(* Title: HOL/Isar_Examples/Group_Context.thy
Author: Makarius
*)
header {* Some algebraic identities derived from group axioms -- theory context version *}
theory Group_Context
imports Main
begin
text {* hypothetical group axiomatization *}
context
fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> '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 "\<dots> = one ** x ** inverse x"
by (simp only: assoc)
also have "\<dots> = inverse (inverse x) ** inverse x ** x ** inverse x"
by (simp only: left_inverse)
also have "\<dots> = inverse (inverse x) ** (inverse x ** x) ** inverse x"
by (simp only: assoc)
also have "\<dots> = inverse (inverse x) ** one ** inverse x"
by (simp only: left_inverse)
also have "\<dots> = inverse (inverse x) ** (one ** inverse x)"
by (simp only: assoc)
also have "\<dots> = inverse (inverse x) ** inverse x"
by (simp only: left_one)
also have "\<dots> = one"
by (simp only: left_inverse)
finally show "x ** inverse x = one" .
qed
lemma right_one: "x ** one = x"
proof -
have "x ** one = x ** (inverse x ** x)"
by (simp only: left_inverse)
also have "\<dots> = x ** inverse x ** x"
by (simp only: assoc)
also have "\<dots> = one ** x"
by (simp only: right_inverse)
also have "\<dots> = x"
by (simp only: left_one)
finally show "x ** one = x" .
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 "\<dots> = (e ** x) ** inverse x"
by (simp only: eq)
also have "\<dots> = e ** (x ** inverse x)"
by (simp only: assoc)
also have "\<dots> = e ** one"
by (simp only: right_inverse)
also have "\<dots> = e"
by (simp only: right_one)
finally show "one = e" .
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 "\<dots> = (x' ** x) ** inverse x"
by (simp only: eq)
also have "\<dots> = x' ** (x ** inverse x)"
by (simp only: assoc)
also have "\<dots> = x' ** one"
by (simp only: right_inverse)
also have "\<dots> = x'"
by (simp only: right_one)
finally show "inverse x = x'" .
qed
end
end