4 

     5 lemma Abs_lst_fcb2:

     6   fixes as bs :: "atom list"

     7     and x y :: "'b :: fs"

     8     and c::"'c::fs"

     9   assumes eq: "[as]lst. x = [bs]lst. y"

    10   and fcb1: "(set as) \<sharp>* f as x c"

    11   and fresh1: "set as \<sharp>* c"

    12   and fresh2: "set bs \<sharp>* c"

    13   and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"

    14   and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"

    15   shows "f as x c = f bs y c"

    16 proof -

    17   have "supp (as, x, c) supports (f as x c)"

    18     unfolding  supports_def fresh_def[symmetric]

    19     by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)

    20   then have fin1: "finite (supp (f as x c))"

    21     by (auto intro: supports_finite simp add: finite_supp)

    22   have "supp (bs, y, c) supports (f bs y c)"

    23     unfolding  supports_def fresh_def[symmetric]

    24     by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)

    25   then have fin2: "finite (supp (f bs y c))"

    26     by (auto intro: supports_finite simp add: finite_supp)

    27   obtain q::"perm" where 

    28     fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 

    29     fr2: "supp q \<sharp>* Abs_lst as x" and 

    30     inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"

    31     using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  

    32       fin1 fin2

    33     by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)

    34   have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp

    35   also have "\<dots> = Abs_lst as x"

    36     by (simp only: fr2 perm_supp_eq)

    37   finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp

    38   then obtain r::perm where 

    39     qq1: "q \<bullet> x = r \<bullet> y" and 

    40     qq2: "q \<bullet> as = r \<bullet> bs" and 

    41     qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"

    42     apply(drule_tac sym)

    43     apply(simp only: Abs_eq_iff2 alphas)

    44     apply(erule exE)

    45     apply(erule conjE)+

    46     apply(drule_tac x="p" in meta_spec)

    47     apply(simp add: set_eqvt)

    48     apply(blast)

    49     done

    50   have "(set as) \<sharp>* f as x c" by (rule fcb1)

    51   then have "q \<bullet> ((set as) \<sharp>* f as x c)"

    52     by (simp add: permute_bool_def)

    53   then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"

    54     apply(simp add: fresh_star_eqvt set_eqvt)

    55     apply(subst (asm) perm1)

    56     using inc fresh1 fr1

    57     apply(auto simp add: fresh_star_def fresh_Pair)

    58     done

    59   then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp

    60   then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"

    61     apply(simp add: fresh_star_eqvt set_eqvt)

    62     apply(subst (asm) perm2[symmetric])

    63     using qq3 fresh2 fr1

    64     apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)

    65     done

    66   then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)

    67   have "f as x c = q \<bullet> (f as x c)"

    68     apply(rule perm_supp_eq[symmetric])

    69     using inc fcb1 fr1 by (auto simp add: fresh_star_def)

    70   also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 

    71     apply(rule perm1)

    72     using inc fresh1 fr1 by (auto simp add: fresh_star_def)

    73   also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp

    74   also have "\<dots> = r \<bullet> (f bs y c)"

    75     apply(rule perm2[symmetric])

    76     using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)

    77   also have "... = f bs y c"

    78     apply(rule perm_supp_eq)

    79     using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)

    80   finally show ?thesis by simp

    81 qed

    82 

    83 lemma Abs_lst1_fcb2:

    84   fixes a b :: "atom"

    85     and x y :: "'b :: fs"

    86     and c::"'c :: fs"

    87   assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"

    88   and fcb1: "a \<sharp> f a x c"

    89   and fresh: "{a, b} \<sharp>* c"

    90   and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"

    91   and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"

    92   shows "f a x c = f b y c"

    93 using e

    94 apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])

    95 apply(simp_all)

    96 using fcb1 fresh perm1 perm2

    97 apply(simp_all add: fresh_star_def)

    98 done

    99 

   100 lemma Abs_lst1_fcb2':

   101   fixes a b :: "'a::at"

   102     and x y :: "'b :: fs"

   103     and c::"'c :: fs"

   104   assumes e: "(Abs_lst [atom a] x) = (Abs_lst [atom b] y)"

   105   and fcb1: "atom a \<sharp> f a x c"

   106   and fresh: "{atom a, atom b} \<sharp>* c"

   107   and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"

   108   and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"

   109   shows "f a x c = f b y c"

   110 using e

   111 apply(drule_tac Abs_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])

   112 using  fcb1 fresh perm1 perm2

   113 apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)

   114 done