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ea76aac | 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 | import OAI.Computability.MatchingCount.ArgRoutine
namespace OAI
section
namespace BinaryMatching.StackCode
open Polynomial
namespace Poly
@[fun_prop] theorem intDivNat : Poly (fun z : ℤ×ℕ => z.1/(z.2:ℤ)) := by
have h : Poly (fun z : ℤ×ℕ => if decide (z.2=0) then (0:ℤ) else
if decide (z.1<0) then -(((z.1.natAbs-1)/z.2+1:ℕ):ℤ)
else ((z.1.natAbs/z.2:ℕ):ℤ)) := by fun_prop
exact h.congr (fun z => by
by_cases hd : z.2=0
· simp [hd]
· have hp : (0:ℤ)<z.2 := by exact_mod_cast (Nat.pos_of_ne_zero hd)
simp only [hd,decide_false,Bool.false_eq_true,ite_false,decide_eq_true_eq]
by_cases ha : z.1<0
· rw [ite_eq_left ha]
change -(((z.1.natAbs-1)/z.2+1:ℕ):ℤ)=Int.ediv z.1 (z.2:ℤ)
rw [Int.ediv_of_neg_of_pos ha hp]
have ha' : 0<z.1.natAbs := Int.natAbs_pos.mpr (ne_of_lt ha)
have habs : (z.1.natAbs:ℤ) = -z.1 := by omega
rw [Nat.cast_add,Nat.cast_one,Int.natCast_ediv,Nat.cast_sub (by omega : 1≤z.1.natAbs),Nat.cast_one,habs]
· rw [ite_eq_right ha,Int.natCast_ediv,Int.natAbs_of_nonneg (le_of_not_gt ha)])
end Poly
def cappedDouble (z : ℕ×ℕ) : ℕ×ℕ := (z.1,min z.1 (2*z.2))
lemma cappedDouble_bound (z : ℕ×ℕ) (k : ℕ) :
(cappedDouble^[k] z).1=z.1 ∧ (cappedDouble^[k] z).2≤z.1+z.2 := by
induction k with
| zero => simp
| succ k ih =>
rw [Function.iterate_succ_apply']
constructor
· exact ih.1
· dsimp only [cappedDouble]
rw [ih.1]
exact (Nat.min_le_left _ _).trans (Nat.le_add_right _ _)
lemma cappedDouble_iterate (cap k : ℕ) :
cappedDouble^[k] (cap,min cap 1)=(cap,min cap (2^k)) := by
induction k with
| zero => simp
| succ k ih =>
rw [Function.iterate_succ_apply',ih]
simp only [cappedDouble,pow_succ]
congr 1
omega
namespace Poly
@[fun_prop] theorem pow2Capped : Poly (fun z : ℕ×ℕ => min z.1 (2^z.2)) := by
have hs : Poly cappedDouble := by unfold cappedDouble; fun_prop
have hi := boundedIterate cappedDouble hs (2*X) (by
intro z j hj
have hb := cappedDouble_bound z.2 j
simp only [encode_prod,pairBits_length,encode_nat,unary_length,eval_mul,eval_ofNat,eval_X]
omega)
have h : Poly (fun z : ℕ×ℕ => (cappedDouble^[z.2] (z.1,min z.1 1)).2) := by fun_prop
exact h.congr (fun z => by rw [cappedDouble_iterate])
end Poly
end BinaryMatching.StackCode
end
end OAI
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