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diff --git a/theories7/Reals/Rfunctions.v b/theories7/Reals/Rfunctions.v deleted file mode 100644 index fe6ccd96..00000000 --- a/theories7/Reals/Rfunctions.v +++ /dev/null @@ -1,832 +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 *) -(************************************************************************) - -(*i $Id: Rfunctions.v,v 1.2.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) - -(*i Some properties about pow and sum have been made with John Harrison i*) -(*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*) - -(********************************************************) -(** Definition of the sum functions *) -(* *) -(********************************************************) - -Require Rbase. -Require Export R_Ifp. -Require Export Rbasic_fun. -Require Export R_sqr. -Require Export SplitAbsolu. -Require Export SplitRmult. -Require Export ArithProp. -Require Omega. -Require Zpower. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. -Open Local Scope nat_scope. -Open Local Scope R_scope. - -(*******************************) -(** Lemmas about factorial *) -(*******************************) -(*********) -Lemma INR_fact_neq_0:(n:nat)~(INR (fact n))==R0. -Proof. -Intro;Red;Intro;Apply (not_O_INR (fact n) (fact_neq_0 n));Assumption. -Qed. - -(*********) -Lemma fact_simpl : (n:nat) (fact (S n))=(mult (S n) (fact n)). -Proof. -Intro; Reflexivity. -Qed. - -(*********) -Lemma simpl_fact:(n:nat)(Rmult (Rinv (INR (fact (S n)))) - (Rinv (Rinv (INR (fact n)))))== - (Rinv (INR (S n))). -Proof. -Intro;Rewrite (Rinv_Rinv (INR (fact n)) (INR_fact_neq_0 n)); - Unfold 1 fact;Cbv Beta Iota;Fold fact; - Rewrite (mult_INR (S n) (fact n)); - Rewrite (Rinv_Rmult (INR (S n)) (INR (fact n))). -Rewrite (Rmult_assoc (Rinv (INR (S n))) (Rinv (INR (fact n))) - (INR (fact n)));Rewrite (Rinv_l (INR (fact n)) (INR_fact_neq_0 n)); - Apply (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1). -Apply not_O_INR;Auto. -Apply INR_fact_neq_0. -Qed. - -(*******************************) -(* Power *) -(*******************************) -(*********) -Fixpoint pow [r:R;n:nat]:R:= - Cases n of - O => R1 - |(S n) => (Rmult r (pow r n)) - end. - -V8Infix "^" pow : R_scope. - -Lemma pow_O: (x : R) (pow x O) == R1. -Proof. -Reflexivity. -Qed. - -Lemma pow_1: (x : R) (pow x (1)) == x. -Proof. -Simpl; Auto with real. -Qed. - -Lemma pow_add: - (x : R) (n, m : nat) (pow x (plus n m)) == (Rmult (pow x n) (pow x m)). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros n0 H' m; Rewrite H'; Auto with real. -Qed. - -Lemma pow_nonzero: - (x:R) (n:nat) ~(x==R0) -> ~((pow x n)==R0). -Proof. -Intro; Induction n; Simpl. -Intro; Red;Intro;Apply R1_neq_R0;Assumption. -Intros;Red; Intro;Elim (without_div_Od x (pow x n0) H1). -Intro; Auto. -Apply H;Assumption. -Qed. - -Hints Resolve pow_O pow_1 pow_add pow_nonzero:real. - -Lemma pow_RN_plus: - (x : R) - (n, m : nat) - ~ x == R0 -> (pow x n) == (Rmult (pow x (plus n m)) (Rinv (pow x m))). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros n0 H' m H'0. -Rewrite Rmult_assoc; Rewrite <- H'; Auto. -Qed. - -Lemma pow_lt: (x : R) (n : nat) (Rlt R0 x) -> (Rlt R0 (pow x n)). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros n0 H' H'0; Replace R0 with (Rmult x R0); Auto with real. -Qed. -Hints Resolve pow_lt :real. - -Lemma Rlt_pow_R1: - (x : R) (n : nat) (Rlt R1 x) -> (lt O n) -> (Rlt R1 (pow x n)). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros H' H'0; ElimType False; Omega. -Intros n0; Case n0. -Simpl; Rewrite Rmult_1r; Auto. -Intros n1 H' H'0 H'1. -Replace R1 with (Rmult R1 R1); Auto with real. -Apply Rlt_trans with r2 := (Rmult x R1); Auto with real. -Apply Rlt_monotony; Auto with real. -Apply Rlt_trans with r2 := R1; Auto with real. -Apply H'; Auto with arith. -Qed. -Hints Resolve Rlt_pow_R1 :real. - -Lemma Rlt_pow: - (x : R) (n, m : nat) (Rlt R1 x) -> (lt n m) -> (Rlt (pow x n) (pow x m)). -Proof. -Intros x n m H' H'0; Replace m with (plus (minus m n) n). -Rewrite pow_add. -Pattern 1 (pow x n); Replace (pow x n) with (Rmult R1 (pow x n)); - Auto with real. -Apply Rminus_lt. -Repeat Rewrite [y : R] (Rmult_sym y (pow x n)); Rewrite <- Rminus_distr. -Replace R0 with (Rmult (pow x n) R0); Auto with real. -Apply Rlt_monotony; Auto with real. -Apply pow_lt; Auto with real. -Apply Rlt_trans with r2 := R1; Auto with real. -Apply Rlt_minus; Auto with real. -Apply Rlt_pow_R1; Auto with arith. -Apply simpl_lt_plus_l with p := n; Auto with arith. -Rewrite le_plus_minus_r; Auto with arith; Rewrite <- plus_n_O; Auto. -Rewrite plus_sym; Auto with arith. -Qed. -Hints Resolve Rlt_pow :real. - -(*********) -Lemma tech_pow_Rmult:(x:R)(n:nat)(Rmult x (pow x n))==(pow x (S n)). -Proof. -Induction n; Simpl; Trivial. -Qed. - -(*********) -Lemma tech_pow_Rplus:(x:R)(a,n:nat) - (Rplus (pow x a) (Rmult (INR n) (pow x a)))== - (Rmult (INR (S n)) (pow x a)). -Proof. -Intros; Pattern 1 (pow x a); - Rewrite <-(let (H1,H2)=(Rmult_ne (pow x a)) in H1); - Rewrite (Rmult_sym (INR n) (pow x a)); - Rewrite <- (Rmult_Rplus_distr (pow x a) R1 (INR n)); - Rewrite (Rplus_sym R1 (INR n)); Rewrite <-(S_INR n); - Apply Rmult_sym. -Qed. - -Lemma poly: (n:nat)(x:R)(Rlt R0 x)-> - (Rle (Rplus R1 (Rmult (INR n) x)) (pow (Rplus R1 x) n)). -Proof. -Intros;Elim n. -Simpl;Cut (Rplus R1 (Rmult R0 x))==R1. -Intro;Rewrite H0;Unfold Rle;Right; Reflexivity. -Ring. -Intros;Unfold pow; Fold pow; - Apply (Rle_trans (Rplus R1 (Rmult (INR (S n0)) x)) - (Rmult (Rplus R1 x) (Rplus R1 (Rmult (INR n0) x))) - (Rmult (Rplus R1 x) (pow (Rplus R1 x) n0))). -Cut (Rmult (Rplus R1 x) (Rplus R1 (Rmult (INR n0) x)))== - (Rplus (Rplus R1 (Rmult (INR (S n0)) x)) - (Rmult (INR n0) (Rmult x x))). -Intro;Rewrite H1;Pattern 1 (Rplus R1 (Rmult (INR (S n0)) x)); - Rewrite <-(let (H1,H2)= - (Rplus_ne (Rplus R1 (Rmult (INR (S n0)) x))) in H1); - Apply Rle_compatibility;Elim n0;Intros. -Simpl;Rewrite Rmult_Ol;Unfold Rle;Right;Auto. -Unfold Rle;Left;Generalize Rmult_gt;Unfold Rgt;Intro; - Fold (Rsqr x);Apply (H3 (INR (S n1)) (Rsqr x) - (lt_INR_0 (S n1) (lt_O_Sn n1)));Fold (Rgt x R0) in H; - Apply (pos_Rsqr1 x (imp_not_Req x R0 - (or_intror (Rlt x R0) (Rgt x R0) H))). -Rewrite (S_INR n0);Ring. -Unfold Rle in H0;Elim H0;Intro. -Unfold Rle;Left;Apply Rlt_monotony. -Rewrite Rplus_sym; - Apply (Rlt_r_plus_R1 x (Rlt_le R0 x H)). -Assumption. -Rewrite H1;Unfold Rle;Right;Trivial. -Qed. - -Lemma Power_monotonic: - (x:R) (m,n:nat) (Rgt (Rabsolu x) R1) - -> (le m n) - -> (Rle (Rabsolu (pow x m)) (Rabsolu (pow x n))). -Proof. -Intros x m n H;Induction n;Intros;Inversion H0. -Unfold Rle; Right; Reflexivity. -Unfold Rle; Right; Reflexivity. -Apply (Rle_trans (Rabsolu (pow x m)) - (Rabsolu (pow x n)) - (Rabsolu (pow x (S n)))). -Apply Hrecn; Assumption. -Simpl;Rewrite Rabsolu_mult. -Pattern 1 (Rabsolu (pow x n)). -Rewrite <-Rmult_1r. -Rewrite (Rmult_sym (Rabsolu x) (Rabsolu (pow x n))). -Apply Rle_monotony. -Apply Rabsolu_pos. -Unfold Rgt in H. -Apply Rlt_le; Assumption. -Qed. - -Lemma Pow_Rabsolu: (x:R) (n:nat) - (pow (Rabsolu x) n)==(Rabsolu (pow x n)). -Proof. -Intro;Induction n;Simpl. -Apply sym_eqT;Apply Rabsolu_pos_eq;Apply Rlt_le;Apply Rlt_R0_R1. -Intros; Rewrite H;Apply sym_eqT;Apply Rabsolu_mult. -Qed. - - -Lemma Pow_x_infinity: - (x:R) (Rgt (Rabsolu x) R1) - -> (b:R) (Ex [N:nat] ((n:nat) (ge n N) - -> (Rge (Rabsolu (pow x n)) b ))). -Proof. -Intros;Elim (archimed (Rmult b (Rinv (Rminus (Rabsolu x) R1))));Intros; - Clear H1; - Cut (Ex[N:nat] (Rge (INR N) (Rmult b (Rinv (Rminus (Rabsolu x) R1))))). -Intro; Elim H1;Clear H1;Intros;Exists x0;Intros; - Apply (Rge_trans (Rabsolu (pow x n)) (Rabsolu (pow x x0)) b). -Apply Rle_sym1;Apply Power_monotonic;Assumption. -Rewrite <- Pow_Rabsolu;Cut (Rabsolu x)==(Rplus R1 (Rminus (Rabsolu x) R1)). -Intro; Rewrite H3; - Apply (Rge_trans (pow (Rplus R1 (Rminus (Rabsolu x) R1)) x0) - (Rplus R1 (Rmult (INR x0) - (Rminus (Rabsolu x) R1))) - b). -Apply Rle_sym1;Apply poly;Fold (Rgt (Rminus (Rabsolu x) R1) R0); - Apply Rgt_minus;Assumption. -Apply (Rge_trans - (Rplus R1 (Rmult (INR x0) (Rminus (Rabsolu x) R1))) - (Rmult (INR x0) (Rminus (Rabsolu x) R1)) - b). -Apply Rle_sym1; Apply Rlt_le;Rewrite (Rplus_sym R1 - (Rmult (INR x0) (Rminus (Rabsolu x) R1))); - Pattern 1 (Rmult (INR x0) (Rminus (Rabsolu x) R1)); - Rewrite <- (let (H1,H2) = (Rplus_ne - (Rmult (INR x0) (Rminus (Rabsolu x) R1))) in - H1); - Apply Rlt_compatibility; - Apply Rlt_R0_R1. -Cut b==(Rmult (Rmult b (Rinv (Rminus (Rabsolu x) R1))) - (Rminus (Rabsolu x) R1)). -Intros; Rewrite H4;Apply Rge_monotony. -Apply Rge_minus;Unfold Rge; Left; Assumption. -Assumption. -Rewrite Rmult_assoc;Rewrite Rinv_l. -Ring. -Apply imp_not_Req; Right;Apply Rgt_minus;Assumption. -Ring. -Cut `0<= (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))`\/ - `(up (Rmult b (Rinv (Rminus (Rabsolu x) R1)))) <= 0`. -Intros;Elim H1;Intro. -Elim (IZN (up (Rmult b (Rinv (Rminus (Rabsolu x) R1)))) H2);Intros;Exists x0; - Apply (Rge_trans - (INR x0) - (IZR (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))) - (Rmult b (Rinv (Rminus (Rabsolu x) R1)))). -Rewrite INR_IZR_INZ;Apply IZR_ge;Omega. -Unfold Rge; Left; Assumption. -Exists O;Apply (Rge_trans (INR (0)) - (IZR (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))) - (Rmult b (Rinv (Rminus (Rabsolu x) R1)))). -Rewrite INR_IZR_INZ;Apply IZR_ge;Simpl;Omega. -Unfold Rge; Left; Assumption. -Omega. -Qed. - -Lemma pow_ne_zero: - (n:nat) ~(n=(0))-> (pow R0 n) == R0. -Proof. -Induction n. -Simpl;Auto. -Intros;Elim H;Reflexivity. -Intros; Simpl;Apply Rmult_Ol. -Qed. - -Lemma Rinv_pow: - (x:R) (n:nat) ~(x==R0) -> (Rinv (pow x n))==(pow (Rinv x) n). -Proof. -Intros; Elim n; Simpl. -Apply Rinv_R1. -Intro m;Intro;Rewrite Rinv_Rmult. -Rewrite H0; Reflexivity;Assumption. -Assumption. -Apply pow_nonzero;Assumption. -Qed. - -Lemma pow_lt_1_zero: - (x:R) (Rlt (Rabsolu x) R1) - -> (y:R) (Rlt R0 y) - -> (Ex[N:nat] (n:nat) (ge n N) - -> (Rlt (Rabsolu (pow x n)) y)). -Proof. -Intros;Elim (Req_EM x R0);Intro. -Exists (1);Rewrite H1;Intros n GE;Rewrite pow_ne_zero. -Rewrite Rabsolu_R0;Assumption. -Inversion GE;Auto. -Cut (Rgt (Rabsolu (Rinv x)) R1). -Intros;Elim (Pow_x_infinity (Rinv x) H2 (Rplus (Rinv y) R1));Intros N. -Exists N;Intros;Rewrite <- (Rinv_Rinv y). -Rewrite <- (Rinv_Rinv (Rabsolu (pow x n))). -Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply Rlt_Rinv. -Assumption. -Apply Rlt_Rinv. -Apply Rabsolu_pos_lt. -Apply pow_nonzero. -Assumption. -Rewrite <- Rabsolu_Rinv. -Rewrite Rinv_pow. -Apply (Rlt_le_trans (Rinv y) - (Rplus (Rinv y) R1) - (Rabsolu (pow (Rinv x) n))). -Pattern 1 (Rinv y). -Rewrite <- (let (H1,H2) = - (Rplus_ne (Rinv y)) in H1). -Apply Rlt_compatibility. -Apply Rlt_R0_R1. -Apply Rle_sym2. -Apply H3. -Assumption. -Assumption. -Apply pow_nonzero. -Assumption. -Apply Rabsolu_no_R0. -Apply pow_nonzero. -Assumption. -Apply imp_not_Req. -Right; Unfold Rgt; Assumption. -Rewrite <- (Rinv_Rinv R1). -Rewrite Rabsolu_Rinv. -Unfold Rgt; Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt. -Assumption. -Rewrite Rinv_R1; Apply Rlt_R0_R1. -Rewrite Rinv_R1; Assumption. -Assumption. -Red;Intro; Apply R1_neq_R0;Assumption. -Qed. - -Lemma pow_R1: - (r : R) (n : nat) (pow r n) == R1 -> (Rabsolu r) == R1 \/ n = O. -Proof. -Intros r n H'. -Case (Req_EM (Rabsolu r) R1); Auto; Intros H'1. -Case (not_Req ? ? H'1); Intros H'2. -Generalize H'; Case n; Auto. -Intros n0 H'0. -Cut ~ r == R0; [Intros Eq1 | Idtac]. -Cut ~ (Rabsolu r) == R0; [Intros Eq2 | Apply Rabsolu_no_R0]; Auto. -Absurd (Rlt (pow (Rabsolu (Rinv r)) O) (pow (Rabsolu (Rinv r)) (S n0))); Auto. -Replace (pow (Rabsolu (Rinv r)) (S n0)) with R1. -Simpl; Apply Rlt_antirefl; Auto. -Rewrite Rabsolu_Rinv; Auto. -Rewrite <- Rinv_pow; Auto. -Rewrite Pow_Rabsolu; Auto. -Rewrite H'0; Rewrite Rabsolu_right; Auto with real. -Apply Rle_ge; Auto with real. -Apply Rlt_pow; Auto with arith. -Rewrite Rabsolu_Rinv; Auto. -Apply Rlt_monotony_contra with z := (Rabsolu r). -Case (Rabsolu_pos r); Auto. -Intros H'3; Case Eq2; Auto. -Rewrite Rmult_1r; Rewrite Rinv_r; Auto with real. -Red;Intro;Absurd ``(pow r (S n0)) == 1``;Auto. -Simpl; Rewrite H; Rewrite Rmult_Ol; Auto with real. -Generalize H'; Case n; Auto. -Intros n0 H'0. -Cut ~ r == R0; [Intros Eq1 | Auto with real]. -Cut ~ (Rabsolu r) == R0; [Intros Eq2 | Apply Rabsolu_no_R0]; Auto. -Absurd (Rlt (pow (Rabsolu r) O) (pow (Rabsolu r) (S n0))); - Auto with real arith. -Repeat Rewrite Pow_Rabsolu; Rewrite H'0; Simpl; Auto with real. -Red;Intro;Absurd ``(pow r (S n0)) == 1``;Auto. -Simpl; Rewrite H; Rewrite Rmult_Ol; Auto with real. -Qed. - -Lemma pow_Rsqr : (x:R;n:nat) (pow x (mult (2) n))==(pow (Rsqr x) n). -Proof. -Intros; Induction n. -Reflexivity. -Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Replace (pow x (S (S (mult (2) n)))) with ``x*x*(pow x (mult (S (S O)) n))``. -Rewrite Hrecn; Reflexivity. -Simpl; Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Qed. - -Lemma pow_le : (a:R;n:nat) ``0<=a`` -> ``0<=(pow a n)``. -Proof. -Intros; Induction n. -Simpl; Left; Apply Rlt_R0_R1. -Simpl; Apply Rmult_le_pos; Assumption. -Qed. - -(**********) -Lemma pow_1_even : (n:nat) ``(pow (-1) (mult (S (S O)) n))==1``. -Proof. -Intro; Induction n. -Reflexivity. -Replace (mult (2) (S n)) with (plus (2) (mult (2) n)). -Rewrite pow_add; Rewrite Hrecn; Simpl; Ring. -Replace (S n) with (plus n (1)); [Ring | Ring]. -Qed. - -(**********) -Lemma pow_1_odd : (n:nat) ``(pow (-1) (S (mult (S (S O)) n)))==-1``. -Proof. -Intro; Replace (S (mult (2) n)) with (plus (mult (2) n) (1)); [Idtac | Ring]. -Rewrite pow_add; Rewrite pow_1_even; Simpl; Ring. -Qed. - -(**********) -Lemma pow_1_abs : (n:nat) ``(Rabsolu (pow (-1) n))==1``. -Proof. -Intro; Induction n. -Simpl; Apply Rabsolu_R1. -Replace (S n) with (plus n (1)); [Rewrite pow_add | Ring]. -Rewrite Rabsolu_mult. -Rewrite Hrecn; Rewrite Rmult_1l; Simpl; Rewrite Rmult_1r; Rewrite Rabsolu_Ropp; Apply Rabsolu_R1. -Qed. - -Lemma pow_mult : (x:R;n1,n2:nat) (pow x (mult n1 n2))==(pow (pow x n1) n2). -Proof. -Intros; Induction n2. -Simpl; Replace (mult n1 O) with O; [Reflexivity | Ring]. -Replace (mult n1 (S n2)) with (plus (mult n1 n2) n1). -Replace (S n2) with (plus n2 (1)); [Idtac | Ring]. -Do 2 Rewrite pow_add. -Rewrite Hrecn2. -Simpl. -Ring. -Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite mult_INR; Rewrite S_INR; Ring. -Qed. - -Lemma pow_incr : (x,y:R;n:nat) ``0<=x<=y`` -> ``(pow x n)<=(pow y n)``. -Proof. -Intros. -Induction n. -Right; Reflexivity. -Simpl. -Elim H; Intros. -Apply Rle_trans with ``y*(pow x n)``. -Do 2 Rewrite <- (Rmult_sym (pow x n)). -Apply Rle_monotony. -Apply pow_le; Assumption. -Assumption. -Apply Rle_monotony. -Apply Rle_trans with x; Assumption. -Apply Hrecn. -Qed. - -Lemma pow_R1_Rle : (x:R;k:nat) ``1<=x`` -> ``1<=(pow x k)``. -Proof. -Intros. -Induction k. -Right; Reflexivity. -Simpl. -Apply Rle_trans with ``x*1``. -Rewrite Rmult_1r; Assumption. -Apply Rle_monotony. -Left; Apply Rlt_le_trans with R1; [Apply Rlt_R0_R1 | Assumption]. -Exact Hreck. -Qed. - -Lemma Rle_pow : (x:R;m,n:nat) ``1<=x`` -> (le m n) -> ``(pow x m)<=(pow x n)``. -Proof. -Intros. -Replace n with (plus (minus n m) m). -Rewrite pow_add. -Rewrite Rmult_sym. -Pattern 1 (pow x m); Rewrite <- Rmult_1r. -Apply Rle_monotony. -Apply pow_le; Left; Apply Rlt_le_trans with R1; [Apply Rlt_R0_R1 | Assumption]. -Apply pow_R1_Rle; Assumption. -Rewrite plus_sym. -Symmetry; Apply le_plus_minus; Assumption. -Qed. - -Lemma pow1 : (n:nat) (pow R1 n)==R1. -Proof. -Intro; Induction n. -Reflexivity. -Simpl; Rewrite Hrecn; Rewrite Rmult_1r; Reflexivity. -Qed. - -Lemma pow_Rabs : (x:R;n:nat) ``(pow x n)<=(pow (Rabsolu x) n)``. -Proof. -Intros; Induction n. -Right; Reflexivity. -Simpl; Case (case_Rabsolu x); Intro. -Apply Rle_trans with (Rabsolu ``x*(pow x n)``). -Apply Rle_Rabsolu. -Rewrite Rabsolu_mult. -Apply Rle_monotony. -Apply Rabsolu_pos. -Right; Symmetry; Apply Pow_Rabsolu. -Pattern 1 (Rabsolu x); Rewrite (Rabsolu_right x r); Apply Rle_monotony. -Apply Rle_sym2; Exact r. -Apply Hrecn. -Qed. - -Lemma pow_maj_Rabs : (x,y:R;n:nat) ``(Rabsolu y)<=x`` -> ``(pow y n)<=(pow x n)``. -Proof. -Intros; Cut ``0<=x``. -Intro; Apply Rle_trans with (pow (Rabsolu y) n). -Apply pow_Rabs. -Induction n. -Right; Reflexivity. -Simpl; Apply Rle_trans with ``x*(pow (Rabsolu y) n)``. -Do 2 Rewrite <- (Rmult_sym (pow (Rabsolu y) n)). -Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Assumption. -Apply Rle_monotony. -Apply H0. -Apply Hrecn. -Apply Rle_trans with (Rabsolu y); [Apply Rabsolu_pos | Exact H]. -Qed. - -(*******************************) -(** PowerRZ *) -(*******************************) -(*i Due to L.Thery i*) - -Tactic Definition CaseEqk name := -Generalize (refl_equal ? name); Pattern -1 name; Case name. - -Definition powerRZ := - [x : R] [n : Z] Cases n of - ZERO => R1 - | (POS p) => (pow x (convert p)) - | (NEG p) => (Rinv (pow x (convert p))) - end. - -Infix Local "^Z" powerRZ (at level 2, left associativity) : R_scope. - -Lemma Zpower_NR0: - (x : Z) (n : nat) (Zle ZERO x) -> (Zle ZERO (Zpower_nat x n)). -Proof. -NewInduction n; Unfold Zpower_nat; Simpl; Auto with zarith. -Qed. - -Lemma powerRZ_O: (x : R) (powerRZ x ZERO) == R1. -Proof. -Reflexivity. -Qed. - -Lemma powerRZ_1: (x : R) (powerRZ x (Zs ZERO)) == x. -Proof. -Simpl; Auto with real. -Qed. - -Lemma powerRZ_NOR: (x : R) (z : Z) ~ x == R0 -> ~ (powerRZ x z) == R0. -Proof. -NewDestruct z; Simpl; Auto with real. -Qed. - -Lemma powerRZ_add: - (x : R) - (n, m : Z) - ~ x == R0 -> (powerRZ x (Zplus n m)) == (Rmult (powerRZ x n) (powerRZ x m)). -Proof. -Intro x; NewDestruct n as [|n1|n1]; NewDestruct m as [|m1|m1]; Simpl; - Auto with real. -(* POS/POS *) -Rewrite convert_add; Auto with real. -(* POS/NEG *) -(CaseEqk '(compare n1 m1 EGAL)); Simpl; Auto with real. -Intros H' H'0; Rewrite compare_convert_EGAL with 1 := H'; Auto with real. -Intros H' H'0; Rewrite (true_sub_convert m1 n1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert m1) (convert n1)) (convert n1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Rewrite Rinv_Rmult; Auto with real. -Rewrite Rinv_Rinv; Auto with real. -Apply lt_le_weak. -Apply compare_convert_INFERIEUR; Auto. -Apply ZC2; Auto. -Intros H' H'0; Rewrite (true_sub_convert n1 m1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert n1) (convert m1)) (convert m1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Apply lt_le_weak. -Change (gt (convert n1) (convert m1)). -Apply compare_convert_SUPERIEUR; Auto. -(* NEG/POS *) -(CaseEqk '(compare n1 m1 EGAL)); Simpl; Auto with real. -Intros H' H'0; Rewrite compare_convert_EGAL with 1 := H'; Auto with real. -Intros H' H'0; Rewrite (true_sub_convert m1 n1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert m1) (convert n1)) (convert n1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Apply lt_le_weak. -Apply compare_convert_INFERIEUR; Auto. -Apply ZC2; Auto. -Intros H' H'0; Rewrite (true_sub_convert n1 m1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert n1) (convert m1)) (convert m1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Rewrite Rinv_Rmult; Auto with real. -Apply lt_le_weak. -Change (gt (convert n1) (convert m1)). -Apply compare_convert_SUPERIEUR; Auto. -(* NEG/NEG *) -Rewrite convert_add; Auto with real. -Intros H'; Rewrite pow_add; Auto with real. -Apply Rinv_Rmult; Auto. -Apply pow_nonzero; Auto. -Apply pow_nonzero; Auto. -Qed. -Hints Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add :real. - -Lemma Zpower_nat_powerRZ: - (n, m : nat) - (IZR (Zpower_nat (inject_nat n) m)) == (powerRZ (INR n) (inject_nat m)). -Proof. -Intros n m; Elim m; Simpl; Auto with real. -Intros m1 H'; Rewrite bij1; Simpl. -Replace (Zpower_nat (inject_nat n) (S m1)) - with (Zmult (inject_nat n) (Zpower_nat (inject_nat n) m1)). -Rewrite mult_IZR; Auto with real. -Repeat Rewrite <- INR_IZR_INZ; Simpl. -Rewrite H'; Simpl. -Case m1; Simpl; Auto with real. -Intros m2; Rewrite bij1; Auto. -Unfold Zpower_nat; Auto. -Qed. - -Lemma powerRZ_lt: (x : R) (z : Z) (Rlt R0 x) -> (Rlt R0 (powerRZ x z)). -Proof. -Intros x z; Case z; Simpl; Auto with real. -Qed. -Hints Resolve powerRZ_lt :real. - -Lemma powerRZ_le: (x : R) (z : Z) (Rlt R0 x) -> (Rle R0 (powerRZ x z)). -Proof. -Intros x z H'; Apply Rlt_le; Auto with real. -Qed. -Hints Resolve powerRZ_le :real. - -Lemma Zpower_nat_powerRZ_absolu: - (n, m : Z) - (Zle ZERO m) -> (IZR (Zpower_nat n (absolu m))) == (powerRZ (IZR n) m). -Proof. -Intros n m; Case m; Simpl; Auto with zarith. -Intros p H'; Elim (convert p); Simpl; Auto with zarith. -Intros n0 H'0; Rewrite <- H'0; Simpl; Auto with zarith. -Rewrite <- mult_IZR; Auto. -Intros p H'; Absurd `0 <= (NEG p)`;Auto with zarith. -Qed. - -Lemma powerRZ_R1: (n : Z) (powerRZ R1 n) == R1. -Proof. -Intros n; Case n; Simpl; Auto. -Intros p; Elim (convert p); Simpl; Auto; Intros n0 H'; Rewrite H'; Ring. -Intros p; Elim (convert p); Simpl. -Exact Rinv_R1. -Intros n1 H'; Rewrite Rinv_Rmult; Try Rewrite Rinv_R1; Try Rewrite H'; - Auto with real. -Qed. - -(*******************************) -(** Sum of n first naturals *) -(*******************************) -(*********) -Fixpoint sum_nat_f_O [f:nat->nat;n:nat]:nat:= - Cases n of - O => (f O) - |(S n') => (plus (sum_nat_f_O f n') (f (S n'))) - end. - -(*********) -Definition sum_nat_f [s,n:nat;f:nat->nat]:nat:= - (sum_nat_f_O [x:nat](f (plus x s)) (minus n s)). - -(*********) -Definition sum_nat_O [n:nat]:nat:= - (sum_nat_f_O [x:nat]x n). - -(*********) -Definition sum_nat [s,n:nat]:nat:= - (sum_nat_f s n [x:nat]x). - -(*******************************) -(** Sum *) -(*******************************) -(*********) -Fixpoint sum_f_R0 [f:nat->R;N:nat]:R:= - Cases N of - O => (f O) - |(S i) => (Rplus (sum_f_R0 f i) (f (S i))) - end. - -(*********) -Definition sum_f [s,n:nat;f:nat->R]:R:= - (sum_f_R0 [x:nat](f (plus x s)) (minus n s)). - -Lemma GP_finite: - (x:R) (n:nat) (Rmult (sum_f_R0 [n:nat] (pow x n) n) - (Rminus x R1)) == - (Rminus (pow x (plus n (1))) R1). -Proof. -Intros; Induction n; Simpl. -Ring. -Rewrite Rmult_Rplus_distrl;Rewrite Hrecn;Cut (plus n (1))=(S n). -Intro H;Rewrite H;Simpl;Ring. -Omega. -Qed. - -Lemma sum_f_R0_triangle: - (x:nat->R)(n:nat) (Rle (Rabsolu (sum_f_R0 x n)) - (sum_f_R0 [i:nat] (Rabsolu (x i)) n)). -Proof. -Intro; Induction n; Simpl. -Unfold Rle; Right; Reflexivity. -Intro m; Intro;Apply (Rle_trans - (Rabsolu (Rplus (sum_f_R0 x m) (x (S m)))) - (Rplus (Rabsolu (sum_f_R0 x m)) - (Rabsolu (x (S m)))) - (Rplus (sum_f_R0 [i:nat](Rabsolu (x i)) m) - (Rabsolu (x (S m))))). -Apply Rabsolu_triang. -Rewrite Rplus_sym;Rewrite (Rplus_sym - (sum_f_R0 [i:nat](Rabsolu (x i)) m) (Rabsolu (x (S m)))); - Apply Rle_compatibility;Assumption. -Qed. - -(*******************************) -(* Distance in R *) -(*******************************) - -(*********) -Definition R_dist:R->R->R:=[x,y:R](Rabsolu (Rminus x y)). - -(*********) -Lemma R_dist_pos:(x,y:R)(Rge (R_dist x y) R0). -Proof. -Intros;Unfold R_dist;Unfold Rabsolu;Case (case_Rabsolu (Rminus x y));Intro l. -Unfold Rge;Left;Apply (Rlt_RoppO (Rminus x y) l). -Trivial. -Qed. - -(*********) -Lemma R_dist_sym:(x,y:R)(R_dist x y)==(R_dist y x). -Proof. -Unfold R_dist;Intros;SplitAbsolu;Ring. -Generalize (Rlt_RoppO (Rminus y x) r); Intro; - Rewrite (Ropp_distr2 y x) in H; - Generalize (Rlt_antisym (Rminus x y) R0 r0); Intro;Unfold Rgt in H; - ElimType False; Auto. -Generalize (minus_Rge y x r); Intro; - Generalize (minus_Rge x y r0); Intro; - Generalize (Rge_ge_eq x y H0 H); Intro;Rewrite H1;Ring. -Qed. - -(*********) -Lemma R_dist_refl:(x,y:R)((R_dist x y)==R0<->x==y). -Proof. -Unfold R_dist;Intros;SplitAbsolu;Split;Intros. -Rewrite (Ropp_distr2 x y) in H;Apply sym_eqT; - Apply (Rminus_eq y x H). -Rewrite (Ropp_distr2 x y);Generalize (sym_eqT R x y H);Intro; - Apply (eq_Rminus y x H0). -Apply (Rminus_eq x y H). -Apply (eq_Rminus x y H). -Qed. - -Lemma R_dist_eq:(x:R)(R_dist x x)==R0. -Proof. -Unfold R_dist;Intros;SplitAbsolu;Intros;Ring. -Qed. - -(***********) -Lemma R_dist_tri:(x,y,z:R)(Rle (R_dist x y) - (Rplus (R_dist x z) (R_dist z y))). -Proof. -Intros;Unfold R_dist; Replace ``x-y`` with ``(x-z)+(z-y)``; - [Apply (Rabsolu_triang ``x-z`` ``z-y``)|Ring]. -Qed. - -(*********) -Lemma R_dist_plus: (a,b,c,d:R)(Rle (R_dist (Rplus a c) (Rplus b d)) - (Rplus (R_dist a b) (R_dist c d))). -Proof. -Intros;Unfold R_dist; - Replace (Rminus (Rplus a c) (Rplus b d)) - with (Rplus (Rminus a b) (Rminus c d)). -Exact (Rabsolu_triang (Rminus a b) (Rminus c d)). -Ring. -Qed. - -(*******************************) -(** Infinit Sum *) -(*******************************) -(*********) -Definition infinit_sum:(nat->R)->R->Prop:=[s:nat->R;l:R] - (eps:R)(Rgt eps R0)-> - (Ex[N:nat](n:nat)(ge n N)->(Rlt (R_dist (sum_f_R0 s n) l) eps)). |