1014 unfolding prod_alpha_def

  1014   unfolding prod_alpha_def

  1015 by (auto intro!: ext)

  1015   by (auto intro!: ext)

  1016 

  1016 

  1017 lemma Abs_lst1_fcb:

  1018   fixes x y :: "'a :: at_base"

  1019     and S T :: "'b :: fs"

  1020   assumes e: "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"

  1021   and f1: "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T \<Longrightarrow> atom x \<sharp> f x T"

  1022   and f2: "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T \<Longrightarrow> atom y \<sharp> f x T"

  1023   and p: "S = (atom x \<rightleftharpoons> atom y) \<bullet> T \<Longrightarrow> x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> S \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S"

  1024   and s: "sort_of (atom x) = sort_of (atom y)"

  1025   shows "f x T = f y S"

  1026   using e

  1027   apply(case_tac "atom x \<sharp> S")

  1028   apply(simp add: Abs1_eq_iff'[OF s s])

  1029   apply(elim conjE disjE)

  1030   apply(simp)

  1031   apply(rule trans)

  1032   apply(rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])

  1033   apply(rule fresh_star_supp_conv)

  1034   apply(simp add: supp_swap fresh_star_def s f1 f2)

  1035   apply(simp add: swap_commute p)

  1036   apply(simp add: Abs1_eq_iff[OF s s])

  1037   done

  1038 

  1039 lemma Abs_res_fcb:

  1040   fixes xs ys :: "('a :: at_base) set"

  1041     and S T :: "'b :: fs"

  1042   assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)"

  1043     and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T"

  1044     and f2: "\<And>x. supp T - atom ` xs = supp S - atom ` ys \<Longrightarrow> x \<in> atom ` ys \<Longrightarrow> x \<in> supp S \<Longrightarrow> x \<sharp> f xs T"

  1045     and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S

  1046                \<Longrightarrow> p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S \<Longrightarrow> p \<bullet> (f xs T) = f ys S"

  1047   shows "f xs T = f ys S"

  1048   using e apply -

  1049   apply(subst (asm) Abs_eq_res_set)

  1050   apply(subst (asm) Abs_eq_iff2)

  1051   apply(simp add: alphas)

  1052   apply(elim exE conjE)

  1053   apply(rule trans)

  1054   apply(rule_tac p="p" in supp_perm_eq[symmetric])

  1055   apply(rule fresh_star_supp_conv)

  1056   apply(drule fresh_star_perm_set_conv)

  1057   apply(rule finite_Diff)

  1058   apply(rule finite_supp)

  1059   apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T")

  1060   apply(metis Un_absorb2 fresh_star_Un)

  1061   apply(subst fresh_star_Un)

  1062   apply(rule conjI)

  1063   apply(simp add: fresh_star_def f1)

  1064   apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")

  1065   apply(simp add: fresh_star_def f2)

  1066   apply(blast)

  1067   apply(simp add: eqv)

  1068   done