| import OAI.Analysis.IrrationalRotation.Model.CircleParameters |
| import Mathlib.Algebra.BigOperators.Intervals |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| namespace Invariant233.CircleParameterData |
|
|
| @[simp] theorem blockEnd_zero (p : CircleParameterData) : p.blockEnd 0 = 0 := by |
| simp [blockEnd] |
|
|
| theorem blockEnd_succ (p : CircleParameterData) (j : ℕ) : |
| p.blockEnd (j + 1) = p.blockEnd j + p.m (j + 1) := by |
| exact Finset.sum_Icc_succ_top (by omega) p.m |
|
|
| theorem card_block (p : CircleParameterData) (j : ℕ) (hj : 1 ≤ j) : |
| (p.block j).card = p.m j := by |
| have hd := p.blockEnd_succ (j - 1) |
| have he : j - 1 + 1 = j := by omega |
| rw [he] at hd |
| simp only [block, Nat.card_Icc] |
| omega |
|
|
| theorem mem_block_iff (p : CircleParameterData) (j i : ℕ) : |
| i ∈ p.block j ↔ p.blockEnd (j - 1) < i ∧ i ≤ p.blockEnd j := by |
| simp only [block, Finset.mem_Icc, Nat.add_one_le_iff] |
|
|
| theorem blockEnd_ge_index (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) (j : ℕ) : j ≤ p.blockEnd j := by |
| induction j with |
| | zero => exact Nat.zero_le _ |
| | succ j ih => |
| rw [blockEnd_succ] |
| have h := hm (j + 1) (by omega) |
| omega |
|
|
| theorem blockEnd_strictMono (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) : StrictMono p.blockEnd := by |
| apply strictMono_nat_of_lt_succ |
| intro j |
| rw [blockEnd_succ] |
| exact Nat.lt_add_of_pos_right (hm (j + 1) (by omega)) |
|
|
| theorem exists_mem_block (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) (i : ℕ) (hi : 1 ≤ i) : |
| ∃ j, 1 ≤ j ∧ i ∈ p.block j := by |
| have hex : ∃ j, i ≤ p.blockEnd j := ⟨i, p.blockEnd_ge_index hm i⟩ |
| let j := Nat.find hex |
| have hj : i ≤ p.blockEnd j := Nat.find_spec hex |
| have hjpos : 1 ≤ j := by |
| by_contra h |
| have he : j = 0 := by omega |
| rw [he, blockEnd_zero] at hj |
| omega |
| refine ⟨j, hjpos, (p.mem_block_iff j i).mpr ⟨?_, hj⟩⟩ |
| have hprev : ¬ i ≤ p.blockEnd (j - 1) := |
| Nat.find_min hex (show j - 1 < j by omega) |
| omega |
|
|
| theorem block_index_unique (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) {j k i : ℕ} |
| (hij : i ∈ p.block j) (hik : i ∈ p.block k) : j = k := by |
| have hj := (p.mem_block_iff j i).mp hij |
| have hk := (p.mem_block_iff k i).mp hik |
| have hmono := (p.blockEnd_strictMono hm).monotone |
| apply le_antisymm |
| · by_contra h |
| have hjk : k ≤ j - 1 := by omega |
| have hd := hmono hjk |
| omega |
| · by_contra h |
| have hjk : j ≤ k - 1 := by omega |
| have hd := hmono hjk |
| omega |
|
|
| theorem blockIndex_eq_of_mem (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) {j i : ℕ} (hi : i ∈ p.block j) : |
| p.blockIndex i = j := by |
| classical |
| have hij := (p.mem_block_iff j i).mp hi |
| have hex : ∃ k, i ≤ p.blockEnd k := ⟨j, hij.2⟩ |
| rw [blockIndex, dite_eq_left hex] |
| apply le_antisymm |
| · exact Nat.find_min' hex hij.2 |
| · by_contra h |
| have hle : Nat.find hex ≤ j - 1 := by omega |
| have hd := (p.blockEnd_strictMono hm).monotone hle |
| have hf := Nat.find_spec hex |
| omega |
|
|
| theorem blockIndex_pos (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) {i : ℕ} (hi : 1 ≤ i) : |
| 1 ≤ p.blockIndex i := by |
| obtain ⟨j, hj, hij⟩ := p.exists_mem_block hm i hi |
| rwa [p.blockIndex_eq_of_mem hm hij] |
|
|
| theorem flattenedCost_eq_of_mem (p : CircleParameterData) |
| (hm : ∀ j, 1 ≤ j → 0 < p.m j) {j n : ℕ} (hn : n + 1 ∈ p.block j) : |
| p.flattenedCost n = p.C j := by |
| simp only [flattenedCost, p.blockIndex_eq_of_mem hm hn] |
|
|
| end Invariant233.CircleParameterData |
|
|
| end OAI |
|
|
