openAI-math-dataset / lean /OAI /Combinatorics /IndependentSets /Expansion /PreprocessingOverlayTables.lean
| import Mathlib | |
| import OAI.Combinatorics.IndependentSets.Expansion.PortTables | |
| import OAI.Combinatorics.IndependentSets.Expansion.ExpanderTables | |
| import OAI.Combinatorics.IndependentSets.Expansion.PoweringLazy | |
| import OAI.Combinatorics.IndependentSets.PCP.Minimum | |
| namespace OAI | |
| namespace IndependentSetsGames.Foundations.PCP.PreprocessingOverlayTables | |
| open PoweringWalks | |
| def overlayPorts (d e : Nat) : (Fin d ⊕ Fin e) ≃ Fin (d + e) := finSumFinEquiv | |
| @[simp] theorem overlayPorts_inl_val (d e : Nat) (i : Fin d) : | |
| (overlayPorts d e (Sum.inl i)).val = i.val := rfl | |
| @[simp] theorem overlayPorts_inr_val (d e : Nat) (i : Fin e) : | |
| (overlayPorts d e (Sum.inr i)).val = d + i.val := rfl | |
| def lazyPorts (d : Nat) : (Bool × Fin d) ≃ Fin (2 * d) := | |
| (Equiv.prodCongr finTwoEquiv.symm (Equiv.refl (Fin d))).trans finProdFinEquiv | |
| @[simp] theorem lazyPorts_false_val (d : Nat) (i : Fin d) : | |
| (lazyPorts d (false, i)).val = i.val := by | |
| simp [lazyPorts, finTwoEquiv, finProdFinEquiv] | |
| @[simp] theorem lazyPorts_true_val (d : Nat) (i : Fin d) : | |
| (lazyPorts d (true, i)).val = i.val + d := by | |
| simp [lazyPorts, finTwoEquiv, finProdFinEquiv] | |
| def materialize {n d : Nat} {D : Type*} | |
| (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label) | |
| (ports : D ≃ Fin d) : PortTables.Table n d := | |
| PortTables.ofPortGraph | |
| (GraphTransport.reindex (Overlay.originalPortGraph G) (Equiv.refl _) ports) | |
| (fun x a b => G.accepts (x.1, ports.symm x.2) a b) | |
| (by | |
| intro x a b | |
| simp only [GraphTransport.reindex, Overlay.originalPortGraph, | |
| Equiv.trans_apply, Equiv.prodCongr_apply, Equiv.prodCongr_symm, | |
| Equiv.refl_apply, Equiv.symm_apply_apply, Prod.map] | |
| convert G.reverse_accepts (x.1, ports.symm x.2) a b using 1 | |
| rfl) | |
| theorem rotation_materialize {n d : Nat} {D : Type*} | |
| (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label) | |
| (ports : D ≃ Fin d) (x : Fin n × Fin d) : | |
| PortTables.rotation (materialize G ports) x = | |
| ((G.reverse (x.1, ports.symm x.2)).1, | |
| ports (G.reverse (x.1, ports.symm x.2)).2) := by | |
| simp only [materialize, PortTables.rotation_ofPortGraph, GraphTransport.reindex, | |
| Overlay.originalPortGraph, Equiv.trans_apply, Equiv.prodCongr_apply, | |
| Equiv.prodCongr_symm, Equiv.refl_apply, Prod.map] | |
| rfl | |
| theorem rotation_materialize_image {n d : Nat} {D : Type*} | |
| (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label) | |
| (ports : D ≃ Fin d) (v : Fin n) (i : D) : | |
| PortTables.rotation (materialize G ports) (v, ports i) = | |
| ((G.reverse (v, i)).1, ports (G.reverse (v, i)).2) := by | |
| rw [rotation_materialize, Equiv.symm_apply_apply] | |
| @[simp] theorem accepts_materialize {n d : Nat} {D : Type*} | |
| (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label) | |
| (ports : D ≃ Fin d) (x : Fin n × Fin d) (a b : PortTables.Label) : | |
| PortTables.accepts (materialize G ports) x a b = | |
| G.accepts (x.1, ports.symm x.2) a b := by | |
| exact PortTables.accepts_ofPortGraph _ _ _ x a b | |
| theorem constraintGraph_ext {V E A : Type*} {G H : ConstraintGraph V E A} | |
| (hreverse : ∀ e, G.reverse e = H.reverse e) | |
| (htail : ∀ e, G.tail e = H.tail e) | |
| (haccepts : ∀ e a b, G.accepts e a b = H.accepts e a b) : G = H := by | |
| rcases G with ⟨r, hi, t, p, hp⟩ | |
| rcases H with ⟨r', hi', t', p', hp'⟩ | |
| have hr : r = r' := Equiv.ext hreverse | |
| cases hr | |
| have ht : t = t' := funext htail | |
| cases ht | |
| have ha : p = p' := funext fun e => funext fun a => funext fun b => haccepts e a b | |
| cases ha | |
| rfl | |
| theorem baseGraph_materialize {n d : Nat} {D : Type*} | |
| (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label) | |
| (ports : D ≃ Fin d) (htail : G.tail = Prod.fst) : | |
| PortTables.baseGraph (materialize G ports) = | |
| G.reindex (Equiv.refl _) (Equiv.prodCongr (Equiv.refl _) ports) (Equiv.refl _) := by | |
| apply constraintGraph_ext | |
| · intro x | |
| rw [PortTables.baseGraph_reverse, rotation_materialize] | |
| rfl | |
| · intro x | |
| change x.1 = G.tail (x.1, ports.symm x.2) | |
| rw [htail] | |
| · intro x a b | |
| rw [PortTables.baseGraph_accepts, accepts_materialize] | |
| rfl | |
| def overlay {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e) : | |
| PortTables.Table n (d + e) := | |
| materialize (Overlay.constraintGraph (PortTables.baseGraph G) (ExpanderTables.graph H)) | |
| (overlayPorts d e) | |
| theorem overlay_rotation_old {n d e : Nat} (G : PortTables.Table n d) | |
| (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin d) : | |
| PortTables.rotation (overlay G H) (v, overlayPorts d e (Sum.inl i)) = | |
| ((PortTables.rotation G (v, i)).1, | |
| overlayPorts d e (Sum.inl (PortTables.rotation G (v, i)).2)) := by | |
| exact rotation_materialize_image _ _ v (Sum.inl i) | |
| theorem overlay_rotation_expander {n d e : Nat} (G : PortTables.Table n d) | |
| (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin e) : | |
| PortTables.rotation (overlay G H) (v, overlayPorts d e (Sum.inr i)) = | |
| ((ExpanderTables.lookup H (v, i)).1, | |
| overlayPorts d e (Sum.inr (ExpanderTables.lookup H (v, i)).2)) := by | |
| exact rotation_materialize_image _ _ v (Sum.inr i) | |
| @[simp] theorem overlay_accepts_old {n d e : Nat} (G : PortTables.Table n d) | |
| (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin d) (a b : PortTables.Label) : | |
| PortTables.accepts (overlay G H) (v, overlayPorts d e (Sum.inl i)) a b = | |
| PortTables.accepts G (v, i) a b := by | |
| simp only [overlay, accepts_materialize, Equiv.symm_apply_apply] | |
| rfl | |
| @[simp] theorem overlay_accepts_expander {n d e : Nat} (G : PortTables.Table n d) | |
| (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin e) (a b : PortTables.Label) : | |
| PortTables.accepts (overlay G H) (v, overlayPorts d e (Sum.inr i)) a b = true := by | |
| simp only [overlay, accepts_materialize, Equiv.symm_apply_apply] | |
| rfl | |
| theorem overlay_semantics {n d e : Nat} (G : PortTables.Table n d) | |
| (H : ExpanderTables.Table n e) : | |
| PortTables.baseGraph (overlay G H) = | |
| (Overlay.constraintGraph (PortTables.baseGraph G) (ExpanderTables.graph H)).reindex | |
| (Equiv.refl _) (Equiv.prodCongr (Equiv.refl _) (overlayPorts d e)) | |
| (Equiv.refl _) := | |
| baseGraph_materialize _ _ rfl | |
| def lazy {n d : Nat} (G : PortTables.Table n d) : PortTables.Table n (2 * d) := | |
| materialize (LazyConstraint.constraintGraph (PortTables.baseGraph G)) (lazyPorts d) | |
| theorem lazy_rotation_false {n d : Nat} (G : PortTables.Table n d) | |
| (v : Fin n) (i : Fin d) : | |
| PortTables.rotation (lazy G) (v, lazyPorts d (false, i)) = | |
| (v, lazyPorts d (false, i)) := by | |
| exact rotation_materialize_image _ _ v (false, i) | |
| theorem lazy_rotation_true {n d : Nat} (G : PortTables.Table n d) | |
| (v : Fin n) (i : Fin d) : | |
| PortTables.rotation (lazy G) (v, lazyPorts d (true, i)) = | |
| ((PortTables.rotation G (v, i)).1, | |
| lazyPorts d (true, (PortTables.rotation G (v, i)).2)) := by | |
| exact rotation_materialize_image _ _ v (true, i) | |
| @[simp] theorem lazy_accepts_false {n d : Nat} (G : PortTables.Table n d) | |
| (v : Fin n) (i : Fin d) (a b : PortTables.Label) : | |
| PortTables.accepts (lazy G) (v, lazyPorts d (false, i)) a b = true := by | |
| simp only [lazy, accepts_materialize, Equiv.symm_apply_apply] | |
| rfl | |
| @[simp] theorem lazy_accepts_true {n d : Nat} (G : PortTables.Table n d) | |
| (v : Fin n) (i : Fin d) (a b : PortTables.Label) : | |
| PortTables.accepts (lazy G) (v, lazyPorts d (true, i)) a b = | |
| PortTables.accepts G (v, i) a b := by | |
| simp only [lazy, accepts_materialize, Equiv.symm_apply_apply] | |
| rfl | |
| theorem lazy_semantics {n d : Nat} (G : PortTables.Table n d) : | |
| PortTables.baseGraph (lazy G) = | |
| (LazyConstraint.constraintGraph (PortTables.baseGraph G)).reindex | |
| (Equiv.refl _) (Equiv.prodCongr (Equiv.refl _) (lazyPorts d)) (Equiv.refl _) := | |
| baseGraph_materialize _ _ rfl | |
| end IndependentSetsGames.Foundations.PCP.PreprocessingOverlayTables | |
| end OAI | |