From 3ef7797ef6fc605dfafb32523261fe1b023aeecb Mon Sep 17 00:00:00 2001 From: Samuel Mimram Date: Fri, 28 Apr 2006 14:59:16 +0000 Subject: Imported Upstream version 8.0pl3+8.1alpha --- theories7/Reals/Rprod.v | 164 ------------------------------------------------ 1 file changed, 164 deletions(-) delete mode 100644 theories7/Reals/Rprod.v (limited to 'theories7/Reals/Rprod.v') diff --git a/theories7/Reals/Rprod.v b/theories7/Reals/Rprod.v deleted file mode 100644 index a524a915..00000000 --- a/theories7/Reals/Rprod.v +++ /dev/null @@ -1,164 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* R;N:nat] : R := Cases N of - O => R1 -| (S p) => ``(prod_f_SO An p)*(An (S p))`` -end. - -(**********) -Lemma prod_SO_split : (An:nat->R;n,k:nat) (le k n) -> (prod_f_SO An n)==(Rmult (prod_f_SO An k) (prod_f_SO [l:nat](An (plus k l)) (minus n k))). -Intros; Induction n. -Cut k=O; [Intro; Rewrite H0; Simpl; Ring | Inversion H; Reflexivity]. -Cut k=(S n)\/(le k n). -Intro; Elim H0; Intro. -Rewrite H1; Simpl; Rewrite <- minus_n_n; Simpl; Ring. -Replace (minus (S n) k) with (S (minus n k)). -Simpl; Replace (plus k (S (minus n k))) with (S n). -Rewrite Hrecn; [Ring | Assumption]. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite S_INR; Rewrite minus_INR; [Ring | Assumption]. -Apply INR_eq; Rewrite S_INR; Repeat Rewrite minus_INR. -Rewrite S_INR; Ring. -Apply le_trans with n; [Assumption | Apply le_n_Sn]. -Assumption. -Inversion H; [Left; Reflexivity | Right; Assumption]. -Qed. - -(**********) -Lemma prod_SO_pos : (An:nat->R;N:nat) ((n:nat)(le n N)->``0<=(An n)``) -> ``0<=(prod_f_SO An N)``. -Intros; Induction N. -Simpl; Left; Apply Rlt_R0_R1. -Simpl; Apply Rmult_le_pos. -Apply HrecN; Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]. -Apply H; Apply le_n. -Qed. - -(**********) -Lemma prod_SO_Rle : (An,Bn:nat->R;N:nat) ((n:nat)(le n N)->``0<=(An n)<=(Bn n)``) -> ``(prod_f_SO An N)<=(prod_f_SO Bn N)``. -Intros; Induction N. -Right; Reflexivity. -Simpl; Apply Rle_trans with ``(prod_f_SO An N)*(Bn (S N))``. -Apply Rle_monotony. -Apply prod_SO_pos; Intros; Elim (H n (le_trans ? ? ? H0 (le_n_Sn N))); Intros; Assumption. -Elim (H (S N) (le_n (S N))); Intros; Assumption. -Do 2 Rewrite <- (Rmult_sym (Bn (S N))); Apply Rle_monotony. -Elim (H (S N) (le_n (S N))); Intros. -Apply Rle_trans with (An (S N)); Assumption. -Apply HrecN; Intros; Elim (H n (le_trans ? ? ? H0 (le_n_Sn N))); Intros; Split; Assumption. -Qed. - -(* Application to factorial *) -Lemma fact_prodSO : (n:nat) (INR (fact n))==(prod_f_SO [k:nat](INR k) n). -Intro; Induction n. -Reflexivity. -Change (INR (mult (S n) (fact n)))==(prod_f_SO ([k:nat](INR k)) (S n)). -Rewrite mult_INR; Rewrite Rmult_sym; Rewrite Hrecn; Reflexivity. -Qed. - -Lemma le_n_2n : (n:nat) (le n (mult (2) n)). -Induction n. -Replace (mult (2) (O)) with O; [Apply le_n | Ring]. -Intros; Replace (mult (2) (S n0)) with (S (S (mult (2) n0))). -Apply le_n_S; Apply le_S; Assumption. -Replace (S (S (mult (2) n0))) with (plus (mult (2) n0) (2)); [Idtac | Ring]. -Replace (S n0) with (plus n0 (1)); [Idtac | Ring]. -Ring. -Qed. - -(* We prove that (N!)²<=(2N-k)!*k! forall k in [|O;2N|] *) -Lemma RfactN_fact2N_factk : (N,k:nat) (le k (mult (2) N)) -> ``(Rsqr (INR (fact N)))<=(INR (fact (minus (mult (S (S O)) N) k)))*(INR (fact k))``. -Intros; Unfold Rsqr; Repeat Rewrite fact_prodSO. -Cut (le k N)\/(le N k). -Intro; Elim H0; Intro. -Rewrite (prod_SO_split [l:nat](INR l) (minus (mult (2) N) k) N). -Rewrite Rmult_assoc; Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Replace (minus (minus (mult (2) N) k) N) with (minus N k). -Rewrite Rmult_sym; Rewrite (prod_SO_split [l:nat](INR l) N k). -Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Apply prod_SO_Rle; Intros; Split. -Apply pos_INR. -Apply le_INR; Apply le_reg_r; Assumption. -Assumption. -Apply INR_eq; Repeat Rewrite minus_INR. -Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply le_trans with N; [Assumption | Apply le_n_2n]. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]. -Apply le_reg_r; Assumption. -Assumption. -Assumption. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]. -Apply le_reg_r; Assumption. -Assumption. -Rewrite <- (Rmult_sym (prod_f_SO [l:nat](INR l) k)); Rewrite (prod_SO_split [l:nat](INR l) k N). -Rewrite Rmult_assoc; Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Rewrite Rmult_sym; Rewrite (prod_SO_split [l:nat](INR l) N (minus (mult (2) N) k)). -Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Replace (minus N (minus (mult (2) N) k)) with (minus k N). -Apply prod_SO_Rle; Intros; Split. -Apply pos_INR. -Apply le_INR; Apply le_reg_r. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. -Assumption. -Apply INR_eq; Repeat Rewrite minus_INR. -Rewrite mult_INR; Do 2 Rewrite S_INR; Ring. -Assumption. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. -Assumption. -Assumption. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. -Assumption. -Assumption. -Elim (le_dec k N); Intro; [Left; Assumption | Right; Assumption]. -Qed. - -(**********) -Lemma INR_fact_lt_0 : (n:nat) ``0<(INR (fact n))``. -Intro; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Elim (fact_neq_0 n); Symmetry; Assumption. -Qed. - -(* We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *) -Lemma C_maj : (N,k:nat) (le k (mult (2) N)) -> ``(C (mult (S (S O)) N) k)<=(C (mult (S (S O)) N) N)``. -Intros; Unfold C; Unfold Rdiv; Apply Rle_monotony. -Apply pos_INR. -Replace (minus (mult (2) N) N) with N. -Apply Rle_monotony_contra with ``((INR (fact N))*(INR (fact N)))``. -Apply Rmult_lt_pos; Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_sym; Apply Rle_monotony_contra with ``((INR (fact k))* - (INR (fact (minus (mult (S (S O)) N) k))))``. -Apply Rmult_lt_pos; Apply INR_fact_lt_0. -Rewrite Rmult_1r; Rewrite <- mult_INR; Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite mult_INR; Rewrite (Rmult_sym (INR (fact k))); Replace ``(INR (fact N))*(INR (fact N))`` with (Rsqr (INR (fact N))). -Apply RfactN_fact2N_factk. -Assumption. -Reflexivity. -Rewrite mult_INR; Apply prod_neq_R0; Apply INR_fact_neq_0. -Apply prod_neq_R0; Apply INR_fact_neq_0. -Apply INR_eq; Rewrite minus_INR; [Rewrite mult_INR; Do 2 Rewrite S_INR; Ring | Apply le_n_2n]. -Qed. -- cgit v1.2.3