| import OAI.Probability.InvariantIsing.Core.Model |
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|
| namespace OAI |
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| /-! Square extrema of finite nonnegative families. -/ |
| noncomputable section |
| open scoped Classical |
| namespace InvariantIsing |
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| lemma finite_sup_square {ι : Type*} [Fintype ι] [Nonempty ι] |
| (f : ι → ℝ) (hf : ∀ i, 0 ≤ f i) (q : ℝ) (hq : 0 ≤ q) : |
| Finset.univ.sup' Finset.univ_nonempty (fun i => q*(f i)^2)= |
| q*(Finset.univ.sup' Finset.univ_nonempty f)^2 := by |
| obtain ⟨a,_,ha⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty f |
| have hle (i : ι) : f i ≤ f a := by |
| rw [← ha] |
| exact Finset.le_sup' f (Finset.mem_univ i) |
| rw [ha] |
| apply le_antisymm |
| · apply Finset.sup'_le |
| intro i _ |
| exact mul_le_mul_of_nonneg_left (sq_le_sq₀ (hf i) (hf a) |>.mpr (hle i)) hq |
| · exact Finset.le_sup' (fun i => q*(f i)^2) (Finset.mem_univ a) |
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| lemma finite_sup_neg_square {ι : Type*} [Fintype ι] [Nonempty ι] |
| (f : ι → ℝ) (hf : ∀ i, 0 ≤ f i) (q : ℝ) (hq : 0 ≤ q) : |
| Finset.univ.sup' Finset.univ_nonempty (fun i => -q*(f i)^2)= |
| -q*(Finset.univ.inf' Finset.univ_nonempty f)^2 := by |
| obtain ⟨a,_,ha⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty f |
| have hle (i : ι) : f a ≤ f i := by |
| rw [← ha] |
| exact Finset.inf'_le f (Finset.mem_univ i) |
| rw [ha] |
| apply le_antisymm |
| · apply Finset.sup'_le |
| intro i _ |
| exact mul_le_mul_of_nonpos_left (sq_le_sq₀ (hf a) (hf i) |>.mpr (hle i)) (neg_nonpos.mpr hq) |
| · exact Finset.le_sup' (fun i => -q*(f i)^2) (Finset.mem_univ a) |
|
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| end InvariantIsing |
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| end |
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| end OAI |
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