--- a/Nominal/Ex/Classical.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Ex/Classical.thy Tue Jul 05 04:18:45 2011 +0200
@@ -47,269 +47,6 @@
thm trm.supp
thm trm.supp[simplified]
-lemma Abs_set_fcb2:
- fixes as bs :: "atom set"
- and x y :: "'b :: fs"
- and c::"'c::fs"
- assumes eq: "[as]set. x = [bs]set. y"
- and fin: "finite as" "finite bs"
- and fcb1: "as \<sharp>* f as x c"
- and fresh1: "as \<sharp>* c"
- and fresh2: "bs \<sharp>* c"
- and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
- shows "f as x c = f bs y c"
-proof -
- have "supp (as, x, c) supports (f as x c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin1: "finite (supp (f as x c))"
- using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
- have "supp (bs, y, c) supports (f bs y c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin2: "finite (supp (f bs y c))"
- using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
- obtain q::"perm" where
- fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and
- fr2: "supp q \<sharp>* ([as]set. x)" and
- inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
- using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"]
- fin1 fin2 fin
- by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
- have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp
- also have "\<dots> = [as]set. x"
- by (simp only: fr2 perm_supp_eq)
- finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp
- then obtain r::perm where
- qq1: "q \<bullet> x = r \<bullet> y" and
- qq2: "q \<bullet> as = r \<bullet> bs" and
- qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs"
- apply(drule_tac sym)
- apply(simp only: Abs_eq_iff2 alphas)
- apply(erule exE)
- apply(erule conjE)+
- apply(drule_tac x="p" in meta_spec)
- apply(simp add: set_eqvt)
- apply(blast)
- done
- have "as \<sharp>* f as x c" by (rule fcb1)
- then have "q \<bullet> (as \<sharp>* f as x c)"
- by (simp add: permute_bool_def)
- then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
- apply(simp add: fresh_star_eqvt set_eqvt)
- apply(subst (asm) perm1)
- using inc fresh1 fr1
- apply(auto simp add: fresh_star_def fresh_Pair)
- done
- then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
- then have "r \<bullet> (bs \<sharp>* f bs y c)"
- apply(simp add: fresh_star_eqvt set_eqvt)
- apply(subst (asm) perm2[symmetric])
- using qq3 fresh2 fr1
- apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
- done
- then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def)
- have "f as x c = q \<bullet> (f as x c)"
- apply(rule perm_supp_eq[symmetric])
- using inc fcb1 fr1 by (auto simp add: fresh_star_def)
- also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
- apply(rule perm1)
- using inc fresh1 fr1 by (auto simp add: fresh_star_def)
- also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
- also have "\<dots> = r \<bullet> (f bs y c)"
- apply(rule perm2[symmetric])
- using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
- also have "... = f bs y c"
- apply(rule perm_supp_eq)
- using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
- finally show ?thesis by simp
-qed
-
-lemma Abs_res_fcb2:
- fixes as bs :: "atom set"
- and x y :: "'b :: fs"
- and c::"'c::fs"
- assumes eq: "[as]res. x = [bs]res. y"
- and fin: "finite as" "finite bs"
- and fcb1: "as \<sharp>* f as x c"
- and fresh1: "as \<sharp>* c"
- and fresh2: "bs \<sharp>* c"
- and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
- shows "f as x c = f bs y c"
-proof -
- have "supp (as, x, c) supports (f as x c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin1: "finite (supp (f as x c))"
- using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
- have "supp (bs, y, c) supports (f bs y c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin2: "finite (supp (f bs y c))"
- using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
- obtain q::"perm" where
- fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and
- fr2: "supp q \<sharp>* ([as]res. x)" and
- inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
- using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"]
- fin1 fin2 fin
- by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
- have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp
- also have "\<dots> = [as]res. x"
- by (simp only: fr2 perm_supp_eq)
- finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp
- then obtain r::perm where
- qq1: "q \<bullet> x = r \<bullet> y" and
- qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and
- qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)"
- apply(drule_tac sym)
- apply(subst(asm) Abs_eq_res_set)
- apply(simp only: Abs_eq_iff2 alphas)
- apply(erule exE)
- apply(erule conjE)+
- apply(drule_tac x="p" in meta_spec)
- apply(simp add: set_eqvt)
- done
- have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *)
- then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)"
- by (simp add: permute_bool_def)
- then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c"
- apply(simp add: fresh_star_eqvt set_eqvt)
- sorry (* perm? *)
- then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2
- apply (simp add: inter_eqvt)
- sorry
- (* rest similar reversing it other way around... *)
- show ?thesis sorry
-qed
-
-
-
-lemma Abs_lst_fcb2:
- fixes as bs :: "atom list"
- and x y :: "'b :: fs"
- and c::"'c::fs"
- assumes eq: "[as]lst. x = [bs]lst. y"
- and fcb1: "(set as) \<sharp>* f as x c"
- and fresh1: "set as \<sharp>* c"
- and fresh2: "set bs \<sharp>* c"
- and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
- shows "f as x c = f bs y c"
-proof -
- have "supp (as, x, c) supports (f as x c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin1: "finite (supp (f as x c))"
- by (auto intro: supports_finite simp add: finite_supp)
- have "supp (bs, y, c) supports (f bs y c)"
- unfolding supports_def fresh_def[symmetric]
- by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
- then have fin2: "finite (supp (f bs y c))"
- by (auto intro: supports_finite simp add: finite_supp)
- obtain q::"perm" where
- fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and
- fr2: "supp q \<sharp>* Abs_lst as x" and
- inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
- 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"]
- fin1 fin2
- by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
- have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
- also have "\<dots> = Abs_lst as x"
- by (simp only: fr2 perm_supp_eq)
- finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
- then obtain r::perm where
- qq1: "q \<bullet> x = r \<bullet> y" and
- qq2: "q \<bullet> as = r \<bullet> bs" and
- qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
- apply(drule_tac sym)
- apply(simp only: Abs_eq_iff2 alphas)
- apply(erule exE)
- apply(erule conjE)+
- apply(drule_tac x="p" in meta_spec)
- apply(simp add: set_eqvt)
- apply(blast)
- done
- have "(set as) \<sharp>* f as x c" by (rule fcb1)
- then have "q \<bullet> ((set as) \<sharp>* f as x c)"
- by (simp add: permute_bool_def)
- then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
- apply(simp add: fresh_star_eqvt set_eqvt)
- apply(subst (asm) perm1)
- using inc fresh1 fr1
- apply(auto simp add: fresh_star_def fresh_Pair)
- done
- then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
- then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
- apply(simp add: fresh_star_eqvt set_eqvt)
- apply(subst (asm) perm2[symmetric])
- using qq3 fresh2 fr1
- apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
- done
- then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
- have "f as x c = q \<bullet> (f as x c)"
- apply(rule perm_supp_eq[symmetric])
- using inc fcb1 fr1 by (auto simp add: fresh_star_def)
- also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
- apply(rule perm1)
- using inc fresh1 fr1 by (auto simp add: fresh_star_def)
- also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
- also have "\<dots> = r \<bullet> (f bs y c)"
- apply(rule perm2[symmetric])
- using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
- also have "... = f bs y c"
- apply(rule perm_supp_eq)
- using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
- finally show ?thesis by simp
-qed
-
-lemma Abs_lst1_fcb2:
- fixes a b :: "atom"
- and x y :: "'b :: fs"
- and c::"'c :: fs"
- assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
- and fcb1: "a \<sharp> f a x c"
- and fresh: "{a, b} \<sharp>* c"
- and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
- shows "f a x c = f b y c"
-using e
-apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
-apply(simp_all)
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def)
-done
-
-lemma supp_zero_perm_zero:
- shows "supp (p :: perm) = {} \<longleftrightarrow> p = 0"
- by (metis supp_perm_singleton supp_zero_perm)
-
-lemma permute_atom_list_id:
- shows "p \<bullet> l = l \<longleftrightarrow> supp p \<inter> set l = {}"
- by (induct l) (auto simp add: supp_Nil supp_perm)
-
-lemma permute_length_eq:
- shows "p \<bullet> xs = ys \<Longrightarrow> length xs = length ys"
- by (auto simp add: length_eqvt[symmetric] permute_pure)
-
-lemma Abs_lst_binder_length:
- shows "[xs]lst. T = [ys]lst. S \<Longrightarrow> length xs = length ys"
- by (auto simp add: Abs_eq_iff alphas length_eqvt[symmetric] permute_pure)
-
-lemma Abs_lst_binder_eq:
- shows "Abs_lst l T = Abs_lst l S \<longleftrightarrow> T = S"
- by (rule, simp_all add: Abs_eq_iff2 alphas)
- (metis fresh_star_zero inf_absorb1 permute_atom_list_id supp_perm_eq
- supp_zero_perm_zero)
-
-lemma in_permute_list:
- shows "py \<bullet> p \<bullet> xs = px \<bullet> xs \<Longrightarrow> x \<in> set xs \<Longrightarrow> py \<bullet> p \<bullet> x = px \<bullet> x"
- by (induct xs) auto
-
-
-
nominal_primrec
crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm" ("_[_\<turnstile>c>_]" [100,100,100] 100)