| import Mathlib |
| import OAI.Combinatorics.IndependentSets.PCP.LabelCoverData |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets |
| |
| instance (priority := 1100) mixedTupleComputableFintype {A B : Type*} [Fintype A] [Fintype B] |
| (n i : ℕ) : Fintype (MixedTuple A B n i) := |
| inferInstanceAs (Fintype {x : Fin n → B ⊕ A // ∀ h, (x h).isLeft = decide (h.val < i)}) |
| |
| instance mixedTupleDecidableEq {A B : Type*} [DecidableEq A] [DecidableEq B] |
| (n i : ℕ) : DecidableEq (MixedTuple A B n i) := |
| inferInstanceAs (DecidableEq {x : Fin n → B ⊕ A // ∀ h, (x h).isLeft = decide (h.val < i)}) |
| |
| namespace LayerImposed |
| |
| abbrev Witness (U V C : Type) (n : ℕ) := |
| Σ h : Fin n, MixedTuple U V n h.val × MixedTuple U V n (h.val+1) × C |
| |
| abbrev Arrow (U V L R : Type) (n : ℕ) := |
| Σ q : LayerQuestion U V n, Σ q' : LayerQuestion U V n, LayerAnswer L R q → LayerAnswer L R q' |
| |
| def valid {U V L R C : Type} (lc : LabelCoverData U V L R C) {n : ℕ} |
| (w : Witness U V C n) : Prop := |
| w.2.1.val w.1 = Sum.inr (lc.left w.2.2.2) ∧ |
| w.2.2.1.val w.1 = Sum.inl (lc.right w.2.2.2) ∧ |
| ∀ k, k ≠ w.1 → w.2.1.val k = w.2.2.1.val k |
| |
| instance valid_decidable {U V L R C : Type} [DecidableEq U] [DecidableEq V] |
| (lc : LabelCoverData U V L R C) {n : ℕ} (w : Witness U V C n) : Decidable (valid lc w) := |
| inferInstanceAs (Decidable (_ ∧ _ ∧ ∀ _, _ → _)) |
| |
| def arrow {U V L R C : Type} (lc : LabelCoverData U V L R C) {n : ℕ} |
| (w : Witness U V C n) : Arrow U V L R n := |
| ⟨⟨w.1.castSucc,w.2.1⟩,⟨w.1.succ,w.2.2.1⟩, mixedProject w.1 (lc.project w.2.2.2)⟩ |
| |
| def arrowImposed {U V L R C : Type} (lc : LabelCoverData U V L R C) {n : ℕ} |
| (a : Arrow U V L R n) : Prop := LayerImposed lc n a.1 a.2.1 a.2.2 |
| |
| lemma iff_witness {U V L R C : Type} (lc : LabelCoverData U V L R C) {n : ℕ} |
| (q q' : LayerQuestion U V n) (π : LayerAnswer L R q → LayerAnswer L R q') : |
| LayerImposed lc n q q' π ↔ |
| ∃ w : Witness U V C n, valid lc w ∧ arrow lc w = ⟨q,q',π⟩ := by |
| constructor |
| · intro h |
| cases h with |
| | adjacent k x y c hl hr ho => exact ⟨⟨k,x,y,c⟩,⟨hl,hr,ho⟩,rfl⟩ |
| · rintro ⟨w,hw,he⟩ |
| have h : arrowImposed lc (arrow lc w) := |
| .adjacent w.1 w.2.1 w.2.2.1 w.2.2.2 hw.1 hw.2.1 hw.2.2 |
| rw [he] at h |
| exact h |
| |
| instance imposed_decidable {U V L R C : Type} [Fintype U] [Fintype V] [Fintype C] |
| [Fintype L] [Fintype R] [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| (lc : LabelCoverData U V L R C) (n : ℕ) (q q' : LayerQuestion U V n) |
| (π : LayerAnswer L R q → LayerAnswer L R q') : Decidable (LayerImposed lc n q q' π) := by |
| letI : Fintype (Witness U V C n) := inferInstance |
| letI : DecidableEq (Arrow U V L R n) := inferInstance |
| exact decidable_of_iff _ (iff_witness lc q q' π).symm |
| |
| end LayerImposed |
| end LargeIndependentSets |
| |
| end OAI |
| |
