feat: Int.subNatNat in grind (#11615)

This PR adds a normalization rule for `Int.subNatNat` to `grind`.

Closes #11543

src/Init/Grind/Norm.lean

@@ -149,6 +149,9 @@ theorem smul_nat_eq_mul {α} [Semiring α] (n : Nat) (a : α) : n • a = NatCas
 theorem smul_int_eq_mul {α} [Ring α] (i : Int) (a : α) : i • a = Int.cast i * a := by
   rw [Ring.zsmul_eq_intCast_mul]
 
+theorem Int.subNatNat_eq (a b : Nat) : Int.subNatNat a b = NatCast.natCast a - NatCast.natCast b := by
+  apply Int.subNatNat_eq_coe
+
 -- Remark: for additional `grind` simprocs, check `Lean/Meta/Tactic/Grind`
 init_grind_norm
   /- Pre theorems -/
@@ -188,7 +191,7 @@ init_grind_norm
   natCast_div natCast_mod natCast_id
   natCast_add natCast_mul natCast_pow
   Int.one_pow
-  Int.pow_zero Int.pow_one
+  Int.pow_zero Int.pow_one Int.subNatNat_eq
   -- Int op folding
   Int.add_def Int.mul_def Int.ofNat_eq_coe
   Int.Linear.sub_fold Int.Linear.neg_fold

tests/lean/run/gring_11543.lean

@@ -0,0 +1,175 @@
+section Mathlib.Algebra.Group.Defs
+
+class Monoid (M : Type u) extends Mul M where
+
+class AddGroup (G : Type u) extends Add G, Neg G, Sub G where
+
+theorem add_sub_assoc {G : Type} [AddGroup G] (a b c : G) : a + b - c = a + (b - c) := by
+  sorry
+
+end Mathlib.Algebra.Group.Defs
+
+abbrev Function.swap {φ : α → β → Sort u₃} (f : ∀ x y, φ x y) : ∀ y x, φ x y := fun y x => f x y
+
+section Mathlib.Algebra.Group.Basic
+
+@[simp]
+theorem add_sub_cancel {G} [AddGroup G](a b : G) : a + (b - a) = b := by
+  sorry
+
+end Mathlib.Algebra.Group.Basic
+
+section Mathlib.Algebra.Order.Monoid.Unbundled.Defs
+
+open Function
+
+variable (M N : Type) (μ : M → N → N) (r : N → N → Prop)
+
+def Covariant : Prop :=
+  ∀ (m) {n₁ n₂}, r n₁ n₂ → r (μ m n₁) (μ m n₂)
+
+class CovariantClass : Prop where
+  protected elim : Covariant M N μ r
+
+abbrev AddLeftMono [Add M] [LE M] : Prop :=
+  CovariantClass M M (· + ·) (· ≤ ·)
+
+abbrev AddRightMono [Add M] [LE M] : Prop :=
+  CovariantClass M M (swap (· + ·)) (· ≤ ·)
+
+instance covariant_swap_add_of_covariant_add [Add N]
+    [CovariantClass N N (· + ·) r] : CovariantClass N N (swap (· + ·)) r where
+  elim := sorry
+
+end Mathlib.Algebra.Order.Monoid.Unbundled.Defs
+
+section Mathlib.Algebra.Order.Monoid.Unbundled.Basic
+
+instance Int.instAddLeftMono : AddLeftMono Int where
+  elim := sorry
+
+end Mathlib.Algebra.Order.Monoid.Unbundled.Basic
+
+section Mathlib.Algebra.Order.Sub.Defs
+
+class OrderedSub (α : Type) [LE α] [Add α] [Sub α] : Prop where
+
+@[simp]
+theorem tsub_le_iff_right {α : Type}  [LE α] [Add α] [Sub α] [OrderedSub α] {a b c : α} :
+    a - b ≤ c ↔ a ≤ c + b := sorry
+
+end Mathlib.Algebra.Order.Sub.Defs
+
+section Mathlib.Algebra.Order.Monoid.Unbundled.Basic
+
+instance AddGroup.toOrderedSub {α : Type} [AddGroup α] [LE α]
+    [AddRightMono α] : OrderedSub α := sorry
+
+end Mathlib.Algebra.Order.Monoid.Unbundled.Basic
+
+section Mathlib.Data.Nat.Cast.Defs
+
+class AddMonoidWithOne (R : Type) extends NatCast R, Add R where
+
+theorem Nat.cast_add [AddMonoidWithOne R] (m n : Nat) : ((m + n : Nat) : R) = m + n := sorry
+
+end Mathlib.Data.Nat.Cast.Defs
+
+section Mathlib.Data.Int.Cast.Defs
+
+class AddGroupWithOne (R : Type) extends IntCast R, AddMonoidWithOne R, AddGroup R where
+
+end Mathlib.Data.Int.Cast.Defs
+
+section Mathlib.Data.Int.Cast.Basic
+
+namespace Int
+
+variable {R : Type} [AddGroupWithOne R]
+
+@[norm_cast]
+theorem cast_subNatNat (m n) : ((Int.subNatNat m n : Int) : R) = m - n := sorry
+
+end Int
+
+end Mathlib.Data.Int.Cast.Basic
+
+section Mathlib.Algebra.Group.Int.Defs
+
+namespace Int
+
+instance instCommMonoid : Monoid Int where
+
+instance instAddCommGroup : AddGroup Int where
+
+instance instCommSemigroup : Mul Int := by infer_instance
+
+end Int
+
+end Mathlib.Algebra.Group.Int.Defs
+
+section Mathlib.Algebra.Ring.Defs
+
+class NonAssocSemiring (α : Type) extends Add α, Mul α, Zero α
+
+class NonAssocRing (α : Type) extends AddGroup α, NonAssocSemiring α
+
+class Ring (R : Type) extends NonAssocSemiring R, AddGroupWithOne R
+
+section NonAssocRing
+
+variable {α : Type} [NonAssocRing α]
+
+theorem mul_sub_left_distrib (a b c : α) : a * (b - c) = a * b - a * c := sorry
+
+end NonAssocRing
+
+section Ring
+
+variable {α : Type} [Ring α]
+
+instance (priority := 100) Ring.toNonAssocRing : NonAssocRing α :=
+  { ‹Ring α› with }
+
+end Ring
+
+class CommRing (α : Type) extends Ring α, Monoid α
+
+end Mathlib.Algebra.Ring.Defs
+
+section Mathlib.Algebra.Ring.Int.Defs
+
+namespace Int
+
+instance instCommRing : CommRing Int where
+  intCast := (·)
+
+end Int
+
+end Mathlib.Algebra.Ring.Int.Defs
+
+example {a b : Nat}
+  (this : (b : Int) + ↑b * (↑a - ↑b) ≤ (↑b + (↑a - ↑b))) :
+    a * b ≤ a + b * b := by
+  simp [mul_sub_left_distrib, ← add_sub_assoc] at this
+  norm_cast at this
+  grind
+
+/--
+info: Try this:
+  [apply] simp only [mul_sub_left_distrib, ← add_sub_assoc, add_sub_cancel, tsub_le_iff_right] at this
+-/
+#guard_msgs in
+example {a b : Nat}
+  (this : (b : Int) + ↑b * (↑a - ↑b) ≤ (↑b + (↑a - ↑b))) :
+    a * b ≤ a + b * b := by
+  simp? [mul_sub_left_distrib, ← add_sub_assoc] at this
+  norm_cast at this
+  grind
+
+example {a b : Nat}
+  (this : (b : Int) + ↑b * (↑a - ↑b) ≤ (↑b + (↑a - ↑b))) :
+    a * b ≤ a + b * b := by
+  simp only [mul_sub_left_distrib, ← add_sub_assoc, add_sub_cancel, tsub_le_iff_right] at this
+  norm_cast at this
+  grind