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110f4f3 | 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 52 53 54 55 56 57 58 59 60 | import Mathlib
namespace OAI
section
noncomputable section
open Set Filter
open scoped Topology
namespace WeakMTWTransport
section FunctionalCoercive
variable {E:Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
lemma functional_rank_one_positive
{L V:E →L[ℝ] E →L[ℝ] ℝ} {f g:E →L[ℝ] ℝ} {e:E} {β m γ:ℝ}
(hLs:∀d e,L d e=L e d) (hL:∀d,0 ≤ L d d)
(hV:∀d,d≠0 → 0 < V d d)
(hVL:∀d,V d d ≤ L d d+β*(f d)^2)
(hker:∀d,L e d=0) (hf:f e≠0) (hg:g e≠0)
(hm:0 < m) (hγ:0 < γ) :
∀d:E,d≠0 → 0 < m*L d d+γ*(g d)^2 := by
intro d hd
have hnon:0 ≤ m*L d d+γ*(g d)^2:=
add_nonneg (mul_nonneg hm.le (hL d)) (mul_nonneg hγ.le (sq_nonneg _))
by_contra H
have hz:m*L d d+γ*(g d)^2=0:=le_antisymm (not_lt.mp H) hnon
have hLd:L d d=0:=by
have :m*L d d=0:=by nlinarith only [hz,mul_nonneg hm.le (hL d),mul_nonneg hγ.le (sq_nonneg (g d))]
exact (mul_eq_zero.mp this).resolve_left hm.ne'
have hgd:g d=0:=by
have :γ*(g d)^2=0:=by rw [hLd,mul_zero,zero_add] at hz; exact hz
exact sq_eq_zero_iff.mp ((mul_eq_zero.mp this).resolve_left hγ.ne')
let a:=f d/f e
let w:=d-a • e
have hfw:f w=0:=by
dsimp only [w,a]
rw [map_sub,map_smul,smul_eq_mul,div_mul_cancel₀ _ hf,sub_self]
have hLw:L w w=0:=by
dsimp only [w]
simp only [map_sub,map_smul,sub_apply,smul_apply,
smul_eq_mul,hker,hLs d e,hLd,mul_zero,sub_zero]
have hw:w=0:=by
by_contra hw
have H:=hVL w
rw [hfw,hLw,pow_two,mul_zero,mul_zero,add_zero] at H
exact (not_lt_of_ge H) (hV w hw)
have hdw:d=a • e:=sub_eq_zero.mp hw
have ha:a=0:=by
rw [hdw,map_smul,smul_eq_mul] at hgd
exact (mul_eq_zero.mp hgd).resolve_right hg
exact hd (by rw [hdw,ha,zero_smul])
end FunctionalCoercive
end WeakMTWTransport
end
end
end OAI
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