File size: 3,969 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.TreePartialChoice
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomial
namespace OAI
namespace MatchingFPRAS.TreeTyped
open TreeData TreeFunction Polynomial
lemma weight_bool (b : Bool) : weight b≤3 := by cases b <;> decide
lemma weight_bits (bs : List Bool) : weight bs≤4*bs.length+1 := TreeData.bits_length bs
lemma weight_nat (n : ℕ) : weight n=weight n.bits := rfl
lemma weight_nat_le (n : ℕ) : weight n≤4*n.size+1 := by
rw [weight_nat,←Nat.size_eq_bits_len]; exact weight_bits _
lemma size_le_weight (n : ℕ) : n.size≤weight n := by
rw [weight_nat,←Nat.size_eq_bits_len]; exact list_length_le_weight _
lemma weight_nat_small {a b : ℕ} (h : a≤b) : weight a≤4*weight b+1 := by
have h1:=Nat.size_le_size h
have h2:=weight_nat_le a
have h3:=size_le_weight b
omega
lemma nat_size_add (a b : ℕ) : (a+b).size≤ max a.size b.size+1 := by
apply Nat.size_le.mpr
have ha : a<2^(max a.size b.size) := (Nat.lt_size_self a).trans_le (Nat.pow_le_pow_right (by decide) (le_max_left _ _))
have hb : b<2^(max a.size b.size) := (Nat.lt_size_self b).trans_le (Nat.pow_le_pow_right (by decide) (le_max_right _ _))
rw [pow_succ]; omega
lemma nat_size_mul (a b : ℕ) : (a*b).size≤a.size+b.size := by
apply Nat.size_le.mpr
rw [pow_add]
exact (Nat.mul_le_mul_left a (Nat.le_of_lt (Nat.lt_size_self b))).trans_lt
(Nat.mul_lt_mul_of_pos_right (Nat.lt_size_self a) (by positivity))
lemma weight_add (a b : ℕ) : weight (a+b)≤4*(weight a+weight b+1)+1 := by
have h:=weight_nat_le (a+b)
have h1:=nat_size_add a b
have h2:=size_le_weight a
have h3:=size_le_weight b
have hm : max a.size b.size≤weight a+weight b := max_le (by omega) (by omega)
omega
lemma polynomial_add : PolynomialTime add := by
refine ⟨20*X+30,fun ⟨a,b⟩=>?_⟩
have ha:=size_le_weight a
have hb:=size_le_weight b
have ho:=weight_add a b
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,weight_prod]
change (2*weight a+5+9*(a.bits.length+b.bits.length)+20)+weight (a+b)≤_
rw [Nat.size_eq_bits_len,Nat.size_eq_bits_len]
omega
lemma polynomial_less : PolynomialTime less := by
refine ⟨8*X+40,fun ⟨a,b⟩=>?_⟩
have ha:=size_le_weight a
have hb:=size_le_weight b
have ho:=weight_bool (decide (a<b))
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,weight_prod]
change (2*weight a+3*(a.bits.length+b.bits.length)+31)+weight (decide (a<b))≤_
rw [Nat.size_eq_bits_len,Nat.size_eq_bits_len]
omega
lemma polynomial_subtract : PolynomialTime subtract := by
refine ⟨50*X+100,fun ⟨a,b⟩=>?_⟩
have ha:=size_le_weight a
have hb:=size_le_weight b
have hc:=weight_nat_small (Nat.sub_le a b)
have ho:=weight_bool (decide (a<b))
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,weight_prod]
change (2*weight a+3*(a.bits.length+b.bits.length)+31)+
(if decide (a<b) then weight (a,b)+2 else
2*weight a+9*(a.bits.length+b.bits.length)+4*weight (a-b)+32)+
7*weight (a,b)+18+weight (a-b)≤_
rw [Nat.size_eq_bits_len,Nat.size_eq_bits_len,weight_prod]
split <;> omega
lemma polynomial_toBits : PolynomialTime toBits := by
refine ⟨X+1,fun _=>?_⟩
simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_one]
change 1+weight _ ≤ weight _+1
rw [weight_nat]; omega
lemma polynomial_listCons {A : Type} [Codec A] : PolynomialTime (listCons (A:=A)) := by
refine ⟨X+1,fun ⟨a,as⟩=>?_⟩
simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_one,weight_prod,weight_cons]
change 1+(1+weight a+weight as)≤(1+weight a+weight as)+1
omega
lemma polynomial_double : PolynomialTime double :=
polynomial_congr _ (polynomial_composition (polynomial_pair polynomial_identity polynomial_identity) polynomial_add)
end MatchingFPRAS.TreeTyped
end OAI
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