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import OAI.Geometry.PeriodicTiling.ShiftMultiplicity
import Mathlib.Tactic.Abel
namespace OAI
universe uΔ uK uA uB
namespace PeriodicTilingThree
structure BlockShiftData (Δ : Type uΔ) (a b : ℕ) where
e : Δ → ZMod a × ZMod b
rhoA : Δ → ZMod a
rhoB : Δ → ZMod b
orderA : Δ ≃ (ZMod b × ZMod a)
orderB : Δ ≃ (ZMod a × ZMod b)
orderA_apply : ∀ δ, orderA δ = ((e δ).2, (e δ).1 + rhoA δ)
orderB_apply : ∀ δ, orderB δ = ((e δ).1, (e δ).2 + rhoB δ)
namespace BlockShiftData
noncomputable def ofMultiplicity {a b : ℕ} [NeZero a] [NeZero b]
(ha : 2 ≤ a) (hb : 2 ≤ b) : BlockShiftData (ShiftIndex a b) a b where
e := shift a b
rhoA := PeriodicTilingThree.rhoA ha hb
rhoB := PeriodicTilingThree.rhoB ha hb
orderA := shiftOrderA ha hb
orderB := shiftOrderB ha hb
orderA_apply := shiftOrderA_apply ha hb
orderB_apply := shiftOrderB_apply ha hb
variable {K : Type uK} {A : Type uA} {B : Type uB} {Δ : Type uΔ} [AddCommGroup K] {a b : ℕ}
def blockForward (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(H : K × Δ → ℤ) (x : ℤ) (k : K) (z : K × Δ) :
(ZMod a × ZMod b) × K :=
(D.C (k - z.1) + S.e z.2, D.T (x - H z) (k - z.1))
def blockInverse (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(x : ℤ) (k : K) (v : (ZMod a × ZMod b) × K) : K × Δ :=
match D.decode v.2 with
| .inl (c, q) =>
let δ := S.orderA.symm (v.1.2, v.1.1 + (x : ZMod a) - q)
(k - D.decode.symm (.inl (c, v.1.1 - (S.e δ).1)), δ)
| .inr (c, q) =>
let δ := S.orderB.symm (v.1.1, v.1.2 + (x : ZMod b) - q)
(k - D.decode.symm (.inr (c, v.1.2 - (S.e δ).2)), δ)
theorem blockInverse_inl (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : A) (q : ZMod a) :
blockInverse D S x k ((α, ξ), D.decode.symm (.inl (c, q))) =
let δ := S.orderA.symm (ξ, α + (x : ZMod a) - q)
(k - D.decode.symm (.inl (c, α - (S.e δ).1)), δ) := by
simp only [blockInverse, Equiv.apply_symm_apply]
theorem blockInverse_inr (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : B) (q : ZMod b) :
blockInverse D S x k ((α, ξ), D.decode.symm (.inr (c, q))) =
let δ := S.orderB.symm (α, ξ + (x : ZMod b) - q)
(k - D.decode.symm (.inr (c, ξ - (S.e δ).2)), δ) := by
simp only [blockInverse, Equiv.apply_symm_apply]
theorem blockForward_inl (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(H : K × Δ → ℤ) (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(x : ℤ) (k : K) (c : A) (y : ZMod a) (δ : Δ) :
blockForward D S H x k (k - D.decode.symm (.inl (c, y)), δ) =
((y + (S.e δ).1, (S.e δ).2),
D.decode.symm (.inl (c, y + (x : ZMod a) - S.rhoA δ))) := by
simp [blockForward, Int.cast_sub, hA, add_sub_assoc]
apply Prod.ext <;> simp
theorem blockForward_inr (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(H : K × Δ → ℤ) (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
(x : ℤ) (k : K) (c : B) (y : ZMod b) (δ : Δ) :
blockForward D S H x k (k - D.decode.symm (.inr (c, y)), δ) =
(((S.e δ).1, y + (S.e δ).2),
D.decode.symm (.inr (c, y + (x : ZMod b) - S.rhoB δ))) := by
simp [blockForward, Int.cast_sub, hB, add_sub_assoc]
apply Prod.ext <;> simp
theorem recoverA (S : BlockShiftData Δ a b) (δ : Δ) (y t : ZMod a) :
S.orderA.symm ((S.e δ).2, (y + (S.e δ).1) + t -
(y + t - S.rhoA δ)) = δ := by
apply S.orderA.injective
rw [Equiv.apply_symm_apply, S.orderA_apply]
apply Prod.ext
· rfl
· abel_nf
theorem recoverB (S : BlockShiftData Δ a b) (δ : Δ) (y t : ZMod b) :
S.orderB.symm ((S.e δ).1, (y + (S.e δ).2) + t -
(y + t - S.rhoB δ)) = δ := by
apply S.orderB.injective
rw [Equiv.apply_symm_apply, S.orderB_apply]
apply Prod.ext
· rfl
· abel_nf
theorem blockInverse_forward_inl (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(x : ℤ) (k : K) (c : A) (y : ZMod a) (δ : Δ) :
blockInverse D S x k
(blockForward D S H x k (k - D.decode.symm (.inl (c, y)), δ)) =
(k - D.decode.symm (.inl (c, y)), δ) := by
rw [blockForward_inl D S H hA, blockInverse_inl]
simp only [recoverA]
simp
theorem blockInverse_forward_inr (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
(x : ℤ) (k : K) (c : B) (y : ZMod b) (δ : Δ) :
blockInverse D S x k
(blockForward D S H x k (k - D.decode.symm (.inr (c, y)), δ)) =
(k - D.decode.symm (.inr (c, y)), δ) := by
rw [blockForward_inr D S H hB, blockInverse_inr]
simp only [recoverB]
simp
theorem blockInverse_forward (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
(x : ℤ) (k : K) :
Function.LeftInverse (blockInverse D S x k) (blockForward D S H x k) := by
rintro ⟨l, δ⟩
cases he : D.decode (k - l) with
| inl v =>
rcases v with ⟨c, y⟩
have hu : D.decode.symm (.inl (c, y)) = k - l := by
simpa only [he] using D.decode.symm_apply_apply (k - l)
simpa only [hu, sub_sub_cancel] using blockInverse_forward_inl D S H hA x k c y δ
| inr v =>
rcases v with ⟨c, y⟩
have hu : D.decode.symm (.inr (c, y)) = k - l := by
simpa only [he] using D.decode.symm_apply_apply (k - l)
simpa only [hu, sub_sub_cancel] using blockInverse_forward_inr D S H hB x k c y δ
theorem blockForward_inverse_inl (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : A) (q : ZMod a) :
blockForward D S H x k
(blockInverse D S x k ((α, ξ), D.decode.symm (.inl (c, q)))) =
((α, ξ), D.decode.symm (.inl (c, q))) := by
let δ := S.orderA.symm (ξ, α + (x : ZMod a) - q)
have horder : ((S.e δ).2, (S.e δ).1 + S.rhoA δ) =
(ξ, α + (x : ZMod a) - q) :=
(S.orderA_apply δ).symm.trans (S.orderA.apply_symm_apply _)
have he : (S.e δ).2 = ξ := congrArg Prod.fst horder
have hr : (S.e δ).1 + S.rhoA δ = α + (x : ZMod a) - q :=
congrArg Prod.snd horder
have hlabel : (α - (S.e δ).1) + (x : ZMod a) - S.rhoA δ = q := by
calc
_ = (α + (x : ZMod a)) - ((S.e δ).1 + S.rhoA δ) := by abel
_ = (α + (x : ZMod a)) - (α + (x : ZMod a) - q) := by rw [hr]
_ = q := sub_sub_cancel _ _
rw [blockInverse_inl]
change blockForward D S H x k
(k - D.decode.symm (.inl (c, α - (S.e δ).1)), δ) = _
rw [blockForward_inl D S H hA]
simp only [sub_add_cancel, he, hlabel]
theorem blockForward_inverse_inr (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
(x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : B) (q : ZMod b) :
blockForward D S H x k
(blockInverse D S x k ((α, ξ), D.decode.symm (.inr (c, q)))) =
((α, ξ), D.decode.symm (.inr (c, q))) := by
let δ := S.orderB.symm (α, ξ + (x : ZMod b) - q)
have horder : ((S.e δ).1, (S.e δ).2 + S.rhoB δ) =
(α, ξ + (x : ZMod b) - q) :=
(S.orderB_apply δ).symm.trans (S.orderB.apply_symm_apply _)
have he : (S.e δ).1 = α := congrArg Prod.fst horder
have hr : (S.e δ).2 + S.rhoB δ = ξ + (x : ZMod b) - q :=
congrArg Prod.snd horder
have hlabel : (ξ - (S.e δ).2) + (x : ZMod b) - S.rhoB δ = q := by
calc
_ = (ξ + (x : ZMod b)) - ((S.e δ).2 + S.rhoB δ) := by abel
_ = (ξ + (x : ZMod b)) - (ξ + (x : ZMod b) - q) := by rw [hr]
_ = q := sub_sub_cancel _ _
rw [blockInverse_inr]
change blockForward D S H x k
(k - D.decode.symm (.inr (c, ξ - (S.e δ).2)), δ) = _
rw [blockForward_inr D S H hB]
simp only [sub_add_cancel, he, hlabel]
theorem blockForward_inverse (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
(x : ℤ) (k : K) :
Function.RightInverse (blockInverse D S x k) (blockForward D S H x k) := by
rintro ⟨⟨α, ξ⟩, β⟩
cases he : D.decode β with
| inl v =>
rcases v with ⟨c, q⟩
have hβ : D.decode.symm (.inl (c, q)) = β := by
simpa only [he] using D.decode.symm_apply_apply β
simpa only [hβ] using blockForward_inverse_inl D S H hA x k α ξ c q
| inr v =>
rcases v with ⟨c, q⟩
have hβ : D.decode.symm (.inr (c, q)) = β := by
simpa only [he] using D.decode.symm_apply_apply β
simpa only [hβ] using blockForward_inverse_inr D S H hB x k α ξ c q
def blockInverseEquiv (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
(H : K × Δ → ℤ) (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2) (x : ℤ) (k : K) :
(K × Δ) ≃ ((ZMod a × ZMod b) × K) where
toFun := blockForward D S H x k
invFun := blockInverse D S x k
left_inv := blockInverse_forward D S H hA hB x k
right_inv := blockForward_inverse D S H hA hB x k
@[simp] theorem blockInverseEquiv_apply (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2) (x : ℤ) (k : K) (z : K × Δ) :
blockInverseEquiv D S H hA hB x k z =
(D.C (k - z.1) + S.e z.2, D.T (x - H z) (k - z.1)) := rfl
@[simp] theorem blockInverseEquiv_symm_apply (D : LabeledBlocks K A B a b)
(S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
(hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
(hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
(x : ℤ) (k : K) (v : (ZMod a × ZMod b) × K) :
(blockInverseEquiv D S H hA hB x k).symm v = blockInverse D S x k v := rfl
end BlockShiftData
end PeriodicTilingThree
end OAI
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