--- a/Nominal/Ex/Lambda.thy Mon Jun 27 22:51:42 2011 +0100
+++ b/Nominal/Ex/Lambda.thy Tue Jun 28 00:30:30 2011 +0100
@@ -2,93 +2,116 @@
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 :: "'a :: at"
- and S T :: "'b :: fs"
- and c::"'c::fs"
- assumes e: "(Abs_lst [atom a] T) = (Abs_lst [atom b] S)"
- and fcb1: "atom a \<sharp> f a T c"
- and fcb2: "atom b \<sharp> f b S c"
+ 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 T c) = f (p \<bullet> a) (p \<bullet> T) c"
- and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b S c) = f (p \<bullet> b) (p \<bullet> S) c"
- shows "f a T c = f b S c"
-proof -
- have fin1: "finite (supp (f a T c))"
- apply(rule_tac S="supp (a, T, c)" in supports_finite)
- apply(simp add: supports_def)
- apply(simp add: fresh_def[symmetric])
- apply(clarify)
- apply(subst perm1)
- apply(simp add: supp_swap fresh_star_def)
- apply(simp add: swap_fresh_fresh fresh_Pair)
- apply(simp add: finite_supp)
- done
- have fin2: "finite (supp (f b S c))"
- apply(rule_tac S="supp (b, S, c)" in supports_finite)
- apply(simp add: supports_def)
- apply(simp add: fresh_def[symmetric])
- apply(clarify)
- apply(subst perm2)
- apply(simp add: supp_swap fresh_star_def)
- apply(simp add: swap_fresh_fresh fresh_Pair)
- apply(simp add: finite_supp)
- done
- obtain d::"'a::at" where fr: "atom d \<sharp> (a, b, S, T, c, f a T c, f b S c)"
- using obtain_fresh'[where x="(a, b, S, T, c, f a T c, f b S c)"]
- apply(auto simp add: finite_supp supp_Pair fin1 fin2)
- done
- have "(a \<leftrightarrow> d) \<bullet> (Abs_lst [atom a] T) = (b \<leftrightarrow> d) \<bullet> (Abs_lst [atom b] S)"
- apply(simp (no_asm_use) only: flip_def)
- apply(subst swap_fresh_fresh)
- apply(simp add: Abs_fresh_iff)
- using fr
- apply(simp add: Abs_fresh_iff)
- apply(subst swap_fresh_fresh)
- apply(simp add: Abs_fresh_iff)
- using fr
- apply(simp add: Abs_fresh_iff)
- apply(rule e)
- done
- then have "Abs_lst [atom d] ((a \<leftrightarrow> d) \<bullet> T) = Abs_lst [atom d] ((b \<leftrightarrow> d) \<bullet> S)"
- apply (simp add: swap_atom flip_def)
- done
- then have eq: "(a \<leftrightarrow> d) \<bullet> T = (b \<leftrightarrow> d) \<bullet> S"
- by (simp add: Abs1_eq_iff)
- have "f a T c = (a \<leftrightarrow> d) \<bullet> f a T c"
- unfolding flip_def
- apply(rule sym)
- apply(rule swap_fresh_fresh)
- using fcb1
- apply(simp)
- using fr
- apply(simp add: fresh_Pair)
- done
- also have "... = f d ((a \<leftrightarrow> d) \<bullet> T) c"
- unfolding flip_def
- apply(subst perm1)
- using fresh fr
- apply(simp add: supp_swap fresh_star_def fresh_Pair)
- apply(simp)
- done
- also have "... = f d ((b \<leftrightarrow> d) \<bullet> S) c" using eq by simp
- also have "... = (b \<leftrightarrow> d) \<bullet> f b S c"
- unfolding flip_def
- apply(subst perm2)
- using fresh fr
- apply(simp add: supp_swap fresh_star_def fresh_Pair)
- apply(simp)
- done
- also have "... = f b S c"
- apply(rule flip_fresh_fresh)
- using fcb2
- apply(simp)
- using fr
- apply(simp add: fresh_Pair)
- done
- finally show ?thesis by simp
-qed
+ 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
@@ -133,7 +156,6 @@
apply(auto)
apply (erule_tac c="()" in Abs_lst1_fcb2)
apply(simp add: supp_removeAll fresh_def)
-apply(simp add: supp_removeAll fresh_def)
apply(simp add: fresh_star_def fresh_Unit)
apply(simp add: eqvt_at_def removeAll_eqvt atom_eqvt)
apply(simp add: eqvt_at_def removeAll_eqvt atom_eqvt)
@@ -163,7 +185,6 @@
apply(simp)
apply(erule_tac c="()" in Abs_lst1_fcb2)
apply(simp add: fresh_minus_atom_set)
- apply(simp add: fresh_minus_atom_set)
apply(simp add: fresh_star_def fresh_Unit)
apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)
apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)
@@ -461,8 +482,7 @@
apply (auto simp add: fresh_star_def)[3]
apply(simp_all)
apply(erule conjE)+
- apply (erule Abs_lst1_fcb2)
- apply (simp add: fresh_star_def)
+ apply (erule_tac Abs_lst1_fcb2')
apply (simp add: fresh_star_def)
apply (simp add: fresh_star_def)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
@@ -487,8 +507,7 @@
apply(rule_tac y="a" and c="b" in lam.strong_exhaust)
apply(auto simp add: fresh_star_def)[3]
apply(erule conjE)
- apply(erule Abs_lst1_fcb2)
- apply(simp add: pure_fresh fresh_star_def)
+ apply(erule Abs_lst1_fcb2')
apply(simp add: pure_fresh fresh_star_def)
apply(simp add: pure_fresh fresh_star_def)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
@@ -557,8 +576,7 @@
apply (auto simp add: fresh_star_def fresh_at_list)[3]
apply(simp_all)
apply(erule conjE)
- apply (erule_tac c="xsa" in Abs_lst1_fcb2)
- apply (simp add: pure_fresh)
+ apply (erule_tac c="xsa" in Abs_lst1_fcb2')
apply (simp add: pure_fresh)
apply(simp add: fresh_star_def fresh_at_list)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq db_in_eqvt)
@@ -661,19 +679,15 @@
apply(simp_all)
apply(erule_tac c="()" in Abs_lst1_fcb2)
apply (simp add: Abs_fresh_iff)
- apply (simp add: Abs_fresh_iff)
apply(simp add: fresh_star_def fresh_Unit)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
apply(erule conjE)
- apply(erule_tac c="t2a" in Abs_lst1_fcb2)
+ apply(erule_tac c="t2a" in Abs_lst1_fcb2')
apply (erule fresh_eqvt_at)
apply (simp add: finite_supp)
apply (simp add: fresh_Inl var_fresh_subst)
- apply (erule fresh_eqvt_at)
- apply (simp add: finite_supp)
- apply (simp add: fresh_Inl var_fresh_subst)
- apply(simp add: fresh_star_def fresh_Unit)
+ apply(simp add: fresh_star_def)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq subst_eqvt)
apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq subst_eqvt)
done