fix: equality propagation and simplification in the comm ring procedure (#8137)

This PR improves equality propagation (also known as theory combination)
and polynomial simplification for rings that do not implement the
`NoZeroNatDivisors` class. With these fixes, `grind` can now solve:
```lean
example [CommRing α] (a b c : α) (f : α → Nat)
  : a + b + c = 3 →
    a^2 + b^2 + c^2 = 5 →
    a^3 + b^3 + c^3 = 7 →
    f (a^4 + b^4) + f (9 - c^4) ≠ 1 := by
  grind +ring
```
This example uses the commutative ring procedure, the linear integer
arithmetic solver, and congruence closure.
For rings that implement `NoZeroNatDivisors`, a polynomial is now also
divided by the greatest common divisor (gcd) of its coefficients when it
is inserted into the basis.

src/Lean/Meta/Tactic/Grind/Arith/CommRing/EqCnstr.lean

@@ -46,12 +46,13 @@ Remark: if the current ring does not satisfy the property
 then the leading coefficient of the equation must also divide `k`
 -/
 def _root_.Lean.Grind.CommRing.Mon.findSimp? (k : Int) (m : Mon) : RingM (Option EqCnstr) := do
+  let checkCoeff ← checkCoeffDvd
   let noZeroDiv ← noZeroDivisors
   let rec go : Mon → RingM (Option EqCnstr)
     | .unit => return none
     | .mult pw m' => do
       for c in (← getRing).varToBasis[pw.x]! do
-        if noZeroDiv || (c.p.lc ∣ k) then
+        if !checkCoeff || noZeroDiv || (c.p.lc ∣ k) then
         if c.p.divides m then
           return some c
       go m'
@@ -156,7 +157,6 @@ def EqCnstr.simplifyAndCheck (c : EqCnstr) : RingM (Option EqCnstr) := do
 def addToBasisCore (c : EqCnstr) : RingM Unit := do
   let .add _ m _ := c.p | return ()
   let .mult pw _ := m | return ()
-  trace_goal[grind.ring.assert.basis] "{← c.denoteExpr}"
   modifyRing fun s => { s with
     varToBasis := s.varToBasis.modify pw.x (c :: ·)
     recheck := true
@@ -207,6 +207,9 @@ if the ring has a nonzero characteristic `p` and `gcd k p = 1`, then
 `k` has an inverse.
 
 It also handles the easy case where `k` is `-1`.
+
+Remark: if the ring implements the class `NoZeroNatDivisors`, then
+the coefficients are divided by the gcd of all coefficients.
 -/
 def EqCnstr.toMonic (c : EqCnstr) : RingM EqCnstr := do
   let k := c.p.lc
@@ -218,17 +221,20 @@ def EqCnstr.toMonic (c : EqCnstr) : RingM EqCnstr := do
       -- `α*k = 1 (mod p)`
       let α := if α < 0 then α % p else α
       return { c with p := c.p.mulConstC α p, h := .mul α c }
-    else
-      return c
-  else if k == -1 then
+  if (← noZeroDivisors) then
+    let g : Int := c.p.gcdCoeffs
+    if g != 1 then
+      let g := if k < 0 then -g else g
+      return { c with p := c.p.divConst g, h := .div g c }
+  if k < 0 then
     return { c with p := c.p.mulConst (-1), h := .mul (-1) c }
-  else
-    return c
+  return c
 
 def EqCnstr.addToBasisAfterSimp (c : EqCnstr) : RingM Unit := do
   let c ← c.toMonic
   c.simplifyBasis
   c.superposeWith
+  trace_goal[grind.ring.assert.basis] "{← c.denoteExpr}"
   addToBasisCore c
 
 def EqCnstr.addToBasis (c : EqCnstr) : RingM Unit := do
@@ -251,8 +257,10 @@ def DiseqCnstr.checkConstant (c : DiseqCnstr) : RingM Bool := do
     trace_goal[grind.ring.assert.trivial] "{← c.denoteExpr}"
   return true
 
-def DiseqCnstr.simplify (c : DiseqCnstr) : RingM DiseqCnstr := do
-  return { c with d := (← c.d.simplify) }
+def DiseqCnstr.simplify (c : DiseqCnstr) : RingM DiseqCnstr :=
+  withCheckCoeffDvd do
+    -- We must enable `checkCoeffDvd := true`. See comments at `PolyDerivation`.
+    return { c with d := (← c.d.simplify) }
 
 def saveDiseq (c : DiseqCnstr) : RingM Unit := do
   trace_goal[grind.ring.assert.store] "{← c.denoteExpr}"
@@ -316,9 +324,15 @@ private def propagateEqs : RingM Unit := do
   TODO: optimize
   -/
   let mut map : PropagateEqMap := {}
+  for a in (← getRing).vars do
+    if (← checkMaxSteps) then return ()
+    let some ra ← toRingExpr? a | unreachable!
+    map ← process map a ra
   for (a, ra) in (← getRing).denote do
     if (← checkMaxSteps) then return ()
-    let a := a.expr
+    map ← process map a.expr ra
+where
+  process (map : PropagateEqMap) (a : Expr) (ra : RingExpr) : RingM PropagateEqMap := do
     let d : PolyDerivation := .input (← ra.toPolyM)
     let d ← d.simplify
     let k := d.getMultiplier
@@ -329,10 +343,17 @@ private def propagateEqs : RingM Unit := do
         let p ← (ra.sub rb).toPolyM
         let d : PolyDerivation := .input p
         let d ← d.simplify
+        if d.getMultiplier != 1 then
+          unless (← noZeroDivisors) do
+            -- Given the multipiler `k' = d.getMultiplier`, we have that `k*(a - b) = 0`,
+            -- but we cannot eliminate the `k` because we don't have `noZeroDivisors`.
+            trace_goal[grind.ring.impEq] "skip: {← mkEq a b}, k: {k}, noZeroDivisors: false"
+            return map.insert (k, d.p) (a, ra)
         trace_goal[grind.ring.impEq] "{← mkEq a b}, {k}, {← p.denoteExpr}"
         propagateEq a b ra rb d
+      return map
     else
-      map := map.insert (k, d.p) (a, ra)
+      return map.insert (k, d.p) (a, ra)
 
 def checkRing : RingM Bool := do
   unless (← needCheck) do return false

src/Lean/Meta/Tactic/Grind/Arith/CommRing/Poly.lean

@@ -195,4 +195,20 @@ def Poly.checkNoUnitMon : Poly → Bool
   | .add _ .unit _ => false
   | .add _ _ p => p.checkNoUnitMon
 
+def Poly.gcdCoeffs : Poly → Nat
+  | .num k => k.natAbs
+  | .add k _ p => go p k.natAbs
+where
+  go (p : Poly) (acc : Nat) : Nat :=
+    if acc == 1 then
+      acc
+    else match p with
+      | .num k => Nat.gcd acc k.natAbs
+      | .add k _ p => go p (Nat.gcd acc k.natAbs)
+
+def Poly.divConst (p : Poly) (a : Int) : Poly :=
+  match p with
+  | .num k => .num (k / a)
+  | .add k m p => .add (k / a) m (divConst p a)
+
 end Lean.Grind.CommRing

src/Lean/Meta/Tactic/Grind/Arith/CommRing/Util.lean

@@ -21,14 +21,29 @@ def checkMaxSteps : GoalM Bool := do
 def incSteps : GoalM Unit := do
   modify' fun s => { s with steps := s.steps + 1 }
 
+structure RingM.Context where
+  ringId : Nat
+  /--
+  If `checkCoeffDvd` is `true`, then when using a polynomial `k*m - p`
+  to simplify `.. + k'*m*m_2 + ...`, the substitution is performed IF
+  - `k` divides `k'`, OR
+  - Ring implements `NoZeroNatDivisors`.
+
+  We need this check when simplifying disequalities. In this case, if we perform
+  the simplification anyway, we may end up with a proof that `k * q = 0`, but
+  we cannot deduce `q = 0` since the ring does not implement `NoZeroNatDivisors`
+  See comment at `PolyDerivation`.
+  -/
+  checkCoeffDvd : Bool := false
+
 /-- We don't want to keep carrying the `RingId` around. -/
-abbrev RingM := ReaderT Nat GoalM
+abbrev RingM := ReaderT RingM.Context GoalM
 
 abbrev RingM.run (ringId : Nat) (x : RingM α) : GoalM α :=
-  x ringId
+  x { ringId }
 
 abbrev getRingId : RingM Nat :=
-  read
+  return (← read).ringId
 
 def getRing : RingM Ring := do
   let s ← get'
@@ -42,6 +57,12 @@ def getRing : RingM Ring := do
   let ringId ← getRingId
   modify' fun s => { s with rings := s.rings.modify ringId f }
 
+abbrev withCheckCoeffDvd (x : RingM α) : RingM α :=
+  withReader (fun ctx => { ctx with checkCoeffDvd := true }) x
+
+def checkCoeffDvd : RingM Bool :=
+  return (← read).checkCoeffDvd
+
 def getTermRingId? (e : Expr) : GoalM (Option Nat) := do
   return (← get').exprToRingId.find? { expr := e }
 

tests/lean/run/grind_ring_2.lean

@@ -65,3 +65,64 @@ example [CommRing α] (a b c : α)
     a^3 + b^3 + c^3 = 7 →
     a^4 + b^4 + c^4 = 9 := by
   grind +ring
+
+/--
+info: [grind.ring.assert.basis] a + b + c + -3 = 0
+[grind.ring.assert.basis] 2 * b ^ 2 + 2 * (b * c) + 2 * c ^ 2 + -6 * b + -6 * c + 4 = 0
+[grind.ring.assert.basis] 6 * c ^ 3 + -18 * c ^ 2 + 12 * c + 4 = 0
+-/
+#guard_msgs (info) in
+example [CommRing α] (a b c : α)
+  : a + b + c = 3 →
+    a^2 + b^2 + c^2 = 5 →
+    a^3 + b^3 + c^3 = 7 →
+    a^4 + b^4 = 9 - c^4 := by
+  set_option trace.grind.ring.assert.basis true in
+  grind +ring
+
+/--
+info: [grind.ring.assert.basis] a + b + c + -3 = 0
+[grind.ring.assert.basis] b ^ 2 + b * c + c ^ 2 + -3 * b + -3 * c + 2 = 0
+[grind.ring.assert.basis] 3 * c ^ 3 + -9 * c ^ 2 + 6 * c + 2 = 0
+-/
+#guard_msgs (info) in
+example [CommRing α] [NoZeroNatDivisors α] (a b c : α)
+  : a + b + c = 3 →
+    a^2 + b^2 + c^2 = 5 →
+    a^3 + b^3 + c^3 = 7 →
+    a^4 + b^4 = 9 - c^4 := by
+  set_option trace.grind.ring.assert.basis true in
+  grind +ring
+
+example [CommRing α] (a b : α) (f : α → Nat) : a - b = 0 → f a = f b := by
+  grind +ring
+
+example (a b : BitVec 8) (f : BitVec 8 → Nat) : a - b = 0 → f a = f b := by
+  grind +ring
+
+example (a b c : BitVec 8) (f : BitVec 8 → Nat) : c = 255 → - a + b - 1 = c → f a = f b := by
+  grind +ring
+
+example (a b c : BitVec 8) (f : BitVec 8 → Nat) : c = 255 → - a + b - 1 = c → f (2*a) = f (b + a) := by
+  grind +ring
+
+/-- info: [grind.ring.impEq] skip: b = a, k: 2, noZeroDivisors: false -/
+#guard_msgs (info) in
+example (a b c : BitVec 8) (f : BitVec 8 → Nat) : 2*a = 1 → 2*b = 1 → f (a) = f (b) := by
+  set_option trace.grind.ring.impEq true in
+  fail_if_success grind +ring
+  sorry
+
+example (a b c : Int) (f : Int → Nat)
+  : a + b + c = 3 →
+    a^2 + b^2 + c^2 = 5 →
+    a^3 + b^3 + c^3 = 7 →
+    f (a^4 + b^4) + f (9 - c^4) ≠ 1 := by
+  grind +ring
+
+example [CommRing α] (a b c : α) (f : α → Nat)
+  : a + b + c = 3 →
+    a^2 + b^2 + c^2 = 5 →
+    a^3 + b^3 + c^3 = 7 →
+    f (a^4 + b^4) + f (9 - c^4) ≠ 1 := by
+  grind +ring