diff options
Diffstat (limited to 'theories/Reals/Rfunctions.v')
| -rw-r--r-- | theories/Reals/Rfunctions.v | 113 |
1 files changed, 107 insertions, 6 deletions
diff --git a/theories/Reals/Rfunctions.v b/theories/Reals/Rfunctions.v index 0a49d498..77e53147 100644 --- a/theories/Reals/Rfunctions.v +++ b/theories/Reals/Rfunctions.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*i Some properties about pow and sum have been made with John Harrison i*) @@ -416,8 +418,9 @@ Proof. simpl; apply Rabs_R1. replace (S n) with (n + 1)%nat; [ rewrite pow_add | ring ]. rewrite Rabs_mult. - rewrite Hrecn; rewrite Rmult_1_l; simpl; rewrite Rmult_1_r; - rewrite Rabs_Ropp; apply Rabs_R1. + rewrite Hrecn; rewrite Rmult_1_l; simpl; rewrite Rmult_1_r. + change (-1) with (-(1)). + rewrite Rabs_Ropp; apply Rabs_R1. Qed. Lemma pow_mult : forall (x:R) (n1 n2:nat), x ^ (n1 * n2) = (x ^ n1) ^ n2. @@ -531,6 +534,36 @@ Qed. (*******************************) (*i Due to L.Thery i*) +Section PowerRZ. + +Local Coercion Z_of_nat : nat >-> Z. + +(* the following section should probably be somewhere else, but not sure where *) +Section Z_compl. + +Local Open Scope Z_scope. + +(* Provides a way to reason directly on Z in terms of nats instead of positive *) +Inductive Z_spec (x : Z) : Z -> Type := +| ZintNull : x = 0 -> Z_spec x 0 +| ZintPos (n : nat) : x = n -> Z_spec x n +| ZintNeg (n : nat) : x = - n -> Z_spec x (- n). + +Lemma intP (x : Z) : Z_spec x x. +Proof. + destruct x as [|p|p]. + - now apply ZintNull. + - rewrite <-positive_nat_Z at 2. + apply ZintPos. + now rewrite positive_nat_Z. + - rewrite <-Pos2Z.opp_pos. + rewrite <-positive_nat_Z at 2. + apply ZintNeg. + now rewrite positive_nat_Z. +Qed. + +End Z_compl. + Definition powerRZ (x:R) (n:Z) := match n with | Z0 => 1 @@ -657,6 +690,74 @@ Proof. auto with real. Qed. +Local Open Scope Z_scope. + +Lemma pow_powerRZ (r : R) (n : nat) : + (r ^ n)%R = powerRZ r (Z_of_nat n). +Proof. + induction n; [easy|simpl]. + now rewrite SuccNat2Pos.id_succ. +Qed. + +Lemma powerRZ_ind (P : Z -> R -> R -> Prop) : + (forall x, P 0 x 1%R) -> + (forall x n, P (Z.of_nat n) x (x ^ n)%R) -> + (forall x n, P ((-(Z.of_nat n))%Z) x (Rinv (x ^ n))) -> + forall x (m : Z), P m x (powerRZ x m)%R. +Proof. + intros ? ? ? x m. + destruct (intP m) as [Hm|n Hm|n Hm]. + - easy. + - now rewrite <- pow_powerRZ. + - unfold powerRZ. + destruct n as [|n]; [ easy |]. + rewrite Nat2Z.inj_succ, <- Zpos_P_of_succ_nat, Pos2Z.opp_pos. + now rewrite <- Pos2Z.opp_pos, <- positive_nat_Z. +Qed. + +Lemma powerRZ_inv x alpha : (x <> 0)%R -> powerRZ (/ x) alpha = Rinv (powerRZ x alpha). +Proof. + intros; destruct (intP alpha). + - now simpl; rewrite Rinv_1. + - now rewrite <-!pow_powerRZ, ?Rinv_pow, ?pow_powerRZ. + - unfold powerRZ. + destruct (- n). + + now rewrite Rinv_1. + + now rewrite Rinv_pow. + + now rewrite <-Rinv_pow. +Qed. + +Lemma powerRZ_neg x : forall alpha, x <> R0 -> powerRZ x (- alpha) = powerRZ (/ x) alpha. +Proof. + intros [|n|n] H ; simpl. + - easy. + - now rewrite Rinv_pow. + - rewrite Rinv_pow by now apply Rinv_neq_0_compat. + now rewrite Rinv_involutive. +Qed. + +Lemma powerRZ_mult_distr : + forall m x y, ((0 <= m)%Z \/ (x * y <> 0)%R) -> + (powerRZ (x*y) m = powerRZ x m * powerRZ y m)%R. +Proof. + intros m x0 y0 Hmxy. + destruct (intP m) as [ | | n Hm ]. + - now simpl; rewrite Rmult_1_l. + - now rewrite <- !pow_powerRZ, Rpow_mult_distr. + - destruct Hmxy as [H|H]. + + assert(m = 0) as -> by now omega. + now rewrite <- Hm, Rmult_1_l. + + assert(x0 <> 0)%R by now intros ->; apply H; rewrite Rmult_0_l. + assert(y0 <> 0)%R by now intros ->; apply H; rewrite Rmult_0_r. + rewrite !powerRZ_neg by assumption. + rewrite Rinv_mult_distr by assumption. + now rewrite <- !pow_powerRZ, Rpow_mult_distr. +Qed. + +End PowerRZ. + +Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope. + (*******************************) (* For easy interface *) (*******************************) |