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import OAI.Probability.LogConcave.Numerics.ScalarPicard
namespace OAI
section
noncomputable section
namespace LogConcaveSampling.MeanTree
open scoped BigOperators
lemma flowWork_bound (p N y : ℕ) : flowWork p N 0 y+1≤(y+1)*(p+1)^N := by
induction N with
| zero => simp [flowWork,scalarPicard]
| succ N ih =>
have hp : 1≤(p+1)^N := Nat.one_le_pow N _ (by omega)
have hy : y+1≤(y+1)*(p+1)^N := by nlinarith
have he := Nat.mul_le_mul_left p ih
change y+p*(1+0+flowWork p N 0 y)+1≤_
rw [pow_succ]
nlinarith
lemma kernelWork_bound (p q N y g : ℕ) :
kernelWork p q N 0 y g+1≤(q+1)^N*(p+1)^(2*N+1)*(y+g+1) := by
let U := flowWork p N 0 y
let V := flowWork q N 0 (U+g)
let W := flowWork p N 0 V
have hP : 1≤(p+1)^N := Nat.one_le_pow N _ (by omega)
have hQ : 1≤(q+1)^N := Nat.one_le_pow N _ (by omega)
have hU : U+g+1≤(y+g+1)*(p+1)^N := by
have he := flowWork_bound p N y
change U+1≤_ at he
nlinarith
have hV : V+1≤(y+g+1)*(p+1)^N*(q+1)^N :=
(flowWork_bound q N (U+g)).trans (Nat.mul_le_mul_right _ hU)
have hW : W+1≤(y+g+1)*(p+1)^N*(q+1)^N*(p+1)^N :=
(flowWork_bound p N V).trans (Nat.mul_le_mul_right _ hV)
have hone : 1≤(y+g+1)*(p+1)^N*(q+1)^N*(p+1)^N := Nat.one_le_iff_ne_zero.mpr (by positivity)
have hm := Nat.mul_le_mul_left p hW
change p*(1+0+W)+1≤_
have he : (q+1)^N*(p+1)^(2*N+1)*(y+g+1)=
(p+1)*((y+g+1)*(p+1)^N*(q+1)^N*(p+1)^N) := by
rw [pow_add,pow_one,show 2*N=N+N by omega,pow_add]
ring
rw [he]
nlinarith
def velocityWorkCoefficient (q m N : ℕ) : ℕ := 4+(m+1)*(q+1)^N
lemma velocityWork_bound (p q m N : ℕ) (z : ℕ × ℕ) :
(velocityWork p q m N 0 z).1+(velocityWork p q m N 0 z).2+1≤
velocityWorkCoefficient q m N*(p+1)^(2*N+2)*(z.1+z.2+1) := by
have hk := kernelWork_bound p q N z.1 z.2
have hp : 1≤(p+1)^(2*N+1) := Nat.one_le_pow _ _ (by omega)
have hh : kernelWork p q N 0 z.1 z.2≤(q+1)^N*(p+1)^(2*N+1)*(z.1+z.2+1) := by omega
have hmul := Nat.mul_le_mul_left (p*(m+1)) hh
have hg : (z.1+z.2+1)≤(p+1)^(2*N+1)*(z.1+z.2+1) := by nlinarith
have hc : p*(m+1)*(q+1)^N+3≤(4+(m+1)*(q+1)^N)*(p+1) := by nlinarith
have hh' := Nat.mul_le_mul_right ((p+1)^(2*N+1)*(z.1+z.2+1)) hc
have hz : z.1+3≤3*(p+1)^(2*N+1)*(z.1+z.2+1) := by nlinarith
change p*(m+1)*kernelWork p q N 0 z.1 z.2+(2+2*0+z.1)+1≤_
unfold velocityWorkCoefficient
rw [show 2*N+2=(2*N+1)+1 by omega,pow_add,pow_one]
nlinarith
lemma vectorPicard_zero_bound (p q m N k : ℕ) (hk : 1≤k) (Nc : ℕ) :
(vectorPicard k (velocityWork p q m N 0) (0,0) Nc).1+
(vectorPicard k (velocityWork p q m N 0) (0,0) Nc).2+1≤
(k*velocityWorkCoefficient q m N)^Nc*(p+1)^(Nc*(2*N+2)) := by
induction Nc with
| zero => simp [vectorPicard]
| succ Nc ih =>
let z := vectorPicard k (velocityWork p q m N 0) (0,0) Nc
have hv := velocityWork_bound p q m N z
have hm := Nat.mul_le_mul_left (k*velocityWorkCoefficient q m N*(p+1)^(2*N+2)) ih
have hx := Nat.mul_le_mul_left k hv
change 0+k*(velocityWork p q m N 0 z).1+(0+k*(velocityWork p q m N 0 z).2)+1≤_
rw [pow_succ,Nat.succ_mul,pow_add]
change z.1+z.2+1≤_ at ih
nlinarith
lemma meanWork_polynomial (p q m N nc Nc : ℕ) :
meanWork p q m N nc Nc 0 0 0≤
(nc+1)*((nc+1)*velocityWorkCoefficient q m N)^Nc*(p+1)^(Nc*(2*N+2)) := by
have he := vectorPicard_zero_bound p q m N (nc+1) (by omega) Nc
have hm := Nat.mul_le_mul_left (nc+1) he
unfold meanWork
nlinarith
end LogConcaveSampling.MeanTree
end
end
end OAI
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