Quelle Group_Notepad.thy Sprache: Isabelle

(*  Title:      HOL/Isar_Examples/Group_Notepad.thy
    Author:     Makarius
*)

section \<open>Some algebraic identities derived from group axioms -- proof notepad version\<close>

theory Group_Notepad
  imports Main
begin

notepad
begin
  txt \<open>hypothetical group axiomatization\<close>

  fix prod :: "'a \ 'a \ 'a" (infixl \\\ 70)
    and one :: "'a"
    and inverse :: "'a \ 'a"
  assume assoc: "(x \ y) \ z = x \ (y \ z)"
    and left_one: "one \ x = x"
    and left_inverse: "inverse x \ x = one"
    for x y z

  txt \<open>some consequences\<close>

  have right_inverse: "x \ inverse x = one" for x
  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

  have right_one: "x \ one = x" for 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

  have one_equality: "one = e" if eq: "e \ x = x" for e x
  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

  have inverse_equality: "inverse x = x'" if eq: "x' \ x = one" for 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

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