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2fca478 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 | import OAI.Geometry.PeriodicTiling.SharedSeedSizeTransport
import OAI.Geometry.PeriodicTiling.ChannelDigitBlocks
import OAI.Geometry.PeriodicTiling.ActivationOffsets
namespace OAI
noncomputable section
namespace PeriodicTilingThree
open SharedSeed
variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
theorem seedOneSize
{p : ℕ} [NeZero p] (E : EncodingParameters p)
: E.a (.inr (0 : Fin 2)) = 2 := by
simpa using E.seed_a (0 : Fin 2)
theorem seedTwoSize
{p : ℕ} [NeZero p] (E : EncodingParameters p)
: E.a (.inr (1 : Fin 2)) = 3 := by
simpa using E.seed_a (1 : Fin 2)
def seedUseful (t : Fin 2) : Residues p → K E (.inr t) → P E (.inr t) :=
Fin.cases
(fun s u => usefulOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) u
(residueEquiv E.p_prime E.p_large s))
(fun j => by
have hj : j = 0 := Subsingleton.elim _ _
subst j
exact fun s u => usefulTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) u
(residueEquiv E.p_prime E.p_large s)) t
def seedHigh (t : Fin 2) : ℤ → Residues p → K E (.inr t) → K E (.inr t) :=
Fin.cases
(fun x s u => highOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) x u
(residueEquiv E.p_prime E.p_large s))
(fun j => by
have hj : j = 0 := Subsingleton.elim _ _
subst j
exact fun x s u => highTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) x u
(residueEquiv E.p_prime E.p_large s)) t
@[simp] theorem seedUseful_zero (s : Residues p) (u : K E (.inr (0 : Fin 2))) :
seedUseful E 0 s u =
usefulOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) u
(residueEquiv E.p_prime E.p_large s) := rfl
@[simp] theorem seedUseful_one (s : Residues p) (u : K E (.inr (1 : Fin 2))) :
seedUseful E 1 s u =
usefulTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) u
(residueEquiv E.p_prime E.p_large s) := rfl
@[simp] theorem seedHigh_zero (x : ℤ) (s : Residues p) (u : K E (.inr (0 : Fin 2))) :
seedHigh E 0 x s u =
highOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) x u
(residueEquiv E.p_prime E.p_large s) := rfl
@[simp] theorem seedHigh_one (x : ℤ) (s : Residues p) (u : K E (.inr (1 : Fin 2))) :
seedHigh E 1 x s u =
highTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) x u
(residueEquiv E.p_prime E.p_large s) := rfl
def seedC (t : Fin 2) (x : Plane) (k : Input E) : P E (.inr t) :=
seedUseful E t (residue p x) (k (.inr t))
def seedBeta (t : Fin 2) (x : Plane) (k : Input E) : K E (.inr t) :=
seedHigh E t x.1 (residue p x) (k (.inr t))
theorem seedC_eq_of_residue_low (t : Fin 2) {x y : Plane} {k l : Input E}
(hx : residue p x = residue p y) (hk : k (.inr t) = l (.inr t)) :
seedC E t x k = seedC E t y l := by
simp only [seedC, hx, hk]
def seedSourceInput (t : Fin 2) (k : Input E) (j : SeedActivationIndex E t) : Input E :=
Function.update k (.inr t) (k (.inr t) - j.1.1)
@[simp] theorem seedSourceInput_same (t : Fin 2) (k : Input E)
(j : SeedActivationIndex E t) :
seedSourceInput E t k j (.inr t) = k (.inr t) - j.1.1 := by
simp [seedSourceInput]
theorem seedSourceInput_low (t : Fin 2) (k : Input E) (j : SeedActivationIndex E t) :
seedSourceInput E t k j (.inr t) =
k (.inr t) - ((seedOffset E t j).1 : K E (.inr t)) := by
rw [seedSourceInput_same, seedOffset_low]
theorem seed_source_residue (t : Fin 2) (x : Plane) (j : SeedActivationIndex E t) :
residue p (x - seedOffset E t j) = residue p x - j.2 := by
have hs : residue p (seedOffset E t j) = j.2 := seedOffset_residue E t j
have hfirst : ((seedOffset E t j).1 : ZMod (p ^ 2)) = j.2.1 := congrArg Prod.fst hs
have hsecond : ((seedOffset E t j).2 : ZMod (p ^ 2)) = j.2.2 := congrArg Prod.snd hs
apply Prod.ext
· change ((x.1 - (seedOffset E t j).1 : ℤ) : ZMod (p ^ 2)) =
(x.1 : ZMod (p ^ 2)) - j.2.1
rw [Int.cast_sub, hfirst]
· change ((x.2 - (seedOffset E t j).2 : ℤ) : ZMod (p ^ 2)) =
(x.2 : ZMod (p ^ 2)) - j.2.2
rw [Int.cast_sub, hsecond]
def seedActualForward (t : Fin 2) (x : Plane) (k : Input E)
(j : SeedActivationIndex E t) : (P E (.inr t) × K E (.inr t)) × ZMod (D p) :=
let x' := x - seedOffset E t j
let k' := seedSourceInput E t k j
((seedC E t x' k' + shift (E.a (.inr t)) (E.b (.inr t)) j.1.2,
seedBeta E t x' k'), commonZ E.p_prime E.p_large x' k')
theorem seedActualForward_zero_eq (x : Plane) (k : Input E) :
seedActualForward E 0 x k =
oneForwardOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ())
(BlockShiftData.ofMultiplicity (E.a_two_le (.inr 0)) (E.b_two_le (.inr 0)))
(residueEquiv E.p_prime E.p_large) (outputEquiv E.p_large)
(fun j => (seedOffset E 0 j).1) x.1 (residue p x) (k (.inr 0)) := by
funext j
simp only [seedActualForward, seedC, seedBeta, seedSourceInput_same,
seed_source_residue, seedUseful_zero, seedHigh_zero, commonZ,
oneForwardOfEq, BlockShiftData.ofMultiplicity, Prod.fst_sub]
theorem seedActualForward_one_eq (x : Plane) (k : Input E) :
seedActualForward E 1 x k =
twoForwardOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ())
(BlockShiftData.ofMultiplicity (E.a_two_le (.inr 1)) (E.b_two_le (.inr 1)))
(residueEquiv E.p_prime E.p_large) (outputEquiv E.p_large)
(fun j => (seedOffset E 1 j).1) x.1 (residue p x) (k (.inr 1)) := by
funext j
simp only [seedActualForward, seedC, seedBeta, seedSourceInput_same,
seed_source_residue, seedUseful_one, seedHigh_one, commonZ,
twoForwardOfEq, BlockShiftData.ofMultiplicity, Prod.fst_sub]
theorem seedActualForward_bijective (t : Fin 2) (x : Plane) (k : Input E) :
Function.Bijective (seedActualForward E t x k) := by
fin_cases t
· change Function.Bijective (seedActualForward E (0 : Fin 2) x k)
rw [seedActualForward_zero_eq]
apply oneForwardOfEq_bijective
· intro j
exact seedOffset_rhoA E 0 j
· intro j
exact seedOffset_rhoB E 0 j
· intro j
have h := seedOffset_other E (0 : Fin 2) j
have ha : E.a (.inr (otherSeed (0 : Fin 2))) = 3 := by
change E.a (.inr (1 : Fin 2)) = 3
exact seedTwoSize E
exact Eq.mp (congrArg (fun modulus : ℕ =>
((seedOffset E (0 : Fin 2) j).1 : ZMod modulus) = 0) ha) h
· change Function.Bijective (seedActualForward E (1 : Fin 2) x k)
rw [seedActualForward_one_eq]
apply twoForwardOfEq_bijective
· intro j
exact seedOffset_rhoA E 1 j
· intro j
exact seedOffset_rhoB E 1 j
· intro j
have h := seedOffset_other E (1 : Fin 2) j
have ha : E.a (.inr (otherSeed (1 : Fin 2))) = 2 := by
change E.a (.inr (0 : Fin 2)) = 2
exact seedOneSize E
exact Eq.mp (congrArg (fun modulus : ℕ =>
((seedOffset E (1 : Fin 2) j).1 : ZMod modulus) = 0) ha) h
theorem both_seed_tests (x : Plane) (k : Input E) :
Function.Bijective (seedActualForward E 0 x k) ∧
Function.Bijective (seedActualForward E 1 x k) :=
⟨seedActualForward_bijective E 0 x k, seedActualForward_bijective E 1 x k⟩
def seedActiveRegion (p : ℕ) (t : Fin 2) : Set (Residues p) :=
if t = 0 then firstRegion p else secondRegion p
theorem seedUseful_active_iff (t : Fin 2) (s : Residues p) :
(∃ u v : K E (.inr t), u - v ∈ digitSubgroup E (.inr t) () ∧
seedUseful E t s u ≠ seedUseful E t s v) ↔ s ∈ seedActiveRegion p t := by
classical
fin_cases t
· change (∃ u v : K E (.inr (0 : Fin 2)),
u - v ∈ digitSubgroup E (.inr (0 : Fin 2)) () ∧
seedUseful E 0 s u ≠ seedUseful E 0 s v) ↔
s ∈ seedActiveRegion p (0 : Fin 2)
have hC := (channelDigitBlocks_active_iff E (.inr (0 : Fin 2)) () ()).mpr rfl
have hlabel : ActiveOne (residueEquiv E.p_prime E.p_large s) ↔ s ∈ firstRegion p := by
rw [← residueEquiv_first_iff E.p_prime E.p_large s]
cases hs : residueEquiv E.p_prime E.p_large s <;> simp [ActiveOne]
have h := usefulOneOfEq_activity_iff (seedOneSize E) (channelDigitBlocks E (.inr 0) ())
(fun u v => u - v ∈ digitSubgroup E (.inr 0) ()) hC
(residueEquiv E.p_prime E.p_large s)
rw [seedActiveRegion, ite_eq_left (rfl : (0 : Fin 2) = 0)]
simpa only [seedUseful_zero] using h.trans hlabel
· change (∃ u v : K E (.inr (1 : Fin 2)),
u - v ∈ digitSubgroup E (.inr (1 : Fin 2)) () ∧
seedUseful E 1 s u ≠ seedUseful E 1 s v) ↔
s ∈ seedActiveRegion p (1 : Fin 2)
have hC := (channelDigitBlocks_active_iff E (.inr (1 : Fin 2)) () ()).mpr rfl
have hlabel : ActiveTwo (residueEquiv E.p_prime E.p_large s) ↔ s ∈ secondRegion p := by
rw [← residueEquiv_second_iff E.p_prime E.p_large s]
cases hs : residueEquiv E.p_prime E.p_large s with
| inl y => simp [ActiveTwo]
| inr u => cases u <;> simp [ActiveTwo]
have h := usefulTwoOfEq_activity_iff (seedTwoSize E) (channelDigitBlocks E (.inr 1) ())
(fun u v => u - v ∈ digitSubgroup E (.inr 1) ()) hC
(residueEquiv E.p_prime E.p_large s)
simpa only [seedUseful_one, seedActiveRegion, ite_eq_right (by decide : (1 : Fin 2) ≠ 0)]
using h.trans hlabel
end PeriodicTilingThree
end
end OAI
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