Theory Lambda

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theory Lambda
imports Main

(*  Title:      HOL/Lambda/Lambda.thy
Author: Tobias Nipkow
Copyright 1995 TU Muenchen
*)


header {* Basic definitions of Lambda-calculus *}

theory Lambda imports Main begin


subsection {* Lambda-terms in de Bruijn notation and substitution *}

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 {* Optimized versions of @{term subst} and @{term lift}. *}

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 {* Beta-reduction *}

inductive beta :: "[dB, dB] => bool" (infixl "->β" 50)
where
beta [simp, intro!]: "Abs s ° t ->β s[t/0]"
| appL [simp, intro!]: "s ->β t ==> s ° u ->β t ° u"
| appR [simp, intro!]: "s ->β t ==> u ° s ->β u ° t"
| abs [simp, intro!]: "s ->β t ==> Abs s ->β Abs t"


abbreviation
beta_reds :: "[dB, dB] => bool" (infixl "->>" 50) where
"s ->> t == beta^** s t"


notation (latex)
beta_reds (infixl "->β*" 50)


inductive_cases beta_cases [elim!]:
"Var i ->β t"
"Abs r ->β s"
"s ° t ->β u"


declare if_not_P [simp] not_less_eq [simp]
-- {* don't add @{text "r_into_rtrancl[intro!]"} *}



subsection {* Congruence rules *}

lemma rtrancl_beta_Abs [intro!]:
"s ->β* s' ==> Abs s ->β* Abs s'"

by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+

lemma rtrancl_beta_AppL:
"s ->β* s' ==> s ° t ->β* s' ° t"

by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+

lemma rtrancl_beta_AppR:
"t ->β* t' ==> s ° t ->β* s ° t'"

by (induct set: rtranclp) (blast intro: rtranclp.rtrancl_into_rtrancl)+

lemma rtrancl_beta_App [intro]:
"[| s ->β* s'; t ->β* t' |] ==> s ° t ->β* s' ° t'"

by (blast intro!: rtrancl_beta_AppL rtrancl_beta_AppR intro: rtranclp_trans)


subsection {* Substitution-lemmas *}

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 {* Equivalence proof for optimized substitution *}

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 {* Preservation theorems *}

text {* Not used in Church-Rosser proof, but in Strong
Normalization. \medskip *}


theorem subst_preserves_beta [simp]:
"r ->β s ==> r[t/i] ->β s[t/i]"

by (induct arbitrary: t i set: beta) (simp_all add: subst_subst [symmetric])

theorem subst_preserves_beta': "r ->β* s ==> r[t/i] ->β* 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 ->β s ==> lift r i ->β lift s i"

by (induct arbitrary: i set: beta) auto

theorem lift_preserves_beta': "r ->β* s ==> lift r i ->β* 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 ->β s ==> t[r/i] ->β* 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 ->β* s ==> t[r/i] ->β* t[s/i]"
apply (induct set: rtranclp)
apply (rule rtranclp.rtrancl_refl)
apply (erule rtranclp_trans)
apply (erule subst_preserves_beta2)
done

end