side-by-side tests of lets with single assignment; deep-binder case works if the recursion is avoided using an auxiliary function
--- a/Fun-Paper/Paper.thy Fri Jul 01 17:46:15 2011 +0900
+++ b/Fun-Paper/Paper.thy Sat Jul 02 00:27:47 2011 +0100
@@ -7,9 +7,15 @@
section {* Introduction *}
+text {*
+mention Russo paper which concludes that technology is not
+ready beyond core-calculi.
+*}
+
+
(*<*)
end
(*>*)
\ No newline at end of file
--- a/Nominal/Ex/Let.thy Fri Jul 01 17:46:15 2011 +0900
+++ b/Nominal/Ex/Let.thy Sat Jul 02 00:27:47 2011 +0100
@@ -39,8 +39,9 @@
thm trm_assn.fresh
thm trm_assn.exhaust
thm trm_assn.strong_exhaust
+thm trm_assn.perm_bn_simps
-lemma alpha_bn_inducts_raw:
+lemma alpha_bn_inducts_raw[consumes 1]:
"\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
\<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
\<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
@@ -49,7 +50,7 @@
(ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto
-lemmas alpha_bn_inducts = alpha_bn_inducts_raw[quot_lifted]
+lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]
@@ -66,6 +67,17 @@
shows "bn x = bn y \<longrightarrow> x = y"
by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)
+lemma bn_inj2:
+ assumes a: "alpha_bn x y"
+ shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"
+using a
+apply(induct rule: alpha_bn_inducts)
+apply(simp add: trm_assn.perm_bn_simps)
+apply(simp add: trm_assn.perm_bn_simps)
+apply(simp add: trm_assn.bn_defs)
+apply(simp add: atom_eqvt)
+done
+
(*lemma alpha_bn_permute:
assumes a: "alpha_bn x y"
and b: "q \<bullet> bn x = r \<bullet> bn y"
@@ -85,6 +97,205 @@
using bn_inj by simp
*)
+
+lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
+ by (simp add: permute_pure)
+
+
+lemma Abs_lst_fcb2:
+ fixes as bs :: "'a :: fs"
+ and x y :: "'b :: fs"
+ assumes eq: "[ba as]lst. x = [ba bs]lst. y"
+ and ctxt: "finite (supp c)"
+ and fcb1: "set (ba as) \<sharp>* f as x c"
+ and fresh1: "set (ba as) \<sharp>* c"
+ and fresh2: "set (ba 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"
+(* What we would like in this proof, and lets this proof finish *)
+ and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> pn q as = pn r bs"
+ 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]
+ apply (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+ sorry
+ then have fin1: "finite (supp (f as x c))"
+ by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)
+ have "supp (bs, y, c) supports (f bs y c)"
+ unfolding supports_def fresh_def[symmetric]
+ apply (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+ sorry
+ then have fin2: "finite (supp (f bs y c))"
+ by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)
+ obtain q::"perm" where
+ fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and
+ fr2: "supp q \<sharp>* ([ba as]lst. x)" and
+ inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
+ using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)"
+ and x="[ba as]lst. x"] fin1 fin2
+ by (auto simp add: supp_Pair finite_supp ctxt Abs_fresh_star dest: fresh_star_supp_conv)
+ have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
+ also have "\<dots> = [ba as]lst. x"
+ by (simp only: fr2 perm_supp_eq)
+ finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
+ then obtain r::perm where
+ qq1: "q \<bullet> x = r \<bullet> y" and
+ qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and
+ qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 (ba as)) \<sharp>* f as x c" by (rule fcb1)
+ then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
+ by (simp add: permute_bool_def)
+ then have "set (q \<bullet> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj
+ apply simp
+ sorry
+ then have "r \<bullet> ((set (ba 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 (ba 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 ba_inj
+ apply(simp)
+ sorry
+ 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
+
+nominal_primrec
+ height_trm :: "trm \<Rightarrow> nat"
+and height_assn :: "assn \<Rightarrow> nat"
+where
+ "height_trm (Var x) = 1"
+| "height_trm (App l r) = max (height_trm l) (height_trm r)"
+| "height_trm (Lam v b) = 1 + (height_trm b)"
+| "height_trm (Let as b) = max (height_assn as) (height_trm b)"
+| "height_assn ANil = 0"
+| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)"
+ apply (simp only: eqvt_def height_trm_height_assn_graph_def)
+ apply (rule, perm_simp, rule, rule TrueI)
+ apply (case_tac x)
+ apply (case_tac a rule: trm_assn.exhaust(1))
+ apply (auto)[4]
+ apply (drule_tac x="assn" in meta_spec)
+ apply (drule_tac x="trm" in meta_spec)
+ apply (simp add: alpha_bn_refl)
+ apply (case_tac b rule: trm_assn.exhaust(2))
+ apply (auto)[2]
+ apply(simp_all del: trm_assn.eq_iff)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ apply(simp)
+ apply (erule_tac c="()" and pn="permute" in Abs_lst_fcb2)
+ apply(simp add: finite_supp)
+ apply (simp_all add: pure_fresh fresh_star_def)[3]
+ apply (simp add: eqvt_at_def)
+ apply (simp add: eqvt_at_def)
+ apply(auto)[1]
+ --"other case"
+ apply (simp only: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
+ apply (simp only: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
+ apply (subgoal_tac "eqvt_at height_assn as")
+ apply (subgoal_tac "eqvt_at height_assn asa")
+ apply (subgoal_tac "eqvt_at height_trm b")
+ apply (subgoal_tac "eqvt_at height_trm ba")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")
+ defer
+ apply (simp add: eqvt_at_def height_trm_def)
+ apply (simp add: eqvt_at_def height_trm_def)
+ apply (simp add: eqvt_at_def height_assn_def)
+ apply (simp add: eqvt_at_def height_assn_def)
+ apply (subgoal_tac "height_assn as = height_assn asa")
+ apply (subgoal_tac "height_trm b = height_trm ba")
+ apply simp
+ apply(simp)
+ apply(erule conjE)
+ apply (erule_tac c="()" and pn="permute_bn" in Abs_lst_fcb2)
+ apply(simp add: finite_supp)
+ apply (simp_all add: pure_fresh fresh_star_def)[3]
+ apply (simp_all add: eqvt_at_def)[2]
+ apply(simp add: bn_inj2)
+ apply(simp)
+ apply(erule conjE)
+ thm trm_assn.fv_defs
+ (*apply(simp add: Abs_eq_iff alphas)*)
+ apply (erule_tac c="()" and pn="permute_bn" and ba="bn" in Abs_lst_fcb2)
+ defer
+ apply (simp_all add: pure_fresh fresh_star_def)[3]
+ defer
+ defer
+ apply (simp_all add: eqvt_at_def)[2]
+ apply (rule bn_inj)
+ prefer 2
+ apply (simp add: eqvts)
+ oops
+
+nominal_primrec
+ subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
+and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"
+where
+ "subst s t (Var x) = (if (s = x) then t else (Var x))"
+| "subst s t (App l r) = App (subst s t l) (subst s t r)"
+| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
+| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (substa s t as) (subst s t b)"
+| "substa s t ANil = ANil"
+| "substa s t (ACons v t' as) = ACons v (subst v t t') as"
+(*unfolding eqvt_def subst_substa_graph_def
+ apply (rule, perm_simp)*)
+ defer
+ apply (rule TrueI)
+ apply (case_tac x)
+ apply (case_tac a)
+ apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
+ apply (auto simp add: fresh_star_def)[3]
+ apply (drule_tac x="assn" in meta_spec)
+ apply (simp add: Abs1_eq_iff alpha_bn_refl)
+ apply (case_tac b)
+ apply (case_tac c rule: trm_assn.exhaust(2))
+ apply (auto)[2]
+ apply blast
+ apply blast
+ apply auto
+ apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
+ apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
+ prefer 2
+ apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
+ apply (simp_all add: fresh_star_Pair)
+ prefer 6
+ apply (erule alpha_bn_inducts)
+ oops
+
+
lemma lets_bla:
"x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
by (simp add: trm_assn.eq_iff)
@@ -126,188 +337,4 @@
apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)
-
-lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
- by (simp add: permute_pure)
-
-
-lemma Abs_lst_fcb2:
- fixes as bs :: "'a :: fs"
- and x y :: "'b :: fs"
- and c::"'c::fs"
- assumes eq: "[ba as]lst. x = [ba bs]lst. y"
- and fcb1: "set (ba as) \<sharp>* f as x c"
- and fresh1: "set (ba as) \<sharp>* c"
- and fresh2: "set (ba 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"
-(* What we would like in this proof, and lets this proof finish *)
- and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> q \<bullet> as = r \<bullet> bs"
-(* What the user can supply with the help of alpha_bn *)
-(* and bainj: "ba as = ba bs \<Longrightarrow> as = bs"*)
- 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 (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and
- fr2: "supp q \<sharp>* ([ba as]lst. x)" and
- inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
- using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)"
- and x="[ba as]lst. x"] fin1 fin2
- by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
- have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
- also have "\<dots> = [ba as]lst. x"
- by (simp only: fr2 perm_supp_eq)
- finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
- then obtain r::perm where
- qq1: "q \<bullet> x = r \<bullet> y" and
- qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and
- qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 (ba as)) \<sharp>* f as x c" by (rule fcb1)
- then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
- by (simp add: permute_bool_def)
- then have "set (q \<bullet> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj
- by simp
- then have "r \<bullet> ((set (ba 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 (ba 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 ba_inj 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
-
-nominal_primrec
- height_trm :: "trm \<Rightarrow> nat"
-and height_assn :: "assn \<Rightarrow> nat"
-where
- "height_trm (Var x) = 1"
-| "height_trm (App l r) = max (height_trm l) (height_trm r)"
-| "height_trm (Lam v b) = 1 + (height_trm b)"
-| "height_trm (Let as b) = max (height_assn as) (height_trm b)"
-| "height_assn ANil = 0"
-| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)"
- apply (simp only: eqvt_def height_trm_height_assn_graph_def)
- apply (rule, perm_simp, rule, rule TrueI)
- apply (case_tac x)
- apply (case_tac a rule: trm_assn.exhaust(1))
- apply (auto)[4]
- apply (drule_tac x="assn" in meta_spec)
- apply (drule_tac x="trm" in meta_spec)
- apply (simp add: alpha_bn_refl)
- apply (case_tac b rule: trm_assn.exhaust(2))
- apply (auto)[2]
- apply(simp_all)
- apply (erule_tac c="()" in Abs_lst_fcb2)
- apply (simp_all add: pure_fresh fresh_star_def)[3]
- apply (simp add: eqvt_at_def)
- apply (simp add: eqvt_at_def)
- apply assumption
- apply(erule conjE)
- apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
- apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
- apply (subgoal_tac "eqvt_at height_assn as")
- apply (subgoal_tac "eqvt_at height_assn asa")
- apply (subgoal_tac "eqvt_at height_trm b")
- apply (subgoal_tac "eqvt_at height_trm ba")
- apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
- apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")
- apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")
- apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")
- defer
- apply (simp add: eqvt_at_def height_trm_def)
- apply (simp add: eqvt_at_def height_trm_def)
- apply (simp add: eqvt_at_def height_assn_def)
- apply (simp add: eqvt_at_def height_assn_def)
- apply (subgoal_tac "height_assn as = height_assn asa")
- apply (subgoal_tac "height_trm b = height_trm ba")
- apply simp
- apply (erule_tac c="()" in Abs_lst_fcb2)
- apply (simp_all add: pure_fresh fresh_star_def)[3]
- apply (simp_all add: eqvt_at_def)[2]
- apply assumption
- apply (erule_tac c="()" and ba="bn" in Abs_lst_fcb2)
- apply (simp_all add: pure_fresh fresh_star_def)[3]
- apply (simp_all add: eqvt_at_def)[2]
- apply (rule bn_inj)
- prefer 2
- apply (simp add: eqvts)
- oops
-
-nominal_primrec
- subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
-and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"
-where
- "subst s t (Var x) = (if (s = x) then t else (Var x))"
-| "subst s t (App l r) = App (subst s t l) (subst s t r)"
-| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
-| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (substa s t as) (subst s t b)"
-| "substa s t ANil = ANil"
-| "substa s t (ACons v t' as) = ACons v (subst v t t') as"
-(*unfolding eqvt_def subst_substa_graph_def
- apply (rule, perm_simp)*)
- defer
- apply (rule TrueI)
- apply (case_tac x)
- apply (case_tac a)
- apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
- apply (auto simp add: fresh_star_def)[3]
- apply (drule_tac x="assn" in meta_spec)
- apply (simp add: Abs1_eq_iff alpha_bn_refl)
- apply (case_tac b)
- apply (case_tac c rule: trm_assn.exhaust(2))
- apply (auto)[2]
- apply blast
- apply blast
- apply auto
- apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
- apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
- prefer 2
- apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
- apply (simp_all add: fresh_star_Pair)
- prefer 6
- apply (erule alpha_bn_inducts)
- oops
-
-
-end
-
-
-
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/LetSimple1.thy Sat Jul 02 00:27:47 2011 +0100
@@ -0,0 +1,207 @@
+theory LetSimple1
+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 =
+ Var "name"
+| App "trm" "trm"
+| Let x::"name" "trm" t::"trm" bind x in t
+
+print_theorems
+
+thm trm.fv_defs
+thm trm.eq_iff
+thm trm.bn_defs
+thm trm.bn_inducts
+thm trm.perm_simps
+thm trm.induct
+thm trm.inducts
+thm trm.distinct
+thm trm.supp
+thm trm.fresh
+thm trm.exhaust
+thm trm.strong_exhaust
+thm trm.perm_bn_simps
+
+nominal_primrec
+ height_trm :: "trm \<Rightarrow> nat"
+where
+ "height_trm (Var x) = 1"
+| "height_trm (App l r) = max (height_trm l) (height_trm r)"
+| "height_trm (Let x t s) = max (height_trm t) (height_trm s)"
+ apply (simp only: eqvt_def height_trm_graph_def)
+ apply (rule, perm_simp, rule, rule TrueI)
+ apply (case_tac x rule: trm.exhaust(1))
+ apply (auto)[3]
+ apply(simp_all)[5]
+ apply (subgoal_tac "height_trm_sumC t = height_trm_sumC ta")
+ apply (subgoal_tac "height_trm_sumC s = height_trm_sumC sa")
+ apply simp
+ apply(simp)
+ apply(erule conjE)
+ apply(erule_tac c="()" in Abs_lst1_fcb2)
+ apply(simp_all add: fresh_star_def pure_fresh)[2]
+ apply(simp_all add: eqvt_at_def)[2]
+ apply(simp)
+ done
+
+termination
+ by lexicographic_order
+
+
+nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")
+ frees_set :: "trm \<Rightarrow> atom set"
+where
+ "frees_set (Var x) = {atom x}"
+| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"
+| "frees_set (Let x t s) = (frees_set s) - {atom x} \<union> (frees_set t)"
+ apply(simp add: eqvt_def frees_set_graph_def)
+ apply(rule, perm_simp, rule)
+ apply(erule frees_set_graph.induct)
+ apply(auto)[3]
+ apply(rule_tac y="x" in trm.exhaust)
+ apply(auto)[3]
+ apply(simp_all)[5]
+ apply(simp)
+ apply(erule conjE)
+ apply(subgoal_tac "frees_set_sumC s - {atom x} = frees_set_sumC sa - {atom xa}")
+ apply(simp)
+ apply(erule_tac c="()" in Abs_lst1_fcb2)
+ 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)
+ done
+
+termination
+ by lexicographic_order
+
+
+nominal_primrec
+ subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_ [_ ::= _]" [90, 90, 90] 90)
+where
+ "(Var x)[y ::= s] = (if x = y then s else (Var x))"
+| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"
+| "atom x \<sharp> (y, s) \<Longrightarrow> (Let x t t')[y ::= s] = Let x (t[y ::= s]) (t'[y ::= s])"
+ apply(simp add: eqvt_def subst_graph_def)
+ apply (rule, perm_simp, rule)
+ apply(rule TrueI)
+ apply(auto)[1]
+ apply(rule_tac y="a" and c="(aa, b)" in trm.strong_exhaust)
+ apply(blast)+
+ apply(simp_all add: fresh_star_def fresh_Pair_elim)[1]
+ apply(blast)
+ apply(simp_all)[5]
+ apply(simp (no_asm_use))
+ apply(simp)
+ apply(erule conjE)+
+ apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
+ apply(simp add: Abs_fresh_iff)
+ apply(simp add: fresh_star_def fresh_Pair)
+ 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)
+done
+
+termination
+ by lexicographic_order
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/LetSimple2.thy Sat Jul 02 00:27:47 2011 +0100
@@ -0,0 +1,497 @@
+theory Let
+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 =
+ Var "name"
+| App "trm" "trm"
+| Let as::"assn" t::"trm" bind "bn as" in t
+and assn =
+ Assn "name" "trm"
+binder
+ bn
+where
+ "bn (Assn x t) = [atom x]"
+
+print_theorems
+
+
+
+thm bn_raw.simps
+thm permute_bn_raw.simps
+thm trm_assn.perm_bn_alpha
+thm trm_assn.permute_bn
+
+thm trm_assn.fv_defs
+thm trm_assn.eq_iff
+thm trm_assn.bn_defs
+thm trm_assn.bn_inducts
+thm trm_assn.perm_simps
+thm trm_assn.induct
+thm trm_assn.inducts
+thm trm_assn.distinct
+thm trm_assn.supp
+thm trm_assn.fresh
+thm trm_assn.exhaust
+thm trm_assn.strong_exhaust
+thm trm_assn.perm_bn_simps
+
+thm alpha_bn_raw.cases
+
+
+lemmas alpha_bn_cases[consumes 1] = alpha_bn_raw.cases[quot_lifted]
+
+lemma alpha_bn_refl: "alpha_bn x x"
+ by (induct x rule: trm_assn.inducts(2))
+ (rule TrueI, auto simp add: trm_assn.eq_iff)
+lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x"
+ apply(erule alpha_bn_cases)
+ apply(auto)
+ done
+
+lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"
+ sorry
+
+lemma k: "A \<Longrightarrow> A \<and> A" by blast
+
+lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
+ by (simp add: permute_pure)
+
+
+section {* definition with helper functions *}
+
+function
+ apply_assn
+where
+ "apply_assn f (Assn x t) = (f t)"
+apply(case_tac x)
+apply(simp)
+apply(case_tac b rule: trm_assn.exhaust(2))
+apply(blast)
+apply(simp)
+done
+
+termination
+ by lexicographic_order
+
+function
+ apply_assn2
+where
+ "apply_assn2 f (Assn x t) = Assn x (f t)"
+apply(case_tac x)
+apply(simp)
+apply(case_tac b rule: trm_assn.exhaust(2))
+apply(blast)
+apply(simp)
+done
+
+termination
+ by lexicographic_order
+
+lemma [eqvt]:
+ shows "p \<bullet> (apply_assn f as) = apply_assn (p \<bullet> f) (p \<bullet> as)"
+apply(induct f as rule: apply_assn.induct)
+apply(simp)
+apply(perm_simp)
+apply(rule)
+done
+
+lemma [eqvt]:
+ shows "p \<bullet> (apply_assn2 f as) = apply_assn2 (p \<bullet> f) (p \<bullet> as)"
+apply(induct f as rule: apply_assn.induct)
+apply(simp)
+apply(perm_simp)
+apply(rule)
+done
+
+
+nominal_primrec
+ height_trm :: "trm \<Rightarrow> nat"
+where
+ "height_trm (Var x) = 1"
+| "height_trm (App l r) = max (height_trm l) (height_trm r)"
+| "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)"
+ apply (simp only: eqvt_def height_trm_graph_def)
+ apply (rule, perm_simp)
+ apply(rule)
+ apply(rule TrueI)
+ apply (case_tac x rule: trm_assn.exhaust(1))
+ apply (auto simp add: alpha_bn_refl)[3]
+ apply (drule_tac x="assn" in meta_spec)
+ apply (drule_tac x="trm" in meta_spec)
+ apply(simp add: alpha_bn_refl)
+ apply(simp_all)[5]
+ apply(simp)
+ apply(erule conjE)+
+ thm alpha_bn_cases
+ apply(erule alpha_bn_cases)
+ apply(simp)
+ apply (subgoal_tac "height_trm_sumC b = height_trm_sumC ba")
+ apply simp
+ apply(simp add: trm_assn.bn_defs)
+ apply(erule_tac c="()" in Abs_lst_fcb2)
+ apply(simp_all add: pure_fresh fresh_star_def)[3]
+ apply(simp_all add: eqvt_at_def)
+ done
+
+
+nominal_primrec
+ subst_trm :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_ [_ ::= _]" [90, 90, 90] 90)
+where
+ "(Var x)[y ::= s] = (if x = y then s else (Var x))"
+| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"
+| "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow>
+ (Let as t)[y ::= s] = Let (apply_assn2 (\<lambda>t. t[y ::=s]) as) (t[y ::= s])"
+ apply (simp only: eqvt_def subst_trm_graph_def)
+ apply (rule, perm_simp)
+ apply(rule)
+ apply(rule TrueI)
+ apply(case_tac x)
+ apply(simp)
+ apply (rule_tac y="a" and c="(b,c)" in trm_assn.strong_exhaust(1))
+ apply (auto simp add: alpha_bn_refl)[3]
+ apply(simp_all)[5]
+ apply(simp)
+ apply(erule conjE)+
+ thm alpha_bn_cases
+ apply(erule alpha_bn_cases)
+ apply(simp)
+ apply(simp add: trm_assn.bn_defs)
+ apply(erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
+ apply(simp add: Abs_fresh_iff fresh_star_def)
+ apply(simp add: fresh_star_def)
+ apply(simp_all add: eqvt_at_def perm_supp_eq fresh_star_Pair)[2]
+ done
+
+
+section {* direct definitions --- problems *}
+
+
+nominal_primrec
+ height_trm :: "trm \<Rightarrow> nat"
+and height_assn :: "assn \<Rightarrow> nat"
+where
+ "height_trm (Var x) = 1"
+| "height_trm (App l r) = max (height_trm l) (height_trm r)"
+| "height_trm (Let as b) = max (height_assn as) (height_trm b)"
+| "height_assn (Assn x t) = (height_trm t)"
+ apply (simp only: eqvt_def height_trm_height_assn_graph_def)
+ apply (rule, perm_simp, rule, rule TrueI)
+ apply (case_tac x)
+ apply(simp)
+ apply (case_tac a rule: trm_assn.exhaust(1))
+ apply (auto simp add: alpha_bn_refl)[3]
+ apply (drule_tac x="assn" in meta_spec)
+ apply (drule_tac x="trm" in meta_spec)
+ apply(simp add: alpha_bn_refl)
+ apply(simp)
+ apply (case_tac b rule: trm_assn.exhaust(2))
+ apply (auto)[1]
+ apply(simp_all)[7]
+ prefer 2
+ apply(simp)
+ --"let case"
+ apply (simp only: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
+ apply (simp only: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
+ apply (subgoal_tac "eqvt_at height_assn as")
+ apply (subgoal_tac "eqvt_at height_assn asa")
+ apply (subgoal_tac "eqvt_at height_trm b")
+ apply (subgoal_tac "eqvt_at height_trm ba")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")
+ apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")
+ defer
+ apply (simp add: eqvt_at_def height_trm_def)
+ apply (simp add: eqvt_at_def height_trm_def)
+ apply (simp add: eqvt_at_def height_assn_def)
+ apply (simp add: eqvt_at_def height_assn_def)
+ prefer 2
+ apply (subgoal_tac "height_assn as = height_assn asa")
+ apply (subgoal_tac "height_trm b = height_trm ba")
+ apply simp
+ apply(simp)
+ apply(erule conjE)+
+ apply(erule alpha_bn_cases)
+ apply(simp)
+ apply(simp add: trm_assn.bn_defs)
+ thm Abs_lst_fcb2
+ apply(erule_tac c="()" in Abs_lst_fcb2)
+ apply(simp_all add: fresh_star_def pure_fresh)[3]
+ apply(simp add: eqvt_at_def)
+ apply(simp add: eqvt_at_def)
+ defer
+ apply(simp)
+ apply(frule Inl_inject)
+ apply(subst (asm) trm_assn.eq_iff)
+ apply(drule Inl_inject)
+ apply(clarify)
+ apply(erule alpha_bn_cases)
+ apply(simp del: trm_assn.eq_iff)
+ apply(rename_tac as s as' s' t' t x x')
+ apply(simp only: trm_assn.bn_defs)
+ (* HERE *)
+ oops
+
+
+lemma ww1:
+ shows "finite (fv_trm t)"
+ and "finite (fv_bn as)"
+apply(induct t and as rule: trm_assn.inducts)
+apply(simp_all add: trm_assn.fv_defs supp_at_base)
+done
+
+text {* works, but only because no recursion in as *}
+
+nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")
+ frees_set :: "trm \<Rightarrow> atom set"
+where
+ "frees_set (Var x) = {atom x}"
+| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"
+| "frees_set (Let as t) = (frees_set t) - (set (bn as)) \<union> (fv_bn as)"
+ apply(simp add: eqvt_def frees_set_graph_def)
+ apply(rule, perm_simp, rule)
+ apply(erule frees_set_graph.induct)
+ apply(auto simp add: ww1)[3]
+ apply(rule_tac y="x" in trm_assn.exhaust(1))
+ apply(auto simp add: alpha_bn_refl)[3]
+ apply(drule_tac x="assn" in meta_spec)
+ apply(drule_tac x="trm" in meta_spec)
+ apply(simp add: alpha_bn_refl)
+ apply(simp_all)[5]
+ apply(simp)
+ apply(erule conjE)
+ apply(erule alpha_bn_cases)
+ apply(simp add: trm_assn.bn_defs)
+ apply(simp add: trm_assn.fv_defs)
+ (* apply(erule_tac c="(trm_rawa)" in Abs_lst1_fcb2) *)
+ apply(subgoal_tac " frees_set_sumC t - {atom name} = frees_set_sumC ta - {atom namea}")
+ apply(simp)
+ apply(erule_tac c="()" in Abs_lst1_fcb2)
+ 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)
+ done
+
+termination
+ by lexicographic_order
+
+lemma test:
+ assumes a: "\<exists>y. f x = Inl y"
+ shows "(p \<bullet> (Sum_Type.Projl (f x))) = Sum_Type.Projl ((p \<bullet> f) (p \<bullet> x))"
+using a
+apply clarify
+apply(frule_tac p="p" in permute_boolI)
+apply(simp (no_asm_use) only: eqvts)
+apply(subst (asm) permute_fun_app_eq)
+back
+apply(simp)
+done
+
+
+nominal_primrec (default "sum_case (\<lambda>x. Inl undefined) (\<lambda>x. Inr undefined)")
+ subst_trm :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_ [_ ::trm= _]" [90, 90, 90] 90) and
+ subst_assn :: "assn \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> assn" ("_ [_ ::assn= _]" [90, 90, 90] 90)
+where
+ "(Var x)[y ::trm= s] = (if x = y then s else (Var x))"
+| "(App t1 t2)[y ::trm= s] = App (t1[y ::trm= s]) (t2[y ::trm= s])"
+| "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow> (Let as t)[y ::trm= s] = Let (ast[y ::assn= s]) (t[y ::trm= s])"
+| "(Assn x t)[y ::assn= s] = Assn x (t[y ::trm= s])"
+apply(subgoal_tac "\<And>p x r. subst_trm_subst_assn_graph x r \<Longrightarrow> subst_trm_subst_assn_graph (p \<bullet> x) (p \<bullet> r)")
+apply(simp add: eqvt_def)
+apply(rule allI)
+apply(simp add: permute_fun_def permute_bool_def)
+apply(rule ext)
+apply(rule ext)
+apply(rule iffI)
+apply(drule_tac x="p" in meta_spec)
+apply(drule_tac x="- p \<bullet> x" in meta_spec)
+apply(drule_tac x="- p \<bullet> xa" in meta_spec)
+apply(simp)
+apply(drule_tac x="-p" in meta_spec)
+apply(drule_tac x="x" in meta_spec)
+apply(drule_tac x="xa" in meta_spec)
+apply(simp)
+--"Eqvt One way"
+defer
+ apply(rule TrueI)
+ apply(case_tac x)
+ apply(simp)
+ apply(case_tac a)
+ apply(simp)
+ apply(rule_tac y="aa" and c="(b, c)" in trm_assn.strong_exhaust(1))
+ apply(blast)+
+ apply(simp)
+ apply(case_tac b)
+ apply(simp)
+ apply(rule_tac y="a" in trm_assn.exhaust(2))
+ apply(simp)
+ apply(blast)
+ apply(simp_all)[7]
+ prefer 2
+ apply(simp)
+ prefer 2
+ apply(simp)
+ apply(simp)
+ apply (simp only: meta_eq_to_obj_eq[OF subst_trm_def, symmetric, unfolded fun_eq_iff])
+ apply (simp only: meta_eq_to_obj_eq[OF subst_assn_def, symmetric, unfolded fun_eq_iff])
+ apply (subgoal_tac "eqvt_at (\<lambda>ast. subst_assn ast ya sa) ast")
+ apply (subgoal_tac "eqvt_at (\<lambda>asta. subst_assn asta ya sa) asta")
+ apply (subgoal_tac "eqvt_at (\<lambda>t. subst_trm t ya sa) t")
+ apply (subgoal_tac "eqvt_at (\<lambda>ta. subst_trm ta ya sa) ta")
+ apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inr (ast, ya, sa))")
+ apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inr (asta, ya, sa))")
+ apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inl (t, ya, sa))")
+ apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inl (ta, ya, sa))")
+ defer
+ defer
+ defer
+ defer
+ defer
+ defer
+ apply(rule conjI)
+ apply (subgoal_tac "subst_assn ast ya sa= subst_assn asta ya sa")
+ apply (subgoal_tac "subst_trm t ya sa = subst_trm ta ya sa")
+ apply(simp)
+ apply(erule_tac conjE)+
+ apply(erule alpha_bn_cases)
+ apply(simp add: trm_assn.bn_defs)
+ apply(rotate_tac 7)
+ apply(drule k)
+ apply(erule conjE)
+ apply(subst (asm) Abs1_eq_iff)
+ apply(rule sort_of_atom_eq)
+ apply(rule sort_of_atom_eq)
+ apply(erule disjE)
+ apply(simp)
+ apply(rotate_tac 12)
+ apply(drule sym)
+ apply(rule sym)
+ apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
+ apply(erule fresh_eqvt_at)
+
+ thm fresh_eqvt_at
+ apply(simp add: Abs_fresh_iff)
+ apply(simp add: fresh_star_def fresh_Pair)
+ 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(simp_all add: fresh_star_def fresh_Pair_elim)[1]
+ apply(blast)
+ apply(simp_all)[5]
+ apply(simp (no_asm_use))
+ apply(simp)
+ apply(erule conjE)+
+ apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
+ apply(simp add: Abs_fresh_iff)
+ apply(simp add: fresh_star_def fresh_Pair)
+ 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)
+done
+
+
+end
\ No newline at end of file