File size: 2,412 Bytes
ea76aac | 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 | import Mathlib
import OAI.Computability.MinUncut.Analysis.GaussianQuadrature
namespace OAI
namespace MinUncut.Preprocess
open MinUncut.FiniteGaussian
private theorem encode_ofLeftInjection_canonical {α β : Type*} (e : α ≃ ℕ)
(f : β → α) (g : α → Option β) (hfg : ∀ b, g (f b) = some b)
(hgf : ∀ a b, g a = some b → f b = a) (n : ℕ) (b : β)
(h : @Encodable.decode β
(@Encodable.ofLeftInjection α β (Encodable.ofEquiv ℕ e) f g hfg) n = some b) :
@Encodable.encode β
(@Encodable.ofLeftInjection α β (Encodable.ofEquiv ℕ e) f g hfg) b = n := by
change (some (e.symm n)).bind g = some b at h
change e (f b) = n
rw [hgf _ _ (by simpa using h), e.apply_symm_apply]
abbrev oldGridEncode (g : GridData) : ℕ := @Encodable.encode GridData instEncodableGridData g
abbrev oldGridDecode (n : ℕ) : Option GridData := @Encodable.decode GridData instEncodableGridData n
lemma oldGridEncode_eq (g : GridData) : oldGridEncode g=Nat.pair (Nat.pair 0 0+1)
(Nat.pair (Nat.pair 0 g.T+1) (Nat.pair (Nat.pair 0 g.L+1) (Nat.pair (Nat.pair 0 g.r+1) (Nat.pair 0 0)))) := rfl
lemma oldGridDecode_canonical (n : ℕ) (g : GridData) (h : oldGridDecode n=some g) :
oldGridEncode g=n := by
apply encode_ofLeftInjection_canonical (h := h)
intro tree result htree
cases tree with
| nat _ => cases htree
| cons tag fields =>
cases tag with
| cons _ _ => cases htree
| nat tag =>
cases tag with
| succ _ => cases htree
| zero =>
cases fields with
| nat _ => cases htree
| cons t fields =>
cases t with
| cons _ _ => cases htree
| nat t =>
cases fields with
| nat _ => cases htree
| cons l fields =>
cases l with
| cons _ _ => cases htree
| nat l =>
cases fields with
| nat _ => cases htree
| cons r fields =>
cases r with
| cons _ _ => cases htree
| nat r =>
cases fields with
| cons _ _ => cases htree
| nat last =>
cases last with
| succ _ => cases htree
| zero =>
cases htree
rfl
end MinUncut.Preprocess
end OAI
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