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Try doing some basic maths questions in the Lean Theorem Prover. Functions, real numbers, equivalence relations and groups. Click on README.md and then on "Open in CoCalc with one click".

Project: Xena
Views: 18536
License: APACHE
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/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon

Free monoid over a given alphabet
-/
import algebra.group.to_additive

@[to_additive free_add_monoid]
def free_monoid (α) := list α

@[to_additive]
instance {α} : monoid (free_monoid α) :=
{ one := [],
  mul := λ x y, (x ++ y : list α),
  mul_one := by intros; apply list.append_nil,
  one_mul := by intros; refl,
  mul_assoc := by intros; apply list.append_assoc }

@[to_additive]
instance {α} : inhabited (free_monoid α) := ⟨1⟩

@[simp, to_additive free_add_monoid.zero_def]
lemma free_monoid.one_def {α} : (1 : free_monoid α) = [] := rfl

@[simp, to_additive free_add_monoid.add_def]
lemma free_monoid.mul_def {α} (xs ys : list α) : (xs * ys : free_monoid α) = (xs ++ ys : list α) :=
rfl