statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
pos_of_pos {x : ℝ} (hx : 0 < x) : 0 < exp_neg_inv_glue x | by simp [exp_neg_inv_glue, not_le.2 hx, exp_pos] | lemma | exp_neg_inv_glue.pos_of_pos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp_neg_inv_glue"
] | The function `exp_neg_inv_glue` is positive on `(0, +∞)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonneg (x : ℝ) : 0 ≤ exp_neg_inv_glue x | begin
cases le_or_gt x 0,
{ exact ge_of_eq (zero_of_nonpos h) },
{ exact le_of_lt (pos_of_pos h) }
end | lemma | exp_neg_inv_glue.nonneg | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp_neg_inv_glue",
"ge_of_eq"
] | The function exp_neg_inv_glue` is nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.smooth_transition (x : ℝ) : ℝ | exp_neg_inv_glue x / (exp_neg_inv_glue x + exp_neg_inv_glue (1 - x)) | def | real.smooth_transition | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp_neg_inv_glue"
] | An infinitely smooth function `f : ℝ → ℝ` such that `f x = 0` for `x ≤ 0`,
`f x = 1` for `1 ≤ x`, and `0 < f x < 1` for `0 < x < 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_denom (x) : 0 < exp_neg_inv_glue x + exp_neg_inv_glue (1 - x) | (zero_lt_one.lt_or_lt x).elim
(λ hx, add_pos_of_pos_of_nonneg (pos_of_pos hx) (nonneg _))
(λ hx, add_pos_of_nonneg_of_pos (nonneg _) (pos_of_pos $ sub_pos.2 hx)) | lemma | real.smooth_transition.pos_denom | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp_neg_inv_glue"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_of_one_le (h : 1 ≤ x) : smooth_transition x = 1 | (div_eq_one_iff_eq $ (pos_denom x).ne').2 $ by rw [zero_of_nonpos (sub_nonpos.2 h), add_zero] | lemma | real.smooth_transition.one_of_one_le | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_eq_one_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_nonpos (h : x ≤ 0) : smooth_transition x = 0 | by rw [smooth_transition, zero_of_nonpos h, zero_div] | lemma | real.smooth_transition.zero_of_nonpos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : smooth_transition 0 = 0 | zero_of_nonpos le_rfl | lemma | real.smooth_transition.zero | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one : smooth_transition 1 = 1 | one_of_one_le le_rfl | lemma | real.smooth_transition.one | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_Icc :
smooth_transition (proj_Icc (0 : ℝ) 1 zero_le_one x) = smooth_transition x | begin
refine congr_fun (Icc_extend_eq_self zero_le_one smooth_transition (λ x hx, _) (λ x hx, _)) x,
{ rw [smooth_transition.zero, zero_of_nonpos hx.le] },
{ rw [smooth_transition.one, one_of_one_le hx.le] }
end | lemma | real.smooth_transition.proj_Icc | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"zero_le_one"
] | Since `real.smooth_transition` is constant on $(-∞, 0]$ and $[1, ∞)$, applying it to the
projection of `x : ℝ` to $[0, 1]$ gives the same result as applying it to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_one (x : ℝ) : smooth_transition x ≤ 1 | (div_le_one (pos_denom x)).2 $ le_add_of_nonneg_right (nonneg _) | lemma | real.smooth_transition.le_one | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg (x : ℝ) : 0 ≤ smooth_transition x | div_nonneg (exp_neg_inv_glue.nonneg _) (pos_denom x).le | lemma | real.smooth_transition.nonneg | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_nonneg",
"exp_neg_inv_glue.nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_of_lt_one (h : x < 1) : smooth_transition x < 1 | (div_lt_one $ pos_denom x).2 $ lt_add_of_pos_right _ $ pos_of_pos $ sub_pos.2 h | lemma | real.smooth_transition.lt_one_of_lt_one | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_pos (h : 0 < x) : 0 < smooth_transition x | div_pos (exp_neg_inv_glue.pos_of_pos h) (pos_denom x) | lemma | real.smooth_transition.pos_of_pos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_pos",
"exp_neg_inv_glue.pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff {n} : cont_diff ℝ n smooth_transition | exp_neg_inv_glue.cont_diff.div
(exp_neg_inv_glue.cont_diff.add $ exp_neg_inv_glue.cont_diff.comp $
cont_diff_const.sub cont_diff_id) $
λ x, (pos_denom x).ne' | lemma | real.smooth_transition.cont_diff | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff",
"cont_diff_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at {x n} : cont_diff_at ℝ n smooth_transition x | smooth_transition.cont_diff.cont_diff_at | lemma | real.smooth_transition.cont_diff_at | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous smooth_transition | (@smooth_transition.cont_diff 0).continuous | lemma | real.smooth_transition.continuous | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at : continuous_at smooth_transition x | smooth_transition.continuous.continuous_at | lemma | real.smooth_transition.continuous_at | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_bump (c : E) | (r R : ℝ)
(r_pos : 0 < r)
(r_lt_R : r < R) | structure | cont_diff_bump | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | `f : cont_diff_bump c`, where `c` is a point in a normed vector space, is a
bundled smooth function such that
- `f` is equal to `1` in `metric.closed_ball c f.r`;
- `support f = metric.ball c f.R`;
- `0 ≤ f x ≤ 1` for all `x`.
The structure `cont_diff_bump` contains the data required to construct the function:
real n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_bump_base (E : Type*) [normed_add_comm_group E] [normed_space ℝ E] | (to_fun : ℝ → E → ℝ)
(mem_Icc : ∀ (R : ℝ) (x : E), to_fun R x ∈ Icc (0 : ℝ) 1)
(symmetric : ∀ (R : ℝ) (x : E), to_fun R (-x) = to_fun R x)
(smooth : cont_diff_on ℝ ⊤ (uncurry to_fun) ((Ioi (1 : ℝ)) ×ˢ (univ : set E)))
(eq_one : ∀ (R : ℝ) (hR : 1 < R) (x : E) (hx : ‖x‖ ≤ 1), to_fun R x = 1)
(support : ∀ (R : ℝ... | structure | cont_diff_bump_base | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_on",
"metric.ball",
"normed_add_comm_group",
"normed_space",
"smooth"
] | The base function from which one will construct a family of bump functions. One could
add more properties if they are useful and satisfied in the examples of inner product spaces
and finite dimensional vector spaces, notably derivative norm control in terms of `R - 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cont_diff_bump (E : Type*) [normed_add_comm_group E] [normed_space ℝ E] : Prop | (out : nonempty (cont_diff_bump_base E)) | class | has_cont_diff_bump | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump_base",
"normed_add_comm_group",
"normed_space"
] | A class registering that a real vector space admits bump functions. This will be instantiated
first for inner product spaces, and then for finite-dimensional normed spaces.
We use a specific class instead of `nonempty (cont_diff_bump_base E)` for performance reasons. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
some_cont_diff_bump_base (E : Type*) [normed_add_comm_group E] [normed_space ℝ E]
[hb : has_cont_diff_bump E] : cont_diff_bump_base E | nonempty.some hb.out | def | some_cont_diff_bump_base | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump_base",
"has_cont_diff_bump",
"nonempty.some",
"normed_add_comm_group",
"normed_space"
] | In a space with `C^∞` bump functions, register some function that will be used as a basis
to construct bump functions of arbitrary size around any point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cont_diff_bump_of_inner_product_space
(E : Type*) [normed_add_comm_group E] [inner_product_space ℝ E] : has_cont_diff_bump E | let e : cont_diff_bump_base E :=
{ to_fun := λ R x, real.smooth_transition ((R - ‖x‖) / (R - 1)),
mem_Icc := λ R x, ⟨real.smooth_transition.nonneg _, real.smooth_transition.le_one _⟩,
symmetric := λ R x, by simp only [norm_neg],
smooth := begin
rintros ⟨R, x⟩ ⟨(hR : 1 < R), hx⟩,
apply cont_diff_at.cont_di... | instance | has_cont_diff_bump_of_inner_product_space | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_at.cont_diff_within_at",
"cont_diff_at.div",
"cont_diff_at_const",
"cont_diff_bump_base",
"div_nonpos_of_nonpos_of_nonneg",
"div_pos",
"eq_or_ne",
"has_cont_diff_bump",
"inner_product_space",
"nhds_prod_eq",
"normed_add_comm_group",
"one_le_div",
"one_lt_div",
"real.smooth_trans... | Any inner product space has smooth bump functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
R_pos {c : E} (f : cont_diff_bump c) : 0 < f.R | f.r_pos.trans f.r_lt_R | lemma | cont_diff_bump.R_pos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_R_div_r {c : E} (f : cont_diff_bump c) : 1 < f.R / f.r | begin
rw one_lt_div f.r_pos,
exact f.r_lt_R
end | lemma | cont_diff_bump.one_lt_R_div_r | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump",
"one_lt_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun {c : E} (f : cont_diff_bump c) : E → ℝ | λ x, (some_cont_diff_bump_base E).to_fun (f.R / f.r) (f.r⁻¹ • (x - c)) | def | cont_diff_bump.to_fun | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump",
"some_cont_diff_bump_base"
] | The function defined by `f : cont_diff_bump c`. Use automatic coercion to
function instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
«def» (x : E) :
f x = (some_cont_diff_bump_base E).to_fun (f.R / f.r) (f.r⁻¹ • (x - c)) | rfl | lemma | cont_diff_bump.«def» | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"some_cont_diff_bump_base"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub (x : E) : f (c - x) = f (c + x) | by simp [f.def, cont_diff_bump_base.symmetric] | lemma | cont_diff_bump.sub | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (f : cont_diff_bump (0 : E)) (x : E) : f (- x) = f x | by simp_rw [← zero_sub, f.sub, zero_add] | lemma | cont_diff_bump.neg | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_of_mem_closed_ball (hx : x ∈ closed_ball c f.r) :
f x = 1 | begin
apply cont_diff_bump_base.eq_one _ _ f.one_lt_R_div_r,
simpa only [norm_smul, norm_eq_abs, abs_inv, abs_of_nonneg f.r_pos.le, ← div_eq_inv_mul,
div_le_one f.r_pos] using mem_closed_ball_iff_norm.1 hx
end | lemma | cont_diff_bump.one_of_mem_closed_ball | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"abs_inv",
"abs_of_nonneg",
"div_eq_inv_mul",
"div_le_one",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg : 0 ≤ f x | (cont_diff_bump_base.mem_Icc ((some_cont_diff_bump_base E)) _ _).1 | lemma | cont_diff_bump.nonneg | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"some_cont_diff_bump_base"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg' (x : E) : 0 ≤ f x | f.nonneg | lemma | cont_diff_bump.nonneg' | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | A version of `cont_diff_bump.nonneg` with `x` explicit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_one : f x ≤ 1 | (cont_diff_bump_base.mem_Icc ((some_cont_diff_bump_base E)) _ _).2 | lemma | cont_diff_bump.le_one | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"some_cont_diff_bump_base"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mem_ball (hx : x ∈ ball c f.R) : 0 < f x | begin
refine lt_iff_le_and_ne.2 ⟨f.nonneg, ne.symm _⟩,
change (f.r)⁻¹ • (x - c) ∈ support ((some_cont_diff_bump_base E).to_fun (f.R / f.r)),
rw cont_diff_bump_base.support _ _ f.one_lt_R_div_r,
simp only [dist_eq_norm, mem_ball] at hx,
simpa only [norm_smul, mem_ball_zero_iff, norm_eq_abs, abs_inv, abs_of_non... | lemma | cont_diff_bump.pos_of_mem_ball | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"abs_inv",
"abs_of_nonneg",
"div_eq_inv_mul",
"div_lt_div_right",
"norm_smul",
"some_cont_diff_bump_base"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_le_dist (hx : f.R ≤ dist x c) : f x = 0 | begin
rw dist_eq_norm at hx,
suffices H : (f.r)⁻¹ • (x - c) ∉ support ((some_cont_diff_bump_base E).to_fun (f.R / f.r)),
by simpa only [mem_support, not_not] using H,
rw cont_diff_bump_base.support _ _ f.one_lt_R_div_r,
simp [norm_smul, norm_eq_abs, abs_inv, abs_of_nonneg f.r_pos.le, ← div_eq_inv_mul],
ex... | lemma | cont_diff_bump.zero_of_le_dist | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"abs_inv",
"abs_of_nonneg",
"div_eq_inv_mul",
"div_le_div_of_le",
"norm_smul",
"not_not",
"some_cont_diff_bump_base"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_eq : support (f : E → ℝ) = metric.ball c f.R | begin
ext x,
suffices : f x ≠ 0 ↔ dist x c < f.R, by simpa [mem_support],
cases lt_or_le (dist x c) f.R with hx hx,
{ simp only [hx, (f.pos_of_mem_ball hx).ne', ne.def, not_false_iff]},
{ simp only [hx.not_lt, f.zero_of_le_dist hx, ne.def, eq_self_iff_true, not_true] }
end | lemma | cont_diff_bump.support_eq | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsupport_eq : tsupport f = closed_ball c f.R | by simp_rw [tsupport, f.support_eq, closure_ball _ f.R_pos.ne'] | lemma | cont_diff_bump.tsupport_eq | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"closure_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support [finite_dimensional ℝ E] : has_compact_support f | by simp_rw [has_compact_support, f.tsupport_eq, is_compact_closed_ball] | lemma | cont_diff_bump.has_compact_support | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_one_of_mem_ball (h : x ∈ ball c f.r) :
f =ᶠ[𝓝 x] 1 | ((is_open_lt (continuous_id.dist continuous_const) continuous_const).eventually_mem h).mono $
λ z hz, f.one_of_mem_closed_ball (le_of_lt hz) | lemma | cont_diff_bump.eventually_eq_one_of_mem_ball | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous_const",
"is_open_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_one : f =ᶠ[𝓝 c] 1 | f.eventually_eq_one_of_mem_ball (mem_ball_self f.r_pos) | lemma | cont_diff_bump.eventually_eq_one | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cont_diff_at.cont_diff_bump {c g : X → E}
{f : ∀ x, cont_diff_bump (c x)} {x : X}
(hc : cont_diff_at ℝ n c x) (hr : cont_diff_at ℝ n (λ x, (f x).r) x)
(hR : cont_diff_at ℝ n (λ x, (f x).R) x)
(hg : cont_diff_at ℝ n g x) : cont_diff_at ℝ n (λ x, f x (g x)) x | begin
rcases eq_or_ne (g x) (c x) with hx|hx,
{ have : (λ x, f x (g x)) =ᶠ[𝓝 x] (λ x, 1),
{ have : dist (g x) (c x) < (f x).r, { simp_rw [hx, dist_self, (f x).r_pos] },
have := continuous_at.eventually_lt (hg.continuous_at.dist hc.continuous_at) hr.continuous_at
this,
exact eventually_of_me... | lemma | cont_diff_at.cont_diff_bump | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_at",
"cont_diff_bump",
"continuous_at.eventually_lt",
"dist_self",
"eq_or_ne",
"is_open_univ",
"le_top",
"some_cont_diff_bump_base"
] | `cont_diff_bump` is `𝒞ⁿ` in all its arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.cont_diff.cont_diff_bump {c g : X → E} {f : ∀ x, cont_diff_bump (c x)}
(hc : cont_diff ℝ n c) (hr : cont_diff ℝ n (λ x, (f x).r)) (hR : cont_diff ℝ n (λ x, (f x).R))
(hg : cont_diff ℝ n g) : cont_diff ℝ n (λ x, f x (g x)) | by { rw [cont_diff_iff_cont_diff_at] at *, exact λ x, (hc x).cont_diff_bump (hr x) (hR x) (hg x) } | lemma | cont_diff.cont_diff_bump | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff",
"cont_diff_bump",
"cont_diff_iff_cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff : cont_diff ℝ n f | cont_diff_const.cont_diff_bump cont_diff_const cont_diff_const cont_diff_id | lemma | cont_diff_bump.cont_diff | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff",
"cont_diff_const",
"cont_diff_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at : cont_diff_at ℝ n f x | f.cont_diff.cont_diff_at | lemma | cont_diff_bump.cont_diff_at | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at {s : set E} : cont_diff_within_at ℝ n f s x | f.cont_diff_at.cont_diff_within_at | lemma | cont_diff_bump.cont_diff_within_at | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous f | cont_diff_zero.mp f.cont_diff | lemma | cont_diff_bump.continuous | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed (μ : measure E) : E → ℝ | λ x, f x / ∫ x, f x ∂μ | def | cont_diff_bump.normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | A bump function normed so that `∫ x, f.normed μ x ∂μ = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_def {μ : measure E} (x : E) : f.normed μ x = f x / ∫ x, f x ∂μ | rfl | lemma | cont_diff_bump.normed_def | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_normed (x : E) : 0 ≤ f.normed μ x | div_nonneg f.nonneg $ integral_nonneg f.nonneg' | lemma | cont_diff_bump.nonneg_normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_normed {n : ℕ∞} : cont_diff ℝ n (f.normed μ) | f.cont_diff.div_const _ | lemma | cont_diff_bump.cont_diff_normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_normed : continuous (f.normed μ) | f.continuous.div_const _ | lemma | cont_diff_bump.continuous_normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_sub (x : E) : f.normed μ (c - x) = f.normed μ (c + x) | by simp_rw [f.normed_def, f.sub] | lemma | cont_diff_bump.normed_sub | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_neg (f : cont_diff_bump (0 : E)) (x : E) : f.normed μ (- x) = f.normed μ x | by simp_rw [f.normed_def, f.neg] | lemma | cont_diff_bump.normed_neg | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable : integrable f μ | f.continuous.integrable_of_has_compact_support f.has_compact_support | lemma | cont_diff_bump.integrable | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_normed : integrable (f.normed μ) μ | f.integrable.div_const _ | lemma | cont_diff_bump.integrable_normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_pos : 0 < ∫ x, f x ∂μ | begin
refine (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr _,
rw [f.support_eq],
refine is_open_ball.measure_pos _ (nonempty_ball.mpr f.R_pos)
end | lemma | cont_diff_bump.integral_pos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_normed : ∫ x, f.normed μ x ∂μ = 1 | begin
simp_rw [cont_diff_bump.normed, div_eq_mul_inv, mul_comm (f _), ← smul_eq_mul,
integral_smul],
exact inv_mul_cancel (f.integral_pos.ne')
end | lemma | cont_diff_bump.integral_normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump.normed",
"div_eq_mul_inv",
"inv_mul_cancel",
"mul_comm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_normed_eq : support (f.normed μ) = metric.ball c f.R | by simp_rw [cont_diff_bump.normed, support_div, f.support_eq,
support_const f.integral_pos.ne', inter_univ] | lemma | cont_diff_bump.support_normed_eq | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump.normed",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsupport_normed_eq : tsupport (f.normed μ) = metric.closed_ball c f.R | by simp_rw [tsupport, f.support_normed_eq, closure_ball _ f.R_pos.ne'] | lemma | cont_diff_bump.tsupport_normed_eq | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"closure_ball",
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support_normed : has_compact_support (f.normed μ) | by simp_rw [has_compact_support, f.tsupport_normed_eq, is_compact_closed_ball] | lemma | cont_diff_bump.has_compact_support_normed | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_support_normed_small_sets {ι} {φ : ι → cont_diff_bump c} {l : filter ι}
(hφ : tendsto (λ i, (φ i).R) l (𝓝 0)) :
tendsto (λ i, support (λ x, (φ i).normed μ x)) l (𝓝 c).small_sets | begin
simp_rw [normed_add_comm_group.tendsto_nhds_zero, real.norm_eq_abs,
abs_eq_self.mpr (φ _).R_pos.le] at hφ,
rw [tendsto_small_sets_iff],
intros t ht,
rcases metric.mem_nhds_iff.mp ht with ⟨ε, hε, ht⟩,
refine (hφ ε hε).mono (λ i hi, subset_trans _ ht),
simp_rw [(φ i).support_normed_eq],
exact ball... | lemma | cont_diff_bump.tendsto_support_normed_small_sets | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff_bump",
"filter",
"real.norm_eq_abs",
"subset_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_normed_smul [complete_space X] (z : X) :
∫ x, f.normed μ x • z ∂μ = z | by simp_rw [integral_smul_const, f.integral_normed, one_smul] | lemma | cont_diff_bump.integral_normed_smul | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"complete_space",
"integral_smul_const",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_choose_succ_mul {R : Type*} [semiring R] (f : ℕ → ℕ → R) (n : ℕ) :
∑ i in range (n+2), ((n+1).choose i : R) * f i (n + 1 - i) =
∑ i in range (n+1), (n.choose i : R) * f i (n + 1 - i)
+ ∑ i in range (n+1), (n.choose i : R) * f (i + 1) (n - i) | begin
have A : ∑ i in range (n + 1), (n.choose (i+1) : R) * f (i + 1) (n - i) + f 0 (n + 1)
= ∑ i in range (n+1), n.choose i * f i (n + 1 - i),
{ rw [finset.sum_range_succ, finset.sum_range_succ'],
simp only [nat.choose_succ_self, algebra_map.coe_zero, zero_mul, add_zero,
nat.succ_sub_succ_eq_sub, nat... | lemma | finset.sum_choose_succ_mul | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"algebra_map.coe_one",
"algebra_map.coe_zero",
"finset.range",
"nat.cast_add",
"nat.choose_succ_self",
"nat.choose_succ_succ",
"nat.choose_zero_right",
"one_mul",
"semiring",
"tsub_zero",
"zero_mul"
] | The sum of `(n+1).choose i * f i (n+1-i)` can be split into two sums at rank `n`,
respectively of `n.choose i * f i (n+1-i)` and `n.choose i * f (i+1) (n-i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_antidiagonal_choose_succ_mul {R : Type*} [semiring R] (f : ℕ → ℕ → R) (n : ℕ) :
∑ ij in nat.antidiagonal (n + 1), ((n + 1).choose ij.1 : R) * f ij.1 ij.2 =
∑ ij in nat.antidiagonal n, (n.choose ij.1 : R) * f ij.1 (ij.2 + 1)
+ ∑ ij in nat.antidiagonal n, (n.choose ij.2 : R) * f (ij.1 + 1) ij.2 | begin
convert sum_choose_succ_mul f n using 1,
{ exact nat.sum_antidiagonal_eq_sum_range_succ (λ i j, ((n+1).choose i : R) * f i j) (n+1) },
congr' 1,
{ rw nat.sum_antidiagonal_eq_sum_range_succ (λ i j, (n.choose i : R) * f i (j + 1)) n,
apply finset.sum_congr rfl (λ i hi, _),
have : n + 1 - i = n - i +... | lemma | finset.sum_antidiagonal_choose_succ_mul | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"nat.choose_symm_of_eq_add",
"semiring"
] | The sum along the antidiagonal of `(n+1).choose i * f i j` can be split into two sums along the
antidiagonal at rank `n`, respectively of `n.choose i * f i (j+1)` and `n.choose j * f (i+1) j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_zero_fun {n : ℕ} :
iterated_fderiv 𝕜 n (λ x : E, (0 : F)) = 0 | begin
induction n with n IH,
{ ext m, simp },
{ ext x m,
rw [iterated_fderiv_succ_apply_left, IH],
change (fderiv 𝕜 (λ (x : E), (0 : (E [×n]→L[𝕜] F))) x : E → (E [×n]→L[𝕜] F)) (m 0) (tail m) = _,
rw fderiv_const,
refl }
end | lemma | iterated_fderiv_zero_fun | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"fderiv",
"fderiv_const",
"iterated_fderiv",
"iterated_fderiv_succ_apply_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_zero_fun :
cont_diff 𝕜 n (λ x : E, (0 : F)) | begin
apply cont_diff_of_differentiable_iterated_fderiv (λm hm, _),
rw iterated_fderiv_zero_fun,
exact differentiable_const (0 : (E [×m]→L[𝕜] F))
end | lemma | cont_diff_zero_fun | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_of_differentiable_iterated_fderiv",
"differentiable_const",
"iterated_fderiv_zero_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_const {c : F} : cont_diff 𝕜 n (λx : E, c) | begin
suffices h : cont_diff 𝕜 ∞ (λx : E, c), by exact h.of_le le_top,
rw cont_diff_top_iff_fderiv,
refine ⟨differentiable_const c, _⟩,
rw fderiv_const,
exact cont_diff_zero_fun
end | lemma | cont_diff_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_top_iff_fderiv",
"cont_diff_zero_fun",
"fderiv_const",
"le_top"
] | Constants are `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_const {c : F} {s : set E} :
cont_diff_on 𝕜 n (λx : E, c) s | cont_diff_const.cont_diff_on | lemma | cont_diff_on_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_const {c : F} :
cont_diff_at 𝕜 n (λx : E, c) x | cont_diff_const.cont_diff_at | lemma | cont_diff_at_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_const {c : F} :
cont_diff_within_at 𝕜 n (λx : E, c) s x | cont_diff_at_const.cont_diff_within_at | lemma | cont_diff_within_at_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_of_subsingleton [subsingleton F] :
cont_diff 𝕜 n f | by { rw [subsingleton.elim f (λ _, 0)], exact cont_diff_const } | lemma | cont_diff_of_subsingleton | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_of_subsingleton [subsingleton F] :
cont_diff_at 𝕜 n f x | by { rw [subsingleton.elim f (λ _, 0)], exact cont_diff_at_const } | lemma | cont_diff_at_of_subsingleton | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_of_subsingleton [subsingleton F] :
cont_diff_within_at 𝕜 n f s x | by { rw [subsingleton.elim f (λ _, 0)], exact cont_diff_within_at_const } | lemma | cont_diff_within_at_of_subsingleton | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"cont_diff_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_of_subsingleton [subsingleton F] :
cont_diff_on 𝕜 n f s | by { rw [subsingleton.elim f (λ _, 0)], exact cont_diff_on_const } | lemma | cont_diff_on_of_subsingleton | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"cont_diff_on_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_succ_const (n : ℕ) (c : F) : iterated_fderiv 𝕜 (n + 1) (λ (y : E), c) = 0 | begin
ext x m,
simp only [iterated_fderiv_succ_apply_right, fderiv_const, pi.zero_apply,
iterated_fderiv_zero_fun, continuous_multilinear_map.zero_apply,
continuous_linear_map.zero_apply],
end | lemma | iterated_fderiv_succ_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous_linear_map.zero_apply",
"continuous_multilinear_map.zero_apply",
"fderiv_const",
"iterated_fderiv",
"iterated_fderiv_succ_apply_right",
"iterated_fderiv_zero_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_const_of_ne {n : ℕ} (hn : n ≠ 0) (c : F) :
iterated_fderiv 𝕜 n (λ (y : E), c) = 0 | begin
cases nat.exists_eq_succ_of_ne_zero hn with k hk,
rw [hk, iterated_fderiv_succ_const],
end | lemma | iterated_fderiv_const_of_ne | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"iterated_fderiv",
"iterated_fderiv_succ_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.cont_diff (hf : is_bounded_linear_map 𝕜 f) :
cont_diff 𝕜 n f | begin
suffices h : cont_diff 𝕜 ∞ f, by exact h.of_le le_top,
rw cont_diff_top_iff_fderiv,
refine ⟨hf.differentiable, _⟩,
simp_rw [hf.fderiv],
exact cont_diff_const
end | lemma | is_bounded_linear_map.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_const",
"cont_diff_top_iff_fderiv",
"is_bounded_linear_map",
"le_top"
] | Unbundled bounded linear functions are `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.cont_diff (f : E →L[𝕜] F) : cont_diff 𝕜 n f | f.is_bounded_linear_map.cont_diff | lemma | continuous_linear_map.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.cont_diff (f : E ≃L[𝕜] F) : cont_diff 𝕜 n f | (f : E →L[𝕜] F).cont_diff | lemma | continuous_linear_equiv.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.cont_diff (f : E →ₗᵢ[𝕜] F) : cont_diff 𝕜 n f | f.to_continuous_linear_map.cont_diff | lemma | linear_isometry.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_equiv.cont_diff (f : E ≃ₗᵢ[𝕜] F) : cont_diff 𝕜 n f | (f : E →L[𝕜] F).cont_diff | lemma | linear_isometry_equiv.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_id : cont_diff 𝕜 n (id : E → E) | is_bounded_linear_map.id.cont_diff | lemma | cont_diff_id | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | The identity is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_id {s x} : cont_diff_within_at 𝕜 n (id : E → E) s x | cont_diff_id.cont_diff_within_at | lemma | cont_diff_within_at_id | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_id {x} : cont_diff_at 𝕜 n (id : E → E) x | cont_diff_id.cont_diff_at | lemma | cont_diff_at_id | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_id {s} : cont_diff_on 𝕜 n (id : E → E) s | cont_diff_id.cont_diff_on | lemma | cont_diff_on_id | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.cont_diff (hb : is_bounded_bilinear_map 𝕜 b) :
cont_diff 𝕜 n b | begin
suffices h : cont_diff 𝕜 ∞ b, by exact h.of_le le_top,
rw cont_diff_top_iff_fderiv,
refine ⟨hb.differentiable, _⟩,
simp [hb.fderiv],
exact hb.is_bounded_linear_map_deriv.cont_diff
end | lemma | is_bounded_bilinear_map.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_top_iff_fderiv",
"is_bounded_bilinear_map",
"le_top"
] | Bilinear functions are `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.continuous_linear_map_comp (g : F →L[𝕜] G)
(hf : has_ftaylor_series_up_to_on n f p s) :
has_ftaylor_series_up_to_on n (g ∘ f) (λ x k, g.comp_continuous_multilinear_map (p x k)) s | begin
set L : Π m : ℕ, (E [×m]→L[𝕜] F) →L[𝕜] (E [×m]→L[𝕜] G) :=
λ m, continuous_linear_map.comp_continuous_multilinear_mapL 𝕜 (λ _, E) F G g,
split,
{ exact λ x hx, congr_arg g (hf.zero_eq x hx) },
{ intros m hm x hx,
convert (L m).has_fderiv_at.comp_has_fderiv_within_at x (hf.fderiv_within m hm x h... | lemma | has_ftaylor_series_up_to_on.continuous_linear_map_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous.comp_continuous_on",
"continuous_linear_map.comp_continuous_multilinear_mapL",
"has_fderiv_at.comp_has_fderiv_within_at",
"has_ftaylor_series_up_to_on"
] | If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `g ∘ f` admits a Taylor
series whose `k`-th term is given by `g ∘ (p k)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.continuous_linear_map_comp (g : F →L[𝕜] G)
(hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (g ∘ f) s x | begin
assume m hm,
rcases hf m hm with ⟨u, hu, p, hp⟩,
exact ⟨u, hu, _, hp.continuous_linear_map_comp g⟩,
end | lemma | cont_diff_within_at.continuous_linear_map_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | Composition by continuous linear maps on the left preserves `C^n` functions in a domain
at a point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.continuous_linear_map_comp (g : F →L[𝕜] G) (hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (g ∘ f) x | cont_diff_within_at.continuous_linear_map_comp g hf | lemma | cont_diff_at.continuous_linear_map_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at.continuous_linear_map_comp"
] | Composition by continuous linear maps on the left preserves `C^n` functions in a domain
at a point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.continuous_linear_map_comp (g : F →L[𝕜] G) (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (g ∘ f) s | λ x hx, (hf x hx).continuous_linear_map_comp g | lemma | cont_diff_on.continuous_linear_map_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | Composition by continuous linear maps on the left preserves `C^n` functions on domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.continuous_linear_map_comp {f : E → F} (g : F →L[𝕜] G)
(hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λx, g (f x)) | cont_diff_on_univ.1 $ cont_diff_on.continuous_linear_map_comp
_ (cont_diff_on_univ.2 hf) | lemma | cont_diff.continuous_linear_map_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on.continuous_linear_map_comp"
] | Composition by continuous linear maps on the left preserves `C^n` functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.iterated_fderiv_within_comp_left
{f : E → F} (g : F →L[𝕜] G) (hf : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s)
{i : ℕ} (hi : (i : ℕ∞) ≤ n) :
iterated_fderiv_within 𝕜 i (g ∘ f) s x =
g.comp_continuous_multilinear_map (iterated_fderiv_within 𝕜 i f s x) | (((hf.ftaylor_series_within hs).continuous_linear_map_comp g).eq_ftaylor_series_of_unique_diff_on
hi hs hx).symm | lemma | continuous_linear_map.iterated_fderiv_within_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"iterated_fderiv_within",
"unique_diff_on"
] | The iterated derivative within a set of the composition with a linear map on the left is
obtained by applying the linear map to the iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.iterated_fderiv_comp_left
{f : E → F} (g : F →L[𝕜] G) (hf : cont_diff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
iterated_fderiv 𝕜 i (g ∘ f) x = g.comp_continuous_multilinear_map (iterated_fderiv 𝕜 i f x) | begin
simp only [← iterated_fderiv_within_univ],
exact g.iterated_fderiv_within_comp_left hf.cont_diff_on unique_diff_on_univ (mem_univ x) hi,
end | lemma | continuous_linear_map.iterated_fderiv_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | The iterated derivative of the composition with a linear map on the left is
obtained by applying the linear map to the iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.iterated_fderiv_within_comp_left
(g : F ≃L[𝕜] G) (f : E → F) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s) (i : ℕ) :
iterated_fderiv_within 𝕜 i (g ∘ f) s x =
(g : F →L[𝕜] G).comp_continuous_multilinear_map (iterated_fderiv_within 𝕜 i f s x) | begin
induction i with i IH generalizing x,
{ ext1 m,
simp only [iterated_fderiv_within_zero_apply, continuous_linear_equiv.coe_coe,
continuous_linear_map.comp_continuous_multilinear_map_coe, embedding_like.apply_eq_iff_eq] },
{ ext1 m,
rw iterated_fderiv_within_succ_apply_left,
have Z : fderiv_... | lemma | continuous_linear_equiv.iterated_fderiv_within_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.comp_continuous_multilinear_mapL_apply",
"continuous_linear_map.coe_comp'",
"continuous_linear_map.comp_continuous_multilinear_map_coe",
"embedding_like.apply_eq_iff_eq",
"fderiv_within",
"fderiv_within_congr'",
"iterated_fderiv_within",
"i... | The iterated derivative within a set of the composition with a linear equiv on the left is
obtained by applying the linear equiv to the iterated derivative. This is true without
differentiability assumptions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.norm_iterated_fderiv_within_comp_left
{f : E → F} (g : F →ₗᵢ[𝕜] G) (hf : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s)
{i : ℕ} (hi : (i : ℕ∞) ≤ n) :
‖iterated_fderiv_within 𝕜 i (g ∘ f) s x‖ = ‖iterated_fderiv_within 𝕜 i f s x‖ | begin
have : iterated_fderiv_within 𝕜 i (g ∘ f) s x =
g.to_continuous_linear_map.comp_continuous_multilinear_map (iterated_fderiv_within 𝕜 i f s x),
from g.to_continuous_linear_map.iterated_fderiv_within_comp_left hf hs hx hi,
rw this,
apply linear_isometry.norm_comp_continuous_multilinear_map
end | lemma | linear_isometry.norm_iterated_fderiv_within_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"iterated_fderiv_within",
"linear_isometry.norm_comp_continuous_multilinear_map",
"unique_diff_on"
] | Composition with a linear isometry on the left preserves the norm of the iterated
derivative within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.norm_iterated_fderiv_comp_left
{f : E → F} (g : F →ₗᵢ[𝕜] G) (hf : cont_diff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
‖iterated_fderiv 𝕜 i (g ∘ f) x‖ = ‖iterated_fderiv 𝕜 i f x‖ | begin
simp only [← iterated_fderiv_within_univ],
exact g.norm_iterated_fderiv_within_comp_left hf.cont_diff_on unique_diff_on_univ (mem_univ x) hi
end | lemma | linear_isometry.norm_iterated_fderiv_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | Composition with a linear isometry on the left preserves the norm of the iterated
derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.norm_iterated_fderiv_within_comp_left
(g : F ≃ₗᵢ[𝕜] G) (f : E → F) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s) (i : ℕ) :
‖iterated_fderiv_within 𝕜 i (g ∘ f) s x‖ = ‖iterated_fderiv_within 𝕜 i f s x‖ | begin
have : iterated_fderiv_within 𝕜 i (g ∘ f) s x =
(g : F →L[𝕜] G).comp_continuous_multilinear_map (iterated_fderiv_within 𝕜 i f s x),
from g.to_continuous_linear_equiv.iterated_fderiv_within_comp_left f hs hx i,
rw this,
apply linear_isometry.norm_comp_continuous_multilinear_map g.to_linear_isometr... | lemma | linear_isometry_equiv.norm_iterated_fderiv_within_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"iterated_fderiv_within",
"linear_isometry.norm_comp_continuous_multilinear_map",
"unique_diff_on"
] | Composition with a linear isometry equiv on the left preserves the norm of the iterated
derivative within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.norm_iterated_fderiv_comp_left
(g : F ≃ₗᵢ[𝕜] G) (f : E → F) (x : E) (i : ℕ) :
‖iterated_fderiv 𝕜 i (g ∘ f) x‖ = ‖iterated_fderiv 𝕜 i f x‖ | begin
rw [← iterated_fderiv_within_univ, ← iterated_fderiv_within_univ],
apply g.norm_iterated_fderiv_within_comp_left f unique_diff_on_univ (mem_univ x) i
end | lemma | linear_isometry_equiv.norm_iterated_fderiv_comp_left | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | Composition with a linear isometry equiv on the left preserves the norm of the iterated
derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.comp_cont_diff_within_at_iff
(e : F ≃L[𝕜] G) :
cont_diff_within_at 𝕜 n (e ∘ f) s x ↔ cont_diff_within_at 𝕜 n f s x | ⟨λ H, by simpa only [(∘), e.symm.coe_coe, e.symm_apply_apply]
using H.continuous_linear_map_comp (e.symm : G →L[𝕜] F),
λ H, H.continuous_linear_map_comp (e : F →L[𝕜] G)⟩ | lemma | continuous_linear_equiv.comp_cont_diff_within_at_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | Composition by continuous linear equivs on the left respects higher differentiability at a
point in a domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.comp_cont_diff_at_iff (e : F ≃L[𝕜] G) :
cont_diff_at 𝕜 n (e ∘ f) x ↔ cont_diff_at 𝕜 n f x | by simp only [← cont_diff_within_at_univ, e.comp_cont_diff_within_at_iff] | lemma | continuous_linear_equiv.comp_cont_diff_at_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at_univ"
] | Composition by continuous linear equivs on the left respects higher differentiability at a
point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.comp_cont_diff_on_iff
(e : F ≃L[𝕜] G) :
cont_diff_on 𝕜 n (e ∘ f) s ↔ cont_diff_on 𝕜 n f s | by simp [cont_diff_on, e.comp_cont_diff_within_at_iff] | lemma | continuous_linear_equiv.comp_cont_diff_on_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | Composition by continuous linear equivs on the left respects higher differentiability on
domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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