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f18e7de | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import OAI.MathematicalPhysics.AlternatingFlow.DerivativeBounds
import OAI.MathematicalPhysics.AlternatingFlow.Effective
import OAI.MathematicalPhysics.AlternatingFlow.Quantitative
namespace OAI
section EffectiveProfilesDevelopment
open scoped BigOperators ContDiff
namespace AlternatingNS
namespace Effective
lemma factorial : Primrec Nat.factorial := by
have hs : Primrec₂ (fun n a : ℕ => (n + 1) * a) :=
Primrec.nat_mul.comp (Primrec.succ.comp Primrec.fst) Primrec.snd
have h := Primrec.nat_rec₁ 1 hs
refine h.of_eq ?_
intro n
induction n with
| zero => rfl
| succ n ih => simpa only [Nat.rec_add_one, ih] using (Nat.factorial_succ n).symm
lemma sum_range {A : Type*} [Primcodable A] {f : A → ℕ → ℕ} (hf : Primrec₂ f) :
Primrec₂ (fun a n => ∑ i ∈ Finset.range n, f a i) := by
have hs : Primrec₂ (fun a : A => fun z : ℕ × ℕ => z.2 + f a z.1) :=
Primrec.nat_add.comp (Primrec.snd.comp Primrec.snd)
(hf.comp Primrec.fst (Primrec.fst.comp Primrec.snd))
have h := Primrec.nat_rec (Primrec.const (0 : ℕ) : Primrec (fun _ : A => (0 : ℕ))) hs
refine h.of_eq ?_
intro a n
induction n with
| zero => simp
| succ n ih => simp only [Finset.sum_range_succ, ih]
lemma choose : Primrec₂ Nat.choose := by
apply Primrec.ite (Primrec.nat_le.comp Primrec.snd Primrec.fst)
(Primrec.nat_div.comp (factorial.comp Primrec.fst)
(Primrec.nat_mul.comp (factorial.comp Primrec.snd)
(factorial.comp (Primrec.nat_sub.comp Primrec.fst Primrec.snd)))) (Primrec.const 0) |>.of_eq
rintro ⟨n,k⟩
dsimp only
split_ifs with h
· exact (Nat.choose_eq_factorial_div_factorial h).symm
· exact (Nat.choose_eq_zero_of_lt (by omega)).symm
lemma rhoBound : Primrec Quantitative.rhoBound := by
exact Primrec.nat_mul.comp (factorial.comp (Primrec.nat_mul.comp (Primrec.const 2) Primrec.id))
(nat_pow.comp (Primrec.nat_add.comp (Primrec.nat_mul.comp (Primrec.const 2) Primrec.id)
(Primrec.const 1)) Primrec.id)
lemma reciprocalBound {A : Type*} [Primcodable A] {q : A → ℕ} {B : A → ℕ → ℕ}
(hq : Primrec q) (hB : Primrec₂ B) :
Primrec₂ (fun a r => Quantitative.reciprocalBound (q a) (B a) r) := by
let hr : Primrec (fun z : A × ℕ => z.2) := Primrec.snd
let hfac := factorial.comp hr
exact Primrec.nat_mul.comp
(Primrec.nat_mul.comp hfac
(Primrec.nat_mul.comp hfac (nat_pow.comp (hq.comp Primrec.fst) (Primrec.succ.comp hr))))
(nat_pow.comp (Primrec.nat_add.comp (Primrec.const 1)
((sum_range hB).comp Primrec.fst (Primrec.succ.comp hr))) hr)
lemma transitionInvBound : Primrec Quantitative.transitionInvBound := by
have hs : Primrec₂ (fun _ : ℕ => fun r : ℕ => 2 * Quantitative.rhoBound r) :=
Primrec.nat_mul.comp (Primrec.const 2) (rhoBound.comp Primrec.snd)
exact (reciprocalBound (Primrec.const 9 : Primrec (fun _ : ℕ => (9 : ℕ))) hs).comp
Primrec.id Primrec.id
lemma transitionBound : Primrec Quantitative.transitionBound := by
have hs : Primrec₂ (fun r i : ℕ => r.choose i * Quantitative.rhoBound i *
Quantitative.transitionInvBound (r - i)) :=
Primrec.nat_mul.comp (Primrec.nat_mul.comp choose (rhoBound.comp Primrec.snd))
(transitionInvBound.comp (Primrec.nat_sub.comp Primrec.fst Primrec.snd))
exact (sum_range hs).comp Primrec.id Primrec.succ
theorem profile_bound_programs :
∃ cρ cH : Program,
(∀ r : ℕ, Returns cρ r (Quantitative.rhoBound r) ∧
∀ x : ℝ, ‖iteratedFDeriv ℝ r expNegInvGlue x‖ ≤ Quantitative.rhoBound r) ∧
(∀ r : ℕ, Returns cH r (Quantitative.transitionBound r) ∧
∀ x : ℝ, ‖iteratedFDeriv ℝ r Real.smoothTransition x‖ ≤ Quantitative.transitionBound r) := by
obtain ⟨cρ, hρ⟩ := program_of_computable Quantitative.rhoBound rhoBound.to_comp
obtain ⟨cH, hH⟩ := program_of_computable Quantitative.transitionBound transitionBound.to_comp
exact ⟨cρ, cH, fun r => ⟨hρ r, Quantitative.rho_jet_bound r⟩,
fun r => ⟨hH r, Quantitative.transition_jet_bound r⟩⟩
end Effective
end AlternatingNS
end EffectiveProfilesDevelopment
end OAI
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