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472a116 | 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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 | import OAI.Computability.SpaceSimulation.Register.Fold
namespace OAI
/-! Arithmetic operators in the bit-space closure. Many are
intentionally very slow. -/
namespace Fifths
namespace Comp
variable {X I : Type} {x : I → List X} {S a u v b c : I → ℕ}
open Bin
theorem add (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => b i + c i) := by
refine fold (fun i j => b i+j) g h ?_ ?_
· let f : Through c → I := fun i => i.1.1
exact (h.target.pull f).add ((g.target.pull f).mono fun i => i.2)
· intro e _ _ l
exact l.plus1
theorem sub (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => b i - c i) := by
refine fold (fun i j => b i-j) g h ?_ ?_
· let f : Through c → I := fun i => i.1.1
exact (h.target.pull f).mono (fun i => Nat.sub_le _ _)
· intro e _ _ l
exact l.minus1.congr (fun i => by
show b i.1.1 - i.1.2 - 1 = _
omega)
theorem mul (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => b i * c i) := by
refine fold (fun i j => b i*j) g ((Comp.constant h.source 0).congr (fun i => by simp)) ?_ ?_
· let f : Through c → I := fun i => i.1.1
exact (h.target.pull f).mul ((g.target.pull f).mono fun i => i.2)
· intro e p _ l
let f : Before c → I := fun i => i.1.1
exact (l.add (p.trans (h.pull f))).congr (fun i => by simp [mul_add,f])
theorem pow (h : Comp x S a b) (g : Comp x S a c)
(p : Fits S (fun i => b i ^ c i)) : Comp x S a (fun i => b i ^ c i) := by
refine fold (fun i j => b i^j) g ((Comp.constant h.source 1).congr (fun i => by simp)) ?_ ?_
· let f : Through c → I := fun i => i.1.1
refine ((p.pull f).add (Fits.const _ 1)).mono (fun i => ?_)
by_cases e : b (f i) = 0
· change b (f i) ^ i.1.2 ≤ b (f i) ^ c (f i) + 1
rw [e]
by_cases a : i.1.2 = 0
· simp [a]
· simp [a]
· apply le_trans (Nat.pow_le_pow_right (Nat.pos_of_ne_zero e) i.2)
exact Nat.le_succ _
· intro e r _ l
let f : Before c → I := fun i => i.1.1
exact (l.mul (r.trans (h.pull f))).congr (fun i => by simp [pow_succ,f])
-- Boolean tests and selections
variable {B C : I → Bool}
theorem flip (h : Comp x S a (fun i => (B i).toNat)) :
Comp x S a (fun i => (!B i).toNat) :=
((Comp.constant h.source 1).sub h).congr fun i => by cases B i <;> rfl
theorem both (h : Comp x S a (fun i => (B i).toNat))
(g : Comp x S a (fun i => (C i).toNat)) :
Comp x S a (fun i => (B i && C i).toNat) :=
(h.mul g).congr fun i => by cases B i <;> cases C i <;> rfl
theorem either (h : Comp x S a (fun i => (B i).toNat))
(g : Comp x S a (fun i => (C i).toNat)) :
Comp x S a (fun i => (B i || C i).toNat) :=
(flip (h.flip.both g.flip)).congr fun i => by cases B i <;> cases C i <;> rfl
theorem lt (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => (decide (b i < c i)).toNat) := by
refine (g.sub h).testnz.congr (fun i => ?_)
by_cases l : b i < c i
· simp [notZero,Nat.ne_of_gt (Nat.sub_pos_of_lt l),l]
· simp [notZero,Nat.sub_eq_zero_of_le (le_of_not_gt l),l]
theorem le (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => (decide (b i ≤ c i)).toNat) := by
refine flip (B := fun i => decide (c i < b i)) (g.lt h) |>.congr (fun i => ?_)
by_cases l : b i ≤ c i <;> simp [l]
omega
theorem eq (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => (decide (b i = c i)).toNat) := by
refine ((h.le g).both (g.le h)).congr (fun i => ?_)
rcases lt_trichotomy (b i) (c i) with l|l|l
· simp [l.le,l.ne,not_le_of_gt l]
· simp [l]
· simp [l.le,l.ne.symm,not_le_of_gt l]
theorem min (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => min (b i) (c i)) :=
((h.le g).iteB h g).congr fun i => by
by_cases l : b i ≤ c i
· simp [l]
· simp [l, min_eq_right (le_of_not_ge l)]
theorem max (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => max (b i) (c i)) :=
((h.le g).iteB g h).congr fun i => by
by_cases l : b i ≤ c i
· simp [l]
· simp [l,max_eq_left (le_of_not_ge l)]
private theorem div_step (j n : ℕ) :
(if notZero n && decide ((j/n+1)*n ≤ j+1) then
j/n+1 else j/n) = (j+1)/n := by
by_cases h : n = 0
· subst n; simp [notZero]
have p := Nat.pos_of_ne_zero h
have hb : j < (j/n+1)*n := (Nat.div_lt_iff_lt_mul p).mp (Nat.lt_succ_self _)
by_cases l : (j/n+1)*n ≤ j+1
· have e : (j/n+1)*n = j+1 := by omega
have v : n ∣ j+1 := ⟨j/n+1, by simpa [mul_comm] using e.symm⟩
simp [notZero,h,l,Nat.succ_div,v]
· have a : (j+1)/n < j/n+1 := (Nat.div_lt_iff_lt_mul p).mpr (lt_of_not_ge l)
have g : j/n ≤ (j+1)/n := Nat.div_le_div_right (Nat.le_succ _)
simpa only [decide_eq_false l,Bool.and_false, Bool.false_eq_true, ite_false] using
Nat.le_antisymm g (by omega)
theorem div (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => b i / c i) := by
refine fold (fun i j => j / c i) h ((Comp.constant h.source 0).congr
(fun i => by simp)) ?_ ?_
· let f : Through b → I := fun i => i.1.1
exact (h.target.pull f).mono (fun i => le_trans (Nat.div_le_self _ _) i.2)
· intro e r k l
let f : Before b → I := fun i => i.1.1
have q := r.trans (g.pull f)
have v := q.testnz.both ((l.plus1.mul q).le k.plus1)
exact (v.iteB l.plus1 l).congr (fun i => div_step _ _)
theorem mod (h : Comp x S a b) (g : Comp x S a c) :
Comp x S a (fun i => b i % c i) := by
apply (h.sub ((h.div g).mul g)).congr
intro i
have l := Nat.mod_add_div (b i) (c i)
rw [mul_comm]
omega
theorem size (h : Comp x S a b) :
Comp x S a (fun i => (b i).size) := by
let n := fun i => (b i).size
let w := fun i (j : ℕ) => Bin.pair j (b i / 2^j)
have e : Comp x S a (fun i => w i 0) :=
((Comp.constant h.source 0).pair h).congr (fun i => by simp [w])
have p : Fits (fun i : Through n => S i.1.1)
(fun i => w i.1.1 i.1.2) := by
let f : Through n → I := fun i => i.1.1
refine ((h.target.pull f).mono ?_).pair ((h.target.pull f).mono ?_)
· exact fun i => le_trans i.2 (Bin.size_le_self _)
· exact fun _ => Nat.div_le_self _ _
have g : Comp x S (fun i => w i 0) (fun i => w i (n i)) := by
refine loop n w ?_ ?_
· let f : Before n → Through n := fun i => ⟨i.1,i.2.le⟩
have r : Comp (fun i : Before n => x i.1.1) (fun i => S i.1.1)
(fun i => w i.1.1 i.1.2) (fun i => w i.1.1 i.1.2) :=
Comp.refl (p.pull f)
exact (r.left.plus1.pair (r.right.div (Comp.constant r.source 2))).congr
(fun i => by simp [w,pow_succ,Nat.div_div_eq_div_mul])
· have r : Comp (fun i : Through n => x i.1.1) (fun i => S i.1.1)
(fun i => w i.1.1 i.1.2) (fun i => w i.1.1 i.1.2) :=
Comp.refl p
apply r.right.testnz.congr
intro i
by_cases l : i.1.2 < n i.1.1
· have a : 2^i.1.2 ≤ b i.1.1 := Nat.lt_size.mp l
simp [notZero,l,Nat.ne_of_gt (Nat.div_pos a (by positivity))]
· have a := Nat.size_le.mp (le_of_not_gt l)
simp [notZero,l,Nat.div_eq_of_lt a]
exact (e.trans g).left
end Comp
end Fifths
end OAI
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