in *; clearbody n'. + rewrite Zabs_eq by romega. + red; simpl. + rewrite Z2P_correct by auto. + unfold Qinv; simpl; destruct n'; simpl; auto; try discriminate. + rewrite Zpos_mult_morphism, N_to_Z2P; auto. + (* 0 > n *) + simpl. + destr_neq_bool; nzsimpl; intros. + intros; rewrite Zabs_non_eq in *; romega. + intros; rewrite Zabs_non_eq in *; romega. + clear H1. + red; nzsimpl; rewrite H0; compute; auto. + clear H1. + set (n':=Z.to_Z n) in *; clearbody n'. + red; simpl; nzsimpl. + rewrite Zabs_non_eq by romega. + rewrite Z2P_correct by romega. + unfold Qinv; simpl; destruct n'; simpl; auto; try discriminate. + assert (T : forall x, Zneg x = Zopp (Zpos x)) by auto. + rewrite T, Zpos_mult_morphism, N_to_Z2P; auto; ring. + Qed. + + Definition inv_norm (x: t): t := + match x 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 + | Eq => zero + | 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 + | Lt => Qq (Z.opp (Z_of_N d)) (Zabs_N n) + | _ => Qz (Z.opp (Z_of_N d)) + end + end + end. + + Theorem spec_inv_norm : forall x, [inv_norm x] == /[x]. + Proof. + intros. + rewrite <- spec_inv. + destruct x as [ z | n d ]. + (* Qz z *) + simpl. + rewrite Zcompare_spec_alt; destr_zcompare; auto with qarith. + (* Qq n d *) + simpl; nzsimpl; destr_neq_bool. + destr_zcompare; simpl; auto with qarith. + destr_neq_bool; nzsimpl; auto with qarith. + intros _ Hd; rewrite Hd; auto with qarith. + destr_neq_bool; nzsimpl; auto with qarith. + intros _ Hd; rewrite Hd; auto with qarith. + (* 0 < n *) + destr_zcompare; auto with qarith. + destr_zcompare; nzsimpl; simpl; auto with qarith; intros. + destr_neq_bool; nzsimpl; [ intros; rewrite Zabs_eq in *; romega | intros _ ]. + rewrite H0; auto with qarith. + romega. + (* 0 > n *) + destr_zcompare; nzsimpl; simpl; auto with qarith. + destr_neq_bool; nzsimpl; [ intros; rewrite Zabs_non_eq in *; romega | intros _ ]. + rewrite H0; auto with qarith. + romega. + Qed. + + Theorem strong_spec_inv_norm : forall x, Reduced x -> Reduced (inv_norm x). + Proof. + unfold Reduced. + intros. + destruct x as [ z | n d ]. + (* Qz *) + simpl; nzsimpl. + rewrite strong_spec_red, Qred_iff. + destr_zcompare; simpl; nzsimpl; auto. + destr_neq_bool; nzsimpl; simpl; auto. + destr_neq_bool; nzsimpl; simpl; auto. + (* Qq n d *) + rewrite strong_spec_red, Qred_iff in H; revert H. + simpl; nzsimpl. + destr_neq_bool; nzsimpl; auto with qarith. + destr_zcompare; simpl; nzsimpl; auto; intros. + (* 0 < n *) + destr_zcompare; simpl; nzsimpl; auto. + destr_neq_bool; nzsimpl; simpl; auto. + rewrite Zabs_eq; romega. + intros _. + rewrite strong_spec_norm; simpl; nzsimpl. + destr_neq_bool; nzsimpl. + rewrite Zabs_eq; romega. + intros _. + rewrite Qred_iff. + simpl. + rewrite Zabs_eq; auto with zarith. + rewrite N_to_Z2P in *; auto. + rewrite Z2P_correct; auto with zarith. + rewrite Zgcd_sym; auto. + (* 0 > n *) + destr_neq_bool; nzsimpl; simpl; auto; intros. + destr_zcompare; simpl; nzsimpl; auto. + destr_neq_bool; nzsimpl. + rewrite Zabs_non_eq; romega. + intros _. + rewrite strong_spec_norm; simpl; nzsimpl. + destr_neq_bool; nzsimpl. + rewrite Zabs_non_eq; romega. + intros _. + rewrite Qred_iff. + simpl. + rewrite N_to_Z2P in *; auto. + rewrite Z2P_correct; auto with zarith. + intros. + rewrite Zgcd_sym, Zgcd_Zabs, Zgcd_sym. + apply Zis_gcd_gcd; auto with zarith. + apply Zis_gcd_minus. + rewrite Zopp_involutive, <- H1; apply Zgcd_is_gcd. + rewrite Zabs_non_eq; romega. + Qed. + + Definition div x y := mul x (inv y). + + Theorem spec_div x y: [div x y] == [x] / [y]. + Proof. + intros x y; unfold div; rewrite spec_mul; auto. + unfold Qdiv; apply Qmult_comp. + apply Qeq_refl. + apply spec_inv; auto. + Qed. + + Definition div_norm x y := mul_norm x (inv_norm y). + + Theorem spec_div_norm x y: [div_norm x y] == [x] / [y]. + Proof. + intros x y; unfold div_norm; rewrite spec_mul_norm; auto. + unfold Qdiv; apply Qmult_comp. + apply Qeq_refl. + apply spec_inv_norm; auto. + Qed. + + Theorem strong_spec_div_norm : forall x y, + Reduced x -> Reduced y -> Reduced (div_norm x y). + Proof. + intros; unfold div_norm. + apply strong_spec_mul_norm; auto. + apply strong_spec_inv_norm; auto. + Qed. + + Definition square (x: t): t := + match x with + | Qz zx => Qz (Z.square zx) + | Qq nx dx => Qq (Z.square nx) (N.square dx) + end. + + Theorem spec_square : forall x, [square x] == [x] ^ 2. + Proof. + destruct x as [ z | n d ]. + simpl; rewrite Z.spec_square; red; auto. + simpl. + destr_neq_bool; nzsimpl; intros. + apply Qeq_refl. + rewrite N.spec_square in *; nzsimpl. + contradict H; elim (Zmult_integral _ _ H0); auto. + rewrite N.spec_square in *; nzsimpl. + 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. + red; simpl. + rewrite Zpos_mult_morphism; rewrite !Z2P_correct; auto. + apply Zmult_lt_0_compat; auto. + Qed. + + Definition power_pos (x : t) p : t := + match x with + | 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. + (* Qz *) + simpl. + rewrite Z.spec_power_pos. + rewrite Qpower_decomp. + red; simpl; f_equal. + rewrite Zpower_pos_1_l; auto. + (* Qq *) + simpl. + rewrite Z.spec_power_pos. + destr_neq_bool; nzsimpl; intros. + apply Qeq_sym; apply Qpower_positive_0. + rewrite N.spec_power_pos in *. + assert (0 < N.to_Z d ^ ' p)%Z. + apply Zpower_gt_0; auto with zarith. + generalize (N.spec_pos d); romega. + romega. + rewrite N.spec_power_pos, H in *. + rewrite Zpower_0_l in H0; [ elim H0; auto | discriminate ]. + rewrite Qpower_decomp. + red; simpl; do 3 f_equal. + rewrite Z2P_correct by (generalize (N.spec_pos d); romega). + rewrite N.spec_power_pos. auto. + Qed. + + Theorem strong_spec_power_pos : forall x p, + Reduced x -> Reduced (power_pos x p). + Proof. + destruct x as [z | n d]; simpl; intros. + red; simpl; auto. + red; simpl; intros. + rewrite strong_spec_norm; simpl. + destr_neq_bool; nzsimpl; intros. + simpl; auto. + rewrite Qred_iff. + revert H. + unfold Reduced; rewrite strong_spec_red, Qred_iff; simpl. + destr_neq_bool; nzsimpl; simpl; intros. + rewrite N.spec_power_pos in H0. + elim H0; rewrite H; rewrite Zpower_0_l; auto; discriminate. + rewrite N_to_Z2P in *; auto. + rewrite N.spec_power_pos, Z.spec_power_pos; auto. + rewrite Zgcd_1_rel_prime in *. + apply rel_prime_Zpower; auto with zarith. + Qed. + + 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) + end. + + Theorem spec_power : forall x z, [power x z] == [x]^z. + Proof. + destruct z. + simpl; nzsimpl; red; auto. + apply spec_power_pos. + simpl. + rewrite spec_inv, spec_power_pos; apply Qeq_refl. + Qed. + + 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) + end. + + Theorem spec_power_norm : forall x z, [power_norm x z] == [x]^z. + Proof. + destruct z. + simpl; nzsimpl; red; auto. + apply spec_power_pos. + simpl. + rewrite spec_inv_norm, spec_power_pos; apply Qeq_refl. + Qed. + + Theorem strong_spec_power_norm : forall x z, + Reduced x -> Reduced (power_norm x z). + Proof. + destruct z; simpl. + intros _; unfold Reduced; rewrite strong_spec_red. + unfold one. + simpl to_Q; nzsimpl; auto. + intros; apply strong_spec_power_pos; auto. + intros; apply strong_spec_inv_norm; apply strong_spec_power_pos; auto. + Qed. + + + (** Interaction with [Qcanon.Qc] *) + + Open Scope Qc_scope. + + Definition of_Qc q := of_Q (this q). + + Definition to_Qc q := !! [q]. + + Notation "[[ x ]]" := (to_Qc x). + + Theorem strong_spec_of_Qc : forall q, [of_Qc q] = q. + Proof. + intros (q,Hq); intros. + unfold of_Qc; rewrite strong_spec_of_Q; auto. + Qed. + + Lemma strong_spec_of_Qc_bis : forall q, Reduced (of_Qc q). + Proof. + intros; red; rewrite strong_spec_red, strong_spec_of_Qc. + destruct q; simpl; auto. + Qed. + + Theorem spec_of_Qc: forall q, [[of_Qc q]] = q. + Proof. + intros; apply Qc_decomp; simpl; intros. + rewrite strong_spec_of_Qc; auto. + Qed. + + Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. + Proof. + intros q; unfold Qcopp, to_Qc, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + rewrite spec_opp, <- Qred_opp, Qred_correct. + apply Qeq_refl. + Qed. + + Theorem spec_oppc_bis : forall q : Qc, [opp (of_Qc q)] = - q. + Proof. + intros. + rewrite <- strong_spec_opp_norm by apply strong_spec_of_Qc_bis. + rewrite strong_spec_red. + symmetry; apply (Qred_complete (-q)%Q). + rewrite spec_opp, strong_spec_of_Qc; auto with qarith. + Qed. + + Theorem spec_comparec: forall q1 q2, + compare q1 q2 = ([[q1]] ?= [[q2]]). + Proof. + unfold Qccompare, to_Qc. + intros q1 q2; rewrite spec_compare; simpl; auto. + apply Qcompare_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_addc x y: + [[add x y]] = [[x]] + [[y]]. + Proof. + intros x y; unfold to_Qc. + apply trans_equal with (!! ([x] + [y])). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_add; auto. + unfold Qcplus, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qplus_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_add_normc x y: + [[add_norm x y]] = [[x]] + [[y]]. + Proof. + intros x y; unfold to_Qc. + apply trans_equal with (!! ([x] + [y])). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_add_norm; auto. + unfold Qcplus, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qplus_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_add_normc_bis : forall x y : Qc, + [add_norm (of_Qc x) (of_Qc y)] = x+y. + Proof. + intros. + rewrite <- strong_spec_add_norm by apply strong_spec_of_Qc_bis. + rewrite strong_spec_red. + symmetry; apply (Qred_complete (x+y)%Q). + rewrite spec_add_norm, ! strong_spec_of_Qc; auto with qarith. + Qed. + + Theorem spec_subc x y: [[sub x y]] = [[x]] - [[y]]. + Proof. + intros x y; unfold sub; rewrite spec_addc; auto. + rewrite spec_oppc; ring. + Qed. + + Theorem spec_sub_normc x y: + [[sub_norm x y]] = [[x]] - [[y]]. + Proof. + intros x y; unfold sub_norm; rewrite spec_add_normc; auto. + rewrite spec_oppc; ring. + Qed. + + Theorem spec_sub_normc_bis : forall x y : Qc, + [sub_norm (of_Qc x) (of_Qc y)] = x-y. + Proof. + intros. + rewrite <- strong_spec_sub_norm by apply strong_spec_of_Qc_bis. + rewrite strong_spec_red. + symmetry; apply (Qred_complete (x+(-y)%Qc)%Q). + rewrite spec_sub_norm, ! strong_spec_of_Qc. + unfold Qcopp, Q2Qc; rewrite Qred_correct; auto with qarith. + Qed. + + Theorem spec_mulc x y: + [[mul x y]] = [[x]] * [[y]]. + Proof. + intros x y; unfold to_Qc. + apply trans_equal with (!! ([x] * [y])). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_mul; auto. + unfold Qcmult, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qmult_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_mul_normc x y: + [[mul_norm x y]] = [[x]] * [[y]]. + Proof. + intros x y; unfold to_Qc. + apply trans_equal with (!! ([x] * [y])). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_mul_norm; auto. + unfold Qcmult, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qmult_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_mul_normc_bis : forall x y : Qc, + [mul_norm (of_Qc x) (of_Qc y)] = x*y. + Proof. + intros. + rewrite <- strong_spec_mul_norm by apply strong_spec_of_Qc_bis. + rewrite strong_spec_red. + symmetry; apply (Qred_complete (x*y)%Q). + rewrite spec_mul_norm, ! strong_spec_of_Qc; auto with qarith. + Qed. + + Theorem spec_invc x: + [[inv x]] = /[[x]]. + Proof. + intros x; unfold to_Qc. + apply trans_equal with (!! (/[x])). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_inv; auto. + unfold Qcinv, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qinv_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_inv_normc x: + [[inv_norm x]] = /[[x]]. + Proof. + intros x; unfold to_Qc. + apply trans_equal with (!! (/[x])). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_inv_norm; auto. + unfold Qcinv, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qinv_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_inv_normc_bis : forall x : Qc, + [inv_norm (of_Qc x)] = /x. + Proof. + intros. + rewrite <- strong_spec_inv_norm by apply strong_spec_of_Qc_bis. + rewrite strong_spec_red. + symmetry; apply (Qred_complete (/x)%Q). + rewrite spec_inv_norm, ! strong_spec_of_Qc; auto with qarith. + Qed. + + Theorem spec_divc x y: [[div x y]] = [[x]] / [[y]]. + Proof. + intros x y; unfold div; rewrite spec_mulc; auto. + unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. + apply spec_invc; auto. + Qed. + + Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. + Proof. + intros x y; unfold div_norm; rewrite spec_mul_normc; auto. + unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. + apply spec_inv_normc; auto. + Qed. + + Theorem spec_div_normc_bis : forall x y : Qc, + [div_norm (of_Qc x) (of_Qc y)] = x/y. + Proof. + intros. + rewrite <- strong_spec_div_norm by apply strong_spec_of_Qc_bis. + rewrite strong_spec_red. + symmetry; apply (Qred_complete (x*(/y)%Qc)%Q). + rewrite spec_div_norm, ! strong_spec_of_Qc. + unfold Qcinv, Q2Qc; rewrite Qred_correct; auto with qarith. + Qed. + + Theorem spec_squarec x: [[square x]] = [[x]]^2. + Proof. + intros x; unfold to_Qc. + apply trans_equal with (!! ([x]^2)). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_square; auto. + simpl Qcpower. + replace (!! [x] * 1) with (!![x]); try ring. + simpl. + unfold Qcmult, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + apply Qmult_comp; apply Qeq_sym; apply Qred_correct. + Qed. + + Theorem spec_power_posc x p: + [[power_pos x p]] = [[x]] ^ nat_of_P p. + Proof. + intros x p; unfold to_Qc. + apply trans_equal with (!! ([x]^Zpos p)). + unfold Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete; apply spec_power_pos; auto. + induction p using Pind. + simpl; ring. + rewrite nat_of_P_succ_morphism; simpl Qcpower. + rewrite <- IHp; clear IHp. + unfold Qcmult, Q2Qc. + apply Qc_decomp; intros _ _; unfold this. + apply Qred_complete. + setoid_replace ([x] ^ ' Psucc p)%Q with ([x] * [x] ^ ' p)%Q. + apply Qmult_comp; apply Qeq_sym; apply Qred_correct. + simpl. + rewrite Pplus_one_succ_l. + rewrite Qpower_plus_positive; simpl; apply Qeq_refl. + Qed. + +End Make. + diff --git a/theories/Numbers/Rational/BigQ/QMake_base.v b/theories/Numbers/Rational/BigQ/QMake_base.v deleted file mode 100644 index 547e74b7..00000000 --- a/theories/Numbers/Rational/BigQ/QMake_base.v +++ /dev/null @@ -1,34 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(* $Id: QMake_base.v 10964 2008-05-22 11:08:13Z letouzey $ *) - -(** * An implementation of rational numbers based on big integers *) - -Require Export BigN. -Require Export BigZ. - -(* Basic type for Q: a Z or a pair of a Z and an N *) - -Inductive q_type := - | Qz : BigZ.t -> q_type - | Qq : BigZ.t -> BigN.t -> q_type. - -Definition print_type x := - match x with - | Qz _ => Z - | _ => (Z*Z)%type - end. - -Definition print x := - match x return print_type x with - | Qz zx => BigZ.to_Z zx - | Qq nx dx => (BigZ.to_Z nx, BigN.to_Z dx) - end. diff --git a/theories/Numbers/Rational/BigQ/QbiMake.v b/theories/Numbers/Rational/BigQ/QbiMake.v deleted file mode 100644 index 699f383e..00000000 --- a/theories/Numbers/Rational/BigQ/QbiMake.v +++ /dev/null @@ -1,1066 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(*i $Id: QbiMake.v 11027 2008-06-01 13:28:59Z letouzey $ i*) - -Require Import Bool. -Require Import ZArith. -Require Import Znumtheory. -Require Import BigNumPrelude. -Require Import Arith. -Require Export BigN. -Require Export BigZ. -Require Import QArith. -Require Import Qcanon. -Require Import Qpower. -Require Import QMake_base. - -Module Qbi. - - Import BinInt Zorder. - Open Local Scope Q_scope. - Open Local Scope Qc_scope. - - (** The notation of a rational number is either an integer x, - interpreted as itself or a pair (x,y) of an integer x and a naturel - number y interpreted as x/y. The pairs (x,0) and (0,y) are all - interpreted as 0. *) - - Definition t := q_type. - - Definition zero: t := Qz BigZ.zero. - Definition one: t := Qz BigZ.one. - Definition minus_one: t := Qz BigZ.minus_one. - - Definition of_Z x: t := Qz (BigZ.of_Z x). - - - Definition of_Q q: t := - match q with x # y => - Qq (BigZ.of_Z x) (BigN.of_N (Npos y)) - end. - - Definition of_Qc q := of_Q (this q). - - Definition to_Q (q: t) := - match q with - Qz x => BigZ.to_Z x # 1 - |Qq x y => if BigN.eq_bool y BigN.zero then 0%Q - else BigZ.to_Z x # Z2P (BigN.to_Z y) - end. - - Definition to_Qc q := !!(to_Q q). - - Notation "[[ x ]]" := (to_Qc x). - - Notation "[ x ]" := (to_Q x). - - Theorem spec_to_Q: forall q: Q, [of_Q q] = q. - intros (x,y); simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigN.spec_of_pos; intros HH; discriminate HH. - rewrite BigZ.spec_of_Z; simpl. - rewrite (BigN.spec_of_pos); auto. - Qed. - - Theorem spec_to_Qc: forall q, [[of_Qc q]] = q. - intros (x, Hx); unfold of_Qc, to_Qc; simpl. - apply Qc_decomp; simpl. - intros; rewrite spec_to_Q; auto. - Qed. - - Definition opp (x: t): t := - match x with - | Qz zx => Qz (BigZ.opp zx) - | Qq nx dx => Qq (BigZ.opp nx) dx - end. - - Theorem spec_opp: forall q, ([opp q] = -[q])%Q. - intros [z | x y]; simpl. - rewrite BigZ.spec_opp; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigZ.spec_opp; auto. - Qed. - - Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. - intros q; unfold Qcopp, to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - rewrite spec_opp. - rewrite <- Qred_opp. - rewrite Qred_involutive; auto. - Qed. - - - Definition compare (x y: t) := - match x, y with - | Qz zx, Qz zy => BigZ.compare zx zy - | Qz zx, Qq ny dy => - if BigN.eq_bool dy BigN.zero then BigZ.compare zx BigZ.zero - else - match BigZ.cmp_sign zx ny with - | Lt => Lt - | Gt => Gt - | Eq => BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny - end - | Qq nx dx, Qz zy => - if BigN.eq_bool dx BigN.zero then BigZ.compare BigZ.zero zy - else - match BigZ.cmp_sign nx zy with - | Lt => Lt - | Gt => Gt - | Eq => BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx)) - end - | Qq nx dx, Qq ny dy => - match BigN.eq_bool dx BigN.zero, BigN.eq_bool dy BigN.zero with - | true, true => Eq - | true, false => BigZ.compare BigZ.zero ny - | false, true => BigZ.compare nx BigZ.zero - | false, false => - match BigZ.cmp_sign nx ny with - | Lt => Lt - | Gt => Gt - | Eq => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) - end - end - end. - - Theorem spec_compare: forall q1 q2, - compare q1 q2 = ([q1] ?= [q2])%Q. - intros [z1 | x1 y1] [z2 | x2 y2]; - unfold Qcompare, compare, to_Q, Qnum, Qden. - repeat rewrite Zmult_1_r. - generalize (BigZ.spec_compare z1 z2); case BigZ.compare; intros H; auto. - rewrite H; rewrite Zcompare_refl; auto. - rewrite Zmult_1_r. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - rewrite Zmult_1_r; generalize (BigZ.spec_compare z1 BigZ.zero); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH1; rewrite HH1; rewrite Zcompare_refl; auto. - set (a := BigZ.to_Z z1); set (b := BigZ.to_Z x2); - set (c := BigN.to_Z y2); fold c in HH. - assert (F: (0 < c)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos y2)); fold c; auto. - intros H1; case HH; rewrite <- H1; auto. - rewrite Z2P_correct; auto with zarith. - generalize (BigZ.spec_cmp_sign z1 x2); case BigZ.cmp_sign; fold a b c. - intros _; generalize (BigZ.spec_compare (z1 * BigZ.Pos y2)%bigZ x2); - case BigZ.compare; rewrite BigZ.spec_mul; simpl; fold a b c; auto. - intros H1; rewrite H1; rewrite Zcompare_refl; auto. - intros (H1, H2); apply sym_equal; change (a * c < b)%Z. - apply Zlt_le_trans with (2 := H2). - change 0%Z with (0 * c)%Z. - apply Zmult_lt_compat_r; auto with zarith. - intros (H1, H2); apply sym_equal; change (a * c > b)%Z. - apply Zlt_gt. - apply Zlt_le_trans with (1 := H2). - change 0%Z with (0 * c)%Z. - apply Zmult_le_compat_r; auto with zarith. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare BigZ.zero z2); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH1; rewrite <- HH1; rewrite Zcompare_refl; auto. - set (a := BigZ.to_Z z2); set (b := BigZ.to_Z x1); - set (c := BigN.to_Z y1); fold c in HH. - assert (F: (0 < c)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos y1)); fold c; auto. - intros H1; case HH; rewrite <- H1; auto. - rewrite Zmult_1_r; rewrite Z2P_correct; auto with zarith. - generalize (BigZ.spec_cmp_sign x1 z2); case BigZ.cmp_sign; fold a b c. - intros _; generalize (BigZ.spec_compare x1 (z2 * BigZ.Pos y1)%bigZ); - case BigZ.compare; rewrite BigZ.spec_mul; simpl; fold a b c; auto. - intros H1; rewrite H1; rewrite Zcompare_refl; auto. - intros (H1, H2); apply sym_equal; change (b < a * c)%Z. - apply Zlt_le_trans with (1 := H1). - change 0%Z with (0 * c)%Z. - apply Zmult_le_compat_r; auto with zarith. - intros (H1, H2); apply sym_equal; change (b > a * c)%Z. - apply Zlt_gt. - apply Zlt_le_trans with (2 := H1). - change 0%Z with (0 * c)%Z. - apply Zmult_lt_compat_r; auto with zarith. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - rewrite Zcompare_refl; auto. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare BigZ.zero x2); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH2; rewrite <- HH2; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare x1 BigZ.zero)%bigZ; case BigZ.compare; - auto; rewrite BigZ.spec_0. - intros HH2; rewrite <- HH2; rewrite Zcompare_refl; auto. - set (a := BigZ.to_Z x1); set (b := BigZ.to_Z x2); - set (c1 := BigN.to_Z y1); set (c2 := BigN.to_Z y2). - fold c1 in HH; fold c2 in HH1. - assert (F1: (0 < c1)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos y1)); fold c1; auto. - intros H1; case HH; rewrite <- H1; auto. - assert (F2: (0 < c2)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos y2)); fold c2; auto. - intros H1; case HH1; rewrite <- H1; auto. - repeat rewrite Z2P_correct; auto. - generalize (BigZ.spec_cmp_sign x1 x2); case BigZ.cmp_sign. - intros _; generalize (BigZ.spec_compare (x1 * BigZ.Pos y2)%bigZ - (x2 * BigZ.Pos y1)%bigZ); - case BigZ.compare; rewrite BigZ.spec_mul; simpl; fold a b c1 c2; auto. - rewrite BigZ.spec_mul; simpl; fold a b c1; intros HH2; rewrite HH2; - rewrite Zcompare_refl; auto. - rewrite BigZ.spec_mul; simpl; auto. - rewrite BigZ.spec_mul; simpl; auto. - fold a b; intros (H1, H2); apply sym_equal; change (a * c2 < b * c1)%Z. - apply Zlt_le_trans with 0%Z. - change 0%Z with (0 * c2)%Z. - apply Zmult_lt_compat_r; auto with zarith. - apply Zmult_le_0_compat; auto with zarith. - fold a b; intros (H1, H2); apply sym_equal; change (a * c2 > b * c1)%Z. - apply Zlt_gt; apply Zlt_le_trans with 0%Z. - change 0%Z with (0 * c1)%Z. - apply Zmult_lt_compat_r; auto with zarith. - apply Zmult_le_0_compat; auto with zarith. - Qed. - - - Definition do_norm_n n := - match n with - | BigN.N0 _ => false - | BigN.N1 _ => false - | BigN.N2 _ => false - | BigN.N3 _ => false - | BigN.N4 _ => false - | BigN.N5 _ => false - | BigN.N6 _ => false - | _ => true - end. - - Definition do_norm_z z := - match z with - | BigZ.Pos n => do_norm_n n - | BigZ.Neg n => do_norm_n n - end. - -(* Je pense que cette fonction normalise bien ... *) - Definition norm n d: t := - if andb (do_norm_z n) (do_norm_n d) then - let gcd := BigN.gcd (BigZ.to_N n) d in - match BigN.compare BigN.one gcd with - | Lt => - let n := BigZ.div n (BigZ.Pos gcd) in - let d := BigN.div d gcd in - match BigN.compare d BigN.one with - | Gt => Qq n d - | Eq => Qz n - | Lt => zero - end - | Eq => Qq n d - | Gt => zero (* gcd = 0 => both numbers are 0 *) - end - else Qq n d. - - Theorem spec_norm: forall n q, - ([norm n q] == [Qq n q])%Q. - intros p q; unfold norm. - case do_norm_z; simpl andb. - 2: apply Qeq_refl. - case do_norm_n. - 2: apply Qeq_refl. - assert (Hp := BigN.spec_pos (BigZ.to_N p)). - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; auto; rewrite BigN.spec_1; rewrite BigN.spec_gcd; intros H1. - apply Qeq_refl. - generalize (BigN.spec_pos (q / BigN.gcd (BigZ.to_N p) q)%bigN). - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; auto; rewrite BigN.spec_1; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; intros H2 HH. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H3; simpl; - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - auto with zarith. - generalize H2; rewrite H3; - rewrite Zdiv_0_l; auto with zarith. - generalize H1 H2 H3 (BigN.spec_pos q); clear H1 H2 H3. - rewrite spec_to_N. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 H4; rewrite Z2P_correct; auto with zarith. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H3; simpl. - case H3. - generalize H1 H2 H3 HH; clear H1 H2 H3 HH. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 HH. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto with zarith. - case (Zle_lt_or_eq _ _ HH); auto with zarith. - intros HH1; rewrite <- HH1; ring. - generalize (Zgcd_is_gcd a b); intros HH1; inversion HH1; auto. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; intros H3. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H4. - case H3; rewrite H4; rewrite Zdiv_0_l; auto with zarith. - simpl. - assert (FF := BigN.spec_pos q). - rewrite Z2P_correct; auto with zarith. - rewrite <- BigN.spec_gcd; rewrite <- BigN.spec_div; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; auto with zarith. - simpl; rewrite BigZ.spec_div; simpl. - rewrite BigN.spec_gcd; auto with zarith. - generalize H1 H2 H3 H4 HH FF; clear H1 H2 H3 H4 HH FF. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 H4 HH FF. - rewrite spec_to_N; fold a. - rewrite Zgcd_div_swap; auto with zarith. - rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_gcd; auto with zarith. - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (BigN.gcd (BigZ.to_N p) q))); - rewrite BigN.spec_gcd; auto with zarith. - intros; apply False_ind; auto with zarith. - intros HH2; assert (FF1 := Zgcd_inv_0_l _ _ (sym_equal HH2)). - assert (FF2 := Zgcd_inv_0_l _ _ (sym_equal HH2)). - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2; simpl. - rewrite spec_to_N. - rewrite FF2; ring. - Qed. - - Definition add (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (BigZ.add zx zy) - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - end - | Qq nx dx => - if BigN.eq_bool dx BigN.zero then y - else match y with - | Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - if BigN.eq_bool dx dy then - let n := BigZ.add nx ny in - Qq n dx - else - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - Qq n d - end - end. - - - - Theorem spec_add x y: - ([add x y] == [x] + [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qplus; simpl. - rewrite BigZ.spec_add; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - assert (F1:= BigN.spec_pos dy). - rewrite Zmult_1_r; red; simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH; simpl; try ring. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH1; simpl; try ring. - case HH; auto. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl; auto. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH; simpl; try ring. - rewrite Zmult_1_r; apply Qeq_refl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH1; simpl; try ring. - case HH; auto. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl; auto. - rewrite Zmult_1_r; rewrite Pmult_1_r. - apply Qeq_refl. - assert (F1:= BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - apply Qeq_refl. - case HH2; auto. - simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - case HH2; auto. - case HH1; auto. - rewrite Zmult_1_r; apply Qeq_refl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - simpl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - case HH; auto. - rewrite Zmult_1_r; rewrite Zplus_0_r; rewrite Pmult_1_r. - apply Qeq_refl. - simpl. - generalize (BigN.spec_eq_bool (dx * dy)%bigN BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_mul; - rewrite BigN.spec_0; intros HH2. - (case (Zmult_integral _ _ HH2); intros HH3); - [case HH| case HH1]; auto. - generalize (BigN.spec_eq_bool dx dy); - case BigN.eq_bool; intros HH3. - rewrite <- HH3. - assert (Fx: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - red; simpl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH4. - case HH; auto. - simpl; rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; repeat rewrite BigZ.spec_mul; simpl. - ring. - assert (Fx: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (Fy: (0 < BigN.to_Z dy)%Z). - generalize (BigN.spec_pos dy); auto with zarith. - red; simpl; rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_mul; - rewrite BigN.spec_0; intros H3; simpl. - absurd (0 < 0)%Z; auto with zarith. - rewrite BigZ.spec_add; repeat rewrite BigZ.spec_mul; simpl. - repeat rewrite Z2P_correct; auto with zarith. - apply Zmult_lt_0_compat; auto. - Qed. - - Theorem spec_addc x y: - [[add x y]] = [[x]] + [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition add_norm (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (BigZ.add zx zy) - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - end - | Qq nx dx => - if BigN.eq_bool dx BigN.zero then y - else match y with - | Qz zy => norm (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - if BigN.eq_bool dx dy then - let n := BigZ.add nx ny in - norm n dx - else - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - norm n d - end - end. - - Theorem spec_add_norm x y: - ([add_norm x y] == [x] + [y])%Q. - intros x y; rewrite <- spec_add; auto. - case x; case y; clear x y; unfold add_norm, add. - intros; apply Qeq_refl. - intros p1 n p2. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - simpl. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - apply Qeq_refl. - apply Qeq_refl. - intros p1 p2 n. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - apply Qeq_refl. - intros p1 q1 p2 q2. - generalize (BigN.spec_eq_bool q2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - apply Qeq_refl. - generalize (BigN.spec_eq_bool q1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; intros HH3; - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end; apply Qeq_refl. - Qed. - - Theorem spec_add_normc x y: - [[add_norm x y]] = [[x]] + [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add_norm; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition sub x y := add x (opp y). - - Theorem spec_sub x y: - ([sub x y] == [x] - [y])%Q. - intros x y; unfold sub; rewrite spec_add; auto. - rewrite spec_opp; ring. - Qed. - - Theorem spec_subc x y: [[sub x y]] = [[x]] - [[y]]. - intros x y; unfold sub; rewrite spec_addc; auto. - rewrite spec_oppc; ring. - Qed. - - Definition sub_norm x y := add_norm x (opp y). - - Theorem spec_sub_norm x y: - ([sub_norm x y] == [x] - [y])%Q. - intros x y; unfold sub_norm; rewrite spec_add_norm; auto. - rewrite spec_opp; ring. - Qed. - - Theorem spec_sub_normc x y: - [[sub_norm x y]] = [[x]] - [[y]]. - intros x y; unfold sub_norm; rewrite spec_add_normc; auto. - rewrite spec_oppc; ring. - Qed. - - Definition mul (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => Qq (BigZ.mul nx ny) (BigN.mul dx dy) - end. - - Theorem spec_mul x y: ([mul x y] == [x] * [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl. - rewrite BigZ.spec_mul; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH1. - red; simpl; ring. - rewrite BigZ.spec_mul; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH1. - red; simpl; ring. - rewrite BigZ.spec_mul; rewrite Pmult_1_r. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_mul; - intros HH1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH2. - red; simpl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH3. - red; simpl; ring. - case (Zmult_integral _ _ HH1); intros HH. - case HH2; auto. - case HH3; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH2. - case HH1; rewrite HH2; ring. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH3. - case HH1; rewrite HH3; ring. - rewrite BigZ.spec_mul. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - apply Qeq_refl. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dy); auto with zarith. - Qed. - - Theorem spec_mulc x y: - [[mul x y]] = [[x]] * [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition mul_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => mul (Qz ny) (norm zx dy) - | Qq nx dx, Qz zy => mul (Qz nx) (norm zy dx) - | Qq nx dx, Qq ny dy => mul (norm nx dy) (norm ny dx) - end. - - Theorem spec_mul_norm x y: - ([mul_norm x y] == [x] * [y])%Q. - intros x y; rewrite <- spec_mul; auto. - unfold mul_norm; case x; case y; clear x y. - intros; apply Qeq_refl. - intros p1 n p2. - repeat rewrite spec_mul. - match goal with |- ?Z == _ => - match Z with context id [norm ?X ?Y] => - let y := context id [Qq X Y] in - apply Qeq_trans with y; [repeat apply Qmult_comp; - repeat apply Qplus_comp; repeat apply Qeq_refl; - apply spec_norm | idtac] - end - end. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH; simpl; ring. - intros p1 p2 n. - repeat rewrite spec_mul. - match goal with |- ?Z == _ => - match Z with context id [norm ?X ?Y] => - let y := context id [Qq X Y] in - apply Qeq_trans with y; [repeat apply Qmult_comp; - repeat apply Qplus_comp; repeat apply Qeq_refl; - apply spec_norm | idtac] - end - end. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH; simpl; try ring. - rewrite Pmult_1_r; auto. - intros p1 n1 p2 n2. - repeat rewrite spec_mul. - repeat match goal with |- ?Z == _ => - match Z with context id [norm ?X ?Y] => - let y := context id [Qq X Y] in - apply Qeq_trans with y; [repeat apply Qmult_comp; - repeat apply Qplus_comp; repeat apply Qeq_refl; - apply spec_norm | idtac] - end - end. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; try ring. - repeat rewrite Zpos_mult_morphism; ring. - Qed. - - Theorem spec_mul_normc x y: - [[mul_norm x y]] = [[x]] * [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul_norm; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition inv (x: t): t := - match x with - | Qz (BigZ.Pos n) => Qq BigZ.one n - | Qz (BigZ.Neg n) => Qq BigZ.minus_one n - | Qq (BigZ.Pos n) d => Qq (BigZ.Pos d) n - | Qq (BigZ.Neg n) d => Qq (BigZ.Neg d) n - end. - - - Theorem spec_inv x: - ([inv x] == /[x])%Q. - intros [ [x | x] | [nx | nx] dx]; unfold inv, Qinv; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - rewrite H1; apply Qeq_refl. - generalize H1 (BigN.spec_pos x); case (BigN.to_Z x); auto. - intros HH; case HH; auto. - intros; red; simpl; auto. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - rewrite H1; apply Qeq_refl. - generalize H1 (BigN.spec_pos x); case (BigN.to_Z x); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl; auto. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - apply Qeq_refl. - rewrite H1; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - rewrite H2; red; simpl; auto. - generalize H1 (BigN.spec_pos nx); case (BigN.to_Z nx); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - apply Qeq_refl. - rewrite H1; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - rewrite H2; red; simpl; auto. - generalize H1 (BigN.spec_pos nx); case (BigN.to_Z nx); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p _ HH; case HH; auto. - Qed. - - Theorem spec_invc x: - [[inv x]] = /[[x]]. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition inv_norm (x: t): t := - match x with - | Qz (BigZ.Pos n) => - if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one n - | Qz (BigZ.Neg n) => - if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one n - | Qq (BigZ.Pos n) d => - if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos d) n - | Qq (BigZ.Neg n) d => - if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg d) n - end. - - Theorem spec_inv_norm x: ([inv_norm x] == /[x])%Q. - intros x; rewrite <- spec_inv; generalize x; clear x. - intros [ [x | x] | [nx | nx] dx]; unfold inv_norm, inv; - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; try apply Qeq_refl; - red; simpl; - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; auto; - case H2; auto. - Qed. - - Theorem spec_inv_normc x: - [[inv_norm x]] = /[[x]]. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv_norm; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - - Definition div x y := mul x (inv y). - - Theorem spec_div x y: ([div x y] == [x] / [y])%Q. - intros x y; unfold div; rewrite spec_mul; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Theorem spec_divc x y: [[div x y]] = [[x]] / [[y]]. - intros x y; unfold div; rewrite spec_mulc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - Definition div_norm x y := mul_norm x (inv y). - - Theorem spec_div_norm x y: ([div_norm x y] == [x] / [y])%Q. - intros x y; unfold div_norm; rewrite spec_mul_norm; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. - intros x y; unfold div_norm; rewrite spec_mul_normc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - - Definition square (x: t): t := - match x with - | Qz zx => Qz (BigZ.square zx) - | Qq nx dx => Qq (BigZ.square nx) (BigN.square dx) - end. - - - Theorem spec_square x: ([square x] == [x] ^ 2)%Q. - intros [ x | nx dx]; unfold square. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - simpl Qpower. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H. - red; simpl. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_square; - intros H1. - case H1; rewrite H; auto. - red; simpl. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_square; - intros H1. - case H; case (Zmult_integral _ _ H1); auto. - simpl. - rewrite BigZ.spec_square. - rewrite Zpos_mult_morphism. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dx); auto with zarith. - Qed. - - Theorem spec_squarec x: [[square x]] = [[x]]^2. - intros x; unfold to_Qc. - apply trans_equal with (!! ([x]^2)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_square; auto. - simpl Qcpower. - replace (!! [x] * 1) with (!![x]); try ring. - simpl. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition power_pos (x: t) p: t := - match x with - | Qz zx => Qz (BigZ.power_pos zx p) - | Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p) - end. - - Theorem spec_power_pos x p: ([power_pos x p] == [x] ^ Zpos p)%Q. - Proof. - intros [x | nx dx] p; unfold power_pos. - unfold power_pos; red; simpl. - generalize (Qpower_decomp p (BigZ.to_Z x) 1). - unfold Qeq; simpl. - rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Zmult_1_r. - intros H; rewrite H. - rewrite BigZ.spec_power_pos; simpl; ring. - simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_power_pos; intros H1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2. - elim p; simpl. - intros; red; simpl; auto. - intros p1 Hp1; rewrite <- Hp1; red; simpl; auto. - apply Qeq_refl. - case H2; generalize H1. - elim p; simpl. - intros p1 Hrec. - change (xI p1) with (1 + (xO p1))%positive. - rewrite Zpower_pos_is_exp; rewrite Zpower_pos_1_r. - intros HH; case (Zmult_integral _ _ HH); auto. - rewrite <- Pplus_diag. - rewrite Zpower_pos_is_exp. - intros HH1; case (Zmult_integral _ _ HH1); auto. - intros p1 Hrec. - rewrite <- Pplus_diag. - rewrite Zpower_pos_is_exp. - intros HH1; case (Zmult_integral _ _ HH1); auto. - rewrite Zpower_pos_1_r; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2. - case H1; rewrite H2; auto. - simpl; rewrite Zpower_pos_0_l; auto. - assert (F1: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (F2: (0 < BigN.to_Z dx ^ ' p)%Z). - unfold Zpower; apply Zpower_pos_pos; auto. - unfold power_pos; red; simpl. - generalize (Qpower_decomp p (BigZ.to_Z nx) - (Z2P (BigN.to_Z dx))). - unfold Qeq; simpl. - repeat rewrite Z2P_correct; auto. - unfold Qeq; simpl; intros HH. - rewrite HH. - rewrite BigZ.spec_power_pos; simpl; ring. - Qed. - - Theorem spec_power_posc x p: - [[power_pos x p]] = [[x]] ^ nat_of_P p. - intros x p; unfold to_Qc. - apply trans_equal with (!! ([x]^Zpos p)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_power_pos; auto. - pattern p; apply Pind; clear p. - simpl; ring. - intros p Hrec. - rewrite nat_of_P_succ_morphism; simpl Qcpower. - rewrite <- Hrec. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; - unfold this. - apply Qred_complete. - assert (F: [x] ^ ' Psucc p == [x] * [x] ^ ' p). - simpl; case x; simpl; clear x Hrec. - intros x; simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - intros nx dx. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - unfold Qpower_positive. - assert (tmp: forall p, pow_pos Qmult 0%Q p = 0%Q). - intros p1; elim p1; simpl; auto; clear p1. - intros p1 Hp1; rewrite Hp1; auto. - intros p1 Hp1; rewrite Hp1; auto. - repeat rewrite tmp; intros; red; simpl; auto. - intros H1. - assert (F1: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - repeat rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto. - 2: apply Zpower_pos_pos; auto. - 2: apply Zpower_pos_pos; auto. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - rewrite F. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - -End Qbi. diff --git a/theories/Numbers/Rational/BigQ/QifMake.v b/theories/Numbers/Rational/BigQ/QifMake.v deleted file mode 100644 index 1d8ecc94..00000000 --- a/theories/Numbers/Rational/BigQ/QifMake.v +++ /dev/null @@ -1,979 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(*i $Id: QifMake.v 11027 2008-06-01 13:28:59Z letouzey $ i*) - -Require Import Bool. -Require Import ZArith. -Require Import Znumtheory. -Require Import BigNumPrelude. -Require Import Arith. -Require Export BigN. -Require Export BigZ. -Require Import QArith. -Require Import Qcanon. -Require Import Qpower. -Require Import QMake_base. - -Module Qif. - - Import BinInt. - Open Local Scope Q_scope. - Open Local Scope Qc_scope. - - (** The notation of a rational number is either an integer x, - interpreted as itself or a pair (x,y) of an integer x and a naturel - number y interpreted as x/y. The pairs (x,0) and (0,y) are all - interpreted as 0. *) - - Definition t := q_type. - - Definition zero: t := Qz BigZ.zero. - Definition one: t := Qz BigZ.one. - Definition minus_one: t := Qz BigZ.minus_one. - - Definition of_Z x: t := Qz (BigZ.of_Z x). - - Definition of_Q q: t := - match q with x # y => - Qq (BigZ.of_Z x) (BigN.of_N (Npos y)) - end. - - Definition of_Qc q := of_Q (this q). - - Definition to_Q (q: t) := - match q with - Qz x => BigZ.to_Z x # 1 - |Qq x y => if BigN.eq_bool y BigN.zero then 0%Q - else BigZ.to_Z x # Z2P (BigN.to_Z y) - end. - - Definition to_Qc q := !!(to_Q q). - - Notation "[[ x ]]" := (to_Qc x). - - Notation "[ x ]" := (to_Q x). - - Theorem spec_to_Q: forall q: Q, [of_Q q] = q. - intros (x,y); simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigN.spec_of_pos; intros HH; discriminate HH. - rewrite BigZ.spec_of_Z; simpl. - rewrite (BigN.spec_of_pos); auto. - Qed. - - Theorem spec_to_Qc: forall q, [[of_Qc q]] = q. - intros (x, Hx); unfold of_Qc, to_Qc; simpl. - apply Qc_decomp; simpl. - intros; rewrite spec_to_Q; auto. - Qed. - - Definition opp (x: t): t := - match x with - | Qz zx => Qz (BigZ.opp zx) - | Qq nx dx => Qq (BigZ.opp nx) dx - end. - - Theorem spec_opp: forall q, ([opp q] = -[q])%Q. - intros [z | x y]; simpl. - rewrite BigZ.spec_opp; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigZ.spec_opp; auto. - Qed. - - Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. - intros q; unfold Qcopp, to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - rewrite spec_opp. - rewrite <- Qred_opp. - rewrite Qred_involutive; auto. - Qed. - - Definition compare (x y: t) := - match x, y with - | Qz zx, Qz zy => BigZ.compare zx zy - | Qz zx, Qq ny dy => - if BigN.eq_bool dy BigN.zero then BigZ.compare zx BigZ.zero - else BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny - | Qq nx dx, Qz zy => - if BigN.eq_bool dx BigN.zero then BigZ.compare BigZ.zero zy - else BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx)) - | Qq nx dx, Qq ny dy => - match BigN.eq_bool dx BigN.zero, BigN.eq_bool dy BigN.zero with - | true, true => Eq - | true, false => BigZ.compare BigZ.zero ny - | false, true => BigZ.compare nx BigZ.zero - | false, false => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) - end - end. - - Theorem spec_compare: forall q1 q2, - compare q1 q2 = ([q1] ?= [q2])%Q. - intros [z1 | x1 y1] [z2 | x2 y2]; - unfold Qcompare, compare, to_Q, Qnum, Qden. - repeat rewrite Zmult_1_r. - generalize (BigZ.spec_compare z1 z2); case BigZ.compare; intros H; auto. - rewrite H; rewrite Zcompare_refl; auto. - rewrite Zmult_1_r. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - rewrite Zmult_1_r; generalize (BigZ.spec_compare z1 BigZ.zero); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH1; rewrite HH1; rewrite Zcompare_refl; auto. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (z1 * BigZ.Pos y2) x2)%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare BigZ.zero z2); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH1; rewrite <- HH1; rewrite Zcompare_refl; auto. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y1); auto with zarith. - rewrite Zmult_1_r. - generalize (BigZ.spec_compare x1 (z2 * BigZ.Pos y1))%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - rewrite Zcompare_refl; auto. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare BigZ.zero x2); - case BigZ.compare; auto. - rewrite BigZ.spec_0; intros HH2; rewrite <- HH2; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - rewrite Zmult_0_l; rewrite Zmult_1_r. - generalize (BigZ.spec_compare x1 BigZ.zero)%bigZ; case BigZ.compare; - auto; rewrite BigZ.spec_0. - intros HH2; rewrite <- HH2; rewrite Zcompare_refl; auto. - repeat rewrite Z2P_correct. - 2: generalize (BigN.spec_pos y1); auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (x1 * BigZ.Pos y2) - (x2 * BigZ.Pos y1))%bigZ; case BigZ.compare; - repeat rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - Qed. - - Definition do_norm_n n := - match n with - | BigN.N0 _ => false - | BigN.N1 _ => false - | BigN.N2 _ => false - | BigN.N3 _ => false - | BigN.N4 _ => false - | BigN.N5 _ => false - | BigN.N6 _ => false - | _ => true - end. - - Definition do_norm_z z := - match z with - | BigZ.Pos n => do_norm_n n - | BigZ.Neg n => do_norm_n n - end. - -(* Je pense que cette fonction normalise bien ... *) - Definition norm n d: t := - if andb (do_norm_z n) (do_norm_n d) then - let gcd := BigN.gcd (BigZ.to_N n) d in - match BigN.compare BigN.one gcd with - | Lt => - let n := BigZ.div n (BigZ.Pos gcd) in - let d := BigN.div d gcd in - match BigN.compare d BigN.one with - | Gt => Qq n d - | Eq => Qz n - | Lt => zero - end - | Eq => Qq n d - | Gt => zero (* gcd = 0 => both numbers are 0 *) - end - else Qq n d. - - Theorem spec_norm: forall n q, - ([norm n q] == [Qq n q])%Q. - intros p q; unfold norm. - case do_norm_z; simpl andb. - 2: apply Qeq_refl. - case do_norm_n. - 2: apply Qeq_refl. - assert (Hp := BigN.spec_pos (BigZ.to_N p)). - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; auto; rewrite BigN.spec_1; rewrite BigN.spec_gcd; intros H1. - apply Qeq_refl. - generalize (BigN.spec_pos (q / BigN.gcd (BigZ.to_N p) q)%bigN). - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; auto; rewrite BigN.spec_1; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; intros H2 HH. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H3; simpl; - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - auto with zarith. - generalize H2; rewrite H3; - rewrite Zdiv_0_l; auto with zarith. - generalize H1 H2 H3 (BigN.spec_pos q); clear H1 H2 H3. - rewrite spec_to_N. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 H4; rewrite Z2P_correct; auto with zarith. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H3; simpl. - case H3. - generalize H1 H2 H3 HH; clear H1 H2 H3 HH. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 HH. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto with zarith. - case (Zle_lt_or_eq _ _ HH); auto with zarith. - intros HH1; rewrite <- HH1; ring. - generalize (Zgcd_is_gcd a b); intros HH1; inversion HH1; auto. - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith; intros H3. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H4. - case H3; rewrite H4; rewrite Zdiv_0_l; auto with zarith. - simpl. - assert (FF := BigN.spec_pos q). - rewrite Z2P_correct; auto with zarith. - rewrite <- BigN.spec_gcd; rewrite <- BigN.spec_div; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; auto with zarith. - simpl; rewrite BigZ.spec_div; simpl. - rewrite BigN.spec_gcd; auto with zarith. - generalize H1 H2 H3 H4 HH FF; clear H1 H2 H3 H4 HH FF. - set (a := (BigN.to_Z (BigZ.to_N p))). - set (b := (BigN.to_Z q)). - intros H1 H2 H3 H4 HH FF. - rewrite spec_to_N; fold a. - rewrite Zgcd_div_swap; auto with zarith. - rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_div; - rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_gcd; auto with zarith. - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (BigN.gcd (BigZ.to_N p) q))); - rewrite BigN.spec_gcd; auto with zarith. - intros; apply False_ind; auto with zarith. - intros HH2; assert (FF1 := Zgcd_inv_0_l _ _ (sym_equal HH2)). - assert (FF2 := Zgcd_inv_0_l _ _ (sym_equal HH2)). - red; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H2; simpl. - rewrite spec_to_N. - rewrite FF2; ring. - Qed. - - - Definition add (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (BigZ.add zx zy) - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - end - | Qq nx dx => - if BigN.eq_bool dx BigN.zero then y - else match y with - | Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - Qq n d - end - end. - - - Theorem spec_add x y: - ([add x y] == [x] + [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qplus; simpl. - rewrite BigZ.spec_add; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - assert (F1:= BigN.spec_pos dy). - rewrite Zmult_1_r; red; simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH; simpl; try ring. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH1; simpl; try ring. - case HH; auto. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl; auto. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH; simpl; try ring. - rewrite Zmult_1_r; apply Qeq_refl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; - rewrite BigN.spec_0; intros HH1; simpl; try ring. - case HH; auto. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl; auto. - rewrite Zmult_1_r; rewrite Pmult_1_r. - apply Qeq_refl. - assert (F1:= BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - apply Qeq_refl. - case HH2; auto. - simpl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - case HH2; auto. - case HH1; auto. - rewrite Zmult_1_r; apply Qeq_refl. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - simpl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - case HH; auto. - rewrite Zmult_1_r; rewrite Zplus_0_r; rewrite Pmult_1_r. - apply Qeq_refl. - simpl. - generalize (BigN.spec_eq_bool (dx * dy)%bigN BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_mul; - rewrite BigN.spec_0; intros HH2. - (case (Zmult_integral _ _ HH2); intros HH3); - [case HH| case HH1]; auto. - rewrite BigZ.spec_add; repeat rewrite BigZ.spec_mul; simpl. - assert (Fx: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (Fy: (0 < BigN.to_Z dy)%Z). - generalize (BigN.spec_pos dy); auto with zarith. - red; simpl; rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto with zarith. - apply Zmult_lt_0_compat; auto. - Qed. - - Theorem spec_addc x y: - [[add x y]] = [[x]] + [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition add_norm (x y: t): t := - match x with - | Qz zx => - match y with - | Qz zy => Qz (BigZ.add zx zy) - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - end - | Qq nx dx => - if BigN.eq_bool dx BigN.zero then y - else match y with - | Qz zy => norm (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq ny dy => - if BigN.eq_bool dy BigN.zero then x - else - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - norm n d - end - end. - - Theorem spec_add_norm x y: - ([add_norm x y] == [x] + [y])%Q. - intros x y; rewrite <- spec_add; auto. - case x; case y; clear x y; unfold add_norm, add. - intros; apply Qeq_refl. - intros p1 n p2. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - simpl. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - apply Qeq_refl. - apply Qeq_refl. - intros p1 p2 n. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - apply Qeq_refl. - intros p1 q1 p2 q2. - generalize (BigN.spec_eq_bool q2 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH1. - apply Qeq_refl. - generalize (BigN.spec_eq_bool q1 BigN.zero); - case BigN.eq_bool; rewrite BigN.spec_0; intros HH2. - apply Qeq_refl. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - apply Qeq_refl. - Qed. - - Theorem spec_add_normc x y: - [[add_norm x y]] = [[x]] + [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add_norm; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition sub x y := add x (opp y). - - - Theorem spec_sub x y: - ([sub x y] == [x] - [y])%Q. - intros x y; unfold sub; rewrite spec_add; auto. - rewrite spec_opp; ring. - Qed. - - Theorem spec_subc x y: [[sub x y]] = [[x]] - [[y]]. - intros x y; unfold sub; rewrite spec_addc; auto. - rewrite spec_oppc; ring. - Qed. - - Definition sub_norm x y := add_norm x (opp y). - - Theorem spec_sub_norm x y: - ([sub_norm x y] == [x] - [y])%Q. - intros x y; unfold sub_norm; rewrite spec_add_norm; auto. - rewrite spec_opp; ring. - Qed. - - Theorem spec_sub_normc x y: - [[sub_norm x y]] = [[x]] - [[y]]. - intros x y; unfold sub_norm; rewrite spec_add_normc; auto. - rewrite spec_oppc; ring. - Qed. - - Definition mul (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => Qq (BigZ.mul nx ny) (BigN.mul dx dy) - end. - - - Theorem spec_mul x y: ([mul x y] == [x] * [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl. - rewrite BigZ.spec_mul; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH1. - red; simpl; ring. - rewrite BigZ.spec_mul; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH1. - red; simpl; ring. - rewrite BigZ.spec_mul; rewrite Pmult_1_r. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; rewrite BigN.spec_mul; - intros HH1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH2. - red; simpl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH3. - red; simpl; ring. - case (Zmult_integral _ _ HH1); intros HH. - case HH2; auto. - case HH3; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH2. - case HH1; rewrite HH2; ring. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros HH3. - case HH1; rewrite HH3; ring. - rewrite BigZ.spec_mul. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - apply Qeq_refl. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dy); auto with zarith. - Qed. - - Theorem spec_mulc x y: - [[mul x y]] = [[x]] * [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - - Definition mul_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => norm (BigZ.mul zx ny) dy - | Qq nx dx, Qz zy => norm (BigZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => norm (BigZ.mul nx ny) (BigN.mul dx dy) - end. - - Theorem spec_mul_norm x y: - ([mul_norm x y] == [x] * [y])%Q. - intros x y; rewrite <- spec_mul; auto. - unfold mul_norm, mul; case x; case y; clear x y. - intros; apply Qeq_refl. - intros p1 n p2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end; apply Qeq_refl. - intros p1 p2 n. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end; apply Qeq_refl. - intros p1 n1 p2 n2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end; apply Qeq_refl. - Qed. - - Theorem spec_mul_normc x y: - [[mul_norm x y]] = [[x]] * [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul_norm; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - - - Definition inv (x: t): t := - match x with - | Qz (BigZ.Pos n) => Qq BigZ.one n - | Qz (BigZ.Neg n) => Qq BigZ.minus_one n - | Qq (BigZ.Pos n) d => Qq (BigZ.Pos d) n - | Qq (BigZ.Neg n) d => Qq (BigZ.Neg d) n - end. - - Theorem spec_inv x: - ([inv x] == /[x])%Q. - intros [ [x | x] | [nx | nx] dx]; unfold inv, Qinv; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - rewrite H1; apply Qeq_refl. - generalize H1 (BigN.spec_pos x); case (BigN.to_Z x); auto. - intros HH; case HH; auto. - intros; red; simpl; auto. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - rewrite H1; apply Qeq_refl. - generalize H1 (BigN.spec_pos x); case (BigN.to_Z x); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl; auto. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - apply Qeq_refl. - rewrite H1; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - rewrite H2; red; simpl; auto. - generalize H1 (BigN.spec_pos nx); case (BigN.to_Z nx); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - apply Qeq_refl. - rewrite H1; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H2; simpl; auto. - rewrite H2; red; simpl; auto. - generalize H1 (BigN.spec_pos nx); case (BigN.to_Z nx); simpl; - auto. - intros HH; case HH; auto. - intros; red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p _ HH; case HH; auto. - Qed. - - Theorem spec_invc x: - [[inv x]] = /[[x]]. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - -Definition inv_norm (x: t): t := - match x with - | Qz (BigZ.Pos n) => - match BigN.compare n BigN.one with - Gt => Qq BigZ.one n - | _ => x - end - | Qz (BigZ.Neg n) => - match BigN.compare n BigN.one with - Gt => Qq BigZ.minus_one n - | _ => x - end - | Qq (BigZ.Pos n) d => - match BigN.compare n BigN.one with - Gt => Qq (BigZ.Pos d) n - | Eq => Qz (BigZ.Pos d) - | Lt => Qz (BigZ.zero) - end - | Qq (BigZ.Neg n) d => - match BigN.compare n BigN.one with - Gt => Qq (BigZ.Neg d) n - | Eq => Qz (BigZ.Neg d) - | Lt => Qz (BigZ.zero) - end - end. - - Theorem spec_inv_norm x: ([inv_norm x] == /[x])%Q. - intros [ [x | x] | [nx | nx] dx]; unfold inv_norm, Qinv. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H. - simpl; rewrite H; apply Qeq_refl. - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); simpl. - generalize H; case BigN.to_Z. - intros _ HH; discriminate HH. - intros p; case p; auto. - intros p1 HH; discriminate HH. - intros p1 HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; discriminate HH. - intros HH; rewrite <- HH. - apply Qeq_refl. - generalize H; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - rewrite H1; intros HH; discriminate. - generalize H; case BigN.to_Z. - intros HH; discriminate HH. - intros; red; simpl; auto. - intros p HH; discriminate HH. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H. - simpl; rewrite H; apply Qeq_refl. - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); simpl. - generalize H; case BigN.to_Z. - intros _ HH; discriminate HH. - intros p; case p; auto. - intros p1 HH; discriminate HH. - intros p1 HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; discriminate HH. - intros HH; rewrite <- HH. - apply Qeq_refl. - generalize H; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - rewrite H1; intros HH; discriminate. - generalize H; case BigN.to_Z. - intros HH; discriminate HH. - intros; red; simpl; auto. - intros p HH; discriminate HH. - simpl Qnum. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; simpl. - case BigN.compare; red; simpl; auto. - rewrite H1; auto. - case BigN.eq_bool; auto. - simpl; rewrite H1; auto. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H2. - rewrite H2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - red; simpl. - rewrite Zmult_1_r; rewrite Pmult_1_r; rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - generalize H2 (BigN.spec_pos nx); case (BigN.to_Z nx). - intros; apply Qeq_refl. - intros p; case p; clear p. - intros p HH; discriminate HH. - intros p HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - simpl; generalize H2; case (BigN.to_Z nx). - intros HH; discriminate HH. - intros p Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H4. - rewrite H4 in H2; discriminate H2. - red; simpl. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - intros p HH; discriminate HH. - simpl Qnum. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1; simpl. - case BigN.compare; red; simpl; auto. - rewrite H1; auto. - case BigN.eq_bool; auto. - simpl; rewrite H1; auto. - match goal with |- context[BigN.compare ?X ?Y] => - generalize (BigN.spec_compare X Y); case BigN.compare - end; rewrite BigN.spec_1; intros H2. - rewrite H2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zmult_1_r; rewrite Pmult_1_r; rewrite Z2P_correct; auto. - generalize (BigN.spec_pos dx); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - generalize H2 (BigN.spec_pos nx); case (BigN.to_Z nx). - intros; apply Qeq_refl. - intros p; case p; clear p. - intros p HH; discriminate HH. - intros p HH; discriminate HH. - intros HH; discriminate HH. - intros p _ HH; case HH; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H3. - case H1; auto. - simpl; generalize H2; case (BigN.to_Z nx). - intros HH; discriminate HH. - intros p Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H4. - rewrite H4 in H2; discriminate H2. - red; simpl. - assert (tmp: forall x, Zneg x = Zopp (Zpos x)); auto. - rewrite tmp. - rewrite Zpos_mult_morphism. - rewrite Z2P_correct; auto. - ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p HH; discriminate HH. - Qed. - - Theorem spec_inv_normc x: - [[inv_norm x]] = /[[x]]. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv_norm; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - - Definition div x y := mul x (inv y). - - Theorem spec_div x y: ([div x y] == [x] / [y])%Q. - intros x y; unfold div; rewrite spec_mul; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Theorem spec_divc x y: [[div x y]] = [[x]] / [[y]]. - intros x y; unfold div; rewrite spec_mulc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - Definition div_norm x y := mul_norm x (inv y). - - Theorem spec_div_norm x y: ([div_norm x y] == [x] / [y])%Q. - intros x y; unfold div_norm; rewrite spec_mul_norm; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. - intros x y; unfold div_norm; rewrite spec_mul_normc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - - Definition square (x: t): t := - match x with - | Qz zx => Qz (BigZ.square zx) - | Qq nx dx => Qq (BigZ.square nx) (BigN.square dx) - end. - - Theorem spec_square x: ([square x] == [x] ^ 2)%Q. - intros [ x | nx dx]; unfold square. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - simpl Qpower. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; intros H. - red; simpl. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_square; - intros H1. - case H1; rewrite H; auto. - red; simpl. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_square; - intros H1. - case H; case (Zmult_integral _ _ H1); auto. - simpl. - rewrite BigZ.spec_square. - rewrite Zpos_mult_morphism. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dx); auto with zarith. - Qed. - - Theorem spec_squarec x: [[square x]] = [[x]]^2. - intros x; unfold to_Qc. - apply trans_equal with (!! ([x]^2)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_square; auto. - simpl Qcpower. - replace (!! [x] * 1) with (!![x]); try ring. - simpl. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - -End Qif. diff --git a/theories/Numbers/Rational/BigQ/QpMake.v b/theories/Numbers/Rational/BigQ/QpMake.v deleted file mode 100644 index ac3ca47a..00000000 --- a/theories/Numbers/Rational/BigQ/QpMake.v +++ /dev/null @@ -1,901 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(*i $Id: QpMake.v 11027 2008-06-01 13:28:59Z letouzey $ i*) - -Require Import Bool. -Require Import ZArith. -Require Import Znumtheory. -Require Import BigNumPrelude. -Require Import Arith. -Require Export BigN. -Require Export BigZ. -Require Import QArith. -Require Import Qcanon. -Require Import Qpower. -Require Import QMake_base. - -Notation Nspec_lt := BigNAxiomsMod.NZOrdAxiomsMod.spec_lt. -Notation Nspec_le := BigNAxiomsMod.NZOrdAxiomsMod.spec_le. - -Module Qp. - - (** The notation of a rational number is either an integer x, - interpreted as itself or a pair (x,y) of an integer x and a naturel - number y interpreted as x/(y+1). *) - - Definition t := q_type. - - Definition zero: t := Qz BigZ.zero. - Definition one: t := Qz BigZ.one. - Definition minus_one: t := Qz BigZ.minus_one. - - Definition of_Z x: t := Qz (BigZ.of_Z x). - - Definition d_to_Z d := BigZ.Pos (BigN.succ d). - - Definition of_Q q: t := - match q with x # y => - Qq (BigZ.of_Z x) (BigN.pred (BigN.of_N (Npos y))) - end. - - Definition of_Qc q := of_Q (this q). - - Definition to_Q (q: t) := - match q with - Qz x => BigZ.to_Z x # 1 - |Qq x y => BigZ.to_Z x # Z2P (BigN.to_Z (BigN.succ y)) - end. - - Definition to_Qc q := !!(to_Q q). - - Notation "[[ x ]]" := (to_Qc x). - - Notation "[ x ]" := (to_Q x). - - Theorem spec_to_Q: forall q: Q, [of_Q q] = q. - intros (x,y); simpl. - rewrite BigZ.spec_of_Z; auto. - rewrite BigN.spec_succ; simpl. simpl. - rewrite BigN.spec_pred; rewrite (BigN.spec_of_pos). - replace (Zpos y - 1 + 1)%Z with (Zpos y); auto; ring. - red; auto. - Qed. - - Theorem spec_to_Qc: forall q, [[of_Qc q]] = q. - intros (x, Hx); unfold of_Qc, to_Qc; simpl. - apply Qc_decomp; simpl. - intros; rewrite spec_to_Q; auto. - Qed. - - Definition opp (x: t): t := - match x with - | Qz zx => Qz (BigZ.opp zx) - | Qq nx dx => Qq (BigZ.opp nx) dx - end. - - - Theorem spec_opp: forall q, ([opp q] = -[q])%Q. - intros [z | x y]; simpl. - rewrite BigZ.spec_opp; auto. - rewrite BigZ.spec_opp; auto. - Qed. - - - Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. - intros q; unfold Qcopp, to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - rewrite spec_opp. - rewrite <- Qred_opp. - rewrite Qred_involutive; auto. - Qed. - - Definition compare (x y: t) := - match x, y with - | Qz zx, Qz zy => BigZ.compare zx zy - | Qz zx, Qq ny dy => BigZ.compare (BigZ.mul zx (d_to_Z dy)) ny - | Qq nx dy, Qz zy => BigZ.compare nx (BigZ.mul zy (d_to_Z dy)) - | Qq nx dx, Qq ny dy => - BigZ.compare (BigZ.mul nx (d_to_Z dy)) (BigZ.mul ny (d_to_Z dx)) - end. - - Theorem spec_compare: forall q1 q2, - compare q1 q2 = ([q1] ?= [q2])%Q. - intros [z1 | x1 y1] [z2 | x2 y2]; unfold Qcompare; simpl. - repeat rewrite Zmult_1_r. - generalize (BigZ.spec_compare z1 z2); case BigZ.compare; intros H; auto. - rewrite H; rewrite Zcompare_refl; auto. - rewrite Zmult_1_r. - rewrite BigN.spec_succ. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (z1 * d_to_Z y2) x2)%bigZ; case BigZ.compare; - intros H; rewrite <- H. - rewrite BigZ.spec_mul; unfold d_to_Z; simpl. - rewrite BigN.spec_succ. - rewrite Zcompare_refl; auto. - rewrite BigZ.spec_mul; unfold d_to_Z; simpl. - rewrite BigN.spec_succ; auto. - rewrite BigZ.spec_mul; unfold d_to_Z; simpl. - rewrite BigN.spec_succ; auto. - rewrite Zmult_1_r. - rewrite BigN.spec_succ. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y1); auto with zarith. - generalize (BigZ.spec_compare x1 (z2 * d_to_Z y1))%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; unfold d_to_Z; simpl; - rewrite BigN.spec_succ; intros H; auto. - rewrite H; rewrite Zcompare_refl; auto. - repeat rewrite BigN.spec_succ; auto. - repeat rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y1); auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (x1 * d_to_Z y2) - (x2 * d_to_Z y1))%bigZ; case BigZ.compare; - repeat rewrite BigZ.spec_mul; unfold d_to_Z; simpl; - repeat rewrite BigN.spec_succ; intros H; auto. - rewrite H; auto. - rewrite Zcompare_refl; auto. - Qed. - - - Theorem spec_comparec: forall q1 q2, - compare q1 q2 = ([[q1]] ?= [[q2]]). - unfold Qccompare, to_Qc. - intros q1 q2; rewrite spec_compare; simpl. - apply Qcompare_comp; apply Qeq_sym; apply Qred_correct. - Qed. - -(* Inv d > 0, Pour la forme normal unique on veut d > 1 *) - Definition norm n d: t := - if BigZ.eq_bool n BigZ.zero then zero - else - let gcd := BigN.gcd (BigZ.to_N n) d in - if BigN.eq_bool gcd BigN.one then Qq n (BigN.pred d) - else - let n := BigZ.div n (BigZ.Pos gcd) in - let d := BigN.div d gcd in - if BigN.eq_bool d BigN.one then Qz n - else Qq n (BigN.pred d). - - Theorem spec_norm: forall n q, - ((0 < BigN.to_Z q)%Z -> [norm n q] == [Qq n (BigN.pred q)])%Q. - intros p q; unfold norm; intros Hq. - assert (Hp := BigN.spec_pos (BigZ.to_N p)). - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; auto; rewrite BigZ.spec_0; intros H1. - red; simpl; rewrite H1; ring. - case (Zle_lt_or_eq _ _ Hp); clear Hp; intros Hp. - case (Zle_lt_or_eq _ _ - (Zgcd_is_pos (BigN.to_Z (BigZ.to_N p)) (BigN.to_Z q))); intros H4. - 2: generalize Hq; rewrite (Zgcd_inv_0_r _ _ (sym_equal H4)); auto with zarith. - 2: red; simpl; auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_1; intros H2. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_1. - red; simpl. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite Zmult_1_r. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite Z2P_correct; auto with zarith. - rewrite spec_to_N; intros; rewrite Zgcd_div_swap; auto. - rewrite H; ring. - intros H3. - red; simpl. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - assert (F: (0 < BigN.to_Z (q / BigN.gcd (BigZ.to_N p) q)%bigN)%Z). - rewrite BigN.spec_div; auto with zarith. - rewrite BigN.spec_gcd. - apply Zgcd_div_pos; auto. - rewrite BigN.spec_gcd; auto. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite Z2P_correct; auto. - rewrite Z2P_correct; auto. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite spec_to_N; apply Zgcd_div_swap; auto. - case H1; rewrite spec_to_N; rewrite <- Hp; ring. - Qed. - - Theorem spec_normc: forall n q, - (0 < BigN.to_Z q)%Z -> [[norm n q]] = [[Qq n (BigN.pred q)]]. - intros n q H; unfold to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_norm; auto. - Qed. - - Definition add (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.add zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.add (BigZ.mul zx (d_to_Z dy)) ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.add nx (BigZ.mul zy (d_to_Z dx))) dx - | Qq nx dx, Qq ny dy => - let dx' := BigN.succ dx in - let dy' := BigN.succ dy in - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy')) (BigZ.mul ny (BigZ.Pos dx')) in - let d := BigN.pred (BigN.mul dx' dy') in - Qq n d - end. - - Theorem spec_d_to_Z: forall dy, - (BigZ.to_Z (d_to_Z dy) = BigN.to_Z dy + 1)%Z. - intros dy; unfold d_to_Z; simpl. - rewrite BigN.spec_succ; auto. - Qed. - - Theorem spec_succ_pos: forall p, - (0 < BigN.to_Z (BigN.succ p))%Z. - intros p; rewrite BigN.spec_succ; - generalize (BigN.spec_pos p); auto with zarith. - Qed. - - Theorem spec_add x y: ([add x y] == [x] + [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qplus; simpl. - rewrite BigZ.spec_add; repeat rewrite Zmult_1_r; auto. - apply Qeq_refl; auto. - assert (F1:= BigN.spec_pos dy). - rewrite Zmult_1_r. - simpl; rewrite Z2P_correct; rewrite BigN.spec_succ; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul. - rewrite spec_d_to_Z; apply Qeq_refl. - assert (F1:= BigN.spec_pos dx). - rewrite Zmult_1_r; rewrite Pmult_1_r. - simpl; rewrite Z2P_correct; rewrite BigN.spec_succ; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul. - rewrite spec_d_to_Z; apply Qeq_refl. - repeat rewrite BigN.spec_succ. - assert (Fx: (0 < BigN.to_Z dx + 1)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (Fy: (0 < BigN.to_Z dy + 1)%Z). - generalize (BigN.spec_pos dy); auto with zarith. - repeat rewrite BigN.spec_pred. - rewrite BigZ.spec_add; repeat rewrite BigN.spec_mul; - repeat rewrite BigN.spec_succ. - assert (tmp: forall x, (x-1+1 = x)%Z); [intros; ring | rewrite tmp; clear tmp]. - repeat rewrite Z2P_correct; auto. - repeat rewrite BigZ.spec_mul; simpl. - repeat rewrite BigN.spec_succ. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto; apply Qeq_refl. - rewrite BigN.spec_mul; repeat rewrite BigN.spec_succ; auto with zarith. - apply Zmult_lt_0_compat; auto. - Qed. - - Theorem spec_addc x y: [[add x y]] = [[x]] + [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition add_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.add zx zy) - | Qz zx, Qq ny dy => - let d := BigN.succ dy in - norm (BigZ.add (BigZ.mul zx (BigZ.Pos d)) ny) d - | Qq nx dx, Qz zy => - let d := BigN.succ dx in - norm (BigZ.add (BigZ.mul zy (BigZ.Pos d)) nx) d - | Qq nx dx, Qq ny dy => - let dx' := BigN.succ dx in - let dy' := BigN.succ dy in - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy')) (BigZ.mul ny (BigZ.Pos dx')) in - let d := BigN.mul dx' dy' in - norm n d - end. - - Theorem spec_add_norm x y: ([add_norm x y] == [x] + [y])%Q. - intros x y; rewrite <- spec_add. - unfold add_norm, add; case x; case y. - intros; apply Qeq_refl. - intros p1 n p2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X (BigN.pred Y)]); - [apply spec_norm | idtac] - end. - rewrite BigN.spec_succ; generalize (BigN.spec_pos n); auto with zarith. - simpl. - repeat rewrite BigZ.spec_add. - repeat rewrite BigZ.spec_mul; simpl. - rewrite BigN.succ_pred; try apply Qeq_refl; apply lt_0_succ. - intros p1 n p2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X (BigN.pred Y)]); - [apply spec_norm | idtac] - end. - rewrite BigN.spec_succ; generalize (BigN.spec_pos p2); auto with zarith. - simpl. - repeat rewrite BigZ.spec_add. - repeat rewrite BigZ.spec_mul; simpl. - rewrite BinInt.Zplus_comm. - rewrite BigN.succ_pred; try apply Qeq_refl; apply lt_0_succ. - intros p1 q1 p2 q2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X (BigN.pred Y)]); - [apply spec_norm | idtac] - end; try apply Qeq_refl. - rewrite BigN.spec_mul. - apply Zmult_lt_0_compat; apply spec_succ_pos. - Qed. - - Theorem spec_add_normc x y: [[add_norm x y]] = [[x]] + [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add_norm. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition sub (x y: t): t := add x (opp y). - - Theorem spec_sub x y: ([sub x y] == [x] - [y])%Q. - intros x y; unfold sub; rewrite spec_add. - rewrite spec_opp; ring. - Qed. - - Theorem spec_subc x y: [[sub x y]] = [[x]] - [[y]]. - intros x y; unfold sub; rewrite spec_addc. - rewrite spec_oppc; ring. - Qed. - - Definition sub_norm x y := add_norm x (opp y). - - Theorem spec_sub_norm x y: ([sub_norm x y] == [x] - [y])%Q. - intros x y; unfold sub_norm; rewrite spec_add_norm. - rewrite spec_opp; ring. - Qed. - - Theorem spec_sub_normc x y: [[sub_norm x y]] = [[x]] - [[y]]. - intros x y; unfold sub_norm; rewrite spec_add_normc. - rewrite spec_oppc; ring. - Qed. - - - Definition mul (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => - Qq (BigZ.mul nx ny) (BigN.pred (BigN.mul (BigN.succ dx) (BigN.succ dy))) - end. - - Theorem spec_mul x y: ([mul x y] == [x] * [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl. - rewrite BigZ.spec_mul; repeat rewrite Zmult_1_r; auto. - apply Qeq_refl; auto. - rewrite BigZ.spec_mul; apply Qeq_refl. - rewrite BigZ.spec_mul; rewrite Pmult_1_r; auto. - apply Qeq_refl; auto. - assert (F1:= spec_succ_pos dx). - assert (F2:= spec_succ_pos dy). - rewrite BigN.succ_pred. - rewrite BigN.spec_mul; rewrite BigZ.spec_mul. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto; apply Qeq_refl. - rewrite Nspec_lt, BigN.spec_0, BigN.spec_mul; auto. - apply Zmult_lt_0_compat; apply spec_succ_pos. - Qed. - - Theorem spec_mulc x y: [[mul x y]] = [[x]] * [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition mul_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => - if BigZ.eq_bool zx BigZ.zero then zero - else - let d := BigN.succ dy in - let gcd := BigN.gcd (BigZ.to_N zx) d in - if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zx ny) dy - else - let zx := BigZ.div zx (BigZ.Pos gcd) in - let d := BigN.div d gcd in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul zx ny) - else Qq (BigZ.mul zx ny) (BigN.pred d) - | Qq nx dx, Qz zy => - if BigZ.eq_bool zy BigZ.zero then zero - else - let d := BigN.succ dx in - let gcd := BigN.gcd (BigZ.to_N zy) d in - if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zy nx) dx - else - let zy := BigZ.div zy (BigZ.Pos gcd) in - let d := BigN.div d gcd in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul zy nx) - else Qq (BigZ.mul zy nx) (BigN.pred d) - | Qq nx dx, Qq ny dy => - norm (BigZ.mul nx ny) (BigN.mul (BigN.succ dx) (BigN.succ dy)) - end. - - Theorem spec_mul_norm x y: ([mul_norm x y] == [x] * [y])%Q. - intros x y; rewrite <- spec_mul. - unfold mul_norm, mul; case x; case y. - intros; apply Qeq_refl. - intros p1 n p2. - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; unfold zero, to_Q; repeat rewrite BigZ.spec_0; intros H. - rewrite BigZ.spec_mul; rewrite H; red; auto. - assert (F: (0 < BigN.to_Z (BigZ.to_N p2))%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos (BigZ.to_N p2))); auto. - intros H1; case H; rewrite spec_to_N; rewrite <- H1; ring. - assert (F1: (0 < BigN.to_Z (BigN.succ n))%Z). - rewrite BigN.spec_succ; generalize (BigN.spec_pos n); auto with zarith. - assert (F2: (0 < Zgcd (BigN.to_Z (BigZ.to_N p2)) (BigN.to_Z (BigN.succ n)))%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos (BigN.to_Z (BigZ.to_N p2)) - (BigN.to_Z (BigN.succ n)))); intros H3; auto. - generalize F; rewrite (Zgcd_inv_0_l _ _ (sym_equal H3)); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; intros H1. - intros; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; - auto with zarith. - intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite spec_to_N. - rewrite Zmult_1_r; repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p1)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite (spec_to_N p2). - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (BigN.succ n / - BigN.gcd (BigZ.to_N p2) - (BigN.succ n)))%bigN); intros F3. - rewrite BigN.succ_pred; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p1)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto; try ring. - rewrite Nspec_lt, BigN.spec_0; auto. - apply False_ind; generalize F1. - rewrite (Zdivide_Zdiv_eq - (Zgcd (BigN.to_Z (BigZ.to_N p2)) (BigN.to_Z (BigN.succ n))) - (BigN.to_Z (BigN.succ n))); auto. - generalize F3; rewrite BigN.spec_div; rewrite BigN.spec_gcd; - auto with zarith. - intros HH; rewrite <- HH; auto with zarith. - assert (FF:= Zgcd_is_gcd (BigN.to_Z (BigZ.to_N p2)) - (BigN.to_Z (BigN.succ n))); inversion FF; auto. - intros p1 p2 n. - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; unfold zero, to_Q; repeat rewrite BigZ.spec_0; intros H. - rewrite BigZ.spec_mul; rewrite H; red; simpl; ring. - assert (F: (0 < BigN.to_Z (BigZ.to_N p1))%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos (BigZ.to_N p1))); auto. - intros H1; case H; rewrite spec_to_N; rewrite <- H1; ring. - assert (F1: (0 < BigN.to_Z (BigN.succ n))%Z). - rewrite BigN.spec_succ; generalize (BigN.spec_pos n); auto with zarith. - assert (F2: (0 < Zgcd (BigN.to_Z (BigZ.to_N p1)) (BigN.to_Z (BigN.succ n)))%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos (BigN.to_Z (BigZ.to_N p1)) - (BigN.to_Z (BigN.succ n)))); intros H3; auto. - generalize F; rewrite (Zgcd_inv_0_l _ _ (sym_equal H3)); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; intros H1. - intros; repeat rewrite BigZ.spec_mul; rewrite Zmult_comm; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; - auto with zarith. - intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite spec_to_N. - rewrite Zmult_1_r; repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p2)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite (spec_to_N p1). - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (BigN.succ n / - BigN.gcd (BigZ.to_N p1) - (BigN.succ n)))%bigN); intros F3. - rewrite BigN.succ_pred; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p2)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto; try ring. - rewrite Nspec_lt, BigN.spec_0; auto. - apply False_ind; generalize F1. - rewrite (Zdivide_Zdiv_eq - (Zgcd (BigN.to_Z (BigZ.to_N p1)) (BigN.to_Z (BigN.succ n))) - (BigN.to_Z (BigN.succ n))); auto. - generalize F3; rewrite BigN.spec_div; rewrite BigN.spec_gcd; - auto with zarith. - intros HH; rewrite <- HH; auto with zarith. - assert (FF:= Zgcd_is_gcd (BigN.to_Z (BigZ.to_N p1)) - (BigN.to_Z (BigN.succ n))); inversion FF; auto. - intros p1 n1 p2 n2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X (BigN.pred Y)]); - [apply spec_norm | idtac] - end; try apply Qeq_refl. - rewrite BigN.spec_mul. - apply Zmult_lt_0_compat; rewrite BigN.spec_succ; - generalize (BigN.spec_pos n1) (BigN.spec_pos n2); auto with zarith. - Qed. - - Theorem spec_mul_normc x y: [[mul_norm x y]] = [[x]] * [[y]]. - intros x y; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul_norm. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition inv (x: t): t := - match x with - | Qz (BigZ.Pos n) => - if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one (BigN.pred n) - | Qz (BigZ.Neg n) => - if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one (BigN.pred n) - | Qq (BigZ.Pos n) d => - if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos (BigN.succ d)) (BigN.pred n) - | Qq (BigZ.Neg n) d => - if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg (BigN.succ d)) (BigN.pred n) - end. - - Theorem spec_inv x: ([inv x] == /[x])%Q. - intros [ [x | x] | [nx | nx] dx]; unfold inv. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z x)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); auto with zarith. - unfold to_Q; rewrite BigZ.spec_1. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - red; unfold Qinv; simpl. - generalize F; case BigN.to_Z; auto with zarith. - intros p Hp; discriminate Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z x)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); auto with zarith. - red; unfold Qinv; simpl. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - generalize F; case BigN.to_Z; simpl; auto with zarith. - intros p Hp; discriminate Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z nx)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos nx)); auto with zarith. - red; unfold Qinv; simpl. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite BigN.spec_succ; rewrite Z2P_correct; auto with zarith. - generalize F; case BigN.to_Z; auto with zarith. - intros p Hp; discriminate Hp. - generalize (BigN.spec_pos dx); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z nx)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos nx)); auto with zarith. - red; unfold Qinv; simpl. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite BigN.spec_succ; rewrite Z2P_correct; auto with zarith. - generalize F; case BigN.to_Z; auto with zarith. - simpl; intros. - match goal with |- (?X = Zneg ?Y)%Z => - replace (Zneg Y) with (-(Zpos Y))%Z; - try rewrite Z2P_correct; auto with zarith - end. - rewrite Zpos_mult_morphism; - rewrite Z2P_correct; auto with zarith; try ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p Hp; discriminate Hp. - generalize (BigN.spec_pos dx); auto with zarith. - Qed. - - Theorem spec_invc x: [[inv x]] = /[[x]]. - intros x; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - -Definition inv_norm x := - match x with - | Qz (BigZ.Pos n) => - if BigN.eq_bool n BigN.zero then zero else - if BigN.eq_bool n BigN.one then x else Qq BigZ.one (BigN.pred n) - | Qz (BigZ.Neg n) => - if BigN.eq_bool n BigN.zero then zero else - if BigN.eq_bool n BigN.one then x else Qq BigZ.minus_one (BigN.pred n) - | Qq (BigZ.Pos n) d => let d := BigN.succ d in - if BigN.eq_bool n BigN.zero then zero else - if BigN.eq_bool n BigN.one then Qz (BigZ.Pos d) - else Qq (BigZ.Pos d) (BigN.pred n) - | Qq (BigZ.Neg n) d => let d := BigN.succ d in - if BigN.eq_bool n BigN.zero then zero else - if BigN.eq_bool n BigN.one then Qz (BigZ.Neg d) - else Qq (BigZ.Neg d) (BigN.pred n) - end. - - Theorem spec_inv_norm x: ([inv_norm x] == /[x])%Q. - intros x; rewrite <- spec_inv. - (case x; clear x); [intros [x | x] | intros nx dx]; - unfold inv_norm, inv. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - apply Qeq_refl. - assert (F: (0 < BigN.to_Z x)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; intros H1. - red; simpl. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite Z2P_correct; try rewrite H1; auto with zarith. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - apply Qeq_refl. - assert (F: (0 < BigN.to_Z x)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; intros H1. - red; simpl. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite Z2P_correct; try rewrite H1; auto with zarith. - apply Qeq_refl. - case nx; clear nx; intros nx. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; intros H1. - red; simpl. - rewrite BigN.succ_pred; try rewrite H1; auto with zarith. - rewrite Nspec_lt, BigN.spec_0, H1; auto with zarith. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; intros H1. - red; simpl. - rewrite BigN.succ_pred; try rewrite H1; auto with zarith. - rewrite Nspec_lt, BigN.spec_0, H1; auto with zarith. - apply Qeq_refl. - Qed. - - - Definition div x y := mul x (inv y). - - Theorem spec_div x y: ([div x y] == [x] / [y])%Q. - intros x y; unfold div; rewrite spec_mul; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Theorem spec_divc x y: [[div x y]] = [[x]] / [[y]]. - intros x y; unfold div; rewrite spec_mulc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - Definition div_norm x y := mul_norm x (inv y). - - Theorem spec_div_norm x y: ([div_norm x y] == [x] / [y])%Q. - intros x y; unfold div_norm; rewrite spec_mul_norm; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - Qed. - - Theorem spec_div_normc x y: [[div_norm x y]] = [[x]] / [[y]]. - intros x y; unfold div_norm; rewrite spec_mul_normc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - Qed. - - - Definition square (x: t): t := - match x with - | Qz zx => Qz (BigZ.square zx) - | Qq nx dx => Qq (BigZ.square nx) (BigN.pred (BigN.square (BigN.succ dx))) - end. - - Theorem spec_square x: ([square x] == [x] ^ 2)%Q. - intros [ x | nx dx]; unfold square. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - assert (F: (0 < BigN.to_Z (BigN.succ dx))%Z). - rewrite BigN.spec_succ; - case (Zle_lt_or_eq _ _ (BigN.spec_pos dx)); auto with zarith. - assert (F1 : (0 < BigN.to_Z (BigN.square (BigN.succ dx)))%Z). - rewrite BigN.spec_square; apply Zmult_lt_0_compat; - auto with zarith. - rewrite BigN.succ_pred by (rewrite Nspec_lt, BigN.spec_0; auto). - rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto with zarith. - repeat rewrite BigN.spec_succ; auto with zarith. - rewrite BigN.spec_square; auto with zarith. - repeat rewrite BigN.spec_succ; auto with zarith. - Qed. - - Theorem spec_squarec x: [[square x]] = [[x]]^2. - intros x; unfold to_Qc. - apply trans_equal with (!! ([x]^2)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_square. - simpl Qcpower. - replace (!! [x] * 1) with (!![x]); try ring. - simpl. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition power_pos (x: t) p: t := - match x with - | Qz zx => Qz (BigZ.power_pos zx p) - | Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.pred (BigN.power_pos (BigN.succ dx) p)) - end. - - - Theorem spec_power_pos x p: ([power_pos x p] == [x] ^ Zpos p)%Q. - Proof. - intros [x | nx dx] p; unfold power_pos. - unfold power_pos; red; simpl. - generalize (Qpower_decomp p (BigZ.to_Z x) 1). - unfold Qeq; simpl. - rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Zmult_1_r. - intros H; rewrite H. - rewrite BigZ.spec_power_pos; simpl; ring. - assert (F1: (0 < BigN.to_Z (BigN.succ dx))%Z). - rewrite BigN.spec_succ; - generalize (BigN.spec_pos dx); auto with zarith. - assert (F2: (0 < BigN.to_Z (BigN.succ dx) ^ ' p)%Z). - unfold Zpower; apply Zpower_pos_pos; auto. - unfold power_pos; red; simpl. - rewrite BigN.succ_pred, BigN.spec_power_pos. - rewrite Z2P_correct; auto. - generalize (Qpower_decomp p (BigZ.to_Z nx) - (Z2P (BigN.to_Z (BigN.succ dx)))). - unfold Qeq; simpl. - repeat rewrite Z2P_correct; auto. - unfold Qeq; simpl; intros HH. - rewrite HH. - rewrite BigZ.spec_power_pos; simpl; ring. - rewrite Nspec_lt, BigN.spec_0, BigN.spec_power_pos; auto. - Qed. - - Theorem spec_power_posc x p: [[power_pos x p]] = [[x]] ^ nat_of_P p. - intros x p; unfold to_Qc. - apply trans_equal with (!! ([x]^Zpos p)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_power_pos. - pattern p; apply Pind; clear p. - simpl; ring. - intros p Hrec. - rewrite nat_of_P_succ_morphism; simpl Qcpower. - rewrite <- Hrec. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; - unfold this. - apply Qred_complete. - assert (F: [x] ^ ' Psucc p == [x] * [x] ^ ' p). - simpl; case x; simpl; clear x Hrec. - intros x; simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - intros nx dx; simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - assert (F1: (0 < BigN.to_Z (BigN.succ dx))%Z). - rewrite BigN.spec_succ; generalize (BigN.spec_pos dx); - auto with zarith. - repeat rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto. - 2: apply Zpower_pos_pos; auto. - 2: apply Zpower_pos_pos; auto. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - rewrite F. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - -End Qp. diff --git a/theories/Numbers/Rational/BigQ/QvMake.v b/theories/Numbers/Rational/BigQ/QvMake.v deleted file mode 100644 index 4523e241..00000000 --- a/theories/Numbers/Rational/BigQ/QvMake.v +++ /dev/null @@ -1,1151 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -(*i $Id: QvMake.v 11027 2008-06-01 13:28:59Z letouzey $ i*) - -Require Import Bool. -Require Import ZArith. -Require Import Znumtheory. -Require Import BigNumPrelude. -Require Import Arith. -Require Export BigN. -Require Export BigZ. -Require Import QArith. -Require Import Qcanon. -Require Import Qpower. -Require Import QMake_base. - -Module Qv. - - Import BinInt Zorder. - Open Local Scope Q_scope. - Open Local Scope Qc_scope. - - (** The notation of a rational number is either an integer x, - interpreted as itself or a pair (x,y) of an integer x and a naturel - number y interpreted as x/y. All functions maintain the invariant - that y is never zero. *) - - Definition t := q_type. - - Definition zero: t := Qz BigZ.zero. - Definition one: t := Qz BigZ.one. - Definition minus_one: t := Qz BigZ.minus_one. - - Definition of_Z x: t := Qz (BigZ.of_Z x). - - Definition wf x := - match x with - | Qz _ => True - | Qq n d => if BigN.eq_bool d BigN.zero then False else True - end. - - Definition of_Q q: t := - match q with x # y => - Qq (BigZ.of_Z x) (BigN.of_N (Npos y)) - end. - - Definition of_Qc q := of_Q (this q). - - Definition to_Q (q: t) := - match q with - Qz x => BigZ.to_Z x # 1 - |Qq x y => BigZ.to_Z x # Z2P (BigN.to_Z y) - end. - - Definition to_Qc q := !!(to_Q q). - - Notation "[[ x ]]" := (to_Qc x). - - Notation "[ x ]" := (to_Q x). - - Theorem spec_to_Q: forall q: Q, [of_Q q] = q. - intros (x,y); simpl. - rewrite BigZ.spec_of_Z; simpl. - rewrite (BigN.spec_of_pos); auto. - Qed. - - Theorem spec_to_Qc: forall q, [[of_Qc q]] = q. - intros (x, Hx); unfold of_Qc, to_Qc; simpl. - apply Qc_decomp; simpl. - intros; rewrite spec_to_Q; auto. - Qed. - - Definition opp (x: t): t := - match x with - | Qz zx => Qz (BigZ.opp zx) - | Qq nx dx => Qq (BigZ.opp nx) dx - end. - - Theorem wf_opp: forall x, wf x -> wf (opp x). - intros [zx | nx dx]; unfold opp, wf; auto. - Qed. - - Theorem spec_opp: forall q, ([opp q] = -[q])%Q. - intros [z | x y]; simpl. - rewrite BigZ.spec_opp; auto. - rewrite BigZ.spec_opp; auto. - Qed. - - Theorem spec_oppc: forall q, [[opp q]] = -[[q]]. - intros q; unfold Qcopp, to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - rewrite spec_opp. - rewrite <- Qred_opp. - rewrite Qred_involutive; auto. - Qed. - - (* Les fonctions doivent assurer que si leur arguments sont valides alors - le resultat est correct et valide (si c'est un Q) - *) - - Definition compare (x y: t) := - match x, y with - | Qz zx, Qz zy => BigZ.compare zx zy - | Qz zx, Qq ny dy => BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny - | Qq nx dx, Qz zy => BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx)) - | Qq nx dx, Qq ny dy => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) - end. - - Theorem spec_compare: forall q1 q2, wf q1 -> wf q2 -> - compare q1 q2 = ([q1] ?= [q2])%Q. - intros [z1 | x1 y1] [z2 | x2 y2]; - unfold Qcompare, compare, to_Q, Qnum, Qden, wf. - repeat rewrite Zmult_1_r. - generalize (BigZ.spec_compare z1 z2); case BigZ.compare; intros H; auto. - rewrite H; rewrite Zcompare_refl; auto. - rewrite Zmult_1_r. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool. - intros _ _ HH; case HH. - rewrite BigN.spec_0; intros HH _ _. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (z1 * BigZ.Pos y2) x2)%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH _ _. - rewrite Z2P_correct; auto with zarith. - 2: generalize (BigN.spec_pos y1); auto with zarith. - rewrite Zmult_1_r. - generalize (BigZ.spec_compare x1 (z2 * BigZ.Pos y1))%bigZ; case BigZ.compare; - rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - generalize (BigN.spec_eq_bool y1 BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH1. - generalize (BigN.spec_eq_bool y2 BigN.zero); - case BigN.eq_bool. - intros _ _ HH; case HH. - rewrite BigN.spec_0; intros HH2 _ _. - repeat rewrite Z2P_correct. - 2: generalize (BigN.spec_pos y1); auto with zarith. - 2: generalize (BigN.spec_pos y2); auto with zarith. - generalize (BigZ.spec_compare (x1 * BigZ.Pos y2) - (x2 * BigZ.Pos y1))%bigZ; case BigZ.compare; - repeat rewrite BigZ.spec_mul; simpl; intros H; apply sym_equal; auto. - rewrite H; rewrite Zcompare_refl; auto. - Qed. - - Theorem spec_comparec: forall q1 q2, wf q1 -> wf q2 -> - compare q1 q2 = ([[q1]] ?= [[q2]]). - unfold Qccompare, to_Qc. - intros q1 q2 Hq1 Hq2; rewrite spec_compare; simpl; auto. - apply Qcompare_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition norm n d: t := - if BigZ.eq_bool n BigZ.zero then zero - else - let gcd := BigN.gcd (BigZ.to_N n) d in - if BigN.eq_bool gcd BigN.one then Qq n d - else - let n := BigZ.div n (BigZ.Pos gcd) in - let d := BigN.div d gcd in - if BigN.eq_bool d BigN.one then Qz n - else Qq n d. - - Theorem wf_norm: forall n q, - (BigN.to_Z q <> 0)%Z -> wf (norm n q). - intros p q; unfold norm, wf; intros Hq. - assert (Hp := BigN.spec_pos (BigZ.to_N p)). - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; auto; rewrite BigZ.spec_0; intros H1. - simpl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_1. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - set (a := BigN.to_Z (BigZ.to_N p)). - set (b := (BigN.to_Z q)). - assert (F: (0 < Zgcd a b)%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos a b)); auto. - intros HH1; case Hq; apply (Zgcd_inv_0_r _ _ (sym_equal HH1)). - rewrite BigN.spec_div; rewrite BigN.spec_gcd; auto; fold a; fold b. - intros H; case Hq; fold b. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite H; auto with zarith. - assert (F1:= Zgcd_is_gcd a b); inversion F1; auto. - Qed. - - Theorem spec_norm: forall n q, - ((0 < BigN.to_Z q)%Z -> [norm n q] == [Qq n q])%Q. - intros p q; unfold norm; intros Hq. - assert (Hp := BigN.spec_pos (BigZ.to_N p)). - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; auto; rewrite BigZ.spec_0; intros H1. - red; simpl; rewrite H1; ring. - case (Zle_lt_or_eq _ _ Hp); clear Hp; intros Hp. - case (Zle_lt_or_eq _ _ - (Zgcd_is_pos (BigN.to_Z (BigZ.to_N p)) (BigN.to_Z q))); intros H4. - 2: generalize Hq; rewrite (Zgcd_inv_0_r _ _ (sym_equal H4)); auto with zarith. - 2: red; simpl; auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_1; intros H2. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_1. - red; simpl. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite Zmult_1_r. - rewrite Z2P_correct; auto with zarith. - rewrite spec_to_N; intros; rewrite Zgcd_div_swap; auto. - rewrite H; ring. - intros H3. - red; simpl. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - assert (F: (0 < BigN.to_Z (q / BigN.gcd (BigZ.to_N p) q)%bigN)%Z). - rewrite BigN.spec_div; auto with zarith. - rewrite BigN.spec_gcd. - apply Zgcd_div_pos; auto. - rewrite BigN.spec_gcd; auto. - rewrite Z2P_correct; auto. - rewrite Z2P_correct; auto. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; auto with zarith. - rewrite spec_to_N; apply Zgcd_div_swap; auto. - case H1; rewrite spec_to_N; rewrite <- Hp; ring. - Qed. - - Theorem spec_normc: forall n q, - (0 < BigN.to_Z q)%Z -> [[norm n q]] = [[Qq n q]]. - intros n q H; unfold to_Qc, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_norm; auto. - Qed. - - Definition add (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.add zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx - | Qq nx dx, Qq ny dy => - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - Qq n d - end. - - Theorem wf_add: forall x y, wf x -> wf y -> wf (add x y). - intros [zx | nx dx] [zy | ny dy]; unfold add, wf; auto. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_mul. - intros H1 H2 H3. - case (Zmult_integral _ _ H1); auto with zarith. - Qed. - - Theorem spec_add x y: wf x -> wf y -> - ([add x y] == [x] + [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qplus; simpl. - rewrite BigZ.spec_add; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - assert (F1:= BigN.spec_pos dy). - rewrite Zmult_1_r. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool. - intros _ _ HH; case HH. - rewrite BigN.spec_0; intros HH _ _. - rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul. - simpl; apply Qeq_refl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH _ _. - assert (F1:= BigN.spec_pos dx). - rewrite Zmult_1_r; rewrite Pmult_1_r. - simpl; rewrite Z2P_correct; auto with zarith. - rewrite BigZ.spec_add; rewrite BigZ.spec_mul; simpl. - apply Qeq_refl. - generalize (BigN.spec_eq_bool dx BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH1. - generalize (BigN.spec_eq_bool dy BigN.zero); - case BigN.eq_bool. - intros _ _ HH; case HH. - rewrite BigN.spec_0; intros HH2 _ _. - assert (Fx: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (Fy: (0 < BigN.to_Z dy)%Z). - generalize (BigN.spec_pos dy); auto with zarith. - rewrite BigZ.spec_add; repeat rewrite BigN.spec_mul. - red; simpl. - rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto. - repeat rewrite BigZ.spec_mul; simpl; auto. - apply Zmult_lt_0_compat; auto. - Qed. - - Theorem spec_addc x y: wf x -> wf y -> - [[add x y]] = [[x]] + [[y]]. - intros x y H1 H2; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition add_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.add zx zy) - | Qz zx, Qq ny dy => - norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy - | Qq nx dx, Qz zy => - norm (BigZ.add (BigZ.mul zy (BigZ.Pos dx)) nx) dx - | Qq nx dx, Qq ny dy => - let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in - let d := BigN.mul dx dy in - norm n d - end. - - Theorem wf_add_norm: forall x y, wf x -> wf y -> wf (add_norm x y). - intros [zx | nx dx] [zy | ny dy]; unfold add_norm; auto. - intros HH1 HH2; apply wf_norm. - generalize HH2; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros HH1 HH2; apply wf_norm. - generalize HH1; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros HH1 HH2; apply wf_norm. - rewrite BigN.spec_mul; intros HH3. - case (Zmult_integral _ _ HH3). - generalize HH1; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - generalize HH2; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - Qed. - - Theorem spec_add_norm x y: wf x -> wf y -> - ([add_norm x y] == [x] + [y])%Q. - intros x y H1 H2; rewrite <- spec_add; auto. - generalize H1 H2; unfold add_norm, add, wf; case x; case y; clear H1 H2. - intros; apply Qeq_refl. - intros p1 n p2 _. - generalize (BigN.spec_eq_bool n BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH _. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - generalize (BigN.spec_pos n); auto with zarith. - simpl. - repeat rewrite BigZ.spec_add. - repeat rewrite BigZ.spec_mul; simpl. - apply Qeq_refl. - intros p1 n p2. - generalize (BigN.spec_eq_bool p2 BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH _ _. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end. - generalize (BigN.spec_pos p2); auto with zarith. - simpl. - repeat rewrite BigZ.spec_add. - repeat rewrite BigZ.spec_mul; simpl. - rewrite Zplus_comm. - apply Qeq_refl. - intros p1 q1 p2 q2. - generalize (BigN.spec_eq_bool q2 BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH1 _. - generalize (BigN.spec_eq_bool q1 BigN.zero); - case BigN.eq_bool. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH2 _. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end; try apply Qeq_refl. - rewrite BigN.spec_mul. - apply Zmult_lt_0_compat. - generalize (BigN.spec_pos q2); auto with zarith. - generalize (BigN.spec_pos q1); auto with zarith. - Qed. - - Theorem spec_add_normc x y: wf x -> wf y -> - [[add_norm x y]] = [[x]] + [[y]]. - intros x y Hx Hy; unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_add_norm; auto. - unfold Qcplus, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qplus_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition sub x y := add x (opp y). - - Theorem wf_sub x y: wf x -> wf y -> wf (sub x y). - intros x y Hx Hy; unfold sub; apply wf_add; auto. - apply wf_opp; auto. - Qed. - - Theorem spec_sub x y: wf x -> wf y -> - ([sub x y] == [x] - [y])%Q. - intros x y Hx Hy; unfold sub; rewrite spec_add; auto. - rewrite spec_opp; ring. - apply wf_opp; auto. - Qed. - - Theorem spec_subc x y: wf x -> wf y -> - [[sub x y]] = [[x]] - [[y]]. - intros x y Hx Hy; unfold sub; rewrite spec_addc; auto. - rewrite spec_oppc; ring. - apply wf_opp; auto. - Qed. - - Definition sub_norm x y := add_norm x (opp y). - - Theorem wf_sub_norm x y: wf x -> wf y -> wf (sub_norm x y). - intros x y Hx Hy; unfold sub_norm; apply wf_add_norm; auto. - apply wf_opp; auto. - Qed. - - Theorem spec_sub_norm x y: wf x -> wf y -> - ([sub_norm x y] == [x] - [y])%Q. - intros x y Hx Hy; unfold sub_norm; rewrite spec_add_norm; auto. - rewrite spec_opp; ring. - apply wf_opp; auto. - Qed. - - Theorem spec_sub_normc x y: wf x -> wf y -> - [[sub_norm x y]] = [[x]] - [[y]]. - intros x y Hx Hy; unfold sub_norm; rewrite spec_add_normc; auto. - rewrite spec_oppc; ring. - apply wf_opp; auto. - Qed. - - Definition mul (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx - | Qq nx dx, Qq ny dy => - Qq (BigZ.mul nx ny) (BigN.mul dx dy) - end. - - Theorem wf_mul: forall x y, wf x -> wf y -> wf (mul x y). - intros [zx | nx dx] [zy | ny dy]; unfold mul, wf; auto. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_mul. - intros H1 H2 H3. - case (Zmult_integral _ _ H1); auto with zarith. - Qed. - - Theorem spec_mul x y: wf x -> wf y -> ([mul x y] == [x] * [y])%Q. - intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl. - rewrite BigZ.spec_mul; repeat rewrite Zmult_1_r; auto. - intros; apply Qeq_refl; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros _ _ HH; case HH. - rewrite BigN.spec_0; intros HH1 _ _. - rewrite BigZ.spec_mul; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros _ HH; case HH. - rewrite BigN.spec_0; intros HH1 _ _. - rewrite BigZ.spec_mul; rewrite Pmult_1_r. - apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros _ HH; case HH. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros _ _ _ HH; case HH. - rewrite BigN.spec_0; intros H1 H2 _ _. - rewrite BigZ.spec_mul; rewrite BigN.spec_mul. - assert (tmp: - (forall a b, 0 < a -> 0 < b -> Z2P (a * b) = (Z2P a * Z2P b)%positive)%Z). - intros [|a|a] [|b|b]; simpl; auto; intros; apply False_ind; auto with zarith. - rewrite tmp; auto. - apply Qeq_refl. - generalize (BigN.spec_pos dx); auto with zarith. - generalize (BigN.spec_pos dy); auto with zarith. - Qed. - - Theorem spec_mulc x y: wf x -> wf y -> - [[mul x y]] = [[x]] * [[y]]. - intros x y Hx Hy; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition mul_norm (x y: t): t := - match x, y with - | Qz zx, Qz zy => Qz (BigZ.mul zx zy) - | Qz zx, Qq ny dy => - if BigZ.eq_bool zx BigZ.zero then zero - else - let gcd := BigN.gcd (BigZ.to_N zx) dy in - if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zx ny) dy - else - let zx := BigZ.div zx (BigZ.Pos gcd) in - let d := BigN.div dy gcd in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul zx ny) - else Qq (BigZ.mul zx ny) d - | Qq nx dx, Qz zy => - if BigZ.eq_bool zy BigZ.zero then zero - else - let gcd := BigN.gcd (BigZ.to_N zy) dx in - if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zy nx) dx - else - let zy := BigZ.div zy (BigZ.Pos gcd) in - let d := BigN.div dx gcd in - if BigN.eq_bool d BigN.one then Qz (BigZ.mul zy nx) - else Qq (BigZ.mul zy nx) d - | Qq nx dx, Qq ny dy => norm (BigZ.mul nx ny) (BigN.mul dx dy) - end. - - Theorem wf_mul_norm: forall x y, wf x -> wf y -> wf (mul_norm x y). - intros [zx | nx dx] [zy | ny dy]; unfold mul_norm; auto. - intros HH1 HH2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - rewrite BigN.spec_1; rewrite BigZ.spec_0. - intros H1 H2; unfold wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - rewrite BigN.spec_0. - set (a := BigN.to_Z (BigZ.to_N zx)). - set (b := (BigN.to_Z dy)). - assert (F: (0 < Zgcd a b)%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos a b)); auto. - intros HH3; case H2; rewrite spec_to_N; fold a. - rewrite (Zgcd_inv_0_l _ _ (sym_equal HH3)); try ring. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; fold a; fold b; auto. - intros H. - generalize HH2; simpl wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - rewrite BigN.spec_0; intros HH; case HH; fold b. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite H; auto with zarith. - assert (F1:= Zgcd_is_gcd a b); inversion F1; auto. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - rewrite BigN.spec_1; rewrite BigN.spec_gcd. - intros HH1 H1 H2. - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; auto. - rewrite BigN.spec_1; rewrite BigN.spec_gcd. - intros HH1 H1 H2. - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; auto. - rewrite BigZ.spec_0. - intros HH2. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - set (a := BigN.to_Z (BigZ.to_N zy)). - set (b := (BigN.to_Z dx)). - assert (F: (0 < Zgcd a b)%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos a b)); auto. - intros HH3; case HH2; rewrite spec_to_N; fold a. - rewrite (Zgcd_inv_0_l _ _ (sym_equal HH3)); try ring. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; fold a; fold b; auto. - intros H; unfold wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - rewrite BigN.spec_0. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; fold a; fold b; auto. - intros HH; generalize H1; simpl wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - rewrite BigN.spec_0. - intros HH3; case HH3; fold b. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite HH; auto with zarith. - assert (F1:= Zgcd_is_gcd a b); inversion F1; auto. - intros HH1 HH2; apply wf_norm. - rewrite BigN.spec_mul; intros HH3. - case (Zmult_integral _ _ HH3). - generalize HH1; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - generalize HH2; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - Qed. - - Theorem spec_mul_norm x y: wf x -> wf y -> - ([mul_norm x y] == [x] * [y])%Q. - intros x y Hx Hy; rewrite <- spec_mul; auto. - unfold mul_norm, mul; generalize Hx Hy; case x; case y; clear Hx Hy. - intros; apply Qeq_refl. - intros p1 n p2 Hx Hy. - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; unfold zero, to_Q; repeat rewrite BigZ.spec_0; intros H. - rewrite BigZ.spec_mul; rewrite H; red; auto. - assert (F: (0 < BigN.to_Z (BigZ.to_N p2))%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos (BigZ.to_N p2))); auto. - intros H1; case H; rewrite spec_to_N; rewrite <- H1; ring. - assert (F1: (0 < BigN.to_Z n)%Z). - generalize Hy; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto. - intros _ HH; case HH. - rewrite BigN.spec_0; generalize (BigN.spec_pos n); auto with zarith. - set (a := BigN.to_Z (BigZ.to_N p2)). - set (b := BigN.to_Z n). - assert (F2: (0 < Zgcd a b )%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos a b)); intros H3; auto. - generalize F; fold a; rewrite (Zgcd_inv_0_l _ _ (sym_equal H3)); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; try rewrite BigN.spec_gcd; - fold a b; intros H1. - intros; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; - auto with zarith; fold a b; intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite spec_to_N; fold a; fold b. - rewrite Zmult_1_r; repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p1)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - intros H2; red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite Z2P_correct; auto with zarith. - rewrite (spec_to_N p2); fold a b. - rewrite Z2P_correct; auto with zarith. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p1)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto; try ring. - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (n / - BigN.gcd (BigZ.to_N p2) - n))%bigN); - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - intros H3. - apply False_ind; generalize F1. - generalize Hy; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; auto with zarith. - intros HH; case HH; fold b. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite <- H3; ring. - assert (FF:= Zgcd_is_gcd a b); inversion FF; auto. - intros p1 p2 n Hx Hy. - match goal with |- context[BigZ.eq_bool ?X ?Y] => - generalize (BigZ.spec_eq_bool X Y); case BigZ.eq_bool - end; unfold zero, to_Q; repeat rewrite BigZ.spec_0; intros H. - rewrite BigZ.spec_mul; rewrite H; red; simpl; ring. - set (a := BigN.to_Z (BigZ.to_N p1)). - set (b := BigN.to_Z n). - assert (F: (0 < a)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos (BigZ.to_N p1))); auto. - intros H1; case H; rewrite spec_to_N; rewrite <- H1; ring. - assert (F1: (0 < b)%Z). - generalize Hx; unfold wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; auto with zarith. - generalize (BigN.spec_pos n); fold b; auto with zarith. - assert (F2: (0 < Zgcd a b)%Z). - case (Zle_lt_or_eq _ _ (Zgcd_is_pos a b)); intros H3; auto. - generalize F; rewrite (Zgcd_inv_0_l _ _ (sym_equal H3)); auto with zarith. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1; rewrite BigN.spec_gcd; fold a b; intros H1. - intros; repeat rewrite BigZ.spec_mul; rewrite Zmult_comm; apply Qeq_refl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_1. - rewrite BigN.spec_div; rewrite BigN.spec_gcd; - auto with zarith. - fold a b; intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite spec_to_N; fold a b. - rewrite Zmult_1_r; repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p2)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto with zarith. - rewrite H2; ring. - intros H2. - red; simpl. - repeat rewrite BigZ.spec_mul. - rewrite BigZ.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - rewrite Z2P_correct; auto with zarith. - rewrite (spec_to_N p1); fold a b. - case (Zle_lt_or_eq _ _ - (BigN.spec_pos (n / BigN.gcd (BigZ.to_N p1) n))%bigN); intros F3. - rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_div; simpl; rewrite BigN.spec_gcd; - fold a b; auto with zarith. - repeat rewrite <- Zmult_assoc. - rewrite (Zmult_comm (BigZ.to_Z p2)). - repeat rewrite Zmult_assoc. - rewrite Zgcd_div_swap; auto; try ring. - apply False_ind; generalize F1. - rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - generalize F3; rewrite BigN.spec_div; rewrite BigN.spec_gcd; fold a b; - auto with zarith. - intros HH; rewrite <- HH; auto with zarith. - assert (FF:= Zgcd_is_gcd a b); inversion FF; auto. - intros p1 n1 p2 n2 Hn1 Hn2. - match goal with |- [norm ?X ?Y] == _ => - apply Qeq_trans with ([Qq X Y]); - [apply spec_norm | idtac] - end; try apply Qeq_refl. - rewrite BigN.spec_mul. - apply Zmult_lt_0_compat. - generalize Hn1; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; auto with zarith. - generalize (BigN.spec_pos n1) (BigN.spec_pos n2); auto with zarith. - generalize Hn2; simpl. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; auto with zarith. - generalize (BigN.spec_pos n1) (BigN.spec_pos n2); auto with zarith. - Qed. - - Theorem spec_mul_normc x y: wf x -> wf y -> - [[mul_norm x y]] = [[x]] * [[y]]. - intros x y Hx Hy; unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_mul_norm; auto. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - Definition inv (x: t): t := - match x with - | Qz (BigZ.Pos n) => - if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one n - | Qz (BigZ.Neg n) => - if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one n - | Qq (BigZ.Pos n) d => - if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos d) n - | Qq (BigZ.Neg n) d => - if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg d) n - end. - - - Theorem wf_inv: forall x, wf x -> wf (inv x). - intros [ zx | nx dx]; unfold inv, wf; auto. - case zx; clear zx. - intros nx. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_mul. - intros nx. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0; rewrite BigN.spec_mul. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - intros _ HH; case HH. - intros H1 _. - case nx; clear nx. - intros nx. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; simpl; auto. - intros nx. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; simpl; auto. - Qed. - - Theorem spec_inv x: wf x -> - ([inv x] == /[x])%Q. - intros [ [x | x] _ | [nx | nx] dx]; unfold inv. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z x)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); auto with zarith. - unfold to_Q; rewrite BigZ.spec_1. - red; unfold Qinv; simpl. - generalize F; case BigN.to_Z; auto with zarith. - intros p Hp; discriminate Hp. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z x)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos x)); auto with zarith. - red; unfold Qinv; simpl. - generalize F; case BigN.to_Z; simpl; auto with zarith. - intros p Hp; discriminate Hp. - simpl wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - intros HH; case HH. - intros _. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z nx)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos nx)); auto with zarith. - red; unfold Qinv; simpl. - rewrite Z2P_correct; auto with zarith. - generalize F; case BigN.to_Z; auto with zarith. - intros p Hp; discriminate Hp. - generalize (BigN.spec_pos dx); auto with zarith. - simpl wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H1. - intros HH; case HH. - intros _. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; rewrite BigN.spec_0; intros H. - unfold zero, to_Q; rewrite BigZ.spec_0. - unfold BigZ.to_Z; rewrite H; apply Qeq_refl. - assert (F: (0 < BigN.to_Z nx)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos nx)); auto with zarith. - red; unfold Qinv; simpl. - rewrite Z2P_correct; auto with zarith. - generalize F; case BigN.to_Z; auto with zarith. - simpl; intros. - match goal with |- (?X = Zneg ?Y)%Z => - replace (Zneg Y) with (Zopp (Zpos Y)); - try rewrite Z2P_correct; auto with zarith - end. - rewrite Zpos_mult_morphism; - rewrite Z2P_correct; auto with zarith; try ring. - generalize (BigN.spec_pos dx); auto with zarith. - intros p Hp; discriminate Hp. - generalize (BigN.spec_pos dx); auto with zarith. - Qed. - - Theorem spec_invc x: wf x -> - [[inv x]] = /[[x]]. - intros x Hx; unfold to_Qc. - apply trans_equal with (!! (/[x])). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_inv; auto. - unfold Qcinv, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qinv_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - - Definition div x y := mul x (inv y). - - Theorem wf_div x y: wf x -> wf y -> wf (div x y). - intros x y Hx Hy; unfold div; apply wf_mul; auto. - apply wf_inv; auto. - Qed. - - Theorem spec_div x y: wf x -> wf y -> - ([div x y] == [x] / [y])%Q. - intros x y Hx Hy; unfold div; rewrite spec_mul; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - apply wf_inv; auto. - Qed. - - Theorem spec_divc x y: wf x -> wf y -> - [[div x y]] = [[x]] / [[y]]. - intros x y Hx Hy; unfold div; rewrite spec_mulc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - apply wf_inv; auto. - Qed. - - Definition div_norm x y := mul_norm x (inv y). - - Theorem wf_div_norm x y: wf x -> wf y -> wf (div_norm x y). - intros x y Hx Hy; unfold div_norm; apply wf_mul_norm; auto. - apply wf_inv; auto. - Qed. - - Theorem spec_div_norm x y: wf x -> wf y -> - ([div_norm x y] == [x] / [y])%Q. - intros x y Hx Hy; unfold div_norm; rewrite spec_mul_norm; auto. - unfold Qdiv; apply Qmult_comp. - apply Qeq_refl. - apply spec_inv; auto. - apply wf_inv; auto. - Qed. - - Theorem spec_div_normc x y: wf x -> wf y -> - [[div_norm x y]] = [[x]] / [[y]]. - intros x y Hx Hy; unfold div_norm; rewrite spec_mul_normc; auto. - unfold Qcdiv; apply f_equal2 with (f := Qcmult); auto. - apply spec_invc; auto. - apply wf_inv; auto. - Qed. - - Definition square (x: t): t := - match x with - | Qz zx => Qz (BigZ.square zx) - | Qq nx dx => Qq (BigZ.square nx) (BigN.square dx) - end. - - Theorem wf_square: forall x, wf x -> wf (square x). - intros [ zx | nx dx]; unfold square, wf; auto. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigN.spec_square; intros H1 H2; case H2. - case (Zmult_integral _ _ H1); auto. - Qed. - - Theorem spec_square x: wf x -> ([square x] == [x] ^ 2)%Q. - intros [ x | nx dx]; unfold square. - intros _. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - unfold wf. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - intros _ HH; case HH. - intros H1 _. - red; simpl; rewrite BigZ.spec_square; auto with zarith. - assert (F: (0 < BigN.to_Z dx)%Z). - case (Zle_lt_or_eq _ _ (BigN.spec_pos dx)); auto with zarith. - assert (F1 : (0 < BigN.to_Z (BigN.square dx))%Z). - rewrite BigN.spec_square; apply Zmult_lt_0_compat; - auto with zarith. - rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto with zarith. - rewrite BigN.spec_square; auto with zarith. - Qed. - - Theorem spec_squarec x: wf x -> [[square x]] = [[x]]^2. - intros x Hx; unfold to_Qc. - apply trans_equal with (!! ([x]^2)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_square; auto. - simpl Qcpower. - replace (!! [x] * 1) with (!![x]); try ring. - simpl. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - - - Definition power_pos (x: t) p: t := - match x with - | Qz zx => Qz (BigZ.power_pos zx p) - | Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p) - end. - - Theorem wf_power_pos: forall x p, wf x -> wf (power_pos x p). - intros [ zx | nx dx] p; unfold power_pos, wf; auto. - repeat match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - rewrite BigN.spec_power_pos; simpl. - intros H1 H2 _. - case (Zle_lt_or_eq _ _ (BigN.spec_pos dx)); auto with zarith. - intros H3; generalize (Zpower_pos_pos _ p H3); auto with zarith. - Qed. - - Theorem spec_power_pos x p: wf x -> ([power_pos x p] == [x] ^ Zpos p)%Q. - Proof. - intros [x | nx dx] p; unfold power_pos. - intros _; unfold power_pos; red; simpl. - generalize (Qpower_decomp p (BigZ.to_Z x) 1). - unfold Qeq; simpl. - rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Zmult_1_r. - intros H; rewrite H. - rewrite BigZ.spec_power_pos; simpl; ring. - unfold wf. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - intros _ HH; case HH. - intros H1 _. - assert (F1: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - assert (F2: (0 < BigN.to_Z dx ^ ' p)%Z). - unfold Zpower; apply Zpower_pos_pos; auto. - unfold power_pos; red; simpl. - rewrite Z2P_correct; rewrite BigN.spec_power_pos; auto. - generalize (Qpower_decomp p (BigZ.to_Z nx) - (Z2P (BigN.to_Z dx))). - unfold Qeq; simpl. - repeat rewrite Z2P_correct; auto. - unfold Qeq; simpl; intros HH. - rewrite HH. - rewrite BigZ.spec_power_pos; simpl; ring. - Qed. - - Theorem spec_power_posc x p: wf x -> - [[power_pos x p]] = [[x]] ^ nat_of_P p. - intros x p Hx; unfold to_Qc. - apply trans_equal with (!! ([x]^Zpos p)). - unfold Q2Qc. - apply Qc_decomp; intros _ _; unfold this. - apply Qred_complete; apply spec_power_pos; auto. - pattern p; apply Pind; clear p. - simpl; ring. - intros p Hrec. - rewrite nat_of_P_succ_morphism; simpl Qcpower. - rewrite <- Hrec. - unfold Qcmult, Q2Qc. - apply Qc_decomp; intros _ _; - unfold this. - apply Qred_complete. - assert (F: [x] ^ ' Psucc p == [x] * [x] ^ ' p). - simpl; generalize Hx; case x; simpl; clear x Hx Hrec. - intros x _; simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - intros nx dx. - match goal with |- context[BigN.eq_bool ?X ?Y] => - generalize (BigN.spec_eq_bool X Y); case BigN.eq_bool - end; auto; rewrite BigN.spec_0. - intros _ HH; case HH. - intros H1 _. - assert (F1: (0 < BigN.to_Z dx)%Z). - generalize (BigN.spec_pos dx); auto with zarith. - simpl; repeat rewrite Qpower_decomp; simpl. - red; simpl; repeat rewrite Zpower_pos_1_l; simpl Z2P. - rewrite Pplus_one_succ_l. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - repeat rewrite Zpos_mult_morphism. - repeat rewrite Z2P_correct; auto. - 2: apply Zpower_pos_pos; auto. - 2: apply Zpower_pos_pos; auto. - rewrite Zpower_pos_is_exp. - rewrite Zpower_pos_1_r; auto. - rewrite F. - apply Qmult_comp; apply Qeq_sym; apply Qred_correct. - Qed. - -End Qv. - |