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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has_fpower_series_on_ball.eventually_has_sum_sub (hf : has_fpower_series_on_ball f p x r) :
∀ᶠ y in 𝓝 x, has_sum (λn:ℕ, p n (λ(i : fin n), y - x)) (f y) | by filter_upwards [emetric.ball_mem_nhds x hf.r_pos] with y using hf.has_sum_sub | lemma | has_fpower_series_on_ball.eventually_has_sum_sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball_mem_nhds",
"has_fpower_series_on_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.eventually_has_sum_sub (hf : has_fpower_series_at f p x) :
∀ᶠ y in 𝓝 x, has_sum (λn:ℕ, p n (λ(i : fin n), y - x)) (f y) | let ⟨r, hr⟩ := hf in hr.eventually_has_sum_sub | lemma | has_fpower_series_at.eventually_has_sum_sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.eventually_eq_zero
(hf : has_fpower_series_on_ball f (0 : formal_multilinear_series 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 | by filter_upwards [hf.eventually_has_sum_sub] with z hz using hz.unique has_sum_zero | lemma | has_fpower_series_on_ball.eventually_eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_on_ball",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.eventually_eq_zero
(hf : has_fpower_series_at f (0 : formal_multilinear_series 𝕜 E F) x) :
∀ᶠ z in 𝓝 x, f z = 0 | let ⟨r, hr⟩ := hf in hr.eventually_eq_zero | lemma | has_fpower_series_at.eventually_eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball_const {c : F} {e : E} :
has_fpower_series_on_ball (λ _, c) (const_formal_multilinear_series 𝕜 E c) e ⊤ | begin
refine ⟨by simp, with_top.zero_lt_top, λ y hy, has_sum_single 0 (λ n hn, _)⟩,
simp [const_formal_multilinear_series_apply hn]
end | lemma | has_fpower_series_on_ball_const | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"const_formal_multilinear_series",
"const_formal_multilinear_series_apply",
"has_fpower_series_on_ball",
"has_sum_single",
"with_top.zero_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at_const {c : F} {e : E} :
has_fpower_series_at (λ _, c) (const_formal_multilinear_series 𝕜 E c) e | ⟨⊤, has_fpower_series_on_ball_const⟩ | lemma | has_fpower_series_at_const | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"const_formal_multilinear_series",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at_const {v : F} : analytic_at 𝕜 (λ _, v) x | ⟨const_formal_multilinear_series 𝕜 E v, has_fpower_series_at_const⟩ | lemma | analytic_at_const | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on_const {v : F} {s : set E} : analytic_on 𝕜 (λ _, v) s | λ z _, analytic_at_const | lemma | analytic_on_const | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at_const",
"analytic_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.add
(hf : has_fpower_series_on_ball f pf x r) (hg : has_fpower_series_on_ball g pg x r) :
has_fpower_series_on_ball (f + g) (pf + pg) x r | { r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg),
r_pos := hf.r_pos,
has_sum := λ y hy, (hf.has_sum hy).add (hg.has_sum hy) } | lemma | has_fpower_series_on_ball.add | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.add
(hf : has_fpower_series_at f pf x) (hg : has_fpower_series_at g pg x) :
has_fpower_series_at (f + g) (pf + pg) x | begin
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩,
exact ⟨r, hr.1.add hr.2⟩
end | lemma | has_fpower_series_at.add | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at.add (hf : analytic_at 𝕜 f x) (hg : analytic_at 𝕜 g x) :
analytic_at 𝕜 (f + g) x | let ⟨pf, hpf⟩ := hf, ⟨qf, hqf⟩ := hg in (hpf.add hqf).analytic_at | lemma | analytic_at.add | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.neg (hf : has_fpower_series_on_ball f pf x r) :
has_fpower_series_on_ball (-f) (-pf) x r | { r_le := by { rw pf.radius_neg, exact hf.r_le },
r_pos := hf.r_pos,
has_sum := λ y hy, (hf.has_sum hy).neg } | lemma | has_fpower_series_on_ball.neg | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.neg
(hf : has_fpower_series_at f pf x) : has_fpower_series_at (-f) (-pf) x | let ⟨rf, hrf⟩ := hf in hrf.neg.has_fpower_series_at | lemma | has_fpower_series_at.neg | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at.neg (hf : analytic_at 𝕜 f x) : analytic_at 𝕜 (-f) x | let ⟨pf, hpf⟩ := hf in hpf.neg.analytic_at | lemma | analytic_at.neg | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.sub
(hf : has_fpower_series_on_ball f pf x r) (hg : has_fpower_series_on_ball g pg x r) :
has_fpower_series_on_ball (f - g) (pf - pg) x r | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | has_fpower_series_on_ball.sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.sub
(hf : has_fpower_series_at f pf x) (hg : has_fpower_series_at g pg x) :
has_fpower_series_at (f - g) (pf - pg) x | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | has_fpower_series_at.sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at.sub (hf : analytic_at 𝕜 f x) (hg : analytic_at 𝕜 g x) :
analytic_at 𝕜 (f - g) x | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | analytic_at.sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on.mono {s t : set E} (hf : analytic_on 𝕜 f t) (hst : s ⊆ t) :
analytic_on 𝕜 f s | λ z hz, hf z (hst hz) | lemma | analytic_on.mono | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on.add {s : set E} (hf : analytic_on 𝕜 f s) (hg : analytic_on 𝕜 g s) :
analytic_on 𝕜 (f + g) s | λ z hz, (hf z hz).add (hg z hz) | lemma | analytic_on.add | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on.sub {s : set E} (hf : analytic_on 𝕜 f s) (hg : analytic_on 𝕜 g s) :
analytic_on 𝕜 (f - g) s | λ z hz, (hf z hz).sub (hg z hz) | lemma | analytic_on.sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.coeff_zero (hf : has_fpower_series_on_ball f pf x r)
(v : fin 0 → E) : pf 0 v = f x | begin
have v_eq : v = (λ i, 0) := subsingleton.elim _ _,
have zero_mem : (0 : E) ∈ emetric.ball (0 : E) r, by simp [hf.r_pos],
have : ∀ i ≠ 0, pf i (λ j, 0) = 0,
{ assume i hi,
have : 0 < i := pos_iff_ne_zero.2 hi,
exact continuous_multilinear_map.map_coord_zero _ (⟨0, this⟩ : fin i) rfl },
have A := ... | lemma | has_fpower_series_on_ball.coeff_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_multilinear_map.map_coord_zero",
"emetric.ball",
"has_fpower_series_on_ball",
"has_sum_single",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.coeff_zero (hf : has_fpower_series_at f pf x) (v : fin 0 → E) :
pf 0 v = f x | let ⟨rf, hrf⟩ := hf in hrf.coeff_zero v | lemma | has_fpower_series_at.coeff_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_linear_map.comp_has_fpower_series_on_ball
(g : F →L[𝕜] G) (h : has_fpower_series_on_ball f p x r) :
has_fpower_series_on_ball (g ∘ f) (g.comp_formal_multilinear_series p) x r | { r_le := h.r_le.trans (p.radius_le_radius_continuous_linear_map_comp _),
r_pos := h.r_pos,
has_sum := λ y hy, by simpa only [continuous_linear_map.comp_formal_multilinear_series_apply,
continuous_linear_map.comp_continuous_multilinear_map_coe, function.comp_app]
using g.has_sum (h.has_sum hy) } | lemma | continuous_linear_map.comp_has_fpower_series_on_ball | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_linear_map.comp_continuous_multilinear_map_coe",
"continuous_linear_map.comp_formal_multilinear_series_apply",
"has_fpower_series_on_ball",
"has_sum"
] | If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_map.comp_analytic_on {s : set E}
(g : F →L[𝕜] G) (h : analytic_on 𝕜 f s) :
analytic_on 𝕜 (g ∘ f) s | begin
rintros x hx,
rcases h x hx with ⟨p, r, hp⟩,
exact ⟨g.comp_formal_multilinear_series p, r, g.comp_has_fpower_series_on_ball hp⟩,
end | lemma | continuous_linear_map.comp_analytic_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_on"
] | If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.uniform_geometric_approx' {r' : ℝ≥0}
(hf : has_fpower_series_on_ball f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0), (∀ y ∈ metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partial_sum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n) | begin
obtain ⟨a, ha, C, hC, hp⟩ : ∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0), ∀ n, ‖p n‖ * r' ^n ≤ C * a^n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le),
refine ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), λ y hy n, _⟩,
have yr' : ‖y‖ < r', by { rw ball_zero_eq at hy, exact hy },
have hr'0 : 0 < (r... | lemma | has_fpower_series_on_ball.uniform_geometric_approx' | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_multilinear_map.le_op_norm",
"div_le_div_of_le_left",
"div_le_one_of_le",
"div_nonneg",
"div_pos",
"emetric.ball",
"has_fpower_series_on_ball",
"metric.ball",
"mul_assoc",
"mul_div_right_comm",
"mul_le_mul_of_nonneg_right",
"mul_le_of_le_one_right",
"mul_pow",
"mul_right_comm",... | If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`has_fpower_series_on_ball.uniform_geometric_approx` for a weaker ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.uniform_geometric_approx {r' : ℝ≥0}
(hf : has_fpower_series_on_ball f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0), (∀ y ∈ metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partial_sum n y‖ ≤ C * a ^ n) | begin
obtain ⟨a, ha, C, hC, hp⟩ : ∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0),
(∀ y ∈ metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partial_sum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n),
from hf.uniform_geometric_approx' h,
refine ⟨a, ha, C, hC, λ y hy n, (hp y hy n).trans _⟩,
have yr' : ‖y‖ < r', by rwa ball_zero_eq at hy,
ref... | lemma | has_fpower_series_on_ball.uniform_geometric_approx | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"div_le_one_of_le",
"div_nonneg",
"has_fpower_series_on_ball",
"metric.ball",
"mul_le_mul_of_nonneg_left",
"mul_le_of_le_one_right",
"pow_le_pow_of_le_left"
] | If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.is_O_sub_partial_sum_pow (hf : has_fpower_series_at f p x) (n : ℕ) :
(λ y : E, f (x + y) - p.partial_sum n y) =O[𝓝 0] (λ y, ‖y‖ ^ n) | begin
rcases hf with ⟨r, hf⟩,
rcases ennreal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩,
obtain ⟨a, ha, C, hC, hp⟩ : ∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0),
(∀ y ∈ metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partial_sum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n),
from hf.uniform_geometric_approx' h,
refine is_... | lemma | has_fpower_series_at.is_O_sub_partial_sum_pow | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"div_mul_eq_mul_div",
"has_fpower_series_at",
"metric.ball",
"metric.ball_mem_nhds",
"mul_assoc",
"mul_div_assoc",
"mul_pow"
] | Taylor formula for an analytic function, `is_O` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal
(hf : has_fpower_series_on_ball f p x r) (hr : r' < r) :
(λ y : E × E, f y.1 - f y.2 - (p 1 (λ _, y.1 - y.2))) =O[𝓟 (emetric.ball (x, x) r')]
(λ y, ‖y - (x, x)‖ * ‖y.1 - y.2‖) | begin
lift r' to ℝ≥0 using ne_top_of_lt hr,
rcases (zero_le r').eq_or_lt with rfl|hr'0,
{ simp only [is_O_bot, emetric.ball_zero, principal_empty, ennreal.coe_zero] },
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0), ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n,
from p.norm_mul_pow_le_mul_po... | lemma | has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"abs_of_pos",
"div_nonneg",
"emetric.ball",
"emetric.ball_prod_same",
"emetric.ball_subset_ball",
"emetric.ball_zero",
"emetric.mem_ball",
"ennreal.coe_lt_coe",
"ennreal.coe_zero",
"fintype.card_fin",
"has_fpower_series_on_ball",
"has_sum",
"has_sum.mul_left",
"has_sum_coe_mul_geometric_of... | If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (λ _, y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `is_O` and
`filter.principal` on `E × E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.image_sub_sub_deriv_le
(hf : has_fpower_series_on_ball f p x r) (hr : r' < r) :
∃ C, ∀ (y z ∈ emetric.ball x r'),
‖f y - f z - (p 1 (λ _, y - z))‖ ≤ C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖ | by simpa only [is_O_principal, mul_assoc, norm_mul, norm_norm, prod.forall,
emetric.mem_ball, prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E]
using hf.is_O_image_sub_image_sub_deriv_principal hr | lemma | has_fpower_series_on_ball.image_sub_sub_deriv_le | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"and_imp",
"emetric.ball",
"emetric.mem_ball",
"forall_swap",
"has_fpower_series_on_ball",
"max_lt_iff",
"mul_assoc",
"norm_mul",
"norm_norm",
"prod.edist_eq"
] | If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (λ _, y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub (hf : has_fpower_series_at f p x) :
(λ y : E × E, f y.1 - f y.2 - (p 1 (λ _, y.1 - y.2))) =O[𝓝 (x, x)]
(λ y, ‖y - (x, x)‖ * ‖y.1 - y.2‖) | begin
rcases hf with ⟨r, hf⟩,
rcases ennreal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩,
refine (hf.is_O_image_sub_image_sub_deriv_principal h).mono _,
exact le_principal_iff.2 (emetric.ball_mem_nhds _ r'0)
end | lemma | has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball_mem_nhds",
"has_fpower_series_at"
] | If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (λ _, y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.tendsto_uniformly_on {r' : ℝ≥0}
(hf : has_fpower_series_on_ball f p x r) (h : (r' : ℝ≥0∞) < r) :
tendsto_uniformly_on (λ n y, p.partial_sum n y)
(λ y, f (x + y)) at_top (metric.ball (0 : E) r') | begin
obtain ⟨a, ha, C, hC, hp⟩ : ∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0),
(∀ y ∈ metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partial_sum n y‖ ≤ C * a ^ n),
from hf.uniform_geometric_approx h,
refine metric.tendsto_uniformly_on_iff.2 (λ ε εpos, _),
have L : tendsto (λ n, (C : ℝ) * a^n) at_top (𝓝 ((C : ℝ) * 0)) :=
... | lemma | has_fpower_series_on_ball.tendsto_uniformly_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"gt_mem_nhds",
"has_fpower_series_on_ball",
"metric.ball",
"mul_zero",
"tendsto_pow_at_top_nhds_0_of_lt_1",
"tendsto_uniformly_on"
] | If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partial_sum n y` there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.tendsto_locally_uniformly_on
(hf : has_fpower_series_on_ball f p x r) :
tendsto_locally_uniformly_on (λ n y, p.partial_sum n y) (λ y, f (x + y))
at_top (emetric.ball (0 : E) r) | begin
assume u hu x hx,
rcases ennreal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩,
have : emetric.ball (0 : E) r' ∈ 𝓝 x :=
is_open.mem_nhds emetric.is_open_ball xr',
refine ⟨emetric.ball (0 : E) r', mem_nhds_within_of_mem_nhds this, _⟩,
simpa [metric.emetric_ball_nnreal] using hf.tendsto_uniforml... | lemma | has_fpower_series_on_ball.tendsto_locally_uniformly_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball",
"emetric.is_open_ball",
"has_fpower_series_on_ball",
"is_open.mem_nhds",
"mem_nhds_within_of_mem_nhds",
"metric.emetric_ball_nnreal",
"tendsto_locally_uniformly_on"
] | If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partial_sum n y` there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.tendsto_uniformly_on' {r' : ℝ≥0}
(hf : has_fpower_series_on_ball f p x r) (h : (r' : ℝ≥0∞) < r) :
tendsto_uniformly_on (λ n y, p.partial_sum n (y - x)) f at_top (metric.ball (x : E) r') | begin
convert (hf.tendsto_uniformly_on h).comp (λ y, y - x),
{ simp [(∘)] },
{ ext z, simp [dist_eq_norm] }
end | lemma | has_fpower_series_on_ball.tendsto_uniformly_on' | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball",
"metric.ball",
"tendsto_uniformly_on"
] | If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partial_sum n (y - x)` there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.tendsto_locally_uniformly_on'
(hf : has_fpower_series_on_ball f p x r) :
tendsto_locally_uniformly_on (λ n y, p.partial_sum n (y - x)) f at_top (emetric.ball (x : E) r) | begin
have A : continuous_on (λ (y : E), y - x) (emetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuous_on,
convert (hf.tendsto_locally_uniformly_on).comp (λ (y : E), y - x) _ A,
{ ext z, simp },
{ assume z, simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub] }
end | lemma | has_fpower_series_on_ball.tendsto_locally_uniformly_on' | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_const",
"continuous_on",
"emetric.ball",
"has_fpower_series_on_ball",
"tendsto_locally_uniformly_on"
] | If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partial_sum n (y - x)` there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.continuous_on
(hf : has_fpower_series_on_ball f p x r) : continuous_on f (emetric.ball x r) | hf.tendsto_locally_uniformly_on'.continuous_on $ eventually_of_forall $ λ n,
((p.partial_sum_continuous n).comp (continuous_id.sub continuous_const)).continuous_on | lemma | has_fpower_series_on_ball.continuous_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_const",
"continuous_on",
"emetric.ball",
"has_fpower_series_on_ball"
] | If a function admits a power series expansion on a disk, then it is continuous there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.continuous_at (hf : has_fpower_series_at f p x) :
continuous_at f x | let ⟨r, hr⟩ := hf in hr.continuous_on.continuous_at (emetric.ball_mem_nhds x (hr.r_pos)) | lemma | has_fpower_series_at.continuous_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_at",
"emetric.ball_mem_nhds",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at.continuous_at (hf : analytic_at 𝕜 f x) : continuous_at f x | let ⟨p, hp⟩ := hf in hp.continuous_at | lemma | analytic_at.continuous_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at",
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on.continuous_on {s : set E} (hf : analytic_on 𝕜 f s) :
continuous_on f s | λ x hx, (hf x hx).continuous_at.continuous_within_at | lemma | analytic_on.continuous_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_on",
"continuous_at.continuous_within_at",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formal_multilinear_series.has_fpower_series_on_ball [complete_space F]
(p : formal_multilinear_series 𝕜 E F) (h : 0 < p.radius) :
has_fpower_series_on_ball p.sum p 0 p.radius | { r_le := le_rfl,
r_pos := h,
has_sum := λ y hy, by { rw zero_add, exact p.has_sum hy } } | lemma | formal_multilinear_series.has_fpower_series_on_ball | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"complete_space",
"formal_multilinear_series",
"has_fpower_series_on_ball",
"has_sum",
"le_rfl"
] | In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.sum (h : has_fpower_series_on_ball f p x r)
{y : E} (hy : y ∈ emetric.ball (0 : E) r) : f (x + y) = p.sum y | (h.has_sum hy).tsum_eq.symm | lemma | has_fpower_series_on_ball.sum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formal_multilinear_series.continuous_on [complete_space F] :
continuous_on p.sum (emetric.ball 0 p.radius) | begin
cases (zero_le p.radius).eq_or_lt with h h,
{ simp [← h, continuous_on_empty] },
{ exact (p.has_fpower_series_on_ball h).continuous_on }
end | lemma | formal_multilinear_series.continuous_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"complete_space",
"continuous_on",
"continuous_on_empty",
"emetric.ball"
] | The sum of a converging power series is continuous in its disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
asymptotics.is_O.continuous_multilinear_map_apply_eq_zero {n : ℕ} {p : E [×n]→L[𝕜] F}
(h : (λ y, p (λ i, y)) =O[𝓝 0] (λ y, ‖y‖ ^ (n + 1))) (y : E) :
p (λ i, y) = 0 | begin
obtain ⟨c, c_pos, hc⟩ := h.exists_pos,
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (is_O_with_iff.mp hc),
obtain ⟨δ, δ_pos, δε⟩ := (metric.is_open_iff.mp t_open) 0 z_mem,
clear h hc z_mem,
cases n,
{ exact norm_eq_zero.mp (by simpa only [fin0_apply_norm, norm_eq_zero, norm_zero, zero_pow',... | lemma | asymptotics.is_O.continuous_multilinear_map_apply_eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"em",
"finset.card_fin",
"finset.prod_const",
"inv_mul_cancel",
"inv_mul_cancel_right₀",
"inv_smul_smul₀",
"metric.mem_ball_self",
"mul_le_mul_of_nonneg_left",
"mul_lt_mul_of_pos_right",
"mul_pow",
"mul_zero",
"norm_eq_zero",
"norm_mul",
"norm_norm",
"norm_pow",
"norm_smul",
"normed_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.apply_eq_zero {p : formal_multilinear_series 𝕜 E F} {x : E}
(h : has_fpower_series_at 0 p x) (n : ℕ) :
∀ y : E, p n (λ i, y) = 0 | begin
refine nat.strong_rec_on n (λ k hk, _),
have psum_eq : p.partial_sum (k + 1) = (λ y, p k (λ i, y)),
{ funext z,
refine finset.sum_eq_single _ (λ b hb hnb, _) (λ hn, _),
{ have := finset.mem_range_succ_iff.mp hb,
simp only [hk b (this.lt_of_ne hnb), pi.zero_apply, zero_apply] },
{ exact fal... | lemma | has_fpower_series_at.apply_eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"asymptotics.is_O_neg_left",
"formal_multilinear_series",
"has_fpower_series_at",
"lt_add_one"
] | If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (λ i, y)` appearing the in sum are zero for any `n : ℕ`, `y : E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.eq_zero {p : formal_multilinear_series 𝕜 𝕜 E} {x : 𝕜}
(h : has_fpower_series_at 0 p x) : p = 0 | by { ext n x, rw ←mk_pi_field_apply_one_eq_self (p n), simp [h.apply_eq_zero n 1] } | lemma | has_fpower_series_at.eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_at"
] | A one-dimensional formal multilinear series representing the zero function is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.eq_formal_multilinear_series
{p₁ p₂ : formal_multilinear_series 𝕜 𝕜 E} {f : 𝕜 → E} {x : 𝕜}
(h₁ : has_fpower_series_at f p₁ x) (h₂ : has_fpower_series_at f p₂ x) :
p₁ = p₂ | sub_eq_zero.mp (has_fpower_series_at.eq_zero (by simpa only [sub_self] using h₁.sub h₂)) | theorem | has_fpower_series_at.eq_formal_multilinear_series | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_at",
"has_fpower_series_at.eq_zero"
] | One-dimensional formal multilinear series representing the same function are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.eq_formal_multilinear_series_of_eventually
{p q : formal_multilinear_series 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : has_fpower_series_at f p x)
(hq : has_fpower_series_at g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) :
p = q | (hp.congr heq).eq_formal_multilinear_series hq | lemma | has_fpower_series_at.eq_formal_multilinear_series_of_eventually | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.eq_zero_of_eventually {p : formal_multilinear_series 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : has_fpower_series_at f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 | (hp.congr hf).eq_zero | lemma | has_fpower_series_at.eq_zero_of_eventually | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_at"
] | A one-dimensional formal multilinear series representing a locally zero function is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.exchange_radius
{p₁ p₂ : formal_multilinear_series 𝕜 𝕜 E} {f : 𝕜 → E} {r₁ r₂ : ℝ≥0∞} {x : 𝕜}
(h₁ : has_fpower_series_on_ball f p₁ x r₁) (h₂ : has_fpower_series_on_ball f p₂ x r₂) :
has_fpower_series_on_ball f p₁ x r₂ | h₂.has_fpower_series_at.eq_formal_multilinear_series h₁.has_fpower_series_at ▸ h₂ | theorem | has_fpower_series_on_ball.exchange_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_on_ball"
] | If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : formal_multilinear_series 𝕜 𝕜 E} (h : has_fpower_series_on_ball f p x r)
(h' : ∀ (r' : ℝ≥0) (hr : 0 < r'),
∃ p' : formal_multilinear_series 𝕜 𝕜 E, has_fpower_series_on_ball f p' x r') :
has_fpower_series_on_ball f p x ∞ | { r_le := ennreal.le_of_forall_pos_nnreal_lt $ λ r hr hr',
let ⟨p', hp'⟩ := h' r hr in (h.exchange_radius hp').r_le,
r_pos := ennreal.coe_lt_top,
has_sum := λ y hy, let ⟨r', hr'⟩ := exists_gt ‖y‖₊, ⟨p', hp'⟩ := h' r' hr'.ne_bot.bot_lt
in (h.exchange_radius hp').has_sum $ mem_emetric_ball_zero_iff.mpr (ennre... | theorem | has_fpower_series_on_ball.r_eq_top_of_exists | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.coe_lt_top",
"ennreal.le_of_forall_pos_nnreal_lt",
"formal_multilinear_series",
"has_fpower_series_on_ball",
"has_sum"
] | If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
change_origin_series_term (k l : ℕ) (s : finset (fin (k + l))) (hs : s.card = l) :
E [×l]→L[𝕜] E [×k]→L[𝕜] F | continuous_multilinear_map.curry_fin_finset 𝕜 E F hs
(by erw [finset.card_compl, fintype.card_fin, hs, add_tsub_cancel_right]) (p $ k + l) | def | formal_multilinear_series.change_origin_series_term | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"add_tsub_cancel_right",
"continuous_multilinear_map.curry_fin_finset",
"finset",
"finset.card_compl",
"fintype.card_fin"
] | A term of `formal_multilinear_series.change_origin_series`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.change_origin x` is a formal multilinear series such that
`p.sum (x+y) = (p.change_origin x).sum y` when this makes sense. Each term of `p.change_origin x`
is itself an analy... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
change_origin_series_term_apply (k l : ℕ) (s : finset (fin (k + l))) (hs : s.card = l)
(x y : E) :
p.change_origin_series_term k l s hs (λ _, x) (λ _, y) =
p (k + l) (s.piecewise (λ _, x) (λ _, y)) | continuous_multilinear_map.curry_fin_finset_apply_const _ _ _ _ _ | lemma | formal_multilinear_series.change_origin_series_term_apply | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_multilinear_map.curry_fin_finset_apply_const",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_change_origin_series_term (k l : ℕ) (s : finset (fin (k + l)))
(hs : s.card = l) :
‖p.change_origin_series_term k l s hs‖ = ‖p (k + l)‖ | by simp only [change_origin_series_term, linear_isometry_equiv.norm_map] | lemma | formal_multilinear_series.norm_change_origin_series_term | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_change_origin_series_term (k l : ℕ) (s : finset (fin (k + l)))
(hs : s.card = l) :
‖p.change_origin_series_term k l s hs‖₊ = ‖p (k + l)‖₊ | by simp only [change_origin_series_term, linear_isometry_equiv.nnnorm_map] | lemma | formal_multilinear_series.nnnorm_change_origin_series_term | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset",
"linear_isometry_equiv.nnnorm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_change_origin_series_term_apply_le (k l : ℕ) (s : finset (fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.change_origin_series_term k l s hs (λ _, x) (λ _, y)‖₊ ≤ ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k | begin
rw [← p.nnnorm_change_origin_series_term k l s hs, ← fin.prod_const, ← fin.prod_const],
apply continuous_multilinear_map.le_of_op_nnnorm_le,
apply continuous_multilinear_map.le_op_nnnorm
end | lemma | formal_multilinear_series.nnnorm_change_origin_series_term_apply_le | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_multilinear_map.le_of_op_nnnorm_le",
"continuous_multilinear_map.le_op_nnnorm",
"fin.prod_const",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_origin_series (k : ℕ) : formal_multilinear_series 𝕜 E (E [×k]→L[𝕜] F) | λ l, ∑ s : {s : finset (fin (k + l)) // finset.card s = l}, p.change_origin_series_term k l s s.2 | def | formal_multilinear_series.change_origin_series | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset",
"finset.card",
"formal_multilinear_series"
] | The power series for `f.change_origin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.change_origin x` is a formal multilinear series such that
`p.sum (x+y) = (p.change_origin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.change_origin_series k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_change_origin_series_le_tsum (k l : ℕ) :
‖p.change_origin_series k l‖₊ ≤
∑' (x : {s : finset (fin (k + l)) // s.card = l}), ‖p (k + l)‖₊ | (nnnorm_sum_le _ _).trans_eq $ by simp only [tsum_fintype, nnnorm_change_origin_series_term] | lemma | formal_multilinear_series.nnnorm_change_origin_series_le_tsum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset",
"tsum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_change_origin_series_apply_le_tsum (k l : ℕ) (x : E) :
‖p.change_origin_series k l (λ _, x)‖₊ ≤
∑' s : {s : finset (fin (k + l)) // s.card = l}, ‖p (k + l)‖₊ * ‖x‖₊ ^ l | begin
rw [nnreal.tsum_mul_right, ← fin.prod_const],
exact (p.change_origin_series k l).le_of_op_nnnorm_le _
(p.nnnorm_change_origin_series_le_tsum _ _)
end | lemma | formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"fin.prod_const",
"finset",
"nnreal.tsum_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_origin (x : E) : formal_multilinear_series 𝕜 E F | λ k, (p.change_origin_series k).sum x | def | formal_multilinear_series.change_origin | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series"
] | Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.change_origin x).sum y` when this makes sense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
change_origin_index_equiv :
(Σ k l : ℕ, {s : finset (fin (k + l)) // s.card = l}) ≃ Σ n : ℕ, finset (fin n) | { to_fun := λ s, ⟨s.1 + s.2.1, s.2.2⟩,
inv_fun := λ s, ⟨s.1 - s.2.card, s.2.card, ⟨s.2.map
(fin.cast $ (tsub_add_cancel_of_le $ card_finset_fin_le s.2).symm).to_equiv.to_embedding,
finset.card_map _⟩⟩,
left_inv :=
begin
rintro ⟨k, l, ⟨s : finset (fin $ k + l), hs : s.card = l⟩⟩,
dsimp only [... | def | formal_multilinear_series.change_origin_index_equiv | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"add_tsub_cancel_right",
"card_finset_fin_le",
"equiv.refl_to_embedding",
"fin.cast",
"fin.cast_refl",
"fin.cast_to_equiv",
"finset",
"finset.card_map",
"finset.map_refl",
"heq_iff_eq",
"inv_fun",
"order_iso.refl_to_equiv",
"subtype.coe_mk",
"tsub_add_cancel_of_le"
] | An auxiliary equivalence useful in the proofs about
`formal_multilinear_series.change_origin_series`: the set of triples `(k, l, s)`, where `s` is a
`finset (fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`finset (fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
change_origin_series_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
summable (λ s : Σ k l : ℕ, {s : finset (fin (k + l)) // s.card = l},
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1) | begin
rw ← change_origin_index_equiv.symm.summable_iff,
dsimp only [(∘), change_origin_index_equiv_symm_apply_fst,
change_origin_index_equiv_symm_apply_snd_fst],
have : ∀ n : ℕ, has_sum
(λ s : finset (fin n), ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n),
{ int... | lemma | formal_multilinear_series.change_origin_series_summable_aux₁ | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"card_finset_fin_le",
"fin.sum_pow_mul_eq_add_pow",
"finset",
"has_sum",
"has_sum_fintype",
"mul_assoc",
"summable",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_origin_series_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
summable (λ s : Σ l : ℕ, {s : finset (fin (k + l)) // s.card = l}, ‖p (k + s.1)‖₊ * r ^ s.1) | begin
rcases ennreal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩,
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne']
using ((nnreal.summable_sigma.1
(p.change_origin_series_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
end | lemma | formal_multilinear_series.change_origin_series_summable_aux₂ | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset",
"mul_inv_cancel_right₀",
"pow_pos",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_origin_series_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
summable (λ l : ℕ, ‖p.change_origin_series k l‖₊ * r ^ l) | begin
refine nnreal.summable_of_le (λ n, _)
(nnreal.summable_sigma.1 $ p.change_origin_series_summable_aux₂ hr k).2,
simp only [nnreal.tsum_mul_right],
exact mul_le_mul' (p.nnnorm_change_origin_series_le_tsum _ _) le_rfl
end | lemma | formal_multilinear_series.change_origin_series_summable_aux₃ | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"le_rfl",
"mul_le_mul'",
"nnreal.summable_of_le",
"nnreal.tsum_mul_right",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_change_origin_series_radius (k : ℕ) :
p.radius ≤ (p.change_origin_series k).radius | ennreal.le_of_forall_nnreal_lt $ λ r hr,
le_radius_of_summable_nnnorm _ (p.change_origin_series_summable_aux₃ hr k) | lemma | formal_multilinear_series.le_change_origin_series_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.le_of_forall_nnreal_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_change_origin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.change_origin x k‖₊ ≤
∑' s : Σ l : ℕ, {s : finset (fin (k + l)) // s.card = l}, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 | begin
refine tsum_of_nnnorm_bounded _ (λ l, p.nnnorm_change_origin_series_apply_le_tsum k l x),
have := p.change_origin_series_summable_aux₂ h k,
refine has_sum.sigma this.has_sum (λ l, _),
exact ((nnreal.summable_sigma.1 this).1 l).has_sum
end | lemma | formal_multilinear_series.nnnorm_change_origin_le | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset",
"has_sum",
"has_sum.sigma",
"tsum_of_nnnorm_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_origin_radius : p.radius - ‖x‖₊ ≤ (p.change_origin x).radius | begin
refine ennreal.le_of_forall_pos_nnreal_lt (λ r h0 hr, _),
rw [lt_tsub_iff_right, add_comm] at hr,
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius, from (le_add_right le_rfl).trans_lt hr,
apply le_radius_of_summable_nnnorm,
have : ∀ k : ℕ, ‖p.change_origin x k‖₊ * r ^ k ≤
(∑' s : Σ l : ℕ, {s : finset (fin (k + l... | lemma | formal_multilinear_series.change_origin_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.le_of_forall_pos_nnreal_lt",
"finset",
"le_rfl",
"lt_tsub_iff_right",
"mul_le_mul_right'",
"nnreal.summable_of_le",
"nnreal.tsum_mul_right"
] | The radius of convergence of `p.change_origin x` is at least `p.radius - ‖x‖`. In other words,
`p.change_origin x` is well defined on the largest ball contained in the original ball of
convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball_change_origin (k : ℕ) (hr : 0 < p.radius) :
has_fpower_series_on_ball (λ x, p.change_origin x k) (p.change_origin_series k) 0 p.radius | have _ := p.le_change_origin_series_radius k,
((p.change_origin_series k).has_fpower_series_on_ball (hr.trans_le this)).mono hr this | lemma | formal_multilinear_series.has_fpower_series_on_ball_change_origin | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_origin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.change_origin x).sum y = (p.sum (x + y)) | begin
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h,
have x_mem_ball : x ∈ emetric.ball (0 : E) p.radius,
from mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h),
have y_mem_ball : y ∈ emetric.ball (0 : E) (p.change_origin x).radius,
{ refine mem_emetric_ball_zero_iff.2 (lt_of_l... | theorem | formal_multilinear_series.change_origin_eval | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuous_multilinear_map.curry_fin_finset_apply_const",
"continuous_multilinear_map.has_sum_eval",
"continuous_multilinear_map.le_op_nnnorm",
"continuous_multilinear_map.sum_apply",
"emetric.ball",
"fin.cast",
"finset",
"has_sum",
"has_sum.sigma_of_has_sum",
"has_sum_fintype",
"le_rfl",
"lt... | Summing the series `p.change_origin x` at a point `y` gives back `p (x + y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.change_origin
(hf : has_fpower_series_on_ball f p x r) (h : (‖y‖₊ : ℝ≥0∞) < r) :
has_fpower_series_on_ball f (p.change_origin y) (x + y) (r - ‖y‖₊) | { r_le := begin
apply le_trans _ p.change_origin_radius,
exact tsub_le_tsub hf.r_le le_rfl
end,
r_pos := by simp [h],
has_sum := λ z hz, begin
convert (p.change_origin y).has_sum _,
{ rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz,
rw [p.change_origin_eval (hz.trans_le hf.... | theorem | has_fpower_series_on_ball.change_origin | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball_subset_ball",
"has_fpower_series_on_ball",
"has_sum",
"le_rfl",
"lt_tsub_iff_right",
"tsub_le_tsub"
] | If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.change_origin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.analytic_at_of_mem
(hf : has_fpower_series_on_ball f p x r) (h : y ∈ emetric.ball x r) :
analytic_at 𝕜 f y | begin
have : (‖y - x‖₊ : ℝ≥0∞) < r, by simpa [edist_eq_coe_nnnorm_sub] using h,
have := hf.change_origin this,
rw [add_sub_cancel'_right] at this,
exact this.analytic_at
end | lemma | has_fpower_series_on_ball.analytic_at_of_mem | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at",
"emetric.ball",
"has_fpower_series_on_ball"
] | If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.analytic_on (hf : has_fpower_series_on_ball f p x r) :
analytic_on 𝕜 f (emetric.ball x r) | λ y hy, hf.analytic_at_of_mem hy | lemma | has_fpower_series_on_ball.analytic_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_on",
"emetric.ball",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_analytic_at : is_open {x | analytic_at 𝕜 f x} | begin
rw is_open_iff_mem_nhds,
rintro x ⟨p, r, hr⟩,
exact mem_of_superset (emetric.ball_mem_nhds _ hr.r_pos) (λ y hy, hr.analytic_at_of_mem hy)
end | lemma | is_open_analytic_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at",
"emetric.ball_mem_nhds",
"is_open",
"is_open_iff_mem_nhds"
] | For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at_iff : has_fpower_series_at f p z₀ ↔
∀ᶠ z in 𝓝 0, has_sum (λ n, z ^ n • p.coeff n) (f (z₀ + z)) | begin
refine ⟨λ ⟨r, r_le, r_pos, h⟩, eventually_of_mem (emetric.ball_mem_nhds 0 r_pos)
(λ _, by simpa using h), _⟩,
simp only [metric.eventually_nhds_iff],
rintro ⟨r, r_pos, h⟩,
refine ⟨p.radius ⊓ r.to_nnreal, by simp, _, _⟩,
{ simp only [r_pos.lt, lt_inf_iff, ennreal.coe_pos, real.to_nnreal_pos, and_true... | lemma | has_fpower_series_at_iff | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"and_imp",
"edist_lt_coe",
"emetric.ball_mem_nhds",
"emetric.mem_ball",
"ennreal",
"ennreal.coe_pos",
"formal_multilinear_series.le_radius_of_tendsto",
"has_fpower_series_at",
"has_sum",
"lt_inf_iff",
"metric.eventually_nhds_iff",
"mul_comm",
"norm_smul",
"normed_field.exists_norm_lt",
"... | A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`has_fpower_series_at` depends on `p.radius`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at_iff' : has_fpower_series_at f p z₀ ↔
∀ᶠ z in 𝓝 z₀, has_sum (λ n, (z - z₀) ^ n • p.coeff n) (f z) | begin
rw [← map_add_left_nhds_zero, eventually_map, has_fpower_series_at_iff],
congrm ∀ᶠ z in (𝓝 0 : filter 𝕜), has_sum (λ n, _) (f (z₀ + z)),
rw add_sub_cancel'
end | lemma | has_fpower_series_at_iff' | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"filter",
"has_fpower_series_at",
"has_fpower_series_at_iff",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_composition
(p : formal_multilinear_series 𝕜 E F) {n : ℕ} (c : composition n) :
(fin n → E) → (fin (c.length) → F) | λ v i, p (c.blocks_fun i) (v ∘ (c.embedding i)) | def | formal_multilinear_series.apply_composition | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series"
] | Given a formal multilinear series `p`, a composition `c` of `n` and the index `i` of a
block of `c`, we may define a function on `fin n → E` by picking the variables in the `i`-th block
of `n`, and applying the corresponding coefficient of `p` to these variables. This function is
called `p.apply_composition c v i` for ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_composition_ones (p : formal_multilinear_series 𝕜 E F) (n : ℕ) :
p.apply_composition (composition.ones n) =
λ v i, p 1 (λ _, v (fin.cast_le (composition.length_le _) i)) | begin
funext v i,
apply p.congr (composition.ones_blocks_fun _ _),
intros j hjn hj1,
obtain rfl : j = 0, { linarith },
refine congr_arg v _,
rw [fin.ext_iff, fin.coe_cast_le, composition.ones_embedding, fin.coe_mk],
end | lemma | formal_multilinear_series.apply_composition_ones | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition.length_le",
"composition.ones",
"composition.ones_blocks_fun",
"composition.ones_embedding",
"fin.cast_le",
"fin.coe_cast_le",
"fin.coe_mk",
"fin.ext_iff",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_composition_single (p : formal_multilinear_series 𝕜 E F) {n : ℕ} (hn : 0 < n)
(v : fin n → E) : p.apply_composition (composition.single n hn) v = λ j, p n v | begin
ext j,
refine p.congr (by simp) (λ i hi1 hi2, _),
dsimp,
congr' 1,
convert composition.single_embedding hn ⟨i, hi2⟩,
cases j,
have : j_val = 0 := le_bot_iff.1 (nat.lt_succ_iff.1 j_property),
unfold_coes,
congr; try { assumption <|> simp },
end | lemma | formal_multilinear_series.apply_composition_single | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition.single",
"composition.single_embedding",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
remove_zero_apply_composition
(p : formal_multilinear_series 𝕜 E F) {n : ℕ} (c : composition n) :
p.remove_zero.apply_composition c = p.apply_composition c | begin
ext v i,
simp [apply_composition, zero_lt_one.trans_le (c.one_le_blocks_fun i), remove_zero_of_pos],
end | lemma | formal_multilinear_series.remove_zero_apply_composition | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_composition_update
(p : formal_multilinear_series 𝕜 E F) {n : ℕ} (c : composition n)
(j : fin n) (v : fin n → E) (z : E) :
p.apply_composition c (function.update v j z) =
function.update (p.apply_composition c v) (c.index j)
(p (c.blocks_fun (c.index j))
(function.update (v ∘ (c.embedding... | begin
ext k,
by_cases h : k = c.index j,
{ rw h,
let r : fin (c.blocks_fun (c.index j)) → fin n := c.embedding (c.index j),
simp only [function.update_same],
change p (c.blocks_fun (c.index j)) ((function.update v j z) ∘ r) = _,
let j' := c.inv_embedding j,
suffices B : (function.update v j z)... | lemma | formal_multilinear_series.apply_composition_update | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series",
"function.update_comp_eq_of_not_mem_range"
] | Technical lemma stating how `p.apply_composition` commutes with updating variables. This
will be the key point to show that functions constructed from `apply_composition` retain
multilinearity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_map_apply_composition {n : ℕ}
(p : formal_multilinear_series 𝕜 F G) (f : E →L[𝕜] F) (c : composition n) (v : fin n → E) :
(p.comp_continuous_linear_map f).apply_composition c v = p.apply_composition c (f ∘ v) | by simp [apply_composition] | lemma | formal_multilinear_series.comp_continuous_linear_map_apply_composition | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_along_composition {n : ℕ}
(p : formal_multilinear_series 𝕜 E F) (c : composition n)
(f : continuous_multilinear_map 𝕜 (λ (i : fin c.length), F) G) :
continuous_multilinear_map 𝕜 (λ i : fin n, E) G | { to_fun := λ v, f (p.apply_composition c v),
map_add' := λ _ v i x y, by
{ cases subsingleton.elim ‹_› (fin.decidable_eq _),
simp only [apply_composition_update, continuous_multilinear_map.map_add] },
map_smul' := λ _ v i c x, by
{ cases subsingleton.elim ‹_› (fin.decidable_eq _),
simp only [apply_... | def | continuous_multilinear_map.comp_along_composition | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"cont",
"continuous_apply",
"continuous_multilinear_map",
"continuous_multilinear_map.map_add",
"continuous_multilinear_map.map_smul",
"continuous_pi",
"formal_multilinear_series"
] | Given a formal multilinear series `p`, a composition `c` of `n` and a continuous multilinear
map `f` in `c.length` variables, one may form a continuous multilinear map in `n` variables by
applying the right coefficient of `p` to each block of the composition, and then applying `f` to
the resulting vector. It is called ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_along_composition_apply {n : ℕ}
(p : formal_multilinear_series 𝕜 E F) (c : composition n)
(f : continuous_multilinear_map 𝕜 (λ (i : fin c.length), F) G) (v : fin n → E) :
(f.comp_along_composition p c) v = f (p.apply_composition c v) | rfl | lemma | continuous_multilinear_map.comp_along_composition_apply | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"continuous_multilinear_map",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_along_composition {n : ℕ}
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(c : composition n) : continuous_multilinear_map 𝕜 (λ i : fin n, E) G | (q c.length).comp_along_composition p c | def | formal_multilinear_series.comp_along_composition | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"continuous_multilinear_map",
"formal_multilinear_series"
] | Given two formal multilinear series `q` and `p` and a composition `c` of `n`, one may
form a continuous multilinear map in `n` variables by applying the right coefficient of `p` to each
block of the composition, and then applying `q c.length` to the resulting vector. It is
called `q.comp_along_composition p c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_along_composition_apply {n : ℕ}
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(c : composition n) (v : fin n → E) :
(q.comp_along_composition p c) v = q c.length (p.apply_composition c v) | rfl | lemma | formal_multilinear_series.comp_along_composition_apply | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) :
formal_multilinear_series 𝕜 E G | λ n, ∑ c : composition n, q.comp_along_composition p c | def | formal_multilinear_series.comp | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series"
] | Formal composition of two formal multilinear series. The `n`-th coefficient in the composition
is defined to be the sum of `q.comp_along_composition p c` over all compositions of
`n`. In other words, this term (as a multilinear function applied to `v_0, ..., v_{n-1}`) is
`∑'_{k} ∑'_{i₁ + ... + iₖ = n} qₖ (p_{i_1} (...)... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_coeff_zero (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(v : fin 0 → E) (v' : fin 0 → F) :
(q.comp p) 0 v = q 0 v' | begin
let c : composition 0 := composition.ones 0,
dsimp [formal_multilinear_series.comp],
have : {c} = (finset.univ : finset (composition 0)),
{ apply finset.eq_of_subset_of_card_le; simp [finset.card_univ, composition_card 0] },
rw [← this, finset.sum_singleton, comp_along_composition_apply],
symmetry, co... | lemma | formal_multilinear_series.comp_coeff_zero | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.ones",
"composition_card",
"finset",
"finset.card_univ",
"finset.eq_of_subset_of_card_le",
"finset.univ",
"formal_multilinear_series",
"formal_multilinear_series.comp"
] | The `0`-th coefficient of `q.comp p` is `q 0`. Since these maps are multilinear maps in zero
variables, but on different spaces, we can not state this directly, so we state it when applied to
arbitrary vectors (which have to be the zero vector). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_coeff_zero'
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (v : fin 0 → E) :
(q.comp p) 0 v = q 0 (λ i, 0) | q.comp_coeff_zero p v _ | lemma | formal_multilinear_series.comp_coeff_zero' | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_coeff_zero'' (q : formal_multilinear_series 𝕜 E F)
(p : formal_multilinear_series 𝕜 E E) :
(q.comp p) 0 = q 0 | by { ext v, exact q.comp_coeff_zero p _ _ } | lemma | formal_multilinear_series.comp_coeff_zero'' | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"formal_multilinear_series"
] | The `0`-th coefficient of `q.comp p` is `q 0`. When `p` goes from `E` to `E`, this can be
expressed as a direct equality | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_coeff_one (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(v : fin 1 → E) : (q.comp p) 1 v = q 1 (λ i, p 1 v) | begin
have : {composition.ones 1} = (finset.univ : finset (composition 1)) :=
finset.eq_univ_of_card _ (by simp [composition_card]),
simp only [formal_multilinear_series.comp, comp_along_composition_apply, ← this,
finset.sum_singleton],
refine q.congr (by simp) (λ i hi1 hi2, _),
simp only [apply_composi... | lemma | formal_multilinear_series.comp_coeff_one | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.ones",
"composition_card",
"finset",
"finset.eq_univ_of_card",
"finset.univ",
"formal_multilinear_series",
"formal_multilinear_series.comp"
] | The first coefficient of a composition of formal multilinear series is the composition of the
first coefficients seen as continuous linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
remove_zero_comp_of_pos (q : formal_multilinear_series 𝕜 F G)
(p : formal_multilinear_series 𝕜 E F) {n : ℕ} (hn : 0 < n) :
q.remove_zero.comp p n = q.comp p n | begin
ext v,
simp only [formal_multilinear_series.comp, comp_along_composition,
continuous_multilinear_map.comp_along_composition_apply, continuous_multilinear_map.sum_apply],
apply finset.sum_congr rfl (λ c hc, _),
rw remove_zero_of_pos _ (c.length_pos_of_pos hn)
end | lemma | formal_multilinear_series.remove_zero_comp_of_pos | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"continuous_multilinear_map.comp_along_composition_apply",
"continuous_multilinear_map.sum_apply",
"formal_multilinear_series",
"formal_multilinear_series.comp"
] | Only `0`-th coefficient of `q.comp p` depends on `q 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_remove_zero (q : formal_multilinear_series 𝕜 F G)
(p : formal_multilinear_series 𝕜 E F) :
q.comp p.remove_zero = q.comp p | by { ext n, simp [formal_multilinear_series.comp] } | lemma | formal_multilinear_series.comp_remove_zero | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"formal_multilinear_series",
"formal_multilinear_series.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_along_composition_bound {n : ℕ}
(p : formal_multilinear_series 𝕜 E F) (c : composition n)
(f : continuous_multilinear_map 𝕜 (λ (i : fin c.length), F) G) (v : fin n → E) :
‖f.comp_along_composition p c v‖ ≤
‖f‖ * (∏ i, ‖p (c.blocks_fun i)‖) * (∏ i : fin n, ‖v i‖) | calc ‖f.comp_along_composition p c v‖ = ‖f (p.apply_composition c v)‖ : rfl
... ≤ ‖f‖ * ∏ i, ‖p.apply_composition c v i‖ : continuous_multilinear_map.le_op_norm _ _
... ≤ ‖f‖ * ∏ i, ‖p (c.blocks_fun i)‖ *
∏ j : fin (c.blocks_fun i), ‖(v ∘ (c.embedding i)) j‖ :
begin
apply mul_le_mul_of_nonneg_left _ (norm... | lemma | formal_multilinear_series.comp_along_composition_bound | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"continuous_multilinear_map",
"continuous_multilinear_map.le_op_norm",
"finset.prod_le_prod",
"finset.prod_mul_distrib",
"finset.prod_sigma",
"finset.univ_sigma_univ",
"formal_multilinear_series",
"mul_assoc",
"mul_le_mul_of_nonneg_left"
] | The norm of `f.comp_along_composition p c` is controlled by the product of
the norms of the relevant bits of `f` and `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_along_composition_norm {n : ℕ}
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(c : composition n) :
‖q.comp_along_composition p c‖ ≤ ‖q c.length‖ * ∏ i, ‖p (c.blocks_fun i)‖ | continuous_multilinear_map.op_norm_le_bound _
(mul_nonneg (norm_nonneg _) (finset.prod_nonneg (λ i hi, norm_nonneg _)))
(comp_along_composition_bound _ _ _) | lemma | formal_multilinear_series.comp_along_composition_norm | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"continuous_multilinear_map.op_norm_le_bound",
"finset.prod_nonneg",
"formal_multilinear_series"
] | The norm of `q.comp_along_composition p c` is controlled by the product of
the norms of the relevant bits of `q` and `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_along_composition_nnnorm {n : ℕ}
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(c : composition n) :
‖q.comp_along_composition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocks_fun i)‖₊ | by { rw ← nnreal.coe_le_coe, push_cast, exact q.comp_along_composition_norm p c } | lemma | formal_multilinear_series.comp_along_composition_nnnorm | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"formal_multilinear_series",
"nnreal.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : formal_multilinear_series 𝕜 E E | | 0 := 0
| 1 := (continuous_multilinear_curry_fin1 𝕜 E E).symm (continuous_linear_map.id 𝕜 E)
| _ := 0 | def | formal_multilinear_series.id | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"continuous_linear_map.id",
"continuous_multilinear_curry_fin1",
"formal_multilinear_series"
] | The identity formal multilinear series, with all coefficients equal to `0` except for `n = 1`
where it is (the continuous multilinear version of) the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply_one (v : fin 1 → E) : (formal_multilinear_series.id 𝕜 E) 1 v = v 0 | rfl | lemma | formal_multilinear_series.id_apply_one | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"formal_multilinear_series.id"
] | The first coefficient of `id 𝕜 E` is the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply_one' {n : ℕ} (h : n = 1) (v : fin n → E) :
(id 𝕜 E) n v = v ⟨0, h.symm ▸ zero_lt_one⟩ | begin
subst n,
apply id_apply_one
end | lemma | formal_multilinear_series.id_apply_one' | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [] | The `n`th coefficient of `id 𝕜 E` is the identity when `n = 1`. We state this in a dependent
way, as it will often appear in this form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply_ne_one {n : ℕ} (h : n ≠ 1) : (formal_multilinear_series.id 𝕜 E) n = 0 | by { cases n, { refl }, cases n, { contradiction }, refl } | lemma | formal_multilinear_series.id_apply_ne_one | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"formal_multilinear_series.id"
] | For `n ≠ 1`, the `n`-th coefficient of `id 𝕜 E` is zero, by definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_id (p : formal_multilinear_series 𝕜 E F) : p.comp (id 𝕜 E) = p | begin
ext1 n,
dsimp [formal_multilinear_series.comp],
rw finset.sum_eq_single (composition.ones n),
show comp_along_composition p (id 𝕜 E) (composition.ones n) = p n,
{ ext v,
rw comp_along_composition_apply,
apply p.congr (composition.ones_length n),
intros,
rw apply_composition_ones,
re... | theorem | formal_multilinear_series.comp_id | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.ones",
"composition.ones_length",
"continuous_multilinear_map.map_coord_zero",
"continuous_multilinear_map.zero_apply",
"fin.coe_cast_le",
"fin.coe_mk",
"fin.ext_iff",
"finset.univ",
"formal_multilinear_series",
"formal_multilinear_series.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (p : formal_multilinear_series 𝕜 E F) (h : p 0 = 0) : (id 𝕜 F).comp p = p | begin
ext1 n,
by_cases hn : n = 0,
{ rw [hn, h],
ext v,
rw [comp_coeff_zero', id_apply_ne_one _ _ zero_ne_one],
refl },
{ dsimp [formal_multilinear_series.comp],
have n_pos : 0 < n := bot_lt_iff_ne_bot.mpr hn,
rw finset.sum_eq_single (composition.single n n_pos),
show comp_along_composit... | theorem | formal_multilinear_series.id_comp | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.eq_single_iff_length",
"composition.single",
"composition.single_length",
"finset.univ",
"formal_multilinear_series",
"formal_multilinear_series.comp",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_summable_nnreal
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F)
(hq : 0 < q.radius) (hp : 0 < p.radius) :
∃ r > (0 : ℝ≥0),
summable (λ i : Σ n, composition n, ‖q.comp_along_composition p i.2‖₊ * r ^ i.1) | begin
/- This follows from the fact that the growth rate of `‖qₙ‖` and `‖pₙ‖` is at most geometric,
giving a geometric bound on each `‖q.comp_along_composition p op‖`, together with the
fact that there are `2^(n-1)` compositions of `n`, giving at most a geometric loss. -/
rcases ennreal.lt_iff_exists_nnreal_btw... | theorem | formal_multilinear_series.comp_summable_nnreal | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"add_tsub_cancel_right",
"composition",
"composition_card",
"div_eq_mul_inv",
"ennreal.coe_lt_one_iff",
"ennreal.coe_pos",
"finset.card_univ",
"finset.prod_le_prod'",
"finset.prod_mul_distrib",
"finset.prod_pow_eq_pow_sum",
"formal_multilinear_series",
"has_sum",
"has_sum_fintype",
"le_rfl... | If two formal multilinear series have positive radius of convergence, then the terms appearing
in the definition of their composition are also summable (when multiplied by a suitable positive
geometric term). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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