(* Title: Reduction.thy ID: $Id: Reduction.thy,v 1.11 2007/10/07 13:49:26 wenzelm Exp $ Author: Ole Rasmussen Copyright 1995 University of Cambridge Logic Image: ZF *) theory Reduction imports Residuals begin (**** Lambda-terms ****) consts lambda :: "i" unmark :: "i=>i" abbreviation Apl :: "[i,i]=>i" where "Apl(n,m) == App(0,n,m)" inductive domains "lambda" <= redexes intros Lambda_Var: " n ∈ nat ==> Var(n) ∈ lambda" Lambda_Fun: " u ∈ lambda ==> Fun(u) ∈ lambda" Lambda_App: "[|u ∈ lambda; v ∈ lambda|] ==> Apl(u,v) ∈ 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 ∈ redexes ==> unmark(u) ∈ lambda" by (erule redexes.induct, simp_all) lemma lambda_unmark: "u ∈ lambda ==> unmark(u) = u" by (erule lambda.induct, simp_all) (* ------------------------------------------------------------------------- *) (* lift and subst preserve lambda *) (* ------------------------------------------------------------------------- *) lemma liftL_type [rule_format]: "v ∈ lambda ==> ∀k ∈ nat. lift_rec(v,k) ∈ lambda" by (erule lambda.induct, simp_all add: lift_rec_Var) lemma substL_type [rule_format, simp]: "v ∈ lambda ==> ∀n ∈ nat. ∀u ∈ lambda. subst_rec(u,v,n) ∈ 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" abbreviation Sred1_rel (infixl "-1->" 50) where "a -1-> b == <a,b> ∈ Sred1" abbreviation Sred_rel (infixl "--->" 50) where "a ---> b == <a,b> ∈ Sred" abbreviation Spar_red1_rel (infixl "=1=>" 50) where "a =1=> b == <a,b> ∈ Spar_red1" abbreviation Spar_red_rel (infixl "===>" 50) where "a ===> b == <a,b> ∈ Spar_red" inductive domains "Sred1" <= "lambda*lambda" intros beta: "[|m ∈ lambda; n ∈ lambda|] ==> Apl(Fun(m),n) -1-> n/m" rfun: "[|m -1-> n|] ==> Fun(m) -1-> Fun(n)" apl_l: "[|m2 ∈ lambda; m1 -1-> n1|] ==> Apl(m1,m2) -1-> Apl(n1,m2)" apl_r: "[|m1 ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ lambda; m':lambda; n ∈ 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 ∈ 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 ==> ∃v. m|>v = n & m~v & regular(v)" by (erule Spar_red1.induct, force+) (* ------------------------------------------------------------------------- *) (* commuting of unmark and subst *) (* ------------------------------------------------------------------------- *) lemma unmmark_lift_rec: "u ∈ redexes ==> ∀k ∈ 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 ∈ redexes ==> ∀k ∈ nat. ∀u ∈ 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 ∈ lambda; u~v; regular(v)|] ==> u =1=> unmark(u|>v)" by (drule completeness_l, simp_all add: lambda_unmark) end
lemma unmark_type:
u ∈ redexes ==> unmark(u) ∈ lambda
lemma lambda_unmark:
u ∈ lambda ==> unmark(u) = u
lemma liftL_type:
[| v ∈ lambda; k ∈ nat |] ==> lift_rec(v, k) ∈ lambda
lemma substL_type:
[| v ∈ lambda; n ∈ nat; u ∈ lambda |] ==> subst_rec(u, v, n) ∈ lambda
lemma red_typechecks:
[| v ∈ lambda; n ∈ nat; u ∈ lambda |] ==> subst_rec(u, v, n) ∈ lambda
[| n ∈ nat; a ∈ C(0); !!m z. [| m ∈ nat; z ∈ C(m) |] ==> b(m, z) ∈ C(succ(m)) |]
==> rec(n, a, b) ∈ C(n)
0 ∈ nat
1 ∈ nat
n ∈ nat ==> succ(n) ∈ nat
Ord(nat)
n ∈ nat ==> Var(n) ∈ lambda
u ∈ lambda ==> Fun(u) ∈ lambda
[| u ∈ lambda; v ∈ lambda |] ==> Apl(u, v) ∈ lambda
1 ∈ bool
0 ∈ bool
[| b ∈ bool; c ∈ A(1); d ∈ A(0) |] ==> cond(b, c, d) ∈ A(b)
a ∈ bool ==> not(a) ∈ bool
[| a ∈ bool; b ∈ bool |] ==> a and b ∈ bool
[| a ∈ bool; b ∈ bool |] ==> a or b ∈ bool
[| a ∈ bool; b ∈ bool |] ==> a xor b ∈ bool
lemma red1D1:
a -1-> b ==> a ∈ lambda
lemma red1D2:
a -1-> b ==> b ∈ lambda
lemma redD1:
a ---> b ==> a ∈ lambda
lemma redD2:
a ---> b ==> b ∈ lambda
lemma par_red1D1:
a =1=> b ==> a ∈ lambda
lemma par_red1D2:
a =1=> b ==> b ∈ lambda
lemma par_redD1:
a ===> b ==> a ∈ lambda
lemma par_redD2:
a ===> b ==> b ∈ lambda
lemma red_Fun:
m ---> n ==> Fun(m) ---> Fun(n)
lemma red_Apll:
[| n ∈ lambda; m ---> m' |] ==> Apl(m, n) ---> Apl(m', n)
lemma red_Aplr:
[| n ∈ lambda; m ---> m' |] ==> Apl(n, m) ---> Apl(n, m')
lemma red_Apl:
[| m ---> m'; n ---> n' |] ==> Apl(m, n) ---> Apl(m', n')
lemma red_beta:
[| m ∈ lambda; m' ∈ lambda; n ∈ lambda; n' ∈ lambda; m ---> m'; n ---> n' |]
==> Apl(Fun(m), n) ---> n' / m'
lemma refl_par_red1:
m ∈ lambda ==> m =1=> m
lemma red1_par_red1:
m -1-> n ==> m =1=> n
lemma red_par_red:
m ---> n ==> m ===> n
lemma par_red_red:
m ===> n ==> m ---> n
lemma simulation:
m =1=> n ==> ∃v. m |> v = n ∧ m ~ v ∧ regular(v)
lemma unmmark_lift_rec:
u ∈ redexes ==> ∀k∈nat. unmark(lift_rec(u, k)) = lift_rec(unmark(u), k)
lemma unmmark_subst_rec:
v ∈ redexes
==> ∀k∈nat.
∀u∈redexes.
unmark(subst_rec(u, v, k)) = subst_rec(unmark(u), unmark(v), k)
lemma completeness_l:
[| u ~ v; regular(v) |] ==> unmark(u) =1=> unmark(u |> v)
lemma completeness:
[| u ∈ lambda; u ~ v; regular(v) |] ==> u =1=> unmark(u |> v)