| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.ArraySum |
|
|
| namespace OAI |
|
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| variable {m n : ℕ} |
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| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Polynomial Filter |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory WithLp |
| open scoped BigOperators RealInnerProductSpace |
| namespace MinUncut.Inner |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| def hermiteSupport (I : Point m n → ℕ) : Finset (Point m n) := Finset.univ.filter (fun u => 0 < I u) |
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| def matchingAxes (x u : Point m n) : Finset (Fin m) := |
| Finset.univ.filter (fun i => face u i = face x i) |
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| def forbiddenAxes (x : Point m n) (I : Point m n → ℕ) : Finset (Fin m) := |
| (hermiteSupport I).biUnion (matchingAxes x) |
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| def goodAxes (x : Point m n) (I : Point m n → ℕ) : Finset (Fin m) := |
| Finset.univ \ forbiddenAxes x I |
|
|
| lemma eq_of_two_faces {x u : Point m n} {i j : Fin m} (hij : i ≠ j) |
| (hi : face u i = face x i) (hj : face u j = face x j) : u=x := by |
| funext t |
| by_cases ht : t=i |
| · subst t; exact congrFun hj ⟨i,hij⟩ |
| · exact congrFun hi ⟨t,ht⟩ |
|
|
| lemma matchingAxes_card {x u : Point m n} (hux : u ≠ x) : (matchingAxes x u).card ≤ 1 := by |
| apply Finset.card_le_one.mpr |
| intro i hi j hj |
| by_contra hij |
| exact hux (eq_of_two_faces hij (Finset.mem_filter.mp hi).2 (Finset.mem_filter.mp hj).2) |
|
|
| lemma hermiteSupport_card (I : Point m n → ℕ) : (hermiteSupport I).card ≤ ∑ u, I u := by |
| calc |
| _ = ∑ u ∈ hermiteSupport I, 1 := by simp |
| _ ≤ ∑ u ∈ hermiteSupport I, I u := Finset.sum_le_sum (fun u hu => (Finset.mem_filter.mp hu).2) |
| _ ≤ ∑ u, I u := Finset.sum_le_univ_sum_of_nonneg (fun _ => Nat.zero_le _) |
|
|
| lemma forbiddenAxes_card (x : Point m n) (I : Point m n → ℕ) (hx : I x=0) : |
| (forbiddenAxes x I).card ≤ ∑ u, I u := by |
| have h := Finset.card_biUnion_le_card_mul (hermiteSupport I) (matchingAxes x) 1 (by |
| intro u hu |
| apply matchingAxes_card |
| intro he; subst u |
| have := (Finset.mem_filter.mp hu).2 |
| simp [hx] at this) |
| exact (h.trans (by simpa using hermiteSupport_card I)) |
|
|
| lemma goodAxes_card (x : Point m n) (I : Point m n → ℕ) (hx : I x=0) : |
| m-(∑ u, I u) ≤ (goodAxes x I).card := by |
| have h := forbiddenAxes_card x I hx |
| rw [goodAxes, Finset.card_sdiff_of_subset (Finset.subset_univ _), Finset.card_univ, Fintype.card_fin] |
| omega |
|
|
| lemma goodAxis_no_support (x : Point m n) (I : Point m n → ℕ) {i : Fin m} |
| (hi : i ∈ goodAxes x I) (u : Point m n) (hu : 0 < I u) : face u i ≠ face x i := by |
| intro hf |
| exact (Finset.mem_sdiff.mp hi).2 (Finset.mem_biUnion.mpr |
| ⟨u,Finset.mem_filter.mpr ⟨Finset.mem_univ _,hu⟩,Finset.mem_filter.mpr ⟨Finset.mem_univ _,hf⟩⟩) |
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| lemma face_rowFill (x : Point m n) (r : Row m n) : face (rowFill x r) r.1 = r.2 := by |
| funext t |
| simp [face, rowFill, t.property] |
|
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| lemma rowFill_axis (x : Point m n) (r : Row m n) : rowFill x r r.1 = x r.1 := by |
| simp [rowFill] |
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| def rowObstructions (x : Point m n) (r : Row m n) : Finset (Fin m) := |
| Finset.univ.filter (fun j => r.1 ≠ j ∧ r.2 ≠ face x r.1 ∧ doubleMatch r.1 j x (rowFill x r)) |
|
|
| lemma obstruction_matching (x : Point m n) (r : Row m n) {j : Fin m} |
| (hj : j ∈ rowObstructions x r) : j ∈ matchingAxes x (rowFill x r) := by |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_univ _, ?_⟩ |
| funext t |
| by_cases ht : t.1=r.1 |
| · change rowFill x r t.1 = x t.1 |
| rw [ht, rowFill_axis] |
| · exact (Finset.mem_filter.mp hj).2.2.2 t.1 ht t.property |
|
|
| lemma rowObstructions_card (x : Point m n) (r : Row m n) : (rowObstructions x r).card ≤ 1 := by |
| by_cases hlocal : r.2 = face x r.1 |
| · simp [rowObstructions, hlocal] |
| · refine (Finset.card_le_card (fun _ hj => obstruction_matching x r hj)).trans (matchingAxes_card ?_) |
| intro he |
| have hh := congrArg (fun u => face u r.1) he |
| rw [face_rowFill] at hh |
| exact hlocal hh |
|
|
| def forbiddenPairs (x : Point m n) (S : Finset (Row m n)) : Finset (Finset (Fin m)) := |
| S.biUnion (fun r => (rowObstructions x r).image (fun j => {r.1,j})) |
|
|
| lemma forbiddenPairs_card (x : Point m n) (S : Finset (Row m n)) : |
| (forbiddenPairs x S).card ≤ S.card := by |
| have h := Finset.card_biUnion_le_card_mul S |
| (fun r => (rowObstructions x r).image (fun j => ({r.1,j} : Finset (Fin m)))) 1 |
| (fun r _ => Finset.card_image_le.trans (rowObstructions_card x r)) |
| simpa [forbiddenPairs] using h |
|
|
| lemma pair_not_forbidden_local (x : Point m n) (S : Finset (Row m n)) |
| {i j : Fin m} (hij : i ≠ j) (hp : ({i,j} : Finset (Fin m)) ∉ forbiddenPairs x S) |
| (y : {t : Fin m // t ≠ i} → Fin n) (hy : (⟨i,y⟩ : Row m n) ∈ S) |
| (hc : doubleMatch i j x (rowFill x ⟨i,y⟩)) : y = face x i := by |
| by_contra hn |
| apply hp |
| apply Finset.mem_biUnion.mpr |
| refine ⟨⟨i,y⟩,hy,Finset.mem_image.mpr ⟨j,?_,rfl⟩⟩ |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,hij,hn,hc⟩ |
|
|
| theorem admissible_pair (x : Point m n) (I : Point m n → ℕ) (S : Finset (Row m n)) |
| (hx : I x=0) (hsize : S.card < (m-(∑ u, I u)).choose 2) : |
| ∃ i j : Fin m, i ≠ j ∧ |
| (∀ u, 0 < I u → face u i ≠ face x i ∧ face u j ≠ face x j) ∧ |
| (∀ y, (⟨i,y⟩ : Row m n) ∈ S → doubleMatch i j x (rowFill x ⟨i,y⟩) → y=face x i) ∧ |
| (∀ y, (⟨j,y⟩ : Row m n) ∈ S → doubleMatch j i x (rowFill x ⟨j,y⟩) → y=face x j) := by |
| have hcard : (forbiddenPairs x S).card < ((goodAxes x I).powersetCard 2).card := by |
| calc |
| _ ≤ S.card := forbiddenPairs_card x S |
| _ < (m-(∑ u, I u)).choose 2 := hsize |
| _ ≤ _ := by rw [Finset.card_powersetCard]; exact Nat.choose_le_choose 2 (goodAxes_card x I hx) |
| have hn : ¬ (goodAxes x I).powersetCard 2 ⊆ forbiddenPairs x S := by |
| intro hh; exact (Nat.not_le_of_lt hcard) (Finset.card_le_card hh) |
| obtain ⟨p,hp,hpn⟩ := Finset.not_subset.mp hn |
| obtain ⟨i,j,hij,he⟩ := Finset.card_eq_two.mp (Finset.mem_powersetCard.mp hp).2 |
| subst p |
| have hi := (Finset.mem_powersetCard.mp hp).1 (by simp : i ∈ ({i,j} : Finset (Fin m))) |
| have hj := (Finset.mem_powersetCard.mp hp).1 (by simp : j ∈ ({i,j} : Finset (Fin m))) |
| refine ⟨i,j,hij,?_,?_,?_⟩ |
| · intro u hu |
| exact ⟨goodAxis_no_support x I hi u hu,goodAxis_no_support x I hj u hu⟩ |
| · exact fun y hy hc => pair_not_forbidden_local x S hij hpn y hy hc |
| · exact fun y hy hc => pair_not_forbidden_local x S hij.symm (by simpa [Finset.pair_comm] using hpn) y hy hc |
| end MinUncut.Inner |
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| end |
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| end OAI |
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