| import OAI.Probability.InvariantIsing.Core.FiniteMinmax |
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| namespace OAI |
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| /-! Finite extrema transport under shifts, order and continuous evaluation. -/ |
| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators |
| namespace InvariantIsing |
| variable {U V : Type*} [Fintype U] [Nonempty U] [Fintype V] [Nonempty V] |
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| omit [Fintype V] [Nonempty V] in |
| lemma finiteRowMax_add_constant (H : U × V → ℝ) (a : ℝ) (v : V) : |
| finiteRowMax (fun x => H x+a) v = finiteRowMax H v+a := by |
| obtain ⟨u,hu,he⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun u => H (u,v)) |
| apply le_antisymm |
| · apply Finset.sup'_le |
| intro u _ |
| exact add_le_add (Finset.le_sup' (fun u => H (u,v)) (Finset.mem_univ u)) (le_refl a) |
| · change (Finset.univ.sup' Finset.univ_nonempty (fun u => H (u,v)))+a ≤ _ |
| rw [he] |
| exact Finset.le_sup' (fun u => H (u,v)+a) hu |
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| lemma finiteMinmax_add_constant (H : U × V → ℝ) (a : ℝ) : |
| finiteMinmax (fun x => H x+a) = finiteMinmax H+a := by |
| obtain ⟨v,hv,he⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (finiteRowMax H) |
| apply le_antisymm |
| · calc |
| _ ≤ finiteRowMax (fun x => H x+a) v := finiteMinmax_le_row _ v |
| _ = finiteRowMax H v+a := finiteRowMax_add_constant H a v |
| _ = finiteMinmax H+a := by rw [← he]; rfl |
| · apply Finset.le_inf' |
| intro v _ |
| rw [finiteRowMax_add_constant] |
| exact add_le_add (finiteMinmax_le_row H v) (le_refl a) |
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| lemma finiteMinmax_mono {H J : U × V → ℝ} (h : ∀ x, H x ≤ J x) : |
| finiteMinmax H ≤ finiteMinmax J := by |
| apply Finset.le_inf' |
| intro v _ |
| apply (finiteMinmax_le_row H v).trans |
| apply Finset.sup'_le |
| intro u _ |
| exact (h (u,v)).trans (Finset.le_sup' (fun u => J (u,v)) (Finset.mem_univ u)) |
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| lemma finiteMinmax_abs_le (H : U × V → ℝ) (K : ℝ) (h : ∀ x, |H x| ≤ K) : |
| |finiteMinmax H| ≤ K := by |
| obtain ⟨v,_,hv⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (finiteRowMax H) |
| obtain ⟨u,_,hu⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun u => H (u,v)) |
| change |Finset.univ.inf' Finset.univ_nonempty _| ≤ K |
| rw [hv] |
| change |Finset.univ.sup' Finset.univ_nonempty _| ≤ K |
| rw [hu] |
| exact h (u,v) |
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| lemma continuous_finiteMinmax {S : Type*} [TopologicalSpace S] (H : S → U × V → ℝ) |
| (hH : ∀ x, Continuous (fun s => H s x)) : Continuous (fun s => finiteMinmax (H s)) := |
| Continuous.finset_inf'_apply Finset.univ_nonempty (fun v _ => |
| Continuous.finset_sup'_apply Finset.univ_nonempty (fun u _ => hH (u,v))) |
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| end InvariantIsing |
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| end |
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| end OAI |
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