| import Mathlib |
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| namespace OAI |
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| namespace PiExponent.PowerCover |
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| noncomputable section |
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| variable {σ : Type*} [Fintype σ] |
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| def scale (w : σ → ℕ) (q : σ →₀ ℕ) : σ →₀ ℕ := |
| Finsupp.equivFunOnFinite.symm (fun i => w i * q i) |
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| def quotient (w : σ → ℕ) (a : σ →₀ ℕ) : σ →₀ ℕ := |
| Finsupp.equivFunOnFinite.symm (fun i => a i / w i) |
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| 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)⟩ |
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| 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) |
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| @[simp] theorem scale_apply (w : σ → ℕ) (q : σ →₀ ℕ) (i : σ) : |
| scale w q i = w i * q i := by simp [scale] |
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| @[simp] theorem quotient_apply (w : σ → ℕ) (a : σ →₀ ℕ) (i : σ) : |
| quotient w a i = a i / w i := by simp [quotient] |
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| 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 |
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| @[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] |
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| @[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) |
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| @[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] |
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| @[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] |
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| 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] |
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| @[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 |
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| @[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 |
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| @[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] |
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| @[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] |
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| @[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] |
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| @[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] |
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| @[simp] theorem scale_zero (w : σ → ℕ) : scale w 0 = 0 := by |
| ext i |
| simp |
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| @[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] |
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| end |
| end PiExponent.PowerCover |
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| end OAI |
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