theory Specifications_with_bundle_mixins
imports "HOL-Combinatorics.Perm"
begin
locale involutory = opening permutation_syntax +
fixes f :: \<open>'a perm\<close>
assumes involutory: \<open>\<And>x. f \<langle>$\<rangle> (f \<langle>$\<rangle> x) = x\<close>
begin
lemma
\<open>f * f = 1\<close>
using involutory
by (simp add: perm_eq_iff apply_sequence)
end
context involutory
begin
thm involutory (*syntax from permutation_syntax only present in locale specification and initial block*)
end
class at_most_two_elems = opening permutation_syntax +
assumes swap_distinct: \<open>a \<noteq> b \<Longrightarrow> \<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> c \<noteq> c\<close>
begin
lemma
\<open>card (UNIV :: 'a set) \<le> 2\<close>
proof (rule ccontr)
fix a :: 'a
assume \<open>\<not> card (UNIV :: 'a set) \<le> 2\<close>
then have c0: \<open>card (UNIV :: 'a set) \<ge> 3\<close>
by simp
then have [simp]: \<open>finite (UNIV :: 'a set)\<close>
using card.infinite by fastforce
from c0 card_Diff1_le [of UNIV a]
have ca: \<open>card (UNIV - {a}) \<ge> 2\<close>
by simp
then obtain b where \<open>b \<in> (UNIV - {a})\<close>
by (metis all_not_in_conv card.empty card_2_iff' le_zero_eq)
with ca card_Diff1_le [of \<open>UNIV - {a}\<close> b]
have cb: \<open>card (UNIV - {a, b}) \<ge> 1\<close> and \<open>a \<noteq> b\<close>
by simp_all
then obtain c where \<open>c \<in> (UNIV - {a, b})\<close>
by (metis One_nat_def all_not_in_conv card.empty le_zero_eq nat.simps(3))
then have \<open>a \<noteq> c\<close> \<open>b \<noteq> c\<close>
by auto
with swap_distinct [of a b c] show False
by (simp add: \<open>a \<noteq> b\<close>)
qed
end
thm swap_distinct (*syntax from permutation_syntax only present in class specification and initial block*)
end