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
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lhopital_zero_right_on_Ioo
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfa : tendsto f (𝓝[>] a) (𝓝 0)) (hga : tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[>] a) l) :
tendsto (λ x, (f x) / (g x)) (�... | begin
have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := λ x hx, Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2),
have hg : ∀ x ∈ (Ioo a b), g x ≠ 0,
{ intros x hx h,
have : tendsto g (𝓝[<] x) (𝓝 0),
{ rw [← h, ← nhds_within_Ioo_eq_nhds_within_Iio hx.1],
exact ((hgg' x hx).continuous_at.continuous_within_a... | theorem | has_deriv_at.lhopital_zero_right_on_Ioo | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"continuous_at.tendsto",
"eventually_nhds_within_of_forall",
"exists_has_deriv_at_eq_zero'",
"exists_ratio_has_deriv_at_eq_ratio_slope'",
"has_deriv_at",
"mul_comm",
"nhds_within_Ioo_eq_nhds_within_Iio",
"nhds_within_Ioo_eq_nhds_within_Ioi",
"tendsto_const_nhds",
"tendsto_nhds_within_congr",
"te... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_right_on_Ico
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hcf : continuous_on f (Ico a b)) (hcg : continuous_on g (Ico a b))
(hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfa : f a = 0) (hga : g a = 0)
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[>] a) l) :
... | begin
refine lhopital_zero_right_on_Ioo hab hff' hgg' hg' _ _ hdiv,
{ rw [← hfa, ← nhds_within_Ioo_eq_nhds_within_Ioi hab],
exact ((hcf a $ left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto },
{ rw [← hga, ← nhds_within_Ioo_eq_nhds_within_Ioi hab],
exact ((hcg a $ left_mem_Ico.mpr hab).mono Ioo_subs... | theorem | has_deriv_at.lhopital_zero_right_on_Ico | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"continuous_on",
"has_deriv_at",
"nhds_within_Ioo_eq_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_left_on_Ioo
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfb : tendsto f (𝓝[<] b) (𝓝 0)) (hgb : tendsto g (𝓝[<] b) (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[<] b) l) :
tendsto (λ x, (f x) / (g x)) (𝓝... | begin
-- Here, we essentially compose by `has_neg.neg`. The following is mostly technical details.
have hdnf : ∀ x ∈ -Ioo a b, has_deriv_at (f ∘ has_neg.neg) (f' (-x) * (-1)) x,
from λ x hx, comp x (hff' (-x) hx) (has_deriv_at_neg x),
have hdng : ∀ x ∈ -Ioo a b, has_deriv_at (g ∘ has_neg.neg) (g' (-x) * (-1))... | theorem | has_deriv_at.lhopital_zero_left_on_Ioo | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"has_deriv_at_neg",
"mul_comm",
"mul_neg",
"mul_one",
"neg_div_neg_eq",
"neg_eq_neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_left_on_Ioc
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hcf : continuous_on f (Ioc a b)) (hcg : continuous_on g (Ioc a b))
(hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfb : f b = 0) (hgb : g b = 0)
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[<] b) l) :
... | begin
refine lhopital_zero_left_on_Ioo hab hff' hgg' hg' _ _ hdiv,
{ rw [← hfb, ← nhds_within_Ioo_eq_nhds_within_Iio hab],
exact ((hcf b $ right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto },
{ rw [← hgb, ← nhds_within_Ioo_eq_nhds_within_Iio hab],
exact ((hcg b $ right_mem_Ioc.mpr hab).mono Ioo_sub... | theorem | has_deriv_at.lhopital_zero_left_on_Ioc | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"continuous_on",
"has_deriv_at",
"nhds_within_Ioo_eq_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_at_top_on_Ioi
(hff' : ∀ x ∈ Ioi a, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioi a, has_deriv_at g (g' x) x)
(hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
(hftop : tendsto f at_top (𝓝 0)) (hgtop : tendsto g at_top (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) at_top l) :
tendsto (λ x, (f x) / (g x)) at_top l | begin
obtain ⟨ a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' :=
⟨1 + max a 0, ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩,
have fact1 : ∀ (x:ℝ), x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := λ _ hx, (ne_of_lt hx.1).symm,
have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹,
... | theorem | has_deriv_at.lhopital_zero_at_top_on_Ioi | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"has_deriv_at_inv",
"inv_inv",
"inv_ne_zero",
"lt_inv",
"lt_one_add",
"mul_div_mul_right",
"mul_ne_zero",
"pow_ne_zero",
"self_mem_nhds_within",
"tendsto_inv_at_top_zero'",
"tendsto_inv_zero_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_at_bot_on_Iio
(hff' : ∀ x ∈ Iio a, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Iio a, has_deriv_at g (g' x) x)
(hg' : ∀ x ∈ Iio a, g' x ≠ 0)
(hfbot : tendsto f at_bot (𝓝 0)) (hgbot : tendsto g at_bot (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) at_bot l) :
tendsto (λ x, (f x) / (g x)) at_bot l | begin
-- Here, we essentially compose by `has_neg.neg`. The following is mostly technical details.
have hdnf : ∀ x ∈ -Iio a, has_deriv_at (f ∘ has_neg.neg) (f' (-x) * (-1)) x,
from λ x hx, comp x (hff' (-x) hx) (has_deriv_at_neg x),
have hdng : ∀ x ∈ -Iio a, has_deriv_at (g ∘ has_neg.neg) (g' (-x) * (-1)) x,
... | theorem | has_deriv_at.lhopital_zero_at_bot_on_Iio | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"has_deriv_at_neg",
"mul_comm",
"mul_neg",
"mul_one",
"neg_div_neg_eq",
"neg_eq_neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_right_on_Ioo
(hdf : differentiable_on ℝ f (Ioo a b)) (hg' : ∀ x ∈ Ioo a b, deriv g x ≠ 0)
(hfa : tendsto f (𝓝[>] a) (𝓝 0)) (hga : tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝[>] a) l) :
tendsto (λ x, (f x) / (g x)) (𝓝[>] a) l | begin
have hdf : ∀ x ∈ Ioo a b, differentiable_at ℝ f x,
from λ x hx, (hdf x hx).differentiable_at (Ioo_mem_nhds hx.1 hx.2),
have hdg : ∀ x ∈ Ioo a b, differentiable_at ℝ g x,
from λ x hx, classical.by_contradiction (λ h, hg' x hx (deriv_zero_of_not_differentiable_at h)),
exact has_deriv_at.lhopital_zero_... | theorem | deriv.lhopital_zero_right_on_Ioo | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"Ioo_mem_nhds",
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at.lhopital_zero_right_on_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_right_on_Ico
(hdf : differentiable_on ℝ f (Ioo a b))
(hcf : continuous_on f (Ico a b)) (hcg : continuous_on g (Ico a b))
(hg' : ∀ x ∈ (Ioo a b), (deriv g) x ≠ 0)
(hfa : f a = 0) (hga : g a = 0)
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝[>] a) l) :
tendsto (λ x, (f x) / (g x)) (𝓝... | begin
refine lhopital_zero_right_on_Ioo hab hdf hg' _ _ hdiv,
{ rw [← hfa, ← nhds_within_Ioo_eq_nhds_within_Ioi hab],
exact ((hcf a $ left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto },
{ rw [← hga, ← nhds_within_Ioo_eq_nhds_within_Ioi hab],
exact ((hcg a $ left_mem_Ico.mpr hab).mono Ioo_subset_Ico... | theorem | deriv.lhopital_zero_right_on_Ico | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"continuous_on",
"deriv",
"differentiable_on",
"nhds_within_Ioo_eq_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_left_on_Ioo
(hdf : differentiable_on ℝ f (Ioo a b))
(hg' : ∀ x ∈ (Ioo a b), (deriv g) x ≠ 0)
(hfb : tendsto f (𝓝[<] b) (𝓝 0)) (hgb : tendsto g (𝓝[<] b) (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝[<] b) l) :
tendsto (λ x, (f x) / (g x)) (𝓝[<] b) l | begin
have hdf : ∀ x ∈ Ioo a b, differentiable_at ℝ f x,
from λ x hx, (hdf x hx).differentiable_at (Ioo_mem_nhds hx.1 hx.2),
have hdg : ∀ x ∈ Ioo a b, differentiable_at ℝ g x,
from λ x hx, classical.by_contradiction (λ h, hg' x hx (deriv_zero_of_not_differentiable_at h)),
exact has_deriv_at.lhopital_zero_... | theorem | deriv.lhopital_zero_left_on_Ioo | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"Ioo_mem_nhds",
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at.lhopital_zero_left_on_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_at_top_on_Ioi
(hdf : differentiable_on ℝ f (Ioi a))
(hg' : ∀ x ∈ (Ioi a), (deriv g) x ≠ 0)
(hftop : tendsto f at_top (𝓝 0)) (hgtop : tendsto g at_top (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) at_top l) :
tendsto (λ x, (f x) / (g x)) at_top l | begin
have hdf : ∀ x ∈ Ioi a, differentiable_at ℝ f x,
from λ x hx, (hdf x hx).differentiable_at (Ioi_mem_nhds hx),
have hdg : ∀ x ∈ Ioi a, differentiable_at ℝ g x,
from λ x hx, classical.by_contradiction (λ h, hg' x hx (deriv_zero_of_not_differentiable_at h)),
exact has_deriv_at.lhopital_zero_at_top_on_I... | theorem | deriv.lhopital_zero_at_top_on_Ioi | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"Ioi_mem_nhds",
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at.lhopital_zero_at_top_on_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_at_bot_on_Iio
(hdf : differentiable_on ℝ f (Iio a))
(hg' : ∀ x ∈ (Iio a), (deriv g) x ≠ 0)
(hfbot : tendsto f at_bot (𝓝 0)) (hgbot : tendsto g at_bot (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) at_bot l) :
tendsto (λ x, (f x) / (g x)) at_bot l | begin
have hdf : ∀ x ∈ Iio a, differentiable_at ℝ f x,
from λ x hx, (hdf x hx).differentiable_at (Iio_mem_nhds hx),
have hdg : ∀ x ∈ Iio a, differentiable_at ℝ g x,
from λ x hx, classical.by_contradiction (λ h, hg' x hx (deriv_zero_of_not_differentiable_at h)),
exact has_deriv_at.lhopital_zero_at_bot_on_I... | theorem | deriv.lhopital_zero_at_bot_on_Iio | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"Iio_mem_nhds",
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at.lhopital_zero_at_bot_on_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lhopital_zero_nhds_right
(hff' : ∀ᶠ x in 𝓝[>] a, has_deriv_at f (f' x) x)
(hgg' : ∀ᶠ x in 𝓝[>] a, has_deriv_at g (g' x) x)
(hg' : ∀ᶠ x in 𝓝[>] a, g' x ≠ 0)
(hfa : tendsto f (𝓝[>] a) (𝓝 0)) (hga : tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[>] a) l) :
tendsto (λ x, (f x) / (g ... | begin
rw eventually_iff_exists_mem at *,
rcases hff' with ⟨s₁, hs₁, hff'⟩,
rcases hgg' with ⟨s₂, hs₂, hgg'⟩,
rcases hg' with ⟨s₃, hs₃, hg'⟩,
let s := s₁ ∩ s₂ ∩ s₃,
have hs : s ∈ 𝓝[>] a := inter_mem (inter_mem hs₁ hs₂) hs₃,
rw mem_nhds_within_Ioi_iff_exists_Ioo_subset at hs,
rcases hs with ⟨u, hau, hu⟩,... | theorem | has_deriv_at.lhopital_zero_nhds_right | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"mem_nhds_within_Ioi_iff_exists_Ioo_subset"
] | L'Hôpital's rule for approaching a real from the right, `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds_left
(hff' : ∀ᶠ x in 𝓝[<] a, has_deriv_at f (f' x) x)
(hgg' : ∀ᶠ x in 𝓝[<] a, has_deriv_at g (g' x) x)
(hg' : ∀ᶠ x in 𝓝[<] a, g' x ≠ 0)
(hfa : tendsto f (𝓝[<] a) (𝓝 0)) (hga : tendsto g (𝓝[<] a) (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[<] a) l) :
tendsto (λ x, (f x) / (g x... | begin
rw eventually_iff_exists_mem at *,
rcases hff' with ⟨s₁, hs₁, hff'⟩,
rcases hgg' with ⟨s₂, hs₂, hgg'⟩,
rcases hg' with ⟨s₃, hs₃, hg'⟩,
let s := s₁ ∩ s₂ ∩ s₃,
have hs : s ∈ 𝓝[<] a := inter_mem (inter_mem hs₁ hs₂) hs₃,
rw mem_nhds_within_Iio_iff_exists_Ioo_subset at hs,
rcases hs with ⟨l, hal, hl⟩,... | theorem | has_deriv_at.lhopital_zero_nhds_left | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"mem_nhds_within_Iio_iff_exists_Ioo_subset"
] | L'Hôpital's rule for approaching a real from the left, `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds'
(hff' : ∀ᶠ x in 𝓝[≠] a, has_deriv_at f (f' x) x)
(hgg' : ∀ᶠ x in 𝓝[≠] a, has_deriv_at g (g' x) x)
(hg' : ∀ᶠ x in 𝓝[≠] a, g' x ≠ 0)
(hfa : tendsto f (𝓝[≠] a) (𝓝 0)) (hga : tendsto g (𝓝[≠] a) (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) (𝓝[≠] a) l) :
tendsto (λ x, (f x) / (g x)) (... | begin
simp only [←Iio_union_Ioi, nhds_within_union, tendsto_sup, eventually_sup] at *,
exact ⟨lhopital_zero_nhds_left hff'.1 hgg'.1 hg'.1 hfa.1 hga.1 hdiv.1,
lhopital_zero_nhds_right hff'.2 hgg'.2 hg'.2 hfa.2 hga.2 hdiv.2⟩
end | theorem | has_deriv_at.lhopital_zero_nhds' | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"nhds_within_union"
] | L'Hôpital's rule for approaching a real, `has_deriv_at` version. This
does not require anything about the situation at `a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds
(hff' : ∀ᶠ x in 𝓝 a, has_deriv_at f (f' x) x)
(hgg' : ∀ᶠ x in 𝓝 a, has_deriv_at g (g' x) x)
(hg' : ∀ᶠ x in 𝓝 a, g' x ≠ 0)
(hfa : tendsto f (𝓝 a) (𝓝 0)) (hga : tendsto g (𝓝 a) (𝓝 0))
(hdiv : tendsto (λ x, f' x / g' x) (𝓝 a) l) :
tendsto (λ x, f x / g x) (𝓝[≠] a) l | begin
apply @lhopital_zero_nhds' _ _ _ f' _ g';
apply eventually_nhds_within_of_eventually_nhds <|> apply tendsto_nhds_within_of_tendsto_nhds;
assumption
end | theorem | has_deriv_at.lhopital_zero_nhds | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"eventually_nhds_within_of_eventually_nhds",
"has_deriv_at",
"tendsto_nhds_within_of_tendsto_nhds"
] | **L'Hôpital's rule** for approaching a real, `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_at_top
(hff' : ∀ᶠ x in at_top, has_deriv_at f (f' x) x)
(hgg' : ∀ᶠ x in at_top, has_deriv_at g (g' x) x)
(hg' : ∀ᶠ x in at_top, g' x ≠ 0)
(hftop : tendsto f at_top (𝓝 0)) (hgtop : tendsto g at_top (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) at_top l) :
tendsto (λ x, (f x) / (g x)) at_top l | begin
rw eventually_iff_exists_mem at *,
rcases hff' with ⟨s₁, hs₁, hff'⟩,
rcases hgg' with ⟨s₂, hs₂, hgg'⟩,
rcases hg' with ⟨s₃, hs₃, hg'⟩,
let s := s₁ ∩ s₂ ∩ s₃,
have hs : s ∈ at_top := inter_mem (inter_mem hs₁ hs₂) hs₃,
rw mem_at_top_sets at hs,
rcases hs with ⟨l, hl⟩,
have hl' : Ioi l ⊆ s := λ x h... | theorem | has_deriv_at.lhopital_zero_at_top | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at"
] | L'Hôpital's rule for approaching +∞, `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_at_bot
(hff' : ∀ᶠ x in at_bot, has_deriv_at f (f' x) x)
(hgg' : ∀ᶠ x in at_bot, has_deriv_at g (g' x) x)
(hg' : ∀ᶠ x in at_bot, g' x ≠ 0)
(hfbot : tendsto f at_bot (𝓝 0)) (hgbot : tendsto g at_bot (𝓝 0))
(hdiv : tendsto (λ x, (f' x) / (g' x)) at_bot l) :
tendsto (λ x, (f x) / (g x)) at_bot l | begin
rw eventually_iff_exists_mem at *,
rcases hff' with ⟨s₁, hs₁, hff'⟩,
rcases hgg' with ⟨s₂, hs₂, hgg'⟩,
rcases hg' with ⟨s₃, hs₃, hg'⟩,
let s := s₁ ∩ s₂ ∩ s₃,
have hs : s ∈ at_bot := inter_mem (inter_mem hs₁ hs₂) hs₃,
rw mem_at_bot_sets at hs,
rcases hs with ⟨l, hl⟩,
have hl' : Iio l ⊆ s := λ x h... | theorem | has_deriv_at.lhopital_zero_at_bot | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at"
] | L'Hôpital's rule for approaching -∞, `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds_right
(hdf : ∀ᶠ x in 𝓝[>] a, differentiable_at ℝ f x)
(hg' : ∀ᶠ x in 𝓝[>] a, deriv g x ≠ 0)
(hfa : tendsto f (𝓝[>] a) (𝓝 0)) (hga : tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝[>] a) l) :
tendsto (λ x, (f x) / (g x)) (𝓝[>] a) l | begin
have hdg : ∀ᶠ x in 𝓝[>] a, differentiable_at ℝ g x,
from hg'.mp (eventually_of_forall $
λ _ hg', classical.by_contradiction (λ h, hg' (deriv_zero_of_not_differentiable_at h))),
have hdf' : ∀ᶠ x in 𝓝[>] a, has_deriv_at f (deriv f x) x,
from hdf.mp (eventually_of_forall $ λ _, differentiable_at.... | theorem | deriv.lhopital_zero_nhds_right | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_at.has_deriv_at",
"has_deriv_at",
"has_deriv_at.lhopital_zero_nhds_right"
] | **L'Hôpital's rule** for approaching a real from the right, `deriv` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds_left
(hdf : ∀ᶠ x in 𝓝[<] a, differentiable_at ℝ f x)
(hg' : ∀ᶠ x in 𝓝[<] a, deriv g x ≠ 0)
(hfa : tendsto f (𝓝[<] a) (𝓝 0)) (hga : tendsto g (𝓝[<] a) (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝[<] a) l) :
tendsto (λ x, (f x) / (g x)) (𝓝[<] a) l | begin
have hdg : ∀ᶠ x in 𝓝[<] a, differentiable_at ℝ g x,
from hg'.mp (eventually_of_forall $
λ _ hg', classical.by_contradiction (λ h, hg' (deriv_zero_of_not_differentiable_at h))),
have hdf' : ∀ᶠ x in 𝓝[<] a, has_deriv_at f (deriv f x) x,
from hdf.mp (eventually_of_forall $ λ _, differentiable_at.... | theorem | deriv.lhopital_zero_nhds_left | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_at.has_deriv_at",
"has_deriv_at",
"has_deriv_at.lhopital_zero_nhds_left"
] | **L'Hôpital's rule** for approaching a real from the left, `deriv` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds'
(hdf : ∀ᶠ x in 𝓝[≠] a, differentiable_at ℝ f x)
(hg' : ∀ᶠ x in 𝓝[≠] a, deriv g x ≠ 0)
(hfa : tendsto f (𝓝[≠] a) (𝓝 0)) (hga : tendsto g (𝓝[≠] a) (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝[≠] a) l) :
tendsto (λ x, (f x) / (g x)) (𝓝[≠] a) l | begin
simp only [←Iio_union_Ioi, nhds_within_union, tendsto_sup, eventually_sup] at *,
exact ⟨lhopital_zero_nhds_left hdf.1 hg'.1 hfa.1 hga.1 hdiv.1,
lhopital_zero_nhds_right hdf.2 hg'.2 hfa.2 hga.2 hdiv.2⟩,
end | theorem | deriv.lhopital_zero_nhds' | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"differentiable_at",
"nhds_within_union"
] | **L'Hôpital's rule** for approaching a real, `deriv` version. This
does not require anything about the situation at `a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_nhds
(hdf : ∀ᶠ x in 𝓝 a, differentiable_at ℝ f x)
(hg' : ∀ᶠ x in 𝓝 a, deriv g x ≠ 0)
(hfa : tendsto f (𝓝 a) (𝓝 0)) (hga : tendsto g (𝓝 a) (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) (𝓝 a) l) :
tendsto (λ x, (f x) / (g x)) (𝓝[≠] a) l | begin
apply lhopital_zero_nhds';
apply eventually_nhds_within_of_eventually_nhds <|> apply tendsto_nhds_within_of_tendsto_nhds;
assumption
end | theorem | deriv.lhopital_zero_nhds | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"differentiable_at",
"eventually_nhds_within_of_eventually_nhds",
"tendsto_nhds_within_of_tendsto_nhds"
] | **L'Hôpital's rule** for approaching a real, `deriv` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_at_top
(hdf : ∀ᶠ (x : ℝ) in at_top, differentiable_at ℝ f x)
(hg' : ∀ᶠ (x : ℝ) in at_top, deriv g x ≠ 0)
(hftop : tendsto f at_top (𝓝 0)) (hgtop : tendsto g at_top (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) at_top l) :
tendsto (λ x, (f x) / (g x)) at_top l | begin
have hdg : ∀ᶠ x in at_top, differentiable_at ℝ g x,
from hg'.mp (eventually_of_forall $
λ _ hg', classical.by_contradiction (λ h, hg' (deriv_zero_of_not_differentiable_at h))),
have hdf' : ∀ᶠ x in at_top, has_deriv_at f (deriv f x) x,
from hdf.mp (eventually_of_forall $ λ _, differentiable_at.ha... | theorem | deriv.lhopital_zero_at_top | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_at.has_deriv_at",
"has_deriv_at",
"has_deriv_at.lhopital_zero_at_top"
] | **L'Hôpital's rule** for approaching +∞, `deriv` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lhopital_zero_at_bot
(hdf : ∀ᶠ (x : ℝ) in at_bot, differentiable_at ℝ f x)
(hg' : ∀ᶠ (x : ℝ) in at_bot, deriv g x ≠ 0)
(hfbot : tendsto f at_bot (𝓝 0)) (hgbot : tendsto g at_bot (𝓝 0))
(hdiv : tendsto (λ x, ((deriv f) x) / ((deriv g) x)) at_bot l) :
tendsto (λ x, (f x) / (g x)) at_bot l | begin
have hdg : ∀ᶠ x in at_bot, differentiable_at ℝ g x,
from hg'.mp (eventually_of_forall $
λ _ hg', classical.by_contradiction (λ h, hg' (deriv_zero_of_not_differentiable_at h))),
have hdf' : ∀ᶠ x in at_bot, has_deriv_at f (deriv f x) x,
from hdf.mp (eventually_of_forall $ λ _, differentiable_at.ha... | theorem | deriv.lhopital_zero_at_bot | analysis.calculus | src/analysis/calculus/lhopital.lean | [
"analysis.calculus.mean_value",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"differentiable_at.has_deriv_at",
"has_deriv_at",
"has_deriv_at.lhopital_zero_at_bot"
] | **L'Hôpital's rule** for approaching -∞, `deriv` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_tangent_cone_at (s : set E) (x : E) : set E | {y : E | ∃(c : ℕ → ℝ) (d : ℕ → E), (∀ᶠ n in at_top, x + d n ∈ s) ∧
(tendsto c at_top at_top) ∧ (tendsto (λn, c n • d n) at_top (𝓝 y))} | def | pos_tangent_cone_at | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [] | "Positive" tangent cone to `s` at `x`; the only difference from `tangent_cone_at`
is that we require `c n → ∞` instead of `‖c n‖ → ∞`. One can think about `pos_tangent_cone_at`
as `tangent_cone_at nnreal` but we have no theory of normed semifields yet. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_tangent_cone_at_mono : monotone (λ s, pos_tangent_cone_at s a) | begin
rintros s t hst y ⟨c, d, hd, hc, hcd⟩,
exact ⟨c, d, mem_of_superset hd $ λ h hn, hst hn, hc, hcd⟩
end | lemma | pos_tangent_cone_at_mono | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"monotone",
"pos_tangent_cone_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pos_tangent_cone_at_of_segment_subset {s : set E} {x y : E} (h : segment ℝ x y ⊆ s) :
y - x ∈ pos_tangent_cone_at s x | begin
let c := λn:ℕ, (2:ℝ)^n,
let d := λn:ℕ, (c n)⁻¹ • (y-x),
refine ⟨c, d, filter.univ_mem' (λn, h _),
tendsto_pow_at_top_at_top_of_one_lt one_lt_two, _⟩,
show x + d n ∈ segment ℝ x y,
{ rw segment_eq_image',
refine ⟨(c n)⁻¹, ⟨_, _⟩, rfl⟩,
exacts [inv_nonneg.2 (pow_nonneg zero_le_two _),
in... | lemma | mem_pos_tangent_cone_at_of_segment_subset | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"filter.univ_mem'",
"inv_le_one",
"mul_inv_cancel",
"one_le_pow_of_one_le",
"one_le_two",
"one_lt_two",
"one_smul",
"pos_tangent_cone_at",
"pow_ne_zero",
"pow_nonneg",
"segment",
"segment_eq_image'",
"smul_smul",
"tendsto_const_nhds",
"tendsto_pow_at_top_at_top_of_one_lt",
"two_ne_zero... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pos_tangent_cone_at_of_segment_subset' {s : set E} {x y : E}
(h : segment ℝ x (x + y) ⊆ s) :
y ∈ pos_tangent_cone_at s x | by simpa only [add_sub_cancel'] using mem_pos_tangent_cone_at_of_segment_subset h | lemma | mem_pos_tangent_cone_at_of_segment_subset' | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"mem_pos_tangent_cone_at_of_segment_subset",
"pos_tangent_cone_at",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_tangent_cone_at_univ : pos_tangent_cone_at univ a = univ | eq_univ_of_forall $ λ x, mem_pos_tangent_cone_at_of_segment_subset' (subset_univ _) | lemma | pos_tangent_cone_at_univ | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"mem_pos_tangent_cone_at_of_segment_subset'",
"pos_tangent_cone_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.has_fderiv_within_at_nonpos {s : set E} (h : is_local_max_on f s a)
(hf : has_fderiv_within_at f f' s a) {y} (hy : y ∈ pos_tangent_cone_at s a) :
f' y ≤ 0 | begin
rcases hy with ⟨c, d, hd, hc, hcd⟩,
have hc' : tendsto (λ n, ‖c n‖) at_top at_top,
from tendsto_at_top_mono (λ n, le_abs_self _) hc,
refine le_of_tendsto (hf.lim at_top hd hc' hcd) _,
replace hd : tendsto (λ n, a + d n) at_top (𝓝[s] (a + 0)),
from tendsto_inf.2 ⟨tendsto_const_nhds.add (tangent_cone... | lemma | is_local_max_on.has_fderiv_within_at_nonpos | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_within_at",
"is_local_max_on",
"le_abs_self",
"le_of_tendsto",
"mul_nonpos_of_nonneg_of_nonpos",
"pos_tangent_cone_at",
"smul_eq_mul",
"tangent_cone_at.lim_zero"
] | If `f` has a local max on `s` at `a`, `f'` is the derivative of `f` at `a` within `s`, and
`y` belongs to the positive tangent cone of `s` at `a`, then `f' y ≤ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max_on.fderiv_within_nonpos {s : set E} (h : is_local_max_on f s a)
{y} (hy : y ∈ pos_tangent_cone_at s a) :
(fderiv_within ℝ f s a : E → ℝ) y ≤ 0 | if hf : differentiable_within_at ℝ f s a
then h.has_fderiv_within_at_nonpos hf.has_fderiv_within_at hy
else by { rw fderiv_within_zero_of_not_differentiable_within_at hf, refl } | lemma | is_local_max_on.fderiv_within_nonpos | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"is_local_max_on",
"pos_tangent_cone_at"
] | If `f` has a local max on `s` at `a` and `y` belongs to the positive tangent cone
of `s` at `a`, then `f' y ≤ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max_on.has_fderiv_within_at_eq_zero {s : set E} (h : is_local_max_on f s a)
(hf : has_fderiv_within_at f f' s a) {y} (hy : y ∈ pos_tangent_cone_at s a)
(hy' : -y ∈ pos_tangent_cone_at s a) :
f' y = 0 | le_antisymm (h.has_fderiv_within_at_nonpos hf hy) $
by simpa using h.has_fderiv_within_at_nonpos hf hy' | lemma | is_local_max_on.has_fderiv_within_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_within_at",
"is_local_max_on",
"pos_tangent_cone_at"
] | If `f` has a local max on `s` at `a`, `f'` is a derivative of `f` at `a` within `s`, and
both `y` and `-y` belong to the positive tangent cone of `s` at `a`, then `f' y ≤ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max_on.fderiv_within_eq_zero {s : set E} (h : is_local_max_on f s a)
{y} (hy : y ∈ pos_tangent_cone_at s a) (hy' : -y ∈ pos_tangent_cone_at s a) :
(fderiv_within ℝ f s a : E → ℝ) y = 0 | if hf : differentiable_within_at ℝ f s a
then h.has_fderiv_within_at_eq_zero hf.has_fderiv_within_at hy hy'
else by { rw fderiv_within_zero_of_not_differentiable_within_at hf, refl } | lemma | is_local_max_on.fderiv_within_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"is_local_max_on",
"pos_tangent_cone_at"
] | If `f` has a local max on `s` at `a` and both `y` and `-y` belong to the positive tangent cone
of `s` at `a`, then `f' y = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min_on.has_fderiv_within_at_nonneg {s : set E} (h : is_local_min_on f s a)
(hf : has_fderiv_within_at f f' s a) {y} (hy : y ∈ pos_tangent_cone_at s a) :
0 ≤ f' y | by simpa using h.neg.has_fderiv_within_at_nonpos hf.neg hy | lemma | is_local_min_on.has_fderiv_within_at_nonneg | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_within_at",
"is_local_min_on",
"pos_tangent_cone_at"
] | If `f` has a local min on `s` at `a`, `f'` is the derivative of `f` at `a` within `s`, and
`y` belongs to the positive tangent cone of `s` at `a`, then `0 ≤ f' y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min_on.fderiv_within_nonneg {s : set E} (h : is_local_min_on f s a)
{y} (hy : y ∈ pos_tangent_cone_at s a) :
(0:ℝ) ≤ (fderiv_within ℝ f s a : E → ℝ) y | if hf : differentiable_within_at ℝ f s a
then h.has_fderiv_within_at_nonneg hf.has_fderiv_within_at hy
else by { rw [fderiv_within_zero_of_not_differentiable_within_at hf], refl } | lemma | is_local_min_on.fderiv_within_nonneg | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"is_local_min_on",
"pos_tangent_cone_at"
] | If `f` has a local min on `s` at `a` and `y` belongs to the positive tangent cone
of `s` at `a`, then `0 ≤ f' y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min_on.has_fderiv_within_at_eq_zero {s : set E} (h : is_local_min_on f s a)
(hf : has_fderiv_within_at f f' s a) {y} (hy : y ∈ pos_tangent_cone_at s a)
(hy' : -y ∈ pos_tangent_cone_at s a) :
f' y = 0 | by simpa using h.neg.has_fderiv_within_at_eq_zero hf.neg hy hy' | lemma | is_local_min_on.has_fderiv_within_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_within_at",
"is_local_min_on",
"pos_tangent_cone_at"
] | If `f` has a local max on `s` at `a`, `f'` is a derivative of `f` at `a` within `s`, and
both `y` and `-y` belong to the positive tangent cone of `s` at `a`, then `f' y ≤ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min_on.fderiv_within_eq_zero {s : set E} (h : is_local_min_on f s a)
{y} (hy : y ∈ pos_tangent_cone_at s a) (hy' : -y ∈ pos_tangent_cone_at s a) :
(fderiv_within ℝ f s a : E → ℝ) y = 0 | if hf : differentiable_within_at ℝ f s a
then h.has_fderiv_within_at_eq_zero hf.has_fderiv_within_at hy hy'
else by { rw fderiv_within_zero_of_not_differentiable_within_at hf, refl } | lemma | is_local_min_on.fderiv_within_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"is_local_min_on",
"pos_tangent_cone_at"
] | If `f` has a local min on `s` at `a` and both `y` and `-y` belong to the positive tangent cone
of `s` at `a`, then `f' y = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min.has_fderiv_at_eq_zero (h : is_local_min f a) (hf : has_fderiv_at f f' a) :
f' = 0 | begin
ext y,
apply (h.on univ).has_fderiv_within_at_eq_zero hf.has_fderiv_within_at;
rw pos_tangent_cone_at_univ; apply mem_univ
end | lemma | is_local_min.has_fderiv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_at",
"is_local_min",
"pos_tangent_cone_at_univ"
] | **Fermat's Theorem**: the derivative of a function at a local minimum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min.fderiv_eq_zero (h : is_local_min f a) : fderiv ℝ f a = 0 | if hf : differentiable_at ℝ f a then h.has_fderiv_at_eq_zero hf.has_fderiv_at
else fderiv_zero_of_not_differentiable_at hf | lemma | is_local_min.fderiv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"differentiable_at",
"fderiv",
"fderiv_zero_of_not_differentiable_at",
"is_local_min"
] | **Fermat's Theorem**: the derivative of a function at a local minimum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max.has_fderiv_at_eq_zero (h : is_local_max f a) (hf : has_fderiv_at f f' a) :
f' = 0 | neg_eq_zero.1 $ h.neg.has_fderiv_at_eq_zero hf.neg | lemma | is_local_max.has_fderiv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_at",
"is_local_max"
] | **Fermat's Theorem**: the derivative of a function at a local maximum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max.fderiv_eq_zero (h : is_local_max f a) : fderiv ℝ f a = 0 | if hf : differentiable_at ℝ f a then h.has_fderiv_at_eq_zero hf.has_fderiv_at
else fderiv_zero_of_not_differentiable_at hf | lemma | is_local_max.fderiv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"differentiable_at",
"fderiv",
"fderiv_zero_of_not_differentiable_at",
"is_local_max"
] | **Fermat's Theorem**: the derivative of a function at a local maximum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr.has_fderiv_at_eq_zero (h : is_local_extr f a) :
has_fderiv_at f f' a → f' = 0 | h.elim is_local_min.has_fderiv_at_eq_zero is_local_max.has_fderiv_at_eq_zero | lemma | is_local_extr.has_fderiv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_fderiv_at",
"is_local_extr",
"is_local_max.has_fderiv_at_eq_zero",
"is_local_min.has_fderiv_at_eq_zero"
] | **Fermat's Theorem**: the derivative of a function at a local extremum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr.fderiv_eq_zero (h : is_local_extr f a) : fderiv ℝ f a = 0 | h.elim is_local_min.fderiv_eq_zero is_local_max.fderiv_eq_zero | lemma | is_local_extr.fderiv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"fderiv",
"is_local_extr",
"is_local_max.fderiv_eq_zero",
"is_local_min.fderiv_eq_zero"
] | **Fermat's Theorem**: the derivative of a function at a local extremum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min.has_deriv_at_eq_zero (h : is_local_min f a) (hf : has_deriv_at f f' a) :
f' = 0 | by simpa using continuous_linear_map.ext_iff.1
(h.has_fderiv_at_eq_zero (has_deriv_at_iff_has_fderiv_at.1 hf)) 1 | lemma | is_local_min.has_deriv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_deriv_at",
"is_local_min"
] | **Fermat's Theorem**: the derivative of a function at a local minimum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min.deriv_eq_zero (h : is_local_min f a) : deriv f a = 0 | if hf : differentiable_at ℝ f a then h.has_deriv_at_eq_zero hf.has_deriv_at
else deriv_zero_of_not_differentiable_at hf | lemma | is_local_min.deriv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"is_local_min"
] | **Fermat's Theorem**: the derivative of a function at a local minimum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max.has_deriv_at_eq_zero (h : is_local_max f a) (hf : has_deriv_at f f' a) :
f' = 0 | neg_eq_zero.1 $ h.neg.has_deriv_at_eq_zero hf.neg | lemma | is_local_max.has_deriv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_deriv_at",
"is_local_max"
] | **Fermat's Theorem**: the derivative of a function at a local maximum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_max.deriv_eq_zero (h : is_local_max f a) : deriv f a = 0 | if hf : differentiable_at ℝ f a then h.has_deriv_at_eq_zero hf.has_deriv_at
else deriv_zero_of_not_differentiable_at hf | lemma | is_local_max.deriv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"is_local_max"
] | **Fermat's Theorem**: the derivative of a function at a local maximum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr.has_deriv_at_eq_zero (h : is_local_extr f a) :
has_deriv_at f f' a → f' = 0 | h.elim is_local_min.has_deriv_at_eq_zero is_local_max.has_deriv_at_eq_zero | lemma | is_local_extr.has_deriv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"has_deriv_at",
"is_local_extr",
"is_local_max.has_deriv_at_eq_zero",
"is_local_min.has_deriv_at_eq_zero"
] | **Fermat's Theorem**: the derivative of a function at a local extremum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr.deriv_eq_zero (h : is_local_extr f a) : deriv f a = 0 | h.elim is_local_min.deriv_eq_zero is_local_max.deriv_eq_zero | lemma | is_local_extr.deriv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"deriv",
"is_local_extr",
"is_local_max.deriv_eq_zero",
"is_local_min.deriv_eq_zero"
] | **Fermat's Theorem**: the derivative of a function at a local extremum equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_Ioo_extr_on_Icc (hab : a < b) (hfc : continuous_on f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, is_extr_on f (Icc a b) c | begin
have ne : (Icc a b).nonempty, from nonempty_Icc.2 (le_of_lt hab),
-- Consider absolute min and max points
obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x,
from is_compact_Icc.exists_forall_le ne hfc,
obtain ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f C,
from is_compact_I... | lemma | exists_Ioo_extr_on_Icc | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"continuous_on",
"exists_between",
"is_extr_on"
] | A continuous function on a closed interval with `f a = f b` takes either its maximum
or its minimum value at a point in the interior of the interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_local_extr_Ioo (hab : a < b) (hfc : continuous_on f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, is_local_extr f c | let ⟨c, cmem, hc⟩ := exists_Ioo_extr_on_Icc f hab hfc hfI
in ⟨c, cmem, hc.is_local_extr $ Icc_mem_nhds cmem.1 cmem.2⟩ | lemma | exists_local_extr_Ioo | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"Icc_mem_nhds",
"continuous_on",
"exists_Ioo_extr_on_Icc",
"is_local_extr"
] | A continuous function on a closed interval with `f a = f b` has a local extremum at some
point of the corresponding open interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_has_deriv_at_eq_zero (hab : a < b) (hfc : continuous_on f (Icc a b)) (hfI : f a = f b)
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) :
∃ c ∈ Ioo a b, f' c = 0 | let ⟨c, cmem, hc⟩ := exists_local_extr_Ioo f hab hfc hfI in
⟨c, cmem, hc.has_deriv_at_eq_zero $ hff' c cmem⟩ | lemma | exists_has_deriv_at_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"continuous_on",
"exists_local_extr_Ioo",
"has_deriv_at"
] | **Rolle's Theorem** `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_deriv_eq_zero (hab : a < b) (hfc : continuous_on f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, deriv f c = 0 | let ⟨c, cmem, hc⟩ := exists_local_extr_Ioo f hab hfc hfI in
⟨c, cmem, hc.deriv_eq_zero⟩ | lemma | exists_deriv_eq_zero | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"continuous_on",
"deriv",
"exists_local_extr_Ioo"
] | **Rolle's Theorem** `deriv` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_has_deriv_at_eq_zero' (hab : a < b)
(hfa : tendsto f (𝓝[>] a) (𝓝 l)) (hfb : tendsto f (𝓝[<] b) (𝓝 l))
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) :
∃ c ∈ Ioo a b, f' c = 0 | begin
have : continuous_on f (Ioo a b) := λ x hx, (hff' x hx).continuous_at.continuous_within_at,
have hcont := continuous_on_Icc_extend_from_Ioo hab.ne this hfa hfb,
obtain ⟨c, hc, hcextr⟩ : ∃ c ∈ Ioo a b, is_local_extr (extend_from (Ioo a b) f) c,
{ apply exists_local_extr_Ioo _ hab hcont,
rw eq_lim_at_ri... | lemma | exists_has_deriv_at_eq_zero' | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"Ioo_mem_nhds",
"continuous_at.continuous_within_at",
"continuous_on",
"continuous_on_Icc_extend_from_Ioo",
"eq_lim_at_left_extend_from_Ioo",
"eq_lim_at_right_extend_from_Ioo",
"exists_local_extr_Ioo",
"extend_from",
"extend_from_extends",
"has_deriv_at",
"is_local_extr"
] | **Rolle's Theorem**, a version for a function on an open interval: if `f` has derivative `f'`
on `(a, b)` and has the same limit `l` at `𝓝[>] a` and `𝓝[<] b`, then `f' c = 0`
for some `c ∈ (a, b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_deriv_eq_zero' (hab : a < b)
(hfa : tendsto f (𝓝[>] a) (𝓝 l)) (hfb : tendsto f (𝓝[<] b) (𝓝 l)) :
∃ c ∈ Ioo a b, deriv f c = 0 | classical.by_cases
(assume h : ∀ x ∈ Ioo a b, differentiable_at ℝ f x,
show ∃ c ∈ Ioo a b, deriv f c = 0,
from exists_has_deriv_at_eq_zero' hab hfa hfb (λ x hx, (h x hx).has_deriv_at))
(assume h : ¬∀ x ∈ Ioo a b, differentiable_at ℝ f x,
have h : ∃ x, x ∈ Ioo a b ∧ ¬differentiable_at ℝ f x, by { push_... | lemma | exists_deriv_eq_zero' | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"exists_has_deriv_at_eq_zero'",
"has_deriv_at"
] | **Rolle's Theorem**, a version for a function on an open interval: if `f` has the same limit
`l` at `𝓝[>] a` and `𝓝[<] b`, then `deriv f c = 0` for some `c ∈ (a, b)`. This version
does not require differentiability of `f` because we define `deriv f c = 0` whenever `f` is not
differentiable at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_roots_to_finset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
p.roots.to_finset.card ≤ (p.derivative.roots.to_finset \ p.roots.to_finset).card + 1 | begin
cases eq_or_ne p.derivative 0 with hp' hp',
{ rw [eq_C_of_derivative_eq_zero hp', roots_C, multiset.to_finset_zero, finset.card_empty],
exact zero_le _ },
have hp : p ≠ 0, from ne_of_apply_ne derivative (by rwa [derivative_zero]),
refine finset.card_le_diff_of_interleaved (λ x hx y hy hxy hxy', _),
... | lemma | polynomial.card_roots_to_finset_le_card_roots_derivative_diff_roots_succ | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"eq_or_ne",
"exists_deriv_eq_zero",
"finset.card_empty",
"finset.card_le_diff_of_interleaved",
"multiset.mem_to_finset",
"multiset.to_finset_zero",
"ne_of_apply_ne"
] | The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_roots_to_finset_le_derivative (p : ℝ[X]) :
p.roots.to_finset.card ≤ p.derivative.roots.to_finset.card + 1 | p.card_roots_to_finset_le_card_roots_derivative_diff_roots_succ.trans $
add_le_add_right (finset.card_mono $ finset.sdiff_subset _ _) _ | lemma | polynomial.card_roots_to_finset_le_derivative | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"finset.card_mono",
"finset.sdiff_subset"
] | The number of roots of a real polynomial is at most the number of roots of its derivative plus
one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_roots_le_derivative (p : ℝ[X]) : p.roots.card ≤ p.derivative.roots.card + 1 | calc p.roots.card = ∑ x in p.roots.to_finset, p.roots.count x :
(multiset.to_finset_sum_count_eq _).symm
... = ∑ x in p.roots.to_finset, (p.roots.count x - 1 + 1) :
eq.symm $ finset.sum_congr rfl $ λ x hx, tsub_add_cancel_of_le $ nat.succ_le_iff.2 $
multiset.count_pos.2 $ multiset.mem_to_finset.1 hx
... = ∑ x i... | lemma | polynomial.card_roots_le_derivative | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"finset.card_eq_sum_ones",
"finset.disjoint_sdiff",
"finset.subset_union_right",
"finset.union_sdiff_self_eq_union",
"multiset.count_eq_zero",
"multiset.count_pos",
"multiset.mem_to_finset",
"multiset.to_finset_sum_count_eq",
"tsub_add_cancel_of_le"
] | The number of roots of a real polynomial (counted with multiplicities) is at most the number of
roots of its derivative (counted with multiplicities) plus one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_root_set_le_derivative {F : Type*} [comm_ring F] [algebra F ℝ] (p : F[X]) :
fintype.card (p.root_set ℝ) ≤ fintype.card (p.derivative.root_set ℝ) + 1 | by simpa only [root_set_def, finset.coe_sort_coe, fintype.card_coe, derivative_map]
using card_roots_to_finset_le_derivative (p.map (algebra_map F ℝ)) | lemma | polynomial.card_root_set_le_derivative | analysis.calculus | src/analysis/calculus/local_extr.lean | [
"analysis.calculus.deriv.polynomial",
"topology.algebra.order.extend_from",
"topology.algebra.polynomial"
] | [
"algebra",
"algebra_map",
"comm_ring",
"finset.coe_sort_coe",
"fintype.card",
"fintype.card_coe"
] | The number of real roots of a polynomial is at most the number of roots of its derivative plus
one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB'... | begin
change Icc a b ⊆ {x | f x ≤ B x},
set s := {x | f x ≤ B x} ∩ Icc a b,
have A : continuous_on (λ x, (f x, B x)) (Icc a b), from hf.prod hB,
have : is_closed s,
{ simp only [s, inter_comm],
exact A.preimage_closed_of_closed is_closed_Icc order_closed_topology.is_closed_le' },
apply this.Icc_subset_o... | lemma | image_le_of_liminf_slope_right_lt_deriv_boundary' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"Icc_mem_nhds_within_Ioi",
"Ioc_mem_nhds_within_Ioi",
"Ioi_mem_nhds",
"bound",
"continuous_fst",
"continuous_on",
"continuous_snd",
"div_le_div_right",
"exists_between",
"has_deriv_within_at",
"has_deriv_within_at_iff_tendsto_slope'",
"is_closed",
"is_closed_Icc",
"is_open.mem_nhds",
"is... | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded abo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(b... | image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound | lemma | image_le_of_liminf_slope_right_lt_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_at.continuous_within_at",
"continuous_on",
"has_deriv_at",
"has_deriv_within_at",
"image_le_of_liminf_slope_right_lt_deriv_boundary'",
"slope"
] | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded above by a functio... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
-- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
(bound : ∀ x ∈ I... | begin
have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a),
{ intros x hx r hr,
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound,
{ rwa [sub_self, mul_zero, add_zero] },
{ exact hB.add (continuous_on_const.mul
(continuous_id.continuous_on.sub continuous_on_const)) },
{ a... | lemma | image_le_of_liminf_slope_right_le_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"continuous_on_const",
"continuous_within_at",
"continuous_within_at_const",
"has_deriv_within_at",
"has_deriv_within_at_id",
"image_le_of_liminf_slope_right_lt_deriv_boundary'",
"mul_one",
"mul_zero",
"slope"
] | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded abo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → 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)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a... | image_le_of_liminf_slope_right_lt_deriv_boundary' hf
(λ x hx r hr, (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound | lemma | image_le_of_deriv_right_lt_deriv_boundary' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"has_deriv_within_at",
"image_le_of_liminf_slope_right_lt_deriv_boundary'"
] | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x < B' x` whenever `f x... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → 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)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B ... | image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound | lemma | image_le_of_deriv_right_lt_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_at.continuous_within_at",
"continuous_on",
"has_deriv_at",
"has_deriv_within_at",
"image_le_of_deriv_right_lt_deriv_boundary'"
] | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x < B' x` whenever `f x = B x`.
Then ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → 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)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a ... | image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' $
assume x hx r hr, (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) | lemma | image_le_of_deriv_right_le_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"has_deriv_within_at",
"image_le_of_liminf_slope_right_le_deriv_boundary"
] | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x ≤ B' x` on `[a, b)`.
Then `f x ≤ B ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*}
[normed_add_comm_group E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : continuous_on f (Icc a b))
-- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r →
∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r)
{B B' :... | image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuous_on hf) hf'
ha hB hB' bound | lemma | image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"has_deriv_within_at",
"image_le_of_liminf_slope_right_lt_deriv_boundary'",
"normed_add_comm_group",
"slope"
] | General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `‖f a‖ ≤ B a`;
* `B` has right derivative at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)`
is bounded ab... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a... | image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf
(λ x hx r hr, (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound | lemma | image_norm_le_of_norm_deriv_right_lt_deriv_boundary' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"has_deriv_within_at",
"image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary"
] | General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `‖f a‖ ≤ B a`;
* `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`;
* the norm of `f'` is strictly less than `B'` whene... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ‖f x... | image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound | lemma | image_norm_le_of_norm_deriv_right_lt_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_at.continuous_within_at",
"continuous_on",
"has_deriv_at",
"has_deriv_within_at",
"image_norm_le_of_norm_deriv_right_lt_deriv_boundary'"
] | General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `‖f a‖ ≤ B a`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* `B` has derivative `B'` everywhere on `ℝ`;
* the norm of `f'` is strictly less th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a... | image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuous_on hf) ha hB hB' $
(λ x hx r hr, (hf' x hx).liminf_right_slope_norm_le (lt_of_le_of_lt (bound x hx) hr)) | lemma | image_norm_le_of_norm_deriv_right_le_deriv_boundary' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"has_deriv_within_at",
"image_le_of_liminf_slope_right_le_deriv_boundary"
] | General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `‖f a‖ ≤ B a`;
* `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`;
* we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`.
T... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x | image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound | lemma | image_norm_le_of_norm_deriv_right_le_deriv_boundary | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_at.continuous_within_at",
"continuous_on",
"has_deriv_at",
"has_deriv_within_at",
"image_norm_le_of_norm_deriv_right_le_deriv_boundary'"
] | General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `‖f a‖ ≤ B a`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* `B` has derivative `B'` everywhere on `ℝ`;
* we have `‖f' x‖ ≤ B x` everywhere on... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
(bound : ∀x ∈ Ico a b, ‖f' x‖ ≤ C) :
∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) | begin
let g := λ x, f x - f a,
have hg : continuous_on g (Icc a b), from hf.sub continuous_on_const,
have hg' : ∀ x ∈ Ico a b, has_deriv_within_at g (f' x) (Ici x) x,
{ assume x hx,
simpa using (hf' x hx).sub (has_deriv_within_at_const _ _ _) },
let B := λ x, C * (x - a),
have hB : ∀ x, has_deriv_at B C... | theorem | norm_image_sub_le_of_norm_deriv_right_le_segment | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"continuous_on",
"continuous_on_const",
"has_deriv_at",
"has_deriv_at_const",
"has_deriv_at_id",
"has_deriv_within_at",
"has_deriv_within_at_const",
"image_norm_le_of_norm_deriv_right_le_deriv_boundary",
"mul_zero"
] | A function on `[a, b]` with the norm of the right derivative bounded by `C`
satisfies `‖f x - f a‖ ≤ C * (x - a)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ}
(hf : ∀ x ∈ Icc a b, has_deriv_within_at f (f' x) (Icc a b) x)
(bound : ∀x ∈ Ico a b, ‖f' x‖ ≤ C) :
∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) | begin
refine norm_image_sub_le_of_norm_deriv_right_le_segment
(λ x hx, (hf x hx).continuous_within_at) (λ x hx, _) bound,
exact (hf x $ Ico_subset_Icc_self hx).nhds_within (Icc_mem_nhds_within_Ici hx)
end | theorem | norm_image_sub_le_of_norm_deriv_le_segment' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"Icc_mem_nhds_within_Ici",
"bound",
"continuous_within_at",
"has_deriv_within_at",
"nhds_within",
"norm_image_sub_le_of_norm_deriv_right_le_segment"
] | A function on `[a, b]` with the norm of the derivative within `[a, b]`
bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `has_deriv_within_at`
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : differentiable_on ℝ f (Icc a b))
(bound : ∀x ∈ Ico a b, ‖deriv_within f (Icc a b) x‖ ≤ C) :
∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) | begin
refine norm_image_sub_le_of_norm_deriv_le_segment' _ bound,
exact λ x hx, (hf x hx).has_deriv_within_at
end | theorem | norm_image_sub_le_of_norm_deriv_le_segment | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"differentiable_on",
"has_deriv_within_at",
"norm_image_sub_le_of_norm_deriv_le_segment'"
] | A function on `[a, b]` with the norm of the derivative within `[a, b]`
bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `deriv_within`
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ}
(hf : ∀ x ∈ Icc (0:ℝ) 1, has_deriv_within_at f (f' x) (Icc (0:ℝ) 1) x)
(bound : ∀x ∈ Ico (0:ℝ) 1, ‖f' x‖ ≤ C) :
‖f 1 - f 0‖ ≤ C | by simpa only [sub_zero, mul_one]
using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) | theorem | norm_image_sub_le_of_norm_deriv_le_segment_01' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"has_deriv_within_at",
"mul_one",
"norm_image_sub_le_of_norm_deriv_le_segment'",
"zero_le_one"
] | A function on `[0, 1]` with the norm of the derivative within `[0, 1]`
bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `has_deriv_within_at`
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
(hf : differentiable_on ℝ f (Icc (0:ℝ) 1))
(bound : ∀x ∈ Ico (0:ℝ) 1, ‖deriv_within f (Icc (0:ℝ) 1) x‖ ≤ C) :
‖f 1 - f 0‖ ≤ C | by simpa only [sub_zero, mul_one]
using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) | theorem | norm_image_sub_le_of_norm_deriv_le_segment_01 | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"differentiable_on",
"mul_one",
"norm_image_sub_le_of_norm_deriv_le_segment",
"zero_le_one"
] | A function on `[0, 1]` with the norm of the derivative within `[0, 1]`
bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `deriv_within` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
constant_of_has_deriv_right_zero (hcont : continuous_on f (Icc a b))
(hderiv : ∀ x ∈ Ico a b, has_deriv_within_at f 0 (Ici x) x) :
∀ x ∈ Icc a b, f x = f a | by simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using
λ x hx, norm_image_sub_le_of_norm_deriv_right_le_segment
hcont hderiv (λ y hy, by rw norm_le_zero_iff) x hx | theorem | constant_of_has_deriv_right_zero | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"has_deriv_within_at",
"norm_image_sub_le_of_norm_deriv_right_le_segment",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_of_deriv_within_zero (hdiff : differentiable_on ℝ f (Icc a b))
(hderiv : ∀ x ∈ Ico a b, deriv_within f (Icc a b) x = 0) :
∀ x ∈ Icc a b, f x = f a | begin
have H : ∀ x ∈ Ico a b, ‖deriv_within f (Icc a b) x‖ ≤ 0 :=
by simpa only [norm_le_zero_iff] using λ x hx, hderiv x hx,
simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using
λ x hx, norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx,
end | theorem | constant_of_deriv_within_zero | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv_within",
"differentiable_on",
"norm_image_sub_le_of_norm_deriv_le_segment",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_has_deriv_right_eq
(derivf : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
(derivg : ∀ x ∈ Ico a b, has_deriv_within_at g (f' x) (Ici x) x)
(fcont : continuous_on f (Icc a b)) (gcont : continuous_on g (Icc a b))
(hi : f a = g a) :
∀ y ∈ Icc a b, f y = g y | begin
simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢,
exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont)
(λ y hy, by simpa only [sub_self] using (derivf y hy).sub (derivg y hy)),
end | theorem | eq_of_has_deriv_right_eq | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"constant_of_has_deriv_right_zero",
"continuous_on",
"has_deriv_within_at"
] | If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`,
then they are equal everywhere on `[a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_deriv_within_eq (fdiff : differentiable_on ℝ f (Icc a b))
(gdiff : differentiable_on ℝ g (Icc a b))
(hderiv : eq_on (deriv_within f (Icc a b)) (deriv_within g (Icc a b)) (Ico a b))
(hi : f a = g a) :
∀ y ∈ Icc a b, f y = g y | begin
have A : ∀ y ∈ Ico a b, has_deriv_within_at f (deriv_within f (Icc a b) y) (Ici y) y :=
λ y hy, (fdiff y (mem_Icc_of_Ico hy)).has_deriv_within_at.nhds_within
(Icc_mem_nhds_within_Ici hy),
have B : ∀ y ∈ Ico a b, has_deriv_within_at g (deriv_within g (Icc a b) y) (Ici y) y :=
λ y hy, (gdiff y (me... | theorem | eq_of_deriv_within_eq | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"Icc_mem_nhds_within_Ici",
"deriv_within",
"differentiable_on",
"eq_of_has_deriv_right_eq",
"has_deriv_within_at",
"has_deriv_within_at.nhds_within"
] | If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere
on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_has_fderiv_within_le
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (bound : ∀x∈s, ‖f' x‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | begin
letI : normed_space ℝ G := restrict_scalars.normed_space ℝ 𝕜 G,
/- By composition with `t ↦ x + t • (y-x)`, we reduce to a statement for functions defined
on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`.
We just have to check the differentiability of the composition and... | theorem | convex.norm_image_sub_le_of_norm_has_fderiv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"has_deriv_at",
"has_deriv_at_id",
"has_deriv_within_at",
"has_fderiv_within_at",
"norm_image_sub_le_of_norm_deriv_le_segment_01'",
"normed_space",
"one_smul",
"restrict_scalars",
"segment_eq_image'",
"zero_smul"
] | The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
the function is `C`-Lipschitz. Version with `has_fderiv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_of_nnnorm_has_fderiv_within_le {C : ℝ≥0}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (bound : ∀x∈s, ‖f' x‖₊ ≤ C)
(hs : convex ℝ s) : lipschitz_on_with C f s | begin
rw lipschitz_on_with_iff_norm_sub_le,
intros x x_in y y_in,
exact hs.norm_image_sub_le_of_norm_has_fderiv_within_le hf bound y_in x_in
end | theorem | convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"has_fderiv_within_at",
"lipschitz_on_with"
] | The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
`s`, then the function is `C`-Lipschitz on `s`. Version with `has_fderiv_within` and
`lipschitz_on_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt
(hs : convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, has_fderiv_within_at f (f' y) s y)
(hcont : continuous_within_at f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) :
∃ t ∈ 𝓝[s] x, lipschitz_on_with K f t | begin
obtain ⟨ε, ε0, hε⟩ :
∃ ε > 0, ball x ε ∩ s ⊆ {y | has_fderiv_within_at f (f' y) s y ∧ ‖f' y‖₊ < K},
from mem_nhds_within_iff.1 (hder.and $ hcont.nnnorm.eventually (gt_mem_nhds hK)),
rw inter_comm at hε,
refine ⟨s ∩ ball x ε, inter_mem_nhds_within _ (ball_mem_nhds _ ε0), _⟩,
exact (hs.inter (convex... | lemma | convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_within_at",
"convex",
"convex_ball",
"gt_mem_nhds",
"has_fderiv_within_at",
"inter_mem_nhds_within",
"lipschitz_on_with"
] | Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function
differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is
continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is
`K`-Lipschitz on some neighborhood of `x` w... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at
(hs : convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, has_fderiv_within_at f (f' y) s y)
(hcont : continuous_within_at f' s x) :
∃ K (t ∈ 𝓝[s] x), lipschitz_on_with K f t | (exists_gt _).imp $
hs.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt hder hcont | lemma | convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_within_at",
"convex",
"has_fderiv_within_at",
"lipschitz_on_with"
] | Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function
differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is
continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz
on some neighborhood of `x` withi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_fderiv_within_le
(hf : differentiable_on 𝕜 f s) (bound : ∀x∈s, ‖fderiv_within 𝕜 f s x‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | hs.norm_image_sub_le_of_norm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at)
bound xs ys | theorem | convex.norm_image_sub_le_of_norm_fderiv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_on",
"has_fderiv_within_at"
] | The mean value theorem on a convex set: if the derivative of a function within this set is
bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_of_nnnorm_fderiv_within_le {C : ℝ≥0}
(hf : differentiable_on 𝕜 f s) (bound : ∀ x ∈ s, ‖fderiv_within 𝕜 f s x‖₊ ≤ C)
(hs : convex ℝ s) : lipschitz_on_with C f s | hs.lipschitz_on_with_of_nnnorm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at) bound | theorem | convex.lipschitz_on_with_of_nnnorm_fderiv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_on",
"has_fderiv_within_at",
"lipschitz_on_with"
] | The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
`s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv_within` and
`lipschitz_on_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_fderiv_le
(hf : ∀ x ∈ s, differentiable_at 𝕜 f x) (bound : ∀x∈s, ‖fderiv 𝕜 f x‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | hs.norm_image_sub_le_of_norm_has_fderiv_within_le
(λ x hx, (hf x hx).has_fderiv_at.has_fderiv_within_at) bound xs ys | theorem | convex.norm_image_sub_le_of_norm_fderiv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_at",
"has_fderiv_at.has_fderiv_within_at"
] | The mean value theorem on a convex set: if the derivative of a function is bounded by `C`,
then the function is `C`-Lipschitz. Version with `fderiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_of_nnnorm_fderiv_le {C : ℝ≥0}
(hf : ∀ x ∈ s, differentiable_at 𝕜 f x) (bound : ∀x∈s, ‖fderiv 𝕜 f x‖₊ ≤ C)
(hs : convex ℝ s) : lipschitz_on_with C f s | hs.lipschitz_on_with_of_nnnorm_has_fderiv_within_le
(λ x hx, (hf x hx).has_fderiv_at.has_fderiv_within_at) bound | theorem | convex.lipschitz_on_with_of_nnnorm_fderiv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_at",
"has_fderiv_at.has_fderiv_within_at",
"lipschitz_on_with"
] | The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on
`s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `lipschitz_on_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_has_fderiv_within_le'
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (bound : ∀x∈s, ‖f' x - φ‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ | begin
/- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem
applies, `convex.norm_image_sub_le_of_norm_has_fderiv_within_le`. Then, we just need to glue
together the pieces, expressing back `f` in terms of `g`. -/
let g := λy, f y - φ y,
have hg : ∀ x ∈ s, has_fder... | theorem | convex.norm_image_sub_le_of_norm_has_fderiv_within_le' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"convex.norm_image_sub_le_of_norm_has_fderiv_within_le",
"has_fderiv_within_at"
] | Variant of the mean value inequality on a convex set, using a bound on the difference between
the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with
`has_fderiv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_fderiv_within_le'
(hf : differentiable_on 𝕜 f s) (bound : ∀x∈s, ‖fderiv_within 𝕜 f s x - φ‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ | hs.norm_image_sub_le_of_norm_has_fderiv_within_le' (λ x hx, (hf x hx).has_fderiv_within_at)
bound xs ys | theorem | convex.norm_image_sub_le_of_norm_fderiv_within_le' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_on",
"has_fderiv_within_at"
] | Variant of the mean value inequality on a convex set. Version with `fderiv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_fderiv_le'
(hf : ∀ x ∈ s, differentiable_at 𝕜 f x) (bound : ∀x∈s, ‖fderiv 𝕜 f x - φ‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ | hs.norm_image_sub_le_of_norm_has_fderiv_within_le'
(λ x hx, (hf x hx).has_fderiv_at.has_fderiv_within_at) bound xs ys | theorem | convex.norm_image_sub_le_of_norm_fderiv_le' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_at",
"has_fderiv_at.has_fderiv_within_at"
] | Variant of the mean value inequality on a convex set. Version with `fderiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_const_of_fderiv_within_eq_zero (hs : convex ℝ s) (hf : differentiable_on 𝕜 f s)
(hf' : ∀ x ∈ s, fderiv_within 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) :
f x = f y | have bound : ∀ x ∈ s, ‖fderiv_within 𝕜 f s x‖ ≤ 0,
from λ x hx, by simp only [hf' x hx, norm_zero],
by simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm]
using hs.norm_image_sub_le_of_norm_fderiv_within_le hf bound hx hy | theorem | convex.is_const_of_fderiv_within_eq_zero | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_on",
"dist_le_zero",
"fderiv_within",
"zero_mul"
] | If a function has zero Fréchet derivative at every point of a convex set,
then it is a constant on this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_const_of_fderiv_eq_zero (hf : differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0)
(x y : E) :
f x = f y | convex_univ.is_const_of_fderiv_within_eq_zero hf.differentiable_on
(λ x _, by rw fderiv_within_univ; exact hf' x) trivial trivial | theorem | is_const_of_fderiv_eq_zero | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"differentiable",
"fderiv",
"fderiv_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_of_fderiv_within_eq (hs : convex ℝ s)
(hf : differentiable_on 𝕜 f s) (hg : differentiable_on 𝕜 g s) (hs' : unique_diff_on 𝕜 s)
(hf' : ∀ x ∈ s, fderiv_within 𝕜 f s x = fderiv_within 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) :
s.eq_on f g | begin
intros y hy,
suffices : f x - g x = f y - g y,
{ rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this },
refine hs.is_const_of_fderiv_within_eq_zero (hf.sub hg) _ hx hy,
intros z hz,
rw [fderiv_within_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz],
end | theorem | convex.eq_on_of_fderiv_within_eq | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"convex",
"differentiable_on",
"fderiv_within",
"fderiv_within_sub",
"unique_diff_on"
] | If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at
one point in that set, then they are equal on that set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.eq_of_fderiv_eq (hf : differentiable 𝕜 f) (hg : differentiable 𝕜 g)
(hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x)
(x : E) (hfgx : f x = g x) :
f = g | suffices set.univ.eq_on f g, from funext $ λ x, this $ mem_univ x,
convex_univ.eq_on_of_fderiv_within_eq hf.differentiable_on hg.differentiable_on
unique_diff_on_univ (λ x hx, by simpa using hf' _) (mem_univ _) hfgx | theorem | eq_of_fderiv_eq | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"differentiable",
"fderiv",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_image_sub_le_of_norm_has_deriv_within_le {C : ℝ}
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) (bound : ∀x∈s, ‖f' x‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | convex.norm_image_sub_le_of_norm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at)
(λ x hx, le_trans (by simp) (bound x hx)) hs xs ys | theorem | convex.norm_image_sub_le_of_norm_has_deriv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"convex.norm_image_sub_le_of_norm_has_fderiv_within_le",
"has_deriv_within_at",
"has_fderiv_within_at"
] | The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C`, then the function is `C`-Lipschitz. Version with `has_deriv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_of_nnnorm_has_deriv_within_le {C : ℝ≥0} (hs : convex ℝ s)
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) (bound : ∀x∈s, ‖f' x‖₊ ≤ C) :
lipschitz_on_with C f s | convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at)
(λ x hx, le_trans (by simp) (bound x hx)) hs | theorem | convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le",
"has_deriv_within_at",
"has_fderiv_within_at",
"lipschitz_on_with"
] | The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
Version with `has_deriv_within` and `lipschitz_on_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_within_le {C : ℝ}
(hf : differentiable_on 𝕜 f s) (bound : ∀x∈s, ‖deriv_within f s x‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | hs.norm_image_sub_le_of_norm_has_deriv_within_le (λ x hx, (hf x hx).has_deriv_within_at)
bound xs ys | theorem | convex.norm_image_sub_le_of_norm_deriv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_on",
"has_deriv_within_at"
] | The mean value theorem on a convex set in dimension 1: if the derivative of a function within
this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv_within` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_of_nnnorm_deriv_within_le {C : ℝ≥0} (hs : convex ℝ s)
(hf : differentiable_on 𝕜 f s) (bound : ∀x∈s, ‖deriv_within f s x‖₊ ≤ C) :
lipschitz_on_with C f s | hs.lipschitz_on_with_of_nnnorm_has_deriv_within_le (λ x hx, (hf x hx).has_deriv_within_at) bound | theorem | convex.lipschitz_on_with_of_nnnorm_deriv_within_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_on",
"has_deriv_within_at",
"lipschitz_on_with"
] | The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
Version with `deriv_within` and `lipschitz_on_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_norm_deriv_le {C : ℝ}
(hf : ∀ x ∈ s, differentiable_at 𝕜 f x) (bound : ∀x∈s, ‖deriv f x‖ ≤ C)
(hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | hs.norm_image_sub_le_of_norm_has_deriv_within_le
(λ x hx, (hf x hx).has_deriv_at.has_deriv_within_at) bound xs ys | theorem | convex.norm_image_sub_le_of_norm_deriv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_at",
"has_deriv_at.has_deriv_within_at"
] | The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_of_nnnorm_deriv_le {C : ℝ≥0}
(hf : ∀ x ∈ s, differentiable_at 𝕜 f x) (bound : ∀x∈s, ‖deriv f x‖₊ ≤ C)
(hs : convex ℝ s) : lipschitz_on_with C f s | hs.lipschitz_on_with_of_nnnorm_has_deriv_within_le
(λ x hx, (hf x hx).has_deriv_at.has_deriv_within_at) bound | theorem | convex.lipschitz_on_with_of_nnnorm_deriv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"convex",
"differentiable_at",
"has_deriv_at.has_deriv_within_at",
"lipschitz_on_with"
] | The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
Version with `deriv` and `lipschitz_on_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.lipschitz_with_of_nnnorm_deriv_le {C : ℝ≥0} (hf : differentiable 𝕜 f)
(bound : ∀ x, ‖deriv f x‖₊ ≤ C) : lipschitz_with C f | lipschitz_on_univ.1 $ convex_univ.lipschitz_on_with_of_nnnorm_deriv_le (λ x hx, hf x)
(λ x hx, bound x) | theorem | lipschitz_with_of_nnnorm_deriv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"bound",
"differentiable",
"lipschitz_with"
] | The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`,
then the function is `C`-Lipschitz. Version with `deriv` and `lipschitz_with`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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