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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le_comp_radius_of_summable
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (r : ℝ≥0)
(hr : summable (λ i : (Σ n, composition n), ‖q.comp_along_composition p i.2‖₊ * r ^ i.1)) :
(r : ℝ≥0∞) ≤ (q.comp p).radius | begin
refine le_radius_of_bound_nnreal _
(∑' i : (Σ n, composition n), ‖comp_along_composition q p i.snd‖₊ * r ^ i.fst) (λ n, _),
calc ‖formal_multilinear_series.comp q p n‖₊ * r ^ n ≤
∑' (c : composition n), ‖comp_along_composition q p c‖₊ * r ^ n :
begin
rw [tsum_fintype, ← finset.sum_mul],
... | theorem | formal_multilinear_series.le_comp_radius_of_summable | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"finset.sum_mul",
"formal_multilinear_series",
"le_rfl",
"mul_le_mul'",
"nnreal.tsum_comp_le_tsum_of_inj",
"sigma_mk_injective",
"summable",
"tsum_fintype"
] | Bounding below the radius of the composition of two formal multilinear series assuming
summability over all compositions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_partial_sum_source (m M N : ℕ) : finset (Σ n, (fin n) → ℕ) | finset.sigma (finset.Ico m M) (λ (n : ℕ), fintype.pi_finset (λ (i : fin n), finset.Ico 1 N) : _) | def | formal_multilinear_series.comp_partial_sum_source | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"finset",
"finset.Ico",
"finset.sigma",
"fintype.pi_finset"
] | Source set in the change of variables to compute the composition of partial sums of formal
power series.
See also `comp_partial_sum`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_comp_partial_sum_source_iff (m M N : ℕ) (i : Σ n, (fin n) → ℕ) :
i ∈ comp_partial_sum_source m M N ↔
(m ≤ i.1 ∧ i.1 < M) ∧ ∀ (a : fin i.1), 1 ≤ i.2 a ∧ i.2 a < N | by simp only [comp_partial_sum_source, finset.mem_Ico, fintype.mem_pi_finset, finset.mem_sigma,
iff_self] | lemma | formal_multilinear_series.mem_comp_partial_sum_source_iff | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"finset.mem_Ico",
"finset.mem_sigma",
"fintype.mem_pi_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_change_of_variables (m M N : ℕ) (i : Σ n, (fin n) → ℕ)
(hi : i ∈ comp_partial_sum_source m M N) : (Σ n, composition n) | begin
rcases i with ⟨n, f⟩,
rw mem_comp_partial_sum_source_iff at hi,
refine ⟨∑ j, f j, of_fn (λ a, f a), λ i hi', _, by simp [sum_of_fn]⟩,
obtain ⟨j, rfl⟩ : ∃ (j : fin n), f j = i, by rwa [mem_of_fn, set.mem_range] at hi',
exact (hi.2 j).1
end | def | formal_multilinear_series.comp_change_of_variables | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"set.mem_range"
] | Change of variables appearing to compute the composition of partial sums of formal
power series | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_change_of_variables_length
(m M N : ℕ) {i : Σ n, (fin n) → ℕ} (hi : i ∈ comp_partial_sum_source m M N) :
composition.length (comp_change_of_variables m M N i hi).2 = i.1 | begin
rcases i with ⟨k, blocks_fun⟩,
dsimp [comp_change_of_variables],
simp only [composition.length, map_of_fn, length_of_fn]
end | lemma | formal_multilinear_series.comp_change_of_variables_length | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition.length"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_change_of_variables_blocks_fun
(m M N : ℕ) {i : Σ n, (fin n) → ℕ} (hi : i ∈ comp_partial_sum_source m M N) (j : fin i.1) :
(comp_change_of_variables m M N i hi).2.blocks_fun
⟨j, (comp_change_of_variables_length m M N hi).symm ▸ j.2⟩ = i.2 j | begin
rcases i with ⟨n, f⟩,
dsimp [composition.blocks_fun, composition.blocks, comp_change_of_variables],
simp only [map_of_fn, nth_le_of_fn', function.comp_app],
apply congr_arg,
exact fin.eta _ _
end | lemma | formal_multilinear_series.comp_change_of_variables_blocks_fun | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition.blocks_fun",
"fin.eta"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_partial_sum_target_set (m M N : ℕ) : set (Σ n, composition n) | {i | (m ≤ i.2.length) ∧ (i.2.length < M) ∧ (∀ (j : fin i.2.length), i.2.blocks_fun j < N)} | def | formal_multilinear_series.comp_partial_sum_target_set | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition"
] | Target set in the change of variables to compute the composition of partial sums of formal
power series, here given a a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_partial_sum_target_subset_image_comp_partial_sum_source
(m M N : ℕ) (i : Σ n, composition n) (hi : i ∈ comp_partial_sum_target_set m M N) :
∃ j (hj : j ∈ comp_partial_sum_source m M N), i = comp_change_of_variables m M N j hj | begin
rcases i with ⟨n, c⟩,
refine ⟨⟨c.length, c.blocks_fun⟩, _, _⟩,
{ simp only [comp_partial_sum_target_set, set.mem_set_of_eq] at hi,
simp only [mem_comp_partial_sum_source_iff, hi.left, hi.right, true_and, and_true],
exact λ a, c.one_le_blocks' _ },
{ dsimp [comp_change_of_variables],
rw composi... | lemma | formal_multilinear_series.comp_partial_sum_target_subset_image_comp_partial_sum_source | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.blocks_fun",
"composition.sigma_eq_iff_blocks_eq",
"subtype.coe_eta"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_partial_sum_target (m M N : ℕ) : finset (Σ n, composition n) | set.finite.to_finset $ ((finset.finite_to_set _).dependent_image _).subset $
comp_partial_sum_target_subset_image_comp_partial_sum_source m M N | def | formal_multilinear_series.comp_partial_sum_target | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"finset",
"finset.finite_to_set",
"set.finite.to_finset"
] | Target set in the change of variables to compute the composition of partial sums of formal
power series, here given a a finset.
See also `comp_partial_sum`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_comp_partial_sum_target_iff {m M N : ℕ} {a : Σ n, composition n} :
a ∈ comp_partial_sum_target m M N ↔
m ≤ a.2.length ∧ a.2.length < M ∧ (∀ (j : fin a.2.length), a.2.blocks_fun j < N) | by simp [comp_partial_sum_target, comp_partial_sum_target_set] | lemma | formal_multilinear_series.mem_comp_partial_sum_target_iff | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_change_of_variables_sum {α : Type*} [add_comm_monoid α] (m M N : ℕ)
(f : (Σ (n : ℕ), fin n → ℕ) → α) (g : (Σ n, composition n) → α)
(h : ∀ e (he : e ∈ comp_partial_sum_source m M N),
f e = g (comp_change_of_variables m M N e he)) :
∑ e in comp_partial_sum_source m M N, f e = ∑ e in comp_partial_sum_targe... | begin
apply finset.sum_bij (comp_change_of_variables m M N),
-- We should show that the correspondance we have set up is indeed a bijection
-- between the index sets of the two sums.
-- 1 - show that the image belongs to `comp_partial_sum_target m N N`
{ rintros ⟨k, blocks_fun⟩ H,
rw mem_comp_partial_sum_... | lemma | formal_multilinear_series.comp_change_of_variables_sum | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"add_comm_monoid",
"composition",
"composition.blocks_fun",
"composition.blocks_fun_congr",
"composition.length"
] | `comp_change_of_variables m M N` is a bijection between `comp_partial_sum_source m M N`
and `comp_partial_sum_target m M N`, yielding equal sums for functions that correspond to each
other under the bijection. As `comp_change_of_variables m M N` is a dependent function, stating
that it is a bijection is not directly po... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_partial_sum_target_tendsto_at_top :
tendsto (λ N, comp_partial_sum_target 0 N N) at_top at_top | begin
apply monotone.tendsto_at_top_finset,
{ assume m n hmn a ha,
have : ∀ i, i < m → i < n := λ i hi, lt_of_lt_of_le hi hmn,
tidy },
{ rintros ⟨n, c⟩,
simp only [mem_comp_partial_sum_target_iff],
obtain ⟨n, hn⟩ : bdd_above ↑(finset.univ.image (λ (i : fin c.length), c.blocks_fun i)) :=
fins... | lemma | formal_multilinear_series.comp_partial_sum_target_tendsto_at_top | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"bdd_above",
"bot_le",
"finset.bdd_above",
"finset.mem_coe",
"finset.mem_image_of_mem",
"finset.mem_univ",
"lt_add_one"
] | The auxiliary set corresponding to the composition of partial sums asymptotically contains
all possible compositions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_partial_sum
(q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (N : ℕ) (z : E) :
q.partial_sum N (∑ i in finset.Ico 1 N, p i (λ j, z)) =
∑ i in comp_partial_sum_target 0 N N, q.comp_along_composition p i.2 (λ j, z) | begin
-- we expand the composition, using the multilinearity of `q` to expand along each coordinate.
suffices H : ∑ n in finset.range N, ∑ r in fintype.pi_finset (λ (i : fin n), finset.Ico 1 N),
q n (λ (i : fin n), p (r i) (λ j, z)) =
∑ i in comp_partial_sum_target 0 N N, q.comp_along_composition p i.2 (λ j... | lemma | formal_multilinear_series.comp_partial_sum | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"continuous_multilinear_map.map_sum_finset",
"finset.Ico",
"finset.range",
"finset.range_eq_Ico",
"fintype.pi_finset",
"formal_multilinear_series",
"formal_multilinear_series.partial_sum"
] | Composing the partial sums of two multilinear series coincides with the sum over all
compositions in `comp_partial_sum_target 0 N N`. This is precisely the motivation for the
definition of `comp_partial_sum_target`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.comp {g : F → G} {f : E → F}
{q : formal_multilinear_series 𝕜 F G} {p : formal_multilinear_series 𝕜 E F} {x : E}
(hg : has_fpower_series_at g q (f x)) (hf : has_fpower_series_at f p x) :
has_fpower_series_at (g ∘ f) (q.comp p) x | begin
/- Consider `rf` and `rg` such that `f` and `g` have power series expansion on the disks
of radius `rf` and `rg`. -/
rcases hg with ⟨rg, Hg⟩,
rcases hf with ⟨rf, Hf⟩,
/- The terms defining `q.comp p` are geometrically summable in a disk of some radius `r`. -/
rcases q.comp_summable_nnreal p Hg.radius_... | theorem | has_fpower_series_at.comp | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"cauchy_seq",
"cauchy_seq_finset_of_norm_bounded",
"composition",
"continuous_at",
"continuous_at.comp",
"continuous_id",
"continuous_multilinear_map.le_op_norm",
"continuous_multilinear_map.sum_apply",
"emetric.ball",
"emetric.ball_mem_nhds",
"emetric.ball_subset_ball",
"emetric.is_open_ball"... | If two functions `g` and `f` have power series `q` and `p` respectively at `f x` and `x`, then
`g ∘ f` admits the power series `q.comp p` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_at.comp {g : F → G} {f : E → F} {x : E}
(hg : analytic_at 𝕜 g (f x)) (hf : analytic_at 𝕜 f x) : analytic_at 𝕜 (g ∘ f) x | let ⟨q, hq⟩ := hg, ⟨p, hp⟩ := hf in (hq.comp hp).analytic_at | theorem | analytic_at.comp | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"analytic_at"
] | If two functions `g` and `f` are analytic respectively at `f x` and `x`, then `g ∘ f` is
analytic at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_composition_eq_iff (i j : Σ (a : composition n), composition a.length) :
i = j ↔ i.1.blocks = j.1.blocks ∧ i.2.blocks = j.2.blocks | begin
refine ⟨by rintro rfl; exact ⟨rfl, rfl⟩, _⟩,
rcases i with ⟨a, b⟩,
rcases j with ⟨a', b'⟩,
rintros ⟨h, h'⟩,
have H : a = a', by { ext1, exact h },
induction H, congr, ext1, exact h'
end | lemma | composition.sigma_composition_eq_iff | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition"
] | Rewriting equality in the dependent type `Σ (a : composition n), composition a.length)` in
non-dependent terms with lists, requiring that the blocks coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_pi_composition_eq_iff
(u v : Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i)) :
u = v ↔ of_fn (λ i, (u.2 i).blocks) = of_fn (λ i, (v.2 i).blocks) | begin
refine ⟨λ H, by rw H, λ H, _⟩,
rcases u with ⟨a, b⟩,
rcases v with ⟨a', b'⟩,
dsimp at H,
have h : a = a',
{ ext1,
have : map list.sum (of_fn (λ (i : fin (composition.length a)), (b i).blocks)) =
map list.sum (of_fn (λ (i : fin (composition.length a')), (b' i).blocks)), by rw H,
simp only... | lemma | composition.sigma_pi_composition_eq_iff | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.length",
"composition.of_fn_blocks_fun",
"heq_iff_eq",
"list.sum"
] | Rewriting equality in the dependent type
`Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i)` in
non-dependent terms with lists, requiring that the lists of blocks coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gather (a : composition n) (b : composition a.length) : composition n | { blocks := (a.blocks.split_wrt_composition b).map sum,
blocks_pos :=
begin
rw forall_mem_map_iff,
intros j hj,
suffices H : ∀ i ∈ j, 1 ≤ i, from
calc 0 < j.length : length_pos_of_mem_split_wrt_composition hj
... ≤ j.sum : length_le_sum_of_one_le _ H,
intros i hi,
apply a.one_le... | def | composition.gather | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition"
] | When `a` is a composition of `n` and `b` is a composition of `a.length`, `a.gather b` is the
composition of `n` obtained by gathering all the blocks of `a` corresponding to a block of `b`.
For instance, if `a = [6, 5, 3, 5, 2]` and `b = [2, 3]`, one should gather together
the first two blocks of `a` and its last three ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length_gather (a : composition n) (b : composition a.length) :
length (a.gather b) = b.length | show (map list.sum (a.blocks.split_wrt_composition b)).length = b.blocks.length,
by rw [length_map, length_split_wrt_composition] | lemma | composition.length_gather | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"list.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_composition_aux (a : composition n) (b : composition a.length)
(i : fin (a.gather b).length) :
composition ((a.gather b).blocks_fun i) | { blocks := nth_le (a.blocks.split_wrt_composition b) i
(by { rw [length_split_wrt_composition, ← length_gather], exact i.2 }),
blocks_pos := assume i hi, a.blocks_pos
(by { rw ← a.blocks.join_split_wrt_composition b,
exact mem_join_of_mem (nth_le_mem _ _ _) hi }),
blocks_sum := by simp only [comp... | def | composition.sigma_composition_aux | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.blocks_fun",
"composition.gather"
] | An auxiliary function used in the definition of `sigma_equiv_sigma_pi` below, associating to
two compositions `a` of `n` and `b` of `a.length`, and an index `i` bounded by the length of
`a.gather b`, the subcomposition of `a` made of those blocks belonging to the `i`-th block of
`a.gather b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length_sigma_composition_aux (a : composition n) (b : composition a.length)
(i : fin b.length) :
composition.length (composition.sigma_composition_aux a b ⟨i, (length_gather a b).symm ▸ i.2⟩) =
composition.blocks_fun b i | show list.length (nth_le (split_wrt_composition a.blocks b) i _) = blocks_fun b i,
by { rw [nth_le_map_rev list.length, nth_le_of_eq (map_length_split_wrt_composition _ _)], refl } | lemma | composition.length_sigma_composition_aux | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.blocks_fun",
"composition.length",
"composition.sigma_composition_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blocks_fun_sigma_composition_aux (a : composition n) (b : composition a.length)
(i : fin b.length) (j : fin (blocks_fun b i)) :
blocks_fun (sigma_composition_aux a b ⟨i, (length_gather a b).symm ▸ i.2⟩)
⟨j, (length_sigma_composition_aux a b i).symm ▸ j.2⟩ = blocks_fun a (embedding b i j) | show nth_le (nth_le _ _ _) _ _ = nth_le a.blocks _ _,
by { rw [nth_le_of_eq (nth_le_split_wrt_composition _ _ _), nth_le_drop', nth_le_take'], refl } | lemma | composition.blocks_fun_sigma_composition_aux | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
size_up_to_size_up_to_add (a : composition n) (b : composition a.length)
{i j : ℕ} (hi : i < b.length) (hj : j < blocks_fun b ⟨i, hi⟩) :
size_up_to a (size_up_to b i + j) = size_up_to (a.gather b) i +
(size_up_to (sigma_composition_aux a b ⟨i, (length_gather a b).symm ▸ hi⟩) j) | begin
induction j with j IHj,
{ show sum (take ((b.blocks.take i).sum) a.blocks) =
sum (take i (map sum (split_wrt_composition a.blocks b))),
induction i with i IH,
{ refl },
{ have A : i < b.length := nat.lt_of_succ_lt hi,
have B : i < list.length (map list.sum (split_wrt_composition a.bloc... | lemma | composition.size_up_to_size_up_to_add | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.blocks_pos'",
"fin.coe_mk",
"list.sum",
"lt_add_one"
] | Auxiliary lemma to prove that the composition of formal multilinear series is associative.
Consider a composition `a` of `n` and a composition `b` of `a.length`. Grouping together some
blocks of `a` according to `b` as in `a.gather b`, one can compute the total size of the blocks
of `a` up to an index `size_up_to b i ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_equiv_sigma_pi (n : ℕ) :
(Σ (a : composition n), composition a.length) ≃
(Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i)) | { to_fun := λ i, ⟨i.1.gather i.2, i.1.sigma_composition_aux i.2⟩,
inv_fun := λ i, ⟨
{ blocks := (of_fn (λ j, (i.2 j).blocks)).join,
blocks_pos :=
begin
simp only [and_imp, list.mem_join, exists_imp_distrib, forall_mem_of_fn_iff],
exact λ i j hj, composition.blocks_pos _ hj
end,
... | def | composition.sigma_equiv_sigma_pi | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"and_imp",
"composition",
"composition.blocks_pos'",
"composition.gather",
"composition.length",
"composition.length_gather",
"composition.length_pos_of_pos",
"composition.of_fn_blocks_fun",
"composition.sigma_composition_aux",
"composition.sum_blocks_fun",
"exists_imp_distrib",
"fin.heq_fun_i... | Natural equivalence between `(Σ (a : composition n), composition a.length)` and
`(Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i))`, that shows up as a
change of variables in the proof that composition of formal multilinear series is associative.
Consider a composition `a` of `n` and a compos... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_assoc (r : formal_multilinear_series 𝕜 G H) (q : formal_multilinear_series 𝕜 F G)
(p : formal_multilinear_series 𝕜 E F) :
(r.comp q).comp p = r.comp (q.comp p) | begin
ext n v,
/- First, rewrite the two compositions appearing in the theorem as two sums over complicated
sigma types, as in the description of the proof above. -/
let f : (Σ (a : composition n), composition a.length) → H :=
λ c, r c.2.length (apply_composition q c.2 (apply_composition p c.1 v)),
let g ... | theorem | formal_multilinear_series.comp_assoc | analysis.analytic | src/analysis/analytic/composition.lean | [
"analysis.analytic.basic",
"combinatorics.composition"
] | [
"composition",
"composition.embedding",
"composition.length_gather",
"continuous_multilinear_map.map_sum",
"continuous_multilinear_map.sum_apply",
"formal_multilinear_series",
"formal_multilinear_series.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
formal_multilinear_series 𝕜 F E | | 0 := 0
| 1 := (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm
| (n+2) := - ∑ c : {c : composition (n+2) // c.length < n + 2},
have (c : composition (n+2)).length < n+2 := c.2,
(left_inv (c : composition (n+2)).length).comp_along_composition
(p.comp_continuous_linear_map i.symm) c | def | formal_multilinear_series.left_inv | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"continuous_multilinear_curry_fin1",
"formal_multilinear_series"
] | The left inverse of a formal multilinear series, where the `n`-th term is defined inductively
in terms of the previous ones to make sure that `(left_inv p i) ∘ p = id`. For this, the linear term
`p₁` in `p` should be invertible. In the definition, `i` is a linear isomorphism that should
coincide with `p₁`, so that one ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_inv_coeff_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.left_inv i 0 = 0 | by rw left_inv | lemma | formal_multilinear_series.left_inv_coeff_zero | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv_coeff_one (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.left_inv i 1 = (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm | by rw left_inv | lemma | formal_multilinear_series.left_inv_coeff_one | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"continuous_multilinear_curry_fin1",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv_remove_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.remove_zero.left_inv i = p.left_inv i | begin
ext1 n,
induction n using nat.strong_rec' with n IH,
cases n, { simp }, -- if one replaces `simp` with `refl`, the proof times out in the kernel.
cases n, { simp }, -- TODO: why?
simp only [left_inv, neg_inj],
refine finset.sum_congr rfl (λ c cuniv, _),
rcases c with ⟨c, hc⟩,
ext v,
dsimp,
sim... | lemma | formal_multilinear_series.left_inv_remove_zero | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"formal_multilinear_series",
"nat.strong_rec'"
] | The left inverse does not depend on the zeroth coefficient of a formal multilinear
series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_inv_comp (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) :
(left_inv p i).comp p = id 𝕜 E | begin
ext n v,
cases n,
{ simp only [left_inv, continuous_multilinear_map.zero_apply, id_apply_ne_one, ne.def,
not_false_iff, zero_ne_one, comp_coeff_zero']},
cases n,
{ simp only [left_inv, comp_coeff_one, h, id_apply_one, continuous_linear_equiv.coe_apply,
continuous_linear_equiv.symm_apply_appl... | lemma | formal_multilinear_series.left_inv_comp | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"composition.length",
"composition.length_le",
"composition.ones",
"composition.ones_length",
"continuous_linear_equiv.coe_apply",
"continuous_linear_equiv.symm_apply_apply",
"continuous_multilinear_curry_fin1",
"continuous_multilinear_curry_fin1_symm_apply",
"continuous_multilinear... | The left inverse to a formal multilinear series is indeed a left inverse, provided its linear
term is invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
formal_multilinear_series 𝕜 F E | | 0 := 0
| 1 := (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm
| (n+2) :=
let q : formal_multilinear_series 𝕜 F E := λ k, if h : k < n + 2 then right_inv k else 0 in
- (i.symm : F →L[𝕜] E).comp_continuous_multilinear_map ((p.comp q) (n+2)) | def | formal_multilinear_series.right_inv | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"continuous_multilinear_curry_fin1",
"formal_multilinear_series"
] | The right inverse of a formal multilinear series, where the `n`-th term is defined inductively
in terms of the previous ones to make sure that `p ∘ (right_inv p i) = id`. For this, the linear
term `p₁` in `p` should be invertible. In the definition, `i` is a linear isomorphism that should
coincide with `p₁`, so that on... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_inv_coeff_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.right_inv i 0 = 0 | by rw right_inv | lemma | formal_multilinear_series.right_inv_coeff_zero | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inv_coeff_one (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.right_inv i 1 = (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm | by rw right_inv | lemma | formal_multilinear_series.right_inv_coeff_one | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"continuous_multilinear_curry_fin1",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inv_remove_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.remove_zero.right_inv i = p.right_inv i | begin
ext1 n,
induction n using nat.strong_rec' with n IH,
rcases n with _|_|n,
{ simp only [right_inv_coeff_zero] },
{ simp only [right_inv_coeff_one] },
simp only [right_inv, neg_inj],
rw remove_zero_comp_of_pos _ _ (add_pos_of_nonneg_of_pos (n.zero_le) zero_lt_two),
congr' 2 with k,
by_cases hk : k... | lemma | formal_multilinear_series.right_inv_remove_zero | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"formal_multilinear_series",
"nat.strong_rec'",
"zero_lt_two"
] | The right inverse does not depend on the zeroth coefficient of a formal multilinear
series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right_inv_aux1 {n : ℕ} (hn : 0 < n)
(p : formal_multilinear_series 𝕜 E F) (q : formal_multilinear_series 𝕜 F E) (v : fin n → F) :
p.comp q n v =
(∑ (c : composition n) in {c : composition n | 1 < c.length}.to_finset,
p c.length (q.apply_composition c v)) + p 1 (λ i, q n v) | begin
have A : (finset.univ : finset (composition n))
= {c | 1 < composition.length c}.to_finset ∪ {composition.single n hn},
{ refine subset.antisymm (λ c hc, _) (subset_univ _),
by_cases h : 1 < c.length,
{ simp [h] },
{ have : c.length = 1,
by { refine (eq_iff_le_not_lt.2 ⟨ _, h⟩).symm, e... | lemma | formal_multilinear_series.comp_right_inv_aux1 | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"composition.eq_single_iff_length",
"composition.length",
"composition.single",
"composition.single_length",
"disjoint",
"finset",
"finset.univ",
"formal_multilinear_series",
"formal_multilinear_series.comp",
"set.to_finset_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_inv_aux2
(p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (v : fin (n + 2) → F) :
∑ (c : composition (n + 2)) in {c : composition (n + 2) | 1 < c.length}.to_finset,
p c.length (apply_composition (λ (k : ℕ), ite (k < n + 2) (p.right_inv i k) 0) c v) =
∑ (c : composition (n + 2)) in {c... | begin
have N : 0 < n + 2, by dec_trivial,
refine sum_congr rfl (λ c hc, p.congr rfl (λ j hj1 hj2, _)),
have : ∀ k, c.blocks_fun k < n + 2,
{ simp only [set.mem_to_finset, set.mem_set_of_eq] at hc,
simp [← composition.ne_single_iff N, composition.eq_single_iff_length, ne_of_gt hc] },
simp [apply_compositio... | lemma | formal_multilinear_series.comp_right_inv_aux2 | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"composition.eq_single_iff_length",
"composition.ne_single_iff",
"formal_multilinear_series",
"set.mem_to_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
p.comp (right_inv p i) = id 𝕜 F | begin
ext n v,
cases n,
{ simp only [h0, continuous_multilinear_map.zero_apply, id_apply_ne_one, ne.def, not_false_iff,
zero_ne_one, comp_coeff_zero']},
cases n,
{ simp only [comp_coeff_one, h, right_inv, continuous_linear_equiv.apply_symm_apply, id_apply_one,
continuous_linear_equiv.coe_apply, co... | lemma | formal_multilinear_series.comp_right_inv | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"continuous_linear_equiv.apply_symm_apply",
"continuous_linear_equiv.coe_apply",
"continuous_multilinear_curry_fin1",
"continuous_multilinear_curry_fin1_symm_apply",
"continuous_multilinear_map.zero_apply",
"formal_multilinear_series",
"set.to_finset_set_of",
"zero_ne_one"
] | The right inverse to a formal multilinear series is indeed a right inverse, provided its linear
term is invertible and its constant term vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_inv_coeff (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (hn : 2 ≤ n) :
p.right_inv i n = - (i.symm : F →L[𝕜] E).comp_continuous_multilinear_map
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition n)),
p.comp_along_composition (p.right_inv i) c) | begin
cases n, { exact false.elim (zero_lt_two.not_le hn) },
cases n, { exact false.elim (one_lt_two.not_le hn) },
simp only [right_inv, neg_inj],
congr' 1,
ext v,
have N : 0 < n + 2, by dec_trivial,
have : (p 1) (λ (i : fin 1), 0) = 0 := continuous_multilinear_map.map_zero _,
simp [comp_right_inv_aux1 ... | lemma | formal_multilinear_series.right_inv_coeff | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"composition.length",
"continuous_multilinear_map.map_zero",
"finset",
"formal_multilinear_series",
"set.to_finset_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv_eq_right_inv_aux (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
left_inv p i = right_inv p i | calc
left_inv p i = (left_inv p i).comp (id 𝕜 F) : by simp
... = (left_inv p i).comp (p.comp (right_inv p i)) : by rw comp_right_inv p i h h0
... = ((left_inv p i).comp p).comp (right_inv p i) : by rw comp_assoc
... = (id 𝕜 E).comp (right_inv p i) : by rw left_inv_comp p i h
... = right_inv p i : by simp | lemma | formal_multilinear_series.left_inv_eq_right_inv_aux | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"continuous_multilinear_curry_fin1",
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv_eq_right_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) :
left_inv p i = right_inv p i | calc
left_inv p i = left_inv p.remove_zero i : by rw left_inv_remove_zero
... = right_inv p.remove_zero i : by { apply left_inv_eq_right_inv_aux; simp [h] }
... = right_inv p i : by rw right_inv_remove_zero | theorem | formal_multilinear_series.left_inv_eq_right_inv | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"continuous_multilinear_curry_fin1",
"formal_multilinear_series",
"left_inv_eq_right_inv"
] | The left inverse and the right inverse of a formal multilinear series coincide. This is not at
all obvious from their definition, but it follows from uniqueness of inverses (which comes from the
fact that composition is associative on formal multilinear series). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radius_right_inv_pos_of_radius_pos_aux1
(n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) :
∑ k in Ico 2 (n + 1), a ^ k *
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
r ^ c.length * ∏ j, p (c.blocks_fun j))
≤ ∑ j in Ico 2 (n + 1), r ^ j * (... | calc
∑ k in Ico 2 (n + 1), a ^ k *
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
r ^ c.length * ∏ j, p (c.blocks_fun j))
= ∑ k in Ico 2 (n + 1),
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
∏ j, r * (a ^ (c.blocks_fun j) * p (c.blocks_fun j)... | lemma | formal_multilinear_series.radius_right_inv_pos_of_radius_pos_aux1 | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"composition.eq_single_iff_length",
"composition.length",
"composition.ne_single_iff",
"composition.single",
"composition.sum_blocks_fun",
"fin.heq_fun_iff",
"finset",
"fintype.pi_finset",
"mul_pow",
"multilinear_map.map_sum_finset",
"multilinear_map.mk_pi_algebra",
"multiline... | First technical lemma to control the growth of coefficients of the inverse. Bound the explicit
expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before,
in a general abstract setup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radius_right_inv_pos_of_radius_pos_aux2
{n : ℕ} (hn : 2 ≤ n + 1) (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
{r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) (hC : 0 ≤ C) (hp : ∀ n, ‖p n‖ ≤ C * r ^ n) :
(∑ k in Ico 1 (n + 1), a ^ k * ‖p.right_inv i k‖) ≤
‖(i.symm : F →L[𝕜] E)‖ * a + ‖(i.symm : F →L[𝕜] E)... | let I := ‖(i.symm : F →L[𝕜] E)‖ in calc
∑ k in Ico 1 (n + 1), a ^ k * ‖p.right_inv i k‖
= a * I + ∑ k in Ico 2 (n + 1), a ^ k * ‖p.right_inv i k‖ :
by simp only [linear_isometry_equiv.norm_map, pow_one, right_inv_coeff_one,
nat.Ico_succ_singleton, sum_singleton, ← sum_Ico_consecutive _ one_le_two hn]... | lemma | formal_multilinear_series.radius_right_inv_pos_of_radius_pos_aux2 | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"composition",
"composition.length",
"continuous_linear_map.norm_comp_continuous_multilinear_map_le",
"finset",
"formal_multilinear_series",
"linear_isometry_equiv.norm_map",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_pow",
"nat.Ico_succ_singleton"... | Second technical lemma to control the growth of coefficients of the inverse. Bound the explicit
expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before,
in the specific setup we are interesting in, by reducing to the general bound in
`radius_right_inv_pos_of_radius_pos_aux1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radius_right_inv_pos_of_radius_pos (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(hp : 0 < p.radius) : 0 < (p.right_inv i).radius | begin
obtain ⟨C, r, Cpos, rpos, ple⟩ : ∃ C r (hC : 0 < C) (hr : 0 < r), ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n :=
le_mul_pow_of_radius_pos p hp,
let I := ‖(i.symm : F →L[𝕜] E)‖,
-- choose `a` small enough to make sure that `∑_{k ≤ n} aᵏ Qₖ` will be controllable by
-- induction
obtain ⟨a, apos, ha1, ha2⟩ : ∃ a (apo... | theorem | formal_multilinear_series.radius_right_inv_pos_of_radius_pos | analysis.analytic | src/analysis/analytic/inverse.lean | [
"analysis.analytic.composition",
"tactic.congrm"
] | [
"div_le_div",
"ennreal",
"formal_multilinear_series",
"geom_sum_Ico'",
"inf_le_left",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"nat.le_induction",
"nnreal",
"pow_le_pow_of_le_left",
"pow_nonneg",
"ring",
"self_mem_nhds_within",
"sq_nonneg",
... | If a a formal multilinear series has a positive radius of convergence, then its right inverse
also has a positive radius of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_at_zero (a : ℕ → E) : has_sum (λ n, (0:𝕜) ^ n • a n) (a 0) | by convert has_sum_single 0 (λ b h, _); simp [nat.pos_of_ne_zero h] <|> simp | lemma | has_sum.has_sum_at_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"has_sum",
"has_sum_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_has_sum_smul_of_apply_eq_zero (hs : has_sum (λ m, z ^ m • a m) s)
(ha : ∀ k < n, a k = 0) :
∃ t : E, z ^ n • t = s ∧ has_sum (λ m, z ^ m • a (m + n)) t | begin
obtain rfl | hn := n.eq_zero_or_pos,
{ simpa },
by_cases h : z = 0,
{ have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using has_sum_at_zero a),
exact ⟨a n, by simp [h, hn, this], by simpa [h] using has_sum_at_zero (λ m, a (m + n))⟩ },
{ refine ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul], _⟩,
have... | lemma | has_sum.exists_has_sum_smul_of_apply_eq_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"finset.range",
"has_sum",
"has_sum_nat_add_iff'",
"inv_pow",
"pow_add",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_dslope_fslope (hp : has_fpower_series_at f p z₀) :
has_fpower_series_at (dslope f z₀) p.fslope z₀ | begin
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv,
have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1,
simp only [has_fpower_series_at_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢,
refine hp.mono (λ x hx, _),
by_cases h : x = 0,
{ convert has_sum_single 0 _; intros; simp [*] },
{ have hxx : ∀ (n : ℕ)... | lemma | has_fpower_series_at.has_fpower_series_dslope_fslope | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"deriv",
"dslope",
"has_fpower_series_at",
"has_fpower_series_at_iff",
"has_sum",
"has_sum_nat_add_iff'",
"has_sum_single",
"pow_succ'",
"slope",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_iterate_dslope_fslope (n : ℕ) (hp : has_fpower_series_at f p z₀) :
has_fpower_series_at ((swap dslope z₀)^[n] f) (fslope^[n] p) z₀ | begin
induction n with n ih generalizing f p,
{ exact hp },
{ simpa using ih (has_fpower_series_dslope_fslope hp) }
end | lemma | has_fpower_series_at.has_fpower_series_iterate_dslope_fslope | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"dslope",
"has_fpower_series_at",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_dslope_fslope_ne_zero (hp : has_fpower_series_at f p z₀) (h : p ≠ 0) :
(swap dslope z₀)^[p.order] f z₀ ≠ 0 | begin
rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1],
simpa [coeff_eq_zero] using apply_order_ne_zero h
end | lemma | has_fpower_series_at.iterate_dslope_fslope_ne_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"dslope",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_order_mul_iterate_dslope (hp : has_fpower_series_at f p z₀) :
∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • ((swap dslope z₀)^[p.order] f z) | begin
have hq := has_fpower_series_at_iff'.mp (has_fpower_series_iterate_dslope_fslope p.order hp),
filter_upwards [hq, has_fpower_series_at_iff'.mp hp] with x hx1 hx2,
have : ∀ k < p.order, p.coeff k = 0,
from λ k hk, by simpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk,
obtain ⟨s, hs1, hs2⟩ := has_... | lemma | has_fpower_series_at.eq_pow_order_mul_iterate_dslope | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"dslope",
"has_fpower_series_at",
"has_sum.exists_has_sum_smul_of_apply_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_ne_zero (hp : has_fpower_series_at f p z₀) (h : p ≠ 0) :
∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 | begin
rw [eventually_nhds_within_iff],
have h2 := (has_fpower_series_iterate_dslope_fslope p.order hp).continuous_at,
have h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h),
filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3] with z e1 e2 e3,
simpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_... | lemma | has_fpower_series_at.locally_ne_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"continuous_at",
"eventually_nhds_within_iff",
"has_fpower_series_at",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_zero_iff (hp : has_fpower_series_at f p z₀) :
(∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 | ⟨λ hf, hp.eq_zero_of_eventually hf, λ h, eventually_eq_zero (by rwa h at hp)⟩ | lemma | has_fpower_series_at.locally_zero_iff | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_zero_or_eventually_ne_zero (hf : analytic_at 𝕜 f z₀) :
(∀ᶠ z in 𝓝 z₀, f z = 0) ∨ (∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0) | begin
rcases hf with ⟨p, hp⟩,
by_cases h : p = 0,
{ exact or.inl (has_fpower_series_at.eventually_eq_zero (by rwa h at hp)) },
{ exact or.inr (hp.locally_ne_zero h) }
end | theorem | analytic_at.eventually_eq_zero_or_eventually_ne_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_at",
"has_fpower_series_at.eventually_eq_zero"
] | The *principle of isolated zeros* for an analytic function, local version: if a function is
analytic at `z₀`, then either it is identically zero in a neighborhood of `z₀`, or it does not
vanish in a punctured neighborhood of `z₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_eq_or_eventually_ne (hf : analytic_at 𝕜 f z₀) (hg : analytic_at 𝕜 g z₀) :
(∀ᶠ z in 𝓝 z₀, f z = g z) ∨ (∀ᶠ z in 𝓝[≠] z₀, f z ≠ g z) | by simpa [sub_eq_zero] using (hf.sub hg).eventually_eq_zero_or_eventually_ne_zero | lemma | analytic_at.eventually_eq_or_eventually_ne | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_zero_iff_eventually_zero {f : 𝕜 → E} {w : 𝕜} (hf : analytic_at 𝕜 f w) :
(∃ᶠ z in 𝓝[≠] w, f z = 0) ↔ (∀ᶠ z in 𝓝 w, f z = 0) | ⟨hf.eventually_eq_zero_or_eventually_ne_zero.resolve_right,
λ h, (h.filter_mono nhds_within_le_nhds).frequently⟩ | lemma | analytic_at.frequently_zero_iff_eventually_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_at",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_eq_iff_eventually_eq (hf : analytic_at 𝕜 f z₀) (hg : analytic_at 𝕜 g z₀) :
(∃ᶠ z in 𝓝[≠] z₀, f z = g z) ↔ (∀ᶠ z in 𝓝 z₀, f z = g z) | by simpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg) | lemma | analytic_at.frequently_eq_iff_eventually_eq | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_zero_of_preconnected_of_frequently_eq_zero
(hf : analytic_on 𝕜 f U) (hU : is_preconnected U)
(h₀ : z₀ ∈ U) (hfw : ∃ᶠ z in 𝓝[≠] z₀, f z = 0) :
eq_on f 0 U | hf.eq_on_zero_of_preconnected_of_eventually_eq_zero hU h₀
((hf z₀ h₀).frequently_zero_iff_eventually_zero.1 hfw) | theorem | analytic_on.eq_on_zero_of_preconnected_of_frequently_eq_zero | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_on",
"is_preconnected"
] | The *principle of isolated zeros* for an analytic function, global version: if a function is
analytic on a connected set `U` and vanishes in arbitrary neighborhoods of a point `z₀ ∈ U`, then
it is identically zero in `U`.
For higher-dimensional versions requiring that the function vanishes in a neighborhood of `z₀`,
se... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_zero_of_preconnected_of_mem_closure (hf : analytic_on 𝕜 f U) (hU : is_preconnected U)
(h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀})) :
eq_on f 0 U | hf.eq_on_zero_of_preconnected_of_frequently_eq_zero hU h₀
(mem_closure_ne_iff_frequently_within.mp hfz₀) | theorem | analytic_on.eq_on_zero_of_preconnected_of_mem_closure | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_on",
"closure",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_of_preconnected_of_frequently_eq (hf : analytic_on 𝕜 f U) (hg : analytic_on 𝕜 g U)
(hU : is_preconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) :
eq_on f g U | begin
have hfg' : ∃ᶠ z in 𝓝[≠] z₀, (f - g) z = 0 := hfg.mono (λ z h, by rw [pi.sub_apply, h, sub_self]),
simpa [sub_eq_zero] using
λ z hz, (hf.sub hg).eq_on_zero_of_preconnected_of_frequently_eq_zero hU h₀ hfg' hz
end | theorem | analytic_on.eq_on_of_preconnected_of_frequently_eq | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_on",
"is_preconnected"
] | The *identity principle* for analytic functions, global version: if two functions are
analytic on a connected set `U` and coincide at points which accumulate to a point `z₀ ∈ U`, then
they coincide globally in `U`.
For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
see `eq_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_of_preconnected_of_mem_closure (hf : analytic_on 𝕜 f U) (hg : analytic_on 𝕜 g U)
(hU : is_preconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀})) :
eq_on f g U | hf.eq_on_of_preconnected_of_frequently_eq hg hU h₀ (mem_closure_ne_iff_frequently_within.mp hfg) | theorem | analytic_on.eq_on_of_preconnected_of_mem_closure | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_on",
"closure",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_frequently_eq [connected_space 𝕜]
(hf : analytic_on 𝕜 f univ) (hg : analytic_on 𝕜 g univ)
(hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : f = g | funext (λ x, eq_on_of_preconnected_of_frequently_eq hf hg is_preconnected_univ
(mem_univ z₀) hfg (mem_univ x)) | theorem | analytic_on.eq_of_frequently_eq | analysis.analytic | src/analysis/analytic/isolated_zeros.lean | [
"analysis.analytic.basic",
"analysis.calculus.dslope",
"analysis.calculus.fderiv_analytic",
"analysis.calculus.formal_multilinear_series",
"analysis.analytic.uniqueness"
] | [
"analytic_on",
"connected_space"
] | The *identity principle* for analytic functions, global version: if two functions on a normed
field `𝕜` are analytic everywhere and coincide at points which accumulate to a point `z₀`, then
they coincide globally.
For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
see `eq_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fpower_series (f : E →L[𝕜] F) (x : E) : formal_multilinear_series 𝕜 E F | | 0 := continuous_multilinear_map.curry0 𝕜 _ (f x)
| 1 := (continuous_multilinear_curry_fin1 𝕜 E F).symm f
| _ := 0 | def | continuous_linear_map.fpower_series | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"continuous_multilinear_curry_fin1",
"continuous_multilinear_map.curry0",
"formal_multilinear_series"
] | Formal power series of a continuous linear map `f : E →L[𝕜] F` at `x : E`:
`f y = f x + f (y - x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fpower_series_apply_add_two (f : E →L[𝕜] F) (x : E) (n : ℕ) :
f.fpower_series x (n + 2) = 0 | rfl | lemma | continuous_linear_map.fpower_series_apply_add_two | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fpower_series_radius (f : E →L[𝕜] F) (x : E) : (f.fpower_series x).radius = ∞ | (f.fpower_series x).radius_eq_top_of_forall_image_add_eq_zero 2 $ λ n, rfl | lemma | continuous_linear_map.fpower_series_radius | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball (f : E →L[𝕜] F) (x : E) :
has_fpower_series_on_ball f (f.fpower_series x) x ∞ | { r_le := by simp,
r_pos := ennreal.coe_lt_top,
has_sum := λ y _, (has_sum_nat_add_iff' 2).1 $
by simp [finset.sum_range_succ, ← sub_sub, has_sum_zero] } | theorem | continuous_linear_map.has_fpower_series_on_ball | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"ennreal.coe_lt_top",
"has_fpower_series_on_ball",
"has_sum",
"has_sum_nat_add_iff'",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at (f : E →L[𝕜] F) (x : E) :
has_fpower_series_at f (f.fpower_series x) x | ⟨∞, f.has_fpower_series_on_ball x⟩ | theorem | continuous_linear_map.has_fpower_series_at | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at (f : E →L[𝕜] F) (x : E) : analytic_at 𝕜 f x | (f.has_fpower_series_at x).analytic_at | theorem | continuous_linear_map.analytic_at | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_bilinear (f : E →L[𝕜] F →L[𝕜] G) : (E × F) [×2]→L[𝕜] G | @continuous_linear_map.uncurry_left 𝕜 1 (λ _, E × F) G _ _ _ _ _ $
(↑(continuous_multilinear_curry_fin1 𝕜 (E × F) G).symm : (E × F →L[𝕜] G) →L[𝕜] _).comp $
f.bilinear_comp (fst _ _ _) (snd _ _ _) | def | continuous_linear_map.uncurry_bilinear | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"continuous_linear_map.uncurry_left",
"continuous_multilinear_curry_fin1"
] | Reinterpret a bilinear map `f : E →L[𝕜] F →L[𝕜] G` as a multilinear map
`(E × F) [×2]→L[𝕜] G`. This multilinear map is the second term in the formal
multilinear series expansion of `uncurry f`. It is given by
`f.uncurry_bilinear ![(x, y), (x', y')] = f x y'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry_bilinear_apply (f : E →L[𝕜] F →L[𝕜] G) (m : fin 2 → E × F) :
f.uncurry_bilinear m = f (m 0).1 (m 1).2 | rfl | lemma | continuous_linear_map.uncurry_bilinear_apply | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fpower_series_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
formal_multilinear_series 𝕜 (E × F) G | | 0 := continuous_multilinear_map.curry0 𝕜 _ (f x.1 x.2)
| 1 := (continuous_multilinear_curry_fin1 𝕜 (E × F) G).symm (f.deriv₂ x)
| 2 := f.uncurry_bilinear
| _ := 0 | def | continuous_linear_map.fpower_series_bilinear | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"continuous_multilinear_curry_fin1",
"continuous_multilinear_map.curry0",
"formal_multilinear_series"
] | Formal multilinear series expansion of a bilinear function `f : E →L[𝕜] F →L[𝕜] G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fpower_series_bilinear_radius (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
(f.fpower_series_bilinear x).radius = ∞ | (f.fpower_series_bilinear x).radius_eq_top_of_forall_image_add_eq_zero 3 $ λ n, rfl | lemma | continuous_linear_map.fpower_series_bilinear_radius | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
has_fpower_series_on_ball (λ x : E × F, f x.1 x.2) (f.fpower_series_bilinear x) x ∞ | { r_le := by simp,
r_pos := ennreal.coe_lt_top,
has_sum := λ y _, (has_sum_nat_add_iff' 3).1 $
begin
simp only [finset.sum_range_succ, finset.sum_range_one, prod.fst_add, prod.snd_add,
f.map_add_add],
dsimp, simp only [add_comm, sub_self, has_sum_zero]
end } | theorem | continuous_linear_map.has_fpower_series_on_ball_bilinear | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"ennreal.coe_lt_top",
"has_fpower_series_on_ball",
"has_sum",
"has_sum_nat_add_iff'",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
has_fpower_series_at (λ x : E × F, f x.1 x.2) (f.fpower_series_bilinear x) x | ⟨∞, f.has_fpower_series_on_ball_bilinear x⟩ | theorem | continuous_linear_map.has_fpower_series_at_bilinear | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
analytic_at 𝕜 (λ x : E × F, f x.1 x.2) x | (f.has_fpower_series_at_bilinear x).analytic_at | theorem | continuous_linear_map.analytic_at_bilinear | analysis.analytic | src/analysis/analytic/linear.lean | [
"analysis.analytic.basic"
] | [
"analytic_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_eq_liminf : p.radius = liminf (λ n, 1/((‖p n‖₊) ^ (1 / (n : ℝ)) : ℝ≥0)) at_top | begin
have : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n →
((r : ℝ≥0∞) ≤ 1 / ↑(‖p n‖₊ ^ (1 / (n : ℝ))) ↔ ‖p n‖₊ * r ^ n ≤ 1),
{ intros r n hn,
have : 0 < (n : ℝ) := nat.cast_pos.2 hn,
conv_lhs {rw [one_div, ennreal.le_inv_iff_mul_le, ← ennreal.coe_mul,
ennreal.coe_le_one_iff, one_div, ← nnreal.rpow_one r, ← mul_in... | lemma | formal_multilinear_series.radius_eq_liminf | analysis.analytic | src/analysis/analytic/radius_liminf.lean | [
"analysis.analytic.basic",
"analysis.special_functions.pow.nnreal"
] | [
"ennreal.coe_le_one_iff",
"ennreal.coe_mul",
"ennreal.le_inv_iff_mul_le",
"ennreal.le_of_forall_nnreal_lt",
"le_abs_self",
"mul_comm",
"mul_inv_cancel",
"nnreal.mul_rpow",
"nnreal.one_rpow",
"nnreal.rpow_le_rpow_iff",
"nnreal.rpow_mul",
"nnreal.rpow_nat_cast",
"nnreal.rpow_one",
"one_div",... | The radius of a formal multilinear series is equal to
$\liminf_{n\to\infty} \frac{1}{\sqrt[n]{‖p n‖}}$. The actual statement uses `ℝ≥0` and some
coercions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_zero_of_preconnected_of_eventually_eq_zero_aux [complete_space F]
{f : E → F} {U : set E} (hf : analytic_on 𝕜 f U) (hU : is_preconnected U) {z₀ : E}
(h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) : eq_on f 0 U | begin
/- Let `u` be the set of points around which `f` vanishes. It is clearly open. We have to show
that its limit points in `U` still belong to it, from which the inclusion `U ⊆ u` will follow
by connectedness. -/
let u := {x | f =ᶠ[𝓝 x] 0},
suffices main : closure u ∩ U ⊆ u,
{ have Uu : U ⊆ u, from
... | theorem | analytic_on.eq_on_zero_of_preconnected_of_eventually_eq_zero_aux | analysis.analytic | src/analysis/analytic/uniqueness.lean | [
"analysis.analytic.linear",
"analysis.analytic.composition",
"analysis.normed_space.completion"
] | [
"analytic_on",
"closure",
"complete_space",
"emetric.ball",
"ennreal.add_halves",
"ennreal.coe_ne_top",
"ennreal.half_le_self",
"ennreal.half_pos",
"ennreal.le_sub_of_add_le_left",
"has_fpower_series_at",
"has_fpower_series_on_ball",
"has_sum",
"has_sum.unique",
"has_sum_zero",
"is_open_... | If an analytic function vanishes around a point, then it is uniformly zero along
a connected set. Superseded by `eq_on_zero_of_preconnected_of_locally_zero` which does not assume
completeness of the target space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_zero_of_preconnected_of_eventually_eq_zero
{f : E → F} {U : set E} (hf : analytic_on 𝕜 f U) (hU : is_preconnected U) {z₀ : E}
(h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
eq_on f 0 U | begin
let F' := uniform_space.completion F,
set e : F →L[𝕜] F' := uniform_space.completion.to_complL,
have : analytic_on 𝕜 (e ∘ f) U := λ x hx, (e.analytic_at _).comp (hf x hx),
have A : eq_on (e ∘ f) 0 U,
{ apply eq_on_zero_of_preconnected_of_eventually_eq_zero_aux this hU h₀,
filter_upwards [hfz₀] wit... | theorem | analytic_on.eq_on_zero_of_preconnected_of_eventually_eq_zero | analysis.analytic | src/analysis/analytic/uniqueness.lean | [
"analysis.analytic.linear",
"analysis.analytic.composition",
"analysis.normed_space.completion"
] | [
"analytic_on",
"is_preconnected",
"uniform_space.completion",
"uniform_space.completion.coe_injective",
"uniform_space.completion.to_complL"
] | The *identity principle* for analytic functions: If an analytic function vanishes in a whole
neighborhood of a point `z₀`, then it is uniformly zero along a connected set. For a one-dimensional
version assuming only that the function vanishes at some points arbitrarily close to `z₀`, see
`eq_on_zero_of_preconnected_of_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_of_preconnected_of_eventually_eq
{f g : E → F} {U : set E} (hf : analytic_on 𝕜 f U) (hg : analytic_on 𝕜 g U)
(hU : is_preconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g) :
eq_on f g U | begin
have hfg' : (f - g) =ᶠ[𝓝 z₀] 0 := hfg.mono (λ z h, by simp [h]),
simpa [sub_eq_zero] using
λ z hz, (hf.sub hg).eq_on_zero_of_preconnected_of_eventually_eq_zero hU h₀ hfg' hz,
end | theorem | analytic_on.eq_on_of_preconnected_of_eventually_eq | analysis.analytic | src/analysis/analytic/uniqueness.lean | [
"analysis.analytic.linear",
"analysis.analytic.composition",
"analysis.normed_space.completion"
] | [
"analytic_on",
"is_preconnected"
] | The *identity principle* for analytic functions: If two analytic functions coincide in a whole
neighborhood of a point `z₀`, then they coincide globally along a connected set.
For a one-dimensional version assuming only that the functions coincide at some points
arbitrarily close to `z₀`, see `eq_on_of_preconnected_of_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_eventually_eq {f g : E → F} [preconnected_space E]
(hf : analytic_on 𝕜 f univ) (hg : analytic_on 𝕜 g univ) {z₀ : E} (hfg : f =ᶠ[𝓝 z₀] g) :
f = g | funext (λ x, eq_on_of_preconnected_of_eventually_eq hf hg is_preconnected_univ
(mem_univ z₀) hfg (mem_univ x)) | theorem | analytic_on.eq_of_eventually_eq | analysis.analytic | src/analysis/analytic/uniqueness.lean | [
"analysis.analytic.linear",
"analysis.analytic.composition",
"analysis.normed_space.completion"
] | [
"analytic_on",
"preconnected_space"
] | The *identity principle* for analytic functions: If two analytic functions on a normed space
coincide in a neighborhood of a point `z₀`, then they coincide everywhere.
For a one-dimensional version assuming only that the functions coincide at some points
arbitrarily close to `z₀`, see `eq_of_frequently_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_with (c : ℝ) (l : filter α) (f : α → E) (g : α → F) : Prop | ∀ᶠ x in l, ‖ f x ‖ ≤ c * ‖ g x ‖ | def | asymptotics.is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | This version of the Landau notation `is_O_with C l f g` where `f` and `g` are two functions on
a type `α` and `l` is a filter on `α`, means that eventually for `l`, `‖f‖` is bounded by `C * ‖g‖`.
In other words, `‖f‖ / ‖g‖` is eventually bounded by `C`, modulo division by zero issues that are
avoided by this definition... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_with_iff : is_O_with c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ | by rw is_O_with | lemma | asymptotics.is_O_with_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | Definition of `is_O_with`. We record it in a lemma as `is_O_with` is irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O (l : filter α) (f : α → E) (g : α → F) : Prop | ∃ c : ℝ, is_O_with c l f g | def | asymptotics.is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | The Landau notation `f =O[l] g` where `f` and `g` are two functions on a type `α` and `l` is
a filter on `α`, means that eventually for `l`, `‖f‖` is bounded by a constant multiple of `‖g‖`.
In other words, `‖f‖ / ‖g‖` is eventually bounded, modulo division by zero issues that are avoided
by this definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_iff_is_O_with : f =O[l] g ↔ ∃ c : ℝ, is_O_with c l f g | by rw is_O | lemma | asymptotics.is_O_iff_is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | Definition of `is_O` in terms of `is_O_with`. We record it in a lemma as `is_O` is
irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ | by simp only [is_O, is_O_with] | lemma | asymptotics.is_O_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | Definition of `is_O` in terms of filters. We record it in a lemma as we will set
`is_O` to be irreducible at the end of this file. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O.of_bound (c : ℝ) (h : ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖) : f =O[l] g | is_O_iff.2 ⟨c, h⟩ | lemma | asymptotics.is_O.of_bound | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.of_bound' (h : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖) : f =O[l] g | is_O.of_bound 1 $ by { simp_rw one_mul, exact h } | lemma | asymptotics.is_O.of_bound' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.bound : f =O[l] g → ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ | is_O_iff.1 | lemma | asymptotics.is_O.bound | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o (l : filter α) (f : α → E) (g : α → F) : Prop | ∀ ⦃c : ℝ⦄, 0 < c → is_O_with c l f g | def | asymptotics.is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | The Landau notation `f =o[l] g` where `f` and `g` are two functions on a type `α` and `l` is
a filter on `α`, means that eventually for `l`, `‖f‖` is bounded by an arbitrarily small constant
multiple of `‖g‖`. In other words, `‖f‖ / ‖g‖` tends to `0` along `l`, modulo division by zero
issues that are avoided by this de... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_iff_forall_is_O_with : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → is_O_with c l f g | by rw is_o | lemma | asymptotics.is_o_iff_forall_is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | Definition of `is_o` in terms of `is_O_with`. We record it in a lemma as we will set
`is_o` to be irreducible at the end of this file. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_iff : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ | by simp only [is_o, is_O_with] | lemma | asymptotics.is_o_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | Definition of `is_o` in terms of filters. We record it in a lemma as we will set
`is_o` to be irreducible at the end of this file. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o.def (h : f =o[l] g) (hc : 0 < c) : ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ | is_o_iff.1 h hc | lemma | asymptotics.is_o.def | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.def' (h : f =o[l] g) (hc : 0 < c) : is_O_with c l f g | is_O_with_iff.2 $ is_o_iff.1 h hc | lemma | asymptotics.is_o.def' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.is_O (h : is_O_with c l f g) : f =O[l] g | by rw is_O; exact ⟨c, h⟩ | theorem | asymptotics.is_O_with.is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.is_O_with (hgf : f =o[l] g) : is_O_with 1 l f g | hgf.def' zero_lt_one | theorem | asymptotics.is_o.is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.is_O (hgf : f =o[l] g) : f =O[l] g | hgf.is_O_with.is_O | theorem | asymptotics.is_o.is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.is_O_with : f =O[l] g → ∃ c : ℝ, is_O_with c l f g | is_O_iff_is_O_with.1 | lemma | asymptotics.is_O.is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.weaken (h : is_O_with c l f g') (hc : c ≤ c') : is_O_with c' l f g' | is_O_with.of_bound $ mem_of_superset h.bound $ λ x hx,
calc ‖f x‖ ≤ c * ‖g' x‖ : hx
... ≤ _ : mul_le_mul_of_nonneg_right hc (norm_nonneg _) | theorem | asymptotics.is_O_with.weaken | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.exists_pos (h : is_O_with c l f g') :
∃ c' (H : 0 < c'), is_O_with c' l f g' | ⟨max c 1, lt_of_lt_of_le zero_lt_one (le_max_right c 1), h.weaken $ le_max_left c 1⟩ | theorem | asymptotics.is_O_with.exists_pos | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.exists_pos (h : f =O[l] g') : ∃ c (H : 0 < c), is_O_with c l f g' | let ⟨c, hc⟩ := h.is_O_with in hc.exists_pos | theorem | asymptotics.is_O.exists_pos | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.exists_nonneg (h : is_O_with c l f g') :
∃ c' (H : 0 ≤ c'), is_O_with c' l f g' | let ⟨c, cpos, hc⟩ := h.exists_pos in ⟨c, le_of_lt cpos, hc⟩ | theorem | asymptotics.is_O_with.exists_nonneg | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.exists_nonneg (h : f =O[l] g') :
∃ c (H : 0 ≤ c), is_O_with c l f g' | let ⟨c, hc⟩ := h.is_O_with in hc.exists_nonneg | theorem | asymptotics.is_O.exists_nonneg | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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