feat: simp +arith sorts linear atoms (#7040)

This PR ensures that terms such as `f (2*x + y)` and `f (y + x + x)`
have the same normal form when using `simp +arith`

src/Init/Data/Nat/Linear.lean

@@ -35,11 +35,11 @@ inductive Expr where
   deriving Inhabited
 
 def Expr.denote (ctx : Context) : Expr → Nat
-  | Expr.add a b  => Nat.add (denote ctx a) (denote ctx b)
-  | Expr.num k    => k
-  | Expr.var v    => v.denote ctx
-  | Expr.mulL k e => Nat.mul k (denote ctx e)
-  | Expr.mulR e k => Nat.mul (denote ctx e) k
+  | .add a b  => Nat.add (denote ctx a) (denote ctx b)
+  | .num k    => k
+  | .var v    => v.denote ctx
+  | .mulL k e => Nat.mul k (denote ctx e)
+  | .mulR e k => Nat.mul (denote ctx e) k
 
 abbrev Poly := List (Nat × Var)
 
@@ -146,17 +146,17 @@ 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
+    | .num k    => bif k == 0 then id else ((coeff * k, fixedVar) :: ·)
+    | .var i    => ((coeff, i) :: ·)
+    | .add a b  => go coeff a ∘ go coeff b
+    | .mulL k a
+    | .mulR a k => bif k == 0 then id else go (coeff * k) a
 
 def Expr.toNormPoly (e : Expr) : Poly :=
   e.toPoly.norm
 
 def Expr.inc (e : Expr) : Expr :=
-   Expr.add e (Expr.num 1)
+   .add e (.num 1)
 
 structure PolyCnstr  where
   eq  : Bool
@@ -244,21 +244,21 @@ def Certificate.denote (ctx : Context) (c : Certificate) : Prop :=
 
 def monomialToExpr (k : Nat) (v : Var) : Expr :=
   bif v == fixedVar then
-    Expr.num k
+    .num k
   else bif k == 1 then
-    Expr.var v
+    .var v
   else
-    Expr.mulL k (Expr.var v)
+    .mulL k (.var v)
 
 def Poly.toExpr (p : Poly) : Expr :=
   match p with
-  | [] => Expr.num 0
+  | [] => .num 0
   | (k, v) :: p => go (monomialToExpr k v) p
 where
   go (e : Expr) (p : Poly) : Expr :=
     match p with
     | [] => e
-    | (k, v) :: p => go (Expr.add e (monomialToExpr k v)) p
+    | (k, v) :: p => go (.add e (monomialToExpr k v)) p
 
 def PolyCnstr.toExpr (c : PolyCnstr) : ExprCnstr :=
   { c with lhs := c.lhs.toExpr, rhs := c.rhs.toExpr }

src/Lean/Meta/Tactic/LinearArith/Int/Basic.lean

@@ -5,6 +5,7 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Init.Data.Int.Linear
+import Lean.Util.SortExprs
 import Lean.Meta.Check
 import Lean.Meta.Offset
 import Lean.Meta.IntInstTesters
@@ -31,6 +32,24 @@ def PolyCnstr.toExprCnstr : PolyCnstr → ExprCnstr
   | .eq p => .eq p.toExpr (.num 0)
   | .le p => .le p.toExpr (.num 0)
 
+/-- Applies the given variable permutation to `e` -/
+def Expr.applyPerm (perm : Lean.Perm) (e : Expr) : Expr :=
+  go e
+where
+  go : Expr → Expr
+    | .num v    => .num v
+    | .var i    => .var (perm[(i : Nat)]?.getD i)
+    | .neg a    => .neg (go a)
+    | .add a b  => .add (go a) (go b)
+    | .sub a b  => .sub (go a) (go b)
+    | .mulL k a => .mulL k (go a)
+    | .mulR a k => .mulR (go a) k
+
+/-- Applies the given variable permutation to the given expression constraint. -/
+def ExprCnstr.applyPerm (perm : Lean.Perm) : ExprCnstr → ExprCnstr
+  | .eq a b => .eq (a.applyPerm perm) (b.applyPerm perm)
+  | .le a b => .le (a.applyPerm perm) (b.applyPerm perm)
+
 end Int.Linear
 
 namespace Lean.Meta.Linear.Int
@@ -187,7 +206,24 @@ def run (x : M α) : MetaM (α × Array Expr) := do
 
 end ToLinear
 
-export ToLinear (toLinearCnstr? toLinearExpr)
+def toLinearExpr (e : Expr) : MetaM (LinearExpr × Array Expr) := do
+  let (e, atoms) ← ToLinear.run (ToLinear.toLinearExpr e)
+  if atoms.size == 1 then
+    return (e, atoms)
+  else
+    let (atoms, perm) := sortExprs atoms
+    let e := e.applyPerm perm
+    return (e, atoms)
+
+def toLinearCnstr? (e : Expr) : MetaM (Option (LinearCnstr × Array Expr)) := do
+  let (some c, atoms) ← ToLinear.run (ToLinear.toLinearCnstr? e)
+    | return none
+  if atoms.size <= 1 then
+    return some (c, atoms)
+  else
+    let (atoms, perm) := sortExprs atoms
+    let c := c.applyPerm perm
+    return some (c, atoms)
 
 def toContextExpr (ctx : Array Expr) : Expr :=
   if h : 0 < ctx.size then

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

@@ -44,7 +44,7 @@ def Int.Linear.PolyCnstr.getConst : PolyCnstr → Int
 namespace Lean.Meta.Linear.Int
 
 def simpCnstrPos? (e : Expr) : MetaM (Option (Expr × Expr)) := do
-  let (some c, atoms) ← ToLinear.run (ToLinear.toLinearCnstr? e) | return none
+  let some (c, atoms) ← toLinearCnstr? e | return none
   withAbstractAtoms atoms ``Int fun atoms => do
     let lhs ← c.toArith atoms
     let p := c.toPoly
@@ -127,13 +127,13 @@ def simpCnstr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
     simpCnstrPos? e
 
 def simpExpr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
-  let (e, ctx) ← ToLinear.run (ToLinear.toLinearExpr e)
+  let (e, atoms) ← toLinearExpr e
   let p  := e.toPoly
   let e' := p.toExpr
   if e != e' then
     -- We only return some if monomials were fused
-    let p := mkApp4 (mkConst ``Int.Linear.Expr.eq_of_toPoly_eq) (toContextExpr ctx) (toExpr e) (toExpr e') reflBoolTrue
-    let r ← LinearExpr.toArith ctx e'
+    let p := mkApp4 (mkConst ``Int.Linear.Expr.eq_of_toPoly_eq) (toContextExpr atoms) (toExpr e) (toExpr e') reflBoolTrue
+    let r ← LinearExpr.toArith atoms e'
     return some (r, p)
   else
     return none

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

@@ -4,12 +4,32 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 prelude
+import Lean.Util.SortExprs
 import Lean.Meta.Check
 import Lean.Meta.Offset
 import Lean.Meta.AppBuilder
 import Lean.Meta.KExprMap
 import Lean.Data.RArray
 
+namespace Nat.Linear
+
+/-- Applies the given variable permutation to `e` -/
+def Expr.applyPerm (perm : Lean.Perm) (e : Expr) : Expr :=
+  go e
+where
+  go : Expr → Expr
+    | .num v    => .num v
+    | .var i    => .var (perm[(i : Nat)]?.getD i)
+    | .add a b  => .add (go a) (go b)
+    | .mulL k a => .mulL k (go a)
+    | .mulR a k => .mulR (go a) k
+
+/-- Applies the given variable permutation to the given expression constraint. -/
+def ExprCnstr.applyPerm (perm : Lean.Perm) : ExprCnstr → ExprCnstr
+  | { eq, lhs, rhs } => { eq, lhs := lhs.applyPerm perm, rhs := rhs.applyPerm perm }
+
+end Nat.Linear
+
 namespace Lean.Meta.Linear.Nat
 
 deriving instance Repr for Nat.Linear.Expr
@@ -140,7 +160,24 @@ def run (x : M α) : MetaM (α × Array Expr) := do
 
 end ToLinear
 
-export ToLinear (toLinearCnstr? toLinearExpr)
+def toLinearExpr (e : Expr) : MetaM (LinearExpr × Array Expr) := do
+  let (e, atoms) ← ToLinear.run (ToLinear.toLinearExpr e)
+  if atoms.size == 1 then
+    return (e, atoms)
+  else
+    let (atoms, perm) := sortExprs atoms
+    let e := e.applyPerm perm
+    return (e, atoms)
+
+def toLinearCnstr? (e : Expr) : MetaM (Option (LinearCnstr × Array Expr)) := do
+  let (some c, atoms) ← ToLinear.run (ToLinear.toLinearCnstr? e)
+    | return none
+  if atoms.size <= 1 then
+    return some (c, atoms)
+  else
+    let (atoms, perm) := sortExprs atoms
+    let c := c.applyPerm perm
+    return some (c, atoms)
 
 def toContextExpr (ctx : Array Expr) : Expr :=
   if h : 0 < ctx.size then
@@ -148,4 +185,4 @@ def toContextExpr (ctx : Array Expr) : Expr :=
   else
     RArray.toExpr (mkConst ``Nat) id (RArray.leaf (mkNatLit 0))
 
-end Lean.Meta.Linear.Nat
+namespace Lean.Meta.Linear.Nat

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

@@ -10,7 +10,8 @@ import Lean.Meta.Tactic.LinearArith.Nat.Basic
 namespace Lean.Meta.Linear.Nat
 
 def simpCnstrPos? (e : Expr) : MetaM (Option (Expr × Expr)) := do
-  let (some c, atoms) ← ToLinear.run (ToLinear.toLinearCnstr? e) | return none
+  let some (c, atoms) ← toLinearCnstr? e
+    | return none
   withAbstractAtoms atoms ``Nat fun atoms => do
     let lhs ← c.toArith atoms
     let c₁ := c.toPoly
@@ -67,7 +68,7 @@ def simpCnstr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
     simpCnstrPos? e
 
 def simpExpr? (e : Expr) : MetaM (Option (Expr × Expr)) := do
-  let (e, ctx) ← ToLinear.run (ToLinear.toLinearExpr e)
+  let (e, ctx) ← toLinearExpr e
   let p  := e.toPoly
   let p' := p.norm
   if p'.length < p.length then

src/Lean/Util.lean

@@ -35,3 +35,4 @@ import Lean.Util.SafeExponentiation
 import Lean.Util.NumObjs
 import Lean.Util.NumApps
 import Lean.Util.FVarSubset
+import Lean.Util.SortExprs

src/Lean/Util/SortExprs.lean

@@ -0,0 +1,23 @@
+/-
+Copyright (c) 2025 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Lean.Expr
+
+namespace Lean
+
+abbrev Perm := Std.HashMap Nat Nat
+
+/--
+Sorts the given expressions using `Expr.lt`, and creates a "permutation map" storing the new position of each expression.
+-/
+def sortExprs (es : Array Expr) : Array Expr × Perm :=
+  let es := es.mapIdx fun i e => (e, i)
+  let es := es.qsort fun (e₁, _) (e₂, _) => e₁.lt e₂
+  let (_, perm) := es.foldl (init := (0, Std.HashMap.empty)) fun (i, perm) (_, j) => (i+1, perm.insert j i)
+  let es := es.map (·.1)
+  (es, perm)
+
+end Lean

tests/lean/run/simp_int_arith.lean

@@ -171,18 +171,18 @@ fun x y z f =>
         id
           (Int.Linear.ExprCnstr.eq_true_of_isValid
             (Lean.RArray.branch 1 (Lean.RArray.leaf x)
-              (Lean.RArray.branch 2 (Lean.RArray.leaf x_1) (Lean.RArray.leaf z)))
+              (Lean.RArray.branch 2 (Lean.RArray.leaf z) (Lean.RArray.leaf x_1)))
             (Int.Linear.ExprCnstr.le
-              ((((((Int.Linear.Expr.var 0).add (Int.Linear.Expr.var 1)).add (Int.Linear.Expr.num 2)).add
-                        (Int.Linear.Expr.var 1)).add
-                    (Int.Linear.Expr.var 2)).add
-                (Int.Linear.Expr.var 2))
-              (((((((Int.Linear.Expr.var 1).add (Int.Linear.Expr.mulL 3 (Int.Linear.Expr.var 2))).add
+              ((((((Int.Linear.Expr.var 0).add (Int.Linear.Expr.var 2)).add (Int.Linear.Expr.num 2)).add
+                        (Int.Linear.Expr.var 2)).add
+                    (Int.Linear.Expr.var 1)).add
+                (Int.Linear.Expr.var 1))
+              (((((((Int.Linear.Expr.var 2).add (Int.Linear.Expr.mulL 3 (Int.Linear.Expr.var 1))).add
                                 (Int.Linear.Expr.num 1)).add
                             (Int.Linear.Expr.num 1)).add
                         (Int.Linear.Expr.var 0)).add
-                    (Int.Linear.Expr.var 1)).sub
-                (Int.Linear.Expr.var 2)))
+                    (Int.Linear.Expr.var 2)).sub
+                (Int.Linear.Expr.var 1)))
             (Eq.refl true)))
       (f y))
 -/
@@ -256,3 +256,12 @@ example (x : Int) : (11*x ≤ 10) ↔ (x ≤ 0) := by
 
 example (x : Int) : (11*x > 10) ↔ (x ≥ 1) := by
   simp +arith only
+
+example (x y : Int) : (2*x + y + y = 4) ↔ (y + x = 2) := by
+  simp +arith
+
+example (x y : Int) : (2*x + y + y ≤ 3) ↔ (y + x ≤ 1) := by
+  simp +arith
+
+example (f : Int → Int) (x y : Int) : f (2*x + y) = f (y + x + x) := by
+  simp +arith

tests/lean/run/simp_nat_arith.lean

@@ -0,0 +1,8 @@
+example (x y : Nat) : (2*x + y = 4) ↔ (y + x + x = 4) := by
+  simp +arith
+
+example (x y : Nat) : (2*x + y ≤ 3) ↔ (y + x + x ≤ 3) := by
+  simp +arith
+
+example (f : Nat → Nat) (x y : Nat) : f (2*x + y) = f (y + x + x) := by
+  simp +arith