Theory Group_Context

theory Group_Context
imports Main
(*  Title:      HOL/Isar_Examples/Group_Context.thy
    Author:     Makarius
*)

section ‹Some algebraic identities derived from group axioms -- theory context version›

theory Group_Context
  imports Main
begin

text ‹hypothetical group axiomatization›

context
  fixes prod :: "'a ⇒ 'a ⇒ 'a"  (infixl "⊙" 70)
    and one :: "'a"
    and inverse :: "'a ⇒ 'a"
  assumes assoc: "(x ⊙ y) ⊙ z = x ⊙ (y ⊙ z)"
    and left_one: "one ⊙ x = x"
    and left_inverse: "inverse x ⊙ x = one"
begin

text ‹some consequences›

lemma right_inverse: "x ⊙ inverse x = one"
proof -
  have "x ⊙ inverse x = one ⊙ (x ⊙ inverse x)"
    by (simp only: left_one)
  also have "… = one ⊙ x ⊙ inverse x"
    by (simp only: assoc)
  also have "… = inverse (inverse x) ⊙ inverse x ⊙ x ⊙ inverse x"
    by (simp only: left_inverse)
  also have "… = inverse (inverse x) ⊙ (inverse x ⊙ x) ⊙ inverse x"
    by (simp only: assoc)
  also have "… = inverse (inverse x) ⊙ one ⊙ inverse x"
    by (simp only: left_inverse)
  also have "… = inverse (inverse x) ⊙ (one ⊙ inverse x)"
    by (simp only: assoc)
  also have "… = inverse (inverse x) ⊙ inverse x"
    by (simp only: left_one)
  also have "… = one"
    by (simp only: left_inverse)
  finally show ?thesis .
qed

lemma right_one: "x ⊙ one = x"
proof -
  have "x ⊙ one = x ⊙ (inverse x ⊙ x)"
    by (simp only: left_inverse)
  also have "… = x ⊙ inverse x ⊙ x"
    by (simp only: assoc)
  also have "… = one ⊙ x"
    by (simp only: right_inverse)
  also have "… = x"
    by (simp only: left_one)
  finally show ?thesis .
qed

lemma one_equality:
  assumes eq: "e ⊙ x = x"
  shows "one = e"
proof -
  have "one = x ⊙ inverse x"
    by (simp only: right_inverse)
  also have "… = (e ⊙ x) ⊙ inverse x"
    by (simp only: eq)
  also have "… = e ⊙ (x ⊙ inverse x)"
    by (simp only: assoc)
  also have "… = e ⊙ one"
    by (simp only: right_inverse)
  also have "… = e"
    by (simp only: right_one)
  finally show ?thesis .
qed

lemma inverse_equality:
  assumes eq: "x' ⊙ x = one"
  shows "inverse x = x'"
proof -
  have "inverse x = one ⊙ inverse x"
    by (simp only: left_one)
  also have "… = (x' ⊙ x) ⊙ inverse x"
    by (simp only: eq)
  also have "… = x' ⊙ (x ⊙ inverse x)"
    by (simp only: assoc)
  also have "… = x' ⊙ one"
    by (simp only: right_inverse)
  also have "… = x'"
    by (simp only: right_one)
  finally show ?thesis .
qed

end

end