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b797a5b | 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 | import Mathlib
namespace OAI
namespace PiExponent.PowerCover
noncomputable section
variable {σ : Type*} [Fintype σ]
def scale (w : σ → ℕ) (q : σ →₀ ℕ) : σ →₀ ℕ :=
Finsupp.equivFunOnFinite.symm (fun i => w i * q i)
def quotient (w : σ → ℕ) (a : σ →₀ ℕ) : σ →₀ ℕ :=
Finsupp.equivFunOnFinite.symm (fun i => a i / w i)
def residue (w : σ → ℕ) (hw : ∀ i, 0 < w i) (a : σ →₀ ℕ) :
∀ i, Fin (w i) :=
fun i => ⟨a i % w i, Nat.mod_lt _ (hw i)⟩
def combine (w : σ → ℕ) (p : (σ →₀ ℕ) × (∀ i, Fin (w i))) : σ →₀ ℕ :=
Finsupp.equivFunOnFinite.symm (fun i => w i * p.1 i + (p.2 i).val)
@[simp] theorem scale_apply (w : σ → ℕ) (q : σ →₀ ℕ) (i : σ) :
scale w q i = w i * q i := by simp [scale]
@[simp] theorem quotient_apply (w : σ → ℕ) (a : σ →₀ ℕ) (i : σ) :
quotient w a i = a i / w i := by simp [quotient]
omit [Fintype σ] in
@[simp] theorem residue_apply (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a : σ →₀ ℕ) (i : σ) :
(residue w hw a i).val = a i % w i := rfl
@[simp] theorem combine_apply (w : σ → ℕ)
(p : (σ →₀ ℕ) × (∀ i, Fin (w i))) (i : σ) :
combine w p i = w i * p.1 i + (p.2 i).val := by simp [combine]
@[simp] theorem combine_quotient_residue (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a : σ →₀ ℕ) :
combine w (quotient w a, residue w hw a) = a := by
ext i
simp only [combine_apply, quotient_apply, residue_apply]
exact Nat.div_add_mod (a i) (w i)
@[simp] theorem quotient_combine (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(p : (σ →₀ ℕ) × (∀ i, Fin (w i))) :
quotient w (combine w p) = p.1 := by
ext i
simp only [quotient_apply, combine_apply]
rw [Nat.mul_add_div (hw i)]
simp [Nat.div_eq_of_lt (p.2 i).isLt]
@[simp] theorem residue_combine (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(p : (σ →₀ ℕ) × (∀ i, Fin (w i))) :
residue w hw (combine w p) = p.2 := by
funext i
apply Fin.ext
simp [Nat.add_mod, Nat.mod_eq_of_lt (p.2 i).isLt]
def exponentEquiv (w : σ → ℕ) (hw : ∀ i, 0 < w i) :
(σ →₀ ℕ) ≃ ((σ →₀ ℕ) × (∀ i, Fin (w i))) where
toFun a := (quotient w a, residue w hw a)
invFun := combine w
left_inv := combine_quotient_residue w hw
right_inv p := by simp [quotient_combine w hw]
@[simp] theorem exponentEquiv_apply (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a : σ →₀ ℕ) : exponentEquiv w hw a = (quotient w a, residue w hw a) := rfl
@[simp] theorem exponentEquiv_symm_apply (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(p : (σ →₀ ℕ) × (∀ i, Fin (w i))) :
(exponentEquiv w hw).symm p = combine w p := rfl
@[simp] theorem quotient_add_scale (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a q : σ →₀ ℕ) :
quotient w (a + scale w q) = quotient w a + q := by
ext i
simp only [quotient_apply, Finsupp.add_apply, scale_apply]
simp [Nat.add_mul_div_left, hw i]
@[simp] theorem residue_add_scale (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a q : σ →₀ ℕ) :
residue w hw (a + scale w q) = residue w hw a := by
funext i
apply Fin.ext
simp [Nat.add_mod]
@[simp] theorem exponentEquiv_add_scale (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a q : σ →₀ ℕ) :
exponentEquiv w hw (a + scale w q) = (quotient w a + q, residue w hw a) := by
simp [quotient_add_scale w hw]
@[simp] theorem combine_add (w : σ → ℕ) (q q' : σ →₀ ℕ)
(r : ∀ i, Fin (w i)) :
combine w (q + q', r) = combine w (q, r) + scale w q' := by
ext i
simp [Nat.mul_add, Nat.add_assoc, Nat.add_comm]
@[simp] theorem scale_zero (w : σ → ℕ) : scale w 0 = 0 := by
ext i
simp
@[simp] theorem scale_add (w : σ → ℕ) (q q' : σ →₀ ℕ) :
scale w (q + q') = scale w q + scale w q' := by
ext i
simp [Nat.mul_add]
end
end PiExponent.PowerCover
end OAI
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