File size: 7,855 Bytes
e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | 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
|