--- a/src/ZF/Resid/Reduction.thy Fri Dec 21 23:20:29 2001 +0100
+++ b/src/ZF/Resid/Reduction.thy Sat Dec 22 19:42:35 2001 +0100
@@ -5,56 +5,246 @@
Logic Image: ZF
*)
-Reduction = Terms+
+theory Reduction = Residuals:
+
+(**** Lambda-terms ****)
consts
- Sred1, Sred, Spar_red1,Spar_red :: i
- "-1->","--->","=1=>", "===>" :: [i,i]=>o (infixl 50)
+ lambda :: "i"
+ unmark :: "i=>i"
+ Apl :: "[i,i]=>i"
translations
- "a -1-> b" == "<a,b>:Sred1"
- "a ---> b" == "<a,b>:Sred"
- "a =1=> b" == "<a,b>:Spar_red1"
- "a ===> b" == "<a,b>:Spar_red"
+ "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"
- intrs
- 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_intrs "red_typechecks"
+ 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"
- intrs
- one_step "[|m-1->n|] ==> m--->n"
- refl "m \\<in> lambda==>m --->m"
- trans "[|m--->n; n--->p|]==>m--->p"
- type_intrs "[Sred1.dom_subset RS subsetD]@red_typechecks"
+ 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"
- intrs
- 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_intrs "red_typechecks"
+ 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)
+apply 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)
+apply 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)
+apply 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
- inductive
- domains "Spar_red" <= "lambda*lambda"
- intrs
- one_step "[|m =1=> n|] ==> m ===> n"
- trans "[|m===>n; n===>p|]==>m===>p"
- type_intrs "[Spar_red1.dom_subset RS subsetD]@red_typechecks"
+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)
+apply (drule_tac psi = "Fun (?u) =1=> ?w" in asm_rl)
+apply auto
+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