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diff --git a/theories7/Reals/Rtrigo_def.v b/theories7/Reals/Rtrigo_def.v new file mode 100644 index 00000000..0897416b --- /dev/null +++ b/theories7/Reals/Rtrigo_def.v @@ -0,0 +1,357 @@ +(************************************************************************) +(* 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: Rtrigo_def.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_fun. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*****************************) +(* Definition of exponential *) +(*****************************) +Definition exp_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``/(INR (fact i))*(pow x i)`` l). + +Lemma exp_cof_no_R0 : (n:nat) ``/(INR (fact n))<>0``. +Intro. +Apply Rinv_neq_R0. +Apply INR_fact_neq_0. +Qed. + +Lemma exist_exp : (x:R)(SigT R [l:R](exp_in x l)). +Intro; Generalize (Alembert_C3 [n:nat](Rinv (INR (fact n))) x exp_cof_no_R0 Alembert_exp). +Unfold Pser exp_in. +Trivial. +Defined. + +Definition exp : R -> R := [x:R](projT1 ? ? (exist_exp x)). + +Lemma pow_i : (i:nat) (lt O i) -> (pow R0 i)==R0. +Intros; Apply pow_ne_zero. +Red; Intro; Rewrite H0 in H; Elim (lt_n_n ? H). +Qed. + +(*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. +Exists O. +Intros; Replace (sum_f_R0 ([i:nat]``/(INR (fact i))*(pow R0 i)``) n) with R1. +Unfold R_dist; Replace ``1-1`` with R0; [Rewrite Rabsolu_R0; Assumption | Ring]. +Induction n. +Simpl; Rewrite Rinv_R1; Ring. +Rewrite tech5. +Rewrite <- Hrecn. +Simpl. +Ring. +Unfold ge; Apply le_O_n. +Defined. + +Lemma exp_0 : ``(exp 0)==1``. +Cut (exp_in R0 (exp R0)). +Cut (exp_in R0 R1). +Unfold exp_in; Intros; EApply unicity_sum. +Apply H0. +Apply H. +Exact (projT2 ? ? exist_exp0). +Exact (projT2 ? ? (exist_exp R0)). +Qed. + +(**************************************) +(* Definition of hyperbolic functions *) +(**************************************) +Definition cosh : R->R := [x:R]``((exp x)+(exp (-x)))/2``. +Definition sinh : R->R := [x:R]``((exp x)-(exp (-x)))/2``. +Definition tanh : R->R := [x:R]``(sinh x)/(cosh x)``. + +Lemma cosh_0 : ``(cosh 0)==1``. +Unfold cosh; Rewrite Ropp_O; Rewrite exp_0. +Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | DiscrR]. +Qed. + +Lemma sinh_0 : ``(sinh 0)==0``. +Unfold sinh; Rewrite Ropp_O; Rewrite exp_0. +Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Apply Rmult_Ol. +Qed. + +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)))))). +Intro; Unfold cos_n; Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Replace ``(pow ( -1) n)*(pow ( -1) (S O))*/(INR (fact (mult (S (S O)) (plus n (S O)))))*(/(pow ( -1) n)*(INR (fact (mult (S (S O)) n))))`` with ``((pow ( -1) n)*/(pow ( -1) n))*/(INR (fact (mult (S (S O)) (plus n (S O)))))*(INR (fact (mult (S (S O)) n)))*(pow (-1) (S O))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Unfold pow; Rewrite Rmult_1r. +Replace (mult (S (S O)) (plus n (S O))) with (S (S (mult (S (S O)) n))); [Idtac | Ring]. +Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rinv_Rmult; Try (Apply not_O_INR; Discriminate). +Rewrite <- (Rmult_sym ``-1``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Replace (S (mult (S (S O)) n)) with (plus (mult (S (S O)) n) (S O)); [Idtac | Ring]. +Rewrite mult_INR; Rewrite Rinv_Rmult. +Ring. +Apply not_O_INR; Discriminate. +Replace (plus (mult (S (S O)) n) (S O)) with (S (mult (S (S O)) n)); [Apply not_O_INR; Discriminate | Ring]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Apply pow_nonzero; DiscrR. +Apply INR_fact_neq_0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Qed. + +Lemma archimed_cor1 : (eps:R) ``0<eps`` -> (EX N : nat | ``/(INR N) < eps``/\(lt O N)). +Intros; Cut ``/eps < (IZR (up (/eps)))``. +Intro; Cut `0<=(up (Rinv eps))`. +Intro; Assert H2 := (IZN ? H1); Elim H2; Intros; Exists (max x (1)). +Split. +Cut ``0<(IZR (INZ x))``. +Intro; Rewrite INR_IZR_INZ; Apply Rle_lt_trans with ``/(IZR (INZ x))``. +Apply Rle_monotony_contra with (IZR (INZ x)). +Assumption. +Rewrite <- Rinv_r_sym; [Idtac | Red; Intro; Rewrite H5 in H4; Elim (Rlt_antirefl ? H4)]. +Apply Rle_monotony_contra with (IZR (INZ (max x (1)))). +Apply Rlt_le_trans with (IZR (INZ x)). +Assumption. +Repeat Rewrite <- INR_IZR_INZ; Apply le_INR; Apply le_max_l. +Rewrite Rmult_1r; Rewrite (Rmult_sym (IZR (INZ (max x (S O))))); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Repeat Rewrite <- INR_IZR_INZ; Apply le_INR; Apply le_max_l. +Rewrite <- INR_IZR_INZ; Apply not_O_INR. +Red; Intro;Assert H6 := (le_max_r x (1)); Cut (lt O (1)); [Intro | Apply lt_O_Sn]; Assert H8 := (lt_le_trans ? ? ? H7 H6); Rewrite H5 in H8; Elim (lt_n_n ? H8). +Pattern 1 eps; Rewrite <- Rinv_Rinv. +Apply Rinv_lt. +Apply Rmult_lt_pos; [Apply Rlt_Rinv; Assumption | Assumption]. +Rewrite H3 in H0; Assumption. +Red; Intro; Rewrite H5 in H; Elim (Rlt_antirefl ? H). +Apply Rlt_trans with ``/eps``. +Apply Rlt_Rinv; Assumption. +Rewrite H3 in H0; Assumption. +Apply lt_le_trans with (1); [Apply lt_O_Sn | Apply le_max_r]. +Apply le_IZR; Replace (IZR `0`) with R0; [Idtac | Reflexivity]; Left; Apply Rlt_trans with ``/eps``; [Apply Rlt_Rinv; Assumption | Assumption]. +Assert H0 := (archimed ``/eps``). +Elim H0; Intros; Assumption. +Qed. + +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). +Elim H0; Intros; Exists x. +Intros; Rewrite simpl_cos_n; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_Ropp; Rewrite Rabsolu_right. +Rewrite mult_INR; Rewrite Rinv_Rmult. +Cut ``/(INR (mult (S (S O)) (S n)))<1``. +Intro; Cut ``/(INR (plus (mult (S (S O)) n) (S O)))<eps``. +Intro; Rewrite <- (Rmult_1l eps). +Apply Rmult_lt; Try Assumption. +Change ``0</(INR (plus (mult (S (S O)) n) (S O)))``; Apply Rlt_Rinv; Apply lt_INR_0. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Apply Rlt_R0_R1. +Cut (lt x (plus (mult (2) n) (1))). +Intro; Assert H5 := (lt_INR ? ? H4). +Apply Rlt_trans with ``/(INR x)``. +Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply lt_INR_0. +Elim H1; Intros; Assumption. +Apply lt_INR_0; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Assumption. +Elim H1; Intros; Assumption. +Apply lt_le_trans with (S n). +Unfold ge in H2; Apply le_lt_n_Sm; Assumption. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Idtac | Ring]. +Apply le_n_S; Apply le_n_2n. +Apply Rlt_monotony_contra with (INR (mult (S (S O)) (S n))). +Apply lt_INR_0; Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Apply lt_O_Sn. +Replace (S n) with (plus n (1)); [Idtac | Ring]. +Ring. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace R1 with (INR (1)); [Apply lt_INR | Reflexivity]. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Apply lt_n_S; Apply lt_O_Sn. +Replace (S n) with (plus n (1)); [Ring | Ring]. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Replace (plus (mult (S (S O)) n) (S O)) with (S (mult (2) n)); [Apply not_O_INR; Discriminate | Ring]. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Apply lt_INR_0. +Replace (mult (mult (2) (S n)) (plus (mult (2) n) (1))) with (S (S (plus (mult (4) (mult n n)) (mult (6) n)))). +Apply lt_O_Sn. +Apply INR_eq. +Repeat Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite mult_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Replace (INR O) with R0; [Ring | Reflexivity]. +Qed. + +Lemma cosn_no_R0 : (n:nat)``(cos_n n)<>0``. +Intro; Unfold cos_n; Unfold Rdiv; Apply prod_neq_R0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0. +Apply INR_fact_neq_0. +Qed. + +(**********) +Definition cos_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(cos_n i)*(pow x i)`` l). + +(**********) +Lemma exist_cos : (x:R)(SigT R [l:R](cos_in x l)). +Intro; Generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos). +Unfold Pser cos_in; Trivial. +Qed. + +(* Definition of cosinus *) +(*************************) +Definition cos : R -> R := [x:R](Cases (exist_cos (Rsqr x)) of (Specif.existT a b) => a end). + + +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)))))). +Intro; Unfold sin_n; Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Replace ``(pow ( -1) n)*(pow ( -1) (S O))*/(INR (fact (plus (mult (S (S O)) (plus n (S O))) (S O))))*(/(pow ( -1) n)*(INR (fact (plus (mult (S (S O)) n) (S O)))))`` with ``((pow ( -1) n)*/(pow ( -1) n))*/(INR (fact (plus (mult (S (S O)) (plus n (S O))) (S O))))*(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow (-1) (S O))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Unfold pow; Rewrite Rmult_1r; Replace (plus (mult (S (S O)) (plus n (S O))) (S O)) with (S (S (plus (mult (S (S O)) n) (S O)))). +Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. +Rewrite <- (Rmult_sym ``-1``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Replace (S (plus (mult (S (S O)) n) (S O))) with (mult (S (S O)) (plus n (S O))). +Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. +Ring. +Apply not_O_INR; Discriminate. +Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. +Apply not_O_INR; Discriminate. +Apply prod_neq_R0. +Apply not_O_INR; Discriminate. +Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. +Apply not_O_INR; Discriminate. +Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. +Rewrite mult_plus_distr_r; Cut (n:nat) (S n)=(plus n (1)). +Intros; Rewrite (H (plus (mult (2) n) (1))). +Ring. +Intros; Ring. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Cut (n:nat) (S (S n))=(plus n (2)); [Intros; Rewrite (H (plus (mult (2) n) (1))); Ring | Intros; Ring]. +Apply pow_nonzero; DiscrR. +Apply INR_fact_neq_0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Qed. + +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. +Intros; Rewrite simpl_sin_n; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_Ropp; Rewrite Rabsolu_right. +Rewrite mult_INR; Rewrite Rinv_Rmult. +Cut ``/(INR (mult (S (S O)) (S n)))<1``. +Intro; Cut ``/(INR (plus (mult (S (S O)) (S n)) (S O)))<eps``. +Intro; Rewrite <- (Rmult_1l eps); Rewrite (Rmult_sym ``/(INR (plus (mult (S (S O)) (S n)) (S O)))``); Apply Rmult_lt; Try Assumption. +Change ``0</(INR (plus (mult (S (S O)) (S n)) (S O)))``; Apply Rlt_Rinv; Apply lt_INR_0; Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. +Apply Rlt_R0_R1. +Cut (lt x (plus (mult (2) (S n)) (1))). +Intro; Assert H5 := (lt_INR ? ? H4); Apply Rlt_trans with ``/(INR x)``. +Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply lt_INR_0; Elim H1; Intros; Assumption. +Apply lt_INR_0; Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. +Assumption. +Elim H1; Intros; Assumption. +Apply lt_le_trans with (S n). +Unfold ge in H2; Apply le_lt_n_Sm; Assumption. +Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Idtac | Ring]. +Apply le_S; Apply le_n_2n. +Apply Rlt_monotony_contra with (INR (mult (S (S O)) (S n))). +Apply lt_INR_0; Replace (mult (2) (S n)) with (S (S (mult (2) n))); [Apply lt_O_Sn | Replace (S n) with (plus n (1)); [Idtac | Ring]; Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace R1 with (INR (1)); [Apply lt_INR | Reflexivity]. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Apply lt_n_S; Apply lt_O_Sn. +Replace (S n) with (plus n (1)); [Ring | Ring]. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Left; Change ``0</(INR (mult (plus (mult (S (S O)) (S n)) (S O)) (mult (S (S O)) (S n))))``; Apply Rlt_Rinv. +Apply lt_INR_0. +Replace (mult (plus (mult (2) (S n)) (1)) (mult (2) (S n))) with (S (S (S (S (S (S (plus (mult (4) (mult n n)) (mult (10) n)))))))). +Apply lt_O_Sn. +Apply INR_eq; Repeat Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite mult_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Replace (INR O) with R0; [Ring | Reflexivity]. +Qed. + +Lemma sin_no_R0 : (n:nat)``(sin_n n)<>0``. +Intro; Unfold sin_n; Unfold Rdiv; Apply prod_neq_R0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Qed. + +(**********) +Definition sin_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(sin_n i)*(pow x i)`` l). + +(**********) +Lemma exist_sin : (x:R)(SigT R [l:R](sin_in x l)). +Intro; Generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin). +Unfold Pser sin_n; Trivial. +Qed. + +(***********************) +(* Definition of sinus *) +Definition sin : R -> R := [x:R](Cases (exist_sin (Rsqr x)) of (Specif.existT a b) => ``x*a`` end). + +(*********************************************) +(* PROPERTIES *) +(*********************************************) + +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_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. + +Lemma sin_0 : ``(sin 0)==0``. +Unfold sin; Case (exist_sin (Rsqr R0)). +Intros; Ring. +Qed. + +Lemma exist_cos0 : (SigT R [l:R](cos_in R0 l)). +Apply Specif.existT with R1. +Unfold cos_in; Unfold infinit_sum; Intros; Exists O. +Intros. +Unfold R_dist. +Induction n. +Unfold cos_n; Simpl. +Unfold Rdiv; Rewrite Rinv_R1. +Do 2 Rewrite Rmult_1r. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Rewrite tech5. +Replace ``(cos_n (S n))*(pow 0 (S n))`` with R0. +Rewrite Rplus_Or. +Apply Hrecn; Unfold ge; Apply le_O_n. +Simpl; Ring. +Defined. + +(* Calculus of (cos 0) *) +Lemma cos_0 : ``(cos 0)==1``. +Cut (cos_in R0 (cos R0)). +Cut (cos_in R0 R1). +Unfold cos_in; Intros; EApply unicity_sum. +Apply H0. +Apply H. +Exact (projT2 ? ? exist_cos0). +Assert H := (projT2 ? ? (exist_cos (Rsqr R0))); Unfold cos; Pattern 1 R0; Replace R0 with (Rsqr R0); [Exact H | Apply Rsqr_O]. +Qed. |