| import Mathlib.Topology.ContinuousMap.Compact | |
| import Mathlib.Topology.ContinuousMap.Algebra | |
| import Mathlib.Topology.CompactOpen | |
| namespace OAI | |
| /-! A continuous running maximum for nonnegative continuous functions. | |
| Using the norm of a curried map avoids any selection of maximizers. -/ | |
| open Set | |
| open scoped Topology unitInterval | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| def runningMaximum (k : C(unitInterval, ℝ)) : C(unitInterval, ℝ) := | |
| let F : C(unitInterval × unitInterval, ℝ) := | |
| ⟨fun p => k (min p.1 p.2), k.continuous.comp (continuous_fst.min continuous_snd)⟩ | |
| ⟨fun t => ‖F.curry t‖, continuous_norm.comp F.curry.continuous⟩ | |
| lemma runningMaximum_nonneg (k : C(unitInterval, ℝ)) (t : unitInterval) : | |
| 0 ≤ runningMaximum k t := by | |
| dsimp only [runningMaximum, ContinuousMap.coe_mk] | |
| exact norm_nonneg _ | |
| lemma le_runningMaximum (k : C(unitInterval, ℝ)) {s t : unitInterval} (hst : s ≤ t) : | |
| k s ≤ runningMaximum k t := by | |
| let F : C(unitInterval × unitInterval, ℝ) := | |
| ⟨fun p => k (min p.1 p.2), k.continuous.comp (continuous_fst.min continuous_snd)⟩ | |
| have hh := (F.curry t).norm_coe_le_norm s | |
| change ‖k (min t s)‖ ≤ runningMaximum k t at hh | |
| rw [min_eq_right hst, Real.norm_eq_abs] at hh | |
| exact (le_abs_self _).trans hh | |
| lemma runningMaximum_le (k : C(unitInterval, ℝ)) (hk : ∀ u, 0 ≤ k u) | |
| {t : unitInterval} {c : ℝ} (hc : 0 ≤ c) (h : ∀ u ≤ t, k u ≤ c) : | |
| runningMaximum k t ≤ c := by | |
| apply (ContinuousMap.norm_le _ hc).mpr | |
| intro u | |
| change ‖k (min t u)‖ ≤ c | |
| rw [Real.norm_eq_abs, abs_of_nonneg (hk _)] | |
| exact h _ (min_le_left _ _) | |
| lemma runningMaximum_mono (k : C(unitInterval, ℝ)) (hk : ∀ u, 0 ≤ k u) : | |
| Monotone (runningMaximum k) := by | |
| intro s t hst | |
| exact runningMaximum_le k hk (runningMaximum_nonneg k t) | |
| (fun u hu => le_runningMaximum k (hu.trans hst)) | |
| lemma runningMaximum_eq_of_constant (k : C(unitInterval, ℝ)) (hk : ∀ u, 0 ≤ k u) | |
| {s t : unitInterval} (hst : s ≤ t) (heq : ∀ u ∈ Icc s t, k u = k s) : | |
| runningMaximum k s = runningMaximum k t := by | |
| apply le_antisymm (runningMaximum_mono k hk hst) | |
| apply runningMaximum_le k hk (runningMaximum_nonneg k s) | |
| intro u hut | |
| by_cases hus : u ≤ s | |
| · exact le_runningMaximum k hus | |
| · rw [heq u ⟨le_of_not_ge hus, hut⟩] | |
| exact le_runningMaximum k le_rfl | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |