lean/OAI/Combinatorics/MatchingCount/Complexity/TreePolynomialLoop.lean

import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeListOps
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialLists

namespace OAI

namespace MatchingFPRAS.TreeTyped
open Polynomial

variable {A B : Type} [Codec A] [Codec B]
lemma polynomial_predecessor : PolynomialTime predecessor :=
  polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant 1)) polynomial_subtract
lemma polynomial_repeatNext {f : ℕ×B → B} {F : Realizer f} (hF : PolynomialTime F) :
    PolynomialTime (Repeat.nextRealizer F) :=
  polynomial_pair (polynomial_composition polynomial_first polynomial_predecessor) hF

omit [Codec A] [Codec B] in
lemma countDown_invariant {f : ℕ×B → B} (I : ℕ → B → Prop) {n : ℕ} {b : B}
    (hi : I n b) (hs : ∀ k<n,∀ z,I (k+1) z → I k (f (k+1,z))) : I 0 (countDown f n b) := by
  induction n generalizing b with
  | zero => exact hi
  | succ n ih => exact ih (hs n (by omega) b hi) (fun k hk z hz=>hs k (by omega) z hz)

lemma polynomial_boundedRepeat {init : A → ℕ×B} {f : ℕ×B → B}
    {Init : Realizer init} {F : Realizer f} (hInit : PolynomialTime Init) (hF : PolynomialTime F)
    (p q : Polynomial ℕ) (I : A → ℕ → B → Prop)
    (hn : ∀ a,(init a).1≤p.eval (weight a))
    (hi : ∀ a,I a (init a).1 (init a).2)
    (hs : ∀ a k,(k<(init a).1) → ∀ z,I a (k+1) z → I a k (f (k+1,z)))
    (hw : ∀ a k z,I a k z → weight z≤q.eval (weight a)) :
    PolynomialTime (composition Init (iterateDown F)) := by
  obtain ⟨r,hr⟩ := hInit
  obtain ⟨t,ht⟩ := polynomial_repeatNext hF
  refine ⟨r+p*(t.comp (2+4*p+q)+4*(4*p+1)+12)+10+q,fun a=>?_⟩
  let S:=weight a
  let N:=(init a).1
  let T:=t.eval (2+4*p.eval S+q.eval S)
  have hN : N≤p.eval S := hn a
  have hsz : ∀ k<N,weight (k+1)≤4*p.eval S+1 := by
    intro k hk
    have hh:=weight_nat_le (k+1)
    have hbit : (k+1).size≤k+1 := Nat.size_le.mpr Nat.lt_two_pow_self
    omega
  have hc:=Repeat.cost_bound (f:=f) (Repeat.nextRealizer F).cost (I a) N (init a).2
    T (4*p.eval S+1) (hi a) (by
      intro k hk z hz
      refine ⟨hs a k hk z hz,?_⟩
      have h1:=ht (k+1,z)
      have h2:=hsz k hk
      have h3:=hw a (k+1) z hz
      have h4:=polynomial_eval_monotone t (show weight (k+1,z)≤2+4*p.eval S+q.eval S by
        rw [weight_prod]; change weight z≤q.eval S at h3; omega)
      dsimp only at h4
      dsimp only [T]
      omega) hsz
  have hout:=hw a 0 (countDown f N (init a).2)
    (countDown_invariant (I a) (hi a) (hs a))
  have hinit:=hr a
  change _≤r.eval S at hinit
  simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_one,
    Polynomial.eval_comp]
  change Init.cost a+Repeat.cost f (Repeat.nextRealizer F).cost N (init a).2+
    weight (countDown f N (init a).2)≤r.eval S+p.eval S*(T+4*(4*p.eval S+1)+12)+10+q.eval S
  change _≤q.eval S at hout
  have hm:=Nat.mul_le_mul_right (T+4*(4*p.eval S+1)+12) hN
  omega

lemma polynomial_listLength : PolynomialTime (listLength (A:=A)) := by
  unfold listLength
  apply polynomial_congr
  apply polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant 0))
  apply polynomial_fold (polynomial_composition (polynomial_pair polynomial_second (polynomial_constant 1)) polynomial_add)
    (4*X+5)
  intro as n pre post he
  have hf : pre.foldl (fun z (_ : A)=>z+1) n=n+pre.length := by
    clear he
    induction pre generalizing n with
    | nil => simp
    | cons a pre ih => simp only [List.foldl_cons,ih,List.length_cons]; omega
  change weight (pre.foldl (fun z (_ : A)=>z+1) n)≤_
  rw [hf]
  have h:=weight_nat_le (n+pre.length)
  have hs:=nat_size_add n pre.length
  have hl : pre.length.size≤pre.length := Nat.size_le.mpr Nat.lt_two_pow_self
  have hn:=size_le_weight n
  have hp:pre.length≤weight as := (list_length_le_weight pre).trans (weight_prefix he)
  have hm : max n.size pre.length.size≤weight as+weight n := max_le (by omega) (by omega)
  simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,weight_prod]
  omega
end MatchingFPRAS.TreeTyped

end OAI