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a89d282 | 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 46 47 48 49 50 51 | import OAI.Probability.InvariantIsing.Magnetic.MagneticMinimumCalculus
import Mathlib.Topology.Order.Compact
namespace OAI
/-! The scalar minimum argument on a compact spatial state space. -/
noncomputable section
open Filter Set
open scoped Topology
namespace InvariantIsing
lemma compact_parabolic_nonneg {K : Type*} [TopologicalSpace K] [CompactSpace K] [Nonempty K]
{T : ℝ} (hT : 0 ≤ T) (W : ℝ → K → ℝ) (Wt : ℝ × K → ℝ) (B : ℝ → K → ℝ)
(hc : ContinuousOn (fun z : ℝ × K => W z.1 z.2) (Icc (0 : ℝ) T ×ˢ Set.univ))
(hi : ∀ x, 0 ≤ W 0 x)
(hd : ∀ t ∈ Ioc (0 : ℝ) T, ∀ x, HasDerivAt (fun s => W s x) (Wt (t,x)) t)
(hpde : ∀ t ∈ Ioc (0 : ℝ) T, ∀ x, B t x-W t x ≤ Wt (t,x))
(hspace : ∀ t ∈ Icc (0 : ℝ) T, ∀ x, (∀ y, W t x ≤ W t y) → 0 ≤ B t x) :
∀ t ∈ Icc (0 : ℝ) T, ∀ x, 0 ≤ W t x := by
intro t ht x
by_contra hn
have hneg : W t x < 0 := lt_of_not_ge hn
let x₀ : K := Classical.choice inferInstance
obtain ⟨q,hq,hm⟩ := (isCompact_Icc.prod (isCompact_univ : IsCompact (Set.univ : Set K))).exists_isMinOn
⟨(0,x₀),⟨⟨le_rfl,hT⟩,Set.mem_univ x₀⟩⟩ hc
have hqneg : W q.1 q.2 < 0 := (hm (show (t,x) ∈ Icc (0 : ℝ) T ×ˢ (Set.univ : Set K) from
⟨ht,Set.mem_univ x⟩)).trans_lt hneg
have hqt : 0 < q.1 := by
by_contra h
have he : q.1=0 := le_antisymm (le_of_not_gt h) hq.1.1
rw [he] at hqneg
exact (not_lt_of_ge (hi q.2)) hqneg
have hleft : ∀ᶠ s in 𝓝[<] q.1, W q.1 q.2 ≤ W s q.2 := by
have hpos : ∀ᶠ s in 𝓝[<] q.1, 0 < s :=
(show ∀ᶠ s : ℝ in 𝓝 q.1, 0 < s from Ioi_mem_nhds hqt).filter_mono
nhdsWithin_le_nhds
filter_upwards [hpos,self_mem_nhdsWithin] with s hs hs'
exact hm (show (s,q.2) ∈ Icc (0 : ℝ) T ×ˢ (Set.univ : Set K) from
⟨⟨hs.le,hs'.le.trans hq.1.2⟩,Set.mem_univ q.2⟩)
have htime := derivative_nonpos_at_left_min (hd q.1 ⟨hqt,hq.1.2⟩ q.2) hleft
have hsp := hspace q.1 hq.1 q.2 (fun y => hm (show (q.1,y) ∈ Icc (0 : ℝ) T ×ˢ (Set.univ : Set K) from
⟨hq.1,Set.mem_univ y⟩))
have hp := hpde q.1 ⟨hqt,hq.1.2⟩ q.2
linarith
end InvariantIsing
end
end OAI
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