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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 | import Mathlib
import OAI.Combinatorics.IndependentSets.Encoding.BinaryNameTables
namespace OAI
namespace LargeIndependentSets.GraphPrinter
open IndependentSetsCut.CounterMachine
open IndependentSetsGames.Foundations.Complexity
open scoped Classical
noncomputable section
lemma matrix_eq (G : Graph) :
(List.finRange G.vertices).flatMap (fun i => (List.finRange G.vertices).map (G.adj i)) =
List.ofFn (fun i : Fin (G.vertices*G.vertices) => G.adj i.divNat i.modNat) := by
rw [List.ofFn_mul]
simp only [List.ofFn_eq_map,List.flatMap_def]
congr 1
apply List.map_congr_left
intro i _
apply List.map_congr_left
intro j _
congr 1
· apply Fin.ext
simp only [Fin.divNat]
rw [Nat.add_comm (i.val * G.vertices),Nat.add_mul_div_right _ _ G.nonempty,Nat.div_eq_of_lt j.isLt,Nat.zero_add]
· apply Fin.ext
simp [Fin.modNat,Nat.mod_eq_of_lt j.isLt]
lemma bits_get (G : Graph) (w : ℕ) (hw : w<(graphBits G).length) :
((graphBits G)[w]?).getD false=
if h : w<(nameBits G.vertices).length then ((nameBits G.vertices)[w]?).getD false
else G.adj
(⟨w-(nameBits G.vertices).length,by
apply Nat.sub_lt_left_of_lt_add (Nat.le_of_not_gt h)
simpa only [graphBits,List.length_append,matrix_eq,List.length_ofFn] using hw⟩ : Fin (G.vertices*G.vertices)).divNat
(⟨w-(nameBits G.vertices).length,by
apply Nat.sub_lt_left_of_lt_add (Nat.le_of_not_gt h)
simpa only [graphBits,List.length_append,matrix_eq,List.length_ofFn] using hw⟩ : Fin (G.vertices*G.vertices)).modNat := by
simp only [graphBits,matrix_eq]
split_ifs with h
· rw [List.getElem?_append_left h]
· rw [List.getElem?_append_right (Nat.le_of_not_gt h),List.getElem?_ofFn]
have hb : w-(nameBits G.vertices).length<G.vertices*G.vertices := by
apply Nat.sub_lt_left_of_lt_add (Nat.le_of_not_gt h)
simpa only [graphBits,List.length_append,matrix_eq,List.length_ofFn] using hw
simp only [dite_eq_left hb,Option.getD_some]
def lengthExpr (n : Expr) : Expr := .add (Expr.framedLen n) (.mul n n)
def bitExpr (n adj : Expr) : Expr :=
Expr.cond (Expr.lt (.arg 0) (Expr.framedLen n)) (Expr.framedBit n (.arg 0))
(adj.subst (fun _ => .sub (.arg 0) (Expr.framedLen n)))
lemma printer_spec (n adj : Expr) (s : List Bool) (G : Graph)
(hn : ∀ a, n.eval s a=G.vertices)
(ha : ∀ i : Fin (G.vertices*G.vertices),
decide (adj.eval s (fun _ => i.val)≠0)=G.adj i.divNat i.modNat) :
table (lengthExpr n) (bitExpr n adj) s=graphBits G := by
have hlen : (lengthExpr n).eval s (fun _ => 0)=(graphBits G).length := by
simp only [lengthExpr,Expr.eval,Expr.framedLen_eval,hn,graphBits,List.length_append,
matrix_eq,List.length_ofFn,nameBits,frame_length,Nat.size_eq_bits_len]
apply List.ext_getElem
· simpa only [table,List.length_ofFn] using hlen
· intro w hw hg
simp only [table,List.getElem_ofFn]
have hget : ((graphBits G)[w]?).getD false=(graphBits G)[w] := by
rw [List.getElem?_eq_getElem hg]; rfl
rw [← hget]
rw [bits_get G w hg]
have hheader : (nameBits G.vertices).length=2*G.vertices.size+1 := by
simp [nameBits,frame_length,Nat.size_eq_bits_len]
simp only [bitExpr,Expr.cond_eval,Expr.lt_eval,Expr.eval,Expr.framedLen_eval,hn,
Expr.subst_eval,Expr.framedBit_eval]
by_cases h : w<2*G.vertices.size+1
· simp only [hheader,h,ite_true,dite_eq_left]
rw [ite_eq_left (show (1:ℕ)≠0 by decide)]
simpa only [nameBits,BinaryNames.frame_eq,Nat.one_ne_zero,ite_true] using (frame_bits_get G.vertices w).symm
· simp only [hheader,h,ite_false,ne_eq,not_true_eq_false]
have hb : w-(2*G.vertices.size+1)<G.vertices*G.vertices := by
have hg' := hg
simp only [graphBits,List.length_append,matrix_eq,List.length_ofFn,hheader] at hg'
omega
exact ha ⟨w-(2*G.vertices.size+1),hb⟩
noncomputable def encodedComputer {α : Type} (ea : α → List Bool) (f : α → Graph)
(n adj : Expr) (hn : ∀ x a, n.eval (ea x) a=(f x).vertices)
(ha : ∀ x (i : Fin ((f x).vertices*(f x).vertices)),
decide (adj.eval (ea x) (fun _ => i.val)≠0)=(f x).adj i.divNat i.modNat) :
Turing.TM2ComputableInPolyTime ea graphBits f where
tm := (tableComputer (lengthExpr n) (bitExpr n adj)).tm
inputAlphabet := (tableComputer (lengthExpr n) (bitExpr n adj)).inputAlphabet
outputAlphabet := (tableComputer (lengthExpr n) (bitExpr n adj)).outputAlphabet
time := (tableComputer (lengthExpr n) (bitExpr n adj)).time
outputsFun x := by
simpa only [id_eq,printer_spec n adj (ea x) (f x) (hn x) (ha x)] using
(tableComputer (lengthExpr n) (bitExpr n adj)).outputsFun (ea x)
lemma encodedComputer_finiteAlphabet {α : Type} (ea : α → List Bool) (f : α → Graph)
(n adj : Expr) (hn) (ha) :
MachineFiniteAlphabet.FiniteAlphabet (encodedComputer ea f n adj hn ha).tm :=
tableComputer_finiteAlphabet _ _
end
end LargeIndependentSets.GraphPrinter
end OAI
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