| import OAI.Geometry.KahlerSplitting.Model |
|
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| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| namespace RectangleGrid |
| variable {P Q X : Type*} |
|
|
| def band (k i j : ℕ) : Prop := i ≤ j+k ∧ j ≤ i+k |
|
|
| structure Band (T : Set (P × Q → X)) (D : ℕ → P × Q → X) |
| (lo hi : ℕ → P) (bot top : ℕ → Q) (k : ℕ) where |
| tile : ℕ → ℕ → P × Q → X |
| member : ∀ i j, band k i j → tile i j ∈ T |
| diagonal : ∀ i, tile i i=D i |
| hseam : ∀ i j, band k i j → band k (i+1) j → |
| ∀ y, tile i j (hi i,y)=tile (i+1) j (lo (i+1),y) |
| vseam : ∀ i j, band k i j → band k i (j+1) → |
| ∀ x, tile i j (x,top j)=tile i (j+1) (x,bot (j+1)) |
| |
| theorem band_extend (H : Set (P → X)) (V : Set (Q → X)) (T : Set (P × Q → X)) |
| (horizontal : ∀ f ∈ T, ∀ q, (fun p ↦ f (p,q)) ∈ H) |
| (vertical : ∀ f ∈ T, ∀ p, (fun q ↦ f (p,q)) ∈ V) |
| (fill : ∀ a ∈ H, ∀ b ∈ V, ∀ p q, a p=b q → |
| ∃ f ∈ T, (∀ x, f (x,q)=a x) ∧ (∀ y, f (p,y)=b y)) |
| (D : ℕ → P × Q → X) (lo hi : ℕ → P) (bot top : ℕ → Q) |
| (join : ∀ i, D i (hi i,top i)=D (i+1) (lo (i+1),bot (i+1))) |
| (k : ℕ) (G : Band T D lo hi bot top k) : |
| ∃ F : Band T D lo hi bot top (k+1), ∀ i j, band k i j → F.tile i j=G.tile i j := by |
| classical |
| have up (i j : ℕ) (h : band (k+1) i j) (hn : ¬band k i j) (hij : i < j) : |
| ∃ f ∈ T, (∀ x, f (x,bot j)=G.tile i (j-1) (x,top (j-1))) ∧ |
| (∀ y, f (hi i,y)=G.tile (i+1) j (lo (i+1),y)) := by |
| have b₁ : band k i (j-1) := by unfold band at * |
| have b₂ : band k (i+1) j := by unfold band at * |
| apply fill _ (horizontal _ (G.member _ _ b₁) _) _ (vertical _ (G.member _ _ b₂) _) |
| by_cases he : j=i+1 |
| · subst j |
| simpa only [Nat.add_sub_cancel,G.diagonal] using join i |
| · have b₃ : band k (i+1) (j-1) := by unfold band at * |
| have e₁ := G.hseam i (j-1) b₁ b₃ (top (j-1)) |
| have e₂ := G.vseam (i+1) (j-1) b₃ (by simpa only [Nat.sub_add_cancel (by omega : 1≤ j)] using b₂) (lo (i+1)) |
| exact e₁.trans (by simpa only [Nat.sub_add_cancel (by omega : 1≤ j)] using e₂) |
| have down (i j : ℕ) (h : band (k+1) i j) (hn : ¬band k i j) (hij : j < i) : |
| ∃ f ∈ T, (∀ x, f (x,top j)=G.tile i (j+1) (x,bot (j+1))) ∧ |
| (∀ y, f (lo i,y)=G.tile (i-1) j (hi (i-1),y)) := by |
| have b₁ : band k i (j+1) := by unfold band at * |
| have b₂ : band k (i-1) j := by unfold band at * |
| apply fill _ (horizontal _ (G.member _ _ b₁) _) _ (vertical _ (G.member _ _ b₂) _) |
| by_cases he : i=j+1 |
| · subst i |
| simpa only [Nat.add_sub_cancel,G.diagonal] using (join j).symm |
| · have b₃ : band k (i-1) (j+1) := by unfold band at * |
| have e₁ := G.hseam (i-1) (j+1) b₃ (by simpa only [Nat.sub_add_cancel (by omega : 1≤ i)] using b₁) (bot (j+1)) |
| have e₂ := G.vseam (i-1) j b₂ b₃ (hi (i-1)) |
| simp only [Nat.sub_add_cancel (by omega : 1≤ i)] at e₁ |
| exact e₁.symm.trans e₂.symm |
| let f (i j : ℕ) : P × Q → X := |
| if h : band k i j then G.tile i j else |
| if hb : band (k+1) i j then |
| if hij : i < j then (up i j hb h hij).choose |
| else (down i j hb h (by unfold band at * |
| else D 0 |
| have old (i j : ℕ) (h : band k i j) : f i j=G.tile i j := by simp only [f,dite_eq_left h] |
| have upper (i j : ℕ) (h : band (k+1) i j) (hn : ¬band k i j) (hij : i < j) : |
| f i j ∈ T ∧ (∀ x, f i j (x,bot j)=G.tile i (j-1) (x,top (j-1))) ∧ |
| (∀ y, f i j (hi i,y)=G.tile (i+1) j (lo (i+1),y)) := by |
| simpa only [f,dite_eq_right hn,dite_eq_left h,dite_eq_left hij] using (up i j h hn hij).choose_spec |
| have lower (i j : ℕ) (h : band (k+1) i j) (hn : ¬band k i j) (hij : j < i) : |
| f i j ∈ T ∧ (∀ x, f i j (x,top j)=G.tile i (j+1) (x,bot (j+1))) ∧ |
| (∀ y, f i j (lo i,y)=G.tile (i-1) j (hi (i-1),y)) := by |
| simpa only [f,dite_eq_right hn,dite_eq_left h,dite_eq_right (not_lt.mpr (Nat.le_of_lt hij))] using |
| (down i j h hn hij).choose_spec |
| refine ⟨⟨f,?_,?_,?_,?_⟩,old⟩ |
| · intro i j h |
| by_cases ho : band k i j |
| · rw [old i j ho] |
| · by_cases hij : i < j |
| · exact (upper i j h ho hij).1 |
| · exact (lower i j h ho (by unfold band at * |
| · intro i |
| rw [old i i (by unfold band; omega),G.diagonal] |
| · intro i j ha hb y |
| by_cases hA : band k i j |
| · rw [old i j hA] |
| by_cases hB : band k (i+1) j |
| · rw [old (i+1) j hB] |
| · have e := (lower (i+1) j hb hB (by unfold band at * |
| simpa only [Nat.add_sub_cancel] using e.symm |
| · have hB : band k (i+1) j := by unfold band at * |
| rw [old (i+1) j hB] |
| exact (upper i j ha hA (by unfold band at * |
| · intro i j ha hb x |
| by_cases hA : band k i j |
| · rw [old i j hA] |
| by_cases hB : band k i (j+1) |
| · rw [old i (j+1) hB] |
| · have e := (upper i (j+1) hb hB (by unfold band at * |
| simpa only [Nat.add_sub_cancel] using e.symm |
| · have hB : band k i (j+1) := by unfold band at * |
| rw [old i (j+1) hB] |
| exact (lower i j ha hA (by unfold band at * |
|
|
| theorem diagonal_grid (H : Set (P → X)) (V : Set (Q → X)) (T : Set (P × Q → X)) |
| (horizontal : ∀ f ∈ T, ∀ q, (fun p ↦ f (p,q)) ∈ H) |
| (vertical : ∀ f ∈ T, ∀ p, (fun q ↦ f (p,q)) ∈ V) |
| (fill : ∀ a ∈ H, ∀ b ∈ V, ∀ p q, a p=b q → |
| ∃ f ∈ T, (∀ x, f (x,q)=a x) ∧ (∀ y, f (p,y)=b y)) |
| (D : ℕ → P × Q → X) (hD : ∀ i, D i ∈ T) |
| (lo hi : ℕ → P) (bot top : ℕ → Q) |
| (join : ∀ i, D i (hi i,top i)=D (i+1) (lo (i+1),bot (i+1))) : |
| ∃ f : ℕ → ℕ → P × Q → X, (∀ i j, f i j ∈ T) ∧ |
| (∀ i, f i i=D i) ∧ |
| (∀ i j y, f i j (hi i,y)=f (i+1) j (lo (i+1),y)) ∧ |
| (∀ i j x, f i j (x,top j)=f i (j+1) (x,bot (j+1))) := by |
| classical |
| let seed : Band T D lo hi bot top 0 := |
| ⟨fun i _ ↦ D i,fun i _ _ ↦ hD i,fun _ ↦ rfl, |
| fun i j ha hb ↦ by unfold band at ha hb |
| fun i j ha hb ↦ by unfold band at ha hb |
| have ext := band_extend H V T horizontal vertical fill D lo hi bot top join |
| let seq : (k : ℕ) → Band T D lo hi bot top k := |
| Nat.rec seed (fun k G ↦ (ext k G).choose) |
| have stable (k i j : ℕ) (h : band k i j) : |
| (seq (k+1)).tile i j=(seq k).tile i j := (ext k (seq k)).choose_spec i j h |
| have stable' (k l : ℕ) (hkl : k≤ l) (i j : ℕ) (h : band k i j) : |
| (seq l).tile i j=(seq k).tile i j := by |
| induction l, hkl using Nat.le_induction with |
| | base => rfl |
| | succ l hkl ih => |
| exact (stable l i j (by unfold band at *; omega)).trans ih |
| let f (i j : ℕ) := (seq (i+j)).tile i j |
| have atBand (k i j : ℕ) (h : i+j≤ k) : f i j=(seq k).tile i j := |
| (stable' (i+j) k h i j (by unfold band; omega)).symm |
| refine ⟨f,fun i j ↦ (seq (i+j)).member i j (by unfold band; omega), |
| fun i ↦ (seq (i+i)).diagonal i,?_,?_⟩ |
| · intro i j y |
| rw [atBand (i+j+1) i j (by omega),atBand (i+j+1) (i+1) j (by omega)] |
| exact (seq (i+j+1)).hseam i j (by unfold band; omega) (by unfold band; omega) y |
| · intro i j x |
| rw [atBand (i+j+1) i j (by omega),atBand (i+j+1) i (j+1) (by omega)] |
| exact (seq (i+j+1)).vseam i j (by unfold band; omega) (by unfold band; omega) x |
|
|
| end RectangleGrid |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology |
| open scoped unitInterval |
| namespace IntervalChain |
| variable {X : Type*} |
|
|
| def cell (t : ℕ → I) (i : ℕ) := Icc (t i) (t (i+1)) |
|
|
| lemma overlap (t : ℕ → I) (ht : Monotone t) |
| (f : ∀ i, cell t i → X) |
| (seam : ∀ i, f i ⟨t (i+1),ht (Nat.le_succ i),le_rfl⟩ = |
| f (i+1) ⟨t (i+1),le_rfl,ht (Nat.le_succ (i+1))⟩) |
| (i j : ℕ) (x : I) (hi : x ∈ cell t i) (hj : x ∈ cell t j) : |
| f i ⟨x,hi⟩=f j ⟨x,hj⟩ := by |
| have half : ∀ j i, i ≤ j → ∀ (x : I) (hi : x ∈ cell t i) (hj : x ∈ cell t j), |
| f i ⟨x,hi⟩=f j ⟨x,hj⟩ := by |
| intro j |
| induction j with |
| | zero => intro i hij x hi hj; have he : i=0 := Nat.eq_zero_of_le_zero hij; subst i; rfl |
| | succ j ih => |
| intro i hij x hi hj |
| by_cases he : i=j+1 |
| · subst i |
| have hij' : i ≤ j := by omega |
| have hx : x=t (j+1) := le_antisymm (hi.2.trans (ht (by omega))) hj.1 |
| subst x |
| exact (ih i hij' _ hi ⟨ht (Nat.le_succ j),le_rfl⟩).trans (seam j) |
| rcases le_total i j with h|h |
| · exact half j i h x hi hj |
| · exact (half i j h x hj hi).symm |
| |
| lemma covers (t : ℕ → I) (ht0 : t 0=0) (ht : Monotone t) |
| (N : ℕ) (hN : t N=1) (x : I) : ∃ i ≤ N, x ∈ cell t i := by |
| by_cases hx : x=1 |
| · subst x |
| refine ⟨N,le_rfl,?_,?_⟩ |
| · simp only [hN,le_refl] |
| · exact (hN ▸ ht (Nat.le_succ N)) |
| have hx1 : x < 1 := lt_of_le_of_ne le_top hx |
| have hm : ∃ i, x < t (i+1) := ⟨N,lt_of_lt_of_le hx1 (hN ▸ ht (Nat.le_succ N))⟩ |
| let i := Nat.find hm |
| have hiN : i ≤ N := Nat.find_min' hm (lt_of_lt_of_le hx1 (hN ▸ ht (Nat.le_succ N))) |
| refine ⟨i,hiN,?_,(Nat.find_spec hm).le⟩ |
| cases hi : i with |
| | zero => simp [ht0] |
| | succ j => |
| have hn := Nat.find_min hm (show j < i from by omega) |
| exact le_of_not_gt (by simpa only [hi] using hn) |
| |
| end IntervalChain |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open Set Filter Topology |
| open scoped unitInterval |
| namespace PathSubdivision |
| variable {P X : Type*} [TopologicalSpace P] |
| |
| theorem parameterize (α : I → X) (Q : (P → X) → Prop) |
| (local_param : ∀ x : I, ∃ U : Set I, IsOpen U ∧ x ∈ U ∧ |
| ∃ f : P → X, Q f ∧ ∃ a : U → P, Continuous a ∧ ∀ y, f (a y)=α y) : |
| ∃ t : ℕ → I, t 0=0 ∧ Monotone t ∧ (∃ N, ∀ m ≥ N, t m=1) ∧ |
| ∃ f : ℕ → P → X, (∀ i, Q (f i)) ∧ |
| ∃ a : ∀ i, IntervalChain.cell t i → P, (∀ i, Continuous (a i)) ∧ |
| ∀ i y, f i (a i y)=α y := by |
| classical |
| choose U hU hx f hf a ha he using local_param |
| have cover : univ ⊆ ⋃ x, U x := fun x _ ↦ mem_iUnion.mpr ⟨x,hx x⟩ |
| obtain ⟨t,ht0,ht,hN,hsub⟩ := exists_monotone_Icc_subset_open_cover_unitInterval hU cover |
| choose k hk using hsub |
| refine ⟨t,ht0,ht,hN,fun i ↦ f (k i),fun i ↦ hf (k i), |
| fun i y ↦ a (k i) ⟨y,hk i y.property⟩,?_,?_⟩ |
| · intro i |
| exact (ha (k i)).comp (continuous_subtype_val.subtype_mk _) |
| · intro i y |
| exact he (k i) _ |
| |
| end PathSubdivision |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open Set |
| namespace RectangleGrid |
| variable {P Q X : Type*} |
| |
| theorem row (H : Set (P → X)) (V : Set (Q → X)) (T : Set (P × Q → X)) |
| (vertical : ∀ f ∈ T, ∀ p, (fun q ↦ f (p,q)) ∈ V) |
| (fill : ∀ a ∈ H, ∀ b ∈ V, ∀ p q, a p=b q → |
| ∃ f ∈ T, (∀ x, f (x,q)=a x) ∧ (∀ y, f (p,y)=b y)) |
| (a : ℕ → P → X) (ha : ∀ i, a i ∈ H) (lo hi : ℕ → P) |
| (join : ∀ i, a i (hi i)=a (i+1) (lo (i+1))) |
| (b : Q → X) (hb : b ∈ V) (q : Q) (corner : a 0 (lo 0)=b q) : |
| ∃ f : ℕ → P × Q → X, (∀ i, f i ∈ T) ∧ |
| (∀ i x, f i (x,q)=a i x) ∧ (∀ y, f 0 (lo 0,y)=b y) ∧ |
| (∀ i y, f i (hi i,y)=f (i+1) (lo (i+1),y)) := by |
| classical |
| let State (i : ℕ) := {c : Q → X // c ∈ V ∧ a i (lo i)=c q} |
| have exist (i : ℕ) (c : State i) := fill (a i) (ha i) c.val c.property.1 (lo i) q c.property.2 |
| let tile (i : ℕ) (c : State i) : P × Q → X := (exist i c).choose |
| have tileT (i : ℕ) (c : State i) : tile i c ∈ T := (exist i c).choose_spec.1 |
| have bottom (i : ℕ) (c : State i) (x : P) : tile i c (x,q)=a i x := |
| (exist i c).choose_spec.2.1 x |
| have left (i : ℕ) (c : State i) (y : Q) : tile i c (lo i,y)=c.val y := |
| (exist i c).choose_spec.2.2 y |
| let next (i : ℕ) (c : State i) : State (i+1) := |
| ⟨fun y ↦ tile i c (hi i,y),vertical _ (tileT i c) _,((bottom i c (hi i)).trans (join i)).symm⟩ |
| let seq : (i : ℕ) → State i := Nat.rec ⟨b,hb,corner⟩ next |
| refine ⟨fun i ↦ tile i (seq i),fun i ↦ tileT i _,fun i x ↦ bottom i _ x,left 0 _,?_⟩ |
| intro i y |
| exact (left (i+1) (seq (i+1)) y).symm |
| |
| theorem grid (H : Set (P → X)) (V : Set (Q → X)) (T : Set (P × Q → X)) |
| (horizontal : ∀ f ∈ T, ∀ q, (fun p ↦ f (p,q)) ∈ H) |
| (vertical : ∀ f ∈ T, ∀ p, (fun q ↦ f (p,q)) ∈ V) |
| (fill : ∀ a ∈ H, ∀ b ∈ V, ∀ p q, a p=b q → |
| ∃ f ∈ T, (∀ x, f (x,q)=a x) ∧ (∀ y, f (p,y)=b y)) |
| (a : ℕ → P → X) (ha : ∀ i, a i ∈ H) (lo hi : ℕ → P) |
| (joinA : ∀ i, a i (hi i)=a (i+1) (lo (i+1))) |
| (b : ℕ → Q → X) (hb : ∀ j, b j ∈ V) (bot top : ℕ → Q) |
| (joinB : ∀ j, b j (top j)=b (j+1) (bot (j+1))) |
| (corner : a 0 (lo 0)=b 0 (bot 0)) : |
| ∃ f : ℕ → ℕ → P × Q → X, (∀ i j, f i j ∈ T) ∧ |
| (∀ i x, f i 0 (x,bot 0)=a i x) ∧ |
| (∀ j y, f 0 j (lo 0,y)=b j y) ∧ |
| (∀ i j y, f i j (hi i,y)=f (i+1) j (lo (i+1),y)) ∧ |
| (∀ i j x, f i j (x,top j)=f i (j+1) (x,bot (j+1))) := by |
| classical |
| let State (j : ℕ) := {c : ℕ → P → X // |
| (∀ i, c i ∈ H) ∧ (∀ i, c i (hi i)=c (i+1) (lo (i+1))) ∧ c 0 (lo 0)=b j (bot j)} |
| have exist (j : ℕ) (c : State j) := |
| row H V T vertical fill c.val c.property.1 lo hi c.property.2.1 |
| (b j) (hb j) (bot j) c.property.2.2 |
| let tiles (j : ℕ) (c : State j) := (exist j c).choose |
| have ht (j : ℕ) (c : State j) (i : ℕ) : tiles j c i ∈ T := (exist j c).choose_spec.1 i |
| have bottom (j : ℕ) (c : State j) (i : ℕ) (x : P) : |
| tiles j c i (x,bot j)=c.val i x := (exist j c).choose_spec.2.1 i x |
| have left (j : ℕ) (c : State j) (y : Q) : |
| tiles j c 0 (lo 0,y)=b j y := (exist j c).choose_spec.2.2.1 y |
| have seam (j : ℕ) (c : State j) (i : ℕ) (y : Q) : |
| tiles j c i (hi i,y)=tiles j c (i+1) (lo (i+1),y) := |
| (exist j c).choose_spec.2.2.2 i y |
| let next (j : ℕ) (c : State j) : State (j+1) := |
| ⟨fun i x ↦ tiles j c i (x,top j),fun i ↦ horizontal _ (ht j c i) _, |
| fun i ↦ seam j c i _,(left j c (top j)).trans (joinB j)⟩ |
| let seq : (j : ℕ) → State j := Nat.rec ⟨a,ha,joinA,corner⟩ next |
| refine ⟨fun i j ↦ tiles j (seq j) i,fun i j ↦ ht j _ i, |
| bottom 0 _,fun j y ↦ left j _ y,fun i j y ↦ seam j _ i y,?_⟩ |
| intro i j x |
| exact (bottom (j+1) (seq (j+1)) i x).symm |
| |
| end RectangleGrid |
| end |
| end |
| |
| end OAI |
| |
