| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.ScalarPicard |
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|
| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling.MeanTree |
| open scoped BigOperators |
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| 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 |
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|
| def velocityWorkCoefficient (q m N : ℕ) : ℕ := 4+(m+1)*(q+1)^N |
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| 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 |
|
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| end LogConcaveSampling.MeanTree |
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| end |
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| end |
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| end OAI |
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