| import OAI.Probability.CriticalZ3.Peierls |
|
|
| namespace OAI |
|
|
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical Filter Topology Function |
|
|
| open scoped ENNReal BigOperators Matrix |
|
|
| namespace CriticalZ3 |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical |
| namespace Coarse |
| abbrev Point := PlanarBoundary.Point |
| abbrev Direction := PlanarBoundary.Direction |
| abbrev outward := PlanarBoundary.outward |
|
|
| def center (r : ℕ) (v : Point) : Vertex := ![(20*(r:ℤ)+1)*v.1,(20*(r:ℤ)+1)*v.2,0] |
| def Q (r : ℕ) (v : Point) : Box := Box.cube (center r v) (10*r) |
| def B (r : ℕ) (v : Point) : Box := Box.cube (center r v) (4*r) |
| def PairRegion (r : ℕ) (v x : Point) : Set Vertex := {y | (Q r v).Contains y ∨ (Q r x).Contains y} |
|
|
| @[simp] theorem center_zero (r : ℕ) : center r 0 = 0 := by ext i; fin_cases i <;> simp [center] |
|
|
| theorem cube_unique (r : ℕ) {v w : Point} {x : Vertex} |
| (hv : (Q r v).Contains x) (hw : (Q r w).Contains x) : v = w := by |
| have hg (a b t : ℤ) (h0 : (20*(r:ℤ)+1)*a-(10*r:ℕ) ≤ t) |
| (h1 : t ≤ (20*(r:ℤ)+1)*a+(10*r:ℕ)) |
| (h2 : (20*(r:ℤ)+1)*b-(10*r:ℕ) ≤ t) |
| (h3 : t ≤ (20*(r:ℤ)+1)*b+(10*r:ℕ)) : a = b := by |
| push_cast at h0 h1 h2 h3 |
| have hr : (0:ℤ) ≤ r := Int.natCast_nonneg _ |
| by_contra h |
| rcases lt_or_gt_of_ne h with hh|hh |
| · have : a+1 ≤ b := by omega |
| nlinarith |
| · have : b+1 ≤ a := by omega |
| nlinarith |
| apply Prod.ext |
| · exact hg v.1 w.1 (x 0) (hv 0).1 (hv 0).2 (hw 0).1 (hw 0).2 |
| · exact hg v.2 w.2 (x 1) (hv 1).1 (hv 1).2 (hw 1).1 (hw 1).2 |
|
|
| theorem Q_disjoint (r : ℕ) {v w : Point} (h : v ≠ w) : Disjoint (Q r v).vertices (Q r w).vertices := by |
| apply Finset.disjoint_left.mpr |
| intro x hx hy |
| exact h (cube_unique r ((Box.mem_vertices _ _).mp hx) ((Box.mem_vertices _ _).mp hy)) |
|
|
| theorem B_subset_Q (r : ℕ) (v : Point) : ∀ x, (B r v).Contains x → (Q r v).Contains x := by |
| intro x hx |
| apply Box.contains_cube_iff.mpr |
| exact cube_mono (center r v) (by omega) (Box.contains_cube_iff.mp hx) |
|
|
| theorem zero_in_Q (r : ℕ) : (Q r 0).Contains 0 := by |
| rw [Q,center_zero] |
| exact Box.center_mem_cube _ _ |
|
|
| theorem zero_outside_pair (r : ℕ) {v w : Point} (hv : v ≠ 0) (hw : w ≠ 0) : |
| (0:Vertex) ∉ PairRegion r v w := by |
| rintro (h|h) |
| · exact hv (cube_unique r h (zero_in_Q r)) |
| · exact hw (cube_unique r h (zero_in_Q r)) |
|
|
| noncomputable def interface (r : ℕ) (v w : Point) : Finset Bond := |
| (((Q r v).vertices ∪ (Q r w).vertices) ×ˢ Finset.univ).filter |
| (fun e => ((Q r v).Contains e.1 ∧ (Q r w).Contains (step e.1 e.2)) ∨ |
| ((Q r w).Contains e.1 ∧ (Q r v).Contains (step e.1 e.2))) |
|
|
| @[simp] theorem mem_interface (r : ℕ) (v w : Point) (e : Bond) : e ∈ interface r v w ↔ |
| ((Q r v).Contains e.1 ∧ (Q r w).Contains (step e.1 e.2)) ∨ |
| ((Q r w).Contains e.1 ∧ (Q r v).Contains (step e.1 e.2)) := by |
| simp only [interface,Finset.mem_filter,Finset.mem_product,Finset.mem_union, |
| Box.mem_vertices,Finset.mem_univ,and_true] |
| exact ⟨fun h => h.2,fun h => ⟨h.elim (fun h => Or.inl h.1) (fun h => Or.inr h.1),h⟩⟩ |
|
|
| theorem interface_comm (r : ℕ) (v w : Point) : interface r v w = interface r w v := by |
| ext e |
| simp only [mem_interface,or_comm] |
|
|
| theorem interface_eq_endpoints (r : ℕ) {v w a b : Point} {e : Bond} |
| (h : e ∈ interface r v w) (h' : e ∈ interface r a b) : |
| (v = a ∧ w = b) ∨ (v = b ∧ w = a) := by |
| rw [mem_interface] at h h' |
| rcases h with h|h <;> rcases h' with h'|h' |
| · exact Or.inl ⟨cube_unique r h.1 h'.1,cube_unique r h.2 h'.2⟩ |
| · exact Or.inr ⟨cube_unique r h.1 h'.1,cube_unique r h.2 h'.2⟩ |
| · exact Or.inr ⟨cube_unique r h.2 h'.2,cube_unique r h.1 h'.1⟩ |
| · exact Or.inl ⟨cube_unique r h.2 h'.2,cube_unique r h.1 h'.1⟩ |
|
|
| theorem internal_not_interface (r : ℕ) {u v w : Point} (hvw : v ≠ w) : |
| Disjoint (Q r u).bonds (interface r v w) := by |
| refine Finset.disjoint_left.mpr ?_ |
| intro e he hi |
| have hh := (Box.mem_bonds _ _).mp he |
| rw [mem_interface] at hi |
| rcases hi with hi|hi |
| · exact hvw ((cube_unique r hi.1 hh.1).trans (cube_unique r hh.2 hi.2)) |
| · exact hvw ((cube_unique r hi.2 hh.2).trans (cube_unique r hh.1 hi.1)) |
|
|
| theorem internal_pair_bond (r : ℕ) (v w : Point) (e : Bond) : |
| Model.Internal .bond (PairRegion r v w) e ↔ |
| e ∈ (Q r v).bonds ∪ (Q r w).bonds ∪ interface r v w := by |
| simp only [Model.Internal,PairRegion,Set.mem_ofPred_eq,Finset.mem_union,Box.mem_bonds,mem_interface] |
| tauto |
|
|
| def frame (r : ℕ) (v : Point) (d : Direction) : SignedSymmetry where |
| perm := if d = 0 ∨ d = 2 then Equiv.swap 0 1 else Equiv.refl _ |
| signs := fun i => if d = 0 then decide (i ≠ 1) else if d = 3 then decide (i ≠ 0) else true |
| shift := center r v |
|
|
| @[simp] theorem frame_zero (r : ℕ) (v : Point) (d : Direction) : (frame r v d).vertex 0 = center r v := by |
| ext i |
| simp [frame,SignedSymmetry.vertex] |
|
|
| @[simp] theorem frame_next (r : ℕ) (v : Point) (d : Direction) : |
| (frame r v d).vertex (LatticeRelay.axisCenter (20*(r:ℤ)+1)) = center r (v+outward d) := by |
| ext i |
| fin_cases d <;> fin_cases i <;> simp [frame,SignedSymmetry.vertex,LatticeRelay.axisCenter, |
| center,outward,PlanarBoundary.outward,Equiv.swap_apply_def] <;> ring |
|
|
| theorem relay_cube_eq (c : Vertex) (n : ℕ) : LatticeRelay.Cube c n = cube c n := by |
| ext x |
| rw [mem_cube] |
| constructor <;> intro h i |
| · have hh := h i |
| change |x i-c i| ≤ (n:ℤ) at hh |
| rw [abs_le] at hh |
| omega |
| · have hh := h i |
| change |x i-c i| ≤ (n:ℤ) |
| rw [abs_le] |
| omega |
|
|
| theorem image_cube (S : SignedSymmetry) (c : Vertex) (n : ℕ) : S.vertex '' cube c n = cube (S.vertex c) n := by |
| ext x |
| constructor |
| · rintro ⟨y,hy,rfl⟩ |
| exact (S.cube_map c y n).mp hy |
| · intro hx |
| exact ⟨S.vertex.symm x,(S.cube_map c _ n).mpr (by simpa only [Equiv.apply_symm_apply] using hx),S.vertex.apply_symm_apply x⟩ |
|
|
| theorem prism_eq_cubes (r : ℕ) : LatticeRelay.Prism r = |
| LatticeRelay.Cube 0 (10*r) ∪ LatticeRelay.Cube (LatticeRelay.axisCenter (20*(r:ℤ)+1)) (10*r) := by |
| ext x |
| have hr : (0:ℤ) ≤ r := Int.natCast_nonneg _ |
| constructor |
| · intro hx |
| by_cases h : x 0 ≤ 10*(r:ℤ) |
| · left |
| intro i |
| by_cases hi : i = 0 |
| · subst i |
| change |x 0-0| ≤ _ |
| rw [sub_zero,abs_le] |
| exact ⟨hx.1,h⟩ |
| · simpa only [Pi.zero_apply,sub_zero] using hx.2.2 i hi |
| · right |
| intro i |
| by_cases hi : i = 0 |
| · subst i |
| change |x 0-(20*(r:ℤ)+1)| ≤ _ |
| change |x 0-(20*(r:ℤ)+1)| ≤ 10*(r:ℤ) |
| rw [abs_le] |
| have hh := hx.2.1 |
| omega |
| · simpa only [LatticeRelay.axisCenter,hi,ite_false,sub_zero] using hx.2.2 i hi |
| · rintro (hx|hx) |
| · have hh := hx 0 |
| change |x 0-0| ≤ 10*(r:ℤ) at hh |
| rw [sub_zero,abs_le] at hh |
| refine ⟨hh.1,by omega,?_⟩ |
| intro i _ |
| simpa only [Pi.zero_apply,sub_zero] using hx i |
| · have hh := hx 0 |
| change |x 0-(20*(r:ℤ)+1)| ≤ 10*(r:ℤ) at hh |
| rw [abs_le] at hh |
| refine ⟨by omega,by omega,?_⟩ |
| intro i hi |
| simpa only [LatticeRelay.axisCenter,hi,ite_false,sub_zero] using hx i |
|
|
| theorem frame_prism (r : ℕ) (v : Point) (d : Direction) : |
| (frame r v d).vertex '' LatticeRelay.Prism r = PairRegion r v (v+outward d) := by |
| rw [prism_eq_cubes,Set.image_union] |
| have he : (10:ℤ)*r = (10*r:ℕ) := by push_cast; rfl |
| rw [he,relay_cube_eq,relay_cube_eq,image_cube,image_cube,frame_zero,frame_next] |
| ext x |
| simp only [Set.mem_union,PairRegion,Set.mem_ofPred_eq,Q,Box.contains_cube_iff] |
|
|
| theorem neighboring_relay {model : Model} (s : ScaleData model) (G : LatticeGraph) |
| (p : model.Bit → Set.Icc (0:ℝ) 1) (v : Point) (d : Direction) |
| (hG : G.ContainsSet (PairRegion s.r v (v+outward d))) |
| (hp : model.UniformOn p s.q (PairRegion s.r v (v+outward d))) |
| (o : G.Node) (ho : o.1 ∉ PairRegion s.r v (v+outward d)) : |
| (ProductCylinder.law p).real {ω | G.Meets model ω o {x | (B s.r v).Contains x} ∧ |
| ¬G.Meets model ω o {x | (B s.r (v+outward d)).Contains x}} ≤ 13*s.η := by |
| have h := s.relay_in_frame G p (frame s.r v d) (by rwa [frame_prism]) (by rwa [frame_prism]) o |
| (by rwa [frame_prism]) |
| simpa only [frame_zero,frame_next,B,Box.contains_cube_iff,Set.ofPred_mem_eq] using h |
| end Coarse |
| end CriticalZ3 |
|
|
| namespace CriticalZ3 |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical |
| namespace LatticeGraph |
|
|
| def Subgraph (G H : LatticeGraph) : Prop := G.vertices ⊆ H.vertices ∧ G.edges ⊆ H.edges |
|
|
| theorem conn_inclusion {G H : LatticeGraph} (h : G.Subgraph H) (m : Model) {ω : m.Config} |
| {x y : G.Node} (hc : G.Conn m ω x y) : |
| H.Conn m ω ⟨x.1,h.1 x.2⟩ ⟨y.1,h.1 y.2⟩ := by |
| rw [G.conn_iff] at hc |
| rw [H.conn_iff] |
| refine ⟨?_,hc.2.lift (fun z : G.Node => (⟨z.1,h.1 z.2⟩ : H.Node)) ?_⟩ |
| · cases m <;> exact hc.1 |
| · intro a b hab |
| cases m |
| · obtain ⟨e,he,hi,ho⟩ := hab |
| exact ⟨e,h.2 he,hi,ho⟩ |
| · obtain ⟨⟨e,he,hi⟩,ha,hb⟩ := hab |
| exact ⟨⟨e,h.2 he,hi⟩,ha,hb⟩ |
|
|
| theorem meets_inclusion {G H : LatticeGraph} (h : G.Subgraph H) (m : Model) {ω : m.Config} |
| {o : G.Node} {T : Set Vertex} (hc : G.Meets m ω o T) : |
| H.Meets m ω ⟨o.1,h.1 o.2⟩ T := by |
| obtain ⟨y,hy,hc⟩ := hc |
| exact ⟨⟨y.1,h.1 y.2⟩,hy,conn_inclusion h m hc⟩ |
|
|
| noncomputable def ofBox (B : Box) : LatticeGraph where |
| vertices := B.vertices |
| edges := B.bonds |
| left_mem := fun _ he => (B.mem_vertices _).mpr ((B.mem_bonds _).mp he).1 |
| right_mem := fun _ he => (B.mem_vertices _).mpr ((B.mem_bonds _).mp he).2 |
|
|
| @[simp] theorem ofBox_contains (B : Box) : (ofBox B).ContainsBox B := ⟨fun _ h => h,fun _ h => h⟩ |
| end LatticeGraph |
|
|
| namespace Coarse |
| noncomputable def linkBits : (m : Model) → ℕ → Point → Point → Finset m.Bit |
| | .bond,r,v,w => interface r v w |
| | .site,_,_,_ => ∅ |
| noncomputable def block (m : Model) (r : ℕ) (v parent : Point) : Finset m.Bit := |
| (Q r v).bits m ∪ linkBits m r v parent |
| noncomputable def rootBits (m : Model) (r : ℕ) : Finset m.Bit := |
| (Q r 0).bits m ∪ Finset.univ.biUnion (fun d : Direction => linkBits m r 0 (outward d)) |
| noncomputable def exposed (m : Model) (r : ℕ) (D : Finset Point) : Finset m.Bit := |
| rootBits m r ∪ D.biUnion (fun v => (Q r v).bits m) ∪ |
| (D ×ˢ D).biUnion (fun e => linkBits m r e.1 e.2) |
|
|
| theorem bits_disjoint (m : Model) (r : ℕ) {v w : Point} (h : v ≠ w) : |
| Disjoint ((Q r v).bits m) ((Q r w).bits m) := by |
| cases m |
| · apply Finset.disjoint_left.mpr |
| intro e he hf |
| exact h (cube_unique r ((Box.mem_bonds _ _).mp he).1 ((Box.mem_bonds _ _).mp hf).1) |
| · exact Q_disjoint r h |
|
|
| theorem bits_link_disjoint (m : Model) (r : ℕ) (u : Point) {v w : Point} (h : v ≠ w) : |
| Disjoint ((Q r u).bits m) (linkBits m r v w) := by |
| cases m |
| · exact internal_not_interface r h |
| · exact Finset.disjoint_empty_right _ |
|
|
| theorem block_disjoint (m : Model) (r : ℕ) {D : Finset Point} {v w a b : Point} |
| (hv : v ∉ D) (hw : w ∉ D) (ha : a ∈ D) (hb : b ∈ D) (hvw : v ≠ w) : |
| Disjoint (block m r v a) (block m r w b) := by |
| have hva : v ≠ a := fun h => hv (h ▸ ha) |
| have hwb : w ≠ b := fun h => hw (h ▸ hb) |
| rw [block,block,Finset.disjoint_union_left,Finset.disjoint_union_right, |
| Finset.disjoint_union_right] |
| refine ⟨⟨bits_disjoint m r hvw,bits_link_disjoint m r v hwb⟩, |
| ⟨(bits_link_disjoint m r w hva).symm,?_⟩⟩ |
| cases m |
| · apply Finset.disjoint_left.mpr |
| intro e he hf |
| rcases interface_eq_endpoints r he hf with h|h |
| · exact hvw h.1 |
| · exact hv (h.1 ▸ hb) |
| · exact Finset.disjoint_empty_left _ |
|
|
| theorem block_subset_exposed_insert (m : Model) (r : ℕ) {D : Finset Point} |
| {v a : Point} (ha : a ∈ D) : block m r v a ⊆ exposed m r (insert v D) := by |
| intro e he |
| rw [block,Finset.mem_union] at he |
| rcases he with he|he |
| · exact Finset.mem_union_left _ (Finset.mem_union_right _ |
| (Finset.mem_biUnion.mpr ⟨v,Finset.mem_insert_self _ _,he⟩)) |
| · exact Finset.mem_union_right _ (Finset.mem_biUnion.mpr |
| ⟨(v,a),Finset.mem_product.mpr ⟨Finset.mem_insert_self _ _,Finset.mem_insert_of_mem ha⟩,he⟩) |
|
|
| theorem exposed_mono (m : Model) (r : ℕ) {D D' : Finset Point} (h : D ⊆ D') : |
| exposed m r D ⊆ exposed m r D' := by |
| intro e he |
| simp only [exposed,Finset.mem_union,Finset.mem_biUnion,Finset.mem_product] at he ⊢ |
| rcases he with (he|⟨v,hv,he⟩)|⟨⟨v,w⟩,⟨hv,hw⟩,he⟩ |
| · exact Or.inl (Or.inl he) |
| · exact Or.inl (Or.inr ⟨v,h hv,he⟩) |
| · exact Or.inr ⟨(v,w),⟨h hv,h hw⟩,he⟩ |
|
|
| theorem internal_pair_not_exposed (m : Model) (r : ℕ) {D : Finset Point} |
| (h0 : 0 ∈ D) {v w : Point} (hv : v ∉ D) (hw : w ∉ D) {e : m.Bit} |
| (he : m.Internal (PairRegion r v w) e) : e ∉ exposed m r D := by |
| have hv0 : v ≠ 0 := fun h => hv (h ▸ h0) |
| have hw0 : w ≠ 0 := fun h => hw (h ▸ h0) |
| cases m |
| · change e ∉ _ |
| rw [internal_pair_bond] at he |
| simp only [Finset.mem_union] at he |
| simp only [exposed,rootBits,linkBits,Box.bits,Finset.mem_union,Finset.mem_biUnion, |
| Finset.mem_product,Finset.mem_univ,true_and] |
| intro hb |
| have noInternal (a : Point) (ha : a ∈ D) (h : e ∈ (Q r a).bonds) : False := by |
| have hl := ((Box.mem_bonds _ _).mp h).1 |
| rcases he with (he|he)|he |
| · exact hv ((cube_unique r ((Box.mem_bonds _ _).mp he).1 hl) ▸ ha) |
| · exact hw ((cube_unique r ((Box.mem_bonds _ _).mp he).1 hl) ▸ ha) |
| · rcases (mem_interface r v w e).mp he with he|he |
| · exact hv ((cube_unique r he.1 hl) ▸ ha) |
| · exact hw ((cube_unique r he.1 hl) ▸ ha) |
| have noLink (a b : Point) (ha : a ∈ D) (h : e ∈ interface r a b) : False := by |
| rcases (mem_interface r a b e).mp h with h|h |
| · have hl := h.1 |
| rcases he with (he|he)|he |
| · exact hv ((cube_unique r ((Box.mem_bonds _ _).mp he).1 hl) ▸ ha) |
| · exact hw ((cube_unique r ((Box.mem_bonds _ _).mp he).1 hl) ▸ ha) |
| · rcases (mem_interface r v w e).mp he with he|he |
| · exact hv ((cube_unique r he.1 hl) ▸ ha) |
| · exact hw ((cube_unique r he.1 hl) ▸ ha) |
| · have hl := h.2 |
| rcases he with (he|he)|he |
| · exact hv ((cube_unique r ((Box.mem_bonds _ _).mp he).2 hl) ▸ ha) |
| · exact hw ((cube_unique r ((Box.mem_bonds _ _).mp he).2 hl) ▸ ha) |
| · rcases (mem_interface r v w e).mp he with he|he |
| · exact hw ((cube_unique r he.2 hl) ▸ ha) |
| · exact hv ((cube_unique r he.2 hl) ▸ ha) |
| rcases hb with ((hb|⟨d,hd⟩)|⟨a,ha,hb⟩)|⟨⟨a,b⟩,⟨ha,_⟩,hb⟩ |
| · exact noInternal 0 h0 hb |
| · exact noLink 0 (outward d) h0 hd |
| · exact noInternal a ha hb |
| · exact noLink a b ha hb |
| · change (Q r v).Contains e ∨ (Q r w).Contains e at he |
| simp only [exposed,rootBits,linkBits,Box.bits,Finset.mem_union,Finset.mem_biUnion, |
| Box.mem_vertices,Finset.notMem_empty,and_false,exists_false,or_false] |
| rintro (hh|⟨a,ha,hh⟩) |
| · rcases he with he|he |
| · exact hv0 (cube_unique r he hh) |
| · exact hw0 (cube_unique r he hh) |
| · rcases he with he|he |
| · exact hv ((cube_unique r he hh) ▸ ha) |
| · exact hw ((cube_unique r he hh) ▸ ha) |
|
|
| noncomputable def extendGraph (r : ℕ) (G : LatticeGraph) (v x : Point) |
| (hv : G.ContainsBox (Q r v)) : LatticeGraph where |
| vertices := G.vertices ∪ (Q r x).vertices |
| edges := G.edges ∪ (Q r x).bonds ∪ interface r v x |
| left_mem := by |
| intro e he |
| rcases Finset.mem_union.mp he with he|he |
| · rcases Finset.mem_union.mp he with he|he |
| · exact Finset.mem_union_left _ (G.left_mem _ he) |
| · exact Finset.mem_union_right _ ((Box.mem_vertices _ _).mpr ((Box.mem_bonds _ _).mp he).1) |
| · rcases (mem_interface _ _ _ _).mp he with he|he |
| · exact Finset.mem_union_left _ (hv.1 ((Box.mem_vertices _ _).mpr he.1)) |
| · exact Finset.mem_union_right _ ((Box.mem_vertices _ _).mpr he.1) |
| right_mem := by |
| intro e he |
| rcases Finset.mem_union.mp he with he|he |
| · rcases Finset.mem_union.mp he with he|he |
| · exact Finset.mem_union_left _ (G.right_mem _ he) |
| · exact Finset.mem_union_right _ ((Box.mem_vertices _ _).mpr ((Box.mem_bonds _ _).mp he).2) |
| · rcases (mem_interface _ _ _ _).mp he with he|he |
| · exact Finset.mem_union_right _ ((Box.mem_vertices _ _).mpr he.2) |
| · exact Finset.mem_union_left _ (hv.1 ((Box.mem_vertices _ _).mpr he.2)) |
|
|
| theorem extendGraph_subgraph (r : ℕ) (G : LatticeGraph) (v x : Point) |
| (hv : G.ContainsBox (Q r v)) : G.Subgraph (extendGraph r G v x hv) := |
| ⟨Finset.subset_union_left,fun _ h => Finset.mem_union_left _ (Finset.mem_union_left _ h)⟩ |
|
|
| theorem extendGraph_contains (r : ℕ) (G : LatticeGraph) (v x : Point) |
| (hv : G.ContainsBox (Q r v)) : (extendGraph r G v x hv).ContainsBox (Q r x) := |
| ⟨Finset.subset_union_right,fun _ h => Finset.mem_union_left _ (Finset.mem_union_right _ h)⟩ |
|
|
| theorem extendGraph_bits (m : Model) (r : ℕ) (G : LatticeGraph) (v x : Point) |
| (hv : G.ContainsBox (Q r v)) : |
| (extendGraph r G v x hv).bits m = G.bits m ∪ block m r x v := by |
| cases m |
| · dsimp [LatticeGraph.bits,extendGraph,block,Box.bits,linkBits] |
| rw [interface_comm r x v,Finset.union_assoc] |
| · simp only [LatticeGraph.bits,extendGraph,block,Box.bits,linkBits,Finset.union_empty] |
|
|
| theorem extendGraph_pair (r : ℕ) (G : LatticeGraph) (v x : Point) |
| (hv : G.ContainsBox (Q r v)) : (extendGraph r G v x hv).ContainsSet (PairRegion r v x) := by |
| constructor |
| · intro y hy |
| rcases hy with hy|hy |
| · exact Finset.mem_union_left _ (hv.1 ((Box.mem_vertices _ _).mpr hy)) |
| · exact Finset.mem_union_right _ ((Box.mem_vertices _ _).mpr hy) |
| · intro e he hf |
| have hh := (internal_pair_bond r v x e).mp ⟨he,hf⟩ |
| rcases Finset.mem_union.mp hh with hh|hh |
| · rcases Finset.mem_union.mp hh with hh|hh |
| · exact Finset.mem_union_left _ (Finset.mem_union_left _ (hv.2 hh)) |
| · exact Finset.mem_union_left _ (Finset.mem_union_right _ hh) |
| · exact Finset.mem_union_right _ hh |
| end Coarse |
| end CriticalZ3 |
|
|
| end OAI |
|
|
