Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
statement stringlengths 1 1.23k | proof stringlengths 0 42.4k | type stringclasses 16
values | symbolic_name stringlengths 1 36 | library stringclasses 2
values | filename stringclasses 28
values | imports listlengths 4 20 | deps listlengths 0 64 | docstring stringclasses 266
values | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
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 | pos_rat | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Rdiv_lt_0_compat",
"apply"
] | This file describes an experiment: most 18-year old French
students pass an exam called Baccalaureate which ends the high school
and is required for attending the university. We took the 2013
mathematics test of the scientific Baccalaureate at the same time as
the students. The pdf of the test is available
#<a href="ht... | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 | sign_0_lt | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"sign"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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. | Lemma | sign_lt_0 | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"sign"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
fab (a b x : R) : R | := (a + b * ln x) / x. | Definition | fab | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [] | 8:14 | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 /zero /one /=.
field.
by apply Rgt_not_eq.
Qed. | Lemma | Dfab | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"apply",
"evar_last",
"fab",
"is_derive",
"is_derive_Reals",
"is_derive_const",
"is_derive_div",
"is_derive_id",
"is_derive_plus",
"is_derive_scal",
"one",
"plus",
"zero"
] | 1.b | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 1) / 1 ^ 2) in Hdf.
rewrite ln_1 /= in Hdf.
field_simp... | Lemma | Val_a_b | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Derive",
"Dfab",
"Rdiv_1",
"apply",
"fab",
"is_derive_unique"
] | 1.c | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
f (x : R) : R | := fab 2 2 x. | Definition | f | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"fab"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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.
apply pow2_gt_0.
by apply Rgt_not_eq.
Qed. | Lemma | Signe_df | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Derive",
"Dfab",
"Rdiv_lt_0_compat",
"apply",
"is_derive_unique",
"pow2_gt_0",
"sign",
"sign_eq_1",
"sign_mult"
] | 2.a. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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'.
case: Rle_dec (Rlt_le _ _ Rlt_0_2) => // H ... | Lemma | filterlim_f_0 | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Rbar_locally",
"Rbar_mult'",
"apply",
"at_right",
"fab",
"filterlim",
"filterlim_Rbar_mult",
"filterlim_Rbar_plus",
"filterlim_Rinv_0_right",
"filterlim_comp_2",
"filterlim_const",
"is_Rbar_mult",
"is_lim_ln_0"
] | 2.b. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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_Rb... | Lemma | Lim_f_p_infty | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Rbar_plus'",
"apply",
"fab",
"is_Rbar_plus",
"is_lim",
"is_lim_div_ln_p",
"is_lim_ext_loc",
"is_lim_id",
"is_lim_inv",
"is_lim_plus",
"is_lim_scal_l"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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_cancel ; rewrite Ropp_0 Ropp_involutive.
rew... | Lemma | Variation_1 | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Dfab",
"Signe_df",
"apply",
"incr_function",
"is_derive_unique",
"sign_0_lt"
] | 2.c. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 Ropp_lt_cancel ; rewrite Ropp_0 Ropp_i... | Lemma | Variation_2 | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Dfab",
"Signe_df",
"apply",
"incr_function",
"is_derive_opp",
"is_derive_unique",
"sign_lt_0"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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) / 1 ^ 2) ; apply is_deri... | Lemma | f_eq_1_0_1 | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"AbsRing_ball",
"Dfab",
"IVT_Rbar_incr",
"Rabs_lt_between'",
"Rbar_finite_eq",
"Rminus_eq_0",
"Rminus_lt_0",
"apply",
"ball",
"continuity_pt_filterlim",
"continuous_Rabs",
"fab",
"filterlim_Rabs_0",
"filterlim_comp",
"filterlim_f_0",
"is_derive_Reals",
"is_lim_comp",
"is_lim_contin... | 3.a | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 Rlt_0_1.
rewrite /f /fab ln_1 /... | Lemma | f_eq_1_1_p_infty | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Dfab",
"IVT_Rbar_incr",
"Lim_f_p_infty",
"Rminus_lt_0",
"apply",
"evar_last",
"fab",
"is_derive_Reals",
"is_lim_continuity",
"is_lim_opp"
] | 3.b. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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.
by apply Rgt_not_eq.
Qed. | Lemma | If | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"apply",
"evar_last",
"fab",
"is_derive",
"is_derive_Reals",
"is_derive_plus",
"is_derive_pow",
"plus"
] | 5.b. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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_conti... | Lemma | RInt_f | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Dfab",
"If",
"apply",
"continuity_pt_filterlim",
"evar_last",
"is_RInt",
"is_RInt_derive",
"is_derive_Reals",
"minus",
"opp",
"plus"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
u (n : nat) : R | :=
match n with
| O => 2
| S n => 2/3 * u n + 1/3 * (INR n) + 1
end. | Fixpoint | u | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [] | 10:36 | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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. | Lemma | Q2a | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Rminus_le_0",
"apply"
] | 2.a | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
Q2b : forall n, u (S n) - u n = 1/3 * (INR n + 3 - u n). | Proof.
move => n ; simpl.
field.
Qed. | Lemma | Q2b | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [] | 2.b. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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. | Lemma | Q2c | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Q2a",
"Q2b",
"Rminus_le_0",
"apply"
] | 2.c. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
v (n : nat) : R | := u n - INR n. | Definition | v | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [] | 10:49 | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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. | Lemma | Q3a | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [] | 3.a. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
Q3b : forall n, u n = 2 * (2/3)^n + INR n. | Proof.
move => n.
rewrite -Q3a /v ; ring.
Qed. | Lemma | Q3b | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Q3a"
] | 3.b. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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. | Lemma | Q3c | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Q3b",
"apply",
"is_lim_seq",
"is_lim_seq_INR",
"is_lim_seq_const",
"is_lim_seq_ext",
"is_lim_seq_geom",
"is_lim_seq_mult",
"is_lim_seq_plus"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
Su (n : nat) : R | := sum_f_R0 u n. | Definition | Su | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [] | 11:00 | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
Tu (n : nat) : R | := Su n / (INR n) ^ 2. | Definition | Tu | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Su"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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_lt_0 ; ring_simplify ; by apply Rlt_0_1.
... | Lemma | Q4a | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Q3b",
"Rlt_div_l",
"Rminus_lt_0",
"Su",
"apply",
"sum_INR"
] | 4.a. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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.
ap... | Lemma | Q4b | examples | examples/BacS2013.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"RInt",
"Continuity",
"Lim_seq",
"ElemFct",
"RInt_analysis"
] | [
"Q4a",
"Tu",
"apply",
"is_Rbar_div_p_infty",
"is_Rbar_minus",
"is_Rbar_mult_p_infty_pos",
"is_Rbar_mult_sym",
"is_Rbar_plus",
"is_lim_seq",
"is_lim_seq_INR",
"is_lim_seq_const",
"is_lim_seq_div",
"is_lim_seq_ext_loc",
"is_lim_seq_geom",
"is_lim_seq_inv",
"is_lim_seq_minus",
"is_lim_s... | 4.b. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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. | Fixpoint | v | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [] | 1. Exprimer v (S n) et c (S n) en fonction de v n et c n | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
A : matrix 2 2 | :=
[[95/100, 1/100 ] ,
[ 5/100, 99/100]]. | Definition | A | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"matrix"
] | 2. Définition de la matrice A | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
X (n : nat) : matrix 2 1 | :=
[[v n],[c n]]. | Definition | X | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"matrix"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 /plus /mult //=.
Qed. | Lemma | Q2 | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"Mmult",
"apply",
"coeff_mat_bij",
"coeff_mat_ext",
"mult",
"plus",
"scal",
"sum_O",
"sum_Sn"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
P : matrix 2 2 | :=
[[1,-1], [5,1]]. | Definition | P | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"matrix"
] | 3. Diagonalisation | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
Q : matrix 2 2 | :=
[[1,1],[-5,1]]. | Definition | Q | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"matrix"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
P' : matrix 2 2 | :=
[[1 / 6,1 / 6],[-5 / 6,1 / 6]]. | Definition | P' | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"matrix"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 /one /= ; try field) ;
rewrit... | Lemma | Q3a | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"Mone",
"P'",
"apply",
"coeff_mat",
"coeff_mat_bij",
"coeff_mat_ext_aux",
"mult",
"one",
"plus",
"sum_O",
"sum_Sn",
"zero"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
D : matrix 2 2 | := [[1,0],[0,94 / 100]]. | Definition | D | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"matrix"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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) ;
rewrite sum_Sn ... | Lemma | Q3b | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"P'",
"apply",
"coeff_mat",
"coeff_mat_bij",
"coeff_mat_ext_aux",
"mult",
"one",
"plus",
"sum_O",
"sum_Sn",
"zero"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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. | Lemma | Q3c | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"P'",
"Q3a",
"Q3b",
"apply",
"mult",
"mult_assoc",
"mult_one_l",
"pow_n"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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] //=.
rewrite IH.
apply (coeff_mat_ext_aux 0 0) => i j Hi Hj.
rewrite co... | Lemma | Q4 | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"Q2",
"Q3c",
"apply",
"coeff_mat",
"coeff_mat_bij",
"coeff_mat_ext",
"coeff_mat_ext_aux",
"mult",
"one",
"plus",
"pow_n",
"scal",
"scal_assoc",
"scal_one",
"sum_O",
"sum_Sn",
"zero"
] | 4. Terme général et limite de la suite v n | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 Rabs_pos_eq ; ... | Lemma | lim_v | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"Finite",
"Q4",
"apply",
"is_lim_seq",
"is_lim_seq_const",
"is_lim_seq_ext",
"is_lim_seq_geom",
"is_lim_seq_minus",
"is_lim_seq_mult",
"is_lim_seq_plus"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | lim_c | examples | examples/BacS2013_bonus.v | [
"Coq",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Hierarchy",
"PSeries",
"Rbar",
"Lim_seq"
] | [
"Finite",
"apply",
"is_lim_seq",
"is_lim_seq_const",
"is_lim_seq_ext",
"is_lim_seq_minus",
"lim_v"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
Bessel1_seq (n k : nat) | :=
(-1)^(k)/(INR (fact (k)) * INR (fact (n + (k)))). | Definition | Bessel1_seq | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [] | This file is an example of how to use power series. It defines and
gives properties of the Bessel functions. | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 | Bessel1_seq_neq_0 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1_seq",
"apply"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 /
(INR (S p) * INR (fact p) * (INR (S (n + ... | Lemma | CV_Bessel1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1_seq",
"Bessel1_seq_neq_0",
"CV_radius",
"CV_radius_infinite_DAlembert",
"Finite",
"Rabs_div",
"Rbar_inv",
"apply",
"is_lim_seq_INR",
"is_lim_seq_ext",
"is_lim_seq_incr_1",
"is_lim_seq_incr_n",
"is_lim_seq_inv",
"is_lim_seq_mult"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
ex_Bessel1 (n : nat) (x : R) :
ex_pseries (Bessel1_seq n) x. | Proof.
apply CV_radius_inside.
by rewrite CV_Bessel1.
Qed. | Lemma | ex_Bessel1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1_seq",
"CV_Bessel1",
"CV_radius_inside",
"apply",
"ex_pseries"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
Bessel1 (n : nat) (x : R) | :=
(x/2)^n * PSeries (Bessel1_seq n) ((x/2)^2). | Definition | Bessel1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1_seq",
"PSeries"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 /Rdiv ; simpl ; field.
by rewrite CV_Bessel1.
Qed. | Lemma | is_derive_Bessel1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"CV_Bessel1",
"Derive_PSeries",
"PS_derive",
"PSeries",
"apply",
"auto_derive",
"ex_derive_PSeries",
"is_derive"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 ... | Proof.
rewrite plus_INR ?mult_INR ; simpl INR.
eapply is_derive_ext.
move => y ;
by apply sym_eq, is_derive_unique, is_derive_Bessel1.
auto_derive.
repeat split.
apply ex_derive_PSeries.
by rewrite CV_radius_derive CV_Bessel1.
apply ex_derive_PSeries.
by rewrite CV_Bessel1.
rewrite !Derive_PSeries... | Lemma | is_derive_2_Bessel1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"CV_Bessel1",
"CV_radius_derive",
"Derive_PSeries",
"PS_derive",
"PSeries",
"apply",
"auto_derive",
"ex_derive_PSeries",
"is_derive_Bessel1",
"is_derive_ext",
"is_derive_n",
"is_derive_unique"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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.
set y := x/2 ; replace x with (2 * y) by (unfold y ; field).
replace (_ + _)
with (4 * y^S (S n) * (y^2 * PSeries (PS_der... | Lemma | Bessel1_correct | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"CV_Bessel1",
"CV_disk_inside",
"CV_radius_derive",
"CV_radius_incr_1",
"CV_radius_inside",
"CV_radius_plus",
"CV_radius_scal",
"Derive",
"Derive_n",
"PS_derive",
"PS_incr_1",
"PS_plus",
"PS_scal",
"PSeries",
"PSeries_const_0",
"PSeries_ext",
"PSeries_in... | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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.
unfold y ; contradict Hx.
replace x with (2 * (x/2)) by field ; rewrite Hx ; ring.
case: n => [ | n] ; simp... | Lemma | Bessel1_equality_1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"Derive",
"PS_derive",
"PS_scal",
"PSeries",
"PSeries_ext",
"PSeries_scal",
"Rdiv_1",
"apply",
"is_derive_Bessel1",
"is_derive_unique",
"scal"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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) +
(x / 2) ^ n * PSeries (Bessel1_seq n) ((x / 2) ^ 2))
with ((x/2)^n *
((x/2)^2 * PSeries (Bessel1_seq (S (S n... | Lemma | Bessel1_equality_2 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"PS_incr_1",
"PS_plus",
"PS_scal",
"PSeries",
"PSeries_ext",
"PSeries_incr_1",
"PSeries_plus",
"PSeries_scal",
"apply",
"ex_Bessel1",
"ex_pseries_incr_1",
"mult",
"plus",
"plus_zero_l",
"scal"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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) * PSeries (Bessel1_seq (S (S n))) ((x / 2) ^ 2) -
(x / 2) ^ n * PSeries (Bessel1_seq n) ((x / 2) ^ 2))
with... | Lemma | Bessel1_equality_3 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"CV_Bessel1",
"Derive",
"PS_derive",
"PS_incr_1",
"PS_minus",
"PS_scal",
"PSeries",
"PSeries_ext",
"PSeries_incr_1",
"PSeries_minus",
"PSeries_scal",
"apply",
"ex_Bessel1",
"ex_pseries_derive",
"ex_pseries_incr_1",
"ex_pseries_scal",
"is_derive_Bessel1",... | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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)... | Proof.
move => Ha H.
cut (forall k,
(PS_plus (PS_plus (PS_incr_n (PS_derive_n 2 a) 2)
(PS_incr_1 (PS_derive a))) (PS_plus (PS_incr_n a 2) (PS_scal (- INR n ^ 2) a))) k = 0).
intros Haux.
split ; [move: (Haux 0%nat) | move: (fun k => Haux (S k))] => {} Haux.
(* n = 0 *)
rewrite /PS_plus /= /PS_incr_1... | Lemma | Bessel1_uniqueness_aux_0 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"CV_radius",
"CV_radius_const_0",
"CV_radius_derive",
"CV_radius_derive_n",
"CV_radius_ext",
"CV_radius_incr_1",
"CV_radius_inside",
"CV_radius_plus",
"CV_radius_scal",
"Derive",
"Derive_PSeries",
"Derive_n",
"Derive_n_PSeries",
"PS_derive",
"PS_derive_n",
"PS_incr_1",
"PS_incr_n",
... | * Unicity | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 = B... | Proof.
intros Ha0 Ha1 Ha.
assert (forall k, S (S k) <> n -> a (S (S k)) = - a k / (INR (S (S k)) ^ 2 - INR n ^ 2)).
intros k Hk.
replace (a k) with (- (INR (S (S k)) ^ 2 - INR n ^ 2) * a (S (S k))).
field.
replace (INR (S (S k)) ^ 2 - INR n ^ 2)
with ((INR (S (S k)) - INR n) * (INR (S (S k)) + I... | Lemma | Bessel1_uniqueness_aux_1 | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1_seq",
"apply",
"ind_0_1_SS",
"lt_neq"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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%nat) /\
(forall k, (INR (S (S k)) ^ 2 - INR n ^ 2) * a (S (S k)) + a k = 0)).
by apply Bessel1_uniqueness_aux_0.
assert ((forall k : nat, (k < n)%nat -> a k = 0)
/\ (forall p : nat, a (n + 2 * p + 1)%nat = 0)
/\ (forall ... | Lemma | Bessel1_uniqueness | examples | examples/Bessel.v | [
"Coq",
"Arith",
"Reals",
"Psatz",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Rbar",
"Hierarchy",
"Derive",
"Series",
"PSeries",
"Lim_seq",
"AutoDerive"
] | [
"Bessel1",
"Bessel1_seq",
"Bessel1_seq_neq_0",
"Bessel1_uniqueness_aux_0",
"Bessel1_uniqueness_aux_1",
"CV_radius",
"Derive",
"Derive_n",
"PS_decr_n",
"PS_scal",
"PSeries",
"PSeries_const_0",
"PSeries_decr_n_aux",
"PSeries_ext",
"PSeries_odd_even",
"PSeries_scal",
"R_AbsRing",
"R_N... | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 ;
... | Ltac | auto_derive_2 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"apply",
"auto_derive",
"auto_derive_fun",
"is_derive",
"is_derive_ext",
"is_derive_n",
"is_derive_unique"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
c : R. | Parameter | c | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
Zc : c <> 0. | Hypothesis | Zc | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
u0 : R -> R. | Parameter | u0 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
Du0 : forall x, ex_derive (fun u => u0 u) x. | Hypothesis | Du0 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"ex_derive",
"u0"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
D2u0 : forall x, ex_derive_n (fun u => u0 u) 2 x. | Hypothesis | D2u0 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"ex_derive_n",
"u0"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
alpha x t | := 1/2 * (u0 (x + c * t) + u0 (x - c * t)). | Definition | alpha | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"u0"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
alpha20 x t | := 1/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)). | Definition | alpha20 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Derive_n",
"u0"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
alpha02 x t | := c^2/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)). | Definition | alpha02 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Derive_n",
"u0"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | alpha_20_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"D2u0",
"Derive_n",
"Du0",
"alpha",
"alpha20",
"apply",
"auto_derive_2",
"is_derive_n"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | alpha_02_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"D2u0",
"Derive_n",
"Du0",
"alpha",
"alpha02",
"apply",
"auto_derive_2",
"is_derive_n"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
u1 : R -> R. | Parameter | u1 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
Du1 : forall x, ex_derive (fun u => u1 u) x. | Hypothesis | Du1 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"ex_derive",
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
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 | Cu1 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Du1",
"apply",
"is_derive_Reals",
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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; exact H1.
intros x _; apply H.
Qed. | Lemma | continuity_implies_ex_Rint | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"apply",
"ex_RInt",
"ex_RInt_Reals_1",
"ex_RInt_swap"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
Iu1: forall a b, ex_RInt (fun u => u1 u) a b. | intros a b.
apply continuity_implies_ex_Rint.
apply Cu1.
Qed. | Lemma | Iu1 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Cu1",
"apply",
"continuity_implies_ex_Rint",
"ex_RInt",
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
beta (x t : R) | := 1/(2*c) * RInt (fun u => u1 u) (x - c * t) (x + c * t). | Definition | beta | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"RInt",
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
beta20 x t | := 1/(2*c) * (Derive (fun u => u1 u) (x + c * t) - Derive (fun u => u1 u) (x - c * t)). | Definition | beta20 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Derive",
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
beta01 x t | := 1/2 * (u1 (x + c * t) + u1 (x - c * t)). | Definition | beta01 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
beta02 x t | := c/2 * (Derive (fun u => u1 u) (x + c * t) - Derive (fun u => u1 u) (x - c * t)). | Definition | beta02 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Derive",
"u1"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | beta20_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Cu1",
"Du1",
"Iu1",
"apply",
"auto_derive_2",
"beta",
"beta20",
"filter_forall",
"is_derive_n"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | beta01_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Cu1",
"Iu1",
"apply",
"auto_derive",
"beta",
"beta01",
"filter_forall",
"is_derive"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | beta02_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Cu1",
"Du1",
"Iu1",
"apply",
"auto_derive_2",
"beta",
"beta02",
"filter_forall",
"is_derive_n"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
f : R -> R -> R. | Hypothesis | f | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | ||
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 | gamma | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"RInt"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | gamma20 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Derive",
"RInt"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | gamma02 | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"Derive",
"RInt"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 -> 1D *)
apply filter_forall => y.
admit. (* cont 2D -> 1D *)
apply filter_fora... | Lemma | gamma20_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"RInt_ext",
"apply",
"auto_derive_2",
"continuity_implies_ex_Rint",
"filter_forall",
"gamma",
"gamma20",
"is_derive_n"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 (mkposreal _ Rlt_0_1).
simpl.
apply filter_forall => y.
apply continuity_implies_ex_Rin... | Lemma | gamma02_lim | examples | examples/DAlembert.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot",
"Rcomplements",
"Derive",
"RInt",
"Hierarchy",
"Derive_2d",
"AutoDerive"
] | [
"RInt_ext",
"RInt_point",
"RInt_scal",
"R_ModuleSpace",
"Zc",
"apply",
"auto_derive_2",
"continuity_implies_ex_Rint",
"filter_forall",
"gamma",
"gamma02",
"is_derive_n",
"locally_2d_forall",
"mult",
"scal"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 -!Cmod_norm.
apply Lf.
Qed. | Lemma | is_linear_C_R | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_NormedModule",
"C_R_NormedModule",
"Cmod_norm",
"apply",
"is_linear",
"scal_R_Cmult"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | is_linear_C_id_1 | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"AbsRing_NormedModule",
"C_AbsRing",
"C_NormedModule",
"apply",
"is_linear"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | is_linear_C_id_2 | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"AbsRing_NormedModule",
"C_AbsRing",
"C_NormedModule",
"apply",
"is_linear"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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.
ap... | Lemma | is_linear_RtoC | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"Cmod_2Rmax",
"Cmod_norm",
"RtoC",
"RtoC_plus",
"apply",
"is_linear",
"prod_scal",
"scal",
"scal_zero_r"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
continuous_RtoC x : continuous RtoC x. | Proof.
apply filterlim_locally.
intros eps ; exists eps => /= y Hy.
split => //=.
by apply ball_center.
Qed. | Lemma | continuous_RtoC | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"RtoC",
"apply",
"ball_center",
"continuous",
"filterlim_locally"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | continuous_C_id_1 | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"AbsRing_UniformSpace",
"C_AbsRing",
"C_UniformSpace",
"apply",
"continuous",
"locally_C"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | continuous_C_id_2 | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"AbsRing_UniformSpace",
"C_AbsRing",
"C_UniformSpace",
"apply",
"continuous",
"locally_C"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | continuous_C | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"AbsRing_UniformSpace",
"C_AbsRing",
"C_UniformSpace",
"apply",
"continuous",
"locally_C"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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_locally_norm.
case: (fun Hy => locally_norm_le_locally _ _ (Hf y Hy eps)) => {Hf} /= delta Hf => //.
apply locally_C, Hy.
by apply locally_C, Hf.
exists delta => /= z Hz.
rewrite -!... | Lemma | is_derive_filterdiff_C_R | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_NormedModule",
"C_R_NormedModule",
"Cmod_norm",
"Cmult_1_r",
"apply",
"filterdiff",
"is_derive",
"is_linear",
"is_linear_C_R",
"locally",
"locally_C",
"locally_le_locally_norm",
"locally_norm_le_locally",
"minus"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 => locally_norm_le_locally _ _ (Df y Hy eps)) => {Df} /= delta Df => //.
apply locally_C, Hy.
by apply locally_C, Df.
exists delta => /= z Hz.
rewrite /norm /= /abs /= !Cmod_norm.
... | Lemma | filterdiff_C_R_is_derive | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_NormedModule",
"C_R_NormedModule",
"Cmod_norm",
"abs",
"apply",
"filterdiff",
"is_derive",
"is_linear_scal_l",
"locally",
"locally_C",
"locally_le_locally_norm",
"locally_norm_le_locally",
"mult",
"norm"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | C_RInt | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"RInt"
] | * Intégrale le long d’un segment | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
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 | is_C_RInt_unique | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_RInt",
"RInt_fct_extend_pair",
"apply",
"is_RInt",
"is_RInt_unique"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | C_RInt_correct | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_RInt",
"apply",
"ex_RInt",
"is_C_RInt_unique",
"is_RInt"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | C_RInt_ext | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_RInt",
"RInt_ext",
"apply"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 | |
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 | C_RInt_swap | examples | examples/Wasow.v | [
"Coq",
"Reals",
"ssreflect",
"Coquelicot"
] | [
"C_RInt",
"apply"
] | https://gitlab.inria.fr/coquelicot/coquelicot | a31242c6efeb77a4874ddbac40eeeb8f1acb1280 |
End of preview. Expand in Data Studio
Coq-Coquelicot
Structured dataset from Coquelicot — Classical real analysis.
Source
- Repository: https://gitlab.inria.fr/coquelicot/coquelicot
- Commit:
a31242c6efeb77a4874ddbac40eeeb8f1acb1280 - Files: 29
- License: lgpl-3.0
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 2,424
- With proof: 2,357 (97.2%)
- With docstring: 332 (13.7%)
- Libraries: 2
By type
| Type | Count |
|---|---|
| Lemma | 1,863 |
| Definition | 326 |
| Canonical | 49 |
| Instance | 37 |
| Fixpoint | 32 |
| Hypothesis | 32 |
| Coercion | 23 |
| Record | 18 |
| Notation | 15 |
| Ltac | 9 |
| Inductive | 6 |
| Class | 5 |
| Theorem | 4 |
| Parameter | 3 |
| Structure | 1 |
| Let | 1 |
Example
pos_rat
:=
repeat ( apply Rdiv_lt_0_compat
|| apply Rplus_lt_0_compat
|| apply Rmult_lt_0_compat) ;
try by apply Rlt_0_1.
- type: Ltac | symbolic_name:
pos_rat| examples/BacS2013.v
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{coq_coquelicot_dataset,
title = {Coq-Coquelicot},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://gitlab.inria.fr/coquelicot/coquelicot, commit a31242c6efeb},
url = {https://huggingface.co/datasets/phanerozoic/Coq-Coquelicot}
}
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