(* Title: HOL/Nominal/Examples/Standardization.thy
Author: Stefan Berghofer and Tobias Nipkow
Copyright 2005, 2008 TU Muenchen
*)
header {* Standardization *}
theory Standardization
imports Nominal
begin
text {*
The proof of the standardization theorem, as well as most of the theorems about
lambda calculus in the following sections, are taken from @{text "HOL/Lambda"}.
*}
subsection {* Lambda terms *}
atom_decl name
nominal_datatype lam =
Var "name"
| App "lam" "lam" (infixl "\<degree>" 200)
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [0, 10] 10)
instantiation lam :: size
begin
nominal_primrec size_lam
where
"size (Var n) = 0"
| "size (t \<degree> u) = size t + size u + 1"
| "size (Lam [x].t) = size t + 1"
apply finite_guess+
apply (rule TrueI)+
apply (simp add: fresh_nat)
apply fresh_guess+
done
instance ..
end
nominal_primrec
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [300, 0, 0] 300)
where
subst_Var: "(Var x)[y::=s] = (if x=y then s else (Var x))"
| subst_App: "(t\<^isub>1 \<degree> t\<^isub>2)[y::=s] = t\<^isub>1[y::=s] \<degree> t\<^isub>2[y::=s]"
| subst_Lam: "x \<sharp> (y, s) \<Longrightarrow> (Lam [x].t)[y::=s] = (Lam [x].(t[y::=s]))"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)
apply(fresh_guess)+
done
lemma subst_eqvt [eqvt]:
"(pi::name prm) \<bullet> (t[x::=u]) = (pi \<bullet> t)[(pi \<bullet> x)::=(pi \<bullet> u)]"
by (nominal_induct t avoiding: x u rule: lam.strong_induct)
(perm_simp add: fresh_bij)+
lemma subst_rename:
"y \<sharp> t \<Longrightarrow> ([(y, x)] \<bullet> t)[y::=u] = t[x::=u]"
by (nominal_induct t avoiding: x y u rule: lam.strong_induct)
(simp_all add: fresh_atm calc_atm abs_fresh)
lemma fresh_subst:
"(x::name) \<sharp> t \<Longrightarrow> x \<sharp> u \<Longrightarrow> x \<sharp> t[y::=u]"
by (nominal_induct t avoiding: x y u rule: lam.strong_induct)
(auto simp add: abs_fresh fresh_atm)
lemma fresh_subst':
"(x::name) \<sharp> u \<Longrightarrow> x \<sharp> t[x::=u]"
by (nominal_induct t avoiding: x u rule: lam.strong_induct)
(auto simp add: abs_fresh fresh_atm)
lemma subst_forget: "(x::name) \<sharp> t \<Longrightarrow> t[x::=u] = t"
by (nominal_induct t avoiding: x u rule: lam.strong_induct)
(auto simp add: abs_fresh fresh_atm)
lemma subst_subst:
"x \<noteq> y \<Longrightarrow> x \<sharp> v \<Longrightarrow> t[y::=v][x::=u[y::=v]] = t[x::=u][y::=v]"
by (nominal_induct t avoiding: x y u v rule: lam.strong_induct)
(auto simp add: fresh_subst subst_forget)
declare subst_Var [simp del]
lemma subst_eq [simp]: "(Var x)[x::=u] = u"
by (simp add: subst_Var)
lemma subst_neq [simp]: "x \<noteq> y \<Longrightarrow> (Var x)[y::=u] = Var x"
by (simp add: subst_Var)
inductive beta :: "lam \<Rightarrow> lam \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>\<beta>" 50)
where
beta: "x \<sharp> t \<Longrightarrow> (Lam [x].s) \<degree> t \<rightarrow>\<^sub>\<beta> s[x::=t]"
| appL [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"
| appR [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> u \<degree> s \<rightarrow>\<^sub>\<beta> u \<degree> t"
| abs [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> (Lam [x].s) \<rightarrow>\<^sub>\<beta> (Lam [x].t)"
equivariance beta
nominal_inductive beta
by (simp_all add: abs_fresh fresh_subst')
lemma better_beta [simp, intro!]: "(Lam [x].s) \<degree> t \<rightarrow>\<^sub>\<beta> s[x::=t]"
proof -
obtain y::name where y: "y \<sharp> (x, s, t)"
by (rule exists_fresh) (rule fin_supp)
then have "y \<sharp> t" by simp
then have "(Lam [y]. [(y, x)] \<bullet> s) \<degree> t \<rightarrow>\<^sub>\<beta> ([(y, x)] \<bullet> s)[y::=t]"
by (rule beta)
moreover from y have "(Lam [x].s) = (Lam [y]. [(y, x)] \<bullet> s)"
by (auto simp add: lam.inject alpha' fresh_prod fresh_atm)
ultimately show ?thesis using y by (simp add: subst_rename)
qed
abbreviation
beta_reds :: "lam \<Rightarrow> lam \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>\<beta>\<^sup>*" 50) where
"s \<rightarrow>\<^sub>\<beta>\<^sup>* t \<equiv> beta\<^sup>*\<^sup>* s t"
subsection {* Application of a term to a list of terms *}
abbreviation
list_application :: "lam \<Rightarrow> lam list \<Rightarrow> lam" (infixl "\<degree>\<degree>" 150) where
"t \<degree>\<degree> ts \<equiv> foldl (op \<degree>) t ts"
lemma apps_eq_tail_conv [iff]: "(r \<degree>\<degree> ts = s \<degree>\<degree> ts) = (r = s)"
by (induct ts rule: rev_induct) (auto simp add: lam.inject)
lemma Var_eq_apps_conv [iff]: "(Var m = s \<degree>\<degree> ss) = (Var m = s \<and> ss = [])"
by (induct ss arbitrary: s) auto
lemma Var_apps_eq_Var_apps_conv [iff]:
"(Var m \<degree>\<degree> rs = Var n \<degree>\<degree> ss) = (m = n \<and> rs = ss)"
apply (induct rs arbitrary: ss rule: rev_induct)
apply (simp add: lam.inject)
apply blast
apply (induct_tac ss rule: rev_induct)
apply (auto simp add: lam.inject)
done
lemma App_eq_foldl_conv:
"(r \<degree> s = t \<degree>\<degree> ts) =
(if ts = [] then r \<degree> s = t
else (\<exists>ss. ts = ss @ [s] \<and> r = t \<degree>\<degree> ss))"
apply (rule_tac xs = ts in rev_exhaust)
apply (auto simp add: lam.inject)
done
lemma Abs_eq_apps_conv [iff]:
"((Lam [x].r) = s \<degree>\<degree> ss) = ((Lam [x].r) = s \<and> ss = [])"
by (induct ss rule: rev_induct) auto
lemma apps_eq_Abs_conv [iff]: "(s \<degree>\<degree> ss = (Lam [x].r)) = (s = (Lam [x].r) \<and> ss = [])"
by (induct ss rule: rev_induct) auto
lemma Abs_App_neq_Var_apps [iff]:
"(Lam [x].s) \<degree> t \<noteq> Var n \<degree>\<degree> ss"
by (induct ss arbitrary: s t rule: rev_induct) (auto simp add: lam.inject)
lemma Var_apps_neq_Abs_apps [iff]:
"Var n \<degree>\<degree> ts \<noteq> (Lam [x].r) \<degree>\<degree> ss"
apply (induct ss arbitrary: ts rule: rev_induct)
apply simp
apply (induct_tac ts rule: rev_induct)
apply (auto simp add: lam.inject)
done
lemma ex_head_tail:
"\<exists>ts h. t = h \<degree>\<degree> ts \<and> ((\<exists>n. h = Var n) \<or> (\<exists>x u. h = (Lam [x].u)))"
apply (induct t rule: lam.induct)
apply (rule_tac x = "[]" in exI)
apply (simp add: lam.inject)
apply clarify
apply (rename_tac ts1 ts2 h1 h2)
apply (rule_tac x = "ts1 @ [h2 \<degree>\<degree> ts2]" in exI)
apply (simp add: lam.inject)
apply simp
apply blast
done
lemma size_apps [simp]:
"size (r \<degree>\<degree> rs) = size r + foldl (op +) 0 (map size rs) + length rs"
by (induct rs rule: rev_induct) auto
lemma lem0: "(0::nat) < k \<Longrightarrow> m \<le> n \<Longrightarrow> m < n + k"
by simp
lemma subst_map [simp]:
"(t \<degree>\<degree> ts)[x::=u] = t[x::=u] \<degree>\<degree> map (\<lambda>t. t[x::=u]) ts"
by (induct ts arbitrary: t) simp_all
lemma app_last: "(t \<degree>\<degree> ts) \<degree> u = t \<degree>\<degree> (ts @ [u])"
by simp
lemma perm_apps [eqvt]:
"(pi::name prm) \<bullet> (t \<degree>\<degree> ts) = ((pi \<bullet> t) \<degree>\<degree> (pi \<bullet> ts))"
by (induct ts rule: rev_induct) (auto simp add: append_eqvt)
lemma fresh_apps [simp]: "(x::name) \<sharp> (t \<degree>\<degree> ts) = (x \<sharp> t \<and> x \<sharp> ts)"
by (induct ts rule: rev_induct)
(auto simp add: fresh_list_append fresh_list_nil fresh_list_cons)
text {* A customized induction schema for @{text "\<degree>\<degree>"}. *}
lemma lem:
assumes "\<And>n ts (z::'a::fs_name). (\<And>z. \<forall>t \<in> set ts. P z t) \<Longrightarrow> P z (Var n \<degree>\<degree> ts)"
and "\<And>x u ts z. x \<sharp> z \<Longrightarrow> (\<And>z. P z u) \<Longrightarrow> (\<And>z. \<forall>t \<in> set ts. P z t) \<Longrightarrow> P z ((Lam [x].u) \<degree>\<degree> ts)"
shows "size t = n \<Longrightarrow> P z t"
apply (induct n arbitrary: t z rule: nat_less_induct)
apply (cut_tac t = t in ex_head_tail)
apply clarify
apply (erule disjE)
apply clarify
apply (rule assms)
apply clarify
apply (erule allE, erule impE)
prefer 2
apply (erule allE, erule impE, rule refl, erule spec)
apply simp
apply (rule lem0)
apply force
apply (rule elem_le_sum)
apply force
apply clarify
apply (subgoal_tac "\<exists>y::name. y \<sharp> (x, u, z)")
prefer 2
apply (rule exists_fresh')
apply (rule fin_supp)
apply (erule exE)
apply (subgoal_tac "(Lam [x].u) = (Lam [y].([(y, x)] \<bullet> u))")
prefer 2
apply (auto simp add: lam.inject alpha' fresh_prod fresh_atm)[]
apply (simp (no_asm_simp))
apply (rule assms)
apply (simp add: fresh_prod)
apply (erule allE, erule impE)
prefer 2
apply (erule allE, erule impE, rule refl, erule spec)
apply simp
apply clarify
apply (erule allE, erule impE)
prefer 2
apply (erule allE, erule impE, rule refl, erule spec)
apply simp
apply (rule le_imp_less_Suc)
apply (rule trans_le_add1)
apply (rule trans_le_add2)
apply (rule elem_le_sum)
apply force
done
theorem Apps_lam_induct:
assumes "\<And>n ts (z::'a::fs_name). (\<And>z. \<forall>t \<in> set ts. P z t) \<Longrightarrow> P z (Var n \<degree>\<degree> ts)"
and "\<And>x u ts z. x \<sharp> z \<Longrightarrow> (\<And>z. P z u) \<Longrightarrow> (\<And>z. \<forall>t \<in> set ts. P z t) \<Longrightarrow> P z ((Lam [x].u) \<degree>\<degree> ts)"
shows "P z t"
apply (rule_tac t = t and z = z in lem)
prefer 3
apply (rule refl)
using assms apply blast+
done
subsection {* Congruence rules *}
lemma apps_preserves_beta [simp]:
"r \<rightarrow>\<^sub>\<beta> s \<Longrightarrow> r \<degree>\<degree> ss \<rightarrow>\<^sub>\<beta> s \<degree>\<degree> ss"
by (induct ss rule: rev_induct) auto
lemma rtrancl_beta_Abs [intro!]:
"s \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<Longrightarrow> (Lam [x].s) \<rightarrow>\<^sub>\<beta>\<^sup>* (Lam [x].s')"
by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
lemma rtrancl_beta_AppL:
"s \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<Longrightarrow> s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t"
by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
lemma rtrancl_beta_AppR:
"t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s \<degree> t'"
by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
lemma rtrancl_beta_App [intro]:
"s \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<Longrightarrow> t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t'"
by (blast intro!: rtrancl_beta_AppL rtrancl_beta_AppR intro: rtranclp_trans)
subsection {* Lifting an order to lists of elements *}
definition
step1 :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"step1 r =
(\<lambda>ys xs. \<exists>us z z' vs. xs = us @ z # vs \<and> r z' z \<and> ys =
us @ z' # vs)"
lemma not_Nil_step1 [iff]: "\<not> step1 r [] xs"
apply (unfold step1_def)
apply blast
done
lemma not_step1_Nil [iff]: "\<not> step1 r xs []"
apply (unfold step1_def)
apply blast
done
lemma Cons_step1_Cons [iff]:
"(step1 r (y # ys) (x # xs)) =
(r y x \<and> xs = ys \<or> x = y \<and> step1 r ys xs)"
apply (unfold step1_def)
apply (rule iffI)
apply (erule exE)
apply (rename_tac ts)
apply (case_tac ts)
apply fastsimp
apply force
apply (erule disjE)
apply blast
apply (blast intro: Cons_eq_appendI)
done
lemma append_step1I:
"step1 r ys xs \<and> vs = us \<or> ys = xs \<and> step1 r vs us
\<Longrightarrow> step1 r (ys @ vs) (xs @ us)"
apply (unfold step1_def)
apply auto
apply blast
apply (blast intro: append_eq_appendI)
done
lemma Cons_step1E [elim!]:
assumes "step1 r ys (x # xs)"
and "\<And>y. ys = y # xs \<Longrightarrow> r y x \<Longrightarrow> R"
and "\<And>zs. ys = x # zs \<Longrightarrow> step1 r zs xs \<Longrightarrow> R"
shows R
using assms
apply (cases ys)
apply (simp add: step1_def)
apply blast
done
lemma Snoc_step1_SnocD:
"step1 r (ys @ [y]) (xs @ [x])
\<Longrightarrow> (step1 r ys xs \<and> y = x \<or> ys = xs \<and> r y x)"
apply (unfold step1_def)
apply (clarify del: disjCI)
apply (rename_tac vs)
apply (rule_tac xs = vs in rev_exhaust)
apply force
apply simp
apply blast
done
subsection {* Lifting beta-reduction to lists *}
abbreviation
list_beta :: "lam list \<Rightarrow> lam list \<Rightarrow> bool" (infixl "[\<rightarrow>\<^sub>\<beta>]\<^sub>1" 50) where
"rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss \<equiv> step1 beta rs ss"
lemma head_Var_reduction:
"Var n \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> v \<Longrightarrow> \<exists>ss. rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss \<and> v = Var n \<degree>\<degree> ss"
apply (induct u \<equiv> "Var n \<degree>\<degree> rs" v arbitrary: rs set: beta)
apply simp
apply (rule_tac xs = rs in rev_exhaust)
apply simp
apply (atomize, force intro: append_step1I iff: lam.inject)
apply (rule_tac xs = rs in rev_exhaust)
apply simp
apply (auto 0 3 intro: disjI2 [THEN append_step1I] simp add: lam.inject)
done
lemma apps_betasE [case_names appL appR beta, consumes 1]:
assumes major: "r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> s"
and cases: "\<And>r'. r \<rightarrow>\<^sub>\<beta> r' \<Longrightarrow> s = r' \<degree>\<degree> rs \<Longrightarrow> R"
"\<And>rs'. rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 rs' \<Longrightarrow> s = r \<degree>\<degree> rs' \<Longrightarrow> R"
"\<And>t u us. (x \<sharp> r \<Longrightarrow> r = (Lam [x].t) \<and> rs = u # us \<and> s = t[x::=u] \<degree>\<degree> us) \<Longrightarrow> R"
shows R
proof -
from major have
"(\<exists>r'. r \<rightarrow>\<^sub>\<beta> r' \<and> s = r' \<degree>\<degree> rs) \<or>
(\<exists>rs'. rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 rs' \<and> s = r \<degree>\<degree> rs') \<or>
(\<exists>t u us. x \<sharp> r \<longrightarrow> r = (Lam [x].t) \<and> rs = u # us \<and> s = t[x::=u] \<degree>\<degree> us)"
apply (nominal_induct u \<equiv> "r \<degree>\<degree> rs" s avoiding: x r rs rule: beta.strong_induct)
apply (simp add: App_eq_foldl_conv)
apply (split split_if_asm)
apply simp
apply blast
apply simp
apply (rule impI)+
apply (rule disjI2)
apply (rule disjI2)
apply (subgoal_tac "r = [(xa, x)] \<bullet> (Lam [x].s)")
prefer 2
apply (simp add: perm_fresh_fresh)
apply (drule conjunct1)
apply (subgoal_tac "r = (Lam [xa]. [(xa, x)] \<bullet> s)")
prefer 2
apply (simp add: calc_atm)
apply (thin_tac "r = ?t")
apply simp
apply (rule exI)
apply (rule conjI)
apply (rule refl)
apply (simp add: abs_fresh fresh_atm fresh_left calc_atm subst_rename)
apply (drule App_eq_foldl_conv [THEN iffD1])
apply (split split_if_asm)
apply simp
apply blast
apply (force intro!: disjI1 [THEN append_step1I] simp add: fresh_list_append)
apply (drule App_eq_foldl_conv [THEN iffD1])
apply (split split_if_asm)
apply simp
apply blast
apply (clarify, auto 0 3 intro!: exI intro: append_step1I)
done
with cases show ?thesis by blast
qed
lemma apps_preserves_betas [simp]:
"rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss \<Longrightarrow> r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> r \<degree>\<degree> ss"
apply (induct rs arbitrary: ss rule: rev_induct)
apply simp
apply simp
apply (rule_tac xs = ss in rev_exhaust)
apply simp
apply simp
apply (drule Snoc_step1_SnocD)
apply blast
done
subsection {* Standard reduction relation *}
text {*
Based on lecture notes by Ralph Matthes,
original proof idea due to Ralph Loader.
*}
declare listrel_mono [mono_set]
lemma listrelp_eqvt [eqvt]:
assumes xy: "listrelp f (x::'a::pt_name list) y"
shows "listrelp ((pi::name prm) \<bullet> f) (pi \<bullet> x) (pi \<bullet> y)" using xy
apply induct
apply simp
apply (rule listrelp.intros)
apply simp
apply (rule listrelp.intros)
apply (drule_tac pi=pi in perm_boolI)
apply perm_simp
apply assumption
done
inductive
sred :: "lam \<Rightarrow> lam \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>s" 50)
and sredlist :: "lam list \<Rightarrow> lam list \<Rightarrow> bool" (infixl "[\<rightarrow>\<^sub>s]" 50)
where
"s [\<rightarrow>\<^sub>s] t \<equiv> listrelp op \<rightarrow>\<^sub>s s t"
| Var: "rs [\<rightarrow>\<^sub>s] rs' \<Longrightarrow> Var x \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> rs'"
| Abs: "x \<sharp> (ss, ss') \<Longrightarrow> r \<rightarrow>\<^sub>s r' \<Longrightarrow> ss [\<rightarrow>\<^sub>s] ss' \<Longrightarrow> (Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> ss'"
| Beta: "x \<sharp> (s, ss, t) \<Longrightarrow> r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t \<Longrightarrow> (Lam [x].r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
equivariance sred
nominal_inductive sred
by (simp add: abs_fresh)+
lemma better_sred_Abs:
assumes H1: "r \<rightarrow>\<^sub>s r'"
and H2: "ss [\<rightarrow>\<^sub>s] ss'"
shows "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> ss'"
proof -
obtain y::name where y: "y \<sharp> (x, r, r', ss, ss')"
by (rule exists_fresh) (rule fin_supp)
then have "y \<sharp> (ss, ss')" by simp
moreover from H1 have "[(y, x)] \<bullet> (r \<rightarrow>\<^sub>s r')" by (rule perm_boolI)
then have "([(y, x)] \<bullet> r) \<rightarrow>\<^sub>s ([(y, x)] \<bullet> r')" by (simp add: eqvts)
ultimately have "(Lam [y]. [(y, x)] \<bullet> r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [y]. [(y, x)] \<bullet> r') \<degree>\<degree> ss'" using H2
by (rule sred.Abs)
moreover from y have "(Lam [x].r) = (Lam [y]. [(y, x)] \<bullet> r)"
by (auto simp add: lam.inject alpha' fresh_prod fresh_atm)
moreover from y have "(Lam [x].r') = (Lam [y]. [(y, x)] \<bullet> r')"
by (auto simp add: lam.inject alpha' fresh_prod fresh_atm)
ultimately show ?thesis by simp
qed
lemma better_sred_Beta:
assumes H: "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
shows "(Lam [x].r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
proof -
obtain y::name where y: "y \<sharp> (x, r, s, ss, t)"
by (rule exists_fresh) (rule fin_supp)
then have "y \<sharp> (s, ss, t)" by simp
moreover from y H have "([(y, x)] \<bullet> r)[y::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
by (simp add: subst_rename)
ultimately have "(Lam [y].[(y, x)] \<bullet> r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
by (rule sred.Beta)
moreover from y have "(Lam [x].r) = (Lam [y]. [(y, x)] \<bullet> r)"
by (auto simp add: lam.inject alpha' fresh_prod fresh_atm)
ultimately show ?thesis by simp
qed
lemmas better_sred_intros = sred.Var better_sred_Abs better_sred_Beta
lemma refl_listrelp: "\<forall>x\<in>set xs. R x x \<Longrightarrow> listrelp R xs xs"
by (induct xs) (auto intro: listrelp.intros)
lemma refl_sred: "t \<rightarrow>\<^sub>s t"
by (nominal_induct t rule: Apps_lam_induct) (auto intro: refl_listrelp better_sred_intros)
lemma listrelp_conj1: "listrelp (\<lambda>x y. R x y \<and> S x y) x y \<Longrightarrow> listrelp R x y"
by (erule listrelp.induct) (auto intro: listrelp.intros)
lemma listrelp_conj2: "listrelp (\<lambda>x y. R x y \<and> S x y) x y \<Longrightarrow> listrelp S x y"
by (erule listrelp.induct) (auto intro: listrelp.intros)
lemma listrelp_app:
assumes xsys: "listrelp R xs ys"
shows "listrelp R xs' ys' \<Longrightarrow> listrelp R (xs @ xs') (ys @ ys')" using xsys
by (induct arbitrary: xs' ys') (auto intro: listrelp.intros)
lemma lemma1:
assumes r: "r \<rightarrow>\<^sub>s r'" and s: "s \<rightarrow>\<^sub>s s'"
shows "r \<degree> s \<rightarrow>\<^sub>s r' \<degree> s'" using r
proof induct
case (Var rs rs' x)
then have "rs [\<rightarrow>\<^sub>s] rs'" by (rule listrelp_conj1)
moreover have "[s] [\<rightarrow>\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
ultimately have "rs @ [s] [\<rightarrow>\<^sub>s] rs' @ [s']" by (rule listrelp_app)
hence "Var x \<degree>\<degree> (rs @ [s]) \<rightarrow>\<^sub>s Var x \<degree>\<degree> (rs' @ [s'])" by (rule sred.Var)
thus ?case by (simp only: app_last)
next
case (Abs x ss ss' r r')
from Abs(4) have "ss [\<rightarrow>\<^sub>s] ss'" by (rule listrelp_conj1)
moreover have "[s] [\<rightarrow>\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
ultimately have "ss @ [s] [\<rightarrow>\<^sub>s] ss' @ [s']" by (rule listrelp_app)
with `r \<rightarrow>\<^sub>s r'` have "(Lam [x].r) \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> (ss' @ [s'])"
by (rule better_sred_Abs)
thus ?case by (simp only: app_last)
next
case (Beta x u ss t r)
hence "r[x::=u] \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s t \<degree> s'" by (simp only: app_last)
hence "(Lam [x].r) \<degree> u \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s t \<degree> s'" by (rule better_sred_Beta)
thus ?case by (simp only: app_last)
qed
lemma lemma1':
assumes ts: "ts [\<rightarrow>\<^sub>s] ts'"
shows "r \<rightarrow>\<^sub>s r' \<Longrightarrow> r \<degree>\<degree> ts \<rightarrow>\<^sub>s r' \<degree>\<degree> ts'" using ts
by (induct arbitrary: r r') (auto intro: lemma1)
lemma listrelp_betas:
assumes ts: "listrelp op \<rightarrow>\<^sub>\<beta>\<^sup>* ts ts'"
shows "\<And>t t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> t \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<degree>\<degree> ts'" using ts
by induct auto
lemma lemma2:
assumes t: "t \<rightarrow>\<^sub>s u"
shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* u" using t
by induct (auto dest: listrelp_conj2
intro: listrelp_betas apps_preserves_beta converse_rtranclp_into_rtranclp)
lemma lemma3:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "s \<rightarrow>\<^sub>s s' \<Longrightarrow> r[x::=s] \<rightarrow>\<^sub>s r'[x::=s']" using r
proof (nominal_induct avoiding: x s s' rule: sred.strong_induct)
case (Var rs rs' y)
hence "map (\<lambda>t. t[x::=s]) rs [\<rightarrow>\<^sub>s] map (\<lambda>t. t[x::=s']) rs'"
by induct (auto intro: listrelp.intros Var)
moreover have "Var y[x::=s] \<rightarrow>\<^sub>s Var y[x::=s']"
by (cases "y = x") (auto simp add: Var intro: refl_sred)
ultimately show ?case by simp (rule lemma1')
next
case (Abs y ss ss' r r')
then have "r[x::=s] \<rightarrow>\<^sub>s r'[x::=s']" by fast
moreover from Abs(8) `s \<rightarrow>\<^sub>s s'` have "map (\<lambda>t. t[x::=s]) ss [\<rightarrow>\<^sub>s] map (\<lambda>t. t[x::=s']) ss'"
by induct (auto intro: listrelp.intros Abs)
ultimately show ?case using Abs(6) `y \<sharp> x` `y \<sharp> s` `y \<sharp> s'`
by simp (rule better_sred_Abs)
next
case (Beta y u ss t r)
thus ?case by (auto simp add: subst_subst fresh_atm intro: better_sred_Beta)
qed
lemma lemma4_aux:
assumes rs: "listrelp (\<lambda>t u. t \<rightarrow>\<^sub>s u \<and> (\<forall>r. u \<rightarrow>\<^sub>\<beta> r \<longrightarrow> t \<rightarrow>\<^sub>s r)) rs rs'"
shows "rs' [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss \<Longrightarrow> rs [\<rightarrow>\<^sub>s] ss" using rs
proof (induct arbitrary: ss)
case Nil
thus ?case by cases (auto intro: listrelp.Nil)
next
case (Cons x y xs ys)
note Cons' = Cons
show ?case
proof (cases ss)
case Nil with Cons show ?thesis by simp
next
case (Cons y' ys')
hence ss: "ss = y' # ys'" by simp
from Cons Cons' have "y \<rightarrow>\<^sub>\<beta> y' \<and> ys' = ys \<or> y' = y \<and> ys [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ys'" by simp
hence "x # xs [\<rightarrow>\<^sub>s] y' # ys'"
proof
assume H: "y \<rightarrow>\<^sub>\<beta> y' \<and> ys' = ys"
with Cons' have "x \<rightarrow>\<^sub>s y'" by blast
moreover from Cons' have "xs [\<rightarrow>\<^sub>s] ys" by (iprover dest: listrelp_conj1)
ultimately have "x # xs [\<rightarrow>\<^sub>s] y' # ys" by (rule listrelp.Cons)
with H show ?thesis by simp
next
assume H: "y' = y \<and> ys [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ys'"
with Cons' have "x \<rightarrow>\<^sub>s y'" by blast
moreover from H have "xs [\<rightarrow>\<^sub>s] ys'" by (blast intro: Cons')
ultimately show ?thesis by (rule listrelp.Cons)
qed
with ss show ?thesis by simp
qed
qed
lemma lemma4:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "r' \<rightarrow>\<^sub>\<beta> r'' \<Longrightarrow> r \<rightarrow>\<^sub>s r''" using r
proof (nominal_induct avoiding: r'' rule: sred.strong_induct)
case (Var rs rs' x)
then obtain ss where rs: "rs' [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss" and r'': "r'' = Var x \<degree>\<degree> ss"
by (blast dest: head_Var_reduction)
from Var(1) [simplified] rs have "rs [\<rightarrow>\<^sub>s] ss" by (rule lemma4_aux)
hence "Var x \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> ss" by (rule sred.Var)
with r'' show ?case by simp
next
case (Abs x ss ss' r r')
from `(Lam [x].r') \<degree>\<degree> ss' \<rightarrow>\<^sub>\<beta> r''` show ?case
proof (cases rule: apps_betasE [where x=x])
case (appL s)
then obtain r''' where s: "s = (Lam [x].r''')" and r''': "r' \<rightarrow>\<^sub>\<beta> r'''" using `x \<sharp> r''`
by (cases rule: beta.strong_cases) (auto simp add: abs_fresh lam.inject alpha)
from r''' have "r \<rightarrow>\<^sub>s r'''" by (blast intro: Abs)
moreover from Abs have "ss [\<rightarrow>\<^sub>s] ss'" by (iprover dest: listrelp_conj1)
ultimately have "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r''') \<degree>\<degree> ss'" by (rule better_sred_Abs)
with appL s show "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s r''" by simp
next
case (appR rs')
from Abs(6) [simplified] `ss' [\<rightarrow>\<^sub>\<beta>]\<^sub>1 rs'`
have "ss [\<rightarrow>\<^sub>s] rs'" by (rule lemma4_aux)
with `r \<rightarrow>\<^sub>s r'` have "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> rs'" by (rule better_sred_Abs)
with appR show "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s r''" by simp
next
case (beta t u' us')
then have Lam_eq: "(Lam [x].r') = (Lam [x].t)" and ss': "ss' = u' # us'"
and r'': "r'' = t[x::=u'] \<degree>\<degree> us'"
by (simp_all add: abs_fresh)
from Abs(6) ss' obtain u us where
ss: "ss = u # us" and u: "u \<rightarrow>\<^sub>s u'" and us: "us [\<rightarrow>\<^sub>s] us'"
by cases (auto dest!: listrelp_conj1)
have "r[x::=u] \<rightarrow>\<^sub>s r'[x::=u']" using `r \<rightarrow>\<^sub>s r'` and u by (rule lemma3)
with us have "r[x::=u] \<degree>\<degree> us \<rightarrow>\<^sub>s r'[x::=u'] \<degree>\<degree> us'" by (rule lemma1')
hence "(Lam [x].r) \<degree> u \<degree>\<degree> us \<rightarrow>\<^sub>s r'[x::=u'] \<degree>\<degree> us'" by (rule better_sred_Beta)
with ss r'' Lam_eq show "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s r''" by (simp add: lam.inject alpha)
qed
next
case (Beta x s ss t r)
show ?case
by (rule better_sred_Beta) (rule Beta)+
qed
lemma rtrancl_beta_sred:
assumes r: "r \<rightarrow>\<^sub>\<beta>\<^sup>* r'"
shows "r \<rightarrow>\<^sub>s r'" using r
by induct (iprover intro: refl_sred lemma4)+
subsection {* Terms in normal form *}
lemma listsp_eqvt [eqvt]:
assumes xs: "listsp p (xs::'a::pt_name list)"
shows "listsp ((pi::name prm) \<bullet> p) (pi \<bullet> xs)" using xs
apply induct
apply simp
apply (rule listsp.intros)
apply simp
apply (rule listsp.intros)
apply (drule_tac pi=pi in perm_boolI)
apply perm_simp
apply assumption
done
inductive NF :: "lam \<Rightarrow> bool"
where
App: "listsp NF ts \<Longrightarrow> NF (Var x \<degree>\<degree> ts)"
| Abs: "NF t \<Longrightarrow> NF (Lam [x].t)"
equivariance NF
nominal_inductive NF
by (simp add: abs_fresh)
lemma Abs_NF:
assumes NF: "NF ((Lam [x].t) \<degree>\<degree> ts)"
shows "ts = []" using NF
proof cases
case (App us i)
thus ?thesis by (simp add: Var_apps_neq_Abs_apps [THEN not_sym])
next
case (Abs u)
thus ?thesis by simp
qed
text {*
@{term NF} characterizes exactly the terms that are in normal form.
*}
lemma NF_eq: "NF t = (\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t')"
proof
assume H: "NF t"
show "\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t'"
proof
fix t'
from H show "\<not> t \<rightarrow>\<^sub>\<beta> t'"
proof (nominal_induct avoiding: t' rule: NF.strong_induct)
case (App ts t)
show ?case
proof
assume "Var t \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta> t'"
then obtain rs where "ts [\<rightarrow>\<^sub>\<beta>]\<^sub>1 rs"
by (iprover dest: head_Var_reduction)
with App show False
by (induct rs arbitrary: ts) (auto del: in_listspD)
qed
next
case (Abs t x)
show ?case
proof
assume "(Lam [x].t) \<rightarrow>\<^sub>\<beta> t'"
then show False using Abs
by (cases rule: beta.strong_cases) (auto simp add: abs_fresh lam.inject alpha)
qed
qed
qed
next
assume H: "\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t'"
then show "NF t"
proof (nominal_induct t rule: Apps_lam_induct)
case (1 n ts)
then have "\<forall>ts'. \<not> ts [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ts'"
by (iprover intro: apps_preserves_betas)
with 1(1) have "listsp NF ts"
by (induct ts) (auto simp add: in_listsp_conv_set)
then show ?case by (rule NF.App)
next
case (2 x u ts)
show ?case
proof (cases ts)
case Nil
from 2 have "\<forall>u'. \<not> u \<rightarrow>\<^sub>\<beta> u'"
by (auto intro: apps_preserves_beta)
then have "NF u" by (rule 2)
then have "NF (Lam [x].u)" by (rule NF.Abs)
with Nil show ?thesis by simp
next
case (Cons r rs)
have "(Lam [x].u) \<degree> r \<rightarrow>\<^sub>\<beta> u[x::=r]" ..
then have "(Lam [x].u) \<degree> r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> u[x::=r] \<degree>\<degree> rs"
by (rule apps_preserves_beta)
with Cons have "(Lam [x].u) \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta> u[x::=r] \<degree>\<degree> rs"
by simp
with 2 show ?thesis by iprover
qed
qed
qed
subsection {* Leftmost reduction and weakly normalizing terms *}
inductive
lred :: "lam \<Rightarrow> lam \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>l" 50)
and lredlist :: "lam list \<Rightarrow> lam list \<Rightarrow> bool" (infixl "[\<rightarrow>\<^sub>l]" 50)
where
"s [\<rightarrow>\<^sub>l] t \<equiv> listrelp op \<rightarrow>\<^sub>l s t"
| Var: "rs [\<rightarrow>\<^sub>l] rs' \<Longrightarrow> Var x \<degree>\<degree> rs \<rightarrow>\<^sub>l Var x \<degree>\<degree> rs'"
| Abs: "r \<rightarrow>\<^sub>l r' \<Longrightarrow> (Lam [x].r) \<rightarrow>\<^sub>l (Lam [x].r')"
| Beta: "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>l t \<Longrightarrow> (Lam [x].r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>l t"
lemma lred_imp_sred:
assumes lred: "s \<rightarrow>\<^sub>l t"
shows "s \<rightarrow>\<^sub>s t" using lred
proof induct
case (Var rs rs' x)
then have "rs [\<rightarrow>\<^sub>s] rs'"
by induct (iprover intro: listrelp.intros)+
then show ?case by (rule sred.Var)
next
case (Abs r r' x)
from `r \<rightarrow>\<^sub>s r'`
have "(Lam [x].r) \<degree>\<degree> [] \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> []" using listrelp.Nil
by (rule better_sred_Abs)
then show ?case by simp
next
case (Beta r x s ss t)
from `r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t`
show ?case by (rule better_sred_Beta)
qed
inductive WN :: "lam \<Rightarrow> bool"
where
Var: "listsp WN rs \<Longrightarrow> WN (Var n \<degree>\<degree> rs)"
| Lambda: "WN r \<Longrightarrow> WN (Lam [x].r)"
| Beta: "WN ((r[x::=s]) \<degree>\<degree> ss) \<Longrightarrow> WN (((Lam [x].r) \<degree> s) \<degree>\<degree> ss)"
lemma listrelp_imp_listsp1:
assumes H: "listrelp (\<lambda>x y. P x) xs ys"
shows "listsp P xs" using H
by induct auto
lemma listrelp_imp_listsp2:
assumes H: "listrelp (\<lambda>x y. P y) xs ys"
shows "listsp P ys" using H
by induct auto
lemma lemma5:
assumes lred: "r \<rightarrow>\<^sub>l r'"
shows "WN r" and "NF r'" using lred
by induct
(iprover dest: listrelp_conj1 listrelp_conj2
listrelp_imp_listsp1 listrelp_imp_listsp2 intro: WN.intros
NF.intros)+
lemma lemma6:
assumes wn: "WN r"
shows "\<exists>r'. r \<rightarrow>\<^sub>l r'" using wn
proof induct
case (Var rs n)
then have "\<exists>rs'. rs [\<rightarrow>\<^sub>l] rs'"
by induct (iprover intro: listrelp.intros)+
then show ?case by (iprover intro: lred.Var)
qed (iprover intro: lred.intros)+
lemma lemma7:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "NF r' \<Longrightarrow> r \<rightarrow>\<^sub>l r'" using r
proof induct
case (Var rs rs' x)
from `NF (Var x \<degree>\<degree> rs')` have "listsp NF rs'"
by cases simp_all
with Var(1) have "rs [\<rightarrow>\<^sub>l] rs'"
proof induct
case Nil
show ?case by (rule listrelp.Nil)
next
case (Cons x y xs ys)
hence "x \<rightarrow>\<^sub>l y" and "xs [\<rightarrow>\<^sub>l] ys" by (auto del: in_listspD)
thus ?case by (rule listrelp.Cons)
qed
thus ?case by (rule lred.Var)
next
case (Abs x ss ss' r r')
from `NF ((Lam [x].r') \<degree>\<degree> ss')`
have ss': "ss' = []" by (rule Abs_NF)
from Abs(4) have ss: "ss = []" using ss'
by cases simp_all
from ss' Abs have "NF (Lam [x].r')" by simp
hence "NF r'" by (cases rule: NF.strong_cases) (auto simp add: abs_fresh lam.inject alpha)
with Abs have "r \<rightarrow>\<^sub>l r'" by simp
hence "(Lam [x].r) \<rightarrow>\<^sub>l (Lam [x].r')" by (rule lred.Abs)
with ss ss' show ?case by simp
next
case (Beta x s ss t r)
hence "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>l t" by simp
thus ?case by (rule lred.Beta)
qed
lemma WN_eq: "WN t = (\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')"
proof
assume "WN t"
then have "\<exists>t'. t \<rightarrow>\<^sub>l t'" by (rule lemma6)
then obtain t' where t': "t \<rightarrow>\<^sub>l t'" ..
then have NF: "NF t'" by (rule lemma5)
from t' have "t \<rightarrow>\<^sub>s t'" by (rule lred_imp_sred)
then have "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" by (rule lemma2)
with NF show "\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" by iprover
next
assume "\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
then obtain t' where t': "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" and NF: "NF t'"
by iprover
from t' have "t \<rightarrow>\<^sub>s t'" by (rule rtrancl_beta_sred)
then have "t \<rightarrow>\<^sub>l t'" using NF by (rule lemma7)
then show "WN t" by (rule lemma5)
qed
end