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(* Title: HOL/Proofs/Lambda/Lambda.thy
Author: Tobias Nipkow
Copyright 1995 TU Muenchen
*)
section \<open>Basic definitions of Lambda-calculus\<close>
theory Lambda
imports Main
begin
declare [[syntax_ambiguity_warning = false]]
subsection \<open>Lambda-terms in de Bruijn notation and substitution\<close>
datatype dB =
Var nat
| App dB dB (infixl "\" 200)
| Abs dB
primrec
lift :: "[dB, nat] => dB"
where
"lift (Var i) k = (if i < k then Var i else Var (i + 1))"
| "lift (s \ t) k = lift s k \ lift t k"
| "lift (Abs s) k = Abs (lift s (k + 1))"
primrec
subst :: "[dB, dB, nat] => dB" ("_[_'/_]" [300, 0, 0] 300)
where (* FIXME base names *)
subst_Var: "(Var i)[s/k] =
(if k < i then Var (i - 1) else if i = k then s else Var i)"
| subst_App: "(t \ u)[s/k] = t[s/k] \ u[s/k]"
| subst_Abs: "(Abs t)[s/k] = Abs (t[lift s 0 / k+1])"
declare subst_Var [simp del]
text \<open>Optimized versions of \<^term>\<open>subst\<close> and \<^term>\<open>lift\<close>.\<close>
primrec
liftn :: "[nat, dB, nat] => dB"
where
"liftn n (Var i) k = (if i < k then Var i else Var (i + n))"
| "liftn n (s \ t) k = liftn n s k \ liftn n t k"
| "liftn n (Abs s) k = Abs (liftn n s (k + 1))"
primrec
substn :: "[dB, dB, nat] => dB"
where
"substn (Var i) s k =
(if k < i then Var (i - 1) else if i = k then liftn k s 0 else Var i)"
| "substn (t \ u) s k = substn t s k \ substn u s k"
| "substn (Abs t) s k = Abs (substn t s (k + 1))"
subsection \<open>Beta-reduction\<close>
inductive beta :: "[dB, dB] => bool" (infixl "\\<^sub>\" 50)
where
beta [simp, intro!]: "Abs s \ t \\<^sub>\ s[t/0]"
| appL [simp, intro!]: "s \\<^sub>\ t ==> s \ u \\<^sub>\ t \ u"
| appR [simp, intro!]: "s \\<^sub>\ t ==> u \ s \\<^sub>\ u \ t"
| abs [simp, intro!]: "s \\<^sub>\ t ==> Abs s \\<^sub>\ Abs t"
abbreviation
beta_reds :: "[dB, dB] => bool" (infixl "\\<^sub>\\<^sup>*" 50) where
"s \\<^sub>\\<^sup>* t == beta\<^sup>*\<^sup>* s t"
inductive_cases beta_cases [elim!]:
"Var i \\<^sub>\ t"
"Abs r \\<^sub>\ s"
"s \ t \\<^sub>\ u"
declare if_not_P [simp] not_less_eq [simp]
\<comment> \<open>don't add \<open>r_into_rtrancl[intro!]\<close>\<close>
subsection \<open>Congruence rules\<close>
lemma rtrancl_beta_Abs [intro!]:
"s \\<^sub>\\<^sup>* s' ==> Abs s \\<^sub>\\<^sup>* Abs s'"
by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
lemma rtrancl_beta_AppL:
"s \\<^sub>\\<^sup>* s' ==> s \ t \\<^sub>\\<^sup>* s' \ t"
by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
lemma rtrancl_beta_AppR:
"t \\<^sub>\\<^sup>* t' ==> s \ t \\<^sub>\\<^sup>* s \ t'"
by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+
lemma rtrancl_beta_App [intro]:
"[| s \\<^sub>\\<^sup>* s'; t \\<^sub>\\<^sup>* t' |] ==> s \ t \\<^sub>\\<^sup>* s' \ t'"
by (blast intro!: rtrancl_beta_AppL rtrancl_beta_AppR intro: rtranclp_trans)
subsection \<open>Substitution-lemmas\<close>
lemma subst_eq [simp]: "(Var k)[u/k] = u"
by (simp add: subst_Var)
lemma subst_gt [simp]: "i < j ==> (Var j)[u/i] = Var (j - 1)"
by (simp add: subst_Var)
lemma subst_lt [simp]: "j < i ==> (Var j)[u/i] = Var j"
by (simp add: subst_Var)
lemma lift_lift:
"i < k + 1 \ lift (lift t i) (Suc k) = lift (lift t k) i"
by (induct t arbitrary: i k) auto
lemma lift_subst [simp]:
"j < i + 1 \ lift (t[s/j]) i = (lift t (i + 1)) [lift s i / j]"
by (induct t arbitrary: i j s)
(simp_all add: diff_Suc subst_Var lift_lift split: nat.split)
lemma lift_subst_lt:
"i < j + 1 \ lift (t[s/j]) i = (lift t i) [lift s i / j + 1]"
by (induct t arbitrary: i j s) (simp_all add: subst_Var lift_lift)
lemma subst_lift [simp]:
"(lift t k)[s/k] = t"
by (induct t arbitrary: k s) simp_all
lemma subst_subst:
"i < j + 1 \ t[lift v i / Suc j][u[v/j]/i] = t[u/i][v/j]"
by (induct t arbitrary: i j u v)
(simp_all add: diff_Suc subst_Var lift_lift [symmetric] lift_subst_lt
split: nat.split)
subsection \<open>Equivalence proof for optimized substitution\<close>
lemma liftn_0 [simp]: "liftn 0 t k = t"
by (induct t arbitrary: k) (simp_all add: subst_Var)
lemma liftn_lift [simp]: "liftn (Suc n) t k = lift (liftn n t k) k"
by (induct t arbitrary: k) (simp_all add: subst_Var)
lemma substn_subst_n [simp]: "substn t s n = t[liftn n s 0 / n]"
by (induct t arbitrary: n) (simp_all add: subst_Var)
theorem substn_subst_0: "substn t s 0 = t[s/0]"
by simp
subsection \<open>Preservation theorems\<close>
text \<open>Not used in Church-Rosser proof, but in Strong
Normalization. \medskip\<close>
theorem subst_preserves_beta [simp]:
"r \\<^sub>\ s ==> r[t/i] \\<^sub>\ s[t/i]"
by (induct arbitrary: t i set: beta) (simp_all add: subst_subst [symmetric])
theorem subst_preserves_beta': "r \\<^sub>\\<^sup>* s ==> r[t/i] \\<^sub>\\<^sup>* s[t/i]"
apply (induct set: rtranclp)
apply (rule rtranclp.rtrancl_refl)
apply (erule rtranclp.rtrancl_into_rtrancl)
apply (erule subst_preserves_beta)
done
theorem lift_preserves_beta [simp]:
"r \\<^sub>\ s ==> lift r i \\<^sub>\ lift s i"
by (induct arbitrary: i set: beta) auto
theorem lift_preserves_beta': "r \\<^sub>\\<^sup>* s ==> lift r i \\<^sub>\\<^sup>* lift s i"
apply (induct set: rtranclp)
apply (rule rtranclp.rtrancl_refl)
apply (erule rtranclp.rtrancl_into_rtrancl)
apply (erule lift_preserves_beta)
done
theorem subst_preserves_beta2 [simp]: "r \\<^sub>\ s ==> t[r/i] \\<^sub>\\<^sup>* t[s/i]"
apply (induct t arbitrary: r s i)
apply (simp add: subst_Var r_into_rtranclp)
apply (simp add: rtrancl_beta_App)
apply (simp add: rtrancl_beta_Abs)
done
theorem subst_preserves_beta2': "r \\<^sub>\\<^sup>* s ==> t[r/i] \\<^sub>\\<^sup>* t[s/i]"
apply (induct set: rtranclp)
apply (rule rtranclp.rtrancl_refl)
apply (erule rtranclp_trans)
apply (erule subst_preserves_beta2)
done
end
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