| import OAI.Combinatorics.CliqueFree.FiniteEntropy |
| import OAI.Combinatorics.CliqueFree.WeightedGraph |
|
|
| namespace OAI |
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| noncomputable section |
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|
| open scoped BigOperators |
| open Finset |
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| attribute [local instance] Classical.propDecidable |
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| namespace CliqueFreeIndependence.WeightedGraph |
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| universe u v |
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| variable {V : Type u} [Fintype V] |
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| / |
| abbrev EdgeState (G : SimpleGraph V) := {e : V × V // G.Adj e.1 e.2} |
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| noncomputable instance edgeStateFintype (G : SimpleGraph V) : Fintype (EdgeState G) := |
| Subtype.fintype _ |
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| local instance (priority := 10000) edgeStateDecEq (G : SimpleGraph V) : DecidableEq (EdgeState G) := |
| Classical.decEq _ |
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|
| namespace EdgeState |
| variable {G : SimpleGraph V} |
| def src (e : EdgeState G) : V := e.val.1 |
| def dst (e : EdgeState G) : V := e.val.2 |
| omit [Fintype V] in |
| @[simp] lemma adj (e : EdgeState G) : G.Adj e.src e.dst := e.property |
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| def rev (e : EdgeState G) : EdgeState G := ⟨(e.dst, e.src), e.adj.symm⟩ |
| omit [Fintype V] in |
| @[simp] lemma src_rev (e : EdgeState G) : e.rev.src = e.dst := rfl |
| omit [Fintype V] in |
| @[simp] lemma dst_rev (e : EdgeState G) : e.rev.dst = e.src := rfl |
| omit [Fintype V] in |
| @[simp] lemma rev_rev (e : EdgeState G) : e.rev.rev = e := by cases e; rfl |
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| def revEquiv : EdgeState G ≃ EdgeState G where |
| toFun := rev |
| invFun := rev |
| left_inv := rev_rev |
| right_inv := rev_rev |
|
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| lemma sum_eq (f : V → V → ℝ) : |
| (∑ e : EdgeState G, f e.src e.dst) = ∑ u, ∑ v ∈ neighbors G u, f u v := by |
| classical |
| change (∑ e : {e : V × V // G.Adj e.1 e.2}, f e.val.1 e.val.2) = _ |
| rw [← Finset.sum_subtype (univ.filter fun e : V × V ↦ G.Adj e.1 e.2) |
| (by simp) (fun e : V × V ↦ f e.1 e.2)] |
| simp only [sum_filter, Fintype.sum_prod_type, neighbors] |
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| lemma sum_fiber (u : V) (f : V → ℝ) : |
| (∑ e : EdgeState G, if e.src = u then f e.dst else 0) = ∑ v ∈ neighbors G u, f v := by |
| classical |
| rw [sum_eq (fun a b ↦ if a = u then f b else 0)] |
| simp |
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| lemma sum_fiber' (u : V) (f : V → ℝ) : |
| (∑ e : EdgeState G, if u = e.src then f e.dst else 0) = ∑ v ∈ neighbors G u, f v := by |
| classical |
| simpa only [eq_comm] using sum_fiber u f |
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| end EdgeState |
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| lemma growth_nonneg {x s : ℝ} (hx : 0 ≤ x) (hs : 0 ≤ s) : 0 ≤ growth x s := |
| mul_nonneg hs (add_nonneg hx (le_max_left _ _)) |
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| @[simp] lemma growth_zero (x : ℝ) : growth x 0 = 0 := by simp [growth] |
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| lemma growth_small {x s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) : |
| growth x s = s * (x - Real.log s) := by |
| have hl : Real.log s ≤ 0 := Real.log_nonpos hs.le hs1 |
| simp [growth, Real.log_inv, max_eq_right (neg_nonneg.2 hl), sub_eq_add_neg] |
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| lemma growth_large {x s : ℝ} (hs : 1 ≤ s) : growth x s = s * x := by |
| have hl : 0 ≤ Real.log s := Real.log_nonneg hs |
| simp [growth, Real.log_inv, max_eq_left (neg_nonpos.2 hl)] |
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| lemma growth_mono {x : ℝ} (hx : 1 ≤ x) : MonotoneOn (growth x) (Set.Ici 0) := by |
| intro a ha b hb hab |
| change 0 ≤ a at ha |
| change 0 ≤ b at hb |
| rcases ha.eq_or_lt with ha | ha |
| · rw [← ha, growth_zero] |
| exact growth_nonneg (by linarith) hb |
| have hb0 : 0 < b := ha.trans_le hab |
| by_cases hb1 : b ≤ 1 |
| · rw [growth_small ha (hab.trans hb1), growth_small hb0 hb1] |
| have hrel := FiniteEntropy.relTerm_nonneg ha.le hb0.le (fun _ ↦ hb0) |
| rw [Real.log_div ha.ne' hb0.ne'] at hrel |
| have hlog : Real.log b ≤ 0 := Real.log_nonpos hb0.le hb1 |
| have hmul := mul_nonneg (sub_nonneg.2 hab) (show 0 ≤ x - 1 - Real.log b by linarith) |
| nlinarith |
| by_cases ha1 : 1 ≤ a |
| · rw [growth_large ha1, growth_large (le_of_not_ge hb1)] |
| exact mul_le_mul_of_nonneg_right hab (by linarith) |
| · rw [growth_small ha (le_of_not_ge ha1), growth_large (le_of_not_ge hb1)] |
| have hrel := FiniteEntropy.relTerm_nonneg ha.le (by norm_num : (0:ℝ) ≤ 1) |
| (fun _ ↦ by norm_num) |
| simp only [div_one] at hrel |
| have hamul := mul_nonneg (sub_nonneg.2 (le_of_not_ge ha1)) (show 0 ≤ x - 1 by linarith) |
| have hbmul := mul_nonneg (sub_nonneg.2 (le_of_not_ge hb1)) (show 0 ≤ x by linarith) |
| nlinarith |
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|
| lemma growth_le_twice {x s : ℝ} (hx : 1 ≤ x) (hs : 0 ≤ s) : |
| growth x s ≤ 2 * x * (s + Real.exp (-x)) := by |
| have hx0 : 0 ≤ x := by linarith |
| have hexp := Real.exp_pos (-x) |
| have hi (b : ℝ) (hb : Real.exp (-x) ≤ b) : growth x b ≤ 2 * x * b := by |
| have hb0 := hexp.trans_le hb |
| have hl : -Real.log b ≤ x := by |
| have := Real.log_le_log hexp hb |
| rw [Real.log_exp] at this |
| linarith |
| have hmax : max 0 (-Real.log b) ≤ x := max_le hx0 hl |
| simp only [growth, one_div, Real.log_inv] |
| nlinarith [mul_le_mul_of_nonneg_left hmax hb0.le] |
| by_cases h : Real.exp (-x) ≤ s |
| · have h' := hi s h |
| nlinarith [mul_nonneg hx0 hexp.le] |
| · have h' := (growth_mono hx hs hexp.le (le_of_not_ge h)).trans (hi _ le_rfl) |
| nlinarith [mul_nonneg hx0 hs] |
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|
| omit [Fintype V] in |
| lemma mass_union_le {w : V → ℝ} (hw : ∀ v, 0 ≤ w v) (A B : Finset V) : |
| mass w (A ∪ B) ≤ mass w A + mass w B := by |
| have h := sum_union_inter (s₁ := A) (s₂ := B) (f := w) |
| have hn := mass_nonneg hw (A ∩ B) |
| change 0 ≤ ∑ u ∈ A ∩ B, w u at hn |
| change (∑ u ∈ A ∪ B, w u) ≤ (∑ u ∈ A, w u) + ∑ u ∈ B, w u |
| linarith |
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| end CliqueFreeIndependence.WeightedGraph |
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| end |
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| end OAI |
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