From 2090e9124ceca157183059ff08be5cad07b1afa4 Mon Sep 17 00:00:00 2001
From: Leonardo de Moura <leonardo@microsoft.com>
Date: Tue, 1 Dec 2015 12:35:36 -0700
Subject: [PATCH] test(tests/lean/run): add more tests from algebra

---
 tests/lean/run/blast_ematch8.lean | 49 +++++++++++++++++++++++++++++++
 1 file changed, 49 insertions(+)
 create mode 100644 tests/lean/run/blast_ematch8.lean

diff --git a/tests/lean/run/blast_ematch8.lean b/tests/lean/run/blast_ematch8.lean
new file mode 100644
index 000000000..386951248
--- /dev/null
+++ b/tests/lean/run/blast_ematch8.lean
@@ -0,0 +1,49 @@
+import algebra.group
+open algebra
+
+variables {A : Type}
+variables [s : group A]
+include s
+
+set_option blast.ematch true
+set_option blast.subst  false
+set_option blast.simp   false
+attribute inv_inv mul.left_inv mul.assoc one_mul mul_one [forward]
+
+theorem mul.right_inv (a : A) : a * a⁻¹ = 1 :=
+calc
+    a * a⁻¹ = (a⁻¹)⁻¹ * a⁻¹ : by blast
+        ... = 1             : by blast
+
+theorem mul.right_inv₂ (a : A) : a * a⁻¹ = 1 :=
+by blast
+
+theorem mul_inv_cancel_left (a b : A) : a * (a⁻¹ * b) = b :=
+calc
+    a * (a⁻¹ * b) = a * a⁻¹ * b : by blast
+      ... = 1 * b               : by blast
+      ... = b                   : by blast
+
+theorem mul_inv_cancel_left₂ (a b : A) : a * (a⁻¹ * b) = b :=
+by blast
+
+theorem mul_inv (a b : A) : (a * b)⁻¹ = b⁻¹ * a⁻¹ :=
+inv_eq_of_mul_eq_one
+  (calc
+    a * b * (b⁻¹ * a⁻¹) = a * (b * (b⁻¹ * a⁻¹)) : by blast
+      ... = 1                                   : by blast)
+
+theorem eq_of_mul_inv_eq_one {a b : A} (H : a * b⁻¹ = 1) : a = b :=
+calc
+    a = a * b⁻¹ * b : by blast
+      ... = 1 * b   : by blast
+      ... = b       : by blast
+
+
+-- This is another theorem that can be easily proved using superposition,
+-- but cannot to be proved using E-matching.
+-- To prove it using E-matching, we must provide the following auxiliary step using calc.
+theorem eq_of_mul_inv_eq_one₂ {a b : A} (H : a * b⁻¹ = 1) : a = b :=
+calc
+  a = a * b⁻¹ * b : by blast
+... = b           : by blast