(* Title: Reduction.thy
ID: $Id$
Author: Ole Rasmussen
Copyright 1995 University of Cambridge
Logic Image: ZF
*)
theory Reduction = Residuals:
(**** Lambda-terms ****)
consts
lambda :: "i"
unmark :: "i=>i"
Apl :: "[i,i]=>i"
translations
"Apl(n,m)" == "App(0,n,m)"
inductive
domains "lambda" <= redexes
intros
Lambda_Var: " n \<in> nat ==> Var(n) \<in> lambda"
Lambda_Fun: " u \<in> lambda ==> Fun(u) \<in> lambda"
Lambda_App: "[|u \<in> lambda; v \<in> lambda|] ==> Apl(u,v) \<in> lambda"
type_intros redexes.intros bool_typechecks
declare lambda.intros [intro]
primrec
"unmark(Var(n)) = Var(n)"
"unmark(Fun(u)) = Fun(unmark(u))"
"unmark(App(b,f,a)) = Apl(unmark(f), unmark(a))"
declare lambda.intros [simp]
declare lambda.dom_subset [THEN subsetD, simp, intro]
(* ------------------------------------------------------------------------- *)
(* unmark lemmas *)
(* ------------------------------------------------------------------------- *)
lemma unmark_type [intro, simp]:
"u \<in> redexes ==> unmark(u) \<in> lambda"
by (erule redexes.induct, simp_all)
lemma lambda_unmark: "u \<in> lambda ==> unmark(u) = u"
by (erule lambda.induct, simp_all)
(* ------------------------------------------------------------------------- *)
(* lift and subst preserve lambda *)
(* ------------------------------------------------------------------------- *)
lemma liftL_type [rule_format]:
"v \<in> lambda ==> \<forall>k \<in> nat. lift_rec(v,k) \<in> lambda"
by (erule lambda.induct, simp_all add: lift_rec_Var)
lemma substL_type [rule_format, simp]:
"v \<in> lambda ==> \<forall>n \<in> nat. \<forall>u \<in> lambda. subst_rec(u,v,n) \<in> lambda"
by (erule lambda.induct, simp_all add: liftL_type subst_Var)
(* ------------------------------------------------------------------------- *)
(* type-rule for reduction definitions *)
(* ------------------------------------------------------------------------- *)
lemmas red_typechecks = substL_type nat_typechecks lambda.intros
bool_typechecks
consts
Sred1 :: "i"
Sred :: "i"
Spar_red1 :: "i"
Spar_red :: "i"
"-1->" :: "[i,i]=>o" (infixl 50)
"--->" :: "[i,i]=>o" (infixl 50)
"=1=>" :: "[i,i]=>o" (infixl 50)
"===>" :: "[i,i]=>o" (infixl 50)
translations
"a -1-> b" == "<a,b> \<in> Sred1"
"a ---> b" == "<a,b> \<in> Sred"
"a =1=> b" == "<a,b> \<in> Spar_red1"
"a ===> b" == "<a,b> \<in> Spar_red"
inductive
domains "Sred1" <= "lambda*lambda"
intros
beta: "[|m \<in> lambda; n \<in> lambda|] ==> Apl(Fun(m),n) -1-> n/m"
rfun: "[|m -1-> n|] ==> Fun(m) -1-> Fun(n)"
apl_l: "[|m2 \<in> lambda; m1 -1-> n1|] ==> Apl(m1,m2) -1-> Apl(n1,m2)"
apl_r: "[|m1 \<in> lambda; m2 -1-> n2|] ==> Apl(m1,m2) -1-> Apl(m1,n2)"
type_intros red_typechecks
declare Sred1.intros [intro, simp]
inductive
domains "Sred" <= "lambda*lambda"
intros
one_step: "m-1->n ==> m--->n"
refl: "m \<in> lambda==>m --->m"
trans: "[|m--->n; n--->p|] ==>m--->p"
type_intros Sred1.dom_subset [THEN subsetD] red_typechecks
declare Sred.one_step [intro, simp]
declare Sred.refl [intro, simp]
inductive
domains "Spar_red1" <= "lambda*lambda"
intros
beta: "[|m =1=> m'; n =1=> n'|] ==> Apl(Fun(m),n) =1=> n'/m'"
rvar: "n \<in> nat ==> Var(n) =1=> Var(n)"
rfun: "m =1=> m' ==> Fun(m) =1=> Fun(m')"
rapl: "[|m =1=> m'; n =1=> n'|] ==> Apl(m,n) =1=> Apl(m',n')"
type_intros red_typechecks
declare Spar_red1.intros [intro, simp]
inductive
domains "Spar_red" <= "lambda*lambda"
intros
one_step: "m =1=> n ==> m ===> n"
trans: "[|m===>n; n===>p|] ==> m===>p"
type_intros Spar_red1.dom_subset [THEN subsetD] red_typechecks
declare Spar_red.one_step [intro, simp]
(* ------------------------------------------------------------------------- *)
(* Setting up rule lists for reduction *)
(* ------------------------------------------------------------------------- *)
lemmas red1D1 [simp] = Sred1.dom_subset [THEN subsetD, THEN SigmaD1]
lemmas red1D2 [simp] = Sred1.dom_subset [THEN subsetD, THEN SigmaD2]
lemmas redD1 [simp] = Sred.dom_subset [THEN subsetD, THEN SigmaD1]
lemmas redD2 [simp] = Sred.dom_subset [THEN subsetD, THEN SigmaD2]
lemmas par_red1D1 [simp] = Spar_red1.dom_subset [THEN subsetD, THEN SigmaD1]
lemmas par_red1D2 [simp] = Spar_red1.dom_subset [THEN subsetD, THEN SigmaD2]
lemmas par_redD1 [simp] = Spar_red.dom_subset [THEN subsetD, THEN SigmaD1]
lemmas par_redD2 [simp] = Spar_red.dom_subset [THEN subsetD, THEN SigmaD2]
declare bool_typechecks [intro]
inductive_cases [elim!]: "Fun(t) =1=> Fun(u)"
(* ------------------------------------------------------------------------- *)
(* Lemmas for reduction *)
(* ------------------------------------------------------------------------- *)
lemma red_Fun: "m--->n ==> Fun(m) ---> Fun(n)"
apply (erule Sred.induct)
apply (rule_tac [3] Sred.trans, simp_all)
done
lemma red_Apll: "[|n \<in> lambda; m ---> m'|] ==> Apl(m,n)--->Apl(m',n)"
apply (erule Sred.induct)
apply (rule_tac [3] Sred.trans, simp_all)
done
lemma red_Aplr: "[|n \<in> lambda; m ---> m'|] ==> Apl(n,m)--->Apl(n,m')"
apply (erule Sred.induct)
apply (rule_tac [3] Sred.trans, simp_all)
done
lemma red_Apl: "[|m ---> m'; n--->n'|] ==> Apl(m,n)--->Apl(m',n')"
apply (rule_tac n = "Apl (m',n) " in Sred.trans)
apply (simp_all add: red_Apll red_Aplr)
done
lemma red_beta: "[|m \<in> lambda; m':lambda; n \<in> lambda; n':lambda; m ---> m'; n--->n'|] ==>
Apl(Fun(m),n)---> n'/m'"
apply (rule_tac n = "Apl (Fun (m'),n') " in Sred.trans)
apply (simp_all add: red_Apl red_Fun)
done
(* ------------------------------------------------------------------------- *)
(* Lemmas for parallel reduction *)
(* ------------------------------------------------------------------------- *)
lemma refl_par_red1: "m \<in> lambda==> m =1=> m"
by (erule lambda.induct, simp_all)
lemma red1_par_red1: "m-1->n ==> m=1=>n"
by (erule Sred1.induct, simp_all add: refl_par_red1)
lemma red_par_red: "m--->n ==> m===>n"
apply (erule Sred.induct)
apply (rule_tac [3] Spar_red.trans)
apply (simp_all add: refl_par_red1 red1_par_red1)
done
lemma par_red_red: "m===>n ==> m--->n"
apply (erule Spar_red.induct)
apply (erule Spar_red1.induct)
apply (rule_tac [5] Sred.trans)
apply (simp_all add: red_Fun red_beta red_Apl)
done
(* ------------------------------------------------------------------------- *)
(* Simulation *)
(* ------------------------------------------------------------------------- *)
lemma simulation: "m=1=>n ==> \<exists>v. m|>v = n & m~v & regular(v)"
by (erule Spar_red1.induct, force+)
(* ------------------------------------------------------------------------- *)
(* commuting of unmark and subst *)
(* ------------------------------------------------------------------------- *)
lemma unmmark_lift_rec:
"u \<in> redexes ==> \<forall>k \<in> nat. unmark(lift_rec(u,k)) = lift_rec(unmark(u),k)"
by (erule redexes.induct, simp_all add: lift_rec_Var)
lemma unmmark_subst_rec:
"v \<in> redexes ==> \<forall>k \<in> nat. \<forall>u \<in> redexes.
unmark(subst_rec(u,v,k)) = subst_rec(unmark(u),unmark(v),k)"
by (erule redexes.induct, simp_all add: unmmark_lift_rec subst_Var)
(* ------------------------------------------------------------------------- *)
(* Completeness *)
(* ------------------------------------------------------------------------- *)
lemma completeness_l [rule_format]:
"u~v ==> regular(v) --> unmark(u) =1=> unmark(u|>v)"
apply (erule Scomp.induct)
apply (auto simp add: unmmark_subst_rec)
done
lemma completeness: "[|u \<in> lambda; u~v; regular(v)|] ==> u =1=> unmark(u|>v)"
by (drule completeness_l, simp_all add: lambda_unmark)
end