Theory Specifications_with_bundle_mixins

theory Specifications_with_bundle_mixins
  imports "HOL-Combinatorics.Perm"
begin

locale involutory = opening permutation_syntax +
  fixes f :: ‹'a perm›
  assumes involutory: ‹⋀x. f ⟨$⟩ (f ⟨$⟩ x) = x›
begin

lemma
  ‹f * f = 1›
  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: ‹a ≠ b ⟹ ⟨a ↔ b⟩ ⟨$⟩ c ≠ c›
begin

lemma
  ‹card (UNIV :: 'a set) ≤ 2›
proof (rule ccontr)
  fix a :: 'a
  assume ‹¬ card (UNIV :: 'a set) ≤ 2›
  then have c0: ‹card (UNIV :: 'a set) ≥ 3›
    by simp
  then have [simp]: ‹finite (UNIV :: 'a set)›
    using card.infinite by fastforce
  from c0 card_Diff1_le [of UNIV a]
  have ca: ‹card (UNIV - {a}) ≥ 2›
    by simp
  then obtain b where ‹b ∈ (UNIV - {a})›
    by (metis all_not_in_conv card.empty card_2_iff' le_zero_eq)
  with ca card_Diff1_le [of ‹UNIV - {a}› b]
  have cb: ‹card (UNIV - {a, b}) ≥ 1› and ‹a ≠ b›
    by simp_all
  then obtain c where ‹c ∈ (UNIV - {a, b})›
    by (metis One_nat_def all_not_in_conv card.empty le_zero_eq nat.simps(3))
  then have ‹a ≠ c› ‹b ≠ c›
    by auto
  with swap_distinct [of a b c] show False
    by (simp add: ‹a ≠ b›) 
qed

end

thm swap_distinct (*syntax from permutation_syntax only present in class specification and initial block*)

end