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760f90e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.Computability.Scheduling.TransitionCosts
namespace OAI
section
namespace ThreeMachine.StackCompiler.Costs
variable {J : Type} (n : J → ℕ)
variable (U : ∀ j, Universe (n j)) (tab : ∀ j, Table (n j))
variable (u : ∀ j, Algorithm.State (Fin (n j)) × Algorithm.Block (Fin (n j)))
variable (v : ∀ j, Algorithm.State (Fin (n j)))
theorem tryW
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryW.time (n j) (U j,(tab j,(u j,v j)))) 30002 := by
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
poly_auto
theorem tryL
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryL.time (n j) (U j,(tab j,(u j,v j)))) 30002 := by
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
poly_auto
theorem tryP
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryP.time (n j) (U j,(tab j,(u j,v j)))) 30002 := by
have hW := tryW n U tab u v hl ht hb
have hL := tryL n U tab u v hl ht hb
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
poly_auto
theorem tryDiff
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryDiff.time (n j) (U j,(tab j,(u j,v j)))) 30002 := by
have hW := tryW n U tab u v hl ht hb
have hL := tryL n U tab u v hl ht hb
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
poly_auto
theorem tryO
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryO.time (n j) (U j,(tab j,(u j,v j)))) 60002 := by
have hD := tryDiff n U tab u v hl ht hb
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
poly_auto
theorem tryWork
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryWork.time (n j) (U j,(tab j,(u j,v j)))) 60002 := by
have hO := tryO n U tab u v hl ht hb
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV : Poly n (fun j => volume (u j).2) 2 := Poly.volumeSmallBlock _ (Poly.size n) hb
have hL : Poly n (fun j => volume (Algorithm.lookup (tab j) ((v j).left \ (u j).1.upto))) 2 :=
Poly.volumeSmallLookup tab _ (Poly.size n) ht
have ha := advanceLast (fun p : Poly.ListPool (fun j => (Algorithm.lookup (tab j)
((v j).left \ (u j).1.upto)).toList) => n p.1) (fun p => U p.1)
(fun p => u p.1) (fun p => p.2.1) (fun p => hb p.1) (fun p =>
ht p.1 _ _ (Algorithm.lookup_mem (by simpa using p.2.2)))
have hAv : Poly (fun p : Poly.ListPool (fun j => (Algorithm.lookup (tab j)
((v j).left \ (u j).1.upto)).toList) => n p.1)
(fun p => volume ((u p.1).1.advance (u p.1).2 p.2.1)) 2 := by
apply Poly.volumeAdvance _ _ _ (Poly.size _) (fun p => hb p.1)
intro p
exact ht p.1 _ _ (Algorithm.lookup_mem (by simpa using p.2.2))
poly_auto
theorem tryAdvance
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.tryAdvance.time (n j) (U j,(tab j,(u j,v j)))) 60002 := by
have hp := tryP n U tab u v hl ht hb
have hw := tryWork n U tab u v hl ht hb
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
have hAv : Poly (fun p : Poly.ListPool (fun j => (Algorithm.lookup (tab j)
((v j).left \ (u j).1.upto)).toList) => n p.1)
(fun p => volume ((u p.1).1.advance (u p.1).2 p.2.1)) 2 := by
apply Poly.volumeAdvance _ _ _ (Poly.size _) (fun p => hb p.1)
intro p
exact ht p.1 _ _ (Algorithm.lookup_mem (by simpa using p.2.2))
have hWV := Poly.volumeOptionMap (s := n) (fun j c => (u j).1.advance (u j).2 c)
(fun j => Algorithm.lookup (tab j) ((v j).left \ (u j).1.upto)) hAv
poly_auto
end ThreeMachine.StackCompiler.Costs
end
section
namespace ThreeMachine.StackCompiler.Costs
variable {J : Type} (n : J → ℕ) (U : ∀ j, Universe (n j)) (tab : ∀ j, Table (n j))
variable (u : ∀ j, Algorithm.State (Fin (n j)) × Algorithm.Block (Fin (n j)))
theorem finishing
(hl : Poly n (fun j => (tab j).length) 30000)
(ht : ∀ j, TableSmall (tab j)) (hb : ∀ j, BlockSmall (u j).2) :
Poly n (fun j => Uniform.finishing.time (n j) (u j,(U j,tab j))) 60002 := by
have htv := Poly.volumeTable tab (Poly.size n) hl ht
have hbV := Poly.volumeSmallBlock (fun j => (u j).2) (Poly.size n) hb
have hL : Poly n (fun j => volume (Algorithm.lookup (tab j) (u j).1.right)) 2 :=
Poly.volumeSmallLookup tab _ (Poly.size n) ht
have ha := advanceLast (fun p : Poly.ListPool (fun j => (Algorithm.lookup (tab j)
(u j).1.right).toList) => n p.1) (fun p => U p.1)
(fun p => u p.1) (fun p => p.2.1) (fun p => hb p.1) (fun p =>
ht p.1 _ _ (Algorithm.lookup_mem (by simpa using p.2.2)))
have hAv : Poly (fun p : Poly.ListPool (fun j => (Algorithm.lookup (tab j)
(u j).1.right).toList) => n p.1)
(fun p => volume ((u p.1).1.advance (u p.1).2 p.2.1)) 2 := by
apply Poly.volumeAdvance _ _ _ (Poly.size _) (fun p => hb p.1)
intro p
exact ht p.1 _ _ (Algorithm.lookup_mem (by simpa using p.2.2))
poly_auto
end ThreeMachine.StackCompiler.Costs
end
section
namespace ThreeMachine.StackCompiler.Uniform
abbrev SolveInput (n : ℕ) := Universe n × (Matrix n ×
(Table n × (List (Finset (Fin n)) × (List (Finset (Fin n)) × Finset (Fin n)))))
def solveU : Uniform (fun n (x : SolveInput n) => x.1) := fst
def solveM : Uniform (fun n (x : SolveInput n) => x.2.1) := snd.comp fst
def solveT : Uniform (fun n (x : SolveInput n) => x.2.2.1) := snd.comp (snd.comp fst)
def solveF : Uniform (fun n (x : SolveInput n) => x.2.2.2.1) := snd.comp (snd.comp (snd.comp fst))
def solveZ : Uniform (fun n (x : SolveInput n) => x.2.2.2.2.1) := snd.comp (snd.comp (snd.comp (snd.comp fst)))
def solveW : Uniform (fun n (x : SolveInput n) => x.2.2.2.2.2) := snd.comp (snd.comp (snd.comp (snd.comp snd)))
def solveS : Uniform (fun n (x : SolveInput n) =>
Algorithm.states (matrixRel x.2.1) x.2.2.2.1 x.2.2.2.2.1 x.2.2.2.2.2) := (solveU.pair (solveM.pair (solveF.pair (solveZ.pair solveW)))).comp states
def solveMarks : Uniform (fun n (x : SolveInput n) =>
Algorithm.markings x.2.2.1 (Algorithm.states (matrixRel x.2.1) x.2.2.2.1 x.2.2.2.2.1 x.2.2.2.2.2) n) := ((solveU.comp universeNumber).pair (solveU.pair (solveT.pair solveS))).comp markings
def solveOut : Uniform (fun n (x : SolveInput n) =>
Algorithm.firstResult (fun u => (Algorithm.lookup x.2.2.1 u.1.right).map (u.1.finish u.2))
(Algorithm.markings x.2.2.1 (Algorithm.states (matrixRel x.2.1) x.2.2.2.1 x.2.2.2.2.1 x.2.2.2.2.2) n)) := (solveMarks.pair (solveU.pair solveT)).comp finishing.firstResultA
theorem time_solve (n : ℕ) (x : SolveInput n) :
solve.time n x = ((solveW.comp setIsEmpty).ite (solveU.comp (blockEmpty.comp some)) solveOut).time n x := by
simp only [solve, time_congr, solveOut, solveMarks, solveS, solveU, solveM, solveT, solveF, solveZ, solveW]
end ThreeMachine.StackCompiler.Uniform
namespace ThreeMachine.StackCompiler.Poly
variable {J : Type} {s n : J → ℕ} {d : ℕ}
theorem volumeSolve (M : ∀ j, Matrix (n j)) (tab : ∀ j, Table (n j))
(fs zs : ∀ j, List (Finset (Fin (n j)))) (W : ∀ j, Finset (Fin (n j)))
(hn : Poly s n d) (ht : ∀ j, Algorithm.TableGood (matrixRel (M j)) (tab j))
(hb : ∀ j, Algorithm.TableBounded (tab j)) :
Poly s (fun j => volume (Algorithm.solve (matrixRel (M j)) (tab j) (fs j) (zs j) (W j))) (2*d) := by
apply volumeOption _ (show Poly s (fun j => 10*(n j+1)^2) (2*d) from by poly_bound)
intro j b h
have hs := blockSmall_of_good (Algorithm.solve_good _ (ht j) _ _ h)
(Algorithm.solve_bounded _ (hb j) _ _ h)
exact volume_block_le b hs.1 hs.2
end ThreeMachine.StackCompiler.Poly
end
section
namespace ThreeMachine.StackCompiler.Poly
variable {J : Type} {s n : J → ℕ} {d : ℕ}
theorem volumeSolveOut (M : ∀ j, Matrix (n j)) (tab : ∀ j, Table (n j))
(fs zs : ∀ j, List (Finset (Fin (n j)))) (W : ∀ j, Finset (Fin (n j)))
(hn : Poly s n d) (ht : ∀ j, Algorithm.TableGood (matrixRel (M j)) (tab j))
(hb : ∀ j, Algorithm.TableBounded (tab j)) :
Poly s (fun j => volume (Algorithm.firstResult
(fun u => (Algorithm.lookup (tab j) u.1.right).map (u.1.finish u.2))
(Algorithm.markings (tab j) (Algorithm.states (matrixRel (M j)) (fs j) (zs j) (W j)) (n j)))) (2*d) := by
apply volumeOption _ (show Poly s (fun j => 100*(n j+2)^2) (2*d) from by poly_bound)
intro j b h
obtain ⟨⟨u,bb⟩,hmem,hm⟩ := Algorithm.firstResult_some h
obtain ⟨t,ht',rfl⟩ := Option.map_eq_some_iff.mp hm
have hmark : MarksSmall (Algorithm.markings (tab j)
(Algorithm.states (matrixRel (M j)) (fs j) (zs j) (W j)) (n j)) := markings_small (matrixRel (M j)) (ht j) (hb j)
(fun _ hu => Algorithm.states_compatible _ hu) (n j)
have htSmall : TableSmall (tab j) := fun a b h => blockSmall_of_good (ht j a b h) (hb j a b h)
exact volume_advance_le u bb t (hmark u bb hmem) (htSmall _ _ (Algorithm.lookup_mem ht'))
end ThreeMachine.StackCompiler.Poly
end
end OAI
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