diff options
author | letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2011-06-20 17:18:39 +0000 |
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committer | letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2011-06-20 17:18:39 +0000 |
commit | ca96d3477993d102d6cc42166eab52516630d181 (patch) | |
tree | 073b17efe149637da819caf527b23cf09f15865d /theories/Numbers | |
parent | ca1848177a50e51bde0736e51f506e06efc81b1d (diff) |
Arithemtic: more concerning compare, eqb, leb, ltb
Start of a uniform treatment of compare, eqb, leb, ltb:
- We now ensure that they are provided by N,Z,BigZ,BigN,Nat and Pos
- Some generic properties are derived in OrdersFacts.BoolOrderFacts
In BinPos, more work about sub_mask with nice implications
on compare (e.g. simplier proof of lt_trans).
In BinNat/BinPos, for uniformity, compare_antisym is now
(y ?= x) = CompOpp (x ?=y) instead of the symmetrical result.
In BigN / BigZ, eq_bool is now eqb
In BinIntDef, gtb and geb are kept for the moment, but
a comment advise to rather use ltb and leb. Z.div now uses
Z.ltb and Z.leb.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14227 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Numbers')
-rw-r--r-- | theories/Numbers/Integer/Abstract/ZAxioms.v | 10 | ||||
-rw-r--r-- | theories/Numbers/Integer/BigZ/BigZ.v | 9 | ||||
-rw-r--r-- | theories/Numbers/Integer/BigZ/ZMake.v | 72 | ||||
-rw-r--r-- | theories/Numbers/Integer/SpecViaZ/ZSig.v | 50 | ||||
-rw-r--r-- | theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v | 64 | ||||
-rw-r--r-- | theories/Numbers/Natural/Abstract/NAxioms.v | 4 | ||||
-rw-r--r-- | theories/Numbers/Natural/BigN/BigN.v | 9 | ||||
-rw-r--r-- | theories/Numbers/Natural/BigN/NMake.v | 60 | ||||
-rw-r--r-- | theories/Numbers/Natural/Peano/NPeano.v | 29 | ||||
-rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSig.v | 48 | ||||
-rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v | 62 | ||||
-rw-r--r-- | theories/Numbers/Rational/BigQ/QMake.v | 92 |
12 files changed, 332 insertions, 177 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v index 237ac93ef..a5be98aab 100644 --- a/theories/Numbers/Integer/Abstract/ZAxioms.v +++ b/theories/Numbers/Integer/Abstract/ZAxioms.v @@ -111,14 +111,12 @@ Module Type ZQuot' (Z:ZAxiomsMiniSig) := QuotRem' Z <+ QuotRemSpec Z. (** Let's group everything *) -Module Type ZAxiomsSig := - ZAxiomsMiniSig <+ HasCompare <+ HasEqBool <+ HasAbs <+ HasSgn - <+ NZParity.NZParity +Module Type ZAxiomsSig := ZAxiomsMiniSig <+ OrderFunctions + <+ HasAbs <+ HasSgn <+ NZParity.NZParity <+ NZPow.NZPow <+ NZSqrt.NZSqrt <+ NZLog.NZLog2 <+ NZGcd.NZGcd <+ ZDiv <+ ZQuot <+ NZBits.NZBits. -Module Type ZAxiomsSig' := - ZAxiomsMiniSig' <+ HasCompare <+ HasEqBool <+ HasAbs <+ HasSgn - <+ NZParity.NZParity +Module Type ZAxiomsSig' := ZAxiomsMiniSig' <+ OrderFunctions' + <+ HasAbs <+ HasSgn <+ NZParity.NZParity <+ NZPow.NZPow' <+ NZSqrt.NZSqrt' <+ NZLog.NZLog2 <+ NZGcd.NZGcd' <+ ZDiv' <+ ZQuot' <+ NZBits.NZBits'. diff --git a/theories/Numbers/Integer/BigZ/BigZ.v b/theories/Numbers/Integer/BigZ/BigZ.v index 236c56b9f..71d275601 100644 --- a/theories/Numbers/Integer/BigZ/BigZ.v +++ b/theories/Numbers/Integer/BigZ/BigZ.v @@ -56,6 +56,9 @@ Infix "*" := BigZ.mul : bigZ_scope. Infix "/" := BigZ.div : bigZ_scope. Infix "^" := BigZ.pow : bigZ_scope. Infix "?=" := BigZ.compare : bigZ_scope. +Infix "=?" := BigZ.eqb (at level 70, no associativity) : bigZ_scope. +Infix "<=?" := BigZ.leb (at level 70, no associativity) : bigZ_scope. +Infix "<?" := BigZ.ltb (at level 70, no associativity) : bigZ_scope. Infix "==" := BigZ.eq (at level 70, no associativity) : bigZ_scope. Notation "x != y" := (~x==y) (at level 70, no associativity) : bigZ_scope. Infix "<" := BigZ.lt : bigZ_scope. @@ -112,7 +115,7 @@ symmetry. apply BigZ.add_opp_r. exact BigZ.add_opp_diag_r. Qed. -Lemma BigZeqb_correct : forall x y, BigZ.eq_bool x y = true -> x==y. +Lemma BigZeqb_correct : forall x y, (x =? y) = true -> x==y. Proof. now apply BigZ.eqb_eq. Qed. Definition BigZ_of_N n := BigZ.of_Z (Z_of_N n). @@ -127,11 +130,11 @@ induction p; simpl; intros; BigZ.zify; rewrite ?IHp; auto. Qed. Lemma BigZdiv : div_theory BigZ.eq BigZ.add BigZ.mul (@id _) - (fun a b => if BigZ.eq_bool b 0 then (0,a) else BigZ.div_eucl a b). + (fun a b => if b =? 0 then (0,a) else BigZ.div_eucl a b). Proof. constructor. unfold id. intros a b. BigZ.zify. -generalize (Zeq_bool_if [b] 0); destruct (Zeq_bool [b] 0). +case Z.eqb_spec. BigZ.zify. auto with zarith. intros NEQ. generalize (BigZ.spec_div_eucl a b). diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v index da501e9ef..8e53e4d62 100644 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ b/theories/Numbers/Integer/BigZ/ZMake.v @@ -109,21 +109,49 @@ Module Make (N:NType) <: ZType. exfalso. omega. Qed. - Definition eq_bool x y := + Definition eqb x y := match compare x y with | Eq => true | _ => false end. - Theorem spec_eq_bool: - forall x y, eq_bool x y = Zeq_bool (to_Z x) (to_Z y). + Theorem spec_eqb x y : eqb x y = Z.eqb (to_Z x) (to_Z y). Proof. - unfold eq_bool, Zeq_bool; intros; rewrite spec_compare; reflexivity. + apply Bool.eq_iff_eq_true. + unfold eqb. rewrite Z.eqb_eq, <- Z.compare_eq_iff, spec_compare. + split; [now destruct Z.compare | now intros ->]. Qed. Definition lt n m := to_Z n < to_Z m. Definition le n m := to_Z n <= to_Z m. + + Definition ltb (x y : t) : bool := + match compare x y with + | Lt => true + | _ => false + end. + + Theorem spec_ltb x y : ltb x y = Z.ltb (to_Z x) (to_Z y). + Proof. + apply Bool.eq_iff_eq_true. + rewrite Z.ltb_lt. unfold Z.lt, ltb. rewrite spec_compare. + split; [now destruct Z.compare | now intros ->]. + Qed. + + Definition leb (x y : t) : bool := + match compare x y with + | Gt => false + | _ => true + end. + + Theorem spec_leb x y : leb x y = Z.leb (to_Z x) (to_Z y). + Proof. + apply Bool.eq_iff_eq_true. + rewrite Z.leb_le. unfold Z.le, leb. rewrite spec_compare. + destruct Z.compare; split; try easy. now destruct 1. + Qed. + 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. @@ -372,12 +400,12 @@ Module Make (N:NType) <: ZType. (Pos q, Pos r) | Pos nx, Neg ny => let (q, r) := N.div_eucl nx ny in - if N.eq_bool N.zero r + if N.eqb N.zero r then (Neg q, zero) else (Neg (N.succ q), Neg (N.sub ny r)) | Neg nx, Pos ny => let (q, r) := N.div_eucl nx ny in - if N.eq_bool N.zero r + if N.eqb N.zero r then (Neg q, zero) else (Neg (N.succ q), Pos (N.sub ny r)) | Neg nx, Neg ny => @@ -401,32 +429,32 @@ Module Make (N:NType) <: ZType. (* Pos Neg *) generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite N.spec_eq_bool, N.spec_0, Hr; + try (injection 1; intros Hr Hq; rewrite N.spec_eqb, N.spec_0, Hr; simpl; rewrite Hq, N.spec_0; auto). change (- Zpos py) with (Zneg py). assert (GT : Zpos py > 0) by (compute; auto). generalize (Z_div_mod (Zpos px) (Zpos py) GT). unfold Zdiv_eucl. destruct (Zdiv_eucl_POS px (Zpos py)) as (q',r'). intros (EQ,MOD). injection 1. intros Hr' Hq'. - rewrite N.spec_eq_bool, N.spec_0, Hr'. + rewrite N.spec_eqb, N.spec_0, Hr'. break_nonneg r pr EQr. subst; simpl. rewrite N.spec_0; auto. - subst. lazy iota beta delta [Zeq_bool Zcompare]. + subst. lazy iota beta delta [Z.eqb]. rewrite N.spec_sub, N.spec_succ, EQy, EQr. f_equal. omega with *. (* Neg Pos *) generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite N.spec_eq_bool, N.spec_0, Hr; + try (injection 1; intros Hr Hq; rewrite N.spec_eqb, N.spec_0, Hr; simpl; rewrite Hq, N.spec_0; auto). change (- Zpos px) with (Zneg px). assert (GT : Zpos py > 0) by (compute; auto). generalize (Z_div_mod (Zpos px) (Zpos py) GT). unfold Zdiv_eucl. destruct (Zdiv_eucl_POS px (Zpos py)) as (q',r'). intros (EQ,MOD). injection 1. intros Hr' Hq'. - rewrite N.spec_eq_bool, N.spec_0, Hr'. + rewrite N.spec_eqb, N.spec_0, Hr'. break_nonneg r pr EQr. subst; simpl. rewrite N.spec_0; auto. - subst. lazy iota beta delta [Zeq_bool Zcompare]. + subst. lazy iota beta delta [Z.eqb]. rewrite N.spec_sub, N.spec_succ, EQy, EQr. f_equal. omega with *. (* Neg Neg *) generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y) as (q,r). @@ -464,7 +492,7 @@ Module Make (N:NType) <: ZType. end. Definition rem x y := - if eq_bool y zero then x + if eqb y zero then x else match x, y with | Pos nx, Pos ny => Pos (N.modulo nx ny) @@ -483,8 +511,8 @@ Module Make (N:NType) <: ZType. Lemma spec_rem : forall x y, to_Z (rem x y) = Zrem (to_Z x) (to_Z y). Proof. - intros x y. unfold rem. rewrite spec_eq_bool, spec_0. - assert (Hy := Zeq_bool_if (to_Z y) 0). destruct Zeq_bool. + intros x y. unfold rem. rewrite spec_eqb, spec_0. + case Z.eqb_spec; intros Hy. now rewrite Hy, Zrem_0_r. destruct x as [x|x], y as [y|y]; simpl in *; symmetry; rewrite N.spec_modulo, ?Zrem_opp_r, ?Zrem_opp_l, ?Zopp_involutive; @@ -551,7 +579,7 @@ Module Make (N:NType) <: ZType. Definition norm_pos z := match z with | Pos _ => z - | Neg n => if N.eq_bool n N.zero then Pos n else z + | Neg n => if N.eqb n N.zero then Pos n else z end. Definition testbit a n := @@ -623,19 +651,17 @@ Module Make (N:NType) <: ZType. Lemma spec_norm_pos : forall x, to_Z (norm_pos x) = to_Z x. Proof. intros [x|x]; simpl; trivial. - rewrite N.spec_eq_bool, N.spec_0. - assert (H := Zeq_bool_if (N.to_Z x) 0). - destruct Zeq_bool; simpl; auto with zarith. + rewrite N.spec_eqb, N.spec_0. + case Z.eqb_spec; simpl; auto with zarith. Qed. Lemma spec_norm_pos_pos : forall x y, norm_pos x = Neg y -> 0 < N.to_Z y. Proof. intros [x|x] y; simpl; try easy. - rewrite N.spec_eq_bool, N.spec_0. - assert (H := Zeq_bool_if (N.to_Z x) 0). - destruct Zeq_bool; simpl; try easy. - inversion 1; subst. generalize (N.spec_pos y); auto with zarith. + rewrite N.spec_eqb, N.spec_0. + case Z.eqb_spec; simpl; try easy. + inversion 2. subst. generalize (N.spec_pos y); auto with zarith. Qed. Ltac destr_norm_pos x := diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v index 9981fab71..0f2862df7 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSig.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSig.v @@ -8,7 +8,7 @@ (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) -Require Import ZArith Znumtheory. +Require Import BinInt. Open Scope Z_scope. @@ -33,7 +33,9 @@ Module Type ZType. Parameter spec_of_Z: forall x, to_Z (of_Z x) = x. Parameter compare : t -> t -> comparison. - Parameter eq_bool : t -> t -> bool. + Parameter eqb : t -> t -> bool. + Parameter ltb : t -> t -> bool. + Parameter leb : t -> t -> bool. Parameter min : t -> t -> t. Parameter max : t -> t -> t. Parameter zero : t. @@ -71,10 +73,12 @@ Module Type ZType. Parameter lxor : t -> t -> t. Parameter div2 : t -> t. - Parameter spec_compare: forall x y, compare x y = Zcompare [x] [y]. - Parameter spec_eq_bool: forall x y, eq_bool x y = Zeq_bool [x] [y]. - Parameter spec_min : forall x y, [min x y] = Zmin [x] [y]. - Parameter spec_max : forall x y, [max x y] = Zmax [x] [y]. + Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]). + Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]). + Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]). + Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]). + Parameter spec_min : forall x y, [min x y] = Z.min [x] [y]. + Parameter spec_max : forall x y, [max x y] = Z.max [x] [y]. Parameter spec_0: [zero] = 0. Parameter spec_1: [one] = 1. Parameter spec_2: [two] = 2. @@ -89,27 +93,27 @@ Module Type ZType. Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n. Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z_of_N n. Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n]. - Parameter spec_sqrt: forall x, [sqrt x] = Zsqrt [x]. - Parameter spec_log2: forall x, [log2 x] = Zlog2 [x]. + Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x]. + Parameter spec_log2: forall x, [log2 x] = Z.log2 [x]. Parameter spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y]. + let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y]. Parameter spec_div: forall x y, [div x y] = [x] / [y]. Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y]. Parameter spec_quot: forall x y, [quot x y] = [x] รท [y]. - Parameter spec_rem: forall x y, [rem x y] = Zrem [x] [y]. - Parameter spec_gcd: forall a b, [gcd a b] = Zgcd [a] [b]. - Parameter spec_sgn : forall x, [sgn x] = Zsgn [x]. - Parameter spec_abs : forall x, [abs x] = Zabs [x]. - Parameter spec_even : forall x, even x = Zeven_bool [x]. - Parameter spec_odd : forall x, odd x = Zodd_bool [x]. - Parameter spec_testbit: forall x p, testbit x p = Ztestbit [x] [p]. - Parameter spec_shiftr: forall x p, [shiftr x p] = Zshiftr [x] [p]. - Parameter spec_shiftl: forall x p, [shiftl x p] = Zshiftl [x] [p]. - Parameter spec_land: forall x y, [land x y] = Zand [x] [y]. - Parameter spec_lor: forall x y, [lor x y] = Zor [x] [y]. - Parameter spec_ldiff: forall x y, [ldiff x y] = Zdiff [x] [y]. - Parameter spec_lxor: forall x y, [lxor x y] = Zxor [x] [y]. - Parameter spec_div2: forall x, [div2 x] = Zdiv2 [x]. + Parameter spec_rem: forall x y, [rem x y] = Z.rem [x] [y]. + Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. + Parameter spec_sgn : forall x, [sgn x] = Z.sgn [x]. + Parameter spec_abs : forall x, [abs x] = Z.abs [x]. + Parameter spec_even : forall x, even x = Z.even [x]. + Parameter spec_odd : forall x, odd x = Z.odd [x]. + Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p]. + Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p]. + Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. + Parameter spec_land: forall x y, [land x y] = Z.land [x] [y]. + Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y]. + Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y]. + Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y]. + Parameter spec_div2: forall x, [div2 x] = Z.div2 [x]. End ZType. diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v index a055007f4..e3fc512e7 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -Require Import Bool ZArith Nnat ZAxioms ZSig. +Require Import Bool ZArith OrdersFacts Nnat ZAxioms ZSig. (** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig] *) @@ -15,9 +15,9 @@ Module ZTypeIsZAxioms (Import Z : ZType'). Hint Rewrite spec_0 spec_1 spec_2 spec_add spec_sub spec_pred spec_succ spec_mul spec_opp spec_of_Z spec_div spec_modulo spec_sqrt - spec_compare spec_eq_bool spec_max spec_min spec_abs spec_sgn - spec_pow spec_log2 spec_even spec_odd spec_gcd spec_quot spec_rem - spec_testbit spec_shiftl spec_shiftr + spec_compare spec_eqb spec_ltb spec_leb spec_max spec_min + spec_abs spec_sgn spec_pow spec_log2 spec_even spec_odd spec_gcd + spec_quot spec_rem spec_testbit spec_shiftl spec_shiftr spec_land spec_lor spec_ldiff spec_lxor spec_div2 : zsimpl. @@ -138,36 +138,66 @@ Qed. (** Order *) -Lemma compare_spec : forall x y, CompareSpec (x==y) (x<y) (y<x) (compare x y). +Lemma eqb_eq x y : eqb x y = true <-> x == y. Proof. - intros. zify. destruct (Zcompare_spec [x] [y]); auto. + zify. apply Z.eqb_eq. Qed. -Definition eqb := eq_bool. +Lemma leb_le x y : leb x y = true <-> x <= y. +Proof. + zify. apply Z.leb_le. +Qed. + +Lemma ltb_lt x y : ltb x y = true <-> x < y. +Proof. + zify. apply Z.ltb_lt. +Qed. + +Lemma compare_eq_iff n m : compare n m = Eq <-> n == m. +Proof. + intros. zify. apply Z.compare_eq_iff. +Qed. + +Lemma compare_lt_iff n m : compare n m = Lt <-> n < m. +Proof. + intros. zify. reflexivity. +Qed. + +Lemma compare_le_iff n m : compare n m <> Gt <-> n <= m. +Proof. + intros. zify. reflexivity. +Qed. -Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y. +Lemma compare_antisym n m : compare m n = CompOpp (compare n m). Proof. - intros. zify. symmetry. apply Zeq_is_eq_bool. + intros. zify. apply Z.compare_antisym. Qed. +Include BoolOrderFacts Z Z Z [no inline]. + Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare. Proof. -intros x x' Hx y y' Hy. rewrite 2 spec_compare, Hx, Hy; intuition. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. Qed. -Instance lt_wd : Proper (eq ==> eq ==> iff) lt. +Instance eqb_wd : Proper (eq ==> eq ==> Logic.eq) eqb. Proof. -intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. Qed. -Theorem lt_eq_cases : forall n m, n <= m <-> n < m \/ n == m. +Instance ltb_wd : Proper (eq ==> eq ==> Logic.eq) ltb. Proof. -intros. zify. omega. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. Qed. -Theorem lt_irrefl : forall n, ~ n < n. +Instance leb_wd : Proper (eq ==> eq ==> Logic.eq) leb. Proof. -intros. zify. omega. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. +Qed. + +Instance lt_wd : Proper (eq ==> eq ==> iff) lt. +Proof. +intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition. Qed. Theorem lt_succ_r : forall n m, n < (succ m) <-> n <= m. @@ -491,5 +521,5 @@ Qed. End ZTypeIsZAxioms. Module ZType_ZAxioms (Z : ZType) - <: ZAxiomsSig <: HasCompare Z <: HasEqBool Z <: HasMinMax Z + <: ZAxiomsSig <: OrderFunctions Z <: HasMinMax Z := Z <+ ZTypeIsZAxioms. diff --git a/theories/Numbers/Natural/Abstract/NAxioms.v b/theories/Numbers/Natural/Abstract/NAxioms.v index 09438628d..45a2cf3e1 100644 --- a/theories/Numbers/Natural/Abstract/NAxioms.v +++ b/theories/Numbers/Natural/Abstract/NAxioms.v @@ -32,11 +32,11 @@ End NDivSpecific. (** We now group everything together. *) -Module Type NAxiomsSig := NAxiomsMiniSig <+ HasCompare <+ HasEqBool +Module Type NAxiomsSig := NAxiomsMiniSig <+ OrderFunctions <+ NZParity.NZParity <+ NZPow.NZPow <+ NZSqrt.NZSqrt <+ NZLog.NZLog2 <+ NZGcd.NZGcd <+ NZDiv.NZDiv <+ NZBits.NZBits. -Module Type NAxiomsSig' := NAxiomsMiniSig' <+ HasCompare <+ HasEqBool +Module Type NAxiomsSig' := NAxiomsMiniSig' <+ OrderFunctions' <+ NZParity.NZParity <+ NZPow.NZPow' <+ NZSqrt.NZSqrt' <+ NZLog.NZLog2 <+ NZGcd.NZGcd' <+ NZDiv.NZDiv' <+ NZBits.NZBits'. diff --git a/theories/Numbers/Natural/BigN/BigN.v b/theories/Numbers/Natural/BigN/BigN.v index d2c93bbfd..b06e42ca2 100644 --- a/theories/Numbers/Natural/BigN/BigN.v +++ b/theories/Numbers/Natural/BigN/BigN.v @@ -62,6 +62,9 @@ Infix "*" := BigN.mul : bigN_scope. Infix "/" := BigN.div : bigN_scope. Infix "^" := BigN.pow : bigN_scope. Infix "?=" := BigN.compare : bigN_scope. +Infix "=?" := BigN.eqb (at level 70, no associativity) : bigN_scope. +Infix "<=?" := BigN.leb (at level 70, no associativity) : bigN_scope. +Infix "<?" := BigN.ltb (at level 70, no associativity) : bigN_scope. Infix "==" := BigN.eq (at level 70, no associativity) : bigN_scope. Notation "x != y" := (~x==y) (at level 70, no associativity) : bigN_scope. Infix "<" := BigN.lt : bigN_scope. @@ -94,7 +97,7 @@ exact BigN.mul_1_l. exact BigN.mul_0_l. exact BigN.mul_comm. exact BigN.mul_assoc. exact BigN.mul_add_distr_r. Qed. -Lemma BigNeqb_correct : forall x y, BigN.eq_bool x y = true -> x==y. +Lemma BigNeqb_correct : forall x y, (x =? y) = true -> x==y. Proof. now apply BigN.eqb_eq. Qed. Lemma BigNpower : power_theory 1 BigN.mul BigN.eq BigN.of_N BigN.pow. @@ -107,11 +110,11 @@ induction p; simpl; intros; BigN.zify; rewrite ?IHp; auto. Qed. Lemma BigNdiv : div_theory BigN.eq BigN.add BigN.mul (@id _) - (fun a b => if BigN.eq_bool b 0 then (0,a) else BigN.div_eucl a b). + (fun a b => if b =? 0 then (0,a) else BigN.div_eucl a b). Proof. constructor. unfold id. intros a b. BigN.zify. -generalize (Zeq_bool_if [b] 0); destruct (Zeq_bool [b] 0). +case Z.eqb_spec. BigN.zify. auto with zarith. intros NEQ. generalize (BigN.spec_div_eucl a b). diff --git a/theories/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v index 23cfec5e9..aabbf87f2 100644 --- a/theories/Numbers/Natural/BigN/NMake.v +++ b/theories/Numbers/Natural/BigN/NMake.v @@ -301,20 +301,48 @@ Module Make (W0:CyclicType) <: NType. intros. subst. apply Zcompare_antisym. Qed. - Definition eq_bool (x y : t) : bool := + Definition eqb (x y : t) : bool := match compare x y with | Eq => true | _ => false end. - Theorem spec_eq_bool : forall x y, eq_bool x y = Zeq_bool [x] [y]. + Theorem spec_eqb x y : eqb x y = Z.eqb [x] [y]. Proof. - intros. unfold eq_bool, Zeq_bool. rewrite spec_compare; reflexivity. + apply eq_iff_eq_true. + unfold eqb. rewrite Z.eqb_eq, <- Z.compare_eq_iff, spec_compare. + split; [now destruct Z.compare | now intros ->]. Qed. Definition lt (n m : t) := [n] < [m]. Definition le (n m : t) := [n] <= [m]. + Definition ltb (x y : t) : bool := + match compare x y with + | Lt => true + | _ => false + end. + + Theorem spec_ltb x y : ltb x y = Z.ltb [x] [y]. + Proof. + apply eq_iff_eq_true. + rewrite Z.ltb_lt. unfold Z.lt, ltb. rewrite spec_compare. + split; [now destruct Z.compare | now intros ->]. + Qed. + + Definition leb (x y : t) : bool := + match compare x y with + | Gt => false + | _ => true + end. + + Theorem spec_leb x y : leb x y = Z.leb [x] [y]. + Proof. + apply eq_iff_eq_true. + rewrite Z.leb_le. unfold Z.le, leb. rewrite spec_compare. + destruct Z.compare; split; try easy. now destruct 1. + Qed. + Definition min (n m : t) : t := match compare n m with Gt => m | _ => n end. Definition max (n m : t) : t := match compare n m with Lt => m | _ => n end. @@ -560,7 +588,7 @@ Module Make (W0:CyclicType) <: NType. (** * General Division *) Definition div_eucl (x y : t) : t * t := - if eq_bool y zero then (zero,zero) else + if eqb y zero then (zero,zero) else match compare x y with | Eq => (one, zero) | Lt => (zero, x) @@ -572,8 +600,8 @@ Module Make (W0:CyclicType) <: NType. ([q], [r]) = Zdiv_eucl [x] [y]. Proof. intros x y. unfold div_eucl. - rewrite spec_eq_bool, spec_compare, spec_0. - generalize (Zeq_bool_if [y] 0); case Zeq_bool. + rewrite spec_eqb, spec_compare, spec_0. + case Z.eqb_spec. intros ->. rewrite spec_0. destruct [x]; auto. intros H'. assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith). @@ -685,7 +713,7 @@ Module Make (W0:CyclicType) <: NType. (** * General Modulo *) Definition modulo (x y : t) : t := - if eq_bool y zero then zero else + if eqb y zero then zero else match compare x y with | Eq => zero | Lt => x @@ -696,8 +724,8 @@ Module Make (W0:CyclicType) <: NType. forall x y, [modulo x y] = [x] mod [y]. Proof. intros x y. unfold modulo. - rewrite spec_eq_bool, spec_compare, spec_0. - generalize (Zeq_bool_if [y] 0). case Zeq_bool. + rewrite spec_eqb, spec_compare, spec_0. + case Z.eqb_spec. intros ->; rewrite spec_0. destruct [x]; auto. intro H'. assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith). @@ -1157,16 +1185,16 @@ Module Make (W0:CyclicType) <: NType. Definition log2 : t -> t := Eval red_t in let log2 := iter_t log2n in - fun x => if eq_bool x zero then zero else log2 x. + fun x => if eqb x zero then zero else log2 x. Lemma log2_fold : - log2 = fun x => if eq_bool x zero then zero else iter_t log2n x. + log2 = fun x => if eqb x zero then zero else iter_t log2n x. Proof. red_t; reflexivity. Qed. Lemma spec_log2_0 : forall x, [x] = 0 -> [log2 x] = 0. Proof. intros x H. rewrite log2_fold. - rewrite spec_eq_bool, H. rewrite spec_0. simpl. exact spec_0. + rewrite spec_eqb, H. rewrite spec_0. simpl. exact spec_0. Qed. Lemma head0_zdigits : forall n (x : dom_t n), @@ -1193,8 +1221,8 @@ Module Make (W0:CyclicType) <: NType. 2^[log2 x] <= [x] < 2^([log2 x]+1). Proof. intros x H. rewrite log2_fold. - rewrite spec_eq_bool. rewrite spec_0. - generalize (Zeq_bool_if [x] 0). destruct Zeq_bool. + rewrite spec_eqb. rewrite spec_0. + case Z.eqb_spec. auto with zarith. clear H. destr_t x as (n,x). intros H. @@ -1229,8 +1257,8 @@ Module Make (W0:CyclicType) <: NType. [log2 x] = Zpos (digits x) - [head0 x] - 1. Proof. intros. rewrite log2_fold. - rewrite spec_eq_bool. rewrite spec_0. - generalize (Zeq_bool_if [x] 0). destruct Zeq_bool. + rewrite spec_eqb. rewrite spec_0. + case Z.eqb_spec. auto with zarith. intros _. revert H. rewrite digits_fold, head0_fold. destr_t x as (n,x). rewrite ZnZ.spec_sub_carry. diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index 0cf9ae441..8a26ec6e3 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.v @@ -14,6 +14,31 @@ Require Import (** Functions not already defined *) +Fixpoint leb n m := + match n, m with + | O, _ => true + | _, O => false + | S n', S m' => leb n' m' + end. + +Definition ltb n m := leb (S n) m. + +Infix "<=?" := leb (at level 70) : nat_scope. +Infix "<?" := ltb (at level 70) : nat_scope. + +Lemma leb_le n m : (n <=? m) = true <-> n <= m. +Proof. + revert m. + induction n. split; auto with arith. + destruct m; simpl. now split. + rewrite IHn. split; auto with arith. +Qed. + +Lemma ltb_lt n m : (n <? m) = true <-> n < m. +Proof. + unfold ltb, lt. apply leb_le. +Qed. + Fixpoint pow n m := match m with | O => 1 @@ -681,6 +706,8 @@ Definition sub := minus. Definition mul := mult. Definition lt := lt. Definition le := le. +Definition ltb := ltb. +Definition leb := leb. Definition min := min. Definition max := max. @@ -692,6 +719,8 @@ Definition min_r := min_r. Definition eqb_eq := beq_nat_true_iff. Definition compare_spec := nat_compare_spec. Definition eq_dec := eq_nat_dec. +Definition leb_le := leb_le. +Definition ltb_lt := ltb_lt. Definition Even := Even. Definition Odd := Odd. diff --git a/theories/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v index f186c55b4..662648432 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSig.v +++ b/theories/Numbers/Natural/SpecViaZ/NSig.v @@ -8,7 +8,7 @@ (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) -Require Import ZArith Znumtheory. +Require Import BinInt. Open Scope Z_scope. @@ -35,7 +35,9 @@ Module Type NType. Definition le n m := [n] <= [m]. Parameter compare : t -> t -> comparison. - Parameter eq_bool : t -> t -> bool. + Parameter eqb : t -> t -> bool. + Parameter ltb : t -> t -> bool. + Parameter leb : t -> t -> bool. Parameter max : t -> t -> t. Parameter min : t -> t -> t. Parameter zero : t. @@ -67,39 +69,41 @@ Module Type NType. Parameter lxor : t -> t -> t. Parameter div2 : t -> t. - Parameter spec_compare: forall x y, compare x y = Zcompare [x] [y]. - Parameter spec_eq_bool: forall x y, eq_bool x y = Zeq_bool [x] [y]. - Parameter spec_max : forall x y, [max x y] = Zmax [x] [y]. - Parameter spec_min : forall x y, [min x y] = Zmin [x] [y]. + Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]). + Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]). + Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]). + Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]). + Parameter spec_max : forall x y, [max x y] = Z.max [x] [y]. + Parameter spec_min : forall x y, [min x y] = Z.min [x] [y]. Parameter spec_0: [zero] = 0. Parameter spec_1: [one] = 1. Parameter spec_2: [two] = 2. Parameter spec_succ: forall n, [succ n] = [n] + 1. Parameter spec_add: forall x y, [add x y] = [x] + [y]. - Parameter spec_pred: forall x, [pred x] = Zmax 0 ([x] - 1). - Parameter spec_sub: forall x y, [sub x y] = Zmax 0 ([x] - [y]). + Parameter spec_pred: forall x, [pred x] = Z.max 0 ([x] - 1). + Parameter spec_sub: forall x y, [sub x y] = Z.max 0 ([x] - [y]). Parameter spec_mul: forall x y, [mul x y] = [x] * [y]. Parameter spec_square: forall x, [square x] = [x] * [x]. Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n. Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z_of_N n. Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n]. - Parameter spec_sqrt: forall x, [sqrt x] = Zsqrt [x]. - Parameter spec_log2: forall x, [log2 x] = Zlog2 [x]. + Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x]. + Parameter spec_log2: forall x, [log2 x] = Z.log2 [x]. Parameter spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y]. + let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y]. Parameter spec_div: forall x y, [div x y] = [x] / [y]. Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y]. - Parameter spec_gcd: forall a b, [gcd a b] = Zgcd [a] [b]. - Parameter spec_even: forall x, even x = Zeven_bool [x]. - Parameter spec_odd: forall x, odd x = Zodd_bool [x]. - Parameter spec_testbit: forall x p, testbit x p = Ztestbit [x] [p]. - Parameter spec_shiftr: forall x p, [shiftr x p] = Zshiftr [x] [p]. - Parameter spec_shiftl: forall x p, [shiftl x p] = Zshiftl [x] [p]. - Parameter spec_land: forall x y, [land x y] = Zand [x] [y]. - Parameter spec_lor: forall x y, [lor x y] = Zor [x] [y]. - Parameter spec_ldiff: forall x y, [ldiff x y] = Zdiff [x] [y]. - Parameter spec_lxor: forall x y, [lxor x y] = Zxor [x] [y]. - Parameter spec_div2: forall x, [div2 x] = Zdiv2 [x]. + Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. + Parameter spec_even: forall x, even x = Z.even [x]. + Parameter spec_odd: forall x, odd x = Z.odd [x]. + Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p]. + Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p]. + Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. + Parameter spec_land: forall x y, [land x y] = Z.land [x] [y]. + Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y]. + Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y]. + Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y]. + Parameter spec_div2: forall x, [div2 x] = Z.div2 [x]. End NType. diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 84682820d..878e46fea 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v @@ -7,7 +7,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -Require Import ZArith Nnat Ndigits NAxioms NDiv NSig. +Require Import ZArith OrdersFacts Nnat Ndigits NAxioms NDiv NSig. (** * The interface [NSig.NType] implies the interface [NAxiomsSig] *) @@ -15,8 +15,8 @@ Module NTypeIsNAxioms (Import N : NType'). Hint Rewrite spec_0 spec_1 spec_2 spec_succ spec_add spec_mul spec_pred spec_sub - spec_div spec_modulo spec_gcd spec_compare spec_eq_bool spec_sqrt - spec_log2 spec_max spec_min spec_pow_pos spec_pow_N spec_pow + spec_div spec_modulo spec_gcd spec_compare spec_eqb spec_ltb spec_leb + spec_sqrt spec_log2 spec_max spec_min spec_pow_pos spec_pow_N spec_pow spec_even spec_odd spec_testbit spec_shiftl spec_shiftr spec_land spec_lor spec_ldiff spec_lxor spec_div2 spec_of_N : nsimpl. @@ -126,36 +126,66 @@ Qed. (** Order *) -Lemma compare_spec : forall x y, CompareSpec (x==y) (x<y) (y<x) (compare x y). +Lemma eqb_eq x y : eqb x y = true <-> x == y. Proof. - intros. zify. destruct (Zcompare_spec [x] [y]); auto. + zify. apply Z.eqb_eq. Qed. -Definition eqb := eq_bool. +Lemma leb_le x y : leb x y = true <-> x <= y. +Proof. + zify. apply Z.leb_le. +Qed. + +Lemma ltb_lt x y : ltb x y = true <-> x < y. +Proof. + zify. apply Z.ltb_lt. +Qed. + +Lemma compare_eq_iff n m : compare n m = Eq <-> n == m. +Proof. + intros. zify. apply Z.compare_eq_iff. +Qed. + +Lemma compare_lt_iff n m : compare n m = Lt <-> n < m. +Proof. + intros. zify. reflexivity. +Qed. + +Lemma compare_le_iff n m : compare n m <> Gt <-> n <= m. +Proof. + intros. zify. reflexivity. +Qed. -Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y. +Lemma compare_antisym n m : compare m n = CompOpp (compare n m). Proof. - intros. zify. symmetry. apply Zeq_is_eq_bool. + intros. zify. apply Z.compare_antisym. Qed. +Include BoolOrderFacts N N N [no inline]. + Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare. Proof. -intros x x' Hx y y' Hy. rewrite 2 spec_compare, Hx, Hy; intuition. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. Qed. -Instance lt_wd : Proper (eq ==> eq ==> iff) lt. +Instance eqb_wd : Proper (eq ==> eq ==> Logic.eq) eqb. Proof. -intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. Qed. -Theorem lt_eq_cases : forall n m, n <= m <-> n < m \/ n == m. +Instance ltb_wd : Proper (eq ==> eq ==> Logic.eq) ltb. Proof. -intros. zify. omega. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. Qed. -Theorem lt_irrefl : forall n, ~ n < n. +Instance leb_wd : Proper (eq ==> eq ==> Logic.eq) leb. Proof. -intros. zify. omega. +intros x x' Hx y y' Hy. zify. now rewrite Hx, Hy. +Qed. + +Instance lt_wd : Proper (eq ==> eq ==> iff) lt. +Proof. +intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition. Qed. Theorem lt_succ_r : forall n m, n < (succ m) <-> n <= m. @@ -474,5 +504,5 @@ Qed. End NTypeIsNAxioms. Module NType_NAxioms (N : NType) - <: NAxiomsSig <: HasCompare N <: HasEqBool N <: HasMinMax N + <: NAxiomsSig <: OrderFunctions N <: HasMinMax N := N <+ NTypeIsNAxioms. diff --git a/theories/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v index 75a548ca9..d4db2c66d 100644 --- a/theories/Numbers/Rational/BigQ/QMake.v +++ b/theories/Numbers/Rational/BigQ/QMake.v @@ -57,7 +57,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. 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 + | Qq x y => if N.eqb y N.zero then 0 else Z.to_Z x # Z2P (N.to_Z y) end. @@ -79,15 +79,13 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. *) Ltac destr_eqb := match goal with - | |- context [Z.eq_bool ?x ?y] => - rewrite (Z.spec_eq_bool x y); - generalize (Zeq_bool_if (Z.to_Z x) (Z.to_Z y)); - case (Zeq_bool (Z.to_Z x) (Z.to_Z y)); + | |- context [Z.eqb ?x ?y] => + rewrite (Z.spec_eqb x y); + case (Z.eqb_spec (Z.to_Z x) (Z.to_Z y)); destr_eqb - | |- context [N.eq_bool ?x ?y] => - rewrite (N.spec_eq_bool x y); - generalize (Zeq_bool_if (N.to_Z x) (N.to_Z y)); - case (Zeq_bool (N.to_Z x) (N.to_Z y)); + | |- context [N.eqb ?x ?y] => + rewrite (N.spec_eqb x y); + case (Z.eqb_spec (N.to_Z x) (N.to_Z y)); [ | let H:=fresh "H" in try (intro H;generalize (N_to_Z_pos _ H); clear H)]; destr_eqb @@ -145,13 +143,13 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. 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 + if N.eqb 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 + if N.eqb 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 + match N.eqb dx N.zero, N.eqb dy N.zero with | true, true => Eq | true, false => Z.compare Z.zero ny | false, true => Z.compare nx Z.zero @@ -301,15 +299,15 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. match y with | Qz zy => Qz (Z.add zx zy) | Qq ny dy => - if N.eq_bool dy N.zero then x + if N.eqb dy N.zero then x else Qq (Z.add (Z.mul zx (Z_of_N dy)) ny) dy end | Qq nx dx => - if N.eq_bool dx N.zero then y + if N.eqb 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 + if N.eqb 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 @@ -333,15 +331,15 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. match y with | Qz zy => Qz (Z.add zx zy) | Qq ny dy => - if N.eq_bool dy N.zero then x + if N.eqb dy N.zero then x else norm (Z.add (Z.mul zx (Z_of_N dy)) ny) dy end | Qq nx dx => - if N.eq_bool dx N.zero then y + if N.eqb 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 + if N.eqb 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 @@ -378,8 +376,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Proof. intros [z | x y]; simpl. rewrite Z.spec_opp; auto. - match goal with |- context[N.eq_bool ?X ?Y] => - generalize (N.spec_eq_bool X Y); case N.eq_bool + match goal with |- context[N.eqb ?X ?Y] => + generalize (N.spec_eqb X Y); case N.eqb end; auto; rewrite N.spec_0. rewrite Z.spec_opp; auto. Qed. @@ -429,26 +427,29 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. | Qq nx dx, Qq ny dy => Qq (Z.mul nx ny) (N.mul dx dy) end. + Ltac nsubst := + match goal with E : N.to_Z _ = _ |- _ => rewrite E in * 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, Z2P_correct; auto. destruct (Zmult_integral (N.to_Z dx) (N.to_Z dy)); intuition. - rewrite H0 in H1; auto with zarith. - rewrite H0 in H1; auto with zarith. - rewrite H in H1; nzsimpl; auto with zarith. + nsubst; auto with zarith. + nsubst; auto with zarith. + nsubst; nzsimpl; auto with zarith. rewrite Zpos_mult_morphism, 2 Z2P_correct; auto. Qed. Definition norm_denum n d := - if N.eq_bool d N.one then Qz n else Qq n d. + if N.eqb d N.one then Qz n else Qq n d. Lemma spec_norm_denum : forall n d, [norm_denum n d] == [Qq n d]. Proof. unfold norm_denum; intros; simpl; qsimpl. congruence. - rewrite H0 in *; auto with zarith. + nsubst; auto with zarith. Qed. Definition irred n d := @@ -528,7 +529,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Qed. Definition mul_norm_Qz_Qq z n d := - if Z.eq_bool z Z.zero then zero + if Z.eqb z Z.zero then zero else let gcd := N.gcd (Zabs_N z) d in match N.compare gcd N.one with @@ -561,7 +562,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite spec_norm_denum. qsimpl. rewrite Zdiv_gcd_zero in GT; auto with zarith. - rewrite H in *. rewrite Zdiv_0_l in *; discriminate. + nsubst. rewrite Zdiv_0_l in *; discriminate. rewrite <- Zmult_assoc, (Zmult_comm (Z.to_Z n)), Zmult_assoc. rewrite Zgcd_div_swap0; try romega. ring. @@ -637,13 +638,15 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite spec_norm_denum. qsimpl. - destruct (Zmult_integral _ _ H0) as [Eq|Eq]. + match goal with E : (_ * _ = 0)%Z |- _ => + destruct (Zmult_integral _ _ E) as [Eq|Eq] end. rewrite Eq in *; simpl in *. rewrite <- Hg2' in *; auto with zarith. rewrite Eq in *; simpl in *. rewrite <- Hg2 in *; auto with zarith. - destruct (Zmult_integral _ _ H) as [Eq|Eq]. + match goal with E : (_ * _ = 0)%Z |- _ => + destruct (Zmult_integral _ _ E) as [Eq|Eq] end. rewrite Hz' in Eq; rewrite Eq in *; auto with zarith. rewrite Hz in Eq; rewrite Eq in *; auto with zarith. @@ -691,13 +694,13 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. rewrite Zgcd_1_rel_prime in *. apply bezout_rel_prime. - destruct (rel_prime_bezout _ _ H4) as [u v Huv]. + destruct (rel_prime_bezout (Z.to_Z ny) (N.to_Z dy)) as [u v Huv]; trivial. 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]. + destruct (rel_prime_bezout (Z.to_Z nx) (N.to_Z dx)) as [u v Huv]; trivial. apply Bezout_intro with (u*g)%Z (v*g')%Z. rewrite <- Huv, <- Hg2', <- Hg1. ring. Qed. @@ -755,10 +758,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destr_eqb; nzsimpl; intros. intros; rewrite Zabs_eq in *; romega. intros; rewrite Zabs_eq in *; romega. - clear H1. - rewrite H0. - compute; auto. - clear H1. + nsubst; compute; auto. set (n':=Z.to_Z n) in *; clearbody n'. rewrite Zabs_eq by romega. red; simpl. @@ -770,9 +770,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destr_eqb; 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. + nsubst; compute; auto. set (n':=Z.to_Z n) in *; clearbody n'. red; simpl; nzsimpl. rewrite Zabs_non_eq by romega. @@ -791,7 +789,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. | Gt => Qq Z.minus_one (Zabs_N z) end | Qq n d => - if N.eq_bool d N.zero then zero else + if N.eqb d N.zero then zero else match Z.compare Z.zero n with | Eq => zero | Lt => @@ -928,9 +926,9 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destr_eqb; nzsimpl; intros. apply Qeq_refl. rewrite N.spec_square in *; nzsimpl. - elim (Zmult_integral _ _ H0); romega. - rewrite N.spec_square in *; nzsimpl. - rewrite H in H0; romega. + match goal with E : (_ * _ = 0)%Z |- _ => + elim (Zmult_integral _ _ E); romega end. + rewrite N.spec_square in *; nzsimpl; nsubst; romega. rewrite Z.spec_square, N.spec_square. red; simpl. rewrite Zpos_mult_morphism; rewrite !Z2P_correct; auto. @@ -961,8 +959,9 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. assert (0 < N.to_Z d ^ ' p)%Z by (apply Zpower_gt_0; auto with zarith). romega. - rewrite N.spec_pow_pos, H in *. - rewrite Zpower_0_l in H0; [romega|discriminate]. + exfalso. + rewrite N.spec_pow_pos in *. nsubst. + rewrite Zpower_0_l in *; [romega|discriminate]. rewrite Qpower_decomp. red; simpl; do 3 f_equal. rewrite Z2P_correct by (generalize (N.spec_pos d); romega). @@ -981,8 +980,9 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. revert H. unfold Reduced; rewrite strong_spec_red, Qred_iff; simpl. destr_eqb; nzsimpl; simpl; intros. - rewrite N.spec_pow_pos in H0. - rewrite H, Zpower_0_l in *; [romega|discriminate]. + exfalso. + rewrite N.spec_pow_pos in *. nsubst. + rewrite Zpower_0_l in *; [romega|discriminate]. rewrite Z2P_correct in *; auto. rewrite N.spec_pow_pos, Z.spec_pow_pos; auto. rewrite Zgcd_1_rel_prime in *. |