add support for Lean Theorem Prover

samples/Lean/binary.lean

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+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Module: algebra.binary
+Authors: Leonardo de Moura, Jeremy Avigad
+
+General properties of binary operations.
+-/
+
+import logic.eq
+open eq.ops
+
+namespace binary
+  section
+    variable {A : Type}
+    variables (op₁ : A → A → A) (inv : A → A) (one : A)
+
+    local notation a * b := op₁ a b
+    local notation a ⁻¹  := inv a
+    local notation 1     := one
+
+    definition commutative := ∀a b, a * b = b * a
+    definition associative := ∀a b c, (a * b) * c = a * (b * c)
+    definition left_identity := ∀a, 1 * a = a
+    definition right_identity := ∀a, a * 1 = a
+    definition left_inverse := ∀a, a⁻¹ * a = 1
+    definition right_inverse := ∀a, a * a⁻¹ = 1
+    definition left_cancelative := ∀a b c, a * b = a * c → b = c
+    definition right_cancelative := ∀a b c, a * b = c * b → a = c
+
+    definition inv_op_cancel_left := ∀a b, a⁻¹ * (a * b) = b
+    definition op_inv_cancel_left := ∀a b, a * (a⁻¹ * b) = b
+    definition inv_op_cancel_right := ∀a b, a * b⁻¹ * b =  a
+    definition op_inv_cancel_right := ∀a b, a * b * b⁻¹ = a
+
+    variable (op₂ : A → A → A)
+
+    local notation a + b := op₂ a b
+
+    definition left_distributive := ∀a b c, a * (b + c) = a * b + a * c
+    definition right_distributive := ∀a b c, (a + b) * c = a * c + b * c
+  end
+
+  context
+    variable {A : Type}
+    variable {f : A → A → A}
+    variable H_comm : commutative f
+    variable H_assoc : associative f
+    infixl `*` := f
+    theorem left_comm : ∀a b c, a*(b*c) = b*(a*c) :=
+    take a b c, calc
+      a*(b*c) = (a*b)*c  : H_assoc
+        ...   = (b*a)*c  : H_comm
+        ...   = b*(a*c)  : H_assoc
+
+    theorem right_comm : ∀a b c, (a*b)*c = (a*c)*b :=
+    take a b c, calc
+      (a*b)*c = a*(b*c) : H_assoc
+        ...   = a*(c*b) : H_comm
+        ...   = (a*c)*b : H_assoc
+  end
+
+  context
+    variable {A : Type}
+    variable {f : A → A → A}
+    variable H_assoc : associative f
+    infixl `*` := f
+    theorem assoc4helper (a b c d) : (a*b)*(c*d) = a*((b*c)*d) :=
+    calc
+      (a*b)*(c*d) = a*(b*(c*d)) : H_assoc
+              ... = a*((b*c)*d) : H_assoc
+  end
+
+end binary