diff options
author | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-01-22 16:30:04 +0000 |
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committer | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-01-22 16:30:04 +0000 |
commit | 096310efb4266fe89455d473f704ec6c7ed5a97c (patch) | |
tree | 17a6ed5acd135748f43ecb944e268657731a55a2 /theories/Reals/Rtrigo_def.v | |
parent | cb7cf50289fbf1c5fe60fd915f63128c88dc6d68 (diff) |
Renommages dans Rtrigo_def
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3589 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Rtrigo_def.v')
-rw-r--r-- | theories/Reals/Rtrigo_def.v | 19 |
1 files changed, 8 insertions, 11 deletions
diff --git a/theories/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v index 038f1efb7..554a2059e 100644 --- a/theories/Reals/Rtrigo_def.v +++ b/theories/Reals/Rtrigo_def.v @@ -38,7 +38,7 @@ Intros; Apply pow_ne_zero. Red; Intro; Rewrite H0 in H; Elim (lt_n_n ? H). Qed. -(*i Calcul de $e^0$ *) +(*i Calculus of $e^0$ *) Lemma exist_exp0 : (SigT R [l:R](exp_in R0 l)). Apply Specif.existT with R1. Unfold exp_in; Unfold infinit_sum; Intros. @@ -81,7 +81,6 @@ Unfold sinh; Rewrite Ropp_O; Rewrite exp_0. Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Apply Rmult_Ol. Qed. -(* TG de la série entière définissant COS *) Definition cos_n [n:nat] : R := ``(pow (-1) n)/(INR (fact (mult (S (S O)) n)))``. Lemma simpl_cos_n : (n:nat) (Rdiv (cos_n (S n)) (cos_n n))==(Ropp (Rinv (INR (mult (mult (2) (S n)) (plus (mult (2) n) (1)))))). @@ -142,7 +141,6 @@ Assert H0 := (archimed ``/eps``). Elim H0; Intros; Assumption. Qed. -(* Convergence de la SE de TG cos_n *) Lemma Alembert_cos : (Un_cv [n:nat]``(Rabsolu (cos_n (S n))/(cos_n n))`` R0). Unfold Un_cv; Intros. Assert H0 := (archimed_cor1 eps H). @@ -207,11 +205,11 @@ Intro; Generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos). Unfold Pser cos_in; Trivial. Qed. -(* Définition du cosinus *) +(* Definition of cosinus *) (*************************) Definition cos : R -> R := [x:R](Cases (exist_cos (Rsqr x)) of (Specif.existT a b) => a end). -(* TG de la série entière définissant SIN *) + Definition sin_n [n:nat] : R := ``(pow (-1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))``. Lemma simpl_sin_n : (n:nat) (Rdiv (sin_n (S n)) (sin_n n))==(Ropp (Rinv (INR (mult (plus (mult (2) (S n)) (1)) (mult (2) (S n)))))). @@ -250,7 +248,6 @@ Apply pow_nonzero; DiscrR. Apply Rinv_neq_R0; Apply INR_fact_neq_0. Qed. -(* Convergence de la SE de TG sin_n *) Lemma Alembert_sin : (Un_cv [n:nat]``(Rabsolu (sin_n (S n))/(sin_n n))`` R0). Unfold Un_cv; Intros; Assert H0 := (archimed_cor1 eps H). Elim H0; Intros; Exists x. @@ -306,20 +303,20 @@ Unfold Pser sin_n; Trivial. Qed. (***********************) -(* Définition du sinus *) +(* Definition of sinus *) Definition sin : R -> R := [x:R](Cases (exist_sin (Rsqr x)) of (Specif.existT a b) => ``x*a`` end). (*********************************************) -(* PROPRIETES *) +(* PROPERTIES *) (*********************************************) -Lemma cos_paire : (x:R) ``(cos x)==(cos (-x))``. +Lemma cos_sym : (x:R) ``(cos x)==(cos (-x))``. Intros; Unfold cos; Replace ``(Rsqr (-x))`` with (Rsqr x). Reflexivity. Apply Rsqr_neg. Qed. -Lemma sin_impaire : (x:R)``(sin (-x))==-(sin x)``. +Lemma sin_antisym : (x:R)``(sin (-x))==-(sin x)``. Intro; Unfold sin; Replace ``(Rsqr (-x))`` with (Rsqr x); [Idtac | Apply Rsqr_neg]. Case (exist_sin (Rsqr x)); Intros; Ring. Qed. @@ -346,7 +343,7 @@ Apply Hrecn; Unfold ge; Apply le_O_n. Simpl; Ring. Defined. -(* Calcul de (cos 0) *) +(* Calculus of (cos 0) *) Lemma cos_0 : ``(cos 0)==1``. Cut (cos_in R0 (cos R0)). Cut (cos_in R0 R1). |