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2f887db | 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 86 87 88 89 90 91 92 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackMultiplier
namespace OAI
section
namespace ExactQuantumFactoring.BitStackProgram
def gcdStep (x : ℕ×ℕ) : ℕ×ℕ := if x.2=0 then x else (x.2,x.1%x.2)
def gcdRound (x : ℕ×ℕ) : ℕ×ℕ := gcdStep (gcdStep x)
lemma gcdStep_value (x : ℕ×ℕ) : (gcdStep x).1.gcd (gcdStep x).2=x.1.gcd x.2 := by
unfold gcdStep
split
· rfl
· dsimp only
rw [Nat.gcd_comm x.2]
exact (Nat.gcd_rec x.2 x.1).symm.trans (Nat.gcd_comm x.2 x.1)
lemma mod_half {b r : ℕ} (hr : 0<r) (hlt : r<b) : b%r≤b/2 := by
by_cases h : 2*r≤b
· have ht:=Nat.mod_lt b hr
omega
· rw [Nat.mod_eq_sub_mod (by omega),Nat.mod_eq_of_lt (by omega : b-r<r)]
omega
lemma gcdRound_half (x : ℕ×ℕ) : (gcdRound x).2≤x.2/2 := by
by_cases hb : x.2=0
· simp [gcdRound,gcdStep,hb]
· have hp : 0<x.2:=by omega
have hr:=Nat.mod_lt x.1 hp
by_cases hz : x.1%x.2=0
· simp [gcdRound,gcdStep,hb,hz]
· simp only [gcdRound,gcdStep,hb,ite_false,hz]
exact mod_half (by omega) hr
lemma gcdRound_value (x : ℕ×ℕ) : (gcdRound x).1.gcd (gcdRound x).2=x.1.gcd x.2 := by
rw [gcdRound,gcdStep_value,gcdStep_value]
lemma gcdRound_iterate_bound (i : ℕ) (x : ℕ×ℕ) :
(gcdRound^[i] x).2≤x.2/2^i := by
induction i with
| zero=>simp
| succ i ih=>
rw [Function.iterate_succ_apply']
calc
(gcdRound (gcdRound^[i] x)).2≤(gcdRound^[i] x).2/2:=gcdRound_half _
_≤(x.2/2^i)/2:=Nat.div_le_div_right ih
_=x.2/2^(i+1):=by rw [Nat.div_div_eq_div_mul,pow_succ]
lemma gcdRound_iterate_value (i : ℕ) (x : ℕ×ℕ) :
(gcdRound^[i] x).1.gcd (gcdRound^[i] x).2=x.1.gcd x.2 := by
induction i with
| zero=>rfl
| succ i ih=>rw [Function.iterate_succ_apply',gcdRound_value,ih]
lemma gcdRound_result (x : ℕ×ℕ) : (gcdRound^[x.2.bits.length] x).1=x.1.gcd x.2 := by
have hz : (gcdRound^[x.2.bits.length] x).2=0:=by
have hh:=gcdRound_iterate_bound x.2.bits.length x
have hzdiv : x.2/2^x.2.bits.length=0:=by
rw [Nat.size_eq_bits_len];exact Nat.div_eq_of_lt (Nat.lt_size_self x.2)
rw [hzdiv] at hh
omega
simpa only [hz,Nat.gcd_zero_right] using gcdRound_iterate_value x.2.bits.length x
lemma gcdStep_sizeSum (x : ℕ×ℕ) :
(gcdStep x).1.bits.length+(gcdStep x).2.bits.length≤x.1.bits.length+x.2.bits.length := by
have hm:=bits_length_mono (Nat.mod_le x.1 x.2)
unfold gcdStep
split
· exact le_rfl
· change x.2.bits.length+(x.1%x.2).bits.length≤x.1.bits.length+x.2.bits.length
omega
lemma gcdRound_iterate_sizeSum (i : ℕ) (x : ℕ×ℕ) :
(gcdRound^[i] x).1.bits.length+(gcdRound^[i] x).2.bits.length≤
x.1.bits.length+x.2.bits.length := by
induction i with
| zero=>rfl
| succ i ih=>
rw [Function.iterate_succ_apply']
exact ((gcdStep_sizeSum _).trans (gcdStep_sizeSum _)).trans ih
end ExactQuantumFactoring.BitStackProgram
end
end OAI
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