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Diffstat (limited to 'theories7/Reals/Rsigma.v')
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diff --git a/theories7/Reals/Rsigma.v b/theories7/Reals/Rsigma.v deleted file mode 100644 index f9e8e92b..00000000 --- a/theories7/Reals/Rsigma.v +++ /dev/null @@ -1,117 +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: Rsigma.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) - -Require Rbase. -Require Rfunctions. -Require Rseries. -Require PartSum. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. -Open Local Scope R_scope. - -Set Implicit Arguments. - -Section Sigma. - -Variable f : nat->R. - -Definition sigma [low,high:nat] : R := (sum_f_R0 [k:nat](f (plus low k)) (minus high low)). - -Theorem sigma_split : (low,high,k:nat) (le low k)->(lt k high)->``(sigma low high)==(sigma low k)+(sigma (S k) high)``. -Intros; Induction k. -Cut low = O. -Intro; Rewrite H1; Unfold sigma; Rewrite <- minus_n_n; Rewrite <- minus_n_O; Simpl; Replace (minus high (S O)) with (pred high). -Apply (decomp_sum [k:nat](f k)). -Assumption. -Apply pred_of_minus. -Inversion H; Reflexivity. -Cut (le low k)\/low=(S k). -Intro; Elim H1; Intro. -Replace (sigma low (S k)) with ``(sigma low k)+(f (S k))``. -Rewrite Rplus_assoc; Replace ``(f (S k))+(sigma (S (S k)) high)`` with (sigma (S k) high). -Apply Hreck. -Assumption. -Apply lt_trans with (S k); [Apply lt_n_Sn | Assumption]. -Unfold sigma; Replace (minus high (S (S k))) with (pred (minus high (S k))). -Pattern 3 (S k); Replace (S k) with (plus (S k) O); [Idtac | Ring]. -Replace (sum_f_R0 [k0:nat](f (plus (S (S k)) k0)) (pred (minus high (S k)))) with (sum_f_R0 [k0:nat](f (plus (S k) (S k0))) (pred (minus high (S k)))). -Apply (decomp_sum [i:nat](f (plus (S k) i))). -Apply lt_minus_O_lt; Assumption. -Apply sum_eq; Intros; Replace (plus (S k) (S i)) with (plus (S (S k)) i). -Reflexivity. -Apply INR_eq; Do 2 Rewrite plus_INR; Do 3 Rewrite S_INR; Ring. -Replace (minus high (S (S k))) with (minus (minus high (S k)) (S O)). -Apply pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Do 4 Rewrite S_INR; Ring. -Apply lt_le_S; Assumption. -Apply lt_le_weak; Assumption. -Apply lt_le_S; Apply lt_minus_O_lt; Assumption. -Unfold sigma; Replace (minus (S k) low) with (S (minus k low)). -Pattern 1 (S k); Replace (S k) with (plus low (S (minus k low))). -Symmetry; Apply (tech5 [i:nat](f (plus low i))). -Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite minus_INR. -Ring. -Assumption. -Apply minus_Sn_m; Assumption. -Rewrite <- H2; Unfold sigma; Rewrite <- minus_n_n; Simpl; Replace (minus high (S low)) with (pred (minus high low)). -Replace (sum_f_R0 [k0:nat](f (S (plus low k0))) (pred (minus high low))) with (sum_f_R0 [k0:nat](f (plus low (S k0))) (pred (minus high low))). -Apply (decomp_sum [k0:nat](f (plus low k0))). -Apply lt_minus_O_lt. -Apply le_lt_trans with (S k); [Rewrite H2; Apply le_n | Assumption]. -Apply sum_eq; Intros; Replace (S (plus low i)) with (plus low (S i)). -Reflexivity. -Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. -Replace (minus high (S low)) with (minus (minus high low) (S O)). -Apply pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Do 2 Rewrite S_INR; Ring. -Apply lt_le_S; Rewrite H2; Assumption. -Rewrite H2; Apply lt_le_weak; Assumption. -Apply lt_le_S; Apply lt_minus_O_lt; Rewrite H2; Assumption. -Inversion H; [ - Right; Reflexivity -| Left; Assumption]. -Qed. - -Theorem sigma_diff : (low,high,k:nat) (le low k) -> (lt k high )->``(sigma low high)-(sigma low k)==(sigma (S k) high)``. -Intros low high k H1 H2; Symmetry; Rewrite -> (sigma_split H1 H2); Ring. -Qed. - -Theorem sigma_diff_neg : (low,high,k:nat) (le low k) -> (lt k high)-> ``(sigma low k)-(sigma low high)==-(sigma (S k) high)``. -Intros low high k H1 H2; Rewrite -> (sigma_split H1 H2); Ring. -Qed. - -Theorem sigma_first : (low,high:nat) (lt low high) -> ``(sigma low high)==(f low)+(sigma (S low) high)``. -Intros low high H1; Generalize (lt_le_S low high H1); Intro H2; Generalize (lt_le_weak low high H1); Intro H3; Replace ``(f low)`` with ``(sigma low low)``. -Apply sigma_split. -Apply le_n. -Assumption. -Unfold sigma; Rewrite <- minus_n_n. -Simpl. -Replace (plus low O) with low; [Reflexivity | Ring]. -Qed. - -Theorem sigma_last : (low,high:nat) (lt low high) -> ``(sigma low high)==(f high)+(sigma low (pred high))``. -Intros low high H1; Generalize (lt_le_S low high H1); Intro H2; Generalize (lt_le_weak low high H1); Intro H3; Replace ``(f high)`` with ``(sigma high high)``. -Rewrite Rplus_sym; Cut high = (S (pred high)). -Intro; Pattern 3 high; Rewrite H. -Apply sigma_split. -Apply le_S_n; Rewrite <- H; Apply lt_le_S; Assumption. -Apply lt_pred_n_n; Apply le_lt_trans with low; [Apply le_O_n | Assumption]. -Apply S_pred with O; Apply le_lt_trans with low; [Apply le_O_n | Assumption]. -Unfold sigma; Rewrite <- minus_n_n; Simpl; Replace (plus high O) with high; [Reflexivity | Ring]. -Qed. - -Theorem sigma_eq_arg : (low:nat) (sigma low low)==(f low). -Intro; Unfold sigma; Rewrite <- minus_n_n. -Simpl; Replace (plus low O) with low; [Reflexivity | Ring]. -Qed. - -End Sigma. |