| import Mathlib |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace SKRatioGaussian.PathBridge |
| open Real Set |
| open scoped BigOperators NNReal |
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| theorem finite_time_mesh {T δ₀ : ℝ} (hT : 0<T) (hδ₀ : 0<δ₀) : |
| ∃ m : ℕ, 0 < m ∧ ∃ δ : ℝ≥0, 0<δ ∧ (δ:ℝ)≤δ₀ ∧ (m:ℝ)*(δ:ℝ)=T ∧ |
| (∀ k : Fin m,(k:ℝ)*(δ:ℝ)∈Icc 0 T) ∧ |
| ∀ t∈Icc 0 T,∃ k : Fin m,∃ u∈Icc (0:ℝ) 1,t=(k:ℝ)*(δ:ℝ)+(δ:ℝ)*u := by |
| obtain ⟨m,hm⟩ := exists_nat_gt (T/δ₀) |
| have hm0 : 0 < m := by |
| exact_mod_cast lt_trans (div_pos hT hδ₀) hm |
| have hmr : 0<(m:ℝ) := by exact_mod_cast hm0 |
| let δ : ℝ≥0 := ⟨T/m,(div_pos hT hmr).le⟩ |
| have hδ : 0<(δ:ℝ) := div_pos hT hmr |
| have he : (m:ℝ)*(δ:ℝ)=T := by |
| change (m:ℝ)*(T/(m:ℝ))=T |
| field_simp |
| refine ⟨m,hm0,δ,hδ,?_,he,?_,?_⟩ |
| · change T/(m:ℝ)≤δ₀ |
| rw [div_le_iff₀ hmr] |
| have h := (div_lt_iff₀ hδ₀).mp hm |
| nlinarith |
| · intro k |
| constructor |
| · positivity |
| · have hk : (k:ℝ)<(m:ℝ) := by exact_mod_cast k.isLt |
| nlinarith |
| · intro t ht |
| by_cases htop : t=T |
| · let k : Fin m := ⟨m-1,by omega⟩ |
| refine ⟨k,1,by norm_num,?_⟩ |
| have hk : (k:ℝ)=(m:ℝ)-1 := by |
| dsimp [k] |
| rw [Nat.cast_sub (by omega)] |
| norm_num |
| rw [htop,hk] |
| nlinarith |
| · have hlt : t<T := lt_of_le_of_ne ht.2 htop |
| have hx : 0≤t/(δ:ℝ) := div_nonneg ht.1 hδ.le |
| have hxm : t/(δ:ℝ) < m := by rw [div_lt_iff₀ hδ];nlinarith |
| have hfloor : ⌊t/(δ:ℝ)⌋₊ < m := (Nat.floor_lt hx).mpr hxm |
| let k : Fin m := ⟨⌊t/(δ:ℝ)⌋₊,hfloor⟩ |
| let u := t/(δ:ℝ)-(k:ℝ) |
| refine ⟨k,u,⟨?_,?_⟩,?_⟩ |
| · exact sub_nonneg.mpr (Nat.floor_le hx) |
| · have hf := Nat.lt_floor_add_one (t/(δ:ℝ)) |
| dsimp [u,k] |
| linarith |
| · dsimp [u] |
| field_simp |
| ring |
| end SKRatioGaussian.PathBridge |
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| end |
| end |
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| section |
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| noncomputable section |
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| end |
| end |
| end |
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| end OAI |
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