openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexArithmetic /ProgramComposition.lean
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexArithmetic.Programs | |
| namespace OAI | |
| /-! Complex arithmetic programs and asymptotic matrix multiplication costs. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation.Arithmetic | |
| namespace Gate | |
| variable {Input Input' Register : Type*} | |
| def mapInput (f : Input → Input') : Gate Input Register → Gate Input' Register | |
| | .constant z => .constant z | |
| | .input i => .input (f i) | |
| | .add i j => .add i j | |
| | .sub i j => .sub i j | |
| | .mul i j => .mul i j | |
| @[simp] theorem eval_mapInput (f : Input → Input') (g : Gate Input Register) | |
| (inputs : Input' → ℂ) (registers : Register → ℂ) : | |
| (g.mapInput f).eval inputs registers = g.eval (inputs ∘ f) registers := by | |
| cases g <;> rfl | |
| @[simp] theorem cost_mapInput (f : Input → Input') (g : Gate Input Register) : | |
| (g.mapInput f).cost = g.cost := by | |
| cases g <;> rfl | |
| def mapSource (f : Input → Input' ⊕ ℂ) : Gate Input Register → Gate Input' Register | |
| | .constant z => .constant z | |
| | .input i => match f i with | |
| | .inl j => .input j | |
| | .inr z => .constant z | |
| | .add i j => .add i j | |
| | .sub i j => .sub i j | |
| | .mul i j => .mul i j | |
| @[simp] theorem eval_mapSource (f : Input → Input' ⊕ ℂ) (g : Gate Input Register) | |
| (inputs : Input' → ℂ) (registers : Register → ℂ) : | |
| (g.mapSource f).eval inputs registers = | |
| g.eval (fun i => Sum.elim inputs id (f i)) registers := by | |
| cases g with | |
| | input i => cases h : f i <;> simp [mapSource, h, eval] | |
| | constant => rfl | |
| | add => rfl | |
| | sub => rfl | |
| | mul => rfl | |
| @[simp] theorem cost_mapSource (f : Input → Input' ⊕ ℂ) (g : Gate Input Register) : | |
| (g.mapSource f).cost = g.cost := by | |
| cases g with | |
| | input i => cases h : f i <;> simp [mapSource, h, cost] | |
| | constant => rfl | |
| | add => rfl | |
| | sub => rfl | |
| | mul => rfl | |
| end Gate | |
| namespace Program | |
| variable {Input Input' Mid : Type*} | |
| def mapInput (f : Input → Input') : {r : ℕ} → Program Input r → Program Input' r | |
| | 0, .nil => .nil | |
| | _ + 1, .step p g => (p.mapInput f).step (g.mapInput f) | |
| @[simp] theorem eval_mapInput (f : Input → Input') {r : ℕ} (p : Program Input r) | |
| (inputs : Input' → ℂ) : | |
| ∀ i : Fin r, (p.mapInput f).eval inputs i = p.eval (inputs ∘ f) i := by | |
| induction p with | |
| | nil => intro i; exact Fin.elim0 i | |
| | @step r p g ih => | |
| intro i | |
| refine Fin.cases ?_ (fun j => ?_) i | |
| · simp only [mapInput, eval_step_zero, Gate.eval_mapInput] | |
| rw [funext ih] | |
| · exact ih j | |
| @[simp] theorem cost_mapInput (f : Input → Input') {r : ℕ} (p : Program Input r) : | |
| (p.mapInput f).cost = p.cost := by | |
| induction p with | |
| | nil => rfl | |
| | step p g ih => simp only [mapInput, cost_step, Gate.cost_mapInput, ih] | |
| def mapSource (f : Input → Input' ⊕ ℂ) : {r : ℕ} → Program Input r → Program Input' r | |
| | 0, .nil => .nil | |
| | _ + 1, .step p g => (p.mapSource f).step (g.mapSource f) | |
| @[simp] theorem eval_mapSource (f : Input → Input' ⊕ ℂ) {r : ℕ} | |
| (p : Program Input r) (inputs : Input' → ℂ) : | |
| ∀ i : Fin r, (p.mapSource f).eval inputs i = | |
| p.eval (fun j => Sum.elim inputs id (f j)) i := by | |
| induction p with | |
| | nil => intro i; exact Fin.elim0 i | |
| | @step r p g ih => | |
| intro i | |
| refine Fin.cases ?_ (fun j => ?_) i | |
| · simp only [mapSource, eval_step_zero, Gate.eval_mapSource] | |
| rw [funext ih] | |
| · exact ih j | |
| @[simp] theorem cost_mapSource (f : Input → Input' ⊕ ℂ) {r : ℕ} | |
| (p : Program Input r) : (p.mapSource f).cost = p.cost := by | |
| induction p with | |
| | nil => rfl | |
| | step p g ih => simp only [mapSource, cost_step, Gate.cost_mapSource, ih] | |
| structure Substituted {r s : ℕ} (p : Program Input r) (q : Program Mid s) | |
| (wires : Mid → Fin r) where | |
| registers : ℕ | |
| program : Program Input registers | |
| oldOutput : Fin r → Fin registers | |
| output : Fin s → Fin registers | |
| correctOld : ∀ inputs i, program.eval inputs (oldOutput i) = p.eval inputs i | |
| correct : ∀ inputs i, program.eval inputs (output i) = | |
| q.eval (fun j => p.eval inputs (wires j)) i | |
| cost_eq : program.cost = p.cost + q.cost | |
| namespace Substituted | |
| def step {r s : ℕ} {p : Program Input r} {q : Program Mid s} | |
| {wires : Mid → Fin r} (a : Substituted p q wires) | |
| (g : Gate Mid (Fin s)) (h : Gate Input (Fin a.registers)) | |
| (heval : ∀ inputs, h.eval inputs (a.program.eval inputs) = | |
| g.eval (fun j => p.eval inputs (wires j)) | |
| (q.eval (fun j => p.eval inputs (wires j)))) | |
| (hcost : h.cost = g.cost) : Substituted p (q.step g) wires where | |
| registers := a.registers + 1 | |
| program := a.program.step h | |
| oldOutput := fun i => (a.oldOutput i).succ | |
| output := Fin.cases 0 (fun i => (a.output i).succ) | |
| correctOld := by | |
| intro inputs i | |
| exact a.correctOld inputs i | |
| correct := by | |
| intro inputs i | |
| refine Fin.cases ?_ (fun j => ?_) i | |
| · exact heval inputs | |
| · exact a.correct inputs j | |
| cost_eq := by | |
| simp only [cost_step, a.cost_eq, hcost, Nat.add_assoc] | |
| end Substituted | |
| def substitute {r : ℕ} (p : Program Input r) : | |
| {s : ℕ} → (q : Program Mid s) → (wires : Mid → Fin r) → Substituted p q wires | |
| | 0, .nil, wires => | |
| { registers := r | |
| program := p | |
| oldOutput := id | |
| output := Fin.elim0 | |
| correctOld := fun _ _ => rfl | |
| correct := fun _ i => Fin.elim0 i | |
| cost_eq := by simp only [cost, Nat.add_zero] } | |
| | _ + 1, .step q g, wires => | |
| let a := substitute p q wires | |
| match g with | |
| | .input j => | |
| { registers := a.registers | |
| program := a.program | |
| oldOutput := a.oldOutput | |
| output := Fin.cases (a.oldOutput (wires j)) a.output | |
| correctOld := a.correctOld | |
| correct := by | |
| intro inputs i | |
| refine Fin.cases ?_ (fun k => ?_) i | |
| · exact a.correctOld inputs (wires j) | |
| · exact a.correct inputs k | |
| cost_eq := by | |
| simpa only [cost_step, Gate.cost, Nat.add_zero] using a.cost_eq } | |
| | .constant z => | |
| a.step (.constant z) (.constant z) (fun _ => rfl) rfl | |
| | .add i j => | |
| a.step (.add i j) (.add (a.output i) (a.output j)) | |
| (by intro inputs; simp only [Gate.eval, a.correct]) rfl | |
| | .sub i j => | |
| a.step (.sub i j) (.sub (a.output i) (a.output j)) | |
| (by intro inputs; simp only [Gate.eval, a.correct]) rfl | |
| | .mul i j => | |
| a.step (.mul i j) (.mul (a.output i) (a.output j)) | |
| (by intro inputs; simp only [Gate.eval, a.correct]) rfl | |
| end Program | |
| end MatrixMultiplication.Foundation.Arithmetic | |
| end | |
| end OAI | |