File size: 3,205 Bytes
e5a97c6 | 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeProgramUnique
namespace OAI
namespace MatchingFPRAS.TreeData
lemma encode_prefix_injective (a : Data) : ∀ b xs ys,
encode a++xs=encode b++ys → a=b ∧ xs=ys := by
induction a with
| nil=>
intro b xs ys h
cases b with
| nil=>simpa [encode] using h
| cons b c=>simp [encode] at h
| cons a c ia ic=>
intro b xs ys h
cases b with
| nil=>simp [encode] at h
| cons b d=>
simp only [encode,List.cons_append,List.cons.injEq,true_and,List.append_assoc] at h
obtain ⟨hab,he⟩:=ia b _ _ h
obtain ⟨hcd,hxy⟩:=ic d xs ys he
exact ⟨congrArg₂ Data.cons hab hcd,hxy⟩
lemma encode_injective : Function.Injective encode := by
intro a b h
exact (encode_prefix_injective a b [] [] (by simpa using h)).1
lemma ofList_injective : Function.Injective ofList := by
intro a
induction a with
| nil=>intro b h; cases b <;> simp_all [ofList]
| cons a as ih=>
intro b h
cases b with
| nil=>simp [ofList] at h
| cons b bs=>
obtain ⟨rfl,he⟩:=Data.cons.inj h
rw [ih he]
lemma bit_injective : Function.Injective bit := by
intro a b h
cases a <;> cases b <;> simp_all [bit]
lemma nat_injective : Function.Injective nat := by
intro a b h
have hh: a.bits=b.bits:=(List.map_injective_iff.mpr bit_injective) (ofList_injective h)
have := congrArg StackAddition.value hh
simpa only [StackAddition.value_bits] using this
end MatchingFPRAS.TreeData
namespace MatchingFPRAS.TreePost
open TreeTyped TreeData TreeFunction TreeMachine RegisterProc
variable {A B C D : Type} [Codec A] [Codec B] [Codec C] [Codec D]
lemma sameProgram_encode {f : A→B} {g : C→D} (F : Realizer f) (G : Realizer g)
(h : F.program=G.program) (a : A) (b : C) (hi : TreeTyped.encode a=TreeTyped.encode b) :
TreeTyped.encode (f a)=TreeTyped.encode (g b) := by
have hf:=F.correct a .nil
have hg:=G.correct b .nil
change Exec F.program (inputStacks a) (inputStacks (f a)) _ at hf
change Exec G.program (inputStacks b) (inputStacks (g b)) _ at hg
rw [h,stacks_eq_of_encode hi] at hf
exact TreeData.encode_injective (congrFun (hf.unique hg) (.inl 0))
end MatchingFPRAS.TreePost
namespace MatchingFPRAS.TreeTyped
open TreeData
lemma list_encode_injective {A : Type} [Codec A] (hf : Function.Injective (encode (A:=A))) :
Function.Injective (encode (A:=List A)) := by
intro a b h
exact (List.map_injective_iff.mpr hf) (ofList_injective h)
lemma nat_encode_injective : Function.Injective (encode (A:=ℕ)) := TreeData.nat_injective
lemma finsetNat_encode_injective : Function.Injective (encode (A:=Finset ℕ)) := by
intro a b h
change encode (a.sort (·≤·))=encode (b.sort (·≤·)) at h
have hh:=list_encode_injective nat_encode_injective h
have := congrArg List.toFinset hh
simpa using this
lemma pair_encode_injective {A B : Type} [Codec A] [Codec B]
(ha : Function.Injective (encode (A:=A))) (hb : Function.Injective (encode (A:=B))) :
Function.Injective (encode (A:=A×B)) := by
intro a b h
exact Prod.ext (ha (TreeData.Data.cons.inj h).1) (hb (TreeData.Data.cons.inj h).2)
end MatchingFPRAS.TreeTyped
end OAI
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