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
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|---|---|---|---|---|---|---|---|---|---|---|
has_compact_support.cont_diff_convolution_left [μ.is_add_left_invariant] [μ.is_neg_invariant]
{n : ℕ∞} (hcf : has_compact_support f) (hf : cont_diff 𝕜 n f) (hg : locally_integrable g μ) :
cont_diff 𝕜 n (f ⋆[L, μ] g) | by { rw [← convolution_flip], exact hcf.cont_diff_convolution_right L.flip hg hf } | lemma | has_compact_support.cont_diff_convolution_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff",
"convolution_flip"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_convolution
(f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F) (ν : measure ℝ . volume_tac) : ℝ → F | indicator (Ioi (0:ℝ)) (λ x, ∫ t in 0..x, L (f t) (g (x - t)) ∂ν) | def | pos_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | The forward convolution of two functions `f` and `g` on `ℝ`, with respect to a continuous
bilinear map `L` and measure `ν`. It is defined to be the function mapping `x` to
`∫ t in 0..x, L (f t) (g (x - t)) ∂ν` if `0 < x`, and 0 otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_convolution_eq_convolution_indicator
(f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F) (ν : measure ℝ . volume_tac) [has_no_atoms ν] :
pos_convolution f g L ν = convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L ν | begin
ext1 x,
rw [convolution, pos_convolution, indicator],
split_ifs,
{ rw [interval_integral.integral_of_le (le_of_lt h),
integral_Ioc_eq_integral_Ioo,
←integral_indicator (measurable_set_Ioo : measurable_set (Ioo 0 x))],
congr' 1 with t : 1,
have : (t ≤ 0) ∨ (t ∈ Ioo 0 x) ∨ (x ≤ t),
{... | lemma | pos_convolution_eq_convolution_indicator | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous_linear_map.map_zero",
"continuous_linear_map.zero_apply",
"convolution",
"interval_integral.integral_of_le",
"measurable_set",
"measurable_set_Ioo",
"pos_convolution"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_pos_convolution {f : ℝ → E} {g : ℝ → E'} {μ ν : measure ℝ}
[sigma_finite μ] [sigma_finite ν] [is_add_right_invariant μ] [has_no_atoms ν]
(hf : integrable_on f (Ioi 0) ν) (hg : integrable_on g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
integrable (pos_convolution f g L ν) μ | begin
rw ←integrable_indicator_iff (measurable_set_Ioi : measurable_set (Ioi (0:ℝ))) at hf hg,
rw pos_convolution_eq_convolution_indicator f g L ν,
exact (hf.convolution_integrand L hg).integral_prod_left,
end | lemma | integrable_pos_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"measurable_set",
"measurable_set_Ioi",
"pos_convolution",
"pos_convolution_eq_convolution_indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_pos_convolution [complete_space E] [complete_space E'] {μ ν : measure ℝ}
[sigma_finite μ] [sigma_finite ν] [is_add_right_invariant μ] [has_no_atoms ν]
{f : ℝ → E} {g : ℝ → E'} (hf : integrable_on f (Ioi 0) ν)
(hg : integrable_on g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
∫ x:ℝ in Ioi 0, (∫ t:ℝ in 0..x, L... | begin
rw ←integrable_indicator_iff (measurable_set_Ioi : measurable_set (Ioi (0:ℝ))) at hf hg,
simp_rw ←integral_indicator measurable_set_Ioi,
convert integral_convolution L hf hg using 2,
apply pos_convolution_eq_convolution_indicator,
end | lemma | integral_pos_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"complete_space",
"integral_convolution",
"measurable_set",
"measurable_set_Ioi",
"pos_convolution_eq_convolution_indicator"
] | The integral over `Ioi 0` of a forward convolution of two functions is equal to the product
of their integrals over this set. (Compare `integral_convolution` for the two-sided convolution.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_div_pow_pos {α : Type*} [linear_ordered_semifield α] {a b : α} (ha : 0 < a)
(hb : 0 < b) (k : ℕ) : 0 < a/b^k | div_pos ha (pow_pos hb k) | lemma | pos_div_pow_pos | analysis | src/analysis/hofer.lean | [
"analysis.specific_limits.basic"
] | [
"div_pos",
"linear_ordered_semifield",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hofer {X: Type*} [metric_space X] [complete_space X]
(x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) :
∃ (ε' > 0) (x' : X), ε' ≤ ε ∧
d x' x ≤ 2*ε ∧
ε * ϕ(x) ≤ ε' * ϕ x' ∧
∀ y, d x' y ≤ ε' → ϕ y ≤ 2*ϕ x' | begin
by_contradiction H,
have reformulation : ∀ x' (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2^k * ϕ x ≤ ϕ x',
{ intros x' k,
rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm],
positivity },
-- Now let's specialize to `ε/2^k`
replace H : ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2^k *... | lemma | hofer | analysis | src/analysis/hofer.lean | [
"analysis.specific_limits.basic"
] | [
"by_contradiction",
"cauchy_seq",
"cauchy_seq_of_le_geometric",
"complete_space",
"cont",
"continuous",
"dist_comm",
"dist_le_range_sum_dist",
"div_mul_eq_mul_div",
"geom_le",
"inv_pow",
"le_div_iff",
"lim",
"metric_space",
"mul_assoc",
"mul_comm",
"mul_le_mul_left",
"mul_le_mul_of... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_add_comm_group : seminormed_add_comm_group (matrix m n α) | pi.seminormed_add_comm_group | def | matrix.seminormed_add_comm_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"seminormed_add_comm_group"
] | Seminormed group instance (using sup norm of sup norm) for matrices over a seminormed group. Not
declared as an instance because there are several natural choices for defining the norm of a
matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le_iff {r : ℝ} (hr : 0 ≤ r) {A : matrix m n α} :
‖A‖ ≤ r ↔ ∀ i j, ‖A i j‖ ≤ r | by simp [pi_norm_le_iff_of_nonneg hr] | lemma | matrix.norm_le_iff | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_le_iff {r : ℝ≥0} {A : matrix m n α} :
‖A‖₊ ≤ r ↔ ∀ i j, ‖A i j‖₊ ≤ r | by simp [pi_nnnorm_le_iff] | lemma | matrix.nnnorm_le_iff | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_lt_iff {r : ℝ} (hr : 0 < r) {A : matrix m n α} :
‖A‖ < r ↔ ∀ i j, ‖A i j‖ < r | by simp [pi_norm_lt_iff hr] | lemma | matrix.norm_lt_iff | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_lt_iff {r : ℝ≥0} (hr : 0 < r) {A : matrix m n α} :
‖A‖₊ < r ↔ ∀ i j, ‖A i j‖₊ < r | by simp [pi_nnnorm_lt_iff hr] | lemma | matrix.nnnorm_lt_iff | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_entry_le_entrywise_sup_norm (A : matrix m n α) {i : m} {j : n} :
‖A i j‖ ≤ ‖A‖ | (norm_le_pi_norm (A i) j).trans (norm_le_pi_norm A i) | lemma | matrix.norm_entry_le_entrywise_sup_norm | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_entry_le_entrywise_sup_nnnorm (A : matrix m n α) {i : m} {j : n} :
‖A i j‖₊ ≤ ‖A‖₊ | (nnnorm_le_pi_nnnorm (A i) j).trans (nnnorm_le_pi_nnnorm A i) | lemma | matrix.nnnorm_entry_le_entrywise_sup_nnnorm | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_map_eq (A : matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
‖A.map f‖₊ = ‖A‖₊ | by simp_rw [pi.nnnorm_def, matrix.map_apply, hf] | lemma | matrix.nnnorm_map_eq | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_map_eq (A : matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖ = ‖a‖) :
‖A.map f‖ = ‖A‖ | (congr_arg (coe : ℝ≥0 → ℝ) $ nnnorm_map_eq A f $ λ a, subtype.ext $ hf a : _) | lemma | matrix.norm_map_eq | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_transpose (A : matrix m n α) : ‖Aᵀ‖₊ = ‖A‖₊ | by { simp_rw [pi.nnnorm_def], exact finset.sup_comm _ _ _ } | lemma | matrix.nnnorm_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"finset.sup_comm",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_transpose (A : matrix m n α) : ‖Aᵀ‖ = ‖A‖ | congr_arg coe $ nnnorm_transpose A | lemma | matrix.norm_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_conj_transpose [star_add_monoid α] [normed_star_group α] (A : matrix m n α) :
‖Aᴴ‖₊ = ‖A‖₊ | (nnnorm_map_eq _ _ nnnorm_star).trans A.nnnorm_transpose | lemma | matrix.nnnorm_conj_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"nnnorm_star",
"normed_star_group",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_conj_transpose [star_add_monoid α] [normed_star_group α] (A : matrix m n α) :
‖Aᴴ‖ = ‖A‖ | congr_arg coe $ nnnorm_conj_transpose A | lemma | matrix.norm_conj_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_star_group",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_col (v : m → α) : ‖col v‖₊ = ‖v‖₊ | by simp [pi.nnnorm_def] | lemma | matrix.nnnorm_col | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_col (v : m → α) : ‖col v‖ = ‖v‖ | congr_arg coe $ nnnorm_col v | lemma | matrix.norm_col | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_row (v : n → α) : ‖row v‖₊ = ‖v‖₊ | by simp [pi.nnnorm_def] | lemma | matrix.nnnorm_row | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_row (v : n → α) : ‖row v‖ = ‖v‖ | congr_arg coe $ nnnorm_row v | lemma | matrix.norm_row | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_diagonal [decidable_eq n] (v : n → α) : ‖diagonal v‖₊ = ‖v‖₊ | begin
simp_rw pi.nnnorm_def,
congr' 1 with i : 1,
refine le_antisymm (finset.sup_le $ λ j hj, _) _,
{ obtain rfl | hij := eq_or_ne i j,
{ rw diagonal_apply_eq },
{ rw [diagonal_apply_ne _ hij, nnnorm_zero],
exact zero_le _ }, },
{ refine eq.trans_le _ (finset.le_sup (finset.mem_univ i)),
rw ... | lemma | matrix.nnnorm_diagonal | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"eq_or_ne",
"finset.le_sup",
"finset.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_diagonal [decidable_eq n] (v : n → α) : ‖diagonal v‖ = ‖v‖ | congr_arg coe $ nnnorm_diagonal v | lemma | matrix.norm_diagonal | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group [normed_add_comm_group α] :
normed_add_comm_group (matrix m n α) | pi.normed_add_comm_group | def | matrix.normed_add_comm_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_add_comm_group"
] | Normed group instance (using sup norm of sup norm) for matrices over a normed group. Not
declared as an instance because there are several natural choices for defining the norm of a
matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space : normed_space R (matrix m n α) | pi.normed_space | def | matrix.normed_space | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_space",
"pi.normed_space"
] | Normed space instance (using sup norm of sup norm) for matrices over a normed space. Not
declared as an instance because there are several natural choices for defining the norm of a
matrix. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linfty_op_seminormed_add_comm_group [seminormed_add_comm_group α] :
seminormed_add_comm_group (matrix m n α) | (by apply_instance : seminormed_add_comm_group (m → pi_Lp 1 (λ j : n, α))) | def | matrix.linfty_op_seminormed_add_comm_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_normed_add_comm_group [normed_add_comm_group α] :
normed_add_comm_group (matrix m n α) | (by apply_instance : normed_add_comm_group (m → pi_Lp 1 (λ j : n, α))) | def | matrix.linfty_op_normed_add_comm_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_add_comm_group",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_normed_space [normed_field R] [seminormed_add_comm_group α]
[normed_space R α] :
normed_space R (matrix m n α) | (by apply_instance : normed_space R (m → pi_Lp 1 (λ j : n, α))) | def | matrix.linfty_op_normed_space | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_field",
"normed_space",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_def (A : matrix m n α) :
‖A‖ = ((finset.univ : finset m).sup (λ i : m, ∑ j : n, ‖A i j‖₊) : ℝ≥0) | by simp [pi.norm_def, pi_Lp.nnnorm_eq_sum ennreal.one_ne_top] | lemma | matrix.linfty_op_norm_def | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"ennreal.one_ne_top",
"finset",
"finset.univ",
"matrix",
"pi_Lp.nnnorm_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_nnnorm_def (A : matrix m n α) :
‖A‖₊ = (finset.univ : finset m).sup (λ i : m, ∑ j : n, ‖A i j‖₊) | subtype.ext $ linfty_op_norm_def A | lemma | matrix.linfty_op_nnnorm_def | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"finset",
"finset.univ",
"matrix",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_nnnorm_col (v : m → α) :
‖col v‖₊ = ‖v‖₊ | begin
rw [linfty_op_nnnorm_def, pi.nnnorm_def],
simp,
end | lemma | matrix.linfty_op_nnnorm_col | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_col (v : m → α) :
‖col v‖ = ‖v‖ | congr_arg coe $ linfty_op_nnnorm_col v | lemma | matrix.linfty_op_norm_col | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_nnnorm_row (v : n → α) :
‖row v‖₊ = ∑ i, ‖v i‖₊ | by simp [linfty_op_nnnorm_def] | lemma | matrix.linfty_op_nnnorm_row | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_row (v : n → α) :
‖row v‖ = ∑ i, ‖v i‖ | (congr_arg coe $ linfty_op_nnnorm_row v).trans $ by simp [nnreal.coe_sum] | lemma | matrix.linfty_op_norm_row | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"nnreal.coe_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_nnnorm_diagonal [decidable_eq m] (v : m → α) :
‖diagonal v‖₊ = ‖v‖₊ | begin
rw [linfty_op_nnnorm_def, pi.nnnorm_def],
congr' 1 with i : 1,
refine (finset.sum_eq_single_of_mem _ (finset.mem_univ i) $ λ j hj hij, _).trans _,
{ rw [diagonal_apply_ne' _ hij, nnnorm_zero] },
{ rw [diagonal_apply_eq] },
end | lemma | matrix.linfty_op_nnnorm_diagonal | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"finset.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_diagonal [decidable_eq m] (v : m → α) :
‖diagonal v‖ = ‖v‖ | congr_arg coe $ linfty_op_nnnorm_diagonal v | lemma | matrix.linfty_op_norm_diagonal | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_nnnorm_mul (A : matrix l m α) (B : matrix m n α) : ‖A ⬝ B‖₊ ≤ ‖A‖₊ * ‖B‖₊ | begin
simp_rw [linfty_op_nnnorm_def, matrix.mul_apply],
calc finset.univ.sup (λ i, ∑ k, ‖∑ j, A i j * B j k‖₊)
≤ finset.univ.sup (λ i, ∑ k j, ‖A i j‖₊ * ‖B j k‖₊) :
finset.sup_mono_fun $ λ i hi, finset.sum_le_sum $ λ k hk, nnnorm_sum_le_of_le _ $ λ j hj,
nnnorm_mul_le _ _
... = finset.univ.sup (λ ... | lemma | matrix.linfty_op_nnnorm_mul | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"finset.le_sup",
"finset.mul_sum",
"finset.sup_mono_fun",
"matrix",
"matrix.mul_apply",
"mul_le_mul_of_nonneg_left",
"nnnorm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_mul (A : matrix l m α) (B : matrix m n α) : ‖A ⬝ B‖ ≤ ‖A‖ * ‖B‖ | linfty_op_nnnorm_mul _ _ | lemma | matrix.linfty_op_norm_mul | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_nnnorm_mul_vec (A : matrix l m α) (v : m → α) : ‖A.mul_vec v‖₊ ≤ ‖A‖₊ * ‖v‖₊ | begin
rw [←linfty_op_nnnorm_col (A.mul_vec v), ←linfty_op_nnnorm_col v],
exact linfty_op_nnnorm_mul A (col v),
end | lemma | matrix.linfty_op_nnnorm_mul_vec | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_mul_vec (A : matrix l m α) (v : m → α) : ‖matrix.mul_vec A v‖ ≤ ‖A‖ * ‖v‖ | linfty_op_nnnorm_mul_vec _ _ | lemma | matrix.linfty_op_norm_mul_vec | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_non_unital_semi_normed_ring [non_unital_semi_normed_ring α] :
non_unital_semi_normed_ring (matrix n n α) | { norm_mul := linfty_op_norm_mul,
.. matrix.linfty_op_seminormed_add_comm_group,
.. matrix.non_unital_ring } | def | matrix.linfty_op_non_unital_semi_normed_ring | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.linfty_op_seminormed_add_comm_group",
"non_unital_semi_normed_ring",
"norm_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_norm_one_class [semi_normed_ring α] [norm_one_class α] [decidable_eq n]
[nonempty n] : norm_one_class (matrix n n α) | { norm_one := (linfty_op_norm_diagonal _).trans norm_one } | instance | matrix.linfty_op_norm_one_class | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"norm_one_class",
"semi_normed_ring"
] | The `L₁-L∞` norm preserves one on non-empty matrices. Note this is safe as an instance, as it
carries no data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linfty_op_semi_normed_ring [semi_normed_ring α] [decidable_eq n] :
semi_normed_ring (matrix n n α) | { .. matrix.linfty_op_non_unital_semi_normed_ring,
.. matrix.ring } | def | matrix.linfty_op_semi_normed_ring | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.linfty_op_non_unital_semi_normed_ring",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_non_unital_normed_ring [non_unital_normed_ring α] :
non_unital_normed_ring (matrix n n α) | { ..matrix.linfty_op_non_unital_semi_normed_ring } | def | matrix.linfty_op_non_unital_normed_ring | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.linfty_op_non_unital_semi_normed_ring",
"non_unital_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_normed_ring [normed_ring α] [decidable_eq n] :
normed_ring (matrix n n α) | { ..matrix.linfty_op_semi_normed_ring } | def | matrix.linfty_op_normed_ring | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.linfty_op_semi_normed_ring",
"normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linfty_op_normed_algebra [normed_field R] [semi_normed_ring α] [normed_algebra R α]
[decidable_eq n] :
normed_algebra R (matrix n n α) | { ..matrix.linfty_op_normed_space } | def | matrix.linfty_op_normed_algebra | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.linfty_op_normed_space",
"normed_algebra",
"normed_field",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_seminormed_add_comm_group [seminormed_add_comm_group α] :
seminormed_add_comm_group (matrix m n α) | (by apply_instance : seminormed_add_comm_group (pi_Lp 2 (λ i : m, pi_Lp 2 (λ j : n, α)))) | def | matrix.frobenius_seminormed_add_comm_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_normed_add_comm_group [normed_add_comm_group α] :
normed_add_comm_group (matrix m n α) | (by apply_instance : normed_add_comm_group (pi_Lp 2 (λ i : m, pi_Lp 2 (λ j : n, α)))) | def | matrix.frobenius_normed_add_comm_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_add_comm_group",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_normed_space [normed_field R] [seminormed_add_comm_group α] [normed_space R α] :
normed_space R (matrix m n α) | (by apply_instance : normed_space R (pi_Lp 2 (λ i : m, pi_Lp 2 (λ j : n, α)))) | def | matrix.frobenius_normed_space | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_field",
"normed_space",
"pi_Lp",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_def (A : matrix m n α) :
‖A‖₊ = (∑ i j, ‖A i j‖₊ ^ (2 : ℝ)) ^ (1/2 : ℝ) | by simp_rw [pi_Lp.nnnorm_eq_of_L2, nnreal.sq_sqrt, nnreal.sqrt_eq_rpow, nnreal.rpow_two] | lemma | matrix.frobenius_nnnorm_def | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"nnreal.rpow_two",
"nnreal.sq_sqrt",
"nnreal.sqrt_eq_rpow",
"pi_Lp.nnnorm_eq_of_L2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_def (A : matrix m n α) :
‖A‖ = (∑ i j, ‖A i j‖ ^ (2 : ℝ)) ^ (1/2 : ℝ) | (congr_arg coe (frobenius_nnnorm_def A)).trans $ by simp [nnreal.coe_sum] | lemma | matrix.frobenius_norm_def | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"nnreal.coe_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_map_eq (A : matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
‖A.map f‖₊ = ‖A‖₊ | by simp_rw [frobenius_nnnorm_def, matrix.map_apply, hf] | lemma | matrix.frobenius_nnnorm_map_eq | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_map_eq (A : matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖ = ‖a‖) :
‖A.map f‖ = ‖A‖ | (congr_arg (coe : ℝ≥0 → ℝ) $ frobenius_nnnorm_map_eq A f $ λ a, subtype.ext $ hf a : _) | lemma | matrix.frobenius_norm_map_eq | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_transpose (A : matrix m n α) : ‖Aᵀ‖₊ = ‖A‖₊ | by { rw [frobenius_nnnorm_def, frobenius_nnnorm_def, finset.sum_comm], refl } | lemma | matrix.frobenius_nnnorm_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_transpose (A : matrix m n α) : ‖Aᵀ‖ = ‖A‖ | congr_arg coe $ frobenius_nnnorm_transpose A | lemma | matrix.frobenius_norm_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_conj_transpose [star_add_monoid α] [normed_star_group α]
(A : matrix m n α) : ‖Aᴴ‖₊ = ‖A‖₊ | (frobenius_nnnorm_map_eq _ _ nnnorm_star).trans A.frobenius_nnnorm_transpose | lemma | matrix.frobenius_nnnorm_conj_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"nnnorm_star",
"normed_star_group",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_conj_transpose [star_add_monoid α] [normed_star_group α]
(A : matrix m n α) : ‖Aᴴ‖ = ‖A‖ | congr_arg coe $ frobenius_nnnorm_conj_transpose A | lemma | matrix.frobenius_norm_conj_transpose | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_star_group",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_normed_star_group [star_add_monoid α] [normed_star_group α] :
normed_star_group (matrix m m α) | ⟨frobenius_norm_conj_transpose⟩ | instance | matrix.frobenius_normed_star_group | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"normed_star_group",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_row (v : m → α) : ‖row v‖ = ‖(pi_Lp.equiv 2 _).symm v‖ | begin
rw [frobenius_norm_def, fintype.sum_unique, pi_Lp.norm_eq_of_L2, real.sqrt_eq_rpow],
simp only [row_apply, real.rpow_two, pi_Lp.equiv_symm_apply],
end | lemma | matrix.frobenius_norm_row | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"pi_Lp.equiv",
"pi_Lp.equiv_symm_apply",
"pi_Lp.norm_eq_of_L2",
"real.rpow_two",
"real.sqrt_eq_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_row (v : m → α) : ‖row v‖₊ = ‖(pi_Lp.equiv 2 _).symm v‖₊ | subtype.ext $ frobenius_norm_row v | lemma | matrix.frobenius_nnnorm_row | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"pi_Lp.equiv",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_col (v : n → α) : ‖col v‖ = ‖(pi_Lp.equiv 2 _).symm v‖ | begin
simp_rw [frobenius_norm_def, fintype.sum_unique, pi_Lp.norm_eq_of_L2, real.sqrt_eq_rpow],
simp only [col_apply, real.rpow_two, pi_Lp.equiv_symm_apply]
end | lemma | matrix.frobenius_norm_col | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"pi_Lp.equiv",
"pi_Lp.equiv_symm_apply",
"pi_Lp.norm_eq_of_L2",
"real.rpow_two",
"real.sqrt_eq_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_col (v : n → α) : ‖col v‖₊ = ‖(pi_Lp.equiv 2 _).symm v‖₊ | subtype.ext $ frobenius_norm_col v | lemma | matrix.frobenius_nnnorm_col | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"pi_Lp.equiv",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_diagonal [decidable_eq n] (v : n → α) :
‖diagonal v‖₊ = ‖(pi_Lp.equiv 2 _).symm v‖₊ | begin
simp_rw [frobenius_nnnorm_def, ←finset.sum_product', finset.univ_product_univ,
pi_Lp.nnnorm_eq_of_L2],
let s := (finset.univ : finset n).map ⟨λ i : n, (i, i), λ i j h, congr_arg prod.fst h⟩,
rw ←finset.sum_subset (finset.subset_univ s) (λ i hi his, _),
{ rw [finset.sum_map, nnreal.sqrt_eq_rpow],
d... | lemma | matrix.frobenius_nnnorm_diagonal | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"finset",
"finset.mem_univ",
"finset.subset_univ",
"finset.univ",
"finset.univ_product_univ",
"nnreal.rpow_two",
"nnreal.sqrt_eq_rpow",
"nnreal.zero_rpow",
"pi_Lp.equiv",
"pi_Lp.nnnorm_eq_of_L2",
"prod.ext",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_diagonal [decidable_eq n] (v : n → α) :
‖diagonal v‖ = ‖(pi_Lp.equiv 2 _).symm v‖ | (congr_arg coe $ frobenius_nnnorm_diagonal v : _).trans rfl | lemma | matrix.frobenius_norm_diagonal | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_one [decidable_eq n] [seminormed_add_comm_group α] [has_one α] :
‖(1 : matrix n n α)‖₊ = nnreal.sqrt (fintype.card n) * ‖(1 : α)‖₊ | begin
refine (frobenius_nnnorm_diagonal _).trans _,
simp_rw [pi_Lp.nnnorm_equiv_symm_const ennreal.two_ne_top, nnreal.sqrt_eq_rpow],
simp only [ennreal.to_real_div, ennreal.one_to_real, ennreal.to_real_bit0],
end | lemma | matrix.frobenius_nnnorm_one | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"ennreal.one_to_real",
"ennreal.to_real_bit0",
"ennreal.to_real_div",
"ennreal.two_ne_top",
"fintype.card",
"matrix",
"nnreal.sqrt",
"nnreal.sqrt_eq_rpow",
"pi_Lp.nnnorm_equiv_symm_const",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nnnorm_mul (A : matrix l m α) (B : matrix m n α) : ‖A ⬝ B‖₊ ≤ ‖A‖₊ * ‖B‖₊ | begin
simp_rw [frobenius_nnnorm_def, matrix.mul_apply],
rw [←nnreal.mul_rpow, @finset.sum_comm _ n m, finset.sum_mul_sum, finset.sum_product],
refine nnreal.rpow_le_rpow _ one_half_pos.le,
refine finset.sum_le_sum (λ i hi, finset.sum_le_sum $ λ j hj, _),
rw [← nnreal.rpow_le_rpow_iff one_half_pos, ← nnreal.rp... | lemma | matrix.frobenius_nnnorm_mul | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"finset.sum_mul_sum",
"is_R_or_C.inner_apply",
"matrix",
"matrix.mul_apply",
"mul_div_cancel'",
"nnnorm_inner_le_nnnorm",
"nnnorm_star",
"nnreal.mul_rpow",
"nnreal.rpow_le_rpow",
"nnreal.rpow_le_rpow_iff",
"nnreal.rpow_mul",
"nnreal.rpow_one",
"nnreal.rpow_two",
"nnreal.sqrt_eq_rpow",
"o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_norm_mul (A : matrix l m α) (B : matrix m n α) : ‖A ⬝ B‖ ≤ ‖A‖ * ‖B‖ | frobenius_nnnorm_mul A B | lemma | matrix.frobenius_norm_mul | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_normed_ring [decidable_eq m] : normed_ring (matrix m m α) | { norm := has_norm.norm,
norm_mul := frobenius_norm_mul,
..matrix.frobenius_seminormed_add_comm_group } | def | matrix.frobenius_normed_ring | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.frobenius_seminormed_add_comm_group",
"norm_mul",
"normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_normed_algebra [decidable_eq m] [normed_field R] [normed_algebra R α] :
normed_algebra R (matrix m m α) | { ..matrix.frobenius_normed_space } | def | matrix.frobenius_normed_algebra | analysis | src/analysis/matrix.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.pi_Lp",
"analysis.inner_product_space.pi_L2"
] | [
"matrix",
"matrix.frobenius_normed_space",
"normed_algebra",
"normed_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i in s, (z i) ^ (w i)) ≤ ∑ i in s, w i * z i | begin
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0,
{ rcases A with ⟨i, his, hzi, hwi⟩,
rw [prod_eq_zero his],
{ exact sum_nonneg (λ j hj, mul_nonneg (hw j hj) (hz j hj)) },
{ rw hzi, exact zero_rpow hwi } },... | theorem | real.geom_mean_le_arith_mean_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"eq_or_lt_of_le",
"exp_sum",
"mul_comm",
"not_and",
"not_exists",
"not_not",
"set.mem_univ",
"smul_eq_mul"
] | AM-GM inequality: the **geometric mean is less than or equal to the arithmetic mean**, weighted
version for real-valued nonnegative functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
(∏ i in s, (z i) ^ (w i)) = x | calc (∏ i in s, (z i) ^ (w i)) = ∏ i in s, x ^ w i :
begin
refine prod_congr rfl (λ i hi, _),
cases eq_or_ne (w i) 0 with h₀ h₀,
{ rw [h₀, rpow_zero, rpow_zero] },
{ rw hx i hi h₀ }
end
... = x :
begin
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one],
have : (∑ i in s, w i) ≠ 0,
{ rw hw',... | theorem | real.geom_mean_weighted_of_constant | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"eq_or_ne",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ)
(hw' : ∑ i in s, w i = 1) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∑ i in s, w i * z i = x | calc ∑ i in s, w i * z i = ∑ i in s, w i * x :
begin
refine sum_congr rfl (λ i hi, _),
cases eq_or_ne (w i) 0 with hwi hwi,
{ rw [hwi, zero_mul, zero_mul] },
{ rw hx i hi hwi },
end
... = x : by rw [←sum_mul, hw', one_mul] | theorem | real.arith_mean_weighted_of_constant | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"eq_or_ne",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
(∏ i in s, (z i) ^ (w i)) = ∑ i in s, w i * z i | by rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant]; assumption | theorem | real.geom_mean_eq_arith_mean_weighted_of_constant | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) :
(∏ i in s, (z i) ^ (w i:ℝ)) ≤ ∑ i in s, w i * z i | by exact_mod_cast real.geom_mean_le_arith_mean_weighted _ _ _ (λ i _, (w i).coe_nonneg)
(by assumption_mod_cast) (λ i _, (z i).coe_nonneg) | theorem | nnreal.geom_mean_le_arith_mean_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"real.geom_mean_le_arith_mean_weighted"
] | The geometric mean is less than or equal to the arithmetic mean, weighted version
for `nnreal`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
w₁ + w₂ = 1 → p₁ ^ (w₁:ℝ) * p₂ ^ (w₂:ℝ) ≤ w₁ * p₁ + w₂ * p₂ | by simpa only [fin.prod_univ_succ, fin.sum_univ_succ, finset.prod_empty, finset.sum_empty,
fintype.univ_of_is_empty, fin.cons_succ, fin.cons_zero, add_zero, mul_one]
using geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂] | theorem | nnreal.geom_mean_le_arith_mean2_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"fin.cons_succ",
"fin.cons_zero",
"fin.prod_univ_succ",
"finset.prod_empty",
"fintype.univ_of_is_empty",
"mul_one"
] | The geometric mean is less than or equal to the arithmetic mean, weighted version
for two `nnreal` numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
w₁ + w₂ + w₃ = 1 → p₁ ^ (w₁:ℝ) * p₂ ^ (w₂:ℝ) * p₃ ^ (w₃:ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ | by simpa only [fin.prod_univ_succ, fin.sum_univ_succ, finset.prod_empty, finset.sum_empty,
fintype.univ_of_is_empty, fin.cons_succ, fin.cons_zero, add_zero, mul_one, ← add_assoc, mul_assoc]
using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃] ![p₁, p₂, p₃] | theorem | nnreal.geom_mean_le_arith_mean3_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"fin.cons_succ",
"fin.cons_zero",
"fin.prod_univ_succ",
"finset.prod_empty",
"fintype.univ_of_is_empty",
"mul_assoc",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
w₁ + w₂ + w₃ + w₄ = 1 → p₁ ^ (w₁:ℝ) * p₂ ^ (w₂:ℝ) * p₃ ^ (w₃:ℝ)* p₄ ^ (w₄:ℝ) ≤
w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ | by simpa only [fin.prod_univ_succ, fin.sum_univ_succ, finset.prod_empty, finset.sum_empty,
fintype.univ_of_is_empty, fin.cons_succ, fin.cons_zero, add_zero, mul_one, ← add_assoc, mul_assoc]
using geom_mean_le_arith_mean_weighted univ ![w₁, w₂, w₃, w₄] ![p₁, p₂, p₃, p₄] | theorem | nnreal.geom_mean_le_arith_mean4_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"fin.cons_succ",
"fin.cons_zero",
"fin.prod_univ_succ",
"finset.prod_empty",
"fintype.univ_of_is_empty",
"mul_assoc",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ | nnreal.geom_mean_le_arith_mean2_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ $
nnreal.coe_eq.1 $ by assumption | theorem | real.geom_mean_le_arith_mean2_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"nnreal.geom_mean_le_arith_mean2_weighted"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
(hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ | nnreal.geom_mean_le_arith_mean3_weighted
⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ $ nnreal.coe_eq.1 hw | theorem | real.geom_mean_le_arith_mean3_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"nnreal.geom_mean_le_arith_mean3_weighted"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
(hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
(hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ | nnreal.geom_mean_le_arith_mean4_weighted ⟨w₁, hw₁⟩ ⟨w₂, hw₂⟩ ⟨w₃, hw₃⟩ ⟨w₄, hw₄⟩
⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ ⟨p₃, hp₃⟩ ⟨p₄, hp₄⟩ $ nnreal.coe_eq.1 $ by assumption | theorem | real.geom_mean_le_arith_mean4_weighted | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"nnreal.geom_mean_le_arith_mean4_weighted"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hpq : p.is_conjugate_exponent q) :
a * b ≤ a^p / p + b^q / q | by simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, div_eq_inv_mul]
using geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg
(rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj | theorem | real.young_inequality_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"div_eq_inv_mul"
] | Young's inequality, a version for nonnegative real numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.is_conjugate_exponent q) :
a * b ≤ |a|^p / p + |b|^q / q | calc a * b ≤ |a * b| : le_abs_self (a * b)
... = |a| * |b| : abs_mul a b
... ≤ |a|^p / p + |b|^q / q :
real.young_inequality_of_nonneg (abs_nonneg a) (abs_nonneg b) hpq | theorem | real.young_inequality | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_mul",
"abs_nonneg",
"le_abs_self",
"real.young_inequality_of_nonneg"
] | Young's inequality, a version for arbitrary real numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) :
a * b ≤ a^(p:ℝ) / p + b^(q:ℝ) / q | real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg ⟨hp, nnreal.coe_eq.2 hpq⟩ | theorem | nnreal.young_inequality | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"real.young_inequality_of_nonneg"
] | Young's inequality, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.is_conjugate_exponent q) :
a * b ≤ a ^ p / real.to_nnreal p + b ^ q / real.to_nnreal q | begin
nth_rewrite 0 ← real.coe_to_nnreal p hpq.nonneg,
nth_rewrite 0 ← real.coe_to_nnreal q hpq.symm.nonneg,
exact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal,
end | theorem | nnreal.young_inequality_real | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"real.coe_to_nnreal",
"real.to_nnreal"
] | Young's inequality, `ℝ≥0` version with real conjugate exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.is_conjugate_exponent q) :
a * b ≤ a ^ p / ennreal.of_real p + b ^ q / ennreal.of_real q | begin
by_cases h : a = ⊤ ∨ b = ⊤,
{ refine le_trans le_top (le_of_eq _),
repeat { rw div_eq_mul_inv },
cases h; rw h; simp [h, hpq.pos, hpq.symm.pos], },
push_neg at h, -- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
rw [←coe_to_nnreal h.left, ←coe_to_nnreal h.right, ←coe_mul... | theorem | ennreal.young_inequality | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"div_eq_mul_inv",
"ennreal.of_real",
"le_top",
"nnreal.young_inequality_real",
"real.to_nnreal"
] | Young's inequality, `ℝ≥0∞` version with real conjugate exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.is_conjugate_exponent q) (hf : ∑ i in s, (f i) ^ p ≤ 1) (hg : ∑ i in s, (g i) ^ q ≤ 1) :
∑ i in s, f i * g i ≤ 1 | begin
have hp_ne_zero : real.to_nnreal p ≠ 0, from (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm,
have hq_ne_zero : real.to_nnreal q ≠ 0, from (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm,
calc ∑ i in s, f i * g i
≤ ∑ i in s, ((f i) ^ p / real.to_nnreal p + (g i) ^ q / real.to_nnreal q) :
finset.s... | lemma | nnreal.inner_le_Lp_mul_Lp_of_norm_le_one | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"div_le_iff",
"div_mul_cancel",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.is_conjugate_exponent q) (hf : ∑ i in s, (f i) ^ p = 0) :
∑ i in s, f i * g i ≤ (∑ i in s, (f i) ^ p) ^ (1 / p) * (∑ i in s, (g i) ^ q) ^ (1 / q) | begin
simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul, inv_eq_zero,
ne.def, not_false_iff, le_zero_iff, mul_eq_zero],
intros i his,
left,
rw sum_eq_zero_iff at hf,
exact (rpow_eq_zero_iff.mp (hf i his)).left,
end | lemma | nnreal.inner_le_Lp_mul_Lp_of_norm_eq_zero | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"inv_eq_zero",
"le_zero_iff",
"mul_eq_zero",
"one_div",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ}
(hpq : p.is_conjugate_exponent q) :
∑ i in s, f i * g i ≤ (∑ i in s, (f i) ^ p) ^ (1 / p) * (∑ i in s, (g i) ^ q) ^ (1 / q) | begin
by_cases hF_zero : ∑ i in s, (f i) ^ p = 0,
{ exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero, },
by_cases hG_zero : ∑ i in s, (g i) ^ q = 0,
{ calc ∑ i in s, f i * g i
= ∑ i in s, g i * f i : by { congr' with i, rw mul_comm, }
... ≤ (∑ i in s, (g i) ^ q) ^ (1 / q) * (∑ i in s, (f i... | theorem | nnreal.inner_le_Lp_mul_Lq | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"div_le_iff",
"div_mul_div_comm",
"div_self",
"inv_mul_cancel",
"mul_comm",
"mul_ne_zero",
"not_and_distrib",
"one_div",
"one_mul"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.is_conjugate_exponent q)
(hf : summable (λ i, (f i) ^ p)) (hg : summable (λ i, (g i) ^ q)) :
summable (λ i, f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, (f i) ^ p) ^ (1 / p) * (∑' i, (g i) ^ q) ^ (1 / q) | begin
have H₁ : ∀ s : finset ι, ∑ i in s, f i * g i
≤ (∑' i, (f i) ^ p) ^ (1 / p) * (∑' i, (g i) ^ q) ^ (1 / q),
{ intros s,
refine le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le),
{ rw nnreal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos),
exact sum_le_tsum _ (λ _ _, zero_le _)... | theorem | nnreal.inner_le_Lp_mul_Lq_tsum | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"bdd_above",
"bot_le",
"finset",
"has_sum_of_is_lub",
"is_lub_csupr",
"mul_le_mul",
"nnreal.rpow_le_rpow_iff",
"set.range",
"sum_le_tsum",
"summable",
"tsum_le_of_sum_le"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `nnreal`-valued
functions. For an alternative version, convenient if the infinite sums are already expressed as
`p`-th powers, see `inner_le_Lp_mul_Lq_h... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.is_conjugate_exponent q)
(hf : summable (λ i, (f i) ^ p)) (hg : summable (λ i, (g i) ^ q)) :
summable (λ i, f i * g i) | (inner_le_Lp_mul_Lq_tsum hpq hf hg).1 | theorem | nnreal.summable_mul_of_Lp_Lq | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.is_conjugate_exponent q)
(hf : summable (λ i, (f i) ^ p)) (hg : summable (λ i, (g i) ^ q)) :
∑' i, f i * g i ≤ (∑' i, (f i) ^ p) ^ (1 / p) * (∑' i, (g i) ^ q) ^ (1 / q) | (inner_le_Lp_mul_Lq_tsum hpq hf hg).2 | theorem | nnreal.inner_le_Lp_mul_Lq_tsum' | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_le_Lp_mul_Lq_has_sum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
(hpq : p.is_conjugate_exponent q) (hf : has_sum (λ i, (f i) ^ p) (A ^ p))
(hg : has_sum (λ i, (g i) ^ q) (B ^ q)) :
∃ C, C ≤ A * B ∧ has_sum (λ i, f i * g i) C | begin
obtain ⟨H₁, H₂⟩ := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable,
have hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p),
{ rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero] },
have hB : B = (∑' (i : ι), g i ^ q) ^ (1 / q),
{ rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero] },
refine ⟨∑' i, f i * g i, ... | theorem | nnreal.inner_le_Lp_mul_Lq_has_sum | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"has_sum"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `nnreal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s) ^ (p - 1) * ∑ i in s, (f i) ^ p | begin
cases eq_or_lt_of_le hp with hp hp,
{ simp [← hp] },
let q : ℝ := p / (p - 1),
have hpq : p.is_conjugate_exponent q,
{ rw real.is_conjugate_exponent_iff hp },
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero,
have hq : 1 / q * p = (p - 1),
{ rw [← hpq.div_conj_eq_sub_one],
ring },
... | theorem | nnreal.rpow_sum_le_const_mul_sum_rpow | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"eq_or_lt_of_le",
"nat.smul_one_eq_coe",
"nnreal.mul_rpow",
"nnreal.rpow_le_rpow",
"nnreal.rpow_mul",
"one_div_mul_cancel",
"one_mul",
"pi.one_apply",
"real.is_conjugate_exponent_iff",
"ring"
] | For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_greatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.is_conjugate_exponent q) :
is_greatest ((λ g : ι → ℝ≥0, ∑ i in s, f i * g i) ''
{g | ∑ i in s, (g i)^q ≤ 1}) ((∑ i in s, (f i)^p) ^ (1 / p)) | begin
split,
{ use λ i, ((f i) ^ p / f i / (∑ i in s, (f i) ^ p) ^ (1 / q)),
by_cases hf : ∑ i in s, (f i)^p = 0,
{ simp [hf, hpq.ne_zero, hpq.symm.ne_zero] },
{ have A : p + q - q ≠ 0, by simp [hpq.ne_zero],
have B : ∀ y : ℝ≥0, y * y^p / y = y^p,
{ refine λ y, mul_div_cancel_left_of_imp (λ ... | theorem | nnreal.is_greatest_Lp | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"div_eq_iff",
"div_le_iff",
"div_mul_cancel",
"is_greatest",
"mul_div_assoc",
"mul_div_cancel_left_of_imp",
"mul_le_mul_left'",
"mul_one",
"nnreal.rpow_le_one",
"one_mul"
] | The `L_p` seminorm of a vector `f` is the greatest value of the inner product
`∑ i in s, f i * g i` over functions `g` of `L_q` seminorm less than or equal to one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, (f i) ^ p) ^ (1 / p) + (∑ i in s, (g i) ^ p) ^ (1 / p) | begin
-- The result is trivial when `p = 1`, so we can assume `1 < p`.
rcases eq_or_lt_of_le hp with rfl|hp, { simp [finset.sum_add_distrib] },
have hpq := real.is_conjugate_exponent_conjugate_exponent hp,
have := is_greatest_Lp s (f + g) hpq,
simp only [pi.add_apply, add_mul, sum_add_distrib] at this,
rcas... | theorem | nnreal.Lp_add_le | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"eq_or_lt_of_le",
"real.is_conjugate_exponent_conjugate_exponent"
] | Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `nnreal`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : summable (λ i, (f i) ^ p))
(hg : summable (λ i, (g i) ^ p)) :
summable (λ i, (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, (f i) ^ p) ^ (1 / p) + (∑' i, (g i) ^ p) ^ (1 / p) | begin
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp,
have H₁ : ∀ s : finset ι, ∑ i in s, (f i + g i) ^ p
≤ ((∑' i, (f i)^p) ^ (1/p) + (∑' i, (g i)^p) ^ (1/p)) ^ p,
{ intros s,
rw ← nnreal.rpow_one_div_le_iff pos,
refine le_trans (Lp_add_le s f g hp) (add_le_add _ _);
rw nnreal.rpow_le_rpow_iff... | theorem | nnreal.Lp_add_le_tsum | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"bdd_above",
"finset",
"has_sum_of_is_lub",
"is_lub_csupr",
"nnreal.rpow_le_rpow_iff",
"nnreal.rpow_one_div_le_iff",
"set.range",
"sum_le_tsum",
"summable",
"tsum_le_of_sum_le",
"zero_lt_one"
] | Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `nnreal`-valued functions. For an alternative version, convenient if the
infinite sums are already expressed as `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : summable (λ i, (f i) ^ p))
(hg : summable (λ i, (g i) ^ p)) :
summable (λ i, (f i + g i) ^ p) | (Lp_add_le_tsum hp hf hg).1 | theorem | nnreal.summable_Lp_add | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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