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c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import OAI.Probability.InvariantIsing.Core.Model
namespace OAI
/-! Square extrema of finite nonnegative families. -/
noncomputable section
open scoped Classical
namespace InvariantIsing
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)
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)
end InvariantIsing
end
end OAI
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