| import OAI.NumberTheory.Ostmann.Tree.CycleQuartetActionsClassification |
| import OAI.NumberTheory.Ostmann.Tree.QuartetLocal |
| import OAI.NumberTheory.Ostmann.Tree.QuartetRemainingMeans |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace Ostmann.Tree.CycleQuartet |
| open Quartet Ostmann.FiniteField |
| variable {p : ℕ} [Fact p.Prime] |
| local instance cycleLocalMajorantCharDecEq : DecidableEq (MulChar (ZMod p) ℂ) := Classical.decEq _ |
|
|
| def bottomSign (P : Parameters (ZMod p) 2) (side freeRight : Bool) : Bool := |
| match P with |
| | .branch _ _ _ _ L R => parameterSign (if side then R else L) freeRight |
|
|
| def orientedCharacter (inverse : Bool) (ρ : MulChar (ZMod p) ℂ) : MulChar (ZMod p) ℂ := |
| if inverse then ρ⁻¹ else ρ |
|
|
| def selectedMajorant (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (κ : Kind) (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : ℝ := |
| match κ with |
| | .same side inverse => FiniteField.sameLocalMajorant g |
| (bottomSign P side false) (bottomSign P (!side) false) side (orientedCharacter inverse ρ) y |
| | .cross a b inverse => FiniteField.crossLocalMajorant g |
| (bottomSign P false a) (bottomSign P true b) a b (orientedCharacter inverse ρ) y |
|
|
| def unselectedMajorant (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : ℝ := |
| if ρ=1 then FiniteField.untouchedLocalMajorant g (bottomSign P false false) |
| (bottomSign P true false) y else 0 |
|
|
| theorem selectedMajorant_nonneg (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (κ : Kind) (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : |
| 0≤ selectedMajorant P g κ ρ y := by |
| cases κ with |
| | same side inverse => exact FiniteField.sameLocalMajorant_nonneg _ _ _ _ _ _ |
| | cross a b inverse => exact FiniteField.crossLocalMajorant_nonneg _ _ _ _ _ _ _ |
|
|
| theorem unselectedMajorant_nonneg (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : 0≤ unselectedMajorant P g ρ y := by |
| unfold unselectedMajorant |
| split_ifs |
| · exact FiniteField.untouchedLocalMajorant_nonneg _ _ _ _ |
| · exact le_rfl |
|
|
| theorem selectedMajorant_mean_small (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (κ : Kind) (ρ : MulChar (ZMod p) ℂ) (hg0 : g 0=0) (hg : l2Sq g≤1) : |
| unitMean (selectedMajorant P g κ ρ)≤25600*treeError g := by |
| cases κ with |
| | same side inverse => |
| exact (FiniteField.sameLocalMajorant_mean_small _ _ _ _ _ hg0 hg).trans |
| (mul_le_mul_of_nonneg_right (by norm_num) (treeError_nonneg g)) |
| | cross a b inverse => exact FiniteField.crossLocalMajorant_mean_small _ _ _ _ _ _ hg0 hg |
|
|
| omit [Fact p.Prime] in |
| theorem sum_orientedCharacter (inverse : Bool) (H : MulChar (ZMod p) ℂ → ℝ) : |
| (∑ ρ : MulChar (ZMod p) ℂ,H (orientedCharacter inverse ρ)) = ∑ ρ : MulChar (ZMod p) ℂ,H ρ := by |
| cases inverse |
| · rfl |
| · exact (Equiv.inv (MulChar (ZMod p) ℂ)).bijective.sum_comp H |
|
|
| theorem selectedMajorant_mean_total (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (κ : Kind) (hg0 : g 0=0) (hg : l2Sq g≤1) : |
| (∑ ρ : MulChar (ZMod p) ℂ,unitMean (selectedMajorant P g κ ρ))≤256 := by |
| cases κ with |
| | same side inverse => |
| change (∑ ρ : MulChar (ZMod p) ℂ,unitMean (FiniteField.sameLocalMajorant g |
| (bottomSign P side false) (bottomSign P (!side) false) side (orientedCharacter inverse ρ)))≤256 |
| rw [sum_orientedCharacter inverse (fun ρ => unitMean (FiniteField.sameLocalMajorant g |
| (bottomSign P side false) (bottomSign P (!side) false) side ρ))] |
| exact FiniteField.sameLocalMajorant_mean_total _ _ _ _ hg0 hg |
| | cross a b inverse => |
| change (∑ ρ : MulChar (ZMod p) ℂ,unitMean (FiniteField.crossLocalMajorant g |
| (bottomSign P false a) (bottomSign P true b) a b (orientedCharacter inverse ρ)))≤256 |
| rw [sum_orientedCharacter inverse (fun ρ => unitMean (FiniteField.crossLocalMajorant g |
| (bottomSign P false a) (bottomSign P true b) a b ρ))] |
| exact (FiniteField.crossLocalMajorant_mean_total _ _ _ _ _ hg0 hg).trans (by norm_num) |
|
|
| theorem unselectedMajorant_mean_total (P : Parameters (ZMod p) 2) (g : ZMod p → ℂ) |
| (hg0 : g 0=0) (hg : l2Sq g≤1) : |
| (∑ ρ : MulChar (ZMod p) ℂ,unitMean (unselectedMajorant P g ρ))≤256 := by |
| classical |
| have he (ρ : MulChar (ZMod p) ℂ) : unitMean (unselectedMajorant P g ρ) = |
| if ρ=1 then unitMean (FiniteField.untouchedLocalMajorant g (bottomSign P false false) |
| (bottomSign P true false)) else 0 := by |
| by_cases hρ : ρ=1 <;> simp [unselectedMajorant,hρ,unitMean] |
| simp_rw [he] |
| simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
| exact (FiniteField.untouchedLocalMajorant_mean _ _ _ hg0 hg).trans (by norm_num) |
|
|
| theorem selectedMajorant_same_node (N : NodeInput (ZMod p) 1) (g : ZMod p → ℂ) |
| (side inverse : Bool) (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : |
| selectedMajorant N.parameters g (.same side inverse) ρ y = |
| N.sameLocalMajorant g side (orientedCharacter inverse ρ) y := by |
| cases side <;> rfl |
|
|
| theorem selectedMajorant_cross_node (N : NodeInput (ZMod p) 1) (g : ZMod p → ℂ) |
| (a b inverse : Bool) (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : |
| selectedMajorant N.parameters g (.cross a b inverse) ρ y = |
| N.crossLocalMajorant g a b (orientedCharacter inverse ρ) y := by |
| simp only [selectedMajorant, bottomSign, NodeInput.parameters, Bool.false_eq_true, |
| ↓reduceIte, FiniteField.crossLocalMajorant, NodeInput.crossLocalMajorant, |
| norm_div, Complex.norm_natCast] |
|
|
| theorem unselectedMajorant_node (N : NodeInput (ZMod p) 1) (g : ZMod p → ℂ) |
| (ρ : MulChar (ZMod p) ℂ) (y : ZMod p) : |
| unselectedMajorant N.parameters g ρ y = if ρ=1 then N.untouchedLocalMajorant g y else 0 := rfl |
|
|
| end Ostmann.Tree.CycleQuartet |
| end |
|
|
| end OAI |
|
|
