lean/OAI/Computability/QuantumFactoring/BitStackComposition.lean

import Mathlib
import OAI.Computability.QuantumFactoring.BitStackRelabel
import OAI.Computability.QuantumFactoring.BitStackLists
import OAI.Computability.QuantumFactoring.BitStackProcedures

namespace OAI



section

namespace ExactQuantumFactoring.BitStackProgram
variable {K L : Type} [DecidableEq K] [DecidableEq L]

omit [DecidableEq K] [DecidableEq L] in
lemma overlay_empty (f : K ↪ L) : overlay f (fun _=>[]) (fun _=>[])=fun _=>[] := by
  funext j
  by_cases h : ∃ i,f i=j
  · rcases h with ⟨i,rfl⟩; exact overlay_at f _ _ i
  · exact overlay_outside f _ _ j (by simpa only [not_exists] using h)
lemma overlay_singleton (f : K ↪ L) (k : K) (xs : List Bool) :
    overlay f (fun _=>[]) (singletonStore k xs)=singletonStore (f k) xs := by
  rw [singletonStore,overlay_update,overlay_empty]
  rfl

/-- Move an entire word without reversing its order. Each symbol is pushed
and popped twice; this is not an atomic bulk-copy instruction. -/
def moveWord (a b t : K) : Program K := .seq (reverseMove a t) (reverseMove t b)
lemma moveWord_runs (a b t : K) (hab : a≠b) (hat : a≠t) (hbt : b≠t) (xs : List Bool) :
    Runs (moveWord a b t) (singletonStore a xs) (singletonStore b xs) (4*xs.length+2) := by
  let s := singletonStore a xs
  let u := Function.update (Function.update s a []) t ((s a).reverse++s t)
  have hh := Runs.seq (reverseMove_runs a t hat s) (reverseMove_runs t b hbt.symm u)
  change Runs (.seq _ _) _ _ _
  convert hh using 1
  · funext k
    by_cases ha : k=a <;> by_cases hb : k=b <;> by_cases ht : k=t <;>
      simp_all [u,s,singletonStore,Function.update,Ne.symm hat,Ne.symm hab,Ne.symm hbt]
  · simp only [u,Function.update_self,s,singletonStore,Function.update_self]
    simp only [Function.update_of_ne hat.symm,List.append_nil,List.length_reverse]
    omega

namespace Procedure
variable {α β γ : Type} {ea : α→List Bool} {eb : β→List Bool} {ec : γ→List Bool}
  {f : α→β} {g : β→γ}

def joinLeft (A B : Type) : A ↪ A ⊕ (B ⊕ Unit) := ⟨Sum.inl,Sum.inl_injective⟩
def joinRight (A B : Type) : B ↪ A ⊕ (B ⊕ Unit) :=
  ⟨Sum.inr ∘ Sum.inl,Sum.inr_injective.comp Sum.inl_injective⟩

lemma eval_nat_mono (p : Polynomial ℕ) : Monotone p.eval := by
  intro a b hab
  change p.eval a ≤ p.eval b
  rw [Polynomial.eval_eq_sum,Polynomial.eval_eq_sum]
  exact Finset.sum_le_sum (fun i _ => Nat.mul_le_mul_left _ (Nat.pow_le_pow_left hab i))

/-- Checked composition for actual finite Boolean-stack procedures. This
constructs the combined program and accounts for the intermediate-word copy;
it does NOT invoke Mathlib's unproved TM2 polytime-composition declaration. -/
noncomputable def comp (q : Procedure eb ec g) (p : Procedure ea eb f) :
    Procedure ea ec (g ∘ f) where
  K := p.K ⊕ (q.K ⊕ Unit)
  finiteK := inferInstance
  decideK := inferInstance
  input := Sum.inl p.input
  output := Sum.inr (Sum.inl q.output)
  program := .seq (p.program.relabel (joinLeft p.K q.K))
    (.seq (moveWord (Sum.inl p.output) (Sum.inr (Sum.inl q.input)) (Sum.inr (Sum.inr ())))
      (q.program.relabel (joinRight p.K q.K)))
  bound := p.bound+(Polynomial.C 4*(Polynomial.X+p.bound)+Polynomial.C 2)+
    q.bound.comp (Polynomial.X+p.bound)
  runs a := by
    obtain ⟨cp,hcp,hp⟩ := p.runs a
    obtain ⟨cq,hcq,hq⟩ := q.runs (f a)
    have hleft := hp.relabel (joinLeft p.K q.K) (fun _=>[])
    have hright := hq.relabel (joinRight p.K q.K) (fun _=>[])
    simp only [overlay_singleton] at hleft hright
    have hmove := moveWord_runs (K:=p.K ⊕ (q.K ⊕ Unit)) (Sum.inl p.output)
      (Sum.inr (Sum.inl q.input)) (Sum.inr (Sum.inr ()))
      (by simp) (by simp) (by simp) (eb (f a))
    refine ⟨cq+(4*(eb (f a)).length+2)+cp,?_,Runs.seq hleft (Runs.seq hmove hright)⟩
    have hlen := p.output_length_le a
    have hmono := eval_nat_mono q.bound hlen
    dsimp only at hmono
    simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_C,
      Polynomial.eval_X,Polynomial.eval_comp]
    omega
end Procedure
end ExactQuantumFactoring.BitStackProgram

end



end OAI