A push of an arbitrary (e.g. multiplying) expression, with the quadratic cost bound. -/ theorem Sp.pushQ…

A push of an arbitrary (e.g. multiplying) expression, with the quadratic cost bound. -/ theorem Sp.pushQ {i : Fin HS} {e : Ex HS} {B : ℕ} (hi : i.val < 56) (hc : e.Clean) (hok : e.ok (encK s f)) (hB : e.Bnd (encK s f) B) : Sp (.push i e) (encK s f) (encK s (Function.update f i (e.eval (encK s f) :: f i))) ((e.size + 1) * 10 ^ 7 * (B + 2) ^ 2) := by refine ⟨⟨hi, hc, hok⟩, ?, ?⟩ · simp only [run] rw [← encK_upd]; simp [ints] · exact pushCost_le e _ B hB theorem Sp.pop {i : Fin HS} {a : ℤ} {l : List ℤ} {B : ℕ} (hi : i.val < 56) (hf : f i = a :: l) (ha : zsz a ≤ B) : Sp (.pop i) (encK s f) (encK s (Function.update f i l)) (2 * B + 2) := by refine ⟨⟨hi, isInt_encK hf⟩, ?, ?⟩ · simp only [run] rw [← encK_upd]; simp [hf] · simp only [cost, top_encK hf]; omega theorem Sp.asg {i : Fin HS} {e : Ex HS} {a : ℤ} {l : List ℤ} {B : ℕ} (hi : i.val < 56) (hf : f i = a :: l) (ha : zsz a ≤ B) (hc : e.Clean) (hok : e.ok (encK s f)) (hl : e.Lin) (hB : e.Bnd (encK s f) B) : Sp (.asg i e) (encK s f) (encK s (Function.update f i (e.eval (encK s f) :: l))) ((e.size + 1) * 2000 * (B + 2)) := by refine ⟨⟨hi, hc, hok, isInt_encK hf⟩, ?, ?⟩ · simp only [run] rw [← encK_upd]; simp [hf, ints] · exact asgCost_le_lin i e _ B hl hB (by rw [top_encK hf]; exact ha) end /-! ## Moving integers between stacks

The following holds (Lean statement, in the scope of module Solutions.KILBase):

theorem NLogN.KIL.Sp.mvK {X Y C : Fin HS} (hXY : X ≠ Y) (hXC : X ≠ C) (hYC : Y ≠ C) (hX5 : X.val < 56)
    (hY5 : Y.val < 56) (hC5 : C.val < 56) (s : IState HS) (f : Fin HS → List ℤ) (n : Ex HS)
    (hnc : n.Clean) (hno : n.ok (NLogN.KIL.encK s f)) (L l : List ℤ) (hX : f X = L ++ l)
    (hn : n.eval (NLogN.KIL.encK s f) = L.length) (B : ℕ) (hL : Bnd B L) (hnl : n.Lin)
    (hnb : n.Bnd (NLogN.KIL.encK s f) B) :
    Sp (NLogN.mvK X Y C n) (NLogN.KIL.encK s f)
      (NLogN.KIL.encK s (Function.update (Function.update f X l) Y (L.reverse ++ f Y)))
      ((n.size + 1) * 2000 * (B + 2) + L.length * (14000 * (B + 2)) + 32)

Context. The lemma belongs to the part KIL base: register programs on "managed" stacks: encK s f is the state s with the list of integers f j pushed on top of every stack j (f j = [] for unmanaged stacks). All programs of the KIL files are specified on such states; updates are Function.updates of f, which keeps the bookkeeping small.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.