File size: 3,738 Bytes
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 | import OAI.MathematicalPhysics.AlternatingFlow.ProfileNames
import OAI.MathematicalPhysics.AlternatingFlow.DecoderBounds
namespace OAI
section DecoderNamesDevelopment
open scoped BigOperators Topology ContDiff
open Filter
namespace AlternatingNS.Effective
attribute [local instance] Arithmetic.rationalCoding
lemma decoderBound : Primrec₂ Quantitative.decoderBound :=
Primrec.nat_add.comp (Primrec.const 1)
(Primrec.nat_mul.comp
(nat_pow.comp (Primrec.nat_mul.comp (Primrec.const 3)
(Primrec.nat_sub.comp Primrec.fst (Primrec.const 1))) Primrec.snd)
(transitionBound.comp Primrec.snd))
lemma rat_fract : Primrec (fun q : ℚ => Int.fract q) :=
Arithmetic.rat_sub.comp Primrec.id (Arithmetic.rat_intCast.comp Arithmetic.rat_floor)
lemma frac_cast (q : ℚ) : ((Int.fract q : ℚ) : ℝ) = Int.fract (q : ℝ) := by
simp only [Int.fract, Rat.cast_sub, Rat.cast_intCast, Rat.floor_cast]
lemma periodizer_named : Named (fun z : (ℚ × ℚ) × ℚ => Profiles.periodizer z.1.1 z.1.2 z.2) := by
have hv : Computable (fun z : (ℚ × ℚ) × ℚ => Int.fract z.2) := rat_fract.to_comp.comp Computable.snd
have hA : Computable (fun z : (ℚ × ℚ) × ℚ => (Int.fract z.2 - z.1.1)/(z.1.2-z.1.1)) :=
Arithmetic.rat_div.to_comp.comp
(Arithmetic.rat_sub.to_comp.comp hv (Computable.fst.comp Computable.fst))
(Arithmetic.rat_sub.to_comp.comp (Computable.snd.comp Computable.fst)
(Computable.fst.comp Computable.fst))
exact ((Named.rational hv).sub (transition_named.comp hA)).congr (by
intro z
simp only [Function.comp_def, Profiles.periodizer, Rat.cast_div, Rat.cast_sub]
rw [frac_cast])
lemma decoder_named : Named (fun z : ℕ × ℚ => Profiles.decoder z.1 (z.2 : ℝ)) := by
let m (b : ℕ) : ℚ := ((b : ℚ)-2)/((b : ℚ)-1)
have hm : Computable m := Arithmetic.rat_div.to_comp.comp
(Arithmetic.rat_sub.to_comp.comp Arithmetic.rat_natCast.to_comp (Computable.const 2))
(Arithmetic.rat_sub.to_comp.comp Arithmetic.rat_natCast.to_comp (Computable.const 1))
have ha : Computable (fun z : ℕ × ℚ => (2*m z.1+1)/3) := Arithmetic.rat_div.to_comp.comp
(Arithmetic.rat_add.to_comp.comp
(Arithmetic.rat_mul.to_comp.comp (Computable.const 2) (hm.comp Computable.fst))
(Computable.const 1)) (Computable.const 3)
have hc : Computable (fun z : ℕ × ℚ => (m z.1+2)/3) := Arithmetic.rat_div.to_comp.comp
(Arithmetic.rat_add.to_comp.comp (hm.comp Computable.fst) (Computable.const 2)) (Computable.const 3)
exact (periodizer_named.comp ((ha.pair hc).pair Computable.snd)).congr (by
intro z; simp [Profiles.decoder, Encoding.tailMax, m])
lemma Named.decoder {A : Type*} [Primcodable A] (b : A → ℕ) (hb : Computable b)
(hbase : ∀ x, 2 ≤ b x) {f : A → ℝ} (hf : Named f) :
Named (fun x => Profiles.decoder (b x) (f x)) := by
let L (x : A) : ℚ := Quantitative.decoderBound (b x) 1
have hL : Computable L := Arithmetic.rat_natCast.to_comp.comp
(decoderBound.to_comp.comp hb (Computable.const 1))
have hg : Named (fun z : A × ℚ => Profiles.decoder (b z.1) (z.2 : ℝ)) :=
Named.comp (g := fun z : A × ℚ => (b z.1, z.2)) decoder_named
((hb.comp Computable.fst).pair Computable.snd)
have hlip (a : A) (x y : ℝ) : |Profiles.decoder (b a) x - Profiles.decoder (b a) y| ≤
(L a : ℝ) * |x-y| := by
simpa only [L, Rat.cast_natCast] using jet_one_lipschitz (Profiles.decoder_smooth _ (hbase a))
(Quantitative.decoderBound (b a) 1) (Quantitative.decoder_jet_bound (b a) (hbase a) 1) x y
exact Named.apply_lipschitz (fun x => Profiles.decoder (b x)) L hL
(fun _ => Nat.cast_nonneg _) hlip hg hf
end AlternatingNS.Effective
end DecoderNamesDevelopment
end OAI
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