feat: simplify nested arith expressions

src/Init/Data/Nat/Linear.lean

@@ -137,8 +137,11 @@ def Expr.toPoly : Expr → Poly
   | Expr.mulL k a => a.toPoly.mul k
   | Expr.mulR a k => a.toPoly.mul k
 
+def Poly.norm (p : Poly) : Poly :=
+  p.sort.fuse
+
 def Expr.toNormPoly (e : Expr) : Poly :=
-  e.toPoly.sort.fuse
+  e.toPoly.norm
 
 def Expr.inc (e : Expr) : Expr :=
    Expr.add e (Expr.num 1)
@@ -490,7 +493,7 @@ theorem Expr.denote_toPoly (ctx : Context) (e : Expr) : e.toPoly.denote ctx = e.
 attribute [local simp] Expr.denote_toPoly
 
 theorem Expr.eq_of_toNormPoly (ctx : Context) (a b : Expr) (h : a.toNormPoly = b.toNormPoly) : a.denote ctx = b.denote ctx := by
-  simp [toNormPoly] at h
+  simp [toNormPoly, Poly.norm] at h
   have h := congrArg (Poly.denote ctx) h
   simp at h
   assumption
@@ -498,13 +501,13 @@ theorem Expr.eq_of_toNormPoly (ctx : Context) (a b : Expr) (h : a.toNormPoly = b
 theorem Expr.of_cancel_eq (ctx : Context) (a b c d : Expr) (h : Poly.cancel a.toNormPoly b.toNormPoly = (c.toPoly, d.toPoly)) : (a.denote ctx = b.denote ctx) = (c.denote ctx = d.denote ctx) := by
   have := Poly.denote_eq_cancel_eq ctx a.toNormPoly b.toNormPoly
   rw [h] at this
-  simp [toNormPoly, Poly.denote_eq] at this
+  simp [toNormPoly, Poly.norm, Poly.denote_eq] at this
   exact this.symm
 
 theorem Expr.of_cancel_le (ctx : Context) (a b c d : Expr) (h : Poly.cancel a.toNormPoly b.toNormPoly = (c.toPoly, d.toPoly)) : (a.denote ctx ≤ b.denote ctx) = (c.denote ctx ≤ d.denote ctx) := by
   have := Poly.denote_le_cancel_eq ctx a.toNormPoly b.toNormPoly
   rw [h] at this
-  simp [toNormPoly, Poly.denote_le] at this
+  simp [toNormPoly, Poly.norm,Poly.denote_le] at this
   exact this.symm
 
 theorem Expr.of_cancel_lt (ctx : Context) (a b c d : Expr) (h : Poly.cancel a.inc.toNormPoly b.toNormPoly = (c.inc.toPoly, d.toPoly)) : (a.denote ctx < b.denote ctx) = (c.denote ctx < d.denote ctx) :=
@@ -526,8 +529,8 @@ theorem ExprCnstr.denote_toNormPoly (ctx : Context) (c : ExprCnstr) : c.toNormPo
   cases c; rename_i eq lhs rhs
   simp [ExprCnstr.denote, PolyCnstr.denote, ExprCnstr.toNormPoly]
   by_cases h : eq = true <;> simp [h]
-  . rw [Poly.denote_eq_cancel_eq]; simp [Poly.denote_eq, Expr.toNormPoly]
-  . rw [Poly.denote_le_cancel_eq]; simp [Poly.denote_le, Expr.toNormPoly]
+  . rw [Poly.denote_eq_cancel_eq]; simp [Poly.denote_eq, Expr.toNormPoly, Poly.norm]
+  . rw [Poly.denote_le_cancel_eq]; simp [Poly.denote_le, Expr.toNormPoly, Poly.norm]
 
 attribute [local simp] ExprCnstr.denote_toNormPoly
 
@@ -671,4 +674,9 @@ theorem ExprCnstr.eq_of_toNormPoly_eq (ctx : Context) (c d : ExprCnstr) (h : c.t
   simp at h
   assumption
 
+theorem Expr.eq_of_toNormPoly_eq (ctx : Context) (e e' : Expr) (h : e.toNormPoly == e'.toPoly) : e.denote ctx = e'.denote ctx := by
+  have h := congrArg (Poly.denote ctx) (eq_of_beq h)
+  simp [Expr.toNormPoly, Poly.norm] at h
+  assumption
+
 end Nat.Linear

src/Lean/Meta/Tactic/LinearArith/Nat.lean

@@ -200,4 +200,17 @@ def simpCnstr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
   else
     simpCnstrPos? e
 
+def simpExpr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
+  let (e, ctx) ← ToLinear.run (ToLinear.toLinearExpr e)
+  let p  := e.toPoly
+  let p' := p.norm
+  if p'.length < p.length then
+    -- We only return some if monomials were fused
+    let e' : LinearExpr := p'.toExpr
+    let p := mkApp4 (mkConst ``Nat.Linear.Expr.eq_of_toNormPoly_eq) (← toContextExpr ctx) (toExpr e) (toExpr e') reflTrue
+    let r ← e'.toArith ctx
+    return some (r, p)
+  else
+    return none
+
 end Lean.Meta.Linear.Nat

src/Lean/Meta/Tactic/LinearArith/Simp.lean

@@ -7,6 +7,15 @@ import Lean.Meta.Tactic.LinearArith.Nat
 
 namespace Lean.Meta.Linear
 
+/-- Quick filter simpExpr? -/
+private partial def isSimpExprTarget (e : Expr) : Bool :=
+  let f := e.getAppFn
+  if !f.isConst then
+    false
+  else
+    let n := f.constName!
+    n == ``HAdd.hAdd || n == ``HMul.hMul || n == ``HSub.hSub || n == ``Nat.succ
+
 /-- Quick filter simpCnstr? -/
 private partial def isSimpCnstrTarget (e : Expr) : Bool :=
   let f := e.getAppFn
@@ -21,10 +30,18 @@ private partial def isSimpCnstrTarget (e : Expr) : Bool :=
     else
       false
 
-def simpCnstr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
+private def parentIsTarget (parent? : Option Expr) : Bool :=
+  match parent? with
+  | none => false
+  | some parent => isSimpExprTarget parent || isSimpCnstrTarget parent
+
+def simp? (e : Expr) (parent? : Option Expr) : MetaM (Option (Expr × Expr)) := do
+  -- TODO: add support for `Int` and arbitrary ordered comm rings
   if isSimpCnstrTarget e then
-    -- TODO: add support for `Int` and arbitrary ordered comm rings
     Nat.simpCnstr? e
+  else if isSimpExprTarget e && !parentIsTarget parent? then
+    trace[Meta.Tactic.simp] "arith expr: {e}"
+    Nat.simpExpr? e
   else
     return none
 

src/Lean/Meta/Tactic/Simp/Rewrite.lean

@@ -199,7 +199,7 @@ def rewriteUsingDecide? (e : Expr) : MetaM (Option Result) := withReducibleAndIn
 
 def simpArith? (e : Expr) : SimpM (Option Step) := do
   if !(← read).config.arith then return none
-  let some (e', h) ← Linear.simpCnstr? e | return none
+  let some (e', h) ← Linear.simp? e (← read).parent? | return none
   return Step.visit { expr := e', proof? := h }
 
 def rewritePre (e : Expr) (discharge? : Expr → SimpM (Option Expr)) : SimpM Step := do

tests/lean/run/simpArith1.lean

@@ -16,3 +16,9 @@ theorem ex5 (h : a + d + b > b + 1 + (a + (c + c) + d)) : False := by
   simp_arith at h
 
 #print ex5
+
+theorem ex6 (p : Nat → Prop) (h : p (a + 1 + a + 2 + b)) : p (2*a + b + 3) := by
+  simp_arith at h
+  assumption
+
+#print ex6

tests/lean/run/simpCnstr1.lean

@@ -4,7 +4,7 @@ open Lean in open Lean.Meta in
 def test (declName : Name) : MetaM Unit := do
   let info ← getConstInfo declName
   forallTelescope info.type fun _ e => do
-    let some (e', p) ← Linear.simpCnstr? e | throwError "failed to simplify{indentExpr e}"
+    let some (e', p) ← Linear.simp? e none | throwError "failed to simplify{indentExpr e}"
     check p
     unless (← isDefEq (← inferType p) (← mkEq e e')) do
       throwError "invalid proof"