perf: optimize Nat.Linear.Expr.toPoly (#6062)

src/Init/Data/Nat/Linear.lean

@@ -146,12 +146,16 @@ def Poly.combineAux (fuel : Nat) (p₁ p₂ : Poly) : Poly :=
 def Poly.combine (p₁ p₂ : Poly) : Poly :=
   combineAux hugeFuel p₁ p₂
 
-def Expr.toPoly : Expr → Poly
-  | Expr.num k    => bif k == 0 then [] else [ (k, fixedVar) ]
-  | Expr.var i    => [(1, i)]
-  | Expr.add a b  => a.toPoly ++ b.toPoly
-  | Expr.mulL k a => a.toPoly.mul k
-  | Expr.mulR a k => a.toPoly.mul k
+def Expr.toPoly (e : Expr) := go 1 e []
+where
+  -- Implementation note: This assembles the result using difference lists
+  -- to avoid `++` on lists.
+  go (coeff : Nat) : Expr → (Poly → Poly)
+    | Expr.num k    => bif k == 0 then id else ((coeff * k, fixedVar) :: ·)
+    | Expr.var i    => ((coeff, i) :: ·)
+    | Expr.add a b  => go coeff a ∘ go coeff b
+    | Expr.mulL k a
+    | Expr.mulR a k => bif k == 0 then id else go (coeff * k) a
 
 def Poly.norm (p : Poly) : Poly :=
   p.sort.fuse
@@ -516,13 +520,25 @@ theorem Poly.denote_combine (ctx : Context) (p₁ p₂ : Poly) : (p₁.combine p
 
 attribute [local simp] Poly.denote_combine
 
+theorem Expr.denote_toPoly_go (ctx : Context) (e : Expr) :
+  (toPoly.go k e p).denote ctx = k * e.denote ctx + p.denote ctx := by
+  induction k, e using Expr.toPoly.go.induct generalizing p with
+  | case1 k k' =>
+    simp only [toPoly.go]
+    by_cases h : k' == 0
+    · simp [h, eq_of_beq h]
+    · simp [h, Var.denote]
+  | case2 k i => simp [toPoly.go]
+  | case3 k a b iha ihb => simp [toPoly.go, iha, ihb]
+  | case4 k k' a ih
+  | case5 k a k' ih =>
+    simp only [toPoly.go, denote, mul_eq]
+    by_cases h : k' == 0
+    · simp [h, eq_of_beq h]
+    · simp [h, cond_false, ih, Nat.mul_assoc]
+
 theorem Expr.denote_toPoly (ctx : Context) (e : Expr) : e.toPoly.denote ctx = e.denote ctx := by
-  induction e with
-  | num k => by_cases h : k == 0 <;> simp [toPoly, h, Var.denote]; simp [eq_of_beq h]
-  | var i => simp [toPoly]
-  | add a b iha ihb => simp [toPoly, iha, ihb]
-  | mulL k a ih => simp [toPoly, ih, -Poly.mul]
-  | mulR k a ih => simp [toPoly, ih, -Poly.mul]
+  simp [toPoly, Expr.denote_toPoly_go]
 
 attribute [local simp] Expr.denote_toPoly
 
@@ -554,8 +570,8 @@ theorem ExprCnstr.denote_toPoly (ctx : Context) (c : ExprCnstr) : c.toPoly.denot
   cases c; rename_i eq lhs rhs
   simp [ExprCnstr.denote, PolyCnstr.denote, ExprCnstr.toPoly];
   by_cases h : eq = true <;> simp [h]
-  · simp [Poly.denote_eq, Expr.toPoly]
-  · simp [Poly.denote_le, Expr.toPoly]
+  · simp [Poly.denote_eq]
+  · simp [Poly.denote_le]
 
 attribute [local simp] ExprCnstr.denote_toPoly