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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 | import OAI.Geometry.PeriodicTiling.OrdinaryActivation
import OAI.Geometry.PeriodicTiling.CommonSeedOutputs
import OAI.Geometry.PeriodicTiling.Dependence
namespace OAI
noncomputable section
namespace PeriodicTilingThree.CommonModel
variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
def outputs : GraphOutputs E where
c b i := match i with
| .inl n => ordinaryUseful E n b.1 (b.2 (.inl n))
| .inr t => seedC E t b.1 b.2
beta b i := match i with
| .inl n => ordinaryHigh E n b.1 (b.2 (.inl n))
| .inr t => seedBeta E t b.1 b.2
z b := SharedSeed.commonZ E.p_prime E.p_large b.1 b.2
def commonGraph : Set (Ambient E) := graph (outputs E)
@[simp] theorem outputs_c_ordinary (n : Column p) (x : Plane) (k : Input E) :
(outputs E).c (x,k) (.inl n) = ordinaryUseful E n x (k (.inl n)) := rfl
@[simp] theorem outputs_c_seed (t : Fin 2) (x : Plane) (k : Input E) :
(outputs E).c (x,k) (.inr t) = seedC E t x k := rfl
@[simp] theorem outputs_beta_ordinary (n : Column p) (x : Plane) (k : Input E) :
(outputs E).beta (x,k) (.inl n) = ordinaryHigh E n x (k (.inl n)) := rfl
@[simp] theorem outputs_beta_seed (t : Fin 2) (x : Plane) (k : Input E) :
(outputs E).beta (x,k) (.inr t) = seedBeta E t x k := rfl
@[simp] theorem outputs_z (x : Plane) (k : Input E) :
(outputs E).z (x,k) = SharedSeed.commonZ E.p_prime E.p_large x k := rfl
@[simp] theorem usefulAt_ordinary (n : Column p) (x : Plane) (w : K E (.inl n)) :
usefulAt E (outputs E) (.inl n) x w = ordinaryUseful E n x w := by
simp [usefulAt]
@[simp] theorem usefulAt_seed (t : Fin 2) (x : Plane) (w : K E (.inr t)) :
usefulAt E (outputs E) (.inr t) x w =
seedUseful E t (SharedSeed.residue p x) w := by
simp [usefulAt, seedC]
theorem outputs_hasDependence : HasDependence E (outputs E) := by
constructor
· intro n x k
simp only [outputs_c_ordinary, ordinaryFactor, usefulAt_ordinary]
apply ordinaryUseful_eq_of_lineValue_eq E n
simp
· intro t x k
simp only [outputs_c_seed, seedFactor, usefulAt_seed, seedC]
have hr : SharedSeed.residue p (seedRepresentative p (seedResidue p x)) =
SharedSeed.residue p x :=
seedResidue_representative (p := p) (seedResidue p x)
rw [hr]
theorem active_ordinary_iff (n : Column p) (x : Plane) (j : Symbol p) :
Active E (outputs E) (.inl n) x j ↔ j = ordinarySymbol E n x := by
simpa only [Active, DigitActive, usefulAt_ordinary] using
ordinaryUseful_active_iff E n x j
theorem active_seed_iff (t : Fin 2) (x : Plane) :
Active E (outputs E) (.inr t) x () ↔
SharedSeed.residue p x ∈ seedActiveRegion p t := by
simpa only [Active, DigitActive, usefulAt_seed] using
seedUseful_active_iff E t (SharedSeed.residue p x)
theorem active_word_allowed (x : Plane) (w : Word p)
(hw : ∀ n, Active E (outputs E) (.inl n) x (w n)) : Allowed p w := by
have he : w = ordinaryWord E x :=
funext fun n => (active_ordinary_iff E n x (w n)).mp (hw n)
rw [he]
exact ordinaryWord_allowed E x
theorem active_seed_forces_symbol (t : Fin 2) (n : Column p) (x : Plane)
(j : Symbol p) (ht : Active E (outputs E) (.inr t) x ())
(hj : Active E (outputs E) (.inl n) x j) :
j = lastDigit p E.p_prime ((t.val : ℤ) + 1) := by
have hj' := (active_ordinary_iff E n x j).mp hj
have hr := (active_seed_iff E t x).mp ht
fin_cases t
· have hfirst : SharedSeed.residue p x ∈ SharedSeed.firstRegion p := by
simpa [seedActiveRegion] using hr
simpa using hj'.trans (ordinarySymbol_firstRegion E n x hfirst)
· have hsecond : SharedSeed.residue p x ∈ SharedSeed.secondRegion p := by
simpa [seedActiveRegion] using hr
simpa using hj'.trans (ordinarySymbol_secondRegion E n x hsecond)
theorem commonGraph_tiles_kernel : Tiles (kernelTile E) (commonGraph E) :=
kernelTile_graph E (outputs E)
end PeriodicTilingThree.CommonModel
end
end OAI
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