(* Title: HOL/Lambda/Lambda.thy ID: $Id: Lambda.thy,v 1.38 2007/07/11 09:23:24 berghofe Exp $ 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 consts subst :: "[dB, dB, nat] => dB" ("_[_'/_]" [300, 0, 0] 300) lift :: "[dB, nat] => dB" primrec "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 (* 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}. *} consts substn :: "[dB, dB, nat] => dB" liftn :: "[nat, dB, nat] => dB" primrec "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 (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
lemma rtrancl_beta_Abs:
s ->> s' ==> Abs s ->> Abs s'
lemma rtrancl_beta_AppL:
s ->> s' ==> s ° t ->> s' ° t
lemma rtrancl_beta_AppR:
t ->> t' ==> s ° t ->> s ° t'
lemma rtrancl_beta_App:
s ->> s' ==> t ->> t' ==> s ° t ->> s' ° t'
lemma subst_eq:
Var k[u/k] = u
lemma subst_gt:
i < j ==> Var j[u/i] = Var (j - 1)
lemma subst_lt:
j < i ==> Var j[u/i] = Var j
lemma lift_lift:
i < k + 1 ==> lift (lift t i) (Suc k) = lift (lift t k) i
lemma lift_subst:
j < i + 1 ==> lift (t[s/j]) i = lift t (i + 1)[lift s i/j]
lemma lift_subst_lt:
i < j + 1 ==> lift (t[s/j]) i = lift t i[lift s i/j + 1]
lemma subst_lift:
lift t k[s/k] = t
lemma subst_subst:
i < j + 1 ==> t[lift v i/Suc j][u[v/j]/i] = t[u/i][v/j]
lemma liftn_0:
liftn 0 t k = t
lemma liftn_lift:
liftn (Suc n) t k = lift (liftn n t k) k
lemma substn_subst_n:
substn t s n = t[liftn n s 0/n]
theorem substn_subst_0:
substn t s 0 = t[s/0]
theorem subst_preserves_beta:
r ->β s ==> r[t/i] ->β s[t/i]
theorem subst_preserves_beta':
r ->> s ==> r[t/i] ->> s[t/i]
theorem lift_preserves_beta:
r ->β s ==> lift r i ->β lift s i
theorem lift_preserves_beta':
r ->> s ==> lift r i ->> lift s i
theorem subst_preserves_beta2:
r ->β s ==> t[r/i] ->> t[s/i]
theorem subst_preserves_beta2':
r ->> s ==> t[r/i] ->> t[s/i]