| import Mathlib |
| import OAI.Computability.QuantumFactoring.BitStackMultiplier |
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| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram |
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| 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) |
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| 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) |
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| 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 |
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| 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 |
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| 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] |
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| 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] |
|
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| 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] |
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| 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 |
|
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| 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 |
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| 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 |
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| end ExactQuantumFactoring.BitStackProgram |
| end |
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| end OAI |
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