statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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gronwall_bound_ε0_δ0 (K x : ℝ) : gronwall_bound 0 K 0 x = 0 | by simp only [gronwall_bound_ε0, zero_mul] | lemma | gronwall_bound_ε0_δ0 | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"gronwall_bound",
"gronwall_bound_ε0",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gronwall_bound_continuous_ε (δ K x : ℝ) : continuous (λ ε, gronwall_bound δ K ε x) | begin
by_cases hK : K = 0,
{ simp only [gronwall_bound_K0, hK],
exact continuous_const.add (continuous_id.mul continuous_const) },
{ simp only [gronwall_bound_of_K_ne_0 hK],
exact continuous_const.add ((continuous_id.mul continuous_const).mul continuous_const) }
end | lemma | gronwall_bound_continuous_ε | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"continuous",
"continuous_const",
"gronwall_bound",
"gronwall_bound_K0",
"gronwall_bound_of_K_ne_0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_gronwall_bound_of_liminf_deriv_right_le {f f' : ℝ → ℝ} {δ K ε : ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r →
∃ᶠ z in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r)
(ha : f a ≤ δ) (bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε) :
∀ x ∈ Icc a b, f x ≤ gronwall_bound δ K ε (x - a) | begin
have H : ∀ x ∈ Icc a b, ∀ ε' ∈ Ioi ε, f x ≤ gronwall_bound δ K ε' (x - a),
{ assume x hx ε' hε',
apply image_le_of_liminf_slope_right_lt_deriv_boundary hf hf',
{ rwa [sub_self, gronwall_bound_x0] },
{ exact λ x, has_deriv_at_gronwall_bound_shift δ K ε' x a },
{ assume x hx hfB,
rw [← hfB... | theorem | le_gronwall_bound_of_liminf_deriv_right_le | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"bound",
"closure_Ioi",
"continuous_on",
"continuous_within_at",
"gronwall_bound",
"gronwall_bound_continuous_ε",
"gronwall_bound_x0",
"has_deriv_at_gronwall_bound_shift",
"image_le_of_liminf_slope_right_lt_deriv_boundary"
] | A Grönwall-like inequality: if `f : ℝ → ℝ` is continuous on `[a, b]` and satisfies
the inequalities `f a ≤ δ` and
`∀ x ∈ [a, b), liminf_{z→x+0} (f z - f x)/(z - x) ≤ K * (f x) + ε`, then `f x`
is bounded by `gronwall_bound δ K ε (x - a)` on `[a, b]`.
See also `norm_le_gronwall_bound_of_norm_deriv_right_le` for a versi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le_gronwall_bound_of_norm_deriv_right_le {f f' : ℝ → E} {δ K ε : ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b)) (hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
(ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) :
∀ x ∈ Icc a b, ‖f x‖ ≤ gronwall_bound δ K ε (x - a) | le_gronwall_bound_of_liminf_deriv_right_le (continuous_norm.comp_continuous_on hf)
(λ x hx r hr, (hf' x hx).liminf_right_slope_norm_le hr) ha bound | theorem | norm_le_gronwall_bound_of_norm_deriv_right_le | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"bound",
"continuous_on",
"gronwall_bound",
"has_deriv_within_at",
"le_gronwall_bound_of_liminf_deriv_right_le"
] | A Grönwall-like inequality: if `f : ℝ → E` is continuous on `[a, b]`, has right derivative
`f' x` at every point `x ∈ [a, b)`, and satisfies the inequalities `‖f a‖ ≤ δ`,
`∀ x ∈ [a, b), ‖f' x‖ ≤ K * ‖f x‖ + ε`, then `‖f x‖` is bounded by `gronwall_bound δ K ε (x - a)`
on `[a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_approx_trajectories_ODE_of_mem_set {v : ℝ → E → E} {s : ℝ → set E}
{K : ℝ} (hv : ∀ t, ∀ x y ∈ s t, dist (v t x) (v t y) ≤ K * dist x y)
{f g f' g' : ℝ → E} {a b : ℝ} {εf εg δ : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ t ∈ Ico a b, has_deriv_within_at f (f' t) (Ici t) t)
(f_bound : ∀ t ∈ Ico a b... | begin
simp only [dist_eq_norm] at ha ⊢,
have h_deriv : ∀ t ∈ Ico a b, has_deriv_within_at (λ t, f t - g t) (f' t - g' t) (Ici t) t,
from λ t ht, (hf' t ht).sub (hg' t ht),
apply norm_le_gronwall_bound_of_norm_deriv_right_le (hf.sub hg) h_deriv ha,
assume t ht,
have := dist_triangle4_right (f' t) (g' t) (v... | theorem | dist_le_of_approx_trajectories_ODE_of_mem_set | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"continuous_on",
"dist_triangle4_right",
"gronwall_bound",
"has_deriv_within_at",
"norm_le_gronwall_bound_of_norm_deriv_right_le"
] | If `f` and `g` are two approximate solutions of the same ODE, then the distance between them
can't grow faster than exponentially. This is a simple corollary of Grönwall's inequality, and some
people call this Grönwall's inequality too.
This version assumes all inequalities to be true in some time-dependent set `s t`,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_approx_trajectories_ODE {v : ℝ → E → E}
{K : ℝ≥0} (hv : ∀ t, lipschitz_with K (v t))
{f g f' g' : ℝ → E} {a b : ℝ} {εf εg δ : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ t ∈ Ico a b, has_deriv_within_at f (f' t) (Ici t) t)
(f_bound : ∀ t ∈ Ico a b, dist (f' t) (v t (f t)) ≤ εf)
(hg : continuous_... | have hfs : ∀ t ∈ Ico a b, f t ∈ (@univ E), from λ t ht, trivial,
dist_le_of_approx_trajectories_ODE_of_mem_set (λ t x hx y hy, (hv t).dist_le_mul x y)
hf hf' f_bound hfs hg hg' g_bound (λ t ht, trivial) ha | theorem | dist_le_of_approx_trajectories_ODE | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"continuous_on",
"dist_le_of_approx_trajectories_ODE_of_mem_set",
"gronwall_bound",
"has_deriv_within_at",
"lipschitz_with"
] | If `f` and `g` are two approximate solutions of the same ODE, then the distance between them
can't grow faster than exponentially. This is a simple corollary of Grönwall's inequality, and some
people call this Grönwall's inequality too.
This version assumes all inequalities to be true in the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_trajectories_ODE_of_mem_set {v : ℝ → E → E} {s : ℝ → set E}
{K : ℝ} (hv : ∀ t, ∀ x y ∈ s t, dist (v t x) (v t y) ≤ K * dist x y)
{f g : ℝ → E} {a b : ℝ} {δ : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ t ∈ Ico a b, has_deriv_within_at f (v t (f t)) (Ici t) t)
(hfs : ∀ t ∈ Ico a b, f t ∈ s t)
(hg... | begin
have f_bound : ∀ t ∈ Ico a b, dist (v t (f t)) (v t (f t)) ≤ 0,
by { intros, rw [dist_self] },
have g_bound : ∀ t ∈ Ico a b, dist (v t (g t)) (v t (g t)) ≤ 0,
by { intros, rw [dist_self] },
assume t ht,
have := dist_le_of_approx_trajectories_ODE_of_mem_set hv hf hf' f_bound hfs hg hg' g_bound
... | theorem | dist_le_of_trajectories_ODE_of_mem_set | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"continuous_on",
"dist_le_of_approx_trajectories_ODE_of_mem_set",
"dist_self",
"exp",
"gronwall_bound_ε0",
"has_deriv_within_at"
] | If `f` and `g` are two exact solutions of the same ODE, then the distance between them
can't grow faster than exponentially. This is a simple corollary of Grönwall's inequality, and some
people call this Grönwall's inequality too.
This version assumes all inequalities to be true in some time-dependent set `s t`,
and a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_trajectories_ODE {v : ℝ → E → E}
{K : ℝ≥0} (hv : ∀ t, lipschitz_with K (v t))
{f g : ℝ → E} {a b : ℝ} {δ : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ t ∈ Ico a b, has_deriv_within_at f (v t (f t)) (Ici t) t)
(hg : continuous_on g (Icc a b))
(hg' : ∀ t ∈ Ico a b, has_deriv_within_at g (v t (g t)... | have hfs : ∀ t ∈ Ico a b, f t ∈ (@univ E), from λ t ht, trivial,
dist_le_of_trajectories_ODE_of_mem_set (λ t x hx y hy, (hv t).dist_le_mul x y)
hf hf' hfs hg hg' (λ t ht, trivial) ha | theorem | dist_le_of_trajectories_ODE | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"continuous_on",
"dist_le_of_trajectories_ODE_of_mem_set",
"exp",
"has_deriv_within_at",
"lipschitz_with"
] | If `f` and `g` are two exact solutions of the same ODE, then the distance between them
can't grow faster than exponentially. This is a simple corollary of Grönwall's inequality, and some
people call this Grönwall's inequality too.
This version assumes all inequalities to be true in the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ODE_solution_unique_of_mem_set {v : ℝ → E → E} {s : ℝ → set E}
{K : ℝ} (hv : ∀ t, ∀ x y ∈ s t, dist (v t x) (v t y) ≤ K * dist x y)
{f g : ℝ → E} {a b : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ t ∈ Ico a b, has_deriv_within_at f (v t (f t)) (Ici t) t)
(hfs : ∀ t ∈ Ico a b, f t ∈ s t)
(hg : continuous_on... | begin
assume t ht,
have := dist_le_of_trajectories_ODE_of_mem_set hv hf hf' hfs hg hg' hgs
(dist_le_zero.2 ha) t ht,
rwa [zero_mul, dist_le_zero] at this
end | theorem | ODE_solution_unique_of_mem_set | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"continuous_on",
"dist_le_of_trajectories_ODE_of_mem_set",
"dist_le_zero",
"has_deriv_within_at",
"zero_mul"
] | There exists only one solution of an ODE \(\dot x=v(t, x)\) in a set `s ⊆ ℝ × E` with
a given initial value provided that RHS is Lipschitz continuous in `x` within `s`,
and we consider only solutions included in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ODE_solution_unique {v : ℝ → E → E}
{K : ℝ≥0} (hv : ∀ t, lipschitz_with K (v t))
{f g : ℝ → E} {a b : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ t ∈ Ico a b, has_deriv_within_at f (v t (f t)) (Ici t) t)
(hg : continuous_on g (Icc a b))
(hg' : ∀ t ∈ Ico a b, has_deriv_within_at g (v t (g t)) (Ici t) t)
(... | have hfs : ∀ t ∈ Ico a b, f t ∈ (@univ E), from λ t ht, trivial,
ODE_solution_unique_of_mem_set (λ t x hx y hy, (hv t).dist_le_mul x y)
hf hf' hfs hg hg' (λ t ht, trivial) ha | theorem | ODE_solution_unique | analysis.ODE | src/analysis/ODE/gronwall.lean | [
"analysis.special_functions.exp_deriv"
] | [
"ODE_solution_unique_of_mem_set",
"continuous_on",
"has_deriv_within_at",
"lipschitz_with"
] | There exists only one solution of an ODE \(\dot x=v(t, x)\) with
a given initial value provided that RHS is Lipschitz continuous in `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_picard_lindelof
{E : Type*} [normed_add_comm_group E] (v : ℝ → E → E) (t_min t₀ t_max : ℝ) (x₀ : E)
(L : ℝ≥0) (R C : ℝ) : Prop | (ht₀ : t₀ ∈ Icc t_min t_max)
(hR : 0 ≤ R)
(lipschitz : ∀ t ∈ Icc t_min t_max, lipschitz_on_with L (v t) (closed_ball x₀ R))
(cont : ∀ x ∈ closed_ball x₀ R, continuous_on (λ (t : ℝ), v t x) (Icc t_min t_max))
(norm_le : ∀ (t ∈ Icc t_min t_max) (x ∈ closed_ball x₀ R), ‖v t x‖ ≤ C)
(C_mul_le_R : (C : ℝ) * linear_order.max... | structure | is_picard_lindelof | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"cont",
"continuous_on",
"lipschitz_on_with",
"normed_add_comm_group"
] | `Prop` structure holding the hypotheses of the Picard-Lindelöf theorem.
The similarly named `picard_lindelof` structure is part of the internal API for convenience, so as
not to constantly invoke choice, but is not intended for public use. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
picard_lindelof (E : Type*) [normed_add_comm_group E] [normed_space ℝ E] | (to_fun : ℝ → E → E)
(t_min t_max : ℝ)
(t₀ : Icc t_min t_max)
(x₀ : E)
(C R L : ℝ≥0)
(is_pl : is_picard_lindelof to_fun t_min t₀ t_max x₀ L R C) | structure | picard_lindelof | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"is_picard_lindelof",
"normed_add_comm_group",
"normed_space"
] | This structure holds arguments of the Picard-Lipschitz (Cauchy-Lipschitz) theorem. It is part of
the internal API for convenience, so as not to constantly invoke choice. Unless you want to use one
of the auxiliary lemmas, use `exists_forall_deriv_within_Icc_eq_of_lipschitz_of_continuous` instead
of using this structure... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t_min_le_t_max : v.t_min ≤ v.t_max | v.t₀.2.1.trans v.t₀.2.2 | lemma | picard_lindelof.t_min_le_t_max | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_Icc : (Icc v.t_min v.t_max).nonempty | nonempty_Icc.2 v.t_min_le_t_max | lemma | picard_lindelof.nonempty_Icc | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with {t} (ht : t ∈ Icc v.t_min v.t_max) :
lipschitz_on_with v.L (v t) (closed_ball v.x₀ v.R) | v.is_pl.lipschitz t ht | lemma | picard_lindelof.lipschitz_on_with | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on :
continuous_on (uncurry v) (Icc v.t_min v.t_max ×ˢ closed_ball v.x₀ v.R) | have continuous_on (uncurry (flip v)) (closed_ball v.x₀ v.R ×ˢ Icc v.t_min v.t_max),
from continuous_on_prod_of_continuous_on_lipschitz_on _ v.L v.is_pl.cont v.is_pl.lipschitz,
this.comp continuous_swap.continuous_on (preimage_swap_prod _ _).symm.subset | lemma | picard_lindelof.continuous_on | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuous_on",
"continuous_on_prod_of_continuous_on_lipschitz_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le {t : ℝ} (ht : t ∈ Icc v.t_min v.t_max) {x : E} (hx : x ∈ closed_ball v.x₀ v.R) :
‖v t x‖ ≤ v.C | v.is_pl.norm_le _ ht _ hx | lemma | picard_lindelof.norm_le | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t_dist : ℝ | max (v.t_max - v.t₀) (v.t₀ - v.t_min) | def | picard_lindelof.t_dist | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | The maximum of distances from `t₀` to the endpoints of `[t_min, t_max]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t_dist_nonneg : 0 ≤ v.t_dist | le_max_iff.2 $ or.inl $ sub_nonneg.2 v.t₀.2.2 | lemma | picard_lindelof.t_dist_nonneg | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_t₀_le (t : Icc v.t_min v.t_max) : dist t v.t₀ ≤ v.t_dist | begin
rw [subtype.dist_eq, real.dist_eq],
cases le_total t v.t₀ with ht ht,
{ rw [abs_of_nonpos (sub_nonpos.2 $ subtype.coe_le_coe.2 ht), neg_sub],
exact (sub_le_sub_left t.2.1 _).trans (le_max_right _ _) },
{ rw [abs_of_nonneg (sub_nonneg.2 $ subtype.coe_le_coe.2 ht)],
exact (sub_le_sub_right t.2.2 _).... | lemma | picard_lindelof.dist_t₀_le | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"abs_of_nonneg",
"abs_of_nonpos",
"real.dist_eq",
"subtype.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj : ℝ → Icc v.t_min v.t_max | proj_Icc v.t_min v.t_max v.t_min_le_t_max | def | picard_lindelof.proj | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | Projection $ℝ → [t_{\min}, t_{\max}]$ sending $(-∞, t_{\min}]$ to $t_{\min}$ and $[t_{\max}, ∞)$
to $t_{\max}$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_coe (t : Icc v.t_min v.t_max) : v.proj t = t | proj_Icc_coe _ _ | lemma | picard_lindelof.proj_coe | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_of_mem {t : ℝ} (ht : t ∈ Icc v.t_min v.t_max) : ↑(v.proj t) = t | by simp only [proj, proj_Icc_of_mem _ ht, subtype.coe_mk] | lemma | picard_lindelof.proj_of_mem | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_proj : continuous v.proj | continuous_proj_Icc | lemma | picard_lindelof.continuous_proj | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuous",
"continuous_proj_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_space | (to_fun : Icc v.t_min v.t_max → E)
(map_t₀' : to_fun v.t₀ = v.x₀)
(lipschitz' : lipschitz_with v.C to_fun) | structure | picard_lindelof.fun_space | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"lipschitz_with"
] | The space of curves $γ \colon [t_{\min}, t_{\max}] \to E$ such that $γ(t₀) = x₀$ and $γ$ is
Lipschitz continuous with constant $C$. The map sending $γ$ to
$\mathbf Pγ(t)=x₀ + ∫_{t₀}^{t} v(τ, γ(τ))\,dτ$ is a contracting map on this space, and its fixed
point is a solution of the ODE $\dot x=v(t, x)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz : lipschitz_with v.C f | f.lipschitz' | lemma | picard_lindelof.fun_space.lipschitz | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous f | f.lipschitz.continuous | lemma | picard_lindelof.fun_space.continuous | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map : v.fun_space ↪ C(Icc v.t_min v.t_max, E) | ⟨λ f, ⟨f, f.continuous⟩, λ f g h, by { cases f, cases g, simpa using h }⟩ | def | picard_lindelof.fun_space.to_continuous_map | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | Each curve in `picard_lindelof.fun_space` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing_to_continuous_map : uniform_inducing (@to_continuous_map _ _ _ v) | ⟨rfl⟩ | lemma | picard_lindelof.fun_space.uniform_inducing_to_continuous_map | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_to_continuous_map :
range to_continuous_map =
{f : C(Icc v.t_min v.t_max, E) | f v.t₀ = v.x₀ ∧ lipschitz_with v.C f} | begin
ext f, split,
{ rintro ⟨⟨f, hf₀, hf_lip⟩, rfl⟩, exact ⟨hf₀, hf_lip⟩ },
{ rcases f with ⟨f, hf⟩, rintro ⟨hf₀, hf_lip⟩, exact ⟨⟨f, hf₀, hf_lip⟩, rfl⟩ }
end | lemma | picard_lindelof.fun_space.range_to_continuous_map | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_t₀ : f v.t₀ = v.x₀ | f.map_t₀' | lemma | picard_lindelof.fun_space.map_t₀ | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball (t : Icc v.t_min v.t_max) : f t ∈ closed_ball v.x₀ v.R | calc dist (f t) v.x₀ = dist (f t) (f.to_fun v.t₀) : by rw f.map_t₀'
... ≤ v.C * dist t v.t₀ : f.lipschitz.dist_le_mul _ _
... ≤ v.C * v.t_dist : mul_le_mul_of_nonneg_left (v.dist_t₀_le _) v.C.2
... ≤ v.R : v.is_pl.C_mul_le_R | lemma | picard_lindelof.fun_space.mem_closed_ball | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
v_comp (t : ℝ) : E | v (v.proj t) (f (v.proj t)) | def | picard_lindelof.fun_space.v_comp | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | Given a curve $γ \colon [t_{\min}, t_{\max}] → E$, `v_comp` is the function
$F(t)=v(π t, γ(π t))$, where `π` is the projection $ℝ → [t_{\min}, t_{\max}]$. The integral of this
function is the image of `γ` under the contracting map we are going to define below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
v_comp_apply_coe (t : Icc v.t_min v.t_max) : f.v_comp t = v t (f t) | by simp only [v_comp, proj_coe] | lemma | picard_lindelof.fun_space.v_comp_apply_coe | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_v_comp : continuous f.v_comp | begin
have := (continuous_subtype_coe.prod_mk f.continuous).comp v.continuous_proj,
refine continuous_on.comp_continuous v.continuous_on this (λ x, _),
exact ⟨(v.proj x).2, f.mem_closed_ball _⟩
end | lemma | picard_lindelof.fun_space.continuous_v_comp | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuous",
"continuous_on.comp_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_v_comp_le (t : ℝ) : ‖f.v_comp t‖ ≤ v.C | v.norm_le (v.proj t).2 $ f.mem_closed_ball _ | lemma | picard_lindelof.fun_space.norm_v_comp_le | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_apply_le_dist (f₁ f₂ : fun_space v) (t : Icc v.t_min v.t_max) :
dist (f₁ t) (f₂ t) ≤ dist f₁ f₂ | @continuous_map.dist_apply_le_dist _ _ _ _ _ f₁.to_continuous_map f₂.to_continuous_map _ | lemma | picard_lindelof.fun_space.dist_apply_le_dist | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuous_map.dist_apply_le_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_of_forall {f₁ f₂ : fun_space v} {d : ℝ} (h : ∀ t, dist (f₁ t) (f₂ t) ≤ d) :
dist f₁ f₂ ≤ d | (@continuous_map.dist_le_iff_of_nonempty _ _ _ _ _ f₁.to_continuous_map f₂.to_continuous_map _
v.nonempty_Icc.to_subtype).2 h | lemma | picard_lindelof.fun_space.dist_le_of_forall | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuous_map.dist_le_iff_of_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_v_comp (t₁ t₂ : ℝ) :
interval_integrable f.v_comp volume t₁ t₂ | (f.continuous_v_comp).interval_integrable _ _ | lemma | picard_lindelof.fun_space.interval_integrable_v_comp | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next (f : fun_space v) : fun_space v | { to_fun := λ t, v.x₀ + ∫ τ : ℝ in v.t₀..t, f.v_comp τ,
map_t₀' := by rw [integral_same, add_zero],
lipschitz' := lipschitz_with.of_dist_le_mul $ λ t₁ t₂,
begin
rw [dist_add_left, dist_eq_norm,
integral_interval_sub_left (f.interval_integrable_v_comp _ _)
(f.interval_integrable_v_comp _ ... | def | picard_lindelof.fun_space.next | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | The Picard-Lindelöf operator. This is a contracting map on `picard_lindelof.fun_space v` such
that the fixed point of this map is the solution of the corresponding ODE.
More precisely, some iteration of this map is a contracting map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
next_apply (t : Icc v.t_min v.t_max) : f.next t = v.x₀ + ∫ τ : ℝ in v.t₀..t, f.v_comp τ | rfl | lemma | picard_lindelof.fun_space.next_apply | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_next (t : Icc v.t_min v.t_max) :
has_deriv_within_at (f.next ∘ v.proj) (v t (f t)) (Icc v.t_min v.t_max) t | begin
haveI : fact ((t : ℝ) ∈ Icc v.t_min v.t_max) := ⟨t.2⟩,
simp only [(∘), next_apply],
refine has_deriv_within_at.const_add _ _,
have : has_deriv_within_at (λ t : ℝ, ∫ τ in v.t₀..t, f.v_comp τ) (f.v_comp t)
(Icc v.t_min v.t_max) t,
from integral_has_deriv_within_at_right (f.interval_integrable_v_comp... | lemma | picard_lindelof.fun_space.has_deriv_within_at_next | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"fact",
"has_deriv_within_at",
"has_deriv_within_at.const_add",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_next_apply_le_of_le {f₁ f₂ : fun_space v} {n : ℕ} {d : ℝ}
(h : ∀ t, dist (f₁ t) (f₂ t) ≤ (v.L * |t - v.t₀|) ^ n / n! * d) (t : Icc v.t_min v.t_max) :
dist (next f₁ t) (next f₂ t) ≤ (v.L * |t - v.t₀|) ^ (n + 1) / (n + 1)! * d | begin
simp only [dist_eq_norm, next_apply, add_sub_add_left_eq_sub,
← interval_integral.integral_sub (interval_integrable_v_comp _ _ _)
(interval_integrable_v_comp _ _ _), norm_integral_eq_norm_integral_Ioc] at *,
calc ‖∫ τ in Ι (v.t₀ : ℝ) t, f₁.v_comp τ - f₂.v_comp τ‖
≤ ∫ τ in Ι (v.t₀ : ℝ) t, v.L *... | lemma | picard_lindelof.fun_space.dist_next_apply_le_of_le | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"continuity",
"continuous.integrable_on_uIoc",
"div_eq_mul_inv",
"integral_pow_abs_sub_uIoc",
"interval_integral.integral_sub",
"measurable_set_Ioc",
"measure_theory.integral_mul_left",
"measure_theory.integral_mul_right",
"mul_assoc",
"mul_inv",
"mul_pow",
"nat.cast_mul",
"nat.cast_succ",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_iterate_next_apply_le (f₁ f₂ : fun_space v) (n : ℕ) (t : Icc v.t_min v.t_max) :
dist (next^[n] f₁ t) (next^[n] f₂ t) ≤ (v.L * |t - v.t₀|) ^ n / n! * dist f₁ f₂ | begin
induction n with n ihn generalizing t,
{ rw [pow_zero, nat.factorial_zero, nat.cast_one, div_one, one_mul],
exact dist_apply_le_dist f₁ f₂ t },
{ rw [iterate_succ_apply', iterate_succ_apply'],
exact dist_next_apply_le_of_le ihn _ }
end | lemma | picard_lindelof.fun_space.dist_iterate_next_apply_le | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"div_one",
"nat.cast_one",
"nat.factorial_zero",
"one_mul",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_iterate_next_le (f₁ f₂ : fun_space v) (n : ℕ) :
dist (next^[n] f₁) (next^[n] f₂) ≤ (v.L * v.t_dist) ^ n / n! * dist f₁ f₂ | begin
refine dist_le_of_forall (λ t, (dist_iterate_next_apply_le _ _ _ _).trans _),
have : 0 ≤ dist f₁ f₂ := dist_nonneg,
have : |(t - v.t₀ : ℝ)| ≤ v.t_dist := v.dist_t₀_le t,
mono*; simp only [nat.cast_nonneg, mul_nonneg, nnreal.coe_nonneg, abs_nonneg, *]
end | lemma | picard_lindelof.fun_space.dist_iterate_next_le | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"abs_nonneg",
"dist_nonneg",
"nat.cast_nonneg",
"nnreal.coe_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_contracting_iterate :
∃ (N : ℕ) K, contracting_with K ((fun_space.next : v.fun_space → v.fun_space)^[N]) | begin
rcases ((real.tendsto_pow_div_factorial_at_top (v.L * v.t_dist)).eventually
(gt_mem_nhds zero_lt_one)).exists with ⟨N, hN⟩,
have : (0 : ℝ) ≤ (v.L * v.t_dist) ^ N / N!,
from div_nonneg (pow_nonneg (mul_nonneg v.L.2 v.t_dist_nonneg) _) (nat.cast_nonneg _),
exact ⟨N, ⟨_, this⟩, hN,
lipschitz_with.o... | lemma | picard_lindelof.exists_contracting_iterate | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"contracting_with",
"div_nonneg",
"gt_mem_nhds",
"nat.cast_nonneg",
"pow_nonneg",
"real.tendsto_pow_div_factorial_at_top",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_fixed : ∃ f : v.fun_space, f.next = f | let ⟨N, K, hK⟩ := exists_contracting_iterate v in ⟨_, hK.is_fixed_pt_fixed_point_iterate⟩ | lemma | picard_lindelof.exists_fixed | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_solution :
∃ f : ℝ → E, f v.t₀ = v.x₀ ∧ ∀ t ∈ Icc v.t_min v.t_max,
has_deriv_within_at f (v t (f t)) (Icc v.t_min v.t_max) t | begin
rcases v.exists_fixed with ⟨f, hf⟩,
refine ⟨f ∘ v.proj, _, λ t ht, _⟩,
{ simp only [(∘), proj_coe, f.map_t₀] },
{ simp only [(∘), v.proj_of_mem ht],
lift t to Icc v.t_min v.t_max using ht,
simpa only [hf, v.proj_coe] using f.has_deriv_within_at_next t }
end | lemma | picard_lindelof.exists_solution | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"has_deriv_within_at",
"lift"
] | Picard-Lindelöf (Cauchy-Lipschitz) theorem. Use
`exists_forall_deriv_within_Icc_eq_of_is_picard_lindelof` instead for the public API. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_picard_lindelof.norm_le₀ {E : Type*} [normed_add_comm_group E]
{v : ℝ → E → E} {t_min t₀ t_max : ℝ} {x₀ : E} {C R : ℝ} {L : ℝ≥0}
(hpl : is_picard_lindelof v t_min t₀ t_max x₀ L R C) : ‖v t₀ x₀‖ ≤ C | hpl.norm_le t₀ hpl.ht₀ x₀ $ mem_closed_ball_self hpl.hR | lemma | is_picard_lindelof.norm_le₀ | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"is_picard_lindelof",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_forall_deriv_within_Icc_eq_of_is_picard_lindelof
[complete_space E] {v : ℝ → E → E} {t_min t₀ t_max : ℝ} (x₀ : E) {C R : ℝ} {L : ℝ≥0}
(hpl : is_picard_lindelof v t_min t₀ t_max x₀ L R C) :
∃ f : ℝ → E, f t₀ = x₀ ∧ ∀ t ∈ Icc t_min t_max,
has_deriv_within_at f (v t (f t)) (Icc t_min t_max) t | begin
lift C to ℝ≥0 using (norm_nonneg _).trans hpl.norm_le₀,
lift t₀ to Icc t_min t_max using hpl.ht₀,
exact picard_lindelof.exists_solution
⟨v, t_min, t_max, t₀, x₀, C, ⟨R, hpl.hR⟩, L, { ht₀ := t₀.property, ..hpl }⟩
end | theorem | exists_forall_deriv_within_Icc_eq_of_is_picard_lindelof | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"complete_space",
"has_deriv_within_at",
"is_picard_lindelof",
"lift",
"picard_lindelof.exists_solution"
] | Picard-Lindelöf (Cauchy-Lipschitz) theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_picard_lindelof_const_of_cont_diff_on_nhds
{s : set E} (hv : cont_diff_on ℝ 1 v s) (hs : s ∈ 𝓝 x₀) :
∃ (ε > (0 : ℝ)) L R C, is_picard_lindelof (λ t, v) (t₀ - ε) t₀ (t₀ + ε) x₀ L R C | begin
-- extract Lipschitz constant
obtain ⟨L, s', hs', hlip⟩ := cont_diff_at.exists_lipschitz_on_with
((hv.cont_diff_within_at (mem_of_mem_nhds hs)).cont_diff_at hs),
-- radius of closed ball in which v is bounded
obtain ⟨r, hr : 0 < r, hball⟩ := metric.mem_nhds_iff.mp (inter_sets (𝓝 x₀) hs hs'),
have h... | lemma | exists_is_picard_lindelof_const_of_cont_diff_on_nhds | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"cont",
"cont_diff_at",
"cont_diff_at.exists_lipschitz_on_with",
"cont_diff_on",
"continuous_on_const",
"div_pos",
"half_pos",
"is_picard_lindelof",
"mem_of_mem_nhds",
"mul_div_cancel'",
"mul_ite",
"mul_one",
"subset_trans",
"zero_lt_one"
] | A time-independent, locally continuously differentiable ODE satisfies the hypotheses of the
Picard-Lindelöf theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_deriv_at_Ioo_eq_of_cont_diff_on_nhds
{s : set E} (hv : cont_diff_on ℝ 1 v s) (hs : s ∈ 𝓝 x₀) :
∃ (ε > (0 : ℝ)) (f : ℝ → E), f t₀ = x₀ ∧
∀ t ∈ Ioo (t₀ - ε) (t₀ + ε), f t ∈ s ∧ has_deriv_at f (v (f t)) t | begin
obtain ⟨ε, hε, L, R, C, hpl⟩ := exists_is_picard_lindelof_const_of_cont_diff_on_nhds t₀ x₀ hv hs,
obtain ⟨f, hf1, hf2⟩ := exists_forall_deriv_within_Icc_eq_of_is_picard_lindelof x₀ hpl,
have hf2' : ∀ t ∈ Ioo (t₀ - ε) (t₀ + ε), has_deriv_at f (v (f t)) t :=
λ t ht, (hf2 t (Ioo_subset_Icc_self ht)).has_de... | theorem | exists_forall_deriv_at_Ioo_eq_of_cont_diff_on_nhds | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"Icc_mem_nhds",
"cont_diff_on",
"continuous_at.preimage_mem_nhds",
"exists_forall_deriv_within_Icc_eq_of_is_picard_lindelof",
"exists_is_picard_lindelof_const_of_cont_diff_on_nhds",
"has_deriv_at",
"metric.ball_subset_ball",
"metric.mem_nhds_iff",
"set.mem_of_mem_of_subset"
] | A time-independent, locally continuously differentiable ODE admits a solution in some open
interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_forall_deriv_at_Ioo_eq_of_cont_diff
(hv : cont_diff ℝ 1 v) : ∃ (ε > (0 : ℝ)) (f : ℝ → E), f t₀ = x₀ ∧
∀ t ∈ Ioo (t₀ - ε) (t₀ + ε), has_deriv_at f (v (f t)) t | let ⟨ε, hε, f, hf1, hf2⟩ := exists_forall_deriv_at_Ioo_eq_of_cont_diff_on_nhds t₀ x₀ hv.cont_diff_on
(is_open.mem_nhds is_open_univ (mem_univ _)) in ⟨ε, hε, f, hf1, λ t ht, (hf2 t ht).2⟩ | theorem | exists_forall_deriv_at_Ioo_eq_of_cont_diff | analysis.ODE | src/analysis/ODE/picard_lindelof.lean | [
"analysis.special_functions.integrals",
"topology.metric_space.contracting"
] | [
"cont_diff",
"exists_forall_deriv_at_Ioo_eq_of_cont_diff_on_nhds",
"has_deriv_at",
"is_open.mem_nhds",
"is_open_univ"
] | A time-independent, continuously differentiable ODE admits a solution in some open interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arsinh (x : ℝ) | log (x + sqrt (1 + x^2)) | def | real.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | `arsinh` is defined using a logarithm, `arsinh x = log (x + sqrt(1 + x^2))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_arsinh (x : ℝ) : exp (arsinh x) = x + sqrt (1 + x^2) | begin
apply exp_log,
rw [← neg_lt_iff_pos_add'],
calc -x ≤ sqrt (x ^ 2) : le_sqrt_of_sq_le (neg_pow_bit0 _ _).le
... < sqrt (1 + x ^ 2) : sqrt_lt_sqrt (sq_nonneg _) (lt_one_add _)
end | lemma | real.exp_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"exp",
"lt_one_add",
"neg_pow_bit0",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_zero : arsinh 0 = 0 | by simp [arsinh] | lemma | real.arsinh_zero | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_neg (x : ℝ) : arsinh (-x) = -arsinh x | begin
rw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh],
apply eq_inv_of_mul_eq_one_left,
rw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, ← sq_sub_sq, sq_sqrt, add_sub_cancel],
exact add_nonneg zero_le_one (sq_nonneg _)
end | lemma | real.arsinh_neg | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"eq_inv_of_mul_eq_one_left",
"exp_eq_exp",
"exp_neg",
"mul_comm",
"neg_sq",
"sq_nonneg",
"sq_sub_sq",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_arsinh (x : ℝ) : sinh (arsinh x) = x | by { rw [sinh_eq, ← arsinh_neg, exp_arsinh, exp_arsinh, neg_sq], field_simp } | lemma | real.sinh_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"neg_sq"
] | `arsinh` is the right inverse of `sinh`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cosh_arsinh (x : ℝ) : cosh (arsinh x) = sqrt (1 + x^2) | by rw [← sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh] | lemma | real.cosh_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_surjective : surjective sinh | left_inverse.surjective sinh_arsinh | lemma | real.sinh_surjective | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | `sinh` is surjective, `∀ b, ∃ a, sinh a = b`. In this case, we use `a = arsinh b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sinh_bijective : bijective sinh | ⟨sinh_injective, sinh_surjective⟩ | lemma | real.sinh_bijective | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | `sinh` is bijective, both injective and surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arsinh_sinh (x : ℝ) : arsinh (sinh x) = x | right_inverse_of_injective_of_left_inverse sinh_injective sinh_arsinh x | lemma | real.arsinh_sinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | `arsinh` is the left inverse of `sinh`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sinh_equiv : ℝ ≃ ℝ | { to_fun := sinh,
inv_fun := arsinh,
left_inv := arsinh_sinh,
right_inv := sinh_arsinh } | def | real.sinh_equiv | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"inv_fun"
] | `real.sinh` as an `equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sinh_order_iso : ℝ ≃o ℝ | { to_equiv := sinh_equiv,
map_rel_iff' := @sinh_le_sinh } | def | real.sinh_order_iso | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | `real.sinh` as an `order_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sinh_homeomorph : ℝ ≃ₜ ℝ | sinh_order_iso.to_homeomorph | def | real.sinh_homeomorph | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | `real.sinh` as a `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arsinh_bijective : bijective arsinh | sinh_equiv.symm.bijective | lemma | real.arsinh_bijective | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_injective : injective arsinh | sinh_equiv.symm.injective | lemma | real.arsinh_injective | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_surjective : surjective arsinh | sinh_equiv.symm.surjective | lemma | real.arsinh_surjective | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_strict_mono : strict_mono arsinh | sinh_order_iso.symm.strict_mono | lemma | real.arsinh_strict_mono | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_inj : arsinh x = arsinh y ↔ x = y | arsinh_injective.eq_iff | lemma | real.arsinh_inj | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_le_arsinh : arsinh x ≤ arsinh y ↔ x ≤ y | sinh_order_iso.symm.le_iff_le | lemma | real.arsinh_le_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_lt_arsinh : arsinh x < arsinh y ↔ x < y | sinh_order_iso.symm.lt_iff_lt | lemma | real.arsinh_lt_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_eq_zero_iff : arsinh x = 0 ↔ x = 0 | arsinh_injective.eq_iff' arsinh_zero | lemma | real.arsinh_eq_zero_iff | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_nonneg_iff : 0 ≤ arsinh x ↔ 0 ≤ x | by rw [← sinh_le_sinh, sinh_zero, sinh_arsinh] | lemma | real.arsinh_nonneg_iff | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_nonpos_iff : arsinh x ≤ 0 ↔ x ≤ 0 | by rw [← sinh_le_sinh, sinh_zero, sinh_arsinh] | lemma | real.arsinh_nonpos_iff | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_pos_iff : 0 < arsinh x ↔ 0 < x | lt_iff_lt_of_le_iff_le arsinh_nonpos_iff | lemma | real.arsinh_pos_iff | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arsinh_neg_iff : arsinh x < 0 ↔ x < 0 | lt_iff_lt_of_le_iff_le arsinh_nonneg_iff | lemma | real.arsinh_neg_iff | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_arsinh (x : ℝ) : has_strict_deriv_at arsinh (sqrt (1 + x ^ 2))⁻¹ x | begin
convert sinh_homeomorph.to_local_homeomorph.has_strict_deriv_at_symm (mem_univ x)
(cosh_pos _).ne' (has_strict_deriv_at_sinh _),
exact (cosh_arsinh _).symm
end | lemma | real.has_strict_deriv_at_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_arsinh (x : ℝ) : has_deriv_at arsinh (sqrt (1 + x ^ 2))⁻¹ x | (has_strict_deriv_at_arsinh x).has_deriv_at | lemma | real.has_deriv_at_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_arsinh : differentiable ℝ arsinh | λ x, (has_deriv_at_arsinh x).differentiable_at | lemma | real.differentiable_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_arsinh {n : ℕ∞} : cont_diff ℝ n arsinh | sinh_homeomorph.cont_diff_symm_deriv (λ x, (cosh_pos x).ne') has_deriv_at_sinh cont_diff_sinh | lemma | real.cont_diff_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_arsinh : continuous arsinh | sinh_homeomorph.symm.continuous | lemma | real.continuous_arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.arsinh {α : Type*} {l : filter α} {f : α → ℝ} {a : ℝ}
(h : tendsto f l (𝓝 a)) : tendsto (λ x, arsinh (f x)) l (𝓝 (arsinh a)) | (continuous_arsinh.tendsto _).comp h | lemma | filter.tendsto.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.arsinh (h : continuous_at f a) : continuous_at (λ x, arsinh (f x)) a | h.arsinh | lemma | continuous_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.arsinh (h : continuous_within_at f s a) :
continuous_within_at (λ x, arsinh (f x)) s a | h.arsinh | lemma | continuous_within_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.arsinh (h : continuous_on f s) : continuous_on (λ x, arsinh (f x)) s | λ x hx, (h x hx).arsinh | lemma | continuous_on.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.arsinh (h : continuous f) : continuous (λ x, arsinh (f x)) | continuous_arsinh.comp h | lemma | continuous.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.arsinh (hf : has_strict_fderiv_at f f' a) :
has_strict_fderiv_at (λ x, arsinh (f x)) ((sqrt (1 + (f a) ^ 2))⁻¹ • f') a | (has_strict_deriv_at_arsinh _).comp_has_strict_fderiv_at a hf | lemma | has_strict_fderiv_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.arsinh (hf : has_fderiv_at f f' a) :
has_fderiv_at (λ x, arsinh (f x)) ((sqrt (1 + (f a) ^ 2))⁻¹ • f') a | (has_deriv_at_arsinh _).comp_has_fderiv_at a hf | lemma | has_fderiv_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.arsinh (hf : has_fderiv_within_at f f' s a) :
has_fderiv_within_at (λ x, arsinh (f x)) ((sqrt (1 + (f a) ^ 2))⁻¹ • f') s a | (has_deriv_at_arsinh _).comp_has_fderiv_within_at a hf | lemma | has_fderiv_within_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.arsinh (h : differentiable_at ℝ f a) :
differentiable_at ℝ (λ x, arsinh (f x)) a | (differentiable_arsinh _).comp a h | lemma | differentiable_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.arsinh (h : differentiable_within_at ℝ f s a) :
differentiable_within_at ℝ (λ x, arsinh (f x)) s a | (differentiable_arsinh _).comp_differentiable_within_at a h | lemma | differentiable_within_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.arsinh (h : differentiable_on ℝ f s) :
differentiable_on ℝ (λ x, arsinh (f x)) s | λ x hx, (h x hx).arsinh | lemma | differentiable_on.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.arsinh (h : differentiable ℝ f) :
differentiable ℝ (λ x, arsinh (f x)) | differentiable_arsinh.comp h | lemma | differentiable.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.arsinh (h : cont_diff_at ℝ n f a) :
cont_diff_at ℝ n (λ x, arsinh (f x)) a | cont_diff_arsinh.cont_diff_at.comp a h | lemma | cont_diff_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.arsinh (h : cont_diff_within_at ℝ n f s a) :
cont_diff_within_at ℝ n (λ x, arsinh (f x)) s a | cont_diff_arsinh.cont_diff_at.comp_cont_diff_within_at a h | lemma | cont_diff_within_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.arsinh (h : cont_diff ℝ n f) : cont_diff ℝ n (λ x, arsinh (f x)) | cont_diff_arsinh.comp h | lemma | cont_diff.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.arsinh (h : cont_diff_on ℝ n f s) : cont_diff_on ℝ n (λ x, arsinh (f x)) s | λ x hx, (h x hx).arsinh | lemma | cont_diff_on.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.arsinh (hf : has_strict_deriv_at f f' a) :
has_strict_deriv_at (λ x, arsinh (f x)) ((sqrt (1 + (f a) ^ 2))⁻¹ • f') a | (has_strict_deriv_at_arsinh _).comp a hf | lemma | has_strict_deriv_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.arsinh (hf : has_deriv_at f f' a) :
has_deriv_at (λ x, arsinh (f x)) ((sqrt (1 + (f a) ^ 2))⁻¹ • f') a | (has_deriv_at_arsinh _).comp a hf | lemma | has_deriv_at.arsinh | analysis.special_functions | src/analysis/special_functions/arsinh.lean | [
"analysis.special_functions.trigonometric.deriv",
"analysis.special_functions.log.basic"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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