chore: add failing grind tests for noncommutative/non-negation rings (#8396)

Author: kim-em

tests/lean/grind/ring_normalization.lean

@@ -0,0 +1,37 @@
+-- Tests for `grind` as a ring normalization tactic, when only `Semiring`, `CommSemiring`, or `Ring` is available.
+-- Note that in these cases we *do not* support hypotheses: there's no (good) analogue of Grobner bases here.
+
+open Lean.Grind
+
+section Semiring
+
+variable (R : Type u) [Semiring R]
+
+example (a b c : R) : a * (b + c) = a * c + a * b := by grind
+example (a b : R) : (a + b)^2 = a^2 + a * b + b * a + b^2 := by grind
+example (a b : R) : (a + 2 * b)^2 = a^2 + 2 * a * b + 2 * b * a + 4 * b^2 := by grind
+example (a b : R) : (a + 2 * b)^2 = a^2 + 2 * a * b + b * 2 * a + 4 * b^2 := by grind
+
+end Semiring
+
+section CommSemiring
+
+variable (R : Type u) [Semiring R]
+
+example (a b c : R) : a * (b + c) = a * c + b * a := by grind
+example (a b : R) : (a + b)^2 = a^2 + 2 * a * b + b^2 := by grind
+example (a b : R) : (a + 2 * b)^2 = a^2 + 4 * a * b + 4 * b^2 := by grind
+example (a b : R) : (a + 2 * b)^2 = 4 * b^2 + b * 4 * a + a^2 := by grind
+
+end CommSemiring
+
+section Ring
+
+variable (R : Type u) [Ring R]
+
+example (a b c : R) : a * (b - c) = - a * c + a * b := by grind
+example (a b : R) : (a - b)^2 = a^2 - a * b - b * a + b^2 := by grind
+example (a b : R) : (a + 2 * b)^2 = a^2 - 2 * a * b - 2 * b * a + 4 * b^2 := by grind
+example (a b : R) : (a + 2 * b)^2 = a^2 - 2 * a * b + -b * (-2) * -a + 4 * b^2 := by grind
+
+end Ring