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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B_mem_nhds_within_Ioi {K : set F} {r s ε x : ℝ} (hx : x ∈ B f K r s ε) :
B f K r s ε ∈ 𝓝[>] x | begin
obtain ⟨L, LK, hL₁, hL₂⟩ : ∃ (L : F), L ∈ K ∧ x ∈ A f L r ε ∧ x ∈ A f L s ε,
by simpa only [B, mem_Union, mem_inter_iff, exists_prop] using hx,
filter_upwards [A_mem_nhds_within_Ioi hL₁, A_mem_nhds_within_Ioi hL₂] with y hy₁ hy₂,
simp only [B, mem_Union, mem_inter_iff, exists_prop],
exact ⟨L, LK, hy₁,... | lemma | right_deriv_measurable_aux.B_mem_nhds_within_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_B {K : set F} {r s ε : ℝ} : measurable_set (B f K r s ε) | measurable_set_of_mem_nhds_within_Ioi (λ x hx, B_mem_nhds_within_Ioi hx) | lemma | right_deriv_measurable_aux.measurable_set_B | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"measurable_set",
"measurable_set_of_mem_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
A_mono (L : F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) :
A f L r ε ⊆ A f L r δ | begin
rintros x ⟨r', r'r, hr'⟩,
refine ⟨r', r'r, λ y hy z hz, (hr' y hy z hz).trans (mul_le_mul_of_nonneg_right h _)⟩,
linarith [hy.1, hy.2, r'r.2],
end | lemma | right_deriv_measurable_aux.A_mono | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mem_A {r ε : ℝ} {L : F} {x : ℝ} (hx : x ∈ A f L r ε)
{y z : ℝ} (hy : y ∈ Icc x (x + r/2)) (hz : z ∈ Icc x (x + r/2)) :
‖f z - f y - (z-y) • L‖ ≤ ε * r | begin
rcases hx with ⟨r', r'mem, hr'⟩,
have A : x + r / 2 ≤ x + r', by linarith [r'mem.1],
exact hr' _ ((Icc_subset_Icc le_rfl A) hy) _ ((Icc_subset_Icc le_rfl A) hz),
end | lemma | right_deriv_measurable_aux.le_of_mem_A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
(hx : differentiable_within_at ℝ f (Ici x) x) :
∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (deriv_within f (Ici x) x) r ε | begin
have := hx.has_deriv_within_at,
simp_rw [has_deriv_within_at_iff_is_o, is_o_iff] at this,
rcases mem_nhds_within_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩,
refine ⟨m - x, by linarith [show x < m, from xm], λ r hr, _⟩,
have : r ∈ Ioc (r/2) r := ⟨half_lt_self hr.1, le_rfl⟩,
refin... | lemma | right_deriv_measurable_aux.mem_A_of_differentiable | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"differentiable_within_at",
"half_pos",
"has_deriv_within_at_iff_is_o",
"mul_le_mul_of_nonneg_left",
"real.norm_of_nonneg",
"ring",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sub_le_of_mem_A
{r x : ℝ} (hr : 0 < r) (ε : ℝ) {L₁ L₂ : F}
(h₁ : x ∈ A f L₁ r ε) (h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ε | begin
suffices H : ‖(r/2) • (L₁ - L₂)‖ ≤ (r / 2) * (4 * ε),
by rwa [norm_smul, real.norm_of_nonneg (half_pos hr).le, mul_le_mul_left (half_pos hr)] at H,
calc
‖(r/2) • (L₁ - L₂)‖
= ‖(f (x + r/2) - f x - (x + r/2 - x) • L₂) - (f (x + r/2) - f x - (x + r/2 - x) • L₁)‖ :
by simp [smul_sub]
... ≤ ‖f (... | lemma | right_deriv_measurable_aux.norm_sub_le_of_mem_A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"half_pos",
"mul_le_mul_left",
"norm_smul",
"real.norm_of_nonneg",
"ring",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_set_subset_D :
{x | differentiable_within_at ℝ f (Ici x) x ∧ deriv_within f (Ici x) x ∈ K} ⊆ D f K | begin
assume x hx,
rw [D, mem_Inter],
assume e,
have : (0 : ℝ) < (1/2) ^ e := pow_pos (by norm_num) _,
rcases mem_A_of_differentiable this hx.1 with ⟨R, R_pos, hR⟩,
obtain ⟨n, hn⟩ : ∃ (n : ℕ), (1/2) ^ n < R :=
exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ)/2 < 1),
simp only [mem_Union, mem_Inter... | lemma | right_deriv_measurable_aux.differentiable_set_subset_D | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"differentiable_within_at",
"exists_pow_lt_of_lt_one",
"pow_le_pow_of_le_one",
"pow_pos"
] | Easy inclusion: a differentiability point with derivative in `K` belongs to `D f K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
D_subset_differentiable_set {K : set F} (hK : is_complete K) :
D f K ⊆ {x | differentiable_within_at ℝ f (Ici x) x ∧ deriv_within f (Ici x) x ∈ K} | begin
have P : ∀ {n : ℕ}, (0 : ℝ) < (1/2) ^ n := pow_pos (by norm_num),
assume x hx,
have : ∀ (e : ℕ), ∃ (n : ℕ), ∀ p q, n ≤ p → n ≤ q → ∃ L ∈ K,
x ∈ A f L ((1/2) ^ p) ((1/2) ^ e) ∩ A f L ((1/2) ^ q) ((1/2) ^ e),
{ assume e,
have := mem_Inter.1 hx e,
rcases mem_Union.1 this with ⟨n, hn⟩,
refine ... | lemma | right_deriv_measurable_aux.D_subset_differentiable_set | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"Icc_mem_nhds_within_Ici",
"cauchy_seq",
"cauchy_seq_tendsto_of_is_complete",
"deriv_within",
"differentiable_within_at",
"dist_comm",
"div_nonneg",
"div_pos",
"eq_or_lt_of_le",
"exists_nat_pow_near_of_lt_one",
"exists_pow_lt_of_lt_one",
"ge_iff_le",
"has_deriv_within_at",
"has_deriv_withi... | Harder inclusion: at a point in `D f K`, the function `f` has a derivative, in `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_set_eq_D (hK : is_complete K) :
{x | differentiable_within_at ℝ f (Ici x) x ∧ deriv_within f (Ici x) x ∈ K} = D f K | subset.antisymm (differentiable_set_subset_D _) (D_subset_differentiable_set hK) | theorem | right_deriv_measurable_aux.differentiable_set_eq_D | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"differentiable_within_at",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_of_differentiable_within_at_Ici_of_is_complete
{K : set F} (hK : is_complete K) :
measurable_set {x | differentiable_within_at ℝ f (Ici x) x ∧ deriv_within f (Ici x) x ∈ K} | by simp [differentiable_set_eq_D K hK, D, measurable_set_B, measurable_set.Inter,
measurable_set.Union] | theorem | measurable_set_of_differentiable_within_at_Ici_of_is_complete | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"differentiable_within_at",
"is_complete",
"measurable_set",
"measurable_set.Inter",
"measurable_set.Union"
] | The set of right differentiability points of a function, with derivative in a given complete
set, is Borel-measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_set_of_differentiable_within_at_Ici :
measurable_set {x | differentiable_within_at ℝ f (Ici x) x} | begin
have : is_complete (univ : set F) := complete_univ,
convert measurable_set_of_differentiable_within_at_Ici_of_is_complete f this,
simp
end | theorem | measurable_set_of_differentiable_within_at_Ici | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"complete_univ",
"differentiable_within_at",
"is_complete",
"measurable_set",
"measurable_set_of_differentiable_within_at_Ici_of_is_complete"
] | The set of right differentiability points of a function taking values in a complete space is
Borel-measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_deriv_within_Ici [measurable_space F] [borel_space F] :
measurable (λ x, deriv_within f (Ici x) x) | begin
refine measurable_of_is_closed (λ s hs, _),
have : (λ x, deriv_within f (Ici x) x) ⁻¹' s =
{x | differentiable_within_at ℝ f (Ici x) x ∧ deriv_within f (Ici x) x ∈ s} ∪
({x | ¬differentiable_within_at ℝ f (Ici x) x} ∩ {x | (0 : F) ∈ s}) :=
set.ext (λ x, mem_preimage.trans deriv_within_mem_iff),
... | lemma | measurable_deriv_within_Ici | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"borel_space",
"deriv_within",
"deriv_within_mem_iff",
"differentiable_within_at",
"measurable",
"measurable_of_is_closed",
"measurable_set.const",
"measurable_set_of_differentiable_within_at_Ici",
"measurable_set_of_differentiable_within_at_Ici_of_is_complete",
"measurable_space",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_deriv_within_Ici [second_countable_topology F] :
strongly_measurable (λ x, deriv_within f (Ici x) x) | by { borelize F, exact (measurable_deriv_within_Ici f).strongly_measurable } | lemma | strongly_measurable_deriv_within_Ici | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"measurable_deriv_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_measurable_deriv_within_Ici [measurable_space F] [borel_space F]
(μ : measure ℝ) : ae_measurable (λ x, deriv_within f (Ici x) x) μ | (measurable_deriv_within_Ici f).ae_measurable | lemma | ae_measurable_deriv_within_Ici | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"ae_measurable",
"borel_space",
"deriv_within",
"measurable_deriv_within_Ici",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_strongly_measurable_deriv_within_Ici [second_countable_topology F] (μ : measure ℝ) :
ae_strongly_measurable (λ x, deriv_within f (Ici x) x) μ | (strongly_measurable_deriv_within_Ici f).ae_strongly_measurable | lemma | ae_strongly_measurable_deriv_within_Ici | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"strongly_measurable_deriv_within_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_of_differentiable_within_at_Ioi :
measurable_set {x | differentiable_within_at ℝ f (Ioi x) x} | by simpa [differentiable_within_at_Ioi_iff_Ici]
using measurable_set_of_differentiable_within_at_Ici f | theorem | measurable_set_of_differentiable_within_at_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"differentiable_within_at",
"differentiable_within_at_Ioi_iff_Ici",
"measurable_set",
"measurable_set_of_differentiable_within_at_Ici"
] | The set of right differentiability points of a function taking values in a complete space is
Borel-measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_deriv_within_Ioi [measurable_space F] [borel_space F] :
measurable (λ x, deriv_within f (Ioi x) x) | by simpa [deriv_within_Ioi_eq_Ici] using measurable_deriv_within_Ici f | lemma | measurable_deriv_within_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"borel_space",
"deriv_within",
"deriv_within_Ioi_eq_Ici",
"measurable",
"measurable_deriv_within_Ici",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_deriv_within_Ioi [second_countable_topology F] :
strongly_measurable (λ x, deriv_within f (Ioi x) x) | by { borelize F, exact (measurable_deriv_within_Ioi f).strongly_measurable } | lemma | strongly_measurable_deriv_within_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"measurable_deriv_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_measurable_deriv_within_Ioi [measurable_space F] [borel_space F]
(μ : measure ℝ) : ae_measurable (λ x, deriv_within f (Ioi x) x) μ | (measurable_deriv_within_Ioi f).ae_measurable | lemma | ae_measurable_deriv_within_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"ae_measurable",
"borel_space",
"deriv_within",
"measurable_deriv_within_Ioi",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_strongly_measurable_deriv_within_Ioi [second_countable_topology F] (μ : measure ℝ) :
ae_strongly_measurable (λ x, deriv_within f (Ioi x) x) μ | (strongly_measurable_deriv_within_Ioi f).ae_strongly_measurable | lemma | ae_strongly_measurable_deriv_within_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv_within",
"strongly_measurable_deriv_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.taylor_approx_two_segment
{v w : E} (hv : x + v ∈ interior s) (hw : x + v + w ∈ interior s) :
(λ h : ℝ, f (x + h • v + h • w) - f (x + h • v) - h • f' x w
- h^2 • f'' v w - (h^2/2) • f'' w w) =o[𝓝[>] 0] (λ h, h^2) | begin
-- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
-- small enough `h`, for any positive `ε`.
apply is_o.trans_is_O (is_o_iff.2 (λ ε εpos, _)) (is_O_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _),
-- consider a ball of radius `δ` around `x` in which the Taylor approxima... | lemma | convex.taylor_approx_two_segment | analysis.calculus | src/analysis/calculus/fderiv_symmetric.lean | [
"analysis.calculus.mean_value"
] | [
"abs_mul",
"abs_of_nonneg",
"abs_pow",
"bit0_eq_zero",
"continuous_const",
"continuous_linear_map.add_apply",
"continuous_linear_map.coe_smul'",
"continuous_linear_map.coe_sub'",
"continuous_linear_map.le_op_norm",
"continuous_linear_map.map_add",
"continuous_linear_map.map_smul",
"continuous_... | Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one can
Taylor-expand to order two the function `f` on the segment `[x + h v, x + h (v + w)]`, giving a
bilinear estimate for `f (x + hv + hw... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.is_o_alternate_sum_square
{v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
(λ h : ℝ, f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
- f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h^2 • f'' v w) =o[𝓝[>] 0] (λ h, h^2) | begin
have A : (1 : ℝ)/2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩,
have B : (1 : ℝ)/2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩,
have C : ∀ (w : E), (2 : ℝ) • w = 2 • w := λ w, by simp only [two_smul],
have h2v2w : x + (2 : ℝ) • v + (2 : ℝ) • w ∈ interior s,
{ convert s_conv.interior.add_smul_sub_mem... | lemma | convex.is_o_alternate_sum_square | analysis.calculus | src/analysis/calculus/fderiv_symmetric.lean | [
"analysis.calculus.mean_value"
] | [
"add_smul",
"bit0_eq_zero",
"continuous_linear_map.add_apply",
"continuous_linear_map.coe_smul'",
"continuous_linear_map.map_add",
"continuous_linear_map.map_smul",
"interior",
"inv_smul_smul₀",
"one_div",
"one_ne_zero",
"pi.smul_apply",
"smul_add",
"smul_smul",
"smul_sub",
"two_smul"
] | One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
In a setting where `f` is not guaranteed to be continuous at `f`, we can still
get this if we use a quadrilateral based at `h v + h w`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.second_derivative_within_at_symmetric_of_mem_interior
{v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
f'' w v = f'' v w | begin
have A : (λ h : ℝ, h^2 • (f'' w v- f'' v w)) =o[𝓝[>] 0] (λ h, h^2),
{ convert (s_conv.is_o_alternate_sum_square hf xs hx h4v h4w).sub
(s_conv.is_o_alternate_sum_square hf xs hx h4w h4v),
ext h,
simp only [add_comm, smul_add, smul_sub],
abel },
have B : (λ h : ℝ, f'' w v - f'' v w) =... | lemma | convex.second_derivative_within_at_symmetric_of_mem_interior | analysis.calculus | src/analysis/calculus/fderiv_symmetric.lean | [
"analysis.calculus.mean_value"
] | [
"has_lt.lt.ne'",
"interior",
"one_smul",
"self_mem_nhds_within",
"smul_add",
"smul_smul",
"smul_sub"
] | Assume that `f` is differentiable inside a convex set `s`, and that its derivative `f'` is
differentiable at a point `x`. Then, given two vectors `v` and `w` pointing inside `s`, one
has `f'' v w = f'' w v`. Superseded by `convex.second_derivative_within_at_symmetric`, which
removes the assumption that `v` and `w` poin... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.second_derivative_within_at_symmetric
{s : set E} (s_conv : convex ℝ s) (hne : (interior s).nonempty)
{f : E → F} {f' : E → (E →L[ℝ] F)} {f'' : E →L[ℝ] (E →L[ℝ] F)}
(hf : ∀ x ∈ interior s, has_fderiv_at f (f' x) x)
{x : E} (xs : x ∈ s) (hx : has_fderiv_within_at f' f'' (interior s) x) (v w : E) :
f'' v... | begin
/- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
that `f''` is symmetric, after cancellin... | theorem | convex.second_derivative_within_at_symmetric | analysis.calculus | src/analysis/calculus/fderiv_symmetric.lean | [
"analysis.calculus.mean_value"
] | [
"continuous_at_const",
"continuous_linear_map.add_apply",
"continuous_linear_map.coe_smul'",
"continuous_linear_map.map_add",
"continuous_linear_map.map_smul",
"continuous_linear_map.map_zero",
"continuous_linear_map.zero_apply",
"convex",
"filter.tendsto",
"has_fderiv_at",
"has_fderiv_within_at... | If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_derivative_symmetric_of_eventually
{f : E → F} {f' : E → (E →L[ℝ] F)} {f'' : E →L[ℝ] (E →L[ℝ] F)}
(hf : ∀ᶠ y in 𝓝 x, has_fderiv_at f (f' y) y)
(hx : has_fderiv_at f' f'' x) (v w : E) :
f'' v w = f'' w v | begin
rcases metric.mem_nhds_iff.1 hf with ⟨ε, εpos, hε⟩,
have A : (interior (metric.ball x ε)).nonempty,
by rwa [metric.is_open_ball.interior_eq, metric.nonempty_ball],
exact convex.second_derivative_within_at_symmetric (convex_ball x ε) A
(λ y hy, hε (interior_subset hy)) (metric.mem_ball_self εpos) hx.... | theorem | second_derivative_symmetric_of_eventually | analysis.calculus | src/analysis/calculus/fderiv_symmetric.lean | [
"analysis.calculus.mean_value"
] | [
"convex.second_derivative_within_at_symmetric",
"convex_ball",
"has_fderiv_at",
"interior",
"interior_subset",
"metric.ball",
"metric.mem_ball_self",
"metric.nonempty_ball"
] | If a function is differentiable around `x`, and has two derivatives at `x`, then the second
derivative is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_derivative_symmetric
{f : E → F} {f' : E → (E →L[ℝ] F)} {f'' : E →L[ℝ] (E →L[ℝ] F)}
(hf : ∀ y, has_fderiv_at f (f' y) y)
(hx : has_fderiv_at f' f'' x) (v w : E) :
f'' v w = f'' w v | second_derivative_symmetric_of_eventually (filter.eventually_of_forall hf) hx v w | theorem | second_derivative_symmetric | analysis.calculus | src/analysis/calculus/fderiv_symmetric.lean | [
"analysis.calculus.mean_value"
] | [
"filter.eventually_of_forall",
"has_fderiv_at",
"second_derivative_symmetric_of_eventually"
] | If a function is differentiable, and has two derivatives at `x`, then the second
derivative is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formal_multilinear_series (𝕜 : Type*) (E : Type*) (F : Type*)
[ring 𝕜]
[add_comm_group E] [module 𝕜 E] [topological_space E] [topological_add_group E]
[has_continuous_const_smul 𝕜 E]
[add_comm_group F] [module 𝕜 F] [topological_space F] [topological_add_group F]
[has_continuous_const_smul 𝕜 F] | Π (n : ℕ), (E [×n]→L[𝕜] F) | def | formal_multilinear_series | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"add_comm_group",
"has_continuous_const_smul",
"module",
"ring",
"topological_add_group",
"topological_space"
] | A formal multilinear series over a field `𝕜`, from `E` to `F`, is given by a family of
multilinear maps from `E^n` to `F` for all `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff {p q : formal_multilinear_series 𝕜 E F} : p = q ↔ ∀ n, p n = q n | function.funext_iff | lemma | formal_multilinear_series.ext_iff | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series",
"function.funext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_iff {p q : formal_multilinear_series 𝕜 E F} : p ≠ q ↔ ∃ n, p n ≠ q n | function.ne_iff | lemma | formal_multilinear_series.ne_iff | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series",
"function.ne_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_zero (p : formal_multilinear_series 𝕜 E F) : formal_multilinear_series 𝕜 E F | | 0 := 0
| (n + 1) := p (n + 1) | def | formal_multilinear_series.remove_zero | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | Killing the zeroth coefficient in a formal multilinear series | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_zero_coeff_zero (p : formal_multilinear_series 𝕜 E F) :
p.remove_zero 0 = 0 | rfl | lemma | formal_multilinear_series.remove_zero_coeff_zero | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_zero_coeff_succ (p : formal_multilinear_series 𝕜 E F) (n : ℕ) :
p.remove_zero (n+1) = p (n+1) | rfl | lemma | formal_multilinear_series.remove_zero_coeff_succ | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_zero_of_pos (p : formal_multilinear_series 𝕜 E F) {n : ℕ} (h : 0 < n) :
p.remove_zero n = p n | by { rw ← nat.succ_pred_eq_of_pos h, refl } | lemma | formal_multilinear_series.remove_zero_of_pos | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr (p : formal_multilinear_series 𝕜 E F) {m n : ℕ} {v : fin m → E} {w : fin n → E}
(h1 : m = n) (h2 : ∀ (i : ℕ) (him : i < m) (hin : i < n), v ⟨i, him⟩ = w ⟨i, hin⟩) :
p m v = p n w | by { cases h1, congr' with ⟨i, hi⟩, exact h2 i hi hi } | lemma | formal_multilinear_series.congr | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | Convenience congruence lemma stating in a dependent setting that, if the arguments to a formal
multilinear series are equal, then the values are also equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_map (p : formal_multilinear_series 𝕜 F G) (u : E →L[𝕜] F) :
formal_multilinear_series 𝕜 E G | λ n, (p n).comp_continuous_linear_map (λ (i : fin n), u) | def | formal_multilinear_series.comp_continuous_linear_map | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | Composing each term `pₙ` in a formal multilinear series with `(u, ..., u)` where `u` is a fixed
continuous linear map, gives a new formal multilinear series `p.comp_continuous_linear_map u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_map_apply
(p : formal_multilinear_series 𝕜 F G) (u : E →L[𝕜] F) (n : ℕ) (v : fin n → E) :
(p.comp_continuous_linear_map u) n v = p n (u ∘ v) | rfl | lemma | formal_multilinear_series.comp_continuous_linear_map_apply | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (p : formal_multilinear_series 𝕜' E F) :
formal_multilinear_series 𝕜 E F | λ n, (p n).restrict_scalars 𝕜 | def | formal_multilinear_series.restrict_scalars | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series",
"restrict_scalars"
] | Reinterpret a formal `𝕜'`-multilinear series as a formal `𝕜`-multilinear series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift : formal_multilinear_series 𝕜 E (E →L[𝕜] F) | λn, (p n.succ).curry_right | def | formal_multilinear_series.shift | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | Forgetting the zeroth term in a formal multilinear series, and interpreting the following terms
as multilinear maps into `E →L[𝕜] F`. If `p` corresponds to the Taylor series of a function, then
`p.shift` is the Taylor series of the derivative of the function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unshift (q : formal_multilinear_series 𝕜 E (E →L[𝕜] F)) (z : F) :
formal_multilinear_series 𝕜 E F | | 0 := (continuous_multilinear_curry_fin0 𝕜 E F).symm z
| (n + 1) := continuous_multilinear_curry_right_equiv' 𝕜 n E F (q n) | def | formal_multilinear_series.unshift | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"continuous_multilinear_curry_fin0",
"continuous_multilinear_curry_right_equiv'",
"formal_multilinear_series"
] | Adding a zeroth term to a formal multilinear series taking values in `E →L[𝕜] F`. This
corresponds to starting from a Taylor series for the derivative of a function, and building a Taylor
series for the function itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_formal_multilinear_series (f : F →L[𝕜] G) (p : formal_multilinear_series 𝕜 E F) :
formal_multilinear_series 𝕜 E G | λ n, f.comp_continuous_multilinear_map (p n) | def | continuous_linear_map.comp_formal_multilinear_series | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | Composing each term `pₙ` in a formal multilinear series with a continuous linear map `f` on the
left gives a new formal multilinear series `f.comp_formal_multilinear_series p` whose general term
is `f ∘ pₙ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_formal_multilinear_series_apply
(f : F →L[𝕜] G) (p : formal_multilinear_series 𝕜 E F) (n : ℕ) :
(f.comp_formal_multilinear_series p) n = f.comp_continuous_multilinear_map (p n) | rfl | lemma | continuous_linear_map.comp_formal_multilinear_series_apply | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_formal_multilinear_series_apply'
(f : F →L[𝕜] G) (p : formal_multilinear_series 𝕜 E F) (n : ℕ) (v : fin n → E) :
(f.comp_formal_multilinear_series p) n v = f (p n v) | rfl | lemma | continuous_linear_map.comp_formal_multilinear_series_apply' | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order (p : formal_multilinear_series 𝕜 E F) : ℕ | Inf { n | p n ≠ 0 } | def | formal_multilinear_series.order | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | The index of the first non-zero coefficient in `p` (or `0` if all coefficients are zero). This
is the order of the isolated zero of an analytic function `f` at a point if `p` is the Taylor
series of `f` at that point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_zero : (0 : formal_multilinear_series 𝕜 E F).order = 0 | by simp [order] | lemma | formal_multilinear_series.order_zero | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_order_ne_zero (hp : p.order ≠ 0) : p ≠ 0 | λ h, by simpa [h] using hp | lemma | formal_multilinear_series.ne_zero_of_order_ne_zero | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_eq_find [decidable_pred (λ n, p n ≠ 0)] (hp : ∃ n, p n ≠ 0) :
p.order = nat.find hp | by simp [order, Inf, hp] | lemma | formal_multilinear_series.order_eq_find | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_eq_find' [decidable_pred (λ n, p n ≠ 0)] (hp : p ≠ 0) :
p.order = nat.find (formal_multilinear_series.ne_iff.mp hp) | order_eq_find _ | lemma | formal_multilinear_series.order_eq_find' | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_eq_zero_iff (hp : p ≠ 0) : p.order = 0 ↔ p 0 ≠ 0 | begin
classical,
have : ∃ n, p n ≠ 0 := formal_multilinear_series.ne_iff.mp hp,
simp [order_eq_find this, hp]
end | lemma | formal_multilinear_series.order_eq_zero_iff | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_eq_zero_iff' : p.order = 0 ↔ p = 0 ∨ p 0 ≠ 0 | by { by_cases h : p = 0; simp [h, order_eq_zero_iff] } | lemma | formal_multilinear_series.order_eq_zero_iff' | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_order_ne_zero (hp : p ≠ 0) : p p.order ≠ 0 | begin
classical,
let h := formal_multilinear_series.ne_iff.mp hp,
exact (order_eq_find h).symm ▸ nat.find_spec h
end | lemma | formal_multilinear_series.apply_order_ne_zero | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_order_ne_zero' (hp : p.order ≠ 0) : p p.order ≠ 0 | apply_order_ne_zero (ne_zero_of_order_ne_zero hp) | lemma | formal_multilinear_series.apply_order_ne_zero' | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_zero_of_lt_order (hp : n < p.order) : p n = 0 | begin
by_cases p = 0,
{ simp [h] },
{ classical,
rw [order_eq_find' h] at hp,
simpa using nat.find_min _ hp }
end | lemma | formal_multilinear_series.apply_eq_zero_of_lt_order | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff (p : formal_multilinear_series 𝕜 𝕜 E) (n : ℕ) : E | p n 1 | def | formal_multilinear_series.coeff | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | The `n`th coefficient of `p` when seen as a power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pi_field_coeff_eq (p : formal_multilinear_series 𝕜 𝕜 E) (n : ℕ) :
continuous_multilinear_map.mk_pi_field 𝕜 (fin n) (p.coeff n) = p n | (p n).mk_pi_field_apply_one_eq_self | lemma | formal_multilinear_series.mk_pi_field_coeff_eq | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_prod_smul_coeff : p n y = (∏ i, y i) • p.coeff n | begin
convert (p n).to_multilinear_map.map_smul_univ y 1,
funext; simp only [pi.one_apply, algebra.id.smul_eq_mul, mul_one],
end | lemma | formal_multilinear_series.apply_eq_prod_smul_coeff | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"algebra.id.smul_eq_mul",
"mul_one",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_zero : p.coeff n = 0 ↔ p n = 0 | by rw [← mk_pi_field_coeff_eq p, continuous_multilinear_map.mk_pi_field_eq_zero_iff] | lemma | formal_multilinear_series.coeff_eq_zero | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_pow_smul_coeff : p n (λ _, z) = z ^ n • p.coeff n | by simp | lemma | formal_multilinear_series.apply_eq_pow_smul_coeff | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_apply_eq_norm_coef : ‖p n‖ = ‖coeff p n‖ | by rw [← mk_pi_field_coeff_eq p, continuous_multilinear_map.norm_mk_pi_field] | lemma | formal_multilinear_series.norm_apply_eq_norm_coef | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"continuous_multilinear_map.norm_mk_pi_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fslope (p : formal_multilinear_series 𝕜 𝕜 E) : formal_multilinear_series 𝕜 𝕜 E | λ n, (p (n + 1)).curry_left 1 | def | formal_multilinear_series.fslope | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"formal_multilinear_series"
] | The formal counterpart of `dslope`, corresponding to the expansion of `(f z - f 0) / z`. If `f`
has `p` as a power series, then `dslope f` has `fslope p` as a power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_fslope : p.fslope.coeff n = p.coeff (n + 1) | begin
have : @fin.cons n (λ _, 𝕜) 1 (1 : fin n → 𝕜) = 1 := fin.cons_self_tail 1,
simp only [fslope, coeff, continuous_multilinear_map.curry_left_apply, this],
end | lemma | formal_multilinear_series.coeff_fslope | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"continuous_multilinear_map.curry_left_apply",
"fin.cons",
"fin.cons_self_tail"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_iterate_fslope (k n : ℕ) :
(fslope^[k] p).coeff n = p.coeff (n + k) | by induction k with k ih generalizing p; refl <|> simpa [ih] | lemma | formal_multilinear_series.coeff_iterate_fslope | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_formal_multilinear_series (𝕜 : Type*) [nontrivially_normed_field 𝕜]
(E : Type*) [normed_add_comm_group E] [normed_space 𝕜 E] [has_continuous_const_smul 𝕜 E]
[topological_add_group E] {F : Type*} [normed_add_comm_group F] [topological_add_group F]
[normed_space 𝕜 F] [has_continuous_const_smul 𝕜 F] (c ... | | 0 := continuous_multilinear_map.curry0 _ _ c
| _ := 0 | def | const_formal_multilinear_series | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"continuous_multilinear_map.curry0",
"formal_multilinear_series",
"has_continuous_const_smul",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space",
"topological_add_group"
] | The formal multilinear series where all terms of positive degree are equal to zero, and the term
of degree zero is `c`. It is the power series expansion of the constant function equal to `c`
everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_formal_multilinear_series_apply [nontrivially_normed_field 𝕜]
[normed_add_comm_group E] [normed_add_comm_group F] [normed_space 𝕜 E] [normed_space 𝕜 F]
{c : F} {n : ℕ} (hn : n ≠ 0) :
const_formal_multilinear_series 𝕜 E c n = 0 | nat.cases_on n (λ hn, (hn rfl).elim) (λ _ _, rfl) hn | lemma | const_formal_multilinear_series_apply | analysis.calculus | src/analysis/calculus/formal_multilinear_series.lean | [
"analysis.normed_space.multilinear"
] | [
"const_formal_multilinear_series",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_function_data (𝕜 : Type*) [nontrivially_normed_field 𝕜]
(E : Type*) [normed_add_comm_group E] [normed_space 𝕜 E] [complete_space E]
(F : Type*) [normed_add_comm_group F] [normed_space 𝕜 F] [complete_space F]
(G : Type*) [normed_add_comm_group G] [normed_space 𝕜 G] [complete_space G] | (left_fun : E → F)
(left_deriv : E →L[𝕜] F)
(right_fun : E → G)
(right_deriv : E →L[𝕜] G)
(pt : E)
(left_has_deriv : has_strict_fderiv_at left_fun left_deriv pt)
(right_has_deriv : has_strict_fderiv_at right_fun right_deriv pt)
(left_range : range left_deriv = ⊤)
(right_range : range right_deriv = ⊤)
(is_compl_ker : ... | structure | implicit_function_data | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"complete_space",
"has_strict_fderiv_at",
"is_compl",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space"
] | Data for the general version of the implicit function theorem. It holds two functions
`f : E → F` and `g : E → G` (named `left_fun` and `right_fun`) and a point `a` (named `pt`)
such that
* both functions are strictly differentiable at `a`;
* the derivatives are surjective;
* the kernels of the derivatives are complem... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_fun (x : E) : F × G | (φ.left_fun x, φ.right_fun x) | def | implicit_function_data.prod_fun | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | The function given by `x ↦ (left_fun x, right_fun x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_fun_apply (x : E) : φ.prod_fun x = (φ.left_fun x, φ.right_fun x) | rfl | lemma | implicit_function_data.prod_fun_apply | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at :
has_strict_fderiv_at φ.prod_fun
(φ.left_deriv.equiv_prod_of_surjective_of_is_compl φ.right_deriv φ.left_range φ.right_range
φ.is_compl_ker : E →L[𝕜] F × G) φ.pt | φ.left_has_deriv.prod φ.right_has_deriv | lemma | implicit_function_data.has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_homeomorph : local_homeomorph E (F × G) | φ.has_strict_fderiv_at.to_local_homeomorph _ | def | implicit_function_data.to_local_homeomorph | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"local_homeomorph"
] | Implicit function theorem. If `f : E → F` and `g : E → G` are two maps strictly differentiable
at `a`, their derivatives `f'`, `g'` are surjective, and the kernels of these derivatives are
complementary subspaces of `E`, then `x ↦ (f x, g x)` defines a local homeomorphism between
`E` and `F × G`. In particular, `{x | f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_function : F → G → E | function.curry $ φ.to_local_homeomorph.symm | def | implicit_function_data.implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | Implicit function theorem. If `f : E → F` and `g : E → G` are two maps strictly differentiable
at `a`, their derivatives `f'`, `g'` are surjective, and the kernels of these derivatives are
complementary subspaces of `E`, then `implicit_function_of_is_compl_ker` is the unique (germ of a)
map `φ : F → G → E` such that `f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_homeomorph_coe : ⇑(φ.to_local_homeomorph) = φ.prod_fun | rfl | lemma | implicit_function_data.to_local_homeomorph_coe | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_homeomorph_apply (x : E) :
φ.to_local_homeomorph x = (φ.left_fun x, φ.right_fun x) | rfl | lemma | implicit_function_data.to_local_homeomorph_apply | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pt_mem_to_local_homeomorph_source :
φ.pt ∈ φ.to_local_homeomorph.source | φ.has_strict_fderiv_at.mem_to_local_homeomorph_source | lemma | implicit_function_data.pt_mem_to_local_homeomorph_source | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pt_mem_to_local_homeomorph_target :
(φ.left_fun φ.pt, φ.right_fun φ.pt) ∈ φ.to_local_homeomorph.target | φ.to_local_homeomorph.map_source $ φ.pt_mem_to_local_homeomorph_source | lemma | implicit_function_data.map_pt_mem_to_local_homeomorph_target | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_implicit_function :
∀ᶠ (p : F × G) in 𝓝 (φ.prod_fun φ.pt), φ.prod_fun (φ.implicit_function p.1 p.2) = p | φ.has_strict_fderiv_at.eventually_right_inverse.mono $ λ ⟨z, y⟩ h, h | lemma | implicit_function_data.prod_map_implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_map_implicit_function :
∀ᶠ (p : F × G) in 𝓝 (φ.prod_fun φ.pt), φ.left_fun (φ.implicit_function p.1 p.2) = p.1 | φ.prod_map_implicit_function.mono $ λ z, congr_arg prod.fst | lemma | implicit_function_data.left_map_implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_map_implicit_function :
∀ᶠ (p : F × G) in 𝓝 (φ.prod_fun φ.pt), φ.right_fun (φ.implicit_function p.1 p.2) = p.2 | φ.prod_map_implicit_function.mono $ λ z, congr_arg prod.snd | lemma | implicit_function_data.right_map_implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_function_apply_image :
∀ᶠ x in 𝓝 φ.pt, φ.implicit_function (φ.left_fun x) (φ.right_fun x) = x | φ.has_strict_fderiv_at.eventually_left_inverse | lemma | implicit_function_data.implicit_function_apply_image | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_eq : map φ.left_fun (𝓝 φ.pt) = 𝓝 (φ.left_fun φ.pt) | show map (prod.fst ∘ φ.prod_fun) (𝓝 φ.pt) = 𝓝 (φ.prod_fun φ.pt).1,
by rw [← map_map, φ.has_strict_fderiv_at.map_nhds_eq_of_equiv, map_fst_nhds] | lemma | implicit_function_data.map_nhds_eq | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"map_fst_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_function_has_strict_fderiv_at
(g'inv : G →L[𝕜] E) (hg'inv : φ.right_deriv.comp g'inv = continuous_linear_map.id 𝕜 G)
(hg'invf : φ.left_deriv.comp g'inv = 0) :
has_strict_fderiv_at (φ.implicit_function (φ.left_fun φ.pt)) g'inv (φ.right_fun φ.pt) | begin
have := φ.has_strict_fderiv_at.to_local_inverse,
simp only [prod_fun] at this,
convert this.comp (φ.right_fun φ.pt)
((has_strict_fderiv_at_const _ _).prod (has_strict_fderiv_at_id _)),
simp only [continuous_linear_map.ext_iff, continuous_linear_map.coe_comp', function.comp_app]
at hg'inv hg'invf ⊢... | lemma | implicit_function_data.implicit_function_has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"continuous_linear_equiv.eq_symm_apply",
"continuous_linear_map.coe_comp'",
"continuous_linear_map.ext_iff",
"continuous_linear_map.id",
"has_strict_fderiv_at",
"has_strict_fderiv_at_const",
"has_strict_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_function_data_of_complemented (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
implicit_function_data 𝕜 E F (ker f') | { left_fun := f,
left_deriv := f',
right_fun := λ x, classical.some hker (x - a),
right_deriv := classical.some hker,
pt := a,
left_has_deriv := hf,
right_has_deriv := (classical.some hker).has_strict_fderiv_at.comp a
((has_strict_fderiv_at_id a).sub_const a),
left_range := hf',
right_range := linea... | def | has_strict_fderiv_at.implicit_function_data_of_complemented | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at.comp",
"has_strict_fderiv_at_id",
"implicit_function_data",
"linear_map.is_compl_of_proj",
"linear_map.range_eq_of_proj"
] | Data used to apply the generic implicit function theorem to the case of a strictly
differentiable map such that its derivative is surjective and has a complemented kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_to_local_homeomorph_of_complemented (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
local_homeomorph E (F × (ker f')) | (implicit_function_data_of_complemented f f' hf hf' hker).to_local_homeomorph | def | has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at",
"local_homeomorph"
] | A local homeomorphism between `E` and `F × f'.ker` sending level surfaces of `f`
to vertical subspaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_function_of_complemented (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
F → (ker f') → E | (implicit_function_data_of_complemented f f' hf hf' hker).implicit_function | def | has_strict_fderiv_at.implicit_function_of_complemented | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | Implicit function `g` defined by `f (g z y) = z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_to_local_homeomorph_of_complemented_fst (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) (x : E) :
(hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker x).fst = f x | rfl | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_fst | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_to_local_homeomorph_of_complemented_apply
(hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤)
(hker : (ker f').closed_complemented) (y : E) :
hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker y =
(f y, classical.some hker (y - a)) | rfl | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_apply | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_to_local_homeomorph_of_complemented_apply_ker
(hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤)
(hker : (ker f').closed_complemented) (y : ker f') :
hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker (y + a) = (f (y + a), y) | by simp only [implicit_to_local_homeomorph_of_complemented_apply, add_sub_cancel,
classical.some_spec hker] | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_apply_ker | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_to_local_homeomorph_of_complemented_self
(hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker a = (f a, 0) | by simp [hf.implicit_to_local_homeomorph_of_complemented_apply] | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_self | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_implicit_to_local_homeomorph_of_complemented_source (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
a ∈ (hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker).source | mem_to_local_homeomorph_source _ | lemma | has_strict_fderiv_at.mem_implicit_to_local_homeomorph_of_complemented_source | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_implicit_to_local_homeomorph_of_complemented_target (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
(f a, (0 : ker f')) ∈ (hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker).target | by simpa only [implicit_to_local_homeomorph_of_complemented_self] using
((hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker).map_source $
(hf.mem_implicit_to_local_homeomorph_of_complemented_source hf' hker)) | lemma | has_strict_fderiv_at.mem_implicit_to_local_homeomorph_of_complemented_target | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_implicit_function_of_complemented_eq (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
∀ᶠ (p : F × (ker f')) in 𝓝 (f a, 0),
f (hf.implicit_function_of_complemented f f' hf' hker p.1 p.2) = p.1 | ((hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker).eventually_right_inverse $
hf.mem_implicit_to_local_homeomorph_of_complemented_target hf' hker).mono $ λ ⟨z, y⟩ h,
congr_arg prod.fst h | lemma | has_strict_fderiv_at.map_implicit_function_of_complemented_eq | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | `implicit_function_of_complemented` sends `(z, y)` to a point in `f ⁻¹' z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_implicit_function_of_complemented (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
∀ᶠ x in 𝓝 a, hf.implicit_function_of_complemented f f' hf' hker (f x)
(hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker x).snd = x | (implicit_function_data_of_complemented f f' hf hf' hker).implicit_function_apply_image | lemma | has_strict_fderiv_at.eq_implicit_function_of_complemented | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | Any point in some neighborhood of `a` can be represented as `implicit_function`
of some point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_function_of_complemented_apply_image (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
hf.implicit_function_of_complemented f f' hf' hker (f a) 0 = a | begin
convert (hf.implicit_to_local_homeomorph_of_complemented f f' hf' hker).left_inv
(hf.mem_implicit_to_local_homeomorph_of_complemented_source hf' hker),
exact congr_arg prod.snd (hf.implicit_to_local_homeomorph_of_complemented_self hf' hker).symm
end | lemma | has_strict_fderiv_at.implicit_function_of_complemented_apply_image | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_implicit_function_of_complemented (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (hker : (ker f').closed_complemented) :
has_strict_fderiv_at (hf.implicit_function_of_complemented f f' hf' hker (f a))
(ker f').subtypeL 0 | begin
convert (implicit_function_data_of_complemented f f' hf hf'
hker).implicit_function_has_strict_fderiv_at (ker f').subtypeL _ _,
swap,
{ ext, simp only [classical.some_spec hker, implicit_function_data_of_complemented,
continuous_linear_map.coe_comp', submodule.coe_subtypeL', submodul... | lemma | has_strict_fderiv_at.to_implicit_function_of_complemented | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"continuous_linear_map.coe_comp'",
"continuous_linear_map.coe_id'",
"continuous_linear_map.zero_apply",
"has_strict_fderiv_at",
"linear_map.map_coe_ker",
"submodule.coe_subtype",
"submodule.coe_subtypeL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_to_local_homeomorph (hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) :
local_homeomorph E (F × (ker f')) | by haveI := finite_dimensional.complete 𝕜 F; exact
hf.implicit_to_local_homeomorph_of_complemented f f' hf'
f'.ker_closed_complemented_of_finite_dimensional_range | def | has_strict_fderiv_at.implicit_to_local_homeomorph | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"finite_dimensional.complete",
"has_strict_fderiv_at",
"local_homeomorph"
] | Given a map `f : E → F` to a finite dimensional space with a surjective derivative `f'`,
returns a local homeomorphism between `E` and `F × ker f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_function (hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) :
F → (ker f') → E | function.curry $ (hf.implicit_to_local_homeomorph f f' hf').symm | def | has_strict_fderiv_at.implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | Implicit function `g` defined by `f (g z y) = z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_to_local_homeomorph_fst (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) (x : E) :
(hf.implicit_to_local_homeomorph f f' hf' x).fst = f x | rfl | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_fst | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_to_local_homeomorph_apply_ker
(hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) (y : ker f') :
hf.implicit_to_local_homeomorph f f' hf' (y + a) = (f (y + a), y) | by apply implicit_to_local_homeomorph_of_complemented_apply_ker | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_apply_ker | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
implicit_to_local_homeomorph_self
(hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) :
hf.implicit_to_local_homeomorph f f' hf' a = (f a, 0) | by apply implicit_to_local_homeomorph_of_complemented_self | lemma | has_strict_fderiv_at.implicit_to_local_homeomorph_self | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_implicit_to_local_homeomorph_source (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) :
a ∈ (hf.implicit_to_local_homeomorph f f' hf').source | mem_to_local_homeomorph_source _ | lemma | has_strict_fderiv_at.mem_implicit_to_local_homeomorph_source | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_implicit_to_local_homeomorph_target (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) :
(f a, (0 : ker f')) ∈ (hf.implicit_to_local_homeomorph f f' hf').target | by apply mem_implicit_to_local_homeomorph_of_complemented_target | lemma | has_strict_fderiv_at.mem_implicit_to_local_homeomorph_target | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_implicit_function (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) {α : Type*} {l : filter α} {g₁ : α → F} {g₂ : α → ker f'}
(h₁ : tendsto g₁ l (𝓝 $ f a)) (h₂ : tendsto g₂ l (𝓝 0)) :
tendsto (λ t, hf.implicit_function f f' hf' (g₁ t) (g₂ t)) l (𝓝 a) | begin
refine ((hf.implicit_to_local_homeomorph f f' hf').tendsto_symm
(hf.mem_implicit_to_local_homeomorph_source hf')).comp _,
rw [implicit_to_local_homeomorph_self],
exact h₁.prod_mk_nhds h₂
end | lemma | has_strict_fderiv_at.tendsto_implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"filter",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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