| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.GridBudget |
| import OAI.Computability.VertexCover.Analysis.PowerValueRepetition |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
|
|
| structure Iso (Φ Ψ : LabelCover) where |
| u : Fin Φ.u ≃ Fin Ψ.u |
| v : Fin Φ.v ≃ Fin Ψ.v |
| a : Fin Φ.qU ≃ Fin Ψ.qU |
| b : Fin Φ.qV ≃ Fin Ψ.qV |
| e : Fin Φ.M ≃ Fin Ψ.M |
| left : ∀ c, Ψ.left (e c) = u (Φ.left c) |
| right : ∀ c, Ψ.right (e c) = v (Φ.right c) |
| projection : ∀ c s, Ψ.projection (e c) (a s) = b (Φ.projection c s) |
|
|
| namespace Iso |
| variable {Φ Ψ : LabelCover} (E : Iso Φ Ψ) |
| def labeling (A : Φ.Labeling) : Ψ.Labeling := |
| (fun x => E.a (A.1 (E.u.symm x)),fun y => E.b (A.2 (E.v.symm y))) |
| theorem satisfies (A : Φ.Labeling) (c : Fin Φ.M) : |
| Ψ.Satisfies (E.labeling A) (E.e c) ↔ Φ.Satisfies A c := by |
| simp only [Satisfies,labeling,E.left,E.right,Equiv.symm_apply_apply,E.projection,Equiv.apply_eq_iff_eq] |
| theorem count (A : Φ.Labeling) : Ψ.satisfiedCount (E.labeling A) = Φ.satisfiedCount A := by |
| classical |
| have h := Fintype.card_congr (E.e.subtypeEquiv (fun c => (E.satisfies A c).symm)) |
| simpa only [Fintype.card_subtype,satisfiedCount] using h.symm |
| include E |
| theorem value_le : Φ.value ≤ Ψ.value := by |
| have hM : Φ.M = Ψ.M := by simpa using Fintype.card_congr E.e |
| apply value_le_of_counts |
| intro A |
| rw [← E.count A,hM] |
| exact Ψ.count_le_of_value_le le_rfl (E.labeling A) |
| def symm : Iso Ψ Φ where |
| u := E.u.symm |
| v := E.v.symm |
| a := E.a.symm |
| b := E.b.symm |
| e := E.e.symm |
| left c := by |
| apply E.u.injective |
| simpa only [Equiv.apply_symm_apply] using (E.left (E.e.symm c)).symm |
| right c := by |
| apply E.v.injective |
| simpa only [Equiv.apply_symm_apply] using (E.right (E.e.symm c)).symm |
| projection c s := by |
| apply E.b.injective |
| simpa only [Equiv.apply_symm_apply] using (E.projection (E.e.symm c) (E.a.symm s)).symm |
| theorem value_eq : Φ.value = Ψ.value := le_antisymm E.value_le E.symm.value_le |
| theorem perfect (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) : |
| ∀ c, Ψ.Satisfies (E.labeling A) c := by |
| intro c |
| rw [← E.e.apply_symm_apply c] |
| exact (E.satisfies A _).mpr (hA _) |
| end Iso |
|
|
| def numberedPower (Φ : LabelCover) (n : ℕ) : LabelCover where |
| u := Φ.u^n |
| v := Φ.v^n |
| qU := Φ.qU^n |
| qV := Φ.qV^n |
| M := Φ.M^n |
| qU_pos := pow_pos Φ.qU_pos _ |
| qV_pos := pow_pos Φ.qV_pos _ |
| M_pos := pow_pos Φ.M_pos _ |
| left c := finFunctionFinEquiv (fun i => Φ.left (finFunctionFinEquiv.symm c i)) |
| right c := finFunctionFinEquiv (fun i => Φ.right (finFunctionFinEquiv.symm c i)) |
| projection c s := finFunctionFinEquiv |
| (fun i => Φ.projection (finFunctionFinEquiv.symm c i) (finFunctionFinEquiv.symm s i)) |
| noncomputable def numberIso (Φ : LabelCover) (n : ℕ) : Iso (Φ.numberedPower n) (Φ.parallelPower n) where |
| u := finFunctionFinEquiv.symm.trans (tupleEquiv Φ.u n) |
| v := finFunctionFinEquiv.symm.trans (tupleEquiv Φ.v n) |
| a := finFunctionFinEquiv.symm.trans (tupleEquiv Φ.qU n) |
| b := finFunctionFinEquiv.symm.trans (tupleEquiv Φ.qV n) |
| e := finFunctionFinEquiv.symm.trans (tupleEquiv Φ.M n) |
| left c := by |
| change (tupleEquiv Φ.u n) (fun i => Φ.left ((tupleEquiv Φ.M n).symm |
| ((tupleEquiv Φ.M n) (finFunctionFinEquiv.symm c)) i)) = |
| (tupleEquiv Φ.u n) (finFunctionFinEquiv.symm (finFunctionFinEquiv |
| (fun i => Φ.left (finFunctionFinEquiv.symm c i)))) |
| simp only [Equiv.symm_apply_apply] |
| right c := by |
| change (tupleEquiv Φ.v n) (fun i => Φ.right ((tupleEquiv Φ.M n).symm |
| ((tupleEquiv Φ.M n) (finFunctionFinEquiv.symm c)) i)) = |
| (tupleEquiv Φ.v n) (finFunctionFinEquiv.symm (finFunctionFinEquiv |
| (fun i => Φ.right (finFunctionFinEquiv.symm c i)))) |
| simp only [Equiv.symm_apply_apply] |
| projection c s := by |
| change (tupleEquiv Φ.qV n) (fun i => Φ.projection |
| ((tupleEquiv Φ.M n).symm ((tupleEquiv Φ.M n) (finFunctionFinEquiv.symm c)) i) |
| ((tupleEquiv Φ.qU n).symm ((tupleEquiv Φ.qU n) (finFunctionFinEquiv.symm s)) i)) = |
| (tupleEquiv Φ.qV n) (finFunctionFinEquiv.symm (finFunctionFinEquiv |
| (fun i => Φ.projection (finFunctionFinEquiv.symm c i) (finFunctionFinEquiv.symm s i)))) |
| simp only [Equiv.symm_apply_apply] |
| end VertexCover.LabelCover |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
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| end OAI |
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