(* Title: Subst/Subst.thy
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
header{*Substitutions on uterms*}
theory Subst
imports AList UTerm
begin
consts
"=$=" :: "[('a*('a uterm)) list,('a*('a uterm)) list] => bool" (infixr 52)
"<|" :: "'a uterm => ('a * 'a uterm) list => 'a uterm" (infixl 55)
"<>" :: "[('a*('a uterm)) list, ('a*('a uterm)) list]
=> ('a*('a uterm)) list" (infixl 56)
sdom :: "('a*('a uterm)) list => 'a set"
srange :: "('a*('a uterm)) list => 'a set"
syntax (xsymbols)
"op =$=" :: "[('a*('a uterm)) list,('a*('a uterm)) list] => bool"
(infixr "\<doteq>" 52)
"op <|" :: "'a uterm => ('a * 'a uterm) list => 'a uterm" (infixl "\<lhd>" 55)
"op <>" :: "[('a*('a uterm)) list, ('a*('a uterm)) list]
=> ('a*('a uterm)) list" (infixl "\<lozenge>" 56)
primrec
subst_Var: "(Var v \<lhd> s) = assoc v (Var v) s"
subst_Const: "(Const c \<lhd> s) = Const c"
subst_Comb: "(Comb M N \<lhd> s) = Comb (M \<lhd> s) (N \<lhd> s)"
defs
subst_eq_def: "r =$= s == ALL t. t \<lhd> r = t \<lhd> s"
comp_def: "al \<lozenge> bl == alist_rec al bl (%x y xs g. (x,y \<lhd> bl)#g)"
sdom_def:
"sdom(al) == alist_rec al {}
(%x y xs g. if Var(x)=y then g - {x} else g Un {x})"
srange_def:
"srange(al) == Union({y. \<exists>x \<in> sdom(al). y = vars_of(Var(x) \<lhd> al)})"
subsection{*Basic Laws*}
lemma subst_Nil [simp]: "t \<lhd> [] = t"
by (induct_tac "t", auto)
lemma subst_mono [rule_format]: "t \<prec> u --> t \<lhd> s \<prec> u \<lhd> s"
by (induct_tac "u", auto)
lemma Var_not_occs [rule_format]:
"~ (Var(v) \<prec> t) --> t \<lhd> (v,t \<lhd> s) # s = t \<lhd> s"
apply (case_tac "t = Var v")
apply (erule_tac [2] rev_mp)
apply (rule_tac [2] P =
"%x. x \<noteq> Var v --> ~(Var v \<prec> x) --> x \<lhd> (v,t\<lhd>s) #s = x\<lhd>s"
in uterm.induct)
apply auto
done
lemma agreement: "(t\<lhd>r = t\<lhd>s) = (\<forall>v \<in> vars_of t. Var v \<lhd> r = Var v \<lhd> s)"
by (induct_tac "t", auto)
lemma repl_invariance: "~ v: vars_of(t) ==> t \<lhd> (v,u)#s = t \<lhd> s"
by (simp add: agreement)
lemma Var_in_subst [rule_format]:
"v \<in> vars_of(t) --> w \<in> vars_of(t \<lhd> (v,Var(w))#s)"
by (induct_tac "t", auto)
subsection{*Equality between Substitutions*}
lemma subst_eq_iff: "r \<doteq> s = (\<forall>t. t \<lhd> r = t \<lhd> s)"
by (simp add: subst_eq_def)
lemma subst_refl [iff]: "r \<doteq> r"
by (simp add: subst_eq_iff)
lemma subst_sym: "r \<doteq> s ==> s \<doteq> r"
by (simp add: subst_eq_iff)
lemma subst_trans: "[| q \<doteq> r; r \<doteq> s |] ==> q \<doteq> s"
by (simp add: subst_eq_iff)
lemma subst_subst2:
"[| r \<doteq> s; P (t \<lhd> r) (u \<lhd> r) |] ==> P (t \<lhd> s) (u \<lhd> s)"
by (simp add: subst_eq_def)
lemmas ssubst_subst2 = subst_sym [THEN subst_subst2]
subsection{*Composition of Substitutions*}
lemma [simp]:
"[] \<lozenge> bl = bl"
"((a,b)#al) \<lozenge> bl = (a,b \<lhd> bl) # (al \<lozenge> bl)"
"sdom([]) = {}"
"sdom((a,b)#al) = (if Var(a)=b then (sdom al) - {a} else sdom al Un {a})"
by (simp_all add: comp_def sdom_def)
lemma comp_Nil [simp]: "s \<lozenge> [] = s"
by (induct "s", auto)
lemma subst_comp_Nil: "s \<doteq> s \<lozenge> []"
by simp
lemma subst_comp [simp]: "(t \<lhd> r \<lozenge> s) = (t \<lhd> r \<lhd> s)"
apply (induct_tac "t")
apply (simp_all (no_asm_simp))
apply (induct "r", auto)
done
lemma comp_assoc: "(q \<lozenge> r) \<lozenge> s \<doteq> q \<lozenge> (r \<lozenge> s)"
by (simp add: subst_eq_iff)
lemma subst_cong:
"[| theta \<doteq> theta1; sigma \<doteq> sigma1|]
==> (theta \<lozenge> sigma) \<doteq> (theta1 \<lozenge> sigma1)"
by (simp add: subst_eq_def)
lemma Cons_trivial: "(w, Var(w) \<lhd> s) # s \<doteq> s"
apply (simp add: subst_eq_iff)
apply (rule allI)
apply (induct_tac "t", simp_all)
done
lemma comp_subst_subst: "q \<lozenge> r \<doteq> s ==> t \<lhd> q \<lhd> r = t \<lhd> s"
by (simp add: subst_eq_iff)
subsection{*Domain and range of Substitutions*}
lemma sdom_iff: "(v \<in> sdom(s)) = (Var(v) \<lhd> s ~= Var(v))"
apply (induct "s")
apply (case_tac [2] a, auto)
done
lemma srange_iff:
"v \<in> srange(s) = (\<exists>w. w \<in> sdom(s) & v \<in> vars_of(Var(w) \<lhd> s))"
by (auto simp add: srange_def)
lemma empty_iff_all_not: "(A = {}) = (ALL a.~ a:A)"
by (unfold empty_def, blast)
lemma invariance: "(t \<lhd> s = t) = (sdom(s) Int vars_of(t) = {})"
apply (induct_tac "t")
apply (auto simp add: empty_iff_all_not sdom_iff)
done
lemma Var_in_srange [rule_format]:
"v \<in> sdom(s) --> v \<in> vars_of(t \<lhd> s) --> v \<in> srange(s)"
apply (induct_tac "t")
apply (case_tac "a \<in> sdom s")
apply (auto simp add: sdom_iff srange_iff)
done
lemma Var_elim: "[| v \<in> sdom(s); v \<notin> srange(s) |] ==> v \<notin> vars_of(t \<lhd> s)"
by (blast intro: Var_in_srange)
lemma Var_intro [rule_format]:
"v \<in> vars_of(t \<lhd> s) --> v \<in> srange(s) | v \<in> vars_of(t)"
apply (induct_tac "t")
apply (auto simp add: sdom_iff srange_iff)
apply (rule_tac x=a in exI, auto)
done
lemma srangeD: "v \<in> srange(s) ==> \<exists>w. w \<in> sdom(s) & v \<in> vars_of(Var(w) \<lhd> s)"
by (simp add: srange_iff)
lemma dom_range_disjoint:
"sdom(s) Int srange(s) = {} = (\<forall>t. sdom(s) Int vars_of(t \<lhd> s) = {})"
apply (simp add: empty_iff_all_not)
apply (force intro: Var_in_srange dest: srangeD)
done
lemma subst_not_empty: "~ u \<lhd> s = u ==> (\<exists>x. x \<in> sdom(s))"
by (auto simp add: empty_iff_all_not invariance)
lemma id_subst_lemma [simp]: "(M \<lhd> [(x, Var x)]) = M"
by (induct_tac "M", auto)
end