statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
upper_mem : I.upper ∈ I | λ i, right_mem_Ioc.2 $ I.lower_lt_upper i | lemma | box_integral.box.upper_mem | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem : ∃ x, x ∈ I | ⟨_, I.upper_mem⟩ | lemma | box_integral.box.exists_mem | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_coe : set.nonempty (I : set (ι → ℝ)) | I.exists_mem | lemma | box_integral.box.nonempty_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_empty : (I : set (ι → ℝ)) ≠ ∅ | I.nonempty_coe.ne_empty | lemma | box_integral.box.coe_ne_empty | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_ne_coe : ∅ ≠ (I : set (ι → ℝ)) | I.coe_ne_empty.symm | lemma | box_integral.box.empty_ne_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def : I ≤ J ↔ ∀ x ∈ I, x ∈ J | iff.rfl | lemma | box_integral.box.le_def | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tfae :
tfae [I ≤ J,
(I : set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper,
J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | begin
tfae_have : 1 ↔ 2, from iff.rfl,
tfae_have : 2 → 3,
{ intro h,
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h },
tfae_have : 3 ↔ 4, from Icc_subset_Icc_iff I.lower_le_upper,
tfae_have : 4 → 2, from λ h x hx i, Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i),
tfae_finish
end | lemma | box_integral.box.le_tfae | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"closure_mono",
"closure_pi_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subset_coe : (I : set (ι → ℝ)) ⊆ J ↔ I ≤ J | iff.rfl | lemma | box_integral.box.coe_subset_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_bounds : I ≤ J ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper | (le_tfae I J).out 0 3 | lemma | box_integral.box.le_iff_bounds | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_coe : injective (coe : box ι → set (ι → ℝ)) | begin
rintros ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h,
simp only [subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h,
congr,
exacts [le_antisymm h.2.1 h.1.1, le_antisymm h.1.2 h.2.2]
end | lemma | box_integral.box.injective_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj : (I : set (ι → ℝ)) = J ↔ I = J | injective_coe.eq_iff | lemma | box_integral.box.coe_inj | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (H : ∀ x, x ∈ I ↔ x ∈ J) : I = J | injective_coe $ set.ext H | lemma | box_integral.box.ext | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_of_disjoint_coe (h : disjoint (I : set (ι → ℝ)) J) : I ≠ J | mt coe_inj.2 $ h.ne I.coe_ne_empty | lemma | box_integral.box.ne_of_disjoint_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc : box ι ↪o set (ι → ℝ) | order_embedding.of_map_le_iff (λ I : box ι, Icc I.lower I.upper) (λ I J, (le_tfae I J).out 2 0) | def | box_integral.box.Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"order_embedding.of_map_le_iff"
] | Closed box corresponding to `I : box_integral.box ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Icc_def : I.Icc = Icc I.lower I.upper | rfl | lemma | box_integral.box.Icc_def | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_mem_Icc (I : box ι) : I.upper ∈ I.Icc | right_mem_Icc.2 I.lower_le_upper | lemma | box_integral.box.upper_mem_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_mem_Icc (I : box ι) : I.lower ∈ I.Icc | left_mem_Icc.2 I.lower_le_upper | lemma | box_integral.box.lower_mem_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_Icc (I : box ι) : is_compact I.Icc | is_compact_Icc | lemma | box_integral.box.is_compact_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_eq_pi : I.Icc = pi univ (λ i, Icc (I.lower i) (I.upper i)) | (pi_univ_Icc _ _).symm | lemma | box_integral.box.Icc_eq_pi | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_Icc : I ≤ J ↔ I.Icc ⊆ J.Icc | (le_tfae I J).out 0 2 | lemma | box_integral.box.le_iff_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_lower : antitone (λ I : box ι, I.lower) | λ I J H, (le_iff_bounds.1 H).1 | lemma | box_integral.box.antitone_lower | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_upper : monotone (λ I : box ι, I.upper) | λ I J H, (le_iff_bounds.1 H).2 | lemma | box_integral.box.monotone_upper | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subset_Icc : ↑I ⊆ I.Icc | λ x hx, ⟨λ i, (hx i).1.le, λ i, (hx i).2⟩ | lemma | box_integral.box.coe_subset_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_coe : has_coe_t (with_bot (box ι)) (set (ι → ℝ)) | ⟨λ o, o.elim ∅ coe⟩ | instance | box_integral.box.with_bot_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ((⊥ : with_bot (box ι)) : set (ι → ℝ)) = ∅ | rfl | lemma | box_integral.box.coe_bot | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe : ((I : with_bot (box ι)) : set (ι → ℝ)) = I | rfl | lemma | box_integral.box.coe_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"coe_coe",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_some_iff : ∀ {I : with_bot (box ι)}, I.is_some ↔ (I : set (ι → ℝ)).nonempty | | ⊥ := by { erw option.is_some, simp }
| (I : box ι) := by { erw option.is_some, simp [I.nonempty_coe] } | lemma | box_integral.box.is_some_iff | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_coe_eq_coe (I : with_bot (box ι)) :
(⋃ (J : box ι) (hJ : ↑J = I), (J : set (ι → ℝ))) = I | by induction I using with_bot.rec_bot_coe; simp [with_bot.coe_eq_coe] | lemma | box_integral.box.bUnion_coe_eq_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"with_bot",
"with_bot.coe_eq_coe",
"with_bot.rec_bot_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_coe_subset_iff {I J : with_bot (box ι)} :
(I : set (ι → ℝ)) ⊆ J ↔ I ≤ J | begin
induction I using with_bot.rec_bot_coe, { simp },
induction J using with_bot.rec_bot_coe, { simp [subset_empty_iff] },
simp
end | lemma | box_integral.box.with_bot_coe_subset_iff | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"with_bot",
"with_bot.rec_bot_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_coe_inj {I J : with_bot (box ι)} :
(I : set (ι → ℝ)) = J ↔ I = J | by simp only [subset.antisymm_iff, ← le_antisymm_iff, with_bot_coe_subset_iff] | lemma | box_integral.box.with_bot_coe_inj | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (l u : ι → ℝ) : with_bot (box ι) | if h : ∀ i, l i < u i then ↑(⟨l, u, h⟩ : box ι) else ⊥ | def | box_integral.box.mk' | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"mk'",
"with_bot"
] | Make a `with_bot (box ι)` from a pair of corners `l u : ι → ℝ`. If `l i < u i` for all `i`,
then the result is `⟨l, u, _⟩ : box ι`, otherwise it is `⊥`. In any case, the result interpreted
as a set in `ι → ℝ` is the set `{x : ι → ℝ | ∀ i, x i ∈ Ioc (l i) (u i)}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i | by { rw mk', split_ifs; simpa using h } | lemma | box_integral.box.mk'_eq_bot | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | begin
cases I with lI uI hI, rw mk', split_ifs,
{ simp [with_bot.coe_eq_coe] },
{ suffices : l = lI → u ≠ uI, by simpa,
rintro rfl rfl, exact h hI }
end | lemma | box_integral.box.mk'_eq_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"mk'",
"with_bot.coe_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk' (l u : ι → ℝ) : (mk' l u : set (ι → ℝ)) = pi univ (λ i, Ioc (l i) (u i)) | begin
rw mk', split_ifs,
{ exact coe_eq_pi _ },
{ rcases not_forall.mp h with ⟨i, hi⟩,
rw [coe_bot, univ_pi_eq_empty], exact Ioc_eq_empty hi }
end | lemma | box_integral.box.coe_mk' | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (I J : with_bot (box ι)) : (↑(I ⊓ J) : set (ι → ℝ)) = I ∩ J | begin
induction I using with_bot.rec_bot_coe, { change ∅ = _, simp },
induction J using with_bot.rec_bot_coe, { change ∅ = _, simp },
change ↑(mk' _ _) = _,
simp only [coe_eq_pi, ← pi_inter_distrib, Ioc_inter_Ioc, pi.sup_apply, pi.inf_apply, coe_mk',
coe_coe]
end | lemma | box_integral.box.coe_inf | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"coe_coe",
"mk'",
"pi.inf_apply",
"pi.sup_apply",
"with_bot",
"with_bot.rec_bot_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_with_bot_coe {I J : with_bot (box ι)} :
disjoint (I : set (ι → ℝ)) J ↔ disjoint I J | by { simp only [disjoint_iff_inf_le, ← with_bot_coe_subset_iff, coe_inf], refl } | lemma | box_integral.box.disjoint_with_bot_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"disjoint",
"disjoint_iff_inf_le",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_coe : disjoint (I : with_bot (box ι)) J ↔ disjoint (I : set (ι → ℝ)) J | disjoint_with_bot_coe.symm | lemma | box_integral.box.disjoint_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"disjoint",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_disjoint_coe_iff_nonempty_inter :
¬disjoint (I : with_bot (box ι)) J ↔ (I ∩ J : set (ι → ℝ)).nonempty | by rw [disjoint_coe, set.not_disjoint_iff_nonempty_inter] | lemma | box_integral.box.not_disjoint_coe_iff_nonempty_inter | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"disjoint",
"set.not_disjoint_iff_nonempty_inter",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
face {n} (I : box (fin (n + 1))) (i : fin (n + 1)) : box (fin n) | ⟨I.lower ∘ fin.succ_above i, I.upper ∘ fin.succ_above i, λ j, I.lower_lt_upper _⟩ | def | box_integral.box.face | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"fin.succ_above"
] | Face of a box in `ℝⁿ⁺¹ = fin (n + 1) → ℝ`: the box in `ℝⁿ = fin n → ℝ` with corners at
`I.lower ∘ fin.succ_above i` and `I.upper ∘ fin.succ_above i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
face_mk {n} (l u : fin (n + 1) → ℝ) (h : ∀ i, l i < u i) (i : fin (n + 1)) :
face ⟨l, u, h⟩ i = ⟨l ∘ fin.succ_above i, u ∘ fin.succ_above i, λ j, h _⟩ | rfl | lemma | box_integral.box.face_mk | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"fin.succ_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
face_mono {n} {I J : box (fin (n + 1))} (h : I ≤ J) (i : fin (n + 1)) :
face I i ≤ face J i | λ x hx i, Ioc_subset_Ioc ((le_iff_bounds.1 h).1 _) ((le_iff_bounds.1 h).2 _) (hx _) | lemma | box_integral.box.face_mono | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_face {n} (i : fin (n + 1)) : monotone (λ I, face I i) | λ I J h, face_mono h i | lemma | box_integral.box.monotone_face | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_insert_nth_face_Icc {n} (I : box (fin (n + 1))) {i : fin (n + 1)} {x : ℝ}
(hx : x ∈ Icc (I.lower i) (I.upper i)) :
maps_to (i.insert_nth x) (I.face i).Icc I.Icc | λ y hy, fin.insert_nth_mem_Icc.2 ⟨hx, hy⟩ | lemma | box_integral.box.maps_to_insert_nth_face_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_insert_nth_face {n} (I : box (fin (n + 1))) {i : fin (n + 1)} {x : ℝ}
(hx : x ∈ Ioc (I.lower i) (I.upper i)) :
maps_to (i.insert_nth x) (I.face i) I | λ y hy, by simpa only [mem_coe, mem_def, i.forall_iff_succ_above, hx, fin.insert_nth_apply_same,
fin.insert_nth_apply_succ_above, true_and] | lemma | box_integral.box.maps_to_insert_nth_face | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"fin.insert_nth_apply_same",
"fin.insert_nth_apply_succ_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_face_Icc {X} [topological_space X] {n} {f : (fin (n + 1) → ℝ) → X}
{I : box (fin (n + 1))} (h : continuous_on f I.Icc) {i : fin (n + 1)} {x : ℝ}
(hx : x ∈ Icc (I.lower i) (I.upper i)) :
continuous_on (f ∘ i.insert_nth x) (I.face i).Icc | h.comp (continuous_on_const.fin_insert_nth i continuous_on_id) (I.maps_to_insert_nth_face_Icc hx) | lemma | box_integral.box.continuous_on_face_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"continuous_on",
"continuous_on_id",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo : box ι →o set (ι → ℝ) | { to_fun := λ I, pi univ (λ i, Ioo (I.lower i) (I.upper i)),
monotone' := λ I J h, pi_mono $ λ i hi, Ioo_subset_Ioo ((le_iff_bounds.1 h).1 i)
((le_iff_bounds.1 h).2 i) } | def | box_integral.box.Ioo | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | The interior of a box. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ioo_subset_coe (I : box ι) : I.Ioo ⊆ I | λ x hx i, Ioo_subset_Ioc_self (hx i trivial) | lemma | box_integral.box.Ioo_subset_coe | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_subset_Icc (I : box ι) : I.Ioo ⊆ I.Icc | I.Ioo_subset_coe.trans coe_subset_Icc | lemma | box_integral.box.Ioo_subset_Icc | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ioo_of_tendsto [finite ι] {I : box ι} {J : ℕ → box ι} (hJ : monotone J)
(hl : tendsto (lower ∘ J) at_top (𝓝 I.lower)) (hu : tendsto (upper ∘ J) at_top (𝓝 I.upper)) :
(⋃ n, (J n).Ioo) = I.Ioo | have hl' : ∀ i, antitone (λ n, (J n).lower i),
from λ i, (monotone_eval i).comp_antitone (antitone_lower.comp_monotone hJ),
have hu' : ∀ i, monotone (λ n, (J n).upper i),
from λ i, (monotone_eval i).comp (monotone_upper.comp hJ),
calc (⋃ n, (J n).Ioo) = pi univ (λ i, ⋃ n, Ioo ((J n).lower i) ((J n).upper i)) :
Un... | lemma | box_integral.box.Union_Ioo_of_tendsto | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"Union_Ioo_of_mono_of_is_glb_of_is_lub",
"antitone",
"finite",
"is_glb_of_tendsto_at_top",
"is_lub_of_tendsto_at_top",
"monotone",
"pi_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_seq_mono_tendsto (I : box ι) : ∃ J : ℕ →o box ι, (∀ n, (J n).Icc ⊆ I.Ioo) ∧
tendsto (lower ∘ J) at_top (𝓝 I.lower) ∧ tendsto (upper ∘ J) at_top (𝓝 I.upper) | begin
choose a b ha_anti hb_mono ha_mem hb_mem hab ha_tendsto hb_tendsto
using λ i, exists_seq_strict_anti_strict_mono_tendsto (I.lower_lt_upper i),
exact ⟨⟨λ k, ⟨flip a k, flip b k, λ i, hab _ _ _⟩,
λ k l hkl, le_iff_bounds.2 ⟨λ i, (ha_anti i).antitone hkl, λ i, (hb_mono i).monotone hkl⟩⟩,
λ n x hx i h... | lemma | box_integral.box.exists_seq_mono_tendsto | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"antitone",
"exists_seq_strict_anti_strict_mono_tendsto",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion (I : box ι) : ℝ≥0 | finset.univ.sup $ λ i : ι, nndist I.lower I.upper / nndist (I.lower i) (I.upper i) | def | box_integral.box.distortion | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [] | The distortion of a box `I` is the maximum of the ratios of the lengths of its edges.
It is defined as the maximum of the ratios
`nndist I.lower I.upper / nndist (I.lower i) (I.upper i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distortion_eq_of_sub_eq_div {I J : box ι} {r : ℝ}
(h : ∀ i, I.upper i - I.lower i = (J.upper i - J.lower i) / r) :
distortion I = distortion J | begin
simp only [distortion, nndist_pi_def, real.nndist_eq', h, map_div₀],
congr' 1 with i,
have : 0 < r,
{ by_contra hr,
have := div_nonpos_of_nonneg_of_nonpos (sub_nonneg.2 $ J.lower_le_upper i) (not_lt.1 hr),
rw ← h at this,
exact this.not_lt (sub_pos.2 $ I.lower_lt_upper i) },
simp_rw [nnreal.... | lemma | box_integral.box.distortion_eq_of_sub_eq_div | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"by_contra",
"div_div_div_cancel_right",
"div_nonpos_of_nonneg_of_nonpos",
"map_div₀",
"map_ne_zero",
"nndist_pi_def",
"nnreal.finset_sup_div",
"real.nnabs",
"real.nndist_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_le_distortion_mul (I : box ι) (i : ι) :
nndist I.lower I.upper ≤ I.distortion * nndist (I.lower i) (I.upper i) | calc nndist I.lower I.upper =
(nndist I.lower I.upper / nndist (I.lower i) (I.upper i)) * nndist (I.lower i) (I.upper i) :
(div_mul_cancel _ $ mt nndist_eq_zero.1 (I.lower_lt_upper i).ne).symm
... ≤ I.distortion * nndist (I.lower i) (I.upper i) :
mul_le_mul_right' (finset.le_sup $ finset.mem_univ i) _ | lemma | box_integral.box.nndist_le_distortion_mul | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"div_mul_cancel",
"finset.le_sup",
"finset.mem_univ",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_distortion_mul (I : box ι) (i : ι) :
dist I.lower I.upper ≤ I.distortion * (I.upper i - I.lower i) | have A : I.lower i - I.upper i < 0, from sub_neg.2 (I.lower_lt_upper i),
by simpa only [← nnreal.coe_le_coe, ← dist_nndist, nnreal.coe_mul, real.dist_eq,
abs_of_neg A, neg_sub] using I.nndist_le_distortion_mul i | lemma | box_integral.box.dist_le_distortion_mul | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"abs_of_neg",
"dist_nndist",
"nnreal.coe_le_coe",
"nnreal.coe_mul",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_Icc_le_of_distortion_le (I : box ι) (i : ι) {c : ℝ≥0} (h : I.distortion ≤ c) :
diam I.Icc ≤ c * (I.upper i - I.lower i) | have (0 : ℝ) ≤ c * (I.upper i - I.lower i),
from mul_nonneg c.coe_nonneg (sub_nonneg.2 $ I.lower_le_upper _),
diam_le_of_forall_dist_le this $ λ x hx y hy,
calc dist x y ≤ dist I.lower I.upper : real.dist_le_of_mem_pi_Icc hx hy
... ≤ I.distortion * (I.upper i - I.lower i) : I.dist_le_distortion_mul i
... ≤ c * ... | lemma | box_integral.box.diam_Icc_le_of_distortion_le | analysis.box_integral.box | src/analysis/box_integral/box/basic.lean | [
"data.set.intervals.monotone",
"topology.algebra.order.monotone_convergence",
"topology.metric_space.basic"
] | [
"mul_le_mul_of_nonneg_right",
"real.dist_le_of_mem_pi_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_center_box (I : box ι) (s : set ι) : box ι | { lower := s.piecewise (λ i, (I.lower i + I.upper i) / 2) I.lower,
upper := s.piecewise I.upper (λ i, (I.lower i + I.upper i) / 2),
lower_lt_upper := λ i, by { dunfold set.piecewise, split_ifs;
simp only [left_lt_add_div_two, add_div_two_lt_right, I.lower_lt_upper] } } | def | box_integral.box.split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [
"add_div_two_lt_right",
"left_lt_add_div_two",
"set.piecewise"
] | For a box `I`, the hyperplanes passing through its center split `I` into `2 ^ card ι` boxes.
`box_integral.box.split_center_box I s` is one of these boxes. See also
`box_integral.partition.split_center` for the corresponding `box_integral.partition`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_split_center_box {s : set ι} {y : ι → ℝ} :
y ∈ I.split_center_box s ↔ y ∈ I ∧ ∀ i, (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s | begin
simp only [split_center_box, mem_def, ← forall_and_distrib],
refine forall_congr (λ i, _),
dunfold set.piecewise,
split_ifs with hs; simp only [hs, iff_true, iff_false, not_lt],
exacts [⟨λ H, ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩,
λ H, ⟨H.2, H.1.2⟩⟩,
⟨λ H, ⟨⟨H.1, H... | lemma | box_integral.box.mem_split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [
"forall_and_distrib",
"set.piecewise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_center_box_le (I : box ι) (s : set ι) : I.split_center_box s ≤ I | λ x hx, (mem_split_center_box.1 hx).1 | lemma | box_integral.box.split_center_box_le | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_split_center_box (I : box ι) {s t : set ι} (h : s ≠ t) :
disjoint (I.split_center_box s : set (ι → ℝ)) (I.split_center_box t) | begin
rw disjoint_iff_inf_le,
rintro y ⟨hs, ht⟩, apply h,
ext i,
rw [mem_coe, mem_split_center_box] at hs ht,
rw [← hs.2, ← ht.2]
end | lemma | box_integral.box.disjoint_split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [
"disjoint",
"disjoint_iff_inf_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_split_center_box (I : box ι) : injective I.split_center_box | λ s t H, by_contra $ λ Hne, (I.disjoint_split_center_box Hne).ne (nonempty_coe _).ne_empty (H ▸ rfl) | lemma | box_integral.box.injective_split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_split_center_box {I : box ι} {x : ι → ℝ} :
(∃ s, x ∈ I.split_center_box s) ↔ x ∈ I | ⟨λ ⟨s, hs⟩, I.split_center_box_le s hs,
λ hx, ⟨{i | (I.lower i + I.upper i) / 2 < x i}, mem_split_center_box.2 ⟨hx, λ i, iff.rfl⟩⟩⟩ | lemma | box_integral.box.exists_mem_split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_center_box_emb (I : box ι) : set ι ↪ box ι | ⟨split_center_box I, injective_split_center_box I⟩ | def | box_integral.box.split_center_box_emb | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [] | `box_integral.box.split_center_box` bundled as a `function.embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Union_coe_split_center_box (I : box ι) :
(⋃ s, (I.split_center_box s : set (ι → ℝ))) = I | by { ext x, simp } | lemma | box_integral.box.Union_coe_split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_sub_lower_split_center_box (I : box ι) (s : set ι) (i : ι) :
(I.split_center_box s).upper i - (I.split_center_box s).lower i = (I.upper i - I.lower i) / 2 | by by_cases hs : i ∈ s; field_simp [split_center_box, hs, mul_two, two_mul] | lemma | box_integral.box.upper_sub_lower_split_center_box | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [
"mul_two",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subbox_induction_on' {p : box ι → Prop} (I : box ι)
(H_ind : ∀ J ≤ I, (∀ s, p (split_center_box J s)) → p J)
(H_nhds : ∀ z ∈ I.Icc, ∃ (U ∈ 𝓝[I.Icc] z), ∀ (J ≤ I) (m : ℕ), z ∈ J.Icc → J.Icc ⊆ U →
(∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
p I | begin
by_contra hpI,
-- First we use `H_ind` to construct a decreasing sequence of boxes such that `∀ m, ¬p (J m)`.
replace H_ind := λ J hJ, not_imp_not.2 (H_ind J hJ),
simp only [exists_imp_distrib, not_forall] at H_ind,
choose! s hs using H_ind,
set J : ℕ → box ι := λ m, (λ J, split_center_box J (s J))^[m... | lemma | box_integral.box.subbox_induction_on' | analysis.box_integral.box | src/analysis/box_integral/box/subbox_induction.lean | [
"analysis.box_integral.box.basic",
"analysis.specific_limits.basic"
] | [
"antitone",
"antitone_nat_of_succ_le",
"by_contra",
"csupr_mem_Inter_Icc_of_antitone_Icc",
"div_div",
"exists_imp_distrib",
"monotone.comp_antitone",
"not_forall",
"one_lt_two",
"pow_succ'",
"tendsto_at_top_csupr",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within",
"tendsto_pow_at_t... | Let `p` be a predicate on `box ι`, let `I` be a box. Suppose that the following two properties
hold true.
* `H_ind` : Consider a smaller box `J ≤ I`. The hyperplanes passing through the center of `J` split
it into `2 ^ n` boxes. If `p` holds true on each of these boxes, then it true on `J`.
* `H_nhds` : For each `z... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
box_additive_map (ι M : Type*) [add_comm_monoid M] (I : with_top (box ι)) | (to_fun : box ι → M)
(sum_partition_boxes' : ∀ J : box ι, ↑J ≤ I → ∀ π : prepartition J, π.is_partition →
∑ Ji in π.boxes, to_fun Ji = to_fun J) | structure | box_integral.box_additive_map | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [
"add_comm_monoid",
"with_top"
] | A function on `box ι` is called box additive if for every box `J` and a partition `π` of `J`
we have `f J = ∑ Ji in π.boxes, f Ji`. A function is called box additive on subboxes of `I : box ι`
if the same property holds for `J ≤ I`. We formalize these two notions in the same definition
using `I : with_bot (box ι)`: the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (f : ι →ᵇᵃ[I₀] M) : f.to_fun = f | rfl | lemma | box_integral.box_additive_map.to_fun_eq_coe | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f h) : ⇑(mk f h : ι →ᵇᵃ[I₀] M) = f | rfl | lemma | box_integral.box_additive_map.coe_mk | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : injective (λ (f : ι →ᵇᵃ[I₀] M) x, f x) | by { rintro ⟨f, hf⟩ ⟨g, hg⟩ (rfl : f = g), refl } | lemma | box_integral.box_additive_map.coe_injective | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {f g : ι →ᵇᵃ[I₀] M} : (f : box ι → M) = g ↔ f = g | coe_injective.eq_iff | lemma | box_integral.box_additive_map.coe_inj | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_partition_boxes (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) {π : prepartition I}
(h : π.is_partition) :
∑ J in π.boxes, f J = f I | f.sum_partition_boxes' I hI π h | lemma | box_integral.box_additive_map.sum_partition_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_split_add (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) (i : ι) (x : ℝ) :
(I.split_lower i x).elim 0 f + (I.split_upper i x).elim 0 f = f I | by rw [← f.sum_partition_boxes hI (is_partition_split I i x), sum_split_boxes] | lemma | box_integral.box_additive_map.map_split_add | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (f : ι →ᵇᵃ[I₀] M) (I : with_top (box ι)) (hI : I ≤ I₀) : ι →ᵇᵃ[I] M | ⟨f, λ J hJ, f.2 J (hJ.trans hI)⟩ | def | box_integral.box_additive_map.restrict | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [
"with_top"
] | If `f` is box-additive on subboxes of `I₀`, then it is box-additive on subboxes of any
`I ≤ I₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_map_split_add [fintype ι] (f : box ι → M) (I₀ : with_top (box ι))
(hf : ∀ I : box ι, ↑I ≤ I₀ → ∀ {i x}, x ∈ Ioo (I.lower i) (I.upper i) →
(I.split_lower i x).elim 0 f + (I.split_upper i x).elim 0 f = f I) :
ι →ᵇᵃ[I₀] M | begin
refine ⟨f, _⟩,
replace hf : ∀ I : box ι, ↑I ≤ I₀ → ∀ s, ∑ J in (split_many I s).boxes, f J = f I,
{ intros I hI s,
induction s using finset.induction_on with a s ha ihs, { simp },
rw [split_many_insert, inf_split, ← ihs, bUnion_boxes, sum_bUnion_boxes],
refine finset.sum_congr rfl (λ J' hJ', _),... | def | box_integral.box_additive_map.of_map_split_add | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [
"finset.induction_on",
"fintype",
"with_top"
] | If `f : box ι → M` is box additive on partitions of the form `split I i x`, then it is box
additive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : ι →ᵇᵃ[I₀] M) (g : M →+ N) :
ι →ᵇᵃ[I₀] N | { to_fun := g ∘ f,
sum_partition_boxes' := λ I hI π hπ, by rw [← g.map_sum, f.sum_partition_boxes hI hπ] } | def | box_integral.box_additive_map.map | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | If `g : M → N` is an additive map and `f` is a box additive map, then `g ∘ f` is a box additive
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_boxes_congr [finite ι] (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) {π₁ π₂ : prepartition I}
(h : π₁.Union = π₂.Union) :
∑ J in π₁.boxes, f J = ∑ J in π₂.boxes, f J | begin
rcases exists_split_many_inf_eq_filter_of_finite {π₁, π₂} ((finite_singleton _).insert _)
with ⟨s, hs⟩, simp only [inf_split_many] at hs,
rcases ⟨hs _ (or.inl rfl), hs _ (or.inr rfl)⟩ with ⟨h₁, h₂⟩, clear hs,
rw h at h₁,
calc ∑ J in π₁.boxes, f J = ∑ J in π₁.boxes, ∑ J' in (split_many J s).boxes, f J'... | lemma | box_integral.box_additive_map.sum_boxes_congr | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [
"finite"
] | If `f` is a box additive function on subboxes of `I` and `π₁`, `π₂` are two prepartitions of
`I` that cover the same part of `I`, then `∑ J in π₁.boxes, f J = ∑ J in π₂.boxes, f J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_smul (f : ι →ᵇᵃ[I₀] ℝ) : ι →ᵇᵃ[I₀] (E →L[ℝ] E) | f.map (continuous_linear_map.lsmul ℝ ℝ).to_linear_map.to_add_monoid_hom | def | box_integral.box_additive_map.to_smul | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [
"continuous_linear_map.lsmul"
] | If `f` is a box-additive map, then so is the map sending `I` to the scalar multiplication
by `f I` as a continuous linear map from `E` to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_smul_apply (f : ι →ᵇᵃ[I₀] ℝ) (I : box ι) (x : E) : f.to_smul I x = f I • x | rfl | lemma | box_integral.box_additive_map.to_smul_apply | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u} upper_sub_lower {G : Type u} [add_comm_group G]
(I₀ : box (fin (n + 1))) (i : fin (n + 1)) (f : ℝ → box (fin n) → G)
(fb : Icc (I₀.lower i) (I₀.upper i) → fin n →ᵇᵃ[I₀.face i] G)
(hf : ∀ x (hx : x ∈ Icc (I₀.lower i) (I₀.upper i)) J, f x J = fb ⟨x, hx⟩ J) :
fin (n + 1) →ᵇᵃ[I₀] G | of_map_split_add
(λ J : box (fin (n + 1)), f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) I₀
begin
intros J hJ j,
rw with_top.coe_le_coe at hJ,
refine i.succ_above_cases _ _ j,
{ intros x hx,
simp only [box.split_lower_def hx, box.split_upper_def hx, update_same,
← with_bot.some_... | def | box_integral.box_additive_map.upper_sub_lower | analysis.box_integral.partition | src/analysis/box_integral/partition/additive.lean | [
"analysis.box_integral.partition.split",
"analysis.normed_space.operator_norm"
] | [
"add_comm_group",
"fin.succ_above_ne",
"option.elim",
"update_comp_eq_of_injective",
"update_noteq",
"update_same",
"with_bot.some_eq_coe",
"with_top",
"with_top.coe_le_coe"
] | Given a box `I₀` in `ℝⁿ⁺¹`, `f x : box (fin n) → G` is a family of functions indexed by a real
`x` and for `x ∈ [I₀.lower i, I₀.upper i]`, `f x` is box-additive on subboxes of the `i`-th face of
`I₀`, then `λ J, f (J.upper i) (J.face i) - f (J.lower i) (J.face i)` is box-additive on subboxes of
`I₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prepartition (I : box ι) | (boxes : finset (box ι))
(le_of_mem' : ∀ J ∈ boxes, J ≤ I)
(pairwise_disjoint : set.pairwise ↑boxes (disjoint on (coe : box ι → set (ι → ℝ)))) | structure | box_integral.prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"finset",
"set.pairwise"
] | A prepartition of `I : box_integral.box ι` is a finite set of pairwise disjoint subboxes of
`I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_boxes : J ∈ π.boxes ↔ J ∈ π | iff.rfl | lemma | box_integral.prepartition.mem_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk {s h₁ h₂} : J ∈ (mk s h₁ h₂ : prepartition I) ↔ J ∈ s | iff.rfl | lemma | box_integral.prepartition.mem_mk | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_coe_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (h : J₁ ≠ J₂) :
disjoint (J₁ : set (ι → ℝ)) J₂ | π.pairwise_disjoint h₁ h₂ h | lemma | box_integral.prepartition.disjoint_coe_of_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_mem_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hx₁ : x ∈ J₁) (hx₂ : x ∈ J₂) :
J₁ = J₂ | by_contra $ λ H, (π.disjoint_coe_of_mem h₁ h₂ H).le_bot ⟨hx₁, hx₂⟩ | lemma | box_integral.prepartition.eq_of_mem_of_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_le_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle₁ : J ≤ J₁) (hle₂ : J ≤ J₂) :
J₁ = J₂ | π.eq_of_mem_of_mem h₁ h₂ (hle₁ J.upper_mem) (hle₂ J.upper_mem) | lemma | box_integral.prepartition.eq_of_le_of_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle : J₁ ≤ J₂) : J₁ = J₂ | π.eq_of_le_of_le h₁ h₂ le_rfl hle | lemma | box_integral.prepartition.eq_of_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mem (hJ : J ∈ π) : J ≤ I | π.le_of_mem' J hJ | lemma | box_integral.prepartition.le_of_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_le_lower (hJ : J ∈ π) : I.lower ≤ J.lower | box.antitone_lower (π.le_of_mem hJ) | lemma | box_integral.prepartition.lower_le_lower | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_le_upper (hJ : J ∈ π) : J.upper ≤ I.upper | box.monotone_upper (π.le_of_mem hJ) | lemma | box_integral.prepartition.upper_le_upper | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_boxes : function.injective (boxes : prepartition I → finset (box ι)) | by { rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂), refl } | lemma | box_integral.prepartition.injective_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (h : ∀ J, J ∈ π₁ ↔ J ∈ π₂) : π₁ = π₂ | injective_boxes $ finset.ext h | lemma | box_integral.prepartition.ext | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single (I J : box ι) (h : J ≤ I) : prepartition I | ⟨{J}, by simpa, by simp⟩ | def | box_integral.prepartition.single | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | The singleton prepartition `{J}`, `J ≤ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_single {J'} (h : J ≤ I) : J' ∈ single I J h ↔ J' = J | mem_singleton | lemma | box_integral.prepartition.mem_single | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def : π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∃ J' ∈ π₂, J ≤ J' | iff.rfl | lemma | box_integral.prepartition.le_def | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top : J ∈ (⊤ : prepartition I) ↔ J = I | mem_singleton | lemma | box_integral.prepartition.mem_top | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_boxes : (⊤ : prepartition I).boxes = {I} | rfl | lemma | box_integral.prepartition.top_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_bot : J ∉ (⊥ : prepartition I) | id | lemma | box_integral.prepartition.not_mem_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_boxes : (⊥ : prepartition I).boxes = ∅ | rfl | lemma | box_integral.prepartition.bot_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_on_set_of_mem_Icc_set_of_lower_eq (x : ι → ℝ) :
inj_on (λ J : box ι, {i | J.lower i = x i}) {J | J ∈ π ∧ x ∈ J.Icc} | begin
rintros J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : {i | J₁.lower i = x i} = {i | J₂.lower i = x i}),
suffices : ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).nonempty,
{ choose y hy₁ hy₂,
exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂ },
intro i,
simp only [set.ext_iff, mem_set_of_eq] at H,
cas... | lemma | box_integral.prepartition.inj_on_set_of_mem_Icc_set_of_lower_eq | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"set.ext_iff",
"set.nonempty_Ioc",
"sup_idem"
] | An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ fintype.card ι` closed boxes of a prepartition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_filter_mem_Icc_le [fintype ι] (x : ι → ℝ) :
(π.boxes.filter (λ J : box ι, x ∈ J.Icc)).card ≤ 2 ^ fintype.card ι | begin
rw [← fintype.card_set],
refine finset.card_le_card_of_inj_on (λ J : box ι, {i | J.lower i = x i})
(λ _ _, finset.mem_univ _) _,
simpa only [finset.mem_filter] using π.inj_on_set_of_mem_Icc_set_of_lower_eq x
end | lemma | box_integral.prepartition.card_filter_mem_Icc_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.card_le_card_of_inj_on",
"finset.mem_filter",
"finset.mem_univ",
"fintype",
"fintype.card",
"fintype.card_set"
] | The set of boxes of a prepartition that contain `x` in their closures has cardinality
at most `2 ^ fintype.card ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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