| import OAI.Geometry.PeriodicTiling.CycleTests |
| import OAI.Geometry.PeriodicTiling.SeedSymbols |
| import Mathlib.Data.Finset.Union |
| import Mathlib.Tactic.FinCases |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PeriodicTilingThree |
|
|
| variable {p : ℕ} [NeZero p] |
|
|
| def ordinaryCycle (p : ℕ) [NeZero p] : ZMod (p ^ 2) ↪ Channel p where |
| toFun j := .inl ((ZMod.finEquiv (p ^ 2)).symm j) |
| inj' _ _ h := (ZMod.finEquiv (p ^ 2)).symm.injective (Sum.inl.inj h) |
|
|
| def ordinaryCycleLabels (w : Word p) (j : ZMod (p ^ 2)) : |
| Label p (ordinaryCycle p j) := |
| w ((ZMod.finEquiv (p ^ 2)).symm j) |
|
|
| def seedCycle (t : Fin 2) (n : Column p) : ZMod 2 ↪ Channel p where |
| toFun j := Fin.cases (.inr t) (fun _ => .inl n) j |
| inj' := by |
| intro i j h |
| fin_cases i <;> fin_cases j |
| · rfl |
| · change Sum.inr t = Sum.inl n at h |
| cases h |
| · change Sum.inl n = Sum.inr t at h |
| cases h |
| · rfl |
|
|
| @[simp] theorem seedCycle_zero |
| {p : ℕ} [NeZero p] |
| (t : Fin 2) (n : Column p) : |
| seedCycle t n 0 = .inr t := rfl |
|
|
| @[simp] theorem seedCycle_one |
| {p : ℕ} [NeZero p] |
| (t : Fin 2) (n : Column p) : |
| seedCycle t n 1 = .inl n := rfl |
|
|
| def seedCycleLabels (t : Fin 2) (n : Column p) (j : Symbol p) (z : ZMod 2) : |
| Label p (seedCycle t n z) := |
| Fin.cases (motive := fun z : Fin 2 => Label p (seedCycle t n z)) |
| () (fun _ => j) z |
|
|
| @[simp] theorem seedCycleLabels_zero |
| {p : ℕ} [NeZero p] |
| (t : Fin 2) (n : Column p) (j : Symbol p) : |
| seedCycleLabels t n j 0 = () := rfl |
|
|
| @[simp] theorem seedCycleLabels_one |
| {p : ℕ} [NeZero p] |
| (t : Fin 2) (n : Column p) (j : Symbol p) : |
| seedCycleLabels t n j 1 = j := rfl |
|
|
| variable (E : EncodingParameters p) |
|
|
| theorem ordinaryCycle_exclusion_iff (w : Word p) (o : GraphOutputs E) |
| (hdep : HasDependence E o) : |
| (∀ F ∈ cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w), |
| Tiles F (graph o)) ↔ |
| ∀ x, ¬ (∀ n : Column p, Active E o (.inl n) x (w n)) := by |
| classical |
| rw [cycle_exclusion_iff E (ordinaryCycle p) (ordinaryCycleLabels w) o hdep] |
| constructor |
| · intro h x hall |
| obtain ⟨z, hz⟩ := h x |
| exact hz (hall ((ZMod.finEquiv (p ^ 2)).symm z)) |
| · intro h x |
| obtain ⟨n, hn⟩ := not_forall.mp (h x) |
| refine ⟨(ZMod.finEquiv (p ^ 2)) n, ?_⟩ |
| exact Eq.mpr |
| (congrArg (fun m : Column p => ¬ Active E o (.inl m) x (w m)) |
| ((ZMod.finEquiv (p ^ 2)).symm_apply_apply n)) hn |
|
|
| theorem seedCycle_exclusion_iff (t : Fin 2) (n : Column p) (j : Symbol p) |
| (o : GraphOutputs E) (hdep : HasDependence E o) : |
| (∀ F ∈ cycleTiles E (seedCycle t n) (seedCycleLabels t n j), |
| Tiles F (graph o)) ↔ |
| ∀ x, ¬ (Active E o (.inr t) x () ∧ Active E o (.inl n) x j) := by |
| classical |
| rw [cycle_exclusion_iff E (seedCycle t n) (seedCycleLabels t n j) o hdep] |
| constructor |
| · intro h x hx |
| obtain ⟨z, hz⟩ := h x |
| fin_cases z |
| · exact hz hx.1 |
| · exact hz hx.2 |
| · intro h x |
| by_cases hs : Active E o (.inr t) x () |
| · refine ⟨1, ?_⟩ |
| change ¬ Active E o (.inl n) x j |
| intro hj |
| exact h x ⟨hs, hj⟩ |
| · exact ⟨0, hs⟩ |
|
|
| def wordConstraintTiles : Finset (Finset (Ambient E)) := by |
| classical |
| exact (Finset.univ.filter (fun w : Word p => w ∉ allowedWords p)).biUnion |
| (fun w => cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w)) |
|
|
| def seedConstraintTiles : Finset (Finset (Ambient E)) := by |
| classical |
| exact (Finset.univ.filter (fun q : Fin 2 × Column p × Symbol p => |
| q.2.2 ≠ seedSymbol E q.1)).biUnion |
| (fun q => cycleTiles E (seedCycle q.1 q.2.1) (seedCycleLabels q.1 q.2.1 q.2.2)) |
|
|
| theorem nonempty_of_mem_wordConstraintTiles {F : Finset (Ambient E)} |
| (hF : F ∈ wordConstraintTiles E) : F.Nonempty := by |
| classical |
| obtain ⟨w, _, hF⟩ := Finset.mem_biUnion.mp hF |
| exact nonempty_of_mem_cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w) hF |
|
|
| theorem nonempty_of_mem_seedConstraintTiles {F : Finset (Ambient E)} |
| (hF : F ∈ seedConstraintTiles E) : F.Nonempty := by |
| classical |
| obtain ⟨q, _, hF⟩ := Finset.mem_biUnion.mp hF |
| exact nonempty_of_mem_cycleTiles E (seedCycle q.1 q.2.1) |
| (seedCycleLabels q.1 q.2.1 q.2.2) hF |
|
|
| theorem wordConstraintTiles_iff (o : GraphOutputs E) (hdep : HasDependence E o) : |
| (∀ F ∈ wordConstraintTiles E, Tiles F (graph o)) ↔ |
| ∀ x w, (∀ n : Column p, Active E o (.inl n) x (w n)) → Allowed p w := by |
| classical |
| constructor |
| · intro h x w hw |
| by_contra hnot |
| have hcycle : ∀ F ∈ cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w), |
| Tiles F (graph o) := by |
| intro F hF |
| apply h F |
| apply Finset.mem_biUnion.mpr |
| exact ⟨w, by simp [hnot], hF⟩ |
| exact (ordinaryCycle_exclusion_iff E w o hdep).mp hcycle x hw |
| · intro h F hF |
| obtain ⟨w, hw, hF⟩ := Finset.mem_biUnion.mp hF |
| have hnot : ¬ Allowed p w := by simpa using (Finset.mem_filter.mp hw).2 |
| apply (ordinaryCycle_exclusion_iff E w o hdep).mpr ?_ F hF |
| intro x hx |
| exact hnot (h x w hx) |
|
|
| theorem seedConstraintTiles_iff (o : GraphOutputs E) (hdep : HasDependence E o) : |
| (∀ F ∈ seedConstraintTiles E, Tiles F (graph o)) ↔ |
| ∀ t n x j, Active E o (.inr t) x () → Active E o (.inl n) x j → |
| j = seedSymbol E t := by |
| classical |
| constructor |
| · intro h t n x j hseed hord |
| by_contra hj |
| have hcycle : ∀ F ∈ cycleTiles E (seedCycle t n) (seedCycleLabels t n j), |
| Tiles F (graph o) := by |
| intro F hF |
| apply h F |
| apply Finset.mem_biUnion.mpr |
| exact ⟨(t, n, j), Finset.mem_filter.mpr ⟨Finset.mem_univ _, hj⟩, hF⟩ |
| exact (seedCycle_exclusion_iff E t n j o hdep).mp hcycle x ⟨hseed, hord⟩ |
| · intro h F hF |
| obtain ⟨⟨t, n, j⟩, hq, hF⟩ := Finset.mem_biUnion.mp hF |
| have hj : j ≠ seedSymbol E t := (Finset.mem_filter.mp hq).2 |
| apply (seedCycle_exclusion_iff E t n j o hdep).mpr ?_ F hF |
| intro x hx |
| exact hj (h t n x j hx.1 hx.2) |
|
|
| end PeriodicTilingThree |
|
|
| end |
|
|
| end OAI |
|
|
