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Diffstat (limited to 'theories/Reals/Binomial.v')
| -rw-r--r-- | theories/Reals/Binomial.v | 204 |
1 files changed, 204 insertions, 0 deletions
diff --git a/theories/Reals/Binomial.v b/theories/Reals/Binomial.v new file mode 100644 index 00000000..e31b623c --- /dev/null +++ b/theories/Reals/Binomial.v @@ -0,0 +1,204 @@ +(************************************************************************) +(* 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: Binomial.v,v 1.9.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import PartSum. +Open Local Scope R_scope. + +Definition C (n p:nat) : R := + INR (fact n) / (INR (fact p) * INR (fact (n - p))). + +Lemma pascal_step1 : forall n i:nat, (i <= n)%nat -> C n i = C n (n - i). +intros; unfold C in |- *; replace (n - (n - i))%nat with i. +rewrite Rmult_comm. +reflexivity. +apply plus_minus; rewrite plus_comm; apply le_plus_minus; assumption. +Qed. + +Lemma pascal_step2 : + forall n i:nat, + (i <= n)%nat -> C (S n) i = INR (S n) / INR (S n - i) * C n i. +intros; unfold C in |- *; replace (S n - i)%nat with (S (n - i)). +cut (forall n:nat, fact (S n) = (S n * fact n)%nat). +intro; repeat rewrite H0. +unfold Rdiv in |- *; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr. +ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply prod_neq_R0. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +intro; reflexivity. +apply minus_Sn_m; assumption. +Qed. + +Lemma pascal_step3 : + forall n i:nat, (i < n)%nat -> C n (S i) = INR (n - i) / INR (S i) * C n i. +intros; unfold C in |- *. +cut (forall n:nat, fact (S n) = (S n * fact n)%nat). +intro. +cut ((n - i)%nat = S (n - S i)). +intro. +pattern (n - i)%nat at 2 in |- *; rewrite H1. +repeat rewrite H0; unfold Rdiv in |- *; repeat rewrite mult_INR; + repeat rewrite Rinv_mult_distr. +rewrite <- H1; rewrite (Rmult_comm (/ INR (n - i))); + repeat rewrite Rmult_assoc; rewrite (Rmult_comm (INR (n - i))); + repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +ring. +apply not_O_INR; apply minus_neq_O; assumption. +apply not_O_INR; discriminate. +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 not_O_INR; discriminate. +apply INR_fact_neq_0. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +apply INR_fact_neq_0. +rewrite minus_Sn_m. +simpl in |- *; reflexivity. +apply lt_le_S; assumption. +intro; reflexivity. +Qed. + +(**********) +Lemma pascal : + forall n i:nat, (i < n)%nat -> C n i + C n (S i) = C (S n) (S i). +intros. +rewrite pascal_step3; [ idtac | assumption ]. +replace (C n i + INR (n - i) / INR (S i) * C n i) with + (C n i * (1 + INR (n - i) / INR (S i))); [ idtac | ring ]. +replace (1 + INR (n - i) / INR (S i)) with (INR (S n) / INR (S i)). +rewrite pascal_step1. +rewrite Rmult_comm; replace (S i) with (S n - (n - i))%nat. +rewrite <- pascal_step2. +apply pascal_step1. +apply le_trans with n. +apply le_minusni_n. +apply lt_le_weak; assumption. +apply le_n_Sn. +apply le_minusni_n. +apply lt_le_weak; assumption. +rewrite <- minus_Sn_m. +cut ((n - (n - i))%nat = i). +intro; rewrite H0; reflexivity. +symmetry in |- *; apply plus_minus. +rewrite plus_comm; rewrite le_plus_minus_r. +reflexivity. +apply lt_le_weak; assumption. +apply le_minusni_n; apply lt_le_weak; assumption. +apply lt_le_weak; assumption. +unfold Rdiv in |- *. +repeat rewrite S_INR. +rewrite minus_INR. +cut (INR i + 1 <> 0). +intro. +apply Rmult_eq_reg_l with (INR i + 1); [ idtac | assumption ]. +rewrite Rmult_plus_distr_l. +rewrite Rmult_1_r. +do 2 rewrite (Rmult_comm (INR i + 1)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym; [ idtac | assumption ]. +ring. +rewrite <- S_INR. +apply not_O_INR; discriminate. +apply lt_le_weak; assumption. +Qed. + +(*********************) +(*********************) +Lemma binomial : + forall (x y:R) (n:nat), + (x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n. +intros; induction n as [| n Hrecn]. +unfold C in |- *; simpl in |- *; unfold Rdiv in |- *; + repeat rewrite Rmult_1_r; rewrite Rinv_1; ring. +pattern (S n) at 1 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add; rewrite Hrecn. +replace ((x + y) ^ 1) with (x + y); [ idtac | simpl in |- *; ring ]. +rewrite tech5. +cut (forall p:nat, C p p = 1). +cut (forall p:nat, C p 0 = 1). +intros; rewrite H0; rewrite <- minus_n_n; rewrite Rmult_1_l. +replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl in |- *; reflexivity ]. +induction n as [| n Hrecn0]. +simpl in |- *; do 2 rewrite H; ring. +(* N >= 1 *) +set (N := S n). +rewrite Rmult_plus_distr_l. +replace (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (N - i)) N * x) with + (sum_f_R0 (fun i:nat => C N i * x ^ S i * y ^ (N - i)) N). +replace (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (N - i)) N * y) with + (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (S N - i)) N). +rewrite (decomp_sum (fun i:nat => C (S N) i * x ^ i * y ^ (S N - i)) N). +rewrite H; replace (x ^ 0) with 1; [ idtac | reflexivity ]. +do 2 rewrite Rmult_1_l. +replace (S N - 0)%nat with (S N); [ idtac | reflexivity ]. +set (An := fun i:nat => C N i * x ^ S i * y ^ (N - i)). +set (Bn := fun i:nat => C N (S i) * x ^ S i * y ^ (N - i)). +replace (pred N) with n. +replace (sum_f_R0 (fun i:nat => C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n) + with (sum_f_R0 (fun i:nat => An i + Bn i) n). +rewrite plus_sum. +replace (x ^ S N) with (An (S n)). +rewrite (Rplus_comm (sum_f_R0 An n)). +repeat rewrite Rplus_assoc. +rewrite <- tech5. +fold N in |- *. +set (Cn := fun i:nat => C N i * x ^ i * y ^ (S N - i)). +cut (forall i:nat, (i < N)%nat -> Cn (S i) = Bn i). +intro; replace (sum_f_R0 Bn n) with (sum_f_R0 (fun i:nat => Cn (S i)) n). +replace (y ^ S N) with (Cn 0%nat). +rewrite <- Rplus_assoc; rewrite (decomp_sum Cn N). +replace (pred N) with n. +ring. +unfold N in |- *; simpl in |- *; reflexivity. +unfold N in |- *; apply lt_O_Sn. +unfold Cn in |- *; rewrite H; simpl in |- *; ring. +apply sum_eq. +intros; apply H1. +unfold N in |- *; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ]. +intros; unfold Bn, Cn in |- *. +replace (S N - S i)%nat with (N - i)%nat; reflexivity. +unfold An in |- *; fold N in |- *; rewrite <- minus_n_n; rewrite H0; + simpl in |- *; ring. +apply sum_eq. +intros; unfold An, Bn in |- *; replace (S N - S i)%nat with (N - i)%nat; + [ idtac | reflexivity ]. +rewrite <- pascal; + [ ring + | apply le_lt_trans with n; [ assumption | unfold N in |- *; apply lt_n_Sn ] ]. +unfold N in |- *; reflexivity. +unfold N in |- *; apply lt_O_Sn. +rewrite <- (Rmult_comm y); rewrite scal_sum; apply sum_eq. +intros; replace (S N - i)%nat with (S (N - i)). +replace (S (N - i)) with (N - i + 1)%nat; [ idtac | ring ]. +rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl in |- *; ring ]; + ring. +apply minus_Sn_m; assumption. +rewrite <- (Rmult_comm x); rewrite scal_sum; apply sum_eq. +intros; replace (S i) with (i + 1)%nat; [ idtac | ring ]; rewrite pow_add; + replace (x ^ 1) with x; [ idtac | simpl in |- *; ring ]; + ring. +intro; unfold C in |- *. +replace (INR (fact 0)) with 1; [ idtac | reflexivity ]. +replace (p - 0)%nat with p; [ idtac | apply minus_n_O ]. +rewrite Rmult_1_l; unfold Rdiv in |- *; rewrite <- Rinv_r_sym; + [ reflexivity | apply INR_fact_neq_0 ]. +intro; unfold C in |- *. +replace (p - p)%nat with 0%nat; [ idtac | apply minus_n_n ]. +replace (INR (fact 0)) with 1; [ idtac | reflexivity ]. +rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym; + [ reflexivity | apply INR_fact_neq_0 ]. +Qed.
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