theory Let+ −
imports "../Nominal2" + −
begin+ −
+ −
atom_decl name+ −
+ −
nominal_datatype trm =+ −
Var "name"+ −
| App "trm" "trm"+ −
| Lam x::"name" t::"trm" bind x in t+ −
| Let as::"assn" t::"trm" bind "bn as" in t+ −
and assn =+ −
ANil+ −
| ACons "name" "trm" "assn"+ −
binder+ −
bn+ −
where+ −
"bn ANil = []"+ −
| "bn (ACons x t as) = (atom x) # (bn as)"+ −
+ −
print_theorems+ −
+ −
thm alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.intros+ −
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+ −
+ −
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;+ −
P3 assn_raw assn_rawa\<rbrakk>+ −
\<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)+ −
(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[consumes 1] = alpha_bn_inducts_raw[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"+ −
sorry+ −
lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"+ −
sorry+ −
+ −
lemma bn_inj[rule_format]:+ −
assumes a: "alpha_bn x y"+ −
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 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)+ −
+ −
function+ −
apply_assn :: "(trm \<Rightarrow> nat) \<Rightarrow> assn \<Rightarrow> nat"+ −
where+ −
"apply_assn f ANil = (0 :: nat)"+ −
| "apply_assn f (ACons x t as) = max (f t) (apply_assn f as)"+ −
apply(case_tac x)+ −
apply(case_tac b rule: trm_assn.exhaust(2))+ −
apply(simp_all)+ −
apply(blast)+ −
done+ −
+ −
termination by lexicographic_order+ −
+ −
lemma [eqvt]:+ −
"p \<bullet> (apply_assn f a) = apply_assn (p \<bullet> f) (p \<bullet> a)"+ −
apply(induct f a rule: apply_assn.induct)+ −
apply simp_all+ −
apply(perm_simp)+ −
apply rule+ −
apply(perm_simp)+ −
apply simp+ −
done+ −
+ −
lemma alpha_bn_apply_assn:+ −
assumes "alpha_bn as bs"+ −
shows "apply_assn f as = apply_assn f bs"+ −
using assms+ −
apply (induct rule: alpha_bn_inducts)+ −
apply simp_all+ −
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 (Lam v b) = 1 + (height_trm b)"+ −
| "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, rule, rule TrueI)+ −
apply (case_tac x 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(simp_all)+ −
apply (erule_tac c="()" in Abs_lst1_fcb2)+ −
apply (simp_all add: pure_fresh fresh_star_def eqvt_at_def)[4]+ −
apply (erule conjE)+ −
apply (subst alpha_bn_apply_assn)+ −
apply assumption+ −
apply (rule arg_cong) back+ −
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]+ −
done+ −
+ −
definition "height_assn = apply_assn height_trm"+ −
+ −
function+ −
apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"+ −
where+ −
"apply_assn2 f ANil = ANil"+ −
| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"+ −
apply(case_tac x)+ −
apply(case_tac b rule: trm_assn.exhaust(2))+ −
apply(simp_all)+ −
apply(blast)+ −
done+ −
+ −
termination by lexicographic_order+ −
+ −
lemma [eqvt]:+ −
"p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"+ −
apply(induct f a rule: apply_assn2.induct)+ −
apply simp_all+ −
apply(perm_simp)+ −
apply rule+ −
done+ −
+ −
lemma bn_apply_assn2: "bn (apply_assn2 f as) = bn as"+ −
apply (induct as rule: trm_assn.inducts(2))+ −
apply (rule TrueI)+ −
apply (simp_all add: trm_assn.bn_defs)+ −
done+ −
+ −
nominal_primrec+ −
subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"+ −
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 (apply_assn2 (subst s t) as) (subst s t b)"+ −
apply (simp only: eqvt_def subst_graph_def)+ −
apply (rule, perm_simp, rule)+ −
apply (rule TrueI)+ −
apply (case_tac x)+ −
apply (rule_tac y="c" and c="(a,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 simp_all[7]+ −
prefer 2+ −
apply(simp)+ −
apply(simp)+ −
apply(erule conjE)++ −
apply (erule_tac c="(sa, ta)" in Abs_lst1_fcb2)+ −
apply (simp add: Abs_fresh_iff)+ −
apply (simp add: fresh_star_def)+ −
apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]+ −
apply (simp add: bn_apply_assn2)+ −
apply(erule conjE)++ −
apply(rule conjI)+ −
apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)+ −
apply (simp add: fresh_star_def Abs_fresh_iff)+ −
apply assumption++ −
apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)[2]+ −
apply (erule alpha_bn_inducts)+ −
apply simp_all+ −
done+ −
+ −
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)+ −
+ −
lemma lets_ok:+ −
"(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"+ −
apply (simp add: trm_assn.eq_iff Abs_eq_iff )+ −
apply (rule_tac x="(x \<leftrightarrow> y)" in exI)+ −
apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)+ −
done+ −
+ −
lemma lets_ok3:+ −
"x \<noteq> y \<Longrightarrow>+ −
(Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>+ −
(Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"+ −
apply (simp add: trm_assn.eq_iff)+ −
done+ −
+ −
lemma lets_not_ok1:+ −
"x \<noteq> y \<Longrightarrow>+ −
(Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>+ −
(Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"+ −
apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)+ −
done+ −
+ −
lemma lets_nok:+ −
"x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>+ −
(Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>+ −
(Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"+ −
apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)+ −
done+ −
+ −
lemma+ −
fixes a b c :: name+ −
assumes x: "a \<noteq> c" and y: "b \<noteq> c"+ −
shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"+ −
apply (rule_tac x="(a \<leftrightarrow> b)" in exI)+ −
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)+ −
+ −
end+ −