lean/OAI/Combinatorics/MatchingCount/Complexity/TreePolynomialLists.lean

import Mathlib
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialArithmetic

namespace OAI

namespace MatchingFPRAS.TreeTyped
open Polynomial

variable {A B C D : Type} [Codec A] [Codec B] [Codec C] [Codec D]
lemma polynomial_fst {f : A → B×C} {F : Realizer f} (h : PolynomialTime F) :
    PolynomialTime (composition F first) := polynomial_composition h polynomial_first
lemma polynomial_snd {f : A → B×C} {F : Realizer f} (h : PolynomialTime F) :
    PolynomialTime (composition F second) := polynomial_composition h polynomial_second
lemma weight_append (as bs : List A) : weight (as++bs)+1=weight as+weight bs := by
  induction as with
  | nil => simp only [List.nil_append,weight_nil]; omega
  | cons a as ih => simp only [List.cons_append,weight_cons]; omega
lemma weight_reverse (as : List A) : weight as.reverse=weight as := by
  simp only [weight,size_encode_list,List.length_reverse,List.map_reverse,List.sum_reverse]
lemma weight_prefix {as pre post : List A} (h : as=pre++post) : weight pre≤weight as := by
  have := weight_append pre post
  have := weight_pos post
  rw [h]; omega
omit [Codec A] in
lemma weight_map_bound {f : A → B} {as : List A} {N : ℕ}
    (h : ∀ a∈as,weight (f a)≤N) : weight (as.map f)≤as.length*(N+1)+1 := by
  induction as with
  | nil => simp
  | cons a as ih =>
    have ht:=ih (fun b hb=>h b (by simp [hb]))
    have ha:=h a (by simp)
    simp only [List.map_cons,weight_cons,List.length_cons]
    nlinarith
lemma polynomial_reverse : PolynomialTime (reverse (A:=A)) := by
  unfold reverse
  apply polynomial_congr
  apply polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant []))
  apply polynomial_fold polynomial_listCons (2*X+1)
  intro as bs pre post he
  have hp := weight_prefix he
  have h:=weight_append pre.reverse bs
  rw [weight_reverse] at h
  simp only [List.foldl_flip_cons_eq_append']
  simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,weight_prod]
  omega
lemma polynomial_mulStep : PolynomialTime mulStepRealizer := by
  unfold mulStepRealizer
  apply polynomial_congr
  apply polynomial_pair (polynomial_composition polynomial_second polynomial_first)
  apply polynomial_composition _ polynomial_add
  apply polynomial_pair
  · exact polynomial_composition (polynomial_composition polynomial_second polynomial_second) polynomial_double
  · exact polynomial_choose polynomial_first
      (polynomial_composition polynomial_second polynomial_first) (polynomial_constant 0)
lemma mulStep_size (b : Bool) (n c : ℕ) : (mulStep (b,(n,c))).2.size≤c.size+n.size+3 := by
  have hh:=nat_size_mul 2 c
  have ha:=nat_size_add (2*c) (if b then n else 0)
  have hb : (if b then n else 0).size≤n.size := by cases b <;> simp
  have h2 : Nat.size 2=2 := by decide
  have hm : max (2*c).size (if b then n else 0).size≤c.size+n.size+2 := max_le (by omega) (by omega)
  change (2*c+(if b then n else 0)).size≤_
  omega
lemma mulStep_fold_first (bs : List Bool) (n c : ℕ) :
    (bs.foldl (fun z b=>mulStep (b,z)) (n,c)).1=n := by
  induction bs generalizing c with
  | nil => rfl
  | cons b bs ih => exact ih _
lemma mulStep_fold_size (bs : List Bool) (n c : ℕ) :
    (bs.foldl (fun z b=>mulStep (b,z)) (n,c)).2.size≤c.size+bs.length*(n.size+3) := by
  induction bs generalizing c with
  | nil => simp
  | cons b bs ih =>
    have hs:=mulStep_size b n c
    have ht:=ih (2*c+(if b then n else 0))
    change (bs.foldl _ (mulStep (b,(n,c)))).2.size≤_
    simp only [mulStep] at hs
    simp only [List.length_cons]
    change (bs.foldl _ (n,2*c+(if b then n else 0))).2.size≤_
    nlinarith
lemma polynomial_mulFold : PolynomialTime (fold mulStepRealizer) := by
  apply polynomial_fold polynomial_mulStep (10*(X+1)^2)
  rintro bs ⟨n,c⟩ pre post he
  let S:=weight (bs,(n,c))
  have hpre : pre.length≤S := (list_length_le_weight pre).trans ((weight_prefix he).trans (by dsimp [S]; simp only [weight_prod]; omega))
  have hn : n.size≤S := by have h:=size_le_weight n; dsimp [S]; simp only [weight_prod]; omega
  have hc : c.size≤S := by have h:=size_le_weight c; dsimp [S]; simp only [weight_prod]; omega
  have hn' : weight n≤S := by dsimp [S]; simp only [weight_prod]; omega
  have hs:=mulStep_fold_size pre n c
  have ho:=weight_nat_le (pre.foldl (fun z b=>mulStep (b,z)) (n,c)).2
  have hr:=mulStep_fold_first pre n c
  rw [←Prod.eta (pre.foldl (fun z b=>mulStep (b,z)) (n,c)),weight_prod,hr]
  simp only [Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_pow,Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_one]
  change _≤10*(S+1)^2
  have hmul : pre.length*(n.size+3)≤S*(S+3) := Nat.mul_le_mul hpre (by omega)
  nlinarith
lemma polynomial_multiply : PolynomialTime multiply := by
  unfold multiply
  apply polynomial_congr
  apply polynomial_composition _ polynomial_second
  apply polynomial_composition _ polynomial_mulFold
  apply polynomial_pair
  · exact polynomial_composition (polynomial_composition polynomial_first polynomial_toBits) polynomial_reverse
  · exact polynomial_pair polynomial_second (polynomial_constant 0)
end MatchingFPRAS.TreeTyped

end OAI