--- 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)
--- a/Nominal/Ex/Lambda.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Ex/Lambda.thy Tue Jul 05 04:18:45 2011 +0200
@@ -2,117 +2,6 @@
imports "../Nominal2"
begin
-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 Abs_lst1_fcb2':
- fixes a b :: "'a::at"
- and x y :: "'b :: fs"
- and c::"'c :: fs"
- assumes e: "(Abs_lst [atom a] x) = (Abs_lst [atom b] y)"
- and fcb1: "atom a \<sharp> f a x c"
- and fresh: "{atom a, atom 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_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])
-using fcb1 fresh perm1 perm2
-apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)
-done
-
atom_decl name
--- a/Nominal/Ex/Let.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Ex/Let.thy Tue Jul 05 04:18:45 2011 +0200
@@ -77,104 +77,6 @@
apply(simp add: atom_eqvt)
done
-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>* c \<Longrightarrow> (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"
- apply(rule fcb1)
- apply(rule fresh1)
- done
- 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[OF fresh1] 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> c \<Longrightarrow> 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 max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
by (simp add: permute_pure)
--- a/Nominal/Ex/LetSimple1.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Ex/LetSimple1.thy Tue Jul 05 04:18:45 2011 +0200
@@ -2,102 +2,6 @@
imports "../Nominal2"
begin
-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
-
-
atom_decl name
nominal_datatype trm =
--- a/Nominal/Ex/LetSimple2.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Ex/LetSimple2.thy Tue Jul 05 04:18:45 2011 +0200
@@ -2,105 +2,6 @@
imports "../Nominal2"
begin
-
-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>* c \<Longrightarrow> (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"
- apply(rule fcb1)
- apply(rule fresh1)
- done
- 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[OF fresh1] 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> c \<Longrightarrow> 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
-
atom_decl name
nominal_datatype trm =
@@ -116,8 +17,6 @@
print_theorems
-
-
thm bn_raw.simps
thm permute_bn_raw.simps
thm trm_assn.perm_bn_alpha
--- a/Nominal/Nominal2.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Nominal2.thy Tue Jul 05 04:18:45 2011 +0200
@@ -1,6 +1,6 @@
theory Nominal2
imports
- Nominal2_Base Nominal2_Abs
+ Nominal2_Base Nominal2_Abs Nominal2_FCB
uses ("nominal_dt_rawfuns.ML")
("nominal_dt_alpha.ML")
("nominal_dt_quot.ML")
--- a/Nominal/Nominal2_Abs.thy Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Nominal2_Abs.thy Tue Jul 05 04:18:45 2011 +0200
@@ -1013,112 +1013,6 @@
unfolding prod_alpha_def
by (auto intro!: ext)
-lemma Abs_lst1_fcb:
- fixes x y :: "'a :: at_base"
- and S T :: "'b :: fs"
- assumes e: "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"
- 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"
- 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"
- 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"
- and s: "sort_of (atom x) = sort_of (atom y)"
- shows "f x T = f y S"
- using e
- apply(case_tac "atom x \<sharp> S")
- apply(simp add: Abs1_eq_iff'[OF s s])
- apply(elim conjE disjE)
- apply(simp)
- apply(rule trans)
- apply(rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])
- apply(rule fresh_star_supp_conv)
- apply(simp add: supp_swap fresh_star_def s f1 f2)
- apply(simp add: swap_commute p)
- apply(simp add: Abs1_eq_iff[OF s s])
- done
-
-lemma Abs_lst_fcb:
- fixes xs ys :: "'a :: fs"
- and S T :: "'b :: fs"
- assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)"
- and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T"
- and f2: "\<And>x. supp T - set (ba xs) = supp S - set (ba ys) \<Longrightarrow> x \<in> set (ba ys) \<Longrightarrow> x \<sharp> f xs T"
- and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> p \<bullet> ba xs = ba ys \<Longrightarrow> supp p \<subseteq> set (ba xs) \<union> set (ba ys) \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
- shows "f xs T = f ys S"
- using e apply -
- apply(subst (asm) Abs_eq_iff2)
- apply(simp add: alphas)
- apply(elim exE conjE)
- apply(rule trans)
- apply(rule_tac p="p" in supp_perm_eq[symmetric])
- apply(rule fresh_star_supp_conv)
- apply(drule fresh_star_perm_set_conv)
- apply(rule finite_Diff)
- apply(rule finite_supp)
- apply(subgoal_tac "(set (ba xs) \<union> set (ba ys)) \<sharp>* f xs T")
- apply(metis Un_absorb2 fresh_star_Un)
- apply(subst fresh_star_Un)
- apply(rule conjI)
- apply(simp add: fresh_star_def f1)
- apply(simp add: fresh_star_def f2)
- apply(simp add: eqv)
- done
-
-lemma Abs_set_fcb:
- fixes xs ys :: "'a :: fs"
- and S T :: "'b :: fs"
- assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)"
- and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T"
- and f2: "\<And>x. supp T - ba xs = supp S - ba ys \<Longrightarrow> x \<in> ba ys \<Longrightarrow> x \<sharp> f xs T"
- and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> p \<bullet> ba xs = ba ys \<Longrightarrow> supp p \<subseteq> ba xs \<union> ba ys \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
- shows "f xs T = f ys S"
- using e apply -
- apply(subst (asm) Abs_eq_iff2)
- apply(simp add: alphas)
- apply(elim exE conjE)
- apply(rule trans)
- apply(rule_tac p="p" in supp_perm_eq[symmetric])
- apply(rule fresh_star_supp_conv)
- apply(drule fresh_star_perm_set_conv)
- apply(rule finite_Diff)
- apply(rule finite_supp)
- apply(subgoal_tac "(ba xs \<union> ba ys) \<sharp>* f xs T")
- apply(metis Un_absorb2 fresh_star_Un)
- apply(subst fresh_star_Un)
- apply(rule conjI)
- apply(simp add: fresh_star_def f1)
- apply(simp add: fresh_star_def f2)
- apply(simp add: eqv)
- done
-
-lemma Abs_res_fcb:
- fixes xs ys :: "('a :: at_base) set"
- and S T :: "'b :: fs"
- assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)"
- and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T"
- 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"
- and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S
- \<Longrightarrow> p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
- shows "f xs T = f ys S"
- using e apply -
- apply(subst (asm) Abs_eq_res_set)
- apply(subst (asm) Abs_eq_iff2)
- apply(simp add: alphas)
- apply(elim exE conjE)
- apply(rule trans)
- apply(rule_tac p="p" in supp_perm_eq[symmetric])
- apply(rule fresh_star_supp_conv)
- apply(drule fresh_star_perm_set_conv)
- apply(rule finite_Diff)
- apply(rule finite_supp)
- apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T")
- apply(metis Un_absorb2 fresh_star_Un)
- apply(subst fresh_star_Un)
- apply(rule conjI)
- apply(simp add: fresh_star_def f1)
- apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")
- apply(simp add: fresh_star_def f2)
- apply(blast)
- apply(simp add: eqv)
- done
end