| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Tapes.TreeEncodingInjective |
| import OAI.Combinatorics.MatchingCount.Tapes.BlockMap |
| import OAI.Combinatorics.MatchingCount.Tapes.Packed |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.LiteralPacked |
| open TreeTyped TreeData TreePost |
|
|
| lemma symbol_encode_injective : Function.Injective (TreeTyped.encode (A:=Symbol)) := by |
| intro a b h |
| have hh : TreeTyped.encode (LiteralCodec.parts a)=TreeTyped.encode (LiteralCodec.parts b) := h |
| have he := (TreeData.Data.cons.inj hh).1 |
| have hs : (LiteralCodec.parts a).1=(LiteralCodec.parts b).1 := |
| list_encode_injective (show Function.Injective (TreeTyped.encode (A:=Bool)) from bit_injective) he |
| apply Fin.ext |
| simpa only [LiteralCodec.parts,List.length_replicate] using congrArg List.length hs |
| lemma symbolBits_injective : Function.Injective symbolBits := by |
| intro a b h |
| have h1:=List.append_cancel_right (show _++[true]=_++[true] from h) |
| exact symbol_encode_injective (TreeData.encode_injective (List.reverse_injective h1)) |
| lemma encoding_injective (Γ : Type) [Fintype Γ] [Inhabited Γ] : |
| Function.Injective (fun a : Γ=>(FixedEncoding.enc Γ a).toList.map BoolPostCompiler.up) := by |
| intro a b h |
| have hb : (FixedEncoding.enc Γ a).toList=(FixedEncoding.enc Γ b).toList := |
| List.map_injective_iff.mpr (Function.LeftInverse.injective BoolPostCompiler.down_up) h |
| exact FixedEncoding.injective Γ (Subtype.ext hb) |
| lemma block_injective : Function.Injective block := encoding_injective Alphabet |
| lemma flatMap_injective_fixed {A B : Type} (f : A→List B) (w : ℕ) (hw : 0<w) |
| (hl : ∀ a,(f a).length=w) (hi : Function.Injective f) : Function.Injective (List.flatMap f) := by |
| intro as |
| induction as with |
| | nil => |
| intro bs h |
| cases bs with |
| | nil => rfl |
| | cons b bs => have hh:=congrArg List.length h;simp only [List.flatMap_nil,List.flatMap_cons,List.length_nil,List.length_append,hl] at hh;omega |
| | cons a as ih => |
| intro bs h |
| cases bs with |
| | nil => have hh:=congrArg List.length h;simp only [List.flatMap_nil,List.flatMap_cons,List.length_nil,List.length_append,hl] at hh;omega |
| | cons b bs => |
| have h1 := List.append_inj_left h ((hl a).trans (hl b).symm) |
| have h2 := List.append_inj_right h ((hl a).trans (hl b).symm) |
| exact congrArg₂ List.cons (hi h1) (ih h2) |
| lemma width_pos : 0<width := Fintype.card_pos |
| lemma letter_injective : Function.Injective letter := by |
| intro a b h |
| have hh:=congrArg (fun x : Alphabet=>x.2 (Sum.inl 0)) h |
| simpa only [letter,bitRow,ite_true,Option.some.injEq] using hh |
| lemma chunk_injective : Function.Injective chunk := by |
| intro a b h |
| have hi : Function.Injective (fun b=>block (letter b)) := by |
| intro b c he |
| exact letter_injective (block_injective he) |
| have hh : (symbolBits a).flatMap (fun b=>block (letter b)) = |
| (symbolBits b).flatMap (fun b=>block (letter b)) := h |
| exact symbolBits_injective (flatMap_injective_fixed (fun b=>block (letter b)) width width_pos |
| (fun b=>block_length (letter b)) hi hh) |
| lemma encoding_blank (Γ : Type) [Fintype Γ] [Inhabited Γ] : |
| (FixedEncoding.enc Γ default).toList.map BoolPostCompiler.up= |
| List.replicate (Fintype.card Γ) (0:Symbol) := by |
| rw [FixedEncoding.blank] |
| change (List.replicate (Fintype.card Γ) false).map _ = _ |
| rw [List.map_replicate] |
| rfl |
| lemma block_blank : block default=List.replicate width 0 := encoding_blank Alphabet |
| lemma block_ne_zero (a : Alphabet) (ha : a≠default) : block a≠List.replicate width 0 := by |
| intro he |
| exact ha (block_injective (he.trans block_blank.symm)) |
| lemma chunk_nonzero (a : Symbol) : ∃ s∈chunk a,s≠0 := by |
| classical |
| by_contra h |
| push Not at h |
| have h1 : ∀ s∈block (letter true),s=(0:Symbol) := by |
| intro s hs |
| exact h s (List.mem_flatMap.mpr ⟨true,by simp [symbolBits],hs⟩) |
| have he : block (letter true)=List.replicate width 0 := by |
| rw [←block_length (letter true)] |
| exact List.eq_replicate_of_mem h1 |
| apply block_ne_zero (letter true) _ he |
| intro h2 |
| have hh:=congrArg (fun x : Alphabet=>x.2 (Sum.inl 0)) h2 |
| change (some true : Option Bool)=none at hh |
| exact Option.some_ne_none true hh |
| noncomputable def chunkFn (a : Symbol) : LiteralBlockMap.Block (18*width) := |
| fun i=>(chunk a)[i.val]'(by rw [chunk_length];exact i.isLt) |
| lemma chunkFn_word (a : Symbol) : List.ofFn (chunkFn a)=chunk a := by |
| apply List.ext_getElem |
| · simp only [List.length_ofFn,chunk_length] |
| · intro i hi hj; simp only [List.getElem_ofFn,chunkFn] |
| noncomputable def unchunk (b : LiteralBlockMap.Block (18*width)) : List Symbol := |
| [Function.invFun chunk (List.ofFn b)] |
| lemma unchunk_chunkFn (a : Symbol) : unchunk (chunkFn a)=[a] := by |
| rw [unchunk,chunkFn_word,Function.leftInverse_invFun chunk_injective] |
| lemma chunkFn_nonzero (a : Symbol) : chunkFn a≠fun _=>0 := by |
| intro he |
| have hh:=congrArg List.ofFn he |
| rw [chunkFn_word,List.ofFn_const] at hh |
| obtain ⟨s,hs,hn⟩:=chunk_nonzero a |
| rw [hh] at hs |
| exact hn (List.eq_of_mem_replicate hs) |
| end MatchingFPRAS.LiteralPacked |
|
|
| end OAI |
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