Proof Assistant Projects
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Digesting proof assistant libraries for AI ingestion. • 84 items • Updated • 3
fact stringlengths 6 40.2k | type stringclasses 14
values | library stringclasses 2
values | imports listlengths 2 4 | filename stringclasses 28
values | symbolic_name stringlengths 1 36 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
pos_rat := repeat ( apply Rdiv_lt_0_compat || apply Rplus_lt_0_compat || apply Rmult_lt_0_compat) ; try by apply Rlt_0_1. | Ltac | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | pos_rat | |
sign_0_lt : forall x, 0 < x <-> 0 < sign x. Proof. intros x. unfold sign. destruct total_order_T as [[H|H]|H] ; lra. Qed. | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | sign_0_lt | |
sign_lt_0 : forall x, x < 0 <-> sign x < 0. Proof. intros x. unfold sign. destruct total_order_T as [[H|H]|H] ; lra. Qed. (** * Exercice 2 *) (* 8:14 *) | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | sign_lt_0 | |
fab (a b x : R) : R := (a + b * ln x) / x. (** ** Questions 1 *) (** 1.a. On voit sur le graphique que l'image de 1 par f correspond au point B(1,2). On a donc f(1) = 2. Comme la tangente (BC) à la courbe en ce point admet pour coefficient directeur 0, f'(1) = 0 *) (** 1.b *) | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | fab | |
Dfab (a b : R) : forall x, 0 < x -> is_derive (fab a b) x (((b - a) - b * ln x) / x ^ 2). Proof. move => x Hx. evar_last. apply is_derive_div. apply @is_derive_plus. apply is_derive_const. apply is_derive_scal. now apply is_derive_Reals, derivable_pt_lim_ln. apply is_derive_id. by apply Rgt_not_eq. rewrite /Rdiv /plus ... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Dfab | |
Val_a_b (a b : R) : fab a b 1 = 2 -> Derive (fab a b) 1 = 0 -> a = 2 /\ b = 2. Proof. move => Hf Hdf. rewrite /fab in Hf. rewrite ln_1 in Hf. rewrite Rdiv_1 in Hf. rewrite Rmult_0_r in Hf. rewrite Rplus_0_r in Hf. rewrite Hf in Hdf |- * => {a Hf}. split. reflexivity. replace (Derive (fab 2 b) 1) with (((b - 2) - b * ln... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Val_a_b | |
f (x : R) : R := fab 2 2 x. (** ** Questions 2 *) (* 8:38 *) (** 2.a. *) | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | f | |
Signe_df : forall x, 0 < x -> sign (Derive f x) = sign (- ln x). Proof. move => x Hx. rewrite (is_derive_unique f x _ (Dfab 2 2 x Hx)). replace ((2 - 2 - 2 * ln x) / x ^ 2) with (2 / x ^ 2 * (- ln x)) by (field ; now apply Rgt_not_eq). rewrite sign_mult sign_eq_1. apply Rmult_1_l. apply Rdiv_lt_0_compat. apply Rlt_0_2.... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Signe_df | |
filterlim_f_0 : filterlim f (at_right 0) (Rbar_locally m_infty). Proof. unfold f, fab. eapply (filterlim_comp_2 _ _ Rmult). eapply filterlim_comp_2. apply filterlim_const. eapply filterlim_comp_2. apply filterlim_const. by apply is_lim_ln_0. apply (filterlim_Rbar_mult 2 m_infty m_infty). unfold is_Rbar_mult, Rbar_mult'... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | filterlim_f_0 | |
Lim_f_p_infty : is_lim f p_infty 0. Proof. apply is_lim_ext_loc with (fun x => 2 / x + 2 * (ln x / x)). exists 0. move => y Hy. rewrite /f /fab. field. by apply Rgt_not_eq. eapply is_lim_plus. apply is_lim_scal_l. apply is_lim_inv. by apply is_lim_id. by []. apply is_lim_scal_l. by apply is_lim_div_ln_p. unfold is_Rbar... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Lim_f_p_infty | |
Variation_1 : forall x y, 0 < x -> x < y -> y < 1 -> f x < f y. Proof. apply (incr_function _ 0 1 (fun x => (2 - 2 - 2 * ln x) / x ^ 2)). move => x H0x Hx1. by apply (Dfab 2 2 x). move => x H0x Hx1. apply sign_0_lt. rewrite -(is_derive_unique _ _ _ (Dfab 2 2 x H0x)). rewrite Signe_df. apply -> sign_0_lt. apply Ropp_lt_... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Variation_1 | |
Variation_2 : forall x y, 1 < x -> x < y -> f x > f y. Proof. move => x y H1x Hxy. apply Ropp_lt_cancel. apply (incr_function (fun x => - f x) 1 p_infty (fun z => - ((2 - 2 - 2 * ln z) / z ^ 2))). move => z H1z _. apply: is_derive_opp. apply (Dfab 2 2 z). by apply Rlt_trans with (1 := Rlt_0_1). move => z H1z _. apply R... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Variation_2 | |
f_eq_1_0_1 : exists x, 0 < x <= 1 /\ f x = 1. Proof. case: (IVT_Rbar_incr (fun x => f (Rabs x)) 0 1 m_infty 2 1). eapply filterlim_comp. apply filterlim_Rabs_0. by apply filterlim_f_0. apply is_lim_comp with 1. replace 2 with (f 1). apply is_lim_continuity. apply derivable_continuous_pt. exists (((2 - 2) - 2 * ln 1) / ... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | f_eq_1_0_1 | |
f_eq_1_1_p_infty : exists x, 1 <= x /\ f x = 1. Proof. case: (IVT_Rbar_incr (fun x => - f x) 1 p_infty (-2) 0 (-1)). replace (-2) with (-f 1). apply (is_lim_continuity (fun x => - f x)). apply continuity_pt_opp. apply derivable_continuous_pt. exists (((2 - 2) - 2 * ln 1) / 1 ^ 2) ; apply is_derive_Reals, Dfab. by apply... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | f_eq_1_1_p_infty | |
If : forall x, 0 < x -> is_derive (fun y : R => 2 * ln y + (ln y) ^ 2) x (f x). Proof. move => y Hy. evar_last. apply @is_derive_plus. apply is_derive_Reals. apply derivable_pt_lim_scal. by apply derivable_pt_lim_ln. apply is_derive_pow. by apply is_derive_Reals, derivable_pt_lim_ln. rewrite /f /fab /plus /= ; field. b... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | If | |
RInt_f : is_RInt f ( / exp 1) 1 1. Proof. have Haux1: (0 < /exp 1). apply Rinv_0_lt_compat. apply exp_pos. evar_last. apply: is_RInt_derive. move => x Hx. apply If. apply Rlt_le_trans with (2 := proj1 Hx). apply Rmin_case. by apply Haux1. by apply Rlt_0_1. move => x Hx. apply continuity_pt_filterlim. apply derivable_co... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | RInt_f | |
u (n : nat) : R := match n with | O => 2 | S n => 2/3 * u n + 1/3 * (INR n) + 1 end. (** ** Questions 1 *) (** 1.a. *) (** 1.b. *) (** ** Questions 2 *) (* 10:40 *) (** 2.a *) | Fixpoint | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | u | |
Q2a : forall n, u n <= INR n + 3. Proof. elim => [ | n IH] ; rewrite ?S_INR /=. apply Rminus_le_0 ; ring_simplify ; apply Rle_0_1. eapply Rle_trans. apply Rplus_le_compat_r. apply Rplus_le_compat_r. apply Rmult_le_compat_l. lra. by apply IH. lra. Qed. (** 2.b. *) | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q2a | |
Q2b : forall n, u (S n) - u n = 1/3 * (INR n + 3 - u n). Proof. move => n ; simpl. field. Qed. (** 2.c. *) | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q2b | |
Q2c : forall n, u n <= u (S n). Proof. move => n. apply Rminus_le_0. rewrite Q2b. apply Rmult_le_pos. lra. apply (Rminus_le_0 (u n)). by apply Q2a. Qed. (** ** Question 3 *) (* 10:49 *) | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q2c | |
v (n : nat) : R := u n - INR n. (** 3.a. *) | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | v | |
Q3a : forall n, v n = 2 * (2/3) ^ n. Proof. elim => [ | n IH]. rewrite /v /u /= ; ring. replace (2 * (2 / 3) ^ S n) with (v n * (2/3)) by (rewrite IH /= ; ring). rewrite /v S_INR /=. field. Qed. (** 3.b. *) | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q3a | |
Q3b : forall n, u n = 2 * (2/3)^n + INR n. Proof. move => n. rewrite -Q3a /v ; ring. Qed. | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q3b | |
Q3c : is_lim_seq u p_infty. Proof. apply is_lim_seq_ext with (fun n => 2 * (2/3)^n + INR n). move => n ; by rewrite Q3b. eapply is_lim_seq_plus. eapply is_lim_seq_mult. by apply is_lim_seq_const. apply is_lim_seq_geom. rewrite Rabs_pos_eq. lra. lra. by []. apply is_lim_seq_INR. by []. Qed. (** ** Questions 4 *) (* 11:0... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q3c | |
Su (n : nat) : R := sum_f_R0 u n. | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Su | |
Tu (n : nat) : R := Su n / (INR n) ^ 2. (** 4.a. *) | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Tu | |
Q4a : forall n, Su n = 6 - 4 * (2/3)^n + INR n * (INR n + 1) / 2. Proof. move => n. rewrite /Su. rewrite -(sum_eq (fun n => (2/3)^n * 2 + INR n)). rewrite sum_plus. rewrite -scal_sum. rewrite tech3. rewrite sum_INR. simpl ; field. apply Rlt_not_eq, Rlt_div_l. repeat apply Rplus_lt_0_compat ; apply Rlt_0_1. apply Rminus... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q4a | |
Q4b : is_lim_seq Tu (1/2). Proof. apply is_lim_seq_ext_loc with (fun n => (6 - 4 * (2/3)^n) / (INR n ^2) + / (2 * INR n) + /2). exists 1%nat => n Hn ; rewrite /Tu Q4a. simpl ; field. apply Rgt_not_eq, (lt_INR O) ; intuition. eapply is_lim_seq_plus. eapply is_lim_seq_plus. eapply is_lim_seq_div. eapply is_lim_seq_minus.... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive RInt Continuity Lim_seq ElemFct RInt_analysis."
] | examples/BacS2013.v | Q4b | |
v (n : nat) : R := match n with | O => 7 / 10 * 250000 | S n => 95 / 100 * v n + 1 / 100 * c n end with c (n : nat) : R := match n with | O => 3 / 10 * 250000 | S n => 5 / 100 * v n + 99 / 100 * c n end. (** 2. Définition de la matrice A *) | Fixpoint | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | v | |
A : matrix 2 2 := [[95/100, 1/100 ] , [ 5/100, 99/100]]. | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | A | |
X (n : nat) : matrix 2 1 := [[v n],[c n]]. | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | X | |
Q2 : forall n, X (S n) = scal A (X n). Proof. intros n. rewrite /scal /= /Mmult. apply (coeff_mat_ext 0). case ; [ | case => //]. case ; [ | case => //] ; rewrite coeff_mat_bij /= ; (try lia) ; rewrite sum_Sn sum_O /plus /mult //=. case ; [ | case => //] ; rewrite coeff_mat_bij /= ; (try lia) ; rewrite sum_Sn sum_O /pl... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | Q2 | |
P : matrix 2 2 := [[1,-1], [5,1]]. | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | P | |
Q : matrix 2 2 := [[1,1],[-5,1]]. Goal mult P Q = [[6,0],[0,6]]. apply (coeff_mat_ext_aux 0 0) => i j Hi Hj. rewrite coeff_mat_bij => //. rewrite /coeff_mat /= /mult /plus /=. (destruct i as [ | i] ; destruct j as [ | j] ; rewrite /zero /= ; try ring) ; (try (destruct i as [ | i]) ; try (destruct j as [ | j]) ; rewrite... | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | Q | |
P' : matrix 2 2 := [[1 / 6,1 / 6],[-5 / 6,1 / 6]]. | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | P' | |
Q3a : mult P P' = Mone /\ mult P' P = Mone. Proof. split. apply (coeff_mat_ext_aux 0 0) => i j Hi Hj. rewrite coeff_mat_bij => //. rewrite /coeff_mat /= /mult /plus /=. (destruct i as [ | i] ; destruct j as [ | j] ; rewrite /zero /= ; try field) ; (try (destruct i as [ | i]) ; try (destruct j as [ | j]) ; rewrite /zero... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | Q3a | |
D : matrix 2 2 := [[1,0],[0,94 / 100]]. | Definition | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | D | |
Q3b : mult P' (mult A P) = D. Proof. apply (coeff_mat_ext_aux 0 0) => i j Hi Hj. rewrite coeff_mat_bij => //. rewrite /coeff_mat /= /mult /plus /=. (destruct i as [ | i] ; destruct j as [ | j] ; rewrite /zero /= ; try field) ; (try (destruct i as [ | i]) ; try (destruct j as [ | j]) ; rewrite /zero /one /= ; try field)... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | Q3b | |
Q3c : forall n, pow_n A n = mult P (mult (pow_n D n) P'). Proof. elim => /= [ | n IH]. rewrite mult_one_l. apply sym_eq, Q3a. by rewrite -{1}Q3b !mult_assoc (proj1 Q3a) mult_one_l -!mult_assoc IH. Qed. (** 4. Terme général et limite de la suite v n *) | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | Q3c | |
Q4 : forall n, v n = 1 / 6 * (1 + 5 * (94 / 100) ^ n) * v 0 + 1 / 6 * (1 - (94 / 100) ^ n) * c 0. Proof. intros n. assert (X n = scal (pow_n A n) (X 0)). elim: n => [ | n IH] /=. by rewrite scal_one. rewrite -scal_assoc -IH. by apply Q2. assert (pow_n D n = [[1,0], [0,(94 / 100)^n]]). elim: (n) => [ | m IH] //=. rewrit... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | Q4 | |
lim_v : is_lim_seq v (41666 + 2 / 3). Proof. eapply is_lim_seq_ext. intros n ; apply sym_eq, Q4. eapply is_lim_seq_plus. eapply is_lim_seq_mult. eapply is_lim_seq_mult. apply is_lim_seq_const. eapply is_lim_seq_plus. apply is_lim_seq_const. eapply is_lim_seq_mult. apply is_lim_seq_const. apply is_lim_seq_geom. rewrite ... | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | lim_v | |
lim_c : is_lim_seq c (208333 + 1 / 3). Proof. assert (forall n, c n = 250000 - v n). elim => [ | n /= ->] /= ; field. eapply is_lim_seq_ext. intros n ; apply sym_eq, H. eapply is_lim_seq_minus. apply is_lim_seq_const. by apply lim_v. apply (f_equal (fun x => Some (Finite x))) ; simpl ; field. Qed. | Lemma | examples | [
"From Coq Require Import Reals Psatz ssreflect.",
"From Coquelicot Require Import Hierarchy PSeries Rbar Lim_seq."
] | examples/BacS2013_bonus.v | lim_c | |
Bessel1_seq (n k : nat) := (-1)^(k)/(INR (fact (k)) * INR (fact (n + (k)))). | Definition | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_seq | |
Bessel1_seq_neq_0 (n : nat) : forall k, Bessel1_seq n k <> 0. Proof. move => k. apply Rmult_integral_contrapositive_currified. apply pow_nonzero, Ropp_neq_0_compat, R1_neq_R0. apply Rinv_neq_0_compat, Rmult_integral_contrapositive_currified ; apply INR_fact_neq_0. Qed. | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_seq_neq_0 | |
CV_Bessel1 (n : nat) : CV_radius (Bessel1_seq n) = p_infty. Proof. apply CV_radius_infinite_DAlembert. by apply Bessel1_seq_neq_0. apply is_lim_seq_ext with (fun p => / (INR (S p) * INR (S (n + p)))). move => p ; rewrite /Bessel1_seq -plus_n_Sm /fact -/fact !mult_INR. simpl ((-1)^(S p)). field_simplify (-1 * (-1) ^ p /... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | CV_Bessel1 | |
ex_Bessel1 (n : nat) (x : R) : ex_pseries (Bessel1_seq n) x. Proof. apply CV_radius_inside. by rewrite CV_Bessel1. Qed. | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | ex_Bessel1 | |
Bessel1 (n : nat) (x : R) := (x/2)^n * PSeries (Bessel1_seq n) ((x/2)^2). | Definition | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1 | |
is_derive_Bessel1 (n : nat) (x : R) : is_derive (Bessel1 n) x ((x / 2) ^ S n * PSeries (PS_derive (Bessel1_seq n)) ((x / 2) ^ 2) + (INR n)/2 * (x / 2) ^ pred n * PSeries (Bessel1_seq n) ((x / 2) ^ 2)). Proof. rewrite /Bessel1. auto_derive. apply ex_derive_PSeries. by rewrite CV_Bessel1. rewrite Derive_PSeries. rewrite ... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | is_derive_Bessel1 | |
is_derive_2_Bessel1 (n : nat) (x : R) : is_derive_n (Bessel1 n) 2 x (((x/2)^(S (S n)) * PSeries (PS_derive (PS_derive (Bessel1_seq n))) ((x / 2) ^ 2)) + ((INR (2*n+1)/2) * (x/2)^n * PSeries (PS_derive (Bessel1_seq n)) ((x / 2) ^ 2)) + (INR (n * pred n) / 4 * (x / 2) ^ pred (pred n) * PSeries (Bessel1_seq n) ((x / 2) ^ ... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | is_derive_2_Bessel1 | |
Bessel1_correct (n : nat) (x : R) : x^2 * Derive_n (Bessel1 n) 2 x + x * Derive (Bessel1 n) x + (x^2 - (INR n)^2) * Bessel1 n x = 0. Proof. rewrite (is_derive_unique _ _ _ (is_derive_Bessel1 _ _)) ; rewrite /Derive_n (is_derive_unique _ _ _ (is_derive_2_Bessel1 _ _)) ; rewrite /Bessel1 plus_INR ?mult_INR ; simpl INR. s... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_correct | |
Bessel1_equality_1 (n : nat) (x : R) : x <> 0 -> Bessel1 (S n)%nat x = INR n * Bessel1 n x / x - Derive (Bessel1 n) x. Proof. move => Hx. rewrite (is_derive_unique _ _ _ (is_derive_Bessel1 _ _)) /Bessel1. set y := (x / 2). replace x with (2 * y) by (unfold y ; field). (* Supprimer les PSeries *) have Hy : y <> 0. unfol... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_equality_1 | |
Bessel1_equality_2 (n : nat) (x : R) : (0 < n)%nat -> x<>0 -> Bessel1 (S n)%nat x + Bessel1 (pred n)%nat x = (2*INR n)/x * Bessel1 n x. Proof. case: n => [ | n] Hn Hx. by apply Nat.lt_irrefl in Hn. clear Hn ; simpl pred. rewrite /Bessel1 S_INR. replace ((x / 2) ^ S (S n) * PSeries (Bessel1_seq (S (S n))) ((x / 2) ^ 2) ... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_equality_2 | |
Bessel1_equality_3 (n : nat) (x : R) : (0 < n)%nat -> Bessel1 (S n)%nat x - Bessel1 (pred n)%nat x = - 2 * Derive (Bessel1 n) x. Proof. move => Hn. rewrite (is_derive_unique _ _ _ (is_derive_Bessel1 _ _)) /Bessel1. case: n Hn => [ | n] Hn. by apply Nat.lt_irrefl in Hn. clear Hn ; simpl pred. replace ((x / 2) ^ S (S n) ... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_equality_3 | |
Bessel1_uniqueness_aux_0 (a : nat -> R) (n : nat) : Rbar_lt 0 (CV_radius a) -> (forall x : R, Rbar_lt (Rabs x) (CV_radius a) -> x^2 * Derive_n (PSeries a) 2 x + x * Derive (PSeries a) x + (x^2 - (INR n)^2) * PSeries a x = 0) -> (a 0%nat = 0 \/ n = O) /\ (a 1%nat = 0 \/ n = 1%nat) /\ (forall k, (INR (S (S k)) ^ 2 - INR ... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_uniqueness_aux_0 | |
Bessel1_uniqueness_aux_1 (a : nat -> R) (n : nat) : (a 0%nat = 0 \/ n = O) -> (a 1%nat = 0 \/ n = 1%nat) -> (forall k, (INR (S (S k)) ^ 2 - INR n ^ 2) * a (S (S k)) + a k = 0) -> (forall k : nat, (k < n)%nat -> a k = 0) /\ (forall p : nat, a (n + 2 * p + 1)%nat = 0) /\ (forall p : nat, a (n + 2 * p)%nat = Bessel1_seq n... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_uniqueness_aux_1 | |
Bessel1_uniqueness (a : nat -> R) (n : nat) : (Rbar_lt 0 (CV_radius a)) -> (forall x : R, x^2 * Derive_n (PSeries a) 2 x + x * Derive (PSeries a) x + (x^2 - (INR n)^2) * PSeries a x = 0) -> {b : R | forall x, PSeries a x = b * Bessel1 n x}. Proof. intros Hcv_a Ha. assert ((a 0%nat = 0 \/ n = O) /\ (a 1%nat = 0 \/ n = 1... | Lemma | examples | [
"From Coq Require Import Arith Reals Psatz ssreflect.",
"From Coquelicot Require Import Rcomplements Rbar Hierarchy Derive Series PSeries Lim_seq AutoDerive."
] | examples/Bessel.v | Bessel1_uniqueness | |
auto_derive_2 := match goal with | |- is_derive_n ?f 2 ?x ?d => auto_derive_fun f ; match goal with | |- (forall x, _ -> is_derive _ x (@?d x)) -> _ => let H := fresh "H" in let u := fresh "u" in intro H ; apply (is_derive_ext d) ; [ intro u ; apply sym_eq, is_derive_unique ; apply H | auto_derive ] ; clear H end end. | Ltac | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | auto_derive_2 | |
c : R. Hypothesis Zc : c <> 0. | Parameter | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | c | |
u0 : R -> R. Hypothesis Du0 : forall x, ex_derive (fun u => u0 u) x. Hypothesis D2u0 : forall x, ex_derive_n (fun u => u0 u) 2 x. | Parameter | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | u0 | |
alpha x t := 1/2 * (u0 (x + c * t) + u0 (x - c * t)). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | alpha | |
alpha20 x t := 1/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | alpha20 | |
alpha02 x t := c^2/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | alpha02 | |
alpha_20_lim : forall x t, is_derive_n (fun u => alpha u t) 2 x (alpha20 x t). Proof. intros x t. unfold alpha. auto_derive_2. repeat split ; apply Du0. repeat split ; apply D2u0. unfold alpha20, Derive_n, Rminus. ring. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | alpha_20_lim | |
alpha_02_lim : forall x t, is_derive_n (fun u => alpha x u) 2 t (alpha02 x t). Proof. intros x t. unfold alpha. auto_derive_2. repeat split ; apply Du0. repeat split ; apply D2u0. unfold alpha02, Derive_n, Rminus, Rdiv. ring. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | alpha_02_lim | |
u1 : R -> R. Hypothesis Du1 : forall x, ex_derive (fun u => u1 u) x. | Parameter | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | u1 | |
Cu1 : forall x, continuity_pt (fun u => u1 u) x. intros x. destruct (Du1 x) as (l,Hl). apply derivable_continuous_pt. unfold derivable_pt, derivable_pt_abs. exists l. now apply is_derive_Reals. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | Cu1 | |
continuity_implies_ex_Rint : forall f a b, (forall x, continuity_pt f x) -> ex_RInt f a b. intros f a b H. case (Rle_or_lt a b); intros H1. apply ex_RInt_Reals_1. apply continuity_implies_RiemannInt. exact H1. intros x _; apply H. apply ex_RInt_swap. apply ex_RInt_Reals_1. apply continuity_implies_RiemannInt. left; exa... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | continuity_implies_ex_Rint | |
Iu1 : forall a b, ex_RInt (fun u => u1 u) a b. intros a b. apply continuity_implies_ex_Rint. apply Cu1. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | Iu1 | |
beta (x t : R) := 1/(2*c) * RInt (fun u => u1 u) (x - c * t) (x + c * t). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta | |
beta20 x t := 1/(2*c) * (Derive (fun u => u1 u) (x + c * t) - Derive (fun u => u1 u) (x - c * t)). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta20 | |
beta01 x t := 1/2 * (u1 (x + c * t) + u1 (x - c * t)). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta01 | |
beta02 x t := c/2 * (Derive (fun u => u1 u) (x + c * t) - Derive (fun u => u1 u) (x - c * t)). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta02 | |
beta20_lim : forall x t, is_derive_n (fun u => beta u t) 2 x (beta20 x t). Proof. intros x t. unfold beta. auto_derive_2. (* . *) split. apply Iu1. repeat split. apply filter_forall. apply Cu1. apply filter_forall. apply Cu1. repeat split ; apply Du1. unfold beta20, Rminus. ring. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta20_lim | |
beta01_lim : forall x t, is_derive (fun u => beta x u) t (beta01 x t). Proof. intros x t. unfold beta. auto_derive. split. apply Iu1. repeat split. apply filter_forall. apply Cu1. apply filter_forall. apply Cu1. unfold beta01, Rminus, Rdiv. now field. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta01_lim | |
beta02_lim : forall x t, is_derive_n (fun u => beta x u) 2 t (beta02 x t). Proof. intros x t. unfold beta. auto_derive_2. split. apply Iu1. repeat split. apply filter_forall. apply Cu1. apply filter_forall. apply Cu1. repeat split ; apply Du1. unfold beta02, Rminus, Rdiv. now field. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | beta02_lim | |
gamma x t := 1/(2*c) * RInt (fun tau => RInt (fun xi => f xi tau) (x - c * (t - tau)) (x + c * (t - tau))) 0 t. | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | gamma | |
gamma20 x t := 1/(2*c) * RInt (fun tau => Derive (fun u => f u tau) (x + c * (t - tau)) - Derive (fun u => f u tau) (x - c * (t - tau))) 0 t. | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | gamma20 | |
gamma02 x t := (f x t + c/2 * RInt (fun tau => Derive (fun u => f u tau) (x + c * (t - tau)) - Derive (fun u => f u tau) (x - c * (t - tau))) 0 t). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | gamma02 | |
gamma20_lim : forall x t, is_derive_n (fun u => gamma u t) 2 x (gamma20 x t). Proof. intros x t. unfold gamma. auto_derive_2. repeat split. exists (mkposreal _ Rlt_0_1). simpl. intros t' u' _ _. repeat split. apply continuity_implies_ex_Rint => y. admit. (* cont 2D -> 1D *) apply filter_forall => y. admit. (* cont 2D -... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | gamma20_lim | |
gamma02_lim : forall x t, is_derive_n (fun u => gamma x u) 2 t (gamma02 x t). Proof. intros x t. unfold gamma. auto_derive_2. repeat split. apply locally_2d_forall => y z. admit. intros t' _. admit. apply filter_forall => y. admit. apply filter_forall => y. apply continuity_implies_ex_Rint => z. admit. exists (mkposrea... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Rcomplements Derive RInt Hierarchy Derive_2d AutoDerive."
] | examples/DAlembert.v | gamma02_lim | |
is_linear_C_R (l : C -> C) : is_linear (U := C_NormedModule) (V := C_NormedModule) l -> is_linear (U := C_R_NormedModule) (V := C_R_NormedModule) l. Proof. intros Lf. - split. intros ; apply Lf. simpl ; intros. rewrite !scal_R_Cmult ; by apply Lf. case: Lf => _ _ [M Lf]. exists M ; split. by apply Lf. intros. rewrite -... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_linear_C_R | |
is_linear_C_id_1 : is_linear (U := C_NormedModule) (V := AbsRing_NormedModule C_AbsRing) (fun y : C => y). Proof. split => //. exists 1 ; split. by apply Rlt_0_1. intros x ; apply Req_le. rewrite Rmult_1_l ; reflexivity. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_linear_C_id_1 | |
is_linear_C_id_2 : is_linear (U := AbsRing_NormedModule C_AbsRing) (V := C_NormedModule) (fun y : C_NormedModule => y). Proof. split => //. exists 1 ; split. by apply Rlt_0_1. intros x ; apply Req_le. rewrite Rmult_1_l ; reflexivity. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_linear_C_id_2 | |
is_linear_RtoC : is_linear RtoC. Proof. split => //=. by intros ; rewrite RtoC_plus. intros ; rewrite {2}/scal /= /prod_scal /= scal_zero_r. reflexivity. exists (sqrt 2) ; split. apply Rlt_sqrt2_0. intros. eapply Rle_trans. rewrite -Cmod_norm. apply Cmod_2Rmax. simpl. rewrite Rabs_R0. rewrite Rmax_left. apply Rle_refl.... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_linear_RtoC | |
continuous_RtoC x : continuous RtoC x. Proof. apply filterlim_locally. intros eps ; exists eps => /= y Hy. split => //=. by apply ball_center. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | continuous_RtoC | |
continuous_C_id_1 (x : C) : continuous (T := C_UniformSpace) (U := AbsRing_UniformSpace C_AbsRing) (fun y => y) x. Proof. intros P HP. by apply locally_C. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | continuous_C_id_1 | |
continuous_C_id_2 (x : C) : continuous (T := AbsRing_UniformSpace C_AbsRing) (U := C_UniformSpace) (fun y => y) x. Proof. intros P HP. by apply locally_C. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | continuous_C_id_2 | |
continuous_C (f : C -> C) (x : C) : continuous (T := C_UniformSpace) (U := C_UniformSpace) f x <-> continuous (T := AbsRing_UniformSpace C_AbsRing) (U := AbsRing_UniformSpace C_AbsRing) f x. Proof. split => H. - intros P HP. by apply locally_C, H, locally_C. - intros P HP. by apply locally_C, H, locally_C. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | continuous_C | |
is_derive_filterdiff_C_R (f : C -> C) (x : C) (df : C -> C) : is_linear df -> is_derive (V := C_NormedModule) f x (df 1) -> filterdiff (U := C_R_NormedModule) (V := C_R_NormedModule) f (locally x) df. Proof. move => Hdf [Lf Hf]. split => //. apply is_linear_C_R. split ; apply Hdf. intros y Hy eps. apply: locally_le_loc... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_derive_filterdiff_C_R | |
filterdiff_C_R_is_derive (f : C -> C) (x : C) (df : C) : filterdiff (U := C_R_NormedModule) (V := C_R_NormedModule) f (locally x) (fun u => mult u df) -> is_derive (V := C_NormedModule) f x df. Proof. intros (Lf,Df). split. apply is_linear_scal_l. intros y Hy eps. apply: locally_le_locally_norm. case: (fun Hy => locall... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | filterdiff_C_R_is_derive | |
C_RInt (f : R -> C) (a b : R) : C := (RInt (fun t => fst (f t)) a b, RInt (fun t => snd (f t)) a b). | Definition | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt | |
is_C_RInt_unique (f : R -> C) (a b : R) (l : C) : is_RInt f a b l -> C_RInt f a b = l. Proof. intros Hf. apply RInt_fct_extend_pair with (3 := Hf). by apply is_RInt_unique. by apply is_RInt_unique. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_C_RInt_unique | |
C_RInt_correct (f : R -> C) (a b : R) : ex_RInt f a b -> is_RInt f a b (C_RInt f a b). Proof. case => l Hf. replace (C_RInt f a b) with l. by []. by apply sym_eq, is_C_RInt_unique. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt_correct | |
C_RInt_ext (f g : R -> C) (a b : R) : (forall x, Rmin a b <= x <= Rmax a b -> g x = f x) -> C_RInt g a b = C_RInt f a b. Proof. intros Heq. apply injective_projections ; simpl ; apply RInt_ext => x Hx ; by rewrite Heq. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt_ext | |
C_RInt_swap (f : R -> C) (a b : R) : - C_RInt f a b = C_RInt f b a. Proof. apply injective_projections ; simpl ; apply RInt_swap. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt_swap | |
C_RInt_scal_R (f : R -> C) (a b : R) (k : R) : C_RInt (fun t => scal k (f t)) a b = scal k (C_RInt f a b). Proof. apply injective_projections ; simpl ; apply RInt_scal. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt_scal_R | |
C_RInt_const c a b : C_RInt (fun _ => c) a b = scal (b - a) c. Proof. apply injective_projections ; simpl ; rewrite RInt_const ; ring. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt_const | |
is_C_RInt_scal f a b (k : C) l : is_RInt f a b l -> is_RInt (fun t => k * f t) a b (k * l). Proof. intros H. move: (is_RInt_fct_extend_fst _ _ _ _ H) => /= H1. move: (is_RInt_fct_extend_snd _ _ _ _ H) => /= {H} H2. apply is_RInt_fct_extend_pair ; simpl. by apply: is_RInt_minus ; apply: is_RInt_scal. by apply: is_RInt_p... | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | is_C_RInt_scal | |
ex_C_RInt_scal f k a b : ex_RInt f a b -> ex_RInt (fun t => k * f t) a b. Proof. intros [lf If]. eexists. apply is_C_RInt_scal ; eassumption. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | ex_C_RInt_scal | |
C_RInt_scal (f : R -> C) (k : C) (a b : R) : ex_RInt f a b -> C_RInt (fun t => k * f t) a b = k * C_RInt f a b. Proof. intros Hf. apply is_C_RInt_unique. apply is_C_RInt_scal. by apply C_RInt_correct. Qed. | Lemma | examples | [
"From Coq Require Import Reals ssreflect.",
"From Coquelicot Require Import Coquelicot."
] | examples/Wasow.v | C_RInt_scal |
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Coq-Coquelicot
Structured dataset from Coquelicot — Classical real analysis.
2,448 declarations extracted from Coq source files.
Applications
- Training language models on formal proofs
- Fine-tuning theorem provers
- Retrieval-augmented generation for proof assistants
- Learning proof embeddings and representations
Source
- Repository: https://gitlab.inria.fr/coquelicot/coquelicot
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Declaration body |
| type | string | Lemma, Definition, Theorem, etc. |
| library | string | Source module |
| imports | list | Required imports |
| filename | string | Source file path |
| symbolic_name | string | Identifier |
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