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feat(analysis/normed_space/box_subadditive): define box_(sub|sup|)additive_on
#4913
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urkud
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🎉 Great news! Looks like all the dependencies have been resolved:
💡 To add or remove a dependency please update this issue/PR description. Brought to you by Dependent Issues (:robot: ). Happy coding! |
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| A function `f : (ι → ℝ) → (ι → ℝ) → M` is called | ||
| `box_subadditive_on`/`box_additive_on`/`box_supadditive_on` a set `s` if for any `l u : ι → ℝ`, `l ≤ | ||
| u`, `Icc l u ⊆ s` and any hyperplane `x i = c`, `l i ≤ c ≤ u i`, the sum of the values of `f` on the | ||
| two subboxes `[l, u] ∩ {x | x i ≤ c}` and `[l, u] ∩ {x | c ≤ x i}` is greater than or | ||
| equal/equal/less than or equal to its value on `[l, u]`. |
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This docstring is hard to understand. Do you mean that you think of the value f l u as the value of the function on the box with lower corner l and upper corner u, i.e., the set Icc l u, that you denote with the usual interval notation [l, u]?
| The main result of this file is theorem `box_subadditive_on.eq_zero_of_forall_is_o_prod`. It says | ||
| that `f l u = 0` provided that `f` is `box_subadditive_on` the interval `[l, u]`, `l ≤ u`, and | ||
| for any `p ∈ [l, u]` we have `f l' u' = o(volume [l', u'])` as both `l'` tends to `p` along | ||
| `[l, p]`, `u'` tends to `p` along `[p, u]`, and the subbox `[l', u']` is homothetic to `[l, u]`. |
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Could you give a motivation for this statement, and maybe also a sketch of proof?
| tendsto_inf.2 ⟨tendsto_right_nhds_fix hf hg I, tendsto_principal.2 $ eventually_of_forall $ | ||
| λ n, ⟨(fix_mem_seq hf hg I n).2, seq_right_mono hf hg I (zero_le n)⟩⟩ | ||
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| lemma le_mul_of_forall_eventually_le_mul (hle : l ≤ u) |
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docstring? (This is really more general than the next one, so it deserves a docstring IMHO)
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| open asymptotics function | ||
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| lemma eq_zero_of_forall_is_o (hle : l ≤ u) (hf : box_subadditive_on (λ x y, ∥f x y∥) (Icc l u)) |
| (hg : box_supadditive_on (λ x y, ∥g x y∥) (Icc l u)) | ||
| (Hc : ∀ (b ∈ Icc l u), is_o (λ p : _ × ℝ, uncurry f p.1) (λ p, uncurry g p.1) | ||
| ((𝓝[Icc l b] b ×ᶠ 𝓝[Icc b u] b ×ᶠ 𝓝[Ioi 0] 0) ⊓ 𝓟 {p | p.1.2 - p.1.1 = p.2 • (u - l)})) | ||
| : f l u = 0 := |
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move colon to the previous line
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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norm_castlemmas, golf proofs #4911