import Mathlib
import OAI.Combinatorics.MatchingCount.Arithmetic.TreeFraction
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialDivision
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialLoop

namespace OAI

namespace MatchingFPRAS.TreeTyped
open Polynomial

lemma polynomial_boolNot : PolynomialTime boolNot := by
  unfold boolNot
  exact polynomial_congr _ (polynomial_choose polynomial_identity (polynomial_constant false) (polynomial_constant true))
lemma polynomial_boolAnd : PolynomialTime boolAnd := by
  unfold boolAnd
  exact polynomial_congr _ (polynomial_choose polynomial_first polynomial_second (polynomial_constant false))
lemma polynomial_natEqual : PolynomialTime natEqual := by
  unfold natEqual
  apply polynomial_congr
  apply polynomial_choose polynomial_less (polynomial_constant false)
  exact polynomial_choose (polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_less)
    (polynomial_constant false) (polynomial_constant true)
lemma polynomial_quotient : PolynomialTime quotient := polynomial_composition polynomial_division polynomial_first
lemma polynomial_remainder : PolynomialTime remainder := polynomial_composition polynomial_division polynomial_second
lemma polynomial_digits : PolynomialTime digits := by
  refine ⟨X+1,?_⟩
  intro a
  change 1+weight a≤_
  simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_one]
  omega
lemma polynomial_bitSize : PolynomialTime bitSize := by
  unfold bitSize
  exact polynomial_congr _ (polynomial_composition polynomial_digits polynomial_listLength)
lemma polynomial_sortedPair : PolynomialTime sortedPairRealizer := by
  unfold sortedPairRealizer
  exact polynomial_congr _ (polynomial_choose polynomial_less (polynomial_pair polynomial_second polynomial_first) polynomial_identity)
lemma polynomial_gcdStep : PolynomialTime gcdStepRealizer := by
  unfold gcdStepRealizer
  exact polynomial_congr _ (polynomial_choose
    (polynomial_composition (polynomial_pair polynomial_second (polynomial_constant 0)) polynomial_natEqual)
    polynomial_identity (polynomial_pair polynomial_second polynomial_remainder))
lemma gcdStep_size_le (x : ℕ×ℕ) : (gcdStep x).1.size+(gcdStep x).2.size≤x.1.size+x.2.size := by
  dsimp only [gcdStep]
  split_ifs with h
  · exact le_rfl
  · simp only
    have hm:=Nat.size_le_size (Nat.mod_le x.1 x.2)
    omega
lemma polynomial_gcd : PolynomialTime gcd := by
  unfold gcd
  apply polynomial_congr
  apply polynomial_composition _ polynomial_first
  apply polynomial_boundedRepeat
    (polynomial_pair (polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_bitSize)
      (polynomial_composition polynomial_second polynomial_bitSize)) polynomial_add) polynomial_sortedPair)
    (polynomial_composition polynomial_second polynomial_gcdStep) X (4*X+3)
    (fun x _ z=>z.1.size+z.2.size≤x.1.size+x.2.size)
  · intro x
    simp only [Polynomial.eval_X,Function.comp_apply]
    have h1:=size_le_weight x.1
    have h2:=size_le_weight x.2
    rw [←Prod.eta x,weight_prod]
    try dsimp only
    omega
  · intro x
    exact (sortedPair_size x).le
  · intro x _iteration _bounded z hz
    exact (gcdStep_size_le z).trans hz
  · intro x _iteration z hz
    have ha:=weight_nat_le z.1
    have hb:=weight_nat_le z.2
    have hx:=size_le_weight x.1
    have hy:=size_le_weight x.2
    rw [←Prod.eta z,weight_prod]
    simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X]
    rw [←Prod.eta x,weight_prod]
    try dsimp only
    omega
end MatchingFPRAS.TreeTyped

namespace MatchingFPRAS.TreeFraction
open TreeTyped

lemma polynomial_normalize : PolynomialTime normalizeRealizer := by
  unfold normalizeRealizer
  apply polynomial_congr
  apply polynomial_choose
  · exact polynomial_composition (polynomial_pair polynomial_second (polynomial_constant 0)) polynomial_natEqual
  · exact polynomial_constant (0,1)
  · exact polynomial_pair
      (polynomial_composition (polynomial_pair polynomial_first polynomial_gcd) polynomial_quotient)
      (polynomial_composition (polynomial_pair polynomial_second polynomial_gcd) polynomial_quotient)
lemma polynomial_mul : PolynomialTime mulRealizer := by
  unfold mulRealizer
  exact polynomial_composition (polynomial_pair
    (polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_fst polynomial_second)) polynomial_multiply)
    (polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)) polynomial_normalize
lemma polynomial_add : PolynomialTime addRealizer := by
  unfold addRealizer
  apply polynomial_composition _ polynomial_normalize
  apply polynomial_pair
  · apply polynomial_composition _ TreeTyped.polynomial_add
    exact polynomial_pair
      (polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)
      (polynomial_composition (polynomial_pair (polynomial_fst polynomial_second) (polynomial_snd polynomial_first)) polynomial_multiply)
  · exact polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply
lemma polynomial_div : PolynomialTime divRealizer := by
  unfold divRealizer
  exact polynomial_composition (polynomial_pair
    (polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)
    (polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_fst polynomial_second)) polynomial_multiply)) polynomial_normalize
lemma polynomial_less : PolynomialTime lessRealizer := by
  unfold lessRealizer
  exact polynomial_composition (polynomial_pair
    (polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)
    (polynomial_composition (polynomial_pair (polynomial_fst polynomial_second) (polynomial_snd polynomial_first)) polynomial_multiply)) TreeTyped.polynomial_less
end MatchingFPRAS.TreeFraction

end OAI