--- 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