fix: missing nonlinear / and % in grind cutsat (#10020)

This PR fixes a missing case for PR #10010.

src/Init/Data/Int/Linear.lean

@@ -2193,6 +2193,9 @@ theorem mul_eq_zero_right (a b : Int) (h : b = 0) : a*b = 0 := by simp [*]
 theorem div_eq (a b k : Int) (h : b = k) : a / b = a / k := by simp [*]
 theorem mod_eq (a b k : Int) (h : b = k) : a % b = a % k := by simp [*]
 
+theorem div_eq' (a b b' k : Int) (h₁ : b = b') (h₂ : k == a/b') : a / b = k := by simp_all
+theorem mod_eq' (a b b' k : Int) (h₁ : b = b') (h₂ : k == a%b') : a % b = k := by simp_all
+
 end Int.Linear
 
 theorem Int.not_le_eq (a b : Int) : (¬a ≤ b) = (b + 1 ≤ a) := by

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

@@ -224,10 +224,13 @@ private def propagateNonlinearDiv (x : Var) : GoalM Bool := do
   let_expr HDiv.hDiv _ _ _ i a b := e | return false
   unless (← isInstHDivInt i) do return false
   let some (k, c) ← isExprEqConst? b | return false
-  let div' ← shareCommon (mkIntDiv a (mkIntLit k))
-  internalize div' (← getGeneration e)
-  let y ← mkVar div'
-  let c' := { p := .add 1 x (.add (-1) y (.num 0)), h := .div k y c : EqCnstr }
+  let c' ← if let some a ← getIntValue? a then
+    pure { p := .add 1 x (.num (-(a/k))), h := .div k none c : EqCnstr }
+  else
+    let div' ← shareCommon (mkIntDiv a (mkIntLit k))
+    internalize div' (← getGeneration e)
+    let y ← mkVar div'
+    pure { p := .add 1 x (.add (-1) y (.num 0)), h := .div k (some y) c : EqCnstr }
   c'.assert
   return true
 
@@ -236,10 +239,13 @@ private def propagateNonlinearMod (x : Var) : GoalM Bool := do
   let_expr HMod.hMod _ _ _ i a b := e | return false
   unless (← isInstHModInt i) do return false
   let some (k, c) ← isExprEqConst? b | return false
-  let mod' ← shareCommon (mkIntMod a (mkIntLit k))
-  internalize mod' (← getGeneration e)
-  let y ← mkVar mod'
-  let c' := { p := .add 1 x (.add (-1) y (.num 0)), h := .mod k y c : EqCnstr }
+  let c' ← if let some a ← getIntValue? a then
+    pure { p := .add 1 x (.num (-(a%k))), h := .mod k none c : EqCnstr }
+  else
+    let mod' ← shareCommon (mkIntMod a (mkIntLit k))
+    internalize mod' (← getGeneration e)
+    let y ← mkVar mod'
+    pure { p := .add 1 x (.add (-1) y (.num 0)), h := .mod k (some y) c : EqCnstr }
   c'.assert
   return true
 

src/Lean/Meta/Tactic/Grind/Arith/Cutsat/Proof.lean

@@ -281,20 +281,34 @@ partial def EqCnstr.toExprProof (c' : EqCnstr) : ProofM Expr := caching c' do
   | .mul a? cs =>
     let .add _ x _ := c'.p | c'.throwUnexpected
     mkMulEqProof x a? cs c'
-  | .div k y c =>
+  | .div k y? c =>
     let .add _ x _ := c'.p | c'.throwUnexpected
     let_expr HDiv.hDiv _ _ _ _ a b := (← getCurrVars)[x]! | c'.throwUnexpected
     let bVar ← getVarOf b
     let h := mkApp6 (mkConst ``Int.Linear.var_eq) (← getContext) (← mkVarDecl bVar) (toExpr k) (← mkPolyDecl c.p) eagerReflBoolTrue (← c.toExprProof)
-    let h := mkApp4 (mkConst ``Int.Linear.div_eq) a b (toExpr k) h
-    return mkApp6 (mkConst ``Int.Linear.of_var_eq_var) (← getContext) (← mkVarDecl x) (← mkVarDecl y) (← mkPolyDecl c'.p) eagerReflBoolTrue h
-  | .mod k y c =>
+    if let some y := y? then
+      let h := mkApp4 (mkConst ``Int.Linear.div_eq) a b (toExpr k) h
+      return mkApp6 (mkConst ``Int.Linear.of_var_eq_var) (← getContext) (← mkVarDecl x) (← mkVarDecl y) (← mkPolyDecl c'.p) eagerReflBoolTrue h
+    else
+      let b' := k
+      let some aVal ← getIntValue? a | unreachable!
+      let k := aVal / b'
+      let h := mkApp6 (mkConst ``Int.Linear.div_eq') a b (toExpr b') (toExpr k) h eagerReflBoolTrue
+      return mkApp6 (mkConst ``Int.Linear.of_var_eq) (← getContext) (← mkVarDecl x) (toExpr k) (← mkPolyDecl c'.p) eagerReflBoolTrue h
+  | .mod k y? c =>
     let .add _ x _ := c'.p | c'.throwUnexpected
     let_expr HMod.hMod _ _ _ _ a b := (← getCurrVars)[x]! | c'.throwUnexpected
     let bVar ← getVarOf b
     let h := mkApp6 (mkConst ``Int.Linear.var_eq) (← getContext) (← mkVarDecl bVar) (toExpr k) (← mkPolyDecl c.p) eagerReflBoolTrue (← c.toExprProof)
-    let h := mkApp4 (mkConst ``Int.Linear.mod_eq) a b (toExpr k) h
-    return mkApp6 (mkConst ``Int.Linear.of_var_eq_var) (← getContext) (← mkVarDecl x) (← mkVarDecl y) (← mkPolyDecl c'.p) eagerReflBoolTrue h
+    if let some y := y? then
+      let h := mkApp4 (mkConst ``Int.Linear.mod_eq) a b (toExpr k) h
+      return mkApp6 (mkConst ``Int.Linear.of_var_eq_var) (← getContext) (← mkVarDecl x) (← mkVarDecl y) (← mkPolyDecl c'.p) eagerReflBoolTrue h
+    else
+      let b' := k
+      let some aVal ← getIntValue? a | unreachable!
+      let k := aVal % b'
+      let h := mkApp6 (mkConst ``Int.Linear.mod_eq') a b (toExpr b') (toExpr k) h eagerReflBoolTrue
+      return mkApp6 (mkConst ``Int.Linear.of_var_eq) (← getContext) (← mkVarDecl x) (toExpr k) (← mkPolyDecl c'.p) eagerReflBoolTrue h
 
 partial def mkMulEqProof (x : Var) (a? : Option Expr) (cs : Array (Expr × Int × EqCnstr)) (c' : EqCnstr) : ProofM Expr := do
   let h ← go (← getCurrVars)[x]!

src/Lean/Meta/Tactic/Grind/Arith/Cutsat/Types.lean

@@ -103,8 +103,18 @@ inductive EqCnstrProof where
   | defnCommRing (e : Expr) (p : Poly) (re : CommRing.RingExpr) (rp : CommRing.Poly) (p' : Poly)
   | defnNatCommRing (h : Expr) (x : Var) (e' : Int.Linear.Expr) (p : Poly) (re : CommRing.RingExpr) (rp : CommRing.Poly) (p' : Poly)
   | mul (a? : Option Expr) (cs : Array (Expr × Int × EqCnstr))
-  | div (k : Int) (y : Var) (c : EqCnstr)
-  | mod (k : Int) (y : Var) (c : EqCnstr)
+  | /--
+    Linearization proof for `/`
+    - If `?y = some y`, then it is a proof for `a / b = y / k` where `c` is a proof that `b = k`
+    - If `?y = none`, then it is a proof for `a / b = a/k` where `c` is a proof that `b = k`. `a` is a numeral in this case.
+    -/
+    div (k : Int) (y? : Option Var) (c : EqCnstr)
+  | /--
+    Linearization proof for `%`
+    - If `?y = some y`, then it is a proof for `a % b = y%k` where `c` is a proof that `b = k`
+    - If `?y = none`, then it is a proof for `a % b = a%k` where `c` is a proof that `b = k`. `a` is a numeral in this case.
+    -/
+    mod (k : Int) (y? : Option Var) (c : EqCnstr)
 
 /-- A divisibility constraint and its justification/proof. -/
 structure DvdCnstr where

tests/lean/run/grind_linearize.lean

@@ -63,3 +63,33 @@ example (a b c d : Nat) : b > 0 → d = 1 → b = d + 1 → a % b = 1 → a = 2
 
 example (a b c d : Nat) : b > 1 → d = 1 → b ≤ d + 1 → a % b = 1 → a = 2 * c → False := by
   grind
+
+example (b : Int) : 4 % b = 1 → b = 2 → False := by
+  grind
+
+example (b : Int) : b = 2 → 4 % b = 1 → False := by
+  grind
+
+example (b : Nat) : 4 % b = 1 → b = 2 → False := by
+  grind
+
+example (b : Int) : 4 / b = 1 → b = 2 → False := by
+  grind
+
+example (b : Nat) : 4 / b = 1 → b = 2 → False := by
+  grind
+
+example (b : Int) : 4 % b = 1 → b ≤ 2 → 2 ≤ b → False := by
+  grind
+
+example (b : Int) : b ≤ 2 → 2 ≤ b → 4 % b = 1 → False := by
+  grind
+
+example (b : Nat) : 4 % b = 1 → b ≤ 2 → 2 ≤ b → False := by
+  grind
+
+example (b : Int) : 4 / b = 1 → b ≤ 2 → 2 ≤ b → False := by
+  grind
+
+example (b : Nat) : 4 / b = 1 → b ≤ 2 → 2 ≤ b → False := by
+  grind